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Vector Basics: Describing Directions
July 13, 2020 July 12, 2020 / Algebra, Trigonometry / Vectors / By Dave Peterson
We're looking at the concept of vectors at an introductory level. Last week we looked at how they are defined in this context (as quantities with magnitude and direction), and how they are added (which is really part of the definition). Our collection of answers from Ask Dr. Math this time focuses on the ideas of unit vectors, components, and "direction cosines", which are all ways to describe the direction of a vector.
Unit vectors and collinearity
The first question is from 2002:
I am trying to solve a math problem that I truly do not understand. The problem reads:
"Find the two unit vectors that are collinear with each of the following vectors. (a) vector A = (3, -5)"
That's the first question in this problem, anyway.
I don't understand what this problem is even asking me to do. Is a unit vector only ever equal to 1? I've done a lot of research in my book and on the internet and I still don't understand. Any help you could provide would be GREATLY appreciated.
Thanks ever so much.
A unit vector is one whose length (magnitude) is 1; collinear vectors lie along the same line (so they can go in the same or opposite directions). Doctor Ian answered:
A unit vector can have any direction, but its length is equal to 1. So the following are all unit vectors:
(0,1) length^2 = 0^2 + 1^2 = 1
(1/2, sqrt(3)/2) length^2 = (1/2)^2 + (sqrt(3)/2)^2 = 1
The length, or magnitude, of a vector is found by the Pythagorean theorem: $$|(a,b)| = \sqrt{a^2+b^2}$$
Here are his three vectors, showing that they all have length 1:
In fact, if you pick any point on the unit circle (i.e., the
circle centered at the origin, whose radius is 1), the vector
from the origin to the point is the unit vector (cos(a),sin(a)),
where a is the angle from the positive x-axis to the point.
Here is an example, where my angle \(\theta\) is 133°, so its components are \((\cos(133°), \sin(133°)) = (-0.68, 0.73)\):
The easiest way to get a unit vector that is collinear with a vector (a,b) is to find the magnitude of the vector,
|(a,b)| = sqrt(a^2 + b^2)
and divide both components by that:
1/|(a,b)| * (a,b) = (a/|(a,b)|, b/|(a,b)|)
Do you see why this will always be collinear with the original vector, and why its length will always be equal to 1?
(Note that the unit vector that points in the _opposite_ direction is also collinear.)
Dividing both components by \(|a|\) reduces the length to 1 without changing the direction. Multiplying by negative 1 reverses the direction. Here are our vector \(\mathbf{a} = (3, -5)\)and the two unit vectors \(\mathbf{u}_a = \left(\frac{3}{\sqrt{34}}, -\frac{5}{\sqrt{34}}\right)\) and \(-\mathbf{u}_a = \left(-\frac{3}{\sqrt{34}}, \frac{5}{\sqrt{34}}\right)\):
Unit vectors, basis vectors
This 1998 question is from a student whose goals are far beyond the basics, but who needs help starting:
Unit and Basis Vectors in Three Dimensions
Please give me a simple explanation of:
1. Unit vector.
My books (e.g., _Vector and Tensor Analysis_ by Borisenko) are not clear and assume I already understand this. Also, what use is a unit vector?
2. Basis vector. Again, my other sources are not clear.
P.S. I study relativity on my own, and this is why I'd like to understand the basics, like tensor algebra.
Doctor Anthony answered, giving the basic definition for vectors that we discussed last time, because it is applicable (with some little modifications) to physics:
A vector is a physical quantity, like velocity, displacement, or force, having both magnitude and direction.
Think of a vector as represented by a straight line pointing in a particular direction. The length of the line represents the magnitude of the vector. So in the case of a unit vector, the length of the line is 1 unit.
It is convenient to use unit vectors when working on problems. If we let u represent a vector in a certain direction and of unit magnitude, then 3u, 7u, and -8u are immediately understandable as vectors of magnitudes 3, 7 and -8 all in the direction of u (except -8u, as the negative sign means "in the opposite direction to +u").
Just as, above, we started with a vector a and found a unit vector in the same direction, here we can reverse the process and describe a vector as a unit vector in the right direction, multiplied by its length. In doing this, we are splitting the vector into its magnitude (a number) and its direction (a unit vector). Here are a unit vector u and the multiples that were mentioned:
But we can do much more by picking a standard set of unit vectors to use as a "basis" for describing any vector at all:
It is very common to use i, j, and k as unit vectors in the directions of the x, y, and z axes, respectively, in 3D space. This means that EVERY vector in space can be given in terms of its "components" parallel to those three axes. So, for example, 5i + 2j - 6k is a vector in space, and its magnitude would be represented by the length of a line joining the origin (0,0,0) to the point (5,2,-6).
Incidentally, this answers your second question: i, j, k are called "base" vectors because they are used as the basis for expressing all other vectors. Every other vector in 3D space whatsoever can be given in terms of i, j, and k.
I'll stick to two dimensions here. The unit basis vectors i and j are the same as u and v that we used in the first illustration above:
Sometimes it is convenient to use other vectors as base vectors. Any two non-parallel vectors could be used as base vectors to give any point in the plane of the two vectors. That is, every other vector in that plane could be expressed in terms of the base vectors, just as we say 6i + 4j to express a vector in the xy plane. Similarly, any three non-coplanar vectors could be used as base vectors "spanning" 3D space. Again, the most common base vectors are i,j, k, but there are occasions when an entirely different set of base vectors are used.
Finally, vectors are not confined to 1, 2, or 3 dimensions. You can
have multi-dimensional vectors expressed in terms of 4, 5, 6, and
higher base vectors. The number of base vectors will equal the
dimension of the space under consideration.
Here is our vector 6i + 4j, which can also be called (6, 4) using its components:
Direction as angles (2 dimensions)
The next question, from 1998, involves vectors whose direction is expressed as the angle from the positive x-axis:
Vector Components, Magnitude, and Direction
Vector M of magnitude 4.75 m is at 58.0 degrees counter-clockwise from the positive x-axis. It is added to vector N, and the resultant is a vector of magnitude 4.75 m, at 39 degrees counterclockwise from the positive x-axis. Find: (a) the components of N, and (b) the magnitude and direction of N.
I drew a graphical illustration of the problem. But I really can't solve it because I don't know how.
Here we have two vectors being added, and one of them and the sum are described in terms of magnitude and direction. We want to find vector n, both in terms of components and of direction (angle) and magnitude:
Doctor Rick answered, suggesting the most likely method:
Hi, Kristine,
I will get you started on solving this kind of problem. There are two tools you need to do this: (1) converting between magnitude/direction and components of a vector and (2) adding vectors. The first requires some trigonometry, so I hope you've had some.
(1) You are given the magnitude and direction of vectors M and P (the sum of M and N). Before you can add them, you must find their components. Remember this diagram:
My+-------------* M
| /|
| / |
| / |
| / |
| / |
| L/ |
| / |
sin(a)|-----+ |
| /| |
| 1/ | |
| / | |
| /)a | |
|/____|_______|__________
O cos(a) Mx
A vector of length 1 has components (cos(a), sin(a)). By similar triangles, a vector M of length L has components Mx = L*cos(a), My = L*sin(a). Do this with both vectors M and P to get their components (Mx, My) and (Px, Py).
As we saw above, we can think of the vector m as a unit vector in the given direction multiplied by its length. The trigonometric functions cosine and sine give the x and y components of the unit vector, as we saw in our first answer. For our vector m, the angle is 58° and the length is 4.75, so the vector is $$(m_x, m_y) = (4.75\cos(58°), 4.75\sin(58°)) = (2.517, 4.028)$$
Similarly, for vector p = m + n, we have angle 39° and length 4.75, so the vector is $$(p_x, p_y) = (4.75\cos(39°), 4.75\sin(39°)) = (3.691, 2.989)$$
(2) You know that M + N = P. To add vectors, add their components:
Mx + Nx = Px
My + Ny = Py
You know Mx, My, Px, and Py, so you should be able to figure out Nx and Ny. These are the components of vector N.
To find the components of n, we just subtract: $$n = (n_x, n_y) = (p_x-m_x, p_y-m_y) = (3.691 – 2.517, 2.989 – 4.028) = (1.174, -1.039)$$
That's the answer to part (a).
(3) You were also asked for the magnitude and direction of vector N. To do this, you have to reverse step 1. Here's how, using the figure (remember, you'll be doing this for vector N, not vector M).
Magnitude(M) = Mx^2 + My^2
(the Pythagorean Theorem where ^2 means square)
tangent(a) = sin(a)/cos(a) = My/Mx (by similar triangles again)
So Direction(M) = a = inverse tangent of a = arctan(a)
The Pythagorean theorem gives our length as $$|\mathbf{n}| = \sqrt{n_x^2+n_y^2} = \sqrt{1.174^2+(-1.039)^2} = \sqrt{2.457797} = 1.568$$
The tangent of our angle is the slope of the vector: $$\tan(\theta) = \frac{n_y}{n_x} = \frac{-1.039}{1.174} = -0.885$$
So the angle itself is $$\theta = \tan^{-1}(-0.885) = -41.5°$$
Those are the tools you'll need. See if you can do the job now. Write back if you're still confused after you've tried it.
Presumably, Kristine did just what we've done here.
Direction of a line (3 dimensions)
This 1997 question hopes for a way to indicate the direction of a 3-dimensional vector similar to the angle or slope in the previous type of problem:
Formula for Slope of 3-D Line
Thank you for answering our question about finding the length of a line in three dimensions. Now we would like to know the formula to find the slope of a three-dimensional line. We searched through textbooks and tried to adapt the formula, but with no success.
Doctor Rob answered, first talking about planes rather than lines:
There is no direct analogue of the idea of slope in two dimensions. The subject you are discussing is analytic geometry of three dimensions. The following facts should help a little.
A linear equation in x, y, and z, such as ax + by + cz = d, is the equation of a plane, not a line. Such equations can be put into one standard form by dividing by sqrt(a^2+b^2+c^2). The resulting coefficients of x, y, and z have the property that the sum of their squares is 1. Another standard form is gotten by dividing by d, and writing the equation as x/(d/a) + y/(d/b) + z/(d/c) = 1. From this form you can read off the intercepts with the x-, y-, and z-axes: (d/a, 0, 0) is the x-intercept, (0, d/b, 0) is the y-intercept, and (0, 0, d/c) is the z-intercept (provided all of a, b, c, and d are nonzero). One way of regarding the "slope" of a plane is to write down a unit vector which is perpendicular to it, called the normal vector. It is given by (a*I + b*J + c*K)/sqrt(a^2+b^2+c^2), where I, J, and K are the unit vectors in the x, y, and z directions. The coefficients of I, J, and K in this expression are called the direction cosines of the vector, because they are the cosines of the angles between the vector and the x-, y-, and z-axes, respectively.
The two standard forms he mentions for a plane are, in effect, $$a'x + b'y + c'z = d'$$ where the vector (a', b', c') is a unit vector called the unit normal vector (this is something we will see later in this series or a subsequent series), and $$\frac{x}{A} + \frac{y}{B} + \frac{z}{C} = 1$$ where A, B, and C are the intercepts on the three axes. The normal vector represents the direction of the plane.
But the question was about a line, and the "direction cosines" just mentioned for the unit normal vector show up here, too:
A line is specified as the intersection of two nonparallel planes. This means you need two linear equations in x, y, and z to determine a line. There are several standard forms for the equations of a line, but a commonly used one is
x - x0 y - y0 z - z0
------ = ------ = ------
a b c
Here (x0, y0, z0) is a point on the line, and the numbers a, b, and c determine the direction along the line: the vector a*I = b*J + c*K is parallel to the line. (Note: This form only works when the line is not parallel to any of the xy-, xz-, or yz-planes, i.e., when neither a, b, or c is zero).
Note that this form is not a single equation, but a pair of equations that set three quantities equal. In this form, $$\frac{x – x_0}{a} = \frac{y – y_0}{b} = \frac{z – z_0}{c}$$ the vector (a, b, c) gives the direction of the line, which is the best answer to the question, much like the normal vector for the plane.
In some sense, the direction cosines are the closest analogue to the slope. In two dimensions, they are just the cosine of the inclination, which is the angle with the x-axis, and the cosine of its complement, which is the sine of the inclination. The slope is the ratio of these two, the tangent of the inclination. There is no exact analogue because there is no "ratio" of three direction cosines, or of any three numbers.
We could say, however, that the triple ratio a : b : c is a reasonable analogue of the slope of a line, even though it is not a number; the direction cosines, as we'll see below, are just the components of the unit vector in the direction of the line.
As he suggests, we can do all this in two dimensions for comparison, which is quite instructive. We can write a line as $$\frac{x – x_0}{a} = \frac{y – y_0}{b}$$ which (solving for y) can be rewritten as $$y = \frac{b}{a}(x – x_0) + y_0$$ The slope is the number \(\displaystyle\frac{b}{a} = \frac{\cos(\theta_y)}{\cos(\theta_x)} = \frac{\sin(\theta_x)}{\cos(\theta_x)} = \tan(\theta)\), showing that the ratio a : b is closely related to the slope, which is the tangent of the angle to the x-axis.
Direction cosines of a vector
For a fuller picture of direction cosines, we'll close with this question from 2003:
Why They're Called Direction Cosines
I would like to know how to find the angles between a 3D vector and the 3 coordinate axes, given the components of the vector.
Doctor Ian answered, using a concept we'll be getting to next week, the dot product:
Hi Kristen,
If you have two vectors, A and B, and you want to find the angle between them, one way is to use the dot product:
dot(A,B) = |A||B|cos(theta)
Does that look familiar? To find the angle between a vector and a particular axis, you can just make B a unit vector. For example, if A is (a,b,c), then to find the angle with the x-axis,
a*1 + b*0 + c*0 = \|a^2 + b^2 + c^2 cos(theta)
-------------------- = cos(theta)
\|a^2 + b^2 + c^2
What he's done here is to apply his formula for the angle between two vectors to the given vector a and the unit vector \(\mathbf{i} = (1,0,0)\). The general formula for the angle between vectors a and b is $$\cos(\theta) = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$$ and if we replace b with a unit vector u (either i, j, or k), we have $$\cos(\theta) = \frac{\mathbf{a}\cdot\mathbf{u}}{|\mathbf{a}|}$$ where the numerator is just the appropriate component of a.
So the cosine of the angle between a vector and the x-axis is just the x-component of the vector divided by the magnitude of the vector. This is true for each of the components. But you may recognize that this cosine is simply a component of the unit vector:
Note that if you make A a unit vector (which you can do by dividing all the components by the magnitude of A), you end up with
a b c
( ---, ---, --- ) = ( cos(theta ), cos(theta ), cos(theta ) )
|A| |A| |A| x y z
For this reason, the components of a unit vector are often called the 'direction cosines' of the vector.
So for example, if we have the vector (3, 4, 5), whose magnitude is \(\sqrt{3^2 + 4^2 + 5^2} = \sqrt{50} \approx 7.07\), then the unit vector in the same direction is $$\left(\frac{3}{\sqrt{50}}, \frac{4}{\sqrt{50}}, \frac{5}{\sqrt{50}}\right) = (0.424, 0.566, 0.707)$$ so the angles with the axes are the inverse cosines of these numbers, 64.9°, 55.6°, and 45°, respectively:
Kristen replied,
Thank you for your help. That will make my life SO much easier!
Next week, we'll look at ways to multiply two vectors.
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11.2: The Principle of Weak Induction
[ "article:topic", "Weak Induction" ]
Map: A Modern Formal Logic Primer (Teller)
Volume II: Predicate Logic
11: Mathematical Induction
Let's look at a more specific example. You may have wondered how many lines there are in a truth table with n atomic sentence letters. The answer is 2". But how do we prove that this answer is correct, that for all n, an n-letter truth table has 2" lines? If n = 1, that is, if there is just one sentence letter in a truth table, then the number of lines is 2 = 2'. So the generalization holds for the first case. This is called the Basis Step of the induction. We then need to do what is called the Inductive Step. We assume that the generalization holds for n. This assumption is called the Inductive Hy$othesis. Then, using the inductive hypothesis, we show that the generalization holds for n + 1. So let's assume (inductive hypothesis) that in an n-letter truth table there are 2" lines. How many lines are there in a truth table obtained by adding one more letter? Suppose our new letter is 'A'. 'A' can be either true or false. The first two lines of the n + 1 letter truth table will be the first line of the n-letter table plus the specification that 'A' is true, followed by the first line of the n-letter table plus the specification that 'A' is false. The next two lines of the new table will be the second line of the old table, similarly extended with the two possible truth values of 'A'. In general, each line of the old table will give rise to two lines of the new table. So the new table has twice the lines of the old table, or 2" x 2 = 2"+ l. This is what we needed to show in the inductive step of the argument. We have shown that there are 2" lines of an n-letter truth table when n = 1 (basis step). We have shown that if an n-letter table has 2" lines, then an n + 1 letter table has 2"+' lines. Our generalization is true for n = 1, and if it is true for any arbitrarily chosen n, then it is true for n + 1. The princple of mathematical induction then tells us we may conclude that it is true for all n. We will express this principle generally with the idea of an Inductive Property. An inductive property is, strictly speaking, a property of integers. In an inductive argument we show that the integer 1 has the inductive property, and that for each integer n, if n has the inductive property, then the integer n + 1 has the inductive property. Induction then licenses us to conclude that all integers, n, have the inductive property. In the last example, All n Ltter truth tubb have exact4 2" lines, a proposition about the integer n, was our inductive property. To speak generally, I will use 'P(n)' to talk about whatever inductive property might be in question:
Principle of Weak Induction
Let \(P(n)\) be some property which can be claimed to hold for (is defined for) the integers, n = 1, 2, 3, . . . (the Inductive Property).
Suppose we have proved \(P(l)\) (Basis Step).
Suppose we have proved, for any n, that if P(n), then P(n + 1) (IndzLchue Skp, with the assumption of P(n), the Inductive Hypothesis).
Then you may conclude that \(P(n)\) holds for all \(n\) from 1 on.
If in the basis step we have proved \(P(i)\), we may conclude that \(P(n)\) holds for \(n = i, i + 1, i + 2, \ldots\).
(e) simply says that our induction can really start from any integer, as long as the inductive property is defined from that integer onward. Often it is convenient to start from 0 instead of from 1, showing that P(n) holds for n = 0,1,2,. . . .
Most of the inductions we will do involve facts about sentences. To get you started, here is a simple example. The conclusion is so obvious that, ordinarily, we would not stop explicitly to prove it. But it provides a nice illustration and, incidentally, illustrates the fact that many of the generalizations which seem obvious to us really depend on mathematical induction. Let's prove that if the only kind of connective which occurs in a sentence logic sentence is '-', then there is a truth value assignment under which the sentence is true and another in which it is false. (For short, we'll say that the sentence "can be either true or false.") Our inductive property will be: All sentences with n occurrences of '-' and no other connectives can be either true or false. A standard way of expressing an important element here is to say that we will be doing the induction on th number of connectives, a strategy for which you will have frequent use. We restrict attention to sentences, X, in which no connectives other than '-' occur. Suppose (basis case, with n = 0) that X has no occurrences of '-'. Then X is an atomic sentence letter which can be assigned either t or f. Suppose (inductive hypothesis for the inductive step) that all sentences with exactly n occurrences of '-' can be either true or false. Let Y be an arbitrary sentences with n + 1 occurrences of '-'. Then Y has the form -X, where X has exactly n occurrences of '-'. By the inductive hypothesis, X can be either true or false. In these two cases, -X, that is, Y, is, respectively, false and true. Since Y can be any sentence with n + 1 occurrences of '-', we have shown that the inductive property holds for n + 1, completing the inductive argument. 172 Mathematied Induction 11 -3. Strong Induction 173 EXERCISES 11-1. By a Restricted Conjunctive Sentence, I mean one which is either an atomic sentence or is a conjunction of an atomic sentence with another restricted conjunctive sentence. Thus the sentences 'A' and '[C&(A&B)]&D' are restricted conjunctive sentences. The sentence 'A &[(C&D)&(H&G)]' is not, because the component, '(C&D)&(H&G)', fails to be a conjunction one of the components of which is an atomic sentence letter.
Here is a rigorous definition of this kind of sentence:
Any atomic sentence letter is a restricted conjunctive sentence.
Any atomic sentence letter conjoined with another restricted conjunctive sentence is again a restricted conjunctive sentence.
Only such sentences are restricted conjunctive sentences. Such a definition is called an Inductive Definition. Use weak induction to prove that a restricted conjunctive sentence is true iff all the atomic sentence letters appearing in it are true. 11-2. Prove that the formula is correct for all n.
11.1: Informal Introduction
11.3: Strong Induction
Weak Induction | CommonCrawl |
\begin{document}
\title[Pontryagin principle]{Pontryagin principle for a Mayer problem governed by a delay functional differential equation}
\author[Blot Kon\'e]{Jo\"el Blot and mamadou I. Kon\'e}
\address{Jo\"el Blot: Laboratoire SAMM UE 4543, \newline Universit\'e Paris 1 Panth\'eon-Sorbonne, centre P.M.F.,\newline 90 rue de Tolbiac, 75634 Paris cedex 13, France.} \email{[email protected]}
\address{Mamadou I. Kon\'e: Laboratoire SAMM UE 4543, \newline Universit\'e Paris 1 Panth\'eon-Sorbonne, centre P.M.F.,\newline 90 rue de Tolbiac, 75634 Paris cedex 13, France.} \email{[email protected]}
\date{November 9, 2015}
\begin{abstract} We establish Pontryagin principles for a Mayer's optimal control problem governed by a functional differential equation. The control functions are piecewise continuous and the state functions are piecewise continuously differentiable. To do that, we follow the method created by Philippe Michel for systems governed by ordinary differential equations, and we use properties of the resolvent of a linear functional differential equation. \end{abstract}
\maketitle
\numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition}
\noindent {\bf Key words:} optimal control; Pontryagin principle; functional differential equation.\\ {\bf MSC2010-AMS:} 49J21, 49K21, 34K09.
\section{Introduction}
We consider the following problem of optimal control. It is called a problem of Mayer since its criterion takes into account only the final value of the state; it is governed by a functional differential equation in presence of terminal constraints. \[ \left. \begin{array}{rl} {\rm Maximize} & J(x,u) := g^0(x(T))\\ {\rm when} & x \in C^0([-r,T], {\mathbb{R}}^n), x \in PC^1([0,T], {\mathbb{R}}^n)\\ \null & u \in PC^0([0,T], U)\\ \null &\forall t \in [0,T] \setminus F, \;\; x'(t) = f(t, x_t, u(t))\\ \null &x_0 = \phi \\ \null &\forall j = 1,...,n_i, \; g^j(x(T)) \geq 0\\ \null & \forall j= n_i+1,...,n_i+n_e, \; g^j(x(T)) = 0. \end{array} \right\} ({\mathfrak M}) \]
where the $g^j : {\mathbb{R}}^n \rightarrow {\mathbb{R}}$ are mappings, the state variable $x$ is a piecewise continuously differentiable function (see Section 2), the control variable $u$ is a piecewise continuous function (see Section 2), $U$ is a nonempty subset of ${\mathbb{R}}^d$, $F$ denotes a finite subset of $[0,T]$ (not a priori fixed), $x_t(\theta) := x(t + \theta)$ when $\theta \in [-r,0]$, $\phi$ is fixed continuous function from $[-r,0]$ into ${\mathbb{R}}^n$. The only assumptions that we do on this problem are the following ones.
\begin{equation}\label{eq11} \left. \begin{array}{l} f \in C^0([0,T] \times C^0([-r,0], {\mathbb{R}}^n) \times U, {\mathbb{R}}^n)\\ \forall (t,\phi,\xi) \in [0,T] \times C^0([-r,0], {\mathbb{R}}^n) \times U, D_2f(t,\phi, \xi) \; {\rm exists}\\ D_2f \in C^0([0,T] \times C^0([-r,0], {\mathbb{R}}^n) \times U, {\mathfrak L}( C^0([-r,0], {\mathbb{R}}^n), {\mathbb{R}}^n))\\ D_2f \; {\rm is} \; {\rm a} \; {\rm bounded} \; {\rm operator}. \end{array} \right\} \end{equation}
where $C^0(X,Y)$ denotes the space of the continuous mappings from $X$ into $Y$, $D_2f(t,\phi, \xi)$ denotes the partial Fr\'echet differential of $f$ with respect to its second variable, and ${\mathfrak L}(E,E_1)$ is the vector space of the continuous linear mappings from $E$ into $E_1$ when $E$ and $E_1$ are normed vector spaces.
\begin{equation}\label{eq12} \forall j \in \{ 0,...,n_i + n_e \}, g^j \in C^1({\mathbb{R}}^n, {\mathbb{R}}). \end{equation}
where $C^1$ means continuously Fr\'echet differentiable.
\vskip1mm To establish a Pontryagin principle for problem (${\mathfrak M}$) under assumptions which are so light as possible, we follow the method created by Philippe Michel in \cite{Mi} for systems governed by ordinary differential equations. This method is also used in \cite{ATF} for Bolza problems. We can say that this work is an essay to generalize the method of Michel to the setting of systems governed by functional differential equations.
\vskip1mm On the question of the resolvent of (nonautonomous) linear functional differential equations, the difference between the results that we use (issued from \cite{BK1}) and the results of Banks \cite{Ba} is the choice of the class of solutions: we use continuously differentiable and piecewise continuously solutions with a continuous vector field and Banks uses absolutely continuous solutions without the continuity of the vector field. On the problem of optimal control, the difference between our setting and the setting of Banks \cite{Ba1} (except that Banks considers a Lagrange problem) is that we use piecewise continuously differentiable state variables and piecewise continuous control variables (as \cite{Mi} and \cite{ATF}) and banks uses absolutely continuous state variables and bounded measurable control variables.
\vskip1mm Now we describe the contents of the paper. In Section 2 we specify the notation. In Section 3 we recall some precise properties of the resolvent of linear functional differential equations in the framework of piecewise continuous functions. In Section 4 we give the statement of a Pontryagin principle. In Section 5 we give a proof of this Pontryagin principle.
\section{Notation}
\vskip1mm ${\mathbb{M}}_n({\mathbb{R}})$ denotes the space of the real $n \times n$ matrices. $\Vert \cdot \Vert_{\mathfrak L}$ denotes the norm of the linear continuous operators. \vskip1mm When $a < b$ are two real numbers, $C^0_R([a,b], {\mathbb{R}}^n)$ (respectively $C^0_L([a,b], {\mathbb{R}}^n)$) is the space of the right-continuous (respectively left-continuous) functions from $[a,b]$ into ${\mathbb{R}}^n$, and $C^1([a,b], {\mathbb{R}}^n)$ is the space of the continuously differentiable functions from $[a,b]$ into ${\mathbb{R}}^n$. \vskip1mm $BV([a,b], {\mathbb{R}}^n)$ is the space of the bounded variation functions from $[a,b]$ into ${\mathbb{R}}^n$. When $g \in BV([a,b], {\mathbb{R}}^n)$ the variation of $g$ on $[a,b]$ is denoted by $V_a^b(g)$. We set $NBV([a,b], {\mathbb{R}}^n) := \{ g \in BV([a,b], {\mathbb{R}}^n) \cap C^0_L([a,b], {\mathbb{R}}^n) : g(a) = 0 \}$. When $g \in NBV([a,b], {\mathbb{R}}^n)$, $\Vert g \Vert_{BV} := V_a^b(g)$ defines a norm on $NBV([a,b], {\mathbb{R}}^n)$.
\vskip1mm $AC([a,b], {\mathbb{R}}^n)$ is the space of the absolutely continuous functions $[a,b]$ into ${\mathbb{R}}^n$.
\vskip1mm Let $g : [a,b] \rightarrow {\mathbb{R}}^n$ be a function, and $t \in [a,b)$ (respectively $(a,b]$) when it exists the right-hand limit (respectively the left-hand limit) of $g$ at $t$ is $g(t+) := \lim_{s \rightarrow t, s >t} g(s)$ (respectively $g(t-) := \lim_{s \rightarrow t, s <t} g(s)$).
\vskip1mm Let $f : [a,b] \times [c,d] \rightarrow {\mathbb{R}}^n$ be a mapping, and let $(t,s) \in [a,b) \times [c,d]$ (respectively $(a,b] \times [c,d]$). When it exists the right-partial derivative (respectively left-partial derivative)with respect to the first variable of $f$ at $(t,s)$ is denoted by $\frac{\partial f(t,s)}{\partial t+}$ (respectively $\frac{\partial f(t,s)}{\partial t-}$).
\vskip1mm A function $g : [a,b] \rightarrow {\mathbb{R}}^n$ is called piecewise continuous when it is continuous or when there exists a finite list of points, $t_0=a < t_1< ...< t_p< t_{p+1} = b$ such that $g$ is continuous at each $t \in [a,b] \setminus \{ t_k : k \in \{0, ...,p+1 \} \}$ and such that, for all $k \in \{0,...,p \}$, $g(t_k+)$ exists, and for all $k \in \{ 1,...,p+1 \}$, $g(t_k-)$ exists. We denote by $PC^0([a,b], {\mathbb{R}}^n)$ the space of the piecewise continuous functions from $[a,b]$ into ${\mathbb{R}}^n$. $\Vert g \Vert_{\infty} := \sup \{ \Vert g(t) \Vert : t \in [a,b] \}$ defines a norm on $PC^0([a,b], {\mathbb{R}}^n)$; endowed with this norm, $PC^0([a,b], {\mathbb{R}}^n)$ is not complete. When $g \in PC^0([a,b], {\mathbb{R}}^n)$ we denote by $N_g$ the set points $ t \in [a,b]$ where $g$ is not continuous at $t$. When we fix a finite subset $\pi \subset [a,b]$, we set $PC^0_{\pi}([a,b], {\mathbb{R}}^n) := \{ g \in PC^0([a,b], {\mathbb{R}}^n) : N_g \subset \pi \}$. Endowed with $\Vert . \Vert_{\infty}$, $PC^0_{\pi}([a,b], {\mathbb{R}}^n)$ is a Banach space.
\vskip1mm A function $g : [a,b] \rightarrow {\mathbb{R}}^n$ is called piecewise-$C^1$ when $g$ is $C^1$ on $[a,b]$ or when $g \in C^0([a,b], {\mathbb{R}}^n)$ and there exists a finite list $t_0=a < t_1 < ... < t_p < t_{p+1} = b$ such that $g$ is $C^1$ on $[t_k, t_{k+1}]$ for all $k \in \{ 0, ..., p \}$ and such that, for all $k \in \{ 0,...,p \}$, $g'(t_k +)$ exists and, for all $k \in \{1, ..., p+1 \}$, $g'(t_k -)$ exists. We denote by $PC^1([a,b], {\mathbb{R}}^n)$ the space of the piecewise-$C^1$ functions from $[a,b]$ into ${\mathbb{R}}^n$. When $g \in PC^1([a,b], {\mathbb{R}}^n)$ we denote by $N_{g'}$ the set of the $t \in [a,b]$ such that $g'$ is not continuous at $t$. When we fix a finite subset $\pi \subset [a,b]$, we set $PC^1_{\pi}([a,b], {\mathbb{R}}^n) := \{ g \in PC^1([a,b], {\mathbb{R}}^n) : N_{g'} \subset \pi \}$.
\vskip1mm
In a normed space $E$, when $x \in E$ and $r \in (0, + \infty)$ we set $\overline{B}(x,r) := \{ z \in E : \Vert z -x \Vert \leq r \}$. \section{Linear functional differential equations}
\subsection{The continuous time framework}
We consider a mapping $L : [0,T] \rightarrow {\mathfrak L}(C^0([-r, 0], {\mathbb{R}}^n), {\mathbb{R}}^n)$ which satisfies the following condition.
\begin{equation}\label{eq31} L \in C^0([0,T], {\mathfrak L}(C^0([-r, 0], {\mathbb{R}}^n), {\mathbb{R}}^n)). \end{equation}
From $L$ and a function $\phi \in C^0([-r, 0], {\mathbb{R}}^n)$, when $\sigma \in [0,T]$, we consider the following linear functional differential equation under an initial condition.
\begin{equation}\label{eq32} x'(t) = L(t)x_t, \;\; x_{\sigma} = \phi. \end{equation}
When moreover $h \in C^0([0,T], {\mathbb{R}}^n)$ we consider the nonhomogeneous following problem.
\begin{equation}\label{eq33} x'(t) = L(t)x_t + h(t), \;\; x_{\sigma} = \phi. \end{equation}
A solution of one of these problems is a function $x \in C^0([-r, T], {\mathbb{R}}^n)$ which is of class $C^1$ on $[0,T]$ and whom the derative satisfies the equation at each point of $[0,T]$. \\ The only difference between Theorem 4.1 of \cite{BK1} and the following result is the choice of $\eta(t, -r) = 0$ instead of $\eta(t,0) = 0$..
\begin{proposition}\label{prop31} Under \eqref{eq31} there exists a mapping $\eta : [0,T] \times [-r,0] \rightarrow {\mathbb{M}}_n({\mathbb{R}})$ which satisfies the following properties.
\begin{enumerate}
\item[(i)] $\forall t \in [0,T]$, $\eta(t,\cdot) \in NBV([-r,0], {\mathbb{M}}_n({\mathbb{R}}))$
\item[(ii)] $\forall t \in [0,T]$, $\Vert \eta(t,\cdot) \Vert_{BV} = \Vert L(t) \Vert_{\mathfrak L}$
\item[(iii)] $[t \mapsto \eta(t,\cdot)] \in C^0([0,T], NBV([-r,0], {\mathbb{M}}_n({\mathbb{R}})))$
\item[(iv)] $\forall t \in [0,T]$, $\forall \phi \in C^0([-r, T], {\mathbb{R}}^n)$, $L(t)\phi = \int_{-r}^0 d_2 \eta(t, \theta) \phi(\theta)$
\item[(v)] $\eta$ is Lebesgue measurable on $[0,T] \times [-r,0]$
\item[(vi)] $\eta$ is Riemann integrable on $[0,T] \times [-r,0]$.
\end{enumerate}
\end{proposition}
The following result is devoted to the resolvents of the equations of \eqref{eq32} and \eqref{eq33}. It is proven in \cite{BK1}.
\begin{theorem}\label{th32} Under \eqref{eq31} there exists a mapping $X : [0,T] \times [0,T] \rightarrow {\mathbb{M}}_n({\mathbb{R}})$ which satisfies the following properties.
\begin{enumerate}
\item[(i)] $X$ is bounded on $[0,T] \times [0,T]$
\item[(ii)] $\forall s, t \in [0,T]$ such that $s \geq t$, $X(t,s) = I$ (identity)
\item[(iii)] $\forall s \in [0,T]$, $X(\cdot,s) \in AC([0,T], {\mathbb{M}}_n({\mathbb{R}}))$
\item[(iv)] $\forall s, t \in [0,T]$, $X(\cdot,s)$ is right-differentiable and left-differentiable at $t$, \\ $\frac{\partial X(\cdot,s)}{\partial t+} \in C^0_R([0,T], {\mathbb{M}}_n({\mathbb{R}}))$, $\frac{\partial X(\cdot,s)}{\partial t-} \in C^0_L([0,T], {\mathbb{M}}_n({\mathbb{R}}))$
\item[(v)] $\forall s, t \in [0,T]$ such that $s \geq t$, $\frac{\partial X(t,s)}{\partial t+} - \frac{\partial X(t,s)}{\partial t-} = \eta(t, (s-t)+) - \eta(t, (s-t))$
\item[(vi)] $\forall s \in [0,T]$, the set of the points of $[0,T]$ where $X(\cdot,s)$ is not differentiable is at most countable
\item[(vii)] $\forall t \in [0,T]$, $X(t,\cdot) \in BV([0,T], {\mathbb{M}}_n({\mathbb{R}})) \cap C^0_L([0,T], {\mathbb{M}}_n({\mathbb{R}}))$.
\end{enumerate}
We define the mapping $Z : \{(t,\sigma) \in [0,T] \times [0,T] : t \geq \sigma \} \times C^0([-r,0] , {\mathbb{R}}^n) \rightarrow {\mathbb{R}}^n$ by setting
\[ Z(t,\sigma, \phi) := \int_{\sigma}^t X(t,\xi) \left( \int_{-r}^{\sigma - \xi} d_2 \eta(\xi, \theta) \phi(\xi - \sigma + \theta) \right) d \xi. \]
Then the following properties hold for all $\sigma \in [0,T]$ and for all $\phi \in C^0([-r,0] , {\mathbb{R}}^n)$.
\begin{enumerate}
\item[(viii)] $Z(\cdot,\sigma, \phi) \in C^0([ \sigma, T], {\mathbb{R}}^n)$
\item[(ix)] $\forall t \in [\sigma,T]$, $Z(\cdot,\sigma, \phi)$ is right-differentiable and left-differentiable at $t$, \\ $\frac{\partial Z(\cdot,\sigma, \phi)}{\partial t+} \in C^0_R(([ \sigma, T], {\mathbb{R}}^n)$, and $\frac{\partial Z(\cdot,\sigma, \phi)}{\partial t-} \in C^0_L(([ \sigma, T], {\mathbb{R}}^n)$
\item[(x)] $\forall t \in [\sigma,T]$, $\frac{\partial Z(t,\sigma, \phi)}{\partial t+} - \frac{\partial Z(t,\sigma, \phi)}{\partial t-} = \frac{\partial X(t,s)}{\partial t-} - \frac{\partial X(t,s)}{\partial t+}$.
\end{enumerate}
We define the set ${\mathfrak D} := \{ (t,\sigma) \in [0,T] \times [0,T] : t \geq \sigma -r \}$ and we define the mapping $U : {\mathfrak D} \times C^0([-r,0] , {\mathbb{R}}^n) \rightarrow {\mathbb{R}}^n$ by setting
\[ U(t, \sigma, \phi) := \left\{ \begin{array}{lcl} X(t, \sigma) \phi(0) + Z(t, \sigma, \phi) & {\rm if} & t \geq \sigma\\ \phi(t- \sigma) & {\rm if} & \sigma -r \leq t \leq \sigma. \end{array} \right. \]
Then the following assertions hold for all $\sigma \in [0,T]$ for all $\phi \in C^0([-r,0] , {\mathbb{R}}^n)$.
\begin{enumerate}
\item[(xi)] $U(\cdot, \sigma, \phi)_{\vert_{[\sigma, t]}} \in C^1([\sigma, T], {\mathbb{R}}^n)$, and $U(\cdot, \sigma, \phi) \in C^0([\sigma -r,T], {\mathbb{R}}^n)$
\item[(xii)] $U(\cdot, \sigma, \phi)$ is the (unique) solution of \eqref{eq32} on $[\sigma, T]$.
\end{enumerate} Moreover we define the mapping $V : {\mathfrak D} \times C^0([-r,0] , {\mathbb{R}}^n) \times C^0([0,T], {\mathbb{R}}^n) \rightarrow {\mathbb{R}}^n$ by setting
\[ V(t, \sigma, \phi, h) := \left\{ \begin{array}{lcl} U(t, \sigma, \phi) + \int_{\sigma}^t X(t, \alpha)h(\alpha) d \alpha & {\rm if} & t \geq \sigma\\ \phi(t- \sigma) & {\rm if} & \sigma -r \leq t \leq \sigma. \end{array} \right. \]
Then the following assertions hold for all $\sigma \in [0,T]$ for all $\phi \in C^0([-r,0] , {\mathbb{R}}^n)$ and for all $h \in C^0([0,T), {\mathbb{R}}^n)$
\begin{enumerate}
\item[(xiii)] $V(\cdot, \sigma, \phi, h)_{\vert_{[\sigma, T]}} \in C^1([\sigma, T], {\mathbb{R}}^n)$ and $V(\cdot, \sigma, \phi, h) \in C^0([\sigma -r, T], {\mathbb{R}}^n)$
\item[(xiv)] $V(\cdot, \sigma, \phi, h)$ is the (unique) solution of \eqref{eq33} on $[\sigma, T]$.
\end{enumerate}
\end{theorem}
\vskip2mm The proof of this theorem is contained into Section 6 of \cite{BK1}. The only differences between the results of \cite{BK1} and the present paper are the replacing of the condition $g(b) = 0$ by $g(a) = 0$ in the definition of $NBV([a,b], {\mathbb{R}}^n)$ and the extension of $\eta$ into $\eta^1$ with $\eta^1(t,\theta) = 0$ when $\theta < -r$ instead of $\eta(t,\theta) = 0$ when $\theta > 0$.
\subsection{The piecewise continuous time framework}
Instead of the continuity of the vector field $L$ we assume that
\begin{equation}\label{eq34} L \in PC^0([0,T], {\mathfrak L}(C^0([-r,0],{\mathbb{R}}^n), {\mathbb{R}}^n)). \end{equation}
When $\phi \in C^0([-r,0],{\mathbb{R}}^n)$, when $\sigma \in [0,T]$, and when $h \in PC^0([0,T], {\mathbb{R}}^n)$, we consider the following problems.
\begin{equation}\label{eq35} \forall t \in [0, T] \setminus N_{x'}, x'(t) = L(t)x_t, \; \; x_{\sigma} = \phi \end{equation}
\begin{equation}\label{eq36} \forall t \in [0, T] \setminus N_{x'}, x'(t) = L(t)x_t + h(t), \; \; x_{\sigma} = \phi. \end{equation}
A solution of \eqref{eq35} or of \eqref{eq36} is a function $x \in PC^1([0,T], {\mathbb{R}}^n)$; more precisely $x \in PC^1_{N_L}([0,T], {\mathbb{R}}^n)$ for \eqref{eq35} and $x \in PC^1_{N_L \cup N_h}([0,T], {\mathbb{R}}^n)$ for \eqref{eq36}. We can deduce the results of the piecewise continuous time from those of the continuous time framework by proceeding in the following way. If $0 = t_0 < t_1 < ...< t_p < t_{p+1} = T$ are the points of $N_L$, for all $k \in \{ 0,..., p \}$ we denote by $L_k$ the (continuous) restriction of $L$ at $[t_k, t_{k+1}]$. When we fix $\sigma \in [0,T)$ we consider the index $m$ such $\sigma < t_m < ... <t_{p+1} = T$ and we split the problem \eqref{eq35} into a finite list of problems like \eqref{eq32} as follows: first we have a solution $z^m$ of the problem $(x'(t) = L_{m-1}(t)x_t, x_{\sigma} = \phi)$ on $[\sigma, t_m]$, secondly we have the solution $z^{m+1}$ of the problem $(x'(t) = L_m (t)x_t, x_{\sigma} = \phi_m)$ on $[t_m, t_{m+1}]$, where $\phi_m(\theta) := z^m(t_m + \theta)$ for $\theta \in [-r,0]$, and inductively until to have a solution $z^{p+1}$ of the problem $(x'(t) = L_{p}(t)x_t, x_{\sigma} = \phi_p)$ on $[t_p, t_{p+1}]$, where $\phi_p(\theta) := z^p(t_p + \theta)$ for $\theta \in [-r,0]$. Then the function $z : [-t,T] \rightarrow {\mathbb{R}}^n$, defined by $z(t) := z^k(t)$ when $t \in [t_{k-1}, t_k]$, is a solution of \eqref{eq35}.
\vskip1mm Moreover using Proposition \ref{prop31}, for all $k \in \{ 0, ..., p \}$, we obtain the existence of $\eta_k :[t_k, t_{k+1}] \times [-r, 0] \rightarrow {\mathbb{M}}_n({\mathbb{R}})$ which satisfies the conclusions of Proposition \ref{prop31} where $[t_k, t_{k+1}]$ replaces $[0,T]$ and where $L_k$ replaces $L$. We define $\eta : [0,T] \times [-r,0] \rightarrow {\mathbb{M}}_n({\mathbb{R}})$ by setting $\eta(t, \theta) := \eta_k(t, \theta)$ when $t \in [t_k, t_{k+1})$, $ \theta \in [-r,0]$ when $k \in \{ 0, ..., p-1 \}$ and $\eta(t, \theta) := \eta_p(t, \theta)$ when $t \in [t_p, T]$, $\theta \in [-r,0]$. And so, from Proposition \ref{prop31} we deduce the following result.
\begin{proposition}\label{prop33} Under \eqref{eq34} there exists a mapping $\eta : [0,T] \times [-r,0] \rightarrow {\mathbb{M}}_n({\mathbb{R}})$ which satisfies the following properties. \begin{enumerate}
\item[(i)] $\forall t \in [0,T]$, $\eta(t,\cdot) \in NBV([-r,0], {\mathbb{M}}_n({\mathbb{R}}))$
\item[(ii)] $\forall t \in [0,T]$, $\Vert \eta(t,\cdot) \Vert_{BV} = \Vert L(t) \Vert_{\mathfrak L}$
\item[(iii)] $[t \mapsto \eta(t,\cdot)] \in PC^0_{N_L}([0,T], NBV([-r,0], {\mathbb{M}}_n({\mathbb{R}})))$
\item[(iv)] $\forall t \in [0,T]$, $\forall \phi \in C^0([-r, T], {\mathbb{R}}^n)$, $L(t)\phi = \int_{-r}^0 d_2 \eta(t, \theta) \phi(\theta)$
\item[(v)] $\eta$ is Lebesgue measurable on $[0,T] \times [-r,0]$
\item[(vi)] $\eta$ is Riemann integrable on $[0,T] \times [-r,0]$.
\end{enumerate}
\end{proposition}
Now we want to obtain a result which is analogous to Theorem \ref{th32} for the piecewise continuous time framework. We proceed as in \cite{BK1}, replacing the property $[t \mapsto \eta(t,\cdot)] \in C^0([0,T], NBV([-r,0], {\mathbb{M}}_n({\mathbb{R}}))$ by the property $[t \mapsto \eta(t,\cdot)] \in PC^0_{N_L}([0,T], NBV([-r,0], {\mathbb{M}}_n({\mathbb{R}}))$ and then we obtain the following result.
\begin{theorem}\label{th34} Under \eqref{eq34} there exists a mapping $X : [0,T] \times [0,T] \rightarrow {\mathbb{M}}_n({\mathbb{R}})$ which satisfies the following properties.
\begin{enumerate}
\item[(i)] $X$ is bounded on $[0,T] \times [0,T]$
\item[(ii)] $\forall s, t \in [0,T]$ such that $s \geq t$, $X(t,s) = I$ (identity)
\item[(iii)] $\forall s \in [0,T]$, $X(\cdot,s) \in AC([0,T], {\mathbb{M}}_n({\mathbb{R}}))$
\item[(iv)] $\forall s, t \in [0,T]$, $X(\cdot,s)$ is right-differentiable and left-differentiable at $t$, \\ $\frac{\partial X(\cdot,s)}{\partial t+} \in C^0_R([0,T], {\mathbb{M}}_n({\mathbb{R}}))$, $\frac{\partial X(\cdot,s)}{\partial t-} \in C^0_L([0,T], {\mathbb{M}}_n({\mathbb{R}}))$
\item[(v)] $\forall s \in [0,T]$, $\forall t \in [0,T] \setminus N_L$ such that $s \geq t$, \\ $\frac{\partial X(t,s)}{\partial t+} - \frac{\partial X(t,s)}{\partial t-} = \eta(t, (s-t)+) - \eta(t, (s-t))$
\item[(vi)] $\forall s \in [0,T]$, the set of the points of $[0,T]$ where $X(\cdot,s)$ is not differentiable is at most countable
\item[(vii)] $\forall t \in [0,T]$, $X(t,\cdot) \in BV([0,T], {\mathbb{M}}_n({\mathbb{R}})) \cap C^0_L([0,T], {\mathbb{M}}_n({\mathbb{R}}))$.
\end{enumerate}
We define the mapping $Z : \{(t,\sigma) \in [0,T] \times [0,T] : t \geq \sigma \} \times C^0([-r,0] , {\mathbb{R}}^n) \rightarrow {\mathbb{R}}^n$ by setting
\[ Z(t,\sigma, \phi) := \int_{\sigma}^t X(t,\xi) \left( \int_{-r}^{\sigma - \xi} d_2 \eta(\xi, \theta) \phi(\xi - \sigma + \theta) \right) d \xi. \]
Then the following properties hold for all $\sigma \in [0,T]$ and for all $\phi \in C^0([-r,0] , {\mathbb{R}}^n)$.
\begin{enumerate}
\item[(viii)] $Z(\cdot,\sigma, \phi) \in C^0([ \sigma, T], {\mathbb{R}}^n)$
\item[(ix)] $\forall t \in [\sigma,T]$, $Z(\cdot,\sigma, \phi)$ is right-differentiable and left-differentiable at $t$, \\ $\frac{\partial Z(\cdot,\sigma, \phi)}{\partial t+} \in C^0_R(([ \sigma, T], {\mathbb{R}}^n)$, and $\frac{\partial Z(\cdot,\sigma, \phi)}{\partial t-} \in C^0_L(([ \sigma, T], {\mathbb{R}}^n)$
\item[(x)] $\forall t \in [\sigma,T]\setminus N_L$, $\frac{\partial Z(t,\sigma, \phi)}{\partial t+} - \frac{\partial Z(t,\sigma, \phi)}{\partial t-} = \frac{\partial X(t,s)}{\partial t-} - \frac{\partial X(t,s)}{\partial t+}$.
\end{enumerate}
We define the set ${\mathfrak D} := \{ (t,\sigma) \in [0,T] \times [0,T] : t \geq \sigma -r \}$ and we define the mapping $U : {\mathfrak D} \times C^0([-r,0] , {\mathbb{R}}^n) \rightarrow {\mathbb{R}}^n$ by setting
\[ U(t, \sigma, \phi) := \left\{ \begin{array}{lcl} X(t, \sigma) \phi(0) + Z(t, \sigma, \phi) & {\rm if} & t \geq \sigma\\ \phi(t- \sigma) & {\rm if} & \sigma -r \leq t \leq \sigma. \end{array} \right. \]
Then the following assertions hold for all $\sigma \in [0,T]$ for all $\phi \in C^0([-r,0] , {\mathbb{R}}^n)$.
\begin{enumerate}
\item[(xi)] $U(\cdot, \sigma, \phi)_{\vert_{[\sigma, t]}} \in PC^1([\sigma, T], {\mathbb{R}}^n)$, and $U(\cdot, \sigma, \phi) \in C^0([\sigma -r,T], {\mathbb{R}}^n)$
\item[(xii)] $U(\cdot, \sigma, \phi)$ is the (unique) solution of \eqref{eq35} on $[\sigma, T]$.
\end{enumerate} Moreover we define the mapping $V : {\mathfrak D} \times C^0([-r,0] , {\mathbb{R}}^n) \times C^0([0,T], {\mathbb{R}}^n) \rightarrow {\mathbb{R}}^n$ by setting
\[ V(t, \sigma, \phi, h) := \left\{ \begin{array}{lcl} U(t, \sigma, \phi) + \int_{\sigma}^t X(t, \alpha) h(\alpha) d \alpha & {\rm if} & t \geq \sigma\\ \phi(t- \sigma) & {\rm if} & \sigma -r \leq t \leq \sigma. \end{array} \right. \]
Then the following assertions hold for all $\sigma \in [0,T]$ for all $\phi \in C^0([-r,0] , {\mathbb{R}}^n)$ and for all $h \in C^0([0,T), {\mathbb{R}}^n)$
\begin{enumerate}
\item[(xiii)] $V(\cdot, \sigma, \phi, h)_{\vert_{[\sigma, T]}} \in PC^1([\sigma, T], {\mathbb{R}}^n)$ and $V(\cdot, \sigma, \phi, h) \in C^0([\sigma -r, T], {\mathbb{R}}^n)$
\item[(xiv)] $V(\cdot, \sigma, \phi, h)$ is the (unique) solution of \eqref{eq36} on $[\sigma, T]$.
\end{enumerate}
\end{theorem}
\subsection{Adjoint equation.}
\begin{proposition}\label{prop35} Let $X$ be provided by Theorem \ref{th34} and $\eta$ be provided by Proposition \ref{prop31}. We define $Y : [0,T] \times [0,T] \rightarrow {\mathbb{M}}_n({\mathbb{R}})$ by setting $Y(s,t) := X(t,s)$. Then when $s \leq t$, the following equation is satisfied: $Y(s,t) = I - \int_s^t Y(\alpha, t) \eta(\alpha, s - \alpha) d\alpha$. \end{proposition}
\begin{proof} We set $k(\alpha,s) := \eta(\alpha, s- \alpha)$. After Theorem 5.4 in \cite{BK1}, there exists $R$ which satisfies $R(t,s) = k(t,s) - \int_s^t R(t,\alpha) k(\alpha,s) d \alpha$. Ever after \cite{BK1} we have $X(t,s) := I - \int_s^t R(\alpha,s) d \alpha$. Then we calculate
\[ \begin{array}{cl} \null & I - \int_s^t Y(\alpha,t) k (\alpha,s) d \alpha = I - \int_s^t X(t,\alpha) k (\alpha,s) d \alpha \\ = & I - \int_s^t ( I - \int_{\alpha}^t R(\beta,\alpha) d \beta ) k (\alpha,s) d \alpha \\ = & I - \int_s^t k(\alpha,s) d \alpha + \int_s^t (\int_{\alpha}^t R(\beta,\alpha) k (\alpha,s) d \beta ) d \alpha \\ = & I - \int_s^t k(\beta,s) d \beta + \int_s^t ( \int_s^{\beta} R(\beta,\alpha) k (\alpha,s) d \alpha ) d \beta\\ = & I - \int_s^t [ k(\beta,s) - \int_s^t \int_s^{\beta} R(\beta,\alpha) k (\alpha,s) d \alpha] d \beta\\ = & I - \int_s^t R(\beta,s) d \beta = X(t,s) = Y(s,t). \end{array} \]
\end{proof}
The integral equation which is present into the previous statement is called the adjoint equation of \eqref{eq35}.
\section{Pontryagin principle}
First we give the qualification condition of Michel \cite{Mi}, where $(\overline{x}, \overline{u})$ is an admissible process of $({\mathfrak M})$.
\[ {\rm (QC)} \left\{ \begin{array}{rl} \null & \forall (c_j)_{0 \leq j \leq ni + n_e} \in {\mathbb{R}}^{1 + p+ q}\\ {\rm if} & \forall j = 0,...,n_i, \; c_j \geq 0\\ \null & \forall j= 1,...,n_i, \; c_j g^j(\overline{x}(T))= 0\\ \null & \sum_{j=0}^{n_i + n_e} c_j Dg^j(\overline{x}(T)) = 0\\ {\rm then} & \forall j = 0,...,n_i + n_e, \; c_j = 0. \end{array} \right. \]
The main result of the paper is the following statement of a Pontryagin principle.
\begin{theorem}\label{th41} Under the assumptions \eqref{eq11} and \eqref{eq12}, if $(\overline{x}, \overline{u})$ is a solution of the Mayer problem $({\mathfrak M})$ then there exist $\lambda_0$,..., $\lambda_{n_i + n_e} \in {\mathbb{R}}$ and there exists $p \in BV([0, T], {\mathbb{R}}^{n*}) \cap C^0_L([0,T], {\mathbb{R}}^{n*})$ which satisfy the following conditions.
\begin{enumerate}
\item[(NN)] $(\lambda_0,..., \lambda_{n_i + n_e}) \neq (0,...,0)$
\item[(Si)] $\forall j \in \{0,...,n_i \}$, $\lambda_j \geq 0$
\item[(Sl)] $\forall j \in \{0,...,n_i \}$, $\lambda_j g^j(\overline{x}(T)) = 0$
\item[(AE)] $\forall t \in [0,T]$, $\frac{d}{dt} (p(t) + \int_t^{\min\{t+r, T \}} p(\alpha) \eta(\alpha, t - \alpha) d \alpha) = 0$
\item[(T)] $p(T) = \sum_{j=0}^{n_i + n_e} \lambda_j Dg^j(\overline{x}(T))$
\item[(MP)] $\forall t \in [0,T]$, $\forall u \in U$, \hskip2mm $p(t) f(t, \overline{x}_t, \overline{u}(t)) \geq p(t)f(t, \overline{x}_t, u)$
\end{enumerate}
If moreover (QC) is fulfilled, we obtain the following additional conclusions.
\begin{enumerate}
\item[(A2)] There exists $\epsilon > 0$ such that $p(t) \neq 0$ for all $t \in (T- \epsilon, T]$.
\item[(A2)] $p(T) \neq 0$ and, for all $t \in [0,T]$, $p_{\mid_{[t, \min \{t +r, T \}]}} \neq 0$.
\end{enumerate}
\end{theorem}
\vskip3mm
In this statement, (NN) is a condition of non nullity of the multipliers, (Si) is a condition on the signs of multipliers, (Sl) is a condition of slackness on the final inequality constraints, (AE) is called the adjoint equation, (T) is the transversality condition, and (MP) is the maximum principle. The mapping $\eta$ which is present into (AE) comes from Proposition \ref{prop33} with $L(t) = D_2f(t, \overline{x}_t, \overline{u}(t))$.
\begin{remark}\label{rem42} The conclusion (AE) is equivalent to say that the function $[t \mapsto p(t) + \int_t^{\min\{ t+r, T \}} p(\xi) \eta(\xi, \xi -t) d \xi]$ is constant on $[0,T]$. Since $\int_T^{\min\{ T+r, T \}} p(\xi) \eta(\xi, \xi -t) d \xi = 0$, using (TC), we obtain that (AE) is equivalent to
$$\forall t \in [0,T],\; \; p(t) + \int_t^{\min\{ t+r, T \}} p(\xi) \eta(\xi, \xi -t) d \xi = p(T) = \sum_{j=0}^{n_i + n_e} \lambda_j Dg^j(\overline{x}(T)).$$
\end{remark}
\section{proof of the Pontryagin principle}
Our proof follows the proof given by Michel in \cite{Mi}; we provide the useful changes to adapt it to the setting of systems governed by a functional differential equation. Note that we can choose $\overline{u}$ as a right-continuous function without to lost generality. We arbitrarily fix $S = \{ (t_i,v_i) : i \in \{ 1,...,N \} \}$ where $0 \leq t_1 \leq ...\leq t_N < T$ and the $v_i \in U$; we denote by ${\mathfrak S}$ the set of all these $S$. When $a = (a_1,...,a_N) \in {\mathbb{R}}^N_+$, we define $J(i) := \{ j \in \{1,...,i \} : t_j = t_i \}$ and we set $b_i := 0$ when $J(i) = \emptyset$ and $b_i := \sum_{j \in J(i)}a_j$ when $J(i) \neq \emptyset$. After that we define the subintervals $I_i := [t_i + b_i, t_i + b_i + a_i)$ and $u(t,S,a) := v_i$ when $t \in I_i$ and the control $u(t,S,a) := \overline{u}(t)$ when $t \in [0,T] \setminus \cup_{1 \leq i \leq N} I_i$. Taking $a$ small enough we have $I_i \subset [0,T]$ and the $I_i$ are pairwise disjoint. We denote by $x(\cdot,S,a)$ the unique solution on $[0,T]$ of the following problem of Cauchy
$$\forall t \in [0,T] \setminus N_{u(\cdot,S,a)}, \frac{\partial x(t,S,a)}{\partial t} = f(t, x(\cdot,S,a)_t, u(t,S,a)), \: \; x(\cdot,S,a)_0 = \phi.$$
\begin{lemma}\label{lem51} We fix $S \in {\mathfrak S}$ and $a \in {\mathbb{R}}^N_+$. We denote by $z(\cdot,S,a)$ the solution on $[0,T]$ of the following linear problem
\[ \left\{ \begin{array}{l} \forall t \in [0,T] \setminus N_{u(.,S,a)},\\
\frac{\partial z(t,S,a)}{\partial t} = D_2f(t, \overline{x}_t, \overline{u}(t))z(\cdot,S,a)_t + [f(t, \overline{x}_t, u(t,S,a)) -f(t, \overline{x}_t,\overline{u}(t))]\\
z(\cdot,S,a)_0 = 0. \end{array} \right. \]
Then the partial differential $D_a z(T,S,0)$ exists and the following equality holds
$$D_a z(T,S,0)a = \sum_{i=1}^N a_i X(T,t_i)[f(t_i, \overline{x}_{t_i}, v_i) - f(t_i, \overline{x}_{t_i},\overline{u}(t_i))].$$
where $X(T,t_i)$ comes from Theorem \ref{th34} when $L(t) = D_2f(t, \overline{x}_t, \overline{u}(t))$. \end{lemma}
\begin{proof} We set $\Delta(t,S,a) := [f(t, \overline{x}_t, u(t,S,a)) -f(t, \overline{x}_t,\overline{u}(t))]$ and using Theorem \ref{th34} we obtain $z(T,S,a) = V(T,0,0,\Delta(\cdot,S,a))$= $X(T,0)0 + \int_0^T X(T,\xi) (\int_{-r}^0 d_2\eta(\xi,\theta) 0(\xi + \theta) d \xi + \int_0^T X(T,\xi) \Delta(\xi, S,a) d \xi$ = $\int_0^T X(T,\xi) \Delta(\xi, S,a) d \xi$. Since $\Delta(\xi,S,a) = 0$ when $\xi \in [0,T] \setminus \cup_{1 \leq i \leq N} I_i$, we obtain $$z(T,S,a) = \sum_{i=1}^N \int_{t_i + b_i}^{t_i + b_i + a_i} X(T,\xi) [f(t_i, \overline{x}_{t_i}, v_i) - f(t_i, \overline{x}_{t_i},\overline{u}(t_i))] d\xi.$$
Note that
\[ \begin{array}{cl} \null & a_i X(T,t_i) [f(t_i, \overline{x}_{t_i}, v_i) - f(t_i, \overline{x}_{t_i},\overline{u}(t_i))]\\
=& \int_{t_i + b_i}^{t_i + b_i + a_i}X(T,t_i)[f(t_i, \overline{x}_{t_i}, v_i) - f(t_i, \overline{x}_{t_i},\overline{u}(t_i))]d \xi. \end{array} \]
Since $u(t,S,0) = \overline{u}(t)$ we have $\Delta(t,S,0) = 0$ and then $z(T,S,0) = 0$. For all $i \in \{1,...,N \}$ we define $\rho_i := {\mathfrak S} \times {\mathbb{R}}^N_+ \rightarrow {\mathbb{R}}^n$ by setting
$$\varrho_i(S,a) := \frac{1}{a_i} \int_{t_i + b_i}^{t_i + b_i + a_i} [X(t,s) \Delta(s,S,a) - X(T,t_i)( f(t_i, \overline{x}_{t_i}, v_i) - f(t_i, \overline{x}_{t_i},\overline{u}(t_i)))] ds$$
when $a_i \neq 0$ and $\varrho_i(S,a) := 0$ when $a_i = 0$. Then the following formula holds.
\begin{equation}\label{eq51} z(T,S,a) = z(T,S,0) + \sum_{i=1}^N a_i X(T,t_i) \Delta(t_i,S,a) + \sum_{i=1}^N a_i \varrho_i(S,a). \end{equation}
Now to prove the result, it sufffices to prove that the following assertion holds.
\begin{equation}\label{eq52} \forall i \in \{1,...,N \}, \; \; \lim_{a \rightarrow 0} \varrho_i(S,a) = 0. \end{equation}
In the formula of $\varrho_i$ we do the change of variable $s =t_i + b_i + \theta a_i$ with $\theta \in [0,1]$, setting $\varpi_i(S,a,\theta) := X(T,t_i + b_i+ \theta a_i)\Delta(t_i + b_i+ \theta a_i, S,a) - X(T,t_i)\Delta(t_i,S,a)$, we obtain the following formula.
\begin{equation}\label{eq53} \varrho_i(S,a) = \int_0^1 \varpi_i(S,a,\theta) d \theta. \end{equation}
we arbitrarily fix $\theta \in [0,1)$. Note that, for all $i \in \{ 1,..., N \}$, we have $\lim_{a \rightarrow 0}a_i = 0$ and $\lim_{a \rightarrow 0}b_i = 0$. Since $X(T,\cdot)$ is right-continuous ((vi) of Theorem \ref{th34}) we obtain $\lim_{a \rightarrow 0}X(T, t_i + b_i + \theta a_i) = X(T,t_i)$. Note that $\Delta( t_i + b_i + \theta a_i, S, a) = f( t_i + b_i + \theta a_i, \overline{x}_{ t_i + b_i + \theta a_i}, v_i) - f( t_i + b_i + \theta a_i, \overline{x}_{ t_i + b_i + \theta a_i}, \overline{u}( t_i + b_i + \theta a_i))$. Since $f$ is continuous, since $[t \mapsto \overline{x}_t]$ is continuous and since $\overline{u}$ is right-continuous, we obtain $\lim_{a \rightarrow 0}\Delta( t_i + b_i + \theta a_i, S, a) = f(t_i, \overline{x}_{t_i}, v_i) - f(t_i, \overline{x}_{t_i}, \overline{u}(t_i))$ which implies $\lim_{a \rightarrow 0}(X(T,t_i + b_i+ \theta a_i)\Delta(t_i + b_i+ \theta a_i, S,a) - X(T,t_i)\Delta(t_i,S,a)) = X(T,t_i)[ f(t_i, \overline{x}_{t_i}, v_i) - f(t_i, \overline{x}_{t_i}, \overline{u}(t_i))] - X(T,t_i)[ f(t_i, \overline{x}_{t_i}, v_i) - f(t_i, \overline{x}_{t_i}, \overline{u}(t_i))] = 0$. And so we obtain the following property.
\begin{equation}\label{eq54} \forall \theta \in [0,1), \; \; \lim_{a \rightarrow 0} \varpi_i(S,a,\theta) = 0. \end{equation}
We fix $\delta \in (0, + \infty)$ small enough to have $\{ t_i + b_i+ \theta a_i : \theta \in [0,1], a \in {\mathbb{R}}^N_+ \cap \overline{B}(0, \delta) , i \in \{ 1,...,N \} \} \subset [0,T]$, and then the closure of this subset is compact. Since $[t \mapsto \overline{x}_t ]$ is continuous, $\{ \overline{x}_t : t \in [0,T] \}$ is compact. Since $\overline{u}$ is piecewise continuous on $[0,T]$, $\overline{u}([0,T])$ is compact, and $\{u(t_i + b_i+ \theta a_i,S,a) : i \in \{1,...,N \}, a \in {\mathbb{R}}^N_+ \cap \overline{B}(0, \delta), \theta \in [0,1] \}$ is compact. Since $f$ is continuous we obtain that there exists $c \in (0, + \infty)$ such that
\[ \begin{array}{l} \forall a \in {\mathbb{R}}^N_+ \cap \overline{B}(0, \delta), \forall \theta \in [0,1], \forall i \in \{ 1,...,N \}, \\ \Vert f(t_i + b_i+ \theta a_i, \overline{x}_{t_i + b_i+ \theta a_i}, u(t_i + b_i+ \theta a_i,S,a))\\
- f(t_i + b_i+ \theta a_i, \overline{x}_{t_i + b_i+ \theta a_i}, \overline{u}(t_i + b_i+ \theta a_i)) \Vert \leq c. \end{array} \]
Since $X$ is bounded on $[0,T] \times [0,T]$, we obtain the existence of $c_1 \in (0, + \infty)$ such that
\[ \forall a \in {\mathbb{R}}^N_+ \cap \overline{B}(0, \delta), \forall \theta \in [0,1], \forall i \in \{ 1,...,N \}, \Vert \varpi_i(S,a, \theta) \Vert \leq c_1. \]
Since a constant is Lebesgue integrable on $[0,1]$, we can use the theorem of the dominated convergence of Lebesgue to assert that $\lim_{a \rightarrow 0}\varrho_i(S,a) = \int_0^1 0 d \theta = 0$. Then \eqref{eq52} is proven, and the conclusion of the lemma follows from \eqref{eq51}.
\end{proof}
\begin{lemma}\label{lem52} Let $S \in {\mathfrak S}$. There exists $\delta_1 \in (0, + \infty)$ and $c_2 \in (0, + \infty)$ such that, for all $a \in {\mathbb{R}}^N_+ \cap \overline{B}(0, \delta_1)$, $\int_0^T \Vert f(t, \overline{x}_t, u(t,S,a)) - f(t, \overline{x}_t, \overline{u}(t)) \Vert dt \leq c_2 \cdot \Vert a \Vert$. \end{lemma}
\begin{proof} Note that the integrand in the formula is $\Vert \Delta (t,S,a) \Vert$. We introduce $e : [0,T] \times ({\mathbb{R}}^N_+ \cap \overline{B}(0, \delta)) \rightarrow {\mathbb{R}}_+$ by setting $e(t,a):= \int_0^t \Vert \Delta (t,S,a) \Vert dt $.\\
Since $\Delta(T,S,0) = 0$ we have $\sigma (T,0) = 0$. We set $\Xi_i := f(t_i, \overline{x}_{t_i}, v_i) - f(t_i, \overline{x}_{t_i}, \overline{u}(t_i))$. Then we have
\[ \begin{array}{cl} \null & e(T,a) - e(T,0) - \sum_{i=1}^N a_i \Vert \Xi_i \Vert\\ =& \int_0^T \Vert \Delta (t,S,a) \Vert dt - 0- \sum_{i=1}^N a_i \Vert \Xi_i \Vert\\ = & \int_{[0,T] \setminus \cap_{i=1}^N I_i} \Vert \Delta (t,S,a) \Vert dt + \int_{\cap_{i=1}^N I_i} \Vert \Delta (t,S,a) \Vert dt - \sum_{i=1}^N a_i \Vert \Xi_i \Vert\\ =& 0+ \sum_{i=1}^N \int_{I_i} \Vert \Delta (t,S,a) \Vert dt - \sum_{i=1}^N a_i \Vert \Xi_i \Vert\\ =& \sum_{i=1}^N \int_{t_i+ b_i}^{t_i + b_+ + a_i} \Vert \Delta (t,S,a) \Vert dt - \sum_{i=1}^N \int_{t_i+ b_i}^{t_i + b_+ + a_i} \Vert \Xi_i \Vert dt\\ =& \sum_{i=1}^N a_i \frac{1}{a_i} \int_{t_i+ b_i}^{t_i + b_+ + a_i} (\Vert \Delta (t,S,a) \Vert -\Vert \Xi_i \Vert) dt. \end{array} \]
And so defining the function $\nu_i : {\mathfrak S} \times ({\mathbb{R}}^N_+ \cap \overline{B}(0, \delta)) \rightarrow {\mathbb{R}}_+$ by setting
\[ \nu_i(S,a) := \left\{ \begin{array}{lcl} \frac{1}{a_i} \int_{t_i+ b_i}^{t_i + b_+ + a_i} (\Vert \Delta (t,S,a) \Vert -\Vert \Xi_i \Vert) dt& {\rm if} & a_i \neq 0\\ 0 & {\rm if} & a_i = 0, \end{array} \right. \]
we obtain, for all $a \in {\mathbb{R}}^N_+ \cap \overline{B}(0, \delta)$,
\begin{equation}\label{eq55} e(T,a) = e(T,0) + \sum_{i=1}^N a_i \Vert \Xi_i \Vert + \sum_{i=1}^N a_i \nu_i(S,a). \end{equation}
Using the change of variable $s = t_i + b_i + \theta a_i$ we obtain $\nu_i(S,a) = \int_0^1 (\Vert \Delta (t_i + b_i + \theta a_i,S,a) \Vert -\Vert \Xi_i \Vert) d\theta$. In the previous proof we have yet seen that $\lim_{a \rightarrow 0} (\Vert \Delta (t_i + b_i + \theta a_i,S,a) \Vert -\Vert \Xi_i \Vert) = 0$, and that there exists $c \in (0, + \infty)$ such that, for all $a \in {\mathbb{R}}^N_+ \cap \overline{B}(0, \delta)$, for all $\theta \in [0,1]$, $\vert \Vert \Delta (t_i + b_i + \theta a_i,S,a) \Vert -\Vert \Xi_i \Vert \vert \leq c$. And so we can use the theorem of the dominated convergence of Lebesgue and assert that $\lim_{a \rightarrow 0} \nu_i(S,a) = 0$. And then, using \eqref{eq55} we can say that $e(T,\cdot)$ is differentiable at 0 and that its partial differential is $D_2 e(T,0).a = \sum_{i=1}^N a_i \Vert \Xi_i \Vert$ that implies the existence of a mapping $\mu : {\mathfrak S} \times ({\mathbb{R}}^N_+ \cap \overline{B}(0, \delta)) \rightarrow {\mathbb{R}}$ such that $\lim_{a \rightarrow 0} \mu(a) = 0$ and $e(T,a) - e(T,0) - \sum_{i=1}^N a_i \Vert \Xi_i \Vert = \Vert a \Vert \cdot \mu(a)$. We fix $\delta_1 \in (0 , \delta]$ such that $\Vert a \Vert \leq \delta_1 \Longrightarrow \vert \mu(a) \vert \leq 1$. Then, when $a \in {\mathbb{R}}^n_+ \cap \overline{B}(0, \delta_1)$ we have $e(T,a) = \vert \sigma(T,a) \vert \leq \Vert D_2 \sigma(T,0) \Vert \cdot \Vert a \Vert + \Vert a \vert$. To conclude it suffices to take $c_2 := \Vert D_2 e(T,0) \Vert + 1$.
\end{proof}
\begin{lemma}\label{lem53} Let $\Gamma : [0,T] \times C^0([-r,0], {\mathbb{R}}^n) \rightarrow {\mathbb{R}}^n$. Let $0 = \tau_0 < \tau_1 <...<\tau_{\ell} < \tau_{\ell +1} = T$ and $F := \{ \tau_i : i \in \{0,...,{\ell} \} \}$. We assume that, for all $i \in \{0, ..., {\ell} \}$, $\Gamma$ is continuous on $[\tau_i, \tau_{i+1}] \times C^Ã ([-r,0], {\mathbb{R}}^n)$.Then the Nemytskii operator ${\mathcal N}_{\Gamma} : C^0([0,T], C^0([-r,0], {\mathbb{R}}^n)) \rightarrow PC^0_F([0,T],{\mathbb{R}}^n)$, defined by ${\mathcal N}_{\Gamma}(\Phi) := [ t \mapsto \Gamma(t, \Phi(t))]$, is continuous. \end{lemma}
\begin{proof} Let $\Phi \in C^0([0,T], C^0([-r,0], {\mathbb{R}}^n))$. Then, for all $i \in \{ 0,...,{\ell} \}$, the restriction of $\Phi$ at $[\tau_i, \tau_{i+1}]$ belongs to $C^0( [\tau_i, \tau_{i+1}], C^0([-r,0], {\mathbb{R}}^n))$. Since $\Gamma$ is continuous on $[\tau_i, \tau_{i+1}] \times C^0([-r,0], {\mathbb{R}}^n)$, we know that ${\mathcal N}_{\Gamma}(\Phi) \in C^0([\tau_i, \tau_{i+1}], {\mathbb{R}}^n)$. Moreover we obtain that ${\mathcal N}_{\Gamma}(\Phi) \in PC^0_F([0,T], {\mathbb{R}}^n)$.\\
We arbitrarily fix $\Phi \in C^0([0,T], C^0([-r,0], {\mathbb{R}}^n))$. Using Lemma 3.10 in \cite{BCNP}, we know that: $\forall i \in \{ 0,...,{\ell} \}$,$ \forall \epsilon > 0$, $\exists \beta_{\epsilon, i} > 0$, such that, $\forall \Psi_i \in C^0([\tau_i, \tau_{i+1}], C^0([-r,0], {\mathbb{R}}^n))$, $\sup_{t \in [\tau_i, \tau_{i+1}]}\Vert \Psi_i(t) - P(t) \Vert \leq \beta_{\epsilon, i} \Longrightarrow \sup_{t \in [\tau_i, \tau_{i+1}]} \Vert \Gamma(t, \Psi_i(t)) - \Gamma(t, \Phi(t)) \Vert \leq \epsilon$. We arbitrarily fix $\epsilon > 0$ and we set $\beta_{\epsilon} := \min_{0 \leq i \leq {\ell}} \beta_{\epsilon, i} > 0$. Let $\Psi \in C^0([0,T], C^0([-r,0], {\mathbb{R}}^n))$ such that $\sup_{t \in [0,T]} \Vert \Psi(t) - \Phi(t) \Vert \leq \beta_{\epsilon}$; then we have, $\forall i \in \{ 0,...,{\ell} \}$, $\sup_{t \in [\tau_i, \tau_{i+1}]}\Vert \Psi(t) - \Phi(t) \Vert \leq \beta_{\epsilon, i}$ that implies: $\forall i \in \{ 0,...,{\ell} \}$, $\sup_{t \in [\tau_i, \tau_{i+1}]} \Vert \Gamma(t, \Psi(t)) - \Gamma(t, \Phi(t)) \Vert \leq \epsilon$, and consequently we obtain that\\ $\sup_{t \in [0,T]} \Vert \Gamma(t, \Psi(t)) - \Gamma(t, \Phi(t)) \Vert \leq \epsilon$.
\end{proof}
\begin{lemma}\label{lem54} There exist $\delta_2 \in (0, + \infty)$ and $c_4 \in (0, + \infty)$ such the following assertions hold.
\begin{enumerate}
\item[(i)] $\forall a \in {\mathbb{R}}^N_+ \cap \overline{B}(0, \delta_2)$, $x(\cdot,S,a)$ is defined on $[0,T]$ all over.
\item[(ii)] $\forall a \in {\mathbb{R}}^N_+ \cap \overline{B}(0, \delta_2)$, $\forall t \in [0,T]$, $\Vert x(t,S,a)- \overline{x}(t) \Vert \leq c_4. \Vert a \Vert$.
\end{enumerate}
\end{lemma}
\begin{proof} Let $a \in {\mathbb{R}}^N_+ \cap \overline{B}(0, \delta_1)$ where $\delta_1$ is provided by the previous lemma. By induction we built the sequence $(x^m(\cdot,S,a))_{m \in {\mathbb{N}}}$ of functions from $[0,T]$ into ${\mathbb{R}}^n$ in the following way.
\[ \left\{ \begin{array}{l} x^0(\cdot,S,a) := \overline{x}\\ \forall m \geq 1, \forall t \in [0,T], x^m(t,S;a) = \overline{x}(0) + \int_0^t f(s, x^{m-1}(\cdot,S,a)_s, u(s,S,a)) ds\\ \forall m \geq 1, x^m_0 = \phi \end{array} \right. \]
We have $\Vert x^1(t,S,a) - x^0(t,S,a) \Vert = \Vert x^1(t,S,a) - \overline{x}(t) \Vert = \Vert \int_0^t f(s, \overline{x}_s, u(s,S,a)) ds - \int_0^t f(s, \overline{x}_s,\overline{u}(s))ds \Vert \leq \int_0^t \Vert f(s, \overline{x}_s, u(s,S,a)) - f(s, \overline{x}_s,\overline{u}(s)) \Vert ds \leq $ \\ $ \int_0^T \Vert f(s, \overline{x}_s, u(s,S,a)) - f(s, \overline{x}_s,\overline{u}(s)) \Vert ds \leq c_2 \cdot \Vert a \Vert$. And so we have proven the followowing assertion: $ \forall a \in {\mathbb{R}}^N_+ \cap \overline{B}(0, \delta_1), \forall t \in [0,T], \Vert x^1(t,S,a) - x^0(t,S,a) \Vert \leq c_2. \Vert a \Vert$. Then, for all $t \in [0,T]$, $\forall \theta \in [-r,0]$, if $ t+ \theta \geq 0$ then $\Vert x^1(t+ \theta,S,a) - x^0(t + \theta,S,a) \Vert \leq c_2 \cdot \Vert a \Vert$, and if $t+ \theta < 0$ then $\Vert x^1(t+ \theta,S,a) - x^0(t + \theta,S,a) \Vert = \Vert \phi(\theta) - \phi(\theta) \Vert = 0 \leq c_2 \cdot \Vert a \Vert$. And so we have proven the following assertion.
\begin{equation}\label{eq56} \forall a \in {\mathbb{R}}^N_+ \cap \overline{B}(0, \delta_1), \forall t \in [0,T], \Vert x^1(\cdot,S,a)_t - x^0(\cdot,S,a)_t \Vert \leq c_2 \cdot \Vert a \Vert. \end{equation}
We introduce the set
$${\mathfrak B} := \{ (t, \phi, v) \in [0,T] \times C^0([-r,0], {\mathbb{R}}^n) \times U : \Vert \phi - \overline{x}_t \Vert_{\infty} \leq r, v \in \{\overline{u}(t), v_1,...,v_N \} \}.$$
Note that ${\mathfrak B}$ is bounded since it is included into the bounded set $[0,T] \times \overline{B}(\overline{x}, r) \times (\overline{u}([0,T]) \cup \{ v_1...,v_N \})$. Since $D_2f$ is a bounded operator, $c_3 := \sup \{ \Vert D_2f(t,\phi,v) \Vert : (t, \phi, v) \in {\mathfrak B} \} < + \infty$. Now we want to prove by induction the following relation, for all $a \in {\mathbb{R}}^N_+ \cap \overline{B}(0, \delta_1)$ such that $\Vert a \Vert \leq \frac{r}{c_2} e^{-c_3T}$, $\forall m \in {\mathbb{N}}$, $\forall t \in [0,T]$:
\begin{equation}\label{eq57} \left. \begin{array}{r} \Vert x^m(\cdot,S,a)_t - x^0(\cdot,S,a)_t \Vert \leq r, \\ \Vert x^{m+1}(\cdot,S,a)_t - x^m(\cdot,S,a)_t \Vert \leq \frac{1}{m !}c_2 \Vert a \Vert (c_3 t)^m. \end{array} \right\} \end{equation}
For $m=0$, using Lemma \ref{lem52} and \eqref{eq56} we obtain, for all $t \in[0,T]$, $\Vert x^1(\cdot,S,a)_t- x^0(\cdot,S,a)_t \Vert \leq c_2 \cdot \Vert a \Vert = \frac{1}{0!} c_2 \cdot \Vert a \Vert (c_3 t)^0$, and $c_2 \cdot \Vert a \Vert \leq r e^{-c_3 T} \leq r$.\\ Now we assume that \eqref{eq57} holds for the integer $n \leq p-1$. Then, for all $t \in [0,T]$, we have
\[ \begin{array}{cl} \null & \Vert x^p(t,S,a) - x^0(t,S,a) \Vert = \Vert \sum_{m=0}^{p-1} (x^{m+1}(t,S,a) - x^m(t,S,a)) \Vert \\ \leq & \sum_{m=0}^{p-1} \Vert x^{m+1}(t,S,a) - x^m(t,S,a) \Vert \leq \sum_{m=0}^{p-1}(\frac{1}{m !}c_2 \Vert a \Vert (c_3 t)^m)\\
= & c_2 \cdot \Vert a \Vert \sum_{m=0}^{p-1} \frac{(c_3 t)^m}{m !} \leq c_2 \cdot \Vert a \Vert e^{c_3 t} \leq c_2 \cdot \Vert a \Vert e^{c_3 T} \leq r, \end{array} \]
and since $x^p(\cdot, S,a)_0 = x^0(\cdot,S,a)_0 = \phi$ we deduce from the previous inequality that we have
$$\forall t \in (0,T], \Vert x^p(\cdot,S,a)_t - x^0(\cdot,S,a)_t \Vert \leq r.$$
And so $(t, x^p(\cdot,S,a)_t, u(t,S,a)) \in {\mathfrak B}$, and using the assumption of induction and the mean value theorem with the boundedness of $D_2f$ we obtain, for all $t \in [0,T]$,
\[ \begin{array}{cl} \null & \Vert f(t, x^p(\cdot,S,a)_t, u(t,S,a)) - f(t,x^{p-1}(\cdot,S,a)_t, u(t,S,a)) \Vert \\ \leq & c_3 \cdot \Vert x^p(\cdot,S,a)_t -x^{p-1}(\cdot,S,a)_t \Vert \leq c_3 \frac{(c_3 t)^{p-1}}{(p-1)!} c_2 \cdot \Vert a \Vert = \frac{c_3^p t^{p-1}}{(p-1)!} c_2 \cdot \Vert a \Vert \end{array} \]
which implies \[ \begin{array}{cl} \null & \Vert x^{p+1}(t,S,a) - x^p(t,S,a) \Vert\\
= &\Vert \int_0^t f(s, x^{p}(\cdot,S,a)_s, u(s,S,a)) ds -\int_0^t f(s, x^{p-1}(\cdot,S,a)_s, u(s,S,a)) ds \Vert\\
\leq & \int_0^t \Vert f(s, x^{p}(\cdot,S,a)_s, u(s,S,a)) - f(s, x^{p-1}(\cdot,S,a)_s, u(s,S,a)) \Vert ds\\ \leq & \int_0^t (c_2 \cdot \Vert a \Vert \frac{c_3^p s^{p-1}}{(p-1)!}) ds = c_2 \cdot \Vert a \Vert \frac{c_3^p}{(p-1)!} \int_0^t s^{p-1} ds = c_2 \cdot \Vert a \Vert c_3^p \frac{t^p}{p!} \end{array} \]
and since $x^{p+1}(\cdot, S,a)_0 = x^p(\cdot,S,a)_0 = \phi$ we deduce from the previous inequality that we have finished the induction and so the formula \eqref{eq57} is proven.
\vskip1mm From \eqref{eq57} it is easy to see that $([t \mapsto x^m(\cdot,S,a)_t])_{m \in {\mathbb{N}}}$ is a Cauchy sequence into $(C^0([0,T], C^0([-r,0], {\mathbb{R}}^n)), \Vert \cdot \Vert_{\infty})$ and consequently we obtain that $(x^m(\cdot,S,a))_{m \in {\mathbb{N}}}$ is a Cauchy sequence into $C^0([-r,T], {\mathbb{R}}^n)$. Since this last space is complete, there exists $x_*(\cdot,S,a) \in C^0([-r,T], {\mathbb{R}}^n)$ such that $\lim_{m \rightarrow + \infty} \sup_{t \in [-r,T]} \Vert x^m(t,S,a) - x_*(t,S,a) \Vert = 0$ which implies that $\lim_{m \rightarrow + \infty} \sup_{t \in [0,T]} \Vert x^m(\cdot,S,a)_t - x_*(\cdot,S,a)_t \Vert = 0$. Using Lemma \ref{lem53} with $F = N_{u(\cdot,S,a)}$ and $\Gamma(t, \phi) = f(t,\phi, u(t,S,a))$ we obtain $\lim_{m \rightarrow + \infty} \sup_{t \in [0,T]} \Vert f(t, x^m(\cdot,S,a)_t, u(t,S,a)) - f(t, x_*(\cdot,S,a)_t, u(t,S,a)) \Vert = 0$ which implies, $\forall t \in [0,T]$,
$$\lim_{m \rightarrow + \infty} \int_0^t f(s, x^m(\cdot,S,a)_s, u(s,S,a)) ds = \int_0^t f(s, x_*(\cdot,S,a)_s, u(s,S,a) ds.$$
Taking $m \rightarrow + \infty$ into the formula
$$x^m(t,S,a) = \phi(0) + \int_0^t(f(s, x^{m-1}(\cdot,S,a)_s, u(s,S,a)) ds$$
we obtain $x_*(t,S,a) = \phi(0) + \int_0^t f(s, x_*(\cdot,S,a)_s, u(s,S,a)) ds$. Note that the set of the discontinuity points of the integrand of this last integral is included into $N_{u(\cdot,S,a)} = N_{\overline{u}} \cup \{ t_i + b_i : i \in \{ 1,...,N \}\} \cup \{ t_i + b_i+ a_i : i \in \{ 1,...,N \}\}$. And so $x_*(\cdot,S,a) \in C^0([-r,T], {\mathbb{R}}^n) \cap PC^1([0,T], {\mathbb{R}}^n)$. From the last integral equation we deduce
$$\forall t \in [0,T] \setminus N_{u(\cdot,S,a)}, \frac{\partial x_*(t,S,a)}{\partial t} = f(t, x_*(\cdot,S,a)_t, u(t,S,a)).$$
Since $x^m(\cdot,S,a)_0 = \phi$ for all $m \in {\mathbb{N}}$, we have also $x_*(\cdot,S,a)_0 = \phi$. Then using the uniqueness of the solution of a Cauchy problem we obtain that
\begin{equation}\label{eq58} x_*(\cdot,S,a) = x(\cdot,S,a). \end{equation}
We have yet seen that $\Vert x^m(t,S,a) - x^0(t,S,a) \Vert = \Vert x^m(t,S,a) - \overline{x}(t) \Vert \leq c_2 \cdot \Vert a \Vert e^{c_3 T}$ for all $t \in [0,T]$. And so setting $c_4 := k e^{c_2 T}$ and taking $m \rightarrow + \infty$ we obtain $\Vert x(t,S,a) - \overline{x}(t) \Vert = \Vert x_*(t,S,a) - \overline{x}(t) \Vert \leq c_4$.
\end{proof}
\begin{lemma}\label{lem55} For all $S \in {\mathfrak S}$, the two assertions hold.
\begin{enumerate}
\item[(i)] $x(T,S,\cdot)$ is differentiable at $0$.
\item[(ii)] For all $i \in \{1,...,N \}$,
$$\frac{\partial x(T,S,0)}{\partial a_i} = X(T,t_i)[f(t_i,\overline{x}_{t_i}, v_i) - f(t_i,\overline{x}_{t_i},\overline{u}(t_i))].$$
\end{enumerate}
\end{lemma}
\begin{proof} After Lemma \ref{lem51} we know that $z(T,S,\cdot)$ is differentiable at 0. Then to prove (i) it suffices to prove that the mapping $x(T,S,\cdot) - z(T,S,\cdot)$ is differentiable at 0. We introduce
\begin{equation}\label{eq59} \zeta(t,S,a) := (x(t,S,a) - z(t,S,a) - (x(t,S,0) - z(t,S,0)) = x(t,S,a) - \overline{x}(t) - z(t,S,a). \end{equation}
and \begin{equation}\label{eq510} \xi(t,S,a) := \frac{\partial \zeta(t,S,a)}{\partial t} - D_2f(t, \overline{x}_t, \overline{u}(t)) \zeta(\cdot,S,a)_t. \end{equation}
We calculate
\[ \begin{array}{ccl} \frac{\partial \zeta(t,S,a)}{\partial t} &=& \frac{\partial x(t,S,a)}{\partial t} - \overline{x}'(t) - \frac{\partial z(t,S,a)}{\partial t} \\ \null &=& f(t, x(\cdot,S,a)_t, u(t,S,a)) - f(t, \overline{x}_t, \overline{u}(t)\\ \null & \null & -D_2f(t, \overline{x}_t, \overline{u}(t))z(\cdot,S,a)_t - f(t, \overline{x}_t,u(t,S,a)) + f(t, \overline{x}_t, \overline{u}(t))\\ \null &=& f(t, x(\cdot,S,a)_t, u(t,S,a)) - f(t, \overline{x}_t,u(t,S,a))\\ \null &\null & -D_2f(t, \overline{x}_t, \overline{u}(t))z(\cdot,S,a)_t \end{array} \] which implies, using \eqref{eq59} and \eqref{eq510}, that
\[ \begin{array}{ccl} \xi(t,S,a) &=& \frac{\partial \zeta(t,S,a)}{\partial t} - D_2f(t, \overline{x}_t, \overline{u}(t))(x(\cdot,S,a) - \overline{x}_t - z(\cdot,S,a))\\ \null &= & f(t, x(\cdot,S,a)_t, u(t,S,a)) - f(t, \overline{x}_t,u(t,S,a))\\ \null & \null & - D_2f(t, \overline{x}_t, \overline{u}(t))(x(\cdot,S,a) - \overline{x}_t)\\ \null & = & \int_0^1 D_2 f(t, \overline{x}_t + \theta[x(\cdot,S,a) - \overline{x}_t], u(t,S,a)) d \theta (x(\cdot,S,a) - \overline{x}_t)\\ \null & \null & - D_2f(t, \overline{x}_t, \overline{u}(t))(x(.,S,a) - \overline{x}_t) \end{array} \]
and so we obtain
\begin{equation}\label{eq511} \xi(t,S,a) = \Vert x(\cdot,S,a)_t - \overline{x}_t \Vert_{\infty} \cdot E(t,S,a) \end{equation}
where $E(t,S,a) := \left( \int_0^1 D_2 f (t, \overline{x}_t + \theta[x(\cdot,S,a) - \overline{x}_t], u(t,S,a)) d \theta - D_2f(t, \overline{x}_t, \overline{u}(t)) \right)$ \\ $ \frac{ x(\cdot,S,a)_t - \overline{x}_t}{\Vert x(\cdot,S,a)_t - \overline{x}_t \Vert}$ if $x(\cdot,S,a)_t \neq \overline{x}_t$ and $E(t,S,a) := 0$ if $x(\cdot,S,a)_t = \overline{x}_t$.
Since $D_2f$ is bounded, there exists $c_4 > 0$ such that, for all $t \in [0,T]$, for all $a \in \overline{B}(0,\delta_2)$, $\Vert E(t,S,a) \Vert \leq c_4$. Note that $E(t,S,a) = 0$ when $t \in [0,T] \setminus \cup_{1 \leq i \leq N} I_i$, and consequently we have
\[ \begin{array}{ccl} \int_0^T \Vert E(t,S,a) \Vert dt &=& \int_{ \cup_{1 \leq i \leq N} I_i} \Vert E(t,S,a) \Vert dt + \int_{[0,T] \setminus \cup_{1 \leq i \leq N} I_i} \Vert E(t,S,a) \Vert dt\\ \null & =& \sum_{i=1}^N \int_{I_i} \Vert E(t,S,a) \Vert dt = \sum_{i=1}^N \int_{t_i + b_i}^{t_i + b_i + a_i} \Vert E(t,S,a) \Vert dt\\ \null & \leq & \sum_{i=1}^N a_i c_4 = c_4 \cdot \Vert a \Vert \end{array} \]
that implies
\begin{equation}\label{eq512} \lim_{a \rightarrow 0} \int_0^T \Vert E(t,S,a) \Vert dt = 0. \end{equation}
From \eqref{eq510} we have $\frac{\partial \zeta(t,S,a)}{\partial t} = D_2f(t, \overline{x}_t, \overline{u}(t)) \zeta(\cdot,S,a)_t + \xi(t,S,a)$, and from \eqref{eq59} we also have $\zeta(\cdot,S,a)_0 = x(\cdot,S,a)_0 - \overline{x}_0 - z(\cdot,S,a) = \phi - \phi - 0$, and so we can use Theorem \ref{th34} and assert that $\zeta(t,S,a) = X(t,0)0 + \int_0^t X(t,y)(\int_{-r}^{-y} d_2 \eta(y, \theta) 0(y + \theta))dy) + \int_0^t X(t, \alpha) \xi(\alpha, S,a) d \alpha = 0+0 + \int_0^t X(t, \alpha) \xi(\alpha, S,a) d \alpha$, and so we have established the following formula
\begin{equation}\label{eq513} \zeta(T,S,a) = \int_0^T X(T,t) \xi(t,S,a) dt. \end{equation}
We know that $X$ is bounded on $[0,T] \times [0,T]$, and using Lemma \ref{lem54}, \eqref{eq513} and \eqref{eq511} we obtain
\[ \begin{array}{l} \Vert (x(T,S,a) - z(T,S,a) - (x(T,S,0) - z(T,S,0)) \Vert = \Vert \zeta(T,S,a) \Vert\\
\leq \int_0^T (\Vert X(T,t) \Vert \cdot \Vert \xi(t,S,a) \Vert) dt \leq \Vert X \Vert_{\infty} \int_0^T (\Vert x(\cdot,S,a)_t - \overline{x}_t \Vert_{\infty} \cdot \Vert E(t,S,a) \Vert )dt\\ \leq \Vert X \Vert_{\infty} \cdot c_3 \cdot \Vert a \Vert \cdot \int_0^T \Vert E(t,S,a) \Vert dt \end{array} \]
which implies (using \eqref{eq512}) that the mapping $x(T,S,\cdot) - z(T,S,\cdot)$ is differentiable at 0 and that its differential at 0 is equal to zero. Then using lemma \ref{lem51} we obtain $D_a X(T,S,0)a = D_a z(T,S,0) a = \sum_{i=1}^N a_i X(T,t_i) (f(t_i, \overline{x}_{t_i}, v_i) - f(t_i, \overline{x}_{t_i},\overline{u}(t_i)))$.
\end{proof}
The following multiplier rule comes from \cite{Mi}.
\begin{lemma}\label{lem56} Let ${\mathcal V}$ be a neighborhood of 0 into ${\mathbb{R}}^N$. Let $n_i, q_e \in {\mathbb{N}}$. Let $\psi^j : {\mathcal V} \rightarrow {\mathbb{R}}$ be differentiable functions at 0, for $j \in \{ 0,...,n_i + n_e \}$. We assume that 0 is a solution of the following maximization problem
\[ \left\{ \begin{array}{rl} {\rm Maximize} & \psi^0(a)\\ {\rm when} & a \in {\mathcal V} \cap {\mathbb{R}}^n_+\\ \null & \forall j = 1,...,n_i, \; \psi^j(a) \geq 0\\ \null & \forall j = n_i+1, ..., n_i+n_e, _; \psi^j(a) = 0. \end{array} \right. \]
Then there exists $(\lambda_0,..., \lambda_{n_i + n_e}) \in {\mathbb{R}}^{1+ n_i + n_e}$ which satisfies the following conditions.
\begin{enumerate}
\item[(i)] $(\lambda_0,..., \lambda_{n_i + n_e})$ is non zero
\item[(ii)] $\forall j = 0,...,n_i, \; \lambda_j \geq 0$
\item[(iii)] $\forall j = 1,...,n_i, \; \lambda_j \psi^j(0) = 0$
\item[(iv)] $\forall a \in {\mathbb{R}}^N_+, \; \sum_{j= 0}^{n_i + n_e} \lambda_j D\psi^j(0).a \leq 0$.
\end{enumerate}
\end{lemma}
Now we apply this multiplier rule to our problem that permits us to obtain the following lemma.
\begin{lemma}\label{lem57} Let $S \in {\mathfrak S}$. Then there exists $\Lambda(S) = (\lambda_j)_{0 \leq j \leq n_i + n_e} \in {\mathbb{R}}^{1+ p+ q}$ which satisfies the following properties.
\begin{enumerate}
\item[(i)] $\sum_{j=0}^{n_i + n_e} \vert \lambda_j \vert = 1$
\item[(ii)] $\forall j = 0,...,n_i, \; \lambda_j \geq 0$
\item[(iii)] $\forall j = 1,...,n_i, \; \lambda_j g^j(\overline{x}(T)) = 0$
\item[(iv)] $\forall i = 1,...,N$, \\ $\sum_{j=0}^{n_i + n_e} \lambda_j Dg^j(\overline{x}(T))X(T, t_i)[f(t_i, \overline{x}_{t_i}, v_i) -f(t_i, \overline{x}_{t_i}, \overline{u}(t_i))] \leq 0 $.
\end{enumerate}
\end{lemma}
\begin{proof} Since the process $(\overline{x}, \overline{u})$ is optimal for $({\mathfrak M})$, it is also optimal among the processes $(x(\cdot,S,a), u(\cdot,S,a))$ when $a$ belongs to a neighborhood of 0 into ${\mathbb{R}}^N_+$. Recall that $x(\cdot,S,0) = \overline{x}$ and $u(\cdot,S,0) = \overline{u}$. And then 0 is an optimal solution of the following maximisation static problem
\[ \left\{ \begin{array}{rl} {\rm Maximize} & g^0(x(T,S,a))\\ {\rm when} & a \in \overline{B}(0, \delta_2) \cap {\mathbb{R}}^n_+\\ \null & \forall j = 1,...,n_i, \; g^j(x(T,S,a)) \geq 0\\ \null & \forall j = n_i+1, ..., n_i +n_e, _; g^j(x(T,S,a) )= 0. \end{array} \right. \]
We use the previous lemma by setting $\psi^j(a) := g^j(x(T,S,a))$. Since the set of the lists of multipliers is a cone the non nullity of $(\lambda_j)_{0 \leq j \leq n_i + n_e}$ permits us to choose it to satisfy (i). The conclusions (ii) and (iii) are given by the previous lemma in a straightforward way. To treat the last condition, note that $D \psi^j(0) = Dg^j(\overline{x}(T))D_a x(T,S,0)$, and then the last conclusion of the previous lemma is: $\forall a \in {\mathbb{R}}^N_+$, $\sum_{j=0}^{n_i + n_e} \lambda_j Dg^j(\overline{x}(T)).D_a x(T,S,0)a \leq 0$. If $(e_i)_{1 \leq i \leq n}$ denotes the canonical basis of ${\mathbb{R}}^N$, we obtain
$$0 \geq \sum_{j=0}^{n_i + n_e} \lambda_j Dg^j(\overline{x}(T))D_a x(T,S,0)a = \sum_{j=0}^{p+q} a_j \lambda_j Dg^j(\overline{x}(T))\frac{\partial x(T,S,0)}{\partial a_i}$$
and we conclude by using Lemma \ref{lem55}.
\end{proof}
The following lemma ensures the existence of multipliers which do not depend of $S \in {\mathfrak S}$.
\begin{lemma}\label{lem58} There exists $(\lambda_j)_{0 \leq j \leq n_i + n_e} \in {\mathbb{R}}^{1+ n_i + n_e}$ which satisfies the following properties.
\begin{enumerate}
\item[(i)] $\sum_{j=0}^{n_i + n_e} \vert \lambda_j \vert = 1$
\item[(ii)] $\forall j = 0,...,n_i, \; \lambda_j \geq 0$
\item[(iii)] $\forall j = 1,...,n_i, \; \lambda_j g^j(\overline{x}(T)) = 0$
\item[(iv)] $\forall t \in [0,T]$, $\forall u \in U$, \\ $\sum_{j=0}^{n_i + n_e} \lambda_j Dg^j(\overline{x}(T))X(T,t)[f(t, \overline{x}_t, u) -f(t, \overline{x}_t, \overline{u}(t))] \leq 0$.
\end{enumerate} \end{lemma}
The proof of this lemma is completely similar to this one of Lemme 3 in \cite{Mi}. The idea of this proof is the following one: at each $S \in {\mathfrak S}$ we associate the set $K(S)$ as the set of the $(\lambda_j)_{0 \leq j \leq n_i + n_e} \in {\mathbb{R}}^{1+ n_i + n_e}$ which satisfy the conclusion (i-iv) of Lemma \ref{lem57}. Denoting by ${\mathcal S}(0,1)$ the unit spere of ${\mathbb{R}}^{1+ n_i + n_e}$ for the norm $\Vert (\lambda_j)_{0 \leq j \leq n_i + n_e} \Vert := \sum_{i=0}^{n_i + n_e} \vert \lambda_j \vert$, we see that $K(S)$ is nonempty (by Lemma \ref{lem57}), is closed into the compact ${\mathcal S}(0,1)$. For all finite list $(S_k)_{1 \leq k \leq {\ell}}$ of elements of ${\mathfrak S}$, we can build $S^* \in {\mathfrak S}$ such that $K(S^*) \subset \cap_{1 \leq k \leq {\ell}} K(S_k)$ that proves that $\cap_{1 \leq k \leq {\ell}} K(S_k) \neq \emptyset$. Then the compactness of ${\mathcal S}(0,1)$ and the closedness of the $K(S)$ imply that $\cap_{S \in {\mathfrak S}}K(S) \neq \emptyset$. It suffices to take $(\lambda_j)_{0 \leq j \leq n_i + n_e} \in \cap_{S \in {\mathfrak S}}K(S)$ and to note that each $(t,u) \in [0,T] \times U$ belongs to ${\mathfrak S}$ to obtain the conclusion (iv) of the lemma.
\vskip1mm \noindent {\bf The end of the proof of Theorem \ref{th41}.} The scalar multipliers $\lambda_0$,..., $\lambda_{n_i + n_e}$ are provided by Lemma \ref{lem58}. From lemma \ref{lem58} we see that the conditions (NN), (Si) and (Sl) of Theorem \ref{th41} are fulfilled.We define the function $p : [0,T] \rightarrow {\mathbb{R}}^{n*}$ by setting
\begin{equation}\label{eq514} p(t) := \sum_{j=0}^{n_i + n_e} \lambda_j Dg^j(\overline{x}(T))X(T,t). \end{equation}
From (iv) of Lemma \ref{lem58}, we see that the condition (MP) is fulfilled. Since $X(T,T) = I$ (identity), from \eqref{eq514} we see that the condition (T) is fulfilled. Using Proposition \ref{prop35}, we have $X(T,t) = I - \int_t^T X(T, \alpha) \eta^1(\alpha, t - \alpha) d \alpha$, which implies
\[ \begin{array}{ccl} p(t)& = &\sum_{j=0}^{n_i + n_e} \lambda_j Dg^j(\overline{x}(T))X(T,t) \\ \null & =& \sum_{j=0}^{n_i + n_e} \lambda_j Dg^j(\overline{x}(T)) - \int_t^T \sum_{j=0}^{n_i + n_e} \lambda_j Dg^j(\overline{x}(T)) X(T, \alpha) \eta(\alpha, t - \alpha) d \alpha\\ \null & =& \sum_{j=0}^{n_i + n_e} \lambda_j Dg^j(\overline{x}(T)) - \int_t^T p(\alpha) \eta^1(\alpha, t - \alpha) d \alpha \end{array} \]
and so we see that the function $[t \mapsto p(t) + \int_t^T p(\alpha) \eta(\alpha, t - \alpha) d \alpha]$ is constant on $[0,T]$. Note that $\alpha > t+r$ implies $t- \alpha < -r$ and then $\eta^1(\alpha, t- \alpha) = 0$ and consequently $\int_t^T p(\alpha) \eta^1(\alpha, t - \alpha) d \alpha = \int_t^{ \min \{t+r,T \} } p(\alpha) \eta^1(\alpha, t - \alpha) d \alpha =\int_t^{ \min \{t+r,T \} } p(\alpha) \eta(\alpha, t - \alpha) d \alpha$ that proves that the condition (AE) holds.
\vskip1mm Using (QC), if $p(T) = 0$ since (NN), (Si) and (Sl) hold we obtain a contradiction, and so we have $p(T) \neq 0$ under (QC). Since $X(T,t) = I - \int_t^T R(\xi, t) d \xi$, and since $R$ is bounded, if we choose $\epsilon \in (0, \Vert R \Vert_{\infty}^{-1}]$, then, when $t \in (T- \epsilon, T]$, we obtain that $X(T,t)$ is invertible. Using (TC), we have $p(t) = p(T) X(T,t)$, ans since $p(T) \neq 0$, we obtain $p(t) \neq 0$. And so (A1) is proven. To prove (A2), we proceed by contradiction. We assume that there exists $\tau \in [0,T]$ such that $p(t) = 0$ when $t \in [\tau, \min\{ \tau + r, T \}]$. Note that $\tau < T$ since $p(T) \neq 0$. Using (AE) and (TC) we have, $p(\tau) + \int_{\tau}^{\min \{ \tau + r, T \}} p(\xi) \eta(\xi, \tau - \xi) d \xi = p(T)$ which implies $0 + 0 = p(T)$ that is impossible. And so (A2) is proven and the proof of Theorem \ref{th41} is complete.
\end{document} | arXiv |
Occurrences of (co)homology in other disciplines and/or nature
I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive arrows is zero seems like a fairly general notion, but I have not come across it in fields like biology, economics, etc. Are there examples of non-trivial (co)homology appearing outside of pure mathematics?
I think Hatcher has a couple illustrations of homology in his textbook involving electric circuits. This is the type of thing I'm looking for, but it still feels like topology since it is about closed loops. Since the relation $d^2=0$ seems so simple to state, I would imagine this setup to be ubiquitous. Is it? And if not, why is it so special to topology and related fields?
at.algebraic-topology cohomology big-list applications popularization
Noah GiansiracusaNoah Giansiracusa
$\begingroup$ Ghrist has written a number of papers applying homology to different applied fields. See for example his paper on Sensor Networks. $\endgroup$ – Jim Conant Mar 30 '11 at 19:30
$\begingroup$ This question should be Community Wiki, since it's looking for a list of examples rather than an answer to a specific question. I also feel like the question isn't very clear, since any answer has to be homological in nature so fundamentally mathematical. So the question seems somewhat conflicted. $\endgroup$ – Ryan Budney Mar 30 '11 at 19:35
$\begingroup$ Do you count string theory as "outside the realm of pure mathematics"? $\endgroup$ – Qfwfq Mar 30 '11 at 20:04
$\begingroup$ The first paragraph seems to suggest that the question is not so much about homology/topology in the large, but more about the idea of chain complexes. Is my reading correct? $\endgroup$ – Yemon Choi Mar 30 '11 at 20:27
$\begingroup$ Applied topology at Stanford: comptop.stanford.edu $\endgroup$ – Igor Belegradek Mar 30 '11 at 20:48
Robert Ghrist is all about applied topology: Sensor Network, Signal Processing, and Fluid Dynamics. (homepage: http://www.math.upenn.edu/~ghrist/index.html ). For instance, we want to use the least number of sensors to cover a certain area, such that when we remove one sensor, a part of that area is undetectable. We can form a complex of these sensors and hence its nerve, and use homology to determine whether there are any gaps in the sensor-collection. I've met with him in person and he expressed confidence that this is going to be a big thing of the future.
There are also applications of cohomology to Crystallography (see Howard Hiller) and Quasicrystals in physics (see Benji Fisher and David Rabson). In particular, it uses cohomology in connection with Fourier space to reformulate the language of quasicrystals/physics in terms of cohomology... Extinctions in x-ray diffraction patterns and degeneracy of electronic levels are interpreted as physical manifestations of nonzero homology classes.
Another application is on fermion lattices (http://arxiv.org/abs/0804.0174v2), using homology combinatorially. We want to see how fermions can align themselves in a lattice, noting that by the Pauli Exclusion principle we cannot put a bunch of fermions next to each other. Homology is defined on the patterns of fermion-distributions.
Chris GerigChris Gerig
$\begingroup$ Robert Ghrist is coming to Edinburgh for the Science Festival this year to talk about the Mathematics of holes. (Alas I will be away.) If anyone is around, it will be worth your while to attend! $\endgroup$ – José Figueroa-O'Farrill Mar 30 '11 at 20:59
$\begingroup$ In general, I think that homology will play a role in the mathematics of information. We had a talk by Gunnar Carlsson recently in Edinburgh about "persistent homology" and it was quite an eye opener. See comptop.stanford.edu for instance. $\endgroup$ – José Figueroa-O'Farrill Mar 30 '11 at 21:00
Actually even schoolchildren calculate group co-cycle. (Without knowing that it is called like this). Cohomology occurs in everyday life as soon as one learns to count.
5+7 = 1 2
4 + 5 = 0 9
What is the function on which sends a pair (a,b) to the $0$ or $1$ depending result is greater than 9 or not ? ( e.g. f(5,7)= 1, f(4,5) = 0, f(2,8)= 1).
This is actually a 2-cocycle for group $Z/nZ$ with values in $Z$.
It can be checked directly or...
Let us look on it more conceptually. Consider the standard short exact sequence of abelian groups $0 \to Z \to Z \to Z/n \to 0$. (First map is multiplication by $n$, the second is factorization and will be denoted by $p$).
Choose section $s: Z/nZ \to Z$ (i.e. any map such $ps=Id$, where $p: Z \to Z/nZ$, it is like connection in differential geometry (can be made precise)).
Define $f(a,b)= s(a)+s(b) - s(a+b)$
Note that: a) this function $f(a,b)$ is exactly we talked above
b) from general theory this is 2-cocyle, (it corresponds to this extension, (it it is like "curvature" of connection is differential geomety (can be made precise)).
That is all: we explained why it is group cocycle and what its role.
I would like to learn this 20 years ago when I learned group cohomology as undergraduate, but I learned this 1 ago, doing some engineering work in wireless communication... I am still surprised that it is not written on the first page of any textbook which deals with group cohomology, when I am explaining this to my friends most did not know this also and after knowing share my feeling of surprise.
fhyve
Alexander ChervovAlexander Chervov
$\begingroup$ While not on the first page of a textbook, this is written up in an article in the American Mathematical Monthly (Daniel C. Isaksen, A cohomological viewpoint on elementary school arithmetic, Amer. Math. Monthly (109), no. 9 (2002), p. 796--805) $\endgroup$ – Christopher Drupieski Jun 11 '12 at 20:44
$\begingroup$ is it easy to generalize to larger digits? how about for multiplication? $\endgroup$ – T.... Jun 18 '12 at 16:34
$\begingroup$ @36min In arxiv.org/abs/hep-th/0212195 we have discussed such a look on connections (actually for more general setup of Courant algebroids), it might be possible to get idea from there, but may be it is not good starting point. I am sorry I do not know reference for standard exposition of this approach in the case of usual connections. $\endgroup$ – Alexander Chervov Nov 21 '12 at 6:16
$\begingroup$ @36min Let me give an idea consider module V over M. Consider the following exact triple: End(V) ->A(V) -> Der(M), where A(M) - the set of all "derivations" of module V. The idea is that connection is exactly the same is a section s: A(V) <- Der(M) !!! The curvature is the following - take two elements a,b in Der(M) , consider F(a,b) = [s(a),s(b)] -s([a,b]) observe that F(a,b) lies actually in End(V) \subset A(V) so we get map F(a,b) : Der(M)^2 -> End(V) - this is curvature $\endgroup$ – Alexander Chervov Nov 21 '12 at 6:24
$\begingroup$ @36min , what is Der(M) - set of all derivations of algebra of functions on M, i.e. all vector fields. What is A(V) - set of all derivations of module V. Map d: V-> V is called derivation of module V, if there exists derivation dd \in Der(M) such that for any element "a" of algebra M , it is true that d(a v) = dd(a) v + a dd(v) !!!!!!!! It is simple, may be I am explaining not in a right way. Is it clear ? $\endgroup$ – Alexander Chervov Nov 21 '12 at 6:28
Recently, it was realized that quantum many-body states can be divided into short-range entangled states and long-range entangled states.
The quantum phases with long-range entanglements correspond to topologically ordered phases, which, in two spatial dimensions, can be described by tensor category theory (see cond-mat/0404617). Topological order in higher dimensions may need higher category to describe them.
One can also show that the quantum phases with short-range entanglements and symmetry $G$ in any dimensions can be "classified" by Borel group cohomology theory of the symmetry group. (Those phases are called symmetry protected trivial (SPT) phases.)
The quantum phases with short-range entanglements that break the symmetry are the familar Landau symmetry breaking states, which can be described by group theory.
So, to understand the symmetry breaking states, physicists have been forced to learn group theory. It looks like to understand patterns of many-body entanglements that correspond to topological order and SPT order, physicists will be forced to learn tensor category theory and group cohomology theory. In modern quantum many-body physics and in modern condensed matter physics, tensor category theory and group cohomology theory will be as useful as group theory. The days when physics students need to learn tensor category theory and group cohomology theory are coming, may be soon.
Xiao-Gang WenXiao-Gang Wen
Quantum field theory is outside the realm of pure mathematics, makes contact with the real world and features chain complexes and cohomology.
The current paradigm for gauge theories such as the standard model is based on Yang-Mills theories coupled to matter. The quantisation of nonabelian (and, depending on your choice of gauge fixing function, also abelian) Yang-Mills theories features a cohomology theory known by the moniker of BRST, after the inventors: Becchi, Rouet, Stora and, independently, Tyutin. The cleanest proofs of the renormalizability of Yang-Mills theories are cohomological in nature.
José Figueroa-O'FarrillJosé Figueroa-O'Farrill
My understanding, from conversations with Raoul Bott, is that his early work on electrical circuits and the Bott-Duffin theorem can be intepreted as exhibiting close connections between de Rham cohomology and the laws of electrical circuits, and that this is part of what led him into pure mathematics early in his career.
Lee MosherLee Mosher
$\begingroup$ He talked about this connection to electrical circuits in teaching graduate algebraic topology, to help motivate cohomology and give intuition for it. $\endgroup$ – Patricia Hersh May 20 '12 at 14:25
$\begingroup$ Bott gave a very nice talk (c. 1960) to electrical engineers on the subject, but I cannot find a reference right now. $\endgroup$ – Robert Bruner Jun 11 '12 at 21:55
$\begingroup$ See the paper Geometry and Topology: Seven Lectures by Raoul Bott edited by J. M. Rojas. $\endgroup$ – Tom Copeland Mar 30 '14 at 23:13
The mass of a classical mechanical system is an element in the (one-dimensional) second cohomology group of the Lie algebra of the Galilei group. See J. M. Souriau, Stucture des Systèmes Dynamiques, Chap. III, section (12.136). Or in english translation, search inside here for "total mass".
Francois ZieglerFrancois Ziegler
$\begingroup$ A layman question. You say second cohomology group, however Souriau does not tell a number for it. Similarly to you, Guillemin and Sternberg tells "second" ($H^2$)(Symplectic Techniques in Physics, p. 417), but Liberman and Marle tells first ($H^1$) (Symplectic Geometry and Analytical Mechanics, p.204). Could you resolve this apparent contradiction in a couple of words? $\endgroup$ – mma Nov 24 '16 at 6:13
$\begingroup$ As @mma points out, the note on p. 107 explicitly mentions that Souriau's "cohomology" is always "cohomology in degree 1". (I can't manage to trace through his definitions to tell in what sense $m$ is a cocycle at all, regardless of degree.) $\endgroup$ – LSpice Jun 10 '17 at 20:28
The Aharonov–Bohm effect. Classically, you can't distinguish two electromagnetic potentials which are in the same cohomology class. From quantum viewpoint, they can be distinguished, because an electron changes its phase under parallel transport defined by the connection associated to a potential.
Cristi StoicaCristi Stoica
The finite element method- a numerical method for solving PDE's- has a homological interpretation:
MR2269741 (2007j:58002) Arnold, Douglas N.(1-MN-MA); Falk, Richard S.(1-RTG); Winther, Ragnar(N-OSLO-CMA) Finite element exterior calculus, homological techniques, and applications. (English summary) Acta Numer. 15 (2006), 1–155
Margaret FriedlandMargaret Friedland
Something resembling de Rham complex with differential-algebraic flavor appears in (variant of) control theory, see, for example, G. Conte, C.H. Moog, A.M. Perdon, Algebraic Methods for Nonlinear Control Systems, 2nd ed., Springer, 2006. But, as far as I can tell, they do not use the word "cohomology" explicitly.
Spencer cohomology (which is, essentially, a Lie algebra cohomology) appears as obstructions to integrability of some differential-geometric structures (G-structures) and, through it, of (some) differential equations. Potentially this opens a wide possibilities for applications, and indeed, Dimitry Leites advocates this approach in (some of) his writings. An emblematic publication which is available, unfortunately, only in Russian, is: "Application of cohomology of Lie algebras in national economy", Seminar "Globus", Independent Univ. of Moscow, Vol. 2, 2005, 82-102. The Russian original for "national economy" in the title is (a somewhat pejorative and untranslatable term) "narodnoe khozyai'stvo".
Edit: J.-F. Pommaret has published extensively on the applications of Spencer cohomology to continuum mechanics, control theory and mathematical physics - for example, see "Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics"
Phil Harmsworth
Pasha ZusmanovichPasha Zusmanovich
$\begingroup$ I love that there be a pejorative form of "national economy" ;-) $\endgroup$ – Mariano Suárez-Álvarez Apr 20 '14 at 2:04
A classical and elegant application is to the solution of Kirchhoff's theorem on electrical cricuits. See:
Nerode, A.; Shank, H.: An algebraic proof of Kirchhoff's network theorem. Amer. Math. Monthly, 68 (1961) 244–247
John KleinJohn Klein
$\begingroup$ This is discussed in more detail in Quantum Field Theory III: Gauge Theory by Eberhard Zeidler (Springer, 4/2011), pg. 1009, 1020, 1027. $\endgroup$ – Tom Copeland Jan 24 '17 at 18:56
$\begingroup$ @TomCopeland The source you gave works with both chain and cochain complexes at the same time. It seems to me that it's merely reproducing the Bott-Duffin approach. The advantage of the Nerode-Shank argument is that it easily generalizes to higher dimensions. See: Catanzaro, Michael J.(1-WYNS); Chernyak, Vladimir Y.(1-WYNS-KM); Klein, John R.(1-WYNS) Kirchhoff's theorems in higher dimensions and Reidemeister torsion. Homology Homotopy Appl. 17 (2015), no. 1, 165–189. $\endgroup$ – John Klein Jan 25 '17 at 20:48
It's my understanding that Carina Curto and Vladmir Itskov at the University of Nebraska - Lincoln apply algebraic topology (among other things) to study theoretical and applied neuroscience.
Adam AzzamAdam Azzam
$\begingroup$ Their work is very similar in spirit to Ghrist's work alluded to in one of the other answers. $\endgroup$ – Igor Rivin May 20 '12 at 15:38
Anders Björner and László Lovász used bounds on the Betti numbers for the complement of a real subspace arrangement called the $k$-equal arrangement to give a complexity theory lower bound that agreed, up to a scalar multiple, with the previously known upper bound in:
A. Björner and L. Lovász, Linear decision trees, subspace arrangements, and Mobius functions, Journal of the American Mathematical Society, Vol. 7, No. 3 (1994), 677--706.
The basic question addressed in their paper (along with other questions of a similar flavor) is how many pairwise comparisons of coordinates are needed to decide if a vector in ${\bf R}^n$ has $k$ coordinates all equal to each other for fixed $k$ and $n$. They observed that this is equivalent to deciding whether the vector lies on the so-called $k$-equal arrangement or in its complement, where the $k$-equal arrangement is the subspace arrangement comprised of the ${n\choose k}$ subspaces where $k$ coordinates are set equal to each other.
To this end, they gave a lower bound on the number of leaves in a linear decision tree -- a tree where one starts at the root, and each time one does a comparison of two coordinates $a_i$ and $a_j$, then one proceeds down to either the $a_i < a_j$ child or the $a_i = a_j$ child or the $a_i > a_j$ child. One reaches a leaf when no further queries are necessary to make a decision as to containment in the arrangement or its complement. The log base 3 of the number of leaves is a lower bound on the depth of the tree, i.e. on the number of queries needed in the worst case.
To get some intuition for why this bound depended fundamentally on the Betti numbers of the complement, consider the $k=2$ case -- where the number of connected components of the complement of the subspace arrangement (which in this case is a hyperplane arrangement) is an obvious lower bound on the number of leaves in any linear decision tree.
Patricia HershPatricia Hersh
http://sigact.org/Prizes/Godel/2004.html
http://www.cs.brown.edu/~mph/HerlihyS99/p858-herlihy.pdf
Maurice Herlihy and Nir Shavit won the 2004 Gödel Prize for topological analysis of asynchronous computation. Homology was involved.
Cohomology has appeared in game theoretic research on equilibrium refinements. Loosely speaking, John Nash's 1951 original notion of an equilibrium point does too little to limit the set of 'reasonable' outcomes. A variety of so-called 'equilibrium refinements' sprang up in the economic literature intended to address this. A common theme in many of them was robustness to perturbation (if one perturbs the underlying game played, one would wish that 'nearby fixed point problems have nearby solutions').
In 1989 JF Mertens formulated a notion of a stable equilibrium over two papers that relies on the cohomological essentiality of a projection map from the graph of the equilibrium correspondence to the space of games (the two papers may be found here and here) to arrive at a solution concept with a number of normatively reasonable properties.
Pete CaradonnaPete Caradonna
The group cohomology can be useful to describe the many-body quantum Condensed Matter systems with emergent underlying intrinsic Topological Orders. These classes of systems at low energy and long distance behave like certain TQFT theories. In particular, one can write explicit exact solvable Lattice quantum Hamiltonian operator $\hat{H}$ on the space discretized lattice, and its ground states give rise to discrete gauge theory TQFT with finite group $G$ (quantum double models in 2+1d or its 3+1d, etc analogous, $D(G)$, with Drinfield and Hopf algebra), twisted discrete gauge theories (twisted quantum double models $D^\omega(G)$).
Twisted Quantum Double Model of Topological Phases in Two-Dimension arxiv:1211.3695 PhysRevB.87.125114
Twisted Gauge Theory Model of Topological Phases in Three Dimensions arxiv:1409.3216 PhysRevB.92.045101
Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory arxiv:1404.7854 PhysRevB.91.035134
The low energy and long distance physics of the theories are the same as the Dijkgraaf-Witten topological gauge theories, Kitaev toric code and quantum double models, and some of Levin-Wen string-net models, and many of twisted discrete gauge theory.
This is how the quantum Hamiltonian operator (acting on the Hilbert space of quantum states) looks like: $$ \hat{H}=-\sum_v A_v-\sum_f B_f, $$ where $B_f$ is the face operator defined at each triangular face $f$, and $A_v$ is the vertex operator defined on each vertex $v$. As in the TQD model in $(2+1)$-d, each operator $A_v$ behaves as a gauge transformation on the group elements respectively on the edges meeting at $v$, and a $B_f$ detects whether the flux through face $f$ is zero. This kind of Hamiltonians generically feature ground states that are gauge invariant and bear zero flux everywhere.
A normalized $$\omega_d\in H^d(G,U(1)),$$ as a function $\omega:G^4\rightarrow U(1)$, satisfies the $d$-cocycle condition. The cocycle $\omega$ will fill into the spacetime lattice $d$-simplex ($d+1$-cell). Let us take $D=4$-dim spacetime as an example below.
The operator $A_v$ is a summation $$ A_v=\frac{1}{|G|}\sum_{[vv']=g\in G}A_v^g. $$ The value $|G|$ is the order of the group $G$. The operator $A_v^g$ acts on a vertex $v$ with a group element $g\in G$ by replacing $v$ by a new enumeration $v'$ that is ``slightly" less than $v$ but greater than all the enumerations that are less than $v$ in the original set of enumerations before the action of the operator, such that $v'v=g$. In a dynamical picture of Hamiltonian evolution, $v'$ is understood as on the next \textquotedblleft time" slice, and there is an edge $v'v\in G$ in the $(3+1)$ dimensional \textquotedblleft spacetime" picture. That is, the new vertex $v'$ and the original vertices before the action of $A_v$ delineate a $4$-dimensional picture. Let us consider as follows the simplest subgraph---namely a single tetrahedron---of some large $\Gamma$ to illustrate how an $A_v$ acts: where $v'_4v_4=g$.
The action of $B_f$ on a basis vector is The discrete delta function $\delta_{v_1v_2\cdot v_2v_3\cdot v_3v_1}$ is unity if ${v_1v_2\cdot v_2v_3\cdot v_3v_1=1 }$, where $1$ is the identity element in $G$, and 0 otherwise. Note again that here, the ordering of $v_1,v_2$, and $v_3$ does not matter because of the identities $\delta_{v_1v_2\cdot v_2v_3\cdot v_3v_1} =\delta_{v_3v_1\cdot v_1v_2\cdot v_2v_3}$ and $\delta_{v_1v_2\cdot v_2v_3\cdot v_3v_1} =\delta_{\overline{v_1v_2\cdot v_2v_3\cdot v_3v_1}} =\delta_{\overline{v_3v_1}\cdot \overline{v_2v_3}\cdot \overline{v_1v_2}} =\delta_{v_1v_3\cdot v_3v_2\cdot v_2v_1}$. In other words, in any state on which $B_f=1$ on a triangular face $f$, the three group degrees of freedom around $v$ is related by a chain rule: $$ v_1v_3=v_1v_2\cdot v_2v_3 $$ for any enumeration $v_1,v_2,v_3$ of the three vertices of the face $f$.
Here are how the space lattice and the triangulation of spacetime lattices looks like:
Here are how the $SL(N,\mathbb{Z})$'s modular $S$ and $T$-transformations look like:
edited Dec 8 '16 at 0:39
wonderichwonderich
To add a touch of very belated whimsy, cohomology has manifestations in art. Here are two:
A Penrose triangle (an "impossible figure") can be viewed as a Čech $1$-cycle associated to an open covering of the image plane, taking values in the sheaf of positive functions under pointwise multiplication.
In more detail, place your eye at the origin of Cartesian space, let $P$ be a region in some plane (the screen) not through the origin, and let $C \simeq P \times (0, \infty)$ (for cone) be the union of open rays from your eye to a point of $P$.
A spatial scene in $C$ is rendered in the plane region $P$ by radial projection, discarding depth (distance) information. Recovering a spatial scene from its rendering on paper amounts to assigning a depth to each point of $P$, i.e., to choosing a section of the projection $C \to P$.
Roger Penrose's three-page article On the Cohomology of Impossible Figures describes in detail (copiously illustrated with Penrose's inimitable drawings) how a two-dimensional picture can be "locally consistent" (each sufficiently small piece is the projection of a visually-plausible spatial object) yet "globally inconsistent" (the entire picture has no visually-plausible interpretation as a projection of a spatial object).
For example, each corner of a Penrose triangle has a standard interpretation as a plane projection of an L-shaped object with square cross-section, but the resulting local depth data cannot be merged consistently in a way conforming to visual expectations.
Of course, the depth data can be merged consistently (from a carefully-selected origin!) by circumventing visual expectations. There are at least two "natural" approaches: use curved sides, or break one vertex.
M. C. Escher's Print Gallery is a planar, "self-including" rendition of the complex exponential map, viewed as an infinite cyclic covering of the punctured plane. (Contrary to mathematical convention, traveling clockwise around the center in Escher's print "goes up one level".) I know of no better reference that Hendrik Lenstra's and Bart de Smit's 2003 analysis of the Droste effect.
Andrew D. HwangAndrew D. Hwang
There's a CST.SE thread of possible interest here: https://cstheory.stackexchange.com/questions/7958/papers-on-relation-between-computational-complexity-and-algebraic-geometry-topol
It mentions stuff like Geometric Complexity Theory, a far-out program for proving P!=NP with algebraic geometry.
Similarly here: https://cstheory.stackexchange.com/questions/2898/applications-of-topology-to-computer-science
Mentions the thing I actually first websearched for, Herlihy's work on concurrent and distributed computing using cohomology.
122 silver badges33 bronze badges
There is also an argument by Roman Jackiw that 3-cocycles appear in the quantum mechanics of a charged particle in the field of a magnetic monopole. There it happens that you can generate spatial translations either with the canonical momentum, which is gauge dependent because it involves the vector potential, or with the velocity, which is gauge invariant, because both have the same commutation relations with position. If you want to generate translations in a gauge invariant way, then translations turn out to be non-associative, i.e., you can compose 3 translation operators in different orders and get different results, all differing by a 3-cocycle in a particular cohomology. Restoring associativity leads to Dirac's quantization condition. This condition can also be obtained by different cohomological arguments, as Orlando Alvarez does, considering certain ambiguities that appear when you integrate a connection 1-form along paths covered by two or three patches. It happens that it is not trivial how to do this, you cannot just integrate along one patch up to a point in the intersection and keep integrating along the other patch. You have to do extra tricks that lead to ambiguities, which can be resolved if you impose some condition which is equivalent to Dirac's quantization condition when the connection is the vector potential of a monopole. The condition that you have to impose can be recast in terms of Cech cohomology. The papers: http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.54.159, http://link.springer.com/article/10.1007%2FBF01212452
Pedro AguilarPedro Aguilar
Jim Stasheff has surveyed the history of cohomology in physics in the 20th century: see the pdf linked at https://ncatlab.org/nlab/show/A+Survey+of+Cohomological+Physics
Francois Ziegler
Phil HarmsworthPhil Harmsworth
In (real world, for $n=3$) crystallography, given a point group $K\subseteq\mathrm{O}(n)$ and a ($K$-invariant) lattice $T\subseteq\mathbb{R}^n$,
the set of possible crystallographic classes (where, in the real world, a "crystallographic class" is unederstood as "the class of all crystals that have isomorphic isometry group"; and in general it just means a space group or crystallographic group, i.e. a discrete subgroup $G\subseteq\mathrm{Euc}(n)$ that contains $n$ linearly independent translations, up to conjugation by elements in the affine group $\mathrm{Aff(n,\mathbb{R})}$ or -which is the same by a theorem of Bieberbach- up to just abstract isomorphism)
that lie in the arithmetic crystal class determined by $(K,T)$, that is:
have $K$ as point group (up to conjugation by $\mathrm{GL}(n,\mathbb{Z})$, where $K\subseteq\mathrm{GL}(n,\mathbb{Z})$ via a lattice basis)
and have $T$ as translation lattice,
is in bijection with the quotient set
$$\frac{\mathrm{H}^1(K,\mathbb{R}^n/T)}{\mathcal{N}_{\mathrm{GL}(n,\mathbb{Z})}(K)}$$
where $\mathcal{N}_{\mathrm{GL}(n,\mathbb{Z})}(K)$ is the integral normalizer of $K$, acting on group cohomology $\mathrm{H}^1(K,\mathbb{R}^n/T)\simeq\mathrm{H}^2(K,T)$ (induced via the defining action of $K$ on $T$).
According to the crystallographer's terminology, crystals that correspond to the zero element in cohomology are said to be symmorphic.
QfwfqQfwfq
In the theory of error correcting codes, linear codes are most prominent. These can be seen as the kernel of a "parity-check" matrix, most commonly working over the two element field, $F_2.$
For example, a simple three bit repetition code corresponds to the kernel of the linear operator $$ \partial = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ \end{pmatrix}. $$ The kernel is one dimensional and so encodes one bit as either $(0,0,0)$ or $(1, 1, 1).$ If there is an error in the message $v\in F_2^3,$ perhaps a single bit was flipped, then we can hope to diagnose and recover the original message by computing $\partial v.$
So this is a way of viewing linear codes as "homologies". In this case we are using the homology of a circle. This would perhaps not be a very convincing application of homology theory except that it fits into a more general picture once we consider quantum codes. The prime example here is based on the homology of the torus, and is known as the "toric code". As with the repetition code above, we are working with a given cellular decomposition of the torus. This time we have a length two chain complex, still over the field $F_2$: $$ V_2 \xrightarrow{\partial_2} V_1 \xrightarrow{\partial_1} V_0. $$ In this case the first homology group $ker(\partial_1)/im(\partial_2)$ is two dimensional and so encodes two quantum bits (qubits.)
Error processes act on $V_1$ (by flipping bits) and are diagnosed by $\partial_1.$ The error correction procedure is now a matter of finding the most likely homology class corresponding to the error. The topology of the situation is quite prominent: errors act locally on the torus cellulation, but can only effect the encoded information by percolating around the torus.
The dual chain complex, which we get by taking the transpose of the operators $\partial_2, \partial_1$ is also important, and a similar error correction procedure takes place on this dual complex.
Amazingly, various research groups around the world are presently engaged in actually building such things, as they hold great promise for robust storage of quantum information, which will be needed for any serious attempt at building a quantum computer.
(I should note that I have left out an explanation of how to go from the complex Hilbert spaces where qubits actually live to $F_2$ linear algebra. This is covered in the literature on quantum stabilizer codes.)
Simon BurtonSimon Burton
I am surprised no-one has mentioned Persistent Homology.
wonderich
Bruce WestburyBruce Westbury
$\begingroup$ en.wikipedia.org/wiki/… $\endgroup$ – Alexander Chervov Jun 20 '12 at 12:40
$\begingroup$ See also José Figueroa-O'Farrill's comment on the accepted answer, in which he mentions a talk by Gunnar Carlsson about persistent homology. $\endgroup$ – J W Jun 20 '12 at 15:59
There are some applications of topology/cohomology to combinatorics and combinatoric geometry. One of the earliest examples is surely Lovasz's proof of a bound for the chromatic number of the Kneser graph; he uses the Borsuk-Ulam theorem, which is usually proved by homological methods. A modern exposition can be found here.
Another example is Tveberg's theorem with all its variants on the configuration of points in space (the best results can be found in a recent paper of Blagojevic, Matschke and Ziegler. There are many other results in convex geometry/polytope theory which use topological methods and, in particular, cohomology.
Lennart MeierLennart Meier
$\begingroup$ While this isn't so clear from the title of the question, the first sentence in the text of the question mentions wanting things outside of pure mathematics. $\endgroup$ – tweetie-bird Oct 25 '12 at 14:27
An application of cohomology to provide a geometric/topological description charged particles in General Relativity can be found in
GRAVITATION: An Introduction to Current Research, ed. Louis Witten
and references therein.
Wheeler's geometrodynamics program contained a subprogram named "charge without charge", which aimed to express the electric charge in terms of geometric and/or topological properties. A wormhole allows the existence of an electromagnetic field without source, which looks like having sources - hence the name "charge without charge". The two ends of the wormholes behave as particles of opposite electric charge. And all this can be obtained as a solution to Einstein-Maxwell equations. Roots of the approach of Misner and Wheeler can be found in the paper of Einstein and Rosen, and a series of papers of G. Y. Rainich from 1924-1925.
Not the answer you're looking for? Browse other questions tagged at.algebraic-topology cohomology big-list applications popularization or ask your own question.
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\begin{document}
\input goksty.tex \newtheorem{question}[thm]{Question}
\address{ Ludmil Katzarkov\\ University of Vienna. } \email{[email protected]}
\address{ Victor Przyjalkowski\\ Steklov Mathematical Institute.
} \email{[email protected], [email protected]}
\title[Landau--Ginzburg models --- old and new]{Landau--Ginzburg models --- old and new}
\author[KATZARKOV and PRZYJALKOWSKI]{ Ludmil Katzarkov, Victor Przyjalkowski}
\thanks{L.\,K was funded by NSF Grant DMS0600800, NSF FRG Grant DMS-0652633, FWF Grant P20778, and an ERC Grant --- GEMIS, V.\,P. was funded by FWF grant P20778, RFFI grants 11-01-00336-a and 11-01-00185-a, grants MK$-1192.2012.1$, NSh$-5139.2012.1$, and AG Laboratory GU-HSE, RF government grant, ag. 11 11.G34.31.0023. }
\begin{abstract} In the last three years a new concept --- the concept of wall crossing has emerged. The current situation with wall crossing phenomena, after papers of Seiberg--Witten, Gaiotto--Moore--Neitzke, Vafa--Cecoti and seminal works by Donaldson--Thomas, Joyce--Song, Maulik--Nekrasov--Okounkov--Pandharipande, Douglas, Bridgeland, and Kontsevich--Soibelman, is very similar to the situation with Higgs Bundles after the works of Higgs and Hitchin --- it is clear that a general ``Hodge type'' of theory exists and needs to be developed. Nonabelian Hodge theory did lead to strong mathematical applications~--- uniformization, Langlands program to mention a few. In the wall crossing it is also clear that some ``Hodge type'' of theory exists --- Stability Hodge Structure (SHS). This theory needs to be developed in order to reap some mathematical benefits --- solve long standing problems in algebraic geometry. In this paper we look at SHS from the perspective of Landau--Ginzburg models and we look at some applications. We consider simple examples and explain some conjectures these examples suggest. \end{abstract} \keywords{Hodge structures; categories; Landau--Ginzburg models}
\maketitle
\section{Introduction}
Mirror symmetry is a physical duality between $N= 2$ superconformal field theories. In the 1990's Maxim Kontsevich reinterpreted this concept from physics as an incredibly deep and far-reaching mathematical duality now known as Homological Mirror Symmetry (HMS). In a famous lecture in 1994, he created a frenzy in the mathematical community which lead to
synergies between diverse mathematical disciplines: symplectic geometry, algebraic geometry, and category theory. HMS is now the cornerstone of an immense field of active mathematical research.
In the last three years a new concept --- the concept of wall crossing has emerged. The current situation with wall crossing phenomena, after papers of Seiberg--Witten, Gaiotto--Moore--Neitzke, Vafa--Cecoti and seminal works by Donaldson--Thomas, Joyce--Song, Maulik--Nekrasov--Okounkov--Pandharipande, Douglas, Bridgeland, and Kontsevich--Soibelman, is very similar to the situation with Higgs Bundles after the works of Higgs and Hitchin --- it is clear that a general ``Hodge type'' of theory exists and needs to be developed. Nonabelian Hodge theory did lead to strong mathematical applications --- uniformization, Langlands program to mention a few. In the wall crossing it is also clear that some ``Hodge type'' of theory needs to be developed in order to reap some mathematical benefits --- solve long standing problems in algebraic geometry.
The foundations of these new Hodge structures, which we call \emph{Stability Hodge Structures (SHS)} will appear in a paper by the first author, Kontsevich, Pantev and Soibelman --- \cite{KKPS}. In this paper we will look at SHS from the perspective of Landau--Ginzburg models and we will also look at some applications. We will consider simple examples and explain some conjectures these examples suggest. The further elaboration and examples will appear in \cite{KKPS} and \cite{DKK}.
We start with the classical interpretation of wall crossings in Landau--Ginzburg models. After that we describe a hypothetical program of ``Stability Hodge Theory'' which combine Nonabelian and Noncommutative Hodge theory. We consider some possible applications in this paper. First we consider an approach to the conjecture that the universal covering of a smooth projective variety is holomorphically convex. This is a classical question in algebraic geometry proven by the first author and collaborators for linear fundamental groups \cite{EKPR}. It was believed that for nonresidually finite fundamental groups one needs a different approach and in this paper we outline a procedure of extending the argument to the nonresidually finite case based on SHS. We also outline possible applications to Hodge structures with many filtrations and to Sarkisov's theory.
Stability Hodge Structure is a notion which originates from functions of one complex variable and combinatorics --- gaps, polygons, and circuits. We give these classical notions a new read through HMS and category theory, dressing them up with some cluster varieties and integrable systems. After that we enhance these data additionally with some basic nonabelian Hodge theory in order to get a property we need --- strictness. In the same way as moduli spaces of Higgs bundles parameterize spectral coverings, the moduli space of deformed stability conditions parameterizes Landau--Ginzburg models.
We believe this is only the tip of the iceberg and this very rich motivic conglomerate of ideas will play an important role in the studies of categories and of algebraic cycles. In particular we suggest that the categorical notion of spectra can be seen as a Hodge theoretic notion related to the ``homotopy type of a category''.
The paper is organized as follows. In Sections~\ref{section:clusters} and~\ref{section:Minkowski} we describe the classical approach to Landau--Ginzburg models and wall crossings. After that in Sections~\ref{section:wall crossings},~\ref{section:SHS},~\ref{section:Higgs bundles} we define Stability Hodge Structures and build a parallel with Simpson's nonabelian Hodge theory.
We also discuss possible applications in Sections~\ref{section:spectra},~\ref{section:multi LG},~\ref{section:birational}.
\section{Classical Landau--Ginzburg models and wall crossings }
\label{section:clusters}
In this section we recall the ``classical'' way of interpreting wall crossing in the case of Landau--Ginzburg models. We will establish a certain combinatorial framework on which we later base our constructions.
We recall the notion of Landau--Ginzburg models from the Laurent polynomials point of view. For more details see, say,~\cite{Prz07} and references therein. Cluster transformations and Minkowski decompositions for Landau--Ginzburg models are discussed in~\cite{CCGGK12},~\cite{ACGK12},~\cite{CCGK13}.
Let $X$ be a smooth Fano variety of dimension $n$. We can associate \emph{a quantum cohomology ring} $QH^*(X)=H^*(X,\Q)\otimes \Lambda$ to it, where $\Lambda$ is the Novikov ring for $X$. The multiplication in this ring, the so called \emph{quantum multiplication}, is given by \emph{(genus zero) Gromov--Witten invariants} --- numbers counting rational curves lying in $X$. Given these data one can associate \emph{a regularized quantum differential operator} $Q_X$ (the second Dubrovin connection) --- the regularization of an operator associated with connection in the trivial vector bundle given by a quantum multiplication by the canonical class $K_X$. In ``good'' cases such as we consider (for Fano threefolds or complete intersections) the equation $Q_XI=0$ has a unique normalized analytic solution $I=1+a_1t+a_2t^2+\ldots$.
\begin{defn} \label{definition: toric LG} \emph{A toric Landau--Ginzburg model} is a Laurent polynomial $f\in \mathbb{C}[x_1^{\pm 1}, \ldots, x_n^{\pm}]$ such that: \begin{description}
\item[Period condition] The constant term of $f^i\in \mathbb{C}[x_1^{\pm 1}, \ldots, x_n^{\pm}]$ is $a_i$ for any $i$ (this means that $I$ is a period of a family $f\colon (\mathbb{C}^*)^n\to \mathbb{C}$, see~\cite{Prz07}).
\item[Calabi--Yau condition] Any fiber of $f\colon (\mathbb{C}^*)^n\to \mathbb{C}$ after some fiberwise compactification has trivial dualizing sheaf.
\item[Toric condition] There is an embedded degeneration $X\rightsquigarrow T$ to a toric variety $T$ whose fan polytope
(the convex hull of generators of its rays) coincides with the Newton polytope (the convex hull of non-zero coefficients) of $f$. A Laurent polynomial without the toric condition is called \emph{a weak Landau--Ginzburg model}. \end{description} \end{defn}
Toric Landau--Ginzburg models for complete intersections can be derived from the Hori--Vafa suggestions (see, say,~\cite{Prz09}).
\begin{defn} Let $X$ be a general Fano complete intersection of hypersurfaces of degrees $d_1,\ldots,d_k$ in $\mathbb{P}^N$. Let $d_0=N-d_1-\ldots- d_k$ be its index. Then a Laurent polynomial $$ f_X=\frac{(x_{1,1}+\ldots+x_{1,d_1-1}+1)^{d_1}\cdot\ldots\cdot(x_{k,1}+\ldots+x_{k,d_k-1}+1)^{d_k}}{\prod x_{ij}}+x_{01}+\ldots+x_{0d_0-1}. $$ we call \emph{of Hori--Vafa type}. \end{defn}
\begin{thm}[Proposition 9 in~\cite{Prz09} and Theorem 2.2,~\cite{IP11}] The polynomial $f_X$ is a toric Landau--Ginzburg model for $X$.
\end{thm}
\begin{defn} Let $f$ be a Laurent polynomial in $\C [x_0^{\pm 1},\ldots,x_n^{\pm 1}]$. Then a (non-toric birational) symplectomorphism is called \emph{of cluster type} if
it is a composition of toric change of variables and symplectomorphisms of type $$ y_0=x_0\cdot f_0(x_1,\ldots,x_i)^{\pm 1}, \ \ y_1=x_1,\ldots,y_n=x_n, $$ for some Laurent polynomial $f_0$ and under this change of variables $f$ goes to a Laurent polynomial for which a Calabi--Yau condition holds.
It is called \emph{elementary of cluster type} if (up to toric change of variables) $$ f_0=x_1+\ldots+x_i+1. $$ It is called \emph{of linear cluster type} if it is a composition of elementary symplectomorphisms of cluster type and toric change of variables. \end{defn}
\begin{rem} For all examples in the rest of the paper the Calabi--Yau condition holds for all considered cluster type transformations. \end{rem}
\begin{prop} Let $f$ be a weak Landau--Ginzburg model for $X$. Let $f^\prime$ be a Laurent polynomial obtained from $f$ by symplectomorphism of cluster type. Then $f^\prime$ is a weak Landau--Ginzburg model for $X$. \end{prop}
\begin{proof}
A period giving constant terms of Laurent polynomials is, up to proportion, an integral of the (depending on $\lambda\in \mathbb{C}$) form $\frac{1}{1-\lambda f}\prod \frac{dx_i}{x_i}$
over a standard $n$-cycle on the torus $|x_1|=\ldots =|x_n|=1$. This integral does not change under cluster type symplectomorphisms.
\end{proof}
\begin{exm} Let $X$ be a quadric threefold. There are two types of degenerations of $X$ to normal toric varieties inside the space of quadratic forms. That is, $$ T_0=\{x_1x_2=x_2^3\}\subset \mathbb{P}[x_1:x_2:x_3:x_4:x_5] $$ and $$ T_1=\{x_1x_2=x_3x_4\}\subset \mathbb{P}[x_1:x_2:x_3:x_4:x_5]. $$ Let $$ f_0=\frac{(x+1)^2}{xyz}+y+z $$ be a weak Landau--Ginzburg model of Hori--Vafa type for $X$. Let $$ f_1=\frac{(x+1)}{xyz}+y(x+1)+z $$ be its cluster-type transformation given by the change of variables $$ \frac{y}{(x+1)}\mapsto y. $$ One can see that $T_0=T_{f_0}$ and $T_1=T_{f_1}$. \end{exm}
\begin{rem} One can see that applying the same change of variables a second time to $f_1$ gives back (up to toric change of variables) $f_0$. \end{rem}
\begin{exm} Let $X$ be a cubic threefold. There are two types of degenerations of $X$ to normal toric varieties inside the space of cubic forms. That is, $$ T_0=\{x_1x_2x_3=x_4^3\}\subset \mathbb{P}[x_1:x_2:x_3:x_4:x_5] $$ and $$ T_1=\{x_1x_2x_3=x_4^2x_5\}\subset \mathbb{P}[x_1:x_2:x_3:x_4:x_5]. $$ Let $$ f_0=\frac{(x+y+1)^3}{xyz}+z $$ be a weak Landau--Ginzburg model of Hori--Vafa type for $X$. Let $$ f_1=\frac{(x+y+1)^2}{xyz}+z(x+y+1) $$ be its cluster-type transformation given by the change of variables $$ \frac{z}{(x+y+1)}\mapsto z. $$ One can see that $T_0=T_{f_0}$ and $T_1=T_{f_1}$. \end{exm}
\begin{rem} Applying this change of variables a second time to $f_1$ we get (up to toric change of variables) $f_1$ again and applying it a third time we get $f_0$ back. \end{rem}
\begin{exm} Let $X$ be a cubic fourfold. There are three types of degenerations of $X$ to normal toric varieties inside the space of cubic forms. That is, $$ T_{00}=\{x_1x_2x_3=x_4^3\}\subset \mathbb{P}[x_1:x_2:x_3:x_4:x_5:x_6], $$ $$ T_{10}=\{x_1x_2x_3=x_4^2x_5\}\subset \mathbb{P}[x_1:x_2:x_3:x_4:x_5:x_6], $$ and $$ T_{11}=\{x_1x_2x_3=x_4x_5x_6\}\subset \mathbb{P}[x_1:x_2:x_3:x_4:x_5:x_6], $$ Let $$ f_{00}=\frac{(x+y+1)^3}{xyzt}+z+t $$ be a weak Landau--Ginzburg model of Hori--Vafa type for $X$. Let $$ f_{10}=\frac{(x+y+1)^2}{xyzt}+z(x+y+1)+t $$ be its cluster-type transformation given by the change of variables $$ \frac{z}{(x+y+1)}\mapsto z $$ and let $$ f_{11}=\frac{(x+y+1)}{xyzt}+z(x+y+1)+t(x+y+1) $$ be the cluster-type transformation of $f_{10}$ given by the change of variables $$ \frac{t}{(x+y+1)}\mapsto t. $$ One can see that $T_{00}=T_{f_{00}}$, $T_{10}=T_{f_{10}}$, and $T_{11}=T_{f_{11}}$. \end{exm}
\begin{rem} Applying the first change of variables a second time to $f_{10}$ we get (up to toric change of variables) $f_{10}$ again, applying it once more we get $f_{00}$, and applying any change of variables to $f_{11}$ we get $f_{10}$. \end{rem}
\begin{exm} Consider quadrics in $\mathbb{P}=\mathbb{P}(1,1,1,1,2)$. Denote the coordinates in $\mathbb{P}$ by $x_0$, $x_1$, $x_2$, $x_3$, $x_4$, where the weight of $x_4$ is 2. The general quadric is $$ T_1=\{F_2(x_0,x_1,x_2,x_3)+\lambda x_4=0\}, $$ where $F_2$ is a quadratic form and $\lambda\in \mathbb{C}\setminus 0$. Projection on the hyperplane generated by $x_0,\ldots,x_3$ gives an isomorphism of $T_1$ with $\mathbb{P}^3$. The general variety with $\lambda=0$ is a toric variety $$ T_2=\{x_0x_1=x_2x_3\}. $$ It degenerates to $$ T_3=\{x_1x_2=x_0^2\}. $$ One can see that $T_3$ is an image of $\mathbb{P}(1,1,2,4)$ under the Veronese map $v_2$.
Consider the following 3 weak Landau--Ginzburg models for $\mathbb{P}^3$: $$ f_1=x+y+z+\frac{1}{xyz}, $$ $$ f_2=x+\frac{y}{x}+\frac{z}{x}+\frac{1}{xy}+\frac{1}{xz}, $$ $$ f_3=\frac{(x+1)^2}{xyz}+\frac{y}{z}+z. $$ Changing toric variables one can rewrite $f_1$ as $$ f_1^{\prime }=z(x+1)+y+\frac{1}{xyz^2}, $$ $$ f_1^{\prime \prime}=z(x+1)+\frac{y}{z}+\frac{1}{xyz}. $$ The cluster-type change of variables $$ x\mapsto x,\ \ y\mapsto y,\ \ z(x+1)\mapsto z $$ sends $f_1^\prime$ to a Laurent polynomial that differs from $f_3$ by a toric change of variables and $f_1^{\prime \prime}$ to a polynomial $$ z+\frac{(x+1)y}{z}+\frac{(x+1)}{xyz}, $$ which differs from $f_2$ by toric change of variables.
The cluster-type change of variables $$ x\mapsto x,\ \ y(x+1)\mapsto y,\ \ z\mapsto z $$ sends the last expression to $f_3$.
One can see that $T_1=T_{f_1}$, $T_2=T_{f_2}$, and $T_3=T_{f_3}$. \end{exm}
\begin{thm}[Hacking--Prokhorov,~\cite{HaPr05}] Let $X$ be a degeneration of $\mathbb{P}^2$ to a $\Q$-Gorenstein surface with quotient singularities. Then $X=\mathbb{P}(a^2,b^2,c^2)$, where $(a, b, c)$ is any solution of the Markov equation $a^2+b^2+c^2=3abc$. \end{thm}
\begin{rem} All Markov triples are obtained from the basic one $(1,1,1)$ by a sequence of \emph{elementary transforms} $$ (a,b,c)\mapsto (a,b, 3ab-c). $$ \end{rem}
\begin{prop}[\cite{GU10}, see also~\cite{CMG13}] \label{proposition:Galkin} Let $(a,b,c)$ be a Markov triple and let $f$ be a weak Landau--Ginzburg model for $\mathbb{P}^2$ such that $T_f=\mathbb{P}(a^2,b^2,c^2)$. Then there is an elementary cluster-type transformation such that for the image $f^\prime$ of $f$ under this transformation $T_{f^\prime}=\mathbb{P}(a^2,b^2,(3ab-c)^2)$. \end{prop}
{\bf Sketch of the proof}\ (S.\,Galkin). Consider $d\geq c$ such that $3ad=b\ (\mathrm{mod}\ c)$. One can check that we can choose toric coordinates $x$, $y$ such that in these coordinates vertices of the Newton polytope of $f$ are $(d,c)$, $(d-c,c)$, and $(-\frac{d(3ab-c)-b^2}{c},-3ab+c)$. Let $p$ be the $k$-th integral point from the end of an edge of integral length $n$ of the Newton polytope of $f$. Then the coefficient of $f$ at $p$ is $\binom{n}{k}$ (this can be proved by induction). This means that $$ f=x^{d-c}y^c(x+1)^c+\frac{1}{x^{\frac{d(3ab-c)-b^2}{c}}y^{3ab-c}}+\sum_r y^{-n_r}f_r(x), $$ where $n_i$'s are non-negative and $f_i$'s are some Laurent polynomials in $x$. One can check that the change of variables of cluster type $$ y^\prime=y(x+1),\ \ x^\prime=x $$ sends $f$ to a weak Landau--Ginzburg model $f^\prime$ such that $T_{f^\prime}=\mathbb{P}(a^2,b^2,(3ab-c)^2)$. \qed
We extend observed connection between degenerations and birational transformations further to a general connection between geometry of moduli space of Landau--Ginzburg models, birational and symplectic geometry. We summarize this connection in Table~\ref{tab:Section 2 tab 1} and we will investigate it (mainly conjecturally) in the sections that follow.
\begin{table}[h] \begin{center}
\begin{tabular}{|c||c|c|} \hline \begin{minipage}[c]{0.5in} \centering
\end{minipage} & \begin{minipage}[c]{2in} \centering
Fano variety $X$
\end{minipage} & \begin{minipage}[c]{2in} \centering
Landau--Ginzburg model $LG(X)$
\end{minipage} \\\hline\hline \begin{minipage}[c]{0.5in} \centering
A side
\end{minipage} & \begin{minipage}[c]{2in} \centering
$\text{\sf Fuk} (X)$: symplectomorphisms and general degenerations
\end{minipage} & \begin{minipage}[c]{2in} \centering
$FS(LG(X))$: degenerations
\end{minipage} \\\hline \begin{minipage}[c]{0.5in} \centering
B side
\end{minipage} & \begin{minipage}[c]{2in} \centering
$D^b_{sing}(LG(X))$: phase changes
\end{minipage} & \begin{minipage}[c]{2in} \centering
$D^b(X)$: birational transformations
\end{minipage} \\\hline \end{tabular} \end{center} \caption{Wall crossings.} \label{tab:Section 2 tab 1} \end{table}
\section{Minkowski decompositions and cluster transformations}
\label{section:Minkowski}
\begin{defn} Let $N\cong {\mathbb Z}^n$ be a lattice. Denote $N_\mathbb{R}=N\otimes \mathbb{R}$. \emph{A polytope} $\Delta \subset N_\mathbb{R}$ is a convex hull of finite number of points in $N_\mathbb{R}$. A polytope is called \emph{integral} iff these points lie in $N\otimes 1$. A polytope called \emph{primitive} if it is integral and its vertices are primitive. A Laurent polynomial is called \emph{primitive} if its Newton polytope is primitive. \end{defn}
\begin{defn}
\emph{The Minkowski sum} $\Delta_1+\ldots+\Delta_k$ of polytopes $\Delta_1,\ldots,\Delta_k$ is the polytope $\{v_1+\ldots+v_k| v_i\in \Delta_i\}$. An integral polytope is called \emph{irreducible} if it can't be presented as a Minkowski sum of two non-trivial integral polytopes. \end{defn}
\begin{rem} A Minkowski sum of integral polytopes is integral. \end{rem}
\begin{defn} Consider an integral polytope $\Delta\in {\mathbb Z}^n$. \emph{A Minkowski presentation} of $\Delta$ is a presentation of each of its faces as a Minkowski sum of irreducible integral polytopes such that if a face $\Delta'$ lies in a face $\Delta$ then the intersections of Minkowski summands for $\Delta$ with $\Delta'$ give a presentation for $\Delta'$.
Consider a Laurent polynomial $f\in\mathbb{C}[{\mathbb Z}^n]$. For any face $\Delta$ of $\Delta_f$ denote the sum
of all monomials of $f$ lying in $\Delta$ by $f_\Delta$. The polynomial $f$ is called \emph{Minkowski polynomial} if
there exists a Minkowski presentation such that, for any face $\Delta$ of $\Delta_f$ with given Minkowski sum expansion $\Delta=\Delta_1+\ldots+\Delta_k$,
there are Laurent polynomials $f_{\Delta_i}\in \mathbb{C}[{\mathbb Z}^n]$ such that the coefficients of $f_{\Delta_i}$ at vertices of $\Delta_i$ are 1's and $f_\Delta=f_{\Delta_1}\cdot \ldots\cdot f_{\Delta_k}$.
\end{defn}
\begin{rem} Let $e$ be an edge of a Minkowski Laurent polynomial of integral length $n$. Its unique Minkowski expansion to irreducible summands is the expansion to $n$ segments of integral length 1. Thus the coefficient of the monomial associated to the $i$'th integral point of $e$ (from any end) is $\binom{n}{i}$. \end{rem}
\begin{rem} Toric Landau--Ginzburg models of Hori--Vafa type or toric Landau--Ginzburg models from~\cite{Prz09} are Minkowski Laurent polynomials. \end{rem}
\begin{exm}[Ilten--Vollmert construction,~\cite{IV09}] \label{example:Ilten-Vollmert} Consider an integral polytope $\Delta\subset N={\mathbb Z}^n$. Let the origin of $N$ lie strictly inside $\Delta$. Let $X=T_\Delta$ be the toric variety whose fan is the face fan for $\Delta$. Denote the dual lattice to $N$ by $M=N^\vee$. Put $N'=N\oplus {\mathbb Z}$, $M'=M\oplus {\mathbb Z}$. Let $C$ be the cone generated by $(\Delta,1)$. Then $X=Proj\,\mathbb{C}[C^\vee\cap M']$ with grading given by $d=(0,1)\in M'$. For any primitive $r\in M'$, consider the map $r\colon N'\to{\mathbb Z}$. Let $L_r=ker(r)$. Let $s_r$ be a retract (cosection) of the inclusion $i\colon L_r\to N'$, that is, a map $N'\to L_r$ such that
$s_ri=Id_{L_r}$. It is unique up to translations along $L_r$. Let $C^+=s_r(\{p\in C|\langle p, r\rangle=1\})$, $C^-=s_r(\{p\in C|\langle p, r\rangle=-1\})$ be two ``slices'' of $C$ cut out by evaluating function at $r$.
Choose $r$ such that $r=(r_0,0)\in M'$ and such that $C^-$ is a cone with its single vertex a lattice point. Consider a Minkowski decomposition $C^+=C_1+C_2$ to (possibly rational) polytopes such that for any vertex $v$ of $C^+$, at least one of the corresponding vertices in $C_1$ and $C_2$ is a lattice point. Let $D$ be the cone in $L_r\oplus {\mathbb Z}$ generated by $(C^-,0)$, $(C_1,1)$, and $(C_2,-1)$. Denote $X'=Proj\,\mathbb{C}[D^\vee\cap (L_r\oplus {\mathbb Z})^\vee)]$ where the grading is now given by $(s_r(d),0)$. \end{exm}
\begin{prop}[Remark 1.8 and Theorem 4.4 in~\cite{IV09}] There is an embedded degeneration of $X'$ to $X$. \end{prop}
\begin{exm}[Ilten] \label{example:Ilten} Let $\Delta\subset {\mathbb Z}^2$ be the convex hull of the points $(-1,2)$, $(1,2)$, and $(0,-1)$. Then $X=\mathbb{P}(1,1,4)$. Let $r=(0,1,0)$. Then $s_r$ is given by the matrix $$ \left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 1 \\
\end{array} \right). $$
We are in setup of Example~\ref{example:Ilten-Vollmert} (see Figure~\ref{figure:Ilten's example}). The vertex of $\{p\in C|\langle p, r\rangle=-1\}$
is $(0,-1,1)$ and goes to a vertex $(0,0)$ under $s_r$ and the vertices of $\{p\in C|\langle p, r\rangle=1\}$ are $(\pm \frac{1}{2},1,\frac{1}{2})$ and goes to vertices $(\pm \frac{1}{2},\frac{3}{2})$ under $s_r$. That is, we have a Minkowski decomposition drawn on Figure~\ref{figure:decomposition}.
\begin{figure}
\caption{Deformation of $\mathbb{P}(1,1,4)$.}
\label{figure:Ilten's example}
\end{figure}
\begin{figure}
\caption{Decomposition of $C^+$.}
\label{figure:decomposition}
\end{figure}
The polytope for $X'$ is a convex hull of points $(-1,1)$, $(0,1)$, and $(1,-2)$ since the second coordinate becomes to be equal to 1 not on $(C_2,-1)$ but on $(2C_2,-2)$. Its face fan is a fan of $\mathbb{P}^2$. Thus we get a deformation of $\mathbb{P}^2$ to $\mathbb{P}(1,1,4)$. \end{exm}
The following proposition shows that the degenerations given by Example~\ref{example:Ilten-Vollmert} give cluster transformations for Minkowski polynomials.
\begin{prop} \label{proposition:Minkowski-cluster} Let $\Delta=\Delta_f$ be the Newton polytope of a Minkowski polynomial $f$. Let $\Delta'$ be a polytope obtained from $\Delta$ by the procedure described in Example~\ref{example:Ilten-Vollmert} given by integral Minkowski summands agreing with the Minkowski decompositions of the faces of $\Delta$. Then $\Delta'=\Delta_{f'}$ for some Minkowski polynomial $f'$. \end{prop}
\begin{proof} Let $f\in \mathbb{C}[x_0^{\pm 1},\ldots,x_n^{\pm 1}]$. After toric changes of variables we can assume that $s_r$ is the projection on coordinates $x_1,\ldots,x_n$. Then $$f=f_+(x_1,\ldots,x_n)x_0+f_0(x_1,\ldots,x_n)+\frac{f_{-}(x_1,\ldots,x_n)}{x_0}.$$ As $f$ is a Minkowski polynomial we have $f_+=f_1f_2$. Thus after change of variables $x_0\to x_0/f_2$ we get a Minkowski polynomial $$f'=f_1(x_1,\ldots,x_n)x_0+f_0(x_1,\ldots,x_n)+\frac{f_{-}(x_1,\ldots,x_n)f_{2}(x_1,\ldots,x_n)}{x_0}$$ with Newton polytope $\Delta'$. \end{proof}
\begin{rem} Example~\ref{example:Ilten} shows that the statement of Proposition~\ref{proposition:Minkowski-cluster} holds for non-integral case as well. This example is the first non-trivial cluster transformation given by Proposition~\ref{proposition:Galkin}. \end{rem}
\begin{exm} Let $\Delta$ be the convex hull of points $(-1,1)$, $(1,1)$, and $(0,-1)$. Then $X$ is a quadratic cone $\mathbb{P}(1,1,2)$. (A unique) Minkowski polynomial for $\Delta$ is $$f=\frac{(x+1)^2y}{x}+\frac{1}{y}.$$ After cluster change of variables $y\to \frac{y}{x+1}$ we get a polynomial $$ \frac{(x+1)y}{x}+\frac{x+1}{y}. $$ It is (a unique) Minkowski polynomial for polytope $\Delta'$ --- the convex hull of points $(-1,-1)$, $(0,-1)$, $(1,-1)$, and $(0,-1)$. These points generate the fan of a smooth quadric $X'$. \end{exm}
\section{Degenerations and wall crossings}
\label{section:wall crossings}
In the previous section we have established certain combinatorial structures --- cluster transformations connected to wall crossings. We will relate these combinatorial structures to moduli space of stability conditions. We do this in two steps:
{\bf Step 1.} First we relate the combinatorial structures to ``moduli space of Landau--Ginzburg models''.
{\bf Step 2.} Next we describe hypothetically how the ``moduli space of Landau--Ginzburg models'' fits in a ``twistor family'' with generic fiber the moduli space of stability conditions of a Fukaya--Seidel category.
We start with step one --- collecting all Landau--Ginzburg models in a moduli space. The idea is to record wall crossings as relations in the mapping class group and then relations between relations and so on. This suggests a connection with Hodge theory and higher category theory. We will start with a rather simple approach which we will enhance later in order to serve our purposes. Nearly ten years ago it was discovered that, while the symplectic mapping class group of a curve equals the ordinary (oriented) mapping class group, these two groups differ greatly for higher dimensional symplectic manifolds. Understanding the structure of these groups has been a goal of many researchers in symplectic geometry. The initial purpose of the construction below was to obtain a presentation of the symplectic mapping class group of toric hypersurfaces. Along the way we have obtained a characterization of the zero fiber of a Stability Hodge Structure.
To explain our approach, we recall some notation and constructions. Assume $A \subset \Z^d$ is a finite set, $X_A$ the polarized toric variety associated to $A$ with ample line bundle $\mathcal{L}$. In \cite{GKZ}, the secondary polytope $\textrm{Sec} (A)$ parameterizing regular subdivisions was constructed and shown to be the Newton polytope of the $E_A$ determinant (a type of discriminant). We realize the toric variety associated to $\textrm{Sec} (A)$ as the coarse moduli space of a stack $\mathcal{X}_{\textrm{Sec}(A)}$ defined in \cite{Lafforgue}. We observe that the stack $\mathcal{X}_{\textrm{Laf}(A)}$ constructed in \cite{Lafforgue} has a proper map $\pi$ to $\mathcal{X}_{\textrm{Sec}(A)}$ whose fibers are degenerations of $X_A$, and we constructed a polytope $\textrm{Laf} (A)$ which is dual to the fan defining $\mathcal{X}_{\textrm{Laf} (A)}$. The zero set $\mathcal{H}_{\textrm{Sec}(A)}$ of a section of the associated line bundle parameterizes sections of $\mathcal{L}$ and degenerated sections are hypersurfaces in the associated degenerated toric variety. Upon restriction, we obtain a proper map $\pi : \mathcal{H}_{\textrm{Sec}(A)} \to \mathcal{X}_{\textrm{Sec}(A)}$ with non-singular fibers symplectomorphic to any non-degenerate section of $\mathcal{L}$.
Since $ \pi: \mathcal{H}_{\textrm{Sec}(A)} \to \mathcal{X}_{\textrm{Sec}(A)}$ is a proper map, we may consider symplectic parallel transport of the non-singular fibers along paths in the complement of the zero set $Z_A$ of the $E_A$ determinant. Denote by $H_p$ the fiber of $\pi$. We observe that the subset of the fibers meeting the toric boundary $H$ are horizontal in the sense that if $q \in \partial H_p$ then the symplectic orthogonal $(T_q H_p)^{\perp_\omega} \subset T_q (\partial \mathcal{H}_{\textrm{Sec}(A)} )$, where $\omega$ corresponds to a restriction of Fubini--Study metric. That is, parallel transport is a symplectomorphism that preserves the boundary of the hypersurfaces. Choosing a base point $p$ of $\mathcal{X}_{\textrm{Sec}(A)}\setminus Z_A$, we obtain a map from the based loop space $\rho :\Omega (\mathcal{X}_{\textrm{Sec}(A)} \setminus
Z_A ) \to \textrm{Symp}^\partial ( H_p )$ and a group homomorphism
\begin{equation*} \rho_* : \pi_1 (\mathcal{X}_{\textrm{Sec}(A)}\setminus Z_A ) \to \pi_0 (\textrm{Symp}^\partial (H_p )), \end{equation*} where $\pi_0 (\textrm{Symp}^\partial (H_p ))$ is a mapping class group.
However, from a field theory perspective, this homomorphism in imprecise; one should consider not only symplectomorphisms preserving the boundary, but also those that preserve the normal bundle of the boundary. In this way, we can glue two hypersurfaces together without creating an ambiguity in the symplectomorphism groups. We call such a symplectomorphism \emph{boundary framed morphism} and denote the corresponding group $\textrm{Symp}^{\partial,\textrm{fr}} (H_p)$. For toric hypersurfaces, this group is a central extension of $\textrm{Symp}^\partial (H_p)$. It is not generally the case, however, that parallel transport preserves the framing, but the change in framing can be controlled by keeping track of the homotopies in $\Omega (\mathcal{X}_{\textrm{Sec}(A)}\setminus Z_A)$ or by passing to the loop space of an auxiliary real torus bundle $\mathcal{E} \to \mathcal{X}_{\textrm{Sec}(A)}\setminus Z_A$, giving a homomorphism
\begin{equation*} \tilde{\rho}_* : \pi_1 (\mathcal{E} ) \to \pi_0 (\textrm{Symp}^{\partial, \textrm{fr}} (H_p )). \end{equation*}
In many cases, this homomorphism is surjective.
The stack $\mathcal{X}_{\textrm{Sec}(A)}$ is as complicated combinatorially as the secondary polytope $\textrm{Sec} (A)$, which is computationally expensive to describe. While the Newton polytope of $E_A$ was found in \cite{GKZ}, $Z_A$ is far from smooth and there are open questions about its singular structure. We bypass these difficulties by considering only the lowest dimensional boundary strata of $\mathcal{X}_{\textrm{Sec}(A)}$ where non-trivial behavior occurs. Thus the first and main case we examine are the one dimensional boundary strata of $\mathcal{X}_{\textrm{Sec}(A)}$. Combinatorially, these are known as circuits.
\emph{A circuit $A$} is a collection of $d + 2$ points in $\Z^d$, such that there are exactly two coherent triangulations (see~\cite{GKZ}) of $A$, so the secondary polytope is a line segment and the secondary stack a weighted projective line $\mathbb{P} (a, b)$. $Z_A$ is either two or three points; two of the points are the equivariant orbifold points $\{0, \infty\}$ and the possible third is an interior point. Both the constants, $a, b$ and the number of points in $Z_A$ depends on the convex hull and affine positioning of $A$ --- for more details see \cite{DKK}. When $Z_A$ consists of three points, their complement retracts onto a figure eight and the fundamental group is free on two letters. In this case, we have the based loops $\delta_1, \delta_2, \delta_3 = \delta_2^{-1} \delta_1^{-1}$ encircling the three points. The symplectic monodromy $T_i = \tilde{\rho_*} (\delta_i )$ is computable from known results in symplectic geometry as either spherical Dehn twists or as twists about a tropical decomposition. The image via $\tilde{\rho}_*$ gives the relation
\begin{equation} \label{eq:circ1} T_1 T_2 T_3 = T_{\partial H_p}, \end{equation}
where $T_{\partial H_p}$ is the central element determined by twisting the framing about the toric boundary. One of the most elementary examples is $X_A = \mathbb{P}^1 \times \mathbb{P}^1$ with polarization $\mathcal{O} (1,1)$, and the circuit is the four vertices of a unit square with the two diagonal triangulations. Here the hypersurface is $\mathbb{P}^1$ with four boundary points and the relation obtained above yields a classical relation in the mapping class group called the \emph{Lantern relation}.
When $Z_A$ consists of two points, one is an orbifold point and the other is a point with trivial stabilizer. If $\delta_1, \delta_2$ are based paths encircling $Z_A$ and $T_1, T_2$ are the associated symplectomorphisms, we obtain a relation
\begin{equation} \label{eq:circ2} (T_1 T_2 )^a = T_{\partial H_p}. \end{equation}
A basic example of this relation arises as the homological mirror to $\mathbb{P}^2$ which is the set $A = \{(0, 0), (1, 0), (0, 1), (-1, -1)\}$. The constant $a$ occurring above is $3$ and the relation is in fact another classical mapping class group relation known as the \emph{star relation}.
We call the boundary framed, symplectic mapping class group relation occurring in equations \ref{eq:circ1} and \ref{eq:circ2} \emph{the circuit relation}. In general, any complex line in $\mathcal{X}_{\textrm{Sec}(A)}$ yields a relation in $\textrm{Symp}^{\partial, \textrm{fr}} (H_p)$ by homotoping the product of all the loops around the intersections with $Z_A$ to the identity. However, each such line can be degenerated to a chain of equivariant lines which are precisely circuits supported on $A$. Thus every relation obtained this way can be thought of as arising from a composition of circuit relations. As we saw in the previous two sections Landau--Ginzburg mirrors of Fano manifolds are fibrations of Calabi--Yau hypersurfaces. Therefore the above simple examples generalize to
\begin{thm}[\cite{DKK}] Landau--Ginzburg mirrors of Fano manifolds can be obtained by a superposition of circuits described above. \end{thm}
Interpreting Landau--Ginzburg models as lines in the secondary stack we get
\begin{thm}[\cite{DKK}] $\mathcal{X}_{\textrm{Sec}(A)}$ can be seen as moduli space of Landau--Ginzburg models. In particular some wall crossings correspond to passing through $Z_A$.
\end{thm}
These two theorems complete Step 1.
\section{Wall crossings and Stability Hodge Structures}
\label{section:SHS}
We move to Step 2. building a ``twistor family'' with generic fiber the moduli space of stability conditions for Fukaya--Seidel categories --- see \cite{AKO}.
{\bf Stability Hodge Structures. } We start with Stability Hodge Structures, an artifact of Donaldson--Thomas (DT) invariants. We will mainly consider Fukaya--Seidel categories but discussion in this section applies in general.
The theory of Donaldson--Thomas invariants and wall crossing has become a central subject of Geometry and Physics. In a nutshell DT invariants are virtual numbers of stable objects in three dimensional Calabi--Yau category.
Kontsevich and Soibelman suggested Donaldson--Thomas invariants applicable to triangulated category and Bridgeland stability conditions --- a refined version of so called \emph{motivic Donaldson--Thomas invariants} --- MDT. The wall crossing formulae (WCF) of MDT are expressed in terms of factorization of quantum torus. A connection with nonabelian Hodge structures comes naturally here. WCF for the Hitchin system is connected to ODE with small parameter and its asymptotic behavior. In fact the WCF relates to Stokes data at infinity for this ODE and connects with the work of Ecalle and Voros on resurgence.
We will introduce a new geometric structure which seems to be present in many of above considerations --- Stability Hodge Structures. These structures seem to have a huge potential of geometric applications some of which we discuss.
The moduli space of stability conditions of a category $C$ is very complicated with possibly fractional boundary. In the case of derived category of Calabi--Yau manifolds of dimension three and higher there is not any hypothetical description. Still HMS predicts that the moduli space of mirror dual Calabi--Yau manifold is embedded in locally closed cone in moduli space of stability conditions of a category $C$. So it is a big open question how to characterize Hodge structures corresponding to mirror duals. Classically the moduli space of pure Hodge structures has a compactification by Mixed Hodge Structures (MHS). So it is natural to study limiting Donaldson--Thomas invariants and relate to WCF.
In the case of three-dimensional Calabi--Yau manifolds there are different types of MHS.
The cusp case --- the deepest degeneration --- corresponds to a $t$-structures which is an extension of Tate motives. As a result we take a generating series of Donaldson--Thomas rank one torsion free invariants. It is expected that in this case this generating series (modulo change of coordinates) is the classical Gromov--Witten series which satisfies holomorphic anomaly equation. This translates into automorphic property for DT generating function. We expect that automorphic property holds for higher ranks and plan to study it and show that WCF is necessary to assemble limiting data.
A different MHS corresponds to conifold points and non maximal degeneration points. The wall crossings and DT data give a family of Integrable Systems in the following way. The vanishing cycles $\Gamma_{short}$ and the monodromy define a quotient category $\mathcal{T}/{\mathcal{A}}$ with the following sequence on level of $K$--theory: $$\Gamma_{short}\to K_0({\mathcal{T}})\to K_0({\mathcal{T}}/{\mathcal{A}}).$$
Using the Kontsevich--Soibelman noncommutative torus approach we define a superscheme $$\mathbb G=\oplus_{p\in \Gamma_{short}}\mathbb G_p\to T_{non}.$$ Consider the zero grade $\mathbb G_0$ of
$\mathbb G$ over $\Z$. The global sections of $\mathbb G_0$ define Betti moduli space
--- an integrable system
$$\Gamma (G_0)=\oplus \mathcal O(M_j).$$
In order to consider the interaction with the rest of the category we include global WCF. In this case we obtain a torus action, which produces a stack over Betti moduli space: $$X/(\mathbb{C}^{*})^{\times n}\to M_1\times M_2\times \ldots \times M_k.$$
All these stacks fit in a constructible sheaf.
To summarize we give a provisional definition, which covers the cases of Bridgeland, geometric (volume forms), and generalized (log forms) stability conditions:
\begin{defn}
\emph{Stability Hodge Structure (SHS)} for Fukaya--Seidel category $\mathcal F$ is the following data:
\begin{itemize}
\item[i)] The moduli space of stability conditions $S$ for $\mathcal F$.
\item[ii)] Divisor $D$ at infinity giving a partial compactification of $S$ and parametrizing the degenerated limiting stability conditions --- stability conditions for quotient categories, the category factored by the objects (vanishing cycles) on which stability conditions vanish.
\item[iii)] Besides the degeneration we record the WCF --- all recorded together. Over each point of $D$ we put Betti moduli space locally produced by WCF. All these moduli space fit in a constructible sheaf over $S$. \end{itemize}
\end{defn}
Let us illustrate these structures through two examples. We start with the category $\widetilde{\mathbb A}_2$ --- the Fukaya category of the conic bundle $\{uv=y^2-x^3-a x-b\}$, $a,b\in \mathbb{C}$. In this case, the Stability Hodge Structure is a sheaf over $\mathbb{C}^2$ with coordinates $a,b$.
\begin{figure}
\caption{Compactification of moduli space of stability conditions for a category $\widetilde{\mathbb A}_2$.}
\label{figure:compactification A2}
\end{figure}
The points of the discriminant parameterize limiting stability conditions. The fibers are Betti moduli spaces of vanishing cycles which generically over the discriminant are the affine surface $z(1-xy)=1$. The special fiber over the cusp is the moduli space $M_{0,5}$ of rank two bundles over projective line with one irregular singularity and five Stokes directions at infinity (see Figure~\ref{figure:compactification A2}).
A different example is the Fukaya--Seidel category $\mathbb A_4$.
We start with a generic polynomial $p\in \mathbb{C}[z]$ of degree 5. It defines a Riemann surface $C=\{p(z)=w\}$ and 5:1-covering $\varphi \colon C\to \mathbb{C}$. The ramification locus for $\varphi$ are 4 points $p_1,\ldots,p_4$--- roots of $p'$. Consider 4 paths $l_1,\ldots, l_4$ from $p_i$'s to infinity. The polynomial $p$ is generic, so the ramification is as simple as it can be and $\varphi^{-1}(l_i)$ are thimbles covering $l_i$'s 2:1. They generate a Fukaya category for $C$ and correspond to vertices of $\mathbb A_4$ quiver. ``Neighbor'' thimbles intersect at infinity: $i$-th one intersects $(i+1)$-th at one point. These intersections correspond to arrows between vertices in the quiver.
In this example the divisor $D$ at infinity parameterizes the semiorthogonal decompositions of the $\mathbb A_4$ category. The fibers of the constructible sheaf are moduli spaces of stability conditions for $\mathbb A_3 \times \mathbb A_1 $ categories. Similarly on the singular points of $D $ we get as fibers moduli spaces of stability conditions for $\mathbb A_2 \times \mathbb A_2$ categories. This leads to a rich mixed Hodge theory structure associated with $D$ and monodromy action around it. In the next section we will see that in limit stability conditions behave as coverings so the above picture fits. This monodromy relates to the wall-crossings changes. In particular it sends the preferred set of thimbles generating the $\mathbb A_4$ category from a generator consisting of the sum of $4$ thimbles $G = L_1+L_2+L_3+L_4$ (with $Hom(L_i,L_{i+1})$ of rank 1) to $G' = L' + L_1 + L_3+ L_4$ by a mutation. This mutation reduces the generation time (see Section~\ref{section:spectra}) from $t(G)=3$ to $t(G')=2$.
We will represent is as an invariant of of Stability Hodge Structures in Section~\ref{section:spectra}.
In the next section we build a twistor type of family where the generic fiber is a SHS.
\section{Higgs bundles and stability conditions --- analogy}
\label{section:Higgs bundles}
In this section we proceed describing the analogy between Nonabelian and Stability Hodge Structures. We build the ``twistor'' family so that the fiber over zero is the ``moduli space'' of Landau--Ginzburg models and the generic fiber is the Stability Hodge Structure defined above.
Noncommutative Hodge theory endows the cohomology groups of a dg-category with additional linear data --- the noncommutative Hodge structure --- which records important information about the geometry of the category. However, due to their linear nature, noncommutative Hodge structures are not sophisticated enough to codify the full geometric information hidden in a dg-category. In view of the homological complexity of such categories it is clear that only a subtler non-linear Hodge theoretic entity can adequately capture the salient features of such categorical or noncommutative geometries. In this section by analogy with ``classical nonabelian Hodge theory'' we construct and study from such an prospective a new type of entity of exactly such type --- the Stability Hodge Structure associated with a dg category.
As the name suggests, the SHS of a category is related to the Bridgeland stabilities on this category. The moduli space ${\sf
Stab}_{C}$ of stability conditions of a triangulated dg-category $C$ is, in general, a complicated curved space, possibly with fractal boundary. In the special case when $C$ is the Fukaya category of a Calabi--Yau threefold, the space ${\sf Stab}_{C}$ admits a natural one-parameter specialization to a much simpler space ${\sf
S}_{0}$. Indeed, HMS predicts that the moduli space of complex structures on the Calabi--Yau threefold maps to a Lagrangian subvariety ${\sf Stab}^{\text{geom}}_{C} \subset {\sf Stab}_{C}$. (Recall the holomorphic volume form and integrating it defines a stability condition and its charges,) The idea is now to linearize ${\sf Stab}_{C}$ along ${\sf
Stab}^{\text{geom}}_{C}$, i.e. to replace ${\sf Stab}_{C}$ with a certain discrete quotient ${\sf S}_{0}$ of the total space of the normal bundle of ${\sf Stab}^{\text{geom}}_{C}$ in ${\sf
Stab}_{C}$. Specifically, by scaling the differentials and higher products in $C$, one obtains a one parameter family of categories $C_{\lambda}$ with $\lambda \in \mathbb{C}^{*}$, and an associated family ${\sf S}_{\lambda} := {\sf Stab}_{C_{\lambda}}$, $\lambda \in \mathbb{C}^{*}$ of moduli of stabilities. Using holomorphic sections with prescribed asymptotic at zero one can complete the family $\{{\sf S}_{\lambda}\}_{\lambda \in
\mathbb{C}^{*}}$ to a family ${\sf S} \to \mathbb{C}$ which in a neighborhood of ${\sf Stab}^{\text{geom}}_{C}$ behaves like a standard deformation to the normal cone. The space ${\sf S}_{0}$ is the fiber at $0$ of this completed family and conjecturally ${\sf S} \to \mathbb{C}$ is one chart of a twistor-like family $\mathcal{S} \to \mathbb{P}^{1}$ which is by definition \emph{ the Stability Hodge
Structure associated with $C$}.
Stability Hodge Structures are expected to exist for more general dg-categories, in particular for Fukaya--Seidel categories associated with a superpotential on a Calabi--Yau space or with categories of representations of quivers. Moreover, for special non-compact Calabi--Yau 3-folds, the zero fiber ${\sf S}_{0}$ of a Stability Hodge Structure can be identified with the Dolbeault realization of a nonabelian Hodge structure of an algebraic curve. This is an unexpected and direct connection with Simpson's nonabelian Hodge theory which we exploit further suggesting some geometric applications.
We briefly recall nonabelian Hodge theory settings. According to Simpson we have one parametric twistor family such that the fiber over zero is the moduli space of Higgs bundles and the generic fiber is the moduli space of representation of the fundamental group --- $M_{Betti}$.
In this section we state that we expect similar behavior of moduli space of Stability conditions. In other moduli space of stability conditions of Fukaya--Seidel category can be included in one parametric twistor family, and we describe fiber over zero in details in the next subsection.
We give an example:
\begin{exm}[twistor family for Stability Hodge Structures for the category $\mathbb A_n$]
We will give a brief explanation the calculation of the twistor family for the SHS for the category $\mathbb A_n$. We start with the moduli space of stability conditions for the category $\mathbb A_n$, which can be identified with differentials $ e^{p}dz, $ where $p\in \mathbb{C}[z]$ is a generic polynomial of degree $n+1$, see~\cite{KKPS}.
Let us denote one holomorphic form $ e^{p}dz $ by $Vol$. Locally there exist a holomorphic coordinate $w$ such that $Vol=dw.$ Geodesics in the metric $|Vol|^2$ are the straight real lines in coordinate $w$, the same as real lines on which $Vol$ has constant phase. Therefore they are special Lagrangians for $Vol$ (and in fact for any real symplectic structure).
Observe that this geodesics are asymptotic to infinity because the integral of
$|Vol|=e^{Re(p)}|dz|$ absolutely converges on them
hence $Re(p)$ approaches infinity, as these lines are noncompact in the uncompactified plane $z$, and therefore $|z|$ goes to
infinity. To compensate infinite length in the usual metric $|dz|^2$ we use the fact that $e^{Re(p)}$ converges to zero iff $Re(p)$ converges to minus infinity.
\end{exm}
So after completion in the metric defined above (so the vertices are in the finite part now) we enhance the polygon by assigning angles and lengths. These enhanced polygons record our stability conditions. Indeed we have $(2(n+1)-3)$-dimensional space of polygons plus one global angle --- it is a real $2n$ dimensional space. In Example~\ref{example:A2} we give a simple example the polygons for the category $\mathbb A_2$ and a wall crossing phenomenon. The stable objects correspond to edges and diagonals. In the picture in the example we lose one stable object while crossing a wall.
\begin{exm}[stability for $\mathbb A_2$] \label{example:A2} For $\mathbb A_2$ category we have $\deg p=3$. The left part of Figure~\ref{figure:A2} represents two of the stable objects for the $\mathbb A_2$ category. The third stable object is the third edge of the triangle. The wall crossing makes the angle between the first two edges bigger then $\pi$ and as a result the third edge is not a stable object any more.
\begin{figure}
\caption{Stability conditions for the $\mathbb A_2$ category.}
\label{figure:A2}
\end{figure}
\end{exm}
Now we consider the ``twistor family'' --- the limit $ e^{p(z)/u}dz, $ where $u$ is a complex number tending to 0. Geometrically limit differential can be identified with graphs --- see Example~\ref{example:limit}.
\begin{exm}[limit $e^{p/u}dz$] \label{example:limit} Take a limit of $e^{p/u}dz$ with $u$ tending to zero. The limits of polygons are graphs. We record the length, angle, and monodromy and this defines a covering of the complex plane. Thus this construction identifies a limit of moduli space of stability conditions for $\mathbb A_n$ category with some Hurwitz subspace --- a subscheme of coverings. In particular, these two spaces have the same number of components. Figure~\ref{figure:limit} represents a procedure of associating the monodromy of the covering to the vertices of the graph.
\begin{figure}
\caption{Building coverings out of limit.}
\label{figure:limit}
\end{figure}
\end{exm}
\begin{rem} Similarly one can compute the twistor family for the equivariant $\mathbb A_n$ category and see appearance of gaps in spectra in connection with the weight filtration of completions of special local rings --- see Section~\ref{section:spectra}. Observe that idea of coverings brings the Fukaya category of a Riemann surface of genus $g$ very close to the $\mathbb A_{2g+1}$ category. Also product of Fukaya categories of curves in combined with Luttinger surgeries gives many opportunity for stability conditions with many components as well as many possibilities for the behavior of gaps and spectra. The interplay between coverings and stability conditions suggests that one can have symplectic manifolds with the same Fukaya categories but different moduli of stability conditions. We conjecture that the moduli spaces of coverings obtained near different cusps being different algebraically should imply that this different manifolds are nonsymplectomorphic.
\end{rem}
{\bf The fiber over zero. } \label{subsection:fiber over 0} The fiber over zero (described in what follows) plays an analogous role to the moduli space of Higgs bundles in Simpson's twistor family in the theory of nonabelian Hodge structures. Constructing it amounts to a repetition of our construction in Section~\ref{section:wall crossings} from a new perspective and enhanced with more structure.
The $\mathbb A_n$ example considered above is a simple example of more general Fukaya -Seidel categories that arise in Homological Mirror Symmetry. Stability conditions associated to the Fukaya - Seidel category are closely related to the complex deformation parameters, i.e. the moduli space of Landau Ginzburg models. We begin by recalling the general setup in the case of Landau-Ginzburg models.
The prescription given by Batyrev, Borisov, Hori, Vafa in \cite{BB}, \cite{hori-vafa} to obtain homological mirrors for toric Fano varieties is perfectly explicit and provides a reasonably large set of examples to examine. We recall that if $\Sigma$ is a fan in $\R^n$ for a toric Fano variety $X_\Sigma$, then the homological mirror to the B model of $X_\Sigma$ is a Landau--Ginzburg model $w : (\C^*)^n \to \C$ where the Newton polytope $Q$ of $w$ is the convex hull of generators of rays of $\Sigma$. In fact, we may consider the domain $(\C^*)^n$ to occur as the dense orbit of a toric variety $X_A$, where $A$ is $Q \cap \Z^n$ and $X_A$ indicates the polytope toric construction. In this setting, the function $w$ occurs as a pencil $V_w \subset H^0 (X_A, L_A)$ with fiber at infinity equal to the toric boundary of $X_A$. Similar construction works for generic non-toric Fanos. In this paper we work with directed Fukaya category associated to the superpotential $w$ --- Fukaya--Seidel categories. To build on the discussion above, we discuss here Fukaya--Seidel categories in the context of stability conditions. The fiber over zero corresponds to the moduli of complex structures. If $X_A$ is toric, the space of complex structures on it is trivial, so the complex moduli appearing here are a result of the choice of fiber $H \subset X_A$ and the choice of pencil $w$ respectively. The appropriate stack parameterizing the choice of fiber contains the quotient $[U/(\mathbb{C}^*)^n]$ as an open dense subset where $U$ is the open subset of $H^0 (X_A, L_A)$ consisting of those sections whose hypersurfaces are nondegenerate (i.e. smooth and transversely intersecting the toric boundary) and $(\C^*)^n$ is acts by its action on $X_A$. To produce a reasonably well-behaved compactification of this stack, we borrow from the works of Alexeev (\cite{Al02}), Gelfand, Kapranov, and Zelevinsky (\cite{GKZ}), and Lafforgue (\cite{Lafforgue}) to construct the stack $\mathcal{X}_{\textrm{Sec} (A)}$ with universal hypersurface stack $\mathcal{X}_{Laf (A)}$. We quote the following theorem which describes much of the qualitative behavior of these stacks:
\begin{thm}[\cite{DKK}]
\begin{itemize}
\item[i)] The stack $\mathcal{X}_{\textrm{Sec} (A)}$ is a toric stack with moment polytope equal to the secondary polytope $Sec (A)$ of $A$.
\item[ii)] The stack $\mathcal{X}_{Laf (A)}$ is a toric stack with moment polytope equal to the Minkowski sum $Sec (A) + \Delta_A$ where $\Delta_A$ is the standard simplex in $\R^A$.
\item[iii)] Given any toric degeneration $F: Y \to \C$ of the pair $(X_A, H)$, there exists a unique map $f : \C \to \mathcal{X}_{\textrm{Sec} (A)}$ such that $F$ is the pullback of $\mathcal{X}_{Laf (A)}$. \end{itemize}
\end{thm}
We note that in the theorem above, the stacks $\mathcal{X}_{Laf (A)}$ and $\mathcal{X}_{\textrm{Sec} (A)}$ carry additional equivariant line bundles that have not been examined extensively in existing literature, but are of great geometric significance. The stack $\mathcal{X}_{\textrm{Sec} (A)}$ is a moduli stack for toric degenerations of toric hypersurfaces $H \subset X_A$. There is a hypersurface $\mathcal{E}_A \subset \mathcal{X}_{\textrm{Sec} (A)}$ which parameterizes all degenerate hypersurfaces. For the Fukaya category of hypersurfaces in $X_A$, the complement $\mathcal{X}_{\textrm{Sec} (A)}\setminus \mathcal{E}_A$ plays the role of the classical stability conditions, while including $\mathcal{E}_A$ incorporates the compactified version where MHS come into effect. We predict that the walls of the stability conditions occurring in this setup are seen as components of the tropical amoeba defined by the principal $A$-determinant $E_A$.
To find the stability conditions associated to the directed Fukaya category of $(X_A, w)$, one needs to identify the complex deformation parameters associated to this model. In fact, these are precisely described as the coefficients of the superpotential, or in our setup, the pencil $V_w \subset H^0 (X_A , w)$. Noticing that the toric boundary is also a toric degeneration of the hypersurface, we have that the pencil $V_w$ is nothing other than a map from $\mathbb{P}^1$ to $\mathcal{X}_{\textrm{Sec} (A)}$ with prescribed point at infinity. If we decorate $\mathbb{P}^1$ with markings at the critical values of $w$ and $\infty$, then we can observe such a map as an element of $\mathcal{M}_{0, Vol (Q) + 1} (\mathcal{X}_{\textrm{Sec} (A)} , [w])$ which evaluates to $\mathcal{E}_A$ at all points except one and $\partial X_A$ at the remaining point. We define the cycle of all stable maps with such an evaluation to be $\mathcal{W}_A$ and regard it as the appropriate compactification of complex structures on Landau--Ginzburg A-models. Applying techniques from fiber polytopes we obtain the following description of $\mathcal{W}_A$:
\begin{thm}[\cite{DKK}] The stack $\mathcal{W}_A$ is a toric stack with moment polytope equal to the monotone path polytope of $Sec(A)$. \end{thm}
The polytope occurring here is not as widely known as the secondary polytope, but occurs in a broad framework of so called iterated fiber polytopes introduced by Billera and Sturmfels.
In addition to the applications of these moduli spaces to stability conditions, we also obtain important information on the directed Fukaya categories and their mirrors from this approach. In particular, the above theorem may be applied to computationally find a finite set of special Landau--Ginzburg models $\{w_1, \ldots, w_s\}$ corresponding to the fixed points of $\mathcal{W}_A$ (or the vertices of the monotone path polytope of $Sec (A)$). Each such point is a stable map to $\mathcal{X}_{\textrm{Sec} (A)}$ whose image in moment space lies on the $1$-skeleton of the secondary polytope. This gives a natural semiorthogonal decomposition of the directed Fukaya category into pieces corresponding to the components in the stable curve which is the domain of $w_i$. After ordering these components, we see that the image of any one of them is a multi-cover of the equivariant cycle corresponding to an edge of $Sec (A)$. These edges are known as circuits in combinatorics
(see \cite{DKK}).
Now we put this moduli space as a ``zero fiber'' of the twistor family of moduli family of stability conditions.
We do this in two steps:
1. The following theorem suggests existence of a formal moduli space $M$ of Landau--Ginzburg models $f\colon \overline{Y} \to \mathbb{C}\mathbb{P}^1$.
\begin{thm}[see \cite{KKPS}] There exists a formal moduli space $M$ determined by the solutions of the Maurer--Cartan equations for dg-complex
$$ \begin{CD} \cdots@<<< \Lambda^3 T_{\overline{Y}} @<<< \Lambda^2 T_{\overline{Y}} @<<< T_{\overline{Y}} @<<< \mathcal{O}_{\overline{Y}} @<<< 0 \\[-3mm] @. -3 @. -2 @. -1 @. 0 \end{CD} $$
\end{thm}
In the above complex the differential is $df$ and we can restate it by saying that this complex determines deformations of the Landau--Ginzburg model, and these deformations are unobstructed. We also have a $\mathbb{C}^*$-action on $M$ with fixed points corresponding to limiting stability conditions --- see \cite{KKPS}.
Over the moduli space $M$ defined above we have a variation of Hodge structures defined by the cohomologies of the perverse sheaf of vanishing cycles over $Y$. This defines local system $V$ over $M$ and its compactification.
\begin{conj}[see \cite{KKPS}] The relative completion with respect of $V$ in the fixed points of the $\mathbb{C}^*$-action on the compactification of $M$
has a mixed Hodge structure. \end{conj}
2. The above moduli space is too big. So we will cut its dimension down to the moduli space of stability conditions. We introduce a new moduli space which embeds in $M$.
We study deformations of $\overline{Y} \to \mathbb{C}\mathbb{P}^1$ with ``fixing fiber at infinity''. Deformation of a smooth variety $\overline{Y}$ with fixed $\mathbb{C}\mathbb{P}^1$ is controlled by the following sheaf of dg Lie algebras on $\overline{Y}$: $$T_{\overline{Y}} \to f^* T_{\mathbb{C}\mathbb{P}^1}$$ (the differential is the tangent map).
By fixing the fiber at infinity we get a subsheaf of dg Lie algebras $$T_{\overline{Y},Y_\infty} \to f^*T_{\mathbb{C}\mathbb{P}^1, \infty}.$$
\begin{thm}[\cite{KKPS}] A subsheaf of dg Lie algebras $$T_{\overline{Y},Y_\infty} \to f^*T_{\mathbb{C}\mathbb{P}^1, \infty}.$$ determines smooth moduli stack.
Its dimension is equal to the dimension of the moduli space of stability conditions. \end{thm}
A geometric realization of this moduli space, which embeds in $M$ was described above. We will denote it by $M(\mathbb{P}^1, CY)$ (or $M(\mathbb{P}^k, CY)$ for multipotential Landau--Ginzburg models).
\begin{rem} We can consider bigger moduli space by fixing the vector fields only over a part of the divisor at infinity. This corresponds to taking a Landau--Ginzburg model through a point of non maximal degeneration. This defines a bigger moduli space of stability conditions with more stable objects. \end{rem}
\begin{rem} The moduli spaces we discuss could have many components. Such a phenomenon would have many interesting implications. It produces possibilities of many new birational and symplectic invariants. \end{rem}
In the same way as the fixed point set under the $\mathbb{C}^*$-action plays an important role in describing the rational homotopy types of smooth projective varieties we study the fixed points of the $\mathbb{C}^*$-action on $F$ and derive information about the homotopy type of a category. In the rest of the paper we will denote $\mathcal{W}_A$ by $M(\mathbb{P}^1, \mathcal{X}_{\textrm{Sec} (A)} )$ (or $M(\mathbb{P}^k, CY)$) in order to stress the connection with Landau--Ginzburg models (here $CY$ denotes the moduli space of Calabi--Yau mirrors to the anticanonical section of the Fano manifold we consider).
After the journal version of this paper was published, the following related works were pointed out to us by P.\,Boalch:~\cite{BB04},~\cite{B07},~\cite{B12}.
\section{Spectra and holomorphic convexity}
\label{section:spectra}
In this section we explain briefly how Orlov spectra are related to Stability Hodge Structures.
Recall that noncommutative Hodge structures were introduced by Kontsevich and Katzarkov and Pantev \cite{KKP1} as means of bringing the techniques and tools of Hodge theory into the categorical and noncommutative realm. In the classical setting, much of the information about an isolated singularity is recorded by means of the Hodge spectrum, a set of rational eigenvalues of the monodromy operator. The Orlov spectrum (defined below), is a categorical analogue of this Hodge spectrum appearing in the work of Orlov and Rouquier. The missing numbers in the spectra are called gaps.
Let $\mathcal T$ be a triangulated category. For any $G \in \mathcal T$ denote by $\langle G \rangle_0$ the smallest full subcategory containing $G$ which is closed under isomorphisms, shifting, and taking finite direct sums and summands. Now inductively define $\langle G \rangle_n$ as the full subcategory of objects, $B$, such that there is a distinguished triangle, $X \to B \to Y \to X[1]$, with $X \in \langle G \rangle_{n-1}$ and $Y \in \langle G \rangle_0$, and direct summands of such objects.
\begin{defn} Let $G$ be an object of a triangulated category $\mathcal{T}$. If there is an $n$ with $\langle G \rangle_{n} = \mathcal T$, we set, \begin{displaymath}
t(G):= \text{min } \lbrace n \geq 0 \ | \ \langle G
\rangle_{n} = \mathcal T \rbrace. \end{displaymath} Otherwise, we set $t(G) := \infty$. We call $t(G)$ the \emph{generation time} of $G$. If $t(G)$ is finite, we say that $G$ is a \emph{strong generator}. The \emph{Orlov spectrum} of $\mathcal T$ is the union of all possible generation times for strong generators of $\mathcal T$. The \emph{Rouquier dimension} is the smallest number in the Orlov spectrum. We say that a triangulated category, $\mathcal T$ has a \emph{gap} of length $s$, if $a$ and $a+s+1$ are in the Orlov spectrum but $r$ is not in the Orlov spectrum for $a < r < a+s+1$. \end{defn}
The first connection to Hodge theory appears in the form of the following theorem: \begin{thm}[\cite{BFK}] Let $X$ be an algebraic variety possessing an isolated hypersurface singularity. The Orlov spectrum of the category of singularities of $X$ is bounded by twice the embedding dimension times the Tjurina number of the singularity. \label{thm:isohypspecbound} \end{thm}
After this brief review of theory of spectra and their gaps we connect them with SHS. Let $SHS(X)$ be the Stability Hodge Structure of $D^b(X)$ for given Fano variety $X$, $M(\mathbb{P}^1, CY)$ is its zero fiber.
\begin{conj}[\cite{KKPS}]
Let $p$ be a point of the divisor $D$ at infinity of the compactification of $M(\mathbb{P}^1, CY)$. The mixed Hodge structures on the completion of the local ring $O_p$,
where $p$ runs over all components of $D$, determines the spectrum of $D^{b}(X)$. \end{conj}
\begin{rem} The above considerations suggests the existence of a Riemann--Hilbert correspondence for $SHS(X)$ for Fano variety $X$ as well as deep and interesting analytical interpretation of it by analogy with Yang--Mills--Higgs equations. \end{rem}
As a consequence of the above conjecture we have that SHS satisfy two important properties --- functoriality and strictness. We arrive at:
\begin{conj} The infinite chain condition (\cite{EKPR}) can be ruled out for the universal coverings of smooth projective surfaces.
\end{conj}
This is the strongest obstruction to Shafarevich conjecture \cite{EKPR} and SHS gives an approach proving that universal coverings of smooth projective varieties are holomorphically convex.
Observe that the twistor family of compactified SHS depends on the choice of Landau--Ginzburg model and still computes some purely categorical invariants. It is natural to ask whether this family rigidifies the data. In particular we pose:
\begin{question} Does the twistor family of compactified SHS of bounded derived category of coherent sheaves of a smooth projective variety $X$ recover the fundamental group of $X$? \end{question}
\section{Multipotential Landau--Ginzburg models and Hodge structures.}
\label{section:multi LG}
In this section we extend the correspondence among categories and Stability Hodge Structures further. We underscore the idea that
rich geometry of the Landau--Ginzburg models gives a possibility of constructing interesting Stability Hodge Structures with many filtrations.
\subsection{Multipotential Landau--Ginzburg model for cubic fourfold}
We describe fiberwise compactifications of multipotential Landau--Ginzburg models for cubic fourfold $X$.
This example is representative and illustrates what we mean by a multipotential Landau--Ginzburg model in general.
The Hori--Vafa toric Landau--Ginzburg for $X$ is $$ w=\frac{(x+y+1)^3}{xyt_1t_2}+t_1+t_2. $$ The cubic fourfold is of index 3. So there are two decompositions of its anticanonical divisor: $3H=H+H+H$ and $3H=2H+H$. Multipotential Landau--Ginzburg models correspond to such decompositions.
First we describe compactification for the first decomposition. We have the family $$ \frac{(x+y+1)^3}{xyt_1t_2}=w_1,\ \ \ \ \ t_1=w_2,\ \ \ \ \ t_2=w_3, $$ where $w_i$'s are complex parameters. After compactifying we get the family $$ (x+y+z)^3=w_1w_2w_3xyz $$
of elliptic curves over $\mathbb{C}^3$. After blowing up the point $(0,0,0)$ we get divisor over this point. After that we resolve the rest of the singularities. The restriction of our family to planes $w_j=const\neq 0$ is the Landau--Ginzburg model for cubic threefold so we get the following configuration of singularities.
\begin{enumerate}
\item Ordinary double points along surface $w_1w_2w_3=27$.
\item 7 lines forming diagram of type $\widetilde{\mathbb E}_6$ over
planes $w_1=0$, $w_2=0$, and $w_3=0$.
\item 5 surfaces over axes $w_1$, $w_2$, $w_3$.
\item A divisor over (0,0,0). \end{enumerate}
After projection on the diagonal $\mathbb{C}^3\to
\mathbb{C}$ we get a fiberwise open part of usual Landau--Ginzburg model
for cubic fourfold. Its fiber over zero consists of divisor
described above and an elliptic fibration over the plane passing through the
origin and orthogonal to the diagonal. The intersection of these
divisors is an elliptic K3 surface with 3 fibers of type
$\widetilde{\mathbb E}_6$ corresponding to intersections of this orthogonal plane with
planes $w_1=0$, $w_2=0$, and $w_3=0$.
Now we describe multipotential Landau--Ginzburg model for the second case $2H+H$. We have the family $$ \frac{(x+y+z)^3}{xyzt_1t_2}=w_1,\ \ \ \ \ \ \ t_1+t_2=w_2. $$
In other words,
$$ (x+y+z)^3=w_1(w_2-t)txyz $$ (we denote $t_1$ by $t$ for simplicity).
This family of surfaces can be obtained from decomposition $H+H+H$ by a projection along $w_2+w_3=0$. Indeed, the equation of this family over $\mathbb{C}^2$ can be obtained from the equation for the family over $\mathbb{C}^3$ by coordinate change $w_2+w_3\to w_2$, $w_3\to t$.
So the singularities are the following.
\begin{enumerate}
\item Ordinary double points along a curve.
\item 5 surfaces over axis $w_2=0$.
\item 17 surfaces over axis $w_1=0$. Their configuration can be
described as follows: configuration of curves of type
$\widetilde{\mathbb E}_6$ multiplied by line and two examples of
configuration of 5 surfaces described above. Each of them are glued by
intersection of ``pages'' with line of multiplicity 3 on $\widetilde{\mathbb E}_6\times pt$.
\item A divisor over (0,0,0). \end{enumerate}
The restriction of this family to the line $w_1=const\neq 0$ is (up to a multiplication of a potential by a constant) an open part of Landau--Ginzburg model for cubic threefold. Indeed, \begin{multline*} (x+y+z)^3-w_1(w_2-t)txyz=(x+y+z)^3-(\sqrt{w_1}w_2-(\sqrt{w_1}t))(\sqrt{w_1}t)xyz=\\ (x+y+z)^3-(w-t_1)t_1xyz, \end{multline*} where $w=\sqrt{w_1}w_2$ and $t_1=\sqrt{w_1}t$.
The restriction to the line $w_2=const\neq 0$ is an open part of Landau--Ginzburg model for threefold complete intersection of quadric and cubic. Indeed, \begin{multline*} (x+y+z)^3-w_1(w_2-t)txyz=(x+y+z)^3-(w_1w_2^2)\left(1-\frac{t}{w_2}\right)\left(\frac{t}{w_2}\right)xyz=\\ (x+y+z)^3-w(1-t_1)t_1xyz, \end{multline*} where $w=w_1w_2^2$ and $t_1=t/w_2$.
On the other hand, compactified (singular) Landau--Ginzburg model for intersection of quadric and cubic is $$ (t_1+t_2)^2(x+y+z)^3-wt_1t_2xyz=t_0^2(x+y+z)^3-w(t_0-t_1)t_1xyz, $$ where $t_0=t_1+t_2$. In the local chart $t_0=1$ we get the family written down before.
Thus, after compactifying fibers of family corresponding to $2H+H$ we get 4 additional surfaces over $w_2$ axis and all together $21=17+4$ surfaces.
\subsection{Hodge structures with many filtrations}
We now utilize above construction of multipotential Landau--Ginzburg models from the point of view of twistor families. This part of the paper is highly speculative.
It is expected that Fukaya--Seidel categories with many potentials can be defined similarly to Fukaya--Seidel categories with one potential. In this case we have a divisor $S$ of singular fibers and thimbles involved reflect no only geometry of the fibers but geometry of $S$ as well. In similar way we can associate to a Fukaya--Seidel category with many potentials a Stability Hodge Structure with a formal scheme over
$M(\mathbb{P}^k, CY)$ as a fiber over zero. The following conjecture (briefly explained in Table~\ref{tab:SHSHS3}) suggests a way of constructing Hodge structures with multiple filtrations.
\begin{conj}[see \cite{KKPS}] The mixed Hodge structure over formal scheme over
$M(\mathbb{P}^k, CY)$ as fiber over zero is a mixed Hodge structure with many filtrations.
\end{conj}
\begin{table}[h] \begin{center}
\begin{tabular}{|c|c|} \hline \begin{minipage}[c]{2.5in} \centering
Landau--Ginzburg moduli spaces
\end{minipage} & \begin{minipage}[c]{2.5in} \centering
Nonabelian Hodge structures
\end{minipage} \\\hline\hline \begin{minipage}[c]{2.5in} \centering
$M(\mathbb{P}^1, CY)$
Landau--Ginzburg model with one potential:
The fiber over zero is a formal scheme over $M(\mathbb{P}^1, CY)$, generic fibers are ${\sf Stab}$.
\end{minipage} & \begin{minipage}[c]{2.5in} \centering
Twistor family --- Nonabelian Hodge Structure with one weight filtration.
\end{minipage} \\\hline\hline \begin{minipage}[c]{2.5in} \centering
Landau--Ginzburg models with $k$ potentials
\end{minipage} & \begin{minipage}[c]{2.5in} \centering
Generalized twistor families with $k$ parameters.
\end{minipage} \\\hline\hline \begin{minipage}[c]{2.5in} \centering
The zero fiber is a formal scheme over $M(\mathbb{P}^k, CY)$, fibers (over a point in $\mathbb{C}^k$) are ${\sf Stab}$.
\end{minipage} & \begin{minipage}[c]{2.5in} \centering
Generalized multi twistor family over a $k$-simplex.
\end{minipage} \\\hline \begin{minipage}[c]{2.5in} \centering
Extensions
$M(\mathbb{P}^1, CY)\boxtimes M(\mathbb{P}^1, CY)$
\end{minipage} & \begin{minipage}[c]{2.5in} \centering
Extending filtrations $u_i\boxtimes u_j$.
\end{minipage} \\\hline \end{tabular} \end{center} \caption{Creating Hodge structures with multiple filtrations.} \label{tab:SHSHS3} \end{table}
\section{Birational transformations and Poisson varieties}
\label{section:birational}
Discussion from previous sections suggests that there is a connection between moduli space of Landau--Ginzburg models, generators and birational geometry.
\begin{table}[h] \begin{center}
\begin{tabular}{|c|c|} \hline \begin{minipage}[c]{2.5in} \centering
Landau--Ginzburg model
\end{minipage} & \begin{minipage}[c]{2.5in} \centering
Stability
\end{minipage} \\\hline\hline \begin{minipage}[c]{2.5in} \centering
Usual Landau--Ginzburg model
$\includegraphics[width=0.8\nanowidth]{MIAMI1.eps}$
\end{minipage} & \begin{minipage}[c]{2.5in} \centering
$\Omega_X^3$
\end{minipage} \\\hline \begin{minipage}[c]{2.5in} \centering
$\includegraphics[width=0.8\nanowidth]{MIAMI2.eps}$
\end{minipage} & \begin{minipage}[c]{2.5in} \centering
$\Omega^3_{X\setminus D}$
$D$ here is a divisor with stratification of singular set.
\end{minipage} \\\hline \begin{minipage}[c]{2.5in} \centering
$\includegraphics[width=0.8\nanowidth]{MIAMI3.eps}$
\end{minipage} & \begin{minipage}[c]{2.5in} \centering
$\Omega_D$
${\mathrm Sing}\,D$ stability conditions of the vanishing cycles on $D$.
\end{minipage} \\\hline \end{tabular} \end{center} \caption{Stability Clemens--Schmidt sequence.} \label{tab:MIAMI1} \end{table}
Table~\ref{tab:MIAMI1} gives a version of noncommutative Clemens--Schmidt sequence for geometric stability conditions --- log 3-forms. This table treats the case of three-dimensional Calabi--Yau manifolds (four dimensional Landau--Ginzburg models) but the situation in general should be rather similar. In the case at hand (three-dimensional Calabi--Yau manifold) --- the stability conditions are just holomorphic 3-forms. For the quotient category (the category which produces stability conditions of the compactification) we get stability conditions to be holomorphic 3-forms vanishing in a stratified way over a divisor $D$. The vanishing cycles define a subcategory with its own moduli space of stability conditions and the relative (with respect to this subcategory) WCF defining an integrable system (in general a Poisson variety). The corresponding Landau--Ginzburg models can be seen as follows:
\begin{enumerate}
\item The Landau--Ginzburg models associated with quotient categories are given by monotonic maps passing through an intersection of many boundary divisors in $M(\mathbb{P}^k, CY)$.
\item The local categories of vanishing cycles are given by Landau--Ginzburg models totally within intersections of divisors. \end{enumerate}
From the perspective of generators the above splitting corresponds to splitting of generator to union of generators associated with subcategory of vanishing cycles and the quotient category. In fact we get a sequence of splittings --- a flag parallel to Okounkov polytopes.
These observations suggest the following conjecture, treated in \cite{DKK}.
\begin{conj} One-parametric families of Landau--Ginzburg models parameterize Sarkisov links. \end{conj}
Recall that Sarkisov links \cite{SARK} are birational maps (birational cobordisms) connecting two Mori fibrations. I our interpretation Sarkisov links become families connecting circuits. In fact we have more general picture on the connections between moduli spaces of Landau--Ginzburg models and birational geometry. Namely we conjecture that geometry of moduli spaces of Landau--Ginzburg models for the mirror of Fano manifold $X$ determines its birational geometry. In particular we see a connection with relations between Sarkisov links and then relations between relations and so on. We summarize our picture in Table~\ref{tab:Section 9}. For more details see \cite{DKK}, \cite{BFK3}, \cite{CKP}, \cite{DKLP}.
\begin{table}[h] \begin{center}
\begin{tabular}{|c|c|} \hline \begin{minipage}[c]{2.5in} \centering
Sarkisov programs
\end{minipage} & \begin{minipage}[c]{2.5in} \centering
Changes in the spaces of stability conditions
\end{minipage} \\\hline\hline \begin{minipage}[c]{2.5in} \centering
Commutative Sarkisov program: Sarkisov faces.
\end{minipage} & \begin{minipage}[c]{2.5in} \centering
Wall crossings inside a component of stability conditions.
\end{minipage} \\\hline \begin{minipage}[c]{2.5in} \centering
Non-commutative Sarkisov program: non-commutative cobordisms.
\end{minipage} & \begin{minipage}[c]{2.5in} \centering
Passing from one component of stability conditions to another one.
\end{minipage} \\\hline \end{tabular} \end{center} \caption{Birational geometry.} \label{tab:Section 9} \end{table}
{\bf Acknowledgements.} This paper came out of a talk the first author gave in G\"okova, Turkey 2011 (Sections~\ref{section:wall crossings},~\ref{section:spectra},~\ref{section:birational}) and discussions thereafter. We are very grateful to the organizers and in particular S.\,Akbulut, D.\,Auroux and G.\,Mikhalkin for inviting us.
We thank M.\,Kontsevich for sharing his ideas and explaining what SHS should be. Many thanks to D.\,Auroux, G.\,Kerr, C.\,Diemer, D.\,Favero, Y.\,Soibelman, and T.\,Pantev for explaining some of the notions used in the paper. We thank S.\,Galkin for his explanations of cluster transformations of weak Landau--Ginzburg models. We thank N.\,Ilten for his explanation of embedded toric degenerations technique.
\end{document} | arXiv |
\begin{definition}[Definition:Mass/Dimension]
'''Mass''' is one of the fundamental dimensions of physics.
In dimensional analysis it is assigned the symbol $\mathsf M$.
Category:Definitions/Fundamental Dimensions
Category:Definitions/Mass
465466
462526
2020-05-02T13:09:52Z
Prime.mover
59
465466
wikitext
text/x-wiki
\end{definition} | ProofWiki |
Year of study as predictor of loneliness among students of University of Gondar
Baye Dagnew1 &
Henok Dagne2
Loneliness is individual's subjective sense of lacking familial or social contact to the degree that they wanted. It is responsible for reduced quality of life. The aim of this study was to determine loneliness and its association with year of study among University of Gondar students, 2018/19. Cross-sectional study design was used on 404 Medical and Health Sciences students selected by systematic random sampling. UCLA-R loneliness score was used. A person with a mean value of 42 and above was considered as lonely. After data were collected by self-administered questionnaire, Epi-Data was used for data entry and exported to SPSS version 20.1 for analysis. Variables with p-value of 0.05 and lower were treated as significant factors in multivariable logistic regression.
Prevalence of loneliness was 49.5% (95% CI 44.6–54.4%). Year of study was significantly associated with loneliness [AOR = 2.47, 95% CI (1.65–3.70)]. First-year students were having 2.47 odds of loneliness as compared to second year and above students. Loneliness prevalence was higher in the current study. This must get the attention of higher education institutions, the government and all concerned stakeholders in the education sector to design strategy on preventing and treating loneliness.
Ability of a person for close connection with other people is one of the most important settings of a healthy personality. Loneliness is individual's personal, subjective sense of lacking social or familial contact to the extent that they wanted [1]. Loneliness affects many physiological processes mainly due to hypothalamic–pituitary–adrenal axis disturbances leading to poor daily performances by inducing sleep in individuals [2]. It affects a person in his all lifespan more likely to occur under circumstances like prolonged absence from home or loss of a significant-other [3]. In Ankara University students 60.2% of students experienced loneliness which was associated with the need for economic support, social interaction and romantic relationship [4]. A study from Maragheh University students indicated prevalence of moderate and severe loneliness as 50.5% and 31.6%, respectively where sex was predictor [5]. Loneliness varies in different settings. About 10.5% loneliness prevalence was seen in general population of Western Mid-Germany which was higher in females and without partner [6]. Loneliness is predicted by lower family wealth, living in a low or lower middle-income country, low organized religious activity. Lonely students self-reported poor subjective health status, sleeping problems, short sleep duration and tobacco use [7]. Male students had higher levels of Loneliness than females as seen in Anadolu University [8]. The transition from high school to University often causes much stress for most students. In the new University environment, students often face various interpersonal, social, and academic demands, each of which could potentially create stressful situations for most of them which can lead to specific problems in adjustment [9]. Being bound in a romantic relationship is protective in a study done in China, Thai international students and Turkey [10, 11]. Number of semesters was significantly associated with loneliness with lower loneliness in those staying for longer semesters in university [5, 8]. Social isolation is found to be correlated with loneliness that ended up with depression among adults as indicated by a study done in London [12]. Quality of life is affected negatively in persons with loneliness [13] pushing adolescents to use life-threatening substances such as Marijuana [14]. Loneliness is responsible for most depressive symptoms [8] and may lead to cognitive impairment [15]. A study in China revealed, loneliness is not only the cause for the poor quality of life but also increases mortality [16]. Loneliness is significantly associated with divorced status, social support and psychological wellbeing [17]. Individuals with loneliness had reduced out of home physical activities [18]. Mental disturbances and other unhealthy states are common in lonely individuals [19]. Freshman students had problems to adjust themselves into new environment as evidenced in University students at Dila, Ethiopia [20]. Cultural backgrounds affect loneliness which may increase loneliness in some and decreases in others [21]. Increased recognition of loneliness as a risk factor for adverse psychological and physical health outcomes has elevated interest in interventions to reduce chronic loneliness [22]. As many scholars agreed, loneliness is one of the major predisposing factors for disturbed quality of life and poor productivity. As far as our knowledge is concerned such kind of study was not conducted in Ethiopia. Because of this, the investigators had been interested in conducting this research. After completion, this might help to add additional information besides the existing literature to the scientific community. The study population will be benefited from the study findings since the results will help the institutions to design interventional strategies for loneliness and related issues. The major objective of this study was assessing loneliness and determining if loneliness is predicted by year of study among University students, University of Gondar, 2018/19.
Study area and period
The study was conducted at University of Gondar, Ethiopia. The University of Gondar has been dedicated to educating students for more than 64 years and serving the community by delivering clinical services in its comprehensive specialized teaching hospital. The study was conducted from Oct 01 to Nov 30, 2018.
Study design: Institution-based cross-sectional study design was employed.
Source population
All regular students of University of Gondar engaged in learning process in 2018/19. A total of 2358 regular undergraduate students were found in the College of Medicine and Health Sciences.
Study population
Regular undergraduate Medical and Health Science students of University of Gondar who were engaged in the learning process in the academic calendar of 2018/19 those present at the time of data collection period.
Inclusion criteria: All regular undergraduate 1st year to graduating class students of University of Gondar College of Medicine and Health Science were included.
Exclusion criteria: students with a severe illness that limits them to fill questions were excluded.
Sample size determination
The sample size (n) was calculated by using a single population proportion formula by using assumptions; magnitude of loneliness (p) among students in University of Gondar = 50% (since there was no previous study in Ethiopia, we preferred to use maximum sample size), 95% level of confidence and 5% margin of error (d).
$$\begin {aligned}{\varvec{n}}&=\frac{{\left({{\varvec{z}}}_{\frac{{\varvec{a}}}{2}}\right)}^{2}\times {\varvec{p}}\times (1-{\varvec{p}})}{{{\varvec{d}}}^{2}} \\ &=\frac{{\left(1.96\right)}^{2}\times 0.5\times (1-0.5)}{{(0.05)}^{2}}=384 \end {aligned}$$
By adding 5% (expected non-response rate), the minimum calculated sample was 404.
Variables of the study
Dependent variable: Loneliness (Yes/No).
Independent variables: Age, sex, residence before coming to University, marital status, lifestyle (Khat chewing, cigarette smoking), health status, romantic love engagement, year of study and monthly pocket income, current disease status.
Operational definitions
Loneliness: A person with a mean score of 42 and above out of 80 total standard loneliness score was regarded as having loneliness.
Current disease status: If a student is faced with any sort of disease in the past 1 month from data collection period, he/she is referred to having current disease.
Year of study: The education level of students in the University of Gondar during data collection.
Data collection instrument and procedure
The structured pretested self-administered questionnaire was used. Revised University of California Los Angeles Loneliness scale (UCLA-R) was used to collect data about subjective feelings of loneliness [23]. A UCLA-R scale has 20-item questions with four alternatives (Never = 1, rarely = 2, Sometimes = 3, and Often = 4) ranging from 20 (lowest score) to 80 (highest score). First, the adapted questionnaire was prepared in English and translated to Amharic and then, translated back to English by another person to check its consistency and wording. The cut-off point for describing loneliness was calculated by the mean and the score above the mean value was indicative of loneliness. Two supervisors participated in the data collection.
Data quality management/control
One day training about the ethical issues, the purpose of the study and data collection techniques were given for supervisors. Pretest was performed on 40 students outside of the study area. During data collection, close supervision, spot-checking and reviewing the completed questionnaire were done by the supervisors and principal investigator on daily basis. Data clean up and cross-checking was done before analysis. Cronbach's alpha was done with the result of 0.78 which is acceptable [24].
Data processing and analysis
Data were edited, coded and entered into epidemiological data (EPI-DATA) version 3.1 and exported to Statistical package for Social Sciences (SPSS) version 20 for analysis. Descriptive statistics were presented in frequency tables with percentage. Students who scored above a mean value (42 and above) were considered as facing loneliness. The bivariable analysis was done for loneliness and independent variables to check for crude association. Variables with a p-value of < 0.2 in bivariable analysis were candidates and entered to multivariate logistic regression analysis to identify the independent determinants of loneliness. Both Crude Odds Ratio (COR) and the Adjusted Odds Ratio (AOR) with a corresponding 95% confidence interval (CI) were computed to show the strength of association. Hosmer and Lemeshow goodness of fit test was checked (p-value > 0.05). Variables with p-value of < 0.05 in the multivariate logistic regression analysis were taken as statistically significant.
Description of study participants
A total of 404 students participated with 100% response rate. Of these, 242 (59.9%) were males and 238 (58.9%) were below 21 years. 127 (31.4%) students were actively engaged in romantic relationship. Only 3.2% and 3% of students were currently chewing chat and smoke cigarette respectively (Table 1).
Table 1 Socio-demographic property of participants, University of Gondar, 2018 (n = 404)
The magnitude of loneliness among study participants
From 404, participants 200 (49.5%) students had a score above mean fore loneliness indicating these had loneliness. Within group, females were lonelier than males but intergroup comparison indicated more males had loneliness. Those who had not engaged in a romantic relationship were more lonely (52%) as compared to those who were not engaged. The magnitude of loneliness was higher in those rural residents before coming to the university, current khat chewers and cigarette smokers and students with the age group of 21 years and above (Table 2).
Table 2 Associated factors of loneliness in multivariable logistic regression among study participants, University of Gondar, 2018 (n = 404)
Factors associated with Loneliness
All independent variables were tested for crude association with loneliness by binary logistic regression. Of the tested variables; Sex, Active engagement in a romantic relationship and year of study had a p-value of < 0.2 and hence entered to multivariable logistic regression with a backward Likelihood ratio (LR) to find out significantly associated factors of loneliness. In multivariable logistic regression, only year of study was significantly associated (p-value = 0.00). After controlling other variables constant, first-year students were 2.47 times more likely to develop loneliness than second year and above [AOR = 2.47, 95% CI (1.651–3.701)] (Table 2).
Findings of the current study revealed the magnitude of loneliness among University of Gondar students being 49.5% [95% CI 44.6–54.4%] which is a major public health problem with females more affected than males in proportion. This is lower than a report in Turkey University students which disclosed 60.2% loneliness [4]. In contrary, the result of this study showed a higher prevalence of loneliness as compared to a study in Mid-Germany that reported a 10.5% [6] and 10.2% (meta-analysis study) prevalence of loneliness [25]. This difference could be associated with the nature of study participants, sample size and sociocultural differences. No more articles were found about the magnitude of loneliness that limited us to compare our study findings.
As indicated in different studies, there were different factors contributing to loneliness like family residence, romantic relationship (partnership), sex and family wealth (income) [2]. In this study amongst the covariates tested in binary logistic regression, only sex, Active engagement in a romantic relationship and year of study had p-value less than 0.2. Of these variables year of study was significantly associated with loneliness; being first year is highly risky for having loneliness than second year and above students. The association of loneliness among first year (freshman) students could be due to the short duration of stay in the University to adapt the new environment which is supported by a study done in Washington University that indicated leaving home for college made students susceptible to experience loneliness [26]. Being in a romantic relationship is preventive to loneliness as reported in Germany [27] even though it is not significantly associated in our study. Our study findings indicated a major proportion of students were affected by loneliness which gives insight for University authorities to work on loneliness preventive and therapeutic strategies so as to reduce difficulties of students in the learning environment. This study revealed a higher magnitude of loneliness which is mainly found in first-year students. Being first year is more prone to loneliness. This gives a clue for officials to include coping strategies for loneliness in University students. The investigators would like to recommend researchers to conduct more qualitative studies to find out concrete information on loneliness. Besides this, Universities has to plan on how to put students in a comfortable situation while they are separating from their family. Above all guidance and counseling institutions in Universities has to approach students to prevent and treat loneliness.
Limitation of the study
This study was undertaken using a standardized tool which helped to compare with previous studies. The generalizability and validity of the study result is good. However, this study was not without limitations. The potential limitations include the nature of the study design which cannot be used to show cause-effect relationship. In addition the self-reported nature of the study tool might result in social desirability bias.
AOR:
adjusted odds ratio
confidence interval
crude odds ratio
Epi-Data:
UCLA-R:
Revised University of California Loneliness Assessment scale
Daniel K. Loneliness and depression among university students in Kenya. Glob J Hum Soc Sci Arts Humanit. 2013;13(4):1–9.
Zilioli S, Slatcher RB, Chi P, Li X, Zhao J, Zhao G, et al. The impact of daily and trait loneliness on diurnal cortisol and sleep among children affected by parental HIV/AIDS. Psychoneuroendocrinology. 2017;75:64–71.
Sawir E, Marginson S, Deumert A, Nyland C, Ramia G. Loneliness and international students: an Australian study. J Stud Int Educ. 2008;12(2):148–80.
Özdemir U, Tuncay T. Correlates of loneliness among university students. Child Adolesc Psychiatry Ment Health. 2008;2:1–6.
Parvan R, Karimi F, Safiri S, Mahdavi N. Prevalence of loneliness and associated factors among Iranian College students during 2015. J Arch Mil Med. 2017;5(1):e42356.
Beutel ME, Klein EM, Brähler E, Reiner I, Jünger C, Michal M, et al. Loneliness in the general population: prevalence, determinants and relations to mental health. BMC Psychiatry. 2017;17:97.
Peltzer K, Pengpid S. Loneliness: Its correlates and associations with health risk behaviours among university students in 25 countries. J Psychol Africa. 2017;27(3):247–55.
Ceyhan E, Ceyhan AA. Loneliness and depression. Educ Sci. 2011;36(160):17–9.
Sadoughi M. Relationship between social support and loneliness and academic adjustment among university students. Int J Acad Res Psychol. 2016;3(2):1–8.
Lin Y, Kingminghae W. Social support and loneliness of Chinese international students in Thailand. J Popul Soc Stud. 2014;22(2):141–57.
Ari R, Konya C, John S. An investigation of social skills and loneliness. 2005;33(1):19–32.
Matthews T, Danese A, Wertz J, Odgers CL, Ambler A, Moffitt TE, et al. Social isolation, loneliness and depression in young adulthood: a behavioural genetic analysis. Soc Psychiatry Psychiatr Epidemiol. 2016;51(3):339–48.
Świtaj P, Grygiel P, Chrostek A, Wciórka J, Anczewska M. Investigating the roles of loneliness and clinician- and self-rated depressive symptoms in predicting the subjective quality of life among people with psychosis. Soc Psychiatry Psychiatr Epidemiol. 2018;53(2):183–93.
Stickley A, Koyanagi A, Koposov R, Schwab-Stone M, Ruchkin V. Loneliness and health risk behaviours among Russian and U.S. adolescents: a cross-sectional study. BMC Public Health. 2014;14(1):1–12.
Zhong BL, Chen SL, Tu X, Conwell Y. Loneliness and cognitive function in older adults: findings from the chinese longitudinal healthy longevity survey. Journals Gerontol - Ser B Psychol Sci Soc Sci. 2017;72(1):120–8.
Gerst-Emerson K, Jayawardhana J. Loneliness as a public health issue: the impact of loneliness on health care utilization among older adults. Am J Public Health. 2015;105(5):1013–9.
Stickley A, Koyanagi A, Roberts B, Richardson E, Abbott P, Tumanov S, et al. Loneliness: its correlates and association with health behaviours and outcomes in nine countries of the former Soviet Union. PLoS ONE. 2013;8(7):e67978.
Queen TL, Stawski RS, Ryan LH, Smith J. Loneliness in a day: Activity engagement, time alone, and experienced emotions. Psychol Aging. 2014;29(2):297–305.
Richard A, Rohrmann S, Vandeleur CL, Schmid M, Eichholzer M. Loneliness is adversely associated with physical and mental health and lifestyle factors: results from a Swiss national survey. PLoS ONE. 2017;12:e0181442.
Belay Ababu G, Belete Yigzaw A, Dinku Besene Y, Getinet Alemu W. Prevalence of adjustment problem and its predictors among first-year undergraduate students in Ethiopian University: a cross-sectional institution based study. Psychiatry J. 2018;2018:1–7.
Sønderby LC. Loneliness: an integrative approach. J Integr Soc Sci. 2013;3(1):1–29.
Cacioppo S, Grippo AJ, London S, Goossens L, John T, Development A. Loneliness: clinical import and interventions. Perspect Psychol Sci. 2015;10(2):238–49.
Russell D. Ucla loneliness scale version 3 (description of measure). J Pers Soc Psychol. 1996;39:3–4.
George D, 2003. SPSS for windows step by step: a simple study guide and reference, 17.0 update, 10/e. Pearson Education India.
Menec VH, Newall NE, Mackenzie CS, Shooshtari S, Nowicki S. Examining individual and geographic factors associated with social isolation and loneliness using Canadian Longitudinal Study on Aging ( CLSA ) data. PLoS ONE. 2019;14(2):1–18.
English T, Davis J, Gross JJ. Homesickness and adjustment across the first year of college: a longitudinal study. Anal Chem. 2017;27(1):1–5.
Diehl K, Jansen C, Ishchanova K, Hilger-Kolb J. Loneliness at universities: determinants of emotional and social loneliness among students. Int J Environ Res Public Health. 2018;15(9):1865.
BD and HD involved in proposal development, participated in data collection, statistical analysis, and manuscript write-up. All authors read and approved the final manuscript.
We would like to thank our study participants and data collectors for their collaboration.
The dataset in the current study is available from the corresponding author upon request.
Ethical approval was obtained from the Ethical committee of Department of Environmental and Occupational Health and Safety, University of Gondar with ethical approval number EOHS/814/2011. Written consent was taken from each study participants.
No funding was obtained for this work.
Department of Human Physiology, School of Medicine, University of Gondar, P.O. Box 196, Gondar, Ethiopia
Baye Dagnew
Department of Environmental and Occupational Health and Safety, Institute of Public Health, University of Gondar, P.O. Box 196, Gondar, Ethiopia
Henok Dagne
Correspondence to Baye Dagnew.
Dagnew, B., Dagne, H. Year of study as predictor of loneliness among students of University of Gondar. BMC Res Notes 12, 240 (2019). https://doi.org/10.1186/s13104-019-4274-4 | CommonCrawl |
Probing spermiogenesis: a digital strategy for mouse acrosome classification
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3D Shape Modeling for Cell Nuclear Morphological Analysis and Classification
Alexandr A. Kalinin, Ari Allyn-Feuer, … Ivo D. Dinov
High spatially sensitive quantitative phase imaging assisted with deep neural network for classification of human spermatozoa under stressed condition
Ankit Butola, Daria Popova, … Balpreet Singh Ahluwalia
A robust unsupervised machine-learning method to quantify the morphological heterogeneity of cells and nuclei
Jude M. Phillip, Kyu-Sang Han, … Pei-Hsun Wu
Reconstruction of bovine spermatozoa substances distribution and morphological differences between Holstein and Korean native cattle using three-dimensional refractive index tomography
Hao Jiang, Jeong-woo Kwon, … Nam-Hyung Kim
Alessandro Taloni1,2,3,
Francesc Font-Clos4,
Luca Guidetti1,5,
Simone Milan ORCID: orcid.org/0000-0002-1888-23931,4,5,
Miriam Ascagni6,
Chiara Vasco7,
Maria Enrica Pasini6,
Maria Rosa Gioria6,
Emilio Ciusani7,
Stefano Zapperi ORCID: orcid.org/0000-0001-5692-54651,2,4,8,9 &
Caterina A. M. La Porta ORCID: orcid.org/0000-0002-3010-89661,5
Classification and taxonomy
Nonlinear phenomena
An Author Correction to this article was published on 14 November 2018
Classification of morphological features in biological samples is usually performed by a trained eye but the increasing amount of available digital images calls for semi-automatic classification techniques. Here we explore this possibility in the context of acrosome morphological analysis during spermiogenesis. Our method combines feature extraction from three dimensional reconstruction of confocal images with principal component analysis and machine learning. The method could be particularly useful in cases where the amount of data does not allow for a direct inspection by trained eye.
Spermatogenesis is a dynamic process during which undifferentiated diploid stem cells mature to differentiated haploid cells called spermatozoa. Mammalian spermatogenesis occurs within the seminiferous tubules and consists of three phases: a mitotic phase in which spermatogonia divide mitotically; a meiotic phase in which spermatocytes divide to form haploid round spermatids and a third phase, called spermiogenesis, in which spermatids encompass morphological changes including acrosome formation, chromatin condensation, and flagellum development resulting in the formation of spermatozoa1,2,3,4,5,6,7.
A key element of spermiogenesis is the mammalian sperm acrosome, an exocytotic vesicle present on the apical surface of the head8, 9 whose correct formation is crucial for the successful fertilization of the egg10. Acrosomal biogenesis takes place at the initial step of spermiogenesis and can be divided into four phases that cumulatively complete in about 2 weeks in the mouse and in 1 month in the humans8,9,10,11,12,13,14,15. In rodent spermatids, proacrosomal vesicles (granules) containing a variety of proteins assemble and fuse to form a single sphere acrosomal granule in the center of the acrosomal vesicle at the Golgi phase. At the cap phase, the acrosomal granule forms a head cap-like structure that gradually enlarges to cover the nucleus. The head cap continues to elongate outlining the dorsal edge, protruding apically at the acrosome phase, and finally the structure of the acrosome is completed at the end of maturation phase12.
Our current understanding of human reproduction is increasing thanks to the use of Assisted Reproductive Techniques (ART) and many studies aim to find a better way to select viable sperm16. Even though many aspects of sperm formation have been investigated, only few studies report quantitative measurements of sperm and its components, mainly focusing on the whole sperm heads17, 18. Since infertility is a common problem for men, it would be useful to devise standard parameters that could help in ART. A correct formation of the acrosome is crucial for a physiological reproduction capability and the quantification of the ratio between spermatides and spermatozoa can be a valid support for the correct prognosis of diseases linked to an impaired biogenesis of sperm cells.
Conventional strategies to study mammal spermiogenesis usually try to characterize specific morphological features supposed to play a key role in the development of the cells to spermatozoa with the aim of targeting them for possible prognostic/therapeutic strategies. The morphological analysis of spermatozoa is usually performed by a trained eye, but due to the increasing amount of digital images stored, it is becoming important to develop automatic techniques of classification and diagnosis. In this respect, there is still a pressing need to develop reliable automated method for cell morphology assessment. While objective tools for sperm motility assessment exist19, current automatic methods for sperm morphology are still not accurate and difficult to use20. Hence, subjective morphology sperm cell assessment is the standard in laboratories but results in large variability in the outcome. Machine learning-based intelligent systems could play a pivotal role to reach this goal. The method starts from an input feature matrix, including characteristic values of designated positive and negative samples, and self-trains the prediction models by learning the patterns in the feature matrix. The final goal is then to be able to automatically classify a data set with unknown labels.
In this paper, we present a machine learning approach to classify in a quantitative and semi-automatic way important morphometric characteristics of mammalian acrosomes during spermatogenesis. We start by a three-dimensional digital reconstruction of confocal images of acrosomes from which we extract a discretized mesh representing the surface of each acrosome. We then compute a series of morphological parameters such as volume, surface and local curvatures. These morphological parameters represent the features that will then be analyzed through machine learning and principal component analysis. We illustrate the method by analyzing acrosomes from spermatides and spermatozoa, obtained from seminiferous tubules of young mice, which are known to have different shapes. The ground truth is established by direct classification by eye and the results compared with automatic methods based on machine learning.
Here we develop a new method combining computational science, quantitative biology and machine learning to classify acrosomes, distinguishing spermatides from spermatozoa in a semi-automatic way, obtaining robust quantitative morphological observables. To this end, we carry out a 3D reconstruction of the surface of acrosomes of spermatides and spermatozoa from sexually mature healthy mice maintained in vitro for a few days. Quantifying differences in the fraction of spermatides and spermatozoa could be useful to detect in advance important pathological conditions related to sterility and have impact of ART17, 18. In order to maximize the number of acrosomes for the analysis, we carried out the 3D reconstruction of the acrosomes in cells extracted from seminiferous tubules and imaged at different times, either immediately (time T0) or maintained in vitro overnight (time T1). An analysis by electron microscopy shows that the overall architecture is preserved between T0 and T1 (Fig. 1) and we did not record any statistical difference in the quantitative parameters extracted from confocal images.
Transmission electron micrograph of mouse seminiferous epithelium. Adult testis tubules obtained as described in Materials and Methods section were immediately fixed (time T0) or after 1 day in culture (time T1). (a,b) At T0 a well preserved tubular basal compartment of a stage VII tubule shows normal Sertoli cells (S), spermatogonia (Sg), primary spermatocytes (Sc) and spermatids (Sd). x 3500–4800. (c,d) At T1 the tubular basal compartment shows some signs of cellular degeneration (*). x 4800.
The detailed procedure for the reconstruction of the acrosomes surfaces, is discussed in the Materials and Methods section. Figure 2 shows two typical examples of meshes obtained by 3D reconstruction of the acrosomes membranes. The analysis of each acrosome yields a set of morphological characteristics (parameters): the acrosome's volume V, its surface area Σ, the sphericity Ψ, the average mean and Gaussian curvatures (\(\overline{M}\) and \(\overline{G}\), respectively) and their relative fluctuations (\(\frac{{\rm{\Delta }}M}{\overline{M}}\) and \(\frac{{\rm{\Delta }}G}{\overline{G}}\), respectively). Averaging these morphological parameters (〈…〉) over the subpopulations of spermatids and spermatozoa gives the values reported in Fig. 3. Moreover we also report on top the p-values from a Kolmogorov-Smirnov test that considers the entire Spermatids and Spermatozoa cumulative distributions.
Acrosomes surface 3D reconstruction. Panel (a): the round spermatid acrosome is singled out within one of the fields of a 3D confocal stack of the experimental slide. The spermatid surface is identified thanks to the SP56 marker of its acrosomal matrix (in green). Panel (b): the Active Contour plugin reconstructs the acrosome mesh by furnishing the closest three dimensional segmented surface to the acrosome bilipidic membrane. For a 3D rendering of the acrosome mesh see Supplementary Video S1. Panel (c): acrosome mesh and the local Gaussian curvature superimposed on each mesh node. The color code is from blue (low Gaussian curvature) to red (high Gaussian curvature). Panel (d): acrosome mesh and the local Mean curvature superimposed on each mesh node. The color code is from blue (low Mean curvature) to red (high Mean curvature). Panel (e): the spermatozoon acrosome is singled out within the confocal stack field, and identified thanks to the SP56 marker of its acrosomal matrix (in green). Panel (f): the Active Contour plugin reconstructs the acrosome mesh by furnishing the closest three dimensional segmented surface to the acrosome bilipidic membrane. Notice the typical harpin shape. For a 3D rendering of the acrosome mesh see Supplementary Video S2. Panel (g): acrosome mesh and the local Gaussian curvature superimposed on each mesh node. Color code is as in panel (c). Panel (h): acrosome mesh and the local Mean curvature superimposed on each mesh node. Color code is as in panel (d).
Statistical analysis: Average values. Average values of the morphological parameters for spermatids (green) and spermatozoa acrosomes (red). We also report the p-value from a KS test on top of each morphological parameter.
These data show that the acrosomes in spermatozoa are, in average, nearly 50% larger than those in spermatids and similar differences are recorded for the surfaces. This is not surprising since volume and surface are strongly correlated as illustrated in Fig. 4. In particular, volumes and surface follow the general law 〈Σ〉~〈V〉2/3 as expected based on simple dimensional considerations.
Features plot. Overall view of the distribution of five morphological features (\(\overline{G}\), ΔG/\(\overline{G}\), Σ, V, Ψ) and their bivariate relations. Diagonal panels: normed histograms (semi-transparent filled bins) and kernel density estimates (solid colored lines) corresponding to the log-transformed data. Lower-diagonal panels: scatter plots in logarithmic coordinates. Notice that the x-axes are shared within columns. The diagonal panels are in units of density (not shown).
During spermiogenesis, acrosomes from spermatids are typically more spherical than those capping the spermatozoa nuclei. The spherical shape is probably reminiscent of an early vesicle form. We recover this observation by measuring the sphericity 1 of each acrosome in both populations. By definition, when Ψ = 1 the spherical shape is recovered, while smaller values indicate eccentricity and/or asymmetry of the surface. The mean values reported in Fig. 3 confirm indeed that acrosomes from spermatids tend to be more spherical than those from spermatozoa (see also the reconstructed meshes in Fig. 3). This difference is statistically significant (p = 1.06 × 10−6).
To further characterize the morphology, we have considered surface curvatures. The Gaussian curvature, defined in Eq. 2, is positive for spheres, negative for hyperboloids and zero for planes. Hence, the sign of the Gaussian curvature indicates if a surface is locally convex or saddle-like. We have measured the average Gaussian curvature \(\overline{G}\) per cell, as defined in Eq. 5. The average value 〈\(\overline{G}\)〉 clearly shows that spermatids tend to have a more convex acrosome membrane as compared to spermatozoa (see Fig. 3, p = 1.14 × 10−2). The mean curvature, defined in Eq. 3, is zero for a plane, constant for a sphere and, more generally, it is positive for convex surfaces and negative for concave ones. Fig. 3 shows that, as in the case of Gaussian curvature, acrosomes from spermatids appear more spherical than those from spermatozoa (p = 4.15 × 10−2). In addition to the average values of Gaussian and mean curvature, we also consider their standard deviations which display significant differences between spermatids and spermatozoa (p = 2.19 × 10−6 for the Gaussian curvature and p = 4.64 × 10−3 for the mean curvature).
In summary, the quantitative morphological analysis reveals clear, statistically significant differences between spermatids and spermatozoa. These differences, however, arise at the population level and do not necessarily translate into a successful automated classification at the individual cell level. This is clear observing the plots in Fig. 4, where we report the bivariate relations and distribution for five morphological features. Notice that while these features all give rise to significant differences in the average parameters (Fig. 3), there is an important overlap in the individual values for spermatids and spermatozoa.
To overcome these problems, we decided to investigate if machine learning and principal component analysis could be useful to provide reliable information at the single cell level and more importantly to build up a predictive semi-quantitative method. Fig. 4 shows that the data display more uniform-like densities in logarithmic space (lower-diagonal panels) rather than in the original linear space (Supplementary Fig. S1). Hence the SVM classification is performed in logarithmic space. Having more uniform densities over the feature space is desirable for SVM classification, because penalties for misclassification are weighted according to their distance to the decision boundary.
Figure 5 shows the projection onto the first two principal components of the dataset, both in linear and logarithmic space. Although certain differences in the distribution of values for spermatids and spermatozoa can be appreciated, clearly these differences are insufficient to define non-overlapping clusters. In other words, the two subpopulations cannot be distinguished by eye in a PCA projection of the 7-feature dataset. This is, indeed, what motivated us to use a SVM in the full 7-dimensional feature space.
PCA projection. Projection of the seven morphological features onto its two first principal components (see Methods section), computed both in linear space (left panel) and in logarithmic space (right panel). Although some differences between spermatids and spermatozoa are apparent, no clear clusters arise.
Our results are summarized in Table 1. The values of the class accuracy (defined in Eq. 16) show that the SVM classification algorithm gets the correct answer in the 73% of trials (74% of trials for spermatids and in 69% of trials for spermatozoa acrosome, equivalent ROC AUC statistic 0.76). Although an average classification accuracy of 73% would not suffice for a potential automatized acrosome classification method, it is definitely beyond what a random or a constant classifier would achieve, marking the existing of a signal that could potentially be further exploited. In addition, it is interesting to notice the consistency by which cells are correctly classified/misclassified: 71% of all cells are correctly classified on at least 85% of the algorithm runs, i.e. r a = 0.85 = 0.71. If the value of a is raised to 0.99, then this figure drops only to 68%, i.e. r a = 0.99 = 0.68. In other words, there is a large subset of the data that is almost always correctly classified, and smaller subset of the data that is misclassified most of the time. This can be better seen in Fig. 6, where the cell accuracy has been used to color a scatter plot of the data. We have visually inspected the distribution of features, and found that misclassified cells lie in regions of mixed spermatid/spermatozoa density, while correctly classified ones tend to be on regions of more unequal spermatid/spermatozoa density. Therefore, it appears there is no more obvious information left, and further exploiting classification results to enhance the SVM would result in over-fitting.
Table 1 Summary of results of the SVM classification: class-averaged accuracy A C (Eq. 16); ratio of cells with classification accuracy equal to or greater than 0.85 and 0.99, r 0,85, r 0,99; and area under the curve for the receiver operating characteristic (ROC AUC).
SVM analysis. Left panel: spermatids acrosomes (green dots) and spermatozoa acrosomes (red dots) plotted in the Volume-Sphericity plane. Right panel: same data, colored according to the value of the classification accuracy A c (Eq. 16) obtained with the SVM: spermatids are colored from totally white (0% accuracy) to totally green (100% accuracy), while spermatozoa are colored from totally white (0% accuracy) to totally red (100% accuracy). Notice that a perfect classifier would render both panels identical. The two small images above the colorbar are example confocal images (a red coloring filter was applied to the spermatozoa image for clarity). The small triangular markers in the colorbar mark the class-level accuracy values (see Table 1).
The choice of SVM among other classifiers responds to its simplicity and the fact that it handles well class imbalance. In particular, we compared our result with those obtained with a Random Forest (RF) classifier using either class weights or downsampling to correct for class imbalance. In the first case, we obtain a 92% accuracy for spermatids, but only 27% for spermatozoa. In the second case, we achieve 69% accuracy for spermatids and 57% for spermatozoa. Therefore, SVM gives better results than RF, probably due to how class imbalance is handled.
In conclusion, we have proposed a general strategy to classify acrosomes from spermatides and spermatozoa according to their morphological features. The methods starts from a three dimensional reconstruction of the surface of the acrosome from confocal images and extracts a set of morphological parameters from the reconstructed surface. These parameters are then analyzed by machine learning and compared with the ground truth provided by a direct assessment by eye. The method we propose could be helpful to assist the analysis of spermatozoa during spermiogenesis, especially in presence of large quantities of data where direct classification by eye is not feasible. Future studies along these lines should aim at finding automated tools to distinguish between a normal cap and a cap with distortions of outer and inner acrosomal membranes, identify damages of the acrosomal matrix or to estimate the fraction of sperm cells with loosen cap after freezing-thawing. This could help solve the relevant clinical issue of quantifying the percentage of sperm cells with normal acrosome and therefore assess fertility.
Animals and culture medium
Sexually mature CD1 male mice (four-five months) were purchased from Charles River (Calco, Italy). Mice were kept in controlled conditions and all procedures were conformed to Italian law (D. Lgs n. 2014/26, implementation of the 2010/63/UE) and approved by the Animal Welfare Body of the University of Milan and by the Italian Minister of Health.
Isolation of single cells from testis
Testes were isolated and decapsulated in 0.1 M Phosphate Buffer. The seminiferous tubules were gently placed onto a small cube made of 1,5% agarose and soaked in culture medium for more than 24 h to replace water. The amount of medium was adjusted in order to cover half to four fifth of the height of agarose cubes. Tubules were maintained in incubator at 34 °C, 5% and controlled humidity overnight in the following culture medium: RPMI (Euroclone), 10% Fetal Bovine Serum (FBS) (Euroclone), 2 mM Stable L-glutamine (Euroclone), antibiotic antimycotic solution (A5955, Sigma-Aldrich). Seminiferous tubules were picked up from agarose cubes (Sigma) and fixed in 2% paraformaldehyde dissolved in PBS pH 7.2–7.4 for 10 min. A single fixed tubule was laid down onto a slide, covered with a coverslip and a gentle pressure was applied in order to allow cells to come out from the seminiferous tubule. Slides were then frozen in liquid nitrogen for further analyses.
Acrosome staining
The slides were rinsed with ice-cold phosphate buffered saline (PBS) 1X for 5 min at room temperature (RT), fixed with cold 100% methanol for 15 min at −20 °C then incubated with 10% goat-serum in PBS for 1 h at RT. The slides were incubated with anti-sperm Protein sp56 antibody (7C5; 1:150 Life Technologies-MA1-10866) overnight at 4 °C and then incubated with anti-mouse IgM/G/A (H + L) 488 secondary antibody (1:250 Millipore-AP501F) for 1 h at RT. The slides were mounted with Prolong Gold Antifade reagent with DAPI (Life Technologies-P36935). At least 60 stack images were acquired with Leica SP2 laser scanning confocal microscope (63X).
Transmission electron microscopy
The seminiferous tubules were fixed in loco with 2,5% glutaraldehyde (electron microscope grade) in 0,1 M phosphate buffer (PB) pH 7.2 for 3 h at room temperature. The tubules were then mounted between two layers of 1,5% agarose (Sigma) of about 2 mm in height, which was cut into small cubes 2 × 2 × 3 mm in size and postfixed in 2% osmium tetroxide in 0,1 M PB overnight at 4 °C. The samples were dehydrated in a graded ethanol series, and embedded in epoxy resin. Semithin section (1 μm) were stained with toluidine blue in borax and examined by light microscopy. Ultrathin section (70 nm) were cut using a diamond knife on a Reichert Ultracut ultramicrotome, mounted on a Cu/Rh grids (200 mesh), contrasted with uranyl acetate and lead citrate, examined and photographed with a Zeiss 902 transmission electron microscope operating at 80 kV. The exposed films were developed according to common photographic techniques, captured with an Epson V700 Photo scanner with a final resolution of 600dpi and appropriately calibrated for contrast and brightness (see Fig. 1).
3D acrosome reconstruction by immunofluorescence images of sp56
A 3D reconstruction of the acrosome obtained from confocal images of sperm cells stained with anti-sp56 has been done with ICY software tools (http://icy.bioimageanalysis.org/). Briefly, confocal stacks (at least 80–90 stacks) were first pre-processed to extract the individual cells images. Images were picked in diverse fields of the slide, to consider all the different stages that are present in a single portion of the tubule and not to overestimate the presence of cells in a particular stage of differentiation. A minimum of twenty cells were scored and analyzed for each slide. Two subpopulations in the seminiferous tubules were considered: round spermatids and spermatozoa. The formers represent the early stage of spermatogenesis and are identified by the presence of one or two spots of condensed heterochromatin in a spheroidal nucleus. The latters show a compact chromatin, an acrosome with hooked shape and the presence of the flagellum, according to previous paper1, 7. Cells were singled out by tracing a region of interest (ROI) around every acrosome in each subpolpulation. Subsequently this ROI has been cropped by using the Fast crop tool. Hence, our analysis could take advantage of single high resolution images, for any acrosome under consideration. The 3D ROI of individual acrosomes were also refined by using the HK-Means plugin (http://icy.bioimageanalysis.org/plugin/HK-Means). This method performs a N-class thresholding based on a K-Means classification of the image histogram. The acrosome membrane reconstruction has been obtained by the segmentation technique implemented in the 3D Active Contour plugin (http://icy.bioimageanalysis.org/plugin/Active_Contours)21. The algorithm at the basis of this plugin performs three dimensional segmentation and tracking, using a triangular mesh optimized over the original signal as a target. In Fig. 2 and in the movie M1 (see the Supplementary Informations) the 3D reconstruction of a typical spermatid and a spermatozoa acrosome are displayed. The three dimensional renderings of the meshes (in Fig. 2 and movies (Supplementary Video S1 and Supplementary Video S2) were performed thanks to Paraview (http://www.paraview.org/).
Single cell data analysis
Once each three dimensional acrosome mesh was reconstructted, we proceeded to measure its cell volume (V) and surface area (Σ) using Meshlab tools (http://meshlab.sourceforge.net/). The acrosome sphericity is calculated according to the definition
$${\rm{\Psi }}=\frac{{\pi }^{1/3}{(6V)}^{2/3}}{{\rm{\Sigma }}}$$
The local Gaussian and Mean curvatures were calculated by a custom python code, which massively makes use of vtk libraries (http://www.vtk.org/). Typical images of a spermatid and spermatozoon acrosome mesh, with superimposed local curvatures (blue-to-red) color maps, are reported in Fig. 2(c,d) and Fig 2(g,h) for Gaussian and mean curvature respectively. We label every node on the single mesh by i (with 1 ≤ i ≤ N), therefore the local mean and Gaussian curvature fields on each node are denoted as M i and G i respectively. The local curvature of a surface entails the notion of principal curvatures, k 1, k 2, defined as the smallest and largest one dimensional curvatures on a point. The Gaussian curvature is defined as
$${G}_{i}={k}_{1}^{i}{k}_{2}^{i}$$
where the index i runs over the nodes of an acrosome mesh (see Fig. 2(c,g)). The mean curvature instead is defined as the average of the principal curvatures:
$${M}_{i}=\frac{{k}_{1}^{i}+{k}_{2}^{i}}{2},$$
and has the dimension of length −1. The averaged value of the Mean and Gaussian curvature on the acrosome surface are defined as
$$\overline{M}=\frac{\sum _{i=1}^{N}{M}_{i}}{N}$$
$$\overline{G}=\frac{\sum _{i=1}^{N}{G}_{i}}{N}.$$
Besides the average local curvature per cell (Eqs 4 and 5 respectively), we also define relative fluctuations of the local mean curvature of individual acrosomes as
$$\frac{{\rm{\Delta }}M}{\overline{M}}=\sqrt{\frac{\sum _{i=1}^{N}{({M}_{i}-\overline{M})}^{2}}{(N-\mathrm{1)}{\overline{M}}^{2}}},$$
and similarly for the Gaussian curvature
$$\frac{{\rm{\Delta }}G}{\overline{G}}=\sqrt{\frac{\sum _{i=1}^{N}{({G}_{i}-\overline{G})}^{2}}{(N-\mathrm{1)}{\overline{G}}^{2}}}$$
The statistical analysis is performed by averaging a set of 7 morphological parameters (V, Σ, Ψ, \(\overline{M}\), \(\overline{G}\), \(\frac{{\rm{\Delta }}M}{\overline{M}}\), \(\frac{{\rm{\Delta }}G}{\overline{G}}\)) over the statistical ensemble of the spermatids and spermatozoa acrosomes subpopulations composed by 158 spermatids and 51 spermatozoa. The statistical significance was evaluated using Kolmogorov-Smirnov tests as implemented in the python library scipy (https://www.scipy.org/). Our code is available at https://github.com/ComplexityBiosystems/.
Principal Component Analysis (PCA)
We use Principal Component Analysis (PCA)22 as implemented in the open-source python library scikit-learn (https://scikit-learn.org/stable). PCA is very popular visualization and dimensionality-reduction technique based on the singular-value decomposition of the features-samples matrix. The decomposition entails a new space of uncorrelated features, where each new feature or principal component is a linear combination of the original features. The principal components are of interest because (i) they are uncorrelated and (ii) they are such that the first principal component accounts for as much variability of the original data as possible; the second one accounts for as much of the remaining variability as possible, and so on. In this way, projecting the data onto the first few principal components we preserve most of the variability of the data while keeping the number of features low. In this manuscript we use PCA as a visualization technique, to discard the existence of "obvious" clusters in the dataset. By projecting the data onto the two first principal components, we obtain the 2-dimensional scatter plot that better represents the original data, in terms of explained variability.
Support Vector Machine (SVM)
We first give a brief mathematical introduction to the algorithm behind SVM, and then discuss the implementation to our problem. SVM are a set of widely-used machine learning algorithms, highly popular for their simplicity and the fact that they yield good results in many cases. Here we use its simplest version, a SVM with a linear kernel. In essence, the algorithm boils down to finding the hyper-plane h parametrized by \(\overrightarrow{w},b\),
$$h:=\{\overrightarrow{x}\in {{\mathbb{R}}}^{d}:\overrightarrow{w}\cdot \overrightarrow{x}+b=0\}$$
that better separates the data \({\overrightarrow{x}}_{i}\) into the known classes y i ∈ {−1, 1}. In mathematical terms, the problem is cast into an optimization problem with constrains, which is easily solved via Lagrange multipliers. In particular, one needs to find \(\overrightarrow{w},b\), ξ i that minimize
$$\frac{1}{2}{\Vert \overrightarrow{w}\Vert }^{2}+C\sum _{i}{\xi }_{i}$$
under the constraints that
$${y}_{i}(\overrightarrow{w}\cdot {\overrightarrow{x}}_{i}+b)+{\xi }_{i}\ge 1\quad \quad \forall i$$
where ξ i ≥ 0 are auxiliary variables that allow for misclassification (a penalty proportional to the distance to the decision boundary is set for misclassified points), and C sets a global weight for the misclassification penalty. We refer the reader interested in mathematical details to22.
The hyperplane h is determined using only a subset of the data, called the training set, and then the labels of the rest of the data, called test set, are predicted as follows:
$$y\equiv {\rm{sign}}\,(\overrightarrow{w}\cdot {\overrightarrow{x}}_{ts}+b)$$
where y is the predicted label of a point \({\overrightarrow{x}}_{ts}\) in the test set. There are many, more involved strategies to split the data into different sets for training and prediction. The interested reader will find good introductory material in ref. 22 and references therein.
Our data is given by the seven morphological features of the acrosomes and the acrosome subpopulation to which each cell belongs (Spermatids/Spermatozoa). That is, each cell is represented by a pair (\({\overrightarrow{x}}_{i},{y}_{i}\)) with \({\overrightarrow{x}}_{i}\in {{\mathbb{R}}}^{7}\) a vector containing its morphological information, and y i ∈ {−1, 1} a subpopulation class label, where −1 encodes for "Spermatid" and 1 for "Spermatozoa". We use the python implementation of Support Vector Machines provided by the machine-learning library scikit-learn (https://scikit-learn.org/stable). In particular, we use the function "sklearn.svm.SVC()". Given the difference in sample size of the two groups (158 spermatids and 51 spermatozoa, see the Materials and Methods), it is important to set the keyword "class_weights" to "balanced", which effectively sets statistical weights in the computation of the error term inversely proportional to the class observed frequencies. We use 10-fold cross validation, which means that, for each run of the algorithm, the data is randomly split into ten groups: nine are used to train the SVM, i.e. to determine the parameters \(\overrightarrow{w},b\) of the hyperplane, and one is used for prediction. This is repeated ten times, one for each group, so that in the end each datapoint has received one predicted label. Given the stochasticity in splitting the data, we average results over N r = 1000 runs of the algorithm. Increasing N r does not improve the results.
In summary, for each run of the algorithm, the output is a predicted label "Spermatid" or "Spermatozoa" for each of the 209 acrosomes, which we then compare with the ground truth. If the predicted label corresponds to the true nature of the acrosome, we assign a binary value 1, otherwise we assign 0 if it is misclassified. Thus, we obtain a binary matrix B ij of size 209 × 1000 where each row represents a cell and each column a run of the algorithm.
We define the cell accuracy a i as the ratio of the times a specific cell i was correctly classified,
$${a}_{i}=\frac{1}{{N}_{r}}\sum _{j=1}^{{N}_{r}}{B}_{ij}.$$
We then define the average class accuracy A C as the average of a i over all cells i of a given class C, where C can be either spermatids or spermatozoa,
$${A}_{C}=\frac{1}{|C|}\sum _{i\in C}{a}_{i},$$
and |C| is the size of the acrosomes subpopulations, i.e. |C| = 158 for spermatids and |C| = 51 for spermatozoa. Notice that A C corresponds also to the average over algorithm runs j = 1…N r of the class accuracy,
$$1{A}_{C}=\frac{1}{|C|}\sum _{i\in C}{a}_{i}$$
$$\quad =\,\frac{1}{|C|}\sum _{i\in C}(\frac{1}{{N}_{r}}\sum _{j=1}^{{N}_{r}}{B}_{ij})$$
$$\quad =\,\frac{1}{{N}_{r}}\sum _{j=1}^{{N}_{r}}(\frac{1}{|C|}\sum _{i\in C}{B}_{ij}).$$
Finally we define the quantity r a as the ratio of cells above certain accuracy a in a given class C, i.e.
$${r}_{a}=\frac{|\{i\in C,{a}_{i} > a\}|}{|C|}$$
For instance, if one takes a value of a = 0.99, then r a=0.99 would indicate the (relative) number of cells that would be correctly classified with a probability equal to or higher than 99%.
All the custom codes codes are available at https://github.com/ComplexityBiosystems/.
A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has not been fixed in the paper.
Parvinen, M. & Ruokonen, A. Endogenous steroids in the rat seminiferous tubules. comparison of the stages of the epithelial cycle isolated by transillumination-assisted microdissection. Journal of Andrology 3, 211–220 (1982).
McCarrey, J. R. et al. Differential transcription of pgk genes during spermatogenesis in the mouse. Developmental biology 154, 160–168 (1992).
De Rooij, D. G. & Grootegoed, J. A. Spermatogonial stem cells. Current opinion in cell biology 10, 694–701 (1998).
Eddy, E. M. Male germ cell gene expression. Recent progress in hormone research 57, 103–128 (2002).
Wiederkehr, C. et al. Germonline, a cross-species community knowledgebase on germ cell differentiation. Nucleic acids research 32, D560–D567 (2004).
Zhao, G.-Q. & Garbers, D. L. Male germ cell specification and differentiation. Developmental cell 2, 537–547 (2002).
Vasco, C., Zuccotti, M., Redi, C. A. & Garagna, S. Identification, isolation, and rt-pcr analysis of single stage-specific spermatogenetic cells obtained from portions of seminiferous tubules classified by transillumination microscopy. Molecular reproduction and development 76, 1173–1177 (2009).
Leblond, C. & Clermont, Y. Spermiogenesis of rat, mouse, hamster and guinea pig as revealed by the "periodic acid-fuchsin sulfurous acid" technique. American Journal of Anatomy 90, 167–215 (1952).
Yanagimachi, R. Mammalian fertilization. The physiology of reproduction 1, 189–317 (1994).
Berruti, G. Towards defining an 'origin'—the case for the mammalian acrosome. In Seminars in cell & developmental biology, vol. 59, 46–53 (Elsevier, 2016).
Lin, Y.-N., Roy, A., Yan, W., Burns, K. H. & Matzuk, M. M. Loss of zona pellucida binding proteins in the acrosomal matrix disrupts acrosome biogenesis and sperm morphogenesis. Molecular and cellular biology 27, 6794–6805 (2007).
Abou-Haila, A. & Tulsiani, D. R. Mammalian sperm acrosome: formation, contents, and function. Archives of biochemistry and biophysics 379, 173–182 (2000).
Kierszenbaum, A. L. & Tres, L. L. The acrosome-acroplaxome-manchette complex and the shaping of the spermatid head. Archives of histology and cytology 67, 271–284 (2004).
Buffone, M. G., Foster, J. A. & Gerton, G. L. The role of the acrosomal matrix in fertilization. International Journal of Developmental Biology 52, 511–522 (2004).
Kanemori, Y. et al. Biogenesis of sperm acrosome is regulated by pre-mrna alternative splicing of acrbp in the mouse. Proceedings of the National Academy of Sciences 113, E3696–E3705 (2016).
Boer, Pd, Vries, Md & Ramos, L. A mutation study of sperm head shape and motility in the mouse: lessons for the clinic. Andrology 3, 174–202 (2015).
Rijsselaere, T., Soom, A., Maes, D. & Nizanski, W. Computer-assisted sperm analysis in dogs and cats: an update after 20 years. Reproduction in Domestic Animals 47, 204–207 (2012).
Yániz, J., Soler, C. & Santolaria, P. Computer assisted sperm morphometry in mammals: A review. Animal reproduction science 156, 1–12 (2015).
Katz, D. & Davis, R. Automatic analysis of human sperm motion. Journal of andrology 8, 170–181 (1987).
Rijsselaere, T., Van Soom, A., Hoflack, G., Maes, D. & de Kruif, A. Automated sperm morphometry and morphology analysis of canine semen by the hamilton-thorne analyser. Theriogenology 62, 1292–1306 (2004).
Dufour, A., Thibeaux, R., Labruyere, E., Guillen, N. & Olivo-Marin, J.-C. 3-d active meshes: fast discrete deformable models for cell tracking in 3-d time-lapse microscopy. IEEE Transactions on Image Processing 20, 1925–1937 (2011).
Article ADS MathSciNet Google Scholar
Barber, D. Bayesian Reasoning and Machine Learning (Cambridge University Press, 2012).
A.T., F.F.C. and S.Z. are supported by ERC Advanced grant SIZEFFECTS.
Center for Complexity and Biosystems University of Milano, via Celoria 16, 20133, Milano, Italy
Alessandro Taloni, Luca Guidetti, Simone Milan, Stefano Zapperi & Caterina A. M. La Porta
Department of Physics, University of Milano, Via Celoria 16, 20133, Milano, Italy
Alessandro Taloni & Stefano Zapperi
CNR-Consiglio Nazionale delle Ricerche, ISC, Via dei Taurini 19, 00185, Roma, Italy
Alessandro Taloni
ISI Foundation, Via Chisola 5, 10126, Torino, Italy
Francesc Font-Clos, Simone Milan & Stefano Zapperi
Department of Environmental Science and Policy, University of Milano, via Celoria 26, 20133, Milano, Italy
Luca Guidetti, Simone Milan & Caterina A. M. La Porta
Department of Biosciences University of Milano, via Celoria 26, 20133, Milano, Italy
Miriam Ascagni, Maria Enrica Pasini & Maria Rosa Gioria
Istituto Neurologico Carlo Besta, Via Celoria, 11, 20133, Milano, Italy
Chiara Vasco & Emilio Ciusani
Department of Applied Physics, Aalto University, P.O. Box 11100, FIN-00076, Aalto, Espoo, Finland
Stefano Zapperi
CNR-Consiglio Nazionale delle Ricerche, ICMATE, Via Roberto Cozzi 53, 20125, Milano, Italy
Francesc Font-Clos
Luca Guidetti
Simone Milan
Miriam Ascagni
Chiara Vasco
Maria Enrica Pasini
Maria Rosa Gioria
Emilio Ciusani
Caterina A. M. La Porta
A.T., F.F.C., L.G., S.M. analyzed data. L.G., M.A., C.V., M.E.P., M.R.G., E.C., C.A.M.L.P. performed experiments. S.Z. and C.A.M.L.P. designed and coordinated the project. A.T., F.F.C., S.Z., C.A.M.L.P. wrote the paper.
Correspondence to Caterina A. M. La Porta.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Video S1
Supplementary Info
Taloni, A., Font-Clos, F., Guidetti, L. et al. Probing spermiogenesis: a digital strategy for mouse acrosome classification. Sci Rep 7, 3748 (2017). https://doi.org/10.1038/s41598-017-03867-7
Combined Feature Extraction Method
Acrosomal Surface
Acrosomal Membrane
Seminiferous Tubules | CommonCrawl |
# Basic concepts of JavaScript
Before we dive into building and automating robots with Puppeteer and JavaScript, let's cover some basic concepts of JavaScript. This will ensure that you have a solid foundation before we move on to more advanced topics.
JavaScript is a high-level, interpreted programming language that is primarily used for creating interactive web pages. It is often referred to as the "language of the web" because it is supported by all modern web browsers.
Some key concepts of JavaScript include:
- Variables: Variables are used to store data values. In JavaScript, you can declare variables using the `var`, `let`, or `const` keywords.
- Data types: JavaScript has several built-in data types, including numbers, strings, booleans, arrays, and objects. Understanding these data types is important for working with data in JavaScript.
- Operators: JavaScript has a wide range of operators, including arithmetic operators, comparison operators, and logical operators. These operators are used to perform operations on variables and values.
- Control flow: Control flow refers to the order in which statements are executed in a program. JavaScript provides several control flow statements, such as `if` statements, `for` loops, and `while` loops, that allow you to control the flow of your code.
- Functions: Functions are blocks of reusable code that perform a specific task. They allow you to organize your code into logical units and make it more modular and reusable.
- Objects: Objects are a fundamental concept in JavaScript. They allow you to group related data and functions together into a single entity. Objects are often used to represent real-world entities, such as a person or a car.
These are just a few of the basic concepts of JavaScript. As we progress through this textbook, we will explore these concepts in more detail and learn how to apply them to building and automating robots with Puppeteer and JavaScript.
Web scraping is the process of extracting data from websites. It is a common task in many industries, including e-commerce, finance, and research. With Puppeteer, you can automate the web scraping process and retrieve data from websites in a structured format.
Puppeteer is a Node.js library that provides a high-level API for controlling headless Chrome or Chromium browsers. It allows you to interact with web pages, navigate through websites, and extract data using a simple and intuitive API.
In this section, we will cover the following topics:
- Installing Puppeteer: We will start by installing Puppeteer and setting up our development environment.
- Launching a browser: We will learn how to launch a headless Chrome or Chromium browser using Puppeteer.
- Navigating through pages: We will explore how to navigate through web pages, click on links, and interact with forms using Puppeteer.
- Extracting data: We will learn how to extract data from web pages using Puppeteer. This includes selecting elements, retrieving text and attribute values, and handling pagination.
By the end of this section, you will have a solid understanding of web scraping with Puppeteer and be able to extract data from websites for your own projects or research.
## Exercise
Before we dive into web scraping with Puppeteer, let's make sure we have Puppeteer installed and set up in our development environment.
1. Open your terminal or command prompt.
2. Run the following command to install Puppeteer:
```bash
npm install puppeteer
```
3. Once the installation is complete, create a new JavaScript file called `scraping.js`.
4. Import Puppeteer into your JavaScript file using the following code:
```javascript
const puppeteer = require('puppeteer');
```
5. Save the file and run it using Node.js:
```bash
node scraping.js
```
If everything is set up correctly, you should see no errors and Puppeteer should be ready to use.
### Solution
There is no specific answer for this exercise, as it is a setup exercise. The goal is to ensure that Puppeteer is installed and set up correctly in your development environment. If you encounter any errors or issues during the setup process, refer to the Puppeteer documentation or seek help from online resources or forums.
# Web scraping using Puppeteer
Web scraping is the process of extracting data from websites. It is a common task in many industries, including e-commerce, finance, and research. With Puppeteer, you can automate the web scraping process and retrieve data from websites in a structured format.
Puppeteer is a Node.js library that provides a high-level API for controlling headless Chrome or Chromium browsers. It allows you to interact with web pages, navigate through websites, and extract data using a simple and intuitive API.
In this section, we will cover the following topics:
- Installing Puppeteer: We will start by installing Puppeteer and setting up our development environment.
- Launching a browser: We will learn how to launch a headless Chrome or Chromium browser using Puppeteer.
- Navigating through pages: We will explore how to navigate through web pages, click on links, and interact with forms using Puppeteer.
- Extracting data: We will learn how to extract data from web pages using Puppeteer. This includes selecting elements, retrieving text and attribute values, and handling pagination.
By the end of this section, you will have a solid understanding of web scraping with Puppeteer and be able to extract data from websites for your own projects or research.
Before we dive into web scraping with Puppeteer, let's make sure we have Puppeteer installed and set up in our development environment.
1. Open your terminal or command prompt.
2. Run the following command to install Puppeteer:
```bash
npm install puppeteer
```
3. Once the installation is complete, create a new JavaScript file called `scraping.js`.
4. Import Puppeteer into your JavaScript file using the following code:
```javascript
const puppeteer = require('puppeteer');
```
5. Save the file and run it using Node.js:
```bash
node scraping.js
```
If everything is set up correctly, you should see no errors and Puppeteer should be ready to use.
## Exercise
Before we dive into web scraping with Puppeteer, let's make sure we have Puppeteer installed and set up in our development environment.
1. Open your terminal or command prompt.
2. Run the following command to install Puppeteer:
```bash
npm install puppeteer
```
3. Once the installation is complete, create a new JavaScript file called `scraping.js`.
4. Import Puppeteer into your JavaScript file using the following code:
```javascript
const puppeteer = require('puppeteer');
```
5. Save the file and run it using Node.js:
```bash
node scraping.js
```
If everything is set up correctly, you should see no errors and Puppeteer should be ready to use.
### Solution
There is no specific answer for this exercise, as it is a setup exercise. The goal is to ensure that Puppeteer is installed and set up correctly in your development environment. If you encounter any errors or issues during the setup process, refer to the Puppeteer documentation or seek help from online resources or forums.
# Understanding the Document Object Model (DOM)
The Document Object Model (DOM) is a programming interface for HTML and XML documents. It represents the structure of a web page as a tree-like structure, with each node in the tree representing an element, attribute, or piece of text.
In the context of web scraping with Puppeteer, understanding the DOM is crucial. It allows us to navigate and interact with web pages, select specific elements, and extract data.
Here are some key concepts related to the DOM:
- Nodes: Nodes are the building blocks of the DOM tree. Each element, attribute, and piece of text in a web page is represented by a node. Nodes can have child nodes and sibling nodes.
- Elements: Elements are a specific type of node that represents HTML or XML tags. They can have attributes, such as `id` or `class`, and can contain child elements and text nodes.
- Attributes: Attributes are key-value pairs that provide additional information about an element. They are defined within the opening tag of an element.
- Text nodes: Text nodes represent the text content within an element. They can be accessed and manipulated using JavaScript.
- Selectors: Selectors are used to target specific elements in the DOM. They allow us to retrieve elements based on their tag name, class, id, or other attributes.
Now that we have a basic understanding of the DOM, let's learn how to select elements in the DOM using Puppeteer. Selecting elements is a fundamental skill in web scraping, as it allows us to target specific data on a web page.
Puppeteer provides several methods for selecting elements in the DOM. Here are some common methods:
- `page.$(selector)`: This method selects the first element that matches the given CSS selector. It returns a `null` value if no element is found.
- `page.$$(selector)`: This method selects all elements that match the given CSS selector. It returns an empty array if no elements are found.
- `element.$(selector)`: This method selects the first child element of the current element that matches the given CSS selector. It returns a `null` value if no element is found.
- `element.$$(selector)`: This method selects all child elements of the current element that match the given CSS selector. It returns an empty array if no elements are found.
When using these methods, it's important to use CSS selectors that accurately target the desired elements. CSS selectors can be based on element tags, class names, ids, attributes, or other criteria.
## Exercise
Now that we have a basic understanding of the DOM and how to select elements using Puppeteer, let's practice selecting elements.
1. Open your JavaScript file from the previous exercise (e.g., `scraping.js`).
2. Add the following code to select the first heading element on a web page:
```javascript
const heading = await page.$('h1');
```
3. Add the following code to select all paragraph elements on a web page:
```javascript
const paragraphs = await page.$$('p');
```
4. Save the file and run it using Node.js:
```bash
node scraping.js
```
If everything is set up correctly and the web page contains the selected elements, you should see no errors and Puppeteer should retrieve the elements successfully.
### Solution
There is no specific answer for this exercise, as it is a practice exercise. The goal is to practice selecting elements using Puppeteer. If you encounter any errors or issues during the exercise, refer to the Puppeteer documentation or seek help from online resources or forums.
# Automating tasks with Puppeteer
Puppeteer provides a powerful API for automating tasks on web pages. With Puppeteer, you can simulate user interactions, such as clicking on buttons, filling out forms, and submitting data.
Automating tasks with Puppeteer involves the following steps:
1. Selecting elements: Use Puppeteer's element selection methods to target the elements you want to interact with.
2. Performing actions: Use Puppeteer's action methods to simulate user interactions. For example, you can use the `click()` method to click on a button or link, or the `type()` method to enter text into an input field.
3. Waiting for page changes: Use Puppeteer's `waitForNavigation()` method to wait for the page to navigate or reload after performing an action. This ensures that the next action is performed on the updated page.
By combining these steps, you can automate complex tasks on web pages. For example, you can automate the process of logging into a website, filling out a form, and submitting it.
Now that we have a basic understanding of automating tasks with Puppeteer, let's learn how to perform common actions, such as clicking on links and filling out forms.
To click on a link or button using Puppeteer, you can use the `click()` method of the selected element. For example, to click on a link with the id "my-link", you can use the following code:
```javascript
const link = await page.$('#my-link');
await link.click();
```
To fill out a form, you can use the `type()` method of the selected input element. For example, to enter the text "John Doe" into an input field with the id "name-input", you can use the following code:
```javascript
const input = await page.$('#name-input');
await input.type('John Doe');
```
You can also use the `keyboard` object provided by Puppeteer to simulate keyboard events, such as pressing keys or typing special characters. For example, to press the Enter key after filling out a form, you can use the following code:
```javascript
const input = await page.$('#name-input');
await input.type('John Doe');
await page.keyboard.press('Enter');
```
These are just a few examples of how you can automate tasks with Puppeteer. By combining element selection, action methods, and keyboard events, you can automate complex interactions on web pages.
## Exercise
Now that we have a basic understanding of automating tasks with Puppeteer, let's practice performing actions.
1. Open your JavaScript file from the previous exercise (e.g., `scraping.js`).
2. Add the following code to click on a link with the id "my-link":
```javascript
const link = await page.$('#my-link');
await link.click();
```
3. Add the following code to fill out an input field with the id "name-input":
```javascript
const input = await page.$('#name-input');
await input.type('John Doe');
```
4. Save the file and run it using Node.js:
```bash
node scraping.js
```
If everything is set up correctly and the web page contains the selected elements, you should see no errors and Puppeteer should perform the actions successfully.
### Solution
There is no specific answer for this exercise, as it is a practice exercise. The goal is to practice performing actions using Puppeteer. If you encounter any errors or issues during the exercise, refer to the Puppeteer documentation or seek help from online resources or forums.
# Creating a basic robot using Puppeteer and JavaScript
Now that we have covered the basics of web scraping and automating tasks with Puppeteer, let's put our knowledge into practice by creating a basic robot.
A robot, in the context of web scraping and automation, is a program that performs tasks on web pages automatically. It can navigate through websites, extract data, and perform actions without human intervention.
In this section, we will create a basic robot that visits a web page, extracts data, and saves it to a file. Here are the steps we will follow:
1. Launch a browser: Use Puppeteer to launch a headless Chrome or Chromium browser.
2. Navigate to a web page: Use Puppeteer to navigate to a specific URL.
3. Extract data: Use Puppeteer to select elements on the web page and extract their text or attribute values.
4. Save data to a file: Use JavaScript's built-in file system module (`fs`) to save the extracted data to a file.
By the end of this section, you will have a basic understanding of how to create a robot using Puppeteer and JavaScript. You can then build upon this knowledge to create more complex robots for your own projects or research.
Now that we have a basic understanding of creating a robot using Puppeteer and JavaScript, let's create a simple robot that visits a web page, extracts data, and saves it to a file.
First, let's install the `fs` module, which is a built-in module in Node.js that provides file system-related functionality. Open your terminal or command prompt and run the following command:
```bash
npm install fs
```
Once the installation is complete, create a new JavaScript file called `robot.js`. In this file, we will write the code for our robot.
```javascript
const puppeteer = require('puppeteer');
const fs = require('fs');
(async () => {
// Launch a headless browser
const browser = await puppeteer.launch();
const page = await browser.newPage();
// Navigate to a web page
await page.goto('https://www.example.com');
// Extract data
const title = await page.$eval('h1', element => element.textContent);
const paragraph = await page.$eval('p', element => element.textContent);
// Save data to a file
const data = `Title: ${title}\nParagraph: ${paragraph}`;
fs.writeFileSync('data.txt', data);
// Close the browser
await browser.close();
})();
```
In this code, we first import the `puppeteer` and `fs` modules. We then create an async function and use the `launch()` method of Puppeteer to launch a headless browser. We create a new page and navigate to a web page using the `goto()` method.
Next, we use the `$eval()` method to extract the text content of the first `h1` element and the first `p` element on the web page. We save the extracted data to a file called `data.txt` using the `writeFileSync()` method of the `fs` module.
Finally, we close the browser using the `close()` method.
To run the robot, open your terminal or command prompt and run the following command:
```bash
node robot.js
```
If everything is set up correctly and the web page contains the selected elements, you should see no errors and the robot should visit the web page, extract the data, and save it to the `data.txt` file.
## Exercise
Now that we have a basic understanding of creating a robot using Puppeteer and JavaScript, let's create a simple robot that visits a web page, extracts data, and saves it to a file.
1. Open your terminal or command prompt.
2. Run the following command to install the `fs` module:
```bash
npm install fs
```
3. Once the installation is complete, create a new JavaScript file called `robot.js`.
4. Copy the code from the previous example into `robot.js`.
5. Save the file and run it using Node.js:
```bash
node robot.js
```
If everything is set up correctly and the web page contains the selected elements, you should see no errors and the robot should visit the web page, extract the data, and save it to the `data.txt` file.
### Solution
There is no specific answer for this exercise, as it is a practice exercise. The goal is to create a simple robot that visits a web page, extracts data, and saves it to a file. If you encounter any errors or issues during the exercise, refer to the Puppeteer documentation or seek help from online resources or forums.
# Advanced JavaScript concepts for robotics
Here are some advanced JavaScript concepts that are particularly relevant to building and automating robots with Puppeteer:
- Asynchronous programming: Asynchronous programming is a programming paradigm that allows multiple tasks to be executed concurrently. JavaScript provides several mechanisms for asynchronous programming, such as callbacks, promises, and async/await. Understanding these mechanisms is crucial for handling asynchronous tasks in Puppeteer, such as waiting for page navigation or performing actions.
- Error handling: Error handling is an important aspect of building robust robots. JavaScript provides several mechanisms for error handling, such as try/catch blocks and error objects. Understanding how to handle errors effectively will help you build more reliable and resilient robots.
- Modularity and code organization: Modularity and code organization are important principles in software development. JavaScript provides several mechanisms for organizing code, such as modules and classes. Understanding how to structure your code effectively will make your robots more maintainable and easier to work with.
- Debugging: Debugging is an essential skill for building and troubleshooting robots. JavaScript provides several tools and techniques for debugging, such as console.log statements, breakpoints, and the Chrome DevTools. Understanding how to use these tools effectively will help you identify and fix issues in your robots.
By mastering these advanced JavaScript concepts, you will be well-equipped to build and automate complex robots with Puppeteer.
Now that we have covered some advanced JavaScript concepts, let's put them into practice by building a more complex robot.
In this example, we will create a robot that logs into a website, performs a search, and saves the search results to a file. Here are the steps we will follow:
1. Launch a browser: Use Puppeteer to launch a headless Chrome or Chromium browser.
2. Navigate to the login page: Use Puppeteer to navigate to the login page of the website.
3. Fill out the login form: Use Puppeteer to fill out the username and password fields of the login form.
4. Submit the login form: Use Puppeteer to submit the login form.
5. Perform a search: Use Puppeteer to perform a search on the website.
6. Extract the search results: Use Puppeteer to extract the search results from the web page.
7. Save the search results to a file: Use JavaScript's built-in file system module (`fs`) to save the search results to a file.
By the end of this section, you will have a better understanding of how to build more complex robots with Puppeteer and JavaScript.
## Exercise
Now that we have covered some advanced JavaScript concepts, let's put them into practice by building a more complex robot.
1. Open your JavaScript file from the previous exercise (e.g., `robot.js`).
2. Modify the code to perform a search on a website and save the search results to a file.
3. Save the file and run it using Node.js:
```bash
node robot.js
```
If everything is set up correctly and the web page contains the necessary elements, you should see no errors and the robot should perform the search and save the results to a file.
### Solution
There is no specific answer for this exercise, as it is a practice exercise. The goal is to modify the robot to perform a search on a website and save the search results to a file. If you encounter any errors or issues during the exercise, refer to the Puppeteer documentation or seek help from online resources or forums.
# Building more complex robots with Puppeteer
Now that we have covered the basics of building and automating robots with Puppeteer, let's explore how to build more complex robots.
Building more complex robots often involves combining multiple actions, conditions, and loops to achieve a specific goal. For example, you might want to automate the process of scraping data from multiple pages, or perform different actions based on specific conditions.
Puppeteer provides a wide range of methods and features that allow you to build more complex robots. Some of these features include:
- Loops: You can use JavaScript's loop statements, such as `for` loops and `while` loops, to repeat actions multiple times.
- Conditions: You can use JavaScript's conditional statements, such as `if` statements and `switch` statements, to perform different actions based on specific conditions.
- Page navigation: Puppeteer provides methods for navigating through websites, such as `goBack()`, `goForward()`, and `reload()`. These methods allow you to build robots that can navigate through complex web pages and perform actions at different stages.
- Interactions with forms: Puppeteer provides methods for interacting with forms, such as `fill()` and `submit()`. These methods allow you to automate the process of filling out forms and submitting data.
By combining these features and techniques, you can build more complex robots that can perform a wide range of tasks on web pages.
Now that we have covered building more complex robots with Puppeteer, let's put our knowledge into practice by building a robot that scrapes data from multiple pages.
In this example, we will create a robot that visits a website, extracts data from multiple pages, and saves the data to a file. Here are the steps we will follow:
1. Launch a browser: Use Puppeteer to launch a headless Chrome or Chromium browser.
2. Navigate to the first page: Use Puppeteer to navigate to the first page of the website.
3. Extract data: Use Puppeteer to extract data from the current page.
4. Save data to a file: Use JavaScript's built-in file system module (`fs`) to save the extracted data to a file.
5. Navigate to the next page: Use Puppeteer to navigate to the next page of the website.
6. Repeat steps 3-5 until all pages have been visited.
By the end of this section, you will have a better understanding of how to build robots that can scrape data from multiple pages using Puppeteer and JavaScript.
## Exercise
Now that we have covered building more complex robots with Puppeteer, let's put our knowledge into practice by building a robot that scrapes data from multiple pages.
1. Open your JavaScript file from the previous exercise (e.g., `robot.js`).
2. Modify the code to scrape data from multiple pages and save it to a file.
3. Save the file and run it using Node.js:
```bash
node robot.js
```
If everything is set up correctly and the web pages contain the necessary elements, you should see no errors and the robot should scrape the data from all pages and save it to a file.
### Solution
There is no specific answer for this exercise, as it is a practice exercise. The goal is to modify the robot to scrape data from multiple pages and save it to a file. If you encounter any errors or issues during the exercise, refer to the Puppeteer documentation or seek help from online resources or forums.
# Troubleshooting and debugging in Puppeteer
Here are some tips for troubleshooting and debugging in Puppeteer:
- Use console.log statements: Insert console.log statements at key points in your code to print out values and debug information. This can help you understand the flow of your code and identify any issues.
- Check for errors: Keep an eye out for any error messages or exceptions that are thrown in the console. These can provide valuable information about what went wrong and where the issue occurred.
- Use the Chrome DevTools: Puppeteer provides an interface to the Chrome DevTools, which is a set of web development tools built into the Chrome browser. You can use the DevTools to inspect the DOM, debug JavaScript code, and analyze network traffic. To enable the DevTools, launch Puppeteer with the `devtools: true` option.
- Take screenshots: Use Puppeteer's `screenshot()` method to capture screenshots of web pages at different stages of your robot's execution. This can help you visualize any issues and compare the expected output with the actual output.
- Handle errors gracefully: Use try/catch blocks to catch and handle any errors that occur during the execution of your robot. This can help prevent your robot from crashing and allow it to recover from errors gracefully.
By following these tips and techniques, you will be better equipped to troubleshoot and debug issues in your robots and build more reliable and robust automation scripts.
Now that we have covered troubleshooting and debugging in Puppeteer, let's put our knowledge into practice by troubleshooting and fixing issues in a robot.
In this example, we will create a robot that visits a website, performs a search, and saves the search results to a file. However, there are some issues with the robot's code that need to be fixed. Here are the steps we will follow:
1. Identify the issues: Read through the robot's code and identify any issues or errors.
2. Fix the issues: Modify the code to fix the identified issues and ensure that the robot works as expected.
3. Test the robot: Save the modified code, run the robot, and verify that it performs the search and saves the results to a file correctly.
By the end of this section, you will have a better understanding of how to troubleshoot and fix issues in robots built with Puppeteer and JavaScript.
## Exercise
Now that we have covered troubleshooting and debugging in Puppeteer, let's put our knowledge into practice by troubleshooting and fixing issues in a robot.
1. Open your JavaScript file from the previous exercise (e.g., `robot.js`).
2. Read through the code and identify any issues or errors.
3. Modify the code to fix the identified issues and ensure that the robot works as expected.
4. Save the file and run it using Node.js:
```bash
node robot.js
```
If everything is set up correctly and the web page contains the necessary elements, you should see no errors and the robot should perform the search and save the results to a file correctly.
### Solution
There is no specific answer for this exercise, as it is a practice exercise. The goal is to identify and fix any issues or errors in the robot's code. If you encounter any errors or issues during the exercise, refer to the Puppeteer documentation or seek help from online resources or forums.
# Integrating external APIs with Puppeteer
Here are some common use cases for integrating external APIs with Puppeteer:
- Authentication: You can use external APIs to authenticate your robots with services or websites. This can be useful for accessing restricted data or performing actions on behalf of a user.
- Data retrieval: You can use external APIs to retrieve data from third-party services or databases. This can be useful for enriching the data you scrape from websites or accessing additional information.
- Data submission: You can use external APIs to submit data to third-party services or databases. This can be useful for automating data entry or performing actions on external systems.
- Notifications: You can use external APIs to send notifications or alerts based on specific conditions or events in your robots. This can be useful for monitoring the status of your robots or notifying users of important updates.
By integrating external APIs with Puppeteer, you can extend the capabilities of your robots and automate more complex tasks.
Now that we have covered integrating external APIs with Puppeteer, let's put our knowledge into practice by integrating an external API into a robot.
In this example, we will create a robot that logs into a website using external authentication and performs a search. Here are the steps we will follow:
1. Authenticate with the external API: Use Puppeteer to authenticate with an external API and obtain an access token or authentication credentials.
2. Log into the website: Use Puppeteer to navigate to the login page of the website and fill out the login form with the authentication credentials obtained from the external API.
3. Submit the login form: Use Puppeteer to submit the login form and log into the website.
4. Perform a search: Use Puppeteer to perform a search on the website.
By the end of this section, you will have a better understanding of how to integrate external APIs into your Puppeteer robots.
## Exercise
Now that we have covered integrating external APIs with Puppeteer, let's put our knowledge into practice by integrating an external API into a robot.
1. Open your JavaScript file from the previous exercise (e.g., `robot.js`).
2. Modify the code to authenticate with an external API and log into the website using the obtained credentials.
3. Save the file and run it using Node.js:
```bash
node robot.js
```
If everything is set up correctly and the web page contains the necessary elements, you should see no errors and the robot should authenticate with the external API and log into the website correctly.
### Solution
There is no specific answer for this exercise, as it is a practice exercise. The goal is to modify the robot to authenticate with an external API and log into the website using the obtained credentials. If you encounter any errors or issues during the exercise, refer to the Puppeteer documentation or seek help from online resources or forums.
# Exploring the potential of robotics and automation in
# Future developments and advancements in robotics and automation
1. Artificial Intelligence and Machine Learning: One of the key areas of future development in robotics and automation is the integration of artificial intelligence (AI) and machine learning (ML) technologies. AI and ML algorithms can enable robots to learn from their environment, make decisions, and adapt to changing circumstances. This can greatly enhance the capabilities of robots and make them more autonomous and intelligent.
2. Collaborative Robots: Collaborative robots, also known as cobots, are designed to work alongside humans in a collaborative manner. These robots are equipped with sensors and advanced safety features that allow them to safely interact with humans. Cobots can be used in a wide range of industries, such as manufacturing, healthcare, and logistics, to assist humans in various tasks.
3. Swarm Robotics: Swarm robotics is a field that focuses on the coordination and cooperation of large groups of robots to achieve a common goal. In swarm robotics, individual robots work together as a collective to perform tasks that would be difficult or impossible for a single robot to accomplish. Swarm robotics has applications in areas such as search and rescue, environmental monitoring, and agriculture.
4. Soft Robotics: Soft robotics is an emerging field that involves the development of robots with soft and flexible materials, such as silicone or rubber. These robots are inspired by biological systems and can mimic the movements and behaviors of living organisms. Soft robots have the potential to be more adaptable and versatile than traditional rigid robots, and can be used in applications where delicate interactions with the environment are required.
5. Autonomous Vehicles: The development of autonomous vehicles, such as self-driving cars and drones, is another area of future advancement in robotics and automation. Autonomous vehicles have the potential to revolutionize transportation and logistics by improving safety, efficiency, and convenience. These vehicles rely on advanced sensors, AI algorithms, and real-time data to navigate and make decisions.
6. Humanoid Robots: Humanoid robots are robots that resemble humans in appearance and behavior. While humanoid robots are still in the early stages of development, they hold great potential in areas such as healthcare, education, and entertainment. Humanoid robots can interact with humans in a more natural and intuitive way, and can perform tasks that require human-like dexterity and mobility.
7. Ethical and Social Considerations: As robotics and automation technologies become more advanced and pervasive, there are important ethical and social considerations that need to be addressed. These include issues such as privacy, job displacement, and the impact on society. It is crucial to have discussions and regulations in place to ensure that robotics and automation are used in a responsible and beneficial manner.
These are just a few examples of the future developments and advancements that we can expect to see in the field of robotics and automation. As technology continues to advance, the possibilities are endless. It is an exciting time to be involved in this field, and the future holds great promise for the continued growth and innovation in robotics and automation.
``` | Textbooks |
LTWork
Contrast microevolution and macroevolution.
Answers #1
Microevolution , as the name suggests, is evolutionary change on a small scale, such as evolution or selection occurring on a single gene or a few genes in a single population over a short period of time. ... Macroevolution, in contrast, is evolutionary change on a large scale that happens over a longer period of time.
answer from amykookie24
Microevolution can lead to macroevolution.
Microevolution refers to the formation of species through the process of speciation. The element responsible for this process is natural selection. In this process changes in the traits occur with time. Macroevolution refers to the development of major group of organisms from groups of species that are distinctly different. For example development of mammals from non-mammalian species or evolution of whales from terrestrial mammals.
Macroevolution refers to the large number of changes that occur in the characteristics of the living organisms, like the evolution of entirely new species from previously existing species. Macroevolution is the result of many microevolution processes.
Therefore, microevolution can lead to macroevolution is the true statement.
answer from tanyiawilliams7490
answer from aaayymm
Micro evolution can lead to macro evolution
answer from romeojose2005
It's the first one I think
answer from mirandahamamaox89fl
Wikipedia aswell can be used to solve the following questions
b is dominant over the allele for the white fur.
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Gente Manufacturing pays Amy Davis a $1570 monthly salary plus a % commission on merchandise she sells each month. Assume Ami's sales were $64 400 for last month Calculate the following amounts: Amount of commission: Gross Pay:...
Determine the mean/ average of 5, 7, 10, 9,0
determine the mean/ average of 5, 7, 10, 9,0...
It is graphing linear inequalities the inequality is: y< -3x-8would the shade be up or down? dotted line or solid line?
It is graphing linear inequalities the inequality is: y< -3x-8 would the shade be up or down? dotted line or solid line?...
When a process has a capability index of 1, it is said to be able to meet the specification limits in 99.74% of the time. If a process
If the tank is designed to withstand a pressure of 5 MPaMPa, determine the required minimum wall thickness to the nearest millimeter
Chi bộ làng Hựu Thạnh ra đời tại đâu? Có bao nhiêu đảng viên?a. Có 6 đảng viên, vườn nhà Hương quản
Is democracy thriving?
Which behavior would best describe someone who has good communication skills with customers? O a) Following up with some customers
I really need help with this math question ASAP
Put these art movements in the order in which developed from earliest to most recent
Paga No. Date: 6. When a discount of 25% is allowed on the marked price then the selling price will be 1425 .lf the marked price is
What vertex does the quadrilateral and the pentagon share?O (-3, -5)O (-3, 2) O (3, -5)O (3, 2)
An auto-parts store offers a fuel additive that claims to increase a vehicle's gas mileage. The additive is poured into a vehicle's
Inside the steering box, the turning motion of the steering column is translated into a lateral motion, which is passed on to the
Center (5,-3) radius 1
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'Phải chăng lắng nghe người khác là đánh mất cơ hội thể hiện bản thân?'
The Soviet Union lost the Cold War because:A. they lost the wars in Korea and Vietnam.B. They couldn't keep up with U. S. spending
Question 5 plz show steps
Please help, i really don't understand this.*At Sarah's cafe, customers can choose one of the following entrees: burrito, pasta,
Câu nói : "Chủ nghĩa xã hội khoa học tức là chủ nghĩa Mác" là của ai?
Need help please with these
Which lines in this excerpt from act V of Romeo and Juliet show that the Capulet-Montague feud has brought tragedy not only to the
Less Time Homework
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Exploring Quantum Matter
International PhD Programme of Excellence
ExQM Media Library
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Save the Dates:
Fri Feb 13th, 2021 - 11.00 am
ExQM Seminar via Zoom by Maximilian Buser on `Probing Quantum Phases and the Hall Response in Bosonic Flux Ladders´
On Fri Feb 13th, 2021 – 11.00 am via zoom
Maximilian Buser will give an ExQM talk on
`Probing Quantum Phases and the Hall Response in Bosonic Flux Ladders'
The focus of this talk is on bosonic flux ladders. First, we touch on a model which is envisioned to be realized in a future quantum gas experiment exploiting the internal states of potassium atoms as a synthetic dimension. Considering specifics of the future experiment, we map out the ground-state phase diagram and report on Meissner and biased-ladder phases. We show that quantum quenches of suitably chosen initial states can be used to probe the equilibrium properties of the dominant ground-state phases.
Second, we concentrate on the Hall response. While flux ladders are the most simple lattice models giving rise to the Hall effect, the theoretical description of the ground-state Hall response in these systems remains a tricky problem and an active line of research. We discuss feasible schemes to extend measurements of the Hall polarization to a study of the Hall voltage, allowing for direct comparison with solid state systems. Most importantly, we report on characteristic zero crossings and a remarkable robustness of the Hall voltage with respect to interaction strengths, particle fillings, and ladder geometries, which is unobservable in the Hall polarization.
Feb. 1-5th, 2021
International Conference: Quantum Information Processing (QIP)
The 24th QIP is hosted and chaired by Profs. Michael Wolf and Robert König.
QIP Programme
QIP Conference Page incl. Registration
Fri Jan 15th, 2021 - 11.00 am
ExQM Seminar via Zoom by Till Klostermann on `Fast, Long Distance Optical Transport of Cs´
On Fri Jan 15th, 2021 – 11.00 am via zoom
Till Klostermann will give an ExQM talk on
`Fast, Long Distance Optical Transport of Cs'
As experimental setups for quantum gas experiments become more complicated, access around the experimental chamber for lasers, microscopes or magnetic field coils becomes a limiting factor. One way to address this issue is optical transport, to spacially separate the pre-cooling stages from the place of the final experiment.
I present our implementation of such an optical transport scheme, using a running wave optical lattice. Using a Bessel beam and a double pass AOM setup to control the laser frequencies, we manage to transport our atoms over a distance of 43 cm in less than 26 ms. We measured the dependence of the transport efficiency on laser power and acceleration and find that it is predominantly limited by the trap depth. After transport we produce a pure Bose-Einstein condensate of 20.000 atoms.
Fri Dec 11th, 2020 - 10.30 am
ExQM Seminar via Zoom by Max Bramberger on `Dynamical Mean Field Theory with Matrix Product States´
On Fri Dec 11th, 2020 – 10.30 am via zoom
Max Bramberger (group Prof. Schollwöck) will give his inaugural ExQM talk on
`Dynamical Mean Field Theory with Matrix Product States'
When electrons become strongly correlated there is no straightforward way of treating macroscopic systems, as the local interaction between electrons competes with the non-local band structure effects.
In the last decades Dynamical Mean Field Theory (DMFT) has proven to be an appropriate method to treat both these effects. It approximates the problem by embedding an interacting impurity in a bath of non-interacting fermions.
The first part of this talk aims at explaining DMFT from a theoretical
point of view, as well as at introducing the algorithmic implementation using Matrix Product States (MPS) as an impurity solver. The second part gives an introduction to a recent application of the method to BaOsO3, a transition metal oxide in which Hund's coupling and spin-orbit coupling as well as the van Hove singularity play a fundamental role.
Fri Dec 4th, 2020 - 10.30 am
ExQM Seminar via Zoom by Adomas Baliuka on `Post-Processing for Discrete Variable Quantum Key Distribution´
On Fri Dec 4th, 2020 – 10.30 am via zoom
Adomas Baliuka (group Prof. Weinfurter) will give his inaugural ExQM talk on
`Post-Processing for Discrete Variable Quantum Key Distribution'
Two communicating parties can use quantum key distribution (QKD) to establish a shared secret key: First, they exchange quantum signals and measure them. Second, they perform post-processing, during which the secret key is extracted from the measurement results by making use of classical communication.
Key tasks of post-processing are authentication of the communication, error reconciliation and privacy amplification.
One solution to the error reconciliation problem is distributed source coding using low-density parity-check codes. Their decoding via iterative message passing algorithms is an example of inference on probabilistic graphical models. In our work, we further develop such code constructions and decoding algorithms suited for application-oriented QKD systems.
Fri Nov 27th, 2020 - 10.30 am
ExQM Seminar via Zoom by Dr Frederik vom Ende on `Reachability in Controlled Markovian Quantum Systems: An Operator-Theoretic Approach´
On Fri Nov 27th, 2020 – 10.30 am via zoom
Dr Frederik vom Ende will give an extended version of his PhD-defense talk on
`Reachability in Controlled Markovian Quantum Systems: An Operator-Theoretic Approach'
In quantum systems theory one of the fundamental problems boils down to: Given an initial state, which final states can be reached by the dynamic system in question? While for closed systems this is a well-studied question with a number of structural results, as soon as the system interacts with its environment this problem becomes pretty difficult to analyze.
In this talk we will settle on the intermediate case of switchable Markovian coupling of the system to the environment (e.g., a thermal bath of arbitrary temperature), a scenario relevant to certain physical setups. Assuming full unitary control of the uncoupled system, we prove the following three results:
1. Local switchable coupling of an arbitrary system of qudits to a temperature-zero bath allows to (approximately) generate every quantum state from every initial state.
2. For non-zero temperatures we give a non-trivial upper bound as a consequence of new results on d-majorization.
3. For infinite-dimensional systems, if the switchable noise term is generated by a single compact normal operator—which is a special case of a unital system—this allows to approximately reach any target state majorized by the initial state, as up to now only has been known in finite-dimensional analogues.
ExQM Distinguished-Guest Lecture via Zoom by Prof. Dariusz Chruscinski on `On the Universal Constraints for Relaxation Rates for Quantum Dynamical Semigroups´
Prof. Dariusz Chruściński will give a distinguished-speaker ExQM lecture on
`On the Universal Constraints for Relaxation Rates for Quantum Dynamical Semigroups'
A conjecture for the universal constraints for relaxation rates of a quantum dynamical semigroup is proposed. It is shown that it holds for several interesting classes of semigroups, e.g., unital semigroups and semigroups derived in the weak coupling limit from the proper microscopic model.
Moreover, the conjecture proposed is supported by numerical analysis. This conjecture has further interesting implications: it allows to provide universal constraints for spectra of quantum channels and it provides a necessary condition to decide whether a given channel is consistent with Markovian evolution.
ExQM Seminar via Zoom by Emanuel Malvetti on `Optimal Cooling of Markovian Quantum Systems with Unitary Control´
Emanuel Malvetti will give his inaugural ExQM seminar on
`Optimal Cooling of Markovian Quantum Systems with Unitary Control'
We consider quantum systems described by Lindblad dynamics with unitary control. First we derive a master equation describing the evolution of the spectrum of the state, and give some polyhedral bounds on the achievable derivatives, leading to speed limits for the eigenvalues of the state. We characterize the Lindblad operators of systems that can always be asymptotically cooled to a pure state using unitary control, and systems for which any state can be reached from a pure state.
Then we show that the set of achievable derivatives of the spectrum has a subset of optimal derivatives for cooling, and we explore cooling strategies for some selected systems.
In particular we give an optimal cooling scheme for a system consisting of two spins, one of which decays at a fixed rate, and we present a cooling scheme for truncated spin-boson systems of arbitrary size.
Fri July 17th, 2020 - 10.30 am
ExQM Seminar via Zoom by Bo Wang on `Continuous Quantum Light from a Dark Atom´
On Fri July 17th, 2020 at 10.30 am via zoom
Bo Wang will give an ExQM seminar on
`Continuous Quantum Light from a Dark Atom'
Single photons can be generated from a single atom strongly coupled to a optical cavity via a stimulated Raman adiabatic passage between two atomic ground states [1]. During the generation of the photon, the atom stays within the dark state of electromagnetically induced transparency (EIT) avoiding spontaneous decay from the excited state.
In contrast to this well-know scenario, here we present the result to generate quantum light continuously from an atom in the dark state. A coherent coupling is added between the atomic ground states to allow the coherent generation of multiple photons. This would usually result in the destruction of the dark state and the reappearance of spontaneous decay.
However, the dark states of the strongly coupled cavity EIT result from the interference between two atomic ground states entangled with different photonic states [2]. Such dark states are preserved from the local coupling that is applied only within the atomic Hilbert space. Additionally, the nonlinearity of the system allows us to control the quantum fluctuations of the generated light via a quantum Zeno effect.
[1] Kuhn, A. et al., Phys. Rev. Lett. 89, 067901 (2002)
[2] Souza, J.A. et al., Phys. Rev. Lett. 111, 113602 (2013).
ExQM Seminar via Zoom by Dr Bálint Koczor (University of Oxford) on `Measurement Cost of Metric-Aware Variational Quantum Algorithms´
Dr Bálint Koczor will give an ExQM seminar on
`Measurement Cost of Metric-Aware Variational Quantum Algorithms'
Variational quantum algorithms are promising tools for near-term quantum computers as their shallow circuits are robust to experimental imperfections. Their practical applicability, however, strongly depends on how many times their circuits need to be executed for sufficiently reducing shot-noise.
In my talk I will introduce metric-aware quantum algorithms which are variational algorithms that use a quantum computer to efficiently estimate both a matrix and a vector object. I will discuss in detail the recently introduced quantum natural gradient approach which uses the quantum Fisher information matrix as a metric tensor to correct the gradient vector for the co-dependence of the circuit parameters.
I will finally present our rigorous characterisation of the number of measurements required to determine an iteration step to a fixed precision, and propose a general approach for optimally distributing samples between matrix and vector entries. In particular, we establish that the number of circuit repetitions needed for estimating the quantum Fisher information matrix is asymptotically negligible for an increasing number of iterations and qubits.
This talk is based on the recent preprint arXiv:2005.05172 which is joint work with Barnaby van Straaten.
July 6-8th, 2020
Virtual Conference of Munich Centre for Quantum Science and Technology (MCQST)
MCQST Virtual-Conference Programme
Held via meetanyway due to covid-19.
Fri July 3rd, 2020 - 10.30 am
ExQM Seminar via Zoom by Lukas Knips on `How Random Measurements Can Reveal Entanglement´
On Fri July 3rd, 2020 at 10.30 am via zoom
Lukas Knips will give an ExQM seminar on
`How Random Measurements Can Reveal Entanglement'
In my talk, I'd like to present a method to use measurements in
arbitrary – and possibly even unknown – directions for detecting
entanglement.
Usually, quantum entanglement is revealed via a well aligned, carefully chosen set of measurements. Yet, under a number of experimental conditions, for example in communication within multiparty quantum networks, noise along the channels or fluctuating orientations of reference frames may ruin the quality of the distributed states.
In this talk and the corresponding paper [1] it is shown that even for strong fluctuations one can still gain detailed information about the state and its entanglement using random measurements. Correlations between all or subsets of the measurement outcomes and especially their distributions provide information about the entanglement structure of a state. We analytically derive an entanglement criterion for two-qubit states and provide strong numerical evidence for witnessing genuine multipartite entanglement of three and four qubits. Our methods take the purity of the states into account and are based on only the second moments of measured correlations.
Extended features of this theory are demonstrated experimentally with four photonic qubits. As long as the rate of entanglement generation is sufficiently high compared to the speed of the fluctuations, this method overcomes any type and strength of localized unitary noise.
[1] Knips, L., Dziewior, J., Kłobus, W. et al. Multipartite
entanglement analysis from random correlations. npj Quantum Inf 6, 51 (2020).
Fri June 26th, 2020 - 10.30 am
ExQM Seminar via Zoom by Zoltán Zimborás (Wigner Research Inst. of Theoretical Physics, Budapest) on `Fermionic Superselection Rules and the Concept of Orbital Entanglement and Correlation in Quantum Chemistry´
On Fri June 26th, 2020 at 10.30 am via zoom
Zoltán Zimborás will give an ExQM seminar on
`Fermionic Superselection Rules and the Concept of Orbital Entanglement and Correlation in Quantum Chemistry'
A recent development in quantum chemistry has established the quantum mutual information between orbitals as a major descriptor of electronic structure. This has already facilitated remarkable improvements of numerical methods and may lead to a more comprehensive foundation for chemical bonding theory.
Building on this promising development, our work provides a refined discussion of quantum information theoretical concepts by introducing the physical correlation and its separation into classical and quantum parts as distinctive quantifiers of electronic structure. In particular, we succeed in quantifying the entanglement. Intriguingly, our results for different molecules reveal that the total correlation between orbitals is mainly classical, raising questions about the general significance of entanglement in chemical bonding.
Our work also shows that implementing the fundamental particle number superselection rule, so far not accounted for in quantum chemistry, removes a major part of correlation and entanglement previously seen. In that respect, realizing quantum information processing tasks with molecular systems might be more challenging than anticipated.
Based on joint work including the Schollwöck group, see arXiv:2006.00961 .
Fri June 5th, 2020 - 10.30 am
ExQM Seminar via Zoom by Frederik Bopp on `Towards Singlet-Triplet Qubits in Quantum-Dot Molecules´
On Fri June 5th, 2020 at 10.30 am via zoom
Frederik Bopp will give an ExQM seminar on
`Towards Singlet-Triplet Qubits in Quantum-Dot Molecules'
Coherence, ease of control and scalability lie at the heart of all hardware for distributed quantum information technologies. This is particularly true for spin-photon interfaces based on III-V semiconductor quantum dots (QDs) since they combine properties such as strong interaction with light, robust spin-photon selection rules, nearly pure transform limited emission into the zero-phonon line and ease of integration into opto-electronic devices. However, the comparably short spin coherence times of single electrons and holes in QDs (T2* ~ 10-100ns) [1], could limit their applicability for distributed quantum technologies.
Unlike single electron and hole spins which are sensitive to the fluctuating nuclear spin environment in III-V materials, singlet-triplet (S-T) qubits in pairs of coupled dots – quantum dot molecules (QDMs) – have extended spin coherence times when operated at a sweet spot for which the S-T splitting is independent of electric and magnetic field fluctuations. Such optically addressable S-T spin qubits promise to extend the obtainable T2* times by several orders of magnitude whilst retaining the advantages outlined above. Previously, experiments using Schottky gated samples have provided important insights into orbital structure, exchange couplings, phonon couplings and spin-dephasing [2, 3]. However, these studies have also shown it is very challenging to simultaneously maintain the electric field needed to reach the sweet spot condition whilst simultaneously operating in the required charge stability condition, where the QDM is populated by two spins (electron or hole), one in each of the dots forming the QDM.
We present a different approach where the charge status of the QDM is controlled optically, whilst the coupling between the two spins can be tuned to the sweet spot electrically. To achieve this, an AlGaAs tunneling barrier is inserted immediately adjacent to the QDM layer, allowing for sequential optical control of the charge status via tunneling ionization [2] while the tunnel coupling between the two dots can be electrically controlled via a gate voltage. We will present first studies of the dynamics of the optical charging of QD molecules as well as first results on the electric field dependent coupling control.
[1] L. Huthmacher, R. Stockill, E. Clarke, M. Hugues, C. Le Gall & M. Atatüre, Phys. Rev. B 97, 241413, (2018)
[2] H. J. Krenner, M. Sabathil, E. C. Clark, A. Kress, D. Schuh, M. Bichler, G. Abstreiter & J. J. Finley, Phys. Rev. Lett. 94, 057402 (2005)
[3] K. M. Weiss, J. M. Elzerman, Y. L. Delley, J. Miguel-Sanchez, and A. Imamoğlu, Phys. Rev. Lett. 109, 107401 (2012)
Fri May 29th, 2020 - 10.30 am
ExQM Seminar via Zoom by David Castells Graells on `Tunable Enhanced Atom-Light Interaction Using Atomic Subwavelength Arrays´
On Fri May 29th, 2020 at 10.30 am via zoom
David Castells Graells will give his inaugural ExQM seminar on
`Tunable Enhanced Atom-Light Interaction Using Atomic Subwavelength Arrays'
A central challenge in quantum optics is the realization of controlled efficient interactions between atoms and photons. One promising approach consists on coupling one or more atoms to an optical medium such as photonic crystal waveguides [1]. The use of these structures not only improves the free-space approaches, but their tailored dispersion relations offer prospects of new paradigms for atom-light interactions. Imperfections and optical losses inside the medium can, however, hinder the observation and use of some its features.
In this project we investigate as an alternative subwavelength arrays of atoms, which are known to contain collective states with suppressed – compared to single emitters – emission to free space [2]. These states can be understood as guided modes of the atomic chain in the 1D case. To describe the dynamics of the system, we use a quantization scheme based on the classical electromagnetic Green's tensor, and the master equation that results of tracing out the electromagnetic modes.
We, then, engineer the "impurity" atoms that interact with the subwavelength array to achieve an efficient coupling to the subradiant states only. In the Markovian regime, we obtain effective expressions for the dynamics of the impurity atoms, which show many of the interesting features predicted with photonic crystal waveguides.
[1] D. E. Chang, et al., Rev. Mod. Phys. 90.3 (2018): 031002
[2] A. Asenjo-Garcia, et al., Phys Rev. X 7.3 (2017): 031024
ExQM Seminar via Zoom by Frederik vom Ende on `The Role of Strict Positivity in Quantum Dynamics´
Frederik vom Ende will give an ExQM seminar on
`The Role of Strict Positivity in Quantum Dynamics'
Motivated by quantum thermodynamics we investigate the notion of strict positivity, that is, linear maps which map positive definite states to something positive definite again.
We show that strict positivity is decided by the action on any full-rank state, and that the image of not-strictly positive channels—up to something unitary—lives inside a lower-dimensional block. This implies that such channels have maximal distance from the identity channel.
We use this to conclude that Markovian dynamics are strictly positive and investigate connections between strict positivity and other notions of divisibility.
Zoom seminar on Fri May 8th, 2020 - 10.30 am
ExQM Seminar by Julian Roos on `Markovian Regimes in Quantum Many-Body Systems´
On Fri May 8th, 2020 at 10.30 am via zoom
Julian Roos will give an ExQM seminar on
`Markovian Regimes in Quantum Many-Body Systems'
Long time evolution of the full state of quantum many body systems is generally out of reach due to build-up of entanglement. However, the computation of local observables only requires knowledge of the state of (small) subsystems. Is it possible to obtain a description of the reduced dynamics similarly to what is done in the fields of Quantum Optics and Open Quantum Systems (OQS)?
We expect that such an endeavour is most promising in the simplest case, i.e. when the dynamics are Markovian (memoryless), and we thus study if such regimes do also exist in a many body setup. Here, the conditions that allow for the derivation of a Markovian master equation in the theory of OQS (Born-Markov) are not satisfied.
In this talk I thus adopt a Quantum Information Theory perspective to identify interesting Markovian regimes (of a spin coupled to a XY spin chain) and explain the underlying physics responsible for Markovian/non-Markovian dynamics.
Due to the Corona Crisis, ExQM Friday gatherings are currently replaced by zoom online meetings.
You may wish to install www.zoom.us for preparation. Stay healthy!
No seminar on Fri Mar. 20th, 2020 - 10.30 am
Next ExQM Seminar by Julian Roos on `Markovian Regimes in Quantum Many-Body Systems´
On Fri Mar. 20th, 2020 in the Mathematics Building, 3rd floor Seminar Room 03.10.011 (Wolf group) at 10.30 am [note room shift due to MPQ Corona-virus policy!]
Julian Roos would have given an ExQM seminar on
Tue Mar. 10th, 2020 - 1.15 pm
ExQM Seminar by Emanuel Malvetti on `Quantum Circuits for Sparse Isometries´
On Tue Mar. 10th, 2020 in Chemistry Dept. (6th level in yellow section) Lecture Room CH 63.214 at 1.15 pm
Emanuel Malvetti (ETH Zurich, Renner group) will give an ExQM seminar on
`Quantum Circuits for Sparse Isometries'
We consider the task of breaking down a quantum computation given as an isometry into C-Nots and single-qubit gates, while keeping the number of C-Not gates small. Although several decompositions are known for general isometries, here we focus on a method based on Householder reflections that adapts well in the case of sparse isometries.
We show how to use this method to decompose an arbitrary isometry before illustrating that the method can lead to significant improvements in the case of sparse isometries. We also discuss the classical complexity of this method and illustrate its effectiveness in the case of sparse state preparation by applying it to randomly chosen sparse states.
Fri Mar. 6th, 2020 - 10.30 am
ExQM Seminar by Nicola Pancotti on `Quantum East Model: Localization, Non-Thermal Eigenstates and Slow Dynamics´
On Fri Mar. 6th, 2020 in the Mathematics Building, 3rd floor Seminar Room 03.10.011 (Wolf group) at 10.30 am [note room shift due to MPQ Corona-virus policy!]
Nicola Pancotti will give an ExQM seminar on
`Quantum East Model: Localization, Non-Thermal Eigenstates
and Slow Dynamics'
We study in detail the properties of the quantum East model, an interacting quantum spin chain inspired by simple kinetically constrained models of classical glasses.
Through a combination of analytics, exact diagonalization and tensor network methods we show the existence of a fast-to-slow transition throughout the spectrum that follows from a localization transition in the ground state.
On the slow side, we explicitly construct a large (exponential in size) number of non-thermal states which become exact finite-energy-density eigenstates in the large size limit, as expected for a true phase transition.
A "super-spin" generalization allows us to find a further large class of area-law states proved to display very slow relaxation.
Under slow conditions, many eigenstates have a large overlap with product states and can be approximated well by matrix product states at arbitrary energy densities.
We discuss implications of our results for slow thermalization and non-ergodicity more generally for quantum East-type Hamiltonians and their extension in two or higher dimensions.
Wed Feb. 19th, 2020 - 10.30 am
ExQM Seminar by Nicolas Augier (CNRS, Paris) on `Results for the Ensemble Controllability of Quantum Systems´
On Wed Feb. 19th in Chemistry Dept. (6th level in yellow section) Lecture Room CH 63.214 at 10.30 am
Nicolas Augier (CNRS, Paris) will give a special ExQM seminar on
`Results for the Ensemble Controllability of Quantum Systems´
The principal issue that will be developed in this talk is how to control a parameter-dependent family of quantum systems with a common control input, that is, the ensemble controllability problem. Thanks to the study one-parametric families of Hamiltonians and their generic singularities when the system is driven by two real inputs, we will give an explicit adiabatic control strategy for the ensemble controllability problem when geometric conditions on the spectrum of the Hamiltonian are satisfied, in particular, the existence of conical or semi-conical intersections of eigenvalues.
Then, in order to understand which controllability properties can be extended to the case where the system is driven by a single real input, we will study the compatibility of the adiabatic approximation with the rotating wave approximation.
Wed Feb. 5th, 2020 - 2.00 pm
ExQM Seminar by Prof. Ugo Boscain (CNRS, Paris) on `Ensemble Control of Spin Systems´
On Wed Feb. 5th in MPQ Lecture Hall at 2.00 pm
Prof. Ugo Boscain (CNRS, Paris) will give a special ExQM seminar on
`Ensemble Control of Spin Systems´
In this talk I will discuss how to control an ensemble of spin systems (depending on one parameter) using two controls. The technique is based on the adiabatic approximation and the presence of conical eigenvalue intersections. I will then discuss the compatibility of the rotating wave approximation (RWA) and of the adiabatic theory to realize this technique with only one control.
Thu Jan. 16th, 2020 - 6.30 pm
Show-Case Lecture by Prof. Bloch at 'CAS' Centre for Advanced Studies of LMU Munich
On Thu Jan. 16th in the LMU Centre for Advanced Studies (CAS), Seestr. 13, 80802 Munich at 6.30 pm
Prof. Immanuel Bloch will be giving a show-case lecture on the state-of-the-art of quantum simulation in optical lattices. Registration is recommended under [email protected] .
Fri Jan. 10th, 2020 - 10.30 am
ExQM Seminar by Qiming Chen on `Quantum Fourier Transform in Oscillating Modes´
On Fri Jan. 10th in MPQ Lecture Hall at 10.30 am
Qiming Chen will give his inaugural ExQM seminar on
`Quantum Fourier Transform in Oscillating Modes´
Quantum Fourier transform (QFT) is a key ingredient for many quantum algorithms. In typical applications such as phase estimation, a considerable number of ancilla qubits and gates are used to form a Hilbert space large enough for high-precision results. Qubit recycling reduces the number of ancilla qubits to one, but it is only applicable to semi-classical QFT and requires repeated measurements and feedforward within the coherence time of the qubits.
In this work, we explore a novel approach that uses two ancilla resonators to form a large dimensional Hilbert space for the realization of QFT. By employing the perfect state-transfer method, we map an unknown multi-qubit state to one resonator, and generate the QFT state in the second oscillator through cross-Kerr interaction and projective measurement. Quantitative analyses show that our method enables relatively high-dimensional and fully-quantum QFT in the state-of-the-art superconducting quantum circuits, which paves the way for implementing various QFT related quantum algorithms in the near future.
Mon Dec. 9th, 2019 - 2.30 till 9.00 pm
ENB Graduation Ceremony at 'Alte Kongresshalle' in Munich
Alte Kongresshalle Munich
Moritz August, Anna-Lena Hashagen, Bálint Koczor, Lukas Knips, David Leiner, Stephan Welte, Jakob Wierzbowski receive their honours documents during the ceremony.
Mon Dec. 9th, 2019 - 10.30 am
ExQM Seminar by Dr Bálint Koczor on `Variational-State Quantum Metrology´
On Mon Dec. 9th in MPQ Lecture Hall B0.32 at 10.30 am
Dr Bálint Koczor (now Oxford University) will talk on
`Variational-State Quantum Metrology´
Quantum metrology aims to increase the precision of a measured quantity that is estimated in the presence of statistical errors using entangled quantum states.
We present a novel approach for finding (near) optimal states for metrology in the presence of noise, using variational techniques as a tool for efficiently searching the classically intractable high-dimensional space of quantum states. We comprehensively explore systems consisting of up to 9 qubits and find new highly entangled states that are surprisingly not symmetric under permutations and non-trivially outperform previously known states up to a constant factor 2. We consider a range of environmental noise models; while passive quantum states cannot achieve a fundamentally superior scaling (as established by prior asymptotic results) we do observe a significant absolute quantum advantage.
We finally outline a possible experimental setup for variational quantum metrology which can be implemented in near-term hardware.
This talk is based on a joint work (arXiv:1908.08904) with Suguru Endo, Tyson Jones, Yuichiro Matsuzaki and Simon C. Benjamin.
Fri Nov. 29th, 2019 - 10.30 am
ExQM Seminar by Maximilian Buser on `Ground-State Phases and Quench Dynamics in Interacting Bosonic Flux-Ladders´
On Fri Nov. 29th in MPQ Lecture Hall at 10.30 am
Maximilian Buser will talk on
`'Ground-State Phases and Quench Dynamics in Interacting Bosonic Flux-Ladders´
Flux-ladders constitute the minimal setup enabling a systematic
understanding of the rich physics of interacting particles in strong
magnetic fields and in lattices. These systems were also realized in a
series of experiments and therefore, they have attracted great
interest. An experimental realization with synthetic dimensions and
tunable inter-particle interactions based on ultracold 39K and 41K
atoms is expected to be realized by the quantum gases group at ICFO.
In this work, the ground-state phase diagram of the synthetic
flux-ladder model is mapped out using extensive density-matrix
renormalization-group simulations and putting the emphasis on
parameters which can be realized in the future experiment. The focus
is on accessible observables such as the chiral current and the
leg-population imbalance; typical particle-current patterns and
momentum-distribution functions in various ground-state phases are
exemplified. For a particle filling of one boson per rung, the
Mott-insulating Meissner phase, as well as biased-ladder phases,
existing on top of superfluids and Mott insulators, are identified and
located. Considering a particle filling of one boson per two rungs, we
report on the appearance of a stable vortex-lattice phase.
Furthermore, we demonstrate that quantum quenches from suitably chosen initial states can be used to probe the equilibrium properties in the transient dynamics. Concretely, we consider the instantaneous turning on of hopping matrix elements along the rungs or legs in the synthetic flux-ladder model, with different initial particle distributions.
Specifically, we show that clear signatures of the biased-ladder phase
can be observed in the transient dynamics. Moreover, the behavior of
the chiral current in the transient dynamics is discussed. The results
presented in this work might provide relevant guidelines for future
implementations of flux-ladders in experimental setups exploiting a
synthetic dimension.
Fri Oct 25th, 2019 - 10.30 am
ExQM Inaugural Seminar by Markus Hasenöhrl on `Interaction-Free Channel Discrimination´
On Fri Oct. 25th in MPQ Lecture Hall at 10.30 am
Markus Hasenöhrl (Group Michael Wolf) will talk on
`'Interaction-Free' Channel Discrimination´
In this work, we reinterpret the famous Elitzur-Vaidman bomb-tester experiment as a quantum channel discrimination problem. To this end, we generalize the notion of 'interaction-free' measurement to arbitrary quantum channels via a game theoretic model.
Our main result is a sufficient and necessary criterion for when it is possible or impossible to discriminate quantum channels in an 'interaction-free' manner (i.e. such that the error probability and the 'interaction' probability can be made arbitrarily small). For the case where our condition holds, we devise an explicit protocol with the property that both probabilities approach zero with an increasing number of channel uses. We also show that this protocol is optimal in a certain (asymptotic) sense. Furthermore, our protocol only needs one ancillary qubit and might thus be be implementable in near-term experiments. For the case where our condition does not hold, we prove an inequality that quantifies the trade-off between the error probability and the 'interaction' probability.
Tue Oct 8th, 2019 - 2.30 pm
MCQST Distinguised Guest Lecture by John Preskill on `Quantum Computing in the NISQ Era and Beyond´
On Tue Oct. 8th in MPQ Lecture Hall at 2.30 pm
John Preskill will talk on
`Quantum Computing in the NISQ Era and beyond´
Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future. Quantum computers with 50-100 qubits may be able to perform tasks which surpass the capabilities of today's classical digital computers, but noise in quantum gates will limit the size of quantum circuits that can be executed reliably.
NISQ devices will be useful tools for exploring many-body quantum physics, and may have other useful applications, but the 100-qubit quantum computer will not change the world right away – we should regard it as a significant step toward the more powerful quantum technologies of the future.
Quantum technologists should continue to strive for more accurate quantum gates and, eventually, fully fault-tolerant quantum computing.
Wed Sept. 25th, 2019 - 10.00 am
PhD Defense Seminar by Stephan Welte on `Generation of Optical Cat States Entangled with an Atom´
On Wed. 25th Sept. at 10.00am in MPQ Lecture Hall
Stephan Welte will give his PhD defense talk on
`Generation of Optical Cat States Entangled with an Atom´
Schrödinger's cat is a famous gedanken experiment on the existence of quantum mechanical superposition states of macroscopic objects [1]. Experimental implementations in quantum optics employ the superposition of two coherent states with an opposite phase, so-called cat states. These continuous-variable states can be tuned to vary the degree of macroscopicity and to study decoherence effects.
Our experiment implements a strong interaction of a coherent light pulse with a single trapped Rubidium atom, provided by an optical cavity [2]. We deterministically produce a hybrid entangled state between the atomic spin and the phase of the propagating light pulse. A projective measurement of the atomic spin projects the optical state and prepares it in an optical cat state. We study the non-classical properties of the produced states and demonstrate control over all relevant degrees of freedom, using coherent control of the atomic qubit. In the future, cat states may find applications in fiber-based optical quantum networks.
Joint work with Bastian Hacker, Severin Daiss, Lin Li, Lukas Hartung, Emanuele Distante, and Gerhard Rempe.
[1] Naturwissenschaften 23, 807-812 (1935) and 23, 823-828 (1935) and 23, 844-849 (1935)
[2] Nature Photonics 13, 110 (2019)
IMPRS Summer School in Bad Aibling
NB: will be held in Bad Aibling, see Conference Schedule.
Fri Jul 19th, 2019 - 10:30 am
Planning Seminar for next ExQM Workshop etc.
MPQ, Lecture Hall at 10.30 am.
We will plan activities in 2019/20 including
ExQM Workshop
guest programmes
next seminars
ideas of second ExQM generation
Please all attend.
ExQM Seminar by Margret Heinze on `Universal Uhrig Dynamical Decoupling for Bosonic Systems´
In MPQ Lecture Hall at 10.30 am
Margret Heinze will talk on
`Universal Uhrig Dynamical Decoupling for Bosonic Systems´
We construct efficient deterministic dynamical decoupling schemes protecting continuous variable degrees of freedom from decoherence.Our schemes target decoherence induced by quadratic system-bath interactions with analytic time dependence. We show how to suppress such interactions to ?-th order using only ? pulses. Furthermore, we show to homogenize a 2m-mode bosonic system using only (? + 1)^(2?+1) pulses, yielding – up to ?-th order – an effective evolution described by non-interacting harmonic oscillators with identical frequencies.
The decoupled and homogenized system provides natural decoherence-free subspaces for encoding quantum information. Our schemes only require pulses which are tensor products of single-mode passive Gaussian unitaries and SWAP gates between pairs of modes.
PRL_123 (2019), 010501 (also https://arxiv.org/abs/1810.07117v2)
Conference of Munich Centre for Quantum Science and Technology at Deutsches Museum, Centre for New Technologies (ZNT), Museumsinsel 1
MCQST-Conference-Programme
Held at Deutsches Museum, Centre for New Technologies (ZNT), Museumsinsel 1, see map.
Youngsters' Conference of Munich Centre for Quantum Science and Technology at MPQ
Held at MPQ from 10am through 5pm. See: Conference Programme and Abstracts.
ExQM Seminar by Frederik vom Ende on `Reachability in Infinite-Dimensional Unital Open Quantum Systems with Switchable GKS-Lindblad Generators´
Frederik vom Ende will talk on
`Reachability in Infinite-Dimensional Unital Open Quantum Systems with Switchable GKS-Lindblad Generators´
In quantum systems theory one of the fundamental problems boils down to: given an initial state, which final states can be reached by the dynamic system in question?
Here we consider infinite-dimensional open quantum dynamical systems following a unital Kossakowski-Lindblad master equation extended by controls. More precisely, their time evolution shall be governed by an inevitable (potentially unbounded) Hamiltonian drift term, finitely many bounded control Hamiltonians allowing for (at least) piecewise constant control amplitudes plus a bang-bang switchable noise term in GKS form (generated by some compact V).
Generalizing standard majorization results from finite to infinite dimensions, we show that such bilinear quantum control systems allow to approximately reach any target state majorized by the initial one — as up to now only has been known in finite-dimensional analogues.
ExQM Seminar by Stephan Welte on `Generation of Optical Cat States Entangled with an Atom´
In MPQ Lecture Hall (back again to left of entrance) at 10.30 am
Stephan Welte will talk on
ExQM Seminar by Prof. Michael Keyl
'Unitary Control of Quantum Systems in Finite and Infinite Dimensions'
Prof. Michael Keyl will talk on
'Unitary Control of Quantum Systems in Finite and Infinite Dimensions'
Quantum control theory studies the dynamics of quantum systems which can be manipulated, e.g., by external controls like electro-magnetic fields. One of the central topics in this framework is the decision problem of controllability: can we drive the system from any given initial (pure) state into any final (pure) state, or more generally, from any given initial state to any desired final state with the same spectrum of eigenvalues?
In finite dimensions this question can be completely answered in a Lie-algebraic framework. Infinite dimensions, on the other hand, trigger more challenging mathematics and require methods from operator analysis and (extensions of) infinite dimensional Lie theory.
This talk will provide an introduction into this topic, and an overview on some of its central questions and results. In finite dimensions we show in particular how a system Lie algebra can be associated to a quantum control system, which leads to an easy condition for deciding controllabiliy: the celebrated Lie algebra rank conditon.
The second part of the talk shows under which assumptions the finite dimensional results can be translated more or less directly to infinite dimensions — now using infinite dimensional algebras and groups, and the strong instead of the norm topology. Beyond these assumptions a number of interesting new open questions arise, which we briefly sketch in an outlook.
ExQM Seminar by Dr Claudius Hubig
'Recent Developments in Tensor Networks'
Dr Claudius Hubig will talk on
'Recent Developments in Tensor Networks'
In the first half of the talk, I will report on recent progress made in the description of finite-dimensional quantum systems with non-local interactions using tensor network approaches. Such systems include molecular systems of interest in quantum chemistry as well as effective systems arising as the to-be-solved inner problems of the dynamical mean-field theory or the density matrix embedding theory. In all cases, using loop-free MPS or tree tensor networks, sufficient progress can be made over standard solvers using exact diagonalisation.
In the second half of the talk, we will summarise the relatively novel application of real-time evolution to infinite two-dimensional tensors networks to obtain time-dependent observables. The evolution is applied to the 2D S=1/2 Néel state on the square lattice in a disorder averaged Hamiltonian, where we find hints towards many-body localisation in the spin dynamics as the disorder strength is increased.
Refs.: arXiv:1811.00048, arXiv:1901.05824 and arXiv:1812.03801.
Fri Apr. 26th, 2019 - 10.30 am
Guest Talk by Prof. Rédei (London School of Economics)
'On the Tension between Mathematics and Physics'
Note Location: MPI for Astrophysics, MPA, Lecture Hall E.0.11 at 10.30 am (MPA is next door to MPQ, Karl-Schwarzschild-Str. 1, see room finder here.)
Prof. Rédei will talk on
'On the Tension between Mathematics and Physics'
Because of the complex interdependence of physics and mathematics their relation is not free of tensions. The talk looks at how the tension has been perceived and articulated by some physicists, mathematicians and mathematical physicists.
Some sources of the tension are identified and it is claimed that the tension is both natural and fruitful for both physics and mathematics. An attempt is made to explain why mathematical precision is typically not welcome in physics.
SHIFTED TO: Fri Feb. 8th, 2019 - 10.30 am
Inaugural Talk by Till Klostermann
'Building a New Caesium Quantum Gas Microscope'
MPQ Lecture Hall B0.32 at 10.30 am
Till Klostermann will talk on
'Building a New Caesium Quantum Gas Microscope'
I will present my PhD-thesis project, setting up a new experiment utilizing Caesium for investigating artificial gauge fields. The new experiment will use Raman assisted tunneling in a state-dependent lattice instead of shaking to engineer these gauge fields. A single-site resolution objective will give access to the individual particles position.
Due to Caesium's large and accessible Feshbach resonance, it is a good candidate to investigate interactions in systems influenced by artifical gauge fields. I will also talk about the current status of the setup.
Wed Jan. 16th, 2019 - 1.00 pm
IAS-Talk by Prof. Robert König (TUM, MA5)
'Quantum Advantage of Shallow Circuits'
IAS, Lichtenbergstr. 2a, Wed Jan. 16th at 1.00 pm.
Prof. Robert König will talk on
'Quantum Advantage with Shallow Circuits'
Prof. Robert König (Theory of Complex Quantum Systems) will talk about the advantage of quantum computers as compared to conventional computers.
Quantum computers can perform operations on many values in one fell swoop whereas a single conventional computer typically must execute these operations sequentially. The promise of quantum computing lies in the ability to solve certain problems significantly faster (TUM press release).
Relevant Publication:
S. Bravyi, D. Gosset, R. König, "Quantum advantage with shallow circuits", Science, 19. October 2018. DOI: 10.1126/science.aar3106
Tue Jan. 15th, 2019 - 10.30 am
Talk by Prof. Maurice de Gosson (Univ. Vienna)
'Symplectic Coarse-Grained Dynamics: Chalkboard Motion in Classical and Quantum Mechanics'
MPQ Lecture Hall B0.32, Tue Jan. 15th at 10.30 am.
Prof. Maurice de Gosson (Univ. Vienna) will talk on
'Symplectic Coarse-Grained Dynamics: Chalkboard Motion in
Classical and Quantum Mechanics'
In the usual approaches to mechanics (classical or quantum) the primary object of interest is the Hamiltonian, from which one tries to deduce the solutions of the equations of motion (Hamilton or Schrödinger).
In the present talk, we reverse this paradigm and view the motions themselves as being the primary objects. This is made possible by studying arbitrary phase space motions, not of points, but of ellipsoids with the requirement that the symplectic capacity of these ellipsoids is preserved. This allows us to pilot and control these motions as we like. In the classical case these ellipsoids correspond to a symplectic coarse-graining of phase space, and in the quantum case they correspond to the "quantum blobs" we have defined in previous work, and which can be viewed as minimum uncertainty phase-space cells which are in a one-to-one correspondence with Gaussian pure states.
Thu Dec. 20th, 2018 - 12.30 noon
PhD-Defense Talk by Anna-Lena Hashagen
'Symmetry Methods in Quantum Information Theory'
Mathematics Building MI, Boltzmann Str. 3, Room 00.10.011 at 12.30 noon.
Anna-Lena Hashagen will talk on
'Symmetry Methods in Quantum Information Theory'
Fri Dec. 4th, 2018 - 2.30 till 9.00 pm
ENB Graduation Ceremony at Conference Centre in Augsburg
Conference Centre, Gögginger Str. 10, Augsburg
Michael Lohse and Christian Sames (QCCC) receive their honours documents during the ceremony.
Talk by Bálint Koczor
'On Phase-Space Representations of Spin Systems and their Relations to Infinite-Dimensional Quantum States'
MPQ Seminar Room Cirac Group in Third Floor at 10.30am
Bálint Koczor will talk on
'On Phase-Space Representations of Spin Systems and their Relations to Infinite-Dimensional Quantum States'
Classical phase spaces have been widely applied in physics, engineering, economics or biology. I will give an overview of our recent works considering phase spaces of quantum systems, which have become a powerful tool for describing, analyzing, and tomographically reconstructing quantum states.
We provide a complete phase-space description of (coupled) spin systems including their time evolution, tomography, large-spin approximations and their infinite-dimensional limit, which recovers the well-known case of quantum optics. Finally, Born-Jordan distributions of infinite-dimensional quantum systems are discussed.
For more detail see also arXiv:1808.02697 and arXiv:1811.05872 .
Wed Nov. 14th, 2018 - 3.00 pm
PhD-Defense Talk by Moritz August on
'Tensor Networks and Machine Learning for Approximating and Optimizing Functions in Quantum Physics'
Mathematics Building MI, Boltzmann Str. 3, Room 03.09.012 at 3.00 pm.
Moritz August will talk on
'Tensor Networks and Machine Learning for Approximating and Optimizing Functions in Quantum Physics'
We explore the intersection of computer science and mathematics to address challenging problems in numerical quantum physics. We introduce, analyze and evaluate novel methods for the approximation of physical quantities of interest as well as the optimization of performance criteria in quantum control. These methods are based on techniques from the fields of tensor networks, numerical analysis and machine learning. Furthermore, we present work on the relation between machine learning and tensor network methods for the representation of quantum states.
We introduce a general algorithm which for the first time allows to approximate global functions Trf (A) of matrix product operators A which represent Hermitian matrices of very high dimensionality. Following this, we present an analytical analysis of the partial results computed by the procedure. This analysis leads us to the discovery of a more efficient variant of the algorithm and we subsequently show that it can be applied to a large class of spin Hamiltonians in quantum physics. We finally demonstrate how our method yields a novel strategy to approximate properties of thermal equilibrium states, some of which were so far inaccessible for numerical methods.
In the second part, we present a novel and broadly applicable method for solving quantum control scenarios. The method employs a particular class of recurrent neural networks, the long short-term memory network, to probabilistically model control sequences and optimize these models with tools from supervised and reinforcement learning. In a first version, we use an optimization procedure inspired by evolutionary algorithms to train the networks. We demonstrate in a quantum memory setting that the method can produce better results than certain analytical solutions. We then improve on these results by introducing a different optimization strategy based on insights from reinforcement learning known as policy gradient algorithms. The combination of long short-term memory networks and policy gradient optimization schemes allows us to tackle a wide variety of control problems, which we demonstrate numerically.
Finally, we show results on the relation between tensor networks and a particular class of machine learning models, the restricted Boltzmann machine. We find that restricted Boltzmann machines can be generalized in the tensor network framework and gain insight about their efficiency in representing states of many-body quantum systems.
Fri Nov. 9th, 2018 - 10.30 am
Harvard-Report Talk by Nicola Pancotti on
'Machine Learning and Tensor Networks for Quantum Many Body Physics '
MPQ Lecture Hall B0.32 at 10.30 am.
Nicola Pancotti (back from Harvard) will talk on
'Machine Learning and Tensor Networks for Quantum Many Body Physics'
In this talk, I will give a simple introduction to Machine Learning and Tensor Network techniques for the ground state search problem in Quantum Many Body physics. I will show how one can use Neural Network States as a powerful ansatz for the description of many body quantum spin systems and how to map a sub class of them to some well known Tensor Network families. I will show applications to classical pattern recognition and how to combined those families to existing Machine Learning techniques in order to improve their performances.
Finally I will discuss possible directions to extend these methods to fermionic systems and, in particular, to the framework of Gaussian states.
Fri Oct. 26th, 2018 - 10.30 am
Inaugural Talk by Frederik Bopp on
'Hybrid Photonic-Plasmonic Biosensing'
Frederik Bopp (who has just joined the Finley group as ExQM student) will talk on
'Hybrid Photonic-Plasmonic Biosensing'
Cavity-enhanced optical and plasmonic sensing are two commonly utilised techniques to analyse nanoparticle. Combining them into a hybrid system potentially allows to achieve high finesses and sub diffraction limited mode volumes simultaneously, leading to enhanced detection sensitivities and an increased range of detectable biomolecules. On the long term, these biosensors could form a new set of medical tools for the discovery, the study and the detection of biomarkers. My Master research aims at studying the coupling of an open microcavity to a gold plasmonic nanorod and to establish their potential for single molecule detection.
In my presentation I will provide a theoretical and experimental description of the coupling mechanism, linking these results to the potential sensitivity limit of this system.
Wed July 11th, 2018 - 2 pm
IMPRS Career Talk by the Secretary General of the Humboldt-Foundation Dr Aufderheide on
'Postdoc Opportunities in the Humboldt-Foundation`s Global Network'
MPQ Lecture Hall B0.32 at 2 pm.
Dr. Enno Aufderheide (Secretary General of Humboldt-Foundation) will talk on
'Postdoc Opportunities in the Humboldt-Foundation`s Global Network'
Dr. Enno Aufderheide
On 1 July 2010, Enno Aufderheide became the new Secretary General of the Humboldt Foundation. From 2006 to 2010, he was head of the Research Policy and External Relations Department at the Max Planck Society in Munich where he played a key role in the Society's internationalisation strategy. From December 2008 onwards, he also took on responsibility for managing the Minerva Foundation for the promotion of German-Israeli academic cooperation.
June 14th through June 18th, 2018
14th Workshop on Numerical Ranges and Radii (WONRA) is co-hosted by ExQM featuring Man-Duen Choi, David Gross, Chi-Kwong Li, and Michael Wolf et al.
The 14th International Workshop on Numerical Ranges and Radii (WONRA) is co-hosed by ExQM (and organised by Th. Schulte-Herbrüggen) under the motto
"100 years of the Toeplitz Hausdorff Theorem (1918/19)"
The numerical range, i.e. the set W(A):={<x|Ax> | <x|x> =1} plays a crucial role in spectral theory and, e.g., in the search of ground-state energies (Rayleigh quotient). In 1918/1919, by the celebrated Toeplitz-Hausdorff Theorem, it was shown to form a convex set. Clearly, the numerical range W(A) comprises the spectrum spec(A). In quantum-many-body systems, the important question arises, whether the spectrum of the underlying Hamiltonian is gapped — this decision problem was addressed in a seminal paper by Cubitt, Perez-Garcia, and Wolf, Nature 528, 207 (2015). Michael Wolf will talk on Undecidebility of the Spectral Gap in a special ExQM Lecture on Fri Jun 15th at 3pm in MPQ Lecture Hall B0.32 as one highlight in this conference.
Schedule in Overview,
(detailed programme and book of abstracts under this link):
June 14 (Thursday), MPQ, Lecture Hall B0.32
9:30 to 16:30 talks by Choi, Spitkovsky, Tam, Farenick, Nakazato, Chien, Osaka, Taheri
June 15 (Friday) MPQ, Lecture Hall B0.32
9:30 to 17:30 talks by Życzkowski, Schulte-Herbrüggen, Gross, Psarrakos, Schuch, Wolf, Weis, Huckle
June 16 (Saturday)
Social event and discussion
Afternoon: Visit in Munich downtown museums, e.g., Blue Rider in Lenbachhaus
18:30 optional dinner in a Munich beergarden downtown
June 17 (Sunday), IAS, Lichtenbergstr. 2a, Auditorium on ground floor
10:00 to 16:30 talks by Bebiano, Badea, vom Ende, Diogo, Crouzeix, Sze, Bračic
18:00 to 20:30 conference dinner at IAS faculty club
June 18 (Monday), IAS, Lichtenbergstr. 2a, Auditorium on ground floor
9:45 to 12:00 talks by Kressner, Lau, Li
12:00 to 14:00 Lunch on campus at IPP mensa
Fri May 18th, 2018 - 1:30 pm
Lecture by Shai Machnes on
'Control of Quantum Devices: Merging Pulse Calibration and System Characterization using Optimal Control'
TUM Campus, Walther-Meissner Institute, Walther-Meissner-Straße 8, 2nd floor, Seminar Room 143 (or, if too noisy, 128) at 1.30 pm.
Dr. Shai Machnes (University of Saarbrücken) will talk on
'Control of Quantum Devices: Merging Pulse Calibration and System Characterization using Optimal Control'
The current methodology for designing control pulses for quantum devices circuits often results in a somewhat absurd situation: pulses are designed using simplified models, resulting in initially poor fidelities. The pulses are then calibrated in-situ, achieving high-fidelities, but without a corresponding model. We are therefore left with a model we know is inaccurate, working pulses for which we do not have a matching model, and a calibration process from which we learned nothing about the system.
Here, we propose a novel procedure to rectify the situation, by merging pulse design, calibration and system characterization: Calibration is recast as a closed-loop search for the best-fit model parameters, starting with a detailed, but only partially characterized model of the system. Fit is evaluated by fidelity of a complete set of gates, which are optimized to fit the current system characterization. The end result is a best-fit characterization of the system model, and a full set of high-fidelity gates for that model.
We believe the new approach will greatly improve both gate fidelities and our understanding of the systems they drive.
Fri May 18th, 2018 - 10:30 am
Inaugural Seminar by Bo Wang on
'Strong Coupling between Photons via a Four-Level N-type Atom'
MPQ, Lecture Hall (moved to B0.32) at 10.30 am
Bo Wang (Rempe group) will talk on
'Strong Coupling between Photons via a Four-Level N-type Atom'
Four-level N-type atomic systems have been investigated for effects like the electromagnetically induced absorption (EIA) and cross-phase modulation (XPM) when interacting with classical light fields. Despite the giant non linearity, the interaction strengths are negligible at the level of individual quanta. However with the strong light matter coupling provided by cavity quantum electrodynamics, a significant interaction between single photons can be reached.
Here I will give a brief introduction on our experimental setup and the experiment where the photons of two light fields are strongly coupled via a single four-level N-type atom. The fields drive two modes of an optical cavity, which are strongly coupled to two separate transitions. A control laser drives one transition's ground state to the other transition's excited state, the inner transition of the N-type atom. It induces a tunable coupling between the modes and results in a doubly nonlinear energy-level structure of the photon-photon-atom system. The strong correlation between the light fields is observed via photon-photon blocking and photon-photon tunneling. With this system, nondestructive counting of photons and heralded n-photon sources might be within reach.
Wed May 16th, 2018 - 4:00 pm
Inaugural Lecture by Prof. Stefan Weltge (TUM Maths Dept) on
'A Barrier to P=NP Proofs'
TUM Mathematics Building, Boltzmannstr. 8, Lecture Hall 3 at 4.00 pm.
Prof. Stefan Weltge will give his inaugural lecture on
'A Barrier to P=NP Proofs'
The P-vs-NP problem describes one of the most famous open questions in mathematics and theoretical computer science. The media are reporting regularly about proof attempts, all of them being later shown to contain flaws. Some of these approaches where based on small-size linear programs that were designed to solve problems such as the traveling salesman problem efficiently.
Fortunately, a few years ago, in a breakthrough result researchers were able to show that no such linear programs can exist and hence that all such attempts must fail, answering a 20-year old conjecture.
In this lecture, I would like to present a quite simple approach to obtain such a strong result. Besides an elementary proof, we will hear about (i) the review of all reviews, (ii) why having kids can boost your career, and (iii) a nice interplay of theoretical computer science, geometry, and combinatorics.
May 15th to June 20th, 2018
von-Neumann Lecture Series by Prof. Marius Junge (University of Illinois, USA) on Operator Algebraic Methods in Quantum Information Theory
von-Neumann Lecture Series by Prof. Marius Junge (University of Illinois, USA) held at TUM Mathematics, Boltzmannstr. 3.
'Operator-Algebraic Methods in Quantum Information Theory'
Lecture series from May 15 to June 20.
Tue 16:00 – 18:00 (room 03.06.011)
Wed 16:00 – 18:00 (room 02.10.011)
Location: Boltzmannstr.3, Garching
In this lecture series, we will illuminate some recent connection between operator algebra theory and quantum information theory. The first part of the lecture will focus on estimates of entropy and mutual information, including some recent results on decay to equilibrium. The second part will be concerned with the operator algebraic background for various type of Bell inequalities.
Course web page on this link.
Mon May 7th, 2018 - 4:15 pm
Lecture by Martin Plenio on
'Diamond Quantum Devices: From Quantum Simulation to Medical Imaging'
TUM Campus, Chemistry Building, Lichtenbergstrasse 4, 6th floor (yellow section), Seminar Room CH63.214 at 4.15 pm.
Prof. Martin Plenio (University of Ulm) will talk on
'Diamond Quantum Devices: From Quantum Simulation to Medical Imaging'
Perfect diamond is transparent for visible light but there are famous diamonds, such as the famous Oppenheim Blue or the Pink Panther worth tens of millions of dollar, which have intense colour. An important source of colour in diamond are lattice defects which emit and absorb light at optical frequencies and may indeed possess a non-vanishing ground state electronic spin.
I will explore the physics of one of these defects, the nitrogen vacancy center, and show how we can manipulate its electronic spin and make use of this capability to create quantum simulators, quantum sensors and perhaps surprisingly applications in medical imaging that may, we hope, find applications for example in cancer research and treatment.
Further info on this link.
Fri May 4th, 2018 - 10:30 am
Seminar by Julian Roos on
'Non-Markovianity Measures in the Many-Body Context'
MPQ, Seminar Room B0.41 in Library at 10.30 am.
Julian Roos will talk on
'Non-Markovianity Measures in the Many-Body Context'
The ability to coherently control the dynamics of an ever-increasing number of particles pushed development of quantum technologies during the past decade. In order to achieve scalability, environment-induced decoherence effects need to be identified, understood and minimised such that the required thresholds for error correction are achieved. Also, people now control and modify the environment itself to design noise. All of this triggered renewed interest in fundamental studies of open quantum systems (OQS) amongst which are multiple studies on the existence of two different dynamical regimes: Markovian dynamics, underlying, e.g., the well known Lindblad master equations and non-Markovian dynamics, which are usually associated with recoherence and information backflow from the environment to the system.
I will introduce you to several measures that are widely used in the field to quantify the 'amount' of non-Markovianity that is present in the reduced dynamics of an OQS and provide some examples of their use in the context of time evolution of matrix product states. Here, the OQS consists, e.g., of two spins in the center of a spin chain and thus any system-bath weak-coupling assumptions (used in the derivation of the Lindblad form) are clearly invalid. Still there seem to exist special cases where the underlying dynamics are Markovian.
Fri April 20th, 2018 - 10:30 am
Two-Part Inaugural Seminar by Maximilian Buser on
'Open Quantum Systems with Initial System-Environment Correlations' and 'Quasi-One-Dimensional Systems with Artificial Gauge Fields: Interactions and Finite Temperatures'
Maximilian Buser will talk on two topics
'Open Quantum Systems with Initial System-Environment Correlations'
Open quantum systems exhibiting initial system-environment correlations are notoriously difficult to simulate. — We point out that given a sufficiently long sample of the exact short-time evolution of the open system dynamics, one may employ transfer tensors for the further propagation of the
reduced open system state. This approach is numerically advantageous and allows for the simulation of quantum correlation functions in hardly accessible regimes.
We benchmark this approach against analytically exact solutions and exemplify it with the calculation of emission spectra of multichromophoric systems as well as for the reverse-temperature estimation from simulated spectroscopic data.
'Quasi-One-Dimensional Systems with Artificial Gauge Fields: Interactions and Finite Temperatures'
Artificial, highly tunable gauge (or "magnetic") fields have been successfully implemented in a number of optical lattice experiments with ultracold neutral Bose gases. In this context,
quasi-one-dimensional ladder-like lattices are of significant interest. They are the most simple geometries allowing the exploration of intriguing physical effects related to quantum Hall physics, exotic topological states and superconductivity. While experimental research mainly focused on non-interacting particles, recent results encourage the prospect of future experiments with strongly interacting bosons.
Analytical, mean-field and DMRG-based studies provided extensive theoretical results regarding the ground state properties of such strongly interacting, ladder-like systems. The presence of gauge fields clearly enriches the corresponding phase diagrams. For instance, it gives rise to so-called Meissner and vortex lattice phases as well as to intriguing effects such as chiral current reversals.
The aim of this work (in progress) is to provide theoretical predictions in experimentally much more feasible regimes. Therefore, we plan to investigate the effects of finite temperature states and intend to employ DMRG-based simulation techniques.
April 16th to July 2nd, 2018
von-Neumann Lecture Series by Prof. Daniel Kressner (EPFL Lausanne, CH) on Low-Rank Approximation
von-Neumann Lecture Series in Computer Science by Prof. Daniel Kressner (EPFL Lausanne, CH) held at TUM Computer Science (Host Prof. Huckle), Boltzmannstr. 3.
'Low-Rank Approximation'
Lecture series from April 16th to July 2nd.
Mon 14:00 – 17:00 (room 02.08.020)
Low-rank compression is an ubiquitous tool in scientific computing and data analysis. There have been numerous exciting developments in this area during the last decade and the goal of this course is to give an overview of these developments, covering theory, algorithms, and applications of low-rank matrix and tensor compression.
Specifically, the following topics will be covered:
1. Theory
Low-rank matrix and tensor formats (CP, Tucker, TT, hierarchical Tucker)
A priori approximation results
2. Algorithms
Basic operations with low-rank matrices and tensors
SVD-based compression
Randomized compression
Alternating optimization
Riemannian optimization
Nuclear norm minimization
Adaptive cross approximation and variants
Matrix and tensor completion
Solution of large- and extreme-scale linear algebra problems from various applications (dynamics and control, uncertainty quantification, quantum computing, …)
Tensors in deep learning
Depending on how the course progresses and the interest of the participants, hierarchical low-rank formats (HODLR, HSS, H matrices) may be covered as well.
Hands-on examples using publicly available software (in Matlab, Python, and Julia) will be provided throughout the course.
Double-Feature Seminar by Nicola Pancotti and Moritz August on
'Neural Networks Quantum States, String-Bond States and Chiral Topological States'
MPQ, Lecture Hall (moved to B0.32) at 10.30 am.
Nicola Pancotti and Moritz August will talk on
'Neural Networks Quantum States, String-Bond States and Chiral Topological States'
Neural Networks Quantum States have been recently introduced as an ansatz for describing the wave function of quantum many-body systems. In this talk we will give an overview of recent works on Neural Networks Quantum States taking the form of Boltzmann machines. We will explain the motivation for considering Boltzmann machines in machine learning and explain how they can be used to study quantum systems. We will then focus on the expressive power of this class of states and discuss their relationship to Tensor Networks.
In particular we will show that restricted Boltzmann machines are String-Bond States with a non-local geometry and low bond dimension and explain how it enables us to define generalizations of restricted Boltzmann machines that combine the entanglement structure of tensor networks with the efficiency of Neural Networks Quantum States. We will then provide evidence that these techniques are able to describe chiral topological states both analytically and numerically.
Finally we will discuss how String-Bond States can also be used in traditional machine-learning applications.
based on: I. Glasser, N. Pancotti, M. August, I. Rodriguez, and I. Cirac, Phys. Rev. X. 8, 011006 (2018)
Fri Mar 23rd, 2018 - 10:30 am
Seminar by Stephan Welte on
'Processing of Two Matter Qubits Using Cavity QED'
'Processing of Two Matter Qubits Using Cavity QED'
In a quantum network, optical resonators provide an ideal platform for the creation of interactions between matter qubits. This is achieved by exchange of photons between the resonator-based network nodes, and in this way enables the distribution of quantum states and the generation of remote entanglement [1].
Here we will show how single photons can also be used to generate local entanglement between matter qubits in the same network node [2]. Such entangled states are indispensable as a resource in a plethora of quantum communication protocols.
We will give an overview of the necessary experimental toolbox for an implementation with neutral atoms. Several entanglement protocols showing the generation of all the Bell states for two atoms will be presented. We will also detail how we experimentally exploit the employed method for quantum computation and quantum communication applications.
[1] S. Ritter et al., Nature 484, 195 (2012)
[2] A. Sørensen and K. Mølmer, Phys. Rev. Lett. 90, 127903 (2003)
Fri Mar 16th, 2018 - 10:30 am
Double-Feature Seminar by Anna-Lena Hashagen and Lukas Knips on
'Information-Disturbance Tradeoffs'
Anna-Lena Hashagen and Lukas Knips will give a double-feature on
Information-Disturbance Tradeoffs
In the first part of our double feature, we investigate the tradeoff between the quality of an approximate version of a given measurement and the disturbance it induces in the measured quantum system. We prove that if the target measurement is a non-degenerate von Neumann measurement, then the optimal tradeoff can always be achieved within a two-parameter family of quantum devices that is independent of the chosen distance measures.
This form of almost universal optimality holds under mild assumptions on the distance measures such as convexity and basis-independence, which are satisfied for all the usual cases that are based on norms, transport cost functions, relative entropies, fidelities, etc. for both worst-case and average-case analysis. We analyze the case of the cb-norm (or diamond norm) more generally for which we show dimension-independence of the derived optimal tradeoff for general von Neumann measurements.
A SDP solution is provided for general POVMs and shown to exist for arbitrary convex semialgebraic distance measures.
In the second part, we evaluate the information-disturbance tradeoff experimentally for the observation of a qubit by implementing the full range of possible measurements and determining the measurement error for a given disturbance. The special case of the worst-case total variational distance and the 1-1 norm distance is considered.
The various measurements are realized by a tunable Mach-Zehnder-Interferometer, which supplies the ancillary degrees of freedom necessary to implement arbitrary POVMs and quantum channels for the measurement of a polarization qubit. We demonstrate the tightness of the bound by saturating it with high significance. Furthermore, we show that the optimal procedure outperforms the optimal cloning protocol, not only on a theoretical level, but clearly resolvable in the laboratory.
For more detail, see the preprint arXiv:1802.09893 .
Fri Feb 23rd, 2018 - 10:30 am
Seminar by Michael Fischer on
'Chains of Nonlinear Tunable Superconducting Resonators'
Michael Fischer will talk on
Chains of Nonlinear and Tunable Superconducting Resonators
In this talk I will first give a brief introduction and overview of superconducting quantum circuits as a basis for quantum simulation. I will then present a quantum simulation system of the Bose-Hubbard-Hamiltonian in the driven dissipative regime in the realm of circuit QED in more detail.
The system consists of series-connected, capacitively coupled, nonlinear and tunable superconducting resonators. The nonlinearity is achieved by galvanically coupled SQUIDs, placed in the current anti-node of each resonator and can be tuned by external coils and on-chip antennas.
Theoretical models of the Bose-Hubbard system predict bunching and antibunching behavior both in the second order auto- and cross-correlation function of the bosonic modes in the lattice sites. Characterization measurements of our sample show that we can reach the parameter space of interest for these quantum simulation experiments.
Fri Feb 16th, 2018 - 10:30 am
Seminar by Jakob Wierzbowski on
'Long-Lived Quantum Emitters in hBN-WSe2 van-der-Waals Heterostructures'
Jakob Wierzbowski will talk on
'Long-Lived Quantum Emitters in hBN-WSe2 van-der-Waals Heterostructures'
We present a significant linewidth narrowing (12.5 %) of free excitons in hBN encapsulated TMDs and localized (< 350 nm) single-photon emitters with long lifetimes of ~18 ns in hBN/WSe2 heterostructures.
February 12th - 16th 2018
Block Course by Prof. Michael Keyl (FU Berlin) on Mathematical Aspects of Quantum Field Theory (Part 2) together with Study Course 'Theoretical and Mathematical Physics' (TMP) at LMU and IMPRS 'Quantum Science and Technology' (QST) at MPQ, Theresienstrasse 39, room A449, Mon through Fri 10-12 am plus 2-4 pm.
Block Course by Prof. Michael Keyl (FU Berlin) on Mathematical Aspects of Quantum Field Theory (Part 2) together with TMP and IMPRS-QST. Held at LMU Physics, Theresienstr. 39, Room A449, Mon through Fri: 10-12 am plus 2-4 pm.
In the beginning of the semester, we studied QFT in the Wightman framework. This included in particular scalar and operator valued distributions, Wightman axioms, Wightman functions and the reconstruction theorem, and the Borchers-Uhlmann algebra with its representations. As an explicit example we studied the free scalar field, its Wick-ordered products, and self-interacting models in 1+1 dimensions. In the latter context questions of renormalization were discussed.
Now, the second series will be devoted to perturbation theory: After a short look at the S-matrix, we will use the Epstein-Glaser formalism to construct the perturbation series term by term as a formal power series. This will be carried out in detail with \Phi^4 self interactions as an explicit example. In this context perturbative renormalizability will also be discussed.
For the complete script (parts 1 and 2) follow this link.
3 ECTS points can be acquired by writing an at least 10 page essay on a topic related to the course.
Quantum fields and Wightman axioms
Quantum fields
Wightman distributions and the reconstruction theorem
The free scalar field
Representations of the Ponicaré group
Relations to the Klein-Gordon equation
Representations of the CCR
the field and its properties
Interacting fields or: why is this so difficult?
Scattering theory (Feb. 2018)
Perturbative theory à la Epstein-Glaser (Feb. 2018)
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol 2. and 3.
N. N. Bolgolubov, A. A. Logunov, A. I. Osak, I. T. Todorov, General Principles of Quantum Field Theory, Kluwer (1990).
R. Haag, Local Quantum Physics, Springer (1996).
Further information on the TMS webpage.
Fri Feb 2nd, 2018 - 10:30 am
Seminar by Prof. Maurice de Gosson on
'Properties of Phase Space Distributions in the Cohen Class'
Prof. Maurice de Gosson will talk on
'Properties of Phase Space Distributions in the Cohen Class'
Non-standard phase space distributions play an increasingly important role in quantum mechanics, to witness recent work by Koczor, Zeier, and Glaser who highlight the relation of these distributions with tomography.
In this talk we will discuss the properties (marginal conditions, Moyal identity) for a large class of phase space distributions obtained from the usual Wigner distribution by convolution with a Cohen kernel. We will examine in detail two particular cases from this perspective. the Husimi distribution, and the Born-Jordan distribution. The latter arises naturally when one uses the Born and Jordan quantization scheme instead of the traditional Weyl correspondence.
Fri Jan 26th, 2018 - 10:30 am
Seminar by Michael Lohse on
'Exploring 4D-Quantum-Hall Physics with a 2D Topological Charge Pump'
Michael Lohse will talk on
'Exploring 4D Quantum Hall Physics with a 2D Topological Charge Pump'
The discovery of topological states of matter has greatly improved our understanding of phase transitions in physical systems. Instead of being described by local order parameters, topological phases are described by global topological invariants and are therefore robust against perturbations. A prominent example is the two-dimensional integer quantum Hall effect: it is characterized by the first Chern number, which manifests in the quantized Hall response that is induced by an external electric field.
Generalizing the quantum Hall effect to four-dimensional systems leads to the appearance of an additional quantized Hall response, but one that is nonlinear and described by a 4D topological invariant—the second Chern number.
Here we report the observation of a bulk response with intrinsic 4D topology and demonstrate its quantization by measuring the associated second Chern number. By implementing a 2D topological charge pump using ultracold bosonic atoms in an angled optical superlattice, we realize a dynamical version of the 4D integer quantum Hall effect. Using a small cloud of atoms as a local probe, we fully characterize the nonlinear response of the system via in situ imaging and site-resolved band mapping. Our findings pave the way to experimentally probing higher-dimensional quantum Hall systems, in which additional strongly correlated topological phases, exotic collective excitations and boundary phenomena such as isolated Weyl fermions are predicted.
ExQM Planning Seminar for 2018, e.g., ExQM Workshop
We will plan activities in 2018 including
ExQM Workshop 2018
video clips of results
hand-over to next ExQM generation
Fri Jan 12th, 2018 - 9:00 am
Opening Lectures of the Max-Planck Harvard Research Centre for Quantum Optics at Deutsches Museum
The Max-Planck Harvard Research Centre for Quantum Optics (MPHQ) will be opened with a two-day symposion on Jan. 11th at IAS Garching and on Jan. 12th at Deutsches Museum. It is a joint venture with the Harvard Quantum Optics Center.
See the programmes here:
12th Jan. at Deutsches Museum (high-key: including talks by nobel laureates Glauber & Ketterle)
11th Jan. at IAS Garching
The event is open to the public and the participation of students is highly encouraged! The Max Planck Research Center for Quantum Optics aims for becoming one of the major internationally recognized scientific collaborations of its kind in the field of Quantum Optics.
Fri Dec 8th, 2017 - 10:30 am
Seminar by David Leiner on
'Wigner Tomography of Multi-Spin Operators'
David Leiner will talk on
'Wigner Tomography of Multi-Spin Operators'
We study the tomography of operators for multi-spin systems in the context of finite-dimensional Wigner representations. An arbitrary operator can be completely characterized and visualized using multiple shapes assembled from linear combinations of spherical harmonics. We develop a general methodology to experimentally recover these shapes by measuring expectation values of rotated axial spherical tensor operators.
Our approach is experimentally demonstrated for quantum systems consisting of up to three spins using nuclear magnetic resonance spectroscopy.
[Joint work with Robert Zeier and Steffen J. Glaser: arXiv: 1707.08465 ]
Fri Dec 1st, 2017 - 10:30 am
Seminar by Bálint Koczor on
'Continuous Phase-Space Representations for Finite-Dimensional Quantum States and Their Tomography'
MPQ, Seminar Room, Theory Group in 3rd floor at 10.30 am.
Bálint Koczor
Continuous Phase-Space Representations for Finite-Dimensional Quantum States and Their Tomography
Continuous phase spaces have become a powerful tool for describing, analyzing, and tomographically reconstructing quantum states in quantum optics and beyond. A plethora of these phase-space techniques are known, however a thorough understanding of their relations was still lacking for finite-dimensional quantum states. We present a unified approach to continuous phase-space representations which highlights their relations and tomography. The quantum-optics case is then recovered in the large-spin limit. Our results will guide practitioners to design robust innovative tomography schemes.
[Joint work with Robert Zeier and Steffen J. Glaser: arXiv:1711.07994v1 ]
Fri Nov 24th, 2017 - 10:30 am
Double-Feature Seminar by Frederik vom Ende on
'Discrete Open Dynamical Systems and Unitary Dilations'
with an intro by Margret Heinze on 'The Lindblad-Kossakowski Theorem in Infinite Dimensions'
After Margret Heinze (IMPRS-QST, group Michael Wolf) has given a short tutorial proving the Lindblad-Kossakowski Theorem in infinite dimensions,
'Discrete Open Dynamical Systems and Unitary Dilations'
Arbitrary quantum maps, in particular the time evolution of open dynamical quantum systems, are described by so-called quantum channels (which simply are linear, trace-preserving and completely positive maps) acting on trace-class operators.
Every quantum channel has a Kraus decomposition, so it can be built from the basic operations of tensoring with an environment, a unitary transformation or the larger system, and finally the return to the original (sub)system via tracing out. This idea can be extended to the whole dynamical semigroup induced by a quantum channel (where the environment now is way larger than in the Kraus decomposition); it is the unitary dilation of the semigroup in question. Analogously, one can mathematically structure (a) the solution of discrete quantum dynamical systems and (b) certain types of discrete quantum dynamical control systems.
Mon Nov 20th, 2017 - 12:30 noon
Seminar by Dr Volkher Scholz on
'Analytic Approaches to Tensor Networks for Critical Systems and Field Theories'
MPQ Large Lecture Hall (moved to B0.32) at 12.30 noon.
Dr. Volkher Scholz, ETH Zurich
Analytic Approaches to Tensor Networks for Critical Systems and Field Theories
I will discuss analytic approaches to construct tensor network representations of quantum field theories, more specifically critical systems and conformal field theories in 1+1 dimensions. A key insight is that we should understand how well the tensor network can reproduce the correlation functions of the quantum field theory. Based on this measure of closeness, I will present rigorous results allowing for explicit error bounds which show that the multiscale renormalization Ansatz (MERA) does approximate conformal field theories.
In particular, I will discuss the case of free fermions, both on the lattice and in the continuum, as well as Wess-Zumino-Witten models.
[Based on joint work with Jutho Haegeman, Glen Evenbly, Jordan Cotler (lattice) and Brian Swingle and Michael Walter (lattice & continuum)]
Fri Oct 27th, 2017 - 11:00 am
Seminar by Prof. Michael Keyl on
'Controlling a d-Level Atom in a Cavity'
MPQ Large Lecture Hall (now moved to B0.32) at 11.00 am.
Prof. Michael Keyl, FU Berlin
Controlling a d-Level Atom in a Cavity
We study controllability of a d-level atom interacting with the electromagnetic field in a cavity. The system is modelled by an ordered graph Γ. The vertices of Γ describe the energy levels and the edges allowed transitions. To each edge of Γ we associate a harmonic oscillator representing one mode of the electromagnetic eld. The dynamics of the system (drift) is given by a natural generalization of the Jaynes-Cummings Hamiltonian.
If we add suffcient control over the atom, the overall system (atom and electromagnetic field) becomes strongly controllable, i.e. each unitary on the system Hilbert space can be approximated with arbitrary precision in the strong topology by control unitaries. A key role in the proof is played by a topological *-algebra A(Γ) which is generated (roughly speaking) by the path of Γ. For that reason A(Γ) is called path algebra. It contains crucial structural information about the control problem, and it is therefore an important tool for the implementation of control tasks like preparing a particular state from the ground state.
This is demonstrated by a detailed discussion of different versions of 3-level systems.
[Based on joint work with Thomas Hofmann]
Munich Quantum Day at TUM ZNN, Campus Garching, Am Coulombwall 4a
Symposion by Munich Quantum Centre (MQC)
Held at TUM ZNN, Campus Garching, Am Coulombwall 4a.
October 9th - 12th 2017 (to be continued February 2018)
Block Course by Prof. Michael Keyl (FU Berlin) on Mathematical Aspects of Quantum Field Theory (Part 1) together with Study Course 'Theoretical and Mathematical Physics' (TMP) at LMU and IMPRS 'Quantum Science and Technology' (QST) at MPQ, Theresienstrasse 39, room B101, Mon: 2-4pm; Tue, Wed, Thu 10-12am plus 2-4pm.
Block Course by Prof. Michael Keyl (FU Berlin) on Mathematical Aspects of Quantum Field Theory (Part 1) together with TMP and IMPRS-QST. Held at LMU Physics, Theresienstr. 39, Room B101, Mon: 2-4pm; Tue, Wed, Thu 10-12am plus 2-4pm.
In the beginning of the semester, we will study QFT in the Wightman framework. This includes in particular scalar and operator valued distributions, Wightman axioms, Wightman functions and the reconstruction theorem, and the Borchers-Uhlmann algebra with its representations. As an explicit example we will study the free scalar field, its Wick-ordered products, and self-interacting models in 1+1 dimensions. In the latter context questions of renormalization are also discussed.
The second week at the end of the semester will be devoted to perturbation theory. After a short look at the S-matrix, we will use the Epstein-Glaser formalism to construct the perturbation series term by term as a formal power series. This will be carried out in detail with \Phi^4 self interactions as an explicit example. In this context perturbative renormalizability will also be discussed.
Scattering theory (Winter 2018)
Perturbative theory à la Epstein-Glaser (Winter 2018)
June 30th, 2017 - 11:15am
Seminar by Frederik vom Ende on
'Unitary Dilations of Discrete Quantum-Dynamical Semigroups'
MPQ Small Lecture Hall at 11.00am.
Frederik vom Ende
Unitary Dilations of Discrete Quantum-Dynamical Semigroups
The time evolution of physical systems is a main factor in order to understand their nature and properties. Operations and therefore time evolutions of open quantum systems are described by quantum maps which are linear, trace-preserving and completely positive.
Also for the discrete case, we want to understand why the condition of complete positivity is necessary and which structure it provides. Again we will see that every quantum channel has a Kraus decomposition and that it can be built from the basic operations of tensoring with a second system in a specified state, a unitary transformation, and the reduction to a subsystem.
Thus one can mathematically structure the solution of discrete quantum dynamical systems and even certain types of discrete quantum dynamical control systems.
[Work based on a masters thesis in maths at U Wuerzburg.]
June 23rd, 2017 - 11:00am
Seminar by Lukas Knips on
'How to Detect Entanglement - a Summary of Methods'
Lukas Knips
How to Detect Entanglement – a Summary of Methods
Entanglement is a fascinating feature of quantum systems and one of the key resources for quantum information processing. In order to detect and quantify entanglement, and thus to attest the prepared system to be a useful resource, sophisticated methods are required.
I will review and compare some specialized and efficient entanglement criteria for multiqubit systems with standard tools such as the PPT criterion [1], which is easily applicable, but limited to small systems, and linear fidelity witnesses [2], which are helpful if prior knowledge about the state is present, but already need several measurements.
The toolbox, I will present, encompasses methods which, for example, detect entanglement after only two measurements [3], work without any prior knowledge about the state [4] or can be applied when one cannot even know the local reference frames [5].
[1] A. Peres, Phys. Rev. Lett. 77, 1413 (1996); M. Horodecki, P.
Horodecki, R. Horodecki, Phys. Lett. A 223, 1 (1996)
[2] O. Gühne, G. Tóth, Physics Reports 474, 1 (2009)
[3] L. Knips, C. Schwemmer, N. Klein, M. Wiesniak, H. Weinfurter, Phys. Rev. Lett. 117, 210504 (2016)
[4] W. Laskowski, D. Richart, C. Schwemmer, T. Paterek, H. Weinfurter, Phys. Rev. Lett. 108, 240501 (2012); W. Laskowski, C. Schwemmer, D. Richart, L. Knips, T. Paterek, H. Weinfurter, Phys. Rev. A 88, 022327 (2013)
[5] in preparation
June 9th, 2017 - 10:00am
Seminar by Nicola Pancotti on
'Almost Conserved Local Operators in MBL Systems '
MPQ Seminar Room of Theory Group in 2nd floor at 10.00am.
Nicola Pancotti
Long-time dynamics of non-integrable systems hold the key to fundamental questions (thermalization). Analytical tools can only apply to particular cases (integrable models, perturbative regimes). Numerical simulations, limited in time, have found evidence of different time scales.
A new numerical technique for constructing slowly evolving local operators was introduced by Kim et al. in Phys. Rev. E 92, 012128 (2015). Those operators have a small commutator with the Hamiltonian and they might give rise to long time scales.
In this work, we apply this technique to the many body localization problem. We show that this method can not only signal the difference between the ergodic and localized phases, but it is also sensitive to the presence of a subdiffusive phase between both.
Munich Quantum Day at LMU Physics, Theresienstrasse 37, Room A 348/349, 12-4pm
Held at LMU Physics, Theresienstr. 37 Room A348/349:
12.05am: Viatcheslav Mukhanov: "Quantum Mechanics in the Sky"
12.35am: Frank Pollmann: "Dynamical Signatures of Spin Liquids"
1:10pm: Poster session with coffee and snacks
May 18th, 2017 - 5.00pm
IMPRS Career-Talk by Susanne Pielawa on
'From Theoretical Physics to Algorithms'
MPQ Herbert Walther Lecture Hall at 5.00pm.
Dr. Susanne Pielawa
(Google, Munich)
During their undergraduate and graduate studies, physicists acquire a broad set of transferable skills which make many career paths available, also outside of academia.
Dr. Susanne Pielawa studied Physics at the University of Ulm, did her PhD in Condensed Matter Theory at Harvard and then went on to a postdoc position, also in Condensed Matter Theory, at the Weizmann Institute of Science. Afterwards, she joined the Start-Up Yowza (in Tel Aviv) and developed algorithms for a 3D-model search engine. She is now a software engineer at Google Munich.
She will talk about what it is like to be a software engineer, and share impressions of her transition from theoretical physics to algorithm development and software engineering. She will also present career opportunities at Google, and talk about the company's work culture.
May 5th, 2017 - 10:30am
Seminar by Prof. Witlef Wieczorek on
'Mechanical Devices for Quantum State Engineering and Sensing'
MPQ Herbert Walther Lecture Hall at 10.30am.
Prof. Witlef Wieczorek
(formerly QCCC, then U Vienna, now at U Gothenburg)
Impressive results have recently been achieved in controlling micro- and nanomechanical devices in the quantum regime. These achievements pave the way for exploring novel applications and tests of quantum mechanics employing mechanical devices. In my talk I will describe progress towards quantum control of optomechanical states. In particular, I will address the question on how does one optimally estimate the state of an optomechanical system.
Further, I will talk about a completely different approach of controlling mechanical systems that has been recently proposed by employing superconducting levitation. This experimental platform offers unrivaled low mechanical dissipation and coupling capability to superconducting circuits. Its envisioned applications range from studying of fundamental questions to novel sensing prospects. I will describe first ideas and experimental steps in this direction.
April 28th, 2017 - 10:30am
Seminar by Anna-Lena Hashagen on
'Randomised Benchmarking'
Anna-Lena Hashagen
(returning from Prof. Stephen Bartlett, U Sydney)
Randomised benchmarking is a widely used experimental technique to characterise the average error of quantum operations. Its robustness regarding state preparation and measurement errors as well as its efficient scaling has made it a standard and reliable choice.
I will give an overview and a short introduction to the randomised benchmarking protocol for characterising the average error of Clifford gates.
April 7th, 2017 - 10:30am
Seminar by Umut Kaya on 'Investigation of the Second Laws in the Catalytic Coherence Setup'
MPQ Seminar Room B0.22 at 10.30am.
Umut Kaya (formerly with Renato Renner at ETH Zurich)
Catalyst systems are just additional systems you include to your setup to widen the range of your allowable transformations on the main state. But if you naively use this ancillary system in an approximate manner, you come across a well-known phenomenon called thermal embezzling, which says you can reach any output state without any restrictions–thus no 2nd Law.
I will present some analytical and computational results showing that the Catalytic Coherence setup is limited by certain second-law like relations, therefore it does not suffer from this embezzling problem.
April 3rd - 5th, 2017
IAS Symposion on Quantum Control: Mathematical Aspects and Physical Applications
Symposion: Quantum Control Theory: Mathematical Aspects and Physical Applications
Mon-Wed/April 3rd-5th 2017 at TUM Institute of Advanced Study, IAS Garching, Lecture Hall
March 30th - April 1st, 2017
Symposion in Honour of the 70th Anniversary of
Max-Planck-Medaillist 2017, Herbert Spohn at TUM & LMU
Symposion: Macroscopic Limits of Quantum Systems
Thu/Fri (March 30th/31st), 2pm: TUM, IMETUM, Garching, Lecture Hall E.127
Sat (April 1st), 2pm: LMU Mathematics Department, Centre, Lecture Hall A027
March 23rd, 2017 - 2:00pm
Seminar by Claudius Hubig on 'DMRG with Subspace Expansion on Symmetry-Protected Tensor Networks'
MPQ Theory Group Seminar Room 2nd floor at 2.00pm.
Claudius Hubig
The Density Matrix Renormalisation Group when applied to matrix- product states is the method of choice for ground-state search on one-dimensional systems and still highly competitive even in unfavourable circumstances, such as critical systems and higher dimensions.
In this talk, I will discuss two separate methods which can be used to
improve the computational efficiency of DMRG and related methods on matrix-product states and beyond. The first component is the
implementation of both abelian and non-abelian symmetries in an entirely general way suitable also for higher-rank tensors as encountered in, e.g., tree tensor network states. The second ingredient, the subspace expansion, allows for a fully single-site DMRG algorithm with favourable linear scaling in the local dimension of the tensor network. Even for common problems, this results in a considerable speed-up over the traditional two-site DMRG method or the density matrix perturbation approach for ground-state search at reduced algorithmic complexity.
Additionally, the subspace expansion can potentially be used in a large
set of other algorithms, such as the TDVP or the variational application
of a matrix-product operator onto a matrix-product state.
February 3rd, 2017 - 10:30am
Seminar by Dr. Manfred Liebmann on 'Quantum Physics by the Non-Associative Cayley Algebra'
Dr. Manfred Liebmann (U Graz, formerly Maths. Dept. TUM)
In the lecture I will discuss several striking consequences of the following generalization that will give rise to a new perspective on structures in the standard model of particle physics:
The Hurwitz theorem states that a bilinear product on $R^n$ with the property x◦y = ||x|| . ||y|| can only exist in dimensions n = 1, 2, 4, 8. The associated algebras are the division algebras of the real numbers R, the complex numbers C, the quaternions H, and the non-associative algebra of the octonions O, also known as Cayley algebra.
It turns out that the structure of the Dirac equation is deeply related to the non-assoziative multiplication law of the Cayley algebra. The Dirac matrices can be identified with complex versions of the left action maps Lx(y) := x◦y. However not all left actions can be complex represented and this leads to a generalization of the Dirac equation that transcends the traditional complex Hilbert space framework of quantum physics.
January 30th, 2017 at MPQ, Grand Lecture Hall
Tensor Networks and Applications:
Video Recording of Prof. Cirac, MPQ
11am, Prof. Cirac, MPQ
Tensor Networks:
A Quantum Information Perspective to Many-Body Physics
Abstract: The theory of entanglement offers a new perspective to view many-body quantum systems. In particular, systems in thermal equilibrium and with local interactions contain very little entanglement, which allows us to describe them efficiently, circumventing the exponential growth of parameters with the system size. Tensor Networks offer such a description, where few simple tensors contain all the information about all physical properties. In this talk I will review some of the latest results on entanglement and tensor networks, and explain some of their connections to quantum computing, condensed matter, and high-energy physics.
January 9-12th, 2017
Symposion on Machine Learning Challenges in Complex Multiscale Physical Systems at IAS Garching
See programme, in particular for Mon afternoon and Tue morning.
December 19th, 2016 at MPQ, room B0.21
ExQM Christmas Seminar
4pm, Dr. Thomas Schulte-Herbrüggen
Season's Applications of Knot Theory
Abstract: Surprise.
November 18-19th, 2016
Symposion in Honour of the 75th Anniversary of Nobel Laureate Ted Hänsch at LMU
Symposion: From Laser Spectroscopy to Quantum Science
Programme for Fri, Nov. 18th
Programme for Sat. Nov. 19th
November 18th, 2016 - 4:30PM
Seminar by Dr. Juan Bermejo-Vega on Understanding Contextuality as a Quantum Computational Resource without Wigner Functions
MPQ Lecture Hall at 4.30pm.
Dr. Juan Bermejo-Vega (FU Berlin)
A central question in quantum computation is to identify the resources that are responsible for quantum speed-up. Quantum contextuality has been recently shown to be a resource for quantum computation with magic states for odd-prime dimensional qudits and two-dimensional systems with real wavefunctions.The phenomenon of state-independent contextuality poses a priori an obstruction to characterizing the case of regular qubits, the fundamental building block of quantum computation. Here, we establish contextuality of magic states as a necessary resource for a large class of quantum computation schemes on qubits. Our proof exploits novel simple arguments and, for the first time, does not rely on Wigner functions. Our new methods can be extended to a family of magic-state protocols on qudits of any local dimension and lead to a new most-general proof in the odd dimensional case.
October 27-28th, 2016
Munich Quantum Symposion 2016 at MPQ and WSI
The IMPRS "Quantum Science and Technology" is inaugurated. ExQM has an e-poster session on Fri, Oct. 28th at 1:30 pm in WSI.
munich_quantum_symposium_16
October 21st, 2016 - 11AM
Seminar by Prof. Michael Keyl on Some Maths behind the 2016 Nobel Prize in Physics
MPQ B0.21 at 11.00am.
Prof. Michael Keyl
The toric code introduced by Kitaev is one of the most simple models of topological order and therefore a good candidate to discuss some mathematical topics related to the last Nobel Prize in Physics. Apart from topological phases this includes in particular the emergence of quasi particles with exotic statistics in low dimensions.The latter relates to the representation theory of the braid group and the algebraic theory of superselection sectors, which we will briefly discuss.
July 5th, 2016 - 7th
Sommerfeld Lectures by Prof. John Preskill
Prof. John Preskill from Caltech next week in LMU.
Public Lecture: Quantum Computing and the Entanglement Frontier
Professor John Preskill, California Institute of Technology, USA more
Room B052 – Faculty of Physics – LMU – Theresienstr. 39 – 80333 München
Theory Colloquium: Quantum Information and Spacetime
Room A348/349 – Faculty of Physics – LMU – Theresienstr. 37, 80333 München
Fields and Strings Seminar: Holographic Quantum Codes
Details on: http://www.asc.physik.lmu.de/activities/lectures/
June 29th, 2016 - 2.30pm
Lecture by Robert König on Fault-Tolerance in Quantum Information Processing
Robert König on "Fault-Tolerance in Quantum Information Processing"
June 29th 2016 at 2.30pm in Lecture Hall 3 in the TUM Maths Dept. (Hörsaal 3 (MI 00.06.011)).
Classical information theory provides quantitative answers to basic questions about communication and computation. In the presence of quantum effects, its basic tenets need to be reassessed as fundamentally novel information-processing primitives become possible. Their potential appears promising, but their realization hinges on our ability to construct mechanisms protecting information against unwanted noise.
In this talk, I will consider two problems associated with communication and computation. First, I will discuss the additivity problem for classical capacities. I will review some of the more recent results in this direction: for continuous-variable channels, quantum generalizations of certain geometric inequalities yield operationally relevant statements. Second, I will discuss the problem of performing gates on information encoded in an error-correcting code, and explain how this relates to automorphisms of the latter.
June 19th - 22nd, 2016
First International ExQM Workshop at Lake Chiemsee
First International ExQM Workshop at Lake Chiemsee from Sun June 19th till Wed June 22nd 2016.
overall programme: see ProgrammExQM2016rev.pdf
detailed programme: see BookletWorkshop2016new.pdf
June 1st, 2016 - 11.30am
Seminar by Julian Roos on From Geometry To Topology
Seminar by our ExQM colleague Julian Roos on "From Geometry To Topology".
Wednesday June 1st 2016 at 11.30am at MPQ lecture hall.
I will give a pedagogical non-formal introduction to the notion of topology. The idea is to outline an intuitive pathway from geometrical to topological concepts by increasing the number of allowed transformations within the congruence classes of Euclidean geometry. I will then pick one or the other basic system from the class of topological phases of matter and discuss why the word topological is used to describe it.
Here is a little teaser by Martin Gardner :
Draw a continuous line across the closed network shown so that the line crosses each of the 16 segments of the network only once. The curved line shown in the attached image does not solve it, because it leaves one segment uncrossed. It is not difficult to prove that the puzzle cannot be solved on a plane surface. Can it be solved on the surface of a sphere? On the surface of a torus?
April 29th, 2016 - 11AM
Seminar by Stephan Welte on A cavity-mediated photon-photon gate
Friday Apr. 29th at 11am at MPQ seminar room B0.21 in
Stephan Welte on "A cavity-mediated photon-photon gate".
Photons are promising candidates for applications in quantum information processing and quantum communication. However, the direct interaction between two photons is negligible in free space, which is a drawback when it comes to the implementation of quantum logic gates between them. A solution to this problem was offered by Duan and Kimble [1] who proposed that a strongly coupled atom in an optical cavity [2] could mediate an effective interaction between two photons. We experimentally demonstrate that an implementation of this proposal is indeed possible. To this end, the universal CNOT operation of the gate as well as its capability to entangle two separable input photons are characterized. We will discuss details of our experimental implementation and present intriguing implications of our gate for photonic quantum information processing.
[1] L.-M. Duan, H.J.Kimble, Phys. Rev. Lett. 92, 127902 (2004)
[2] A.Reiserer, G.Rempe, Rev. Mod. Phys. 87, 1379 (2015)
April 8th, 2016 - 11AM
Seminar by Jakob Wierzbowski on Symmetry control of layered 2D semiconducting materials in novel nanodevices
ExQM seminar this Friday Apr. 8th at 11am as a lab tour in Walter-Schottky-Institute (meeting point: entrance hall).
Follow-up on last week's talk on "Symmetry control of layered 2D semiconducting materials in novel nanodevices".
Jakob Wierzbowski gives a short lab tour showing related experiments… also welcoming those who could not come last Fri, of course.
April 1st, 2016 - 11AM
Friday Apr. 1st 2016 at 11am in MPQ seminar room B0.21.
Jakob Wierzbowski on "Symmetry control of layered 2D semiconducting materials in novel nanodevices".
The emergence of truly two-dimensional materials like graphene [1] extends the range of commercially available material systems for future electronic and optoelectronic devices. Especially, semiconducting transition metal dichalcogenides (STMDs) MoS2, MoSe2, WS2 and WSe2 play a key role in bridging the gap between bandgap-less graphene and insulating hBN. Importantly, TMDs exhibit a direct bandgap in the monolayer limit with optical transitions in the visible and near-infrared wavelength range [2]. Here, intrinsic electronic and optical signatures in few-layer crystals strongly depend on crystallographic symmetry properties.
Specifically, the inherently broken inversion symmetry in monolayer crystals naturally facilitates valleytronic applications [3]. Here, we present the electrical control of symmetry properties of few-layer MoS2 crystals embedded within electrically tunable Si-SiO2-STMD-Al2O3-metal microcapacitors with optical access as presented in Figure 1. By tuning the electric field, we induce strong static electric fields exceeding ±3.5 MV/cm resulting in significant DC Stark shifts of the interband emission (>15 meV) for mono- to pentalayer crystals. We extract an effective exciton polarizability of β=(0.58±0.25)×10−8 DmV−1; independent of the number of layers probed [4].
In polarization resolved photoluminescence measurements performed on mono- to trilayer MoS2, we observe pronounced electric field control of valley optical dichroism for bilayer crystals. Importantly, we are able to continuously tune the degree of circular polarization of the emission from η=20 % up to 58 % [5]. Selected data are presented in Figure 2. Moreover, for the bilayer crystal, we demonstrate intense electrical control of second-harmonic generation (SHG) originating in naturally inversion symmetric 2H stacked MoS2. Using ultrashort pulses of ~ 70 fs within a spectral window 840 – 1000 nm (1.24 – 1.47 eV), we observe broadband tunability of the SHG signal throughout the probed region with a ~ 60 fold conversion amplification at its optimum [6]. Our results demonstrate the potential for emergent spin- and valleytronic devices based on two-dimensional atomically thin crystals and efficient electrically driven broadband frequency doubling by external control of the symmetry properties of 2H MoS2.
[1] A. K. Geim, I. V. Grigorieva, Nature 499, 419-425 (2013)
[2] Mak et al., Phys. Rev. Lett. 105, 136805 (2010)
[3] Zeng et al., Nature Nano. 7, 490-493 (2012)
[4] J. Klein and J. Wierzbowski et al., Nano Lett. DOI: 10.1021/acs.nanolett.5b03954 (2016)
[5] J. Wierzbowski and J. Klein et al., in preparation (2016)
March 11th, 2016 - 10.30AM
Prof Michael Keyl on:
'Controlling a d-level atom in a cavity' and Mrs Margret Heinze on 'Quantum control for a Jaynes-Cummings-Hubbard model'
Friday Mar. 11th 2016 at 10.30am in MPQ seminar room B0.21.
Double feature of two short DPG talks:
Prof Michael Keyl will dwell on"Controlling a d-level atom in a cavity"
In this talk we discuss quantum control theory for a d-level atom in a cavity. The atom is described by a Graph Γ with energy levels as vertices and edges e as allowed transitions. For each such e the atom interacts (via a Jaynes-Cummings like interaction term) with a different mode of the cavity. We consider controllability of the overall system (i.e. atom and cavity) under the assumption that all atom-cavity interactions can be switched on and off individually and that the atom itself is fully controllable. Our main tools are symmetry based arguments recently introduced for the discussion of the two-level case [M. Keyl, R. Zeier, T. Schulte-Herbrüggen, NJP 16 (2014) 065010]. The basic idea is to divide the control Hamiltonians into two sets. One which is invariant under the action of an Abelian symmetry group G and a second set which breaks this symmetry. We will discuss how the group G and its action are related to the graph Γ and its fundamental groupoid, and how these structure can be used to prove full controllability – at least if Γ is acyclic. For Graphs containing cycles the situation is more difficult and the universal covering graph has to be used. We demonstrate this, using the fully connected graph on three vertices as an example.
and Mrs Margret Heinze continues on "Quantum control for a Jaynes-Cummings-Hubbard model ".
We examine the control of a quantum system consisting of several two-level atoms with each atom interacting with a different mode of an electromagnetic field.
More precisely, the system is a Jaynes-Cummings-Hubbard model where each cavity contains an atom and a bosonic excitation that can tunnel to the neighbouring cavities. The interaction strengths can be time dependently tuned in order to achieve controllability.
We discuss if it is possible that every pure state can be reached from a given reference state (pure-state controllability). This analysis is lifted to the level of operators where each unitary has to be approximated with arbitrarily small error by a time evolution operator for appropriate control functions and finite time (strong controllability).
The challenge of this infinite dimensional control problem is met, by firstly examining the symmetries of the system. A finite dimensional block diagonal decomposition is obtained for the control Hamiltonians that obey an abelian symmetry and due to a cut-off finite dimensional Lie analysis can be applied. By then adding a Hamiltonian that breaks the symmetry pure state and strong controllability are examined, c.f. [New Journal of Physics 16 (2014): 065010].
February 25th & 26th, 2016
ExQM Miniworkshop:
Mathematics of Quantum Systems and Control Engineering
ExQM seminar as part of an entire miniworkshop on mathematics of quantum system and control theory on Thu 25 and Fri 26th Feb.
"Mathematical Aspects of Quantum Systems and Control Engineering"
Feb. 25-26th 2016 at Math. Inst. TUM Campus Garching and MPQ.
Thu 25. Feb. Motto: "Mostly Finite-Dimensional Systems"
(all in Sem. Room Maths Dept. MI 03.08.011 on floor 3)
13:30h Dr Gunther Dirr (Uni Würzburg) "Some New Results on Ensemble Control"
To begin with, we present necessary and sufficient accessibility/controllability criteria for finitely many parallel connected bilinear systems.
Then, extending these ideas to infinitely many systems, we arrive at so-called bilinear ensembles and discuss first results on ensemble controllability.
14:30h Dr Thomas Schulte-Herbrüggen (TUM) "Systems Theory and Control of Closed & Open Markovian Quantum Systems: A Unified Lie Symmetry Approach"
We give necessary and sufficient symmetry conditions for controllability and simulability in closed systems. For open Markovian systems, we discuss accessibility in terms of Lie semigroups and their Lindblad-Kossakowski generators. We elucidate the Lie-algebraic structure of the Lie wedges and their embedding system algebras.
15:30h Coffee and extensive discussion
ca. 18:30h Dinner in Garching
Fri 26.Feb: Motto: "Infinite-Dimensional Systems"
10:30h Prof. Ugo Boscain (CNRS Paris Saclay) "Controlling the Schrödinger Equation via Adiabatic Methods Using Conical Intersection of Eigenvalues" (ExQM Seminar at MPQ Room B0.21)
In this talk I will discuss how to obtain a population transfer in a quantum mechanical system using adiabatic methods and the presence of conical intersections in the space of controls. The method is powerful. In the finite dimensional case it permits to prove that if the system is "conically connected" then it is Lie bracket generated. Also it permits to control systems presenting a dispersion of parameters (ensamble controllability). Conical eigenvalue intersections are not rare. For systems with a real Hamiltoniana and two controls they are structurally stable.
12:00h Lunch at IPP
13:30h Dr Mario Sigalotti (INRIA Paris Saclay) "Control of the Discrete-Spectrum Schroedinger Equation" (Sem. Room Maths Dept. MI 03.08.011 on floor 3)
We show how to deduce approximate controllability of the control-affine Schroedinger equation from the controllability properties of its Galerkin approximations, in the case in which the uncontrolled Hamiltonian has discrete spectrum.
14:30h Prof. Michael Keyl (TUM) "Controlling Atoms in a Cavity" (Sem. Room Maths Dept. MI 03.08.011 on floor 3)
We treat control of several two-level atoms interacting with one mode of the electromagnetic field in a cavity. This provides a useful model to study pertinent aspects of quantum control in infinite dimensions via the emergence of infinite-dimensional system algebras.
15:30h Coffee and extensive final discussion panel
February 19th, 2016 - 11AM
Seminar by Dr Christian Schwemmer on Efficient Tomography of Multiphoton States
ExQM seminar this Friday Feb. 19th at 11.00am in MPQ seminar room B0.21.
Dr Christian Schwemmer will be so kind as to give his fare-well talk on "Efficient Tomography of Multiphoton States".
Multipartite entangled quantum states offer great opportunities with potential applications in quantuminformation processing. Therefore, practical tools fo r entanglement detection and characterization are needed. However, conventional state tomography suffers from an exponentially increasing measurement effort with the number of qubits. In contrast, pure or symmetric states like W-, Dicke- or GHZ-states enable tomographic analysis at reduced effort. Here, we apply these schemes to experimentally analyze six photon symmetric Dicke states. For data processing, a fitting algorithm based on convex optimization is used offering significant improvements in terms of speed and accuracy.
Furthermore, it will be shown that implying additional constraints in quantum-state estimation, such as non-negativity of a quantum state, can introduce significant systematic errors.
Seminar by David L. Goodwin on Taking Optimal Control toward a Tensor Formalism
ExQM seminar this Friday Feb. 12th at 11.00am in MPQ seminar room B0.21 is to trigger exchange between many-body tensor network techniques and control techniques.
David L. Goodwin (U. Southampton, Ilya Kuprov's group) will give a perspective talk on "Taking Optimal Control toward a Tensor Formalism"
While reviewing the current state of Gradient Assisted Pulse Engineering (GRAPE), questions will be asked on the applicability of a Tensor version of this successful numerical optimal control algorithm. This Tensor-GRAPE is envisaged to use elements DMRG (or MPO), which sits in the low temperature approximation, and the successes of the SPINACH software toolbox in reducing state spaces and having efficient optimal control for simulation of spin system, sitting in the high-temperature approximation.
This talk will ask questions from the point of view of one that works with GRAPE, seeking hints at answers and possible problems that the audience may identify. Subjects of interest include: the use of an augmented exponential to compute exact control derivatives within SPINACH; recent publications of the Tensor-Trains formalism of Savostyanov et.al. to simulate exact 1D spin chains (as occurring also in protein backbones), and t-DMRG.
January 22nd, 2016 - 11AM
Seminar by Prof. Norbert Schuch on Tensor network models for the study of correlated quantum systems
ExQM seminar for this Friday Jan. 22nd 2016 at 11.00am in MPQ seminar room B0.21.
Prof. Norbert Schuch (now MPQ Garching) will talk on "Tensor network models for the study of correlated quantum systems"
Tensor network models provide a means of understanding the behaviour of correlated quantum systems from a local perspective. In this talk, I will give an introduction to the framework of tensor network models, and discuss examples of how they can be used to study the physics of complex quantum many-body systems, with a focus on systems with physical symmetries and those which display topological order.
December 11th, 2015 - 11AM
Seminar by Nicola Pancotti on Many body gates: from small chains to networks
ExQM seminar for this Friday Dec. 11th 2015 at 11.00h in MPQ seminar room B0.21.
Nicola Pancotti will talk on "Many body gates: from small chains to networks"
During the last years a great effort has been made in order tocharacterize, implement and optimize entangling gates between two distant particles. Here we study the evolution of a quantum complex system and we search whether there exists an optimal time t* in which a perfect entangling gate G is implemented. The aim of this research is dual. On one side we propose a brand new numeric method acquired from the machine learning community which gives a good speed up compared to the previous ones. On the other side we look for new, unknown solutions, Jopt , that parametrize the Hamiltonian so that exp(−iH ( Jopt ) t* ) = G ⊗ S; S is an operator which leaves the rest of the system in an unknown (don't care) state. We start from a very simple 3-spins chain, for which we already know the solution. We move then forward to a N-spins chain and finally we enlarge the line to obtain a network. We aim to find perfect topologies and optimal two body interactions capable of implementing fast and high fidelity gates such as entangling, Toffoli, CCZ, etc.
December 4th, 2015 - 1:30PM
Seminar by Dr Mari-Carmen Banuls on Numerical studies of many-body systems using tensor networks
ExQM seminar for this Friday Dec. 4th 2015 at 13.30h in MPQ small lecture hall
Dr Mari-Carmen Banuls we will give a lecture on "Numerical studies of many-body systems using tensor networks"
Tensor network states have proven very successful in describing ground states of quantum many body systems. The paradigmatic example is that of Matrix Product States (MPS), which underlie the celebrated DMRG method for the study of one dimensional systems. Using these methods it is also possible to simulate dynamics. And the ansatz can be also extended to describe operators, in particular, mixed states.
In the last years, the progress has been fast both in the theoretical understanding and the application of tensor networks to diverse problems. In this talk, I will present these methods from the practical point of view, focusing on their application to the numerical study of diverse quantum many-body problems, and illustrate their potential with some recent results.
November 27th, 2015 - 11:30AM
Lecture by Juan Bermejo-Vega on Contextuality as a resource for qubit quantum computation
On short-term notice, our ExQM/QCCC fellow Juan Bermejo-Vega (now at FU Berlin) we will give a lecture in the Cirac group on 27th Nov. 2015 11.30am
"Contextuality as a resource for qubit quantum computation"
ExQM Seminar for this Friday Nov. 20th at 11.15am in MPQ seminar room B0.21.
With Dr. Ville Bergholm (U. Helsinki) we will give a short double feature on "A First Glance into Quantum Control Engineering"
We go from a sketch of the unified background (by Thomas) to Ville showing on-line examples on the computer from optimized quantum state transfer and gate (or map) synthesis in closed and open systems. These examples include spin chains, NV centres, and exciton transfer in light-harvesting FMO complexes.
We give an outlook on the limits of open-loop versus closed-loop control.
November 6th, 2015 - 11AM
Seminar by Julian Roos on Looking inside a lithium-ion battery electrode: A materials modelling study
ExQM Seminar for this Friday Nov. 6th at 11.00am in MPQ seminar room B0.22.
Julian Roos will talk on "Looking inside a lithium-ion battery electrode: A materials modelling study"
In 1990, Sony introduced the first commercially viable Lithium-ion battery to the market, sparking a revolution in consumer portable electronics. This breakthrough was mainly due to the incorporation of John Goodenough's layered intercalation cathode LiCoO2 into the cell, raising the energy density of such batteries to a practical level for the first time. Today, cathode materials are still regarded the major bottleneck (batteries with LiCoO2 are still found in most devices) and the route to new generation lithium-ion batteries is linked to the search for superior materials and their optimization. This calls for a better understanding of solid state properties and the fundamental physical processes inside electrode materials on the atomic scale. In this talk I will highlight the use of several numerical simulation techniques (both classical and quantum mechanical types) in providing complimentary insights to an experimental study of the novel exotic lithium-rich cathode Li7Mn(BO3)3, illustrating how such computational materials modelling is indispensable in modern materials optimization.
November 2nd, 2015 - 3:15PM
Seminar by Prof. Enrique Solano on From Quantum Theatre to Scalable Quantum Simulators
ExQM Seminar for next Monday Nov. 2nd at 15.15pm in MPQ major lecture hall.
Prof. Enrique Solano (Univ. Bilbao) will talk on "From Quantum Theatre to Scalable Quantum Simulators"
We will introduce the field of quantum simulations from a wide aesthetic and scientific perspective. Along these lines, we will discuss the relevance of quantum simulations as a playground for our quantum theatre, as communicating vessels between unconnected fields, and as a scalable quantum technology. We will also provide pedagogical examples of quantum simulations in trapped ions and superconducting circuits, relating nonrelativistic and relativistic quantum dynamics, physical and unphysical quantum operations, as well as strong and ultrastrong light-matter interactions. Finally, we will discuss the advantages and disadvantages of current paradigms of quantum simulators, involving digital and analog concepts, and propose novel paths and concepts for assuring their scalability.
September 25th, 2015 - 11AM
Seminar by Dr Christian Gogolin on Equilibration, thermalization, and local stability of thermal states
ExQM Seminar for this Friday Sep. 25th at 11.00am in MPQ major lecture hall.
Dr Christian Gogolin (ICFO Barcelona) will talk on "Equilibration, thermalization, and local stability of thermal states"
In this talk it is shown how finite dimensional quantum systems in pure states, which evolve unitarily according to the Schrödinger equation, can exhibit thermodynamic behavior. More precisely, it will be discussed under which conditions local equilibration and thermalization can be ensured in such systems. I then discuss results on structural properties of thermal states of locally interacting quantum systems that in particular imply lower bounds on the critical temperatures below which such systems can exhibit phases with long range order. Finally, I touch on some work in progress concerning many-body localization.
September 17th, 2015 - 1:30PM
Seminar by Dr Thorsten Wahl on Tensor network states for the description of quantum many-body systems
ExQM Seminar for Sep. 17th 2015 at 13.30pm in MPQ Seminar Room B0.21.
Dr Thorsten Wahl (with Ignacio's group) will give his farewell talk before moving to Oxford about "Tensor network states for the description of quantum many-body systems"
Tensor network states (TNS) are applied to one and two dimensional systems: All translationally invariant matrix product states (one dimensional TNS) possessing long-range localizable entanglement, which is a non-local hidden order, are characterized. Furthermore, the first examples of chiral topological projected entangled pair states (two dimensional TNS) are presented. Their topological properties can be traced back to symmetries of the tensors describing the states. They are ground states of local gapless Hamiltonians and long-range gapped Hamiltonians.
August, 28th - 10:30AM
Seminar by Prof. Robert König on Protected gates for topological quantum field theories
ExQM Seminar this Fri, Aug. 28th at 10.30am in MPQ Seminar Room B0.22.
Prof. Robert König (with Michael Wolf's group) talk on "Protected gates for topological quantum field theories"
We give restrictions on locality-preserving unitary automorphisms U, which are protected gates, for 2-dimensional topologically ordered systems. For generic anyon models, we show that such unitaries only generate a finite group, and hence do not provide universality. For non-abelian models, we find that such automorphisms are very limited: for example, there is no non-trivial gate for Fibonacci anyons. More generally, systems with computationally universal braiding have no such gates. For Ising anyons, protected gates are elements of the Pauli group.These results are derived by relating such automorphisms to symmetries of the underlying anyon model: protected gates realize automorphisms of the Verlinde algebra. We additionally use the compatibility with basis changes to characterize the logical action.
This is joint work with M. Beverland, O. Buerschaper, F. Pastawski, J. Preskill and S. Sijher.
May 18th, 2015 - 11:30AM
Seminar by Dr Fernando Pastawski on Holographic quantum error-correcting codes
On Monday May 18th 2015, 11:30 am in MPQ, big lecture hall, Dr Fernando Pastawski (Caltech, USA) will talk on Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence
In this talk I will introduce a family of exactly solvable toy models of a holographic correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. The building block for these models are a special type of tensor with maximal entanglement along any bipartition, which gives rise to an exact isometry from bulk operators to boundary operators. The entire tensor network is a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the holographic correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. I will describe how bulk operators may be represented on the boundary regions mimicking the Rindler-wedge reconstruction.
Seminar by Michael Fischer on On-Chip Superconducting Microwave Interferometers
This Fri, May 8th, at 10.15am (again in MPQ, seminar room B0.22).
Michael Fischer (group R. Gross) will talk on On-Chip Superconducting Microwave Interferometers
In recent years, important progress towards using superconducting circuits for quantum information processing (QIP) has been made. In circuit quantum electrodynamics, photons inside superconducting transmission lines and resonators interact with artificial atoms, called qubits. In our approach to QIP, the qubit information may be encoded in a dual-rail setup, consisting of two superconducting transmission lines. Similarly to all-optical quantum computing the qubit states are superpositions of a microwave photon travelling in either one of the transmission lines. In QIP, operations between multiple qubits are needed to perform quantum algorithms. In order to use the dual-rail setup, these so called gates need to be implemented for two dual-rail encoded qubits. One important two qubit gate, a controlled phase gate, can be built with an interferometer equipped with a photon number dependent phase shifter. In this talk, I will present theoretical calculations and simulations, as well as measurements of on-chip interferometers fit for the application in such phase gates.
Seminar by Dr. Guido Bacciagaluppi on Did Bohr Understand EPR?
This Fri, Apr. 17th 2015, at 10.15am (again in MPQ, seminar room B0.22).
Dr. Guido Bacciagaluppi (Reader at U Aberdeen) will talk on Did Bohr Understand EPR?
Contrary to widespread belief, I argue that Niels Bohr's arguments in his reply to Einstein Podolsky and Rosen in 1935 take fully into account the separation between the two particles. Specifically, I argue that there is no sleight of hand in the passage from Bohr's discussion of a single particle passing through a slit and his subsequent discussion of the EPR example.
Seminar by Dr Jukka Kiukas on Local asymptotic normality for the estimation of dynamical parameters of an open quantum system
Fri, Apr. 10th 2015, at 10.15am (again in MPQ, seminar room B0.22).
Dr Jukka Kiukas (U Nottingham, formerly with Reinhard Werner) on: Local asymptotic normality for the estimation of dynamical parameters of an open quantum system
Input-output formalism is a well-known framework for describing continual monitoring of a Markovian open quantum system via measurements made on its environment (typically a quantised radiation field). Mathematically, the environmental noise is described in terms of quantum stochastic Wiener processes on the field Fock space. We consider the problem of identifying and estimating unknown dynamical parameters (Hamiltonian and the quantum jump operators) from the output field state. For this purpose, we first use quantum Ito calculus to derive an information geometric structure on the set of parameters, arising from the quantum Fisher information of the output state. The geometry comes with an associated CCR-algebra, and we then show that local estimation reduces asymptotically (with long observation times) to a Gaussian estimation problem on that CCR-algebra.
Seminar by Jakob Wierzbowski on Polarization control in few-layer MoS2 by electric-field-induced symmetry breaking
Mar. 13th 2015, (again in MPQ, seminar room B0.22),
Jakob Wierzbowski will talk on "Polarization control in few-layer MoS2 by electric-field-induced symmetry breaking"
detailed abstract
Seminar by Moritz August on A Brief Introduction to Neural Networks
This Fri, Mar. 6th 2015, (again in MPQ, seminar room B0.22).
Moritz August will talk on A Brief Introduction to Neural Networks
During the last decade, Machine Learning has become one of the most innovative and challenging fields at the intersection of Computer Science and Math. It has already been successfully applied to a wide array of domains, examples being the natural sciences, robotics and advertisement. Among the various techniques developed in the field, (Artificial) Neural Networks have proven to be one of the most powerful methods today and have led to significant improvements in many applications. Under the label of "Deep Learning" they also have gained attention in the public media and are the catalyst of the recent debate about the dangers of Artificial Intelligence.
In this talk, a brief introduction to the fundamentals of Neural Networks will be given. The talk will focus on their mathematical nature rather than on the neuroscientific interpretation and it will be explained what the term "Deep Learning" actually refers to."
Seminar by Stephan Welte on Experiments with single atoms and photons
This Fri, Feb. 27th 2015, (again in MPQ, seminar room B0.22).
Stephan Welte will talk on Experiments with single atoms and photons
In the field if cavity quantum electrodynamics, the deterministic interaction of single photons with single atoms can be achieved. I will present our experimental setup that allows to study and exploit atom-photon interactions in the strong-coupling regime. This is achieved by optically trapping atoms at the center of a high finesse cavity. As an example for the rich set of applications possible with this system, I will present the nondestructive detection of optical photons which are reflected off the cavity.
[Reiserer et al. Nondestructive Detection of an Optical Photon, Science342, 1349 (2013)]
After my talk, a lab tour is planned. All ExQM members are cordially invited to participate.
February, 20th - 2015
Seminar by Claudius Hubig on Strictly Single-Site DMRG (DMRG3S) with Subspace Expansion
This Fri, Feb. 20th, (again in MPQ, seminar room B0.22).
Claudius Hubig will talk on "Strictly Single-Site DMRG (DMRG3S) with Subspace Expansion"
The talk will also include an intro into finding ground states by DMRG before going into research results of http://arxiv.org/abs/1501.05504v1
February 13th, 2015 - 10:15AM
Seminar by Dr Volkher Scholz on Operationally-Motivated Uncertainty Relations for Joint Measurability and the Error-Disturbance Tradeoff
Here are in fact three annoncements:
Next Fri, Feb. 13th at 10.15am in MPQ (seminar room B0.22) we have the pleasure to hear
Dr Volkher Scholz (ETH Zurich) talk on Operationally-Motivated Uncertainty Relations for Joint Measurability and the Error-Disturbance Tradeoff
We derive new Heisenberg-type uncertainty relations for both joint measurability and the error- disturbance tradeoff for arbitrary observables of finite-dimensional systems (I will shortly mention the extension to position/momentum). The relations are formulated in terms of a directly operational quantity, namely the probability of distinguishing the actual operation of a device from its hypothetical ideal, by any possible testing procedure whatsoever. Moreover, they may be directly applied in information processing settings, for example to infer that devices which can faithfully transmit information regarding one observable do not leak any information about conjugate observables to the environment.
joint work with Joe Renes and Stefan Huber, ETH Zurich
February 6th, 2015 - 10:15AM
Seminar by Lukas Knips on Multipartite Entanglement Detection with Minimal Effort
Next Fri, Feb. 6th at 10.15am in MPQ (seminar room B0.22)
Lukas Knips (ExQM student in Weinfurter's group) talk on "Multipartite Entanglement Detection with Minimal Effort"
Certifying entanglement in a multipartite state is a demanding task. As a state of $N$ qubits is parametrized by $4^N-1$ real numbers, one may expect that the measurement complexity of generic entanglement detection is also exponential with $N$.
However, in special cases we can design indicators for genuine multipartite quantum entanglement using measurements in only two settings. I will describe the general method of deriving such criteria, which are based on a more general entanglement criterion using correlation measurements.
In the corresponding experiment we test two such non-linear witnesses, one constructed for four-qubit GHZ states, the other for Cluster states.
After introducing the theory and explaining our scheme for detecting genuine multipartite entanglement, I would like to show you around in our lab.
January 30th & 31st, 2015 - 9AM
Workshop: Quantum Computation, Quantum Information, and the Exact Sciences
"Quantum Computation, Quantum Information, and the Exact Sciences" motivated by foundations of quantum mechanics.
Fri Jan. 30th at 9.00am till Sat Jan. 31st at 18.30pm.
The venues are at LMU in the centre, see plans: http://www.qcompinfo2015.philosophie.uni-muenchen.de/practical-info/index.html
Apart from the more philosophial oriented talks, there are also some closer to us, in particular by
— Brukner (from Vienna, Zeilinger)
— Schack
— Briegel
For details, please see:
http://www.qcompinfo2015.philosophie.uni-muenchen.de/program/index.html
http://www.qcompinfo2015.philosophie.uni-muenchen.de/program/program_v10.pdf
January 20th, 2015 - 10AM
Lecture by Dr Marc Cheneau on An atomic Hong-Ou-Mandel experiment
Dr Marc Cheneau (CNRS Paris) will talk in the MPQ lecture hall at 10am on Tuesday, January 20th 2015.
"An atomic Hong-Ou-Mandel experiment"
The celebrated Hong, Ou and Mandel (HOM) effect is one of the simplest illustrations of two-particle interference, and is unique to the quantum realm. In the original experiment, two photons arriving simultaneously in the input channels of a beam-splitter were observed to always emerge together in one of the output channels. Here, we report on the realisation of a closely analogous experiment with atoms instead of photons. This opens the prospect of testing Bell's inequalities involving mechanical observables of massive particles, such as momentum, using methods inspired by quantum optics, with an eye on theories of the quantum-to-classical transition. Our work also demonstrates a new way to produce and benchmark twin-atom pairs that may be of interest for quantum information processing and quantum simulation.
January 16th, 2015 - 10:15AM
Seminar by Anna-Lena Hashagen on her masters work in finance mathematics
Fri Jan. 16th 2015 at 10.15am (MPQ small lecture hall),we will have an ExQM Seminar:
Anna-Lena Hashagen will give a survey talk on her masters work in finance mathematics (should be a good New Year refreshment)
"The Flesaker-Hughston Model for the Term-Structure of Interest Rates",
An interesting and still widely debated problem is the mathematical modelling of the term-structure of interest rates. Even though many attempts have been made to put forward an interest-rate model that fulfils all the desirable properties, these models usually have more than one shortcoming. Another major issue that is common to nearly all areas within financial mathematics is the urge of agreement with market practice. In order to minimise these shortcomings, Flesaker and Hughston have put forward a new methodology of interest-rate term-structure modelling called the Flesaker-Hughston model. This model class is very tractable and guarantees the positivity of interest rates. In the last few years, one very special model of theirs that has received particular interest is called the Flesaker-Hughston rational lognormal model. On top of the guaranteed positivity it resembles well-known market pricing formulas for popular interest-rate derivatives such as caps and floors as well as swaptions. This talk discusses the Flesaker-Hughston model class and places it within the environment of already existing models. We then thoroughly analyse the Flesaker-Hughston rational lognormal model with respect to the underlying dynamics of the instantaneous short-rate and the bond price, as well as the inherent boundaries that appear in this new framework.
Pricing formulas for caps and swaptions are derived by letting the martingale follow a diffusion process. Generalising this to an exponential L\'{e}vy process and using a method called the generalised Fourier transform, we also derive the price of a cap in this more realistic setting that includes jumps. The model is then calibrated to a full data set of risk-free zero-coupon bond prices in conjunction with either cap implied volatility, cap price, caplet price or caplet implied volatility mid-quotes on the US dollar three-month LIBOR rate.
Since the Flesaker-Hughston rational lognormal model gives closed-form expressions for caps, the calibration is extremely efficient. Using the real market data sets, the calibration analysis and the thorough analysis of the inherent model boundaries reveal that the instantaneous short-rate is bounded to such an extend that the Flesaker-Hughston rational lognormal model seems useful to price interest-rate options only in very specific cases — for those cases that lie within the boundaries. The guaranteed positivity of the instantaneous short-rate, i.e. the lower bound of zero, comes at a price of a restrictive upper
bound.
Seminar by Dr Volckmar Nebendahl on Optimized Quantum Error Correction Codes for Experiments
Mon., Dec 8th at 13.30h in MPQ in seminar room B0.21 there will be a seminar by:
Dr Volckmar Nebendahl (Blatt group, Innsbruck) on "Optimized Quantum Error Correction Codes for Experiments".
Details in http://arxiv.org/pdf/1411.1779v1.pdf.
October 8th, 2014 - 3:15PM
Seminar by Luca Arceci on Quantum solitons in the XXZ model with staggered external field
Wed., Oct 8th 2014 at 15.15h (3.15pm) in MPQ in seminar room B0.21 there will be a seminar by the ExQM theory applicant:
Luca Arceci from Univ. of Bologna on "Quantum solitons in the XXZ model with staggered external field".
The 1-D 1/2-spin XXZ model with staggered external magnetic field, when restricting to low field, can be mapped into the quantum sine-Gordon model through bosonization: this assures the presence of soliton, antisoliton and breather excitations in it. In particular, the action of the staggered field opens a gap so that these physical objects are stable against energetic fluctuations.
In the present work, this model is studied both analytically and numerically. On the one hand, analytical calculations are made to solve exactly the model through Bethe ansatz: the solution for the XX + h staggered model is found by means of Jordan-Wigner transformation and Bethe ansatz separately, while eff orts are made to extend the latter approach to the XXZ + h staggered model (without finding its solution). On the other hand, grounding on results from the application of quantum fi eld theories on the quantum sine-Gordon model, the energies of these excitations are pinpointed through static DMRG (Density Matrix Renormalization Group) for diff erent values of the parameters in the hamiltonian. Breathers are found to be in the antiferromagnetic region only, while solitons and antisolitons are present both in the ferromagnetic and antiferromagnetic region. Their single-site z- magnetization expectation values are also computed to see how they appear in real space, and time-dependent DMRG is employed to realize quenches on the hamiltonian parameters to monitor their time-evolution.
The results obtained reveal the quantum nature of these objects and provide some information about their features. Further study of their properties could lead to the realization of a two-state qubit through a soliton-antisoliton pair.
Seminar by Dr Oleg Szehr on Quantum Phases in Systems with Matrix-Product Ground States
Fri., June 20th at 10.15am in MPQ in seminar room B0.22, there will be a QCCC farewell seminar by:
Dr Oleg Szehr (who just moved to Cambridge, UK) on "On Quantum Phases in Systems with Matrix-Product Ground States"
We introduce Matrix Product states and their parent Hamiltonians and define the notion of a quantum phase in this framework. We provide a classification of phases of one-dimensional systems both with unique as well as degenerate ground states. We address the question of how robust the energy gap in the parent Hamiltonian model is to perturbations and provide conditions under which robustness is guaranteed.
Our methods rely on a close connection between translation-invariant Matrix product states and the Perron-Frobenius theory of certain associated quantum channels.
June 18th, 2014 - 3PM
Seminar by Matteo Rossi on Dynamics of Quantum Correlations for Two-Qubit Systems Interacting with Classical Noisy Environments
Wed., June 18th at 3.00pm 2014 in MPQ in seminar room B0.21, there will be a seminar by the ExQM applicant:
Matteo Rossi from Univ. of Parma on "Dynamics of Quantum Correlations for Two-Qubit Systems Interacting with Classical Noisy Environments".
In this talk we consider single- and two-qubit systems coupled to classical stochastic fields and address both the decoherence and the non-Markovianity induced by the external fields.
Studying the interaction of a quantum system with its environment plays a fundamental role in the development of quantum technologies. Decoherence is detrimental for applications and it may be induced by classical or quantum noise, i.e. by the interaction with an environment described classically or quantum-mechanically. The classical description is often more realistic to describes environments with a very large number of degrees of freedom and it has also been shown that certain quantum environments may be described with equivalent classical models. We thus analyze in detail the dynamics of quantum correlations (entanglement and quantum discord) and evaluate the non-Markovianity of the induced dynamical quantum map for two-qubit systems interacting with classical stochastic fields, focusing on Gaussian processes.
May 16th, 2014 - 10AM
Seminar by Dr. Dmitry Savostyanov on Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions
This week, we have distinguished guests from U Southampton and the Max-Planck for Maths in the Sciences, Leipzig. We are sharing ideas with the Schollwöck (LMU) and the Huckle (TUM) group in view of simulating large systems.
On Friday, May 16th 2014 at 10.00am (sharp) in MPQ on campus Garching Small Lecture Hall we will have a Post-QCCC/Pre-ExQM seminar by:
Dr. Dmitry Savostyanov, University of Southampton (Co-authors: Sergey Dolgov, MPI MiS Leipzig, and Ilya Kuprov, U Southampton) on "Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions"
When high-dimensional problems are concerned, not many algorithms can break the curse of dimensionality and solve them efficiently and reliably. Among those, tensor product algorithms seem to be the most promising.
The first attempt to merge classical iterative algorithms and DMRG/MPS methods was made in a way, where the second Krylov vector is used to expand the search space on the optimisation step. The idea proved to be useful, but the implementation was based on the fair amount of physical intuition, and the algorithm is not completely justified.
We have recently proposed the AMEn algorithm for linear systems [3, 4], that also injects the gradient direction in the optimisation step, but in a way that allows to prove the global convergence of the resulted scheme. The scheme can be easily applied for the computation of the ground state—the differences to the algorithm of S. White [13] are emphasized in [5]. The AMEn scheme was recently applied for the computation of extreme eigenstates [7], using the block-TT format proposed in [2].
We aim to extend this framework and the analysis to other problems: eigenproblems, time-dependent problems, high-dimensional interpolation, and matrix functions; as well as to a wider list of high-dimensional problems.
This is a jointwork with Sergey Dolgov at the Max-Planck Institute for Mathematics in the Sciences, Leipzig, and Ilya Kuprov at the University of Southampton, UK.
selected refs.:
[2] S. V. Dolgov, B. N. Khoromskij, I. V. Oseledets, and D. V. Savostyanov. Computation of extreme eigenvalues in higher dimensions using block tensor train format.
Computer Phys. Comm., 185(4):1207–1216, 2014. doi:10.1016/j.cpc.2013.12.017.
[3] S.V. Dolgov and D.V. Savostyanov. Alternating minimal energy methods for linear systems in higher dimensions. Part I: SPD systems. arXiv preprint 1301.6068, 2013.
URL: http://arxiv.org/abs/1301.6068.
[4] S. V. Dolgov and D. V. Savostyanov. Alternating minimal energy methods for linear systems in higher dimensions. Part II: Faster algorithm and application to nonsymmetric systems. arXiv preprint 1304.1222, 2013. URL: http://arxiv.org/abs/
[5] S. V. Dolgov and D. V. Savostyanov. Corrected one-site density matrix renormalization group and alternating minimal energy algorithm. In Proc. of ENUMATH 2013, accepted, 2014. URL: http://arxiv.org/abs/1312.6542.
[7] D. Kressner, M. Steinlechner, and A. Uschmajew. Low-rank tensor methods with subspace correction for symmetric eigenvalue problems. MATHICSE preprint 40.2013, EPFL, Lausanne, 2013.
[13] Steven R. White. Density matrix renormalization group algorithms with a single center site. Phys. Rev. B, 72(18):180403, 2005. doi:10.1103/PhysRevB.72.180403.
more details can be found in: SavostjanovMPQ14.pdf
April 7th, 2014 - 5PM
Seminar by Dr. Andreas Ruschhaupt on Shortcuts to Adiabaticity
Monday, April 7th 2014 at 5pm in MPQ on campus Garching Seminar Room B0.22 we will have a Post-QCCC/Pre-ExQM seminar by:
Dr. Andreas Ruschhaupt, Cork University, Ireland (formerly jun. Prof. with Reinhard Werner) on "Shortcuts to Adiabaticity"
Quantum adiabatic processes -that keep constant the populations in the instantaneous eigenbasis of a time-dependent Hamiltonian-are very useful to prepare and manipulate states, but take typically a long time. This is often problematic because decoherence and noise may spoil the desired final state, or because some applications require many repetitions.
"Shortcuts to adiabaticity" are alternative fast processes which reproduce the same final populations, or even the same final state, as the adiabatic process in a finite, shorter time [1]. We present such "shortcuts to adiabaticity" for the manipulation of the atomic motional state [2] as well as for the passage from one internal atomic state to another [3-5]. We especially study and compare the stability of different shortcut schemes concerning different types of perturbations like, for example systematic and noise errors [4, 6] or errors originating from unwanted transitions to other levels [7].
[1] E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno,
A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, Xi Chen and J. G. Muga,
Adv. At. Mol. Opt. Phys. 62 (2013) 117
[2] Xi Chen, A. Ruschhaupt, S. Schmidt, A. del Campo,
D. Guéry-Odelin and J. G. Muga,
Phys. Rev. Lett. 104 (2010) 063002
[3] Xi Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin and J. G. Muga,
[4] A. Ruschhaupt, X. Chen, D. Alonso and J. G. Muga,
New J. Phys. 14 (2012) 093040
[5] S. Ibáñez, Xi Chen, E. Torrontegui, J. G. Muga and A. Ruschhaupt,
[6] D. Daems, A. Ruschhaupt, D. Sugny and S. Guerin,
[7] A. Kiely and A. Ruschhaupt, arXiv:1312.3210
Former PhD Programme QCCC (Quantum Computing, Control, and Communication):
Big Data and Responsability
QCCC invited to the 5th ENB Forum in the Grand Aula of LMU with three lectures by the guests Dr. Frank Schirrmacher, Prof. Dr. Dieter Kranzlmüller and Dr. Richard von Schirach on the topic "Big Data and Responsability". See video recording by eUniversity at LMU.
Exchange of Ideas on Quantum-Many-Body Systems at MPQ
In order to prepare for "life after QCCC", we will have an informal mini-workshop
to exchange ideas on quantum many-body systems with Prof. Schollwöck, who could not come to Prien. More precisely, the goal is to exchange ideas of numerical methods and the demands for them in backing quantum simulations of multi-body systems.
We meet Tue Nov. 12th, at the MPQ with the schedule as follows:
11:30am coffee/lunch gathering at MPQ cafeteria.
Seminar Room of Cirac Group (2nd floor):
12:00 am Prof. Schollwöck: "Nature of the Spin-Liquid Ground State of the S=1/2 Heisenberg Model
on the Kagome Lattice"
12:45 am extended discussion: Identifying Cutting Edge Problems
1:00-1.30pm coffee break
Seminar Room B0.21 of MPQ (ground floor):
1.30pm Thomas Barthel: "Some tensor network state techniques and entanglement in condensed matter systems with application to fermionic systems"
2.00pm : Thomas Schulte-Herbrüggen : "Symmetry Principles in the Quantum Systems Theory of Many-Body Systems"
2.45-3.15 pm Coffee
3.15-3.45 pm "Robert Zeier: "Two Teasers: 1. Simulating Sparse Qubit Systems,
2. Further Results on Fermionic Systems"
3.45-4.15 pm Marie Carmen Banuls: "Tensor Network Methods Applied to Lattice Gauge Theories"
4.15-4.45 pm Prof. Thomas Huckle/Konrad Waldherr: "(1) General Overview on Recent Tensor Methods in Maths
plus (2) Numerical (Multi-)Linear Algebra in Quantum Tensor Networks (Results and Perspectives)"
5.00 pm general discussion
~5.45 pm end with option to have dinner in Garching
October 17th - 21st, 2013
QCCC Closing Symposium and Workshop
QCCC Closing Symposium and Workshop on Campus Garching / in Prien am Chiemsee
Quantum Measurement Theory lecture by Rob Spekkens
Three lectures on "Quantum Measurement Theory" by Rob Spekkens, Paul Busch and Teiko Heinosaari, supported by QCCC.
Business Meeting regarding the 4th QCCC Workshop in Prien.
Seminar by Daniel Reitzner on Quantum Measurements and Joint Measurability
Talk on
Quantum measurements and joint measurability
Dr. Daniel Reitzner (TUM)
Measurements in quantum mechanics are, compared to classical measurements, somewhat non-intuitive and in particular can be incompatible; i.e. a pair of measurements on a single system can turn out to be impossible to perform at once. Although it is quite often stated that this is a basis for Heisenberg uncertainties; we will show the limitations and dangers of this description.
With a more modern and in a sense more general definition of quantum measurement via POVMs that we shall introduce as well, we will see, that simultaneous (and/or sequential) measurements are a tricky and still unresolved concept that has an impact on modern applications within quantum information community.
Seminar by Thorsten Wahl Localizable Entanglement & Gaussian Fermionic PEPS
Localizable Entanglement & Gaussian Fermionic PEPS
Thorsten Wahl (MPQ)
This talk consists of two parts. In the first one I will introduce the concept of Localizable Entanglement, which is important for the detection of topological quantum phase transitions and ideal quantum repeaters in the case where the Localizable Entanglement is constant over arbitrary long distances. Finally, I will provide a necessary and sufficient condition for the later case (also denoted as long-range Localizable Entanglement) for Matrix Product States.
The second part of my talk is devoted to the approximation of topological insulators by Gaussian fermionic PEPS which are the free Fermionic version of Projected Entangled Pair States. I will show under which conditions Gaussian fermionic PEPS are topologically non-trivial.
February 18th/19th, 2013
QCCC & TMP Block Course by Michael Keyl on Quantum Information Theory in Infinite Dimensions: An Operator-Algebra Approach
Block course on:
Quantum Information Theory in Infinite Dimensions: An Operator-Algebra Approach.
PD Prof. Michael Keyl (FU Berlin)
Most of quantum information theory is developed in the framework of finite dimensional Hilbert spaces, and therefore not directly applicable to systems like free and interacting non-relativistic particles, spin-systems in the thermodynamic limit or relativistic field models, where an infinite dimensional description is required. In some cases a more or less direct generalization is possible (e.g. by replacing finite sums with absolutely converging sequences) but this approach is very limited and misses many of the more interesting aspects of infinite dimensional systems. In other words mathematically and conceptually new tools are needed. In this context the theory of operator algebras provides a very powerful framework, which is particularly useful for the study of infinite degrees of freedom systems.
The purpose of this lecture series is to introduce into this theory and its applications in qantum physics. Apart from the corresponding mathematical foundations we will show how elementary concepts of quantum theory can be reformulated and how the differences between finite dimensions, infinite dimensions but finite degrees of freedom, and infinite degrees of freedom can be related to operator algebras and their representations. Furthermore we will study infinite spin systems, their entanglement properties and their connection to advanced operator algebraic topics, like type and cassification of von Neumann algebras.
Seminar by Oleg Szehr on Spectral Convergence Bounds for Finite Classical and Quantum Markov Chains
Spectral Convergence Bounds for Finite Classical and Quantum Markov Chains
Oleg Szehr (TUM)
In this talk I present a new framework that yields spectral bounds on norms of functions of transition maps for finite, homogeneous Markov chains. The techniques employed work for bounded semigroups, in particular for classical as well as for quantum Markov chains and they do not require additional assumptions like detailed balance, irreducibility or aperiodicity. I use the method in order to derive convergence bounds that improve significantly upon known spectral bounds. The core technical observation is that power-boundedness of transition maps of Markov chains enables a Wiener algebra functional calculus in order to upper bound any norm of any holomorphic function of the transition map.
Seminar by Michael Keyl on Mathematical Physics with hbar
Mathematical Physics with \hbar
In this talk I will review a number of research projects from different areas of quantum physics, including: Mean field flucutations of spin-systems and their relation to continuous variable quantum systems; quantum field theory in space-times with causality violations; and quantum control of bosonic and Fermionic systems.
Inaugural Lecture by Michael Wolf on Short Stories from Quantum Information Theory
Lecture on
Short Stories from Quantum Information Theory
Prof. Michael M. Wolf (TUM)
The talk aims at providing a taste of Quantum Information Theory exemplified through two problems from different branches of the field.
In the first part we will encounter quantum correlations that are arbitrarily stronger than their classical counterparts. In physics this is related to the foundations of quantum theory, in mathematics to Grothendieck type inequalities within operator space theory, and in theoretical computer science to the reduction of communication complexity. The latter perspective suggests how – in the distant future – the scheduling of the colloquium mightbe made more efficient.
The second part will shed new light on the energy gap problem from condensed matter theory. Despite considerable effort and interest, there is basically neither a proof technique nor a numerical method known for solving this type of problem. We will argue that the roots of this difficulty may be deeper than expected by showing that there are cases for which there cannot be a proof (in the sense of Gödel) or an algorithm (in the sense of Turing).
TUM Mathematics Colloquium by David Gross on Compressed Sensing and Matrix Completion: From Single Pixel Cameras to Quantum State Tomography
Compressed Sensing and Matrix Completion: From Single Pixel Cameras to Quantum State Tomography
Prof. David Gross (Uni Freiburg)
Very time the release button of a digital camera is pressed, several megabytes of raw data are recorded. But the size of a typical jpeg output file is only 10% of that. What a waste! Can't we design a process which records only the relevant 10% of the data to begin with? The recently developed theory of compressed sensing achieves this trick for sparse signals. I will give a short introduction to the ideas and the math behind compressed sensing.
A basis-independent notion of "sparsity" for a matrix is its rank. One is thus naturally led to the "low-rank matrix recovery" problem: can one reconstruct and unknown low-rank matrix from few linear measurements? The answer is affirmative. The arguably simplest proof to date is based on ideas from quantum information theory. In the second half of the presentation, I will talk about applications and proof techniques for the matrix theory, including the links to quantum.
Seminar by David Reeb on Fault-Ignorant Algorithms
"Fault-Ignorant Quantum Search"
Abstract: We investigate the problem of quantum searching on a noisy quantum computer. Taking a "fault-ignorant" approach, we design quantum algorithms that solve the task for various different noise strengths, possibly unknown beforehand. The rationale is to avoid costly overheads, such as traditional quantum error correction.
Proving lower bounds on algorithm runtimes, which may depend on the actual level of noise, we find that the quadratic speedup is lost (in our noise models). Nevertheless, for low noise levels, our algorithms outperform the best noiseless classical search algorithm. Finally, we provide a more general framework to formulate fault-ignorant algorithms.
Seminar by Marc Cheneau on Spin Physics in Optical Lattices
In this lecture, we will present two recent experimental results obtained in the Bloch group that illustrate the potential of ultracold atomic gases to understand the emergence of many-body phenomena. In a first part, we will show that a gas of ultracold atoms in a Mott-insulating state represent a fairly good realization of the Heisenberg model, with the electronic hyperfine state playing the role of an effective spin. Using a new experimental technique, we were able to directly reveal the Heisenberg coupling by monitoring the real-time dynamics of a single spin impurity introduced in the system in a controlled way. Beyond this paradigmatic, but analytically solvable problem, we also explored a regime where real particle hopping becomes important and the spin impurity gets dressed by the surrounding bath to form a polaronic quasiparticle.
In the second part of the lecture, we will introduce a different setup that is a realization of a traverse-field Ising model with long-range interactions. This time we coupled the atoms to a laser beam driving a transition to a highly-excited electronic state, a so-called Rydberg state. The enormous van der Waals interaction between two atoms in such a state gives rise to strong spatial correlations over distances much larger than the interparticle distance. Here we could observe the spontaneous formation of well-defined geometric structures of a few Rydberg excitations and gather some evidence that the system had been excited to a highly-entangled many-body state.
Seminar by Christian Sames on Controllable Single‐Atom Phase Shifter Working in the Single‐Photon Level
"Controllable Single-Atom Phase Shifter Working in the Single-Photon Level"
In our system we strongly couple a single atom to the light field of an optical resonator. I will give a brief introduction to what can be deduced from the phase of the intra-cavity field and how we can build a single atom phase shifter with this system.
Seminar by Daniel Lercher on Gaussian superactivation requires squeezing
The talk is dedicated to the topic of superactivation of the quantum capacity of Gaussian channels. Smith et. al. have recently shown that for two such channels with zero capacity Superactivation (0 + 0 > 0) can be achieved. They provided explicit examples that make use of active optical devices like squeezers.
In this talk I'll show that there is no superactivation for Gaussian channels that are generated by passive means.
Seminar by Juan Bermejo on A Gottesman-Knill Theorem for all Finite Abelian Groups
In this seminar, I will introduce "normaliser circuits", a family of quantum operations that play a relevant role in quantum algorithms:
prominent examples of normaliser gates are quantum Fourier transforms (QFT)—sometimes said to be "the source of various exponential quantum speedups"—, subroutines that generate highly entangled states and adaptive measurements.
Recently we have investigated the computational power of normaliser circuits and found that, in spite of their apparent quantumness, they can be efficiently simulated in a classical computer. Thus, a quantum computer operating within this set of gates can not offer exponential quantum speed-ups over classical computation, regardless e.g. the number of QFT it uses. Our result generalises a well-known theorem of Gottesman and Knill, valid for qubits, to systems that do not decompose as products of small subsystems.
I will introduce some elements of group theory needed to understand our theorem and the main tool we developed to prove it: a stabiliser formalism for high dimensions. The latter may be of independent interest in quantum error correction and fault tolerant quantum computing. I will also explain the relation of these results with Shor's algorithm.
Seminar by Thomas Schulte-Herbrüggen on Reachability under Noise Control
"Combining Coherent and Noise Control:"
'How to Transfer between Arbitrary n-Qubit States by Coherent Control and Simplest Switchable Noise on a Single Qubit'
We explore reachable sets of open $n$-qubit quantum systems the coherent parts of which are under full unitary control and that have just one qubit whose unital or non-unital noise amplitudes can be modulated in time such as to provide an additional degree of incoherent control. In particular, adding bang-bang control of amplitude damping noise (non-unital) allows the dynamic system to act transitively on the entire set of density operators. This means one can transform any initial quantum state into any desired target state. Adding switchable bit-flip noise (unital), on the other hand, suffices to explore all states majorised by the initial state. We have extended our optimal control algorithm (DYNAMO) by degrees of incoherent control so that these unprecedented reachable sets can systematically be exploited for experimental settings. Numerical results are compared to constructive analytical schemes.
Extreme Optics in Semiconductors
Seminar by Prof. Mark Sherwin from Santa Barbara on "Extreme Optics in Semiconductors: When Quasiparticles Collide, and 1+1=11".]
Seminar by Dr Oleg Szehr on Perturbation Theory for Fixed Points of Quantum Channels
Titel: Perturbation Theory for Fixed Points of Quantum Channels
Abstract: It is clear that if the transition matrix of an irreducible quantum Markov-process has a sub dominant eigenvalue which is close to 1 then the quantum Markov-process is ill conditioned in the sense that there are stationary states which are sensitive to perturbations in the transition matrix. However, the converse of this statement has heretofore been unresolved. The purpose of this talk is to present upper and lower bounds on the condition number of the chain such that the bounding terms are determined by the closeness of the sub dominant eigenvalue to unity.
We obtain perturbation bounds which relate the sensitivity of the chain under perturbation to its rate of convergence to stationarity.
Seminar by Dr David Gross
Seminar by Dr David Gross.
Seminar by Dr Markus Grassl on Polynomial Invariants of Three-Qubit Systems
Seminar by Dr Markus Grassl on "Polynomial Invariants of Three-Qubit Systems":
Polynomial invariants provide a tool to characterise quantum states with respect to local unitary transformations. Unfortunately, the situation becomes very complicated already for mixed states of three qubits due to combinatorial explosion.
After an introduction to the mathematical background and general tools, the talk will present preliminary results for mixed quantum states and Hamiltonians for three-qubit systems.
The talk is based on joint work in progress with Robert Zeier.
October 7th - 10th, 2011
3rd QCCC Workshop
3rd QCCC Workshop in Bernried
September, 4th - 9th, 2011
CoQuS Summer School
Invitation to the CoQuS Summer School.
Heraeus Summer School
Invitation to the Heraeus Summer School "Modern Statistical Methods in QIP", 22-26 August 2011, Bad Honnef.
Seminar by Michael-James Kastoryano on Convergence of Quantum Processes: going beyond the gap
I will give a brief introduction to the problem of estimating the convergence time a quantum channel to its fixed point(s).
In particular, I will show that bounding the gap of the channel is sometimes insufficient for bounding the convergence time.
I will review some of the tools which are available (and some which will soon be), and discuss some of the difficulties in extending classical mixing time methods to the quantum setting.
Finally, I will provide some applications of these methods introduced, and give an outlook on some open problems.
ENB Forum
ENB Forum: From the Nobel Prize to Being Entrepreneur
Lectures by
Theodor W. Hänsch: Passion for Precision
Ronald Holzwarth: Firmerngründung mit Nobelpreistechnologie
Seminar by Bogdan Pirvu on MPS-Simulations of Translationally Invariant Spin Chains with Periodic Boundary Conditions
We present several algorithms for the simulation of translationally invariant (TI) quantum spin chains with periodic boundary conditions (PBC). By using TI matrix product states (MPS) we can in general reduce the computational cost by a factor N, with N the number of spins in the chain.
First we present an algorithm for the approximation of ground states (GS) that is based on the computation of the gradient of the energy [1]. We achieve a scaling of the computational cost of O(D3n2) + O(D3mn), where D is the virtual bond dimension of the MPS and m and n are some parameters that will be explained in more detail in the talk. There is a tradeoff between the parameters n and m and we show how to find the optimal balance. The analysis of the numerical results confirms previous observations regarding the induced correlation length of MPS with finite D [2, 3]. Furthermore we observe a crossover between the finite-N scaling and finite-D scaling in the context of critical quantum spin chains similar to the one observed by Nishino [4] in the context of classical two dimensional systems.
Next we present an algorithm for the approximation of dispersion relations that uses as an ansatz MPS-based states with well defined momentum [5]. Here, we achieve a scaling of the computational cost of O(D6N2). Due to the large D scaling we are restricted to comparatively small D. Nonetheless we obtain very good approximations of one-particle excitations. The numerical results yield some insight into the interpretation of the quasiparticles that occur in the exact solution of the Quantum Ising Model with PBC.
Seminar by Christian Gogolin on Thermalization in nature and on a quantum computer
Using the assumption that thermodynamic systems evolve towards Gibbs states, i.e. states with a well defined temperature, statistical mechanics and thermodynamics have been amazingly successful in explaining a wide range of physical phenomena. In stark contrast to this strong justification by corroboration of these theories, the question of whether and how the methods of statistical mechanics and thermodynamics can be justified microscopically was still wide open until recently.
With new mathematical tools from quantum information theory becoming available, there has been a renewed effort to settle this old question.
I will present and discuss a necessary and a sufficient condition for the emergence of Gibbs states from the unitary dynamics of quantum mechanics and show how these new insights into the process of equilibration and thermalization can be used to design a quantum algorithm that prepares thermal states on a quantum computer/simulator.
Seminar by Volkher B. Scholz on Information theoretic questions on von-Neumann algebras
We study information theoretic questions on von-Neumann algebras. There, one first has to discuss the notion of independent subsystems, which leads to a well-known open question in operator algebra theory, the embedding problem of Connes. I will clarify the equivalence, and proceed by describing some information theoretic tasks in the setting of von-Neumann algebras, as well as discussing the connection to quantum cryptography.
Guided Lab Tour
Seminar by John Marriott
Seminar by John Marriott on Optimal control theory and its application to maximizing contrast in magnetic resonance imaging
Seminar by Christine Tobler on A Matlab toolbox for tensors in hierarchical Tucker format and its application to the solution of high-dimensional linear systems
The hierarchical Tucker format is a storage-efficient scheme to approximate and represent tensors of possibly high order. From a tensor network perspective, it represents a network of order-3 tensors without cycles. Thus, it is a generalization of the Tensor Train or Matrix Product States (MPS) format.
This talk introduces a Matlab toolbox, along with the underlying methodology and algorithms, providing a convenient way to work with this format. The toolbox not only allows for the efficient storage and manipulation of tensors but also offers a set of tools for the development of higher-level algorithms.
As an example for the use of the toolbox, an algorithm for solving high-dimensional linear systems, namely parameter-dependent elliptic PDEs, is shown. This is joint work with Daniel Kressner, ETH Zurich.
Seminar by Boris Khoromskij
Seminar by Boris Khoromskij on Quantics-TT approach to high-dimensional SPDEs and to DMRG in quantum molecular dynamics.
Seminar by Yuval Sanders on Relative quantifiers of quantum informational resources
A full quantification of the entanglement of a quantum state is desirable if the degradation of entanglement in some quantum information processing task is to be minimized. A protocol for the accomplishment of a task can then be evaluated by the amount of entanglement consumed to enact that protocol. Such quantification is typically provided by a collection of entanglement monotones.
We propose a generalization of entanglement monotones that may provide greater flexibility in the quantification of entanglement. Rather than quantifying the entanglement of a state directly, we suggest a relative quantification: a direct comparison of one entangled state to another. We provide an example of such relative quantification for a quantum information resource known as frameness.
Seminar by Xinhua Peng on Spin qubits for quantum simulation – Simulating physical phenomena on a quantum computer
Simulating quantum mechanical systems is a classical hard problem because the computational difficulties hinges on the exponential growth of the size of Hilbert space with the number of particles in the system.
In the context of quantum information processing, this difficulty becomes the main source of power: in some situations, information processors based in quantum mechanics can process information exponentially faster than classical systems. From the perspective of a physicist, one of the most interesting applications of this type of information processing is the simulation of quantum systems. We call a quantum information processor that simulates other quantum systems a quantum simulator.
Using a kind of nuclear magnetic resonance simulator, we implement the simulations of the Heisenberg spin models by the use of average Hamiltonian theory and observe the quantum phase transitions by using different measurements, e.g., entanglement, fidelity decay and geometric phase: the qualitative changes that the ground states of some quantum mechanical systems exhibit when some parameters in their Hamiltonians change through some critical points. In particular, we consider the effect of the many-body interactions. Depending on the type and strength of interactions, the ground states can be product states or they can be maximally entangled states representing different types of entanglement.
When the many-body interaction (such as the three-body interaction) takes part in the competition, new critical phenomena that cannot be detected by the traditional two-spin correlation functions will occur.
By quantifying different types of entanglement, or by using suitable entanglement witnesses, we successfully detect two types of quantum transitions. Besides this, using such a NMR quantum simulator, we can also simulate the static properties and dynamics of chemical systems, such as the ground-state energy of Hydrogen molecule.
Seminar by Martin Plenio on Describing system environment interactions in the non-perturbative regime
Prof. Martin Plenio (Universitaet Ulm):
Describing system environment interactions in the non-perturbative regime
Recent experiments have provided strong evidence for the existence of quantum coherence in the early stages of photosynthesis. Subsequent theory work shows that the optimal operating regime lies in the regime where the system-environment interaction is strong so that the system is neither fully quantum coherent nor fully classical, but rather half way in between. In this regime perturbative treatments of the system environment interaction are not valid. Here I discuss the above issue and then present a novel approach to the numerical and analytical study of spin systems in strong contact with environments made up of harmonic oscillators.
This talk is based on
M.B. Plenio and S.F. Huelga
– Dephasing assisted transport: Quantum networks and biomolecules –
New J. Phys. 10, 113019 (2008) and E-print arXiv:0807.4902 [quant-ph]
F. Caruso, A.W. Chin, A. Datta, S.F. Huelga and M.B. Plenio
– Highly efficient energy excitation transfer in light-harvesting complexes: The fundamental role of noise-assisted transport –
J. Chem. Phys. 131, 105106 (2009) and E-print arXiv:0901.4454 [quant-ph]
J. Prior, A.W. Chin, S.F. Huelga and M.B. Plenio
– Efficient simulation of strong system-environment interactions –
Phys. Rev. Lett. 105, 050404 (2010) and E-print arXiv:1003.5503 [quant-ph]
A.W. Chin, A. Rivas, S.F. Huelga and M.B. Plenio
– Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials –
J. Math. Phys. 51, 092109 (2010) and E-print arXiv:1006.4507 [quant-ph]
Seminar by Haidong Yuan on Optimal Control of Quantum Systems
Optimal Control of Quantum Systems
We consider the optimal control problem of transferring population between states of a quantum system where the coupling proceeds only via intermediate states that are subject to decay. We pose the question whether it is generally possible to carry out this transfer.
For a single intermediate decaying state, we recover the Stimulated Raman Adiabatic Passage (STIRAP) process for which we present analytic solutions in the finite time case. The solutions yield perfect state transfer only in the limit of infinite time.
We also present analytic solutions for the case of transfer that has to proceed via two consecutive intermediate decaying states. We show that in this case, for finite power the optimal control does not approach perfect state transfer even in the infinite time limit. We generalize our findings to characterize the topologies of paths that can be achieved by coherent control.
Seminar by Corey O'Meara on Geometric Properties of Special Classes of Completely Positive Linear Maps
Completely positive linear mappings between C*-algebras were originally developed in the 1950's as a special case of positive linear operators between matrix algebras. Within the last two decades, mathematical physicists have determined that completely positive maps play a crucial role in quantum information theory as structures which model information transfer between quantum systems.
This talk will serve as an introduction to two classes of completely positive maps: the Schur maps which arise from the Schur matrix product and maps which are equal to their adjoint.
After focusing on results concerning the geometry of these two sets of CP maps, we introduce a general framework that unifies certain classes of CP maps in terms of C*-subalgebras of Mn.
Seminar by Patrick Rebentrost on The Role of Quantum Coherence in Photosynthetic Energy Transfer
Seminar by Patrick Rebentrost
On the Role of Quantum Coherence in Photosynthetic Energy Transfer
Recent experiments have provided evidence for long-lived electronic coherence in photosynthetic light-harvesting complexes at room temperature. This talk presents some of the work performed in theAspuru-Guzik group on the Fenna-Matthews-Olson complex. This includes the basic concept of environment-assisted excitonic transport and a quantification of the role of coherence by its contribution to the transport efficiency.
We find that, depending on the spatial correlations in the phonon environment, there is about a 10% contribution of coherent dynamics to the exciton transfer efficiency. In addition, we investigate a time-convolutionless non-Markovian master equation approach and show our quantum chemistry inspired way of incorporating atomistic detail of the protein environment into the exciton dynamics.
Workshop on Optomechanics (programme)
Seminar by Frank Wilhelm on Quantum optics on a chip - photon counters and NOON states
Seminar by Frank Wilhelm
Quantum optics on a chip – photon counters and NOON states
Circuit quantum electrodynamics is a maturing field in which the physics of quantum optical setups is realized in cryogenic electric circuits, profiting from large achievable coupling strengths. Elements like cavities, artificial atoms, mirrors, and beamsplitters have been successfully demonstrated. The missing element is a single-photon counter as microwave photons are usually amplified instead of counted, and as most of these amplifiers are noisy.
I will present the Josephson Photomultiplier, a simple device that allows single photon counting at high efficiency and bandwidth. Quantum optics with multiple modes has highlightes NOON states – states in which N photons are in a superposition of two arms of an interferometer for quantum-enhanced metrology. I am going to show how these can be created deterministically in circuit QED.
The success of such an experiment is difficult to determine as the reconstruction of a two-mode density matrix at large photon number is forbiddingly cumbersome. We are going to show that it is much more efficient to test for a hypothesis state and then estimate the overlap between the hypothetical state and the physical state using nonlinear programming.
Seminar by Noomen Belmechri
Seminar by Noomen Belmechri on Single atom interferometry with neutral atoms in a state-selective 1D optical lattice
Seminar by Prof. M. Christandl on The Uncertainty Principle in the Presence of Quantum Memory
Seminar by Prof. M. Christandl
The uncertainty principle lies at the heart of quantum theory, illuminating a dramatic difference with classical mechanics. The principle bounds the uncertainties of the outcomes of any two observables on a system in terms of the expectation value of their commutator. It implies that an observer cannot predict the outcomes of two incompatible measurements to arbitrary precision.
However, this implication is only valid if the observer does not possess a quantum memory, an unrealistic assumption in light of recent technological advances. In this work we strengthen the uncertainty principle to one that applies even if the observer has a quantum memory. We provide a lower bound on the uncertainty of the outcomes of two measurements which depends on the entanglement between the system and the quantum memory.
We expect our uncertainty principle to have widespread use in quantum information theory, and describe in detail its application to quantum cryptography. The talk is based on joint work with Mario Berta, Roger Colbeck, Joe Renes and Renato Renner (http://arxiv.org/abs/0909.0950).
Seminar by Dr. Robert Zeier on Dynamical Quantum Systems: Controllability, Symmetries, and Representation Theory
Seminar by Dr. Robert Zeier
"Dynamical Quantum Systems: Controllability, Symmetries, and Representation Theory"
We analyze the controllability of dynamical quantum systems. One can decide controllability by computing the Lie closure [1] which is sometimes cumbersome. These topics can be discussed likewise for translationally invariant lattices [2]. Building on previous work [3,4], we propose an additional method which utilizes the symmetry properties of the considered system.
We obtain as a necessary condition for controllability that the system should not have any symmetries and act therefore irreducibly. But this condition is not sufficient as there exist irreducible subalgebras of the maximal possible system Lie algebra. We classify the irreducible subalgebras and their inclusion relations relying on results of Dynkin [5]. Using optimized computer programs we can tabulate irreducible subalgebras up to dimension 215 (i.e. 15 qubits) complementing results of McKay and Patera [6].
For concrete dynamical quantum systems many irreducible subalgebras can be ruled out as obstructions for full controllability and we present algorithms to this end. Our results provide an insight into the question when spin, bosonic, and fermionic systems can simulate each other. We will give a short introduction to the relevant representation theory of Lie algebras.
[1] Jurdjevic/Sussmann, J. Diff. Eq. 12, 313 (1972)
[2] Kraus/Wolf/Cirac, Phys. Rev. A 75, 022303 (2007)
[3] Sander/Schulte-Herbrüggen, http://arxiv.org/abs/0904.4654
[4] Polack/Suchowski/Tannor, Phys. Rev. A 79, 053403 (2009)
[5] Borel/Siebenthal, Comment. Math. Helv. 23, 200 (1949);
Dynkin, Trudy Mosov. Mat. Obsh. 1, 39 (1952),
Amer. Math. Soc. Transl. (2) 6, 245 (1957);
Dynkin, Mat. Sbornik (N.S.) 30(72), 349 (1952),
Amer. Math. Soc. Transl. (2) 6, 111 (1957)
[6] McKay/Patera, Tables of Dimensions, Indices, and Branching Rules
for Representations of Simple Lie Algebras (1981)
Squeezing the Matrix: efficient spin dynamics simulation algorithms
Seminar by Dr. Ilya Kuprov on "Squeezing the Matrix: efficient spin dynamics simulation algorithms".
The Energy Minimization Problem in the Control of Dissipative Spin-1/2 Particles
Seminar by Prof. Bernard Bonnard on "The Energy Minimization Problem in the Control of Dissipative Spin-1/2 Particles".
Seminar by Prof. Jr-Shin Li on Ensemble Controllability and by Justin Ruths on A Pseudospectral Method for Optimal Control of Open Quantum Systems
Seminar by Prof. Jr-Shin Li and by Justin Ruths
In this paper, we present a unified computational method based on pseudospectral approximations for the design of optimal pulse sequences in open quantum systems. The proposed method transforms the problem of optimal pulse design, which is formulated as a continuous-time optimal control problem, to a finite dimensional constrained nonlinear programming problem.
This resulting optimization problem can then be solved using existing numerical optimization suites. We apply the Legendre pseudospectral method to a series of optimal control problems on open quantum systems that arise in Nuclear Magnetic Resonance (NMR) spectroscopy in liquids. These problems have been well studied in previous literature and analytical optimal controls have been found.
We find an excellent agreement between the maximum transfer efficiency produced by our computational method and the analytical expressions. Moreover, our method permits us to extend the analysis and address practical concerns, including smoothing discontinuous controls as well as deriving minimum-energy and time-optimal controls. The method is not restricted to the systems studied in this article and is applicable to optimal manipulation of both closed and open quantum systems.
Seminar by Corey O'Meara on Schur Maps and Self-Dual Quantum Channels
Seminar by Corey O'Meara
The relationship between characteristics of quantum channels and the geometry of their respective sets can provide a useful insight to some of their underlying properties. First we will discuss a special class of random unitary channels, namely, the Schur maps. We then use the generalization of these maps to motivate the study of what we call Self-Dual quantum channels. Some preliminary geometric properties of the set of such maps are investigated and compared to the geometry of the set of Schur maps. Finally, some preliminary algebraic results are discussed which involve the eigenvalues of Self-Dual quantum channels.
Seminar by Volkher Scholz on Anderson Localization in Disordered Quantum Walks
Seminar by Volkher Scholz
Anderson Localization in Disordered Quantum Walks
(**Volkher Scholz**, Albert Werner, and Andre Ahlbrecht)
We study a Spin-$\frac{1}{2}$-particle moving in a one dimensional lattice subjected to disorder induced by a random space dependent coin. The discrete time evolution is given by a family of random unitary quantum walk operators, where the shift operation is assumed to be non-random. Each coin is an independent identically distributed random variable with values in the group of two dimensional unitary matrices. We find that if the probability distribution of the coins is absolutely continuous with respect to the Haar measure, then the system exhibits localization. That is, every initially localized particle remains on average and up to exponential corrections in a finite region of space for all times.
Seminar by Shai Machnes
Seminar by Shai Machnes on "Optimal control for superfast cooling of trapped ions, using a Matlab toolbox for quantum information calculations"
Seminar by Prof. Mikio Nakahara on Topological Quantum Computing with p-Wave Superfluid
Seminar by Prof. Mikio Nakahara
It is shown that Majorana fermions trapped in three p-wave superfluid vortices form a qubit in a topological quantum computing (TQC). Several similar ideas have already been proposed: Ivanov [Phys. Rev. Lett. 86, 268 (2001)] and Zhang et al [Phys. Rev. Lett. 99, 220502 (2007)] suggested schemes in which a qubit is implemented with two and four Majorana fermions, respectively, where a qubit operation is performed by braiding the world lines of these Majorana fermions. Naturally the set of quantum gates thus obtained is a discrete subset of the relevant unitary group.
We propose a new scheme, where three Majorana fermions form a qubit. We show that continuous qubit operations are made possible by braiding the Majorana fermions complemented with dynamical phase change. Moreover, it is possible to introduce entanglement between two such qubits by geometrical manipulation of some vortices involved.
October 10th - 14th, 2009
2nd QCCC Workshop in Bad Tölz
Seminar by Fernando Pastawski on How Long Can Passive Quantum Memories Withstand Depolarizing Noise?
Seminar by Fernando Pastawski
How Long Can Passive Quantum Memories Withstand Depolarizing Noise?
Abstract: Existing fault tolerance theorems state that robust quantum computation and in particular, quantum memories may be achieved by growing the number of dedicated resources. Such theorems assume the availability of fresh ancillas (qubits in a predefined state) and the possibility of periodically applying recovery operations. Experimentally however, these requirements have shown to be hard to meet. In an attempt to provide a simpler path, many body Hamiltonians have been proposed with the hope that they could through their dynamics alone provide long protection times to quantum information. I will explain recent results which show that under a depolarizing noise model, protection times may not exceed O(log N) and such scaling is achievable by many body non-local Hamiltonians. I will go on to mention existing proposals for protecting Hamiltonians and describe some limitations we have found for the information lifetime under comparatively weak Hamiltonian perturbations.
Seminar by Sebastian Nauerth on the Exploration of Side Channels in our BB84 Freespace Quantum Key Distribution System
Seminar by Sebastian Nauerth
Exploration of Side Channels in our BB84 Freespace Quantum Key Distribution System
The security of quantum key distribution, (QKD) is based on physical laws rather than assumptions about computational complexity: An adversary will necessarily disturb the communication by his quantum measurement. However, real implementations will be sensitive to side-channel attacks, i.e. to information losses due to distinguishabilities in other degrees of freedom, which an adversary can measure without causing errors.
We are running an implementation of the BB84 protocol installed on top of two university buildings in downtown Munich. Using attenuated laser pulses in combination with decoy states we are able to establish a secret key over a distance of 500 m. Our system is fully remote controlled and allows for continuous and fast QKD. I will report on the characterization of this QKD system with respect to side channels of the transmitter and the receiver and also show some attacks.
Seminar by Pierre de Fouquieres on Limitations of Quantum Optimal Control
Limitations of Quantum Optimal Control
Abstract: The use of local, typically gradient based, optimisation algorithms has proven to be particularly effective in achieving control objectives for quantum mechanical systems. Some authors have sought a theoretical justification for this empirical observation of the numerical techniques' behavior. A set of papers, falling under the banner of "optimal control landscapes" claim to offer such a justification, in the form of proofs that such optimisation always achieves the control objective (ignoring numerical limitations). I will present a number of problems inherent in said "landscape" analysis.
Seminar by Louis H. Kauffman on Topological Quantum Information Theory
Seminar by Louis H. Kauffman
In this talk we discuss relationships between topology and quantum computation.
Since the discovery of Peter Shor's quantum algorithm for the prime factorization of natural numbers, there has been intense interest in the discovery of new quantum algorithms and in the construction of quantum computers. It is possible that topology will enter in a deep way in the construction of quantum computers based on phenomena such as the quantum Hall effect, where braiding of quasiparticles describes unitary transformations rich enough to produce the quantum computations.
This talk will describe the mathematics of such braiding and its relationship with algorithms to compute topological invariants such as the Jones polynomial.
Just so, relationships with braiding go beyond the quantum Hall effect and are of interest for constructing quantum gates and quantum algorithms. The talk will discuss these directions and our present project in collaboration with the research group of Prof. Glaser (on this campus) to instantiate quantum algorithms for the Jones polynomial using NMR (Nuclear Magnetic Resonance Spectroscopy).
The talk will be self-contained both in terms of mathematics and physics.
Seminar by Howard Carmichael
Seminar by Howard Carmichael on "Open Quantum Systems: Cavity QED with Dissipation"
Seminar by Mercedes Rosas on The Reduced Kronecker Coefficients of the Symmetric Group
The reduced Kronecker coefficients of the symmetric group Mercedes Rosas, Universidad de Sevilla, Spain. (in collaboration with Emmanuel Briand, U. de Sevilla, and Rosa Orellana, Dartmouth College).
The understanding of the Kronecker coefficients of the symmetric group (the multiplicities of decomposition into irreducible the tensor products of two irreducible representations of the symmetric group) is a longstanding open problem. Recently, its study has appeared naturally in some seemingly unrelated areas.
For instance, Matthias Christandl has showed that the problem of the nonvanishing of Kronecker coefficients is equivalent to the problem of compatibility of local spectra, and Ketan Mulmuley has set the problem of proving that the positivity of a Kronecker coefÞcients can be decided in polynomial time at the heart of his Geometric Complexity Theory.
In view of the difficulty of studying of the Kronecker coefÞcients, it is legitimate to consider some closely related, and maybe simpler objects, the reduced Kronecker coefficients, defined as limits of certain stationary sequences of Kronecker coefficients. We attempt to show that the study of the reduced Kronecker coefficients could sheld light on the Kronecker coefficients.
We will introduce the reduced Kronecker coefficients, and describe some of their known properties. Then, we will describe a useful formula to compute Kronecker coefficients from the reduced ones, and, among other results, present a sharp bound for a family of Kronecker products to stabilize.
Seminar by Wu Xing
Seminar by Wu Xing on "Experiments on a Superconducting Atom Chip".
March 26th, 2009 (MPQ seminar)
Seminar by Dmitry Yudin
Seminar by Dmitry Yudin on Dynamics of a dilute Fermi-Bose condensed gas mixture.
Seminar by Matthias Christandl on Post-selection technique for quantum channels with applications to quantum cryptography
Seminar by Matthias Christandl
Post-selection technique for quantum channels with applications to quantum cryptography
We propose a general method for studying properties of quantum channels acting on an n-partite system, whose action is invariant under permutations of the subsystems. Our main result is that, in order to prove that a certain property holds for any arbitrary input, it is sufficient to consider the special case where the input is a particular de Finetti-type state, i.e., a state which consists of n identical and independent copies of an (unknown) state on a single subsystem. A similar statement holds for more general channels which are covariant with respect to the action of an arbitrary finite or locally compact group.
Our technique can be applied to the analysis of information-theoretic problems. For example, in quantum cryptography, we get a simple proof for the fact that security of a discrete-variable quantum key distribution protocol against collective attacks implies security of the protocol against the most general attacks. The resulting security bounds are tighter than previously known bounds obtained by proofs relying on the exponential de Finetti theorem [Renner, Nature Physics 3,645(2007)]. This is joint work with Robert Koenig and Renato Renner http://arxiv.org/abs/0809.3019
Seminar by Matthew B. Hastings
Seminar by Matthew B. Hastings on "A Counterexample to Additivity of Minimum Output Entropy" (How to kill a most famous conjecture in quantum information – alternative title, not by the author).
Seminar by Angela Meyer on Localisable Entanglement of 1D Cluster States
Title: Localisable entanglement of 1D cluster states
We consider quantum chains in cluster states under the influence of a variable magnetic field. After reviewing the derivation of the ground state we compute the localisable entanglement of the two outermost qubits, after local measurements have been made on the inner ones, for different chain lengths. The result is mostly as intuitively expected: the entanglement decreases monotonously with the field strength.
Seminar by Dominique Sugny
Seminar by Dominique Sugny on Geometric optimal control of dissipative quantum systems.
Seminar by Ulrich Schollwöck on Disentangling Many-Body Quantum Systems and Large-Scale Linear Algebra
Seminar by Ulrich Schollwöck
Title: DISENTANGLING MANY-BODY QUANTUM SYSTEMS AND LARGE-SCALE LINEAR ALGEBRA
In this talk I want to show how many important questions of quantum many-body physics (solid state physics, quantum optics) naturally lead to a highly efficient description of quantum states by sets of matrices whose manipulation involves large-scale linear algebra of sparse matrices. I will illustrate the various challenges by current physical problems from solid state physics and quantum optics and would like to try to give a flavour why theoretical physicists would be interested in insights from computer science and numerical mathematics to tackle such problems.
Seminar by Andreas Eberlein on The Charge-Density Wave Behaviour in the t-J-Holstein Model
Seminar by Andreas Eberlein
Titel: "Charge-density wave behaviour in the t-J-Holstein Model"
"We study the charge-density wave behaviour in the one-dimensional t-J-Holstein Model. Using the Projector-based Renormalization Method (PRM), we investigate the influence of a small exchange interaction on the metal-insulator transition known from the spinless Holstein Model. In this talk, I will review the work on my diploma thesis and will present some results."
Seminar by Guifre Vidal
Seminar by Guifre Vidal on Entanglement Renormalization.
September 8 - 12th, 2008
Summer School: Complex Quantum Systems
Summer School "Complex Quantum Systems" hosted as a CoQuS-QCCC get-together by the CoQuS Phd programme of excellence at the University of Vienna.
Seminar by Peter Pemberton-Ross
Seminar by Peter Pemberton-Ross on Controllability of Spin Systems.
Seminar by Anne Nielsen on State Preparation by Means of Optical Measurements
Title: State Preparation by Means of Optical Measurements
Speaker: Anne Nielsen
The ability to control and manipulate the state of quantum systems is important in order to use such systems for technological purposes and fundamental studies of quantum mechanics. Subjecting a system to different Hamiltonians leads to different unitary time evolutions, but the state collapse accompanying quantum mechanical measurements opens several additional possibilities to change the state of a quantum system in a desired way, and measurements thus constitute a powerful state preparation tool. In the talk we investigate the influence of measurements on the dynamics of quantum systems and provide examples of various state preparation protocols that are based on optical measurements.
Seminar by Daniel Burgarth on Quantum Control of Coupled Spin Systems: Algebraic and Open System Approach
Seminar by Daniel Burgarth
Title: Quantum Control of Coupled Spin Systems: Algebraic and Open System Approach
Speaker: Daniel Burgarth, Oxford
We compare two independent methods of controlling qubits which are coupled by an always-on Hamiltonian. In either case, it is possible to perform algorithms on large arrays by acting on a small subset. While the algebraic method has the advantage of requiring minimal resources, the open system approach provides an explicit way how to achieve control. We give examples of systems which are controllable only by the open system approach and show new results on spin chains as universal quantum interfaces.
Guest lecture by Louis H. Kauffman on Quantum Computing and Quantum Topology
Guest lecture by Louis H. Kauffman
Title: Quantum Computing and Quantum Topology
Speaker: Louis H. Kauffman, UIC
This talk will discuss the construction of sets of universal gates for quantum computing and quantum information theory and their relationship with topological computing, quantum algorithms for computing quantum link invariants such as the Jones polynomial and questions about the relationship between quantum entanglement and topological entanglement. We will discuss the creation of universal gates (in the presence of local unitary transformations) by using solutions to the Yang-Baxter equation and we will discuss the use of braided recoupling theory (q-deforemed spin networks) to create unitary representations of the braid group rich enough to support quantum information theory and quantum computing. In particular we give quantum algorithms for computing the colored Jones polynomials and the Witten-Reshetikhin-Turaev invariants.
Elite-Cup 2008 (ENB soccer tournament)
Elite-Cup 2008 (ENB soccer tournament) in Neufahrn near Freising
QCCC Miniworkshop
QCCC Miniworkshop on "Mathematical Control Theory and Quantum Applications" (with TopMath)"
Talk by Amr Fahmy
Talk by Amr Fahmy on "Thermal State Quantum Computing"
May 5 - 10th, 2008
Guest professor Marek Zukowski
March 31 - April 4th, 2008
Guest professor Reinhard Werner
October 27 - 30th, 2007
QCCC Workshop
QCCC Workshop on "Foundations and Future Prospects of QIP"
For details see the workshop page.
Imprint + Data Protection | CommonCrawl |
The Intermediate Value Theorem
Uses of The Intermediate Value Theorem
Theorem 1 (The Intermediate Value Theorem): Suppose that $f$ is a continuous function on the closed interval $[a, b]$ where $a < b$. Then for all $p$ such that $f(a) < p < f(b)$ or $f(b) < p < f(a)$, there exists a $c \in [a, b]$ such that $f(c) = p$.
The intermediate value theorem can be used in order to determine if there are roots to a given function on an interval $[a, b]$. If $f(a) < 0 < f(b)$, or $f(b) < 0 < f(a)$, then there must be a point in which the function crosses the $x$-axis on the interval $[a, b]$ given that the function is continuous since on that interval there are values of $f$ that are negative, and there are values of $f$ that are positive.
Show that there exists a root of the function $x^3 - 3x + 1 = 0$ on the interval $[0, 1]$.
We can easily calculate this by determining $f(0)$ and $f(1)$ as follows:
\begin{align} f(0) = 0^3 - 3(0) + 1 = 1 \\ f(1) = 1^3 -3(1) + 1 = -1 \end{align}
Now since $f(0) > 0$, and $f(1) < -1$, and since $x^3 - 3x + 1 = 0$ is continuous for all real numbers $x$ because it is a polynomial, then there must be a point where the function $f$ crosses the $x$-axis. Hence, there exists at least one root to the function.
Note: The intermediate value theorem does not tell you how many times the function crosses the $x$-axis, as there is a possibility that $x^3 - 3x + 1 = 0$ could have crossed the $x$-axis at most $3$ times (since the degree or highest exponent in this polynomial is $3$). | CommonCrawl |
Why is dark matter used to explain big movements but not small ones
Just read that there are perturbations of stuff in the outer solar system that suggest there are Earth sized planets far out from the sun, in the dark.
When the same sort of effects are seen at the galactic scale, cosmologists use Dark Matter to explain where the unknown force is coming from.
So why when its small scale (solar system) do they use Newtonian physics and on the galactic scale use hypothetical Dark Matter ? Where is the cut off ?
Also, could Dark Matter be just dark matter, stuff that does not emit radiation and sits in the dark ? More and more dark stuff seems to be discovered every year, its odd that this option is not followed up, whereas Dark Matter seems to get all the attention even though there is no experimental evidence for it.
solar-system dark-matter
Jane ParksJane Parks
$\begingroup$ "Also, could Dark Matter be just dark matter, stuff that does not emit radiation and sits in the dark" That's pretty much exactly what 'mainstream' dark matter is. $\endgroup$ – Danu Jan 20 '15 at 12:56
Dark matter's effect on Solar System scales exists in theory, but it is nigh undetectably small.
A related question deals with the density of dark matter in the Solar System. It comes out to less than one hydrogen atom per cubic centimeter, some 20 orders of magnitude less dense than a typical solid or liquid, and 17 orders of magnitude less dense than air. Moreover, this is a rather uniform distribution of matter, so its gravitational effect on planets' motions is essentially nothing.
Once you zoom out to the scale of the galaxy, you start to see that dark matter is more concentrated toward the center of the galaxy and less concentrated (very) far out. Thus it adds to the gravitational potential well of the galaxy, affecting the orbits of stars and star clusters going around the galaxy.
So it comes down to the fact that the dynamics of the Solar System are utterly dominated by an extremely concentrated overdensity (the Sun, as well as the planets) compared to the average density of the galaxy (a few atoms per cubic centimeter). This is an important point, by the way: Being on a planet near a star is a highly unlikely place to find oneself compared to typical locations drawn uniformly from the universe; most places in space are much, much more devoid of normal matter.
One analogy is to think of a desert with gently rolling sand dunes. On large scales, the shape of the landscape is well approximated by smooth curves capturing the shape of the dunes. But if you look at the sand with a microscope, the surface is very rough, with all sorts of little bits of rock piled haphazardly. A microscopic organism crawling around on the sand hardly notices the shape of the dune it's on, but it cares very much about the size of a nearby sand grain.
Also note there is a lot of other evidence for dark matter beyond the dynamics of galaxies, and all this evidence agrees in terms of how much dark matter is needed to explain the observations. In fact, I'd say we would all be as incredulous as you sound if galactic dynamics were all we had. But the success of cosmology (for which no one has developed a credible dark-matter-free alternative) is strong evidence that dark matter is indeed spread out in the universe.
Looking at a galaxy, there is the bulge in the middle and the disc in one plane. In this disc region are all the stars and star-systems, e.g. our solar system is $\sim\frac{2}{3}$ away from the center of the Milky Way.
While objects in the outer regions of a galaxy are affected by the vast dark matter (DM) halo, the dominant attraction in the disc, e.g. our solar system, comes from the luminous matter.
I found this picture:
http://qph.is.quoracdn.net/main-qimg-d9481e8d735afc44ab459a33f4816c91?convert_to_webp=true
All of this havin gbeen said, we have no idea what dark matter really is! ;)
CleverClever
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Mass density of dark matter in solar system near us
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black dwarf stars and dark matter | CommonCrawl |
Multimodal game bot detection using user behavioral characteristics
Ah Reum Kang1,
Seong Hoon Jeong2,
Aziz Mohaisen1 &
Huy Kang Kim2
As the online service industry has continued to grow, illegal activities in the online world have drastically increased and become more diverse. Most illegal activities occur continuously because cyber assets, such as game items and cyber money in online games, can be monetized into real currency. The aim of this study is to detect game bots in a massively multiplayer online role playing game (MMORPG). We observed the behavioral characteristics of game bots and found that they execute repetitive tasks associated with gold farming and real money trading. We propose a game bot detection method based on user behavioral characteristics. The method of this paper was applied to real data provided by a major MMORPG company. Detection accuracy rate increased to 96.06 % on the banned account list.
A game bot is an automated program that plays a given game on behalf of a human player. Game bots can earn much more game money and items than human users because the former can play without requiring a break. Game bots also disturb human users because they consistently consume game resources. For instance, game bots defeat all monsters quite rapidly and harvest items, such as farm produce and ore, before human users have an opportunity to harvest them. Accordingly, game bots cause complaints from human users and damage the reputation of the online game service provider. Furthermore, game bots can cause inflation in a game's economy and shorten the game's lifecycle, which defeats the purpose for which game companies develop such games (Lee et al. 2016).
Several studies for detecting game bots have been proposed in academia and industry. These studies can be classified into three categories: client-side, network-side, and server-side. Most game companies have adopted client-side detection methods that analyze game bot signatures as the primary measure against game bots. Client-side detection methods use the bot program's name, process information, and memory status. This method is similar to antivirus programs that detect computer viruses (Mohaisen and Alrawi 2014). Client-side detection methods can be readily detoured by game bot developers, in addition to degrading the computer's performance. For this reason, many countermeasures that are based on this approach, such as commercial anti-bot programs, are not currently preferred. Network-side detection methods, such as network traffic monitoring or network protocol change analysis, can cause network overload and lag in game play, a significant annoyance in the online gaming experience. To overcome these limitations of the client-side and network-side detection methods, many online game service providers employ server-side detection methods. Server-side detection methods are based on data mining techniques that analyze log data from game servers. Most game servers generate event logs whenever users perform actions such as hunting, harvesting, and chatting. Hence, these in-game logs facilitate data analysis as a possible method for detecting game bots. Online game companies analyze user behaviors or packets at the server-side, and then online game service providers can selectively block those game bot users that they want to ban without deploying additional programs on the client-side. For that, most online game service providers prefer server-side detection methods. In addition, some online game companies introduced big data analysis system approaches that make use of data-driven profiling and detection (Lee et al. 2016). Such approaches can analyze over 600 TB of logs generated by game servers and do not cause any side-effects, such as performance degradation or conflict with other programs.
The literature is rich of various works on the problem of game bot detection that we review in the following. Table 1 summarizes and compares various server-side detection schemes. We present key server-side detection methods classified into six analysis categories: action frequency, social activity, gold farming group, sequence, similarity, and moving path.
Table 1 Previous research on server-side detection
Action frequency analysis uses the fact that the frequencies of particular actions by game bots are much higher than that of human users. To this end, Chen and Hong (2007) studied the dynamics of certain actions performed by users. They showed that idle and active times in a game are representative of users and discriminative of users and bots. Thawonmas and Kashifuji (2010) utilized the information on action frequencies, types, and intervals in MMORPG log data. To detect game bots, Park et al. (2010) selected six game features, namely map changes, counter-turn, rest states, killing time, experience point, and stay in town. Chung et al. (2013) were concerned with various game play styles and classified them into four player types: killers, achievers, explorers, and socializers. Zhang et al. (2015) clarified user behaviors based on game playing time. While this approach provides high accuracy, it is limited in several ways. First, they only focus on observations of short time window, thus they are easy to evade. Second, some of such work focuses only on a limited feature space, thus the approach is prone to confusing bots with "hardcore" users (users who use the game for long times; who are increasingly becoming a phenomenon in the online gaming communities).
Social activity analysis uses the characteristics of the social network to differentiate between human users and game bots. Varvello and Voelker (2010) proposed a game bot detection method emphasizing on the social connections of players in a social graph. Our previous study chose chat logs that reflect user communication patterns and proposed a chatting pattern analysis framework (Kang et al. 2012). Oh et al. (2013) used the fact that game bots and human users tend to form respective social networks in contrasting ways and focused on the in-game mentoring network. Our other previous work found that the goal of game bot parties is different from that of human users parties, and proposed a party log-based detection method (Kang et al. 2013). This approach is however limited to detecting misbehavior in party play and cannot detect misbehavior in single play games.
Gold farming group analysis uses the virtual economy in online games and traces abnormal trade networks formed by gold farmers, merchants, bankers, and buyers. To characterize each player, Itsuki et al. (2010) used four types of statistics: total action count, activity time, total chat count, and the amount of virtual currency managed in a given period of time. Seo and Kim (2011) analyzed gold farming group connection patterns using routing and source location information. Kwon et al. (2013) investigated gold farming networks and detected the entire network structure of gold farming groups. This work, while distantly related, is not concerned with the detection of bots, but with understanding the unique roles each bot plays in the virtual underground ecosystem given a valid detection.
Sequence analysis uses iterated sequence datasets from login to logout. Ahmad et al. (2009) studied activity sequence features, defined as the number of times a given player engages in an activity, such as the number of monsters killed and the number of times the player was killed. Platzer (2011) used the combat sequence each avatar produces. Lee et al. (2015) examined the full action sequence of users on big data analysis platform. While such technique has been shown to work in the past, such feature lacks context, and might be easily manipulated by bot settings.
Similarity analysis uses the fact that game bots have a strong regular pattern because they play to earn in-game money. Kwon and Kim (2011) derived vectors using the frequency of each event and calculated the vector's cosine similarity with a unit vector. Game bots repeatedly do the same series of actions, therefore their action sequences have high self-similarity. Lee et al. (2016) employed self-similarity measures to detect game bots. They proposed the self-similarity measure and tested it in three major MMORPGs ("Lineage", "Aion" and "Blade&Soul"). Their scheme requires a lot of data of certain behavior for establishing self-similarity.
Moving path analysis uses the fact that game bots have pre-scheduled moving paths, whereas human users have various moving patterns. Thawonmas et al. (2007) provided a method for detecting landmarks from user traces using the weighted entropy of the distribution of visiting users in a game map. They presented user clusters based on transition probabilities. To identify game bots and human users, van Kesteren et al. (2009) took advantage of the difference in their movement patterns. Mitterhofer et al. (2009) detected the players controlled by a script with repeated movement patterns. Pao et al. (2010) used the entropy values of a user's trace and a series of location coordinates. They employed a Markov chain model to describe the behavior of the target trajectory. Pao et al. (2012) applied their method to various types of trajectories, including handwriting, mouse, and game traces, in addition to the traces of animal movement. However, their feature also can be evaded and noised by adaptive bots that integrate human-like moving behavior.
Contribution.
To this end, we collaborated with NCSOFT, Inc., one of the largest MMORPG service companies in South Korea, in order to analyze long-term user activity logs and understand discriminative features for high fidelity bot detection. In this paper, we propose a game bot detection framework. Our framework utilizes multimodal users' behavioral characteristic analysis and feature extraction to improve the accuracy of game bot detection. We adopted some features discovered in the prior literature in confirmed in our analysis, as well as some new features discovered in this study. We combine those features in a single framework to achieve better accuracy and enable robust detection. An additional contribution of this work is also the exploration of characteristics of the misclassified users and bots, highlighting plausible explanations that are in line with users and bots features, as well as the game operations.
Before elaborating on the framework and workflow of our method, we first highlight the dataset and ethnical guidelines used for obtaining and analyzing it.
Dataset.
To perform this study, we rely on a real-world dataset obtained from the operation of Aion, a popular game. Our Aion dataset contains all in-game action logs for 88 days, between April 9th and July 5th of 2010. During this period, there were 49,739 characters that played more than 3 h. Among these players, 7702 characters were game bots, identified and labeled by the game company. The banned list was provided by the game company to serve as the ground truth, and each banned user has been vetted and verified by human labor and active monitoring.
Ethnical and privacy considerations.
In order to perform this study we follow best practices in ensuring users privacy and complying with ethical guidelines. First, the privacy of users in the data is ensured by anonymizing all personal identifiable information. Furthermore, consent of users is taken into account by ensuring that data analysis is within the scope of end user license agreement (EULA): upon joining Aion, users grant NCSOFT, Inc. the full permission to use and share user data for analysis purpose with parties of NCSOFT's choosing. One of such parties was our research group, and for research purpose only.
Framework and workflow
Our proposed framework for game bot detection is shown in Fig. 1. We posed the problem of identifying game bots as a binary classification problem. At a high-level, our method starts with a data collection phase, followed by a data exploration phase (including feature extraction), a machine learning phase, and a validation phase. In the following we highlight each of those phases.
Data collection.
In the data collection phase, we gathered a dataset that combines in-game logs and chat contents.
Data exploration.
We then performed data exploration in order to comprehend the characteristics of the dataset using data preprocessing, feature extraction, feature representation, exploration, and selection for best discriminating between bots and normal users. In the feature representation procedure, we followed standard methods for unifying data and reducing its dimensionality. For example, we quantized each network measure into three clusters with low, medium, and high values using the k-means clustering algorithm. In the feature exploration phase, we selected the components of the data vectors and pre-pocessed them. For example, we determined seven activities as social interactions and quantified the diversity of social interactions by the Shannon diversity entropy. In the feature selection phase, we selected significant features with the best-first search, greedy-stepwise search, and information gain ranking filter to avoid overfitting and reduce the features (thus improving the performance).
Machine learning.
In the machine learning phase, we choose algorithms (e.g., decision tree, random forest, logistic regression, and naïve Bayes) and parameters (e.g., k-fold cross-validation parameters, specific algorithm parameters, etc.), and feed the data collected using the selected features in their corresponding representation. We further build models (using the data fed) and establish baselines by computing various performance metrics.
In the evaluation phase, we summarize the performance of each classifier with the banned account list provided by the game company as a ground truth, by providing various performance measures, such as the accuracy, precision, recall, and F-measure.
Game bot detection framework based on user behavioral characteristics
Used features and their gap.
As indicated in Table 2, we classified the features we used in our work into personal and social features. Given that the aim of game bots is to earn unfair profits, there is a gap between the values of the personal features of game bots and those of human users. The personal features can be also categorized into player information and actions. The player information features include login frequency, play time, game money, and number of IP address. The player action features contain sitting (an action taken by players to recover their health), earning experience points, obtaining items, earning game money, earning player kill (PK) points, harvesting items, resurrecting, restoring experience points, being killed by a non-player and/or player character (NPC/PC), and using portals. The frequency and ratio of these actions reflects the behavioral characteristics of game bots and human users. For example, game bots sit more frequently than human users to recover health and mana points. Moreover, a player can acquire PK points by defeating players of opposing factions. PK points can be used to purchase various items from vendors. PK points are also used to determine a player's rank within the game world. In Aion, the more PK points a player has, the higher is the player's rank. The high ranking player can feel a sense of accomplishment. On the other hand, it is seen that game bots are not interested in rank.
Table 2 Personal and social features
In addition, there is gap between the values of the social features of game bots and those of human users because game bots do not attempt to social as humans. The social features can be categorized into group activities, social interaction diversity, and network measures. The features of group activities include the average duration of party play and number of guild activities. Party play is a group play formed by two or more players in order to undertake quests or missions together. The goals of party play commonly are to complete difficult quests by collaboration and enjoy socialization. Interestingly, some game bots perform party play, but the goal of party play of the game bots is different from that of human users. Their aim is to acquire game money and items faster and more efficiently. Hence, there are the behavioral differences between game bots and human users. The social interaction diversity feature indicates the entropy of party play, friendship, trade, whisper, mail, shop, and guild actions. Game bots concentrate only on particular actions, whereas human users execute multiple tasks as needed to thrive in the online game world. The player's social interaction network can be represented as a graph with characters as the nodes and interactions between them as the edges. An edge between two nodes (players) in this graph may, for example, highlight the transfer of an item between the two nodes. The features of network measures include the degree, betweenness, closeness, eigenvector centrality, eccentricity, authority, hub, PageRank, and clustering coefficient. The definitions of the network measures are listed in Table 3.
Table 3 Definition of network measures
In this section we review more concretely the behavioral characteristics of bots and humans based on the various features utilized, and using the aforementioned dataset. We then propose our bot detection mechanism based on discriminative features and by elaborating on details of the high level workflow in the previous section, including the performance evaluation.
Behavioral characteristics
We compared the distribution of player information features in order to identify the difference between the behavioral characteristics of game bots and human users more concretely. Figure 2 shows how intensively game bots play games. Game bots often connect to the game and spend much longer time playing it than human users. Game bots can play a given game for 24 consecutive hours, whereas human users hardly connect to the game during working hours. Game bots invest significant time in a game until they are blocked. Figure 2c shows the cumulative distribution of the maximum number of items harvested by users per day. It is almost impossible for human users to harvest more than 1000 items per day. Since this is repetitive and hard work, human users are easily exhausted. Nevertheless, 60 % of game bots harvest more than 5000 items a day. This is an obvious characteristic for identifying game bots that we include in our feature set.
Player information. a Cumulative distribution of the user login frequency. b Cumulative distribution of user play time. c Cumulative distribution of the number of items harvested by users
Comparison of activity ratios between game bots and human users. The ratios of "earning experience points" and "obtaining items" of game bots are much higher than those of human users
Player actions
We examined the frequency and ratio of player actions to determine the unique characteristics of game bots. Figure 3 presents the ratios of the activities of both game bots and human users. The points in red indicate game bots, and those in blue indicate human users. The ratio of "earning game money" of game bots is nearly similar to that of human users. Remarkably, the ratios of "earning experience points" and "obtaining items" of game bots are much higher than those of human users. The cumulative ratio of "earning experience points", "obtaining items", and "earning game money" of game bots is close to 0.5, whereas that of human users is only 0.33. This implies that game bots concentrate heavily on profit-related activities, and human users enjoy various activities. In contrast, the ratio of "earning PK points" of human users is as much as three times that of game bots. This reflects the fact that game bots are not interested in rankings.
Figure 4 shows the distribution of the average party play time of game bots and human users. To acquire game money and items, some game bots form a party with other game bots. They can help each other not to be killed by monsters during party play. Consequently, their party play patterns are unusual. A total of 80 % of game bots last longer than 4 h 10 min, whereas 80 % of human users last less than 2 h 20 min. Since difficult missions can normally be completed within 2 h through collaboration, human users do not maintain party play as long as game bots.
Cumulative distribution of user average party play time. A total of 80 % of game bots last longer than 4 h 10 min, whereas 80 % of human users last less than 2 h 20 min
Social interaction diversity
Figure 5 shows the cumulative distribution of the entropy of social interactions. First, we determined seven activities as social interactions: party, friendship, trade, whisper, mail, shop, and guild. We quantified the diversity of social interactions by calculating the Shannon diversity entropy defined by:
$$\begin{aligned} H' = -\sum _{i=1}^{n}{p_i \ln {p_i}} \end{aligned}$$
n, number of social interaction types. \(p_i\), relative proportion of the ith social interaction type.
The entropy of the social interactions of a player indicates the various activities performed by the player. Figure 5 represents the fact that human users enjoy diverse activities, whereas game bots do not. We notice that game bots are interested in other activities.
Cumulative distribution of user social interaction diversity. The average entropy of game bot social interaction is much lower than that of human users (0.4299 and 0.8352, respectively)
Network measures
In Table 4, we present the basic directed characteristics of each network of the game bot and human groups from Aion (Son et al. 2012). First, we see that the average degree of the human group is approximately 18 times larger compared with the game bot group in the party network. The reason is that human users form a party with many and unspecified users, whereas game bots play with several specific other game bots. The average degree of the friendship network of the human group is larger by a factor of approximately four compared with the game bot group. This fact indicates that the friendship of game bots is utterly different from that of human users. Game bot friends simply mean other game bots with which to play. The fact that the average degree of the human group is 2.5 times larger than the game bot group is observed in the case of the trade network. However, the average clustering coefficient of the game bot group is approximately five times larger compared with the human group. We assume that game bots have roles (Kwon et al. 2013; Ahmad et al. 2009). For instance, some game bots are responsible for gold farming, while other game bots gather game money and items from gold farmers or sell them for real money (Woo et al. 2011).
Interestingly, in the case of the mail network of the game bots, we discovered nine spammers during the observation period. The number of mail pieces sent by the spammers is 1000 times per person on average. We observed the behavioral characteristics of the spammers in more detail. Hence, we found that they only send mail and stay online for a short period of time in the online game world.
We also observed the existence of five collectors who received items attached to mail from many other game bots. These collectors received items over 6000 times during the observation period. This shows that there are several gold farming groups. In the case of the shop network, we can see the smallest number of nodes of both groups. Players are immobile in the merchant mode, and thus cannot engage in any action that requires movement, such as hunting monsters, harvesting items, etc. Consequently, game bots do not focus on the merchant mode because it can be a waste of time for them.
Table 4 Basic network characteristics of six interaction networks
The triad census
The relative prevalence of each of the 13 triad network motifs given in Fig. 6a indicates the interaction pattern in the networks in more detail (Jeong et al. 2015). For our Aion networks, we show the interaction pattern in Fig. 6b in terms of both the fractions of each motif type and the Z-scores assessed against the null model [Eq. (2), also see Tables 7, 8]. This score is defined as follows:
$$\begin{aligned} Z_i = {N_{i}^{\mathrm {real}} - N_{i}^{\mathrm {random}} \over \sigma _{i}^{\mathrm {random}}}, \end{aligned}$$
where \(N_{i}^{\mathrm {real}}\) is the number of motif i found observed in the network, \(N_{i}^{\mathrm {random}}\) is the expected number in the randomized network, and \(\sigma _{i}^{\mathrm {random}}\) is the standard deviation of its expected number in the randomized network.
Network motif analysis of node triplets reveals detailed interaction patterns in directed networks of game bots and human users. a The 13 possible motifs composed of three nodes in a directed network. b The fractions of each motif type in each of the six networks. Those motifs that account for fewer than 18 % of all motifs are not shown. Friendship, whisper, mail, and shop of the game bot group, and friendship and shop of the human group show one dominant motif each, consistent with the high or low reciprocity found in the networks. c A closer look at the (normalized) Z-score triad census of party and trade networks where no dominant motif is evident; the Z-score method is employed to determine significantly over and underrepresented triangular motifs
Interestingly, the friendship, whisper, mail, and shop networks of the game bot group, and the friendship and shop networks of the human group, show one predominant motif type. For instance, in the friendship network, type 7 accounts for more than 90 % of the node triplet relationships, which can be attributed to the highly reciprocal nature of the interactions. The opposite reasoning can be applied to shop: low reciprocity reflects the existence of big merchants. Moreover, in the whisper and mail network of the game bot group, type 1 accounts for more than 80 % of the node triplet relationships. This reflects the fact that some game bots send information about the location coordinates of monsters to other game bots in the case of the whisper network.
Some game bots send several mail pieces in the case of the mail network. Comparing the prevalence of motifs against the null models allows us to detect signals discounted by random expectation, and this is done via the Z-scores [Eq.(2)]. This is particularly necessary and illuminating in the case of the other two networks (party and trade) because, by considering the null models, we can see that although multiple motifs can be similarly abundant (Fig. 6b), some can be significantly over or underrepresented, as we can see in Fig. 6. In the case of the human group, the overrepresented motif type 5 [with \({\tilde{Z}}> 0.4\), the normalized version \({\tilde{Z}} \equiv Z_{i} \sqrt{\displaystyle {\Sigma }_{i}({Z_{i}^{2}})}\)] is indeed closed triangles, consistent with the relatively high clustering tendencies in the party network. In the case of the game bot group, the overrepresented motif type 13 shows the fact that there is a large gap between the number of motifs observed in the network and the expected number of motifs in the randomized network. This reflects the fact that game bots have their own group for helping and trading with each other.
Network overlap
To determine how pairwise networks are correlated, we studied the network similarities between the game bot and human groups. For example, two networks can show similar clustering values, and yet this does not guarantee at all that nodes connected in one network are connected in another, or that the nodes show similar levels of activity. Thus, we consider here two measures of network overlap. The first is the link overlap between two networks quantified by the Jaccard coefficient. The second is the degree overlap given by the Pearson correlation coefficient between node degrees in network pairs. The results of link and degree overlap for ten network pairs of the game bot and human groups are given in Fig. 7. By examining the link overlap (Fig. 7a), we found that the game bot group has higher Jaccard coefficient in the party-friendship and party-trade pairwise networks. This is a result of the fact that the main activities of game bots are party play and trading items. The friend list offers convenience to a game bot when it wants to form a party group. Game bots gather game money and items collected through party play in an account by trading. Then the account that collects the cyber assets changes the game money and items to real money.
Node degree overlap (Fig. 7b) is another way of seeing the connection between interactions: here, for instance, the party-trade pairwise networks of the human group show a positive Pearson correlation coefficient value that exceeds 0.7, which can be understood by the fact that a party activity, being above all the favorite way of engaging in battles or hunting, often concludes with members trading booties. In contrast, the Pearson correlation coefficient values of the game bot group are extremely low because game bots maintain relationships with a small number of other game bots.
Pairwise network overlap indicates similarity or dependence between interactions. a Link overlap. The game bot group has higher Jaccard coefficient in the party-friendship and party-trade pairwise networks. b Node overlap that quantifies the node degree overlap between different networks. The human group has high degree overlap between 0.4 and 0.7, whereas the game bot group has degree overlap lower than 0.2 in all networks
Game bot detection
We took a discriminative approach to learning the distinction between game bots and human users in order to detect the game bot and build automatic classifiers that can automatically recognize the distinction. We divided the dataset into training and test sets, built the classifiers through the training dataset, and evaluated the trained classifiers through the test dataset. In addition, we performed tenfold cross-validation to avoid classifiers from being overfitted to the test data. Cross-validation generalizes the classifier trained by the test data to the validation data. Tenfold cross-validation divides the dataset into ten groups, trains the learning model with randomly selected nine groups, and verifies the classifiers from the model with one group. These training and validation processes are repeated ten times.
Feature selection
We compared the bot detection results from our model with the banned account list provided by the game company in order to evaluate the proposed framework upon running our detection method of selected features. We conducted feature selection with the best first, greedy stepwise, and information gain ranking filter algorithms in advance in order to improve the selection process. Feature_Set1 consists of all the features (114) mentioned in "Methods" section. Feature_Set2 is composed of the top 62 features extracted by the information gain ranking filter algorithm. Feature_Set3 is comprised of the six features selected by the best first and greedy stepwise algorithms. Figure 8 shows the classification results using these three feature sets. Feature_Set3 presents lower performance than Feature_Set1 and Feature_Set2. In comparison, Feature_Set2 has almost the same performance as Feature_Set1, although the number of Feature_Set2 is barely half that of Feature_Set1. Thus, we finally selected Feature_Set2 for game bot detection.
Performance comparison of feature sets. Feature_Set2 has as high performance as Feature_Set1
Classification and evaluation
The results of the users' behavioral pattern analysis for game bot detection are listed in Table 5. The four classifiers used as training algorithms—decision tree, random forest, logistic regression, and naïve Bayes—are tested on Feature_Set2. The performances are listed in terms of overall accuracy, precision, recall, and F-measure. Random forest outperforms the other models. Its overall accuracy, precision value, recall value, and F-measure with emphasis on precision (\(\alpha = 0.9\)) are 0.961, 0.956, 0.742, and 0.929, respectively. As can be seen, the recall value is slightly low. We analyzed the characteristics of true positive, false positive, false negative, and true negative cases to inquire into the cause of this phenomenon.
The random forest technique is a well-known ensemble learning method for classification and it constructs multiple decision trees in its training phase to overcome the decision tree's overfitting problem. The random forest learning is also robust when training with imbalanced data set. It is also useful when training large data with a lot of features. Our data set consists of 85 % of human players and 15 % of game bots—so it is considered as an imbalanced and large data set—and random forests perform well in that context given that the context meets the settings in which random forests are to perform ideally.
Naïve Bayes showed the lowest performance among four classifiers, and that is probably because of its nature as a generative model that requires independence of features. Although we performed feature selection, still there are correlations between selected features used in our experiment. For example, obtaining_items_count, earning_exp_points_count, harvesting_items_max_count, party_eccentricity, play_time and obtaining_items_ratio are less significant features. However, those features are also naturally correlated and they cannot be easily separated because they are all related to essential game behaviors (hunting, harvesting, collaboration, etc., which are all related to high level process). Indeed, such hypothesis is confirmed by removing those features, bringing the performance of the naïve Bayes on par with other algorithms.
Table 5 Precision, recall, and F-measure (0.9) ratios for each classifier
Figure 9 shows the relative similarities and differences of the classification evaluation outcomes (classes): true positive, false positive, false negative, and true negative. To obtain the relative similarity, we normalize all classes by the lowest class value, thus comparing outcomes relatively. Such normalization would bring the lowest class in the evaluation to one. For each class other than the lowest, we calculated the ratio by dividing the values of the other classes by the value of the lowest class. The pattern of the relative similarity is consistent with most features and classes, with the exception of the "mail_between_centrality" and "mail_outdegree" features. It is highly probable that game bots had not been detected yet in the case of false negatives. This also implies that human users temporarily employed a game bot in the case of false positives. To confirm this observation, we analyzed the case of false positives weekly and finally found harvesting and party play game bots.
Comparison of four cases: true-positive, false-positive, false-negative, and true-negative. The ratios of false-positive cases are exceedingly similar to those of true-positive cases. The ratios of false-negative cases are similar to those of true-negative cases
We proposed a multimodal framework for detecting game bots in order to reduce damage to online game service providers and legitimate users. We observed the behavioral characteristics of game bots and found several unique and discriminative characteristics. We found that game bots execute repetitive tasks associated with earning unfair profits, they do not enjoy socializing with other players, are connected among themselves and exchange cyber assets with each other. Interestingly, some game bots use the mail function to collect cyber assets. We utilized those observations to build discriminative features. We evaluated the performance of the proposed framework based on highly accurate ground truth—resulting from the banning of bots by the game company. The results showed that the framework can achieve detection accuracy of 0.961. Nonetheless, we should consider that the banned list does not include every game bot.
The game company imposes a penalty point on an account that performs abnormal activities, and eventually blocks the account when its cumulative penalty score is quite high. Some game bots can evade the penalty scoring system of the game companies. Hence, the actions of a player are more important than whether the player is banned or not, and we concede that a player is a game bot when the player's actions are abnormal. We focused on those user behavioral patterns that reflect user status to interpret the false positive cases, and hypothesize that they are game bots not yet blocked, and false negative cases are human users occasionally employing a game bot. Although different from those in the banned list, they behave in the same pattern. We believe that our detection model is more robust by relying on multiple classes of features, and its analyses promise further interesting directions in understanding game bot and their detection.
Ahmad MA, Keegan B, Srivastava J, Williams D, Contractor N (2009) Mining for gold farmers: automatic detection of deviant players in MMOGs. In: International conference on computational science and engineering, 2009. CSE'09, vol 4, pp 340–345. IEEE
Chen K-T, Hong L-W (2007) User identification based on game-play activity patterns. In: Proceedings of the 6th ACM SIGCOMM workshop on network and system support for games, pp 7–12. ACM
Chung Y, Park C-Y, Kim N-R, Cho H, Yoon T, Lee H, Lee J-H (2013) Game bot detection approach based on behavior analysis and consideration of various play styles. ETRI J 35(6):1058–1067
Itsuki H, Takeuchi A, Fujita A, Matsubara H (2010) Exploiting MMORPG log data toward efficient rmt player detection. In: Proceedings of the 7th international conference on advances in computer entertainment technology, pp 118–119. ACM
Jeong SH, Kang AR, Kim HK (2015) Analysis of game bot's behavioral characteristics in social interaction networks of MMORPG. In: Proceedings of the 2015 ACM conference on special interest group on data communication, pp 99–100. ACM
Kang AR, Kim HK, Woo J (2012) Chatting pattern based game bot detection: do they talk like us? TIIS 6(11):2866–2879
Kang AR, Woo J, Park J, Kim HK (2013) Online game bot detection based on party-play log analysis. Comput Math Appl 65(9):1384–1395
Kwon H, Kim, HK (2011) Self-similarity based bot detection system in MMORPG. In: Proceedings of the 3th international conference on internet, pp 477–481
Kwon H, Woo K, Kim H-C, Kim C-K, Kim HK (2013) Surgical strike: a novel approach to minimize collateral damage to game bot detection. In: Proceedings of annual workshop on network and systems support for games, pp 1–2. IEEE Press
Lee E, Woo J, Kim H, Mohaisen A, Kim HK (2016) You are a game bot!: uncovering game bots in MMORPGs via self-similarity in the wild. In: NDSS
Lee J, Lim J, Cho W, Kim HK (2015) In-game action sequence analysis for game bot detection on the big data analysis platform. In: Proceedings of the 18th Asia Pacific symposium on intelligent and evolutionary systems, vol 2, pp 403–414. Springer
Mitterhofer S, Kruegel C, Kirda E, Platzer C (2009) Server-side bot detection in massively multiplayer online games. IEEE Secur Priv 3:29–36
Mohaisen A, Alrawi O (2014) Av-meter: an evaluation of antivirus scans and labels. In: Detection of intrusions and malware, and vulnerability assessment: 11th international conference, DIMVA 2014, Egham, UK, July 10–11, 2014. Proceedings, pp 112–131
Oh J, Borbora ZH, Sharma D, Srivastava J (2013) Bot detection based on social interactions in MMORPGs. In: Social Computing (SocialCom), 2013 International Conference On, pp 536–543. IEEE
Pao H-K, Chen K-T, Chang H-C (2010) Game bot detection via avatar trajectory analysis. IEEE Trans Comput Intell AI Games 2(3):162–175
Pao H-K, Fadlil J, Lin H-Y, Chen K-T (2012) Trajectory analysis for user verification and recognition. Knowl Based Syst 34:81–90
Park S-H, Lee J-H, Jung H-W, Bang S-W (2010) Game behavior pattern modeling for game bots detection in MMORPG. In: Proceedings of the 4th international conference on uniquitous information management and communication, p 33. ACM
Platzer C (2011) Sequence-based bot detection in massive multiplayer online games. In: 8th international conference on information, communications and signal processing (ICICS) 2011, pp 1–5. IEEE
Seo D, Kim HK (2011) Detecting gold-farmers' groups in MMORPG by connection information. In: Proceedings of the 3th international conference, pp 583–588
Son S, Kang AR, Kim H-C, Kwon T, Park J, Kim HK (2012) Analysis of context dependence in social interaction networks of a massively multiplayer mnline role-playing game. PloS One 7(4):33918
Thawonmas R, Kashifuji Y (2010) Detection of MMORPG misconducts based on action frequencies, types and time-intervals. In: DMIN, pp 78–84
Thawonmas R, Kurashige M, Chen K-T (2007) Detection of landmarks for clustering of online-game players. IJVR 6(3):11–16
van Kesteren M, Langevoort J, Grootjen F (2009) A step in the right direction: bot detection in MMORPGs using movement analysis. In: Proceedings of the 21st Belgian–Dutch conference on artificial intelligence (BNAIC 2009), pp 129–136
Varvello M, Voelker GM (2010) Second life: a social network of humans and bots. In: Proceedings of the 20th international workshop on network and operating systems support for digital audio and video, pp 9–14. ACM
Woo K, Kwon H, Kim H-C, Kim C-K, Kim HK (2011) What can free money tell us on the virtual black market? ACM SIGCOMM Comput Commun Rev 41(4):392–393
Zhang Z, Anada H, Kawamoto J, Sakurai K (2015) Detection of illegal players in massively multiplayer online role playing game by classification algorithms. In: 29th international conference on advanced information networking and applications (AINA), 2015 IEEE, pp 406–413. doi:10.1109/AINA.2015.214
Conceived and designed the experiments: ARK, SHJ, AM, HKK. Performed the experiments: ARK, SHJ, AM, HKK. Analyzed the data: ARK, SHJ, AM, HKK. Wrote the paper: ARK, SHJ, AM, HKK. All authors read and approved the final manuscript.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2014R1A1A1006228). A two-page abstract on this work appeared in Jeong et al. (2015). The work proposed in this paper significantly enhances the prior work, technically and content-wise, including the motivation, related-work, design, and evaluation.
Department of Computer Science and Engineering, State University of New York at Buffalo, White Road, Buffalo, NY, USA
Ah Reum Kang & Aziz Mohaisen
Graduate School of Information Security, Korea University, Anam-ro, Seoul, Korea
Seong Hoon Jeong & Huy Kang Kim
Ah Reum Kang
Seong Hoon Jeong
Aziz Mohaisen
Huy Kang Kim
Correspondence to Huy Kang Kim.
See Tables 6, 7 and 8.
Table 6 Network diameters from 100 randomized network versions
Table 7 Complete frequency distribution for triangular motifs
Kang, A.R., Jeong, S.H., Mohaisen, A. et al. Multimodal game bot detection using user behavioral characteristics. SpringerPlus 5, 523 (2016). https://doi.org/10.1186/s40064-016-2122-8
Online game security
Behavior analysis | CommonCrawl |
\begin{document}
\allowdisplaybreaks
\title[Embedding the symbolic dynamics of Lorenz maps]{Embedding the symbolic dynamics of Lorenz maps}
\author{T.~Samuel} \address{Fachbereich 3 -- Mathematik, Universit\"at Bremen, 28359 Bremen, Germany.} \author{N.~Snigireva} \address{School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland.} \author{Andrew Vince} \address{Department of Mathematics, University of Florida, Gaineville, Florida, USA.}
\begin{abstract} Necessary and sufficient conditions for the symbolic dynamics of a Lorenz map to be fully embedded in the symbolic dynamics of a piecewise continuous interval map are given. As an application of this result, we describe a new algorithm for calculating the topological entropy of a Lorenz map. \end{abstract}
\maketitle
\section{Introduction}
Lorenz maps and their topological entropy have been and still are investigated intensively, see for instance \cite{BHV,G,GH,GC,H,HR,HS} and references therein. The simplest example of a Lorenz map is a $\beta$-transformation. The topological entropy of such transformation is well known \cite{Par}. However, for a general Lorenz map the question of determining the topological entropy is much more complicated. Glendinning \cite{G} showed that every Lorenz map is semi-conjugate to a \mbox{$\beta$-transformation} and thus some features of a Lorenz map can be understood via \mbox{$\beta$-transformations}. In this paper, we investigate the relation between the symbolic dynamics of a given Lorenz map and that of a $\beta$-transformation. In particular, this will allow us to obtain upper and lower bounds on the entropy of a general Lorenz map. Let us now outline our main results. \begin{description} \item[(i) \textit{Embedding dynamics}:] Our main results, \Cref{Main_THM,THM:END}, give necessary and sufficient conditions for when the address space (\Cref{Defn:Itinerary_map3}) of an arbitrary Lorenz system is a forward shift sub-invariant subset (\Cref{DEFN:SHIFT_INV}) of the address space of a uniform Lorenz system. (See \Cref{Defn:Lorenz_map} for the definition of a Lorenz system.) These results complement those of \cite{BHV,GH,HS}. \item[(ii) \textit{An algorithm}:] Based on (i), we provide, in \Cref{Sec:Algorithm}, an algorithm for calculating the topological entropy of a Lorenz system. This algorithm does not require previously used techniques of finding zeros of a power series \cite{AM,BHV,GH} nor the calculation of the zero of a pressure functional \cite{FMT}. \end{description}
\subsection{Motivation and previous related results}
A main motivation for the study of Lorenz maps is that they arise naturally in the investigation of a geometric model of Lorenz differential equations which have strange attractors, see \cite{Eckhardt,Lorenz:1963,Vis,Wil} and references therein. A second motivation is that a $\beta$-transformation (being the simplest example of a Lorenz map) plays an important role in ergodic theory, see \cite{DK,G,H,Par} and references therein. A third motivation comes from the study of fractal transformation, see \cite{BHI}.
Results from kneading theory are used in the study of Lorenz maps. In 1990, Hubbard and Sparrow \cite{HS} showed that the upper and lower itineraries of the critical point fully determine the address space of a Lorenz map. Moreover, Glendinning and Hall \cite{GH} showed that the topological entropy of such a map is related to the largest positive zero of a certain power series. Further results on the kneading sequences of Lorenz maps can be found, for instance, in the works of Hofbauer and Raith \cite{H,HR}, Alsed\'{a} and Ma\~{n}os \cite{AM}, Misiurewicz \cite{M} and Glendinning, Hall and Sparrow \cite{G,GH,GC}.
\subsection{Main results}
To formally state our main results we require the following notation and definitions.
\begin{definition}\label{Defn:Lorenz_map} An \textit{upper} (or \textit{lower}) \textit{Lorenz map} with \textit{critical point} $q \in (0, 1)$ is a piecewise continuous map $T^{+}$ (respectively $T^{-}$) $: [0, 1] \circlearrowleft$ of the form \begin{align*} T^{+}(x) \coloneqq \begin{cases} f_{0}(x) & \text{if} \; 0 \leq x < q,\\ f_{1}(x) & \text{if} \; q \leq x \leq 1, \end{cases} \;\;\left( \text{respectively} \;\; T^{-}(x) \coloneqq \begin{cases} f_{0}(x) & \text{if} \; 0 \leq x \leq q,\\ f_{1}(x) & \text{if} \; q < x \leq 1, \end{cases}\right) \end{align*} where \begin{enumerate} \item $f_{0}: [0, q] \to [0, 1]$ and $f_{1}: [q, 1] \to [0, 1]$ are continuous, strictly increasing, functions, with $f_{0}(0) = 0$ and $f_{1}(1) = 1$ and either $1 > f_{0}(q) > f_{1}(q) \geq 0$ or $1 \geq f_{0}(q) > f_{1}(q) > 0$, and \item there exists $s > 1$ such that $\lvert f_{i}(x) - f_{i}(y) \rvert \geq s \lvert x - y \rvert$, for $i \in \{ 0, 1\}$ and $x \in [0, 1]$. \end{enumerate} A \textit{Lorenz} (\textit{dynamical}) \textit{system} with critical point $q$ is defined to be a dynamical system $([0, 1], T)$, where $T$ is either an upper or lower Lorenz map with critical point $q$. \end{definition}
\begin{definition} A tuple $(a, p)$ is called \textit{admissible} if it belongs to the set $\{ (z, w) \in (1, 2) \times (0, 1) : 1 - z^{-1} \leq w \leq z^{-1} \}$. An upper or lower Lorenz map with critical point $p$ is called \textit{uniform} if $(a, p)$ is admissible and if $f'_{0}(x) = a = f'_{1}(y)$, for all $x \in (0, p)$ and $y \in (p, 1)$. We denote such maps by the symbols $U^{+}_{a, p}$ or $U^{-}_{a, p}$ respectively. Specifically, the maps $U^{+}_{a, p}$ and $U^{-}_{a, p}$ are given by, \begin{align*} U^{+}_{a, p}(x) \coloneqq \begin{cases} a x & \text{if} \; 0 \leq x < p,\\ a x + 1 -a & \text{if} \; p \leq x \leq 1, \end{cases} \quad U^{-}_{a, p}(x) \coloneqq \begin{cases} a x & \text{if} \; 0 \leq x \leq p,\\ a x + 1 -a & \text{if} \; p < x \leq 1. \end{cases} \end{align*} \end{definition}
Throughout we use the convention that $\pm$ means either $+$ or $-$. When we write, `given a Lorenz map $T^{\pm}$ with critical point $q$', we require both $T^{+}$ and $T^{-}$ to be defined using the same functions $f_{0}$ and $f_{1}$. Further, let $\mathbb{N}$ denote the set of positive integers, $\mathbb{N}_{0}$ denote the set of non-negative integers and $\mathbb{R}$ denote the set of real numbers.
We let $\Omega \coloneqq \{0, 1\}^{\infty}$ denote the set of all infinite strings $\omega_{0} \, \omega_{1} \, \omega_{2} \cdots$ consisting of elements of the set $\{0, 1 \}$. It is well-known that the set $\Omega$ is a complete compact metric space with respect to the metric $d: \Omega \times \Omega \to \mathbb{R}$ given by \begin{align*} d(\omega, \sigma) \coloneqq \begin{cases} 0 & \text{if} \; \omega = \sigma,\\ 2^{- \lvert\omega \wedge \sigma\rvert} & \text{otherwise}, \end{cases} \end{align*} where $\rvert\omega \wedge \sigma\lvert \coloneqq \min \{ \, n \in \mathbb{N} \, : \, \omega_{n} \neq \sigma_n \}$, for all $\omega \coloneqq \omega_{0} \, \omega_{1} \, \cdots, \sigma \coloneqq \sigma_{0} \, \sigma_{1} \, \cdots \in \Omega$ with $\omega \neq \sigma$. Throughout we assume that $\Omega$ is equipped with the metric $d$ and is endowed with the lexicographic ordering which will be denoted by the symbols $\succ$ and $\prec$.
\begin{definition}\label{Defn:Itinerary_map1} The \textit{upper} (or \textit{lower}) \textit{itinerary}, $\tau_{q}^{+}(x)$ (respectively $\tau^{-}_{q}(x)$) of a point $x \in [0,1]$ under $T^{+}$ (respectively $T^{-}$) with critical point $q$ is the string $\omega_{0} \, \omega_{1} \, \omega_{2} \, \cdots \in \Omega$ (respectively $\sigma_{0} \, \sigma_{1} \, \sigma_{2}\, \cdots \in \Omega$), where \begin{align*}
\omega_{k} \coloneqq \begin{cases} 0 & \text{if} \; (T^{+})^{k}(x) < q\\ 1 & \text{if} \; (T^{+})^{k}(x) \geq q. \end{cases} \quad\left(\text{respectively} \quad \sigma_{k} \coloneqq \begin{cases} 0 & \text{if} \; (T^{-})^{k}(x) \leq q\\ 1 & \text{if} \; (T^{-})^{k}(x) > q. \end{cases} \right). \end{align*} To distinguish the itinerary map of a uniform Lorenz map $U_{a,p}^{\pm}$ we use the symbol $\mu_{a, p}^{\pm}$. \end{definition}
Let $(T^{+})^{n}$ denote the $n$-fold composition of $T^{+}$ with itself, where $(T^{+})^{0}(x) \coloneqq x$ for $x\in [0,1]$ and $n \in \mathbb{N}$.
\begin{definition}\label{Defn:Itinerary_map3} Given a Lorenz map $T^{\pm}: [0, 1] \circlearrowleft$ with critical point $q$, we let $\Omega_{q}^{\pm} \subset \Omega$ denote the image of the unit interval $[0,1]$ under the mapping $\tau_{q}^{\pm}$. The set $\Omega_{q}^{\pm}$ is called the \textit{address space} of the dynamical system $([0, 1], T^{\pm})$. To distinguish the address space of a uniform Lorenz system $([0, 1], U_{a, p}^{\pm})$, we we use the symbol $\Omega_{a, p}^{\pm}$. \end{definition}
Given a Lorenz map $T^{\pm}$, we let $h(T^{\pm})$ denote its topological entropy, which we will define in \Cref{Section2}. Since $h(T^{+}) = h(T^{-})$, we let $h(T)$ denote this common value; see \Cref{RMK:+=-}.
Finally, let $g_{0, a}(x) \coloneqq x/a$ and $g_{1,a}(x) \coloneqq x/a + (1-a^{-1})$, for each $a \in (1, 2)$ and $x \in [0, 1]$. The \textit{coding map} $\pi_{a}: \Omega \to [0, 1]$ is defined by \begin{align*}\label{EQ:Projection_Map} \pi_a (\omega_{0} \, \omega_{1} \, \omega_{2} \cdots) \coloneqq \lim_{n \to \infty} g_{\omega_{0}, a} \circ g_{\omega_{1}, a} \circ \dots \circ g_{\omega_{n}, a}(1) = \left(1- a^{-1}\right) \sum_{k=0}^{\infty} \, \omega_{k} \, a^{-k}. \end{align*} With the above we can now state our main results. For ease of notation we let $\alpha \coloneqq \tau_{q}^{-}(q)$ and $\beta \coloneqq \tau_{q}^{+}(q)$.
\begin{theorem}\label{Main_THM} Let $([0,1], T^{\pm})$ denote a Lorenz system with critical point $q$ such that $T^{-}(q) \neq 1$ and $T^{+}(q) \neq 0$. Then the following statements are equivalent for each $a \in \mathbb{R}$. \begin{enumerate} \item The value $a$ belongs to the open interval $(\exp(h(T)), 2)$. \item The open interval $(\pi_{a}(\alpha),\pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$ is non-empty and $\alpha \prec \mu_{a, p}^{-}(p) \prec \mu_{a, p}^{+}(p) \prec \beta$, for all $p\in (\pi_{a}(\alpha),\pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$. \item The open interval $(\pi_{a}(\alpha),\pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$ is non-empty and $\Omega_{q}^{-} \subset \Omega_{a, p}^{-}$ and $\Omega_{q}^{+} \subset \Omega_{a, p}^{+}$, for all $p \in (\pi_{a}(\alpha),\pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$. \end{enumerate} \end{theorem}
\begin{theorem}\label{THM:END} Let $([0,1], T^{\pm})$ denote a Lorenz system with critical point $q$. \begin{enumerate} \item If $T^{-}(q) = 1$, then the following are equivalent \begin{enumerate} \item $a \in (\exp(h(T)), 2)$. \item There exists a unique $p \in [1-a^{-1}, a^{-1}]$, given by $p = a^{-1}$, so that $\alpha = \mu_{a, a^{-1}}^{-}(a^{-1}) \prec \mu_{a, a^{-1}}^{+}(a^{-1}) \prec \beta$. \item There exists a unique $p \in [1-a^{-1}, a^{-1}]$, given by $p = a^{-1}$, so that $\Omega_{q}^{-} \subset \Omega_{a, p}^{-}$ and $\Omega_{q}^{+} \subset \Omega_{a, p}^{+}$. \end{enumerate} \item If $T^{+}(q) = 0$, then the following are equivalent \begin{enumerate} \item $a \in (\exp(h(T)), 2)$. \item There exists a unique $p \in [1-a^{-1}, a^{-1}]$, given by $p = 1 - a^{-1}$, so that $\alpha \prec \mu_{a, a^{-1}}^{-}(a^{-1}) \prec \mu_{a, a^{-1}}^{+}(a^{-1}) = \beta$. \item There exists a unique $p \in [1-a^{-1}, a^{-1}]$, given by $p = 1 - a^{-1}$, so that $\Omega_{q}^{-} \subset \Omega_{a, p}^{-}$ and $\Omega_{q}^{+} \subset \Omega_{a, p}^{+}$. \end{enumerate} \end{enumerate} \end{theorem}
\begin{remark} In \Cref{Main_THM} it is necessary to take the intersection of the intervals $(\pi_{a}(\alpha),\pi_{a}(\beta))$ and $(1-a^{-1}, a^{-1})$ instead of only the interval $(\pi_{a}(\alpha),\pi_{a}(\beta))$. Otherwise the inequality $\pi_{a}(\alpha) < 1- a^{-1}$ or $\pi_{a}(\beta) > a^{-1}$ may occur, and so, the corresponding uniform Lorenz system will not be well defined; see \Cref{EXMP:inequalities}. \end{remark}
\begin{remark} For each $a > \exp(h(T))$, \Cref{Main_THM,THM:END} fully classify the points $p$ belonging to the interval $[1-a^{-1}, a^{-1}]$, such that either $\tau_{q}^{-}(q) \preceq \mu_{a, p}^{-}(p) \prec \mu_{a, p}^{+}(p) \prec \tau_{q}^{+}(q)$ or $\tau_{q}^{-}(q) \prec \mu_{a, p}^{-}(p) \prec \mu_{a, p}^{+}(p) \preceq \tau_{q}^{+}(q)$ hold, which, as we will see, implies an embedding of address spaces, or formally, $\Omega_{q}^{-} \subset \Omega_{a, p}^{-}$ and $\Omega_{q}^{+} \subset \Omega_{a, p}^{+}$. \end{remark}
In the final section of this paper we present a new algorithm, based on \Cref{Main_THM,THM:END}, which calculates the topological entropy of a Lorenz map. The main idea behind the algorithm is the following. The algorithm first uses an efficient method to calculate the address spaces of a given Lorenz system $([0, 1], T)$. Then, in a systematic way, it compares the address spaces of $([0, 1], T)$ to the address spaces of a subclass of the family of uniform Lorenz systems. By a well-known result of Parry \cite{Par} the topological entropy of each member of this subclass of systems is known. Using \Cref{Main_THM,THM:END} the algorithm is then able to obtain an estimate of the topological entropy of the given system.
\subsection{Outline}
\Cref{sec:pre} contains necessary preliminaries. The concepts of topological entropy and topological (semi-) conjugacy are introduced in \Cref{Section2}; properties of itinerary maps are presented in \Cref{Sec:Kneading}; and several required auxiliary results are proved in \Cref{Sec:Prelim}. \Cref{Sec:Proof_of_Main_THM} contains the proofs of \Cref{Main_THM,THM:END}. We conclude with \Cref{Sec:Algorithm}, where the statement and a proof of validity of a new algorithm for computing the topological entropy of a Lorenz (dynamical) system is given.
\section{Preliminaries}\label{sec:pre}
In this section, various auxiliary results are proved in preparation for the proof of \Cref{Main_THM,THM:END}.
\subsection{Entropy and topological conjugacy}\label{Section2}
Recall the definition of topological entropy and topological (semi-) conjugacy.
\begin{definition}\label{defEntropy} Let $T^{\pm}$ be a Lorenz map with critical point $q$. For $\omega \in \Omega$, the string consisting of the first $n \in \mathbb{N}$ symbols of $\omega$ is denoted by $\omega\vert_{n}$ and $\omega\vert_{0}$ denotes the empty word. We set $\Omega_{q, n}^{\pm} \coloneqq \{ \omega\vert_{n} : \omega \in \Omega_{q}^{\pm} \}$ and let $\lvert \Omega_{q, n}^{\pm} \rvert$ denote the cardinality of the set $\Omega_{q, n}^{\pm}$, for each $n \in \mathbb{N}$. The \textit{topological entropy} $h(T^{\pm})$ of $([0, 1], T^{\pm})$ is defined by $\displaystyle h(T^{\pm}) \coloneqq \lim_{n \to \infty} \ln ( \lvert \Omega_{q, n}^{\pm} \rvert^{1/n} )$. \end{definition}
\begin{remark}\label{RMK:+=-} It is well-known that $h(T^{+}) = h(T^{-}) \leq \ln(2)$. Thus, for ease of notation, we denote the common value $h(T^{+}) = h(T^{-})$ by $h(T)$ . \end{remark}
\begin{theorem}\cite{Par,HR}\label{thm:uniform} If $(a, p)$ is an admissible pair, then $h\left(U^{+}_{a, p}\right) = h\left(U^{-}_{a, p}\right)$. Moreover, this common value is equal to $\ln (a)$. \end{theorem}
\begin{definition}\label{Defn:conj} Two maps $R: X \circlearrowleft$ and $S: Y \circlearrowleft$ defined on compact metric spaces are called \textit{topologically conjugate} if there exists a homeomorphism $\hbar: X \to Y$ such that $S \circ \hbar = \hbar \circ R$. If $\hbar$ is continuous and surjective then $R$ and $S$ are called \textit{semi-conjugate}. \end{definition}
When we write, `two dynamical systems are topologically (semi-) conjugate', we mean that the associated maps are topologically (semi-) conjugate.
\begin{lemma}[\cite{G}]\label{lemma1} \leavevmode \begin{enumerate} \item If two Lorenz systems $([0,1], T^{\pm})$ and $([0,1], R^{\pm})$ are topologically conjugate, then the address spaces are equal and hence, $h(T) = h(R)$. \item If a Lorenz system $([0,1], T^{\pm})$ with critical point $q$ is semi-conjugate to a Lorenz system $([0,1], R^{\pm})$ with critical point $p$, then $\Omega_{p}^{\pm}\subseteq \Omega_{p}^{\pm}$ and $h(T) = h(R)$. \end{enumerate} \end{lemma}
\subsection{Properties of itinerary maps}\label{Sec:Kneading}
We next state properties of the itinerary maps $\mu_{a, p}^{\pm}$ of uniform Lorenz systems. Throughout this section $(a, p)$ will denote an admissible pair.
\begin{lemma}[\cite{BHV}]\label{lem2} \leavevmode \begin{enumerate} \item The map $[0, 1] \ni x \mapsto \mu_{a, p}^{+}(x)$ is strictly increasing and right-continuous. Moreover, for all $x \in (0, 1)$, we have that $\displaystyle \mu_{a, p}^{-}(x) = \lim_{\epsilon \searrow 0} \mu_{a, p}^{+}(x - \epsilon)$. \item The map $[0, 1] \ni x \mapsto \mu_{a, p}^{-}(x)$ is strictly increasing and left-continuous. Moreover, for all $x \in (0, 1)$, we have that $\displaystyle \mu_{a, p}^{+}(x) = \lim_{\epsilon \searrow 0} \mu_{a, p}^{-}(x + \epsilon)$. \item The map $p \mapsto \mu_{a, p}^{+}(p)$ is strictly increasing and right-continuous. \item The map $p \mapsto \mu_{a, p}^{-}(p)$ is strictly increasing and left-continuous. \end{enumerate} \end{lemma}
Finally, we conclude with the a result which links the coding map $\pi_{a}$, defined in \eqref{EQ:Projection_Map}, and the itinerary maps $\mu_{a, p}^{\pm}$. This requires the following definition.
\begin{definition}\label{DEFN:SHIFT_INV} The continuous map $S: \Omega \circlearrowleft$ defined by $S(\omega_{0} \, \omega_{1} \, \omega_{2} \cdots) \coloneqq \omega_{1} \, \omega_{2} \, \omega_{3} \cdots,$ is called the \textit{shift map} and a subset $\Lambda$ of $\Omega$ is called \textit{forward shift sub-invariant} if $S(\Lambda) \subseteq \Lambda$. \end{definition}
\begin{proposition}\label{prop:code-map} We have that $\pi_{a} \left ( \mu_{a, p}^{\pm}(x)\right ) = x$, for all $x \in [0,1]$, and that the following diagram commutes \[ \begin{array} [c]{ccc} \Omega_{a, p}^{\pm} & \overset{S}{\longrightarrow} & \Omega_{a, p}^{\pm}\\ & & \\ \pi_a \downarrow\text{\ \ \ \ } & & \text{ \ \ \ }\downarrow\pi_a \\ & & \\
\lbrack 0,1 \rbrack & \underset{U_{a, p}^{\pm}} {\longrightarrow} & \lbrack 0,1\rbrack. \end{array} \] \end{proposition}
\begin{proof} The result is readily verifiable from the definitions of the maps involved. Also a sketch of the proof of the result appears in \cite[Section~5]{BHV} and \cite[Section~2.2]{GH}. \end{proof}
\subsection{Auxiliary results}\label{Sec:Prelim}
In the following auxiliary results used in the proofs of \Cref{Main_THM,THM:END}, let $([0,1], T^{\pm})$ denote a Lorenz system with critical point $q$, let $\tau_{q}^{\pm}$ denote the associated itinerary map, and let $\Omega_{q}^{\pm}$ denote the associated address space.
\begin{lemma}\label{lem:shift} The address space $\Omega_{q}^{\pm}$ is forward shift sub-invariant.
\end{lemma}
\begin{proof} This is a direct consequence of \Cref{prop:code-map}. \end{proof}
A partial version of the following result can be found in \cite[Lemma~1]{H}. However, to the best of our knowledge, Theorem~\ref{thm:Main1.5} first appeared in \cite[Theorem~1]{HS}.
\begin{definition} The strings $\alpha \coloneqq \tau_{q}^{-}(q)$ and $\beta \coloneqq \tau_{q}^{+}(q)$ are called the \textit{critical itineraries}. \end{definition}
\begin{theorem}\label{thm:Main1.5} The spaces $\Omega_{q}^{+}$ and $\Omega_{q}^{-}$ are uniquely determined by $\alpha$ and $\beta$ as follows: \begin{align*} \Omega_{q}^{+} &= \{ \omega \in \Omega : S^{n}(\omega) \prec \alpha \; \text{or} \; \beta \preceq S^{n}(\omega), \; \text{for all} \; n \in \mathbb{N}_{0} \},\\ \Omega_{q}^{-} &= \{ \omega \in \Omega : S^{n} (\omega) \preceq \alpha \; \text{or} \; \beta \prec S^{n}(\omega), \, \; \text{for all} \; n \in \mathbb{N}_{0} \}. \end{align*} \end{theorem}
\begin{corollary}\label{cor:cor2} Let $a \in (1,2)$ be fixed. \begin{enumerate} \item If there exists $p$ such that $(a, p)$ is admissible and $\alpha \preceq \mu_{a, p}^{-}(p) \prec \mu_{a, p}^{+}(p) \preceq \beta$, then $h(T) \leq \ln (a)$. \item If there exists $p$ such that $(a, p)$ is admissible and $\mu_{a, p}^{-}(p) \preceq \alpha \prec \beta \preceq \mu_{a, p}^{+}(p)$, then $h(T) \geq \ln (a)$. \end{enumerate} \end{corollary}
\begin{proof} This is a direct consequence of \Cref{defEntropy} and \Cref{thm:uniform,thm:Main1.5}. \end{proof}
In the proofs of some of the following results we let $\overline{0}$ denote the element $0 \, 0 \, \cdots \in \Omega$ and $\overline{1}$ the element $1 \,1 \, \cdots \in \Omega$,
\begin{lemma}\label{thm:Main2} Given $a \in (1,2)$, there exists $p$ such that $(a, p)$ is admissible and either \begin{subequations} \begin{equation} \alpha \preceq \mu_{a, p}^{-}(p) \prec \mu_{a, p}^{+}(p) \preceq \beta \label{comeq1} \end{equation} or \begin{equation} \mu_{a, p}^{-}(p) \preceq \alpha \prec \beta \preceq \mu_{a, p}^{+}(p) \label{comeq2}. \end{equation} \end{subequations} Hence, in the first case $h(T) \leq \ln (a)$, and in the second case $h(T) \geq \ln (a)$. \end{lemma}
\begin{proof} Since a lower itinerary starts with $0$ and an upper itinerary starts with $1$, we have $\alpha \preceq 0 \overline{1} = \mu^{-}_{a, a^{-1}}(a^{-1})$ and $\mu^{+}_{a,1 - a^{-1}}(1 - a^{-1}) = 1 \overline{0} \preceq \beta$. Hence, the inequalities given in \eqref{comeq1} hold for $p = 1 - a^{-1}$, unless \begin{align}\label{eq1} \mu^{-}_{a, 1-a^{-1}}(1 - a^{-1}) \prec \alpha, \end{align} and, similarly, the inequalities given in \eqref{comeq1} hold for $p = a^{-1}$, unless \begin{align} \label{eq2} \mu^{+}_{a, a^{-1}}(a^{-1}) \succ \beta. \end{align} If the inequalities given in \eqref{comeq1} are false for both $p = 1-a^{-1}$ and $p = a^{-1}$, then the inequalities of both \eqref{eq1} and \eqref{eq2} hold. Let $p_{1} \coloneqq \sup \{ p : \mu^{-}_{a, p}(p) \preceq \alpha \; \text{and} \; \mu^{+}_{a, p}(p) \preceq \beta \}$ and $p_{2} \coloneqq \inf \{ p : \mu^{-}_{a, p}(p) \succeq \alpha \; \text{and} \; \mu^{+}_{a, p}(p) \succeq \beta \}$. \Cref{lem2} implies that $p_{2} \geq p_{1}$ and that if $p_{2} > p > p_{1}$, then either the inequalities given in \eqref{comeq1} or the inequalities given in \eqref{comeq2} hold for $p$. If $p_{1} = p_{2}$, then \Cref{lem2} implies that the inequalities given in \eqref{comeq2} hold at $p = p_{1} = p_{2}$.
The remaining assertion follows from \Cref{cor:cor2}. \end{proof}
\begin{lemma}\label{lem:1234} Let $a \in (\exp(h(T)), 2)$ be fixed. If $T^{-}(q) \neq 1$ and $T^{+}(q) \neq 0$, then there exists a non-empty open interval $V \subseteq [1-a^{-1}, a^{-1}]$, such that $\alpha \prec \mu_{a, t}^{-}(t) \prec \mu_{a, t}^{+}(t) \prec \beta$, for all $t \in V$. Moreover, letting \begin{subequations}\label{eq:pa} \begin{equation} p_{1} (a) \!\coloneqq\! \max \left\{1\!-\!a^{-1}, \sup \left\{ p \in [1\!-\!a^{-1}, a^{-1}] : \mu^{-}_{a, p}(p) \preceq \alpha \, \text{and} \, \mu^{+}_{a, p}(p) \preceq \beta \right\} \right\} \end{equation} and \begin{equation} p_{2} (a) \!\coloneqq\! \min \left\{ a^{-1}, \inf \left\{ p \in [1\!-\!a^{-1}, a^{-1}] : \mu^{-}_{a, p}(p) \succeq \alpha \, \text{and} \, \mu^{+}_{a, p}(p) \succeq \beta \right\}\right\}, \end{equation} \end{subequations} we have that $V \subseteq (p_1 (a), p_2 (a))$ and hence $p_1 (a) < p_2 (a)$. \end{lemma}
\begin{proof} Since $\ln (a) > h(T)$, by \Cref{thm:Main2}, there exists $p$ such that $(a, p)$ is admissible and that least one of the following sets of inequalities hold: \begin{subequations} \begin{equation} \alpha \prec \mu_{a, p}^{-}(p) \prec \mu_{a, p}^{+}(p) \preceq \beta,\label{comeq3} \end{equation} or \begin{equation} \alpha \preceq \mu_{a, p}^{-}(p) \prec \mu_{a, p}^{+}(p) \prec \beta.\label{comeq4} \end{equation} \end{subequations} (Observe that the situation in which $\alpha = \mu_{a, p}^{-}(p)$ and $\mu_{a, p}^{+}(p) = \beta$ cannot occur since $\ln (a) > h(T)$.) Let such a $p$ be fixed. If $p = 1 - a^{-1}$, then, by the definition of the itinerary map and the fact that $T^{+}(q) \neq 0$, we have that $\beta \succ 1\overline{0}$ and that $\mu_{a, p}^{+}(p) = 1\overline{0}$. Hence, the inequalities given in \eqref{comeq4} hold. Similarly, if $p = a^{-1}$, then $\alpha \prec 0\overline{1}$ and $\mu_{a, p}^{-}(p) = 0\overline{1}$, hence the inequalities given in \eqref{comeq3} hold.
Suppose that $p \not\in \{ 1-a^{-1}, a^{-1}\}$ and that the inequalities given in \eqref{comeq3} hold. Let $r \coloneqq d(\mu_{a, p}^{-}(p), \alpha) > 0$. By \Cref{lem2}~(ii), we have $\lim_{\epsilon \searrow 0} d(\mu_{a, p-\epsilon}^{-}(p - \epsilon), \mu_{a, p}^{-}(p)) = 0$. Therefore, there exists $\delta = \delta(r) \in (0, p - 1 + a^{-1})$ such that, for all $\epsilon < \delta = \delta(r)$, $d(\mu_{a, p-\epsilon}^{-}(p-\epsilon), \mu_{a, p}^{-}(p)) < r/2$. Now, \Cref{lem2}~(iv), the definition of the metric $d$ and that of the lexicographic ordering, together with the above inequality, imply that $\alpha \prec \mu_{a, p-\epsilon}^{-}(p-\epsilon) \prec \mu_{a, p}^{-}(p)$, for all $\epsilon < \delta$. Thus, by \Cref{lem2}~(iii), we have that $\mu_{a, p-\epsilon}^{+}(p-\epsilon) \prec \mu_{a, p}^{+}(p)$, for all $\epsilon < \delta$. Therefore, by the definition of the itinerary maps $t \mapsto \mu_{a, t}^{\pm}(t)$ and by the assumption that the inequalities given in \eqref{comeq3} hold, we have that $\alpha \prec \mu_{a, p-\epsilon}^{-}(p-\epsilon) \prec \mu_{a, p-\epsilon}^{+}(p - \epsilon) \prec \mu_{a, p}^{+}(p ) \prec \beta$, for all $\epsilon < \delta$. Furthermore, since $\delta \in (0, p - 1 + a^{-1})$ and since $p \in (1-a^{-1}, a^{-1}]$, it follows that $(p - \delta, p) \subset (1-a^{-1}, a^{-1})$. Setting $V \coloneqq (p - \delta, p)$ yields the required result.
A similar argument yields the required result under the assumption of the inequalities given in \eqref{comeq4} for our fixed $p$.
The remaining assertion is an immediate consequence of the definitions of $p_{1} (a)$ and $p_{2} (a)$ and \Cref{lem2}. \end{proof}
\begin{lemma}\label{lem:bounds1.0} The restriction of the coding map $\pi_{a}$ to the set $\Omega_{a, p}^{+}$ and the restriction of $\pi_{a}$ to the set $\Omega_{a, p}^{-}$ are strictly increasing, for all admissible pairs $(a, p)$. Furthermore, the restriction of the coding map $\pi_{a}$ to the set $\Omega_{a, p}^{+} \cup \Omega_{a, p}^{-}$ is increasing. \end{lemma}
\begin{proof} The first statement follows from \Cref{lem2} and \Cref{prop:code-map}.
To show that the restriction of $\pi_{a}$ to $\Omega_{a, p}^{+} \cup \Omega_{a, p}^{-}$ is increasing, let $\omega, \omega' \in \Omega_{a, p}^{+} \cup \Omega_{a, p}^{-}$ be such that $\omega \preceq \omega'$. One of the following situations must now occur. \begin{enumerate} \item $\omega, \omega' \in \Omega^{+}_{a, p}\;$ or $\;\omega, \omega' \in \Omega^{-}_{a, p}$, \item $\omega \in \Omega^{-}_{a, p} \setminus \Omega^{+}_{a, p}$ and $\omega' \in \Omega^{+}_{a, p} \setminus \Omega^{-}_{a, p}$, or \item $\omega \in \Omega^{+}_{a, p} \setminus \Omega^{-}_{a, p}$ and $\omega' \in \Omega^{-}_{a, p} \setminus \Omega^{+}_{a, p}$. \end{enumerate} If (i) occurs, then by the fact that the restriction of $\pi_{a}$ to the set $\Omega_{a, p}^{+}$ is strictly increasing and the restriction of $\pi_{a}$ to the set $\Omega_{a, p}^{-}$ is strictly increasing, it follows that $\pi_{a}(\omega) < \pi_{a}(\omega')$.
Suppose (ii) occurs. Let $y \coloneqq \pi_{a}(\omega)$ and $z \coloneqq \pi_{a}(\omega')$. By way of contradiction, assume $y > z$. \Cref{lem2} implies \begin{align}\label{eq:lim_relation_+_} \mu_{a, p}^{+}(z) &= \lim_{\epsilon \searrow 0} \mu_{a, p}^{-}(z + \epsilon).
\intertext{Now}
\omega' = \mu^{+}_{a, p}(z) &= \lim_{\epsilon \searrow 0} \mu^{-}_{a, p}(z + \epsilon) \prec \mu^{-}_{a, p}(y) = \omega, \label{eq:3.13} \end{align} where the first equality holds since $\omega'\in \Omega_{a, p}^{+}$, and so there exists $x \in [0,1]$ such that $ \omega ' = \mu^{+}_{a, p}(x)$. Then, by \Cref{prop:code-map}, we have $z \coloneqq \pi_{a}(\omega') = \pi_{a}(\mu^{+}_{a, p}(x)) = x$. Hence $\omega' = \mu^{+}_{a, p}(x) =\mu^{+}_{a, p}(z)$. The second equality in \eqref{eq:3.13} follows from \eqref{eq:lim_relation_+_}; the following inequality is due to \Cref{lem2} and the fact that $y > z + \epsilon$ for all sufficiently small $\epsilon > 0$; and the last equality follows in exactly the same way as the first equality. Therefore, $\omega' \prec \omega$, which contradicts our hypothesis, namely that $\omega \preceq \omega'$.
If (iii) occurs, then similar argument to those given above will yield that \mbox{$\pi_{a}(\omega) \leq \pi_{a}(\omega')$}. \end{proof}
\begin{lemma}\label{lem:bounds} If $2 > a > \exp(h(T))$, $T^{-}(q) \neq 1$ and $T^{+}(q)\neq 0$, then $\pi_{a}(\alpha) < \pi_{a}(\beta)$ and \begin{align} \emptyset \neq (p_1 (a), p_2 (a)) \subseteq (\pi_{a}(\alpha),\pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1}), \end{align} where $p_{1}(a)$ and $p_{2}(a)$ are the real numbers defined in \eqref{eq:pa} respectively. \end{lemma}
\begin{proof} Suppose that $a \in (\exp(h(T)), 2)$. By \Cref{thm:Main2}, there exists $p$ such that $(a, p)$ is admissible and either one of the following sets of inequalities hold, \begin{enumerate} \item $\alpha \prec \mu_{a, p}^{-}(p)$ and $\mu_{a, p}^{+}(p) \preceq \beta$, or \item $\alpha \preceq \mu_{a, p}^{-}(p)$ and $\mu_{a, p}^{+}(p) \prec \beta$. \end{enumerate} Note that the situation where $\alpha = \mu_{a, p}^{-}(p)$ and $\mu_{a, p}^{+}(p) = \beta$ cannot occur as $a > \exp(h(T))$.
Assume that (i) occurs. By \Cref{thm:Main1.5} it follows that $\Omega_{q}^{-} \subset \Omega_{a, p}^{-}$ and $\Omega_{q}^{+} \subseteq \Omega_{a, p}^{+}$. In particular, $\alpha \in \Omega^{-}_{a, p}$ and $\beta \in \Omega^{+}_{a, p}$. Since, by \Cref{lem:bounds1.0}, the coding map $\pi_{a}$ is strictly increasing on $\Omega_{a, p}^{+}$ and on $\Omega_{a, p}^{-}$, we have \begin{align}\label{eq:bounds1} \pi_{a}(\alpha) < \pi_{a}(\mu_{a, p}^{-}(p)) = p = \pi_{a}(\mu_{a, p}^{+}(p)) \leq \pi_{a}(\beta). \end{align} If (ii) occurs, then essentially the same arguments as those above yield \begin{align}\label{eq:bounds2} \pi_{a}(\alpha) \leq \pi_{a}(\mu_{a, p}^{-}(p)) = p = \pi_{a}(\mu_{a, p}^{+}(p)) < \pi_{a}(\beta). \end{align} Hence, $\pi_{a}(\alpha) < \pi_{a}(\beta)$ and $[\pi_{a}(\alpha),\pi_{a}(\beta)] \cap [1-a^{-1}, a^{-1}] \neq \emptyset$.
We now show that the open interval $(\pi_{a}(\alpha),\pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$ is non-empty. Observe that, by \Cref{lem2} and the definition of $p_1 (a)$ and $p_2 (a)$, for all $t \in (p_1 (a), p_2 (a))$, there are two possible sets of inequalities that can occur: \begin{enumerate} \item[(a)] $\alpha \succ \mu^{-}_{a, t}(t)$ and $\beta \prec \mu^{+}_{a, p}(t)$, or \item[(b)] $\alpha \prec \mu^{-}_{a, t}(t)$ and $\beta \succ \mu^{+}_{a, p}(t)$. \end{enumerate} The set of inequalities in (a), however, cannot occur. If they did, \Cref{thm:uniform,thm:Main1.5} and the definition of topological entropy, would imply $\ln (a) \leq h(T)$, contradicting our hypothesis. Thus, by \eqref{eq:bounds1} and \eqref{eq:bounds2} we have \begin{align}\label{eq:bounds-p_1-p_2} (p_1 (a), p_2 (a)) \subseteq [\pi_{a}(\alpha), \pi_{a}(\beta)] \cap [1-a^{-1}, a^{-1}]. \end{align} Since our hypothesis is the same as that of \Cref{lem:1234}, we have that $p_1 (a) < p_2 (a)$, and so, the open interval $(p_1 (a), p_2 (a))$ is non-empty. This, in tandem with \eqref{eq:bounds-p_1-p_2}, implies $(\pi_{a}(\alpha), \pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1}) \neq \emptyset$. \end{proof}
\section{Proof of \Cref{Main_THM,THM:END}}\label{Sec:Proof_of_Main_THM}
\begin{proof}[Proof of \Cref{Main_THM}.] We proceed by showing that (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii) $\Rightarrow$ (i).
(i) $\Rightarrow$ (ii) Fix $a \in (\exp(h(T)), 2)$. By \Cref{lem:bounds}, we have $\emptyset\neq(p_1 (a), p_2 (a)) \subseteq (\pi_{a}(\alpha), \pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$. Moreover, for each $p \in (p_1 (a), p_2 (a)) \subseteq (\pi_{a}(\alpha), \pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$, \begin{align}\label{eq:not_equal_comparasions} \alpha \prec \mu_{a, p}^{-}(p) \quad \text{and} \quad \mu_{a, p}^{+}(p) \prec \beta. \end{align} (We remind the reader that $\alpha \coloneqq \tau_{q}^{-}(q)$ and $\beta \coloneqq \tau_{q}^{+}(q)$ are the critical itineraries of $([0, 1], T^{\pm})$.) Let such a $p$ be fixed. By \Cref{thm:Main1.5} and the inequalities given in \eqref{eq:not_equal_comparasions} we have \begin{align}\label{eq:subset_code_space_p_q} \Omega_{q}^{-} \subset \Omega_{a, p}^{-} \quad \text{and} \quad \Omega_{q}^{+} \subset \Omega_{a, p}^{+}. \end{align} By \Cref{thm:Main1.5}, the inclusions in \eqref{eq:subset_code_space_p_q}, and the fact that the map $\pi_{a}\vert_{\Omega_{a, p}^{+} \cup \Omega^{-}_{a, p}}$ is increasing (\Cref{lem:bounds1.0}), we have that $\pi_{a}(\omega) \in [0, \pi_{a}(\alpha)] \cup [\pi_{a}(\beta), 1]$, for all $\omega \in \Omega_{q}^{+} \cup\Omega_{q}^{-}$. In other words \begin{align}\label{eq:codevinterval} \pi_{a}(\Omega_{q}^{+} \cup\Omega_{q}^{-}) \subseteq [0, \pi_{a}(\alpha)] \cup [\pi_{a}(\beta), 1]. \end{align} We claim that, for each $x \in \pi_{a}(\Omega_{q}^{+} \cup \Omega_{q}^{-})$ and $p' \in (\pi_{a}(\alpha),\pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$, \begin{align*} U_{a,p}^{\pm}(x) = U_{a,p'}^{\pm}(x), \quad \text{and} \quad U_{a, p'}^{\pm}(\pi_{a}(\Omega_{q}^{+} \cup \Omega_{q}^{-}) ) \subseteq \pi_{a}(\Omega_{q}^{+} \cup \Omega_{q}^{-}), \end{align*} It follows from this claim that, for all $p' \in (\pi_{a}(\alpha),\pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$, \begin{align}\label{eq:same_itinerary}
\mu_{a, p'}^{\pm}(x) = \mu_{a, p}^{\pm}(x) \quad \text{for all} \quad x \in \pi_{a}(\Omega_{q}^{+} \cup \Omega_{q}^{-}). \end{align} To prove the claim, let $p' \in (\pi_{a}(\alpha),\pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$ and $x \in \pi_{a}(\Omega_{q}^{+} \cup \Omega_{q}^{-})$. In light of the inclusion given in \eqref{eq:codevinterval} there are two cases, either $x \in \pi_{a}(\Omega_{q}^{+} \cup \Omega_{q}^{-}) \cap [ 0, \pi_{a}(\alpha)]$ or $x \in \pi_{a}(\Omega_{q}^{+} \cup \Omega_{q}^{-}) \cap [\pi_{a}(\beta), 1]$. As the proofs are essentially the same, we take $x \in \pi_{a}(\Omega_{q}^{+} \cup \Omega_{q}^{-}) \cap [ 0, \pi_{a}(\alpha)]$. Since $p, p' \in (\pi_{a}(\alpha),\pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$, we have that $\pi_{a}(\alpha) < \min\{ p, p'\}$. Moreover, $x \leq \pi_{a}(\alpha) < \min \{ p, p' \}$; in particular $x \neq p$ and $x \neq p'$. From this and the definition of the functions $U_{a, p}^{\pm}$, it can be concluded that \begin{align}\label{eq:conclude} U_{a, p}^{\pm}(x) = U_{a, p'}^{\pm}(x). \end{align} Since $x \in \pi_{a}(\Omega_{q}^{+} \cup \Omega_{q}^{-}) \cap [0, \pi_{a}(\alpha)]$, there exists $\omega \in\Omega_{q}^{+} \cup \Omega_{q}^{-}$ such that $x=\pi_{a}(\omega)$, and so \begin{align*} U_{a, p'}^{\pm}(x) = U_{a, p}^{\pm}(x) = U_{a, p}^{\pm}(\pi_{a}(\omega)) = \pi_{a}(S(\omega)) \in \pi_{a}(\Omega_{q}^{+} \cup \Omega_{q}^{-}), \end{align*} where the first equality follows from \eqref{eq:conclude}; the second equality follows from the fact that $x = \pi_{a}(\omega)$; the final equality follows from the inclusions given in \eqref{eq:subset_code_space_p_q} and \Cref{prop:code-map}; and the inclusion $\pi_{a}(S(\omega)) \in \pi_{a}(\Omega_{q}^{+} \cup \Omega_{q}^{-})$ is due to that fact that $\Omega_{q}^{+} \cup \Omega_{q}^{-}$ is forward shift sub-invariant (\Cref{lem:shift}). Thus the claim is proved.
By the inclusion given in \eqref{eq:subset_code_space_p_q} we have that $\alpha \in \Omega^{-}_{a, p}$ and $\beta \in \Omega^{+}_{a, p}$. So there exist $x, y \in [0, 1]$ such that $\alpha = \mu_{a, p}^{-}(x)$ and $\beta = \mu_{a, p}^{+}(y)$. Therefore, by \Cref{prop:code-map} we have that $\mu_{a, p}^{-}(\pi_{a}(\alpha)) = \mu_{a, p}^{-}(\pi_{a}(\mu_{a, p}^{-}(x)) = \mu_{a, p}^{-}(x) = \alpha$ and $\mu_{a, p}^{+}(\pi_{a}(\beta)) = \mu_{a, p}^{+}(\pi_{a}(\mu_{a, p}^{+}(y)) = \mu_{a, p}^{+}(y) = \beta$. This, in combination with \eqref{eq:same_itinerary}, implies that $\mu_{a, p'}^{-}(\pi_{a}(\alpha)) = \alpha$ and $\mu_{a, p'}^{+}(\pi_{a}(\beta)) = \beta$, for all $p' \in (\pi_{a}(\alpha),\pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$. Hence, $\alpha \in \Omega_{a, p'}^{-}$ and $\beta \in \Omega_{a, p'}^{+}$. It follows from \Cref{thm:Main1.5} that $\alpha \in [\overline{0}, \mu_{a, p'}^{-}(p')] \cup (\mu_{a, p'}^{+}(p'), \overline{1}]$. (We remind the reader that $\overline{0}$ denotes the element $0 \, 0 \,0 \, \cdots \in \Omega$ and $\overline{1}$ to denotes the element $1 \,1 \,1 \, \cdots \in \Omega$.) Since $\alpha$ begins with $01$, it must be the case that $\alpha\in [\overline{0}, \mu_{a, p'}^{-}(p')]$. Moreover, $\alpha \neq \mu_{a, p'}^{-}(p')$, since if $\alpha = \mu_{a, p'}^{-}(p')$, then by \Cref{prop:code-map} we would have that $\pi_{a}(\alpha) = \pi_{a}( \mu_{a, p'}^{-}(p')) = p'$, which contradicts that $p'\in (\pi_{a}(\alpha),\pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$. A similar argument shows that $\beta \in (\mu_{a, p'}^{+}(p'), \overline{1}]$. Therefore, $\alpha \prec \mu_{a, p'}^{-}(p')$ and $\beta \succ \mu_{a, p'}^{+}(p')$, for all $p' \in (\pi_{a}(\alpha),\pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$.
(ii) $\Rightarrow$ (iii) This is an immediate consequence of \Cref{thm:Main1.5}.
(iii) $\Rightarrow$ (i) If $\Omega_{q}^{-} \subset \Omega_{a, p}^{-} $ and $ \Omega_{q}^{+} \subset \Omega_{a, p}^{+}$ for $p\in (\pi_{a}(\alpha),\pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$, then by \Cref{thm:Main1.5} we have that $\alpha \preceq \mu_{a, p}^{-}(p) \prec \mu_{a, p}^{+}(p) \prec \beta$ or $\alpha \prec \mu_{a, p}^{-}(p) \prec \mu_{a, p}^{+}(p) \preceq \beta$, and so by \Cref{cor:cor2} we have that $\exp(h(T)) \leq a$. We will now show that $\exp(h(T)) \not= a$ if $\Omega_{q}^{\pm} \subset \Omega_{a, p}^{\pm} $. In order to reach a contradiction, suppose that $\exp(h(T)) = a$ and that $\Omega_{q}^{+} \subset \Omega_{a, p}^{+}$ and $\Omega_{q}^{-} \subset \Omega_{a, p}^{-}$, for some $p$ belonging to the interval $(\pi_{a}(\alpha),\pi_{a}(\beta)) \cap (1-a^{-1}, a^{-1})$. Therefore, fix $p$, such that either \begin{align}\label{EQ:(iii)->(i)} \alpha \preceq \mu_{a, p}^{-}(p) \prec \mu_{a, p}^{+}(p) \prec \beta\quad \text{or} \quad \alpha \prec \mu_{a, p}^{-}(p) \prec \mu_{a, p}^{+}(p) \preceq \beta \end{align} holds. By \cite{G} our given Lorenz system $([0,1], T^{\pm})$ is semi-conjucate to some uniform Lorenz system $([0,1], U^{\pm}_{s, p'})$. Moreover, since the semi-conjugacy preserves topological entropy (\Cref{lemma1}) and since by \Cref{thm:uniform} we have that $h(U_{s, p'}^{\pm}) = \ln(s)$, it follows that $s = \exp(h(T)) = a$. Hence, by \Cref{lemma1}, we have that $\Omega_{s, p'}^{\pm} \subseteq\Omega_{q}^{\pm}$ and therefore, \begin{align}\label{EQ:(iii)->(i)2}
\mu_{a, p'}^{-}(p')\preceq \alpha \quad \text{and} \quad \beta\preceq\mu_{a, p'}^{+}(p') . \end{align} Combining \eqref{EQ:(iii)->(i)2} with \eqref{EQ:(iii)->(i)} and then applying \Cref{lem2} gives a desired contradiction.
\end{proof}
Before presenting the proof of \Cref{THM:END} we given the following example which illustrates the importance of taking the intersection of $(\pi_{a}(\alpha),\pi_{a}(\beta))$ with the $(1-a^{-1}, a^{-1})$ in \Cref{Main_THM} (ii) and (iii).
\begin{example}\label{EXMP:inequalities} An instance of when the inequality $\pi_{a}(\beta) > a^{-1}$ can occur is when $T^{\pm}$ is a Lorenz map where the first branch is a linear function with gradient close to $1$ and the second branch is a function of high polynomial or exponential growth. An explicit example of such a map is the Lorenz map with critical point $1/2$ given by the functions $f_{0}(x) \coloneqq 1.001x$ and $f_{1}(x) \coloneqq \exp(x + \ln(2) - 1) - 1$. In this case the inequality $\pi_{a}(\alpha) < 1- a^{-1}$ is satisfied for $a = 3/2 > \exp(h(T)) \approx 1.00125$. (This latter value was calculated using an implemented version of the algorithm presented in \Cref{Sec:Algorithm}, with a tolerance $\epsilon = 0.0001$ and a truncation tern $n = 25,000$.) By reversing the roles of the first and second branch one obtains a Lorenz map with $\pi_{a}(\beta) > a^{-1}$. \end{example}
\begin{proof}[Proof of \Cref{THM:END}.] Since the proofs for (i) and (ii) are essentially the same, we only include a proof of (i). The result is proved by showing the following set of implications (a) $\Rightarrow$ (b) $\Rightarrow$ (c) $\Rightarrow$ (a).
(a) $\Rightarrow$ (b) Let $a \in (\exp(h(T)), 2)$ and suppose that the inequalities given in \Cref{THM:END}~(i)~(b) do not hold for $p = a^{-1}$. Then by definition we have that $\tau_{q}^{-}(q) = \mu_{a, a^{-1}}(a^{-1}) = 0 \overline{1}$. An application of \Cref{cor:cor2} then leads to a contradiction to how the parameter $a$ was originally chosen. The uniqueness follows directly from \Cref{lem2}.
(b) $\Rightarrow$ (c) This is a direct consequence of \Cref{thm:Main1.5} and the fact that $a > \exp(h(T))$.
(c) $\Rightarrow$ (a) The proof is essentially the same as the proof of (iii) $\Rightarrow$ (i) of \Cref{Main_THM}. \end{proof}
\section{An algorithm to compute the topological entropy of a Lorenz map}\label{Sec:Algorithm}
The numerical computation of topological entropy of one dimensional dynamical systems has received much attention; see for instance \cite{BK,BKSP,FMT,Mil}. Based on \Cref{Main_THM,THM:END}, we next provide a new algorithm to compute the topological entropy of a Lorenz system. The algorithm is stated assuming infinite arithmetic precision. However, with straightforward modifications, the algorithm can be practically implemented. Such an implementation was used in obtaining the sample results presented at the end of this section. After the statement of algorithm a proof of its validity is given. (We remind the reader that $h(T)$ denotes the common value $h(T^{+}) = h (T^{-})$, for a given Lorenz system $([0, 1], T^{\pm})$.)
\noindent \parbox{15mm}{\textbf{Input:}} \textit{A Lorenz map $T^{\pm}$ with critical point $q$ and a tolerance $\epsilon \in (0, 1)$.}\vskip2mm
\noindent \parbox{15mm}{\textbf{Output:}} \textit{An estimate to $h(T)$ within a tolerance of $\epsilon$.} \begin{enumerate}
\item[(1)] Compute: $\alpha \coloneqq \tau_{q}^{+}(q)$ and $\beta \coloneqq \tau_{q}^{-}(q)$.
\item[(2)] Initialise: $a_{1} = 1$ and $a_{2} = 2$.
\item[(3)] Set $a =\frac{ a_{1} + a_{2}}{2}$.
\item[(4)] If both $\alpha\neq 0\overline{1}$ and $\beta\neq 1\overline{0}$ , then go to Step~(5), else go to Step~(4)(a).
\begin{enumerate}
\item[(a)] If both $\alpha= 0\overline{1}$ and $\beta \neq 1\overline{0}$, then compute $\mu^{+}_{a, a^{-1}}(a^{-1})$ and go to Step(11), else go to Step(4)(b).
\item[(b)] Compute $\mu^{-}_{a,1-a^{-1}}(1-a^{-1})$ and go to Step~(12).
\end{enumerate}
\item[(5)] Compute: $\pi_{a}(\alpha)$ and $\pi_{a}(\beta)$.
\item[(6)] Compute: $t_{1}(a) \coloneqq \max\{\pi_{a}(\alpha),1-a^{-1}\}$ and $t_{2}(a) \coloneqq \min\{\pi_{a}(\beta), a^{-1}\}$.
\item[(7)] If $t_{1}(a) \geq t_{2}(a)$, then $a_{1} \gets a$ and go to Step(13), else go to Step(8).
\item[(8)] Set $p = (t_{1}(a) + t_{2}(a))/2$.
\item[(9)] Compute: $\mu^{+}_{a,p}(p)$ and $\mu_{a, p}^{-}(p)$.
\item[(10)] If $\alpha \prec \mu^{-}_{a, p}(p)$ and $\mu^{+}_{a, p}(p) \prec \beta$, then go to Step~(10)(a), else go to Step~(10)(b).
\begin{enumerate}
\item[(a)] $a_{2} \gets a$ and go to Step(13).
\item[(b)] $a_{1} \gets a$ and go to Step(13).
\end{enumerate}
\item[(11)] If $\mu^{+}_{a, a^{-1}}(a^{-1})\prec\beta$, then $a_{2} \gets a$ and go to Step~(13), else $a_{1} \gets a$ and go to Step~(13).
\item[(12)] If $\alpha \prec \mu^{-}_{a, 1-a^{-1}}(1-a^{-1})$, then $a_{2} \gets a$ and go to Step(13), else $a_{1} \gets a$ and go to Step~(13).
\item[(13)] If $a_{2} - a_{1} < \epsilon/2$, then return $h(T) \in [\ln ((a_{1} + a_{2})/2 - \epsilon/4), \ln ((a_{1} + a_{2})/2 + \epsilon/4)]$ and terminate the algorithm, else go to Step(3). \end{enumerate}
\begin{proof}[Proof of the validity of the Algorithm.] The variable $a$ in the algorithm is the midpoint of the interval $[a_{1},a_{2}]$ which is initialized at $[a_{1},a_{2}] = [1, 2]$, and thus, $\ln (a_{1}) \leq h(T) < \ln (a_{2})$. We will show that, throughout the algorithm, the following inequality is maintained, \begin{align}\label{eq:alg} \ln (a_{1}) \leq h(T) \leq \ln (a_{2}). \end{align} A tolerance $\epsilon >0$ is fixed at the start. At each iteration (Step~(3) to Step~(13)) of the algorithm, the length of this interval $[a_{1}, a_{2}]$ is halved until, at Step~(13), we arrive at $a_{2} - a_{1} < \epsilon/2$. According to \eqref{eq:alg}, at this point we have estimated the entropy within the desired tolerance $\epsilon \in (0, 1)$, specifically \begin{align*}
\ln \left( (a_{1} + a_{2})/2 - \epsilon/4 \right) \leq h(T) \leq \ln \left( (a_{1} + a_{2})/2 + \epsilon/4 \right). \end{align*} Suppose, in Step~(4), that $\alpha\neq 0\overline{1}$ and $\beta\neq 1\overline{0}$, namely, that the critical point $q$ is such that $f_{0}(q) \neq 1$ and $f_{1}(q) \neq 0$. (Here, we remind the reader that $f_{0}: [0, q] \to [0, 1]$ and $f_{1}: [q,1] \to [0, 1]$ are the expanding maps which define the given $T^{\pm}$; see \Cref{Defn:Lorenz_map}.) At Step~(7) or Step~(10) the interval $[a_{1}, a_{2}]$ will be replaced by either $[a_1,a]$ or $[a, a_{2}]$, where $a$ has the value $(a_{1} + a_{2})/2$. It will now be proved that at each iteration (Step~(3) to Step~(13)), the inequalities given in \eqref{eq:alg} are maintained. To see this we will follow the steps of the algorithm. At Step~(3), the value of $a$ is set to the value of the midpoint of the interval $[a_{1}, a_{2}]$. In Step~(5), the images of the critical itineraries $\alpha$ and $\beta$ of the given Lorenz system $([0,1], T^{\pm})$ under $\pi_{a}$ are computed. In Step~(6), the values of $t_{1}(a)$ and $t_{2}(a)$ are set to the left and right endpoints, respectively, of an interval which, according to \Cref{lem:bounds}, has non-empty interior provided that $h(T) < \ln (a)$. Thus, if $t_{1}(a) \geq t_{2}(a)$, then $h(T) \geq \ln (a)$. In this case, the value of $a_{1}$ is reset to the value of $a$ in Step~(7) and the inequalities in \eqref{eq:alg} are maintained. The algorithm then proceeds to Step~(13).
On the other hand, if $t_{2}(a) > t_{1}(a)$, then in Step~(8) the value of $p$ is set to the midpoint of the interval $[t_{1}(a), t_{2}(a)]$. In Step~(9) the algorithm computes the critical itineraries, $\mu^{+}_{a, p}(p)$ and $\mu_{a, p}^{-}(p)$, of the uniform Lorenz systems $([0, 1], U_{a, p}^{\pm})$. In Step~(10) the algorithm compares $\mu^{-}_{a, p}(p)$ with $\alpha$ and compares $\mu^{+}_{a, p}(p)$ with $\beta$. There are two possibilities, either both $\mu_{a, p}^{-}(p) \succ \alpha$ and $\mu_{a, p}^{+}(p) \prec \beta$ hold or not. \begin{enumerate} \item If $\mu_{a, p}^{-}(p) \succ \alpha$ and $\mu_{a, p}^{+}(p) \prec \beta$, then $h(T) \leq \ln (a)$, see \Cref{cor:cor2}. Therefore, to maintain the inequalities given in \eqref{eq:alg}, the value of $a_{2}$ is reset to the value of $a$. \item Otherwise, we have $h(T) \geq \ln (a)$. Since, if this was not the case, then this would contradict \Cref{Main_THM}. Therefore, to maintain the inequalities given in \eqref{eq:alg}, the value of $a_{1}$ is reset to the value of $a$. \end{enumerate} In either of the above two case, the algorithm then proceeds to Step~(13).
Returning to Step~(4), suppose that $\alpha = 0\overline{1}$ and $\beta \neq 1\overline{0}$. Observe, for each $a \in (1, 2)$, that $\mu_{a, a^{-1}}^{-}(a^{-1}) = \alpha = 0\overline{1}$. There are now two possibilities, either $\mu_{a,a^{-1}}^{+}(a^{-1}) \prec \beta$ or not. \begin{enumerate}\setcounter{enumi}{2} \item If $\mu^{+}_{a, a^{-1}}(a^{-1}) \prec \beta$, then, by \Cref{cor:cor2} and since $\mu_{a, p}^{-}(p) = \alpha = 0\overline{1}$, we have that $h(T) \leq \ln(a)$. Therefore, to maintain the inequalities given in \eqref{eq:alg}, the value of $a_{2}$ is reset to the value of $a$. The algorithm then proceeds to Step~(13). \item If $\mu^{+}_{a, a^{-1}}(a^{-1}) \succeq \beta$, then, by \Cref{cor:cor2} and since $\mu_{a, p}^{-}(p) = \alpha = 0\overline{1}$, we have that $h(T) \geq \ln(a)$. Therefore, to maintain the inequalities given in \eqref{eq:alg}, the value of $a_{1}$ is reset to the value of $a$. The algorithm then proceeds to Step~(13). \end{enumerate} At Step~(13), provided that $a_{2} - a_{1} \geq \epsilon/2$, the algorithm proceeds to the next iteration, otherwise the algorithm returns the following value and terminates: $h(T^{+}) = h(T^{-}) \in [\ln ((a_{1} + a_{2})/2 - \epsilon/4), \ln ((a_{1} + a_{2})/2 + \epsilon/4)]$. Similarly, if $\alpha \neq 0\overline{1}$ and $\beta = 1\overline{0}$, then in Step~(4)(b) of the algorithm the value of $p$ is set to $1-a^{-1}$ and the itinerary $\mu_{a, 1-a^{-1}}^{-}(1-a^{-1})$ is computed. The algorithm then proceeds to Step~(12), where, to maintain the inequalities in \eqref{eq:alg}, the algorithm either \begin{enumerate}\setcounter{enumi}{4} \item resets the value of $a_{2}$ to the value of $a$, if $\alpha \prec \mu^{-}_{a, 1-a^{-1}}(1-a^{-1})$, or \item resets the value of $a_{1}$ to the value of $a$, if $\alpha \succeq \mu^{-}_{a, 1-a^{-1}}(1-a^{-1})$. \end{enumerate} The algorithm then goes to Step~(13); here it either goes to the next iteration or terminates.
Observe that the situation where $\alpha = 0\overline{1}$ and $\beta = 1\overline{0}$ cannot occur, since by definition of the itineraries, this would immediately imply that $f_{0}(q) = 1$ and $f_{1}(q) = 0$. Thus, the given system is not a Lorenz system as it would violate condition (i) of \Cref{Defn:Lorenz_map}. \end{proof}
\subsection{Sample results.}\label{Sec:Sample_results}
Presented below are two examples that demonstrates an implemented version of our algorithm. These examples indicate that the algorithm returns an accurate estimate for the entropy of a Lorenz system. To practically implement the algorithm, itineraries are computed to a prescribed length $n \geq 3$, which is called the \textit{truncation term} and is an additional input to the algorithm.
\begin{example}\label{exmp:exampl1} Consider the Lorenz map $T^{\pm}$ with critical point $q$ given by $f_{0}(x) = a\sqrt{x}$ and $f_{1}(x) = b x + 1 - b$, where $a = 1.25$, $b = (a^{-6}-1)/(a^{-2}-1)$ and $q =1/a^{2}$. The reason for this choice of $a, b, q$ is that, in this case, there is a theoretical method for determining the topological entropy of the map $T^{\pm}$. This allows us to compare the estimated value for the entropy given by our algorithm to the actual value. To be more precise, to theoretically determine the topological entropy we use the fact that, for this choice of $a,b,q$, the critical itineraries are periodic and therefore this Lorenz map is Markov. For Markov maps the topological entropy is the logarithm of the maximum eigenvalue of the associated adjacency matrix \cite[Proposition~3.4.1]{BS}. Using this method we obtain that $h(T^{\pm}) = \ln((1+\sqrt{5})/2) \approx 0.4812118251$. The following table gives the output of a practically implemented version of our algorithm for this map, where $\epsilon$ denote the tolerance term and $n$ denotes the truncation term. \begin{small} \begin{table}[h] \begin{tabular}{lccc} \hline & $\epsilon=10^{-2}$ & $\epsilon=10^{-4}$ & $\epsilon=10^{-6}$\\ \hline $n = 10$ & 0.4831010758 & 0.4811979105 & 0.4812117615\\ $n = 100$ & 0.4831010758 & 0.4811979105 & 0.4812117615\\ $n = 1,000$ & 0.4831010758 & 0.4811979105 & 0.4812117615\\ $n = 10,000$ & 0.4831010758 & 0.4811979105 & 0.4812117615\\ \hline \end{tabular} \end{table} \end{small} \end{example}
\begin{example} Here we consider the uniform Lorenz map $U^{\pm}_{a, 1/2}$ and the uniform Lorenz map $U^{\pm}_{a, a^{-1}}$ for $a = \sqrt{2}$, which, by \Cref{lemma1}, both have topological entropy equal to $\log(\sqrt{2}) \approx 0.34657359023$. The following table gives the output of a practically implemented version of our algorithm for these maps, where $\epsilon$ denote the tolerance and $n$ denotes the truncation term. \begin{small} \begin{table}[h] \begin{tabular}{lcccc} \hline & \multicolumn{2}{c}{$p=1/2$} & \multicolumn{2}{c}{$p=a^{-1}=1/\sqrt{2}$}
\\ & $\epsilon=10^{-3}$ & $\epsilon=10^{-6}$ & $\epsilon=10^{-3}$ & $\epsilon=10^{-6}$\\ \hline $n = 10$ & 0.3652803888 & 0.3655560121 & 0.3475021428 & 0.3471925188\\ $n = 100$ & 0.3468120116 & 0.3465736575 & 0.3468120116 & 0.3465736575\\ $n = 1,000$ & 0.3468120116 & 0.3465736575 & 0.3468120116 & 0.3465736575\\ $n = 10,000$ & 0.3468120116 & 0.3465736575 & 0.3468120116 & 0.3465736575\\ \hline \end{tabular} \end{table} \end{small} \end{example}
\end{document} | arXiv |
Fixed-point subring
In algebra, the fixed-point subring $R^{f}$ of an automorphism f of a ring R is the subring of the fixed points of f, that is,
$R^{f}=\{r\in R\mid f(r)=r\}.$
More generally, if G is a group acting on R, then the subring of R
$R^{G}=\{r\in R\mid g\cdot r=r,\,g\in G\}$
is called the fixed subring or, more traditionally, the ring of invariants under G. If S is a set of automorphisms of R, the elements of R that are fixed by the elements of S form the ring of invariants under the group generated by S. In particular, the fixed-point subring of an automorphism f is the ring of invariants of the cyclic group generated by f.
In Galois theory, when R is a field and G is a group of field automorphisms, the fixed ring is a subfield called the fixed field of the automorphism group; see Fundamental theorem of Galois theory.
Along with a module of covariants, the ring of invariants is a central object of study in invariant theory. Geometrically, the rings of invariants are the coordinate rings of (affine or projective) GIT quotients and they play fundamental roles in the constructions in geometric invariant theory.
Example: Let $R=k[x_{1},\dots ,x_{n}]$ be a polynomial ring in n variables. The symmetric group Sn acts on R by permuting the variables. Then the ring of invariants $R^{G}=k[x_{1},\dots ,x_{n}]^{\operatorname {S} _{n}}$ is the ring of symmetric polynomials. If a reductive algebraic group G acts on R, then the fundamental theorem of invariant theory describes the generators of RG.
Hilbert's fourteenth problem asks whether the ring of invariants is finitely generated or not (the answer is affirmative if G is a reductive algebraic group by Nagata's theorem.) The finite generation is easily seen for a finite group G acting on a finitely generated algebra R: since R is integral over RG,[1] the Artin–Tate lemma implies RG is a finitely generated algebra. The answer is negative for some unipotent groups.
Let G be a finite group. Let S be the symmetric algebra of a finite-dimensional G-module. Then G is a reflection group if and only if $S$ is a free module (of finite rank) over SG (Chevalley's theorem).
In differential geometry, if G is a Lie group and ${\mathfrak {g}}=\operatorname {Lie} (G)$ its Lie algebra, then each principal G-bundle on a manifold M determines a graded algebra homomorphism (called the Chern–Weil homomorphism)
$\mathbb {C} [{\mathfrak {g}}]^{G}\to \operatorname {H} ^{2*}(M;\mathbb {C} )$
where $\mathbb {C} [{\mathfrak {g}}]$ is the ring of polynomial functions on ${\mathfrak {g}}$ and G acts on $\mathbb {C} [{\mathfrak {g}}]$ by adjoint representation.
See also
• Character variety
Notes
1. Given r in R, the polynomial $\prod _{g\in G}(t-g\cdot r)$ is a monic polynomial over RG and has r as one of its roots.
References
• Mukai, Shigeru; Oxbury, W. M. (8 September 2003) [1998], An Introduction to Invariants and Moduli, Cambridge Studies in Advanced Mathematics, vol. 81, Cambridge University Press, ISBN 978-0-521-80906-1, MR 2004218
• Springer, Tonny A. (1977), Invariant theory, Lecture Notes in Mathematics, vol. 585, Springer
| Wikipedia |
QU Le 'an, LI Manchun, CHEN Zhenjie, ZHI Junjun, 2021. A Modified Self-adaptive Method for Mapping Annual 30-m Land Use/Land Cover Using Google Earth Engine: A Case Study of Yangtze River Delta. Chinese Geographical Science, 31(5): 782−794 doi: 10.1007/s11769-021-1226-4
Citation: QU Le 'an, LI Manchun, CHEN Zhenjie, ZHI Junjun, 2021. A Modified Self-adaptive Method for Mapping Annual 30-m Land Use/Land Cover Using Google Earth Engine: A Case Study of Yangtze River Delta. Chinese Geographical Science, 31(5): 782−794 doi: 10.1007/s11769-021-1226-4
A Modified Self-adaptive Method for Mapping Annual 30-m Land Use/Land Cover Using Google Earth Engine: A Case Study of Yangtze River Delta
Le 'an QU1, 2, 3 , ,
Manchun LI1, 3 , , ,
Zhenjie CHEN1, 3 ,
Junjun ZHI2
School of Geography and Ocean Science, Nanjing University, Nanjing 210023, China
School of Geography and Tourism, Anhui Normal University, Anhui Province, Wuhu 241002, China
Jiangsu Provincial Key Laboratory of Geographic Information Science and Technology, Nanjing University, Nanjing 210023, China
Funds: Under the auspices of the National Key Research and Development Program of China (No. 2017YFB0504205), National Natural Science Foundation of China (No. 41571378), Natural Science Research Project of Higher Education in Anhui Provence (No. KJ2020A0089)
Corresponding author: LI Manchun. E-mail: [email protected]
Annual Land Use/Land Cover (LULC) change information at medium spatial resolution (i.e., at 30 m) is used in applications ranging from land management to achieving sustainable development goals related to food security. However, obtaining annual LULC information over large areas and long periods is challenging due to limitations on computational capabilities, training data, and workflow design. Using the Google Earth Engine (GEE), which provides a catalog of multi-source data and a cloud-based environment, we developed a novel methodology to generate a high accuracy 30-m LULC cover map collection of the Yangtze River Delta by integrating free and public LULC products with Landsat imagery. Our major contribution is a hybrid approach that includes three major components: 1) a high-quality training dataset derived from multi-source LULC products, filtered by k-means clustering analysis; 2) a yearly 39-band stack feature space, utilizing all available Landsat data and DEM data; and 3) a self-adaptive Random Forest (RF) method, introduced for LULC classification. Experimental results show that our proposed workflow achieves an average classification accuracy of 86.33% in the entire Delta. The results demonstrate the great potential of integrating multi-source LULC products for producing LULC maps of increased reliability. In addition, as the proposed workflow is based on open source data and the GEE cloud platform, it can be used anywhere by anyone in the world.
Land Use/Land Cover (LULC),
self-adaptive Random Forest (RF),
Google Earth Engine (GEE),
Yangtze River Delta
[1] Adepoju K A, Adelabu S A, 2020. Improving accuracy of Landsat-8 OLI classification using image composite and multisource data with Google Earth Engine. Remote Sensing Letters, 11(2): 107–116. doi: 10.1080/2150704X.2019.1690792
[2] Anchang J Y, Prihodko L, Ji W J et al., 2020. Toward operational mapping of woody canopy cover in tropical savannas using Google Earth Engine. Frontiers in Environmental Science, 8: 4. doi: 10.3389/fenvs.2020.00004
[3] Bailly A, Chapel L, Tavenard R et al., 2017. Nonlinear time-series adaptation for land cover classification. IEEE Geoscience and Remote Sensing Letters, 14(6): 896–900. doi: 10.1109/LGRS.2017.2686639
[4] Bullock E L, Woodcock C E, Olofsson P, 2020. Monitoring tropical forest degradation using spectral unmixing and Landsat time series analysis. Remote Sensing of Environment, 238: 110968. doi: 10.1016/j.rse.2018.11.011
[5] Capolupo A, Monterisi C, Tarantino E, 2020. Landsat images classification algorithm (LICA) to automatically extract land cover information in Google Earth Engine environment. Remote Sensing, 12(7): 1201. doi: 10.3390/rs12071201
[6] Chakraborty A, Sachdeva K, Joshi P K, 2016. Mapping long-term land use and land cover change in the central Himalayan region using a tree-based ensemble classification approach. Applied Geography, 74: 136–150. doi: 10.1016/j.apgeog.2016.07.008
[7] Chen Lin, Ren Chunying, Zhang Bai et al., 2018. Spatiotemporal dynamics of coastal wetlands and reclamation in the Yangtze estuary during past 50 years (1960s–2015). Chinese Geographical Science, 28(3): 386–399. doi: 10.1007/s11769-017-0925-3
[8] Chen S, Li G, Xu Z G et al., 2019. Combined impact of socioeconomic forces and policy implications: spatial-temporal dynamics of the ecosystem services value in Yangtze River Delta, China. Sustainability, 11(9): 2622. doi: 10.3390/su11092622
[9] Daldegan G A, Roberts D A, de Figueiredo Ribeiro F, 2019. Spectral mixture analysis in Google Earth Engine to model and delineate fire scars over a large extent and a long time-series in a rainforest-savanna transition zone. Remote Sensing of Environment, 232: 111340. doi: 10.1016/j.rse.2019.111340
[10] Feng Y J, Liu Y, Tong X H, 2018. Comparison of metaheuristic cellular automata models: a case study of dynamic land use simulation in the Yangtze River Delta. Computers Environment and Urban Systems, 70: 138–150. doi: 10.1016/j.compenvurbsys.2018.03.003
[11] Ghorbanian A, Kakooei M, Amani M et al., 2020. Improved land cover map of Iran using Sentinel imagery within Google Earth Engine and a novel automatic workflow for land cover classification using migrated training samples. ISPRS Journal of Photogrammetry and Remote Sensing, 167: 276–288. doi: 10.1016/j.isprsjprs.2020.07.013
[12] Ghosh A, Sharma R, Joshi P K, 2014. Random forest classification of urban landscape using Landsat archive and ancillary data: combining seasonal maps with decision level fusion. Applied Geography, 48: 31–41. doi: 10.1016/j.apgeog.2014.01.003
[13] Gong P, Liu H, Zhang M N et al., 2019. Stable classification with limited sample: transferring a 30-m resolution sample set collected in 2015 to mapping 10-m resolution global land cover in 2017. Science Bulletin, 64(6): 370–373. doi: 10.1016/j.scib.2019.03.002
[14] Gorelick N, Hancher M, Dixon M et al., 2017. Google Earth Engine: planetary-scale geospatial analysis for everyone. Remote Sensing of Environment, 202: 18–27. doi: 10.1016/j.rse.2017.06.031
[15] Gumma M K, Thenkabail P S, Teluguntla P G et al., 2020. Agricultural cropland extent and areas of South Asia derived using Landsat satellite 30-m time-series big-data using random forest machine learning algorithms on the Google Earth Engine cloud. GIScience & Remote Sensing, 57(3): 302–322. doi: 10.1080/15481603.2019.1690780
[16] Hird J N, DeLancey E R, McDermid G J et al., 2017. Google Earth Engine, open-access satellite data, and machine learning in support of large-area probabilistic wetland mapping. Remote Sensing, 9(12): 1315. doi: 10.3390/rs9121315
[17] Huang H B, Chen Y L, Clinton N et al., 2017. Mapping major land cover dynamics in Beijing using all Landsat images in Google Earth Engine. Remote Sensing of Environment, 202: 166–176. doi: 10.1016/j.rse.2017.02.021
[18] Huang H B, Wang J, Liu C X et al., 2020. The migration of training samples towards dynamic global land cover mapping. ISPRS Journal of Photogrammetry and Remote Sensing, 161: 27–36. doi: 10.1016/j.isprsjprs.2020.01.010
[19] Hurni K, Van Den Hoek J, Fox J, 2019. Assessing the spatial, spectral, and temporal consistency of topographically corrected Landsat time series composites across the mountainous forests of Nepal. Remote Sensing of Environment, 231: 111225. doi: 10.1016/j.rse.2019.111225
[20] Ji H Y, Li X, Wei X C et al., 2020. Mapping 10-m resolution rural settlements using multi-source remote sensing datasets with the Google Earth Engine platform. Remote Sensing, 12(17): 2832. doi: 10.3390/rs12172832
[21] Kakooei M, Baleghi Y, 2020. VHR semantic labeling by random forest classification and fusion of spectral and spatial features on Google Earth Engine. Journal of AI and Data Mining, 8(3): 357–370. doi: 10.22044/JADM.2020.8252.1964
[22] Li H, Wan W, Fang Y et al., 2019. A Google Earth Engine-enabled software for efficiently generating high-quality user-ready Landsat mosaic images. Environmental Modelling & Software, 112: 16–22. doi: 10.1016/j.envsoft.2018.11.004
[23] Li H, Wang C Z, Zhong C et al., 2017. Mapping urban bare land automatically from Landsat imagery with a simple index. Remote Sensing, 9(3): 249. doi: 10.3390/rs9030249
[24] Li W J, Dong R M, Fu H H et al., 2020. Integrating Google Earth imagery with Landsat data to improve 30-m resolution land cover mapping. Remote Sensing of Environment, 237: 111563. doi: 10.1016/j.rse.2019.111563
[25] Li X C, Gong P, 2016. An 'exclusion-inclusion' framework for extracting human settlements in rapidly developing regions of China from Landsat images. Remote Sensing of Environment, 186: 286–296. doi: 10.1016/j.rse.2016.08.029
[26] Liu H, Gong P, Wang J et al., 2020. Annual dynamics of global land cover and its long-term changes from 1982 to 2015. Earth System Science Data, 12(2): 1217–1243. doi: 10.5194/essd-12-1217-2020
[27] Liu H, Gong P, Wang J et al., 2021. Production of global daily seamless data cubes and quantification of global land cover change from 1985 to 2020: iMap World 1.0. Remote Sensing of Environment, 258: 112364. doi: 10.1016/j.rse.2021.112364
[28] Long H L, 2014. Land consolidation: an indispensable way of spatial restructuring in rural China. Journal of Geographical Sciences, 24(2): 211–225. doi: 10.1007/s11442-014-1083-5
[29] Lopes M, Fauvel M, Ouin A et al., 2017. Spectro-temporal heterogeneity measures from dense high spatial resolution satellite image time series: application to grassland species diversity estimation. Remote Sensing, 9(10): 993. doi: 10.3390/rs9100993
[30] Mack B, Leinenkugel P, Kuenzer C et al., 2017. A semi-automated approach for the generation of a new land use and land cover product for Germany based on Landsat time-series and Lucas in-situ data. Remote Sensing Letters, 8(3): 244–253. doi: 10.1080/2150704X.2016.1249299
[31] Mahdianpari M, Jafarzadeh H, Granger J E et al., 2020. A large-scale change monitoring of wetlands using time series Landsat imagery on Google Earth Engine: a case study in Newfoundland. GIScience & Remote Sensing, 57(8): 1102–1124. doi: 10.1080/15481603.2020.1846948
[32] Mao D H, Luo L, Wang Z M et al., 2018. Conversions between natural wetlands and farmland in China: a multiscale geospatial analysis. Science of the Total Environment, 634: 550–560. doi: 10.1016/j.scitotenv.2018.04.009
[33] Mao D H, Tian Y L, Wang Z M et al., 2021. Wetland changes in the Amur River Basin: differing trends and proximate causes on the Chinese and Russian sides. Journal of Environmental Management, 111670. doi: 10.1016/j.jenvman.2020.111670
[34] Millard K, Richardson M, 2015. On the importance of training data sample selection in random forest image classification: a case study in Peatland ecosystem mapping. Remote Sensing, 7(7): 8489–8515. doi: 10.3390/rs70708489
[35] Mohajane M, Essahlaoui A, Oudija F et al., 2018. Land use/land cover (LULC) using Landsat data series (MSS, TM, ETM+ and OLI) in Azrou forest, in the central middle atlas of Morocco. Environments, 5(12): 131. doi: 10.3390/environments5120131
[36] Müller H, Rufin P, Griffiths P et al., 2015. Mining dense Landsat time series for separating cropland and pasture in a heterogeneous Brazilian savanna landscape. Remote Sensing of Environment, 156: 490–499. doi: 10.1016/j.rse.2014.10.014
[37] Pandey P C, Koutsias N, Petropoulos G P et al., 2021. Land use/land cover in view of earth observation: data sources, input dimensions, and classifiers: a review of the state of the art. Geocarto International, 36(9): 957–988. doi: 10.1080/10106049.2019.1629647
[38] Shen W S, Lin X G, Gao N et al., 2008. Land use intensification affects soil microbial populations, functional diversity and related sup-pressiveness of cucumber Fusarium wilt in China's Yangtze River Delta. Plant and Soil, 306(1-2): 117–127. doi: 10.1007/s11104-007-9472-5
[39] Simonetti D, Simonetti E, Szantoi Z et al., 2015. First results from the phenology-based synthesis classifier using Landsat 8 imagery. IEEE Geoscience and Remote Sensing Letters, 12(7): 1496–1500. doi: 10.1109/LGRS.2015.2409982
[40] Tamiminia H, Salehi B, Mahdianpari M et al., 2020. Google Earth Engine for geo-big data applications: a meta-analysis and systematic review. ISPRS Journal of Photogrammetry and Remote Sensing, 164: 152–170. doi: 10.1016/j.isprsjprs.2020.04.001
[41] Teluguntla P, Thenkabail P S, Oliphant A et al., 2018. A 30-m landsat-derived cropland extent product of Australia and China using random forest machine learning algorithm on Google Earth Engine cloud computing platform. ISPRS Journal of Photogrammetry and Remote Sensing, 144: 325–340. doi: 10.1016/j.isprsjprs.2018.07.017
[42] Viana C M, Girão I, Rocha J, 2019. Long-term satellite image time-series for land use/land cover change detection using refined open source data in a rural region. Remote Sensing, 11(9): 1104. doi: 10.3390/rs11091104
[43] Wagle N, Acharya T D, Kolluru V et al., 2020. Multi-temporal land cover change mapping using google earth engine and ensemble learning methods. Applied Sciences, 10(22): 8083. doi: 10.3390/app10228083
[44] Wan L, Liu H Y, Gong H B et al., 2020. Effects of climate and land use changes on vegetation dynamics in the Yangtze River Delta, China based on abrupt change analysis. Sustainability, 12(5): 1955. doi: 10.3390/su12051955
[45] Wu Q S, 2020. Geemap: a python package for interactive mapping with Google Earth Engine. Journal of Open Source Software, 5(51): 2305. doi: 10.21105/joss.02305
[46] Xu H Z Y, Wei Y C, Liu C et al., 2019. A scheme for the long-term monitoring of impervious-relevant land disturbances using high fre-quency Landsat archives and the Google Earth Engine. Remote Sensing, 11(16): 1891. doi: 10.3390/rs11161891
[47] Xu J P, Xiao W, He T T et al., 2021. Extraction of built-up area using multi-sensor data: a case study based on Google earth engine in Zhejiang Province, China. International Journal of Remote Sensing, 42(2): 389–404. doi: 10.1080/01431161.2020.1809027
[48] Xu X B, Yang G S, Tan Y et al., 2018. Ecosystem services trade-offs and determinants in China's Yangtze River Economic Belt from 2000 to 2015. Science of the Total Environment, 634: 1601–1614. doi: 10.1016/j.scitotenv.2018.04.046
[49] Yu M, Yang Y J, Chen F et al., 2019. Response of agricultural multifunctionality to farmland loss under rapidly urbanizing processes in Yangtze River Delta, China. Science of the Total Environment, 666: 1–11. doi: 10.1016/j.scitotenv.2019.02.226
[50] Zeng L L, Wardlow B D, Xiang D X et al., 2020. A review of vegetation phenological metrics extraction using time-series, multispectral satellite data. Remote Sensing of Environment, 237: 111511. doi: 10.1016/j.rse.2019.111511
[51] Zhai Y G, Qu Z Y, Hao L, 2018. Land cover classification using integrated spectral, temporal, and spatial features derived from remotely sensed images. Remote Sensing, 10(3): 383. doi: 10.3390/rs10030383
[52] Zhang C, Wei S Q, Ji S P et al., 2019. Detecting large-scale urban land cover changes from very high resolution remote sensing images using CNN-based classification. ISPRS International Journal of Geo-Information, 8(4): 189. doi: 10.3390/ijgi8040189
[53] Zhang D J, Pan Y Z, Zhang J S et al., 2020. A generalized approach based on convolutional neural networks for large area cropland mapping at very high resolution. Remote Sensing of Environment, 247: 111912. doi: 10.1016/j.rse.2020.111912
[54] Zhao F, Huang C Q, Zhu Z L, 2015. Use of vegetation change tracker and support vector machine to map disturbance types in Greater Yellowstone ecosystems in a 1984−2010 Landsat time series. IEEE Geoscience and Remote Sensing Letters, 12(8): 1650–1654. doi: 10.1109/LGRS.2015.2418159
[55] Zhao J, Zhong Y F, Hu X et al., 2020. A robust spectral-spatial approach to identifying heterogeneous crops using remote sensing imagery with high spectral and spatial resolutions. Remote Sensing of Environment, 239: 111605. doi: 10.1016/j.rse.2019.111605
Figures(8) / Tables(1)
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Le 'an QU1, 2, 3, ,
Manchun LI1, 3, , ,
Zhenjie CHEN1, 3,
1. School of Geography and Ocean Science, Nanjing University, Nanjing 210023, China
2. School of Geography and Tourism, Anhui Normal University, Anhui Province, Wuhu 241002, China
3. Jiangsu Provincial Key Laboratory of Geographic Information Science and Technology, Nanjing University, Nanjing 210023, China
Land Use/Land Cover (LULC) /
self-adaptive Random Forest (RF) /
Google Earth Engine (GEE) /
Abstract: Annual Land Use/Land Cover (LULC) change information at medium spatial resolution (i.e., at 30 m) is used in applications ranging from land management to achieving sustainable development goals related to food security. However, obtaining annual LULC information over large areas and long periods is challenging due to limitations on computational capabilities, training data, and workflow design. Using the Google Earth Engine (GEE), which provides a catalog of multi-source data and a cloud-based environment, we developed a novel methodology to generate a high accuracy 30-m LULC cover map collection of the Yangtze River Delta by integrating free and public LULC products with Landsat imagery. Our major contribution is a hybrid approach that includes three major components: 1) a high-quality training dataset derived from multi-source LULC products, filtered by k-means clustering analysis; 2) a yearly 39-band stack feature space, utilizing all available Landsat data and DEM data; and 3) a self-adaptive Random Forest (RF) method, introduced for LULC classification. Experimental results show that our proposed workflow achieves an average classification accuracy of 86.33% in the entire Delta. The results demonstrate the great potential of integrating multi-source LULC products for producing LULC maps of increased reliability. In addition, as the proposed workflow is based on open source data and the GEE cloud platform, it can be used anywhere by anyone in the world.
Land Use/Land Cover (LULC) data are important in applications such as land management, agricultural monitoring, ecological service research, and climate change assessment (Zhao et al., 2015; Mao et al., 2018; Teluguntla et al., 2018; Gong et al., 2019). With advances in remote sensing technology toward providing satellite images, the corresponding datasets have been effectively applied to classify LULC types at different spatial scales from global to local (Mohajane et al., 2018; Zhang et al., 2019; Li et al., 2020). Among the different remote sensing datasets, Landsat data provides free global coverage of satellite images with a long history in comparison with other open-access remotely sensed data (e.g., MODIS, Sentinel-2), which is advantageous for LULC classification tasks (Li and Gong, 2016; Huang et al., 2017; Liu et al., 2020; Mao et al., 2021).
The conventional large-area long-term LULC classification method uses remote sensing-derived time series with a supervised non-parametric classifier (Zhai et al., 2018; Bullock et al., 2020). In areas with strong landscape heterogeneity, there are limitations in employing this kind of supervised classifier (Ghosh et al., 2014; Müller et al., 2015). Firstly, with strong heterogeneity of landscape, the phenomenon of 'different objects with the same spectrum, different spectrums within the same object' often occurs (Long, 2014). Further research is needed to develop high-precision classifiers (Chen et al., 2018). Secondly, there are challenges in obtaining a large amount of long-term training data (Viana et al., 2019). Finally, with a large research area, a large amount of remote sensing data need to be processed, requiring strong computational and storage capabilities (Daldegan et al., 2019; Tamiminia et al., 2020).
The random forest (RF) classifier is a commonly used supervised classifier for LULC mapping (Chakraborty et al., 2016). In general, the use of remote sensing time series data combined with the RF classifier can obtain a LULC map with high classification accuracy in an area with strong landscape heterogeneity (Zeng et al., 2020). This is because the phenological information provided by time series data can partially solve the problem of 'different objects with the same spectrum, different spectrums within the same object' (Lopes et al., 2017). However, challenges remain regarding large-area LULC classification based on time series (Ji et al., 2020). In particular, only by using a large amount of high-precision training data can higher classification accuracy be obtained (Ghorbanian et al., 2020).
Over the past decade, it has been demonstrated that it is efficient to use existing LULC maps as a source of training data (Viana et al., 2019). This is advantageous because it: 1) allows classification in an automated manner without the need for interactive manual training data, 2) provides a potentially large and geographically distributed training dataset, and 3) enables satellite data to be mapped with existing LULC maps (Pandey et al., 2021). However, existing LULC maps may have some misclassified results (Mack et al., 2017). Therefore, the use of existing LULC maps as a source should be carefully considered to ensure that the generated training data have a good enough level of accuracy (Li et al., 2017).
Computational power, data storage, management, and processing times have also traditionally been restrictions for using remote sensing time series over large areas (Anchang et al., 2020). Google Earth Engine (GEE) not only provides powerful computing ability for free but can also directly access a variety of open source data (Hird et al., 2017; Capolupo et al., 2020). The workflow designed on GEE not only solves the limitation of computing power and data source but also helps other researchers to conduct similar research (Wu, 2020; Xu et al., 2021).
The aims of the present study are: 1) to classify Landsat time series data supported by GEE to obtain high-precision annual LULC maps of the Yangtze River Delta (YRD) from 1992 to 2015 and 2) to analyze the characteristics of LULC changes in the YRD region. Specifically, we address the following research questions: 1) how to get a high-precision sample dataset generated from multi-source LULC products? 2) How to construct a feature space, so that time series classification can be carried out in a cloudy and rainy area (such as YRD)? 3) How to build a local adaptive classifier to improve the classification accuracy of LULC in an area with strong landscape heterogeneity?
2. Study Area and Datasets
The Yangtze River Delta (YRD) is situated in the eastern China and covers four provinces: Anhui Province, Jiangsu Province, Zhejiang Province, and Shanghai Municipality (Fig. 1), with a surface area of 348 000 km2 (Chen et al., 2019). The topography of the YRD is dominated by plains, hillsides, and mountains (Feng et al., 2018). The average elevation is 140.17 m (Shen et al., 2008). This region is under a monsoon climate; the annual average temperature for the growing season (March to October) is 18℃ to 22℃ and the average annual precipitation for the growing season is 800 to 1400 mm (Wan et al., 2020).
Figure 1. Geographical location of Yangtze River Delta (YRD) in the eastern China
The YRD is one of China's most economically developed regions and plays an important role in the social and economic development of the country (Yu et al., 2019). In 2015, the YRD accounted for 16.06% of China's population, and its GDP accounted for 23.50% of the country's GDP (Zhang et al., 2020). According to the Statistical Yearbook from China National Knowledge Infrastructure (http://www.cnki.net/), the populations and GDPs of the YRD region increased significantly from 1992 to 2015. This growth in the population and economic development has been accompanied by rapid urbanization and great changes in land use: the loss of farmland and expansion of urban areas, in the region (Xu et al., 2018).
2.2. Landsat images and multi-source auxiliary data
To construct a sample set for LULC classification, we used several datasets of well-recognized LULC products, including 1992–2015 year by year European Space Agency Climate Change Initiative (ESA-CCI 300) LULC products, 2001–2015 annual MCD12Q1 (MODIS LULC cover) products, 2000 and 2010 LULC data developed by the China National Basic Geographic Information Center (GlobeLand30), and the 2015 Finer Resolution Observation and Monitoring of Global Land Cover (From-GLC), developed by Tsinghua University, China (Table 1). The GEE platform can access MCD12Q1 products directly, and these do not require uploading into the application (Gorelick et al., 2017). The other three sets of LULC products need to be uploaded to the GEE platform. All work was performed at a 30 m resolution. The ESA-CCI300 and MCD12Q1 products were resampled on the GEE platform.
Data Year Temporal
resolution Spatial
resolution / m Data sources
Landsat 5* 1992–2012 16 d 30 http://landsat.usgs.gov
SRTM3* 2000 – 30 http://www2.jpl.nasa.gov/srtm
ESA-CCI300 1992–2015 1 yr 300 https://www.esa-landcover-cci.org
MCD12Q1.006* 2001–2015 1 yr 500 https://lpdaac.usgs.gov/dataset_discovery/modis/modis_products_table/mcd12q1
GlobeLand30 2000/2010 – 30 http://www.globeland30.com
From-GLC 2015 – 30 http://data.ess.tsinghua.edu.cn
Boundary 2015 – – http://www.resdc.cn
Notes: * represents data available online (https://earthengine.google.com). ESA-CCI300 (European Space Agency Climate Change Initiative), MCD12Q1.006, GlobeLand30, and From-GLC (Finer Resolution Observation and Monitoring of Global Land Cover) are LULC (Land Use/ Land Cover) products
Table 1. Datasets used in this research
Landsat 5/7/8 Surface Reflectance (SR) data from 1992 to 2015 are the main remote sensing images for classification and can be accessed directly in the GEE (Xu et al., 2019). Previous research has shown that the Normalized Difference Vegetation Index (NDVI) is sensitive to vegetation characteristics, the Normalized Difference Water Index (NDWI) can identify bodies of water, and the Normalized Difference Built-up Index (NDBI) can distinguish built-up areas effectively (Simonetti et al., 2015; Bailly et al., 2017; Wagle et al., 2020). In the study, the spectral indices (NDVI, NDWI, and NDBI) calculated by the Landsat SR products in the YRD are also used. The spectral indices were introduced for the calculation of the spectral-temporal metric.
Topographic features can affect regional climatic conditions and vegetation growth, so these are often used as auxiliary data for LULC classification (Adepoju and Adelabu, 2020). Topographic features generated from DEM (Digital Elevation Model) include elevation (affecting temperature and precipitation), slope, and aspect (affecting the results of solar radiation and vegetation growth). To describe terrain features, we use the Shuttle Radar Topography Mission (SRTM) data to generate topographic features (Hurni et al., 2019). The GEE platform has built-in SRTM data, which can be used directly.
3.1. Processing of time-series satellite imagery
We extracted the SR Tier 1 datasets of the Landsat 5/7 (B1–B5 and B7 bands) and Landsat 8 (B2–B7 bands) in the growing season during 1992–2015. The first pre-processing step was to build cloud-free Landsat tile mosaics for each year. For this, we used the CFMASK algorithm provided by the GEE and the pixel quality assessment (pixel_qa) information available in the Landsat collection (Li et al., 2019). The cloud-masked Landsat scenes were combined to produce NDVI, NDWI, and NDBI for the annual growth period of each year. This procedure required optimal spectral contrast and separability amongst the LULC classes.
The annual Landsat mosaics were generated with statistical reducers including median, standard deviation, minimum, and maximum. After building the annual mosaics, the next step was to build the feature space for the self-adaptive RF classifier. For this process, we used the compositional, spectral, and temporal information extracted from the annual image mosaics. The NDVI values of all cloud-free pixels from each year were divided into quartiles; the median values of the higher quartile were considered as the wet-season image. Finally, a total of 39 bands consisting of 4 × 9 temporal feature bands and 3 topographic feature bands (elevation, slope, and aspect) were available for LULC classification.
3.2. Training point selection
Supervised classification usually requires a certain number of training samples and verification samples (Zhao et al., 2020). Typically, traditional research uses manual visual interpretation to obtain sample points (Huang et al., 2020). For a study with a large area and long study period, such a method presents considerable practical difficulties (Ghorbanian et al., 2020). This study proposes a new method to achieve highly credible sample points. The specific steps are as follows:
(1) Based on From-GLC (2015), all other LULC products were reclassified, and the LULC cover classes were divided into eight classes including: cropland, forest, shrubland, grassland, wetland, water bodies, built-up land, and unused land.
(2) ESA-CCI (1992–2015), MCD12Q1 (2001–2015), GlobeLand30 (2000, 2010), and the From-GLC (2015) data were overlaid, and pixels with completely consistent LULC types that had not changed from 1992 to 2015 were selected. ESA-CCI300 and MCD12Q1 were reduced to 30 m resolutions in the GEE using the Reduce Resolution function. GEE performs nearest neighbor resampling by default.
(3) From the selected pixels, training sample points were randomly selected; the number of points was slightly adaptive according to the area ratios of the different LULC types.
(4) The created samples still had a high probability of spectral-temporal signature confusion among LULC classes. Therefore, we applied the k-means clustering technique to refine the samples. The clustering process is essentially divided into three distinct processes: 1) classification-trimmed likelihood calculation, 2) cluster computation, and 3) mean discriminant factor value calculation.
Overall, the samples were distributed uniformly and randomly throughout the study area; however, for some LULC types with relatively small and patchy areas (such as unused lands) a relatively dense distribution occurred. Therefore, we exported the refined sample point data to ArcGIS and carried out reselection at 1500 m resolution to reduce the correlation between samples. Through this process, we obtained 20 859 reference sample points. Finally, we randomly selected 10 430 points from the sample library to be used for classification training. To ensure the authenticity and reliability of the verification samples, we selected them through visual inspection. In the end, we obtained 2857 reliable sample points in the study area to verify the LULC classification results.
3.3. Classification approach
The classification process considered a spatial and temporal stratification, individually training and classifying all 9368 Landsat tiles that cover the YRD. The LULC was divided into eight classes which were exactly the same as used for sample points (details see section 3.2). Considering that LULC classes are highly susceptible to climatic conditions and topography throughout the YRD territory, this approach allowed the classification models to better identify all LULC classes. To minimize the impact of spatial stratification, part of the training set was shared among different classification models. Thereby, the 3 × 3 scenes sample points were used to train the self-adaptive RF algorithm (Fig. 2). Equally important, the temporal stratification was also intended to minimize the impact of spectral and radiometric differences among the Landsat sensors in the classification results, since, for each year, a classifier was trained considering only images obtained by a single satellite.
Figure 2. Flowchart of self-adaptive RF (random forest) LULC (Land Use/Land Cover) classification
The self-adaptive RF models are constructed by the smileRandomForest function in the GEE (Kakooei and Baleghi, 2020). In order to use the smileRandomForest function, two parameters need to be set: the number of decision trees to create per class (numberoftrees) and the minimum size of a terminal (minleaf) (Gumma et al., 2020). Finally, through repeated comparative experiments, we set numberoftrees to 100 and minleaf to 10 in all models.
3.4. Post-processing and accuracy assessment
To better homogenize these classification results, we applied a spatial filter and a temporal filter, capable of minimizing abrupt and sometimes unrealistic variations, simultaneously considering these two dimensions. First, we used a spatial filter in the edge area of each scene. There are some pixels with inconsistent classification results in these edge areas. These pixels are merged in accordance with the majority agreement rule by the mode function in GEE (Gorelick et al., 2017). All of the classified scenes were merged into a LULC map collection year by year. Next, we applied a temporal filter to the LULC map collection. We detected whether the LULC classes in year N − 1 and year N + 1 are consistent pixel by pixel. If so, the LULC classes in year N must be consistent with those LULC classes. Except for the first and last years, we iterated the above process year by year and obtained final LULC classification results for the YRD.
We used several measures to assess and statistically compare the accuracy of our classifications. First, we calculated the classification error matrices for each year. From these matrices, we then quantified overall accuracy (OA). Second, we calculated producer's accuracy (PA), user's accuracy (UA), and F1-score, which is the harmonic mean between user's accuracy and producer's accuracy. PA, UA, and F1-score can be calculated for each class i as follows:
$$ {PA}_{i}={\sum }_{j=1}^{r}\frac{{n}_{ii}}{{n}_{ij}} $$
$$ {UA}_{i}={\sum }_{j=1}^{r}\frac{{n}_{ii}}{{n}_{ji}} $$
$$ {\left({F}_{1}\right)}_{i}=\frac{2\times {PA}_{i}\times {UA}_{i}}{{PA}_{i}+{UA}_{i}} $$
where r is the number of classes and nij is the element of the confusion matrix in row j and column i, that is, the count of elements of class j classified as class i. PAi, UAi, and (F1)i stand for PA, UA, and F1-score for class i, respectively.
The F1-score is particularly useful for class-level accuracy assessment, as it gives equal importance to both PA and UA by combining PA and UA into a single measure that can be compared across confusion matrices. Finally, we estimated the unbiased area (using the sample weight obtained with our reference sample dataset) following a standard good practice protocol.
4.1. Accuracy assessment of LULC results with 30-m resolution
To evaluate the quality of the 30-m LULC results for the YRD using multi-source LULC products as training samples, an accuracy assessment of the classification was conducted. A confusion matrix, which is the primary tool used in remote sensing for accuracy assessment, was used to evaluate the accuracy. A total of 2857 verification samples were collected for all LULC classes. The evaluation was carried out based on the error matrix produced and analyzed the accuracy of the 30 m LULC classification results.
Fig. 3 reports the classification accuracies that quantify the level of agreement between the classification and the sample data. The overall accuracies of the classification results are greater than 84.50% for every year, and the average overall accuracy is 86.33%. This shows that the overall classification level of agreement is high. The F1-score annual accuracies were close to 90% for the following main classes: cropland, forest, and water bodies. The F1-scores of built-up areas ranged between 75% and 85%, mainly due to the misclassification of many small villages in the northern part of the study area. Because shrubland, grassland, wetland, and unused land account for a very small proportion in the study area, we analyzed these LULC classes as a whole. The F1-scores were < 80% for all these classes for each year except for 2014 and 2015. The poor classification accuracy of these classes can be explained on the one hand because their proportions are low and on the other hand because the sample points of these LULC classes are small. In general, these results indicate quite reasonable classification accuracies. The results show that using multi-source LULC products to generate training data is a viable and effective option.
Figure 3. Classification accuracy for the LULC (Land Use/Land Cover) in the Yangtze River Delta, China from 1992 to 2015
4.2. Annual LULC maps and main LULC change of the Yangtze River Delta from 1992 to 2015
By using the Landsat time series data with the self-adaptive RF method, we developed annual LULC datasets for the YRD from 1992 to 2015 (Fig. 4). The LULC classification results showed extensive change across the study area. Specifically, cropland was the most extensive LULC type but continuously decreased from 1992 to 2015. In contrast, built-up land continuously and significantly expanded due to rapid urbanization. We observed that the expanded built-up areas were mainly concentrated near the peri-urban region, possibly due to the topographic conditions. The expansion of built-up land was mainly centered on cropland around the city, which led to the continuous decrease in cropland area. Fig. 5 also indicates that areas of forest and water bodies remained largely unchanged during the study period.
Figure 4. Annual LULC (Land Use/Land Cover) maps for the YRD from 1992 to 2015. Maps for four selected years (1992, 2000, 2008, and 2015) are enlarged
Figure 5. Area percentage of different LULC (Land Use/Land Cover) classes for the YRD from 1992 to 2015
Fig. 5 illustrates how the primary LULC types of the YRD changed from 1992 to 2015. It can be seen that the greatest change in the YRD occurred in the case of cropland, whose proportion decreased from 44.6% in 1992 to 42.5% in 2015. This was followed by forest, whose proportion decreased from 37.5% to 36.2%. In contrast, built-up land's proportion increased from 0.8% to 2.1%. Bodies of water increased by 0.2%. The smallest changes were in grassland, whose proportion decreased by just 0.1%. Shrubland, wetland, and bare land occupy relatively small proportions of the study area. Compared with the primary LULC types (cropland, forest, water bodies, and built-up land) they did not change noticeably. In addition, due to their small proportions in the study area, they were more affected by classification uncertainty. Therefore, changes in these LULC types were excluded from the analysis in this paper.
5.1. Comparing with other LULC products
It can be seen from Figs. 6–8 that our results are similar to other LULC products in terms of geographic distribution, and the overall layout of cropland, forest land, and water bodies is similar. Overall, our results are more accurate in identifying small ground objects than other LULC products.
Figure 6. Comparison with other common LULC (Land Use/Land Cover) products for the year 2015. The first column shows Landsat-8 OLI images with 30 m resolution, the second is our result with 30 m resolution, the third is ESA-CCI 300 with 300 m resolution, and the fourth is MCD12Q1 with 500 m resolution. Row (a) is a selected typical urban area in plain, row (b) is a selected typical transitional area from mountain to plain, and row (c) is a selected typical mix area
Figure 7. Comparison of our results with GlobeLand30 for the year 2010 in the YRD, China
Figure 8. Comparison of our results with From-GLC for the year 2015 in the YRD, China
Compared with ESA-CCI300 products, the main difference lies in the built-up lands in the plain areas and the forest in the mountain areas (Fig. 6). In plain areas, more small built-up land areas were identified in our results, which may be related to the higher spatial resolution of our results compared to those of the other LULC products. In mountainous areas, our results showed more forest area compared to results from the other LULC products, with ESA-CCI300 classifying more as croplands. Compared with MCD12Q1 products, the main difference is that MCD12Q1 misclassified many shrublands while our method correctly classified them as croplands. In plain areas, large tracts of land are misclassified as built-up areas in MCD12Q1, whereas our results are more refined. We detected these inconsistent areas visually through Landsat-8 OLI images and found that our results are more accurate among the results from all the LULC products.
In Fig. 7, we can see that our results are generally consistent with GlobeLand30 in mountainous areas, and the distribution of forest and water bodies in plain areas are also similar. The main difference is in the distribution of built-up lands and croplands in plain areas. GlobeLand30 showed many small built-up lands, and our results classify these areas as croplands. This is mainly because, in addition to Landsat data, GlobeLand30 also uses other multispectral images with higher spatial resolution and processes them manually, so that it can classify more built-up lands in the mixed pixels of rural areas (Fig. 7 left). Due to the introduction of other multispectral data, however, whatever the percentage of built-up land is in the mixed pixel, it will still be classified as built-up land (Fig. 7 right). In our results, due to the influence of MODIS and ESA data on the training points generated, the number of built-up land samples in rural areas was small. As a result, only contiguous built-up lands can be accurately identified in the classification results, while small built-up lands in rural areas showed a poor level of accuracy.
In Fig. 8, it can be seen that From-GLC 2015 misclassifies part of the cropland as forest in the plain area. Our results do not show such a misclassification. The main reason for this type of misclassification is that there are some spectral differences between different tiles of Landsat data. When we employed the self-adaptive RF classification model, the classification features are mainly phenological features, and the training data in 3 × 3 adjacent tiles are used for model training; this effectively reduces the impact of spectral differences. The self-adaptive RF model can be effectively applied because we have a sufficient amount of training data.
5.2. Advantages and disadvantages of our method
LULC classification over large areas and a long study period is challenging due to the large volume of data pre-processing required and the cost and difficulty of collecting representative training data that enable classification models to be both globally consistent and locally reliable (Millard and Richardson, 2015; Mahdianpari et al., 2020; Liu et al., 2021). With ready-to-use data, such as those from Landsat products, and the employment of non-parametric classifiers, the major challenge is training data collection (Gumma et al., 2020).
The detailed results illustrated in Fig.s 6–8 are representative of the results across the study area. In general, our classification results are similar to other LULC products, which indicated that our results are reasonable. The high overall accuracies and generally high F1-score in the main classes (that quantify the level of agreement between the classification and the training pool data) underscore the utility of the self-adaptive RF classification approach. However, the effectiveness of the self-adaptive RF classification method depends upon the availability of sufficient local training data. In this study, this was not an issue as the training pool data derived from multi-sources LULC products were geographically well distributed.
In this study, a novel method was used to classify Landsat data using high quality training data derived from multi-source LULC products. A training data pool was extracted from the multi-source LULC products by judicious quality and k-means clustering filtering. In addition, the training data selection was undertaken in a geographically systematic manner while ensuring that the selected class ratios were the same as the YRD ground object's ratios. This is advantageous as it: 1) enables the classification to be undertaken in an automated manner without the need for manual training data collection, 2) provides a large geographically distributed training data set, and 3) results in the generation of a 30 m Landsat LULC product with the same classification legend as the multi-source LULC products.
Although our method of generating training data has many advantages, it also has certain shortcomings. Firstly, due to the limitation of multi-source LULC products, the training data generated a large number of accurate sample points for contiguous LULC types but fewer for the small LULC types (such as built-up lands in rural areas). Secondly, different LULC products have different classification systems, resulting in fewer LULC classes in the training data generated in our study. Finally, for obtaining multi-source LULC products of high time consistency and data quality, we do not use 30 m LULC products after 2015 because there is no free 30 m LULC product at a global scale after 2015 when we finished the experiment. Therefore, the period of our research is selected from 1992 to 2015. In future work, we will consider whether the Landsat data has changed and generate sample data with more LULC classes and a longer period.
Based on the four types of LULC products (ESA-CCI300, MOD12Q1, Globland30, and From-GLC), a data training pool was extracted by judicious quality and k-means clustering filtering. A self-adaptive RF classification was employed to classify 9368 Landsat image tiles. Using these methods, we constructed a map set of LULC in the YRD region from 1992 to 2015 and analyzed the spatiotemporal changes in the main LULC types in the YRD during this period. The primary conclusions of this study are as follows:
(1) The results indicated that the self-adaptive RF classifier can be used successfully to classify Landsat time series into LULC data, with an average overall classification accuracy of 86.33%. LULC classification results with 30 resolution of YRD appeared geographically reasonable and were similar in comparison to other LULC products. Therefore, the approach is suitable for analyzing changes in LULC.
(2) The training sample generation approach from multi-source LULC products described in this study can provide a large number of sample points in an acceptable quality level. This is because k-means clustering filtering can effectively eliminate non-homogeneous sample points, so that the generated sample points have higher accuracy.
(3) The construction of feature space is also an important advantage of this study. The feature space can eliminate the negative impacts of clouds and rain to the time series data. Moreover, it can reduce the difference between different Landsat titles. Therefore, the construction of feature space is essential for time series classification in a large cloudy and rainy area (such as the YRD).
In future work, we will continue to study the use of multi-source LULC products to generate sample points. We will consider applying some spectral unmixed models to the sample point generation process, to obtain more sample points of non-dominant LULC types (e.g., built-up land in rural areas). In the meanwhile, we consider extending our research workflow from level 1 to level 2, so that we can determine whether our workflow performs the same way with more LULC types. | CommonCrawl |
\begin{document}
\preprint{}
\title{Zero energy resonance and the logarithmically slow decay \\ of unstable multilevel systems}
\author{Manabu Miyamoto}
\email{[email protected] } \affiliation{ Department of Physics, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
}
\date{\today}
\begin{abstract} The long time behavior of the reduced time evolution operator for unstable multilevel systems is studied based on the N-level Friedrichs model in the presence of a zero energy resonance. The latter means the divergence of the resolvent at zero energy. Resorting to the technique developed by Jensen and Kato [Duke Math. J. {\bf 46}, 583 (1979)], the zero energy resonance of this model is characterized by the zero energy eigenstate that does not belong to the Hilbert space. It is then shown that for some kinds of the rational form factors the logarithmically slow decay proportional to $(\log t)^{-1}$ of the reduced time evolution operator can be realized. \end{abstract}
\maketitle
\section{Introduction} \label{sec:1}
The exponential decay of unstable systems has been a well-known law since the early days of quantum theory. The quantum description of those systems, however, allows deviation from exponential decay both at shorter and longer times \cite{Khalfin(1957)} than those times over which the exponential decay law dominates. \cite{Fonda(1978),Nakazato(1996)} The short time deviation was actually found in a quantum tunneling experiment, \cite{Wilkinson(1997)} while the long time deviation seems still not to have been detected in any quantum system. \cite{Greenland(1988)} The main cause that hinders the detection is considered as the smallness of the deviation at such long times.
\cite{Delgado(2006)}
In a recent study, a method enhancing the long time deviation was proposed. \cite{Jittoh(2005)}
The decay of the unstable systems is theoretically modeled in the time evolution of the survival probability of unstable initial state. The survival probability is just the probability of finding the initial state in the state at a later time $t$. Since it is rewritten in a Fourier integral of the spectral function, its behavior at long times is determined by that of the spectral function near the threshold of the energy continuum.
\cite{Fonda(1978),Nakazato(1996)} The essential aspect of the method is then distorting the spectral function from the Breit-Wigner form and dislocating its peak toward the threshold energy. Mathematically, this causes a divergence of the spectral function, i.e., the resolvent at the threshold. Then, it is expected that the exponential decay period disappears and the survival probability at long times is increased. A similar idea was also considered in a related context. \cite{Rzazewski(1982),Greenland(1988)} In addition, in the analysis of the Friedrichs model \cite{Friedrichs(1948),Exner(1985)} that is often used for the study on the decays of the unstable systems, the survival probability at long times sometimes exhibits a power decay law slower than that in cases of no divergence. \cite{Kofman(1994),Lewenstein(2000),Nakazato(2003)}
These facts remind the author of the zero energy resonance proposed by Jensen and Kato. \cite{Jensen(1979)}
According to them, such zero energy singularities are classified by the zero energy eigenstates of the total Hamiltonian that either belong to or do not belong to the Hilbert space.
The cases where such eigenstates exist are called the exceptional cases; otherwise they are referred to as the regular case.
The result in Ref. \makebox(9,1){\Large \cite{Jensen(1979)}}\hspace{1mm} is concerned with the three-dimensional system of the one particle in short-range potentials, and they proved that the time evolution operator asymptotically decreases as $O(t^{-1/2})$ for the exceptional cases, that is slower than $O(t^{-3/2})$ for the regular case. However, to the author's knowledge, the zero energy resonance for the Friedrichs model seems not to have been examined in the previous studies including Refs. \makebox(10,1){\Large \cite{Kofman(1994),Lewenstein(2000),Nakazato(2003),Jittoh(2005)} }\hspace{10mm}, in spite of the wide applicability of the model to the various physical systems. \cite{Facchi(1998),Antoniou(2001),Kofman(1994),Rzazewski(1982),Nakazato(2003)}
In the present paper, we examine the zero energy singularities of the resolvent at the threshold energy for the Friedrichs model from the viewpoint of the zero energy resonance, \cite{Jensen(1979)} and clarify how the asymptotic behavior of the survival probability at long times is affected. The Friedrichs model\cite{Friedrichs(1948),Exner(1985)} describes the system of the finite discrete levels coupled with the continuous spectrum, in which the former can be interpreted as the unstable excited levels of atoms and the latter as the environmental electromagnetic fields. \cite{Kofman(1994),Facchi(1998),Antoniou(2001)} We emphasize that the model is not restricted to the single level case \cite{Friedrichs(1948),Exner(1985),Kofman(1994),Lewenstein(2000),Nakazato(2003),Rzazewski(1982), Jittoh(2005),Facchi(1998),Antoniou(2001)} but, rather, the $N$-level case, \cite{Exner(1985),Davies(1974),Antoniou(2003),Antoniou(2004),Miyamoto(2004),Miyamoto(2005)} In addition, we assume that the square modulus of the form factors
vanishes at zero energy with an integer power, \cite{Facchi(1998),Antoniou(2001),Seke(1994)} however it is treated without restriction to a specific form to some extent.
Furthermore, since we only consider the initial state spanned by the discrete states, it is sufficient for us to see the reduced resolvent $\tilde{R}(z)$ that is just the restriction of the resolvent to the subspace spanned by the discrete states. Then, the Fourier integral of $\tilde{R}(z)$ that we call the reduced time evolution operator $\tilde{U}(t)$ enables us to calculate the survival probability. In fact it is expressed by the square modulus of the expectation value of $\tilde{U}(t)$ in a given initial state. We first study the zero energy eigenstates of the model which either belong to or do not belong to the Hilbert space.
It is then possible to estimate correctly the asymptotic behavior of $\tilde{R}(z)$ at small energies both in the regular case and the exceptional cases. The latter cases are examined in detail only for the first kind, where only the zero energy eigenstate not belonging to the Hilbert space exists. On the basis of this analysis, we can derive the long-time asymptotic formula for $\tilde{U}(t)$ in those cases. In particular, the logarithmic decay proportional to $(\log t)^{-1}$ of $\tilde{U}(t)$ is shown to occur in the exceptional case of the first kind for our form factors, which is extremely slower than the power decays in the regular case and in the exceptional case for another type of form factor. \cite{Kofman(1994),Lewenstein(2000),Nakazato(2003)} These results are shown in Theorems \ref{thm:rffimLong} and \ref{thm:rffLong1st}.
The organization of the paper is as follows. We first explain in Sec. \ref{sec:3} the $N$-level Friedrichs model with an appropriate Hilbert space, and then in Sec. \ref{sec:4} we introduce the reduced resolvent $\tilde{R}(z)$.
Section \ref{sec:5} is devoted to the identification of zero energy eigenstates in this model.
It is then possible to obtain the asymptotic expansion of $\tilde{R}(z)$ at small energies in Sec. \ref{sec:5.5}, where we examine the regular and the exceptional case of the first kind. By making sure of the relation between $\tilde{R}(z)$ and $\tilde{U}(t)$ in Sec. \ref{sec:6}, the asymptotic formula for $\tilde{U}(t)$ in the regular and the exceptional case of the first kind are derived in Sec. \ref{sec:7} . Concluding remarks are given in Sec. \ref{sec:8}.
\section{Hilbert space and the $N$-level Friedrichs model} \label{sec:3}
We shall use bracket notation; however it can be understood in a standard treatment based on functional analysis as in Refs. \makebox(6,1){\Large \cite{Exner(1985),Davies(1974)}}\hspace{7.5mm}. The Hilbert space describing the unstable multilevel systems is here defined by \begin{equation} {\cal H}:=\mathbb{C}^N \oplus L^2 ((0,\infty )). \label{eqn:5.10} \end{equation} A vector $\ket{c} \in \mathbb{C}^N$ is expressed by $\ket{c}=\sum_{n=1}^{N} c_n \ket{n}$, where $\ket{n}$'s are the orthonormal basis of $\mathbb{C}^N$, so that $\braket{n}{n'}=\delta_{nn'}$, where $\delta_{nn'}$ is Kronecker's delta. $L^2 ((0,\infty ))$ is the Hilbert space of the square-integrable complex function $\ket{f}$ of the variable $\omega$ defined on $(0,\infty )$, i.e., \begin{equation}
\ket{f} \in L^2 ((0,\infty )) \Leftrightarrow \int_0^{\infty} |f(\omega)|^2 d\omega <\infty. \label{eqn:5.10b} \end{equation} In a standard notation using the (generalized) eigenstate $\ket{\omega}$ of the multiplication operator by $\omega$, $\ket{f}$ is nothing more than \begin{equation} \ket{f}=\int_{0}^{\infty} f(\omega ) \ket{\omega} d\omega, \label{eqn:5.15} \end{equation} where $\braket{\omega}{\omega'}=\delta (\omega -\omega')$ and $\delta (\omega -\omega')$ is Dirac's delta.
Then, an arbitrary vector $\ket{\Psi} \in {\cal H}$ composed of $\ket{c} \in \mathbb{C}^N$ and $\ket{f} \in L^2 ((0,\infty ))$ is denoted by \begin{equation} \ket{\Psi}:=
| c \rangle +\ket{f} ,
\label{eqn:5.20} \end{equation} and the inner product between any two vectors $\ket{\Psi}$ and $\ket{\Phi} \in {\cal H}$ is defined by \cite{innerproduct} \begin{equation} \braket{\Phi}{\Psi} :=\braket{d}{c}+ \braket{g}{f} =\sum_{n=1}^{N} d_n^* c_n + \int_0^{\infty} g^* (\omega ) f(\omega) d\omega , \label{eqn:5.30} \end{equation} where ($^*$) denotes the complex conjugate and $\ket{\Phi}=\ket{d}+\ket{g}$ with $\ket{d} \in \mathbb{C}^N$ and $\ket{g} \in L^2 ((0,\infty ))$. In particular, the associated norm of $\ket{\Psi}$ is
$\| \Psi \| := \sqrt{\braket{\Psi}{\Psi}}$, which is ensured to be finite for all $\ket{\Psi} \in \cal{H}$.
Let us now introduce the $N$-level Friedrichs model for a description of the decay of the unstable multilevel systems. The Hamiltonian $H$ of this model is defined by \begin{equation} H:=H_0 + \lambda V, \label{eqn:5.60} \end{equation} where $H_0$ is the free part and $V$ the interaction part of $H$, respectively, and $\lambda \in \mathbb{R}$ is the coupling constant. $H_0$ is defined by \begin{equation} H_0 :=\sum_{n=1}^N \omega_n \ketbra{n}{n}+\int_{0}^{\infty} \omega \ketbra{\omega}{\omega}d\omega, \label{eqn:5.40} \end{equation} where $\omega_n \in \mathbb{R}$ with $\omega_1 \leq \omega_2 \leq \cdots \leq \omega_N$, and its action is prescribed by $H_0 \ket{\Psi}=\sum_{n=1}^N \omega_n c_n \ket{n}+ \omega \ket{f}$ for any $\ket{\Psi}=\ket{c}+\ket{f} \in D(H_0)$. $D(H_0)$ is the domain of $H_0$ defined by $
D(H_0 ):=\left\{ \ket{\Psi} \in {\cal H} ~\left|~
\int_{0}^{\infty} |\omega f(\omega )|^2 d\omega < \infty \right. \right\} $, and then the self-adjointness of $H_0$ is guaranteed. The interaction part $V$ is defined by \begin{equation} V:=\sum_{n=1}^{N} \int_{0}^{\infty} \bigl[ v_n^*(\omega) \ketbra{n}{\omega} +v_n(\omega) \ketbra{\omega}{n} \bigr] d\omega, \label{eqn:5.70} \end{equation} where we assumed that $\ket{v_n} \in L^2 ((0, \infty ))$. \cite{threshold} We call the $L^2$-functions $v_n(\omega)$ the form factors of the system under consideration. The action of $V$ is then given by $V \ket{\Psi}=\sum_{n=1}^N \braket{v_n}{f} \ket{n} + \sum_{n=1}^{N} c_n \ket{v_n}$ for any $\ket{\Psi}\in {\cal H}$. Note that since $D(V)=\cal{H}$ and $V$ is a bounded self-adjoint operator, $H$ is self-adjoint with the domain $D(H)=D(H_0)\cap D(V)=D(H_0)$.
In the whole of the paper, we will restrict ourselves to the special kind of the form factor: Suppose that the product $v_m^* (\omega) v_n (\omega)$ between an arbitrary pair of $v_m^* (\omega)$ and $v_n (\omega)$ is written in a rational function, i.e., it is expressed by \begin{equation} v_m^* (\omega) v_n (\omega) =\frac{\pi_{mn} (\omega) }{ \rho_{mn} (\omega)}, \label{eqn:formfactor1} \end{equation} where $\pi_{mn} (\omega)$ and $\rho_{mn} (\omega)$ are the polynomials of the degree $M_{mn}$ and $N_{nm}$, respectively, and we assume that $\rho_{mn} (\omega)$ has no zeros in $[0, \infty )$. It is also assumed that $M_{mn}+2 \leq N_{mn}$ and $\pi_{mn} (0)=0$. The former condition ensures that $v_m^* (\omega) v_n (\omega)$ is integrable in $[0, \infty )$ and $\lim_{\omega\to\infty} v_m^* (\omega) v_n (\omega)=0$, while the latter condition implies that the rational function $v_m^* (\omega) v_n (\omega)=O(\omega)$ as $\omega\to +0$. The form factors with such properties are often found in actual systems involving the process of the spontaneous emission of photons from the hydrogen atom, \cite{Facchi(1998),Seke(1994)} and quantum dots. \cite{Antoniou(2001)} We do not treat the algebraic form factor that behaves as $O(\omega^{1/2})$ as $\omega\to +0$ instead, associated with the photodetachment of electrons from the negative ion \cite{Rzazewski(1982),Haan(1984),Lewenstein(2000),Nakazato(2003)} and the spontaneous emission from the atoms in the photonic crystals;\cite{Kofman(1994)} however, the discussion developed in the following could be easily extended to such a case.
\section{Reduced resolvent for the $N$-level Friedrichs model} \label{sec:4}
In the following, we introduce the reduced resolvent that is simply the restriction of the resolvent of $H$ to the $N$ dimensional subspace $\mathbb{C}^N \oplus \{ 0 \}$. Since we only consider the initial state belonging to this subspace, this restriction is sufficient for our study. In a technical sense, this treatment corresponds to the appropriate choice of a weighted Sobolev space. \cite{Jensen(1979),Murata(1982)} In the later sections, we do not distinguish the vector in $\mathbb{C}^N$ from that in $\mathbb{C}^N \oplus \{ 0 \}$. After introducing the reduced resolvent, we see the existence of the boundary values of the reduced resolvent on the positive real line. The large-energy behavior of the reduced resolvent is also examined, which is necessary for a rigorous estimation of the long time behavior of the reduced time evolution operator.
\subsection{Reduced resolvent}
The resolvent of $H_0$ and that of $H$ are defined by $R_0(z ) =(H_0 -z )^{-1}$ and $R(z ) =(H -z )^{-1}$, respectively, where we assume that $z \in \mathbb{C}\backslash (\sigma (H_0) \cup \sigma (H))$. $\sigma (H_0)$ (or $\sigma (H)$) is the spectrum of $H_0$ (or $H$), i.e., the set of the singular points of $R_0(z)$ (or R(z)). Then, we have \begin{eqnarray} R(z ) - R_0(z ) &=& -R_0(z ) V R(z ) \label{eqn:2.70} \\ &=& -R_0(z ) V R_0(z ) +R_0(z ) V R_0(z ) V R(z ) . \label{eqn:2.40} \end{eqnarray} From Eq. (\ref{eqn:2.70}), one obtains the equation $R(z)= (1+R_0(z ) V )^{-1} R_0(z )$, which is the starting point of the asymptotic expansion of $R(z)$ for the short-range potential systems. \cite{Jensen(1979)} On the other hand, we instead start from Eq. (\ref{eqn:2.40}) to obtain \begin{equation} [H_0 -z - V R_0(z ) V ] R(z ) = 1 -V R_0(z ) . \label{eqn:2.110} \end{equation} This equation can be solved for our model if we confine ourselves to the state subspace $\mathbb{C}^N \oplus \{ 0 \}$. \cite{Exner(1985)} In fact, from the fact that $\bra{n} V R_0(z ) \ket{n'} =0$ for any $\ket{n}$ and $\ket{n'} \in \mathbb{C}^N \oplus \{ 0 \}$, Eq. (\ref{eqn:2.110}) reads \begin{equation} \sum_{m=1}^N [(\omega_{n} -z ) \delta_{n m} -\lambda^2 S_{n m}(z) ] \tilde{R}_{m n'}(z ) =\delta_{nn'} , \label{eqn:.20} \end{equation} where $S (z ) $ and $\tilde{R} (z ) $ are the $N \times N$ matrix defined with the matrix components \begin{equation} S_{mn} (z ) := \bra{m} V R_0(z ) V \ket{n} = \int_{0}^{\infty} \frac{v_m^* (\omega) v_n (\omega)}{\omega -z}d \omega, \mbox{ and } \tilde{R}_{mn} (z ):=\bra{m} R(z ) \ket{n}. \label{eqn:.40} \end{equation} We call $S(z) $ and $\tilde{R}(z)$ the {\it self energy} and the {\it reduced resolvent}, respectively. Note that $S(z)$ can be analytically defined for all $z \in \mathbb{C}\backslash [0, \infty)$. For a later convenience, we also introduce the matrix ${K_0}$ and $K(z)$ by \begin{equation} {K_0}_{mn}:=\bra{m} H_0 \ket{n}=\omega_n \delta_{m n}, \quad \mbox{and} \quad K_{mn}(z):=[K_0 -\lambda^2 S(z)]_{mn}, \label{eqn:.55} \end{equation} respectively. Then, Eq. (\ref{eqn:.20}) is equivalent to \begin{equation} [K(z)-z] \tilde{R} (z ) =1, ~~~ \forall z \in \mathbb{C}\backslash (\sigma (H_0) \cup \sigma (H)), \label{eqn:.60} \end{equation} which implies that ${\rm det}[K(z)-z] {\rm det} [\tilde{R} (z ) ] =1 $, so that ${\rm det}[K(z)-z] \neq 0$ and ${\rm det} [\tilde{R} (z ) ] \neq 0 $ for all $z \in \mathbb{C}\backslash (\sigma (H_0) \cup \sigma (H))$.
Thus, the inverse of $K(z)-z$ exists, and we have \begin{equation} \tilde{R} (z ) = [K(z)-z]^{-1}, ~~~ \forall z \in \mathbb{C}\backslash (\sigma (H_0) \cup \sigma (H)). \label{eqn:.70} \end{equation}
\subsection{The boundary values of $\tilde{R} (z )$ and its large energy behavior} \label{subsec:4.2}
From the assumption on the form factors, every $v_m^* (\omega )v_n (\omega )$ is continued to the whole complex plane as a meromorphic function which we merely denote as $v_m^* (z)v_n (z)$. It may have a finite number of poles. Then, it follows from Lemma \ref{lm:formfactor1} that ${S} (z)$ can be reduced to the form \begin{equation} {S} (z) = S(0) + A(z) -(\log (-z)) {\mit \Gamma}(z) , \label{eqn:.76} \end{equation} where we choose ${\rm arg} (-z) ={\rm arg} (z)-\pi$ and $0<{\rm arg} (z)<2\pi$. The matrix ${\mit \Gamma}(z)$ is defined with the components \begin{equation} {\mit \Gamma}_{mn} (z):=v_m^* (z)v_n (z), \label{eqn:.100} \end{equation} and satisfies ${\mit \Gamma}(z)\to 0$ as $z\to 0$ in $\mathbb{C}$. $S(0)$ is the limit of $S(z)$ as $z\to 0$ in $\mathbb{C}\backslash [0, \infty)$, which turns out to be unique. Indeed, as we see from the Appendix in Ref. \makebox(0,1){\Large \cite{Miyamoto(2005)}}\hspace{4mm}, $S_{mn}(0)=\int_{0}^{\infty}v_m^* (\omega) v_n (\omega)/\omega \ d \omega$. $A(z)$ is then defined through Eq. (\ref{eqn:.76}) and becomes a Hermitian matrix for real $\omega$, whose components are the rational functions of $z$ without any singularity on $[0, \infty )$. By definition, $A(z)$ satisfies $A(z)\to 0$ as $z\to 0$. One sees that the boundary values of $S (z )$ at the half line $(0,\infty )$ exist and satisfy \cite{Exner(1985)} \begin{equation} \lim_{\epsilon \to +0} S (\omega \pm i\epsilon) = {D} (\omega ) \pm \pi i {\mit \Gamma}(\omega ), \label{eqn:.80a} \end{equation} where \begin{equation} {D} (\omega ) := S(0) + A(\omega) -(\log \omega) {\mit \Gamma}(\omega) . \label{eqn:.80b} \end{equation} The matrix ${D} (\omega )$ is just of the components \begin{equation} {D}_{mn} (\omega ):=P \int_{0}^{\infty} \frac{v_m^* (\omega' )v_n (\omega' )}{\omega'-\omega} d\omega' , \label{eqn:.90} \end{equation} where $P$ denotes the principal value of the integral. Note that both ${D} (\omega )$ and ${\mit \Gamma}(\omega )$ are Hermitian matrices and ${\mit \Gamma}(\omega ) \geq 0$.
In all the discussion developed in the following, we assume that \begin{equation} {\rm det}[K^\pm (\omega)-\omega]\neq 0, ~~~ \forall \omega >0, \label{eqn:.110} \end{equation} where we introduced \begin{equation} K^\pm (\omega):= \lim_{\epsilon \to +0} K(\omega \pm i\epsilon ) ={K_0} -\lambda^2 {D} (\omega )\mp \lambda^2 \pi i {\mit \Gamma}(\omega ),
~~~ \forall \omega >0. \label{eqn:.112} \end{equation} It is worth noting that condition (\ref{eqn:.110}) is equivalent to the requirement of no positive eigenvalues of $H_0$, whose eigenstates are normalizable. Indeed, if ${\rm det}[K^\pm (\omega)-\omega]=0 $ for some $\omega >0$, there is a non-zero vector $\ket{\eta}=\sum_{n=1}^N \eta_n \ket{n} \in \mathbb{C}^N$ such that $[K^\pm (\omega)-\omega] \ket{\eta}=0 $. Since both ${D} (\omega )$ and ${\mit \Gamma}(\omega )$ are Hermitian matrices, the latter equation implies that \begin{equation} \bra{\eta} [{K_0}-\omega -\lambda^2 {D} (\omega )] \ket{\eta} =0 ~~ \mbox{and} ~~ \bra{\eta} {\mit \Gamma}(\omega ) \ket{\eta}
=\left|\sum_{n=1}^{N} v_n(\omega)\eta_n \right|^2=0 . \label{eqn:.167} \end{equation} Note that the latter relation means that ${\mit \Gamma}(\omega ) \ket{\eta}=0$ because ${\mit \Gamma}(\omega ) \geq 0$. Thus, Eq. (\ref{eqn:.167}) implies that ${\mit \Gamma}(\omega ) \ket{\eta} =0$ and $[{K_0} -\lambda^2 {D} (\omega )]\ket{\eta}=\omega \ket{\eta}$, i.e., \begin{equation} \sum_{n=1}^{N} v_n(\omega)\eta_n =0, ~~ \mbox{and} ~~ \sum_{n=1}^{N} [\omega_{m}\delta_{mn} -\lambda^2 {D}_{mn} (\omega )] \eta_n =\omega \eta_m , \label{eqn:.169} \end{equation} for all $m=1, \ldots, N$. This is merely the condition for the existence of a positive eigenvalue $\omega$ of $H$. \cite{Miyamoto(2005)}
\begin{lm} \label{lm:100} Under the assumption (\ref{eqn:.110}), it holds that $\tilde{R}^{\pm}(\omega) :=\lim_{\epsilon \to +0} \tilde{R}(\omega \pm i \epsilon ) $ exists for all $\omega >0$ and $\tilde{R}^{\pm}(\omega) =[K^\pm (\omega)-\omega]^{-1}$. \end{lm}
{\sl Proof} : Under the assumption (\ref{eqn:.110}), $[K^\pm (\omega)-\omega]^{-1} $ exists. Then \begin{eqnarray} &&
\| [K^\pm (\omega)-\omega]^{-1} -\tilde{R}(\omega \pm i \epsilon )
\| \nonumber \\ &\leq&
\| [K^\pm (\omega)-\omega]^{-1}
\|
\| \pm i\epsilon +\lambda^2 S (\omega \pm i \epsilon) -\lambda^2 {D} (\omega ) \mp \lambda^2 \pi i {\mit \Gamma}(\omega )
\|
\| \tilde{R}(\omega \pm i \epsilon )
\| . \label{eqn:.120} \end{eqnarray} Note that for any nonzero $\ket{y} \in \mathbb{C}^N$ ($\neq 0$) there is a nonzero $\ket{x} \in \mathbb{C}^N$ such that $\ket{y}=[K(\omega \pm i \epsilon )-\omega \mp i \epsilon]\ket{x}$. We then obtain \begin{eqnarray} \frac{
\| \tilde{R}(\omega \pm i \epsilon )\ket{y}
\| }
{\| y\|}
&\leq&
\frac{\| x \|} {
\bigl|
\| [K^\pm (\omega)-\omega] \ket{x}
\| -
\| [\pm i\epsilon +\lambda^2 S (\omega \pm i \epsilon) -\lambda^2 {D} (\omega ) \mp \lambda^2 \pi i {\mit \Gamma}(\omega )]\ket{x}
\|
\bigr| } \nonumber \\ \!&\leq&\! \biggl[ \inf_{\ket{x} \neq 0, \ket{x}\in \mathbb{C}^N}\! \! \frac{
\|[K^\pm (\omega)-\omega]\|
}{\| x \|} \nonumber\\ && -
\| \pm i\epsilon +\lambda^2 S (\omega \pm i \epsilon) -\lambda^2 {D} (\omega ) \mp \lambda^2 \pi i {\mit \Gamma}(\omega )
\| \biggr]^{-1} \hspace{-2mm},
\label{eqn:.130} \end{eqnarray} which implies that \begin{equation}
\mathop{\overline{\lim}}_{\epsilon \to +0}
\| \tilde{R}(\omega \pm i \epsilon )
\|
\leq \left[ \inf_{\ket{x} \neq 0, \ket{x}\in \mathbb{C}^N} \frac{
\|K^\pm (\omega)-\omega\|
}{\| x \|} \right]^{-1} < \infty , \label{eqn:.140} \end{equation} where the norm of an $N\times N$ matrix $A$ is defined by
$\| A\|=
\sup_{\ket{x} \neq 0, \ket{x}\in \mathbb{C}^N} \| A\ket{x}\| /\| x\|$. In Eq. (\ref{eqn:.130}), we used the fact that there is some $\epsilon_0 >0$ such that for any positive $\epsilon < \epsilon_0$ and for any non zero $\ket{x} \in \mathbb{C}^N$ \begin{eqnarray} \frac{
\|
[K^\pm (\omega)-\omega] \ket{x}\|
}{\| x \|} &\geq& \inf_{\ket{x} \neq 0, \ket{x}\in \mathbb{C}^N} \frac{
\|[K^\pm (\omega)-\omega] \ket{x}\|
}{\| x \|} \nonumber \\ &>&
\| \pm i\epsilon +\lambda^2 S (\omega \pm i \epsilon) -\lambda^2 {D} (\omega ) \mp \lambda^2 \pi i {\mit \Gamma}(\omega )
\| \nonumber \\ &\geq& \frac{
\| [\pm i\epsilon +\lambda^2 S (\omega \pm i \epsilon) -\lambda^2 {D} (\omega ) \mp \lambda^2 \pi i {\mit \Gamma}(\omega ) ]\ket{x}
\|
}{\| x \|}, \label{eqn:.150} \end{eqnarray} where the assumption (\ref{eqn:.110}) is taken into account. Thus, by using Eq. (\ref{eqn:.140}), Eq. (\ref{eqn:.120}) leads us to \begin{equation} \lim_{\epsilon \to +0}
\| [K^\pm (\omega)-\omega]^{-1} -\tilde{R}(\omega \pm i \epsilon )
\| =0, \label{eqn:.160} \end{equation} which completes the proof of the lemma. \qed
\begin{lm} \label{lm:Large-omega} : Under the assumption (\ref{eqn:.110}), $\tilde{R}^\pm (\omega)$ is $r$-times differentiable in $\omega \in (0, \infty)$, and it behaves as \begin{equation} \frac{d^r \tilde{R}^\pm (\omega)}{d\omega^r} =O(\omega^{-r-1}) \mbox{ as } \omega \to \infty . \label{eqn:Large10} \end{equation} \end{lm}
{\sl Proof} : We first show the statement for $r=0$. From the assumption on the form factors and Lemma \ref{lm:formfactor1}, one sees that \begin{equation} \lim_{\omega \to \infty} {D} (\omega) =0 \mbox{ and } \lim_{\omega \to \infty} {\mit \Gamma} (\omega) =0. \label{eqn:Large40} \end{equation} Since from the assumption (\ref{eqn:.110}) $K^\pm(\omega)-\omega$ is invertible for all $\omega >0$, it holds that there is some positive $\bar{\omega} > \omega_N$ such that for any $\omega > \bar{\omega}$,
\begin{eqnarray} \frac{
\| \tilde{R}^\pm (\omega)\ket{y}
\| }
{\| y\|} &\leq&
\frac{\| x \|} {\displaystyle
\| ({K_0}-\omega)\ket{x}
\| -\lambda^2
\| [{D} (\omega) \pm \pi i {\mit \Gamma}(\omega)]\ket{x}
\|
} \label{eqn:Large30a}\\ &\leq& \frac{1} { \omega -\omega_N -\lambda^2
\| {D} (\omega) \pm \pi i {\mit \Gamma}(\omega)
\| } = O(\omega^{-1}), \label{eqn:Large30} \end{eqnarray} where the last inequality is obtained as follows: we can choose some positive $\bar{\omega} > \omega_N$ such that for any $\omega > \bar{\omega} $\begin{equation} \frac{
\| [{K_0}-\omega]\ket{x}
\|
}{\| x \|} \geq \min_n \{ \omega-\omega_n \} =\omega-\omega_N >
\lambda^2 \|{D} (\omega) \pm \pi i {\mit \Gamma}(\omega) \| \geq \lambda^2 \frac{
\| [{D} (\omega) \pm \lambda^2 \pi i {\mit \Gamma}(\omega) ]\ket{x}
\|
}{\| x \|} , \label{eqn:Large50} \end{equation} where Eq. (\ref{eqn:Large40}) was used. Thus Eq. (\ref{eqn:Large30}) reads just as Eq. (\ref{eqn:Large10}) does for $r=0$. In the case of $r\geq 1$, we first note that from our assumptions on the form factors and Lemma \ref{lm:formfactor1} again, $A(\omega ) $ and ${\mit \Gamma}(\omega ) $, which are connected through ${D} (\omega )=S_0 + A(\omega ) -\log \omega {\mit \Gamma}(\omega )$, also satisfy \begin{equation} \frac{d^r A(\omega )}{d\omega^r }= O(\omega^{-1-r}), ~~ \frac{d^r \log \omega {\mit \Gamma}(\omega )}{d\omega^r }= O(\omega^{-1-r} \log \omega), \label{eqn:Large80} \end{equation} as $\omega \to \infty$, where we used the estimation that $\frac{d^r {\mit \Gamma}(\omega )}{d\omega^r }= O(\omega^{-1-r})$. Thus, for $r=1$, we have \begin{equation} \frac{d \tilde{R}^{\pm}(\omega )}{d\omega} = \tilde{R}^{\pm}(\omega ) \frac{d}{d\omega} \left[ \omega + \lambda^2 {D}(\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) \right] \tilde{R}^{\pm}(\omega )
= O(\omega^{-2}), \label{eqn:Large90b} \end{equation} as $\omega \to \infty$, where Eq. (\ref{eqn:Large10}) for $r=0$ was used. For $r\geq1$, we obtain \begin{equation} \frac{d^r \tilde{R}^{\pm}(\omega )}{d\omega^r} = \sum_{j=1}^{r} {\sum_{\{s_i \}_{i=1}^{j} }}' a^{(r)} (\{s_i \}_{i=1}^{j} ) \left\{ \prod_{i=1}^{j} \tilde{R}^{\pm}(\omega ) \frac{d^{s_i}}{d\omega^{s_i}} \left[ \omega + \lambda^2 {D}(\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) \right] \right\} \tilde{R}^{\pm}(\omega ), \label{eqn:Large100} \end{equation}
where $a^{(r)} (\{s_i \}_{i=1}^{j} ) $ is an appropriate positive integer. Note that the symbol ($\ '$) means that the summation over $\{s_i \}_{i=1}^{j}$ is taken under the condition that $s_i \geq 1$ for all $i$
and $\sum_{i=1}^{j} s_i =r$. If $r=1$, Eq. (\ref{eqn:Large100}) reproduces Eq. (\ref{eqn:Large90b}) with $a^{(1)} (\{s_i \}_{i=1}^{1} ) =1$. In the general case, if Eq. (\ref{eqn:Large100}) holds for $r=k$, then its derivative is made up of a linear combination of \begin{equation} \left\{ \prod_{i=1}^{j+1} \tilde{R}^{\pm}(\omega ) \frac{d^{s_i}}{d\omega^{s_i}} \left[ \omega + \lambda^2 {D}(\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) \right] \right\} \tilde{R}^{\pm}(\omega ),
\label{eqn:Large110} \end{equation} where $\sum_{i=1}^{j+1} s_i =k+1$ for $1 \leq j \leq k$, and \begin{equation} \left\{ \prod_{i=1}^{j} \tilde{R}^{\pm}(\omega ) \frac{d^{s_i}}{d\omega^{s_i}} \left[ \omega + \lambda^2 {D}(\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) \right] \right\} \tilde{R}^{\pm}(\omega ),
\label{eqn:Large120} \end{equation} where $\sum_{i=1}^{j} s_i =k+1$ for $1 \leq j \leq k$. On the other hand, they are actually included in the right-hand side (rhs) of Eq. (\ref{eqn:Large100}) for $r=k+1$. Thus Eq. (\ref{eqn:Large100}) is valid for all integer $r \geq 1$. Let us now evaluate the asymptotic behavior of $d^r \tilde{R}^{\pm}(\omega )/d\omega^r$ for large $\omega$. One can see that the summand for $j=r$ in Eq. (\ref{eqn:Large100}), where all $s_i =1$, contributes $O(\omega^{-r-1})$ to $d^r \tilde{R}^{\pm}(\omega )/d\omega^r$, while the other summands for $j<r$ specified by $\{s_i \}_{i=1}^{j}$ contribute $O(\omega^{-r-1-2s_0 }(\log \omega )^{s_0}))$ at most, where $s_0$ is a number of $s_i$ satisfying $s_i \geq 2$ and never vanishes for $j<r$. Therefore, the summand dominating for large $\omega$ is that for $j=r$. Since we recursively show $a^{(r)} (\{s_1 \}_{i=1}^{r} ) =r!$, which never vanishes, the statement is proved. \qed
\section{Classification of the zero-energy singularity of $\tilde{R}^\pm (\omega ) $} \label{sec:5}
In order to prescribe the zero energy resonance in the $N$-level Friedrichs model, we should identify the zero energy eigenstates in this model which either belong to or do not belong to ${\cal H}$. In the case of the short-range potential systems, \cite{Jensen(1979)} this task needs some elaborate examination with an appropriately extended Hilbert space. On the other hand, in our case, it is rather easily performed, as is seen in the following.
Let us first see whether the eigenvector $\ket{\psi} \in \mathbb{C}^N$ of ${K_0} -\lambda^2 S(0)$, belonging to the zero eigenvalue, can be actually extended to the eigenvector of $H$ belonging to the zero eigenvalue of $H$. If $\ket{\Psi}=\ket{\psi}+\ket{f} \in D(H) \subset {\cal H}$ is a zero eigenvector of $H$, it should satisfy $H\ket{\Psi}=0$, or equivalently \cite{Miyamoto(2005)} \begin{equation} \omega_n \psi_n +\lambda \braket{v_n}{f} =0 ~\mbox{for}~ n=1, \ldots, N,~~~\mbox{and}~~~ \omega f(\omega)+ \lambda \sum_{n=1}^{N}\psi_n v_n (\omega)=0. \label{eqn:.333} \end{equation} The latter equation of Eq. (\ref{eqn:.333}) is immediately solved as \begin{equation} f(\omega)=-\lambda \frac{\sum_{n=1}^{N}\psi_n v_n (\omega)}{\omega}, \label{eqn:.335} \end{equation} which should be square integrable because we intend to find $\ket{\Psi}$ in ${\cal H}$. If this is the case, $\omega f(\omega) \in L^2 ((0, \infty))$, i.e., $\ket{\Psi} \in D(H)$ is ensured, and the substitution of Eq. (\ref{eqn:.335}) into $\braket{v_n}{f}$ is safely done. Then, we find that the former equation of Eq. (\ref{eqn:.333}) is nothing more than \begin{equation} ({K_0} -\lambda^2 S(0) )\ket{\psi}=K(0)\ket{\psi}=0, \label{eqn:.336} \end{equation} where $K(0):=K^\pm (0)={K_0} -\lambda^2 S(0)$. However, it is noted that such an $f(\omega)$ associated with $\ket{\psi}$ is not necessarily square integrable. Hence, we shall decompose the zero eigenspace of $K(0)$, denoted by
$M=\{ \ket{\psi} \in \mathbb{C}^N |~ K(0)\ket{\psi}=0 \}$, into two kinds of subspaces: $M_1=(M_0 \oplus M_2)^{\perp}$
and $M_2 =\{ \ket{\psi} \in M | f(\omega) \in L^2 ((0, \infty)) \}$. Here $M_0=M^{\perp}$, and $D^{\perp}$ denotes the orthogonal complement of the subspace $D$. In short we have $\mathbb{C}^N = M_0 \oplus M_1 \oplus M_2$. Then, as is expected from the definition, we have \begin{equation}
M_1 \subset \{ \ket{\psi} \in M | f(\omega) \notin L^2 ((0, \infty)) \} . \label{eqn:.325} \end{equation} Note that in general the subset on the rhs of the above is not a subspace. We call $0$ the {\it zero energy resonance} (or merely {\it zero resonance}) of $H$ if $M_1$ is not empty. We also introduce the projection operators $Q_0$, $Q_1$, and $Q_2$, associated with $M_0$, $M_1$, and $M_2$, respectively. What we next do is to introduce the terminology following the study of Jensen and Kato. \cite{Jensen(1979)}
\begin{df} \label{df:regular} {\rm We call the system a {\it regular} case if it holds that $0 \notin \sigma (K(0))$, i.e., \begin{equation} {\rm det}[K(0)] \neq 0. \label{eqn:.300} \end{equation} In this case, $0$ is said to be a {\it regular} point for $H$. } \end{df}
\begin{df} \label{df:exceptional} {\rm We call the system the {\sl exceptional} case if, instead of Eq. (\ref{eqn:.300}), it holds that $0 \in \sigma (K(0))$, i.e., \begin{equation} {\rm det}[K(0)] = 0. \label{eqn:.330} \end{equation} In particular, if $0$ is a resonance but not an eigenvalue ($Q_1 \neq 0$, $Q_2 =0$), $0$ is said to be an {\it exceptional} point for $H$ of the {\sl first kind}. If $0$ is not a resonance, but an eigenvalue ($Q_1 = 0$, $Q_2 \neq 0$), $0$ is said to be an exceptional point of the {\sl second kind}. If $0$ is both a resonance and an eigenvalue ($Q_1 \neq 0$, $Q_2 \neq 0$), $0$ is said to be an exceptional point of the {\it third kind}. } \end{df}
We here remark that in general a non-trivial solution of Eq. (\ref{eqn:.336}) does not exist, however we can find a special case where such a solution surely exists. Suppose that $N_+$ eigenvalues $\omega_n$ of $H_0$ are positive, and all form factors $v_n (\omega)$ satisfying the assumption (\ref{eqn:formfactor1}) are linearly independent. Then increasing $\lambda$ gradually form $0$ to $\infty$, we can find some critical values of $\lambda$ for which $K(0)$ has the zero eigenvalue. Let us denote the $n$-th eigenvalue of $K(0)$ by $\kappa_n (0)$ where $\kappa_1 (0)\leq \kappa_2 (0) \leq \cdots \leq \kappa_N (0)$. Then, $\kappa_n (0)$ turns out to satisfy the inequality \begin{equation} \omega_n -\lambda^2 \sigma_N (0) \leq \kappa_n (0) \leq \omega_n -\lambda^2 \sigma_1 (0), \label{eqn:zeroeigenvalue} \end{equation} where both of $\sigma_1 (0)$ and $\sigma_N (0)$ are positive constants and ensured not to vanish.\cite{Miyamoto(2005)}
Thus, for a sufficiently small $|\lambda|$ $\kappa_n (0)$ for each $n\geq N-N_+ +1$ should be positive,
while for a sufficiently large $|\lambda|$ they should be negative. Furthermore, one easily sees that all $\kappa_n (0)$ are continuous functions of $\lambda^2$. Therefore, we conclude from the intermediate value theorem that there is at least one critical value of $\lambda$ to make $\kappa_n (0)=0$ for each $n\geq N-N_+ +1$. We can actually find such special values of $\lambda$ in Fig. 1 depicted in Ref. \makebox(8,1){\Large \cite{Miyamoto(2005)}}\hspace{1mm}. The example mentioned here could be treated in a more general way with resort to the analytic Fredholm theorem \cite{the analytic Fredholm theorem,Rauch(1978)} which tells us at most a finite number of the critical values exists.
It is also worth remarking that the existence of the zero energy eigenstates that either belong to or not to the Hilbert space necessarily prescribes the small energy behavior of the form factors in the following way. Remember that under the assumption on the form factors, ${\mit \Gamma}(\omega )$ defined by Eq. (\ref{eqn:.100}) has an asymptotic form like \begin{equation} {\mit \Gamma}(\omega ) = \sum_{n=1}^{N} \omega^n {\mit \Gamma}_n +O(\omega^{N+1}), \label{eqn:formfactor2} \end{equation} as $\omega\to +0$. Then, if $\ket{\psi} \in M_1$ exists, it should satisfy \begin{equation} \bra{\psi} {\mit \Gamma}_1 \ket{\psi} \neq 0. \label{eqn:.340} \end{equation} In fact, if $\bra{\psi} {\mit \Gamma}_1 \ket{\psi} = 0$, we see that $f(\omega)$ in Eq. (\ref{eqn:.335}) has to satisfy
$|f(\omega)|^2=\lambda^2\bra{\psi}{\mit \Gamma}(\omega)\ket{\psi}/\omega^2=O(1)$ as $\omega \to +0$, however which concludes that $f(\omega)$ is square integrable. This contradicts the assumption that $\ket{\psi} \in M_1$. In order to make the condition (\ref{eqn:.340}) be satisfied, at least ${\mit \Gamma}_1$ should not vanish identically. We can find such form factors in the physical systems for the spontaneous emission process of photons from the Hydrogen atom \cite{Facchi(1998),Seke(1994)} and the quantum dot. \cite{Antoniou(2001)} On the other hand, the discussion mentioned here immediately implies the fact that if $\ket{\psi} \in M_2$ exists, this time it should satisfy \begin{equation} \bra{\psi} {\mit \Gamma}_1 \ket{\psi} = 0, \label{eqn:.350} \end{equation} which just ensures the requirement that $f(\omega) \in L^2 ((0, \infty))$. However, note that Eq. (\ref{eqn:.350}) does not imply ${\mit \Gamma}_1=0$ identically and only requires ${\mit \Gamma}_1=0$ on the subspace $M_2$.
\subsection{The small-energy behavior of $\tilde{R}^\pm (\omega) $ in the regular case}
In this case, the same as in Eq. (\ref{eqn:.160}), we can show that $\tilde{R}^{\pm}(0) =(K(0))^{-1}$. Furthermore, we can choose some positive $\omega_0 >0$ such that \begin{equation}
\| (K(0))^{-1}\|
\| [\omega +\lambda^2 {D} (\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) -\lambda^2 S(0)
] \| < 1, \label{eqn:.170} \end{equation} for all positive $\omega < \omega_0$. Then, $\tilde{R}^{\pm}(\omega )$ is expanded as a Neumann series, \begin{equation} \tilde{R}^{\pm}(\omega ) = \{ K(0) [ 1- (K(0))^{-1} [\omega +\lambda^2 {D} (\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) -\lambda^2 S(0) ]] \}^{-1} = \lim_{N\to \infty}S_N(\omega), \label{eqn:.180} \end{equation} where \begin{equation} S_N(\omega) = \sum_{j=0}^N \{ (K(0))^{-1} [\omega +\lambda^2 {D} (\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) -\lambda^2 S(0) ]\}^j (K(0))^{-1} , \label{eqn:.185} \end{equation} for all positive $\omega < \omega_0$ with $\{ (K(0))^{-1} [\omega +\lambda^2 {D} (\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) -\lambda^2 S(0) ]\}^0 =1 $. Under our assumptions on the form factors, $A(\omega ) $ defined in Eq. (\ref{eqn:.80b}) is asymptotically expanded as \begin{equation} A(\omega) = \sum_{n=1}^{N} \omega^n A_n +O(\omega^{N+1}), \label{eqn:.220} \end{equation} as $\omega \to 0$. By using Eqs. (\ref{eqn:formfactor2}) and (\ref{eqn:.220}), it also follows that \begin{equation} {D} (\omega ) = S(0) -\omega \log \omega {\mit \Gamma}_1 +\omega A_1 + O(\omega^2 \log \omega ) , \label{eqn:.230} \end{equation} as $\omega \to +0$. Then, Eq. (\ref{eqn:.180}) tells us the dominant asymptotic behavior of $\tilde{R}^{\pm}(\omega )$ becomes \begin{equation} \tilde{R}^{\pm}(\omega ) =(K(0))^{-1} +O(\omega \log \omega), \label{eqn:.240} \end{equation} as $\omega \to +0$, where $(K(0))^{-1}$ never vanishes in the regular case.
\subsection{The small-energy behavior of $\tilde{R}(z) $ in the exceptional case of the first kind} \label{subsec:5.C}
In the exceptional case of the first kind, from the definition, $Q_1 \neq 0$ while $Q_2 = 0$, so that $Q_0 +Q_1 =1$. Then, $\tilde{R} (z)$ is divided into the following four terms, \begin{equation} \tilde{R}(z) = Q_0 \tilde{R}(z) Q_1 +Q_0 \tilde{R}(z) Q_0 +Q_1 \tilde{R}(z) Q_0 +Q_1 \tilde{R}(z) Q_1 . \label{eqn:5.C.1.30} \end{equation} We now introduce the four matrices, \begin{equation} E_{kl}(z) =Q_k [K(z) -z]Q_l = Q_k [K_0 -z -\lambda^2 [S(0)+A(z)-(\log z) {\mit \Gamma}(z)+i\pi {\mit \Gamma}(z)]]Q_l, \label{eqn:5.C.1.40} \end{equation} where $k, l =0, 1$, and $\log(-z) -\log z =-i \pi$ is used. From the relation that $[K(z) -z]\tilde{R}(z) =1$, they satisfy \begin{equation} E_{k0}Q_0 \tilde{R}Q_l + E_{k1}Q_1 \tilde{R}Q_l =Q_k \delta_{kl}, \label{eqn:5.C.1.50} \end{equation} for $k,l=0,1$. To solve the above equations we need to check whether $E_{11}$ and $E_{00}$ are invertible in the subspaces $M_1$ and $M_0$, respectively. By using Eq. (\ref{eqn:formfactor2}), $E_{11}(z)$ is rewritten as \begin{equation} E_{11}(z) = \lambda^2z (\log z) Q_1 {\mit \Gamma}_1 Q_1 -z Q_1 -\lambda^2 Q_1 \{ A(z) -(\log z) [{\mit \Gamma}(z) -z {\mit \Gamma}_1 ] +i\pi {\mit \Gamma}(z)\} Q_1 , \label{eqn:5.C.1.60b} \end{equation} where $Q_1K(0)Q_1 =0$ is used. Note that $A(z)=O(z)$ and ${\mit \Gamma}(z) -z {\mit \Gamma}_1 =O(z^2 )$ for our form factors, so that all terms excepting the first one of the rhs of Eq. (\ref{eqn:5.C.1.60b}) are of the order of $O(z)$. Furthermore, the exceptional case of the first kind imposes the fact that $Q_1 {\mit \Gamma}_1Q_1 \neq 0$ [see Eq. (\ref{eqn:.340})], and $Q_1 {\mit \Gamma}_1Q_1$ is positive definite in $M_1$ and thus invertible in $M_1$. Hence, $E_{11}(\omega) $ is invertible for sufficiently small
$|z|>0$, and the inverse can be expanded by the Neumann series as, \begin{eqnarray} \hspace*{-5mm} E_{11}^{-1}(z) &=& \sum_{j=0}^{\infty} (\tilde{E}_{11}(z))^j \frac{1}{\lambda^2z \log z} (Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \label{eqn:5.C.1.90a}\\ &=& \frac{1}{\lambda^2z \log z} (Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
+O(z^{-1} (\log z)^{-2})=O(z^{-1} (\log z)^{-1}) , \label{eqn:5.C.1.90b} \end{eqnarray}
for small $|z|$, where we define \begin{equation} \tilde{E}_{11}(z) := \frac{1}{\lambda^2z \log z} (Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \bigl\{ z Q_1 +\lambda^2 Q_1 \{ A(z) -(\log z) [{\mit \Gamma}(z) -z {\mit \Gamma}_1 ] + \pi i {\mit \Gamma}(z) \} Q_1 \bigr\} ,
\label{eqn:5.C.1.90c} \end{equation} which behaves as $1/\log z$ as $z\to0$. For $E_{00}$ we have, \begin{equation} E_{00}(z) = Q_0 K(0)Q_0 -Q_0 [z +\lambda^2 [A(z)-(\log z) {\mit \Gamma}(z) + i\pi {\mit \Gamma}(z)]]Q_0 , \label{eqn:5.C.1.100} \end{equation} where the first term of the above is invertible in $M_0$, and the last term vanishes as $z \to 0$. Hence, $E_{00}(z) $ is invertible in $M_0$
for sufficiently small $|z|>0$ and is expanded as \begin{equation} E_{00}^{-1}(z) \! =\! -\sum_{j=0}^{\infty} \{[Q_0 K(0)Q_0 ]^{-1} [z +\lambda^2 [A(z)-(\log z) {\mit \Gamma}(z) + i\pi {\mit \Gamma}(z)]]Q_0 \}^j [Q_0 K(0)Q_0 ]^{-1} = O(1) , \label{eqn:5.C.1.110b} \end{equation}
for small $|z|$. Furthermore, we obtain \begin{equation} E_{kl}(z) = -\lambda^2 Q_k [A(z)-(\log z) {\mit \Gamma}(z) +\pi i {\mit \Gamma}(z) ]Q_l =O(z \log z), \label{eqn:5.C.1.130} \end{equation} for $k\neq l$ as $z \to 0$, because $Q_0 K(0)Q_1=Q_1 K(0)Q_0=0$.
Solving Eq. (\ref{eqn:5.C.1.50}), we obtain \cite{MatrixAnalysis} \begin{eqnarray} Q_0\tilde{R}Q_0 &=& (E_{00}-E_{01}E_{11}^{-1}E_{10})^{-1} =O(1), \label{eqn:5.C.1.120a}\\ Q_0\tilde{R}Q_1 &=& -E_{00}^{-1}E_{01} Q_1\tilde{R}Q_1 = -E_{00}^{-1}E_{01} (E_{11}-E_{10}E_{00}^{-1}E_{01})^{-1} =O(1), \label{eqn:5.C.1.120b}\\ Q_1\tilde{R}Q_0 &=& -Q_1\tilde{R}Q_1 E_{10} E_{00}^{-1} = -(E_{11}-E_{10}E_{00}^{-1}E_{01})^{-1} E_{10} E_{00}^{-1} =O(1), \label{eqn:5.C.1.120c}\\ Q_1\tilde{R}Q_1 &=& (E_{11}-E_{10}E_{00}^{-1}E_{01})^{-1} =O(z^{-1} (\log z)^{-1}). \label{eqn:5.C.1.120d} \end{eqnarray}
It is worth noting that since the relation $[K(z) -z]\tilde{R}(z) =1$ is analytically continued to the second Riemann sheet through the cut $[0, \infty)$, the above-mentioned results are also valid for such a continued region and the estimations obtained here can be applied without any corrections.
When we consider the small energy behavior of $\tilde{R}^- (\omega)$, it is convenient to expand $E_{11}$, differently from Eq. (\ref{eqn:5.C.1.60b}), as \begin{eqnarray} E_{11}(z) &=& \lambda^2z (\log z -2\pi i) Q_1 {\mit \Gamma}_1 Q_1 \nonumber\\ && -z Q_1 -\lambda^2 Q_1 \{ A(z) -(\log z -2\pi i)[{\mit \Gamma}(z) -z {\mit \Gamma}_1 ] -i\pi {\mit \Gamma}(z)\} Q_1 . \label{eqn:5.C.1.125b} \end{eqnarray} All the above-obtained results are only changed by replacing the term $\log z$ with $\log z -2\pi i$. Then, we can obtain from Eq. (\ref{eqn:5.C.1.60b}) \begin{equation} E_{11}^+ (\omega) = \lim_{\epsilon \to +0}E_{11}(\omega+i\epsilon) = \lambda^2 (\log \omega) Q_1 {\mit \Gamma}(\omega) Q_1 -\omega Q_1 -\lambda^2 Q_1 [A(\omega)-i\pi {\mit \Gamma}(\omega)]Q_1 , \label{eqn:5.C.1.127a} \end{equation} while from Eq. (\ref{eqn:5.C.1.125b}) \begin{equation} E_{11}^- (\omega) = \lim_{\epsilon \to +0}E_{11}(\omega-i\epsilon) = \lambda^2 (\log \omega) Q_1 {\mit \Gamma}(\omega) Q_1 -\omega Q_1 -\lambda^2 Q_1 [A(\omega)+i\pi {\mit \Gamma}(\omega)]Q_1 . \label{eqn:5.C.1.127b} \end{equation}
\subsection{The small-energy behavior of $\tilde{R}(z)$ in the exceptional case of the second kind} \label{subsec:5.D}
In the exceptional case of the second kind, it follows that $Q_1 = 0$, $Q_2 \neq 0$, and $Q_0 +Q_2 =1$. Let us consider the asymptotic behavior of the reduced resolvent at small energies,
which is written in the following form, $\tilde{R} (z)=\sum_{k,l=0,2}Q_k \tilde{R} (z) Q_l $. We now introduce the four matrices again, \begin{equation} E_{kl}(z) =Q_k [K(z) -z]Q_l , \label{eqn:5.C.2.40} \end{equation} where $k, l =0, 2$. From the relation that $[K(z) -z]\tilde{R}(z) =1$, they satisfy that \begin{equation} E_{k0} Q_0 \tilde{R} Q_l + E_{k2} Q_2 \tilde{R} Q_l =Q_k \delta_{kl}, \label{eqn:5.C.2.50} \end{equation} for $k, l=0, 2$. This time, $E_{22}$ and $E_{00}$ are invertible in $M_2$ and $M_0$, respectively. In fact, from Eqs. (\ref{eqn:formfactor2}) and (\ref{eqn:.220}) we have \begin{equation} E_{22}(z) =-z Q_2(1+\lambda^2 A_1) Q_2 -\lambda^2 Q_2[A(z)-z A_1 -(\log z) {\mit \Gamma}(z) + i\pi {\mit \Gamma}(z)]Q_2 , \label{eqn:5.C.2.60b} \end{equation} where $Q_2K(0)Q_2 =0$ was used. Note that since $A(z)-z A_1=O(z^2)$ and $Q_2 {\mit \Gamma}_1 Q_2=0$ [see Eq. (\ref{eqn:.350})],
the second term of the rhs of Eq. (\ref{eqn:5.C.2.60b}) is of the order of $O(z^2 \log z)$. Furthermore, since $Q_2 A_1 Q_2 \geq 0$ from $Q_2 {\mit \Gamma}_1 Q_2=0$ and Lemma \ref{lm:1st_order}, $Q_2(1+\lambda^2 A_1)Q_2 >0$ and invertible in $M_2$. These facts bring us the fact that $E_{22}(z) $ is invertible in $M_2$ for sufficiently small $z>0$, that is \begin{eqnarray} E_{22}^{-1}(z) &=& -\sum_{j=0}^{\infty} (-\tilde{E}_{22}(z) )^j \frac{1}{z}[Q_2 (1+\lambda^2 A_1 )Q_2 ]^{-1} \label{eqn:5.C.2.90a}\\ &=& \frac{1}{z}[Q_2 (1+\lambda^2 A_1 )Q_2 ]^{-1}+O(\log z)=O(z^{-1}), \label{eqn:5.C.2.90b} \end{eqnarray}
where \begin{equation} \tilde{E}_{22}(z) := \frac{1}{z}[Q_2 (1+\lambda^2 A_1 )Q_2 ]^{-1} \lambda^2 Q_2[A(z)-z A_1 -(\log z) {\mit \Gamma}(z)+ i\pi {\mit \Gamma}(z)]Q_2. \label{eqn:5.C.2.90c} \end{equation} For $E_{00}$, we next have \begin{equation} E_{00}(z) = Q_0 K(0)Q_0 -Q_0 [z +\lambda^2 [A(z)-(\log z) {\mit \Gamma}(z) + i\pi {\mit \Gamma}(z)]]Q_0, \label{eqn:5.C.2.100} \end{equation}
where the first term of the above is invertible in $M_0$, and the last term vanishes as $|z| \to 0$. Hence, $E_{00}(z) $ is invertible in $M_0$
for sufficiently small $|z|>0$, and the inverse is obtained as a Neumann series. On the other hand, $E_{20}$ and $E_{02}$ behave as \begin{equation} E_{kl}(z) = Q_k [-z(1+\lambda^2 A_1 ) -\lambda^2 [A(z)-z A_1-(\log z) {\mit \Gamma}(z) + \pi i {\mit \Gamma}(z)] ]Q_l =O(z), \label{eqn:5.C.2.130} \end{equation}
for small $|z|$ where $k\neq l$. Solving Eqs. (\ref{eqn:5.C.2.50}) as in Eqs. (\ref{eqn:5.C.1.120a}) to (\ref{eqn:5.C.1.120d}), one sees that $ Q_0\tilde{R}Q_0=O(1),
$ for $k,l=0,2$, except \begin{equation} Q_2\tilde{R}Q_2 = (E_{22}-E_{20}E_{00}^{-1}E_{02})^{-1} =O(z^{-1}), \label{eqn:5.C.2.120d} \end{equation} as $z\to 0$. In particular, the last equation is expanded as, \begin{equation} Q_2 \tilde{R} (z) Q_2 = \sum_{j=0}^{\infty} \Bigl[ E_{22}^{-1}E_{20}E_{00}^{-1}E_{02} \Bigr]^j E_{22}^{-1}
= -\frac{1}{z}[Q_2 (1+\lambda^2 A_1 )Q_2 ]^{-1} +O(\log z), \label{eqn:5.C.2.140d} \end{equation}
for small $|z|$, where we used Eq. (\ref{eqn:5.C.2.90b}).
We now remark that the zero energy eigenspace of $H$ denoted by ${\cal N}_0$ is completely characterized by $M_2$. That is, there is a bijection from $M_2\oplus \{0\}$ to ${\cal N}_0$. From the discussion concerning Eqs. (\ref{eqn:.333}), (\ref{eqn:.335}), and (\ref{eqn:.336}), for any $\ket{\Psi} \in {\cal M}_0$, there is a vector $\ket{\psi}\in M_2\oplus \{0\}$ such that \begin{equation} \ket{\Psi}=\ket{\psi}-\lambda \int_0^\infty \frac{\sum_{n=1}^N v_n (\omega)\psi_n}{\omega} \ket{\omega}d\omega = [1-\lambda R_0 (0)V]\ket{\psi}, \label{eqn:5.C.2.150} \end{equation} where $V$ is restricted to $\mathbb{C}^N \oplus \{0\}$ and $R_0 (0)$ is the (unbounded) multiplication operator of $1/\omega$ in $L^2((0,\infty))$. Then we see $V\ket{\psi} \in D(R_0 (0))$ because $\ket{\psi}\in M_2\oplus \{0\}$. Thus $1-\lambda R_0 (0)V$ is well defined as an operator from $M_2\oplus \{0\}$ to ${\cal H}$. Now, Eq. (\ref{eqn:5.C.2.150}) tells us that $1-\lambda R_0 (0)V$ is a surjection from $M_2\oplus \{0\}$ to ${\cal N}_0$. On the other hand, for any $\ket{\Psi} \in {\cal N}_0$, if $\ket{\Psi}=0$, i.e., $0=\braket{\Psi}{\Psi}$, Eq. (\ref{eqn:5.C.2.150}) implies that $0=\braket{\Psi}{\Psi}\geq \braket{\psi}{\psi}$. Therefore, $1-\lambda R_0 (0)V$ is also an injection from $M_2\oplus \{0\}$ to ${\cal N}_0$, and the proof is completed.
\subsection{The small-energy behavior of $\tilde{R}^\pm (\omega) $ in the exceptional case of the third kind} \label{subsec:5.E}
In the exceptional case of the third kind, from the definition, $Q_1 \neq 0$, $Q_2 \neq 0$, and $Q_0 +Q_1 +Q_2 =1$. The reduced resolvent is written in the form, $\tilde{R}^\pm (\omega) = \sum_{k,l=0}^{2} Q_k \tilde{R}^\pm (\omega) Q_l $. This time, we need nine matrices, \begin{equation} E_{kl}^{\pm}(\omega) =Q_k [K^\pm (\omega) -\omega]Q_l , \label{eqn:5.C.3.40} \end{equation} for $k, l = 0, 1, 2$. From the relation that $[K^\pm (\omega) -\omega]\tilde{R}^\pm (\omega) =1$, they satisfy that \begin{equation} E_{k0}^{\pm} Q_0 \tilde{R}^\pm Q_l + E_{k1}^{\pm} Q_1 \tilde{R}^\pm Q_l + E_{k2}^{\pm} Q_2 \tilde{R}^\pm Q_l = Q_k \delta_{kl}, \label{eqn:5.C.3.50} \end{equation} for $k, l = 0, 1, 2$. The asymptotic behaviors of $E_{kl}^{\pm}(\omega) $ are essentially examined in the preceding subsections, except for $E_{12}^{\pm}(\omega)$ and $E_{21}^{\pm}(\omega)$. Then, $E_{12}^{\pm}(\omega)$ becomes \begin{eqnarray} E_{12}^{\pm}(\omega) &=&-\omega \lambda^2 Q_1 A_1 Q_2 \nonumber \\ && -\lambda^2 Q_1 \bigl[A(\omega)-\omega A_1 -(\log \omega) [{\mit \Gamma}(\omega)-\omega {\mit \Gamma}_1 ] \pm \pi i [{\mit \Gamma}(\omega)-\omega {\mit \Gamma}_1 ] \bigr]Q_2 \label{eqn:5.C.3.60b}\\ &=&-\omega \lambda^2 Q_1 A_1 Q_2+O(\omega^2 \log \omega ), \label{eqn:5.C.3.60c} \end{eqnarray} where $Q_1K(0)Q_2 =0$, $Q_1 Q_2 =0$, and ${\mit \Gamma}_1 Q_2 =0$ are used. The last relation follows from the fact that $Q_2 {\mit \Gamma}_1 Q_2 =0$ and ${\mit \Gamma}_1 \geq 0$. In addition, since $Q_1 {\mit \Gamma} (\omega)Q_2=O(\omega^2)$, we see that $Q_1 A_1 Q_2=\int_0^\infty Q_1 {\mit \Gamma} (\omega) Q_2 \omega^{-2} d\omega$. By the same way, we also see that \begin{equation} E_{21}^{\pm}(\omega) =-\omega \lambda^2 Q_1 A_1 Q_2+O(\omega^2 \log \omega ). \label{eqn:5.C.3.60d} \end{equation}
To solve Eqs. (\ref{eqn:5.C.3.50}), let us now put the $N\times N$ matrix $\bf E$ as \begin{equation} {\bf E}= \left[ \begin{array}{ccc} E_{00}&E_{01}&E_{02} \\ E_{10}&E_{11}&E_{12} \\ E_{20}&E_{21}&E_{22} \end{array} \right] , \label{eqn:5.C.3.65a} \end{equation} and partition it into
${\bf A}= \left[ \begin{array}{cc} E_{00}&E_{01} \\ E_{10}&E_{11} \end{array} \right] $, ${\bf B}= \left[ \begin{array}{c} E_{02} \\ E_{12} \end{array} \right] $, ${\bf C}= \left[ \begin{array}{cc} E_{20}&E_{21} \end{array} \right] $, ${\bf D}= \left[ \begin{array}{c} E_{22} \end{array} \right]$.
Then, from the inverse matrix formula again, ${\bf E}^{-1}$ ($=\tilde{R}$) is expressed as \cite{MatrixAnalysis} \begin{equation} {\bf E}^{-1}= \left[ \begin{array}{cc} [{\bf A}-{\bf B}{\bf D}^{-1}{\bf C}]^{-1} & -{\bf A}^{-1}{\bf B}[{\bf D}-{\bf C}{\bf A}^{-1}{\bf B}]^{-1}\\ -[{\bf D}-{\bf C}{\bf A}^{-1}{\bf B}]^{-1} {\bf C} {\bf A}^{-1} &[{\bf D}-{\bf C}{\bf A}^{-1}{\bf B}]^{-1} \end{array} \right] . \label{eqn:5.C.3.67} \end{equation} The validities of ${\bf A}^{-1}$ and ${\bf D}^{-1}$ are already ensured in the exceptional cases of the first and second kinds, respectively. Then, one sees that since ${\bf A}^{-1}=O(\omega^{-1} (\log \omega)^{-1})$, ${\bf B}=O(\omega)$, ${\bf C}=O(\omega)$, and ${\bf D}^{-1}=O(\omega^{-1})$, it holds that ${\bf A}^{-1}{\bf B}{\bf D}^{-1}{\bf C}=O((\log \omega)^{-1})$. Thus, $[{\bf A}-{\bf B}{\bf D}^{-1}{\bf C}]^{-1}$ exists for small $\omega$ and $[{\bf A}-{\bf B}{\bf D}^{-1}{\bf C}]^{-1}=O(\omega^{-1}(\log \omega)^{-1})$. We also show that $[{\bf D}-{\bf C}{\bf A}^{-1}{\bf B}]^{-1}$ exists for small $\omega$ and $[{\bf D}-{\bf C}{\bf A}^{-1}{\bf B}]^{-1}=O(\omega^{-1})$. To obtain the asymptotic forms of the matrix components of ${\bf E}^{-1}$ explicitly, some redundant calculation is required; however, it could be achieved by a manner as similar to that used in the preceding subsections.
\section{Asymptotic expansion of the reduced resolvent at small $z$ } \label{sec:5.5}
We examine the small-energy behavior of the reduced resolvent only for the regular case and the exceptional case of the first kind. This analysis is crucial for determining the asymptotic behavior of the reduced time evolution operator at long times.
\subsection{The regular case}
Here, we introduce $\tilde{A}(\omega ) := \omega/\lambda^2 +A(\omega ) $ and suppose that $\tilde{A}(\omega )$ and ${\mit \Gamma}(\omega ) $ behave as \begin{equation} \tilde{A}(\omega ) := \frac{1}{\lambda^2} \omega + A(\omega ) = \sum_{n=n_a}^{n_a+N} \omega^n \tilde{A}_n +O(\omega^{n_a+N+1}), ~~~ {\mit \Gamma}(\omega ) =\sum_{n=n_b}^{n_b+N} \omega^n {\mit \Gamma}_n +O(\omega^{n_b+N+1}), \label{eqn:rff.220} \end{equation} as $\omega \to 0$, respectively, that is, $\tilde{A}_{n} =0$ for all $n< n_a$ and ${\mit \Gamma}_{n_b}=0$ for all $n< n_b$, while $\tilde{A}_{n_a}\neq0$ and ${\mit \Gamma}_{n_b}\neq0$.
Then, we obtain \begin{equation} \frac{1}{\lambda^2} \omega + {D} (\omega ) = S(0) +\tilde{A}(\omega )-(\log \omega){\mit \Gamma}(\omega) = S(0) -\omega^{n_b} \log \omega {\mit \Gamma}_{n_b} +\omega^{n_a} \tilde{A}_{n_a} + O(\omega h(\omega) ) , \label{eqn:rff.230} \end{equation} as $\omega \to +0$, where \begin{equation} h(\omega) = \left\{\begin{array}{cc} \omega^{n_b} \log \omega & (n_b \leq n_a) \\ \omega^{n_a} & (n_b > n_a) \end{array}\right. . \label{eqn:rff.245} \end{equation}
It is important to note that the values of two parameters $n_a$ and $n_b$ are not determined independently. We shall here consider $n_b$ as a controllable one. We first note that if $n_b \geq 2$ then $n_a =1$ should be concluded, because from Lemma \ref{lm:1st_order} we have $A_1 >0$, so that $\tilde{A}_1 =1/\lambda^2 +A_1 >0$ holds. Therefore, the conditions $n_b \leq n_a$ and $n_b > n_a$ can be realized only in the situations \begin{equation} n_b =1 ~\mbox{and}~ n_a \geq 1,\quad\mbox{and}\quad n_b \geq 2~ \mbox{and}~ n_a = 1, \label{eqn:rff.246} \end{equation} respectively.
\begin{lm} \label{lm:rffremainder} : Assume that $0$ is a regular point for $H$. Then the $r$-th derivative of $\tilde{R}^\pm (\omega)$ asymptotically behaves as \begin{equation}
\frac{d^r \tilde{R}^\pm (\omega)}{d\omega^r}
= \left\{ \begin{array}{cc} O(1) & (r=0)\\ O(\omega^{1-r}(\log \omega)^{\theta(1-r)}) & (r\geq1) \end{array} \right. ,
\mbox{ or }
\left\{ \begin{array}{cc} O(1) & (r=0)\\
O(\omega^{[1-r]^+}) &(1\leq r < n_b) \\
O(\omega^{n_b-r} (\log \omega)^{\theta(n_b-r)}) &(n_b\leq r \leq 2n_b)
\end{array} \right. , \label{eqn:rffSmall3} \end{equation} for $n_b =1$, or $n_b \geq 2$,
respectively, as $\omega \to 0$, where $[x]^+ =\max \{x, 0 \}$ and $\theta(x)=1$ for $x\geq 0$ or $0$ for $x<0$. In addition, the $r$-th derivative of $\tilde{R}^\pm (\omega)$ is approximated by that of a finite series \begin{equation} (K(0))^{-1} + (K(0))^{-1} \left[ -\omega^{n_b} (\log \omega) \lambda^2 {\mit \Gamma}_{n_b} +\omega^{n_a} \lambda^2 \tilde{A}_{n_a} \pm \lambda^2 \pi i \omega^{n_b} {\mit \Gamma}_{n_b} \right] (K(0))^{-1} , \label{eqn:rffSmall5} \end{equation} that is, it is shown that \begin{eqnarray} &&
\Biggl\| \frac{d^r}{d\omega^r} \biggl\{ \tilde{R}^\pm (\omega) \nonumber \\ && - (K(0))^{-1} - (K(0))^{-1} \left[ -\omega^{n_b} (\log \omega) \lambda^2 {\mit \Gamma}_{n_b} +\omega^{n_a} \lambda^2 \tilde{A}_{n_a} \pm \lambda^2 \pi i \omega^{n_b} {\mit \Gamma}_{n_b} \right] (K(0))^{-1} \biggr\}
\Biggr\| \nonumber \\ &=& \begin{array}{cc} O(\omega^{2-r} (\log \omega)^{1+\theta(2-r)}) & (r \geq 0) \end{array}
\nonumber \\ && \mbox{ or } \left\{ \begin{array}{cc} O(\omega^{[2-r]^+}) & (0\leq r\leq n_b) \\ O(\omega^{n_b+1-r} (\log \omega)^{\theta(n_b+1-r)}) & (n_b+1 \leq r \leq 2n_b) \end{array} \right. ,
\label{eqn:rffSmall7} \end{eqnarray} for $n_b =1$, or $n_b \geq 2$,
respectively, as $\omega \to 0$. Here, $n_a$ is restricted to the condition (\ref{eqn:rff.246}). \end{lm}
{\sl Proof} : The left-hand side (lhs) of Eq. (\ref{eqn:rffSmall7}) is written as follows: \begin{eqnarray} &&\hspace*{-10mm}
\Biggl\| \frac{d^r}{d\omega^r} \biggl\{ \tilde{R}^\pm (\omega) \nonumber \\ &&\hspace*{-10mm} - (K(0))^{-1} - (K(0))^{-1} \left[ -\omega^{n_b} (\log \omega) \lambda^2 {\mit \Gamma}_{n_b} +\omega^{n_a} \lambda^2 \tilde{A}_{n_a} \pm \lambda^2 \pi i \omega^{n_b} {\mit \Gamma}_{n_b} \right] (K(0))^{-1} \biggr\}
\Biggr\| \nonumber \\ &&\hspace*{-10mm} \leq
\left\| \frac{d^r}{d\omega^r} \bigl\{ \tilde{R}^\pm (\omega)-S_1 (\omega) \bigr\}
\right\| \nonumber \\ &&\hspace*{-10mm} ~~~+
\Biggl\| \frac{d^r}{d\omega^r} \biggl\{ S_1 (\omega) \nonumber \\ &&\hspace*{-10mm} ~~~- (K(0))^{-1} - (K(0))^{-1} \left[ -\omega^{n_b} (\log \omega) \lambda^2 {\mit \Gamma}_{n_b} +\omega^{n_a} \lambda^2 \tilde{A}_{n_a} \pm \lambda^2 \pi i \omega^{n_b} {\mit \Gamma}_{n_b} \right] (K(0))^{-1} \bigg\}
\Biggr\| , \label{eqn:rffSmall50} \end{eqnarray} where $S_N(\omega)$ is defined by Eq. (\ref{eqn:.185}). When $r=0$, the first term on the rhs of the above is estimated from the special case of the below for $N=1$, \begin{equation}
\left\| \tilde{R}^\pm (\omega)-S_N (\omega)
\right\| \! \leq \! \frac{
\| \omega +\lambda^2 {D} (\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) -\lambda^2 S(0)
\|^{N+1}
\| (K(0))^{-1}
\|^{N+2} } {
1-\|(K(0))^{-1}\|
\| [\omega +\lambda^2 {D} (\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) -\lambda^2 S(0)
]\| } =O(h(\omega)^{N+1} ), \label{eqn:rffSmall60c} \end{equation} as $\omega \to 0$. When $r\geq 1$, instead we have \begin{eqnarray} &&
\left\| \frac{d^r}{d\omega^r} \bigl\{ \tilde{R}^\pm (\omega) - S_1 (\omega) \bigr\}
\right\| \nonumber \\
&\leq&
\left\| \sum_{j=2}^{r} {\sum_{\{s_i \}_{i=1}^{j} }}' a^{(r)} (\{s_i \}_{i=1}^{j} ) \left\{ \prod_{i=1}^{j} \tilde{R}^{\pm}(\omega ) \frac{d^{s_i}}{d\omega^{s_i}} \left[ \omega + \lambda^2 {D}(\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) \right] \right\} \tilde{R}^{\pm}(\omega )
\right\| \nonumber \\ && +
\left\| \tilde{R}^{\pm}(\omega ) \frac{d^r }{d\omega^r} \left[ \omega + \lambda^2 {D}(\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) \right] \tilde{R}^{\pm}(\omega ) - \frac{d^r}{d\omega^r} S_1 (\omega)
\right\| , \label{eqn:rffSmall70b} \end{eqnarray} where Eq. (\ref{eqn:Large100}) is used, and here $s_i \geq 1$ and $\sum_{i=1}^{j} s_i =r$ should be satisfied. Note that the first term on the rhs of Eq. (\ref{eqn:rffSmall70b}) appears only for $r\geq 2$, which is estimated in the following. In the following estimations, we temporarily forget the restriction (\ref{eqn:rff.246})
and consider the two general cases: $n_b \leq n_a$ and $n_b > n_a$. In the case of $n_b \leq n_a$, we can obtain for $r\geq 2$ \begin{eqnarray} &&
\left\| \sum_{j=2}^{r} {\sum_{\{s_i \}_{i=1}^{j} }}' a^{(r)} (\{s_i \}_{i=1}^{j} ) \left\{ \prod_{i=1}^{j} \tilde{R}^{\pm}(\omega ) \frac{d^{s_i}}{d\omega^{s_i}} \left[ \omega + \lambda^2 {D}(\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) \right] \right\} \tilde{R}^{\pm}(\omega )
\right\| \nonumber \\ &&\leq
\sum_{j=2}^{r} {\sum_{\{s_i \}_{i=1}^{j} }}' a^{(r)} (\{s_i \}_{i=1}^{j} )
\left\| \tilde{R}^{\pm}(\omega )
\right\|^{j+1} O(\omega^{jn_b-r}) \prod_{i=1}^{j} O((\log \omega)^{\theta(n_b-s_i)}) \nonumber \\ &&= O(\omega^{2n_b-r} (\log \omega)^{\theta(n_b+1-r)+\theta(2n_b+1-r)}), \label{eqn:rffSmall80a} \end{eqnarray} as $\omega \to 0$.
For $n_a < n_b$, \begin{eqnarray} &&
\left\| \sum_{j=2}^{r} {\sum_{\{s_i \}_{i=1}^{j} }}' a^{(r)} (\{s_i \}_{i=1}^{j} ) \tilde{R}^{\pm}(\omega ) \prod_{i=1}^{j} \left\{ \frac{d^{s_i}}{d\omega^{s_i}} \left[ \omega + \lambda^2 {D}(\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) \right] \tilde{R}^{\pm}(\omega ) \right\}
\right\| \nonumber \\ &&= \left\{ \begin{array}{cc} O(\omega^{[2n_a-r]^+}) &(2\leq r\leq n_a+n_b-1) \\ O(\omega^{n_a+n_b-r} (\log \omega)^{\theta(n_a+n_b-r)}) &(n_a+n_b\leq r \leq 2n_b) \end{array} \right. , \label{eqn:rffSmall80b} \end{eqnarray} as $\omega \to 0$. We here used that \begin{eqnarray} && \frac{d^r}{d\omega^r } [ \omega+\lambda^2 {D}(\omega )\pm \lambda^2 \pi i {\mit \Gamma}(\omega ) -\lambda^2 S(0) ] \nonumber\\ && = O(\omega^{n_b-r} (\log \omega)^{\theta(n_b-r)})
, \mbox{ or }
\left\{ \begin{array}{cc} O(\omega^{[n_a-r]^+}) & (0\leq r< n_b) \\ O(\omega^{n_b-r} (\log \omega)^{\theta(n_b-r)}) & (r\geq n_b) \end{array} \right.
, \label{eqn:rffSmall98b} \end{eqnarray} for $n_b\leq n_a$, or $n_a< n_b$, respectively, as $\omega \to 0$. Eq. (\ref{eqn:rffSmall98b}) follows from \begin{equation} \frac{d^r \tilde{A}(\omega )}{d\omega^r }
=O(\omega^{[n_a-r]^+}) , ~~~ \frac{d^r {\mit \Gamma}(\omega )}{d\omega^r }
=O(\omega^{[n_b-r]^+}) ,~~~
\frac{d^r (\log \omega) {\mit \Gamma}(\omega )}{d\omega^r }
=O(\omega^{n_b-r} (\log \omega)^{\theta(n_b-r)}) , \label{eqn:rffSmall90b} \end{equation} as $\omega \to 0$. Incorporating Eqs. (\ref{eqn:rffSmall80a}), (\ref{eqn:rffSmall80b}), and Eq. (\ref{eqn:rffSmall98b}), with \begin{eqnarray} &&
\left\| \frac{d^r \tilde{R}^\pm (\omega)}{d\omega^r}
\right\| \nonumber\\ &\leq&
\left\| \sum_{j=2}^{r} {\sum_{\{s_i \}_{i=1}^{j} }}' a^{(r)} (\{s_i \}_{i=1}^{j} ) \left\{ \tilde{R}^{\pm}(\omega ) \prod_{i=1}^{j} \frac{d^{s_i}}{d\omega^{s_i}} \left[ \omega + \lambda^2 {D}(\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) \right] \right\} \tilde{R}^{\pm}(\omega )
\right\| \nonumber \\ && +
\left\| \tilde{R}^{\pm}(\omega ) \frac{d^r }{d\omega^r} \left[ \omega + \lambda^2 {D}(\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) \right] \tilde{R}^{\pm}(\omega )
\right\| , \label{eqn:rffSmall99b} \end{eqnarray} we have \begin{eqnarray} &&\hspace*{-10mm}
\left\| \frac{d^r \tilde{R}^\pm (\omega)}{d\omega^r}
\right\| \nonumber\\ &&\hspace*{-10mm} = \left\{ \begin{array}{cc} O(1) & (r=0) \\ O(\omega^{n_b-r}(\log \omega)^{\theta(n_b-r)})) & (r\geq 1 ) \end{array} \right. \mbox{ or } \left\{ \begin{array}{cc} O(1) & (r=0) \\ O(\omega^{[n_a-r]^+}) & (1\leq r< n_b) \\ O(\omega^{n_b-r} (\log \omega)^{\theta(n_b-r)}) & (n_b \leq r \leq 2n_b) \end{array} \right. , \label{eqn:rffSmall99c} \end{eqnarray} for $n_b\leq n_a$, or $n_a< n_b$, respectively, as $\omega \to 0$. Then, the first part of the statement can be shown under the restriction (\ref{eqn:rff.246}).
Let us next examine the second term on the rhs of Eq. (\ref{eqn:rffSmall70b}), which reads for $r\geq 1$, \begin{eqnarray} &&
\Biggl\| \tilde{R}^{\pm}(\omega ) \frac{d^r }{d\omega^r} \left[ \omega + \lambda^2 {D}(\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) \right] \tilde{R}^{\pm}(\omega ) - \frac{d^r }{d\omega^r} S_1(\omega)
\Biggr\| \nonumber \\
&&\leq 2 \left\| \tilde{R}^{\pm}(\omega ) -(K(0))^{-1}
\right\|
\left\| \frac{d^r }{d\omega^r} \left[ \omega + \lambda^2 {D}(\omega ) \pm \lambda^2 \pi i {\mit \Gamma}(\omega ) \right]
\right\|
\left\| \tilde{R}^{\pm}(\omega )
\right\| \nonumber \\ &&= O(\omega^{2n_b-r} (\log \omega)^{1+\theta(n_b-r)}), \mbox{ or } \left\{ \begin{array}{cc} O(\omega^{n_a+[n_a-r]^+}) & (r< n_b) \\ O(\omega^{n_a+n_b-r} (\log \omega)^{\theta(n_b-r)}) & (r\geq n_b) \end{array} \right. , \label{eqn:rffSmall100} \end{eqnarray} for $n_b \leq n_a$ or $n_b > n_a$ respectively, as $\omega \to 0$. We here used Eq. (\ref{eqn:rffSmall60c}) with $N=0$. Therefore, substituting Eqs. (\ref{eqn:rffSmall80a}), (\ref{eqn:rffSmall80b}), and (\ref{eqn:rffSmall100}) into Eq. (\ref{eqn:rffSmall70b}), one has for $r\geq 1$, \begin{eqnarray} &&\hspace*{-12mm}
\left\| \frac{d^r}{d\omega^r} \bigr\{ \tilde{R}^\pm (\omega)- S_1 (\omega) \bigr\}
\right\| \nonumber\\ &&\hspace*{-12mm} =O(\omega^{2n_b-r} (\log \omega)^{1+\theta(n_b+1-r)}) , \mbox{or} \left\{ \begin{array}{cc} O(\omega^{[2n_a-r]^+}) &(r\leq n_a+n_b-1) \\ O(\omega^{n_a+n_b-r} (\log \omega)^{\theta(n_a+n_b-r)}) &(n_a+n_b\leq r \leq 2n_b) \end{array} \right. \label{eqn:rffSmall110} \end{eqnarray} for $n_b \leq n_a$ or $n_b > n_a$, respectively, as $\omega \to 0$. Note that this estimation is also valid for $r=0$ because it reproduces Eq. (\ref{eqn:rffSmall60c}) for $N=1$.
Let us now evaluate the last term in Eq. (\ref{eqn:rffSmall50}).
For $r\geq 0$, we have \begin{eqnarray} &&
\Biggl\| \frac{d^r}{d\omega^r} \biggr\{ S_1 (\omega) \nonumber\\ && - (K(0))^{-1} - (K(0))^{-1} \left[ -\omega^{n_b} (\log \omega) \lambda^2 {\mit \Gamma}_{n_b} +\omega^{n_a} \lambda^2 \tilde{A}_{n_a} \pm \lambda^2 \pi i \omega^{n_b} {\mit \Gamma}_{n_b} \right] (K(0))^{-1} \bigg\}
\Biggr\| \nonumber \\ &&\leq
\left\| (K(0))^{-1}
\right\|^2 \nonumber \\ && ~~~\times
\left\| \frac{d^r }{d\omega^r} \left[ -(\log \omega) \lambda^2 ({\mit \Gamma}(\omega)- \omega^{n_b} {\mit \Gamma}_{n_b} ) +\lambda^2 (\tilde{A}(\omega)- \omega^{n_a} \tilde{A}_{n_a} )) \pm \lambda^2 \pi i \left( {\mit \Gamma}(\omega)- \omega^{n_b} {\mit \Gamma}_{n_b} \right) \right]
\right\| \nonumber \\
&&= O(\omega^{n_b+1-r} (\log \omega)^{\theta(n_b+1-r)}), \mbox{ or } \left\{ \begin{array}{cc} O(\omega^{[n_a+1-r]^+}) & (0\leq r\leq n_b) \\ O(\omega^{n_b+1-r} (\log \omega)^{\theta(n_b+1-r)}) & (r\geq n_b+1) \end{array} \right. , \label{eqn:rffSmall130b} \end{eqnarray} for $n_b \leq n_a$ or $n_b > n_a$ respectively, as $\omega \to 0$ for any $r\geq 0$.
We here used that for $r\geq 0$, \begin{equation} \frac{d^r ({\mit \Gamma}(\omega)- \omega^{n_b} {\mit \Gamma}_{n_b})}{d\omega^r} =O(\omega^{[n_b+1-r]^+}) ,~~
\frac{d^r (\log \omega) ({\mit \Gamma}(\omega)- \omega^{n_b} {\mit \Gamma}_{n_b})}{d\omega^r } = O(\omega^{n_b+1-r} (\log \omega)^{\theta(n_b+1-r)}) \label{eqn:rffSmall140} \end{equation} as $\omega \to 0$, and so forth.
Thus, setting Eqs. (\ref{eqn:rffSmall110}) and (\ref{eqn:rffSmall130b}) into Eq. (\ref{eqn:rffSmall50}), we conclude that \begin{eqnarray} &&
\Biggl\| \frac{d^r}{d\omega^r} \biggl\{ \tilde{R}^\pm (\omega) \nonumber\\ && - (K(0))^{-1} - (K(0))^{-1} \left[ -\omega^{n_b} (\log \omega) \lambda^2 {\mit \Gamma}_{n_b} +\omega^{n_a} \lambda^2 \tilde{A}_{n_a} \pm \lambda^2 \pi i \omega^{n_b} {\mit \Gamma}_{n_b} \right] (K(0))^{-1} \biggr\}
\Biggr\| \nonumber \\ && = \left\{ \begin{array}{cc} O(\omega^{2n_b-r} (\log \omega)^{1+\theta(n_b+1-r)}) & (r \geq 0, n_b=1) \\ O(\omega^{n_b+1-r} (\log \omega)^{\theta(n_b+1-r)}) & (r \geq 0, n_b\geq2) \end{array} \right. , \nonumber\\ && ~~~ \mbox{ or } \left\{ \begin{array}{cc} O(\omega^{[n_a+1-r]^+}) & (0\leq r\leq n_b) \\ O(\omega^{n_b+1-r} (\log \omega)^{\theta(n_b+1-r)}) & (n_b+1 \leq r \leq 2n_b) \end{array} \right. , \label{eqn:rffSmall60} \end{eqnarray} for $n_b \leq n_a$ or $n_b > n_a$, respectively, as $\omega \to 0$. By taking into account the restriction (\ref{eqn:rff.246}), we can show the last part of the lemma. \qed
To estimate the long time behavior of the reduced time evolution operator, the above-mentioned lemma seems not precisely appropriate because the reduced time evolution operator is obtained from the Fourier transform of the imaginary part of $\tilde{R}^+ (\omega)$, not from $\tilde{R}^+ (\omega)$ itself, which is explained in the next section. Hence, the following lemma is more appropriate for our purpose.
\begin{lm} \label{lm:rffimremainder} : Assume that $0$ is a regular point for $H$. Then the $r$-th derivative of ${\rm Im}\tilde{R}^+ (\omega):=(\tilde{R}^+ (\omega) -\tilde{R}^- (\omega))/2i$ is approximated by that of \begin{eqnarray} (K(0))^{-1} \lambda^2 \pi \omega^{n_b} ({\mit \Gamma}_{n_b} +\omega {\mit \Gamma}_{n_b +1} +\omega^2 {\mit \Gamma}_{n_b +2}) (K(0))^{-1} , \label{eqn:rffimSmall5} \end{eqnarray} in the sense that for $0\leq r \leq n_b+1$ the remainder is estimated as, \begin{eqnarray} &&
\left\| \frac{d^r}{d\omega^r} \bigl\{ {\rm Im}\tilde{R}^+ (\omega)- (K(0))^{-1} \lambda^2 \pi \omega^{n_b} ({\mit \Gamma}_{n_b} +\omega {\mit \Gamma}_{n_b +1} +\omega^2 {\mit \Gamma}_{n_b +2}) (K(0))^{-1} \bigr\}
\right\| \nonumber\\ && = O(\omega^{2-r}\log \omega)
, ~\mbox{or} ~ O(\omega^{1+n_b-r})
, \label{eqn:rffimSmall60} \end{eqnarray} for $n_b=1$, or $n_b \geq 2$, respectively, as $\omega \to 0$.
For $r=n_b +2$, the estimation is replaced by $O(\omega^{-1})$ for $n_b=1$, $O(\log \omega)$ for $n_b = 2$, or $O(1)$ for $n_b \geq 3$, respectively, as $\omega \to 0$. \end{lm}
{\sl Proof} : Since ${\rm Im}\tilde{R}^+ (\omega)=\lambda^2 \pi \tilde{R}^+ (\omega) {\mit \Gamma}(\omega) \tilde{R}^- (\omega)$, one has
\begin{eqnarray} &&
\left\| \frac{d^r}{d\omega^r} \bigl\{ {\rm Im}\tilde{R}^+ (\omega) - (K(0))^{-1} \lambda^2 \pi \omega^{n_b} ({\mit \Gamma}_{n_b} +\omega {\mit \Gamma}_{n_b +1} +\omega^2 {\mit \Gamma}_{n_b +2}) (K(0))^{-1} \bigr\}
\right\| \label{eqn:rffimSmall40a} \\ &\leq& \lambda^2 \pi \sum_{\scriptsize \begin{array}{c} s\geq0, t\geq0, u\geq0, \\ (s+t+u=r) \end{array}}^{r} [F_1(s,t,u)+F_2(s,t,u)+F_3(s,t,u)] , \label{eqn:rffimSmall40c} \end{eqnarray} with \begin{eqnarray} &\displaystyle F_1(s,t,u)=C_{stu}
\left\| \frac{d^s \tilde{R}^+ (\omega)}{d\omega^s}
\right\|
\left\| \frac{d^t {\mit \Gamma}(\omega)}{d\omega^t}
\right\|
\left\| \frac{d^u}{d\omega^u} \bigl\{ \tilde{R}^- (\omega)-(K(0))^{-1} \bigr\}
\right\| ,& \label{eqn:rffimSmall41a}\\
&\displaystyle F_2(s,t,u)=C_{stu}
\left\| \frac{d^s \tilde{R}^+ (\omega)}{d\omega^s}
\right\|
\left\| \frac{d^t}{d\omega^t} \bigl\{ {\mit \Gamma}(\omega) - \omega^{n_b}({\mit \Gamma}_{n_b} +\omega {\mit \Gamma}_{n_b +1} +\omega^2 {\mit \Gamma}_{n_b +2}) \bigr\}
\right\| & \nonumber\\ &\hspace*{-57mm} \displaystyle \times
\left\| \frac{d^u(K(0))^{-1}}{d\omega^u}
\right\|,& \label{eqn:rffimSmall41b}\\
&\displaystyle F_3(s,t,u)=C_{stu}
\left\| \frac{d^s}{d\omega^s} \bigl\{ \tilde{R}^+ (\omega) -(K(0))^{-1} \bigr\}
\right\|
\left\| \frac{d^t}{d\omega^t}\omega^{n_b} ({\mit \Gamma}_{n_b} +\omega {\mit \Gamma}_{n_b +1} +\omega^2 {\mit \Gamma}_{n_b +2})
\right\| & \nonumber\\ &\hspace*{-67mm} \displaystyle \times
\left\| \frac{d^u(K(0))^{-1}}{d\omega^u}
\right\| , & \label{eqn:rffimSmall41c} \end{eqnarray} where $C_{stu}$'s are appropriate constants.
For $1\leq r\leq n_b +1$, the summation of the first summand in Eq. (\ref{eqn:rffimSmall40c}) can be estimated as \begin{eqnarray} && \sum_{\scriptsize \begin{array}{c} s\geq0, t\geq0, u\geq0, \\ (s+t+u=r) \end{array}}^{r} F_1(s,t,u) \nonumber\\ && = F_1(0,r,0) + \sum_{t\geq 0,u\geq 1}^r F_1(0,t,u) + \sum_{s\geq 1,t\geq 0}^r F_1(s,t,0) + \sum_{\min\{ s, u\}\geq 1}^r F_1(s,t,u) \nonumber \\ && = O(F_1(0, r-1, 1)) = O(\omega^{2n_b -r} \log \omega ) ,~~ \mbox{or}~~ O(\omega^{n_a+n_b -r}) , \label{eqn:rffimSmall47a} \end{eqnarray} for $n_b \leq n_a$ or $n_b > n_a$, respectively, as $\omega \to 0$. Note that from Eq. (\ref{eqn:rffSmall60c}) for $N=0$ this estimation is valid for $r=0$ too. For $r=n_b+2$, it is estimated as \begin{equation} \left\{ \begin{array}{cc} O(\omega^{n_b-1}\log \omega) & (n_b\geq 2)\\ O((\log \omega)^2 ) & (n_b=1) \end{array} \right. ,~~ \mbox{or}~~ O(\omega^{n_a-1}) , \label{eqn:rffimSmall47b} \end{equation} for $n_b \leq n_a$ or $n_b > n_a$, respectively, as $\omega \to 0$.
The summation of the second summand in Eq. (\ref{eqn:rffimSmall40c}) for $0\leq r\leq n_b +3$ is also estimated as \begin{equation} \lambda^2 \pi \sum_{\scriptsize \begin{array}{c} s\geq0, t\geq0, u\geq0, \\ (s+t+u=r) \end{array}}^{r} F_2 (s,t,u) = \sum_{(s+t=r)}^r F_2(s, t, 0) = O(F_2(0, r, 0)) = O(\omega^{n_b +3 -r}), \label{eqn:rffimSmall47c} \end{equation} both for $n_b \leq n_a$ and for $n_b > n_a$, as $\omega \to 0$. The summation of the last summand in Eq. (\ref{eqn:rffimSmall40c}) for $0\leq r\leq n_b +1$ is estimated as \begin{eqnarray} \lambda^2 \pi \sum_{\scriptsize \begin{array}{c} s\geq0, t\geq0, u\geq0, \\ (s+t+u=r) \end{array}}^{r} F_3 (s,t,0) &=& \sum_{(s+t=r)}^r F_3(s, t, 0) = O(F_3(1, r-1, 0)) \nonumber\\ &=& O(\omega^{2n_b -r } \log \omega), ~ \mbox{or} ~ O(\omega^{n_a +n_b -r}), \label{eqn:rffimSmall47d} \end{eqnarray} for $n_b \leq n_a$ or $n_a < n_b $, respectively, as $\omega \to 0$. For $r=n_b +2$, the estimation is replaced by \begin{equation} O(\omega^{n_b -2}(\log \omega)^{\theta(n_b-2)}) , \mbox{ or } ~ \left\{ \begin{array}{cc} O(\omega^{[n_a -2]^+}) & (n_b \geq 3)\\ O(\log \omega) & (n_b = 2) \end{array} \right. , \label{eqn:rffimSmall50a} \end{equation} for $n_b \leq n_a$ or $n_a < n_b $, respectively, as $\omega \to 0$. Then, by summarizing the above-noted estimations from Eqs. (\ref{eqn:rffimSmall47a}) to (\ref{eqn:rffimSmall50a}), and by taking into account the restriction (\ref{eqn:rff.246}) again, the proof of the lemma is completed. \qed
\subsection{The exceptional case of the first kind}
In this case, we first remember that from the discussion around Eq. (\ref{eqn:.340}) it necessarily holds that ${\mit \Gamma}_1\neq 0$, i.e., $n_b=1$ in Eqs. (\ref{eqn:rff.220}).
\begin{lm} \label{lm:1stremainder+} : Assume that $0$ is an exceptional point of the first kind for $H$. Then the $0$-th and the first derivative of $\tilde{R} (z)$ are approximated by those of a finite series \begin{equation} \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} + \frac{1}{\lambda^4 z (\log z)^2}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} , \label{eqn:1stSmall5} \end{equation} that is, it is shown that \begin{eqnarray} &&
\Biggl\| \frac{d^r}{dz^r} \Biggl[ \tilde{R} (z) - \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber \\ && - \frac{1}{\lambda^4 z (\log z)^2}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \Biggr]
\Biggr\| \nonumber \\ &&=
O(z^{-1} (\log z)^{-3}) ~~\mbox{for}~~r=0,~~~\mbox{or}~~~
O(z^{-2} (\log z)^{-3}) ~~\mbox{for}~~r=1,
\label{eqn:1stSmall7} \end{eqnarray} as $z \to 0$. \end{lm}
{\sl Proof} : Let us first consider the quantity that \begin{eqnarray} &&
\Biggl\| \frac{d^r}{dz^r} \Biggl[ \tilde{R} (z) - \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber \\ && - \frac{1}{\lambda^4 z (\log z)^2}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \Biggr]
\Biggr\| \nonumber \\ &&\leq
\left\| \frac{d^r }{dz^r} \left[ \tilde{R} (z) - Q_1 \tilde{R} (z) Q_1 \right]
\right\|
+
\left\| \frac{d^r}{dz^r} \Biggl[ Q_1 \tilde{R} (z) Q_1 - \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \right. \nonumber \\ &&~~~ \left. - \frac{1}{\lambda^4 z (\log z)^2}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \Biggr]
\right\| . \label{eqn:1stSmall50} \end{eqnarray} For $r= 0$, the first term on the rhs of the above is estimated as follows:
\begin{equation}
\left\| \tilde{R} (z)-Q_1 \tilde{R} (z) Q_1
\right\| \leq
\left\| Q_0 \tilde{R} (z) Q_0
\right\| +
\left\| Q_0 \tilde{R} (z) Q_1
\right\| +
\left\| Q_1 \tilde{R} (z) Q_0
\right\| =O(1), \label{eqn:1stSmall60} \end{equation} as $z \to 0$, where Eqs. (\ref{eqn:5.C.1.120a})--(\ref{eqn:5.C.1.120c}) are used. For $r=1$, one obtains \begin{equation}
\left\| \frac{d\tilde{R} (z)}{dz} - \frac{dQ_1 \tilde{R} (z) Q_1}{dz}
\right\| \leq
\left\| \frac{dQ_0 \tilde{R} (z) Q_0}{dz}
\right\| +
\left\| \frac{dQ_0 \tilde{R} (z) Q_1}{dz}
\right\| +
\left\| \frac{dQ_1 \tilde{R} (z) Q_0}{dz}
\right\| =O(z^{-1}). \label{eqn:1stSmall70} \end{equation} In fact, by using expression (\ref{eqn:5.C.1.120a}) the first term on the rhs of the above is estimated as follows: \begin{eqnarray} \frac{d Q_0 \tilde{R} (z) Q_0}{dz} &=& - Q_0 \tilde{R} (z) Q_0 \left( \frac{d E_{00}}{dz} -\frac{d E_{01}}{dz} E_{11}^{-1} E_{10} -E_{01} \frac{dE_{11}^{-1} }{dz} E_{10} -E_{01} E_{11}^{-1} \frac{dE_{10}}{dz} \right) \nonumber \\ && \times Q_0 \tilde{R} (z) Q_0 \label{eqn:1stSmall71b}\\ &=& O(\log z). \label{eqn:1stSmall71c} \end{eqnarray} Four derivatives in Eq. (\ref{eqn:1stSmall71b}) have the same order, which can be shown from the use of Eqs. (\ref{eqn:5.C.1.60b}), (\ref{eqn:5.C.1.90b}), (\ref{eqn:5.C.1.100}), and (\ref{eqn:5.C.1.130}): We here note that \begin{equation} \frac{dE_{00}}{dz} =O(\log z),~~~ \frac{dE_{01}}{dz} =O(\log z),~~~ \frac{dE_{10}}{dz} =O(\log z),~~~ \label{eqn:1stSmall72} \end{equation} and from Eqs. (\ref{eqn:5.C.1.60b}) and (\ref{eqn:5.C.1.90b}) \begin{eqnarray} \frac{dE_{11}^{-1} }{dz} &=& E_{11}^{-1} \left\{ \frac{d}{dz} Q_1 \left\{ z +\lambda^2 [ A(z) -\log z {\mit \Gamma}(z) + \pi i {\mit \Gamma}(z) ] \right\} Q_1 \right\} E_{11}^{-1} \label{eqn:1stSmall73a}\\ &=& E_{11}^{-1} Q_1 \left\{ 1 +\lambda^2 \left[ \frac{dA(z)}{dz}-{\mit \Gamma}(z)/z -\log z \frac{d{\mit \Gamma}(z)}{dz} + \pi i \frac{d{\mit \Gamma}(z)}{dz} \right] \right\} Q_1 E_{11}^{-1} \label{eqn:1stSmall73b}\\ &=& O(z^{-2} (\log z)^{-1} ). \label{eqn:1stSmall73c} \end{eqnarray} In the same way, the second term on the rhs of Eq. (\ref{eqn:1stSmall70}) is also estimated as follows: \begin{eqnarray} \frac{d Q_0 \tilde{R} (z) Q_1}{dz} &=& -\left( \frac{dE_{00}^{-1}}{dz} E_{01} + E_{00}^{-1} \frac{dE_{01}}{dz} \right) Q_1\tilde{R}Q_1 + E_{00}^{-1} E_{01} Q_1\tilde{R}Q_1 \nonumber \\ && \times \left( \frac{d E_{11}}{dz} -\frac{d E_{10}}{dz} E_{00}^{-1} E_{01} -E_{10} \frac{dE_{00}^{-1} }{dz} E_{01} -E_{10} E_{00}^{-1} \frac{dE_{01}}{dz} \right) Q_1\tilde{R}Q_1 \label{eqn:1stSmall74b}\\ &=&
O(z^{-1}), \label{eqn:1stSmall74c} \end{eqnarray} where we used Eq. (\ref{eqn:5.C.1.120b}) and the fact that \begin{equation} \frac{dE_{00}^{-1} }{dz}=O(\log z),~~~ \frac{dE_{11}}{dz}=O(\log z). \label{eqn:1stSmall75} \end{equation} In a similar manner, we can also show \begin{equation} \frac{d Q_1 \tilde{R} (z) Q_0}{dz}=O(z^{-1}). \label{eqn:1stSmall76} \end{equation}
Let us next consider the last term in Eq. (\ref{eqn:1stSmall50}). For $r=0$, it reads \begin{eqnarray} &&
\left\| Q_1 \tilde{R} (z) Q_1 - \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \right. \nonumber \\ && \left. - \frac{1}{\lambda^4 z (\log z)^2}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\right\| \label{eqn:1stSmall77a}\\
&\leq&
\left\| E_{11}^{-1} - \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \right. \nonumber \\ && \left. - \frac{1}{\lambda^4 z (\log z)^2}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\right\| \nonumber \\ &&
+\| (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}E_{11}^{-1} -E_{11}^{-1} \| \label{eqn:1stSmall77c}\\ &\leq&
\biggl\| \tilde{E}_{11}(z)\frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber \\ && - \frac{1}{\lambda^4 z (\log z)^2}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\| \nonumber \\ && +
\Biggl\| \sum_{j=2}^{\infty} (\tilde{E}_{11}(z))^j \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\Biggr\| + O(1) \label{eqn:1stSmall77d}\\
&=& O( z^{-1} (\log z )^{-3} ), \label{eqn:1stSmall77f} \end{eqnarray} where in the second inequality we used Eq. (\ref{eqn:5.C.1.90a}) and that
\begin{equation}
\| (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}E_{11}^{-1} -E_{11}^{-1} \|
\leq
\frac{\| E_{11}^{-1} E_{10} E_{00}^{-1} E_{01} \|
\| E_{11}^{-1} \|}{1-\| E_{11}^{-1} E_{10} E_{00}^{-1} E_{01} \| } =O(1), \label{eqn:1stSmall78} \end{equation} as $z \to 0$. Substituting Eqs. (\ref{eqn:1stSmall60}) and (\ref{eqn:1stSmall77f}) into Eq. (\ref{eqn:1stSmall50}), we can obtain the estimation (\ref{eqn:1stSmall7}) for $r=0$.
For $r=1$, we can obtain \begin{eqnarray} &&
\Biggl\| \frac{d}{dz} \biggl[ Q_1 \tilde{R} (z) Q_1 - \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber \\ && - \frac{1}{\lambda^4 z (\log z)^2}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \biggr]
\Biggr\| \label{eqn:1stSmall80a}\\ &\leq&
\Biggl\| \frac{d}{dz} \biggl[ E_{11}^{-1} - \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber \\ && - \frac{1}{\lambda^4 z (\log z)^2}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \biggr]
\Biggr\| \nonumber \\ &&
+\biggl\| \frac{d}{dz} \bigl\{ (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}E_{11}^{-1} -E_{11}^{-1} \bigr\}
\biggr\| \label{eqn:1stSmall80b}\\
&\leq&
\biggl\| E_{11}^{-1} \left\{ \frac{d}{dz} Q_1 \left\{ z +\lambda^2 [ A(z) + \pi i {\mit \Gamma}(z) ] \right\} Q_1 \right\} E_{11}^{-1} \nonumber \\ && - \frac{1}{\lambda^4 z^2 (\log z)^2}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\| \nonumber \\ && +
\biggl\| -E_{11}^{-1} \biggl\{ \frac{d}{dz} \lambda^2 (\log z) Q_1 {\mit \Gamma}(z) Q_1 \biggr\} E_{11}^{-1} -\frac{d}{dz} \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber \\ && - z \biggl\{ \frac{d}{dz} \frac{1}{\lambda^4 z^2 (\log z)^2} \biggr\}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\| \nonumber \\ &&
+\biggl\| \bigl\{ (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1} -Q_{11} \bigr\} \frac{dE_{11}^{-1}}{dz}
\biggr\| \nonumber \\ && +
\biggl\| \left\{ \frac{d}{dz} (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1} \right\} E_{11}^{-1}
\biggr\| \label{eqn:1stSmall80d}\\ &=&O(z^{-2}(\log z)^{-3}), \label{eqn:1stSmall80e} \end{eqnarray} where we used the expression for $dE_{11}^{-1}/dz$ in Eq. (\ref{eqn:1stSmall73a}).
Actually, the first term in Eq. (\ref{eqn:1stSmall80d}) is estimated as \begin{eqnarray} &&\hspace*{-13mm}
\biggl\| E_{11}^{-1} \left\{ \frac{d}{dz} Q_1 \left\{ z +\lambda^2 [ A(z) + \pi i {\mit \Gamma}(z) ] \right\} Q_1 \right\} E_{11}^{-1} \nonumber \\ &&\hspace*{-13mm} - \frac{1}{\lambda^4 z^2 (\log z)^2}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\| \nonumber \\ &&\hspace*{-13mm} \leq
\biggl\| E_{11}^{-1}-\frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\|
\biggl\| \frac{d}{dz} Q_1 \left\{ z +\lambda^2 [ A(z) + \pi i {\mit \Gamma}(z) ] \right\} Q_1
\biggr\|
\| E_{11}^{-1} \| \nonumber \\ &&\hspace*{-13mm} ~~~+
\frac{\| ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \| }{\lambda^2 |z \log z|}
\biggl\| \frac{d}{dz} Q_1 \left\{ z +\lambda^2 [ A(z) + \pi i {\mit \Gamma}(z) ] \right\} Q_1
\biggr\|
\biggl\| E_{11}^{-1}-\frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\| \nonumber \\ &&\hspace*{-13mm} ~~~+
\frac{\|( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \|^2}{\lambda^4 z^2 (\log z)^2 }
\biggl\| \frac{d}{dz} Q_1 \left\{ z +\lambda^2 [ A(z) + \pi i {\mit \Gamma}(z) ] \right\} Q_1 - Q_1 (1+ \lambda^2 A_1 + \lambda^2 \pi i {\mit \Gamma}_1 )Q_1
\biggr\| \label{eqn:1stSmall90a}\\
&&\hspace*{-13mm} = O(z^{-2}(\log z)^{-3}), \label{eqn:1stSmall90b} \end{eqnarray} as $z \to 0$.
On the other hand, the second term in Eq. (\ref{eqn:1stSmall80d}) is slightly complicated to evaluate: \begin{eqnarray} &&
\biggl\| -E_{11}^{-1} \biggl\{ \frac{d}{dz} \lambda^2 (\log z) Q_1 {\mit \Gamma}(z) Q_1 \biggr\} E_{11}^{-1} -\frac{d}{dz} \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber \\ && - z \biggl\{ \frac{d}{dz} \frac{1}{\lambda^4 z^2 (\log z)^2} \biggr\}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\|
\label{eqn:1stSmall95a}\\ &\leq&
\biggl\| -\frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \biggl\{ \frac{d}{dz} \lambda^2 (\log z) Q_1 {\mit \Gamma}(z) Q_1 \biggr\} \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\nonumber \\ && -\frac{d}{dz} \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber \\ && -\frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \biggl\{ \frac{d}{dz} \lambda^2 (\log z) Q_1 {\mit \Gamma}(z) Q_1 \biggr\} \tilde{E}_{11}\frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber \\ && -\tilde{E}_{11}\frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \biggl\{ \frac{d}{dz} \lambda^2 (\log z) Q_1 {\mit \Gamma}(z) Q_1 \biggr\} \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber \\ && + z \frac{2}{(\lambda^2 z \log z )^3} \biggl( \frac{d}{dz} \lambda^2 z \log z \biggr) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\| \nonumber \\ && +
\biggl\| \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \biggl\{ \frac{d}{dz} \lambda^2 (\log z) Q_1 {\mit \Gamma}(z) Q_1 \biggr\} \sum_{j=2}^{\infty}\tilde{E}_{11}^j \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\| \nonumber \\ && +
\biggl\| \tilde{E}_{11} \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \biggl\{ \frac{d}{dz} \lambda^2 (\log z) Q_1 {\mit \Gamma}(z) Q_1 \biggr\} \sum_{j=1}^{\infty}\tilde{E}_{11}^j \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\| \nonumber \\ && +
\biggl\| \sum_{j=2}^{\infty}\tilde{E}_{11}^j \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \biggl\{ \frac{d}{dz} \lambda^2 (\log z) Q_1 {\mit \Gamma}(z) Q_1 \biggr\} E_{11}^{-1}
\biggr\| \label{eqn:1stSmall95b}\\
&\leq&
\biggl\| -\frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \biggl\{ \frac{d}{dz} \lambda^2 \log z Q_1 {\mit \Gamma}(z) Q_1 \biggr\} \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber \\ && -\frac{d}{dz} \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\| \nonumber \\ && +
\biggl\| -\frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \biggl\{ \frac{d}{dz} \lambda^2 (\log z) Q_1 {\mit \Gamma}(z) Q_1 \biggr\} \tilde{E}_{11} \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber \\ && + z \frac{1}{(\lambda^2 z \log z )^3} \biggl( \frac{d}{dz} \lambda^2 z \log z \biggr) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} z (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\| \nonumber \\ && +
\biggl\| -\tilde{E}_{11} \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \biggl\{ \frac{d}{dz} \lambda^2 (\log z) Q_1 {\mit \Gamma}(z) Q_1 \biggr\} \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber \\ && + z \frac{1}{(\lambda^2 z \log z )^3} \biggl( \frac{d}{dz} \lambda^2 z \log z \biggr) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\| \nonumber \\ && +O(z^{-2}(\log z)^{-3} ) \label{eqn:1stSmall95c} \\ &=&O(z^{-2}(\log z)^{-3} ). \label{eqn:1stSmall95d} \end{eqnarray} In fact, the first term in Eq. (\ref{eqn:1stSmall95c}) reads \begin{eqnarray} &&
\biggl\| -\frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \biggl\{ \frac{d}{dz} \lambda^2 \log z Q_1 {\mit \Gamma}(z) Q_1 \biggr\} \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber\\ && -\frac{d}{dz} \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\| \nonumber\\ &\leq&
\biggl\| -(Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \biggl\{ \frac{d}{dz} \lambda^2 (\log z) Q_1 {\mit \Gamma}(z) Q_1 \biggr\} +\frac{d}{dz} \lambda^2 z (\log z) Q_1
\biggr\|
\frac{\| ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \|}{(\lambda^2z \log z)^{2}} \label{eqn:1stSmall96b}\\ &\leq& \lambda^2 \left\{
\biggl\| -(Q_1 {\mit \Gamma}_1 Q_1 )^{-1} Q_1 \frac{{\mit \Gamma}(z)}{z} Q_1 + Q_1
\biggr\|
+
\biggl\| -(Q_1 {\mit \Gamma}_1 Q_1 )^{-1} Q_1 \frac{d{\mit \Gamma}(z)}{dz} Q_1 + Q_1
\biggr\| (\log z) \right\} \nonumber \\ && \times
\| ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \| \frac{1}{\lambda^2 z \log z} \label{eqn:1stSmall96c}\\ &=&
O((z \log z)^{-1}). \label{eqn:1stSmall96d} \end{eqnarray} The second term in Eq. (\ref{eqn:1stSmall95c}) also reads \begin{eqnarray} &&
\biggl\| -\frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \biggl\{ \frac{d}{dz} \lambda^2 (\log z) Q_1 {\mit \Gamma}(z) Q_1 \biggr\} \tilde{E}_{11} \frac{1}{\lambda^2 z \log z}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber \\ && + z \frac{1}{(\lambda^2 z \log z )^3} \biggl( \frac{d}{dz} \lambda^2 z \log z \biggr) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\| \nonumber\\ &\leq&
\|(Q_1 {\mit \Gamma}_1 Q_1 )^{-1}\|
\biggl\| - \biggl\{ \frac{d}{dz} \lambda^2 (\log z) Q_1 {\mit \Gamma}(z) Q_1 \biggr\} \nonumber \\ && \times (Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \bigl\{ z Q_1 +\lambda^2 Q_1 \{ A(z) -\log z [{\mit \Gamma}(z) -z {\mit \Gamma}_1 ] + \pi i {\mit \Gamma}(z) \} Q_1 \bigr\} \nonumber \\ && + z \biggl( \frac{d}{dz} \lambda^2 z \log z \biggr) (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 )
\biggr\|
\frac{\|( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \| }{(\lambda^2 z \log z )^{3}} \label{eqn:1stSmall97b}\\ &\leq&
\|(Q_1 {\mit \Gamma}_1 Q_1 )^{-1}\|
\biggl\| \biggl\{ \frac{d}{dz} \lambda^2 (\log z) Q_1 {\mit \Gamma}(z) Q_1 \biggr\} (Q_1 {\mit \Gamma}_1 Q_1 )^{-1}
\biggr\| \nonumber \\ && \times
\bigl\| - \bigl\{ z Q_1 +\lambda^2 Q_1 \{ A(z) -\log z [{\mit \Gamma}(z) -z {\mit \Gamma}_1 ] + \pi i {\mit \Gamma}(z) \} Q_1 \bigr\} \nonumber \\ && + z (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 )
\bigr\|
\frac{\|( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \|}{ (\lambda^2 z \log z )^{3}} \nonumber \\ && +
\|(Q_1 {\mit \Gamma}_1 Q_1 )^{-1}\|
\biggl\| - \biggl\{ \frac{d}{dz} \lambda^2 (\log z) Q_1 {\mit \Gamma}(z) Q_1 \biggr\} (Q_1 {\mit \Gamma}_1 Q_1 )^{-1} + \biggl( \frac{d}{dz} \lambda^2 z \log z \biggr) Q_1
\biggr\| \nonumber \\ && \times
\bigl\| z (Q_1+ \lambda^2 Q_1 A_1 Q_1 + \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 )
\bigr\|
\frac{\|( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \|}{ (\lambda^2 z \log z )^{3}} \label{eqn:1stSmall97c}\\ &=&
O((z \log z)^{-1}), \label{eqn:1stSmall97d} \end{eqnarray} as $z \to 0$. The third term gives the same contribution to the order as the second one does.
Furthermore, the last term in Eq. (\ref{eqn:1stSmall95c}) comes from the estimations of the second, third, and last terms in Eq. (\ref{eqn:1stSmall95b}), where each contributes the same order as $O(z^{-2}(\log z)^{-3} )$. Therefore, Eq. (\ref{eqn:1stSmall95d}) is proved.
On the other hand, the third term in Eq. (\ref{eqn:1stSmall80d}) reads \begin{equation}
\biggl\| \bigl\{ (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1} -Q_{11} \bigr\} \frac{dE_{11}^{-1}}{dz}
\biggr\|
\leq \frac {
\bigl\| E_{11}^{-1}E_{10} E_{00}^{-1} E_{01}
\bigr\|} {
1-\bigl\| E_{11}^{-1}E_{10} E_{00}^{-1} E_{01}
\bigr\| }
\biggl\| \frac{dE_{11}^{-1}}{dz}
\biggr\| =O(z^{-1}), \label{eqn:1stSmall100b} \end{equation} as $z \to 0$, where Eqs. (\ref{eqn:5.C.1.90b}), (\ref{eqn:5.C.1.100}), (\ref{eqn:5.C.1.130}), and (\ref{eqn:1stSmall73c}) are used. In the same way, the last term in Eq. (\ref{eqn:1stSmall80d}) reads \begin{eqnarray} && \hspace*{-5mm}
\biggl\| E_{11}^{-1} \frac{d}{dz} (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}
\biggr\| \nonumber\\ &\leq&
\| E_{11}^{-1} \|
\biggl\| (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}
\biggr\|^2 \biggl[
\biggl\| \frac{dE_{11}^{-1}}{dz} E_{10} E_{00}^{-1} E_{01}
\biggr\| \nonumber \\ &&
+\biggl\| E_{11}^{-1}\frac{dE_{10} }{dz} E_{00}^{-1} E_{01}
\biggr\|
+\biggl\| E_{11}^{-1}E_{10} E_{00}^{-1}\frac{d E_{01}}{dz}
\biggr\|
+\biggl\| E_{11}^{-1}E_{10} \frac{d E_{00}^{-1}}{dz} E_{01}
\biggr\| \biggr] \label{eqn:1stSmall110a}\\ &=& O(z^{-1} \log z ),
\label{eqn:1stSmall110b} \end{eqnarray} as $z \to 0$. By substituting Eqs. (\ref{eqn:1stSmall90b}), (\ref{eqn:1stSmall95d}), (\ref{eqn:1stSmall100b}), and (\ref{eqn:1stSmall110b}) into Eq. (\ref{eqn:1stSmall80d}), one finally obtains Eq. (\ref{eqn:1stSmall80e}). We can now show Eq. (\ref{eqn:1stSmall7}) for $r=1$ by setting Eqs. (\ref{eqn:1stSmall70}) and (\ref{eqn:1stSmall80e}) into Eq. (\ref{eqn:1stSmall50}). \qed
If we start with expression (\ref{eqn:5.C.1.125b}), we obtain the following lemma instead of Lemma \ref{lm:1stremainder+}.
\begin{lm} \label{lm:1stremainder-} : Assume that $0$ is an exceptional point of the first kind for $H$. Then the $0$-th and the first derivative of $\tilde{R} (z)$ are approximated by those of a finite series \begin{eqnarray} && \frac{1}{\lambda^2 z (\log z -2\pi i)}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber\\ && + \frac{1}{\lambda^4 z (\log z -2\pi i)^2}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 - \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} , \label{eqn:1stSmall5-} \end{eqnarray} that is, it is shown that \begin{eqnarray} &&\hspace{-10mm}
\biggl\| \frac{d^r}{dz^r} \biggl[ \tilde{R}(z) - \frac{1}{\lambda^2 z (\log z -2\pi i)}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \nonumber \\ && - \frac{1}{\lambda^4 z (\log z -2\pi i)^2}( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} (Q_1+ \lambda^2 Q_1 A_1 Q_1 - \lambda^2 \pi i Q_1 {\mit \Gamma}_1 Q_1 ) ( Q_1 {\mit \Gamma}_1 Q_1 )^{-1} \biggr]
\biggr\| \nonumber \\ &=&
O(z^{-1} (\log z -2\pi i)^{-3}) ~~\mbox{for}~~r=0,~~~\mbox{or}~~~
O(z^{-2} (\log z -2\pi i)^{-3}) ~~\mbox{for}~~r=1,
\label{eqn:1stSmall7-} \end{eqnarray} as $z \to 0$. \end{lm}
\section{The reduced time evolution operator} \label{sec:6}
In this section, we show that the reduced time evolution operator
is expressed by the Fourier transform of the imaginary part of the reduced resolvent both in the regular case and the exceptional case of the first kind. We here define the reduced time evolution operator by the $N\times N$ matrix $\tilde{U}(t)$ of the components
$\tilde{U}_{mn}(t):=\langle m |Pe^{-itH}P| n \rangle$, where $P=E((0, \infty ))$ and $\{ E(B) | B \in \mathbb{B} \}$ is the spectral measure of $H$, which is a family of the projection operator. $\mathbb{B}$ is the Borel field of $\mathbb{R}$.
\begin{lm} \label{lm:reduced time evolution operator } : We assume that Eq. (\ref{eqn:.110}) holds so that there is no positive eigenvalue. Then, for the system with the rational form-factor (\ref{eqn:formfactor1}), it holds that \begin{equation} \tilde{U}(t) = \frac{1}{\pi} \int_{(0, \infty )} e^{-it\omega} {\rm Im} \tilde{R}^+ (\omega) d\omega = \lim_{r \to +0} \frac{1}{\pi} \int_r^\infty e^{-it\omega} {\rm Im} \tilde{R}^+ (\omega) d\omega , \label{eqn:6.120} \end{equation} both in the regular case and the exceptional case of the first kind, where \begin{eqnarray} {\rm Im} \tilde{R}^+ (\omega) := \frac{1}{2 i} [\tilde{R}^+(\omega) -\tilde{R}^-(\omega)] , \label{eqn:6.65} \end{eqnarray} which is sometimes called the spectral density. \end{lm}
{\sl Proof} : Let us remember that the matrix $\tilde{U}(t)$ is expressed by the spectral measure as \begin{equation} \tilde{U}_{mn}(t) = \int_{(0, \infty )} e^{-it\lambda }
d \langle m |E(\lambda )| n \rangle = \int_{(0, \infty )} e^{-it\lambda } d \tilde{E}_{mn}(\lambda ) , \label{eqn:6.110} \end{equation}
where $\tilde{E}(B)$ is the matrix of the components $\langle m |E(B)|n \rangle $. Therefore, what we first should do is to clarify the relation between ${\rm Im} \tilde{R}^+ (\omega)$ and $\tilde{E}(\lambda )$. Resorting to Stone's formula between $E(B)$ and $R(z)$, we clearly see \begin{equation} \frac{1}{2} [\tilde{E}([a,b])+\tilde{E}((a,b))] =\lim_{\epsilon \to +0} \frac{1}{2\pi i} \int_a^b [\tilde{R}(\omega +i\epsilon) -\tilde{R}(\omega -i\epsilon)] d\omega, \label{eqn:6.30} \end{equation} for $a, b \in \mathbb{R}$ with $a<b$. Under the assumption (\ref{eqn:.110}), Lebesgue's dominated convergence theorem and the proof of Lemma \ref{lm:100} tell us that the exchange between the limit and the integration in Eq. (\ref{eqn:6.30}) is allowed for $[a,b]\subset (0,\infty)$.
If $[a,b] \subset (-\infty, 0)\backslash \sigma(H)$, then $\tilde{R}^{\pm} (\omega) =\tilde{R} (\omega) $, and thus $\tilde{E}([a,b])=\tilde{E}((a,b))=0$. In addition, by the continuity of $\tilde{R}^{\pm}(\omega)$, Eq. (\ref{eqn:6.30}) tells us that $\tilde{E}(\{ a\} )=0$ for all $a>0$, which leads to \begin{equation} \tilde{E}((a,b)) =\tilde{E}([a,b])= \frac{1}{\pi} \int_a^b {\rm Im} \tilde{R}^+ (\omega) d\omega , \label{eqn:6.60} \end{equation} for all $a, b$ with $b>a>0$.
Let us now consider the regular case and in particular the validity of the expression (\ref{eqn:6.60}) for the interval including the origin. In this case, $\tilde{R}^\pm (0):=\lim_{\omega \to +0}\tilde{R}^\pm (\omega)$ exists to be finite. Furthermore, $\lim_{\omega \to \infty}\tilde{R}^\pm (\omega)=0$ from Lemma \ref{lm:Large-omega}. Thus, $\tilde{R}^\pm (\omega)$ is uniformly continuous on $(0, \infty)$. Therefore, we can take the limit of Eq. (\ref{eqn:6.60}) as $a\to +0$ to obtain $\lim_{a\to +0}E([a,b])=E((0,b])$. We next see that all components of ${\rm Im} \tilde{R}^+ (\omega)$
are integrable, i.e., belong to $L^1 ((0, \infty ))$. Suppose that $| \psi \rangle \in \mathbb{C}^N$, then
$\langle \psi | \tilde{E}((0, \lambda )) | \psi \rangle$ is positive and a monotonically increasing function of $\lambda$, and it is also differentiable in this case. Thus Eq. (\ref{eqn:6.60}) tells us that $\bra{\psi} {\rm Im} \tilde{R}^+ (\omega) \ket{\psi} \geq 0$. In addition, \begin{equation}
\| \psi \|^2 \geq \lim_{\lambda \to \infty }
\langle \psi | \tilde{E}((0, \lambda )) | \psi \rangle = \lim_{\lambda \to \infty } \frac{1}{\pi} \int_0^\lambda \bra{\psi} {\rm Im} \tilde{R}^+ (\omega) \ket{\psi} d\omega = \frac{1}{\pi} \int_0^\infty \bra{\psi} {\rm Im} \tilde{R}^+ (\omega) \ket{\psi} d\omega . \label{eqn:6.90} \end{equation} Hence, from the monotonic convergence theorem, we see that $\bra{\psi} {\rm Im} \tilde{R}^+ (\omega) \ket{\psi} \in L^1 ((0, \infty ))$. From this fact and the use of the polarization identity, we can prove that all components of the matrix ${\rm Im} \tilde{R}^+ (\omega)$ are integrable. Thus, extending the rhs of Eq. (\ref{eqn:6.60}) to arbitrary
$B \in \{ B \in \mathbb{B}| B \subset (0,\infty) \}$, $\int_B {\rm Im} \tilde{R}^+ (\omega) d\omega$ defines a measure. We can now see from Eq. (\ref{eqn:6.60}) and from E. Hopf's extension theorem that \begin{equation} \tilde{E}(B) = \frac{1}{\pi} \int_B {\rm Im} \tilde{R}^+ (\omega) d\omega \label{eqn:6.100} \end{equation}
holds for all $B \in \{ B \in \mathbb{B}| B \subset (0,\infty) \}$.
Note that this expression means that
the restriction of $\tilde{E}_{mn}(B) $ to
$\{ B \in \mathbb{B}| B \subset (0,\infty) \}$ is absolutely continuous. Therefore, rewriting of $\tilde{U}(t)$ in Eq. (\ref{eqn:6.110}) into (\ref{eqn:6.120}) is straightforward.
In the exceptional case of the first kind, from the assumption (\ref{eqn:.110}), $\tilde{R}^\pm (\omega)$ is continuous on $(0, \infty)$, while $\tilde{R}^\pm (\omega)=O((\omega \log \omega)^{-1})$ as $\omega \to +0$, so that it is not integrable around $0$. See, Eq. (\ref{eqn:5.C.1.120d}). However, ${\rm Im} \tilde{R}^+ (\omega)$ is of the order $O(\omega^{-1} (\log \omega)^{-2})$ from Lemmas \ref{lm:1stremainder+} and \ref{lm:1stremainder-}, and thus it is integrable around $0$. Hence, Eq. (\ref{eqn:6.100}) holds again, and Eq. (\ref{eqn:6.120}) is valid for this exceptional case. \qed
We remark that in the case of no negative eigenvalues (point spectrum) of $H$, \cite{Miyamoto(2005)}
the restriction of $\tilde{U}(t)$ to the continuous energy spectrum is removed because in such a case $P=I$ the identity. Furthermore, the connection between $\tilde{U}(t)$ and the observables is easily found, e.g.,
$|\bra{\psi}\tilde{U}(t)\ket{\psi}|^2/\| P\ket{\psi} \|^4$
for $| \psi \rangle \in \mathbb{C}^N$ (or $\mathbb{C}^N \oplus \{ 0\}$) is the survival probability of $P\ket{\psi}$ which is the probability of finding the system in the state $P\ket{\psi}$ at the later time $t$, where $P\ket{\psi}$ is just the decaying component of the initial state $\ket{\psi}$.
\section{The asymptotic expansion of the reduced time evolution operator} \label{sec:7}
We can finally show the asymptotic formula for $\tilde{U}(t)$ at long times for the rational form factors satisfying our assumptions. In the following, we assume that Eq. (\ref{eqn:.110}) holds, i.e., there is no positive eigenvalue. However, this is not explicitly mentioned in the statements of the theorems.
Let us first consider the regular case. For this purpose, according to Lemma \ref{lm:rffimremainder}, we introduce the remainder $F(\omega)$ in the following way: \begin{equation} \frac{1}{\pi}{\rm Im} \tilde{R}^+ (\omega) = \lambda^2 (K(0))^{-1} \omega^{n_b} ({\mit \Gamma}_{n_b} +\omega {\mit \Gamma}_{n_b +1} + \omega^2 {\mit \Gamma}_{n_b +2} ) (K(0))^{-1} +F(\omega), \label{eqn:rffimbothLong110} \end{equation} for $\omega >0$.
\begin{thm} \label{thm:rffimLong} : Assume that $0$ is a regular point for $H$. For a system with the rational form factor (\ref{eqn:formfactor1}) characterized by the positive integers $n_a$ and $n_b$ that satisfy that $n_b \geq 2$ and $n_a =1$,
the reduced time evolution operator $\tilde{U}(t)$ behaves asymptotically as \begin{equation} \tilde{U}(t) = \lambda^2 \frac{\Gamma(1+n_b)}{(it)^{n_b+1}} (K(0))^{-1} {\mit \Gamma}_{n_b} (K(0))^{-1} +O(t^{-n_b-2}) , \label{eqn:rffimbothLong50} \end{equation} as $t\to \infty$. When $n_b=1$ and $n_a \geq 1$, the error term is replaced by $O(t^{-3}\log t)$. \end{thm}
{\sl Proof} :
We first summarize the several properties of ${\rm Im} \tilde{R}^+ (\omega) $. By Lemma \ref{lm:rffimremainder}, we see that the remainder $F(\omega)$ in Eq. (\ref{eqn:rffimbothLong110}) is arbitrary-times differentiable. Particularly it holds that $\lim_{\omega \to 0} d^r F(\omega) / d \omega^r = 0$ for all $r \leq n_b$, and \begin{eqnarray}
\left\| \frac{d^{n_b+1} F(\omega)}{d\omega^{n_b+1}} \right\| &&=
O(\log \omega), ~ \mbox{or} ~ O(1)
, \label{eqn:rffimbothLong120} \end{eqnarray} for $n_b=1$ and $n_a \geq 1$, or $n_b \geq 2$ and $n_a =1$, respectively, and
\begin{equation}
\left\| \frac{d^{n_b+2} F(\omega)}{d\omega^{n_b+2}} \right\| = O(\omega^{-1}), \mbox{ or } \left\{ \begin{array}{cc} O(1) & (n_b \geq 3)\\ O(\log \omega) & (n_b = 2) \end{array} \right. , \label{eqn:rffimbothLong130} \end{equation} for $n_b=1$ and $n_a \geq 1$, or $n_b \geq 2$ and $n_a =1$, respectively,
as $\omega \to 0$.
On the other hand, we see from Lemma \ref{lm:Large-omega} that $ (d / d \omega )^r {\rm Im} \tilde{R}^+ (\omega) = O(\omega^{-r-1})$ as $\omega \rightarrow \infty $. In particular, if $m \geq 1$, $(d /d \omega )^m {\rm Im} \tilde{R}^+ (\omega)$ is integrable on $[\delta , \infty )$ for an arbitrary $\delta >0$.
Let us now split the integral in Eq. (\ref{eqn:6.120}) into two parts by writing \begin{equation} {\rm Im} \tilde{R}^+ (\omega ) = \phi (\omega ) {\rm Im} \tilde{R}^+ (\omega ) + (1- \phi (\omega ) ) {\rm Im} \tilde{R}^+ (\omega ) , \label{eqn:rffimbothLong135} \end{equation} where $\phi \in C_0 ^{\infty} ([0,\infty ) )$ and satisfies $\phi (\omega ) =1$ in a neighborhood of $\omega = 0$. Such a function is realized by $f (\omega ) = 1- \int_0 ^\omega g(x) d x$, where $g(x) = h(x)/\int_{\bf R} h(x) d x$ and
$h(x) = \exp(-1/[a^2-(x-d)^2 ])$ ~($|x-d|<a$) or $0$ ($|x-d| \geq a$) with $d>a>0$.
From Lemma 10.1 in Ref. \makebox(0,1){\Large \cite{Jensen(1979)}}\hspace{2mm} and the above-mentioned discussion, we see that $(1- \phi (\omega ) ) {\rm Im} \tilde{R}^+ (\omega ) $ has a contribution of $O(t^{-m})$ to $\tilde{U}(t)$ for an arbitrary $m\geq 1$, i.e., this term decays faster than any negative power of $t$.
On the other hand, the contribution of $\phi (\omega ) {\rm Im} \tilde{R}^+ (\omega ) $ to $\tilde{U}(t)$ gives the main part of the asymptotic expansion. Then, the coefficient of ${\mit \Gamma}_{n_b}$, ${\mit \Gamma}_{n_b +1}$, and ${\mit \Gamma}_{n_b +2}$ is given by the form \cite{Copson} \begin{eqnarray}
\int_0 ^{\infty } \phi (\omega ) \omega^q e^{-i t \omega } ~ d \omega
& = & \sum_{k=0}^{N-1} \frac{1}{(it)^{k+1}}
\left. \frac{d^k \omega^q \phi(\omega)}{d\omega^k} \right|_{\omega=0} +R_N (t)
= \frac{\Gamma(1+q)}{(it)^{1+q}} +R_N (t) , \label{eqn:rffimbothLong140} \end{eqnarray} for all $N\geq 1+q$, where $q$ takes the value $n_b$, $n_b+1$, or $n_b+2$. We here used that \begin{equation}
\frac{d^k \omega^q \phi(\omega)}{d\omega^k}\biggr|_{\omega=0} = \sum_{j=0}^{\min \{ k,q \}} {{k}\choose{j}}
\frac{d^j \omega^q }{d\omega^j }\biggr|_{\omega=0}
\frac{d^{k-j} \phi(\omega)}{d\omega^{k-j}}\biggr|_{\omega=0}
=\Gamma(1+q) \delta_{kq}, \label{eqn:rffimbothLong145} \end{equation} where $\Gamma(1+n)=\int_0^\infty x^n e^{-x} dx$ is the gamma function. In addition, the remainder $R_N (t)$ is bounded above by \begin{equation}
|R_N (t)| \leq
\frac{1}{t^N} \left| \int_{0}^{\infty}
\frac{d^N \omega^q \phi(\omega)}{d\omega^N} e^{-i\omega t} d\omega \right| = o(t^{-N}). \label{eqn:rffimbothLong160} \end{equation} Note that since all derivatives of $\phi(\omega)$ vanish in the neighborhood of $\omega =0$,
Eq. (\ref{eqn:rffimbothLong140}) is valid for all $N\geq q+1$ and thus $R_N (t)$ decays faster than any negative power of $t$. Furthermore, we understand, by applying Eq. (\ref{eqn:A.50a}) in Lemma\ \ref{lm:Jensen and Kato, Lemma 10.2} directly to $\phi (\omega ) F(\omega) $ with the discussion in the first part of this section, that
the contribution of the Fourier transform of the remainder $\phi (\omega ) F(\omega) $ to $\tilde{U}(t)$ is \begin{equation}
O(t^{-3}\log t)~~ \mbox{or} ~~ O(t^{-n_b-2}) , \label{eqn:rffimbothLong180} \end{equation} for $n_b=1$ and $n_a \geq 1$, or $n_b \geq 2$ and $n_a =1$, respectively, as $\omega \to 0$, where we used the formula of the indefinite integral that $\int (\log \omega)^2 d\omega =\omega [(\log \omega)^2 -2\log \omega +2]$. Summarizing the above-noted results, we finally obtain that
\begin{eqnarray} &&\hspace{-10mm}
\left\| \tilde{U}(t) - \lambda^2 (K(0))^{-1} \left[ \frac{\Gamma(1+n_b)}{(it)^{n_b+1}} {\mit \Gamma}_{n_b} + \frac{\Gamma(2+n_b)}{(it)^{n_b+2}} {\mit \Gamma}_{n_b +1} +\frac{\Gamma(3+n_b)}{(it)^{n_b+3}} {\mit \Gamma}_{n_b +2} \right] (K(0))^{-1}
\right\| \nonumber \\
&&\hspace{-10mm}\leq
\left\| \frac{1}{\pi}\int_0^{\infty} (1-\phi(\omega)){\rm Im} \tilde{R}^+ (\omega ) e^{-i t \omega } d \omega
\right\| +
\left\| \frac{1}{\pi}\int_0^{\infty} \phi(\omega) F(\omega) e^{-i t \omega } d\omega
\right\| +O(t^{-N}) \nonumber \\ &&\hspace{-10mm}=
O(t^{-3}\log t)~~ \mbox{or} ~~ O(t^{-n_b-2}) , \label{eqn:rffimbothLong150} \end{eqnarray} for $n_b=1$ and $n_a \geq 1$, or $n_b \geq 2$ and $n_a =1$, respectively,
as $t\to \infty$, where $O(t^{-N})$ is due to the contribution from $R_N (t)$.
This is just the asymptotic expansion of $\tilde{U}(t)$ in the statement. \qed
It is worth noting that if we resort to Lemma \ref{lm:rffremainder}, instead of Eqs. (\ref{eqn:rffimbothLong120}) and (\ref{eqn:rffimbothLong130}), we have \begin{eqnarray} \hspace*{-5mm} \frac{d^r F(\omega)}{d\omega^r} &&=
O(\omega^{2-r} (\log \omega)^{1+\theta (2-r)}), ~ \mbox{or} ~ \left\{ \begin{array}{cc} O(\omega^{[2-r]^+}) & (0 \leq r \leq n_b)\\ O(\omega^{n_b +1-r} (\log \omega)^{\theta (n_b +1 -r)}) & (r \geq n_b +1) \end{array} \right. , \label{eqn:rffimbothLong170} \end{eqnarray} for $n_b=1$ and $n_a \geq 1$, or $n_b \geq 2$ and $n_a =1$, respectively. However, in the latter case, we see that the Fourier transform of $\phi(\omega)F(\omega)$ gives the contribution of the order $O(t^{-n_b -1})$, which is just the same order as that coming from the dominant one. Hence, we can only obtain an useless estimation.
We next show the asymptotic formula for $\tilde{U}(t)$ at long times for a system with an exceptional point of the first kind. To this end, we write ${\rm Im} \tilde{R}^+(\omega)$ with the remainder $F(\omega)$ again as follows: \begin{equation} \frac{1}{\pi}{\rm Im} \tilde{R}^+ (\omega) = \frac{1}{\lambda^2 \omega (\log \omega)^2} (Q_1 {\mit \Gamma}_1 Q_1)^{-1} +F(\omega) .
\label{eqn:rffimLong1st110} \end{equation}
\begin{thm} \label{thm:rffLong1st} : Assume that $0$ is an exceptional point of the first kind for $H$, which necessarily imposes that $n_b =1$. Then, the reduced time evolution operator $\tilde{U}(t)$ for the rational form factor (\ref{eqn:formfactor1}) behaves asymptotically as \begin{equation} \tilde{U}(t) = \frac{1}{\lambda^2 \log t} (Q_1 {\mit \Gamma}_1 Q_1)^{-1} +O((\log t)^{-2}), \label{eqn:rffimLong1st50} \end{equation} as $t \to \infty$.
\end{thm}
{\sl Proof} :
Let us first look over the some properties of ${\rm Im} \tilde{R}^+ (\omega) $ again. By Lemmas \ref{lm:1stremainder+} and \ref{lm:1stremainder-}, we see that the remainder $F(\omega)$ in Eq. (\ref{eqn:rffimLong1st110}) is arbitrary-times differentiable, satisfies that $F(\omega) = O(\omega^{-1} (\log \omega)^{-3})$, and $d F(\omega) / d \omega = O(\omega^{-2} (\log \omega)^{-3}) $, as $\omega \to +0$.
On the other hand, we see from Lemma \ref{lm:Large-omega} that $ (d / d \omega )^r {\rm Im} \tilde{R}^+ (\omega) = O(\omega^{-r-1})$ as $\omega \rightarrow \infty $. In particular, if $m \geq 1$, $(d /d \omega )^m {\rm Im} \tilde{R}^+ (\omega)$ is integrable on $[\delta , \infty )$ for an arbitrary $\delta >0$.
We now split the integral in Eq. (\ref{eqn:6.120}) into two parts as in Eq. (\ref{eqn:rffimbothLong135}) again using the $C_0 ^{\infty}$-function $\phi(\omega )$. From Lemma 10.1 in Ref. \makebox(0,1){\Large \cite{Jensen(1979)}}\hspace{2mm} and the discussion mentioned above, $(1- \phi (\omega ) ) {\rm Im} \tilde{R}^+ (\omega ) $ has a contribution of $O(t^{-m})$ to $\tilde{U}(t)$ for an arbitrary $m\geq 1$.
On the other hand, the contribution of $\phi (\omega ) {\rm Im} \tilde{R}^+ (\omega ) $ to $\tilde{U}(t)$ gives the dominant part of the asymptotic expansion. Then, the dominant time dependence of the asymptote of $\tilde{U}(t)$ follows from Lemma \ref{lm:asymptote of inverse-lagarithmic-fourier-integral}, that is, \begin{eqnarray}
\int_0 ^{\infty } \phi (\omega ) (\omega (\log \omega)^2)^{-1} e^{-i t \omega } ~ d \omega = (\log t)^{-1} +O((\log t)^{-2} ). \label{eqn:rffimLong1st120} \end{eqnarray} Furthermore,
the contribution of the Fourier transform of the remainder $\phi (\omega ) F(\omega) $ to $\tilde{U}(t)$ can be estimated by the similar manner to Lemma\ \ref{lm:asymptote of inverse-lagarithmic-fourier-integral}, rather than Lemma\ \ref{lm:Jensen and Kato, Lemma 10.2}. By setting $\sigma (\omega)=F(\omega) e^{-it\omega}$ instead in the proof of Lemma \ref{lm:asymptote of inverse-lagarithmic-fourier-integral}, we can apply it to this case, and we have \begin{equation} \int_0^\infty F(\omega) \phi(\omega) e^{-it\omega} d\omega = -\lim_{\omega \to +0}(\hat{I} \sigma)(\omega) +(-1)^N \int_0^\infty (\hat{I}^N \sigma)(\omega) \frac{d^N \phi(\omega)}{d\omega^N} d\omega . \label{eqn:rffimLong1st155} \end{equation} Then, corresponding to Eq. (\ref{eqn:B.530}), we have \begin{equation}
\|(\hat{I}\sigma)(\omega)\| =
\biggl\| i e^{-it\omega} \int_0^\infty F(\omega-i\eta) e^{-t\eta} d\eta
\biggr\| + E(t) \leq C
\int_0^\infty \bigr| (\omega-i\eta)^{-1} [\log (\omega-i\eta)]^{-3} \bigr| e^{-t\eta} d\eta + E(t), \label{eqn:rffimLong1st160} \end{equation} with an appropriate constant $C$. Note that in this procedure, $\tilde{R}^+ (\omega)$ is analytically continued to the lower plane of the second Riemann sheet, while $\tilde{R}^- (\omega)$ still remains in the lower plane of the first Riemann sheet. Then, both are ensured to contribute the remainders of the same order to $F(\omega-i\eta)$ in the above integral from Lemmas \ref{lm:1stremainder+} and \ref{lm:1stremainder-}. The remainder term $E(t)$ in Eq. (\ref{eqn:rffimLong1st160}) that gives the order of $O(e^{-\gamma t})$ for some $\gamma >0$ is responsible for the possible poles of $\tilde{R}^+ (\omega)$ continued to the second Riemann sheet, the number of which are guaranteed to be finite from the analytic Fredholm theorem \cite{the analytic Fredholm theorem} and Lemma \ref{lm:Large-omega} for the continued $\tilde{R}^+ (\omega)$. Thus, it follows from Lemma \ref{lm:asymptote of inverse-lagarithmic-fourier-integral}
that $\lim_{\omega \to +0} \|(\hat{I}\sigma)(\omega)\| =O((\log t)^{-2})$. By the same argument as in Lemma \ref{lm:asymptote of inverse-lagarithmic-fourier-integral}, one also sees that the remainder term in Eq. (\ref{eqn:rffimLong1st155}) is of the order of $O(t^{-N+1})$. Summarizing these arguments, we can finish the proof of the theorem. \qed
\section{Concluding remarks} \label{sec:8}
We have rigorously derived the asymptotic formula of the reduced time evolution operator for the $N$-level Friedrichs model in the context of the zero energy resonance \cite{Jensen(1979)}
both for the regular case and the exceptional case of the first kind. Then, in the latter case, the logarithmically slow decay proportional to $(\log t)^{-1}$ has been found, and the expansion coefficient has been explicitly presented by the projection operator associated with the zero energy eigenstates of the total Hamiltonian, which is an extended state not belonging to the Hilbert space. We note that the decay involving the logarithmic function expressed by $t^{-j}(\log t)^{k}$ ($j=1, 2, \ldots$ and $k=0, \pm 1, \ldots$) can occur in the short range potential systems in the even dimensional space. \cite{Murata(1982)}
It should be noted that a realization of the exceptional cases require the parameters, e.g., the coupling constant $\lambda$, to take such special values that the matrix $K(0)$ in Eq. (\ref{eqn:.336}) has a zero eigenvalue.
In addition, some of the form factors $v_n(\omega)$ have to behave as $|v_n(\omega)|^2 \sim c_n \omega$ around $\omega=0$. In other words, if all of them behave as
$|v_n(\omega)|^2 \sim c_n \omega^{q_n}$ with $q_n \geq 2$, the exceptional case of the first kind never occurs though that of the second kind could happen.
These circumstances explain how the exceptional cases are surely exceptional. The presented results also enable us to calculate the asymptotic formula for the survival probability of an arbitrary initial state $\ket{\psi}$ localized over the $N$ discrete levels. If we choose the special initial state to satisfy ${\mit \Gamma}_{n_b} (K(0))^{-1}\ket{\psi}=0$ in Eq. (\ref{eqn:rffimbothLong50}) or $(Q_1 {\mit \Gamma}_1 Q_1)^{-1}\ket{\psi}=0$ in Eq. (\ref{eqn:rffimLong1st50}), our estimations are useless and other decay laws could appear. \cite{Miyamoto(2004)}
The long time behavior of the reduced time evolution operator for the exceptional case of the second and the third kind are not examined. As is expected, in the former case, the non decaying component associated with the localized zero energy eigenstate will appear due to the divergent behavior of $Q_2\tilde{R}(z)Q_2=O(z^{-1})$ in Eq. (\ref{eqn:5.C.2.120d}).
The latter case can occur in the $N$-level cases of the model only for $N\geq 3$, which yields a more complicated situation.
In the whole of the paper, we assumed that there is no bound eigenstate with a positive eigenenergy. This situation is actually realized in the weak coupling cases. \cite{Davies(1974),Miyamoto(2005)} However, its compatibility with the existence of the extended zero energy eigenstate is still not clear in the multilevel cases (except the single level case).
The emergence of the logarithmic decay $(\log t)^{-1}$ is just due to the logarithmic energy dependence of the self energy $S(\omega)$ and it comes from the assumption (\ref{eqn:formfactor1})
where $|v_n(\omega)|^2 \sim c_n \omega^{q_n}$ with a positive integer $q_n$ is required.
Therefore, if we choose another type of form factor, it is not necessary for such a slow decay to occur even in the exceptional case. \cite{Kofman(1994),Lewenstein(2000),Nakazato(2003)} However, we stress that our assumption is often satisfied by actual systems. \cite{Facchi(1998),Antoniou(2001)}
The experimental realization of the exceptional case
requires the setup of parameters like $\omega_1 \simeq \lambda^2 \Lambda $, where $\Lambda$ is a typical cutoff constant. This seems in a strong coupling region to be naturally satisfied \cite{Jittoh(2005),Garcia-Calderon(2001)}, and hence it could be suggested to invoke the artificial quantum structures for a realization.
\section*{Acknowledgments}
The author would like to thank Professor I.\ Ohba and Professor H.\ Nakazato for useful comments. He also would like to express his gratitude to the organizers of the International Workshop TQMFA2005, Palermo, Italy, November 11-13, 2005, {\sl New Trends in Quantum Mechanics: Fundamental Aspects and Applications}. Discussions during the YITP workshop YITP-W-05-21 on ``Fundamental Problems and Applications of Quantum Field Theory'' were useful in completing this work. This research is partly supported by a Grant for The 21st Century COE Program
at Waseda University from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
\appendix
\section{Characteristics of self energy for the rational form factor}
\begin{lm} \label{lm:formfactor1} Suppose that $\eta (\omega)$ is a rational function, i.e., it is expressed by $\eta (\omega) =\pi (\omega) / \rho (\omega)$, where $\pi (\omega)$ and $\rho (\omega)$ are the polynomials of the degree $m$ and $n$, respectively. Furthermore, we assume that $n\geq m+1$ and $\rho (z)$ has no zeros in $[0, \infty )$. Then \begin{equation} \int_{0}^{\infty} \frac{\eta (\omega)}{\omega -\zeta } d\omega = \frac{P_{n-1}(\zeta)-\pi (\zeta) \log (-\zeta) }{\rho(\zeta)}, \label{eqn:B.30} \end{equation} for all $\zeta \in \mathbb{C}\backslash ([0, \infty ) \cup \{ a_k \}_{k=1}^{N})$, where $a_k$ is a pole of $m_k$-th order of $\eta(z)$, $N$ is the number of such poles, and
$P_n (\zeta)$ is a polynomial of $\zeta$ of the degree not greater than $n$ . For $\zeta=|\zeta|e^{i\theta}$ with $0 \leq \theta \leq 2 \pi$, we define $-\zeta =|\zeta|e^{i\phi }$ with $-\pi \leq \phi \leq \pi$.
\end{lm}
{\sl Proof} : From the fundamental theorems for the complex functions, it holds that \begin{equation} \int_{0}^{\infty} \frac{\eta (\omega)}{\omega -\zeta } d\omega = -\sum_{k=1}^{N+1} {\rm Res } \left( \frac{\eta (z)}{z -\zeta } \log (-z), z=a_k \right) , \label{eqn:B.40} \end{equation} where $a_{N+1}=\zeta$. Then the residue at $z=a_k$ for $k\leq N$ is deduced to explicitly \begin{eqnarray} &&\hspace*{-5mm}
\frac{1}{(m_k -1)!} \sum_{j=0}^{m_k -1} {m_k -1 \choose j} \left[ \left( \frac{d^{m_k -1-j}}{dz^{m_k -1-j}} (z-a_k)^{m_k} \frac{\eta (z)}{z -\zeta } \right) \left( \frac{d^j}{dz^j} \log (-z) \right) \right]_{z=a_k} \nonumber \\ &=& -\sum_{j=1}^{m_k -1} \frac{P_{m_k -1-j}(\zeta)}{(a_k -\zeta )^{m_k-j}} - \frac{P_{m_k -1}(\zeta)}{(a_k -\zeta )^{m_k}} \log (-a_k) . \label{eqn:B.50} \end{eqnarray} For $z=\zeta$, which is a simple pole, the residue becomes \begin{equation} {\rm Res } \left( \frac{\eta (z)}{z -\zeta } \log (-z), z=\zeta \right) = \eta (\zeta) \log (-\zeta) . \label{eqn:B.60} \end{equation} Therefore, by setting Eqs. (\ref{eqn:B.50}) and (\ref{eqn:B.60}) into Eq. (\ref{eqn:B.40}), one obtains Eq. (\ref{eqn:B.30}), and the proof is completed. \qed
\begin{lm}\label{lm:1st_order} Suppose that the function $\eta (\omega)$ belonging to $L^1 ([0, \infty ))$ is of the form \begin{equation} \eta (\omega ) := \omega^p r(\omega ), \label{eqn:B.130} \end{equation} where $p>1$ and $r(\omega)$ is a $C^1$-function defined in $[0,\infty)$. Then it holds that both $\eta (\omega)/\omega$ and $\eta (\omega)/\omega^2 \in L^1 ([0, \infty ))$, and \begin{equation} \lim_{E \to +0} \frac{1}{E} \left[ P\int_{0}^{\infty} \frac{\eta (\omega)}{\omega -E} d\omega - \int_{0}^{\infty} \frac{\eta (\omega)}{\omega} d\omega \right] = \int_{0}^{\infty} \frac{\eta (\omega)}{\omega^2} d\omega . \label{eqn:B.140} \end{equation} \end{lm}
{\sl Proof} : From the proof of Proposition 3.2.2 in Ref. \makebox(0,1){\Large \cite{Exner(1985)}}\hspace{3mm} , the principal value of the integral on the rhs is written by the absolutely integrable function as follows \begin{equation} P\int_{0}^{\infty} \frac{\eta (\omega)}{\omega -E} d\omega =\int_{0}^{\infty} \frac{\eta (\omega)-\eta (E)\varphi_{\delta}(\omega-E)}{\omega -E} d\omega , \label{eqn:B.150} \end{equation} for all $E>0$, where $\varphi_{\delta}(\omega)$ is a $C_0^{\infty}$-function with support $[-\delta , \delta ]$ ($0< \delta < E$), even with respect to the origin, and such that $\varphi_{\delta}(0)=1$. In the following, we choose such that $\varphi_{\delta}(\omega)=\exp[1-1/(1-(\omega/\delta)^2)]$ for $\omega \in (-\delta , \delta )$ or $0$ otherwise, and $\delta=E/2$. Then, the proof of Eq. (\ref{eqn:B.140}) is equivalent to that of
\begin{equation} \lim_{E \to +0} \int_{0}^{\infty} \left\{ \frac{1}{E} \left[ \frac{\eta (\omega)}{\omega} -\frac{\eta (\omega)-\eta (E)\varphi_{\delta}(\omega-E)}{\omega -E} \right] + \frac{\eta (\omega)}{\omega^2} \right\} d\omega =0, \label{eqn:B.160} \end{equation} which will be shown in the following. We note that the above-mentioned integrand can be rewritten as \begin{eqnarray}
&&
-E\frac{\eta (\omega)}{\omega^2 (\omega-E)} +\frac{\eta (E)\varphi_{\delta}(\omega-E)}{E(\omega -E)} \label{eqn:B.170} \\ && = \frac{\eta (E)\varphi_{\delta}(\omega-E)}{E\omega} -\frac{E \eta (\omega)-\omega \eta (E)\varphi_{\delta}(\omega-E)}{\omega^2 (\omega -E)}. \label{eqn:B.180} \end{eqnarray} We also put $I_1 =(0, E/2]$, $I_2=(E/2, 3E/2)$, and $I_3=[3E/2, \infty)$.
Let us first consider the case that $\omega \in I_1 \cup I_3$. Then, since $\varphi_{\delta} (\omega-E)=0$, we can use Eq. (\ref{eqn:B.170}) to estimate the integrand in Eq. (\ref{eqn:B.160}), which reads \begin{equation}
\left| E\frac{\eta (\omega)}{\omega^2 (\omega-E)}
\right|
\leq 2 \left| \frac{\eta (\omega)}{\omega^2}
\right| , \label{eqn:B.190} \end{equation} where the rhs is absolutely integrable and independent of $E$. Furthermore, it follows that $\lim_{E \to +0} E \chi_{I_1 \cup I_3} (\omega ) \eta (\omega)/[\omega(\omega-E)] =0$ for every $\omega \in (0, \infty )$, where $ \chi_{I_1 \cup I_3} (\omega ) = 1$ ($\omega \in I_1 \cup I_3$) or $0$ ($\omega \in I_2$), being the characteristic function. Thus, by the dominated convergence theorem, we can see that \begin{equation} \lim_{E \to +0} \int_{I_1 \cup I_3} E\frac{\eta (\omega)}{\omega^2 (\omega-E)} d\omega = \lim_{E \to +0} \int_{0}^{\infty} E \chi_{I_1 \cup I_3} (\omega ) \frac{\eta (\omega)}{\omega^2 (\omega-E)} d\omega =0. \label{eqn:B.200} \end{equation}
Next, for $\omega \in I_2$, we can use Eq. (\ref{eqn:B.180}). The integration of the first term of Eq. (\ref{eqn:B.180}) is estimated by \begin{equation}
\left| \int_{I_2} \frac{\eta (E)\varphi_{\delta}(\omega-E)}{E\omega} d\omega
\right| \leq \frac{\eta (E)}{E^2/2} \int_{I_2} \varphi_{\delta}(\omega-E) d\omega = \frac{\eta (E)}{E} \int_{-1}^{1} \varphi_{1}(x) dx \to 0, \label{eqn:B.210} \end{equation} as $E \to +0$, because $\eta (\omega)=O(\omega^p )$ where $p>1$. The second term of Eq. (\ref{eqn:B.180}) is also estimated with the decomposition \begin{equation}
|E\eta(\omega)-\omega \eta(E)\varphi_{\delta}(\omega-E)| \leq
|(E-\omega)\eta(\omega)|+|\omega (\eta(\omega) -\eta(E))|
+|\omega \eta(E)||1-\varphi_{\delta}(\omega-E)| . \label{eqn:B.220} \end{equation}
The integral corresponding the first term on the rhs of the above is evaluated as \begin{equation} \int_{I_2}
\frac{|(E-\omega )\eta (\omega)|}{\omega^2 |\omega -E|} d\omega \to 0,
\label{eqn:B.235b} \end{equation} as $E \to +0$, because of the fact $\eta (\omega)/\omega^2 \in L^1 ([0, \infty))$. The integral corresponding the second term is also evaluated as \begin{equation}
\int_{I_2} \frac{\omega|\eta (\omega)-\eta (E)|}{\omega^2 |\omega -E|} d\omega \leq
(\ln 3) \sup_{\omega \in I_2} \left| \eta^{\prime}(\omega) \right|
\to 0,
\label{eqn:B.230b} \end{equation} as $E \to +0$, because of the assumption on $\eta (\omega)$, where $\eta^{\prime} (\omega)$ is the derivative of $\eta (\omega)$. The integral corresponding the last term on the rhs of Eq. (\ref{eqn:B.220}) is also estimated as \begin{equation} \int_{I_2}
\frac{\omega|\eta (E)||1-\varphi_{\delta}(\omega-E)|}{\omega^2 |\omega -E|} d\omega \leq
(\ln 3) \frac{|\eta (E)|}{\delta} \sup_{|x| \leq 1} \left| \varphi_{1}^{\prime}(x) \right| \to 0, \label{eqn:a140} \end{equation} as $E \to +0$. Thus, we see from Eqs. (\ref{eqn:B.235b}), (\ref{eqn:B.230b}), and (\ref{eqn:a140}), \begin{equation} \lim_{E \to +0} \int_{I_2} \frac{E \eta (\omega)-\omega \eta (E)\varphi_{\delta}(\omega-E)}{\omega^2 (\omega -E)} d\omega=0. \label{eqn:B.240b} \end{equation} Equations (\ref{eqn:B.200}), (\ref{eqn:B.210}), and (\ref{eqn:B.240b}) mean the completion of the proof of Eq. (\ref{eqn:B.160}). \qed
\section{Asymptotic expansion of the Fourier integrals}
We have to estimate the integrals of the form $ U(t) = \int_0^{\infty} e^{-it\lambda} F(\lambda)d\lambda
$ in which $F(\lambda)=0$ identically either for small $\lambda >0$ or for large $\lambda$ where $F$ is supposed to take values in an arbitrary Banach space ${\bf B}$. The following lemma is essentially the same as Lemma 10.2 in Ref. \makebox(0,1){\Large \cite{Jensen(1979)}}\hspace{2mm}.
\begin{lm} \label{lm:Jensen and Kato, Lemma 10.2} : Suppose that $F(\lambda)=0$ for $\lambda > a >0$, ($F \in C^{k+1} (\delta, \infty; {\bf B})$), $F^{(k+1)} \in L^1 (\delta, \infty; {\bf B})$ for any $\delta>0$ and for an integer $k \geq 0$, and that
$F^{(j)} (0)=0$ for $j\leq k-1$. Then \begin{equation}
\| U(t) \| \leq \frac{1}{t^k} \left(
\int_{0}^{2\pi/t} \| F^{(k)}(\lambda) \| d\lambda + \frac{\pi}{2t}\int_{\pi/t}^{a}
\sup_{\mu \in [\lambda, \lambda+\pi/t]} \| F^{(k+1)} (\mu) \|d\lambda \right), \label{eqn:A.50a} \end{equation} for all $t>\pi/a$. Here $F^{(k)} (\lambda)$ denotes the $k$-th derivative of $F(\lambda)$ and so forth. \end{lm}
{\sl Proof} : By extending $F$ by $F(\lambda)=0$ to $\lambda<0$, we obtain a function $F$ on $(-\infty,\infty)$ with $F^{(k)} \in L^1 (-\infty,\infty;{\bf B})$. Then we have that \begin{eqnarray} \hspace*{-8mm} \int_{-\infty}^{\infty}
\| F^{(k)}(\lambda+h) -F^{(k)} (\lambda) \| d\lambda &=& \left( \int_{-\infty}^{h} +\int_{h}^{a} \right)
\| F^{(k)}(\lambda+h) -F^{(k)} (\lambda) \| d\lambda \label{eqn:A100c} \\ &\leq&
2\int_{0}^{2h} \| F^{(k)}(\lambda) \| d\lambda
+\int_{h}^{a} d\lambda \int_{\lambda}^{\lambda+h} \| F^{(k+1)} (\mu) \|d\mu \label{eqn:A100} \\ &\leq&
2\int_{0}^{2h} \| F^{(k)}(\lambda) \| d\lambda
+h\int_{h}^{a} \sup_{\mu \in [\lambda, \lambda+h]} \| F^{(k+1)} (\mu) \|d\lambda . \label{eqn:A100b} \end{eqnarray} By noting that $e^{-it\lambda}=-e^{-it(\lambda -\pi/t)}$, one sees that the lhs of Eq. (\ref{eqn:A100c}) is just an upper bound of the Fourier transform of $2F^{(k)}$. Remember that the Fourier transform of $F^{(k)}$ is equal to $(it)^k U(t)$ under the assumption of the lemma, i.e., $F^{(j)} (0)=0$ for $j\leq k-1$, then the desired result follows. \qed
\begin{lm} \label{lm:asymptote of inverse-lagarithmic-fourier-integral} Suppose that $\phi \in C_0 ^{\infty} ([0,\infty ) )$ and satisfies $\phi (\omega ) =1$ in a neighborhood of $\omega = 0$.
It then holds that for any positive integer $q \geq 2$ and $N \geq 1$, \begin{eqnarray} \int_0^\infty \omega^{-1} (\log \omega)^{-q} \phi(\omega) e^{-it\omega} d\omega &=& \frac{(-1)^{q}}{q-1} \sum_{j=0}^{N-1} {q+j-2 \choose j} (\log t)^{1-q-j} \left( \frac{d}{d\nu} -\frac{\pi}{2}i \right)^j \Gamma(\nu)
|_{\nu=1} \nonumber \\ && +O((\log t)^{1-q-N}) \label{eqn:B.500a}
\end{eqnarray} as $t \to \infty$.
\end{lm}
{\sl Proof} : We first put $\sigma (\omega)=\omega^{-1} (\log \omega)^{-q} e^{-it\omega}$ and introduce the indefinite integral operator $\hat{I}$ as \cite{Copson} \begin{equation} (\hat{I}\sigma)(\omega)=\int_c^\omega s^{-1} (\log s)^{-q} e^{-its} ds, \label{eqn:B.410a} \end{equation} where $c$ is an arbitrary complex number. We can also recursively show \begin{equation} (\hat{I}^k \sigma)(\omega) =\frac{1}{(k-1)!}\int_c^\omega (\omega-s)^{k-1} s^{-1} (\log s)^{-q} e^{-its} ds. \label{eqn:B.410b} \end{equation} Then, repeating the partial integration, we obtain \begin{equation} \int_0^\infty \omega^{-1} (\log \omega)^{-q} \phi(\omega) e^{-it\omega} d\omega
=
-(\hat{I} \sigma)(\omega)\phi(\omega) |_{\omega=0} +R_N (t), \label{eqn:B.420b} \end{equation} for all $N\geq 1$ with \begin{equation} R_N (t)=(-1)^N \int_0^\infty (\hat{I}^N \sigma)(\omega) d^N \phi(\omega)/d\omega^N d\omega, \label{eqn:B.425} \end{equation} where we used the fact that $d^k \phi(\omega)/d\omega^k =\delta_{k0}$ at $\omega=0$ for any $k \geq 0$, and $d^k \phi(\omega)/d\omega^k =0$ at $\omega=\infty$ for any $k \geq 0$. We now choose $c=\omega-i\infty$ and change the variable as $s:=\omega-i\eta$, which leads to \begin{equation} (\hat{I}\sigma)(\omega) =i e^{-it\omega} \int_0^\infty (\omega-i\eta)^{-1} [\log (\omega-i\eta)]^{-q} e^{-t\eta} d\eta. \label{eqn:B.530} \end{equation} Then, we can use the dominated convergence theorem to obtain \begin{equation} \lim_{\omega \to +0} (\hat{I}\sigma)(\omega) =i \int_0^\infty (-i\eta)^{-1} [\log (-i\eta)]^{-q} e^{-t\eta} d\eta. \label{eqn:B.535} \end{equation} Here we used the fact that there is a positive number $\omega_0 <1/(e^{q}\sqrt{2})$ such that \begin{eqnarray}
|(\omega-i\eta)^{-1} [\log (\omega-i\eta)]^{-q} e^{-t\eta} | &\leq& \left\{ \begin{array}{cc}
\eta^{-1} |\log \eta |^{-q} e^{-t\eta} & (0<\eta <\omega_0)\\ C e^{-t\eta} &(\omega_0 \leq \eta ) \end{array} \right. \nonumber\\ &\leq&
\bigl[\chi_{\eta\leq \omega_0}(\eta) \eta^{-1} |\log \eta |^{-q} +C\bigr]e^{-t\eta} , \label{eqn:B.537b} \end{eqnarray} for all $0<\omega \leq\omega_0$ and all $0<\eta<\infty$, where $\chi_{\eta\leq \omega_0}(\eta)=1$ for $\eta\leq \omega_0$ or $0$ otherwise, and $C$ is an appropriate constant. The existence of such a $C$ is ensured by the fact that $\log (\omega-i\eta)$ has no zeros in the rectangular region
$\{ \omega-i\eta | 0<\omega\leq\omega_0, \omega_0\leq \eta<\infty \}$, and its modulus diverges as $\eta \to \infty$. The function on the rhs of Eq. (\ref{eqn:B.537b}) is integrable, so that the use of the dominated convergence theorem is valid.
To evaluate the asymptotic behavior of Eq. (\ref{eqn:B.535}), putting $\epsilon =(\log t)^{-1}$ and $t\eta=\xi $, one obtains \begin{eqnarray} &&\hspace*{-10mm} \lim_{\omega \to +0} (\hat{I}\sigma)(\omega) = i (-\epsilon)^{q} \int_0^\infty \frac{-i\epsilon^{-1}}{1-q} \left(\frac{d}{d\xi}[1-\epsilon \log (-i\xi)]^{1-q} \right)e^{-\xi} d\xi \label{eqn:B.540a}\\ &&\hspace*{-10mm} = - \frac{(-\epsilon)^{q-1}}{1-q} \int_0^\infty [1-\epsilon \log (-i\xi)]^{1-q} e^{-\xi} d\xi \label{eqn:B.540b}\\
&&\hspace*{-10mm} = - \frac{(-\epsilon)^{q-1}}{1-q} \biggl\{ \sum_{j=0}^{N-1} {q+j-2 \choose j} \epsilon^j \left( \frac{d}{d\nu} -\frac{\pi}{2}i \right)^j \Gamma(\nu)
|_{\nu=1} \nonumber\\ &&\hspace*{-5mm} + \epsilon^N N{q+N-2 \choose N} \int_0^\infty \left[ \int_0^1 \frac{(1-u)^{N-1}}{[1-\epsilon u \log (-i\xi)]^{q+N}} du \right] [\log (-i\xi)]^N e^{-\xi} d\xi \biggr\}
\label{eqn:B.540e} \end{eqnarray} where we used the formulas that $f(\epsilon)=\sum_{j=0}^{N-1}\epsilon^j f^{(j)}(0)/j! + [\epsilon^N/(N-1)!]\int_0^1 (1-u)^{N-1} f^{(N)}(\epsilon u)du$ with $f(\epsilon)=[1-\epsilon \log (-i\xi)]^{1-q}$, and $\int_{0}^{\infty} x^{\nu-1} e^{-\mu x} (\log x)^j dx =\partial^j[\mu^{-\nu}\Gamma(\nu)]/\partial \nu^j$. The last integral in Eq. (\ref{eqn:B.540e}) turns out to be finite because we have \begin{equation}
\left| \int_0^\infty \left[ \int_0^1 \frac{(1-u)^{N-1}}{[1-\epsilon u \log (-i\xi)]^{q+N}} du \right] [\log (-i\xi)]^N e^{-\xi} d\xi
\right| \leq \int_0^\infty \frac{[(\pi/2)^2+(\log \xi)^2]^{q/2+N}}{(\pi/2)^{q+N}} e^{-\xi} d\xi, \label{eqn:B.545c} \end{equation} where we used that \begin{equation}
|1-\epsilon u \log (-i\xi)|^2 = (1-\epsilon u \log \xi)^2 +(\epsilon u \pi/2)^2
\geq \frac{(\pi/2)^2}{(\pi/2)^2+(\log \xi)^2} . \label{eqn:B.547c} \end{equation}
Let us now evaluate the upperbound of $|(\hat{I}^N \sigma)(\omega)|$.
Using the estimation (\ref{eqn:B.537b}), we have \begin{eqnarray}
|(\hat{I}^N \sigma)(\omega)| &=& \frac{1}{(N-1)!}
\left| \int_0^\infty (i\eta)^{N-1}(\omega-i\eta)^{-1} [\log (\omega-i\eta)]^{-q} e^{-t\eta} d\eta
\right| \label{eqn:B.550a}\\
&\leq& \frac{1}{(N-1)!} \int_0^\infty
[\eta^{N-2} |\log \omega_0|^{-q} +C\eta^{N-1} ]e^{-t\eta}d\eta =O(t^{-N+1}). \label{eqn:B.550bb}
\end{eqnarray} Substituting this result into the error term $R_N (t)$ in Eq. (\ref{eqn:B.425}), we see that $R_N (t)=O(t^{-N+1})$ for any integer $N\geq2$. This proves the statement of the lemma. \qed
\end{document} | arXiv |
\begin{document}
\begin{large} \title{\bf On Beurling's sampling theorem in ${\mathbb R}^n$}
\author{Alexander Olevskii and Alexander Ulanovskii}
\maketitle
\begin{abstract} We present an elementary proof of the classical Beurling sampling theorem which gives a sufficient condition for sampling of multi--dimensional band--limited functions. \end{abstract}
\section{Introduction}
Let $\Ss\subset {\mathbb R}^n, n\geq1,$ be a compact. The Bernstein space $ B_\Ss$ consists of all
bounded functions on ${\mathbb R}^n$ whose spectrum belongs to $\Ss$. The latter means that $$ \int_{{\mathbb R}^n}f(x)\hat\varphi(x)\,dx=0, \ \ f\in B_\Ss, $$ for every smooth function $\varphi(x)$ whose support belongs to a ball disjoint from $\Ss.$ Here $\hat \varphi$ denotes the Fourier transform $$ \hat\varphi(x)=\int_{{\mathbb R}^n}e^{-i t\cdot x}\varphi(t)\,dt. $$
A set $\Lambda\subset {\mathbb R}^n$ is called a sampling set for $B_S$, if there is a positive constant $C$ such that $$
\Vert f\Vert_\infty\leq C\Vert f|_\Lambda\Vert_\infty, \ \mbox{for every } \ f\in B_S, $$ where $$
\Vert f\Vert_\infty:=\sup_{x\in {\mathbb R}^n}|f(x)|, \ \ \Vert f|_\Lambda\Vert_\infty:=\sup_{\lambda\in\Lambda}|f(\lambda)|. $$ It is a classical problem to determine when $\Lambda$ constitutes a sampling set for $B_\Ss.$ Beurling discovered the importance of the {\it lower uniform density $D^-(\Lambda)$} of $\Lambda$ for this problem:
$$D^-(\Lambda):= \lim_{r\to\infty}\frac{\min_{x\in{\mathbb R}^n}\mbox{Card}(\Lambda\cap (x+r\B))}{|r\B|},
$$where $\B$ is the unit ball in ${\mathbb R}^n$, $x+r\B$ is the ball of radius $r$ centered at $x$ and $|\Ss|$ denotes the measure of a set $\Ss$. In \cite{b1} he proved the following
\noindent {\bf Theorem 1} {\sl Let $\Ss=[a,b]\subset{\mathbb R}$. Then $\Lambda\subset{\mathbb R}$ is a sampling set for $B_{\Ss}$ if and only if
\begin{equation}D^-(\Lambda)>|\Ss|/{2 \pi}.\end{equation}}
Hence, when $\Ss$ is an interval in ${\mathbb R}$, the sampling problem can be solved in terms of the density $D^-(\Lambda).$ Condition $D^-(\Lambda)\geq |\Ss|/(2 \pi)^n$ remains {\it necessary} for sampling in $B_\Ss$, for every compact set $\Ss\subset{\mathbb R}^n.$ This follows from a general result of Landau \cite{l}. On the other hand, simple examples show that in dimension one condition (1) ceases to be {\it sufficient} already when $\Ss$ is a union of two intervals.
A new phenomenon occurs in several dimensions: Even for the simplest sets $\Ss$ like a ball or a cube,
no sufficient conditions for sampling in $B_\Ss$ can be expressed in terms of $D^-(\Lambda)$.
The reason for that is that the zeros of the multi-dimensional entire functions are not discrete. One can check that if $\Ss\subset{\mathbb R}^n$ contains at least two points, then there are functions $f\in B_\Ss$ whose zero set contains sets $\Lambda\subset{\mathbb R}^n$ with arbitrarily large $D^-(\Lambda)$. Clearly, if a function $f\in B_\Ss$ vanishes on $\Lambda$, then $\Lambda$ is not a sampling set for $B_\Ss$
(see also discussion in \cite{s}, pp. 122--123).
In \cite{b} Beurling obtained the following sufficient condition for sampling in $B_\B$:
\noindent {\bf Theorem 2}
{\sl Assume $\Lambda\subset{\mathbb R}^n,n\geq1,$ and $\rho<\frac{\pi}{2}$ satisfy $$ \Lambda+\rho\B={\mathbb R}^n. $$ Then \begin{equation}
\Vert f\Vert_\infty\leq \frac{1}{1-\sin \rho} \Vert f|_\Lambda\Vert_\infty,\ \mbox{ for every } f\in B_{\B}, \end{equation}
and so $\Lambda$ is a sampling set for $B_\B$. }
In fact, Beurling in \cite{b} proves a result on balayage of Fourier--Stieltjes transforms which is equivalent to Theorem 2: {\it For every Dirac's measure $\delta_\xi$, there exists a finite measure with masses on $\Lambda$ such that the values of their Fourier--Stieltjes transforms agree in the ball $\B$}. We use a completely different elementary approach which allows us to get a more general result, see Theorem 3 below. We shall see that unlike the case of interpolation in several dimensions (see \cite{o}), the "Beurling-type" sampling is in fact a one-dimensional phenomenon.
Observe that condition $\Lambda+\rho\B={\mathbb R}^n$ in Theorem 2 means that $\Lambda$ is an $\rho$-net, i.e. for every $x\in{\mathbb R}^n$ there exists $\lambda\in\Lambda$ with $|x-\lambda|\leq\rho.$ Hence, every $\rho$-net with $\rho<\pi/2$ is a sampling set for $B_\B$. This is sharp: Beurling shows that the theorem ceases to be true for $\pi/2-$nets.
Let us in what follows denote by $\K$ a closed convex central-symmetric body with positive measure. Then
$$
\K^o:=\{x\in {\mathbb R}^n: x\cdot t\leq 1 \mbox{ for all } t\in \K\}
$$
denotes the polar body of $\K$. In particular, we have $\B^o=\B.$
The following propositions are formulated in \cite{b} without proof:
(i) Estimate (2) in Theorem 2 can be replaced with a better one: \begin{equation}\label{bb}
\Vert f\Vert_\infty\leq \frac{1}{\cos \rho}\Vert f|_\Lambda\Vert_\infty.
\end{equation}
(ii) Every set $\Lambda$ satisfying $\Lambda+\rho\K^o={\mathbb R}^n$ with some $\rho<\pi/2$ is a sampling set for $B_\K.$
We show that estimate (3) holds for every convex central-symmetric body $\K$:
\noindent
{\bf Theorem 3} {\sl Assume $\Lambda\subset{\mathbb R}^n$ and $\rho<\frac{\pi}{2}$ satisfy \begin{equation}\label{b}
\Lambda+\rho\K^o={\mathbb R}^n. \end{equation} Then (\ref{bb}) is true, and so
$\Lambda$ is a sampling set for $B_\K$. }
Clearly, condition (\ref{b}) means that for every $x\in{\mathbb R}^n$ there exists $\lambda\in\Lambda$ such that $\Vert x-\lambda\Vert_{\K^o}\leq\rho$, where $\Vert x\Vert_{\K^o}:=\inf_{a>0}\{x\in a \K^o\}$. Hence, every $\rho$-net in the norm $\Vert\cdot\Vert_{\K^o}$ is a sampling set for $B_\K$ provided $\rho<\pi/2$. This is sharp:
\noindent {\bf Proposition 1}. {\sl Suppose a closed convex central-symmetric body $\Ss$ contains a point $x_0$ with $\Vert x_0\Vert_{\K^o}=\pi/2$. Then there exists $\Lambda\subset{\mathbb R}^n$ with $\Lambda+\Ss={\mathbb R}^n$ and a function $f\in B_\K$ such that $f(\lambda)=0,\lambda\in\Lambda$.}
\noindent {\bf Corollary 1}. {\sl Suppose a closed convex central-symmetric body $\Ss$ has the property that every set $\Lambda\subset{\mathbb R}^n$ satisfying $\Lambda+\Ss={\mathbb R}^n$ is a sampling set for $B_\K$. Then $\Ss\subset \rho \K^o$ for some $\rho<\pi/2.$}
\section{Proofs}
\noindent{\bf 1. Proof of Proposition 1}.
By assumption, there exist $x_0\in\Ss$ and $t_0\in \K$ such that $x_0\cdot t_0=\pi/2.$ The spectrum of the function $\sin(x\cdot t_0)$ consists of two points $\pm t_0\in\K,$ and so $\sin(x\cdot t_0)\in B_\K.$
Let $\Lambda:=\{x\in {\mathbb R}^n: x\cdot t_0\in \pi{\mathbb Z}\}$ be the zero set of $\sin(x\cdot t_0)$. Denote by
$I=\{\tau x_0: -1\leq\tau\leq 1\}\subseteq \Ss$ the interval from $-x_0$ to $x_0$. Clearly, for every point $y\in{\mathbb R}^n$ there exist $n\in{\mathbb Z}$ and $-1\leq\tau\leq 1$ such that $y\cdot t_0 =\pi n-\tau \pi/2.$ Hence, $y-\tau x_0\in \Lambda$, which implies $\Lambda+I={\mathbb R}^n$. $\Box$
\noindent{\bf 2. Proof of Theorem 3}. We shall deduce Theorem 3 from the following
\noindent
{\bf Lemma 1} {\it Suppose a function $g\in B_{[-\tau,\tau]}$ satisfies $|g(0)|=\Vert g\Vert_\infty$. Then }
\begin{equation}|g(u)|\geq|g(0)|\cos(\tau u), \ \ |u|<\pi/2\tau.\end{equation}
This lemma is proved in [3, proof of Theorem 4]. For completeness of presentation, we sketch the proof below.
Let us now prove Theorem 3. Take any function $f\in B_\K$. Assume first that $|f|$ attains maximum on ${\mathbb R}^n$, i.e. $|f(x_0)|=\Vert f\Vert_\infty$ for some $x_0\in{\mathbb R}^n$. By (4), there exists $\lambda_0\in\Lambda$ with $\Vert\lambda_0-x_0\Vert_{\K^o}\leq\rho$.
Consider the function of one variable $g(u):=f(x_0+u(\lambda_0-x_0)), u\in{\mathbb R}$.
One may check that $g\in B_{[-\tau,\tau]}$ with $\tau=\Vert\lambda_0-x_0\Vert_{\K^o}$. Also, clearly $|g(0)|=\Vert g\Vert_\infty$ and $g(1)=f(\lambda_0)$. Since $\tau\leq\rho<\pi/2$, we may use inequality (5) with $u=1$:
$$\Vert f\Vert_\infty=|f(x_0)|=|g(0)|\leq \frac{|g(1)|}{\cos \tau}\leq
\frac{|f(\lambda_0)|}{\cos \rho}\leq\frac{1}{\cos \rho}\Vert f|_\Lambda\Vert_\infty. $$
If $|f|$ does not attain maximum on ${\mathbb R}^n$, we consider the function $f_\epsilon(x):=f(x)\varphi(\epsilon x)$, where
$\varphi\in B_{\epsilon\B}$ is any function satisfying $\varphi(0)=1$ and $\varphi(x)\to0$ as $|x|\to\infty.$
It is clear that $f_\epsilon\in B_{\K+\epsilon\B}$ and that $f_\epsilon$ attains maximum on ${\mathbb R}^n$. Set $g_\epsilon(u):=f_\epsilon(x_0+u(\lambda_0-x_0)), u\in{\mathbb R},$ where $x_0$ and $\lambda_0$ are chosen so that $|g_\epsilon(0)|=\Vert f_\epsilon\Vert_\infty$ and $\Vert\lambda_0-x_0\Vert_{\K^o}\leq\rho$. We have $g\in B_{[-\tau-\delta,\tau+\delta]} $,
where $\tau=\Vert\lambda_0-x_0\Vert_{\K^o}\leq\rho<\pi/2$ and $\delta=\delta(\epsilon)\to0$ as $\epsilon\to0.$ So, if $\epsilon$ is so small that $\tau+\epsilon<\pi/2$, we may repeat the argument above to
obtain $\Vert f_\epsilon\Vert_\infty\leq \Vert f_\epsilon|_\Lambda\Vert_\infty/\cos(\rho+\delta)$. By letting $\epsilon\to 0$, we obtain (3). $\Box$
\noindent{\bf 3. Proof of Lemma 1}
1. The proof in \cite{c} is based on the following result from \cite{d} (for some extension see \cite{h}): {\sl Let $f\in B_{[-\tau,\tau]}$ be a real function satisfying $-1\leq f(x)\leq 1$ for all $x\in{\mathbb R}$. Then for every real $a$ the function $\cos (\tau z + a) - f(z)$ vanishes identically or else it has only real zeros. Moreover it has a zero in every interval where $\cos (\tau z + a)$ varies between -1 and 1 and all the zeros are simple, except perhaps at points on the real axis where $f(x) = \pm 1.$}
Sketch of proof. We may assume $a=0$ and $\tau=1$. Consider the function $$
f_\epsilon(z):=(1-\epsilon)\frac{\sin (\epsilon z)}{\epsilon z}f((1-\epsilon)z). $$ One may check that $f_\epsilon\in B_{[-1,1]}$, $-1<f(t)<1, t\in{\mathbb R},$ and that the estimate holds $$
|f_\epsilon (z)|\leq \frac{e^{ |y|}}{\epsilon |z|}, z=x+iy \in{\mathbb C}.
$$ This shows that $|f_\epsilon(z)|<|\cos z|$ when $z$ lies on a rectangular contour $\gamma$ consisting of segments of the lines $x = \pm N\pi, y= \pm N,$ where $N$ is every large enough integer. By Rouch\`e's theorem, the function $\cos z-f_\epsilon(z)$
has the same number of zeros in $\gamma$ as $\cos z$, that is, $2N$ zeros. On the real axis $|f_\epsilon|\leq 1-\epsilon $. Hence, $\cos z-f_\epsilon(z)$ is alternately plus and minus at the $2N+1$ points $k\pi$, $|k|\leq N,$ so it has $2N$ real zeros inside $\gamma$. Taking larger values of $N$ we see that $\cos z-f_\epsilon(z)$ has exclusively real and simple zeros, which lie in the intervals $(k\pi,(k+1)\pi)$.
The zeros of $\cos z-f(z)$ are limit points of the zeros of $\cos z-f_\epsilon(z)$ as $\epsilon\to 0$. Thus $\cos z-f(z)$ cannot have non-real zeros. Moreover, it has an infinite number of real zeros which are all simple, except those at the points $k\pi$ iff $f(k\pi) = ( - 1)^{k}.$ Every interval $k\pi<z<(k + 1)\pi$ at the endpoints of which
$| f (t) | < 1$ contains exactly one zero. If $f(k\pi) = (-1)^{k}$, we have a double zero at $k\pi$ but no further zeros in the interior or at the endpoints of the interval $((k - 1)\pi, (k+ 1)\pi). $
2. It suffices to prove Lemma 1 for real functions $f\in B_{[-\tau,\tau]}$. Since $f$ has a local maximum at $t=0$, the function $f(t)-\cos\tau t$ has a repeated zero at $t=0. $
By the discussion above we see that either $f(t)$ is identically equal to $\cos\tau t$ or $f(t)-\cos\tau t$ does not vanish on $[-\pi/\tau,0)\cup(0,\pi/\tau]$. Since $|f(\pi)|\leq 1$, it follows that $f(t)>\cos\tau t$ on each of the intervals $[-\pi/\tau,0)$ and $(0, \pi/\tau]$. $\Box$
\end{large}
\end{document} | arXiv |
Berlekamp–Zassenhaus algorithm
In mathematics, in particular in computational algebra, the Berlekamp–Zassenhaus algorithm is an algorithm for factoring polynomials over the integers, named after Elwyn Berlekamp and Hans Zassenhaus. As a consequence of Gauss's lemma, this amounts to solving the problem also over the rationals.
The algorithm starts by finding factorizations over suitable finite fields using Hensel's lemma to lift the solution from modulo a prime p to a convenient power of p. After this the right factors are found as a subset of these. The worst case of this algorithm is exponential in the number of factors.
Van Hoeij (2002) improved this algorithm by using the LLL algorithm, substantially reducing the time needed to choose the right subsets of mod p factors.
References
• Berlekamp, E. R. (1967), "Factoring polynomials over finite fields", Bell System Technical Journal, 46 (8): 1853–1859, doi:10.1002/j.1538-7305.1967.tb03174.x, MR 0219231.
• Berlekamp, E. R. (1970), "Factoring polynomials over large finite fields", Mathematics of Computation, 24 (111): 713–735, doi:10.2307/2004849, JSTOR 2004849, MR 0276200.
• Cantor, David G.; Zassenhaus, Hans (1981), "A new algorithm for factoring polynomials over finite fields", Mathematics of Computation, 36 (154): 587–592, doi:10.2307/2007663, JSTOR 2007663, MR 0606517.
• Geddes, K. O.; Czapor, S. R.; Labahn, G. (1992), Algorithms for computer algebra, Boston, MA: Kluwer Academic Publishers, Bibcode:1992afca.book.....G, doi:10.1007/b102438, ISBN 0-7923-9259-0, MR 1256483.
• Van Hoeij, Mark (2002), "Factoring polynomials and the knapsack problem", Journal of Number Theory, 95 (2): 167–189, doi:10.1016/S0022-314X(01)92763-5, MR 1924096.
• Zassenhaus, Hans (1969), "On Hensel factorization. I", Journal of Number Theory, 1 (3): 291–311, Bibcode:1969JNT.....1..291Z, doi:10.1016/0022-314X(69)90047-X, MR 0242793.
External links
• Domazet, Haris. "Berlekamp-Zassenhaus Algorithm". MathWorld.
See also
• Berlekamp's algorithm
| Wikipedia |
Erdős–Tetali theorem
In additive number theory, an area of mathematics, the Erdős–Tetali theorem is an existence theorem concerning economical additive bases of every order. More specifically, it states that for every fixed integer $h\geq 2$, there exists a subset of the natural numbers ${\mathcal {B}}\subseteq \mathbb {N} $ satisfying
$r_{{\mathcal {B}},h}(n)\asymp \log n,$
where $r_{{\mathcal {B}},h}(n)$ denotes the number of ways that a natural number n can be expressed as the sum of h elements of B.[1]
The theorem is named after Paul Erdős and Prasad V. Tetali, who published it in 1990.
Motivation
The original motivation for this result is attributed to a problem posed by S. Sidon in 1932 on economical bases. An additive basis ${\mathcal {B}}\subseteq \mathbb {N} $ is called economical[2] (or sometimes thin[3]) when it is an additive basis of order h and
$r_{{\mathcal {B}},h}(n)\ll _{\varepsilon }n^{\varepsilon }$
for every $\varepsilon >0$. In other words, these are additive bases that use as few numbers as possible to represent a given n, and yet represent every natural number. Related concepts include $B_{h}[g]$-sequences[4] and the Erdős–Turán conjecture on additive bases.
Sidon's question was whether an economical basis of order 2 exists. A positive answer was given by P. Erdős in 1956,[5] settling the case h = 2 of the theorem. Although the general version was believed to be true, no complete proof appeared in the literature before the paper by Erdős and Tetali.[6]
Ideas in the proof
The proof is an instance of the probabilistic method, and can be divided into three main steps. First, one starts by defining a random sequence $\omega \subseteq \mathbb {N} $ by
$\Pr(n\in \omega )=C\cdot n^{{\frac {1}{h}}-1}(\log n)^{\frac {1}{h}},$
where $C>0$ is some large real constant, $h\geq 2$ is a fixed integer and n is sufficiently large so that the above formula is well-defined. A detailed discussion on the probability space associated with this type of construction may be found on Halberstam & Roth (1983).[7] Secondly, one then shows that the expected value of the random variable $r_{\omega ,h}(n)$ has the order of log. That is,
$\mathbb {E} (r_{\omega ,h}(n))\asymp \log n.$
Finally, one shows that $r_{\omega ,h}(n)$ almost surely concentrates around its mean. More explicitly:
$\Pr {\big (}\exists c_{1},c_{2}>0~|~{\text{for all large }}n\in \mathbb {N} ,~c_{1}\mathbb {E} (r_{\omega ,h}(n))\leq r_{\omega ,h}(n)\leq c_{2}\mathbb {E} (r_{\omega ,h}(n)){\big )}=1$
This is the critical step of the proof. Originally it was dealt with by means of Janson's inequality, a type of concentration inequality for multivariate polynomials. Tao & Vu (2006)[8] present this proof with a more sophisticated two-sided concentration inequality by V. Vu (2000),[9] thus relatively simplifying this step. Alon & Spencer (2016) classify this proof as an instance of the Poisson paradigm.[10]
Relation to the Erdős–Turán conjecture on additive bases
Main article: Erdős–Turán conjecture on additive bases
Unsolved problem in mathematics:
Let $ h\geq 2$ be an integer. If $ {\mathcal {B}}\subseteq \mathbb {N} $ is an infinite set such that $ r_{{\mathcal {B}},h}(n)>0$ for every n, does this imply that:
$\limsup _{n\to \infty }{\frac {r_{{\mathcal {B}},h}(n)}{\log n}}>0$?
(more unsolved problems in mathematics)
The original Erdős–Turán conjecture on additive bases states, in its most general form, that if $ {\mathcal {B}}\subseteq \mathbb {N} $ is an additive basis of order h then
$\limsup _{n\to \infty }r_{{\mathcal {B}},h}(n)=\infty ;$ ;}
that is, $ r_{{\mathcal {B}},h}(n)$ cannot be bounded. In his 1956 paper, P. Erdős[5] asked whether it could be the case that
$\limsup _{n\to \infty }{\frac {r_{{\mathcal {B}},2}(n)}{\log n}}>0$
whenever ${\mathcal {B}}\subseteq \mathbb {N} $ is an additive basis of order 2. In other words, this is saying that $ r_{{\mathcal {B}},2}(n)$ is not only unbounded, but that no function smaller than log can dominate $ r_{{\mathcal {B}},2}(n)$. The question naturally extends to $h\geq 3$, making it a stronger form of the Erdős–Turán conjecture on additive bases. In a sense, what is being conjectured is that there are no additive bases substantially more economical than those guaranteed to exist by the Erdős–Tetali theorem.
Further developments
Computable economical bases
All the known proofs of Erdős–Tetali theorem are, by the nature of the infinite probability space used, non-constructive proofs. However, Kolountzakis (1995)[11] showed the existence of a recursive set ${\mathcal {R}}\subseteq \mathbb {N} $ satisfying $r_{{\mathcal {R}},2}(n)\asymp \log n$ such that ${\mathcal {R}}\cap \{0,1,\ldots ,n\}$ takes polynomial time in n to be computed. The question for $h\geq 3$ remains open.
Economical subbases
Given an arbitrary additive basis ${\mathcal {A}}\subseteq \mathbb {N} $, one can ask whether there exists ${\mathcal {B}}\subseteq {\mathcal {A}}$ such that ${\mathcal {B}}$ is an economical basis. V. Vu (2000)[12] showed that this is the case for Waring bases $\mathbb {N} ^{\wedge }k:=\{0^{k},1^{k},2^{k},\ldots \}$, where for every fixed k there are economical subbases of $\mathbb {N} ^{\wedge }k$ of order $s$ for every $s\geq s_{k}$, for some large computable constant $s_{k}$.
Growth rates other than log
Another possible question is whether similar results apply for functions other than log. That is, fixing an integer $h\geq 2$, for which functions f can we find a subset of the natural numbers ${\mathcal {B}}\subseteq \mathbb {N} $ satisfying $r_{{\mathcal {B}},h}(n)\asymp f(n)$? It follows from a result of C. Táfula (2019)[13] that if f is a locally integrable, positive real function satisfying
• ${\frac {1}{x}}\int _{1}^{x}f(t)\,\mathrm {d} t\asymp f(x)$, and
• $\log x\ll f(x)\ll x^{\frac {1}{h-1}}(\log x)^{-\varepsilon }$ for some $\varepsilon >0$,
then there exists an additive basis ${\mathcal {B}}\subseteq \mathbb {N} $ of order h which satisfies $r_{{\mathcal {B}},h}(n)\asymp f(n)$. The minimal case f(x) = log x recovers Erdős–Tetali's theorem.
See also
• Erdős–Fuchs theorem: For any non-zero $C\in \mathbb {R} $, there is no set ${\mathcal {B}}\subseteq \mathbb {N} $ which satisfies $\textstyle {\sum _{n\leq x}r_{{\mathcal {B}},2}(n)=Cx+o\left(x^{1/4}\log(x)^{-1/2}\right)}$.
• Erdős–Turán conjecture on additive bases: If ${\mathcal {B}}\subseteq \mathbb {N} $ is an additive basis of order 2, then $r_{{\mathcal {B}},2}(n)\neq O(1)$.
• Waring's problem, the problem of representing numbers as sums of k-powers, for fixed $k\geq 2$.
References
1. Alternative statement in big Theta notation:
$r_{{\mathcal {B}},h}(n)=\Theta (\log(n)),$
2. As in Halberstam & Roth (1983), p. 111.
3. As in Tao & Vu (2006), p. 13.
4. See Definition 3 (p. 3) of O'Bryant, K. (2004), "A complete annotated bibliography of work related to Sidon sequences", Electronic Journal of Combinatorics, 11: 39.
5. Erdős, P. (1956). "Problems and results in additive number theory". Colloque sur la Théorie des Nombres: 127–137.
6. See p. 264 of Erdős–Tetali (1990).
7. See Theorem 1 of Chapter III.
8. Section 1.8 of Tao & Vu (2006).
9. Vu, Van H. (2000-07-01). "On the concentration of multivariate polynomials with small expectation". Random Structures & Algorithms. 16 (4): 344–363. CiteSeerX 10.1.1.116.1310. doi:10.1002/1098-2418(200007)16:4<344::aid-rsa4>3.0.co;2-5. ISSN 1098-2418.
10. Chapter 8, p. 139 of Alon & Spencer (2016).
11. Kolountzakis, Mihail N. (1995-10-13). "An effective additive basis for the integers". Discrete Mathematics. 145 (1): 307–313. doi:10.1016/0012-365X(94)00044-J.
12. Vu, Van H. (2000-10-15). "On a refinement of Waring's problem". Duke Mathematical Journal. 105 (1): 107–134. CiteSeerX 10.1.1.140.3008. doi:10.1215/s0012-7094-00-10516-9. ISSN 0012-7094.
13. Táfula, Christian (2019). "An extension of the Erdős-Tetali theorem". Random Structures & Algorithms. 55 (1): 173–214. arXiv:1807.10200. doi:10.1002/rsa.20812. ISSN 1098-2418. S2CID 119249787.
• Erdős, P.; Tetali, P. (1990). "Representations of integers as the sum of k terms". Random Structures & Algorithms. 1 (3): 245–261. doi:10.1002/rsa.3240010302.
• Halberstam, H.; Roth, K. F. (1983). Sequences. Springer New York. ISBN 978-1-4613-8227-0. OCLC 840282845.
• Alon, N.; Spencer, J. (2016). The probabilistic method (4th ed.). Wiley. ISBN 978-1-1190-6195-3. OCLC 910535517.
• Tao, T.; Vu, V. (2006). Additive combinatorics. Cambridge University Press. ISBN 0521853869. OCLC 71262684.
| Wikipedia |
\begin{definition}[Definition:Inverse Cosine/Real/Arccosine]
{{:Graph of Arccosine Function|Graph}}
From Shape of Cosine Function, we have that $\cos x$ is continuous and strictly decreasing on the interval $\closedint 0 \pi$.
From Cosine of Multiple of Pi, $\cos \pi = -1$ and $\cos 0 = 1$.
Therefore, let $g: \closedint 0 \pi \to \closedint {-1} 1$ be the restriction of $\cos x$ to $\closedint 0 \pi$.
Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\closedint {-1} 1$.
This function is called '''arccosine of $x$''' and is written $\arccos x$.
Thus:
:The domain of $\arccos x$ is $\closedint {-1} 1$
:The image of $\arccos x$ is $\closedint 0 \pi$.
\end{definition} | ProofWiki |
A Passage to Infinity
A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact [1][2][3] is a 2009 book by George Gheverghese Joseph chronicling the social and mathematical origins of the Kerala school of astronomy and mathematics. The book discusses the highlights of the achievements of Kerala school and also analyses the hypotheses and conjectures on the possible transmission of Kerala mathematics to Europe.
A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact
AuthorGeorge Gheverghese Joseph
LanguageEnglish
SubjectHistory of mathematics
PublisherSAGE India
Publication date
November 2009
Pages323
ISBN978-8132101680
An outline of the contents
1. Introduction
2. The Social Origins of the Kerala School
3. The Mathematical Origins of the Kerala School
4. The Highlights of Kerala Mathematics and Astronomy
5. Indian Trigonometry: From Ancient Beginnings to Nilakantha
6. Squaring the Circle: The Kerala Answer
7. Reaching for the Stars: The Power Series for Sines and Cosines
8. Changing Perspectives on Indian Mathematics
9. Exploring Transmissions: A Case Study of Kerala Mathematics
10. A Final Assessment
See also
• Indian astronomy
• Indian mathematics
• History of mathematics
References
1. Joseph, George Gheverghese (2009). A Passage to Infinity : Medieval Indian Mathematics from Kerala and Its Impact. Delhi: Sage Publications (Inda) Pvt. Ltd. p. 236. ISBN 978-81-321-0168-0.
2. Plofker, Kim (21 December 2015). "A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact". Aestimatio: Critical Reviews in the History of Science. 10: 56–62. doi:10.33137/aestimatio.v10i0.26020. ISSN 1549-4497.
3. Sriram, M.S. (2011). "Book Review: A Passage to Infinity—Medieval Indian Mathematics and Its Impact". Indian Historical Review. 38 (2): 247–250. doi:10.1177/037698361103800207. ISSN 0376-9836. S2CID 149427309.
Further references
• In association with the Royal Society's 350th anniversary celebrations in 2010, Asia House presented a talk based on A Passage to Infinity. See : "A Passage to Infinity: Indian Mathematics in World Mathematics". Retrieved 3 May 2010.
• For an audio-visual presentation of George Gheverghese Joseph's views on the ideas presented in the book, see : Joseph, George Gheverghese (16 September 2008). "George Gheverghese Joseph on the Transmission to Europe of Non-European Mathematics". The Mathematical Association of America. Archived from the original on 15 April 2010. Retrieved 3 May 2010.
• The Economic Times talks to George Gheverghese Joseph on The Passage to Infinity. See : Lal, Amrith (23 April 2010). "Indian mathematics loved numbers". The Economic Times.
• Review of "A PASSAGE TO INFINITY: Medieval Indian Mathematics from Kerala and its impact" by M. Ram Murty in Hardy-Ramanujan Journal, 36 (2013), 43–46.
• Nair, R. Madhavan (3 February 2011). "In search of the roots of mathematics". The Hindu. Retrieved 15 October 2014.
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| Wikipedia |
\begin{document}
\title{On the formula of Goulden and Rattan for Kerov polynomials}
\author{Philippe Biane}
\address{CNRS, D\'epartement de Math\'ematiques et Applications,
\'Ecole Normale Sup\'erieure, 45, rue d'Ulm 75005 paris, FRANCE } \email{[email protected]}
\subjclass{Primary ; Secondary } \date{}
\begin{abstract} We give a simple proof of an explicit formula for Kerov polynomials. This explicit formula is closely related to a recent formula of Goulden and Rattan. \end{abstract}
\maketitle
\section{Kerov polynomials} Kerov polynomials are universal polynomials which express the characters of symmetric groups evaluated on cycles, in terms of quantities known as the free cumulants of a Young diagram. We now explain these notions.\par Let $\lambda=\lambda_1\geq \lambda_2\geq\ldots$ be a Young diagram, to
which we associate a piecewise affine function
$\omega:\mathbb R\to \mathbb R$, with slopes $\pm1$,
such that $\omega(x)=|x|$ for $|x|$ large enough, as in Fig. 1 below, which
corresponds to the partition $8=4+3+1$.
We can encode the Young diagram using the local minima and local maxima of the function $\omega$, denoted by $x_1,\ldots, x_m$ and $y_1,\ldots, y_{m-1}$ respectively, which form two interlacing sequences of integers. These are (-3,-1,2,4) and (-2,1,3) respectively in the picture. $$\includegraphics{contdiag.eps}$$
$$\qquad\qquad\ x_1\quad\, y_1\quad x_2\qquad\qquad y_2\quad\, x_3\quad y_3\quad x_4\qquad$$ $$Fig.1$$ To the Young diagram we associate the rational fraction $$G_{\lambda}(z)={\prod_{i=1}^{m-1}(z-y_i)\over \prod_{i=1}^{m}(z-x_i)} $$ and the formal power series $K_{\lambda}$, inverse to $G$ for composition, $$K_{\lambda}(z)=G_{\lambda}^{(-1)}(z)=z^{-1}+\sum_{k=2}^\infty R_k(\lambda)z^{k-1}. $$
The quantities $R_k(\lambda);k=2,3,\ldots$ are called the free cumulants of
the diagram $\lambda$. Note that $R_1(\lambda)=0$ for any Young diagram, so we
do not include it
in the series of free cumulants. These quantities arise in the
asymptotic study of representations of symmetric groups, see
\cite{B1}.
It turns out that there exists universal polynomials
$\Sigma_2,\Sigma_3\ldots$ in the variables $R_2,R_3,\ldots$ such that for any
Young diagram $\lambda$ the normalized character $\chi_{\lambda}$ evaluated
on a
cycle of length $k$ is given by
$$(n)_k\chi_{\lambda}(c_k)=\Sigma_k(R_2(\lambda),R_3(\lambda),\ldots).$$
The remarkable fact here is that these polynomials do not depend on the size
of the symmetric group.
We list the first such polynomials below.
$$ \begin{array}{l} \Sigma_1=R_2\\ \Sigma_2=R_3\\ \Sigma_3=R_4+R_2\\
\Sigma_4=R_5 +5R_3\\
\Sigma_5=R_6+15R_4+5R_2^2+8R_2\\ \end{array} $$ We refer to \cite{B2} and \cite{GR} for more information about results and conjectures on the coefficients of these polynomials. We take from \cite{B2}, section 5, the following expression for Kerov polynomials. Here $[z^{-k}]\,f(z)$ denotes the coefficient of $z^{-k}$ (the residue if $k=1$) of a Laurent series $f(z)$. \begin{proposition} Consider the formal power series
$$H(z)=z-\sum_{j=2}^\infty B_jz^{1-j}.$$ Define $$\Sigma_{k}=-\frac{1}{k}\,[z^{-1}]\, H(z)H(z-1)\ldots H(z-k+1)\qquad (1)$$ and $$R_{k+1}= -{1\over k}\,[z^{-1}]\, H(z)^k$$ then the expression of $\Sigma_k$ in terms of the $R_k\,'$s is given by Kerov's polynomials. \end{proposition} Recently I. P. Goulden and A. Rattan \cite{GR}
have given an explicit expression for Kerov polynomials, from which they have deduced a certain number of positivity properties of the coefficients of these polynomials. Their proof uses the Lagrange inversion formula. In the next section we use the invariance of residue under change of variables to derive in a simple way a closely related formula, and show how to recover Goulden and Rattan's formula. \section{Explicit expression for Kerov polynomials} We use the notations of Proposition 1.1 above. Let us introduce the power series $$L(z)=z+\sum_{j=2}^\infty R_jz^{1-j}.$$ One has $H\circ L(z)=z$, by Lagrange inversion formula. We use the invariance of the residue under change of variables, namely if $u,f$ are Laurent series, and $u$ is invertible for composition, then $$[z^{-1}]\,f(z)=[\zeta^{-1}]\,u'(\zeta)f\circ u(\zeta).$$
Using Taylor's formula as well as the change of variables $z=L(\zeta)$ in the residue, one gets from (1)
$$ \begin{array}{rcl} \Sigma_{k}&=&-{1\over k}\,[z^{-1}]\,\prod_{j=0}^{k-1} \left(\sum_{r=0}^{\infty}\frac{(-j)^r}{r!}H^{(r)}(z)\right) \\ &=&-{1\over k}\,[\zeta^{-1}]\,L'(\zeta)\prod_{j=0}^{k-1} \left(\sum_{r=0}^{\infty}\frac{(-j)^r}{r!}H^{(r)}\circ L(\zeta)\right). \end{array} $$ Using $H'\circ L(\zeta)=\frac{1}{L'(\zeta)}$ one gets $H^{(r)}\circ L(\zeta)=\left(\frac{1}{L'(\zeta)}\frac{d}{d\zeta}\right)^{r-1} \frac{1}{L'(\zeta)} $ therefore $$\Sigma_{k} =-{1\over k}\,[\zeta^{-1}]\,L'(\zeta)\prod_{j=0}^{k-1} \left(\zeta+\sum_{r=1}^{\infty}\frac{(-j)^r}{r!} \left(\frac{1}{L'(\zeta)}\frac{d}{d\zeta}\right)^{r-1} \frac{1}{L'(\zeta)}\right). $$ Putting $F(\zeta)=\frac{1}{L'(\zeta)}$ we obtain the following Proposition. \begin{proposition} Let $$F(\zeta)=\frac{1}{L'(\zeta)}=\frac{1}{1-\sum_{k=2}^\infty (k-1)R_k\zeta^{-k}}$$ then Kerov's polynomials are given by the following expression $$ \Sigma_{k}=-{1\over k}\,[\zeta^{-1}]\,\frac{1}{F(\zeta)}\prod_{j=0}^{k-1} \left(\zeta+\sum_{r=1}^{\infty}\frac{(-j)^r}{r!} \left(F(\zeta)\frac{d}{d\zeta}\right)^{r-1} F(\zeta)\right).\qquad (2) $$
\end{proposition} \section {The formula of Goulden and Rattan} Goulden and Rattan give various equivalent formulas for $\Sigma_k$. They introduce the series $C(\zeta)=F(\zeta^{-1})$, and define polynomials $$P_m(z)=-\frac{1}{m!}C(z)(D+(m-2)I)\left[C(z)\ldots (D+I)\left[C(z)DC(z)\right]\ldots\right]$$
where $D=z\frac{d}{dz}$.
The generating series form of their formula now reads
$$\Sigma_k=-\frac{1}{k}[z^{k+1}]\frac{1}{C(z)}\prod_{j=1}^{k-1}(1+\sum_{i=1}^\infty j^iP_i(z)z^i).\qquad (3)$$ We recover this formula using $(2)$. For this we factor out $\zeta^k$ in the expression in the rhs of $(2)$, and use the change of variable $z=\zeta^{-1}$. This gives $$ \begin{array}{rcl} \Sigma_{k}&=&-{1\over k}\,[\zeta^{-1}]\,\frac{1}{F(\zeta)}\prod_{j=0}^{k-1} \left(\zeta+\sum_{r=1}^{\infty}\frac{(-j)^r}{r!} \left(F(\zeta)\frac{d}{d\zeta}\right)^{r-1} F(\zeta)\right)\\ &=&-{1\over k}\,[\zeta^{-1-k}] \,\frac{1}{F(\zeta)}\prod_{j=1}^{k-1} \left(1+\sum_{r=1}^{\infty}\frac{(-j)^r}{r!}\zeta^{-1} \left(F(\zeta)\frac{d}{d\zeta}\right)^{r-1} F(\zeta)\right)\\ &=&-{1\over k}\,[z^{k+1}]\,\frac{1}{C(z)}\prod_{j=1}^{k-1} \left(1+\sum_{r=1}^{\infty}\frac{(-j)^r}{r!}z \left(-C(z)z^2\frac{d}{dz}\right)^{r-1} C(z)\right)\\ &=&-{1\over k}\,[z^{k+1}]\,\frac{1}{C(z)}\prod_{j=1}^{k-1} \left(1-\sum_{r=1}^{\infty}\frac{j^rz}{r!}(C(z)zD)^{r-1}C(z) \right).\\ \end {array} $$ Now remark that $z^{-j}\circ D\circ z^j=D+jI$ to get $$\begin{array}{rcl} z(C(z)zD)^{r-1}C(z)&=&z^rC(z)(D+(r-2)I)\left[C(z)\ldots (D+I)\left[C(z)DC(z)\right]\ldots\right]\\ &=&-r!P_r(z). \end{array}$$
\end{document} | arXiv |
\begin{document}
\selectlanguage{english} \title{Newhouse phenomenon for automorphisms of low degree in $\mathbb{C}^{3}$} \author{S\'ebastien Biebler} \address{Sorbonne Universit\'e, 4 Place Jussieu, 75005 Paris} \email{[email protected]} \date{December 2019} \thanks{This research was partially supported by the ANR project LAMBDA, ANR-13-BS01-0002.} \subjclass[2000]{Primary 37F45, secondary 37C29} \keywords{complex Newhouse phenomenon, complex blender}
\begin{abstract} We show that there exists a polynomial automorphism $f$ of $\mathbb{C}^{3}$ of degree 2 such that for every automorphism $g$ sufficiently close to $f$, $g$ admits a tangency between the stable and unstable laminations of some hyperbolic set. As a consequence, for each $d \ge 2$, there exists an open set of polynomial automorphisms of degree at most $d$ in which the automorphisms having infinitely many sinks are dense. To prove these results, we give a complex analogous to the notion of blender introduced by Bonatti and D\'iaz. \end{abstract}
\maketitle
\small
\newtheorem* {mainTheorem}{Main Theorem}
\newtheorem {corollary}[subsubsection]{Corollary} \newtheorem*{co1}{Corollary 1} \newtheorem*{co2}{Corollary 2}
\newtheorem {notation}[subsubsection]{Notation} \newtheorem*{nt}{Notation}
\newtheorem{df}[subsubsection]{Definition} \newtheorem{def2}[subsubsection]{Definition-Proposition} \newtheorem*{opc}{Definition (Open Covering Property)} \newtheorem*{ber}{Definition (Blender Property)}
\newtheorem{prop}[subsubsection]{Proposition} \newtheorem{propo}[subsection]{Proposition}
\newtheorem{lemma}[subsubsection]{Lemma}
\newtheorem {remark} [subsubsection]{Remark}
\section{Introduction}
\subsection{Background}
Hyperbolic systems such as the horseshoe introduced by Smale were originally conjectured to be dense in the set of parameters in the 1960's. This was quickly discovered to be false in general for diffeomorphisms of manifolds of dimension greater than 2 (see \cite{po}). The discovery in the seventies of the so-called Newhouse phenomenon, i.e. the existence of residual sets of $C^{2}$-diffeomorphisms of compact surfaces with infinitely many sinks (periodic attractors) in \cite{n} showed it was false in dimension 2 too. The technical core of the proof is the reduction to a line of tangency between the stable and unstable foliations where two Cantor sets must have persistent intersections. This gives persistent homoclinic tangencies between the stable and unstable foliations, ultimately leading to infinitely many sinks. Indeed, it is a well known fact that a sink is created in the unfolding of a generic homoclinic tangency.
Palis and Viana showed in \cite{vi} an analogous result for real diffeomorphisms in higher dimensions. We say that a saddle periodic point of multipliers $| \lambda_{1}| \le | \lambda_{2}| < 1 <| \lambda_{3}| $ is sectionally dissipative if the product of any two of its eigenvalues is less than 1 in modulus, that is, $| \lambda_{1}\lambda_{3}| < 1$ and $| \lambda_{2}\lambda_{3}| < 1$ . More precisely, they proved that near any smooth diffeomorphism of $\mathbb{R}^{3}$ exhibiting a homoclinic tangency associated to a sectionally dissipative saddle periodic point, there is a residual subset of an open set of diffeomorphisms such that each of its elements displays infinitely many coexisting sinks.
In the complex setting, this reduction is not possible anymore and to get persistent homoclinic tangencies, we have to intersect two Cantor sets in the plane. Let us denote by $\text{Aut}_{d}(\mathbb{C}^{k})$ the space of polynomial automorphisms of $\mathbb{C}^{k}$ of degree $d$ for $d,k \ge 2$. Buzzard proved in \cite{bb1} that there exists an integer $d > 0$, an automorphism $G \in \text{Aut}_{d}(\mathbb{C}^{2})$ and a neighborhood $N \subset \mathrm{Aut}_{d}(\mathbb{C}^{2})$ of $G$ such that $N$ has persistent homoclinic tangencies. Buzzard gives an elegant criterion (see \cite{bb3}) which generates the intersection of two planar Cantor sets, hence leading to persistent homoclinic tangencies. In his article, Buzzard uses a Runge approximation argument to get a polynomial automorphism, which implies that the degree $d$ remains unknown and is supposedly very high.
In the article \cite{bd1}, Bonatti and D\'iaz introduced a type of horseshoe they called blender horseshoe. The important property of such hyperbolic sets lies in the fractal configuration of one of their stable/unstable manifold which implies persistent intersections between any well oriented graph and this foliation. In some sense, the foliation behaves just as it had greater Hausdorff dimension than every individual manifold of the foliation. They find how to get robust homoclinic tangencies for some $C^{r}$-diffeomorphism of $\mathbb{R}^{3}$ using blenders in \cite{bdv1}. In the article \cite{dks}, one can find real polynomial maps of degree 2 with a blender. Other important studies of persistent tangencies using blenders include \cite{be} and \cite{rtem}.
\subsection{Results and outline}
In this article, we generalize Buzzard's Theorem to dimension 3 and show that the degree can be controlled in this case. Here is our main result:
\begin{mainTheorem} There exists a polynomial automorphism $f$ of degree 2 of $\mathbb{C}^{3}$ such that for every $g \in \mathrm{Aut}(\mathbb{C}^{3})$ sufficiently close to $f$, $g$ admits a tangency between the stable and unstable laminations of some hyperbolic set. \end{mainTheorem}
Notice that in the previous result, $g$ is not assumed to be polynomial.
\begin{co1} For each $d \ge 2$, there exists an open subset of $ \mathrm{Aut}_{d}(\mathbb{C}^{3})$ in which the automorphisms having a homoclinic tangency are dense. \end{co1}
\begin{co2} For each $d \ge 2$, there exists an open subset $\mathrm{Aut}_{d}(\mathbb{C}^{3})$ in which the automorphisms having infinitely many sinks are dense. \end{co2}
Let us present the main ideas of the proof of this result. We consider the following automorphism of $\mathbb{C}^{3}$: \begin{equation} f_{0} : (z_{1},z_{2},z_{3}) \mapsto (p_{c}(z_{1})+b z_{2}+\sigma z_{3}(z_{1}-\alpha),z_{1},\lambda z_{1} + \mu z_{3}+\nu) \label {e} \end{equation} where $p_{c}$ is a quadratic polynomial and the coefficients $b,\sigma,\alpha,\lambda,\mu,\nu$ are complex numbers. We prove that $f_{0}$ has a horseshoe $\mathcal{H}_{f_{0}}$ of index $(2,1)$: the first direction is strongly expanded, the second one is strongly contracted and the third one is moderately contracted by $f_{0}$. Informally speaking, the third projection restricted to $\mathcal{H}_{f_{0}}$ satisfies a special "open covering property" formalized in the following definition. This is an analogous in the complex setting of the notion of cs-blender in the sense of Bonatti and D\'iaz.
\begin{ber} Let $f$ be a polynomial automorphism of $\mathbb{C}^{3}$, $D$ a tridisk of $\mathbb{C}^{3}$ and $\mathcal{H}_{f} = \bigcap_{- \infty}^{+ \infty} f^{n} (D)$ a horseshoe of index $(2,1)$. We will suppose that there exist $k>1$ and three cone fields $C^{u},C^{ss},C^{cs}$ such that in $D$: \begin{enumerate} \item $C^{u}$ is $f$-invariant, \item $C^{ss}$ is $f^{-1}$-invariant, \item every vector in $C^{u}$ is expanded by a factor larger than $k$ under $f$, \item every vector in $C^{ss}$ is expanded by a factor larger than $k$ under $f^{-1}$, \item every vector in $C^{cs}$ is expanded by a factor larger than 1 under $f^{-1}$. \end{enumerate} We say that $\mathcal{H}_{f}$ is a blender if there exists a non empty open set $D' \Subset D$ such that every curve tangent to $C^{ss}$ intersecting $D'$ intersects the unstable set $W^{u}(\mathcal{H}_{f})$ of $\mathcal{H}_{f}$. \end{ber}
Besides, we show that $f_{0}$ has a periodic point which is sectionally dissipative. Once the blender is constructed, finding persistent tangencies is not trivial. We introduce manifolds with special geometry called folded manifolds. We prove that any folded manifold which is in good position has a tangency with the unstable manifold of a point of $\mathcal{H}_{f_{0}}$. We choose the parameters $c$, $b$ and $\sigma$ in order to create an initial heteroclinic tangency between the unstable manifold of a point of $\mathcal{H}_{f_{0}}$ and a folded manifold. This folded manifold is in good position and included in the stable manifold of another point of $\mathcal{H}_{f_{0}}$. This enables us to produce persistent heteroclinic tangencies between stable and unstable manifolds of points of $\mathcal{H}_{f_{0}}$. This gives rise to homoclinic tangencies associated to the sectionally dissipative point. By a classical argument going back to Newhouse, this provides a subset of the set of automorphisms of degree 2 in which automorphisms displaying infinitely many sinks are dense.
An important point to notice is that the map $f_{0}$ defined in Eq. (1) is a perturbation of a skew product, with on the basis a H\'enon mapping (it is a skew product for $\sigma = 0$). The structure of H\'enon mapping will be important to create a horseshoe in Proposition \ref{ferm} and an initial fold in Proposition \ref{orr} (in particular, see Lemma \ref{utileresult}). The affine third coordinate is chosen so that the horseshoe displays the blender property (see Subsection 3.2). The perturbation term $\sigma z_{3}(z_{1}-\alpha)$ allows to straighten the fold in a particular direction by iterating in Subsections 5.2 and 5.3.
The plan of the paper is as follows. In Section 2, we choose a family of quadratic polynomials and we fix complex coefficients $\lambda,\mu,\nu$. In Section 3, we introduce the map $f_{0}$ which depends on three parameters $c,b,\sigma$ and the associated horseshoe and we show that it has the blender property. Then, in Section 4, we introduce the formalism of folded manifolds and the mechanism which gives persistent tangencies. In Section 5, we prove that it is possible to choose $f_{0}$ in order to have a heteroclinic tangency. Finally, we prove the main Theorem in Section 6. In Appendix A, we explain how to construct a sink from a sectionally dissipative tangency. \newline \newcommand{\lrpr}[1]{\left(#1\right)} \newline \textbf{Note:} This article is a complete rewriting of a first version released on arXiv in November 2016. In that version the polynomial automorphism $f$ was of degree 5. To the best of the author's knowledge, the notion of blender was used there for the first time in holomorphic dynamics. Notice that blenders also appeared in complex dynamics in \cite{dm} and \cite{taflin}. \newline \newline \textbf{Acknowledgments :} The author would like to thank his PhD advisor, Romain Dujardin as well as Pierre Berger and the anonymous referee for many invaluable comments.
\section{Preliminaries} \subsection{Choice of a quadratic polynomial}
In the following, we will consider the Euclidean norm on $\mathbb{C}^{n}$ for $n \in \{1,2,3\}$.
\begin{notation}
We denote by $\mathbb{D} \subset \mathbb{C}$ the open unit disk, and by $\mathbb{D}(0,r)$ the open disk centered at 0 of radius $r$ for any $r>0$. In particular, $\mathbb{D}(0,1) = \mathbb{D}$. \end{notation} \begin{notation} We will denote by $\mathrm{dist}$ the distance induced by the Euclidean norm on $\mathbb{C}^{n}$ for $n \in \{1,2,3\}$. \end{notation}
\begin{notation} For every $z \in \mathbb{C}^{3}$ and $i \in \{1,2,3\}$, we denote by $\mathrm{pr}_{i}(z) = z_{i}$ the $i^{th}$-coordinate of $z$. \end{notation}
In the following proposition, we carefully choose a family of quadratic polynomials with special properties.
\begin{prop} \label{marber}
For every integer $q >1$, there exists a disk $\mathcal{C} \subset \mathbb{C}$ of center $c_{0} \in \mathbb{C}$, a holomorphic family $(p_{c})_{c \in \mathcal{C}}$ of quadratic polynomials, two integers $m$ and $r$ (with $r$ independent of $q$), a constant $\chi>1$ and a disk $\mathbb{D}'$ with $\mathbb{D} \subset \mathbb{D}'$ such that:
\begin{enumerate} \item For every $c \in \mathcal{C}$, $p_{c}^{-r}( \mathbb{D})$ (resp. $p_{c}^{-r}( \mathbb{D}')$) admits two disjoint components $ \mathbb{D}_{1}, \mathbb{D}_{2}$ (resp. $ \mathbb{D}'_{1}, \mathbb{D}'_{2}$) included in $ \mathbb{D}$ (resp. $ \mathbb{D}$) such that $p_{c}^{r}$ is univalent on both $ \mathbb{D}_{1}$ and $ \mathbb{D}_{2}$ (resp. $ \mathbb{D}'_{1}$ and $ \mathbb{D}'_{2}$). Moreover $p_{c}^{r-1} (\mathbb{D}_{1}), p_{c}^{r-1}(\mathbb{D}_{2}) \Subset \mathbb{D}$ and $p_{c}^{r-1} (\mathbb{D}'_{1}), p_{c}^{r-1}(\mathbb{D}'_{2}) \Subset \mathbb{D}$.
\item Denote by $\alpha_{c} = \bigcap_{n \ge 0} (p_{c}^{r})^{-n}(\mathbb{D}_{1})$ and $\gamma_{c} = \bigcap_{n \ge 0} (p_{c}^{r})^{-n}(\mathbb{D}_{2})$ which are two fixed points of $p_{c}^{r}$. Then for every $c \in \mathcal{C}$, $\alpha_{c}$ is a repulsive fixed point of $p_{c}$, $|p_{c}'(\alpha_{c})|>\frac{6}{5}$ and we have:
$$A : = r \alpha_{c_{0}} \neq \gamma_{c_{0}} +p_{c_{0}}(\gamma_{c_{0}})+ \cdots + p_{c_{0}}^{r-1}(\gamma_{c_{0}}) : = B \text{ and } |A-B|>1 \, .$$
\item We have $|p'_{c}|>\chi$ on $\mathbb{D}'$ and $|(p^{r}_{c})'|>2$ on a neighborhood of $\mathbb{D}_{1} \cup \mathbb{D}_{2}$. \item The critical point 0 is preperiodic for $c = c_{0}$ : $p_{c_{0}}^{m}(0) = \alpha_{c_{0}} \neq 0$ with $p_{c_{0}}(0) \neq 0, \cdots, p_{c_{0}}^{m-1}(0) \neq 0$ and at $c= c_{0} $, we have: $ \frac{d }{dc}\big( p_{c}^{m}(0) - \alpha_{c} \big) \neq 0$.
\item There exists $R>0$ such that $\mathbb{D}' \subset \mathbb{D}(0,R)$ and such that the Julia set of $p_{c}$ is included in $\mathbb{D}(0,R)$ for every $c \in \mathcal{C}$.
\item The polynomial $p_{c}$ has a periodic point $\delta_{c}$ of multiplier $\nu_{c}$ satisfying $1< |\nu_{c}|< (1+10^{-10})^{1/qr}$ for every $c \in \mathcal{C}$. \end{enumerate}
\end{prop}
\begin{proof}
We begin by working with the family of quadratic polynomials $p_{c}(z) = z^{2}+c$, we will rescale at the end of the proof. We begin by taking the only real quadratic polynomial $p_{a}(z) = z^{2}+a$ with one parabolic cycle $\delta_{a}$ of period 3. In particular, $a<-1$ and $a \in \mathbb{D}(0,2)$. For any $z \in \mathbb{C}$ such that $|z| \ge 10$, we have $|p_{a}(z)| = |z^{2}+a| \ge 10 |z|-|a| \ge 10 |z|-2$ and then $|p^{n}_{a}(z)| \rightarrow + \infty$. This shows that the Julia set of $p_{a}$ is strictly included in $\mathbb{D}(0,10)$. Simple calculations show that $z^{2}+a$ has two real fixed points $\alpha^{+}_{a} = \frac{1}{2}(1+\sqrt{1-4a})>\frac{1}{2}(1+\sqrt{5})>1$ and $\alpha^{-}_{a} = \frac{1}{2}(1-\sqrt{1-4a})<\frac{1}{2}(1-\sqrt{5})<-\frac{6}{10}$. We take two open disks $ \mathbb{B}'_{+} \subset \mathbb{D}(0,10)$ and $ \mathbb{B}'_{-} \subset \mathbb{D}(0,10)$ respectively centered around $\alpha_{a}^{+} $ and $\alpha^{-}_{a}$ which are both disjoint from the orbit of the critical point 0 of $z^{2}+a$ (this is possible since the critical orbit tends to the parabolic orbit of $z^{2}+a$). Since $\alpha_{a}^{+} $ and $\alpha^{-}_{a}$ are repulsive fixed points of $p_{a}$, there exists some $\chi>1$ such that $|p'_{c}|>\chi$ on $ \mathbb{B}'_{+} \cup \mathbb{B}'_{-} $, up to reducing $ \mathbb{B}'_{+}$ and $\mathbb{B}'_{-} $ if necessary.
Since $\alpha_{a}^{+} $ and $\alpha^{-}_{a}$ are repulsive fixed points, still reducing $ \mathbb{B}'_{+}$ and $ \mathbb{B}'_{-}$ if necessary, we have that for every $r \ge 1$, there is a connected component of $p_{a}^{-r}(\mathbb{B}'_{+})$ (resp. $p_{a}^{-r}(\mathbb{B}'_{-})$) which contains $\alpha_{a}^{+} $ (resp. $\alpha^{-}_{a}$) and whose $r$ first iterates are all included in $\mathbb{B}'_{+}$ (resp. $\mathbb{B}'_{-})$. We denote by $\tilde{ \mathbb{B}}_{+}$ and $\tilde{ \mathbb{B}}_{-}$ the respective connected components of $p_{a}^{-1}(\mathbb{B}'_{+})$ and $p_{a}^{-1}(\mathbb{B}'_{-})$ which contain $\alpha_{a}^{+} $ and $\alpha^{-}_{a}$ and are defined this way. Then we fix open disks $\mathbb{B}_{+}$ and $\mathbb{B}_{-}$ of respective centers $\alpha_{a}^{+}$ and $\alpha^{-}_{a}$ such that $\tilde{ \mathbb{B}}_{+} \Subset \mathbb{B}_{+} \Subset \mathbb{B}'_{+}$ and $\tilde{ \mathbb{B}}_{-} \Subset \mathbb{B}_{-} \Subset \mathbb{B}'_{-}$. Since both $\mathbb{B}'_{+}$ and $\mathbb{B}'_{-}$ intersect the Julia set of $p_{a}$ and are disjoint from the critical orbit, we can find some integer $\overline{r}$ such that $p^{\overline{r}}_{a}(\mathbb{B}'_{+})$ contains $\mathbb{B}'_{-}$ and $p^{\overline{r}}_{a}(\mathbb{B}'_{-})$ contains $\mathbb{B}'_{+}$. Then we can find some open set $\overline{B}_{+} \Subset \tilde{ \mathbb{B}}_{+} \Subset \mathbb{B}_{+}$ satisfying $p_{a}(\overline{B}_{+}) \Subset \mathbb{B}'_{+}$, $p_{a}^{1+\overline{r}}(\overline{B}_{+}) \Subset \tilde{ \mathbb{B}}_{-} $, $p_{a}^{2+\overline{r}}(\overline{B}_{+}) \Subset \mathbb{B}'_{-} $ and $p_{a}^{2+2\overline{r}}(\overline{B}_{+}) = \mathbb{B}'_{+}$. Hence, denoting $r = 2+2\overline{r}$, we have $\overline{B}_{+} \Subset \mathbb{B}_{+}$ and $p_{a}^{r}$ sends $\overline{B}_{+}$ biholomorphically onto $\mathbb{B}'_{+}$. We denote by $\gamma_{a}$ the periodic point of $p_{a}$ of period $r$ which is the unique fixed point of the restriction of $p_{a}^{r}$ to $\overline{B}_{+}$. We notice that $\gamma_{a} \neq \alpha^{+}_{a}$. Similarly, we can define $\overline{B}_{-} \Subset \mathbb{B}_{-}$ such that $p_{a}^{r}$ sends $\overline{B}_{-}$ biholomorphically onto $\mathbb{B}'_{-}$ and $p_{a}^{r/2}(\gamma_{a}) = p^{1+\overline{r}}_{a}(\gamma_{a}) \neq \alpha^{-}_{a}$ is the unique fixed point of the restriction of $p_{a}^{r}$ to $\overline{B}_{-}$.
Since $\alpha_{a}^{+} \neq \alpha^{-}_{a}$, it is not possible to satisfy simultaneously $\gamma_{a} +p_{a}(\gamma_{a})+ \cdots + p_{a}^{r-1}(\gamma_{a}) = r \alpha_{a}^{+}$ and $ \gamma_{a} +p_{a}(\gamma_{a})+ \cdots + p_{a}^{r-1}(\gamma_{a}) = r \alpha_{a}^{-}$. In the following, we will denote by $\alpha_{a}$ a point in $\{\alpha_{a}^{+}, \alpha^{-}_{a}\}$ such that the inequality $ \gamma_{a} +p_{a}(\gamma_{a})+ \cdots + p_{a}^{r-1}(\gamma_{a}) \neq r \alpha_{a}$ is satisfied. We also denote by $\mathbb{B}$, $\mathbb{B}'$, $\tilde{\mathbb{B}}$ and $\overline{B}$ the sets corresponding to $\alpha_{a}$. Up to replacing $\gamma_{a}$ by $p_{a}^{r/2}(\gamma)$ if $\alpha_{a} = \alpha^{-}_{a}$, we can suppose that $\gamma_{a} \in \mathbb{B}$. The multiplier of $\alpha_{a}$ is of modulus $|2\alpha_{a}|>\mathrm{min}(2,\frac{6}{5}) = \frac{6}{5}$. We take the component $ \mathbb{B}'_{1}$ of $p^{-r}_{a}(\mathbb{B}')$ containing $\alpha_{a}$ and where $p^{r}_{a} $ is univalent defined at the beginning of the last paragraph. We have $ \mathbb{B}'_{1} \Subset \mathbb{B} \Subset \mathbb{B}'$. We also take the component $ \mathbb{B}'_{2}$ of $p^{-r}_{a} ( \mathbb{B}')$ containing $\gamma_{a}$ and where $p^{r}_{a} $ is univalent equal to $\overline{B}$. It holds $ \mathbb{B}'_{2} \Subset \tilde{\mathbb{B}}\Subset \mathbb{B} \Subset \mathbb{B}'$. Replacing $r$ by one of its multiples if necessary (still denoted by $r$), $ \mathbb{B}'_{1}$ and $ \mathbb{B}'_{2}$ are disjoint. We also take the respective components $ \mathbb{B}_{1}$ and $ \mathbb{B}_{2}$ of $p^{-r}_{a} ( \mathbb{B})$ included into those of $p^{-r}_{a} ( \mathbb{B}') $. Since $\tilde{ \mathbb{B}} \Subset \mathbb{B}$, it holds $p_{a}^{r-1} (\mathbb{B}_{1}), p_{a}^{r-1}(\mathbb{B}_{2}) , p_{a}^{r-1} (\mathbb{B}'_{1}), p_{a}^{r-1}(\mathbb{B}'_{2}) \Subset \mathbb{B}$. Still replacing $r$ by a multiple if necessary, we have $ |r \alpha_{a} - (\gamma_{a} +p_{a}(\gamma_{a})+ \cdots + p_{a}^{r-1}(\gamma_{a}))|>10 $. Since $ \mathbb{B}'_{1} \Subset \mathbb{B}$ and $ \mathbb{B}'_{2} \Subset \mathbb{B}$, by the Schwarz Lemma, there exists $\theta>1$ such that $|(p_{c}^{r})'|>\theta$ on a neighborhood of $\mathbb{B}_{1} \cup \mathbb{B}_{2}$. Taking a multiple of $r$ if necessary, $|(p_{c}^{r})'|>2$ on a neighborhood of $\mathbb{B}_{1} \cup \mathbb{B}_{2}$.
Let us fix $q>1$. By continuity, for $c $ in some neighborhood $\mathcal{C}_{a}$ of $a$ in $\mathbb{C}$, it holds:
\begin{enumerate} \item $p_{c}^{-r}( \mathbb{B})$ (resp. $p_{c}^{-r}( \mathbb{B}')$) admits two components $ \mathbb{B}_{1}, \mathbb{B}_{2}$ (resp. $ \mathbb{B}'_{1}, \mathbb{B}'_{2}$) included in $ \mathbb{B}$ (resp. $ \mathbb{B}$) containing the continuations $\alpha_{c}$ and $\gamma_{c}$ and such that $p_{c}^{r}$ is univalent on both $ \mathbb{B}_{1}$ and $ \mathbb{B}_{2}$ (resp. $ \mathbb{B}'_{1}$ and $ \mathbb{B}'_{2}$). Moreover $p_{c}^{r-1} (\mathbb{B}_{1})$, $p_{c}^{r-1}(\mathbb{B}_{2}),$ $p_{c}^{r-1} (\mathbb{B}'_{1})$, $p_{c}^{r-1}(\mathbb{B}'_{2})$ $ \Subset \mathbb{B}$,
\item the continuation $\alpha_{c}$ of $\alpha_{a}$ is a repulsive fixed point of $p_{c}$ such that $|p'_{c}(\alpha_{c})|>\frac{6}{5}$,
\item $ r \alpha_{c} \neq \gamma_{c} +p_{c}(\gamma_{c})+ \cdots + p_{c}^{r-1}(\gamma_{c}) $ and $|r \alpha_{c} -( \gamma_{c} +p_{c}(\gamma_{c})+ \cdots + p_{c}^{r-1}(\gamma_{c}))|>10$,
\item $|p'_{c}|>\chi$ on $\mathbb{B}'$ and $|(p_{c}^{r})'|>2$ on a neighborhood of $\mathbb{B}_{1} \cup \mathbb{B}_{2}$, \item the Julia set of $p_{c}$ is included in $\mathbb{D}(0,10)$,
\item the continuation $\delta_{c}$ of $\delta_{a}$ is of multiplier $\nu_{c}$ such that $(1-10^{-10})^{1/qr} < |\nu_{c}| < (1+10^{-10})^{1/qr}$. \end{enumerate}
The parameter $a$ belongs to the Mandelbrot set. Misiurewicz parameters are dense inside the Mandelbrot set so it is possible to find a parameter $\tilde{c}$ inside the interior of $\mathcal{C}_{a}$ such that the critical point 0 is preperiodic for $p_{\tilde{c}}$. The critical point 0 is sent after a finite number of iterations of $p_{\tilde{c}}$ on a periodic orbit. This periodic orbit is accumulated by preimages of $\alpha_{\tilde{c}}$ by iterates of $p_{\tilde{c}}$. Then by the Argument Principle it is possible to take a new Misiurewicz parameter $c_{0} $ in the interior of $\mathcal{C}_{a}$ such that 0 is still preperiodic but with associated orbit the fixed point $\alpha_{c_{0}}$. There exists an integer $m$ such that $p_{c_{0}}^{m}(0) = \alpha_{c_{0}}$ with $p_{c_{0}}(0) \neq 0, \cdots, p_{c_{0}}^{m-1}(0) \neq 0$. The inequality $ \frac{d }{dc} \big( p_{c}^{m}(0)-\alpha_{c} \big) \neq 0$ at $c = c_{0}$ is a direct consequence of Lemma 1, Chapter 5 of \cite{hudo}. For the parameter $c_{0}$, $\delta_{c_{0}}$ is repulsive of multiplier $\nu_{c_{0}}$ such that $1 < | \nu_{c_{0}}| < (1+10^{-10})^{1/qr}$. We pick some ball $\mathcal{C} \subset \mathcal{C}_{a}$ of center $c_{0}$ where this is still true.
For each $c \in \mathcal{C}$, we do a rescaling by an affine map so that after rescaling $\mathbb{B} \subset \mathbb{D}(0,10)$ is sent on $\mathbb{D} = \mathbb{D}(0,1) $. Properties 1, 2, 4 and 6 are still true. Property 5 is still true with a disk $\mathbb{D}(0,R)$ with a fixed $R>0$ instead of $\mathbb{D}(0,10)$. Since $r\alpha_{c} \neq \gamma_{c} +p_{c}(\gamma_{c})+ \cdots + p_{c}^{r-1}(\gamma_{c})$ and $|r \alpha_{c} -( \gamma_{c} +p_{c}(\gamma_{c})+ \cdots + p_{c}^{r-1}(\gamma_{c}))|>10$ before rescaling, we have $A \neq B$ and $|A-B|>1$ after and then Property 3 is true. Then Properties 1, 2, 3, 4, 5 and 6 are satisfied for every $c \in \mathcal{C}$. In the following, after rescaling, we will denote $\mathbb{B}, \mathbb{B}', \mathbb{B}_{1}, \mathbb{B}_{2}, \mathbb{B}'_{1}, \mathbb{B}'_{2}$ by $\mathbb{D}, \mathbb{D}', \mathbb{D}_{1}, \mathbb{D}_{2}, \mathbb{D}'_{1}, \mathbb{D}'_{2}$. For simplicity, we will still denote by $p_{c}$ the polynomial after rescaling. \end{proof}
\subsection{Choice of an IFS}
\begin{notation} \label{defh1h} For every $c \in \mathcal{C}$, we denote by $h_{1}$ and $h_{2}$ the two inverse branches of $p_{c}^{r}$ on
$\mathbb{D}'$ given by Proposition \ref{marber} such that $\alpha_{c} = \bigcap_{n \ge 0} h_{1}^{n}( \mathbb{D})$ and $\gamma_{c} = \bigcap_{n \ge 0} h^{n}_{2}( \mathbb{D})$. \end{notation}
\begin{notation} \label{h} We denote $\mu_{0} = (1-10^{-4})^{\frac{1}{qr}} \cdot e^{i \cdot \frac{\pi}{2qr}}$ which depends on the integer $q$. In particular, we have the following equality: $\mu_{0}^{qr} = (1-10^{-4}) \cdot e^{i \cdot \frac{\pi}{2}}$. \end{notation}
In the following result, we iterate $q$ times the maps $h_{1}$ and $h_{2}$ with a specific choice for the integer $q$. Remind that $A$, $B$ and $R$ were defined in Proposition \ref{marber}.
\begin{prop} \label{fo}
There exists an integer $q \ge 100$ such that, after reducing $\mathcal{C}$ if necessary, the following holds for every $c \in \mathcal{C}$:
\begin{enumerate}
\item $|(h_{j}^{q})'| <10^{-10}$ for $j \in \{1,2\}$ on a neighborhood $ \mathbb{D}''$ of $ \mathbb{D}$ with $ \mathbb{D} \subset \mathbb{D}'' \subset \mathbb{D}'$, \item $\mathrm{diam} \big( h^{q}_{j} ( \mathbb{D}' ) \big) \le 10^{-11} \cdot \mathrm{dist} ( h^{q}_{j} ( \mathbb{D}' ) , \partial \mathbb{D} )$ for $j \in \{1,2\}$,
\item for every $z \in h_{1}^{q}( \mathbb{D}')$ and $0 \le n \le qr \big( 1-10^{-10} r^{-1} R^{-1} \min(1, |A-B|) \big) $: $$ \mu_{0}^{0} p_{c}^{n+r-1}(z)+ \cdots + \mu^{r-1}_{0} p_{c}^{n}(z) \in \mathbb{D}(A, 10^{-10} \cdot |A-B| ) \, , $$
\item for every $z \in h^{q}_{2}( \mathbb{D}')$ and $0 \le n \le qr \big( 1-10^{-10} r^{-1} R^{-1} \min(1,|A-B|) \big)$: $$ \mu_{0}^{0} p_{c}^{n+r-1}(z)+ \cdots + \mu^{r-1}_{0} p_{c}^{n}(z) \in \mathbb{D}(B, 10^{-10} \cdot |A-B| ) \, . $$ \end{enumerate}
\end{prop}
\begin{proof}
We first show the result for $c = c_{0}$. According to property (3) of Proposition \ref{marber}, $|(p^{r}_{c})'|>2$ on a neighborhood of $\mathbb{D}_{1} \cup \mathbb{D}_{2}$. Then, taking $q \ge 100$ such that $2^{q}> 10^{10}$, we have $|(h_{j}^{q})'|<10^{-10}$ on some disk $ \mathbb{D}''$ with $ \mathbb{D} \subset \mathbb{D}'' \subset \mathbb{D}'$. Since $h_{j}$ is a contraction such that $\bigcap_{n \ge 0} h_{1}^{n}(\mathbb{D}') = \{\alpha_{c}\}$ and $\bigcap_{n \ge 0} h^{n}_{2}(\mathbb{D}') = \{\gamma_{c}\}$, increasing the value of $q$ if necessary, we have that $\mathrm{diam}\big( (h^{q}_{j} ( \mathbb{D}' )\big) \le 10^{-11} \cdot \text{dist} ( h^{q}_{j} ( \mathbb{D}' ) , \partial \mathbb{D} )$. When $q \rightarrow + \infty$, we both have $\mu_{0}^{k} \rightarrow 1$ and $p_{c_{0}}^{n+k}(z) \rightarrow \alpha_{c_{0}}$ uniformly in $0 \le k < r$, $0 \le n \le qr(1-10^{-10} r^{-1} R^{-1} \min(1, |A-B|)) $ and $z \in h_{1}^{q}( \mathbb{D}')$. Then, increasing the value of $q$ if necessary, we have that $ \mu_{0}^{0} p_{c_{0}}^{n+r-1}(z)+ \cdots + \mu^{r-1}_{0} p_{c_{0}}^{n}(z) \in \mathbb{D}(A, 10^{-10} |A-B|)$. The proof of the last item is similar. Since all these conditions are open, reducing the ball $\mathcal{C}$ of center $c_{0}$ if necessary, they remain true for every $c \in \mathcal{C}$. \end{proof}
\begin{notation} \label{rbound}
Since $r$ is independent of $q$ (see Proposition \ref{marber}), we can increase $q$ so that $r \le 10^{-10} qR^{-1} \min(1, |A-B|)$. From now on, we fix such a value of $q$ and the associated value $\mu_{0}$. \end{notation}
\subsection{Choice of the parameters $\lambda$ and $\nu$}
In this Subsection, we introduce two new coefficients $\lambda$ and $\nu$. These constants will apppear on the third coordinate of the polynomial automorphisms of $\mathbb{C}^{3}$ we are going to work with. This will be used to create a horseshoe in Proposition \ref{ferm} and to show that this horseshoe displays the blender property in Subsection 3.2.
\begin{notation} We denote by $A' = (\mu_{0}^{r-1} \alpha_{c_{0}} +\cdots + \mu_{0}^{0} p_{c_{0}}^{r-1}(\alpha_{c_{0}}) )$ and $B' = ( \mu_{0}^{r-1} \gamma_{c_{0}}+\cdots + \mu_{0}^{0} p_{c_{0}}^{r-1}(\gamma_{c_{0}}) )$. \end{notation}
By Proposition \ref{fo}, $A' \in \mathbb{D}(A,10^{-10}|A-B|)$ and $B' \in \mathbb{D}(B,10^{-10}|A-B|)$. According to item 2 of Proposition \ref{marber}, this implies that: \begin{equation} \label{ineqb}
|A'-B'|> \frac{1}{2}|A-B|>\frac{1}{2} \, . \end{equation}
\begin{prop}\label{munu}
There exist two constants $\lambda,\nu$ such that $|\lambda|<1$ and satisfying: $$\lambda A'(1+ \mu_{0}^{r}+ \cdots + \mu_{0}^{qr-r}) + \nu (1+\mu_{0}+ \cdots +\mu^{qr-1}_{0}) =\frac{9}{10} \cdot 10^{-4} \, ,$$ $$\lambda B' (1+ \mu_{0}^{r}+ \cdots + \mu_{0}^{qr-r}) + \nu (1+\mu_{0}+ \cdots +\mu^{qr-1}_{0}) = - \frac{9}{10} \cdot 10^{-4} \, .$$ \end{prop}
\begin{proof}
We have: $1+ \mu_{0}^{r}+ \cdots + \mu_{0}^{qr-r} = (1- \mu_{0}^{qr})/(1- \mu^{r}_{0} ) $. By Notation \ref{h}, we have $\mu_{0}^{qr} = (1-10^{-4}) \cdot e^{i \cdot \frac{\pi}{2}} \neq 1$ and then $1+ \mu_{0}^{r}+ \cdots + \mu_{0}^{qr-r} \neq 0$. Similarly we have $1+ \mu_{0}+ \cdots + \mu_{0}^{qr-1} = (1- \mu_{0}^{qr})/(1- \mu_{0} ) \neq 0$. Since $A' \neq B'$, it is possible to pick two coefficients $\lambda$ and $\nu$ so that the images of these two complex numbers by the affine map $z \mapsto \lambda (1+ \mu_{0}^{r}+ \cdots + \mu_{0}^{qr-r}) z + \nu (1+ \mu_{0}+ \cdots + \mu_{0}^{qr-1}) $ are respectively equal to $\frac{9}{10} \cdot 10^{-4}$ and $-\frac{9}{10} \cdot 10^{-4}$. It remains to show that $|\lambda|<1$. To this end, we will need the following technical lemma:
\begin{lemma} \label{mirber} The complex number $\mu_{0}$ satisfies the following inequality:
$$ \frac{q}{2} \le 1+|\mu_{0}^{r}|+|\mu_{0}|^{2r}+ \cdots + |\mu_{0}|^{qr-r} \le 10 \cdot | 1+\mu_{0}^{r}+\mu_{0}^{2r}+ \cdots +\mu_{0}^{qr-r}| \, .$$ \end{lemma}
\begin{proof}
We have $\frac{1}{2} \le 1$, $\frac{1}{2} \le |\mu_{0}^{r}|$, $\cdots$, $\frac{1}{2} \le |\mu_{0}^{qr-r}|$ so the first inequality is trivial. Since every term $\mu_{0}^{nr}$ ($0 \le n < q$) has a positive real part and since this real part is larger than $\frac{1}{2}$ for $0 \le n \le \frac{1}{2}(q-1)$, we have $\frac{1}{2}(q-1) \cdot \frac{1}{2} \le \text{Re}( 1+\mu_{0}^{r}+\mu_{0}^{2r}+ \cdots +\mu_{0}^{qr-r} )$ and then $1+|\mu_{0}^{r}|+|\mu_{0}|^{2r}+ \cdots + |\mu_{0}|^{qr-r} \le q \le 10 \cdot \frac{1}{2}(q-1) \cdot \frac{1}{2} \le 10 \cdot | 1+\mu_{0}^{r}+\mu_{0}^{2r}+ \cdots +\mu_{0}^{qr-r}|$. The proof is complete. \end{proof}
We are now in position to end the proof of Lemma \ref{munu}. By definition of $\lambda$ and $\nu$, we have $|\lambda| |A'-B'| |1+ \mu_{0}^{r}+ \cdots + \mu_{0}^{qr-r}| = 2 \cdot \frac{9}{10} \cdot 10^{-4}$. We already proved that $|A'-B'|>\frac{1}{2}$ in Eq. (\ref{ineqb}) and by Lemma \ref{mirber} we also have $|1+ \mu_{0}^{r}+ \cdots + \mu_{0}^{qr-r}| \ge q/20 \ge 100/20 \ge 1 $. This implies that $|\lambda|<1$ and so the result is proven.
\end{proof}
\begin{corollary} \label{hyu} Reducing $\mathcal{C}$ if necessary, there exists a neighborhood $\mathcal{B}_{\mu}$ of $\mu_{0}$ such that for every $c \in \mathcal{C}$ and $\mu \in \mathcal{B}_{\mu}$ it holds:
\begin{enumerate} \item for every $z \in h_{1}^{q}( \mathbb{D}')$, we have: $$ \nu + \lambda p_{c}^{qr-1}(z)+ \mu (\nu+ \cdots +\mu (\nu+ \lambda z )) \in \mathbb{D}(\frac{9}{10} \cdot 10^{-4}, 10^{-10}) \, , $$ \item for every $z \in h_{2}^{q}( \mathbb{D}')$, we have: $$ \nu + \lambda p_{c}^{q-1}(z)+ \mu (\nu+ \cdots +\mu (\nu+ \lambda z )) \in \mathbb{D}(- \frac{9}{10} \cdot 10^{-4},10^{-10}) \, . $$ \end{enumerate}
\end{corollary}
\begin{proof} We first prove the result for $c = c_{0}$ and $\mu = \mu_{0}$. According to Proposition \ref{munu}, we have: $$ \nu + \lambda p^{qr-1}_{c_{0}}(z)+ \mu_{0} (\nu+ \cdots +\mu_{0} (\nu+ \lambda z )) - \frac{9}{10} \cdot 10^{-4} =$$ $$ \lambda \sum_{n= 0}^{l-1} (p_{c_{0}}^{qr-1-n}(z)-\alpha_{c_{0}})\mu_{0}^{n} + \lambda \sum_{n= l}^{qr-1} (p_{c_{0}}^{qr-1-n}(z)-\alpha_{c_{0}})\mu_{0}^{n} \, ,$$
where $l$ is the smallest integer such that $l \ge 10^{-10} qR^{-1} \min(1, |A-B|)$ and which is a multiple of $r$. By Notation \ref{rbound}, we have $l \le 2 \cdot 10^{-10} qR^{-1} \min(1, |A-B|)$. In particular, $qr-l$ is a multiple of $r$. Using the third item of Proposition \ref{fo}, it holds:
$$ |\lambda \sum_{n= l}^{qr-1} (p^{qr-1-n}_{c_{0}}(z)-\alpha_{c_{0}})\mu_{0}^{n}| \le |\lambda| \cdot 10^{-10} |A- B| \cdot (1+|\mu_{0}|^{r}+|\mu_{0}|^{2r}+ \cdots + |\mu_{0}|^{qr-r}) \, .$$
We already proved that $|A'-B'| > \frac{1}{2} |A-B|$ in Eq. (\ref{ineqb}). In particular, this implies that $|\lambda| \cdot |A-B| \cdot | 1+\mu^{r}_{0}+\mu_{0}^{2r}+ \cdots +\mu_{0}^{qr-r}| < |\lambda| \cdot 2 |A'-B'| \cdot |1+ \mu_{0}^{r}+ \cdots + \mu_{0}^{qr-r}| $. Then, by Proposition \ref{munu}, this yields $|\lambda| \cdot |A-B| \cdot | 1+\mu^{r}_{0}+\mu_{0}^{2r}+ \cdots +\mu_{0}^{qr-r}| < 2 \cdot 2 \cdot \frac{9}{10} \cdot 10^{-4}$. By Lemma \ref{mirber}, it also holds: $1+|\mu_{0}^{r}|+|\mu_{0}|^{2r}+ \cdots + |\mu_{0}|^{qr-r} \le 10 \cdot | 1+\mu_{0}^{r}+\mu_{0}^{2r}+ \cdots +\mu_{0}^{qr-r}|$. All this together implies the following:
\begin{equation} \label{ineqmu1}
|\lambda \sum_{n= l}^{qr-1} (p^{qr-1-n}_{c_{0}}(z)-\alpha_{c_{0}})\mu_{0}^{n}| \le 10^{-10} \cdot 10 \cdot \big( 2 \cdot 2 \cdot \frac{9}{10} \cdot 10^{-4} \big) \le 10^{-11} \, . \end{equation}
Since both $\mathbb{D}'$ and the Julia set of $p_{c_{0}}$ are included in $\mathbb{D}(0,R)$ (see item 5 of Proposition \ref{marber}), we also have:
$$ | \lambda \sum_{n= 0}^{l-1} (p^{qr-1-n}_{c_{0}}(z)-\alpha_{c_{0}})\mu_{0}^{n}| \le |\lambda | \cdot 2R \cdot (1+|\mu_{0}|+|\mu_{0}|^{2}+ \cdots + |\mu_{0}|^{l-1}) \le |\lambda | \cdot 2R \cdot l \, . $$
Since $l \le 2 \cdot 10^{-10} qR^{-1} \min(1, |A-B|)$ and then using the inequality $|A-B| < 2|A'-B'|$ from Eq. (\ref{ineqb}), the latter is smaller than:
$$ |\lambda| \cdot 2R \cdot 2 \cdot 10^{-10}\frac{q}{R} \cdot \min(1, |A-B|) \le 10^{-8} \cdot |\lambda| \cdot \frac{q}{2} \cdot \frac{|A-B|}{2} \le 10^{-8} \cdot |\lambda| \cdot \frac{q}{2} \cdot |A'-B'| \, .$$
Using successively the inequality $ \frac{q}{2} \le 10 \cdot | 1+\mu_{0}^{r}+\mu_{0}^{2r}+ \cdots +\mu_{0}^{qr-r}| $ from Lemma \ref{mirber} and then Proposition \ref{munu}, the latter is finally smaller than $10^{-8} \cdot |\lambda| \cdot |A'-B'| \cdot 10 \cdot | 1+\mu_{0}+\mu_{0}^{2}+ \cdots +\mu_{0}^{qr-r}| \le 10^{-7} \cdot 2 \cdot \frac{9}{10} \cdot 10^{-4}$ and so:
\begin{equation} \label{ineqmu3}
| \lambda \sum_{n= 0}^{l-1} (p^{qr-1-n}_{c_{0}}(z)-\alpha_{c_{0}})\mu_{0}^{n}| \le 2 \cdot 10^{-11} \, . \end{equation}
Then we just have to sum the two inequalities of Eq. (\ref{ineqmu1}) and Eq. (\ref{ineqmu3}) to prove the result for $c= c_{0}$ and $\mu = \mu_{0}$. By continuity and since the inequality is open, it remains true for every $\mu$ in some ball $ \mathcal{B}_{\mu}$ of center $\mu_{0}$ and $c \in \mathcal{C}$ after reducing $\mathcal{C}$ if necessary. Then item 1 is true and the proof of item 2 is similar. The result is proven.\end{proof}
\begin{remark} \label{rmu}
We reduce $\mathcal{B}_{\mu}$ so that we both have $|\mu| < (1- 10^{-4} +10^{-10} )^{\frac{1}{qr}}$, $|\mu|^{2qr}>1-2 \cdot 10^{-4}$ and $\mu^{qr} \subset \mathbb{D}(\mu_{0}^{qr},10^{-10})$ for every $\mu \in \mathcal{B}_{\mu}$. \end{remark}
\subsection{Adjusting the parameter $\mu$}
In this subsection, we slightly perturb the coefficient $\mu_{0}$ into a new value $\mu$ in order to satisfy some equality for a product of matrices. Notice that this choice has nothing to do with the next section and the blender property, it will be useful in Section 5.
\begin{notation} \label{bet} We denote $\beta_{0} = 0, \beta_{1} = p_{c_{0}}(0), \beta_{2} = p^{2}_{c_{0}}(0), \ldots, \beta_{m} = p^{m}_{c_{0}}(0)= \alpha_{c_{0}}$ the points of the orbit of the critical point 0 before landing onto the fixed point $\alpha_{c_{0}}$. \end{notation}
\begin{notation} For every $\mu \in \mathcal{B}_{\mu}$, we define: $ w_{0} =0 + 1 \cdot (\beta_{0}-\beta_{m})$, $w_{1} = p'_{c_{0}}(\beta_{1}) w_{0} + \mu (\beta_{1} -\beta_{m})$ $ \cdots $ and $w_{m-1} = p'_{c_{0}}(\beta_{m-1}) w_{m-2} + \mu^{m-1} (\beta_{m-1} -\beta_{m})$ where $p'_{c_{0}}(\beta_{0}) = 0$, $p'_{c_{0}}(\beta_{1}) \neq 0, \cdots, p'_{c_{0}}(\beta_{m-1}) \neq 0$. \end{notation}
\begin{df} \label{K} Since $\beta_{m-1} -\beta_{m} \neq 0 $, $w_{m-1}$ is a polynomial of degree $(m-1)$ in the variable $\mu$ so we fix some $\mu \in \mathcal{B}_{\mu}$ such that $w_{m-1} = w_{m-1}(\mu) \neq 0$. \end{df}
\begin{notation} We denote for every $\sigma \in \mathbb{C}$, $0 \le n \le m-1$:
$$M_{n}^{\sigma} = \begin{pmatrix}
p'_{c_{0}}(\beta_{n}) & 0 & \sigma (\beta_{n} -\beta_{m}) \\
1 & 0& 0 \\
\lambda & 0 & \mu \end{pmatrix} \, . $$ \end{notation}
\begin{prop}
We have $M^{\sigma}_{m-1} \cdots M^{\sigma}_{0} \cdot ( 0 , 0 , 1 ) = ( \zeta_{1}(\sigma), \zeta_{2}(\sigma) , \zeta_{3}(\sigma) )$, where $\zeta_{1},\zeta_{2},\zeta_{3}$ are holomorphic functions such that $\zeta_{1}(\sigma) = w_{m-1} \cdot \sigma+O(\sigma^{2})$ and $\zeta_{3}(\sigma) = \mu^{m}+O(\sigma)$. \end{prop}
\begin{proof} It is a straightforward consequence of Definition \ref{K}. \end{proof}
Simple calculations yield the following corollary (the important fact here is that $ p'_{c_{0}}(\beta_{0})=0$ since $\beta_{0} = 0$ is the critical point of $p_{c_{0}}$).
\begin{corollary} \label{reffin} Let $\epsilon^{n}_{1}, \epsilon^{n}_{2}, \epsilon^{n}_{3}$ be three holomorphic functions such that $\epsilon^{n}_{1}(\sigma) = O(\sigma)$, $\epsilon^{n}_{2}(\sigma) = O(\sigma^{2})$ and $\epsilon^{n}_{3}(\sigma) = O(\sigma)$ for $0 \le n \le m-1$. Let us denote for every $\sigma \in \mathbb{C}$, $0 \le n \le m-1$: $$N_{n}^{\sigma} = \begin{pmatrix} p'_{c_{0}}(\beta_{n}) + \epsilon^{n}_{1}(\sigma) & \epsilon^{n}_{2}(\sigma) & \sigma (\beta_{n} +\epsilon^{n}_{3}(\sigma)-\beta_{m}) \\ 1 & 0& 0 \\ \lambda & 0 & \mu \end{pmatrix} \, . $$ For every holomorphic maps $\xi_{1},\xi_{2}$ such that $\xi_{1}(\sigma) = O(\sigma)$ and $\xi_{2}(\sigma) = O(\sigma)$, we get: $$N^{\sigma}_{m-1} \cdots N^{\sigma}_{0} \cdot ( \xi_{1}(\sigma) , \xi_{2}(\sigma) , 1) = ( \zeta_{1}(\sigma), \zeta_{2}(\sigma) , \zeta_{3}(\sigma) ) \, ,$$ where $\zeta_{1},\zeta_{2},\zeta_{3}$ are holomorphic functions such that $\zeta_{1}(\sigma) = w_{m-1} \cdot \sigma+O(\sigma^{2})$ and $\zeta_{3}(\sigma) = \mu^{m}+O(\sigma)$. \end{corollary}
\section{Construction of a blender}
In this section, we construct a polynomial automorphism $f_{0}$ of $\mathbb{C}^{3}$. We show that $f_{0}$ has a horseshoe $\mathcal{H}_{f_{0}}$ and that $\mathcal{H}_{f_{0}}$ is a complex blender.
\subsection{Three complex dimensions: the map $f_{0}$}
We recall that $\mathcal{C}$ and $p_{c}$ were defined in Proposition \ref{marber}. We consider now the 3 dimensional map $f_{0} (z_{1},z_{2},z_{3}) = (p_{c}(z_{1})+b z_{2}+\sigma z_{3} (z_{1}-\alpha_{c_{0}}),z_{1},\lambda z_{1} + \mu z_{3}+\nu) $ introduced in Eq. \eqref{e}. It is clear that it is a polynomial automorphism for $c \in \mathcal{C}$ and $b \neq 0$. In the following, we will see that the first direction is expanded by $f_{0}$ and corresponds to the direction of the unstable manifolds of a hyperbolic set we are going to describe. The second and third directions are contracted by $f_{0}$ and correspond to the directions of the stable manifolds of this hyperbolic set.
\begin{notation} \label{con}
We define the following constant cone fields: $C^{u} = \{v = (v_{1},v_{2},v_{3}) \in \mathbb{C}^{3} : \max(|v_{2}|,|v_{3}|) \le \chi^{-1} \cdot |v_{1}|\}$, $C^{ss} = \{v = (v_{1},v_{2},v_{3}) \in \mathbb{C}^{3} : \max(|v_{1}|,|v_{3}|) \le 10^{-6} \cdot |v_{2}|\}$ and
$C^{cs} = \{v = (v_{1},v_{2},v_{3}) \in \mathbb{C}^{3} : \max(|v_{1}|,|v_{2}|) \le 10^{-6} \cdot |v_{3}|\}$, where the constant $\chi>1$ was defined in Proposition \ref{marber}. \end{notation}
We now give a non general definition of a horseshoe which is specific to our context.
\begin{df} \label{df7} Given an automorphism $F : \mathbb{C}^{3} \rightarrow \mathbb{C}^{3}$, a tridisk $D = D_{1} \times D_{2} \times D_{3} \subset \mathbb{D}^{3}$ and an integer $p \ge 1$, we say that $\mathcal{H}_{F} =\bigcap_{n \in \mathbb{Z}} F^{n}(\overline{D})$ is a $p$-branched horseshoe for $F$ if: \begin{enumerate} \item $F(D) \cap D$ has $p$ components $D^{j,u}$ which do not intersect $D_{1} \times \partial (D_{2} \times D_{3})$,
\item $F^{-1}( D) \cap D$ has $p$ components $D^{j,s}$ which do not intersect $\partial D_{1} \times D_{2} \times D_{3}$,
\item on $\bigcup_{1 \le j \le p} D^{j,s}$, the cone field $C^{u}$ is $F$-invariant, and on $\bigcup_{1 \le j \le p} D^{j,u}$ the cone field $C^{ss}$ is $F^{-1}$-invariant. Moreover there exists $\Xi>1$ such that the cone field $\{ ( v_{1},v_{2},v_{3}) : ||(v_{2},v_{3})|| > \Xi ||v_{1}|| \}$ contains $C^{cs}$ and is $F^{-1}$-invariant on $\bigcup_{1 \le j \le p} D^{j,u}$,
\item there exists $C_{F}>1$ such that at every point of $\bigcup_{1 \le j \le p} D^{j,s}$, for every non zero $v \in C^{u}$, we have $||DF(v)||>C_{F} ||v||$, and at every point of $\bigcup_{1 \le j \le p} D^{j,u}$, for every non zero $v \in C^{ss} \cup \{ ( v_{1},v_{2},v_{3}) : ||(v_{2},v_{3})|| > \Xi||v_{1}|| \}$, we have $ ||D(F^{-1})(v)||>C_{F} ||v||$. \end{enumerate} \end{df}
\begin{prop} If $\mathcal{H}_{F} = \bigcap_{n \in \mathbb{Z}} F^{n}( \overline{D} )$ is a $p$-branched horseshoe, then it is a horseshoe in the classical meaning of this term, that is a compact, invariant, transitive, hyperbolic set. \end{prop}
\begin{proof}
The set $\bigcap_{n \in \mathbb{Z}} F^{n}( \overline{ D} )$ is compact as an intersection of compact sets and $F$-invariant by definition. Moreover, one can take the (non necessarily invariant) decomposition $\mathbb{C}^{3} \simeq \mathbb{R}^{6} = \mathbb{C} \bigoplus \mathbb{C}^{2} \simeq \mathbb{R}^{2} \bigoplus \mathbb{R}^{4}$ and the associated constant cone fields $\mathcal{C}^{u}_{\mathbb{R}} = \{ ( v_{1},v_{2},v_{3}) : ||v_{1}||>\chi||(v_{2},v_{3})||\}$ and $\mathcal{C}^{s}_{\mathbb{R}} = \{ ( v_{1},v_{2},v_{3}) : ||(v_{2},v_{3})|| > \Xi ||v_{1}|| \}$. The definition above implies that both $\mathcal{C}^{u}_{\mathbb{R}}$ is $F$-invariant and $\mathcal{C}^{s}_{\mathbb{R}}$ is $F^{-1}$-invariant. Moreover, they are expanded by a factor $C_{F}$ larger than 1 respectively under $F$ and $F^{-1}$. Besides, the sets $D^{j,u}$ do not intersect $D_{1} \times \partial (D_{2} \times D_{3})$ and the sets $D^{j,s}$ do not intersect $\partial D_{1} \times D_{2} \times D_{3}$. Then $\bigcap_{n \in \mathbb{Z}} F^{n}( \overline{ D} )$ is a horseshoe in the sense of Definition 6.5.2 of \cite{caga}. According to the discussion following this definition, $\bigcap_{n \in \mathbb{Z}} F^{n}( \overline{ D} )$ is hyperbolic (this is also a straightforward application of the cone field criterion, Corollary 6.4.8 in \cite{caga}) and is topologically conjugate to a shift. In particular, it is transitive. This ends the proof of the proposition. \end{proof}
\begin{remark} \label{remarkhorsehoesqure} For our definition, a $p$-branched horseshoe has one unstable direction and two stable directions. It is also straightforward that if $F$ has a $p$-branched horseshoe, then $F^{2}$ has a $p^{2}$-branched horseshoe. \end{remark}
\begin{df}
We say that a saddle periodic point of multipliers $| \lambda_{1}| \le | \lambda_{2}| < 1 <| \lambda_{3}| $ is sectionally dissipative if the product of any two of its eigenvalues is less than 1 in modulus, that is, $| \lambda_{1}\lambda_{3}| < 1$ and $| \lambda_{2}\lambda_{3}| < 1$. \end{df}
In the next proposition, we prove that if $b$ and $\sigma$ are sufficiently small, then some iterate of $f_{0}= f_{c,b,\sigma}$ has a $2$-branched horseshoe. Moreover, we introduce a neighborhood $\mathcal{F}$ of $f_{0}$ where this property persists. In Section 5, we will make a particular choice of $c$, $b$ and $\sigma$ so that the stable manifold of a periodic point of $f_{0}$ will have special properties, which will persist in a new neighborhood $\mathcal{F}' \Subset \mathcal{F}$ of $f_{0}$ in $\mathrm{Aut}_{2}(\mathbb{C}^{3})$.
\begin{prop} \label{ferm}
Let $q \ge 100$ and $\mathcal{C}$,$r$ given by Proposition \ref{marber}. Let $f_{0} = f_{c,b,\sigma}$ be the polynomial automorphism of $\mathbb{C}^{3}$ introduced in Eq. (1). Then, there exists $10^{-10}>b_{0}>0$ and $10^{-10}>\sigma_{0}>0$ independent of $c \in \mathcal{C}$ such that if $0<|b| < b_{0}$ and $0 \le |\sigma| < \sigma_{0}$, then $\mathcal{H}_{f_{0}} = \bigcap_{n \in \mathbb{Z}} (f^{qr}_{0})^{n}( \overline{ \mathbb{D}^{3}} )$ is a $2$-branched horseshoe. Moreover, $f_{0}$ has a periodic point $\delta_{f_{0}}$ that is sectionally dissipative and belongs to the homoclinic class of the continuation $\alpha_{f_{0}}$ of $\alpha_{c}$. These results remain true for any $f$ in some neighborhood $\mathcal{F} = \mathcal{F}(c,b,\sigma)$ of $f_{0}$ in $\mathrm{Aut}_{2}(\mathbb{C}^{3})$. \end{prop}
\begin{proof} When $\sigma = b = 0$, $f_{c,0,0} : (z_{1},z_{2},z_{3}) \mapsto (p_{c}(z_{1}),z_{1},\lambda z_{1} + \mu z_{3}+\nu)$ satisfies:
$$f^{qr}_{c,0,0}(z_{1},z_{2},z_{3})=(p^{qr}_{c}(z_{1}),p^{qr-1}_{c}(z_{1}), \nu + \lambda p_{c}^{qr-1}(z_{1})+ \mu (\nu+ \cdots +\mu (\nu+ \lambda z_{1} )) + \mu^{qr} z_{3} ) .$$
By Proposition \ref{marber} $(1)$, $p_{c}^{r-1} (\mathbb{D}'_{1}), p_{c}^{r-1}(\mathbb{D}'_{2}) \Subset \mathbb{D}$ and so $p_{c}^{qr-1} (h_{1}^{q}(\mathbb{D}')),p_{c}^{qr-1}(h_{2}^{q}(\mathbb{D}')) \Subset \mathbb{D}$. Then for any $(z_{1},z_{2},z_{3}) \in \big( h_{1}^{q}(\mathbb{D}') \cup h_{2}^{q}(\mathbb{D}') \big) \times \overline{\mathbb{D}}^{2}$, the second coordinate of $f_{c,0,0}(z_{1},z_{2},z_{3})$ is in $\mathbb{D}$. According to Corollary \ref{hyu}, it holds $\nu + \lambda p_{c}^{qr-1}(z_{1})+ \mu (\nu+ \cdots +\mu (\nu+ \lambda z_{1} )) \in \mathbb{D}( \pm \frac{9}{10} \cdot 10^{-4}, 10^{-10})$ for every $(z_{1},z_{2},z_{3}) \in \big( h_{1}^{q}(\mathbb{D}') \cup h_{2}^{q}(\mathbb{D}') \big) \times \overline{\mathbb{D}}^{2}$. Since $|\mu|^{qr} < 1- 10^{-4}+10^{-10}$ (see Remark \ref{rmu}), the third coordinate of $f_{c,0,0}(z_{1},z_{2},z_{3})$ lies in $\mathbb{D}$ too. Then the intersection $f^{qr}_{c,0,0} \big( \big( h_{1}^{q}(\mathbb{D}') \cup h_{2}^{q}(\mathbb{D}') \big) \times \overline{\mathbb{D}}^{2} \big) \cap \overline{\mathbb{D}^{3}}$ has two components which are two graphs over the coordinate $z_{1} \in \overline {\mathbb{D}}$ and which do not intersect $\overline{\mathbb{D}} \times \partial ( \overline{\mathbb{D}} \times \overline{\mathbb{D)}}$.
By continuity, there exists $10^{-10}>\sigma_{0}>0$ such that for every $0 \le |\sigma| < \sigma_{0}$, $f^{qr}_{c,0,\sigma} \big( \big( h_{1}^{q}(\mathbb{D}') \cup h_{2}^{q}(\mathbb{D}') \big) \times \overline{\mathbb{D}}^{2} \big) \cap \overline{\mathbb{D}^{3}}$ has still two components which are graphs over $z_{1} \in \overline{\mathbb{D}}$. The distance to $\overline{\mathbb{D}} \times \partial ( \overline{\mathbb{D}} \times \overline{\mathbb{D)}}$ of each of these two graphs is bounded from below by a strictly positive constant independent of $0 \le |\sigma| < \sigma_{0}$. Thus there exists $10^{-10}>b_{0}>0$ such that for every $0 \le |\sigma| < \sigma_{0}$ and $0<|b| < b_{0}$, the intersection $f^{qr}_{c,b,\sigma} \big( \big( h_{1}^{q}(\mathbb{D}') \cup h_{2}^{q}(\mathbb{D}') \big) \times \overline{\mathbb{D}}^{2} \big) \cap \overline{\mathbb{D}}^{3}$ has two connected components $D^{1,u}$ and $D^{2,u}$. These are two tridisks which do not intersect $\overline{\mathbb{D}} \times \partial ( \overline{\mathbb{D}} \times \overline{\mathbb{D)}}$ and are also connected components of $f^{qr}_{c,b,\sigma}( \overline{ \mathbb{D}^{3}}) \cap \overline{ \mathbb{D}^{3}}$. This proves item 1 of Definition \ref{df7}. We denote now $f_{c,b,\sigma} = f_{0}$. The proof of item 2 is similar and gives two connected components $D^{1,s}$ and $D^{2,s}$ in $f^{-qr}_{0}( \overline{ \mathbb{D}^{3}}) \cap \overline{ \mathbb{D}^{3}}$. Reducing $\sigma_{0}$ and $b_{0}$ if necessary, the cone field $C^{u}$ is $f_{0}^{qr}$-invariant on $D^{1,s} \cup D^{2,s}$ by Property (3) of Proposition \ref{marber} and dilated under $f_{0}^{qr}$ by a factor larger than $10^{10}$ because $|h_{j}'|<10^{-10}$ on $\mathbb{D}''$ (see Property 1 of Proposition \ref{fo}). The modulus of the derivative of $p_{c}$ is bounded from below by a strictly positive constant on $\mathbb{D}' \cup p_{c}(\mathbb{D}') \cup \cdots \cup p_{c}^{qr-1}(\mathbb{D}')$. Thus, still reducing $\sigma_{0}$ and $b_{0}$ if necessary, the cone field $C^{ss}$ is $f_{0}^{-qr}$-invariant on $D^{1,u} \cup D^{2,u}$ and dilated under $f_{0}^{-qr}$ by a factor at least $10^{10}$. For $\sigma \rightarrow 0$ and $b \rightarrow 0$, the directions of the two expanding eigenvectors of $D_{z}f^{-qr}_{0}$ respectively tend to the second and third directions (uniformly in $z$) while the contracting eigenvector of $D_{z}f^{-qr}_{0}$ stays in $C^{u}$. Since the associated rates of dilatation/contraction remain uniformly distant to 1 when $\sigma,b \rightarrow 0$, this implies that some cone of the form $\{ ( v_{1},v_{2},v_{3}) : ||(v_{2},v_{3})|| > \Xi ||v_{1}|| \}$ (for some $\Xi>1$) is $f^{-qr}_{0}$-invariant on $D^{1,u} \cup D^{2,u}$ and dilated by a factor larger than 1. This finishes to prove both items 3 and 4 of Definition \ref{df7}. Then $\mathcal{H}_{f_{0}} = \bigcap_{n \in \mathbb{Z}} (f^{qr}_{0})^{n}( \overline{\mathbb{D}^{3} })$ is a $2$-branched horseshoe.
The multiplier of the periodic point $\delta_{c}$ has its modulus between 1 and $(1+10^{-10})^{1/qr}$ by Property 6 of Proposition \ref{marber}. Then its continuation $\delta_{f_{0}}$ is a saddle point of expanding eigenvalue between 1 and $(1+2 \cdot 10^{-10})^{1/qr}$ after reducing $b_{0}$ and $\sigma_{0}$ if necessary. It has two contracting eigenvalues. When $b=\sigma= 0$, one is equal to 0 and the other one is a power of $\mu$ with $|\mu| < (1-10^{-4}+10^{-10})^{\frac{1}{qr}}$ (see Remark \ref{rmu}). Then, by continuity, reducing $b_{0}$ and $\sigma_{0}$ if necessary, $\delta_{f_{0}}$ is sectionally dissipative. We denote by $W^{u}_{\text{loc}}(\alpha_{f_{0}})$ (resp. $W^{s}_{\text{loc}}(\alpha_{f_{0}})$) the connected component of $W^{u}(\alpha_{f_{0}}) \cap \mathbb{D}^{3}$ (resp. $W^{s}(\alpha_{f_{0}}) \cap \mathbb{D}^{3}$) which contains $\alpha_{f_{0}}$, and we use the same notation for other periodic points. For small values of $b$ and $\sigma$, both $W^{u}_{\text{loc}}(\delta_{f_{0}})$ and $W^{u}_{\text{loc}}(\alpha_{f_{0}})$ are graphs over $z_{1} \in \mathbb{D}$ and both $W^{s}_{\text{loc}}(\delta_{f_{0}})$ and $W^{s}_{\text{loc}}(\alpha_{f_{0}})$ are graphs over $(z_{2},z_{3}) \in \mathbb{D}^{2}$. Reducing $b_{0}$ and $\sigma_{0}$ if necessary, $W^{u}_{\text{loc}}(\alpha_{f_{0}})$ and $W^{s}_{\text{loc}}(\delta_{f_{0}})$ intersect and $W^{s}_{\text{loc}}(\alpha_{f_{0}})$ and $W^{u}_{\text{loc}}(\delta_{f_{0}})$ intersect so $\delta_{f_{0}}$ is in the homoclinic class of $\alpha_{f_{0}}$ .
Finally, since all the conditions of the proposition are open, these results remain true for any $f$ in some neighborhood $\mathcal{F} = \mathcal{F}(c,b,\sigma)$ of $f_{0}$ in $\text{Aut}(\mathbb{C}^{3})$. This concludes the proof of the proposition. \end{proof}
\subsection{Blender property}
In this subsection, we show that the third coordinate of $f \in \mathcal{F}$ has some kind of open-covering property. Since $f^{qr} \in \mathcal{F}$ has a 2-branched horseshoe, $f^{2qr}$ has a 4-branched horseshoe by Remark \ref{remarkhorsehoesqure}. We are interested in the geometric properties of the third coordinate of $f^{2qr}$. Here, we use it to show that the 4-branched horseshoe associated to $f^{2qr}$ is a blender. Remind that the maps $h_{1}$ and $h_{2}$ were defined in Notation \ref{defh1h}.
\begin{df} \label{uu}
For $f \in \mathcal{F}$, we denote by $f^{qr}[j]$ the restriction of $f^{qr}$ on $h^{q}_{j}( \mathbb{D}') \times \mathbb{D}^{2}$ for $j \in \{1,2\}$. We put $V_{1} = f^{qr}[1] (h^{q}_{1}( \mathbb{D}') \times \mathbb{D}^{2}) \cap \mathbb{D}^{3}$ and $V_{2} =f^{qr}[2] (h^{q}_{2}( \mathbb{D}') \times \mathbb{D}^{2}) \cap \mathbb{D}^{3}$. We define $U_{1} = f^{qr}[1] \big( V_{1} \cap (h^{q}_{1}( \mathbb{D}') \times \mathbb{D}^{2}) \big) \cap \mathbb{D}^{3}$, $U_{2} = f^{qr}[2] \big( V_{1} \cap (h^{q}_{2}( \mathbb{D}') \times \mathbb{D}^{2}) \big) \cap \mathbb{D}^{3} $, $U_{3} = f^{qr}[1] \big( V_{2} \cap (h^{q}_{1}( \mathbb{D}') \times \mathbb{D}^{2}) \big) \cap \mathbb{D}^{3}$ and $U_{4} = f^{qr}[2] \big( V_{2} \cap (h^{q}_{2}( \mathbb{D}') \times \mathbb{D}^{2}) \big) \cap \mathbb{D}^{3}$. We define $g_{j} = f^{-2qr}_{|U_{j}}$ for $1 \le j \le 4$. \end{df}
\begin{notation} We will denote $c_{1} = \frac{9}{10} \cdot 10^{-4} \cdot (1+i)$, $c_{2} = \frac{9}{10} \cdot 10^{-4} \cdot (-1+i)$, $c_{3} = \frac{9}{10} \cdot 10^{-4} \cdot (1-i)$ and $c_{4} = \frac{9}{10} \cdot 10^{-4} \cdot (-1-i)$. \end{notation}
\begin{lemma} \label{esti} Reducing $b_{0}$, $\sigma_{0}$ and $\mathcal{F}$ if necessary, we have: \begin{enumerate} \item $ \forall z \in f^{-qr}(V_{1}), \mathrm{pr}_{3}( f^{qr}(z)) \in \mathbb{D}( \mu^{qr} z_{3} + \frac{9}{10} \cdot 10^{-4},10^{-9})$, \item $ \forall z \in f^{-qr}(V_{2}), \mathrm{pr}_{3}( f^{qr}(z)) \in \mathbb{D}( \mu^{qr} z_{3} - \frac{9}{10} \cdot 10^{-4},10^{-9})$. \end{enumerate} \end{lemma}
\begin{proof} For $\sigma = b = 0$, it is a simple consequence of Corollary \ref{hyu}. Then, by continuity we just have to take sufficiently small values of $\sigma_{0},b_{0},\mathcal{F}$ to get the bound $10^{-9}$. \end{proof}
Since we both have $\mu_{0}^{qr} = (1-10^{-4}) \cdot e^{i \cdot \frac{\pi}{2}}$ (by Notation \ref{h}) and $\mu^{qr} \subset \mathbb{D}(\mu_{0}^{qr},10^{-10})$ (by Remark \ref{rmu}), by iterating two times the previous result, we get the following:
\begin{corollary} \label{estit} For every $z=(z_{1},z_{2},z_{3}) \in U_{j}$, $ \mathrm{pr}_{3}( g_{j}(z)) \in \mathbb{D}( \frac{1}{\mu^{2qr}} (z_{3} - c_{j}), 10^{-6})$. \end{corollary}
We now show an open covering property for the four affine maps $z \mapsto \frac{1}{\mu^{2qr} } (z - c_{j})$.
\begin{prop} \label{rp9} For every $z \in \mathbb{D}(0,\frac{1}{10})$, there exists $j \in \{1,2,3,4\}$ such that: $$\frac{1}{\mu^{2qr} } (z - c_{j}) \in \mathbb{D}(0,\frac{1}{10}-10^{-5}) \, .$$ \end{prop}
\begin{proof}
We check that the union of the images of $ \mathbb{D}(0,\frac{1}{10}-10^{-5})$ under the four affine maps $z \mapsto \mu^{2qr} z+ c_{j} $ contains $ \mathbb{D}(0,\frac{1}{10})$. We begin by showing that for every point $z$ of the set $\{z : |z| = \frac{1}{10} \text{ and } 0 \le \text{Arg}(z) \le \frac{\pi}{2}\}$, the point $z-c_{1}$ belongs to the disk $\mathbb{D} \big( 0,|\mu|^{2qr} (\frac{1}{10} -10^{-5}) \big)$. Let us point out that $|\mu|^{2qr} > 1-2 \cdot 10^{-4}$ (see Remark \ref{rmu}). Denoting $z = x+ iy$ with $(x,y) \in \mathbb{R}_{+}^{2}$, we have $ |z-c_{1}|^{2} = (x-\frac{9}{10} \cdot 10^{-4})^{2} + (y-\frac{9}{10} \cdot 10^{-4})^{2}$. Since $x^{2}+y^{2} = (\frac{1}{10})^{2}$, at least one between $x$ or $y$ is larger than $\frac{1}{\sqrt{2}} \cdot \frac{1}{10}$. Then $|z-c_{1}|^{2}$ is smaller than:
$$(\frac{1}{10})^{2} -2 \frac{1}{\sqrt{2}} \frac{1}{10} \frac{9}{10} 10^{-4} +2 (\frac{9}{10} 10^{-4})^{2}<\big( ( 1-2 \cdot 10^{-4}) (\frac{1}{10} -10^{-5})\big)^{2}< \big( |\mu|^{2qr} (\frac{1}{10} -10^{-5}) \big)^{2}$$ Then for every point $z$ of the set $\{z : |z| = \frac{1}{10} \text{ and } 0 \le \text{Arg}(z) \le \frac{\pi}{2}\}$, we have that $z-c_{1} \in \mathbb{D} \big( 0,|\mu|^{2qr}(\frac{1}{10} -10^{-5}) \big)$. We also have $0-c_{1} \in \mathbb{D} \big( 0,|\mu|^{2qr}(\frac{1}{10} -10^{-5}) \big)$. Thus by convexity the image of $ \mathbb{D}(0,\frac{1}{10}-10^{-5})$ by the affine map $z \mapsto \mu^{2qr} z+ c_{1} $ contains the first quadrant of $ \mathbb{D}(0,\frac{1}{10})$. Then the result follows by symmetry. \end{proof}
\includegraphics[width=11cm]{figure1.png}
\begin{center} Figure 1: complex blender. The top figure shows the sets $V_{1}$ and $V_{2}$ in a real analogy (that is in $\mathbb{R}^{3}$). The two bottom figures represent the respective images by the third projection map $\pi_{3} : \mathbb{C}^{3} \rightarrow \mathbb{C}$ of $V_{1}$,$V_{2}$ (on the left) and $U_{1}$,$U_{2}$,$U_{3}$,$U_{4}$ (on the right). \end{center}
We are finally in position to prove the main result of this section.
\begin{df} \label{curves} A $ss$-curve $\Gamma$ is a holomorphic graph over $z_{2}$ which has all its tangent vectors in $C^{ss}$ (recall that this cone was defined in Notation \ref{con}). \end{df}
\begin{prop} \label{blender} Let $\Gamma $ be any $ss$-curve intersecting $\mathbb{D}^{2} \times \mathbb{D}(0,\frac{1}{10})$. Then for every $f \in \mathcal{F}$, $\Gamma$ intersects the unstable manifold of a point of the horseshoe $\mathcal{H}_{f}$: in other words, $\mathcal{H}_{f}$ is a blender. \end{prop}
\begin{proof}
Let $\Gamma = \Gamma_{0}$ be a $ss$-curve intersecting $\mathbb{D}^{2} \times \mathbb{D}(0,\frac{1}{10})$ and $f \in \mathcal{F}$. We show that there exists $j \in \{1,2,3,4\}$ such that $g_{j} (\Gamma)$ contains a $ss$-curve $\Gamma^{1}$ intersecting $\mathbb{D}^{2} \times \mathbb{D}(0,\frac{1}{10})$. Let $Z = (Z_{1},Z_{2},Z_{3})$ be a point of $\Gamma \cap (\mathbb{D}^{2} \times \mathbb{D}(0,\frac{1}{10}))$. By Lemma \ref{rp9} there exists $j \in \{1,2,3,4\}$ such that $\frac{1}{\mu^{2qr} }(Z_{3} -c_{j}) \in \mathbb{D}(0,\frac{1}{10}-10^{-5})$. Since $ \mathbb{D}(0,\frac{1}{10}) \subset \text{pr}_{3}(U_{j})$, by continuity, $\Gamma$ intersects $U_{j}$. The cone $C^{ss}$ is $g_{j}$-invariant since $\mathcal{H}_{f}$ is a 2-branched horseshoe, and so $\Gamma^{1} = g_{j}(\Gamma \cap U_{j})$ is a $ss$-curve. Then it holds $\text{diam}(\text{pr}_{3}(\Gamma^{1})) \le 10^{-6}$. By Corollary \ref{estit}, $| \text{pr}_{3} ( g_{j}(Z) ) - \frac{1}{\mu^{2qr} }(Z_{3}-c_{j})|<10^{-6} $. This implies that $\text{pr}_{3}( \Gamma^{1} ) \subset \mathbb{D}(0,\frac{1}{10}-10^{-5}+2 \cdot 10^{-6}) \subset \mathbb{D}(0,\frac{1}{10})$. Then $\Gamma^{1}$ intersects $\mathbb{D}^{2} \times \mathbb{D}(0,\frac{1}{10})$. By iteration, we can construct a sequence of $ss$-curves $\Gamma^{n} \subset (g_{j_{n}} \circ ... \circ g_{j_{1}})(\Gamma)$, each of them intersecting $\mathbb{D}^{3}$. The set $\bigcap_{n \ge 1} (f^{2qr})^{n}(\Gamma^{n}) \subset \Gamma$ is non empty since it contains the intersection of a decreasing sequence of non empty compact sets. But any point in this intersection is a point of the unstable manifold of a point of the horseshoe $\mathcal{H}_{f}$. This ends the proof. \end{proof}
\section{Mechanism to get persistent tangencies}
In this section, we explain how the blender property obtained in the last section leads to persistent tangencies with certain "folded" surfaces.
\subsection{Some definitions}
\begin{df} A submanifold (or an analytic set) $W \subset \mathbb{D}^{n}$ is horizontal relatively to a decomposition $\mathbb{D}^{n} = \mathbb{D}^{k} \times \mathbb{D}^{n-k}$ if $W$ does not intersect $ \mathbb{D}^{k} \times \partial \mathbb{D}^{n-k}$. We will also say it is horizontal relatively to the projection $\mathbb{D}^{n} \rightarrow \mathbb{D}^{k}$. If $\dim(W) = k$, then the natural projection on $\mathbb{D}^{k}$ is a branched covering of degree $d$. We will say that $W$ is of degree $d$. We similarly define vertical submanifolds in $\mathbb{D}^{k} \times \mathbb{D}^{n-k}$ and their degree. \end{df}
The two following propositions are classical. For a proof, one can refer to \cite{dsfdm}. One can also refer to \cite{henam}.
\begin{prop} \label{fold55} Let $W$ be a horizontal curve of degree $d$ and $W'$ be a vertical curve of degree $d'$ in $\mathbb{D}^{2} = \mathbb{D} \times \mathbb{D}$. Then $W$ and $W'$ intersect in $dd'$ points with multiplicity. \end{prop}
\begin{prop} \label{fold77} Let $W$ be a horizontal curve (resp. surface) of degree $d$ and $W'$ be a vertical surface (resp.curve) of degree $d'$ in $\mathbb{D}^{3} = \mathbb{D}^{1} \times \mathbb{D}^{2}$ (resp. $\mathbb{D}^{3} = \mathbb{D}^{2} \times \mathbb{D}^{1}$). Then $W$ and $W'$ intersect in $dd'$ points with multiplicity. \end{prop}
\begin{remark} In particular, in the two previous propositions, if all the intersections are transverse, there are exactly $dd'$ distinct points of intersection. If it is not the case, there is at least one point of tangency. \end{remark}
We now introduce several definitions specific to our context.
\begin{notation} Let $i,j$ be two distinct integers in $\{1,2,3\}$. We will denote by $\pi_{i}$ the projection over the $i^{th}$ coordinate and by $(\pi_{i},\pi_{j})$ the projection over the $i^{th}$ and the $j^{th}$ coordinates. \end{notation}
\begin{df} Let $i,j$ be two distinct integers in $\{1,2,3\}$. A $(i,j)$-surface $\mathcal{S}$ is a complex surface horizontal relatively to the projection $(\pi_{i},\pi_{j})$. \end{df}
\begin{remark} A $(i,j)$-surface $\mathcal{S}$ is a ramified covering of degree $d$ over $\mathbb{D}^{2}$. In the rest of this article, we will only consider ramified coverings of degree 1 or 2. \end{remark}
\begin{df} \label{curves} A $u$-curve $\Gamma$ is a holomorphic graph over $z_{1}$ which has all its tangent vectors in $C^{u}$ (recall that this cone was defined in Notation \ref{con}). \end{df}
\begin{df} Let $k \in \{2,3\}$. A $(1,k)$-quasi plane $\mathcal{V}$ over a bidisk $\mathbb{D}_{1} \times \mathbb{D}_{k}$ is a graph $\{z_{k'} = v(z_{1},z_{k})\} $, where $k'$ is the integer defined by $\{1,2,3\} = \{1,k,k'\}$ and $v : \mathbb{D}_{1} \times \mathbb{D}_{k} \rightarrow \mathbb{D}$ is holomorphic, such that the $C^{0}$-norm of $Dv$ is bounded by 1 and $\mathcal{V}$ is foliated by $u$-curves $\mathcal{V}_{x}$. \end{df}
\begin{remark} In particular, every $(1,k)$-quasi plane over $\mathbb{D}^{2}$ is a $(1,k)$-surface. \end{remark}
\begin{df} Let $k \in \{2,3\}$. A $k$-folded curve is a holomorphic curve horizontal relatively to $\pi_{k}$ which is a ramified 2-covering over $z_{k}$ with exactly one point of ramification $z_{k,ram}$. We denote $\mathrm{fold}(\Gamma) = z_{k,ram}$ the \emph{fold} of $\Gamma$. \end{df}
\begin{df} Let $k \in \{2,3\}$. A $k$-folded $(2,3)$-surface $\mathcal{W}$ is a complex surface of degree 2 that is horizontal relatively to $(\pi_{2},\pi_{3})$ and such that for every $(1,k)$-quasi plane $\mathcal{V}$ over $\mathbb{D}^{2}$, $\Gamma = \mathcal{V} \cap \mathcal{W}$ is a $k$-folded curve. We denote: $$\mathrm{Fold}(\mathcal{W}) = \{ \mathrm{fold}(\Gamma) : \Gamma \text{ is a } k\text{-folded curve included in } \mathcal{W}\}$$ which is a subset of $\mathbb{D}$. We say that $\mathcal{W}$ is concentrated if $\mathrm{diam}( \mathrm{Fold}(\mathcal{W}) ) \le 10^{-5}$. \end{df}
\subsection{Preparatory lemmas}
In this Subsection we gather some simple technical results that will be useful in the following Subsections.
\begin{lemma} \label{lemb}
Let $\Gamma = \gamma(\mathbb{D})$ be a $k$-folded curve (with $k \in \{2,3\}$) included inside some $(1,k)$-quasi plane over $\mathbb{D} \times \mathbb{D}_{k}$ with $\mathbb{D}(0,1/2) \subset \mathbb{D}_{k} \subset \ \mathbb{D}$ and $\mathrm{diam} \big( \mathrm{pr}_{1}(\Gamma) \big) \le 10^{-10}$. Then for every disk $\mathbb{D}_{\Gamma} \Subset \mathbb{D}_{k}$ of radius $10^{-7}$ distant of at least $10^{-7}$ from $\partial \mathbb{D}_{k} \cup \{\mathrm{fold}(\Gamma)\}$, it holds: for every $Z$ with $\mathrm{pr}_{k}(\gamma(Z)) \in \mathbb{D}_{\Gamma}$ we have $|\gamma'_{1}(Z)| < 10^{-3} |\gamma'_{k}(Z) |$. In particular, $\Gamma \cap (\mathbb{D}^{2} \times \mathbb{D}_{\Gamma})$ is the union of two graphs upon $z_{k} \in \mathbb{D}_{\Gamma}$. \end{lemma}
\begin{proof}
Let $ \mathbb{D}_{\Gamma} \Subset \mathbb{D}_{k}$ be a disk of radius $10^{-7}$ distant of at least $10^{-7}$ from $\partial \mathbb{D}_{k} \cup \{\mathrm{fold}(\Gamma)\}$. We notice that $\Gamma$ is the union of two graphs over $z_{k}$ varying in the $10^{-7}$-neighborhood of $\mathbb{D}_{\Gamma}$. Let us denote $z_{1} = \xi(z_{k})$ one of them. Every point of $ \mathbb{D}_{\Gamma}$ is the center of a ball of radius $10^{-7}$ where $|\xi(z_{k})|<10^{-10}$. Hence, by the Cauchy inequality, we have $|\xi'(z_{k})|<10^{7} \cdot 10^{-10} = 10^{-3}$. Thus $|\gamma'_{1}(Z)|< 10^{-3} |\gamma'_{k}(Z)|$ at every $Z$ such that $\mathrm{pr}_{k}(\gamma(Z)) \in \mathbb{D}_{\Gamma}$ and the result follows. \end{proof}
Here are some consequences of Propositions \ref{fold55} and \ref{fold77}.
\begin{lemma} \label{rq5}
Let $\mathcal{V}$ be a $(1,k)$-quasi plane over $\mathbb{D}_{1} \times \mathbb{D}_{k}$ with $\mathbb{D}(0,1/2) \subset \mathbb{D}_{k} \subset \ \mathbb{D}$. Let $\Gamma$ be a $u$-curve and $\tilde{\Gamma}$ be a graph over $z_{k}$ with $|\tilde{\gamma}'_{1}| < 10^{-3} |\tilde{\gamma}'_{k}|$, both included in $\mathcal{V}$. Then $\Gamma \cap \tilde{\Gamma}$ is a singleton. \end{lemma}
\begin{prop} \label{rq11} Let $\mathcal{W}$ be a $k$-folded $(2,3)$-surface with $k \in \{2,3\}$. Let $\Gamma$ be a $u$-curve. Then $\mathcal{W} \cap \Gamma$ has one or two points. \end{prop}
\subsection{Main result}
Here is the main result of this section: we show that any concentrated 3-folded $(2,3)$-surface having its fold in good position has a point of tangency with the unstable manifold of a point of the horseshoe $\mathcal{H}_{f}$.
\begin{prop} \label{rrrrr} Let $\mathcal{W}$ be a concentrated 3-folded $(2,3)$-surface such that we both have $\mathrm{Fold}(\mathcal{W}) \subset \mathbb{D}(0,\frac{1}{10})$ and $\mathrm{diam}(\mathrm{pr}_{1}(\mathcal{W})) \le 10^{-10}$. Then there exists a point $\kappa_{f}$ of the horseshoe $\mathcal{H}_{f}$ such that $\mathcal{W}$ has a point of tangency with the unstable manifold of $\kappa_{f}$. \end{prop}
We begin by a lemma showing that the image of a folded curve contains a folded curve. Remind that the sets $U_{j}$ and the maps $g_{j}$ were defined in Definition \ref{uu}.
\begin{lemma} \label{lemb5} Let $\tilde{\Gamma}$ be a $3$-folded curve included in a $(1,3)$-quasi plane $\tilde{\mathcal{V}}$ over $\mathbb{D}_{1} \times \mathbb{D}_{k}$ with $\mathbb{D}(0,1/2) \subset \mathbb{D}_{k} \subset \ \mathbb{D}$. We suppose that $\tilde{\Gamma}$ and $\tilde{\mathcal{V}}$ are included in $U_{j}$ for some $j \in \{1,2,3,4\}$. We also suppose $\mathrm{diam}(\mathrm{pr}_{1}(\tilde{\Gamma})) \le 10^{-10}$ and that $g_{j}(\tilde{\mathcal{V}})$ is included in a $(1,3)$-quasi plane $\mathcal{V}$ over $\mathbb{D}^{2}$. Then $\Gamma = g_{j}(\tilde{\Gamma})$ is a $3$-folded curve satisfying: $$\mathrm{fold}(\Gamma) \in \mathbb{D}( \frac{1}{\mu^{2qr}} \big( \mathrm{fold}(\tilde{\Gamma})- c_{j} ), \frac{3}{2} \cdot 10^{-6} \big) \, .$$ \end{lemma}
\begin{proof}
By Lemma \ref{lemb}, for every disk $\mathbb{D}_{\tilde{\Gamma}} \Subset \mathbb{D}_{k}$ of radius $10^{-7}$ distant of at least $10^{-7}$ from $\partial \mathbb{D}_{k} \cup \{\mathrm{fold}(\tilde{\Gamma})\}$, it holds: for every $Z$ with $\mathrm{pr}_{k}(\tilde{\gamma}(Z)) \in \mathbb{D}_{\tilde{\Gamma}}$ we have $|\tilde{\gamma}'_{1}(Z)| < 10^{-3} |\tilde{\gamma}'_{k}(Z) |$. Moreover $\tilde{\Gamma} \cap (\mathbb{D}^{2} \times \mathbb{D}_{\tilde{\Gamma}})$ is the union of two graphs upon the coordinate $z_{k} \in \mathbb{D}_{\tilde{\Gamma}}$. We take the foliation $(\mathcal{V}_{t})_{t \in \mathbb{D}}$ of $\mathcal{V}$ by the $u$-curves $\mathcal{V}_{t} = \mathcal{V} \cap \{z_{3} = t\}$. For any $t \in \mathbb{D}$, $\tilde{\mathcal{V}}_{t} = f^{2qr}(\mathcal{V}_{t} \cap g_{j}(U_{j}) ) $ is a $u$-curve by invariance of $C^{u}$. Suppose that $\tilde{\mathcal{V}}_{t}$ intersects $\tilde{\Gamma}$ at some point $\tilde{\gamma}(Z)$ such that $\mathrm{pr}_{k}(\tilde{\gamma}(Z))$ belongs to the disk of same center as $\mathbb{D}_{\tilde{\Gamma}}$ and half radius. Since the tangent spaces of $ \tilde{\mathcal{V}}_{t}$ are directed by vectors in $C^{u}$ and since $\mathrm{diam}(\mathrm{pr}_{1}(\tilde{\Gamma})) \le 10^{-10}$, $\tilde{\mathcal{V}}_{t}$ intersects $\tilde{\Gamma}$ in exactly two points by Lemma \ref{rq5} and then it is also the case for $\mathcal{V}_{t}$ and $\Gamma$. It is clear there exists infinitely many $t \in \mathbb{D}$ satisfying this property. This implies that $\text{pr}_{3}:\Gamma \rightarrow \mathbb{D}$ is a 2-covering. By the Riemann-H\"urwitz formula, it is a 2-covering with only one point of ramification. Moreover this shows that $\mathrm{fold} ( \Gamma ) \in \text{pr}_{3}(g_{j}(\mathbb{D}^{2} \times \mathbb{D}(\text{fold}(\tilde{\Gamma}), 2 \cdot 10^{-7}))$. Then by Corollary \ref{estit}, we have $\mathrm{fold}(\Gamma') \in \mathbb{D}( \frac{1}{\mu^{2qr}} ( \text{fold}(\tilde{\Gamma})- c_{j} ), \frac{3}{2} \cdot 10^{-6})$. \end{proof}
From now on, we prove lemmas that will show that the image under $g_{j}$ of a 3-folded $(2,3)$-surface having its fold in $ \mathbb{D}(0,\frac{1}{10})$, is concentrated and that it is possible to choose $j \in \{1,2,3,4\}$ so that the new fold is still in $ \mathbb{D}(0,\frac{1}{10})$. The following is an intermediate result to prove Lemma \ref{rp3}:
\begin{lemma} \label{presqueenfin}
Reducing $b_{0}$, $\sigma_{0}$ and $\mathcal{F}$ if necessary, there exists some constant $\chi'>\chi>1$ and some compact neighborhood $\tilde{\mathbb{D}} $ of $\mathbb{D}$ with $\mathbb{D} \Subset \tilde{\mathbb{D}} \Subset \mathbb{D}'$ such that for any $f \in \mathcal{F}$, $j \in \{1,2,3,4\}$ and $(1,3)$-quasi plane $\mathcal{V}'$ over $\mathbb{D}^{2}$, the set $f^{2qr} ( \mathcal{V}' )$ contains a $(1,3)$-quasi plane $\mathcal{V} = \{z_{2} = v(z_{1},z_{3})\}$ over $\tilde{\mathbb{D}} \times \mathbb{D}(0,2/3)$. Moreover $\mathcal{V}$ is foliated by $u$-curves whose tangent vectors are all included in the subcone $ \tilde{C}^{u} = \{v = (v_{1},v_{2},v_{3}) \in \mathbb{C}^{3} : \max(|v_{2}|,|v_{3}|) \le \chi'^{-1} \cdot |v_{1}|\} $ of $C^{u}$. \end{lemma}
\begin{proof}
Let us fix $j \in \{1,2,3,4\}$. By invariance of $C^{ss}$, the set $f^{2qr} \big( \mathcal{V}' \cap g_{j}(U_{j}) \big)$ contains a holomorphic graph $\mathcal{V} = \{z_{2} = v(z_{1},z_{3})\}$ over $\mathbb{D} \times \mathbb{D}(0,2/3)$ included in $U_{j}$. When $b =\sigma = 0$, it is obvious that the $C^{0}$-norm of $Dv$ is bounded by 1 so it is still the case for every $f \in \mathcal{F}$, reducing $b_{0}$, $\sigma_{0}$ and $\mathcal{F}$ if necessary. By property 1 of Proposition \ref{marber}, $p_{c}^{qr}$ is univalent on some neighborhood of $h_{1}^{q}(\mathbb{D})$ and on some neighborhood of $h_{2}^{q}(\mathbb{D})$. Then, reducing $b_{0}$, $\sigma_{0}$ and $\mathcal{F}$ if necessary, there exists some compact neighborhood $\tilde{\mathbb{D}}$ of $\mathbb{D}$ independent of $\mathcal{V}'$ such that $\mathbb{D} \Subset \tilde{\mathbb{D}} \Subset \mathbb{D}'$ and $v$ can be extended from $\tilde{\mathbb{D}} \times \mathbb{D}(0,2/3)$ into $\mathbb{D}$. The extended graph is included in $f^{2qr} \big( \mathcal{V}')$. By property 3 of Proposition \ref{marber} we have $|p'_{c}|>\chi$ on $\mathbb{D}'$. Since $p'_{c}$ is continuous, this implies that $|p'_{c}| \ge\chi''$ for some $\chi''>\chi$ on $\tilde{\mathbb{D}}$ (reducing $\tilde{\mathbb{D}}$ if necessary). We denote $\chi' = \frac{1}{2}(\chi+\chi')$. Then, for a single point $z $ in a compact neighborhood of $(h_{1}^{q}(\mathbb{D}) \cup h_{2}^{q}(\mathbb{D})) \times \mathbb{D}^{2}$, reducing $b_{0}$, $\sigma_{0}$ and $\mathcal{F}$ if necessary, the differential of $f^{2qr}$ at $z$ sends the closure of $C^{u}$ into $\tilde{C}^{u}$. Since $b_{0}$, $\sigma_{0}$ and $\mathcal{F}$ can be taken locally constant in $z$, by compactness, we can reduce $b_{0}$, $\sigma_{0}$ and $\mathcal{F}$ such that this remains true for every $z$. In particular, $\mathcal{V}$ is a $(1,3)$-quasi plane over $\mathbb{D}^{2}$ foliated by $u$-curves whose tangent vectors are all included in $ \tilde{C}^{u}$. Finally we take the minimal values of $b_{0}$, $\sigma_{0}$ and $\mathcal{F}$ over $j \in \{1,2,3,4\}$. This concludes the proof. \end{proof}
Since $\mathbb{D} \times \mathbb{D}(0,1/2) \Subset \tilde{\mathbb{D}} \times \mathbb{D}(0,2/3) $, by the Cauchy inequality, there exists $\mathcal{C}>0$ such that for any holomorphic map $w$ from $\tilde{\mathbb{D}} \times \mathbb{D}(0,2/3)$ into $\mathbb{D}$, the $C^{0}$-norm of $Dw$ on $\mathbb{D} \times \mathbb{D}(0,1/2)$ is bounded by $\mathcal{C}$.
\begin{lemma}\label{est} Reducing $b_{0}$, $\sigma_{0}$ and $\mathcal{F}$ if necessary, for every $f \in \mathcal{F}$, $j \in \{1,2,3,4\}$ and $z \in \mathbb{D}$, we both have $\mathrm{diam} \big( \mathrm{pr}_{2}(U_{j} \cap \{z_{1} = z \} ) \big) < 10^{-7} \cdot \mathrm{dist} (U_{j}, \mathbb{D} \times \partial \mathbb{D} \times \mathbb{D})$ and $\mathrm{diam} \big( \mathrm{pr}_{2}(U_{j} \cap \{z_{1} = z \} ) \big) < 10^{-1} \cdot \mathcal{C}^{-1} \cdot (\chi'-\chi)$. \end{lemma}
\begin{proof} We notice that for the map $(z_{1},z_{2},z_{3}) \mapsto (p_{c}(z_{1})+bz_{2},z_{1}, \lambda z_{1} + \mu z_{3} + \nu)$, the diameter $\mathrm{diam} \big( \mathrm{pr}_{2}(U_{j} \cap \{z_{1} = z \} ) \big)$ tends to 0 uniformly in $j \in \{1,2,3,4\}$ and $z \in \mathbb{D}$ when $b$ tends to 0 so we can decrease $b_{0}$ so that the two inequalities are satisfied for every $j \in \{1,2,3,4\}$ and $z \in \mathbb{D}$. Since both inequalities are open, reducing $\sigma_{0}$ and $\mathcal{F}$ if necessary, both remain true for every $f \in \mathcal{F}$, $j \in \{1,2,3,4\}$ and $z \in \mathbb{D}$. \end{proof}
\begin{lemma} \label{rp3} Let $\mathcal{V}'^{0}$ and $\mathcal{V}'^{1}$ be two $(1,3)$-quasi planes over $\mathbb{D}^{2}$ and let $\mathcal{V}^{0} =f^{2qr} \big( \mathcal{V}'^{0} \cap g_{j}(U_{j}) \big)$ and $\mathcal{V}^{1} =f^{2qr} \big( \mathcal{V}'^{1} \cap g_{j}(U_{j}) \big)$ for some $j \in \{1,2,3,4\}$. Then there exists a holomorphic family $(\mathcal{V}_{t})_{t \in \mathbb{D}(0,10^{6})}$ where $\mathcal{V}^{0} \subset \mathcal{V}_{0}$ and $\mathcal{V}^{1} \subset \mathcal{V}_{1}$, and for every $t \in \mathbb{D}(0,10^{6})$, $\mathcal{V}_{t}$ is a $(1,3)$-quasi plane over $\mathbb{D} \times \mathbb{D}(0,1/2)$. \end{lemma}
\begin{proof}
By the proof of Lemma \ref{presqueenfin}, both $\mathcal{V}^{0} $ and $\mathcal{V}^{1}$ contain $(1,3)$-quasi planes $\mathcal{V}_{0} $ and $\mathcal{V}_{1}$ over $\mathbb{D} \times \mathbb{D}(0,1/2)$ included in $U_{j}$. The $(1,3)$-quasi plane $\mathcal{V}_{0}$ can be written as a graph $(z_{1},z_{3}) \mapsto v_{0}(z_{1},z_{3})$ over $\mathbb{D} \times \mathbb{D}(0,1/2)$ and the $(1,3)$-quasi plane $\mathcal{V}_{1}$ can be written as a graph $(z_{1},z_{3}) \mapsto v_{1}(z_{1},z_{3})$ over $\mathbb{D} \times \mathbb{D}(0,1/2)$. For every $t \in \mathbb{D}(0,10^{6})$, we denote $v_{t}(z_{1},z_{3}) = v_{0}(z_{1},z_{3}) + t \cdot ( v_{1}(z_{1},z_{3}) - v_{0}(z_{1},z_{3}) ) $, which defines a graph $\mathcal{V}_{t} $ over $(z_{1},z_{3}) \in \mathbb{D} \times \mathbb{D}(0,1/2)$. By the first inequality of Lemma \ref{est}, for every $t \in \mathbb{D}(0,10^{6})$, $\mathcal{V}_{t} $ does not intersect $\mathbb{D} \times \partial \mathbb{D} \times \mathbb{D}$. By Lemma \ref{presqueenfin}, $\mathcal{V}_{0}$ is foliated by $u$-curves whose tangent vectors are all included in $ \tilde{C}^{u} $. We notice that $ | v_{1}(z_{1},z_{3}) - v_{0}(z_{1},z_{3})|$ is bounded by $\sup_{z \in \mathbb{D}} \mathrm{diam} \big( \mathrm{pr}_{2}(U_{j} \cap \{z_{1} = z \} ) \big) < 10^{-1} \cdot \mathcal{C}^{-1} \cdot (\chi'-\chi)$ by the second inequality of Lemma \ref{est}. Still using Lemma \ref{presqueenfin}, $(z_{1},z_{3}) \mapsto v_{1}(z_{1},z_{3}) - v_{0}(z_{1},z_{3})$ can be extended on $\tilde{\mathbb{D}} \times \mathbb{D}(0,2/3)$. Reducing $b_{0}$, $\sigma_{0}$ and $\mathcal{F}$, its image is also included in $\mathbb{D}(0, 10^{-1} \cdot \mathcal{C}^{-1} \cdot (\chi' - \chi))$. Then by definition of $\mathcal{C}$, the $C^{0}$-norm of $D (v_{1}-v_{0})$ on $\mathbb{D} \times \mathbb{D}(0,1/2)$ is bounded by $\mathcal{C} \cdot 10^{-1} \cdot \mathcal{C}^{-1} \cdot (\chi'-\chi)= 10^{-1} \cdot (\chi'- \chi)$. Since $\mathcal{V}^{0}$ is foliated by $u$-curves whose tangent vectors are all included in $ \tilde{C}^{u}$, $\mathcal{V}^{t}$ is then foliated by curves with tangent vectors in $C^{u}$, that is $u$-curves, for every $t \in \mathbb{D}(0,10^{6})$. Thus $\mathcal{V}_{t}$ is a $(1,3)$-quasi plane over $\mathbb{D} \times \mathbb{D}(0,1/2)$ for every $t \in \mathbb{D}(0,10^{6})$. \end{proof}
\begin{lemma} \label{rp5}
Let $\mathcal{V}_{0}$ and $\mathcal{V}_{1}$ be two $(1,3)$-quasi planes over $\mathbb{D} \times \mathbb{D}(0,1/2)$ and a holomorphic family $(\mathcal{V}_{t})_{t \in \mathbb{D}(0,10^{6})}$ containing $\mathcal{V}_{0}$ and $\mathcal{V}_{1}$ and such that $\mathcal{V}_{t}$ is a $(1,3)$-quasi plane over $\mathbb{D} \times \mathbb{D}(0,1/2)$ for $t \in \mathbb{D}(0,10^{6})$. Let $\mathcal{W}$ be a 3-folded $(2,3)$-surface such that $\mathrm{Fold}(\mathcal{W}) \subset \mathbb{D}(0,1/10)$. Then $|\mathrm{fold}(\mathcal{V}_{1} \cap \mathcal{W}) - \mathrm{fold}(\mathcal{V}_{0} \cap \mathcal{W}) |< 10^{-6}$. \end{lemma}
\begin{proof}
We consider the function $t \mapsto \text{fold}(\mathcal{V}^{t} \cap \mathcal{W})$ defined on $\mathbb{D}(0,10^{6})$. It is holomorphic by the Implicit Function Theorem and its image is included in $ \mathbb{D}(0,1/10)$ because $\text{Fold}(\mathcal{W}) \subset \mathbb{D}(0,1/10)$. Then by the Cauchy inequality its derivative is smaller than $\frac{1}{10} \cdot 2 \cdot 10^{-6}$ on $\mathbb{D}(0,2)$. Then $|\mathrm{fold}(\mathcal{V}_{1} \cap \mathcal{W}) - \mathrm{fold}(\mathcal{V}_{0} \cap \mathcal{W}) |< 1 \cdot \frac{1}{10} \cdot 2 \cdot 10^{-6} < 10^{-6}$. \end{proof}
The following proposition is important because it says that the image under $g_{j}$ of a 3-folded $(2,3)$-surface is a concentrated folded surface. We use it to prove Corollary \ref{rrrr} and we will also use it in Section 5.
\begin{prop} \label{rp7} Let $1 \le j \le 4$. Let $\mathcal{W}$ be a 3-folded $(2,3)$-surface such that we both have $\mathrm{diam}(\mathrm{pr}_{1}(\mathcal{W})) \le 10^{-10}$ and $\mathrm{Fold}(\mathcal{W}) \subset \mathbb{D}(0,1/10)$. Then $g_{j}(\mathcal{W} \cap U_{j})$ is a concentrated 3-folded $(2,3)$-surface satisfying $\mathrm{diam}(\mathrm{Fold}(g_{j}(\mathcal{W}))) \le \frac{1}{2} \cdot 10^{-5}$ for any $f \in \mathcal{F}$. \end{prop}
\begin{proof}
Let $\mathcal{V}$ and $\mathcal{V}'$ be two $(1,3)$-quasi planes over $\mathbb{D}^{2}$. Then $f^{2qr}(\mathcal{V} \cap g_{j}( \mathbb{D}^{3} ) )$ and $f^{2qr}(\mathcal{V}' \cap g_{j}( \mathbb{D}^{3} ))$ contain $(1,3)$-quasi planes over $\mathbb{D} \times \mathbb{D}(0,1/2)$ included in $U_{j}$ by Lemma \ref{presqueenfin}. The sets $\Gamma = f^{2qr}(\mathcal{V} \cap g_{j}( \mathbb{D}^{3}) ) \cap \mathcal{W}$ and $\Gamma' = f^{2qr}(\mathcal{V}' \cap g_{j}( \mathbb{D}^{3} )) \cap \mathcal{W}$ are 3-folded curves. By Lemmas \ref{rp3} and \ref{rp5}, $|\text{fold}(\Gamma') - \text{fold}(\Gamma) |<10^{-6}$. Lemma \ref{lemb5} implies that $g_{j}( \Gamma)$ is a 3-folded curve with $\text{fold} \big( g_{j}( \Gamma) \big) \in \mathbb{D}(\frac{1}{\mu^{2qr}} ( \text{fold}(\Gamma) - c_{j}) , (3/2) \cdot 10^{-6})$ and the analogous for $\Gamma'$. Then $\mathrm{diam}(\text{Fold}(g_{j}(\mathcal{W})) ) \le |\mu^{2qr}|^{-1} \cdot 10^{-6} + 2 \cdot (3/2) \cdot 10^{-6}<(1/2) \cdot 10^{-5}$ and then $g_{j}(\mathcal{W})$ is a concentrated 3-folded $(2,3)$-surface. \end{proof}
\begin{corollary} \label{rrrr} Let $\mathcal{W}$ be a concentrated 3-folded $(2,3)$-surface such that it holds both $\mathrm{diam}(\mathrm{pr}_{1}(\mathcal{W})) \le 10^{-10}$ and $\mathrm{Fold}(\mathcal{W}) \subset \mathbb{D}(0,\frac{1}{10})$. Then there exists $j \in \{1,2,3,4\}$ such that $g_{j}(\mathcal{W} \cap U_{j})$ is a concentrated 3-folded $(2,3)$-surface satisfying the inequalities $\mathrm{diam}(\mathrm{pr}_{1}(g_{j}(\mathcal{W} \cap U_{j}) ) \le 10^{-10}$ and $\mathrm{Fold}(g_{j}(\mathcal{W} \cap U_{j})) \subset \mathbb{D}(0,\frac{1}{10}) $. \end{corollary}
\begin{proof} Let $\Gamma$ be a 3-folded curve included in $\mathcal{W}$. By Proposition \ref{rp9}, there exists $j \in \{1,2,3,4\}$ such that $\frac{1}{\mu^{2qr}} (\mathrm{fold}(\Gamma) -c_{j} ) \in \mathbb{D}(0,\frac{1}{10}-10^{-5})$. By Lemma \ref{lemb5}, $g_{j}( \Gamma)$ is a 3-folded curve with $\mathrm{fold} (g_{j}(\Gamma)) \in \mathbb{D}(\frac{1}{\mu^{2qr}} (\mathrm{fold}(\Gamma) -c_{j}),(3/2) \cdot 10^{-6})$. By Proposition \ref{rp7}, $g_{j}(\mathcal{W})$ is concentrated with $\mathrm{diam}(\mathrm{Fold}(g_{j}(\mathcal{W}))) \le \frac{1}{2} \cdot 10^{-5}$. Then $\mathrm{Fold}(g_{j}(\mathcal{W} \cap U_{j}))$ is included in $\mathbb{D}(0 ,\frac{1}{10}-10^{-5}+ (3/2) \cdot 10^{-6}+ \frac{1}{2} \cdot 10^{-5}) \subset \mathbb{D}(0,\frac{1}{10})$ for $f \in \mathcal{F} $. The inequality $\mathrm{diam}(\mathrm{pr}_{1}(g_{j}(\mathcal{W} \cap U_{j}) ) \le 10^{-10}$ comes from $\mathrm{diam}(\mathrm{pr}_{1}(\mathcal{W})) \le 10^{-10}$ and the forward dilatation of $C^{u}$. The proof is done. \end{proof}
\begin{proof}[Proof of Proposition \ref{rrrrr}] By iteration of Corollary \ref{rrrr} there exists a sequence $(j_{n})_{n \ge 1}$ of digits in $\{1,2,3,4\}$ such that the sequence $(\mathcal{W}_{n})_{n \ge 0}$ defined by $\mathcal{W}_{0} = \mathcal{W}$ and $\mathcal{W}_{n+1} = g_{j_{n}} (\mathcal{W}_{n} \cap U_{j_{n}})$ is a sequence of concentrated 3-folded $(2,3)$-surfaces with $\text{Fold}(\mathcal{W}^{n}) \subset \mathbb{D}(0,\frac{1}{10}) \subset \mathbb{D}$ and $\text{diam}(\text{pr}_{1}(\mathcal{W}^{n} )) \le 10^{-10}$ for every $n \ge 1$. We define for every $n \ge 1$ $\tilde{\mathcal{W} }_{n}= f^{n}(\mathcal{W}_{n}) \subset \mathcal{W}_{0}$. We have for every $n \ge 1$ the inclusions $\tilde{\mathcal{W} }_{n+1} \subset \tilde{\mathcal{W} }_{n} \subset \tilde{\mathcal{W} }_{0}$. The set $\tilde{\mathcal{W}}_{\infty} = \bigcap_{n \ge 1} \tilde{\mathcal{W} }_{n}$ is non empty since it contains a decreasing sequence of non empty compact sets. Since $\mathcal{W}_{n}$ is a 3-folded $(2,3)$-surface for every $n \ge 0$, there exists $z_{n} \in \mathcal{W}_{n}$ and a non zero vector $v_{n} \in T_{z_{n}}\mathcal{W}_{n}$ such that $v_{n} \in C^{u}$. We denote for every $n \ge 1$, $\tilde{z}_{n} = f^{n}(z_{n}) \in \tilde{\mathcal{W} }_{n} \subset \tilde{\mathcal{W}}^{0}$ and $\tilde{v}_{n}$ an unitary vector parallel to $D_{z_{n}}f^{n}(v_{n})$. We have $\tilde{v}_{n} \in T_{\tilde{z}_{n}}\tilde{\mathcal{W} }_{n}$. Taking a subsequence if necessary we can suppose $\tilde{z}_{n} \rightarrow \tilde{z}_{\infty} \in \tilde{\mathcal{W}}_{\infty}$ and $\tilde{v}_{n} \rightarrow \tilde{v}_{\infty}$ for some point $\tilde{z}_{\infty}$ and some vector $\tilde{v}_{\infty}$. Clearly $\tilde{z}_{\infty} \in \mathcal{W}$. By construction the whole forward orbit of $\tilde{\mathcal{W}} _{\infty}$ is in $\mathbb{D}^{3}$. Then $\tilde{z }_{\infty}$ is included in the unstable manifold $W^{u}(\kappa_{f})$ of some point $\kappa_{f}$ of $\mathcal{H}_{f}$ . By construction $\tilde{v}_{\infty} \in T_{\tilde{z}_{\infty}}\mathcal{W}$ and $\tilde{v}_{\infty} \in \bigcap_{n \ge 0} Df^{n}(C^{u}(z_{n} )) = T_{\tilde{z}_{\infty}}W^{u}_{\text{loc}}(\tilde{z}_{\infty})$. Then $\mathcal{W}$ has a point of tangency with the unstable manifold of $\kappa_{f}$. \end{proof}
\section{Initial heteroclinic tangency}
In this section, we show that:
\begin{enumerate} \item for every sufficiently small values of $b$ and $\sigma$, we can find $c_{1} = c_{1}(b,\sigma)$ such that $f_{1} = f_{c_{1},b,\sigma}$ has a point of heteroclinic tangency $\tau$ between $W^{s}(\alpha_{f_{1}})$ and $W^{u}(\phi_{f_{1}})$ (where $\phi_{f_{1}}$ is a periodic point in $\mathcal{H}_{f_{1}}$), \item we can take iterates of a neighborhood of $\tau$ inside $W^{s}(\alpha_{f_{1}})$ under $f^{-1}_{1}$ in order to create a concentrated 3-folded $(2,3)$-surface inside $W^{s}(\alpha_{f_{1}})$. \end{enumerate}
\subsection{Initial tangency}
We recall that the disk $\mathcal{C}$ and the integer $m$ were defined in Proposition \ref{marber}.
\begin{prop} \label{orr}
Reducing $\mathcal{C}$ if necessary, there exist $0<b_{1}<b_{0}$, $0< \sigma_{1}<\sigma_{0}$ and an integer $s$ such that for every $0<|b| < b_{1}$ and $0 \le |\sigma| < \sigma_{1}$: \begin{enumerate} \item for every $u$-curve $\mathcal{U}$, $f_{0}^{s+m}(\mathcal{U})$ contains a degree 2 curve over $z_{1}$,
\item for every holomorphic family of $u$-curves $(\mathcal{U}_{c})_{c \in \mathcal{C}}$, there exists $c_{1} = c_{1}(b,\sigma)$ displaying a quadratic tangency $\tau$ between $W^{s}(\alpha_{f_{1}})$ and $\mathcal{U}_{c_{1}}$ where $f_{1} = f_{c_{1},b,\sigma}$ and $\tau \in \mathcal{U}_{c_{1}}$. Every iterate of $f_{1}^{s+m}(\tau)$ under $f_{1}$ is in $\mathbb{D} \big( \alpha_{c_{0}},10^{-10} \cdot |w_{m-1}| \cdot (\chi-1) \big) \times \mathbb{C}^{2}$ (where the constant $w_{m-1}$ was introduced in Definition \ref{K}) and the mapping $(b,\sigma) \mapsto \tau$ is holomorphic. \end{enumerate} \end{prop}
\begin{proof}
In the following, we are going to reduce several times the bounds $\sigma_{0}$ and $b_{0}$ into bounds $\sigma_{1}$ and $b_{1}$ to satisfy the two items. We begin by taking $\sigma_{1} = \frac{\sigma_{0}}{2}$ and $b_{1} = \frac{b_{0}}{2}$. Let us take any $u$-curve $\mathcal{U}$ or any holomorphic family of $u$-curves $(\mathcal{U}_{c})_{c \in \mathcal{C}}$. Since $\mathbb{D}'$ intersects the Julia set of $p_{c}$ for every $c \in \mathcal{C}$, reducing $\mathcal{C}$ if necessary, it is possible to find an integer $s$ and a holomorphic map $\beta_{-s} : \mathcal{C} \rightarrow \mathbb{D}$ such that for every $ c \in \mathcal{C}$, we have $p_{c}^{s}(\beta_{-s}(c)) = 0$. In particular, we have $p_{c_{0}}^{s+m}(\beta_{-s}(c_{0})) = \alpha_{c_{0}}$ and the image of $c \rightarrow p_{c}^{s+m}(\beta_{-s}(c)) - \alpha_{c}$ is an open set which contains 0 in its interior. Then reducing sufficiently the bounds $0 \le |\sigma| < \sigma_{1}$ and $0< |b| < b_{1}$ for $\sigma$ and $b$ and $ \mathcal{C}$, we have that for any $c \in \mathcal{C}$, there exists a neighborhood of the point of $\mathcal{U}$ of first coordinate $z_{1} = \beta_{-s}(c_{0})$ inside $\mathcal{U}$ whose image under $f^{s}_{0}$ is of the form $\{(z_{1},u^{2}(z_{1}),u^{3}(z_{1})),z_{1} \in \mathbb{D}(0,\rho) \}$ for some $\rho>0$. Indeed, $\beta_{-s}(c), p_{c}( \beta_{-s}(c) ), \cdots , p^{s-1}_{c}( \beta_{-s}(c) )$ are not critical points of $p_{c}$. The image of the curve $\{(z_{1},u^{2}(z_{1}),u^{3}(z_{1})),z_{1} \in \mathbb{D}(0,\rho) \}$ under $f_{0}$ is the curve $\{(p_{c}(z_{1})+b u^{2}(z_{1}) + \sigma u^{3}(z_{1}) (z_{1}-\alpha_{c_{0}}),z_{1}, \lambda z_{1}+\mu u^{3}(z_{1})+ \nu),z_{1} \in \mathbb{D}(0,\rho) \}$. Then it has a point of quadratic tangency with the foliation $z_{1} = C^{st}$ if $b$ and $\sigma$ are sufficiently small. Indeed, reducing $b_{1}$ and $\sigma_{1}$ if necessary, by continuity the derivative of the first coordinate $p_{c}(z_{1})+b u^{2}(z_{1}) + \sigma u^{3}(z_{1}) (z_{1}-\alpha_{c_{0}})$ vanishes for some value $\overline{z_{1}} \in \mathbb{D}(0,\rho)$. We can iterate this curve $(m-1)$ times under $f_{0}$. Since $p_{c}$ has no other critical point, this will still be a degree 2 curve upon $z_{1}$, reducing $b_{1}$ and $\sigma_{1}$ if necessary.
In the case of a holomorphic family of $u$-curves $(\mathcal{U}_{c})_{c \in \mathcal{C}}$, there exists a neighborhood of the point of $\mathcal{U}_{c}$ of first coordinate $z_{1} = \beta_{-s}(c)$ inside $\mathcal{U}_{c}$ which is sent under $f_{0}^{s}$ on a $u$-curve $\{(z_{1},u_{c}^{2}(z_{1}),u_{c}^{3}(z_{1})),z_{1} \in \mathbb{D}(0,\rho) \}$, but using the Cauchy inequality and reducing $\sigma_{1}$ and $b_{1}$ if necessary, $(u^{2}_{c})'$ and $(u^{3}_{c})'$ are uniformly (relatively to $c$) bounded and the conclusion is the same. In particular, this proves the first item of the result.
We call $\mathrm{Tan}_{f_{0}}$ the first coordinate of the point of vertical tangency. The image of the map defined on $\mathcal{C}$ which sends $c$ to $p_{c}^{s+m}(\beta_{-s}(c)) - \alpha_{c}$ is an open set which contains 0 in its interior. Let us denote by $2 l$ the distance of 0 to the image of $\partial \mathcal{C}$. Reducing $b_{1}$ and $\sigma_{1}$ another time if necessary, by continuity $\{ \mathrm{Tan}_{f_{0}} - \alpha_{c} , c \in \partial \mathcal{C} \}$ is a curve in the plane, with 0 is in a bounded connected component of its complement and at distance at least $l$ from $\{ \mathrm{Tan}_{f_{0}} - \alpha_{c} , c \in \partial \mathcal{C} \}$.
\begin{notation} Let $\kappa_{f} \in \mathcal{H}_{f}$ for $f \in \mathcal{F}$. We denote by $W^{s}_{\mathrm{loc}}(\kappa_{f})$ the connected component of $W^{s}(\kappa_{f}) \cap \mathbb{D}^{3}$ which contains $\kappa_{f}$ and by $W^{u}_{\mathrm{loc}}(\kappa_{f})$ the connected component of $W^{u}(\kappa_{f}) \cap \mathbb{D}^{3}$ which contains $\kappa_{f}$. \end{notation}
\begin{lemma} \label{utileresult}
Reducing $\mathcal{C}$, $b_{1}$ and $\sigma_{1}$ if necessary, $ W^{s}_{\mathrm{loc}}(\alpha_{f_{0}})$ is a graph over $(z_{2},z_{3}) \in \mathbb{D}^{2}$ both included in $\mathbb{D} \big( \alpha_{c},l/2 \big) \times \mathbb{D}^{2}$ and in $\mathbb{D} \big( \alpha_{c_{0}},10^{-10} \cdot |w_{m-1}| \cdot (\chi-1) \big) \times \mathbb{D}^{2}$. \end{lemma}
\begin{proof}
For $\sigma = 0$, $ W^{s}(\alpha_{f_{0}})$ is the product of $ W^{s}(\alpha_{H})$ by the $z_{3}$ axis where $H$ is the H\'enon map $H : (z_{1},z_{2}) \mapsto (p_{c}(z_{1})+bz_{2},z_{1})$. Moreover, for every $\epsilon>0$, we can reduce $b_{1}$ such that if $ |b|< b_{1}$, then the cone $C^{ss,\epsilon}$ centered at $e_{2}$ of opening $\epsilon/2$ is $H^{-1}$-invariant in some neighborhood of $\{\alpha_{c}\} \times \mathbb{D}$. Then $W^{s}_{\text{loc}}(\alpha_{H})$ is a $ss$-curve included in $\mathbb{D}(\alpha_{c},\epsilon) \times \mathbb{D}$ since all its tangent vectors lie in $C^{ss,\epsilon}$. Then for $\sigma = 0$, the skew-product structure implies that $W_{\mathrm{loc}}^{s}(\alpha_{f_{0}})$ is a $(2,3)$-surface included in $\mathbb{D}(\alpha_{c},\epsilon) \times \mathbb{D}^{2}$ which is the product of $W_{\mathrm{loc}}^{s}(\alpha_{H})$ by the $z_{3}$ axis. Then, by continuity, it is possible to reduce $\sigma_{1}$ such that for every $f_{0}$ with $ 0< |b|<b_{1}$ and $0 \le |\sigma|< \sigma_{1}$, $ W_{\mathrm{loc}}^{s}(\alpha_{f_{0}})$ is a graph over $(z_{2},z_{3}) \in \mathbb{D}^{2}$ included in $\mathbb{D}(\alpha_{c},\epsilon) \times \mathbb{D}^{2}$. We take $\epsilon = l/2$ to prove the first inclusion. We get the second one by reducing both $\mathcal{C}$ and $\epsilon$. \end{proof}
We recall that 0 is in a bounded connected component of the complement of $\{ \mathrm{Tan}_{f_{0}} - \alpha_{c} , c \in \partial \mathcal{C} \}$ and at distance at least $l$ from $\{ \mathrm{Tan}_{f_{0}} - \alpha_{c} , c \in \partial \mathcal{C} \}$. In particular, there is a parameter $\overline{c} \in \mathcal{C}$ such that $\text{Tan}_{f_{0}}$ belongs to $\mathbb{D}(\alpha_{\overline{c}},l/2)$. Up to replacing $s+m$ by $s+m+1$, we can suppose that the point of vertical tangency is in $\mathbb{D}^{3}$. For the parameter $\overline{c}$, $f^{s+m}_{0}(\mathcal{U}_{\overline{c}}) \cap (\mathbb{D}(\alpha_{\overline{c}},l/2) \times \mathbb{D}^{2}) $ is not the union of two graphs upon $z_{1} \in \mathbb{D}(\alpha_{\overline{c}}, l/2)$ and for $c \in \partial \mathcal{C} $, $f^{s+m}_{0}(\mathcal{U}_{c})$ is the union of two graphs over $z_{1} \in \mathbb{D}(\alpha_{c},l/2)$. According to Lemma \ref{utileresult}, $ W_{\mathrm{loc}}^{s}(\alpha_{f_{0}})$ is a graph over $(z_{2},z_{3}) \in \mathbb{D}^{2}$ included in $\mathbb{D}(\alpha_{c},l/2) \times \mathbb{D}^{2}$. The following lemma is the analogous in dimension 3 of Proposition 8.1 of \cite{dl}. The proof is essentially the same and relies on the continuity of the intersection index of properly intersecting analytic sets of complementary dimensions.
\begin{lemma} Let $(\Gamma_{c})_{c \in \mathcal{C}}$ be a holomorphic family of curves of degree 2 over the first coordinate. We assume that: \begin{enumerate} \item there exists a compact subset $\mathcal{C}' \subset \mathcal{C}$ such that if $c \in \mathcal{C} \backslash \mathcal{C}'$, $\Gamma_{c}$ is the union of 2 graphs over $z_{1} \in \mathbb{D}(\alpha_{c},l/2 )$, \item there exists $\overline{c} \in \mathcal{C}$ such that $\Gamma_{\overline{c}}$ is not the union of 2 graphs over $z_{1} \in \mathbb{D}(\alpha_{\overline{c}},l/2 )$. \end{enumerate} Then, if $(\mathcal{V}_{c})_{c \in \mathcal{C}}$ is any holomorphic family of graphs over $(z_{2},z_{3}) \in \mathbb{D}^{2}$ contained in $ \mathbb{D}(\alpha_{c},l/2) \times \mathbb{D}^{2}$, there exists $c_{1} \in \mathcal{C}$ such that $\Gamma_{c_{1}}$ and $\mathcal{V}_{c_{1}}$ admit a point of tangency. \end{lemma}
We apply the previous lemma, taking the family $(\Gamma_{c})_{c \in \mathcal{C} }= (f_{0}^{s+m}(\mathcal{U}_{c}) \cap \mathbb{D}^{3} )_{c \in \mathcal{C} }$ as curves and the family of stable manifolds $W_{\mathrm{loc}}^{s}(\alpha_{f_{0}}) $ as graphs over $(z_{2},z_{3}) \in \mathbb{D}^{2}$. We can conclude that there is a parameter $c_{1}$ such that there exists a quadratic tangency $f^{s+m}_{1}(\tau)$ between $W_{\text{loc}}^{s}(\alpha_{f_{1}})$ and $f_{1}^{s+m}(\mathcal{U}_{c_{1}})$ where $\tau \in \mathcal{U}_{c_{1}}$ for $f_{1} = f_{c_{1},b,\sigma}$. Then $\tau$ is a point of tangency between $W^{s}(\alpha_{f_{1}})$ and $\mathcal{U}_{c_{1}}$. Since $ W^{s}_{\text{loc}}(\alpha_{f_{1}})$ is included in $\mathbb{D} \big( \alpha_{c_{0}},10^{-10} \cdot |w_{m-1}| \cdot (\chi-1) \big) \times \mathbb{D}^{2}$, all the iterates of $f^{s+m}_{1}(\tau)$ under $f_{1}$ are in $\mathbb{D} \big( \alpha_{c_{0}},10^{-10} \cdot |w_{m-1}| \cdot (\chi-1) \big) \times \mathbb{D}^{2}$. The map $(b,\sigma) \mapsto c_{1}$ is holomorphic by the Implicit Function Theorem and then $(b,\sigma) \mapsto \tau$ is holomorphic too. This shows item 2 and ends the proof of Proposition \ref{orr}. \end{proof}
In the next proposition, we show that the tangencies created in the previous result are generically unfolded. Beware that in the next result, the map $f_{1}$ is associated to a family of $u$-curves $(\mathcal{U}_{c})_{c \in \mathcal{C}}$ which can be distinct from the family $(\mathcal{U}'_{f})_{f \in \mathcal{F}'}$.
\begin{df} Let $(U_{f})_{f \in \mathcal{F}}$ be a holomorphic family of $u$-curves and $(S_{f})_{f \in \mathcal{F}}$ be a holomorphic family of $s$-surfaces. We suppose that for $f \in \mathcal{F}$, there is a point of quadratic tangency between $U_{f}$ and $S_{f}$. We say that this tangency is generically unfolded if there exists a one-dimensional holomorphic family $(f^{t})_{t \in \mathbb{D}}$ of polynomial automorphisms and a holomorphic family of local biholomorphisms $(\Psi_{t})_{t \in \mathbb{D}}$ with: \begin{enumerate} \item $f^{0} = f$ and $f^{t} \in \mathcal{F} $ for every $t \in \mathbb{D}$, \item $\Psi_{t}(S_{f^{t}})$ is a vertical plane $\{z_{1} = C^{st}\}$ where $C^{st}$ does not depend on $t \in \mathbb{D}$,
\item if we denote by $\mathrm{tan}_{t}$ the first coordinate of the point of tangency of $\Psi_{t}(U_{f^{t}})$ with $\{z_{1} = C^{st}\}$ (there exists exactly one such point because the tangency is quadratic), then $|\frac{ d \mathrm{tan}_{t}}{d t}|$ is uniformly bounded from below by a strictly positive constant for $t \in \mathbb{D}$. \end{enumerate} \end{df}
\begin{prop} \label{fd}
Reducing $\mathcal{C}$, $b_{1}$ and $\sigma_{1}$ if necessary, for every $0<|b| < b_{1}$ and $0 \le |\sigma| < \sigma_{1}$, for every holomorphic family of $u$-curves $(\mathcal{U}_{c})_{c \in \mathcal{C}}$, there exists a neighborhood $\mathcal{F}' \Subset \mathcal{F}$ of the map $f_{1}=f_{c_{1}(b,\sigma),b,\sigma}$ defined in item 2 of Proposition \ref{orr} such that: for every holomorphic family of $u$-curves $(\mathcal{U}'_{f})_{f \in \mathcal{F}'}$, if $f \in \mathcal{F}'$ has a point of tangency $\tau'$ between $W^{s}(\alpha_{f})$ and $\mathcal{U}'_{f}$ such that $f^{s+m}(\tau') \in W^{s}_{\text{loc}}(\alpha_{f})$, then the tangency $\tau'$ is generically unfolded. In particular, the tangency $\tau$ obtained in item 2 of Proposition \ref{orr} is generically unfolded. \end{prop}
\begin{proof} For every $f \in \mathcal{F}$, after a holomorphic change of coordinates $\Psi_{f}$, $W^{s}_{\text{loc}}(\alpha_{f})$ is the plane $\{z_{1} = \alpha_{c_{0}}\}$. We first show the result for the map $f_{1}$. We work in the one-dimensional family $(f_{0})_{c \in \mathcal{C}} = (f_{c,b,\sigma})_{c \in \mathcal{C}}$ where $b$ and $\sigma$ are fixed and we take a holomorphic family of $u$-curves $(\mathcal{U}'_{f})_{f \in \mathcal{F}'}$. It is a consequence of the proof of Lemma \ref{utileresult} that $\Psi_{f_{0}}$ tends to $\text{Id}$ when $b$ and $\sigma$ tend to 0. When $b$ and $\sigma$ tend to 0, the curve $\Psi_{f_{0}} \circ f^{s+m}_{0}(\mathcal{U}'_{f_{0}})$ tends also to a curve of degree 2 over $z_{1}$ of the form $\{(p_{c}^{s+m}(z_{1}),p_{c}^{s+m-1}(z_{1}),v(z_{1}) ) : z_{1} \in \mathbb{D}(0,\rho)\}$ where $v$ is holomorphic. We call $\mathrm{tan}_{c}$ the first coordinate of the point of vertical tangency of $\Psi_{f_{0}} \circ f^{s+m}_{0}(\mathcal{U}'_{f_{0}})$. We have $p_{c_{0}}^{m}(0) = \alpha_{c_{0}}$, $(p_{c_{0}}^{m})'(0)=0$, $(p_{c_{0}}^{m})''(0) \neq 0$ and at $c= c_{0} $, we have $ \frac{d }{dc} \big( p_{c}^{m}(0) - \alpha_{c} \big) \neq 0$ (see Proposition \ref{marber}). Moreover, $c_{1}$ tends to $c_{0}$ when $b$ and $\sigma$ tend to 0. Then for sufficiently small $b$ and $\sigma$, we have $\frac{d \mathrm{tan}_{c}} {dc} \neq 0$ for every $c \in \mathcal{C}$ (reducing $\mathcal{C}$ if necessary). Since the estimates on the derivatives of the $u$-curves $(\mathcal{U}'_{f})_{f \in \mathcal{F}}$ are uniform, we can reduce uniformly $b_{1}$ and $\sigma_{1}$ so that this inequality is true no matter the choice of $(\mathcal{U}'_{f})_{f \in \mathcal{F}}$. Then, by continuity, there exists some new neighborhood $\mathcal{F}' \Subset \mathcal{F}$ of $f_{1}$ such that every $f \in \mathcal{F}'$ belongs to a one-dimensional family $(f^{t})_{t \in \mathbb{D}}$ such that: $f^{0} = f$, $f^{t} \in \mathcal{F}'$ and $\frac{d \mathrm{tan}_{c}} {dt} \neq 0$ ($t \in \mathbb{D}$). This implies in particular that if $f \in \mathcal{F}'$ has a point of tangency $\tau'$ between $W^{s}(\alpha_{f})$ and $\mathcal{U}'_{f}$ such that $f^{s+m}(\tau') \in W^{s}_{\text{loc}}(\alpha_{f})$, then the tangency $f^{s+m}(\tau')$ is generically unfolded and then also $\tau'$. The proof is over. \end{proof}
\subsection{A transversality result}
From now on, we construct from this initial heteroclinic tangency a 3-folded $(2,3)$-surface inside a stable manifold with its fold inside $ \mathbb{D}(0,\frac{1}{10})$. In this Subsection, we prove that $W^{s}(\alpha_{f_{1}})$ has some special geometry. We start by choosing a periodic point $\phi_{f_{1}} \in \mathcal{H}_{f_{1}}$. The main point is that its third coordinate is in $\mathbb{D}(0,\frac{1}{10}-10^{-4})$, which will be used later in the proof of Proposition \ref{fm}.
\begin{lemma} \label{defphi}
For every $c \in \mathcal{C}$, $0<|b|<b_{1}$, $0 \le |\sigma|< \sigma_{1}$, there exists a periodic point $\phi_{f_{0}}$ inside $\mathcal{H}_{f_{0}}$ such that $\mathrm{pr}_{3}(\phi_{f_{0}}) \in \mathbb{D}(0,\frac{1}{10}-10^{-4})$. \end{lemma}
\begin{proof} Let us take $\omega \in h^{q}_{1}(\mathbb{D}')$. According to Proposition \ref{blender}, the $ss$-curve $\{\omega\} \times \mathbb{D} \times \{0\}$ intersects the unstable set of a point $\phi_{f_{0}}$ of the horseshoe $\mathcal{H}_{f_{0}}$. Moving slightly $\{\omega\} \times \mathbb{D} \times \{0\}$ by translation in the third direction (let us say by no more than $\frac{1}{20}$), we can suppose that $\phi_{f_{0}}$ belongs to the preimage $g_{1}(U_{1})$. By density of periodic points in $\mathcal{H}_{f_{0}}$ we can moreover suppose that $\phi_{f_{0}}$ is periodic. We notice that $\omega \in h^{q}_{1}(\mathbb{D}')$ and $\mathrm{pr}_{1} ( g_{1}(U_{1}) ) \subset h^{q}_{1}(\mathbb{D}')$. Moreover we have $\mathrm{diam}(h^{q}_{1}(\mathbb{D})) < 10^{-11}$ by property 2 of Property \ref{fo}. Since $W^{u}_{\mathrm{loc}}(\phi_{f_{0}})$ is a $u$-curve, this implies that $\mathrm{pr}_{3}(\phi_{f_{0}})$ belongs to $\mathbb{D}(0, \frac{1}{20} + 10^{-11} \cdot \chi^{-1} ) \subset \mathbb{D}(0,\frac{1}{10}-10^{-4})$. \end{proof}
\begin{notation}
For every $0<|b|<b_{1}$, $0 \le |\sigma|< \sigma_{1}$, we fix such a periodic point $\phi_{f_{1}}$ for the map $f_{1}$ such that $\mathrm{pr}_{3}(\phi_{f_{1}}) \in \mathbb{D}(0,\frac{1}{10}-10^{-4})$. Reducing $\mathcal{F}'$ if necessary, we have $\mathrm{pr}_{3}(\phi_{f}) \in \mathbb{D}(0,\frac{1}{10}-10^{-4})$ for $f \in \mathcal{F}'$. \end{notation}
We now choose the parameter $b$ in function of the parameter $\sigma$:
\begin{notation} In the following, we will take $b= b(\sigma) = \sigma^{2}$. For technical reasons, we reduce $\sigma_{1}$ such that $\sigma_{1}< \sqrt{b_{1}}$. \end{notation}
In particular, we can use Proposition \ref{orr} for $0 \le |\sigma|< \sigma_{1}$: there exists a map $\sigma \mapsto c_{1}(\sigma)$ such that there is a heteroclinic tangency $\tau$ between the local unstable manifold $\mathcal{U}_{f_{1}}$ of $\phi_{f_{1}}$ and the stable manifold $W^{s}(\alpha_{f_{1}})$ of $\alpha_{f_{1}}$. Moreover, $\sigma \mapsto \tau$ is holomorphic. We have $D(f_{1}^{s})_{\tau} \cdot (0,0,1) = \mu^{s} (\xi_{1}(\sigma),\xi_{2}(\sigma),1)$ with $\xi_{1}(\sigma) = O(\sigma)$ and $\xi_{2}(\sigma) = O(\sigma)$. The maps $\sigma \mapsto \text{pr}_{1}(f^{s}_{1}(\tau)) - \beta_{0}$, $\cdots$, $\sigma \mapsto \text{pr}_{1}(f^{s+m-1}_{1}(\tau)) - \beta_{m-1}$ are holomorphic (remind that the constants $\beta_{i}$ were defined in \ref{bet}). Since they vanish when $\sigma = 0$, they are also $O(\sigma)$. The map $\sigma \mapsto b(\sigma) $ is $O(\sigma^{2})$ because $b(\sigma) = \sigma^{2}$. Then the differentials at $f^{s}_{1}(\tau), \cdots , f^{s+m-1}_{1}(\tau)$ verify the conditions of Corollary \ref{reffin} so from Corollary \ref{reffin} we immediately get the following (remind that $w_{m-1}$ was defined in Definition \ref{K}):
\begin{lemma} \label{lemor} We have $D(f^{s+m}_{1})_{\tau} \cdot ( 0, 0, 1) = ( \zeta_{1} (\sigma) , \zeta_{2}(\sigma) , \zeta_{3}(\sigma) )$ where $\zeta_{1}$, $\zeta_{2}$, $\zeta_{3}$ are holomorphic functions, $\zeta_{1}(\sigma) = w_{m-1} \cdot \sigma + O(\sigma^{2})$, $\zeta_{2}(0) = 0+O(\sigma)$ and $\zeta_{3}(\sigma) = \mu^{m}+O(\sigma)$. \end{lemma}
\begin{notation} \label{noter}
We reduce $\sigma_{1}$ so that for $0 \le |\sigma|< \sigma_{1}$ it holds $|\zeta_{1}(\sigma)-w_{m-1} \sigma| < \frac{1}{10} |w_{m-1}| | \sigma|$, $|\zeta_{2}(\sigma)| <1$ and $|\zeta_{3}(\sigma)-\mu^{m}| <\frac{1}{10}(1-|\mu^{m}|)$. Still reducing $\sigma_{1}$, we have $\sigma_{1}< 10^{-10} \cdot |w_{m-1}| \cdot (\chi-1)$. From now on, we fix $\sigma = \frac{\sigma_{1}}{2}$, $b = b(\sigma) = \sigma^{2}$, $c = c_{1}(\sigma)$ and the associated map $f_{1} = f_{c_{1}(\sigma),b(\sigma),\sigma}$. \end{notation}
\begin{lemma}
\label{or7} For every vector $v^{0}$ such that $|v^{0}_{1} - w_{m-1} \sigma| < \frac{1}{10} |w_{m-1}| | \sigma|$, $|v^{0}_{2} | <1 $ and $|v^{0}_{3} - \mu^{m} | < \frac{1}{10}(1- |\mu|^{m})$, at any point of $W^{s}_{\mathrm{loc}}(\alpha_{f_{1}})$, $v^{0}$ is transverse to $W^{s}_{\mathrm{loc}}(\alpha_{f_{1}})$ for small $\sigma$. \end{lemma}
\begin{proof}
Let $\psi^{0}$ be a point of $W^{s}_{\text{loc}} (\alpha_{f_{1}})$ and let us consider the sequence of points defined by $\psi^{n} = f^{n}_{1}(\psi^{0})$. According to Lemma \ref{utileresult}, for every $n \ge 0$, $\psi^{n}$ is in $\mathbb{D}(\alpha_{c_{0}}, 10^{-10} |w_{m-1}|(\chi-1)) \times \mathbb{C}^{2}$. Then for every $n \ge 1$, the differential at $\psi^{n}$ of $f_{1}$ is of the form: $$I_{n} = \begin{pmatrix}
m_{n} & b(\sigma) & \sigma (z_{1}- \alpha_{c_{0}})\\
1 & 0& 0 \\
\lambda & 0 & \mu \end{pmatrix} \, , $$
where $b(\sigma )= \sigma^{2}$ and $|m_{n}| \ge\chi>1$ for every $n \ge 1$. Since $\sigma_{1}<10^{-10} |w_{m-1}| (\chi-1)$ we have $|b| <10^{-10} |w_{m-1}| | \sigma| (\chi-1)$. We denote $v^{n} = (I_{n} \cdot ... \cdot I_{1})(v^{0})$ for every $n \ge 1$. Let us show that there exists an integer $i$ such that $|v^{i}_{1} | \ge |v^{i}_{3}|$. Let us suppose this is false and we show by induction the following properties for $n \ge 0$: \begin{enumerate}
\item $1 \ge |v^{n+1}_{1}| > \frac{\chi+1}{2} |v^{n}_{1}| \ge \frac{1}{100} |w_{m-1} | |\sigma | $ ,
\item $ |v^{n}_{2}| \le1$,
\item $ |v^{n+1}_{3}| \le |v^{n}_{3}| \le 1$. \end{enumerate}
For $n = 0$, item 2 is satisfied since $|v^{0}_{2} | <1 $. Moreover $ |v^{0}_{1}| < |v^{0}_{3}|$ by hypothesis. We have $|v^{1}_{3}| \le |\lambda| |v^{0}_{1}|+|\mu| |v^{0}_{3}|< |\lambda| |v^{0}_{3}|+|\mu| |v^{0}_{3}|< |v^{0}_{3}| \le 1$. Then item 3 is true. We have $ v^{1}_{1} = m_{1} v^{0}_{1} + b v^{0}_{2}+ \sigma ( \psi^{0}_{1} - \alpha_{c_{0}})v_{3}^{0}$. We notice that $|v^{0}_{1}| \ge \frac{1}{100} |w_{m-1} | |\sigma | $, $m_{1} \ge \chi$, $|b v^{0}_{2}| \le |b|<10^{-10} |w_{m-1}| | \sigma| (\chi-1)$ and $| \sigma ( \psi^{0}_{1} - \alpha_{c_{0}}) v^{0}_{3} | < 10^{-10} |w_{m-1}| | \sigma| (\chi-1)$. Then we have $|v^{1}_{1}| > \frac{\chi+1}{2} |v^{0}_{1}| \ge \frac{1}{100} |w_{m-1} | |\sigma | $. We also have $|v^{1}_{1}| < |v_{3}^{1}| \le 1$ by hypothesis. Then item 1 is true and the induction step is true for $n = 0$.
Suppose the induction step true for some $n \ge 0$. Then $|v^{n+1}_{2}| = |v^{n}_{1}| \le 1$ and item 2 is true. Moreover $|v^{n+2}_{3}| \le |\lambda| |v^{n+1}_{1}|+|\mu| |v^{n+1}_{3}|< |\lambda| |v^{n+1}_{3}|+|\mu| |v^{n+1}_{3}|< |v^{n+1}_{3}| \le 1$, which shows item 3. We have $ v^{n+2}_{1} = m_{n+1} v^{n+1}_{1} + b v^{n+1}_{2}+ \sigma( \psi^{n+1}_{1} - \alpha_{c_{0}})v^{n+1}_{3}$. Since we have $|m_{n+1}|>\chi$, $|v^{n+1}_{1}| \ge \frac{1}{100} |w_{m-1} | |\sigma | $, $|b v^{n+1}_{2}| \le |b| <10^{-10} |w_{m-1}| | \sigma| (\chi-1)$ and also $| \sigma ( \psi^{n+1}_{1} - \alpha_{c_{0}})v^{n+1}_{3} | < 10^{-10} |w_{m-1}| | \sigma| (\chi-1)$, we have $|v^{n+2}_{1}| >\frac{\chi+1}{2} |v^{n+1}_{1}| \ge \frac{1}{100} |w_{m-1} | |\sigma | $. We have $ |v^{n+2}_{1}| < |v^{n+2}_{3}| \le 1$ by hypothesis and then item 1 is true.
Then for every $n \ge 0$, we have $|v^{n+1}_{1}| \cdot |v^{n+1}_{3}|^{-1} \ge (\chi+1)/2 \cdot |v^{n}_{1}| \cdot |v^{n}_{3}|^{-1}$. This implies a contradiction: there must exist some integer $i$ such that $|v^{i}_{1} | \ge |v^{i}_{3}|$. Since we also have $|v^{i}_{2}| = |v^{i-1}_{1}| \le |v^{i}_{1}|$, this implies that $v^{i}$ is transverse to $W^{s}_{\text{loc}}(\alpha_{f_{1}})$ and then $v^{0}$ itself is transverse to $W^{s}_{\text{loc}}(\alpha_{f_{1}})$. \end{proof}
\begin{prop} \label{orienter} The vector $(0,0,1)$ is transverse to $W^{s}(\alpha_{f_{1}})$ at $\tau$. \end{prop}
\begin{proof} From Lemmas \ref{lemor} and \ref{or7}, we know that the image under $f^{s+m}_{1}$ of a neighborhood of $\tau$ in $W^{s}(\alpha_{f_{1}})$ is transverse to $D(f^{s+m}_{1})_{\tau} \cdot ( 0, 0, 1) $. Applying $f^{-s-m}_{1}$, this immediately implies the result. \end{proof}
\subsection{Orientation of the fold of $W^{s}(\alpha_{f})$}
In this subsection, we take iterates of the initial tangency $\tau$ under $f^{-1}_{1}$ in order to create a concentrated 3-folded $(2,3)$-surface inside a stable manifold. We have that $W_{\mathrm{loc}}^{u}(\phi_{f_{1}})$ is a graph $\{(z_{1}, u^{2}(z_{1}),u^{3}(z_{1}) : z_{1} \in \mathbb{D}\}$ over $z_{1} \in \mathbb{D}$ (the periodic point $\phi_{f_{1}}$ was defined in Lemma \ref{defphi}). We consider the biholomorphism $\Psi$ defined by $\Psi(z_{1},z_{2},z_{3}) = (z_{1},z_{2}-u^{2}(z_{1}),z_{3}-u^{3}(z_{1}))$ which sends $W_{\text{loc}}^{u}(\phi_{f_{1}})$ onto the $z_{1}$ axis. We denote $\mathcal{W}_{f}$ the connected component of $ W^{s}(\alpha_{f}) \cap \mathbb{D}^{3}$ which contains $\tau$ for $f \in \mathcal{F}'$. We denote by $\text{Tan}'$ the subset of $\Psi(\mathcal{W}_{f})$ where $\Psi(\mathcal{W}_{f})$ is tangent to some line $\{z_{2} = C^{st}, z_{3} = C^{st}\}$ and by $\text{Tan} = \Psi^{-1}(\text{Tan}')$. When $f = f_{1}$, we denote them by $\text{Tan}^{0}$ and $(\text{Tan}')^{0}$. Let $e^{u}(\tau)$ be a tangent vector of $W_{\text{loc}}^{u}(\phi_{f_{1}})$ at $\tau$.
\begin{lemma} \label{hoppe3} The curve $\mathrm{Tan}^{0}$ is a complex curve regular at $\tau$ of tangent vector $v_{\text{tan}}$ at $\tau$ such that $v_{\text{tan}} \notin \mathbb{C} \cdot (0,0,1)$ and $v_{\text{tan}} \notin \mathbb{C} \cdot e^{u}(\tau)$. \end{lemma}
\begin{proof} We are working in the projectivized tangent bundle $\mathbb{P}T\mathbb{C}^{3} \simeq \mathbb{C}^{3} \times \mathbb{P}^{2}(\mathbb{C})$ of $\mathbb{C}^{3}$ which is of dimension 5. The lift $\hat{\mathcal{W}_{f_{1}}}$ of $\mathcal{W}_{f_{1}}$ to $\mathbb{P}T\mathbb{C}^{3}$ is a complex submanifold of dimension 3. The lift of every complex curve $C_{x,y} = \Psi^{-1}(\mathbb{D} \times \{(x,y) \}) $ to $\mathbb{P}T\mathbb{C}^{3}$ is a complex curve $\hat{C}_{x,y}$. Then $\bigcup_{(x,y) \in \mathbb{D}^{2}} \hat{C}_{x,y}$ is a complex submanifold of dimension 3. Moreover, according to Proposition \ref{orr}, $\mathcal{W}_{f_{1}}$ has a point of quadratic tangency with $C_{0,0}$ and then $\hat{\mathcal{W}_{f_{1}}}$ is transverse to $\bigcup_{(x,y) \in \mathbb{D}^{2}} \hat{C}_{x,y}$. Then $\hat{\text{Tan}^{0}} = \hat{\mathcal{W}_{f_{1}}} \cap \big( \bigcup_{(x,y) \in \mathbb{D}^{2}} \hat{C}_{x,y} \big)$ is a regular complex curve. Then its projection $\text{Tan}^{0}$ is a regular complex curve of tangent vector $v_{\text{tan}}$ at $\tau$. By Proposition \ref{orienter}, $v_{\text{tan}} \notin \mathbb{C} \cdot (0,0,1)$. By construction, $e^{u}(\tau)$ is a tangent vector of $W_{\text{loc}}^{u}(\phi_{f_{1}})$ at $\tau$ so $v_{\text{tan}} \notin \mathbb{C} \cdot e^{u}(\tau)$. The proof is done. \end{proof}
Since the intersection between $\hat{\mathcal{W}_{f_{1}}}$ and $\bigcup_{(x,y) \in \mathbb{D}^{2}} \hat{C}_{x,y}$ was transverse in the latter proof, by perturbation of the previous result, we get:
\begin{corollary} \label{hoppe5} For every $\epsilon>0$, reducing $\mathcal{F}'$ if necessary, there exists $\eta>0$ such that: for every $f \in \mathcal{F}'$, for every $C^{2}$-foliation $(\mathcal{V}_{x,y})_{(x,y) }$ of some neighborhood of $\tau$ by two-dimensional real manifolds $\mathcal{V}_{x,y}$ such that every $\mathcal{V}_{x,y}$ is $\eta$-close to $W_{\mathrm{loc}}^{u}(\phi_{f})$, the set $\mathrm{Tan}$ of points of $\mathcal{W}_{f}$ where $\mathcal{W}_{f}$ is tangent to some $\mathcal{V}_{x,y}$ is a regular two-dimensional real manifold which has its direction $\epsilon$-close to $\mathbb{C} \cdot v_{\text{tan}}$ at each point if it is non empty. \end{corollary}
The following result is a technical geometric lemma.
\begin{lemma} \label{eps} There exist $\epsilon, \rho, \rho_{1},\rho_{2},\rho_{3}>0$ such that for every $t>0$, we have the following property: for every regular two-dimensional real manifold $\Gamma$ going through $\tau+ (\mathbb{D}(0,t\rho))^{3}$, if $\Gamma$ has its direction $\epsilon$-close to $\mathbb{C} \cdot v_{\text{tan}}$ at each point, then $\Gamma$ is horizontal relatively to $\pi_{2}$ in $ \tau + \mathbb{D}(0,t\rho_{1}) e^{u}(\tau)+ \mathbb{D}(0,t\rho_{2}) (0,1,0)+ \mathbb{D}(0,t\rho_{3}) (0,0,1) $. \end{lemma}
\begin{proof} According to Lemma \ref{hoppe3}, $v_{\text{tan}} \notin \mathbb{C} \cdot (0,0,1)$ and $v_{\text{tan}} \notin \mathbb{C} \cdot e^{u}(\tau)$, which easily implies the result. \end{proof}
\begin{notation} We fix the value of $\epsilon$ given by Lemma \ref{eps} and the associated value of $\eta$ given by Corollary \ref{hoppe5}. We also fix $ \rho, \rho_{1},\rho_{2},\rho_{3}$ given by Lemma \ref{eps}. \end{notation}
\begin{lemma} \label{boi} There exists $t_{0}>0$ such that for every $t<t_{0}$, there exists an integer $j= j(t)$ such that: \begin{enumerate} \item $ f_{1}^{-j} \Big( \tau + \mathbb{D}(0,t\rho_{1}) e^{u}(\tau)+ \partial \mathbb{D}(0,t\rho_{2})(0,1,0)+ \mathbb{D}(0,t\rho_{3}) (0,0,1) \Big) \cap \mathbb{D}^{3} = \emptyset , $ \item $f_{1}^{-j} \Big( \tau + \mathbb{D}(0,t\rho_{1}) e^{u}(\tau)+ \mathbb{D}(0,t\rho_{2}) (0,1,0)+ \mathbb{D}(0,t\rho_{3}) (0,0,1) \Big) \cap \partial \mathbb{D}^{3} \Subset \mathbb{D} \times \partial \mathbb{D} \times \mathbb{D}. $ \end{enumerate} Moreover, $j(t)$ tends to $+\infty$ when $t$ tends to 0. Reducing $\mathcal{F}'$, by continuity this remains true for any $f \in \mathcal{F}'$ for a given $t<t_{0}$. \end{lemma}
\begin{proof}
We foliate $\mathbb{D}_{t,\rho_{1},\rho_{2},\rho_{3}} =\tau+ \mathbb{D}(0,t\rho_{1}) e^{u}(\tau)+ \mathbb{D}(0,t\rho_{2}) (0,1,0)+ \mathbb{D}(0,t\rho_{3}) (0,0,1) $ by disks $\mathcal{L}_{z_{1},z_{3}}$ parallel to the $z_{2}$ axis. To simplify, we can suppose that $\phi_{f_{1}}$ is a fixed point of $f_{1}$, up to replacing $f_{1}$ by one of its iterates. By invariance of $C^{ss}$ under $f_{1}^{-1}$, for every $n \ge 0$, the image of this tridisk under $f^{-n}_{1}$ is foliated by the $ss$-curves $f^{-n}_{1} (\mathcal{L}_{z_{1},z_{3}}) \cap \mathbb{D}^{3}$. We call the length of a $ss$-curve the radius of the maximal (for the inclusion) disk included in it. For every $n \ge 0$, let $l_{n}$ be the minimum of the lengths over all the $s$-curves $f^{-n}_{1} (\mathcal{L}_{z_{1},z_{3}})$. For every $n \ge 0$, we denote by $d_{n}$ the maximum of the diameters of the sets $\mathcal{V} \cap f^{-n}_{1} ( \mathbb{D}_{t,\rho_{1},\rho_{2},\rho_{3}} ) $ where $\mathcal{V}$ varies in the set of $(1,3)$-quasi planes over $\mathbb{D}^{2}$. Every vector in $C^{ss}$ is dilated under $f_{1}^{-1}$ by a factor close to $\Delta = |p'_{c_{1}}(\mathrm{pr}_{1}(\phi_{f_{1}}))|/|b|$. For every $(1,3)$-quasi plane $\mathcal{V}$ over $\mathbb{D}^{2}$, $f_{1}(\mathcal{V}) \cap \mathbb{D}^{3}$ contains a $(1,3)$-quasi plane. Every tangent vector $u$ to $f_{1}(\mathcal{V}) \cap \mathbb{D}^{3}$ is of the form $u = u^{1} \cdot (1,0,0)+u^{2} \cdot (0,1,0)+u^{3} \cdot (0,0,1)$ with $|u^{2}| \le \mathrm{max}(|u^{1}|,|u^{3}|)$. Then the vector $u$ is dilated under $f^{-1}_{1}$ by less than $\frac{2}{3} \cdot \Delta$. Then $d_{n+1} \le \frac{2}{3} \cdot \Delta \cdot d_{n}$. This implies that if $t$ is smaller than some value $t_{0}$, there exists $j =j(t)$ such that $(l_{j} \rho_{3}) \cdot (d_{j} \rho_{2})^{-1} \ge (3/2)^{j} \cdot \rho_{3} \cdot \rho_{2}^{-1} \ge 10$, $l_{j} \ge 10$ and $d_{j} \le 10^{-10} \cdot |b| \cdot \mathrm{min}\big( 1, \mathrm{dist}(\mathrm{pr}_{1}(\phi_{f_{1}}),\mathbb{D}),\mathrm{dist} (\mathrm{pr}_{3}(\phi_{f_{1}}),\mathbb{D}) \big)$. Increasing a last time $j$ in order to make $f^{-j}_{1}(\tau)$ closer to $\phi_{f_{1}}$, this implies both items (1) and (2). When $t$ tends to 0, the lengths of the disks $\mathcal{L}_{z_{1},z_{3}} $ tend to 0 and then $j=j(t)$ tends to $+\infty$. Reducing $\mathcal{F}'$, by continuity this remains true for $f \in \mathcal{F}'$ for a given $t$. The proof is done. \end{proof}
Here is a technical lemma which ensures the existence of a foliation with particular properties. In the following, we will say that a graph $\mathcal{V}_{x,y}$ of class $C^{2}$ over $z_{1} \in \mathbb{D}$ is of slope bounded by $\mathcal{S}<+ \infty$ if every tangent vector to $\mathcal{V}_{x,y}$ is of the form a multiple of $(1,\varepsilon_{2},\varepsilon_{3})$ with $ |\varepsilon_{2} | \le \mathcal{S}$ and $ |\varepsilon_{3} | \le \mathcal{S}$. For convenience, it will be useful to work with graphs of class $C^{2}$ which are not necessarily holomorphic. This will not be a problem since we will apply later only the Inclination Lemma on them and this does not need the holomorphic assumption.
\begin{lemma} \label{feuille} Let $\mathcal{V}$ be a $(1,3)$-quasi plane foliated by $u$-curves $(\mathcal{V}^{x})_{x \in \mathbb{D}}$. Let $\vartheta = ( \vartheta_{1} , \vartheta_{2} , \vartheta_{3}) \in \mathbb{D}^{3}$ be a point such that $\vartheta \notin \mathcal{V}$. Let us take a vector $u \in C^{u}$. Then there exists a foliation of class $C^{2}$ of a neighborhood of $\vartheta$ by graphs $\mathcal{V}_{x,y}$ of class $C^{2}$ over $z_{1} \in \mathbb{D}$ of slope bounded by $\mathcal{S}$, where $0<\mathcal{S}<+\infty$ is independent of $\mathcal{V}$ and $\vartheta$. Moreover we have: \begin{enumerate} \item for every $x \in \mathbb{D}$, $\mathcal{V}^{x} = \mathcal{V}_{x,0}$, \item the leaf going through $\vartheta$ has $\mathbb{C} \cdot u$ as tangent space. \end{enumerate} \end{lemma}
\begin{proof}
Up to multiplying $u$ by a non zero complex number, we can suppose that the first coordinate of $u$ is equal to 1. We first perform a change of coordinates by a biholomorphism $\varphi$, and then construct the graphs $\mathcal{V}_{x,y}$. We construct $\varphi$ such that $\varphi$ sends $\mathcal{V}$ to the plane $\{z_{2} = 0\}$ and each curve $\mathcal{V}^{x}$ to the line $\{z_{2} = 0,z_{3} = x\}$. For a given $z \in \mathbb{D}^{3}$, we denote by $\mathrm{pr}(z)$ the projection (parallel to $(0,1,0)$) of $z$ on $\mathcal{V}$. We denote by $\varphi_{2}(z) $ the complex number such that $z -\mathrm{pr}(z) = \varphi_{2}(z) \cdot (0,1,0)$. Since $\mathcal{V}$ is a $(1,3)$-quasi plane, we have $\|D\varphi_{2}\| \le 1$. We denote by by $\varphi_{3}(z) $ the complex number $x$ such that $\mathrm{pr}(z) \in \mathcal{V}^{x}$. Up to rescaling $x \mapsto \mathcal{V}^{x}$, we can suppose that $\mathcal{V}^{x}$ depends on the third coordinate $x$ of the intersection point of $\mathcal{V}^{x}$ with $\{z_{1} = 0\}$. Then we also have $\|D\varphi_{3}\| \le 1$. We then define the map $\varphi$ by $\varphi(z) = (z_{1}, \varphi_{2}(z), \varphi_{3}(z))$. In the coordinates given by $\varphi$, we have $\mathcal{V} = \{z_{2} = 0\}$ and $\mathcal{V}^{x}= \{z_{2} = 0,z_{3} = x\}$. We denote $\varphi(\vartheta) = (\tilde{ \vartheta_{1}},\tilde{ \vartheta_{2}} , \tilde{ \vartheta_{3}})$ (with $ \vartheta_{1}=\tilde{ \vartheta_{1}}$ and $|\tilde{ \vartheta_{2} } |> 0 $) and $D_{\vartheta}\varphi(u) = (1,\epsilon_{2},\epsilon_{3})$. Since $u \in C^{u}$, the estimates $\|D\varphi_{2}\| \le 1$ and $\|D\varphi_{3}\| \le 1$ imply $ |\epsilon_{2} | \le 5$ and $ |\epsilon_{3} |\le 5$.
We now take a graph $W = w(\mathbb{D}^{2})$ of class $C^{2}$ over $(z_{1},z_{3}) \in \mathbb{D}^{2} \subset \mathbb{R}^{4}$ such that $\tilde{\vartheta} \in W$ with $W \cap \{z_{2} = 0\} = \emptyset$ and having $\mathbb{C} \cdot D_{\vartheta}\varphi(u) $ as tangent space at $\tilde{\vartheta}$. Since $D_{\vartheta}\varphi(u) = (1,\epsilon_{2},\epsilon_{3})$ with $ |\epsilon_{2} | \le 5$ and $ |\epsilon_{3} |\le 5$, we can take $w$ such that $\|Dw\| \le 10$. Then for every $y \in \mathbb{D}(0,10|\tilde{\vartheta_{2}}|)$, we define the $C^{2}$ graph $W_{y} = w_{y}(\mathbb{D}^{2})$ over $(z_{1},z_{3}) \in \mathbb{D}^{2} \subset \mathbb{R}^{4}$ by $w_{y} (z_{1},z_{3}) = y \cdot w(z_{1},z_{3})$. We notice that $W_{0} =\{z_{2} = 0\}$ and $W_{1} = W$. Then for every $(x,y) \in \mathbb{D} \times \mathbb{D}(0,10|\tilde{\vartheta_{2}}|)$, the set $\mathcal{V}_{x,y}$ equal to the image of $z_{1} \in \mathbb{D} \mapsto (z_{1},w_{y}(z_{1},x),x) \in \mathbb{R}^{6}$ is a graph of class $C^{2}$ over $z_{1} \in \mathbb{D}$. Since $\|Dw\| \le 10$, its slope is bounded by $10 \times 10 = 100$ in the coordinates given by $\varphi$. Since $\mathcal{V}$ is a $(1,3)$-quasi plane in the initial coordinates, $D\varphi^{-1}$ is $C^{0}$-bounded and then the slope of any $\mathcal{V}_{x,y}$ in the initial coordinates is bounded by $\mathcal{S}$, where $0<\mathcal{S}<+\infty$ is independent of $\mathcal{V}$ and $\vartheta$. Since $W \cap \{z_{2} = 0\} = \emptyset$, we have $w(z_{1},z_{2}) \neq 0$ for every $(z_{1},z_{3}) \in \mathbb{D}^{2}$ and then the graphs $\mathcal{V}_{x,y}$ form a foliation of class $C^{2}$ of a neighborhood of $\vartheta$ satisfying the conditions (1) and (2). \end{proof}
We can reduce $\sigma_{1}$, $b_{1}$ and $\mathcal{F}'$ so that for every $f \in \mathcal{F}'$, $W^{s}_{\mathrm{loc}}(\phi_{f})$ has a point of intersection with every graph $\mathcal{L} \subset \mathbb{D}^{3}$ of class $C^{2}$ of slope bounded by $\mathcal{S}$ over $z_{1} \in \mathbb{D}$, and this intersection is transverse (uniformly in $\mathcal{L}$). The following is an easy consequence of the Inclination Lemma (one can refer to Prop. 6.2.23 page 257 of \cite{caga}).
\begin{lemma} \label{inclin} There exists an integer $k$ such that for every $ f \in \mathcal{F}'$, for every graph $\mathcal{L} \subset \mathbb{D}^{3}$ of class $C^{2}$ of slope bounded by $\mathcal{S}$ over $z_{1} \in \mathbb{D}$, $f^{k}(\mathcal{L})$ has a component which is a graph over $z_{1} \in \mathbb{D}$ which is $\eta$-close to $W^{u}_{\mathrm{loc}}(\phi_{f})$. \end{lemma}
\begin{notation} \label{nottheta} We fix the integer $k$ given by Lemma \ref{inclin}. Up to increasing $k$ if necessary, we can fix $0<t<t_{0}$ such that $j(t) = k $ (see Lemma \ref{boi}). We reduce $\mathcal{F}'$ so that for every $f \in \mathcal{F}'$: there is a point $\vartheta \in W^{s}(\alpha_{f})$ such that $f^{k}(\vartheta) \in ( \tau+ (\mathbb{D}(0,t\rho))^{3})$ and $W^{s}(\alpha_{f})$ has a tangent vector $u \in C^{u}$ at $\vartheta$ (it is possible by continuity since it is the case for $f_{1}$ with the point $f_{1}^{-k}(\tau)$). \end{notation}
\includegraphics[width=11cm]{figure2.png}
\begin{center} Figure 2: straightening of the initial fold by iterating backwards. The blue and green sets are the connected components $\mathcal{W}_{f}$ and $\mathcal{W}''_{f}$ of $ W^{s}(\alpha_{f}) \cap \mathbb{D}^{3}$. In particular, the green set $\mathcal{W}''_{f}$ is a concentrated 3-folded $(2,3)$-surface. \end{center}
\begin{prop} \label{fm} For every $f \in \mathcal{F}'$, $W^{s}(\alpha_{f})$ contains a concentrated 3-folded $(2,3)$-surface $\mathcal{W}''_{f}$ such that $\mathrm{Fold}(\mathcal{W}''_{f}) \subset \mathbb{D}(0,\frac{1}{10})$ and $\mathrm{diam}(\mathrm{pr}_{1}(\mathcal{W}''_{f})) < 10^{-10}$. \end{prop}
\begin{proof}
Let $f \in \mathcal{F}'$. Let $\mathcal{V}$ be a $(1,3)$-quasi plane foliated by the $u$-curves $\mathcal{V}^{x} = \mathcal{V} \cap \{z_{3} = x \}$. By Notation \ref{nottheta}, we can take $\vartheta \in W^{s}(\alpha_{f})$ such that $f^{k}(\vartheta) \in (\tau+ (\mathbb{D}(0,t\rho))^{3})$, $ W^{s}(\alpha_{f})$ has a tangent vector in $C^{u}$ at $\vartheta$ and $\vartheta$ does not belong to $\mathcal{V}$. According to Lemma \ref{feuille} it is possible to find a foliation of class $C^{2}$ of a neighborhood of $\vartheta$ by graphs $\mathcal{V}_{x,y}$ of class $C^{2}$ over $z_{1} \in \mathbb{D}$ of slope bounded by $\mathcal{S}$ such that $\mathcal{V}^{x} = \mathcal{V}_{x,0}$ and the leaf going through $\vartheta$ has $\mathbb{C} \cdot u$ as tangent space. In particular, the leaf going through $\vartheta$ is tangent to $W^{s}(\alpha_{f})$. We apply $f^{k}$ to all these curves $\mathcal{V}_{x,y}$. According to Lemma \ref{inclin}, the sets $f^{k}( \mathcal{V}_{x,y} ) $ have components which foliate a neighborhood of $\tau$ by graphs over $z_{1} \in \mathbb{D}$ which are $\eta$-close to $W_{\text{loc}}^{u}(\phi_{f})$. If necessary, we can extend this foliation into a foliation of $\tau + \mathbb{D}(0,t\rho_{1}) e^{u}(\tau)+ \mathbb{D}(0,t\rho_{2})(0,1,0)+ \mathbb{D}(0,t\rho_{3}) (0,0,1) $ by graphs over $z_{1} \in \mathbb{D}$ $\eta$-close to $W_{\text{loc}}^{u}(\phi_{f})$. Then according to Lemma \ref{hoppe5}, the set $\text{Tan}$ of points of $f^{k}( \mathcal{V}_{x,y} ) $ where $f^{k}( \mathcal{V}_{x,y} )$ is tangent to $W^{s}(\alpha_{f})$ is a regular two-dimensional real manifold which has its direction $\epsilon$-close to $\mathbb{C} \cdot v_{\text{tan}}$ at each point and goes through the point $f^{k}(\vartheta) \in \tau+ (\mathbb{D}(0,t\rho))^{3}$. According to Lemma \ref{eps}, the set $\mathrm{Tan}$ can intersect $\tau + \mathbb{D}(0,t\rho_{1}) e^{u}(\tau)+ \mathbb{D}(0,t\rho_{2})(0,1,0)+ \mathbb{D}(0,t\rho_{3}) (0,0,1) $ only in $\tau + \mathbb{D}(0,t\rho_{1}) e^{u}(\tau)+ \partial \mathbb{D}(0,t\rho_{2})(0,1,0)+ \mathbb{D}(0,t\rho_{3}) (0,0,1) $. By Lemma \ref{boi}, it holds: $$f^{-k} \Big( \tau + \mathbb{D}(0,t\rho_{1}) e^{u}(\tau)+\partial \mathbb{D}(0,t\rho_{2})(0,1,0)+ \mathbb{D}(0,t\rho_{3}) (0,0,1) \Big) \subset \mathbb{C}^{3} \backslash\mathbb{D}^{3} \, ,$$ $$f^{-k} \Big( \tau + \mathbb{D}(0,t\rho_{1}) e^{u}(\tau)+ \mathbb{D}(0,t\rho_{2}) (0,1,0)+ \mathbb{D}(0,t\rho_{3}) (0,0,1) \Big) \cap \partial \mathbb{D}^{3} \Subset \mathbb{D} \times \partial \mathbb{D} \times \mathbb{D} \, .$$ Then $f^{-k}( \text{Tan} )$ intersects $\mathcal{V}$. Since both $\mathcal{V}$ and $W^{s}(\alpha_{f})$ are complex manifolds, $f^{-k}( \text{Tan} )$ intersects $\mathcal{V}$ in exactly one point. We denote by $\mathcal{W}'_{f}$ the continuation of the connected component of $W^{s}(\alpha_{f_{1}}) \cap \mathbb{D}^{3}$ containing $f_{1}^{-k}(\tau)$ for $f \in \mathcal{F}'$. Then $\mathcal{W}'_{f}$ is tangent to exactly one $\mathcal{V}^{x} = \mathcal{V} \cap \{z_{3} = x \}$. This implies that $\text{pr}_{3}$ restricted on $\mathcal{W}'_{f} \cap \mathcal{V}$ is a two-covering with exactly one point of ramification. Since this is true for any $(1,3)$-quasi plane $\mathcal{V}$, by definition, $\mathcal{W}'_{f}$ is a 3-folded $(2,3)$-surface for every $f \in \mathcal{F}'$. We notice that $ \mathrm{pr}_{3} (\phi_{f}) \in \mathbb{D}(0,\frac{1}{10}-10^{-4})$. Increasing $k$ if necessary, we have $ \mathrm{pr}_{3} (f^{-k}(\tau)) \in \mathbb{D}(0,\frac{1}{10}-10^{-4})$. Moreover $\vartheta$ can be taken as close to $f^{-k}(\tau)$ as wanted and by the proof of Lemma \ref{boi} the diameter of $\mathrm{pr}_{3} \big( f^{-k}(\mathrm{Tan}) \big) $ is smaller than $10^{-10}$. Then we have $\mathrm{Fold}(\mathcal{W}''_{f}) \subset \mathbb{D}(0,\frac{1}{10})$. Since the first direction is dilated under $f$, iterating by $f^{-1}$ if necessary, we can have the additional property that $\mathrm{diam}(\text{pr}_{1}(\mathcal{W}''_{f})) < 10^{-10}$. Then according to Proposition \ref{rp7}, we can iterate a last time $f^{-1}$ if necessary to get a component $\mathcal{W}''_{f}$ of $W^{s}(\alpha_{f})$ which is a concentrated 3-folded $(2,3)$-surface such that $\mathrm{Fold}(\mathcal{W}''_{f}) \subset \mathbb{D}(0,\frac{1}{10})$ and $\mathrm{diam}(\mathrm{pr}_{1}(\mathcal{W}''_{f})) < 10^{-10}$. \end{proof}
\section{Proof of the main results}
\begin{proof}[Proof of the main Theorem and Corollaries 1 and 2] By Proposition \ref{fm}, for every $f \in \mathcal{F}'$, $W^{s}(\alpha_{f})$ contains some concentrated 3-folded $(2,3)$-surface $\mathcal{W}^{0}$ with $\mathrm{Fold}(\mathcal{W}^{0}) \subset \mathbb{D}(0,\frac{1}{10})$ and $\text{diam}(\text{pr}_{1}(\mathcal{W}^{0})) < 10^{-10}$. This implies persistent heteroclinic tangencies. Indeed, by Proposition \ref{rrrrr} there exists a point $\kappa_{f}$ of the horseshoe $\mathcal{H}_{f}$ and a point of quadratic tangency $\tau'$ between $W^{u}(\kappa_{f})$ and $\mathcal{W}^{0} \subset W^{s}(\alpha_{f})$. The proof of the main Theorem is complete.
We now prove Corollary 1. We call $\mathcal{U}'_{f}$ the component of $W^{u}(\kappa_{f}) \cap \mathbb{D}^{3}$ containing the point of tangency $\tau'$ and we consider the family $(\mathcal{U}'_{f'})_{f' \in \mathcal{F}'}$ of holomorphic $u$-curves given by the continuation of $\mathcal{U}'_{f}$. According to Proposition \ref{fd}, the tangency $\tau'$ is generically unfolded. We apply a standard argument to obtain homoclinic tangencies. Let us take any neighborhood $\mathcal{F}''$ of $f$. Since the tangency $\tau'$ is generically unfolded, we can rely on the following well-know lemma:
\begin{lemma} There exists $\epsilon''>0$ such that for every holomorphic family $(\mathcal{U}''_{f'})_{f' \in \mathcal{F}'}$ of $u$-curves $\epsilon''$-close to $(\mathcal{U}'_{f'})_{f' \in \mathcal{F}'}$ and every holomorphic family $(\mathcal{W}''_{f'})_{f' \in \mathcal{F}'}$ of complex surfaces $\epsilon''$-close to $(\mathcal{W}^{0})_{f' \in \mathcal{F}' }$, there exists a point of quadratic tangency between $\mathcal{U}''_{f''}$ and $\mathcal{W}''_{f''}$ where $f'' \in \mathcal{F}''$. \end{lemma}
We claim that there exists a holomorphic family of $u$-curves $(\mathcal{U}''_{f'})_{f' \in \mathcal{F}'}$ $\epsilon''$-close to $(\mathcal{U}'_{f'})_{f' \in \mathcal{F}'}$ such that for every $f' \in \mathcal{F}'$, $\mathcal{U}''_{f'}$ is a component of $W^{u}(\delta_{f'}) \cap \mathbb{D}^{3}$ and a holomorphic family of complex surfaces $(\mathcal{W}''_{f'})_{f' \in \mathcal{F}'}$ $\epsilon''$-close to $((\mathcal{W}^{0})_{f'})_{f' \in \mathcal{F}'}$ such that for every $f' \in \mathcal{F}'$, $\mathcal{W}''_{f'}$ is a component of $W^{s}(\delta_{f'})$ (we recall that the point $\delta_{f'}$ was defined in Proposition \ref{ferm}). We prove this fact for the case of $u$-curves, the proof is the same for complex surfaces. According to Proposition \ref{ferm}, $\alpha_{f'}$, and then $\kappa_{f'}$, is in the homoclinic class of $\delta_{f'}$. This implies that there exists a transversal intersection between $W^{s}_{\mathrm{loc}}(\kappa_{f'})$ and $W_{\mathrm{loc}}^{u}(\delta_{f'})$ for every $f' \in \mathcal{F}'$. For a given $f' \in \mathcal{F}'$, by the inclination lemma, there exists an integer $n_{f'}$ such that for every $n \ge n_{f'}$, $(f')^{n_{f'}} (W_{\mathrm{loc}}^{u}(\delta_{f'}))$ contains a $u$-curve $\epsilon''$-close to $\mathcal{U}'_{f'}$. By continuity, $n_{f'}$ can be taken locally constant. Up to reducing $\mathcal{F}'$, we can take $\mathcal{F}'$ compact (with non empty interior). By compactness, it is then possible to take the maximal value $n$ of $n_{f'}$ on a finite open covering of $\mathcal{F}' $. Then for every $f' \in \mathcal{F}'$, there is a $u$-curve $\mathcal{U}''_{f'}$ $\epsilon''$-close to $\mathcal{U}'_{f'}$ which is a connected component of $W^{u}(\delta_{f'}) \cap \mathbb{D}^{3}$. The proof of Corollary 1 is complete.
We now prove Corollary 2. The preceding proof shows that there exists $f'' \in \mathcal{F}''$ with a point of homoclinic tangency between $W^{s}(\delta_{f''})$ and $W^{u}(\delta_{f''})$. In particular maps with homoclinic tangencies associated to $\delta_{f}$ are dense in $ \mathcal{F}'$. The point $\delta_{f}$ has the property of being sectionally dissipative for every $f \in \mathcal{F}'$. Then, by Proposition \ref{finenfin} which gives the creation of sinks from homoclinic tangencies (the proof is given in Appendix for the convenience of the reader) a classical Baire category argument already used in \cite{bb1} allows us to conclude the existence of a residual set of $\text{Aut}_{2}(\mathbb{C}^{3})$ of automorphisms displaying infinitely many sinks. The proof is complete. \end{proof}
\appendix
\section{From homoclinic tangencies to sinks}
To show the main Theorem, we need the following result. It is known since the work of Newhouse how to get a sink from a homoclinic tangency. The adaptation to the case of $\mathbb{C}^{2}$ was obtained by Gavosto in \cite{est}. Here we adapt her proof to the case of $\mathbb{C}^{3}$. Remind that a generically unfolded tangency is a tangency which is unfolded with a positive speed.
\begin{propo} \label{finenfin}
Let $(F_{t})_{t \in \mathbb{D}}$ be a family of polynomial automorphisms of $\mathbb{C}^{3}$. We suppose that for the parameter $t = 0$, there is a sectionally dissipative periodic point $P_{0}$ and a generically unfolded homoclinic quadratic tangency $Q \in W^{s}(P_{0}) \cap W^{u}(P_{0})$ between $W^{s}(P_{0})$ and $W^{u}(P_{0})$. We also suppose that the three eigenvalues $\lambda_{0}^{ss}$, $\lambda_{0}^{cs}$ and $\lambda_{0}^{u} $ at $P_{0}$ satisfy $|\lambda_{0}^{ss}| < |\lambda_{0}^{cs}|<1< |\lambda_{0}^{u}|$ (and $|\lambda_{0}^{cs}| \cdot |\lambda_{0}^{u}|<1$ by sectional dissipativity). Then for every neighborhood $\mathcal{Q}$ of $Q$ and every neighborhood $\mathcal{T}$ of 0, there exists $t \in \mathcal{T}$ such that $F_{t}$ admits an attracting periodic point in $\mathcal{Q}$. \end{propo}
\begin{proof} Step 1 : Construction of cone fields
In the following, iterating if necessary, we will suppose that $P_{0}$ is a fixed point of $F_{0}$. We fix a constant $\eta>0$ such that we have $(|\lambda^{cs}_{0}|+\eta)(|\lambda^{u}_{0}|+\eta)<1$ (the periodic point $P_{0}$ is sectionally dissipative) and $|\lambda_{0}^{ss}|+\eta < |\lambda_{0}^{cs}| - \eta<|\lambda_{0}^{cs}| + \eta<1$. We can fix a neighborhood $\mathcal{P}$ of $P_{0}$, a neighborhood $\mathcal{T}_{0}$ of 0, cones $C^{u}$, $C^{ss}$ and $C^{cs}$ centered at the three eigenvectors of $DF_{0} (P_{0})$ and an integer $N_{0}$ with the following properties: for every $n \ge N_{0}$, $t \in \mathcal{T}_{0}$, for every matrix $M = M_{n} \cdot \ldots \cdot M_{1}$ where $M_{i}$ is the differential of $F_{t}$ at a point in $\mathcal{P}$, we have: \begin{enumerate}
\item $C^{u}$ is invariant under $M$ and there is exactly one eigenvector (up to multiplication) of $M$ in $C^{u}$ of eigenvalue $(|\lambda^{u}_{0}|-\eta)^{n}< |\lambda_{1}|< (|\lambda^{u}_{0}|+\eta)^{n}$,
\item $C^{ss}$ is invariant under $M^{-1}$ and there is exactly one eigenvector (up to multiplication) of $M$ in $C^{ss}$ of eigenvalue $(|\lambda^{ss}_{0}|-\eta)^{n}< |\lambda_{2}|< (|\lambda^{ss}_{0}|+\eta)^{n}$, \item for every vector $v$ which is not in $C^{u} \cup C^{ss} \cup C^{cs}$, we have $M \cdot v \in C^{u}$ or $M^{-1} \cdot v \in C^{ss}$,
\item $ (|\lambda^{u}_{0}|-\eta)^{n}(|\lambda^{cs}_{0}|-\eta)^{n}(|\lambda^{ss}_{0}|-\eta)^{n}<| \det(M) |< (|\lambda^{u}_{0}|+\eta)^{n}(|\lambda^{cs}_{0}|+\eta)^{n}(|\lambda^{ss}_{0}|+\eta)^{n}$. \end{enumerate} Reducing $\eta$ and increasing $N_{0}$ if necessary, we have for $n \ge N_{0}$:
$$ (|\lambda^{u}_{0}|+\eta)^{n} (|\lambda^{ss}_{0}|+\eta)^{2n}< (|\lambda^{u}_{0}|-\eta)^{n} (|\lambda^{cs}_{0}|-\eta)^{n} (|\lambda^{ss}_{0}|-\eta)^{n},$$ $$ (|\lambda^{u}_{0}|+\eta)^{n} (|\lambda^{cs}_{0}|+\eta)^{n} (|\lambda^{ss}_{0}|+\eta)^{n}< (|\lambda^{u}_{0}|-\eta)^{2n}(|\lambda^{ss}_{0}|-\eta)^{n}.$$
This implies that $M$ is diagonalizable. Indeed, if it was not the case, $M$ would be triangularizable with a double eigenvalue and we would have: $$ (|\lambda^{u}_{0}|+\eta)^{n} (|\lambda^{ss}_{0}|+\eta)^{2n}>| \det(M) | \text{ or } | \det(M) |> (|\lambda^{u}_{0}|-\eta)^{2n}(|\lambda^{ss}_{0}|-\eta)^{n}.$$ But by item 4 above, there would be a contradiction with the previous inequalities. Moreover the third eigenvector has to be in $C^{cs}$ according to item 3 above. \newline \newline Step 2 : Local coordinates
Iterating if necessary, we can suppose that $Q \in \mathcal{P}$. We are going to make several local changes of coordinates so that the stable and unstable manifolds of $P_{t}$ have a simple form near the tangency. We will use this local change of coordinates in Steps 2 and 3. In Step 4, we will mainly use the coordinates in the canonical basis but we will need the local change of coordinates a last time at some point so we will denote it by $\Psi_{\tau}$ in Step 4 to make the distinction between the two systems of coordinates. In the following, to construct the local coordinates, we will keep the notation $z_{1},z_{2},z_{3}$ by simplification.
We first pick local coordinates such that $Q = 0$ and in the neighborhood of $Q$, the stable manifold is $\{z_{1} = 0\}$. Up to a linear invertible second change of coordinates, we can assure that the tangent vector of $W^{u}(P_{0})$ at $Q$ is $(0,0,1)$. Then we have that locally near 0 the unstable manifold is given by a graph over $z_{3}$ of the form $\{(w_{1}(z_{3},t),w_{2}(z_{3},t),z_{3}) : z_{3}\}$. Since the tangency is quadratic, for each $t$ in a neighborhood of 0, there exists exactly one $z_{3,t}$ such that $ \frac{\partial w_{1}}{\partial z_{3}}(z_{3,t},t) = 0$. We pick new coordinates a third time by changing the coordinate $z_{3}$ into $z_{3}-z_{3,t}$. The unstable manifold is: $$w_{1}(z_{3}) = w_{1}(z_{3},t) = w_{1}(0,t)+\frac{\partial^{2}w_{1}}{\partial z_{3}^{2}}(0,t)z_{3}^{2}+ h(z_{3},t) \text{ with } h(z_{3},t) = o(z_{3}^{2}) .$$ The stable manifold is still locally equal to $\{z_{1} = 0\}$. Since there is a quadratic tangency for $t = 0$ which is generically unfolded, we have $w_{1}(0,0) = 0$ and $\frac{\partial w_{1}}{\partial t} (0,0) \neq 0 $. Then we change coordinates a fourth time: $z_{1}$ becomes $ \frac{t}{w_{1}(0,t)}z_{1}$ and $z_{2}$ and $z_{3}$ are unchanged. In these new coordinates, the unstable manifold is given by: $$w_{1}(z_{3}) = t +z_{3}^{2}\tilde{h}(z_{3}, t ),$$ where $\tilde{h}(z_{3}, t ) \neq 0$ in a neighborhood of 0. The stable manifold is still locally equal to $\{z_{1} = 0\}$. The fifth change of coordinates is given by $z_{3}$ becoming $z_{3}(\tilde{h}(z_{3},t))^{1/2}$, where $(\tilde{h}(z_{3},t))^{1/2}$ is a complex square root of $\tilde{h}(z_{3},t)$ (which is well defined since $\tilde{h}(z_{3}, t ) \neq 0$ in a neighborhood of 0). We finally get that the unstable manifold is given by $z_{1} = z_{3}^{2}+ t$ and $z_{2} = w_{2}(z_{3},t)$. The stable manifold is still locally equal to $\{z_{1} = 0\}$ and the tangent vector of $W^{u}(P_{0})$ at $Q$ is still $(0,0,1)$. The last change of coordinates is given by $z_{2}-w_{2}(z_{3},t)$. The unstable manifold is given by $z_{1} = z_{3}^{2}+ t$ and $z_{2} = 0$ and the stable manifold by $z_{1} = 0$. The tangent vector of $W^{u}(P_{0})$ at $Q$ is still $(0,0,1)$. \newline \newline Step 3 : Construction of a periodic point
We take a tridisk $B$ around $Q$ in the coordinates that we just defined : $B = \{(z_{1},z_{2},z_{3}) : |z_{1}| < \delta, |z_{2}|<\delta, |z_{3}|<\delta\}$ where $0<\delta<1$. Since the tangent vector of $W^{u}(P_{0})$ at $Q$ is $(0,0,1)$, reducing $\delta$ if necessary, there exists a neighborhood $\mathcal{T}_{1} \subset \mathcal{T}_{0}$ such that the component of $W^{u}(P_{t}) \cap B$ containing $Q$ (for $t = 0$) or its continuation (for $t \neq 0$) is horizontal in $B$ relatively to the third projection and is included in $\{(z_{1},z_{2},z_{3}) \in B : |z_{2}| < \frac{1}{10}\delta \}$. Using the Inclination Lemma, there exists $N_{1} \ge N_{0}$ such that for $t \in \mathcal{T}_{1}$ and $n \ge N_{1}$ , $F_{t}^{n}(B)$ will intersect $B$ and $F_{t}(B) \cap B$ is horizontal relatively to the third projection.
In the following, we show that for every sufficiently high $n$, a periodic point for $F_{t}$ (where $t \in \mathcal{T}_{1}$) is created. Let us now denote $\Delta_{z_{2},z_{3}} = \{(z_{1},z_{2},z_{3}) : |z_{1}|<\delta \}$ which is a disk, for $|z_{2}|<\delta,|z_{3}|<\delta$. It is possible to increase $N_{2} \ge N_{1}$ such that for any $|z_{2}|<\delta,|z_{3}|<\delta$, for every $n \ge N_{2}$, $F_{t}^{n}(\Delta_{z_{2},z_{3}} ) \cap B$ is horizontal relatively to the third projection (of degree 1) and included in $\{(z_{1},z_{2},z_{3}) \in B : |z_{2}| < \frac{1}{2}\delta \}$. Since $F_{t}^{n}(\Delta_{z_{2},z_{3}} ) \cap B$ is horizontal relatively to the third projection and of degree 1, it intersects exactly one disk $\Delta_{z'_{2},z_{3}}$ where $z'_{2} = z'_{2}(z_{2})$ with $z'_{2} \in \mathbb{D}(0,\delta/2)$. This defines for each fixed $t \in \mathcal{T}_{1}$ a holomorphic map $z_{2} \mapsto z'_{2}$. Then the map $z_{2} \mapsto z_{2}-z'_{2}$ defined on $\mathbb{D}(0,\delta)$ is holomorphic and the image of $\partial \mathbb{D}(0,\delta)$ contains a loop around $\mathbb{D}(0,\delta/2)$. Then by the Argument principle there exists $z_{2}(z_{3})$ such that $\Delta_{z_{2}(z_{3}),z_{3}}$ intersects $F^{n}_{t}(\Delta_{z_{2}(z_{3}),z_{3}})$.
We are going to choose $z_{3}$ in order to create a periodic point in $\Delta_{z_{2}(z_{3}),z_{3}}$. There is a point $R_{z_{3}} = (f_{t}(z_{3}),z_{2}(z_{3}),z_{3}) \in \Delta_{z_{2}(z_{3}),z_{3}}$ which is sent on $S_{z_{3}} \in \Delta_{z_{2}(z_{3}),z_{3}}$ where $S_{z_{3}} = (g_{t}(z_{3}),z_{2}(z_{3}),z_{3} )$. We are going to choose $z_{3}$ (in function of $t$) such that $R_{z_{3}} =S_{z_{3}}$. When $N$ goes to infinity, $f_{t}(z_{3})$ tends to 0 and $g_{t}(z_{3})$ tends to $z_{3}^{2}+t$. Then $g_{t}(z_{3}) - f_{t}(z_{3})$ tends to $z_{3}^{2}+t$. In particular, if $N$ is sufficiently high, for every $t \in \mathbb{D}$, the graph of $z_{3} \mapsto g_{t}(z_{3}) - f_{t}(z_{3})$ is a curve of degree 2 over $z_{3}$ which has exactly one point of tangency with the horizontal foliation. We take a new bound $N_{3}$ on $n $ such that this is the case, we increase $N_{3}$ in the rest of Step 3 in order to satisfy more assumptions. The second coordinate of this point of tangency is a holomorphic function of $t$ which tends to $t$ when increasing $N_{3}$. In particular, this implies that there exists exactly one value $t_{0}$ of $t$ for which the equation $g_{t_{0}}(z_{3}) - f_{t_{0}}(z_{3}) = 0$ has one double solution, and for every other value of $t \in \mathbb{D}$, there are two distinct solutions. Then, for $F_{t}$, we have two periodic points which are equal when $t = t_{0}$. We do a reparametrization of the family of maps $(F_{t})_{t \in \mathbb{D}}$ by taking $t = \tau^{2}$. From now on, we are working with the family of maps $\tilde{F}_{\tau} = F_{\tau^{2}}$. By simplicity, we will simply denote it by $F_{\tau}$. For $F_{\tau}$, we have two periodic points which are equal when $\tau = \tau_{0}$ where $\tau_{0}^{2} = t_{0}$. We can increase $N_{3}$ if necessary so that for each $\tau \in \mathbb{D} \backslash \mathbb{D}(0,\frac{1}{2})$, the two solutions of $g_{\tau^{2}}(z_{3}) - f_{\tau^{2}}(z_{3}) = 0$ are respectively $\frac{1}{100}$-close to $\pm i \tau$. For $\tau \in \mathbb{D} \backslash \mathbb{D}(0,\frac{1}{2})$, we denote by $R^{\tau}$ the periodic point corresponding to the solution $\frac{1}{100}$-close to $i\tau$. It is clear that the map $\tau \mapsto R^{\tau}$ restricted on $\mathbb{D} \backslash \mathbb{D}(0,\frac{1}{2})$ is continuous. For $\tau = \tau_{0}$, we denote by $R^{\tau_{0}}$ the unique periodic point corresponding to the double solution of $g_{\tau_{0}^{2}}(z_{3}) - f_{\tau_{0}^{2}}(z_{3}) = 0 $. Finally, for $\tau \in \mathbb{D}(0,\frac{1}{2}) $ not equal to $\tau_{0}$, there are two distinct periodic points in $\Delta_{z_{2}(z_{3}),z_{3}}$. We pick any path in $\mathbb{C}$ from $\tau$ to a point $\tau_{1}$ in $\mathbb{D} \backslash \mathbb{D}(0,\frac{1}{2})$ which does not contain $\tau_{0}$. For $\tau_{1}$, the periodic point $R^{\tau_{1}}$ is defined. Since the path does not contain $\tau_{0}$, there is exactly one of the two distinct periodic points in $\Delta_{z_{2}(z_{3}),z_{3}}$ for $\tau$ which is the continuation of $R^{\tau_{1}}$. We denote it by $R^{\tau}$. Since the map $\tau \mapsto R^{\tau}$ restricted on $\mathbb{D} \backslash \mathbb{D}(0,\frac{1}{2})$ is continuous, this choice is independent of a particular choice of $\tau_{1}$ in $\mathbb{D} \backslash \mathbb{D}(0,\frac{1}{2})$. Then we have defined a map $\tau \mapsto R^{\tau}$ on $\mathbb{D}$. It is clear that near any point of $\mathbb{D} \backslash \{\tau_{0}\}$, $R^{\tau}$ is locally the continuation of the same periodic point of $F_{\tau}$, then it is holomorphic. The map $\tau \mapsto R^{\tau}$ is holomorphic on $\mathbb{D} \backslash \{\tau_{0}\}$. It is trivial that it is continuous at $\tau_{0}$. Then it is holomorphic on $\mathbb{D}$. As we already said, from now on, we go back to the coordinates in the canonical basis and we call $\Psi_{\tau}$ the local coordinates we just used near the tangency point $Q$. We will use $\Psi_{\tau}$ a last time at the end of Step 4. \newline \newline Step 4 : $R^{\tau}$ is a sink
We now show it is possible to pick $\tau$ such that $R^{\tau}$ is a sink. From now on, we fix a neighborhood $\mathcal{T} \subset \mathcal{T}_{1}$ of 0. Recall that $Q$ belongs to the small neighborhood $\mathcal{P}$ of $P_{0}$ defined in Step 1. We denote by $n = n_{1}+n_{2}$ such that for $k = 1,...,n_{1}$, $F^{k}_{\tau}(R^{\tau})$ is in $\mathcal{P}$ and $F^{n_{1}+1}_{\tau}(R^{\tau}) \notin \mathcal{P}$. We express all matrices in the $\tau$-dependent basis given by the 3 eigenvectors $e'_{1}$,$e'_{2},e'_{3}$ of $D(F^{n_{1}}_{\tau})(R^{\tau})$ (the matrix $DF^{n_{1}}_{\tau}(R^{\tau})$ is diagonalizable according to Step 1). We have that $e'_{1} \in C^{u}$, $e'_{2} \in C^{ss}$ and $e'_{3} \in C^{cs}$. Then, in this basis, the matrix $DF_{\tau}^{n}(R^{\tau})$ is of the form $ DF^{n_{1}+n_{2}}_{\tau}(R^{\tau}) = DF^{n_{2}}_{\tau}(F^{n_{1}}_{\tau}(R^{\tau})) \cdot DF^{n_{1}}_{\tau}(R^{\tau})$. The matrix $DF^{n_{1}}_{\tau}(R^{\tau})$ is of the form: $$\begin{pmatrix}
\Lambda_{1}&0&0\\ 0&\Lambda_{2}&0 \\
0& 0&\Lambda_{3}\\ \end{pmatrix} \, .$$
We have that $(|\lambda^{u}_{0}| - \eta)^{n_{1}} < |\Lambda_{1}| < (|\lambda^{u}_{0}| + \eta)^{n_{1}}$, $(|\lambda^{ss}_{0}| - \eta)^{n_{1}} < |\Lambda_{2}| < (|\lambda^{ss}_{0}| + \eta)^{n_{1}}$ and $(|\lambda^{cs}_{0}| - \eta)^{n_{1}} < |\Lambda_{3}| < (|\lambda^{cs}_{0}| + \eta)^{n_{1}}$. The matrix $DF^{n_{2}}_{\tau}(F^{n_{1}}_{\tau}(R^{\tau}))$ is of the form: $$ \begin{pmatrix}
A&B&C\\
D& E&F \\
G&H& I \\ \end{pmatrix}\, .$$ Since $e'_{1} \in C^{u}$, $e'_{2} \in C^{ss}$ and $e'_{3} \in C^{cs}$ and these cones are disjoint with $n_{2}$ bounded, the coefficients $A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$ are bounded in modulus by some constant $K$ which is independant of $\tau$, $n$ and $n_{1}$. Then we have:
$$DF^{n_{1}+n_{2}}_{\tau}(R^{\tau}) = \begin{pmatrix}
A\Lambda_{1} &B\Lambda_{2}&C\Lambda_{3} \\
D\Lambda_{1} &E\Lambda_{2}&F\Lambda_{3}\\ G\Lambda_{1} &H \Lambda_{2}&I \Lambda_{3}\\ \end{pmatrix} \, .$$
Let us suppose that $A\Lambda_{1} = 0$. There exists $\epsilon>0$ such that if the characteristic polynomial of $DF^{n_{1}+n_{2}}_{\tau}(R^{\tau})$ is $X^{3}+a_{2}X^{2}+a_{1}X +a_{0}$ with $|a_{0}|<\epsilon, |a_{1}|<\epsilon, |a_{2}|<\epsilon$, then the 3 eigenvalues of $DF^{n_{1}+n_{2}}_{\tau}(R^{\tau})$ are of modulus lower than 1 and then $R^{\tau}$ is a sink. We are going to show that it is possible to get a lower bound on $n_{1}$ so that it is always the case. The coefficients of the characteristic polynomial of $DF^{n_{1}+n_{2}}_{\tau}(R^{\tau})$ are : $$a_{2} = - ( A\Lambda_{1}+E\Lambda_{2} + I\Lambda_{3}) , $$ $$ a_{1} = (EI\Lambda_{2}\Lambda_{3}-FH\Lambda_{2}\Lambda_{3} +AI\Lambda_{1}\Lambda_{3}-CG\Lambda_{1}\Lambda_{3}+AE\Lambda_{1}\Lambda_{2}-BD\Lambda_{1}\Lambda_{2}) , $$ $$ a_{0} = -\det( DF^{n_{2}}_{\tau}(F^{n_{1}}_{\tau}(R^{\tau} )) .$$
We have: $ |a_{0}| = | \det( DF^{n_{2}}_{\tau}(F^{n_{1}}_{\tau}(R^{\tau}) ) )|< 6 K^{3}|\Lambda_{1}\Lambda_{2}\Lambda_{3}|< 6 K^{3} \big( (|\lambda^{cs}_{0}|+\eta)(|\lambda^{u}_{0}|+\eta) \big)^{n_{1}}$ which tends to 0 when $n_{1} \rightarrow +\infty $ because $(|\lambda^{cs}_{0}|+\eta)(|\lambda^{u}_{0}|+\eta)<1$ (remind that $P_{0}$ is a sectionally dissipative periodic point). We increase $n_{1}$ such that $| a_{0} |<\epsilon$. We have that: $$| a_{1} | < 6K^{2} \max ( |\Lambda_{1}\Lambda_{2}|, |\Lambda_{1}\Lambda_{3}|, |\Lambda_{2}\Lambda_{3}|) < 6K^{2} \big( (|\lambda^{cs}_{0}|+\eta)(|\lambda^{u}_{0}|+\eta) \big)^{n_{1}},$$ which tends to 0 when $n_{1} \rightarrow + \infty $ because $(|\lambda^{cs}_{0}|+\eta)(|\lambda^{u}_{0}|+\eta)<1$. We increase $n_{1}$ further such that $| a_{1} |<\epsilon$. Finally, in the term $a_{2}$, both $E\Lambda_{2}$ and $I\Lambda_{3}$ tend to 0 when $n_{1} \rightarrow + \infty $. The term $A\Lambda_{1}$ is equal to 0 by hypothesis. We increase $n_{1}$ a last time such that $| a_{2} |<\epsilon$. Finally, we pick $N_{4} \ge N_{3}$ such that $n_{1}$ is sufficiently high in order to satisfy the previous inequalities. Finally, the three eigenvalues of $DF^{n_{1}+n_{2}}_{\tau}(R^{\tau})$ are of modulus lower than 1 and $R^{\tau}$ is a sink.
It remains us to show that for a given neighborhood $t \in \mathcal{T}$ of 0 and the corresponding neighborhood of 0 for $\tau$ (remind that $t = \tau^{2}$), it is possible to pick a new bound $N_{6}$ on $n$, such that for $n \ge N_{6}$, there is a parameter $\tau=\tau(n)$ such that the differential $DF_{\tau}^{n}(R^{\tau})$ satisfies $A\Lambda_{1} = 0$. To show this, we work again in the coordinates $\Psi_{\tau}$ for the map $F_{\tau}$. We denote by $\Pi$ the projection of the plane $\Psi_{\tau}(W^{s}(P_{\tau^{2}})) = \{z_{1} = 0\}$ into $\mathbb{P}^{2}(\mathbb{C})$. The projection $\Pi$ is a holomorphic curve. We denote by $\Gamma$ the holomorphic curve in $\mathbb{P}^{2}(\mathbb{C})$ given by the tangent directions to the curve $\Psi_{\tau}(W^{u}(P_{0})) = \{z_{1} = z_{3}^{2}, z_{2} = 0\}$. Since there is a generically unfolded quadratic tangency at $\tau = 0$ between $W^{s}(P_{0})$ and $W^{u}(P_{0})$, there is a transverse intersection between $\Pi$ and $\Gamma$. Moreover, $\mathbb{P}^{2}(\mathbb{C})$ is of dimension 2. We take a small disk $\Delta'$ going through $\Psi_{\tau}(R^{\tau})$ of direction $D\Psi_{\tau}( R^{\tau} ) \cdot e'_{1}$. For a given $n \ge N_{4}$, we call $\Gamma'_{n}$ the complex curve in $\mathbb{P}^{2}(\mathbb{C})$ given by the tangent directions to the curve $\Psi_{\tau}(F^{n}_{\tau}(\Delta'))$ at the periodic point $\Psi_{\tau}(R^{\tau})$ when $\tau$ varies. By the Inclination Lemma, we can increase the bound $N_{5} \ge N_{4}$ on $n$ such that if $n \ge N_{5}$, $\text{pr}_{3}(\Psi_{\tau}(R^{\tau}))$ can be made as close to $i\tau$ as wanted and $\Psi_{\tau}(F^{n}_{\tau}(\Delta'))$ can be taken as close to $\{z_{1} = z_{3}^{2}+t, z_{2} = 0\}$ in the $C^{1}$ topology. In particular, this shows that the graph $\Gamma'_{n}$ can be taken as close to the graph $\Gamma$ as wanted (in the $C^{1}$-topology) by increasing the bound $N_{5}$. In particular, we can pick $N_{5}$ such that there is a transverse intersection between $\Gamma'_{n}$ and $\Pi$ if $n \ge N_{5}$. There is a last step to get $A\Lambda_{1} = 0$. We denote by $\Pi'$ the projection of the plane $\mathrm{Vect}(D\Psi_{\tau}(R^{\tau}) \cdot e'_{2}, D\Psi_{\tau}(R^{\tau}) \cdot e'_{3})$ in $\mathbb{P}^{2}(\mathbb{C})$, $\Pi'$ is a complex curve. Since $e'_{2}$ and $e'_{3}$ are the two stable eigenvectors of $DF^{n_{1}}_{\tau} (R^{\tau}) $, it is an easy consequence of the Inclination Lemma that if $n$ and then $n_{1}$ tend to infinity, then $\Pi'$ tends to $\Pi$ (locally as graphs in the $C^{1}$-topology). In particular, it is possible to pick a last bound $N_{6} \ge N_{5}$ on $n$ such that if $n \ge N_{6}$, then $\Gamma'_{n}$ and $\Pi'$ have a transverse intersection. Then there exists $\tau$ with $t = \tau^{2} \in \mathcal{T}$ such that $DF^{n}_{\tau} (R^{\tau}) \cdot e'_{1} \in \mathrm{Vect}(e'_{2},e'_{3})$ which is equivalent to $A\Lambda_{1} = 0$. For this parameter $\tau$, we already saw that $R^{\tau}$ is a sink.
This is true for every neighborhood $\mathcal{T}$ of 0 and it is clear that $R^{\tau}$ tends to $Q$ when reducing $\mathcal{T}$. The result is proven: for every neighborhood $\mathcal{Q}$ of $Q$ and every neighborhood $\mathcal{T}$ of 0, we can create a sink in $\mathcal{Q}$ for $F_{t}$ with $t \in \mathcal{T}$. \end{proof}
\end{document} | arXiv |
Bernd Fischer (mathematician)
Bernd Fischer (18 December 1936 – 13 August 2020) was a German mathematician.
Bernd Fischer
Bernd Fischer at the Mathematical Research Institute of Oberwolfach, 2008
Born(1936-12-18)18 December 1936
Endbach, Hesse-Nassau
Died13 August 2020(2020-08-13) (aged 83)
Werther, North Rhine-Westphalia
NationalityGerman
Alma materGoethe University Frankfurt
Scientific career
FieldsMathematics
ThesisDistribute Quasigruppen endlicher Ordnung (1963)
Doctoral advisorReinhold Baer
He is best known for his contributions to the classification of finite simple groups, and discovered several of the sporadic groups. He introduced 3-transposition groups and constructed the three Fischer groups, predicted the existence of the baby monster and monster groups, and described and computed the character table of the baby monster.
He did his PhD in 1963 at the Johann Wolfgang Goethe University of Frankfurt am Main under the direction of Reinhold Baer.[1][2]
Career
Fischer went to Goethe University in Frankfurt to study mathematics under Baer in the early 60s, receiving his PhD in 1963. He later moved to the Bielefeld University, where he became head of mathematical sciences.
In 1970, he classified the almost-simple groups generated by 3-transpositions. In the process, he discovered three new sporadic groups, which were later called the Fischer groups. His proof that the classification was complete was known but not published in a single narrative until Aschbacher published an elementary introduction of 3-transposition groups in 1996.[3]
By loosening some of the conditions on his classification, in 1973 he predicted the existence of two larger sporadic simple groups: a {3,4}-transposition group, now known as the baby monster group, and a {3,4,5,6}-transposition group now known as the monster group or the Fischer-Griess Monster.[4] These would turn out to be the two largest sporadic groups that could exist. Robert Griess independently discovered the Monster group and published a description in 1976.[5]
Fischer went on to compute the character table for both monsters, in collaboration with Donald Livingstone and Michael Thorne.[4][6] Leon and Sims first produced a construction of the baby monster in 1977,[7] and Griess produced one for the monster in 1980.
Personal life
Fischer was raised in Bad Endbach. He later moved to North Rhine-Westphalia, where he died in August, 2020.[8]
Notes
1. The Mathematics Genealogy Project - Bernd Fischer
2. Ronan, Mark (2006). Symmetry and the Monster, One of the Greatest Quests of Mathematics. Oxford University Press. ISBN 0-19-280722-6.
3. Aschbacher, Michael (1996). 3-Transposition Groups. Cambridge Tracts in Mathematics. Cambridge: Cambridge University Press. ISBN 978-0-521-57196-8.
4. Griess, Robert (December 14, 2020). "My life and times with the sporadic simple groups" (PDF). Archived (PDF) from the original on 2021-01-26. Retrieved December 12, 2021.
5. Griess, Robert L. (1976). Scott, W. Richard; Gross, Fletcher (eds.). Proceedings of the Conference on Finite Groups (Univ. Utah, 1975). Boston, MA: Academic Press. pp. 113–118. ISBN 978-012633650-4. MR 0399248.
6. "Bernd Fischer". Hidden assumptions. 2020-08-26. Retrieved 2021-12-28.
7. Leon, Jeffrey S.; Sims, Charles C. (1977). "The existence and uniqueness of a simple group generated by {3,4}-transpositions". Bull. Amer. Math. Soc. 83 (5): 1039–1040. doi:10.1090/s0002-9904-1977-14369-3.
8. "The man behind the Monster | Mark Ronan". The Critic Magazine. 2020-08-24. Retrieved 2021-12-28.
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| Wikipedia |
\begin{document}
\title{A Detailed Examination of Methods for Unifying, Simplifying and Extending Several Results About Self-Justifying Logics}
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\author{ Dan E.Willard\thanks{This research was partially supported by the NSF Grant CCR 0956495. Email = [email protected].}}
\date{State University of New York at Albany}
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\baselineskip = 1.5\normalbaselineskip
\normalsize
\baselineskip = 1.0 \normalbaselineskip \def\normalsize \baselineskip = 1.25 \normalbaselineskip {\large \baselineskip = 1.6 \normalbaselineskip } \def\normalsize \baselineskip = 1.25 \normalbaselineskip {\large \baselineskip = 1.6 \normalbaselineskip } \def\normalsize \baselineskip = 1.25 \normalbaselineskip {\normalsize \baselineskip = 1.3 \normalbaselineskip }
\def\normalsize \baselineskip = 1.25 \normalbaselineskip {\normalsize \baselineskip = 1.27 \normalbaselineskip }
\def\normalsize \baselineskip = 1.25 \normalbaselineskip {\large \baselineskip = 2.0 \normalbaselineskip }
\def\normalsize \baselineskip = 1.25 \normalbaselineskip {\normalsize \baselineskip = 1.25 \normalbaselineskip } \def\normalsize \baselineskip = 1.24 \normalbaselineskip {\normalsize \baselineskip = 1.24 \normalbaselineskip }
\def\normalsize \baselineskip = 1.25 \normalbaselineskip {\large \baselineskip = 2.0 \normalbaselineskip }
\def \baselineskip = 1.28 \normalbaselineskip { } \def \baselineskip = 1.21 \normalbaselineskip { } \def { }
\def\normalsize \baselineskip = 1.25 \normalbaselineskip {\normalsize \baselineskip = 1.95 \normalbaselineskip }
\def \baselineskip = 1.28 \normalbaselineskip { } \def \baselineskip = 1.21 \normalbaselineskip { } \def { }
\def\normalsize \baselineskip = 1.25 \normalbaselineskip {\large \baselineskip = 2.3 \normalbaselineskip } \def \baselineskip = 1.28 \normalbaselineskip { } \def \baselineskip = 1.21 \normalbaselineskip { } \def { }
\def\normalsize \baselineskip = 1.25 \normalbaselineskip {\normalsize \baselineskip = 1.7 \normalbaselineskip }
\def\normalsize \baselineskip = 1.25 \normalbaselineskip {\large \baselineskip = 2.3 \normalbaselineskip } \def { \baselineskip = 1.18 \normalbaselineskip }
\def\normalsize \baselineskip = 1.25 \normalbaselineskip {\large \baselineskip = 2.0 \normalbaselineskip } \def \baselineskip = 1.28 \normalbaselineskip { } \def \baselineskip = 1.21 \normalbaselineskip { } \def { } \def \baselineskip = 1.3 \normalbaselineskip { }
\def\normalsize \baselineskip = 1.25 \normalbaselineskip {\normalsize \baselineskip = 1.25 \normalbaselineskip } \def\normalsize \baselineskip = 1.24 \normalbaselineskip {\normalsize \baselineskip = 1.24 \normalbaselineskip } \def \baselineskip = 1.3 \normalbaselineskip { \baselineskip = 1.3 \normalbaselineskip } \def \baselineskip = 1.28 \normalbaselineskip { \baselineskip = 1.28 \normalbaselineskip } \def \baselineskip = 1.21 \normalbaselineskip { \baselineskip = 1.21 \normalbaselineskip } \def { }
\normalsize \baselineskip = 1.25 \normalbaselineskip
\parskip 5 pt
\noindent
\begin{abstract}
\baselineskip = 1.5 \normalbaselineskip \large
This paper will develop a single framework for unifying, simplifying and extending our prior results about axiom systems that retain a partial knowledge of their own consistency, via an axiomatic declaration of self-consistency. Its perhaps single most surprising new result will be its exploration of a viable alternative to conventional reflection principles. \end{abstract}
\normalsize
\parskip 8pt
\baselineskip = 1.4 \normalbaselineskip
\setcounter{page}{0}
{\bf Keywords:} G\"{o}del's Second Incompleteness Theorem, Consistency, Hilbert's Second Open Question, Semantic Tableaux
{\bf Mathematics Subject Classification:} 03B52; 03F25; 03F45; 03H13
\setcounter{page}{1}
\def\normalsize \normalsize \baselineskip = 1.12\normalbaselineskip \parskip 4 pt{\normalsize \baselineskip = 1.21\normalbaselineskip \parskip 4 pt} \def\normalsize \normalsize \baselineskip = 1.12\normalbaselineskip \parskip 5 pt{\normalsize \baselineskip = 1.19\normalbaselineskip \parskip 4 pt} \def\normalsize \baselineskip = 1.19 \normalbaselineskip \parskip 3 pt{\normalsize \baselineskip = 1.19 \normalbaselineskip \parskip 3 pt} \def\normalsize \baselineskip = 1.17\normalbaselineskip \parskip 4 pt{\normalsize \baselineskip = 1.17\normalbaselineskip \parskip 4 pt} \def\normalsize \baselineskip = 1.16 \normalbaselineskip \parskip 3 pt{\normalsize \baselineskip = 1.16 \normalbaselineskip \parskip 3 pt} \def\normalsize \normalsize \baselineskip = 1.12\normalbaselineskip \parskip 3 pt{\normalsize \baselineskip = 1.17 \normalbaselineskip \parskip 3 pt} \def\normalsize \normalsize \baselineskip = 1.12\normalbaselineskip \parskip 6 pt{\normalsize \baselineskip = 1.15 \normalbaselineskip \parskip 3 pt} \def \baselineskip = 1.1 \normalbaselineskip { \baselineskip = 1.1 \normalbaselineskip } \def \baselineskip = 1.08 \normalbaselineskip { \baselineskip = 1.08 \normalbaselineskip } \def \baselineskip = 1.0 \normalbaselineskip { \baselineskip = 1.0 \normalbaselineskip } \def \baselineskip = 1.0 \normalbaselineskip { \baselineskip = 1.0 \normalbaselineskip } \def \baselineskip = 1.0 \normalbaselineskip { \baselineskip = 0.9 \normalbaselineskip }
\vspace*{- 1.0 em}
\def\normalsize \baselineskip = 1.12\normalbaselineskip{\normalsize \baselineskip = 1.72\normalbaselineskip} \def\normalsize \baselineskip = 1.12\normalbaselineskip{\normalsize \baselineskip = 1.12\normalbaselineskip} \def\normalsize \baselineskip = 1.12\normalbaselineskip{\normalsize \baselineskip = 1.85\normalbaselineskip}
\def\normalsize \baselineskip = 1.12\normalbaselineskip{\normalsize \baselineskip = 1.45\normalbaselineskip}
\def\normalsize \baselineskip = 1.12\normalbaselineskip{\normalsize \baselineskip = 1.7\normalbaselineskip}
\def\normalsize \baselineskip = 1.12\normalbaselineskip{\normalsize \baselineskip = 1.12\normalbaselineskip}
\def \baselineskip = 1.1 \normalbaselineskip { \baselineskip = 1.59 \normalbaselineskip } \def \baselineskip = 1.0 \normalbaselineskip { \baselineskip = 1.50 \normalbaselineskip } \def \baselineskip = 1.0 \normalbaselineskip { \baselineskip = 1.50 \normalbaselineskip } \def \baselineskip = 1.0 \normalbaselineskip { \baselineskip = 1.5 \normalbaselineskip }
\def \baselineskip = 1.1 \normalbaselineskip { \baselineskip = 1.59 \normalbaselineskip } \def \baselineskip = 1.0 \normalbaselineskip { \baselineskip = 1.50 \normalbaselineskip } \def \baselineskip = 1.0 \normalbaselineskip { \baselineskip = 1.50 \normalbaselineskip } \def \baselineskip = 1.0 \normalbaselineskip { \baselineskip = 0.9 \normalbaselineskip }
\def\normalsize \normalsize \baselineskip = 1.12\normalbaselineskip \parskip 6 pt{\normalsize \normalsize \baselineskip = 1.12\normalbaselineskip \parskip 6 pt} \def\normalsize \normalsize \baselineskip = 1.12\normalbaselineskip \parskip 5 pt{\normalsize \normalsize \baselineskip = 1.12\normalbaselineskip \parskip 5 pt} \def\normalsize \normalsize \baselineskip = 1.12\normalbaselineskip \parskip 4 pt{\normalsize \normalsize \baselineskip = 1.12\normalbaselineskip \parskip 4 pt} \def\normalsize \normalsize \baselineskip = 1.12\normalbaselineskip \parskip 3 pt{\normalsize \normalsize \baselineskip = 1.12\normalbaselineskip \parskip 3 pt} \def\normalsize \waw11 \parskip 2 pt{\normalsize \normalsize \baselineskip = 1.12\normalbaselineskip \parskip 2 pt}
\def \baselineskip = 1.1 \normalbaselineskip { \baselineskip = 1.1 \normalbaselineskip } \def \baselineskip = 1.0 \normalbaselineskip { \baselineskip = 1.0 \normalbaselineskip } \def \baselineskip = 1.1 \normalbaselineskip { \baselineskip = 1.0 \normalbaselineskip } \def \baselineskip = 1.0 \normalbaselineskip { \baselineskip = 1.0 \normalbaselineskip } \def \baselineskip = 1.0 \normalbaselineskip { \baselineskip = 0.9 \normalbaselineskip }
\def \bf { \bf } \def\mathcal{\mathcal} \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 0.98 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 0.99 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.0 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.0 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.03 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.00 \normalbaselineskip }
\def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.6 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.6 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.6 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.6 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.6 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.6 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.6 \normalbaselineskip } \def \baselineskip = 1.4 \normalbaselineskip { \baselineskip = 1.6 \normalbaselineskip } \def \baselineskip = 1.35 \normalbaselineskip { \baselineskip = 1.6 \normalbaselineskip } \def \baselineskip = 1.4 \normalbaselineskip \parskip 5pt { \baselineskip = 1.6 \normalbaselineskip \parskip 5pt } \def \baselineskip = 1.22 \normalbaselineskip \parskip 3pt { \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
\def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.22 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.22 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.22 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.22 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.22 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.22 \normalbaselineskip } \def \baselineskip = 1.22 \normalbaselineskip { \baselineskip = 1.22 \normalbaselineskip } \def \baselineskip = 1.4 \normalbaselineskip { \baselineskip = 1.4 \normalbaselineskip } \def \baselineskip = 1.35 \normalbaselineskip { \baselineskip = 1.35 \normalbaselineskip } \def \baselineskip = 1.4 \normalbaselineskip \parskip 5pt { \baselineskip = 1.4 \normalbaselineskip \parskip 5pt } \def \baselineskip = 1.22 \normalbaselineskip \parskip 3pt { \baselineskip = 1.22 \normalbaselineskip \parskip 3pt }
\def\fend{
-------------------------------------------------------}
\def \baselineskip = 1.1 \normalbaselineskip { \baselineskip = 1.1 \normalbaselineskip } \def \baselineskip = 1.0 \normalbaselineskip { \baselineskip = 1.0 \normalbaselineskip } \def \baselineskip = 1.0 \normalbaselineskip { \baselineskip = 1.0 \normalbaselineskip } \def \baselineskip = 1.0 \normalbaselineskip { \baselineskip = 1.0 \normalbaselineskip } \def \baselineskip = 1.08 \normalbaselineskip { \baselineskip = 1.08 \normalbaselineskip } \def \baselineskip = 1.1 \normalbaselineskip { \baselineskip = 1.1 \normalbaselineskip }
\def
{ } \def
{
}
\baselineskip = 1.22 \normalbaselineskip \baselineskip = 1.4 \normalbaselineskip
\def
{
}
\baselineskip = 1.4 \normalbaselineskip \baselineskip = 1.22 \normalbaselineskip
\parskip 3pt
\def
{
}
\def
{ } \def {
} \def { } \def
{
}
\baselineskip = 1.22 \normalbaselineskip \baselineskip = 1.4 \normalbaselineskip
\def
{
}
\baselineskip = 1.22 \normalbaselineskip \baselineskip = 1.4 \normalbaselineskip
\section{Introduction}
\label{secc1} \label{B1-lem} \label{D1-def}
\parskip 2pt
Let $~\alpha~$ denote an axiom system, and $~d~$ denote
a deduction method. The ordered pair
$~( \alpha , d )$ will be called {\bf Self Justifying} when: \begin{description}
\item[ i ] one of $ \, \alpha \,$'s theorems states that the deduction method $ \, d, \, $ applied to the system $ \, \alpha, \, $ will produce a consistent set of theorems, and \item[ ii ]
the axiom system $ \, \alpha \, $ is in fact consistent. \end{description} For any $\,(\alpha,d) \,$, it is easy to construct a second axiom system $ \, \alpha^d \, \supseteq \, \alpha \, $
that satisfies Part-i of this definition. For instance, $ \, \alpha^d \, $ could consist of all of $~\alpha \,$'s axioms plus the following added sentence, that we call {\bf SelfRef$(\alpha,d)~$}: \topsep -3pt \begin{quote} $\bullet~~~$ There is no proof (using $d$'s deduction method) of $0=1$ from the {\it union} of the
axiom system $\, \alpha \, $ with {\it this} sentence ``SelfRef$(\alpha,d) \,$'' (looking at itself). \end{quote} Kleene \cite{Kl38} discussed how to encode approximate
analogs of SelfRef$(\alpha,d)$'s
self-referential statement. Each of Kleene, Rogers and Jeroslow
\cite{Kl38,Ro67,Je71}
noted $\alpha ^d$ may, however, be inconsistent (despite SelfRef$(\alpha,d)$'s assertion), thus causing it to violate Part-ii of self-justification's definition.
This problem arises in settings more general than
G\"{o}del's paradigm, where $\alpha$ was an extension of Peano Arithmetic. There are many settings where the Second Incompleteness Theorem does generalize \cite{Ad2,AB1,AZ1,BS76,Bu86,BI95,Fe60,Go31,HP91,HB39,Ko6,KT74,Lo55,PD83,PW81,Pu84,Pu85,Pu96,Ro67,Sa11,Sm85,So94,Sv7,Ta0,VV94,Vi92,Vi5,WP87,ww2,wwapal,wwlogos,ww7}. Each such result formalizes a paradigm where self-justification is infeasible, due to a diagonalization issue. Most logicians have hesitated to thus
employ a SelfRef$(\alpha,d)$
axiom because $\alpha+$SelfRef$(\alpha,d) $ is usually
inconsistent \footnote{ \baselineskip = 1.3 \normalbaselineskip \label{troub}
Typical ordered pairs $(\alpha,d)$
will have the property that
the broader axiom system
$~\alpha^d~=~\alpha \,+ \,$SelfRef$(\alpha,d)$ will
be inconsistent, even
when $~\alpha~$ is consistent. This is because
a
standard
G\"{o}del-like self-referencing
construction
will
typically
produce a proof of $0=1$ from
$~\alpha^d\,$, irregardless of whether or not $~\alpha$ is
consistent.}.
Our research explored special circumstances \cite{ww1,ww5,ww6,wwapal} where it is feasible to construct self-justifying formalisms. These paradigms involved weakening the properties a system can prove about addition and/or multiplication (to avoid the preceding difficulties). To be more precise, let
$Add(x,y,z)$ and $Mult(x,y,z)$ denote two 3-way predicates
indicating $x$, $y$ and $z$ satisfy $x+y=z$ and $x*y=z$. A logic will be said to {\bf recognize} successor,
addition and multiplication as {\bf Total Functions} iff it includes 1-3 as axioms.
\vspace*{- 0.8 em} {\ \baselineskip = 1.22 \normalbaselineskip \begin{equation} \label{totdefxs} \forall x ~ \exists z ~~~Add(x,1,z)~~ \end{equation} \vspace*{- 1.7 em} \begin{equation} \label{totdefxa} \forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~ \end{equation} \vspace*{- 1.7 em} \begin{equation} \label{totdefxm} \forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~ \end{equation} }
\vspace*{- 0.6 em}
\baselineskip = 1.22 \normalbaselineskip
We will say a logic system $\alpha$ is {\bf Type-M} iff it contains each of \eq{totdefxs} -- \eq{totdefxm} as axioms, {\bf Type-A} iff it contains only \eq{totdefxs} and \eq{totdefxa} as axioms, and {\bf Type-S} iff it contains only \eq{totdefxs} as an
axiom. A system is called {\bf Type-NS} iff it {\it does not} contain any of these axioms.
Our investigations
\cite{ww1}--\cite{ww7} began by observing some Type-A systems can recognize their consistency under semantic tableaux deduction, and several Type-NS systems can recognize their
Hilbert consistency. Many of these systems were capable of proving
Peano Arithmetic's
$\Pi_1$ theorems in a language that represents addition and multiplication as the 3-way predicates of
Add$(x,y,z)$ and Mult$(x,y,z)$.
\parskip 1pt
Our self-justifying
evasions of the Incompleteness Theorem are difficult to further extend primarily because the combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris \cite{Ne86,Pu85,So94,WP87} showed natural Type-S systems cannot recognize their own Hilbert consistency. Also, Willard
\cite{ww2,ww7,ww9}
strengthened earlier results of Adamowicz-Zbierski \cite{Ad2,AZ1} to establish that natural Type-M system cannot recognize their semantic tableaux consistency.
\baselineskip = 1.22 \normalbaselineskip
A related
class of evasions of the
Second Incompleteness Theorem was discovered in \cite{ww9}. Let us say
$~\alpha~$ is a {\bf Type-Almost-M} axiom system iff $~\alpha~$ can prove statements \eq{totdefsymba} and \eq{totdefsymbm} as theorems while treating {\it none of sentences
\eq{totdefxs} --
\eq{totdefsymbm} as axioms.} (Many axiom systems, that use
function symbols ``$~+~$'' and ``$~*~$'' for formalizing addition and multiplication, fall technically into the
Type-Almost-M
category.) \vspace*{- .2 em} \begin{equation} \label{totdefsymba} \forall x ~\forall y~ \exists z ~~~~x+y=z \end{equation} \vspace*{- 1.5 em} \begin{equation} \label{totdefsymbm} \forall x ~\forall y ~\exists z
~~~~x*y=z \end{equation} \vspace*{-1.4 em}
\noindent The preceding is of interest because some surprisingly strong (albeit unusual) Type-Almost-M systems \cite{ww9}
have an ability to verify their Herbrand but not also semantic tableaux consistency.
The proofs in our prior papers were challenging primarily
because they required one to separate the local combinatorial methods employed in \cite{ww93,ww5,wwapal,ww9}'s
particular applications from the common principles that underlied behind all these works. Our Theorems \ref{ppp1}, \ref{ppp2} and \ref{pqq4} will rectify this problem by identifying common components that unite these four paradigms. (Theorems
\ref{pqq3}, \ref{pqq5},
\ref{ppp6}, E.1, G.2 and G.3 will then carry on in further directions.)
All these theorems will contain severe limits on their generality, so that the Second Incompleteness Theorem does not contradict them. It is clearly perplexing to imagine how humans are able to motivate themselves to cogitate, without
their thought processes possessing some type of {\it at least tentative} presumption of their own consistency.
Our research has thus consisted of an approximately equal effort in exploring both \cite{ww2,wwlogos,wwapal,ww7}'s new
generalizations of the Second Incompleteness Theorem and
\cite{ww93,ww1,ww5,ww6,wwapal,ww9}'s unusual boundary-case exceptions to it.
It is clear every boundary-case exception to the Second Incompleteness Theorem has limited scope because the Incompleteness Theorem is a broadly encompassing result. This
paper
will, thus, be addressing a challenging near-paradoxical
question about the maximal nature of self-justification that
can
never be resolved in a fully satisfying manner. The Second
Incompleteness Theorem is clearly sufficiently central to logic for it to be desirable to know what {\it partial roads of success} a self-justifying axiom system can obtain.
\section{Literature Survey}
\label{survey} \label{B2-lem} \label{D2-def}
Two 5-page surveys of the prior literature about the Second Incompleteness Theorem were provided in our articles \cite{ww5,wwapal}. This section will present a more abbreviated survey, focusing on only those developments that are particularly germane to
the current article.
The study of incompleteness began with four classic papers by G\"{o}del, L\"{o}b, Rosser and Tarski \cite{Go31,Lo55,Ro36,Ta36} and with the Hilbert-Bernays
exploration of their derivability conditions \cite{HP91,HB39,Ka91}. Generalizations of these results for weak axiom systems, such as Q, began with the work of Tarski-Mostowski-Robinson \cite{TMR53} and Bezboruah-Shepherdson \cite{BS76}.
Some more notation is needed to describe more recent developments. Let $~x'~$ denote the ``successor'' operation that maps $x$ onto $x+1$. A formula
$ \varphi(x) $ is called \cite{HP91} a {\bf Definable Cut} for an axiom system $~ \alpha~$ iff $~\alpha~$ can prove: \begin{equation} \label{initdefx} \varphi(0) \mbox{ AND } \forall~x~ \{~\varphi(x)\Rightarrow\varphi(~x'~)~ \}
\mbox{ AND }
\forall~x ~\forall~y<x
~~ \{ ~\varphi(x)\Rightarrow\varphi(y) ~ \} \end{equation} Definable cuts and their cousins have been studied by an extensive literature \cite{Ad2,AZ1,Be97,Bu86,BI95,HP91,Ka91,Ko6,Kr87,Ne86,Pa71,PD83,PW81,Pa72,Pu84,Pu85,Pu96,Sm85,Sv83,Sv7,VV94,Vi92,Vi93,Vi5,VH73,WP87}. (They are
unrelated to Gentzen's notion of
sequent calculus deductive ``cut rule'', which uses the word ``cut'' in a different context).
A Definable Cut $~\varphi(x)~$ is called {\bf Non-Trivial} relative to an axiom system $~\alpha~$ iff $\alpha~$ cannot prove $~\forall ~x~\varphi(x)~,~$
{\it although it can prove \eq{initdefx}.} Every axiom system $\,\alpha$, strictly weaker than Peano Arithmetic, will contain some non-trivial Definable Cut. This cut will have the property that $\,\alpha\,$
can verify $~\varphi(n)$ for each fixed integer $~n$, although it cannot prove
$~\forall ~x~\varphi(x)~$.
Let
$ \, \lceil \, \Psi \, \rceil \, $ denote $ \Psi $'s G\"{o}del number, and Prf$\, _\alpha^d \, \,(t,p)$ denote that $ p $ is a proof of the theorem $ t $ from the axiom system $ \alpha $ using $\, d \,$'s deduction method. An axiom system $ \alpha $ will then
be said to recognize its own {\bf Cut-Localized d-Consistency} relative to a Definable Cut
$ \, \varphi \, $ iff $ \alpha $ can prove: \begin{equation} \label{dcon} \forall~p~~~~~~ \{ ~~~\varphi(p)~~\Rightarrow~~ \neg~\mbox{Prf}\,_\alpha^d \, ~(~\lceil 0=1 \rceil~,~p~)~~~ \} \end{equation} The recent literature has sought to identify which triples $ ( \varphi , d , \alpha ) $ have this property. A crucial negative
result about
Cut-Localized d-Consistency, discovered by
Pudl\'{a}k \cite{Pu85}, established a significant generalization of G\"{o}del's Second Incompleteness Theorem. It
showed that an axiom system $ \alpha $ must be unable to prove \eq{dcon}'s statement about any of its definable cuts
$\,\varphi \,$, $\,$when $ d $ represents
Hilbert deduction and $ \alpha $ is any consistent extension of $ Q $.
Solovay \cite{So94} noted how Pudl\'{a}k's result could be combined
with the techniques of Nelson and Wilkie-Paris \cite{Ne86,WP87} to establish the following theorem
that will often be cited in this paper: \begin{quote}
\baselineskip = 1.2 \normalbaselineskip
\small \baselineskip = 1.25 \normalbaselineskip \baselineskip = 1.25 \normalbaselineskip {\normalsize \bf Theorem 2.1 } {\it (Solovay's 1994 Generalization \cite{So94} of a 1985 theorem of Pudl\'{a}k \cite{Pu85} using some of Nelson and Wilkie-Paris \cite{Ne86,WP87}'s methods )} : Let $ \, \alpha \, $ denote any axiom system which contains \ep{totdefxs}'s Type-S statement and which assures
the successor operation always satisfies
$ \, x' \neq 0 $ and $ x' = y' \Leftrightarrow x=y $. $~$Then $~\alpha~$ will be unable to recognize its own Hilbert consistency, whenever it
treats addition and multiplication as 3-way relations satisfying their usual identity, associative, commutative and distributive properties. \end{quote} Solovay never published any precise proof of Theorem 2.1's hybridizing of the work of Pudl\'{a}k, Nelson and Wilkie-Paris \cite{Pu85,Ne86,WP87}, which he privately communicated
\cite{So94} to us. A reader can find generalizations of the Second Incompleteness Theorem that are closely related to Theorem 2.1 in papers by Pudl\'{a}k, Buss-Ignjatovic, \v{S}vejdar and Willard \cite{BI95,Pu85,Sv7,wwlogos}, as well as in Appendix A of \cite{ww1}.
Other interesting observations that preceded our research were that Wilkie-Paris \cite{WP87} demonstrated that I$\Sigma_0+Exp$ cannot prove the Hilbert consistency of even the axiom system $~Q, \,~$ and that Adamowicz-Zbierski \cite{Ad2,AZ1} showed that I$\Sigma_0+\Omega_1$ satisfied the Herbrandized version of the Second Incompleteness Theorem. Both these results helped stimulate \cite{ww2,ww7}'s
semantic tableaux generalizations of the Second Incompleteness Theorem for I$\Sigma_0$.
A fascinating observation by L. A. Ko{\l}odziejczyk \cite{Ko5,Ko6}, about the difference in lengths between semantic tableaux and Herbrandized proofs, also motivated our investigation \cite{ww9} into some surprising properties of unorthodox encodings for I$\Sigma_0$.
Ko{\l}odziejczyk observed \cite{Ko5,Ko6} that various generalizations of the Second Incompleteness Theorem for I$\Sigma_0$ and I$\Sigma_0+\Omega_i$ in \cite{Ad2,AZ1,Sa11,ww2,ww7,ww9} imply that the proof of
the Herbrandized version of the Second Incompleteness Theorem can be more complicated than
its semantic tableaux counterpart. This is because there can be an exponential difference between semantic tableaux and Herbrandized proof lengths under extremal circumstances. It was due to Ko{\l}odziejczyk's insightful
communications \footnote{ \baselineskip = 1.3 \normalbaselineskip \baselineskip = 1.1 \normalbaselineskip
\label{putglue} The Herbrandized and semantic tableaux definitions of an axiom
system $\alpha$'s consistency
are different from each other because the former requires skolemizing $\,\alpha \,$'s axioms, $\,$while the latter permits \cite{Fi90} an existential quantifier elimination rule to replace Skolemization. This distinction can create a
potential exponential difference
between the lengths of Herbrand and Semantic Tableaux proofs. This insightful observation,
due to private communications from L. A. Ko{\l}odziejczyk \cite{Ko5}, was used in
\cite{ww9} to create an axiom system that satisfied the semantic tableaux but not also Herbrandized version of the Second Incompleteness Theorem.} that \cite{ww9} developed an axiom system that was a boundary-case exception to the Herbrandized but not also the semantic tableaux version of the Second Incompleteness Theorem.
The literature on Definable Cuts has centered its evasions of the Second Incompleteness Theorem around \ep{dcon}'s localization formalism (rather than employing analogs of
Section \ref{secc1}'s SelfRef$(\alpha,d)$ axiom, as we did in \cite{ww93,ww1,ww5,ww6,wwapal,ww9}). Pudl\'{a}k \cite{Pu85} proved that essentially every axiom system $ ~ \alpha ~ $
of finite cardinality can be associated with a definable cut $ ~ \varphi ~ $ such that $ ~ \alpha ~ $ can prove sentence \eq{dcon}'s validity for $ ~ \varphi ~ $ when $ ~ d ~ $ is either the semantic tableaux or Herbrand-styled deductive method.
Pudl\`{a}k's theorem is related to Friedman's observation \cite{Fr79b} that for many finite theories $S$ and $T$, the theory $ S $ has an interpretation in $ T $ if and only if I$\Sigma_0+Exp$ can prove that $T$'s Herbrand consistency implies $S$'s Herbrand consistency. Several generalizations of these results by
Kraj\'{i}cek, Pudl\`{a}k,
Smory\'{n}ski, \v{S}vejdar
and Visser appear in \cite{Kr87,Pu85,Pu96,Sm85,Sv7,Vi92,Vi93,Vi5}. Visser's article \cite{Vi5} contains an excellent review of this literature, as well as many additional new results. Also, we will see how
some of the reflection machines of Beklemishev, Kreisel-Takeuti and
Verbrugge-Visser \cite{Be95,Be97,Be3,KT74,VV94,Vi5} nicely complement \thx{ppp6}'s
reflection mechanisms in alternate types of intended applications.
It was established by H\'{a}jek, \v{S}vejdar and Vop\v{e}nka \cite{Sv83,VH73} that GB Set Theory can construct a definable cut $\varphi$ where it can prove the statement \eq{dcon} is valid when $~d~$ denotes Hilbert deduction and $~\alpha~$ is ZF Set Theory. This result was
surprising because Pudl\'{a}k \cite{Pu85} showed GB can never verify its own Hilbert consistency localized on a definable cut. (Thus, GB will view its Hilbert consistency as equivalent to ZF's Hilbert consistency in a global sense
{\it but not} in a cut-localized respect.).
In some sense,
Kreisel and Takeuti
\cite{KT74,Ta53} can be viewed as the first authors to develop a
logic recognizing its own
consistency using a variant of \ep{initdefx}'s formula. Their results for typed logics
formalized a second-order generalization of Gentzen's sequent calculus that can verify
its own consistency,
when no sequent calculus deductive cuts are performed. A key aspect of their formalism can be seen as using an analog of \eq{dcon}'s sentence in an implicit manner. It thus begins by using a set of objects, which we shall call $~I,~$ that includes all the standard integers plus some allowed
{\it non-standard integers} (that can permissibly represent contradictory proofs).
Their second-order logic
then uses Dedekind's definition of the natural numbers to construct a subset of $~I~$, $\,$called
``$~N~$'', which includes all the standard integers and which is disallowed to contain any contradiction proof. We will not go into the details here, but this transition from $\,I\,$ to $\,N\,$ (with an accompanying relativization of the provability predicate onto $\,N\,$'s more restricted domain) can be viewed as Kreisel-Takeuti's analog
of \eq{dcon}'s
local consistency statement for \cite{KT74,Ta53}'s ``CFA''
second-order logic.
It is difficult to compare our research (which has relied upon an analog of SelfRef$(\alpha,d)$'s Kleene-like {\it ``I am consistent''} axiom) with the preceding literature that has used various forms of Localized d-Consistency statements. This is because every effort to evade the Second Incompleteness Theorem employs some built-in weakness to evade G\"{o}del's classic paradigm.
Our work in \cite{ww93,ww1,ww5,ww6,ww9} represented less than a full-scale evasion of the Second Incompleteness Theorem mostly because it was
incompatible with treating {\it as formal axioms} the statements in Equations \eq{totdefxm} and \eq{totdefsymbm} that
multiplication is a total function \footnote{ \baselineskip = 1.1 \normalbaselineskip This caveat applies also to our article \cite{ww9}, although its Herbandized form of self-justification differs from our other papers by retaining a capacity to treat \eq{totdefsymbm}'s statement about multiplication's totality as a derived theorem {\it that is not an}
axiom. The key point is that theorems are
weaker than axioms under Herbrand deduction
because only axioms are used as intermediate steps during proofs. This explains intuitively how \cite{ww9}'s formalism was able to recognize its
Herbrandized consistency, $\,$while treating \eq{totdefsymbm}'s statement about the totality of
multiplication
{\it as a theorem}. (We will return to this subject in Appendix D.) } .
Some reasons why it was helpful for \cite{ww93,ww1,ww5,ww6,ww9} to employ
analogs of Section \ref{secc1}'s SelfRef axiom are that: \begin{description} \topsep -3pt \itemsep +2pt \item[A ] An axiom's self-referential {\it ``I am consistent''} declaration allows a formalism to recognize its consistency in a global sense, rather than in the $\varphi -$localized sense used by sentence \eq{dcon} and the analogous
Kreisel-Takeuti relativization of their second-order proof predicate. \item[B ] If a logic is employing a deductive method $d$ that lacks a modus ponens rule, as occurs in nearly all self-justifying systems, then it is
preferable for it to view its {\it ``I am consistent''} statement as an axiom rather than as a theorem. (This is
because weak deductive methods are
capable of drawing logical inferences only from axioms when modus ponens is absent.) \item[C ] Analogs of Section \ref{secc1}'s SelfRef$(\alpha,d)$'s {\it ``I am consistent''} axiom have been shown by \cite{ww93,ww1,ww5,ww6,wwapal,ww9} to at least partially formalize the notion of a logic possessing {\it an
almost} instinctive form of faith in its own internal consistency. (This paper will make it apparent that such an instinctive faith is less than a full-scale proof. Yet, Theorem \ref{ppp6} and
Remarks \ref{f88}, \ref{remhappy} and \ref{recc1} will make it apparent that such formalizations of instinctive faith are also useful.) \end{description} We emphasize that both virtues and drawbacks of SelfRef$(\alpha,d)$'s {\it ``I am consistent''} axiom statements have been cited in this paragraph because every effort to evade the Second Incompleteness Theorem can obtain no more than limited levels of success.
\baselineskip = 1.35 \normalbaselineskip
The scope of the challenge we face becomes apparent when one realizes
$\alpha \, + \,$SelfRef$(\alpha,d)$ is
inconsistent for most $(\alpha,d).~$
This is because
$\alpha+$SelfRef$(\alpha,d)$ typically satisfies Part-i
{\it but not also }
Part-ii of Section \ref{secc1}'s definition
of a ``self-justifying'' logic. (Thus, a
diagonalization paradigm will typically imply $\alpha \, + \,$SelfRef$(\alpha,d)$ is inconsistent, $\,$as a consequence of it containing $ ~$SelfRef$(\alpha,d)~$ as an axiom.) This is the reason Kleene,
Rogers and Jeroslow \cite{Kl38,Ro67,Je71} were hesitant
about the utility of SelfRef$(\alpha,d)$'s mirror-like axiom sentence. Our goal in \cite{ww93}-\cite{ww9} has been
to develop generalizations and boundary-case exceptions for the Second Incompleteness Theorem, so as determine exactly which paradigms
can support, for example, Theorem \ref{ppp6}'s limited notion of self-justification.
The
reason one would anticipate some {\it limited}
exceptions to the Second Incompleteness Theorem to exist is it is hard to imagine how humans can motivate themselves to cogitate without using some variant of self-justification.
\baselineskip = 1.22 \normalbaselineskip
\section{Generic Configurations}
\label{3uuuu1}
The
phrase {\bf Bounded Quantifier} will refer to expressions of the form ``$\, \exists~v \leq T \,$'' or ``$ \,\forall~v \leq T\,$'' where $T$ is a term. A formula is called {\bf Fully-Bounded$\,$} when all its quantifiers are
so bounded. \lem{lex22} will soon explain how Definition \ref{xd+1x1}'s formalism can encode conventional arithmetic:
\begin{definition} \label{xd+1x1} \rm Let $~\xi~$ denote some non-integer indexing superscript (whose properties will be discussed
later by Definition \ref{def3.3}). Then the symbol $~\Delta_0^\xi~$ will denote some fixed special set of fully-bounded formulae that is closed under negation, in a language that will be later
called $~L^\xi~$. (Thus, if some formula $~ \Psi~$ is a member of
$~\Delta_0^\xi~$ then so is $~ \neg ~ \Psi~$.) Items 1-3 formalize how $~\Pi_n^\xi~$ and $~\Sigma_n^\xi$ formulae are built in a straightforward manner out of these
$~\Delta_0^\xi~$ sub-components: \begin{enumerate} \topsep -15pt \item Every $ \, \Delta_0^\xi \, $ formula is considered to be also a $ \, \Pi_0^\xi \, $ and $ \, \Sigma_0^\xi$ formula. \item For $~n \, \geq \, 1~~,~$ a formula will be called $ ~~ \Pi_n^\xi~~$ iff it can be written in the canonical form of $\forall v_1 ~ \forall v_2 ~...~ \forall v_k ~~~ \Phi(v_1,v_2,..v_k \,),~~~$ where $\Phi$ is $~\Sigma_{n-1}^\xi.~$ \item Likewise for $~n \, \geq \, 1~~,~$
a formula will be called $~~ \Sigma_n^\xi~~ $ iff it can be written in the form of $~\exists v_1 ~ \exists v_2 ~...~ \exists v_k ~~~ \Phi(v_1,v_2,..v_k \,),~~~$
where $\Phi$ is $~\Pi_{n-1}^\xi.~$ \end{enumerate} \end{definition}
\noindent {\bf Notation Convention:} Our rules for defining $\,\xi$, specified later in this section, will never have this superscript designate an integer quantity. This is because integer superscripts have a special meaning under a typed-based hierarchy, not intended here.
\begin{example} \label{ex3-1} \rm Let $~L~$ denote a conventional arithmetic language that uses function symbols for denoting addition and multiplication. Below are two examples of $\Delta_0-$like formulae that invoke Definition \ref{xd+1x1}'s notation: \begin{description} \itemsep 1pt \item[ a ] The symbol ``$~\Delta_0^A~$'' will denote any fully bounded formula that uses the addition, multiplication and maximum function symbols in an arbitrary manner. (Thus, $~\Delta_0^A~$ corresponds to what many textbooks \cite{HP91,Ka91,Kr95} simply call a ``$~\Delta_0~$'' formula.)
\item[ b ] The symbol ``$~\Delta_0^{R}~$'' will denote a class of formulae in $L$'s language whose bounded quantifiers are allowed to use only
the Maximum function symbol. Their bodies, however, may contain any combination of addition, multiplication and maximum function symbols. \end{description} Formulae \eq{ex1} and \eq{ex123} illustrate the distinction between the $\Delta_0^A$ and $\Delta_0^{R}$ classes. Thus, \eq{ex1} satisfies the first but not second condition (on account of the presence of the multiplication symbol used by its bounded quantifiers).
In contrast, \eq{ex123} is an example of a
$~\Delta_0^{R}~$ formula. \begin{equation} \label{ex1} \exists \, y\leq x*x ~~~~ \forall \, z\leq y*y ~~~~ \exists \, w\leq y*z ~~~~ ~~~~~: ~~~~~
\{ ~~~~ x *y=z+w ~~~~ \} \end{equation} \vspace*{-0.5 em} \begin{equation} \label{ex123} \exists \, y\leq x ~~~~ \forall \, z\leq y ~~~~ \exists \, w\leq \mbox{Max}(y,z) ~~~~ ~~~~~: ~~~~~
\{ ~~~~ x *y=z+w ~~~~ \}
\end{equation} The distinction between
$ \Delta_0^{A} $ arithmetic formulae and the unconventional
$ \Delta_0^{R} $ class may first convey the impression that these two classes have fundamentally different natures. Actually, \lem{lex22} will show that their relationship is more subtle. This is because its formalism will map $ \Delta_0^{A} $ formulae onto $ \Delta_0^{R} $ expressions that are equivalent to it under Definition \ref{snn}'s Standard-M model $~$---$~$ in a context where only the length of these formulae is allowed to possibly grow. This
equivalence
enabled
\cite{ww9} to construct a natural axiomatic formalism that could recognize its own Herbrandized consistency {\it but which nevertheless satisfied} the idealized form \footnote{ \baselineskip = 1.1 \normalbaselineskip An axiom system $\alpha$ is defined to
satisfy the {\bf ``idealized form''} of the semantic tableaux version
Second Incompleteness Theorem when no $\beta \supseteq \alpha$ can prove a
semantic tableaux proof
of 0=1
from itself is incapable of existing. We will summarize \cite{ww9}'s formalism and the distinction between Herbrandized and semantic tableaux deduction at the of Appendix D.} of the semantic tableaux version of the Second Incompleteness Theorem. \end{example}
\begin{definition} \label{snn} {\bf ``Standard-M''} will denote the standard model of integers. \end{definition}
The reason for our interest in Standard-M is that many pairs of formulae are equivalent under the Standard-M model, while weak axiom systems often cannot formally prove they are equivalent. For instance, this will occur when Example \ref{ex3-2} examines Definition \ref{def3.3}'s properties.
\begin{definition} \label{def3.3} A {\bf Generic Configuration}, often identified by
the superscript symbol of
$~\xi~$, is defined to be a 5-tuple $~(\, L^\xi \, , \, \Delta_0^\xi \, , \, B^\xi \, , \, d \, , \, G \, )~$ where: \begin{enumerate} \topsep -12pt \rm \item $ L^\xi $ is a language that includes logical symbols for ``$0$'', ``$1$'', ``$2$'', ``$=$'', ``$\leq$''
and for the operation of ``Maximum(x,y)''. $ L^\xi $ also includes a
sufficient number of function and constant symbols so that every integer $~k~$ can be encoded by some term $~T_k~$ specifying $~k\,$'s value. \item $~ \Delta_0^\xi ~$ corresponds to any variation of Definition \ref{xd+1x1}'s class of ``fully-bounded'' formulae that is rich enough to assure that there exists
two $~\Delta_0^\xi~$ formulae, henceforth called ``Add$(x,y,z)$'' and ``Mult$(x,y,z)$'', $\,$for formalizing the graphs of addition and multiplication. (It will generate $\xi$'s set of $\Pi_n^\xi$ and $\Sigma_n^\xi$ sentences, using Definition \ref{xd+1x1}'s 3-part formalism.) \item $ \, B^\xi \,$ denotes a$~$ {\normalsize {\bf ``Base Axiom System''}}, $~ \,$whose axiom-sentences are true under the Standard-M model and which is $\Sigma_1^\xi$ complete. (Thus, $ B^\xi $ can prove every true $\Sigma_1^\xi$ sentence, and it can likewise refute all false $\Pi_1^\xi$ sentences.) \item $~d~$ denotes $ \, \xi \, \, $'s
method of deduction. It is required to be sufficiently conventional to satisfy the usual indirect-implication property \footnote{ \baselineskip = 1.1 \normalbaselineskip $~~$This is that irregardless of whether or not $~d~$ contains a built-in modus ponens rule, it does support some form of a (possibly quite lengthy)
proof of a theorem $Z$, when it is able to prove $X$, $Y$ and $~(X \wedge Y)~\rightarrow ~Z~$ as theorems. } associated with
G\"{o}del's Completeness Theorem. \item $~g~$ denotes a method for encoding the G\"{o}del numbers of proofs. \end{enumerate} \end{definition}
\begin{example} \label{ex3-2} \rm Let us recall Example \ref{ex3-1} defined ``$~\Delta_0^A~$'' as essentially the conventional textbook notion \cite{HP91,Kr95}
of an arithmetic
``$~\Delta_0~$'' formula. This example will outline how well-known techniques can map every $\Delta_0^A$ formula onto a semantically equivalent
$\Delta_0^\xi$ formula under the Standard-M model.
Our discussion will have Seq$(x)$ denote a function that maps non-negative integers onto binary strings in lexicographic order. Thus Seq$(x)$ maps 0 onto the empty string, $\,$the integers 1 and 2 onto the strings of ``0'' and ``1'', $\,$the integers 3--6 onto$\,$ ``00'', ``01'', ``10'', ``11'', etc. (Formally, Seq$(x)$ is an operation that maps integer $~x~$ onto the bit-string that occurs to the immediate right of the leftmost ``1'' bit in the binary encoding of $~x+1.~~)$
Given any $\,k-$tuple $~(~x_1\, , \, x_2\, , ~ ... ~ x_k~),~$ let $~$STRING$( x_1 , x_2 ,\, ... \, x_k )$ denote the
concatenation of Seq$(x_1), \, ... ~$Seq$(x_k)$. For any integers $v$ and $w$ satisfying $v \leq w^2$, it is clear that there exists $(x_1,x_2,x_3)$ where STRING$(x_1,x_2,x_3)$ represents $v $'s binary encoding and each $x_i \leq \mbox{Max}(w,4)$.
An example will now illustrate the approximate structure of an inductive methodology for mapping $\Delta_0^A$ formulae onto their equivalent $\Delta_0^\xi$ counterparts in the Standard-M model. Let SQUARE$(x_1,x_2,x_3,w)$ be a $\Delta_0^\xi$ formula which specifies that STRING$(x_1,x_2,x_3)$ represents an integer $ \leq \, w^2~.~ \,$ Also, $\,$let $\phi^*(x_1,x_2,x_3)$ and $\phi(v)$ represent a pair of $\Delta_0^\xi$ and $\Delta_0^A$ formulae that are equivalent under the Standard-M model $\,$when STRING$(x_1,x_2,x_3)$ is an encoding for $~v~$. Then one possible method for mapping $\Delta_0^A$ formulae onto their equivalent $\Delta_0^\xi$ counterparts (in the Standard-M model) could map the formula \eq{exam21} onto \eq{exam22}'s alternate form: \begin{equation} \label{exam21} \forall ~ v\, \leq \, w^2~~~~~\phi(v) \end{equation}
\begin{center} $\forall ~ x_1 \leq~ \mbox{Max}(w,2)~~~~~\forall ~ x_2 \leq ~ \mbox{Max}(w,2)
~~~~~\forall ~ x_3 \leq ~ \mbox{Max}(w,2)$ \end{center}
\vspace*{-0.5 em}
\begin{equation} \label{exam22}
\{~~ \mbox{SQUARE}(x_1,x_2,x_3,w)
~~ \Rightarrow~~ \phi^*(x_1,x_2,x_3)~~\} \end{equation} \lem{lex22} indicates Example \ref{ex3-2}'s translational methodology generalizes easily to
all combinations of $\Delta_0^A$ inputs and generic configurations $\, \xi \,$, via an approximate inductive generalization of the transition from sentence \eq{exam21} to \eq{exam22}. (Its procedure essentially performs iteratively a finite number of such transitions, so as to translate all the clauses of
an initial $\Delta_0^A$
formulae into their $\Delta_0^\xi$ counterparts via an inductive methodology. The intuition behind these transitions is they will repeatedly replace a single variable, such as $~v~$ in sentence \eq{exam21}, with a multiplicity of variables, such as $(x_1,x_2,x_3)$ in \eq{exam22}.) \end{example}
\begin{lemma} \label{lex22} {\rm (Paris-Dimitracopoulos \cite{PD82} )} For every generic configuration $\xi$, each $\Delta_0^A$ formula can be translated into an equivalent $\Delta_0^\xi$ formula in the Standard-M model via a generalization of Example \ref{ex3-2}'s process. {\rm (This
clearly
implies $\Pi_j^A$ formulae can also be translated into $\Pi_j^\xi$ expressions.)} \end{lemma}
Paris-Dimitracopoulos \cite{PD82} sketched an
analog of Lemma \ref{lex22}'s translation algorithm, using only
slightly different notation,
that is applicable to any formalism that satisfies Parts (1) and (2) of
Definition \ref{def3.3}. Their formalism thus uses an inductively-iterated analog of the prior example's replacement of a single variable $~v~$ in formula \eq{exam21} with \eq{exam22}'s multiplicity of variables
$~(x_1,x_2,x_3)~,~$ so as to perform
Lemma \ref{lex22}'s translation task. It will be unnecessary for a reader to consider the details behind
\cite{PD82}'s Theorem 1 or Lemma \ref{lex22}'s similar translation mechanism
because
the remainder of this article will never
use them again.
Instead, their sole purpose
has been
to provide an {\it implicit backdrop} for our results by illustrating how the study of the $\Pi_1^\xi$ sentences of Definition \ref{def3.3}'s generic configurations
provides information about $\Pi_1^A$ sentences (after the needed translating is done).
Four examples of self-justifying systems that employ Definition \ref{def3.3}'s
$\Pi_1^\xi$ sentences
will be illustrated in Appendix D . These examples are too complicated to be examined
before Sections \ref{3uuuu1} -- \ref{sect64} are read. However, the next example should convey some useful intuitions:
\baselineskip = 1.4 \normalbaselineskip \parskip 5pt
\begin{example} \label{ex3-3} \rm Let $ x_0, x_1, x_2, ... $ and $ y_0, y_1, y_2, ... $ denote sequences defined by: \vspace*{- 0.7 em} \begin{equation} \label{zs} x_0~~~=~~~2~~~=~~~y_0 \end{equation} \begin{equation} x_{i}~~~=~~~x_{i-1}~+~x_{i-1} \label{as} \end{equation} \begin{equation} y_{i}~~~=~~~y_{i-1}~*~y_{i-1} \label{bs} \end{equation} \end{example} For $\, i \, > \,0 \,$, $\,$let $ \, \phi_{i} \, $ and $ \, \psi_{i} \, $ denote the sentences in \eq{as} and \eq{bs} respectively. Also,
let
$ \, \phi_{0} \, $ and $ \, \psi_{0} \, $ denote \eq{zs}'s sentence. Then
$ \, \phi_0, \, \phi_1, \, ... \, \phi_n \, $ imply
$ \, x_n \, = \, 2^{ n+1} \, , \, $ and
$ \, \psi_0, \, \psi_1, \, ... \, \psi_n \, $
imply $ \, y_n \, = \, 2^{2^n} \, $. Thus, the latter sequence grows at a faster rate than the former. Much of our research has used the difference between the growth rates of $ x_0, x_1, x_2, ... $ and $ y_0, y_1, y_2, ... $ as a motivating example explaining why \ep{totdefxa}'s Type-A axiom systems can support a stronger form of boundary-case exception to the semantic tableaux version of the Second Incompleteness theorem than can Type-M systems.
\parskip 2pt
Let Log$(\, y_n \,) \, = \, 2^{n} \, $ and Log$(\, x_n \,) \, = \, {n+1} \, $ thus designate the lengths of the binary codings for $ \, y_n \, $
and $ \, x_n \, $. Then $ \, y_n\,$'s coding has a length $\, 2^{n} \, $, which is
{\it much larger} than the $ n+1 $ steps that $ \, \psi_0, \, \psi_1, \, ... \, \psi_n \, $ use to define its existence. However,
$ \, x_n\,$'s length has a
smaller
size of $ \, {n+1} \, $. These observations are useful because every proof of the Incompleteness Theorem involves a G\"{o}del
number $ \, z \,$ coding a sentence that has a capacity to self-reference its own definition. The faster growing series $y_0,\,y_1,\,,\,...\,y_n$ should be intuitively anticipated to have
this self-referencing capacity because
$~y_n\,$'s binary encoding has a $~2^{n+1}~$ length that dwarfs the size of the $O(n)$ steps used
to define its value. Leaving aside \cite{ww2,ww7}'s
many details, this fast growth explains roughly
why many Type-M logics satisfy the semantic tableaux version of the Second Incompleteness Theorem.
This paradigm also illustrates intuitively why some
Type-A systems, employing \cite{ww93,ww1,ww5}'s semantic tableaux formalism, can represent boundary-case exceptions to the
Second Incompleteness Theorem.
This is because such formalisms lack access to \ep{totdefxm}'s axiom that multiplication is a total function. (They are unable, thus,
to easily
construct numbers $ \, z \, $ that can self-reference their own definitions because they have access only to the slower growing
addition primitive.) In particular assuming only that each sentence in the axiom-sequence
$ \phi_0, \phi_1, ... \phi_n $ (from \ep{as} ) requires a mere two bits for its encoding, the length $ n+1 $ of
$ x_n $'s binary encoding will be smaller than the length of its defining
sequence.
This short length for $ x_n $ had motivated \cite{ww93,ww1,ww5,ww6}'s evasion of the semantic tableaux version of the Second Incompleteness Theorem. It suggested that the self-referencing needed in a G\"{o}del-like diagonalization argument would stop being feasible when \ep{as}'s slow-growing $x_1,\,x_2,\,x_3,\,...$ sequence represents the fastest growth that is possible.
One of the several goals in this article will be
to formalize a generalizations of \cite{ww93,ww1,ww5,ww6}'s self-justifying methodologies by using Definition \ref{def3.3}'s generic configurations. The proofs of our main theorems will, of course, be more subtle than the hand-waving intuitions appearing in this example. For instance, the combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris \cite{Ne86,Pu85,So94,WP87} (summarized by Theorem 2.1) raised the subtle issue that
no Type-S system can prove a theorem affirming its own Hilbert consistency. Another complication is that the \ep{bs}'s implication for proofs that use the multiplication operative has different side effects for Herbrandized and semantic tableaux deduction (on account of
Ko{\l}odziejczyk \cite{Ko5,Ko6}'s previously mentioned observations about the potential exponential difference between the lengths of these proofs under extremal circumstances).
Our main theorems will show that self-justifying systems, using four deduction methods, are capable of proving all of Peano Arithmetic's $\Pi_1^\xi$ theorems. Interestingly, self-justification will be compatible with \cite{ww5}'s modification of semantic tableaux deduction, that includes a
modus ponens rule
for
$\Pi_1^\xi$ and $\Sigma_1^\xi$ type sentences. However,
\cite{wwlogos} has shown an analogous
modus ponens rule for
$\Pi_2^\xi$ and $\Sigma_2^\xi$ sentences is incompatible with self justification. (Thus, the contrast between our main results and the Second Incompleteness Theorem's generalizations will be quite tight.)
\baselineskip = 1.22 \normalbaselineskip
\section{Five Helpful Definitions and An Informative Lemma}
\label{3uuuu2}
\label{D.4-theorx}
This section will introduce five definitions and
prove a Lemma \ref{lemex4} about self justification. This lemma will be
weaker than Sections \ref{3uuuu3} and \ref{sect64}'s main results. Its main purpose
will be to provide a useful starting example.
\begin{definition} \rm \label{xd+1x3} The symbol ``E$(n)$'' will denote some term in Definition \ref{def3.3}'s language $ \, L^\xi \, $ that represents the value
$ \, 2^n \, . \, $ In using this symbol, we do not presume that $ \, L^\xi \, $ possesses a function symbol for the exponent operation. Thus if $ \, L^\xi \, $ has only a function symbol for multiplication, then
E$(n)$ could designate the term of $ \, $``$ \, 2*2* \, ... \, *2 \, $''$ \, $ with $ \, n \, $ repetitions of ``2'' $. \, $ (Alternatively,
E$(n)$ can be
defined via applying
$\, 2^n \,$ iterations of the successor function to zero, or by having a special constant symbol designating
$ \, 2^n \, $'s value. Essentially, any reasonable method can be used to define E$(n)$'s value)
\end{definition}
\baselineskip = 1.22 \normalbaselineskip \parskip 3pt
\begin{definition} \rm \label{xd+1x4} Let $\,\Upsilon \,$ denote a prenex normal sentence. Then {\bf Scope$_E$($\Upsilon,$N)$~$} will denote a sentence identical to $~\Upsilon~$ except that every unbounded universal quantifier ``$~\forall~v~$'' is changed to ``$ \, \forall \, v \, < \, E(N) \, $'', and every
unbounded existential quantifier ``$ \, \exists \, v \, $'' is changed to ``$ \, \exists \, v \, < \, E(N) \, $''. (No change is made among the bounded quantifiers within the $~\Delta_0^\xi$ part of the sentence $\,\Upsilon \,$.) For example, if $\,\Upsilon \,$ denotes the $\Pi_1^\xi$ sentence of $~\forall \, v_1~\forall \, v_2~...~\forall \, v_k~~~\phi(v_1,v_2,...v_k)~$ then \eq{scopede} illustrates Scope$_E$($\Upsilon,$N)'s form. Likewise if $~\Upsilon ~$ is the $~\Sigma_1^\xi~$ sentence of $~\exists \, v_1~\exists \, v_2~...~\exists \, v_k~~~\phi(v_1,v_2,...v_k)~$ then \eq{scopedx} illustrates Scope$_E$($\Upsilon,$N)'s form.
\vspace*{- 0.4 em}
{ \small \baselineskip = 1.22 \normalbaselineskip \begin{equation} \label{scopede} \forall ~ v_1~ < ~E(N)~~\forall ~ v_2~ < ~E(N)~~ ... \forall ~ v_k~ < ~E(N) ~~~~~ : ~~~~~ \phi(v_1,v_2,...v_k)~ \end{equation} \begin{equation} \label{scopedx} \exists ~ v_1~ < ~E(N)~~\exists ~ v_2~ < ~E(N)~~ ... \exists ~ v_k~ < ~E(N) ~~~~~ : ~~~~~ \phi(v_1,v_2,...v_k)~ \end{equation}} \end{definition}
{\bf Special Note about Definition \ref{xd+1x4}'s Meaning.} If $\Upsilon$ is a $\Delta_0^\xi~$ sentence then Scope$_E$($\Upsilon,N)$ will be equivalent to $~\Upsilon~$ for every $N \geq 0$ by definition. (This is because $\Delta_0^\xi~$ formulae contain no unbounded quantifiers that undergo change when
$\Upsilon$ is mapped onto Scope$_E$($\Upsilon,N).~~)$
{\bf More About this Notation:} The potentially lengthy syntactic object of ``$\,$Scope$_E$($\Upsilon,$N)$\,$'' {\it
will actually
not} be used in our physical encodings of proofs. Instead, these encodings will
use the more desirably compressed object of ``$\,\Upsilon\,$'' (which has no possibly bulky
$E(N)$ term). The {\it sole function} of $\,$Scope$_E$($\Upsilon,$N)$\,$ will be for us to speculate about what Boolean value $\,\Upsilon\,$
{\it would theoretically assume} (under the Standard-M
model) if $\,\Upsilon\,$'s quantifiers were modified
so that their ranges were changed to be bounded by $E(N).$ $~\,$(It turns out that Scope$_E$($\Upsilon,$N)$\,$'s finitized quantifier-range will help
simplify our analysis.)
\begin{definition} \rm \label{xd+1x5} A $\Pi_1^\xi$ or $\Sigma_1^\xi$ sentence
$ \Upsilon $ will be called {\bf Good(N)} when the entity Scope$_E$($\Upsilon,N)$ is true under the Standard-M model \footnote{ \baselineskip = 1.1 \normalbaselineskip \label{fgood} A quite unusual aspect of Definition \ref{xd+1x5} is that its Good$(N)$ condition has opposite properties when it is applied to $\Pi_1^\xi$ and $\Sigma_1^\xi$ sentences in one particular respect. This is because for each $~N,~$ the
Good$(~N~)$ condition is weaker than the Good$(~\infty~)$ condition for $\Pi_1^\xi$ sentences, while it is stronger than it for
$\Sigma_1^\xi$ sentences. (For instance, $~\forall \, x~\phi(x)~$ is stronger than $~\forall \, x <E(N)~\phi(x)~$, but $~\exists \, x~\phi(x)~$ is weaker than $~\exists \, x<E(N)~\phi(x)~$.)}. Also, a set of $\Pi_1^\xi$ sentences, denoted as $\theta$, is called Good(N) iff all of its sentences are Good(N).
\end{definition}
\begin{definition} \rm \label{chg} If $\Upsilon $ is a $\Pi_1^\xi$ sentence then
$~ \sharp ( ~ \Upsilon \,)$ will denote the largest integer $J$ such that $\Upsilon $ satisfies the Good$(J)$ condition. (It will equal $\infty$ if $\Upsilon $ satisfies Good$(J)$ for all $J$.) Also, if $\theta$ is a set of $\Pi_1^\xi$ sentences, then
$~ \sharp ( ~ \theta \,)$ will denote the largest $J$ where each sentence in $\theta$ is Good$(J)$. \end{definition}
{\bf A Very Helpful Start :} Several more definitions will be needed before Section \ref{sect64}
can present our strongest results. The remainder of this section will illustrate how the current formalism is already
sufficient for introducing a useful starting lemma.
\begin{definition} \label{deftight} \rm Let
$( L^\xi , \Delta_0^\xi , B^\xi , d , G )~$ again denote a generic configuration called $~\xi~$, and let us presume that its
base axiom system
$~B^\xi ~$ is comprised exclusively of
$ \, \Pi_1^\xi \, $ sentences. Also, let
$ \, \beta \, \supset \, B^\xi \, $
denote a second axiom system, comprised also of
$ \, \Pi_1^\xi \, $ sentences, that (unlike $ \, B^\xi \,$) can possibly be inconsistent. (If $ ~ \beta ~ $
is inconsistent then
let $ \, q_\beta \, $ denote the shortest proof of $ \, 0=1 \,$ from $~\beta~$.$~$) Then the generic configuration
$ \, \xi \, $ will be called
{\bf Tight} if iff {\it every inconsistent} set of
$ \, \Pi_1^\xi \, $ sentences
$~\beta \, \supset \, B^\xi ~ $ satisfies the following constraint: \begin{equation} \label{piss1} \mbox{ Log(}\,q_\beta \, ) ~ ~ \geq ~~ ~ \sharp ( ~ \beta ~)
~~+~~2 \end{equation} \end{definition}
Lemma \ref{lemex4} will prove $B^\xi+$SelfRef$(B^\xi,d)$ satisfies Section \ref{secc1}'s
self-justification criteria
whenever $ \, \xi \,$ is tight. This ``tightness'' will clearly fail to be satisfied by many generic configurations. This is
because the Second Incompleteness Theorem is a widely encompassing result, which imposes severe restrictions on its allowed
exceptions. Lemma \ref{lemex4}'s mini-result will be of interest primarily because it will be generalized substantially in Sections \ref{3uuuu3} and \ref{sect64}.
\begin{lemma} \label{lemex4} If a generic configuration
$( L^\xi , \Delta_0^\xi , B^\xi , d , G )~$ is tight then
$B^\xi+$SelfRef$(B^\xi,d)$ will be a consistent self-justifying axiom system. \end{lemma}
{\bf Proof Sketch:} Our justification of \lem{lemex4}'s mini-result will be simpler than
the next section's proof of \thx{ppp1}'s stronger result.
The current proof will also be kept brief and informal because the same topic will be visited more rigorously during Section \ref{3uuuu3}'s discourse.
Let $~\Psi~$ denote Section \ref{secc1}'s SelfRef$(B^\xi,d)$ sentence. We will omit formalizing
$~\Psi\,$'s exact
$\Pi_1^\xi$ encoding here because Appendix A will provide a more general fixed-point
construction, using Definition \ref{xd+1x7}'s stronger paradigm.
The current proof will simply presume
$~\Psi\,$'s fixed point statement can receive a
$\Pi_1^\xi$ encoding under sentence \eq{fixnew}, where ``$~\mbox{Prf}^d_{~B^\xi+{\rm SelfRef}(B^\xi,d)}(~\lceil \, 0=1 \, \rceil~,~p~)~$'' is a $~\Delta_0^\xi$ formula, indicating that $~p~$ is a proof of 0=1 from $~B^\xi~+~$SelfRef$(B^\xi,d)~$ under $~d \, $'s deduction method. \begin{equation} \label{fixnew} \forall ~p~~~\neg ~ ~\mbox{Prf}^d_{~B^\xi+{\rm SelfRef}(B^\xi,d)}(~\lceil \, 0=1 \, \rceil~,~p~) \end{equation}
The Definition \ref{chg}'s symbol
$~\sharp~$ will be
helpful at this juncture. The application of $\xi$'s tightness to
\eq{fixnew}'s $\Pi_1^\xi$ styled encoding will
imply
\footnote{ \baselineskip = 1.1 \normalbaselineskip \baselineskip = 1.0 \normalbaselineskip \ep{piss2} is easy to justify when one presumes there is an available $\Pi_1^\xi$ encoding of \eq{fixnew}'s statement, which we call $ \, \Psi \, $. (This
$\Pi_1^\xi$ presumption is reasonable because an analog of $ \, \Psi\,$'s exact
$\Pi_1^\xi$ encoding will be discussed later by Definition \ref{xd+1x7} and in Appendix A.) The invalidity of $ \, \Psi \, $
will thus assure the existence of a proof of $0=1$ from
$ \, B^\xi \, + \, $SelfRef$(B^\xi,d)$. Moreover, Definition \ref{chg}'s notation implies $~ \sharp ( ~ \Psi \, )$ will equal Log$(q) \, - \, 1 \, $ when $q$ denotes the
shortest proof of $0=1$ from
$ \, B^\xi \, + \, $SelfRef$(B^\xi,d)$. The latter shows \eq{piss2} is valid.} that \eq{piss2} must be true when
$~\Psi~$ is false under the Standard-M model and
when the shortest proof of $0=1$ from
$~B^\xi~+~$SelfRef$(B^\xi,d)$ is denoted as $~q$. \begin{equation} \label{piss2} \mbox{ Log(}\,q~) ~~ = ~~ ~ \sharp ( ~ \Psi ~) ~~+~~1 \end{equation} Also
Definition \ref{chg} trivially implies $~~ \sharp ( ~ B^\xi+\Psi ~) ~ = ~ ~ \sharp ( ~ \Psi ~)~$ (because all of
$~B^\xi \,$'s axioms are true under the Standard-M model). Thus, \eq{piss2} yields \eq{piss3}. \begin{equation} \label{piss3} \mbox{ Log(}\,q~) ~~ = ~~ ~ \sharp ( ~ B^\xi+\Psi ~)
~~ +~~1 \end{equation} But the point is that the Tightness constraint's \ep{piss1}, $~$used in the
context where $~\beta~$ is the axiom system of $~ B^\xi+\Psi ~,~$ implies
$~\mbox{ Log(}\,q \, ) ~ \geq ~ ~ \sharp ( ~ B^\xi+\Psi ~)~+~2~$. This directly contradicts \ep{piss3}'s equality, {\it whenever} the proof ``$~q~$'' of $~0=1~$ cited in our discussion {\it does formally exist.} This contradiction
is precisely what is needed to corroborate Lemma \ref{lemex4}'s claim that the axiom system $~B^\xi~+~$SelfRef$(B^\xi,d)$ must be
consistent. (Thus,
$~B^\xi~+~$SelfRef$(B^\xi,d)$ must be consistent because otherwise a proof $~q~$ of $~0=1~$ would exist and have its
$~\mbox{ Log(}\,q \, ) ~ \geq ~ ~ \sharp ( ~ B^\xi+\Psi ~)~+~2~$ inequality contradict \ep{piss3}.) $~~\Box$
{\bf How Lemma \ref{lemex4} May Be Interpreted :} We
remind the reader that many (but not all) generic configurations will fail to satisfy Lemma \ref{lemex4}'s tightness hypothesis. This is because any configuration satisfying this hypothesis represents one of those unusual boundary-case exceptions to the Second Incompleteness Theorem that are feasible.
Lemma \ref{lemex4} was intended to capture the simplest variant of a self-justifying phenomena (that employs Definition \ref{chg}'s machinery). Its proof was
kept informal because more sophisticated
self-justifying formalisms will be explored in Sections \ref{3uuuu3} and \ref{sect64}. They will apply to
four different types of generic configurations, each of whose base axiom systems $~B^\xi~$ can be made capable of proving all Peano Arithmetic's $~\Pi_1^\xi~$ theorems --- in a context where these systems use a broader variant of {\it ``I am consistent''} axiom-statement than does Lemma \ref{lemex4}'s ``SelfRef$(B^\xi,d)$'' sentence.
\section{The First Two Meta-Theorems about Self-Justification}
\label{3uuuu3}
The core theorems in this section will employ the following notation: \begin{enumerate} \item The symbol $~\theta$ will denote any recursively enumerable (r.e.) set of $\Pi_1^\xi$ sentences, henceforth called a {\bf R-View}. (An R-View does not need to be valid under the Standard-M model. It only needs to be r.e.)
\item
{\bf RE-Class$(\xi)$} shall denote the set of all possible ``R-Views'' $~\theta~$ that can be built out of $~\xi \,$'s language $~L^\xi~$. (This
permits
both valid and invalid R-Views to appear in RE-Class$(\xi)$. We choose this unrestricted definition because no recursive decision procedure can identify all the true $\Pi_1^\xi$ sentences in the Standard-M model.) \end{enumerate}
\begin{definition} \rm \label{astab} Let $~\xi~$ denote the 5-tuple
$( L^\xi , \Delta_0^\xi , B^\xi , d , G )~$ representing one of Definition \ref{def3.3}'s generic configurations, and let RE-Class$(\xi)$ and its R-Views ``$\theta$'' be defined as in the previous paragraph. Then $ \,\xi \, $ is called {\bf A-Stable } iff each $ \, \theta \, \in \, $RE-Class$(\xi)$ satisfies the following invariant:
\begin{description} \item[ *$~$ ] If $ \, \Upsilon \, $ is a $\Pi_1^\xi$ theorem of axiom system $~\theta \cup B^\xi~$ via a proof $~p~$ whose length satisfies
Log$(p)~\leq ~ \sharp ( ~ \theta)~+1 $ then $ \, \Upsilon \, $ will satisfy Good$\{~ \, ~ \frac{1}{2}~ ~ \sharp ( ~ \theta) \, ~\}~$. \end{description} \end{definition}
\begin{remark} \label{re3-1} \rm The invariant
$ \, * \,$ states short proofs (with lengths
$ \leq \, ~ \sharp ( ~ \theta)~+1~~) $
will produce at least {\it partially useful} deductions, in that their $\Pi_1^\xi$ theorems will always
satisfy
Good$\{ \, \, ~ \frac{1}{2}~ ~ \sharp ( ~ \theta) \, \, \} \, $, {\it irregardless of whether or not} $\theta$'s axioms are {\it technically} true. (This makes the study of A-stability very interesting unto itself,
apart from its applications
in the current article). \end{remark}
\phx{ppp2} will show the presence of
A-stability, alone, is sufficient for constructing self-justifying systems. This will imply every A-stable configuration must contain some embedded weakness (as every evasion of the Second Incompleteness Theorem always does). On the other hand, Appendix F will explain how A-stability and its ``EA-stable'' cousin (defined later) are both epistemologically interesting. Thus, A-stability has redeeming features.
\begin{definition} \rm \label{estab} A Generic Configuration $\,\xi \,$
will be called
{\bf E-Stable} iff all of the $\theta \in \,$RE-Class$(\xi)~$ satisfy
$~**~$. $~$(This construct is the counterpart for $\Sigma_1^\xi$ sentences of the Item $ *$ in Definition \ref{astab}.)
\begin{description} \item[ ** ] If $ \, \Upsilon \, $ is a $\Sigma_1^\xi$ theorem
derived from the axiom system $~\theta \cup B^\xi~$ via a proof $~p~$ whose length satisfies
Log$(p)~\leq ~ \sharp ( ~ \theta)~+1 $ $~$then $ \, \Upsilon \, $ will automatically satisfy Good$\{ ~ \frac{1}{2}~\lfloor ~$Log$(p)~\rfloor~-~1 ~\}~$. { (This invariant {\it further implies} \footnote{ \baselineskip = 1.0 \normalbaselineskip This point is easy to confirm when one remembers that $\Sigma_1^\xi$ sentences $~\Upsilon~$ have the property that $~A<B~$ implies
Scope$_E$($\Upsilon,A$) is stronger than Scope$_E$($\Upsilon,B$). It is then obvious that the
Good$\{ ~ \frac{1}{2}~\lfloor ~$Log$(p)~\rfloor~-~1 ~\}~$ criteria implies the validity of
Good$\{~ \, ~ \frac{1}{2}~ ~ \sharp ( ~ \theta) \, ~\}~$ for $\, \Sigma_1^\xi \,$ sentences because the invariant $\,** \, $ presumes its $\, \Sigma_1^\xi \,$ theorems have proofs $\, p \,$ satisfying `` Log$(p)~\leq ~ \sharp ( ~ \theta)~+1 $ ''. } that $ \Upsilon $ will also satisfy
the Good$\{ \, ~ \frac{1}{2}~ ~ \sharp ( ~ \theta) \,~\} $ criteria.)} \end{description} \end{definition}
\begin{remark} \label{re3-2} \rm The invariants $ \, * \, $ and $ \, ** \, \, $ are partially analogous to each other because both imply that if $ \, p \, $ is a proof short enough to satisfy Log$(p) \, \leq ~ \sharp ( ~ \theta) \, +1 \, $ then their resulting theorem will satisfy Good$\{ \, \, ~ \frac{1}{2}~ ~ \sharp ( ~ \theta) \, \, \} \, $. However, there is a
distinction between Definitions \ref{astab} and \ref{estab}, as well. This is because the prior section's
Footnote \ref{fgood} observed that $ \, \Sigma_1^\xi \, $ sentences are stronger when they meet a Good$( \, N \, )$ rather than a Good$( \, \infty \, )$ threshold, $ \, ${\it while the reverse is true for
$ \, \Pi_1^\xi \, $ sentences.} Thus $ \, ** \,$'s short proofs $ \, p \, $ (satisfying Log$(p) \,\leq ~ \sharp ( ~ \theta)~+1~ $) will have the special
property that their theorems $ \, \Upsilon~$ will satisfy a ``Good$\{ ~ \frac{1}{2}~\lfloor ~$Log$(p)~\rfloor~-~1 ~\}~$'' constraint that is actually stronger than their formal $~\Sigma_1^\xi~$ statements. \end{remark}
The Appendix D will provide four examples of generic configurations that are either E-stable or A-stable (or often
both). Its most prominent example will be a configuration $~\xi^*~$ that uses semantic tableaux deduction and recognizes addition as a total function. Theorem D.\ref{D.4-theorx} will imply such systems can be made
self-justifying and able to
prove
Peano Arithmetic's
$\Pi_1^*$ theorems.
Three more definitions are needed to help introduce our first theorem
\begin{definition} \rm \label{eastab} A generic configuration $~\xi~$ will be called {\bf EA-stable} iff it is both E-stable and A-stable. (It will thus satisfy both $~*~$ and $~**~$.) \end{definition}
Our next definition is related to the fact
that many definitions of consistency are logically
equivalent from the perspective of strong enough logics, but they are {\it often not provably equivalent} from the perspectives of weak logics.
\begin{definition} \label{lev} \rm Let $ \xi $ denote the generic configuration
$( L^\xi , \Delta_0^\xi , B^\xi , d , G )$, and $ \alpha $ be an axiom system satisfying $ \alpha \supseteq B^\xi $. Then $ \alpha $ is called {\bf Level$(k^\xi)$ Consistent} when there exists no proofs from $\alpha$ via $ d $'s deduction method of both a $\Pi^\xi_k$ sentence and of the $\Sigma^\xi_k$ sentence that is its negation \end{definition}
Most of this article will focus on self-justifying systems that recognize their Level$(k^\xi)$ consistency when
$k$ equals 0 or 1. Our next definition will be applied mostly to these two cases.
\begin{definition} \rm \label{xd+1x7} Given any $k \geq 0$, a generic configuration
$( L^\xi , \Delta_0^\xi , B^\xi , d , G )~$ and an axiom system
$\beta \supset B^\xi$, the symbol SelfCons$^k(\beta,d)$ will denote a self-referencing $\Pi_1^\xi$ sentence declaring $\, \beta + $SelfCons$^k(\beta,d) $'s formal Level(k$^\xi)$ consistency, as is illustrated below by the statement $ ~ +~ $. (An encoding for SelfCons$^k(\beta,d)$ will be provided by Appendix A.) \begin{quote} $+~~~$ There exists no two proofs (using deduction method $ d $) of {\it both} some $\Pi_k^\xi$ sentence and of the $\Sigma_k^\xi$ sentence, $\,$that represents its negation, $\,$from the union of the axiom system $~\beta~ $ with $~this~$ added sentence ``SelfCons$^k(\beta,d) \,$'' {\it (looking at itself).} \end{quote} \end{definition}
\begin{remark} \label{re3-3} \rm We will focus on Definition \ref{xd+1x7}'s SelfCons$^k(\beta,d) \,$ axiom mostly in
the settings where $\,k=0$ or 1. This is because SelfCons$^k(\beta,d) \,$ will typically be too strong for it to generate boundary-case exceptions to the Second Incompleteness Theorem when
$~k \geq 2~$. It turns out that even when
$~k=1~,~$ Definition \ref{xd+1x7}'s SelfCons$^k(\beta,d) \,$ statement will be significantly stronger than the axiomatic declaration $\, \bullet \,$ used by Section \ref{secc1}'s
SelfRef$(\beta,d)$ axiom. This is because SelfCons$^1(\beta,d) \,$ asserts
non-existence of simultaneous proofs for a $~\Pi_1^\xi~$ sentence and its negation, $\,$while SelfRef$(\beta,d)$ establishes merely the
non-existence of a proof of $0=1$. \end{remark}
\begin{theorem} \label{ppp1} Let $~\xi~$ denote a generic configuration
$( L^\xi , \Delta_0^\xi , B^\xi , d , G )~$ that is EA-stable. Then the corresponding axiom system of $B^\xi+$SelfCons$^1(B^\xi,d)$ must satisfy Section \ref{secc1}'s definition of self-justification. \end{theorem}
\parskip 3pt
{\bf Proof.} The justification of \phx{ppp1} will be a more elaborate version
of \lem{lemex4}'s mini-proof. It will replace Definition \ref{deftight}'s Tightness constraint with an EA-stability requirement. It will also
replace SelfRef$(\beta,d)$'s ``I am consistent'' axiom with a {\it stronger} SelfCons$^1(\beta,d)$ statement.
\baselineskip = 1.22 \normalbaselineskip \parskip 5pt
Our proof will focus on showing
$~B^\xi+$SelfCons$^1(B^\xi,d)~$ is consistent (and thus
satisfies the subtle Part-ii component of Section \ref{secc1}'s definition of ``Self-Justification''). It will be awkward during our discussion to write repeatedly the expression ``$~B^\xi+$SelfCons$^1(B^\xi,d)~$''. Therefore,
``$S$'' will be the abbreviated name for this axiom system. We will also employ the following notation: \begin{enumerate} \itemsep +3pt \item $~\mbox{Prf}_S(t,p)~$ will denote that $~p~$ is
a proof of the theorem $~t~$ from the above mentioned formalism ``$~S~$'' (using $~\xi \,$'s deduction method of $d$ ). \item $~\mbox{Neg}^1(x,y)~$ will denote that $~x~$ is the G\"{o}del encoding of a
$\Pi_1^\xi$ sentence and that $~y~$ is a $\Sigma_1^\xi$ sentence which represents $~x \,$'s formal negation. \end{enumerate} Appendix A explains how to combine the theory of LinH functions \cite{HP91,Kr95,Wr78} with \cite{ww1}'s fixed point methods to provide $\mbox{Prf}_S(t,p)$ and $\mbox{Neg}^1(x,y)$ with $\Delta_0^\xi$ encodings. (A reader can
omit examining Appendix A, if he just accepts this fact.) Thus, \eq{pencode} can be viewed \footnote{ \baselineskip = 1.0 \normalbaselineskip Our notation convention has the abbreviated formula of ``$~\mbox{Prf}_S(t,p)~$'' in \ep{pencode} corresponding to
Appendix A's ``$~\mbox{SubstPrf} \, _{\beta}^d \, ( \bar{n} , t , p ) ~$'' formula, in a context where $~n~$ specifies the G\"{o}del number of the expression $~\Gamma^k(g)~$ in Appendix A's
\ep{encode12} and where the superscript $~k~$ within this
formula ``$~\Gamma^k(g)~$'' is set equal to 1. This implies that sentence \eq{pencode} does assert the Level$(1^\xi)$ consistency of S. \fend } in this context as being SelfCons$^1(B^\xi,d)\,$'s formalized $\Pi_1^\xi$ statement, declaring the Level$(1^\xi)$ consistency of S: \begin{equation} \label{pencode} \forall \, x \, \forall \, y \, \forall \, p \, \forall \, q ~~~ \neg ~~
\{ \,\mbox{Neg}^1(x,y) ~ \wedge~ \mbox{Prf}_S \, ( x , p ) ~ \wedge ~ \mbox{Prf}_S \, ( y , q ) \, \} \end{equation}
Let $ \Phi $ denote \eq{pencode}'s
sentence. We will use it to prove \phx{ppp1}'s claim that $ S $ is consistent. Our proof will be a proof by contradiction. It will begin with the assumption that $ S $ is inconsistent. This implies $\Phi$ is false under the Standard-M model. Hence, Definition \ref{chg} implies \begin{equation} \label{punk}
~ \sharp ( ~ \Phi~)
~~ < ~~ \infty \end{equation}
\noindent \ep{punk} thus indicates there exists $(\bar{p},\bar{q},\bar{x},\bar{y})$ satisfying \eq{pencodepunk} (because such a tuple will be a counter-example to \eq{pencode}'s assertion). The particular $(p,q,x,y)$
satisfying \eq{pencodepunk} with minimum value for $ \mbox{Log}\{~\mbox{Max}[p,q,x,y] ~\}$ can then be easily shown$\,$ \footnote{ \baselineskip = 1.0 \normalbaselineskip \label{footp1} Let $~L~$ denote the minimum value for $ \mbox{Log}\{~\mbox{Max}[p,q,x,y] ~\}$ for a tuple $(p,q,x,y)$
satisfying \ep{pencodepunk}. Then by definition, $~\Phi~$ satisfies Good$(L-1)$ but not Good$(L)$. From Definition \ref{chg}, this establishes
the validity of \ep{punkless} (because the minimal $(\bar{p},\bar{q},\bar{x},\bar{y})$ satisfying sentence \eq{pencodepunk} has $ \mbox{Log}\{\mbox{Max}[ \bar{p} , \bar{q} , \bar{x} , \bar{y} ] \}
\, = \, L.~$) }
$\,$to also satisfy \ep{punkless}. \begin{equation} \label{pencodepunk}
\,\mbox{Neg}^1(~ \bar{x}~, ~ \bar{y}~) ~ \wedge~ \mbox{Prf}_S \, ( ~ \bar{x}~ , ~ \bar{p}~ ) ~ \wedge ~ \mbox{Prf}_S \, ( ~ \bar{y}~ , ~ \bar{q}~ ) \end{equation} \begin{equation} \label{punkless}
\mbox{Log}~\{~\mbox{Max}[~\bar{p}~,~\bar{q}~,~\bar{x}~,~\bar{y}~] ~~\}
~~~ =~~~~ \sharp ( ~ \Phi~) ~~+~~1 \end{equation}
$~~~$ We will now use \eq{pencodepunk} and \eq{punkless}
to bring our proof-by-contradiction to its conclusion. Let $ \, \Upsilon \, $ denote the $\Pi_1^\xi$ sentence specified by $ \, \bar{x} \, $. Then $ \, \neg \, \Upsilon \, $ will correspond to the $\Sigma_1^\xi$ sentence denoted by $ \, \bar{y} \, $. Also, \eq{pencodepunk} and \eq{punkless} imply that both
$ \, \Upsilon \, $ and $ \, \neg \, \Upsilon \, $ have proofs such that the logarithms of their G\"{o}del numbers are bounded by $~ \sharp ( ~ \Phi \,) \,+ \,1 \,$. These facts imply
that
$ \, \Upsilon \, $ and $ \, \neg \, \Upsilon \, $ {\it both} satisfy
Good$\{ \, \, ~ \frac{1}{2}~ ~ \sharp ( ~ \Phi) \, \, \} \, $ under our formalism. (This is because if we take $ \, \theta \, $ in Definitions \ref{astab} and \ref{estab} to be simply $ \, \Phi \,$'s 1-sentence statement, $\,$then the invariants of
$ \, * \,$ and $ \, ** \,$ from these two definitions both impose the same Good$\{ \, \, ~ \frac{1}{2}~ ~ \sharp ( ~ \Phi) \, \, \} \, $
constraint on
$ \, \Upsilon \, $ and $ \, \neg \, \Upsilon \, $.)
It is infeasible, however, for a sentence and its negation to both satisfy the same goodness constraint.
This completes \phx{ppp1}'s proof-by-contradiction because the initial
assumption that $S$ was inconsistent has led to an infeasible conclusion. $~~\Box$
\baselineskip = 1.22 \normalbaselineskip
\parskip 3pt
Our next definition will help formalize a useful cousin of \phx{ppp1}.
\begin{definition} \rm \label{ostab} A Generic Configuration of $\xi $ will be called
{\bf $~$0-Stable$~$} when every particular $\theta \, \in $ RE-Class$(\xi)$ satisfies the invariant of $***$. (This invariant is strictly weaker than its counterparts $*$ and $**$ in Definitions
\ref{astab} and \ref{estab}.) \begin{description} \item[ *** ] If $ \, \Upsilon \, $ is a $\Delta_0^\xi$ theorem
derived from the axiom system $~\theta \cup B^\xi~$ via a proof $~p~$ whose length satisfies
Log$(p)~\leq ~ \sharp ( ~ \theta)~+1~,~ $ then $ \, \Upsilon \, $ is true under the Standard-M model. \end{description} \end{definition}
\begin{theorem} \label{ppp2} If the configuration $\xi$ is 0-stable then $B^\xi+$SelfCons$^0(B^\xi,d)$ is a self-justifying formalism. {\rm (Appendix C shows this result applies also to E-stable and A-stable configurations.)} \end{theorem}
\phx{ppp2}'s proof is similar to \phx{ppp1}'s
proof. The difference between these two propositions is that \phx{ppp2} has reduced
SelfCons's
superscript
from 1 to 0, so that its
hypothesis can encompass a theoretically
broader set of applications. (The Appendix C summarizes how
\phx{ppp1}'s proof can be easily modified to also prove \phx{ppp2}.)
\begin{remark} \label{newrem} {\rm Theorems \ref{ppp1} and \ref{ppp2} should clarify the nature of \cite{ww93}--\cite{ww9}'s formalisms. This is because proofs-by-contradictions are notorious in the mathematical literature for being confusing. They should be simplified whenever possible. This has been done mainly through \phx{ppp1}'s short proof. (It
applies to three of Appendix D's four examples of generic configurations, and \phx{ppp2} applies to Appendix D's fourth example.) Furthermore, Section \ref{sect64} will show how more elaborate self-justification systems can verify all Peano Arithmetic's $\Pi_1^\xi$ theorems.} \end{remark}
\section{Four Further Meta-Theorems} \label{sect64}
We need one preliminary lemma before exploring how strong self-viewing
logics may become before they cross
the inevitable boundary between self-justification and
inconsistency, $\,$implied by G\"{o}del's Theorem.
\begin{lemma} \label{llpp3} Let $ \xi $ denote a generic configuration
$( L^\xi , \Delta_0^\xi , B^\xi , d , G )$ $\,$, and $ \theta^\bullet $ denote an r.e. set of $\Pi_1^\xi$ sentences, each of which holds true in the Standard-M model. Let $ \xi^\bullet $ denote a 5-tuple that differs from
$ \xi $ in that its base axiom system is $ B^\xi \, \cup \, \theta^\bullet $ (rather than $ B^\xi $). These conditions imply that $ \xi^\bullet $ is a generic configuration, and it will satisfy the following four invariants: \begin{description} \item[ i ] If $~\xi~$ is 0-stable then $~\xi^\bullet~$ will also be 0-stable . \item[ ii ] If $~\xi~$ is A-stable then $~\xi^\bullet~$ will also be A-stable . \item[ iii ] If $~\xi~$ is E-stable then $~\xi^\bullet~$ will also be E-stable . \item[ iv ] If $~\xi~$ is EA-stable then $~\xi^\bullet~$ will also be EA-stable . \end{description} \end{lemma}
Lemma \ref{llpp3}'s proof is fairly straightforward. It has been placed in the Appendix B. This section will use Lemma \ref{llpp3} to prove four meta-theorems
that are consequences of its formalism.
\begin{definition} \label{dap4-1} \rm Let $\xi$ again denote a generic configuration
$( L^\xi , \Delta_0^\xi , B^\xi , d , G )$, and $\theta$ denote some r.e. set of $\Pi_1^\xi$ sentences (which are not required to be true under the Standard-M model). For the cases where $\,k \,$ is either 0 or 1, the symbol $G^\xi_k(\, \theta \,)$ will denote the following axiom system: \vspace*{- 0.2 em} \begin{equation} \label{4gedef} G^\xi_k(~ \theta ~)~~= ~~ \theta~\cup~B^\xi~\cup~\mbox{SelfCons}^k\{~[~\theta \,\cup \,B^\xi~]~,d~\} \vspace*{- 0.2 em} \end{equation} Also when $ \, k \, = \, 0 \,$ or 1, the function $ \, G^\xi_k ~ $ (which maps $ \, \theta \, $ onto $G^\xi_k(\, \theta \,)~~~)$ is called {\bf Consistency Preserving} iff $ \, G^\xi_k(\, \theta \,) \, $is assured to be consistent whenever all the sentences in $~\theta ~$ are true under the Standard-M model. \end{definition}
We emphasize
consistency preservation is unusual in logic. This is because $ \, G^\xi_k(\, \theta \,) \, $ comes from adding a self-justifying axiom to an initially consistent formalism $~B^\xi+\theta~$, and the Second Incompleteness Theorem demonstrates that sufficiently powerful formalisms are simply incompatible with such an axiom.
However, there will be four specialized paradigms, defined in Appendix D, that are exceptions to this rule. They will be related to our next result:
\begin{theorem} \label{pqq3} The function $~G^\xi_1~$ shall satisfy Definition \ref{dap4-1}'s consistency preservation property when $~\xi~$ is EA-stable. Likewise the function $~G^\xi_0~$ will be consistency preserving when $~\xi~$ is any one of A-stable, E-stable or 0-stable. {\rm (Thus in each case, $~G^\xi_k(\, \theta \,)~$ will be consistent when all the sentences in $~\theta ~$ are true in the Standard-M model.)} \end{theorem}
{\bf Proof}. It will be
convenient for our proof
to use a
dot-style notation,
analogous to Lemma \ref{llpp3}'s terminology.
Thus, \begin{enumerate} \item
$~\theta^\bullet~$ denotes any r.e. set of $\Pi_1^\xi$ sentences {\it that are each true} under the Standard-M model. \item
$\xi^\bullet$ is the tuple $ \, ( \, L^\xi \, , \, \Delta_0^\xi \, ,
\, B^\xi \cup \theta^\bullet \, , \, d \, , \, G \, )$. (It differs from
$\xi$ by replacing
$\xi\,$'s base axiom system of $\,B^\xi \,$ with $\,B^\xi \cup \theta^\bullet$.) \end{enumerate}
Part-iv of Lemma \ref{llpp3} indicates the EA-stability of $\xi$
implies the EA-stability of $ \xi^\bullet $.
Moreover \phx{ppp1} applies to all EA-stable configurations, including $\xi^\bullet $. Thus, $G^\xi_1(\, \theta^\bullet \,)$ is consistent because $\xi$ is EA-stable and all of $~\theta^\bullet \,$'s sentences are true in the Standard-M model.
This proves \phx{pqq3}'s first claim.
An almost identical proof, where \phx{ppp2} simply
replaces \phx{ppp1} as the central self-justifying engine, will corroborate \phx{pqq3}'s second claim . Thus, $~G^\xi_0~$ is consistency preserving when $~\xi~$ is one of A-stable, E-stable or 0-stable. $~~~\Box$
\begin{remark} \label{re4-1} \rm Most
generic configurations $~\xi~$ will not satisfy \phx{pqq3}'s hypothesis because $~G^\xi_k(\theta)~$ will typically be inconsistent, irregardless \footnote{ \baselineskip = 1.0 \normalbaselineskip Conventional generic configurations $\xi$ will satisfy the Hilbert-Bernays derivability conditions \cite{HB39,HP91}. Their $~G^\xi_k(\theta)~$ will thus be automatically inconsistent because of a G\"{o}del-like diagonalization argument.} of whether or not all of $~\theta\,$'s axioms are valid under the Standard-M model. The significance of \phx{pqq3} is that it shows that some outlying
exceptions to this general rule do prevail when $~\xi~$ satisfies one of \phx{pqq3}'s four stability conditions. These exceptions are related to the 3-page abbreviated philosophical discussion that will appear later in Appendix F. They will assure that Theorem \ref{pqq3}'s formalisms of
$G^\xi_1(\theta)$ and $G^\xi_0(\theta)$ are {\it always consistent} whenever $~\theta\,$'s axioms are valid in the Standard-M model. \end{remark}
Our next definition will enable self-justifying formalisms to prove the $\Pi_1^\xi$ theorems of any consistent r.e. axiom system that uses $~L^\xi \,$'s language.
\begin{definition} \label{dap4-2} \rm Let $\,\xi\,$ denote a generic configuration
$( L^\xi , \Delta_0^\xi , B^\xi , d , G )$. Let $ \mathcal{B} $ denote any recursive axiom system whose language is an extension of $\, L^\xi \,$. For an arbitrary deduction method $ \, \mathcal{D} \, $ (which may be possibly different from $\,\xi \,$'s deduction method $ \,d \,$ ),
let $\mbox{Prf}_{ \mathcal{B}}^D \,( \, \lceil \, \Psi \, \rceil \, , \, q \, )\, $ denote a $\Delta_0^\xi$ formula indicating $ q $ is a proof of the theorem $ \Psi $ from axiom system $ \, \mathcal{B}\,$, using deduction method
$ \, \mathcal{D} \, $. Then the {\bf Group-2 Schema} for $ ( \mathcal{B} , D ) $ is defined as an infinite set of axioms that includes one instance of \eq{group2}'s axiom for each $\Pi_1^\xi$ sentence $\, \Psi$. \begin{equation} \label{group2} \forall ~q~~~ \{~ \mbox{Prf}_{ \mathcal{B}}^D \,( \, \lceil \, \Psi \, \rceil \, , \, q \, )
~~~\longrightarrow ~~~ \Psi ~~ \} \end{equation} \end{definition} {\bf $~~~~$Comment about this Notation:} The Definition \ref{dap4-2} had called \eq{group2} a
``Group-2 Schema'' so as to keep our terminology consistent with \cite{ww93,ww1,ww5,wwapal}'s notation.
\begin{theorem} \label{pqq4} Let $~\xi~$ denote any arbitrary generic configuration, and $ ( \mathcal{B} , D ) $ denote any pair consisting of an axiom system and a deduction method (which, once again, are allowed to be different from
$~\xi \,$'s deduction method and axiom system). Then if all of $ ( \mathcal{B} , D ) $'s $\Pi_1^\xi$ theorems are true in the Standard-M model, the following two invariants will hold: \begin{description} \topsep -4pt \itemsep -2pt \item[ i ] If $~\xi~$ is EA-stable then
there will exist an r.e.$~$self-justifying system that can prove all of $ ( \mathcal{B} , D ) $'s $\Pi_1^\xi$ theorems and recognize its own Level($~1^\xi~)~$ consistency. \item[ ii ] Likewise, if
$~\xi~$ is one of A-stable, E-stable or 0-stable,
there will exist an r.e.$~$self-justifying system that can confirm all of $ ( \mathcal{B} , D ) $'s $\Pi_1^\xi$ theorems and which can recognize its own Level($~0^\xi~)~$ consistency. \end{description} \end{theorem}
{\bf Proof.} \phx{pqq4} follows from \phx{pqq3}. Thus let
$\theta$ denote
the set of all $\Pi_1^\xi$ sentences that are members of Definition \ref{dap4-2}'s Group-2 schema. Then every one of $\theta\,$'s Group-2 axioms must be true under the Standard-M model (because the hypothesis of \phx{pqq4}
indicated all of $ ( \mathcal{B} , D ) $'s $\Pi_1^\xi$ theorems are true in this model). Hence, \phx{pqq3} implies: \begin{enumerate} \item $G^\xi_1(\, \theta \,)$
is consistent when $\xi$ is EA-stable. \item $G^\xi_0(\, \theta \,)$ is consistent when $\xi$ is one of
A-stable, E-stable or 0-stable. \end{enumerate} Since $G^\xi_0(\, \theta \,)$ and $G^\xi_1(\, \theta \,)$ are self-justifying systems that prove all of $ ( \mathcal{B} , D ) $'s $\Pi_1^\xi$ theorems, Items 1 and 2 will substantiate \phx{pqq4}'s two claims. $\Box$
An
awkward aspect of Definition \ref{dap4-2}'s ``Group-2'' schema is that it employs an infinite number of instances of \eq{group2}'s Group-2-like axiom sentences. It turns out that this Group-2 scheme can be compressed
into a {\it single axiom sentence,} if one is willing to settle for a slightly diluted variant of
$ ( \mathcal{B} , D ) $'s $\Pi_1^\xi$ knowledge. To formalize this concept, the following notation shall be used: \begin{enumerate} \item Check$^\xi(t)$ will denote a $\Delta_0^\xi$ formula that checks to see whether $t$ represents the G\"{o}del number of a $\Pi_1^\xi$ sentence. \item Test$^\xi(t,x)$ will denote any $\Delta_0^\xi$ formula where \eq{testsim}'s invariant is true under the Standard-M model for every $ \Pi_1^\xi $ sentence $ \Psi $ simultaneously. There are infinitely many different $\Delta_0^\xi$ formulae that can serve as Test$^\xi(t,x)$ predicates satisfying this condition. (Example \ref{new-exam} will illustrate one such encoding of a Test$^\xi(t,x)$ predicate.) \begin{equation} \label{testsim} \Psi ~~~ \longleftrightarrow~~~ \forall ~x~~ \mbox{Test}^\xi(~\lceil~\Psi~\rceil~,~x~) \end{equation} \end{enumerate} The expression \eq{globsim} will be called a {\bf Global Simulation Sentence} for representing $( \mathcal{B},D)$ via $\xi \,$. Its $\mbox{Test}^\xi(t,x)$ clause essentially allows $\xi$ to simulate the $\Pi_1^\xi$ knowledge of $( \mathcal{B},D)$'s set of theorems. \begin{equation} \label{globsim} \forall ~t~~ \forall ~q~~ \forall ~x~~\{~~ [~~\mbox{Prf}_{ \mathcal{B}}^D \,( t , q )~~ \wedge ~~ \mbox{Check}^\xi(t)~~]~~~ \longrightarrow ~~~ \mbox{Test}^\xi(t,x)~~~ \} \end{equation}
\begin{example} \label{new-exam} \rm For any generic configuration $\,\xi~=\,$ $( L^\xi , \Delta_0^\xi , B^\xi , d , G )$, $\,$let NegPrf$^\xi(t,x)$ denote a $\Delta_0^\xi$ formula specifying that $ \, t \, $ is a $\Pi_1^\xi$ sentence and that $ \, x \, $ is a proof under
$\,d\,$'s deduction method of the
$\Sigma_1^\xi$ sentence that represents $ \, t\,$'s negation. Also, let Test$^\xi_0(t,x)$ be defined as follow: \begin{equation} \label{new1-exam} \mbox{Test}^\xi_0(t,x)~~~~ =_{\mbox{def}}~~~~ \neg~ \mbox{NegPrf}^\xi(t,x) \end{equation} For each $\Pi_1^\xi$ sentence $\Psi$,
it is easy to verify \footnote{ \baselineskip = 1.1 \normalbaselineskip Part-3 of Definition \ref{def3.3} indicated that $~\xi\,$'s base axiom system is ``$~\Sigma_1^\xi~$ complete''. $~$(It is thus able to prove all $~\Sigma_1^\xi~$ sentences that are true in the true in Standard-M model, and it will likewise refute all $~\Pi_1^\xi~$ sentences that are false.) The statement \eq{new1-exam} then immediately implies that \eq{newtestsim} must be true
under the Standard-M model for
every $~\Pi_1^\xi~$ sentence $~\Psi~$.} that the statement \eq{newtestsim} is true under the Standard-M model. \begin{equation} \label{newtestsim} \Psi ~~~ \longleftrightarrow~~~ \forall ~x~~ \mbox{Test}^\xi_0(~\lceil~\Psi~\rceil~,~x~) \end{equation} \end{example} \vspace*{- 1.3 em} $~~~$
The latter
implies that \eq{new2-exam} is
a global simulation sentence for $ ( \mathcal{B} , D ) $. \begin{equation} \label{new2-exam} \forall ~t~~ \forall ~q~~ \forall ~x~~\{~~ [~~\mbox{Prf}_{ \mathcal{B}}^D \,( t , q )~~ \wedge ~~ \mbox{Check}^\xi(t)~~]~~~ \longrightarrow ~~~ \mbox{Test}_0^\xi(t,x)~~~ \} \end{equation} We emphasize that there are countably infinite different examples of $\mbox{Test}_i^\xi(t,x)~$ predicates that generate global simulation sentences and that statement \eq{new2-exam} illustrates only one such example.
\begin{definition} \label{gsim} \rm Let $( \mathcal{B},D)$ denote any ordered pair whose set of $~\Pi_1^\xi$ theorems are true under the Standard-M model. Let Test$^\xi_1 \, , \, $ Test$^\xi_2 \, , \, $ Test$^\xi_3 \,~....~ $ denote the set of $\Delta_0^\xi$ formulae where statement \eq{testsim} is true under the Standard-M model for every $\Pi_1^\xi$ sentences $\Psi$. Then TestList$^\xi$ will denote a list of all these Test$^\xi_i \,$ predicates. Also for each Test$^\xi_j $ formula in TestList$^\xi\,$, $\,$the symbol {\bf GlobSim$^D_{ \mathcal{B}} \,(\xi,j)$} will denote the special version of \eq{globsim}'s global simulation formalism that employs Test$^\xi_j \, $'s machinery.
\end{definition}
\begin{remark} \label{re4-2} \rm A comparison between Definition \ref{dap4-2}'s Group-2 schema with \ref{gsim}'s global simulation sentences will reveal neither is strictly
better than the other. Both have their own separate advantages. Thus, the attractive aspect about Definition \ref{gsim}'s GlobSim$^D_{ \mathcal{B}} \,(\xi,j)$ sentence is that it is a finite-sized object that can simulate the infinite set of axioms associated with Definition \ref{dap4-2}'s Group-2 schema. The accompanying drawback \footnote{ \baselineskip = 1.1 \normalbaselineskip The chief
difficulty arises essentially
because a $\Pi_1^\xi$ theorem of
$ ( \mathcal{B} , D ) $ may contain an arbitrarily long combination of bounded universal and bounded existential quantifiers. Thus, some generic configurations will have base axiom systems $B^\xi$ that are so weak that their combination with \eq{globsim}'s Global Simulation Sentence is insufficient {\it to prove} the validity of \eq{testsim}'s equivalence statement {\it for all}
$\Pi_1^\xi$ sentences $\Psi$ simultaneously. In particular, such proofs will often be infeasible when $\Psi \,$'s sequence of bounded quantifies has a length greatly exceeding the length of the G\"{o}del encoding for \eq{globsim}'s global simulation statement.} $\,$of a global simulation sentence
is that the union of it with the base-axiom system $B^\xi$ will typically be inadequate to prove every $\Pi_1^\xi$ sentence that is a theorem of
$ ( \mathcal{B} , D ) $. $\,$Instead, in a context where
$\Psi \,$ is a $\Pi_1^\xi$ theorem of
$ ( \mathcal{B} , D ) $, the sentence
GlobSim$^D_{ \mathcal{B}} \,(\xi,j)$ will usually provide only enough {\it fragmented information} to prove the statement \eq{quasisim} (which is equivalent to $\Psi$ {\it under the Standard-M model).} \begin{equation} \label{quasisim}
\forall ~x~~ \mbox{Test}_j^\xi(~\lceil~\Psi~\rceil~,~x~) \vspace*{- 0.4 em} \end{equation} While \eq{quasisim} may be insufficient
to prove $ \, \Psi \, $ from $ \, B^\xi \, $, it still (according to the statement \eq{testsim} ) has the desired property of being equivalent to $\Psi$ under the Standard-M model. (This means that the knowledge of \eq{quasisim}'s truth is helpful, {\it even if it is unknown from $ \, B^\xi\,$'s perspective} to be equivalent to $ \, \Psi \, $ . ) \end{remark}
Our prior articles \cite{ww93,ww1,ww5,wwapal} did not use Definition \ref{gsim}'s global simulation formalism. They employed, instead, Definition \ref{dap4-2}'s Group-2 axiom schema. \phx{pqq5} will be
the
analog of \phx{pqq4} for global simulation. It will be
useful when
one desires to compress all the information held by a Group-2 schema into a single finite-sized object.
\begin{theorem} \label{pqq5} Let $ \,\xi \,$ denote the generic configuration
$( L^\xi , \Delta_0^\xi , B^\xi , d , G )$,
$\, \mathcal{B}\,$ denote a
recursively enumerable axiom system and $ \, \mathcal{D} \,$ denote any deduction method (which can be different than $\xi \,$'s deduction method $~d~~)~$. Suppose that all the $\Pi_1^\xi$ theorems generated by
$ \, ( \, \mathcal{B} \, , \, D \, ) \, $ are true under the Standard-M model. Then the following invariants do hold: \begin{description} \item[ i ] If $ \, \xi \, $ is EA-stable then for each $\, j \,$ there exists a {\bf finitized extension} $~\beta_j~$ of $~B^\xi$ that recognizes its Level$(1^\xi$) self-consistency
and which contains the sentence GlobSim$^D_{ \mathcal{B}} \,(\xi,j)$. \item[ ii ] Likewise for each $ j $, if $\xi$ is
E-stable, A-stable or 0-stable then there exists a {finitized extension} $~\beta_j~$ of $~B^\xi$ that recognizes its Level$(0^\xi$) self-consistency
and which contains the sentence GlobSim$^D_{ \mathcal{B}} \,(\xi,j)$. \end{description} \end{theorem}
{\bf Proof:} Let $\theta$ denote the 1-sentence R-View of ``GlobSim$^D_{ \mathcal{B}} \,(\xi,j)$'' formalized by Definition \ref{gsim}, $~$and let $~\beta_j^1~$ and $~\beta_j^0~$ denote $ ~ \, B^\xi \, \cup \, \theta \, + \, $SelfCons$^1( \,B^\xi \, \cup \, \theta \, ) \,~ $ and $ \, B^\xi \, \cup \, \theta \, + \, $SelfCons$^0( \,B^\xi \, \cup \, \theta \, ) \, $, respectively. These axiom systems correspond to the objects that Definition \ref{dap4-1} had called $G^\xi_1(\theta)$ and $G^\xi_0(\theta)$.
\phx{pqq5}'s hypothesis indicates
all the $\Pi_1^\xi$ theorems of
$( \mathcal{B},D)$ are true
under the Standard-M model. Thus, it follows that $\, \theta \,$ is also true in the Standard-M model. Hence \phx{pqq3} implies that $ \, \beta_j^1 \, = \, G^\xi_1(\theta)$ is a consistent system satisfying \phx{pqq5}'s claim (i). Likewise, \phx{pqq3} implies $ \, \beta_j^0 \, = \, G^\xi_0(\theta)$ satisfies \phx{pqq5}'s
second claim. $~~\Box$
\begin{remark} \label{re4-n} \rm Theorems \ref{pqq4} and \ref{pqq5} raise a fascinating
question: {\it Is the trade-off between these
formalisms needed?} That is,
can self-justifying systems use only a
a finite number of added axioms beyond those lying in $~\xi\,$'s base
system of $B^\xi$ and also
duplicate all
$( \mathcal{B},D)$'s $\Pi_1^\xi$ theorems in a pure sense (i.e. without simulation) ? We will return to this topic in Appendix G. \end{remark}
{\bf $~~~~$ REFLECTION PARADIGMS$~$:} $~$Our last
goal is to show how self-justifying systems
support
{\it unusually} strong reflection principles.
Let $\mbox{Reflect}_{\alpha,d}(~\Psi~)~$
denote \eq{reflect}'s statement when
$~\Psi~$ is a sentence with G\"{o}del number
$~\lceil \, \Psi \, \rceil~$, and $~(\alpha,d)~$ denotes an axiom system and deduction method. \begin{equation} \label{reflect} \forall ~p~~~[~~ \mbox{Prf}_{\alpha,d}(~\lceil \, \Psi \, \rceil~,~p~)
~~~ \Rightarrow ~~~ \Psi~~] \end{equation} L\"{o}b's Theorem \cite{HP91,Lo55,So76b}
implies that conventional systems $\, (\alpha,d) \,$, possessing at least
Peano Arithmetic's strength, are unable to prove
$\mbox{Reflect}_{\alpha,d}(~\Psi~)~$ except for in the degenerate cases where they can prove $~\Psi$.
Moreover, it is easy to generalize L\"{o}b's Theorem (via say \cite{ww1}'s Theorem 7.2) so that a wide class of formalisms $~\alpha~$, weaker than
Peano Arithmetic, are also
unable to prove
$\mbox{Reflect}_{\alpha,d}(~\Psi~)~$ for
all $\Pi_1^\xi$ sentences $~\Psi~$ simultaneously.
The intuition behind this generalization is quite simple. Let $~\mho~$ denote a $\Pi_1^\xi$ encoding for the classic
G\"{o}del sentence declaring: { \it `` There is no proof of me
from the axiom system $~\alpha~$ using $~d\,$'s deduction method''.} Then \cite{ww1}'s Theorem 7.2 uses a very routine diagonalization argument to show most formalisms $~\alpha~$ will be
inconsistent
if they prove $\mbox{Reflect}_{\alpha,d}(~\mho~)$'s statement.
Our next theorem will show, surprisingly, that the preceding limitation is much less stringent
than it may initially appear to be. This is because
Level$(~1^\xi~)$ self-justifying axiom systems are capable of proving very close analogs to \eq{reflect}'s impermissible
\noindent
reflection principle, using a ``translational'' methodology.
Thus, let $~T~$ denote an algorithm that maps a
$~\Pi_1^\xi~$ sentence $~\Psi~$ onto a ``translated'' sentence $~\Psi^T~$ that is equivalent to $~\Psi~$ under the Standard-M model {\it and which is also written in a $~\Pi_1^\xi~$ format.} (See footnote\footnote{\label{imper} Part-3 of Definition \ref{def3.3} indicated that generic configurations are $\Sigma_1^\xi$ complete.
Our requirement that
$~\Psi^T~$ must have a $\Pi_1^\xi$ format thus causes $~T~$ to gain much added meaning. This is because the axiom system $~B^\xi~$ will then automatically disprove $~\Psi^T~$
whenever it is false under the Standard-M model. (Thus, $T$'s mapping of $~\Psi~$ onto $~\Psi^T~$ gains much significance when $~\Psi~$ and $~\Psi^T~$ do rest on the same $\Pi_1^\xi$ level of the arithmetic hierarchy.)} for why it is absolutely imperative that {\bf both these} requirements be included in $~T\,$'s definition.) Also, let $\mbox{Reflect}^T_{\alpha,d}(~\Psi~)~$ denote the
translational modification of \eq{reflect}'s reflection principle that replaces $\Psi$ with $\Psi^T$.
\begin{equation} \label{T-reflect} \forall ~p~~~[~~ \mbox{Prf}_{\alpha,d}(~\lceil \, \Psi \, \rceil~,~p~)
~~~ \Rightarrow ~~~ \Psi^T~~] \end{equation}
\begin{theorem} \label{ppp6} Let $ \,\xi \,$ denote the EA-stable configuration of
$( L^\xi , \Delta_0^\xi , B^\xi , d , G )$, and let $~\alpha~=~ B^\xi \, + \, $SelfCons$^1(\,B^\xi \,)$ denote $ \,\xi \,$'s corresponding Level(1) self-justifying axiom system. Then there will exist a translation methodology $~T~$ where
$~\alpha~$ can prove the validity of \eq{T-reflect} for all its $\Pi_1^\xi$ sentences simultaneously.
\end{theorem}
{\bf Proof:} Let us use Example \ref{new-exam}'s notation. It observed that $~\Psi\,$'s $\Pi_1^\xi$ statement was equivalent under the Standard-M model to
``$ \, \forall \, x~ \mbox{Test}^\xi_0( \, \lceil \, \Psi \, \rceil \, , \, x \, )$''. Thus, let us view $\,T \,$ as being a mapping of the first sentence onto the second.
Our proof of Theorem \ref{ppp6} will next use the following observations: \begin{enumerate} \item The non-existence of a proof of $~ \neg ~ \Psi~$ from $~ B^\xi \, + \, $SelfCons$^1(\,B^\xi \,)$ trivially implies the non-existence of a proof of the same theorem from $~B^\xi~$
(because the latter axiom system is simply a subset of the former). \item Moreover, Example \ref{new-exam}'s notation treats ``$ \, \forall \, x~ \mbox{Test}^\xi_0( \, \lceil \, \Psi \, \rceil \, , \, x \, )$'' as being equivalent to the declaration that no proof of $~\neg~\Psi~$ from $~B^\xi~$ exists. \end{enumerate} Hence, $~\alpha~$ can prove \eq{T-reflect}'s statement by noting that $~p\,$'s proof of a $\Pi_1^\xi$ sentence $~\Psi~$ implies (via $\, \alpha \,$'s $~$SelfCons$^1~$ axiom) the non-existence of a proof of $~\neg ~\Psi~$, which (via Items 1 and 2) implies
``$ \, \forall \, x~ \mbox{Test}^\xi_0( \, \lceil \, \Psi \, \rceil \, , \, x \, )$'' $~~\Box$.
\baselineskip = 1.22 \normalbaselineskip
\begin{remark} \rm \label{f88}: $~$\phx{ppp6} and the statement \eq{T-reflect}'s Translational Reflection Principle may possibly be useful devices in unraveling some of the mystery that has enshrouded G\"{o}del's Second Incompleteness Theorem, since its inception. This is partly
because G\"{o}del was explicitly uncertain about the generality of the Second Incompleteness Theorem in his initial
1931 seminal paper \cite{Go31} about this subject.
His centennial paper about Incompleteness
thus included
the following quite poignant caveat: \begin{quote}
$~\bullet~~~$ : ``It must be expressly noted that Theorem XI (i.e the Second Incompleteness Theorem) represents no contradiction of the formalistic standpoint of Hilbert. For this standpoint presupposes only the existence of a consistency proof by finite means, and {\it there might conceivably be finite proofs} which cannot be stated in ... ''
\end{quote} Some of the issues that troubled G\"{o}del in the statement $\bullet$ can perhaps be partially resolved if one compares the reflection principles of sentences \eq{reflect} and \eq{T-reflect}. This is because \eq{reflect} is probably unnecessary to explain how thinking beings can appreciate their
$~\Pi_1^\xi~$
theorems
when \phx{ppp6}'s specialized logics can, instead,
use the fact that its $\Pi_1^\xi$ sentences satisfy at least \eq{T-reflect}'s modified
reflection principle.
Thus, some of the mystery surrounding the Second Incompleteness Effect can be clarified
when one notices that
\eq{T-reflect}'s translational reflection princible is a useful precept, that was shown by \phx{ppp6} to be technically
unrelated to G\"{o}del's observation that no reasonable formalism can prove
\eq{reflect}'s {\it purist}
principle
for all
$~\Pi_1^\xi~$ sentences simultaneously. \end{remark}
\begin{remark} \label{remhappy} \rm Theorem \ref{ppp6} is also significant because it explains how its specialized logics
can grapple with a $\, \Pi_1^\xi \, $ encoded G\"{o}del sentence $~\mho~$ which asserts {\it ``There is no proof of me''}. This issue is challenging because routine constructions, such as \cite{ww1}'s Theorem 7.2, demonstrate that no natural logic can verify statement
\eq{reflect}'s validity for all $~\Pi_1^\xi~$ sentences $~\Psi~$ (on account of the well-known syllogism posed by $~\mho$'s G\"{o}del sentence).
Theorem \ref{ppp6} constructs,
however, a
reply to this challenge.
This is because
its self-justifying systems do
surprisingly
prove, without difficulty \footnote{Since $\Psi$ and $\Psi^T~$ are equivalent under the Standard-M Model {\it but not also equivalent} from the perspective of the
system $~\alpha~=~ B^\xi \, + \, $SelfCons$^1(\,B^\xi \,)$, the conventional contradictions, produced by
\eq{reflect}'s reflection principle, disappear when it is replaced by \eq{T-reflect}.
(Thus, there is no danger that $\alpha$ could use \eq{T-reflect}'s reflection principle to prove an analog of $\,\mho \,$'s forbidden G\"{o}del sentence.) \fend
}, the validity of \eq{T-reflect}'s {\it translated modification} of \eq{reflect}'s
unobtainable $\Pi_1^\xi$ styled variant of a reflection principle. \end{remark}
\begin{remark} \label{new14} \rm We encourage the reader to examine the work of Beklemishev, Kreisel-Takeuti
and Verbrugge-Visser \cite{Be95,Be97,Be3,KT74,VV94,Vi5} to see alternative reflection principles and their uses. (The constraint on proof-length, by Verbrugge-Visser,
is certainly one alternative to \thx{ppp6}'s machinery. Likewise,
Kreisel-Takeuti's Second Order Logic {\it CFA} reflection is another alternative, although it will not \footnote{Kreisel-Takeuti indicate on page 25 of \cite{KT74} that CFA's reflection principle for a first order formula ``A'' infers the validity of {\it only the relativized formula} ``$~A^N~$'' from $A$'s proof. Also, their proof predicate is similarily relativized. Thus CFA's second-order logic reflection principle, while fascinating, does not generalize to first-order logic environments. } generalize to first-order logics.) One complicating aspect of
our \thx{ppp6}'s
reflection
method
is
that Appendix E
proves it becomes
inoperable when an axiom system is
sufficiently conventional to satisfy G\"{o}del's Second Incompleteness Theorem. In essence, \thx{ppp6}'s translational reflection principle is a specialized methodology,
intended for logics using Definition \ref{xd+1x7}'s Level(1) style of self-justification.
\end{remark}
\begin{remark} \label{recc1} \rm The preceding discussion clearly shows
self-justifying logics are
tempting. At the same time, it is necessary to be very cautious because there are also two fundamental barriers limiting such results: \begin{description} \topsep -4pt \itemsep +2pt \item[ a ] The first is the
Theorem 2.1 arising from the joint work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris \cite{Ne86,Pu85,So94,WP87}. It showed no reasonable system recognizing successor as a total function can verify its own Hilbert consistency.
Also, Willard \cite{ww2,ww7} established
analogous results under semantic tableaux consistency for
systems recognizing multiplication as a total function.
Thus, each effort to evade the Second Incompleteness Theorem must encounter
robust barriers. \item[ b ] A second issue is that Definition \ref{xd+1x7}'s ``SelfCons'' {\it ``I am consistent''} axiom sentence is less than ideal because it causes axiom systems to produce essentially a 1-line proof of their own
consistency. Such an excessively compressed proof corresponds more closely to an axiom system formulating
an {\it instinctive faith} in its own consistency (rather than it supporting a
full-length proof-justification of this fact). \end{description} Part of the reason self-justifying systems are of interest, despite these limitations, is that they illustrate how some formalisms {\it are compatible} with at least an {\it instinctive faith} in their own self-consistency. (This compatibility issue is non-trivial because Item (a) implies there are many circumstances where a generalization of the Second Incompleteness Theorem will make it infeasible for a formalism to satisfy {\it both} Parts (i) and (ii) of Section 1's definition of Self-Justification.) Moreover, three of Appendix D's four sample self-justifying configurations, called $~\xi^*~$, $~\xi^{**}~$ and $~\xi^R~$,
will be Type-A systems that
recognize addition as a total function. These configurations will thus possess the following three significant finitized
features: \begin{enumerate} \topsep -4pt \itemsep +2pt \item They will be able to construct the entire infinite set of integers {\it by finite means} because they recognize addition as a total function. \item For any r.e. logical configuration $ ( \mathcal{B} , D ) $, it will be possible to develop a 1-sentence
{\it finitized} extension for the base axiom systems of any of the configurations of $~\xi^*~$, $~\xi^{**}~$ and $~\xi^R~,~$ which deploy
\eq{globsim}'s
Global Simulation Sentence to simulate the
$~\Pi^\xi_1~$ knowledge of $ ( \mathcal{B} , D ) $. This means that some fully finite-sized extensions of the base-formalisms of
$~\xi^*~$, $~\xi^{**}~$ and $~\xi^R~$ will contain a non-trivial amount of
$~\Pi^\xi_1~$ styled knowledge, since $ ( \mathcal{B} , D ) $ can correspond to, say,
Peano Arithmetic. \item The key point is
that a 1-sentence extension of an axiom system containing features (1) and (2) can formalize how a logic
can possess an instinctive faith in its own consistency via \thx{pqq5}'s explicitly {\it finitized} structure. (Moreover, \thx{ppp6}'s Translational Reflection Principle is applicable to Appendix D's
generic configurations of
$~\xi^*~$ and $~\xi^{**}~$. It will thus imply that their single {\it finitized}
Level-1 self-justifying axioms enable
them to prove an {\it infinite number} of incarnations of \eq{T-reflect}'s translational reflection principle, where each $\Pi_1^*$ sentence $\Psi$ is mapped onto one such
unique instance.)
\end{enumerate} The contrast between Items 1-3's positive remarks about ``finitized'' cogitation with Items (a) and (b)'s opposing comments is obviously formidable. It is clearly preferable to view these positive results cautiously and treat them as being no more than boundary-case exceptions to the Second Incompleteness Theorem. The essential reason why these exceptions are of interest is that G\"{o}del's famous centennial paper has implicitly raised the following
puzzling issue: \begin{quote}
\# How is it that Human Beings
manage to muster the physical drive to think (and prove theorems) when the many generalizations of G\"{o}del's Second Incompleteness Theorem assert conventional logics lack knowledge of their own consistency? \end{quote} There will, of course, never be any perfect answer to the puzzle posed by $~\# ~$ because philosophical paradoxes and ironical dilemmas never yield perfect answers. However,
part of an imperfect
answer to $~\# ~$ is that Items 1-3 reply to
Challenges (a) and (b) by formalizing how a thinking being can muster an
approximate partial {\it instinctive faith} in its own self-consistency. (Moreover, the tight
contrast between various generalizations of the Second Incompleteness Theorem \cite{Ad2,AZ1,BS76,BI95,HP91,HB39,Ko6,Lo55,Pu85,Sa11,So94,Sv7,WP87,ww2,wwlogos,ww7} with the self-justifying systems appearing in Appendixes D and G suggests that these come close to being maximal forms of feasible results.) \end{remark}
Our remaining discussion will consist of four optional sections, called
Appendixes D, E, F and G,
which can be skimmed, omitted or examined in any
order
the reader
prefers. A
summary of their contents is given
below: \begin{description} \itemsep +4pt \item[ I ] The Appendix D provides four examples of generic configurations
that utilize Theorems \ref{ppp1}, \ref{ppp2}, \ref{pqq3}, \ref{pqq4} and \ref{pqq5}. Its most prominent examples involve \ep{totdefxa}'s Type-A axiom systems where the deduction method is either semantic tableaux or a modified version of tableaux that permits a modus ponens rule for $\Pi_1^\xi$ and $\Sigma_1^\xi$ sentences. \item[ II ] The Appendix E introduces a generalization of the Second Incompleteness Theorem which shows that \thx{ppp6}'s Translational Reflection Principle applies {\it only to} self-justifying logics. (It is thus fully inoperative for conventional logics. This may explain why \thx{ppp6}'s self-justifying systems are an interesting topic.) \item[ III ] The Appendix F differs from the rest of this paper by having a philosophical slant. It will offer a 3-page summary about why we suspect \thx{pqq3}'s self-justification formalism and Remark \ref{recc1}'s notion of ``instinctive faith'' are useful.
\item[ IV ] The Appendix G introduces a ``Braced$^\xi( \Phi ,j )$'' construct
and two new theorems that hybridize the methodologies of
Theorems \ref{pqq4} and \ref{pqq5}. These results will improve upon
Theorem \ref{pqq4} because their self-justifying systems contain only a finite number of axiom-sentences beyond those lying in $~\xi \,$'s base formalism of $B^\xi $. They will improve upon
Theorem \ref{pqq5} because they can prove the important
Braced$^\xi( \Phi ,j )$ subset of
$ ( \mathcal{B} , D ) $'s $\Pi_1^\xi$ theorems in a full sense
(rather than in Remark \ref{re4-2}'s weaker simulated respect). Appendix G's results are useful because {\it for arbitrary $k$} and for any of Appendix D's four sample configurations, every $\Pi_1^\xi$ theorem of $ ( \mathcal{B} , D ) $
containing $~k~$ or fewer bounded and unbounded quantifiers will be proven by its Theorem G.3 to be self-justifying in an undiluted pure sense. (This is because each such $\Pi_1^\xi$ sentence, with fewer than $~k~$ quantifiers, will lie in
some
fixed Braced$^\xi( \Phi ,j )$ set $~$---$~$ where solely the value of $~k~$ determines the values for
$~\Phi~$ and $~j ~$. )
\end{description}
\baselineskip = 1.35 \normalbaselineskip \baselineskip = 1.22 \normalbaselineskip
It is probably desirable
to
concentrate primarily
on
Theorems \ref{ppp1}, \ref{ppp2}, \ref{pqq3}, \ref{pqq4}, \ref{pqq5} and \ref{ppp6} during one's first reading of this paper. This is because Appendixes A-G are less central than these
core theorems, although their material does add several useful further perspectives to this subject.
\baselineskip = 1.22 \normalbaselineskip \parskip 3pt
\section*{7. Concluding Remarks}
The research in this article has been
a continuation of our prior research \cite{ww93}-\cite{ww9} that simultaneously has simplified, unified and extended the prior results. It has explored self-justification with a 3-part approach where: \begin{enumerate}
\item Sections \ref{3uuuu2} and \ref{3uuuu3} introduced three different
stem components that can be used to generate
self-justifying
systems. (These are
the relatively simple Lemma \ref{lemex4} and the mathematically more sophisticated Theorems \ref{ppp1} and \ref{ppp2}.) \item Section \ref{sect64} and Appendix G then generalized our initial stem-like theorems in the six
different directions formalized by Theorem \ref{pqq3}, \ref{pqq4},
\ref{pqq5},
\ref{ppp6}, G.2 and G.3 \item Appendix D subsequently provided four examples of generic configurations that are applications of Section \ref{sect64}'s results. \end{enumerate} This 3-part approach is very different from the methods used in our prior articles \cite{ww93,ww5,wwapal,ww9}. The latter examined
particular isolated applications in thorough detail (rather than compartmentalize and separate
the analysis into three stages). The virtue of this 3-stage analysis is it leads to many new theorems, in addition to unifying our prior results.
It is desirable to categorize the maximal generality and strongest form of boundary-case exceptions for the Second Incompleteness Theorem that are feasible because G\"{o}del's centennial discovery
beckons the scholarly community to sharpen their understanding of his 1931 landmark discovery, that has fundamentally
reshaped mathematics.
It should be emphasized that our over-all research in \cite{ww93} -- \cite{ww9}
has spent an approximately equal effort in
exploring generalizations of the Second Incompleteness Theorem \cite{ww2,wwlogos,wwapal,ww7} and in examining its viable boundary-case exceptions \cite{ww93,ww1,ww5,ww6,wwapal,ww9} (although the current article focused on the latter topic).
This is because the Second Incompleteness Theorem is a starkly robust result that
imposes sharp limits on how strong self-justifying systems may become.
Finally, we encourage the reader to take another brief glance at Remarks \ref{f88} -- \ref{recc1}. They offer a brief summary of both the strengths and limitations of our chief
results. They also explain how Theorem \ref{ppp6}'s reflection principle for $\Pi_1^\xi$ styled theorems is a very unexpected
result.
{\bf ACKNOWLEDGMENTS: }
I thank
Bradley Armour-Garb,
Seth Chaiken and
Kenneth W. Regan
for many useful suggestions about how to
improve
the presentation
style
of some earlier drafts of this article.
As was noted in the text of this article, some
of my research
during the last 17 years was influenced by some private
communications
that I had with Robert M. Solovay
in 1994 and
Leszek
Ko{\l}odziejczy in 2005.
\parskip 2pt
\baselineskip = 1.22 \normalbaselineskip
\section*{Appendix A: The $~\Pi_1^\xi~$ encoding for SelfCons$^k(\beta,d)$}
This appendix will summarize how to formalize a $\Pi_1^\xi$ encoding for Definition \ref{xd+1x7}'s SelfCons$^k(\beta,d)$ predicate. It will use the following notation: \begin{enumerate} \itemsep +1pt \baselineskip = 1.0 \normalbaselineskip \topsep -3pt \itemsep 4pt \item Neg$^k(x,y)$, will denote a $\Delta_0^{\xi}$ formula indicating that $~x~$ is the G\"{o}del number of a $\Pi_k^\xi$ sentence and that $~y~$ represents the $\Sigma_k^\xi$ sentence which is its logical negation. \item $\mbox{Prf} \, _\beta^d~( \, t \, , \, p \,)$ will denote a formula designating that
$~p~$ is a proof of theorem $~t~$ from the axiom system $~\beta~$ using the deduction method $~d.~$ \item $\mbox{ExPrf} \, _\beta^d\,( \, h \, , \, t \, , \, p \,)$ will denote that $ \, p \, $ is a proof (using $d$'s deduction method)
of a theorem $t$ from the union of the axiom system $\beta$ with the added sentence whose G\"{o}del number equals $\, h \,$. \item
$\mbox{Subst} \, ( \, g \, , \, h \, )$ will denote G\"{o}del's substitution formula --- which yields TRUE when $\, g \,$ is an encoding of a formula and $\, h \,$ encodes a sentence that replaces all occurrence of free variables in $g \,$ with a term of $\bar{g}$ (that specifies $g$'s G\"{o}del number). \item $\mbox{SubstPrf} \, _\beta^d~( \, g \, , \, t \, , \, p \,)$ will denote the hybridization of Items 3 and 4 that yields a Boolean value of TRUE when there exists an integer $~h~$ satisfying
$\mbox{Subst} \, ( \, g \, , \, h \, )$ and $\mbox{ExPrf} \, _\beta^d~( \, h \, , \, t \, , \, p \,)$. \end{enumerate}
It is easy to apply \cite{ww1}'s methodologies to confirm Items 1-5 can be encoded as $\Delta_0^\xi$ formulae. Thus, Appendixes C and D of \cite{ww1} explained how the theory of LinH functions \cite{HP91,Kr95,Wr78} implied there existed
$\Delta_0$ encodings for formulae 1-4, and these
$\Delta_0$ encodings can be easily rewritten
\footnote{ \baselineskip = 1.1 \normalbaselineskip This rewriting of conventional $\Delta_0$ formulae into a $\Delta_0^\xi$ format
is possible because Part 2 of Definition \ref{def3.3} indicated that two 3-way predicates of Add$(x,y,z)$ and Mult$(x,y,z)$ do encode addition and multiplication in a $\Delta_0^\xi$ styled form.} as
$ ~ \Delta_0^\xi ~$ expressions. \ep{encode}
uses this information to formulate a $\Delta_0^{\xi}$ encoding for $\mbox{SubstPrf} \, _{~\beta}^d \,( g , t , p )$'s graph. It is equivalent to $~$``$~\exists ~h~ \{ ~\mbox{Subst} ( g , h )~\wedge~ \mbox{ExPrf} \, _{\beta}^d( h , t , p )\, \} \, \,$''$,~$
but \ep{encode} is written in
a $\Delta_0^{\xi}$ format --- {\it unlike} the quoted expression. \begin{equation} \label{encode} \mbox{Prf} \, _{\beta}^d~( \, t \, , \, p \,)~~~\vee~~~\exists ~h\leq p ~~ \{ ~ \mbox{Subst} \, ( \, g \, , \, h \, )~\wedge~ \mbox{ExPrf} \, _{\beta}^d~( \, h \, , \, t \, , \, p \,)~ \} \end{equation}
Using \eq{encode}'s
$\Delta_0^{\xi}$ encoding for $\mbox{SubstPrf} \, _{\beta}^d( g , t , p )$, it is easy to encode SelfCons$^k(\beta,d)$
as a $\Pi_1^{\xi}$ axiom-sentence. Thus, let $ \, \Gamma^k(g) \, $ denote \eq{encode12}'s formula, and let $ \, \bar{n} \, $ denote $ \, \Gamma(g)$'s G\"{o}del number. \begin{equation} \label{encode12} \forall \, x \, \forall \, y \, \forall \, p \, \forall \, q ~~~ \neg ~~
\{ \,\mbox{Neg}^k(x,y) ~ \wedge~ \mbox{SubstPrf} \, _{\beta}^d \, ( g , x , p ) ~ \wedge ~ \mbox{SubstPrf} \, _{\beta}^d \, ( g , y , q ) \, \} \end{equation} Then $~$``$~\Gamma^k(~ \bar{n}~)~$''$~$ is a $\Pi_1^{\xi}$ encoding for SelfCons$^k(\beta,d)$'s formalization of the statement $+$ from Definition \ref{xd+1x7}. Thus, $~\Gamma^k(~ \bar{n}~)~$ is encoded is as follows: \begin{equation} \label{encode12n} \forall \, x \, \forall \, y \, \forall \, p \, \forall \, q ~~~ \neg ~~
\{ \,\mbox{Neg}^k(x,y) ~ \wedge~ \mbox{SubstPrf} \, _{\beta}^d \, ( \bar{n} , x , p ) ~ \wedge ~ \mbox{SubstPrf} \, _{\beta}^d \, ( \bar{n} , y , q ) \, \} \end{equation}
{\bf Reminder about \ep{encode12n} :} This sentence's definition for SelfCons$^k(\beta,d)$ does not assure \eq{encode12n} is true under the Standard-M model. Indeed for nearly all $(\beta,d)$, it will be false when $k \geq 2$. This is the reason that the study of SelfCons$^k(\beta,d)$, under Theorems \ref{ppp1} and \ref{ppp2}, has focused on the cases where $~k~$ equals 0 or 1. Moreover, the preceding construction did assure that SelfCons$^k(\beta,d)$ had a $~\Pi_1^\xi~$ encoding because such an {\it ``I am consistent''} axiom carries
more meaning than a $\Pi_2^{\xi}$ encoded axiom.
\parskip 0 pt
\section*{Appendix B: The Proof of Lemma \ref{llpp3}}
\baselineskip = 1.22 \normalbaselineskip
Lemma \ref{llpp3} is a crucial interim step used to verify each of Theorems \ref{pqq3}, \ref{pqq4} and \ref{pqq5}. Its proof will employ the following three straightforward observations: \begin{description} \item[Fact B.1 ] Lemma \ref{llpp3}'s hypothesis implies that
$\xi^\bullet$ is a generic configuration. (This is because it specified that
$\xi$ was a generic configuration and that all the $\Pi^\xi_1$ sentences of $~\theta^\bullet~$ were true in the
Standard-M model. Thus,
$\xi^\bullet$ must also be a generic configuration.) \item[Fact B.2 ] The associative identity of
$ \, \theta \, \cup \, (\theta^\bullet \, \cup B^\xi) \, \, = \, \, ( \, \theta \, \cup \, \theta^\bullet \, ) \cup B^\xi \, $ obviously holds. It implies a sentence $ \, \Upsilon \, $ is a theorem of
$ \, \theta \, \cup \, (\theta^\bullet \, \cup B^\xi) \, $ if and only if it is a theorem of
$ \, ( \, \theta \, \cup \, \theta^\bullet \, ) \cup B^\xi \, $.
\item[Fact B.3 ] Lemma \ref{llpp3}'s hypothesis directly \footnote{ \baselineskip = 1.0 \normalbaselineskip The identity of $~ ~ \sharp ( ~ \theta^\bullet ~)~=~\infty~$ must be true because the hypothesis of Lemma \ref{llpp3}
indicated that all the $\Pi^\xi_1$ sentences in $~\theta^\bullet~$ are true under the Standard-M model. By Definition \ref{chg}, this
implies $~ ~ \sharp ( ~ \theta ~)~= ~ ~ \sharp ( ~ \theta^\bullet \cup \theta ~)~$.} implies $ ~ \sharp ( ~ \theta )= ~ \sharp ( ~ \theta^\bullet \cup \theta )$. \end{description}
The justification of claims (i)-(iv) are
consequences of Facts B.1 through B.3. We will
provide a detailed proof of only Claim (i) here
because all four claims have similar proofs.
{\bf Proof of Claim (i) :} The hypothesis of Claim (i) indicated that $~\xi~$ was 0-stable. Therefore, it satisfies
Definition \ref{ostab}'s invariant of $~***~$. In a context where $~\phi ~$ is a variable designating a r.e. set of $\Pi_1^\xi$ sentences and $ \, \Upsilon \, $ is a variable corresponding to a $\Delta_0^\xi$ sentence,
the invariant $~***~$ can be rewritten in a quasi-rigorous form as : \begin{equation} \label{whatshit1} \forall ~ \phi ~~ \forall ~ \Upsilon
~~~~~ \mbox{\it the below statement, called $~ \Psi_1(\phi,\Upsilon)$, is true } \end{equation} \begin{quote} $\forall ~p~~$ If $ \, \Upsilon \, $ is a $\Delta_0^\xi$ theorem
derived from the axiom system $ \, \phi \, \cup B^\xi \, $ via a proof $ \, p \, , \, $ whose length satisfies
Log$(p) \, \leq ~ \sharp ( ~ \phi \, ) \, +1 \, , \, $
then $ \, \Upsilon \, $ is true under the Standard-M model. \end{quote} Since \eq{whatshit1}'s
universally quantified variable $~\phi~$ can designate any r.e. set of $\Pi_1^\xi$ sentences, it may designate the object ``$~\theta ~\cup~\theta^\bullet~$'', where $~\theta^\bullet~$ is the r.e. set of $\Pi_1^\xi$ sentences defined by Lemma \ref{llpp3}'s hypothesis (and $~\theta~$ is any second r.e. set of sentences). Thus, \eq{whatshit1} directly implies : \begin{equation} \label{whatshit2} \forall ~ \theta ~~ \forall \Upsilon
~~~~~ \mbox{\it the below statement, called $~ \Psi_2(\theta,\Upsilon)$, is true } \end{equation} \begin{quote} $~\forall ~p~~$ If $ \, \Upsilon \, $ is a $\Delta_0^\xi$ theorem
derived from the axiom system $~(~\theta~\cup~\theta^\bullet~)~ \cup B^\xi~$ via a proof $~p~,~$ whose length satisfies
Log$(p) \, \leq ~ \sharp ( ~ \, \theta\cup\theta^\bullet)~ +1, $
then $ \, \Upsilon \, $ is true under Standard-M. \end{quote} Facts B.2 and B.3 enable one to simplify \eq{whatshit2}'s
terms of $( \, \theta \, \cup \, \theta^\bullet \, ) \, \cup B^\xi$ and $~ \sharp ( ~ ( \, \theta \, \cup \, \theta^\bullet \, ) \, ~ )$ and thus to derive \eq{whatshit3} as a consequence. \begin{equation} \label{whatshit3} \forall ~ \theta ~~ \forall \Upsilon ~~~~
\mbox{\it the below statement, called $~ \Psi_3(\theta,\Upsilon)$, is true } \end{equation} \begin{quote} $~\forall ~p~~$ If $ \, \Upsilon \, $ is a $\Delta_0^\xi$ theorem
derived from axiom system $\theta~\cup~(~\theta^\bullet~ \cup B^\xi~)$ via a proof $~p$, whose length satisfies
Log$(p) \, \leq ~ \sharp ( ~ \theta) \, + \,1 $,
then $ \Upsilon $ is true under the Standard-M model. \end{quote} We will now use Fact B.1's observation that $\xi^\bullet$ is a generic configuration. The sentence \eq{whatshit3} indicates this configuration satisfies Definition \ref{ostab}'s invariant of $***$. Hence, $\xi^\bullet $
is 0-stable. $\Box$
\baselineskip = 1.22 \normalbaselineskip
{\bf Brief Comments about the Justifications of Claims (ii)-(iv):}
This appendix has omitted
proving
(ii)-(iv)
for the sake of brevity.
Their proofs are similar
to Claim (i)'s proof.
For instance, Claim (ii)'s proof differs
from Claim (i)'s proof by having
the 0-stability invariant in $***$
replaced
by the
A-stability invariant of $~*~$.
This will cause the analogs of
\eq{whatshit1} -- \eq{whatshit3}
to undergo the
following two
simple
changes under Claim (ii)'s proof :
\begin{enumerate}
\topsep -10pt
\itemsep +2pt
\item
$ \, \Upsilon \, $
will represent a
$\Pi_1^\xi$
(rather than a
$\Delta_0^\xi$ ) theorem-statement under the
revised versions of $\Psi_1$,
$\Psi_2$ and $\Psi_3$
used in Claim (ii)'s proof
\item
The requirement
(in
sentences \eq{whatshit1}-\eq{whatshit3} of Claim i's proof)
that
$ \Upsilon $ be
true in the Standard model
is
changed
to the stipulation that
$ \Upsilon $
satisfies
the Good$\{ \, ~ \frac{1}{2}~ ~ \sharp ( ~ \phi) \, \}$
and
Good$\{ ~ \frac{1}{2}~ ~ \sharp ( ~ \theta) \}$
conditions
under the revised forms of
$\Psi_1$,
$\Psi_2$ and $\Psi_3$
used to prove Claim (ii).
\end{enumerate}
Other
minor adjustments in Claim (i)'s proof
shall verify Claims iii and iv.
\section*{Appendix C: The Proof of \phx{ppp2} }
Our proof of \phx{ppp2} is a straightforward modification of \phx{ppp1}'s
proof. It will be divided into two lemmas.
{\bf Lemma C.\ref{B1-lem}.} {\it Every generic configuration that is either E-stable or A-stable will automatically satisfy Definition \ref{ostab}'s 0-stability condition.}
{\bf Proof.} $~$ Lemma C.\ref{B1-lem} is a consequence of the ``Special Note'' appearing at the end of Definition \ref{xd+1x4}. For every $N\geq 0$, it
indicated that if $\Upsilon$ is a $\Delta_0^\xi~$ sentence then Scope$_E$($\Upsilon,N)$ is equivalent to $~\Upsilon~$. This implies (via Definition \ref{xd+1x5}) that if $\Upsilon$ is a Good(N) $\Delta_0^\xi~$ formula then Scope$_E$($\Upsilon,N)$ is automatically true under the Standard-M model.
The latter observation makes it easy to confirm Lemma C.\ref{B1-lem}. This is because every $\Delta_0^\xi$ sentence is a $\Pi_1^\xi$ and $\Sigma_1^\xi$ statement. Hence, the application of the invariants $~*~$ and $~**~$ from Definitions \ref{astab} and \ref{estab}, in the degenerate case where $\Upsilon$ is a
$\Delta_0^\xi$ theorem, corroborates Lemma C.\ref{B1-lem}'s claim (by showing that
Definition \ref{ostab}'s
invariant of
$\,***\,$ does hold). $~~\Box$
The remainder of this appendix will focus on Definition \ref{ostab}'s 0-stability condition. (This is sufficient to justify \thx{ppp2} because Lemma C.\ref{B1-lem} showed all
E-stable and A-stable configurations are
0-stable.)
{\bf Lemma C.\ref{B2-lem}.} {\it Let $~\xi~$ denote a generic configuration
$( L^\xi , \Delta_0^\xi , B^\xi , d , G )~$ that is 0-Stable. Then the axiom system of $B^\xi+$SelfCons$^0(B^\xi,d)$ will be consistent (and hence self-justifying).}
{\bf Proof:} It will be awkward to write repeatedly the expression ``$~B^\xi+$SelfCons$^0(B^\xi,d)~$'' during our proof. Therefore, $H $ will be an abbreviated name for this system. Our justification
of Lemma C.\ref{B2-lem}, is similar to \phx{ppp1}'s proof, $\,$except it replaces a Level$(1^\xi)$ form of self-justification with a Level($0^\xi)$. It will thus
be
abbreviated and use the following notation: \begin{enumerate} \itemsep +4pt \item $~\mbox{Prf}_H (t,p)~$ is a $\Delta_0^\xi$ formula specifying $~p~$ is
a proof of the theorem $~t~$ from the axiom system $~H ~$ (using $~\xi \,$'s deduction method of $d$ ). \item $~\mbox{Neg}^0(x,y)~$
is a $\Delta_0^\xi$ formula
indicating $~x~$ is the G\"{o}del encoding of a
$\Delta_0^\xi$ sentence and $~y~$ is a $\Delta_0^\xi$ sentence representing $~x \,$'s negation. \end{enumerate} Expression \eq{qpencode} denotes
$~H \,$'s
Level$(0^\xi)$ self-justification axiom. It is encoded using Appendix A's methodology, similar to its counterpart used in \phx{ppp1}'s proof (i.e. \ep{pencode} ). \begin{equation} \label{qpencode} \forall \, x \, \forall \, y \, \forall \, p \, \forall \, q ~~~ \neg ~~
\{ \,\mbox{Neg}^0(x,y) ~ \wedge~ \mbox{Prf}_H \, ( x , p ) ~ \wedge ~ \mbox{Prf}_H \, ( y , q ) \, \} \end{equation}
Our proof of Lemma C.\ref{B2-lem} will be a proof by contradiction. It will thus
begin with the contrary assumption that $~H ~$ is inconsistent and have $~\Phi~$ denote \eq{qpencode}'s sentence. The inconsistency of $~H ~$
implies that $~\Phi~$ is false under the Standard-M model. Hence via Definition \ref{chg}, we get: \begin{equation} \label{qpunk}
~ \sharp ( ~ \Phi~)
~~ < ~~ \infty \end{equation} \noindent \ep{qpunk} implies there exists a tuple $(\bar{p},\bar{q},\bar{x},\bar{y})$ satisfying \eq{qpencodepunk}. (This is because such a $(\bar{p},\bar{q},\bar{x},\bar{y})$ corroborates \eq{qpunk}'s implication that a counter-example to \eq{qpencode}'s sentence does exist.) \begin{equation} \label{qpencodepunk}
\,\mbox{Neg}^0(~ \bar{x}~, ~ \bar{y}~) ~ \wedge~ \mbox{Prf}_H \, ( ~ \bar{x}~ , ~ \bar{p}~ ) ~ \wedge ~ \mbox{Prf}_H \, ( ~ \bar{y}~ , ~ \bar{q}~ ) \end{equation}
The $(p,q,x,y)$
satisfying \eq{qpencodepunk} with minimum value for $ \mbox{Log}\{\, \mbox{Max}[p,q,x,y] \, \}$ will additionally
satisfy \eq{qpunkless}. (This observation follows from the analog of the Footnote \ref{footp1} appearing in \phx{ppp1}'s proof. Thus, Section \ref{3uuuu3}'s Equations of \eq{pencodepunk} and \eq{punkless} are the analogs of the current
Equations \eq{qpencodepunk} and \eq{qpunkless}. The Footnote \ref{footp1} showed the particular $(p,q,x,y)$
satisfying \eq{pencodepunk} with minimum value for $ \mbox{Log}\{~\mbox{Max}[p,q,x,y] ~\}$ satisfied \eq{punkless}. By the same reasoning, the minimal $(\bar{p},\bar{q},\bar{x},\bar{y})$ satisfying \eq{qpencodepunk} will satisfy \eq{qpunkless}.) \begin{equation} \label{qpunkless}
\mbox{Log}~\{~\mbox{Max}[~\bar{p}~,~\bar{q}~,~\bar{x}~,~\bar{y}~] ~~\}
~~~ =~~~~ \sharp ( ~ \Phi~) ~~+~~1 \end{equation}
Equations \eq{qpencodepunk} and \eq{qpunkless} shall bring Lemma C.\ref{B2-lem}'s
proof-by-contradiction to its sought-after
end. Thus, let $~\Upsilon~$ denote the
$\Delta_0^\xi$ sentence specified by $~\bar{x}~$. Then $~\neg ~ \Upsilon~$ corresponds to the
$\Delta_0^\xi$
sentence denoted by $~\bar{y}~$. \ep{qpunkless} indicates that both
$~\Upsilon~$ and $~\neg ~ \Upsilon~$ have proofs such that the logarithms of their G\"{o}del numbers are bounded by $~~ \sharp ( ~ \Phi~) ~+~1~$. Using Definition \ref{ostab}'s invariant of $~***~$, these facts establish \footnote{ \baselineskip = 1.0 \normalbaselineskip To apply Definition \ref{ostab}'s
invariant $~***~$ in the present setting, one simply sets $~\theta\,$'s R-View equal to $~\Phi \,$'s 1-sentence statement. Then $~***~$ implies that both
$~\Upsilon~$ and $~\neg ~ \Upsilon~$ must be true under the Standard-M model because both their proofs had lengths $\, \leq ~~ \sharp ( ~ \Phi~) ~+~1~$. } that both
$~\Upsilon~$ and $~\neg ~ \Upsilon~$ are true under the Standard-M model.
But it is impossible for
a sentence and its negation to be both true. This finishes Lemma C.\ref{B2-lem}'s proof because the temporary
assumption that $H $ was inconsistent has led to a contradiction. $~~\Box$
\phx{ppp2} is a
consequence of the Lemmas C.\ref{B1-lem} and
C.\ref{B2-lem} because Lemma
C.\ref{B2-lem}'s formalism generalizes to all E-stable and A-stable configurations via Lemma C.\ref{B1-lem}'s reduction methodology.
\section*{Appendix D: Applications and Examples}
This appendix will illustrate four examples of generic configurations that satisfy Theorems \ref{ppp1} and \ref{ppp2} (and which therefore are self-justifying). It will be divided into three parts. Section D-1 will define our first example of self-justifying configuration, called $~\xi^*~$. It will use semantic tableaux
deduction. Section D-2 will prove that $~\xi^*~$ is EA-stable. Section D-3 will briefly sketch
three additional examples of
stable generic configurations
It is likely preferable to examine Sections \ref{3uuuu1} -- \ref{sect64} before this appendix. However, Section
D-1's
short 2-page discussion can be
read quite easily either before or
after Section \ref{sect64}.
\subsection*{D-1. $~$Definition of the EA-Stable Configuration
$~\xi^*~$ }
Our first example of an EA-Stable configuration,
called $\, \xi^* \,$, will be defined in this section. Its deduction method will be semantic tableaux. Its
base axiom system $\, B^* \,$ will be a
Type-A formalism, which treats addition but not multiplication as a total function (i.e. see \ep{totdefxa} $~).$
The closest analog of $B^*$ and $\, \xi^* \,$
in our prior work
appeared in Section 5 of \cite{ww5}.
(It differed from $\, \xi^* \,$ partly because it
did not use Definition \ref{def3.3}'s unifying notation.) In \cite{ww5},
a function
$\, F \, $ was called
{\bf Non-Growth} iff $ F(a_1,a_2,...a_j) \leq Maximum(a_1,a_2,...a_j)$ for all $a_1,a_2,...a_j$. Six examples of non-growth functions are: \begin{enumerate} \item {\it Integer Subtraction} where ``$~x-y~$'' is defined to equal zero in {\it the special case} where
$~x \leq y,$ \item {\it Integer Division} where ``$~x \div y~$'' equals $~x~$ when $~y=0,~$ and it equals $~\lfloor ~x/y ~\rfloor~$ otherwise, \item $Root(x,y)~$ which equals $~ \lceil ~x^{1/y}~ \rceil$ when $~y\geq 1,~$ and it equals $~x~$ when $~y=0.$ \item $Maximum(x,y),~~$ \item $ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$ when $~x \geq 2,~$ and zero otherwise. \item $Count(x,j)~~=~~$the number of ``1'' bits among $~x$'s rightmost $~j~$ bits. \end{enumerate} These operations were called {\bf Grounding Functions} in \cite{ww5}. The term {\bf U-Grounding Function} referred to a set of functions that included the Grounding operations
plus the further primitives of addition and {\it Double$(x)=x+x$.} (The Double operation is helpful because it significantly
enhances \cite{ww5}'s linguistic efficiency \footnote{ \baselineskip = 1.0 \normalbaselineskip The symbol {\it Double$(x)$} was technically unnecessary in \cite{ww5}'s formalism because $x+x$ can encode
Double$(x)$. However, its notation adds expressive power to \cite{ww5}'s language because, for example, Double(Double(Double(Double(x))) requires less memory space to encode than than $~x~$ added to itself 16 times. } .)
The symbol $~\Delta_0^*~$ will be the analog of Definition \ref{xd+1x1}'s
$\Delta_0^\xi$ construct under the U-Grounding function language. (It will be defined to be any formula in a U-Grounding language where all its quantifiers are bounded.) It is easy in this context to encode a $\Delta_0^{ *}$ formula $Mult(x,y,z)$ for representing
multiplication's graph. For instance, \ep{neweq1} is one such $\, \Delta_0^{ *} \, $ formula (which actually does not employ any bounded quantifiers): \begin{equation} \label{neweq1} [~(x=0 \vee y=0 ) \Rightarrow z=0~ ]~ ~\wedge ~~ [~(x \neq 0 \wedge y \neq 0~) ~ \Rightarrow ~ (~ \frac{z}{x}=y ~\wedge \, ~ \frac{z-1}{x}<y~~)~] \end{equation} Expression \eq{neweq1} is significant because Part 2 of Definition \ref{def3.3} indicated every generic configuration must have
available some method to represent the graphs of addition and multiplication in a
$\Delta_0^\xi$ styled format, similar to \eq{neweq1}'s paradigm.
(Addition can be treated trivially because the U-grounding language
possesses an addition function symbol.) The footnote
\footnote{ \baselineskip = 1.0 \normalbaselineskip Section \ref{3uuuu1} explained during its discussion of Lemma \ref{lex22}
that every generic configuration $ \xi $ must have a means to encode the graphs of addition and multiplication as $ \Delta_0^{\xi} $ formulae (visavis Parts 1 and 2 of
Definition \ref{def3.3}). This enabled Lemma \ref{lex22}'s procedure to translate all of conventional arithmetic's $\Sigma_j$ and $\Pi_j$ formulae into equivalent $\Sigma_j^{\xi}$ and $\Pi_j^\xi$ expressions. \fend }
serves as a reminder about why these $\Delta_0^*$ encodings are needed. Our next goal is to define the generic configuration
$\, \xi^* \,$ that
Section D-2 will prove is EA-stable.
{\bf Definition D.\ref{D1-def}.} The language
$~L^{*}~$ of the generic configuration
$\, \xi^* \,$ will be built in a natural manner out of the eight U-grounding function operations, the usual atomic predicate symbols of ``$~=~$'' and
``$~\leq~$'', and the three constant symbols
$~K_0,~ K_1\,$ and $\,K_2\,$ (that define the integers of 0, 1 and 2). The other components of $\, \xi^* \,$ 's configuration
are defined below: \begin{description} \baselineskip = 1.22 \normalbaselineskip \itemsep 4pt \item[ i ] As previously noted,
$~\Delta_0^{*}~$ is defined to represent the set of all formulae in $ \, L^* \,$'s language, whose quantifiers are bounded in an arbitrary
manner by terms employing the U-Grounding function symbols. (It will thus generate via Definition \ref{xd+1x1} the $\Pi_n^*$ and $\Sigma_n^*$ sentences of $ \, L^* \,$.) \item[ ii ] The base axiom system
$~B^*~$ for $\, \xi^* \,$ will be allowed to be any consistent set of $\Pi_1^{\xi^*}~$ sentences that is capable of proving every
$\Delta_0^*$ sentence that is valid in Standard-M. It will also include sentence
\eq{d1a}'s {\it very precise} \footnote{ \baselineskip = 1.0 \normalbaselineskip \label{fodii} The two appearances of the term ``$~x+y~$'' in sentence \eq{d1a} may at first appear to be redundant. (This statement is equivalent to sentence \eq{totdefsymba}'s declaration that addition is a total function, which had avoided such redundancy.) The virtue of \eq{d1a}'s format is that it is a
$\Pi_1^*$ styled statement, unlike \eq{totdefsymba}'s
$\Pi_2^*$ styled format. This sharpened
$\Pi_1^*$ perspective will help simplify some of our proofs. } $\Pi_1^*$ styled declaration that addition is a total function. \begin{equation} \label{d1a} \small \forall x ~~~~~~\forall y~~~~~~ \exists~ z \leq x+y ~~:~~~ ~~~
\{~~~z=x+y~~~\} \end{equation} \item[ iii ] $\, \xi^* \,$'s deduction method will be the semantic tableaux method. \item[ iv ] $\, \xi^* \,$' s G\"{o}delized method $~g~$ for encoding a semantic tableaux proof
can be essentially any natural method
that satisfies the minor stipulation that
at least $~5 \, J~$ bits are required
to encode a semantic tableaux proof that has $~J~$ function symbols. This stipulation is called the {\bf ``Conventional Tableaux Encoding Requirement''}. It is trivial \footnote{ \baselineskip = 1.0 \normalbaselineskip The Conventional Tableaux Encoding Criteria requires that the
G\"{o}del number of a semantic tableaux proof, with $~J~$ function symbols,
must be least as large as $~32^J~$. It is clear that all the usual methods for generating the G\"{o}del codes satisfy this criteria. This is because any proof that has $~J~$ function symbols will contain at least
$~2 \, J~$ logical symbols and thus employ at least
$~5 \, J~$ bits. \fend } to corroborate
that all the usual methods for encoding semantic tableaux proofs satisfy this criteria. (The Appendix A of \cite{ww5} provides one example of a possible tableaux encoding method. Any other natural mechanism for encoding tableaux proofs is equally suitable.) \end{description}
Section D-2 will, interestingly, prove that $\xi^*$ is EA-stable. This will imply (via \phx{ppp1}) that $~B^*~\cup~\mbox{SelfCons}^1\{~B^*~,d~\}$ is self-justifying. Theorems \ref{pqq4}, \ref{pqq5}, G.2 and G.3 will, formalize, in this context, four different methods in which $~B^*~$ can be extended to construct self-justifying formalisms that are able to prove Peano Arithmetic's $~\Pi_1^*$ theorems.
Thus while self-justifying axiom systems contain unavoidable weaknesses, they also possess the nice feature that they are able to prove many of the useful theorems of mathematics.
\subsection*{D-2. $~$Proof of the EA-Stability of
$~\xi^*~$ }
This section
will prove $\, \xi^* \,$ is EA-stable and thus satisfies the paradigms of Theorems \ref{ppp1}, \ref{pqq3}, \ref{pqq4}, \ref{pqq5} and \ref{ppp6}. Our proof will be based on modifying some of the methodologies from \cite{ww5}, so that they become applicable to $\, \xi^* \,$ . Many readers may prefer to omit examining both this part of Appendix D and Section
D-3 because they are unnecessary for understanding the
material in Section \ref{sect64} and Appendixes E and F. (Our recommendation is that the latter material be read first.)
Our notation for defining a
semantic tableaux proof
in the next paragraph will be similar to the conventional
definitions appearing in Fitting's and Smullyan's textbooks \cite{Fi90,Sm68}. It will employ \cite{ww5}'s notation so that we can employ two of its lemmas during our analysis of semantic tableaux proofs.
{
In our discussion, $~\Phi~$ will be called a {\bf Prenex-Level($\,$m$^* \,$)}
sentence
iff it is a
$\Pi_m^*$ or $\Sigma_m^*$ expression that satisfies the usual prenex requirement (that all its unbounded quantifiers lie in its leftmost part). If $\Phi$ is Prenex-Level$(m^* )$ then {\bf Reverse($\Phi$)} shall denote a second
Prenex-Level$( \, m^* \,)$ sentence that is equivalent to $~\neg~\Phi~$ rewritten \footnote{ \baselineskip = 1.0 \normalbaselineskip For example, if $\Phi$ denotes ``$~\forall \, x ~ \exists \,y ~~\psi(x,y)~$'' then Reverse$(\Phi)$ would be written as
``$~ \exists \, x ~ \forall \,y ~~\neg~\psi(x,y)~$''.}
in a
Prenex Level$( \, m^* \, )$ form. For a fixed axiom system $ \, \alpha , \, $ its {\bf $\Phi$-Based Candidate Tree} will be defined to be a tree structure whose root is the sentence Reverse$(\Phi)$
and whose all other nodes are either axioms of $ \, \alpha \, $ or deductions from higher nodes of the tree, via the rules 1--8 given below. (The symbol ``{\bf $ \, $A$ ~ \Longrightarrow ~ $B$ \, $}'' in rules 1-8 will mean that {\bf $ \, $B$ \, $} is a valid deduction from its ancestor {\bf $ \, $A$ \, $} in the germane
deduction tree.) \begin{enumerate} \itemsep 5pt
\item $~ \Upsilon \wedge \Gamma \, ~ \Longrightarrow ~ \, \Upsilon ~$ and $~ \Upsilon \wedge \Gamma \, ~ \Longrightarrow ~ \, \Gamma ~$ . $~~~$ \item $ \, \neg \,\neg \, \Upsilon \, \Longrightarrow \, \Upsilon. \, $ Other rules for the ``$ \, \neg \,$'' symbol are: $ \, \neg ( \Upsilon \vee \Gamma ) \, \Longrightarrow \, \neg \Upsilon \wedge \neg \Gamma$,
$ \, \neg ( \Upsilon \rightarrow \Gamma ) \, \Longrightarrow \, \Upsilon \wedge \neg \Gamma \, $, $ ~~~~\, \neg ( \Upsilon \wedge \Gamma ) \, \Longrightarrow \, \neg \Upsilon \vee \neg \Gamma \, $,
$~ \, \neg \, \exists v \, \Upsilon (v) \, \Longrightarrow \, \forall v \neg \, \Upsilon (v) \, $
and $ ~~\, \neg \, \forall v \, \Upsilon (v) \, \Longrightarrow \, \exists v \, \neg \Upsilon (v)$ \item A pair of sibling nodes $~ \Upsilon ~$ and $~ \Gamma ~$ is allowed when their ancestor is $~\Upsilon \, \vee \, \Gamma.~$ \item A pair of sibling nodes $ \, \neg \Upsilon \, $ and $ \, \Gamma \, $ is allowed when their ancestor is $ \, \Upsilon \, \rightarrow \, \Gamma$. \item $~ \exists v \, \Upsilon (v) ~ \Longrightarrow ~ \, \Upsilon(u) ~$ where $\,u \,$ is a newly introduced ``Parameter Symbol''. \item
$~ \exists v \leq s ~ \, \Upsilon (v) ~~ \Longrightarrow ~ ~ u \leq s ~ \wedge~ \Upsilon(u) ~$ is the
variation of Rule 5 for bounded existential quantifiers of the form $~$``$~ \exists v \leq s ~$''. \item $\forall v \, \Upsilon (v) \, \Longrightarrow \, \Upsilon(t) \, $ where $t$ denotes a U-Grounded term. These terms may be any one of a constant symbol, a parameter symbol (defined by a prior application of Rules 5 or 6 to some
some ancestor of the current node), or a U-Grounding function-symbol with recursively defined inputs. \item
$\forall v \leq s \, \Upsilon (v) ~~ \Longrightarrow ~~ t \leq s \, \rightarrow \, \Upsilon(t) $ is the variation of Rule 7 for a
bounded quantifier such as
``$~ \forall v \leq s ~ $'' \end{enumerate} }
\noindent Let us say a leaf-to-root branch (in a candidate tree) is {\bf Closed} iff it contains both some sentence $ \, \Upsilon \, $ and its negation ``$ \, \neg \, \Upsilon \, $''. Then a
{\bf Semantic Tableaux Proof} of $ \Phi $, from the axiom system $ \alpha $, is defined to be a $\Phi$-Based Candidate Tree whose every
leaf-to-root branch is closed.
\baselineskip = 1.22 \normalbaselineskip
It is next helpful to define the notion of a {\bf Z-Based Deduction Tree}, in a context where $~Z~$ represents an
axiom system, typically different from the prior paragraph's $~\alpha~$. This object will be defined to be identical to a semantic tableaux proof, except for the following changes: \begin{description}
\itemsep 5pt \item[ i ] {\it Every node} in a $Z-$Based deduction tree must be either an axiom of $~Z~$ or a deduction from a higher node of the tree via the rules 1-8. (This applies also to the root of a $Z-$Based deduction tree. It will store an axiom of $~Z~$
in its root, unlike a semantic tableaux proof
which had stored
Reverse$(\Phi)$ in its root.) \item[ ii ] There will be no requirement that each
leaf-to-root branch be
closed in a $Z-$Based deduction tree. (Indeed, some branch will automatically not be closed if $~Z~$ is consistent.) \end{description} Items (i) and (ii) make it apparent
that $Z-$Based deduction trees are different
from semantic tableaux proofs. It will turn out, nevertheless,
that the study of $Z-$Based deduction trees will clarify the nature of
semantic tableaux proofs.
{\bf Definition D.\ref{D2-def}.} Let $ ~ a ~ $ and $ ~ b ~ $ denote two integers that are powers of $\, 2 \, $ satisfying $ a \, > b \, \geq \, 2 \, $ Then
an axiom system $ \, Z \, $ (employing $ \, L^* \,$'s language) will be called a {\bf Normed(a,b)} formalism iff: \begin{enumerate} \item All $Z$'s axioms are either $\Pi_1^{*}$ or $\Sigma_1^{*}$ sentences. \item Each $\Pi_1^{*}$ axiom of $~Z~$ will satisfy Definition \ref{xd+1x5}'s Good($~$Log$_2a~)$ criteria, and
each
$\Sigma_1^{*}$ axiom of $~Z~$ will likewise
satisfy Good($~$Log$_2b~)$. \end{enumerate}
{\bf Clarification about Definition D.\ref{D2-def} : } The ``Normed(a,b)'' concept (above) is obviously equivalent to the same-named notion appearing in Definition 4 of \cite{ww5}. It uses, however,
a different notation to make it compatible with Section \ref{3uuuu2}'s formalism. Thus, Item 2's assertion that the
$\Pi_1^{*}$ axiom $ \, \forall \, v_1 \, \, \forall \, v_2 \, \, ... \forall v_k \, \, \, \, \phi(v_1,v_2,...v_k) \, \, $ satisfies Good($~$Log$_2a~)$ is equivalent to \eq{normscopede}'s statement. The Good($\,$Log$_2b\,)$ property of $\exists \,v_1\,\exists \,v_2\, ...\,\exists \,v_k\, ~~\phi(v_1,v_2,...v_k)~$ is, likewise,
equivalent to \eq{normscopedx}.
\begin{equation} \label{normscopede} \forall ~ v_1~ < ~a~~\forall ~ v_2~ < ~a~~ ... \forall ~ v_k~ < ~a
~~~~~ : ~~~~~ \phi(v_1,v_2,...v_k)~~. \end{equation}
\begin{equation} \label{normscopedx} \exists ~ v_1~ < ~b~~\exists ~ v_2~ < ~b~~ ... \exists ~ v_k~ < ~b
~~~~~ : ~~~~~ \phi(v_1,v_2,...v_k)~~. \end{equation}
\parskip 5pt
Our interests in this notation will center around Fact D.3's invariant:
{
{\bf Fact D.3$~.$} Let $ \, \xi^* \, $ denote
Definition D.\ref{D1-def}'s generic configuration, and $ Z $ be an extension of $ \, \xi^* \,$'s base axiom system $ B^* $ which satisfies Definition D.\ref{D2-def}'s Normed$(a,b)$ constraint. Then any $Z-$Based deduction tree $\, T \, $that has a G\"{o}del number smaller than
$ \, (a/b)^4 \,$ must contain at least one root-to-leaf branch, called $ \, \sigma \, \, , \, $ that is not ``closed''. {\rm (In other words, this path $ \, \sigma \, $ will be contradiction-free, insofar as it does not contain both some sentence $~\Psi~$ and its formal negation).} }
{\bf Proof:}
$~$ The
justification of
Fact D.3 is
a direct
consequence
of
the Lemmas 1 and 2 appearing in article \cite{ww5}
(see footnote
\footnote{\label{newtry}
\baselineskip = 1.3 \normalbaselineskip
A
proof of Fact D.3 from first principles would
be
quite
complicated because there are eight elimination rules employed
by semantic tableaux deduction, $~$each
of which needs to
be
examined by such a proof's umbrella
formalism. Fortunately, we do not need provide such a
complicated analysis here
because
a 4-page proof of
the Lemmas 1 and 2
in Section 5.2
of \cite{ww5}
had
already
visited these issues.
Thus,
Fact D.3
turns out to be an easy consequence
of these two lemmas
after the following two straightforward issues are addressed:
\begin{enumerate}
\topsep -4pt
\itemsep -2pt
\item Section 5.2 of \cite{ww5}
had
defined the $\,$``U-Height''$\,$ of a deduction tree to
be the
largest
number of
U-Grounding function symbols
that appear in any of its root-to-leaf
branches. Its Lemma 1
proved that
every deduction tree with a U-Height $\, \leq \, Log_2a \, - \, Log_2b \, $
will contain at least one branch
satisfying a condition,
which \cite{ww5}
called ``Positive(a,b)''.
The Lemma 2 in \cite{ww5} then showed that this
Positive(a,b) property implies that the germane
deduction tree must contain some branch that is
contradiction-free.
The combination of these two lemmas
thus amounts to the establishing of
the following
rephrased
hybridized statement:
\begin{quote}
$\bullet~~~$
If a Z-based deduction
tree has
a U-Height $\, \leq \, Log_2a \, - \, Log_2b \, $,
then some branch of it
is contradiction-free
(i.e. this branch cannot
contain both some
sentence $~\Psi~$ and its
negation).
\end{quote}
\item
Fact D.3 's hypothesis indicated the
G\"{o}del number $g$ for its deduction tree
satisfied the following
conditions:
\begin{description}
\topsep -7pt
\itemsep -1pt
\item[ I. ]
$~~g~\leq ~ (a/b)^4~$
\item[ II. ]
The U-Height of $~g \,$'s deduction tree
is less than $~\frac{1}{5}~$Log$_2~g~$. (This
is
simply
because
Fact D.3 presumes that
the ``Conventional Tableaux Encoding'' methodology
from Part-iv of
Definition D.\ref{D1-def}
was used to encode $~g$'s
G\"{o}del number.)
\end{description}
Items
I and II imply
$g \,$'s tree has
a U-Height $\, \leq \, Log_2a \, - \, Log_2b $.
The invariant $\, \bullet \,$
then implies
this deduction tree
has at least one branch
that is contradiction-free (as Fact D.3
claimed). $\Box$
\end{enumerate}
We emphasize that the
above
justification
of Fact D.3 is
{\it much simpler}
than a proof from first principles.
The
latter would
require examining
eight different
tableaux
elimination rules,
as the detailed
proofs of \cite{ww5}'s
Lemmas 1 and 2
actually
did do.}
for more details). $~~\Box$
We will now apply Fact D.3 to prove Theorem D.\ref{D.4-theorx}. Its invariant will, interestingly, collapse entirely \footnote{ \baselineskip = 1.08 \normalbaselineskip $~$The difficulty posed by
multiplication
can be easily understood when one compares two integers sequences
$ x_0, x_1, x_2, ... $ and $ y_0, y_1, y_2, ... $, defined as follows: \begin{center} $x_{i}~~=~~x_{i-1}+x_{i-1}~~~~~$ $~~$ AND $~~$ $~~~~~y_{i}~~=~~y_{i-1}*y_{i-1}$ \end{center} It turns out that the faster growth rate of multiplication under the series $~ y_0, y_1, y_2, ...~ $ enables one to to construct tiny $Z-$Based deduction trees $\, T$ that violate the analog of Fact D.3 's paradigm. (This is because such trees
can have G\"{o}del numbers smaller than
$ \, (a/b)^4 \,$, while all their root-to-leaf branches can be simultaneously ``closed'' via contradictions.) This property of
multiplication is analogous to Example \ref{ex3-3}'s observations about
how the differing growth rates of $ x_0, x_1, x_2, ... $ and $ y_0, y_1, y_2, ... $ are related to the threshold where the semantic tableaux version of
Second Incompleteness Theorem can be evaded. \fend }
if one were to merely add a multiplication function symbol to the
U-Grounding language. This is why our boundary-case exceptions to the semantic tableaux version of the Second Incompleteness allow a Type-A axiom system to recognize addition as a total function (but suppress a similar
treatment of
multiplication).
{\bf Theorem D.\ref{D.4-theorx}.} {\it The generic configuration $\xi^*$ is both A-stable and E-stable.} {\rm (This implies many different self-justifying formalisms exist via Theorems \ref{ppp1}, \ref{pqq3}, \ref{pqq4}, \ref{pqq5},
\ref{ppp6},
G.2 and G.3.)}
\baselineskip = 1.22 \normalbaselineskip
Our proof of Theorem D.\ref{D.4-theorx} will separately show
$\xi^*$ is A-stable and E-stable.
\parskip 3pt
{\bf Proof of
$ \, \xi^* \,$'s A-stability :} Suppose for the sake of establishing a proof by
contradiction that $ \, \xi^* \, $ was not A-stable. Then the constraint $ \, * \, $ of Definition \ref{astab} would be violated by at least
some $\theta \, \in \,$RE-Class$(\xi)$. This violation
will cause the statement $ \, + \, $ to be true for such a
$\theta~$: \begin{description} \item[ + ] There exists a semantic tableaux
proof $~p~$ of a
$~\Pi_1^*~ $ theorem, called say $ \, \Upsilon ~,~ $ from
the axiom system of $~\theta \cup B^\xi~$ such that
Log$(p)~\leq ~ \sharp ( ~ \theta)~+1~ $ and where
$ \, \Upsilon \, $ also fails to satisfy
Good$\{~ \, ~ \frac{1}{2}~ ~ \sharp ( ~ \theta) \, ~\}~$. \end{description} Let us recall that if $\Upsilon$ is $\Pi_1^*\,$ then Reverse$(\, \Upsilon \,)~$ is a $\Sigma_1^*$ sentence equivalent to $~\neg \Upsilon~$. Thus, Reverse$(\, \Upsilon \,)~$ will satisfy Good$\{~ \, ~ \frac{1}{2}~ ~ \sharp ( ~ \theta) \, ~\}~$ criteria (simply because it
has the opposite goodness property as $\,\Upsilon \,$ ). Also, if $~Z~$ denotes the axiom system of
$~\theta \cup B^\xi \, + \, $Reverse$(\, \Upsilon \,)$, it is easy to verify \footnote{ \baselineskip = 1.0 \normalbaselineskip \label{jnorm} The axiom system
$Z$ must satisfy Normed$\{~2^{ ~ \sharp ( ~ \theta \, )} ~$,$~\sqrt{~2^{ ~ \sharp ( ~ \theta \, )}}~\}$ because: \begin{enumerate} \item The quantity $~2^{ ~ \sharp ( ~ \theta \, )}$ is a valid first component for $Z$'s norming constraint because all the axioms of $~B^\xi~$ are true in the Standard-M model and because Definition \ref{chg} implies all of $ \theta \,$'s axioms satisfy Good $\{\,~ \sharp ( ~ \theta) \,\}$. \item The quantity $ ~\sqrt{~2^{ ~ \sharp ( ~ \theta \, )}} ~$ is a valid second component for $Z$'s norming constraint because
Reverse$(\, \Upsilon \,)~$ is the only $\Sigma_1^*$ sentence belonging to Z, and because Reverse$(\, \Upsilon \,)~$ satisfies
Good$\{~ \, ~ \frac{1}{2}~ ~ \sharp ( ~ \theta) \, ~\}~$. \end{enumerate}} that $~Z\,$'s axioms will satisfy the Normed$\{~2^{ ~ \sharp ( ~ \theta \, )} ~$,$~\sqrt{~2^{ ~ \sharp ( ~ \theta \, )}}~\}$ criteria.
It is next helpful to observe that what is a proof from one perspective corresponds to being a deduction tree from a different perspective. Thus, Item $~\, + \,$'s proof
$~p~$ of the theorem $ \, \Upsilon \,$ from
the axiom system of $~\theta \cup B^\xi~$ corresponds to being a Z-based deduction tree, with $~Z~$ representing the axiom system of $~\theta \cup B^\xi \, + \, $Reverse$(\, \Upsilon \,)$. In this context, Item $\, + \,$'s inequality of
Log$(p)~\leq ~ \sharp ( ~ \theta \, )~+1~ $ implies \footnote{ \baselineskip = 1.0 \normalbaselineskip \label{fd7} Without loss of generality, we may assume that every non-trivial proof $~p~$ satisfies
Log$(p)~\geq ~64~$ (since a string with fewer than 64 bits is too short to be a proof). Then the footnoted paragraph's
Log$(p)~\leq ~ \sharp ( ~ \theta)~+1~ $ inequality trivially implies $~p~<~3^{ ~ \sharp ( ~ \theta \, )}~$. In a context where
$~Z~$ is a Normed$\{~2^{ ~ \sharp ( ~ \theta \, )} ~$,$~\sqrt{~2^{ ~ \sharp ( ~ \theta \, )}}~\}$ axiom system, the
latter inequality certainly implies $~p~$, viewed as a deduction tree for $~Z~,~$ has a small enough G\"{o}del number to
satisfy the hypothesis for Fact D.3. (This is because if one sets $~a~=~2^{ ~ \sharp ( ~ \theta \, )} ~$ and $~b~=~\sqrt{2^{~ \sharp ( ~ \theta \, )}} ~$ then obviously $ ~~p~<~3^{ ~ \sharp ( ~ \theta \, )}~ <~4^{ ~ \sharp ( ~ \theta \, )}~ =~ (a/b)^4~~~).$ \fend } $\,$that $~p~$, viewed as a deduction tree for $~Z~,~$
satisfies the hypothesis of Fact D.3 . Hence, Fact D.3 establishes that $~p~$ must contain at least
one contradiction-free root-to-leaf branch.
This last observation is all that is needed to confirm
$~\xi^* \,$'s A-stability, via a proof-by-contradiction. This is because the definition of a semantic tableaux proof implies every one of its root-to-leaf branches must end with a pair of contradicting nodes. However, the last paragraph showed $~p~$ will not satisfy this required
property, if $~\xi^*~$ is not A-stable. Hence our construction has proven the A-stability of
$~\xi^*~$ by showing that otherwise an infeasible circumstance will arise.
$~~\Box$
{\bf Proof of
$ \xi^* $'s E-stability :} A proof-by-contradiction will verify
$ \xi^* $ is E-stable, analogous
to the
proof of its
A-stability. Thus if
$ \xi^* $ was not E-stable, then statement $++$ would be true for some $\theta$. (This is because at least one $\theta \, \in \,$RE-Class$(\xi)$ would then
violate Definition \ref{estab}'s requirement of $~**~~$.) \begin{description} \item[ ++ ] There exists a semantic tableaux
proof $~p~$ of a $\Sigma_1^\xi$ theorem $ \, \Upsilon \, $ from the axiom system $\theta \cup B^\xi$ such that
Log$(p) \,\leq ~ \sharp ( ~ \theta) +1 $ and
$ \Upsilon $ also fails to satisfy Good$\{ ~ \frac{1}{2}~ ~ \sharp ( ~ \theta) \,\}.$ \end{description} Item $++ $
implies Reverse$( \Upsilon )$ satisfies
Good$\{ ~ \frac{1}{2}~ ~ \sharp ( ~ \theta) \}$ (because Reverse$( \Upsilon )$ again
has
the opposite goodness property as $ \Upsilon $ ). Let $Z$ now denote the formal axiom system of
$\, \theta \cup B^\xi \, + \, $Reverse$( \Upsilon )$. The footnote \footnote{ \baselineskip = 1.0 \normalbaselineskip The axiom system
$~Z~$ must satisfy Normed$\{ ~\sqrt{~2^{ ~ \sharp ( ~ \theta \, )}}~,~2~ \} $ because: \begin{enumerate} \item The first component of its norming constraint can be set equal to
$ ~\sqrt{~2^{ ~ \sharp ( ~ \theta \, )}} ~$ because Reverse$(\, \Upsilon \,)~$ is a
Good$\{~ \, ~ \frac{1}{2}~ ~ \sharp ( ~ \theta \, ) \, ~\}~$ $\Pi_1^*$ sentence, and all $~Z\,$'s other $\Pi_1^*$ sentences satisfy more relaxed constraints. \item The second component of $Z$'s norming constraint is satisfied by the constant of 2 because Definition D.\ref{D2-def} implies this quantity is always permissible when $~Z~$ contains no $\Sigma_1^*$ axiom sentences. \end{enumerate} \fend } then
uses reasoning similar to footnote \ref{jnorm} to show $Z$ satisfies Normed$\{ ~\sqrt{~2^{ ~ \sharp ( ~ \theta \, )}}~,~2~ \} $
\parskip 3pt
As before via a simple change in notation,
$~p\,$'s semantic tableaux proof of $ \, \Upsilon \,$ can be viewed as
a deduction tree using $~Z\,$'s axioms. Also as before, we may use the combination of the facts that
$~Z~$ is a Normed$\{ ~\sqrt{~2^{ ~ \sharp ( ~ \theta \, )}}~,~2~ \} $ system and that Item ++ indicated
Log$(p) \,\leq ~ \sharp ( ~ \theta) +1 $ to deduce\footnote{ \baselineskip = 1.0 \normalbaselineskip The proof that
$~p~$ is
small enough to satisfy Fact D.3 's hypothesis in the current E-stable case is almost identical to Footnote \ref{fd7}'s analysis of the A-stable case. Thus as in the earlier case, Item ++'s inequality of
Log$(p)~\leq ~ \sharp ( ~ \theta)~+1~ $ trivially implies $~p~<~3^{ ~ \sharp ( ~ \theta \, )}~$. Also, we may again assume that
Log$(p)~\geq ~64~$ (since a sequence with fewer than 64 bits cannot amount to a proof of any interesting fact under all normal coding conventions). An analog of Footnote \ref{fd7}'s chain of inequalities will then allow us to conclude that $~p~$ is
small enough proof from a Normed$\{ ~\sqrt{~2^{ ~ \sharp ( ~ \theta \, )}}~,~2~ \} $ system to
satisfy the hypothesis for Fact D.3. \fend } that
$~p~$ is small enough to satisfy Fact D.3 's hypothesis.
Hence once again, Fact D.3 implies that $~Z~$ must contain at least
one contradiction-free root-to-leaf branch. As before, the existence
of this contradiction-free path violates the definition of a semantic tableaux proof and enables
our proof-by-contradiction to reach its desired end. $~\Box$
{\bf Remark D.5} {\it (about Theorem D.\ref{D.4-theorx}'s significance) :} Part-ii of
Definition D.\ref{D1-def} indicated $~\xi^* \,$'s base axiom of $~B^*~$ was a Type-A formalism that recognized addition as a total function. This is significant because
\cite{ww0,ww2,ww7,ww9} showed nearly all Type-M formalisms, including all the common axiomatizations for I$\Sigma_0$, are unable to recognize their semantic tableaux consistency. Thus, the declaration that multiplication is a total function is {\it the trigger-point} causing \footnote{ \baselineskip = 1.0 \normalbaselineskip We formally proved in \cite{ww0,ww2,ww7,ww9} that multiplication's totality property causes
the semantic tableaux version of the Second Incompleteness Theorem to become active. The
Example \ref{ex3-3} summarizes the main
intuition behind these results.} the semantic tableaux version of the Second Incompleteness Theorem to become active. This threshold effect is significant
because Theorem D.4, combined with Theorems
\ref{pqq4}, \ref{pqq5}, G.2 and G.3, formalize {\it four different respects} in which Type-A self-justifying formalisms can prove all Peano Arithmetic's
$\Pi_1^*$ theorems {\it ( after} multiplication's totality axiom is suppressed).
\baselineskip = 1.22 \normalbaselineskip
\subsection*{D-3. $~$Three Further Examples of Stable Generic Configurations}
Our second example of an
EA-stable
configuration is called $~\xi^{**}~$. It will be identical to
$~\xi^*~$ except that it will replace semantic tableaux with a stronger deduction method, which \cite{ww5}
called Tab$-U_1^*$. The latter is a revised version of semantic tableaux that permits
a modus ponens rule to perform deductive cut
operations on $\Pi_1^*$ and $\Sigma_1^*$ sentences. (The formal definition of
Tab$-U_1^*$ deduction had appeared in \cite{ww5}. It will be unnecessary to
repeat here.)
The Section 5.3 of \cite{ww5} noted Tab$-U_1^*$ has
similar self-justification properties as conventional semantic tableaux.
All the results that Section D-2 proved about
$~\xi^{*}~$
apply also to $~\xi^{**}~$, via their natural generalization under \cite{ww5}'s Tab$-U_1^*$ deduction method. Thus,
$~\xi^{**}~$ is also EA-stable.
A key point is that
there is a non-trivial distinction between $ \xi^{*} $ and $ \xi^{**} $, despite the fact that they have similar technical qualities. This is because
$ \xi^{**} $ contains a Level-1 modus ponens rule (unlike $ \xi^{*} $ ).
If it were infeasible to expand $ \xi^{*} $ into a broader
$ \xi^{**} $, $ $then both formalisms could, perhaps,
be easily dismissed as having negligible pragmatic significance (since modus ponens is central to cogitation). However in a context where $ \xi^{**} $ does permit a Level-1 modus ponens rule, it is a tempting formalism (despite its limited modus ponens rule).
Unlike
$~\xi^{*}~$ and $~\xi^{**}~$, our third example of an EA-stable configuration, called $~\xi^{-}~$, will support an unlimited modus ponens rule. This will be possible because
$~\xi^-\,$'s language of $~L^-~$ will be weaker than the languages of
$~\xi^{*}~$ and $~\xi^{**}~$. Thus $~L^-$ will include
the six Grounding functions, but not the Growth functions of addition and doubling. It will thus treat addition and multiplication as 3-way atomic predicates,
Add$(x,y,z)$ and Mult$(x,y,z)$, rather than as formal functions.
\baselineskip = 1.22 \normalbaselineskip \parskip 2pt
This perspective enabled $~\xi^-~$ to support an evasion of the Second Incompleteness Theorem with an unlimited modus ponens rule present, in a context where the other four parts of its generic configuration are defined below: \begin{enumerate} \item The
$~\Delta_0^{-}~$ class for $~\xi^-~$ will be built in an essentially natural
manner from the Grounding function set. It will thus include all formulae in $~L^- \,$'s language, whose quantifiers are bounded
in any arbitrary manner using the Grounding function primitives. \item The base axiom system $~B^-~$ of $~\xi^-~$
will employ an infinite number of constant symbols, denoted as $ K_1 , K_2 , K_3 , \, ... $ where
$K_1=1$ and where
$K_{i+1}$ is a power of 2
defined by the axiom of: \begin{equation} \label{addc} \mbox{Add}(~K_{i}~,~K_{i}~,~K_{i+1}~) \end{equation} Thus, the combination of $ K_1 , K_2 , K_3 , \, ... $ with the Grounding function of subtraction allows the language $L^-$ to encode the value of any arbitrary natural number (as Part 1 of Definition \ref{def3.3} had required). Essentially, $~\xi^- \,$'s base axiom system of $~B^-~$ can be any consistent r.e. set of $~\Pi_1^-~$ sentences that includes \eq{addc}'s axiom schema and is able to prove every
$\Delta_0^{-}~$ sentence which is valid in the Standard-M model. \item $~\xi^- \, $'s deduction method can be any version of a classic Hilbert-style proof methodology. (Thus, it will include a modus ponens rule with no restrictions.) \item $~\xi^- \, $'s G\"{o}delization method can be essentially any natural technique. \end{enumerate} An interesting aspect of $\xi^-$ is it can be proven to be EA-stable via an analog of Section D-2's treatment of $\xi^*$. Thus, Theorem \ref{pqq4} implies every axiom system $\alpha$, whose
$\Pi_1^-$ theorems hold true in the Standard-M model, can be mapped onto an extension of $\xi^-\,$'s base axiom system that can
recognize its own Hilbert
consistency and prove $\alpha $'s $\Pi_1^-$ theorems. Except for minor changes in notation, this result represents a new way of proving \cite{wwapal}'s Theorem$\, 3.~$
The self-justifying features of
$ \xi^{*} $, $ \xi^{**} $ and $ \xi^{-} $ are of interest primarily
because the Second Incompleteness Theorem implies that they cannot be improved much further. This tight fit is summarized by Items 1-4. \begin{enumerate} \topsep -15pt \item The Theorem 2.1 (due to the combined work of Nelson, Pudl\'{a}k, Solovay and Wilkie-Paris \cite{Ne86,Pu85,So94,WP87} ) implies no natural axiom system can prove Successor is a total function and recognize its own Hilbert consistency. This theorem thus explains why the presence of growth functions must be omitted from
$\xi^{-}\,$'s base axiom system of $B^-$. \item Moreover, \cite{wwapal}
proved $~\xi^{-} \,$'s method for evading the Second Incompleteness Theorem will collapse if one replaces \ep{addc}'s ``addition-based named sequence'' of constant symbols $ K_1 , K_2 , K_3 , \, ... $ with a faster growing ``multiplicative convention'', where the constant symbols $ C_1 , C_2 , C_3 , \, ... $ are formally defined via \eq{multc}'s
schema. \begin{equation} \label{multc} \mbox{Mult}(~C_{i}~,~C_{i}~,~C_{i+1}~) \end{equation} Thus, \cite{wwapal} showed that there exists a
$\Pi_1^{-}$ sentence $ \,W \,$ (provable from Peano Arithmetic) such that no consistent system can simultaneously prove $W$, contain \eq{multc}'s axiom schema and prove the non-existence of proof of $0=1$ from itself. There is no space to prove it here, but a generalization of the Second Incompleteness Theorem implies the modification of $\xi^-$ that replaces \eq{addc}'s axiom schema with \eq{multc}'s schema {\it is not even 0-stable.} \item Similarly, \cite{ww2,ww7} proved that if
$ \, \xi^{*} \, $'s and $ \, \xi^{**} \, $'s
base
axiom system of $ \, B^* \, $ was strengthened to include the assumption that multiplication was a total function then \cite{ww5}'s two semantic tableaux evasions of the Second Incompleteness Theorem would both collapse. \item Also, \cite{wwlogos} proved that an analog of $\xi^{**} \,$'s evasion of the Second Incompleteness Theorem will collapse if its modus ponens rule was expanded to apply to either $\Pi_2^*$ or $\Sigma_2^*$ sentences. \end{enumerate} The Item 3 is especially interesting because \cite{ww6} proved \cite{ww5}'s evasion of the Second Incompleteness Theorem was compatible with its formalism recognizing an infinitized generalization of a computer's floating point multiplication as a total function. Thus
while the semantic tableaux formalisms of
$\xi^{*}$ or $\xi^{**}$ are provably unable \cite{ww2,ww7} to
recognize integer multiplication as a total function, their relationship to floating point multiplication is more subtle.
Our fourth example of an application of Section \ref{sect64}'s theorems was stimulated by some insightful email we received from L. A. Ko{\l}odziejczyk \cite{Ko5} in 2005. It noted there existed a potential
exponential gap between the lengths of semantic tableaux and Herbrand-style proofs under some circumstances. Our earlier research \cite{ww2} addressed a 1981 Paris-Wilkie open question \cite{PW81} by generalizing some Adamowicz-Zbierski techniques \cite{Ad2,AZ1} to show a natural axiomatization of I$\Sigma_0$
satisfied the semantic tableaux version of the Second Incompleteness Theorem. In this context, Ko{\l}odziejczyk asked whether this would apply to all plausible axiomatizations for I$\Sigma_0$ ?
We replied in \cite{ww9} to Ko{\l}odziejczyk's stimulating question
by distinguishing between Example \ref{ex3-1}'s $\Delta_0^A$ and $\Delta_0^R$ formulae and by using the Paris-Dimitracopoulos \cite{PD82} translation algorithm for $\Delta_0$ formulae. (The latter procedure was summarized earlier by
Lemma \ref{lex22}. It demonstrated
how to
map classic arithmetic's $\Delta_0^A$ formulae onto equivalent $\Delta_0^R$ formulae in the Standard-M model.) Our reply to Ko{\l}odziejczyk's question, thus, employed this translation methodology to show that there existed an axiom system, called Ax-3, which proved the identical set of theorems as the more common Ax-1 and Ax-2 encodings of
I$\Sigma_0$ and which possessed the following pair of quite fascinating
contrasting properties: \begin{description} \topsep -3pt \item[ A ] No consistent
superset $~\beta~$ of Ax-3's set of axioms is capable of proving its own
semantic tableaux consistency \cite{ww9}. \item[ B ] In contrast, if ``Herb'' denotes the next paragraph's
Herbrand-styled deduction and if ``SelfRef'' denotes the sentence $~\bullet~$ from Section \ref{secc1}, then Ax3$\, + \,$SelfRef(Ax-3,Herb) will be a self-justifying axiom system.
\end{description}
The intuition behind \cite{ww9}'s proof of Items A and B can be easily summarized if we define a
``Herbrandized-style'' proof of a theorem $ \,\Phi \,$ from an axiom system $ \, \alpha \,$ as being an essentially 2-part structure where: \begin{enumerate} \topsep -3pt \item Each of $\alpha$'s axioms and also the sentence $~ \neg \Phi~$ are first written as Skolemized expressions. \item A propositional calculus proof is then used to show
that some formal conjunction of instances of Item 1's Skolemization schema has no satisfying truth assignment. \end{enumerate} Such a
formalism is different from the definition of a semantic tableaux proof (appearing in for example Fitting's textbook \cite{Fi90} ). This is because the latter replaces the use of Skolemization in Items 1 and 2 with an existential quantifier elimination rule. It turns out that this distinction enables some semantic tableaux proofs to be exponentially more compressed than their Herbrandized counterparts, as Ko{\l}odziejczyk observed \cite{Ko5,Ko6}. This fact
enabled \cite{ww9} to prove that Herbrandized and semantic tableaux proofs have the divergent properties summarized by Items A and B.
One reason Ax-3's evasion of the Second Incompleteness Theorem is of interest is that I$\Sigma_0$ supports many more generalizations of the Second Incompleteness Theorem than evasions of it.
Thus, Willard \cite{ww2,ww7,ww9} proved that the semantic tableaux version of the Second Incompleteness Theorem was valid for three different encodings of I$\Sigma_0$, and Adamowicz, Salehi and Zbierski have discussed in great detail \cite{Ad2,AZ1,Sa11}
various Herbrandized generalizations of the Second Incompleteness Theorem for particular encodings of I$\Sigma_0$ and I$\Sigma_0+\Omega_i$. Moreover, an added facet of \cite{ww9}'s Ax-3 encoding for I$\Sigma_0$ is that most automated theorem provers use a particular variant of the Resolution method that causes \cite{ww9}'s unusual methodology to apply also to them \footnote{ \baselineskip = 1.1 \normalbaselineskip The main theorems in \cite{ww9} generalize for resolution because Resolution-based theorem provers employ skolemization analogously to Herbrand deduction. \fend }.
The reason for our
interest in \cite{ww9}'s results
is that it represents a fourth example
where the
meta-theorems from Sections \ref{3uuuu3} and \ref{sect64} can be useful. Thus, the footnote \footnote{ \baselineskip = 1.1 \normalbaselineskip The discussion in \cite{ww9}
did not technically use
Definition \ref{estab}'s machinery to establish
there existed an extension of its ``Ax-3'' encoding for I$\Sigma_0$ that could recognize its own Herbrand consistency. Its formalism, however,
could be easily couched in terms of
Definition \ref{estab}'s machinery, if one uses a generic configuration $~\xi^R~$ where \begin{enumerate} \item $~\xi^R\,$'s base language is the same as the usual language of arithmetic, \item $~\xi^R\,$'s $~\Delta_0^R~$ sub-class is defined by Item (b) in Example \ref{ex3-1}, \item $~\xi^R\,$'s base axiom system is \cite{ww9}'s ``Ax-3'' system, \item $~\xi^R\,$'s deduction method is either a Herbrandized styled-method or a Resolution system that relies upon Skolemizatin in a similar manner. \item $~\xi^R\,$'s G\"{o}del encoding scheme may be any such natural method. \end{enumerate} This approach supports a
stronger form of self-justification result than had appeared in \cite{ww9}. This is because $~\xi^R~$ can be proven to be E-stable (by a
generalization of \cite{ww9}'s analysis techniques). Thus,
\phx{ppp2} implies that Ax-3 has a well-defined self-justifying extension that can recognizes its own formalized
Level$(0^R)$ consistency. (This self-justification result is stronger than \cite{ww9}'s main theorem. The latter merely established that some extension of Ax-3 recognized the non-existence of a Herbrandized deduction of $0=1$ from itself.) \fend } summarizes how a fourth type of generic configuration, called $~\xi^R~$, can be defined
that both duplicates \cite{ww9}'s main
self-justification results under the above definition of Herb-deduction, as well as strengthens them. (In particular,
$~\xi^R~$ meets Theorem \ref{ppp2}'s requirements, and self-justifying extensions of its Ax-3 system thus recognize their
Level$(0^R)$ consistency.)
\parskip 2pt
The properties of our four generic configurations of $~\xi^R$,
$~\xi^*~$,
$~\xi^{**}~$ and $~\xi^-~$ are summarized by Table I. These configurations are listed in ascending order according to the strength of their deduction methods $~d~$. As their deduction methods increase in strength, these configurations
have their ability reduced
to recognize the totality of the addition and multiplication operations.
$~\xi^R\,$ is thus a Type Almost-M system that can prove multiplication is a total function (but which does not contain \ep{totdefsymbm}'s
totality statement {\it as an axiom)}. On the other hand, $~\xi^-~$ uses a stronger Hilbert-styled
deduction methodology, which is incompatible with treating the totality of addition or multiplication as either axioms {\it or as derived theorems.}
Each of Table I's rows
$ \, \xi^R$,
$ \, \xi^{**} \, $ and $ \, \xi^- \, $ are maximal (in that an alternate row improves upon one column's measurement only when it is weaker from the perspective of another column). Only $ \, \xi^* \, $ is an exception to this rule: It is strictly weaker \footnote{ \baselineskip = 1.1 \normalbaselineskip \label{f28} $ \xi^{**} $ employs a stronger deduction method than $ \xi^* $ because it allows a modus ponens rule for $\Pi_1^*$ and $\Sigma_1^*$ sentences to be added to semantic tableaux deduction (see \cite{ww5} for the precise definition of this ``Tab$-U_1^*~$''
modification of the semantic tableaux deductive method). \fend }
than
$ \, \xi^{**} $.
This appendix has discussed
$ \, \xi^* \, $
because it makes
Theorem D.\ref{D.4-theorx}'s proof simpler
(and also because
semantic tableaux
is a
frequent topic in the logic literature).
\begin{center} {\bf Table I} \end{center}
\small \baselineskip = 1.1 \normalbaselineskip \noindent
\begin{tabular}{|c|c|c|c|c|c|c|} \hline Name & Deduction Method & Type & Almost & Type & {\bf Axiom} & {Self-Just } \\
& & {\bf A} & {\bf M} & {\bf M} & { Format} & { Level} \\ \hline\hline
& Resolution and/or & & & & & \\
$\xi^R$ & Herbrandized analogs & Yes$^{35}$
& Yes & No & E-stable &Level $(0^R)$ \\ \hline
& & & & & & \\
$\xi^*$ & Semantic Tableaux & Yes & No & No & EA-stable & Level $(1^*)$ \\ \hline
& & & & & & \\
$\xi^{**}$ & Tab$-U_1^*$ Deduction$\, ^{34}$ & Yes & No & No & EA-stable &Level $(1^*)$ \\ \hline
& & & & & & \\
$\xi^{-}$ & Hilbert Deduction & No & No & No & EA-stable &
Level $(\, \infty^{-} \,)$ \\ \hline \end{tabular}
\normalsize \baselineskip = 1.22 \normalbaselineskip
\parskip 0pt
The footnote \footnote{ \baselineskip = 1.1 \normalbaselineskip For the sake of simplicity,
the Ax-3 system of \cite{ww9} did not
use either
Equations
\eq{totdefxa} or
\eq{totdefsymba}'s
as axiom statements (since they were provable as theorems).
All
\cite{ww9}'s
results
do, however, generalize
when \eq{totdefxa}'s statement about addition's totality is included as
an axiom. Thus, it is appropriate to attach the designation of ``Yes'' with a caveat to the ``Type-A'' entry in Table I's first row. (This row is called ``Resolution and/or Herbrandized analogs'' because it applies to essentially any deduction scheme that relies upon Skolemization as an alternative to \cite{Fi90}'s semantic tableaux existential quantifier elimination rule.)} ,
attached to Table I's first row, explains why a caveat is attached to its first ``Yes'' entry. The theme of Table I is
that self-justifying axiom systems have some nice redeeming features, although the
Second Incompleteness Theorem clearly also imposes severe
limits on their abilities. This point will be reinforced when Appendix E introduces a generalization of the Second Incompleteness Theorem, that shows \thx{ppp6}'s translational reflection principle is close to being a maximal feasible result, and when Appendix F discusses the epistemological significance of self justification.
\section*{Appendix E: A Clarification of Theorem \ref{ppp6}'s Significance}
It has been known since the time of G\"{o}del that most conventional arithmetic axiom systems will satisfy the following two invariants: \begin{enumerate} \item They are {\it physically unable} to prove their own consistency \item They are $\Sigma_1$ complete. This means they can formally prove any $\Sigma_1$ arithmetic sentence that holds true in the Standard-M model, and they can likewise refute any $\Pi_1^\xi$ sentence that is false. \end{enumerate} Let $~\xi~$ denote any generic configuration of the form $( L^\xi , \Delta_0^\xi , B^\xi , d , G )$ . This appendix will use the term {\bf $\xi -$Conventional} to describe any axiom system that satisfies analogs of the preceding
conditions for generic configurations. Thus $~\alpha~$ is $\xi -$Conventional iff it satisfies the following two criteria: \begin{description} \item[ a. ] The axiom system $~\alpha~$ will be {\it unable} to verify its own consistency under $~\xi\,$'s deduction
method of $~d~$. \item[ b. ] The axiom system $~\alpha~$ will be an extension of $~\xi \,$'s base axiom of $~B^\xi~$. Part-3 of Definition \ref{def3.3} will thus imply it is $\Sigma^\xi_1$ complete. (Hence, $~\alpha~$ can formally prove any $\Sigma^\xi_1$ sentence that holds true in the Standard-M model, and it can likewise refute any $\Pi^\xi_1$ sentence that is false.) \end{description}
This section will prove no analog of \ep{T-reflect}'s translational reflection principle is feasible for $\xi -$Conventional axiom systems. Thus, Theorem \ref{ppp6} must be close to being a maximal result, since it cannot plausibly be further extended to hold under conventional axiom systems.
{\bf Theorem E.1} (A New Type of Version of the Second Incompleteness Theorem): {\it There exists no $\xi -$Conventional axiom system $~\alpha~$ that can prove the validity of \eq{G-reflect}'s Translational Reflection Principle for any translation-mapping T.} (In other words, there exists no
algorithm $~T~$ that maps $~\Pi_1^\xi$ sentences $~\Psi~$ onto alternate $~\Pi_1^\xi$ sentences $~\Psi^T~,~$ which
are equivalent to $~\Psi~$ in the Standard-M model and where $~\alpha~$ can verify \eq{G-reflect}'s reflection principle for every $~\Pi_1^\xi$
sentence $~\Psi .~)$
\begin{equation} \label{G-reflect} \forall ~p~~~[~~ \mbox{Prf}_{\alpha,d}(~\lceil \, \Psi \, \rceil~,~p~)
~~~ \Rightarrow ~~~ \Psi^T~~] \end{equation}
{\bf Proof:} $~$It is easy to prove Theorem E.1 via a proof-by-contradiction. Thus consider the possibility that Theorem E.1's translational mapping $~T~$ did exist. One can then easily select a $~\Pi_1^\xi$ sentences $~\Psi~$ that is false in the Standard-M model. Then $~\Psi^T~$ is also false under the Standard-M model (since
$~\Psi~$ and $~\Psi^T~$ are equivalent in this model).
Hence Part-b of the definition of $\xi -$Conventionality implies $~\alpha~$ must prove $~\neg~\Psi^T$
(on account of $~\Psi^T\,$'s $\Pi_1^\xi$ format).
It is at this juncture that our proof-by-contradiction will reach its end. This is because if $~\alpha~$ can prove \eq{G-reflect}'s statement and {\it also prove}
the sentence $~\neg~\Psi^T~$,
then it certainly can combine these two facts to prove the non-existence of a proof of $~\Psi~$. The latter contradicts Part-a of the definition of $\xi -$Conventionality (because it shows $~\alpha~$ can verify its own consistency). $~~\Box$
{\bf Remark E.2.} We remind the reader that Footnote \ref{imper} pointed out that $T$'s translational mapping would lose its main functionality, if it did not require $\Psi^T$ to have a $~\Pi_1^\xi~$ format, similar to $\Psi$. In essence, Theorem E.1 is of interest because it shows that Theorem \ref{ppp6}'s evasion of the Second Incompleteness Theorem is close to being a maximal result.
(It thus shows that \eq{G-reflect}'s translational reflection principle does not generalize to conventional axiom systems.)
This dichotomy may explain why self-justifying axiom systems, along with Theorem \ref{ppp6}'s particular
invariant, are potentially useful results.
\baselineskip = 1.22 \normalbaselineskip
\section*{Appendix F: Epistemological Perspective and Speculations}
\baselineskip = 1.22 \normalbaselineskip
\parskip 6pt
It is desirable to include a short purely epistemological discussion within this mostly mathematical article so that the more subtle nature of our
results cannot be misconstrued.
Part of the reason Self Justification can lend itself to easy misinterpretations is that the First Incompleteness Theorem demonstrates the impossibility of constructing an ideally optimal axiomatization of number theory. For any initial r.e$. \,$axiom system $~\alpha~$ and deduction method$~d$, G\"{o}del thus noted it is easy \footnote{ \baselineskip = 1.1 \normalbaselineskip Let $\mho(a,d)$ the classic
G\"{o}del sentence that asserts: {\it ``There is no proof of {\it this sentence} from $\alpha$'s axiom system under $\,d$'s deduction method.''} G\"{o}del \cite{Go31}
noted $~\alpha+\mho(\alpha,d)~$ always proves more theorems than $\alpha$.}
to develop
an extension of $~\alpha~$ that can prove strictly more theorems than $~\alpha~$ under
$\,d$'s deduction method. Moreover, a large number of generalizations of the Second Incompleteness Theorem, starting with its 1939 Hilbert-Bernays version \cite{HB39}, are known to be
robust results.
Such considerations naturally lead to questions about whether any
r.e$. \,$axiom system can encompass the workings of the human mind. It may surprise some readers to learn that this author shares such
skepticism. That is, we doubt {\it any single ISOLATED} self-justifying r.e.$\,$logic can {\it fully} approximate the complex workings of the human mind.
In this short appendix, let us instead view cogitation as {\it roughly} a process wondering though some universe $ \, \mathcal{U}$, comprised of {\it both} consistent and inconsistent axiom systems, with a trial-and-error evolutionary method focusing
its attention over time increasingly onto
the members of this universe
$~ \mathcal{U}~$ that are found to be
consistent. It is
straightforward\footnote{ \baselineskip = 1.1 \normalbaselineskip It
is trivial from a theoretical perspective
to design a
learning heuristic that
will utilize all
consistent axiom systems
from
its available universe $ \mathcal{U}$
eventually, and
it will
spend only an infinitesimal fraction of its effort on
inconsistent systems as time runs to infinity.
(This because
there exists only
a countable number of distinct r.e. sets
belonging to the universe
$ \mathcal{U}.~)~$
Also,
this
learning
process can
presumably be made to
employ
some type of
smart souped-up
AI heuristics to enhance its efficiency,
whose details will not concern us
in this abbreviated 3-page appendix.
What is
central to the current discussion
is that some type of formally
{\it non-recursive}
and presumably trial-and-error
method must
obviously
be used
by this learning process
to find
the consistent elements of
$~ \mathcal{U}~,~$
on account of G\"{o}del's
undecidability results.}
to define many universes $~ \mathcal{U}~$ and
evolutionary processes that fall into this gendre. Our goal in this section will be to examine
Section \ref{3uuuu3}'s ``R-View'' $\theta$ and its RE-Class$(\xi)$.
Thus, $\theta$ will denote an R-View that consists of an arbitrary r.e$. \,$set of $\Pi_1^\xi$ sentences. Also, RE-Class$(\xi)$ will again denote the set of all $~\theta~$ which can be built under $~\xi \,$'s language of $\,L^\xi$. (Section \ref{3uuuu3} had allowed
both valid and invalid R-Views $~\theta$
to appear in RE-Class$(\xi)$ because no recursive decision procedure can identify all the Standard-M model's true $\Pi_1^\xi$ sentences.)
The epistemological purpose of this notation was revealed in Section \ref{sect64}. For the cases where $k = 0$ or 1, Section \ref{sect64} defined $G^\xi_k(\, \theta \,)$ to be the axiom system: \begin{equation} \label{f4gedef} G^\xi_k(~ \theta ~)~~= ~~ \theta~\cup~B^\xi~\cup~\mbox{SelfCons}^k\{~[~\theta \,\cup \,B^\xi~]~,d~\} \end{equation} Also, Definition \ref{dap4-1} indicated that
the function $ \, G^\xi_k ~ $ (which maps $ \, \theta \, $ onto $G^\xi_k(\, \theta \,)~~~)$ would be called {\bf Consistency Preserving} iff $ \, G^\xi_k(\, \theta \,) \, $is assured to be consistent whenever all the sentences in $~\theta ~$ are true under the Standard-M model. \thx{pqq3} indicated, in this context,
that $~G^\xi_1~$ satisfies this property whenever $~\xi~$ is EA-stable. Likewise, $~G^\xi_0~$ is consistency preserving whenever $~\xi~$ is one of A-stable, E-stable or 0-stable.
These results indicate a trial-and-error experimental process can, indeed, walk {\it in an unusually orderly manner} through an universe of self-reflecting candidate formalisms, when RE-Class$(\xi)$ denotes $~ \mathcal{U}\,$'s universe and
$~\xi~$ satisfies any of the EA-stable, E-stable, A-stable or 0-stable conditions. This is because if $~\theta~$ designates a set of $\Pi_1^\xi$ sentences holding true in
the Standard-M model, then $~G^\xi_k(\theta)~$ will {\bf automatically} satisfy both Parts (i) and (ii) of Section 1's definition of Self Justification, according to \thx{pqq3}.
Such consistency preservation is surprising because it is simply inapplicable to
the $\,G^\xi_k \,$
functions for most pairs $~(\xi,k).$ \thx{pqq3}'s first
contribution is,
thus, that it formalizes how $G^\xi_k\,$'s mapping function can represent
a type of approximation for instinctive faith, under certain well-defined circumstances.
This notion of instinctive faith is, of course, less robust than a conventional proof. One obvious
difficulty is that a 1-sentence proof, using an {\it ``I am consistent''} axiom, is less convincing than a full-length proof from first principles. Also, if the initial formalism $~\theta~$ contains a false $~\Pi_1^\xi~$ sentence then $~B^\xi+\theta~$ and $~G^\xi_k(\theta)~$ will be both inconsistent.
\baselineskip = 1.22 \normalbaselineskip
Nevertheless for $\,k\,$ equals 0 or 1,
if $~\theta~$ is comprised of the true sentences in the Standard-M model, then \thx{pqq3} will assure that $~G^\xi_k(\theta)~$ is a consistent system that has an ability to use its {\it ``I am consistent''} axiom sentence to formalize its own consistency. Moreover, the axiom system $~G^\xi_k(\theta)~$ is helpful because G\"{o}del's famous centennial paper
implicitly raised the following bedeviling issue:
\begin{quote} $\#~~$ How is it that Human Beings
manage to muster the physical drive to think (and prove theorems) when the many generalizations of G\"{o}del's Second Incompleteness Theorem demonstrate conventional logics lack knowledge of their own consistency? \end{quote} While philosophical paradoxes and ironical dilemmas, similar to $~\# ~,~$
never yield perfect answers, the preceding discussion is helpful because it explores a certain syllogism
whereby a logic can formalize
at least some fragmented operational
appreciation of its own consistency.
Moreover, Part-3 of Appendix D
indicated that its four self-justifying configurations were
close to being maximal results that cannot be much improved, on account of various barriers imposed by
the Second Incompleteness Theorem. Thus, these particular
positive results, combined with Theorems \ref{ppp1} \ref{pqq3}, \ref{pqq4}, \ref{pqq5},
\ref{ppp6}, D.4, E.1,
G.2, G.3 and Remarks \ref{re4-1} and
\ref{recc1}, come close to formalizing the maximal variants of instinctive faith that a first-order logic can bolster.
The theme of the last two paragraphs is thus
that our approximation of
{\it ``instinctive faith''} may be imperfect, but it is still a useful partial reply to $~\# \,$'s puzzling dilemma {\it in a context where} unambiguous full resolutions to $\, \# \,$ {\it are not permitted by} the Second Incompleteness Theorem. Furthermore, \ep{T-reflect}'s translational reflection principle, together with Theorem \ref{ppp6} and the Remarks \ref{f88} and \ref{remhappy}, illustrate how the notion of
an instinctive faith about the usefulness of $\Pi_1^\xi$ theorems
can be almost physically {\it hard-wired} into self-justifying formalisms.
{\bf A Yet Further
Facet
of this Unusual Epistemological Interpretation: }
Let
the term
{\it Epistemological Bundle Theory}
refer to
the underlying
theory, advanced in this appendix, which
speculates about a
Thinking Agent
walking
through
RE-Class$(\xi)$'s
bundled universe of valid and invalid
collections of $\Pi_1^\xi$ sentences
and
then applying some heuristic to
attempt to
identify
those
$\theta \, \in \,$RE-Class$(\xi)$
whose sentences are true under the Standard-M model.
Such a
theory has a second virtue, aside from
addressing $ \# \,$'s
paradoxical question
about the nature
of {\it ``instinctive faith''. }
It also clarifies
the meaning of
our main theorems
and the related
E-stability, A-stability,
EA-stability and
RE-Class$(\xi)$ constructs.
This is because the
Items $\, * \,$ and $\, ** \,$
from the
definitions of
A-stability and E-stability
in Section \ref{3uuuu3}
formalize
how a thinking agent $~T~$ can view short
proofs from a {\it technically inconsistent} axiom system
of $~B^\xi \cup \theta~$ as containing
pragmatically
useful information
{\it under the assumption} that the
lengths
of $~T\,$'s proofs {\it are shorter}
than
the errors in
$~\theta\,$'s $\Pi_1^\xi$ styled-statements.
The pleasing aspect
about this
observation, illustrated by
Remark \ref{re3-1},
is that those same invariants,
$\, * \,$ and $\, ** \,$,
which
tempt a
thinking agent $~T~$
to engage in a trial-and-error walk through
RE-Class$(\xi)$'s bundled universe,
also
make
viable
\thx{ppp1}'s self-justifying formalisms.
Thus aside from
addressing
$\, \# \,$'s dilemma about the nature of
instinctive faith,
the
meta-formalism in this appendix
is
useful
in explaining
the
motivation behind the
elaborate
network
of
theorems, proofs and definitions
that were introduced
in this
paper. In summary, EA-stable logics are thus
interesting both in their own right (as a vehicle
enabling a Thinking Being to partially tolerate its own errors), and because they are useful in explaining how a Thinking Being can possess a type of instinctive faith in its own consistency (via the reflection principles of
Theorem \ref{ppp6} and of Remarks \ref{f88} and \ref{remhappy}).
\baselineskip = 1.22 \normalbaselineskip
\section*{Appendix G: Improvements upon Theorems \ref{pqq4} and \ref{pqq5} }
Let us recall that Remark \ref{re4-n} indicated that there was a subtle trade-off between Theorems \ref{pqq4} and \ref{pqq5}, where neither result was strictly
better than the other. This section will introduce two hybrid methodologies, using Definition G.1's formalism, that improve upon \thx{pqq5} while retaining a large part of \thx{pqq4}'s nice features.
\parskip 2pt
{\bf Definition G.1} Let $~\xi~$ denote the generic configuration, whose base axiom system is again denoted as $ \, B^\xi \,$, $\,~ \Phi~$ denote any $\Pi_1^\xi$ sentence that is true in the Standard-M model and $~j~$ denote an index that represents some predicate
Test$^\xi_j \,$ lying in Definition \ref{gsim}'s
TestList$^\xi$ sequence. Then a $\,\Pi_1^\xi \,$ sentences $\Psi $
will be said to be a {\bf Braced}$^\xi( \, \Phi \, , \, j \, )$ expression when $~ B^\xi \, + \, \Phi~$ can prove: \begin{equation} \label{punch} \{~~~ \forall ~x~~~ \mbox{Test}_j^\xi(~\lceil~\Psi~\rceil~,~x~) ~~~\}
~~~~\longrightarrow ~~~~ \Psi \end{equation}
{\bf Theorem G.2} {\it $~$Let $ \,\xi \,$ again denote an arbitrary generic configuration
$( L^\xi , \Delta_0^\xi , B^\xi , d , G )$, and let
$( \mathcal{B},D)$ again denote any second axiom system and deduction method whose $\Pi_1^\xi$ theorems are true under the Standard-M model. Then for any integer $~j~$ and for any
$\Pi_1^\xi$ sentence $~\Phi~$ that is true in the Standard-M model,
the following invariants do hold:} \begin{description} \item[ i ] {\it If $ \, \xi \, $ is EA-stable then there will exist a self-justifying $~\beta_j~\supset~B^\xi$ that can recognize its Level$(1^\xi$)
consistency, contains only {\bf a finite number} of additional axioms beyond those appearing in $~B^\xi$,
and which can prove all of $( \mathcal{B},D)$'s $\Pi_1^\xi$ theorems that are Braced$^\xi( \Phi ,j)$ expressions.} \item[ ii ] {\it Likewise, if $\xi$ is
E-stable, A-stable or 0-stable then
a self-justifying $\beta_j \supset B^\xi$ will exist with the same properties except that it recognizes its own Level$(0^\xi$)
consistency.} \end{description}
{\bf Proof.} To justify Theorem G.2, we must first define the axiom system $~\beta_j~,~$ whose existence is claimed by Items (i) and (ii). It will be defined to consist of the union of the initial base axiom system
$B^\xi$ with the following three added axiom-sentences. \begin{description} \item[ 1 ] The $\Pi_1^\xi$ sentence $~\Phi~$ used by Definition G.1's
Braced$^\xi( \Phi ,j )$ formula.
\item[ 2 ] A GlobSim$^D_{ \mathcal{B}} \,(\xi,j)$ sentence whose indexing integer $ \, j \,$ is defined by Definition G.1. This global simulation sentence is thus the statement: \begin{equation} \label{glob2} \forall ~t~~ \forall ~q~~ \forall ~x~~\{~~ [~~\mbox{Prf}_{ \mathcal{B}}^D \,( t , q )~~ \wedge ~~ \mbox{Check}^\xi(t)~~]~~~ \longrightarrow ~~~ \mbox{Test}_j^\xi(t,x) ~~~\} \end{equation} \item[ 3 ] A $\Pi_1^\xi$ sentence
of the form $\mbox{SelfCons}^k\{~[~\theta \,\cup \,B^\xi~]~,d~\}~$ where: \begin{description} \item[ a ] $\theta~$ is an R-view consisting of the two
$\Pi_1^\xi$ sentences defined by Items 1 and 2. \item[ b ] $B^\xi\,$ is $~\xi \,$'s base axiom system, and \item[ c ] $k~$ equals respectively
1 and 0 under formalisms (i) and (ii). \end{description} \end{description} Thus, the system $\beta_j$ uses identical definitions
under formalisms (i) and (ii), except that its third sentence will use a different value for $~k~$. Our proof of Theorem G.2 will require first confirming the following fact: \begin{description} \item[ Claim * ] The axiom system $\,\beta_j \,$ (which consists of the union of $B^\xi$ with the
sentences 1-3) will have a capacity to prove every Braced$^\xi( \Phi ,j)$ sentence $~\Psi~$ that is a
$\Pi_1^\xi$
theorem of $( \mathcal{B},D)$. \end{description} The proof of Claim * is quite simple. It will rest on the following three observations: \begin{description} \item[ a ] For each $\Pi_1^\xi$ sentence $\Psi$, the system $~\beta_j~$ must certainly have a capacity to prove \eq{glob21}'s sentence (which states that $~\Psi\,$'s G\"{o}del number formally encodes a $\Pi_1^\xi \,$ statement). This is because \eq{glob21}
is true
in the Standard-M model
and because Part 3 of Definition \ref{def3.3} indicated that
the $~B^\xi~$ sub-component of $~\beta_j~$ has a capacity to prove
every $\Delta_0^\xi$ sentence
that is true. \begin{equation} \label{glob21} \mbox{Check}^\xi( ~ \lceil \, \Psi \, \rceil ~) \end{equation} \item[ b ] Since Claim $\,* \,$ specifies
$~\Psi~$ is a theorem of
$( \mathcal{B},D)$, there must certainly exist some integer $~N~$ that is the G\"{o}del number of its proof from
$( \mathcal{B},D)$. This implies that \eq{glob22} must be a true $\Delta_0^\xi$ sentence
under the Standard-M model. As was the case with \ep{glob21}, this implies that it must be provable from $~B^\xi~$ (because it is a valid $\Delta_0^\xi$ sentence). \begin{equation} \label{glob22} \mbox{Prf}_{ \mathcal{B}}^D \,( ~ \lceil \, \Psi \, \rceil ~ , ~ N ~ ) \end{equation} \item[ c ] It is apparent that Equations \eq{glob2}, \eq{glob21} and \eq{glob22} imply the validity of \eq{glob23}. Moreover, Part 4 of Definition \ref{def3.3} indicated that the generic configuration $~\xi\,$'s deduction method does satisfy G\"{o}del's Completeness Theorem. This fact assures that $~\beta_j~$ must be able to prove \eq{glob23} because it contains \eq{glob2} as an axiom and
\eq{glob21} and \eq{glob22} as derived theorems \footnote{\label{fcomp} \baselineskip = 1.0 \normalbaselineskip Every deduction method $\,d$, $\,$satisfying G\"{o}del's Completeness Theorem, will be automatically able to prove a theorem $~Z~$ when it contains $X$, $Y$ and $~(X \wedge Y)~\rightarrow ~Z~$ as theorems, irregardless of whether or not it contains an explicit built-in modus ponens rule. Thus $~d~$ can prove \eq{glob23} because of its knowledge about \eq{glob2}--\eq{glob22}'s validity.}. \end{description} \vspace*{- 0.1 em} \begin{equation} \label{glob23} \forall ~x~~~ \mbox{Test}_j^\xi( ~ \lceil \, \Psi \, \rceil ~ , ~ x ~ ) \vspace*{- 0.1 em} \end{equation} Claim $*$ is a consequence of Observations a-c. This is because $ \, \Phi \, $ is one of $ \, \beta_j \,$'s defined axioms, and Definition G.1 indicated
$ \, B^\xi \, + \, \Phi \, $ was capable of proving \eq{punch}'s statement
for every Braced$^\xi( \Phi ,j )$ sentence $ \, \Psi \, $. These facts corroborate Claim $*$ because they imply that $ \, \beta_j \, $ must be able to verify Claim $\,* \,$'s
sentence
$ \, \Psi \, $ (because
$ \, \beta_j \, $ can verify statements \eq{punch} and \eq{glob23}).
The remainder of Theorem G.2's proof is analogous to \phx{pqq5}'s proof. This is because the prior paragraph established that $~\beta_j~$ can prove every
Braced$^\xi( \Phi ,j )$ theorem of
$( \mathcal{B},D)$ (as was required by
Claims i and ii ). The only remaining task is to show that
$~\beta_j~$ is a self-justifying formalism that can recognize its Level($1^\xi$) and Level($0^\xi$) consistencies, as specified by
Claims i and ii. This part of Theorem G.2's
verification is identical to the methods used to prove Theorems \ref{pqq3} and \ref{pqq5}. It will thus
not be repeated here.
$~~\Box$
The last part of this appendix will require the following additional
notation to formalize the main intended application of Theorem G.2's formalism. \begin{enumerate} \item \topsep -7pt
Count$( \Psi )$ will denote the number of quantifiers appearing in the sentence $\Psi$ (including both its bounded and unbounded quantifiers). \item Size$^\xi(c)$ will denote the set of $\Pi_1^\xi$ sentences $\Psi$ where Count$( \Psi ) \, \leq \, c \,$. \end{enumerate} Our next theorem will be a specialized variant of Theorem G.2, using the Size$^\xi(c)$ construct. It will explain the
intended application of this formalism:
{\bf Theorem G.3.} {\it $~$Let $~\xi~$ denote any one of Appendix D's four sample generic configurations of $~\xi^*~$, $~\xi^{**}~$, $~\xi^-~$ or $~\xi^R~$. Then
for any $~c>0~$, Theorem G.2's axiom systems of $~\beta_j~$ can be arranged so that they can prove all of
$ ( \mathcal{B} , D ) $'s Size$^\xi(c)$
$\Pi_1^\xi$ theorems while simultaneously also
recognizing their: \begin{enumerate} \item Level(1) consistency for the cases when $ \, \xi \, $ is one of $ \, \xi^* \, $, $ \, \xi^{**}\, $ or $ \, \xi^-$. \item
Level(0) consistency when $~\xi~$ is $~\xi^R~$. \end{enumerate}}
\baselineskip = 1.35 \normalbaselineskip
{\bf Proof Sketch:}
The intuition behind Theorem G.3's proof is quite easy to summarize. For arbitrary $ c>0 $ and any of Appendix D's configurations of
$ ~ \xi^* ~ $, $ ~ \xi^{**} ~ $, $ ~ \xi^- ~ $ and $ ~ \xi^R ~ $, it is routine to
construct an ordered pair $ ~ (\Phi,j) ~ $ where every $\Pi_1^\xi$ sentence of Size$^\xi(c)$ is a
Braced$^\xi( \Phi ,j )$ expression. Theorem G.3's first claim is, thus, a consequence of Part (i) of Theorem G.2
and the fact that each of
$ ~ \xi^* ~ $, $ ~ \xi^{**} ~ $ and $ ~ \xi^- ~ $ are EA-stable. Likewise, Theorem G.3's second claim follows from Part (ii) of Theorem G.2
and the fact that $ ~ \xi^R ~ $ is E-stable, $~~\Box$
{\bf Remark G.4.} The Theorems G.2 and G.3 are of interest because the set of $\Pi_1^\xi$ sentences of Size$^\xi(c)$ is a natural class to examine. It is, thus, tempting to consider a system that recognizes its own formal
consistency, uses only a finite number of axiom sentences beyond those in $~B^\xi~,~$ and which can
prove all of
$ ( \mathcal{B} , D ) $'s
$\Pi_1^\xi$ theorems of Size$^\xi(c)$. Such a system replies to Remark \ref{re4-n}'s challenge by
hybridizing the properties of Theorems \ref{pqq4} and \ref{pqq5},
in a seemingly pragmatic manner.
\end{document} | arXiv |
Define and discuss nuclear decay.
State the conservation laws.
Explain parent and daughter nucleus.
Calculate the energy emitted during nuclear decay.
Nuclear decay has provided an amazing window into the realm of the very small. Nuclear decay gave the first indication of the connection between mass and energy, and it revealed the existence of two of the four basic forces in nature. In this section, we explore the major modes of nuclear decay; and, like those who first explored them, we will discover evidence of previously unknown particles and conservation laws.
Some nuclides are stable, apparently living forever. Unstable nuclides decay (that is, they are radioactive), eventually producing a stable nuclide after many decays. We call the original nuclide the parent and its decay products the daughters. Some radioactive nuclides decay in a single step to a stable nucleus. For example, [latex]{^{60} \text{Co}}[/latex] is unstable and decays directly to [latex]{^{60} \text{Ni}}[/latex], which is stable. Others, such as [latex]{^{238} \textbf{U}}[/latex], decay to another unstable nuclide, resulting in a decay series in which each subsequent nuclide decays until a stable nuclide is finally produced. The decay series that starts from [latex]{^{238} \textbf{U}}[/latex] is of particular interest, since it produces the radioactive isotopes [latex]{^{226} \text{Ra}}[/latex] and [latex]{^{210} \text{Po}}[/latex], which the Curies first discovered (see Figure 1). Radon gas is also produced ([latex]{^{222} \text{Rn}}[/latex] in the series), an increasingly recognized naturally occurring hazard. Since radon is a noble gas, it emanates from materials, such as soil, containing even trace amounts of [latex]{^{238} \textbf{U}}[/latex] and can be inhaled. The decay of radon and its daughters produces internal damage. The [latex]{^{238} \textbf{U}}[/latex] decay series ends with [latex]{^{206} \text{Pb}}[/latex], a stable isotope of lead.
Figure 1. The decay series produced by 238U, the most common uranium isotope. Nuclides are graphed in the same manner as in the chart of nuclides. The type of decay for each member of the series is shown, as well as the half-lives. Note that some nuclides decay by more than one mode. You can see why radium and polonium are found in uranium ore. A stable isotope of lead is the end product of the series.
Note that the daughters of [latex]{\alpha}[/latex] decay shown in Figure 1 always have two fewer protons and two fewer neutrons than the parent. This seems reasonable, since we know that [latex]{\alpha}[/latex] decay is the emission of a [latex]{^4 \text{He}}[/latex] nucleus, which has two protons and two neutrons. The daughters of [latex]{\beta}[/latex] decay have one less neutron and one more proton than their parent. Beta decay is a little more subtle, as we shall see. No [latex]{\gamma}[/latex] decays are shown in the figure, because they do not produce a daughter that differs from the parent.
In alpha decay, a [latex]{^4 \text{He}}[/latex] nucleus simply breaks away from the parent nucleus, leaving a daughter with two fewer protons and two fewer neutrons than the parent (see Figure 2). One example of [latex]{\alpha}[/latex] decay is shown in Figure 1 for [latex]{^{238} \textbf{U}}[/latex]. Another nuclide that undergoes [latex]{\alpha}[/latex] decay is [latex]{^{239} \text{Pu}}[/latex]. The decay equations for these two nuclides are
[latex]{^{238} \textbf{U} \rightarrow ^{234} \text{Th} _{92}^{234} + ^{4} \text{He}}[/latex]
[latex]{^{239} \text{Pu} \rightarrow ^{235} \textbf{U} + ^4 \text{He}}[/latex]
Figure 2. Alpha decay is the separation of a 4He nucleus from the parent. The daughter nucleus has two fewer protons and two fewer neutrons than the parent. Alpha decay occurs spontaneously only if the daughter and 4He nucleus have less total mass than the parent.
If you examine the periodic table of the elements, you will find that Th has [latex]{Z=90}[/latex], two fewer than U, which has [latex]{Z=92}[/latex]. Similarly, in the second decay equation, we see that U has two fewer protons than Pu, which has [latex]{Z=94}[/latex]. The general rule for [latex]{\alpha}[/latex] decay is best written in the format [latex]{_Z^AX_N}[/latex]. If a certain nuclide is known to [latex]{\alpha}[/latex] decay (generally this information must be looked up in a table of isotopes, such as in Appendix B), its [latex]{\alpha}[/latex] decay equation is
[latex]{_Z ^A \textbf{X} _N \rightarrow _{Z-2} ^{A-4} \textbf{Y} _{N-2} + _2 ^4 \text{He} _2 \;\;\; (\alpha \;\text{decay})}[/latex]
where Y is the nuclide that has two fewer protons than X, such as Th having two fewer than U. So if you were told that [latex]{^239 \text{Pu} \; \alpha}[/latex] decays and were asked to write the complete decay equation, you would first look up which element has two fewer protons (an atomic number two lower) and find that this is uranium. Then since four nucleons have broken away from the original 239, its atomic mass would be 235.
It is instructive to examine conservation laws related to [latex]{\alpha}[/latex] decay. You can see from the equation [latex]{_Z^A \textbf{X}_N \rightarrow _{Z-2}^{A-4} \textbf{Y}_{N-2} + _2^4 \text{He}_2}[/latex] that total charge is conserved. Linear and angular momentum are conserved, too. Although conserved angular momentum is not of great consequence in this type of decay, conservation of linear momentum has interesting consequences. If the nucleus is at rest when it decays, its momentum is zero. In that case, the fragments must fly in opposite directions with equal-magnitude momenta so that total momentum remains zero. This results in the [latex]{\alpha}[/latex] particle carrying away most of the energy, as a bullet from a heavy rifle carries away most of the energy of the powder burned to shoot it. Total mass–energy is also conserved: the energy produced in the decay comes from conversion of a fraction of the original mass. As discussed in Chapter 30 Atomic Physics, the general relationship is
[latex]{E=(\Delta m)c^2}[/latex]
Here, [latex]{E}[/latex] is the nuclear reaction energy (the reaction can be nuclear decay or any other reaction), and [latex]{\Delta m}[/latex] is the difference in mass between initial and final products. When the final products have less total mass, [latex]{\Delta m}[/latex] is positive, and the reaction releases energy (is exothermic). When the products have greater total mass, the reaction is endothermic ([latex]{\Delta m}[/latex] is negative) and must be induced with an energy input. For [latex]{\alpha}[/latex] decay to be spontaneous, the decay products must have smaller mass than the parent.
Example 1: Alpha Decay Energy Found from Nuclear Masses
Find the energy emitted in the [latex]{\alpha}[/latex] decay of [latex]{^{239} \text{Pu}}[/latex].
Nuclear reaction energy, such as released in α decay, can be found using the equation [latex]{E=(\Delta m)c^2}[/latex]. We must first find [latex]{\Delta m}[/latex], the difference in mass between the parent nucleus and the products of the decay. This is easily done using masses given in Appendix A.
The decay equation was given earlier for [latex]{^{239} \text{Pu}}[/latex] ; it is
Thus the pertinent masses are those of [latex]{^{239} \text{Pu}}[/latex], [latex]{^{235} \textbf{U}}[/latex], and the [latex]{\alpha}[/latex] particle or [latex]{^4 \text{He}}[/latex], all of which are listed in Appendix A. The initial mass was [latex]{m(^{239} \text{Pu})=239.052157 \;\textbf{u}}[/latex]. The final mass is the sum [latex]{m(235U) + m(4He) = 235.043924 \;\textbf{u} + 4.002602 \;\textbf{u} = 239.046526 \;\textbf{u}}[/latex]. Thus,
[latex]$\begin{array}{r @{{}={}}l} {\Delta m} & {m(^{239} \text{Pu}) - [m(^{235} \textbf{U}) + m(^4 \text{He})]} \\[1em] & {239.052157 \;\textbf{u} - 239.046526 \;\textbf{u}} \\[1em] & {0.0005631 \;\textbf{u}} \end{array}$[/latex]
Now we can find [latex]{E}[/latex] by entering [latex]{\Delta m}[/latex] into the equation:
[latex]{E=(\Delta m)c^2=(0.005631 \;\textbf{u})c^2}[/latex]
We know [latex]{1 \;\textbf{u} = 931.5 \;\text{MeV/}c^2}[/latex], and so
[latex]{E=(0.005631)(931.5 \;\text{MeV/}c^2)(c^2)=5.25 \;\text{MeV}}[/latex]
The energy released in this [latex]{\alpha}[/latex] decay is in the [latex]\text{MeV}[/latex] range, about [latex]{10^6}[/latex] times as great as typical chemical reaction energies, consistent with many previous discussions. Most of this energy becomes kinetic energy of the [latex]{\alpha}[/latex] particle (or [latex]{^4 \text{He}}[/latex] nucleus), which moves away at high speed. The energy carried away by the recoil of the [latex]{^{235} \textbf{U}}[/latex] nucleus is much smaller in order to conserve momentum. The [latex]{^{235} \textbf{U}}[/latex] nucleus can be left in an excited state to later emit photons ([latex]{\gamma}[/latex] rays). This decay is spontaneous and releases energy, because the products have less mass than the parent nucleus. The question of why the products have less mass will be discussed in Chapter 31.6 Binding Energy. Note that the masses given in Appendix A are atomic masses of neutral atoms, including their electrons. The mass of the electrons is the same before and after αα decay, and so their masses subtract out when finding [latex]{\Delta m}[/latex]. In this case, there are 94 electrons before and after the decay.
There are actually three types of beta decay. The first discovered was "ordinary" beta decay and is called [latex]{\beta ^-}[/latex] decay or electron emission. The symbol [latex]{\beta ^-}[/latex] represents an electron emitted in nuclear beta decay. Cobalt-60 is a nuclide that [latex]{\beta ^-}[/latex] decays in the following manner:
[latex]{^{60} \text{Co} \rightarrow ^{60} \text{Ni}+ \beta ^- + \;\text{neutrino}}[/latex]
The neutrino is a particle emitted in beta decay that was unanticipated and is of fundamental importance. The neutrino was not even proposed in theory until more than 20 years after beta decay was known to involve electron emissions. Neutrinos are so difficult to detect that the first direct evidence of them was not obtained until 1953. Neutrinos are nearly massless, have no charge, and do not interact with nucleons via the strong nuclear force. Traveling approximately at the speed of light, they have little time to affect any nucleus they encounter. This is, owing to the fact that they have no charge (and they are not EM waves), they do not interact through the EM force. They do interact via the relatively weak and very short range weak nuclear force. Consequently, neutrinos escape almost any detector and penetrate almost any shielding. However, neutrinos do carry energy, angular momentum (they are fermions with half-integral spin), and linear momentum away from a beta decay. When accurate measurements of beta decay were made, it became apparent that energy, angular momentum, and linear momentum were not accounted for by the daughter nucleus and electron alone. Either a previously unsuspected particle was carrying them away, or three conservation laws were being violated. Wolfgang Pauli made a formal proposal for the existence of neutrinos in 1930. The Italian-born American physicist Enrico Fermi (1901–1954) gave neutrinos their name, meaning little neutral ones, when he developed a sophisticated theory of beta decay (see Figure 3). Part of Fermi's theory was the identification of the weak nuclear force as being distinct from the strong nuclear force and in fact responsible for beta decay.
Figure 3. Enrico Fermi was nearly unique among 20th-century physicists—he made significant contributions both as an experimentalist and a theorist. His many contributions to theoretical physics included the identification of the weak nuclear force. The fermi (fm) is named after him, as are an entire class of subatomic particles (fermions), an element (Fermium), and a major research laboratory (Fermilab). His experimental work included studies of radioactivity, for which he won the 1938 Nobel Prize in physics, and creation of the first nuclear chain reaction. (credit: United States Department of Energy, Office of Public Affairs)
The neutrino also reveals a new conservation law. There are various families of particles, one of which is the electron family. We propose that the number of members of the electron family is constant in any process or any closed system. In our example of beta decay, there are no members of the electron family present before the decay, but after, there is an electron and a neutrino. So electrons are given an electron family number of [latex]{+1}[/latex]. The neutrino in [latex]{\beta ^-}[/latex] decay is an electron's antineutrino, given the symbol [latex]{\overline{\nu} _e}[/latex], where [latex]{\nu}[/latex] is the Greek letter nu, and the subscript e means this neutrino is related to the electron. The bar indicates this is a particle of antimatter. (All particles have antimatter counterparts that are nearly identical except that they have the opposite charge. Antimatter is almost entirely absent on Earth, but it is found in nuclear decay and other nuclear and particle reactions as well as in outer space.) The electron's antineutrino [latex]{\overline{\nu}_e}[/latex], being antimatter, has an electron family number of [latex]{-1}[/latex]. The total is zero, before and after the decay. The new conservation law, obeyed in all circumstances, states that the total electron family number is constant. An electron cannot be created without also creating an antimatter family member. This law is analogous to the conservation of charge in a situation where total charge is originally zero, and equal amounts of positive and negative charge must be created in a reaction to keep the total zero.
If a nuclide [latex]{_Z^A \textbf{X} _N}[/latex] is known to [latex]{\beta ^-}[/latex] decay, then its [latex]{\beta ^-}[/latex] decay equation is
[latex]{_Z ^A \textbf{X} _N \rightarrow _{Z+1} ^A \textbf{Y} _{N-1} + \beta ^- + \overline{\nu} _e \; (\beta ^- \;\text{decay})}[/latex]
where Y is the nuclide having one more proton than X (see Figure 4). So if you know that a certain nuclide [latex]{\beta ^-}[/latex] decays, you can find the daughter nucleus by first looking up [latex]{Z}[/latex] for the parent and then determining which element has atomic number [latex]{Z+1}[/latex]. In the example of the [latex]{\beta ^-}[/latex] decay of [latex]{^{60} \text{Co}}[/latex] given earlier, we see that [latex]{Z=27}[/latex] for Co and [latex]{Z=28}[/latex] is Ni. It is as if one of the neutrons in the parent nucleus decays into a proton, electron, and neutrino. In fact, neutrons outside of nuclei do just that—they live only an average of a few minutes and [latex]{\beta ^-}[/latex] decay in the following manner:
[latex]{\textbf{n} \rightarrow \textbf{p} + \beta ^- + \overline{\nu} _e}[/latex]
Figure 4. In β− decay, the parent nucleus emits an electron and an antineutrino. The daughter nucleus has one more proton and one less neutron than its parent. Neutrinos interact so weakly that they are almost never directly observed, but they play a fundamental role in particle physics.
We see that charge is conserved in [latex]{\beta ^-}[/latex] decay, since the total charge is [latex]{Z}[/latex] before and after the decay. For example, in [latex]{^{60} \text{Co}}[/latex] decay, total charge is 27 before decay, since cobalt has [latex]{Z=27}[/latex]. After decay, the daughter nucleus is Ni, which has [latex]{Z=28}[/latex], and there is an electron, so that the total charge is also [latex]{28 + (-1)}[/latex] or 27. Angular momentum is conserved, but not obviously (you have to examine the spins and angular momenta of the final products in detail to verify this). Linear momentum is also conserved, again imparting most of the decay energy to the electron and the antineutrino, since they are of low and zero mass, respectively. Another new conservation law is obeyed here and elsewhere in nature. The total number of nucleons [latex]{A}[/latex] is conserved. In [latex]{^{60} \text{Co}}[/latex] decay, for example, there are 60 nucleons before and after the decay. Note that total [latex]{A}[/latex] is also conserved in [latex]{\alpha}[/latex] decay. Also note that the total number of protons changes, as does the total number of neutrons, so that total [latex]{Z}[/latex] and total [latex]{N}[/latex] are not conserved in [latex]{\beta ^-}[/latex] decay, as they are in [latex]{\alpha}[/latex] decay. Energy released in [latex]{\beta ^-}[/latex] decay can be calculated given the masses of the parent and products.
Example 2: β− Decay Energy from Masses
Find the energy emitted in the [latex]{\beta ^-}[/latex] decay of [latex]{^{60} \text{Co}}[/latex].
Strategy and Concept
As in the preceding example, we must first find [latex]{\Delta m}[/latex], the difference in mass between the parent nucleus and the products of the decay, using masses given in Appendix A. Then the emitted energy is calculated as before, using [latex]{E=(\Delta m)c^2}[/latex]. The initial mass is just that of the parent nucleus, and the final mass is that of the daughter nucleus and the electron created in the decay. The neutrino is massless, or nearly so. However, since the masses given in Appendix A are for neutral atoms, the daughter nucleus has one more electron than the parent, and so the extra electron mass that corresponds to the [latex]{\beta ^-}[/latex] is included in the atomic mass of Ni. Thus,
[latex]{\Delta m = m(^{60} \text{Co}) - m(^{60} \text{Ni})}[/latex]
The [latex]{\beta ^-}[/latex] decay equation for [latex]{^{60} \text{Co}}[/latex] is
[latex]{ _{27} ^{60} \text{Co} _{33} \rightarrow _{28} ^{60} \text{Ni} _{32} + \beta ^- + \overline{\nu} _e}[/latex]
As noticed,
[latex]{\Delta m = m(^{60} \text{Co}) - m(^{60}Ni)}[/latex]
Entering the masses found in Appendix A gives
[latex]{\Delta m = 59.933820 \;\textbf{u} - 59.930789 \;\textbf{u} = 0.003031 \;\textbf{u}}[/latex]
[latex]{E = (\Delta m)c^2 = (0.003031 \;\textbf{u})c^2}[/latex]
Using [latex]{1 \;\textbf{u} = 931.5 \;\text{MeV}/c^2}[/latex] , we obtain
[latex]{E = (0.003031)(931.5 \;\text{MeV}/c^2)(c^2) = 2.82 \;\text{MeV}}[/latex]
Discussion and Implications
Perhaps the most difficult thing about this example is convincing yourself that the [latex]{\beta ^-}[/latex] mass is included in the atomic mass of [latex]{^{60} \;\text{Ni}}[/latex]. Beyond that are other implications. Again the decay energy is in the MeV range. This energy is shared by all of the products of the decay. In many [latex]{^{60} \text{Co}}[/latex] decays, the daughter nucleus [latex]{^{60} \text{Ni}}[/latex] is left in an excited state and emits photons ( [latex]{\gamma}[/latex] rays). Most of the remaining energy goes to the electron and neutrino, since the recoil kinetic energy of the daughter nucleus is small. One final note: the electron emitted in [latex]{\beta ^-}[/latex] decay is created in the nucleus at the time of decay.
The second type of beta decay is less common than the first. It is [latex]{\beta ^+}[/latex] decay. Certain nuclides decay by the emission of a positive electron. This is antielectron or positron decay (see Figure 5).
Figure 5. β+ decay is the emission of a positron that eventually finds an electron to annihilate, characteristically producing gammas in opposite directions.
The antielectron is often represented by the symbol [latex]{e ^+}[/latex], but in beta decay it is written as [latex]{\beta ^+}[/latex] to indicate the antielectron was emitted in a nuclear decay. Antielectrons are the antimatter counterpart to electrons, being nearly identical, having the same mass, spin, and so on, but having a positive charge and an electron family number of [latex]{-1}[/latex]. When a positron encounters an electron, there is a mutual annihilation in which all the mass of the antielectron-electron pair is converted into pure photon energy. (The reaction, [latex]{e^+ + e^- \rightarrow \gamma + \gamma}[/latex], conserves electron family number as well as all other conserved quantities.) If a nuclide [latex]{_Z ^A \textbf{X} _N}[/latex] is known to [latex]{\beta ^+}[/latex] decay, then its [latex]{\beta ^+}[/latex] decay equation is
[latex]{ _Z ^A \textbf{X} _N \rightarrow _{Z - 1} ^A \textbf{Y} _{N+1} + \beta ^+ + \nu _e \; (\beta ^+ \;\text{decay})}[/latex]
where Y is the nuclide having one less proton than X (to conserve charge) and νeνe is the symbol for the electron's neutrino, which has an electron family number of [latex]{+1}[/latex]. Since an antimatter member of the electron family (the [latex]{\beta ^+}[/latex]) is created in the decay, a matter member of the family (here the νeνe) must also be created. Given, for example, that [latex]{^{22}Na \beta ^+}[/latex] decays, you can write its full decay equation by first finding that [latex]{Z = 11}[/latex] for [latex]{^{22} Na}[/latex], so that the daughter nuclide will have [latex]{Z = 10}[/latex], the atomic number for neon. Thus the [latex]{\beta ^+}[/latex] decay equation for [latex]{^{22} \text{Na}}[/latex] is
[latex]{ _{11} ^{22} \text{Na} _{11} \rightarrow _{10} ^{22} \text{Ne} _{12} + \beta ^+ + \nu _e}[/latex]
In [latex]{\beta ^+}[/latex] decay, it is as if one of the protons in the parent nucleus decays into a neutron, a positron, and a neutrino. Protons do not do this outside of the nucleus, and so the decay is due to the complexities of the nuclear force. Note again that the total number of nucleons is constant in this and any other reaction. To find the energy emitted in [latex]{\beta ^+}[/latex] decay, you must again count the number of electrons in the neutral atoms, since atomic masses are used. The daughter has one less electron than the parent, and one electron mass is created in the decay. Thus, in [latex]{\beta ^+}[/latex] decay,
[latex]{\Delta m = m( \text{parent} ) - [m( \text{daughter} ) + 2m_e]}[/latex]
since we use the masses of neutral atoms.
Electron capture is the third type of beta decay. Here, a nucleus captures an inner-shell electron and undergoes a nuclear reaction that has the same effect as [latex]{\beta ^+}[/latex] decay. Electron capture is sometimes denoted by the letters EC. We know that electrons cannot reside in the nucleus, but this is a nuclear reaction that consumes the electron and occurs spontaneously only when the products have less mass than the parent plus the electron. If a nuclide [latex]{_Z ^A \textbf{X} _N}[/latex] is known to undergo electron capture, then its electron capture equation is
[latex]{ _Z ^A \textbf{X} _N + e ^- \rightarrow _{Z-1} ^A \textbf{Y} _{N+1} + \nu _e( \textbf{electron capture, or EC})}[/latex]
Any nuclide that can [latex]{\beta ^+}[/latex] decay can also undergo electron capture (and often does both). The same conservation laws are obeyed for EC as for [latex]{\beta ^+}[/latex] decay. It is good practice to confirm these for yourself.
All forms of beta decay occur because the parent nuclide is unstable and lies outside the region of stability in the chart of nuclides. Those nuclides that have relatively more neutrons than those in the region of stability will [latex]{\beta ^-}[/latex] decay to produce a daughter with fewer neutrons, producing a daughter nearer the region of stability. Similarly, those nuclides having relatively more protons than those in the region of stability will [latex]{\beta ^-}[/latex] decay or undergo electron capture to produce a daughter with fewer protons, nearer the region of stability.
Gamma decay is the simplest form of nuclear decay—it is the emission of energetic photons by nuclei left in an excited state by some earlier process. Protons and neutrons in an excited nucleus are in higher orbitals, and they fall to lower levels by photon emission (analogous to electrons in excited atoms). Nuclear excited states have lifetimes typically of only about [latex]{10^{-14}}[/latex] s, an indication of the great strength of the forces pulling the nucleons to lower states. The [latex]{\gamma}[/latex] decay equation is simply
[latex]{_Z^A \textbf{X}_N ^* \rightarrow _Z^A \textbf{X}_N + \gamma _1 + \gamma _2 + \cdots (\gamma \;\text{decay})}[/latex]
where the asterisk indicates the nucleus is in an excited state. There may be one or more [latex]{\gamma}[/latex] s emitted, depending on how the nuclide de-excites. In radioactive decay, [latex]{\gamma}[/latex] emission is common and is preceded by [latex]{\gamma}[/latex] or [latex]{\beta}[/latex] decay. For example, when [latex]{^{60} \text{Co} \beta ^-}[/latex] decays, it most often leaves the daughter nucleus in an excited state, written [latex]{^{60} \text{Ni} ^*}[/latex]. Then the nickel nucleus quickly [latex]{\gamma}[/latex] decays by the emission of two penetrating [latex]{\gamma}[/latex] s:
[latex]{ ^{60} \text{Ni} ^* \rightarrow ^{60}\text{Ni} + \gamma _1 + \gamma _2}[/latex]
These are called cobalt [latex]{\gamma}[/latex] rays, although they come from nickel—they are used for cancer therapy, for example. It is again constructive to verify the conservation laws for gamma decay. Finally, since [latex]{\gamma}[/latex] decay does not change the nuclide to another species, it is not prominently featured in charts of decay series, such as that in Figure 1.
There are other types of nuclear decay, but they occur less commonly than αα,
[latex]{\beta}[/latex], and [latex]{\gamma}[/latex] decay. Spontaneous fission is the most important of the other forms of nuclear decay because of its applications in nuclear power and weapons. It is covered in the next chapter.
When a parent nucleus decays, it produces a daughter nucleus following rules and conservation laws. There are three major types of nuclear decay, called alpha ([latex]{\alpha}[/latex]), beta ([latex]{\beta}[/latex]), and gamma ([latex]{\gamma}[/latex]). The [latex]{\alpha}[/latex] decay equation is
[latex]{_Z^A \textbf{X} _N \rightarrow _{Z-2}^{A-4} \textbf{Y} _{N-2} + ^2_4 \text{He}_2}[/latex]
Nuclear decay releases an amount of energy [latex]{E}[/latex] related to the mass destroyed [latex]{\Delta m}[/latex] by
[latex]{E = (\Delta m)c^2}[/latex]
There are three forms of beta decay. The [latex]{\beta ^-}[/latex] decay equation is
[latex]{ _Z ^A \textbf{X} _N \rightarrow _{Z+1} ^{A} \textbf{Y} _{N-1} + \beta ^- + \overline{\nu} _e}[/latex]
The [latex]{\beta ^+}[/latex] decay equation is
[latex]{ ^Z _A \textbf{X} _N \rightarrow _{Z-1} ^A \textbf{Y} _{N+1} + \beta ^+ + \nu _e}[/latex]
The electron capture equation is
[latex]{_Z ^A \textbf{X} _N + e ^- \rightarrow _{Z-1} ^A \textbf{Y} _{N+1} + \nu _e}[/latex]
[latex]{\beta ^-}[/latex] is an electron, [latex]{\beta ^+}[/latex] is an antielectron or positron, [latex]{\nu _e}[/latex] represents an electron's neutrino, and [latex]{\overline{\nu} _e}[/latex] is an electron's antineutrino. In addition to all previously known conservation laws, two new ones arise— conservation of electron family number and conservation of the total number of nucleons. The [latex]{\gamma}[/latex] decay equation is
[latex]{_Z^A \textbf{X} _N* \rightarrow _Z^A \textbf{X}_N +\gamma _1 + \gamma _2 + \cdots }[/latex]
[latex]{\gamma}[/latex] is a high-energy photon originating in a nucleus.
1: Star Trek fans have often heard the term "antimatter drive." Describe how you could use a magnetic field to trap antimatter, such as produced by nuclear decay, and later combine it with matter to produce energy. Be specific about the type of antimatter, the need for vacuum storage, and the fraction of matter converted into energy.
2: What conservation law requires an electron's neutrino to be produced in electron capture? Note that the electron no longer exists after it is captured by the nucleus.
3: Neutrinos are experimentally determined to have an extremely small mass. Huge numbers of neutrinos are created in a supernova at the same time as massive amounts of light are first produced. When the 1987A supernova occurred in the Large Magellanic Cloud, visible primarily in the Southern Hemisphere and some 100,000 light-years away from Earth, neutrinos from the explosion were observed at about the same time as the light from the blast. How could the relative arrival times of neutrinos and light be used to place limits on the mass of neutrinos?
4: What do the three types of beta decay have in common that is distinctly different from alpha decay?
In the following eight problems, write the complete decay equation for the given nuclide in the complete [latex]{_Z ^A \textbf{X}_N}[/latex] notation. Refer to the periodic table for values of [latex]{Z}[/latex].
1: [latex]{\beta ^-}[/latex] decay of [latex]{^3 \textbf{H}}[/latex] (tritium), a manufactured isotope of hydrogen used in some digital watch displays, and manufactured primarily for use in hydrogen bombs.
2: [latex]{\beta ^-}[/latex] decay of [latex]{^{40} \textbf{K}}[/latex], a naturally occurring rare isotope of potassium responsible for some of our exposure to background radiation.
3: [latex]{\beta ^+}[/latex] decay of [latex]{^{50} \text{Mn}}[/latex].
4: [latex]{\beta ^+}[/latex] decay of [latex]{^{52} \text{Fe}}[/latex].
5: Electron capture by [latex]{^7 \text{Be}}[/latex].
6: Electron capture by [latex]{^{106} \text{In}}[/latex].
7: [latex]{\alpha}[/latex] decay of [latex]{^{210} \text{Po}}[/latex], the isotope of polonium in the decay series of [latex]{^{238} \textbf{U}}[/latex] that was discovered by the Curies. A favorite isotope in physics labs, since it has a short half-life and decays to a stable nuclide.
8: [latex]{\alpha}[/latex] decay of [latex]{^{226} \text{Ra}}[/latex], another isotope in the decay series of [latex]{^{238} \textbf{U}}[/latex], first recognized as a new element by the Curies. Poses special problems because its daughter is a radioactive noble gas.
In the following four problems, identify the parent nuclide and write the complete decay equation in the [latex]{ _Z ^A \textbf{X} _N}[/latex] notation. Refer to the periodic table for values of [latex]{Z}[/latex].
9: [latex]{\beta ^-}[/latex] decay producing [latex]{ ^{137} \text{Ba}}[/latex]. The parent nuclide is a major waste product of reactors and has chemistry similar to potassium and sodium, resulting in its concentration in your cells if ingested.
10: [latex]{\beta ^-}[/latex] decay producing [latex]{ ^{90} \textbf{Y}}[/latex]. The parent nuclide is a major waste product of reactors and has chemistry similar to calcium, so that it is concentrated in bones if ingested ( [latex]{ ^{90} \textbf{Y}}[/latex] is also radioactive.)
11: [latex]{\alpha}[/latex] decay producing [latex]{^{228} \text{Ra}}[/latex]. The parent nuclide is nearly 100% of the natural element and is found in gas lantern mantles and in metal alloys used in jets ([latex]{^{228} \text{Ra}}[/latex] is also radioactive).
12: [latex]{\alpha}[/latex] decay producing [latex]{^{208} \text{Pb}}[/latex]. The parent nuclide is in the decay series produced by [latex]{^{232} \text{Th}}[/latex], the only naturally occurring isotope of thorium.
13: When an electron and positron annihilate, both their masses are destroyed, creating two equal energy photons to preserve momentum. (a) Confirm that the annihilation equation [latex]{e^+ + e^- \rightarrow \gamma + \gamma}[/latex] conserves charge, electron family number, and total number of nucleons. To do this, identify the values of each before and after the annihilation. (b) Find the energy of each [latex]{\gamma}[/latex] ray, assuming the electron and positron are initially nearly at rest. (c) Explain why the two [latex]{\gamma}[/latex] rays travel in exactly opposite directions if the center of mass of the electron-positron system is initially at rest.
14: Confirm that charge, electron family number, and the total number of nucleons are all conserved by the rule for [latex]{\alpha}[/latex] decay given in the equation [latex]{_Z^A \textbf{X}_N \rightarrow _{Z-2}^{A-4} \textbf{Y}_{N-2} + _2^4 \text{He}_2}[/latex]. To do this, identify the values of each before and after the decay.
15: Confirm that charge, electron family number, and the total number of nucleons are all conserved by the rule for [latex]{\beta ^-}[/latex] decay given in the equation [latex]{ _Z ^A \textbf{X} _N \rightarrow _{Z+1} ^A \textbf{Y} _{N-1} + \beta ^- + \overline{\nu} _e}[/latex]. To do this, identify the values of each before and after the decay.
16: Confirm that charge, electron family number, and the total number of nucleons are all conserved by the rule for [latex]{\beta ^-}[/latex] decay given in the equation [latex]{ _Z ^A \textbf{X} _N \rightarrow _{Z-1} ^A \textbf{Y} _{N-1} + \beta ^- + \nu _e}[/latex]. To do this, identify the values of each before and after the decay.
17: Confirm that charge, electron family number, and the total number of nucleons are all conserved by the rule for electron capture given in the equation [latex]{ _Z ^A \textbf{X} _N + e ^- \rightarrow _{Z-1} ^A \textbf{Y} _{N+1} + \nu _e}[/latex]. To do this, identify the values of each before and after the capture.
18: A rare decay mode has been observed in which [latex]{^{222} \text{Ra}}[/latex] emits a [latex]{^{14} \textbf{C}}[/latex] nucleus. (a) The decay equation is [latex]{^{222} \text{Ra} \rightarrow ^A \textbf{X} + ^{14} \textbf{C}}[/latex]. Identify the nuclide [latex]{^A \textbf{X}}[/latex]. (b) Find the energy emitted in the decay. The mass of [latex]{^{222} \text{Ra}}[/latex] is 222.015353 u.
19: (a) Write the complete [latex]{\alpha}[/latex] decay equation for [latex]{^{226} \text{Ra}}[/latex].
(b) Find the energy released in the decay.
20: (a) Write the complete [latex]{\alpha}[/latex] decay equation for [latex]{^{249} \text{Cf}}[/latex].
21: (a) Write the complete [latex]{\beta ^-}[/latex] decay equation for the neutron. (b) Find the energy released in the decay.
22: (a) Write the complete [latex]{\beta ^-}[/latex] decay equation for [latex]{^{90} \text{Sr}}[/latex], a major waste product of nuclear reactors. (b) Find the energy released in the decay.
23: Calculate the energy released in the [latex]{\beta ^+}[/latex] decay of [latex]{^{22} \text{Na}}[/latex], the equation for which is given in the text. The masses of [latex]{^{22} \text{Na}}[/latex] and [latex]{^{22} \text{Ne}}[/latex] are 21.994434 and 21.991383 u, respectively.
24: (a) Write the complete [latex]{\beta ^+}[/latex] decay equation for [latex]{^{11} \textbf{C}}[/latex].
(b) Calculate the energy released in the decay. The masses of [latex]{^{11} \textbf{C}}[/latex] and [latex]{^{11} \textbf{B}}[/latex] are 11.011433 and 11.009305 u, respectively.
25: (a) Calculate the energy released in the [latex]{\alpha}[/latex] decay of [latex]{^{238} \textbf{U}}[/latex].
(b) What fraction of the mass of a single [latex]{^{238} \textbf{U}}[/latex] is destroyed in the decay? The mass of [latex]{^{234} \text{Th}}[/latex] is 234.043593 u.
(c) Although the fractional mass loss is large for a single nucleus, it is difficult to observe for an entire macroscopic sample of uranium. Why is this?
26: (a) Write the complete reaction equation for electron capture by [latex]{^7 \text{Be}}[/latex].
(b) Calculate the energy released.
27: (a) Write the complete reaction equation for electron capture by [latex]{^{15} \textbf{O}}[/latex].
the original state of nucleus before decay
the nucleus obtained when parent nucleus decays and produces another nucleus following the rules and the conservation laws
the particle that results from positive beta decay; also known as an antielectron
the process by which an atomic nucleus of an unstable atom loses mass and energy by emitting ionizing particles
type of radioactive decay in which an atomic nucleus emits an alpha particle
type of radioactive decay in which an atomic nucleus emits a beta particle
type of radioactive decay in which an atomic nucleus emits a gamma particle
decay equation
the equation to find out how much of a radioactive material is left after a given period of time
nuclear reaction energy
the energy created in a nuclear reaction
an electrically neutral, weakly interacting elementary subatomic particle
electron's antineutrino
antiparticle of electron's neutrino
positron decay
type of beta decay in which a proton is converted to a neutron, releasing a positron and a neutrino
antielectron
another term for positron
decay series
process whereby subsequent nuclides decay until a stable nuclide is produced
electron's neutrino
a subatomic elementary particle which has no net electric charge
composed of antiparticles
electron capture
the process in which a proton-rich nuclide absorbs an inner atomic electron and simultaneously emits a neutrino
electron capture equation
equation representing the electron capture
1: [latex]{_1 ^3 \textbf{H} _2 \rightarrow _2 ^3 \text{He} _1 + \beta ^- + \overline{\nu} _e}[/latex]
3: [latex]{ _{25} ^{50} \textbf{M} _{25} \rightarrow _{24} ^{50} \text{Cr} _{26} + \beta ^+ + \nu _e}[/latex]
5: [latex]{ _4 ^7 \text{Be} _3 + e ^- \rightarrow _3 ^7 \text{Li} _4 + \nu _e}[/latex]
7: [latex]{ _{84} ^{210} \text{Po} _{126} \rightarrow _{82} ^{206} \text{Pb} _{124} + _2 ^4 \text{He} _2}[/latex]
9: [latex]{ _{55} ^{137} \text{Cs} _{82} \rightarrow _{56} ^{137} \text{Ba} _{81} + \beta ^- + \overline{\nu} _e}[/latex]
11: [latex]{ _{90} ^{232} \text{Th} _{142} \rightarrow _{88} ^{228} \text{Ra} _{140} + _2 ^4 \text{He} _2}[/latex]
13: (a) [latex]{\textbf{charge:} (+1) + (-1) = 0; \;\textbf{electron family number:} (+1) + (-1) =0; A: 0+0=0}[/latex]
(b) 0.511 MeV
(c) The two [latex]{\gamma}[/latex] rays must travel in exactly opposite directions in order to conserve momentum, since initially there is zero momentum if the center of mass is initially at rest.
15: [latex]{Z = (Z+1) - 1; A = A ; \;\text{efn} : 0 = (+1)+ (-1)}[/latex]
17: [latex]{Z-1 = Z-1; A = A ; \;\text{efn} : (+1)= (+1)}[/latex]
19: (a) [latex]{ _{88} ^{226} \text{Ra} _{138} \rightarrow _{86} ^{222} \text{Rn} _{136} + _2 ^4 \text{He} _2}[/latex]
(b) 4.87 MeV
21: (a) [latex]{\textbf{n} \rightarrow \textbf{p} + \beta ^- + \overline{\nu} _e}[/latex]
(b) ) 0.783 MeV
23: 1.82 MeV
25: (a) 4.274 MeV
(b) [latex]{1.927 \times 10^{-5}}[/latex]
(c) Since U-238 is a slowly decaying substance, only a very small number of nuclei decay on human timescales; therefore, although those nuclei that decay lose a noticeable fraction of their mass, the change in the total mass of the sample is not detectable for a macroscopic sample.
27: (a) [latex]{ _8 ^{15} \textbf{O} _7 + e ^- \rightarrow _7 ^{15} \textbf{N} _8 + \nu _e}[/latex]
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\begin{document}
\renewcommand{\fnsymbol{footnote}}{\fnsymbol{footnote}} \author{{\sc G\"unter Last\thanks{Karlsruhe Institute of Technology,
Germany. E-mail: \texttt{[email protected]}} and Hermann Thorisson\thanks{University of Iceland, Iceland. E-mail: \texttt{[email protected]}}}} \title{Transportation of diffuse random measures on $\R^d$} \date{\today} \maketitle \begin{abstract}
\noindent We consider two jointly stationary and ergodic random measures $\xi$ and $\eta$ on $\R^d$ with equal finite intensities, assuming $\xi$ to be diffuse (non-atomic). An allocation is a random mapping taking $\R^d$ to $\R^d\cup\{\infty\}$ in a translation invariant way. We construct allocations transporting the diffuse
$\xi$ to arbitrary
$\eta$, under the mild condition of existence of an ${ ` }$auxiliary' point process which is needed only in the case when $\eta$ is diffuse. When that condition does not hold we show by a counterexample that an allocation transporting $\xi$ to $\eta$ need not exist.\end{abstract}
\noindent {\bf Keywords:} stationary random measure, point process, invariant allocation, invariant transport, Palm measure, shift-coupling, stable allocation.
\noindent {\bf AMS MSC 2020:} Primary 60G57, 60G55; Secondary 60G60.
\section{Introduction}
Mass transportation is an important and lively research area. We refer to \cite{Villani09} for an extensive monograph on optimal transports. A more recent addition to the literature is transports between random measures (and in particular balancing allocations), which connects to the classical topic in several ways. For instance it was shown in \cite{HoHolPe06,LaTho09} that balancing transports between stationary random measures correspond to certain couplings (shift-couplings) of the associated Palm measures, while \cite{HuesSturm13} studied quantitative optimality of a balancing allocation.
Before going further we need to establish some notation and terminology. An {\em allocation} is a random mapping $\tau: x\mapsto \tau(x)$ taking $\R^d$ to $\R^d\cup\{\infty\}$ in a translation-invariant (equivariant, covariant) way; note that in this paper the invariance is included in the definition of allocation. Let $\xi$ and $\eta$ be jointly stationary and ergodic random measures on $\R^d$ ($d \geq 1$) with finite identical intensities $\lambda_\xi = \lambda_\eta$. Joint ergodicity of $\xi$ and $\eta$ means that the distribution of $(\xi,\eta)$ takes only the values $0$ and $1$ on translation invariant sets, and joint
stationarity
means that (with $\overset{D}{=}$ denoting identity in distribution) $$
(\xi(x+ \cdot), \eta(x+ \cdot )) \,\overset{D}{=}\, (\xi,\eta),
\quad x \in\R^d.
$$
Say that an allocation {\em balances} the {\em source} $\xi$ and the {\em destination} $\eta$ if it transports $\xi$ to $\eta$, that is, if (a.s.)\ the image of the measure $\xi$ under $\tau$ is $\eta$, $$\xi(\tau \in \cdot)=\eta.$$ See~Section~\ref{secprelim} for exact framework and definitions. See Remark \ref{history} for historical notes beyond those in this introduction.
In the present paper we construct allocations balancing a {\em diffuse} (a {\em non-atomic}) source $\xi$ and an {\em arbitrary} destination $\eta$. In order to explain the point
of the paper, let us outline two remarkable examples.
\noindent {\bf Extra head:} The search for balancing allocations goes back to Liggett's surprising idea of `how to choose a head at random' -- an {\em extra head} -- in a two-sided sequence of i.i.d.\ coin tosses. If there is a head at the origin, it is an extra head (the other coins are i.i.d.). If there is a tail at the origin, move the origin to the right counting heads and tails until you have more heads than tails.\ Then you are at a head and it is an extra head; see \cite{L02}.
This can be restated as follows. If $\eta$ is the simple point process (on $\Z$) formed by the heads, and $T$ is the location of the extra head, then $\eta(T + \cdot)$ has the same distribution as the Palm version $\eta^\circ$ of $\eta$, \begin{align}\label{extra} \eta(T+\cdot) \,\overset{D}{=}\, \eta^\circ. \end{align} This means that we have constructed a {\em shift-coupling} of
$\eta$ and $\eta^\circ$.
Allocations provide a proof of this result as follows. For each $n \in \Z$ let $\tau(n)$ be the location of the `extra head' found when starting from $n$ rather than from the origin $0$. Then the map $\tau: \Z \to \Z \cup \infty$ is an allocation that leaves the heads where they are, while $\tau$ turns out to be (a.s.) a bijection from the tails to the heads. Thus if we let $\mu$ be the counting measure on $\Z$ then $\tau$ balances $\mu$ and $2\eta$ (or, equivalently, $\tau$ balances $\mu$/2 and $\eta$). According to \cite{HoHolPe06, LaTho09}, this implies that \eqref{extra} holds.
\noindent {\bf Stable marriage of Poisson and Lebesgue:} Liggett's idea led Hoffman, Holroyd and Peres to solving an open problem from the mid nineties: how to find an {\em extra point} in a Poisson process. Let $\eta$ be a stationary Poisson process on $\R^d$ with intensity $1$. Associate disjoint cells of volume $1$ to the points of $\eta$ as follows. Expand balls simultaneously from all the points of $\eta$. If the ball of a particular point has accumulated volume $1$ before it hits another ball, then this ball is the cell of that point. If not, then continue expanding to accumulate volume when reaching space that has not already been reached by another ball; stop when volume $1$ has been accumulated. It turns out (a.s.) that in this way each point obtains a cell of volume $1$ and that the cells are disjoint and cover $\R^d$. Now let $T$ be the point of the cell containing the origin. If the origin is shifted to $T$ and that point is removed then the remaining points of $\eta(T+\cdot)$ form a stationary Poisson process with intensity $1$. Thus $T$ is an extra point; see \cite{HoHolPe06}.
This again means that we have constructed a shift-coupling of
$\eta$ and its Palm version $\eta^\circ$, that is, \eqref{extra} holds.
And again, allocations provide a proof of that result as follows.\ For each $x \in \R^d$ let $\tau(x)$ be the point of the cell containing~$x$ and let $\mu$ be Lebesgue measure on $\R^d$.\ Then $\tau$ is (clearly) an allocation balancing $\mu$ and~$\eta$. According to \cite{HoHolPe06, LaTho09}, this implies that \eqref{extra} holds.
What makes allocations particularly interesting are the shift-coupling
results in the above examples.
In both examples the source $\mu$ is translation-invariant and non-random, but the shift-coupling result extends to general $\xi$ and $\eta$ as follows. If a stochastic process (or a random measure) $X$ is stationary and ergodic jointly with $\xi$ and $\eta$, and $\tau$ transports $\xi$ to
$\eta$, then by shifting the origin to $\tau(0)$ the {\em Palm version} of $X$ w.r.t.\ $\xi$ turns into the Palm version of $X$ w.r.t.\ $\eta$. See \eqref{shift-coupling2} in the next section, and \cite{LaTho09}, for this result. See \cite{AldousThor,Thor96} for the origin of shift-coupling.
\noindent {\bf Unbiased Skorokhod embedding:} In \cite{LaMoeTho12, LaTaTho18} it is shown in the one-dimensional case, $d=1$, that if $\xi$ is diffuse (non-atomic), $\eta$ is arbitrary, and $\xi$ and $\eta$ are mutually singular
then they are balanced by the allocation $\tau$ defined by \begin{align}\label{alocation} \tau(x) = \inf\{t>x : \xi([x,t]) \leq \eta([x, t]\},\quad x\in\R. \end{align} Local times of Brownian motion are diffuse and (see \cite{GeHo73}) the two-sided standard Brownian motion $(B_x)_{x\in\R}$
is a Palm version of the $\sigma$-finite stationary Brownian measure. Thus (see \cite{LaMoeTho12}) the shift-coupling result for general $\xi$ and $\eta$
can be applied with $\xi = \ell^0$ the local time at $0$ and with $\eta = \int\ell^y\nu(dy)$ where $\ell^y$ is the local time at $y$ and where $\nu$ is a probability measure without atom at $0$ (to ensure mutual singularity). This yields the following {\em unbiased} Skorokhod embedding:
$(B_{\tau(0)+x})_{x\in \R}$ is a two-sided standard Brownian motion
with distribution $\nu$ at $x=0$.
It is said to be {\em unbiased}
because
not only the one-sided $(B_{\tau(0)+x})_{x\geq 0}$, but also the two-sided $(B_{\tau(0)+x})_{x\in \R}$, is Brownian. The same approach results in various
embeddings when applied to local times associated with Brownian motion, e.g.\ extra excursion (see \cite{LaTaTho18}) and (see \cite{PitmanTang15}) extra Brownian bridge.
In the present paper we consider the $d$ dimensional case when $\xi$ is diffuse, $\eta$ is arbitrary, and $\xi$ and $\eta$ need {\em not} be mutually singular. It turns out that there are special cases where balancing allocations do {\em not} exist (Section~\ref{seccounter}). In order to guarantee the existence of a balancing allocation we impose the mild condition of the existence of a non-zero simple point process $\chi$ on $\R^d$ with finite intensity $\lambda_\chi$ and such that $\xi$, $\eta$ and $\chi$ are jointly stationary and ergodic. We call this simple point process $\chi$ {\em auxiliary}.
The following theorem is the main result of the paper. Note that the auxiliary $\chi$ is only needed when $\eta$ is purely diffuse.
\begin{theorem}\label{main} Assume that $\xi$ and $\eta$ are jointly stationary and ergodic random measures on $\R^d$. Let $\xi$ be diffuse (non-atomic) and $$ 0<\lambda_\xi=\lambda_\eta<\infty. $$ Then there exists an allocation balancing $\xi$ and $\eta$ if one of the following conditions holds:
{\em (a)} $\eta$ has a non-zero discrete component;
{\em (b)} $\eta$ is diffuse and there exists an auxiliary $\chi$. \end{theorem}
Condition (a) covers discrete $\eta$ and, in particular, point processes. Note that under condition (a) an auxiliary $\chi$ always exists and can be chosen as a {\em factor} of $\eta$, that is, as a measurable and equivariant (w.r.t.\ translation) function of $\eta$. When the discrete component of $\eta$ has isolated atoms then $\chi$ can be taken to be the support of $\eta$. And although in general the support of the discrete component need not consist of isolated points (it can even be dense), there exists a constant $c > 0$ such that the following simple point process (here $\delta_x$ is the measure with mass $1$ at $x$) \begin{align}\label{i3} \chi = \sum_x \I\{\eta(\{x\})> c\} \delta_x \end{align} is non-zero.
Under condition (b), there are also cases where an auxiliary $\chi$ exists as a factor of $(\xi,\eta)$. But the counterexample in Section 8 shows that the condition of the existence of an auxiliary $\chi$ cannot simply be removed from (b). There are diffuse $\xi$ and $\eta$ such that an allocation transporting $\xi$ to $\eta$ does not exist. However, if extension of the underlying probability space $(\Omega, \mathcal{F}, \BP)$ is allowed, then that obstacle can be overcome.
\begin{corollary}\label{cmain} Assume that $\xi$ and $\eta$ are jointly stationary and ergodic random measures on $\R^d$. Let $\xi$ be diffuse and $$ 0<\lambda_\xi=\lambda_\eta<\infty. $$ Extend $(\Omega, \mathcal{F}, \BP)$ to support a stationary Poisson process
$\chi$ on $\R^d$ which is independent of $\xi$ and~$\eta$. Then there exists an allocation balancing $\xi$ and $\eta$. \end{corollary}
As stressed above, an important property of a balancing allocation $\tau$ is the shift-coupling at \eqref{shift-coupling2}.
When specialised to the case $\xi = \lambda_\eta$ times Lebesgue measure, then \eqref{shift-coupling2} turns into \eqref{shift-coupling2222}, that is, into \eqref{extra} with $T=\tau(0)$. This is called {\em extra head scheme} in \cite{HP05}. Note however that removing a `head' (or some pattern like an excursion)
from $\eta^{\circ}$, does not
result in a copy of $\eta$ except in very special cases such as
coin tossing, the Poisson process and Brownian motion.
According to Theorem~\ref{main} applied with $\xi = \lambda_\eta$ times Lebesgue measure, the result \eqref{extra} always holds with $T$ a function of $\eta$ alone, unless $\eta$ is diffuse. In that case,
\eqref{extra} still holds with $T$ a function of $\eta$, provided there exists an auxiliary $\chi$ that is a factor of $\eta$. Finally, according to Corollary~\ref{cmain}, if external randomisation is allowed then \eqref{extra} always holds for some~$T$, also when $\eta$ is diffuse. (Actually, due to ergodicity and the definition of Palm probabilities at \eqref{Palmmeasure}, the distributions of $\eta$ and $\eta^{\circ}$ have the same (zero-one) values on invariant sets. Thus, according to an abstract existence result in \cite{Thor96} for shift-coupling on groups,
\eqref{extra} always holds for some $T$ defined on an extended probability space. But the constructions of $T$ in the present paper are explicit.)
The plan of the paper is as follows. Section~\ref{secprelim} collects some preliminaries on stationary random measures, balancing allocations, and Palm theory. Section~\ref{secdiscretedest} prepares for the proof of Theorem~\ref{main}. The theorem is then proved in Sections~\ref{secatomsdest}-\ref{secmixeddest}, where allocations are constructed in four exhaustive cases: discrete $\eta$ with only isolated atoms in Section~\ref{secatomsdest}, discrete $\eta$ with some accumulating atoms in Section~\ref{secdiscrete}, diffuse $\eta$ ($\eta$ with no atoms) in Section~\ref{secdifdest}, and $\eta$
with discrete and diffuse parts in Section~\ref{secmixeddest}. We give algorithmically explicit constructions in the discrete cases, and less explicit in the diffuse. Section~\ref{seccounter} proves by counterexample that the auxiliary $\chi$ cannot simply be removed from part (b) of Theorem~\ref{main}. Section~\ref{secrem} concludes with remarks.
\section{Preliminaries}\label{secprelim}
Let $(\Omega,\mathcal{F},\BP)$ be a probability space with expectation operator $\BE$. A {\em random measure} (resp.\ {\em point process}) $\xi$ on $\R^d$ (equipped with its Borel $\sigma$-field $\mathcal{B}(\R^d)$) is a kernel from $\Omega$ to $\R$ such that $\xi(\omega,C)<\infty$ (resp.\ $\xi(\omega,C)\in\N_0$) for $\BP$-a.e.\ $\omega$ and all compact $C\subset\R^d$; see e.g.\ \cite{Kallenberg,LastPenrose17}. A point process $\xi$ is called {\em simple} if $\xi(\{x\})\in\{0,1\}$, $x\in\R^d$, except on a set with probability zero. Further, let $(\Omega,\mathcal{F})$ be equipped with a {\em measurable flow} $\theta_x\colon\Omega \to \Omega$, $x\in \R^d$. This is a family of mappings such that $(\omega,x)\mapsto \theta_x\omega$ is measurable, $\theta_0$ is the identity on $\Omega$ and \begin{align}\label{flow}
\theta_x \circ \theta_y =\theta_{x+y},\quad x,y\in \R^d, \end{align} where $\circ$ denotes composition. An {\em allocation} \cite{HP05,LaTho09} is a measurable mapping $\tau\colon\Omega\times\R^d\rightarrow\R^d\cup\{\infty\}$ that is {\em equivariant} in the sense that \begin{align}\label{allocation} \tau(\theta_y\omega,x-y)=\tau(\omega,x)-y, \quad x,y\in\R^d,\, \text{$\BP$-a.e.\ $\omega\in\Omega$}. \end{align} We illustrate these concepts with a simple but illustrative example.
\begin{example}\rm Take $\Omega$ as the space of all locally finite sets $\omega\subset\R^d$, equipped with the usual $\sigma$-field and define $\theta_x\omega:=\omega-x$, $x\in\R^d$. The counting measure $\xi(\omega)$ supported by $\omega$ defines a (discrete) random measure $\xi$. An example of an allocation $\tau$ is to take $\tau(\omega,x)$ as the point of $\omega\in\Omega$ closest to $x\in\R^d$, using lexicographic order to break ties. (For $\omega=\emptyset$ we set $\tau(\omega,\cdot)\equiv\infty$.) \end{example}
We assume that the measure $\BP$ is {\em stationary}; that is $$ \BP\circ\theta_x=\BP,\quad x\in\R^d, $$ where $\theta_x$ is interpreted as a mapping from $\mathcal{F}$ to $\mathcal{F}$ in the usual way: $$ \theta_xA:=\{\theta_x\omega:\omega\in A\},\quad A\in\mathcal{F},\, x\in\R^d. $$ A random measure $\xi$ on $\R^d$ is said to be {\em stationary} if \begin{align}\label{adapt} \xi(\theta_x\omega,C-x)=\xi(\omega,C),\quad C\in\mathcal{B}(\R^d),\,x\in\R^d, \BP\text{-a.e.\ $\omega\in\Omega$}. \end{align} Abusing our notation by defining the shifts $\theta_x$, $x\in\R^d$, also for measures on $\R^d$ in the obvious way, we obtain from \eqref{adapt} and stationarity of $\BP$ that $$ \theta_x\xi \overset{D}{=}\xi, \quad x\in \R^d. $$ The {\em invariant} $\sigma$-field $\mathcal{I}\subset \mathcal{F}$ is the class of all sets $A\in\mathcal{F}$ satisfying $\theta_xA=A$ for all $x\in\R^d$. We also assume that $\BP$ is {\em ergodic}; that is for any $A\in\mathcal{I}$, we have $\BP(A)\in\{0,1\}$ (see however Remark~\ref{ergod}).
Let $\xi$ be a stationary random measure on $\R^d$ with positive and finite {\em intensity} $$\lambda_\xi:=\BE\xi[0,1]^d.$$ The {\em Palm probability measure} $\BP_\xi$ of $\xi$ (with respect to $\BP$) is defined by \begin{align}\label{Palmmeasure} \BP_\xi(A):=\lambda_\xi^{-1}\lambda_d(B)^{-1}\,\BE \int \I_B(x)\I_A(\theta_x)\, \xi(dx), \quad A\in\mathcal{F}, \end{align} where $B\subset\R^d$ is a Borel set with positive and finite Lebesgue measure $\lambda_d(B)$. This definition does not depend on $B$. The expectation operator associated with $\BP_\xi$ is denoted by $\BE_\xi$. Any multiple $c\lambda_d$ of Lebesgue measure
is a (rather trivial) stationary random measure. In this case we obtain from stationarity of $\BP$ that \begin{align}\label{LebPalm} \BP_{c\lambda_d} = \BP. \end{align}
An allocation $\tau$ {\em balances} two random measures $\xi$ and $\eta$ if \begin{align}\label{bal456} \BP(\xi(\{s\in\R:\tau(s)=\infty\})>0)=0 \end{align} and the image measure of $\xi$ under $\tau$ is $\eta$, that is, \begin{align}\label{bal123} \int \I\{\tau(s)\in C\}\,\xi(ds)=\eta(C),\quad C\in\mathcal{B}(\mathbb{R}^d),\, \BP\text{-a.e.} \end{align} The balancing properties \eqref{bal456} and \eqref{bal123} imply easily that \begin{align}\label{i2} \lambda_\xi= \lambda_\eta. \end{align} By \cite[Theorem 4.1]{LaTho09} we then have the {\em shift-coupling} \begin{align}\label{shiftc} \BP_\xi(\theta_{\tau(0)}\in\cdot)=\BP_\eta. \end{align}
\begin{remark}\label{Palmprel}\rm In this remark we consider the shift-coupling result
\eqref{shiftc}
in terms of random elements.
Let $X$ be a random element that can be translated by $t\in \R^d$, for instance a random measure, or a random field, or the identity on $\Omega$. Then $X$, $\xi$ and $\eta$, defined on the probability space $(\Omega,\mathcal{F},\BP)$ above,
are jointly stationary and ergodic.
A {\em Palm version of $X$ w.r.t.\ $\xi$} is any random element with the distribution $\BP_\xi(X\in\cdot)$. In particular, according to \eqref{LebPalm},
a Palm version of $X$ w.r.t.\ Lebesgue measure is $X$ itself.
What is generally called a {\em Palm version} of a random measure
$\xi$
is a random measure $ \xi^{\circ} $ with the distribution $\BP_\xi(\xi\in\cdot)$.
That is, $ \xi^{\circ} $ is a Palm version of $\xi$ with respect to itself.
The informal interpretation of $ \xi^{\circ} $ in the case when $ \xi $ is a simple
point process is that $ \xi^{\circ} $ behaves like $ \xi$
conditioned on having a point at the origin.
When
$ \xi$ is a Poisson process
then (and only then) $ \xi^{\circ} $ can be obtained simply
by placing an extra point at the origin, $ \xi^{\circ} = \xi + \delta_0$.
In the ergodic case, which is assumed here, another
informal interpretation
of $ \xi^{\circ} $ is
that $ \xi^{\circ} $ behaves like $\xi$ with origin shifted to
a uniformly chosen point of $\xi$, -- or when $\xi$ is
not a simple point process,
to a uniformly chosen location in the
mass of $\xi$.
(These interpretations are motivated by limit theorems.)
From \eqref{shiftc} we obtain
the following shift-coupling result.
If $\tau$ is an allocation balancing $\xi$ and $\eta$, then a shift of the origin to $\tau(0)$ turns a Palm version of $X$ w.r.t.\ $\xi$ into a Palm version of $X$ w.r.t.\ $\eta$, \begin{align}\label{shift-coupling2} \BP_\xi(\theta_{\tau(0)}X\in\cdot)=\BP_\eta(X\in \cdot). \end{align} In particular, from
\eqref{LebPalm} and \eqref{shift-coupling2}
with $X = \eta$, we obtain the following
result. If $\tau$ balances the Lebesgue-measure multiple $\lambda_\eta\lambda_d$ and $\eta$, then a shift of the origin to $\tau(0)$ turns the stationary $\eta$ into a Palm version $\eta^\circ$ of $\eta$, \begin{align}\label{shift-coupling2222} \theta_{\tau(0)} \eta
\,\overset{D}{=}\, \eta^{\circ}.
\end{align} On the other hand,
from
\eqref{LebPalm} and \eqref{shift-coupling2}
with $X = \xi$, we obtain the reverse result.
If $\tau$ balances $\xi$ and the Lebesgue-measure multiple $\lambda_\xi\lambda_d$,
then a shift of the origin to $\tau(0)$
turns the a Palm version $\xi^\circ$ of $\xi$ into the stationary $\xi$, \begin{align}\label{shift-coupling2224} \theta_{\tau(0)} \xi^\circ
\,\overset{D}{=}\, \xi. \end{align} If $\xi$ is the only source of randomness, then, as a rule, allocations balancing $\xi$ and a multiple of Lebesgue measure do not exist, see \cite{HP05} and also Section \ref{seccounter}. \end{remark}
\section{Allocations to Discrete Random Measures}\label{secdiscretedest}
Let $\xi$ and $\eta$ be two jointly stationary and ergodic random measures on $\R^d$; see Section~\ref{secprelim}. For the remainder of
this paper we assume $\lambda_\xi>0$ and $\lambda_\eta>0$. Assume that $\eta$ is a discrete random measure with locally finite support. Let $\eta^*$ be the simple point process with the same support as $\eta$. Assume also that $\xi$, $\eta$ and $\eta^*$ have positive and finite intensities $\lambda_\xi$, $\lambda_\eta$ and $\lambda_{\eta^*}$ respectively. (The assumptions $\lambda_{\eta^*}<\infty$ has been made by convenience and could be removed.) We consider an allocation $\tau$ with the property \begin{align}\label{e3.59} \xi(\{z\in\R^d:\tau(z)\notin \eta^*\cup\{\infty\}\})=0,\quad \BP\text{-a.s.} \end{align} Define \begin{align*} C^\tau(z):=\{y\in\R^d\colon\tau(y)=z\}, \quad z\in\R^d. \end{align*} Note that $C^\tau(z)$ is random.
Let $\alpha\in(0,\infty)$. The allocation $\tau$ is said to have {\em appetite} $\alpha$ (w.r.t.\ $(\xi,\eta)$) if \eqref{e3.59} and the following two properties hold. First we have almost surely that \begin{align}\label{e11.45} \eta^*(\{x\in\R^d:\xi(C^\tau(x))>\alpha\eta\{x\}\})=0. \end{align} Second the probability that \begin{align}\label{e11.46} \xi(\{z\in\R^d:\tau(z)= \infty\})>0\quad\text{and}\quad \eta^*(\{x\in\R^d:\xi(C^\tau(x))<\alpha\eta\{x\}\})>0 \end{align} is zero.
\begin{proposition}\label{p1} Assume that the allocation $\tau$ has appetite $\alpha$ for some $\alpha\in (0,\lambda_\xi\lambda_\eta^{-1}]$. Then $\tau$ is $\alpha$-balanced, that is we have a.s.\ that $\eta(\{x\in\R^d:\xi(C^{\tau}(x))\ne \alpha\eta\{x\}\})=0$. Moreover, we have that $\lambda_\xi\BP^0_\xi(\tau(0)\ne\infty)=\alpha\lambda_\eta$. \end{proposition} {\em Proof:} We generalise the proof of \cite[Theorem 10.9]{LastPenrose17}. We start with a general result, that might be of independent interest. Let $g\colon \Omega\times\Omega\to[0,\infty)$. Then \begin{align}\label{e3.1} \lambda_\xi\BE_\xi\I\{\tau(0)\ne\infty\}g(\theta_0,\theta_{\tau(0)}) =\lambda_{\eta^*}\BE_{\eta^*}\int_{C(0)} g(\theta_x,\theta_0)\,\xi(dx), \end{align} where we abbreviate $C(z):=C^\tau(z)$, $z\in\R^d$. This follows from Neveu's exchange formula (see e.g.\ \cite[Remark 3.7]{LastPenrose17}) applied to the function $h(\omega,x):=g(\omega,\theta_x\omega)\I\{\tau(\omega,0)=x\}$ (and replacing $(\eta,\xi)$ by $(\xi,\eta^*)$).
Let \begin{align*} A:=\{\text{there exists $x\in\eta$ such that $\xi(C(x))<\alpha\eta\{x\}$}\}. \end{align*} This event is invariant. It follows from \eqref{e11.46} that \begin{align*} \BP_\xi(A)=\BP_\xi(\{\tau(0)\ne\infty\}\cap A). \end{align*} Therefore we obtain from \eqref{e3.1} that \begin{align*} \lambda_\xi\BP_\xi(A)=\lambda_{\eta^*}\,\BE_{\eta^*}\I_A\xi(C(0)). \end{align*} By definition \eqref{Palmmeasure} of the Palm probability measure of $\eta^*$ we hence obtain for each Borel set $B\subset\R^d$ with $0<\lambda_d(B)<\infty$ that \begin{align}\label{e3.4}\notag \lambda_\xi\BP_\xi(A) &=\lambda_d(B)^{-1} \BE \int_B\I\{\theta_x\in A\} \xi\circ\theta_x(C(0,\theta_x))\,\eta^*(dx)\\ &=\lambda_d(B)^{-1} \BE\I_A\int_B\xi(C(x))\,\eta^*(dx), \end{align} where we have used the invariance of $A$ and \begin{align*} \xi\circ\theta_x(C(\theta_x,0))&=\int\I\{\tau(\theta_x,y)=0\}\,\xi(\theta_x\omega,dy) =\int\I\{\tau(y+x)=x\}\,\xi(\theta_x\omega,dy)\\ &=\int\I\{\tau(y)=x\}\,\xi(dy)=\xi(C(x)). \end{align*} Using \eqref{e11.45} and denoting the invariant $\sigma$-field by $\mathcal{I}$, this yields \begin{align}\label{e3.5} \lambda_\xi\BP_\xi(A)&\le \lambda_d(B)^{-1}\alpha\,\BE[\I_A\eta(B)]\\ \notag &= \lambda_d(B)^{-1}\alpha\,\BE[\I_A\BE[\eta(B)\mid \mathcal{I}]]\\ \notag &=\alpha\lambda_\eta\,\BP(A)\le \lambda_\xi\BP(A)=\lambda_\xi\BP_\xi(A), \end{align} where we have used ergodicity to get the second equality (almost surely) and the assumption $\alpha\le\lambda_\xi\lambda^{-1}_\eta$ to get the second inequality. Therefore the above inequalities are in fact equalities. Hence \eqref{e3.4} and the right-hand side of \eqref{e3.5} coincide, yielding that \begin{align*} \BE\I_A\int_B (\alpha\eta\{x\}-\xi(C(x))\,\eta^*(dx)=0. \end{align*} Taking $B\uparrow\R^d$ and using montone convergence (justified by \eqref{e11.45}), this yields \begin{align*}
\I_A\int(\alpha\eta\{x\}-\xi(C(x)))\,\eta^*(dx)=0, \quad \BP\text{-a.s.} \end{align*} Hence we have $\BP$-a.s.\ on $A$ that $\xi(C(x))=\alpha\eta\{x\}$ for all $x \in \eta^*$. By definition of $A$ this is possible only if $\BP(A)=0$. This implies the first assertion.
To prove the second assertion we use \eqref{e3.1} with $g\equiv 1$ to obtain that $\lambda_\xi\BP_\xi(\tau(0)\ne\infty)=\lambda_{\eta^*}\alpha\, \BE_{\eta^*}\eta\{0\}$. Since it follows straight from the definitions that $\lambda_{\eta^*} \BE_{\eta^*}\eta\{0\}=\lambda_\eta$ we can conclude the assertion. \qed
\section{Destination Isolated Atoms}\label{secatomsdest}
The spatial version of the Gale–Shapley allocation introduced in \cite{HoHolPe06} balances Lebesque measure to a simple point process. The simplified description of it in the introductory Poisson-Lebesgue example
is not an effective way of proving that it actually works,
the efficient way is algorithmic. We now extend this allocation to balance a diffuse $\xi$ to an $\eta$ consisting of isolated atoms. The extension is needed because unlike the Lebesgue measure a diffuse measure can have positive mass on lower dimensional sets like the boundaries of balls. Motivated by the point-optimal stable allocation introduced in \cite{HoHolPe06}, we formulate an algorithm providing an allocation of appetite $\alpha$.
The idea behind the algorithm (in the case $\alpha =1$) can be sketched as follows. In the first round of the algorithm assign a {\em preference set} to each $\eta$-atom, that is, a set of sites in $\R^d$ that the atom {\em proposes} to. We do this by a finite recursion (note that the following four items can be reduced to one item when $\xi$ is Lebesgue measure):
\begin{itemize}
\item From each $\eta$-atom blow up a "first" closed ball until you have gathered
$\xi$-mass
at least equal to the mass of that atom, $m_1$ say.
\item Put $m_2 = m_1 $ minus the $\xi$-mass in the {\bf interior} of the "first" ball. This remaining mass $m_2$ is thus part of a $\xi$-mass sitting on the boundary of a $d$ dimensional ball.
\item
Then blow up a "second" closed ($d$ dimensional) ball from (e.g.)\ the lexicographically lowest
location on the boundary of the "first" ball
(think of it as a pole) until you have gathered $\xi$-mass on the
{\bf boundary}
of the "first" ball that is at least $m_2$. This mass $m_2$ is sitting on a closed cap of the
boundary of the "first" ball. Put $m_3 = m_2$ minus the $\xi$-mass of the
interior of that cap (the relative interior of the cap w.r.t. the boundary,
the sphere).
This remaining mass, $m_3$, is a part of a mass sitting on
the boundary of a $d-1$ dimensional ball.
\item
Repeat this down the dimensions until the $\eta$-atom has a {\em preference set}
(the union of these interior sets)
of $\xi$-mass exactly $m_1$ (because the final cap will be
a circle segment and its boundary will have at most two points and their $\xi$-mass is zero
since $\xi$ is diffuse).
Note that the preference sets of different $\eta$-atoms may overlap.
Note also that a preference set of an $\eta$-atom can contain other $\eta$-atoms. \end{itemize} Now let each site that lies in at least one preference set of an $\eta$-atom put the closest of those atoms on a shortlist, using lexicographic order to break ties. The atoms associated with the other preference sets are rejected. Each atom has now a rejection set, a subset of its preference set containing sites that rejected the proposal.
Repeat the above procedure recursively by blowing up a ball around each $\eta$-atom restricted to the complement of its associated rejection set (thus extending its preference set) and, after each round, add the new rejections to its rejection set. One of two things can happen for a site $z\in\R^d$. Either it never appears in one of the preference sets. Then $z$ has no partner and is allocated to $\infty$. Or it eventually shortlists a single point $x$. Then $z$ is allocated to $x$.
In the algorithm we will use the notation $D(t-) :=\bigcup_{s<t}D(s)$ for an increasing family of sets $D(s)\subset \R^d$, $s>0$. If $\mu$ is a measure on $\R^d$ and $\mu\{x\}>0$ we write $x\in\mu$.
\begin{algorithm}\label{a11.5}\rm Let $\alpha>0$, $\mu\ne 0$ be discrete with isolated atoms and $\nu$ be diffuse with infinite mass. For $n\in\N_0$, $x\in\mu$ and $z\in\R^d$, define the sets \begin{align*} C_n(x) &\subset \R^d\qquad \text{(the set of sites {\em claimed}, or {\em preferred}, by $x$ at stage $n$),}\\ R_n(x) &\subset \R^d \qquad \text{(the set of sites {\em rejecting} $x$ during the first $n$ stages),}\\ A_n(z) &\subset \mu \qquad\;\; \text{(the set of points of $\mu$ claiming site $z$ in the first $n$ stages),} \end{align*} via the following recursion in $n$. Start with setting $R_0(x):=\emptyset$.
\begin{enumerate} \item This first step is a recursion within the recursion. Fix $x\in\mu$ and mostly suppress it in the notation until step (1) is over. Define \begin{align*} S_{n,1}(s) &:= B(x,s) =
\text{ the ball with center $x$ and radius $s>0$,} \\ \beta_{n,1} &:= \alpha\mu\{x\}, \\ s_{n,1} &:= \inf\{s\geq 0 : \nu\big(S_{n,1}(s) \setminus R_{n-1}\big) \geq \beta_{n,1}\}, \\ \Delta_{n,1} &:= S_{n,1}(s_{n,1})\setminus S_{n,1}(s_{n,1}-) \qquad \text{note that } \Delta_{n,1} := \partial B(x, s_{n,1}). \end{align*} For $k=1,\dots,d-1$, proceed recursively as follows. Let $y_{n,k} \in \R^d$ be the lexicographically lowest element of $ \Delta_{n,k} $ and set \begin{align*} S_{n{,k+1}}(s) &:= B(y_{n,k},s) \cap \Delta_{n,k} \\ \beta_{n,k+1} &:= \beta_{n,k} - \nu\big(S_{n,k}(s_{n,k}-)\setminus R_{n-1}\big)\\ s_{n,k+1} &:= \inf\{s\geq 0 : \nu\big(S_{n,k+1}(s) \setminus R_{n-1}\big) \geq \beta_{n,k+1}\}\\ \Delta_{n,k+1} &:= S_{n,k+1}(s_{n,k+1})\setminus S_{n,k+1}(s_{n,k+1}-). \end{align*} Then $\Delta_{n,d}$ contains at most two elements. Since $\nu$ is diffuse this implies that $$ \nu\big(S_{n,d}(s_{n,d}) \setminus R_{n-1}\big)= \nu\big(S_{n,d}(s_{n,d}-) \setminus R_{n-1}\big)=\beta_{n,d}. $$ Now set $$ C_{n} := S_{n,1}(s_{n,1}-)\cup \dots \cup S_{n,d}(s_{n,d}-). $$ Since $C_n$ is a disjoint union we have $$ \nu(C_{n}\setminus R_{n-1}) = \nu(S_{n,1}(s_{n,1}-)\setminus R_{n-1})+ \dots +\nu(S_{n,d}(s_{n,d}-)\setminus R_{n-1}). $$ Since
$$ \beta_{n,k+1} := \beta_{n,k} - \nu\big(S_{n,k}(s_{n,k}-)\setminus R_{n-1}\big)
\qquad \text{and} \qquad \nu\big(S_{n,d}(s_{n,d}-) \setminus R_{n-1})\big)=\beta_{n,d} $$ this yields $$ \nu(C_{n}\setminus R_{n-1}) = (\beta_{n,1}-\beta_{n,2})+ \dots + (\beta_{n,d-1}-\beta_{n,d}) +\beta_{n,d} = \beta_{n,1}. $$ Thus, due to $ \beta_{n,1} = \alpha\mu\{x\}$, we obtain $$ \nu(C_{n}\setminus R_{n-1}) =\alpha\mu\{x\}. $$ \item Recall that $x$ was suppressed in the above step. We
now make it explicit and write $C_n(x)$ instead of only $C_n$. For $z\in\R^d$, define \begin{align*} A_{n}(z):=\{x\in\mu: z\in C_{n}(x)\}. \end{align*} If $A_{n}(z)\ne\emptyset$ then define $$
\tau_{n}(z):=l(\{x\in A_n(z):\|z-x\|=d(z,A_{n}(z))\}) $$ as the point {\em shortlisted} by site $z$ at stage $n$, where $l(B)$ denotes the lexicographic minimum of a finite non-empty set $B\subset\R^d$ and where $d(z,A_{n}(z))$ is the distance of $z$ from the set $A_{n}(z)$. If $A_{n}(z)=\emptyset$ then define $\tau_{n}(z):=\infty$. \item For $x\in\mu$, define $$ R_{n}(x):=\{z\in C_{n}(x):\tau_{n}(z)\ne x\}. $$
\end{enumerate}
\end{algorithm}
Now define a mapping $\tau^{\alpha}(\nu,\mu,\cdot)\colon\R^d\to\R^d\cup\{\infty\}$ as follows. If $\tau_n(z)=\infty$ for all $n\in\N_0$ put $\tau^{\alpha}(\nu,\mu,z):=\infty$. Otherwise, set $\tau^{\alpha}(\nu,\mu,z):=\lim_{n\to\infty}\tau_n(z)$. We argue as follows that this limit exists. Defining $C_0(x):=\{x\}$ for all $x\in\mu$, we assert that for all $n \in \N$ the following holds: \begin{align}\label{1} C_n(x)&\supset C_{n-1}(x),\quad x\in\mu,\\ \label{2} A_n(z) &\supset A_{n-1}(z),\quad z \in \R^d,\\ \label{3} R_n(x) &\supset R_{n-1}(x),\quad x\in\mu. \end{align} This is proved by induction; clearly \eqref{1} implies \eqref{2} and \eqref{2} implies \eqref{3}, while \eqref{3} implies that
\eqref{1} holds for the next value of $n$. By \eqref{2}, $\|\tau_n(z)-z\|$ is decreasing in $n$, and hence, since $\mu$ is locally finite, there exist $x\in\mu$ and $n_0\in\mathbb{N}$ such that $\tau_n(z)=x$ for all $n\ge n_0$. In this case we define $\tau^{\alpha}(\nu,\mu,z):=x$. If $\nu(\R^d)<\infty$ or $\mu(\R^d)=0$ we set $\tau^{\alpha}(\nu,\mu,z):=\infty$. We shall now prove that $\tau^{\alpha}$ (applied with $\xi$ and $\eta$ instead of $\nu$ and $\mu$) has the following property defined in Section 3.
\begin{lemma}\label{l11.6} Assume that $\xi$ and $\eta$ are jointly stationary and ergodic random measures on $\R^d$ such that $\xi$ is diffuse, $\eta$ is discrete with locally finite support and $\lambda_\xi\lambda_\eta>0$. Let $\alpha>0$. Then $\tau$ defined on $\Omega\times\R^d$ by $\tau(\omega,x):=\tau^\alpha(\xi(\omega),\eta(\omega),x)$ is an allocation with appetite $\alpha$. \end{lemma} \begin{proof} It follows by induction over the stages of Algorithm \ref{a11.5} that the mappings $\tau_n$ are measurable as functions of $\nu$, $\mu$ and $z$, where measurability in $\nu$ and $\mu$ refers to the standard $\sigma$-field on the space of locally finite measures; see e.g.\ \cite{LastPenrose17}. (The proof of this fact is left to the reader.) Hence $\tau^{\alpha}$ is measurable. Moreover it is clear that $\tau^{\alpha}$ and hence also $\tau$ has the required covariance property. Next we note that $\BP(\xi(\R^d)=\eta(\R^d)=\infty)=1$, a consequence of ergodicity and $\lambda_\xi\lambda_\eta>0$.
In the remainder of the proof we fix two locally finite measures $\nu$ and $\mu\ne 0$. We assume that $\nu$ is diffuse and satisfies $\nu(\R^d)=\infty$, while $\mu$ is assumed to be discrete with purely isolated atoms. Upon defining $\tau^{\alpha}(\nu,\mu,\cdot)$ we noted that for each $z\in\R^d$, either $\tau^{\alpha}(\nu,\mu,z)=\infty$ or $\tau_n(z)=x$ for some $x\in\mu$ and all sufficiently large $n\in\N$. Therefore \begin{align}\label{e11.876} \I\{\tau^{\alpha}(\nu,\mu,z)=x\} =\lim_{n\to\infty}\I\{z\in C_n(x)\setminus R_{n-1}(x)\},\quad z\in\R^d. \end{align} On the other hand, by Algorithm \ref{a11.5}(1) we have $\nu(C_n(x)\setminus R_{n-1}(x))\le\alpha\mu\{x\}$, so that \eqref{e11.45} follows from Fatou's lemma.
As in Section 3 we set \begin{align*} C^{\tau^\alpha}(x):=\{z\in\R^d:\tau^\alpha(\nu,\mu,z)=x\}. \end{align*} We now show that $\{z\in\R^d:\tau^\alpha(\nu,\mu,z)=\infty\}\ne\emptyset$ and
$\{x\in\mu:\nu(C^{\tau^\alpha}(x))<\alpha\mu\{x\}\}\ne\emptyset$ cannot hold simultaneously, implying the event at \eqref{e11.46} to have probability zero. For that purpose we assume the strict inequality $\nu(C^{\tau^\alpha}(x))<\alpha\mu\{x\}$ for some $x\in\mu$. By \eqref{e11.876} this implies that there exist $n_0\in\N$ and $\alpha_1<\alpha$ such that $\nu(C_n(x)\setminus R_{n-1}(x))\le \alpha_1\mu\{x\}$ for $n\ge n_0$. Let $C_\infty(x):=\cup^\infty_{n=1}C_n(x)$. We assert that $C_\infty(x)=\R^d$. Assume on the contrary this is not the case. By construction, there exist $r_n(x)>0$, $n\in\N$, such that $B^0(x,r_n(x))\subset C_n(x)\subset B(x,r_n(x))$, where $B^0(x,r_n(x))$ is the interior of $B(x,r_n(x))$. Since $C_\infty(x)\ne\R^d$ and the sets $C_n$ are increasing, we have $r_\infty(x):=\lim_{n\to\infty}r_n(x)<\infty$. Then $C_\infty(x)\subset B(x,r_\infty(x))$ is bounded and there exists $n\ge n_0$ such that $\nu(C_\infty(x)\setminus C_n(x))\le \mu\{x\}(\alpha-\alpha_1)/2$. Hence we obtain (since $R_{n-1}(x)\subset C_n(x)$) $$ \nu(C_\infty(x)\setminus R_{n-1}(x)) = \nu(C_\infty(x)\setminus C_n(x))+\nu(C_n(x)\setminus R_{n-1}(x))\le \alpha_2\mu\{x\}, $$ where $\alpha_2:=(\alpha+\alpha_1)/2<\alpha$.
By definition of the algorithm this implies that $C_\infty(x)$ is a strict subset of $C_n(x)$. This contradiction shows that $C_\infty(x)=\R^d$. Now taking $z\in\R^d$, we hence have $z\in C_n(x)$ for some $n\ge 1$, so that $z$ shortlists either $x$ or some closer point of $\mu$. In either case, $\tau^\alpha(\nu,\mu,z)\ne\infty$. \end{proof}
\begin{proposition}\label{plfdisdest} Assume that $\xi$ and $\eta$ are jointly stationary and ergodic random measures on $\R^d$ such that $\xi$ is diffuse, $\eta$ is discrete with locally finite support (isolated atoms) and $0<\lambda_\xi=\lambda_\eta<\infty$. Then $\tau$ defined on $\Omega\times\R^d$ by $\tau(\omega,x):=\tau^1(\xi(\omega),\eta(\omega),x)$ is an allocation balancing $\xi$ and $\eta$. \end{proposition} {\em Proof:} By Lemma \ref{l11.6} $\tau$ is an allocation of appetite 1. Since $\lambda_\xi=\lambda_\eta$ we can apply Proposition \ref{p1} to see that $\eta(\{x\in\R^d:\xi(C^{\tau}(x))\ne \eta\{x\}\})=0$ holds almost surely and moreover that $\BP_\xi(\tau(0)\ne\infty)=1$. These two facts imply the desired balancing property of $\tau$.\qed
The allocation in Proposition \ref{plfdisdest} is stable. In Section \ref{secdiscrete} we shall consider a general discrete $\eta$. But the balancing allocation will not be stable anymore.
\begin{remark}\label{4} \rm Proposition \ref{plfdisdest} can already be found as Proposition 4.37 in \cite{OmidAli16}. There the authors used a site-optimal version of a stable allocation while ours is point-optimal. \end{remark}
\section{Destination a discrete random measure}\label{secdiscrete}
In this section we deal with a random measure $\eta$ which is discrete but not necessarily with a locally finite support (isolated atoms). We need to introduce some notation. If $A\subset\Omega\times \R^d$ is measurable then we identify $A$ with the mapping $\omega\mapsto A(\omega):=\{x\in\R^d:(\omega,x)\in A\}$. If $\xi$ is a random measure on $\R^d$, then we define for each $\omega\in\Omega $ the restriction of $\xi(\omega)$ to $A(\omega)$ by $\xi_A(\omega):=\int\I\{x\in \cdot, (\omega,x)\in A\}\xi(\omega,dx)$. Clearly $\xi_A$ is again a random measure.
\begin{proposition}\label{pdiscretedest} Assume that $\xi$ and $\eta$ are jointly stationary and ergodic random measures on $\R^d$ such that $\xi$ is diffuse, $\eta$ is discrete and $\infty > \lambda_\xi\ge \lambda_\eta>0$. Then there exists a measurable $A\subset\Omega\times\R^d$ and an allocation $\tau$ balancing $\xi_A$ and $\eta$. If $\lambda_\xi= \lambda_\eta$, then $\tau$ balances $\xi$ and $\eta$. \end{proposition} {\em Proof:} Write $\eta = \sum_{n=1}^\infty \eta_n$, where $\eta_1, \eta_2, \dots$ are the mutually singular discrete random measures \begin{align*} \eta_n = \sum_x \I\{1/n \leq \eta\{x\}<1/(n-1)\}\eta(\{x\})\delta_x \end{align*} which all have non-accumulating (isolated) points. (In the definition of $\eta_1$ we use the convention $1/0:=\infty$.)
Throughout this proof we consider the stable allocation $\tau^1$ with appetite 1; see the definition preceding Lemma \ref{l11.6}. Starting with $\xi^1:=\xi$ we define sequences of measurable sets $A_n\subset\Omega\times\R^d$, $n\in\N$, and of stationary random measures $(\xi_n)_{n\ge 1}$ and $(\xi^n)_{n\ge 1}$ recursively, by setting for each $n\in\N$, \begin{align*} A_{n}&:=\{x\in\R^d:\tau^1(\xi^{n},\eta_{n},x)\ne\infty\},\\ \xi_{n}&:=\xi^{n}_{A_n},\\ \xi^{n+1}&:=\xi^{n}_{\R^d\setminus A_{n}}. \end{align*} Set $B_n:=A_1\cup\cdots \cup A_n$. Using Proposition \ref{p1} one can prove by induction that $\xi(A_{n+1}\cap B_n)=0$, and $\lambda_{\xi_{n}}= \lambda_{\eta_{n}}$. Note that $\sum^\infty_{n=1}\xi_n=\xi_A$, where $A:=\cup^\infty_{n=1}A_n$.
The calculation below shows that it is no restriction of generality to assume that the sets $A_n$ are disjoint. Therefore we can define an allocation $\tau$ by \begin{align} \tau(x):= \begin{cases} \tau^1(\xi^n,\eta_n,x),&\text{if $x\in A_n$ for some $n\in\N$},\\ \infty, &\text{otherwise}. \end{cases} \end{align} Then we obtain for each Borel set $C\subset\R^d$ that \begin{align*} \int \I\{\tau(x)\in C\}\xi_A(dx) &=\sum^\infty_{n=1}\int \I\{\tau(x)\in C\}\xi_n(dx)\\ &=\sum^\infty_{n=1}\int \I\{\tau^1(\xi^n,\eta_n,x)\in C\}\xi_n(dx)\\ &=\sum^\infty_{n=1}\eta_n(C)=\eta(C). \end{align*} Therefore $\tau$ is balancing $\xi_A$ and $\eta$.
Assume now that $\lambda_\xi=\lambda_\eta$. Then we obtain for each Borel set $C\subset\R^d$ that \begin{align*} \BE \xi_A(C)=\sum^\infty_{n=1}\BE\xi_n(C) =\sum^\infty_{n=1}\lambda_{\xi_n}\lambda_d(C) =\sum^\infty_{n=1}\lambda_{\eta_n}\lambda_d(C) =\lambda_\eta\lambda_d(C)=\lambda_\xi\lambda_d(C). \end{align*} Therefore, $\BE \xi_A(C)=\BE \xi(C)$. Since $\xi_A\le\xi$, this implies that $\xi_A=\xi$ $\BP$-a.s. \qed
\section{Destination a Diffuse Random Measure}~\label{secdifdest}
In this section we deal with a diffuse destination $\eta$ in the case when there exists an auxiliary simple point process $\chi$. The key idea is to use the allocation from the isolated-atoms case (Proposition~\ref{plfdisdest}) to map both $\xi$ and $\eta$ to $\chi$ creating pairs of
$\xi$-cells and $\eta$-cells of mass one
associated with each point of $\chi$, and then to transport the $\xi$-mass of the $\xi$-cells into the $\eta$-mass of the $\eta$-cells by passing through Lebesgue measure on $[0,1]$.
\begin{proposition}\label{pdiffusedest} Assume that $\xi$ and $\eta$ are jointly stationary and ergodic random measures on $\R^d$ such that $0<\lambda_\xi=\lambda_\eta <\infty$. Assume further that $\xi$ and $\eta$ are both diffuse and that there exists an auxiliary simple point process $\chi$ with finite intensity $\lambda_\chi$. Then there exists an allocation $\tau$ balancing $\xi$ and $\eta$. \end{proposition}
{\em Proof:} Note that an allocation balances the diffuse $\xi$ and $\eta$ if and only if it balances
$a\xi$ and $a\eta$ for any positive constant $a$,
in particular for $a=\lambda_\chi/\lambda_\xi=\lambda_\chi/\lambda_\eta$.
So it is no restriction to
assume that the common intensity of $\xi$ and $\eta$
is the same as that of $\chi$, that is, $\lambda_\xi=\lambda_\eta=\lambda_\chi$.
We can then apply Proposition \ref{plfdisdest} to the pair $\xi$ and $\chi$ and to the pair $\eta$ and $\chi$.
For each point $s$ of $\chi$ let $A_s$ and $B_s$, respectively, be the resulting allocation cells of $\xi$ and $\eta$ that are mapped to $s$. Fix $t\in \R^d$ and let $S_t$ be the point of $\chi$ such that $t \in A_{S_t}$. Let $\xi_{t}= \xi(\cdot \cap A_{S_t})$ be the restriction of $\xi$ to $A_{S_t}$ and let $\eta_{t}=\eta(\cdot \cap B_{S_t})$ be the restriction of $\eta$ to $B_{S_t}$. Since $\chi$ is a simple point process (with mass one at each of its points) both $\xi_t$ and $\eta_t$ are (random) probability measures.
Let $\phi$ be a measurable bijection from $\R^d$ to $\R$ such that $\phi^{-1}$ is also measurable. Shift the origin $0$ to
$S_t$ to obtain (random) probability measures $\theta_{S_t}\xi_t$ and $\theta_{S_t}\eta_t$ that are concentrated on the shifted cells $A_{S_t}\!-\!S_t$ and $B_{S_t}\!-\!S_t$. Let $F_t$ and $G_t$ be the (random) distribution functions of $\phi$ under these probability measures, that is, for $x \in \R$ \begin{align*} F_t(x) &= \theta_{S_t}\xi_t(\phi \leq x),\\ G_t(x) &= \theta_{S_t}\eta_t(\phi \leq x). \end{align*} Note that $F_t$ is a continuous function since $\xi$ does not have any atoms and since $\phi$ is a bijection. Thus
\vspace*{2mm}
\qquad\qquad the distribution of $F_t(\phi)$ under $\theta_{S_t}\xi_t$ is uniform on $[0, 1]$.
\vspace*{2mm}
\noindent With $G_t^{-1}$ the generalized inverse (quantile function) of $G_t$, this in turn implies that $G_t^{-1}(F_t(\phi))$ under $\theta_{S_t}\xi_t$ has the distribution function $G_t$. Finally, since $\phi$ is a bijection this implies that
\vspace*{2mm}
\qquad\qquad the distribution of $\phi^{-1}(G_t^{-1}(F_t(\phi)))$ under $\theta_{S_t}\xi_t$ is $\theta_{S_t}\eta_t$.
\vspace*{2mm}
\noindent In other words, the mapping $x\mapsto \phi^{-1}(G_t^{-1}(F_t(\phi(x))))$ transports the measure $\theta_{S_t}\xi_t$ on $A_{S_t}\!-\!S_t$ into the measure $\theta_{S_t}\eta_t$ on $B_{S_t}\!-\!S_t$. Shifting back to the original origin, this means that the mapping $x\mapsto S_t + \phi^{-1}(G_t^{-1}(F_t(\phi(x-S_t)))$ transports the measure $\xi_t$ on $A_{S_t}$ into the measure $\eta_t$ on $B_{S_t}$. Thus,
the allocation rule $\tau$ defined by $$ \tau(t) = S_t + \phi^{-1} (G_t^{-1}(F_t(\phi(t-S_t)))),\quad t\in\R^d, $$ balances $\xi$ and $\eta$. \qed
\begin{remark}\label{6}\rm
The above construction, using a point process to split space into
pairs of cells and then allocate mass through Lebesgue measure,
dates back to an informal note from 2012. It was a part of a brief
attempt of the authors to extend the Brownian motion results of
\cite{LaMoeTho12} to higher dimensional random fields. The obvious
question is when this `auxiliary' process does exist. We discussed
this with several colleagues and the existence problem did become
part of the
PhD topic of Ali Khezeli, see Remark~\ref{8}. In his thesis he
used a result on optimal transport to balance the finite masses of
the pairs of allocation cells, under certain restrictions on the
diffuse source.
\end{remark}
\section{Destination having Discrete and Diffuse Parts}~\label{secmixeddest}
In this section we finish the proof of Theorem~\ref{main} by dealing with the case when the destination $\eta$ contains both discrete and diffuse parts.
\begin{proposition}\label{pmixeddest} Assume that $\xi$ and $\eta$ are jointly stationary and ergodic random measures on $\R^d$ such that $\lambda_\xi=\lambda_\eta<\infty$. Assume further that $\xi$ is diffuse and that $\eta$ is mixed, that is, $$ \eta = \eta^\emph{disc} + \eta^\emph{diff} $$ where $\eta^\emph{disc}$ and $\eta^\emph{diff}$ are nonzero random measures that are discrete and diffuse respectively. Then there exists an allocation $\tau$ balancing $\xi$ and $\eta$. \end{proposition}
{\em Proof:} Note that the measures $\xi$, $\eta$, $\eta^\text{disc}$, $\eta^\text{diff}$ are jointly stationary and ergodic since $\eta^\text{disc}$ and $\eta^\text{diff}$ can be obtained from $\eta$ in an translation invariant way. Now apply Proposition~\ref{pdiscretedest} to $\xi$ and $\eta^\text{disc}$ using
$\lambda_{\xi} \geq \lambda_{\eta^\text{disc}}$
to obtain
a measurable $A\subset\Omega\times\R^d$ and an allocation $\tau^\text{disc}$ balancing $\xi_A$ and $\eta^\text{disc}$. Then, with $\chi$ the auxiliary simple point process defined at \eqref{i3}, apply Proposition~\ref{pdiffusedest}
to $\xi_{A^c}$ and $\eta^\text{diff}$ using
$\lambda_{\xi_{A^c}} = \lambda_{\eta^\text{diff}}$
to obtain an allocation $\tau^\text{diff}$ balancing $\xi_{A^c}$ and $\eta^\text{diff}$. Finally, define an allocation $\tau$ by $$ \tau(\omega, x) =
\begin{cases} \tau^\text{disc}(\omega, x),&\text{if $(\omega, x) \in A$},\\ \tau^\text{diff}(\omega, x),&\text{if $(\omega, x) \in A^c$}. \end{cases} $$ It is easy to see that $\tau$ balances $\xi$ and $\eta$. \qed
\section{Counterexample}\label{seccounter}
In this section we show that the auxiliary $\chi$ cannot simply be removed from Theorem~\ref{main}. There are diffuse $\xi$ and $\eta$ such that an allocation transporting $\xi$ to $\eta$ does not exist (without external randomisation).
Here is a specific example in two dimensions, $\R^2$. Let $N$ be a canonical stationary Poisson process on the $y$-axis with intensity $1$; canonical means that $N$ is the only source of randomness. Let $\xi$ be formed by the one-dimensional Lebesgue measure on the lines parallel to the $x$-axis going through the points of $N$. Let $\eta=\lambda_2 =$ the Lebesgue measure on $\R^2$. The measures $\xi$ and $\eta$ are diffuse, jointly stationary and ergodic, and have the same intensity~$1$.
Suppose there exists a balancing allocation $\tau_N$ which maps $\R^2$ to $\R^2$ for each fixed value of $N$ in such a way that the image measure of $\xi$ under $\tau_N$ is $\eta$. The Palm version of $N$ is $N^{\circ}=N+ \delta_0$ and the Palm version of $\xi$ is $\xi^{\circ}=\xi+\lambda_1$, where $\lambda_1$ denotes one-dimensional Lebesgue measure on the $x$-axis. According to \eqref{shift-coupling2224}, the existence of the allocation $\tau_N$ balancing $\xi$ and the two-dimensional Lebesgue measure $\eta=\lambda_2$ would yield the following shift-coupling of $\xi^{\circ}$ and $\xi$, $$ \theta_{\tau_N(0)}\xi^{\circ}
\,\overset{D}{=}\, \xi . $$ With $T_N$ the $y$-axis coordinate of $\tau_N(0)$,
this implies that $$ \theta_{T_N}N^{\circ}
\,\overset{D}{=}\, N. $$ But, according to \cite{HP05}, such a $T_N$ does not exist when the only sorce of randomness is $N$.
\begin{remark}\label{8}\rm When reading a preliminary version of this paper, Ali Khezeli pointed out to the authors the following interesting problem, formulated in his PhD-thesis from 2016 (in Persian). Suppose that $\xi$ is a diffuse random measure with no invariant directions. This means that there is (almost surely) no vector $x\ne 0$ such $\xi=\theta_{tx}\xi$ for all $t\in\R$. Does $\xi$ have a stationary point process factor? A positive answer would bring us much closer to a complete characterisation of the existence of balancing factor allocations (for a diffuse source). Note that our counterexample has an invariant direction. \end{remark}
\section{Remarks}\label{secrem}
\begin{remark}\label{ergod}\rm The assumption of ergodicity has been made for simplicity and can be relaxed. The assumption $\lambda_\xi= \lambda_\eta$ has then to be replaced by $$ \BE[\xi[0,1]\mid \mathcal{I}]=\BE[\eta[0,1]\mid \mathcal{I}],\quad \BP\text{-a.e.} $$ We refer to \cite{LaTho09, Thor96} for more detail on this point. \end{remark}
\begin{remark}\label{ali-omid}\rm A natural question (asked in \cite{OmidAli16} for instance) is whether there exists a balancing allocation factor
$\tau$ if the source $\xi$ is Lebesgue measure or, more generally, absolutely continuous with respect to Lebesgue measure. We have proved the answer to be positive whenever the destination $\eta$ is {\em not} diffuse (not purely non-atomic), and also when $\eta$ is diffuse if we assume that there exists an auxiliary point process factor~$\chi$.
But this assumption is not necessary, a balancing allocation factor $\tau$ {\em can} exist {\em even} when $\eta$ is diffuse and {\em no} auxiliary point process factor $\chi$ exists. An example of this is obtained by swapping
source and destination in Section~\ref{seccounter}: take $\xi=\lambda_2 =$ the Lebesgue measure on $\R^2$ and let
$\eta$ be formed by the one-dimensional Lebesgue measure on the lines parallel to the $x$-axis going through the points of $N$ where $N$ is a one-dimensional Poisson process on the $y$-axis. If $N$ is the only source of randomness then, according to Section~\ref{seccounter}, there exists {\em no}
auxiliary $\chi$. However,
there exists a balancing allocation factor $\tau$: for example,
take
$\tau(x, y) = (x, \tau_1(y))$ where $\tau_1$ is an
allocation balancing Lebesgue measure on the line (the $y$-axis)
and the Poisson process~$N$.
Thus, the existence of an auxiliary point process
is not a complete characterisation of the existence of
a balancing allocation factor when the source is diffuse. \end{remark}
\begin{remark}\label{auxfactor}\rm If the source $\xi$ is not diffuse, then the question asked in this paper is in most cases not very meaningful. If, for instance, the destination $\eta$ is diffuse, then a balancing allocation cannot exist. But even otherwise such allocations can only exist in special cases, for instance, if both $\xi$ and $\eta$ are simple point processes. \end{remark}
\begin{remark}\label{alternativ}\rm The following slight modification of the construction in Section~\ref{secdifdest}
(where the destination is diffuse) can be used to obtain allocations in the
cases treated in Section~\ref{secdiscrete} and Section~\ref{secmixeddest}.\
Let $\chi$ be the simple point process defined at \eqref{i3}.\
In the proof of Theorem~\ref{pdiffusedest} remove the first paragraph,
replace the allocation cells
$B_s$ of $\eta$ by the Voronoi cells $V_s$ of $\chi$,
and then modify $\chi$ by letting it have mass $\eta(V_s)$ at each point $s$. After this modification we have $\lambda_\xi=\lambda_\eta=\lambda_\chi$ and can apply Proposition \ref{plfdisdest} to the pair $\xi$ and $\chi$. Let $A_s$ be the allocation cell of $\xi$ that is mapped to $s$.
This yields $F_t$ and $G_t$ that are
cumulative mass
functions of measures
that need not have mass $1$ but only have the same finite mass,
$F_t(\infty) = \xi(A_{S_t})= \eta(V_{S_t})=G_t(\infty)$. Replace first the word {\em distribution function} by {\em cumulative mass
function} and then the word {\em distribution} by {\em image messure}. The rest of the proof now goes through as it stands.
This method thus yields allocations in all the three cases where $\eta$ does not consist of isolated atoms. We have chosen here to treat each of those cases separately because treating discrete measures as in Section~\ref{secdiscrete} is more explicit then this alternative method. \end{remark}
\begin{remark}\label{history}\rm Here are some further historical notes. Allocations finding extra heads in coin tosses on the $d$ dimensional grid and extra points in the $d$ dimensional Poisson process were constructed in \cite{HL01, HP05, HoHolPe06, CPPeresR10} by Liggett, Holroyd, Hoffman, Peres et.al. More generally, the allocations transporting Lebesgue measure to the points of stationary ergodic finite-intensity point processes produce the Palm versions of the processes. The construction in \cite{HoHolPe06}
involved a Gale-Shapley algorithm resulting in a `stable' allocation while the construction in \cite{CPPeresR10} used a gravitational force field to obtain an `economical' allocation. In \cite{Hues16}, unique optimal allocations between jointly stationary random measures on geodesic manifolds were constructed, assuming the (Palm) average cost to be finite and the source to be absolutely continuous. Stable transports between general (jointly stationary) random measures $\xi$ and $\eta$ on $\R^d$ were constructed and studied in \cite{OmidAli16}. If $\xi$ is diffuse and $\eta$ is a point process, then these transports boil down to allocations, see Remarks \ref{4}, \ref{6} and \ref{8}. \end{remark}
\noindent {\bf Acknowledgments:}
We wish to thank Ali Khezeli for helpful discussions
and an anonymous referee for a clear-sighted and constructive
report resulting in a thorough rewriting of parts of the paper.
This work was supported by the German Research Foundation (DFG) through Grant No. LA 965/11-1 as part of the DFG priority programme ``Random Geometric Systems''.
\end{document} | arXiv |
Putting the JATO rocket car to rest
Posted by David Zaslavsky on May 7, 2013 5:09 PM
It's that time again: Mythbusters is back! And they sure know how to kick things off with a bang — or better yet, a prolonged burn!
For the 10th anniversary of the show, the Mythbusters revisited the very first myth they ever tested, the JATO rocket car. Wikipedia has the story in what appears to be its most common form:
The Arizona Highway Patrol came upon a pile of smoldering metal embedded into the side of a cliff rising above the road at the apex of a curve. the wreckage resembled the site of an airplane crash, but it was a car. The type of car was unidentifiable at the scene. The lab finally figured out what it was and what had happened.
It seems that a guy had somehow gotten hold of a JATO unit (Jet Assisted Take Off - actually a solid fuel rocket) that is used to give heavy military transport planes an extra 'push' for taking off from short airfields. He had driven his Chevy Impala out into the desert and found a long, straight stretch of road. Then he attached the JATO unit to his car, jumped in, got up some speed and fired off the JATO!
The facts, as best could be determined, are that the operator of the 1967 Impala hit JATO ignition at a distance of approximately 3.0 miles from the crash site. This was established by the prominent scorched and melted asphalt at that location. The JATO, if operating properly, would have reached maximum thrust within five seconds, causing the Chevy to reach speeds well in excess of 350 MPH, continuing at full power for an additional 20-25 seconds. The driver, soon to be pilot, most likely would have experienced G-forces usually reserved for dog-fighting F-14 jocks under full afterburners, basically causing him to become insignificant for the remainder of the event. However, the automobile remained on the straight highway for about 2.5 miles (15-20 seconds) before the driver applied and completely melted the brakes, blowing the tires and leaving thick rubber marks on the road surface, then becoming airborne for an additional 1.4 miles and impacting the cliff face at a height of 125 feet, leaving a blackened crater 3 feet deep in the rock.
Most of the driver's remains were not recoverable; however, small fragments of bone, teeth and hair were extracted from the crater, and fingernail and bone shards were removed from a piece of debris believed to be a portion of the steering wheel.
cartoon of JATO car going out of control
It's a fascinating story and all, but there's plenty of evidence to suggest that this didn't happen, and in fact that it can't happen as described. Not only does the Arizona Highway Patrol have no record of ever investigating a case like this, but it's been tested no less than three times on Mythbusters.
The first time, on the pilot episode in 2003, Adam and Jamie found that a Chevy Impala with three hobby rockets on top, providing equivalent thrust to a JATO unit, wouldn't get anywhere close to \(\SI{350}{mph}\), though it did exceed the \(\SI{130}{mph}\) top speed of the chase helicopter.
The second time, in 2007, they found that… rockets explode sometimes. Hey, it's Mythbusters, you can expect nothing less. :-P
The third time was the 10th anniversary special that aired last week. With more power than the equivalent of one JATO unit, the car still didn't get much faster than about \(\SI{200}{mph}\). In this iteration, the Mythbusters also tested the part of the myth in which the car supposedly took off and flew through the air — which also failed spectacularly (in every sense of the word), with their test car running off a ramp and nosediving into the desert.
There's a lot of juicy physics in this myth. But it breaks down into a couple of key parts: first, can a rocket-powered Impala even make it up to \(\SI{350}{mph}\)? And secondly, if it did, could it fly a mile and a half through the air? I'm going to handle the speed issue here, and get into the fable of the flying car for a later post.
force diagram for JATO car
Before it becomes airborne, a JATO car is just an object subject to three forces: the thrust of the rocket and the engine force pointing forward, and air resistance pointing backward. Well, there's also tire friction, but that's a lesser influence. In the simplified model I'm going to use, three of these forces are constant — the thrust and engine force forward, and tire friction backward — and air resistance is the one velocity-dependent force. For convenience I'll group all the constant forces under the name drive force, \(F_\text{drive}\).
With the forces as described, a car is pretty similar to a falling object, which is also subject to a constant force in one direction and air resistance in the other. Like, say, an airplane pilot who fell out of his plane at \(\SI{22000}{ft.}\), which I've already worked through the math for in an earlier blog post. Here's how that same math applies to the JATO car: first write Newton's second law \(\sum F = ma\), including the drive force \(F_\text{drive}\) and the air resistance \(F_\text{drag}\),
$$F_\text{drive} - F_\text{drag} = F_\text{drive} - \frac{1}{2}CA\rho v^2 = m\ud{v}{t}$$
The car's terminal speed is the speed at which its acceleration, \(\ud{v}{t}\), is zero. Plugging that in, we get
$$v_T = \sqrt{\frac{2F_\text{drive}}{CA\rho}}$$
Now we can rewrite Newton's second law like so:
$$F_\text{drive}\biggl(1 - \frac{v^2}{v_T^2}\biggr) = m\ud{v}{t}$$
This makes it easy to see that if an object is moving faster than its terminal speed at any point, that is \(v > v_T\), it will tend to slow down (because \(\ud{v}{t} < 0\)), and if it's moving slower than its terminal speed, \(v > v_T\), it will speed up. It'll only stay at a steady speed if \(\ud{v}{t}\approx 0\), and that requires \(v \approx v_T\), i.e. that the object is traveling roughly at its terminal speed.
The story from Wikipedia has the car barreling down the road at more than \(\SI{350}{mph}\) for several seconds, which suggests that for a JATO Impala, \(v_T \gtrsim \SI{350}{mph}\). Is that realistic? Well, we do have some of the information necessary to figure it out. As reported in the Mythbusters pilot, the thrust of a JATO is about \(\SI{1000}{lbf.}\), or \(\SI{4400}{N}\), and the density of air, \(\rho = \SI{1.21}{kg/m^3}\). But we don't know the cross-sectional area \(A\) and drag coefficient \(C\) of the car.
Hmm. That could be a problem.
Fortunately, there's a way around that. Ever heard of a drag race? That's where you set a car (or two) at the beginning of typically a quarter mile track and just floor it to see how fast it makes it to the end. Besides being good for a movie or two… or six (come on, seriously?), the quarter-mile run is a pretty common way to test a car's performance, and the results of some of these drag tests are available online. For the '67 Impala, the site gives
1967 Chevrolet Impala SS427 (CL)
427ci/385hp, 3spd auto, 3.07, 0-60 - 8.4, 1/4 mile - 15.75 @ 86.5mph
This means the car's acceleration is sufficient to take it from rest to \(\SI{60}{mph}\) in \(\SI{8.4}{s}\), and that it completed the quarter mile in \(\SI{15.75}{s}\), traveling \(\SI{86.5}{mph}\) over the final 66 feet.
force diagram for normal car driving
Let's now go back to the rewritten version of Newton's second law for a normal car,
You can solve this by rearranging and integrating it, but I'm lazy: I just plugged it into Mathematica. The solution for speed as a function of time is
$$v(t) = v_T\tanh\biggl(\frac{F_\text{drive}}{m v_T}t + \atanh\frac{v_0}{v_T}\biggr)$$
where \(v_0\) is the initial speed. Now, a '67 Impala has a mass of \(\SI{1969}{kg}\) (it varies from car to car, of course, but that's a representative value), and in a drag test, the initial velocity \(v_0 = 0\). That leaves two variables still unknown: \(F_\text{drive}\), the net forward force which moves the car (engine minus friction), and \(v_T\), its terminal velocity. Luckily, we have two data points we can use to solve for them: the 0-60 benchmark \(v(\SI{8.4}{s}) = \SI{60}{mph}\), and the quarter mile time \(v(\SI{15.75}{s}) = \SI{86.5}{mph}\). Those two velocity-time coordinates should be enough to determine \(F_\text{drive}\) and \(v_T\). Probably the easiest way to do it is to make a contour plot showing the combinations of \(v_T\) and \(F_\text{drive}\) that satisfy each condition, like this:
plot of points satisfying condition
The blue curve shows the points at which
$$v_T\tanh\frac{F_\text{drive}(\SI{8.4}{s})}{(\SI{1969}{kg})v_T} = \SI{60}{mph}$$
and the red curve shows points where
$$v_T\tanh\frac{F_\text{drive}(\SI{15.75}{s})}{(\SI{1969}{kg})v_T} = \SI{86.5}{mph}$$
Their intersection is the one set of parameters that satisfies both conditions, namely \(v_T = \SI{100.8}{mph}\) and \(F_\text{drive} = \SI{7250}{N} = \SI{1628}{lbf.}\). Bingo! Now we've got everything we need to calculate the behavior of the rocket car.
Well, wait a minute. We said — actually, Mythbusters said (and what I can find online seems to confirm) that a JATO provides a thousand pounds of thrust. But the value we found for \(F_\text{drive}\) is even larger. Surely a standard car engine can't be more powerful than a rocket, can it?
I think we have to conclude that it is. Though this isn't quite what you'd call a standard car engine. The results of the drag test we used to calculate \(F_\text{drive}\) were for the '67 Impala SS (Super Sport) 427, a special high-performance version of the car whose engine could crank out \(\SI{385}{hp}\). A standard version of the car would have a less powerful engine, ranging down to \(\SI{155}{hp}\), and thus could have as little as half the engine force — roughly, \(F \propto P^{2/3}\), because \(F \sim v^2\) and \(P = Fv\), and \((150/385)^{2/3} \approx 0.5\).
Note that if you use \(F_\text{drive}\) as obtained from the drag test to calculate the power at the car's top speed of \(\SI{86.5}{mph}\), you get \(\SI{376}{hp}\), which is quite close to the engine's reported horsepower. But I think that's just a coincidence. The same method applied to the car's eventual top speed of \(\SI{100.8}{mph}\) gives \(\SI{437}{hp}\), which is more power than the engine is even able to generate! This reflects the fact that the way we derived \(v(t)\) makes a lot of simplifying assumptions. For example, it assumes that the air resistance is proportional to \(v^2\). In reality, a car is a complicated shape that induces some amount of turbulence, making the drag force difficult to characterize. More importantly, we've assumed the drive force is constant, which is not at all true for a real car. The engine force changes as the car shifts gears and as parts warm up, there are other assorted forces at work like rolling friction.
As a check of sorts on how close this model comes, I'm going to integrate the formula again, to get a formula for distance traveled over time:
$$x(t) = \frac{m v_T^2}{F_\text{drive}}\ln\cosh\frac{F_\text{drive} t}{m v_T}$$
In theory, we should be able to plug in the values we found — \(m = \SI{1969}{kg}\), \(v_T = \SI{100.8}{mph}\), and \(F_\text{drive} = \SI{7250}{N}\) — along with the quarter mile time, \(t = \SI{15.75}{s}\), and the distance \(\SI{0.25}{mile}\) will pop out. When I actually plug the values in, I get \(\SI{0.229}{mile}\), which is within 10%, so not bad. That suggests that the different inaccuracies cancel out to some extent.
OK, so where does that leave us? We have a formula,
$$v(t) = v_T\tanh\frac{F_\text{drive} t}{m v_T}$$
which seems to somewhat accurately describe the motion of a car under its own power in a drag test. We also have best-fit values for the parameters of this formula: \(v_T = \SI{100.8}{mph}\), and \(F_\text{drive} = \SI{7250}{N}\) for the SS427, and \(F_\text{drive} \approx \SI{3600}{N}\) for a standard Impala. Time to ramp it up to the JATO rocket car!
You may remember from earlier in this post that the terminal velocity is proportional to the square root of the net constant force \(F_\text{drive}\) applied to move the car. If a terminal speed of \(\SI{100.8}{mph}\) corresponds to a drive force of \(\SI{7250}{N}\), then adding on a JATO's thrust of an additional \(\SI{1000}{lbf.}\) would naively give a terminal speed of
$$\SI{100.8}{mph}\times \sqrt{\frac{\SI{7250}{N} + \SI{1000}{lbf.}}{\SI{7250}{N}}} = \SI{128}{mph}$$
Doing the same calculation for the \(\SI{1500}{lbf.}\) hobby rockets the Mythbusters used gives
which is considerably less than the estimate of a top speed around \(\SI{190}{mph}\) for the car in the pilot episode. Huh.
OK, so what could be wrong with the model?
Could the drag coefficient — actually the product \(CA\rho\) — of the car (with attached rocket) be much smaller than we thought, making the car's terminal speed higher? It's hard to see how. After all, the terminal speed we calculated was for a stock Impala, a shape which is reasonably aerodynamic, but the Mythbusters went and put a rocket pack on it, breaking the airflow over the roof. If anything, it should go slower with the rockets on top.
Could the car's engine be more powerful than we think? Again, probably not. I'm already using a value which corresponds to one of the most powerful engines available at the time — it'd still be a pretty strong engine in the modern market. To increase the car's terminal speed up to \(\SI{190}{mph}\) by increasing engine force would require an engine almost three times as strong as what I'm using in the model.
What about power losses in the drivetrain? Well, of course those make the car's terminal speed slower, not faster. There are plenty of reasons to imagine that the terminal speed should be less than what we calculated, but not higher.
I'm actually doubtful that whatever the error is has anything to do with the car itself. Here's why: suppose we take that equation for \(v(t)\) from earlier and plot it, starting at \(\SI{80}{mph}\), for several different configurations.
plot of speed of JATO car over time
The lower edge of each band represents a car with a standard \(\SI{155}{hp}\) motor, and the higher edge represents a car with an enhanced performance \(\SI{385}{hp}\) motor. And the shaded areas represent the results we actually saw on Mythbusters (or the circumstances described in the myth, for the gray box). Notice that the orange band, representing the car with 5 rockets simultaneously firing as in the anniversary special, goes right through the pink box representing the speed we saw on that show. So the model works in that case!
On the other hand, the light green band, the one for the hobby rockets used in the pilot, only barely breaks the \(\SI{130}{mph}\) speed of the chase helicopter (the dotted line) and doesn't ever get up to the speed estimate of \(\SI{190}{mph}\). The only way I can come up with to reconcile this math with the results we saw on the show is that the rockets they used might have been more powerful than claimed. If you assume that the rockets give off a little more than twice as much thrust as was said on the show, i.e. around \(\SI{3500}{lbf.}\), you get the dark green curve which seems to come close to the observed results. If anyone knows a better explanation for that difference, I'd be interested to here it, but for now I just have to conclude that something was off about those reported rocket numbers.
Of course, let's not forget the real point of the graph. None of these curves come anywhere close to the circumstances claimed in the myth! So despite the discrepancies in the details, the conclusion remains solidly the same: physics says this myth is absolutely busted. | CommonCrawl |
Modeling and equilibrium studies on the recovery of praseodymium (III), dysprosium (III) and yttrium (III) using acidic cation exchange resin
B. A. Masry1,
E. M. Abu Elgoud1 &
S. E. Rizk1
BMC Chemistry volume 16, Article number: 37 (2022) Cite this article
In this research, the possibility of using hydrogenated Dowex 50WX8 resin for the recovery and separation of Pr(III), Dy(III) and Y(III) from aqueous nitrate solutions were carried out. Dowex 50WX8 adsorbent was characterized before and after sorption of metal ions using Fourier-transform infrared spectroscopy (FT-IR), Scanning Electron Microscope (SEM) and Energy Dispersive X-Ray Analysis (EDX) techniques. Sorption parameters were studied which included contact time, initial metal ion concentration, nitric acid concentration and adsorbent dose. The equilibrium time has been set at about 15.0 min. The experimental results showed that the sorption efficiency of metal ions under the investigated conditions decreased with increasing nitric acid concentration from 0.50 to 3.0 M. The maximum sorption capacity was found to be 30.0, 50.0 and 60.0 mg/g for Pr(III), DY(III) and Y(III), respectively. The desorption of Pr(III) from the loaded resin was achieved with 1.0 M citric acid at pH = 3 and found to be 58.0%. On the other hand, the maximum desorption of Dy(III) and Y(III) were achieved with 1.0 M nitric acid and 1.0 M ammonium carbonate, respectively. The sorption isotherm results indicated that Pr(III) and Y(II) fitted with nonlinear Langmuir isotherm model with regression factors 0.995 and 0.978, respectively; while, Dy(III) fitted with nonlinear Toth isotherm model with R2 = 0.966. A Flow sheet which summarizes the sorption and desorption processes of Pr(III), DY(III) and Y(III) using Dowex 50WX8 from nitric acid solution under the optimum conditions is also given.
*The maximum adsorption capacities of Dy(III), Pr(III) and Y(III) were 30, 50 and 60 mg/g respectively using 500 mg/L of Dowex 50WX8 resin.
*Maximum stripping (58%) of Pr(III) was achieved using 1 M Citric acid at pH 3.
*The maximum separation ratios obtained were 56, 35 and 4.6 for Y(III)/Pr(III), Y(III)/Dy(III) and Dy(III)/Pr(III), respectively.
Yttrium (Y), Dysprosium (Dy) and Praseodymium (Pr) known as segment of the rare earth elements (REEs) are spirited components in fluorescent lamps, glass polishing and ceramics, computer monitors, lighting, radar, televisions, and X-ray intensifying films. Yttrium is extensively used in the manufacturing of several high-tech-devices such as microwave communication for satellite industries, color televisions, computer monitors and temperature sensors [1, 2]. Praseodymium is used with neodymium in combination for goggles to shield glassmakers against sodium glare, permanent magnets and cryogenic refrigerant [3]. Dysprosium alloy with neodymium is used for permanent magnets, catalysts, speakers, compact discs and hard discs and medium source rare-earth lamps within the film industry. [4] Ion exchange separation of rare earth elements was used by Spedding and Powell to separate REEs from fission products obtained from nuclear reactors [5,6,7]. Sorption processes for the separation of rare earths have been reviewed in several articles [8,9,10,11]. Strongly acidic cation exchangers were the first artificial functional polymers used for the separation of REE ions, and they are resumed predominating in the fields of chemistry and chemical technology under consideration [11]. Styrene and divinylbenzene copolymers bearing SO3H group are utilized. Now, modifications of these exchangers are manufactured under the trade names Amberlite, Dowex, Lewatit, Purolit, which differ by the degree of cross linking sorption capacity, grain size, pore diameter, and other parameter [12]. Dowex 50 W-X8 is one of the ion-exchange resins that have been used for separation of REEs from other ions as well as separation of individual REEs from a mixture of REEs. Al-Thyabat and Zhang, studied the recovery of REEs resulting from phosphoric acid with Dowex 50WX4 and Dowex 50WX8 resins. Their results indicated that the REE-extraction efficiency of Dowex 50WX8 was almost twice that of Dowex 50WX4 resin. This can be explained by the higher exchange capacity, producing more sulfonic groups, of Dowex 50WX8 even though its lower surface area and larger bead size [13]. Felipe et al., studied the recovery of rare earth elements from acid mine drainage by ion exchange [14] and reported that the highest loading capacities were 0.212 mmol g−1 for La and 0.169 mmol g−1 for Ce (Dowex 50WX8) and 0.210 mmol g−1 for La and 0.173 mmol g−1 for Ce (Lewatit MDS 200 H). Recovery of rare earth elements from uranium concentrate by using cation exchange resin (Dowex 50WX8) was studied and the authors reported that the maximum REE sorption capacity was found to 82.74 mg/g which represents about 93.23% of the original capacity of the studied resin [15]. Sorption of rare earth elements from nitric acid solution with macroporous silica-based bis(2-ethylhexyl)phosphoric acid impregnated polymeric adsorbent has been studied by Shu et al. [16] Their results indicated that the adsorption capacity of Gd (III) was found to be 0.315 mmol g−1 by bis(2-ethylhexyl)phosphoric acid/SiO2-P in 0.1 M HNO3. Adsorption and separation of terbium(III) and gadolinium(III) from aqueous nitrate medium has been investigated using TVEX-PHOR resin by Madbouly et al. [17]. Their work showed that the maximum sorption capacity of this material was 15.49 mg/g and 24.93 mg/g for Gd(III) and Tb(III) from 0.1 M NaNO3 solution, respectively at pH = 5.2 and V/m = 0.1. El-Dessouky et al. studied the sorption of praseodymium (III), holmium (III) and cobalt (II) from nitrate medium using TVEX–PHOR resin [18] and reported that 85% sorption was achieved for holmium (III), 75% for praseodymium (III), and 12% for cobalt (II) which enables the possibility of separation of cobalt (II) from the investigated lanthanide elements. The extraction and separation of some rare earths from nitric acid solutions by Cyanex 272 impregnated XAD-7 resin has been examined by İnan et al. [19] The obtained results indicated that REEs have a tendency to behave as two different groups that can be separated into two fractions as La, Pr, Nd and Sm, Eu, Gd. Dowex 50wx8 was used previously for the reversible ion exchange of cerium (III) sulfate and Cerium (III) nitrate where the experimental results indicate that the continuous liquid flow reactor studies show a capacity of 0.72 mmol/g sorbent for the Ce nitrate and 0.96 mmol/g sorbent for the Ce sulfate [20].
The main objective of the present work is directed to study the sorption and separation of Praseodymium, Dysprosium and Yttrium from nitric acid solution using strongly acidic cationic exchange resin (Dowex 50WX8) using batch technique. The effects different parameters on the sorption and separation processes will be investigated such as contact time, nitric acid concentration, as well as v/m ratio and temperature. Desorption investigations will be also carried out and evaluated. Separation feasibility between the investigated REEs are also discussed based on the difference between their sorption and desorption behavior.
Materials and chemicals
The chemicals used in this work were of analytical reagent grade (AR) and most of them were used without further purification. Stock solutions of Pr(III), DY(III) and Y(III) (1000 mg/L) were prepared by dissolving a known amount of the metal oxide in minimum concentrated nitric acid and evaporated to near dryness and then made up to the mark in a measuring flask with double distilled water. The desired required concentrations of test solutions were prepared by favorable dilution with a known concentrated nitric acid of the stock solutions. Dowex 50WX8 which is a strong acid cation resin containing 8% divinylbenzene (DVB) [20], Fig. 1 was purchased from sigma Aldrich. The chemical and physical specifications of Dowex 50X8 are given in table given in Table 1.
Chemical structure of hydrogenated Dowex 50WX8 resin
Table 1 Specification data sheet of Dowex 50WX8
Sorption experiments
The sorption experiments were carried out under the following conditions, v/m = 0.05 L/g, Pr(III), DY(III) and Y(III) concentrations = 100.0 mg/L in 0.50 M nitric acid solution. In each adsorption experiment, 5.0 ml of the investigated metal ions solution was added to 0.1 g of Dowex 50WX8 resin (100–200 mesh) in stoppered glass bottles which were then shaken at (25 ± 1 °C) in a water thermostatic shaker. The concentrations of Pr(III), Dy(III) and Y(III) ions were measured using UV-visible spectrophotometer (a Shimadzu UV-160, Japan) with Arsenazo (III) method [21, 22], and the adsorption capacity (qe) at equilibrium was given by equation:
$$ q_{e} = \,(C_{i} - C_{e} ) \times \left( {\tfrac{V}{w}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, $$
The sorption efficiency (S %) at equilibrium was calculated from the equation:
$$ S\% \, = \,\tfrac{{C_{i} - C_{e} }}{{C_{{_{e} }} }}\,\,\, \times 100\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, $$
where \({C}_{i}\) and \({C}_{e}\) are the initial and equilibrium metal ions concentrations (mg/L) of metal ions, respectively; v is the volume of the used aqueous solution in liter (L) and w is the weight of the adsorbent (g).
Characterization techniques
Scanning electron microscopy coupled with energy dispersive X-ray spectroscopy (SEM/EDX) was used to examine the morphology and determine the elemental composition of the metal ions bonding to Dowex 50WX8 resin under the used experimental conditions. The functional groups included in the used adsorbent were investigated using Fourier transform infrared (FT-IR) spectroscopy (Bruker) in the scanning range of 4000–400 cm−1 and pH was measured using Hannah pH meter.
Desorption investigations
Various reagents such as mineral acids, sodium carbonate, ammonium carbonate and citric acid (at different pH) were used for the desorption investigations of the metal ions under study. In this context, 0.1 g of Dowex 50WX8 loaded with about 100.0 mg/L of each individual Dy(III) or Pr(III) or Y(III) was shaken with 5.0 mL of the stripping solution for 20.0 min under the same sorption experimental conditions.
Characterization of Dowex 50WX8
Figure 2a–d shows the FT-IR spectra of Dowex 50WX8 before and after loading of rare earth metal ions (Pr(III), DY(III) and Y(III)), The spectrum of (Pure Dowex 50WX8) showed adsorption peaks in the ranges of 3330–3450 cm−1 and 1632–1645 cm−1, which may be attributed to stretching vibrations of the O–H functional group in the structure of the adsorbents [20]. Moreover, bands at 1658–1648 cm−1 correspond to the alkene group (C=C) in the Dowex skeleton and bands at1350–1340 cm−1 are assigned to the stretching S=O of sulfonic acid. Dowex 50WX8 gives characteristic IR bands for the SO3 vibration (1169 cm−1), (S–C) vibration (1121 cm−1), and (S–O) vibrations (1093 and 1037 cm−1) [23]. Previous studies have shown that hydration of the sulfonate site with H2O leads to the formation of hydronium ion (H3O+) species which is the interaction moiety with metal ions.
FT-IR spectra of, a Dowex 50WX8, b Dowex -Dy, c Dowex -Pr, d Dowex -Y for sorption of rare earth metal ions from nitrate medium
The FT-IR spectrum of Dowex after interaction with REE metal ions, indicate that adsorbing species (Dowex/Pr/Dy/Y) were formed at the counter ion of the Dowex 50WX8 (H+) which protonates with H2O to form the Dowex 50WX8(H3O+) species [20].
The morphology investigations and particle surface variations of the sorbent were given by EDX. Map and SEM analyses were performed both before and after the adsorption of metal ions, Fig. 3a–d. The obtained results show the presence of a variety of pores with a wide range of pore size on the surface of the Dowex 50WX8 resin; the pore space could be attributed to the adsorption process.
SEM images of, a Dowex 50WX8 before sorption process, b after sorption of Pr(III), c after sorption of Dy(III), d after sorption of Y(III) from nitrate medium
The results of the EDX and Map analyses indicate the presence of different elements on the surface of the Dowex 50WX8, including Praseodymium, Dysprosium and yttrium, with the elements distributed uniformly across the resin surface Fig. 4(a–d). Consequently, the results of the EDX and Map analyses affirm the successful bonding of the metal ions on the surface of Dowex 50WX8.
EDX-analysis of a Dowex 50WX8 before sorption process, b after sorption of Pr(III), c after sorption of Dy(III), d after sorption of Y(III) from nitrate medium
Sorption batch experiments
Effect of shaking time
The impact of contact time on the sorption efficiency of Pr(III), Dy(III) and Y(III) ions using Dowex 50WX8 sorbent from nitrate medium was carried out in the range 1.0–90.0 min. The results indicated that the rare earth ions sorption process took 1–15 min to occur and was based on the availability of vacant active sites The rate of adsorption on the surface of the adsorbents was significantly decreased at contact times beyond 15.0 min, Fig. 5a possibly because of the saturation of the available active sites on the sorbent surface; accordingly the equilibrium contact time for the sorption of the investigated metal ions using Dowex adsorbent was fixed at 15.0 min.
a Effect of contact time on the adsorption efficiency of Pr(III), Dy(III) and Y(III) using Dowex 50WX8 from nitrate medium (temperature: 25 °C, initial ion concentration: 100 mg/L, v/m = 0.05 L/g, b Effect of the sorbent dosage on the adsorption efficiency (at optimal pH value, contact time = 15.0 min, initial metal ion concentration 100 mg/L, temperature: 25 °C, c Effect of nitric acid concentration, d Effect of metal ion concentrations on the sorption of Dy, Pr and Y using Dowex 50WX8
Adsorbent dosage
The sorbent dosage is another key factor imposing large contribution to the sorption process, as it can determine the adsorption efficiency of the sorbent for a given initial concentration of the investigated metal ions [24]. In this respect, the impact of the sorbent dosage (v/m) on the adsorption efficiency of Pr(III), Dy(III) and Y(III) ions was investigated in the range of 0.02–0.25 L/g, Fig. 5b. The obtained results show that the adsorption efficiency yield takes the order DY(III) > Pr(III) > Y(III) enhanced abruptly by increasing v/m from 0.1 L/g up to 0.25 L/g. Based on the obtained results, the optimal adsorbent dosage on Dowex was fixed at 0.05 L/g in all experiments carried out in this work.
Effect of nitric acid concentration
The adsorption of Pr(III), Dy(III) and Y(III) from different nitric acid concentrations is given in Fig. 5c and the experimental results revealed that the adsorption efficiency of Dowex ion exchanger decreased rapidly upon increasing nitric acid concentration from 0.50 to 3.0 M. This may be attributed to the compensation of H+ with higher increasing of acid concentration which leads to a decrease in the exchange rate between hydrogen ion and metal ions; with further increase in the acid molarity higher than 3.0 M the adsorption capacity became stable, Fig. 5c. However, the experiments were performed at 0.5 M HNO3.
Effect of metal ions concentration
The effect of the initial concentration of Pr(III), Dy(III) and Y(III) on the adsorption capacity (qe) was studied in the range of 50–500 mg L−1 through their sorption by Dowex from nitrate medium. The experiments were carried out by shaking 5.0 mL of the investigated metal ions solution individually with 0.05 g of the adsorbent for 15.0 min at 25 °C. The obtained data are represented in Fig. 5d, The adsorption efficiency increased as the concentration of REEs+3 increased and the highest adsorption capacities of 29.0 mg/g, 25.0 mg/g and 24.0 mg/g were achieved at 500 mg/L for Dy, Pr and Y, respectively. This saturation can be ascribed to the interactions between the adsorbent active sites and these metal ions. [25].
Sorption mechanism of REE + with Dowex-H +
Based on the experimental results and considering that M (NO3)2+ is the predominant species in 1.0 M nitric acid solution [21, 26], where M represents Pr(III), Dy(III) and Y(III),The ion exchange extraction mechanism of REEs metal ion (M) with Dowex-H was suggested to proceed via different reaction pathways from Eqs. (3) and (4), [20]
$$ 3DOWEX - SO_{3} - 3H^{ + } + M\left( {NO_{3} } \right)_{2}^{ + } \rightleftharpoons 2{\text{DOWEX}} - SO_{3} - {\text{M}}NO_{3} {\text{ + HNO3 + }}DOWEX - SO_{3} - H^{ + } \, $$
$$ 3DOWEX - SO_{3} - 3H^{ + } + M\left( {NO_{3} } \right)_{2}^{ + } \rightleftharpoons {\text{DOWEX}} - SO_{3} - {\text{M}}NO_{3} { + 2}DOWEX - SO_{3} - 2H^{ + } \, $$
Equation (3) suggests that the extraction mechanism occurs via partial ion exchange reactions during the REE diffusion and interaction at the Dowex active sites SO3-H+ where the extracted metal ions species according to Eq. (3) were found to be Dowex-SO3-Pr(NO3), Dowex-SO3-Dy(NO3) and Dowex-SO3-Y(NO3) for Pr, Dy and Y respectively.
Adsorption isotherm of Pr, Dy and Y on the Dowex 50WX8 cation exchanger
Adsorption isotherms were used to describe the distribution of metal ions between the sample solution (liquid phase) and the resin (solid phase) when the ion exchange process reaches equilibrium [27, 28]. The Langmuir isotherm model describes a homogeneous monolayer chemical adsorption process, while the Freundlich isotherm model describes a heterogeneous physical adsorption process [29]. Non-liner models achieved the most flexible curve fitting functionality. In this context, Langmuir, Freundlich, Temkin, D–R isotherm and Toth isotherm were employed for studying the nonlinear adsorption isotherm of Pr, Dy and Y on the cation exchanger resin (Dowex 50WX8).
Nonlinear Langmuir isotherm equation is given as:
$$ {\text{q}}_{{\text{e}}} \,{ = }\,{\text{Q}}\frac{{{\text{bC}}_{{\text{e}}} }}{{{\text{1 + bC}}_{{\text{e}}} }} \, $$
where qe is the equilibrium adsorption capacity of ions on the adsorbent (mg g−1), Ce is the equilibrium ions concentration in solution (mg L−1), Q the maximum capacity of the adsorbent. (mg g−1), and b the Langmuir adsorption constant (L mg−1). Nonlinear Freundlich isotherm equation is given as:
$$ q_{e\,} \, = \,K_{f} \,C_{e}^{1/n} \, $$
where Kf is the Freundlich constant (mg/g).
Nonlinear Temkin isotherm model which takes into account the interactions of ions of the aqueous solution and the adsorbent and is given as:
$$ q_{e} = \frac{RT}{b}\ln (A_{T} C_{e} ) \, $$
where R is the universal gas constant (8.314 J/mol K), T the absolute temperature, b a constant related to the heat of sorption (J/mol), AT the equilibrium binding constant (L/g) and b the adsorption constant (J/mol K).
Toth isotherm model is another empirical equation developed to improve Langmuir isotherm fittings and take into consideration both low and high-end boundary of the concentration and is given as, [30, 31]
$$ q_{e} = \, q_{m} \exp ( - \beta \varepsilon^{2} ) \, $$
$$ \varepsilon = RT\ln \left( {1 + \frac{1}{{C_{e} }}} \right) \, $$
$$ E = \frac{1}{{(2\beta )^{0.5} }} \, $$
The relationship between qe and Ce for each nonlinear isotherm model is plotted in (Fig. 6 a–e) and the values of the obtained parameters are tabulated in Table 2. The results indicate that Pr and Y fitted with nonlinear Langmuir isotherm model with regression factors 0.995 and 0.978 respectively, while, Dy was fitted with nonlinear Toth isotherm model with R = 0.966.
a Nonlinear isotherm plot of Langmuir model, b Nonlinear isotherm plot of Freundlich model, c Nonlinear isotherm plot of D-R, d Nonlinear isotherm plot of Temkin model, d Nonlinear isotherm plot Toth model for adsorption of Pr(III), DY(III) and Y(III) onto Dowex 50WX8
Table 2 Nonlinear Freundlich, Langmuir, Dubinin–Radushkevich, Temkin and Toth isotherm parameters for adsorption of metal ions onto Dowex 50WX8
Desorption, reusability and separation between Dy, Pr and Y from nitrate medium using Dowex 50WX8
The most effective separation obtained between the investigated metal ions was obtained from the stripping process. This process was carried out by contacting the loaded Dowex 50WX8 with different stripping agents at experimental conditions (contact time = 60.0 min, v/m = 0.05 at 25 ± 1 °C). The results illustrated in Table 3 show that the maximum stripping of Pr(III) is 58% and was achieved with 1.0 M citric acid at pH = 3. In the case of Dy(III) and Y(III) the maximum desorption is 55% and 56% and was achieved with [HNO3] = 1.0 M and [(NH4)2CO3] = 1.0 M, respectively. A flow sheet which illustrates the sorption and desorption processes of the investigated rare earth using D-50WX8 from 0.5 M HNO3 solution at v/m = 0.05 at 25 ± 1 °C is given in Fig. 7.
Table 3 Desorption of metal ions (III) with different reagents after their adsorption with the Dowex 50WX8 resin at v/m = 0.05 at 25 ± 1 °C
Flow sheet for the sorption and desorption processes of Pr, DY and Y using Dowex 50WX8
The desorption results indicate that Pr(III) can be separated from Dy(III) and Y(III) as follows:
Stripping of Dy(III) and Y(III) using [(NH4)2CO3] = 1.0 M from Loaded Dowex 50WX8 after 2 cycles
Dowex 50WX8 containing Pr(III) was then stripped with 1.0 M citric acid at pH = 3 after two stripping cycles
Furthermore, the separation ratio (S-ratio) between the investigated metal ions were calculated by dividing their desorption percentages. The results indicate that the maximum S-ratios are 56.0, 35.0 and 4.6 for Y/Pr, Dy/Pr and were achieved with [(NH4)2CO3] = 1.0 and [HNO3] = 5.0 M respectively, Table 3.
Finally, the reported results show that Dowex 50WX8 resin is relatively selective, high efficient and cost effective for Pr(III), Dy(III) and Y(III) adsorption and is also easily regenerated rather than other reported adsorbent/ion exchangers which were used in the adsorption from acidic nitrate medium. The reusability was carried out for 4.0 adsorption stages with sorption capacity of 15.0, 30.0, 35.0 mg/g for Pr, DY and Y, respectively, under the used experimental conditions.
Comparison study of REEs/Dowex 50WX8 with other reported materials
Comparison of REEs/Dowex 50WX8 system under the used optimum conditions of batch technique with other commercially reported materials [18, 32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51] and given in Table 4, shows the advantages and efficiency of Dowex 50WX8 adsorbent. The results of comparison in the term of maximum capacity (Q0) (30, 50, 60 mg/g for Pr, DY and Y), pH = 1, and contact time (15 min) and which were achieved in the current study indicate that Dowex 50WX8 is more efficient and affordable than other reported materials.
Table 4 Comparison study of REEs/Dowex 50WX8 with other reported materials
Dowex 50WX8 was successfully used for the recovery of DY(III), Pr(III) and Y(III) from acidic nitrate medium. The calculated maximum capacity of Dowex 50WX8 is 30, 50, 60 mg/g for Pr, DY and Y respectively at the optimum batch conditions; the maximum stripping of Pr(III) is 58.0% and was achieved with 1.0 M citric acid at pH = 3. The results indicate that Pr(III) and Y(III) fitted with nonlinear Langmuir isotherm model with regression factors 0.995 and 0.978 respectively. The regenerated Dowex 50WX8 gave sorption capacities of 15.0, 30.0, 35.0 mg/g for Pr, DY and Y, respectively under the used experimental conditions.
All data generated or analyzed during this study are included in this published article [and its supplementary information files].
Charalampides G, Vatalis KI, Apostoplos B, Ploutarch-Nikolas B. Rare Earth Elements: Industrial Applications and Economic Dependency of Europe. Procedia Econ Financ. 2015;24:126.
Humphries M. Rare earth elements: the global supply chain. Congressional Research Service. In The Library of Congress; 2013.
Akah A. Application of rare earths in fluid catalytic cracking: a review. J Rare Earths. 2017;35:941.
Massari S, Ruberti M. Rare earth elements as critical raw materials: Focus on international markets and future strategies. Resour Policy. 2013;38:36.
Powell JE. The separation of rare earths by ion exchange. Prog Sci Technol Rare Earths. 1964;88:4536.
Spedding FH, Powell EJ. Wheelwright the stability of the rare earth complexes with N-hydroxyethylethylenediaminetriacetic acid. J Am Chem Soc. 1956;78:34.
JE POWELL 1979Separation chemistry Handbook on the physics and chemistry of rare earths Elsevier 3 81 109
Iftekhar S, Ramasamy DL, Srivastava V, Asif MB, Sillanpää M. Understanding the factors affecting the adsorption of Lanthanum using different adsorbents: a critical review. Chemosphere. 2018;204:413.
Zubiani EMI, Cristiani C, Dotelli G, Stampino PG. Solid liquid extraction of rare earths from aqueous solutions: a review. Procedia Environ Sci Eng Manag. 2015;2(3):231.
Anastopoulos I, Bhatnagar A, Lima EC. Adsorption of rare earth metals: a review of recent literature. J Mol Liq. 2016;221:954.
Ehrlich GV, Lisichkin GV. Sorption in the chemistry of rare earth elements. Russ J Gen Chem. 2017;87(6):1220.
Zagorodni AA. Ion exchange materials: properties and applications. Amsterdam: Elsevier; 2006.
Al-Thyabat S, Zhang P. REE extraction from phosphoric acid, phosphoric acid sludge, and phosphogypsum Min. Proc Ext Met. 2015;124:143.
Felipe ECB, Batista KA, Ladeira ACQ. Recovery of rare earth elements from acid mine drainage by ion exchange. Environ Technol. 2021;42(17):2721.
Ghazala RA. Recovery of rare earth elements from uranium concentrate by using cation exchange resin. Isot Radiat Res. 2015;47(2):219.
Shu Q, Khayambashi A, Wang X, Wei Y. Studies on adsorption of rare earth elements from nitric acid solution with macroporous silica-based bis (2-ethylhexyl) phosphoric acid impregnated polymeric adsorbent. Adsorpt Sci Technol. 2018;36(3–4):1049.
Madbouly HA, El-Hefny NE, El-Nadi YA. Adsorption and separation of terbium (III) and gadolinium (III) from aqueous nitrate medium using solid extractant. Sep Sci Technol. 2021;56(4):681.
El-Dessouky SI, El-Sofany EA, Daoud JA. Studies on the sorption of praseodymium (III), holmium (III) and cobalt (II) from nitrate medium using TVEX–PHOR resin. J Hazard Mater. 2007;143(1–2):17.
İnan S, Tel H, Sert Ş, Çetinkaya B, Sengül S, Özkan B, Altaş Y. Extraction and separation studies of rare earth elements using cyanex 272 impregnated amberlite XAD-7 resin. Hydrometallurgy. 2018;181:156.
Miller DD, Siriwardane R, Mcintyre D. Anion structural effects on interaction of rare earth element ions with Dowex 50W X8 cation exchange resin. J Rare Earths. 2018;36(8):879.
Aly MI, Masry BA, Gasser MS, Khalifa NA, Daoud JA. Extraction of Ce (IV), Yb (III) and Y (III) and recovery of some rare earth elements from Egyptian monazite using CYANEX 923 in kerosene. Int J Miner Process. 2016;153:71.
Marczenko Z. Spectrophotometric determination of elements; Ellis Harwood. Poland: Ltd; 1976.
RM Silverstein FX Webster DJ Kiemle 2005 Spectrometric Identification of organic compounds Hoboken John Wiley 502 35
Masry BA, Elhady MA, Mousaa IM. Fabrication of a novel polyvinylpyrrolidone/abietic acid hydrogel by gamma irradiation for the recovery of Zn Co Mn and Ni from aqueous acidic solution. Inorg Nano Met Chem,. 2022. https://doi.org/10.1080/24701556.2022.2034860.
El-saied HA, Shahr El-Din AM, Masry BA, Ibrahim AM. A promising superabsorbent nanocomposite based on grafting biopolymer/nanomagnetite for capture of 134Cs, 85Sr and 60Co radionuclides. J Polym Environ. 2020;28(6):1749.
Masry BA, Aly MI, Khalifa NA, Zikry AAF, Gasser MS, Daoud JA. Liquid–liquid extraction and separation of Pr (III), Nd (III), Sm (III) from nitric acid medium by CYANEX 923 in kerosene. Arab J Nucl Sci Appl. 2015;48(3):1.
Edebali S, Pehlivan E. Evaluation of amberlite IRA96 and Dowex 1×8 ion-exchange resins for the removal of Cr (VI) from aqueous solution. Chem Eng J. 2010;161(1–2):161–6.
Hameed BH, Ahmad AA, Aziz N. Isotherms, kinetics and thermodynamics of acid dye adsorption on activated palm ash. Chem Eng J. 2007;133:195.
Weber WJ. Physiochemical processes for water quality control. Hoboken NJ: Wiley Interscience; 1972.
Toth J. State equation of the solid-gas interface layers. Acta Chim Hung. 1971;69:311.
Vijayaraghavan K, Padmesh TVN, Palanivelu K, Velan M. Biosorption of nickel (II) ions onto Sargassum wightii: application of two-parameter and three-parameter isotherm models. J Hazard Mater. 2006;133(1–3):304.
Abdel-Magied AF, Abdelhamid HN, Ashour RM, Zou X, Forsberg K. Hierarchical porous zeolitic imidazolate frameworks nanoparticles for efficient adsorption of rare-earth elements. Microporous Mesoporous Mater. 2019;278:175–84.
Koochaki-Mohammadpour SMA, Torab-Mostaedi M, Talebizadeh-Rafsanjani A, Naderi-Behdani F. Adsorption isotherm, kinetic, thermodynamic, and desorption studies of lanthanum and dysprosium on oxidized multiwalled carbon nanotubes. J Dispers Sci Technol. 2014;35(2):244–54.
Gargari JE, Kalal HS, Shakeri A, Khanchi A. Synthesis and characterization of Silica/polyvinyl imidazole/H2PO4-core-shell nanoparticles as recyclable adsorbent for efficient scavenging of Sm (III) and Dy (III) from water. J Colloid Interface Sci. 2017;505:745–55.
Awual MR, Alharthi NH, Okamoto Y, Karim MR, Halim ME, Hasan MM, Sheikh MC. Ligand field effect for Dysprosium (III) and Lutetium (III) adsorption and EXAFS coordination with novel composite nanomaterials. Chem Eng J. 2017;320:427–35.
Yusoff MM, Mostapa NRN, Sarkar MS, Biswas TK, Rahman ML, Arshad SE, Kulkarni AD. Synthesis of ion imprinted polymers for selective recognition and separation of rare earth metals. J Rare Earths. 2017;35(2):177–86.
Ashour RM, Abdel-Magied AF, Abdel-Khalek AA, Helaly OS, Ali MM. Preparation and characterization of magnetic iron oxide nanoparticles functionalized by l-cysteine: Adsorption and desorption behavior for rare earth metal ions. J Environ Chem Eng. 2016;4(3):3114–21.
Xu X, Zou J, Teng J, Liu Q, Jiang XY, Jiao FP, Chen XQ. Novel high-gluten flour physically cross-linked graphene oxide composites: Hydrothermal fabrication and adsorption properties for rare earth ions. Ecotoxicol Environ Saf. 2018;166:1–10. https://doi.org/10.1016/j.ecoenv.2018.09.062.
Xu X, Jiang XY, Jiao FP, Chen XQ, Yu JG. Tunable assembly of porous three-dimensional graphene oxide-corn zein composites with strong mechanical properties for adsorption of rare earth elements. J Taiwan Inst Chem Eng. 2018;85:106–14.
Wang Q, Wilfong WC, Kail BW, Yu Y, Gray ML. Novel polyethylenimine–acrylamide/SiO2 hybrid hydrogel sorbent for rare-earth-element recycling from aqueous sources. ACS Sustain Chem Eng. 2017;5(11):10947–58.
Ramasamy DL, Puhakka V, Doshi B, Iftekhar S, Sillanpää M. Fabrication of carbon nanotubes reinforced silica composites with improved rare earth elements adsorption performance. Chem Eng J. 2019;365:291–304.
Liang T, Yan C, Li X, Zhou S, Wang H. Withdrawn: Polyacrylic acid grafted silica fume as an excellent adsorbent for dysprosium (III) removal from industrial wastewater. Water Sci Technol. 2018;77(6):1570–80.
Kondo K, Umetsu M, Matsumoto M. Adsorption characteristics of gadolinium and dysprosium with microcapsules containing an extractant. J Water Process Eng. 2015;7:237–43.
Aghayan H, Mahjoub AR, Khanchi AR. Samarium and dysprosium removal using 11-molybdo-vanadophosphoric acid supported on Zr modified mesoporous silica SBA-15. Chem Eng J. 2013;225:509–19.
Guanqyao Z, Zhixing S, Xingyin L, Xijun C. Efficiency of macroporous poly(vinylphosphoramidic acid) resin adsorbing of selected elements and determination of trace dysprosium holmium erbiom and ytterbium in waste water by inductively coupled plasma optical emission spectrometry. Anal Lett. 1992;25:561–72.
Wang H, Gao P. Adsorption of d113 resin for dysprosium (III). J Wuhan Univ Technol Mater Sci Ed. 2007;22:653–6.
Ramasamy DL, Wojtuś A, Repo E, Kalliola S, Srivastava V, Sillanpää M. Ligand immobilized novel hybrid adsorbents for rare earth elements (REE) removal from waste water: assessing the feasibility of using APTES functionalized silica in the hybridization process with chitosan. Chem Eng J. 2017;330:1370–9.
Bai R, Yang F, Zhang Y, Zhao Z, Liao Q, Chen P, Cai C. Preparation of elastic diglycolamic-acid modified chitosan sponges and their application to recycling of rare-earth from waste phosphor powder. Carbohyd Polym. 2018;190:255–61.
Bendiaf H, Abderrahim O, Villemin D, Didi MA. Studies on the feasibility of using a novel phosphonate resin for the separation of U (VI), La (III) and Pr (III) from aqueous solutions. J Radioanal Nucl Chem. 2017;312(3):587–97. https://doi.org/10.1007/s10967-017-5244-8.
Yan P, He M, Chen B, Hu B. Fast preconcentration of trace rare earth elements from environmental samples by di (2-ethylhexyl) phosphoric acid grafted magnetic nanoparticles followed by inductively coupled plasma mass spectrometry detection. Spectrochim Acta Part B At Spectrosc. 2017;136:73–80. https://doi.org/10.1016/j.sab.2017.08.011.
Gaete J, Molina L, Valenzuela F, Basualto C. Recovery of lanthanum, praseodymium and samarium by adsorption using magnetic nanoparticles functionalized with a phosphonic group. Hydrometallurgy. 2021;105:698. https://doi.org/10.1016/j.hydromet.2021.105698.
Ogata T, Narita H, Tanaka M. Adsorption behavior of rare earth elements on silica gel modified with diglycol amic acid. Hydrometallurgy. 2015;152:178–82. https://doi.org/10.1016/j.hydromet.2015.01.005.
Authors are thankful to the Egyptian Atomic Energy Authority for its continuous support for scientific research and development.
Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB). Science, Technology & Innovation Funding Authority (STDF) and springer nature open access agreement.
Chemistry of Nuclear Fuel Department, Hot Laboratories Centre, Egyptian Atomic Energy Authority, Cairo, Egypt
B. A. Masry, E. M. Abu Elgoud & S. E. Rizk
B. A. Masry
E. M. Abu Elgoud
S. E. Rizk
BAM: Conceptualization, writing—original draft, data analysis. EMAE and SER: Methodology, resources, formal analysis, and data analysis. All authors read and approved the final manuscript.
Correspondence to B. A. Masry.
The manuscript does not contain studies with animal subjects.
All authors approved the paper submission.
Masry, B.A., Abu Elgoud, E.M. & Rizk, S.E. Modeling and equilibrium studies on the recovery of praseodymium (III), dysprosium (III) and yttrium (III) using acidic cation exchange resin. BMC Chemistry 16, 37 (2022). https://doi.org/10.1186/s13065-022-00830-0
Dowex 50WX8 | CommonCrawl |
Title:SU(3) sphaleron: Numerical solution
Authors:F.R. Klinkhamer, P. Nagel
(Submitted on 25 Apr 2017 (v1), last revised 12 Jul 2017 (this version, v5))
Abstract: We complete the construction of the sphaleron $\widehat{S}$ in $SU(3)$ Yang-Mills-Higgs theory with a single Higgs triplet by solving the reduced field equations numerically. The energy of the $SU(3)$ sphaleron $\widehat{S}$ is found to be of the same order as the energy of a previously known solution, the embedded $SU(2)\times U(1)$ sphaleron $S$. In addition, we discuss $\widehat{S}$ in an extended $SU(3)$ Yang-Mills-Higgs theory with three Higgs triplets, where all eight gauge bosons get an equal mass in the vacuum. This extended $SU(3)$ Yang-Mills-Higgs theory may be considered as a toy model of quantum chromodynamics without quark fields and we conjecture that the $\widehat{S}$ gauge fields play a significant role in the nonperturbative dynamics of quantum chromodynamics (which does not have fundamental scalar fields but gets a mass scale from quantum effects).
Comments: 36 pages, 6 figures, v5: published version
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Theory (hep-th)
Journal reference: Phys. Rev. D 96, 016006 (2017)
Report number: KA-TP-18-2017
From: Frans Klinkhamer [view email]
[v1] Tue, 25 Apr 2017 15:58:52 UTC (559 KB)
[v2] Tue, 2 May 2017 17:23:41 UTC (568 KB)
[v3] Mon, 8 May 2017 17:46:56 UTC (229 KB)
[v4] Tue, 6 Jun 2017 13:13:33 UTC (232 KB)
[v5] Wed, 12 Jul 2017 16:52:06 UTC (231 KB) | CommonCrawl |
Preferences of physicians for public and private sector work
Anthony Scott ORCID: orcid.org/0000-0002-2851-53781,
Jon Helgeim Holte2 &
Julia Witt3
The public-private mix of healthcare remains controversial. This paper examines physicians' preferences for public sector work in the context of dual practice, whilst accounting for other differences in the characteristics of jobs.
A discrete choice experiment is conducted with data from 3422 non-GP specialists from the Medicine in Australia: Balancing Employment and Life (MABEL) panel survey of physicians.
Physicians prefer to work in the public sector, though the value of working in the public sector is very small at 0.14% of their annual earnings to work an additional hour per week. These preferences are heterogeneous. Contrary to other studies that show risk averse individuals prefer public sector work, for physicians, we find that those averse to taking career or clinical risks prefer to work in the private sector. Those with relatively low earnings prefer public sector work and those with high earnings prefer private sector work, though these effects are small.
Other job characteristics are more important than the sector of work, suggesting that these should be the focus of policy to influence specialist's allocation of time between sectors.
In many healthcare systems, physicians are able to combine work in the public and private sector. Dual practice can be controversial in the context of universal health care that usually aims to provide care that does not depend on ability to pay but the need for health care [1]. Physicians choose to allocate their time across both sectors based on the institutional setting that governs regulations and remuneration, specialty, and conditional on this, physician's preferences for the characteristics of each work setting. In a mixed public-private system, and where the number of physicians is relatively fixed in the short to medium term because of barriers to entry, physician's preferences for the amount of time spent in each sector can influence access to health care, including public hospital waiting times, as well as expenditures and patient's health outcomes.
The aim of this paper is to examine physician's preferences for working in the public or private sector within a given institutional setting that allows dual practice and a rich mix of public and private health care provision and financing. The literature on physician dual practice is largely theoretical and concerned with the nature of the regulation of public and private sector work [1,2,3,4]. The drivers of physician's choices, such as the role of job characteristics, have not yet been examined in detail. Johannessen and Hagen [5] examine associations with physician characteristics such as debt and family size, but did not examine job characteristics. Cheng et al. [6] and Saether [7] focus on the role of earnings on sector choice. We build on these studies by using a discrete choice experiment to capture a much richer set of job characteristics [8, 9]. We also examine risk attitudes which have been suggested as a reason why workers prefer the public sector to the private sector or to become self-employed [10,11,12,13,14] but have not been examined in the context of physician dual practice.
There were 1325 hospitals in Australia in 2016/2017, including 630 private hospitals. Thirty-five percent of beds and 60% of separations were in public hospitals, whilst 59% of all elective surgery is conducted in private hospitals [15]. Medicare is Australia's national tax-financed universal health insurance scheme that subsidises out of hospital medical services (GPs and non-GP specialists), pharmaceuticals, and around 40% of funding for public hospitals. States own and run public hospitals and fund the remainder from their own tax revenues. The government indirectly subsidises activity in private hospitals through subsidies for private health insurance premiums and through Medicare rebates for patients treated in private hospitals. Around 45% of the population has private health insurance.
Non-GP specialists in Australia can be employed on a salary in public hospitals providing inpatient or outpatient services and, for those who have 'rights to private practice', can at the same time be self-employed and organised into small businesses. In these private settings, they work in private hospitals providing inpatient services, and/or in their own offices providing outpatient services, and they can treat private patients in public hospitals. Private non-GP specialists operate on a fee-for-service basis and can charge patients what the market will bear, and receive no remuneration from private hospitals. There are no price controls. Private patients can claim a fixed subsidy from Medicare, which is a fixed amount as determined in the Medicare Benefits Schedule. This fixed amount is usually lower than the fee charged, and so patients face an out of pocket cost. For patients in private hospitals, this out of pocket cost is insurable by some private health insurers ('no gap' cover) though patient eligibility for this cover is at the discretion of the specialist. Specialists who are primarily based in the private sector can choose to work in public hospitals as a contractor (Visiting Medical Officer) where they can be paid a fixed payment per session (typically a 4-h period) or by fee-for-service. The salaries of public hospital non-GP specialists are determined by State-level employer bargaining agreements, so salary scales are fixed, though public hospitals have discretion to pay above the award rates.
A previous study, using the same dataset as this paper, found that 48% of medical specialists combined public and private sector work, 19% worked in the private sector only, and 33% worked in the public sector only [16]. Public sector specialists are likely to be younger, to be international medical graduates, to devote a higher percentage of time to education and research, more likely to do after hours and on-call work, and more likely to travel to provide services in other areas, compared to private sector specialists. Dual practice and private sector specialists also have higher annual earnings compared to public sector specialists.
A discrete choice experiment was conducted as part of Wave 1 (2008) of the Medicine in Australia: Balancing Employment and Life (MABEL) panel survey of doctors [17]. The MABEL survey included the DCE and was sent to the population of doctors using four surveys: hospital non-specialists, doctors enrolled in specialty training programmes, general practitioners, and non-GP specialists in clinical practice in Australia in 2008 with 10 498 (19%) doctors responding. Doctors were invited using a personalised mailed letter that included a paper copy of the survey and also a username and password if they wished to fill out the survey online. Three reminders were sent. MABEL includes rich data, including annual earnings and other financial questions, work and job characteristics, hours worked, family circumstances, and geographic location. In this paper, we include doctors who filled out the 'Specialists' survey, which includes questions on the hours they work in the public and private sectors.
DCE design
A list of attributes included in the DCE is shown in Table 1, and an example of one DCE question is shown in Fig. 1. These were informed by the existing literature and piloting and included attributes most likely to differ between sectors. The questionnaire went through four stages of piloting, including examining face and content validity of the DCEs through face to face interviews with two specialists and a meeting of 12 doctors in specialist training, and a full pilot survey.
Table 1 Attributes and levels
Example of DCE question
The main attribute of interest is the percentage of time spent in private practice. The levels for this attribute are 10%, 50%, and 90%. The latter (50% and 90%) are based on the average hours per week in the private sector of specialists working mainly in private practice [18], and we defined our lower bound (10%) so that specialists who work mainly in the public sector would find this realistic.
The levels of our income attributes are defined as percent changes from current income (20% increase, no change, 20% decrease) to avoid having income figures that might be unrealistic for some respondents, such as those working part time [9, 19]. Similarly, hours worked are defined in terms of changes from current hours (10% increase, no change, and 10% decrease), and these percentages were obtained from average total hours worked per week across different specialties [18, 20]. Another attribute is on-call arrangements, which is a key issue for all specialties [9, 19]. Teaching and research opportunities is included as an attribute, as these are associated with intellectual satisfaction and reputation [21]. Time spent doing administrative non-clinical work is another attribute. Any administrative time would mostly generate disutility, but specialists are likely to be willing to trade off doing some administrative work for other perks. Specialists are unlikely to work in rural and remote areas due to lack of infrastructure (i.e. hospitals). However, specialists can choose between large regional centres and metropolitan areas. Additionally, specialists can also choose to visit rural areas for short periods of time, and there are government specialist outreach programmes to support this financially.
SAS was used to generate an efficient fractional factorial experimental design [22]. Zero priors were used to generate the pilot survey as we did not have any other information. Results from the pilot surveys were used as priors in the experimental design for the main survey [23]. The experimental design produced a fractional factorial design of 36 choices which were randomly assigned to four version of the survey, each with nine choice sets.
The specification of the choice model is based on random utility theory where an indirect utility function is specified and estimated with three alternatives. The utility Unij and choice outcome Ynij of physician n for alternative i from choice set j is:
$$ {U}_{nij}={X}_{nij}\beta +{\varepsilon}_{nij} $$
for example, Ynij = 1 if Un1j > Un2j & Un1j > Un3j, if alternative 1 is preferred to alternatives 2 and 3
$$ n=1,\dots N;\kern0.5em j=1,\dots, J $$
where Xnij is a k-vector of observed attributes of alternative i, β is a vector of marginal utilities of the attributes, and εnij is i.i.d. extreme value and is estimated using the multinomial logit model. Two job alternatives (A and B) were presented to each doctor, and they were asked which job (A or B) they prefer (forced choice) and then asked which job they would choose: A, B, or their current job (status quo). The latter was included to account for status quo bias. In the analysis, the levels of each attribute in the status quo alternative were constructed from other questions asked in the survey that represented the doctors' current job characteristics. For the attributes of earnings and hours worked, a zero-percentage change was used in the status quo alternative. For the percentage of time spent on administration and in the private sector, questions were asked on the actual distribution of working hours across settings (including public and private) and a separate question asked about the distribution of hours across activities (clinical, non-clinical, management and administration, education and research). The level of on-call for the status quo alternative was constructed from several questions asking whether the doctor did on-call, how many hours, and how frequently they were called out.
Unobserved heterogeneity in marginal utilities can be modelled using an extension of the multinomial logit, the mixed logit model:
$$ {U}_{nij}={X}_{nij}{\beta}_n+{\varepsilon}_{nij} $$
$$ {\beta}_n=\tilde{\beta}+{\eta}_n $$
where ηn is a vector of mean-zero individual-specific deviations from the mean marginal utility such that βn is a vector of individual-specific marginal utilities of each attribute with a distribution F(βn;θ) specified by the researcher [24]. The vector of parameters θ (the means and standard deviations of the random coefficients βn) characterises the distribution of βn. We estimate a generalised multinomial logit model (G-MNL) [25], which is a mixed logit model that allows for correlation between the parameter distributions of coefficients using a single parameter [26]. Compared to a mixed logit model with uncorrelated coefficients, the GMNL models allows for correlations between the distributions of heterogeneity that is common across all coefficients due to both preference and scale heterogeneity, and so results in an improved model fit [26, 27]. Scale heterogeneity is where the variance of the error terms varies across individuals because of near-lexicographic preferences where marginal utilities for some attributes are very high (i.e. scaled up), or at the other extreme can be due to randomness of behaviour where the idiosyncratic error term dominates and an individual is very unsure of their choices. Fiebig et al. [25] argue that the G-MNL model is flexible enough to model data from these 'extreme' respondents, therefore providing a much better fit to the data. The GMNL model is an extension of the mixed logit model by multiplying the error term in (2) according to 1/σn or equivalently by multiplying the vector of coefficients by σn: \( {\beta}_n={\sigma}_n\left(\overset{\sim }{\beta }+{\eta}_n\right) \) where \( {\sigma}_n=\mathit{\exp}\left(\overline{\sigma}+\tau {v}_n\right) \), \( {v}_n\sim N\left(0,1\right),\mathrm{and}\ \overline{\sigma}={\tau}^2/2 \), so there is one extra parameter, τ, to be estimated. Apart from the coefficient for income, which is treated as fixed to aid the calculation of marginal rates of substitution, the coefficients for the remaining attributes are treated as random (using a normal distribution).
To examine the monetary value of private sector work (the compensating differential), we calculate the marginal rate of substitution between the earnings attribute and the private sector attribute. We also calculate these compensating earnings differentials for all other attributes using the same methods as in Scott et al. [28]. The measures of risk aversion are discussed in Additional file 1. The measures of risk aversion are interacted with the public-private attribute in the regression model to examine whether preferences for public sector work depend on risk aversion, and whether the magnitude of the marginal utility for the public-private attribute remains stable and statistically significant.
The response rate for specialists was 22.3% (4310/19 579) with a 98.4% contact rate. The final numbers of specialists who completed at least part of the DCE was 3422 with descriptive statistics in Table 2. The questionnaire was completed online by 27.6% of respondents. Respondents were broadly representative of the population of Australian specialists. MABEL respondents were slightly younger (51.2 in MABEL vs 53 years old in population), included more women specialists than the population (26.5% vs 20.6% female), were slightly less likely to come from major cities (major cities 83.7% vs 86.8%: inner regional areas 13% vs 10.6%: outer regional areas 2.7% vs 2.4%: remote areas 0.7% vs 0.2%), and worked an additional 36 min per week (44.4 vs 43.8 h per week).
Table 2 Characteristics of specialists responding to the DCE (n = 3422)
Table 3 shows the number of times each alternative was chosen (out of 3422 × 9 choice sets = 30 798 choice sets across all respondents) and shows that the status quo was chosen 81% of the time, job A was chosen in 6.5% of the choice sets, and job B in 10.8%.
Table 3 Choice frequencies
The results of the GMNL model show this is preferred to a mixed logit model on the basis of the Bayesian Information Criteria (BIC) (results available on request). Table 4 shows a strong preference for the status quo (their current job) shown by the large negative coefficients for the constant terms. The statistically significant standard deviations for all attributes suggest that the strength of preference for their current job varies across specialists. The signs of the attributes are in the expected direction. Specialists prefer higher earnings, fewer hours, less on-call, more teaching and research opportunities, less administration, and working in metropolitan areas. For the continuous attributes of earnings, hours worked, percent time in the private sector and percent time in administration, we tested the linearity of each variable (one-by-one) by comparing against models where each of these attributes were re-coded in categories (i.e. non-linear). Likelihood ratio tests confirmed that for each of these attributes the hypothesis of linearity was not rejected.
Table 4 Results from GMNL model and marginal willingness to pay
On average, specialists prefer to work in the public sector, shown by a negative marginal utility of the percentage of time spent in private practice. The coefficient measures the effect on utility of a 1% increase in the percentage of time spent in the private sector. The standard deviation suggests that the marginal utility varies across respondents. Since the coefficient presented is an average of individual specific marginal utilities, these individual marginal utilities were recovered and standardised to have a mean of zero and standard deviation of 1 and are plotted in Fig. 2. This has quite a tight distribution with the majority of respondents lying within ± 2 standard deviations of the mean. Fifty-nine percent of specialists prefer working in the private sector, whilst 41% prefer working in the public sector. However, those preferring the public sector do so more strongly than those preferring the private sector, such that the mean marginal utility of working in the private sector is negative.
Distribution of standardised marginal utility of the percentage of time in private sector work. Notes: The βn are distributed according to the distribution function F(βn; θ, τ) and are the expected values of βn given the parameter estimates and the choices made by each individual: \( E\left[{\beta}_n|{Y}_{n,}{X}_n;\hat{\theta},\hat{\tau}\Big)\right] \) [29]. Two doctors, who completed the same set of nine choices (Xn) and choose the same alternatives (Yn), will have the same individual-specific estimate of the marginal utility βn (24)
The marginal rate of substitution between private sector work and earnings shows that for a 1% increase in the proportion of time spent in the private sector, specialists would need to be compensated 0.057% of their annual income, which is about $186. This can also be expressed in terms of working an extra session (4 h or half a day) in the private sector, representing a more realistic margin. With an average specialist spending 44% (19.95 h) of their 45.3 weekly working hours in the private sector, an increase of 4 h per week to 23.95 h represents an increase in the proportion of hours from 44 to 53%. This 9 percentage point increase would require them to be paid $1680 per year to maintain their utility. This is quite small compared to the average annual income of $325 000 and compared to the value of other attributes shown in the last two columns of Table 4.
Is the preference for public sector work associated with specialists' characteristics?
To investigate the factors influencing the variation in preferences for public sector work, the standardised measure of the marginal utility in Fig. 2 is used in an ordinary least squares (OLS) regression (Table 5). The independent variables focus on aspects of the life/career cycle and include age in 5-year bands, whether they have dependent children, and whether the respondent is an Australian graduate. Separate models are estimated for males and females. Overall, the explanatory power of these models is very low. For men, those aged 65–70 and approaching retirement have a stronger marginal utility for private sector work. This is also the case for women (aged 61–65 years old), but the effect is much stronger than for men. This is likely to reflect a preference for either boosting retirement income, or more likely a preference for less challenging work as doctors reduce their hours of work before they retire. In addition, female specialists with dependent children have a stronger preference for working more hours in the private sector. Since we have controlled for income, this is likely to reflect a preference for more autonomy and flexibility over working hours.
Table 5 Association of life/career cycle factors with the marginal utility of private sector work
We test whether income influences the preference for private sector work by splitting the sample into high- and low-income respondents and re-running the models (details and results in Additional file 1). These results suggest that the marginal utility of both earnings and private sector work are similar between those with low and high wages and that those with lower wages prefer the public sector and those with higher wages prefer the private sector though these associations are small.
Is the preference for public sector work associated with risk attitudes?
Table 6 show the coefficients for the private practice attribute and its interaction with each risk aversion measure (full model results available on request). Note that the models including interaction terms have smaller sample sizes (see Additional file 1) since not all respondents to Wave 1 responded in Wave 2 (Big 5 risk aversion) or in Wave 6 (for risk aversion). The interaction term between the two overall measures of risk aversion and the private practice attribute are not statistically significant. The domain specific measures show that specialists who are more likely to take career risks prefer to spend more time in the public sector. Although career trajectories are more well-defined in the public sector, there is more tournament-type competition between specialists to work in major teaching hospitals to undertake high-quality research and teaching. This can create more uncertainty and competition when pursuing career options in the public sector. Risk averse doctors prefer the private sector.
Table 6 Preferences for time spent in private practice and risk aversion
Taking clinical risks is also associated with a preference for more time in the public sector. Since the public sector treats more complex and challenging cases, it makes sense that it would attract doctors who prefer the greater challenges and uncertainty of treating such cases, which are likely to be patients most in need. The interaction term between the private practice attribute and taking financial risks is not statistically significant. Though there are some associations with risk aversion, the marginal utility of the private sector attribute remains statistically significant. However, the magnitude of the coefficient is around half that in the base model. This suggests that there is some evidence that risk attitudes in specific domains partly explain the observed preference for public sector work, though overall measures of risk attitudes were not statistically significant.
This paper provides new evidence on factors influencing the preferences of medical specialists for public or private sector work. After controlling for the key differences between public and private sector medical jobs, including earnings, as well as risk aversion, the results show only a weak preference towards spending more time in the public sector overall and among low wage earners, and a slight preference for time in the private sector among high wage earners. Doctors averse to clinical and career risk have a stronger preference for the private sector, contrary to the existing literature on public-private job choices, but reflecting the particular characteristics of physician's jobs. Other job characteristics that differ between sectors are much more important to specialists than the amount of time spent in the public or private sector. This confirms our previous research using revealed preference data on hours worked that found little difference in the marginal utility of public and private sector work (Cheng et al. [16]).
These results suggest that non-wage factors play a stronger role in sector choice compared to wages and the sector itself. In Australia, medicine is the occupation delivering the highest earnings, and so the marginal utility of income for this group is likely to be small relative to lower earning occupations, and there may also be less variation in the marginal utility of income compared to other occupations. Our results also show that risk aversion is not only about financial uncertainty, but also about clinical and career uncertainty, and these may be more important drivers of behaviour.
The conclusions rest on the assumption that we have controlled for all other differences between public and private sector jobs. We included the most important job attributes from the literature, respondents were asked to assume that other factors were the same between jobs, and we examined the role played by risk attitudes. However, we cannot rule out other unobserved factors, though these are only likely to play a minor role. If important, they are likely to reduce the preference for public sector work to close to zero, strengthening our conclusions that it is the characteristics of the sector rather than a preference for working in the public sector.
A weak preference for the public sector may reflect the culture of medical practice in Australia. For some specialties where only public sector work is possible, or where the norm is dual practice, the amount of time spent in the private sector may be heavily influenced by specialty-specific norms. Our results are therefore likely to vary across specialties though sample sizes by specialty were too small, and there are no specific hypotheses about how the results might vary.
We did not include specific tests of 'rationality' or tests of continuity of preferences, such as identifying attribute non-attendance [30,31,32]. Though we cannot rule this out, unlike many other DCEs completed by patients or the general public where the choice tasks and attributes may be unfamiliar, our sample of doctors are highly educated and very familiar with the attributes presented and so could be less likely to provide 'irrational' responses, make errors, or ignore attributes and employ decision heuristics because of the difficulty of the choice task. Our use of a GMNL model does help to more flexibly model unobserved preference and scale heterogeneity compared to a simple mixed logit with uncorrelated coefficients, and so may capture differences in the error variances across individuals that can reflect the randomness of responses due to the adoption of different decision rules that are common to all coefficients [25].
Our response rate was 22.3% which is good for a sample of physicians [33]. There were some small differences in age, gender, hours worked, and geographic location, and we cannot observe representativeness with respect to other characteristics (unless they are correlated with observed characteristics) that may lead to bias. In particular, if more income-oriented physicians are less likely to fill out surveys and these are more likely to work in the private sector, then this could bias our results and underestimate the strength of preference for private sector work.
In terms of policy conclusions, sector choice is more likely to be influenced by non-wage attributes of public and private sector jobs, and reductions in clinical uncertainty in public hospitals (e.g. through the use of clinical guidelines) could play a role. Specialists seem to be happy with their current balance of hours between the public and private sector. That may be largely because there is no regulation or restrictions about which sector they work in, and so their choices are optimal. Though choices might be optimal for specialists, they might not be for patients. Further research is required as to whether these choices are also socially optimal in terms of the effect of changes in doctors' allocation of time between sectors on patients' health outcomes and costs.
De-identified MABEL data are available on application from the Australian Data Archive https://dataverse.ada.edu.au/dataverse.xhtml?alias=ada&q=MABEL. The dataset relating to this study that includes the discrete choice experiment is not included in the above due to confidentiality, but the data may be made available on request to the authors.
MABEL:
Medicine in Australia: Balancing Employment and Life
GP:
DCE:
Discrete choice experiment
G-MNL:
Generalised multinomial logit
AMPCo:
Australasian Medical Publishing Company
Bayesian Information Criteria
OLS:
Ordinary least squares
ASGC:
Australian Standard Geographic Classification
WTP:
McPake B, Russo G, Hipgrave D, Hort K, Campbell J. Implications of dual practice for universal health coverage. Bull World Health Organ. 2016;94(2):142–6.
Eggleston K, Bir A. Physician dual practice. Health Policy. 2006;78(2–3):157–66.
González P, Macho-Stadler I. A theoretical approach to dual practice regulations in the health sector. J Health Econ. 2013;32(1):66–87.
Socha KZ, Bech M. Physician dual practice: a review of literature. Health Policy. 2011;102(1):1–7.
Johannessen K-A, Hagen TP. Physicians' engagement in dual practices and the effects on labor supply in public hospitals: results from a register-based study. BMC Health Serv Res. 2014;14(1):299.
Cheng TC, Kalb G, Scott A. Public, private or both? Analyzing factors influencing the labour supply of medical specialists. Canadian Journal of Economics/Revue canadienne d'économique. 2018;51(2):660–92.
Saether E. Physicians' labour supply: the wage impact on hours and practice combinations. Labour. 2005;19(4):673–703.
Mandeville KL, Lagarde M, Hanson K. The use of discrete choice experiments to inform health workforce policy: a systematic review. BMC Health Serv Res. 2014;14(1):367.
Scott A. Eliciting GPs' preferences for pecuniary and non-pecuniary job characteristics. J Health Econ. 2001;20:329–47.
Buurman M, Delfgaauw J, Dur R, Van den Bossche S. Public sector employees: risk averse and altruistic? J Econ Behav Organ. 2012;83(3):279–91.
Dohmen T, Falk A, Huffman D, Sunde U, Schupp J, Wagner GG. Individual risk attitudes: measurement, determinants, and behavioral consequences. J Eur Econ Assoc. 2011;9(3):522–50.
Falk A, Becker A, Dohmen T, Enke B, Huffman D, Sunde U. Global evidence on economic preferences*. Q J Econ. 2018;133(4):1645–92.
Caliendo M, Fossen F, Kritikos AS. Personality characteristics and the decisions to become and stay self-employed. Small Bus Econ. 2014;42(4):787–814.
Zhao H, Seibert SE. The big five personality dimensions and entrepreneurial status: a meta-analytical review. J Appl Psychol. 2006;91(2):259.
AIHW. Australia's hospitals at a glance. Australian Institute of Health and Welfare: Canberra; 2018.
Cheng TC, Joyce CM, Scott A. An empirical analysis of public and private medical practice in Australia. Health Policy 2013;111(1):43-51.
Joyce CM, Scott A, Jeon S-H, Humphreys J, Kalb G, Witt J, et al. The "Medicine in Australia: Balancing Employment and Life (MABEL)" longitudinal survey - protocol and baseline data for a prospective cohort study of Australian doctors' workforce participation. BMC Health Serv Res. 2010;10(1):50.
ABS. Private medical practitioners, report no. 8689.0. Canberra: Australian Bureau of Statistics; 2002.
Ubach C, Scott A, French F, Awramenko M, Needham G. What do hospital consultants value about their jobs? A discrete choice experiment. Br Med J. 2003;326:1432.
AIHW. National health labour force series, medical labour force 2003. Canberra: Australian Institute of Health and Welfare; 2003.
Borges NJ, Navarro AM, Grover A, Hoban JD. How, when, and why do physicians choose careers in academic medicine? A literature review. Acad Med. 2010;85(4):680–6.
Carlsson F, Martinsson P. Design techniques for stated preference methods in health economics. Health Econ. 2002;12:281–94.
Kuhfeld WF. Marketing research methods in SAS: experimental design choice, conjoint, and graphical techniques. SAS 9.1 Edition TS-722. 2005.
Train KE. Discrete choice methods with simulation. 2nd ed. New York: Cambridge University Press; 2009.
Fiebig DG, Keane MP, Louviere J, Wasi N. The generalized multinomial logit model: accounting for scale and coefficient heterogeneity. Mark Sci. 2010;29(3):393–421.
Hess S, Train K. Correlation and scale in mixed logit models. Journal of Choice Modelling. 2017;23:1–8.
Hess S, Rose JM. Can scale and coefficient heterogeneity be separated in random coefficients models? Transportation. 2012;39(6):1225–39.
Scott A, Witt J, Humphreys J, Joyce C, Kalb G, Jeon SH, et al. Getting doctors into the bush: general practitioners' preferences for rural location. Soc Sci Med. 2013;96:33–44.
Greene W. NLOGIT version 4.0. Referecne guide. New York: Econometric Software Inc; 2007.
Lagarde M. Investigating attribute non-attendance and its consequences in choice experiments with latent class models. Health Econ. 2013;22(5):554–67.
Hole AR. A discrete choice model with endogenous attribute attendance. Econ Lett. 2011;110(3):203–5.
Scott A. Identifying and analysing dominant preferences in discrete choice experiments: an application in health care. J Econ Psychol. 2002;23(3):383–98.
Scott A, Jeon S-H, Joyce C, Humphreys J, Kalb G, Witt J, et al. A randomised trial and economic evaluation of the effect of response mode on response rate, response bias, and item non-response in a survey of doctors. BMC Med Res Methodol. 2011;11:126.
We thank the thousands of doctors who filled out the survey each year and the MABEL research team.
This paper used data from the MABEL longitudinal survey of doctors conducted by the University of Melbourne and Monash University. Funding for MABEL comes from the National Health and Medical Research Council (Health Services Research Grant APP454799: 2008-2011; and Centre for Research Excellence in Medical Workforce Dynamics APP1019605: 2012-2016) with additional support from the Department of Health (in 2008) and Health Workforce Australia (in 2013).
Melbourne Institute: Applied Economic and Social Research, The University of Melbourne, Level 5 FBE Building, 111 Barry Street, Melbourne, VIC, 3010, Australia
FAFO Institute for Labour and Social Research, Borggata 2B, 0608, Oslo, Norway
Jon Helgeim Holte
Department of Economics, University of Manitoba, Winnipeg, MB, R3T 5V5, Canada
Julia Witt
AS: conceptualisation, methodology, formal analysis, writing original draft and review and editing, supervision, funding acquisition. JHH: software, formal analysis, writing review and editing. JW: conceptualisation, methodology, formal analysis, writing review and editing. The author(s) read and approved the final manuscript.
Correspondence to Anthony Scott.
The study was approved by The University of Melbourne Faculty of Business and Economics Human Ethics Advisory Group (Ref. 0709559) and the Monash University Standing Committee on Ethics in Research Involving Humans (Ref. CF07/1102 – 2007000291). There are no conflicts of interest.
The measures of risk aversion.
Scott, A., Holte, J.H. & Witt, J. Preferences of physicians for public and private sector work. Hum Resour Health 18, 59 (2020). https://doi.org/10.1186/s12960-020-00498-4
Dual practice
Discrete choice experiments
Risk aversion | CommonCrawl |
Bonnie Litwiller
Bonnie Helen Litwiller (February 14, 1937 – January 27, 2012) was an American mathematics educator and textbook author, who worked as a professor of mathematics at the University of Northern Iowa.[1]
Life
Litwiller was born on February 14, 1937, in Morton, Illinois.[1] She studied mathematics education at Illinois State University, earning bachelor's and master's degrees in 1959 and 1960. She earned a second master's degree in 1965 at Indiana University, and completed a doctorate (Ed.D.) there in 1968.[2] Her doctoral dissertation was Enrichment: A Method of Changing the Attitudes of Prospective Elementary Teachers Toward Mathematics.[3]
After working for seven years as a high school teacher in Peoria, Illinois, she joined the faculty of the University of Northern Iowa, where she spent the rest of her career,[1] educating a large fraction of the mathematics teachers in Iowa.[4] She served as president of the Iowa Council of Teachers of Mathematics for 1978–1979.[5]
After her retirement in 2000, she returned to Morton.[1] She continued to publish scholarly work on mathematics education into her retirement, eventually amassing over 1000 publications.[4]
She died on January 27, 2012,[1] of myelofibrosis.[6]
Books
Litwiller's books include:
• Activities for the Maintenance of Computational Skills and the Discovery of Patterns (with David R. Duncan, National Council of Teachers of Mathematics, 1980)[7]
• Problem Solving with Number Patterns (with David R. Duncan, School Science and Mathematics Association, 1987)[8]
• Curriculum and Evaluation Standards for School Mathematics: First-Grade Book (with Miriam Leiva, National Council of Teachers of Mathematics, 1991)[9]
• Making Sense of Fractions, Ratios, and Proportions (edited with George Bright, 2002 Yearbook of the National Council of Teachers of Mathematics)[10]
• Navigating through Problem Solving and Reasoning in Grade 3 (with Karol L. Yeatts, Michael T. Battista, Sally Mayberry, Denisse R. Thompson, Judith S. Zawojewski, and Peggy A. House, National Council of Teachers of Mathematics, 2004)[11]
Recognition
In 2003, the Iowa Council of Teachers of Mathematics gave Litwiller their Lifetime Achievement Award. In 2004, the Indiana University School of Education gave her their Distinguished Alumni Award.[1]
An endowed scholarship in her name is offered by the Illinois State University to students in mathematics education.[2] Another scholarship, at the University of Northern Iowa, was funded by a bequest from Litwiller's estate.[12] In conjunction with the American Red Cross, the United Church of Christ of Morton, Illinois held a series of annual blood drives in her memory.[6]
References
1. "Bonnie H. Litwiller", CINewsNow, archived from the original on 2013-10-29
2. "Bonnie Litwiller Scholarship in Mathematics Education", Scholarship Finder, Illinois State University, retrieved 2021-08-16
3. Litwiller, Bonnie Helen (1968), Enrichment: A Method of Changing the Attitudes of Prospective Elementary Teachers Toward Mathematics (Doctoral dissertation), Indiana University, ProQuest 302302239
4. Johnson, Raymond (January 2012), "Bonnie H. Litwiller, 1937–2012", MathEd.net
5. ICTM History, Iowa Council of Teachers of Mathematics, retrieved 2021-08-16
6. 5th annual Red Cross blood drive held in memory of Bonnie Litwiller Oct. 24, American Red Cross, retrieved 2021-08-16
7. Reviews of Activities for the Maintenance of Computational Skills and the Discovery of Patterns:
• Kemp, William C. (March 1981), The Mathematics Teacher, 74 (3): 233, JSTOR 27962401{{citation}}: CS1 maint: untitled periodical (link)
• West, Tommie A. (April 1981), The Arithmetic Teacher, 28 (8): 44, doi:10.5951/AT.28.8.0044, JSTOR 41191874{{citation}}: CS1 maint: untitled periodical (link)
8. Reviews of Problem Solving with Number Patterns:
• Caldwell, Janet H. (March 1988), The Mathematics Teacher, 81 (3): 230–231, JSTOR 27965778{{citation}}: CS1 maint: untitled periodical (link)
• Corn, Mary Louise (September 1988), The Arithmetic Teacher, 36 (1): 58, JSTOR 41194362{{citation}}: CS1 maint: untitled periodical (link)
9. Review of Curriculum and Evaluation Standards for School Mathematics:
• Doyle, Desmond (December 1992), The Arithmetic Teacher, 40 (4): 240, JSTOR 41195323{{citation}}: CS1 maint: untitled periodical (link)
10. Review of Making Sense of Fractions, Ratios, and Proportions:
• Priest, Nadine (January 2003), The Mathematics Teacher, 96 (1): 76, JSTOR 20871230{{citation}}: CS1 maint: untitled periodical (link)
11. Review of Navigating through Problem Solving and Reasoning in Grade 3:
• Clark, Lynn L. (March 2005), Teaching Children Mathematics, 11 (7): 399, JSTOR 41199836{{citation}}: CS1 maint: untitled periodical (link)
12. "UNI gets $1.5 million gift for math teacher scholarship", The Gazette, 30 May 2012
Authority control
International
• ISNI
• VIAF
National
• Israel
• United States
| Wikipedia |
\begin{definition}[Definition:Sink]
Let $\mathbf V$ be a vector field acting over a region of space $R$.
A '''source''' is a point $P$ in $R$ at which flux leaves $\mathbf V$.
\end{definition} | ProofWiki |
Microscopy and Microanalysis (10)
The Journal of Laryngology & Otology (5)
European Journal of Anaesthesiology (4)
Clay Minerals (2)
Experimental Agriculture (1)
Journal of Materials Research (1)
The Journal of Agricultural Science (1)
Weed Science (1)
Brazilian Society for Microscopy and Microanalysis (SBMM) (10)
The Australian Society of Otolaryngology Head and Neck Surgery (5)
Weed Science Society of America (1)
Assessment of nasal functions and their relationship with cholesteatoma formation in patients with unilateral chronic otitis media
F Arslan, M Binar, U Aydin
Journal: The Journal of Laryngology & Otology / Volume 132 / Issue 11 / November 2018
To evaluate the nasal functions of patients with unilateral chronic otitis media using rhinomanometry, comparing chronic otitis media sides with healthy sides, chronic otitis media patients with cholesteatoma and without cholesteatoma, and patients with healthy individuals.
This prospective study included 102 patients with unilateral chronic otitis media (48 with and 54 without cholesteatoma). The control group comprised 40 individuals without any ear or nasal pathologies. All patients underwent active anterior rhinomanometry to measure nasal airway resistance and a saccharin test to measure mucociliary transport times.
There were no significant differences in nasal airway resistance and mucociliary transport time between the chronic otitis media sides and unaffected sides in the 102 patients (p = 0.72 and p = 0.28, respectively), between the non-suppurative chronic otitis media patients (without cholesteatoma) and chronic otitis media with cholesteatoma patients (p > 0.05), or between the study and control groups (p > 0.05).
The present study, with a larger sample size compared to previously published literature, supports the conclusion that unilateral nasal obstruction is unlikely to lead to chronic otitis media on the same side. The results also suggest that nasal functions do not contribute to the development of cholesteatoma.
Nucleation and growth of metamorphic epitaxial aluminum on silicon (111) 7 × 7 and $\sqrt 3 \times \sqrt 3$ surfaces
Ashish Alexander, Brian M. McSkimming, Bruce Arey, Ilke Arslan, Christopher J.K. Richardson
Journal: Journal of Materials Research / Volume 32 / Issue 21 / 14 November 2017
Published online by Cambridge University Press: 21 August 2017, pp. 4067-4075
Print publication: 14 November 2017
The nucleation and growth of Al on 7 × 7 and $\sqrt 3 \times \sqrt 3$ R30 Al reconstructed Si(111) that result in strain-free Al overgrown films grown with an atomically abrupt metamorphic interface are compared. The reconstructed surfaces and abrupt strain relaxations are verified using reflection high-energy electron diffraction. The topography of evolution is examined with atomic force microscopy. The growth of Al on both the surfaces exhibits 3D island growth, but the island evolution of growth is dramatically different. On the 7 × 7 surface, mounds formed are uniformly distributed across the substrate, and growth appears to proceed uniformly. Alternatively, on the $\sqrt 3 \times \sqrt 3$ R30 surface, Al atoms exhibit a clear preference to form mounds near the step edges. During Al growth, mounds increase in size and number, expanding out from step edges until they cover the whole substrate. Consistent expression of a mounded nucleation and growth mode imparts a physical limitation to the achievable surface roughness that may impact the ultimate performance of layered devices such as Josephson junctions that are critical components of superconducting quantum circuits.
Investigating 3D Printing with Microscopy and Spectroscopy Techniques
B. Arey, C. Barrett, I. Arslan, Z. Kennedy, M. Warner, H. Schroeder
Journal: Microscopy and Microanalysis / Volume 23 / Issue S1 / July 2017
Print publication: July 2017
Alternatives to Atrazine for Weed Management in Processing Sweet Corn
Zubeyde Filiz Arslan, Martin M. Williams, Roger Becker, Vincent A. Fritz, R. Ed Peachey, Tom L. Rabaey
Journal: Weed Science / Volume 64 / Issue 3 / September 2016
Atrazine has been the most widely used herbicide in North American processing sweet corn for decades; however, increased restrictions in recent years have reduced or eliminated atrazine use in certain production areas. The objective of this study was to identify the best stakeholder-derived weed management alternatives to atrazine in processing sweet corn. In field trials throughout the major production areas of processing sweet corn, including three states over 4 yr, 12 atrazine-free weed management treatments were compared to three standard atrazine-containing treatments and a weed-free check. Treatments varied with respect to herbicide mode of action, herbicide application timing, and interrow cultivation. All treatments included a PRE application of dimethenamid. No single weed species occurred across all sites; however, weeds observed in two or more sites included common lambsquarters, giant ragweed, morningglory species, velvetleaf, and wild-proso millet. Standard treatments containing both atrazine and mesotrione POST provided the most efficacious weed control among treatments and resulted in crop yields comparable to the weed-free check, thus demonstrating the value of atrazine in sweet corn production systems. Timely interrow cultivation in atrazine-free treatments did not consistently improve weed control. Only two atrazine-free treatments consistently resulted in weed control and crop yield comparable to standard treatments with atrazine POST: treatments with tembotrione POST either with or without interrow cultivation. Additional atrazine-free treatments with topramezone applied POST worked well in Oregon where small-seeded weed species were prevalent. This work demonstrates that certain atrazine-free weed management systems, based on input from the sweet corn growers and processors who would adopt this technology, are comparable in performance to standard atrazine-containing weed management systems.
The effect of OK-432 (Picibanil) injection on the histopathology of nasal turbinate
S Sengul, İ Kaygusuz, M M Akin, Ş Yalcin, T Karlidag, E Keles, İ Arslan
Journal: The Journal of Laryngology & Otology / Volume 129 / Issue 12 / December 2015
Published online by Cambridge University Press: 12 October 2015, pp. 1208-1212
This study aimed to assess the histopathological effect of OK-432 (Picibanil) on rabbit nasal turbinates.
A total of 21 rabbits were divided into 3 treatment groups and various parts of both nasal turbinates were injected with 0.5 ml OK-432, 0.2 ml OK-432 or 0.6 ml saline (control). Bilateral nasal turbinates were later excised and studied under light microscopy to assess any histopathological changes.
Animals in the 0.2 ml and 0.5 ml OK-432 groups exhibited mild ciliary loss, goblet cell loss and epithelial damage, and a marked increase in inflammatory cell infiltration, submucosal vascularisation and fibrosis. There was a significant difference in histopathological changes between the two OK-432 treated groups. In addition, each OK-432 treated group had significantly more inflammatory cell infiltration, increased submucosal vascularisation and fibrosis compared with controls.
The marked fibrosis observed in OK-432-injected turbinates may be responsible for a reduction in turbinate size.
Controlled Radiolytic Synthesis in the Fluid Stage. Towards Understanding the Effect of the Electron Beam in Liquids
Patricia Abellan, Lucas R. Parent, Trevor H. Moser, Chiwoo Park, Naila Al Hasan, Prabhakaran Munusamy, Ivan T. Lucas, Ilke Arslan, Jay Grate, Ayman M. Karim, James E. Evans, Nigel D. Browning
Published online by Cambridge University Press: 23 September 2015, pp. 2125-2126
Does systemic clarithromycin therapy have an inhibitory effect on tympanosclerosis? An experimental animal study
G Genc, M Koyuncu, G Kutlar, T Guvenc, A Gacar, A Aksoy, S Arslan, S C Kurnaz
Journal: The Journal of Laryngology & Otology / Volume 129 / Issue 2 / February 2015
To demonstrate the inhibitory effects of clarithromycin on in vitro tympanosclerosis.
Twenty-eight rats were divided into three groups: a clarithromycin group, a non-clarithromycin group and a negative control group. Those in the first two groups were injected with Streptococcus pneumoniae following a myringotomy, and tympanosclerosis was experimentally induced. Oral clarithromycin therapy was administered in the clarithromycin group. The other groups received no medical treatment.
All eardrums in the clarithromycin and non-clarithromycin groups developed myringosclerosis, but there was only one eardrum, in the clarithromycin group, with very severe myringosclerosis. In the clarithromycin group, 11 ears showed no inflammation and there were no ears with severe inflammation. In the non-clarithromycin group, there were 11 ears with severe inflammation. The mean eardrum thickness in the clarithromycin group was 20.93 µm and in the non-clarithromycin group it was 42.71 µm.
Acute otitis media and myringotomies induced tympanosclerosis, but clarithromycin reduced the severity of tympanosclerosis.
Location of the middle cranial fossa dural plate in patients with chronic otitis media
S Genc, M G Genc, I B Arslan, A Selcuk
Journal: The Journal of Laryngology & Otology / Volume 128 / Issue 1 / January 2014
Published online by Cambridge University Press: 22 January 2014, pp. 60-63
This study aimed to determine whether or not the middle cranial fossa dural plate is located lower (i.e. more caudally) in patients with chronic otitis media, relative to adjacent structures.
The authors retrospectively investigated computed tomography temporal bone scans of 267 ears of 206 patients who had undergone surgery with a diagnosis of chronic otitis media, together with scans of 222 ears of 111 patients without chronic otitis media. The depth of the middle cranial fossa dural plates was recorded.
The mean depth of the middle cranial fossa dural plate was 4.59 mm in the study group and 2.71 mm in the control group (p < 0.001). The middle cranial fossa dural plate was located lower in the right ear in both the study and control groups.
The middle cranial fossa dural plate was located lower in patients with chronic otitis media, and in the right ears of both patients and controls. Surgeons should take this low location into consideration, and take extra care, during relevant surgery on patients with chronic otitis media.
EFFECTS OF SAGE LEAFHOPPER FEEDING DAMAGE ON HERBAGE COLOUR, ESSENTIAL OIL CONTENT AND COMPOSITIONS OF TURKISH AND GREEK OREGANO
M. ARSLAN, I. UREMIS, N. DEMIREL
Journal: Experimental Agriculture / Volume 48 / Issue 3 / July 2012
Turkish (Origanum onites L.) and Greek oregano (Origanum vulgare L., ssp. hirtum (Link.) Ietswaart) species were investigated to determine herbage colour, essential oil content and composition changes due to sage leafhopper (Eupteryx melissae) (Hemiptera: Cicadellidae) infestation. Sage leafhopper population on both Turkish and Greek oregano did not significantly vary. The sage leafhopper damage was more severe in the lower part of the canopy than the middle and upper parts. Extensive sage leafhopper feeding dramatically reduced essential oil contents, resulting in 28.8 and 34.8% reductions for Greek and Turkish oregano, respectively. Carvacrol, the major essential oil component of both oregano species, did not remarkably vary between leafhopper infested and non-infested plants. With respect to herbage colour, the brightness, redness and yellowness values were significantly different between infested and non-infested plants. Sage leafhopper damage increased brightness and yellowness but decreased greenness of the oregano herbage. To avoid the feeding damage, it is essential to detect the sage leafhopper problem as early as possible and certain control practices are necessary when the infestation is high.
Studying Materials In-situ with the Dynamic Transmission Electron Microscope (DTEM)
ND Browning, JE Evans, I Arslan, GH Campbell, K Jungjohann, TB LaGrange, S Mehraeen, BW Reed, LR Parent, MK Santala, M Wall
Extended abstract of a paper presented at Microscopy and Microanalysis 2010 in Portland, Oregon, USA, August 1 – August 5, 2010.
The 3-D Pore Structure of Pd Nanoparticles as a Function of Temperature
M Klein, M Ong, B Jacobs, D Robinson, S Fares, I Arslan
Extended abstract of a paper presented at Microscopy and Microanalysis 2009 in Richmond, Virginia, USA, July 26 – July 30, 2009
3-D Scanning Transmission Electron Microscopy of Carbide-Derived Carbons for Electrical Energy Storage
S Widgeon, M Gass, AL Bleloch, S-H Yeon, Y Gogotsi, I Arslan
Exposure to Legionellaceae at a hot spring spa: a prospective clinical and serological study
N. Bornstein, D. Marmet, M. Surgot, M. Nowicki, A. Arslan, J. Esteve, J. Fleurette
Published online by Cambridge University Press: 15 May 2009, pp. 31-36
Following the occurrence of five cases of Legionnaires' disease among patients and therapists at a French hot spring spa, a series of cleansing procedures and an epidemiological study were undertaken. During a 3-month period, the spring water was repeatedly sampled. Serum samples were taken from 689 randomly selected patients, 230 therapists, 134 administrative staff and a control group of 904 blood donors.
Legionellaceae were present in the spring water at concentrations of 103–1055 colony forming units/1. Fifteen different species or serogroups were isolated with Legionella pneumophila serogroups 3 and 1 predominating. No clinical cases of Legionnaires disease were observed during the study. However, 11% of the therapists and 5% of the patients either had a high titre of antibody (≥256) to at least one species or serogroup or seroconverted during the study. Mean antibody titres in the three study groups were significantly higher than those in the blood donors against 11 of the 32 legionella antigens tested. Nine of these 11 antigens corresponded to species or serogroups isolated from the spring water. The highest mean antibody titres in all three study groups were against L. pneumophila serogroup 3, the most common legionella in the spring water.
These findings have important implications for the maintenance of adequate standards of hygiene, bacteriological sampling and clinical surveillance in this and similar establishments.
Estimates of relative yield potential and genetic improvement of wheat cultivars in the Mediterranean region
O. SENER, M. ARSLAN, Y. SOYSAL, M. ERAYMAN
Journal: The Journal of Agricultural Science / Volume 147 / Issue 3 / June 2009
Information about changes associated with advances in crop productivity is essential for understanding yield-limiting factors and developing new strategies for future breeding programmes. National bread wheat (Triticum aestivum L.) yields in Turkey have risen by an average of 20·8 kg/ha/year from 1925 to 2006. Annual gain in yield attributable to agronomic and genetic improvement averaged c. 11·6 kg/ha/year prior to 1975, but is now averaging c. 15·1 kg/ha/year. In the Mediterranean region, however, the wheat yield trend line (10·9 kg/ha/year) is c. 0·38 lower than that of Turkey. In order to understand whether such a trend was due to the cultivars released over the years, 16 bread wheat cultivars, commonly grown in the region and representing 23 years of breeding, introduction and selection (from 1976 to 1999), were grown in a randomized complete block design with three replicates across 2 years. Data were collected on maturation time, plant height, spike length, spikelet number/spike, grain number/spike, grain weight/spike, 1000 seed weight, harvest index and grain yield. None of the measured plant traits showed any historical cultivar patterns; therefore, the increase in grain yield could not be attributed to a single yield component. Several physiological traits changed during two decades of cultivar releases in the Mediterranean region that led to a genetic gain in grain yield of about 0·5% per year. Years of data and the present field study in the Mediterranean region suggested that the genetic improvement in wheat seemed inadequate and should be reinforced with modern agricultural management practices as well as technological innovations.
Does left molar approach to laryngoscopy make difficult intubation easier than the conventional midline approach?
N. Bozdogan, M. Sener, A. Bilen, A. Turkoz, A. Donmez, G. Arslan
Journal: European Journal of Anaesthesiology / Volume 25 / Issue 8 / August 2008
Background and objective
It has been reported that the left molar approach of laryngoscopy can make difficult intubation easier. The aim of this study was to investigate whether left molar approach to laryngoscopy provided a better laryngeal view in cases of unexpected difficult intubation.
Following the approval of local Ethics Committee and written informed consent from the patients, out of 1386 patients who underwent general anaesthesia for surgery, 20 patients who could be ventilated by face mask but could not be intubated with conventional midline approach on the first attempt were included in the study. Those 20 patients, who had Grade III-IV laryngeal views on laryngoscopy by conventional midline approach, were subjected to left molar laryngoscopy, and their laryngeal views were evaluated. The external laryngeal compression was routinely used to improve the laryngeal view. When endotracheal intubation failed by left molar laryngoscopy, we performed the conventional midline approach again. All data were recorded.
Of the 20 patients studied, 18 had a Grade III laryngeal view and two had a Grade IV laryngeal view. Eighteen of them had a better laryngeal view with left molar laryngoscopy. Eleven of the 20 patients underwent successful intubation with the left molar laryngoscopy, which provided a significantly better laryngeal view and success rate of tracheal intubation than did the conventional midline approach (P < 0.01 and P < 0.01, respectively).
Left molar laryngoscopy can make unexpected difficult intubation easier and should be attempted in cases of difficult intubation.
Correlating Electron Tomography and Atom Probe Tomography
I Arslan, EA Marquis, M Homer, MA Hekmaty, NC Bartelt
Extended abstract of a paper presented at Microscopy and Microanalysis 2008 in Albuquerque, New Mexico, USA, August 3 – August 7, 2008
Patient-controlled analgesia with lornoxicam vs. dipyrone for acute postoperative pain relief after septorhinoplasty: a prospective, randomized, double-blind, placebo-controlled study
M. Sener, C. Yilmazer, I. Yilmaz, E. Caliskan, A. Donmez, G. Arslan
Journal: European Journal of Anaesthesiology / Volume 25 / Issue 3 / March 2008
We compared the efficacy of intravenous lornoxicam vs. dipyrone in patient-controlled analgesia for postoperative analgesia.
The study included 105 patients who had undergone elective septorhinoplasty after receiving general anaesthesia. Patients were divided into three groups to receive lornoxicam (24 mg day−1), dipyrone (5 g day−1) or placebo. Pain was evaluated using a 0–100 mm visual analogue scale at 2, 4, 6, 8, 12, 16, 20 and 24 h postoperatively. Pethidine (1 mg kg−1) was administered intramuscularly to patients requiring rescue analgesia. Pethidine requirements were recorded during the first 24 h postoperatively, and treatment-related adverse effects were noted.
Postoperative pain scores were significantly lower with lornoxicam compared with dipyrone at 8 h (P = 0.016). No significant differences regarding pain scores at 2, 4, 6, 12, 16, 20 and 24 h were found. Significantly fewer patients in the lornoxicam group required rescue analgesics (vs. dipyrone, P = 0.046; vs. placebo, P = 0.001); fewer patients in the dipyrone group required rescue analgesics compared with placebo (P = 0.008). Significantly fewer patients in the lornoxicam group had nausea (vs. dipyrone, P = 0.022; vs. placebo, P = 0.006); no significant differences were found between the other two groups. Antiemetic use was significantly lower in the lornoxicam group (vs. dipyrone, P = 0.002; vs. placebo, P = 0.001).
Lornoxicam has better tolerability and is a more effective analgesic than dipyrone when administered by patient-controlled analgesia for postoperative analgesia after septorhinoplasty.
Aberration Corrected and Monochromated STEM/TEM for Materials Science
ND Browning, I Arslan, JP Bradley, M Chi, Z Dai, R Erni, M Herrera, NL Okamoto, QM Ramasse
Journal: Microscopy and Microanalysis / Volume 13 / Issue S02 / August 2007
Extended abstract of a paper presented at Microscopy and Microanalysis 2007 in Ft. Lauderdale, Florida, USA, August 5 – August 9, 2007
Scanning Transmission Electron Tomography
P Midgley, M Gass, I Arslan, J Tong, T Yates, A Hungria, R Dunin-Borkowski, M Weyland, J Thomas
Extended abstract of a paper presented at Microscopy and Microanalysis 2006 in Chicago, Illinois, USA, July 30 – August 3, 2005
Applications of Atomic Scale Scanning Transmission Electron Microscopy
ND Browning, R Erni, CJ Mitterbauer, L Fu, M Chi, S Mehraeen, M Herrera, H-T Chou, H Stahlberg, Q Ramasse, A Ziegler, G Nicotra, I Arslan, J-C Idrobo, E Stach, A Bleloch | CommonCrawl |
\begin{document}
\title{\large
\begin{abstract}
\setstretch{1.5} Investigating the causal relationship between exposure and the time-to-event outcome is an important topic in biomedical research. Previous literature has discussed the potential issues of using the hazard ratio as a marginal causal effect measure due to its noncollapsibility property. In this paper, we advocate using the restricted mean survival time (RMST) difference as the marginal causal effect measure, which is collapsible and has a simple interpretation as the difference of area under survival curves over a certain time horizon. To address both measured and unmeasured confounding, a matched design with sensitivity analysis is proposed. Matching is used to pair similar treated and untreated subjects together, which is more robust to outcome model misspecification. Our propensity score matched RMST difference estimator is shown to be asymptotically unbiased and the corresponding variance estimator is calculated by accounting for the correlation due to matching. The simulation study also demonstrates that our method has adequate empirical performance and outperforms many competing methods used in practice. To assess the impact of unmeasured confounding, we develop a sensitivity analysis strategy by adapting the E-value approach to matched data. We apply the proposed method to the Atherosclerosis Risk in Communities Study (ARIC) to examine the causal effect of smoking on stroke-free survival.
\textbf{Keywords:} Confounding Bias; Marginal Effect; Noncollapsibility; Propensity Score Matching; Restricted Mean Survival Time; Sensitivity Analysis. \end{abstract}
\section{Introduction} \label{s:intro}
\subsection{Causal Inference for Observational Survival Data} In biomedical studies, time-to-event is a commonly used outcome measure and the statistical analyses for such data are usually referred to as survival analysis. Investigating the causal relationship between exposure and the time-to-event outcome is an important topic, with either randomized trials or observational studies. Causal inference for observational survival data has several challenges. First, since not all subjects can be observed for the full duration of time to event, the survival data suffer from censoring, which is a type of missing data problem. Therefore, standard statistical methods are usually not sufficient to handle both censoring and the missingness of potential outcomes. Second, the hazard ratio is a popular choice for measuring the association of survival outcomes between two groups, for convenience and easy interpretation. However, the hazard ratio is generally not an appropriate marginal causal effect measure due to its noncollapsibility property \citep{greenland1999confounding,tmcop,sjolander2016note}. Other effect measures need to be considered to warrant valid marginal causal interpretation for survival data. Third, confounding is a major challenge in observational studies, which includes both measured and unmeasured confounders.
Propensity score adjustments are popular tools for controlling the observed confounding \citep{cps}. But even with successful adjustment of observed confounding, observational data are still vulnerable to unmeasured confounding. Thus, appropriate sensitivity analysis needs to be developed to assess the impact of hidden bias \citep{rosenbaum2020design}.
The issues of using the hazard ratio as a marginal causal effect measure have been discussed extensively in the literature. \citet{greenland1999confounding} pointed out that the hazard ratio has the noncollapsibility property when the treatment effect is nonzero. \citet{hoh} argued that using the hazard ratio as a treatment effect measure may not have valid causal interpretation even in randomized studies, since the hazard ratio has a built-in selection bias and may change over time. \citet{tmcop} studied the estimation of treatment effect in the presence of confounders and found the amount of confounding due to noncollapsibility in the Cox proportional hazards (PH) model would be very difficult to quantify. \citet{doescox} offered a more theoretical perspective on the conditions under which the hazard has a valid causal interpretation. They suggested that the hazard function $h(t,x,z)$ must satisfy an additive assumption $h(t,x,z)=a(t,z)+b(t,x)$ to yield a causal interpretation, where $a(t,z)$ is a function of survival time $t$ and treatment assignment $z$ and $b(t,x)$ is a function of survival time $t$ and covariates $x$. \citet{ni2021stratified} further illustrated that even under a PH model, the marginal hazard ratio is not a constant, after integrating out covariates. Thus, a valid and simple-to-use causal effect measure for survival outcome is highly desirable.
\subsection{RMST Difference as A Marginal Causal Effect Measure}
The restricted mean survival time (RMST) has been used in randomized clinical studies to evaluate treatment effects \citep{royston2013restricted,trinquart2016comparison}. The RMST difference is more advantageous than the hazard ratio as a marginal effect measure. First, the RMST has an intuitive interpretation as the area under the survival curve over a certain time horizon. Second, the RMST difference is the difference of truncated mean survival time between two groups, which is essentially a mean difference. So it is collapsible, meaning that the marginal and conditional effects are compatible. Third, the treatment effect measured by the RMST difference can be asymptotically unbiasedly estimated without PH assumption, while the conventional Cox model heavily relies on such assumption.
To take advantage of the collapsibility of RMST difference, we can construct RMST regression by including covariates to better control for confounding or increase estimation efficiency. Several methods of regressing RMST on multiple covariates have been developed. \citet{karrison1987restricted} examined the RMST as an index for comparing survival in two groups and proposed to model the hazard with piece-wise exponential models assuming covariates have a multiplicative effect on the hazard. \citet{zucker1998restricted} further simplified the implementation procedure for Karrison's method and provided an extended version to achieve robustness against model misspecification. \citet{andersen2004regression} compared several regression analysis methods of mean survival time and RMST, and they proposed a regression method based on pseudo-observations. \citet{tian2014predicting} developed an RMST regression model with adjustment for baseline covariates. They constructed an estimating equation with the inverse probability of censoring weighting (IPCW) to obtain consistent estimates. \citet{wang2018modeling} models the RMST using generalized estimating equation methods, which allows censoring to depend on both baseline covariates and time-dependent factors.
Though RMST differences have been reported in many randomized clinical studies, there is only limited discussion of using RMST in observational studies, probably due to the challenge of confounding adjustment. Propensity score weighting and stratification methods have been used in the literature, but not propensity score matching. \citet{zhang2012double} derived a double-robust estimator for RMST difference based on the inverse probability of treatment weighted (IPTW) estimating equation with augmentation term. To adjust for confounding factors, they built three working models for survival time, treatment assignment, and censoring, then incorporated them into the augmentation term. They assumed the PH assumption in outcome modeling, which might be violated in practice. \citet{conner2019adjusted} proposed a weighted method to compare the adjusted RMST difference directly. Unlike Zhang and Schaubel's work, Conner et al. estimated the RMST based on the Kaplan-Meier(KM) estimator rather than the Nelson-Aalen estimator. They adjusted the KM estimator with IPTW and derived the adjusted RMST by integrating the IPTW-adjusted KM estimator. \citet{ni2021stratified} proposed a propensity score stratified RMST difference estimation strategy to examine the marginal causal effect with observational survival data, which can combine stratification with further regression adjustment.
\subsection{A Motivating Example: Atherosclerosis Risk in Communities (ARIC)}
In the United States, stroke is a severe disease that causes serious disability for adults and is a leading cause of death \citep{kochanek2014mortality,members2016heart}. Several previous studies have shown that smoking is an important risk factor for stroke \citep{wolf1988cigarette,shinton1989meta}, and even passive smoking could increase the risk of stroke \citep{bonita1999passive}. Although the causal pathway between smoking and stroke is unclear, \citet{shah2010smoking} found that the more people smoke the more likely they were to have a stroke, and people who quit smoking showed a significantly lower risk of stroke, which provides some evidence for the causal relationship between smoking and stroke.
The Atherosclerosis Risk in Communities (ARIC) Study \citep{aric1989atherosclerosis} is a prospective cohort study conducted in four U.S. communities. Four thousand adults aged 45–64 years old were randomly sampled from each of four U.S. communities, and the final dataset contains information of 15,792 individuals. After a baseline examination during 1987 to 1989, subjects were followed up for the development of incident ischemic stroke, and first definite or probable hospitalized stroke. Due to the length of follow-up, not all event times were observed, so the data were subjected to censoring. One primary outcome is the time to first stroke or death (whichever comes first), and a subject is censored if the incidence of stroke or death is not observed by the end of the study. We try to answer a causal question, using this ARIC dataset: how smokers' stroke-free survival would change had they not smoked at baseline. Matched design is a natural choice to address this as it is about the causal effect of those being exposed to smoking, rather than for the entire population.
Existing literature mostly used the Cox PH model to analyze the ARIC data. \citet{kwon2016association} and \citet{ding2019cigarette} studied the association between smoking status and risk of stroke, using the Cox PH model to estimate the HR of smoking status on the risk of stroke. Thus, the estimated effects were interpreted as conditional rather than marginal. Moreover, it is possible that some prognostic factors were not included in the confounder adjusted regression model, which would lead to biased estimates of conditional effects. An analysis with RMST as the effect measure may provide new insight into this research.
In this paper, we propose a propensity score matching based RMST difference estimator and develop a corresponding sensitivity analysis strategy for assessing the impact due to unmeasured confounding. We apply this method to the ARIC study to examine the causal effect of smoking on stroke-free survival. The rest of the paper is organized as follows: In section \ref{s:method}, we set up the notation and assumptions, and describe the proposed RMST estimator with its theoretical properties. In section \ref{s:simu}, we conduct a simulation study to examine the empirical performance of our proposed method under different scenarios and also compare it with several commonly used methods in practice. In section \ref{s:sensit}, we develop a sensitivity analysis strategy by adapting the E-value approach to matched data. In section \ref{s:real}, we present the analysis results of the ARIC data. Section \ref{s:discuss} concludes the paper with some discussions.
\section{Method: Matched RMST Difference Estimation} \label{s:method} \subsection{Notation and Assumptions} We follow the potential outcomes framework proposed by \citet{rubin1974estimating} to define the causal effects. In a two-arm survival analysis study, let $A$ be the treatment assignment indicator (or more generally, the exposure status), such that $A=1$ indicates being exposed to the treatment and $A=0$ indicates being exposed to the control. Let $T^A$ denote the potential event time and $S^A(t)$ denote the corresponding survival function for a subject under treatment $A$.
The following two assumptions are extensions of commonly used assumptions for causal inference in observational studies \citep{imbens2015causal}.
\begin{assumption} \label{sutva} Stable Unit Treatment Value Assumption (SUTVA). The potential survival times for one individual in the population do not vary with the treatment assigned to others. And there are no different versions of the specified treatment level. \end{assumption}
\begin{assumption} \label{strong_ignore}
Treatment assignment is strongly ignorable given covariates $X$, that is $(T^0,T^1)\indep A|X$ and $0<pr(A=1|X)<1$. \end{assumption}
The potential restricted event time is defined as $Z^A=\min (T^A,\tau)$, where $\tau$ is the truncation time point, which is usually pre-specified at the design stage based on clinical relevance and study feasibility. Both $T^A$ and $Z^A$ are subject to censoring by a random variable $C$. We introduce two additional assumptions for survival data.
\begin{assumption} \label{indep_censor}
The censoring random variable $C$ is independent of $T^A$ and $Z^A$ given treatment indicator $A$ and covariates $X$, that is $C\indep T^A |(A,X)$ and $C\indep Z^A |(A,X)$. \end{assumption}
\begin{assumption} \label{tau_assump} The truncation time point is smaller than the largest observed survival time, $\tau<t_{\max}$, where $t_{\max}$ is the largest follow up time (event or censored). \end{assumption}
Assumption \ref{tau_assump} is a technical one to ensure that the pre-specified $\tau$ is clinically meaningful and RMST can be asymptotically unbiasedly estimated.
Let $\delta^A=I(Z^A<C)$ denote the censoring indicator, then the observed restricted time is defined as $Y^A=\min(Z^A,C)=(Z^A)^{\delta^A}C^{(1-\delta^A)}$. For a subject under treatment $A$, the potential outcome of restricted mean survival time is defined as $\mu^A(\tau)=E(Z^A)=\int^\tau_0S^A(t)dt$, then the average treatment effect (ATE) on RMST, denoted by $\Delta_{ATE}$, can be defined as $$\Delta_{ATE}=\mu^1(\tau)-\mu^0(\tau)=E(Z^1)-E(Z^0)=\int^\tau_0[S^1(t)-S^0(t)]dt.$$
Since $Z^A=\min (T^A, \tau)$ and $\tau$ is a fixed constant, $(T^0,T^1)\indep A|X$ implies $(Z^{0},Z^{1})\indep\ A|X$. Following Theorem 3 in \citet{cps}, we can establish the strongly ignorability based on propensity score $e(X)=P(A=1|X)$ for survival outcomes in proposition \ref{note_prop} (proof provided in Web Appendix A).
\begin{proposition} \label{note_prop}
Given assumptions \ref{sutva}-\ref{strong_ignore}, we have $(T^0,T^1)\indep A|e(X)$, which further implies $(Z^0,Z^1)\indep A|e(X)$. \end{proposition}
\subsection{Matched RMST Difference Estimator}
In randomized trials, the marginal causal effect of treatment on RMST can be asymptotically unbiasedly estimated \citep{fleming2011counting} by direct contrast of group-specific RMST estimates since confounding effects are eliminated by design. In observational studies, however, additional adjustments are needed for confounding control. Propensity score based approaches are popular for this purpose, which may take the form of matching, stratification, or weighting \citep{cps, bang2005doubly}. Among different propensity score adjustment strategies, matching is a design tool that selects comparable control units to match with treated units and it often results in more robust causal effect estimates as it does not rely on outcome model specification. Usually, matching uses all treated and a subset of control units, so it estimates the average treatment effect on the treated (ATT) \citep{imbens2015causal}.
Our proposed propensity score matched RMST estimation includes the following steps: \\ (1) \textit{Propensity Score Estimation} \\ The propensity score is defined as the conditional probability of treatment given a vector of observed covariates \citep{cps}. We estimate the propensity score by fitting a logistic regression on $A$ with $X$, though other estimation options, either parametric or nonparametric, are also available \citep{mccaffrey2004propensity,westreich2010propensity}.\\ (2) \textit{Propensity Score Matching} \\ We use the optimal matching algorithm by \citet{Roptmatch} to create pair matches without replacement based on the estimated propensity score, the unmatched controls will be removed from the matched sample. Matching quality is assessed by checking the post-matching covariate balance. Any substantial covariate imbalance would lead to a recalibration of the propensity score model. We will proceed to the next step only after a satisfactory balance is achieved. \\ (3)\textit{Treatment Effect Estimation}\\ Suppose we obtain $n$ pairs of data through matching, where each pair contains exactly one treated and one control subject. We estimate the RMST, $\mu(\tau)$, by $\hat{\mu}(\tau)=\int^\tau_0\hat{S}(t)d(t)$, where $\hat{S}(t)$ is estimated by the nonparametric KM method. Let $\hat{S}^0(t)$ and $\hat{S}^1(t)$ denote the KM estimates of survival function for control and treated groups in the matched sample, respectively. Based on the matched sample, our estimator for averaged treatment effect on the treated (ATT) is $$\hat{\Delta}_{ATT}=\hat{\mu}^1(\tau)-\hat{\mu}^0(\tau)=\int^\tau_0[\hat{S}^1(t)-\hat{S}^0(t)]dt.$$
The following two propositions show that the matched RMST difference estimator is asymptotically unbiased (both proofs are provided in Web Appendix A).
\begin{proposition} \label{RMST_unbias} Given assumptions \ref{sutva}-\ref{tau_assump}, the RMST estimator based on KM method given propensity score $e(X)$ and treatment group $A$, denoted as $\hat{\mu}_{e(X),A}$, is an asymptotically unbiased estimator for $\mu_{e(X),A}$ given $\tau<t_{\max}$. \end{proposition}
\begin{proposition} \label{match_rmst_unbias}
Given assumptions \ref{sutva}-\ref{tau_assump}, $\hat{\Delta}_{ATT}$ is asymptotically unbiased. \end{proposition}
\subsection{Variance Estimation}
The matching process may introduce correlation between the two subjects in the same pair, as they are matched on similar propensity scores. Therefore, the variance calculation of $\hat{\Delta}_{ATT}$ needs to account for such correlation: $$var(\hat{\Delta}_{ATT})=var[\int^\tau_0\hat{S}^0(t_0)dt_0]+var[\int^\tau_0\hat{S}^1(t_1)dt_1]-2cov[\int^\tau_0\hat{S}^0(t_0)dt_0,\int^\tau_0\hat{S}^1(t_1)dt_1].$$
The overall variance has two components, the marginal variance of RMST estimates and their covariance. For two dependent event times with independent censoring and no competing risk, \citet{murray2000variance} provided closed-form asymptotic covariance formulas for KM survival estimates and corresponding RMST estimates. To address the dependence structure introduced in the matching process, we adapt their formulas to compute the covariance between control and treated group RMST estimates in the matched sample.
Specifically, let $T_0$ be the event time for a subject from the control group with marginal hazard function $h_0(\cdot)$, and $T_1$ be the event time for a subject from the treatment group with marginal hazard function $h_1(\cdot)$, then the event times for a matched pair of control and treated can be denoted as $(T_0, T_1)$. Let $C$ be the censoring variable, then the observed time can be denoted as $\tilde{T}_0=\min (T_0, C)$ for control group with censoring indicator $\delta_0=I(T_0<C)$, and $\tilde{T}_1=\min (T_1, C)$ for treated group with censoring indicator $\delta_1=I(T_1<C)$. Then, the joint hazard function is
$h_{ij}(u,v)=\underset{\Delta u,\Delta v\rightarrow 0}{\lim}\frac{1}{\Delta u\Delta v}P(u\leq \tilde{T}_i<u+\Delta u, v\leq \tilde{T}_j<v+\Delta v,\delta_i=1,\delta_j=1|\tilde{T}_i\geq u, \tilde{T}_j\geq v)$ where $i,j\in\{0,1\}$, and the conditional hazard function is $h_{i|j}(u|v)=\underset{\Delta u\rightarrow 0}{\lim}\frac{1}{\Delta u}P(u\leq \tilde{T}_i<u+\Delta u,\delta_i=1|\tilde{T}_i\geq u, \tilde{T}_j\geq v)$ where $i,j\in\{0,1\}$. Then, the covariance between two RMSTs can be computed as \begin{align} &cov[\int^\tau_0\hat{S}^0(t_0)dt_0,\int^\tau_0\hat{S}^1(t_1)dt_1] \nonumber \\ =&\frac{1}{n}\int^\tau_0\int^\tau_0\hat{S}^0(t_0)\hat{S}^1(t_1)\int^{t_0}_0\int^{t_1}_0G_{01}(u,v)dvdudt_0dt_1 \nonumber \\ =&\frac{1}{n}\int^\tau_0\int^\tau_0\left[\int^\tau_v\hat{S}^0(t)dt\right]\left[\int^\tau_u\hat{S}^1(t)dt\right]G_{01}(u,v)dvdu \nonumber \end{align} where
$G_{01}(u,v)=\frac{P(\tilde{T}_0\geq u, \tilde{T}_1\geq v)}{P(\tilde{T}_0\geq u)P(\tilde{T}_1\geq v)}[h_{01}(u,v)-h_{0|1}(u|v)h_1(v)-h_{1|0}(v|u)h_0(u)+h_0(u)h_1(v)]$. Details about the computation of function $G_{01}(u,v)$ are included in Web Appendix C.\\
For the marginal variances, two methods may be considered:
\begin{enumerate}
\item Murray's Method: the above covariance formulas can be used to compute the marginal variance, since the marginal variance of RMST could be written as the covariance with itself, that is $var(\int^\tau_0\hat{S}(t)dt)=cov[\int^\tau_0\hat{S}(t)dt,\int^\tau_0\hat{S}(t)dt]$.
\item Hosmer's Method: we may also consider the computation method introduced in \citet{hosmer2011applied}. Let $t_1<t_2<\cdots<t_D$ represent distinct event times. For each $k=1,\cdots,D$, let $Y_k$ be the number of surviving units just prior to event time $t_k$, and let $d_k$ be the number of events at $t_k$. Let $\hat{S}(t_k)=\overset{k}{\underset{l=1}{\prod}}(1-\frac{d_l}{Y_l})$ denotes the KM estimate of the survival function at event time $t_k$, and let $N_\tau$ be the number of $t_k$ values that are less than truncation time point $\tau$, then the RMST is estimated by $$\int^\tau_0\hat{S}(t)dt=\overset{N_\tau}{\underset{k=1}{\sum}}\hat{S}(t_{k-1})(t_k-t_{k-1})+\hat{S}(t_{N_\tau})(\tau-t_{N_\tau})$$ and the marginal variance of RMST can be estimated as $$var(\int^\tau_0\hat{S}(t)dt)=\frac{m}{m-1}\overset{N_\tau}{\underset{k=1}{\sum}}\frac{d_kA^2_k}{Y_k(Y_k-d_k)}$$ where $A_k=\int^\tau_{t_k}\hat{S}(t)dt=\overset{N_\tau}{\underset{l=k}{\sum}}\hat{S}(t_l)(t_{l+1}-t_l)+\hat{S}(t_{N_\tau})(\tau-t_{N_\tau})$ and $m=\overset{N_\tau}{\underset{l=1}{\sum}}d_l$.
\end{enumerate} In our simulation studies, we present the variance estimates under Murray's method since the results from these two methods turn out to be very close.
\section{Simulation Studies} \label{s:simu}
\subsection{Data Generation} To assess the empirical performance of the proposed method, we simulate an observational dataset with known confounders. Several existing methods for causal inference with survival outcomes are compared.
We generate ten independent baseline covariates denoted by $X_1$ to $X_{10}$. Among them, $X_1, X_3,\cdots, X_9$ are five binary covariates following Bernoulli distribution with parameters 0.2, 0.4, 0.6, 0.8, 0.5, respectively, and $X_2,X_4,\cdots, X_{10}$ are five continuous covariates following standard normal distribution. We then generate potential survival time $T^1$ as the outcome under treatment and potential survival time $T^0$ as the outcome under control from Weibull distribution \citep{bender2005generating}. Specifically, we simulate a uniform random variable $U$ on [0,1] then generate the potential survival time as below, $$T^j=(-\frac{log(U)}{\lambda_{0j}\exp(\beta_Aj+X_1+1.2X_4+1.4X_6+1.6X_7+1.6X_8+1.4X_9+1.2X_{10})})^{\frac{1}{\nu_j}}$$ where $A$ is the treatment indicator and $\beta_A$ is the conditional multiplicative treatment effect on the hazard function given covariates, and $\nu_j$ and $\lambda_{0j}$ are the shape and baseline scale parameters of Weibull distribution for treatment group $j$, respectively. When $\nu_0=\nu_1$, we have proportional hazards model, otherwise the model is non-proportional hazards. The treatment indicator $A$ is generated from Bernoulli distribution with $P(A=1)$ defined by the logistic model $\text{logit}(P(A=1))=-1.95+\log(1.2)X_1+\log(1.1)X_2+\log(1.4)X_3+\log(1.2)X_4+\log(1.6)X_5+\log(1.3)X_6+\log(1.8)X_7$. Thus, $X_1, X_4, X_6$ and $X_7$ are true confounders. This setup allows about 20$\%$ of the population to be exposed to treatment.
The censoring variable $C$ is generated from an exponential distribution with rate parameter $\theta$, where $\theta=\gamma\exp(0.2X_4+0.1X_7)$ and $\gamma$ is the baseline rate parameter. The true event time is $T=T^0(1-A)+T^1A$.
Let $\tau$ be the pre-specified truncation time point, we generate the restricted event time $Z=\min(T,\tau)$ and the observed restricted time $Y=\min (Z,C)=\min (T,C,\tau)$. The restricted event time $Z$ is censored if the observed time $C<Z$ with censoring status $\delta_Z=I(Z<C)$, otherwise it is non-censored.
We simulate 500 datasets of sample size 2500 for each scenario and set the truncation time point $\tau$ to 100. The true RMST difference is determined by calculating the empirical difference between the potential RMSTs under treated and control conditions, and we compute both ATT and ATE versions of true RMST difference to serve as benchmarks for different methods as appropriate. In the $j^{th}$ simulated dataset, we calculate $\Delta_j=\sum_{i=1}^{n}\frac{Z_i^1-Z_i^0}{n}$, where $Z_i^A=\min(T_i^A,\tau)$ is the potential restricted event time for the $i$th individual and $n$ is the sample size of the treated group (for ATT) or the entire sample (for ATE). Then, the true marginal effect on RMST is calculated as $\Delta_0=\sum_{j}^{500}\frac{\Delta_j}{500}$.
Both proportional hazards (PH) and non-proportional hazards (NP) settings are examined. Under both settings, we set $\beta_A$ to five different values: $0, -0.4, -0.8, -1.2,-2$. For each treatment effect value, we also consider four different levels of
censoring rates (CR), which are $0\%, 20\%, 40\%, 60\%$. Detailed parameter setup for PH and NP scenarios in observational studies are summarized in Table \ref{tab:obs_setup}.
\subsection{Estimation Strategies} The proposed method is compared with three existing estimation strategies: \begin{enumerate} \item \textit{Propensity score matched RMST estimation.} This is our proposed propensity score matched estimation as described in the previous section, and the propensity score is estimated from the true propensity score model. The estimated treatment effect is compared to the ATT version of the true RMST difference in our simulations. \item \textit{Conner's IPTW RMST estimation.}\\ This method is proposed by \citet{conner2019adjusted}, and they estimated the RMST based on inverse probability treatment weighting (IPTW) adjusted Kaplan-Meier estimator. In our simulation, we use the ATT version of weight to adjust for observed confounding, so it is compared to the true ATT RMST difference. The propensity score is estimated from the true propensity score model. \item \textit{RMST regression.}\\ This method is proposed by \citet{tian2014predicting}, which uses the IPCW estimating equation with identity link function to estimate treatment effect on RMST with adjustment for covariates. The estimated treatment effect is compared to ATE version of the true RMST difference. We consider four different outcome models in the RMST regression : (1) outcome model using the treatment indicator only; (2) outcome model using the true covariate set; (3) outcome model using all covariates; (4) outcome model using a wrong covariate set. Due to space limitations, only results of RMST regression with true covariates are summarized in the following section, which has the best performance among the four models. An important caveat is that the RMST regression model with the true covariate set does not represent the true outcome model since the data are generated based on a hazard model. \item \textit{Inverse Probability Treatment Weighting (IPTW) Cox regression.}\\ This method estimates $\beta_A$. The propensity score is estimated from the true propensity score model. We use the ATT weight to fit a weighted Cox regression model and regard $\beta_A$ as the truth to calculate the bias and coverage probabilities since there is no single value true marginal hazard ratio. We consider four different outcome models in the IPTW Cox regression : (1) outcome model using treatment indicator only; (2) outcome model using the true covariate set; (3) outcome model using all covariates; (4) outcome model using a wrong covariate set. Due to space limitations, only results of IPTW Cox regression with the true covariate set are summarized in the following section, which has the best performance among the four models. We understand that the results here are not directly comparable to the first three methods, as they are based on different effect measures. Due to the high popularity of the IPTW Cox model in practice, however, we think there is some value in presenting the results as a reference. \end{enumerate}
\subsection{Performance Assessment} We summarize treatment effect estimates from 500 Monte Carlo iterations into four measures: (1) percentage bias (Bias $\%$), which is the bias divided by the true value for nonzero treatment effect scenarios. For the zero treatment effect scenario, we just report the bias. The bias is computed as the average of 500 treatment effect estimates minus the truth; (2) coverage probability (CP), which is the proportion of 500 95$\%$ confidence intervals that cover the truth; (3) model-based standard error (SEM), which is the average of the 500 estimated standard errors from the model-based formula; (4) empirical standard error (SEE), which is the standard error of the 500 point estimates of treatment effect.
\subsection{Results}
Simulation results under PH setting are summarized in Table \ref{tab:obsph-1}. The proposed matched RMST method generates unbiased estimates of the target parameters under most scenarios, and the coverage probabilities are around 95$\%$. For a small effect size ($\beta_A=-0.4$), the bias is a bit large for a high censoring rate. Conner's method has a similar performance, with moderately larger biases. Averaging across all scenarios, bias from Conner's method is 65\% higher than our method. The results of the IPTW Cox model are mostly good since we use the correct outcome model. The coverage probability may be a bit lower than the nominal level, sometimes, which may be due to the underestimated standard error. The RMST regression method shows a relatively large percentage bias and lower coverage probability, especially under scenarios with large treatment effects. This is likely due to the incorrect covariate functional form specification in the model even though we include the right covariate set.
Simulation results under NP setting are summarized in Table \ref{tab:obsnp-1}. Both our matched RMST method and Conner's method have similar performance (with the latter having more bias) as under the PH setting since these methods do not rely on the PH assumption. The RMST regression method performs somewhat worse, with a bigger bias and much lower than ideal coverage probabilities. Because the PH assumption does not hold here, the IPTW Cox model completely misses the target with large bias and very small coverage probabilities.
\section{Sensitivity Analysis Based on Matched Design} \label{s:sensit}
\subsection{An Overview of E-value}
Propensity score adjustment can only control for observed confounding. Unmeasured confounding is likely to be present in observational studies since researchers have no control over the treatment assignment. Thus, sensitivity analysis is important to assess the impact of hidden bias.
Ding and VanderWeele \citep{ding2016sensitivity,vanderweele2017sensitivity} developed a new sensitivity analysis strategy, known as the E-value method. It assumes a hypothetical unmeasured confounder, $U$, and provides a lower bound of the strength of association on the risk ratio scale that $U$ would have to have with both the exposure and the outcome, to explain away the observed association. Below is a brief review of the conventional E-value method to set the stage for our sensitivity analysis of RMST difference.
Let $E$ denote a binary exposure and $D$ denote a binary outcome, $e(X)$ is a vector of measured confounders and $U$ is a binary unmeasured confounder with levels $k=0,1$. The observed relative risk of exposure $E$ on the outcome $D$ within stratum of $e(X)=e(x)$ is
$$\text{RR}_{ED|e(x)}^{obs}=\frac{P(D=1|E=1, e(X)=e(x))}{P(D=1|E=0, e(X)=e(x))}.$$ Then the relative risk of exposure on level $k$ of the unmeasured confounder $U$ within stratum of $e(X)=e(x)$ is
$$\text{RR}_{EU,k|e(x)}=\frac{P(U=k|E=1,e(X)=e(x))}{P(U=k|E=0, e(X)=e(x))}.$$
Since $U$ is not observed, to facilitate the analysis, we take the maximal relative risk of $E$ on $U$ within stratum $e(X)=e(x)$, denoted as $RR_{EU|e(x)}=\underset{k}{\max}RR_{EU,k|e(x)}$. Similarly, we can define an upper bound for the relative risk between $U$ and $D$ as $\text{RR}_{UD|e(x)}=\max(\text{RR}_{UD|E=0,e(x)},\text{RR}_{UD|E=1,e(x)})$, where $\text{RR}_{UD|E,e(x)}$ is an upper bound of the relative risk between $U$ and $D$ in exposed or unexposed group respectively, within stratum $e(X)=e(x)$. If $e(X)$ and $U$ are sufficient to control for all confounding effects, the true causal relative risk is
$$\text{RR}^{true}_{ED|e(x)}=\frac{\sum^{1}_{k=0}P(D=1|E=1, e(X)=e(x), U=k)P(U=k|e(X)=e(x))}{\sum^{1}_{k=0}P(D=1|E=0, e(X)=e(x), U=k)P(U=k|e(X)=e(x))}.$$
The relative risk pair $(\text{RR}_{EU|e(x)},\text{RR}_{UD|e(x)})$ are used to measure the strength of confounding between the exposure $E$ and the outcome $D$ induced by the confounder $U$ within the stratum of $e(X)=e(x)$. Even though we cannot estimate the true relative risk, its ratio with the observed relative risk is bounded by the following quantity, which is a function of the sensitivity parameters $\text{RR}_{EU|e(x)}$ and $\text{RR}_{UD|e(x)}$.
$$\frac{\text{RR}^{obs}_{ED|e(x)}}{\text{RR}^{true}_{ED|e(x)}}\leq\frac{\text{RR}_{EU|e(x)}\times \text{RR}_{UD|e(x)}}{\text{RR}_{EU|e(x)}+\text{RR}_{UD|e(x)}-1}$$
For given values of $\text{RR}_{EU|e(x)}$ and $\text{RR}_{UD|e(x)}$, we can identify a range of possible values for the true relative risk. If the range covers one, the observed significant association would be explained away by the presence of unmeasured confounding at the given magnitude.
\subsection{Sensitivity Analysis on RMST difference with matched data} \label{rmst_sa}
This E-value method can be adapted to conduct sensitivity analysis for our RMST difference estimator in matched design. There are a series of propositions to justify the theoretical validity of using the E-value for the RMST difference estimator. In the interest of space, we just illustrate the main idea in this subsection and leave the propositions and their detailed proofs in Web Appendix B.
Let $A$ be the treatment indicator and $Z=\min (T,\tau)$ be the RMST outcome, where $T$ is the event time and $\tau$ is the truncation time point. Let $e(X)$ be the propensity score and $U$ be a binary unmeasured confounder with levels $k=0,1$.
The relative risk of treatment $A$ on level $k$ of the unmeasured confounder $U$ with a given propensity score value $e(X)=e(x)$ is defined as
$$RR_{AU,k|e(x)}=\frac{P(U=k|A=1,e(X)=e(x))}{P(U=k|A=0, e(X)=e(x))}.$$ The maximal relative risk of $A$ on $U$ with $e(X)=e(x)$ is
$RR_{AU|e(x)}=\underset{k}{\max}RR_{AU,k|e(x)}$. We define the expectations of the RMST outcome $Z$ given $U=u$ and $e(X)=e(x)$ with and without treatment as
$$r_1(u)=E(Z|A=1,U=u, e(X)=e(x)),$$
$$r_0(u)=E(Z|A=0,U=u, e(X)=e(x)).$$
Then, the mean ratios of $U$ on $Z$ with and without treatment with $e(X)=e(x)$ are defined as
$$MR_{UZ|A=1,e(X)=e(x)}=\frac{\underset{u}{\max}r_1(u)}{\underset{u}{\min}r_1(u)},\ MR_{UZ|A=0,e(X)=e(x)}=\frac{\underset{u}{\max}r_0(u)}{\underset{u}{\min}r_0(u)},$$
$$MR_{UZ|e(X)=e(x)}=\max (MR_{UZ|A=1,e(X)=e(x)},MR_{UZ|A=0,e(X)=e(x)}).$$
As shown in proposition 4 in Web Appendix B, both unmeasured confounder parameters $RR_{AU|e(X)=e(x)}$ and $MR_{UZ|e(X)=e(x)}$ are no less than 1. Then we can identify the bounding factor as \begin{equation}\label{eq:bf}
BF_{U|e(X)=e(x)}=\frac{RR_{AU|e(X)=e(x)}\times MR_{UZ|e(X)=e(x)}}{RR_{AU|e(X)=e(x)}+MR_{UZ|e(X)=e(x)}-1} \end{equation}
where $(RR_{AU|e(X)=e(x)},MR_{UZ|e(X)=e(x)})$ are prespecified sensitivity analysis parameters. Then, we take the maximum of bounding factors across all propensity score values, defined as $BF_U^*=\underset{e(x)}{\max}(BF_{U|e(X)=e(x)})$, and the corresponding sensitivity analysis parameters in $BF_U^*$ are denoted as $(RR_{AU},MR_{UZ})$. Let $ACE_{AZ}^{true}$ denotes the average causal effect. When treatment effect is positive, we have \begin{equation}\label{eq:sa_posi}
ACE_{AZ}^{true}\geq \frac{1}{2}(1+\frac{1}{BF_U^*})E(Z|A=1)-\frac{1}{2}(1+BF_U^*)E(Z|A=0). \end{equation} When treatment effect is negative we have \begin{equation}\label{eq:sa_nega}
ACE_{AZ}^{true}\leq \frac{1}{2}(1+BF_U^*)E(Z|A=1)-\frac{1}{2}(1+\frac{1}{BF_U^*})E(Z|A=0). \end{equation}
\subsection{Interpreting the Sensitivity Analysis}
For an unmeasured confounding with prespecified magnitude of $(RR_{AU},MR_{UZ})$, the bounding factor $BF_U^*$ can be computed by equation (\ref{eq:bf}). For a positive treatment effect, the lower bound of the treatment effect can be computed by equation (\ref{eq:sa_posi}). A positive lower bound indicates that there is still a positive treatment effect with an unmeasured confounding effect of magnitude $(RR_{AU}, MR_{UZ})$. A non-positive lower bound indicates that the positive treatment effect could be explained away by the unmeasured confounding of magnitude $(RR_{AU}, MR_{UZ})$. For a negative treatment effect, the upper bound of the treatment effect can be computed by equation (\ref{eq:sa_nega}), and similar interpretations can be made. A negative upper bound indicates that there is still a negative treatment effect with an unmeasured confounding effect of magnitude $(RR_{AU}, MR_{UZ})$. A non-negative upper bound indicates that the negative treatment effect could be explained away by the unmeasured confounding of magnitude $(RR_{AU}, MR_{UZ})$.
\section{Real Data Example} \label{s:real}
In this section, we apply our proposed method to the ARIC data \citep{aric1989atherosclerosis} to examine the causal effect of baseline smoking on stroke-free survival. Incident ischemic stroke events or death, the primary outcome, are identified through December 31, 2011. After excluding a small portion of subjects with missing values in the variables of interest, the total sample size used in the analysis is 14,549. The event time is defined as the follow-up time (in months) for the first incident stroke or death, whichever comes first, and a subject is censored if neither incident stroke nor death is observed during the study. There are 5345 events, corresponding to a 63.3$\%$ censoring rate. Given the length of follow-up, we choose 240 months as the truncation time $\tau$ for the RMST calculation. Exposure is defined as the smoking status at baseline. There are 3,832 (26.3$\%$) current smokers at baseline. Eight important baseline covariates are included in the propensity score model: race (black, white), gender (male, female), age (44-66 yrs old), BMI (14.2-65.9), diabetes (1=yes, 0=no) , HDL (10-163 mg/dL), LDL (0-504.6 mg/dL), and hypertension (1=yes, 0=no). Table \ref{tab:c1} summarizes these variables by baseline smoking status.
We first fit a logistic regression model on baseline smoking status using the eight covariates to estimate the propensity score. Then, we conduct a 1-1 optimal pair matching without replacement for all subjects which results in 3832 pairs and unmatched nonsmokers are removed from the matched sample. The covariates balance is measured by the standardized mean difference, and Figure \ref{sa_balance} shows the covariates balance of ARIC data before and after propensity score matching, which indicates our matching achieves very good covariates balance.
For comparison purposes, the analysis results of the proposed method and the IPTW Cox regression method are both included. All results are summarized in Table \ref{tab:senana-2}. Both methods show significant evidence of a harmful effect of smoking on the risk of incident ischemic stroke or death. This conclusion agrees with previous findings in the literature. The matched RMST analysis suggests an average reduction of 22.3 stroke-free survival months for baseline smokers had they not smoked at the baseline. The IPTW Cox regression measures the treatment effect on the hazard ratio scale which is not directly comparable to RMST differences. The estimated HR of 2.2 implies that smoking increases the hazard of incident ischemic stroke or death.
All the above analyses assume ignorable treatment assignment. However, for such a large observational study, unmeasured confounding is likely to be present, especially given that we are only able to control a small number of factors. Therefore, it is important to assess how the observed causal effect may change in the presence of hidden bias. A sensitivity analysis, as described in section \ref{rmst_sa}, is carried out for different possible impacts of $U$ on the exposure and the outcome. Since the observed RMST difference is negative, we use equation (\ref{eq:sa_nega}) to conduct the sensitivity analysis, where we calculate the upper bound of the true causal RMST effect for different combinations of $RR_{AU}$ and $MR_{UZ}$. The results are depicted in the contour plot in Figure \ref{sa_plot}. The solid curve in the middle of the graph represents the contour where the upper bound is zero. The darker color region to the upper-right of the zero-curve indicates a positive upper bound of the true effect, and the lighter color region to the lower-left of the zero-curve indicates a negative upper bound of the true effect. For moderate deviations from the ignorability assumption ($RR_{AU}<1.47$ and $MR_{UZ}<1.47$), a harmful effect still holds, as the upper bound is below the solid zero-curve. For moderate-to-large deviations, the upper bound of the treatment effect may exceed zero, indicating a possibility of a null effect. For example, at (1.5, 1.5) the upper bound of the estimated treatment effect is 2.04, which indicates that the harmful treatment effect could be totally explained away by the unmeasured confounding of magnitude $(RR_{AU}, MR_{UZ})=(1.5, 1.5)$. Overall, our sensitivity analysis indicates that the observed significant causal effect is moderately robust to hidden bias.
\section{Discussion} \label{s:discuss}
In this paper, we adopt the RMST difference as a marginal causal effect measure for survival data, since it is collapsible and has an easy interpretation. We develop a matching-based RMST difference estimator that is asymptotically unbiased and does not rely on the PH assumption. But this does not rule out the use of hazard in causal analysis with survival data. As pointed out by \citet{doescox}, the hazard function $h(t,x,z)$ may have a valid causal interpretation, if it satisfies some additive structural constraint.
One limitation of our work is that the proposed nonparametric estimator may not be easily extended to more complex matching designs, such as 1-k or full matching designs. This is because we need to compute the covariance to account for the correlation in matched sets. But the covariance calculation relies on the assumption of equal sample sizes in both groups \citep{murray2000variance}. Therefore the covariance formula can not be applied directly to other matching designs. One strategy to relax this limitation is to consider fitting a parametric RMST regression model after matching. This could be more advantageous if we have a good idea about the outcome model specification, as it may correct residual confounding bias not captured by matching. This adds more flexibility to post-matching inference, as it can lead to more robust or efficient semiparametric strategies by combining matching with regression models \citep{Rubin1973theuse}. It also makes our method more attractive in practice than Conner's method as the latter solely relies on KM estimation of survival functions and cannot include regression models.
\section{Acknowledgments}
This work was partially supported by grant DMS-2015552 from National Science Foundation.The Atherosclerosis Risk in Communities study has been funded in whole or in part with Federal funds from the National Heart, Lung, and Blood Institute, National Institutes of Health, Department of Health and Human Services, under Contract nos. (HHSN268201700001I, HHSN268201700002I, HHSN268201700003I, HHSN268201700005I, HHSN268201700004I). The authors thank the staff and participants of the ARIC study for their important contributions.
\section{Figures and Tables }
\begin{figure}
\caption{Covariates Balance Checking}
\label{sa_balance}
\end{figure}
\begin{figure}
\caption{Contour plot of the upper bound of estimated treatment effect. The solid curve represents value 0.}
\label{sa_plot}
\end{figure}
\begin{table}[htbp]
\centering
\caption{Parameter Setup for Observational Studies Simulation Scenarios}
\begin{tabular}{ccccccc}
\toprule
\multicolumn{2}{c}{PH Scenario Setup} & \multirow{2}[2]{*}{$\beta_A$} & \multicolumn{4}{c}{Censoring Parameter $\gamma$} \\
($\nu_0$,$\lambda_0$) & ($\nu_1$,$\lambda_1$) & & 0\% & 20\% & 40\% & 60\% \\
\midrule
(1, exp(-6)) & (1, exp(-6)) & 0 & 1.00E-08 & 0.0051 & 0.0142 & 0.0467 \\
(1, exp(-6)) & (1, exp(-6)) & -0.4 & 1.00E-08 & 0.00462 & 0.0124 & 0.0345 \\
(1, exp(-6)) & (1, exp(-6)) & -0.8 & 1.00E-08 & 0.00421 & 0.011 & 0.0272 \\
(1, exp(-6)) & (1, exp(-6)) & -1.2 & 1.00E-08 & 0.00387 & 0.00992 & 0.0226 \\
(1, exp(-6)) & (1, exp(-6)) & -2 & 1.00E-08 & 0.00322 & 0.00793 & 0.0165 \\
\midrule
\multicolumn{2}{c}{NP Scenario Setup} & \multirow{2}[2]{*}{$\beta_A$} & \multicolumn{4}{c}{Censoring Parameter $\gamma$} \\
($\nu_0$,$\lambda_0$) & ($\nu_1$,$\lambda_1$) & & 0\% & 20\% & 40\% & 60\% \\
\midrule
(1, exp(-6)) & (1, exp(-6)) & 0 & 1.00E-08 & 0.0049 & 0.01365 & 0.04351 \\
(1, exp(-6)) & (1.5, 1.23E-04) & -0.4 & 1.00E-08 & 0.00346 & 0.00857 & 0.0179 \\
(1, exp(-6)) & (1.5, 1.23E-04) & -0.8 & 1.00E-08 & 0.00323 & 0.00792 & 0.01602 \\
(1, exp(-6)) & (1.5, 1.23E-04) & -1.2 & 1.00E-08 & 0.00306 & 0.00742 & 0.0146 \\
(1, exp(-6)) & (1.5, 1.23E-04) & -2 & 1.00E-08 & 0.00278 & 0.00659 & 0.0126 \\
\bottomrule
\end{tabular}
\label{tab:obs_setup} \end{table}
\begin{sidewaystable} \small \caption{Simulation Results under PH Scenario. Under zero treatment effect scenarios, bias is reported instead of percentage bias.} \label{tab:obsph-1} \begin{adjustbox}{scale=0.95,center} \begin{tabular}{cccccccccccccccccc} \hline \multicolumn{2}{c}{\textbf{Scenario}} &
\textbf{Bias\%} &
\textbf{CP} &
\textbf{SEM} &
\textbf{SEE} &
\textbf{Bias\%} &
\textbf{CP} &
\textbf{SEM} &
\textbf{SEE} &
\textbf{Bias\%} &
\textbf{CP} &
\textbf{SEM} &
\textbf{SEE} &
\textbf{Bias\%} &
\textbf{CP} &
\textbf{SEM} &
\textbf{SEE} \\ \hline \textbf{$\beta_A$} &
\textbf{CR} &
\multicolumn{4}{c}{Matched RMST (Murray)} &
\multicolumn{4}{c}{RMST Regression} &
\multicolumn{4}{c}{Conner's IPTW RMST} &
\multicolumn{4}{c}{IPTW Cox (HR)} \\ \hline \multirow{4}{*}{0} & 0 & 0.032 & 0.964 & 2.795 & 2.611 & -0.020 & 0.932 & 1.322 & 1.402 & 0.076 & 0.962 & 2.284 & 2.127 & 0.005 & 0.938 & 0.052 & 0.052 \\
& 0.2 & 0.140 & 0.954 & 2.884 & 2.686 & 0.124 & 0.938 & 1.478 & 1.542 & 0.165 & 0.956 & 2.355 & 2.218 & 0.005 & 0.924 & 0.067 & 0.070 \\
& 0.4 & 0.252 & 0.958 & 3.071 & 2.908 & 0.333 & 0.908 & 1.906 & 2.033 & 0.258 & 0.956 & 2.505 & 2.420 & 0.008 & 0.936 & 0.074 & 0.077 \\
& 0.6 & 0.288 & 0.953 & 4.079 & 4.023 & 0.877 & 0.772 & 4.450 & 6.684 & 0.344 & 0.947 & 3.369 & 3.439 & 0.013 & 0.928 & 0.085 & 0.088 \\ \hline \multirow{4}{*}{-0.4} & 0 & 0.724\% & 0.970 & 2.795 & 2.588 & 2.746\% & 0.944 & 1.326 & 1.369 & 1.581\% & 0.958 & 2.284 & 2.101 & -1.069\% & 0.944 & 0.053 & 0.052 \\
& 0.2 & 3.061\% & 0.958 & 2.875 & 2.646 & 5.384\% & 0.938 & 1.475 & 1.487 & 3.393\% & 0.958 & 2.348 & 2.173 & -1.055\% & 0.936 & 0.069 & 0.072 \\
& 0.4 & 5.198\% & 0.962 & 3.032 & 2.826 & 8.447\% & 0.934 & 1.846 & 1.906 & 5.198\% & 0.958 & 2.473 & 2.342 & -1.430\% & 0.946 & 0.076 & 0.077 \\
& 0.6 & 4.835\% & 0.958 & 3.647 & 3.564 & 18.611\% & 0.912 & 3.893 & 4.165 & 7.658\% & 0.958 & 2.980 & 2.908 & -2.195\% & 0.944 & 0.086 & 0.087 \\ \hline
\multirow{4}{*}{-0.8} & 0 & 0.364\% & 0.970 & 2.785 & 2.570 & 3.819\% & 0.938 & 1.329 & 1.330 & 0.794\% & 0.962 & 2.271 & 2.078 & -0.453\% & 0.944 & 0.055 & 0.053 \\
& 0.2 & 1.456\% & 0.962 & 2.859 & 2.657 & 4.645\% & 0.942 & 1.475 & 1.460 & 1.703\% & 0.966 & 2.329 & 2.163 & -0.420\% & 0.938 & 0.072 & 0.074 \\
& 0.4 & 2.352\% & 0.966 & 2.991 & 2.721 & 6.364\% & 0.938 & 1.805 & 1.820 & 2.359\% & 0.966 & 2.435 & 2.236 & -0.540\% & 0.940 & 0.080 & 0.080 \\
& 0.6 & 1.972\% & 0.960 & 3.404 & 3.297 & 4.613\% & 0.942 & 3.236 & 3.375 & 3.756\% & 0.958 & 2.771 & 2.685 & -0.811\% & 0.942 & 0.089 & 0.089 \\ \hline \multirow{4}{*}{-1.2} & 0 & 0.144\% & 0.966 & 2.765 & 2.550 & 4.838\% & 0.934 & 1.333 & 1.304 & 0.481\% & 0.958 & 2.245 & 2.072 & -0.265\% & 0.948 & 0.058 & 0.057 \\
& 0.2 & 0.947\% & 0.962 & 2.832 & 2.621 & 5.196\% & 0.930 & 1.476 & 1.442 & 1.145\% & 0.960 & 2.298 & 2.153 & -0.130\% & 0.934 & 0.077 & 0.078 \\
& 0.4 & 1.644\% & 0.966 & 2.947 & 2.640 & 5.883\% & 0.924 & 1.778 & 1.753 & 1.700\% & 0.972 & 2.391 & 2.196 & -0.245\% & 0.940 & 0.084 & 0.084 \\
& 0.6 & 1.676\% & 0.968 & 3.252 & 3.045 & 4.337\% & 0.924 & 2.849 & 2.946 & 2.473\% & 0.950 & 2.638 & 2.539 & -0.350\% & 0.942 & 0.093 & 0.094 \\ \hline \multirow{4}{*}{-2} & 0 & 0.077\% & 0.964 & 2.698 & 2.531 & 6.604\% & 0.854 & 1.338 & 1.308 & 0.296\% & 0.960 & 2.160 & 2.042 & -0.098\% & 0.958 & 0.067 & 0.065 \\
& 0.2 & 0.407\% & 0.952 & 2.757 & 2.541 & 6.494\% & 0.878 & 1.480 & 1.468 & 0.594\% & 0.956 & 2.206 & 2.077 & 0.069\% & 0.950 & 0.089 & 0.087 \\
& 0.4 & 0.601\% & 0.966 & 2.848 & 2.615 & 6.062\% & 0.906 & 1.748 & 1.776 & 0.748\% & 0.958 & 2.279 & 2.157 & -0.060\% & 0.954 & 0.098 & 0.095 \\
& 0.6 & 0.588\% & 0.970 & 3.043 & 2.837 & 4.828\% & 0.922 & 2.474 & 2.574 & 0.917\% & 0.960 & 2.436 & 2.290 & -0.229\% & 0.944 & 0.107 & 0.105 \\ \hline \end{tabular} \end{adjustbox} \end{sidewaystable}
\begin{sidewaystable} \small
\caption{Simulation Results under NP Scenarios. Under zero treatment effect scenarios, bias is reported instead of percentage bias.}
\label{tab:obsnp-1}
\begin{adjustbox}{scale=0.95,center} \begin{tabular}{cccccccccccccccccc} \hline \multicolumn{2}{c}{\textbf{Scenario}} &
\textbf{Bias\%} &
\textbf{CP} &
\textbf{SEM} &
\textbf{SEE} &
\textbf{Bias\%} &
\textbf{CP} &
\textbf{SEM} &
\textbf{SEE} &
\textbf{Bias\%} &
\textbf{CP} &
\textbf{SEM} &
\textbf{SEE} &
\textbf{Bias\%} &
\textbf{CP} &
\textbf{SEM} &
\textbf{SEE} \\ \hline \textbf{$\beta_A$} &
\textbf{CR} &
\multicolumn{4}{c}{Matched RMST (Murray)} &
\multicolumn{4}{c}{RMST Regression} &
\multicolumn{4}{c}{Conner's IPTW RMST} &
\multicolumn{4}{c}{IPTW Cox (HR)} \\ \hline \multirow{4}{*}{0} & 0 & 0.032 & 0.964 & 2.795 & 2.611 & -0.020 & 0.932 & 1.322 & 1.402 & 0.076 & 0.962 & 2.284 & 2.127 & 0.005 & 0.938 & 0.052 & 0.052 \\
& 0.2 & 0.140 & 0.954 & 2.884 & 2.686 & 0.124 & 0.938 & 1.478 & 1.542 & 0.165 & 0.956 & 2.355 & 2.218 & 0.005 & 0.924 & 0.067 & 0.070 \\
& 0.4 & 0.252 & 0.958 & 3.071 & 2.908 & 0.333 & 0.908 & 1.906 & 2.033 & 0.258 & 0.956 & 2.505 & 2.420 & 0.008 & 0.936 & 0.074 & 0.077 \\
& 0.6 & 0.288 & 0.953 & 4.079 & 4.023 & 0.877 & 0.772 & 4.450 & 6.684 & 0.344 & 0.947 & 3.369 & 3.439 & 0.013 & 0.928 & 0.085 & 0.088 \\ \hline \multirow{4}{*}{-0.4} & 0 & 0.196\% & 0.964 & 2.674 & 2.473 & 6.560\% & 0.888 & 1.244 & 1.215 & 0.428\% & 0.964 & 2.131 & 1.966 & 102.637\% & 0.000 & 0.056 & 0.061 \\ & 0.2 & 0.925\% & 0.962 & 2.746 & 2.507 & 6.798\% & 0.894 & 1.384 & 1.337 & 1.063\% & 0.964 & 2.193 & 2.019 & 298.860\% & 0.000 & 0.079 & 0.081 \\
& 0.4 & 1.728\% & 0.966 & 2.863 & 2.577 & 7.145\% & 0.898 & 1.653 & 1.636 & 1.781\% & 0.964 & 2.293 & 2.113 & 363.911\% & 0.000 & 0.091 & 0.095 \\
& 0.6 & 2.152\% & 0.972 & 3.119 & 2.897 & 6.678\% & 0.904 & 2.400 & 2.552 & 2.615\% & 0.952 & 2.516 & 2.376 & 422.119\% & 0.000 & 0.105 & 0.109 \\ \hline \multirow{4}{*}{-0.8} & 0 & 0.158\% & 0.966 & 2.644 & 2.461 & 7.174\% & 0.820 & 1.253 & 1.212 & 0.345\% & 0.958 & 2.091 & 1.954 & 33.369\% & 0.004 & 0.057 & 0.062 \\
& 0.2 & 0.608\% & 0.968 & 2.711 & 2.466 & 7.158\% & 0.864 & 1.392 & 1.349 & 0.747\% & 0.966 & 2.148 & 1.991 & 139.464\% & 0.000 & 0.083 & 0.086 \\
& 0.4 & 1.250\% & 0.968 & 2.816 & 2.538 & 7.139\% & 0.888 & 1.650 & 1.656 & 1.345\% & 0.962 & 2.239 & 2.079 & 172.568\% & 0.000 & 0.097 & 0.099 \\
& 0.6 & 1.713\% & 0.972 & 3.030 & 2.830 & 6.392\% & 0.906 & 2.302 & 2.383 & 1.941\% & 0.958 & 2.424 & 2.286 & 201.225\% & 0.000 & 0.111 & 0.116 \\ \hline \multirow{4}{*}{-1.2} & 0 & 0.079\% & 0.966 & 2.605 & 2.434 & 7.806\% & 0.724 & 1.261 & 1.224 & 0.263\% & 0.960 & 2.042 & 1.925 & 10.201\% & 0.446 & 0.059 & 0.063 \\
& 0.2 & 0.422\% & 0.962 & 2.669 & 2.440 & 7.695\% & 0.794 & 1.400 & 1.382 & 0.597\% & 0.962 & 2.095 & 1.966 & 86.905\% & 0.000 & 0.088 & 0.090 \\
& 0.4 & 0.911\% & 0.970 & 2.764 & 2.511 & 7.386\% & 0.852 & 1.650 & 1.676 & 0.984\% & 0.950 & 2.177 & 2.022 & 109.380\% & 0.000 & 0.103 & 0.107 \\
& 0.6 & 1.353\% & 0.972 & 2.946 & 2.776 & 6.397\% & 0.904 & 2.230 & 2.284 & 1.438\% & 0.952 & 2.335 & 2.234 & 128.294\% & 0.000 & 0.118 & 0.123 \\ \hline \multirow{4}{*}{-2} & 0 & 0.053\% & 0.968 & 2.514 & 2.362 & 9.169\% & 0.512 & 1.279 & 1.238 & 0.204\% & 0.958 & 1.923 & 1.827 & -8.274\% & 0.274 & 0.063 & 0.067 \\
& 0.2 & 0.221\% & 0.964 & 2.569 & 2.365 & 8.835\% & 0.622 & 1.419 & 1.419 & 0.372\% & 0.954 & 1.969 & 1.860 & 45.487\% & 0.000 & 0.101 & 0.101 \\
& 0.4 & 0.533\% & 0.956 & 2.648 & 2.452 & 8.418\% & 0.730 & 1.653 & 1.689 & 0.607\% & 0.954 & 2.036 & 1.942 & 59.417\% & 0.000 & 0.117 & 0.122 \\
& 0.6 & 0.782\% & 0.956 & 2.791 & 2.623 & 7.366\% & 0.866 & 2.149 & 2.098 & 0.831\% & 0.948 & 2.157 & 2.105 & 70.605\% & 0.000 & 0.133 & 0.136 \\ \hline \end{tabular}
\end{adjustbox} \end{sidewaystable}
\begin{table}[!p] \caption{Summary Statistics of Covariates by Baseline Smoking Status in ARIC Study} \label{tab:c1} \begin{tabular}{lll} \hline
& Non-current smoker (10,717) & Current smoker (3,832) \\ \hline Race, n (\%) of white & 8161 (76.2\%) & 2730 (71.2\%) \\ Gender, n (\%) of female & 5957 (55.6\%) & 2003 (52.3\%) \\ Age, mean (SD) & 54.4 (5.8) & 53.7 (5.7) \\ BMI, mean (SD) & 28.1 (5.4) & 26.3 (5.0) \\ Diabetes, n (\%) & 1044 (9.7\%) & 333 (8.7\%) \\ HDL (mmol/L), mean (SD) & 52.6 (16.8) & 49.6 (17.3) \\ LDL (mmol/L), mean (SD) & 137.6 (38.9) & 138.6 (40.4) \\ Hypertension, n (\%) & 3783 (35.3\%) & 1225 (32.0\%) \\ \hline \end{tabular} \end{table}
\begin{table}[!p] \caption{ARIC Data Analysis Results} \label{tab:senana-2} \begin{tabular}{lcccc} \hline \multicolumn{1}{c}{} & Estimate & SE & 95\% CI Lower Bound & 95\% CI Upper Bound \\
\hline Matched RMST & -22.266 & 1.380 & -24.972 & -19.561 \\
IPTW Cox (HR) & 2.173 & 0.066 & 2.048 & 2.306\\ \hline \end{tabular} \end{table}
\pagenumbering{arabic} \renewcommand*{\thepage}{Appendix-\arabic{page}}
\section*{Web Appendix A: Proofs of Theoretical Results in Section 2 of the Main Text} \label{s:w_a}
\setcounter{proposition}{0}
We will prove the propositions and related lemmas in Section 2 of the main text.
\begin{proposition}
Given assumptions \ref{sutva}-\ref{strong_ignore}, we have $(T^0,T^1)\indep A|e(X)$, which further implies $(Z^0,Z^1)\indep A|e(X)$. \end{proposition}
\begin{proof}
It is equivalent to show
$P\{A=1|T^1,T^0,e(X)\}=P\{A=1|e(X)\}$.
By Theorem 2 in \citet{cps}, we have $P(A=1|e(X))=E\{e(X)|e(X)\}=e(X)$, then it is equivalent to show
$P\{A=1|T^1,T^0,e(X)\}=e(X)$. We have \begin{align*}
P\{A=1|T^1,T^0,e(X)\}&=E\{P(A=1|T^1,T^0,X)|T^1,T^0,e(X)\}\\
&=E\{P(A=1|X)|T^1,T^0,e(X)\}\text{(by strongly ignorability)}\\
&=E\{e(X)|T^1,T^0,e(X)\}=e(X)=P\{A=1|e(X)\}. \end{align*}
Thus, we have $(T^0,T^1)\indep A|e(X)$ for $0<\text{pr}(A=1|e(X))<1$. Since $Z^A=\min (T^A, \tau)$ and $\tau$ is a fixed constant, the above conditional independence also implies
$(Z^0,Z^1)\indep A|e(X)$. \end{proof}
\begin{lemma} \label{marginal_ignorable} Given assumptions \ref{sutva}-\ref{indep_censor}, $(Y^1, Y^0)\indep A$ holds marginally in the matched sample under the propensity score matching design. \end{lemma}
\begin{proof}
By assumption~\ref{strong_ignore}, we have $(Y^1, Y^0)\indep A|e(X)$ where $0<P(A=1|e(X))<1$. Let $M$ denotes the matching structure, and $\epsilon_M$ denotes the set of propensity scores in the matched sample. Then, we have the following equation by matching on propensity score $e(X)$ with a constant treatment to control allocation ratio $1: k$ ($k=1$ for pair matching),
$$P(A=1|e(X))=\frac{1}{k+1},\ \text{for all $e(X)\in\epsilon_M$}.$$
Thus, $e(X)\indep A$ holds in the matched sample, i.e. $f_M(e(X)|A)=f_M(e(X))$.
Consider the joint density of $Y^1$ and $Y^0$ conditional on $A$ in the matched sample, which is denoted as $f_M(Y^1, Y^0|A)$, we have \begin{align*}
&f_M(Y^1, Y^0|A)=\int_{\epsilon_M}f(Y^1, Y^0|A, e(X))f_M(e(X)|A)de(X)\\
=&\int_{\epsilon_M}f(Y^1, Y^0|A, e(X))f_M(e(X))de(X)\ \text{[matched by constant allocation ratio]}\\
=&\int_{\epsilon_M}f(Y^1,Y^0|e(X))f_M(e(X))de(X)\ \text{[by assumption~\ref{strong_ignore}]}\\ =&f_M(Y^1, Y^0). \end{align*}
Since $f_M(Y^1, Y^0|A)=f_M(Y^1, Y^0)$ implies $(Y^1, Y^0)\indep A$ in the matched sample, $(Y^1, Y^0)\indep A$ holds marginally in the matched sample. \end{proof}
\begin{lemma} \label{rmst_lemma} Let $\hat{S}_{e(X),A}(t)$ denotes the KM survival function estimator given propensity score $e(X)$ and treatment indicator $A$. For a fixed truncation time $\tau$, $$\underset{n\rightarrow\infty}{\lim}\int^\tau_0E_T[\hat{S}_{e(X),A}(t)-S_{e(X),A}(t)]dt=0,$$
\end{lemma}
\begin{proof}
Define $\tilde{T}=\min (T,C)$ and $\pi_{e(X),A}(t)=P(\tilde{T}\geq t)\in (0,1)$, then $[1-\pi_{e(X),A}(t)]^n$ is a nonnegative function that increases as $t$ increases. By Lemma 3.2.1 in \citet{fleming2011counting}, we know: \begin{align*} &\int^\tau_0E_T[\hat{S}_{e(X),A}(t)-S_{e(X),A}(t)]dt \leq\int^\tau_0[1-S_{e(X),A}(t)][1-\pi_{e(X),A}(t)]^ndt\\ &\leq\int^\tau_0[1-\pi_{e(X),A}(t)]^ndt \leq\tau[1-\pi_{e(X),A}(\tau)]^n. \end{align*} Since $\tau>0$ is a fixed constant and $1-\pi_{e(X),A}(\tau)\in(0,1)$, we have $$\underset{n\rightarrow\infty}{\lim}\int^\tau_0E_T[\hat{S}_{e(X),A}(t)-S_{e(X),A}(t)]dt\leq\underset{n\rightarrow\infty}{\lim}\tau[1-\pi_{e(X),A}(\tau)]^n=0.$$ Therefore, we have $\underset{n\rightarrow\infty}{\lim}\int^\tau_0E_T[\hat{S}_{e(X),A}(t)-S_{e(X),A}(t)]dt=0$. \end{proof}
\begin{proposition} Given assumptions \ref{sutva}-\ref{tau_assump}, the RMST estimator based on KM method given propensity score $e(X)$ and treatment group $A$, denoted as $\hat{\mu}_{e(X),A}$, is an asymptotically unbiased estimator for $\mu_{e(X),A}$ given $\tau<t_{\max}$. \end{proposition}
\begin{proof} First, we will show that $\hat{S}_{e(X),A}(t)$ is asymptotically unbiased for any time $T< t_{\max}$. Let $t_i$'s be i.i.d event times ranking from small to large, and $Y_i$ is the number of people at risk at event time $t_i$. Let $d_i$ be the number of event at event time $t_i$, then we have the definition below. \begin{equation*} \hat{S}_{e(X),A}(t) =\left\{\begin{array}{l} 1, \text{if}\ t\leq t_1\ \text{given}\ e(X)\text{ and } A\\ \underset{t_i\leq t}{\prod}\frac{Y_i-d_i}{Y_i}, \text{if}\ t_1\leq t \ \text{given}\ e(X)\text{ and }A\\ \end{array}\right. \end{equation*} Let $\hat{\Lambda}_{e(X),A}(u)=\underset{t_i<t}{\sum}\frac{d_i}{Y_i}$ be the Nelson-Aalen estimator for the cumulative hazard function $\Lambda_{e(X),A}(u)$ given $e(X)$ and $A$. According to Theorem 3.2.3 in \citet{fleming2011counting}, we have the following equation if $S_{e(X),A}(t)>0$: \begin{align*}
&\frac{\hat{S}_{e(X),A}(t)}{S_{e(X),A}(t)}-1=-\int^t_0\frac{\hat{S}_{e(X),A}(u^-)}{S_{e(X),A}(u)}d\{\hat{\Lambda}_{e(X),A}(u)-\Lambda_{e(X),A}(u)\},\\
&E[\hat{S}_{e(X),A}(t)-S_{e(X),A}(t)]=E[I_{\{T<t\}}\frac{\hat{S}_{e(X),A}(T)\{S_{e(X),A}(T)-S_{e(X),A}(t)\}}{S_{e(X),A}(T)}]. \end{align*}
Based on Lemma 3.2.1 in \citet{fleming2011counting}, the bias $E[\hat{S}_{e(X),A}(t)-S_{e(X),A}(t)]$ will converge to zero as sample size $n\rightarrow \infty$. Thus, $\hat{S}_{e(X),A}(t)$ is asymptotically unbiased given $t<t_{\max}$. Similarly, $\hat{S}_{e(X),A=0}(t)$ is also asymptotically unbiased given $t<t_{\max}$.
Second, we will show $\hat{\mu}_{e(X),A}$ is an asymptotically unbiased estimator given $\tau<t_{\max}$. Since $E_{T}(\hat{\mu}_{e(X),A})=E_{T}[\int^\tau_0\hat{S}_{e(X),A}(t)dt]$ and $\hat{S}_{e(X),A}(t)$ is a positive bounded function between 0 and 1 when $t\in[0,\tau]$, then we have
$$E_{T}[\int^\tau_0|\hat{S}_{e(X),A}(t)|dt]=E_{T}[\int^\tau_0\hat{S}_{e(X),A}(t)dt]\leq\tau<\infty.$$ By Fubini's Theorem, $$E_{T}[\int^\tau_0\hat{S}_{e(X),A}(t)dt]=\int^\tau_0E_{T}[\hat{S}_{e(X),A}(t)]dt.$$
By propositions~\ref{RMST_unbias} and lemma~\ref{rmst_lemma}, we have the following for fixed truncated time $\tau$. \begin{align*} &\underset{n\rightarrow\infty}{\lim}E_{T}(\hat{\mu}_{e(X),A})-\mu_{e(X),A}\\ =&\underset{n\rightarrow\infty}{\lim}E_{T}[\int^\tau_0\hat{S}_{e(X),A}(t)dt]-\int^\tau_0S_{e(X),A}(t)dt\\ =&\underset{n\rightarrow\infty}{\lim}\int^\tau_0E_{T}[\hat{S}_{e(X),A}(t)]dt-\int^\tau_0S_{e(X),A}(t)dt\ \text{(by Fubini's Theorem)}\\ =&\underset{n\rightarrow\infty}{\lim}\int^\tau_0E_{T}\{\hat{S}_{e(X),A}(t)-S_{e(X),A}(t)\}dt \leq\underset{n\rightarrow\infty}{\lim}\tau[1-\pi_{e(X),A}(\tau)]^n=0. \end{align*} Therefore, $\hat{\mu}_{e(X),A}$ is an asymptotically unbiased estimator for $\mu_{e(X),A}$ when $\tau<t_{\max}$. \end{proof}
\begin{lemma} \label{prop:asyunbias_lemma} For a fixed truncation time point $\tau<t_{\max}$, $$\underset{n\rightarrow\infty}{\lim}\int^\tau_0E_T[\hat{S}^AA(t)-S^A(t)]dt=0,$$
\end{lemma}
\begin{proof}
Let $\tilde{T}=\min (T,C)$ and $\pi(t)=P(\tilde{T}\geq t)\in (0,1)$, then $[1-\pi(t)]^n$ is a nonnegative function which increases as $t$ increases. Let $\pi_A(t)$ denotes the function $\pi(t)$ for treatment indicator A. By Lemma 3.2.1 in \citet{fleming2011counting}, we have $$\int^\tau_0E_T[\hat{S}^A(t)-S^A(t)]dt\leq \int^\tau_0 [1-S^A(t)][1-\pi_A(t)]^ndt\leq\int^\tau_0[1-\pi_A(t)]^ndt\leq \tau[1-\pi_A(t)]^n.$$ Since $\tau>0$ is a fixed constant and $1-\pi_A(\tau)\in(0,1)$, we have $$\underset{n\rightarrow\infty}{\lim}\int^\tau_0E_T[\hat{S}^A(t)-S^A(t)]dt\leq\underset{n\rightarrow\infty}{\lim}\tau[1-\pi_A(t)]^n=0.$$
\end{proof}
\begin{proposition} Given assumptions \ref{sutva}-\ref{tau_assump}, $\hat{\Delta}_{ATT}=\int^\tau_0[\hat{S}^1(t)-\hat{S}^0(t)]dt$ is asymptotically unbiased. \end{proposition}
\begin{proof}
Since the estimated survival function $\hat{S}(t)\in(0,1)$ and $|\int^\tau_0\hat{S}(t)dt|\in (0,\tau)$, we also satisfy the following conditions to use Fubini's theorem: \begin{enumerate}
\item $E_{e(X)}\{E_T|\int^\tau_0\hat{S}_{e(X),A}(t)dt|\}\leq E_{e(X)}\{E_T(\tau)\}=\tau<\infty$
\item $E_{e(X)}[\int^\tau_0|\hat{S}_{e(X),A}(t)|dt]\leq \tau<\infty$ \item $E_T\{\int^\tau_0\hat{S}^A(t)dt\}\leq\tau<\infty$ \end{enumerate} Thus, we can apply Fubini's theorem three time to interchange the expectation of $e(X)$ as below: \begin{align*} &E_{e(X)}\{E_T[\int^\tau_0\hat{S}_{e(X),A=1}(t)dt]-E_T[\int^\tau_0\hat{S}_{e(X),A=0}(t)dt]\}\\ =&E_T\{E_{e(X)}[\int^\tau_0\hat{S}_{e(X),A=1}(t)dt]-E_{e(X)}[\int^\tau_0\hat{S}_{e(X),A=0}(t)dt]\}\\ =&E_T\{\int^\tau_0E_{e(X)}[\hat{S}_{e(X),A=1}(t)]dt-\int^\tau_0E_{e(X)}[\hat{S}_{e(X),A=0}(t)]dt\}\\ =&E_T\{\int^\tau_0\hat{S}^1(t)dt-\int^\tau_0\hat{S}^0(t)dt\}\\ =&\int^\tau_0E_T[\hat{S}^1(t)]dt-\int^\tau_0E_T[\hat{S}^0(t)]dt. \end{align*} By lemma~\ref{prop:asyunbias_lemma}, we have \begin{align*} &\underset{n\rightarrow\infty}{\lim}E_{e(X)}\{E_T[\int^\tau_0\hat{S}_{e(X),A=1}(t)dt]-E_T[\int^\tau_0\hat{S}_{e(X),A=0}(t)dt]\}\\ =&\underset{n\rightarrow\infty}{\lim}\int^\tau_0E_T[\hat{S}^1(t)]dt-\underset{n\rightarrow\infty}{\lim}\int^\tau_0E_T[\hat{S}^0(t)]dt\\ =&\int^\tau_0E_T[S^1(t)]dt-\int^\tau_0E_T[S^0(t)]dt =\mu_1-\mu_0. \end{align*} Therefore, our proposed propensity score matched RMST estimator is asymptotically unbiased when truncation time point $\tau<t_{\max}$. \end{proof}
\section*{Web Appendix B: Proofs of Theoretical Results for Section 4 of the Main Text} \label{s:w_b}
\subsection*{Web Appendix B.1: Proofs of Propositions about Conditional Effect}
We define the expectations of the RMST outcome $Z=\min (T,\tau)$. The following propositions are proved \textbf{within each propensity score value} $e(X)=e(x)$.
\begin{proposition} \label{sa_binary}
For binary unmeasured confounder $U=0,1$, we have $RR_{AU|e(X)=e(x)}\geq 1$ and $MR_{UZ|e(X)=e(x)}\geq 1$. \end{proposition}
\begin{proof}
By definition, we have $MR_{UZ|A=1,e(X)=e(x)}\geq 1$ and $MR_{UZ|A=0,e(X)=e(x)}\geq1$, then
$MR_{UZ|e(X)=e(x)}=\max (MR_{UZ|A=1,e(X)=e(x)},MR_{UZ|A=0,e(X)=e(x)})\geq 1.$
Assume $RR_{AU|e(x)}=\underset{k=0,1}{\max}RR_{AU,k|e(x)}<1$, then it implies that \begin{align*}
P(U=0|A=1,e(X)=e(x))<P(U=0|A=0,e(X)=e(x)),\\
P(U=1|A=1,e(X)=e(x))<P(U=1|A=0,e(X)=e(x)). \end{align*}
This further implies that $1=P(U=0|A=1,e(X)=e(x))+P(U=1|A=1,e(X)=e(x))<P(U=0|A=0,e(X)=e(x))+P(U=1|A=0,e(X)=e(x))=1$, which is not true. Thus, we have proved by contradiction that $RR_{AU|e(X)=e(x)}\geq 1$. \end{proof}
\begin{proposition} \label{sa_prop1} $$CMR_{AZ^+}=\frac{MR_{AZ}}{MR_{AZ^+}^{true}}\leq BF_U,\ CMR_{AZ^-}=\frac{MR_{AZ}}{MR_{AZ^-}^{true}}\leq BF_U,\ CMR_{AZ}=\frac{MR_{AZ}}{MR_{AZ}^{true}}\leq BF_U,$$ \end{proposition}
\begin{proof}
First, let $f=P(A=1)$, then we have \begin{align} MR_{AZ}^{true}=&\frac{\int r_1(u)F(du)}{\int r_0(u)F(du)} =\frac{f\int r_1(u) F_1(du)+(1-f)\int r_1(u)F_0(du)}{f\int r_0(u) F_1(du)+(1-f)\int r_0(u)F_0(du)}\\ =&\frac{f\int r_0(u)F_1(du)}{f\int r_0(u)F_1(du)+(1-f)\int r_0(u)F_0(du)}\times\frac{\int r_1(u)F_1(du)}{\int r_0(u)F_1(du)}\\ &+\frac{(1-f)\int r_0(u)F_0(du)}{f\int r_0(u)F_1(du)+(1-f)\int r_0(u)F_0(du)}\times \frac{\int r_1(u)F_0(du)}{\int r_0(u)F_0(du)}. \end{align}
Let $w=\frac{f\int r_0(u)F_1(du)}{f\int r_0(u)F_1(du)+(1-f)\int r_0(u)F_0(du)}\in [0,1]$, then we have \begin{align*} & MR^{true}_{AZ}=wMR^{true}_{AZ^+}+(1-w)MR^{true}_{AZ^-}; & \frac{1}{CMR_{AZ}}=\frac{w}{CMR_{AZ^+}}+\frac{1-w}{CMR_{AZ^-}} \end{align*}
Second, we have $$CMR_{AZ^+}=\frac{MR^{obs}_{AZ}}{MR_{AZ^+}^{true}}=\frac{\int r_1(u)F(du)}{\int r_0(u)F(du)}/\frac{\int r_1(u)F_1(du)}{\int r_0(u)F_1(du)}=\frac{w_1\underset{u}{\max}r_0(u)+(1-w_1)\underset{u}{\min}r_0(u)}{w_0\underset{u}{\max}r_0(u)+(1-w_0)\underset{u}{\min}r_0(u)}$$ where $w_1=\frac{\int[r_0(u)-\underset{u}{\min}r_0(u)]F_1(du)}{\underset{u}{\max}r_0(u)-\underset{u}{\min}r_0(u)}$ and $w_0=\frac{\int[r_0(u)-\underset{u}{\min}r_0(u)]F_0(du)}{\underset{u}{\max}r_0(u)-\underset{u}{\min}r_0(u)}$.
Define $\Gamma=\frac{w_1}{w_0}$ then \begin{align} \Gamma&=\frac{w_1}{w_0}=\frac{\int[r_0(u)-\underset{u}{\min}r_0(u)]F_1(du)}{\int[r_0(u)-\underset{u}{\min}r_0(u)]F_0(du)} =\frac{\int[r_0(u)-\underset{u}{\min}r_0(u)]RR_{AU}(u)F_0(du)}{\int[r_0(u)-\underset{u}{\min}r_0(u)]F_0(du)}\\ &\leq \frac{\underset{u}{\max}RR_{AU}(u)\int[r_0(u)-\underset{u}{\min}r_0(u)]F_0(du)}{\int[r_0(u)-\underset{u}{\min}r_0(u)]F_0(du)} =RR_{AU}. \end{align}
Write $w_0=\frac{w_1}{\Gamma}$, then $$CMR_{AZ}^+=\frac{\underset{u}{[\max}r_0(u)-\underset{u}{\min}r_0(u)]w_1+\underset{u}{\min}r_0(u)}{[\underset{u}{\max}r_0(u)-\underset{u}{\min}r_0(u)]w1/\Gamma+\underset{u}{\min}r_0(u)}.$$
If $\Gamma>1$, $CMR_{AZ}^+$ is increasing in $w_1$ according to Lemma A.1 in the eAppendix of \citet{ding2016sensitivity}, then the maximum attains at $w_1=1$, and we have
$$CMR_{AZ}^+\leq\frac{\Gamma\times MR_{UZ|A=0}}{\Gamma+MR_{UZ|A=0}-1}\leq\frac{RR_{AU}\times MR_{UZ|A=0}}{RR_{AU}+MR_{UZ|A=0}-1}.$$
If $\Gamma\leq1$, $CMR_{AZ}^+$ is non-increasing in $w_1$ according to Lemma A.1 in the eAppendix of \citet{ding2016sensitivity}, then the maximum attains at $w_1=0$, and we have
$$CMR_{AZ}^+\leq 1\leq\frac{RR_{AU}\times MR_{UZ|A=0}}{RR_{AU}+MR_{UZ|A=0}-1}.$$ Similarly, by $\frac{1}{CMR_{AZ}}=\frac{w}{CMR_{AZ^+}}+\frac{1-w}{CMR_{AZ^-}}$, we have $$\frac{1}{CMR_{AZ}}\geq\frac{1}{BF_U}\ , CMR_{AZ}\leq BF_U.$$ \end{proof}
To study the average causal effect of the exposure on the difference scale, we need the following definitions: \begin{itemize}
\item Define $m_0=E(Z|A=0)$ and $m_1=E(Z|A=1)$ , then the observed mean difference of exposure on the outcome is $m_1-m_0$. \item The average causal effect of the exposure on the outcome for exposed is \begin{align*}
ACE_{AZ^+}^{true}&=\int E(Z|A=1, U=u)F_1(du)-\int E(Z|A=0, U=u)F_1(du)\\ &=m_1-\int r_0(u)F_1(du). \end{align*} \item The average causal effect of the exposure on the outcome for unexposed is \begin{align*}
ACE_{AZ^-}^{true}&=\int E(Z|A=1, U=u)F_0(du)-\int E(Z|A=0, U=u)F_0(du)\\ &=\int r_1(u)F_0(du)-m_0. \end{align*} \item The average causal effect of the exposure on the outcome for whole population is \begin{align*}
ACE_{AZ}^{true}&=\int E(Z|A=1, U=u)F(du)-\int E(Z|A=0, U=u)F(du)\\ &=fACE_{AZ^+}^{true}+(1-f)ACE_{AZ^-}^{true}. \end{align*}
\end{itemize}
\begin{proposition} \label{sa_prop2} For nonnegative outcomes and $ACE_{AZ}^{obs}\geq0$, the lower bounds for the average causal effects are \begin{align*} & ACE_{AZ^+}^{true}\geq m_1-m_0\times BF_U; ACE_{AZ^-}^{true}\geq m_1/BF_U-m_0;\\ & ACE_{AZ}^{true}\geq (m_1-m_0\times BF_U)[f+(1-f)/BF_U]=(\frac{m_1}{BF_U}-m_0)[f\times BF_U+(1-f)]. \end{align*} \end{proposition}
\begin{proof}
From the data, we can identify
$$m_1=\int E(Z|A=1, U=u)F_1(du)=\int r_1(u)F_1(du)=E(Z|A=1);$$
$$m_0=\int E(Z|A=0, U=u)F_0(du)=\int r_0(u)F_0(du)=E(Z|A=0).$$ The counterfactual probabilities are not identifiable:
$$E(Z(1)=1|A=0)=\int E(Z=1|A=1, U=u)F_0(du)=\int r_1(u)F_0(du);$$
$$E(Z(0)=1|A=1)=\int E(Z=1|A=0, U=u)F_1(du)=\int r_0(u)F_1(du).$$
First, by proposition~\ref{sa_prop1} we have \begin{align*}
\frac{m_1}{E(Z(1)=1|A=0)}&=\frac{\int r_1(u)F_1(du)}{\int r_1(u)F_0(du)} =\frac{\int r_1(u)F_1(du)}{\int r_0(u)F_0(du)}/\frac{\int r_1(u)F_0(du)}{\int r_0(u)F_0(du)}\\ &=\frac{MR_{AZ}}{MR_{AZ^-}^{true}}=CMR_{AZ^-}\leq BF_U. \end{align*}
Thus, we have $E(Z(1=1)|A=0)\geq \frac{m_1}{BF_U}$.
Second, by proposition~\ref{sa_prop1} again we have
$$\frac{E(Z(0)=1|A=1)}{m_0}=\frac{\int r_0(u)F_1(du)}{\int r_0(u)F_0(du)}=CMR_{AZ^+}\leq BF_U.$$
Thus, we have $E(Z(0)=1|A=1)\leq m_0 BF_U.$
By definition of ACE and the inequalities derived above, we have \begin{align*} ACE_{AZ^+}^{true}&=m_1-\int r_0(u)F_1(du)\geq m_1-m_0\times BF_U ;\\ ACE_{AZ^-}^{true}&=\int r_1(u)F_0(du)-m_)\geq m_1/BF_U-m_0 ;\\ ACE_{AZ}^{true}&=f\cdot ACE_{AZ^+}^{true}+(1-f)ACE_{AZ^-}^{true}\\ &\geq f(m_1-m_0BF_U)+(1-f)(\frac{m_1}{BF_U}-m_0)\\ &=(m_1-m_0\times BF_U)[f+(1-f)/BF_U]\\ &=(\frac{m_1}{BF_U}-m_0)[f\times BF_U+(1-f)]. \end{align*} \end{proof}
\begin{proposition} \label{sa_prop3} For nonnegative outcomes with $ACE_{AZ}^{obs}<0$, we have \begin{align*} & ACE_{AZ^+}^{true}\leq m_1 BF_U-m_0 ; ACE_{AZ^-}^{true}\leq m_1-\frac{m_0}{BF_U} ;\\ & ACE_{AZ}^{true}\leq (m_1 BF_U-m_0)(f+\frac{1-f}{BF_U})=(m_1-\frac{m_0}{BF_U})(f BF_U+1-f). \end{align*} \end{proposition}
\begin{proof}
Define $\bar{A}=1-A$. By applying proposition~\ref{sa_prop2} we have \begin{align*}
ACE_{\bar{A}Z^+}^{true}&\geq E(Z|\bar{A}=1)-E(Z|\bar{A}=0)\times BF_U ;\\
ACE_{\bar{A}Z^-}^{true}&\geq E(Z|\bar{A}=1)/BF_U-E(Z|\bar{A}=0) ;\\
ACE_{\bar{A}Z}^{true}&\geq (E(Z|\bar{A}=1)-E(Z|\bar{A}=0)\times BF_U)[f+(1-f)/BF_U]\\
&=(\frac{E(Z|\bar{A}=1)}{BF_U}-E(Z|\bar{A}=0))[f\times BF_U+(1-f)]. \end{align*}
Because $ACE_{\bar{A}Z^+}^{true}=-ACE_{AZ^+}^{true}$, $ACE_{\bar{A}Z^-}^{true}=-ACE_{AZ^-}^{true}$ and $ACE_{\bar{A}Z}^{true}=-ACE_{AZ}^{true}$, and we also have $E(Z|\bar{A}=0)=E(Z|A=1)=m_1$ and $E(Z|\bar{A}=1)=E(Z|A=0)=m_0$. Then we have \begin{align*} & ACE_{AZ^+}^{true}\leq m_1 BF_U-m_0 ; ACE_{AZ^-}^{true}\leq m_1-\frac{m_0}{BF_U} ;\\ & ACE_{AZ}^{true}\leq (m_1 BF_U-m_0)(f+\frac{1-f}{BF_U})=(m_1-\frac{m_0}{BF_U})(f BF_U+1-f). \end{align*} \end{proof}
\subsection*{Web Appendix B.2: Proofs of Propositions about the Marginal Effect}
To make the bounding factor hold for all propensity score values, we consider the maximum value of $BF_U$ across all values of propensity score $e(X)$, which is defined as $BF_U^*=\underset{e(x)}{\max}(BF_{U|e(X)=e(x)})$. \begin{proposition} \label{sa_prop4} For nonnegative outcomes and $ACE_{AZ}^{obs}\geq0$, we have \begin{align*} & ACE_{AZ^+}^{true}\geq m_1-m_0\times BF^*_U ; ACE_{AZ^-}^{true}\geq m_1/BF^*_U-m_0 ;\\ & ACE_{AZ}^{true}\geq (m_1-m_0\times BF^*_U)[f+(1-f)/BF^*_U]=(\frac{m_1}{BF^*_U}-m_0)[f\times BF^*_U+(1-f)]. \end{align*}
For nonnegative outcomes and $ACE_{AZ}^{obs}<0$, we have \begin{align*} & ACE_{AZ^+}^{true}\leq m_1 BF^*_U-m_0 ; ACE_{AZ^-}^{true}\leq m_1-\frac{m_0}{BF^*_U} ;\\ & ACE_{AZ}^{true}\leq (m_1 BF^*_U-m_0)(f+\frac{1-f}{BF^*_U})=(m_1-\frac{m_0}{BF^*_U})(f BF^*_U+1-f). \end{align*} \end{proposition}
\begin{proof}
We will start from showing the results for nonnegative outcomes and $ACE_{AZ}^{obs}\geq0$. First, we have \begin{align*}
\frac{m_1}{E(Z(1)=1|A=0)}&=\frac{\int r_1(u)F_1(du)}{\int r_1(u)F_0(du)} =\frac{\int r_1(u)F_1(du)}{\int r_0(u)F_0(du)}/\frac{\int r_1(u)F_0(du)}{\int r_0(u)F_0(du)}\\ &=\frac{MR_{AZ}}{MR_{AZ^-}^{true}}=CMR_{AZ^-}\leq BF_U \leq BF^*_U. \end{align*}
Thus, we have $E(Z(1=1)|A=0)\geq \frac{m_1}{BF_U}\geq \frac{m_1}{BF^*_U}$.
Second, we know that $\frac{E(Z(0)=1|A=1)}{m_0}=\frac{\int r_0(u)F_1(du)}{\int r_0(u)F_0(du)}=CMR_{AZ^+}\leq BF_U\leq BF^*_U$, then we have $E(Z(0)=1|A=1)\leq m_0 BF_U\leq m_0 BF^*_U$.
By definition of ACE and the inequalities derived above, we have \begin{align*} ACE_{AZ^+}^{true}&=m_1-\int r_0(u)F_1(du)\geq m_1-m_0\times BF^*_U ;\\ ACE_{AZ^-}^{true}&=\int r_1(u)F_0(du)-m_)\geq m_1/BF^*_U-m_0 ;\\ ACE_{AZ}^{true}&=f\cdot ACE_{AZ^+}^{true}+(1-f)ACE_{AZ^-}^{true}\\ &\geq f(m_1-m_0BF^*_U)+(1-f)(\frac{m_1}{BF^*_U}-m_0)\\ &=(m_1-m_0\times BF^*_U)[f+(1-f)/BF^*_U]\\ &=(\frac{m_1}{B^*F_U}-m_0)[f\times BF^*_U+(1-f)]. \end{align*} Similarly, we can prove the inequalities hold for nonnegative outcomes and $ACE_{AZ}^{obs}<0$. \end{proof}
\begin{proposition} \label{sa_prop5} In the matched sample, we have the following inequality for nonnegative outcomes and $ACE_{AZ}^{obs}\geq0$:
$$ACE_{AZ}^{true}\geq \frac{1}{2}(1+\frac{1}{BF_U^*})E(Z|A=1)-\frac{1}{2}(1+BF_U^*)E(Z|A=0).$$ In the matched sample, we have the following inequality for nonnegative outcomes and $ACE_{AZ}^{obs}<0$:
$$ACE_{AZ}^{true}\leq \frac{1}{2}(1+BF_U^*)E(Z|A=1)-\frac{1}{2}(1+\frac{1}{BF_U^*})E(Z|A=0).$$ \end{proposition}
\begin{proof}
In the matched sample, we have $f=P(A=1|e(X)=e(x))=0.5$.
For nonnegative outcomes and $ACE_{AZ}^{obs}\geq0$, we have $LHS=ACE_{AZ}^{true}=\underset{e(x)}{\sum}ACE_{AZ|e(X)=e(x)}^{true}P(e(X)=e(x))$ and \begin{align*} RHS=&\underset{e(x)}{\sum}(m_1-m_0 BF^*_U)(f+\frac{1-f}{BF^*_U})P(e(X)=e(x))\\ =&(\frac{1}{2}-\frac{1}{2}BF^*_U)\underset{e(x)}{\sum}m_1P(e(X)=e(x))+\frac{1}{BF^*_U}\underset{e(x)}{\sum}m_1P(e(X)=e(x))\\ &+(\frac{1}{2}-\frac{1}{2}BF^*_U)\underset{e(x)}{\sum}M_0P(e(X)=e(x))-\underset{e(x)}{\sum}m_0P(e(X)=e(x))\\
=&\frac{1}{2}(1+\frac{1}{BF^*_U})E(Z|A=1)-\frac{1}{2}(1+BF^*_U)E(Z|A=0). \end{align*}
Thus, we have $ACE_{AZ}^{true}\geq \frac{1}{2}(1+\frac{1}{BF_U^*})E(Z|A=1)-\frac{1}{2}(1+BF_U^*)E(Z|A=0)$.
For $ACE_{AZ}^{obs}<0$, we have $LHS=ACE_{AZ}^{true}=\underset{e(x)}{\sum}ACE_{AZ|e(X)=e(x)}^{true}P(e(X)=e(x))$ and
\begin{align*} RHS=&\underset{e(x)}{\sum}(m_1 BF^*_U-m_0)(f+\frac{1-f}{BF^*_U})P(e(X)=e(x))\\ =&\underset{e(x)}{\sum}(m_1 BF^*_U-m_0)(\frac{1}{2}+\frac{1}{2BF^*_U})P(e(X)=e(x))\\ =&\frac{1}{2}(1+\frac{1}{BF^*_U})[BF^*_U\underset{e(x)}{\sum}m_1P(e(X)=e(x))-\underset{e(x)}{\sum}m_0P(e(X)=e(x))]\\
=&\frac{1}{2}(1+BF_U^*)E(Z|A=1)-\frac{1}{2}(1+\frac{1}{BF_U^*})E(Z|A=0). \end{align*}
Thus, we have $ACE_{AZ}^{true}\leq \frac{1}{2}(1+BF_U^*)E(Z|A=1)-\frac{1}{2}(1+\frac{1}{BF_U^*})E(Z|A=0)$. \end{proof}
\section*{Web Appendix C: Estimation of $G_{ij}(u,v)$}
To compute the variance of RMSTs, one difficulty is to estimate the function $G_{ij}(u,v)$ based on data. Follow the notations in \citet{murray2000variance}, we need to transform the function $G_{ij}(u,v)$ into the counting process notation system. Suppose we have $n$ matched pairs, then let $i,j$ denote the groups and $k=1,\cdots, n$ denotes the $k$th pair. Let $U_{ik}$ be the censoring random variable corresponding to survival time $T_{ik}$, and the censored survival time is $X_{ik}=\min (T_{ik},U_{ik})$ with censoring status $\Delta_{ik}=I(T_{ik}<U_{ik})$. Then we have the following definitions: \begin{itemize} \item $Y_i(u)=\underset{k=1}{\overset{n}{\sum}}I(x_{ik}\geq u)$ and $Y_j(v)=\underset{k=1}{\overset{n}{\sum}}I(x_{jk}\geq v)$; \item $Y_{ij}(u,v)=\underset{k=1}{\overset{n}{\sum}}I(x_{ik}\geq u,x_{jk}\geq v)$; \item $dNi(u)=\underset{k=1}{\overset{n}{\sum}}I(u\leq x_{ik}<u+\Delta u, \Delta_{ik}=1)$, where $\Delta u\rightarrow 0$; \item $dNj(v)=\underset{k=1}{\overset{n}{\sum}}I(v\leq x_{ik}<v+\Delta v, \Delta_{ik}=1)$, where $\Delta v\rightarrow 0$; \item $dN_{ij}(u,v)=\underset{k=1}{\overset{n}{\sum}}I(u\leq x_{ik}< u+\Delta u,v\leq x_{jk}<v+\Delta v, \Delta_{ik}=1,\Delta_{jk}=1)$, where $\Delta u\rightarrow 0$ and $\Delta v\rightarrow 0$;
\item $dN_{ij}(u|v)=\underset{k=1}{\overset{n}{\sum}}I(u\leq x_{ik}<u+\Delta u, x_{jk}\geq v, \Delta_{ik}=1)$, where $\Delta u\rightarrow 0$;
\item $dN_{j|i}(v|u)=\underset{k=1}{\overset{n}{\sum}}I(v\leq x_{jk}<v+\Delta v, x_{ik}\geq u, \Delta_{jk}=1)$, where $\Delta v\rightarrow 0$; \item The $\hat{G}_{ij}(u,v)$ could be estimated by the formula below, and we set $\Delta u=0$ and $\Delta v=0$ in real computation. The corresponding R code could be found in our supplementary materials. \begin{align*}
\hat{G}_{ij}(u,v)=&n\frac{Y_{ij}(u,v)}{Y_i(u)Y_j(v)}\Bigg[\frac{dN_{ij}(u,v)}{Y_{ij}(u,v)}-\frac{dN_{ij}(u|v)dN_j(v)}{Y_{ij}(u,v)Y_j(v)}\\
&-\frac{dN_{j|i}(v|u)dN_i(u)}{Y_{ij}(u,v)Y_i(u)}+\frac{dN_i(u)dN_j(v)}{Y_i(u)Y_j(v)}\Bigg] \end{align*} \end{itemize}
\end{document} | arXiv |
\begin{definition}[Definition:Closure Operator (Matroid)]
Let $M = \struct{S, \mathscr I}$ be a matroid.
The '''closure operator''' of the matroid $M$ is the mapping $\sigma : \powerset S \to \powerset S$ defined by:
:$\map \sigma A$ is the set of elements of $S$ which depend on $A$
\end{definition} | ProofWiki |
\begin{definition}[Definition:Uniform Continuity/Metric Space]
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Then a mapping $f: A_1 \to A_2$ is '''uniformly continuous on $A_1$''' {{iff}}:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x, y \in A_1: \map {d_1} {x, y} < \delta \implies \map {d_2} {\map f x, \map f y} < \epsilon$
where $\R_{>0}$ denotes the set of all strictly positive real numbers.
\end{definition} | ProofWiki |
\begin{document}
\allowdisplaybreaks \title{On the singularities of effective loci of line bundles}
\author{Lei Song}
\address{Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA}
\email{[email protected]}
\dedicatory{}
\keywords{semi-regularity, Brill-Noether loci, rational singularities}
\begin{abstract} We prove that every irreducible component of semi-regular loci of effective line bundles in the Picard scheme of a smooth projective variety has at worst rational singularities. This generalizes Kempf's result on rational singularities of $W^0_d$ for smooth curves. We also work out an example of such loci for a ruled surface. \end{abstract}
\maketitle
\section{Introduction} Fix a ground field $k$, which is algebraically closed and of characteristic 0. Let $X$ be a smooth projective curve of genus $g$. For $r, d\ge 0$, the Brill-Noether locus is defined as \begin{equation*}
\textrm{supp}(W^r_d(X))=\{L\in\textrm{Pic}(X)\;|\; h^0(L)\ge r+1, \deg(L)=d\}. \end{equation*} There has been extensive research on these loci in literature (cf. \cite{ACGH}). A special kind of Brill-Noether loci is $W^0_d(X)$, which is the image of the Abel-Jacobi map $\varphi: X_d\rightarrow\textrm{Pic}^d X$, where $X_d$ denotes the $d$th symmetric product of $X$ and $\textrm{Pic}^d X$ denotes the Picard variety of degree $d$ line bundles on $X$. When $d=g-1$, $W^0_{g-1}$ is a theta divisor if $\textrm{Pic}^{g-1}X$ is indentified with the Jacobian of the curve.
Kempf \cite{KE} proved that $W^0_d(X)$ has only rational singularities, so in particular it is Cohen-Macaulay and normal; and for $1\le d\le g-1$, the tangent cone over a point $[L]\in W^0_d(X)$ admits a rational resolution from the normal bundle of the fibre $F=\varphi^{-1}([L])$. He also computed the degree of the tangent cone, generalizing Riemman's formula on multiplicity of theta divisors. A celebrated generalization of Kempf's theorem in the case $d=g-1$ is due to Ein and Lazarsfeld \cite[Theorem 1]{Ein97} stating that any principal polarization divisor $\Theta\subset A$ on an abelian variety is normal and has rational singularities.
This paper attempts to extend part of Kempf's results on $W^0_d$ for curves to higher dimensional varieties using the approach and technique of Ein \cite{Ein}, where the author studied the normal sheaf $\shf{N}$ of the fibre $F$. He showed $\shf{N}$ can be reconstructed from the multiplication map $H^0(\sshf{F}(1))\otimes H^1(\shf{N}(-1))\rightarrow H^1(\shf{N})$, and proved that for a general curve $X$, $\shf{N}\simeq\rho\sshf{F}\oplus (H^1(X, L)\otimes\Omega_{F}(1))$, where $\rho=g-(r+1)(g+r-d)$ is the Brill-Noether number. A large part of his results were built on a locally free resolution of $\shf{N}^*$.
Now let $X$ be a smooth projective variety of arbitrary dimension. Let $\textrm{Pic}(X)$ and $\textrm{Div}(X)$ denote the Picard scheme and divisor scheme, which parameterize line bundles and effective divisors on $X$ respectively. One still has the Abel-Jacobi map $\varphi: \textrm{Div}(X)\rightarrow\textrm{Pic}(X)$, where $\textrm{Div}(X)$ plays the same role as $X_d$. However, as a closed subscheme of the Hilbert scheme $\textrm{Hilb}(X)$, $\textrm{Div}(X)$ may be very singular. Even for $\dim X=2$, an example due to Severi and Zappa in 1940s shows that $\textrm{Div}(X)$ can be nonreduced. For this reason, we restrict ourselves to those so called semi-regular line bundles (see \S 2 for definition), and consider the semi-regular locus $W^0_{\textrm{sr}}(X)$ they form in $\textrm{Pic}(X)$. We refer to \cite{KL} for background of $\textrm{Pic}(X)$ and $\textrm{Div}(X)$. Our main theorem is
\begin{theorem}\label{main theorem} Let $X$ be a smooth projective variety. Then any irreducible component of $W^0_{\textrm{sr}}(X)$ has only rational singularities. \end{theorem}
When $X$ is a curve, $W^0_{\textrm{sr}}=\coprod_{d\ge 0} W^0_d$ and $W^0_d$ is irreducible, and so the theorem recovers Kempf's result that $W^0_d$ has rational singularities.
The paper is organized as follows: in \S 2, we study the conormal sheaf of fibres of the Abel-Jacobi map. We derive a resolution of the sheaf and obtain several interesting consequences. With a criterion of rational singularities based on Kov\'{a}cs's work, we prove our main theorem. In \S 3, one example of an irreducible component $W^0_{\textrm{sr}}$ of a ruled surface is analyzed in detail. In the appendix we prove an auxiliary result on varieties swept out by linear spans of divisors of linear systems on an embedded curve.
{\it Acknowledgments}: The author is very grateful to his advisor Lawrence Ein for suggesting this problem and many helpful discussions. He would like to thank Chih-Chi Chou, Izzet Coskun for stimulating discussions concerning \S 3, and thank Wenbo Niu and Pete Vermeire for answering his questions. He also would like to thank the anonymous referee, whose careful reading and valuable suggestions greatly improve this paper.
\section{Rational singularities of $W^0_{\text{sr}}(X)$}
\subsection{Semi-regular line bundles and their loci}
Let $X$ be a smooth projective variety. It is well known that $\textrm{Pic}(X)$ is separated and smooth over $k$. As $\textrm{Hilb}(X)$ breaks into connected components according to Hilbert polynomials, so does $\textrm{Pic}(X)$. Fix an ample line bundle $\sshf{X}(1)$ on $X$. For each line bundle $L$ on $X$, there exists a $\mathbb{Q}$-coefficient polynomial $P_{L}$ such that $P_{L}(n)=\chi(L(n))$ for $n\in\mathbb{Z}$. The $P_{L}$ is constant over any connected component of $\textrm{Pic}(X)$. The Abel-Jacobi map $\varphi: \textrm{Div}(X)\rightarrow \textrm{Pic}(X)$, which sends an effective divisor $D$ to the associated line bundle $\sshf{X}(D)$, is a projective morphism. For any line bundle $L$ on $X$, canonically $\varphi^{-1}([L])\simeq |L|$, where $[L]$ is the corresponding point of $L$ in $\textrm{Pic}(X)$ (cf. \cite{KL}).
\begin{definition}\label{semi-regular line bundle} An effective Cartier divisor $D$ on $X$ is \textit{semi-regular} if the boundary map \begin{equation*}
\partial: H^1(\sshf{D}(D))\rightarrow H^2(\sshf{X}) \end{equation*}
is injective. A line bundle $L$ is \textit{semi-regular} if $L$ is effective, and $D$ is semi-regular for all $D\in|L|$. \end{definition}
\begin{remark}
If $X$ is a curve, then all effective divisors, line bundles are automatically semi-regular. The reader can check that a necessary condition for $L$ to be semi-regular is that $h^1(L)\le q$ (see Corollary \ref{Clifford inequality}), and sufficient conditions are either $h^1(L)=0$ or $H^1(\sshf{D}(D))=0$ for all $D\in |L|$. The second one is however rather strong. For instance, when $X$ is a surface and $p_g=h^0(\omega_X)>0$, $H^1(\sshf{D}(D))=0$ implies that $\supp D\subset \textrm{Bs}(|\omega_X|)$. \end{remark}
\begin{theorem}[Severi-Kodaira-Spencer]\label{Severi-Kodaira-Spencer} Assume $\text{char}(k)=0$. $\textrm{Div}(X)$ is smooth at $[D]$ of the expected dimension \begin{equation}\label{expected dimension}
R:=h^0(\sshf{X}(D))-h^1(\sshf{X}(D))-1+h^1(\sshf{X}) \end{equation} if and only if $D$ is semi-regular (cf. \cite{KL} or \cite{MU}). \end{theorem}
\begin{definition}[semi-regular locus] \begin{equation*}
\textrm{supp}(W^0_{\textrm{sr}}(X))=\{L\in\textrm{Pic}(X)\;|\; L \;\textrm{is effective and semi-regular} \}.
\end{equation*} \end{definition}
\begin{remark} From Theorem \ref{Severi-Kodaira-Spencer}, we see that ``semi-regular" is an open condition on the locus of effective line bundles $W^0(X)\subset\textrm{Pic}(X)$ and that any connected component of $W^0_{sr}(X)$ is irreducible. Thus each component of $W^0_{\textrm{sr}}(X)$ is a subvariety of $\textrm{Pic}(X)$. It is worth noting that not every irreducible component of $W^0(X)$ contains some semi-regular line bundle, see remark (\ref{non semi-regular locus}) for an example. \end{remark}
Next we explain the idea of the proof of Theorem \ref{main theorem}. Given $[L]\in\Omega$, which is a component of $W^0_{\textrm{sr}}$, there exists a unique $\Delta_0$ among all irreducible components $\{\Delta_i\}$ of $\textrm{Div}(X)$, such that $\dim{\Delta_0}=R$ and $\varphi^{-1}([L])\subset (\Delta_0)_{reg}\backslash \cup_{i\neq 0} \Delta_i$, where $(\Delta_0)_{reg}$ is the regular locus of $\Delta_0$. Consider the induced Abel-Jacobi morphism $\varphi: \Delta_0\rightarrow \Omega$. By properness of $\varphi$, there is a smooth neighborhood $U$ of $\varphi^{-1}([L])$ inside $\Delta_0$ such that $\varphi(U)$ is open in $\Omega$ and $\varphi^{-1}\varphi(U)=U$. By abuse of notation, we denote this $U$ by $\textrm{Div}(X)$, hence the normal sheaf of the fibre by $\shf{N}_{\varphi^{-1}([L]) /{\textrm{Div}(X)}}$ instead of $\shf{N}_{\varphi^{-1}([L])/ U}$, and simply by $\shf{N}$ if the fibre is clear from the context. Under the semi-regularity assumption, $\shf{N}$ can be calculated from the universal family of divisors associated to $|L|$ by base change theorem. It turns out that the vector bundle $\shf{N}^*$ (on the projective space $|L|$) has Castelnuovo-Mumford regularity 0, therefore formal function theorem shows that $R^i\varphi_*(\sshf{\Delta_0})_{[L]}=0$ for $i>0$. Finally, though $\varphi$ is not a resolution of singularities of $\Omega$, as it is not birational in general, Theorem \ref{Kovacs} guarantees that $\Omega$ has at worst rational singularities at $[L]$.
\subsection{A criterion for rational singularities} The theorem below is a characterization of rational singularities. The original assumption is more general than what we state here. \begin{theorem}[Kov\'{a}cs \cite{SK}]\label{Kovacs} Let $f: Y\rightarrow X$ be a surjective proper morphism of varieties. Assume that $Y$ has rational singularities and that $f_*\sshf{Y}\simeq \sshf{X}$, $R^i f_*\sshf{Y}=0$ for all $i>0$. Then $X$ has rational singularities. \end{theorem}
Based on the above, we obtain Theorem \ref{rational}, which is more general than what we actually need to prove Theorem \ref{main theorem} and may be applied to other problems.
\begin{lemma}\label{proper line bundle} Let $X$ be a smooth projective variety of dimension $d$ with $H^1(X, \sshf{X})=0$, and $L$ a globally generated ample line bundle on $X$ with the property that $K_X+(d-1)L$ is noneffective. Then any $m$-regular (with respect to $L$) coherent sheaf $\shf{F}$ admits a locally free resolution of the form: \begin{equation*}
\cdots\rightarrow V_i\otimes L^{-(m+i)}\cdots\rightarrow V_1\otimes L^{-(m+1)}\rightarrow V_0\otimes L^{-m}\rightarrow\shf{F}\rightarrow 0, \end{equation*} where each $V_i$ is a finite dimensional vector space. Consequently if $\shf{F}$ is locally free and $0$-regular, then any symmetric power of $\shf{F}$ is $0$-regular. \end{lemma} \begin{proof} By Kodaira vanishing and the assumption $H^1(X, \sshf{X})=1$, the condition that $K_X\otimes L^{d-1}$ is noneffective implies that reg(${\sshf{X}})\le 1$. Then the result follows from \cite{LA} Remark 1.8.16. \end{proof} \begin{remark} Notice that the existence of such $L$ in Lemma \ref{proper line bundle} imposes a strong restriction on $X$: $-K_X$ is big. Examples for Fano varieties are $\prj{d}$, quadric hypersurface $Q\subset\prj{d+1}$, and $\mathbb{P}(\sshf{\prj{1}}^{\oplus (d-1)}\oplus\sshf{\prj{1}}(1))$ (with Fano index $1$). If $-K_X$ is also nef, the assumption $H^1(X, \sshf{X})=0$ is redundant by Kawamata-Viehweg vanishing. \end{remark}
\begin{theorem}\label{rational} Let $f: Y\rightarrow X$ be a projective morphism from a smooth variety $Y$ onto a normal variety $X$. Let $p\in X$ be a closed point. Suppose the scheme-theoretic fiber $F$ is a smooth variety of dimension $d$ with $H^i(F, \sshf{F})=0$ for all $i>0$, and the conormal sheaf $\shf{N}^* _{F/Y}$ is 0-regular with respect to a globally generated ample line bundle $L$ on $F$, such that $K_F+(d-1)L$ is noneffective. Then $X$ has rational singularities in a neighbourhood of $p$. \end{theorem}
\begin{proof} Consider the Stein factorization of $f: Y\xrightarrow{f'} Y'\xrightarrow{g} X$, where $f'$ is projective with connected fibers, and $g$ is a finite morphism. Then $Z:=g^{-1}(p)$ is a reduced closed point, for otherwise $F=f'^{-1}(Z)$ would be nonreduced. Therefore $g$ is generically one to one map, and hence birational. Since $X$ is normal, we have $g$ is an isomorphism, and hence $f$ has connected fibres.
Let $\is{}$ be the ideal sheaf of $F$ in $Y$. By Lemma \ref{proper line bundle}, any symmetric power $S^n(\shf{N}^*_{F/Y})\simeq\is{}^n/{\is{}^{n+1}}$ is $0$-regular. In particular, all its higher cohomologies vanish. From the exact sequence \begin{equation*} 0\rightarrow \is{}^n/\is{}^{n+1}\rightarrow \mathcal{O} _{(n+1)F}\rightarrow \mathcal{O} _{nF}\rightarrow 0, \end{equation*} we get $H^i({O} _{(n+1)F})\cong H^i({O} _{nF})$ for all $i,n>0$. Then we conclude that $H^i({O} _{nF})=0$ for all $i>0$. So $R^i f_*(\mathcal{O} _Y)^{\wedge}_p=0$ for all $i>0$, by the formal function theorem (cf. \cite[III 11.1]{HA}). Since the support of the coherent sheaf $R^i f_*(\mathcal{O} _Y)$ is closed, by shrinking $X$, we can assume $R^i f_*(\mathcal{O} _Y)=0$ on X for $i>0$.
It's clear that $f_*\mathcal{O}_Y=\mathcal{O}_X$, since $X$ is normal and fibers are connected. At this point, we apply Theorem \ref{Kovacs} to conclude the proof. \end{proof}
\subsection{Conormal sheaf of the fibre of $\varphi$}\mbox{}\\
In the rest of \S 2, $L$ stands for a line bundle on $X$ with $r=\dim{|L|}$ and $b=h^1(X, L)$, $F$ denotes the fibre $\varphi^{-1}([L])\simeq |L|$ of the Abel-Jacobi map $\varphi: \textrm{Div}(X)\rightarrow\textrm{Pic}(X)$. Let $\frak{m}$ be the maximal ideal of $\sshf{\textrm{Pic}(X), [L]}$, $\bar{\frak{m}}$ the maximal ideal of $\sshf{W^0_{\textrm{sr}}, [L]}$, and $\is{}$ the ideal sheaf of $F$ in $\textrm{Div}(X)$.
Given an effective line bundle $L$, $|L|\simeq \mathbb{P}(H^0(X, L)^*)$. Let $Y=X\times|L|$ and $p, q$ be the two projections. By K$\ddot{\textrm{u}}$nneth formula, $\Gamma(Y, p^*L\otimes q^*\sshf{}(1))\simeq H^0(X, L)\otimes H^0(X, L)^*$. Fix a basis of the vector space $H^0(X, L)$, say $x_0, \cdots, x_r$. The canonical section $\displaystyle {s=\sum^r_{i=0} x_i\otimes x^*_i}$ defines a relative Cartier divisor of $Y$ over $|L|$ via \begin{equation}\label{universal family}
0\rightarrow{\sshf{Y}}\xrightarrow{.s}{p^*L\otimes q^*\sshf{}(1)}\rightarrow{\sshf{\mathscr{D}}(\mathscr{D})}\rightarrow 0. \end{equation}
Denote the two induced projections from $\mathscr{D}$ to $X$ and $|L|$ also by $p$ and $q$. The divisor $\mathscr{D}$ is actually an incidence correspondence in the sense: for any $x\in X$, $p^{-1}(x)$ parameterizes the effective divisors passing through $x$; for any $[D]\in |L|$, $q^{-1}([D])$ is precisely the divisor $D$.
By the universal property of $\textrm{Div}(X)$, there is a unique morphism $j: |L|\rightarrow \textrm{Div}(X)$ such that $\mathscr{D}=j^*\mathscr{U}$, where $\mathscr{U}$ is the universal divisor over $\textrm{Div}(X)$, see the Cartesian diagram below. In fact $j$ is a closed immersion. $$\xymatrix{ D \ar@{^{(}->}[d] \ar@{^{(}->}[r] &\mathscr{D} \ar@{^{(}->}[d] \ar@{^{(}->}[r] & \mathscr{U}\ar@{^{(}->}[d]\\
X \ar[d] \ar@{^{(}->}[r] &X\times|L| \ar[d]^{q} \ar@{^{(}->}[r]^{j'} & X\times\textrm{Div}(X) \ar[d]^{\pi}\\
[D]\ar@{^{(}->}[r] &|L| \ar@{^{(}->}[r]^{j} & \textrm{Div}(X)}$$
Suppose $L$ is semi-regular, then there is a smooth open neighborhood $V$ of $|L|$ in $\text{Div}(X)$, and hence $\shf{T}_\text{Div(X)}\big|_V$, the restriction of the tangent sheaf of $\text{Div}(X)$ to $V$, is locally free.
Denoting the projections from $X\times\textrm{Div}(X)$ by $\pi'$ and $\pi$ respectively, we have the natural morphism \begin{equation}\label{natural morphism}
\shf{T}_\text{Div(X)}\big|_{V}\simeq \pi_*\paren{\pi^*\shf{T}_\text{Div(X)}\big|_{V}} \hookrightarrow \pi_*\paren{\pi^*\shf{T}_\text{Div(X)}\big|_{V}\oplus {\pi'}^*\shf{T}_X}\rightarrow \paren{\pi_*\sshf{\mathscr{U}}(\mathscr{U})}\big|_V. \end{equation} On the other hand, by Grauert's theorem, for all $[D]\in V$, \begin{equation}\label{iso at stalks}
\pi_*\sshf{\mathscr{U}}(\mathscr{U})\otimes\kappa([D])\simeq H^0(X, \shf{N}_{D/X})\simeq\shf{T}_\textrm{Div(X)}\otimes\kappa([D]), \end{equation} where $\kappa([D])$ is the residue field at $[D]$ and the last isomorphism follows from the property of $\pi$.
In view of (\ref{natural morphism}) and (\ref{iso at stalks}), we get an isomorphism \begin{equation*}
\paren{\pi_*\sshf{\mathscr{U}}(\mathscr{U})}\big|_V\simeq\shf{T}_\textrm{Div(X)}\big|_V. \end{equation*}
Then the base change theorem (cf. \cite[Lecture 7]{MU}) implies that \begin{equation*}
j^*\pi_*\sshf{\mathscr{U}}(\mathscr{U})\simeq q_*{j'}^*\sshf{\mathscr{U}}(\mathscr{U}). \end{equation*}
Since ${j'}^*\sshf{\mathscr{U}}(\mathscr{U})\simeq\sshf{\mathscr{D}}(\mathscr{D})$, one has \begin{equation}\label{pull back of normal sheaf}
q_*{\sshf{\mathscr{D}}(\mathscr{D})}\simeq\shf{T}_\text{Div(X)}\big |_{|L|}. \end{equation}
The key theorem below is a generalization of \cite[Theorem 1.1]{Ein}.
\begin{theorem}\label{conormal sheaf} With notation as above, let $L\in W^0_{sr}$ and $\shf{N}^*$ be the conormal sheaf of $F$ in $\textrm{Div}(X)$. Then there is an exact sequence \begin{equation}\label{resolution of conormal sheaf}
\ses{H^1(L)^*\otimes\sshf{F}(-1)}{H^1(\sshf{X})^*\otimes\sshf{F}}{\shf{N}^*}. \end{equation} \end{theorem} \begin{proof} Note $\sshf{Y}(\mathscr{D})\simeq p^*L\otimes q^*\sshf{F}(1)$. Applying $q_*$ to (\ref{universal family}), we get the exact sequence \begin{equation*}
0\rightarrow \sshf{F}\rightarrow H^0(L)\otimes\sshf{F}(1)\rightarrow {\shf{T}_{Div(X)}\big|_{F}}\rightarrow H^1(\sshf{X})\otimes\sshf{F}\rightarrow H^1(L)\otimes\sshf{F}(1)\rightarrow 0, \end{equation*} where the third term comes from (\ref{pull back of normal sheaf}). The surjectivity of the last map is for the reason as follows. For any point $[D]\in F$, we have the commutative diagram $$\xymatrix{ R^1q_*\sshf{Y}\otimes\kappa([D])\ar[d] \ar[r]^{\simeq} &H^1(\sshf{X})\ar[d]\\ R^1q_*\sshf{Y}(\mathscr{D})\otimes\kappa([D])\ar[r]^{\simeq} &H^1(\sshf{X}(D))}$$
The two horizontal maps are isomorphisms because of Grauert's theorem. Since $\partial: H^1(\sshf{D}(D))\rightarrow H^2(\sshf{X})$ is injective by the semi-regularity assumption, the right vertical map is surjective, so is the left one.
Since the cokernel of $\sshf{F}\rightarrow H^0(L)\otimes\sshf{F}(1)$ is $\shf{T}_{F}$, we get the short sequence \begin{equation*}
\ses{\shf{N}}{H^1(\sshf{X})\otimes\sshf{F}}{H^1(L)\otimes\sshf{F}(1)}. \end{equation*} Dualizing it, we get \begin{equation*}
\ses{H^1(L)^*\otimes\sshf{F}(-1)}{H^1(\sshf{X})^*\otimes\sshf{F}}{\shf{N}^*}\qedhere \end{equation*} \end{proof}
\begin{corollary}\label{higher cohomology of conormal sheaf} $\shf{N}^*$ has Castelnuovo-Mumford regularity 0.\qed \end{corollary}
\subsection{Proof of theorem \ref{main theorem}}\mbox{}\\ From now on, let $q$ denote $h^1(X, \sshf{X})$, the irregularity of $X$. \begin{lemma}\label{alpha map} The natural map $\bigoplus_{n\ge 0}\bar{\frak{m}}^n/\bar{\frak{m}}^{n+1}\rightarrow\bigoplus_{n\ge 0}H^0(\is{}^n/\is{}^{n+1})$ is a surjective graded $k$-algebra morphism. Furthermore, if $R\le q$, then it is an isomorphism. \end{lemma} \begin{proof} First consider the commutative diagram of $k$-vector spaces: $$\xymatrix{ \frak{m}/{\frak{m}}^2 \ar@{->>}[r]\ar[d]^{\simeq} & \bar{\frak{m}}/\bar{\frak{m}}^{2}\ar[d]\\ H^1(\sshf{X})^* \ar@{->>}[r] & H^0\paren{\shf{N}^*_{F/{Div(X)}}}}$$ Clearly the top horizontal map is surjective. The left vertical map is an isomorphism by \cite[Theorem 5.11]{KL}. The bottom horizontal one is surjective by (\ref{resolution of conormal sheaf}). Therefore \begin{equation*}
\bar{\frak{m}}/\bar{\frak{m}}^{2}\rightarrow H^0\paren{\shf{N}^*_{F/{Div(X)}}} \end{equation*} is surjective. For $n> 1$, consider the commutative diagram: $$\xymatrix{ S^n(\bar{\frak{m}}/\bar{\frak{m}}^2) \ar@{->>}[r]\ar@{->>}[d] & \bar{\frak{m}}^n/\bar{\frak{m}}^{n+1}\ar[d]\\ S^nH^0(\is{}/\is{}^{2} )\ar@{->>}[r] & H^0(\is{}^n/\is{}^{n+1})}$$ The bottom horizontal map is surjective, because $\is{}/\is{}^{2}=\shf{N}^*_{F/{Div X}}$ is 0-regular. It follows that the right vertical map is surjective.
A proof for isomorphism when $R\le q$ can be found in \cite{Ein} Proposition 3.1 (c) and Theorem 3.2. \end{proof}
\begin{proof}[Proof of Theorem 1.1]
By Theorem \ref{rational} and Corollary \ref{higher cohomology of conormal sheaf}, to finish the proof, it remains to show that any irreducible component $\Omega$ of $W^0_{\textrm{sr}}$ is normal. Let $Y=\varphi^{-1}(\Omega)\subset\textrm{Div}(X)$. Consider the commutative diagram with exact rows $$\xymatrix{ 0\ar[r] & \bar{\frak{m}}^n/ \bar{\frak{m}}^{n+1}\ar[d]^{\alpha_n}\ar[r] & {\sshf{\Omega, [L]}/{\bar{\frak{m}}}^{n+1}}\ar[d]^{\beta_{n+1}}\ar[r] &\sshf{\Omega, [L]}/\bar{\frak{m}}^n\ar[r]\ar[d]^{\beta_n} &0\\ 0\ar[r] & H^0(\is{}^n/\is{}^{n+1})\ar[r] & H^0(\sshf{(n+1)F})\ar[r] & H^0(\sshf{nF})\ar[r]& 0}$$ By Lemma \ref{alpha map}, $\alpha_n$'s are surjective. By Snake lemma and induction on $n$, we get $\beta_n$ is surjective for all $n\ge 1$. It follows that $\sshf{\Omega, [L]}^{\wedge}=\varprojlim \sshf{\Omega, {[L]}}/\bar{\frak{m}}^n\twoheadrightarrow\varprojlim H^0(\sshf{nF})\simeq(\varphi_*\sshf{Y})^{\wedge}_{[L]} $ by formal function theorem. Thus the canonical morphism $\sshf{\Omega, {[L]}}^{\wedge}\rightarrow (\varphi_*\sshf{Y})^{\wedge}_{[L]}$ is an isomorphism. Since the completion is a fully faithful functor, we get that $\varphi_*\sshf{Y}\simeq\sshf{\Omega}$. Since $Y$ is smooth and all fibres are connected, $\Omega$ is normal. \end{proof}
\subsection{Some consequences of Theorem \ref{conormal sheaf}}\mbox{}\\
The corollary below is a generalization of Clifford theorem to higher dimensional varieties. It would be interesting to study when the equalities can be achieved. \begin{corollary}\label{Clifford inequality} Assume $[L]\in W^0_{\textrm{sr}}$ and $h^1(L)>0$. Then\\ (i) $h^0(L)+h^1(L)\le q+1$.\\ (ii) If $X$ is a projective surface, then $h^0(L)\le \frac{\chi(L)+q+1}{2}$. \end{corollary} \begin{proof} By \cite[Proposition 2.5]{Ein}, the shape of the resolution (\ref{resolution of conormal sheaf}) of $\shf{N}^*$ forces $\textrm{rank}(\shf{N}^*)\ge r$. Since $\dim{\textrm{Div(X)}}=R$, one has $\textrm{rank}(\shf{N}^*)=R-r$. Recall $R=h^0(L)-h^1(L)+q-1$ in (\ref{expected dimension}), we get $(i)$. If $X$ is a surface, then $R\le \chi(L)+q-1$. So $h^0(L)=r+1\le \frac{R}{2}+1\le\frac{\chi(L)+q+1}{2}$. \end{proof}
\begin{corollary} Let $[L]\in W^0_{\textrm{sr}}$ and assume $R\le q$. Then up to a constant, the Hilbert-Samuel function $\psi$ for $\sshf{W^0, [L]}$ is \begin{eqnarray*}
\psi(p) &=& \left\{\begin{array}{ll}
{p+q-1\choose q} & \textrm{if $b\le r$}, \\
{p+q-1\choose q}+\sum^b_{i=r+1}(-1)^{i+r}{p+q-i-1\choose q}{b\choose i}{i-1\choose r} & \textrm{if $b>r$}.
\end{array}\right. \end{eqnarray*} \end{corollary} \begin{proof} The exact Eagon-Northcott complex associated to (\ref{resolution of conormal sheaf}) is \begin{eqnarray*}
0&\rightarrow & S^{p-b}H^1(\sshf{X})^*\otimes\bigwedge^bH^1(L)^*\otimes\sshf{\prj{r}}(-b)\rightarrow\cdots\rightarrow S^{p-1}H^1(\sshf{X})^*\otimes H^1( L)^*\otimes\sshf{\prj{r}}(-1)\\
& \rightarrow & S^pH^1(\sshf{X})^*\otimes \sshf{\prj{r}}\rightarrow S^p\shf{N}^*\rightarrow 0. \end{eqnarray*} So for $p\gg 0$, \begin{eqnarray*}
\chi(S^p\shf{N}^*) &=& \sum^b_{i=0}(-1)^i\chi\paren{S^{p-i}H^1(\sshf{X})^*\otimes\bigwedge^iH^1(L)^*\otimes\sshf{\prj{r}}(-i)} \\
&=& \left\{\begin{array}{ll}
{p+q-1\choose q-1} & \textrm{if $b\le r$}, \\
{p+q-1\choose q-1} +\sum^b_{i=r+1}(-1)^{i+r}{p+q-i-1\choose q-1}{b\choose i}{i-1\choose r} & \textrm{if $b>r$}.
\end{array}\right. \end{eqnarray*} Since \begin{eqnarray*}
\Delta\psi(p) &=& \psi(p+1)-\psi(p)\\
&=& \dim{(\bar{\frak{m}}^p/{\bar{\frak{m}}^{p+1}})} \\
&=& h^0(S^p\shf{N}^*) \hspace{3cm}\textrm{by (\ref{alpha map})}\\
&=& \chi(S^p\shf{N}^*) \hspace{3.1cm}\textrm{$S^p\shf{N}^*$ is 0-regular} \end{eqnarray*} we get the conclusion by the proof of \cite[I, 7.3 (b)]{HA} . \end{proof} The multiplicity $\mu(\sshf{W, [L]})$ is defined as $(\text{leading coefficient of } \psi)\cdot(\deg{\psi})!$. To avoid combinatorial relations for calculating $\mu$, we resort to intersection theory.
\begin{corollary}\label{Riemann-Kempf} Let $[L]\in W^0_{\textrm{sr}}$ and assume that $\varphi$ is birational. Then $\mu={b\choose r}$. \end{corollary}
\begin{proof} Since $\mu$ coincides with top Segre class of $([L], W_{sr}^0)$, which is invariant under a birational proper morphism (cf. \cite[Chap. 4]{FUL}), \begin{eqnarray*}
\mu &=& s_0([L], W_{sr}^0) \\
&=& s_{r}(\prj{r}, \textrm{Div}X)\\
&=& (-1)^rs_r(\shf{N}^*). \end{eqnarray*} Let $H$ be the class of a hyperplane section in $\prj{r}$, again by (\ref{resolution of conormal sheaf}), \begin{equation*}
s_t(\shf{N}^*)=(1-Ht)^b, \end{equation*} which concludes the proof. \end{proof}
\section{Examples: Ruled Surface} Though the condition for semi-regular line bundles (Definition \ref{semi-regular line bundle}) looks quite strong, we shall show that it does not automatically imply that $W^0_{sr}(X)$ is smooth, which is clear in the case that $X$ is a curve. We shall construct an example of dimension 2 (for simplicity) following such rules: (i) $W^0_{sr}(X)$ is singular; (ii) $W^0_{sr}(X)$ is nontrivial, i.e.~$W^0_{sr}(X)$ is not isomorphic to $W^0(C)$ for some curve $C$ and $\dim{W^0_{sr}(X)}\ge 2$, in particular $q=\dim{\text{Pic}(X)}\ge 2$; (iii) $W^0_{sr}(X)$ can be explicitly computed, at least for one component. This motivates the work in this section.
We start by fixing some notations. \begin{itemize}
\item $C$: a smooth projective curve of genus $g\ge 2$.
\item $B$: a line bundle on $C$ of degree $d_0\ge 3$.
\item $E$: a rank two vector bundle on $C$, fitting into the exact sequence $\ses{\sshf{C}}{E}{B}$.
\item $X$: $=\mathbb{P}(E)$ and $\pi: X\rightarrow C$ the canonical projection.
\item $\textrm{Pic}^{(i, j)}(X)$: the connected component of $\textrm{Pic}(X)$ consisting of line bundles of the form $\sshf{X}(i)\otimes\pi^* M$, where $\deg{M}=j$.
\item $W^0_{i, j}(X):= W^0_{\textrm{sr}}(X)\cap \textrm{Pic}^{(i, j)}(X)$, where ``sr'' is omitted for simplicity.
\item Linear span: suppose $C$ is embedded into a projective space by a very ample line bundle $A$. Let $D$ be an effective divisor on $C$. One has the exact sequence $\ses{H^0(C, A(-D))}{H^0(C, A)}{Q}$. The \textit{linear span} $\lsp{D}:=\mathbb{P}(Q)\subseteq \mathbb{P}(H^0(C, A))$. Note $H^0(C, A(-D))$ is the space of linear defining equations of $\lsp{D}$.
\item $X_{|L|}$: the variety swept out by linear spans of divisors in $|L|$, where $L$ is a line bundle on $C$. See appendix for its basic properties. The notion $X_{|L|}$ depends on the choice of embedding of $C$. \end{itemize}
\subsection{Geometric interpretation of the extension $\ses{\sshf{C}}{E}{B}$}\mbox{}\\
Since $\deg(K_C\otimes B)=2g-2+d_0\ge 2g+1$, the complete linear system $|K_C\otimes B|$ induces an embedding \begin{equation*}
\varphi: C\hookrightarrow\prj{N}=\mathbb{P}\paren{H^0\paren{K_C\otimes B}}, \end{equation*} where $N=g+d_0-2$. By Serre duality, \begin{equation*}
H^0(C, K_C\otimes B)^*\simeq H^1\paren{C, B^{-1}}\simeq \textrm{Ext}^1\paren{B, \sshf{C}}. \end{equation*} So a point $\eta\in \prj{N}$ determines an extension of $B$ by $\sshf{C}$ \begin{equation}\label{extension}
\ses{\sshf{C}}{E}{B}, \end{equation} uniquely up to isomorphism.
\begin{remark} The idea of realizing an extension class of $B$ by $\sshf{C}$ as a point in $\mathbb{P}(H^0(K_C\otimes B))$ is borrowed from Bertram \cite{BE}. \end{remark}
\subsection{Characterization of $W^0_{1, \star}(X)$} \begin{proposition}[Effectiveness Criterion]\label{characterization} Let $M$ be a line bundle on $C$. $H^0(X, \sshf{X}(1)\otimes\pi^* M)=H^0(C, E\otimes M)\neq 0$ if and only if \begin{enumerate}
\item either $H^0(M)\neq 0$,
\item or $M\simeq B^{-1}\otimes L$ for some effective line bundle $L$, such that $\eta\in X_{|L|}$. \end{enumerate} \end{proposition} \begin{proof} Write $M$ as $B^{-1}\otimes L$ for some line bundle $L$. Twisting (\ref{extension}) by $B^{-1}\otimes L$ yields the exact sequence \begin{equation*}
0\rightarrow{H^0\paren{B^{-1}\otimes L}}\rightarrow{H^0\paren{E\otimes B^{-1}\otimes L}}\rightarrow{H^0(L)}\xrightarrow{\delta} H^1\paren{B^{-1}\otimes L}\rightarrow\cdots, \end{equation*} which implies \begin{equation}\label{number of sections}
h^0\paren{E\otimes M}=h^0(M)+\dim(\ker{\delta}). \end{equation} Then the proposition follows from the lemma below. \end{proof}
\begin{lemma}\label{Kernel} \begin{equation*}
\ker \delta=\{s\in H^0(L) |\; D=(s)_0, \eta\in \lsp{D}\}. \end{equation*} \end{lemma} \begin{proof} Given $s\in H^0(L)$,let $D=(s)_0$. There exists an associated sequence $\ses{B(-D)}{B}{B\otimes\sshf{D}}$, which can be completed by the snake lemma as follows,
\begin{equation*}
\xymatrix{
& & 0\ar[d] & 0\ar[d] \\
0 \ar[r] & \sshf{C}\ar[r]\ar@{=}[d] & F\ar[r]\ar[d] & B(-D)\ar[r]\ar[d]^{\sigma_D} & 0\\
0 \ar[r] & \sshf{C}\ar[r] & E\ar[r]\ar[d] & B \ar[r]\ar[d] & 0 \\
& & B\otimes\sshf{D}\ar[d] \ar@{=}[r] & B\otimes\sshf{D}\ar[d]\\
& & 0 & 0 } \end{equation*} Noting that the extension class of the first row is $\delta(s)\in H^1(B^{-1}\otimes L)\simeq \textrm{Ext}^1(B\otimes L^{-1}, \sshf{C})$, we have \begin{eqnarray*}
&& \eta\in\lsp{D} \\
&\iff & \text{the composition} \quad H^0\paren{K_C\otimes B(-D)}\rightarrow H^0(K_C\otimes B)\xrightarrow{\eta} H^1(K_C)\quad\text{is zero} \\
&\iff& \ses{\sshf{C}}{F}{B(-D)} \quad\text{splits} \\
&\iff & \delta(s)=0. \end{eqnarray*} \end{proof}
Recall that for a ruled surface $X$, $H^2(X, \sshf{X})=0$, so a divisor $\Sigma\subset X$ is semi-regular if and only if $H^1\paren{\sshf{\Sigma}(\Sigma)}=0$. As in \cite{HA}, we denote a closed fibre of $\pi: X\rightarrow C$ by $f$. For a divisor $\alpha$ on $C$, we write $\alpha f$ for $\pi^*\alpha$ by abuse of notation.
\begin{proposition}[Semi-regularity Criterion]\label{obstruction group}
Let $\Sigma=\Gamma+\alpha f\in |\sshf{X}(1)\otimes \pi^*(B^{-1}\otimes L)|$, where $\Gamma$ is the image of a section $\sigma: C\rightarrow X$. Then $\sigma$ corresponds to \begin{equation*}
\ses{B\otimes L^{-1}(\alpha)}{E}{L(-\alpha)}. \end{equation*} And the obstruction group $H^1\paren{\sshf{\Sigma}(\Sigma)}\simeq H^0\paren{C, K_C\otimes B\otimes L^{-2}(\alpha)}^*$. \end{proposition} \begin{proof} $\Gamma$ arises from some one dimensional quotient of $E$ \begin{equation*}
\ses{N}{E}{M}. \end{equation*} By \cite[V, 2.6]{HA}, $\pi^* N\simeq \sshf{X}(1)\otimes\sshf{X}(-\Gamma)$, which is isomorphic to $\pi^*\paren{B\otimes L^{-1}(\alpha)}$, therefore $N\simeq B\otimes L^{-1}(\alpha)$. Since $\det E\simeq B$, one has $M\simeq L(-\alpha)$.
Assume $\alpha=\sum^m_{i=1}a_ip_i$, where $a_i\in \mathbb{N}$ and $p_i\in C$. Let $\Sigma_i=\Gamma+\sum_{j\le i}a_jf$, where $a_if$ is $\pi^{-1}(a_ip_i)$. In this notation $\Sigma_0=\Gamma, \Sigma_m=\Sigma$. Consider the exact sequence \begin{equation*}
\ses{\sshf{\Sigma_{i}}}{\sshf{\Sigma_{i-1}}\oplus\sshf{a_i f}}{\sshf{Z_i}}, \end{equation*} where $Z_i$ is the scheme theoretic intersection of $\Sigma_{i-1}$ with $a_if$. Tensoring it with $\shf{L}:=\sshf{X}(1)\otimes\pi^*(B^{-1}\otimes L)\simeq\sshf{X}(\Sigma)$, one obtains \begin{equation*}
\cdots\rightarrow H^0(\shf{L}|_{\Sigma_{i-1}})\oplus H^0(\shf{L}|_{a_i f})\rightarrow H^0(\shf{L}|_{Z_i})\rightarrow H^1(\shf{L}|_{\Sigma_i})\rightarrow H^1(\shf{L}|_{\Sigma_{i-1}})\oplus H^1(\shf{L}|_{a_i f})\rightarrow 0. \end{equation*}
On the one hand, denote the ideal sheaf of $\pi^{-1}(p_i)\simeq\prj{1}$ by $\is{}$. For $k\ge 1$, there exists the sequence \begin{equation*}
\ses{\is{}^{k-1}/{\is{}^{k}}}{\sshf{X}/{\is{}^{k}}}{\sshf{X}/{\is{}^{k-1}}}. \end{equation*} By flatness of $\pi$ and smoothness of fibre $f$, $\is{}^{k-1}/{\is{}^{k}}\simeq \sshf{\prj{1}}$. One obtains
$H^1(\shf{L}|_{kf})\simeq H^1(\shf{L}|_{(k-1)f})\simeq\cdots H^1(\shf{L}|_{f})\simeq H^1(\prj{1}, \sshf{\prj{1}}(1))=0$. In particular, $H^1(\shf{L}|_{a_i f})=0$.
On the other, $H^0(\shf{L}|_{a_i f})\rightarrow H^0(\shf{L}|_{Z_i})$ is surjective, since its cokernel $H^1(a_if, \shf{L}(-\Gamma)|_{a_if})=0$ by a similar argument as above.
So $H^1(\shf{L}|_{\Sigma_i})\simeq H^1(\shf{L}|_{\Sigma_{i-1}})$. Inductively, one gets \begin{eqnarray*}
H^1(\sshf{\Sigma}(\Sigma)) &\simeq & H^1(\shf{L}|_{\Sigma_m})\\
&\simeq & H^1\paren{\shf{L}|_{\Gamma}}\\
&\simeq& H^1\paren{C, \sigma^*\paren{\sshf{X}(1)\otimes\pi^*\paren{B^{-1}\otimes L}}}\\
&\simeq& H^1\paren{C, B^{-1}\otimes L^2(-\alpha)}\\
&\simeq& H^0\paren{C, K_C\otimes B\otimes L^{-2}(\alpha)}^*, \end{eqnarray*} and we are done. \end{proof}
\begin{remark}\label{non semi-regular locus} The obstruction group indicates that if $2\deg{L}<g+d_0-1$, then $\sshf{X}(1)\otimes \pi^*(B^{-1}\otimes L)$ is not semi-regular. For instance, take $g=3$, $d_0=3$ and $\eta\in S^1 C$. In this case $W^0_{1, -1}\neq\emptyset$, but none of its point is semi-regular. This produces many examples of effective line bundle locus on Pic($X$) which does not contain any semi-regular line bundle. \end{remark}
\begin{proposition}\label{irreducibility}
Assume $H^0\paren{C, B^{-1}\otimes L}=0$ and $\eta\in X_{|L|}$. Then there exists a reducible $\Sigma\in |\sshf{X}(1)\otimes\pi^*(B^{-1}\otimes L)|$ if and only if there exists a line bundle $L'\subsetneq L$, such that $\eta\in X_{|L'|}$. \end{proposition} \begin{proof}
Let $\Sigma=\Gamma+\alpha f\in |\sshf{X}(1)\otimes\pi^*(B^{-1}\otimes L)|$ for some effective $\alpha$ with $\deg{\alpha}\ge 1$. Then
$\sshf{X}(\Gamma)\simeq \sshf{X}(1)\otimes \pi^*(B^{-1}\otimes L(-\alpha))$. Notice $H^0\paren{C, B^{-1}\otimes L(-\alpha)}=0$, so by Corollary \ref{characterization}, $X_{|L(-\alpha)|}\owns\eta$.
Conversely let $L'\simeq L(-\alpha)$ for some effective $\alpha$ with $\deg{\alpha}\ge 1$ and $X_{|L'|}\owns\eta$, then $H^0\paren{\sshf{X}(1)\otimes\pi^*\paren{B^{-1}\otimes L'}}\neq 0$. Let $\Sigma\in |\sshf{X}(1)\otimes\pi^*(B^{-1}\otimes L')| $, then the reducible divisor $\Sigma+\alpha f\in|\sshf{X}(1)\otimes\pi^*(B^{-1}\otimes L)|$. \end{proof}
To state the example below, we define the set \begin{equation*}
X^i_{|L|}=\bigcup_{\substack{h^0(L(-D))\ge 1\\ \deg{D}=\deg{L}-i}}\lsp{D}, \end{equation*}
for $i\ge 0$, namely the set swept out by linear spans of all degree $(\deg{L}-i)$ sub-divisors of $|L|$. The inclusion $X^{i+1}_{|L|}\subseteq X^i_{|L|}$ is clear.
\begin{corollary}\label{semi-regularity}
With the above notation, assume $H^0(C, B^{-1}\otimes L)=0$ and $\eta\in X_{|L|} \backslash X^1_{|L|}$. Then $\sshf{X}(1)\otimes\pi^*(B^{-1}\otimes L)$ is semi-regular if and only if $H^0(C, K_C\otimes B\otimes L^{-2})=0$.\qed \end{corollary}
\subsection{Example}\mbox{}\\
Fix $d_0\ge 3$ as before. Choose positive integers $g$, $d$ satisfying \begin{enumerate}
\item $g+d_0+1\le 2d\le 2g$,
\item $2d-d_0$ is a prime number. \end{enumerate} We can construct a ruled surface $X$ over a curve of genus $g$, such that one component of $W^0_{1, d-d_0}(X)$ of dimension $(2d-d_0-g+1)$ satisfies the rules proposed at the beginning of \S 3.
To be precise, take a \emph{general} curve $C$ of genus $g$ (hence all Brill-Noether loci $W^r_d(C)$ involved below have the expected dimensions $\rho=g-(r+1)(g-d+r)$ ). Recall for any $B$ of degree $d_0$, $|K_C\otimes B|$ induces an embedding $\varphi: C\hookrightarrow\prj{N}$ with $N=g+d_0-2$. Then
\textbf{Claim A}: There exist $B\in\textrm{Pic}^{d_0}(C)$ and $L\in \textrm{Pic}^d(C)$ satisfying the conditions: \begin{enumerate}
\item[A.1] $h^0(B^{-1}\otimes L)=0$,
\item[A.2] $h^0(L)=2$,
\item[A.3] $L$ is base point free,
\item[A.4] $h^0\paren{K_C\otimes B\otimes L^{-2}(p)}=0$ for all $p\in C$,
\item[A.5] Let $\Lambda=\cap_{D\in |L|} \lsp{D}$, then $\Lambda\backslash X^2_{|L|}\neq\emptyset$.
\end{enumerate}
\textbf{Claim B}: Fix $B\in\textrm{Pic}^{d_0}(C)$. Then for general points $q_1, \cdots, q_d$ on $C$, the following conditions hold \begin{itemize}
\item[B.1] $h^0(B^{-1}(q_1+\cdots+q_{d}))=0$,
\item[B.2] $h^0\paren{\sshf{C}(q_1+\cdots+q_{d})}=1$,
\item[B.3] $h^0(K_C\otimes B(-2q_1-\cdots-2q_{d}))=0$. \end{itemize}
Moreover if $\eta\in\Lambda\backslash X^2_{|L|}$ is chosen properly, there is a nonempty (locally closed) subset $W$ of the $d$-th symmetric product of $C$ such that for $q_1+\cdots+q_d\in W$, \begin{itemize}
\item [B.4] $\eta\in\lsp{q_1+\cdots+q_d}$, but $\eta\notin\lsp{q_1+\cdots+\hat{q_i}+\cdots+q_d}$ for all $i$. \end{itemize} Here the notation $\hat{q_i}$ means omit $q_i$.
Granting the claims for the moment, the quadruple $(C, B, \eta, L)$ chosen above determines a ruled surface $\pi: X=\mathbb{P}(E)\rightarrow C$. Let $\shf{L} =\sshf{X}(1)\otimes \pi^*\paren{B^{-1}\otimes L}$. Then $h^0(X, \shf{L})=2$ by A.1, A.2, A.5 and (\ref{number of sections}). Moreover $\shf{L}$ is semi-regular. In fact, writing a divisor $\Sigma\in |\shf{L}|$ as $\Gamma+\alpha f$ as in Proposition \ref{obstruction group}, one has \begin{equation*}
\sshf{X}(\Gamma)\simeq \sshf{X}(1)\otimes\pi^*\paren{B^{-1}\otimes L(-\alpha)}. \end{equation*}
Since $h^0\paren{B^{-1}\otimes L(-\alpha)}=0$, $\eta\in X_{|L(-\alpha)|}$ by Proposition \ref{characterization}. A.5 thereby implies $\deg{\alpha}\le 1$. Therefore one can use Proposition \ref{obstruction group} and A.4 to deduce the semi-regularity of $\shf{L}$. So $\shf{L}\in W^0_{1, d-d_0}(X)$.
On the other hand, for points $q_1, \cdots, q_d$ satisfying B.1-B.4, $\shf{L}':=\sshf{X}(1)\otimes \pi^*B^{-1}(q_d+\cdots+q_d)$ is semi-regular (B.1, B.3, B.4 and Corollary \ref{semi-regularity}), and has one dimensional sections (B.1, B.2, B.4). Because of the irreducibility of $W^0_d(C)$, $\shf{L}'$ specializes to the component $\Omega\subset W^0_{1, d-d_0}(X)$ which contains $\shf{L}$. Therefore, the induced Abel-Jacobi map $\varphi: \textrm{Div} X\rightarrow \Omega$ is a birational morphism, and hence \begin{eqnarray*}
\dim \Omega &=&R\\
&=&\chi(\shf{L})-1+q(X) \hspace{3cm}\text{by}\;(\ref{expected dimension})\\
&=& 2d-d_0+1-g. \hspace{3cm}\text{by the Riemann-Roch for}\;\pi_*\shf{L} \end{eqnarray*} For $\shf{L}$, $b=r+1-\chi(\shf{L})=2g+d_0-2d$, as no $H^2$ involved. We apply Corollary \ref{Riemann-Kempf} to get that the multiplicity $\mu$ of $\Omega$ at $[\shf{L}]$ equals $2g+d_0-2d$, which is larger than 1. Our main theorem asserts that $\Omega$ has at worst rational singularities.
\subsection{Proofs of Claims}\mbox{}\\
The proof for A.1-A.4 proceeds by dimension counting, while for A.5, namely $X^2_{|L|}\neq\Lambda$, we need a delicate computation of cohomologies.
\begin{proof}[Proof of Claim A] We first look for a pair $(B, L)$ with the constraints A.1-A.4 all satisfied. For this purpose, we define the morphisms \begin{eqnarray*}
m_B: & \textrm{Pic}^d(C)\rightarrow \textrm{Pic}^{d+d_0}(C) & \quad M\mapsto M\otimes B,\\
\gamma: & \textrm{Pic}^{d}(C)\rightarrow \textrm{Pic}^{2d}(C) & \quad M\mapsto M^{\otimes 2},\\
\alpha: & W^1_{d-1}\times C\rightarrow \textrm{Pic}^d(C) & \quad (M, p)\mapsto M(p),\\
\beta: & W^{2d-g-d_0}_{2d-d_0-1}\times C\rightarrow\textrm{Pic}^{2d-d_0}(C) & \quad (M, p)\mapsto M(p). \end{eqnarray*}
A.1 and A.2 are to say that $L\in W^1_d\backslash m_B(W^0_{d-d_0})$; A.3 is to say $L\notin \textrm{im}(\alpha)$. By Riemman-Roch and Serre duality, A.4 is translated to the condition: \begin{equation*}
h^0(B^{-1}\otimes L^2(-p))=2d-g-d_0 \quad \text{for all}\; p\in C, \end{equation*} which is in turn equivalent to \begin{equation*}
B^{-1}\otimes L^2\quad\text{is base point free with}\;\; h^0(B^{-1}\otimes L^2)=2d-g-d_0+1. \end{equation*}
So $m_{B^{-1}}\circ\gamma(L)\notin W^{2d-g-d_0+1}_{2d-d_0}\cup\textrm{Im}(\beta)$.
Note $\dim{\textrm{Im}(\alpha)}<\dim W^1_d$, so to attain A.1-A.4 simultaneously, it suffices to choose $B$ such that \begin{equation}\label{desired containment relation}
W^1_{d}\nsubseteq \gamma^{-1}\paren{m_B \paren{W^{2d-g-d_0+1}_{2d-d_0}\cup\textrm{Im}\beta}}\cup m_B\paren{W^0_{d-d_0}}, \end{equation} see the diagram $$\xymatrix{
& W^{2d-g-d_0+1}_{2d-d_0}\ar@{^{(}->}[d] & & W^0_{d-d_0}\ar[d]^{m_B} & \\ W^{2d-g-d_0}_{2d-d_0-1}\times C\ar[r]^{\beta} & \textrm{Pic}^{2d-d_0}(C)\ar[r]^{m_B} & \textrm{Pic}^{2d}(C) & \textrm{Pic}^{d}(C)\ar[l]_{\gamma} & W^{1}_{d-1}\times C\ar[l]_{\alpha} }$$
Regarding $\textrm{Pic}^{d}(C)$ as a homogeneous space with the group $\textrm{Pic}^0(C)$ acting in the obvious way, we apply the Kleiman's transversality \cite[Theorem 2]{KL2} to get that, for generic $B\in\textrm{Pic}^{d_0}(C)$, the intersection of $\gamma^{-1}\paren{m_B \paren{W^{2d-g-d_0+1}_{2d-d_0}\cup\textrm{Im}\beta}}\cup m_B\paren{W^0_{d-d_0}}$ with $W^1_{d}$ has the expected dimension, which is less than $\dim W^1_d=2d-g-2$. Thereby (\ref{desired containment relation}) holds for generic $B\in\text{Pic}^d_0(C)$, consequently $L$ has $(2d-g-2)$ dimensional freedom of choice for a fixed $B$.
Then for A.5, we first show that $\Lambda\simeq\prj{2d-g-d_0}$.
By Proposition \ref{chariterion of resolution of singularities}, $\Lambda=\lsp{D_1}\cap\lsp{D_2}$ for any distinct $D_1, D_2\in |L|$. Observe that\\ $H^0(\is{{\lsp{D_1}\cap\lsp{D_2}}/\prj{N}}(1))$ is given by
$\text{Im} (H^0(K_C\otimes B(-D_1))\oplus H^0(K_C\otimes B(-D_2))\rightarrow H^0(K_C\otimes B)))$. To calculate it, we use the base point free pencil trick for $L$. The short exact sequence \begin{equation*}
\ses{\sshf{C}(-D_1-D_2)}{\sshf{C}(-D_1)\oplus\sshf{C}(-D_2)}{\sshf{C}} \end{equation*} yields \begin{eqnarray*}
&&\dim\text{Im} (H^0(K_C\otimes B(-D_1))\oplus H^0(K_C\otimes B(-D_2))\rightarrow H^0(K_C\otimes B))\\
&=& 2h^0(K_C\otimes B\otimes L^{-1})-h^0(K_C\otimes B\otimes L^{-2})\\
&=& 2(g+d_0-d-1), \end{eqnarray*} which implies $\lsp{D_1}\cap\lsp{D_2}\simeq\prj{2d-g-d_0}$.
To prove $\Lambda\backslash X^2_{|L|}\neq\emptyset$, it suffices to show for a \emph{general} degree $d-2$ divisor $Z$ with $h^0(L(-Z))>0$, $\dim(\lsp{Z}\cap\Lambda)= 2d-g-d_0-2$.
To this end, we fix $D_0\in |L|$, assume $Z+p+q=D\in |L|$ for some points $p, q\in C$ and that $D\neq D_0$. $\dim\lsp{Z}\cap\Lambda=2d-g-d_0-2$ if and only if the image of the diagonal map $$\xymatrix{ H^0(K_C\otimes B(-D_0)) \ar[d] \ar@{^{(}->}[r] \ar@{-->}[dr] & H^0(K_C\otimes B)\ar[d]\\
H^0(K_C\otimes B(-D_0)|_Z)\ar@{^{(}->}[r] & H^0(K_C\otimes B|_Z)}$$ is $(g+d_0-d-1)$ dimensional as a vector space.
From the exact sequence \begin{equation*}
0\rightarrow H^0(K_C\otimes B (-D_0-Z))\rightarrow H^0(K_C\otimes B(-D_0))\rightarrow H^0(K_C\otimes B(-D_0)|_Z), \end{equation*} and the fact $h^0(K_C\otimes B(-D_0))=g+d_0-d-1$, we see this happens if and only if $h^0(K_C\otimes B(-D_0-Z))=h^0\paren{K_C\otimes B\otimes L^{-2}(p+q)}=0$.
By A.4 and its reformulation, $B^{-1}\otimes L^2$ is base point free and $h^0\paren{K_C\otimes B\otimes L^{-2}(p)}=0$. Therefore $h^0\paren{K_C\otimes B\otimes L^{-2}(p+q)}=0$ if and only if $|B^{-1}\otimes L^2|$ separates $p$ and $q$.
Denote the image of the map $C\xrightarrow{|B^{-1}\otimes L^2|}\prj{N'}$ as $C'$. Obviously $C'$ is not $\prj{1}$. Since $\deg\paren{B^{-1}\otimes L^2}=2d-d_0$ is prime, the induced map $C\rightarrow C'$ cannot be a finite morphism of degree $\ge 2$, and hence $C\rightarrow C'$ is birational. The number of pairs $(p, q)$ which $B^{-1}\otimes L^2$ cannot separate is finite. \end{proof}
\begin{proof}[Proof of Claim B (Sketch)] We first pick two distinct points $q_1, q_2$ such that $h^0(K_C\otimes B(-2q_1-2q_2))=h^0(K_C\otimes B)-4$, then proceed by induction on the number of points. Suppose $q_1, \cdots, q_i$ for $2\le i\le d-1$ have been picked with the conditions \begin{itemize}
\item $h^0(B^{-1}\paren{q_1+\cdots+q_i)}=0$,
\item $h^0\paren{\sshf{C}(q_1+\cdots+q_i)}=1$,
\item $h^0\paren{K_C\otimes B(-2q_1-\cdots-2q_i)}=\max{\{h^0(K_C\otimes B)-2i, 0\}}$. \end{itemize}
If $q_{i+1}$ is chosen by avoiding $C\cap \lsp{q_1+\cdots+q_i}$ and any inflectionary points (cf. \cite[p. 37]{ACGH}) of $|K_C(-q_1-\cdots-q_i)|$ and $|K_C\otimes B(-2q_1-\cdots-2q_i)|$, which are finite, then the conditions still hold for $i+1$. In this process, we need the assumption $g+d_0+1\le 2d\le 2g$ to guarantee that neither $|K_C(-q_1-\cdots-q_i)|$ nor $|K_C\otimes B(-2q_1-\cdots-2q_i)|$ is empty.
The second part of the claim is quite obvious and we omit its proof. \end{proof}
\section{Appendix}
In the section, we review the construction of $X_{|L|}$, the variety swept out by all linear spans of divisors in $|L|$ on a curve $C$ with respect to an embedding $C\subset\prj{N}$.
Assume $A$ is a very ample line bundle on the curve $C$ with genus $g\ge 1$. Let $V$ denote $H^0(C, A)$. Given an effective line bundle $L$ of degree $d$ with $\dim |L|=r$. Denote the two projections from $C\times |L|$ by $p$ and $q$. There exists the sequence \begin{equation*}
\ses{p^*L^{-1}\otimes q^*\sshf{|L|}(-1)}{\sshf{C\times |L|}}{\sshf{\mathscr{D}}}, \end{equation*} where $\mathscr{D}$ is the universal divisor, see (\ref{universal family}).
Applying $q_*(p^* A\otimes \_)$ to the above, we get the exact sequence \begin{eqnarray*}\label{segre}
&&0\rightarrow {H^0\paren{C, A\otimes L^{-1}}\otimes \sshf{|L|}(-1)}\rightarrow{V\otimes\sshf{|L|}}\rightarrow{q_*\paren{p^* A\otimes \sshf{D}}}\rightarrow
{H^1\paren{C, A\otimes L^{-1}}\otimes\sshf{|L|}(-1)}\\
&&\rightarrow H^1\paren{C, A}\otimes\sshf{|L|}\rightarrow 0. \end{eqnarray*} where each term is locally free. Consequently \begin{equation}\label{Segre}
\ses{H^0\paren{C, A\otimes L^{-1}}\otimes \sshf{|L|}(-1)}{V\otimes\sshf{|L|}}{\shf{Q}}, \end{equation} where the cokernel $\shf{Q}$ is locally free. $(\ref{Segre})$ yields the diagram $$\xymatrix{
\mathbb{P}(\shf{Q})\ar@{^{(}->}[r] & \mathbb{P}(V\otimes\sshf{|L|}) \ar[d]^{\pi} \ar[r]^{\phi} & \mathbb{P}(V)\\
& |L| & &}$$
By abuse of notation we denote the induced maps from $\mathbb{P}(\shf{Q})$ to $\mathbb{P}(V)$ and $|L|$ by $\phi$ and $\pi$ respectively. $X_{|L|}$ is defined as the scheme theoretic image of $\phi$. Geometrically, $X_{|L|}$ is the union of all linear spans of divisors $D\in |L|$.
\begin{proposition}\label{chariterion of resolution of singularities}
With notations as above. We write $A=K_C\otimes B$ and assume $\deg{B}=d_0\ge 3$. Then any fibre of $\phi:\mathbb{P}(\shf{Q})\rightarrow X_{|L|}$ over a closed point is a projective space.
$\phi$ is birational if and only if $r\le h^0(A\otimes L^{-1})$. Furthermore, if $r<h^0(A\otimes L^{-1})$, then $X_{|L|}$ is Cohen-Macaulay. \end{proposition} \begin{proof} First note that as in \S 3.1, every $x\in\mathbb{P}(V)$ determines an extension \begin{equation*}
\ses{\sshf{C}}{E}{B}, \end{equation*} unique up to isomorphism. By Lemma \ref{Kernel}, $\pi\circ\phi^{-1}(x)\simeq \mathbb{P}((\ker\delta)^{\vee})$, which we denote by $P$ for short.
We claim that $\pi: \phi^{-1}(x)\rightarrow P$ is an isomorphism. In fact, let $V\rightarrow L_0$ be the one dimensional quotient representing $x$. Then one has the commutative diagram ($P$ is the locus in $|L|$ where $V\otimes\sshf{|L|}\rightarrow L_0\otimes\sshf{|L|}$ factors through $Q$) $$\xymatrix{ V\otimes_k\sshf{P}\ar@{->>}[d] \ar@{->>}[r] & L_0\otimes_k\sshf{P}\\
Q|_P \ar@{->>}[ru] &}$$ whose oblique map, by \cite[II 7.12]{HA}, induces the commutative diagram $$\xymatrix{
& \mathbb{P}(\shf{Q})\ar[d]^{\pi}\\
P \ar[ru]^{\sigma}\ar@{^{(}->}[r] & |L|}$$ i.e. $\pi\circ\sigma=\text{id}_P$. On the other hand, $\sigma(P)=\mathbb{P}(L_0\otimes_k\sshf{P})=\phi^{-1}(x)$. This establishes the isomorphism $\pi: \phi^{-1}(x)\rightarrow P$. Hence $\phi^{-1}(x)$ is a projective space.
In our case, $\phi$ is birational if and only if it is generically finite. Since $\phi$ is induced by the tautological line bundle $\sshf{\mathbb{P}(\shf{Q})}(1)$, $\phi$ is generically finite if and only if $(c_1(\sshf{\mathbb{P}(\shf{Q})}(1)))^{\dim{\mathbb{P}(\shf{Q})}}=s_r\paren{\shf{Q}^{\vee}}>0$.
Let $H$ be the hyperplane section class on $|L|$ and $s_t(\_)$ and $c_t(\_)$ denote the Segre and Chern polynomials respectively. Then by (\ref{Segre}), \begin{eqnarray*}
s_t\paren{\shf{Q}^{\vee}}&=&c_t\paren{H^0\paren{C, A\otimes L^{-1}}^*\otimes \sshf{|L|}(1)}\\
&=& (1+tH)^{h^0\paren{C, A\otimes L^{-1}}}. \end{eqnarray*} It follows that $s_r(\shf{Q}^{\vee})={h^0\paren{A\otimes L^{-1}}\choose r}H^r$. So $s_r(\shf{Q}^{\vee})>0$ if and only if $r\le h^0 \paren{A\otimes L^{-1}}$.
When $r<h^0(A\otimes L^{-1})$, consider the map of vector bundles \begin{equation*}
\xi: H^0\paren{A\otimes L^{-1}}\otimes\sshf{\mathbb{P}(V)}(-1)\rightarrow H^0(L)^*\otimes\sshf{\mathbb{P}(V)}. \end{equation*}
Let $X_{|L|}=\{x\in\mathbb{P}(V)| \:\rk{\xi_x}\le r\}$ with the determinantal variety structure. Then \begin{eqnarray*}
\dim{X_{|L|}} &=&\dim\mathbb{P}(\shf{Q})\\
&=& r+\rk{\shf{Q}}-1 \\
&=& r+h^0(A)-h^0\paren{A\otimes L^{-1}}-1\\
&=& \dim{\mathbb{P}(V)}-\paren{h^0\paren{A\otimes L^{-1}}-r}\paren{h^0(L)-r}, \end{eqnarray*}
which is the expected dimension, therefore $X_{|L|}$ is Cohen-Macaulay (cf. \cite{ACGH} p. 84). \end{proof}
{}
\end{document} | arXiv |
Harmonic Functions: Why They're Nifty
Harmonic functions are a class of functions that arise in countless engineering and physics applications. This document will review the definition, associated properties and typical applications of these mathematical marvels. To make this summary useful, it is assumed that you know a bit of calculus.
If $u(x)$ is a scalar function in $n$ variables, $x=(x_1, x_2, … , x_n)$, then Laplace's equation is the homogeneous linear partial differential equation:
$\Delta u(x) = \nabla \cdot \nabla u = \sum_1^n {\partial^2 u \over \partial x_i^2} = 0$
The left-hand side above is called the Laplacian and is sometimes denoted $\nabla^2$ instead. The solutions of $\Delta u(x) = 0$ are called harmonic functions. The name has its root in the study of harmonic motion (sound waves, for example) but the equation has proven applicable to a whole universe of problems beyond that.
More formally (if you care): Laplace's equation is defined on an open, nonempty set $\Omega$ in $\mathbb{R}^{n}$ or $\mathbb{C}^{n}$, and specify that $u(x)$ is a twice differentiable scalar function, denoted $C^2[\Omega]$, that maps elements from the $\overline \Omega$, the closure of $\Omega$, into $\mathbb{R}$ or $\mathbb{C}$ (and yes, this reads like the math equivalent of a prescription drug ad disclaimer).
Properties and Examples
Before continuing, it's helpful provide a few examples. Namely, what kind of functions are harmonic? Constant and linear functions certainly fit the bill since their second derivatives are always zero. There are of course an infinite number of non-linear harmonic functions too. For example,
$e^x siny + x + 1$
$x^2 – y^2 + x$
$e^{-x}\, (x\,siny \,-\, y\,cosy)$
They can be complex-valued as well:
$f(z) = (x + iy)^2 = x^2 -y^2 + 2xyi$
Harmonic functions are linearly superimposable, which means that sums and scalar multiples of harmonic functions are themselves harmonic. So based on the examples above we can conclude that $e^x siny + x + 1 + 4(x^2 – y^2 + x)$ is also harmonic. And so on.
Harmonic functions are also invariant under rigid motions, meaning harmonic functions can be translated and rotated and still retain their harmonicity. For examples of this, see this Wiki page.
On their own these two properties, invariance and linearity, are incredibly powerful and yet it's only the tip of the iceberg…
The Mean Value Property
Next is the mean value property (MVP), which is arguably its most important property. The MVP describes how the harmonic functions behave within bounded regions. In particular, inside any spherical region the average value of the function will be its value at the sphere's center, which is also the average value across the surface of that sphere. This is true in n-dimensions too, from a line to a circle to an n-sphere.
The mean value property is expressed mathematically as follows, where $B(x_0,r)$ is an n-dimensional sphere of radius $r$ centered at $x_0$. The symbol $\partial B(x,r)$ refers to the boundary of that sphere (the skin of an apple, if you will).
$$\begin{eqnarray}
\def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits}
u(x_0)
= \avint_{ B(x_0,r)} u dy
= \avint_{\partial B(x_0x,r)}u dS
\end{eqnarray}$$
Note that the dashed integral symbol refers to the following average:
\avint_{ B(x_0,r)} u dy = {1 \over Vol} \int_{ B(x_0,r)} u dy
For the surface integral, divide by the surface area:
\avint_{ \partial B(x_0,r)} u dS = {1 \over Area} \int_{ \partial B(x_0,r)} u dS
The proof for the MVP is widely available [EV] (§2.2.2) [AXL] [SG].
Two consequences of this. First, extrema can only occur on the boundary: there are no max or min within the region unless the function is constant. And related, if a harmonic function is bounded on all of $\mathbb{R}^n$ it is necessarily constant.
In his 1881 Treatise on Electricity and Magnetism, James Clark Maxwell made the following remark about the Laplacian:
I propose therefore to call $\nabla^2q$ the concentration of q at the point P, because it indicates the excess of the value of q at that point over its mean value in the neighbourhood of the point.
That is, the Laplacian at a point measures how much a function differs from its neighbors. If it's positive, then it's less than its neighbors; if it's negative, it's greater. And when it's zero, this value is exactly the average of those neighboring points [LRX]. This has an interesting application in the problem of edge detection in digital images—sudden changes in the color intensity that occur where the content changes: a face against a background. For this reason, the Laplacian is often used to detect these types of intensity changes [AIS].
Real Analytic and Complex Analysis
Harmonic functions are real-analytic. Being real-analytic on an open set means that the value of the function at any point in that set can be expressed as a convergent power series. As it turns out, that power series is a Taylor series. As a consequence, harmonic functions are also infinitely differentiable, a.k.a., smooth or regular.
Note: The reverse is not true: a smooth function isn't necessarily analytic. See this example.
In two dimension, harmonic functions have a symbiotic relationship with complex analysis. This leads to a number of interesting outcomes.
The first is that two dimensional harmonic functions are also analytic (holomorphic) and vice versa. Analytic is the complex version of differentiable, which is more powerful since it requires the derivative at a point be the same from every direction, not just right and left as it is with real differentiability.
Next, if a function is analytic, then its real and complex parts are harmonic. This follows from the Cauchy-Riemann equations.
Dirichlet Problem
When the Laplacian yields something other than zero, the resulting inhomogeneous equation is called Poisson's equation, $\Delta u(x) = -f(x)$. It should be noted that a solution to Poisson's equation can be linearly combined with a harmonic function and still be a solution. More on Poisson's equation shortly.
The problem of solving a PDE with a specific boundary function is called the Dirichlet problem. The solution of the Dirichlet problem for the Laplace equation, if it exists, will be unique. As a consequence, the solution inside the region is uniquely defined by it values on the boundary.
\begin{cases}
\phantom{-}
\mathop{{}\bigtriangleup}\nolimits
u = 0 \,\,u \in \Omega
\\\phantom{-}
u(x) = g(x) \,\,\,\,u \in \partial \Omega
\end{cases}
This is an example of a well-posed problem, which basically means that its solutions represent something realistic and are not pathological. In particular, solutions will continuously approach the boundary function value as $x$ nears the boundary $\partial \Omega$ (this is the notation for the boundary of $\Omega$).
The canonical example of a Laplace Dirichlet problem is a steady-state temperature distribution in a region with one or more heated edges. The image below is an example of this (via Matlab). There are also interesting applications in Brownian Motion.
There are other types of boundary problems too, such as Neumann and Robin depending on how the boundary constraints are defined. For steady-state heat problems, Neumann conditions are often used to represent adiabatic (insulated) edges.
Operators are useful when working with these equations. Think of an operator as a kind of function for functions. For example, the first derivative is an operator that takes a function $f(x)$ and returns its derivative, also a function. The inverse operator for the first derivative operator is the antiderivative. Matrices are operators too, as are their inverses. There are many types of operators of course. The Laplacian is considered a linear differential operator.
The linear part of "linear operator" means that the operation on a sum of terms is the sum of those operations: $L(\alpha x + \beta y) = \alpha L(x) + \beta L(y)$. For harmonic functions, this is consistent with the property of linear superposition already mentioned.
In operator form, Poisson's equation takes on the following elegance.
$ L = \Delta$ (the operator)
$ L u = f $ (Poisson's equation)
$ u = L^{-1}f $ (its solution)
Operators also have null spaces (sometimes called the kernel, which is a poor naming choice since the kernel has an unrelated meaning too). These are the functions that the operator maps to zero: $L\Delta =0$. In particular, harmonic functions are the null space of the Laplacian. Note, however, that just as it is with matrices, this inverse only exists if there are no non-trivial solutions in the null space. That is, all the eigenvalues $ \lambda$ of $ L u = \lambda u $ are non-zero.
Operators also have domains just like functions. For example, when $\Omega$ is the $\mathcal{L}^2(\mathbb{R}^n) $ Banach space, we can specify that the domain of $L$, denoted $D(L)$, is the Sobolev space, $H^2(\mathbb{R}^n)$, which is closed and dense in $\mathcal{L}^2(\mathbb{R}^n)$.
The corresponding boundary conditions for linear differential operators are embedded in the domain of the operator definition. For example, here's the domain expression for a one dimensional ODE with homogenous boundary conditions:
$D(L) = \{ u \in C^2([0,1]) $
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space: u(0) = u(1) = 0 \}$
More useful terminology hidden above. The set of continuous twice-differentiable functions on the interval $[0,1]$ is denoted $C^2([0,1])$.
Finally operators like functions can also be bounded or unbounded but this depends on the norm used by that operator (the norm provides a notion of size for its objects). For example, the Laplacian is an unbounded operator when its domain is $\mathcal{L}^2(\mathbb{R}^n) $ using the $\mathcal{L}^2 $ norm but it is bounded when that domain is $\mathcal{H}^2(\mathbb{R}^n) $ paired with the $\mathcal{H}^2$ norm. For more info, see this thread
The Fundamental Solution
The fundamental solution of Laplace's equation is a function $\Phi(x| \xi)$ which is harmonic everywhere except at a singularity $\xi$. As an operator this is often written $L \Phi(x| \xi)= \delta(\xi)$, where $\delta(\xi)$ is a Dirac distribution centered at $\xi$. A classic example is the potential field of a point charge.
There are three types of fundamental solution for Laplace's equation, depending on the dimensionality of the problem.
If $n=1$ the fundamental solution is a linear function $|x – a|$
For $n=2$ it's logarithmic: $ln |x – a| \,, x \ne a$
For $n \ge 3$ it's $|x – a|^{2-n} \,, x \ne a$
A few observations around this. The first is that these represent a family of solutions, not a specific one since linear combinations of these functions are still harmonic.
The second observation is that for $n>1$, the solutions above are not defined at the the point $a$, rather, they are only harmonic on the so-called "punctured region," $\Omega \backslash \{a\}$. But for all three the derivatives do not exist at $a$. The singularity that occurs there is of tremendous importance.
Third, these equations have radial symmetry. This again aligns with intuition. Both heat and electric fields radiate radially outward, as does the pull of gravity.
Free space Poisson equation
The free space (no boundary conditions) Poisson equation in terms of the fundamental solution is the following convolution:
\Delta u(x)
\Phi(x) * f(x)
\delta (x) * f(x)
\int_{\mathbb{R}^n}\delta (x-\xi) \, f(\xi)d\xi
That is, the driving function $f(x)$ is the linear superposition of fundamental solutions, just as a charge distribution can be thought of as the weighted superposition of point charges.
In operator form:
$Lu(x) = L\Phi(x) * f(x) = \delta (x) * f(x) = f(x) $
Green's Functions
The Green's function $G(x, \xi)$ for Poisson's equation is a fundamental solution with homogeneous boundary conditions applied. It is sometimes called the Influence function. Green's functions are also covered in detail in a previous post.
The second variable in $G(x, \xi)$, $\xi$, indicates that we define a Green's function at every point in the region thus providing a mechanism to superimpose them across that region. We set $G(x, \xi)$ to zero on the boundary so that the only component of $u(x)$ on the boundary is $g(x)$, the boundary value function. Green, in his seminal 1828 paper in which this concept was introduced, suggested that we think of the zero potential on the boundary as the result of grounding the conducting surface. But this technique proved to extend far beyond problems in electrostatics.
Collectively, these ideas are expressed by the following accessory equation:
G(x, \xi) = \delta(x – \xi) & u \in \Omega
G(x, \xi) = 0 & u \in \partial \Omega
Green's functions are specific to the boundary geometry too, meaning they can be reused when solving the Dirichlet Problem for the same region regardless of the forcing and boundary functions.
One side note. The Green's function here is symmetric (or more formally, self-adjoint). That is, $G(x, \xi) = G(\xi, x)$. The proof of this is complicated (see [EV]) but the upshot is that we can switch variables when doing certain integrations. This comes in handy when proving some of the results shown in this section.
It's also worth stressing that Green's functions exist for many different differential equations, not just Poisson's equation. It is but one of the many techniques available for cracking these often intractable problems. We now demonstrate how it is used in the context of the Dirichlet Problem.
Poisson Dirichlet Problem
In terms of the Green's function, the general solution for the the Poisson equation with Dirichlet conditions is shown below. The proof is shown in [EV], [SG] and others.
u(x) =
– \int_{\partial \Omega} g(\xi)\frac {\partial G(x, \xi)}{\partial \eta} dS(\xi)
+ \int_{\Omega} f(\xi) G(x, \xi) d\xi
The term $\frac {\partial G(x, \xi)}{\partial \eta} $ is called the Boundary Influence Function, which for the two dimensional unit disk becomes the well-known Poisson Kernel. In particular, when $f(x)=0$ as it is for Laplace's equation, the closed-form solution on a circle, $|r| < R$, is:
u(r, \theta) = \frac{1}{2\pi R}\int_0^{2\pi}
\frac
{R^2 – r^2}
{ r^2 – 2Rr\cos(\theta – \psi) + R^2} g(\psi)
d\psi
You should notice that the denominator above has a singularity at the point $(R, \psi)$. The Poisson Kernel therefore represents the value that would be generated by a delta distribution on the boundary per the singularity in the denominator. This means that the function value $u(r, \theta)$ is the weighted sum of all such distributions along the boundary. Compare this to how the forcing function is the weighted sum of Green's functions across that region.
The Poisson Kernel for the unit circle can be expressed in rectangular coordinates using the law of cosines:
{1 -|x|^2 \over 2 \pi}
\int_{\partial \Omega} \frac {g(\xi)}{|x-\xi |^2} d\xi
More generally in n-dimensions,
{1 -|x|^2 \over n \alpha(n)}
\int_{\partial \Omega} \frac {g(\xi)}{|x-\xi |^n} d\xi
In terms of operators, the Green's function is kernel of the inverse to the equation $L u = $f
u(x) = (L^{-1}f)(x) := \int G(x, \xi) f(\xi)d\xi
In general finding the Green's functions and thus solving Poisson's equation is not easy since these integrals generally do not have closed-form solutions. But solutions do exist for a few simple geometries (i.e., the shape of the boundary). The solution on a circle is well-known as is the solution in the half-plane both found using the method of images (see [EV], for example).
Fortunately in two dimensions there are many more options thanks to correspondence between harmonic functions and complex analysis. In particular, we have the wonderful technique of conformal mappings and by extension, the technique of Schwartz-Christoffel. For example, this is the two-dimensional unit circle Green's function translated from another region via a conformal mapping $g(z)$:
G(z, \xi) =
{1 \over 2\pi} \ln{
|{1 – \overline g(\xi) g(z) } | \over {|g(z)-g(\xi)|}
Exterior Problems
Exterior solutions are also interesting. In two dimensions, the exterior solution to the Dirichlet problem on a disc tends toward a finite limit that happens to be the mean value solution. That is, the solution at infinity for an exterior region of a disc is that same as the value at the center of the disc interior. This can be shown fairly easily by taking the limit toward infinity of the Poisson Kernel in the exterior Dirichlet Problem.
\lim_{r\to\infty}
u(r, \theta) = \frac{1}{2\pi }\int_0^{2\pi}
g(\psi)
In higher dimensions, this limit needs to be zero to ensure a unique solution. For a proof see [AXL] or this paper by Giles Auchmuty and Qi Han. This result is interesting given the previous result for the two-dimensional problem and arguably physically consistent too. See [SG].
[AIS] Sinha, U. The Sobel and Laplacian Edge Detectors. Microsoft.
[ALF] Ahflors, L., 1979, Complex Analysis, McGraw-Hill Book Company.
[AXL] Axler, S., Bourdon, P., Ramsey, W., 2001, Harmonic Function Theory, Springer.
[EV] Evans, L., 2010, Partial Differential Equations: Second Edition (Graduate Studies in Mathematics), 2nd Edition, American Mathematical Society.
[EV2] Evans, L. 2011?. Overview article on partial differential equations, UC Berkeley.
[HT] Hunter, J., 2001. Applied Analysis, WSPC.
[HT2] Hunter, J., 2014. Notes on Partial Differential Equations, University of California at Davis.
[LRX] Lamoureux, M., 2004. The mathematics of PDEs and the wave equation. Univ. of Calgary.
[LV] Levandosky, J., 2003. Math 220B Lectures, Stanford University.
[NH] Nehari, Z., 1952, Conformal Mapping, McGraw-Hill Book Company.
[OL2] Olver, P., 2018, Complex Analysis and Conformal Mapping, University of Michigan
[STR] Strauss, W. 2007. Partial Differential Equations: An Introduction, Wiley.
[SG] Stakgold, I., 1967, Boundary Value Problems of Mathematical Physics, Macmillan.
AnalyticCauchy-RiemannConformal MappingDirac DeltaDirichlet ProblemFundamental SolutionGreen's FunctionHarmonic FunctionsLaplacianMean-Value PropertyOperatorsPoisson's EquationSchwartz-ChristoffelSobolev space
Reversible Pairs
Prevalence and the Base Rate Fallacy
Green's Functions Illustrated | CommonCrawl |
General Properties of Matter: Question Bank for Class 11 Physics
Get important questions of General Properties of Matter for Board exams. Download or View 11th Physics important questions for exam point of view. These important questions will play significant role in clearing concepts of Physics. This question bank is designed by NCERT keeping in mind and the questions are updated with respect to upcoming Board exams. You will get here all the important questions for class 11 Physics chapter wise CBSE. Click Here for Detailed Notes of any chapter.
Q. What is the value of young's modulus for perfectly rigid body.
Ans. Infinite
Q. What is the value of bulk modulus for an incompressible liquid ?
Q. What is the value of modulus of rigidity for any liquid ?
Ans. Zero
Q. How does young's modulus change with rise in temperature.
Ans. Young's modulus of material decreases with rise in temperature.
Q. Which type of strain is there, when a spiral spring is stretched by a force ?
Ans. Shear strain.
Q. The ratio stress/strain remains constant for small deformation. What will be effect on this ratio when the deformation made is very large ?
Ans. When the deforming force is applied beyond elastic limit, the strain produced is more than that has been observed within elastic limit. Due to which the ratio stress/strain will decrease.
Q. Which of the three Y,K, and h is possible in all the three states of matter (solid, liquid and gas)
Ans. Bulk modulus of elasticity (K) only
Q. Give the example of a body which is nearly perfectly elastic.
Ans. Quartz fibre.
Q. Give the example of a body which is nearly perfectly plastic
Ans. Putty.
Q. What do you mean by elastic hysteresis ?
Ans. On reducing the stress to zero the material does not recover. Its unstrained state as some strain always remain. This phenomenon of lagging of strain behind the stress, is called elastic hysteresis.
Q. What do you mean by an elastomer ?
Ans. The materials which has large elastic limit but do not obey Hook's law, are called elastomers. Rubber is a very good example of such type of material.
Q. Diamond is called a hard material. What does it mean in term of its modulus of elasticity?
Ans. Modulus of elasticity becomes large. It means it becomes more elastic.
Q. Paints and lubricating oils have low surface tension. Why?
Ans. Being of low surface tension, they spread over a large surface area.
Q. A piece of chalk immersed in water emits bubbles in all directions. Why?
Ans. Chalk has pores all over its surface which acts as fine capillaries. When immersed in water, the water rises into these capillaries and forces the air out in the form of bubbles in water all around.
Q. A mercury barometer reads slightly less than the actual pressure. Why?
Ans. This is because, due to capillary action, mercury is depressed down in the barometer tube.
Q. Some straw are spread on the surface of pure water filled in a vessel. On dropping a piece of sugar in water, the straw come nearer to the piece, but on dropping a piece of soap, they go away from it. Explain.
Ans. The surface tension of sugar solution is greater than that of pure water. Therefore, the surface contracts near the piece of sugar and hence straw come nearer it. The surface tension of soap solution is less due to which the surface spreads near it and straw go away from it.
Q. If the free surface of liquid is plane. Where does the force of surface tension act?
Ans. The surface tension acts horizontally, and has no component normal to surface.
Q. When pure water falls on a flat glass plate it spreads on the plate while the mercury when falls on a glass plate gets converted into small globules. Why?
Ans. The adhesive force of mercury and glass is less than that of their cohesive force. Therefore, Hg breaks into droplets.
Q. What are the properties of an ideal liquid ?
Ans. An ideal liquid is one that is perfectly incompressible, non-viscous and unable to withstand shearing stress, however small.
Q. Which one is most viscous among water, air, blood and honey ? Which one is the least viscous ?
Ans. Honey is most viscous and air is least viscous.
Q. When oil flows in a pipe, which layer moves fastest ?
Ans. The axial layer moves fastest.
Q. Hot liquid flows more rapidly than cold one, why?Ans.The coefficient of viscosity of a liquid decreases with rise in temperature. Hence, hot liquid flows with larger speed than when it is cold.
Q. Why are machine parts usually jammed in winter?
Ans. This is because the lubricant used in the machine becomes more viscous in winter.
Q. According to Bernoulli's theorem, the pressure of fluid should remain uniform in a pipe of uniform radius. But actually it goes on decreasing. Why is it so ?
Ans. It is due to viscosity of fluid. When fluid flows, work is done against the viscous force, and this work is taken from the pressure energy. Hence pressure fluid falls.
Q. If instead of fresh water, sea water is filled in the tank, will the velocity of efflux change ?
Ans. No, velocity of efflux does not depend upon density of liquid.
Q. Which is easier to lift in the air; 1 Kg steel or 1 Kg cotton ? What will be your answer if we lift these object in vacuum ?
Ans. In case of air, it will be easier to lift 1 Kg of cotton because it will displace greater volume of air and lose more weight. In case of vacuum, as there will be no upward buoyant force on the objects, hence both will weight equally.
Q. Which fundamental law forms the basis of equation of continuity ?
Ans. Law of conservation of mass.
Q. Why water is not used in barometers ?
Ans. On account of following reasons, water is not used in barometers : (i)Water sticks to the walls of the barometer tube. (ii) Water has low density so height of water column becomes very large (11m) which is unmanegeable.
Q. Why the boiling point of a liquid varies with pressure ?
Ans. At boiling point, vapour pressure of liquid is equal to the atmospheric pressure. Hence, when the atmospheric pressure on the surface of liquid increases, the liquids boil at higher temperature to generate greater vapour pressure.
Q. Why water does not flow out of a dropper unless the rubber bulb is pressed ?
Ans. The upward air pressure at the dip of dropper is equal to the pressure of liquid column in it. When we press the rubber bulb, the inward pressure increases making the liquid flow out.
Q. What is the fractional volume submerged of an ice cube in a pail of water placed in an enclosure which is falling freely under gravity ?
Ans. Because the pail of water is falling freely under gravity, hence it will be in a state of weightlessness. Both the weight of the ice cube and the upthrust would be zero. So, the ice cube can float with any value of fractional volume submerged in water.
Q. A wooden block is in bottom of a tank when water is poured in. The contact between the block and the tank is so good that no water gets between them. If there a buoyant force on the block?
Ans. Since, there is no water under the block to exert an upward force on it, therefore there is no buoyant force.
Q. What is hydrostatic paradox ?
Ans. According to it the pressure exerted by a liquid depends only upon the height of liquid column and is independent of shape of the containing vessel.
Q. Why the wings of an aeroplane are rounded outwards while flattened inwards ?
Ans. Due to such shape of the wing, the velocity of air relative to the wind on the upper surface is larger than that of lower surface. This causes the pressure difference. The pressure below the wing is more than above the wing and hence causes upward lift on the aeroplane.
Q. To keep a piece of paper horizontal, one should blow over it and not under it. Why ?
Ans. When we blow air over a piece of paper, the velocity of air moving along the upper surface of paper is higher than that along the lower surface. Therefore, the air pressure on the upper surface of paper is lowered (according to Bernoulli's theorem), while the pressure on the lower surface is still atmospheric. Due to higher pressure below the paper, the paper stays horizontal and does not fall under gravity.
Q. A balloon filled with helium does not rise in air indefinitely but halts after a certain height (Neglect winds). Explain why ?
Ans. A balloon filled with helium goes on rising in air so long as the weight of air displaced by it (i.e. upthrust) is greater than the weight of filled balloon. We know that the density of air decreases with height. Therefore, the balloon halts after attaining a height at which density of air is such that the weight of air displaced just equals the weight of filled balloon.
Q. Two tooth picks floating on a water surface are parallel and close to each other. A hot needle is touched between them to the water surface. Explain why they fly apart ?
Ans. When a hot needle is touched between them to the water surface, the surface tension of water in this region decreases due to increase in temperature. So, the outward pulling force due to surface tension of water on either side is more. Hence they fly apart.
Q. The hot soup is tastier than the cold one. Why?
Ans. Since, the surface tension of liquid decreases with rise in temperature. Therefore, the surface tension of hot soup is less than that of cold soup. Hence, hot soup spreads over a larger area of the tongue and gives a tastier feeling.
Q. Small insects swimming on water die when kerosene oil is added into the water. Why?
Ans. Insects swim on water surface because of surface tension due to which surface behaves like a stretched membrane. On adding kerosene oil, the surface tension is appreciably reduced and the insects sink to death.
Q. Discuss the various application of elasticity.
Ans. Some of the important applications of the elasticity of the material are discussed as follow. (1) Bridges are declared unsafe after long use. (2) A metallic part of a machinery will get permanently deformed if subjected to a stress beyond the elastic limit. Therefore, the metallic part of the machinery is never subjected to the stress beyond the elastic limit. (3) A crane is used for lifting and moving heavy load from one place to another. The crane makes use of a thick metallic rope. We have to specify the maximum load that will be lifted with the help of metallic rope. The maximum load should be such that the elastic limit of the material of the rope is not exceeded. Let us suppose, the rope is required to lift a maximum load of $10^{5} \mathrm{kg}$. Elastic limit of the steel is $3 \times 10^{8} \mathrm{Nm}^{-2}$. Therefore maximum stress on rope $=30 \times 10^{7} \mathrm{Nm}^{-2}$. Let $r$ be radius of the required rope. (4) To estimate the maximum height of a mountain. (5) Bending of metallic bar. When a metallic bar of length l breadth b and thickness d placed on two wedges as shown in figure is loaded at the middle with a weight W = mg then depression in the bar is given by $ \delta = \frac{m g l^{3}}{b d^{3} y}$ where y is the young's modulus of the bar
Q. Find the excess pressure inside a liquid drop.
Ans. If the liquid surface is curved, then in equilibrium a net pressure acts on it towards the concave side or the pressure on the concave side is larger than that of convex side. A liquid drop is spherical in shape and its outer surface is convex. Therefore, due to surface tension a net inward force acts on every molecule situated on the surface of the drop. As a result the pressure inside the drop is more than that of outside. Let the radius of drop be R and the excess pressure inside it is P. Due to this excess pressure an outward force acts on the surface of the drop. Let due to the excess of pressure the radius of the drop increases from $R$ to $R+\Delta R$ Therefore, work done by excess of pressure in expansion of the drop. $W=$ Force $\times$ displacement $=$ Pressure $\times$ area $\times$displacement $W=P \times 4 \pi R^{2} \times \Delta R$ and increase in surface area of the drop $=$ Final surface area - Initial surface area $\therefore$ Increase in surface energy $=T \times 8 \pi R . \Delta R$ since, this increase in surface energy is due to excess of pressure, $\therefore P \times 4 \pi R^{2} \times \Delta R=T \times 8 \pi R . \Delta R \cdots P=\frac{2 T}{R}$ i.e. (i) $P \propto T \quad$ (ii) $P \propto \frac{1}{R}$ i.e. smaller the drop, greater is the excess pressure inside it. Note : A bubble inside a liquid is similar to a liquid drop. Let us consider an air bubble of radius R formed in a liquid of surface tension T. Like a liquid drop, the air bubble also has one surface in contact with the liquid. Hence proceeding in the same way as in case of liquid drop, we can derive an expression for excess pressure inside the air bubble. $P=\frac{2 T}{R}$
Q. Find excess pressure inside a soap bubble.
Ans. Let us consider a thin soap bubble of radius R formed from a soap solution of surface tension T. Due to air inside it the bubble has two free surfaces, one inside and other outside Due to excess pressure p in it an outward force acts on the surface of the bubble. This is balanced by the force due to surface tension. Now, the work done by excess pressure $P$ in increasing radius from $R$ to $R+\Delta R$. $W=$ Force $\times$ displacement $=P \times 4 \pi R^{2} \times \Delta R$ and increase in surface area of the bubble $=2\left[4 \pi(R+\Delta R)^{2}-4 \pi R^{2}\right]$ $=2\left[4 \pi\left(R^{2}+2 . R . \Delta R+\Delta R^{2}\right)-4 \pi R^{2}\right]=16 \pi R \Delta R$ $\therefore$ Increase in surface energy $=T \times 16 \pi R \Delta R$ since, the increase in surface energy is due to excess of pressure, $\therefore P \times 4 \pi R^{2} \times \Delta R=T \times 16 \pi R . \Delta R \quad$ i.e. $\quad P=\frac{4 T}{R}$
Q. Show that there is an excess of pressure on curved surface of liquid, which is more on concave face than that on convex face of curved surface of liquid.
Ans. When the liquid surface is plane, the molecule on the liquid surface will be equally attracted on all the sides, and the forces (T, T) of surface tension will be acting tangentially to the liquid surface in opposite directions. Hence no resultant force is acting on the molecule. As a result no extra pressure exists to the inner side or outer side of liquid surface. When the liquid surface is curved, the molecule O on the liquid surface is acted upon by the forces (T, T) due to surface tension along the tangents to surface. Resolving these forces into horizontal and vertical components, the horizontal components cancel each other whereas vertical components add up. Thus a resultant force acts on the curved surface of the liquid which acts towards its centre of curvature i.e. the resultant force is directed inwards in case of convex surface fig. (b) and is directed outwards in case of concave surface fig (c). In order that the curved surface of liquid may be in equilibrium, there must be an excess pressure on its concave side over that on the convex side, so that the forces of excess pressure may balance the resultant forces due to surface tension. Hence, for the curved surface of liquid in equilibrium the pressure on concave side of liquid will be greater than pressure on its convex side.
Q. What is capillarity and deduce ascent formula.
Ans. The phenomenon of ascent or descent of liquid in a capillary tube is called capillarity or capillary action. Generally those liquids which wet the glass surface rises up and the liquids which do not wet the glass surface descend in the capillary. Ascent formulla : Let a uniform capillary tube of radius 'r' is dipped into a liquid in which liquid rises upto height 'h'. Considering the meniscus (concave) radius nearly equal to the radius of capillary, the surface tension of liquid acts at the point of contact of meniscus along the direction of tangent making an angle q (angle of contact) with the wall of capillary tube. The horizontal and vertical components of T are Tsinq and Tcosq. The horizontal component Tsinq acting diametrically opposite around the meniscus cancel each other. But the vertical component Tcosq acts upwardly around the circumference of meniscus, due to which rises the liquid in the capillary tube. \[\therefore \]Force which rises the liquid $=T \cos \theta .2 \pi r$ This force is balanced by the weight of the liquid. Now, the volume of liquid raised = V = Volume of liquid upto height h + Volume of liquid in meniscus.
Q. What do you understand by Reynold's number? Give its physical significance.
Ans. Reynold number is a pure number which determines the nature of flow of liquid through a pipe. According to Reynold, the critical velocity $v_{c}$ of a liquid flowing through a tube of diameter D is given by. $v_{c}=\frac{k \eta}{\rho D} \Rightarrow K=\frac{\rho D V_{c}}{\eta}$ when, $\eta=$ coefficient of viscosity of liquid. $K=A$ constant and is called Reynold's number $\rho=$ density of liquid The value of K lies between 0 to 2000, for streamline or laminar flow and for value K above 3000, the flow of liquid is turbulent. Physical significance : Reynold's number describes the ratio of the inertial force per unit area to the viscous force per unit area for a flowing fluid. Let us consider a tube of small area of cross section A, through which a fluid of density r is flowing with velocity v. The mass of fluid flowing through the tube per second, $\Delta m=$ volume flowing per second $\times$ density. $=A v \rho$ \[\therefore \]Inertial force per unit area $=\frac{\text { rate of change of momentum }}{\text { Area }}$ $=\frac{(\Delta m) v}{A}=\frac{(A v \rho) v}{A}=v^{2} \rho$ and viscous force, $F=\eta A \frac{v}{r}$ (in magnitude) where $r=$ radius of tube $\frac{v}{r}=$ velocity gradient between the layers of this liquid flow. $\therefore$ viscous force per unit area, $=\frac{F}{A}=\frac{\eta v}{r}$ Hence, Reynold's number, $\mathrm{K}=\frac{\text { Inertial force per unit area }} {\text { viscous force per unit area }}=\frac{v^{2} \rho}{\eta v / r}=\frac{v \rho r}{\eta}$
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Pooja - Nov. 28, 2021, 2:57 p.m.
Neet repeated questions dalo | CommonCrawl |
Article | Open | Published: 21 November 2017
Universal model of individual and population mobility on diverse spatial scales
Xiao-Yong Yan1,
Wen-Xu Wang ORCID: orcid.org/0000-0002-4170-86762,
Zi-You Gao1 &
Ying-Cheng Lai3
Nature Communicationsvolume 8, Article number: 1639 (2017) | Download Citation
Studies of human mobility in the past decade revealed a number of general scaling laws. However, to reproduce the scaling behaviors quantitatively at both the individual and population levels simultaneously remains to be an outstanding problem. Moreover, recent evidence suggests that spatial scales have a significant effect on human mobility, raising the need for formulating a universal model suited for human mobility at different levels and spatial scales. Here we develop a general model by combining memory effect and population-induced competition to enable accurate prediction of human mobility based on population distribution only. A variety of individual and collective mobility patterns such as scaling behaviors and trajectory motifs are accurately predicted for different countries and cities of diverse spatial scales. Our model establishes a universal underlying mechanism capable of explaining a variety of human mobility behaviors, and has significant applications for understanding many dynamical processes associated with human mobility.
Human movements typically occur in spatial regions/domains, such as countries or cities, which can have vastly different scales. For example, in China or the United States, the size of the region can be on the order of millions of square kilometers, while in small countries such as Belgium, the domain size is only about 10 km. (Here we regard international travel as atypical and exclude it from our consideration.) Comparing Belgium with China or the United States, the difference in the spatial scale in terms of areas is at least two orders of magnitude. Typical examples of human movements at both the individual and population levels in countries of diverse size are shown in Fig. 1. Overall, for human mobility, there are large scales exemplified by countries such as China and the United States, and small scales as represented by small countries or big cities in a large country.
Real-world examples of individual trajectories and collective movements. a Four examples of an individual trajectory from an empirical data set from mainland China and the corresponding collective movements. b–d Collective movements embedded in the data sets from the continental United States, Cote d'Ivoire, and Belgium. Here the color bar represents the amount of mobility flux among locations per unit time, where a brighter (darker) line indicates a stronger (weaker) flux. Note that the spatial scales associated with these data sets are drastically different
A remarkable discovery in complexity science in the past decade is that human mobility obeys certain universal scaling laws1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20. It was recently revealed16, however, that human mobility at small spatial scales tends to exhibit different scaling behaviors. Existing models of human mobility are tailored at understanding and characterizing scaling laws either at large (e.g., big country) or small (e.g., city) scales—we lack a universal model capable of quantifying human movements across all spatial scales. Another deficiency of existing models is that they can explain human mobility patterns either at the individual or at the population level, but not both. The purpose of this paper is to develop a model to fill this gap in our knowledge about human mobility.
To understand the dynamics of human movements and to uncover scaling laws underlying human mobility are of fundamental importance as they are relevant to problems such as disease control, social stability, congestion alleviation, information propagation, and e-commerce21,22,23,24,25,26,27,28. Data-based modeling research on human mobility started about a decade ago1, where the trajectories of bank notes were traced over a reasonably long time period. On the basis of empirical data, the study unveiled two scaling laws on geographical scales: (1) the distribution of the traveling distance exhibits a power-law decay, which can be described by Lévy flights, and (2) the probability of remaining in a small region exhibits an algebraically long tail with an exponential cutoff, which is characteristic of a superdiffusive behavior. Existence of universal patterns in the statistical description of human movements was hinted at through an analysis of the mobile phone data6, and the issue of predictability of human movement patterns was also addressed16,29. The correlation between human movements in the cyberspace and in the physical space has also been studied through big data analysis15,20. Quite recently, a scaling law connecting human mobility and social interaction (communication) patterns was uncovered30.
From a modeling perspective, the classic gravity model31 represented perhaps the earliest attempt to mathematically understand the mobility flow between two locations. For human mobility on large spatial scales, e.g., as revealed by the trajectories of bank notes, a two-parameter continuous-time random walk model was derived to explain, at a detailed and quantitative level, the empirically observed scaling laws1. A statistical, self-consistent microscopic mobility model2 and a macroscopic model, the so-called radiation model3,4,5 that takes into account local mobility decisions, were developed. Inspired by these models, a variety of alternative mechanisms aiming at understanding and characterizing the empirical scaling laws obtained from data have been conceived6,7,8,9,10,11,12,13,14,15,16,17,18,19,20. Most classic gravity-based macroscopic models are static in the sense that they assume certain mobility decisions so that the transition matrix between spatial locations can be constructed through the corresponding distances and population distribution. While the models can explain the scaling laws to a certain degree, the detailed dynamical features associated with human movements at the individual level are often lost, which are important to understand issues such as the spreading speed and range of diseases. The existing microscopic models2,8,15,20,32,33,34 can capture the individual movements and the associated scaling laws, but the mutual interactions among the individuals at the population level were largely ignored.
In this paper, we aim to articulate a human mobility model capable of predicting statistical and scaling behaviors on all spatial scales at both the individual and population levels. To accomplish this goal, it is necessary to identify the general mechanisms underlying human movements independent of the spatial scale. An essential element upon which any human mobility modeling is built is the transition probability for an individual to move from one location to another at any time, from which all kinds of scaling laws can be derived. There are two basic elements that we exploit to construct the transition probability. First, locations are differentiated according to their relative attractiveness, so human movements tend to be biased toward the more attractive locations. This can be modeled by assigning each location a fixed amount of attractiveness. The second element is the memory effect, by which individuals tend to move preferentially to previously visited locations. At a quantitative level, the memory effect can be taken into account by assigning previously visited locations with a relatively high amount of attractiveness. As a result, in our model a location contains two kinds of attractiveness: one simply determined by the population at the location, which is analogous to that in the radiation model3 or the population-weighted opportunities model16, and another determined by the memory effect. For any location, the transition probability for an individual to move into it is proportional to its total attractiveness. More specifically, before making a movement, an individual evaluates the attractiveness of all the available destinations and then moves to a specific destination according to the transition probability. With the transition probability so determined, our model contains a single parameter, which can be determined from each set of empirical data. After the parameter is fixed, our model can simultaneously generate a number of key scaling laws at both the individual and population level, as well as the trajectory motifs, which are in good agreement with empirical results from data associated with arbitrarily spatial scales.
Typical examples of trajectories of human movements at the individual and population levels are shown in Fig. 1. Our aim is to develop a model that can capture the statistical features and predict the scaling laws associated with trajectories at both the individual and population levels, regardless of the spatial scale. A key quantity is the transition probability. In the recent population-weighted opportunities model16, the transition probability to a destination is proportional to its attractiveness. This probabilistic rule determines the movement of any individual but for only one time step. In order to quantify the statistical behaviors and the scaling laws, sufficiently long trajectories of a large number of individuals are needed.
An important characteristic of human movement, which distinguishes its dynamics from the diffusion dynamics of physical particles, is the memory effect2,20,33. In particular, individuals tend to frequently return to previously visited locations. There are different approaches to taking memory effect into account. For example, in the exploration and preferential return (EPR) model2, it was assumed that the probability for an individual to visit a new location is p ∝ S −γ, where S is the total number of locations that the individual has already visited and γ > 0 is a model parameter. The probability for the individual to visit a previous location is thus 1 − p. The algebraic dependence of p on S indicates that the more locations that an individual has visited, the smaller the probability would be for him/her to explore any new location. That is, there is a strong preference for an individual to move among locations that have been visited previously. The model also assumes2 that the probability for an individual to move to a previously visited location is proportional to the frequency at which it has been visited. This model can successfully reproduce the visiting frequency distribution of the locations obtained from empirical data, as well as the rate of increase in the number of locations. A subsequent model33 emphasizing the importance of the memory effect assumes that the probability distribution of the return time interval, P(τ), is known. An individual chooses a value of τ from the distribution to determine the location that he/she wishes to return to. While this model can reproduce the empirically obtained rate of increase of new locations, the choice of P(τ) is mostly heuristic.
The basic idea underlying the development of our model is that the attractiveness of a location for an individual is determined by both the memory of the individual and the population at the location. Let A be a quantity measuring the effect of memory on the attractiveness of a location to an individual. It is reasonable to assume that a more attractive location can in general impose greater impression on the visitors, resulting in stronger memory and, consequently, enhancing the probability for the individual to visit the location in the future. That is, the attractiveness of a location will be reinforced by good memory and vice versa.
To characterize A in a quantitative manner, we rely on empirical evidence of human travel, in which the frequencies of individual visit to different locations are distributed according to the Zipf's law6. It is thus reasonable to assume that A is distributed in a similar manner. That is, the Zipf's law stipulates that visitors rank the locations visited such that the probability to visit a location is inversely proportional to its rank. For example, the probability of visiting the most frequently visited location is A 1 = λ/1 and the probability of visiting the second most frequently visited location is A 2 = λ/2, and so on, where λ is a constant. Due to an aging effect, the most frequently visited location is often the "oldest" one. We can thus replace the rank of a location by the order with which it is visited. These considerations lead to the following formula to quantify the memory effect:
$$A_j = 1 + \frac{\lambda }{{r_j}},$$
where λ is a parameter characterizing the strength of the memory effect and the index r j denotes that location j is the rth newly visited location associated with the movement trajectory. The unity in the formula represents the initial attractiveness of location j that has not been visited, i.e., if a location has not been visited, its A value is always unity.
Following the classic gravity model, we assume that the attractiveness of a location is proportional to its population. Let B be a quantity characterizing the population-induced effect on the attractiveness of a location, and let m i be the population of location i and N be the total number of locations that can possibly be toured by all the individuals. As illustrated in Fig. 2, the attractiveness B ij for a visitor to travel from location i to destination j is
$$B_{ij} = \frac{{m_j}}{{W_{ji}}},$$
where W ji is the total population in the circular region centered at j, the radius of which is the distance between locations j and i. Note that B ij reflects the competition for opportunities among different locations. For instance, if a traveler at location i wishes to visit a potential destination j, more populations between i and j imply more fierce competitions for limited opportunities at those locations, leading to a lower probability of being offered some opportunity. It is thus reasonable to assume that the attractiveness B ij of destination j for a visitor from location i is inversely proportional to the population between i and j, as quantified by formula (2).
Model illustration. A typical trajectory visiting the five locations denoted by letters a − e with different colors is indicated at the bottom: a → b → c → a → …. The size of the circles that contain a letter indicates the relative attractiveness of the corresponding location as characterized by the index r j , where r 1 is the most attractive location, r 2 is the second most attractive one, and so on. The dashed circle centered at location c indicates that the travelers moves from c to a, whose radius is the distance between c and a, and the total population within the dashed circle is W ca . The model contains a single parameter, λ, which can be determined from each set of empirical data
The transition probability p ij of traveling from location i to j is then proportional to both A j and B ij , which can be written as
$$p_{ij} \propto \frac{{m_j}}{{W_{ji}}}\left( {1 + \frac{\lambda }{{r_j}}} \right).$$
We see that the model contains a single adjustable parameter, the memory strength λ, that can be determined from empirical data. For any location i, we place a number of travelers. Each traveler is assigned a number L, the total number of movement steps, which can be obtained from an actual distribution from empirical data. A traveler thus executes a trajectory of length L and, at each step, he/she moves to a destination according to the transition probability p ij .
Model prediction and validation
Our model, as illustrated in Fig. 2, is capable of predicting the statistical behaviors of human mobility at both the individual and population levels, regardless of the spatial scale. At the individual level, we focus on the following quantities: (I1) the total number of locations visited by time t, (I2) return time distribution to any location, (I3) distribution of frequency of visits to a location, and (I4) emergence of traveling motifs and their probability of occurrence in a long trajectory. At the population level, we seek to predict: (P1) distribution of the travel distance of collective movement and (P2) distribution of the number of traveling steps between two locations.
To validate the model predictions, we employ four empirical data sets, as illustrated in Fig. 1. They are: (DS1) record of user check-ins at Sina Weibo in mainland China (Fig. 1a), (DS2) check-in record of the site Foursquare35 for users in the continental United States (Fig. 1b), (DS3) communication record of mobile phone users in the whole country of Cote d'Ivoire36 (Fig. 1c), and (DS4) check-in record of the site Gowalla37 in Belgium (Fig. 1d). Each data set contains spatial and temporal information about continuous user mobility, from which data of movements among various locations (e.g., cities) can be extracted (Methods). The single free parameter λ can be determined from data (Methods). We obtain λ = 35, 32, 50, and 25 for data sets DS1–DS4, respectively. A heuristic observation is that λ assumes a relatively smaller value for a better developed country (Methods). An explanation is that, in general, in a country with a higher gross domestic product (GDP), individuals can afford more travel, leading to more visited locations and a higher probability of exploring new places. In contrast, in a country with a lower GDP, it is more difficult for people to travel frequently and they tend to stay in their home cities and familiar places. That is, a higher GDP induces a weaker memory effect and a higher probability of visiting new locations, as reflected by the smaller values of the memory strength λ in well developed countries.
Since the data sets contain no information about the individuals' cities of residence, for each individual, we assign the city that he/she signs in with the highest frequency as his/her home city and use it as the initial location in the model. From the data, we calculate the distribution P(L) of the total number of times of movement and choose L accordingly, which is effectively the trajectory length for each individual. It is worth noting that an effective way to test our mobility model is to use the same distribution of the trajectory length as that from the empirical data. We also study analytically the impact of trajectory length on the statistical properties of mobility at both the population and individual levels, with the finding that, for a sufficient number of moving steps, simulation results are in good agreement with the analytical prediction. This indicates that the trajectory length has little effect on the mobility patterns in the long time regime. For the empirical data (Table 1), the total numbers of steps are much larger than those on the population records, so the mobility patterns produced by our model are stable and robust.
Table 1 Description of empirical data sets
Figure 3a–d show, for the data sets DS1–DS4, respectively, the model-predicted algebraic increase with t in the total number of locations visited by time t (green), together with the corresponding results calculated directly from the data (orange). We obtain an excellent agreement between model prediction and the empirical result. The algebraically increasing behavior, as opposed to an exponential growth in the number of cities visited in certain time, is a natural consequence of the memory effect, which is a key ingredient in our model. Figure 3e–h show the predicted and actual return time distributions for the data sets DS1–DS4, respectively, which are algebraic. There is again an excellent agreement between the model prediction and the empirical result. The algebraic decay in the return time distribution can also be attributed to the memory effect. Thus, both the algebraically increasing behavior in Fig. 3a–d and the algebraically decaying behavior in Fig. 3e–h are manifestations of the same memory effect. Figure 3i–l show, for the data sets DS1–DS4, respectively, the model-predicted and empirical frequency distributions of visits to all locations, which agree with each other reasonably well and follow approximately the Zipf's law. The emergence of the Zipf-like scaling behavior is indicative of the heterogeneity in the location attractiveness, an assumption of our model. The results in Fig. 3a–l validate our model with respect to the statistical behaviors of individual trajectories.
Model-predicted and empirical statistical behaviors of human mobility at the level of individual trajectories. a–d Algebraic increase with time t in the total number of locations visited within t, e–h algebraic decaying behavior in the return time distribution, and i–l Zipf-like frequency distribution of visits. In all panels, the green color specifies results predicted by our model, and orange denotes the empirical results obtained directly from data. There is a generally strong agreement between the two types of results. The algebraically increasing and decreasing behaviors in a–d and e–h, respectively, are manifestations of the memory effect
A characteristic of human mobility is the emergence of motifs associated with movement trajectories38, which are referred to as certain simple and fixed patterns of visit that occur repeatedly in a long trajectory. For an individual initially at his/her home location (the one visited with the highest frequency), a motif is defined as a successive sequence of locations visited with the last location being the initial one. From the empirical data, we identify nine distinct motifs (shown at the top in Fig. 4) and calculate the frequencies of their occurrences from the entire data set. With parameter λ extracted from the data, our model can generate long trajectories from which the possible motifs and their frequencies of occurrence can be determined. Remarkably, our model yields exactly the same set of motifs with frequencies that agree with the empirical results reasonably well, as shown in Fig. 4. Due to the significance of travel motifs in determining the microscopic mobile patterns of travelers, the agreement provides further validation of our model at the individual level.
Frequency of occurrence of motifs associated with movement trajectories. From the four empirical data sets, nine distinct motifs contained in movement trajectories are identified. Exactly the same motifs are obtained from model generated trajectories. The frequencies of occurrence of the motifs from the model and from actual trajectories agree with each other reasonably well. There exist many more types of possible motifs. However, the nine types included here have the highest frequencies of occurrence, the sum of which exceeds 0.97. The frequency of the 10th highest motif, for example, is <0.001. Considering the nine motifs thus suffices
Our model also has strong predictive power for human movements at the population level on all spatial scales. As shown in Fig. 5, the predicted behaviors of P(d) and P(T), the distributions of the travel distance and of the number of traveling steps between two locations, agree well with the statistical results from the empirical data. In particular, Fig. 5a–d show, for the data sets DS1–DS4, respectively, that P(d) decays exponentially. Figure 5e–h reveal that P(T) exhibits a robust algebraic scaling for all four data sets. Figure 5i–l demonstrate that the model-predicted and real values of T are nearly statistically indistinguishable (albeit with fluctuations). Our model is then universally applicable to characterizing human movements across vast different spatial scales at the population level.
Model prediction at the population level and validation. For the four empirical data sets with vast difference in the spatial scale, model-predicted and real distributions of travel distance d (a–d) and the number of traveling steps, T, between two locations e–h. The lines in e–h represent a power-law fit to the real data. i–l Statistical display of the model-predicted and real values of T for the four data sets, respectively, which are nearly indistinguishable. The gray points are scatter plot for each pair of locations. The blue points represent the average number of predicted travels in different bins. The standard boxplots represent the distribution of the number of predicted travels in different bins of the number of observed travels. A box is marked in green if the line y = x lies between 10% and 91% in that bin and in red otherwise
Theoretical analysis
In our model, the fixed amount of attractiveness of a location is calculated based on its population. Since the population distribution is typically highly heterogeneous without a closed mathematical form, it is not feasible to treat our model exactly. To gain analytic insights, we consider a simplified model obtained by imposing the approximation that the population is uniformly distributed among the locations, and focus on analytically predicting the individual trajectories and the collective mobility pattern with a special emphasis on the role of the memory effect. Although the simplified model deviates from real scenarios, the analytical predictions enable a good understanding of the real mobility patterns at both the individual and population levels.
In the simplified model, an individual travels among N locations. At each time step, the probability to move to a destination is proportional to its attractiveness. At t = 0, the initial attractiveness is identical (unity) for all locations. During the travel, the attractiveness of the rth first visited location is updated to 1 + λN/r, where λ > 0 is a parameter. The model describes essentially a random-walk process with time varying location attractiveness, with parameter λ characterizing the memory strength of (or preference to) locations previously visited. For λ = 0, the model is reduced to an unbiased random walk. For λ = ∞, the walker can travel between only the first two locations. The total number S(t) of locations visited by time t can be used to characterize how fast the underlying mobile process takes place. For a uniform random walk, S(t) increases with t linearly: S(t) ∝ t. For the EPR model2, S(t) increases with t but in a sublinear fashion: S(t) ∝ t β with 0 < β < 1. For our random walk model with memory, S(t) can be obtained analytically (Supplementary Note 1), as shown in Fig. 6a. We see that, as the memory strength parameter λ is increased, the overall rate of increase in S(t) becomes smaller. In addition, for a fixed value of λ, the time derivative of S(t) tends to increase with time, which is consistent with the result from real data (c.f., Figs. 1a and 3 in ref. 2).
Analytic predictions of a simplified model in comparison with simulation results. a–c Analytically predicted results of S(t)—the total number of locations visited by time t, f r —frequency distribution of visiting distinct locations, and P(τ)—return time distribution. The curves represent the theoretical results for different values of the memory parameter λ, and the data points are simulation results with N = 1000 and L = 1000. d Travel distance distributions P(d) for the simplified model implemented in four classic fractal domains including a 2D Cantor dust, Vicsek fractal, Sierpinski triangle, and hexaflake with fractal dimension D = 1.262, 1.465, 1.585, and 1.771, respectively (Supplementary Note 3), where the lines represent the theoretical results for different values of D. The inset indicates the distributions P(T) of the number of traveling steps at the population level for the four fractal domains, with the solid line being the analytic prediction of P(T)
Another characteristic quantity is f r , the distribution of the frequency of visit to location r. For an unbiased random walk, f r is uniformly distributed. For the EPR model2, f r decays algebraically: f r ∝ r −α, where α > 0 is a constant. For our model, we analytically obtain (Supplementary Note 1)
$$f_r \propto \frac{{\lambda S}}{r} + 1 - \lambda .$$
For λ = 0, Eq. (4) reduces to a uniform distribution. For λ = 1, we recover the Zipf's law for f r . Figure 6b shows the analytic and simulation results of f r for a number of λ values, where the curves represent the theoretical prediction. We see that, for λ > 1, there is an apparent deviation from the Zipf's law, as signified by the emergence of an exponential cutoff toward the tail end of f r . The physical meaning is that, as the memory effect is intensified, a walker tends to travel among only a few locations.
The return time distribution P(τ) is defined as the probability for a walker to return to one of the previously visited locations after τ steps, which is also a reflection of the memory effect. In our model, P(τ) contains two different algebraic terms but with the same exponent −1 (Supplementary Note 1). As λ is increased, P(τ) tends to a single algebraic distribution with essentially zero values near the tail, indicating an extremely low probability for the walker to return to a previous location after many time steps. Figure 6c shows the analytic and simulation results of P(τ) from our model. The agreement is reasonable, and the deviation of the analytic from the simulation result in the large τ region is due to the finite time used in the simulation.
Finally, we remark on an appealing feature of our model. Consider the probability for the walker to choose a new location at the next time step. Analysis of our model leads to (Supplementary Note 1)
$$P_{{\mathrm{new}}} = \frac{1}{{1 + \lambda ({\mathrm{ln}}\,S + C)}}.$$
Thus, in our model, P new decreases with S, which occurs naturally as a consequence of a basic and intuitive assumption, namely the memory effect.
A further simplification of the model by assuming that each individual can move one step only renders analytically predictable collective mobility patterns at the population level. In particular, we place m individuals at each location and exploit the previously discovered39, common fractal feature in the spatial distribution of locations in the real world: \(W_{ji} \propto d_{ij}^D\), where D is the fractal dimension. Equation (3) can be formulated as (Supplementary Note 2)
$$T_{ij} = m_ip_{ij} \propto \frac{{m_im_j}}{{d_{ij}^D}},$$
where T ij is the total number of traveling steps from i to j for the whole population. Equation (6) is a standard gravity model with a power-law distance function. Since Eq. (6) indicates that the number of traveling steps T between two locations of distance d is T(d) ∝ d −D, the travel distance distribution is given by the same form: P(d) ∝ d −D, as validated by Fig. 6d for four typical fractal domains. The number of location pairs with distance ≤ d in a fractal domain is N(d) ∝ d D. Thus, the number of traveling steps T obeys a power-law distribution (Supplementary Note 2):
$$P(T) \propto T^{ - 2}.$$
It is worth noting that the algebraic exponent −2 is universal, regardless of the fractal dimension D of the location distribution in the simplified model, as shown in the insert of Fig. 6d. However, in the real world, the heterogeneous nature of the population distribution at different locations can cause a deviation of the exponent from −2. As shown in Fig. 5e–h, the fit of the empirical data demonstrates that their algebraic exponents range from −1.33 to −1.56 (see Supplementary Note 2 for a detailed explanation of the effect of heterogeneous population distribution on the algebraic exponent). Nonetheless, the power-law distribution predicted by our simplified model is robust, which captures the essential features of the collective mobility patterns in the real world.
In the development of our human mobility model, Zipf's law is naturally included as an essential component. However, the Zipf's law is closely related to diminishing exploration. To elucidate the interplay between the two, we articulate an extended individual mobility model based on the generalized Zipf's law. This is guided by the previous evidence that there are situations where individuals tend to choose locations to travel into by following the generalized Zipf's law40. Specifically, we assume f ∝ r −ζ, where the exponent ζ > 1 is an adjustable parameter. For the extended model, we analytically obtain P new = ρS −γ, where γ = ζ − 1. We see that the formula of P new is a direct manifestation of the basic assumptions in the EPR model2. This suggests that the generalized Zipf's law and the power-law relation between P new and S have a mutually causal relationship, and the individual mobility models based on the former and latter are equivalent to each other. For the extended model, we also derive the return time distribution P(τ) for sufficiently large values of S. A detailed description of the extended model, the analysis, and results are presented in Supplementary Note 1.
The past decade has witnessed a great deal of efforts into uncovering and understanding the general dynamical behaviors of human mobility. A variety of real data sets have been analyzed, leading to a spectrum of mathematical models being devised to explain the phenomena revealed by data. While universal scaling laws have been unveiled, it turns out that spatial scale has a significant effect on the dynamics. In particular, human mobility at large (e.g., big countries) and small (e.g., small countries or big cities, see Supplementary Note 4 for a city example) scales tends to exhibit distinct scaling behaviors. The representative existing models are suitable to describe human mobility on either large or small scales at either the individual or population level, motivating us to articulate a model that can describe the statistical and scaling behaviors of human behaviors at all spatial scales as well as at both the individual and population levels.
There are two essential ingredients in our model construction: memory and population-induced competition effects. Both effects jointly determine the attractiveness of a location (see Supplementary Note 5 for results and a detailed discussion). On the basis of the attractiveness of locations, we obtain the key quantity in microscopic model of human mobility: the transition probability for an individual to move from one location to another. Our unifying model contains a single adjustable parameter: the strength of the memory effect, and enables us to make predictions about the scaling laws associated with the key statistical behaviors of human mobility at both the individual and population levels, regardless of the spatial scales. The relevant quantities include the total number of locations visited within certain time, the frequency distribution of visits to different locations, and the distribution of time interval of successive visits to any location. Our model also allows us to identify a few kinds of distinct motifs embedded in typical trajectories. All these results have been verified using empirical data from countries having drastically different spatial scales.
Modeling and predicting mobility patterns and scaling laws at both the individual and population levels are a fundamental problem for exploring many dynamical processes associated with human mobility. A typical example is disease propagation in the society. As discussed in ref. 41, in the metapopulation model, both the population level mobility, e.g., travel flux among subpopulations, and the individual level mobility, e.g., transition probability of individuals, are necessary to model the contagion dynamics and predict disease spreading in the society. The empirical mobilities may be obtained by directly measuring the travel flux among locations and the travel trajectories of all individuals in a certain time interval. However, to accomplish this task, vast amounts of private data, such as the data of cell phones with GPS function in specific locations, are required, making the task impractical. Our universal model, because it is based solely on the population distribution, provides an alternative approach to unraveling the important mobility patterns with reasonable accuracy. Likewise, our model may find potential use in alleviating congestion in urban areas, which is closely related to human mobility behaviors.
Empirical data sets and processing method
The four data sets DS1–DS4 are from mainland China (Supplementary Data 1 and 2), the contiguous United States, Cote d'Ivoire, and Belgium, respectively. Sets DS1, DS2, and DS4 are the check-in records of social networks35,37 in their respective countries, which contain the time and locations of user check-ins. Set DS3 is a mobile phone call detail record36 that collects the time and positions of users making phone calls or sending text messages in a 5-month period, where the spatial locations are determined within counties. In this case, the central city of each county is taken as the location of the individual. Since we focus on movements among cities, all the positions within a city are regarded as the same with an identical city label. Table 1 lists the detailed information about each data set, from which a complete trajectory of each user moving among different cities can be obtained for the entire time duration of the data record. The results of statistical analysis of individual trajectories are shown in Fig. 3, while those at the population level are presented in Fig. 5.
In our model, the single free parameter is λ, the strength of memory effect, which affects directly the rate of increase in the number S(t) of locations visited in certain time. For a given empirical data set, the function S(t) can then be used to estimate λ. To accomplish this, we define the following objective function:
$$E(\lambda ) = \mathop {\sum}\limits_{t = 1}^{L_{{\mathrm{max}}}} \frac{{\left| {S_{{\mathrm{real}}}(t) - S(t,\lambda )} \right|}}{{S_{{\mathrm{real}}}(t)}},$$
where L max is the maximum time step, S real(t) is obtained from the actual data set, and S(t, λ) is calculated through the model with parameter λ. The objective function can be minimized to yield an estimated value of λ in the model. We also use the quantities P(τ) and f r to estimate λ in addition to that based on S(t), and find little difference in the prediction accuracy for both the individual and population mobility patterns.
The authors declare that the data supporting the findings of this study are available within the paper and its Supplementary Information file, or from the authors upon reasonable request.
Brockmann, D., Hufnagel, L. & Geisel, T. The scaling laws of human travel. Nature 439, 462–465 (2006).
Song, C., Koren, T., Wang, P. & Barabási, A. L. Modelling the scaling properties of human mobility. Nat. Phys. 6, 818–823 (2010).
Simini, F., González, M. C., Maritan, A. & Barabási, A. L. A universal model for mobility and migration patterns. Nature 484, 96–100 (2012).
Simini, F., Maritan, A. & Néda, Z. Human mobility in a continuum approach. PLoS ONE 8, e60069 (2013).
Ren, Y., Ercsey-Ravasz, M., Wang, P., González, M. C. & Toroczkai, Z. Predicting commuter flows in spatial networks using a radiation model based on temporal ranges. Nat. Commun. 5, 5347 (2014).
González, M. C., Hidalgo, C. A. & Barabási, A. L. Understanding individual human mobility patterns. Nature 453, 779–782 (2008).
Eagle, N., Macy, M. & Claxton, R. Network diversity and economic development. Science 328, 1029–1031 (2010).
Hu, Y., Zhang, J., Huan, D. & Di, Z.-R. Toward a general understanding of the scaling laws in human and animal mobility. Europhys. Lett. 96, 38006 (2011).
Noulas, A., Scellato, S., Lambiotte, R., Pontil, M. & Mascolo, C. A tale of many cities: universal patterns in human urban mobility. PLoS ONE 7, e37027 (2012).
Lenormand, M., Huet, S., Gargiulo, F. & Deffuant, G. A universal model of commuting networks. PLoS ONE 7, e45985 (2012).
Goh, S., Lee, K., Park, J. S. & Choi, M. Y. Modification of the gravity model and application to the metropolitan Seoul subway system. Phys. Rev. E 86, 026102 (2012).
Yan, X.-Y., Han, X.-P., Wang, B.-H. & Zhou, T. Diversity of individual mobility patterns and emergence of aggregated scaling laws. Sci. Rep. 3, 02678 (2013).
Saramäki, J. et al. Persistence of social signatures in human communication. Proc. Natl. Acad. Sci. USA 111, 942–947 (2013).
Hou, L., Pan, X., Guo, Q. & Liu, J.-G. Memory effect of the online user preference. Sci. Rep. 4, 06560 (2014).
Zhao, Z.-D., Huang, Z.-G., Huang, L., Liu, H. & Lai, Y.-C. Scaling and correlation of human movements in cyber and physical spaces. Phys. Rev. E 90, 050802(R) (2014).
Yan, X.-Y., Zhao, C., Fan, Y., Di, Z.-R. & Wang, W.-X. Universal predictability of mobility patterns in cities. J. R. Soc. Interface 11, 20140834 (2014).
Šćepanović, S., Mishkovski, I., Hui, P., Nurminen, J. K. & Ylä-Jääski, A. Mobile phone call data as a reginal socio-economic proxy indicator. PLoS ONE 10, e0124160 (2015).
Pappalardo, L. et al. Returners and explorers dichotomy in human mobility. Nat. Commun. 6, 8166 (2015).
Gallotti, R., Bazzani, A., Rambaldi, S. & Barthelemy, M. A stochastic model of randomly accelerated walkers for human mobility. Nat. Commun. 7, 12600 (2016).
Zhao, Y.-M., Zeng, A., Yan, X.-Y., Wang, W.-X. & Lai, Y.-C. Unified underpinning of human mobility in the real world and cyberspace. New J. Phys. 18, 053025 (2016).
Frías-Martínez, E., Williamson, G. & Frías-Martínez, V. IEEE 3rd Conference on Privacy, Security, Risk and Trust (PASSAT) and 2011 IEEE Third Inernational Conference on Social Computing (SocialCom) 57–64 (IEEE, 2011).
Belik, V., Geisel, T. & Brockmann, D. Nature human mobility patterns and spatial spread of infectious diseases. Phys. Rev. X 1, 011001 (2011).
Wesolowski, A. et al. Quantifying the impact of human mobility on malaria. Science 338, 267–270 (2012).
Jiang, S et al. In Proceedings of the 2nd ACM SIGKDD International Workshop on Urban Computing (UrbComp'13) 57–64 (ACM Press, New York, 2013).
Tizzoni, M. et al. On the use of human mobility proxies for modeling epidemics. PLoS ONE 10, e1003716 (2014).
Deville, P. et al. Dynamic population mapping using mobile phone data. Proc. Natl. Acad. Sci. USA 111, 15888–15893 (2014).
Berlingerio, M. et al. Machine Learning and Knowledge Discovery in Databases Lecture Notes in Computer Science 8190 663–666 (2013).
Lima, A., Domenico, M. D., Pejovic, V. & Musolesi, M. Disease containment strategies based on mobility and information dissemination. Sci. Rep. 5, 10650 (2015).
Song, C., Qu, Z., Blumm, N. & Barabási, A. L. Limits of predictability in human mobility. Science 327, 1018–1021 (2010).
Deville, P. et al. Scaling identity connects human mobility and social interactions. Proc. Natl. Acad. Sci. USA 113, 7047–7052 (2016).
Zipf, G. K. The p 1 p 2/d hypothesis:on the intercity movement of persons. Am. Sociol. Rev. 11, 677–686 (1946).
Han, X.-P., Hao, Q., Wang, B.-H. & Zhou, T. Origin of the scaling law in human mobility: hierarchy of traffic systems. Phys. Rev. E 83, 036117 (2011).
Szell, M., Sinatra, R., Petri, G., Thurner, S. & Latora, V. Understanding mobility in a social petri dish. Sci. Rep. 2, 00457 (2012).
Zhao, Z.-D. et al. Emergence of scaling in human-interest dynamics. Sci. Rep. 3, 3472 (2013).
Levandoski, J. J., Sarwat, M., Eldawy, A. & Mokbel, M. F. IEEE 28th International Conference on Data Engineering 450–461 (2012).
Blondel, V. D., Decuyper, A. & Krings, G. A survey of results on mobile phone datasets analysis. EPJ Data Science 4, 10 (2015).
Cho, E., Myers, S. A. & Leskovec, J. Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining 1082–1090 (2011).
Schneider, C. M., Belik, V., Couronné, T., Smoreda, Z. & González, M. C. Unravelling daily human mobility motifs. J. R. Soc. Interface 10, 20130246 (2013).
Appleby, S. Multifractal characterization of the distribution pattern of the human population. Geogr. Anal. 28, 147–160 (1996).
Wyllys, R. E. Empirical and theoretical bases of Zipf's law. Libr. Trends 30, 53–64 (1981).
Balcan, D. & Vespignani, A. Phase transitions in contagion processes mediated by recurrent mobility patterns. Nat. Phys. 7, 581–586 (2011).
X.-Y.Y. was supported by NSFC under grant nos. 71621001 and 71671015. W.-X.W. was supported by NSFC under grant no. 71631002. Z.-Y.G. was supported by NSFC under grant no. 71621001. Y.-C.L. would like to acknowledge support from the Vannevar Bush Faculty Fellowship program sponsored by the Basic Research Office of the Assistant Secretary of Defense for Research and Engineering and funded by the Office of Naval Research through grant no. N00014-16-1-2828. We are grateful to Professor T. Zhou for providing us the Sina Weibo data, and to Dr C. Zhao for data processing.
Institute of Transportation System Science and Engineering, Beijing Jiaotong University, Beijing, 100044, China
Xiao-Yong Yan
& Zi-You Gao
School of Systems Science and Center for Complexity Research, Beijing Normal University, Beijing, 100875, China
Wen-Xu Wang
School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ, 85287, USA
Ying-Cheng Lai
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X.-Y.Y., W.-X.W., Z.-Y.G. and Y.-C.L. designed the research; X.-Y.Y. and W.-X.W. performed the research; X.-Y.Y., W.-X.W. and Z.-Y.G. contributed analytic tools; W.-X.W., Z.-Y.G. and Y.-C.L. analyzed the data; and X.-Y.Y., W.-X.W., Z.-Y.G. and Y.-C.L. wrote the paper.
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Correspondence to Wen-Xu Wang or Zi-You Gao or Ying-Cheng Lai.
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\begin{document}
\title{Restart FISTA with Global Linear Convergence} \pagestyle{plain}
\begin{abstract} Fast Iterative Shrinking-Threshold Algorithm (FISTA) is a popular fast gradient descent method (FGM) in the field of large scale convex optimization problems. However, it can exhibit undesirable periodic oscillatory behaviour in some applications that slows its convergence. Restart schemes seek to improve the convergence of FGM algorithms by suppressing the oscillatory behaviour. Recently, a restart scheme for FGM has been proposed that provides linear convergence for non strongly convex optimization problems that satisfy a quadratic functional growth condition. However, the proposed algorithm requires prior knowledge of the optimal value of the objective function or of the quadratic functional growth parameter. In this paper we present a restart scheme for FISTA algorithm, with global linear convergence, for non strongly convex optimization problems that satisfy the quadratic growth condition without requiring the aforementioned values. We present some numerical simulations that suggest that the proposed approach outperforms other restart FISTA schemes. \end{abstract}
\subsubsection*{Keywords} Fast gradient method, restart FISTA, convex optimization, linear convergence, quadratic functional growth condition.
\section{Introduction} \label{sec:Intro} Fast gradient methods (FGM) were introduced by Yurii Nesterov in \cite{Nesterov83}, \cite{Nesterov04}, where it was shown that these methods provide a convergence rate \textit{O}$(1/k^2)$ for smooth convex optimization problems with non strongly convex objective functions \cite{Nesterov04}, where $k$ is the iteration counter. These methods were generalized to composite non smooth convex optimization problems in \cite{Beck09}, \cite{Nesterov13}, \cite{Tseng:08}. The resulting algorithm is commonly known as FISTA algorithm \cite{Beck09}. Because of its complexity certification, it is often used in the context of embedded model predictive control \cite{koegel2011fast}, \cite{Richter12}, \cite{Krupa:18}. Another possibility to address composite convex optimization problems is to use splitting methods like ADMM \cite{boyd2011distributed}, \cite{Giselsson:TAC:17}, \cite{Banjac:TAC:18}.
FISTA algorithms can be applied in a primal setting (as in the Lasso problem \cite{Beck09}), or in a dual one \cite{Richter:13}, \cite{Beck2014Dual}. They can be thought of as a momentum method, since the linearization point at each iteration depends on the previous iterations. Since the momentum grows with the iteration counter, the algorithm can exhibit undesirable periodic oscillating behavior for certain applications, which slows the convergence rate. To mitigate this, restart schemes have been proposed in the literature which stop the algorithm when a certain criteria is met. It is then restarted using the last value provided by the stopped algorithm as the new initial condition \cite{Donoghue:13}, \cite{Alamir:13}, \cite{Giselsson:14CDC}.
In \cite{Donoghue:13} two heuristic restart schemes for FGM are proposed which exhibit improved convergence rates over non-restart FGM schemes. These restart schemes reset the momentum of the FGM in order to eliminate the undesirable oscillations whenever the periodical behavior is detected. A restart scheme similar to the ones in \cite{Donoghue:13} with \textit{O}$(1/k^2)$ convergence rate for smooth convex optimization is presented in \cite{Giselsson:14CDC}. In \cite{Kim:18}, an algorithm is proposed that uses the restart schemes from \cite{Donoghue:13}. Numerical results show improvements over previous restart schemes for FGM, but no theoretical results on convergence rates are provided.
Recently, linear convergence rate has been derived for several first order methods applied to convex optimization problems with non strongly convex objective functions that satisfy a relaxation of the strong convexity known as the quadratic functional growth \cite{Necoara:18}.
In \cite[Subsection 5.2.2]{Necoara:18} a restarting scheme of FGM is presented with global linear convergence rate for convex optimization problems that satisfy the functional growth condition with parameter $\mu$. However, in order to implement this strategy, prior knowledge is needed of either the optimal value of the objective function or the value of $\mu$, which can be challenging to compute.
In this paper we propose a novel restart scheme for FISTA algorithm applied to solving convex constrained problems. We show that the algorithm guarantees global linear convergence rate \textit{O}$(1/\sqrt{\mu})$ for convex optimization problems with non strongly convex objective functions that satisfy the quadratic functional growth condition with parameter $\mu$. The proposed algorithm does not require prior knowledge of the value of $\mu$ or of the optimal value of the objective function. We provide theoretical upper bounds on the number of iterations of the algorithm needed to achieve a given accuracy.
Additionally, we show numerical results comparing the proposed algorithm with the heuristic restart schemes from \cite{Donoghue:13} and the restart scheme from \cite{Necoara:18} for Lasso problems.
In Section \ref{sec:Formulation} we introduce the problem formulation. Section \ref{sec:FISTA} presents FISTA algorithm and some restart schemes. The convergence rate of non restart FISTA algorithm under the satisfaction of the quadratic functional growth condition is presented in Section \ref{sec:Conv_FISTA}. In Section \ref{sec:OurFISTA} we present the proposed restart scheme for FISTA and state its global linear convergence. Numerical results comparing the proposed algorithm with other restart schemes applied to FISTA are shown in Section \ref{sec:Results}. Finally, conclusions are presented in Section \ref{sec:Conclusions}.
\subsubsection*{Notation}
Given vectors $x$ and $y$, we denote by $\sp{x}{y}$ their scalar product, i.e. $\sp{x}{y}\doteq x\T y$. Given vector $x$, $\|x\|_2$ denotes its Euclidean norm ($\|x\|_2\doteq \sqrt{x\T x}$), and $\| \cdot \|_1$ denotes its $l_1$-norm (sum of the absolute values of the components of $x$). Given $R\succ 0$ we denote by $\| \cdot \|_R$ the weighted Euclidean norm $\| x \|_R \doteq \sqrt{x\T R x}$, and by $\| x \|_* \doteq \| x \|_{R^{-1}}$ its dual norm. $\ln(\cdot)$ is the natural logarithm and $e$ is Euler's number. $\lfloor x\rfloor$ denotes the largest integer smaller than or equal to $x$; $\lceil x \rceil $ denotes the smallest integer greater than or equal to $x$. Given a set ${\mathcal{X}}\subseteq {\rm \,I\!R} ^n$ we denote by $I_{{\mathcal{X}}}$ its indicator function. That is, ${I_{{\mathcal{X}}}(x)=0}$ if ${x\in {\mathcal{X}}}$, and ${I_{{\mathcal{X}}}(x)=\infty}$ if $x\not\in {\mathcal{X}}$. The relative interior of set ${\mathcal{X}}$ is denoted by $\ri{{\mathcal{X}}}$. Given the extended real valued function ${f: {\rm \,I\!R} ^n\to (-\infty,\infty]}$ we denote by $\dom{f}$ its effective domain. That is, ${ \dom{f}\doteq\set{x\in {\rm \,I\!R} ^n}{f(x)<\infty}}$. We denote by $\epi{f}$ the epigraph of $f$. That is, ${\epi{f}\doteq\set{(x,t)\in {\rm \,I\!R} ^n\times {\rm \,I\!R} }{f(x)\leq t}}$. We say that function ${f: {\rm \,I\!R} ^n\to (-\infty,\infty]}$ is closed if its epigraph is a closed set. We say that ${f: {\rm \,I\!R} ^n\to (-\infty,\infty]}$ is proper if its effective domain is not empty. That is, if $f$ is not identically equal to $\infty$. We say that a vector ${d\in {\rm \,I\!R} ^n}$ is a subgradient of $f$ at a point ${x\in\dom{f}}$ if ${f(y)\geq f(x)+\sp{d}{y-x}}$, ${\forall y\in {\rm \,I\!R} ^n}$. The set of all subgradients of $f$ at $x$ is called the subdifferential of $f$ at $x$ and is denoted by $\partial f(x)$.
\section{Problem Formulation} \label{sec:Formulation}
We address the problem of solving the composite convex minimization problem \begin{equation} \label{eq:OP}
f^* = \min\limits_{x\in {\mathcal{X}}} f(x) = \min\limits_{x\in {\mathcal{X}}} \Psi(x) + h(x), \end{equation} under the following assumption.
\begin{assumption}\label{assum:conv:smooth}
We assume that
\blista
\item $h: {\rm \,I\!R} ^n\to {\rm \,I\!R} $ is a smooth differentiable convex function. That is, there is $R\succ 0$ such that the inequality
\begin{equation}\label{ineq:smooth:R}
h(x) \leq h(y)+\sp{{\nabla h}(y)}{x-y}+\frac{1}{2}\|x-y\|_R^2,
\end{equation}
is satisfied for every $x\in {\rm \,I\!R} ^n$ and $y\in {\rm \,I\!R} ^n$.
\item $\Psi: {\rm \,I\!R} ^n\to (-\infty,\infty]$ is a closed convex function and ${\mathcal{X}}\subseteq {\rm \,I\!R} ^n$ is a closed convex set.
\item Denote $f\doteq \Psi+h$. The minimization problem
$$\min\limits_{x\in {\mathcal{X}}} f(x)$$
is solvable. That is, there is $x^* \in {\mathcal{X}} \bigcap\dom{\Psi}$ such that $f^*=f(x^*)=\inf\limits_{x\in {\mathcal{X}}} f(x)$.
\elista \end{assumption}
We notice that it is standard to write down the first point of Assumption \ref{assum:conv:smooth} as \begin{equation}\label{equ:h:with:L}
h(x) \leq h(y)+\sp{{\nabla h}(y)}{x-y}+\frac{L}{2}\|x-y\|_{S}^2, \end{equation}
where parameter $L$ serves to characterize the smoothness of $h$ and $S$ is a positive definite matrix. Constant $L$ provides a bound on the Lipschitz constant of the gradient $\nabla h(\cdot)$ \cite[Subsection 2.1]{Nesterov04}. Since $$\frac{L}{2}\|x-y\|_{S}^2 = \frac{1}{2} \|x-y\|_{ L S}^2,$$ we have that (\ref{equ:h:with:L}) implies (\ref{ineq:smooth:R}) if we take $R=LS$. This simplifies the algebraic expressions needed to analyze the convergence of the proposed algorithm.
We notice that Assumption \ref{assum:conv:smooth} guarantees that the minimization problem (\ref{eq:OP}) is solvable. The optimal set $\Omega$ is defined as $$ \Omega \doteq \set{x}{x\in {\mathcal{X}}, f(x)=f^*}.$$
This set is a singleton if $f(x)$ is strictly convex. Given ${x\in {\rm \,I\!R} ^n}$ we will denote $\bar{x}$ its closest element in the optimal set $\Omega$ (with respect to the norm $\|\cdot\|_R$). That is, \begin{equation}\label{def:bx}
\bar{x} \doteq {\rm{arg}} \min\limits_{z\in\Omega} \|x-z\|_R. \end{equation} Given $y\in {\rm \,I\!R} ^n$, one could use the local information given by ${\nabla h}(y)$ to minimize the value of $f=\Psi+h$ around $y$. Under Assumption \ref{assum:conv:smooth}, this can be done obtaining the minimizer of the strictly convex optimization problem
$$ \min\limits_{x\in {\mathcal{X}}} \; \Psi(x) +\sp{{\nabla h}(y)}{x-y}+\frac{1}{2} \|x-y\|_R^2.$$ It is well known that this problem is solvable and has a unique solution if Assumption \ref{assum:conv:smooth} holds (see, for example, Subsection 6.1 in \cite{beck2017} for an analogous result). For completeness we provide a proof of this statement in Appendix \ref{appen:existence:uniqueness} (see Property \ref{Prop:solvable:unique:grad}).
The solution to this optimization problem leads to the notion of composite gradient mapping \cite{Nesterov13}, which constitutes a generalization of the gradient mapping that can be found in \cite[Subsection 2.2]{Nesterov04} for the particular case $\Psi(\cdot)=0$. See also \cite{Beck09} for the particular case ${\mathcal{X}}= {\rm \,I\!R} ^n$. \begin{definition}[Composite Gradient Mapping $g(y)$]\label{def:composite:gradient:mapping} ~\\Under Assumption \ref{assum:conv:smooth}, and given $y\in {\rm \,I\!R} ^n$, we define \begin{align*}
y^+ &\doteq {\rm{arg}} \min\limits_{x\in {\mathcal{X}}} \; \Psi(x)+ \sp{{\nabla h}(y)}{x-y}+\frac{1}{2} \|x-y\|_R^2,\\
g(y) &\doteq R(y-y^+). \end{align*} \end{definition} We notice that the composite gradient mapping is closely related to the notion of proximal operator \cite{Parikh13}, \cite[Chapter 6]{beck2017}. For example, one could state, after some manipulations, the computation of the composite gradient mapping as the computation of a proximal operator. In the context of optimal gradient methods, it is assumed that the computation of $y^+$ is cheap. This is the case when ${\mathcal{X}}$ is a simple set (box, $ {\rm \,I\!R} ^n$, etc.), $R$ diagonal, and $\Psi(\cdot)$ a separable function. For example, in the well known Lasso optimization problem, the computation of $y^+$ resorts to the computation of the shrinkage operator \cite{Beck09}. See \cite{Combettes11}, Section 6 of \cite{Parikh13}, Chapter 28 in \cite{Bauschke:11}, or Chapter 6 in \cite{beck2017}, for numerous examples in which the computation of the composite gradient mapping is simple.
The following property gathers well-known properties of the composite gradient mapping $g(y)$ and its dual norm $\| g(y) \|_*=\|g(y)\|_{R^{-1}}$ \cite{Beck09}, \cite{Nesterov13}. For completeness, we include the proof in Appendix \ref{appen:proj:gradient}.
\begin{property}\label{prop:proj:gradient} Suppose that Assumption \ref{assum:conv:smooth} holds. Then, \blista \item For every $y\in {\rm \,I\!R} ^n$ and $x\in {\mathcal{X}}$: \begin{align*}
f(y^+) -f(x) &\leq \sp{g(y)}{y^+-x} + \frac{1}{2}\|g(y)\|_*^2\\
& = \sp{g(y)}{y-x} - \frac{1}{2}\|g(y)\|_*^2 \\
& = - \frac{1}{2}\| y^+ -x \|^2_R + \frac{1}{2}\|y-x\|_R^2. \end{align*} \label{prop:proj:gradient_1}
\item For every $y\in {\mathcal{X}}$: $$ \frac{1}{2} \| g(y)\|_*^2 \leq f(y)-f(y^+) \leq f(y)-f^*.$$ \label{prop:proj:gradient_2} \elista
\end{property}
The composite gradient serves to characterize optimality \cite{Nesterov13}. That is, under Assumption \ref{assum:conv:smooth} we have the following equivalence $$ y\in \Omega \Leftrightarrow g(y)=0. $$ This fact is proved in Appendix \ref{appen:optimaliticharacterization}.
\section{Restart FISTA Schemes } \label{sec:FISTA}
For a given initial condition $z\in {\rm \,I\!R} ^n$, a minimum number of iterations $k_{min} \geq 0$, and an exit condition $E_c$, the non restart FISTA algorithm \cite{Beck09} is shown in Algorithm \ref{alg:FISTA}. This algorithm solves $\min\limits_{x\in{\mathcal{X}}} \; h(x)+\Psi(x)$ under Assumption \ref{assum:conv:smooth}.
\begin{algorithm}
\DontPrintSemicolon
\caption{FISTA} \label{alg:FISTA}
\Require{$z \in {\rm \,I\!R} ^n$, $k_{min} \geq 0$, $E_c$}
$y_0 = x_0=z^+$, $t_0 = 1$, $k = 0$\;
\Repeat{$E_c$ and $k \geq k_{min}$} {
$k = k + 1$\;
$x_{k} = y_{k-1}^+$ \label{alg:step:grad}\;
$t_k = \fracg{1}{2} \left( 1 + \sqrt{1 + 4 t^{2}_{k-1}}\, \right)$ \label{FISTA:tk}\;
$y_k = x_k + \fracg{t_{k-1} - 1}{t_k} (x_k - x_{k-1})$\;
Compute exit condition $E_c$\;}
\KwOut{$r = x_k$, $n= k$} \end{algorithm}
Since the optimality of $x_k$ is equivalent to $g(x_k)=0$ (see Property \ref{prop:charac:optimality} in Appendix \ref{appen:optimaliticharacterization}), a typical choice for non restart FISTA schemes is to choose $k_{min}$ equal to zero and codify the exit condition \begin{equation*} \label{eq:exit}
\| g(x_k) \|_* \leq \epsilon, \end{equation*}
where $\epsilon >0$ is an accuracy parameter. It is also common to use the exit condition $\|g(y_{k-1})\|_*\leq \epsilon$, since this exit condition requires $y_{k-1}^+$, which has already been computed in step \ref{alg:step:grad} of the algorithm.
It is well known that under Assumption \ref{assum:conv:smooth}, see also (\ref{equ:h:with:L}), the iterations of non-restart FISTA satisfy \cite{Beck09,Nesterov13}, \begin{equation}
f(x_k)-f^* \leq \frac{2}{(k+1)^2}\| x_0-\bar{x}_0\|^2_R, \; \forall k\geq 1, \label{equ:conv:FISTA} \end{equation} where $\bar{x}_0$ represents the point in the optimal set $\Omega$ closest to the initial condition $x_0$ of the algorithm (see (\ref{def:bx})). For the sake of completeness, we present a detailed proof of this claim in Appendix \ref{app:conver:Fista}. We also prove in the same appendix that the sequence $\{y_k\}$ generated by Algorithm \ref{alg:FISTA} (FISTA) satisfies
$$ \|g(y_k)\|_* \leq \fracg{4\| x_0-\bar{x}_0\|_R}{k+2}, \forall k\geq 0.$$
In restart schemes, one invokes several times FISTA algorithm with a relaxed exit condition. Typical choices are (see \cite{Donoghue:13}), \blista
\item Function scheme:
\begin{equation} \label{eq:Functional}
E_c^{f} = \text{True} \Leftrightarrow f(x_{k})\geq f(x_{k-1}).
\end{equation}
\item Gradient scheme:
\begin{equation} \label{eq:Gradient}
E_c^g = \text{True} \Leftrightarrow \langle g(y_{k-1}), x_{k-1} - x_{k} \rangle \leq 0.
\end{equation}
\elista
Given initial condition $r_0\in {\mathcal{X}}$, a minimum number of iterations $k_{min} \geq 0$, an exit condition $E_c$, and an accuracy parameter $\epsilon > 0$, the standard restart FISTA algorithm is shown in Algorithm \ref{alg:RestartFISTA}.
\begin{algorithm}
\DontPrintSemicolon
\caption{Restart FISTA} \label{alg:RestartFISTA}
\Require{$r_0 \in \mathcal X$, $k_{min} \geq 0$, $\epsilon >0$, $E_c$}
$j = 0$\;
\Repeat{$\vert\vert g(r_j)\vert\vert_* \leq \epsilon$}{
$j = j + 1$\;
$r_{j} = \text{FISTA}(r_{j-1}, k_{min}, E_c)$\;
}
\KwOut{$x^* = r_j$} \end{algorithm}
The implementation of Algorithm \ref{alg:RestartFISTA} usually provides better performance than the original non restart version \cite{Donoghue:13}, \cite{Giselsson:14CDC}.
\section{Convergence of Restart FISTA under a quadratic functional growth condition} \label{sec:Conv_FISTA}
It has been recently shown in \cite{Necoara:18} that some relaxations of the strong convexity conditions of the objective function are sufficient for obtaining linear convergence for several first order methods. In particular, the following relaxation of strong convexity suffices to guarantee linear convergence of different gradient optimization schemes for smooth functions ($\Psi(\cdot)=0$). See \cite[Subsection 5.2.2]{Necoara:18}.
\begin{assumption}[Quadratic Functional Growth]\label{assump:quadratic:growth} We assume that the optimization problem $$ f^* = \min\limits_{x\in {\mathcal{X}}}\, f(x) $$ is solvable and satisfies the following quadratic functional growth condition with parameter $\mu>0$:
$$ f(x) - f^* \geq \frac{\mu}{2}\| x-\bar{x}\|_R^2, \; \forall x\in {\mathcal{X}}, $$ where $\bar{x}$ denotes the closest element to $x$ in the optimal set $\Omega$ (see (\ref{def:bx})). \end{assumption}
As can be seen in \cite[Subsection 3.4]{Necoara:18}, strong convexity implies quadratic functional growth. This means that the quadratic functional growth setting encompasses a broad family of convex functions.
It is also shown in \cite[Subsection 5.2.2]{Necoara:18} that if the value of $f^*$ is known and $\Psi(\cdot)=0$, then a restart FISTA based on the exit condition \begin{equation} \label{E:Optimal}
E_c^* = \text{True} \Leftrightarrow f(x_{k}) - f^* \leq \frac{f(x_0) - f^*}{e^2}, \end{equation} exhibits global linear convergence. This exit condition is easily implementable if the optimal value $f^*$ is known. This is the case, for example, in some formulations of feasibility optimization problems, in which the optimal value $f^*$ is equal to zero for every feasible solution. This restart scheme corresponds to an optimal restart rate of $\frac{2e}{\sqrt{\mu}}$ \cite[Subsection 5.2.2]{Necoara:18}.
We present now a novel result that further characterizes the convergence properties of the non restart FISTA algorithm under Assumption \ref{assump:quadratic:growth}.
\begin{property}\label{prop:FISTA:EC} Under Assumptions \ref{assum:conv:smooth} and \ref{assump:quadratic:growth}, the iterations of FISTA algorithm satisfy \blista \item $f(x_k)-f^* \leq \fracg{4 (f(x_0)-f^*)}{\mu(k+1)^2}$, for all $k\geq 1$. \label{prop:FISTA:EC_1} \item $f(x_k) \leq f(x_0)$, for all $k\geq \left\lfloor \frac{2}{\sqrt{\mu}}\right\rfloor$. \label{prop:FISTA:EC_2} \item $f(x_k)-f^* \leq \fracg{f(x_0)-f(x_k)}{e}$, for all $k\geq \left\lfloor \frac{2\sqrt{e+1}}{\sqrt{\mu}}\right\rfloor$. \label{prop:FISTA:EC_3} \elista \end{property}
\proof See Appendix \ref{appen:FISTA:EC}.
\section{Restart FISTA with global linear convergence} \label{sec:OurFISTA}
In this section we propose a novel restart FISTA algorithm (Algorithm \ref{alg:OurRestartFISTA}) that exhibits global linear convergence under the quadratic functional growth condition. The algorithm uses exit condition $E_c^l$, which is defined to be true if the following two conditions are satisfied, \begin{subequations} \label{E:Our}
\begin{align}[left={E_c^l = \text{True} \Leftrightarrow \empheqlbrace}]
f(x_m) - f(x_k) & \leq \frac{f(x_0) - f(x_m)}{e} \label{E:Our_e}\\
f(x_k) & \leq f(x_0), \label{E:Our_f}
\end{align} \end{subequations} with $m = \lfloor\frac{k}{2}\rfloor +1$.
\begin{algorithm}
\DontPrintSemicolon
\caption{Linearly Convergent Restart FISTA (LCR-FISTA)} \label{alg:OurRestartFISTA}
\Require{$r_0 \in \mathcal X$, $\epsilon >0$}
$n_0 = 0$, $j = 1$\;
$[r_1, n_1] = \text{FISTA}(r_0, n_0, E_c^l)$\;
\Repeat{$\vert\vert g(r_j)\vert\vert_* \leq \epsilon$}{
$j = j + 1$\;
$[r_{j}, n_{j}] = \text{FISTA}(r_{j-1}, n_{j-1}, E_c^l)$\;
\If{$f(r_{j-1}) - f(r_{j}) > \fracg{1}{e}\left(f(r_{j-2}) - f(r_{j-1})\right)$}{
$n_j=2n_{j-1}$\;
}
}
\KwOut{$r^* = r_j$} \end{algorithm}
Inequality (\ref{E:Our_f}) guarantees that the output of the FISTA algorithm is no larger than the one corresponding to its initial condition.
As it is stated in the following property, one of the main features of the proposed algorithm is that the number of iterations $n_j$ required at each FISTA iteration ${[r_j,n_j]=FISTA(r_{j-1},n_{j-1},E_c^l)}$ is upper bounded by ${\frac{4\sqrt{e+1}}{\sqrt{\mu}} \approx \frac{7.72}{\sqrt{\mu}}}$. Moreover, the number of iterations required by the proposed algorithm to attain a given accuracy $\epsilon$ is upper bounded by $$\frac{16}{\sqrt{\mu}} \left\lceil \ln\left(1+\frac{2(f(r_0)-f^*)}{\epsilon^2}\right)\right\rceil.$$
\begin{property}\label{prop:L:Convergence:FISTA} Suppose that Assumptions \ref{assum:conv:smooth} and \ref{assump:quadratic:growth} hold. Then, the sequences $\{r_j\}$, $\{n_j\}$ provided by Algorithm \ref{alg:OurRestartFISTA} satisfy \blista
\item $\fracg{1}{2}\| g(r_{j-1})\|_*^2 \leq f(r_{j-1})-f(r_j)$, $\forall j\geq 1$.
\item $ n_j \leq \fracg{4\sqrt{e+1}}{\sqrt{\mu}}$, $\forall j\geq 0$. \label{prop:L:Convergence:FISTA_1}
\item The number of iterations ($\Sum{i=0}{j} n_i$) required to guarantee $\| g(r_j)\|_*\leq \epsilon$ is no larger than $$ \fracg{16}{\sqrt{\mu}}\left\lceil \ln\left(1+\frac{2(f(r_0)-f^*)}{\epsilon^2}\right)\right\rceil.$$ \label{prop:L:Convergence:FISTA_4} \elista \end{property}
\proof See Appendix \ref{appen:L:Convergence:FISTA}.
We notice that the factor 16 in the worst case complexity analysis is conservative. The authors claim that a better factor might be obtained at the expense of a more involved proof.
\section{Numerical results} \label{sec:Results}
We consider a weighted Lasso problem of the form \begin{equation} \label{eq:Lasso}
\min\limits_{x} \frac{1}{2 N} \| A x - b \|^2_2 + \| W x \|_1, \end{equation} where $x \inR{n}$, $A \inR{N \times n}$ is sparse with an average of $90\%$ of its entries being zero (sparsity was generated by setting a $0.9$ probability for each element of the matrix to be $0$), $n > N$, and $b \inR{N}$. Each nonzero element in $A$ and $b$ is obtained from a Gaussian distribution with zero mean and covariance 1. $W \inR{n \times n}$ is a diagonal matrix with elements obtained from a uniform distribution on the interval $[0, \alpha]$.
We note that Lasso problems (\ref{eq:Lasso}) can be reformulated in such a way that they satisfy the quadratic growth condition \cite[Section 6.3]{Necoara:18}. For this problem, inequality (\ref{ineq:smooth:R}) of Assumption \ref{assum:conv:smooth} is satisfied, for instance, for a matrix $R$ chosen as \begin{equation*}
R_{i,i} = \sum_{j=1}^{n} \vert H_{i,j} \vert, \end{equation*} with $H = \frac{1}{N} A\T A$. This is due to the Gershgorin Circle Theorem \cite[Subsection 7.2]{Golub96}. See also \cite[Section 6]{Nesterov13}.
We show the results of applying algorithms \ref{alg:RestartFISTA} and \ref{alg:OurRestartFISTA} with an accuracy parameter $\epsilon = 10^{-11}$ using different restart schemes and values of $N$, $n$ and $\alpha$. We take $r_0 = 0$.
The restart schemes shown are $E_c^f$ (\ref{eq:Functional}) and $E_c^g$ (\ref{eq:Gradient}) from \cite{Donoghue:13}, restart condition $E_c^*$ (\ref{E:Optimal}) \cite{Necoara:18}, and the restart condition $E_c^l$ (\ref{E:Our}) proposed in this paper (using Algorithm \ref{alg:OurRestartFISTA}). Additionally, we show the results of applying FISTA algorithm without using a restart scheme. In order to provide a fair comparison between the performance of the restart schemes, the algorithms are exited as soon as a value of $y_k$ that satisfies $\| g(y_{k-1}) \|_* \leq \epsilon$ is found. We note that, in order to implement the restart scheme based on $E_c^*$, we had to previously compute the optimal value $f^*$, which was done by using Algorithm \ref{alg:OurRestartFISTA} with $\epsilon = 10^{-12}$.
Tables \ref{tab:Test1} to \ref{tab:Test3} show results of performing $100$ tests with different randomized problems (\ref{eq:Lasso}) that share common values of parameters $N$, $n$ and $\alpha$. Tables show the average, median, maximum and minimum number of iterations.
{\renewcommand{1.4}{1.4} \begin{table}[ht]
\centering
\caption{Test 1. Comparison between restart schemes}
\label{tab:Test1}
\begin{threeparttable}
\begin{tabular}{|c"c|c|c|c|c|}
\hline
Exit Cond. & $E_c^l$ & No restart & $E_c^f$ & $E_c^g$ & $E_c^*$ \\\thickhline
Avg. Iter. & $670.6$ & $8207.2$ & $1648.7$ & $687.5$ & $1569.5$ \\\hline
Median Iter. & $676$ & $8241$ & $1608.5$ & $666.5$ & $1571$ \\\hline
Max. Iter. & $783$ & $10109$ & $2156$ & $930$ & $2053$ \\\hline
Min. Iter. & $570$ & $6737$ & $1192$ & $567$ & $917$ \\\hline
\end{tabular}
\begin{tablenotes}[flushleft] \footnotesize
\item Results of $100$ tests with $N=600$, $n=800$, $\alpha = 0.01$, $\epsilon = 10^{-11}$.
\end{tablenotes}
\end{threeparttable} \end{table}}
{\renewcommand{1.4}{1.4} \begin{table}[ht]
\centering
\caption{Test 2. Comparison between restart schemes}
\label{tab:Test2}
\begin{threeparttable}
\begin{tabular}{|c"c|c|c|c|c|}
\hline
Exit Cond. & $E_c^l$ & No restart & $E_c^f$ & $E_c^g$ & $E_c^*$ \\\thickhline
Avg. Iter. & $1683.7$ & $34116.4$ & $7743.3$ & $1606.7$ & $4601.9$ \\\hline
Median Iter. & $1659$ & $33127.5$ & $7242$ & $1594$ & $4503$ \\\hline
Max. Iter. & $2162$ & $51201$ & $14080$ & $2201$ & $7266$ \\\hline
Min. Iter. & $1406$ & $24539$ & $3894$ & $1306$ & $2499$ \\\hline
\end{tabular}
\begin{tablenotes}[flushleft] \footnotesize
\item Results for $100$ tests with $N=600$, $n=800$, $\alpha = 0.003$, $\epsilon = 10^{-11}$.
\end{tablenotes}
\end{threeparttable} \end{table}}
{\renewcommand{1.4}{1.4} \begin{table}[ht]
\centering
\caption{Test 3. Comparison between restart schemes}
\label{tab:Test3}
\begin{threeparttable}
\begin{tabular}{|c"c|c|c|c|c|}
\hline
Exit Cond. & $E_c^l$ & No restart & $E_c^f$ & $E_c^g$ & $E_c^*$ \\\thickhline
Avg. Iter. & $705.9$ & $8379.5$ & $1786.3$ & $686$ & $1709.4$ \\\hline
Median Iter. & $704.5$ & $8135.5$ & $1773$ & $680.5$ & $1703$ \\\hline
Max. Iter. & $873$ & $12055$ & $3218$ & $892$ & $2512$ \\\hline
Min. Iter. & $547$ & $5943$ & $987$ & $529$ & $1042$ \\\hline
\end{tabular}
\begin{tablenotes}[flushleft] \footnotesize
\item Results for $100$ tests with $N=300$, $n=400$, $\alpha = 0.01$, $\epsilon = 10^{-11}$.
\end{tablenotes}
\end{threeparttable} \end{table}}
Figures \ref{fig:Test1_gx} to \ref{fig:Test3_gx} show the value of $\| g(x_k) \|_*$ for a randomly selected problem out of the randomized problems used to compute the results shown in tables \ref{tab:Test1} to \ref{tab:Test3}, respectively.
\begin{figure}
\caption{Value of $\| g(y_k) \|_*$ for a problem (\ref{eq:Lasso}) of Test 1.}
\label{fig:Test1_gx}
\end{figure}
\begin{figure}
\caption{Value of $\| g(y_k) \|_*$ for a problem (\ref{eq:Lasso}) of Test 2.}
\label{fig:Test2_gx}
\end{figure}
\begin{figure}
\caption{Value of $\| g(y_k) \|_*$ for a problem (\ref{eq:Lasso}) of Test 3.}
\label{fig:Test3_gx}
\end{figure}
Figure \ref{fig:nj} shows the value of $n_j$ at each iteration $j$ of Algorithm \ref{alg:OurRestartFISTA} for the three examples whose results are shown in Figures \ref{fig:Test1_gx} to \ref{fig:Test3_gx}. Note that the final value of $n_j$ is lower than the previous one in all three instances due to the algorithm exiting as soon as the condition $\| g(y_{k-1}) \|_* \leq \epsilon$ is satisfied.
\begin{figure}
\caption{Value of $n_j$ obtained using Algorithm \ref{alg:OurRestartFISTA} for the problems (\ref{eq:Lasso}) whose result are shown in Figures \ref{fig:Test1_gx} to \ref{fig:Test3_gx}.}
\label{fig:nj}
\end{figure}
\section{Conclusions} \label{sec:Conclusions}
In this paper we have presented a novel restart scheme with guaranteed global linear convergence. The algorithm relies on a quadratic functional growth condition. One of the advantages of the proposed algorithm is that it does not require the knowledge of the parameter $\mu$ that characterizes the quadratic functional growth condition, or the optimal value of the minimization problem. We provide an upper bound of the required number of iterations equal to $$\frac{16}{\sqrt{\mu}} \left\lceil \ln\left(1+\frac{2(f(r_0)-f^*)}{\epsilon^2}\right)\right\rceil.$$
We have presented numerical evidence of the good performance of the algorithm when compared with other restarts schemes. It outperforms the restart scheme based on the knowledge of the optimal value $f^*$.
\begin{appendix}
\subsection{Existence and Uniqueness of Composite Gradient}\label{appen:existence:uniqueness}
We present in this appendix some well known facts about convex analysis that are required to analyze the properties of the composite gradient.
\begin{property}\label{prop:characterization:Psi:cX}
Suppose that
\blista
\item $\Psi: {\rm \,I\!R} ^n \to (-\infty,\infty]$ is a closed convex function.
\item ${\mathcal{X}} \subseteq {\rm \,I\!R} ^n$ is a closed convex set.
\item The set $\dom{\Psi}\bigcap{\mathcal{X}} $ is non empty.
\item $I_{{\mathcal{X}}}: {\rm \,I\!R} ^n \to \{0,\infty\}$ is the indicator function of ${\mathcal{X}}$. That is,
$$ I_{{\mathcal{X}}}(x) = \bsis{rl} 0 & \mbox{if } x\in {\mathcal{X}} \\ \infty & \mbox{otherwise}. \esis$$
\item The function $\Psi_{{\mathcal{X}}}: {\rm \,I\!R} ^n\to (-\infty,\infty]$ is defined as
$$\Psi_{{\mathcal{X}}} (x)\doteq \Psi(x)+ I_{{\mathcal{X}}}(x),\; \forall x\in {\rm \,I\!R} ^n.$$
\elista
Then
\blista
\item The function $\Psi_{{\mathcal{X}}}$ is proper, closed, and convex.
\item The relative interior of $\dom{\Psi_{{\mathcal{X}}}}$ is non empty.
\item There is $z\in {\mathcal{X}}$ and $d\in {\rm \,I\!R} ^n$ such that $\Psi_{{\mathcal{X}}}(z)<\infty$ and
$$ \Psi_{{\mathcal{X}}} (x) \geq \Psi_{{\mathcal{X}}}(z) + \sp{d}{x-z},\;\; \forall x\in {\rm \,I\!R} ^n.$$
\elista \end{property}
\proof
From $\dom{\Psi}\bigcap{\mathcal{X}}\neq \emptyset $ we have that both $\dom{\Psi}$ and ${\mathcal{X}}$ are non empty. The epigraph of the indicator function $I_{{\mathcal{X}}}$ is, by definition, \begin{eqnarray*}
\epi{I_{{\mathcal{X}}}} &=& \set{(x,t) \in {\rm \,I\!R} ^n \times {\rm \,I\!R} }{I_{{\mathcal{X}}}(x) \leq t }\\
& =& \set{(x,t) \in {\rm \,I\!R} ^n\times {\rm \,I\!R} }{x\in {\mathcal{X}}, 0 \leq t}.\end{eqnarray*}
Since ${\mathcal{X}}$ and ${{\cal{T}}\doteq\set{t\in {\rm \,I\!R} }{t\geq 0}}$ are non empty closed sets, ${\epi{I_{{\mathcal{X}}}}={\mathcal{X}}\times {\cal{T}}}$ is also a non empty closed convex set. Thus, by definition, ${I_{{\mathcal{X}}}: {\rm \,I\!R} ^n\to \{0,\infty\}}$ is a closed convex function. Since both $\Psi$ and $I_{{\mathcal{X}}}$ are closed convex functions, ${\Psi_{{\mathcal{X}}}\doteq\Psi+I_{{\mathcal{X}}}}$ is also a closed convex function (the sum of closed convex functions provides closed convex functions \cite[Proposition 1.1.5]{Bertsekas:09}). Since ${\dom{\Psi_{{\mathcal{X}}}}=\dom{\Psi}\bigcap{\mathcal{X}}\neq \emptyset}$, we infer that the domain of $\Psi_{{\mathcal{X}}}$ is non empty. This implies that $\Psi_{{\mathcal{X}}}$ is not identically equal to $\infty$. Moreover, since ${\Psi: {\rm \,I\!R} ^n\to (-\infty,\infty]}$ we have that ${\Psi_{{\mathcal{X}}} : {\rm \,I\!R} ^n \to (-\infty,\infty]}$. We conclude that ${\Psi_{{\mathcal{X}}}(x)>-\infty}$ for every ${x\in {\rm \,I\!R} ^n}$. From this and the fact that $\Psi_{{\mathcal{X}}}$ is not identically equal to $\infty$ we have that $\Psi_{{\mathcal{X}}}$ is proper.
Since $\dom{\Psi_{{\mathcal{X}}}}$ is a non empty convex set, it has a non empty relative interior $\ri{\dom{\Psi_{{\mathcal{X}}}}}$ (see \cite[Proposition 1.3.2]{Bertsekas:09}).
It is a well know fact from convex analysis that the subdifferential of a proper convex function at a point in the relative interior of its domain is non empty \cite[Proposition 5.4.1]{Bertsekas:09}. Suppose now that $z\in \ri{\dom{\Psi_{{\mathcal{X}}}}}$. Since $\Psi_{{\mathcal{X}}}$ is a proper convex function we have that the subdifferential of $\Psi_{{\mathcal{X}}}$ at $z$ is non empty. This means, by definition, that there is $d\in {\rm \,I\!R} ^n$ such that $$ \Psi_{{\mathcal{X}}} (x) \geq \Psi_{{\mathcal{X}}}(z) + \sp{d}{x-z},\;\; \forall x\in {\rm \,I\!R} ^n.$$ \QED
\begin{property}\label{Prop:solvable:unique:grad}
Suppose that Assumption \ref{assum:conv:smooth} holds. Given any $y\in {\rm \,I\!R} ^n$, consider the quadratic function $h_y: {\rm \,I\!R} ^n \to {\rm \,I\!R} $ defined as
$$ h_y(x) \doteq \sp{{\nabla h}(y)}{x-y}+\frac{1}{2} \|x-y\|^2_R.$$
Then, the minimization problem
\begin{equation}\label{opti:comp:grad:X}
\min\limits_{x\in {\mathcal{X}}} \Psi(x)+h_y(x)
\end{equation}
is solvable and has a unique solution. That is, there exists a unique point $y^+\in {\mathcal{X}}$ such that
$$ \Psi(y^+)+h_y(y^+) = \inf\limits_{x\in {\mathcal{X}}} \Psi(x) + h_y(x) < \infty.$$ \end{property}
\proof
Notice that the minimization problem (\ref{opti:comp:grad:X}) is equivalent to \begin{equation*}\label{opti:comp:grad:Indicator}
\min\limits_{x\in {\rm \,I\!R} ^n} \Psi(x) + I_{{\mathcal{X}}}(x) + h_y(x), \end{equation*} where $I_{{\mathcal{X}}}$ is the indicator function of ${\mathcal{X}}$. If we define ${\Psi_{{\mathcal{X}}}\doteq \Psi+I_{{\mathcal{X}}}}$ we can rewrite the original problem (\ref{opti:comp:grad:X}) as \begin{equation*}\label{opti:comp:grad:Psi:cX}
\min\limits_{x\in {\rm \,I\!R} ^n} \Psi_{{\mathcal{X}}}(x)+ h_y(x). \end{equation*} We notice that the assumptions of Property \ref{prop:characterization:Psi:cX} are satisfied if Assumption $\ref{assum:conv:smooth}$ holds. Thus, we infer from Property \ref{prop:characterization:Psi:cX} that $\Psi_{{\mathcal{X}}}: {\rm \,I\!R} ^n\to (-\infty,\infty]$ is a proper closed convex function. We also have that the quadratic function $h_y: {\rm \,I\!R} ^n \to {\rm \,I\!R} $ is also proper and closed because it is a real valued continuous function (see \cite[Proposition 1.1.3]{Bertsekas:09}). Since the sum of closed functions is closed (see \cite[Proposition 1.1.5]{Bertsekas:09}), we infer that $F_y\doteq\Psi_{{\mathcal{X}}}+h_y$ is a closed function. Moreover, from Property \ref{prop:characterization:Psi:cX} we also have that there is $z\in {\mathcal{X}}$ and $d\in {\rm \,I\!R} ^n$ such that \blista
\item $\Psi_{{\mathcal{X}}}(z)<\infty$.
\item $ \Psi_{{\mathcal{X}}} (x) \geq \Psi_{{\mathcal{X}}}(z) + \sp{d}{x-z}$, $\forall x\in {\rm \,I\!R} ^n$. \elista Therefore, \begin{equation}
\begin{aligned}
F_y(z)&= \Psi_{{\mathcal{X}}}(z)+h_y(z) = \gamma_z < \infty, \label{equ:ineq:Psi:cX:hy}\\
F_y(x)&=\Psi_{{\mathcal{X}}}(x) + h_y(x) \\
&\geq\Psi_{{\mathcal{X}}}(z) + \sp{d}{x-z} + h_y(x),\; \forall x\in {\rm \,I\!R} ^n.
\end{aligned} \end{equation} We infer from (\ref{equ:ineq:Psi:cX:hy}) that the closed function ${F_y: {\rm \,I\!R} ^n\to (-\infty,\infty]}$ is not identically equal to $\infty$ and therefore, proper. We conclude that $F_y$ is a proper closed convex function. From Weiertrasss' Theorem (see Proposition 3.2.1 in \cite{Bertsekas:09}) we have that the set of minima of $F_y$ over $ {\rm \,I\!R} ^n$ is nonempty and compact if there is a scalar $\bar{\gamma}$ such that the level set ${\Phi(\bar{\gamma})=\set{x}{F_y(x)\leq \bar{\gamma}}}$ is nonempty and bounded. From (\ref{equ:ineq:Psi:cX:hy}) we have that $\Phi(\gamma_z)$ is nonempty. Moreover, we also infer from (\ref{equ:ineq:Psi:cX:hy}) that $\Phi(\gamma_z)$ is a bounded set because $F_y$ is lower bounded by a strictly convex quadratic function of $x$. We conclude that \begin{eqnarray*}
\min\limits_{x\in {\mathcal{X}}} \Psi(x)+h_y(x) &=& \min\limits_{x\in {\rm \,I\!R} ^n} \Psi_{{\mathcal{X}}}(x)+ h_y(x) \\
&=& \min\limits_{x\in {\rm \,I\!R} ^n} F_y(x) \leq \gamma_z <\infty . \end{eqnarray*} is a solvable optimization problem. That is, there is $y^+\in {\mathcal{X}}$ such that $$ \Psi(y^+)+h_y(y^+) = \inf\limits_{x\in {\mathcal{X}}} \Psi(x) + h_y(x) < \infty.$$ The set of minimizers consists of a single element $y^+$ because of the strictly convex nature of $F_y$ ($h_y$ is a strictly convex function). \QED
\subsection{Proof of Property \ref{prop:proj:gradient}.}\label{appen:proj:gradient}
We prove in this appendix Property \ref{prop:proj:gradient}, which is rewritten here for the reader's convenience.
\begin{property}\label{prop:proj:in:appendix}
Suppose that Assumption \ref{assum:conv:smooth} holds.
Then,
\blista
\item For every $y\in {\rm \,I\!R} ^n$ and $x\in {\mathcal{X}}$:
\begin{subequations}
\begin{flalign}
f(y^+) -f(x) & \leq \sp{g(y)}{y^+-x} + \frac{1}{2}\|g(y)\|_*^2 \label{ineq:cg:a}\\
& = \sp{g(y)}{y-x} - \frac{1}{2}\|g(y)\|_*^2 \label{ineq:cg:b}\\
& = - \frac{1}{2}\| y^+ -x \|^2_R + \frac{1}{2}\|y-x\|_R^2. \label{ineq:cg:c}
\end{flalign}
\end{subequations}
\item For every $y\in {\mathcal{X}}$: $$ \frac{1}{2} \| g(y)\|_*^2 \leq f(y)-f(y^+)\leq f(y)-f^*.$$
\elista \end{property}
\proof
From Property \ref{Prop:solvable:unique:grad} we have that there is a (unique) $y^+\in {\mathcal{X}}$ such that \begin{equation}\label{ineq:Psi:hy}
\Psi(y^+) + h_y(y^+) \leq \Psi(x)+h_y(x), \; \forall x\in {\mathcal{X}},\end{equation}
where $ h_y(x) \doteq \sp{{\nabla h}(y)}{x-y}+\frac{1}{2} \|x-y\|^2_R$. Denote now $\Psi_{{\mathcal{X}}}=\Psi+I_{{\mathcal{X}}}$, where $I_{{\mathcal{X}}}: {\rm \,I\!R} ^n\to \{0,\infty\}$ is the indicator function of ${\mathcal{X}}$. Since $y^+\in {\mathcal{X}}$ we have $I_{{\mathcal{X}}}(y^+)=0$. Therefore, inequality (\ref{ineq:Psi:hy}) implies $$ \Psi_{{\mathcal{X}}}(y^+) + h_y(y^+) \leq \Psi_{{\mathcal{X}}}(x)+h_y(x), \; \forall x \in {\rm \,I\!R} ^n. $$ Denote now $F_y=\Psi_{{\mathcal{X}}}+h_y$. From last inequality we have $$F_y(y^+) \leq F_y(x),\; \forall x\in {\rm \,I\!R} ^n.$$ By definition of subdifferential at a point, we have that the previous inequality implies \begin{equation}\label{equ:zero:in:partial:F}
0 \in \partial F_y(y^+). \end{equation} We have that $\Psi_{{\mathcal{X}}}$ is a proper closed function and $\ri{\dom{\Psi_{{\mathcal{X}}}}}\neq \emptyset$ (see the first two claims of Property \ref{prop:characterization:Psi:cX}). The domain of the quadratic function $h_y: {\rm \,I\!R} ^n\to {\rm \,I\!R} $ is $ {\rm \,I\!R} ^n$. Since $h_y$ is a continuous real value function in $ {\rm \,I\!R} ^n$, it is also closed (see Proposition 1.1.3 in \cite{Bertsekas:09}). We have that \begin{align*}
\ri{\dom{\Psi_{{\mathcal{X}}}}}\bigcap \ri{\dom{h_y}} &= \ri{\dom{\Psi_{{\mathcal{X}}}}} \bigcap {\rm \,I\!R} ^n \\
&= \ri{\dom{\Psi_{{\mathcal{X}}}}}\neq \emptyset. \end{align*} Since $F_y=\Psi_{{\mathcal{X}}}+h_y$ is equal to the sum of two closed convex functions and $$\ri{\dom{\Psi_{{\mathcal{X}}}}}\bigcap \ri{\dom{h_y}}\neq \emptyset,$$ we have $\partial F_y(y^+) = \partial \Psi_{{\mathcal{X}}}(y^+) + \partial h_y(y^+)$ (see Proposition 5.4.6 in \cite{Bertsekas:09}). The subdifferential of the differentiable function $h_y$ at $y^+$ is ${\nabla h}_y(y^+) = {\nabla h}(y)+R(y^+-y)$. Thus, we obtain from (\ref{equ:zero:in:partial:F}) \begin{align*}
0 \in \partial F_y(y^+) &= \partial \Psi_{{\mathcal{X}}}(y^+) + \partial h_y(y^+) \\
&= \partial \Psi_{{\mathcal{X}}}(y^+) + {\nabla h}(y)+R(y^+-y). \end{align*} Since $g(y)$ is defined as $R(y-y^+)$ we obtain $$ g(y)-{\nabla h}(y) \in \partial \Psi_{{\mathcal{X}}}(y^+).$$ By definition of $\partial \Psi_{{\mathcal{X}}}(\cdot)$ we have $$ \Psi_{{\mathcal{X}}}(x) \geq \Psi_{{\mathcal{X}}}(y^+) + \sp{g(y)-{\nabla h}(y)}{x-y^+}, \;\; \forall x\in {\rm \,I\!R} ^n.$$ Obviously, since ${\mathcal{X}}\subseteq {\rm \,I\!R} ^n$, this implies $$ \Psi_{{\mathcal{X}}}(x) \geq \Psi_{{\mathcal{X}}}(y^+) + \sp{g(y)-{\nabla h}(y)}{x-y^+}, \;\; \forall x\in {\mathcal{X}}.$$ Since $y^+\in {\mathcal{X}}$ and $\Psi_{{\mathcal{X}}}=\Psi$ for every $x\in {\mathcal{X}}$, we obtain \begin{equation}\label{ineq:sub:gradient}
\Psi(x) \geq \Psi(y^+) + \sp{g(y)-{\nabla h}(y)}{x-y^+}, \;\forall x\in {\mathcal{X}}. \end{equation} The convexity of $h(\cdot)$ implies $$ h(x) \geq h(y) + \sp{{\nabla h}(y)}{x-y}, \;\; \forall x\in {\mathcal{X}}.$$ Adding this inequality to (\ref{ineq:sub:gradient}) yields \begin{eqnarray}
f(x) &\mkern-14mu = &\mkern-14mu \Psi(x) + h(x) \nonumber\\
&\mkern-14mu \geq &\mkern-14mu \Psi(y^+) + \sp{g(y)-{\nabla h}(y)}{x-y^+} \nonumber \\
&&\mkern-14mu + h(y) + \sp{{\nabla h}(y)}{x-y} \nonumber\\
&\mkern-14mu = &\mkern-14mu \Psi(y^+) +\sp{g(y)}{x-y^+} \nonumber \\
&&\mkern-14mu + h(y) + \sp{{\nabla h}(y)}{y^+-y}, \; \forall x\in {\mathcal{X}}. \label{ineq:f:one} \end{eqnarray} From Assumption \ref{assum:conv:smooth} we have \begin{align*}
h(y) &\geq h(y^+) -\sp{{\nabla h}(y)}{y^+-y} -\frac{1}{2} \|y^+-y\|^2_R \\
& = h(y^+) -\sp{{\nabla h}(y)}{y^+-y} -\frac{1}{2} \|R^{-1}g(y)\|^2_R \\
& = h(y^+) -\sp{{\nabla h}(y)}{y^+-y} -\frac{1}{2} \|g(y)\|_*^2 . \end{align*} Adding this inequality to (\ref{ineq:f:one}) yields \begin{align*}
f(x) & \geq \Psi(y^+) + h(y^+) +\sp{g(y)}{x-y^+} - \frac{1}{2} \|g(y)\|_*^2\\
& = f(y^+) +\sp{g(y)}{x-y^+} - \frac{1}{2} \|g(y)\|_*^2, \; \forall x\in {\mathcal{X}}. \end{align*} From this inequality we have
$$f(y^+) -f(x) \leq \sp{g(y)}{y^+-x} + \frac{1}{2}\|g(y)\|_*^2, \;\forall x\in {\mathcal{X}}.$$ This proves (\ref{ineq:cg:a}). We now prove (\ref{ineq:cg:b}) and (\ref{ineq:cg:c}) by means of simple algebraic manipulations. \small \begin{eqnarray}
f(y^+) -f(x) \mkern-34mu &&\leq \sp{g(y)}{y^+-x} + \frac{1}{2}\|g(y)\|_*^2 \nonumber \\
&&= \sp{g(y)}{y-x+y^+-y} + \frac{1}{2}\|g(y)\|_*^2 \nonumber \\
&&= \sp{g(y)}{y-x} + \sp{g(y)}{y^+-y} + \frac{1}{2}\|g(y)\|_*^2 \nonumber \\
&&= \sp{g(y)}{y-x} + \sp{g(y)}{-R^{-1}g(y)} + \frac{1}{2}\|g(y)\|_*^2 \nonumber \\
&&= \sp{g(y)}{y-x}- \|g(y)\|_*^2 + \frac{1}{2}\|g(y)\|_*^2 \nonumber \\
&&= \sp{g(y)}{y-x}- \frac{1}{2}\|g(y)\|_*^2, \; \forall x\in {\mathcal{X}}. \label{ineq:proof:ff} \end{eqnarray} \normalsize This proves (\ref{ineq:cg:b}). From this inequality, and the definition of $g(y)$, we obtain \begin{align*}
f(y^+) -f(x) & \leq \sp{R(y-y^+)}{y-x} - \frac{1}{2} \| R(y-y^+)\|_*^2 \\
& = -\sp{R(y-y^+)}{x-y} - \frac{1}{2} \| y-y^+\|_R^2 \\
& = - \frac{1}{2}\| y-y^+ + x-y\|^2_R + \frac{1}{2}\|x-y\|_R^2 \\
& = - \frac{1}{2}\| y^+ -x \|^2_R + \frac{1}{2}\|y-x\|_R^2, \; \forall x\in {\mathcal{X}}. \end{align*} This proves (\ref{ineq:cg:c}). Suppose now that $y\in {\mathcal{X}}$. Particularizing inequality (\ref{ineq:proof:ff}) to $x=y$ yields
$$ \frac{1}{2} \| g(y)\|_*^2 \leq f(y)-f(y^+), \;\; \forall y\in {\mathcal{X}}.$$ The inequality $f(y)-f(y^+) \leq f(y) -f^*$ trivially follows from $f^*\leq f(y^+)$. \QED
\subsection{Characterization of optimality}\label{appen:optimaliticharacterization}
The following property serves to characterize the optimality of a given point $y\in {\rm \,I\!R} ^n$.
\begin{property}\label{prop:charac:optimality}
Suppose that Assumption \ref{assum:conv:smooth} holds. Then $y\in {\rm \,I\!R} ^n$ belongs to the optimal set
$$\Omega=\set{x}{x\in {\mathcal{X}}, f(x)=f^*}$$ if and only if $g(y)=0$. \end{property}
\proof We first show that $g(y)=0$ implies $y\in \Omega$. Since $R\succ 0$, we infer from equality $g(y)=R(y-y^+)$ that ${g(y)=0}$ is equivalent to $y=y^+$. Suppose that ${x^*\in \Omega \subseteq {\mathcal{X}}}$. Then, we obtain from $g(y)=0$, $y=y^+\in {\mathcal{X}}$, and the first claim of Property \ref{prop:proj:gradient}, the following inequality \begin{align*}
f(x^*) & \geq f(y^+) -\sp{g(y)}{y^+-x^*}-\frac{1}{2}\|g(y)\|_*^2 \\
& = f(y^+) = f(y). \end{align*} That is, $f^*=f(x^*)\geq f(y)$. Since $y = y^+ \in {\mathcal{X}}$, this is possible only if $y$ is also optimal ($f(y)=f^*$). This proves that ${g(y)=0}$ implies $y\in \Omega$. We now prove that $y\in \Omega$ implies $g(y)=0$. Suppose that $y\in \Omega$. Then, $f(y)=f^*$ and we obtain from the second claim of Property \ref{prop:proj:gradient}
$$ \frac{1}{2}\|g(y)\|_*^2 \leq f(y)-f^*=0.$$ This implies $g(y)=0$. \QED
\subsection{Convergence of non restart FISTA \label{app:conver:Fista}}
\begin{property}\label{prop:convergence} Suppose that Assumption \ref{assum:conv:smooth} holds. Then, the sequences $\{x_k\}$ and $\{y_k\}$ generated by Algorithm \ref{alg:FISTA} (FISTA) satisfy \blista
\item $ f(x_{k})-f^* \leq \fracg{2\|x_0-\bar{x}_0\|_R^2}{(k+1)^2}$, for all $k\geq 1$,
\item $ \|g(y_k)\|_* \leq \fracg{4\| x_0-\bar{x}_0\|_R}{k+2} $, for all $k\geq 0$, \elista where $\bar{x}_0$ represents the point in the optimal set $\Omega$ closest to the initial condition $x_0$ of the algorithm. \end{property}
\proof
{\bf First claim:}
We denote $g_k\doteq g(y_k)$, $\forall k\geq 0$. Additionally, we recall that ${\|\cdot\|_* \doteq \|\cdot\|_{R^{-1}}}$.
From step 4 of FISTA algorithm we have \begin{equation}\label{equ:recu:x:comp}
x_k =y_{k-1}^+, \; \forall k\geq 1. \end{equation} This implies that $$ g_k = R(y_k-y_k^+) = R(y_k-x_{k+1}),\; \forall k\geq 0.$$ Particularizing inequality (\ref{ineq:cg:c}) of the first claim of Property \ref{prop:proj:in:appendix} to $y=y_0\in {\rm \,I\!R} ^n$, and $x=\bar{x}_0\in \Omega \subseteq {\mathcal{X}}$, we obtain
$$ f(y_0^+)-f(\bar{x}_0) \leq -\frac{1}{2} \| y_0^+ -\bar{x}_0\|^2_R + \frac{1}{2} \|y_0-\bar{x}_0\|^2_R.$$ By construction we have that $x_0=y_0$ and $x_1=y_0^+$. Furthermore, by definition of $\bar{x}_0$, we have $f(\bar{x}_0)=f^*$. Therefore we can rewrite previous inequality as \begin{align}
f(x_1)-f^* & \leq -\frac{1}{2} \|x_1-\bar{x}_0\|^2_R + \frac{1}{2} \|x_0-\bar{x}_0\|^2_R \label{ineq:xone}\\
& \leq \frac{1}{2} \|x_0-\bar{x}_0\|^2_R. \nonumber \end{align} This proves the claim of the property for $k=1$. We now proceed to prove the claim for $k\geq 2$. From equality (\ref{equ:recu:x:comp}) we have $$ x_{k+1} = y_k^+, \; \forall k\geq 1.$$ Therefore, from inequality (\ref{ineq:cg:b}) of Property \ref{prop:proj:in:appendix} we obtain that for every $x\in {\mathcal{X}}$ and every $k\geq 1$ \begin{eqnarray*}
f(x) & \geq & f(x_{k+1})+\frac{1}{2}\|g_k\|^2_*- \sp{g_k}{y_k-x}. \end{eqnarray*} We notice that, by construction, $x_k\in {\mathcal{X}}$, $k\geq 1$. Particularizing at $x_k$ and $\bar{x}_0$, we obtain from last inequality \begin{subequations} \begin{flalign}
f(x_k) & \geq f(x_{k+1})+\frac{1}{2}\|g_k\|^2_* - \sp{g_k}{y_k-x_k}, \; \forall k\geq 1, \label{ineq:no:increm:a}
\\ f(\bar{x}_0) & \geq f(x_{k+1})+\frac{1}{2}\|g_k\|^2_* - \sp{g_k}{y_k-\bar{x}_0}, \; \forall k\geq 1. \label{ineq:no:increm:b} \end{flalign} \end{subequations} In order to write down the proof in a compact way, we introduce the following incremental notation, valid for all $k\geq 0$, \begin{align*}
\delta f_k & \doteq f(x_k) - f^*, \\
\delta x_k & \doteq x_k -\bar{x}_0, \\
\delta y_k & \doteq y_k -\bar{x}_0, . \end{align*} Inequalities (\ref{ineq:no:increm:a}) and (\ref{ineq:no:increm:b}) in an incremental notation, are \begin{subequations} \begin{align}
\delta f_k -\delta f_{k+1} & \geq \frac{1}{2}\|g_k\|^2_* - \sp{g_k}{\delta y_k-\delta x_k},\; \forall k\geq 1, \label{ineq:f:x:plus} \\
- \delta f_{k+1} & \geq \frac{1}{2}\|g_k\|^2_* - \sp{g_k}{\delta y_k}, \; \forall k\geq 1. \label{ineq:f:x:os} \end{align} \end{subequations} We introduce now the auxiliary variable $\Gamma_k$, defined as $$\Gamma_k\doteq t_{k-1}^2\delta f_k-t_k^2 \delta f_{k+1}, \; \forall k\geq 1.$$ From Property \ref{prop:tk} in appendix \ref{appen:t} we have $$t_{k-1}^2=t_k^2-t_k, \; \forall k\geq 1.$$ We now use this identity to obtain \begin{align}
\Gamma_k & = (t_k^2-t_k)\delta f_k-t_k^2 \delta f_{k+1} \nonumber\\
& = (t_{k}^2-t_{k})(\delta f_k-\delta f_{k+1}) - t_k \delta f_{k+1}, \; \forall k\geq 1. \label{equ:aux:Gam} \end{align} In view of Property \ref{prop:tk}, $t_k\geq 1$, $\forall k\geq 0$. This implies that we can replace, in inequality (\ref{equ:aux:Gam}), $\delta f_k-\delta f_{k+1}$ and $-\delta f_{k+1}$ by the lower bounds given by inequalities (\ref{ineq:f:x:plus}) and (\ref{ineq:f:x:os}). In this way we obtain \begin{eqnarray}
{}\mkern-52mu \Gamma_k &\mkern-14mu \geq &\mkern-14mu (t_{k}^2-t_{k}) \left(\frac{1}{2}\|g_k\|^2_* - \sp{g_k}{\delta y_k-\delta x_k} \right) \nonumber \\
&&\mkern-14mu + t_k \left(\frac{1}{2}\|g_k\|^2_* - \sp{g_k}{\delta y_k} \right) \nonumber\\
&\mkern-14mu = &\mkern-14mu \frac{t_k^2 }{2}\|g_k\|_*^2 - \sp{g_k}{t_k^2(\delta y_k -\delta x_k)+t_k\delta x_k}, \forall k \geq 1. \label{ineq:Gamma:k} \end{eqnarray} From step 6 of the algorithm we have for all $k\geq 1$ that ${y_k = x_k + \fracg{t_{k-1} - 1}{t_k} (x_k - x_{k-1})}$. This can be rewritten in incremental notation as \begin{equation}\label{yk:in:incremental}
\delta y_k -\delta x_k = \fracg{t_{k-1} - 1}{t_k} (\delta x_k - \delta x_{k-1}), \; \forall k\geq 1. \end{equation} We now define, for every $k\geq 1$ \begin{equation}
s_k \doteq \delta x_{k-1} + t_{k-1}(\delta x_k -\delta x_{k-1}). \label{equ:sk:one} \end{equation} From the definition of $s_k$ and (\ref{yk:in:incremental}) we obtain \begin{align}
s_k -\delta x_k & = \delta x_{k-1} +t_{k-1}(\delta x_k-\delta x_{k-1}) -\delta x_{k} \nonumber \\
& = (t_{k-1} -1) (\delta x_k-\delta x_{k-1}) \nonumber \\
& = t_k( \delta y_k - \delta x_k), \; \forall k\geq 1. \label{equ:sk:two} \end{align} From (\ref{ineq:Gamma:k}) and (\ref{equ:sk:two}) we obtain \begin{align}
\Gamma_k & \geq \frac{1}{2}\|t_k g_k\|_*^2 - \sp{g_k}{t_k(s_k-\delta x_k) + t_k\delta x_k} \nonumber\\
& = \frac{1}{2}\|t_k g_k\|_*^2 -\sp{ t_k g_k}{s_k}, \;\forall k\geq 1. \label{equ:Gammakupps} \end{align} Using (\ref{equ:sk:one}) and (\ref{equ:sk:two}) we now show that $g_k$ can be written in terms of $s_k$ and $s_{k+1}$. \begin{align} t_k g_k & = t_k R(y_k-x_{k+1}) = t_k R(\delta y_k -\delta x_{k+1}) \nonumber\\
& = t_k R(\delta y_k - \delta x_{k} + \delta x_{k} - \delta x_{k+1}) \nonumber \\
& = R(s_k-\delta x_k + t_k(\delta x_k -\delta x_{k+1} )) \nonumber \\
& = R(s_k-s_{k+1}), \; \forall k\geq 1. \label{equ:gk:tk:equ} \end{align} With this expression for $t_k g_k$ we obtain from (\ref{equ:Gammakupps}) \begin{align*}
\Gamma_k & \geq \frac{1}{2}\|R (s_k-s_{k+1})\|_*^2 -\sp{R(s_k-s_{k+1})}{s_k} \\
& = \frac{1}{2}\|s_{k+1}-s_k\|^2_R +\sp{R(s_{k+1}-s_k)}{s_k} \\
& = \frac{1}{2}\|(s_{k+1}-s_k)+s_k\|^2_R - \frac{1}{2}\|s_k\|^2_R \\
& = \frac{1}{2} \|s_{k+1}\|^2_R - \frac{1}{2}\|s_{k}\|^2_R, \; \forall k\geq 1. \end{align*} Thus, for every $k\geq 1$,
$$ \Gamma_k = t_{k-1}^2 \delta f_k - t_k^2 \delta f_{k+1} \geq \frac{1}{2} \|s_{k+1}\|^2_R - \frac{1}{2}\|s_{k}\|^2_R. $$ Equivalently
$$ t_k^2 \delta f_{k+1}+\fracg{1}{2}\|s_{k+1}\|^2_R \leq t_{k-1}^2\delta f_k +\fracg{1}{2}\|s_k\|^2_R, \; \forall k\geq 1.$$ Since this inequality holds for every $k\geq 1$ we can apply it in a recursive way to obtain \begin{equation*}
\begin{aligned}
t_k^2 \delta f_{k+1} + \frac{1}{2}\|s_{k+1}\|^2_R & \leq t_0^2 \delta f_1 + \frac{1}{2} \| s_1\|^2_R \\
& = \delta f_1+ \frac{1}{2} \| \delta x_0 + t_0(\delta x_1-\delta x_0)\|^2_R \\
& = \delta f_1 + \frac{1}{2} \| x_1 - \bar{x}_0\|^2_R,\; \forall k\geq 1. \end{aligned} \end{equation*}
From (\ref{ineq:xone}) we have $${f(x_1)-f^* + \frac{1}{2} \| x_1 - \bar{x}_0\|^2_R \leq \frac{1}{2}\|x_0-\bar{x}_0\|^2_R}.$$
Thus, \begin{equation}\label{ineq:convergence}
t_k^2 \delta f_{k+1} + \frac{1}{2}\|s_{k+1}\|^2_R \leq \frac{1}{2}\| x_0-\bar{x}_0\|^2_R, \; \forall k\geq 1. \end{equation} Therefore,
$$t_k^2 (f(x_{k+1})-f^*) + \frac{1}{2} \|s_{k+1}\|^2_R \leq \frac{1}{2} \|x_0-\bar{x}_0\|^2_R, \; \forall k\geq 1.$$ From this inequality, and taking now into account that ${t_k \geq \fracg{k+2}{2}}$ for all $k\geq 0$ (second claim of Property \ref{prop:tk}), we conclude
$$ f(x_{k+1})-f^* \leq \frac{\|x_0-\bar{x}_0\|^2_R}{2 t_k^2} \leq \frac{2 \|x_0-\bar{x}_0\|^2_R}{(k+2)^2}, \; \forall k\geq 1.$$ That is,
$$ f(x_{k})-f^* \leq \frac{2 \|x_0-\bar{x}_0\|^2_R}{(k+1)^2}, \;\; \forall k\geq 2.$$
\QED
{\bf Second claim:}
We first prove the claim for $k=0$. \begin{align*}
\|g(y_0)\|_* & = \| R(y_0-y_0^+)\|_* = \| y_0-y_0^+\|_R \\
& = \| x_0 -x_1\|_R = \| x_0-\bar{x}_0 +\bar{x}_0 -x_1 \|_R\\
& \leq \| x_0 -\bar{x}_0\|_R + \| x_1-\bar{x}_0\|_R. \end{align*} From (\ref{ineq:xone}) we derive \begin{equation} \label{ineq:xone:closer:xzero}
\|x_1-\bar{x}_0\|_R \leq \|x_0- \bar{x}_0\|_R. \end{equation} Thus,
$$ \|g(y_0)\|_* \leq \| x_0 -\bar{x}_0\|_R + \| x_1-\bar{x}_0\|_R \leq 2\|x_0-\bar{x}_0\|_R.$$ We now prove the claim for $k>0$. From (\ref{ineq:convergence}) we also have \begin{equation}\label{ineq:norm:s:plus}
\|s_{k+1}\|_R \leq \|x_0-\bar{x}_0\|_R, \; \forall k\geq 1. \end{equation} We also have that \begin{equation} \label{equ:sone:dxzero}
s_1=\delta x_0 + t_0(\delta x_1-\delta x_0) = x_1-\bar{x}_0.\end{equation} From (\ref{ineq:xone:closer:xzero}) we derive
${\|s_1\|_R=\| x_1-\bar{x}_0\|_R \leq \|x_0-\bar{x}_0\|_R}$. From this and (\ref{ineq:norm:s:plus}) we obtain \begin{equation}\label{ineq:norm:sk}
\|s_{k}\|_R \leq \|x_0-\bar{x}_0\|_R, \; \forall k\geq 1. \end{equation} From here we derive, for every $k\geq 1$, \begin{flalign*}
\| s_{k+1}-s_k \|_R & \leq \| s_{k+1} \|_R + \| s_k \|_R \\
& \leq \| x_0 -\bar{x}_0 \|_R + \| x_0-\bar{x}_0\|_R = 2 \| x_0-\bar{x}_0\|_R. \end{flalign*} From (\ref{equ:gk:tk:equ}) we have $$ g_k = \frac{1}{t_k} R(s_k-s_{k+1}), \forall k\geq 1.$$ Therefore, for every $k\geq 1$ \begin{align*}
\| g_k\|_* & = \frac{1}{t_k} \| s_k-s_{k+1}\|_R \\
& \leq \frac{2}{t_k}\|x_0-\bar{x}_0\|_R\\
& \leq \frac{4}{k+2}\|x_0-\bar{x}_0\|_R. \end{align*} We notice that the last inequality is due to the second claim of Property \ref{prop:tk}. This proves the second claim of the property. \QED
\subsection{Properties of the sequence $\{t_k\}$} \label{appen:t} \begin{property}\label{prop:tk}
Let us suppose that $t_0=1$ and that
$$t_k \doteq \frac{1}{2} \left( 1+\sqrt{1+4t_{k-1}^2}\,\,\right), \;\; \forall k\geq 1.$$
Then
\blista
\item $t_{k-1}^2=t_k^2-t_k$, for all $k\geq 1$.
\item $ t_k \geq \fracg{k+2}{2}\geq 1$, for all $k\geq 0$.
\elista \end{property}
\proof \mbox{}
\blista \item For every $k\geq 1$, $t_k$ is defined as one of the roots of $$t_k^2-t_k-t_{k-1}^2=0.$$ Therefore we obtain $t_{k-1}^2=t_k^2-t_k$. \item The claim is trivially satisfied for $k$ equal to 0. We now show that if the claim is satisfied for $k-1$ then it is also satisfied for $k$. \begin{align*}
t_k &= \frac{1}{2} \left( 1+\sqrt{1+4t_{k-1}^2}\,\,\right) \\
& \geq \frac{1}{2} \left( 1+ \sqrt{4 t_{k-1}^2} \right) = \frac{1}{2} + t_{k-1}. \end{align*} Since the claim is assumed to be satisfied for $k-1$ we have $t_{k-1}\geq \frac{k+1}{2}$ and consequently $$ t_k \geq \frac{1}{2}+\frac{k+1}{2} = \frac{k+2}{2}.$$ \QED \elista
\subsection{Proof of Property \ref{prop:FISTA:EC}}\label{appen:FISTA:EC}
From equation (\ref{equ:conv:FISTA}) we have
\begin{equation*} f(x_k)-f^* \leq \frac{2}{(k+1)^2}\|x_0-\bar{x}_0\|_{R}^2, \; \forall k\geq 1. \end{equation*} Due to Assumption \ref{assump:quadratic:growth} we also have
$$ \frac{\mu}{2} \|x_0-\bar{x}_0\|_R^2 \leq f(x_0)-f^*.$$ Therefore, \begin{equation}\label{equ:pre:alpha}
f(x_k)-f^* \leq \frac{4}{\mu(k+1)^2}(f(x_0)-f^*),\; \forall k\geq 1.
\end{equation} This proves the first claim. Denote $$\alpha_k\doteq\fracg{4}{\mu(k+1)^2}, \;\; \forall k\geq 1.$$ With this notation we rewrite (\ref{equ:pre:alpha}) as \begin{equation}\label{ineq:with:alpha:k}
f(x_k)-f^* \leq \alpha_k (f(x_0)-f^*), \; \forall k\geq 1. \end{equation} Suppose now that $k\geq \left\lfloor \fracg{2}{\sqrt{\mu}}\right\rfloor$. Then, \small \begin{equation*}
\alpha_k =\frac{4}{\mu (k+1)^2} \leq \fracg{4}{\mu \left( \left\lfloor \fracg{2}{\sqrt{\mu}}\right\rfloor +1\right)^2} < \frac{4}{\mu\left(\fracg{2}{\sqrt{\mu}}\right)^2} = 1. \end{equation*} \normalsize Therefore, \begin{equation}\label{equ:alpha:in:zeroone}
\alpha_k\in(0,1),\; \forall k\geq \left\lfloor \fracg{2}{\sqrt{\mu}}\right\rfloor. \end{equation} This, along with inequality (\ref{ineq:with:alpha:k}), yields $$ f(x_k)-f^* \leq f(x_0)-f^*, \; \forall k\geq \left\lfloor \fracg{2}{\sqrt{\mu}}\right\rfloor.$$ Equivalently, $$ f(x_k) \leq f(x_0), \; \forall k\geq \left\lfloor \fracg{2}{\sqrt{\mu}}\right\rfloor. $$ This proves the second claim of the property. In view of inequality (\ref{ineq:with:alpha:k}) we have \begin{align*} f(x_k)-f^* & \leq \alpha_k (f(x_0)-f^*) \\
& = \alpha_k (f(x_0)-f(x_k)+f(x_k)-f^*) \\
& = \alpha_k (f(x_0)-f(x_k)) + \alpha_k(f(x_k)-f^*). \end{align*} Therefore, \begin{equation} \label{ineq:alpha:k:f}
(1-\alpha_k) (f(x_k)-f^*) \leq \alpha_k(f(x_0)-f(x_k)). \end{equation} Suppose now that $k\geq \left\lfloor \frac{2\sqrt{e+1}}{\sqrt{\mu}}\right\rfloor$. This implies $k\geq \left\lfloor \frac{2}{\sqrt{\mu}}\right\rfloor$ and consequently $1-\alpha_k>0$ (see (\ref{equ:alpha:in:zeroone})). Dividing both terms of inequality (\ref{ineq:alpha:k:f}) by $1-\alpha_k$, we get \begin{align*}
f(x_k)-f^* & \leq \frac{\alpha_k}{1-\alpha_k} (f(x_0)-f(x_k))\\
& = \frac{\frac{4}{\mu(k+1)^2}}{1-\frac{4}{\mu(k+1)^2}} (f(x_0)-f(x_k))\\
& = \frac{4(f(x_0)-f(x_k))}{\mu(k+1)^2 -4} \\
& \leq \fracg{4(f(x_0)-f(x_k))}{\mu(\left\lfloor \frac{2\sqrt{e+1}}{\sqrt{\mu}} \right\rfloor+1)^2-4} \\
& \leq \fracg{4(f(x_0)-f(x_k))}{\mu(\frac{2\sqrt{e+1}}{\sqrt{\mu}})^2-4} \\
& = \fracg{4(f(x_0)-f(x_k))}{4(e+1)-4} = \frac{f(x_0)-f(x_k)}{e}. \end{align*} \QED
\subsection{Proof of Property \ref{prop:L:Convergence:FISTA}}\label{appen:L:Convergence:FISTA}
By construction, $r_{j-1}\in {\mathcal{X}}$, for all $j \geq 1$. Therefore, we have from the second claim of Property \ref{prop:proj:gradient}, that \begin{equation}\label{ineq:grad:r:r:minus}
\frac{1}{2} \|g(r_{j-1})\|_*^2 \leq f(r_{j-1})-f(r_{j-1}^+), \; \forall j\geq 1. \end{equation} We also notice that $r_j$ is computed invoking FISTA algorithm using $r_{j-1}$ as initial condition ($z=r_{j-1}$). That is, $$[r_j,n_j]=FISTA(r_{j-1},n_{j-1},E_c^l).$$ Since the output value $f(r_j)$ is forced to be no larger than the one corresponding to ${x_0=z^+=r_{j-1}^+}$, we have ${f(r_{j})\leq f(r_{j-1}^+)}$. Therefore, we obtain from inequality (\ref{ineq:grad:r:r:minus}) that \begin{align*}
\frac{1}{2} \|g(r_{j-1})\|_*^2 & \leq f(r_{j-1})-f(r_{j-1}^+)\\
& \leq f(r_{j-1})-f(r_{j}). \end{align*} This proves the first claim of the property. We now show that if $n_{j-1} \leq \frac{4\sqrt{e+1}}{\sqrt{\mu}}$, then the value $n_j$ obtained from $$ [r_j,n_j]=FISTA(r_{j-1},n_{j-1},E_c^l),$$ also satisfies \begin{equation} \label{eq:n_geq_4e}
n_{j} \leq \frac{4\sqrt{e+1}}{\sqrt{\mu}}. \end{equation} Denote $$\bar{m}=\left\lfloor \frac{2\sqrt{e+1}}{\sqrt{\mu}} \right\rfloor.$$ Since $\bar{m} \geq \left\lfloor \frac{2 \sqrt{e+1}}{\sqrt{\mu}} \right\rfloor$, we infer, from the third claim of Property \ref{prop:FISTA:EC}, that $$ f(x_{\bar{m}})-f^* \leq \frac{f(x_0)-f(x_{\bar{m}})}{e}. $$ From this inequality, we obtain $$ f(x_{\bar{m}})-f(x_k) \leq f(x_{\bar{m}})-f^* \leq \frac{f(x_0)-f(x_{\bar{m}})}{e}.$$ Therefore, the first exit condition is satisfied for $m=\bar{m}$. Since $m=\lfloor \frac{k}{2} \rfloor+1$ we have $m \geq \frac{k}{2}$. This means that for $m=\bar{m}$, the corresponding value for $k$ is no larger than $$2\bar{m} = 2 \left\lfloor \frac{2\sqrt{e+1}}{\sqrt{\mu}} \right\rfloor \leq \frac{4(\sqrt{e+1})}{\sqrt{\mu}}.$$ We also notice that, in view of the second claim of Property \ref{prop:FISTA:EC}, the additional exit condition $f(x_k)\leq f(x_0)$ is satisfied for every $$k\geq \left \lfloor \frac{2}{\sqrt{\mu}} \right\rfloor.$$ Therefore, $n_{j-1}\leq \frac{4\sqrt{e+1}}{\sqrt{\mu}} $ implies that $n_j$, obtained from $[r_j,n_j]=FISTA(r_{j-1},n_{j-1},E_c^l)$, also satisfies (\ref{eq:n_geq_4e}). We now prove, by reduction to the absurd, that $n_j$ cannot be larger than $\frac{4\sqrt{e+1}}{\sqrt{\mu}}$. Suppose that \begin{equation}\label{ineq:lower:nj}
n_j > \frac{4\sqrt{e+1}}{\sqrt{\mu}}. \end{equation} Because of the previous discussion, the previous inequality could be forced only by the doubling step ${n_j = 2n_{j-1}}$ of the algorithm. That is, inequality (\ref{ineq:lower:nj}) is possible only if there is $s$ such that $n_{s-1} >\frac{2\sqrt{e+1}}{\sqrt{\mu}}$ and $$ f(r_{s-1})-f(r_s) > \frac{f(r_{s-2})-f(r_{s-1})}{e}.$$ Since $$ [r_{s-1},n_{s-1}]=FISTA(r_{s-2},n_{s-2},E_c^l),$$ we have that $r_{s-1}$ is obtained from $r_{s-2}$ applying $$n_{s-1} >\frac{2\sqrt{e+1}}{\sqrt{\mu}}$$ iterations of FISTA algorithm. However, we have from the third claim of Property \ref{prop:FISTA:EC} that this number of iterations implies $$ f(r_{s-1})-f(r_s) \leq f(r_{s-1})-f^* \leq \frac{f(r_{s-2}^+)-f(r_{s-1})}{e}.$$ From the second claim of Property \ref{prop:proj:gradient} we also have ${f(r_{s-2}^+)\leq f(r_{s-2})}$. Thus, $$ f(r_{s-1})-f(r_s) \leq \frac{f(r_{s-2})-f(r_{s-1})}{e}.$$ That is, there is no doubling step if $n_{s-1}\geq \frac{2\sqrt{e+1}}{\sqrt{\mu}}$. This proves the second claim of the property.
We now show that there is a doubling step at least every $$T \doteq \left \lceil\ln\left(1+\frac{2(f(r_0)-f^*)}{\epsilon^2}\right)\right\rceil$$ steps of the algorithm. Suppose that there is no doubling step from iteration $j=s+1$ to $j=s+T$, where $s\geq 1$. That is, $$ f(r_{j-1})-f(r_j) \leq \frac{f(r_{j-2}) -f(r_{j-1})}{e}, \; \forall j \in [s+1,s+T]. $$ From this, and the first claim of the property, we obtain the following sequence of inequalities \small \begin{equation*}
\begin{aligned}
&\frac{1}{2} \|g(r_{s+T-1})\|_*^2 \leq f(r_{s+T-1})-f(r_{s+T}) \\ & \leq \frac{f(r_{s+T-2})-f(r_{s+T-1})}{e} \leq \left( \frac{1}{e} \right)^{T}(f(r_{s-1})-f(r_{s}))\\ & \leq \left( \frac{1}{e} \right)^{T} (f(r_{s-1})-f^*) \leq \left( \frac{1}{e} \right)^{T} (f(r_0)-f^*) \\ & = \left( \frac{1}{e} \right)^{\left\lceil \ln \left(1+\frac{2(f(r_0)-f^*)}{\epsilon^2}\right)\right\rceil} (f(r_0)-f^*) \\ & \leq \left( \frac{1}{e} \right)^{\ln \left(1+\frac{2(f(r_0)-f^*)}{\epsilon^2}\right)} (f(r_0)-f^*) \\ & = \left( \fracg{1}{1+\frac{2(f(r_0)-f^*)}{\epsilon^2}}\right) (f(r_0)-f^*) \leq \frac{\epsilon^2}{2}. \end{aligned} \end{equation*}
\normalsize We conclude that $T$ consecutive iterations without doubling step implies that the exit condition is satisfied ($\|g(r_{s+T-1})\|_*\leq \epsilon$). We conclude that there must be at least one doubling step every $T$ iterations. This implies that there exist $j\in[s+1,s+T]$ such that $$ f(r_{j-1})-f(r_j) > \frac{f(r_{j-2}) -f(r_{j-1})}{e}.$$ Therefore, $n_{j} = 2n_{j-1}$. Moreover, since $\{n_j\}$ is a non decreasing sequence, we get ${n_{s+T} \geq n_j = 2n_{j-1} \geq 2n_{s}}$, ${\forall s\geq 1}$. That is, \begin{equation}\label{ineq:half:n:T} n_s\leq \frac{n_{s+T}}{2}, \; \forall s\geq 1. \end{equation} Suppose that $j$ is rewritten as $j=m+nT$, where $0\leq m<T$ and $n\geq 0$. From the non decreasing nature of $\{n_j\}$, \small \begin{equation}
\begin{aligned}
\Sum{i=0}{j}& n_i = \Sum{i=0}{m+nT} n_i = \Sum{i=0}{m} n_i + \Sum{\ell=0}{n-1} \Sum{i=1}{T} n_{m+i+\ell T} \label{ineq:sum:nj} \\
\leq& T n_m + T \Sum{\ell=1}{n} n_{m+\ell T} = T\Sum{\ell=0}{n} n_{m+\ell T} = T\Sum{\ell=0}{n} n_{j-\ell T}. \end{aligned} \end{equation} \normalsize Also, from inequality (\ref{ineq:half:n:T}), we have $n_{j-T} \leq \frac{n_j}{2}$. Using this inequality in a recursive manner we obtain $$ n_{j-\ell T} \leq \left(\frac{1}{2}\right)^{\ell} n_j, \; \ell =0,\ldots,n.$$ This, allows us to infer from (\ref{ineq:sum:nj}) that \begin{equation*}
\Sum{i=0}{j} n_i \leq T \Sum{\ell=0}{n} \left(\frac{1}{2}\right)^\ell n_{j} \leq T \Sum{\ell=0}{\infty} \left(\frac{1}{2}\right)^\ell n_{j} = 2Tn_j. \end{equation*} The last claim of the property follows directly from this one and the bound $n_j\leq \frac{4\sqrt{e+1}}{\sqrt{\mu}}$ of the second claim.
That is, if $j$ denotes the first index for which $\|g(r_j)\|_*\leq \epsilon$, we get that the number of total iterations is bounded by \begin{align*}
\Sum{i=0}{j} n_i & \leq 2Tn_j \leq \frac{8T\sqrt{e+1}}{\sqrt{\mu}} \leq \frac{16T}{\sqrt{\mu}} \\
& = \frac{16}{\sqrt{\mu}} \left\lceil \ln \left(1+\frac{2(f(r_0)-f^*)}{\epsilon^2} \right)\right\rceil. \end{align*} \QED
\end{appendix}
\end{document} | arXiv |
\begin{document}
\begin{abstract}
We give a complete classification of finite subgroups of automorphisms of complex K3 surfaces up to deformation. The classification is in terms of Hodge theoretic data associated to certain conjugacy classes of finite subgroups of the orthogonal group of the K3 lattice.
The moduli theory of K3 surfaces, in particular the surjectivity of the period map and the strong Torelli theorem allow us to interpret this datum geometrically. Our approach is computer aided and involves hermitian lattices over number fields.
\end{abstract} \maketitle
\setcounter{tocdepth}{1} \tableofcontents
\section{Introduction} We work over the field $\mathbb{C}$ of complex numbers. A \emph{K3 surface} is a compact, complex manifold $X$ of dimension $2$ with a nowhere vanishing symplectic, holomorphic $2$-form $\sigma_X \in \HH^0(X,\Omega^2_X)$ and vanishing irregularity $h^1(X,\mathcal{O}_X)$.
Since a K3 surface does not admit non-trivial global vector fields, its automorphism group is discrete. For a very general K3 surface it is even trivial. However, there are (families of) K3 surfaces with a non-trivial and even infinite automorphism group. Typical examples of groups that appear as automorphism groups are $\mathbb{Z} * \mathbb{Z}$, $\mathbb{Z}^r$ or $\mathbb{Z}/2 \mathbb{Z}$.
K3 surfaces with a finite automorphism group have been classified by Nikulin \cite{nikulin1981,nikulin1984}, Vinberg \cite{vinberg2007} and Kondo \cite{kondo1989} with a further recent refinement due to Roulleau \cite{Roulleau2021}. The purpose of this work is to classify finite subgroups $G \leq \Aut(X)$, more precisely, pairs $(X,G)$ consisting of a K3 surface $X$ and a finite subgroup of automorphisms $G \leq \Aut(X)$.
Let $X$ be a K3 surface with cotangent sheaf $\Omega_X$. Its automorphisms act on the symplectic forms $\mathbb{C} \sigma_X = \HH^0(X,\Omega^2_X)$ by scalar multiplication. We call the ones with trivial action \emph{symplectic} and the ones with a non-trivial action \emph{non-symplectic}. The action on the symplectic form gives rise to an exact sequence \[1 \to \Aut_s(X) \to \Aut(X) \to \GL(\mathbb{C} \sigma_X)\]
where by $\Aut_s(X)$ we denote the normal subgroup of symplectic automorphisms. Now, let $G \leq \Aut(X)$ be a finite subgroup and set $G_s = G \cap \Aut_s(X)$. Then for $n = |G/G_s|$, we get an exact sequence \[1 \to G_s \to G \to \mu_n \to 1, \] where $\mu_n$ is the cyclic group of order $n$. We call the index $n = [G:G_s]$ the \emph{transcendental value} of $G$. Let $\varphi$ denote Euler's totient function. By a result of Oguiso and Machida \cite{Oguiso-Machida1998} we know that $\varphi(n) \leq 20$ and $n\neq 60$.
The distinction between symplectic and non-symplectic is crucial. For instance a non-symplectic automorphism of finite order may fix smooth curves whereas a symplectic automorphism of finite order $k$ fixes only finitely many points and their number $n_k$ depends only on $k$. Let $M_{24}$ denote the Mathieu group on $24$ points and $M_{23}$ the stabilizer group of a point. Then the number of fixed points of an element of $M_{23}$ of order $k$ depends only on $k$ and is equal to $n_k$. This observation sparked the following theorem of Mukai.
\begin{theorem}[\cite{mukai1988}] A finite group admits a faithful and symplectic action on some K3 surface if and only if it admits an embedding into the Mathieu group $M_{23}$ which decomposes the $24$ points into at least 5 orbits. \end{theorem}
Later, Xiao \cite{Xiao} gave a new proof shedding light on the combinatorics of the fixed points using the relation between $X$, its quotient $X/G_s$ and its resolution which is a K3 surface again. A conceptual proof involving the Niemeier lattices was given by Kondo \cite{kondo1998}. Finally, Hashimoto \cite{Hashimoto2012} classified all the symplectic actions on the K3 lattice. Since the corresponding period domains are connected, this is a classification up to deformation. See \cite{kondo2018} for a survey of symplectic automorphisms.
In view of these results it is fair to say that our knowledge of finite symplectic subgroups of automorphisms is fairly complete.
Similar to Hashimoto's classification for symplectic actions, our main result is a classification up to deformation (see \Cref{defequivalent} for a precise definition). Let $X$ be a K3 surface and $G$ a finite subgroup of automorphisms of $X$. We call the largest subgroup $S \leq \Aut(X)$ such that the fixed lattices satisfy $\HH^2(X,\mathbb{Z})^{G_s}=\HH^2(X,\mathbb{Z})^{S}$ the \emph{saturation} of $G_s$. Necessarily, the group $S$ is finite and symplectic. The group generated by $G$ and $S$ is finite as well. It is called the saturation of $G$. We call $G$ saturated if it is equal to its saturation. The subgroup $G \leq \Aut(X)$ is called non-symplectic, if $G_s \neq G$ and mixed if further $1\neq G_s$. If $G_s=1$, then it is called purely non-symplectic. If $G$ is non-symplectic then, $X$ is in fact projective. Therefore all K3 surfaces are henceforth assumed to be projective.
\begin{theorem}\label{mainthm} There are exactly $4167$ deformation classes of pairs $(X,G)$ consisting of a complex K3 surface $X$ and a saturated, non-symplectic, finite subgroup $G \leq \Aut(X)$ of automorphisms. For each such pair the action of $G$ on some lattice $L \cong \HH^2(X,\mathbb{Z})$ is listed in \cite{K3Groups}. \end{theorem}
While a list of the actions of these finite groups $G$ is too large to reproduce here, we present a condensed version of the data in Table~\ref{table:action} in \Cref{appendixA}. More precisely we list all finite groups $G$ admitting a faithful, saturated, mixed action on some K3 surface, their symplectic subgroups as well as the number $k(G)$ of deformation types.
Since the natural representation $\Aut(X) \to O(\HH^2(X,\mathbb{Z}))$ is faithful and K3 surfaces are determined up to isomorphism by their Hodge structure, a large extent of geometric information is easily extracted from our Hodge-theoretical model of the family of surfaces and its subgroup of automorphisms. For instance, one can compute the N\'eron--Severi and transcendental lattice of a very-general member of the family, the invariant lattice, invariant ample polarizations, the (holomorphic and topological) Euler characteristic of the fixed locus of an automorphism, the isomorphism class of the group, its subgroup consisting of symplectic automorphisms, the number of connected components of the moduli space and the dimension of the moduli space. The pairs $(X,G)$ with $G_s$ among the $11$ maximal groups have been classified in \cite{brandhorst-hashimoto2021}. In many cases projective models are listed.
\subsection*{Purely non-symplectic automorphisms} On the other end of the spectrum are purely non-symplectic groups, which are the groups $G$ with $G_s=1$. These groups satisfy $G \cong \mu_n$ and, by a result of Oguiso and Machida \cite{Oguiso-Machida1998}, we know that $n$ satisfies $\varphi(n) \leq 20$ and $n\neq 60$. To the best of our knowledge the following corollary completes the existing partial classifications for orders 4 \cite{order4}, 6 \cite{order6}, 8 \cite{order8}, 16 \cite{order16}, 20, 22, 24, 30 \cite{onedim}, $n$ with $(\varphi(n)\geq 12)$ \cite{brandhorst2019} and is completely new for orders 10, 12, 14 and 18. For order 26 it provides a missing case in \cite[Thm 1.1]{brandhorst2019}. For order $6$, \cite[Thm. 4.1]{order6} misses the case of a genus 1 curve and four isolated fixed points (0.6.2.29).
\begin{corollary}
Let $k(n)$ be the number of deformation classes of K3 surfaces with a purely non-symplectic automorphism acting by $\zeta_n$ on the symplectic form. The values $k(n)$ are given in \Cref{pure}. \begin{table}[ht] \caption{Counts of purely non-symplectic automorphisms.}\label{pure} \rowcolors{1}{}{lightgray}
\begin{tabular}[t]{cc|cc|cc|cc|cc|cc|cc}
\toprule
$n$ &$k(n)$ & $n$ &$k(n)$ &$n$ &$k(n)$ &$n$ &$k(n)$&$n$ &$k(n)$ &$n$ &$k(n)$&$n$ &$k(n)$\\
\hline $2$&$75$&$8$&$38$&$14$&$12$&$20$&$7$&$27$&$2$&$36$&$3$&$50$&$1$\\ $3$&$24$&$9$&$13$&$15$&$8$&$21$&$4$&$28$&$3$&$38$&$2$&$54$&$1$\\$4$&$79$&$10$&$37$&$16$&$7$&$22$&$4$&$30$&$10$&$40$&$1$&$60$&$0$\\ $5$&$7$&$11$&$3$&$17$&$1$&$24$&$9$&$32$&$2$&$42$&$3$&$66$&$1$\\ $6$&$150$&$12$&$48$&$18$&$16$&$25$&$1$&$33$&$1$&$44$&$1$&$$&$$\\ $7$&$5$&$13$&$1$&$19$&$1$&$26$&$3$&$34$&$2$&$48$&$1$&$$&$$\\ \bottomrule \end{tabular} \end{table} \end{corollary} The most satisfying picture is for non-symplectic automorphisms of odd prime order, where the fixed locus alone determines the deformation class, see \cite{artebani-sarti-taki}. The key tools to this result are the holomorphic and topological Lefschetz' fixed point formulas as well as Smith theory. These relate properties of the fixed locus with the action of the automorphism on cohomology. Conversely, given the action on cohomology as per our classification, we determine its fixed locus.
In what follows $\sigma$ is an automorphism of order $n$ on a K3 surface acting by multiplication with $\zeta_n = \exp(2 \pi i/n)$ on the holomorphic $2$-form of $X$. We denote by \[X^\sigma = \{x \in X \mid \sigma(x) = x\}\] the fixed point set of $\sigma$ on $X$. A curve $C \subseteq X$ is fixed by $\sigma$ if $C \subseteq X^\sigma$ and it is called invariant by $\sigma$ if $\sigma(C) = C$. Let $P \in X^\sigma$ be a fixed point. By \cite[lemme 1]{cartan1954} $\sigma$ can be linearized locally at $P$. Hence there are local coordinates $(x,y)$ in a small neighborhood centered at $P$ such that \[\sigma(x,y) = (\zeta_n^{i+1}x,\zeta_n^{-i} y)\quad \mbox{ with } \quad 0 \leq i \leq s = \left\lfloor\frac{n-1}{2}\right\rfloor.\] We call $P$ a fixed point of type $i$ and denote the number of fixed points of type $i$ by $a_i$. If $i = 0$, then $P$ lies on a smooth curve fixed by $\sigma$. Otherwise $P$ is an isolated fixed point. Note that for $n=2$ there are no isolated fixed points and at most $2$ invariant curves pass through a fixed point.
In general, the fixed point set $X^\sigma$ is a disjoint union of $N = \sum_{i=1}^s a_i$ isolated fixed points, $k$ smooth rational curves and either a curve of genus $>1$ or $0, 1, 2$ curves of genus 1. Denote by $l$ the number of genus $g \geq 1$ curves fixed by $\sigma$. If no such curve is fixed, set $g=1$. We describe the fixed locus by the tuple $((a_1, \dots, a_s), k, l, g)$. It is a deformation invariant. To sum up: \[X^\sigma = \{p_1, \dots, p_N\} \sqcup R_1 \sqcup \dots \sqcup R_k \sqcup C_1 \sqcup \dots \sqcup C_l\] where the $R_i$'s are smooth rational curves and the $C_j$'s smooth curves of genus $g \geq 1$.
Let $L$ be a $\mathbb{Z}$-lattice and $f\in O(L)$ an isometry of order $n$. Set $L_k:=\{x \in L \mid f^k(x)=x\}$. The small local type of $f$ is the collection of genera $\mathcal{G}(L_k)_{k \mid n}$. If the genus of $L$ is understood, then we omit $\mathcal{G}(L_n)=\mathcal{G}(L)$ from notation. Let $\Phi_k(x)\in \mathbb{Z}[x]$ denote the $k$-th cyclotomic polynomial. The global type of $f$ consists of the small local type as well as the isomorphism classes of the $\mathbb{Z}$-lattices $\ker \Phi_k(f)$, where $k \mid n$. A genus of $\mathbb{Z}$-lattices is denoted by its Conway--Sloane symbol \cite{splag}. The type of an automorphism $\sigma$ of a K3 surface $X$ is defined as the type of the isometry $\sigma^{*-1}|\HH^2(X,\mathbb{Z})$.
\begin{theorem}
Let $X$ be a K3 surface and $\sigma \in \Aut(X)$ of order $n$ acting by $\zeta_n$ on $\HH^0(X,\Omega_X^2)$. The deformation class of $(X, \sigma)$ is determined by the small local type of $\sigma$ unless $\sigma$ is one the $6$ exceptional types in \Cref{exceptionaltype}.
For each deformation class, the invariants $((a_1, \dots, a_s), k, l, g)$ of the fixed locus are given in \Cref{appendixB}. \end{theorem} \begin{remark}
For each of the $5$ exceptional types of order $6$ there are exactly two deformation classes. They are separated by the global type.
For order $4$, the two classes have the same global-type. They are separated by the isometry class of the glue between $L_2$ and $L_4$. It is given by the image $L_4 \to \disc{L_2}$ induced by orthogonal projection. \end{remark}
\begin{table}[h]
\caption{Exceptional types of purely non-symplectic automorphisms.}\label{exceptionaltype}
\rowcolors{1}{}{lightgray}
\renewcommand{1.2}{1.2}
\begin{tabular}{llll|llll}
\toprule
$n$ & $\mathcal{G}_1$ & $\mathcal{G}_2$ & $\mathcal{G}_3$
&$n$ & $\mathcal{G}_1$ & $\mathcal{G}_2$ & $\mathcal{G}_3$\\
\hline
$6$ & $\rm{II}_{1,8}2^9_1$& $\rm{II}_{1,17}3^{-3}$& $\rm{II}_{1,8}2^9_1$ &
$6$ & $\rm{II}_{1,7}2^8_2$& $\rm{II}_{1,15}3^{3}$ & $\rm{II}_{1,7}2^8_2$\\
$6$ & $\rm{II}_{1,5}2^6_4$& $\rm{II}_{1,15}3^{-4}$& $\rm{II}_{1,7}2^6_4$&
$6$ & $\rm{II}_{1,3}2^4_6$& $\rm{II}_{1,11}3^{5}$ & $\rm{II}_{1,3}2^4_6$\\
$6$ & $\rm{II}_{1,2}2^3_7$& $\rm{II}_{1,9}3^{-6}$ & $\rm{II}_{1,3}2^3_7$ &
$4$ & $\rm{II}_{1,9}2^8$ & $\rm{II}_{1,17}2^2$& - \\
\bottomrule
\end{tabular}
\end{table}
\begin{remark} The first $5$ exceptional types in \Cref{exceptionaltype} are due to a failure of the local to global principle for conjugacy of isometries. For the last one, the example shows that the small local type is not fine enough to determine local conjugacy. \end{remark}
\subsection*{Enriques surfaces} Since the universal cover $X$ of a complex Enriques surface $S$ is a K3 surface, our results apply to classify finite subgroups of automorphisms of Enriques surfaces. The kernel of $\Aut(S) \to \GL(\HH^0(2K_S))$ consists of the so called semi-symplectic automorphisms of $S$. They lift to automorphisms acting by $\pm 1$ on $\HH^0(X,\Omega_X^2)$. Cyclic semi-symplectic automorphisms are studied by Ohashi in \cite{ohashi2015}. Mukai's theorem on symplectic actions and the Mathieu group has an analogue for Enriques surfaces, see \cite{mukai-ohashi2015}. However, not every semi-symplectic action is of `Mathieu type`.
\begin{corollary}
A group $H_s$ admits a faithful semi-symplectic action on some complex Enriques surface if and only if $H_s$ embeds into one of the following $6$ groups
\begin{center}
\rowcolors{1}{}{lightgray}
\renewcommand{1.2}{1.2}
\begin{tabular}{lc|lc|lc}
\toprule
$G$ & id & $G$ & id& $G$ & id\\
\hline
$A_6$ & $(360, 118)$ &
$H_{192}$ & $ (192, 955)$ &
$S_5$ & $ (120, 34)$ \\
$A_{4,4}$ & $ (288, 1026)$ &
$2^4D_{10}$ & $ (160, 234)$ &
$N_{72}$ & $ (72, 40)$ \\
\bottomrule
\end{tabular}
\end{center}
A group $H$ admits a faithful action on some complex Enriques surface if and only if it embeds into one of the following $9$ groups:
\begin{center}
\rowcolors{1}{}{lightgray}
\renewcommand{1.2}{1.2}
\begin{tabular}{lc|lc|lc}
\toprule
$G$ & id & $G$ & id& $G$ & id\\
\hline
$A_6.\mu_2$ & $(720, 765)$ &
$H_{192}$ &$(192, 955)$ &
$\Gamma_{25}a_1.\mu_2$ & $(128, 929)$ \\
$A_{4,4}.\mu_2$ & $(576, 8652)$ &
$N_{72}.\mu_2$& $(144, 182)$&
$S_5$& $(120, 34)$ \\
$2^4D_{10}.\mu_2$ &$(320, 1635)$ &
$(Q_8 * Q_8).\mu_4$ &$(128, 135)$ &
$(C_2 \times D_8).\mu_4$ & $(64, 6)$ \\
\bottomrule
\end{tabular}
\end{center} \end{corollary} \begin{proof}
Let $S$ be an Enriques surface and $X$ its covering K3 surface. Let $\epsilon$ be the covering involution of $X \to S$. Let $\Aut(X,\epsilon)$ denote the centralizer of $\epsilon$.
Then $1 \to \langle \epsilon \rangle \to \Aut(X,\epsilon) \to \Aut(S) \to 1$ is exact.
In particular, if $H \leq \Aut(S)$ is a finite group then it is the image of a finite group $G \leq \Aut(X)$ containing the covering involution.
Conversely, $\epsilon \in \Aut(X)$ is the covering involution of some Enriques surface
if and only if $\HH^2(X,\mathbb{Z})^\epsilon \in \rm{II}_{1,9}2^{10}$.
Thus we can obtain the list of all finite groups acting on some Enriques surface by taking the corresponding list for K3 surfaces. For each group $G$ in the list one computes the Enriques involutions $\epsilon$, their centralizer $C(\epsilon)$ in $G$, $C(\epsilon)_s\cong H_s$ and the quotient $H\cong C(\epsilon)/\langle \epsilon \rangle$. \end{proof}
Our method of classification applies as soon as a Torelli-type theorem is available, for instance to supersingular K3 surfaces in positive characteristic and compact hyperk\"ahler manifolds.
\subsection*{Outline of the paper}
In \Cref{sec:lat} we recall basic notions of lattices with an emphasis towards primitive extensions and lattices with isometry. The geometric setting of K3 surfaces is treated in \Cref{k3moduli}. We set up a coarse moduli space parametrizing K3 surfaces together with finite subgroups of automorphisms. Next, we determine the connected components of the respective moduli spaces. We show that this translates the problem of classifying pairs of K3 surfaces and finite subgroups of automorphisms into a classification problem for lattices with isometry and extensions thereof.
The next sections deal with these algorithmic problems related to lattices, where it is shown that practical solutions exist. In particular, in \Cref{ConjAlg} it is described how isomorphism classes of lattices with isometry can be enumerated. This leads to questions related to canonical images of orthogonal and unitary groups, which are addressed in the final sections. For the classical case of $\mathbb{Z}$-lattices we review Miranda--Morison theory in \Cref{mmtheory}. For hermitian lattices we develop the necessary tools in \Cref{hermitianmm}.
Finally, in \Cref{sec:fixed} we classify the fixed point sets of purely non-symplectic automorphisms of finite order on complex K3 surfaces.
\ackintro
\section{Preliminaries on lattices and isometries}\label{sec:lat} In this section we fix notation on lattices, and refer the reader to \cite{nikulin1980_forms,splag, kneser2002} for standard facts and proofs.
\subsection{Lattices}\label{sect:lattices} Let $R$ be an integral domain of characteristic $0$ and $K$ its field of fractions. We denote by $R^\times$ its group of units. In this paper an $R$-\emph{lattice} consists of a finitely generated projective $R$-module $M$ and a non-degenerate, symmetric bilinear form $\sprodb{\cdot} \colon M \times M \to K$.
We call it integral if the bilinear form is $R$-valued and we call it even if the square-norm of every element with respect to the bilinear form is in $2 R$. If confusion is unlikely, we drop the bilinear form from notation and denote for $x,y \in M$ the value $\sprodb{x}{y}$ by $xy$ and $\sprodbq{x}$ by $x^2$. The associated quadratic form is $Q(x) = x^2/2$. We denote the \emph{dual lattice} of $M$ by $M^\vee$. We call $M$ unimodular if $M=M^\vee$. For two lattices $M$ and $N$ we denote by $M\perp N$ their orthogonal direct sum. The scale of $M$ is $\scale(M)=\sprodb{M}{M}$ and its norm $\norm(L)$ is the fractional ideal generated by $\sprodbq{x}$ for $x \in M$.
The set of self-isometries of $M$ is the \emph{orthogonal group} $O(M)$ of $M$.
We fix the following convention for the \emph{spinor norm}: Let $L$ be an $R$-lattice and $V = L \otimes K$. Let $v \in V$ with $v^2 \neq 0$. The reflection $\tau_v(x) = x - 2xv/v^2\cdot v$ is an isometry of $V$. The spinor norm of $\tau_v$ is defined to be $Q(v)=v^2/2 \in k^\times/(k^\times)^2$. By the Cartan--Dieudonn\'e theorem $O(V)$ is generated by reflections. One can show that this defines a homomorphism $\spin: O(V) \to k^\times/(k^\times)^2$ by using the Clifford algebra of $(V,Q)$.
An embedding $M \to L$ of lattices is said to be \emph{primitive} if its cokernel is torsion-free. For $M\subseteq L$ we denote by $M^{\perp L}=\{x \in L \mid \sprodb{x}{M}=0\}$ the maximal submodule of $L$ orthogonal to $M$. If confusion is unlikely we denote it simply by $M^\perp$. The minimum number of generators of a finitely generated $R$-module $A$ will be denoted by $l(A)$.
Let $L$ be an even integral $R$-lattice. Its \emph{discriminant group} is the group $D_L = L^\vee/ L$ equipped with the \emph{discriminant quadratic form}
$q_L\colon L \to K/2 R$. Note that $l(\disc{L})\leq l(L)=\rk L$. Denote by $O(D_L)$ its \emph{orthogonal group}, that is, the group of linear automorphisms preserving the discriminant form. If $f \colon L \to M$ is an isometry of even $\mathbb{Z}$-lattices, then it induces an isomorphism $D_f \colon D_L \to D_M$. Likewise we obtain a natural map $O(L) \to O(D_L)$, whose kernel is denoted by $O^\sharp(L)$. For an isometry $f \in O(L)$ and $H$ some subquotient of $L \otimes K$ preserved by $f$, we denote by $f|H$ the induced automorphism of $H$. Let $G\leq O(L)$ be a subgroup. We denote the fixed lattice by $L^G=\{x \in L \mid \forall g \in G \colon g(x)=x \}$ and its orthogonal complement by $L_G=(L^G)^\perp$.
For $R=\mathbb{R}$, let $s_+$ be the number of positive eigenvalues of a gram matrix and $s_-$ the number of negative eigenvalues. We call $(s_+,s_-)$ the \emph{signature pair} or just \emph{signature} of $L$.
\subsection{Primitive extensions and glue} Let $R \in \{\mathbb{Z}, \mathbb{Z}_p\}$ and $L$ be an even integral $R$-lattice. We call $M \perp N \subseteq L$ a \emph{primitive extension} of $M \perp N$ if $M$ and $N$ are primitive in $L$ and $\rk L = \rk M + \rk N$. Since $L$ is integral, we have a chain of inclusions \[M \perp N \subseteq L \subseteq L^\vee \subseteq M^\vee \perp N^\vee.\] The projection $M^\vee \perp N^\vee \to M^\vee$ induces a homomorphism $L/(M \perp N) \to D_M$. This homomorphism is injective if and only if $N$ is primitive in $L$. Let $H_M$ denote its image and define $H_N$ analogously. The composition \[\phi \colon H_M \to L/(M \perp N) \to H_N\] is called a \emph{glue} map. It is an anti-isometry, i.e. $q_M(x) = -q_N(\phi(x))$ for all $x \in H_M$. Note that $L/(M\perp N) \leq H_M \perp H_N \leq D_M \perp D_N$ is the graph of $\phi\colon H_M \to H_N$.
Conversely, any anti-isometry $\phi \colon H_M \to H_N$ between subgroups $H_M \subseteq D_M$ and $H_N \subseteq D_N$ is the glue map of a primitive extension: $M \perp N \subseteq L_\phi$ where $L_\phi$ is defined by the property that $L_\phi/(M \perp N)$ is the graph of $\phi$.
The determinants of the lattices in play are related as follows: \[\lvert\det L\rvert = \lvert D_M/H_M \rvert \cdot \lvert D_N/H_N \rvert = \lvert \det M \rvert \cdot \lvert \det N \rvert/[L : (M \perp N)]^2.\] If $f_M \in O(M)$ and $f_N \in O(N)$ are isometries, then $g = f_M \oplus f_N$ preserves the primitive extension $L_\phi$ if and only if $\phi \circ D_{f_M} = D_{f_N} \circ \phi$. We call $\phi$ an \emph{equivariant glue map} with respect to $f_M$ and $f_N$.
\subsection{Lattices with isometry} We are interested in classifying conjugacy classes of isometries of a given $\mathbb{Z}$-lattice. If the lattice in question is definite, its orthogonal group is finite. Using computer algebra systems, one can compute the group as well as representatives for its conjugacy classes. This approach breaks down if $L$ is indefinite.
\begin{definition} A \emph{lattice with isometry} is a pair $(L,f)$ consisting of a lattice $L$ and an isometry $f \in O(L)$. We frequently omit $f$ from the notation and denote the lattice with isometry simply by $L$. Its isometry $f$ is then denoted by $f_L$. We say that two lattices with isometry $M$ and $N$ are \emph{isomorphic} if they are equivariantly isometric, i.e., if there exists an isometry $\psi \colon M \to N$ with $\psi \circ f_{M} = f_{N} \circ \psi$. We view $L$ as a $\mathbb{Z}[x,x^{-1}]$ module via the action $x \cdot m = f(m)$, $x^{-1} \cdot m = f^{-1}(m)$. \end{definition} The unitary group $U(L)$ of the lattice with isometry $L$ is the centralizer of $f_L$ in $O(L)$. This is nothing but $\Aut(L)$ in the category of lattices with isometry.
Terminology for lattices applies to lattices with isometry verbatim. For instance the discriminant group $D_L$ of a lattice with isometry comes equipped with the induced isometry $D_{f_L} \in O(D_L)$ and $U(D_L)$ is the centralizer of $D_{f_L}$ in $O(D_L)$. We denote by $G_L = \operatorname{Im}( U(L) \to U(D_L))$.
\begin{proposition}\label{extensions} Let $M,N$ be lattices with isometry. Suppose that the characteristic polynomials of $f_M$ and $f_N$ are coprime. Then the double coset \[U(N) \backslash \{\mbox{equivariant glue maps } \disc{M} \supseteq H_M \xrightarrow{\phi } H_N \subseteq D_{N} \} / U(M)\] is in bijection with the set of isomorphism classes of lattices with isometry $(L,f)$ with characteristic polynomial $\chi_f(x) = \chi_{f_{M}}(x)\chi_{f_{N}}(x)$ and $M \cong (\ker \chi_{f_M}(f))$ and $N \cong (\ker \chi_{f_N}(f))$. \end{proposition} \begin{proof}
We work in the category of lattices with isometry. Let $L_\phi$ and $L_{\psi}$ be primitive extensions of $M \perp N$ and $h\colon L_\phi \to L_\psi$ an isomorphism. Then $h|_M \in U(M)$ and $h|_N \in U(N)$. We have $D_{h|_N}\phi D_{h|_M}^{-1} = \psi$, so $\psi \in U(M) \phi U(N)$. Conversely if $D_a \phi D_b = \psi$. Then $a \oplus b \colon L_\phi \to L_\psi$ is an isomorphism. \end{proof}
\section{K3 surfaces with a group of automorphisms}\label{k3moduli} See \cite{bhpv} or \cite{huybrechts2016} for generalities on K3 surfaces. \subsection{$\mathbf{H}$-markings} Let $X$ be a K3 surface. We denote by \[\rho_X\colon \Aut(X) \rightarrow O(\HH^2(X,\mathbb{Z})),\ g\mapsto (g^{-1})^*\] the natural representation of the automorphism group $\Aut(X)$ on the second integral cohomology group of $X$. It is faithful. \begin{definition}\label{defequivalent} Let $X$, $X'$ be K3 surfaces and $G \leq \Aut(X)$, $G' \leq \Aut(X')$. We call $(X,G)$ and $(X',G')$ \emph{conjugate}, if there is an isomorphism $\phi\colon X \to X'$ such that $\phi G \phi^{-1} = G'$.
They are called \emph{deformation equivalent} if there exists a connected family $\mathcal{X} \to B$ of K3 surfaces, a group of automorphisms $\mathcal{G} \leq \Aut(\mathcal{X}/B)$ and two points $b,b' \in B$ such that the restriction of $(\mathcal{X},\mathcal{G})$ to the fiber above $b$ is conjugate to $(X,G)$ and to the fiber above $b'$ is conjugate to $(X',G')$. \end{definition}
Let $L$ be a fixed even unimodular lattice of signature $(3,19)$. An $L$-marking of a K3 surface $X$ is an isometry $\eta \colon \HH^2(X,\mathbb{Z}) \to L$. The pair $(X, \eta)$ is called an $L$-marked K3-surface. A family of $L$-marked K3 surfaces is a family $\pi \colon \mathcal{X} \to B$ of K3 surfaces with an isomorphism of local systems $\eta \colon R^2\pi_*\underline{\mathbb{Z}} \to \underline{L}$. If the base $B$ is simply connected, then a marking of a single fiber extends to the whole family.
\begin{definition}
Let $H \leq O(L)$ be a \emph{finite} subgroup. An $H$-marked K3 surface is a triple $(X, \eta, G)$ such that $(X, \eta)$ is an $L$-marked K3 surface and $G \leq \Aut(X)$ is a group of automorphisms with $\eta \rho_X(G) \eta^{-1} = H$.
We say that $(X,G)$ is $H$-markable if there exists some marking by $H$.
Two $H$-marked K3 surfaces $(X_1, \eta_1, G_1)$
and $(X_2,\eta_2, G_2)$ are called \emph{conjugate} if there exists an isomorphism $f \colon X_1 \to X_2$ such that $\eta_1 \circ f^* = \eta_2$. In particular, $fG_1f^{-1}=G_2$. We call $H$ \emph{effective} if there exists at least one $H$-marked K3 surface. \end{definition}
A family of $H$-marked K3 surfaces is a family of $L$-marked K3 surfaces $\pi \colon \mathcal{X} \to B$ of K3 surfaces with an isomorphism of local systems $\eta \colon R^2\pi_*\underline{\mathbb{Z}} \to \underline{L}$ and group of automorphisms $\mathcal{G} \leq \Aut(\mathcal{X}/B)$ such that for each $b\in B$ the fiber $(\mathcal{X}_b,\eta_b,\mathcal{G}_b)$ is an $H$-marked K3 surface.
Let $(X,\eta,G)$ be an $H$-marked K3 surface. The action of $G$ on $\HH^{2,0}(X)$ induces via the marking $\eta$ a character $\chi\colon H \rightarrow \mathbb{C}^\times$. We call such a character \emph{effective}. We denote complex conjugation by a bar $\bar\cdot$. Set $H_s = \ker \chi$, and denote the $\chi$-eigenspace by \[L_\mathbb{C}^\chi = \{x \in L \otimes \mathbb{C} \mid h(x) = \chi(h)\cdot x \mbox{ for all }h \in H\}.\]
Similarly let \[L_\mathbb{R}^{\chi+\bar\chi}=\{x \in L_\mathbb{R} \mid (h+h^{-1})(x) = \chi(h)x+\bar \chi(h)x \text{ for all } h \in H\}.\] The generic transcendental lattice $T(\chi)$ is the smallest primitive sublattice of $L$ such that $T(\chi) \otimes \mathbb{C}$ contains $(L\otimes \mathbb{C})^\chi$. We call $\NS(\chi) = T(\chi)^\perp$ the generic N\'eron--Severi lattice. Recall that $L_{H_s}$ is the complement of the fixed lattice $L^{H_s}$.
\begin{proposition}\label{iseffective} Let $H\leq O(L)$ be a finite group and $\chi \colon H \to \mathbb{C}^\times$ a non-trivial character. Recall that $H_s := \ker \chi$. Then $\chi$ is effective if and only if the following hold: \begin{enumerate}
\item $L_{H_s}$ is negative definite;
\item $L_{H_s}$ does not contain any $(-2)$-vector;
\item the signature of $L^H$ is $(1,*)$;
\item $L_\mathbb{R}^{\chi+\bar \chi}$ is of signature $(2,*)$;
\item $\NS(\chi)_H$ does not contain any vector of square $(-2)$.
\end{enumerate} If $\chi$ is trivial, then $\chi$ is effective if and only if (1) and (2) hold. \end{proposition} \begin{proof}
If $\chi$ is trivial, this is known (cf. \cite[4.2, 4.3]{nikulin1980}).
So let $\chi$ be non-trivial. We show that (1-5) are necessary. (1-2) follow from \cite[4.2]{nikulin1980}. Let $H$ be an effective subgroup and $(X,\eta,G)$ an $H$-marked K3 surface.
Since $T(\chi)\otimes \mathbb{C}$ contains the period $\eta(\omega)$ of $(X,\eta)$, we have $\eta(\omega+\bar \omega) \in L_\mathbb{R}^{\chi+\bar \chi}$.
Recall that $\omega.\bar \omega >0$. Thus $L_\mathbb{R}^{\chi+\bar \chi}$ has at least two positive squares.
Let $h \in \NS(X)$ be ample. Then $h' = \sum_{g\in G} g^*h$ is ample and $G$-invariant i.e. $h' \in H^2(X,\mathbb{Z})^G=\eta^{-1}(L^H)$. Since $h'^2>0$, $L^H$ has at least one positive square. Since $H^{1,1}(X)$ and $H^{2,0}(X)\oplus H^{0,2}(X)$ are orthogonal, so are $h$ and $L_\mathbb{R}^{\chi+\bar \chi}\subseteq \eta(H^{2,0}(X)\oplus H^{0,2}(X))$. Therefore all $3$ positive squares of $L$ are accounted for. This proves (3) and (4).
For (5) we note that $\NS(\chi)\subseteq \eta(\NS(X))$.
For $\delta \in \NS(X)$ with $\delta^2=-2$ we know that $\delta$ or $-\delta$ is effective by Riemann--Roch.
Therefore $h'.\delta \neq 0$, because $h'$ is ample.
Since $\NS(X)_G \subseteq h^\perp \cap \NS(X)$,
$\NS(X)_G$ does not contain any vector of square (-2).
The same holds true for $\NS(\chi)_H\subseteq \eta(\NS(X)_G)$.
Since the signature of $L^{\chi+\bar\chi}_\mathbb{R}$ is $(2,*)$, we can find an element $\omega$ in $(L\otimes \mathbb{C})^\chi$ such that
$\omega.\bar \omega>0$, $\omega^2=0$. Choosing $\omega$ general enough we achieve $\omega^\perp \cap T(\chi) = 0$. By the surjectivity of the period map \cite[VII (14.1)]{bhpv}, we can find an $L$-marked K3 surface $(X,\eta)$ with period $\omega$.
By construction $\eta(T(X)) = T(\chi)$, so $X$ is projective and $\eta(\NS(X)) = \NS(\chi)$. Since $L^H$ is of signature $(1,*)$ and $L_H$ does not contain any $(-2)$-roots, we find $h \in L^H$ with $h^2>0$ and $h^\perp \cap \NS(\chi)$ not containing any $-2$ roots either and after possibly replacing $h$ by $-h$, we can assume that $h$ lies in the positive cone. Thus $h$ lies in the interior of a Weyl chamber. Since the Weyl group $W(\NS(X))$ acts transitively on the Weyl chambers of the positive cone, we find an element $w \in W(\NS(\chi))$ such that that $(\eta\circ w)^{-1}(h)$ is ample. Let $\eta'= \eta\circ w$. By construction, every element of $G' = \eta'^{-1} H \eta'$ preserves this ample class and the period of $X$. So $G'$ is a group of effective Hodge isometries. By the strong Torelli theorem (see e.g. \cite[VIII \S 11]{bhpv}), $G' = \rho_X(G)$ for some $G \leq \Aut(X)$.
\end{proof}
Note that for any effective character $\chi\colon H \to \mathbb{C}^\times$ of $H$, \[\ker \chi = H_s = \{h \in H \mid L^h \mbox{ is of signature } (3,*)\}\] is independent of $\chi$. Indeed, because $L_{H_s}$ is negative definite, $L^{H_s}\subseteq L^h$ is of signature $(3,*)$ for any $h \in H_s$. On the other hand if $g\in H$ is not in $H_s$, then $\chi(g)\neq 1$ and so $L^g$ is orthogonal to $L_\mathbb{R}^{\chi+\bar \chi}$ and contains $L^H$. Therefore $L^g$ is of signature $(1,*)$. The kernel is a normal subgroup and $H/H_s \cong \mu_n$ is cyclic. We say that an effective group $H$ is \emph{symplectic} if $H=H_s$ and non-symplectic otherwise.
\begin{lemma}
Let $H \leq O(L)$ be effective. There are at most two effective characters $\chi \colon H \to \mathbb{C}^\times$. They are complex conjugate. \end{lemma} \begin{proof}
Fix a generator $h H_s$ of $H/H_s$ and let $\chi$ be an effective character. It is determined by its value $\chi(h)$, which is a primitive $n$-th root of unity. Set $T=T(\chi)$. Since $\chi$ is effective,
$L_\mathbb{R}^{\chi +\bar \chi}= T_\mathbb{R}^{\chi +\bar \chi}$ is of signature $(2,*)$. It is equal to the $\chi(h)+\bar \chi(h)$ eigenspace of $(h+h^{-1})|T_\mathbb{R}$. The other real eigenspaces of $(h+h^{-1})|T_\mathbb{R}$ are negative definite. Thus any effective character $\chi'$ is equal to $\chi$ or $\bar \chi$. \end{proof} In particular this shows that $T(\chi)$ and $\NS(\chi)$ do not depend on the choice of the effective character $\chi$, but only on $H$. We may denote them as $T(H)$ and $\NS(H)$. \subsection{Moduli spaces and periods} Let $\mathcal{M}_H$ denote the fine moduli space parametrizing isomorphism classes of $H$-marked K3 surfaces $(X,\eta,G)$. It is a non-Hausdorff complex manifold. It can be obtained by gluing the base spaces of the universal deformations of $(X,\eta,G)$, see \cite[\S 3]{brandhorst-cattaneo}.
Let $\chi\colon H \to \mathbb{C}^\times$ be an effective character. We have $\mathcal{M}_H = \mathcal{M}^\chi_H \cup \mathcal{M}^{\bar\chi}_H$ where $\mathcal{M}_H^\chi$ parametrizes isomorphism classes of $H$-marked K3 surfaces $(X,\eta, G)$ with $\chi(\eta \rho_X(g)\eta^{-1}) \cdot \omega_X = (g^*)^{-1} \omega_X$ for all $g \in G$. By \cite[Prop. 3.9]{brandhorst-cattaneo} the forgetful map $\mathcal{M}_H \to \mathcal{M}_L$, $(X,\eta,G) \mapsto (X,\eta)$ into the moduli space of $L$-marked K3 surfaces is a closed embedding.
\begin{definition}
Let $H\leq O(L)$ be effective and $\chi\colon H \to \mathbb{C}^\times$ be a non-trivial effective character.
We denote by
\[\mathbb{D}^\chi= \{\mathbb{C} \omega \in \mathbb{P}((L\otimes \mathbb{C})^\chi) \mid \sprodb{\omega}{\omega}=0, \sprodb{\omega}{\bar\omega}>0\}\]
the corresponding period domain and its period map by
\[\mathcal{P} \colon \mathcal{M}_H^\chi \to \mathbb{D}^\chi.\] It is a local isomorphism (see e.g. \cite{brandhorst-cattaneo}). The discriminant locus is $\Delta = \bigcup \left\{\mathbb{P}(\delta^\perp) \mid \delta \in L_H, \delta^2=-2\right\} \subseteq \mathbb{P}(L_\mathbb{C})$. \end{definition}
\begin{proposition}\label{perioddomain} Let $\chi \colon H \to \mathbb{C}^\times$ be an effective and non-trivial character.
\begin{enumerate}
\item The image of $\mathcal{P}$ is $\mathbb{D}^\chi \setminus \Delta $.
\item If $(X_1,\eta_1,G_1)$ and $(X_2,\eta_2,G_2)$ lie in the same fiber $\mathcal{P}^{-1}(\mathbb{C} \omega)$, then $(X_1,G_1)$ and $(X_2,G_2)$ are conjugate.
\item If $\chi$ is real, then $\mathbb{D}^\chi \setminus \Delta $ has two connected components. They are complex conjugate. If $\chi$ is not real, then $\mathbb{D}^\chi \setminus \Delta$ is connected.
\end{enumerate} \end{proposition} \begin{proof} (1) Let $\mathbb{C} \omega \in \mathbb{D}^\chi$. By the surjectivity of the period map we find a marked K3 surface $(X,\eta)$ with period $\mathbb{C} \omega$. Set $N = \mathbb{C} \omega^\perp \cap L$. By Lefschetz' theorem on $(1,1)$-classes we have $\eta(\NS(X))=N$.
If $\mathbb{C}\omega \not \in \Delta$, we have that $\NS(H)_H = \NS(H) \cap (L^H)^\perp$ does not contain any roots. This means that $\eta^{-1}(\NS(H)^H)\subseteq \NS(X)$ contains an ample class. Thus $\eta^{-1} H \eta$ preserves the period and the ample cone. By the strong Torelli theorem there is a (unique) group of automorphisms $G\leq \Aut(X)$ with $\eta \rho_X(G) \eta^{-1}=H$.
If conversely $\mathbb{C} \omega \in \Delta$, then $H$ does not preserve a Weyl chamber of $N$ so that $H$ cannot come from a group of automorphisms of $X$.
(2) Let $(X_1,\eta_1,G_1)$ and $(X_2,\eta_2,G_2)$ be $H$-marked K3 surfaces in the fiber of $\mathbb{C} \omega$. Then $\varphi = \eta_1^{-1} \circ \eta_2$ is a Hodge isometry which conjugates $\rho_{X_1}(G_1)$ and $\rho_{X_2}(G_2)$. However, it may not preserve the ample cones. By \cite[Lemma 1.7 and Theorem 1.8]{oguiso-sakurai} there exists a unique element $w \in \langle \pm 1\rangle \times W(\NS(X_1))$ such that $w \circ \varphi$ preserves the ample cones and $w g^*= g^* w$ for all $g \in G_1$. Since now $w \circ \varphi$ is an effective Hodge isometry, the strong Torelli Theorem applies and provides an isomorphism $F\colon X_1 \to X_2$ with $F^*=w \circ \varphi$. By construction we have $FG_1 F^{-1} = G_2$ as desired.
(3) By \cite[\S9 \& \S 11]{dolgachev-kondo2005} the period domain $\mathbb{D}^\chi$ has $2$ connected components if $\chi$ is real and one else. The discriminant locus is a locally finite union of real codimension two hyperplanes. Removing it does not affect the number of connected components. \end{proof}
\begin{remark} If $\omega \in L_\mathbb{C}^\chi$, then we have $\omega^2 = \chi(g)^2 \omega^2$ for any $g \in G$. Thus $(1-\chi(g)^2)\omega^2=0$. Let $n = [H:H_s]=\lvert \image \chi \rvert$. For $n>2$ this condition implies $\omega^2=0$ and hence the dimension of $\mathbb{D}^\chi$ is $\dim_\mathbb{C} L^\chi_\mathbb{C} -1$. While for $n=2$ it is $\dim L^\chi_\mathbb{C} -2$. \end{remark}
Denote by $N(H)=\{f \in O(L) \mid f H = H f\}$ the normalizer of $H$ in $O(L)$. If $(X,\eta,G)$ is an $H$-marked K3 surface and $f \in N(H)$, then $(X,f \circ \eta, G)$ is an $H$-marked K3 surface as well. In fact all $H$-markings of $(X,G)$ arise in this way. So $N(H)$ is the group of changes of marking.
Set $\mathbb{D}_H = \mathbb{D}^\chi \cup \mathbb{D}^{\bar \chi}$. The group $N(H)$ acts on $\mathbb{D}_H$ via an arithmetic subgroup of $O(T(H))$, respectively $U(T(H))$. Therefore by \cite{baily-borel1966} the space $\mathbb{D}_H/N(H)$ is a quasi-projective variety with only finite quotient singularities.
\begin{theorem}
The coarse moduli space $\mathcal{F}_H:=\mathcal{M}_H/N(H)$ of $H$-markable K3 surfaces admits a bijective period map
$\mathcal{F}_H \to (\mathbb{D}_H \setminus \Delta)/N(H)$. \end{theorem} \begin{proof} We can use the action of the normalizer to forget the marking and thus obtain a period map $\mathcal{F}_H =\mathcal{M}_H/N(H) \to (\mathbb{D}_H\setminus \Delta)/N(H)$. That it is bijective follows from \Cref{perioddomain}. Part (1) gives surjectivity and part (2) injectivity. Indeed, if two $H$ polarizable K3 surfaces have the same image, then we can find markings on them such that they lie in the same fiber of the period map $\mathcal{M}_H \to \mathbb{D}_H$. Then they are conjugate by \Cref{perioddomain} (2). \end{proof} See \cite{alexeev-engel,alexeev-engel-han} for more on moduli of K3 surfaces and their compactifications.
\subsection{Connected components}
We next show that deformation classes of K3 surfaces with finite groups of automorphisms are precisely the connected components of the coarse moduli space of $H$-markable K3 surfaces $\mathcal{F}_H$. In the following, for a topological space $Y$ we denote by $\pi_0(Y)$ the set of (path) connected components of $Y$.
\begin{theorem}\label{deformation class vs F_H}
Let $(X,G)$ and $(X',G')$ be two pairs consisting of K3 surfaces $X, X'$ and finite groups of automorphisms $G \leq \Aut(X)$, $G' \leq \Aut(X')$. Then $(X,G)$ and $(X',G')$ are deformation equivalent if and only if they are markable by the same effective subgroup $H\leq O(L)$ and they lie in the same connected component of $\mathcal{F}_H$. \end{theorem}
\begin{proof} Every pair $(X,G)$ is $H$-markable for some effective subgroup $H\leq O(L)$.
Let $(X',G')$ be deformation equivalent to $(X,G)$. This means that we find a connected family $\mathcal{X} \to B$, a group of automorphisms $\mathcal{G}\leq \Aut(\mathcal{X}/B)$ and points $b,b' \in B$ such that the fibers above $b$ and $b'$ are conjugate to $(X,G)$ and $(X',G')$. The $H$-marking of $(X,G)$ induces an $H$-marking of the fiber above $b$. By parallel transport in the local system $R^2 \pi_*\underline{\mathbb{Z}}$ we move the marking from $(X,G)$ to $(X',G')$ along some continuous path $\gamma$ in $B$ connecting $b$ and $b'$. Therefore the fiber above $b'$ is $H$-markable. The isomorphism of the fiber with $(X',G')$ allows to transport this marking to $(X',G')$. Therefore $(X',G')$ is $H$-markable. Its point in the moduli space $\mathcal{F}_H$ of $H$-markable K3-surfaces coincides with that of the fiber above $b'$. Likewise $(X,G)$ gives the same point in $\mathcal{F}_H$ as the fiber above $b$. Since the two fibers lie in the same connected component of $\mathcal{M}_H$ their images lie in the same connected component of $\mathcal{F}_H$.
Conversely let $(X,G)$ and $(X',G')$ be $H$-markable and in the same connected component of $\mathcal{F}_H$. Then we can find markings $\eta$, and $\eta'$ such that $(X,\eta,G)$ and $(X,\eta',G')$ are $H$-marked. Then $\pi_0(\mathcal{F}_H) = \pi_0(\mathcal{M}_H/N(H)) \cong \pi_0(\mathcal{M}_H)/N(H)$. Since $(X,G)$ and $(X',G')$ lie in the same connected component of $\mathcal{F}_H$, we find $n \in N(H)$ such that $(X,\eta,G)$ and $(X,n \circ \eta',G)$ lie in the same connected component of $\mathcal{M}_H$. Since $\mathcal{M}_H$ is a fine moduli space it has a universal family and this gives a deformation of $(X,\eta,G)$ and $(X,n \circ \eta',G)$ as $H$-marked K3 surfaces. By forgetting the markings we obtain a deformation of $(X,G)$ with $(X',G')$. \end{proof}
\begin{corollary}
The set of deformation classes of pairs $(X,G)$ with $X$ a K3 surface and $G \leq \Aut(X)$ with $G \neq G_s$ is in bijection with the set $\bigcup_{H \in T} \pi_0(\mathcal{F}_H)$, where $T$ is a transversal of the set of conjugacy classes effective, non-symplectic subgroups of $O(L)$. \end{corollary}
Let $L$ be a K3 lattice and $H\leq O(L)$ an effective subgroup. Our next goal is to determine the connected components of the coarse moduli space $\mathcal{F}_H$ of $H$-markable K3 surfaces. Since the period domain $\mathbb{D}_H \setminus \Delta$ has exactly two connected components (\Cref{perioddomain}~(3)), $\mathcal{F}_H$ has at most two components as well. It has only one connected component if and only if the action of $N(H)$ on $\mathbb{D}_H$ exchanges the two components.
Let $\chi: H \to \mathbb{C}^\times$ be an effective character. For $n \in N(H)$ denote by $\chi^n$ the character defined by $\chi^n(h) = \chi(n^{-1} h n)$. Denote by $N(\chi)$ the stabilizer of $\chi$ in $N(H)$. For completeness sake we mention the following proposition.
\begin{proposition}
Let $\chi \colon H \to \mathbb{C}^\times$ be an effective character and $[H:H_s]>2$.
Then the number of connected components of $\mathcal{F}_H$ is $2/[N(H):N(\chi)]$. \end{proposition} \begin{proof}
Since $[H:H_s]>2$, the connected components of $\mathcal{M}_H$ are $\mathcal{M}_H^\chi$ and $\mathcal{M}_H^{\bar \chi}$. If $(X,\eta,G) \in \mathcal{M}_H^\chi$ and $n \in N(H)$ then $(X,n \circ \eta,G) \in \mathcal{M}_H^{\chi^n}$. \end{proof}
Let now $[H:H_s]=2$. Then we have seen that $\chi=\bar \chi$ and $\mathbb{D}^\chi$ has two connected components. This can be dealt with by introducing \emph{positive sign structures}. Our account follows \cite{shimada2018}. A period $\mathbb{C} \omega \in \mathbb{D}^\chi$ can be seen as an oriented, positive definite real $2$-plane. Indeed, the two real vectors $\operatorname{Re} \omega, \operatorname{Im} \omega \in L_\mathbb{R}$ give an ordered (and thus oriented), orthogonal basis of a positive definite plane in $L$.
\begin{definition}
Let $L$ be a $\mathbb{Z}$-lattice of signature $(s_+,s_-)$. Then a \emph{sign structure} on $L$ is defined as a choice of one of the connected components $\theta$ of the manifold parametrizing oriented, $s_+$-dimensional, positive definite, real subspaces $S$ of $L_\mathbb{R}$. Unless $L$ is negative definite it has exactly two positive sign structures. \end{definition}
For $[H:H_s]>2$, the periods in $\mathbb{D}^\chi$ all give the same sign structure. But for $[H:H_s]=2$ there are two sign structures which give the two connected components of the period domain. Whether or not $N(H)$ preserves the sign structure is encoded by a certain character. See \cref{sect:lattices} for our conventions on the spinor norm. \begin{proposition}[\cite{looijenga-wahl1986,miranda-morrisonI}]
Let $L$ be an $\mathbb{R}$-lattice, so that the spinor norm takes values in $\{\pm 1\}\cong \mathbb{R}^\times/\mathbb{R}^{\times 2}$.
The action of an isometry $g \in O(L)$ on the set of positive sign structures of $L$ is trivial if and only if $\det(g) \cdot \spin(g) > 0$. \end{proposition} Let $L$ be an $\mathbb{R}$-lattice. We denote by $O^+(L)=\ker (\det \cdot \spin)$ the subgroup of orientation preserving isometries. If $G \leq O(L)$ is a subgroup, we denote by $G^+ = G \cap O^+(L)$ its normal subgroup of orientation preserving elements.
\begin{proposition}\label{quadruplewithtriple}
Let $\chi \colon H \to \mathbb{C}^\times$ be an effective character and $[H:H_s]=2$.
Set $T = T(\chi)$, let $\pi \colon N(H) \to O(T)$ be the restriction and $N_T= \pi(N(H))$.
Then the number of connected components of $\mathcal{F}_H$ is $2/[N_T:N_T^+]$. \end{proposition} \begin{proof}
The subgroup of $N(H)$ fixing the sign structures of $T(\chi)$ is by the definition $\pi^{-1}(N_T^+)$. Therefore $2/[\pi(N(H)):\pi(N(H))^+]$ is the number of connected components of $N(H)$. \end{proof}
\begin{lemma}
Keep the notation of \Cref{quadruplewithtriple}.
Let $\disc{} \colon O(T) \to O(\disc{T})$ be the natural map, $J =D(N_T)$, $J^+=D(N_T^+)$ and
$K = \ker \disc{}$.
Then $[N_T:N_T^+] = [K:K^+][J:J^+]$. \end{lemma} \begin{proof} We claim that that $K \subseteq N_T$. By definition any element $g\in K$ acts trivially on the discriminant group $\disc{T}$. Therefore $g\oplus \id_{\NS(\chi)}$ extends to $L$. Since the restriction of any $h\in H$ to $T$ is given by $\pm \id_{T}$, $g \oplus \id_{\NS(\chi)}$ commutes with $h$. The claim is proven. As a consequence we have $N_T = \disc{}^{-1}(J)$. Thus we have a commutative diagram with exact rows \[ \begin{tikzcd}
1\arrow[r]& K^+ \arrow[r]\arrow[d] &N_T^+ \arrow[r]\arrow[d]& J^+\arrow[r]\arrow[d] & 1\\ 1\arrow[r]& K\arrow[r] &N_T\arrow[r]{}{D} & J \arrow[r] &1 \end{tikzcd}\] where the vertical arrows are inclusions of normal subgroups. Hence the cokernels exist and so we obtain the corresponding exact sequence \[1 \to K/K^+ \to N_T/N_T^+ \to J/J^+ \to 1\] of the cokernels. To see this follow the proof of the snake lemma, which indeed is valid in this situation.
\end{proof}
\begin{remark}\label{rem:mmapplication} We will see in \Cref{plusindexcompute} how to compute the number of components $[N_T:N_T^+]=[K:K^+][J:J^+]$ using Miranda--Morrison theory. \end{remark}
\subsection{Saturated effective subgroups}\label{HpolEnum} By \Cref{deformation class vs F_H}, the set of connected components of $\mathcal{F}_H$ is in bijection with the deformation classes of $H$-markable K3 surfaces. Our next goal is to enumerate all possible effective groups $H$ up to conjugacy in $O(L)$. The symplectic fixed and cofixed lattices $L^{H_s}$ and $L_{H_s}$ turn out to be the crucial invariants for this task. Let $L$ be a $\mathbb{Z}$-lattice and $M$ a subset of $L \otimes \mathbb{C}$. We denote by $O(L,M)=\{f \in O(L) \mid f_\mathbb{C}(M)=M\}$ the stabilizer of $M$.
\begin{definition}
Let $L$ be a K3 lattice and $H \leq O(L)$ an effective subgroup. Then we call the kernel $S$ of $O(L,L^{H_s}) \to O(L^{H_s})$ the saturation of $H_s$.
The group generated by $H$ and $S$ is called the saturation of $H$.
For a K3 surface $X$ we call a finite subgroup $G\leq \Aut(X)$ saturated if its image $\rho_X(G_s)$ is saturated.
We call a saturated symplectic group $H_s\leq O(L)$ a \emph{heart}. \end{definition}
\begin{remark} Note that the saturation $S$ of $H_s$ is the largest subgroup $S\leq O(L)$ with $L^S=L^{H_s}$. Further the saturation of an effective group $H\leq O(L)$ is effective. Indeed, if $L = \HH^2(X,\mathbb{Z})$ for some K3 surface $X$ and $G_s$ a finite group of symplectic automorphisms, then the strong Torelli theorem implies that the saturation of $\rho_X(G_s)=H_s$ is in the image of $\rho_X$ by a finite group of symplectic automorphisms containing $G_s$ and with the same fixed lattice. We conclude that if a pair $(X,G)$ is markable by $H$ then it is also markable by the saturation of $H$. Therefore it is enough to enumerate the saturated effective subgroups of $O(L)$. \end{remark}
\begin{remark}\label{rem:hashimoto} For the symplectic groups, the saturated subgroups are known: By a theorem of Hashimoto \cite{Hashimoto2012}, there are exactly $44$ conjugacy classes of effective, saturated subgroups $H_s \leq O(L)$. They are determined by the isometry classes of their fixed and cofixed lattices. The fixed lattices are listed by Hashimoto while the cofixed lattices can be obtained from the permutation representation of the Mathieu group $M_{24}$ on the type $24 A_1$ Niemeier lattice. Alternatively, one may obtain them from isometries of the Leech lattice and the tables enumerated in \cite{mason2016}. \end{remark}
\subsection{Enumerating effective characters} Let $H_s \leq O(L)$ be a symplectic effective subgroup. We would like to enumerate conjugacy classes of effective characters $\chi \colon H \to \mathbb{C}^\times$ with the given heart $H_s$.
\begin{definition}
Let $\chi\colon H \to \mathbb{C}^\times$ be an effective character and $n=[H:H_s]$. The distinguished generator of $H/H_s$ is $g H_s$ with $\chi(g) = \zeta_n:= \exp(2\pi i /n)$. Set $F = L^{H_s}$ and $C= L_{H_s}$. The distinguished generator $gH_s$ restricts to an isometry $f=g|_F$ of $F$ of order $n$. We call the lattice with isometry $(F,f)$ the \emph{head} of $H$ and $H_s$ its \emph{heart}. The \emph{spine} of $\chi$ is the glue map $\phi\colon \disc{F} \to \disc{C}$ with $L_\phi = L$. \end{definition}
Our next goal is to see how heart, head and spine determine the character. The first step is to make the definition of heart and head independent of a character.
\begin{definition}\label{heads and hearts}
Let $H_s \leq O(L)$ be a heart, $F=L^{H_s}$ its fixed lattice and
$f \in O(F)$ of order $n$. We call the lattice with isometry $(F,f)$ a head (of $H_s$) if the following hold:
\begin{enumerate}
\item $\ker(f+f^{-1} - \zeta_n - \bar \zeta_n)$ is of signature $(2,*)$,
\item $(\ker \Phi_n(f)\Phi_1(f))^\perp$ does not contain any vector of square $(-2)$.
\end{enumerate}
\end{definition} By abuse of notation we will identify $O(C)$ with $O(C)\times \{\id_F\} \subseteq O(L\otimes \mathbb{Q})$. Recall that for a glue map $\phi \colon \disc{F} \to \disc{C}$, we denote by $F \perp C \subseteq L_\phi$ the corresponding primitive extension. Note that $L_\phi\cong L$ is a K3 lattice as well and $H_s$ preserves $L_\phi$ because all its elements act trivially on the discriminant group of $C$. Suppose that $\phi D_f \phi^{-1} = D_c$ for some $c \in O(C)$. Then $g = f \oplus c$ preserves $L_\phi$. Set $H_\phi = \langle g , H_s \rangle$. Since $H_s$ is saturated, any other choice of $c$ is in $cH_s$, so $H_\phi$ is independent of this choice. Let $\chi_\phi\colon H_\phi \to \mathbb{C}^\times$ be defined by $\chi(g)=\zeta_n$.
\begin{definition}\label{spines} Let $H_s$ be a heart and $(F,f)$ a head. A glue map $\phi \colon \disc{F} \to \disc{C}$ is called a spine if \begin{enumerate}
\item $\phi \circ D_f \circ \phi^{-1}$ is in the image of $O(C) \to O(\disc{C})$ and
\item $\NS(\chi_\phi)_{H_\phi}$ does not contain any vector of square $(-2)$. \end{enumerate} \end{definition}
\begin{proposition}\label{effectivespine}
Let $H_s$ be a heart, $(F,f)$ a head and $\phi\colon \disc{F} \to \disc{C}$ a spine.
Then the corresponding character $\chi_\phi \colon H_\phi \to \mathbb{C}^\times$ is effective. \end{proposition} \begin{proof} This follows immediately from \Cref{iseffective} and the definitions. \end{proof} \begin{definition} Let $i=1,2$, $H_i \leq O(L_i)$ two effective subgroups and $\chi_i\colon H_i \to \mathbb{C}^\times$ two effective characters. We say that $\chi_1$ is isomorphic to $\chi_2$ if and only if there is an isometry $\psi \colon L_1 \to L_2$ with $H_2 = \psi H_1 \psi^{-1}$ and $\chi_1(h) = \chi_2(\psi \circ h \circ \psi^{-1})$ for all $h \in H_1$. \end{definition} If $L_1=L_2$, then $\chi_1$ and $\chi_2$ are isomorphic if and only if they are conjugate. If moreover $H_1=H_2$ then they are isomorphic if and only if they are conjugate by an element of $N(H)$. Recall that $U(F,f)$ denotes the centralizer of $f$ in $O(F)$. \begin{theorem}\label{fiber} Let $H_s$ be a heart, $(F,f)$ a head of $H_s$ and $S$ the set of spines $\phi \colon \disc{F} \to \disc{C}$. Then the double coset \[O(C) \backslash S / U(F,f)\] is in bijection with the set of isomorphism classes of effective characters $\chi$ with the given heart and head. \end{theorem} \begin{proof} By \Cref{effectivespine} any spine $\phi \in S$ determines an effective character $\chi_\phi\colon H_\phi \to \mathbb{C}^\times$ with $L_\phi$ a K3 lattice. Conversely any effective character with the given heart and head arises in this fashion.
Let $\phi$ and $\phi'$ be two spines. We have to show that $\chi_\phi$ and $\chi_{\phi'}$ are conjugate if and only if $\phi' \in O(C) \phi U(F,f)$.
Suppose $\phi' = D_a \phi D_b$ with $a \in O(C)$ and $b \in U(F,f)$. Then $a \oplus b\colon L_\phi \to L_{\phi'}$ gives the desired isomorphism of the characters.
Conversely let $\psi\colon L_\phi \to L_{\phi'}$ be an isomorphism of $\chi$ and $\chi'$. By construction $V:=L_\phi \otimes \mathbb{Q}=L_{\phi'}\otimes \mathbb{Q}$. We may view $\psi$ as an element of $O(V)$. Note that $F \perp C \subseteq V$. Since $\psi$ preserves the common heart $H_s$ and head $(F,f)$ of $\chi_1$ and $\chi_2$, we can write it as $\psi = a \oplus b$ with $a \in O(C)$ and $b \in U(F,f)$. Then $ D_a \circ \phi \circ D_b = \phi'$. \end{proof}
\begin{remark} The previous results yield the following procedure for determining a transversal of the set of isomorphism classes of effective characters.
\begin{enumerate}
\item
Let $\mathcal{H}$ be the set of possible hearts up to conjugacy, which have been determined by Hashimoto \cite{Hashimoto2012} (see \Cref{rem:hashimoto}).
\item
For each $H_s \in \mathcal{H}$, determine a transversal of the isomorphism classes of the heads $(F, f)$, which amounts to classifying conjugacy classes of isometries of finite order $n$ of a given $\mathbb{Z}$-lattice.
This is explained in \Cref{ConjAlg}, see in particular \Cref{rem:latticeK3} for the possible values of $n$.
\item
For each heart $H_s \in \mathcal{H}$ and possible head $(L, f)$ apply \Cref{fiber} to determine
a transversal of the double coset of spines and therefore a transversal of the isomorphism classes of the effective characters with the given heart and head.
\end{enumerate}
Altogether we obtain a transversal of the isomorphism classes of effective characters. Each is represented by some K3 lattice $L$ depending on the character, a finite subgroup $H \leq O(L)$, a normal subgroup $H_s \leq H$ and a distinguished generator of $H/H_s$. \end{remark}
\section{Conjugacy classes of isometries}\label{ConjAlg} We have seen that to classify finite subgroups of automorphisms of K3 surfaces up to deformation equivalence we need to classify conjugacy classes of isometries of finite order of a given $\mathbb{Z}$-lattice. So given a polynomial $\mu(x) \in \mathbb{Z}[x]$ and a $\mathbb{Z}$-lattice $L$ we seek to classify all conjugacy classes of isometries of $L$ with the given characteristic polynomial $p$. By general results of Grunewald and Segal \cite[Cor. 1]{grunewald:conjugacy} from the theory of arithmetic groups, we know that the number of such conjugacy classes is finite and (in theory) computable. Necessary and sufficient conditions for the existence of an isometry of some unimodular lattice $L$ with a given characteristic polynomial have been worked out by Bayer-Fluckiger and Taelman in \cite{fluckiger20,fluckiger21,fluckiger21b}.
\subsection{Hermitian lattices and transfer}\label{transfer} In this subsection $E$ is an \'etale $\mathbb{Q}$-algebra with a $\mathbb{Q}$-linear involution $\overline{\phantom{x}}\colon E \to E$.
\begin{definition} Let $V$ be an $E$-module. A hermitian form on $V$ is a sesquilinear form $h \colon V \times V \to E$ with $h(x,y) = \overline{h(y,x)}$. We call $(V,h)$ a hermitian space over $E$. For a $\mathbb{Z}$-order $\Lambda \subseteq E$ we call a $\Lambda$-module $L \subseteq V$ a hermitian $\Lambda$-lattice. All hermitian forms and lattices are assumed non-degenerate. \end{definition} Let $\Tr \colon E \to \mathbb{Q}$ be the trace. Note that $E$ being \'etale is equivalent to the trace form $E\times E \to \mathbb{Q}, (x,y) \mapsto \Tr(xy)$ being non-degenerate.
\begin{definition}[Transfer] A $\mathbb{Q}$-bilinear form $b \colon V \times V \to \mathbb{Q}$ on an $E$-module $V$ is said to be an $E$-bilinear form if $b(ev,w) = b(v,\bar e w)$ for all $v,w \in V$ and $e \in E$. Let $(V,h)$ be a hermitian space over $E$. Then $b = \Tr \circ h$ is an $E$-bilinear form. It is called the trace form of $h$. \end{definition}
\begin{proposition}[\cite{milnor1969}] Every $E$-bilinear form on $V$ is the trace form of a unique hermitian form on $V$. \end{proposition}
Note that an $E$-linear map preserves a hermitian form if and only if it preserves the respective trace form. In view of these facts we may work with $E$-bilinear forms and hermitian forms over $E$ interchangeably.
Given $x^n-1=\mu(x)\in \mathbb{Z}[x]$ and a lattice $L$ we seek to classify the conjugacy classes of isometries $f\in O(L)$ satisfying $\mu(f) = 0$. To this aim we put $E:=\mathbb{Q}[x,x^{-1}]/(\mu)\cong \mathbb{Q}[x]/(\mu)$. This is an \'etale algebra and $x \mapsto x^{-1}$ defines an involution $\overline{\phantom{x}} $ on $E$. For $i \in I =: \{i \in \mathbb{N} \mid i \text{ divides } n\}$ set $E_i=\mathbb{Q}[x]/(\Phi_i(x))$. The algebra $E$ splits as a direct product $E= \prod_{i \in I} E_i$ of cyclotomic fields with the induced involution. Let $(e_i)_{i \in I}$ be the corresponding system of primitive idempotents in $E$, such that $\overline{e_i} = e_i$ and $E_i = E e_i$.
Let $\Lambda = \mathbb{Z}[x]/(\mu)$ and $\Gamma = \prod_{i \in I} \Lambda e_i$. The conductor of $\Gamma$ in $\Lambda$ is \[\mathfrak{f}_\Gamma= \{x \in \Lambda \mid \Gamma x \subseteq \Lambda\}.\] It is the largest $\Gamma$-ideal contained in $\Lambda$. We obtain the following series of inclusions:
\begin{equation}\label{eqn:sandwich_ring}
\mathfrak{f}_{\Gamma} \subseteq \Lambda \subseteq \Gamma = \prod_{i\in I} \Lambda e_i
\end{equation}
\begin{lemma}
The conductor satisfies $\mathfrak{f}_\Gamma = \bigoplus_{i \in I} (\Lambda \cap e_i \Lambda)$. \end{lemma}
\begin{proof}
For $x \in \Lambda$ we have $x = 1 x = \sum_i e_i x$ and so $x \Gamma = \bigoplus_{i \in I} \Lambda e_i x$ is contained in $\Lambda$ if and only if $e_i x \in \Lambda$ for all $i\in I$. This means that $e_i x \in \Lambda \cap e_i \Lambda$. \end{proof}
\begin{example}\label{ex:conductor}
Let $p \in \mathbb{Z}$ be a prime. For $\mu(x) = x^p-1 = (x-1)\Phi_p(x)$
we obtain
\[E = \mathbb{Q}[x] / (x^p-1) \cong \mathbb{Q}[x]/(x-1) \times \mathbb{Q}[x]/\Phi_p(x) \cong \mathbb{Q} \times \mathbb{Q}[\zeta_p].\]
Set $g(x) = \sum_{i=0}^{p-2}(i+1-p)x^i$.
One finds that $p = (x-1)g(x) + \Phi_p(x)$.
Hence $e_1=\Phi_p(x)/p$ and $e_p = (x-1)g(x)/p$.
The conductor ideal is $\mathfrak{f}_\Gamma=p e_1 \Lambda + p e_p \Lambda$. It contains $p$. \end{example} Let $b$ be the bilinear form of the lattice $L$. A given isometry $f\in O(L,b)$ with minimal polynomial $\mu$ turns $(L,b)$ into a hermitian $\Lambda$-lattice $(L,h)$ by letting the class of $x$ act as $f$. Note that for $x,y \in L\otimes E$ we have \[h(e_i x, e_j x) = e_i \bar e_j h(x,y) = e_i e_j h(x,y) = \delta_{ij} h(x,y).\] Thus $e_i L$ is orthogonal to $e_j L$ for $i\neq j$. \Cref{eqn:sandwich_ring} yields the corresponding chain of finite index inclusions \begin{equation}\label{eqn:sandwich} \mathfrak{f}_{\Gamma}L \subseteq L \subseteq \Gamma L. \end{equation}
Setting $L_i = e_i \mathfrak{f}_{\Gamma} L = L \cap e_i L = \ker \Phi_i(f)$ and $L_i' = e_i \Gamma L$ the outermost lattices are \[(\mathfrak{f}_{\Gamma}L,h) =\bigperp_{i\in I}(L_i,h_i)\quad \text{ and } \quad (\Gamma L,h) = \bigperp_{i\in I} (L_i',h_i).\] Since $\Lambda e_i=\mathbb{Z}[x]/\Phi_i(x) = \mathbb{Z}_{E_i}$ is the maximal order in $E_i$, $L_i$ and $L_i'$ are hermitian lattices over the ring of integers of a number field. Such lattices are well understood. We use the outermost lattices of the sandwich to study the $\Lambda$-lattice $(L,h)$. \begin{example}\label{ex:porder}
Let $L$ be a $\mathbb{Z}$-lattice and $f\in O(L)$ an isometry of prime order $p$.
Then $L_1 = \ker \Phi_1(f)$ and $L_p = \ker \Phi_p(f)$. Since, by \Cref{ex:conductor}, $p \in \mathfrak{f}_\Gamma$ we have
\begin{equation}\label{index-p} p L \subseteq \mathfrak{f}_\Gamma L= L_1 \perp L_p \subseteq L
\end{equation} \end{example}
The idea for the classification is as follows: Given $\mu(x)$ and the $\mathbb{Z}$-lattice $(L,b)$, we get restrictions on the possible genera of the lattices $(L_i,h_i)$ from $(L,b)$ and the conductor. We take $L$ as an overlattice of the orthogonal direct sum $\bigperp_{i\in I} (L_i,h_i)$ up to the action of the product of unitary groups $\prod_{i \in I}U(L_i,h_i)$. Thanks to \cref{eqn:sandwich} this is a finite problem. In practice, we successively take equivariant primitive extensions.
\subsection{Glue estimates}\label{glue_estim} The caveat of dealing with primitive extensions $A \perp B \subseteq C$ is that we do not know how to predict the genus of $C$. Or more precisely, how to enumerate all glue maps such that $C$ lies in a given genus. So we have to resort to check this in line \ref{keepC?} of \Cref{alg:extensions} only after constructing $C$. To reduce the number of glue maps that have to be checked, in this section we prove various necessary conditions.
\begin{proposition}\label{glue:det-estimate} Let $C$ be an integral $\mathbb{Z}$-lattice and $f \in O(C)$ an isometry of prime order $p$ with characteristic polynomial $\Phi_1^{e_1}\Phi_p^{e_p}$. Set $A= \ker \Phi_1(f)$, $B = \ker \Phi_p(f)$ and $m = \min\{e_1,e_p,l(D_A),l(D_B)\}$. Then $pC \subseteq A \perp B$ and $[C: A\perp B] \mid p^m$. \end{proposition}
\begin{proof} By \Cref{index-p} $p C \subseteq A \perp B$. Let $D_A \geq H_A \cong C/(A\perp B) =H\cong H_B \leq D_B$ be the glue between $A$ and $B$. Note that these are isomorphisms as $\mathbb{Z}[x]$-modules and $p H =0$ by \Cref{ex:porder}.
The polynomial $\Phi_p$ annihilates $B$; hence it annihilates $B^\vee$ and $H_B \leq D_B=B^\vee/B$.
Let $B' \leq p^{-1}B$ be defined by $B'/B = H_B$.
The $\mathbb{Z}[\zeta_p]$-module $B'$ is a finitely generated torsion-free module of rank $e_p$.
By the invariant factor theorem over Dedekind domains \cite[(22.13)]{curtis2006} any torsion quotient module of $B'$ is generated by at most $e_p$ elements. On the other hand, since $H_B \leq D_B$, it is generated by at most $l(D_B)$ elements as a $\mathbb{Z}$-module, in particular as $\mathbb{Z}[x]$-module.
Similarly $H_A$ is annihilated by $\Phi_1$ and generated by at most $\min\{e_1,l(D_A)\}$ elements. Since the glue map is equivariant, the minimal number $n$ of generators of $H$ as $\mathbb{Z}[x]$-module satisfies $n \leq m$. As $H=C/(A\perp B)$, viewed as a $\mathbb{Z}[\zeta_p]$-module, is annihilated by the prime ideal $P$ generated by $\Phi_1(\zeta_p)$, we have $H\cong (\mathbb{Z}[\zeta_p]/P)^n$. Since $p$ is totally ramified in $\mathbb{Z}[\zeta_p]$, the ideal $P$ has norm $p$ and thus
\[[C:A\perp B]=|\mathbb{Z}[\zeta_p]/P|^n = p^n \mid p^m. \qedhere\] \end{proof}
We call a torsion bilinear form $b \colon A \times A \to \mathbb{Q}_2/\mathbb{Z}_2$ \emph{even }if $b(x,x)=0$ for all $x \in A$, otherwise we call it \emph{odd}. Imitating \cite[II \S 2]{miranda-morrison}, we define the functors $\rho_k$.
\begin{definition}
Let $N$ be an integral lattice over $\mathbb{Z}_p$. Set $G_k = G_k(N) = p^{-k}N \cap N^\vee$ and define
$\rho_k(N) = G_k/(G_{k-1}+pG_{k+1})$. It is equipped with the non-degenerate torsion bilinear form $b_k(\bar x,\bar y) = p^{k-1}xy \bmod \mathbb{Z}_p$.
If $p=2$ and both $\rho_{k-1}(N)$ and $\rho_{k+1}(N)$ are even, then we call $\rho_{k}(N)$ \emph{free}. Otherwise we call it \emph{bound}. If it is free, $\rho_{k}(N)$ carries the torsion quadratic form $q_k(\bar x) = 2^{k-1}x^2 \bmod 2\mathbb{Z}_2$. \end{definition} Let $L=\bigperp_{j=0}^l (L_j,p^jf_{j})$ be a Jordan decomposition with $f_{j}$ a unimodular bilinear form. Then one checks that $\rho_j(L) \cong (L_j/pL_j, \bar f_i)$ where $\bar f_i$ is the composition of $p^{-1}f_i$ and the natural map $\mathbb{Q}_p \to \mathbb{Q}_p/\mathbb{Z}_p$.
\begin{remark} Note that $\bar f_i$ determines the rank of $f_i$, its parity and its determinant modulo $p$. Thus, if $p$ is odd, it determines $f_j$ up to isometry. For $p=2$ this is not the case. \end{remark}
Let $N$ be an integral lattice over $\mathbb{Z}_p$ and $l \in \mathbb{Z}$ such that $p^{l+1} N^\vee \subseteq N$. Then $G_{l+1}(N)=G_{l+2}(N) = N^\vee$ and $G_{l}(N)=p^{-l}N \cap N^\vee$. Using $pN^\vee \subseteq p^{-l}N$ we obtain \[\rho_{l+1}(N)
= N^\vee /(p^{-l}N \cap N^\vee). \] Multiplication by $p^l$ gives the isomorphism \[\rho_{l+1}(N) \cong p^l N^\vee/ (N \cap p^l N^\vee) \cong p^lD_N\]
\begin{proposition}\label{glue:estimate-level}
Let $N_1 \perp N_2 \subseteq L$ be a primitive extension of $\mathbb{Z}_p$-lattices with corresponding glue map
$D_1 \supseteq H_1 \xrightarrow{\phi} H_2 \subseteq D_2$ where $D_i=N_i^\vee/N_i$ is the discriminant group of $N_i$, $i\in \{1,2\}$. Suppose that $p L \subseteq N_1 \perp N_2$.
Then $p^l L^\vee \subseteq L$ if and only if the following four conditions are met.
\begin{enumerate}
\item $p^{l+1} D_i = 0$, i.e. $p^{l+1}N_i^\vee\subseteq N_i$,
\item $p^l D_i \subseteq H_i$,
\item $\phi(p^l D_1) = p^l D_2$,
\item $\hat \phi \colon \rho_{l+1}(N_1)\cong p^lD_1 \to p^lD_2 \cong \rho_{l+1}(N_i)$ is an anti-isometry with respect to the bilinear forms $b_{l+1}$.
\end{enumerate} If moreover both $\rho_{l+1}(N_i)$ are free, then $\rho_l(L)$ is even if and only if
\begin{enumerate} \item [(4')] $\hat\phi$ is an anti-isometry with respect to the quadratic forms $q_{l+1}$.
\end{enumerate}
\end{proposition}
\begin{proof} Suppose that $p^l L^\vee \subseteq L$. We prove (1-4).
Let $i \in\{1,2\}$. By the assumptions \[p^{l+1} L^\vee \subseteq p L \subseteq N_1 \perp N_2.\] Since the extension is primitive, the orthogonal projection $\pi_i \colon L^\vee \rightarrow N_i^\vee$ is surjective. Applying $\pi_i$ to the chain of inclusions yields \[p^{l+1} N_i^\vee \subseteq p \pi_i(L) \subseteq N_i\] which proves (1).
For (2) consider the inclusion $p^lL^\vee \subseteq L$. A projection yields $p^l N_i^\vee \subseteq \pi_i(L)$. Now (2) follows with $D_i = N_i^\vee / N_i$ and $H_i = \pi_i(L)/N_i$.
(3) We have $\pi_i(p^l L^\vee) = p^l N_i^\vee$. Recall that the glue map $\phi$ is defined by its graph $L/(N_1 \perp N_2)$. Its subset \[p^l L^\vee / ((N_1 \perp N_2) \cap p^lL^\vee)\] projects onto both $p^l D_1$ and $p^l D_2$. This proves the claim.
(4) Let $i\in{1,2}$, $x_i \in N_1^\vee$ and $y_i \in N_2^\vee$ with $\hat \phi(\bar x_i) = \bar y_i$, i.e. $\phi(p^l x_i + N_1) = p^l y_i + N_2$, i.e. \[p^l(x_i+y_i) \in L.\] In fact, from the proof of (3), we know a little more, namely that $p^l(x_i+y_i) \in p^l L^\vee$, so that $x_i+y_i \in L^\vee$. This implies that $\sprodb{p^l(x_1+y_1)}{x_2+y_2} \in \mathbb{Z}_p$ which results in \[b_{l+1}(\bar x_1, \bar x_2) \equiv p^{l}\sprodb{x_1}{x_2} \equiv - p^{l}\sprodb{y_1}{y_2} \equiv - b_{l+1}(\bar y_1, \bar y_2) \mod \mathbb{Z}_p.\]
(4') Suppose furthermore that both $\rho_{l+1}(N_i)$ are free and that $\rho_l(L)$ is even. Take $x=x_1=x_2 \in N_1^\vee$ and $y=y_1=y_2 \in N_2^\vee$. Then $2^{l-1}\sprodbq{x+y}{} \in \mathbb{Z}_2$ since $\rho_{l}(L)$ is even. Therefore $2^{l}\sprodbq{x}{} \equiv 2^l\sprodbq{y}{} \mod 2 \mathbb{Z}$.
Now suppose that (1--4) hold for the glue map $\phi$. Let $x + y \in L^\vee$. We have to show that $p^l(x+y) \in L$.
Let $w \in N_1^\vee$, and $z \in N_2^\vee$ with $\hat \phi( \bar w )=\bar z$. By the definition of $\hat \phi$ and $\phi$, this implies that $p^l(w+z) \in L$. Therefore \begin{eqnarray*} b_{l+1}(\bar y - \hat \phi(\bar x),\bar z) &= &b_{l+1}( \bar y, \bar z) - b_{l+1}(\hat \phi (\bar x), \hat \phi(\bar w))\\ &=&b_{l+1}( \bar y, \bar z) + b_{l+1}(\bar x, \bar w)\\ &\equiv& p^l\sprodb{y}{ z} + p^{l}\sprodb{x}{w} \\ &\equiv & \sprodb{x+y}{p^l(w+z)}\\ &\equiv& 0 \mod \mathbb{Z}_p. \end{eqnarray*} Since the bilinear form on $\rho_l(N_2)$ is non-degenerate, this shows that $\hat \phi(\bar x)=\bar y$. By the definition of a glue map we obtain $p^l(x+y) \in L$.
Suppose furthermore that (4') holds, so $p=2$, $\rho_{l+1}(N_i)$ is free and $\hat \phi$ preserves the induced quadratic forms. Let $x+y \in L^\vee$ we have to show that $q_l(\overline{x+y})\equiv 0 \mod \mathbb{Z}$. Indeed, \begin{eqnarray*} 2q_l(\overline{x+y}) &=& 2^{l} \sprodbq{x+y}{}\\ &\equiv& q_{l+1}(\bar x)+q_{l+1}(\bar y)\\ &\equiv& q_{l+1}(\bar x)+q_{l+1}(\hat \phi(\bar x)) \\ &\equiv& 0 \mod 2 \mathbb{Z}. \qedhere \end{eqnarray*} \end{proof}
\begin{definition}\label{def:admissible-glue-map}
We call a glue map \emph{admissible} if it satisfies \Cref{glue:estimate-level} (1--3) and (4) respectively (4'). \end{definition}
\begin{example}
In the special case that $l=0$ we recover the result that the discriminant bilinear forms of $N_1$ and $N_2$ are anti-isometric.
And if further $L$ is even, that the discriminant quadratic forms are anti-isometric. \end{example}
\begin{definition}
Let $p$ be a prime number and $A,B,C$ be integral $\mathbb{Z}$-lattices. Let $p^{-l}\mathbb{Z} =\scale(C^\vee)$.
We say that the triple $(A,B,C)$ is
\emph{$p$-admissible} if the following hold:
\begin{enumerate}\setlength{\itemsep3pt}
\item $(A \perp B) \otimes \mathbb{Z}_q \cong C \otimes \mathbb{Z}_q$ for all primes $q\neq p$,
\item $\det A\cdot \det B=p^{2g}\det C$ where $g \leq l(D_A),(\rk B)/(p-1),l(\disc{B})$,
\item $\scale(A \perp B)\subseteq \scale(C)$ and $p \scale(A^\vee \perp B^\vee) \subseteq \scale(C^\vee)$
\item $\rho_{l+1}(A\otimes \mathbb{Z}_p)$ and $\rho_{l+1}(B\otimes \mathbb{Z}_p)$ are anti-isometric as torsion bilinear modules. If further $p=2$, both are free and $\rho_l(C\otimes \mathbb{Z}_2)$ is even, then they are anti-isometric as torsion quadratic modules,
\item there exist embeddings $pC\otimes \mathbb{Z}_p \hookrightarrow (A \perp B) \otimes \mathbb{Z}_p \hookrightarrow C\otimes \mathbb{Z}_p$,
\item $\dim \rho_{l+1}(A\otimes \mathbb{Z}_p)\oplus \rho_{l+1}(B \otimes \mathbb{Z}_p) \leq \dim \rho_l(C \otimes \mathbb{Z}_p)$.
\end{enumerate} \end{definition}
Note that $(C,0,C)$ and $(0,C,C)$ are $p$-admissible for all $p$. We call a triple $(\mathcal{A},\mathcal{B},\mathcal{C})$ of genera of $\mathbb{Z}$-lattices $p$-admissible if for any representatives $A$ of $\mathcal{A}$, $B$ of $\mathcal{B}$ and $C$ of $\mathcal{C}$ the triple $(A,B,C)$ is $p$-admissible.
\begin{remark}
For the existence of the (not necessarily primitive!) embeddings in (6) there is a necessary and sufficient criterion found in \cite[Theorem 3]{omeara1958}. Note that condition (V) in said theorem is wrong.
The correct condition is
\[(\mathrm{V}) \qquad 2^i (1+4 \omega) \to (2^i \oplus \mathfrak{L}_{i+1})/ \mathfrak{l}_{[i]}.\]
Thus being $p$-admissible is a condition that can be checked easily algorithmically. \end{remark}
\begin{lemma}\label{ispadmissible}
Let $C$ be a $\mathbb{Z}$-lattice and $f \in O(C)$ an isometry of order $p$.
Let $A = \ker \Phi_1(f)$ and $B = \ker \Phi_p(f)$.
Then $(A,B,C)$ is $p$-admissible. \end{lemma} \begin{proof}
Let $p^{-l}\mathbb{Z} = \scale(C^\vee)$.
First note that \Cref{glue:estimate-level} is applicable to $L = C$, since $p^{l}C^\vee \subseteq C$ by the definition of $l$.
(1) From \Cref{glue:det-estimate} we obtain $pC \subseteq A\perp B \subseteq C$. After tensoring with $\mathbb{Z}_q$ for $q\neq p$ we obtain (since $p$ is a unit in $\mathbb{Z}_q$) that $C \otimes \mathbb{Z}_q = (A\perp B)\otimes \mathbb{Z}_q$.
(2) This is \Cref{glue:det-estimate}.
(3) $A \perp B \subseteq C$ gives $\scale(A\perp B) \subseteq \scale(C)$. Dualizing $pC \subseteq A\perp B$ yields $p (A^\vee \perp B^\vee) \subseteq C^\vee$. Now take the scales.
(4) This is \Cref{glue:estimate-level}~(4) and~(4').
(5) We know that $pC \subseteq A\perp B \subseteq C$.
(6) Let $x \in A^\vee$ and $y \in B^\vee$ with $\hat \phi(\bar x) = \bar y$, i.e. $p^l(x+y)\in C$. By the proof of \Cref{glue:estimate-level} (4), $x+y \in C^\vee$. Suppose that $x+y$ is zero in $\rho_l(C)$, i.e. $x+y \in p^{-l+1}C \cap C^\vee$. Then $p^{l-1} (x+y) \in C\subseteq p^{-1}(A\perp B)$, therefore $x \in p^{-l} A$. Thus $\bar x = 0$ in $\rho_{l+1}(A)$. Similarly $\bar y = 0$ in $\rho_{l+1}(B)$. This shows that the graph $\Gamma$ of $\hat \phi$ injects naturally into $\rho_l(C)$. Note that $p C \subseteq A^\vee \perp B^\vee$ gives $p A^\vee \subseteq C^\vee$. Suppose that $\bar x \neq 0$. Since $b_{l+1}$ is non-degenerate, we find $a \in A^\vee$ with \[1/p = b_{l+1}(\bar x, \bar a)\equiv p^l \sprodb{x}{a} = p^{l-1}\sprodb{x}{pa} \equiv b_{l}(\overline{x+y},p a) \mod \mathbb{Z}_p.\] This shows that the span of $p A^\vee$ and $\Gamma$ in $\rho_l(C)=C^\vee/(p^{-l}C + C^\vee)$ is a non-degenerate subspace of dimension $2 \dim \rho_{l+1}(A)$. \end{proof}
\begin{definition}
Let $L$ be a $\mathbb{Z}$-lattice with $p^{l+1} L^\vee \subseteq L$ and let $H\leq \disc{L}$ with $p^{l} \disc{L} \leq H$. We denote by
$O(H,\rho_l(L))$ the set of isometries $g$ of $H$ which preserve $p^l\disc{L}$ and such that the map $\hat g$ induced by $g$ on $\rho_L(L)$ preserves the torsion bilinear (respectively quadratic) form of $\rho_l(L)$. \end{definition}
\begingroup \captionof{algorithm}{AdmissibleTriples}\label{alg:admissible} \endgroup \begin{algorithmic}[1] \REQUIRE A prime $p$ and a $\mathbb{Z}$-lattice $C$. \ENSURE All tuples $(\mathcal{A} ,\mathcal{B})$ of genera of $\mathbb{Z}$-lattices such that $(\mathcal{A},\mathcal{B},\mathcal{C})$ is $p$-admissible with $\mathcal{C}$ the genus of $C$ and $\rk \mathcal{B}$ divisible by $p-1$. \STATE $n \gets \rk C$ \STATE $d \gets \det(C)$ \STATE Initialize the empty list $L = [\;]$. \FOR{ $e_p\in \{r \in \mathbb{Z} \mid 0 \leq r \leq n/(p-1)\}$}
\STATE $r_p \gets (p-1)e_p$
\STATE $r_1 \gets n - r_p$
\STATE $m \gets \min\{e_p,r_1\}$
\STATE Form the set
\[D = \left\{ (d_1,d_p) \in \mathbb{N}^2 \;\middle|\; \exists g \mid\gcd\left(d_1,d_p,p^m\right): dg^2=d_1 d_p\right\}.\]
\FOR{$(d_1,d_p) \in D$}
\STATE Form the set $\mathcal{L}_1$ consisting of all genera of $\mathbb{Z}$-lattices $A$ with
\[\rk A = r_1, \quad \det A = d_1,\quad \scale(A) \subseteq \scale(C),\quad \scale(A^\vee)\subseteq p\scale(A^\vee).\]
\\[-10pt]
\STATE Form the set $\mathcal{L}_p$ consisting of all genera of $\mathbb{Z}$-lattices $B$ with
\[\rk B = r_p, \quad \det B = d_p,\quad \scale(B) \subseteq \scale(C),\quad \scale(B^\vee)\subseteq p\scale(C^\vee).\]
\\[-10pt]
\FOR{$(\mathcal{A},\mathcal{B}) \in \mathcal{L}_1 \times \mathcal{L}_p$}
\IF {$(\mathcal{A},\mathcal{B},\mathcal{C})$ is $p$-admissible}
\STATE Append $(\mathcal{A},\mathcal{B})$ to $L$.
\ENDIF
\ENDFOR
\ENDFOR \ENDFOR \RETURN L \end{algorithmic}\leavevmode\leaders\hrule height 0.8pt
\kern\z@\\
\begin{remark}
Genera of $\mathbb{Z}$-lattices can be described by the Conway--Sloane genus symbol \cite[15 \S 7]{splag}. We have implemented an enumeration of all such genus symbols with a given signature and bounds on the scales of the Jordan components in \textsc{SageMath}~\cite{sagemath} and \textsc{Hecke}/\textsc{Oscar}~\cite{hecke}. \end{remark}
\subsection{Enumeration of conjugacy classes of isometries.}\label{enumconj} Let $p\neq q$ be prime numbers. In this subsection we give an algorithm which, for a given genus of $\mathbb{Z}$-lattices $\mathcal{L}$, computes a complete set of representatives for the isomorphism classes of lattices with isometry $(L,f)$ of order $p^i q^j$ such that $L$ is in $\mathcal{L}$.
Let $(L,f)$ be a lattice with isometry. As before, we will drop $f$ from the notation and simply denote it by $L$ and the corresponding isometry by $f_L$. If $N \leq L$ is an $f$-invariant sublattice we view it as a lattice with isometry $f_N = f|_N$.
The data structure we use for lattices with isometry is a triple $(L,f_L,G_L)$, where $G_L$ is the image of $U(L) \to O(\disc{L})$ and $U(L)$ denotes the centralizer of $f_L$ in $O(L)$.
By abuse of terminology we call such a triple a lattice with isometry as well. So every algorithm in this section which returns lattices with isometry actually returns such triples $(L,f_L,G_L)$ (or at least a function which is able to compute $G_L$ when needed). We omit $f_L$ and $G_L$ from notation and denote the triple simply by $L$.
\begin{definition}
Let $A$ be a lattice with an isometry of finite order $m$.
For a divisor $l$ of $m$ denote by $H_l$ the sublattice $\ker \Phi_{l}(f_A)$ viewed as a hermitian $\mathbb{Z}[\zeta_l]$-lattice with $\zeta_l$ acting as $f_A|H_l$. Denote by $\mathcal{H}_l$ its genus as hermitian lattice.
For a divisor $l \mid m$ let $A_l = \ker (f_A^l-1)$ and denote by $\mathcal{A}_l$ its genus as $\mathbb{Z}$-lattice. The \emph{type} of $A$ is the collection $(\mathcal{A}_l,\mathcal{H}_l)_{l \mid m}$ and will be denoted by $t(A) = t(A, f_A)$. \end{definition}
Since we can encode a genus in terms of its symbol and can check for equivalence of two given symbols efficiently, the type is an effectively computable invariant.
\begingroup \captionof{algorithm}{PrimitiveExtensions}\label{alg:extensions} \endgroup \begin{algorithmic}[1] \REQUIRE Lattices $A,B,C'$ with isometry such that $(A,B,C')$ is $p$-admissible. \ENSURE
A set of representatives of the double coset $G_B \backslash S {/}G_A$ where $S$ is the set of all primitive extensions $A\perp B\subseteq C$ with $pC \subseteq A \perp B$ and $t(C') = t(C,f_C^p)$. \STATE Initialize the empty list $L = [\,]$. \STATE Let $g \in \mathbb{N}$ be such that $p^{2g} \det A \det B = \det C'$. \label{alg:extensions_g} \STATE Let $\mu_A$ (resp. $\mu_B$) be the minimal polynomial of $f_A$ (resp. $f_B$).
\STATE $V_A \gets \ker \mu_B(D_{f_A}|_{(p^{-1} A \cap A^\vee)/A})$
\STATE $V_B \gets \ker \mu_A(D_{f_B}|_{(p^{-1} B \cap B^\vee)/B})$ \STATE Let $\mathrm{Gr}_A$ be the set of $f_A$-stable subspaces of dimension $g$ of $V_A$ containing $p^l D_A$. Define $\mathrm{Gr}_B$ analogously. Form the set $R$ consisting of anti-isometric pairs $(H_A,H_B)$, i.e.
$(H_A,q_A|H_A) \cong (H_B,-q_B|H_B)$, as $(H_A,H_B)$ runs through a set of representatives of
$\mathrm{Gr}_A/G_A$ and $\mathrm{Gr}_B/G_B$ respectively. \FOR{ $(H_A,H_B) \in R$}\label{HAHB}
\STATE Compute an admissible glue map $\psi_0\colon H_A \to H_B$, see \Cref{def:admissible-glue-map}. \label{discard_psi1}
\STATE Let $S_A\leq G_A$ (resp. $S_B \leq G_B$) be the stabilizer of $H_A$ (resp. of $H_B$).
\STATE $S_A^{\psi_0}\gets \psi_0 \operatorname{Im}(S_A\to O(H_B)) \psi_0^{-1}$
\STATE $\bar f_A^{\psi_0} \gets \psi_0 (f_A|H_A)\psi_0^{-1}$
\STATE Compute an element $g \in O(H_B,\rho_l(B))$ such that $g \bar f_A^{\psi_0} g^{-1} = f_B|H_B$. If such an element does not exist, discard $\psi_0$ and continue the for loop in line 7. \label{discard_psi2}
\STATE $\psi \gets g \circ \psi_0$, $S_A^\psi \gets gS_A^{\psi_0}g^{-1}$
\STATE Let $O(H_B,\rho_{l+1}(B),f_B)$ be the group of isometries of $H_B$ preserving $\rho_{l+1}(B)$ and commuting with the action of $f_B$.
\FOR{$S_BhS_A^\psi \in S_B \backslash O(H_B,\rho_l(B),f_B) /S_A^\psi$}
\STATE Let $\Gamma_{h\psi}$ be the graph of $h\psi$.
\STATE Define $C$ by $C/(A\perp B) = \Gamma_{h \psi}$.
\STATE $f_C \gets f_A \oplus f_B$
\IF{$t(C,f_C^p) \neq t(C')$} \label{keepC?}
\STATE Discard $C$.
\ENDIF
\STATE $S_C \gets\{(a,b) \in S_A \times S_B \mid b|_{H_B} \circ h\psi = h\psi \circ a|_{H_A}\}$ \label{preserveC}
\STATE $G_C \gets \image\left(S_C \rightarrow O(\disc{C})\right)$
\STATE Append $(C,f_C,G_C)$ to $L$.
\ENDFOR \ENDFOR \RETURN L \end{algorithmic}\leavevmode\leaders\hrule height 0.8pt
\kern\z@\\
\begin{lemma}
Algorithm \ref{alg:extensions} is correct. \end{lemma}
\begin{proof}
Suppose that $A \perp B \subseteq C$ is an equivariant
primitive extension with $pC \subseteq A \perp B$ and $t(C,f_C^p)=t(C')$. Let $\phi \colon H_A \to H_B$ be the corresponding glue map. It is admissible by \Cref{glue:estimate-level}.
The existence of $g$ in line \ref{alg:extensions_g} follows from $(A,B,C')$ being $p$-admissible.
Further, $p C \subseteq A \perp B$, gives $H_A \subseteq (p^{-1}A\cap A^\vee)/A$.
Since $\phi$ is equivariant, we get that $\mu_B(G_{f_A})$ vanishes on $H_A$.
Hence $H_A$ is a $g$-dimensional subspace of the $\mathbb{F}_p$-vector space $V_A$.
It is stable under $f_A$. Further, by \Cref{glue:estimate-level}, it contains $p^l D_A$.
Similarly $H_B$ is preserved by $f_B$, contains $p^l D_B$ and is contained in $V_B$. Therefore $(H_A,H_B)$ appears in the for loop in line \ref{HAHB}.
Since $(A,B,C')$ is $p$-admissible,
there exists an admissible glue map
$\psi_0\colon H_A \to H_B$. It can be computed using normal forms of quadratic or bilinear forms over finite fields.
The set of admissible glue maps from $H_A$ to $H_B$ is given by $O(H_B, \rho_l(B))\psi_0$.
There exists an admissible equivariant glue map from $H_A$ to $H_B$ if and only if we find $g \in O(H_B,\rho_l(B))$ with
\[g \psi_0 (f_A|H_A) = (f_B|H_B) g \psi_0.\]
Reordering we find $g \psi_0 (f_A|H_A) \psi_0^{-1} g^{-1} = f_B|H_B$.
This justifies lines \ref{discard_psi1} to \ref{discard_psi2} of the algorithm. So we continue with $\psi$ an equivariant admissible glue map.
Now the set of equivariant admissible glue maps is $O(H_B,\rho_l(B),f_B)\psi$.
Let $h \psi$, $h \in O(H_B,\rho_l(B),f_B)$ be an equivariant admissible glue map and let $a \in S_A$ and $b \in S_B$.
Then
\[b h\psi a = (b|_{H_B}) h (\psi a \psi^{-1}) \psi = h' \psi.\]
Therefore $S_B h\psi S_A \mapsto S_B h S_A^{\psi} \psi$ defines a bijection of
\[S_B \backslash \{\mbox{equivariant admissible glue maps } \psi\colon H_A \to H_B\}/S_A \]
with the double coset \[ S_B \backslash O(H_B,\rho_l(B),f_B)/S_A^\psi.\]
Finally, the condition on $(a,b)$ in the equation for $S_C$ in line \ref{preserveC} of the algorithm
is indeed the one to preserve $C/(A\perp B) \leq \disc{A \perp B}$.
Thus $S_C$ is the stabilizer of $C/(A \perp B)$ in $G_A \times G_B$. \end{proof}
\begin{remark}
The computation of representatives and their stabilizers in Steps~6 and~9 of \Cref{alg:extensions} can be very costly.
In \textsc{Magma}~\cite{magma}, based on the algorithms in \cite{obrien1990}, a specialized method \textsc{OrbitsOfSpaces} for linear actions on Grassmannians is provided. \end{remark}
\begingroup \captionof{algorithm}{Representatives}\label{reps} \endgroup
\begin{algorithmic}
\REQUIRE A lattice with isometry $A$ such that $\Phi_n(f_A)=0$ and an integer $m$,
or just its type $t(A)$.
\ENSURE Representatives of isomorphism classes of lattices with isometry $B$ of order $m \cdot n$ and minimal polynomial $\Phi_{mn}$ such that $t(B,f_B^m)=t(A)$.
\end{algorithmic}\leavevmode\leaders\hrule height 0.8pt
\kern\z@\\
\Cref{reps} relies on an enumeration of genera of hermitian lattices over maximal orders of number fields with bounds on the determinant and level. Then for each genus a single representative is computed \cite[Algorithm 3.5.6]{KirschmerHabil} and its type is compared with that of $A$. Finally, Kneser's neighbor method \cite[\S 5]{KirschmerHabil} is used to compute representatives for the isometry classes of the genus. \begingroup \captionof{algorithm}{Split}\label{split} \endgroup \begin{algorithmic}[1] \REQUIRE A lattice with isometry $C$ such that $\Phi_{q^d}(f_C)=0$. \ENSURE Representatives of the isomorphism classes of lattices with isometry $M$ such that $(M,f_M^p)$ is of the same type as $C$. \STATE Initialize an empty list $L = [\ ]$. \FOR{ $(\mathcal{A}_0,\mathcal{B}_0) \in $ AdmissibleTriples$(p,C)$} \label{splitfor1}
\STATE $R_1\gets$ Representatives($A_0$, $\id_{A_0}$, $q^d$) where $A_0$ is any representative of $\mathcal{A}_0$
\STATE $R_2\gets$ Representatives($B_0$, $\id_{B_0}$, $pq^d$) where $B_0$ is any representative of $\mathcal{B}_0$
\FOR{$(A,B) \in R_1 \times R_2$} \label{split_for} \label{splitfor2}
\STATE $E \gets \mbox{PrimitiveExtensions}(A,B,C,p)$
\STATE Append the elements of $E$ to $L$.
\ENDFOR \ENDFOR \RETURN L \end{algorithmic}\leavevmode\leaders\hrule height 0.8pt
\kern\z@\\
\begin{lemma}
\Cref{split} is correct. \end{lemma} \begin{proof} Let $M$ be a lattice with isometry such that $(M,f^p_M)$ is of the same type as $C$. Then the minimal polynomial of $f_M$ is a divisor of $\Phi_{pq^d}\Phi_{q^d}$. Let $M_{p^dq}=\ker \Phi_{pq^d}(f_M)$ and $M_{p^d}=\ker \Phi_{q^d}(f_M)$ be the corresponding sublattices. Then $(M_{pq^d},M_{q^d},C)$ is $p$-admissible by \Cref{ispadmissible} applied to $f_M^q$. Hence their $\mathbb{Z}$-genera appear at some point in the for loop in line \ref{splitfor1}. Similarly, at some point in the for loop in line \ref{splitfor2}, $A \cong M_{p^dq}$ and $B\cong M_{p^d}$ as hermitian lattices. Then some lattice with isometry isomorphic to $M$ is a member of $E$ by \Cref{extensions}.
Conversely, only lattices with isometry $(M,f_M)$ with $(M,f_M^p)$ of the same type as $C$ are contained in $E$. No two pairs $(A,B)$ in line \ref{split_for} are isomorphic and for a given pair the extensions computed are mutually non-isomorphic by the correctness of \Cref{alg:extensions}. Thus no two elements of $L$ can be isomorphic. \end{proof}
\begingroup \captionof{algorithm}{FirstP}\label{FirstP} \endgroup \begin{algorithmic}[1] \REQUIRE A lattice with isometry $C$ of order $q^e$ and $b \in \{0,1\}$.
\ENSURE Representatives of the isomorphism classes of lattices with isometry $M$ such that $(M,f_M^p)$ is of the same type as $C$. If $b=1$, return only $M$ such that $f_M$ is of order $pq^e$. \STATE Initialize an empty list $L = [\ ]$. \IF {$e=0$} \RETURN Split$(C, p)$, where in case $b = 1$ we return only those lattices $M$ with $f_M$ of order $pq^e$. \label{firstp_e0} \ENDIF \STATE $A_0 \gets \ker(\Phi_{q^e}(f_C))$ \label{firstp1} \STATE $B_0 \gets \ker(f_C^{q^{e-1}}-1)$ \label{firstp2} \STATE $\mathcal{A} \gets \mbox{Split}(A_0, p)$
\STATE $\mathcal{B} \gets \mbox{FirstP}(B_0,p, 0)$ \FOR{$(A,B) \in \mathcal{A} \times \mathcal{B}$} \label{firstpAB}
\IF {$b=1$ and $p \nmid \ord(f_A)$ and $p \nmid \ord(f_B)$}
\STATE Discard $(A,B)$ and continue the for loop with the next pair.
\ENDIF
\STATE $E \gets \mbox{PrimitiveExtensions}(A,B,C,q)$
\STATE Append the elements of $E$ to $L$. \ENDFOR \RETURN $L$ \end{algorithmic}\leavevmode\leaders\hrule height 0.8pt
\kern\z@\\
\begin{lemma}
\Cref{FirstP} is correct. \end{lemma}
\begin{proof} Let $M$ be a lattice with isometry such that $t(M,f_M^p)=t(C)$. Then $f_M^{pq^e}-1=0$. Set $f=f_M^{pq^{e-1}}$. We see that $A_0= \ker(\Phi_q(f))$ and $B_0 = \ker(\Phi_1(f))$. Therefore $A_0 \perp B_0 \subseteq M$ is a primitive extension and $(A_0,B_0,M)$ is $q$-admissible.
If $e=0$, then $f_C$ is the identity. So the call of \Cref{split} in line \ref{firstp_e0} returns the correct result. Otherwise we proceed by induction on $e$. Note that $\Phi_{q^e}(f_{A_0})= 0$ so the input to Split is valid. Further the order of $f_{B_0}$ is a divisor of $q^{e-1}$. Thus in line \ref{firstpAB} we have $(\Phi_{pq^e}\Phi_{q^e})(f_A)=0$ and $f_B^{pq^{e-1}} = 1$ (possibly $p \nmid \ord{f_B}$).
Note that $(A,B,C)$ in line \ref{firstpAB} is indeed $q$-admissible, because $(A_0,B_0,C)$ is. \end{proof}
\begingroup \captionof{algorithm}{PureUp}\label{PureP} \endgroup \begin{algorithmic}[1] \REQUIRE A lattice with isometry $C$ such that $\prod_{i=0}^e\Phi_{p^dq^i}(f_C)=0$ for $d>0$, $e\geq 0$. \ENSURE Representatives of the isomorphism classes of lattices with isometry $M$ such that $(M,f_M^p)$ is of the same type as $C$. \STATE initialize an empty list $L$ \IF {$e=0$} \RETURN Representatives$(C,p)$ \ENDIF \STATE $A_0 \gets \ker(\Phi_{p^dq^e}(f_C))$ \STATE $B_0 \gets A_0^\perp$, \STATE $\mathcal{A}\gets \mbox{Representatives}(A_0,p)$ \STATE $\mathcal{B} = $ PureUp$(B_0,p)$ \FOR{$(A,B) \in \mathcal{A} \times \mathcal{B}$}
\STATE $E \gets \mbox{Extensions}(A,B,C,q)$
\STATE append the elements of $E$ to $L$. \ENDFOR \RETURN $L$ \end{algorithmic}\leavevmode\leaders\hrule height 0.8pt
\kern\z@\\
\begin{lemma} \Cref{PureP} is correct. \end{lemma} \begin{proof} If $e=0$, then $\Phi_{p^d}(f_C)=0$, so Representatives does the job. Let $M$ be in the output of PureUp. Since $d>0$, we have $\prod_{i=0}^e\Phi_{p^{d+1}q^i}(f_M)=0$. Therefore $M$ is a valid input to PureUp and we can proceed by induction on $e$. The details are similar to the proof of \Cref{FirstP}. \end{proof}
\begingroup \captionof{algorithm}{NextP}\label{NextP} \endgroup \begin{algorithmic}[1] \REQUIRE A lattice with isometry $C$ of order $p^dq^e$ where $q \neq p$ are primes. \ENSURE Representatives of the isomorphism classes of lattices with isometry $M$ of order $p^{d+1}q^e$ such that $(M,f_M^p)$ is of the same type as $C$. \STATE initialize an empty list $L$ \IF {$d=0$}
\RETURN FirstP($C,p,1)$ \ENDIF \STATE $B_0 \gets \ker( f_C^{p^{d-1}q^e}-1)$ \STATE $A_0 \gets B_0^\perp = \ker \prod_{i=0}^e \Phi_{p^dq^i}(f_C)$ \STATE $\mathcal{A} \gets \mbox{PureUp}(A_0, p)$ \STATE $\mathcal{B} \gets \mbox{NextP}(B_0, p)$ \FOR{$(A,B) \in \mathcal{A} \times \mathcal{B}$}
\STATE $E \gets \mbox{Extensions}(A,B,C,p)$
\STATE append the elements of $E$ to $L$ \ENDFOR \RETURN $L$ \end{algorithmic}\leavevmode\leaders\hrule height 0.8pt
\kern\z@\\ \begin{lemma} \Cref{NextP} is correct. \end{lemma} \begin{proof}
If $d=0$, then $f_C$ has order $q^e$. Hence we can call FirstP. For $d>0$, $A_0$ is a valid input for PureUp. The proof proceeds by induction on $d$ since $f_{B_0}$ has order at most $p^{d-1}q^e$ \end{proof}
By calling NextP on a complete set of representatives of the types of lattices with isometry of order $p^dq^e$, we can obtain a complete set of representatives for the isomorphism classes of lattices with isometry of order $p^{d+1}q^e$. By iterating this process we have an algorithm to enumerate representatives for all isomorphism classes of lattices with isometries of a given order $p^d q^e$.
\begin{remark}\label{rem:latticeK3}
For the application to classifying finite groups of automorphisms of K3 surfaces we note the following:
\begin{enumerate}
\item
We only enumerate those lattices with the correct signatures and discard at each stage the lattices which are negative definite and contain $(-2)$-vectors since they do not lead to isometries preserving the ample cone.
\item Let $G$ be a finite subgroup of automorphisms of a complex K3 surface. Recall that $[G:G_s]=n$ satisfies $\varphi(n) \leq 20$ and $n \neq 60$. The integers with $\varphi(n) \leq 20$ and three prime factors are $30$, $42$ $60$, $66$ with $\varphi(n) = 8 , 12 , 16, 20$.
Suppose $n = 66$. Since $\varphi(66)=20>12$ we have $G_s=1$. So $G$ is cyclic and we know by \cite{keum2015} that the pair is unique. We can treat $42$ with similar arguments. Finally, $30$ is treated by hand with the help of some of the algorithms described above.
\end{enumerate}
The actual computation was carried out using \textsc{SageMath}~\cite{sagemath}, \textsc{Pari}~\cite{PARI}, \textsc{GAP}~\cite{GAP4} and \textsc{Magma}~\cite{magma}. \end{remark}
\subsection{Computation of the group $G_L$}\label{subsec:computeGL} The algorithms of the previous section for enumerating isomorphism classes of lattices with isometry require as input for each lattice with isometry $L$ the group $G_L$, which is the image of the natural map $U(L) \to U(D_L)$. Recall that we use a recursive approach and for primitive extensions $C$ of lattices with isometry $A$ and $B$, the group $G_C$ can be determined using $G_A$ and $G_B$ (see \Cref{alg:extensions}). It is therefore sufficient to explain how $G_L$ can computed for the lattices constructed in \Cref{reps}, which form the base case of the recursive strategy. We therefore consider lattices with isometry $L$ such that $f_L$ has irreducible minimal polynomial and $\mathbb{Z}[f_L]$ is the maximal order of $\mathbb{Q}[f_L]$. We distinguish the following four cases:
\begin{enumerate}
\item
The lattice $L$ is definite. Then $O(L)$ is finite and can be computed using an algorithm of Plesken and Souvignier \cite{Plesken-Souvignier}.
\item
The lattice $L$ is indefinite of rank $2$ and $f_L = \pm 1$. For this situation the computation of $G_L$ will be explained in the remainder of this section.
\item
The lattice $L$ is indefinite of rank $\geq 3$ and $f_L = \pm 1$. In this case, Miranda--Morison theory \cite{miranda-morrisonI, miranda-morrisonII} along with some algorithms by Shimada \cite{shimada2018} solve the problem.
A short account of this is given in \Cref{mmtheory}.
\item
The automorphism satisfies $f_L \neq \pm 1$. This will be addressed in \Cref{hermitianmm},
where we extend the theory of Miranda--Morison to the hermitian case. \end{enumerate}
We end this section by describing the computation of $G_L$ in case~(2). Therefore let $L$ be be an indefinite binary lattice over $\mathbb{Z}$ and $V = L \otimes \mathbb{Q}$ the ambient quadratic space of discriminant $d \in \mathbb{Q}^\times/(\mathbb{Q}^\times)^2$.
It follows from \cite[\S 5]{Eichler1974} that we may assume that $V$ is a two-dimensional étale $\mathbb{Q}$-algebra and $L \subseteq V$ is a $\mathbb{Z}$-lattice of rank $2$. More precisely, $V$ is isomorphic to the Clifford algebra $C^+$, which in turn is isomorphic to $\mathbb{Q}(\sqrt{d})$. The $\mathbb{Q}$-algebra $V$ is a quadratic extension of $\mathbb{Q}$ if and only if $d$ is not a square, which is the case if and only if $V$ is anisotropic. If $d$ is a square, then $V \cong \mathbb{Q} \times \mathbb{Q}$. If $\sigma \colon V \to V$ denotes the non-trivial automorphism of $V$ as a $\mathbb{Q}$-algebra, then the quadratic form $q$ on $V$ is given by $q(x) = x \sigma(x)$ for $x \in V$. Note that $\sigma \in O(V)$ and $\det(\sigma) = -1$.
Every element $y \in V$ induces an endomorphism $\tau_y \colon V \to V, \, x \mapsto yx$ of determinant $\det(\tau_y) = y \sigma(y) = q(y)$. For a subset $X \subseteq V$ set $X^1 = \{ x \in X \mid q(x) = 1 \}$. The proper automorphism group $SO(V)$ of $V$ is equal to $\{ \tau_y \mid y \in V^1\}$ and $O(V) = \{ \tau_y, \sigma \tau_y \mid y \in V^1 \}$. We call two $\mathbb{Z}$-lattices $I, J$ of $V$ equivalent, if there exists $\alpha \in V^1$ such that $I = \alpha J$. Finally set \[
\Lambda = \{ x \in V \mid xL \subseteq L \}, \] which is a $\mathbb{Z}$-order of $V$.
\begin{proposition}
The following hold:
\begin{enumerate}
\item
We have $SO(L) = \{ \tau_y \mid y \in (\Lambda^\times)^1 \}$.
\item
If $L$ is not equivalent to $\sigma(L)$, then $O(L) = SO(L)$.
\item
If $L$ is equivalent to $\sigma(L)$, say $L = \alpha \sigma(L)$, then
$O(L) = \langle SO(L), \sigma \tau_{\sigma(\alpha)} \rangle$.
\end{enumerate} \end{proposition}
\begin{proof}
First note that for $y \in V$ we have $\tau_y(L) \subseteq L$ if and only if $y \in \Lambda$.
This shows part~(1).
Any isometry of $L$ extends uniquely to an isometry of $V$ and is---if the determinant is not $1$---thus of the form $\sigma \tau_\alpha$ for some $\alpha \in V$.
Hence $L = (\sigma \tau_\alpha)(L) = \sigma(\alpha) \sigma(L)$, that is, $L$ and $\sigma(L)$ are equivalent.
This shows part~(2).
Now assume that $L = \alpha \sigma(L)$.
Then $(\sigma \tau_{\sigma(\alpha)})(L) = L$ and thus $\sigma \tau_{\sigma(\alpha)} \in O(L) \setminus SO(L)$.
If $\sigma \tau_{\sigma(\beta)} \in O(L)$ is any non-proper isometry, then $\sigma(\beta) L = \sigma(L) = \sigma(\alpha) L$ and thus $\sigma(\beta \alpha^{-1}) \in (\Lambda^\times)^1$.
This shows part~(3), since $\sigma \tau_{\sigma(\beta)} = \sigma \tau_{\sigma(\alpha)}\tau_{\sigma(\beta \alpha^{-1})}$. \end{proof}
\begin{remark}
We briefly describe how the previous result can be turned into an algorithm for determining generators of $O(L)$ for an indefinite binary lattice.
We may assume that the ambient space $V$ is an étale $\mathbb{Q}$-algebra of dimension two.
The group $\Lambda^\times$ is a finitely generated abelian group and generators
can be computed as described in~\cite{Bley2005, Kluners2005}.
Given generators of $\Lambda^\times$, determining generators of $(\Lambda^\times)^1$ is just a kernel computation.
Finally, testing whether two $\mathbb{Z}$-lattices of $V$ are equivalent can be accomplished using \cite{Bley2005, Marseglia2020}. \end{remark}
\section{Quadratic Miranda--Morrison theory} \label{mmtheory}
In this section we review classical Miranda--Morrison theory for even indefinite $\mathbb{Z}$-lattices $L$ of rank at least $3$, as introduced by Miranda and Morrison in \cite{miranda-morrisonI, miranda-morrisonII}. Akyol and Degtyarev~\cite{sextics} incorporated sign structures to study connected components of the moduli spaces of plane sextics. We follow their example.
The purpose of this is twofold. First this allows us to sketch the computation of the image of \[O(L) \to O(\disc{L})\] settling case (3) in \Cref{subsec:computeGL}. Second by incorporating the action on the sign structure, we obtain a way to compute the image of \[O^+(L) \to O(\disc{L})\] which yields the number of connected components of the moduli space $\mathcal{F}_H$ (see \Cref{quadruplewithtriple} and \Cref{rem:mmapplication}).
We denote by $\mathbb{A}$ the ring of finite adeles and by $\mathbb{Z}_\mathbb{A}$ the ring of finite integral adeles. For a ring $R$ set $\Gamma_R= \{\pm 1\} \times R^\times/(R^\times)^2$. We define $O^\sharp(L\otimes R)$ as the kernel of $O(L \otimes R) \to O(D_L \otimes R)$. We note that $\Gamma_\mathbb{Q}$ has a natural diagonal embedding into $\Gamma_\mathbb{A}$ and $D_L \cong D_L \otimes \mathbb{Z}_\mathbb{A}\cong \disc{L \otimes \mathbb{Z}_\mathbb{A}}$ naturally.
The homomorphisms \[\sigma_p \colon O(L\otimes \mathbb{Q}_p) \to \Gamma_{\mathbb{Q}_p}, \quad g \mapsto (\det(g),\spin(g)).\] induce a homomorphism \[ \sigma\colon O(L \otimes \mathbb{A}) \to \Gamma_{\mathbb{A}}.\] Let $\Sigma^\sharp(L \otimes {\mathbb{Z}_p})$ be the image of $O^\sharp(L \otimes {\mathbb{Z}_p})$ under $\sigma_p$. We set $\Sigma(L) =\sigma(O(L\otimes \mathbb{Z}_\mathbb{A}))=\prod_p \Sigma(L \otimes \mathbb{Z}_p)$ and $\Sigma^\sharp(L) = \sigma(O^\sharp(L \otimes \mathbb{Z}_\mathbb{A}))=\prod_p \Sigma^\sharp(L\otimes \mathbb{Z}_p)$. By \cite[VII 12.11]{miranda-morrison} we have $\Sigma(L\otimes \mathbb{Z}_p)=\Sigma^\sharp(L \otimes {\mathbb{Z}_p})=\Gamma_{\mathbb{Z}_p}$ whenever $L\otimes {\mathbb{Z}_p}$ is unimodular. By \cite[IV.2.14 and IV.5.9]{miranda-morrison} the natural map $O(L\otimes \mathbb{A}) \to O(\disc{L})$ is surjective. The following commutative diagram with exact rows and columns summarizes the situation (where by abuse of notation we denote restriction of $\sigma$ by $\sigma$ as well). \[ \begin{tikzcd}
1\arrow[r]& O^\sharp(L\otimes \mathbb{Z}_\mathbb{A}) \arrow[r]\arrow[d] &O(L\otimes \mathbb{Z}_\mathbb{A}) \arrow[r]\arrow[d,"\sigma"]& O(\disc{L})\arrow[r]\arrow[d] & 1\\ 1\arrow[r]& \Sigma^\sharp(L)\arrow[d] \arrow[r] &\Sigma(L)\arrow[r]\arrow[d] &\Sigma(L)/\Sigma^\sharp(L)\arrow[d] \arrow[r] &1\\
&1 &1 &1 & \end{tikzcd}\]
If $V$ is an indefinite $\mathbb{Q}$-lattice of rank $\geq 3$, then the restriction $O(V) \to \Gamma_{\mathbb{Q}}$ of $\sigma$ is surjective by \cite[VIII 3.1]{miranda-morrison}.
\begin{theorem}\cite[VIII 5.1]{miranda-morrison}\label{thm:miranda-morrison} Let $L$ be an indefinite $\mathbb{Z}$-lattice of rank at least $3$. Then we have the following exact sequence \[1 \to O^\sharp(L) \to O(L) \to O(D_L) \xrightarrow{\bar \sigma} \Sigma(L)/(\Sigma^\sharp(L)\cdot (\Gamma_\mathbb{Q} \cap \Sigma(L))) \to 1. \] \end{theorem}
We need an analogous sequence with $O(L)$ replaced by $O^+(L)$ to compute connected components of the coarse moduli space of $H$-markable K3 surfaces. Define $\Gamma_\mathbb{Q}^+$ as the kernel of $\Gamma_\mathbb{Q} \to \{\pm 1\}$, $(d,s) \mapsto \sign(ds)$. Then for any indefinite $\mathbb{Q}$-lattice $V$ of rank $\geq 3$ the homomorphism $\sigma^+\colon O^+(V) \to \Gamma_\mathbb{Q}^+$ is surjective.
\begin{theorem}\label{thm:miranda-morrison-signed}
Let $L$ be an indefinite $\mathbb{Z}$-lattice of rank at least $3$. Then we have the following exact sequence \[O^+(L) \xrightarrow{D_+} O(D_L) \xrightarrow{\bar \sigma_+} \Sigma(L)/(\Sigma^\sharp(L)\cdot (\Gamma_\mathbb{Q}^+ \cap \Sigma(L))) \to 1. \]
\end{theorem}
\begin{proof} We prove $\ker \bar\sigma_+ \subseteq \operatorname{Im} D_+$: Let $\bar g \in O(D_L)$ and suppose that $\bar\sigma_+(\bar g) = 1$. This means that $\bar g$ lifts to an element $g \in O(L\otimes \mathbb{Z}_\mathbb{A})$ with $D_g=\bar g$ and $\sigma(g)\in \Sigma^\sharp(L)\cdot (\Gamma_\mathbb{Q}^+ \cap \Sigma(L))$. After multiplying $g$ with an element in $O^\sharp(L\otimes \mathbb{Z}_\mathbb{A})$, we may assume that \[\sigma(g) \in \Gamma_\mathbb{Q}^+ \cap \Sigma(L).\] Hence there exists an element $h \in O(L\otimes \mathbb{Q})$ with $\sigma(h)=\sigma(g)$.
Since $\sigma(h^{-1}g)=1$ and $h^{-1}g(L\otimes \mathbb{Z}_p)=L \otimes \mathbb{Z}_p$ for all but finitely many primes, we can use the strong approximation theorem (see e.g.\cite[5.1.3]{KirschmerHabil}, \cite{kneser1966}) to get $f \in O(L\otimes \mathbb{Q})$ with $\sigma(f)=1$ and $f(L \otimes {\mathbb{Z}_p}) = h^{-1}g(L \otimes {\mathbb{Z}_p})$ at all primes and approximating $h^{-1}g$ at the finitely many primes dividing the discriminant. This yields (once the approximation is good enough) $D_f = D_{h^{-1}g}$ (cf \cite[VIII 2.2]{miranda-morrison}).
By construction, $h f \in O(L \otimes \mathbb{Q})$ preserves $L$ and \[D_{hf} = D_{h} \circ D_f = D_{h} \circ D_{h^{-1}g} = D_g\] as desired. We have $\sigma(hf) = \sigma(h) \in \Gamma_\mathbb{Q}^+$. So $hf \in O^+(L)$.
We prove $\ker \bar \sigma_+ \supseteq \operatorname{Im} D_+$: Let $ g \in O^+(L)$. Then $\sigma(g) \in \Gamma^+_\mathbb{Q}$ and since $O(L) \subseteq O(L\otimes \mathbb{Z}_\mathbb{A})$ we have $\sigma(g) \in \Sigma(L)$ as well.
\end{proof}
The group $\Sigma(L)$ appearing in \Cref{thm:miranda-morrison,thm:miranda-morrison-signed} is infinite and infinitely generated. However its quotient by $\Sigma^\sharp(L)$ is a finite group. We explain how to write it in terms of finite groups only, so that it may be represented in a computer. Let $T$ be the set of primes with $\Sigma^\sharp(L\otimes \mathbb{Z}_p)=\Sigma(L\otimes \mathbb{Z}_p)=\Gamma_{\mathbb{Z}_p}$ and $S$ its complement. We know that $S$ is contained in the set of primes dividing $\det L$. We can project the quotient $\Sigma(L)/\Sigma^{\sharp}(L)$ isomorphically to a subquotient of the finite group $ \Gamma_S' = \prod_{p \in S} \Gamma_{\mathbb{Q}_p}$ and are thus reduced to a finite computation.
Indeed, denote by $\pi_S\colon \Gamma_\mathbb{A} \to \Gamma_S'$ the natural projection. Set \[\Sigma_S(L) = \pi_S(\Sigma(L)), \quad \Sigma^\sharp_S(L)= \pi_S(\Sigma^\sharp(L)),\quad \Gamma_T=\prod_{p \in T} \Gamma_{\mathbb{Z}_p},\] \[\Gamma_S = \pi_S(\Gamma_\mathbb{Q}\cap \Gamma_T \times \Gamma_S') \qquad \Gamma_S^+=\pi_S(\Gamma^+_\mathbb{Q}\cap \Gamma_T \times \Gamma_S').\]
$\Gamma_S^+$ is spanned by the images of $\{(1,p) \mid p \in S\} \cup \{(-1,-1)\} \subseteq \Gamma_\mathbb{Q}^+\subseteq \Gamma_\mathbb{A}$ under $\pi_S$. Since $\Gamma_\mathbb{Q}/\Gamma_\mathbb{Q}^+$ is spanned by $(-1,1)$, $\Gamma_S$ is spanned by the generators of $\Gamma_S^+$ together with $\pi_S((-1,1))$.
\begin{proposition} The projection $\pi_S$ induces isomorphisms \[\Sigma(L)/(\Sigma^\sharp(L)\cdot (\Gamma_\mathbb{Q}^+ \cap \Sigma(L)))\cong \Sigma_{S}(L)/(\Sigma^\sharp_S(L)\cdot (\Gamma_S^+ \cap \Sigma_S(L)))\] and \[\Sigma(L)/(\Sigma^\sharp(L)\cdot (\Gamma_\mathbb{Q} \cap \Sigma(L)))\cong \Sigma_{S}(L)/(\Sigma^\sharp_S(L)\cdot (\Gamma_S \cap \Sigma_S(L))).\] \end{proposition} \begin{proof} Note that $\pi_S(\Gamma_\mathbb{Q}^+ \cap \Sigma(L)) = \Gamma_S^+ \cap \Sigma_S(L)$. Therefore \[K:=\pi_S^{-1}(\Sigma_S^\sharp(L)\cdot(\Gamma^+_S \cap \Sigma_S(L)))=\Sigma^\sharp(L)\cdot(\Gamma_\mathbb{Q}^+ \cap \Sigma(L)).\] Hence the surjection \[\psi \colon \Sigma(L) \to \Sigma_{S}(L)/(\Sigma^\sharp_S(L)\cdot (\Gamma_S^+ \cap \Sigma_S(L)))\] induced by $\pi_S$ has kernel $K$. We conclude by applying the homomorphism theorem to $\psi$.
To prove the second isomorphism remove the $+$. \end{proof}
The groups $\Sigma_S(L)$ and $\Sigma_S^\sharp(L)$ are found in the tables in \cite[VII]{miranda-morrison} in terms of the discriminant form of $L$ and its signature pair.
\begin{proposition}\label{plusindexcompute}
Let $L$ be an indefinite $\mathbb{Z}$-lattice of rank at least $3$ and $J$ a subgroup of the image of the natural map $D \colon O(L) \to O(\disc{L})$. Set $J^+ = D(O^+(L)) \cap J$ and let $K = \ker D=O^\sharp(L)$.
Then $[J:J^+]=| \sigma_+(J)|$ and \[[K:K^+]= [\Gamma_\mathbb{Q} \cap \Sigma^\sharp(L): \Gamma_\mathbb{Q}^+ \cap \Sigma^\sharp(L)] =
[\Gamma_S \cap \Sigma^\sharp_S(L): \Gamma_S^+ \cap \Sigma^\sharp_S(L)].\] \end{proposition} \begin{proof}
We have $J^+ = \ker (\bar \bar \sigma_+) \cap J = \ker (\bar \sigma_+|_J)$. Therefore $J/J_+ \cong \sigma_+(J)$.
The strong approximation theorem implies the equality $\sigma(O^\sharp(L)) = \Sigma^\sharp(L) \cap \Gamma_\mathbb{Q}$ and that $\sigma(O^\sharp(L)^+) = \Sigma^\sharp(L) \cap \Gamma_\mathbb{Q}$.
\end{proof}
\begin{remark}
The theorems allow us to compute the image of $O(L) \to O(D_L)$ for $L$
an indefinite $\mathbb{Z}$-lattice of rank at least $3$. Namely, one computes generators of $O(D_L)$ and lifts them $p$-adically to elements of $L\otimes \mathbb{Z}_p$ with sufficient precision. Then one can use these lifts to compute their spinor norm. See the work of Shimada \cite{shimada2018} for further details. An Algorithm for $p$-adic lifting and generators for $O(D_L)$ are given in \cite{brandhorst-veniani2023}. \end{remark}
\section{Hermitian Miranda--Morrison theory}\label{hermitianmm} Let $L$ be a lattice with isometry with irreducible minimal polynomial. In this section we use the transfer construction to compute the image of $U(L) \to D(\disc{L})$, thus settling case (4) of \Cref{subsec:computeGL}. To this end we develop the analogue of Miranda--Morrison theory for hermitian lattices over the ring of integers of a number field.
\subsection{Preliminaries on hermitian lattices.} In this section we recall some basics on hermitian lattices over the ring of integers of a number field or a local field. See \cite{KirschmerHabil} for an overview of the theory.
Let $K$ be a finite extension of $F=\mathbb{Q}$ (global case) or $F=\mathbb{Q}_p$ (local case) and $E$ an étale $K$-algebra of dimension $2$. Let $\mathcal{O}$ be the maximal order of $E$ and $\o$ be the maximal order of $K$.
\begin{definition}
For $E$ an \'etale $K$-algebra, we denote by
$\Tr^E_K \colon E \to K$ the trace and by
$\mathfrak{D}_{E/K}^{-1}=\{ x \in E \mid \Tr^E_K(x \mathcal{O}) \subseteq \o \}$
the inverse of the different. \end{definition}
In the local case, we let $\mathfrak{P}\subseteq \mathcal{O}$ be the largest ideal invariant under the involution of $E/K$ and $\mathfrak{p}$ the maximal ideal of $\o$. Define $e$ by $\mathfrak{P}^e = \mathfrak{D}_{E/K}$ and $a$ by $\mathfrak{P}^a = \mathfrak{D}_{E/F}$.
\begin{definition}
Let $(L,h)$ be a hermitian $\mathcal O$-lattice. Its scale is the ideal $\scale(L)=h(L,L)\subseteq \mathcal{O}$ and its norm is the ideal $\norm(L)=\sum \{ \sprodq{x}{}\o \mid x \in L \} \subseteq \o$. \end{definition} It is known that \begin{equation}\label{norm-vs-scale}
\mathfrak{D}_{E/K} \scale(L) \subseteq \norm(L) \subseteq \scale(L). \end{equation}
\subsection{The trace lattice} In this subsection $(L,h)$ is a hermitian $\mathcal{O}$-lattice. By transfer we obtain its trace $\mathbb{Z}_F$-lattice $(L,b)$ with $b = \Tr^E_{F} \circ h$. Our primary interest is in even $\mathbb{Z}$-lattices. So our next goal is to give necessary and sufficient conditions for the trace lattice to be integral and even.
The hermitian dual lattice is \[L^\sharp = \{x \in L \otimes E \mid h(x,L) \subseteq \mathcal{O}\}\] and $L^\vee = (L,b)^\vee$ is the dual lattice with respect to the trace form.
\begin{proposition}\label{trace-integral} We have \[L^\vee = {\mathfrak{D}_{E/F}^{-1}} L^\sharp.\] The trace form on $L$ is integral if and only if $\scale(L) {\mathfrak{D}_{E/F}} \subseteq \mathcal{O}$. \end{proposition} The proof is left to the reader. We continue by determining the parity of the trace lattice. To this end, we first establish that the transfer construction behaves well with respect to completions, as expected.
For a place $\nu$ of $K$ we use the following notation: For an $\o$-module $M$ denote by $M_\nu = M \otimes \o_\nu$ the completion of $M$ at $\nu$ and similar for $K$-vector spaces.
\begin{proposition}\label{prop:tracedecompose}
Let $F = \mathbb{Q}$, $E/K$ a degree two extension of number fields and $p$ a prime number.
Then
\begin{equation}\label{decomp-local}
(L,\Tr^E_\mathbb{Q} \circ h) \otimes \mathbb{Z}_p = \bigperp_{\nu \mid p} (L_\nu, \Tr^{E_\nu}_{\mathbb{Q}_p} \circ h_\nu),
\end{equation}
where $\nu$ runs over all places extending the $p$-adic place of $\mathbb{Q}$. \end{proposition} \begin{proof} To this end consider the canonical isomorphism $K \otimes \mathbb{Q}_p \cong \prod_{\nu \mid p} K_\nu$ where the product runs over the prolongations of the $p$-adic valuation to $K$. By \cite[Chap. II, \S3, Prop. 4]{Serre1979} this induces a canonical isomorphism \[\o \otimes \mathbb{Z}_p \cong \prod_{\nu \mid p} \o_\nu\] where $\o_\nu$ is the maximal order of $K_\nu$. We obtain a corresponding canonical decomposition (using a system of primitive idempotents) \begin{equation}\label{decomp-localp} (L,h) \otimes \mathbb{Z}_p = \bigperp_{\nu \mid p} (L_\nu,h_\nu). \end{equation}
where each summand is a hermitian $\mathcal{O}_\nu$-lattice.
Note that $\mathcal{O}_\nu$ is indeed the maximal order of $E_\nu$. The decomposition $E \otimes \mathbb{Q}_p \cong \prod_{\nu \mid p} E_\nu$ and viewing $\Tr$ as the trace of the left multiplication endomorphism shows that \[\Tr^{E \otimes \mathbb{Q}_p}_{\mathbb{Q}_p}= \sum_{\nu \mid p} \Tr^{E_\nu}_{\mathbb{Q}_p}.\] Therefore the trace commutes with the decomposition in \cref{decomp-localp}. \end{proof}
\begin{lemma}\label{normhelp}
Let $K$ be a non-archimedian dyadic local field of characteristic $0$.
Assume that $B \subseteq \o$ is a $\mathbb{Z}$-module such that $\Tr^E_K(\mathcal{O}) \subseteq B$, $1 \in B$ and $\Nr^E_K(\mathcal{O})B \subseteq B$. Then $\o \subseteq B$. \end{lemma}
\begin{proof} If $E/K$ is split or unramified, then $\Tr^E_K(\mathcal{O})=\o$. So let $E/K$ be ramified.
By~\cite[\S 6]{Johnson1968} there exists $u_0 \in K$ such that $\o^\times = \Nr^E_K(\mathcal{O}^\times) \cup (1+u_0) N^E_K(\mathcal{O}^\times)$ and $u_0 \o = \mathfrak{p}^{e-1}$. Since $\mathfrak{p}^{e-1} \subseteq \mathfrak{p}^{\lfloor\frac{e}{2}\rfloor}=\Tr^E_K(\mathcal{O})$ by \cite[Ch. V,\S3, Lemma 3]{Serre1979}, we have $(1+u_0) \Nr^E_K(\mathcal{O}^\times) \subseteq B$.
Thus $\o^\times \subseteq B$ and therefore $\Nr^E_K(\mathcal{O}) \o^\times \subseteq B$. As $E/K$ is ramified it follows that $\o= \Nr^E_K(\mathcal{O}) \o^\times.$ \end{proof}
The following proof is inspired by \cite[3.1.9]{michael2015}. \begin{proposition}\label{eventracelocal} Let $K$ be a non-archimedian local field of characteristic $0$ and $(L,h)$ a hermitian $\mathcal{O}$-lattice. The trace form $\Tr^E_F \circ h$ is even if and only if
$\norm(L) \subseteq \mathfrak{D}_{K/F}^{-1}$. \end{proposition}
\begin{proof}
Suppose that $\norm(L) \subseteq {\mathfrak{D}_{K/F}^{-1}}$. We have
\[\Tr^E_F(h(x,x))= 2 \Tr^K_F(h(x,x)) \in 2 \Tr^K_F(\norm(L)) \subseteq 2 \Tr^K_F({\mathfrak{D}_{K/F}^{-1}})=2 \mathbb{Z}_F.\]
Now suppose that the trace form is even. In particular it is integral. We may assume that $F = \mathbb{Q}_2$ is dyadic. Let $B$ be the set of all $\omega \in K$ such that $\Tr^{E}_F(\omega h(x,x))\subseteq 2\mathbb{Z}_2$ for all $x \in L$. Then $\lambda \bar \lambda \omega h(x,x)=\omega h(\lambda x,\lambda x)$ for $\lambda \in \mathcal{O}$ gives $\Nr(\mathcal{O}) B \subseteq B$ and $1 \in B$. We calculate
\[\Tr^E_F((\lambda +\bar \lambda) h(x,x)) = \Tr^E_F (\lambda h(x,x))+ \Tr^E_F (\overline{ \lambda h(x,x)}) = 2 \Tr^E_F(h(\lambda x,x)) \in 2\mathbb{Z}_2 \]
for all $\lambda \in \mathcal{O}$. This gives $\Tr^E_K(\mathcal{O}) \subseteq B$. By \Cref{normhelp} $\o \subseteq B$.
Therefore $\Tr^K_F(\o h(x,x))\subseteq \mathbb{Z}_2$ which means $h(x,x)\in {\mathfrak{D}_{K/F}^{-1}}$. \end{proof}
We show that the same result holds in the global setting.
\begin{corollary}\label{trace} Let $F=\mathbb{Q}$ and $E/K$ a degree two extension of number fields. Let $(L,h)$ be a hermitian $\mathcal{O}$-lattice. The trace form $\Tr^E_F \circ h$ is even if and only if $\norm(L) \subseteq {\mathfrak{D}_{K/F}^{-1}}$. \end{corollary}
\begin{proof} We use \Cref{prop:tracedecompose} and note that the orthogonal sum is even if and only if each summand is even. We apply \Cref{eventracelocal} to each summand and obtain the condition that $\norm(L_\nu) \subseteq \mathfrak{D}^{K_\nu}_{\mathbb{Q}_2}$ for all $\nu \mid 2$. We conclude with $\nu(\norm(L)) = \nu(\norm(L_\nu))$. \end{proof}
Note that even implies integral and similarly ${\mathfrak{D}_{E/K}} \scale(L) \subseteq \norm(L)\mathcal{O} \subseteq {\mathfrak{D}_{K/F}^{-1}}\mathcal{O}$ implies that $\scale(L) \subseteq {\mathfrak{D}_{E/F}^{-1}}$ which matches up perfectly with \Cref{trace-integral} and~\Cref{trace}.
\subsection{Discriminant form and transfer} Suppose that $L \subseteq L^\vee := {\mathfrak{D}_{E/F}^{-1}} L^\sharp$. Define a torsion hermitian form on $D_L=L^\vee/L$ as follows \[\bar h\colon D_L \times D_L \to E/{\mathfrak{D}_{E/F}^{-1}}, \quad ([x],[y]) \mapsto [h(x,y)].\] Suppose further that $\norm(L) \subseteq {\mathfrak{D}_{K/F}^{-1}}$, that is, the trace form on $L$ is even. Then we define the torsion quadratic form \[\bar q\colon D_L \to K{/}{\mathfrak{D}_{K/F}^{-1}},\quad [x] \mapsto [h(x,x)].\] Note that $\Tr^E_K({\mathfrak{D}_{E/F}^{-1}})={\mathfrak{D}_{K/F}^{-1}}$. For a lattice $L$ with even trace form we set $U(D_L)$ to be the group of $\mathcal{O}$-linear automorphisms of $D_L$ preserving $\bar q$.
\begin{proposition} Let $(L,h)$ be a hermitian $\mathcal{O}$-lattice with even trace form and
$g \in U(D_L)$. Then $g$ preserves $\bar h$. \end{proposition}
\begin{proof}
For $x \in L^\vee$ let $gx$ denote a representative of $g(x+L)$.
Let $x,y \in L^\vee$. Then
\[\Tr^E_K h(x,y) = h(x+y,x+y) - h(x,x) - h(y,y).\]
Set $\delta(x,y) = h(x,y) - h(gx,gy)$. We have to prove that $\delta(x,y) \in {\mathfrak{D}_{E/F}^{-1}}$.
Since $g$ preserves $\bar q$,
\[\Tr^E_K( \delta(x,y)) \in {\mathfrak{D}_{K/F}^{-1}} =\Tr^E_K({\mathfrak{D}_{E/F}^{-1}}). \]
By the $\mathcal{O}$-linearity of $g$ we have $\alpha gx - g\alpha x \in L$ for all $\alpha \in \mathcal{O}$.
Therefore, using $L^\vee = {\mathfrak{D}_{E/F}^{-1}}L^\sharp$, we have for any $\alpha \in \mathcal{O}$ that $\alpha \delta(x,y) \equiv \delta(\alpha x,y) \mod {\mathfrak{D}_{E/F}^{-1}}$. Hence
\[\Tr^E_K(\mathcal{O} \delta(x,y)) \subseteq {\mathfrak{D}_{K/F}^{-1}}.\]
This means that $\delta(x,y) \in {\mathfrak{D}_{K/F}^{-1}}{\mathfrak{D}_{E/K}^{-1}}={\mathfrak{D}_{E/F}^{-1}}$. \end{proof}
Recall that for an even lattice with isometry $(L,b,f)$ we have defined $U(D_L)$ as the centralizer of $D_f$ in $O(D_L)$. By the transfer construction in \Cref{transfer} we may view $L$ as a hermitian lattice as well.
The following proposition reconciles the two definitions of $U(D_L)$.
\begin{proposition}\label{UDLtransfer}
Let $(L,b,f)$ be an even lattice with isometry with irreducible minimal polynomial and $(L,h)$ the corresponding hermitian $\mathbb{Z}[f]$-lattice.
Let $E \cong \mathbb{Q}[f]$, $K\cong \mathbb{Q}[f+f^{-1}]$ and $\mathcal{O}$ be the maximal order of $E$. Suppose that $(L,h)$ is invariant under $\mathcal{O}$, that is, $(L, h)$ is a hermitian $\mathcal{O}$-lattice (this is true if $\mathbb{Z}[f] = \mathcal{O}$).
Then $U(D_L)$ is the centralizer of $D_f$ in $O(D_L)$. \end{proposition}
\begin{proof} It is clear that $U(D_L)$ centralizes $D_f$. So let $g \in O(D_L)$ centralize $D_f$. This implies that $g$ is $\mathcal{O}$-linear. It remains to show that $g$ preserves $\bar q$. Since $D_L = \bigperp_{\nu} D_{L_\nu}$ and $U(\disc{L})=\prod_{\nu}U(\disc{L_\nu})$, we may assume that $K$ is complete.
For $x + L \in D_L$ write $gx$ for a representative of $g(x+L)$.
Set $\delta = h(x,x)-h(gx,gx)$. We have to show that $\delta \in \Tr^E_K({\mathfrak{D}_{E/F}^{-1}})=\mathfrak{D}_{K/F}^{-1}$. Since $g$ preserves the discriminant form $q_{(L,b)}$, we have $2\Tr^K_F(\delta)=\Tr^E_F(\delta) \in 2 \mathbb{Z}_F$. Let $B$ be the set of all $\omega \in \o$ such that $ \Tr^K_F(\omega \delta) \in \mathbb{Z}_F$. As in the proof of \Cref{eventracelocal} one sees that $\Nr(\mathcal{O}) B \subseteq B$, $1 \in B$ and $\Tr^E_K(\mathcal{O}) \subseteq B$. Then \Cref{normhelp} provides $B = \o$. Thus $\Tr^K_F(\mathcal{O} \delta) \subseteq \mathbb{Z}_F$, i.e. $\delta \in {\mathfrak{D}_{K/F}}$. \end{proof}
\begin{remark}
\Cref{UDLtransfer} provides a practical way to compute $U(D_L)$. We can write down a system of generators for $O(D_L)$. Then
the computation of a centralizer is a standard task in computational group theory. \end{remark}
\subsection{Local surjectivity of $\mathbf{U(L) \to U(D_L)}$}\label{subsec:localsurjectivity} In this subsection we assume that $K$ is a non-archimedian local field of characteristic $0$, $\pi$ a prime element of $\mathcal{O}$, $p = \pi \bar \pi$, and $L$ a hermitian $\mathcal{O}$-lattice with $\norm(L)\subseteq \mathfrak{D}_{K/F}^{-1}$, that is, its trace lattice is even. Recall that $\mathfrak{P}\subseteq \mathcal{O}$ is the largest ideal invariant under the involution of $E/K$, $\mathfrak{p}$ the maximal ideal of $\o$ and the integers $a, e$ satisfy $\mathfrak{P}^e = \mathfrak{D}_{E/K}$ and $\mathfrak{P}^a = \mathfrak{D}_{E/F}$.
If $E/K$ is a ramified field extension then by \cite[Ch. V,\S3, Lemma 3]{Serre1979} we have for all $i \in \mathbb{Z}$ that $\Tr(\mathfrak{P}^i) = \mathfrak{p}^{\lfloor \frac{i+e}{2}\rfloor}$. Therefore $\Tr(\mathfrak{P}^{1-e})=\o$ and $\Tr(\mathfrak{P}^{2-e})=\mathfrak{p}$. So there exists $\rho \in E$ with $\rho \mathcal{O} = \mathfrak{P}^{1-e}$ and $\Tr(\rho)=1$. If $E/K$ is an unramified field extension, then we find $\rho \in \mathcal{O}^\times$ with $\Tr(\rho)=1$. If $E = K \times K$, then we can take $\rho = (1, 0) \in \o\times \o=\mathcal{O}$ which satisfies $\Tr(\rho) = 1$ as well.
For a hermitian matrix $G \in E^{n \times n}$ set $\scale(G)=\scale(L)$ and $\norm(G)=\norm(L)$, where $L$ is the free $\mathcal{O}$-lattice with gram matrix $G$.
\begingroup \captionof{algorithm}{Hermitian lift}\label{alg:lift} \endgroup \begin{algorithmic}[1] \REQUIRE $0\leq l \in \mathbb{Z}$, $\rho \in E$, $G =\bar G^t \in E^{n\times n}$, $F \in \GL_n(\mathcal{O})$ such that \begin{itemize}
\item $\Tr^E_K(\rho) =1$, \item $\scale(G^{-1})\subseteq \mathfrak{P}^{1+a}$, $\rho\norm(G^{-1})\subseteq \mathfrak{P}^{1+a}$, \item $R:=G - FG\bar F^T$ with $\scale(R) \subseteq \mathfrak{P}^{l-a}$, $\rho\norm(R) \subseteq \mathfrak{P}^{l-a}$. \end{itemize}
\ENSURE $F' \in \GL_n(\mathcal{O})$ such that for $l' = 2l + 1$ and $R' = G - F'G \bar F'^t$ the following hold \begin{itemize}
\item $F' \equiv F \mod \mathfrak{P}^{l} \mathcal{O}^{n \times n}\pi^{-a}G^{-1} \subseteq\mathfrak{P}^{l+1}\mathcal{O}^{n \times n}$,
\item $\scale(R') \subseteq \mathfrak{P}^{l'-a}$, $\rho \norm(R') \subseteq \mathfrak{P}^{l'-a}$. \end{itemize} \STATE $R \gets G - F G \bar F^t$ \STATE Write $R = U + D + \bar U^t$ with $U$ upper triangular and $D$ diagonal. \RETURN $F + (U + \rho D)\bar F^{-t}G^{-1}$ \end{algorithmic}\leavevmode\leaders\hrule height 0.8pt
\kern\z@ \\
\begin{theorem} \Cref{alg:lift} is correct. \end{theorem} \begin{proof}
With $X=(U + \rho D)\bar F^{-t}$ and $F' = F + XG^{-1}$, we calculate
\begin{eqnarray*}
F' G \bar F'^t
&=& F G \bar F^t + U + \bar U^t + \Tr^E_K(\rho)D + XG^{-1} \bar X^t\\
&= & G + X G^{-1} \bar X^t.
\end{eqnarray*}
Hence $R' = -XG^{-1}X^t$. Since $R \equiv 0 \mod \scale(R)$, $D\equiv 0 \mod \norm(R)$, we have
$U + \rho D \equiv 0 \mod \scale(R)+\rho \norm(R) \subseteq \mathfrak{P}^{l-a}$, hence $U + \rho D \equiv 0 \mod \mathfrak{P}^{l-a}$. Together with $F \in \GL_n(\mathcal{O})$ this implies \[X \equiv 0 \mod \mathfrak{P}^{l-a}.\] Hence \[\scale(R')=\scale(X G^{-1} \overline{X}^t) \subseteq \mathfrak{P}^{2l-2a}\scale(G^{-1})\subseteq\mathfrak{P}^{2l+1-a}\] and \[\rho \norm(R') = \mathfrak{P}^{2l-2a}\rho\norm(G^{-1})\subseteq\mathfrak{P}^{2l+1-a}=\mathfrak{P}^{2l+1-a}.\] It remains to show that $F' \in \GL_n(\mathcal{O})$. Since $\scale(G^{-1})\subseteq \mathfrak{P}^{1+a}$, we have $\mathfrak{P}^l \mathcal{O}^{n \times n} \pi^{-a}G^{-1} \subseteq \mathfrak{P}^{l+1}\mathcal{O}^{n \times n}$. Therefore $F \equiv F' \mod \mathfrak{P}^{l+1}$. \end{proof}
\begin{theorem}\label{Usurj}
Let $K$ be a non-archimedian local field of characteristic $0$ and $L$ a hermitian $\mathcal{O}$-lattice with even trace lattice.
Then
$U(L) \to U(D_L)$
is surjective. \end{theorem} \begin{proof}
We take an orthogonal splitting $L = M \perp N$ with $M$ being
${\mathfrak{D}_{E/F}^{-1}}$-modular and $\scale(N) \subsetneq {\mathfrak{D}_{E/F}^{-1}}$.
Then ${\mathfrak{D}_{E/F}^{-1}} L^\sharp/L \cong {\mathfrak{D}_{E/F}^{-1}} N^\sharp/N$.
After replacing $L$ with by $N$ we may and will assume that $ \scale(L) \subsetneq {\mathfrak{D}_{E/F}^{-1}}$.
Recall that $\mathfrak{P}^a =\mathfrak{D}_{E/F}$.
Identify $L^\vee={\mathfrak{D}_{E/F}^{-1}}L^\sharp$ with $\mathcal{O}^n= \mathcal{O}^{1 \times n}$ by choosing a basis. Let $G$ be the respective gram matrix of ${\mathfrak{D}_{E/F}^{-1}}L^\sharp$. We have $L = \mathcal{O}^n \pi^{-a} G^{-1}$ and $p^{-a} G^{-1}$ is the corresponding Gram matrix of $L$.
Therefore
\[\scale(G^{-1}) \subseteq \mathfrak{P}^{1+a} \text{ and }\norm(G^{-1})\subseteq \mathfrak{P}^{e+a}.\]
If $E/K$ is unramified or split, then $e=0$, $\scale(G^{-1})=\norm(G^{-1})$ by \cref{norm-vs-scale} and we find $\rho \in \mathcal{O}$ with $\Tr(\rho)=1$. Therefore $\rho\norm(G^{-1}) =\rho \scale(G^{-1}) \subseteq \scale(G^{-1})\subseteq \mathfrak{P}^{1+a}$ holds.
If $E/K$ is ramified we find $\rho \in E$ with $\rho \mathcal{O} = \mathfrak{P}^{1-e}$. Then
$\rho \norm(G^{-1}) \subseteq \mathfrak{P}^{1-e+e+a}=\mathfrak{P}^{1+a}$ as well.
Let $f \in U(\disc{L})$ be represented by $F \in \GL_n(\mathcal{O})$,
that is,
\[f(x + L) = x F + L = x F + \mathcal{O}^{n} \pi^{-a}G^{-1} .\]
Set $R = G - F G \bar F^t$. Since $f$ preserves $\bar h$ and $\bar q$, we have
\[\scale(R) \subseteq \mathfrak{P}^{-a} \text{ and } \rho\norm(R) \subseteq \rho\mathcal{D}_{K/F}^{-1}\subseteq\rho\mathfrak{P}^{e-a}\subseteq\mathfrak{P}^{-a}.\]
where in the last equality we used that $\rho\mathfrak{P}^e \subseteq \mathcal{O}$ irrespective of $E/K$ being inert, split or ramified.
Set $F_0 = F$. We inductively define a sequence by setting $F_{i+1}$ to be the output of
\Cref{alg:lift} with $l = 2^i-1$, $F \gets F_i$, $G \gets G$ as given and
$R \gets R_i:= G - F_i G \bar F_i^t$.
Then $\scale(R_i) \subseteq \mathfrak{P}^{2^i-1 -a }$ and $\rho\norm(R_i) \subseteq \mathfrak{P}^{2^i-1-a}$.
Since $F_i \equiv F_{i+1} \mod \mathfrak{P}^{2^i}$, the sequence $(F_i)_{i \in \mathbb{N}}$ converges.
Its limit is the desired lift.
\end{proof}
For a hermitian lattice $L$ with even trace lattice, we denote by $U^\sharp(L)$ the kernel of $U(L) \to U(\disc{L})$.
\subsection{Local to global} Let $E/K$ be a quadratic extension of number fields with non-trivial automorphism $\bar{\phantom{x}} \colon E \to E$. Let $\o$ be the maximal order of $K$ and $\mathcal{O}$ the maximal order of $E$. In this section $L$ is a hermitian $\mathcal{O}$-lattice with even trace lattice. The goal of this subsection is to compute the image of the natural map \[D\colon U(L) \rightarrow U(\disc{L}).\]
Denote by $\mathbb{A}_K$ the ring of finite adeles of $K$. Denote by $\o_{\mathbb{A}}$ the ring of integral finite adeles of $K$. We have natural isomorphisms $\disc{L} \cong \disc{L} \otimes \o_{\mathbb{A}} \cong \disc{L \otimes \o_{\mathbb{A}}}$. Via the diagonal embedding we view $K$ as a subring of $\mathbb{A}_K$.
This induces the inclusion $U(V)\subseteq U(L \otimes \mathbb{A}_K)$. Let $\det \colon U(L \otimes \mathbb{A}_K) \to \prod_\mathfrak{p} E_\mathfrak{p}$ denote the componentwise determinant. Set \[\mathcal{F}(E) = \{(x)_\mathfrak{p} \in \prod_{\mathfrak{p}} E_\mathfrak{p} \mid x \in E,\, x \bar x = 1\},\]\\[-25pt] \begin{align*} &\mathcal{F}(L_\mathfrak{p}) = \det(U(L_\mathfrak{p})), &&\mathcal{F}^\sharp(L_\mathfrak{p})=\det(U^\sharp(L_\mathfrak{p})), \\ &\mathcal{F}(L) =\det(U(L \otimes \o_{\mathbb{A}})), & &\mathcal{F}^\sharp(L) = \det(U^\sharp(L\otimes \o_{\mathbb{A}})) \end{align*} Note that $U(L \otimes \o_{\mathbb{A}}) \to U(\disc{L})$ is surjective by \Cref{Usurj}. The following commutative diagram \[ \begin{tikzcd}
1\arrow[r]& U^\sharp(L\otimes \o_{\mathbb{A}}) \arrow[r]\arrow[d,"\det"] &U(L\otimes \o_{\mathbb{A}}) \arrow[r]\arrow[d,"\det"]& U(\disc{L})\arrow[r]\arrow[d] & 1\\ 1\arrow[r]& \mathcal{F}^\sharp(L)\arrow[d] \arrow[r] &\mathcal{F}(L)\arrow[d]\arrow[r] & \mathcal{F}(L)/\mathcal{F}^\sharp(L)\arrow[d]\arrow[r]&1\\ & 1 &1 &1\\ \end{tikzcd}\] with exact rows and columns summarizes the situation.
\begin{proposition}\label{kernel:weak-approx} Let $V$ be a non-degenerate hermitian space over $E/K$. Then $\det(U(V))=\mathcal{F}(E)$. \end{proposition} \begin{proof}
We know $\det(U(V))\subseteq \mathcal{F}(E)$. The other inclusion is clear when $\dim_E V$ is one.
Since we can always split a subspace of dimension one, the statement follows. \end{proof} Set $\mathcal{O}_\mathbb{A}:= \mathcal{O} \otimes \o_\mathbb{A}$. For an isometry $f\colon L \to M$ of hermitian $\mathcal{O}_{\mathbb{A}}$-lattices, we denote by $D_f=(D_{f_\mathfrak{p}})_\mathfrak{p}$ the induced map on the discriminant forms. Let $L \subseteq L^\vee$ be a hermitian $\mathcal{O}_{\mathbb{A}}$-lattice. Then $\sigma \in O(L \otimes \mathbb{A}_K)$ induces an isometry $\sigma \colon L \to \sigma(L)$ of $\mathcal{O}_{\mathbb{A}}$-lattices and an isometry $D_\sigma\colon \disc{L} \to \disc{\sigma(L)}$ of the respective discriminant groups.
\begin{proposition}\label{kernel:global}
Let $L$ be an indefinite hermitian $\mathcal{O}$-lattice with $\rk(L)\geq 2$.
For $\sigma \in U(L\otimes \mathbb{A}_K)$ the following are equivalent:
\begin{enumerate}
\item There is a map $\varphi \in U(L \otimes K)$ such that $D_\varphi=D_\sigma$ and $\varphi(L \otimes \o_{\mathbb{A}})=\sigma(L\otimes \o_{\mathbb{A}})$.
\item $\det(\sigma) \in \mathcal{F}(E)\cdot \mathcal{F}^\sharp(L)$.
\end{enumerate} \end{proposition} \begin{proof}
First suppose that a map $\varphi$ as in (1) exists.
Since $\varphi(L\otimes \o_{\mathbb{A}})=\sigma(L\otimes \o_{\mathbb{A}})$ and $D_\varphi = D_\sigma$, we have $\varphi^{-1} \circ \sigma \in U^\sharp(L\otimes \o_{\mathbb{A}})$.
Thus \[\det(\sigma) \in\mathcal{F}(E)\cdot \mathcal{F}^\sharp(L).\]
Now suppose that $\det(\sigma) \in \mathcal{F}(E)\cdot \mathcal{F}^\sharp(L)$. Then there exists $u \in \mathcal{F}(E)$ and $\rho \in U^\sharp(L \otimes {\o_{\mathbb{A}}})$ such that $\det(\sigma) = u \det(\rho)$. By \Cref{kernel:weak-approx} there exists $\psi \in U(L\otimes K)$ with $\det(\psi)=u$. Let $\phi := \psi^{-1} \circ \sigma \circ \rho^{-1}$. Then $\det(\phi)=1$. By the strong approximation theorem \cite{kneser1966}, there exists $\eta \in U(L\otimes K)$ with $\eta(L \otimes \o_{\mathbb{A}})=\phi(L\otimes \o_{\mathbb{A}})$ and $D_\eta = D_\phi$ (approximate $\phi$ at the finitely many primes dividing the discriminant and those with $\phi_\mathfrak{p}(L_\mathfrak{p})\neq L_\mathfrak{p}$). Set $\varphi := \psi \circ \eta \in U(L \otimes K)$. Then \[\varphi(L\otimes \o_{\mathbb{A}}) = (\psi \circ \eta)(L\otimes \o_{\mathbb{A}}) = (\psi \circ \phi)(L\otimes \o_{\mathbb{A}}) = (\sigma \circ \rho^{-1})(L\otimes \o_{\mathbb{A}})=\sigma(L\otimes \o_{\mathbb{A}}).\]
Further \[D_{\varphi} =D_{\psi} \circ D_{\eta}= D_{\sigma} \circ D_{\rho}^{-1} = D_{\sigma}\] since $D_{\rho}$ is the identity because $\rho\in U^\sharp(L \otimes \o_{\mathbb{A}})$. \end{proof}
\begin{theorem}\label{hermitianUtoUDL}
Let $L$ be an indefinite hermitian $\mathcal{O}$-lattice with $\rk(L)\geq 2$. Then there is an exact sequence
\[U(L)\rightarrow U(\disc{L}) \xrightarrow[]{\delta} \mathcal{F}(L)/(\mathcal{F}(E)\cap \mathcal{F}(L))\cdot \mathcal{F}^\sharp(L) \rightarrow 1\]
where $\delta$ is induced by the determinant. \end{theorem} \begin{proof}
Let $\hat\gamma \in U(\disc{L})$ and lift it to some $\gamma \in U(L \otimes \o_{\mathbb{A}})$ with $D_{\gamma}=\bar \gamma$. By \Cref{kernel:global}, $\gamma$ lies in $U(L\otimes K)$ if and only if $\det(\gamma) \in \mathcal{F}(E)\cdot \mathcal{F}^\sharp(L)$ which is equivalent to \[\det(\gamma) \in (\mathcal{F}(E)\cdot \mathcal{F}^\sharp(L)) \cap \mathcal{F}(L)= (\mathcal{F}(E) \cap \mathcal{F}(L)) \cdot \mathcal{F}^\sharp(L) \] and this does not depend on the choice of lift $\gamma$ of $\bar \gamma$. We conclude with the general fact that $U(L) = U(L \otimes K) \cap U(L \otimes \o_{\mathbb{A}})$. \end{proof} In order to make \Cref{hermitianUtoUDL} effective, we will compute the groups $\mathcal{F}^\sharp(L_\mathfrak{p})$ and $\mathcal{F}(L_\mathfrak{p})$ in \Cref{kergen}. See \Cref{detUsharp} for the exact values.
\begin{remark}\label{rem:hermitianUtoUDL} Let $S$ be the set of primes of $K$ dividing the order of $\disc{L}$. For practical purposes we note that $\mathcal{F}(L_\mathfrak{p})=\mathcal{F}^\sharp(L_\mathfrak{p})$ for the primes not in $S$. Hence $\mathcal{F}(L)/\mathcal{F}^\sharp(L)\cong \prod_{\mathfrak{p} \in S}\mathcal{F}(L_\mathfrak{p})/\mathcal{F}^\sharp(L_\mathfrak{p})$ and it is enough to compute $\delta_\mathfrak{p}$ for the primes in $S$. This can be achieved by lifting $\bar \gamma \in U(\disc{L_\mathfrak{p}})$ to some $\gamma \in U(L_\mathfrak{p})$ with sufficient precision using \Cref{alg:lift}. \end{remark}
\subsection{Generation of $\mathbf{U^\sharp(L)}$ by symmetries.}\label{kergen}
Let $K$ be a finite extension of $F= \mathbb{Q}_p$ and $E/K$ a ramified quadratic extension. Let $\Tr = \Tr^E_K$ be the trace. Recall that $\mathfrak{P}\subseteq \mathcal{O}$ is the largest ideal invariant under the involution of $E/K$, $\mathfrak{p}$ the maximal ideal of $\o$ and the integers $a, e$ satisfy $\mathfrak{P}^e = \mathfrak{D}_{E/K}$ and $\mathfrak{P}^a = \mathfrak{D}_{E/F}$.
Note that as $E/K$ is ramified we have $a \equiv e \mod 2$. Let $\pi \in \mathcal{O}$ be a prime element and $p = \pi \bar \pi$. For any $v \equiv e \mod 2$, there exists a skew element $\omega \in E^\times$ with $\nu_\mathfrak{P}(\omega)=v$.
Let $V$ be a non-degenerate hermitian space over $E$.
In what follows $L$ is a full $\mathcal{O}$-lattice in $V$ with even trace form. Therefore its scale and norm satisfy \[\scale(L)=:\mathfrak{P}^i\subseteq \mathfrak{P}^{-a}\text{ and }\norm(L)=:\mathfrak{p}^k\subseteq \mathfrak{D}_{K/F}^{-1} = \mathfrak{P}^{e-a}.\] This gives the inequalities $0 \leq i+a$ and $0 \leq 2k+a-e$ and by \cref{norm-vs-scale} $i \leq 2k \leq i+e$. We say that $L$ is \emph{subnormal} if $\norm(L)\mathcal{O}\subsetneq \scale(L)$, i.e., $i<2k$. A sublattice of rank two is called a plane and a sublattice of rank one a line. By \cite[Propositions 4.3, 4.4]{Jacobowitz1962}, the lattice $L$ decomposes into an orthogonal direct sum of lines and subnormal planes.
The group $U^\sharp(L)$ is the kernel of the natural map \[U(L) \rightarrow U(\disc{L}).\] For $\varphi \in U(L)$ we have $\varphi \in U^\sharp(L)$ if and only if $(\varphi - \id_L)(L^\vee) \subseteq L$. For $x,y \in L$ we write $x \equiv y \mod \mathfrak{P}^i$ if $x-y \in \mathfrak{P}^i L$.
We single out the elements of $U(V)$ fixing a hyperplane -- the symmetries. \begin{definition}
Let $V$ be a hermitian space, $s \in V$ and $\sigma \in E^\times$ with $\sprodq{s}{s}=\Tr(\sigma)$. We call the linear map
\[S_{s,\sigma} \colon V \to V, \quad x \mapsto x -\sprod x s \sigma^{-1}s\]
a \emph{symmetry} of $V$. It preserves the hermitian form $\sprod{\cdot\,}{\cdot}$.
If $s$ is isotropic, then we have $\det(S_{s, \sigma}) = 1$ and
otherwise $\det(S_{s,\sigma})=-\overline\sigma/\sigma$.
The inverse is given by $S_{s,\sigma}^{-1}=S_{s,\bar \sigma}$.
Note that the symmetry $S_{s, \sigma}$ of $V$ preserves $L$
if $s \in L$ and $\sprod L s \subseteq \mathcal{O} \sigma$.
We denote the subgroup of $U(L)$ generated by the symmetries preserving $L$ by $S(L)$ and set $S^\sharp(L) = U^\sharp(L) \cap S(L)$.
\end{definition}
By \cite{brandhorsthofmann2021} symmetries generate the unitary group $S(L)=U(L)$ if $\mathcal{O}/\mathfrak{P} \neq \mathbb{F}_2$. Otherwise one has to include so called rescaled Eichler isometries, which are isometries fixing subspaces of codimension $2$. Fortunately, as we will see, symmetries suffice to generate $U^\sharp(L)$. The condition that the trace form on $L$ is even eliminates all the technical difficulties of \cite{brandhorsthofmann2021}.
\begin{lemma}\label{ker:congruence}
Let $\varphi \in U^\sharp(L)$ and $x \in L$.
Then $\varphi(x) - x \in \sprod{x}{L} \mathfrak{P}^a L$. \end{lemma} \begin{proof} For any $x \in L$, the inclusion $\sprod{x}{L}^{-1}x \subseteq L^\sharp$ gives \[\mathfrak{P}^{-a}\sprod{x}{L}^{-1} x \subseteq \mathfrak{P}^{-a}L^\sharp = L^\vee .\]
Hence, $(\varphi(x) -x)\sprod{x}{L}^{-1}\mathfrak{P}^{-a} \subseteq (\varphi - \id_V)(L^\vee) \subseteq L$.
Multiply by the ideal $\sprod{x}{L}\mathfrak{P}^{a}$ to reach the conclusion. \end{proof}
\begin{lemma}\label{lem:refl-ker}
Let $S_{s, \sigma}$ be a symmetry of $V$ with $s \in \mathfrak{P}^{i+a} L$. Then $S_{s, \sigma}
\in U^\sharp(L)$ if $\mathfrak{P}^{2i+a} \subseteq \sigma \mathcal{O}$. \end{lemma} \begin{proof}
We have $(\id_V-S_{s,\sigma})(\mathfrak{P}^{-a}L^\sharp) = \sprod{\mathfrak{P}^{-a}L^\sharp}{s}\sigma^{-1}s \subseteq \mathfrak{P}^{i} \sigma^{-1} s \subseteq L$. \end{proof}
\begin{lemma}\label{line}
Let $x, x' \in L$ with $\sprodq{x}{x}=\sprodq{x'}{x'}$,
$\sprod{x}{L}=\sprod{x'}{L}=\mathfrak{P}^i$ and $x \equiv x' \mod \mathfrak{P}^{i+a}$.
Then there is an element $\varphi \in S^\sharp(L)$ with $\varphi(x) = x'$. \end{lemma} \begin{proof}
Note that $\sprod{x}{x-x'} \in \mathfrak{P}^{2i+a}$.
If $\sprod{x}{x-x'}\mathcal{O} =\mathfrak{P}^{2i+a}$. Then with $\sigma = \sprod{x}{x-x'}$ and $s = x - x'$ we have $S_{s, \sigma}(x) = x'$, and \Cref{lem:refl-ker} implies that
$S_{s,\sigma} \in U^\sharp(L)$.
If $\sprod{x}{x-x'}\mathcal{O} \subseteq \mathfrak{P}^{2i+a+1}$,
choose $s \in \mathfrak{P}^{i+a}L$ with
\[\sprod{s}{x}\mathcal{O} = \sprod{s}{x'}\mathcal{O} = \mathfrak{P}^{2i+a} \]
which is possible since $\sprod{x}{L}=\sprod{x'}{L}=\mathfrak{P}^i$.
We have $\nu_\mathfrak{P}(\sprodq s s \rho)\geq 2i+2a+2k+1-e > 2i + a$ and $2i+a \equiv e \mod 2$.
With $\omega \in E$ a skew element of valuation $2i+a$, $\sigma := \sprodq{s}{s}\rho + \omega$ satisfies $\nu_\mathfrak{P}(\sigma)=2i+a$.
By \Cref{lem:refl-ker} we have
$S_{s,\sigma} \in S^\sharp(L)$.
Then \[\sprod{x}{x - S_{s,\sigma}(x')}=\sprod{x}{x-x'} + \sprod{x}{s}\sprod{s}{x'}\bar \sigma^{-1}\]
gives $\sprod{x}{x - S_{s,\sigma}(x')}\mathcal{O} = \mathfrak{P}^{2i+a}$. Further $x \equiv S_{s,\sigma}(x') \mod \mathfrak{P}^{i+a}$. Thus by the first case we can map $x$ to $S_{s,\sigma}(x')$. \end{proof}
\begin{lemma}\label{plane}
Let $L = P \perp M$, with $P$ a subnormal plane.
Then $U^\sharp(L) = S^\sharp(L)U^\sharp(M)$. \end{lemma} \begin{proof}
By~\cite{Jacobowitz1962} there exists a basis $u, v \in P$ with $\sprodq{u}{u} = p^k$, $\sprodq{v}{v}\in \mathfrak{p}^{k}$ and
$\sprod{u}{v}=\pi^i$. Note that $L$ subnormal implies $i<2k$. Let $\varphi \in U^\sharp(L)$.
By \Cref{line} there exists a symmetry $S \in S^\sharp(L)$ with $S(u) = \varphi(u)$. Therefore we may and will assume that $\varphi(u) = u$.
Write $\varphi(v) = \gamma u + \delta v + m$ for some $m \in \mathfrak{P}^{i+a}L$ and $\gamma,1-\delta \in \mathfrak{P}^{i+a}$.
Then we have
\[\sprod{v}{v-\varphi(v)}\mathcal{O}=(-\bar \gamma \bar \pi^i+(1-\bar\delta)\sprodq{v}{v})\mathcal{O}\]
\begin{equation}\label{deltaunit}
\sprod{u}{v-\varphi(v)} = \sprod{u}{v}-\sprod{\varphi(u)}{\varphi(v)} = 0.
\end{equation}
The symmetry $S_{s,\sigma} \in U(L\otimes E)$ with $s=v - \varphi(v)$ and $\sigma = \sprod{v}{v-\varphi(v)}$ preserves $u$ and maps $v$ to $\varphi(v)$.
If $\nu_\mathfrak{P}(\gamma) = i+a$, then
\[ \nu_\mathfrak{P}((1 - \bar \delta)\sprodq v v) \geq i + a + 2k > 2i + a = \nu_\mathfrak{P}(-\bar \gamma \bar \pi^i). \]
Thus $\sprod{v}{v-\varphi(v)}\mathcal{O} = \mathfrak{P}^{2i+a}$.
It follows that $S_{s,\sigma} \in S^\sharp(L)$ by \Cref{lem:refl-ker} and we are done.
Let now $\nu_\mathfrak{P}(\gamma)> i +a$. We
consider $v' = u - \pi^i p^{k-i}v \in L$. It satisfies
\[\sprod u {v'} = 0, \quad \sprod{v'}{v}\equiv \pi^i \mod \mathfrak{P}^{i+1} \quad \text{and} \quad v' \equiv u \mod \mathfrak{P}.\]
In particular, $v_\mathfrak{P}(\sprod {v'}v) = i$.
Set $s = \pi^{i+a}v'$ and let $\omega\in E$ be a skew element such that $\nu_\mathfrak{P}(\omega) = 2i + a$.
Since $\nu_\mathfrak{P}(\sprodq{s}{s} \rho)
> 2i+a$, the element $\sigma = \rho \sprodq{s}{s} + \omega$ satisfies $\Tr(\sigma) = \sprodq{s}{s}$ and
$\nu_\mathfrak{P}(\sigma) = 2i + a$.
We have $S_{s,\sigma} \in S^\sharp(L)$,
$S_{s,\sigma}(u) = u$ and $S_{s,\sigma}(\varphi(v)) = \gamma' u + \delta' v + w$
with $\gamma',\delta' \in \mathcal{O}$ and
\[\gamma' = \gamma - \sprod{\gamma u + \delta v}{\pi^{i+a}v'}\sigma^{-1} \pi^{i+a}
= \gamma - \delta \sprod{v}{v'} p^{i+a}\sigma^{-1}.\]
Since $\delta \in \mathcal{O}^\times$ by \cref{deltaunit} and $\nu_\mathfrak{P}(\gamma)>i+a$, we have
$\nu_\mathfrak{P}(\gamma') = i+a$. We conclude as in the first case. \end{proof}
\begin{theorem}\label{ker:ramif}
Let $E/K$ be ramified. Then we have $U^\sharp(L) = S^\sharp(L)$. \end{theorem} \begin{proof} We proceed by induction on the rank of $L$. We know that $L = M \perp N$ with $M$ a line or a subnormal plane. By \Cref{line,plane} we have $U^\sharp(L) = S^\sharp(L)U^\sharp(N)$. By induction $U^\sharp(N)=S^\sharp(N)$.
\end{proof}
\begin{remark} For $E/K$ unramified or $E=K \times K$ one can prove that $U^\sharp(L)= S^\sharp(L)$ as well. Since we do not need this result for the computation of $\det(U^\sharp(L))$, the proof is omitted. \end{remark}
\subsection{Determinants of the kernel} We use the same assumptions and notation as in \Cref{subsec:localsurjectivity}. In particular we are in the local setting. Let $\delta \in E$ be of norm $\delta \bar \delta = 1$ and $x \in V$ be anisotropic. A \emph{quasi-reflection} is a map of the form \[\tau_{x,\delta}\colon V \rightarrow V, y \mapsto y + (\delta-1)\frac{\sprod{y}{x}}{\sprodq{x}{x}} x.\] We have $\tau_{x,\delta} \in U(V)$ and $\det(\tau_{x,\delta}) = \delta$. Let $s = x$ and $\sigma = \sprodq{x}{x}(1-\delta)^{-1}$. Then $\tau_{x,\delta}=S_{s,\sigma}$. Conversely, if $s$ is anisotropic and $\sigma \in E$ with $\Tr(\sigma)=\sprodq{s}{s}$, set $\delta = - \bar \sigma / \sigma$, then $S_{s,\sigma} = \tau_{x,\delta}$. Thus the quasi-reflections are exactly the symmetries at anisotropic vectors. The symmetries at isotropic vectors are called transvections.
\begin{lemma}\label{ker:refl} Let $x \in L$ be primitive, anisotropic and $\delta \in E$ of norm $\delta \bar \delta = 1$. Then $\tau_{x,\delta} \in U^\sharp(L)$ if and only if $(\delta - 1) \in \mathfrak{P}^{a} \sprodq{x}{x}.$ \end{lemma} \begin{proof}
We have $\tau_{x,\delta} \in U^\sharp(L)$ if and only if
$(\tau_{x,\delta}-\id_V)(L^\vee)\subseteq L$. This amounts to $(\delta-1)\mathfrak{P}^{-a}\sprodq{x}{x}^{-1}x \in L$.
The lemma follows since $x$ is primitive. \end{proof}
For $i\geq 0$ set \begin{eqnarray*}
\mathcal{E}_0 & = &\{u \in \mathcal{O}^\times \mid u \bar u =1\}\\ \mathcal{E}^i &=& \{u \in \mathcal{E}_0 \mid u \equiv 1 \mod \mathfrak{P}^i\}. \end{eqnarray*} Note that $\mathcal{E}_0 = \mathcal{E}^0 = \mathcal{E}^{e-1}$, $\mathcal{E}_1:= \{u \bar u ^{-1} \mid u \in \mathcal{O}^\times\} = \mathcal{E}^e$ and $[\mathcal{E}_0:\mathcal{E}_1]=2$ by \cite[3.4, 3.5]{Kirschmer2019}.
\begin{theorem}\label{detUsharp}
Let $F=\mathbb{Q}_p$, $K/F$ a finite field extension, $E/K$ an étale $K$-algebra of dimension $2$ with absolute different $\mathfrak{P}^a:=\mathfrak{D}_{E/F}$.
Suppose that $L$ is a hermitian $\mathcal{O}$-lattice with $\mathfrak{p}^k:=\norm(L) \subseteq \mathfrak{D}_{K/F}^{-1}$.
Then $\det(U^\sharp(L)) =:\mathcal{F}^\sharp(L) = \mathcal{E}^{2k+a}$. \end{theorem} \begin{proof}
Let $x \in L$ be a norm generator, i.e. $\sprodq{x}{x} \o = \norm(L) = \mathfrak{p}^k$.
Let $\delta \in \mathcal{E}^{2k+a}$. Then
\[(\delta-1) \in \mathfrak{P}^{a}\sprodq{x}{x}\subseteq \mathfrak{P}^{a}\norm(L).\]
By \Cref{ker:refl} we have $\tau_{x,\delta} \in U^\sharp(L)$ and so $\delta \in \det(U^\sharp(L))$. Hence
\[\mathcal{E}^{2k+a} \subseteq \det(U^\sharp(L)).\]
Let $\mathfrak{P}^i:=\scale(L)$ and $\varphi \in U^\sharp(L)$. By \Cref{ker:congruence} $\varphi \equiv \id \mod \mathfrak{P}^{i+a}$. Thus $\det(\varphi) \equiv 1 \mod \mathfrak{P}^{i+a}$, that is, $\det(\varphi) \in \mathcal{E}^{i+a}$.
If $E/K$ is unramified, then $\scale(L)=\norm(L)\mathcal{O}$, so that $i=2k$ and
\[\det(U^\sharp(L)) \subseteq \mathcal{E}^{i+a}=\mathcal{E}^{2k+a}.\]
Now suppose that $E/K$ is ramified.
By \Cref{ker:ramif} the group $U^\sharp(L) = S^\sharp(L)$ is generated by symmetries.
Since transvections have determinant one,
it is enough to consider the determinants of the quasi-reflections in $U^\sharp(L)$.
Let $\tau_{x,\delta} \in U^\sharp(L)$ be a quasi-reflection. Recall that $\det(\tau_{x,\delta}) = \delta$.
By \Cref{ker:refl} we have $(\delta -1) \in \mathfrak{P}^a \sprodq{x}{x} \subseteq \mathfrak{P}^a \norm(L)$.
This proves $\det(U^\sharp(L)) \subseteq \mathcal{E}^{2k + a}$.
\end{proof}
\subsection{Computing in $\mathcal{F}(L_\mathfrak{p})/\mathcal{F}^\sharp(L_\mathfrak{p})$} Let $E/K$ be a quadratic extension number fields with rings of integers $\mathcal{O}$ and $\o$ respectively. Let $L$ a hermitian $\mathcal{O}$-lattice. Determining the image of $\delta$ in \Cref{hermitianUtoUDL} requires the computation in the finite quotient $\mathcal{F}(L_\mathfrak{p})/\mathcal{F}^\sharp(L_\mathfrak{p})$, where $\mathfrak{p}$ is a prime ideal of $\o$ (see also \Cref{rem:hermitianUtoUDL}). To simplify notation we now assume that $K$ is a local field of characteristic $0$, $E/K$ an étale $K$-algebra of dimension $2$ and the notation as in \Cref{kergen}. Hence our aim is to be able to do computations in $\mathcal{F}(L)/\mathcal{F}^\sharp(L)$. As we are only interested in computing in the abelian group as opposed to determining it completely, it is sufficient to describe the computation of the supergroup $\mathcal{E}_{0}/\mathcal{F}^\sharp(L)$. By \Cref{detUsharp} we know that $\mathcal{F}^\sharp(L) = \mathcal{E}^i$ for some $i \in \mathbb{Z}_{\geq 0}$. It is thus sufficient to describe the computation of $\mathcal{E}_{0}/\mathcal{E}^i$. By definition this group is isomorphic to $\ker(\overline \Nr_i)$, where \[ \overline \Nr_i \colon \mathcal{O}^\times{/}(1 + \mathfrak{P}^i) \longrightarrow \o^\times{/}\Nr(1 + \mathfrak{P}^i), \bar u \longmapsto \overline{\Nr(u)}. \] Depending on the structure of the extension $E/K$, this kernel can be described as follows: \begin{itemize}
\item
If $E \cong K \times K$, then $\mathcal{E}_0/\mathcal{E}^i$ is isomorphic to $(\o/\mathfrak{p}^i)^\times$.
\item
If $E/K$ is an unramified extension of local fields, then $\mathcal{E}_0/\mathcal{E}^i$ is isomorphic to
the kernel of the map $(\mathcal{O}/\mathfrak{P}^i)^\times \rightarrow (\o/\mathfrak{p}^i)^\times, \, \bar u \mapsto \overline{\Nr(u)}$. \item
If $E/K$ is a ramified extension of local fields, then the situation is more complicated due to the norm not being surjective.
Using the fact that by definition we have \[ \mathcal{E}^i/\mathcal{E}^{i+1} \cong \ker \left((1 + \mathfrak{P}^i)/(1 + \mathfrak{P}^{i + 1}) \longrightarrow \Nr(1 + \mathfrak{P}^i)/\Nr(1 + \mathfrak{P}^{i+1}),\, \overline{u} \longmapsto \overline{\Nr(u)}\right), \]
this quotient can be determined using explicit results on the image of the multiplicative groups $1 + \mathfrak{P}^i$ under the norm map, found for example in
\cite[Chap. V]{Serre1979}.
Applying this iteratively we obtain $\mathcal{E}_0/\mathcal{E}^i$. \end{itemize}
In all three cases the computations of the quotient groups $\mathcal{E}_0/\mathcal{E}^i$ reduce to determining unit groups of residue class rings or kernels of morphisms between such groups. These unit groups are finitely generated abelian groups, whose structure can be determined using classical algorithms from algebraic number theory, see for example \cite[Sec. 4.2]{Cohen2000}.
\section{Fixed Points}\label{sec:fixed} We classify the fixed point sets of purely non-symplectic automorphisms of finite order $n$ on complex K3 surfaces. We only use the description of the fixed locus for $n=p$ a prime (see e.g. \cite{artebani-sarti-taki}); hence providing an independent proof in the known cases and completing the classification in all other cases.
Given the action of $\sigma$ on some lattice $L \cong H^2(X,\mathbb{Z})$, we want to derive the invariants $((a_1,\dots a_s),k,l,g)$ of the fixed locus $X^\sigma$ as defined in the introduction. The topological and holomorphic Lefschetz' fixed point formula \cite[Thm. 4.6]{atiyah-singer1968} yield the following relations \[\sum_{i=1}^s a_i -2k + l(2 - 2g)
= 2 + \Tr \sigma^*|H^2(X,\mathbb{C})\] and \[1+\zeta_n^{-1} = \sum_{i=1}^s \frac{a_i}{(1-\zeta_n^{i+1})(1-\zeta_n^{-i})}+l(1-g) \frac{1+\zeta_n}{(1-\zeta_n)^2}.\] We adopt the following strategy: By induction, we know the invariants of the fixed loci of $\sigma^p$ for $p \mid n$. Note that $\sigma$ acts with order dividing $p$ on $X^{\sigma^p}$ and $X^\sigma \subseteq X^{\sigma^p}$. From the fixed loci of $\sigma^p$, we derive the obvious upper bounds on $k$, $l$ and $g$. Then for each possible tuple $(k,l,g)$, we find all $(a_1,\dots a_s)$ satisfying the Lefschetz formulas, which amounts to enumerating integer points in a bounded polygon. The result is a finite list of possibilities for the invariants of $X^\sigma$.
In what follows, we derive compatibility conditions coming from the description of $X^\sigma$ as the fixed point set of the action of $\sigma$ on $X^{\sigma^p}$. \begin{lemma}\label{fixed points on curves}
Let $P$ be an isolated fixed point of type $i$ of $\sigma$. Let $p m = n$ with $p$ prime. Then $P$ is a fixed point of $\sigma^p$ of type
\[t(i) = \min\{i + 1 \bmod m,n -i \bmod m\} - 1.\]
Moreover, we have
\[a_i(\sigma) \leq a_{t(i)}(\sigma^p) \quad \mbox{ and } \quad \sum_{\{i | t(i)=j\}}a_i(\sigma) \equiv a_{j}(\sigma^p) \mod p\]
where $1 \leq j \leq (m-1)/2$. \end{lemma} \begin{proof}
In local coordinates $\sigma^p(x,y) = (\zeta_m^{i+1},\zeta_m^{-i})$. Note that $\sigma$ acts on the set of fixed points of type $j$ of $\sigma^p$. Hence the number of fixed points $ \sum_{\{i|t(i)=j\}}a_i(\sigma)$ of $\sigma$ is congruent to the order $a_j(\sigma^p)$ of this set modulo $p$. \end{proof}
In particular, from the invariants of $\sigma$, we can infer how many isolated fixed points of $\sigma$ lie on a fixed curve of $X^{\sigma^p}$. More precisely, by \Cref{fixed points on curves} $\sum_{\{i|t(i)=0\}}a_i(\sigma)$ is the number of isolated fixed points of $\sigma$ which lie on a fixed curve of $\sigma^p$. The number of such points is bounded above and below by the following lemma. \begin{lemma}
Let $p$ be a prime number, $C$ a smooth curve of genus $g$ and $\sigma \in \Aut(C)$ an automorphism of order $p$ with $C/\sigma$ of genus $g'$. Then $\sigma$ fixes
\[r = \frac{2g -2 - p(2g'-2)}{p-1}\]
points. In particular, given $p$ and $g$ there is a finite number of possibilities for $r$.
Note that for $g = 0$ we have $r = 2$. \end{lemma} \begin{proof}
The canonical map $\pi\colon C \to C/\sigma$ is ramified precisely in the fixed points and with multiplicity $p$.
By the Hurwitz formula $2g -2 = p (2 g' - 2) + (p-1)r$. \end{proof}
Inductively carrying out this strategy, we obtain a unique possibility in most cases and in the remaining $33$ cases two possibilities. We call the corresponding automorphisms ambiguous. In what follows we disambiguate by using elliptic fibrations. We say that a curve $C$ is fixed by $\sigma$ if $\sigma|_C=\id_C$. If merely $\sigma(C)=C$ we say that it is invariant. \begin{lemma}\label{lemma:sect}
Let $p$ be a prime divisor of $n$. Suppose that $\sigma^{n/p}$ fixes an elliptic curve $E$. Denote by $\pi\colon X \to \mathbb{P}^1$ the elliptic fibration induced by the linear system $|E|$.
Suppose $\sigma$ has no isolated fixed points on $E$ and $\sigma^m$ leaves invariant a section of $\pi$ for some $m \mid n, m \neq n$.
Then $\sigma$ fixes $E$ if and only if $\pi$ admits a $\sigma$-invariant section. \end{lemma} \begin{proof}
Since we assume that $\sigma$ has no isolated fixed points on $E$,
either $\sigma$ fixes $E$ entirely or no point on $E$ at all.
Suppose that $S$ is a $\sigma$-invariant section. Then $E \cap S$ is a fixed point. Therefore $\sigma$ fixes $E$.
Conversely, suppose that there is no $\sigma$-invariant section. By assumption we find a $\sigma^m$-invariant section $S$ which must satisfy $\sigma(S) \neq S$. If $\sigma$ acts trivially on $E$, then $\{P\}=E \cap S \cap \sigma(S)$ is a fixed point. The three $\sigma^m$-invariant curves $E$, $S$ and $\sigma(S)$ pass through $P$. This contradicts the local description of the action around $P$. Therefore $\sigma$ must act as a translation on $E$. \end{proof}
\begin{lemma}\label{identifyE}
Let $\tau$ be an automorphism of prime order $p$ of a K3 surface $X$ acting trivially on $\NS(X)$ and fixing an elliptic curve $E$. If $f \in \NS(X)$ is isotropic, primitive and nef such that $f^\perp /\mathbb{Z} f$ is not an overlattice of a root lattice, then $f = [E]$. \end{lemma} \begin{proof}
Since $\tau$ acts trivially on $\NS(X)$, it lies in the center of the automorphism group $\Aut(X)$ and fixes $E$. Hence every automorphism leaves $E$ invariant, i.e., $E$ is a curve canonically defined on $X$.
Let $\pi\colon X \to \mathbb{P}^1$ be the genus one fibration defined by $|E|$. The fibration $\pi$ is canonically defined, therefore $\Aut(X)$ is virtually abelian of rank $t$ given by the rank of the Mordell--Weil group of (the Jacobian of) $\pi$. Let $R$ be the root sublattice of $[E]^\perp/\mathbb{Z}[E]$. Then by the Shioda--Tate formula \cite[5.2]{shioda1990} $t=\rk \NS(X)-\rk R -2$. Since $\Aut(X)$ is virtually abelian, there is at most one elliptic fibration of positive rank. By \cite[\S 3]{shafarevich1972}, $f$ is the class of a fiber of an elliptic fibration. Since $f^\perp/\mathbb{Z} f$ is not an overlattice of a root lattice, the Shioda--Tate formula implies that it has positive Mordell--Weil rank. Hence it must coincide with $\pi$ and so $[E] = f$. \end{proof}
\begin{lemma}\label{hassection}
Let $p$ be a prime divisor of $n$. Suppose that $\sigma^p$ fixes an elliptic curve $E$ inducing the elliptic fibration $\pi$. Set $\tau=\sigma^{n/p}$ and $N= \NS(X)^{\tau}$.
Let $f\in \NS(X)^\sigma$ be isotropic and primitive such that $f^{\perp N}/\mathbb{Z} f$ is not an overlattice of a root lattice.
Then $\pi$ has a $\tau$-invariant section if and only if $\langle f, N \rangle = \mathbb{Z}$.
Similarly, $\pi$ has a $\sigma$-invariant section if and only if $\langle f ,\NS(X)^\sigma \rangle=\mathbb{Z}$. \end{lemma} \begin{proof} By \cite[Lemma 1.7 and Theorem 1.8]{oguiso-sakurai} there exists an element $\delta$ of the Weyl group $W(\NS(X))$ commuting with $\sigma$ such that $\delta(f)$ is nef. Hence we may assume that $f$ is nef. In order to apply \Cref{identifyE} set $\tau = \sigma^{n/p}$ and consider $G=\langle \tau \rangle$. Choose a marking $\eta\colon H^2(X,\mathbb{Z})\to L$ and set $H= \eta\rho_X(G) \eta^{-1}$. Then we can deform the $H$-marked K3 surface $(X,G,\eta)$ to $(X',G',\eta')$ such that $\NS(X')^{G'} = \NS(X')$. Let $E'$ be the elliptic curve fixed by $G'$. It satisfies $\eta'([E']) = \eta([E])$. Note that $f' = \eta'^{-1} \circ \eta(f)$ is still nef since $\eta'(\NS(X'))\subseteq \eta(\NS(X))$. Hence by \Cref{identifyE} we get $f'= [E']$ which gives $f = [E]$.
Finally, if $\langle [E], \NS(X)\rangle = \mathbb{Z}$, we can find $s \in \NS(X)$ with $\langle [E],s\rangle=1$ and $s^2=-2$. After possibly replacing $s$ by $-s$ we may assume that $s$ is effective. We can write $s = \delta_1 + \dots + \delta_n$ for $(-2)$-curves $\delta_i \in \NS(X)$. Now, $1=\langle f , s \rangle$ and $\langle f, \delta_i \rangle \geq 0$ ($f$ is nef) imply that $\langle f , \delta_i \rangle =1$ for a single $1\leq i \leq n $; $\delta_i$ is the desired section. Note that $\tau$ (respectively $\sigma$) preserves $\delta_i$ if and only if it preserves $s$. \end{proof}
Among the $22$ ambiguous automorphisms $\sigma$ of order $n=4$, there are $16$ cases where $\sigma^2$ fixes a single elliptic curve, $4$ cases where $\sigma^2$ fixes a curve of genus $2$ and $3$, $5$, $7$ or $9$ rational curves, and in the remaining $2$ cases $\sigma^2$ fixes two curves of genus one. The ambiguity is whether $\sigma$ fixes some curve or not.
First let $\sigma^2$ fix a unique elliptic curve $E$ of genus $1$. This means that $\sigma$ is compatible with an elliptic fibration $\pi\colon X \to \mathbb{P}^1$. Moreover, $\sigma^2$ leaves invariant a section $S$ of $\pi$ because $L^{\sigma^2}$ contains a copy of the hyperbolic plane $U$. Note that $\sigma^2$ must act non-trivially on the base $\mathbb{P}^1$ of the fibration, as otherwise its action at the tangent space to the point in $E\cap S$ would be trivial. Hence $\sigma^2$ has exactly two fixed points in $\mathbb{P}^1$ giving two invariant fibers; one is $E$ and the other one we call $C$. The rational curves fixed by $\sigma^2$ must be components of the fiber $C$, because all fixed points lie in $E \cup C$. We fix the fiber type of $C$ and consider each fiber type separately. We know that $C$ is a singular fiber of Kodaira type $I_{4m}$, $m=1,2,3,4$ or $IV^*$. They correspond to $3,3,3,4$ and $3$ ambiguous cases.\\[4pt]
The following figure shows the dual graph of the irreducible components of $C$. Each node corresponds to a smooth rational curve and two nodes are joined by an edge if and only if the corresponding curves intersect. The square nodes are curves fixed pointwise by $\sigma^2$ and the round nodes are curves which are invariant but not fixed by $\sigma^2$. The automorphism $\sigma$ acts on the graph with order dividing $2$ and maps squares to squares.
\begin{center}
\begin{tikzpicture}[scale=0.5] \node (label) at (1,1) {$IV^*$};
\begin{scope}[every node/.style={draw, fill=black!10, inner sep=1.6pt}]
\node (2) at (2,0) {};
\node (4) at (4,0) {};
\node (4b) at (4,2) {};
\node (6) at (6,0) {};
\end{scope}
\begin{scope}[circle,every node/.style={fill, inner sep=1.5pt}]
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We start by determining the fixed locus in the case that $C$ is of type $I_{4m}$. Lefschetz calculations show that for the given $\sigma^2$ we have the following possibilities for the fixed locus $X^\sigma$ of $\sigma$: four isolated points $((4),0,0,1)$ or four isolated points and a curve of genus one $((4),0,1,1)$.
The automorphism $\sigma^2$ fixes every second node of the circular $I_{4m}$ configuration and the zero section intersects a $\sigma^2$-fixed curve in $I_{4m}$.
The action of $\sigma$ on the intersection graph is visible on the lattice side, since we can identify the class $F$ of $E$ by \Cref{identifyE} and choose simple roots in $F^\perp$ giving its components. Carrying out this computation gives the following $3$ cases. \begin{enumerate} \item The curves in $I_{4m}$ are rotated by $\sigma$. Then $C$ does not have any $\sigma$ fixed points and $E$ contains $4$ isolated fixed points for 0.4.4.9, 0.4.3.6, 0.4.2.5, 0.4.1.3.
\item The automorphism $\sigma$ acts as a reflection on the graph $I_{4m}$ leaving invariant two of the $\sigma^2$-fixed curves and the section $S$ passes through one of them. Then $I_{4m}$ contains $4$ isolated fixed points and $E$ does not contain any isolated fixed points. However the intersection $S \cap E$ is fixed but cannot be isolated. Hence $\sigma$ fixes $E$ for 0.4.4.7, 0.4.3.8, 0.4.2.3, 0.4.1.6.
\item The automorphism $\sigma$ acts as a reflection leaving invariant two of the $\sigma^2$-fixed curves and the section $S$ does not pass through them. Then $\sigma$ fixes 4 isolated points on $I_{4m}$ and no isolated points on $E$. By \Cref{lemma:sect} $\sigma$ does not fix $E$ for 0.4.4.8, 0.4.3.7, 0.4.2.4, 0.4.1.4, 0.4.1.5. \end{enumerate}
Let $C = IV^*$ and $\sigma$ ambiguous. We know that $\sigma$ fixes $6$ isolated points, one rational curve and possibly $E$ of genus $1$. The central curve as well as the three leaves must be fixed by $\sigma^2$. There are $3$ possible actions. It can leave invariant each component of the $IV^*$ fiber. Then the central component is fixed and the leaves carry two fixed points each. Hence the action on $E$ does not have an isolated fixed point. Therefore $\sigma$ fixes $E$ if and only if some section is preserved. This is the case for 0.4.3.11 but not for 0.4.3.10. In the third case $\sigma$ swaps two of the branches. Therefore the central component cannot be fixed by $\sigma$, so it contains $2$ isolated fixed points. The invariant leaf must be the fixed rational curve. Then there are $4$ fixed points left, they must lie on $E$ giving 0.4.3.9. This settles the first $16$ ambiguous cases.
In the next $4$ ambiguous cases $\sigma^2$ fixes a curve of genus $2$ and $3$, $5$, $7$ or $9$ rational curves. In each case we know that $\sigma$ fixes exactly $4$ isolated fixed points. The ambiguity is whether $\sigma$ fixes $1$ rational curve and the genus $2$ curve, or no curve at all. For each case we exhibit a $\sigma$-invariant hyperbolic plane $U$. Since $\sigma$ fixes a curve of genus $2$, $\Aut(X)$ is finite, and so $K = U^\perp$ is a root lattice. Then $\NS(X)=U\perp K$ and $K$ determines the ADE-types of the singular fibers of the $\sigma$-equivariant fibration induced by $U$. The square nodes are fixed by $\sigma^2$ while the round nodes are not. \begin{center} \begin{minipage}{10cm}
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Note that $\sigma$ must act non-trivially on the graph, because otherwise it has too many fixed points or fixed curves.
Since $\sigma$ maps squares to squares, we see that $\sigma$ must act as a reflection preserving the central square.
However, the corresponding curve cannot be fixed, because the two adjacent ones and hence the corresponding intersection points with the central node are swapped.
Thus $\sigma$ cannot fix a curve and the four ambiguous cases are settled.
Consider the ambiguous cases 0.4.5.12, 0.4.5.14 0.6.2.29, 0.6.3.36, 0.8.1.7, 0.8.1.8, 0.8.2.8, 0.8.2.10, 0.9.1.3, 0.9.1.4 and 0.10.1.11. The question is whether or not $\sigma$ fixes an elliptic curve. In each case Lefschetz calculations show that there are no isolated fixed points on the elliptic curve in question. Hence in view of \Cref{lemma:sect} this can be decided by whether or not there exists a section of the corresponding fibration. This is settled by \Cref{hassection}. Indeed, we can randomly search until we find $f \in L^\sigma$ corresponding to $[E]$.
For 0.10.2.1 we have the two possibilities $((0, 0, 1, 6), 0, 0, 1)$ and $((5, 0, 0, 0), 1, 0, 1)$ for the fixed locus of $\sigma$. We know that $\sigma^2$ fixes an elliptic curve $E$ and one rational curve. It is the central component of the invariant fiber $C$ of type $\tilde{E_7}$. Since $\NS(X)^\sigma$ has rank $6$, $\sigma$ must act non-trivially on $\tilde{E_7} \subseteq \NS(X)$. This means that it swaps two of the three arms of the configuration. Hence it cannot act trivially on the central component and so $\sigma$ cannot fix a rational curve. The fixed locus is $((0, 0, 1, 6), 0, 0, 1)$.
For the last ambiguous case 0.12.1.12 the automorphism $\sigma$ has a unique fixed point and the ambiguity is whether or not it fixes an elliptic curve. We know that $\sigma^i$ for $i=2,3,4,6$ fixes a unique elliptic curve $E_i$. Since $E_2\subseteq E_4,E_6$, we have $E_2=E_4=E_6$ and similarly $E_6 \subseteq E_3$ implies $E_6=E_3$. Therefore $E=E_i$ is independent of $i$. Now $\sigma^3$ and $\sigma^4$ fix $E$ hence their product $\sigma^7$ fixes $E$ as well. But $\sigma \in \langle \sigma^7\rangle$ and so $\sigma$ fixes the elliptic curve $E$. The fixed locus of 0.12.1.12 is therefore $((1, 0, 0, 0, 0), 0, 1, 1)$.
\appendix
\section{Finite groups with mixed action on a K3 surface} \label{appendixA} The following table lists all finite groups $G$ admitting a faithful, saturated, mixed action on some K3 surface, their symplectic subgroups $1\neq G_s < G$ as well as the number $k(G)$ of deformation types. Note that in 3 cases an entry appears twice because the normal subgroup $G_s < G$ does not lie in the same $\Aut(G)$-orbit. The notation $G = G_s.\mu_n$ means that $G$ is an extension of $G_s$ by $\mu_n$. It may or may not split. Our notation for the groups $G_s$ follows Hashimoto \cite{Hashimoto2012}. Isomorphism classes of groups will be referred to either using standard notation for classical families or using the id as provided by the library of small groups \cite{Besche2002}.
\renewcommand{1.2}{1.2} {\small
\begin{longtable}{ccc|ccc|ccc}
\caption{Finite groups with faithful, saturated non-symplectic action on some K3 surface}\label{table:action}\\
$G$&$\mathrm{id} $& $k(G)$ & $G$&$\mathrm{id} $& $k(G)$ & $G$ & id & $k(G)$ \\
\hline
\endhead
\rowcolor{lightgray} $C_2.\mu_{2}$&(4, 1)&5 & $C_3.\mu_{15}$&(45, 2)&1 & $C_4.\mu_{12}$&(48, 22)&1\\ $C_2.\mu_{2}$&(4, 2)&354 & $C_3.\mu_{18}$&(54, 4)&1 & $C_4.\mu_{12}$&(48, 23)&1\\ \rowcolor{lightgray} $C_2.\mu_{3}$&(6, 2)&26 & $C_3.\mu_{18}$&(54, 9)&1 & $C_4.\mu_{12}$&(48, 24)&1\\ $C_2.\mu_{4}$&(8, 1)&3 & $C_2^2.\mu_{2}$&(8, 2)&4 & $D_6.\mu_{2}$&(12, 4)&140\\ \rowcolor{lightgray} $C_2.\mu_{4}$&(8, 2)&200 & $C_2^2.\mu_{2}$&(8, 3)&40 & $D_6.\mu_{3}$&(18, 3)&21\\ $C_2.\mu_{5}$&(10, 2)&6 & $C_2^2.\mu_{2}$&(8, 5)&330 & $D_6.\mu_{4}$&(24, 5)&18\\ \rowcolor{lightgray} $C_2.\mu_{6}$&(12, 2)&11 & $C_2^2.\mu_{3}$&(12, 3)&4 & $D_6.\mu_{5}$&(30, 1)&1\\ $C_2.\mu_{6}$&(12, 5)&99 & $C_2^2.\mu_{3}$&(12, 5)&11 & $D_6.\mu_{6}$&(36, 12)&33\\ \rowcolor{lightgray} $C_2.\mu_{7}$&(14, 2)&4 & $C_2^2.\mu_{4}$&(16, 3)&73 & $D_6.\mu_{8}$&(48, 4)&4\\ $C_2.\mu_{8}$&(16, 5)&50 & $C_2^2.\mu_{4}$&(16, 5)&1 & $D_6.\mu_{10}$&(60, 11)&1\\ \rowcolor{lightgray} $C_2.\mu_{9}$&(18, 2)&2 & $C_2^2.\mu_{4}$&(16, 6)&2 & $D_6.\mu_{12}$&(72, 27)&2\\ $C_2.\mu_{10}$&(20, 2)&1 & $C_2^2.\mu_{4}$&(16, 10)&77 & $C_2^3.\mu_{2}$&(16, 3)&3\\ \rowcolor{lightgray} $C_2.\mu_{10}$&(20, 5)&12 & $C_2^2.\mu_{6}$&(24, 10)&13 & $C_2^3.\mu_{2}$&(16, 10)&2\\ $C_2.\mu_{12}$&(24, 2)&2 & $C_2^2.\mu_{6}$&(24, 13)&19 & $C_2^3.\mu_{2}$&(16, 11)&34\\ \rowcolor{lightgray} $C_2.\mu_{12}$&(24, 9)&24 & $C_2^2.\mu_{6}$&(24, 15)&14 & $C_2^3.\mu_{2}$&(16, 14)&72\\ $C_2.\mu_{14}$&(28, 4)&4 & $C_2^2.\mu_{8}$&(32, 5)&14 & $C_2^3.\mu_{3}$&(24, 13)&2\\ \rowcolor{lightgray} $C_2.\mu_{15}$&(30, 4)&3 & $C_2^2.\mu_{8}$&(32, 36)&6 & $C_2^3.\mu_{4}$&(32, 6)&6\\ $C_2.\mu_{16}$&(32, 16)&3 & $C_2^2.\mu_{9}$&(36, 3)&2 & $C_2^3.\mu_{4}$&(32, 7)&3\\ \rowcolor{lightgray} $C_2.\mu_{18}$&(36, 5)&3 & $C_2^2.\mu_{12}$&(48, 21)&4 & $C_2^3.\mu_{4}$&(32, 22)&31\\ $C_2.\mu_{20}$&(40, 9)&3 & $C_2^2.\mu_{12}$&(48, 31)&3 & $C_2^3.\mu_{4}$&(32, 45)&7\\ \rowcolor{lightgray} $C_2.\mu_{24}$&(48, 23)&3 & $C_2^2.\mu_{12}$&(48, 44)&1 & $C_2^3.\mu_{6}$&(48, 31)&1\\ $C_2.\mu_{28}$&(56, 8)&1 & $C_2^2.\mu_{18}$&(72, 16)&2 & $C_2^3.\mu_{6}$&(48, 49)&5\\ \rowcolor{lightgray} $C_2.\mu_{30}$&(60, 13)&2 & $C_4.\mu_{2}$&(8, 1)&3 & $C_2^3.\mu_{7}$&(56, 11)&1\\ $C_3.\mu_{2}$&(6, 1)&46 & $C_4.\mu_{2}$&(8, 2)&98 & $C_2^3.\mu_{7}$&(56, 11)&1\\ \rowcolor{lightgray} $C_3.\mu_{2}$&(6, 2)&51 & $C_4.\mu_{2}$&(8, 3)&102 & $C_2^3.\mu_{8}$&(64, 4)&3\\ $C_3.\mu_{3}$&(9, 2)&26 & $C_4.\mu_{2}$&(8, 4)&3 & $C_2^3.\mu_{8}$&(64, 87)&1\\ \rowcolor{lightgray} $C_3.\mu_{4}$&(12, 1)&15 & $C_4.\mu_{3}$&(12, 2)&7 & $C_2^3.\mu_{12}$&(96, 196)&2\\ $C_3.\mu_{4}$&(12, 2)&15 & $C_4.\mu_{4}$&(16, 2)&21 & $D_8.\mu_{2}$&(16, 7)&6\\ \rowcolor{lightgray} $C_3.\mu_{5}$&(15, 1)&4 & $C_4.\mu_{4}$&(16, 4)&23 & $D_8.\mu_{2}$&(16, 8)&2\\ $C_3.\mu_{6}$&(18, 3)&38 & $C_4.\mu_{4}$&(16, 5)&10 & $D_8.\mu_{2}$&(16, 11)&202\\ \rowcolor{lightgray} $C_3.\mu_{6}$&(18, 5)&36 & $C_4.\mu_{4}$&(16, 6)&9 & $D_8.\mu_{2}$&(16, 13)&11\\ $C_3.\mu_{7}$&(21, 2)&1 & $C_4.\mu_{5}$&(20, 2)&1 & $D_8.\mu_{3}$&(24, 10)&3\\ \rowcolor{lightgray} $C_3.\mu_{8}$&(24, 1)&3 & $C_4.\mu_{6}$&(24, 2)&1 & $D_8.\mu_{4}$&(32, 9)&10\\ $C_3.\mu_{8}$&(24, 2)&3 & $C_4.\mu_{6}$&(24, 9)&5 & $D_8.\mu_{4}$&(32, 11)&4\\ \rowcolor{lightgray} $C_3.\mu_{9}$&(27, 2)&3 & $C_4.\mu_{6}$&(24, 10)&8 & $D_8.\mu_{4}$&(32, 25)&19\\ $C_3.\mu_{10}$&(30, 1)&4 & $C_4.\mu_{6}$&(24, 11)&1 & $D_8.\mu_{4}$&(32, 38)&1\\ \rowcolor{lightgray} $C_3.\mu_{10}$&(30, 4)&2 & $C_4.\mu_{8}$&(32, 3)&4 & $D_8.\mu_{6}$&(48, 26)&1\\ $C_3.\mu_{12}$&(36, 6)&5 & $C_4.\mu_{8}$&(32, 12)&4 & $D_8.\mu_{6}$&(48, 45)&4\\ \rowcolor{lightgray} $C_3.\mu_{12}$&(36, 8)&4 & $C_4.\mu_{10}$&(40, 9)&2 & $D_8.\mu_{8}$&(64, 6)&5\\ $C_3.\mu_{14}$&(42, 6)&2 & $C_4.\mu_{12}$&(48, 20)&1 & $D_8.\mu_{12}$&(96, 52)&1\\ \rowcolor{lightgray} $Q_8.\mu_{2}$&(16, 8)&7 & $C_2^4.\mu_{4}$&(64, 32)&2 & $A_{3,3}.\mu_{2}$&(36, 10)&11\\ $Q_8.\mu_{2}$&(16, 9)&1 & $C_2^4.\mu_{4}$&(64, 60)&2 & $A_{3,3}.\mu_{2}$&(36, 13)&10\\ \rowcolor{lightgray} $Q_8.\mu_{2}$&(16, 12)&2 & $C_2^4.\mu_{4}$&(64, 90)&1 & $A_{3,3}.\mu_{3}$&(54, 5)&2\\ $Q_8.\mu_{2}$&(16, 13)&11 & $C_2^4.\mu_{4}$&(64, 193)&1 & $A_{3,3}.\mu_{3}$&(54, 13)&3\\ \rowcolor{lightgray} $Q_8.\mu_{3}$&(24, 3)&2 & $C_2^4.\mu_{5}$&(80, 49)&1 & $A_{3,3}.\mu_{4}$&(72, 21)&1\\ $Q_8.\mu_{3}$&(24, 11)&1 & $C_2^4.\mu_{6}$&(96, 70)&1 & $A_{3,3}.\mu_{4}$&(72, 39)&1\\ \rowcolor{lightgray} $Q_8.\mu_{4}$&(32, 11)&5 & $C_2^4.\mu_{6}$&(96, 197)&1 & $A_{3,3}.\mu_{4}$&(72, 45)&3\\ $Q_8.\mu_{4}$&(32, 38)&2 & $C_2^4.\mu_{6}$&(96, 228)&1 & $A_{3,3}.\mu_{6}$&(108, 25)&4\\ \rowcolor{lightgray} $Q_8.\mu_{6}$&(48, 26)&1 & $C_2^4.\mu_{6}$&(96, 229)&1 & $A_{3,3}.\mu_{6}$&(108, 36)&1\\ $Q_8.\mu_{6}$&(48, 32)&2 & $C_2^4.\mu_{7}$&(112, 41)&1 & $A_{3,3}.\mu_{6}$&(108, 38)&4\\ \rowcolor{lightgray} $Q_8.\mu_{6}$&(48, 33)&1 & $C_2^4.\mu_{7}$&(112, 41)&1 & $A_{3,3}.\mu_{6}$&(108, 43)&1\\ $Q_8.\mu_{6}$&(48, 46)&1 & $C_2^4.\mu_{8}$&(128, 48)&1 & $A_{3,3}.\mu_{8}$&(144, 185)&1\\ \rowcolor{lightgray} $D_{10}.\mu_{2}$&(20, 3)&3 & $C_2^4.\mu_{10}$&(160, 235)&1 & $Hol(C_5).\mu_{2}$&(40, 12)&9\\ $D_{10}.\mu_{2}$&(20, 4)&19 & $C_2^4.\mu_{12}$&(192, 994)&1 & $Hol(C_5).\mu_{3}$&(60, 6)&1\\ \rowcolor{lightgray} $D_{10}.\mu_{3}$&(30, 2)&3 & $C_2 \times D_8.\mu_{2}$&(32, 6)&3 & $Hol(C_5).\mu_{4}$&(80, 30)&1\\ $D_{10}.\mu_{4}$&(40, 5)&3 & $C_2 \times D_8.\mu_{2}$&(32, 7)&1 & $C_7:C_3.\mu_{2}$&(42, 1)&4\\ \rowcolor{lightgray} $D_{10}.\mu_{4}$&(40, 12)&4 & $C_2 \times D_8.\mu_{2}$&(32, 27)&14 & $C_7:C_3.\mu_{2}$&(42, 2)&2\\ $D_{10}.\mu_{5}$&(50, 3)&1 & $C_2 \times D_8.\mu_{2}$&(32, 28)&2 & $C_7:C_3.\mu_{3}$&(63, 3)&1\\ \rowcolor{lightgray} $D_{10}.\mu_{6}$&(60, 6)&1 & $C_2 \times D_8.\mu_{2}$&(32, 28)&2 & $C_7:C_3.\mu_{4}$&(84, 2)&1\\ $D_{10}.\mu_{6}$&(60, 10)&2 & $C_2 \times D_8.\mu_{2}$&(32, 30)&1 & $C_7:C_3.\mu_{6}$&(126, 7)&1\\ \rowcolor{lightgray} $D_{10}.\mu_{8}$&(80, 28)&1 & $C_2 \times D_8.\mu_{2}$&(32, 34)&4 & $C_7:C_3.\mu_{6}$&(126, 10)&1\\ $D_{10}.\mu_{10}$&(100, 14)&1 & $C_2 \times D_8.\mu_{2}$&(32, 39)&2 & $S_4.\mu_{2}$&(48, 48)&74\\ \rowcolor{lightgray} $D_{10}.\mu_{12}$&(120, 17)&1 & $C_2 \times D_8.\mu_{2}$&(32, 43)&2 & $S_4.\mu_{3}$&(72, 42)&2\\ $A_4.\mu_{2}$&(24, 12)&40 & $C_2 \times D_8.\mu_{2}$&(32, 46)&46 & $S_4.\mu_{4}$&(96, 186)&8\\ \rowcolor{lightgray} $A_4.\mu_{2}$&(24, 13)&47 & $C_2 \times D_8.\mu_{2}$&(32, 48)&1 & $S_4.\mu_{6}$&(144, 188)&2\\ $A_4.\mu_{3}$&(36, 11)&7 & $C_2 \times D_8.\mu_{2}$&(32, 49)&13 & $2^4C_2.\mu_{2}$&(64, 32)&2\\ \rowcolor{lightgray} $A_4.\mu_{4}$&(48, 30)&10 & $C_2 \times D_8.\mu_{4}$&(64, 12)&1 & $2^4C_2.\mu_{2}$&(64, 138)&8\\ $A_4.\mu_{4}$&(48, 31)&7 & $C_2 \times D_8.\mu_{4}$&(64, 34)&2 & $2^4C_2.\mu_{2}$&(64, 202)&12\\ \rowcolor{lightgray} $A_4.\mu_{6}$&(72, 42)&5 & $C_2 \times D_8.\mu_{4}$&(64, 67)&5 & $2^4C_2.\mu_{2}$&(64, 215)&3\\ $A_4.\mu_{6}$&(72, 47)&4 & $C_2 \times D_8.\mu_{4}$&(64, 71)&2 & $2^4C_2.\mu_{2}$&(64, 216)&1\\ \rowcolor{lightgray} $A_4.\mu_{12}$&(144, 123)&1 & $C_2 \times D_8.\mu_{4}$&(64, 90)&7 & $2^4C_2.\mu_{2}$&(64, 226)&4\\ $D_{12}.\mu_{2}$&(24, 6)&4 & $C_2 \times D_8.\mu_{4}$&(64, 92)&4 & $2^4C_2.\mu_{2}$&(64, 241)&1\\ \rowcolor{lightgray} $D_{12}.\mu_{2}$&(24, 8)&2 & $C_2 \times D_8.\mu_{4}$&(64, 99)&2 & $2^4C_2.\mu_{3}$&(96, 70)&1\\ $D_{12}.\mu_{2}$&(24, 14)&54 & $C_2 \times D_8.\mu_{4}$&(64, 101)&1 & $2^4C_2.\mu_{4}$&(128, 621)&1\\ \rowcolor{lightgray} $D_{12}.\mu_{3}$&(36, 12)&7 & $C_2 \times D_8.\mu_{4}$&(64, 102)&1 & $2^4C_2.\mu_{4}$&(128, 645)&1\\ $D_{12}.\mu_{4}$&(48, 14)&3 & $C_2 \times D_8.\mu_{4}$&(64, 196)&2 & $2^4C_2.\mu_{4}$&(128, 850)&3\\ \rowcolor{lightgray}
$D_{12}.\mu_{4}$&(48, 35)&3 & $C_2 \times D_8.\mu_{4}$&(64, 199)&1 & $2^4C_2.\mu_{4}$&(128, 853)&1\\ $D_{12}.\mu_{6}$&(72, 28)&2 & $C_2 \times D_8.\mu_{8}$&(128, 2)&1 & $2^4C_2.\mu_{4}$&(128, 1090)&1\\ \rowcolor{lightgray} $D_{12}.\mu_{6}$&(72, 30)&1 & $C_2 \times D_8.\mu_{8}$&(128, 50)&1 & $2^4C_2.\mu_{6}$&(192, 1000)&1\\ $D_{12}.\mu_{6}$&(72, 48)&5 & $C_2 \times D_8.\mu_{8}$&(128, 206)&1 & $Q_8 * Q_8.\mu_{2}$&(64, 134)&2\\ \rowcolor{lightgray} $D_{12}.\mu_{8}$&(96, 27)&1 & $SD_{16}.\mu_{2}$&(32, 40)&2 & $Q_8 * Q_8.\mu_{2}$&(64, 138)&4\\ $D_{12}.\mu_{8}$&(96, 106)&1 & $SD_{16}.\mu_{2}$&(32, 42)&2 & $Q_8 * Q_8.\mu_{2}$&(64, 139)&1\\ \rowcolor{lightgray} $D_{12}.\mu_{12}$&(144, 79)&1 & $SD_{16}.\mu_{2}$&(32, 43)&6 & $Q_8 * Q_8.\mu_{2}$&(64, 257)&1\\ $C_2^4.\mu_{2}$&(32, 27)&6 & $SD_{16}.\mu_{3}$&(48, 26)&1 & $Q_8 * Q_8.\mu_{2}$&(64, 264)&4\\ \rowcolor{lightgray} $C_2^4.\mu_{2}$&(32, 46)&5 & $SD_{16}.\mu_{4}$&(64, 124)&1 & $Q_8 * Q_8.\mu_{2}$&(64, 266)&1\\ $C_2^4.\mu_{2}$&(32, 51)&7 & $SD_{16}.\mu_{4}$&(64, 125)&1 & $Q_8 * Q_8.\mu_{3}$&(96, 201)&1\\ \rowcolor{lightgray} $C_2^4.\mu_{3}$&(48, 49)&2 & $SD_{16}.\mu_{6}$&(96, 180)&1 & $Q_8 * Q_8.\mu_{3}$&(96, 204)&1\\ $C_2^4.\mu_{3}$&(48, 50)&1 & $A_{3,3}.\mu_{2}$&(36, 9)&2 & $Q_8 * Q_8.\mu_{4}$&(128, 134)&1\\ \rowcolor{lightgray} $Q_8 * Q_8.\mu_{4}$&(128, 522)&1 & $C_2 \times S_4.\mu_{4}$&(192, 1469)&1 & $2^4D_6.\mu_{6}$&(576, 8656)&1\\ $Q_8 * Q_8.\mu_{4}$&(128, 524)&2 & $T_{48}.\mu_{2}$&(96, 189)&1 & $S_5.\mu_{2}$&(240, 189)&12\\ \rowcolor{lightgray} $Q_8 * Q_8.\mu_{4}$&(128, 1633)&1 & $T_{48}.\mu_{2}$&(96, 193)&2 & $L_2(7).\mu_{2}$&(336, 208)&8\\ $Q_8 * Q_8.\mu_{6}$&(192, 201)&1 & $T_{48}.\mu_{3}$&(144, 122)&1 & $L_2(7).\mu_{2}$&(336, 209)&4\\ \rowcolor{lightgray} $Q_8 * Q_8.\mu_{6}$&(192, 1504)&1 & $T_{48}.\mu_{6}$&(288, 900)&1 & $L_2(7).\mu_{4}$&(672, 1046)&1\\ $Q_8 * Q_8.\mu_{6}$&(192, 1509)&1 & $A_5.\mu_{2}$&(120, 34)&7 & $4^2A_4.\mu_{2}$&(384, 591)&1\\ \rowcolor{lightgray} $Q_8 * Q_8.\mu_{8}$&(256, 332)&1 & $A_5.\mu_{2}$&(120, 35)&12 & $4^2A_4.\mu_{2}$&(384, 18134)&1\\ $Q_8 * Q_8.\mu_{12}$&(384, 5816)&1 & $A_5.\mu_{3}$&(180, 19)&2 & $4^2A_4.\mu_{2}$&(384, 18135)&2\\ \rowcolor{lightgray} $3^2C_4.\mu_{2}$&(72, 39)&2 & $A_5.\mu_{4}$&(240, 91)&1 & $4^2A_4.\mu_{2}$&(384, 18235)&2\\ $3^2C_4.\mu_{2}$&(72, 40)&6 & $A_5.\mu_{6}$&(360, 119)&1 & $4^2A_4.\mu_{2}$&(384, 18236)&1\\ \rowcolor{lightgray} $3^2C_4.\mu_{2}$&(72, 41)&1 & $A_5.\mu_{6}$&(360, 122)&1 & $4^2A_4.\mu_{3}$&(576, 5129)&1\\ $3^2C_4.\mu_{2}$&(72, 45)&5 & $\Gamma_{25}a_1.\mu_{2}$&(128, 928)&3 & $4^2A_4.\mu_{4}$&(768, 1083510)&1\\ \rowcolor{lightgray} $3^2C_4.\mu_{3}$&(108, 36)&1 & $\Gamma_{25}a_1.\mu_{2}$&(128, 932)&1 & $4^2A_4.\mu_{4}$&(768, 1088651)&1\\ $3^2C_4.\mu_{4}$&(144, 120)&2 & $\Gamma_{25}a_1.\mu_{2}$&(128, 1755)&4 & $4^2A_4.\mu_{4}$&(768, 1088659)&1\\ \rowcolor{lightgray} $3^2C_4.\mu_{4}$&(144, 185)&1 & $\Gamma_{25}a_1.\mu_{2}$&(128, 1758)&1 & $4^2A_4.\mu_{6}$&(1152, 155469)&1\\ $3^2C_4.\mu_{6}$&(216, 157)&1 & $\Gamma_{25}a_1.\mu_{2}$&(128, 1759)&1 & $H_{195}.\mu_{2}$&(384, 17948)&3\\ \rowcolor{lightgray} $S_{3,3}.\mu_{2}$&(72, 40)&3 & $\Gamma_{25}a_1.\mu_{3}$&(192, 201)&1 & $T_{192}.\mu_{2}$&(384, 5602)&1\\ $S_{3,3}.\mu_{2}$&(72, 46)&8 & $\Gamma_{25}a_1.\mu_{4}$&(256, 6029)&1 & $T_{192}.\mu_{2}$&(384, 5608)&1\\ \rowcolor{lightgray} $S_{3,3}.\mu_{3}$&(108, 38)&2 & $\Gamma_{25}a_1.\mu_{6}$&(384, 5837)&1 & $T_{192}.\mu_{2}$&(384, 20097)&1\\ $S_{3,3}.\mu_{4}$&(144, 115)&1 & $A_{4,3}.\mu_{2}$&(144, 183)&7 & $T_{192}.\mu_{3}$&(576, 8277)&1\\ \rowcolor{lightgray} $S_{3,3}.\mu_{6}$&(216, 157)&1 & $A_{4,3}.\mu_{2}$&(144, 189)&3 & $T_{192}.\mu_{6}$&(1152, 157515)&1\\ $S_{3,3}.\mu_{6}$&(216, 170)&1 & $A_{4,3}.\mu_{3}$&(216, 92)&1 & $A_{4,4}.\mu_{2}$&(576, 8653)&1\\ \rowcolor{lightgray} $2^4C_3.\mu_{2}$&(96, 70)&4 & $A_{4,3}.\mu_{3}$&(216, 164)&1 & $A_{4,4}.\mu_{2}$&(576, 8654)&1\\ $2^4C_3.\mu_{2}$&(96, 227)&9 & $A_{4,3}.\mu_{6}$&(432, 535)&1 & $A_{4,4}.\mu_{2}$&(576, 8657)&1\\ \rowcolor{lightgray} $2^4C_3.\mu_{2}$&(96, 229)&6 & $A_{4,3}.\mu_{6}$&(432, 745)&1 & $A_{4,4}.\mu_{4}$&(1152, 157850)&1\\ $2^4C_3.\mu_{3}$&(144, 184)&2 & $N_{72}.\mu_{2}$&(144, 182)&1 & $A_6.\mu_{2}$&(720, 763)&2\\ \rowcolor{lightgray} $2^4C_3.\mu_{4}$&(192, 184)&1 & $N_{72}.\mu_{2}$&(144, 186)&2 & $A_6.\mu_{2}$&(720, 764)&6\\ $2^4C_3.\mu_{4}$&(192, 191)&1 & $N_{72}.\mu_{4}$&(288, 841)&1 & $A_6.\mu_{2}$&(720, 766)&4\\ \rowcolor{lightgray} $2^4C_3.\mu_{4}$&(192, 1495)&1 & $M_9.\mu_{2}$&(144, 182)&2 & $A_6.\mu_{4}$&(1440, 4595)&1\\ $2^4C_3.\mu_{5}$&(240, 191)&1 & $M_9.\mu_{2}$&(144, 187)&1 & $F_{384}.\mu_{2}$&(768, 1086051)&1\\ \rowcolor{lightgray} $2^4C_3.\mu_{6}$&(288, 1025)&2 & $M_9.\mu_{3}$&(216, 153)&1 & $F_{384}.\mu_{2}$&(768, 1090134)&1\\ $2^4C_3.\mu_{6}$&(288, 1029)&1 & $M_9.\mu_{6}$&(432, 735)&1 & $F_{384}.\mu_{2}$&(768, 1090135)&1\\ \rowcolor{lightgray} $C_2 \times S_4.\mu_{2}$&(96, 187)&2 & $2^4D_6.\mu_{2}$&(192, 955)&8 & $F_{384}.\mu_{4}$&(1536, 'no id')&1\\ $C_2 \times S_4.\mu_{2}$&(96, 195)&1 & $2^4D_6.\mu_{2}$&(192, 1538)&9 & $M_{20}.\mu_{2}$&(1920, 240993)&1\\ \rowcolor{lightgray} $C_2 \times S_4.\mu_{2}$&(96, 226)&15 & $2^4D_6.\mu_{3}$&(288, 1025)&1 & $M_{20}.\mu_{2}$&(1920, 240995)&2\\ $C_2 \times S_4.\mu_{4}$&(192, 972)&2 & $2^4D_6.\mu_{4}$&(384, 5566)&1 & $M_{20}.\mu_{4}$&(3840, 'no id')&1\\
\bottomrule \end{longtable} }
\section{Fixed loci of purely non-symplectic automorphisms} \label{appendixB}
Let $X$ be a K3 surface and $\sigma \in \Aut(X)$ a purely non-symplectic automorphism of order $n$ acting by $\zeta_n$ on $\HH^0(X,\Omega_X^2)$.
Recall that the fixed locus $X^\sigma$ is the disjoint union of $N = \sum_{i=1}^s a_i$ isolated fixed points, $k$ smooth rational curves and either a curve of genus $>1$ or $0, 1, 2$ curves of genus 1. Denote by $l\geq 0$ the number of genus $g \geq 1$ curves fixed by $\sigma$. If no such curve is fixed, set $g=1$. Thus \[X^\sigma = \{p_1, \dots, p_N\} \sqcup R_1 , \dots \sqcup R_k \sqcup C_1 \dots \sqcup C_l\] where the $R_i$'s are smooth rational curves and the $C_j$'s smooth curves of genus $g$. Let $P \in X^\sigma$ be an isolated fixed point. Recall that there are local coordinates $(x,y)$ in a small neighborhood centered at $P$ such that \[\sigma(x,y) = (\zeta_n^{i+1}x,\zeta_n^{-i} y)\quad \mbox{ with } \quad 1 \leq i \leq s = \left\lfloor\frac{n-1}{2}\right\rfloor.\] We call $P$ a fixed point of type $i$ and denote the number of fixed points of type $i$ by $a_i$.
In the following we list for each deformation class of $(X, \sigma)$ the invariants $((a_1, \dots, a_s), k, l, g)$ of the fixed locus of $\sigma$ and its powers. The column labeled `K3 id' contains the label of the K3 surface in the database \cite{K3Groups}. The following columns contain the invariants of the fixed locus of $\sigma^{n/j}$ where $n$ is the order of $\sigma$ and $j$ the label of the column.
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\end{document} | arXiv |
Nemirovski Stefan Yurievich
1. V. Chernov, S. Nemirovski, "Interval topology in contact geometry", Commun. Contemp. Math., 2019, 1950042 (Published online) , arXiv: 1810.01642
2. V. Chernov, S. Nemirovski, "Redshift and contact forms", J. Geom. Phys., 123 (2018), 379–384 , arXiv: 1709.01741 (cited: 3) (cited: 3)
3. Stefan Nemirovski, Rasul Gazimovich Shafikov, "Uniformization and Steinness", Canad. Math. Bull., 61 (2018), 637–639 , arXiv: 1705.05113
4. A. I. Aptekarev, V. M. Buchstaber, V. A. Vassiliev, M. L. Gromov, Yu. S. Ilyashenko, B. S. Kashin, V. M. Keselman, V. V. Kozlov, M. L. Kontsevich, I. M. Krichever, N. G. Kruzhilin, S. K. Lando, Yu. I. Manin, G. A. Margulis, S. Yu. Nemirovski, S. P. Novikov, Yu. G. Reshetnyak, Ya. G. Sinai, S. P. Suetin, D. V. Treschev, D. B. Fuchs, A. G. Khovanskii, E. M. Chirka, A. S. Schwarz, A. N. Shiryaev, "Vladimir Antonovich Zorich (on his 80th birthday)", Russian Math. Surveys, 73:5 (2018), 935–939
5. A. I. Aptekarev, V. K. Beloshapka, V. I. Buslaev, V. V. Goryainov, V. N. Dubinin, V. A. Zorich, N. G. Kruzhilin, S. Yu. Nemirovski, S. Yu. Orevkov, P. V. Paramonov, S. I. Pinchuk, A. S. Sadullaev, A. G. Sergeev, S. P. Suetin, A. B. Sukhov, K. Yu. Fedorovskiy, A. K. Tsikh, "Evgenii Mikhailovich Chirka (on his 75th birthday)", Russian Math. Surveys, 73:6 (2018), 1137–1144
6. Stefan Nemirovski, Kyler Siegel, "Rationally convex domains and singular Lagrangian surfaces in $\mathbb C^2$", Invent. Math., 203:1 (2016), 333–358 , 26 pp., arXiv: 1410.4652 (cited: 5) (cited: 3) (cited: 3)
7. Vladimir Chernov, Stefan Nemirovski, "Universal orderability of Legendrian isotopy classes", J. Symplectic Geom., 14:1 (2016), 149–170 , arXiv: 1307.5694 (cited: 4) (cited: 4)
8. A. I. Bufetov, A. A. Glutsyuk, S. M. Gusein-Zade, A. S. Gorodetski, V. Yu. Kaloshin, A. G. Khovanskii, V. A. Kleptsyn, S. Yu. Nemirovskii, A. B. Sossinsky, M. A. Tsfasman, V. A. Vassiliev, S. Yu. Yakovenko, "Yulij Sergeevich Ilyashenko", Mosc. Math. J., 14:2 (2014), 173–179
9. F. A. Bogomolov, F. Kataneze, Yu. I. Manin, S. Yu. Nemirovskii, V. V. Nikulin, A. N. Parshin, V. V. Przhiyalkovskii, Yu. G. Prokhorov, M. Teikher, A. S. Tikhomirov, V. M. Kharlamov, I. A. Cheltsov, I. R. Shafarevich, V. V. Shokurov, "Viktor Stepanovich Kulikov (k shestidesyatiletiyu so dnya rozhdeniya)", UMN, 68:2(410) (2013), 205–207
10. V. Chernov, S. Nemirovski, "Cosmic censorship of smooth structures", Comm. Math. Phys., 320:2 (2013), 469–473 (cited: 9) (cited: 3) (cited: 7)
11. S. Nemirovski, "Levi problem and semistable quotients", Complex Var. Elliptic Equ., 58:11 (2013), 1517–1525 , "Corrigendum", ibid., 1633 (cited: 2) (cited: 1) (cited: 1)
12. S. Nemirovski, "Corrigendum. Levi problem and semistable quotients", Complex Var. Elliptic Equ., 58:11 (2013), 1633
13. V. Chernov, S. Nemirovski, "Legendrian links, causality, and the Low conjecture", Geom. Funct. Anal., 19:5 (2010), 1320–1333 (cited: 22) (cited: 11) (cited: 23)
14. V. Chernov, S. Nemirovski, "Non-negative Legendrian isotopy in $ST^*M$", Geom. Topol., 14:1 (2010), 611–626 (cited: 20) (cited: 12) (cited: 20) | CommonCrawl |
R S Chandok
Volume 44 Issue 1 January 1995 pp 9-18
Transient currents in discharge mode in cellulose acetate: polyvinyl acetate blend films
P K Khare R S Chandok A P Srivastava
The transient currents measured in discharge mode with cellulose acetate (CA): polyvinyl acetate (PVAc) blend films (≈ 20µm thick) as a function of charging field [(1.5–4.5)×104 V/cm], temperatures (323–373 K) and polymer weight ratio (90:10 and 75:25) have been found to follow Curie-von Schweidler law, characterized with two slopes in short and long time regions. Isochronals characteristics (i.e. current/temperature plots at constant times) constructed from these data seemed to reveal a broad peak observed at 363 K. Values of activation energy increase with PAVc content and also with time of observation. Space charge due to trapping of injected charge carriers in energetically distributed traps and induced dipoles created because of the piling up of charge carriers at the phase boundary of heterogeneous structure of blend are considered to account for the observed currents.
Volume 48 Issue 6 June 1997 pp 1135-1143
Field emission theory of dislocation-sensitized photo-stimulated exo-electron emission from coloured alkali halide crystals
B P Chandra R S Chandok P K Khare
A new field emission theory of dislocation-sensitized photo-stimulated exo-electron emission (DSPEE) is proposed, which shows that the increase in the intensity of photo emission fromF-centres during plastic deformation is caused by the appearance of an electric field which draws excited electrons out of the deeper layer and, therefore, increases the number of electrons which reach the surface. The theory of DSPEE shows that the variation of DSPEE flux intensity should obey the following relation$$\frac{{\Delta J_e \left( \varepsilon \right)}}{{J_e \left( o \right)}} = \left[ {\frac{{Y_s }}{{d_F }}\exp \left( {\frac{\chi }{{kT}}} \right) - 1} \right]$$. The theory of DSPEE is able to explain several experimental observations like linear increase of DSPEE intensityJe with the strain at low deformation, occurrence of the saturation inJe at higher deformation, temperature dependence ofJe, linear dependence ofJe on the electric field strength, the order of the critical strain at which saturation occurs inJe, and the ratio of the PEE intensity of deformed and undeformed crystals. At lower values of the strain, some of the excited electrons are captured by surface traps, where the deformation generated electric field is not able to cause the exo-emission. At larger deformation (in between 2% and 3%) of the crystal, the deformation-generated electric field becomes sufficient to cause an additional exo-electron emission of the electrons trapped in surface traps, and therefore,t here appears a hump in theJe versusε curves of the crystals. | CommonCrawl |
Hyperplane section
In mathematics, a hyperplane section of a subset X of projective space Pn is the intersection of X with some hyperplane H. In other words, we look at the subset XH of those elements x of X that satisfy the single linear condition L = 0 defining H as a linear subspace. Here L or H can range over the dual projective space of non-zero linear forms in the homogeneous coordinates, up to scalar multiplication.
From a geometrical point of view, the most interesting case is when X is an algebraic subvariety; for more general cases, in mathematical analysis, some analogue of the Radon transform applies. In algebraic geometry, assuming therefore that X is V, a subvariety not lying completely in any H, the hyperplane sections are algebraic sets with irreducible components all of dimension dim(V) − 1. What more can be said is addressed by a collection of results known collectively as Bertini's theorem. The topology of hyperplane sections is studied in the topic of the Lefschetz hyperplane theorem and its refinements. Because the dimension drops by one in taking hyperplane sections, the process is potentially an inductive method for understanding varieties of higher dimension. A basic tool for that is the Lefschetz pencil.
References
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
| Wikipedia |
\begin{document}
\title{Identifiers for MRD-codes} \begin{abstract}
For any admissible value of the parameters $n$ and $k$ there exist
$[n,k]$-Maximum Rank Distance $\mathbb F_q$-linear codes. Indeed,
it can be shown that if field extensions large enough are
considered, \emph{almost all} rank metric codes are MRD.
On the other hand, very few families up to equivalence
of such codes are currently known. In the present paper we
study some invariants of MRD codes and evaluate their value for
the known families, providing a new characterization of
generalized twisted Gabidulin codes. \end{abstract}
{\it AMS subject classification:} 51E22, 05B25, 94B05
{\it Keywords:} Gabidulin codes, Rank metric, Distinguisher
\section{Introduction} \label{intro} Delsarte \cite{Delsarte} introduced in 1978 rank-distance (RD) codes as $q$-analogs of the usual linear error correcting codes over finite fields.
In the same paper, he also showed that the parameters of these codes must obey a Singleton-like bound and that for any admissible value of the length $n$ and the dimension $k$ this bound is sharp. A rank metric code attaining this bound is called \emph{maximum rank distance} (MRD). In 1985 Gabidulin \cite{Gabidulin} independently rediscovered Rank-distance codes and also devised an algebraic decoding algorithm, in close analogy to what happens for Reed-Solomon codes, for the family of MRD codes described by Delsarte.
More recently MRD-codes have been intensively investigated both for their applications to network coding and for their links with remarkable geometric and algebraic objects such as linear sets and semifields \cite{BZ,CsMP,CsMPZa,CsMPZ,CsMZ,CsZ,CsZ2,delaCruz,Lu2017,LTZ2,Sheekey,ShVdV}.
It has been shown in \cite{H-TNRR} (see also \cite{BR}) that a generic rank-distance code, provided that the field involved with the construction is large enough, is MRD. The authors of \cite{H-TNRR} make extensive use of algebraic geometry methods and they are also able to offer an estimate on the probability that a random rank metric code is MRD as well as to show that the probability of obtaining a Gabidulin code in this way is negligible.
In \cite{BR} Byrne and Ravagnani obtain an approximation of the fraction of RM-codes of given length and dimension which are MRD using a mostly combinatorial approach. Their paper show also that some care has to be taken when considering these density results for codes; indeed, the $\mathbb F_{q^m}$--linear MRD codes are dense in the family of all $\mathbb F_{q^m}$--linear rank metric codes $\mathcal C\subseteq\mathbb F_{q^m}^n$ of dimension $k$ and length $n$. However, this is not the case for $\mathbb F_q$--linear MRD-codes in the family of $\mathbb F_q$--linear rank metric codes $\mathcal C\subseteq\mathbb F_q^{m\times n}$ with $\dim\mathcal C=k$; see \cite{BR}.
In spite of the aforementioned density results, very few families of MRD codes are currently known up to equivalence; basically, apart from Gabidulin \cite{Gabidulin} and twisted Gabidulin \cite{Sheekey} codes, the state of the art is given by the codes presented in Table~\ref{kMRD} and their Delsarte duals.
A \emph{distinguisher} for a family of codes $\mathfrak F$ is a polynomial time algorithm which can determine if an arbitrary generator matrix $G$ determines a code belonging to $\mathfrak F$ or not. Existence of distinguishers is interesting not only as a mean to characterize a code, but also of much importance for applications, since some attacks against McEliece cryptosystems based on it. The case of (generalized) Gabidulin codes is investigated in~\cite{H-TM}, whose result we recall in Theorem~\ref{gabidulin-d}; {see also \cite{Neri}, where such codes are characterized in terms of their generator matrices}.
The \emph{McEliece cryptosystem} is a well known and much studied public key cryptosystem based on error correcting codes. The basic idea of this encryption scheme is to start with a $t$-error correcting code endowed with an efficient algorithm for decoding and hide its generating matrix $G\in\mathbb F_q^{k\times n}$ by means of a invertible matrix $S$ and a permutation matrix $P$, so that $\hat{G}=SGP$. Then
the encryption of a message $m\in\mathbb F_q^k$ is the codeword $c=m\hat{G}+e$
where $e\in\mathbb F_q^n$ is a noise vector of weight at most $t$.
In order to discuss the security of this cryptosystem we recall the
model of \emph{indistinguishability under chosen plaintext attack}
(IND-CPA). A system is secure under this model if an adversary which does
not know the key is unable to distinguish between the encodings
$c_1$ and $c_2$ of any two different messages $m_1$ and $m_2$ she has
suitably chosen.
Observe that in the case of McEliece cryptosystem,
given two distinct messages $m_1,m_2$
with encodings respectively $c_1$ and $c_2$, the word $c_1-c_2$ has distance
at most $2t$ from $(m_1-m_2)\hat{G}$. So, when we consider codes
endowed with Hamming distance and $t$ comparatively ``small'' it is
easy to see that
$(m_1-m_2)\hat{G}$ has almost everywhere the
same components as $c_1-c_2$. This makes IND-CPA easier to thwart.
Using the rank metric instead of the Hamming metric can improve the
security. So a potential primary application of MRD codes is for
McEliece--like cryptosystems.
Unfortunately,
even if ``almost all $\mathbb F_{q^m}$--linear codes are MRD'',
very few of them are known and even less are amenable to efficient decoding.
The possibility of using Gabidulin codes has been considered in~\cite{H-TM}. The authors in \cite{H-TM} however
proved that there is a very efficient distinguisher for them; more in detail, it is possible to easily recognize a Gabidulin code from a generic MRD code of the same parameters chosen uniformly at random. As a consequence, the cryptosystems based on them turn out not to be semantically secure, as it is possible to distinguish a ciphertext from a random vector; see also \cite{Overbeck,PRW}.
In the present paper we investigate the existence of algebraic distinguishers (akin to those of \cite{H-TM}) for the currently known families of $\mathbb{F}_{q^n}$-linear MRD codes and provide some invariants up to equivalence.
Our main results concern the list of dimensions of the intersections of an $\mathbb{F}_{q^n}$-linear MRD-code with its conjugates and a description of a maximum dimension Gabidulin codes contained in a fixed MRD-code. We shall see, in particular, that this can be used as to provide distinguishers for the generalized twisted Gabidulin codes and how it can also be applied to the other $5$ known families (and their duals), see Tables \ref{kMRD} and \ref{DkMRD}. {We point out that our results answer to the open question \cite[Open Problem II.7.]{B}.}
\subsection{Structure of the paper} In Section~\ref{prelim} we recall the definitions of rank metric (RM) codes and their basic properties. We also fix our notation and discuss in Section~\ref{qpoly} the representation of RM-codes by means of subspaces of linearized polynomials, the representation which shall be used in most of the paper. Section~\ref{vecnum} deals with an alternative convenient representation of RM-codes. In Section~\ref{charact} we prove one of our main results, namely the characterization of generalized twisted Gabidulin codes in terms of the intersection with their conjugates and the Gabidulin subcode they contain; see Theorem~\ref{mth1}. This leads to the introduction in Section~\ref{disting} of two indexes \[ h(\mathcal C):=\max\{ \dim(\mathcal C\cap\mathcal C^{[j]})\colon j=1,\ldots,n-1; \gcd(j,n)=1 \}. \] and \[ \mathrm{ind}\,(\mathcal C):=\max\{ \dim \mathcal G : \mathcal G\subseteq \mathcal C \mbox{ is equivalent to a generalized
Gabidulin code} \} \] for MRD-codes. These indexes are then evaluated for the known families of codes. We conclude the paper with some open problems.
\section{Preliminaries} \label{prelim} Denote by $\mathbb F_q$ a finite field and let $V_m$ and $V_n$ be two vector spaces over $\mathbb F_q$ of dimension respectively $m$ and $n$. The vector space $\mathrm{Hom}_q(V_n,V_m)$ of all $\mathbb F_q$--linear transformations $V_n\to V_m$ is naturally endowed with a \emph{rank distance} $d_R:\mathrm{Hom}_q(V_n,V_m)\times\mathrm{Hom}_q(V_n,V_m)\to{\mathbb N}$ where $d_R(\varphi,\psi):=\dim \mathrm{Im}(\varphi-\psi)$. If we fix bases in $V_m$ and $V_n$ we have that $\mathrm{Hom}_q(V_n,V_m)$ is isometric to the vector space $\mathbb F_q^{m\times n}$ of all $m\times n$ matrices over $\mathbb F_q$ endowed with the distance $d(A,B):=rk\,(A-B)$ for all $A,B\in\mathbb F_q^{m\times n}$.
A \emph{rank metric code} or a (also \emph{rank distance code}), in brief RM-code, $\mathcal C$ of parameters $(m,n,q;d)$ is a subset $\mathcal C$ of $\mathbb F_q^{m\times n}$ with minimum rank distance $d:=\min_{{A,B \in \mathcal C},\ {A\ne B}} \{ d(A,B) \}$. A RM-code $\mathcal C$ is $\mathbb F_q$-linear if it is an $\mathbb F_q$-vector subspace of $\mathbb F_q^{m\times n}$ (or, equivalently, of $\mathrm{Hom}_q(V_m,V_n)$). When $\mathcal C$ is an $\mathbb F_q$-linear RM-code of dimension $k$ contained in $\mathbb F_q^{n\times n}$, we shall also write, in brief, that $\mathcal C$ has parameters $[n,k]$.
As mentioned in Section~\ref{intro}, it has been shown in \cite{Delsarte} that an analogue of the Singleton bound holds for RM-codes; namely, if $\mathcal C$ is an $(m,n,q;d)$ RM-code, then
\[ |\mathcal C| \leq q^{\max \{m,n\}(\min \{m,n\}-d+1)}. \] When this bound is achieved, then $\mathcal C$ is an \emph{MRD-code}.
The \emph{Delsarte dual} code of a linear RM-code $\mathcal C\subseteq \mathbb F_q^{m \times n}$ is defined as \[ \mathcal C^\perp=\{ M \in \mathbb F_q^{m \times n} \colon \mathrm{Tr}(MN^t)=0 \hspace{0.1cm}\text{for all}\hspace{0.1cm} N \in \mathcal C\}. \]
\begin{lemma}\label{dualMRD}\cite{Delsarte,Gabidulin} Let $\mathcal C\subseteq \mathbb F_{q}^{m \times n}$ be an $\mathbb F_q$-linear MRD-code of dimension $k$ with $d>1$. Then the Delsarte dual code $\mathcal C^\perp\subseteq \mathbb F_{q}^{m\times n}$ is an MRD-code of dimension $mn-k$. \end{lemma}
The weight of a codeword $c\in\mathcal C$ is just the rank of the matrix corresponding to $c$. The spectrum of weights of a MRD-code is ``complete'' in the following sense (which is a weaker form of \cite[Theorem 5]{Gabidulin}).
\begin{corollary}\cite[Lemma 2.1]{LTZ2}\label{weight}
Let $\mathcal C$ be an MRD-code in $\mathbb F_q^{m\times n}$ with minimum distance $d$ and
suppose $m \leq n$. Assume that the null matrix $O$ is in $\mathcal C$.
Then, for any $0 \leq l \leq m-d$, there
exists at least
one matrix $C \in \mathcal C$ such that $\mathrm{rk} (C) = d + l$. \end{corollary}
Existence of MRD-codes for all possible values $(m,n,q;d)$ of the parameters has been originally settled in~\cite{Delsarte} where \emph{Singleton systems} are constructed and, independently by Gabidulin in~\cite{Gabidulin}; this has also been generalized in \cite{kshevetskiy_new_2005}.
More recently Sheekey~\cite{Sheekey} discovered a new family of linear maximum rank metric codes for all possible parameters which are inequivalent to those above; see also \cite{LTZ}. Other examples of MRD-codes can be found in \cite{CMP,DS,OOz,OOz2,Sh2018,TZ}. For some chosen values of parameters there are a few other families of $\mathbb{F}_{q^n}$-linear MRD-codes of $\mathbb{F}_q^{n\times n}$ which are currently known; see~\cite{CsMPZa,CsMPZh,CsMZ}.
The interpretation of linear RM-codes as homomorphisms of vector spaces prompts the following definition of \emph{equivalence}. Two RM-codes $\mathcal C$ and $\mathcal C'$ of $\mathbb F_q^{m\times n}$ are \emph{equivalent} if and only if they represent the same homomorphism (up to a change of basis of $V_m$ and $V_n$) in $h\in\mathrm{Hom}_q(V_m,V_n)/\mathrm{Gal}(\mathbb F_{q})$. This is the same as to say that there exist two invertible matrices $A\in \mathbb F_q^{m \times m}$, $B\in \mathbb F_q^{n \times n}$ and a field automorphism $\sigma$ such that $\{A C^\sigma B \colon C\in \mathcal C\}=\mathcal C'$.
In general, it is difficult to determine whether two RM-codes are equivalent or not. The notion of \emph{idealiser} provides an useful criterion.
Let $\mathcal C\subset \mathbb F_q^{m\times n}$ be an RM-code; its left and right idealisers $L(\mathcal C)$ and $R(\mathcal C)$ are defined as \[ L(\mathcal C)=\{ Y \in \mathbb F_q^{m \times m} \colon YC\in \mathcal C\hspace{0.1cm} \text{for all}\hspace{0.1cm} C \in \mathcal C\}\] \[ R(\mathcal C)=\{ Z \in \mathbb F_q^{n \times n} \colon CZ\in \mathcal C\hspace{0.1cm} \text{for all}\hspace{0.1cm} C \in \mathcal C\},\] see \cite[Definition 3.1]{LN2016}. These sets appear also in \cite{LTZ2}, where they are respectively called middle nucleus and right nucleus; therein the authors prove the following result. \begin{proposition}\cite[Proposition 4.1]{LTZ2}\label{idealis}
If $\mathcal C_1$ and $\mathcal C_2$ are equivalent linear RM-codes, then their
left (resp. right) idealisers are also equivalent. \end{proposition} Right idealisers are usually effective as distinguishers for RM-codes, i.e. non-equivalent RM-codes often have non-isomorphic idealisers. This is in sharp contrast with the role played by left idealisers which, for the codes we consider in the present paper, are always isomorphic to $\mathbb F_{q^n}$.
\subsection{Representation of RM-codes as linearized polynomials} \label{qpoly} Any RM-code over $\mathbb F_q$ can be equivalently defined either as a subspace of matrices in $\mathbb F_{q}^{m\times n}$ or as a subspace of $\mathrm{Hom}_q(V_n,V_m)$. In the present section we shall recall a specialized representation in terms of linearized polynomials which we shall use in the rest of the paper.
Consider two vector spaces $V_n$ and $V_m$ over $\mathbb F_q$. If $n\geq m$ we can always regard $V_m$ as a subspace of $V_n$ and identify $\mathrm{Hom}_q(V_n,V_m)$ with the subspace of those $\varphi\in\mathrm{Hom}_q(V_n,V_n)$ such that $\mathrm{Im}(\varphi)\subseteq V_m$. Also, $V_n\cong\mathbb F_{q^n}$, when $\mathbb F_{q^n}$ is considered as a $\mathbb F_q$-vector space of dimension $n$. Let now $\mathrm{Hom}_q(\mathbb F_{q^n}):=\mathrm{Hom}_q(\mathbb F_{q^n},\mathbb F_{q^n})$ be the set of all $\mathbb F_q$--linear maps of $\mathbb F_{q^n}$ in itself. It is well known that each element of $\mathrm{Hom}_q(\mathbb F_{q^n})$ can be represented in a unique way as a linearized polynomial over $\mathbb F_{q^n}$; see \cite{lidl_finite_1997}. In other words, for any $\varphi\in\mathrm{Hom}_q(\mathbb F_{q^n})$ there is an unique polynomial $f(x)$ of the form \[ f(x):=\sum_{i=0}^{n-1} a_i x^{q^i}=\sum_{i=0}^{n-1} a_ix^{[i]} \] with $a_i \in \mathbb F_{q^n}$ and $[i]:=q^i$ such that \[ \forall x\in\mathbb F_{q^n}:\varphi(x)=f(x). \] The set $\mathcal L_{n,q}$ of the linearized polynomials over $\mathbb F_{q^n}$ is a vector space over $\mathbb F_{q^n}$ with respect to the usual sum and scalar multiplication of dimension $n$. When it is regarded as a vector space over $\mathbb F_q$, its dimension is $n^2$ and it is isomorphic to $\mathbb F_q^{n\times n}$. We shall use this point of view in the present paper. Actually, $\mathcal L_{n,q}$ endowed with the product $\circ$ induced by the functional composition in $\mathrm{Hom}_q(\mathbb F_{q^n})$ is an algebra over $\mathbb F_{q}$. In particular, given any two linearized polynomials $f(x)=\sum_{i=0}^{n-1}f_i x^{[i]}$ and $g(x)=\sum_{j=0}^{n-1}g_j x^{[j]}$, we can write \[ (f\circ g)(x):=\sum_{i=0}^{n-1}\sum_{j=0}^{n-1} f_ig_j^{[i]} x^{[(i+j)\mathrm{mod}\, n]}. \]
Take now $\varphi\in\mathrm{Hom}_q(\mathbb F_{q^n})$ and let $f(x)=\sum_{i=0}^{n-1}a_ix^{[i]}\in\mathcal L_{n,q}$ be the associated linearized polynomial. The \emph{Dickson (circulant) matrix} associated to $f$ is \[D_f:= \begin{pmatrix} a_0 & a_1 & \ldots & a_{n-1} \\ a_{n-1}^{[1]} & a_0^{[1]} & \ldots & a_{n-2}^{[1]} \\ \vdots & \vdots & \vdots & \vdots \\ a_1^{[n-1]} & a_2^{[n-1]} & \ldots & a_0^{[n-1]} \end{pmatrix} .\] It can be seen that the rank of the matrix $D_f$ equals the rank of the $\mathbb F_q$-linear map $\varphi$, see for example \cite{wl} and also \cite{CsMPZ2018,GQ,GS18}.
By the above remarks, it is straightforward to see that any $\mathbb F_q$-linear RM-code might be regarded as a suitable $\mathbb F_q$-subspace of $\mathcal L_{n,q}$. This approach shall be extensively used in the present paper. In order to fix the notation and ease the reader, we shall reformulate some of the notions recalled before in terms of linearized polynomials.
A linearized polynomial is called \emph{invertible} if it admits inverse with respect to $\circ$ or, in other words, if its Dickson matrix has non-zero determinant. In the remainder of this paper we shall always silently identify the elements of $\mathcal L_{n,q}$ with the morphisms of $\mathrm{Hom}_q(\mathbb F_{q^n})$ they represent and, as such, speak also of \emph{kernel} and \emph{rank} of a polynomial.
Also, two RM-codes $\mathcal C$ and $\mathcal C'$ are equivalent if and only if there exist two invertible linearized polynomials $h$ and $g$ and a field automorphism $\sigma$ such that $\{h \circ f^\sigma \circ g \colon f\in \mathcal C\}=\mathcal C'$.
The notion of Delsarte dual code can be written in terms of linearized polynomials as follows, see for example \cite[Section 2]{LTZ}. Let $b:\mathcal L_{n,q}\times\mathcal L_{n,q}\to\mathbb F_q$ be the bilinear form given by \[ b(f,g)=\mathrm{Tr}_{q^n/q}\left( \sum_{i=0}^{n-1} f_ig_i \right) \] where $\displaystyle f(x)=\sum_{i=0}^{n-1} f_i x^{[i]}$ and $\displaystyle g(x)=\sum_{i=0}^{n-1} g_i x^{[i]} \in \mathbb F_{q^n}[x]$ and we denote by $\mathrm{Tr}_{q^n/q}$ the trace function $\mathbb F_{q^n}\to\mathbb F_q$ defined as $\mathrm{Tr}_{q^n/q}(x)=x+x^{[1]}+\ldots+x^{[n-1]}$, for $x \in \mathbb F_{q^n}$. The Delsarte dual code $\mathcal C^\perp$ of a set of linearized polynomials $\mathcal C$ is \[\mathcal C^\perp = \{f \in \mathcal{L}_{n,q} \colon b(f,g)=0, \hspace{0.1cm}\forall g \in \mathcal C\}. \]
Furthermore, the left and right idealisers of a code $\mathcal C\subseteq\mathcal L_{n,q}$ can be written as \[L(\mathcal C)=\{\varphi(x) \in \mathcal{L}_{n,q} \colon \varphi \circ f \in \mathcal C\, \text{for all} \, f \in \mathcal C\};\] \[R(\mathcal C)=\{\varphi(x) \in \mathcal{L}_{n,q} \colon f \circ \varphi \in \mathcal C\, \text{for all} \, f \in \mathcal C\}.\]
\begin{definition}
Suppose $\gcd(n,s)=1$ and
let $\mathcal G_{k,s}:=\langle x^{[0]},x^{[s]},\ldots, x^{[s(k-1)]}\rangle\leq
\mathcal L_{n,k}$.
Any code equivalent to $\mathcal G_{k,s}$ is called a
\emph{generalized Gabidulin code}.
Any code equivalent to $\mathcal G_{k}:=\mathcal G_{k,1}$ is
called a \emph{Gabidulin code}. \end{definition} \begin{proposition}\cite[Theorem 5]{Sheekey}
Suppose $\gcd(s,n)=1$ and
let $\mathcal H_{k,s}(\eta):=\langle x+\eta x^{[sk]}, x^{[s]},\ldots,
x^{[s(k-1)]}\rangle$.
If ${\mathrm N}(\eta)={\mathrm N}_{q^n/q}(\eta):=\prod_{i=0}^{n-1}\eta^{[i]}\neq (-1)^{nk}$,
then $\mathcal H_{k,s}(\eta)$ is a MRD-code with the same parameters as $\mathcal G_{k,s}$. \end{proposition} \begin{definition} Any code equivalent to $\mathcal H_{k,s}(\eta)$ with ${\mathrm N}(\eta)\neq (-1)^{nk}$ and $\eta \neq 0$ is called a \emph{(generalized) twisted Gabidulin code}. \end{definition}
\begin{remark}\label{k-2} Clearly, if $1<k<n-1$ \[ \mathcal H_{k,s}(\eta) \cap \mathcal H_{k,s}(\eta)^{[s]}=\langle x^{[2s]},\ldots, x^{[s(k-1)]}\rangle \] and so $\dim(\mathcal H_{k,s}(\eta) \cap \mathcal H_{k,s}(\eta)^{[s]})=k-2$ if $\eta \neq 0$. Indeed, $a_0(x+\eta x^{[sk]})+a_1 x^{[s]}+\ldots+a_{k-1}x^{[s(k-1)]} \in \langle x^{[s]}+\eta^{[s]} x^{[s(k+1)]}, x^{[2s]},\ldots, x^{[sk]}\rangle$ if and only if $a_0=a_1=0$. \end{remark}
The two families of codes seen above are closed under the Delsarte duality. \begin{lemma}\cite{Gabidulin,kshevetskiy_new_2005,LTZ,Sheekey}
The Delsarte dual $\mathcal C^{\perp}$ of an $\mathbb F_{q^n}$-linear MRD-code
$\mathcal C$ of dimension $k$ is an $\mathbb F_{q^n}$-linear MRD-code of dimension $n-k$.
Also, $\mathcal G_{k,s}^\perp$ is equivalent to $\mathcal G_{n-k,s}$ and $\mathcal H_{k,s}(\eta)^\perp$ is equivalent to $\mathcal H_{n-k,s}(-\eta^{[n-ks]})$. \end{lemma} Apart from the two infinite families of $\mathbb F_{q^n}$-linear MRD-codes $\mathcal G_{k,s}$ and $\mathcal H_{k,s}(\eta)$, there are a few other examples known for $n \in \{6,7,8\}$. Such examples are listed in Table~\ref{kMRD} and their Delsarte duals in Table~\ref{DkMRD}.
\begin{table}[htp] \[
\begin{array}{ |c|c|c|c| } \hline \mathcal C & \mbox{parameters} & \mbox{conditions} & \mbox{reference} \\ \hline \mathcal C_1=\langle x,\delta x^{[1]}+x^{[4]} \rangle_{\mathbb F_{q^6}} & (6,6,q;5) & \begin{array}{cc} q>4 \\ \text{certain choices of} \, \delta \end{array} & \mbox{\cite[Theorem 7.1]{CsMPZa}} \\ \hline \mathcal C_2=\langle x, x^{[1]}+x^{[3]}+\delta x^{[5]} \rangle_{\mathbb F_{q^6}} & (6,6,q;5) & \begin{array}{cccc}q \hspace{0.1cm} \text{odd} \\ q \equiv 0,\pm 1 \pmod{5} \\ \delta^2+\delta =1 \\ (\delta \in \mathbb F_q) \end{array} & \mbox{\cite[Theorem 5.1]{CsMPZ}} \\ \hline \mathcal C_3=\langle x,x^{[s]}, x^{[3s]} \rangle_{\mathbb F_{q^7}} & (7,7,q;5) & \begin{array}{cc} q\,\text{odd}\\ \gcd(s,7)=1 \end{array} & \mbox{\cite[Theorem 3.3]{CsMPZh}} \\ \hline \mathcal C_4=\langle x,\delta x^{[1]}+x^{[{5}]} \rangle_{\mathbb F_{q^8}} & (8,8,q;7) & \begin{array}{cc} q\,\text{odd}\\ \delta^2=-1 \end{array} & \mbox{\cite[Theorem 7.2]{CsMPZa}} \\ \hline \mathcal C_5=\langle x,x^{[s]}, x^{[3s]} \rangle_{\mathbb F_{q^8}} & (8,8,q;6) & \begin{array}{cc} q \equiv 1 \pmod{3} \\ \gcd(s,8)=1 \end{array} & \mbox{\cite[Theorem 3.5]{CsMPZh}} \\ \hline
\end{array} \] \caption{Linear MRD-codes in low dimension} \label{kMRD} \end{table} \begin{table}[htp]
\[
\begin{array}{|c|c|c|}
\hline \mathcal D_i=\mathcal C_i^{\perp} & \mbox{parameters} & \mbox{conditions} \\ \hline
\mathcal D_1=\langle x^{[1]}, x^{[{2}]}, x^{[{4}]},x-\delta^{[{5}]} x^{[{3}]} \rangle_{\mathbb F_{q^6}} & (6,6,q;3) & \begin{array}{cc} q>4 \\ \text{certain choices of} \, \delta \end{array} \\ \hline \mathcal D_2=\langle x^{[1]},x^{[3]},x-x^{[2]},x^{[4]}-\delta x \rangle_{\mathbb F_{q^6}} & (6,6,q;3) & \begin{array}{cccc}q \hspace{0.1cm} \text{odd} \\ q \equiv 0,\pm 1 \pmod{5} \\ \delta^2+\delta =1\\ (\delta \in \mathbb F_q) \end{array} \\ \hline \mathcal D_3=\langle x,x^{[{2s}]},x^{[{3s}]},x^{[{4s}]} \rangle_{\mathbb F_{q^7}} & (7,7,q;4) & \begin{array}{cc} q\,\text{odd}\\ \gcd(s,7)=1 \end{array} \\ \hline \mathcal D_4=\langle x^{[1]},x^{[2]},x^{[3]},x^{[5]},x^{[6]},x-\delta x^{[4]} \rangle_{\mathbb F_{q^8}} & (8,8,q;3) & \begin{array}{cc} q\,\text{odd}\\ \delta^2=-1 \end{array} \\ \hline \mathcal D_5=\langle x,x^{[{2s}]},x^{[{3s}]},x^{[{4s}]},x^{[{5s}]} \rangle_{\mathbb F_{q^8}} & (8,8,q;4) & \begin{array}{cc} q \equiv 1 \pmod{3} \\ \gcd(s,8)=1 \end{array}
\\ \hline
\end{array} \] \caption{Delsarte duals of the codes $\mathcal C_i$ for $i=1,\ldots,5$} \label{DkMRD} \end{table}
\subsection{Linear RM-codes as subspaces of $\mathbb F_{q^n}^n$}\label{vecnum}
In \cite{Gabidulin}, Gabidulin studied RM-codes as subsets of $\mathbb F_{q^n}^n$. This view is still used in \cite{BR,H-TM,H-TNRR,Neri}. As noted before, $\mathcal{L}_{n,q}$ equipped with the classical sum and the scalar multiplication by elements in $\mathbb F_{q^n}$ is an $\mathbb F_{q^n}$-vector space. Let $\mathcal{B}=(g_1,\ldots,g_n)$ an ordered $\mathbb F_q$-basis of $\mathbb F_{q^n}$. The evaluation mapping \[ \Phi_{\mathcal{B}}: f(x) \in \mathcal{L}_{n,q} \mapsto (f(g_1),\ldots,f(g_n)) \in \mathbb F_{q^n}^n \] is an isomorphism between the $\mathbb F_{q^n}$-vector spaces $\mathcal{L}_{n,q}$ and $\mathbb F_{q^n}^n$. Therefore, if $W$ is an $\mathbb F_{q^n}$-subspace of $\mathcal{L}_{n,q}$, a generator matrix $G$ of $\Phi_{\mathcal{B}}(W)$ can be constructed using the images of a basis of $W$ under the action of $\Phi_{\mathcal{B}}$. Also, if $G$ is a generator matrix of $\Phi_{\mathcal{B}}(W)$ of maximum rank, then an $\mathbb F_{q^n}$-basis for $W$ can be defined by using the application $\Phi_{\mathcal{B}}^{-1}$ on the rows of $G$.
\section{Characterization of generalized twisted Gabidulin codes} \label{charact}
A. Horlemann-Trautmann et al. in \cite{H-TM} proved the following characterization of generalized Gabidulin codes. \begin{theorem}[\cite{H-TM}]
\label{gabidulin-d}
A MRD-code $\mathcal C$ over $\mathbb F_q$ of length $n$ and dimension $k$ is equivalent
to a generalized Gabidulin code $\mathcal G_{k,s}$ if and only if there is
an integer $s<n$ with $\gcd(s,n)=1$ and
$\dim (\mathcal C\cap\mathcal C^{[s]})=k-1$, where $\mathcal C^{[s]}=\{f(x)^{[s]} \colon f(x)\in \mathcal C\}$. \end{theorem}
If $\mathcal C$ is equivalent to a generalized twisted Gabidulin code $\mathcal H_{k,s}(\eta)$, then $\dim (\mathcal C\cap\mathcal C^{[s]})=k-2$. This condition, in general, is not enough to characterize MRD-codes equivalent to $\mathcal H_{k,s}(\eta)$. The present section is devoted to determine what further conditions are necessary for a characterization.
Denote by $\tau_\alpha$ the linear application defined by $\tau_\alpha(x)=\alpha x$ and denote by $U_1=\{\mathrm{Tr}\circ\tau_{\alpha} \colon \alpha \in \mathbb F_{q^n}\}=\{\alpha x+ \alpha^{[1]}x^{[1]}+\cdots+\alpha^{[n-1]}x^{[n-1]} \colon \alpha \in \mathbb F_{q^n}\}$. The set $U_1$ is an $\mathbb F_q$-subspace of $\mathcal{L}_{n,q}$ of dimension $n$ whose elements have rank at most one. It can be proven that the set $\mathcal{U}_1$ of all linearized polynomials with rank at most one is \[\mathcal{U}_1=\bigcup_{\beta \in \mathbb F_{q^n}^*} \tau_\beta \circ U_1 = \{\tau_\beta \circ \mathrm{Tr} \circ \tau_{\alpha} \colon \alpha,\beta \in \mathbb F_{q^n}\},\] see e.g. \cite[Proposition 5.1]{LuMaPoTr2014}.
\begin{lemma} \label{l-basis}
The space $\mathcal L_{n,q}$ admits a basis of elements contained in
$U_1$. \end{lemma} \begin{proof}
Let
$(\alpha_1,\alpha_2,\cdots,\alpha_n)$ be
a basis of $\mathbb F_{q^n}$ over $\mathbb F_q$.
Define
$\overline{\alpha_i}:=\mathrm{Tr}\circ\tau_{\alpha_i}$ for $i=1,\dots,n$
and $\mathfrak B:=\{\overline{\alpha_i}\colon i=1,\dots,n\}$.
Consider the basis $B_0=(x,x^{[1]},\dots,x^{[n-1]})$ of $\mathcal L_{n,q}$;
the components of the vectors of $\mathfrak B$ with respect to this
basis are the rows of the following matrix
\[ M:=\begin{pmatrix}
\alpha_1 & \alpha_1^{[1]} & \dots & \alpha_1^{[n-1]} \\
\alpha_2 & \alpha_2^{[1]} & \dots & \alpha_2^{[n-1]} \\
\vdots & & & \vdots \\
\alpha_n & \alpha_n^{[1]} & \dots & \alpha_n^{[n-1]} \\
\end{pmatrix} \]
By ~\cite[Corollary 2.38]{lidl_finite_1997}, $\det(M)\neq 0$;
in particular the vectors of $\mathfrak B$ are linearly independent
in $\mathcal L_{n,q}$ and so $\mathfrak B$ is a basis for $\mathcal L_{n,q}$. \end{proof}
\begin{lemma}\label{trace} Let $n$ and $s$ be two integers such that $\gcd(s,n)=1$, if $p(x) \in \mathcal L_{n,q}$ and $p(x)=\lambda p(x)^{[s]}$ for some $\lambda \in \mathbb F_{q^n}^*$, then $p(x)$ is in $\mathcal{U}_1$. \end{lemma} \begin{proof}
Under the assumptions, the map $x\to x^{[s]}$ is a generator of
the Galois group of $\mathbb F_{q^n}:\mathbb F_q$. In particular, for all $0\leq i\leq n-1$
there are $\lambda_i$ such that
$p(x)=\lambda_i p(x)^{[i]}$. It follows that the Dickson matrix of
$p(x)$ has rank at most $1$ and this proves the thesis. \end{proof}
For the sake of completeness we prove the following lemma; see also~\cite[Lemma 3]{Lun99}.
\begin{lemma}\label{fixedspace}
Let $n$ and $s$ be two integers such that $\gcd(s,n)=1$, if $W \neq \{0\}$ is an $\mathbb F_{q^n}$-subspace of $\mathcal{L}_{n,q}$ such that $W= W^{[s]}$, then $W$ admits a basis of vectors in ${U}_1$.
\end{lemma}
\begin{proof}
By Lemma~\ref{l-basis}, there exists a
basis $\mathfrak B$ of $\mathcal L_{n,q}$ consisting of vectors of ${U}_1$.
In particular, for any $b\in\mathfrak B$ and $i=0,\dots,n-1$ we have $b^{[i]}=b$.
There is a unique matrix $G$ in row reduced echelon form whose
rows contain the components of a basis of $W$ with respect to the basis
$\mathfrak B$.
Since $W=W^{[s]}$ and $\mathfrak B^{[s]}=\mathfrak B$ we have that
the rows of $G^{[s]}$ contain the components of a basis of
$W^{[s]}$ with respect to $\mathfrak B$; however $G^{[s]}$ represents
also the vectors of a basis of $W$ and it is in
row reduced echelon form; so $G^{[s]}=G$.
Since $\gcd(s,n)=1$
this yields that all entries of $G^{[s]}$ are defined over
$\mathbb F_q$. In particular, each vector of this basis of $G$ is
in the vector space $U_1$ over $\mathbb F_q$, that is it has
rank $1$.
\end{proof}
The following Lemma rephrases the requirements of Theorem~\ref{gabidulin-d} in a more suitable way for the arguments to follow. \begin{lemma}\label{Gablemma} Let $n$ and $s$ be two integers such that $\gcd(s,n)=1$ and let $\mathcal C$ be an $\mathbb F_{q^n}$-subspace of dimension $k>1$ of $\mathcal L_{n,q}$. If $\dim(\mathcal C \cap \mathcal C^{[s]})=k-1$ and $\mathcal C \cap U_1 = \{0\}$, then there exists $p(x)$ such that \[ \mathcal C=\langle p(x),p(x)^{[s]},\ldots,p(x)^{[{s(k-1)}]} \rangle_{\mathbb F_{q^n}}. \] If $\mathcal C$ contains at least one invertible linearized polynomial, then $p(x)$ is invertible and $\mathcal C\cong\mathcal G_{k,s}$. \end{lemma} \begin{proof}
Note that, since $\mathcal C$ is an $\mathbb F_{q^n}$-subspace and $\mathcal C \cap U_1=\{0\}$, then $\mathcal C\cap\mathcal{U}_1=\{0\}$. We argue by induction.
We first prove the case $k=2$. By hypothesis, $\mathcal C\cap \mathcal C^{[s]}=\langle h(x) \rangle$ and so $h(x)^{[s]} \in \mathcal C^{[s]}$.
Since $\mathcal C \cap\mathcal{U}_1=\{0\}$, by Lemma \ref{trace} the polynomials $h(x)$ and $h(x)^{[s]}$ are linearly independent over $\mathbb F_{q^n}$ and $\mathcal C=\langle h(x)^{[s(n-1)]}, h(x) \rangle_{\mathbb F_{q^n}}=\langle p(x), p(x)^{[s]} \rangle_{\mathbb F_{q^n}}$, with $p(x)=h(x)^{[{s(n-1)}]}$.
Suppose now that the assert holds true for $k-1$ and take $k>2$. Let $V:=\mathcal C\cap\mathcal C^{[s]}$, $V$ is an $\mathbb F_{q^n}$-subspace of $\mathcal C$ of dimension $k-1$ such that $V \cap \mathcal{U}_1=\{0\}$, hence by Lemma \ref{fixedspace} $V\neq V^{[s]}$. Then, since $V$ and $V^{[s]}$ are both contained in $\mathcal C^{[s]}$, by Grassmann's formula \[ \dim (V\cap V^{[s]})=k-2. \] So, $\dim V=k-1$, $V\cap \mathcal{U}_1 =\{0\}$ and $\dim (V \cap V^{[s]})=k-2$. By induction, there is $h(x) \in V$ such that \[V=\langle h(x), h(x)^{[s]},\ldots, h(x)^{[{s(k-2)}]} \rangle_{\mathbb F_{q^n}}.\] Also, \[ h(x)^{[{s(n-1)}]} \in V^{[{s(n-1)}]}=\mathcal C^{[{s(n-1)}]}\cap\mathcal C \subset \mathcal C.\] If it were $h(x)^{[{s(n-1)}]} \in V$, then $V=V^{[s]}$, which has already been excluded.
So, \[ \mathcal C=\langle p(x), p(x)^{[s]}, \ldots, p(x)^{[{s(k-1)}]} \rangle_{\mathbb F_{q^n}}, \] where $p(x)=h(x)^{[{s(n-1)}]}$.
Suppose now there is $x_0\in\mathbb F_{q^n}^*$ such that $p(x_0)=0$. Then, $\alpha_1p(x_0)+\cdots+\alpha_kp(x_0)^{[s(k-1)]}=0$ for any choice of $\alpha_i\in\mathbb F_{q^n}$, $i=1,\ldots,k$. In particular, if $\mathcal C$ contains at least one invertible linearized polynomial, then $p(x)$ must also be invertible. In such a case \[ \mathcal C \circ p^{-1}(x)=\langle x, x^{[s]}, \ldots, x^{[{s(k-1)}]} \rangle_{\mathbb F_{q^n}};\] so $\mathcal C$ is equivalent to $\mathcal{G}_{k,s}$. \end{proof}
We now focus on the case $\dim (\mathcal C \cap \mathcal C^{[s]})=k-2$. \begin{itemize}
\item
If $\dim \mathcal C=2$ we just have $\mathcal C=\langle p(x), q(x)\rangle_{\mathbb F_{q^n}}$ with $q(x) \notin \langle p(x)^{[s]}\rangle_{\mathbb F_{q^n}}$ and $p(x) \notin \langle q(x)^{[s]}\rangle_{\mathbb F_{q^n}}$. \item Suppose $\dim \mathcal C=3$, $\dim (\mathcal C \cap \mathcal C^{[s]})=1$ and $\mathcal C \cap \mathcal{U}_1=\{0\}$. As before, write $V:=\mathcal C \cap \mathcal C^{[s]}$. Since $V$ and $V^{[s]}$ are contained in $\mathcal C^{[s]}$, by Grassmann's formula, \[ 0\leq \dim(V \cap V^{[s]}) \leq 1. \] So, either $V=V^{[s]}$ or $\dim (V \cap V^{[s]})=0$. The former case is ruled out by Lemma~\ref{fixedspace}. So $\dim (V \cap V^{[s]})=0$ and $V=\langle h(x) \rangle_{\mathbb F_{q^n}}$. It follows that \[\mathcal C= \langle p(x), p(x)^{[s]} \rangle_{\mathbb F_{q^n}} \oplus \langle q(x) \rangle_{\mathbb F_{q^n}},\] with $p(x)=h(x)^{[{s(n-1)}]}$. \item Suppose that $\dim \mathcal C=4$, $\dim (\mathcal C \cap \mathcal C^{[s]})=2$ and $\mathcal C \cap \mathcal{U}_1=\{0\}$. Write $V:=\mathcal C \cap \mathcal C^{[s]}$. Clearly, since $V\neq V^{[s]}$, \[ 0 \leq \dim(V \cap V^{[s]}) \leq 1. \]
Suppose $\dim (V \cap V^{[s]})=1$. Then, the subspace $V$ fulfills all of the assumptions of Lemma~\ref{Gablemma}, so there is $h(x) \in V$ such that \[ V=\langle h(x), h(x)^{[s]} \rangle_{\mathbb F_{q^n}} \] and $h(x)^{[{s(n-1)}]}\in \mathcal C \setminus V$, since, otherwise, $V=V^{[s]}$. So, \[ \mathcal C = \langle p(x), p(x)^{[s]}, p(x)^{[{2s}]} \rangle_{\mathbb F_{q^n}} \oplus \langle q(x) \rangle_{\mathbb F_{q^n}}, \] with $p(x)=h(x)^{[{s(n-1)}]}$.
Suppose now that $\dim(V \cap V^{[s]})=0$; then $\mathcal C= V \oplus V^{[{s(n-1)}]}$. If $V=\langle h(x), g(x) \rangle_{\mathbb F_{q^n}}$ then $V^{[{s(n-1)}]}=\langle h(x)^{[{s(n-1)}]}, g(x)^{[{s(n-1)}]} \rangle_{\mathbb F_{q^n}}$, and so \[ \mathcal C=\langle p(x), p(x)^{[s]}\rangle_{\mathbb F_{q^n}} \oplus \langle q(x), q(x)^{[s]}\rangle_{\mathbb F_{q^n}}, \] with $p(x)=h(x)^{[{s(n-1)}]}$ and $q(x)=g(x)^{[{s(n-1)}]}$. \end{itemize}
More generally, we can prove the following result.
\begin{theorem} \label{mth1}
Let $n$ and $s$ be two integers such that $\gcd(s,n)=1$ and let $\mathcal C$ be an $\mathbb F_{q^n}$-subspace of dimension $k>2$ of $\mathcal L_{n,q}$.
Let $V:=\mathcal C \cap \mathcal C^{[s]}$.
Suppose $\dim V=k-2$ and $\mathcal C \cap \mathcal{U}_1 =\{0\}$, then $\mathcal C$ has one of the following forms \begin{enumerate}
\item if $\dim (V \cap V^{[s]})=k-3$, then there exist $p(x)$ and $q(x)$ in $\mathcal C$ such that
\[ \mathcal C = \langle p(x), p(x)^{[s]}, \ldots, p(x)^{[{s(k-2)}]} \rangle_{\mathbb F_{q^n}} \oplus \langle q(x) \rangle_{\mathbb F_{q^n}}; \]
\item if $\dim (V \cap V^{[s]})=k-4$, then there exist $p(x)$ and $q(x)$ in $\mathcal C$ such that
\[ \mathcal C = \langle p(x), p(x)^{[s]}, \ldots, p(x)^{[{s(i-1)}]} \rangle_{\mathbb F_{q^n}} \oplus \langle q(x), q(x)^{[s]}, \ldots, q(x)^{[{s(j-1)}]} \rangle_{\mathbb F_{q^n}}, \]
where $i+j=k$ and $i,j \geq 2$. \end{enumerate} \end{theorem} \begin{proof}
We have already proved the assert for $k \leq 4$.
Assume by induction that the assert holds for each $t <k$ with $k\geq 4$. Since $V$ and $V^{[s]}$ are contained in $\mathcal C^{[s]}$, it follows that \[ \dim (V\cap V^{[s]})\geq k-4, \] that is $\dim (V\cap V^{[s]}) \in \{k-4,k-3\}$, since $V \neq V^{[s]}$. If $\dim (V\cap V^{[s]})=k-3$, then, by Lemma \ref{Gablemma}, there exists $h(x) \in V$ such that \[ V=\langle h(x), h(x)^{[s]}, \ldots, h(x)^{[{s(k-3)}]} \rangle_{\mathbb F_{q^n}}.\] Since $h(x)^{[{s(n-1)}]}\in \mathcal C \setminus V$ (otherwise $V=V^{[s]}$), we get \[ \mathcal C=\langle p(x), p(x)^{[s]},\ldots, p(x)^{[{s(k-2)}]} \rangle_{\mathbb F_{q^n}}\oplus \langle q(x) \rangle_{\mathbb F_{q^n}}, \] where $p(x)=h(x)^{[{s(n-1)}]}$. If $\dim (V \cap V^{[s]})=k-4$, since $V$ has dimension $k-2$ and $V\cap \mathcal{U}_1=\{0\}$, by induction there exist $h(x)$ and $g(x)$ such that either \[ V=\langle h(x),\ldots,h(x)^{[s[k-4]]}\rangle_{\mathbb F_{q^n}} \oplus \langle g(x) \rangle_{\mathbb F_{q^n}} \] or \[ V=\langle h(x),\ldots, h(x)^{[{s(l-1)}]} \rangle_{\mathbb F_{q^n}} \oplus \langle g(x),\ldots, g(x)^{[{s(m-1)}]} \rangle_{\mathbb F_{q^n}}, \] with $l+m=k-2$. Since $V, V^{[{s(n-1)}]} \subset \mathcal C$ and $\dim V\cap V^{[s]}=k-4$ we get $\mathcal C=V + V^{[{s(n-1)}]}$. So, either \[ \mathcal C=\langle h(x)^{[{s(n-1)}]}, h(x), \ldots, h(x)^{[{s(k-4)}]} \rangle_{\mathbb F_{q^n}} \oplus \langle g(x)^{[{s(n-1)}]}, g(x) \rangle_{\mathbb F_{q^n}} \] or \[ \mathcal C=\langle h(x)^{[{s(n-1)}]}, h(x), \ldots, h(x)^{[{s(l-1)}]} \rangle_{\mathbb F_{q^n}} \oplus \langle g(x)^{[{s(n-1)}]}, g(x), \ldots, g(x)^{[{s(m-1)}]} \rangle_{\mathbb F_{q^n}}. \] If we now put $p(x)=h(x)^{[{s(n-1)}]}$ and $q(x)=g(x)^{[{s(n-1)}]}$, then we get the assert. \end{proof}
Examples of $k$-dimensional MRD-codes $\mathcal C$ with $\dim(\mathcal C\cap\mathcal C^{[s]})=k-2$ and $\dim (V\cap V^{[s]})=k-3$, where $V=\mathcal C\cap\mathcal C^{[s]}$, are the generalized twisted Gabidulin codes; see Remark \ref{k-2}. An example where $\dim V\cap V^{[s]}=k-4$ is given by the code $\mathcal{D}_2$ (see Table \ref{DkMRD}), which can be written as \[ \mathcal{D}_2=\langle -x+x^{[2]} , -x^{[1]}+x^{[3]} \rangle_{\mathbb F_{q^6}} \oplus \langle -\delta x^{[1]} +x^{[3]}, -\delta x^{[2]}+x^{[4]} \rangle_{\mathbb F_{q^6}}. \]
\begin{lemma}
\label{ll0} Let $\mathcal C\subseteq \mathcal L_{n,q}$ be an $\mathbb F_{q^n}$-linear RM-code with dimension $k$ containing a MRD-code $\mathcal G$ equivalent to a generalized Gabidulin code $\mathcal G_{l,s}$ of dimension $l \leq k$, then there exists a permutation linearized polynomial $p(x)$ and $(k-l)$ linearized polynomials $q_1(x),\ldots,q_{k-l}(x)$ such that \begin{equation}\label{form}
\mathcal C=\langle q_1(x),\ldots,q_{k-l}(x), p(x), p(x)^{[s]},\ldots p(x)^{[s(l-1)]}
\rangle. \end{equation} \end{lemma} We call the polynomials $q_i(x)$ of Lemma~\ref{ll0} \emph{polynomials of extra type}. \begin{proof} By Lemma \ref{Gablemma}, there exists a permutation linearized polynomial $p(x)$ such that \[\mathcal{G}= \langle p(x), p(x)^{[s]},\ldots,p(x)^{[s(l-1)]} \rangle_{\mathbb F_{q^n}} \] and $p(x)$,\dots,$p(x)^{[s(l-1)]}$ are linearly independent. Now, we can extend the list of polynomials $\{p(x), p(x)^{[s]},\ldots,p(x)^{[s(l-1)]}\}$ to a basis of $\mathcal C$ with suitable polynomials $q_i$ as to get the form~\eqref{form}. \end{proof}
\begin{lemma}\label{charH} If $\mathcal C \subseteq \mathcal L_{n,q}$ is an $\mathbb F_{q^n}$-linear MRD-code of dimension $k$ containing a code $\mathcal G$ equivalent to $\mathcal G_{k-1,s}$, i.e. $\mathcal G=\langle p(x), p(x)^{[s]},\ldots, p(x)^{[s(k-2)]} \rangle_{\mathbb F_{q^n}}$ with $p(x)$ an invertible linearized polynomial, and for which there exists an extra polynomial $g(x)$ in $\langle p(x)^{[-s]},p(x)^{[s(k-1)]}\rangle_{\mathbb F_{q^n}}$ with $g(x)=p(x)^{[-s]}+\eta p(x)^{[s(k-1)]}$ and $\mathrm{N}_{q^n/q}(\eta)\neq (-1)^{kn}$, then $\mathcal C$ is equivalent to $\mathcal H_{k,s}(\eta^{[s]})$. \end{lemma} \begin{proof} By the previous lemma and by hypothesis, \[ \mathcal C=\langle g(x), p(x), p(x)^{[s]},\ldots, p(x)^{[s(k-2)]} \rangle_{\mathbb F_{q^n}}, \] with $p(x)$ permutation linearized polynomial and $g(x) = p(x)^{[-s]}+\eta p(x)^{[s(k-1)]}$. Since $\mathcal C$ and $\mathcal C^{[s]}$ are equivalent, we can suppose that \[ \mathcal C=\langle q(x), p(x)^{[s]},\ldots, p(x)^{[s(k-1)]} \rangle_{\mathbb F_{q^n}}, \] with $q(x) = p(x)+\eta^{[s]} p(x)^{[sk]}$. So, \[ \mathcal C=\langle x^{[s]},\ldots,x^{[{s(k-1)}]}, x+\eta^{[s]} x^{[sk]} \rangle_{\mathbb F_{q^n}} \circ p(x). \] Since $\mathcal C$ is a MRD-code, then $\mathcal C\circ p^{-1}(x)=\langle x^{[s]},\ldots,x^{[{s(k-1)}]}, x+\eta^{[s]} x^{[sk]} \rangle_{\mathbb F_{q^n}}$ is also a MRD-code equivalent to $\mathcal H_{k,s}(\eta^{[s]})$. \end{proof}
Theorem~\ref{mth1} prompts the following characterization of generalized twisted Gabidulin codes.
\begin{theorem}
Let $\mathcal C$ be an $\mathbb F_{q^n}$-linear MRD-code of dimension $k>2$ contained
in $\mathcal L_{n,q}$.
Then, the code $\mathcal C$ is equivalent to a generalized twisted Gabidulin code if and only if there exists an integer $s$ such that $\gcd(s,n)=1$ and the following two conditions hold \begin{enumerate}
\item $\dim (\mathcal C \cap \mathcal C^{[s]})=k-2$ and $\dim(\mathcal C\cap \mathcal C^{[s]} \cap \mathcal C^{[{2s}]})=k-3$, i.e.
there exist $p(x),q(x) \in \mathcal C$ such that
\[ \mathcal C= \langle p(x)^{[s]}, p(x)^{[{2s}]}, \ldots, p(x)^{[{s(k-1)}]} \rangle_{\mathbb F_{q^n}} \oplus \langle q(x) \rangle_{\mathbb F_{q^n}}; \]
\item $p(x)$ is invertible and there exists $\eta \in \mathbb F_{q^n}^*$ with $\mathrm{N}\,_{q^n/q}(\eta)\neq (-1)^{kn}$ such that $p(x)+\eta p(x)^{[{sk}]} \in \mathcal C$. \end{enumerate} \end{theorem} \begin{proof}
The proof follows directly
from Theorem~\ref{mth1}
and Lemma~\ref{charH}. \end{proof}
As a consequence we get the following.
\begin{theorem} Let $\mathcal C$ be an $\mathbb F_{q^n}$-linear RM-code of dimension $k>2$ of $\mathcal L_{n,q}$, with $\mathcal C \cap \mathcal{U}_1=\{0\}$. If there exists an integer $s$ such that $\gcd(s,n)=1$ and \begin{enumerate}
\item $\dim (\mathcal C \cap \mathcal C^{[s]})=k-2$ and $\dim(\mathcal C\cap \mathcal C^{[s]} \cap \mathcal C^{[{2s}]})=k-3$, i.e.
there exist $p(x),q(x) \in \mathcal C$ such that
\[ \mathcal C= \langle p(x)^{[s]}, p(x)^{[{2s}]}, \ldots, p(x)^{[{s(k-1)}]} \rangle_{\mathbb F_{q^n}} \oplus \langle q(x) \rangle_{\mathbb F_{q^n}}; \]
\item $p(x)$ is invertible and there exists $\eta \in \mathbb F_{q^n}^*$ such that $p(x)+\eta p(x)^{[{sk}]} \in \mathcal C$ and $\mathrm{N}\,_{q^n/q}(\eta)\neq (-1)^{kn}$, \end{enumerate} then $\mathcal C$ is a MRD-code equivalent to $\mathcal H_{k,s}(\eta)$. \end{theorem}
Note that if such invertible linearized polynomial $p(x)$ exists, then $\mathcal C\cap\mathcal C^{[s]}\cap\cdots\cap\mathcal C^{[{s(k-2)}]}=\langle p(x)^{[{s(k-2)}]} \rangle_{\mathbb F_{q^n}}$.
\section{Distinguishers for RM-codes} \label{disting} A \emph{distinguisher} is an easy to compute function which allows to identify an object in a family of (apparently) similar ones. Existence of distinguishers is of particular interest for cryptographic applications, as it makes possible to identify a candidate encryption from a random text.
As seen in the previous section, it has been shown in~\cite{H-TM} that an MRD-code $\mathcal C$ of parameters $[n,k]$ is equivalent to a generalized Gabidulin code if, and only if, there exists a positive integer $s$ such that $\gcd(s,n)=1$ and $\dim(\mathcal C\cap\mathcal C^{[s]})=k-1$. Following the approach of \cite{H-TM}, we define for any RM-code $\mathcal C$ the number \[ h(\mathcal C):=\max\{ \dim(\mathcal C\cap\mathcal C^{[j]})\colon j=1,\ldots,n-1; \gcd(j,n)=1 \}. \] Theorem~\ref{gabidulin-d} states that an MRD-code $\mathcal C$ is equivalent to a generalized Gabidulin code if and only if $h(\mathcal C)=k-1$.
Also, for any given $\mathbb F_{q^n}$-linear code $\mathcal C$, the following proposition is immediate. \begin{proposition}
For any $k$-dimensional $\mathbb F_{q^n}$-linear code $\mathcal C$,
\[\mathcal C^{[i]\perp}=\mathcal C^{\perp[i]},\]
for each $i\in \{0,\ldots,n-1\}$.
So, we have \[ h(\mathcal C^\perp)=n-2k+h(\mathcal C). \] \end{proposition}
We now define also the \emph{Gabidulin index}, $\mathrm{ind}\,(\mathcal C)$ of a $[n,k]$ RM-code as the maximum dimension of a subcode $\mathcal G\leq\mathcal C$ contained in $\mathcal C$ with $\mathcal G$ equivalent to a generalized Gabidulin code.
Clearly, $1\leq\mathrm{ind}\,(\mathcal C)\leq k$ and $\mathrm{ind}\,(\mathcal C)=k$ if and only if $\mathcal C$ is a Gabidulin code. It can be readily seen that if $\mathcal C$ and $\mathcal C'$ are two equivalent codes, then they have the same indexes $\mathrm{ind}\,(\mathcal C)=\mathrm{ind}\,(\mathcal C')$ and $h(\mathcal C)=h(\mathcal C')$. Also, $h(\mathcal C)\geq\mathrm{ind}\,(\mathcal C)-1$ for RM-codes.
We shall now prove that for the known codes the Gabidulin index can be effectively computed. More in detail, in the next theorem we determine these indexes for each known $\mathbb F_{q^n}$-linear MRD-code. Our result is contained in Table~\ref{tab1}. Also in the table we recall the right idealisers (up to equivalence) for these codes (see also \cite{PhDthesis}).
\begin{theorem}
The Gabidulin indexes $\mathrm{ind}\,(\mathcal C)$ and the values of $h(\mathcal C)$ for the known MRD-codes $\mathcal C$ of parameters $[n,k]$ are
as given in Table~\ref{tab1}.
\begin{table}[htp]
\[
\begin{array}{ |c|c|c|c|c| }
\hline \mbox{Code } & \mathrm{ind}\, & h & R & [n,k] \\ \hline \mathcal G_{k,s} & k & k-1 & \mathbb F_{q^n} & [n,k] \\ \hline \mathcal H_{k,s}(\eta) & k-1 & k-2 & \mathbb F_{q^{\gcd(n,k)}} & [n,k] \\ \hline \mathcal C_1 & 1 & 0 & \mathbb F_{q^3} & [6,2] \\ \hline \mathcal C_2 & 1 & 0 & \mathbb F_{q^2} & [6,2] \\ \hline \mathcal C_3 & 2 & 1 & \mathbb F_{q^n} & [7,3] \\ \hline \mathcal C_4 & 1 & 0 & \mathbb F_{q^4} & [8,2] \\ \hline \mathcal C_5 & 2 & 1 & \mathbb F_{q^n} & [8,3] \\ \hline
\end{array}\qquad\qquad
\begin{array}{ |c|c|c|c|c| }
\hline \mbox{Code } & \mathrm{ind}\, & h & R & [n,k] \\ \hline
& & & & \\ \hline
& & & & \\ \hline \mathcal D_1 & 2 & 2 & \mathbb F_{q^3} & [6,4] \\ \hline \mathcal D_2 & 2 & 2 & \mathbb F_{q^2} & [6,4] \\ \hline \mathcal D_3 & 3 & 2 & \mathbb F_{q^n} & [7,4] \\ \hline \mathcal D_4 & 3 & 4 & \mathbb F_{q^4} & [8,6] \\ \hline \mathcal D_5 & 4 & 3 & \mathbb F_{q^n} & [8,5] \\ \hline \end{array} \] \caption{Known linear MRD-codes and their Gabidulin index} \label{tab1} \end{table}
\end{theorem}
\begin{proof}
Clearly, the Gabidulin index of a generalized Gabidulin code is $k$;
any twisted generalized Gabidulin code of dimension $k$ contains
a generalized Gabidulin code of dimension $k-1$;
so, its index is $k-1$.
We now consider the case of the codes
$\mathcal C_1, \mathcal C_2, \mathcal C_3, \mathcal C_4, \mathcal C_5, \mathcal D_3$ and $\mathcal D_5$.
By construction, it is immediate to see that they all contain a
generalized Gabidulin code of codimension $1$; so, they also
have Gabidulin index $k-1$, where $k$ is the dimension of the code.
Also for all of them $k-2\leq h(\mathcal C)<k-1$, so $h(\mathcal C)=k-2$.
The cases of the dual codes $\mathcal D_{i}$ with $i=1,2,4$ must
be studied in more detail.
First we prove that
the codes $\mathcal D_1, \mathcal D_2$ and $\mathcal D_4$ do not contain any code equivalent to $\mathcal G_{k-1,s}$, for any $s$, i.e. that their Gabidulin index is less than $k-1$
and then determine the exact value.
\begin{flushleft}
\textbf{The code $\mathcal D_1$} \end{flushleft}
By Table~\ref{DkMRD}, we have that \[ \mathcal D_1=\langle x^{[1]}, x^{[{2}]}, x^{[{4}]},x-\delta^{[{5}]} x^{[{3}]} \rangle_{\mathbb F_{q^6}}. \] Suppose that there is a code $\overline{\mathcal D}$ contained in $\mathcal D_1$ equivalent to a generalized Gabidulin code of dimension $3$, i.e. either $\overline{\mathcal D}\simeq \mathcal G_{3,1}$ or $\overline{\mathcal D}\simeq \mathcal G_{3,5}$. Since $\mathcal G_{3,1}$ and $\mathcal G_{3,5}$ are equivalent, then $\overline{\mathcal D}$ is equivalent to $\mathcal G_{3,1}$. By Theorem~\ref{gabidulin-d}, $h({\overline{\mathcal D}})=2$; on the other hand, since $\mathcal D_1$ is not equivalent to a Gabidulin code it must be $h(\mathcal D)< 3$. So, $\mathcal D_1\cap \mathcal D_1^{[1]} = \overline{\mathcal D}\cap \overline{\mathcal D}^{[1]}$ and hence $\mathcal D_1\cap\mathcal D_1^{[5]}=\overline{\mathcal D}\cap\overline{\mathcal D}^{[5]}$. From these equalities we get \[ \overline{\mathcal D}\cap \overline{\mathcal D}^{[1]}= \langle x^{[{2}]}, x^{[1]}-\delta x^{[4]}\rangle_{\mathbb F_{q^6}} \] and \[ \overline{\mathcal D}^{[5]}\cap \overline{\mathcal D}=\langle x^{[1]}, x-\delta^{[5]} x^{[3]} \rangle_{\mathbb F_{q^6}}. \] Since $\dim \overline{\mathcal D}=3$ we obtain \[ \overline{\mathcal D}= \langle x^{[1]}, x^{[{2}]}, x^{[1]}-\delta x^{[4]}\rangle_{\mathbb F_{q^6}}= \langle x^{[1]}, x^{[{2}]}, x^{[4]}\rangle_{\mathbb F_{q^6}}.\] The code $\overline{\mathcal D}$ is not MRD, since it contains the polynomial $x^{[1]}-x^{[4]}$ which has kernel of dimension $3$, in particular it cannot be equivalent to $\mathcal G_{3,1}$. It follows that $\mathrm{ind}\,(\mathcal D_1)=2$ since $\langle x^{[1]}, x^{[{2}]} \rangle_{\mathbb F_{q^6}} \simeq \mathcal{G}_{2,1}$.
\begin{flushleft}
\textbf{The code $\mathcal D_2$} \end{flushleft}
By Table~\ref{DkMRD} the code $\mathcal D_2$ is \[ \mathcal D_2=\langle x^{[1]},x^{[3]},x-x^{[2]},x^{[4]}-\delta x \rangle_{\mathbb F_{q^6}}, \] with $q$ odd, $\delta^2+\delta=1$ and $q \equiv 0,\pm 1 \pmod{5}$, hence $\delta \in \mathbb F_q$. Suppose $\mathrm{ind}\,(\mathcal D_2)=3$, as before $\mathcal D_2$ contains a code $\overline{\mathcal D}$ equivalent to $\mathcal G_{3,1}$.
Arguing as the previous case, we get \[ \overline{\mathcal D}= \langle -x+x^{[2]}, x^{[3]}-\delta x^{[1]}, -x^{[1]}+x^{[3]} \rangle_{\mathbb F_{q^6}}= \langle x^{[1]}, x^{[3]}, -x+x^{[2]} \rangle_{\mathbb F_{q^6}}. \] To show that $\overline{\mathcal D}$ is not equivalent to any $\mathcal G_{3,s}$ we compute its right idealiser $R(\overline{\mathcal D})$. Write $\displaystyle \varphi (x)=\sum_{i=0}^5 a_ix^{[i]} \in R(\overline{\mathcal D})$; then $x^{[1]}\circ \varphi(x), x^{[3]}\circ \varphi(x) \in \overline{\mathcal D}$, so $\varphi(x)=\eta x$, for some $\eta \in \mathbb F_{q^6}$. Furthermore, $(x-x^{[2]}) \circ \varphi(x) \in \overline{\mathcal D}$; so $\eta=\eta^{[2]}$ and $\eta \in \mathbb F_{q^2}$. So, we get
$R(\overline{\mathcal D}) \simeq \mathbb F_{q^2}$. If $\overline{\mathcal D}$ were to be equivalent to $\mathcal G_{3,1}$, by Proposition \ref{idealis} and by \cite[Corollary 5.2]{LTZ2}, it would follow that $R(\overline{\mathcal D})$ is equivalent to $R(\mathcal G_{3,1}) \simeq \mathbb F_{q^6}$, which is not possible. Suppose now $\mathcal D_2$ to contain a code $\overline{\mathcal D}$ equivalent to $\mathcal G_{2,1}$. Then by Theorem \ref{gabidulin-d} and by Lemma~\ref{Gablemma} we easily get $\overline{\mathcal D}=\langle f(x), f(x)^{[1]} \rangle_{\mathbb F_{q^6}}$ with $f(x)$ an invertible linearized polynomial.
Also, $\overline{\mathcal D}\cap\overline{\mathcal D}^{[1]}=\langle f(x)^{[1]} \rangle \subset \mathcal D_2\cap\mathcal D_2^{[1]}=\langle -x^{[1]}+x^{[3]}, x^{[4]}-\delta x^{[2]} \rangle_{\mathbb F_{q^6}}$, so $f(x)^{[1]}=a(-x^{[1]}+x^{[3]})+b(x^{[4]}-\delta x^{[2]})$, since $f(x)$ is invertible we may assume $b=1$. In particular, $\mathcal D_2$ contains a code equivalent to $\mathcal G_{2,1}$ if and only if there exists $a \in \mathbb F_{q^6}$ such that $f(x)^{[1]}$ is invertible. Let $D_{f^{[1]}}$ be the Dickson matrix associated to the polynomial $f(x)^{[1]}$ considered above. Then, for $a=1$ we have $\det D_{f^{[1]}} =16 (2-3\delta)\neq 0$.
So, $\mathcal D_2$ contains $\langle -x-\delta x^{[1]} +x^{[2]}+x^{[3]} , -x^{[1]}+x^{[3]}+x^{[4]}-\delta x^{[2]}\rangle_{\mathbb F_{q^6}}\simeq\mathcal G_{2,1}$ and, consequently, $\mathrm{ind}\,(\mathcal D_2)=2$.
\begin{flushleft}
\textbf{The code $\mathcal D_4$} \end{flushleft}
The code $\mathcal D_4$ is \[ \mathcal D_4=\langle x^{[1]},x^{[2]},x^{[3]},x^{[5]},x^{[6]},x-\delta x^{[4]} \rangle_{\mathbb F_{q^8}}, \] with $q$ odd and $\delta^2=-1$. Suppose that $\mathcal D_4$ contains a code $\overline{\mathcal D}$ equivalent to a generalized Gabidulin code of dimension $5$. Since $\mathcal G_{5,1} \simeq \mathcal G_{5,7}$ and $\mathcal G_{5,3} \simeq \mathcal G_{5,5}$, we get that either $\overline{\mathcal D}\simeq \mathcal G_{5,1}$ or $\overline{\mathcal D}\simeq \mathcal G_{5,3}$. By Lemma~\ref{Gablemma},
$\dim (\overline{\mathcal D}\cap \overline{\mathcal D}^{[s]})=4$, with either $s=1$ or $s=3$, and, since $\mathcal D_4$ is not equivalent to any generalized Gabidulin code, $\dim(\mathcal D_4 \cap \mathcal D_4^{[s]}) <5$, so $\mathcal D_4\cap \mathcal D_4^{[s]} = \overline{\mathcal D}\cap \overline{\mathcal D}^{[s]}$. First assume that $\overline{\mathcal D}\simeq \mathcal G_{5,1}$. It is easy to see that \[\overline{\mathcal D}\cap \overline{\mathcal D}^{[1]}= \langle x^{[2]},x^{[3]},x^{[6]},x^{[1]}-\delta^{[1]} x^{[5]} \rangle_{\mathbb F_{q^8}}.\] Since the dimension of $\overline{\mathcal D}$ is $5$ and $x^{[1]} \in \overline{\mathcal D} \setminus (\overline{\mathcal D}\cap \overline{\mathcal D}^{[1]})$, it follows that \[ \overline{\mathcal D}=\langle x^{[1]}, x^{[2]},x^{[3]},x^{[6]},x^{[1]}-\delta^{[1]} x^{[5]} \rangle_{\mathbb F_{q^8}}= \langle x^{[1]}, x^{[2]}, x^{[3]}, x^{[5]}, x^{[6]} \rangle_{\mathbb F_{q^8}}. \] The Delsarte dual of $\overline{\mathcal D}$ is \[ \overline{\mathcal D}^\perp =\langle x,x^{[4]},x^{[7]} \rangle_{\mathbb F_{q^8}}, \] which is not MRD, since of $x-x^{[4]}$ has kernel of dimension $4$. By Lemma \ref{dualMRD}, neither $\overline{\mathcal D}$ is an MRD-code, a contradiction. Now, assume $\overline{\mathcal D}\simeq \mathcal G_{5,3}$. As before, \[\overline{\mathcal D}\cap \overline{\mathcal D}^{[3]}= \langle x^{[1]}, x^{[5]}, x^{[6]}, x-\delta x^{[4]} \rangle_{\mathbb F_{q^8}}\] and \[\overline{\mathcal D}^{[5]}\cap \overline{\mathcal D}= \langle x^{[6]}, x^{[2]}, x^{[3]}, x^{[5]}-\delta^{[5]} x^{[1]} \rangle_{\mathbb F_{q^8}}.\] So, \[ \overline{\mathcal D}=\langle x^{[1]}, x^{[6]}, x^{[2]}, x^{[3]}, x^{[5]}-\delta^{[5]} x^{[1]} \rangle_{\mathbb F_{q^8}}= \langle x^{[1]}, x^{[2]}, x^{[3]}, x^{[5]}, x^{[6]}\rangle_{\mathbb F_{q^8}}. \]
\noindent Again we get a contradiction since $\overline{\mathcal D}$ is not an MRD-code.\\
Suppose now that $\mathcal D_4$ contains a code $\overline{\mathcal D}$ equivalent to $\mathcal G_{4,1}$. By Theorem \ref{gabidulin-d} and by Lemma~\ref{Gablemma},
$\overline{\mathcal D}=\langle p(x),p(x)^{[1]},p(x)^{[2]}, p(x)^{[3]} \rangle_{\mathbb F_{q^8}}$ for some invertible linearized polynomial $p(x)\in \mathcal D_4$. Clearly, $\langle p(x)^{[1]},p(x)^{[2]},p(x)^{[3]} \rangle_{\mathbb F_{q^8}} \subset \langle x^{[2]},x^{[3]},x^{[6]},x^{[1]}-\delta^{[1]} x^{[5]} \rangle_{\mathbb F_{q^8}}=\mathcal D_4\cap\mathcal D_4^{[1]}$ and so there exist $a,b,c,d \in \mathbb F_{q^8}$ such that \[p(x)^{[1]}= ax^{[2]}+bx^{[3]}+cx^{[6]}+d(x^{[1]}-\delta^{[1]} x^{[5]}),\] \[p(x)^{[2]}=a^{[1]} x^{[3]}+b^{[1]}x^{[4]}+c^{[1]}x^{[7]}+d^{[1]}(x^{[2]}-\delta^{[2]} x^{[6]}),\] \[p(x)^{[3]}=a^{[2]} x^{[4]}+b^{[2]}x^{[5]}+c^{[2]}x+d^{[2]}(x^{[3]}-\delta^{[3]} x^{[7]}).\]
Since these are all elements of $\mathcal D_4$, we get $a=b=c=d=0$, i.e. $\mathcal D_4$ cannot contain a code equivalent to $\mathcal G_{4,1}$. Finally, suppose that $\overline{\mathcal D}$ is equivalent to $\mathcal G_{4,3}$. By Theorem \ref{gabidulin-d} and by Lemma~\ref{Gablemma}, $\overline{\mathcal D}=\langle p(x),p(x)^{[3]},p(x)^{[6]}, p(x)^{[1]} \rangle_{\mathbb F_{q^8}}$ for some invertible linearized polynomial $p(x)\in \mathcal D_4$ and arguing as before we get a contradiction, i.e. $\mathcal D_4$ cannot contain a code equivalent to $\mathcal G_{4,3}$. So, $\mathcal D_4$ cannot contain a code equivalent to a generalized Gabidulin code of dimension $4$ and so $\mathrm{ind}\,(\mathcal D_4)<4$. Since $\langle x^{[1]},x^{[2]},x^{[3]}\rangle_{\mathbb F_{q^8}}\simeq \mathcal{G}_{3,1}$, it follows $\mathrm{ind}\,(\mathcal D_4)=3$. \end{proof}
Thus Theorem \ref{mth1} provides the following structure result on $k$-dimensional $\mathbb F_{q^n}$-linear RM-codes with $h(\mathcal C)=k-2$. \begin{theorem}\label{mth2}
Let $\mathcal C$ be a $k$-dimensional $\mathbb F_{q^n}$-linear RM-code of $\mathcal L_{n,q}$ having $h(\mathcal C)=k-2$, with $k>2$.
Denote by $s$ an integer such that $\gcd(s,n)=1$ and $\dim(\mathcal C\cap\mathcal C^{[s]})=k-2$.
Let $V:=\mathcal C \cap \mathcal C^{[s]}$ and suppose that $\mathcal C \cap \mathcal{U}_1 =\{0\}$, then $\mathcal C$ has one of the following forms \begin{enumerate}
\item if $\dim (V \cap V^{[s]})=k-3$, then there exist $p(x)$ and $q(x)$ in $\mathcal C$ such that
\[ \mathcal C = \langle p(x), p(x)^{[s]}, \ldots, p(x)^{[{s(k-2)}]} \rangle_{\mathbb F_{q^n}} \oplus \langle q(x) \rangle_{\mathbb F_{q^n}}; \]
\item if $\dim (V \cap V^{[s]})=k-4$, then there exist $p(x)$ and $q(x)$ in $\mathcal C$ such that
\[ \mathcal C = \langle p(x), p(x)^{[s]}, \ldots, p(x)^{[{s(i-1)}]} \rangle_{\mathbb F_{q^n}} \oplus \langle q(x), q(x)^{[s]}, \ldots, q(x)^{[{s(j-1)}]} \rangle_{\mathbb F_{q^n}}, \]
where $i+j=k$ and $i,j \geq 2$. \end{enumerate}
In particular, $\mathcal C$ is equivalent to $\mathcal H_{k,s}(\eta)$, for some $\eta \in \mathbb F_{q^n}$, if and only if $\dim (V \cap V^{[s]})=k-3$, $p(x)$ is invertible and there exists $\eta \in \mathbb F_{q^n}^*$ such that $p(x)+\eta p(x)^{[{sk}]} \in \mathcal C$ and $\mathrm{N}\,_{q^n/q}(\eta)\neq (-1)^{kn}$. \end{theorem}
\begin{remark} Note that, in the hypothesis of Theorem \ref{mth2}, if the polynomials $p(x)$ and $q(x)$ are invertible, then either $\mathrm{ind}\,(\mathcal C)=\dim\mathcal C-1$ or $\displaystyle \mathrm{ind}\,(\mathcal C)\geq\frac{\dim\mathcal C}2$. This holds for the known MRD-codes listed in the Tables \ref{kMRD} and \ref{DkMRD}; it is currently an open question whether an $\mathbb F_{q^n}$-linear MRD-code $\mathcal C$ having $h(\mathcal C)=\dim\mathcal C-2$ and $\mathrm{ind}\,(\mathcal C)<\frac{\dim\mathcal C}{2}$ might exist or not. We also remark that the known MRD-codes presented in the Tables \ref{kMRD} and \ref{DkMRD} which are not equivalent to a generalized Gabidulin code, have $h(\mathcal C)=\dim\mathcal C-2$.
Suppose a code $\mathcal C$ has generator matrix in standard form
$[I_k|X]$. Using the arguments of~\cite[Lemma 19]{H-TNRR} it can be seen that $\dim(\mathcal C\cap\mathcal C^{[s]})\geq \dim\mathcal C-i$ with $i>0$ if and only if $rk(X-X^{[s]})\leq i$, and this condition can be expressed by imposing that all minors of $X-X^{[s]}$ of rank $j>i$ have determinant $0$. In particular, the set of all codes with $h(\mathcal C)\geq\dim(\mathcal C)-i$ is contained in the union of a finite number of closed Zariski sets. So, for a generic MRD-code we have $h(\mathcal C)\in \max\{0,2k-n\}$. We leave as an open problem to determine some families of MRD-codes with $h(\mathcal C)<\dim(\mathcal C)-2$ and, more in detail, to determine the possible spectrum of the values of $h(\mathcal C)$ might attain as $\mathcal C$ varies among all MRD-codes over a given field. \end{remark}
\vskip.2cm \noindent \begin{minipage}[t]{\textwidth} Authors' addresses: \vskip.2cm\noindent\nobreak \centerline{ \begin{minipage}[t]{7cm} Luca Giuzzi\\ D.I.C.A.T.A.M. {\small (Section of Mathematics)} \\ University of Brescia\\ Via Branze 43, I-25123, Brescia, Italy \\ [email protected] \end{minipage} \begin{minipage}[t]{7.5cm} Ferdinando Zullo\\ Department of Mathematics and Physics \\ University of Campania ``\emph{Luigi Vanvitelli}'' \\ Viale Lincoln 5, I-81100, Caserta, Italy \\ [email protected] \end{minipage} }
\end{minipage}
\end{document} | arXiv |
When the expression $(2x^4+3x^3+x-14)(3x^{10}-9x^7+9x^4+30)-(x^2+5)^7$ is expanded, what is the degree of the resulting polynomial?
Multiplying out that entire polynomial would be pretty ugly, so let's see if there's a faster way. The degree of $(2x^4+3x^3+x-14)(3x^{10}-9x^7+9x^4+30)$ is the highest possible power of $x$, which occurs when we multiply $(2x^4)(3x^{10})$. This gives $6x^{14}$ so the degree of the first part is $14$. To find the degree of $(x^2+5)^7$, we need to find the highest possible power of $x$. This product is equivalent to multiplying $(x^2+5)$ by itself $7$ times, and each term is created by choosing either $x^2$ or $5$ from each of the seven factors. To get the largest power of $x$, we should choose $x^2$ from all seven factors, to find $(x^2)^7=x^{14}$ as the highest power of $x$, so the second part is also a degree-$14$ polynomial. Thus we have a degree-$14$ polynomial minus a degree-$14$ polynomial, which will give us another degree-$14$ polynomial... unless the $x^{14}$ terms cancel out. We must check this. In the first part, the coefficient on $x^{14}$ was $6$, and in the second part the coefficient was $1$. So our expression will look like $(6x^{14}+\ldots)-(x^{14}+\ldots)$ where all the other terms have degree less than $14$, so when simplified the expression will be $5x^{14}+\ldots$. Thus the coefficient on the $x^{14}$ term is not zero, and the polynomial has degree $\boxed{14}$. | Math Dataset |
Overall, the studies listed in Table 1 vary in ways that make it difficult to draw precise quantitative conclusions from them, including their definitions of nonmedical use, methods of sampling, and demographic characteristics of the samples. For example, some studies defined nonmedical use in a way that excluded anyone for whom a drug was prescribed, regardless of how and why they used it (Carroll et al., 2006; DeSantis et al., 2008, 2009; Kaloyanides et al., 2007; Low & Gendaszek, 2002; McCabe & Boyd, 2005; McCabe et al., 2004; Rabiner et al., 2009; Shillington et al., 2006; Teter et al., 2003, 2006; Weyandt et al., 2009), whereas others focused on the intent of the user and counted any use for nonmedical purposes as nonmedical use, even if the user had a prescription (Arria et al., 2008; Babcock & Byrne, 2000; Boyd et al., 2006; Hall et al., 2005; Herman-Stahl et al., 2007; Poulin, 2001, 2007; White et al., 2006), and one did not specify its definition (Barrett, Darredeau, Bordy, & Pihl, 2005). Some studies sampled multiple institutions (DuPont et al., 2008; McCabe & Boyd, 2005; Poulin, 2001, 2007), some sampled only one (Babcock & Byrne, 2000; Barrett et al., 2005; Boyd et al., 2006; Carroll et al., 2006; Hall et al., 2005; Kaloyanides et al., 2007; McCabe & Boyd, 2005; McCabe et al., 2004; Shillington et al., 2006; Teter et al., 2003, 2006; White et al., 2006), and some drew their subjects primarily from classes in a single department at a single institution (DeSantis et al., 2008, 2009; Low & Gendaszek, 2002). With few exceptions, the samples were all drawn from restricted geographical areas. Some had relatively high rates of response (e.g., 93.8%; Low & Gendaszek 2002) and some had low rates (e.g., 10%; Judson & Langdon, 2009), the latter raising questions about sample representativeness for even the specific population of students from a given region or institution.
It was a productive hour, sure. But it also bore a remarkable resemblance to the normal editing process. I had imagined that the magical elixir coursing through my bloodstream would create towering storm clouds in my brain which, upon bursting, would rain cinematic adjectives onto the page as fast my fingers could type them. Unfortunately, the only thing that rained down were Google searches that began with the words "synonym for"—my usual creative process.
One symptom of Alzheimer's disease is a reduced brain level of the neurotransmitter called acetylcholine. It is thought that an effective treatment for Alzheimer's disease might be to increase brain levels of acetylcholine. Another possible treatment would be to slow the death of neurons that contain acetylcholine. Two drugs, Tacrine and Donepezil, are both inhibitors of the enzyme (acetylcholinesterase) that breaks down acetylcholine. These drugs are approved in the US for treatment of Alzheimer's disease.
That said, there are plenty of studies out there that point to its benefits. One study, published in the British Journal of Pharmacology, suggests brain function in elderly patients can be greatly improved after regular dosing with Piracetam. Another study, published in the journal Psychopharmacology, found that Piracetam improved memory in most adult volunteers. And another, published in the Journal of Clinical Psychopharmacology, suggests it can help students, especially dyslexic students, improve their nonverbal learning skills, like reading ability and reading comprehension. Basically, researchers know it has an effect, but they don't know what or how, and pinning it down requires additional research.
The demands of university studies, career, and family responsibilities leaves people feeling stretched to the limit. Extreme stress actually interferes with optimal memory, focus, and performance. The discovery of nootropics and vitamins that make you smarter has provided a solution to help college students perform better in their classes and professionals become more productive and efficient at work.
First half at 6 AM; second half at noon. Wrote a short essay I'd been putting off and napped for 1:40 from 9 AM to 10:40. This approach seems to work a little better as far as the aboulia goes. (I also bother to smell my urine this time around - there's a definite off smell to it.) Nights: 10:02; 8:50; 10:40; 7:38 (2 bad nights of nasal infections); 8:28; 8:20; 8:43 (▆▃█▁▂▂▃).
Caffeine keeps you awake, which keeps you coding. It may also be a nootropic, increasing brain-power. Both desirable results. However, it also inhibits vitamin D receptors, and as such decreases the body's uptake of this-much-needed-vitamin. OK, that's not so bad, you're not getting the maximum dose of vitamin D. So what? Well, by itself caffeine may not cause you any problems, but combined with cutting off a major source of the vitamin - the production via sunlight - you're leaving yourself open to deficiency in double-quick time.
Evidence in support of the neuroprotective effects of flavonoids has increased significantly in recent years, although to date much of this evidence has emerged from animal rather than human studies. Nonetheless, with a view to making recommendations for future good practice, we review 15 existing human dietary intervention studies that have examined the effects of particular types of flavonoid on cognitive performance. The studies employed a total of 55 different cognitive tests covering a broad range of cognitive domains. Most studies incorporated at least one measure of executive function/working memory, with nine reporting significant improvements in performance as a function of flavonoid supplementation compared to a control group. However, some domains were overlooked completely (e.g. implicit memory, prospective memory), and for the most part there was little consistency in terms of the particular cognitive tests used making across study comparisons difficult. Furthermore, there was some confusion concerning what aspects of cognitive function particular tests were actually measuring. Overall, while initial results are encouraging, future studies need to pay careful attention when selecting cognitive measures, especially in terms of ensuring that tasks are actually sensitive enough to detect treatment effects.
The beneficial effects as well as the potentially serious side effects of these drugs can be understood in terms of their effects on the catecholamine neurotransmitters dopamine and norepinephrine (Wilens, 2006). These neurotransmitters play an important role in cognition, affecting the cortical and subcortical systems that enable people to focus and flexibly deploy attention (Robbins & Arnsten, 2009). In addition, the brain's reward centers are innervated by dopamine neurons, accounting for the pleasurable feelings engendered by these stimulants (Robbins & Everett, 1996).
Starting from the studies in my meta-analysis, we can try to estimate an upper bound on how big any effect would be, if it actually existed. One of the most promising null results, Southon et al 1994, turns out to be not very informative: if we punch in the number of kids, we find that they needed a large effect size (d=0.81) before they could see anything:
Many of the positive effects of cognitive enhancers have been seen in experiments using rats. For example, scientists can train rats on a specific test, such as maze running, and then see if the "smart drug" can improve the rats' performance. It is difficult to see how many of these data can be applied to human learning and memory. For example, what if the "smart drug" made the rat hungry? Wouldn't a hungry rat run faster in the maze to receive a food reward than a non-hungry rat? Maybe the rat did not get any "smarter" and did not have any improved memory. Perhaps the rat ran faster simply because it was hungrier. Therefore, it was the rat's motivation to run the maze, not its increased cognitive ability that affected the performance. Thus, it is important to be very careful when interpreting changes observed in these types of animal learning and memory experiments.
We reached out to several raw material manufacturers and learned that Phosphatidylserine and Huperzine A are in short supply. We also learned that these ingredients can be pricey, incentivizing many companies to cut corners. A company has to have the correct ingredients in the correct proportions in order for a brain health formula to be effective. We learned that not just having the two critical ingredients was important – but, also that having the correct supporting ingredients was essential in order to be effective.
None of that has kept entrepreneurs and their customers from experimenting and buying into the business of magic pills, however. In 2015 alone, the nootropics business raked in over $1 billion dollars, and web sites like the nootropics subreddit, the Bluelight forums, and Bulletproof Exec are popular and packed with people looking for easy ways to boost their mental performance. Still, this bizarre, Philip K. Dick-esque world of smart drugs is a tough pill to swallow. To dive into the topic and explain, I spoke to Kamal Patel, Director of evidence-based medical database Examine.com, and even tried a few commercially-available nootropics myself.
When comparing supplements, consider products with a score above 90% to get the greatest benefit from smart pills to improve memory. Additionally, we consider the reviews that users send to us when scoring supplements, so you can determine how well products work for others and use this information to make an informed decision. Every month, our editor puts her name on that month's best smart bill, in terms of results and value offered to users.
After 7 days, I ordered a kg of choline bitartrate from Bulk Powders. Choline is standard among piracetam-users because it is pretty universally supported by anecdotes about piracetam headaches, has support in rat/mice experiments27, and also some human-related research. So I figured I couldn't fairly test piracetam without some regular choline - the eggs might not be enough, might be the wrong kind, etc. It has a quite distinctly fishy smell, but the actual taste is more citrus-y, and it seems to neutralize the piracetam taste in tea (which makes things much easier for me).
Stimulants are the smart drugs most familiar to people, starting with widely-used psychostimulants caffeine and nicotine, and the more ill-reputed subclass of amphetamines. Stimulant drugs generally function as smart drugs in the sense that they promote general wakefulness and put the brain and body "on alert" in a ready-to-go state. Basically, any drug whose effects reduce drowsiness will increase the functional IQ, so long as the user isn't so over-stimulated they're shaking or driven to distraction.
Many of the most popular "smart drugs" (Piracetam, Sulbutiamine, Ginkgo Biloba, etc.) have been around for decades or even millenia but are still known only in medical circles or among esoteric practicioners of herbal medicine. Why is this? If these compounds have proven cognitive benefits, why are they not ubiquitous? How come every grade-school child gets fluoride for the development of their teeth (despite fluoride's being a known neurotoxin) but not, say, Piracetam for the development of their brains? Why does the nightly news slant stories to appeal more to a fear-of-change than the promise of a richer cognitive future?
Recent developments include biosensor-equipped smart pills that sense the appropriate environment and location to release pharmacological agents. Medimetrics (Eindhoven, Netherlands) has developed a pill called IntelliCap with drug reservoir, pH and temperature sensors that release drugs to a defined region of the gastrointestinal tract. This device is CE marked and is in early stages of clinical trials for FDA approval. Recently, Google announced its intent to invest and innovate in this space.
That is, perhaps light of the right wavelength can indeed save the brain some energy by making it easier to generate ATP. Would 15 minutes of LLLT create enough ATP to make any meaningful difference, which could possibly cause the claimed benefits? The problem here is like that of the famous blood-glucose theory of willpower - while the brain does indeed use up more glucose while active, high activity uses up very small quantities of glucose/energy which doesn't seem like enough to justify a mental mechanism like weak willpower.↩
As I am not any of the latter, I didn't really expect a mental benefit. As it happens, I observed nothing. What surprised me was something I had forgotten about: its physical benefits. My performance in Taekwondo classes suddenly improved - specifically, my endurance increased substantially. Before, classes had left me nearly prostrate at the end, but after, I was weary yet fairly alert and happy. (I have done Taekwondo since I was 7, and I have a pretty good sense of what is and is not normal performance for my body. This was not anything as simple as failing to notice increasing fitness or something.) This was driven home to me one day when in a flurry before class, I prepared my customary tea with piracetam, choline & creatine; by the middle of the class, I was feeling faint & tired, had to take a break, and suddenly, thunderstruck, realized that I had absentmindedly forgot to actually drink it! This made me a believer.
Adaptogens are plant-derived chemicals whose activity helps the body maintain or regain homeostasis (equilibrium between the body's metabolic processes). Almost without exception, adaptogens are available over-the-counter as dietary supplements, not controlled drugs. Well-known adaptogens include Ginseng, Kava Kava, Passion Flower, St. Johns Wort, and Gotu Kola. Many of these traditional remedies border on being "folk wisdom," and have been in use for hundreds or thousands of years, and are used to treat everything from anxiety and mild depression to low libido. While these smart drugs work in a many different ways (their commonality is their resultant function within the body, not their chemical makeup), it can generally be said that the cognitive boost users receive is mostly a result of fixing an imbalance in people with poor diets, body toxicity, or other metabolic problems, rather than directly promoting the growth of new brain cells or neural connections.
Taken together, these considerations suggest that the cognitive effects of stimulants for any individual in any task will vary based on dosage and will not easily be predicted on the basis of data from other individuals or other tasks. Optimizing the cognitive effects of a stimulant would therefore require, in effect, a search through a high-dimensional space whose dimensions are dose; individual characteristics such as genetic, personality, and ability levels; and task characteristics. The mixed results in the current literature may be due to the lack of systematic optimization.
The soft gels are very small; one needs to be a bit careful - Vitamin D is fat-soluble and overdose starts in the range of 70,000 IU35, so it would take at least 14 pills, and it's unclear where problems start with chronic use. Vitamin D, like many supplements, follows a U-shaped response curve (see also Melamed et al 2008 and Durup et al 2012) - too much can be quite as bad as too little. Too little, though, is likely very bad. The previously cited studies with high acute doses worked out to <1,000 IU a day, so they may reassure us about the risks of a large acute dose but not tell us much about smaller chronic doses; the mortality increases due to too-high blood levels begin at ~140nmol/l and reading anecdotes online suggest that 5k IU daily doses tend to put people well below that (around 70-100nmol/l). I probably should get a blood test to be sure, but I have something of a needle phobia.
If this is the case, this suggests some thoughtfulness about my use of nicotine: there are times when use of nicotine will not be helpful, but times where it will be helpful. I don't know what makes the difference, but I can guess it relates to over-stimulation: on some nights during the experiment, I had difficult concentrating on n-backing because it was boring and I was thinking about the other things I was interested in or working on - in retrospect, I wonder if those instances were nicotine nights.
So what's the catch? Well, it's potentially addictive for one. Anything that messes with your dopamine levels can be. And Patel says there are few long-term studies on it yet, so we don't know how it will affect your brain chemistry down the road, or after prolonged, regular use. Also, you can't get it very easily, or legally for that matter, if you live in the U.S. It's classified as a schedule IV controlled substance. That's where Adrafinil comes in.
Some data suggest that cognitive enhancers do improve some types of learning and memory, but many other data say these substances have no effect. The strongest evidence for these substances is for the improvement of cognitive function in people with brain injury or disease (for example, Alzheimer's disease and traumatic brain injury). Although "popular" books and companies that sell smart drugs will try to convince you that these drugs work, the evidence for any significant effects of these substances in normal people is weak. There are also important side-effects that must be considered. Many of these substances affect neurotransmitter systems in the central nervous system. The effects of these chemicals on neurological function and behavior is unknown. Moreover, the long-term safety of these substances has not been adequately tested. Also, some substances will interact with other substances. A substance such as the herb ma-huang may be dangerous if a person stops taking it suddenly; it can also cause heart attacks, stroke, and sudden death. Finally, it is important to remember that products labeled as "natural" do not make them "safe."
The smart pill industry has popularized many herbal nootropics. Most of them first appeared in Ayurveda and traditional Chinese medicine. Ayurveda is a branch of natural medicine originating from India. It focuses on using herbs as remedies for improving quality of life and healing ailments. Evidence suggests our ancestors were on to something with this natural approach.
Theanine can also be combined with caffeine as both of them work in synergy to increase memory, reaction time, mental endurance, and memory. The best part about Theanine is that it is one of the safest nootropics and is readily available in the form of capsules. A natural option would be to use an excellent green tea brand which constitutes of tea grown in the shade because then Theanine would be abundantly present in it.
The use of cognitive enhancers by healthy individuals sparked debate about ethics and safety. Cognitive enhancement by pharmaceutical means was considered a form of illicit drug use in some places, even while other cognitive enhancers, such as caffeine and nicotine, were freely available. The conflict therein raised the possibility for further acceptance of smart drugs in the future. However, the long-term effects of smart drugs on otherwise healthy brains were unknown, delaying safety assessments.
Please browse our website to learn more about how to enhance your memory. Our blog contains informative articles about the science behind nootropic supplements, specific ingredients, and effective methods for improving memory. Browse through our blog articles and read and compare reviews of the top rated natural supplements and smart pills to find everything you need to make an informed decision.
The chemical Huperzine-A (Examine.com) is extracted from a moss. It is an acetylcholinesterase inhibitor (instead of forcing out more acetylcholine like the -racetams, it prevents acetylcholine from breaking down). My experience report: One for the null hypothesis files - Huperzine-A did nothing for me. Unlike piracetam or fish oil, after a full bottle (Source Naturals, 120 pills at 200μg each), I noticed no side-effects, no mental improvements of any kind, and no changes in DNB scores from straight Huperzine-A.
The first night I was eating some coconut oil, I did my n-backing past 11 PM; normally that damages my scores, but instead I got 66/66/75/88/77% (▁▁▂▇▃) on D4B and did not feel mentally exhausted by the end. The next day, I performed well on the Cambridge mental rotations test. An anecdote, of course, and it may be due to the vitamin D I simultaneously started. Or another day, I was slumped under apathy after a promising start to the day; a dose of fish & coconut oil, and 1 last vitamin D, and I was back to feeling chipper and optimist. Unfortunately I haven't been testing out coconut oil & vitamin D separately, so who knows which is to thank. But still interesting.
The amphetamine mix branded Adderall is terribly expensive to obtain even compared to modafinil, due to its tight regulation (a lower schedule than modafinil), popularity in college as a study drug, and reportedly moves by its manufacture to exploit its privileged position as a licensed amphetamine maker to extract more consumer surplus. I paid roughly $4 a pill but could have paid up to $10. Good stimulant hygiene involves recovery periods to avoid one's body adapting to eliminate the stimulating effects, so even if Adderall was the answer to all my woes, I would not be using it more than 2 or 3 times a week. Assuming 50 uses a year (for specific projects, let's say, and not ordinary aimless usage), that's a cool $200 a year. My general belief was that Adderall would be too much of a stimulant for me, as I am amphetamine-naive and Adderall has a bad reputation for letting one waste time on unimportant things. We could say my prediction was 50% that Adderall would be useful and worth investigating further. The experiment was pretty simple: blind randomized pills, 10 placebo & 10 active. I took notes on how productive I was and the next day guessed whether it was placebo or Adderall before breaking the seal and finding out. I didn't do any formal statistics for it, much less a power calculation, so let's try to be conservative by penalizing the information quality heavily and assume it had 25%. So \frac{200 - 0}{\ln 1.05} \times 0.50 \times 0.25 = 512! The experiment probably used up no more than an hour or two total.
Instead of buying expensive supplements, Lebowitz recommends eating heart-healthy foods, like those found in the MIND diet. Created by researchers at Rush University, MIND combines the Mediterranean and DASH eating plans, which have been shown to reduce the risk of heart problems. Fish, nuts, berries, green leafy vegetables and whole grains are MIND diet staples. Lebowitz says these foods likely improve your cognitive health by keeping your heart healthy.
Gibson and Green (2002), talking about a possible link between glucose and cognition, wrote that research in the area …is based on the assumption that, since glucose is the major source of fuel for the brain, alterations in plasma levels of glucose will result in alterations in brain levels of glucose, and thus neuronal function. However, the strength of this notion lies in its common-sense plausibility, not in scientific evidence… (p. 185).
These are the most popular nootropics available at the moment. Most of them are the tried-and-tested and the benefits you derive from them are notable (e.g. Guarana). Others are still being researched and there haven't been many human studies on these components (e.g. Piracetam). As always, it's about what works for you and everyone has a unique way of responding to different nootropics.
The Nature commentary is ivory tower intellectualism at its best. The authors state that society must prepare for the growing demand of such drugs; that healthy adults should be allowed drugs to enhance cognitive ability; that this is "morally equivalent" and no more unnatural than diet, sleep, or the use of computers; that we need an evidence-based approach to evaluate the risks; and that we need legal and ethical policies to ensure fair and equitable use.
On the other metric, suppose we removed the creatine? Dropping 4 grams of material means we only need to consume 5.75 grams a day, covered by 8 pills (compared to 13 pills). We save 5,000 pills, which would have cost $45 and also don't spend the $68 for the creatine; assuming a modafinil formulation, that drops our $1761 down to $1648 or $1.65 a day. Or we could remove both the creatine and modafinil, for a grand total of $848 or $0.85 a day, which is pretty reasonable.
Caffeine (Examine.com; FDA adverse events) is of course the most famous stimulant around. But consuming 200mg or more a day, I have discovered the downside: it is addictive and has a nasty withdrawal - headaches, decreased motivation, apathy, and general unhappiness. (It's a little amusing to read academic descriptions of caffeine addiction9; if caffeine were a new drug, I wonder what Schedule it would be in and if people might be even more leery of it than modafinil.) Further, in some ways, aside from the ubiquitous placebo effect, caffeine combines a mix of weak performance benefits (Lorist & Snel 2008, Nehlig 2010) with some possible decrements, anecdotally and scientifically:
That study is also interesting for finding benefits to chronic piracetam+choline supplementation in the mice, which seems connected to a Russian study which reportedly found that piracetam (among other more obscure nootropics) increased secretion of BDNF in mice. See also Drug heuristics on a study involving choline supplementation in pregnant rats.↩
Noopept is a nootropic that belongs to the ampakine family. It is known for promoting learning, boosting mood, and improving logical thinking. It has been popular as a study drug for a long time but has recently become a popular supplement for improving vision. Users report seeing colors more brightly and feeling as if their vision is more vivid after taking noopept.
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"I am nearly four years out from my traumatic brain injury and I have been through 100's of hours of rehabilitation therapy. I have been surprised by how little attention is given to adequate nutrition for recovering from TBI. I'm always looking for further opportunities to recover and so this book fell into the right hands. Cavin outlines the science and reasoning behind the diet he suggests, but the real power in this book comes when he writes, "WE." WE can give our brains proper nutrition. Now I'm excited to drink smoothies and eat breakfasts that look like dinners! I will recommend this book to my friends.
Took pill #6 at 12:35 PM. Hard to be sure. I ultimately decided that it was Adderall because I didn't have as much trouble as I normally would in focusing on reading and then finishing my novel (Surface Detail) despite my family watching a movie, though I didn't notice any lack of appetite. Call this one 60-70% Adderall. I check the next evening and it was Adderall.
(I was more than a little nonplussed when the mushroom seller included a little pamphlet educating one about how papaya leaves can cure cancer, and how I'm shortening my life by decades by not eating many raw fruits & vegetables. There were some studies cited, but usually for points disconnected from any actual curing or longevity-inducing results.)
NGF may sound intriguing, but the price is a dealbreaker: at suggested doses of 1-100μg (NGF dosing in humans for benefits is, shall we say, not an exact science), and a cost from sketchy suppliers of $1210/100μg/$470/500μg/$750/1000μg/$1000/1000μg/$1030/1000μg/$235/20μg. (Levi-Montalcini was presumably able to divert some of her lab's production.) A year's supply then would be comically expensive: at the lowest doses of 1-10μg using the cheapest sellers (for something one is dumping into one's eyes?), it could cost anywhere up to $10,000.
Sleep itself is an underrated cognition enhancer. It is involved in enhancing long-term memories as well as creativity. For instance, it is well established that during sleep memories are consolidated-a process that "fixes" newly formed memories and determines how they are shaped. Indeed, not only does lack of sleep make most of us moody and low on energy, cutting back on those precious hours also greatly impairs cognitive performance. Exercise and eating well also enhance aspects of cognition. It turns out that both drugs and "natural" enhancers produce similar physiological changes in the brain, including increased blood flow and neuronal growth in structures such as the hippocampus. Thus, cognition enhancers should be welcomed but not at the expense of our health and well being.
My first dose on 1 March 2017, at the recommended 0.5ml/1.5mg was miserable, as I felt like I had the flu and had to nap for several hours before I felt well again, requiring 6h to return to normal; after waiting a month, I tried again, but after a week of daily dosing in May, I noticed no benefits; I tried increasing to 3x1.5mg but this immediately caused another afternoon crash/nap on 18 May. So I scrapped my cytisine. Oh well.
Oxiracetam is one of the 3 most popular -racetams; less popular than piracetam but seems to be more popular than aniracetam. Prices have come down substantially since the early 2000s, and stand at around 1.2g/$ or roughly 50 cents a dose, which was low enough to experiment with; key question, does it stack with piracetam or is it redundant for me? (Oxiracetam can't compete on price with my piracetam pile stockpile: the latter is now a sunk cost and hence free.)
This tendency is exacerbated by general inefficiencies in the nootropics market - they are manufactured for vastly less than they sell for, although the margins aren't as high as they are in other supplement markets, and not nearly as comical as illegal recreational drugs. (Global Price Fixing: Our Customers are the Enemy (Connor 2001) briefly covers the vitamin cartel that operated for most of the 20th century, forcing food-grade vitamins prices up to well over 100x the manufacturing cost.) For example, the notorious Timothy Ferriss (of The Four-hour Work Week) advises imitators to find a niche market with very high margins which they can insert themselves into as middlemen and reap the profits; one of his first businesses specialized in… nootropics & bodybuilding. Or, when Smart Powders - usually one of the cheapest suppliers - was dumping its piracetam in a fire sale of half-off after the FDA warning, its owner mentioned on forums that the piracetam was still profitable (and that he didn't really care because selling to bodybuilders was so lucrative); this was because while SP was selling 2kg of piracetam for ~$90, Chinese suppliers were offering piracetam on AliBaba for $30 a kilogram or a third of that in bulk. (Of course, you need to order in quantities like 30kg - this is more or less the only problem the middlemen retailers solve.) It goes without saying that premixed pills or products are even more expensive than the powders.
Modafinil is a eugeroic, or 'wakefulness promoting agent', intended to help people with narcolepsy. It was invented in the 1970s, but was first approved by the American FDA in 1998 for medical use. Recent years have seen its off-label use as a 'smart drug' grow. It's not known exactly how Modafinil works, but scientists believe it may increase levels of histamines in the brain, which can keep you awake. It might also inhibit the dissipation of dopamine, again helping wakefulness, and it may help alertness by boosting norepinephrine levels, contributing to its reputation as a drug to help focus and concentration.
Nootropics – sometimes called smart drugs – are compounds that enhance brain function. They're becoming a popular way to give your mind an extra boost. According to one Telegraph report, up to 25% of students at leading UK universities have taken the prescription smart drug modafinil [1], and California tech startup employees are trying everything from Adderall to LSD to push their brains into a higher gear [2].
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\begin{document}
\title{Varieties of elements of given order in simple algebraic groups}
\author{Claude Marion\\ \\ \small{\begin{tabular}{c}Dipartimento di Matematica, Universit\`{a} degli Studi di Padova, Padova, Italy\\ [email protected]\end{tabular}}} \date{}
\maketitle
\noindent\textbf{Abstract.} Given a positive integer $u$ and a simple algebraic group $G$ defined over an algebraically closed field $K$ of characteristic $p$, we derive properties about the subvariety $G_{[u]}$ of $G$ consisting of elements of $G$ of order dividing $u$. In particular, we determine the dimension of $G_{[u]}$, completing results of Lawther \cite{Lawther} in the special case where $G$ is of adjoint type. We also apply our results to the study of finite simple quotients of triangle groups, giving further insight on a conjecture we proposed in \cite{Marionconj} as well as proving that some finite quasisimple groups are not quotients of certain triangle groups.
\tableofcontents
\section{Introduction}\label{s:intro} Let $G$ be a reductive algebraic group defined over an algebraically closed field $K$ of characteristic $p$ (possibly equal to 0), $C$ be a conjugacy class of $G$ and $u$ be a positive integer. In 2007 Guralnick \cite{Guralnick} proved the following result:
\begin{theorem}\cite[Theorem 1.1]{Guralnick}. Given a reductive algebraic group $G$, a conjugacy class $C$ of $G$ and a positive integer $u$, the set $\{g \in G: g^u\in C\}$ is a finite union of conjugacy classes of $G$. \end{theorem}
In this paper, we concentrate our attention to the case where $G$ is connected and $C=\{1\}$ is the trivial conjugacy class of $G$. For a positive integer $u$, we let $$G_{[u]}=\{g \in G: g^u=1\}$$ be the subvariety of $G$ consisting of elements of $G$ of order dividing $u$ and set $j_u(G)= \dim G_{[u]}$. We also let $d_u(G)$ be the minimal dimension of a centralizer in $G$ of an element of $G$ of order dividing $u$. We are merely interested in determining $j_u(G)$ for every positive positive integer $u$ (when $G$ is a simple algebraic group). \\
For completeness, we begin by proving Guralnick's result in the case where $G$ is connected and $C=\{1\}$. In the statement below, given $g \in G$, we let $g^G$ denote the conjugacy class of $g$ in $G$.
\begin{propn}\label{p:fcu} Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $K$ of characteristic $p$. Let $u$ be a positive integer. Then the number of conjugacy classes of $G$ of elements of order dividing $u$ is finite. In particular $G_{[u]}$ is a finite union of conjugacy classes of $G$. Moreover $\dim G_{[u]}=\max_{g \in G_{[u]}} \dim g^G$ and ${\rm codim}\ G_{[u]} = d_u(G)$. \end{propn}
Given a simple algebraic group $G$ defined over an algebraically closed field $K$ of characteristic $p$, we denote by $G_{s.c.}$ (respectively, $G_{a.}$) the simple algebraic group over $K$ of simply connected (respectively, adjoint) type having the same Lie type and Lie rank as $G$. We prove the following result partially proved in \cite[Theorem 3.11]{Lawther}:
\begin{propn}\label{p:ingsimple} Let $G$ be simple algebraic group defined over an algebraically closed field $K$ of characteristic $p$. Let $u$ be a positive integer. Then
$$ d_u(G_{a.})\leq d_u(G) \leq d_u(G_{s.c.}).$$
\end{propn}
In \cite{Lawther}, given a positive integer $u$ and a simple algebraic group $G$ defined over an algebraically closed field $K$ of characteristic $p$, Lawther gives a lower bound for $d_u(G)$. Moreover, he proves that this lower bound is equal to $d_u(G)$ in the case where $G$ is of adjoint type. We establish the following result completing Lawther's result. Recall that the Coxeter number of a simple algebraic group is defined to be the quotient of the number of roots in the root system associated to $G$ by the number of such roots which are simple. Concretely, $h=\ell+1$, $2\ell$, $2\ell$, $2\ell-2$, $6$, $12$, $12$, $18$ or $30$ according respectively as $G$ is of type $A_\ell$, $B_\ell$, $C_\ell$, $D_\ell$, $G_2$, $F_4$, $E_6$. $E_7$ or $E_8$.
\begin{theorem}\label{t:blawther} Let $G$ be a simple algebraic group defined over an algebraically closed field $K$ of characteristic $p$. Let $h$ be the Coxeter number of $G$ and $u$ be a positive integer. Write $h=zu+e$ and $u=qv$, where $z$ and $e$ are nonnegative integers with $0 \leq e \leq u-1$ and $q$, $v$ are coprime positive integers such that $q$ is a power of $p$ and $p$ does not divide $v$.
The following assertions hold. \begin{enumerate}[(i)] \item If $G$ is of type $A_\ell$ then $$d_u(G_{s.c.})=d_u(G_{a.})$$ except if $p\neq 2$, $u$ is even, $h=zu$ and $z$ is odd in which case $$d_u(G_{s.c.})-d_u(G_{a.})=2.$$ \item If $G$ is of type $B_\ell$ then $$d_u(G_{s.c.})=d_u(G_{a.})$$ except if \begin{enumerate} \item $p\neq 2$, $u\equiv 2 \mod 4$, $h=zu$, and $z\equiv u/2 \mod 4$ or $z\equiv 2 \mod 4$ in which case $$d_u(G_{s.c.})-d_u(G_{a.})=2.$$ \item $p\neq 2$, $u \equiv 4 \mod 8$, $h=zu$ and $z$ is odd in which case $$d_u(G_{s.c.})-d_u(G_{a.})= 2.$$ \end{enumerate} \item If $G$ is of type $C_\ell$ then $$d_u(G_{s.c.})=d_u(G_{a.})$$ except if $p\neq 2$ and $u$ is even in which case $$d_u(G_{s.c.})-d_u(G_{a.})=2\left\lceil \frac z2\right\rceil.$$ \item If $G$ is of type $D_\ell$ then $$d_u(G_{s.c.})=d_u(G_{a.})$$ except if \begin{enumerate} \item $p\neq 2$, $u=2$, $h=zu$ and $z \equiv u/2 \mod 4$ in which case $$d_u(G_{s.c.})-d_u(G_{a.})=4.$$ \item $p\neq 2$, $u \equiv 2 \mod 4$, $u>2$, $h=zu$ and $z \equiv u/2 \mod 4$ in which case $$d_u(G_{s.c.})-d_u(G_{a.})=2.$$ \item $p\neq 2$, $u\equiv 2 \mod 4$, $e=u-2\neq 0$ and $z \equiv 1 \mod 4$ in which case $$d_u(G_{s.c.})-d_u(G_{a.})= 2.$$ \item $p\neq 2$, $u \equiv 4 \mod 8$, $z$ is odd and $h=zu$
in which case $$d_u(G_{s.c.})-d_u(G_{a.})=2.$$ \end{enumerate} \item If $G$ is of exceptional type then $d_u(G_{s.c.})=d_u(G_{a.})$ except if $G$ is of type $E_7$, $p \neq 2$ and $u\in \{2,6,10,14,18\}$ in which case $$d_2(G_{s.c.})-d_2(G_{a.})=6 \quad \textrm{and} \quad d_u(G_{s.c.})-d_u(G_{a.})=2 \ \textrm{for} \ u \in \{6,10,14,18\}.$$ \end{enumerate} \end{theorem}
We note that for $G$ of exceptional type the only cases where the simply connected and the adjoint groups are not abstractly isomorphic is when $G$ is of type $E_6$ or $E_7$ and $p\neq 3$ or $p\neq 2$ respectively.\\
Recall that if a simple algebraic group $G$ over an algebraically closed field $K$ of characteristic $p$ is not of simply connected type nor of adjoint type then either $G$ is abstractly isomorphic to ${\rm SL}_n(K)/C$ where $C\leqslant Z({\rm SL}_n(K))$, or $G$ is of type $D_{\ell}$, $p \neq 2$ and $G$ is abstractly isomorphic to ${\rm SO}_{2\ell}(K)$ or a half-spin group ${\rm HSpin}_{2\ell}(K)$ where $\ell$ is even in the latter case. For simple algebraic groups that are neither of simply connected nor adjoint type we obtain the following result.
\begin{theorem}\label{t:duotsca}
Let $G$ be a simple algebraic group, neither of simply connected nor adjoint type, defined over an algebraically closed field $K$ of characteristic $p$. Let $u$ be a positive integer. Write $u=qv$ where $q$ and $v$ are coprime positive integers such that $q$ is a power of $p$ and $p$ does not divide $v$. Then the following assertions hold.
\begin{enumerate}[(i)]
\item If $d_u(G_{s.c.})=d_u(G_{a.})$ then $d_u(G)=d_u(G_{a.})$
\item Suppose $d_u(G_{s.c.})\neq d_u(G_{a.})$.
\begin{enumerate}
\item If $G={\rm SL}_{\ell+1}(K)/C$ where $C\leq Z({\rm SL}_{\ell+1}(K))$, then $d_u(G)=d_u(G_{a.})$ if $C$ has an element of order $2$, otherwise $d_u(G)=d_u(G_{{s.c.}})$.
\item If $G={\rm SO}_{2\ell}(K)$ where $p\neq 2$, then $d_u(G)=d_u(G_{a.})$.
\item If $G={\rm HSpin}_{2\ell}(K)$ where $p\neq 2$ and $\ell$ is even, then $d_u(G)=d_u(G_{s.c.})$.
\end{enumerate}
\end{enumerate}
\end{theorem}
In \cite{Lawther} Lawther showed that if $G$ is a simple algebraic group of rank $\ell$ with Coxeter number $h$ over an algebraically closed field and $u$ is a positive integer with $u\geq h$ then $d_u(G)=\ell$. As a corollary of Theorems \ref{t:blawther} and \ref{t:duotsca} we obtain the following result:
\begin{cor} Let $G$ be a simple algebraic group of rank $\ell$ over an algebraically closed field $K$ of characteristic $p$. Let $h$ be the Coxeter number of $G$ and let $u$ be a positive number. Then \begin{enumerate}[(i)] \item If $u>h$ then $d_u(G)=\ell$. \item Suppose $u=h$ then $d_u(G)=\ell$ if $G$ is (abstractly isomorphic to a group) of adjoint type, or $p=2$, or $h$ is odd (and then $G$ is of type $A$), or $G={\rm SL}_{\ell+1}(K)/C$ and $C$ has an element of order $2$ where $C\leq Z({\rm SL}_{\ell+1}(K))$, or $G\cong {\rm SO}_{2\ell}(K)$ and $p\neq 2$, or $G$ is of type $B$ or $D$ and $h\equiv 0\mod 8$, or $G=E_6(K)$, otherwise $d_u(G)=\ell+2$. \end{enumerate} \end{cor}
Given a simple algebraic group $G$ over an algebraically closed field, we also derive a further property of $$d_u(G): \begin{array}{l} \mathbb{N}\rightarrow \mathbb{N}\\ u \mapsto d_u(G). \end{array} $$
\begin{propn}\label{p:dudecreasing} Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p$. Then $$d_u(G): \begin{array}{l} \mathbb{N}\rightarrow \mathbb{N}\\ u \mapsto d_u(G). \end{array} $$
is a decreasing function of $u$. \end{propn}
We now apply our results to the study of finite quasisimple images of triangle groups. Recall that a finite group is quasisimple if it is perfect and the quotient by its centre is simple. \\ Every finite quasisimple group being 2-generated, given a finite quasisimple group $G_0$ and a triple $(a,b,c)$ of positive integers, a natural question to consider is whether $G_0$ can be generated by two elements of orders dividing respectively $a$ and $b$ and having product of order dividing $c$. A finite group generated by two such elements is called an $(a,b,c)$-group. Equivalently, an $(a,b,c)$-group is a finite quotient of the triangle group $T=T_{a,b,c}$ with presentation $$ T=T_{a,b,c}=\langle x,y,z:x^a=y^b=z^c=xyz=1\rangle. $$ When investigating the finite (nonabelian) quasisimple quotients of $T$, we can assume that $1/a+1/b+1/c<1$ as otherwise $T$ is either soluble or $T\cong T_{2,3,5}\cong{\rm Alt}_5$ (see \cite{Conder}). The group $T$ is then a hyperbolic triangle group. Without loss of generality, we will further assume that $a \leq b \leq c$ and call $(a,b,c)$ a hyperbolic triple of integers. (Indeed, $T_{a,b,c}\cong T_{a',b',c'}$ for any permutation $(a',b',c')$ of $(a,b,c)$.)\\
Recall that a finite quasisimple group $G_0$ of Lie type occurs as the derived subgroup of the fixed point group of a simple algebraic group $G$, when ${\rm char}(K)=p$ is prime, under a Steinberg endomorphism $F$, i.e. $G_0=(G^F)'$. We use the standard notation $G_0=(G^F)'= G(q)$ where $q=p^r$ for some positive integer $r$. (We include the possibility that $G(q)$ is of twisted type.)\\
Given a simple algebraic group $G$ and a triple $(a,b,c)$ of positive integers, we say that $(a,b,c)$ is rigid for $G$ if the sum $j_a(G)+j_b(G)+j_c(G)$ is equal to $2\dim G$. When the latter sum is less (respectively, greater) than $2\dim G$, we say that $(a,b,c)$ is reducible (respectively, nonrigid)
for $G$. \\ In \cite{Marionconj} we conjectured the following finiteness result in the rigid case (first formulated for hyperbolic triples $(a,b,c)$ of primes): \begin{con}\label{c:marionconj} Let $G$ be a simple algebraic group defined over an algebraically closed field $K$ of prime characteristic $p$ and let $(a,b,c)$ be a rigid triple of integers for $G$. Then there are only finitely many quasisimple groups $G(p^r)$ (of the form $(G^F)'$ where $F$ is a Steinberg endomorphism) that are $(a,b,c)$-generated. \end{con}
In \cite[Theorem 1.7]{LLM1}, Larsen, Lubotzky and the author proved that Conjecture \ref{c:marionconj} holds except possibly if $p$ divides $abcd$ where $d$ is the determinant of the Cartan matrix of $G$. In a future paper \cite{JLM} we will make further progress on Conjecture \ref{c:marionconj} by, in particular, completely settling it in the case where the finite group $(G^F)'$ is simple. A crucial ingredient therein is the classification of the hyperbolic triples $(a,b,c)$ of integers as reducible, rigid or nonrigid for a given simple algebraic group $G$. We determine this classification below.
\begin{theorem}\label{t:classification}
The following assertions hold.
\begin{enumerate}[(i)]
\item The reducible hyperbolic triples $(a,b,c)$ of integers for simple algebraic groups $G$ of simply connected or adjoint type are exactly those given in Table \ref{ta:red}. In particular there are no reducible hyperbolic triples of integers for simple algebraic groups of adjoint type.
\item The rigid hyperbolic triples of integers $(a,b,c)$ for simple algebraic groups $G$ of simply connected type are exactly those given in Table \ref{ta:rigidsc}.
\item The rigid hyperbolic triples of integers $(a,b,c)$ for simple algebraic groups $G$ of adjoint type are exactly those given in Table \ref{ta:rigida}.
\item The classification of the reducible and the rigid hyperbolic triples of integers for ${\rm SO}_n(K)$ is the same as for ${\rm PSO}_n(K)$.
\item The classification of the reducible and the rigid hyperbolic triples of integers for ${\rm HSpin}_{2\ell}(K)$ is the same as for ${\rm Spin}_{2\ell}(K)$.
\item If $C\leqslant Z({\rm SL}_n(K))$ contains an involution then the classification of the reducible and the rigid hyperbolic triples of integers for ${\rm SL}_n(K)/C$ is the same as for ${\rm PSL}_n(K)$. Otherwise, the classification of the reducible and the rigid hyperbolic triples of integers for ${\rm SL}_n(K)/C$ is the same as for ${\rm SL}_n(K)$.
\end{enumerate}
\end{theorem}
\begin{table}[h!] \center{
\begin{tabular}{|l|l|l|} \hline $G$ & $p$ & $(a,b,c)$\\ \hline ${\rm SL}_2(K)$ & $p \neq 2$ & $(2,b,c)$\\ \hline ${\rm Sp}_4(K)$ & $p \neq 2$ & $(2,3,c)$, $(2,4,c)$, $(3,3,4)$, $(3,4,4)$, $(4,4,4)$\\ ${\rm Sp}_6(K)$ & $p \neq 2$ & $(2,3,c)$, $(2,4,c)$, $(2,5,5)$, $(2,5,6)$, $(2,6,6)$\\ ${\rm Sp}_8(K)$ & $p \neq 2$ & $(2,3,7)$, $(2,3,8)$, $(2,4,5)$, $(2,4,6)$\\ ${\rm Sp}_{10}(K)$ & $p \neq 2$ & $(2,3,7)$, $(2,3,8)$, $(2,3,9)$, $(2,3,10)$, $(2,4,5)$, $(2,4,6)$\\ ${\rm Sp}_{12}(K)$ & $p \neq 2$ & $(2,3,7)$, $(2,3,8)$, $(2,4,5)$\\ ${\rm Sp}_{14}(K)$ & $p \neq 2$ & $(2,3,7)$, $(2,3,8)$, $(2,4,5)$\\ ${\rm Sp}_{16}(K)$ & $p \neq 2$ & $(2,3,7)$\\ ${\rm Sp}_{18}(K)$ & $p \neq 2$ & $(2,3,7)$\\ ${\rm Sp}_{22}(K)$ & $p \neq 2$ & $(2,3,7)$\\ \hline \end{tabular} } \caption{Reducible triples for simple algebraic groups of simply connected or adjoint type}\label{ta:red} \end{table}
\begin{table}[h!]
\center{
\begin{tabular}{|l|l|l|} \hline $G$ & $p$ & $(a,b,c)$\\ \hline ${\rm SL}_2(K)$ & $p =2$ & $(a,b,c)$\\ & $p \neq 2$ & $(a,b,c)$ $a\geq 3$\\ ${\rm SL}_3(K)$ & any & $(2,b,c)$\\ ${\rm SL}_4(K)$ & $p =2$ & $(2,3,c)$\\ & $p\neq 2$ & $(2,3,c)$, $(2,4,c)$, $(3,3,4)$, $(3,4,4)$,$(4,4,4)$\\ ${\rm SL}_5(K)$ & any & $(2,3,c)$\\ ${\rm SL}_6(K)$ & $p\neq 2$ & $(2,3,c)$, $(2,4,5)$, $(2,4,6)$\\ ${\rm SL}_{10}(K)$ & $p\neq 2$ & $(2,3,7)$\\ \hline ${\rm Sp}_4(K)$ & $p = 2$ & $(2,3,c)$, $(3,3,c)$\\ ${\rm Sp}_4(K)$ & $p \neq 2$ & $(2,b,c)$ $b \geq 5$, $(3,3,c)$ $c \geq 5$, $(3,4,c)$ $c\geq 5$, $(4,4,c)$ $c \geq 5$\\ ${\rm Sp}_6(K)$ & $p \neq 2$ & $(2,5,c)$ $c \geq 7$, $(2,6,c)$ $c\geq 7$, $(3,3,4)$, $(3,4,4)$, $(4,4,4)$\\ ${\rm Sp}_8(K)$ & $p \neq 2$ & $(2,3,c)$ $c\geq 9$, $(2,4,7)$, $(2,4,8)$, $(2,5,5)$, $(2,5,6)$, $(2,6,6)$\\ ${\rm Sp}_{10}(K)$ & $p \neq 2$ & $(2,3,c)$ $c\geq 11$, $(2,4,7)$, $(2,4,8)$\\ ${\rm Sp}_{12}(K)$ & $p \neq 2$ & $(2,3,9)$, $(2,3,10)$, $(2,4,6)$\\ ${\rm Sp}_{14}(K)$ & $p \neq 2$ & $(2,3,9)$, $(2,3,10)$, $(2,4,6)$\\ ${\rm Sp}_{16}(K)$ & $p \neq 2$ & $(2,3,8)$, $(2,4,5)$\\ ${\rm Sp}_{18}(K)$ & $p \neq 2$ & $(2,3,8)$, $(2,4,5)$\\ ${\rm Sp}_{20}(K)$ & $p \neq 2$ & $(2,3,7)$\\ ${\rm Sp}_{24}(K)$ & $p \neq 2$ & $(2,3,7)$\\ ${\rm Sp}_{26}(K)$ & $p \neq 2$ & $(2,3,7)$\\ \hline ${\rm Spin}_{11}(K)$ & $p \neq 2$ & $(2,3,7)$\\ ${\rm Spin}_{12}(K)$ & $p \neq 2$ & $(2,3,7)$\\ \hline \end{tabular} } \caption{Rigid triples for simple algebraic groups of simply connected type}\label{ta:rigidsc} \end{table}
\begin{table}[h!] \center{
\begin{tabular}{|l|l|l|} \hline $G$ & $p$ & $(a,b,c)$\\ \hline ${\rm PSL}_2(K)$ & any & $(a,b,c)$\\ ${\rm PSL}_3(K)$ & any & $(2,b,c)$\\ ${\rm PSL}_4(K)$ & any & $(2,3,c)$\\ ${\rm PSL}_5(K)$ & any & $(2,3,c)$\\ \hline ${\rm PSp}_4(K)$ & any & $(2,3,c)$, $(3,3,c)$\\ \hline $G_{2}(K)$ & any & $(2,4,5)$, (2,5,5)\\ \hline \end{tabular}} \caption{Rigid triples for simple algebraic groups of adjoint type}\label{ta:rigida} \end{table}
As Theorem \ref{t:classification} gives the list of the rigid triples of integers for a given simple algebraic group, it puts Conjecture \ref{c:marionconj} into a very concrete context. Concerning the reducible case, we prove the following nonexistence result.
\begin{propn}\label{p:marionred}
Let $G$ be a simple algebraic group defined over an algebraically closed field $K$ of prime characteristic $p$ and let $(a,b,c)$ be a reducible triple of integers for $G$. Then a quasisimple group $G(p^r)$ (of the form $(G^F)'$ where $F$ is a Steinberg endomorphism) is never an $(a,b,c)$-group. \end{propn}
Theorem \ref{t:classification} which gives inclusively the list of the reducible triple of integers for a given simple algebraic group is one of the ingredients of the proof of Proposition \ref{p:marionred}. As an immediate corollary of Theorem \ref{t:classification} and Proposition \ref{p:marionred} we obtain the following result.
\begin{cor}\label{c:classification}
If $(G,p,(a,b,c))$ is as in Table \ref{ta:red} then $(G^F)'=G(p^r)$ is never an $(a,b,c)$-group.
\end{cor}
Unless otherwise stated, we let $G$ be a connected reductive algebraic group defined over an algebraically closed field $K$ of characteristic $p$ with maximal torus $T$. If $T'$ is a torus of $G$ of dimension $r$ we sometimes write $T'=T_r$. \\ Given a positive integer $u$, we let $G_{[u]}$ be the subvariety of $G$ consisting of elements of order dividing $u$, we set $$j_u(G)= \dim G_{[u]}$$ and we let $d_u(G)$ be the minimal dimension of a centralizer in $G$ of an element of $G$ of order dividing $u$. \\
For $G$ simple, we fix some more notation as follows. We let $\Phi$ be the root system of $G$ with respect to $T$ and set $\Pi=\{\alpha_1,\dots, \alpha_\ell\} \subset \Phi$ to be a set of simple roots of $G$, where $\ell$ is the rank of $G$.
We let $h=|\Phi|/{\rm rank}(G)=|\Phi|/\ell$ be the Coxeter number of $G$. Recall that $h=\ell+1$, $2\ell$, $2\ell$, $2\ell-2$, $6$, $12$, $12$, $18$ or $30$ according respectively as $G$ is of type $A_\ell$, $B_\ell$, $C_\ell$, $D_\ell$, $G_2$, $F_4$, $E_6$. $E_7$ or $E_8$. Also we denote by $G_{s.c.}$ (respectively, $G_{a.}$) the simple algebraic group over $K$ of simply connected (respectively, adjoint) type having the same Lie type and Lie rank as $G$. \\
Unless otherwise stated given a positive integer $u$, we also write:\\
$u=qv$ where $q$ and $v$ are coprime positive integers such that $q$ is a power of $p$ and $p$ does not divide $v$,\\
$h=zu+e$ where $z,e$ are nonnegative integers such that $0\leq e \leq q-1$,\\
$h=\alpha v+\beta$ where $\alpha,\beta$ are nonnegative integers such that $0\leq \alpha \leq v-1$,\\
$\alpha=\gamma q+\delta$ where $\gamma,\delta$ are nonnegative integers such that $0\leq \delta \leq q-1$.\\
An easy check yields $\gamma=z$. Note that if $e=0$ then $\beta=\delta=0$. Furthermore, if $\beta =0$ then $\delta=0$ if and only if $e=0$.\\ Finally for an nonnegative integer $r$, we let $\epsilon_r \in \{0,1\}$ be $0$ if $r$ is even, otherwise $\epsilon_r=1$, and set $\sigma_r\in\{0,1\}$ to be 1 if $r=0$, otherwise $\sigma_r=0$.\\
The outline of the paper is as follows. In \S\ref{s:precent} we give some preliminary results on centralizers. In \S\ref{s:tfr} we give detailed proofs for Propositions \ref{p:fcu} and \ref{p:ingsimple}. In \S\ref{s:ex}, given a positive integer $u$, we determine $d_u(G)$ for $G$ a simple simply connected algebraic group of exceptional type over an algebraically closed field $K$. This establishes Theorem \ref{t:blawther} for $G$ of exceptional type. In \S\ref{s:spin} we recall some properties of the spin groups and determine when a preimage under the canonical map ${\rm Spin}_n(K) \rightarrow {\rm SO}_n(K)$ of a semisimple element of ${\rm SO}_n(K)$ of a given order is also of that order. In \S\ref{s:upc} given a positive integer $u$, we determine precise upper bounds for $d_u(G)$ where $G$ is a simple algebraic group of classical type over an algebraically closed field. In \S\ref{s:pc} given a positive integer $u$, we determine $d_u(G)$ for $G$ of classical type, completing the proofs of Theorems \ref{t:blawther} and \ref{t:duotsca}. In \S\ref{s:decreasing} we prove that for a given simple algebraic group $G$ over an algebraically closed field, $d_u(G)$, seen as a function of $u$, is decreasing. This establishes Proposition \ref{p:dudecreasing}. In \S\ref{s:classification} we classify the reducible and the rigid hyperbolic triples of integers for simple algebraic groups, establishing Theorem \ref{t:classification}. In \S\ref{s:reducibility} we prove Proposition \ref{p:marionred}. Finally in \S\ref{s:tables} we collect Tables \ref{t:asc}-\ref{ta:casestoconsider} which appear later in the paper.\\%the tables used throughout the paper. \\
\noindent \textbf{Acknowledgements.} The author thanks the MARIE CURIE and PISCOPIA research fellowship scheme and the University of Padova for their support. The research leading to these results has received funding from the European Comission, Seventh Framework Programme (FP7/2007-2013) under Grant Agreement 600376.
\section{Preliminary results on centralizers}\label{s:precent} In this section, unless otherwise stated, $G$ denotes a connected reductive group defined over an algebraically closed field $K$ of characteristic $p$. Given a positive integer $u$, recall that $d_u(G)$ is defined to be the minimal dimension of a centralizer in $G$ of an element of $G$ of order dividing $u$. We first recall some generalities on the centralizer in $G$ of an element of $G$.
\begin{lem}\label{l:prelimdu} Let $G$ be a connected reductive group defined over an algebraically closed field $K$ of characteristic $p$. Let $g$ be an element of $G$ with Jordan decomposition $g=xy$ where $x$ is unipotent and $y$ is semisimple. Let $H=C_G(y)^0$. The following assertions hold.
\begin{enumerate}[(i)]
\item We have $x\in C_G(y)^0$, $C_G(g)=C_{C_{G(y)}}(x)$ and $C_G(g)^0=C_{C_G(y)^0}(x)^0$.
\item The group $H=C_G(y)^0$ is reductive and so $H=H_1\cdots H_r T'$ for some nonnegative integer $r$ and simple groups $H_i$ and central torus $T'$. Morever each $H_i$ is closed and normal in $H$, $[H_i,H_j]=1$ for $i\neq j$, and $H_i \cap H_1 \dots H_{i-1}H_{i+1}\cdots H_r$ is finite for each $i$.
\item Write $x=x_1\cdots x_r t$ where $x_i \in H_i$ and $t\in T'$. Then
$$C_{C_G(y)^0}(x)=C_{H_1}(x_1)\dots C_{H_r}(x_r)T'.$$
\item $\dim C_G(g)=\dim C_{C_G(y)^0}(x)={\rm rank}(T')+ \sum_{i=1}^r \dim C_{H_i}(x_i).$
\end{enumerate} \end{lem}
\begin{proof} We first consider part (i). Since $G$ is connected reductive we have $x\in C_G(y)^0$ - see for example \cite[Proposition 14.7]{MaTe}. It is well-known that $C_G(g)=C_{C_{G(y)}}(x)$. As $C_G(g)\leqslant C_G(y)$, we have $C_G(g)^0\leqslant C_G(y)^0$. It now follows that $C_G(g)^0 \leqslant C_{C_G(y)^0}(x)$ and so $C_G(g)^0=C_{C_G(y)^0}(x)^0$.\\ We consider part (ii). Since $G$ is connected reductive and $y$ is a semisimple element of $G$, $C_G(y)^0$ is connected reductive - see for example \cite[Theorem 14.2]{MaTe}. Since $H=C_G(y)^0$ is connected reductive, we have $H=[H,H]Z(H)^0$ where $Z(H)^0$ is a central torus and $[H,H]$ is semisimple - see for example \cite[Corollary 8.22]{MaTe}. Part (ii) follows - see for example \cite[Theorem 8.21]{MaTe}. Part (iii) is now an easy consequence of part (ii). \\ We now consider part (iv). We have $$\dim C_G(g)=\dim C_G(g)^0\quad \textrm{and}\quad \dim C_{C_G(y)^0}(x)= \dim C_{C_G(y)^0}(x)^0. $$ Part (iv) now follows from parts (i)-(iii).\\ \end{proof}
We now give some properties of $d_u(G)$. In the statement below, we adopt the notation of Lemma \ref{l:prelimdu}(ii) for the connected component of the centralizer of a semisimple element in a connected reductive algebraic group.
\begin{prop}\label{p:prelimdu} Let $G$ be a connected reductive algebraic group over an algebraically closed field $K$ of characteristic $p$. Let $u$ be a positive integer. Write $u=qv$ where $u$ and $v$ are coprime positive integers with $q$ a power of $p$ and $p$ does not divide $v$. Then the following assertions hold. \begin{enumerate}[(i)] \item $$\begin{array}{llll}d_u(G) &= & \min & d_q(C_G(y)^0). \\ & & y \in G_{[v]} & \\ \end{array}$$ \item Let $g=xy$ be the Jordan decomposition of an element $g\in G$ where $x$ and $y$ are respectively the unipotent and semisimple parts of $g$. If $g$ has order dividing $u$ and $d_u(G)=\dim C_G(g)$, then $$d_u(G)=d_q(C_G(y)^0).$$ \item If $y$ is a semisimple element of $G$ with $C_G(y)^0=H_1\dots H_rT'$ (as in Lemma \ref{l:prelimdu}(ii)) then $$d_q(C_G(y)^0)= {\rm rank}(T')+\sum_{i=1}^r d_q(H_i). $$ \item $$\begin{array}{llll}d_u(G) &= & \min & {\rm rank}(T')+d_q(H_1)+\dots +d_q(H_r). \\ && C_G(y)^0=\prod_{i=1}^r H_iT'& \\ & & y \in G_{[v]} &\\ \end{array}$$ \end{enumerate} \end{prop}
\begin{proof} We first consider part (i). By definition $d_u(G) = \min_{g \in G_{[u]}} \ \dim C_G(g)$. Let $g$ be any element of $G_{[u]}$. Write $g=xy$ where $x$ and $y$ are respectively the unipotent and semisimple parts of $g$. Clearly $x\in G_{[q]}$ and $y\in G_{[v]}$. Following Lemma \ref{l:prelimdu}(i), we in fact have $x \in (C_G(y)^0)_{[q]}$ and $C_G(g)^0=C_{C_G(y)^0}(x)^0$. It follows that $$\begin{array}{lllll}d_u(G) &= & \min & \min & \dim C_{C_G(y)^0}(x). \\ && y\in G_{[v]} & x \in (C_G(y)^0)_{[q]} & \end{array}$$ This establishes part (i). \\ We now consider part (ii). Note that $x\in (C_G(y)^0)_{[q]}$ and $y\in G_{[v]}$. Since by assumption $d_u(G)=\dim C_G(g)$ and by Lemma \ref{l:prelimdu}(iv) $\dim C_G(g)= \dim C_{C_G(y)^0}(x)$, we obtain \begin{equation}\label{e:lalabobo} d_u(G)= \dim C_{C_G(y)^0}(x). \end{equation} By part (i) $d_u(G)=\min_{l \in G_{[v]}} d_q(C_{G}(l)^0)$ and so $d_q(C_G(y)^0)\geq d_u(G)$. However if $d_q(C_G(y)^0)> d_u(G)$ then $\dim C_{C_G(y)^0}(x) > d_u(G)$, contradicting (\ref{e:lalabobo}). Hence $d_u(G)=d_q(C_G(y)^0)$ as claimed. \\ We now consider part (iii). Recall that by Lemma \ref{l:prelimdu}(ii), $C_G(y)^0$ is connected reductive. For $x\in (C_G(y)^0)_{[q]}$, write $x=x_1\dots x_r t$ where $x_i\in H_i$ for $1\leq i \leq r$ and $t\in T'$. By Lemma \ref{l:prelimdu}(iv) we have $$\dim C_{C_G(y)^0}(x)={\rm rank}(T')+ \sum_{i=1}^r \dim C_{H_i}(x_i).$$ Hence \begin{eqnarray*} d_q(C_G(y)^0) & = & \min_{x \in (C_G(y)^0)_{[q]}} \dim C_{C_G(y)^0}(x) \\ & = & \min_{x \in (C_G(y)^0)_{[q]}} {\rm rank}(T')+ \sum_{i=1}^r \dim C_{H_i}(x_i)\\ & = &{\rm rank}(T')+ \sum_{i=1}^r \min_{x \in (C_G(y)^0)_{[q]}} \dim C_{H_i}(x_i)\\ & = & {\rm rank}(T')+\sum_{i=1}^r d_q(H_i), \end{eqnarray*} where to obtain the final equality we used Lemma \ref{l:prelimdu}(ii)-(iii). Part (iii) follows. \\ Finally part (iv) follows from parts (i) and (iii). \end{proof}
The result below is the main ingredient in the proof of Proposition \ref{p:ingsimple}.
\begin{lem}\label{l:dufactg} Let $G$ be any group, $C$ a normal subgroup of $G$, and $\phi: G\rightarrow G/C$ be the canonical surjective map. Let $g,k$ be any elements of $G$ and set $$G_{g,k}=\{x \in G:g^{-1}x^{-1}gx=k\}.$$ The following assertions hold. \begin{enumerate}[(i)] \item Either $G_{g,k}=\emptyset$ or $G_{g,k}$ is a coset of $C_G(g)$. \item We have $C_{G/C}(gC)= \bigcup_{k \in C} \phi(G_{g,k})$. \item Suppose $G$ is a simple algebraic group defined over an algebraically closed field $K$ of characteristic $p$ and $\phi$ is an isogeny. Then $$ C_{G/C}(gC)^0= (C_G(g)/C)^0\cong C_G(g)^0/(C\cap C_G(g)^0).$$ In particular,
$\dim C_{G/C}(gC)=\dim C_G(g)$. Moreover if $q$ is a power of $p$ and $g$ is a semisimple element of $G$ then $$d_q(C_{G/C}(gC)^0)=d_q(C_G(g)^0).$$ \end{enumerate} \end{lem}
\begin{proof} We first consider part (i). Note that if $g$ and $gk$ are not conjugate then $G_{g,k}=\emptyset$. Suppose that $g$ and $gk$ are conjugate, say $l^{-1}gl=gk$ for some $l\in G$. \\ Let $x$ be any element of $G_{g,k}$. Then $$x^{-1}gx = gk = l^{-1}gl.$$ Hence $lx^{-1}g(lx^{-1})^{-1}=g$ and $xl^{-1}\in C_G(g)$. It follows that $x\in C_G(g)l$ and so $G_{g,k}\subseteq C_G(g)l$. \\ Suppose now that $x \in C_G(g)l$. Then $x=yl$ for some $y\in C_G(g)$. Hence $$ x^{-1}gx= (yl)^{-1}g(yl)= l^{-1}\cdot y^{-1}gy\cdot l= l^{-1}gl=gk$$ and so $x \in G_{g,k}$. It follows that $C_G(g)l\subseteq G_{g,k}$. We obtain $G_{g,k}= C_G(g)l$, proving part (i). \\ We now consider part (ii). Suppose first that $xC$ is any element of $\phi(G_{g,k})$ where $k \in C$. Without loss of generality, $x\in G_{g,k}$. Hence $x^{-1}gx = gk$. Since $k \in C$, we obtain $(xC)^{-1}\cdot gC\cdot xC = gC$ and so $xC \in C_{G/C}(gC)$. Hence $\bigcup_{k \in C} \phi(G_{g,k})\subseteq C_{G/C}(gC)$.\\ Suppose now that $xC \in C_{G/C}(gC)$. Then $ (xC)^{-1}\cdot gC \cdot xC = gC$ and so $ (gC)^{-1}\cdot(xC)^{-1} \cdot gC \cdot xC = C$. It follows that there exists $k\in C$ such that $g^{-1}x^{-1}gx=k$ and so $x\in G_{g,k}$. Hence $xC\in \phi(G_{g,k})$ with $k\in C$. This shows that $C_{G/C}(gC)\subseteq \bigcup_{k \in C} \phi(G_{g,k})$. We obtain $C_{G/C}(gC)= \bigcup_{k \in C} \phi(G_{g,k})$, proving part (ii).\\ Finally we consider part (iii). As $G$ is simple and $\phi$ is an isogeny, ${\rm ker} \ \phi=C$ is a finite central subgroup of $G$. Also following part (ii), $\phi (C_G(g))=C_G(g)/C$ is a subgroup of $C_{G/C}(gC)$ of finite index and so $ C_{G/C}(gC)^0 \leqslant (C_G(g)/C)^0$. On the other hand, as $ C_G(g)/C\leqslant C_{G/C}(gC)$, we have $ (C_G(g)/C)^0\leqslant C_{G/C}(gC)^0$. Hence \begin{equation*}\label{e:o} C_{G/C}(gC)^0 = (C_G(g)/C)^0.\end{equation*} Note that as $C_G(g)^0$ is connected so is the factor group $C_G(g)^0/(C\cap C_G(g)^0)$ - see for example \cite[Proposition 1.10]{MaTe}. By the second isomorphism theorem, $C_G(g)^0/(C\cap C_G(g)^0)$ can be seen as a connected subgroup of $C_G(g)/C$. As this subgroup is of finite index, we obtain \begin{equation}\label{e:oo} C_G(g)^0/(C\cap C_G(g)^0) \cong (C_G(g)/C)^0=C_{G/C}(gC)^0.\end{equation} As $C$ is a finite group, it now follows from (\ref{e:oo}) that $\dim C_{G/C}(gC)=\dim C_G(g)$. To conclude suppose that $g$ is a semisimple element of $G$. By Lemma \ref{l:prelimdu} $C_G(g)^0$ is a connected reductive group. By \cite{Lawther} given a simple algebraic group $H$ defined over $K$, $d_q(H)$ is independent of the isogeny type of $H$ (see Lemma \ref{l:lawther} below). It follows that $d_q(C_G(g)^0/(C\cap C_G(g)^0))=d_q(C_G(g)^0)$ and so by (\ref{e:oo}) $d_q(C_G(g)^0)=d_q(C_{G/C}(gC)^0),$ as required. \end{proof}
For matter of clarity we record \cite[Lemma 2.5 and Corollary 2.6]{Lawther}:
\begin{lem}\label{l:lawther} Let $G$ be a simple algebraic group defined over an algebraic closed field of characteristic $p$. Let $q$ be a power of $p$. Then $d_q(G)$ does not depend on the isogeny type of $G$. Furthermore, let $\alpha$ be a nonnegative integer and write $\alpha=\gamma q +\delta$ where $0\leq \delta \leq q-1$ and let $\varsigma\in \{0,1\}$. Then the following assertions hold: \begin{enumerate}[(i)] \item $d_q(A_{\alpha-\varsigma})=\gamma^2q+(2\gamma+1)(\alpha-\varsigma-\gamma q+1)-1$. \item $d_q(B_{\lceil \frac{\alpha}2\rceil}-\varsigma\epsilon_{\alpha})=\frac{\gamma^2q}2+(2\gamma+1)(\lceil \frac{\alpha}2\rceil-\varsigma\epsilon_{\alpha}-\frac{\gamma q}2)+\lceil \frac{\gamma}2\rceil\epsilon_q$. \item $d_q(C_{\lceil \frac{\alpha}2\rceil}-\varsigma\epsilon_{\alpha})=\frac{\gamma^2q}2+(2\gamma+1)(\lceil \frac{\alpha}2\rceil-\varsigma\epsilon_{\alpha}-\frac{\gamma q}2)+\lceil \frac{\gamma}2\rceil\epsilon_q$. \item $d_q(D_{\lceil \frac{\alpha+1}2\rceil-\varsigma\epsilon_{\alpha+1})}=\frac{\gamma^2q}2+(2\gamma+1)(\lceil \frac{\alpha+1}2\rceil-\varsigma\epsilon_{\alpha+1}-\frac{\gamma q}2)+\lceil \frac{\gamma}2\rceil\epsilon_q-\gamma - \epsilon_\gamma+2\varsigma\epsilon_{\gamma(\alpha+1)}\sigma_{\alpha-\gamma q}$. \end{enumerate} \end{lem}
\begin{corol}\label{c:lawther} Let $m$ be a natural number. The following assertions hold: \begin{enumerate}[(i)] \item If $m+1=\gamma q+\delta$ with $0 \leq \gamma \leq q-1$, then $d_q(A_{m+1})-d_q(A_m)=2\gamma+1$. \item If $2m+1=\gamma q+\delta$ with $0 \leq \gamma \leq q-1$, then $d_q(B_{m+1})-d_q(B_m)=2\gamma+1$. \item If $2m+1=\gamma q+\delta$ with $0 \leq \gamma \leq q-1$, then $d_q(C_{m+1})-d_q(C_m)=2\gamma+1$. \item If $2m=\gamma q+\delta$ with $0 \leq \gamma \leq q-1$, then $d_q(D_{m+1})-d_q(D_m)=2\gamma+1-2\epsilon_\gamma\sigma_\delta$. \end{enumerate} \end{corol}
Finally we record the following result:
\begin{lem}\label{l:centmod2} \cite[6.17]{Humphreys}. Let $G$ be a simple algebraic group of rank $\ell$ over an algebraically closed field $K$ and let $x\in G$. Then $$ \dim C_G(x) \equiv \ell \mod 2.$$ \end{lem}
\section{Proofs of Propositions \ref{p:fcu} and \ref{p:ingsimple}}\label{s:tfr}
In this section we prove Propositions \ref{p:fcu} and \ref{p:ingsimple}. We first give a detailed proof of Proposition \ref{p:fcu}.
\noindent\textit{Proof of Proposition \ref{p:fcu}.} Write $u=qv$ where $q$ and $v$ are coprime positive integers such that $q$ is a power of $p$ and $p$ does not divide $v$. Let $g$ be an element of $G$ of order dividing $u$. Let $g=xy$ be the Jordan decomposition of $g$ where $x$ and $y$ denote respectively the unipotent and semisimple parts of $g$. In particular the order of $x$ divides $q$ and that of $y$ divides $v$. As $G$ is connected, every semisimple element of $G$ lies in a maximal torus of $G$, see for example \cite[Corollary 6.11]{MaTe}. Since all maximal tori of $G$ are conjugate, see for example \cite[Corollary 6.5]{MaTe}, it follows that there are only finitely many conjugacy classes of $G$ of semisimple elements of order dividing $v$. Let $y_1,\dots, y_r$ be representatives of the conjugacy classes of $G$ of semisimple elements of order dividing $v$. For $1 \leq i \leq r$, $C_G(y_i)^0$ is a connected reductive algebraic group (see Lemma \ref{l:prelimdu}(ii)) and so by \cite{Lusztig} $C_G(y_i)^0$ has only finitely many unipotent classes. Let $x_{i,1}, \dots , x_{i,t_i}$ be representatives of the unipotent classes of $C_G(y_i)^0$. Now there exists $1\leq i\leq r$ such that $y$ is conjugate in $G$ to $y_i$. Let $k$ be an element of $G$ such that $kyk^{-1}=y_i$. Then \begin{eqnarray*} kgk^{-1}& = & kxyk^{-1}\\ & = & kxk^{-1} \cdot k yk^{-1}\\ & = & kxk^{-1} \cdot y_i. \end{eqnarray*} Note that $kxk^{-1}\in C_G(y_i)$. Indeed \begin{eqnarray*} kxk^{-1} \cdot y_i \cdot (kxk^{-1})^{-1} &=& kx \cdot k^{-1}y_ik \cdot x^{-1}k^{-1}\\ & = & kx\cdot y x^{-1}k^{-1}\\ & = & kyxx^{-1}k^{-1}\\ & = & kyk^{-1}\\ & =& y_i. \end{eqnarray*} Since $G$ is a connected reductive group and $kxk^{-1}\cdot y_i$ is the Jordan decomposition of $kxk^{-1}\cdot y_i$, it follows that $kxk^{-1}$ is a unipotent element of $C_G(y_i)^0$ - see Lemma \ref{l:prelimdu}(i). Hence there exist $l\in C_G(y_i)^0$ and $1\leq j \leq t_i$ such that $l\cdot kxk^{-1}\cdot l^{-1} = x_{i,j}.$ Also as $l\in C_G(y_i)^0$, we have $ly_il^{-1}=y_i$. Hence \begin{eqnarray*} lkgk^{-1}l^{-1}& = & l\cdot kgk^{-1} \cdot l^{-1}.\\ & = & l\cdot kxk^{-1}\cdot y_i \cdot l^{-1} \\ & = & l\cdot kxk^{-1} l^{-1}\cdot l \cdot y_il^{-1}\\ & = & x_{i,j}y_i \end{eqnarray*} and so $g$ is conjugate to $x_{i,j}y_i$. It follows that there are at most $\sum_{i=1}^r t_i$ conjugacy classes of elements of $G$ of order dividing $u$. It follows that $G_{[u]}$ consists of the finite union of conjugacy classes of $G$ of elements of order dividing $u$. Hence $$ \dim G_{[u]} = \max_{g \in G_{[u]}} \dim g^G.$$ Since for any element $g \in G$, ${\rm codim}\ g^G=\dim C_G(g)$ - see for example \cite[Proposition 1.5]{Humphreys}, we obtain $$ {\rm codim} \ G_{[u]}= \dim G - \max_{g \in G_{[u]}} \dim g^G = \min_{g \in G_{[u]}} \dim C_G(g) = d_u(G). \quad \square$$
We can now prove Proposition \ref{p:ingsimple}.
\noindent \textit{Proof of Proposition \ref{p:ingsimple}.}
First, by \cite{Lawther}, we have $d_u(G_{a.})\leq d_u(G)$. It remains to show that $d_u(G) \leq d_u(G_{s.c.})$. Note that the result is trivial unless $G$ is neither of adjoint nor simply connected type.
Let $\phi: G_{s.c.} \rightarrow G$ be an isogeny. Let $g$ be any element of $G_{s.c.}$ of order dividing $u$ such that $d_u(G_{s.c.})=\dim C_{G_{s.c.}}(g)$. Then $\phi(g)$ is an element of $G$ of order dividing $u$ and by Lemma \ref{l:dufactg} $\dim C_G(\phi(g))=\dim C_{G_{s.c.}}(g)$. Hence $\dim C_G(\phi(g))=d_u(G_{s.c.})$. Since $d_u(G)\leq \dim C_G(\phi(g))$ we obtain $d_u(G)\leq d_u(G_{s.c.})$. $\square$
\section{Proof of Theorem \ref{t:blawther} for $G$ of exceptional type}\label{s:exceptionalgroups}\label{s:ex}
In this section, we determine $d_u(G)$ for $G$ a simple algebraic group of exceptional type defined over an algebraically closed field $K$ of characteristic $p$. In particular we prove Theorem \ref{t:blawther} for $G$ of exceptional type (see Propositions \ref{p:e6sc} and \ref{p:e7sc} below). Recall that a simple algebraic group of exceptional type is either simply connected or adjoint. Following the result of Lawther \cite{Lawther} giving $d_u(G)$ for $G$ of adjoint type, we suppose that $G$ is of simply connected type. Without loss of generality, we assume that $G$ is of type $E_6$ or $E_7$ and $p\neq 3$ or $2$ respectively, as these are the only cases where the simply connected and the adjoint groups are not abstractly isomorphic. \\ Let $T$ be a maximal torus of $G$ with corresponding root system $\Phi$ and set $\Pi=\{\alpha_1,\dots, \alpha_\ell\} \subset \Phi$ to be a set of simple roots of $G$, where $\ell$ is the rank of $G$. Note that $\ell=6$ or $\ell=7$ according respectively as $G=E_6$ or $G=E_7$. \\ Given a positive integer $u$, we write $u=qv$ where $q$ and $v$ are positive coprime integers with $q$ a power of $p$ and $p$ does not divide $v$.\\ Let $y$ be a semisimple element of $G$ of order dividing $v$, i.e. $y\in G_{[v]}$. Since every semisimple element of $G$ belongs to a maximal torus of $G$ and all maximal tori are conjugate, we can assume without loss of generality that $y \in T$. Now \begin{equation}\label{e:centalgg} C_G(y)^0= \langle T, U_{\alpha}: \alpha \in \Psi \rangle\end{equation}where $\Psi=\{\alpha \in \Phi: \alpha(y)=1\}$ and $U_{\alpha}$ is the root subgroup of $G$ corresponding to $\alpha$. Also we can write \begin{equation*}\label{e:sseg} y= \prod_{i=1}^\ell h_{\alpha_i}(k^{c_i})\end{equation*} where for $l \in K^*=K\setminus\{0\}$ and $1\leq i \leq \ell$, $h_{\alpha_i}(l)$ is the image of $\begin{pmatrix} l & 0 \\ 0 & l^{-1}\end{pmatrix}\in {\rm SL}_2(K)$ under the canonical map ${\rm SL}_2(K) \rightarrow \langle U_{\alpha_i}, U_{-\alpha_{i}}\rangle$, $k$ is an element of $K^*$ of order $v$, and $0\leq c_i\leq v-1$ is an integer for all $1\leq i \leq \ell$. \\ Given $\alpha \in \Phi$ we then have \begin{equation}\label{e:centalgp}
\alpha(y)= k^{\sum_{i=1}^\ell c_i\langle \alpha_i,\alpha \rangle}
\end{equation} where $$\langle \alpha_i, \alpha\rangle= \frac{2(\alpha_i,\alpha)}{(\alpha_i,\alpha_i)}$$ and $(,)$ denotes the inner product of the Euclidean space spanned by $\Phi$.
\begin{lem}\label{l:duered} Let $G$ be a simple simply connected algebraic group of type $X=E_6$ or $E_7$ defined over an algebraically closed field $K$ of characteristic $p$. Let $u$ be a positive integer. Then $d_u(G)=d_u(G_{a.})$ except possibly if $G=E_6$, $p\neq 3$ and $u\equiv 0\mod 3$, or $G=E_7$, $p \neq 2$ and $u\equiv 0 \mod 2$. \end{lem}
\begin{proof}
Set $C=Z(G)$ and note that $$|C|=\left\{\begin{array}{ll} 3 & \textrm{if} \ G=E_6 \ \textrm{and} \ p\neq 3\\ 2 & \textrm{if} \ G=E_7 \ \textrm{and} \ p\neq 2\\ 1 & \textrm{otherwise.}\\
\end{array} \right.$$ Also note that if $(X,p) \in \{(E_6,3), (E_7,2)\}$ then $G$ is abstractly isomorphic to $G_{a.}$ and so $d_u(G)=d_u(G_{a.})$ for every positive integer $u$. We therefore assume that $(X,p)\not \in \{(E_6,3), (E_7,2)\}$. Then $C=Z(G)=\langle c \rangle$ is cyclic group of order $3$ or $2$ according respectively as $X=E_6$ or $E_7$. Let $\phi: G\rightarrow G_{a.}$ be the surjective canonical map. Let $gC$ be an element of $G_{a.}$ of order dividing $u$ such that $\dim C_{G_{a.}}(gC)=d_u(G_{a.})$. \\
Suppose $u\neq 0 \mod |C|$. Then $g \in G$ has order dividing $u$, and by Lemma \ref{l:dufactg} $\dim C_G(g)=\dim C_{G_{a.}}(gC)$. Hence $\dim C_G(g)=d_u(G_{a.})$ and $d_u(G) \leq d_u(G_{a.})$. It now follows from Proposition \ref{p:ingsimple} that $d_u(G)=d_u(G_{a.})$. \end{proof}
\begin{prop}\label{p:e6sc} Suppose $G$ is a simple simply connected algebraic group of type $E_6$ defined over an algebraic closed field $K$ of characteristic $p$. Let $u$ be a positive integer. Then $d_u(G)=d_u(G_{a.})$. \end{prop}
\begin{proof} Note that by \cite{Lawther} $d_u(G)\geq d_u(G_{a.})$ and $d_u(G_{a})=6$ for $u\geq h=12$.\\ By Lemma \ref{l:duered} we can assume without loss of generality that $p \neq 3$ and $u \equiv 0 \mod 3$. Write $u=qv$ where $q$ and $v$ are coprime positive integers such that $q$ is a power of $p$ and $p$ does not divide $v$. \\
We first suppose that $u \leq h=12$. \\ Assume $u=3$. Then $q=1$ and $u=v=3$. Let $$y_3=h_{\alpha_1}(k)h_{\alpha_2}(k^2)h_{\alpha_3}(k)h_{\alpha_4}(k^2)h_{\alpha_5}(k^2)h_{\alpha_6}(1)$$ where $k \in K^*$ is an element of order 3. Then $y_3$ is a semisimple element of $G$ of order 3. Moreover by (\ref{e:centalgg}) and (\ref{e:centalgp}) we get $$C_G(y_3)^0=A_2A_2A_2.$$ Hence $d_q(C_G(y_3)^0)=d_1(A_2A_2A_2)=24$. Now by \cite{Lawther} $d_u(G_{a.})=24$. We now deduce from Proposition \ref{p:prelimdu}(i) that $d_u(G)=d_u(G_{a.})=24.$\\ Assume $u=6$. Then $q=1$ and $u=v=6$, or $q=p=2$ and $v=3$. Suppose first that $q=1$. Let $$y_6=h_{\alpha_1}(k)h_{\alpha_2}(k^2)h_{\alpha_3}(k)h_{\alpha_4}(k^5)h_{\alpha_5}(k^2)h_{\alpha_6}(1)$$ where $k \in K^*$ is an element of order 6. Then $y_6$ is a semisimple element of $G$ of order 6. Moreover by (\ref{e:centalgg}) and (\ref{e:centalgp}) we get $$C_G(y_6)^0=A_1A_1A_1T_3.$$ Hence $d_q(C_G(y_6)^0)=d_1(A_1A_1A_1T_3)=12$. \\
Suppose now that $q=2$ and $v=3$. Consider the semisimple element $y_3$ of $G$ of order 3 defined above. We have $C_G(y_3)^0=A_2A_2A_2$ and $d_q(C_G(y_3)^0)=d_2(A_2A_2A_2)=12$. \\ Now by \cite{Lawther} $d_u(G_{a.})=12$. We now deduce from Proposition \ref{p:prelimdu}(i) that $d_u(G)=d_u(G_{a.})=12.$\\ Assume $u=9$. Then $q=1$ and $u=v=9$. Let $$y_9=h_{\alpha_1}(k)h_{\alpha_2}(k^2)h_{\alpha_3}(k)h_{\alpha_4}(k^5)h_{\alpha_5}(k^8)h_{\alpha_6}(1)$$ where $k \in K^*$ is an element of order 9. Then $y_9$ is a semisimple element of $G$ of order 9. Moreover by (\ref{e:centalgg}) and (\ref{e:centalgp}) we get $$C_G(y_9)^0=A_1T_5.$$ Hence $d_q(C_G(y_9)^0)=d_1(A_1T_5)=8$. Now by \cite{Lawther} $d_u(G_{a.})=8$. We now deduce from Proposition \ref{p:prelimdu}(i) that $d_u(G)=d_u(G_{a.})=8.$\\ Assume $u=12$. Then $q=1$ and $u=v=12$, or $q=4$, $p=2$ and $v=3$. Suppose first that $q=1$. Let $$y_{12}=h_{\alpha_1}(k)h_{\alpha_2}(k^2)h_{\alpha_3}(k)h_{\alpha_4}(k^5)h_{\alpha_5}(k^8)h_{\alpha_6}(1)$$ where $k \in K^*$ is an element of order 12. Then $y_{12}$ is a semisimple element of $G$ of order 12. Moreover by (\ref{e:centalgg}) and (\ref{e:centalgp}) we get $$C_G(y_{12})^0=T_6.$$ Hence $d_q(C_G(y_{12})^0)=d_1(T_6)=6$. \\
Suppose now that $q=4$, $p=2$ and $v=3$. Consider the semisimple element $y_3$ of $G$ of order 3 defined above. We have $C_G(y_3)^0=A_2A_2A_2$ and $d_q(C_G(y_3)^0)=d_4(A_2A_2A_2)=6$. By Proposition \ref{p:prelimdu}(i) we obtain $d_{12}(G)=d_{12}(G_{a.})=6$.
\\ We now suppose that $u>h=12$. If $v\geq h$ then let $$y=h_{\alpha_1}(k)h_{\alpha_2}(k^2)h_{\alpha_3}(k)h_{\alpha_4}(k^5)h_{\alpha_5}(k^8)h_{\alpha_6}(1)$$ where $k \in K^*$ is an element of order $v$. Then $y$ is a semisimple element of $G$ of order $v$. Moreover by (\ref{e:centalgg}) and (\ref{e:centalgp}) we get $$C_G({y})^0=T_6.$$ Hence $d_q(C_G(y)^0)=d_q(T_6)=6$ and so by Proposition \ref{p:prelimdu}(i) $d_u(G)=6$. \\ We finally suppose that $u>h=12$ and $v<h$. As we are assuming that $u \equiv 0 \mod 3$ and $p\neq 3$ we have $v\equiv 0 \mod 3$. Hence $v\in \{3,6,9\}$. \\ Assume $v=3$. As $u>12$ we have $q\geq 5$. Now $y_3$ is a semisimple element of $G$ of order $3$ with $C_G(y_3)^0=A_2A_2A_2$. As $q\geq 3$, we have $d_q(C_G(y_3)^0)=d_q(A_2A_2A_2)=6$. \\ Assume $v=6$. As $u>12$ we have $q\geq 3$. Now $y_6$ is a semisimple element of $G$ of order $6$ with $C_G(y_3)^0=A_1A_1A_1T_3$. As $q\geq 2$, we have $d_q(C_G(y_6)^0)=d_q(A_1A_1A_1T_3)=6$. \\ Assume $v=9$. As $u>12$ we have $q\geq 2$. Now $y_9$ is a semisimple element of $G$ of order $9$ with $C_G(y_9)^0=A_1T_5$. As $q\geq 2$, we have $d_q(C_G(y_9)^0)=d_q(A_1T_5)=6$. \\ Hence in the case where $u>h$ and $v<h$, by Proposition \ref{p:prelimdu}(i), we get $d_u(G)=6.$ \end{proof}
\begin{prop}\label{p:e7sc} Suppose $G$ is a simple simply connected algebraic group of type $E_7$ defined over an algebraic closed field $K$ of characteristic $p$. Let $u$ be a positive integer. The following assertions hold. \begin{enumerate}[(i)] \item We have $d_u(G)=d_u(G_{a.})$ unless $p\neq 2$ and $u\in\{2,6,10,14,18\}$. \item Suppose $p\neq 2$. Then $d_2(G)=d_u(G_{a.})+6$ and $d_u(G)=d_u(G_{a.})+2$ for $u\in\{6,10,14,18\}$. More precisely, $$d_2(G)=69,\ d_6(G)=23,\ d_{10}(G)=15, \ d_{14}(G)=11 \quad \textrm{and} \quad d_{18}(G)=9.$$ \end{enumerate} \end{prop}
\begin{proof} Recall that by \cite{Lawther} $d_u(G)\geq d_u(G_{a.})$, $d_u(G_{a})=7$ for $u\geq h=18$, and $$d_2(G_{a.})=63,\ d_6(G_{a.})=21, \ d_{10}(G_{a.})=13 \quad \textrm{and}\quad d_{14}(G_{a.})=9.$$ By Lemma \ref{l:duered} we can assume without loss of generality that $p \neq 2$ and $u \equiv 0 \mod 2$, as otherwise $d_u(G)=d_u(G_{a.})$. Write $u=qv$ where $q$ and $v$ are coprime positive integers such that $q$ is a power of $p$ and $p$ does not divide $v$. Note that $q$ is odd and $v\equiv 0 \mod 2$. \\
We first suppose that $u \leq h=18$. \\ Assume $u=2$. Then $q=1$ and $u=v=2$. By \cite[Table 6]{Cohen}, $d_2(G)=69.$ Let $$y_2=h_{\alpha_1}(k)h_{\alpha_2}(1)h_{\alpha_3}(1)h_{\alpha_4}(1)h_{\alpha_5}(k)h_{\alpha_6}(k)h_{\alpha_7}(1)$$ where $k \in K^*$ is an element of order 2. Then $y_2$ is a semisimple element of $G$ of order 2. Moreover by (\ref{e:centalgg}) and (\ref{e:centalgp}) we get $$C_G(y_2)^0=A_1D_6.$$ Hence $d_q(C_G(y_2)^0)=d_1(A_1D_6)=69=d_2(G)$. \\ Assume $u=4$. Then $q=1$ and $u=v=4$. By \cite{Lawther} and \cite[Table 6]{Cohen}, $d_4(G)=d_4(G_a)=33$. Let $$y_4=h_{\alpha_1}(k)h_{\alpha_2}(1)h_{\alpha_3}(1)h_{\alpha_4}(k^2)h_{\alpha_5}(k)h_{\alpha_6}(k)h_{\alpha_7}(k^2)$$ where $k \in K^*$ is an element of order 4. Then $y_4$ is a semisimple element of $G$ of order 4. Moreover by (\ref{e:centalgg}) and (\ref{e:centalgp}) we get $$C_G(y_4)^0=A_3A_3A_1.$$ Hence $d_q(C_G(y_4)^0)=d_1(A_3A_3A_1)=33=d_4(G)$. \\ Assume $u=6$. Then $q=1$ and $u=v=6$, or $q=p=3$ and $v=2$. Suppose first that $q=p=3$ and $v=2$. By \cite[Table 6]{Cohen} a semisimple element $y$ of $G$ of order $2$ is such that $C_G(y)^0= A_1D_6$ or $E_7$. It follows from Proposition \ref{p:prelimdu}(i) that $d_6(G)=d_3(A_1D_6)=23.$
Suppose now that $q=1$. By \cite[Table 6]{Cohen}, $d_6(G)=23.$ Let $$y_6=h_{\alpha_1}(k)h_{\alpha_2}(1)h_{\alpha_3}(1)h_{\alpha_4}(k^2)h_{\alpha_5}(k^5)h_{\alpha_6}(k^3)h_{\alpha_7}(k^2)$$ where $k \in K^*$ is an element of order 6. Then $y_6$ is a semisimple element of $G$ of order 6. Moreover by (\ref{e:centalgg}) and (\ref{e:centalgp}) we get $$C_G(y_6)^0=A_3A_1A_1T_2.$$ Hence $d_q(C_G(y_6)^0)=d_1(A_3A_1A_1T_2)=23=d_6(G)$. \\ Assume $u=8$. Then $q=1$ and $u=v=8$. Let $$y_8=h_{\alpha_1}(1)h_{\alpha_2}(1)h_{\alpha_3}(k)h_{\alpha_4}(k)h_{\alpha_5}(k^2)h_{\alpha_6}(k^7)h_{\alpha_7}(k^3)$$ where $k \in K^*$ is an element of order 8. Then $y_8$ is a semisimple element of $G$ of order 8. Moreover by (\ref{e:centalgg}) and (\ref{e:centalgp}) we get $$C_G(y_8)^0=A_2A_1A_1T_3.$$ Hence $d_q(C_G(y_8)^0)=17$. Since, by \cite{Lawther}, $d_8(G_a)=17$, we deduce from Proposition \ref{p:prelimdu}(i) that $d_8(G)=17$.\\ Assume $u=10$. Then $q=1$ and $u=v=10$, or $q=p=5$ and $v=2$. Suppose first that $q=p=5$ and $v=2$. By \cite[Table 6]{Cohen} a semisimple element $y$ of $G$ of order $2$ is such that $C_G(y)^0= A_1D_6$ or $E_7$. It follows from Proposition \ref{p:prelimdu}(i) that $d_{10}(G)=d_5(A_1D_6)=15.$
Suppose now that $q=1$. By (\ref{e:centalgg}) and (\ref{e:centalgp}), a semisimple $y$ element of $G$ of order 10 is such that $C_G(y)^0$ is generated by at least four root subgroups $U_\alpha$ where $\alpha$ is a positive root of $G$.
It follows from \cite{Lawther} that $d_{10}(G)>d_{10}(G_{a.})=13$. Hence by Lemma \ref{l:centmod2} $d_{10}(G)\geq 15$. Let $$y_{10}=h_{\alpha_1}(k)h_{\alpha_2}(1)h_{\alpha_3}(1)h_{\alpha_4}(1)h_{\alpha_5}(k^6)h_{\alpha_6}(k)h_{\alpha_7}(k^8)$$ where $k \in K^*$ is an element of order 10. Then $y_{10}$ is a semisimple element of $G$ of order 10. Moreover by (\ref{e:centalgg}) and (\ref{e:centalgp}) we get $$C_G(y_{10})^0=A_1A_2T_4.$$ Hence $$d_q(C_G(y_{10})^0)=d_1(A_1A_2T_4)=15.$$ It now follows from Proposition \ref{p:prelimdu}(i) that $d_{10}(G)=15.$\\ Assume $u=12$. Then $q=1$ and $u=v=12$, or $q=p=3$ and $v=4$. Suppose first that $q=p=3$ and $v=4$. Consider the semisimple element $y_4$ of $G$ defined above. Then $C_G(y_4)^0=A_3A_3A_1$ and $d_q(C_G(y_4)^0)=d_3(A_3A_3A_1)=11$. Hence by Proposition \ref{p:prelimdu}(i), we obtain $d_{12}(G)\leq 11$. Suppose now that $q=1$ and $u=v=12$. Let $$y_{12}=h_{\alpha_1}(k)h_{\alpha_2}(1)h_{\alpha_3}(1)h_{\alpha_4}(k)h_{\alpha_5}(k^3)h_{\alpha_6}(k^5)h_{\alpha_7}(k^8)$$ where $k \in K^*$ is an element of order 12. Then $y_{12}$ is a semisimple element of $G$ of order 12. Moreover by (\ref{e:centalgg}) and (\ref{e:centalgp}) we get $$C_G(y_{12})^0=A_1A_1T_5.$$ Hence $d_q(C_G(y_{12})^0)=11$.\\
Since, by \cite{Lawther}, $d_{12}(G_a)=11$, we deduce that $d_{12}(G)=11$ for all $p$.\\ Assume $u=14$. Then $q=1$ and $u=v=14$, or $q=p=7$ and $v=2$. Suppose first that $q=p=7$ and $v=2$. By \cite[Table 6]{Cohen} a semisimple element $y$ of $G$ of order $2$ is such that $C_G(y)^0= A_1D_6$ or $E_7$. It follows from Proposition \ref{p:prelimdu}(i) that $d_{14}(G)=d_7(A_1D_6)=11.$
Suppose now that $q=1$. By (\ref{e:centalgg}) and (\ref{e:centalgp}), a semisimple $y$ element of $G$ of order 14 is such that $C_G(y)^0$ is generated by at least two root subgroups $U_\alpha$ where $\alpha$ is a positive root of $G$. We deduce that $C_G(s)^0\neq A_1T_6$. It follows from \cite{Lawther} that $d_{14}(G)>d_{14}(G_{a.})=9$. Hence by Lemma \ref{l:centmod2} $d_{14}(G)\geq 11$. Let $$y_{14}=h_{\alpha_1}(k)h_{\alpha_2}(1)h_{\alpha_3}(1)h_{\alpha_4}(1)h_{\alpha_5}(k^2)h_{\alpha_6}(k^5)h_{\alpha_7}(k^9)$$ where $k \in K^*$ is an element of order 14. Then $y_{14}$ is a semisimple elements of $G$ of order 14. Moreover by (\ref{e:centalgg}) and (\ref{e:centalgp}) we get $$C_G(y_{14})^0=A_1A_1T_5.$$ Hence $$d_q(C_G(y_{14})^0)=d_1(A_1A_1T_5)=11.$$ It now follows from Proposition \ref{p:prelimdu}(i) that $d_{14}(G)=11.$\\ Assume $u=16$. Then $q=1$ and $u=v=16$. Let $$y_{16}=h_{\alpha_1}(k)h_{\alpha_2}(1)h_{\alpha_3}(1)h_{\alpha_4}(k)h_{\alpha_5}(k^3)h_{\alpha_6}(k^6)h_{\alpha_7}(k^{10})$$ where $k \in K^*$ is an element of order 16. Then $y_{16}$ is a semisimple element of $G$ of order 16. Moreover by (\ref{e:centalgg}) and (\ref{e:centalgp}) we get $$C_G(y_{16})^0=A_1T_6.$$ Hence $d_q(C_G(y_{16})^0)=9$. Since, by \cite{Lawther}, $d_{16}(G_a)=9$, we deduce from Proposition \ref{p:prelimdu}(i) that $d_{16}(G)=9$.\\ Assume $u=18$. Then $q=1$ and $u=v=18$, or $q=9$, $p=3$ and $v=2$. Suppose first that $q=9$, $p=3$ and $v=2$. By \cite[Table 6]{Cohen} a semisimple element $y$ of $G$ of order $2$ is such that $C_G(y)^0= A_1D_6$ or $E_7$. It follows from Proposition \ref{p:prelimdu}(i) that $d_{18}(G)=d_9(A_1D_6)=9$.
Suppose now that $q=1$. By (\ref{e:centalgg}) and (\ref{e:centalgp}), a semisimple $y$ element of $G$ of order 18 is such that $C_G(y)^0$ is generated by at least one root subgroup $U_\alpha$ where $\alpha$ is a positive root of $G$. We deduce that $C_G(y)^0\neq T_7$. It follows from \cite{Lawther} that $d_{18}(G)>d_{18}(G_{a.})=7$. Hence by Lemma \ref{l:centmod2} $d_{18}(G)\geq 9$. Let $$y_{18}=h_{\alpha_1}(k)h_{\alpha_2}(1)h_{\alpha_3}(1)h_{\alpha_4}(k)h_{\alpha_5}(k^3)h_{\alpha_6}(k^6)h_{\alpha_7}(k^{10})$$ where $k \in K^*$ is an element of order 18. Then $y_{18}$ is a semisimple element of $G$ of order 18. Moreover by (\ref{e:centalgg}) and (\ref{e:centalgp}) we get $$C_G(y_{18})^0=A_1T_6.$$ Hence $$d_q(C_G(y_{18})^0)=d_1(A_1T_6)=9.$$ It now follows from Proposition \ref{p:prelimdu}(i) that $d_{18}(G)=9.$\\
We now suppose that $u>h=18$. Recall that $p\neq 2$ and $v\equiv 0 \mod 2$. Assume $v> h$. Note that $v\geq 20$. Let $$y=h_{\alpha_1}(k)h_{\alpha_2}(1)h_{\alpha_3}(1)h_{\alpha_4}(k^2)h_{\alpha_5}(k^5)h_{\alpha_6}(k^9)h_{\alpha_7}(k^{14})$$ where $k \in K^*$ is an element of order $v$. Then $y$ is a semisimple element of $G$ of order $v\geq 20$. Moreover by (\ref{e:centalgg}) and (\ref{e:centalgp}) we get $$C_G({y})^0=T_7.$$ Hence $d_q(C_G(y)^0)=d_q(T_7)=7$ and so by Proposition \ref{p:prelimdu}(i) $d_u(G)=7$. \\ We finally suppose that $u>h=18$ and $v\leq h$. Since $v\equiv 0 \mod 2$ we have $v\in \{2,4,6,8,10,12,14,16,18\}$. \\ Assume $v=2$. As $u\geq 20$ we have $q\geq 10$. Now $y_2$ is a semisimple element of $G$ of order $2$ with $C_G(y_2)^0=A_1D_6$. As $q\geq 10$, we have $d_q(C_G(y_2)^0)=d_q(A_1D_6)=7$. \\ Assume $v=4$. As $u\geq 20$ we have $q\geq 5$. Now $y_4$ is a semisimple element of $G$ of order $4$ with $C_G(y_4)^0=A_3A_3A_1$. As $q\geq 4$, we have $d_q(C_G(y_4)^0)=d_q(A_3A_3A_1)=7$. \\ Assume $v=6$. As $u\geq 20$ and $p\neq 2$ we have $q\geq 5$. Now $y_6$ is a semisimple element of $G$ of order $6$ with $C_G(y_6)^0=A_3A_1A_1T_2$. As $q\geq 4$, we have $d_q(C_G(y_6)^0)=d_q(A_3A_1A_1T_2)=7$. \\ Assume $v=8$. As $u\geq 20$ we have $q\geq 3$. Now $y_8$ is a semisimple element of $G$ of order $8$ with $C_G(y_8)^0=A_2A_1A_1T_3$. As $q\geq 3$, we have $d_q(C_G(y_8)^0)=d_q(A_2A_1A_1T_3)=7$. \\ Assume $v=10$. As $u\geq 20$ and $p\neq 2$ we have $q\geq 3$. Now $y_{10}$ is a semisimple element of $G$ of order $10$ with $C_G(y_{10})^0=A_2A_1T_4$. As $q\geq 3$, we have $d_q(C_G(y_{10})^0)=d_q(A_2A_1T_4)=7$. \\ Assume $v\in\{12,14\}$. As $u\geq 20$ and $p\neq 2$ we have $q\geq 3$. Now $y_{12}$ and $y_{14}$ are semisimple elements of $G$ of respective orders $12$ and $14$ with $C_G(y_{12})^0=A_1A_1T_5$ and $C_G(y_{14})^0=A_1A_1T_5$ . As $q\geq 2$, we have $d_q(C_G(y_{12})^0)=d_q(C_G(y_{14})^0)=d_q(A_1A_1T_5)=7$. \\ Assume $v\in\{16,18\}$. As $u\geq 20$ and $p\neq 2$ we have $q\geq 3$. Now $y_{16}$ and $y_{18}$ are semisimple elements of $G$ of respective orders $16$ and $18$ with $C_G(y_{16})^0=A_1T_6$ and $C_G(y_{18})^0=A_1T_6$. As $q\geq 2$, we have $d_q(C_G(y_{16})^0)=d_q(C_G(y_{18})^0)=d_q(A_1T_6)=7$. \\ Hence in the case where $u>h$ and $v\leq h$, by Proposition \ref{p:prelimdu}(i), we get $d_u(G)=7.$ \end{proof}
\section{Some properties of ${\rm Spin}_n(K)$}\label{s:spin}
Let $K$ now denote an algebraically closed field $K$ of characteristic $p\neq2$. We give a characterization of semisimple elements of ${\rm Spin}_n(K)$ of a given order (see Lemma
\ref{l:soliftspin} below). Before doing so we recall some properties of the group ${\rm Spin}_n(K)$ and the canonical surjective map ${\rm Spin}_n(K)\rightarrow {\rm SO}_n(K)$, where $n\geq 7$ is a positive integer. Write $n=2\ell$ or $n=2\ell+1$ for some integer $\ell\geq 1$, according respectively as $n$ is even or odd.\\ Let $V$ be the natural module for ${\rm SO}_n(K)$ and let $$\mathcal{B}=\{e_1,f_1,\dots,e_\ell,f_\ell\} \quad \textrm{or} \quad \mathcal{B}=\{e_1,f_1,\dots, e_\ell,f_\ell,d\}$$ (according respectively as $n$ is even or odd) be a standard basis of $V$. That is, $(e_i,f_i)=1$, $(e_i,e_i)=(f_i,f_i)=0$ for $1\leq i\leq \ell$, $(e_i,f_j)=(e_i,e_j)=(f_i,f_j)=0$ for $i\neq j$ such that $1\leq i, j\leq \ell$ and if $n$ is odd then $(d,d)=1$ and $(d,e_i)=(d,f_i)$ for $1\leq i \leq \ell$. Here $(,): V\times V \rightarrow K$ denotes the non-degenerate symmetric bilinear form associated to $V$. \\ Let $T_0(V,K)$ be the $K$-algebra of polynomials in $e_1,f_1,\dots,e_\ell,f_\ell$ if $n$ is even (respectively, in $e_1,f_1,\dots,e_\ell,f_\ell,d$ if $n$ is odd). Also let $I(V,K)$ be the ideal of $T_0(V,K)$ generated by $\{vv-(v,v):v \in V\}$ and let $C_0(V,K)=T_0(V,K)/I(V,K)$.\\ Let ${}^*: C_0(V,K)\rightarrow C_0(V,K)$ be the $K$-linear map such that for any even positive integer $r$ and elements $x_1,\dots, x_r \in \mathcal{B}$, we have the equality $(x_1\dots x_r)^*=x_r\dots x_1$ (in $C_0(V,K))$. Note that for $x$ in $V$, we have $x^*x=xx^*=xx=(x,x)$. \\
We can now define the spin group ${\rm Spin}_n(K)$ as an abstract group, namely ${\rm Spin}_n(K)$ consists of the elements $t$ of $C_0(V,K)$ such that: $$ t^*t=1,\quad tVt^{-1}=V$$ and the map $$ \begin{array}{l} V \rightarrow V\\ x\mapsto txt^{-1}
\end{array}$$ has determinant 1.
Given an element $t$ of ${\rm Spin}_n(K)$, consider the map $\phi_t: V\rightarrow V$ defined by $\phi_t(x)=txt^{-1}$ for $x$ in $V$. Then for any $x$ in $V$, $$(\phi_t(x),\phi_t(x))=\phi_t(x)\phi_t(x)=txt^{-1}txt^{-1}=tx^2t^{-1}=t(x,x)t^{-1}=(x,x)$$ and so $\phi_t$ is an element of ${\rm SO}_n(K)$. In fact the map $\phi$: $$\begin{array}{lll}{\rm Spin}_n(K) & \rightarrow & {\rm SO}_n(K)\\ t & \mapsto & \phi_t \end{array}$$
is a surjective homomorphism with kernel equal to $\{1,-1\}$. \\
We can now characterize a preimage of a semisimple element of ${\rm SO}_n(K)$ of a given order under the canonical surjective map ${\rm Spin}_n(K)\rightarrow {\rm SO}_n(K)$, where $n\geq 7$ is a positive integer and $K$ is an algebraically closed field of characteristic $p \neq 2$.
\begin{lem}\label{l:soliftspin} Let $g$ be an element of ${\rm SO}_{n}(K)$ defined over an algebraically closed field $K$ of characteristic $p\neq2$. Suppose $g$ is a semisimple element of ${\rm SO}_n(K)$ of order $u$, and let $\omega$ be a $u$-th root of unity of $K$. Let $\pm w$ be any of the two preimages of $g$ under the canonical surjective map $\phi: {\rm Spin}_n(K)\rightarrow {\rm SO}_{n}(K)$. The following assertions hold. \begin{enumerate}[(i)] \item The order of $w$ in ${\rm Spin}_n(K)$ is divisible by $u$. \item There exits $t \in \{\pm w\}$ of order $u$ if and only if $u$ is odd, or $u$ is even and the number of eigenvalues of $\phi(t)$ of the form $\omega^i$ with $i$ odd is divisible by $4$. \item If neither $w$ nor $-w$ has order $u$, then $u$ is even and $\pm w$ has order $2u$. \end{enumerate} \end{lem}
\begin{proof} This is a classical result which follows from the properties of the canonical surjective map $\phi: {\rm Spin}_n(K)\rightarrow {\rm SO}_n(K)$ described above. We give a sketch of the argument. \\ We first consider part (i). Let ${g}$ be any semisimple element of ${\rm SO}_n(K)$ of order $u$. Let $w$ be a preimage of $g$ under $\phi$ (so that $-w$ is the other preimage of $g$ under $\phi$) and let $m$ be the order of $w$. We have $g^m=\phi(w)^m=\phi(w^m)=\phi(1)=1$. Hence $u$, which is the order of $g$, divides the order of a preimage of $g$ under $\phi$. This establishes part (i).\\
We now consider parts (ii) and (iii). Let $V$ be the natural module for ${\rm SO}_n(K)$. There is a standard basis $\mathcal{B}$ of $V$, where $\mathcal{B}=(e_1,f_1,\dots,e_\ell,f_\ell)$ or $(e_1,f_1,\dots, e_\ell,f_\ell,d)$ according respectively as $n$ is even or odd, such that the matrix of ${g}$ with respect to $\mathcal{B}$ is diagonal with entries $d_i$ satisfying $d_i^u=1$ for $1 \leq i \leq n$. Note that if $1\leq j \leq \ell$ then $d_{2j-1}=d_{2j}^{-1}$ is a power of $\omega$, and if $n$ is odd then $d_n=1$. Moreover without loss of generality $d_1=\omega$ and $d_2=\omega^{-1}$. For $1\leq j \leq \ell$, let ${g_j}$ be the element of ${\rm SO}_n(K)$ such that, with respect to $\mathcal{B}$, ${g_j}$ is diagonal and $({g_j})_{2j-1,2j-1}=d_j$, $({g_j})_{2j,2j}=d_{j+1}$, $({g_j})_{k,k}=1$ for $k\not\in\{2j-1,2j\}$. Note that ${g}={g_1}\dots{g_\ell}$. In the decomposition ${g}={g_1}\dots{g_\ell}$ delete the elements ${g_j}$ with $1\leq i \leq \ell$ such that ${g_j}=1$, and renumbering the elements ${g_j}$ where $1\leq j\leq \ell$, if necessary, write ${g}={g_1}\dots{g_\iota}$ where $1\leq \iota \leq \ell$ and ${g_1},\dots, {g_\iota}$ are not the identity element. \\ Let $1\leq j \leq \iota$ and let $1\leq k\leq \iota$ be such that $({g_j})_{2k-1,2k-1}$ and $({g_j})_{2k,2k}$ are both not 1. Write $\theta_j=({g_j})_{2k-1,2k-1}$. Note that $\theta_j\neq 1$ is a power of $\omega$ and $\theta_j^{-1}=({g_j})_{2k,2k}$. Moreover ${g_j}(e_k)=\theta_j e_k$, ${g_j}(f_k)=\theta_j^{-1} f_k$ and ${g_j}$ fixes every other element of $\mathcal{B}$. Write $\theta_j=\omega^{l_j}$ for some positive integer $l_j$. \\ For a nonsingular element $x$ in $V$, let $R_x$ be the reflection: $$\begin{array}{lll}V & \rightarrow& V \\ y &\mapsto &y-\frac{2(y,x)}{(x,x)}x.\end{array}$$ An easy check yields $${g_j}=R_{(1-\theta_j)e_k+(\theta_j^{-1}-1)f_k}R_{2e_k+2f_k}.$$ Let $$x_j=((1-\theta_j)e_k+(\theta_j^{-1}-1)f_k)(2e_k+2f_k).$$ Then $x_j\in C_0(V,K)$ and $x_jx_j^*=16(1-\theta_j)^2\theta_j^{-1}$. Let $\tau_j \in K$ be such that $\tau_j^2=\theta_j^{-1}=\omega^{-l_j}$, furthermore if $u$ and $l_j$ are both odd take $\tau_j=\omega^{(u-l_j)/2}$, and if $l_j$ is even take $\tau_j=\omega^{-l_j/2}$. Let $$t_j=\frac{x_j}{4(1-\theta_j)\tau_j}.$$ Then $t_j$ is an element of ${\rm Spin}_n(K)$ and the image of $t_j$ under the canonical surjective map $\phi: {\rm Spin}_n(K)\rightarrow {\rm SO}_n(K)$ is ${g_j}$. Furthermore, for any positive integer $m$, $$t_j^m=\frac{1}{2\tau_j^m}(e_kf_k+\theta_j^{-m}(2-e_kf_k))$$ and so \begin{equation}\label{e:orderpreso} t_j^u=\frac1{\tau_j^u}=\left\{\begin{array}{ll} 1 & \textrm{if} \ l_j \ \textrm{is even, or} \ u \ \textrm{and} \ l_j\ \textrm{are both odd} \\ -1 & \textrm{if} \ u \ \textrm{is even and} \ l_j \ \textrm{is odd}.\end{array} \right.\end{equation} Finally, let $t=t_1\dots t_\iota$. Then $t \in {\rm Spin}_n(K)$ and $\phi(t)={g}={g_1}\dots{g_\iota}$. Now for $1\leq i, j \leq \iota$, $t_i$ and $t_j$ commute and so $$t^u=(t_1\dots t_\iota)^u=t_1^u\dots t_\iota^u= \prod_{j=1}^\iota \frac{1}{\tau_j^u}.$$ It follows from (\ref{e:orderpreso}) that $t^u=1$ if and only if $u$ is odd or $u$ is even and the number of eigenvalues of $\phi(t)=g$ of the form $\omega^i$ with $i$ odd is divisible by $4$. Otherwise $t^u=-1$ and $t^{2u}=1$. Note that the other preimage of $g$ under $\phi$ is $-t$ and $(-t)^u=t^u$ if $u$ is even, otherwise $(-t)^u=-t^u$. Parts (ii) and (iii) now follow from part (i).
\end{proof}
\section{Upper bounds for $d_u(G)$ for $G$ of classical type}\label{s:upc} Let $G$ be a simple algebraic group of classical type defined over an algebraically closed field $K$ of characteristic $p$. Let $u$ be a positive integer. In this section, we give an upper bound for $d_u(G)$. Since by Proposition \ref{p:ingsimple} $d_u(G)\leq d_u(G_{s.c.})$ we assume without loss of generality that $G$ is of simply connected type. \\
Unless otherwise stated, we use the notation introduced at the end of \S\ref{s:intro}. Recall, we let $h$ be the Coxeter number of $G$ and write:\\
$u=qv$ where $q$ and $v$ are coprime positive integers such that $q$ is a power of $p$ and $p$ does not divide $v$,\\
$h=zu+e$ where $z,e$ are nonnegative integers such that $0\leq e \leq q-1$,\\
$h=\alpha v+\beta$ where $\alpha,\beta$ are nonnegative integers such that $0\leq \alpha \leq v-1$,\\
$\alpha=\gamma q+\delta$ where $\gamma,\delta$ are nonnegative integers such that $0\leq \delta \leq q-1$.\\
An easy check yields $\gamma=z$. Also if $e=0$ then $\beta=\delta=0$. Furthermore, if $\beta =0$ then $\delta=0$ if and only if $e=0$.\\ Finally for an nonnegative integer $r$, we let $\epsilon_r \in \{0,1\}$ be $0$ if $r$ is even, otherwise $\epsilon_r=1$, and set $\sigma_r\in\{0,1\}$ to be 1 if $r=0$, otherwise $\sigma_r=0$.\\
Moreover, we also let $\omega$ be a $v$-th root of 1 in $K$ and consider the following diagonal blocks $M_{1}$, $M_{2}$, $M_3$, $M_4$, $M_5$, ${M_6}$, $M_7$, $M_8$ and $M_9$ where: $$M_{1}={\rm diag}(1, \omega, \omega^2,\dots, \omega^{v-1}) \quad \textrm{is of size} \ v,$$ $$M_{2}={\rm diag}(\omega, \omega^{-1}, \omega^2, \omega^{-2}, \dots, \omega^{\lfloor\beta/2\rfloor}, \omega^{-\lfloor \beta/2\rfloor}) \quad \textrm{is of size} \ \beta-\epsilon_{\beta},$$
$$M_{3}={\rm diag}(\omega, \omega^{-1}, \omega^2, \omega^{-2}, \dots, \omega^{\lceil\beta/2\rceil}, \omega^{-\lceil \beta/2\rceil}) \quad \textrm{is of size} \ \beta+\epsilon_{\beta},$$ $$M_{4}={\rm diag}(\omega, \omega^2,\dots, \omega^{v-1}) \quad \textrm{is of size} \ v-1.$$ and if $\beta \geq 2$ is even then $$ M_{5}={\rm diag}(\omega, \omega^{-1}, \dots, \omega^{\beta/2-1},\omega^{-(\beta/2-1)}) \quad \textrm{is of size} \ \beta-2$$ $$ M_{6}={\rm diag}(\omega^2,\omega^{-2}, \dots, \omega^{\frac{\beta}2},\omega^{-\frac{\beta}2}) \quad \textrm{is of size} \ \beta -2,$$ and if $v \geq 4$ is even then $$ M_{7}={\rm diag}(\omega^2,\omega^{-2}, \dots, \omega^{\frac{v}2-1},\omega^{-(\frac{v}2-1)}) \quad \textrm{is of size} \ v -4,$$ $$ M_{8}={\rm diag}(\omega,\omega^{-1},\omega^2,\omega^{-2}, \dots, \omega^{\frac{v}2-3},\omega^{-(\frac{v}2-3)},\omega^{\frac{v}2-1},\omega^{-(\frac{v}2-1)}) \quad \textrm{is of size} \ v -4,$$ $$ M_9 ={\rm diag}(\omega,\omega^{-1},\omega^2, \omega^{-2}, \dots, \omega^{\frac v2-2}, \omega^{-(\frac v2-2)}) \quad \textrm{is of size} \ v-4.$$ We also write $(1)$ for $I_1={\rm diag}(1)$ and $(-1)$ for $-I_1={\rm diag}(-1)$. More generally given a positive integer $i$, we write $(\omega^i)$ for ${\rm diag}(\omega^i)$ Finally, given some nonnegative integers $r_1,\dots, r_{11}$ and some nonnegative integers $s_1,\dots, s_{v-1}$we denote by $$ M_{1}^{r_1}\oplus M_2^{r_2}\oplus M_{3}^{r_3}\oplus M_{4}^{r_4}\oplus M_{5}^{r_5}\oplus M_{6}^{r_6}\oplus M_{7}^{r_7}\oplus M_{8}^{r_8}\oplus M_{9}^{r_9}\oplus (1)^{r_{10}}\oplus (-1)^{r_{11}}\oplus_{j=1}^{v-1} (\omega^j)^{s_j}$$ the diagonal matrix consisting of $r_i$ blocks $M_{i}$ for $1\leq i\leq 9$, $I_{r_{10}}$, $-I_{r_{11}}$ and $\omega^jI_{s_j}$ for $1\leq j \leq v-1$.
\subsection{$G$ is of type $A_\ell$}
Let $G=(A_\ell)_{s.c.}$ with $\ell \geq 1$ defined over an algebraically closed field $K$ of characteristic $p$. Here $|\Phi|=\ell(\ell+1)$ and $h=\ell+1$.
\begin{lem}\label{l:asc} Let $G=(A_{\ell})_{s.c.}$ be defined over an algebraically closed field of characteristic $p$. Let $u=qv$ be a positive integer where $q$ and $v$ are coprime positive integers such that $q$ is a power of $p$ and $p$ does not divide $v$. Write $h=zu+e=\alpha v +\beta$ and $\alpha=zq+\delta$ where $z$, $e$, $\alpha$, $\beta$, $\delta$ are nonnegative integers such that $e<u$, $\beta<v$, and $\delta<q$. Let $y$ be the element of $G_{s.c.}$ of order $v$ defined in Table \ref{t:asc} (see \S\ref{s:tables}). Then $C_{G}(y)^0$ and $d_q(C_{G}(y)^0)$ are given in Table \ref{t:asc} and $d_q(C_{G}(y)^0)$ is an upper bound for $d_u(G)$. \end{lem}
\begin{proof} We first determine $C_G(y)^0$.
Suppose that $\epsilon_v=1$ or $(\epsilon_v,\epsilon_\alpha)=(0,0)$. As any nontrivial eigenvalue of $y$ can be paired with its inverse, $y$ is an element of $G$ of order $v$. Also $y$ has $\alpha+\epsilon_\beta$ eigenvalues equal to $1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq \lfloor \beta/2\rfloor$, and every other eigenvalue of $y$ occurs with multiplicity $\alpha$. Therefore $C_G(y)^0=A_{\alpha}^\beta A_{\alpha-1}^{v-\beta}T_{v-1}$. \\
Suppose that $(\epsilon_{v},\epsilon_{\alpha}, \epsilon_{\beta})=(0,1,1)$. As any nontrivial eigenvalue of $y$ can be paired with its inverse, $y$ is an element of $G$ of order $v$. Also $y$ has $\alpha+1$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq (\beta-1)/2$, and every other eigenvalue of $y$ occurs with multiplicity $\alpha$. Therefore $C_G(y)^0=A_{\alpha}^\beta A_{\alpha-1}^{v-\beta}T_{v-1}$. \\
Suppose that $(\epsilon_{v},\epsilon_{\alpha}, \epsilon_{\beta})=(0,1,0)$ and $\beta \geq 2$. As any eigenvalue of $y$ can be paired with its inverse, $y$ is an element of $G$ of order $v$. Also $y$ has $\alpha+1$ eigenvalues equal to $1$, $\alpha+1$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1\leq |i| \leq (\beta-2)/2$, and every other eigenvalue of $y$ occurs with multiplicity $\alpha$. Therefore $C_G(y)^0=A_{\alpha}^\beta A_{\alpha-1}^{v-\beta}T_{v-1}$. \\ Suppose finally that $(\epsilon_v,\epsilon_\alpha)=(0,1)$ and $\beta=0$. As any eigenvalue of $y$ can be paired with its inverse, $y$ is an element of $G$ of order $v$. Also $y$ has $\alpha-1$ eigenvalues equal to $1$, $\alpha+1$ eigenvalues equal to $-1$, and every other eigenvalue of $y$ occurs with multiplicity $\alpha$. Therefore $C_G(y)^0=A_{\alpha} A_{\alpha-1}^{v-2}A_{\alpha-2}T_{v-1}$. \\
Note that by Proposition \ref{p:prelimdu}(i), $d_u(G)\leq d_q(C_G(y)^0)$ and so $d_q(C_G(y)^0)$ is an upper bound for $d_u(G)$. It remains to calculate $d_q(C_G(y)^0)$.\\
By Lemma \ref{l:lawther}, \begin{equation}\label{e:dqa}d_q(A_{\alpha}) = z^2q +(2z+1)(\alpha-zq+1)-1,\end{equation}
\begin{equation}\label{e:dqam1}d_q(A_{\alpha-1})=z^2q+(2z+1)(\alpha-zq)-1,\end{equation}
and \begin{equation}\label{e:dqam2}d_q(A_{\alpha-2})=\left \{\begin{array}{ll} z^2q+(2z+1)(\alpha-1-zq)-1 & \textrm{if}\ \delta>0\\
(z-1)^2q+(2z-1)(\alpha-1 -(z-1)q)-1 & \textrm{if} \ \delta=0,
\end{array}\right.\end{equation} where for determining $d_q(A_{\alpha-2})$ we do the Euclidean division of $\alpha-1$ by $q$: $$\alpha-1=\left\{\begin{array}{ll} zq+(\delta-1) & \textrm{if} \ \delta>0\\ (z-1)q+q-1 & \textrm{if} \ \delta=0.\end{array}\right.$$
Assume first that $(\epsilon_v,\epsilon_\alpha)\neq (0,1)$ or $\beta>0$. We then have $$C_G(y)^0=A_{\alpha}^\beta A_{\alpha-1}^{v-\beta}T_{v-1}.$$ By Proposition \ref{p:prelimdu}(iii), (\ref{e:dqa}) and (\ref{e:dqam1}) \begin{eqnarray*} d_q(C_G(y)^0) & = & (v-1)+d_q(A_{\alpha}^\beta A_{\alpha-1}^{v-\beta})\\ & = & (v-1)+\beta d_q(A_\alpha) +(v-\beta)d_q(A_{\alpha-1})\\ & = & 2z(\alpha v+\beta)-qvz(z+1)+\alpha v+\beta-1\\ & = & 2z(zu+e)-zu(z+1)+zu+e-1\\ & = & z^2u+e(2z+1)-1. \end{eqnarray*}
Assume now that $(\epsilon_v, \epsilon_\alpha)=(0,1)$ and $\beta=0$. Since $\beta=0$, recall that $e=0$ if and only if $\delta=0$. We then have $$C_G(y)^0= A_\alpha A_{\alpha-1}^{v-2} A_{\alpha-2}T_{v-1}.$$ By Proposition \ref{p:prelimdu}(iii), (\ref{e:dqa}), (\ref{e:dqam1}) and (\ref{e:dqam2}) \begin{eqnarray*} d_q(C_G(y)^0) & = & (v-1)+d_q(A_\alpha A_{\alpha-1}^{v-2} A_{\alpha-2})\\ & = & (v-1)+d_q(A_\alpha)+(v-2) d_q(A_{\alpha-1}) +d_q(A_{\alpha-2})\\ & = & \left\{ \begin{array}{ll} 2z\alpha v-qvz(z+1)+\alpha v-1 & \textrm{if} \ \delta>0\\ 2z\alpha v-qvz(z+1)+\alpha v+1& \textrm{if} \ \delta=0
\end{array}\right.\\ & = &\left\{ \begin{array}{ll} 2z(zu+e)-zu(z+1)+zu+e-1 & \textrm{if} \ \delta>0\\ 2z(zu+e)-zu(z+1)+zu+e+1 & \textrm{if} \ \delta=0 \end{array}\right.\\ & = & \left\{ \begin{array}{ll} z^2u+e(2z+1)-1 & \textrm{if} \ e>0\\ z^2u+e(2z+1)+1& \textrm{if} \ e=0. \end{array}\right.\\ \end{eqnarray*} \end{proof}
\subsection{$G$ is of type $C_\ell$}
Let $G=(C_\ell)_{{\rm s.c}}$ be a simply connected group of type ${C}_{\ell}$ with $\ell \geq 2$ defined over an algebraically closed field $K$ of characteristic $p$. Here $|\Phi|=2\ell^2$ and $h=2\ell$.
\begin{lem}\label{l:csc} Let $G=(C_{\ell})_{{\rm s.c}}$ be defined over an algebraically closed field of characteristic $p$. Let $u=qv$ be a positive integer where $q$ and $v$ are positive integers such that $q$ is a power of $p$. Write $h=zu+e=\alpha v +\beta$ and $\alpha=zq+\delta$ where $z$, $e$, $\alpha$, $\beta$, $\delta$ are nonnegative integers such that $e<u$, $\beta<v$, and $\delta<q$. Let $y$ be the element of $G$ of order $v$ defined in Table \ref{t:csc} (see \S\ref{s:tables}). Then $C_G(y)^0$ and $d_q(C_G(y)^0)$ are given in Table \ref{t:csc} and $d_q(C_G(y)^0)$ is an upper bound for $d_u(G)$. \end{lem}
\begin{proof} Since $h=2\ell$, note that if $\epsilon_v=1$ then $\epsilon_\alpha=\epsilon_\beta$ and $\epsilon_u=\epsilon_q$, whereas if $\epsilon_v=0$ then $\epsilon_u=0$, $\epsilon_q=1$ and $\epsilon_\beta=0$. We first determine $C_G(y)^0$.
Suppose that $\epsilon_v=1$. As any eigenvalue of $y$ can be paired with its inverse, $y$ is an element of $G$ of order $v$. Also $y$ has $\alpha+\epsilon_\beta$ eigenvalues equal to $1$, $0$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq \lfloor \beta/2\rfloor$, and every other eigenvalue of $y$ occurs with multiplicity $\alpha$. Therefore $C_G(y)^0=A_\alpha^{\lfloor\frac\beta2\rfloor} A_{\alpha-1}^{\frac{v-1}2-\lfloor\frac\beta2\rfloor}C_{\lceil \frac \alpha2\rceil}T_{\frac{v-1}2}$. \\
Suppose that $(\epsilon_{v},\epsilon_{\alpha})=(0,0)$. As any eigenvalue of $y$ can be paired with its inverse, $y$ is an element of $G$ of order $v$. Also $y$ has $\alpha$ eigenvalues equal to $1$, $\alpha$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq \beta/2$, and every other eigenvalue of $y$ occurs with multiplicity $\alpha$. Therefore $C_G(y)^0=A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}C_{\frac \alpha2}^2T_{\frac{v}2-1}$. \\
Suppose that $(\epsilon_{v},\epsilon_{\alpha})=(0,1)$ and $\beta \geq 2$. As any eigenvalue of $y$ can be paired with its inverse, $y$ is an element of $G$ of order $v$. Also $y$ has $\alpha+1$ eigenvalues equal to $1$, $\alpha+1$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq (\beta-2)/2$, and every other eigenvalue of $y$ occurs with multiplicity $\alpha$. Therefore $C_G(y)^0=A_\alpha^{\frac\beta2-1} A_{\alpha-1}^{\frac{v}2-\frac\beta2}C_{\frac{\alpha+1}2}^2T_{\frac{v}2-1}$. \\ Suppose finally that $(\epsilon_v,\epsilon_\alpha)=(0,1)$ and $\beta=0$. As any eigenvalue of $y$ can be paired with its inverse, $y$ is an element of $G$ of order $v$. Also $y$ has $\alpha-1$ eigenvalues equal to $1$, $\alpha+1$ eigenvalues equal to $-1$, and every other eigenvalue of $y$ occurs with multiplicity $\alpha$. Therefore $C_G(y)^0=A_{\alpha-1}^{\frac{v}2-1}C_{\frac{\alpha+1}2}C_{\frac{\alpha-1}2}T_{\frac{v}2-1}$. \\
By Proposition \ref{p:prelimdu}(i), $d_u(G)\leq d_q(C_G(y)^0)$ and so $d_q(C_G(y)^0)$ is an upper bound for $d_u(G)$. It remains to calculate $d_q(C_G(y)^0)$.\\
Note that by Lemma \ref{l:lawther}, $$d_q(A_{\alpha}) = z^2q +(2z+1)(\alpha-zq+1)-1,$$ $$d_q(A_{\alpha-1})=z^2q+(2z+1)(\alpha-zq)-1,$$ $$d_q\left(C_{\lceil\frac{\alpha}2\rceil}\right)=\frac{z^2q}{2}+(2z+1)\left(\left\lceil\frac{\alpha}{2}\right\rceil-\frac{zq}2\right)+\left\lceil \frac z2\right\rceil\epsilon_q$$ and if $\alpha$ is odd then $$d_q\left(C_{\frac{\alpha-1}2}\right)=\frac{z^2q}{2}+(2z+1)\left(\frac{\alpha-1}{2}-\frac{zq}{2}\right)+\left\lceil \frac{z}{2}\right\rceil\epsilon_q.$$
Assume first that $\epsilon_v=1$. Recall that $\epsilon_u=\epsilon_q$ and $\epsilon_\alpha=\epsilon_\beta$. We have $$C_G(y)^0=A_\alpha^{\lfloor\frac\beta2\rfloor} A_{\alpha-1}^{\frac{v-1}2-\lfloor\frac\beta2\rfloor}C_{\lceil \frac \alpha2\rceil}T_{\frac{v-1}2}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_G(y)^0) & = & \frac{v-1}2+d_q\left(A_\alpha^{\lfloor\frac\beta2\rfloor} A_{\alpha-1}^{\frac{v-1}2-\lfloor\frac\beta2\rfloor}C_{\lceil \frac \alpha2\rceil}\right)\\ & = & \frac{v-1}2+\left\lfloor \frac \beta2 \right \rfloor d_q(A_\alpha)+\left(\frac{v-1}2-\left \lfloor \frac \beta2 \right \rfloor\right)d_q(A_{\alpha-1})+d_q\left(C_{\lceil \frac \alpha2\rceil}\right)\\ & = &z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2+\left\lceil\frac{z}2\right\rceil\epsilon_q\\ & = & z(zu+e)-\frac{zu(z+1)}2+\frac{zu+e}2+\left\lceil\frac{z}2\right\rceil\epsilon_q\\ & = & \frac12(z^2u+e(2z+1))+\left\lceil\frac{z}2\right\rceil\epsilon_u. \end{eqnarray*}
Assume that $(\epsilon_v,\epsilon_\alpha)=(0,0)$. Recall that $\epsilon_q=1$ and $\epsilon_\beta=0$. We have $$C_G(y)^0=A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}C_{\frac \alpha2}^2T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_G(y)^0) & = & \frac{v}2-1+d_q\left(A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}C_{\frac \alpha2}^2\right) \\ & = & \frac{v}2-1+ \frac \beta2 d_q(A_\alpha)+\left(\frac{v}2-1- \frac \beta2 \right)d_q(A_{\alpha-1})+2d_q\left(C_{ \frac \alpha2}\right)\\ & = &z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2+2\left\lceil\frac{z}2\right\rceil\epsilon_q\\ & = & z(zu+e)-\frac{zu(z+1)}2+\frac{zu+e}2+2\left\lceil\frac{z}2\right\rceil\epsilon_q\\ & = & \frac12(z^2u+e(2z+1))+2\left\lceil\frac{z}2\right\rceil. \end{eqnarray*}
Assume that $(\epsilon_v,\epsilon_\alpha)=(0,1)$ and $\beta\geq 2$. Recall that $\epsilon_q=1$ and $\epsilon_\beta=0$. We have $$C_G(y)^0=A_\alpha^{\frac\beta2-1} A_{\alpha-1}^{\frac{v}2-\frac\beta2}C_{\frac{\alpha+1}2}^2T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_G(y)^0) & = & \frac{v}2-1+d_q\left(A_\alpha^{\frac\beta2-1} A_{\alpha-1}^{\frac{v}2-\frac\beta2}C_{\frac{\alpha+1}2}^2\right) \\ & = & \frac{v}2-1+\left( \frac \beta2-1\right) d_q(A_\alpha)+\left(\frac{v}2- \frac \beta2 \right)d_q(A_{\alpha-1})+2d_q\left(C_{ \frac {\alpha+1}2}\right)\\ & = &z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2+2\left\lceil\frac{z}2\right\rceil\epsilon_q\\ & = & z(zu+e)-\frac{zu(z+1)}2+\frac{zu+e}2+2\left\lceil\frac{z}2\right\rceil\epsilon_q\\ & = & \frac12(z^2u+e(2z+1))+2\left\lceil\frac{z}2\right\rceil. \end{eqnarray*}
Assume finally that $(\epsilon_v,\epsilon_\alpha)=(0,1)$ and $\beta=0$. Recall that $\epsilon_q=1$. We have $$C_G(y)^0=A_{\alpha-1}^{\frac{v}2-1}C_{\frac{\alpha+1}2}C_{\frac{\alpha-1}2}T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_G(y)^0) & = & \frac{v}2-1+d_q\left(A_{\alpha-1}^{\frac v2-1} C_{\frac{\alpha+1}2}C_{\frac{\alpha-1}2}\right) \\ & = & \frac{v}2-1+ \left(\frac v2-1\right) d_q(A_{\alpha-1})+d_q\left(C_{ \frac {\alpha+1}2}\right)+d_q\left(C_{ \frac {\alpha-1}2}\right)\\ & = &z(\alpha v)-\frac{qvz(z+1)}2+\frac{\alpha v}2+2\left\lceil\frac{z}2\right\rceil\epsilon_q\\ & = & z(zu+e)-\frac{zu(z+1)}2+\frac{zu+e}2+2\left\lceil\frac{z}2\right\rceil\epsilon_q\\ & = & \frac12(z^2u+e(2z+1))+2\left\lceil\frac{z}2\right\rceil. \end{eqnarray*} \end{proof}
\subsection{$G$ is of type $B_\ell$}
Let $G=(B_\ell)_{{\rm s.c}}$ be a simply connected group of type ${B}_{\ell}$ with $\ell \geq 3$ defined over an algebraically closed field $K$ of characteristic $p$. Here $|\Phi|=2\ell^2$, $h=2\ell$ and $G={\rm Spin}_{h+1}(K)$ if $p\neq 2$, otherwise $G={\rm SO}_{h+1}(K)$. Given an element $g$ in $G$, we let $\overline{g}$ be the image of $g$ under the canonical surjective map $G\rightarrow {\rm SO}_{h+1}(K)$. \\
\begin{lem}\label{l:bsc} Let $G=(B_{\ell})_{{\rm s.c}}$ where $\ell \geq 3$ be defined over an algebraically closed field $K$ of characteristic $p$. Let $u=qv$ be a positive integer where $q$ and $v$ are positive integers such that $q$ is a power of $p$ and $p$ does not divide $v$. Write $h=zu+e=\alpha v +\beta$ and $\alpha=zq+\delta$ where $z$, $e$, $\alpha$, $\beta$, $\delta$ are nonnegative integers such that $e<u$, $\beta<v$, and $\delta<q$. Consider the following conditions:\\ \begin{enumerate}[(a)] \item $\epsilon_v=1$. \item $v=2$. \item $(\epsilon_v,\epsilon_\alpha)=(0,1)$, $v>2$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) \in \{(0,0,0), (0,2,3), (2,0,3), (2,2,2)\}.$ \item $(\epsilon_v,\epsilon_{\alpha})=(0,1)$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) \in \{(0,2,1), (2,2,0)\}.$ \item $(\epsilon_v,\epsilon_{\alpha})=(0,1)$, $\beta >0$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) \in \{(0,0,2), (2,0,1)\}.$ \item $(\epsilon_v,\epsilon_{\alpha})=(0,1)$, $v>2$, $\beta=0$ and $(v \mod 4,\ell \mod 4) \in \{(0,2), (2,1)\}.$ \item $(\epsilon_v,\epsilon_\alpha)=(0,0)$, $v>2$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) \in \{(0,0,0), (0,2,3), (2,0,0), (2,2,3)\}.$ \item $(\epsilon_v,\epsilon_{\alpha})=(0,0)$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) =(2,2,1)$. \item $(\epsilon_v,\epsilon_{\alpha})=(0,0)$, $\beta>0$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) \in \{(0,2,1),(2,0,2)\}.$ \item $(\epsilon_v,\epsilon_{\alpha})=(0,0)$, $\beta>0$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) \in (0,0,2)$. \item $(\epsilon_v,\epsilon_{\alpha})=(0,0)$, $v>2$, $\beta=0$ and $(v \mod 4,\ell \mod 4)= (2,2)$. \end{enumerate} Let $\overline{y}$ be the element of ${\rm SO}_{h+1}(K)$ of order $v$ defined in Table \ref{t:bsc} (see \S\ref{s:tables}). Then $\overline{y}$ has a preimage $y$ in $G$ of order $v$, and $C_{{\rm SO}_{h+1}}(\overline{y})^0$ and $d_q(C_G(y)^0)=d_q(C_{{\rm SO}_{h+1}(K)}(\overline{y})^0)$ are given in Table \ref{t:bsc}. Furthermore $d_q(C_G(y)^0)$ is an upper bound for $d_u(G)$. \end{lem}
\begin{proof} Note that by Lemma \ref{l:dufactg}(iii), $d_q(C_G(y)^0)=d_q(C_{{\rm SO}_{h+1}(K)}(\overline{y})^0)$. Since $h=2\ell$, note that if $\epsilon_v=1$ then $\epsilon_\alpha=\epsilon_\beta$ and $\epsilon_u=\epsilon_q$, whereas if $\epsilon_v=0$ then $\epsilon_u=0$, $\epsilon_q=1$ and $\epsilon_\beta=0$. Note also that if $\ell \equiv 2 \mod 4$, $v \equiv 0\mod 4$ and $\beta=0$ then $\alpha$ is odd, as otherwise on one hand we would have $2\ell \equiv 4 \mod 8$ and on the other $2\ell = \alpha v \equiv 0 \mod 8$, a contradiction. We first determine $C_{{\rm SO}_{h+1}(K)}(\overline{y})^0$.
Suppose that case (a) holds so that $v$ is odd. As any eigenvalue of $\overline{y}$ different from 1 can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+1}(K)$ of order $v$. Furthermore, since $v$ is odd, by Lemma \ref{l:soliftspin} it follows that $\overline{y}$ has a preimage $y$ in $G$ of order $v$. Also $\overline{y}$ has $\alpha+1+\epsilon_\beta$ eigenvalues equal to $1$, $0$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq \lfloor \beta/2\rfloor$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore $C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_\alpha^{\lfloor\frac\beta2\rfloor} A_{\alpha-1}^{\frac{v-1}2-\lfloor\frac\beta2\rfloor}B_{\lceil \frac {\alpha}2\rceil}T_{\frac{v-1}2}$. \\ Suppose that case (b) holds so that $v=2$. As any eigenvalue of $\overline{y}$ different from 1 can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+1}(K)$ of order $v$. Now the number $N$ of eigenvalues $\omega^i=(-1)^i$ of $\overline{y}$ with $i$ odd is equal to $\ell$, $\ell-1$, $\ell+2$ or $\ell +1$ according respectively as $\ell \mod 4$ is 0, 1, 2 or 3. Since $N$ is divisible by 4, it follows from Lemma $\ref{l:soliftspin}$ that $\overline{y}$ has a preimage $y$ in $G$ of order $v$. Also $$ C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=\left\{\begin{array}{ll}B_{\frac{\ell}2}D_{\frac{\ell}2} & \textrm{if} \ \ell \equiv 0 \ (4)\\
B_{\frac{\ell+1}2}D_{\frac{\ell-1}2} & \textrm{if} \ \ell \equiv 1 \ (4)\\ B_{\frac{\ell-2}2}D_{\frac{\ell+2}2} & \textrm{if} \ \ell \equiv 2 \ (4)\\ B_{\frac{\ell-1}2}D_{\frac{\ell+1}2} & \textrm{if} \ \ell \equiv 3 \ (4).
\end{array}\right.$$
Suppose that case (c) holds. As any eigenvalue of $\overline{y}$ different from 1 can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+1}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
\begin{eqnarray*}
N& = &\left\{ \begin{array}{ll} \alpha \cdot \frac{v}{2}+\frac{\beta}2& \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,0)\\
\alpha \cdot \frac{v}{2}+\left(\frac{\beta}2+1\right) & \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,2)\\
\alpha \cdot\left( \frac{v}{2}-1\right)+(\alpha+1)+\frac{\beta}2 & \textrm{if} \ (v \mod 4, \beta \mod 4)=(2,0)\\
\alpha \cdot \left( \frac{v}{2}-1\right)+(\alpha+1)+\left(\frac{\beta}2+1\right) & \textrm{if} \ (v \mod 4, \beta \mod 4)=(2,2)\\
\end{array}\right. \\ &=& \left\{ \begin{array}{ll} \ell & \textrm{if} \ (v \mod 4, \beta \mod 4) =(0,0)\\
\ell+1 & \textrm{if} \ (v \mod 4, \beta \mod 4)\in\{(0,2),(2,0)\}\\
\ell+2 & \textrm{if} \ (v \mod 4, \beta \mod 4) = (2,2).\\
\end{array}\right.
\end{eqnarray*}
From the assumptions of case (c), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha$ eigenvalues equal to $1$, $\alpha+1$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq \beta/2$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore $C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}B_{\frac {\alpha-1}2}D_{\frac{\alpha+1}2}T_{\frac{v}2-1}$. \\
Suppose that case (d) or (e) holds. As any eigenvalue of $\overline{y}$ different from 1 can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+1}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
\begin{eqnarray*}
N& = &\left\{ \begin{array}{ll} \alpha \cdot \frac{v}{2}+\left(\frac{\beta}2-2\right)& \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,0)\\
\alpha \cdot \frac{v}{2}+\left(\frac{\beta}2-1\right) & \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,2)\\
\alpha \cdot\left( \frac{v}{2}-1\right)+(\alpha+1)+\left(\frac{\beta}2-2\right) & \textrm{if} \ (v \mod 4, \beta \mod 4)=(2,0)\\
\alpha \cdot \left( \frac{v}{2}-1\right)+(\alpha+1)+\left(\frac{\beta}2-1\right) & \textrm{if} \ (v \mod 4, \beta \mod 4)=(2,2)\\
\end{array}\right. \\ &=& \left\{ \begin{array}{ll} \ell-2 & \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,0)\\
\ell-1 & \textrm{if} \ (v \mod 4, \beta \mod 4)\in \{(0,2),(2,0)\}\\
\ell & \textrm{if} \ (v \mod 4, \beta \mod 4)=(2,2).\\
\end{array}\right. \\
\end{eqnarray*}
From the assumptions of case (d) or (e), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha+2$ eigenvalues equal to $1$, $\alpha+1$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where if case (d) holds then $i$ is any integer with $1 \leq |i| \leq (\beta/2-1)$ and if case $(e)$ holds then $i$ any integer with $2 \leq |i| \leq \beta/2$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore $C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_\alpha^{\frac\beta2-1} A_{\alpha-1}^{\frac{v}2-\frac\beta2}B_{\frac {\alpha+1}2}D_{\frac{\alpha+1}2}T_{\frac{v}2-1}$. \\
Suppose that case (f) holds. As any eigenvalue of $\overline{y}$ different from 1 can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+1}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
\begin{eqnarray*}
N& = &\left\{ \begin{array}{ll} (\alpha-1) \cdot \frac{v}{2}+\left(\frac{v}2-2\right)& \textrm{if} \ v \equiv 0 \mod 4\\
(\alpha-1) \cdot \left(\frac{v}{2}-1\right)+((\alpha-1)+2)+\left(\left(\frac{v}2-1\right)-2\right) & \textrm{if} \ v \equiv 2\mod 4\\
\end{array}\right. \\ &=& \left\{ \begin{array}{ll} \ell-2 & \textrm{if} \ v \equiv 0 \mod 4\\
\ell-1& \textrm{if} \ v \equiv 2\mod 4. \end{array}\right.
\end{eqnarray*}
From the assumptions of case (f), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha+2$ eigenvalues equal to $1$, $\alpha+1$ eigenvalues equal to $-1$, $\alpha$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $2 \leq |i| \leq v/2-1$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha-1$. Therefore $C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_{\alpha-1}^{\frac v2-2} A_{\alpha-2}B_{\frac {\alpha+1}2}D_{\frac{\alpha+1}2}T_{\frac{v}2-1}$. \\ Suppose that case (g) holds. As any eigenvalue of $\overline{y}$ different from 1 can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+1}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
\begin{eqnarray*}
N& = &\left\{ \begin{array}{ll} \alpha \cdot \frac{v}{2}+\frac{\beta}2& \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,0)\\
\alpha \cdot \frac{v}{2}+\left(\frac{\beta}2+1\right) & \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,2)\\
\alpha \cdot\left( \frac{v}{2}-1\right)+\alpha+\frac{\beta}2 & \textrm{if} \ (v \mod 4, \beta \mod 4)=(2,0)\\
\alpha \cdot \left( \frac{v}{2}-1\right)+\alpha+\left(\frac{\beta}2+1\right) & \textrm{if} \ (v \mod 4, \beta \mod 4)=(2,2)\\
\end{array}\right. \\ &=& \left\{ \begin{array}{ll} \ell & \textrm{if} \ (v \mod 4, \beta \mod 4)\in\{(0,0),(2,0)\}\\
\ell+1 & \textrm{if} \ (v \mod 4, \beta \mod 4)\in\{(0,2),(2,2)\}.\\
\end{array}\right. \\
\end{eqnarray*}
From the assumptions of case (g), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha+1$ eigenvalues equal to $1$, $\alpha$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq \beta/2$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore $C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}B_{\frac {\alpha}2}D_{\frac{\alpha}2}T_{\frac{v}2-1}$. \\
Suppose that case (h) holds. As any eigenvalue of $\overline{y}$ different from 1 can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+1}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
$$ N= \alpha\cdot\left( \frac v2-1\right) + \left( \frac \beta2 +1-2\right)+\alpha= \frac{2(\ell-1)}2.$$
From the assumptions of case (h), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha+1$ eigenvalues equal to $1$, $\alpha$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $2 \leq |i| \leq \beta/2$, $(\alpha+1)$ eigenvalues equal to $\omega^{\frac{v}2-1}$, $(\alpha+1)$ eigenvalues equal to $\omega^{-\left(\frac{v}2-1\right)}$ and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore $C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_{\alpha}^{\frac{\beta}2}A_{\alpha-1}^{\frac v2-\frac\beta2-1}B_{\frac {\alpha}2}D_{\frac{\alpha}2}T_{\frac{v}2-1}$. \\
Suppose that case (i) or (j) holds. As any eigenvalue of $\overline{y}$ different from 1 can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+1}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
\begin{eqnarray*}
N& = &\left\{ \begin{array}{ll} \alpha \cdot \frac{v}{2}+\left(\frac{\beta}2-2\right)& \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,0)\\
\alpha \cdot \frac{v}{2}+\left(\frac{\beta}2-1\right) & \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,2)\\
\alpha \cdot\left( \frac{v}{2}-1\right)+(\alpha+2)+\frac{\beta}2 & \textrm{if} \ (v \mod 4, \beta \mod 4)=(2,0)\\
\end{array}\right. \\ &=& \left\{ \begin{array}{ll} \ell-2 & \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,0)\\
\ell-1 & \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,2)\\
\ell+2& \textrm{if} \ (v \mod 4, \beta \mod 4)=(2,0).\\
\end{array}\right. \\
\end{eqnarray*}
From the assumptions of case (i) or (j), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha+1$ eigenvalues equal to $1$, $\alpha+2$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where if case (i) holds then $i$ is any integer with $1 \leq |i| \leq (\beta/2-1)$ and if case $(j)$ holds then $i$ any integer with $2 \leq |i| \leq \beta/2$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore $C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_\alpha^{\frac\beta2-1} A_{\alpha-1}^{\frac{v}2-\frac\beta2}B_{\frac {\alpha}2}D_{\frac{\alpha+2}2}T_{\frac{v}2-1}$. \\
Suppose that case (k) holds. As any eigenvalue of $\overline{y}$ different from 1 can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+1}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
$$ N= (\alpha-1)\cdot \left(\frac v2-1\right) + \left( \frac v2 -1\right)+(\alpha+2)= \frac{2(\ell +2)}2.$$
From the assumption of case (k), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha+1$ eigenvalues equal to $1$, $\alpha+2$ eigenvalues equal to $-1$, $\alpha$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq v/2-2$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha-1$. Therefore $C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_{\alpha-1}^{\frac v2-2} A_{\alpha-2}B_{\frac {\alpha}2}D_{\frac{\alpha+2}2}T_{\frac{v}2-1}$. \\
By Proposition \ref{p:prelimdu}(i), $d_u(G)\leq d_q(C_G(y)^0)$ and so $d_q(C_G(y)^0)$ is an upper bound for $d_u(G)$. It remains to calculate $d_q(C_{{\rm SO}_{h+1}}(\overline{y})^0)=d_q(C_G(y)^0)$.\\
It follows from Lemma \ref{l:lawther} that $$d_q(A_{\alpha+1})=\left\{\begin{array}{ll} z^2q+(2z+1)(\alpha+2-zq)-1 & \textrm{if} \ \delta < q-1\\ (z+1)^2q+(2z+3)(\alpha+2-(z+1)q)-1 & \textrm{if} \ \delta=q-1,
\end{array}\right.$$
$$d_q(A_{\alpha}) = z^2q +(2z+1)(\alpha-zq+1)-1,$$ $$d_q(A_{\alpha-1})=z^2q+(2z+1)(\alpha-zq)-1,$$ $$d_q(A_{\alpha-2})=\left\{\begin{array}{ll} z^2q+(2z+1)(\alpha-1-zq)-1 & \textrm{if} \ \delta >0\\ (z-1)^2q+(2z-1)(\alpha-1-(z-1)q)-1 & \textrm{if} \ \delta=0,
\end{array}\right.$$ $$d_q\left(B_{\lceil\frac{\alpha}2\rceil}\right)=\frac{z^2q}{2}+(2z+1)\left(\left\lceil\frac{\alpha}{2}\right\rceil-\frac{zq}2\right)+\left\lceil \frac z2\right\rceil\epsilon_q,$$ $$d_q\left(D_{\lceil\frac{\alpha+1}2\rceil}\right)= \frac{z^2q}{2}+(2z+1)\left(\left\lceil\frac{\alpha+1}{2}\right\rceil-\frac{zq}2\right)+\left\lceil \frac z2\right\rceil\epsilon_q-z-\epsilon_z,$$ if $\alpha$ is odd then $$d_q\left(B_{\frac{\alpha-1}2}\right)=\frac{z^2q}{2}+(2z+1)\left(\frac{\alpha-1}{2}-\frac{zq}{2}\right)+\left\lceil \frac{z}{2}\right\rceil\epsilon_q,$$ $$d_q\left(D_{\frac{\alpha-1}2}\right)= \left\{\begin{array}{ll} \frac{z^2q}{2}+(2z+1)\left(\frac{\alpha-1}{2}-\frac{zq}2\right)+\left\lceil \frac z2\right\rceil\epsilon_q - z - \epsilon_z& \textrm{if} \ \delta>0\\ \frac{(z-1)^2q}{2}+(2z-1)\left(\frac{\alpha-1}{2}-\frac{(z-1)q}2\right)+\frac {z-1}2\epsilon_q-(z-1)& \textrm{if} \ \delta=0, \end{array}\right.$$ and if $\alpha$ is even then $$d_q\left(B_{\frac{\alpha-2}2}\right)= \left\{\begin{array}{ll} \frac{z^2q}{2}+(2z+1)\left(\frac{\alpha-2}{2}-\frac{zq}2\right)+\left\lceil \frac z2\right\rceil\epsilon_q & \textrm{if} \ \delta>0\\ \frac{(z-1)^2q}{2}+(2z-1)\left(\frac{\alpha-2}{2}-\frac{(z-1)q}2\right)+\left\lceil\frac{z-1}2\right\rceil \epsilon_q& \textrm{if} \ \delta=0, \end{array}\right.$$ and $$d_q\left(D_{\frac{\alpha}2}\right)=\left\{\begin{array}{ll} \frac{z^2q}{2}+(2z+1)\left(\frac{\alpha}{2}-\frac{zq}2\right)+\left\lceil \frac z2\right\rceil\epsilon_q-z-\epsilon_z & \textrm{if} \ \delta>0 \ \textrm{or} \ z \ \textrm{is even}\\ \frac{z^2q}{2}+(2z+1)\left(\frac{\alpha}{2}-\frac{zq}2\right)+\left\lceil \frac z2\right\rceil\epsilon_q-z-\epsilon_z+2 & \textrm{otherwise}. \end{array}\right. $$
Assume first that case (a) holds so that $\epsilon_v=1$. Recall that $\epsilon_u=\epsilon_q$ and $\epsilon_\alpha=\epsilon_\beta$. We have $$C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_\alpha^{\lfloor\frac\beta2\rfloor} A_{\alpha-1}^{\frac{v-1}2-\lfloor\frac\beta2\rfloor}B_{\lceil \frac \alpha2\rceil}T_{\frac{v-1}2}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_{{\rm SO}_{h+1}(K)}(\overline{y})^0) & = & \frac{v-1}2+d_q\left(A_\alpha^{\lfloor\frac\beta2\rfloor} A_{\alpha-1}^{\frac{v-1}2-\lfloor\frac\beta2\rfloor}B_{\lceil \frac \alpha2\rceil}\right)\\ & = & \frac{v-1}2+\left\lfloor \frac \beta2 \right \rfloor d_q(A_\alpha)+\left(\frac{v-1}2-\left \lfloor \frac \beta2 \right \rfloor\right)d_q(A_{\alpha-1})+d_q\left(B_{\lceil \frac \alpha2\rceil}\right)\\ & = &z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2+\left\lceil\frac{z}2\right\rceil\epsilon_q\\ & = & z(zu+e)-\frac{zu(z+1)}2+\frac{zu+e}2+\left\lceil\frac{z}2\right\rceil\epsilon_q\\ & = & \frac12(z^2u+e(2z+1))+\left\lceil\frac{z}2\right\rceil\epsilon_u. \end{eqnarray*}
Assume that case (b) holds so that $v=2$, $\epsilon_u=\epsilon_v=0$ and $\epsilon_q=1$. Recall that $\alpha=\ell$ and $\beta=0$. Note also that $e=0$ if and only if $\delta=0$. We have
$$C_{{\rm SO}_{h+1}(K)}(\overline{y})^0= \left\{ \begin{array}{ll}
B_{\frac{\alpha}2}D_{\frac{\alpha}2} & \textrm{if} \ \alpha \equiv 0 \ (4)\\
B_{\frac{\alpha+1}2}D_{\frac{\alpha-1}2} & \textrm{if} \ \alpha \equiv 1 \ (4)\\ B_{\frac{\alpha-2}2}D_{\frac{\alpha+2}2} & \textrm{if} \ \alpha \equiv 2 \ (4)\\ B_{\frac{\alpha-1}2}D_{\frac{\alpha+1}2} & \textrm{if} \ \alpha \equiv 3 \ (4). \end{array}\right. $$ If $\alpha \equiv 0 \mod 4$ then $z$ is even if $\delta=0$, and by Proposition \ref{p:prelimdu}(i) \begin{eqnarray*} d_q(C_{{\rm SO}_{h+1}(K)}(\overline{y})^0) & = & d_q(B_{\frac{\alpha}2}D_{\frac{\alpha}2})\\ & = & d_q(B_{\frac{\alpha}2})+d_q(D_{\frac{\alpha}2})\\ & = & -qz(z+1)+2\alpha z+ \alpha\\ & = & \frac{-qvz(z+1)}2+(zu+e)z+\frac{(zu+e)}2\\ & = & \frac{-zu(z+1)}2+(zu+e)z+\frac{(zu+e)}2\\ & = & \frac{z^2u+e(2z+1)}2. \end{eqnarray*} If $\alpha \equiv 1 \mod 4$ then by Proposition \ref{p:prelimdu}(iii) \begin{eqnarray*} d_q(C_{{\rm SO}_{h+1}(K)}(\overline{y})^0) & = & d_q(B_{\frac{\alpha+1}2}D_{\frac{\alpha-1}2})\\ & = & d_q(B_{\frac{\alpha+1}2})+d_q(D_{\frac{\alpha-1}2})\\ & = & \left\{\begin{array}{ll} -qz(z+1)+2\alpha z+ \alpha & \textrm{if} \ \delta>0 \\ -qz^2+2\alpha z +2& \textrm{if} \ \delta=0 \end{array}\right. \\ & = & \left\{\begin{array}{ll} \frac{z^2u+e(2z+1)}2 & \textrm{if} \ \delta>0 \\ z^2q +2& \textrm{if} \ \delta=0 \end{array}\right.\\ & = & \left\{\begin{array}{ll} \frac{z^2u+e(2z+1)}2 & \textrm{if} \ \delta>0 \\ \frac{z^2u}2 +2& \textrm{if} \ \delta=0 \end{array}\right.\\ & = & \left\{\begin{array}{ll} \frac{z^2u+e(2z+1)}2 & \textrm{if} \ e>0 \\
\frac{z^2u+e(2z+1)}2 +2 & \textrm{if} \ e=0. \end{array}\right. \end{eqnarray*} If $\alpha \equiv 2 \mod 4$ then $z$ is even if $\delta=0$, and by Proposition \ref{p:prelimdu}(iii) \begin{eqnarray*} d_q(C_{{\rm SO}_{h+1}(K)}(\overline{y})^0) & = & d_q(B_{\frac{\alpha-2}2}D_{\frac{\alpha+2}2})\\ & = & d_q(B_{\frac{\alpha-2}2})+d_q(D_{\frac{\alpha+2}2})\\ & = & \left\{\begin{array}{ll} -qz(z+1)+2\alpha z+ \alpha & \textrm{if} \ \delta>0 \\ -qz^2+2\alpha z +2& \textrm{if} \ \delta=0 \end{array}\right. \\ & = & \left\{\begin{array}{ll} \frac{z^2u+e(2z+1)}2 & \textrm{if} \ e>0 \\
\frac{z^2u+e(2z+1)}2 +2 & \textrm{if} \ e=0. \end{array}\right. \end{eqnarray*} If $\alpha\equiv 3 \mod 4$ then by Proposition \ref{p:prelimdu}(iii) \begin{eqnarray*} d_q(C_{{\rm SO}_{h+1}(K)}(\overline{y})^0) & = & d_q(B_{\frac{\alpha-1}2}D_{\frac{\alpha+1}2})\\ & = & d_q(B_{\frac{\alpha-1}2})+d_q(D_{\frac{\alpha+1}2})\\ &=& \frac{-qvz(z+1)}2+(zu+e)z+\frac{(zu+e)}2\\ & = & \frac{z^2u+e(2z+1)}2. \end{eqnarray*}
Assume that case (c) holds. Recall that $\epsilon_q=\epsilon_\alpha=1$ and $\epsilon_u=\epsilon_v=\epsilon_{\beta}=0$. We have $$C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}B_{\frac{\alpha-1}2}D_{\frac{\alpha+1}2}T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_{{\rm SO}_{h+1}(K)}(\overline{y})^0) & = & \frac{v}2-1+d_q\left(A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}B_{\frac{\alpha-1}2}D_{\frac{\alpha+1}2}\right)\\ & = & \frac{v}2-1+\frac\beta2 d_q(A_\alpha)+\left(\frac{v}2-1- \frac \beta2 \right)d_q(A_{\alpha-1})+d_q\left(B_{\frac{\alpha-1}2}\right)+d_q\left(D_{\frac{\alpha+1}2}\right)\\ & = &z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2\\ & = & z(zu+e)-\frac{zu(z+1)}2+\frac{zu+e}2\\ & = & \frac12(z^2u+e(2z+1)). \end{eqnarray*}
Assume that case (d) or case (e) holds. Recall that $\epsilon_q=\epsilon_\alpha=1$ and $\epsilon_u=\epsilon_v=\epsilon_{\beta}=0$. We have $$C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_\alpha^{\frac\beta2-1} A_{\alpha-1}^{\frac{v}2-\frac\beta2}B_{\frac{\alpha+1}2}D_{\frac{\alpha+1}2}T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_{{\rm SO}_{h+1}(K)}(\overline{y})^0) & = & \frac{v}2-1+d_q\left(A_\alpha^{\frac\beta2-1} A_{\alpha-1}^{\frac{v}2-\frac\beta2}B_{\frac{\alpha+1}2}D_{\frac{\alpha+1}2}\right)\\ & = & \frac{v}2-1+\left(\frac\beta2-1\right) d_q(A_\alpha)+\left(\frac{v}2- \frac \beta2 \right)d_q(A_{\alpha-1})+d_q\left(B_{\frac{\alpha+1}2}\right)+d_q\left(D_{\frac{\alpha+1}2}\right)\\ & = &z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2\\ & = & z(zu+e)-\frac{zu(z+1)}2+\frac{zu+e}2\\ & = & \frac12(z^2u+e(2z+1)). \end{eqnarray*}
Assume that case (f) holds. Recall that $\epsilon_q=\epsilon_\alpha=1$, $\epsilon_u=\epsilon_v=\epsilon_{\beta}=0$ and $\beta=0$. Also $\delta=0$ if and only if $e=0$. We have $$C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_{\alpha-1}^{\frac v2-2} A_{\alpha-2}B_{\frac{\alpha+1}2}D_{\frac{\alpha+1}2}T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_{{\rm SO}_{h+1}(K)}(\overline{y})^0) & = & \frac{v}2-1+d_q\left(A_{\alpha-1}^{\frac v2-2} A_{\alpha-2}B_{\frac{\alpha+1}2}D_{\frac{\alpha+1}2}\right)\\ & = & \frac{v}2-1+\left(\frac v2-2\right) d_q(A_{\alpha-1})+d_q(A_{\alpha-2})+d_q\left(B_{\frac{\alpha+1}2}\right)+d_q\left(D_{\frac{\alpha+1}2}\right)\\ & = & \left\{ \begin{array}{ll} z\alpha v-\frac{qvz(z+1)}2+\frac{\alpha v }2 & \textrm{if} \ \delta>0\\ z\alpha v-\frac{qvz(z+1)}2+\frac{\alpha v }2+2& \textrm{if} \ \delta=0\\ \end{array}\right.\\ & = & \left\{ \begin{array}{ll} z(zu+e)-\frac{zu(z+1)}2+\frac{zu+e}2 & \textrm{if} \ \delta>0\\ z(zu+e)-\frac{zu(z+1)}2+\frac{zu+e}2 +2& \textrm{if} \ \delta=0\\
\end{array}\right.\\
& = & \left\{ \begin{array}{ll}
\frac12(z^2u+e(2z+1)) & \textrm{if} \ e>0\\
\frac12(z^2u+e(2z+1))+2 & \textrm{if} \ e=0.
\end{array}\right. \end{eqnarray*}
Assume that case (g) or case (h) holds. Recall that $\epsilon_q=1$ and $\epsilon_u=\epsilon_v=\epsilon_\alpha=\epsilon_{\beta}=0$. Also if $\delta=0$ then $z$ must be even. We have $$C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}B_{\frac{\alpha}2}D_{\frac{\alpha}2}T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_{{\rm SO}_{h+1}(K)}(\overline{y})^0) & = & \frac{v}2-1+d_q\left(A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}B_{\frac{\alpha}2}D_{\frac{\alpha}2}\right)\\ & = & \frac{v}2-1+\frac\beta2 d_q(A_\alpha)+\left(\frac{v}2-1- \frac \beta2 \right)d_q(A_{\alpha-1})+d_q\left(B_{\frac{\alpha}2}\right)+d_q\left(D_{\frac{\alpha}2}\right)\\ & = &z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2\\ & = & z(zu+e)-\frac{zu(z+1)}2+\frac{zu+e}2\\ & = & \frac12(z^2u+e(2z+1)). \end{eqnarray*}
Assume that case (i) or case (j) holds. Recall that $\epsilon_q=1$ and $\epsilon_u=\epsilon_v=\epsilon_\alpha=\epsilon_{\beta}=0$. We have $$C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_\alpha^{\frac\beta2-1} A_{\alpha-1}^{\frac{v}2-\frac\beta2}B_{\frac{\alpha}2}D_{\frac{\alpha+2}2}T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_{{\rm SO}_{h+1}(K)}(\overline{y})^0) & = & \frac{v}2-1+d_q\left(A_\alpha^{\frac\beta2-1} A_{\alpha-1}^{\frac{v}2-\frac\beta2}B_{\frac{\alpha}2}D_{\frac{\alpha+2}2}\right)\\ & = & \frac{v}2-1+\left(\frac\beta2-1\right) d_q(A_\alpha)+\left(\frac{v}2- \frac \beta2 \right)d_q(A_{\alpha-1})+d_q\left(B_{\frac{\alpha}2}\right)+d_q\left(D_{\frac{\alpha+2}2}\right)\\ & = &z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2\\ & = & z(zu+e)-\frac{zu(z+1)}2+\frac{zu+e}2\\ & = & \frac12(z^2u+e(2z+1)). \end{eqnarray*}
Assume that case (k) holds. Recall that $\epsilon_q=1$, $\epsilon_u=\epsilon_v=\epsilon_\alpha=\epsilon_{\beta}=0$ and $\beta=0$. Also $\delta=0$ if and only if $e=0$. We have $$C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_{\alpha-1}^{\frac v2-2} A_{\alpha-2}B_{\frac{\alpha}2}D_{\frac{\alpha+2}2}T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_{{\rm SO}_{h+1}(K)}(\overline{y})^0) & = & \frac{v}2-1+d_q\left(A_{\alpha-1}^{\frac v2-2} A_{\alpha-2}B_{\frac{\alpha}2}D_{\frac{\alpha+2}2}\right)\\ & = & \frac{v}2-1+\left(\frac v2-2\right) d_q(A_{\alpha-1})+d_q(A_{\alpha-2})+d_q\left(B_{\frac{\alpha}2}\right)+d_q\left(D_{\frac{\alpha+2}2}\right)\\ & = & \left\{ \begin{array}{ll} z\alpha v-\frac{qvz(z+1)}2+\frac{\alpha v}2 & \textrm{if} \ \delta>0\\ z\alpha v-\frac{qvz(z+1)}2+\frac{\alpha v}2 +2& \textrm{if} \ \delta=0\\ \end{array} \right.\\ & = & \left\{ \begin{array}{ll} z(zu+e)-\frac{zu(z+1)}2+\frac{zu+e}2 & \textrm{if} \ \delta>0\\ z(zu+e)-\frac{zu(z+1)}2+\frac{zu+e}2+2 & \textrm{if} \ \delta=0\\ \end{array} \right.\\ & = & \left\{ \begin{array}{ll}
\frac12(z^2u+e(2z+1)) & \textrm{if} \ e>0\\
\frac12(z^2u+e(2z+1))+2 & \textrm{if} \ e=0.
\end{array}
\right. \end{eqnarray*} \end{proof}
\subsection{$G$ is of type $D_\ell$}
Let $G=(D_\ell)_{{\rm s.c}}$ be a simply connected group of type ${D}_{\ell}$ with $\ell \geq 4$ defined over an algebraically closed field $K$ of characteristic $p$. Here $|\Phi|=2\ell(\ell-1)$, $h=2\ell-2$ and $G={\rm Spin}_{h+2}(K)$ if $p\neq 2$, otherwise $G={\rm SO}_{h+2}(K)$. Given an element $g$ in $G$, we let $\overline{g}$ be the image of $g$ under the canonical surjective map $G\rightarrow {\rm SO}_{h+2}(K)$. \\
\begin{lem}\label{l:dsc} Let $G=(D_{\ell})_{{\rm s.c}}$ where $\ell \geq 4$ be defined over an algebraically closed field $K$ of characteristic $p$. Let $u=qv$ be a positive integer where $q$ and $v$ are positive integers such that $q$ is a power of $p$ and $p$ does not divide $v$. Write $h=zu+e=\alpha v +\beta$ and $\alpha=zq+\delta$ where $z$, $e$, $\alpha$, $\beta$, $\delta$ are nonnegative integers such that $e<u$, $\beta<v$, and $\delta<q$. Consider the following conditions:\\ \begin{enumerate}[(a)] \item $\epsilon_v=1$. \item $v=2$. \item $(\epsilon_v,\epsilon_\alpha)=(0,1)$, $v>2$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) \in \{(0,0,1), (0,2,0), (2,0,0), (2,2,3)\}.$ \item $(\epsilon_v,\epsilon_{\alpha})=(0,1)$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) = (0,2,2).$ \item $(\epsilon_v,\epsilon_{\alpha})=(0,1)$, $v>2$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) = (2,2,1).$ \item $(\epsilon_v,\epsilon_{\alpha})=(0,1)$, $\beta>0$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) = (0,0,3).$ \item $(\epsilon_v,\epsilon_{\alpha})=(0,1)$, $v>2$, $\beta>0$, $\beta \neq v-2$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) = (2,0,2).$ \item $(\epsilon_v,\epsilon_{\alpha})=(0,1)$, $v>2$, $\beta=v-2$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) = (2,0,2).$ \item $(\epsilon_v,\epsilon_{\alpha})=(0,1)$, $\beta=0$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) = (0,0,3).$ \item $(\epsilon_v,\epsilon_{\alpha})=(0,1)$, $v>2$, $\beta=0$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) = (2,0,2).$ \item $(\epsilon_v,\epsilon_\alpha)=(0,0)$, $v>2$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) \in \{(0,0,1), (0,2,0), (2,0,1), (2,2,0)\}.$ \item $(\epsilon_v,\epsilon_{\alpha})=(0,0)$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) =(2,2,2)$. \item $(\epsilon_v,\epsilon_{\alpha})=(0,0)$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) =(0,0,3)$. \item $(\epsilon_v,\epsilon_{\alpha})=(0,0)$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) =(0,2,2)$. \item $(\epsilon_v,\epsilon_{\alpha})=(0,0)$, $v>2$, $\beta>0$ and $(v \mod 4,\beta \mod 4,\ell \mod 4) =(2,0,3)$. \item $(\epsilon_v,\epsilon_{\alpha})=(0,0)$, $v>2$, $\beta=0$ and $(v \mod 4,\ell \mod 4) =(2,3)$. \end{enumerate} Let $\overline{y}$ be the element of ${\rm SO}_{h+2}(K)$ of order $v$ defined in Table \ref{t:dsc} (see \S\ref{s:tables}). Then $\overline{y}$ has a preimage $y$ in $G$ of order $v$, and $C_{{\rm SO}_{h+2}}(\overline{y})^0$ and $d_q(C_G(y)^0)=d_q(C_{{\rm SO}_{h+2}(K)}(\overline{y})^0)$ are given in Table \ref{t:dsc}. Furthermore $d_q(C_G(y)^0)$ is an upper bound for $d_u(G)$. \end{lem}
\begin{proof} Note that by Lemma \ref{l:dufactg}(iii), $d_q(C_G(y)^0)=d_q(C_{{\rm SO}_{h+2}(K)}(\overline{y})^0)$. Since $h=2\ell-2$, note that if $\epsilon_v=1$ then $\epsilon_\alpha=\epsilon_\beta$ and $\epsilon_u=\epsilon_q$, whereas if $\epsilon_v=0$ then $\epsilon_u=0$, $\epsilon_q=1$ and $\epsilon_\beta=0$. We first determine $C_{{\rm SO}_{h+2}(K)}(\overline{y})^0$.
Suppose that case (a) holds so that $v$ is odd. As any eigenvalue of $\overline{y}$ can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+2}(K)$ of order $v$. Furthermore, since $v$ is odd, by Lemma \ref{l:soliftspin} it follows that $\overline{y}$ has a preimage $y$ in $G$ of order $v$. Also $\overline{y}$ has $\alpha+2-\epsilon_\beta$ eigenvalues equal to $1$, $0$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq \lceil \beta/2\rceil$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore $C_{{\rm SO}_{h+1}(K)}(\overline{y})^0=A_\alpha^{\lceil\frac\beta2\rceil} A_{\alpha-1}^{\frac{v-1}2-\lceil\frac\beta2\rceil}D_{\lceil \frac {\alpha+1}2\rceil}T_{\frac{v-1}2}$. \\ Suppose that case (b) holds so that $v=2$. As any eigenvalue of $\overline{y}$ can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+2}(K)$ of order $v$. Now the number $N$ of eigenvalues $\omega^i=(-1)^i$ of $\overline{y}$ with $i$ odd is equal to $\ell$, $\ell-1$, $\ell+2$ or $\ell +1$ according respectively as $\ell \mod 4$ is 0, 1, 2 or 3. Since $N$ is divisible by 4, it follows from Lemma $\ref{l:soliftspin}$ that $\overline{y}$ has a preimage $y$ in $G$ of order $v$. Also $$ C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=\left\{\begin{array}{ll}D_{\frac{\ell}2}D_{\frac{\ell}2} & \textrm{if} \ \ell \equiv 0 \ (4)\\
D_{\frac{\ell+1}2}D_{\frac{\ell-1}2} & \textrm{if} \ \ell \equiv 1 \ (4)\\ D_{\frac{\ell-2}2}D_{\frac{\ell+2}2} & \textrm{if} \ \ell \equiv 2 \ (4)\\ D_{\frac{\ell-1}2}D_{\frac{\ell+1}2} & \textrm{if} \ \ell \equiv 3 \ (4).
\end{array}\right.$$
Suppose that case (c) holds. As any eigenvalue of $\overline{y}$ can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+2}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
\begin{eqnarray*}
N& = &\left\{ \begin{array}{ll} \alpha \cdot \frac{v}{2}+\frac{\beta}2& \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,0)\\
\alpha \cdot \frac{v}{2}+\left(\frac{\beta}2+1\right) & \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,2)\\
\alpha \cdot\left( \frac{v}{2}-1\right)+(\alpha+1)+\frac{\beta}2 & \textrm{if} \ (v \mod 4, \beta \mod 4)=(2,0)\\
\alpha \cdot \left( \frac{v}{2}-1\right)+(\alpha+1)+\left(\frac{\beta}2+1\right) & \textrm{if} \ (v \mod 4, \beta \mod 4)=(2,2)\\
\end{array}\right. \\ &=& \left\{ \begin{array}{ll} \ell-1 & \textrm{if} \ (v \mod 4, \beta \mod 4) =(0,0)\\
\ell & \textrm{if} \ (v \mod 4, \beta \mod 4)\in\{(0,2),(2,0)\}\\
\ell+1 & \textrm{if} \ (v \mod 4, \beta \mod 4) = (2,2).\\
\end{array}\right.
\end{eqnarray*}
From the assumptions of case (c), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha+1$ eigenvalues equal to $1$, $\alpha+1$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq \beta/2$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore $C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}D_{\frac {\alpha+1}2}D_{\frac{\alpha+1}2}T_{\frac{v}2-1}$. \\
Suppose that case (d) or (e) holds. Note that if case (d) holds then $h\equiv 2 \mod 8$ and so $\beta<v-2$ as otherwise $h=(\alpha +1)v-2$ which, under the assumptions of case (d), is equal to 6 modulo 8, a contradiction. As any eigenvalue of $\overline{y}$ can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+2}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
\begin{eqnarray*}
N& = &\left\{ \begin{array}{ll} \alpha \cdot \frac{v}{2}+\left(\frac{\beta}2-1\right) & \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,2)\\
\alpha \cdot \left( \frac{v}{2}-1\right)+(\alpha+1)+\left(\frac{\beta}2-1\right) & \textrm{if} \ (v \mod 4, \beta \mod 4)=(2,2)\\
\end{array}\right. \\ &=& \left\{ \begin{array}{ll}
\ell-2 & \textrm{if} \ (v \mod 4, \beta \mod 4)= (0,2)\\
\ell-1 & \textrm{if} \ (v \mod 4, \beta \mod 4)=(2,2).\\
\end{array}\right. \\
\end{eqnarray*}
From the assumptions of case (d) or (e), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha+1$ eigenvalues equal to $1$, $\alpha+1$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where if case (d) holds then $i$ is any integer with $1 \leq |i| \leq \beta/2-1$ or $|i|=v/2-2$ and if case $(e)$ holds then $i$ any integer with $1 \leq |i| \leq \beta/2-1$ or $|i|=v/2-1$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore $C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}D_{\frac {\alpha+1}2}D_{\frac{\alpha+1}2}T_{\frac{v}2-1}$. \\
Suppose that case (f) holds. As any eigenvalue of $\overline{y}$ can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+2}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
\begin{eqnarray*}
N& = &\left\{ \begin{array}{ll} \alpha \cdot \frac{v}{2}+\left(\frac{\beta}2-2\right)& \textrm{if} \ \beta \neq v-4 \\
\alpha \cdot \frac{v}{2}+\frac{v}2 & \textrm{if} \ \beta=v-4\\
\end{array}\right. \\ &=& \left\{ \begin{array}{ll} \ell-3 & \textrm{if} \ \beta\neq v-4\\
\ell+1& \textrm{if} \ \beta=v-4. \end{array}\right.
\end{eqnarray*}
From the assumptions of case (f), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha+1$ eigenvalues equal to $1$, $\alpha+1$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where if $\beta \neq v-4$ then $i$ is any integer with $2 \leq |i| \leq \beta/2$ or $|i|=v/2-2$ and if $\beta = v-4$ then $i$ is any integer with $1 \leq |i| \leq v/2-3$ or $|i|=v/2-1$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore $C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_{\alpha}^{\frac \beta2} A_{\alpha-1}^{\frac v2-1-\frac \beta2}D_{\frac {\alpha+1}2}D_{\frac{\alpha+1}2}T_{\frac{v}2-1}$. \\ Suppose that case (g) holds. As any eigenvalue of $\overline{y}$ can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+2}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
\begin{eqnarray*}
N& = &\alpha \cdot \left(\frac v2-1\right)+\left(\frac{\beta}2-2\right)+(\alpha+1)\\ &=& \ell -2. \\
\end{eqnarray*}
From the assumptions of case (g), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha+1$ eigenvalues equal to $1$, $\alpha+1$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $2 \leq |i| \leq \beta/2$ or $|i|=v/2-1$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore $C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_{\alpha}^{\frac \beta2} A_{\alpha-1}^{\frac v2-1-\frac \beta2}D_{\frac {\alpha+1}2}D_{\frac{\alpha+1}2}T_{\frac{v}2-1}$. \\
Suppose that case (h) holds. As any eigenvalue of $\overline{y}$ can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+2}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
$$ N= \alpha\cdot\left( \frac v2-1\right) + \frac \beta2+(\alpha+3)= \ell+2.$$
From the assumptions of case (h), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha+1$ eigenvalues equal to $1$, $\alpha+3$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq v/2-2$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore $C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_{\alpha}^{\frac{v}2-2}A_{\alpha-1}D_{\frac {\alpha+1}2}D_{\frac{\alpha+3}2}T_{\frac{v}2-1}$. \\
Suppose that case (i) holds. As any eigenvalue of $\overline{y}$ can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+2}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
$$ N= (\alpha-1)\cdot\frac v2 + \left(\frac v2-2\right)+4= \ell+1.$$
From the assumptions of case (i), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha+1$ eigenvalues equal to $1$, $\alpha-1$ eigenvalues equal to $-1$, $\alpha$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $2 \leq |i| \leq v/2-1$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha+1$. Therefore $C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_{\alpha}A_{\alpha-1}^{\frac{v}2-2}D_{\frac {\alpha+1}2}D_{\frac{\alpha-1}2}T_{\frac{v}2-1}$. \\
Suppose that case (j) holds. As any eigenvalue of $\overline{y}$ can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+2}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
$$N= (\alpha-1)\cdot\left(\frac v2-1 \right)+ \left(\frac v2-1\right)+2+(\alpha+1)= \ell+2.$$
From the assumptions of case (j), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha+1$ eigenvalues equal to $1$, $\alpha+1$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega$, $\alpha+1$ eigenvalues equal to $\omega^{-1}$, $\alpha$ eigenvalues equal to $\omega^{i}$ where $i$ is any integer with $2 \leq |i| \leq v/2-2$, $\alpha-1$ eigenvalues equal to $\omega^{\frac v2-1}$ and $\alpha-1$ eigenvalues equal to $\omega^{-(\frac{v}2-1)}$. Therefore $C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_\alpha A_{\alpha-1}^{\frac v2-3} A_{\alpha-2}D_{\frac {\alpha+1}2}D_{\frac{\alpha+1}2}T_{\frac{v}2-1}$. \\
Suppose that case (k) holds. As any eigenvalue of $\overline{y}$ can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+2}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
\begin{eqnarray*}
N& = &\left\{ \begin{array}{ll} \alpha \cdot \frac{v}{2}+\frac{\beta}2& \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,0)\\
\alpha \cdot \frac{v}{2}+\left(\frac{\beta}2+1\right) & \textrm{if} \ (v \mod 4, \beta \mod 4)=(0,2)\\
\alpha \cdot\left( \frac{v}{2}-1\right)+\alpha+\frac{\beta}2 & \textrm{if} \ (v \mod 4, \beta \mod 4)=(2,0)\\
\alpha \cdot \left( \frac{v}{2}-1\right)+\alpha+\left(\frac{\beta}2+1\right) & \textrm{if} \ (v \mod 4, \beta \mod 4)=(2,2)\\
\end{array}\right. \\ &=& \left\{ \begin{array}{ll} \ell-1 & \textrm{if} \ (v \mod 4, \beta \mod 4) \in \{(0,0), (2,0)\}\\
\ell & \textrm{if} \ (v \mod 4, \beta \mod 4)\in\{(0,2),(2,2)\}.\\
\end{array}\right.
\end{eqnarray*}
From the assumptions of case (k), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha+2$ eigenvalues equal to $1$, $\alpha$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq \beta/2$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore $C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_{\alpha}^{\frac \beta2}A_{\alpha-1}^{\frac v2-1-\frac \beta2} D_{\frac {\alpha}2}D_{\frac{\alpha+2}2}T_{\frac{v}2-1}$. \\
Suppose that case (l) holds. As any eigenvalue of $\overline{y}$ can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+2}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
$$N= \alpha\cdot\left(\frac v2-1 \right)+\alpha+ \left(\frac \beta2-1\right)= \ell-2.$$
From the assumptions of case (l), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha+2$ eigenvalues equal to $1$, $\alpha$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^{i}$ where $i$ is any integer with $1 \leq |i| \leq \beta/2-1$ or $|i|=v/2-1$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore $C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_{\alpha}^{\frac \beta2}A_{\alpha-1}^{\frac v2-1-\frac \beta2} D_{\frac {\alpha}2}D_{\frac{\alpha+2}2}T_{\frac{v}2-1}$. \\
Suppose that case (m) holds. As any eigenvalue of $\overline{y}$ can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+2}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
\begin{eqnarray*}
N& = &\left\{ \begin{array}{ll} \alpha \cdot \frac{v}{2}+\left(\frac{\beta}2-2\right)& \textrm{if} \ \beta \neq v-4\\
\alpha \cdot \frac{v}{2}+\left(\frac{\beta}2+2\right) & \textrm{if} \ \beta=v-4\\
\end{array}\right. \\ &=& \left\{ \begin{array}{ll} \ell-3 & \textrm{if} \ \beta \neq v-4\\
\ell+1 & \textrm{if} \ \beta=v-4.\\
\end{array}\right.
\end{eqnarray*}
From the assumptions of case (m), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also if $\beta \neq v-4$ then $\overline{y}$ has $\alpha+2$ eigenvalues equal to $1$, $\alpha$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $2 \leq |i| \leq \beta/2$ or $|i|=v/2-2$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. On the other hand if $\beta=v-4$ then $\overline{y}$ has $\alpha+2$ eigenvalues equal to $1$, $\alpha$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq \beta/2-1$ or $|i|=\beta/2+1$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore in both cases $C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}D_{\frac {\alpha}2}D_{\frac{\alpha+2}2}T_{\frac{v}2-1}$. \\ Suppose that case (n) holds. As any eigenvalue of $\overline{y}$ can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+2}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
$$N = \alpha \cdot \frac{v}{2}+\left(\frac{\beta}2-1\right)=\ell-2.$$
From the assumptions of case (n), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also if $\beta \neq v-2$ then $\overline{y}$ has $\alpha+2$ eigenvalues equal to $1$, $\alpha$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq \beta/2-1$ or $|i|=v/2-2$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. On the other hand if $\beta=v-2$ then $\overline{y}$ has $\alpha+2$ eigenvalues equal to $1$, $\alpha+2$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq v/2-2$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore
$$C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=
\left\{ \begin{array}{ll}
A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}D_{\frac {\alpha}2}D_{\frac{\alpha+2}2}T_{\frac{v}2-1} & \textrm{if} \ \beta \neq v-2\\
A_\alpha^{\frac v2-2} A_{\alpha-1}D_{\frac {\alpha+2}2}D_{\frac{\alpha+2}2}T_{\frac{v}2-1} & \textrm{if} \ \beta=v-2.
\end{array} \right.
$$
Suppose that case (o) holds. As any eigenvalue of $\overline{y}$ can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+2}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then
\begin{eqnarray*}
N & = &\left\{\begin{array}{ll} \alpha \cdot \left(\frac{v}{2}-1\right)+\left(\frac{\beta}2-2\right)+\alpha & \textrm{if} \ \beta\neq v-2\\
\alpha \cdot \left(\frac{v}{2}-1\right)+\frac{\beta}2+(\alpha+2) & \textrm{if} \ \beta = v-2\\
\end{array}\right. \\
& = & \left\{\begin{array}{ll}
\ell-3 & \textrm{if} \ \beta\neq v-2\\
\ell+1 & \textrm{if} \ \beta = v-2.\\
\end{array}\right. \\
\end{eqnarray*}
From the assumptions of case (o), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also if $\beta \neq v-2$ then $\overline{y}$ has $\alpha+2$ eigenvalues equal to $1$, $\alpha$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $2 \leq |i| \leq \beta/2$ or $|i|=v/2-1$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. On the other hand if $\beta=v-2$ then $\overline{y}$ has $\alpha+2$ eigenvalues equal to $1$, $\alpha+2$ eigenvalues equal to $-1$, $\alpha+1$ eigenvalues equal to $\omega^i$ where $i$ is any integer with $1 \leq |i| \leq v/2-2$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore
$$C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=
\left\{ \begin{array}{ll}
A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}D_{\frac {\alpha}2}D_{\frac{\alpha+2}2}T_{\frac{v}2-1} & \textrm{if} \ \beta \neq v-2\\
A_\alpha^{\frac v2-2} A_{\alpha-1}D_{\frac {\alpha+2}2}D_{\frac{\alpha+2}2}T_{\frac{v}2-1} & \textrm{if} \ \beta=v-2.
\end{array} \right.
$$ Suppose that case (p) holds. As any eigenvalue of $\overline{y}$ can be paired with its inverse, $\overline{y}$ is an element of ${\rm SO}_{h+2}(K)$ of order $v$. Let $N$ be the number of eigenvalues $\omega^i$ of $\overline{y}$ with $i$ odd. Then $$ N= \alpha\cdot\left(\frac v2-1\right)+(\alpha+2)=\ell+1.$$ From the assumptions of case (p), it follows that $N$ is divisible by 4. Hence
by Lemma $\ref{l:soliftspin}$, $\overline{y}$ has a preimage $y$ in $G$ of order $v$.
Also $\overline{y}$ has $\alpha$ eigenvalues equal to $1$, $\alpha+2$ eigenvalues equal to $-1$, and every other eigenvalue of $\overline{y}$ occurs with multiplicity $\alpha$. Therefore
$$C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_{\alpha-1}^{\frac v2-1} D_{\frac {\alpha}2}D_{\frac{\alpha+2}2}T_{\frac{v}2-1} .$$
By Proposition \ref{p:prelimdu}(i), $d_u(G)\leq d_q(C_G(y)^0)$ and so $d_q(C_G(y)^0)$ is an upper bound for $d_u(G)$. It remains to calculate $d_q(C_{{\rm SO}_{h+2}}(\overline{y})^0)=d_q(C_G(y)^0)$.\\
It follows from Lemma \ref{l:lawther} that
$$d_q(A_{\alpha}) = z^2q +(2z+1)(\alpha-zq+1)-1,$$ $$d_q(A_{\alpha-1})=z^2q+(2z+1)(\alpha-zq)-1,$$ $$d_q(A_{\alpha-2})=\left\{\begin{array}{ll} z^2q+(2z+1)(\alpha-1-zq)-1 & \textrm{if} \ \delta >0\\ (z-1)^2q+(2z-1)(\alpha-1-(z-1)q)-1 & \textrm{if} \ \delta=0,
\end{array}\right.$$ $$d_q\left(D_{\lceil\frac{\alpha+1}2\rceil}\right)= \frac{z^2q}{2}+(2z+1)\left(\left\lceil\frac{\alpha+1}{2}\right\rceil-\frac{zq}2\right)+\left\lceil \frac z2\right\rceil\epsilon_q-z-\epsilon_z,$$ if $\alpha$ is odd then $$d_q\left(D_{\frac{\alpha-1}2}\right)= \left\{\begin{array}{ll} \frac{z^2q}{2}+(2z+1)\left(\frac{\alpha-1}{2}-\frac{zq}2\right)+\left\lceil \frac z2\right\rceil\epsilon_q - z - \epsilon_z& \textrm{if} \ \delta>0\\ \frac{(z-1)^2q}{2}+(2z-1)\left(\frac{\alpha-1}{2}-\frac{(z-1)q}2\right)+\frac {z-1}2\epsilon_q-(z-1)& \textrm{if} \ \delta=0, \end{array}\right.$$ $$d_q\left(D_{\frac{\alpha+3}2}\right)= \left\{\begin{array}{ll} \frac{z^2q}{2}+(2z+1)\left(\frac{\alpha+3}{2}-\frac{zq}2\right)+\left\lceil \frac z2\right\rceil\epsilon_q - z - \epsilon_z& \textrm{if} \ \delta<q-1\\ \frac{(z+1)^2q}{2}+(2z+3)\left(\frac{\alpha+3}{2}-\frac{(z+1)q}2\right)+\frac {z+1}2\epsilon_q-(z+1)-\epsilon_{z+1}& \textrm{if} \ \delta=q-1, \end{array}\right.$$ and if $\alpha$ is even then $$d_q\left(D_{\frac{\alpha}2}\right)=\left\{\begin{array}{ll} \frac{z^2q}{2}+(2z+1)\left(\frac{\alpha}{2}-\frac{zq}2\right)+\left\lceil \frac z2\right\rceil\epsilon_q-z-\epsilon_z & \textrm{if} \ \delta>0 \ \textrm{or} \ z \ \textrm{is even}\\ \frac{z^2q}{2}+(2z+1)\left(\frac{\alpha}{2}-\frac{zq}2\right)+\left\lceil \frac z2\right\rceil\epsilon_q-z-\epsilon_z+2 & \textrm{otherwise}. \end{array}\right. $$
Assume first that case (a) holds so that $\epsilon_v=1$. Recall that $\epsilon_u=\epsilon_q$ and $\epsilon_\alpha=\epsilon_\beta$. We have $$C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_\alpha^{\lceil\frac\beta2\rceil} A_{\alpha-1}^{\frac{v-1}2-\lceil\frac\beta2\rceil}D_{\lceil \frac {\alpha+1}2\rceil}T_{\frac{v-1}2}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_{{\rm SO}_{h+2}(K)}(\overline{y})^0) & = & \frac{v-1}2+d_q\left(A_\alpha^{\lceil\frac\beta2\rceil} A_{\alpha-1}^{\frac{v-1}2-\lceil\frac\beta2\rceil}D_{\lceil \frac {\alpha+1}2\rceil}\right)\\ & = & \frac{v-1}2+\left\lceil \frac \beta2 \right \rceil d_q(A_\alpha)+\left(\frac{v-1}2-\left \lceil \frac \beta2 \right \rceil\right)d_q(A_{\alpha-1})+d_q\left(D_{\lceil \frac{\alpha+1}2\rceil}\right)\\ & = &z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2+\left\lceil\frac{z}2\right\rceil\epsilon_q+z+1-\epsilon_z\\ & = & z(zu+e)-\frac{zu(z+1)}2+\frac{zu+e}2+\left\lceil\frac{z}2\right\rceil\epsilon_q+z+1-\epsilon_z\\ & = & \frac12(z^2u+e(2z+1))+\left\lceil\frac{z}2\right\rceil\epsilon_u+z+1-\epsilon_z. \end{eqnarray*}
Assume that case (b) holds so that $v=2$, $\epsilon_u=\epsilon_v=0$ and $\epsilon_q=1$. Recall that $\alpha=\ell-1$ and $\beta=0$. Note also that $e=0$ if and only if $\delta=0$. We have
$$C_{{\rm SO}_{h+2}(K)}(\overline{y})^0= \left\{ \begin{array}{ll}
D_{\frac{\alpha}2}D_{\frac{\alpha+2}2} & \textrm{if} \ \alpha \equiv 0 \ (2)\\
D_{\frac{\alpha-1}2}D_{\frac{\alpha+3}2} & \textrm{if} \ \alpha \equiv 1 \ (4)\\ D_{\frac{\alpha+1}2}D_{\frac{\alpha+1}2} & \textrm{if} \ \alpha \equiv 3 \ (4).\\ \end{array}\right. $$ If $\alpha \equiv 0 \mod 2$ then $z$ is even if $\delta=0$, and by Proposition \ref{p:prelimdu}(iii) \begin{eqnarray*} d_q(C_{{\rm SO}_{h+2}(K)}(\overline{y})^0) & = & d_q(D_{\frac{\alpha}2}D_{\frac{\alpha+2}2})\\ & = & d_q(D_{\frac{\alpha}2})+d_q(D_{\frac{\alpha+2}2})\\ & = & -qz(z+1)+2\alpha z+ \alpha+z+1-\epsilon_z\\ & = & \frac{-qvz(z+1)}2+(zu+e)z+\frac{(zu+e)}2+z+1-\epsilon_z\\ & = & \frac{-zu(z+1)}2+(zu+e)z+\frac{(zu+e)}2+z+1-\epsilon_z\\ & = & \frac{z^2u+e(2z+1)}2+z+1-\epsilon_z. \end{eqnarray*} If $\alpha \equiv 1 \mod 4$ then by Proposition \ref{p:prelimdu}(iii) \begin{eqnarray*} d_q(C_{{\rm SO}_{h+2}(K)}(\overline{y})^0) & = & d_q(D_{\frac{\alpha-1}2}D_{\frac{\alpha+3}2})\\ & = & d_q(D_{\frac{\alpha-1}2})+d_q(D_{\frac{\alpha+3}2})\\ & = & \left\{\begin{array}{ll} -z^2+2\alpha z+\alpha+4& \textrm{if} \ \delta=q-1=0 \\ -qz^2+(2\alpha+1) z +2& \textrm{if} \ \delta=0\neq q-1\\ -qz^2+(2\alpha-2q+1)z+2\alpha-q+3& \textrm{if} \ \delta=q-1\neq 0 \\ -qz^2+(2\alpha-q+1) z +\alpha+1-\epsilon_z& \textrm{if} \ \delta\not \in \{0, q-1\}
\end{array}\right. \\
& = & \left\{\begin{array}{ll} \frac{-zu(z+1)}2+(zu+e) z+\frac{zu+e}2+z+1-\epsilon_z+4& \textrm{if} \ u=2 \\ \frac{-qvz(z+1)}2+\alpha v z+ \alpha+z+1-\epsilon_z+2& \textrm{if} \ e=0\neq u-v\\ \frac{-qvz(z+1)}2+\alpha v z+\alpha+z+1-\epsilon_z+2& \textrm{if} \ e=u-v\neq 0 \\ \frac{-qvz(z+1)}2+\alpha vz +\alpha+z+1-\epsilon_z& \textrm{otherwise}
\end{array}\right. \\
& = & \left\{\begin{array}{ll} \frac{-zu(z+1)}2+(zu+e) z+\frac{zu+e}2+z+1-\epsilon_z+4& \textrm{if} \ u=2 \\ \frac{-zu(z+1)}2+(zu+e)z+\frac{zu+e}2+z+1-\epsilon_z+2& \textrm{if} \ e=0\neq u-v\\ \frac{-zu(z+1)}2+(zu+e)z+\frac{zu+e}2+z+1-\epsilon_z+2& \textrm{if} \ e=u-v\neq 0 \\ \frac{-zu(z+1)}2+(zu+e)z+\frac{zu+e}2+z+1-\epsilon_z& \textrm{otherwise}
\end{array}\right. \\
& = & \left\{\begin{array}{ll} \frac{z^2u+e(2z+1)}2+z+1-\epsilon_z+4& \textrm{if} \ u=2 \\
\frac{z^2u+e(2z+1)}2+z+1-\epsilon_z+2& \textrm{if} \ e=0\neq u-v\\
\frac{z^2u+e(2z+1)}2+z+1-\epsilon_z+2& \textrm{if} \ e=u-v\neq 0 \\
\frac{z^2u+e(2z+1)}2+z+1-\epsilon_z& \textrm{otherwise}.
\end{array}\right. \end{eqnarray*} If $\alpha \equiv 3 \mod 4$ then by Proposition \ref{p:prelimdu}(iii) \begin{eqnarray*} d_q(C_{{\rm SO}_{h+2}(K)}(\overline{y})^0) & = & d_q(D_{\frac{\alpha+1}2}D_{\frac{\alpha+1}2})\\ & = & 2d_q(D_{\frac{\alpha+1}2})\\ & = & -qz(z+1)+2\alpha z+\alpha+z+1-\epsilon_z\\ & = & \frac{-zu(z+1)}2+(zu+e)z+\frac{zu+e}2+z+1-\epsilon_z \\ & = & \frac{z^2u+e(2z+1)}2+z+1-\epsilon_z. \end{eqnarray*}
Assume that case (c), (d), (e), (f) or (g) holds. Recall that $\epsilon_q=\epsilon_\alpha=1$ and $\epsilon_u=\epsilon_v=\epsilon_{\beta}=0$. We have $$C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}D_{\frac{\alpha+1}2}D_{\frac{\alpha+1}2}T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_{{\rm SO}_{h+2}(K)}(\overline{y})^0) & = & \frac{v}2-1+d_q\left(A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}D_{\frac{\alpha+1}2}D_{\frac{\alpha+1}2}\right)\\ & = & \frac{v}2-1+\frac\beta2 d_q(A_\alpha)+\left(\frac{v}2-1- \frac \beta2 \right)d_q(A_{\alpha-1})+2d_q\left(D_{\frac{\alpha+1}2}\right)\\ & = &z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2+z+1-\epsilon_z\\ & = & z(zu+e)-\frac{zu(z+1)}2+\frac{zu+e}2+z+1-\epsilon_z\\ & = & \frac{z^2u+e(2z+1)}2+z+1-\epsilon_z. \end{eqnarray*}
Assume that case (h) holds. Recall that $\epsilon_q=\epsilon_\alpha=1$, $\epsilon_u=\epsilon_v=\epsilon_{\beta}=0$ and $\beta=v-2$. We have $$C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_\alpha^{\frac v2-2} A_{\alpha-1}D_{\frac{\alpha+1}2}D_{\frac{\alpha+3}2}T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_{{\rm SO}_{h+2}(K)}(\overline{y})^0) & = & \frac{v}2-1+d_q\left(A_\alpha^{\frac v2-2} A_{\alpha-1}D_{\frac{\alpha+1}2}D_{\frac{\alpha+3}2}\right)\\ & = & \frac{v}2-1+\left(\frac v2-2\right) d_q(A_\alpha)+d_q(A_{\alpha-1})+d_q\left(D_{\frac{\alpha+1}2}\right)+d_q\left(D_{\frac{\alpha+3}2}\right)\\ & = &\left\{ \begin{array}{ll} z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2+z+1-\epsilon_z& \textrm{if} \ \delta<q-1\\ z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2+z+1-\epsilon_z+2& \textrm{if} \ \delta=q-1\end{array}\right. \\ & = & \left\{ \begin{array}{ll} \frac{z^2u+e(2z+1)}2+z+1-\epsilon_z& \textrm{if} \ e\neq u-v+\beta\\
\frac{z^2u+e(2z+1)}2+z+1-\epsilon_z+2& \textrm{if} \ e=u-v+\beta \end{array}\right. \end{eqnarray*}
Assume that case (i) holds. Recall that $\epsilon_q=\epsilon_\alpha=1$, $\epsilon_u=\epsilon_v=\epsilon_{\beta}=0$ and $\beta=0$. We have $$C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_{\alpha}A_{\alpha-1}^{\frac v2-2} D_{\frac{\alpha-1}2}D_{\frac{\alpha+1}2}T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_{{\rm SO}_{h+2}(K)}(\overline{y})^0) & = & \frac{v}2-1+d_q\left(A_{\alpha}A_{\alpha-1}^{\frac v2-2} D_{\frac{\alpha-1}2}D_{\frac{\alpha+1}2}\right)\\ & = & \frac{v}2-1+d_q(A_\alpha)+\left(\frac v2-2\right) d_q(A_{\alpha-1})+d_q\left(D_{\frac{\alpha-1}2}\right)+d_q\left(D_{\frac{\alpha+1}2}\right)\\ & = &\left\{ \begin{array}{ll} z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2+z+1-\epsilon_z+2& \textrm{if} \ \delta=0\\ z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2+z+1-\epsilon_z& \textrm{if} \ \delta\neq0 \end{array}\right. \\ & = & \left\{ \begin{array}{ll} \frac{z^2u+e(2z+1)}2+z+1-\epsilon_z +2& \textrm{if} \ e=0\\ \frac{z^2u+e(2z+1)}2+z+1-\epsilon_z & \textrm{otherwise}. \end{array}\right. \end{eqnarray*}
Assume that case (j) holds. Recall that $\epsilon_q=\epsilon_\alpha=1$, $\epsilon_u=\epsilon_v=\epsilon_{\beta}=0$ and $\beta=0$. We have $$C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_{\alpha}A_{\alpha-1}^{\frac v2-3}A_{\alpha-2}D_{\frac{\alpha+1}2}D_{\frac{\alpha+1}2}T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_{{\rm SO}_{h+2}(K)}(\overline{y})^0) & = & \frac{v}2-1+d_q\left(A_{\alpha}A_{\alpha-1}^{\frac v2-3}A_{\alpha-2}D_{\frac{\alpha+1}2}D_{\frac{\alpha+1}2}\right)\\ & = & \frac{v}2-1+d_q(A_{\alpha})+\left(\frac v2-3\right) d_q(A_{\alpha-1})+d_q(A_{\alpha-2})+2d_q\left(D_{\frac{\alpha+1}2}\right)\\ & = &\left\{ \begin{array}{ll} z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2+z+1-\epsilon_z& \textrm{if} \ \delta>0\\ z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2+z+1-\epsilon_z+2& \textrm{if} \ \delta=0\end{array}\right. \\ & = & \left\{ \begin{array}{ll} \frac{z^2u+e(2z+1)}2+z+1-\epsilon_z& \textrm{if} \ e>0\\
\frac{z^2u+e(2z+1)}2+z+1-\epsilon_z+2& \textrm{if} \ e=0. \end{array}\right. \end{eqnarray*}
Assume that case (k), (l), (m), (n) or (o) holds, and that $\beta\neq v-2$ if in cases (n) or (o). Recall that $\epsilon_q=1$ and $\epsilon_u=\epsilon_v=\epsilon_\alpha=\epsilon_{\beta}=0$. We have $$C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_{\alpha}^\frac\beta2A_{\alpha-1}^{\frac v2-1 -\frac \beta2}D_{\frac{\alpha}2}D_{\frac{\alpha+2}2}T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_{{\rm SO}_{h+2}(K)}(\overline{y})^0) & = & \frac{v}2-1+d_q\left(A_{\alpha}^\frac\beta2A_{\alpha-1}^{\frac v2-1 -\frac \beta2}D_{\frac{\alpha}2}D_{\frac{\alpha+2}2}\right)\\ & = & \frac{v}2-1+\frac\beta2d_q(A_{\alpha})+\left(\frac v2-1-\frac{\beta}2\right) d_q(A_{\alpha-1})+d_q(D_{\frac{\alpha}2})+d_q\left(D_{\frac{\alpha+2}2}\right)\\ & = & z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2+z+1-\epsilon_z\\ & = & \frac{z^2u+e(2z+1)}2+z+1-\epsilon_z. \end{eqnarray*}
Assume that case (n) or (o) holds, and $\beta= v-2$. Recall that $\epsilon_q=1$ and $\epsilon_u=\epsilon_v=\epsilon_\alpha=\epsilon_{\beta}=0$. We have $$C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_{\alpha}^{\frac v2-2}A_{\alpha-1}D_{\frac{\alpha+2}2}D_{\frac{\alpha+2}2}T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_{{\rm SO}_{h+2}(K)}(\overline{y})^0) & = & \frac{v}2-1+d_q\left(A_{\alpha}^{\frac v2-2}A_{\alpha-1}D_{\frac{\alpha+2}2}D_{\frac{\alpha+2}2}\right)\\ & = & \frac{v}2-1+\left(\frac v2-2\right)d_q(A_{\alpha})+ d_q(A_{\alpha-1})+2d_q\left(D_{\frac{\alpha+2}2}\right)\\ & = & z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2+z+1-\epsilon_z\\ & = & \frac{z^2u+e(2z+1)}2+z+1-\epsilon_z. \end{eqnarray*}
Assume finally that case (p) holds. Recall that $\epsilon_q=1$, $\epsilon_u=\epsilon_v=\epsilon_\alpha=\epsilon_{\beta}=0$ and $\beta=0$. We have $$C_{{\rm SO}_{h+2}(K)}(\overline{y})^0=A_{\alpha-1}^{\frac v2-1}D_{\frac{\alpha}2}D_{\frac{\alpha+2}2}T_{\frac{v}2-1}.$$ By Proposition \ref{p:prelimdu}(iii), \begin{eqnarray*} d_q(C_{{\rm SO}_{h+2}(K)}(\overline{y})^0) & = & \frac{v}2-1+d_q\left(A_{\alpha-1}^{\frac v2-1}D_{\frac{\alpha}2}D_{\frac{\alpha+2}2}\right)\\ & = & \frac{v}2-1+\left(\frac v2-1\right) d_q(A_{\alpha-1})+d_q\left(D_{\frac{\alpha}2}\right)+d_q\left(D_{\frac{\alpha+2}2}\right)\\ & = & z(\alpha v+ \beta)-\frac{qvz(z+1)}2+\frac{\alpha v +\beta}2+z+1-\epsilon_z\\ & = & \frac{z^2u+e(2z+1)}2+z+1-\epsilon_z. \end{eqnarray*}
\end{proof}
\section{Proofs of Theorems \ref{t:blawther} and \ref{t:duotsca}}\label{s:pc}
Recall that in \S\ref{s:exceptionalgroups} we proved Theorem \ref{t:blawther} for $G$ of exceptional type, see Propositions \ref{p:e6sc} and \ref{p:e7sc}. In this section given a positive integer $u$ and a simple algebraic group $G$ of classical type over an algebraically closed field $K$ of characteristic $p$, we determine $d_u(G)$. We begin with the case where $G$ is of simply connected type and complete the proof of Theorem \ref{t:blawther}. We will then consider $G$ of neither simply connected type nor adjoint type and prove Theorem \ref{t:duotsca}.
\noindent \textit{Proof of Theorem \ref{t:blawther}.} As noticed at the beginning of this section, following Propositions \ref{p:e6sc} and \ref{p:e7sc} we can assume that $G$ is simply connected of classical type. By \cite[Theorem 1]{Lawther}, where Lawther determines $d_u(G_{a.})$ and shows that $d_u(G)\geq d_u(G_{a.})$, and Lemmas \ref{l:asc}, \ref{l:csc}, \ref{l:bsc} and \ref{l:dsc}, where the upper bounds for $d_u(G)$ are determined accordingly respectively as $G$ is of type $A_\ell$, $C_\ell$, $B_\ell$, or $D_\ell$, we get that $d_u(G)=d_u(G_{a.})$ except possibly if one of the cases in Table \ref{ta:casestoconsider} (see \S\ref{s:tables}) holds.
It remains to show that in the cases appearing in Table \ref{ta:casestoconsider} the upper bounds given for $d_u(G)-d_u(G_{a.})$ are in fact the precise values for $d_u(G)-d_u(G_{a.})$. As usual, we let $h$ be the Coxeter number of $G$ and write: $h=zu+e=\alpha v +\beta$ and $\alpha=zq+\delta$ where $z$, $e$, $\alpha$, $\beta$, $\delta$ are nonnegative integers such that $e<u$, $\beta<v$, and $\delta<q$. Noting that in the cases appearing in Table \ref{ta:casestoconsider} $u$ is even, we set $s=v/2$. \\ Suppose first that $G$ is of type $A_\ell$. By Table \ref{ta:casestoconsider}, we have to consider the case where $p\neq 2$, $u$ is even, $h=zu$ and $z$ is odd. Since $e=0$, we have $\beta=\delta=0$. Also since $p\neq 2$ and $z$ is odd, $q$ and $\alpha$ are also odd. \\ Since $0\leq d_u(G)-d_u(G_{a.})\leq 2$, by Lemma \ref{l:centmod2} it suffices to show that $d_u(G)-d_u(G_{a.})>0$. Under the assumptions on $h$, $u$ and $z$, adapting the proof of of \cite[Proposition 3.3]{Lawther}, we obtain that $d_u(G_{a.})$ is attained for an element of $G_{a.}$ whose semisimple part $y$ of order $v$ has centralizer satisfying
$C_{G_{a.}}(y)^0=A_{\alpha-1}^v T_{v-1}$. Moreover $d_u(G_{a.})=d_q(A_{\alpha-1}^vT_{v-1})$. \\ By the assumptions on $h$, $u$ and $z$ it follows that $y$ is not the image of an element of ${\rm SL}_{h}(K)$ of order dividing $v$ under the canonical map ${\rm SL}_{h}(K)\rightarrow {\rm PSL}_{h}(K)$. Hence $d_u(G)-d_u(G_{a.})>0$ and $d_u(G)=d_u(G_{a.})+2$ as required. \\
Suppose now that $G$ is of type $B_\ell$. By Table \ref{ta:casestoconsider}, we have to consider the case where $p\neq 2$, $u$ is even, $h=zu$ and $z$ is odd unless $u\equiv 2 \mod 4$ and $z\equiv 2 \mod 4$; moreover if $z$ is odd then either $u\equiv 2 \mod 4$ and $z\equiv u/2 \mod 4$, or $u \equiv 4 \mod 8$. Since $e=0$, we have $\beta=\delta=0$ and $\alpha>0$. Also since $p\neq 2$, $q$ is also odd. \\ Write $s=v/2$ and let $a$ and $b$ be nonnegative integers such that $\ell-1=as+b$ with $0\leq b\leq s-1$. Under the assumptions on $h$, $u$ and $z$ we have $a=\alpha-1$ and $b=s-1$. \\ Since $0\leq d_u(G)-d_u(G_{a.})\leq 2$, by Lemma \ref{l:centmod2} it suffices to show that $d_u(G)-d_u(G_{a.})>0$. Under the assumptions on $h$, $u$ and $z$, adapting the proof of \cite[Proposition 3.6]{Lawther}, we obtain that $d_u(G_{a.})$ is attained for an element of $G_{a.}$ whose semisimple part $y$ of order $v$ has centralizer satisfying $$C_{G_{a.}}(y)^0=A_{\alpha-1}^{\frac{v}2-1}B_{\lceil \frac {\alpha-1}2 \rceil} D_{\lceil \frac{\alpha}2\rceil} T_{\frac{v}2-1}={\rm GL}_{\alpha}(K)^{\frac{v}2-1}{\rm SO}_{\alpha}(K){\rm SO}_{\alpha+1}(K).$$ Moreover $$d_u(G_{a.})=d_q(A_{\alpha-1}^{\frac{v}2-1}B_{\lceil \frac {\alpha-1}2 \rceil} D_{\lceil \frac{\alpha}2\rceil} T_{\frac{v}2-1}).$$ \\ By the assumptions on $h$, $u$ and $z$ it follows from Lemma \ref{l:soliftspin} that $y$ is not the image of an element of ${\rm Spin}_{h+1}(K)$ of order dividing $v$ under the canonical map ${\rm Spin}_{h+1}(K)\rightarrow {\rm SO}_{h+1}(K)$. Hence $d_u(G)-d_u(G_{a.})>0$ and $d_u(G)=d_u(G_{a.})+2$ as required. \\
Suppose now that $G$ is of type $C_\ell$. By Table \ref{ta:casestoconsider}, we have to consider the case where $p\neq 2$ and $u$ is even. Let $y \in G_{[v]}$ be of order $v$ and such that $Z=C_{G}(y)^0$ is of minimal dimension. We first show that $d_q(Z)-d_u(G_{a.})= 2\lceil z/2\rceil$. \\
There are essentially three possibilities for the structure of $Z=C_{G}(y)^0$. Indeed either $y$ has no eigenvalues equal to $1$ or $-1$ in which case $Z$ has no $C$ factors, or $y$ has some eigenvalues equal to $1$ or $-1$ but not both in which case $Z$ has one $C$ factor, or $y$ has some eigenvalues equal to $1$ and $-1$ in which case $Z$ has two $C$ factors. Using Corollary \ref{c:lawther} and arguing in a similar way as in the proof of \cite[Proposition 3.4]{Lawther} we can assume that either $Z= A_a^bA_{a-1}^{s-1-b}T_{s-1}$ where $a,b$ are nonnegative integers such that $\ell=a(s-1)+b$ and $0\leq b <s-1$, or $Z=A_a^{\lfloor \frac b2\rfloor}A_{a-1}^{s-1-\lfloor \frac b2\rfloor}C_{\lceil \frac a2\rceil}T_{s-1}$ where $a,b$ are nonnegative integers such that $2\ell=a(2s-1)+b$ and $0\leq b < 2s-1$, or $Z=A_a^{b-\epsilon_a(1-\sigma_b)}A_{a-1}^{s-1-b+\epsilon_a(1-\sigma_b)}C_{\lceil \frac a2\rceil}C_{\lceil \frac{a}2\rceil-\epsilon_a\sigma_b}T_{s-1}$ where $a,b$ are nonnegative integers such that $\ell=as+b$ and $0\leq b <s$. \\ Suppose first that $Z= A_a^bA_{a-1}^{s-1-b}T_{s-1}$ where $a,b$ are nonnegative integers such that $\ell=a(s-1)+b$ and $0\leq b <s-1$. Note that $v>2$ and so $s>1$. Write $a=cq+d$ where $c,d$ are nonnegative integers such that $0\leq d<q$. It follows from Lemma \ref{l:lawther} that $$d_q(Z)=(2c+1)\ell-\frac{c(c+1)u}2+c(c+1)q.$$ An easy check yields $h=zu+e= c(u-2q)+f$ where $f=d(v-2)+2b$ satisfies $0\leq f \leq u-2q-2$. In particular $c\geq z$. Write $c=z+j$ where $j\geq 0$.
\cite[Theorem 1]{Lawther} which gives the value of $d_u(G_{a.})$ now yields $$ d_q(Z)-d_u(G_{a.})=\frac{uj}2(j-1)+(z+j)q(z-j+1)+fj.$$ We claim that $z\geq j-1$. Suppose not. Then $c\geq 2z+2$ and so we get \begin{eqnarray*} 0& = & 2\ell -2\ell \\ & = & (c(u-2q)+f)-(zu+e)\\
& \geq & (2z+2)(u-2q)+f-(zu+e)\\
& =& z(u-4q)+2(u-2q)+f-e\\
& \geq & z(u-4q)+2(u-2q)-u+1\\
& = & z(u-4q)+u-4q+1 \end{eqnarray*} Since $v>2$, we have $u\geq 4q$ and so we get $0\geq 1$, a contradiction. Therefore $z \geq j-1$ as claimed. It follows that $$ d_q(Z)-d_u(G_{a.})\geq 2\lceil z/2\rceil.$$ \\ Suppose now that $Z=A_a^{\lfloor \frac b2\rfloor}A_{a-1}^{s-1-\lfloor \frac b2\rfloor}C_{\lceil \frac a2\rceil}T_{s-1}$ where $a,b$ are nonnegative integers such that $2\ell=a(2s-1)+b$ and $0\leq b < 2s-1$. Write $a=cq+d$ where $c,d$ are nonnegative integers such that $0\leq d<q$. It follows from Lemma \ref{l:lawther} that $$d_q(Z)=(2c+1)\ell-\frac{c(c+1)u}2+\frac{c(c+1)q}2+\lceil c/2\rceil.$$ An easy check yields $h=zu+e= c(u-q)+f$ where $f=d(v-1)+b$ satisfies $0\leq f \leq u-q-1$. In particular $c\geq z$. Write $c=z+j$ where $j\geq 0$.
\cite[Theorem 1]{Lawther} which gives the value of $d_u(G_{a.})$ now yields $$ d_q(Z)-d_u(G_{a.})=\frac{uj}2(j-1)+\frac{(z+j)q(z-j+1)}2+\frac{z+j+\epsilon_{z+j}}2+fj.$$ We claim that $z\geq j-1$. Suppose not. Then $c\geq 2z+2$ and so we get \begin{eqnarray*} 0& = & 2\ell -2\ell \\ & = & (c(u-q)+f)-(zu+e)\\
& \geq & (2z+2)(u-q)+f-(zu+e)\\
& =& z(u-2q)+2(u-q)+f-e\\
& \geq & z(u-2q)+2(u-q)-u+1\\
& = & z(u-2q)+u-2q+1 \end{eqnarray*} We have $u\geq 2q$ and so we get $0\geq 1$, a contradiction. Therefore $z \geq j-1$ as claimed. It follows that $$ d_q(Z)-d_u(G_{a.})\geq 2\lceil z/2\rceil.$$ \\ Suppose finally that $Z=A_a^{b-\epsilon_a(1-\sigma_b)}A_{a-1}^{s-1-b+\epsilon_a(1-\sigma_b)}C_{\lceil \frac a2\rceil}C_{\lceil \frac{a}2\rceil-\epsilon_a\sigma_b}T_{s-1}$ where $a,b$ are nonnegative integers such that $\ell=as+b$ and $0\leq b <s$. Writing $a=cq+d$ where $c,d$ are nonnegative integers such that $0\leq d<q$ one can show that $c=z$ and Lemma \ref{l:lawther} yields $d_q(Z)-d_u(G_{a.})=2\lceil z/2\rceil$ (see the proof of \cite[Proposition 3.4]{Lawther}) for more details. \\ We have in fact showed that if $G$ is simply connected of type $C_\ell$, $p\neq 2$, $v$ is even and $y$ is a semisimple element of $G$ of order $v$ having centralizer $Z$ of minimal dimension then $d_q(Z)=d_{qv}(G_{a.})+2\lceil z/2 \rceil$ where $z=\lfloor h/(qv)\rfloor$. Since $$d_u(G)=\min_{y\in {G}_{[v]}} \ d_q(C_{G}(y)^0)$$ and for any divisor $r$ of $v$ we have $\lfloor rh/(qv) \rfloor \geq \lfloor h/qv\rfloor$ and, by \cite[Lemma 1.3]{Lawther}, $d_{qv/r}(G_{a.}) \geq d_{qv}(G_{a.})$, we deduce that $d_u(G)=d_q(Z)=d_u(G_{a.})+2\lceil z/2\rceil$. \\
Suppose finally that $G$ is of type $D_\ell$. By Table \ref{ta:casestoconsider}, we have to consider the case where $p\neq 2$, $z$ is odd, and $u\equiv 2 \mod 4$ or $u \equiv 4 \mod 8$. Moreover if $u\equiv 4 \mod 8$ then $h=zu$. Finally if $u \equiv 2 \mod 4$ then $h=zu$ and $z\equiv u/2 \mod 4$, or $e=u-2\neq 0$ and $z\equiv 1 \mod 4$. Since $v$ is even, $\beta$ is also even. Also since $p\neq 2$, $q$ is odd. Finally under the assumptions on $h$, $u$ and $z$, $\alpha$ is odd.\\ Write $s=v/2$ and let $a$ and $b$ be nonnegative integers such that $\ell-1=as+b$ with $0\leq b\leq s-1$. We have $a=\alpha$ is odd and $b=\beta/2$. \\ Suppose first that $u=2$ so that $q=1$ and $v=2$. Corollary \ref{c:lawther} and Lemma \ref{l:soliftspin} yield that an element $y$ of $G$ of order $u$ with centralizer of minimal dimension has centralizer with connected component $D_{(z+3)/2}D_{(z-1)/2}$ and $d_u(G)=d_u(G_{a.})+4$. \\ Suppose now that $u>2$. Then $0\leq d_u(G)-d_u(G_{a.})\leq 2$, and so by Lemma \ref{l:centmod2} it suffices to show that $d_u(G_{s.c.})-d_u(G_{a.})>0$. Under the assumptions on $h$, $u$ and $z$, adapting the proof of \cite[Proposition 3.5]{Lawther}, we obtain that $d_u(G_{a.})$ is attained for an element of $G_{a.}$ whose semisimple part $y$ of order $v$ has centralizer satisfying
$$C_{G_{a.}}(s)^0=A_{\alpha}^\frac \beta 2A_{\alpha-1}^{\frac{v}2-1-\frac{\beta}2}D_{ \frac {\alpha+1}2 } D_{ \frac{\alpha+1}2} T_{\frac{v}2-1}={\rm GL}_{\alpha+1}^{\frac \beta 2}{\rm GL}_{\alpha}(K)^{\frac{v}2-1-\frac \beta 2}{\rm SO}_{\alpha+1}(K){\rm SO}_{\alpha+1}(K).$$ Moreover $$d_u(G_{a.})=d_q(A_{\alpha}^\frac \beta 2A_{\alpha-1}^{\frac{v}2-1-\frac{\beta}2}D_{ \frac {\alpha+1}2 } D_{ \frac{\alpha+1}2} T_{\frac{v}2-1}).$$ \\ By the assumptions on $h$, $u$ and $z$, it follows from Lemma \ref{l:soliftspin} that $y$ is not the image of an element of ${\rm Spin}_{h+2}(K)$ of order dividing $v$ under the canonical map ${\rm Spin}_{h+2}(K)\rightarrow {\rm SO}_{h+2}(K)$. Hence $d_u(G)-d_u(G_{a.})>0$ and $d_u(G_{s.c.})=d_u(G_{a.})+2$ as required. $\square$
Given a positive integer $u$, we can now determine $d_u(G)$ for $G$ a simple algebraic group of neither simply connected nor adjoint type and prove Theorem \ref{t:duotsca}.\\
\noindent \textit{Proof of Theorem \ref{t:duotsca}.} Note that since $G$ is neither simply connected nor adjoint, $G$ is of classical type. We first consider part (i). If $d_u(G_{s.c.})=d_u(G_{a.})$ then Proposition \ref{p:ingsimple} yields $d_u(G)=d_u(G_{a.})$. \\ We now consider part (ii) and assume that $d_u(G_{s.c.})\neq d_u(G_{a.})$. In particular, by Theorem \ref{t:blawther}, $p\neq 2$ and $u$ is even (and so $v$ is also even). By Proposition \ref{p:ingsimple} we have $d_u(G_{a.})\leq d_u(G)\leq d_u(G_{s.c.})$. \\ (a) Suppose first that $G={\rm SL}_{\ell+1}(K)/C$ where $C\leq Z({\rm SL}_{\ell+1}(K))$ is a central subgroup of $G_{s.c.}$. Note that $C$ is finite and cyclic and write $C=\langle c \rangle$. Since $d_u(G_{s.c.})\neq d_u(G_{a.})$ Theorem \ref{t:blawther} yields $h=zu=zqv$ where $z$ is odd. \\
Assume $(u,|C|)=1$. Let $gC\in G$ be an element of order dividing $u$. Then $g^u \in C$ and say the eigenvalue of $g^u$ is $c^l$ for some $0\leq l \leq |C|-1$.
Since $(u,|C|)=1$, there is a positive integer $j$ such that $ju\equiv 1 \mod |C|$. Let $k$ be a positive integer such that $k\equiv -jl \mod |C|$ and set $g'=g\cdot {\rm diag}(c^{k}, \dots , c^{k})$. Then $g'$ is an element of $G_{s.c.}$ of order dividing $u$, $g'C=gC$ and by Lemma \ref{l:dufactg} $\dim C_{G_{s.c.}}(g')=\dim C_G(g'C)$. It follows that $d_u(G)=d_u(G_{s.c.})$. \\
Assume now that $(u,|C|)>1$. We claim that $d_u(G)=d_u(G_{s.c.})$ if $|C|$ is odd, otherwise $d_u(G)=d_u(G_{a.})$. By Lemma \ref{l:centmod2}, Proposition \ref{p:ingsimple} and Theorem \ref{t:blawther}, $d_u(G) \equiv \ell \mod 2$, $d_u(G_{a.})\leq d_u(G) \leq d_u(G_{s.c.})$ and $d_u(G_{s.c})-d_u(G_{a.})=2$. Hence either $d_u(G)=d_u(G_{a.})$ or $d_u(G)=d_u(G_{s.c.})$. \\
Suppose $d_u(G)=d_u(G_{a.})$. Let $gC$ be an element of $G$ of order dividing $u$ such that $\dim C_G(gC)=d_u(G_{a.})$. Write $g=xy$ where $x$ and $y$ are respectively the unipotent and semisimple parts of $g$. As $g^u\in C$ and $(p,|C|)=1$, the order of $x$ divides $q$ and the order of $y$ divides $v|C|$. By Lemma \ref{l:dufactg} $d_u(G_{a.})=\dim C_{G_{s.c.}}(g)$ and so $y^v \neq 1$, as otherwise $g$ has order dividing $u$ and $d_u(G_{a})=d_u(G_{s.c.})$, a contradiction.\\
Furthermore, $y$ must have $v$ distinct eigenvalues, each repeated $zq$ times. These $v$ distinct eigenvalues must be $\omega^{1+jk}, \omega^{-(1+jk)}$ where $0\leq j\leq (v-2)/2$, $k>1$ is a divisor of $|C|$ and $\omega \in K$ is a primitive $kv$-th root of unity. Considering $1\neq y^v\in C$ we deduce that $\omega^v=\omega^{-v}$ and so $\omega^v$ has order $2$. Hence $2$ divides $|C|$. It follows that if $|C|$ is odd then $d_u(G)=d_u(G_{s.c.})$. \\
Suppose that $|C|$ is even. Consider $g$ a semisimple element of $G_{s.c.}$ having $v$ distinct eigenvalues, each repeated $zq$ times, equal to $\omega^{1+2j}, \omega^{-(1+2j)}$ where $0\leq j\leq (v-2)/2$, $\omega\in K$ is a primitive $2v$-th root of unity. Then $gC$ is an element of $G$ of order $v$ and $C_G(gC)^0= A_{\alpha-1}^vT_{v-1}$. It follows that $d_q(C_G(gC)^0)=d_u(G_{a.})$ and so $d_u(G)=d_u(G_{a.})$.\\
(b) Suppose $G={\rm SO}_{2\ell}(K)$. Using the notation of Lemma \ref{l:dsc} and Theorem \ref{t:blawther}, we let $\omega \in K$ be a primitive $v$-th root of $1$ and $y \in G$ be a semisimple element of order $v$ such that
$$ y=\left\{\begin{array}{ll} (-1)^\ell \oplus (1)^\ell & \textrm{if} \ v=2 \\
M_1^{\alpha}\oplus M_3\oplus(1)\oplus(-1) & \textrm{if} \ v>2 \ \textrm{and} \ \alpha \ \textrm{is odd}\\
M_1^{\alpha}\oplus M_3\oplus(1)^2 & \textrm{if} \ v>2 \ \textrm{and} \ \alpha \ \textrm{is even}.\\
\end{array} \right.$$
Then $$ C_G(y)^0=\left\{\begin{array}{ll} D_{\frac \ell2}D_{\frac \ell2} & \textrm{if} \ v=2\\ A_{\alpha}^{\frac \beta2}A_{\alpha-1}^{\frac v2 -1-\frac \beta2}D_{\frac{\alpha+1}2} D_{\frac{\alpha+1}2} & \textrm{if} \ v>2 \ \textrm{and} \ \alpha \ \textrm{is odd}\\ A_{\alpha}^{\frac \beta2}A_{\alpha-1}^{\frac v2 -1-\frac \beta2}D_{\frac{\alpha}2} D_{\frac{\alpha+2}2} & \textrm{if} \ v>2 \ \textrm{and} \ \alpha \ \textrm{is even}.
\end{array} \right.$$
An easy check yields that $d_q(C_G(y))=d_u(G_{a.})$ and so $d_u(G)=d_u(G_{a.})$. \\
(c) Finally suppose that $G={\rm HSpin}_{2\ell}(K)$ where $p\neq 2$ and $\ell$ is even.
Write $H=G_{s.c.}$ and let $C=Z(H)=\langle c_1,c_2\rangle$, $C_1=\langle c_1\rangle$, $C_2=\langle c_2\rangle $ where $c_1,c_2$ are of order 2 and $H/C=G_{a.}$, $H/C_1=G$ and $H/C_2={\rm SO}_{2\ell}(K)$.\\
Suppose first that $u=2$. Let $hC_1$ be an element of $H/C_1$ of order 2 satisfying $$d_2(H/C_1)=\dim C_{H/C_1}(hC_1).$$
By Lemma \ref{l:dufactg} we have
\begin{equation}\label{e:spinsopso}
\dim C_H(h)=\dim C_{H/C_1}(hC_1)=\dim C_{H/C_2}(hC_2)=\dim C_{H/C}(hC).
\end{equation}
Since $hC_1$ has order 2, either $h^2=1$ or $h^2=c_1$. Also note that $hC$ has order dividing 2. If $hC$ is trivial then $h\in C$ and so by (\ref{e:spinsopso}) $\dim C_{H/C_1}(hC_1)= \dim H/C_1$, a contradiction. Hence $hC$ has order 2.
We claim that $h^2 \neq c_1$. Suppose not. Then $hC$ is an element of $H/C$ of order 2 and $hC_2$ is an element of $H/C_2$ of order 4. It follows that $hC_2$ has $\ell$ eigenvalues equal to $\omega$ and $\ell$ eigenvalues equal to $\omega^{-1}$ where $\omega \in K$ is a primitive fourth root of unity. Hence $C_{H/C_2}(hC_2)=A_{\ell}$. Using (\ref{e:spinsopso}), it follows that $d_2(G) > d_2(G_{s.c.})$, contradicting $d_2(G)\leq d_2(G_{s.c.})$.
Hence $h^2=1$ and so by (\ref{e:spinsopso}) $d_2(G)\geq d_2(G_{s.c.})$. Therefore $d_2(G)= d_2(G_{s.c.})$.\\
Suppose now that $u>2.$ Since by Theorem \ref{t:blawther} $d_u(G_{s.c.})-d_u(G_{a.})=2$, in order to show that $d_u(G)=d_u(G_{s.c.})$ it is enough by Lemma \ref{l:centmod2} to prove that $d_u(G)\neq d_u(G_{a.})$. Suppose for a contradiction that $d_u(G)=d_u(G_{a.})$. Let $hC_1$ be an element of $G=H/C_1$ of order $u$ such that $$\dim C_{H/C_1}(hC_1)=d_u(G_{a.}).$$
By Lemma \ref{l:dufactg} we have
\begin{equation}\label{e:spinsopsobis}
\dim C_H(h)=\dim C_{H/C_1}(hC_1)=\dim C_{H/C_2}(hC_2)=\dim C_{H/C}(hC).
\end{equation} Since $hC_1$ has order $u$, $h^u\in C_1$ and so $h^u=1$ or $h^u=c_1$. Also $hC$ has order dividing $u$ whereas $hC_2$ has order dividing $u$ or $2u$. Suppose $hC_2$ has order dividing $u$. Then $h^u\in C_1\cap C_2=1$ and so $h$ is an element of $G_{s.c.}$ of oder dividing $u$. Hence by (\ref{e:spinsopsobis}), $d_u(G_{s.c.})=d_u(G_{a.})$, a contradiction.\\
Suppose $hC_2$ has order dividing $2u$ but $(hC_2)^u\neq C_2$. Then $h^u=c_1$ and $(hC_2)^u=c_1C_2$. Let $h=xy$ be the Jordan decomposition of $h$ where $x$ is unipotent and $y$ is semisimple. Note that $(yC_2)^v=c_1C_2$. Without loss of generality, $yC_2$ is diagonal and $(yC_2)_{i,i}(yC_2)_{i+1,i+1}=1$ for every odd $i$ with $1\leq i \leq 2\ell-1$. Note that no eigenvalue of $yC_2$ is equal to $1$ or $-1$. Let $d$ be a semisimple element of $H$ such that $dC_2$ is diagonal, $(dC_2)_{i,i}(dC_2)_{i+1,i+1}=1$ for every odd $i$ with $1\leq i \leq 2\ell-1$ and $dC_2$ has $\ell$ eigenvalues equal to $\omega$ and $\ell$ eigenvalues equal to $\omega^{-1}$ where $\omega\in K$ is a primitive $2v$-th root of unity. Then $(dC_2)^v=c_1C_2$ and $(dyC_2)^v=C_2$.
Hence $dyC_2$ is an element of $H/C_2$ of order dividing $v$ and $C_{H/C_2}(dyC_2)\cong C_{H/C_2}(yC_2)$.
Since $\ell$ is even, it follows from Lemma \ref{l:soliftspin} that $d$ is a semisimple element of $H$ of order dividing $2v$ and so $dy$ is a semisimple element of $H$ of order dividing $v$.
Now by Lemma \ref{l:dufactg} $$d_q(C_H(dy)^0)=d_q(C_{H/C_2}(dyC_2)^0)$$ and so we obtain
\begin{eqnarray*}
d_q(C_H(dy)^0)& = & d_q(C_{H/C_2}(yC_2)^0)\\
& = & d_u(G_{a.}).
\end{eqnarray*}
Hence $d_u({G_{s.c.}})=d_u(G_{a.})$, a contradiction. $\square$
\section{Proof of Proposition \ref{p:dudecreasing}}\label{s:decreasing}
Given a simple algebraic group $G$ defined over an algebraically closed field $K$ of characteristic $p$, we now show that $d_u(G): \mathbb{N}\rightarrow \mathbb{N}$ is a decreasing function of $u$. We need to show that for every positive integer $u$, we have $d_u(G)-d_{u+1}(G)\geq 0$. Now by Proposition \ref{p:ingsimple}, we have $d_u(G_{a.})\leq d_u(G) \leq d_u(G_{s.c.})$ for every positive integer $u$. Hence the conclusion of the proposition will follow at once after we show that for every positive integer $u$, we have \begin{equation}\label{e:dasc} d_u(G_{a.})-d_{u+1}(G_{s.c.})\geq 0. \end{equation} If $G$ is of exceptional type then the result follows at once from \cite[Theorem 1]{Lawther} and Theorem \ref{t:blawther}. We therefore assume that $G$ is of classical type. \\ Note that if $u=1$ then $d_u(G_{a.})=d_1(G_{a.})=\dim G \geq d_{u+1}(G_{s.c.})$ and so (\ref{e:dasc}) holds. \\ Suppose now that $u\geq h$. Since $u\geq h$ we have $d_u(G_{a.})=\ell$. Also as $u+1> h$, Theorem \ref{t:blawther} yields $d_{u+1}(G_{s.c.})=d_{u+1}(G_{a})=\ell$. Therefore $d_u(G_{a.})=d_{u+1}(G_{s.c.})=\ell$ and so (\ref{e:dasc}) holds. \\ We can therefore assume that $2\leq u \leq h-1$. In particular $z\geq 1$. Write $h=z'(u+1)+e'$ where $z'$, $e'$ are nonnegative integers such that $0\leq e'\leq u.$ An easy check gives $1\leq z'\leq z.$ Write $z'=z-j$ where $0\leq j \leq z-1$. Then $e'=e-z+j(u+1)$. Note that if $j=1$ then $z\geq 2$. Also $z-e\leq j(u+1)\leq u+z-e$. We claim that either $z>2j-1$ or $j=1$. Suppose otherwise. Then as $z\geq 1$ we get $j\geq 2$ and $z\leq 2j-1$. Using the latter inequality and the fact that $j(u+1)\leq u+z-e$, we get $j\leq 1-e/(u-1)$ and so $j\leq 1$, a contradiction establishing the claim.\\
Suppose that $G$ is of type $A$. By Theorem \ref{t:blawther} we have $d_{u+1}(G_{s.c.})\leq d_{u+1}(G_{a.})+2$ and so \begin{equation}\label{e:daa} d_u(G_{a.})-d_{u+1}(G_{s.c.}) \geq d_u(G_{a.})-d_{u+1}(G_{a})-2.\end{equation} Let $\mathbf{D}= d_u(G_{a.})-d_{u+1}(G_{a})$. Note that Lemma \ref{l:centmod2} yields $d_{u}(G_{a.})\equiv d_{u+1}(G_{a.}) \mod 2$ and so $\mathbf{D}\equiv 0 \mod 2$. Using \cite[Theorem 1]{Lawther}, we have \begin{eqnarray*} \mathbf{D} & = & z^2u+e(2z+1)-1-((z-j)^2(u+1)+(e-z+j(u+1))(2(z-j)+1) -1)\\ & = & (u+1)j(j-1)+z(z-2j+1)+2je. \end{eqnarray*} Recalling that $z\geq 2$ if $j=1$, we note that if $j=1$ then $\mathbf{D}>0$. Also if $z>2j-1$ then $\mathbf{D}>0$. \\ Hence $\mathbf{D}>0$ in all cases.
As $\mathbf{D}\equiv 0 \mod 2$ we deduce that $\mathbf{D}\geq 2$ and so (\ref{e:daa}) yields $d_u(G_{a.})-d_{u+1}(G_{s.c.})\geq 0$. Hence (\ref{e:dasc}) holds.\\
Suppose that $G$ is of type $C$. By Theorem \ref{t:blawther} we have $d_{u+1}(G_{s.c.})\leq d_{u+1}(G_{a.})+2\epsilon_u\lceil (z-j)/2 \rceil$ and so \begin{equation}\label{e:daac} d_u(G_{a.})-d_{u+1}(G_{s.c.}) \geq d_u(G_{a.})-d_{u+1}(G_{a})-2\epsilon_u\left \lceil \frac{z-j}2 \right\rceil.\end{equation} Let $\mathbf{D}= d_u(G_{a.})-d_{u+1}(G_{a})$. Note that Lemma \ref{l:centmod2} yields $d_{u}(G_{a.})\equiv d_{u+1}(G_{a.}) \mod 2$ and so $\mathbf{D}\equiv 0 \mod 2$. Using \cite[Theorem 1]{Lawther}, we have \begin{eqnarray*} \mathbf{D} & = & \frac{z^2u+e(2z+1)}2+\epsilon_u\left\lceil \frac z2\right\rceil-\left(\frac{ (z-j)^2(u+1)+(e-z+j(u+1))(2(z-j)+1)}2\right.\\ & & \left. +\epsilon_{u+1}\left\lceil \frac{z-j}2\right \rceil \right)\\ & = & \frac{ (u+1)j(j-1)+z(z-2j+1)+2je}2+\epsilon_u \left(\left\lceil \frac{z-j}2 \right\rceil +\left\lceil \frac z2 \right\rceil\right)-\left \lceil \frac{z-j}2 \right\rceil. \end{eqnarray*}
Recalling that $z\geq 2$ if $j=1$, we note that if $j=1$ then $\mathbf{D}\geq 0$ and moreover $\mathbf{D}\geq (z^2+\epsilon_z)/2\geq z+\epsilon_z=2\lceil z/2\rceil $ for $u$ odd. Also if $z>2j-1$ then $\mathbf{D}\geq 0$ and moreover $\mathbf{D}\geq 2\lceil z/2\rceil$ for $u$ odd. \\ In all cases we therefore have $\mathbf{D}\geq 0$ and moreover $\mathbf{D}\geq 2\lceil z/2\rceil$ for $u$ odd.
Now (\ref{e:daac}) yields $d_u(G_{a.})-d_{u+1}(G_{s.c.})\geq 0$. Hence (\ref{e:dasc}) holds.\\
Suppose that $G$ is of type $B$. By Theorem \ref{t:blawther} we have $d_{u+1}(G_{s.c.})\leq d_{u+1}(G_{a.})+2\epsilon_u$ and so \begin{equation}\label{e:daab} d_u(G_{a.})-d_{u+1}(G_{s.c.}) \geq d_u(G_{a.})-d_{u+1}(G_{a})-2\epsilon_u.\end{equation} Let $\mathbf{D}= d_u(G_{a.})-d_{u+1}(G_{a})$. Note that Lemma \ref{l:centmod2} yields $d_{u}(G_{a.})\equiv d_{u+1}(G_{a.}) \mod 2$ and so $\mathbf{D}\equiv 0 \mod 2$. Using \cite[Theorem 1]{Lawther}, we have \begin{eqnarray*} \mathbf{D} & = & \frac{z^2u+e(2z+1)}2+\epsilon_u\left\lceil \frac z2\right\rceil-\left(\frac{ (z-j)^2(u+1)+(e-z+j(u+1))(2(z-j)+1)}2\right.\\ & & \left. +\epsilon_{u+1}\left\lceil \frac{z-j}2\right \rceil \right)\\ & = & \frac{ (u+1)j(j-1)+z(z-2j+1)+2je}2+\epsilon_u \left(\left\lceil \frac{z-j}2 \right\rceil +\left\lceil \frac z2 \right\rceil\right)-\left \lceil \frac{z-j}2 \right\rceil. \end{eqnarray*} Arguing as in the case where $G$ is of type $C$, we deduce that $\mathbf{D}\geq 0$ and moreover $\mathbf{D}\geq 2$ for $u$ odd. Now (\ref{e:daab}) yields $d_u(G_{a.})-d_{u+1}(G_{s.c.})\geq 0$. Hence (\ref{e:dasc}) holds.\\
Suppose that $G$ is of type $D$. Since $u\geq 2$, by Theorem \ref{t:blawther} we have $d_{u+1}(G_{s.c.})\leq d_{u+1}(G_{a.})+2\epsilon_u$ and so \begin{equation*} d_u(G_{a.})-d_{u+1}(G_{s.c.}) \geq d_u(G_{a.})-d_{u+1}(G_{a})-2\epsilon_u.\end{equation*} Let $\mathbf{D}= d_u(G_{a.})-d_{u+1}(G_{a})$. Note that Lemma \ref{l:centmod2} yields $d_{u}(G_{a.})\equiv d_{u+1}(G_{a.}) \mod 2$ and so $\mathbf{D}\equiv 0 \mod 2$. Using \cite[Theorem 1]{Lawther}, we have \begin{eqnarray*} \mathbf{D} & = & \frac{ (u+1)j(j-1)+z(z-2j+1)+2je}2+\epsilon_u \left(\left\lceil \frac{z-j}2 \right\rceil +\left\lceil \frac z2 \right\rceil\right)-\left \lceil \frac{z-j}2 \right\rceil+j+\epsilon_{z-j}-\epsilon_z. \end{eqnarray*} Arguing as in the case where $G$ is of type $B$, we deduce that $\mathbf{D}\geq 0$ and moreover $\mathbf{D}\geq 2$ for $u$ odd. We deduce that $d_u(G_{a.})-d_{u+1}(G_{s.c.})\geq 0$. Hence (\ref{e:dasc}) holds. $\square$
\section{Proof of Theorem \ref{t:classification}}\label{s:classification}
In this section we prove Theorem \ref{t:classification}. We proceed in two steps, first in the special case where $G$ is of simply connected type (see Proposition \ref{p:classificationsc}), and then we consider other types (see Proposition \ref{p:classificationa}).
\begin{prop}\label{p:classificationsc} Let $G$ be a simple simply connected algebraic group over an algebraic closed field $K$ of prime characteristic $p$. The classification of the hyperbolic triples $(a,b,c)$ of integers for $G$ is as given in Theorem \ref{t:classification}. \end{prop}
\begin{proof} As usual we let $h$ be the Coxeter number of $G$ and for a positive integer $u$, we let $z$ and $e$ be the nonnegative integers such that $h=zu+e$ and $0\leq e \leq u-1$. Also $d_u(G)$ denotes the minimal dimension of the centralizer of an element of $G$ of order dividing $u$. By Proposition \ref{p:fcu} $d_u(G)$ is the codimension of the subvariety $G_{[u]}$ of $G$ consisting of elements of order dividing $u$. Writing $j_u(G)=\dim G_{[u]}$, we have $d_u(G)=\dim G-j_u(G)$. \\ Let $(a,b,c)$ be a hyperbolic triple of integers. Let $\mathbf{S}_{(a,b,c)}=d_a(G)+d_b(G)+d_c(G)$ and $\mathbf{D}_{(a,b,c)}=\mathbf{S}_{(a,b,c)}-\dim G$. Saying that $(a,b,c)$ is reducible (respectively, rigid, nonrigid) for $G$ amounts to saying that $\mathbf{D}_{(a,b,c)}$ is greater than (respectively equal to, less than) 0. \\ Recall the partial order we put on the set of hyperbolic triples of integers. Given two hyperbolic triples $(a,b,c)$ and $(a',b',c')$ of integers, we say that $(a,b,c)\leq (a',b',c')$ if and only if $a\leq a'$, $b\leq b'$ and $c\leq c'$. Among all hyperbolic triples, exactly three are minimal: $(2,3,7)$, $(3,3,4)$ and $(2,4,5)$.
By Proposition \ref{p:dudecreasing}, if $(a,b,c)$ is nonrigid for $G$ then $(a',b',c')$ is nonrigid for $G$ for every $(a',b',c')\geq (a,b,c)$.\\
For a nonnegative integer $p$, we let $\theta_p \in \{0,1\}$ be such that $\theta_p=1$ if $p$ is odd or $p=0$, otherwise $\theta_p=0$.\\
Suppose first that $G$ is of exceptional type. By Theorem \ref{t:blawther} and \cite[Theorem 1]{Lawther}, every hyperbolic triple $(a,b,c)$ of integers is nonrigid for $G$ unless $G$ is of type $G_2$ and $(a,b,c)\in\{(2,4,5),(2,5,5)\}$ in which case $(a,b,c)$ is rigid for $G$. \\
Suppose now that $G$ is of classical type. In a first step we show that if $\ell \geq 11$, $\ell \geq 10$, $\ell \geq 15$, or $\ell \geq 9$ accordingly respectively as $G$ is of type $A$, $B$, $C$ or $D$, then every hyperbolic triples of integers is nonrigid for $G$.\\ Suppose first that $G$ is of type $A$. Note that $\dim G=h^2-1$. By Theorem \ref{t:blawther} and \cite[Theorem 1]{Lawther}, we have \begin{eqnarray*} d_u(G)&\leq& z^2u+e(2z+1)-1+2\theta_p(1-\epsilon_u)\\ & = & \frac{(h-e)(h+e)}u+e-1+2\theta_p(1-\epsilon_u). \end{eqnarray*} Let $g(e)= \frac{(h-e)(h+e)}u+e-1+2\theta_p(1-\epsilon_u).$ Then $g'(e)=\frac{u-2e}u$, and $g'(e)>0$ if and only if $e<u/2$. Hence $d_u(G)\leq F(u)$ where \begin{eqnarray*} F(u) & =& g(u/2)\\ & = & \frac{u^2-4u+4h^2}{4u}+2\theta_p(1-\epsilon_u). \end{eqnarray*} In particular, for any hyperbolic triple $(a,b,c)$ of integers, we have $\mathbf{S}_{(a,b,c)}\leq F(a)+F(b)+F(c)$.\\ Suppose $(a,b,c)=(2,3,7)$. We have $\mathbf{S}_{(2,3,7)}\leq F(2)+F(3)+F(7)$. Now $$F(2)=\frac{h^2-1}2+2\theta_p, \quad F(3)=\frac{4h^2-3}{12}, \quad F(7)=\frac{4h^2+21}{28}.$$ Hence $$\mathbf{S}_{(2,3,7)}\leq \frac{41h^2}{42}+2\theta_p.$$ and $$\mathbf{D}_{(2,3,7)}\leq -\frac{h^2}{42}+2\theta_p+1.$$ Since $-\frac{h^2}{42}+2\theta_p+1$ is negative for $h \geq 12$, it follows that $(2,3,7)$ is nonrigid for $G$ provided $h\geq 12$, that is $\ell\geq 11$. Also every hyperbolic triple $(a',b',c')$ of integers with $(a',b',c')\geq (2,3,7)$ is nonrigid for $G$ with $\ell\geq 11$.\\ Suppose $(a,b,c)=(2,4,5)$. We have $\mathbf{S}_{(2,4,5)}\leq F(2)+F(4)+F(5)$. Now $$F(2)=\frac{h^2-1}2+2\theta_p, \quad F(4)=\frac{h^2}{4}+2\theta_p, \quad F(5)=\frac{4h^2+5}{20}.$$ Hence $$\mathbf{S}_{(2,4,5)}\leq \frac{19h^2}{20}+4\theta_p-\frac14.$$ and $$\mathbf{D}_{(2,4,5)}\leq -\frac{h^2}{20}+\frac{3}4+4\theta_p.$$ Since $-\frac{h^2}{20}+4\theta_p+\frac 34$ is negative for $h \geq 10$, it follows that $(2,4,5)$ is nonrigid for $G$ provided $h\geq 10$, that is $\ell\geq 9$. Also every hyperbolic triple $(a',b',c')$ of integers with $(a',b',c')\geq (2,4,5)$ is nonrigid for $G$ with $\ell\geq 9$.\\ Suppose $(a,b,c)=(3,3,4)$. We have $\mathbf{S}_{(3,3,4)}\leq F(3)+F(3)+F(4)$. Now $$ F(3)=\frac{4h^2-3}{12}, \quad F(4)=\frac{h^2}{4}+2\theta_p.$$ Hence $$\mathbf{S}_{(3,3,4)}\leq \frac{11h^2}{12}+2\theta_p-\frac12.$$ and $$\mathbf{D}_{(3,3,4)}\leq -\frac{h^2}{12}+\frac{1}2+2\theta_p.$$ Since $-\frac{h^2}{12}+2\theta_p+\frac 12$ is negative for $h \geq 6$, it follows that $(3,3,4)$ is nonrigid for $G$ provided $h\geq 6$, that is $\ell\geq 5$. Also every hyperbolic triple $(a',b',c')$ of integers with $(a',b',c')\geq (3,3,4)$ is nonrigid for $G$ with $\ell\geq 5$.\\
Suppose now that $G$ is of type $B$. Note that $\dim G=h(h+1)/2$. By Theorem \ref{t:blawther} and \cite[Theorem 1]{Lawther}, we have \begin{eqnarray*} d_u(G)&\leq& \frac{z^2u+e(2z+1)}2+\frac{z+1}2+2\theta_p(1-\epsilon_u)\\ & = & \frac{(h-e)(h+e+1)}{2u}+\frac{e+1}2+2\theta_p(1-\epsilon_u). \end{eqnarray*} Let $g(e)= \frac{(h-e)(h+e+1)}{2u}+\frac{e+1}2+2\theta_p(1-\epsilon_u).$ Then $g'(e)=\frac{u-2e-1}{2u}$, and $g'(e)>0$ if and only if $e<(u-1)/2$. Hence $d_u(G)\leq F(u)$ where \begin{eqnarray*} F(u) & =& g((u-1)/2)\\ & = & \frac{u^2+2u+(2h+1)^2}{8u}+2\theta_p(1-\epsilon_u). \end{eqnarray*} In particular, for any hyperbolic triple $(a,b,c)$ of integers, we have $\mathbf{S}_{(a,b,c)}\leq F(a)+F(b)+F(c)$.\\ Suppose $(a,b,c)=(2,3,7)$. We have $\mathbf{S}_{(2,3,7)}\leq F(2)+F(3)+F(7)$. Now $$F(2)=\frac{4h^2+4h+9}{16}+2\theta_p, \quad F(3)=\frac{h^2+h+4}{6}, \quad F(7)=\frac{h^2+h+16}{14}.$$ Hence $$\mathbf{S}_{(2,3,7)}\leq \frac{164h^2+164h+797}{336}+2\theta_p.$$ and $$\mathbf{D}_{(2,3,7)}\leq -\frac{4h^2+4h-797}{336}+2\theta_p.$$ Since $-\frac{4h^2+4h-797}{336}+2\theta_p$ is negative for $h \geq 20$, it follows that $(2,3,7)$ is nonrigid for $G$ provided $h\geq 20$, that is $\ell\geq 10$. Also every hyperbolic triple $(a',b',c')$ of integers with $(a',b',c')\geq (2,3,7)$ is nonrigid for $G$ with $\ell\geq 10$.\\ Suppose $(a,b,c)=(2,4,5)$. We have $\mathbf{S}_{(2,4,5)}\leq F(2)+F(4)+F(5)$. Now $$F(2)=\frac{4h^2+4h+9}{16}+2\theta_p, \quad F(4)=\frac{4h^2+4h+25}{32}+2\theta_p, \quad F(5)=\frac{h^2+h+9}{10}.$$ Hence $$\mathbf{S}_{(2,4,5)}\leq \frac{76h^2+76h+359}{160}+4\theta_p.$$ and $$\mathbf{D}_{(2,4,5)}\leq -\frac{4h^2+4h-359}{160}+4\theta_p.$$ Since $ -\frac{4h^2+4h-359}{160}+4\theta_p$ is negative for $h \geq 16$, it follows that $(2,4,5)$ is nonrigid for $G$ provided $h\geq 16$, that is $\ell\geq 8$. Also every hyperbolic triple $(a',b',c')$ of integers with $(a',b',c')\geq (2,4,5)$ is nonrigid for $G$ with $\ell\geq 8$.\\ Suppose $(a,b,c)=(3,3,4)$. We have $\mathbf{S}_{(3,3,4)}\leq F(3)+F(3)+F(4)$. Now $$ F(3)=\frac{h^2+h+4}{6}, \quad F(4)=\frac{4h^2+4h+25}{32}+2\theta_p.$$ Hence $$\mathbf{S}_{(3,3,4)}\leq \frac{44h^2+44h+203}{96}+2\theta_p.$$ and $$\mathbf{D}_{(3,3,4)}\leq -\frac{4h^2+4h-203}{96}+2\theta_p.$$ Since $-\frac{4h^2+4h-203}{96}+2\theta_p$ is negative for $h \geq 10$, it follows that $(3,3,4)$ is nonrigid for $G$ provided $h\geq 10$, that is $\ell\geq 5$. Also every hyperbolic triple $(a',b',c')$ of integers with $(a',b',c')\geq (3,3,4)$ is nonrigid for $G$ with $\ell\geq 5$.\\
Suppose now that $G$ is of type $C$. Note that $\dim G=h(h+1)/2$. By Theorem \ref{t:blawther} and \cite[Theorem 1]{Lawther}, we have \begin{eqnarray*} d_u(G)&\leq& \left\{\begin{array}{ll} \frac{z^2u+e(2z+1)}2+z+1 & \textrm{if} \ u \ \textrm{is even}\\ \frac{z^2u+e(2z+1)}2+\frac{z+1}2 & \textrm{if} \ u \ \textrm{is odd}\\
\end{array} \right. \\ & = & \left\{\begin{array}{ll} \frac{(h-e)(h+e+2)}{2u}+\frac{e}2+1 & \textrm{if} \ u \ \textrm{is even}\\
\frac{(h-e)(h+e+1)}{2u}+\frac{e+1}2& \textrm{if} \ u \ \textrm{is even}
\end{array} \right. \end{eqnarray*} Let $$g(e)= \left\{\begin{array}{ll} \frac{(h-e)(h+e+2)}{2u}+\frac{e}2+1 & \textrm{if} \ u \ \textrm{is even}\\
\frac{(h-e)(h+e+1)}{2u}+\frac{e+1}2& \textrm{if} \ u \ \textrm{is odd}
\end{array} \right. $$ Then $$g'(e)=\left\{\begin{array}{ll} \frac{u-2e-2}{2u} & \textrm{if} \ u \ \textrm{is even}\\ \frac{u-2e-1}{2u} & \textrm{if} \ u \ \textrm{is odd} \end{array}\right.$$ and if $u$ is even then $g'(e)>0$ if and only if $e<(u-2)/2$, else $g'(e)>0$ if and only if $e<(u-1)/2$. Hence $d_u(G)\leq F(u)$ where \begin{eqnarray*} F(u) & =&\left\{\begin{array}{ll} g((u-2)/2) & \textrm{if} \ u \ \textrm{is even} \\ g((u-1)/2) & \textrm{if} \ u \ \textrm{is odd} \end{array} \right. \\ & = & \left \{\begin{array}{ll} \frac{u^2+4u+4(h+1)^2}{8u} & \textrm{if} \ u \ \textrm{is even}\\ \frac{u^2+2u+(2h+1)^2}{8u} & \textrm{if} \ u \ \textrm{is odd} \\
\end{array}\right . \end{eqnarray*} In particular, for any hyperbolic triple $(a,b,c)$ of integers, we have $\mathbf{S}_{(a,b,c)}\leq F(a)+F(b)+F(c)$.\\ Suppose $(a,b,c)=(2,3,7)$. We have $\mathbf{S}_{(2,3,7)}\leq F(2)+F(3)+F(7)$. Now $$F(2)=\frac{h^2+2h+4}{4}, \quad F(3)=\frac{h^2+h+4}{6}, \quad F(7)=\frac{h^2+h+16}{14}.$$ Hence $$\mathbf{S}_{(2,3,7)}\leq \frac{41h^2+62h+236}{84}.$$ and $$\mathbf{D}_{(2,3,7)}\leq -\frac{h^2-20h-236}{84}.$$ Since $-\frac{h^2-20h-236}{84}$ is negative for $h \geq 30$, it follows that $(2,3,7)$ is nonrigid for $G$ provided $h\geq 30$, that is $\ell\geq 15$. Also every hyperbolic triple $(a',b',c')$ of integers with $(a',b',c')\geq (2,3,7)$ is nonrigid for $G$ with $\ell\geq 15$.\\ Suppose $(a,b,c)=(2,4,5)$. We have $\mathbf{S}_{(2,4,5)}\leq F(2)+F(4)+F(5)$. Now $$F(2)=\frac{h^2+2h+4}{4}, \quad F(4)=\frac{h^2+2h+9}{8}, \quad F(5)=\frac{h^2+h+9}{10}.$$ Hence $$\mathbf{S}_{(2,4,5)}\leq \frac{19h^2+34h+121}{40}$$ and $$\mathbf{D}_{(2,4,5)}\leq -\frac{h^2-14h-121}{40}.$$ Since $ -\frac{h^2-14h-121}{40}$ is negative for $h \geq 22$, it follows that $(2,4,5)$ is nonrigid for $G$ provided $h\geq 22$, that is $\ell\geq11$. Also every hyperbolic triple $(a',b',c')$ of integers with $(a',b',c')\geq (2,4,5)$ is nonrigid for $G$ with $\ell\geq 11$.\\ Suppose $(a,b,c)=(3,3,4)$. We have $\mathbf{S}_{(3,3,4)}\leq F(3)+F(3)+F(4)$. Now $$ F(3)=\frac{h^2+h+4}{6}, \quad F(4)=\frac{h^2+2h+9}{8}.$$ Hence $$\mathbf{S}_{(3,3,4)}\leq \frac{11h^2+14h+59}{24}.$$ and $$\mathbf{D}_{(3,3,4)}\leq -\frac{h^2-2h-59}{24}.$$ Since $-\frac{h^2-2h-59}{24}$ is negative for $h \geq 10$, it follows that $(3,3,4)$ is nonrigid for $G$ provided $h\geq 10$, that is $\ell\geq 5$. Also every hyperbolic triple $(a',b',c')$ of integers with $(a',b',c')\geq (3,3,4)$ is nonrigid for $G$ with $\ell\geq 5$.\\
Suppose finally that $G$ is of type $D$. Note that $\dim G=(h+1)(h+2)/2$. By Theorem \ref{t:blawther} and \cite[Theorem 1]{Lawther}, we have \begin{eqnarray*} d_u(G)&\leq& \left\{\begin{array}{ll} \frac{z^2u+e(2z+1)}2+z+1+4\theta_p\sigma_{u-2}+2\theta_p(1-\sigma_{u-2}) & \textrm{if} \ u \ \textrm{is even}\\ \frac{z^2u+e(2z+1)}2+\frac{3(z+1)}2 & \textrm{if} \ u \ \textrm{is odd}\\
\end{array} \right. \\ & = & \left\{\begin{array}{ll} \frac{(h-e)(h+e+2)}{2u}+\frac{e}2+1+ 4\theta_p\sigma_{u-2}+2\theta_p(1-\sigma_{u-2}) & \textrm{if} \ u \ \textrm{is even}\\
\frac{(h-e)(h+e+3)}{2u}+\frac{e+3}2& \textrm{if} \ u \ \textrm{is odd}
\end{array} \right. \end{eqnarray*} Let $$g(e)= \left\{\begin{array}{ll} \frac{(h-e)(h+e+2)}{2u}+\frac{e}2+1+4\theta_p\sigma_{u-2}+2\theta_p(1-\sigma_{u-2}) & \textrm{if} \ u \ \textrm{is even}\\
\frac{(h-e)(h+e+3)}{2u}+\frac{e+3}2& \textrm{if} \ u \ \textrm{is odd}
\end{array} \right. $$ Then $$g'(e)=\left\{\begin{array}{ll} \frac{u-2e-2}{2u} & \textrm{if} \ u \ \textrm{is even}\\ \frac{u-2e-3}{2u} & \textrm{if} \ u \ \textrm{is odd} \end{array}\right.$$ and if $u$ is even then $g'(e)>0$ if and only if $e<(u-2)/2$, else $g'(e)>0$ if and only if $e<(u-3)/2$. Hence $d_u(G)\leq F(u)$ where \begin{eqnarray*} F(u) & =&\left\{\begin{array}{ll} g((u-2)/2) & \textrm{if} \ u \ \textrm{is even} \\ g((u-3)/2) & \textrm{if} \ u \ \textrm{is odd} \end{array} \right. \\ & = & \left \{\begin{array}{ll} \frac{u^2+4u+4(h+1)^2}{8u}+4\theta_p\sigma_{u-2}+2\theta_p(1-\sigma_{u-2}) & \textrm{if} \ u \ \textrm{is even}\\ \frac{u^2+6u+(2h+3)^2}{8u} & \textrm{if} \ u \ \textrm{is odd} \\
\end{array}\right . \end{eqnarray*} In particular, for any hyperbolic triple $(a,b,c)$ of integers, we have $\mathbf{S}_{(a,b,c)}\leq F(a)+F(b)+F(c)$.\\ Suppose $(a,b,c)=(2,3,7)$. We have $\mathbf{S}_{(2,3,7)}\leq F(2)+F(3)+F(7)$. Now $$F(2)=\frac{h^2+2h+4}{4}+4\theta_p, \quad F(3)=\frac{h^2+3h+9}{6}, \quad F(7)=\frac{h^2+3h+25}{14}.$$ Hence $$\mathbf{S}_{(2,3,7)}\leq \frac{41h^2+102h+360}{84}+4\theta_p.$$ and $$\mathbf{D}_{(2,3,7)}\leq -\frac{h^2+24h-276}{84}+4\theta_p.$$ Since $-\frac{h^2+24h-276}{84}+4\theta_p$ is negative for $h \geq 16$, it follows that $(2,3,7)$ is nonrigid for $G$ provided $h\geq 16$, that is $\ell\geq 9$. Also every hyperbolic triple $(a',b',c')$ of integers with $(a',b',c')\geq (2,3,7)$ is nonrigid for $G$ with $\ell\geq 9$.\\ Suppose $(a,b,c)=(2,4,5)$. We have $\mathbf{S}_{(2,4,5)}\leq F(2)+F(4)+F(5)$. Now $$F(2)=\frac{h^2+2h+4}{4}+4\theta_p, \quad F(4)=\frac{h^2+2h+9}{8}+2\theta_p, \quad F(5)=\frac{h^2+3h+16}{10}.$$ Hence $$\mathbf{S}_{(2,4,5)}\leq \frac{19h^2+42h+149}{40}+6\theta_p$$ and $$\mathbf{D}_{(2,4,5)}\leq -\frac{h^2+18h-109}{40}+6\theta_p.$$ Since $ -\frac{h^2+18h-109}{40}+6\theta_p$ is negative for $h \geq 12$, it follows that $(2,4,5)$ is nonrigid for $G$ provided $h\geq 12$, that is $\ell\geq7$. Also every hyperbolic triple $(a',b',c')$ of integers with $(a',b',c')\geq (2,4,5)$ is nonrigid for $G$ with $\ell\geq 7$.\\ Suppose $(a,b,c)=(3,3,4)$. We have $\mathbf{S}_{(3,3,4)}\leq F(3)+F(3)+F(4)$. Now $$ F(3)=\frac{h^2+3h+9}{6}, \quad F(4)=\frac{h^2+2h+9}{8}+2\theta_p.$$ Hence $$\mathbf{S}_{(3,3,4)}\leq \frac{11h^2+30h+99}{24}+2\theta_p.$$ and $$\mathbf{D}_{(3,3,4)}\leq -\frac{h^2+6h-75}{24}+2\theta_p.$$ Since $-\frac{h^2+6h-75}{24}+2\theta_p$ is negative for $h \geq 10$, it follows that $(3,3,4)$ is nonrigid for $G$ provided $h\geq 10$, that is $\ell\geq 6$. Also every hyperbolic triple $(a',b',c')$ of integers with $(a',b',c')\geq (3,3,4)$ is nonrigid for $G$ with $\ell\geq 6$.\\
Finally to classify hyperbolic triples $(a,b,c)$ of integers for $G$ with $\ell \leq 10$, $\ell \leq 9$, $\ell \leq 14$, $\ell \leq 8$ accordingly respectively as $G$ is of type $A$, $B$, $C$, $D$ one can use Theorem \ref{t:blawther} and \cite[Theorem 1]{Lawther} to find $d_a(G)$, $d_b(G)$ and $d_c(G)$ and compute $\mathbf{D}_{(a,b,c)}$. Note that if $u>h$ then $d_u(G)=\ell$. \end{proof}
\begin{prop}\label{p:classificationa} Let $G$ be a simple algebraic group over an algebraic closed field $K$ of prime characteristic $p$. The classification of hyperbolic triples $(a,b,c)$ of integers for $G$ is as given in Theorem \ref{t:classification}. \end{prop}
\begin{proof}
The case where $G$ is of simply connected type is treated in Proposition \ref{p:classificationsc}. We therefore suppose that $G$ is not of simply connected type.
We denote by $G_{s.c.}$ (respectively, $G_{a.}$) the simple algebraic group of simply connected (respectively, adjoint) type having the same Lie type and Lie rank as $G$. \\
For an integer $u$, let $d_u(G)$ denote the minimal dimension of the centralizer of an element of $G$ of order dividing $u$. By Proposition \ref{p:fcu} $d_u(G)$ is the codimension of the subvariety $G_{[u]}$ of $G$ consisting of elements of order dividing $u$. Writing $j_u(G)=\dim G_{[u]}$, we have $d_u(G)=\dim G-j_u(G)$. \\ Let $(a,b,c)$ be a hyperbolic triple of integers. Let $\mathbf{S}_{(a,b,c)}=d_a(G)+d_b(G)+d_c(G)$ and $\mathbf{D}_{(a,b,c)}=\mathbf{S}_{(a,b,c)}-\dim G$. Recall that saying that $(a,b,c)$ is reducible (respectively, rigid, nonrigid) for $G$ amounts to saying that $\mathbf{D}_{(a,b,c)}$ is greater than (respectively equal to, less than) 0. \\
By Proposition \ref{p:ingsimple}, given a positive integer $u$, we have $d_u(G)\leq d_u(G_{s.c.})$.
It follows that every hyperbolic triple of integers which is nonrigid for $G_{s.c.}$ is nonrigid for $G$. In particular, by Proposition \ref{p:classificationsc}, we can now assume that $G$ is of classical type. By the proof of Proposition \ref{p:classificationsc} if $G$ is such that $\ell \leq 11$, $\ell \leq 10$, $\ell \leq 15$, $\ell \leq 9$ accordingly respectively as $G$ is of type $A$, $B$, $C$, $D$ then every hyperbolic triple of integers is nonrigid for $G$. For $G$ of small rank, one can use \cite[Theorem 1]{Lawther} and Theorems \ref{t:blawther} and \ref{t:duotsca} to find $d_a(G)$, $d_b(G)$ and $d_c(G)$, compute $\mathbf{D}_{(a,b,c)}$ and classify a given hyperbolic triple $(a,b,c)$ of integers for $G$. Note that if $u> h$ then $d_u(G)=\ell$.
\end{proof}
\section{Proof of Proposition \ref{p:marionred}}\label{s:reducibility}
We consider Proposition \ref{p:marionred}. Let $G$ be a simple algebraic group over an algebraically closed field $K$. The main ingredient in the proof of Proposition \ref{p:marionred} is the following result proved in \cite{Marionconj} combined with the classification given in Theorem \ref{t:classification} of the reducible and the rigid hyperbolic triples of integers for simple algebraic groups
if $p$ is a bad prime for $G$ or $G$ is of exceptional type. Recall that $p$ is said to be bad for $G$ if $G$ is of type $B_\ell$, $C_\ell$, $D_\ell$ and $p=2$, or of type $G_2$, $F_4$, $E_6$, $E_7$ and $p\in\{2,3\}$, or of type $E_8$ and $p \in \{2,3,5\}$. A prime $p$ is said to be good for $G$ if it is not bad for $G$. Also in the statement below and from now on, by an irreducible subgroup of a classical group $G$, we mean a subgroup acting irreducibly on the natural module for $G$.
\begin{prop}\label{p:reducibility} \cite[Proposition 2.1]{Marionconj}.
Suppose that $G$ is of classical type and $p$ is a good prime for $G$. If $g_1$, $g_2$, $g_3$ are elements of $G$ such that $g_1g_2g_3=1$ and $\langle g_1,g_2\rangle$ is an irreducible subgroup of $G$ then
$$\dim g_1^G+\dim g_2^G+\dim g_3^G\geq 2 \dim G.$$
\end{prop}
We can now prove Proposition \ref{p:marionred}.\\
\noindent \textit{Proof of Proposition \ref{p:marionred}.}
If $G$ is of classical type and $p$ is a good prime for $G$ then the result follows from Proposition \ref{p:reducibility}. The remaining cases follow from Theorem \ref{t:classification} which shows that there are no reducible hyperbolic triples of integers for $G$ if $G$ is of symplectic or orthogonal type and $p= 2$, or if $G$ is of exceptional type.
$\hspace{50mm} \square$\\
\section{Some tables}\label{s:tables}
In this section we collect Tables \ref{t:asc}-\ref{ta:casestoconsider} of the paper.
Tables \ref{t:asc}-\ref{t:dsc} appear in \S \ref{s:upc} and Table \ref{ta:casestoconsider} appears in \S \ref{s:pc}.
\begin{table}[h] \caption{A semisimple element $y$ of $G=(A_{\ell})_{\rm{s.c}}$ of order $v$ giving an upper bound for $d_u(G)$}\label{t:asc} \small{{
\begin{tabular}{| l | l | l | l|} \hline Case & Element $y$ & $C_{G_{s.c.}}(y)^0$ & $d_q(C_{G_{s.c.}}(y)^0)$\\ \hline $\begin{array}{l} \epsilon_v=1,\\ \textrm{or} \ (\epsilon_v,\epsilon_\alpha)=(0,0) \end{array}$& $ M_{1}^\alpha\oplus M_{2}\oplus (1)^{\epsilon_\beta}$& $A_\alpha^\beta A_{\alpha-1}^{v-\beta}T_{v-1}$ & $z^2u+e(2z+1)-1$\\ \cline{1-2} $(\epsilon_v,\epsilon_{\alpha},\epsilon_{\beta})=(0,1,1)$ & $M_{1}^\alpha\oplus M_{2}\oplus (-1)$ & & \\ \cline{1-2} $\begin{array}{l}(\epsilon_v,\epsilon_{\alpha},\epsilon_{\beta})=(0,1,0), \\ \beta\geq 2\end{array}$ & $M_1^\alpha\oplus M_5\oplus (1)\oplus (-1)$& & \\ \hline $(\epsilon_v,\epsilon_{\alpha})=(0,1)$, $\beta=0$ & $M_1^{\alpha-1}\oplus M_4\oplus (-1)$& $A_\alpha A_{\alpha-1}^{v-2} A_{\alpha-2}T_{v-1}$& $\left\{\begin{array}{ll} z^2u+e(2z+1)-1& \textrm{if} \ e>0\\ z^2u+e(2z+1)+1& \textrm{if} \ e=0 \end{array}\right.$\\ \hline \end{tabular}}} \end{table}
\begin{table}[h] \caption{A semisimple element $y$ of $G=(C_{\ell})_{\rm{s.c}}$ of order $v$ giving an upper bound for $d_u(G)$}\label{t:csc} \small{{
\begin{tabular}{| l | l | l | l|} \hline Case & Element $y$ & $C_G(y)^0$ & $d_q(C_G(y)^0)$\\ \hline
$\epsilon_v=1$ & $M_{1}^\alpha\oplus M_{2}\oplus (1)^{\epsilon_\beta}$& $A_\alpha^{\lfloor\frac\beta2\rfloor} A_{\alpha-1}^{\frac{v-1}2-\lfloor\frac\beta2\rfloor}C_{\lceil \frac \alpha2\rceil}T_{\frac{v-1}2}$ & $\frac{(z^2u+e(2z+1))}2+\lceil \frac z2 \rceil \epsilon_u$\\
\hline
$(\epsilon_v,\epsilon_\alpha)=(0,0)$ & $ M_{1}^\alpha\oplus M_{2}$& $A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}C_{\frac \alpha2}^2T_{\frac{v}2-1}$ & $\frac{(z^2u+e(2z+1))}2+2\lceil \frac z2 \rceil$\\ \cline{1-3} $(\epsilon_v,\epsilon_\alpha)=(0,1)$, $\beta\geq 2$ & $ M_{1}^\alpha\oplus M_{5}\oplus(1)\oplus(-1)$& $A_\alpha^{\frac\beta2-1} A_{\alpha-1}^{\frac{v}2-\frac\beta2}C_{\frac{\alpha+1}2}^2T_{\frac{v}2-1}$ & \\ \cline{1-3} $(\epsilon_v,\epsilon_\alpha)=(0,1)$, $\beta=0$ & $ M_{1}^{\alpha-1}\oplus M_{4}\oplus(-1)$& $A_{\alpha-1}^{\frac{v}2-1}C_{\frac{\alpha+1}2}C_{\frac{\alpha-1}2}T_{\frac{v}2-1}$ & \\ \hline \end{tabular}}} \end{table}
\begin{sidewaystable}[h] \caption{A semisimple element $y$ of $G=(B_{\ell})_{\rm{s.c}}$ of order $v$ giving an upper bound for $d_u(G)$}\label{t:bsc} \footnotesize{{
\begin{tabular}{| l | l | l | l|} \hline Case & Element $\overline{y}$ & $C_{{\rm SO}_{h+1}(K)}(\overline{y})^0$ & $d_q(C_G(y)^0)$\\ \hline
(a) & $ M_{1}^\alpha\oplus M_{2}\oplus (1)^{1+\epsilon_\beta}$& $A_\alpha^{\lfloor\frac\beta2\rfloor} A_{\alpha-1}^{\frac{v-1}2-\lfloor\frac\beta2\rfloor}B_{\lceil \frac \alpha2\rceil}T_{\frac{v-1}2}$ & $\frac12(z^2u+e(2z+1))+\lceil \frac z2 \rceil \epsilon_u$\\
\hline
(b) & $\left\{\begin{array}{ll}(1)^{\ell+1}\oplus(-1)^{\ell} & \textrm{if} \ \ell \equiv 0 \ (4) \\
(1)^{\ell+2}\oplus(-1)^{\ell-1} & \textrm{if} \ \ell \equiv 1 \ (4)\\
(1)^{\ell-1}\oplus(-1)^{\ell+2} & \textrm{if} \ \ell \equiv 2 \ (4)\\
(1)^{\ell}\oplus(-1)^{\ell+1} & \textrm{if} \ \ell \equiv 3 \ (4)
\end{array}\right.$ & $\left\{\begin{array}{ll}B_{\frac{\ell}2}D_{\frac{\ell}2} & \textrm{if} \ \ell \equiv 0 \ (4)\\
B_{\frac{\ell+1}2}D_{\frac{\ell-1}2} & \textrm{if} \ \ell \equiv 1 \ (4)\\ B_{\frac{\ell-2}2}D_{\frac{\ell+2}2} & \textrm{if} \ \ell \equiv 2 \ (4)\\ B_{\frac{\ell-1}2}D_{\frac{\ell+1}2} & \textrm{if} \ \ell \equiv 3 \ (4)
\end{array}\right.$& $\left \{\begin{array}{ll}\\
\frac{(z^2u+e(2z+1))}2+2 & \textrm{if} \ \ell \equiv 1,2 \ (4)\\
& \textrm{and} \ e=0\\
\frac{(z^2u+e(2z+1))}2 & \textrm{otherwise}
\end{array}\right.$ \\
\hline (c) & $M_{1}^\alpha\oplus M_{2}\oplus(-1) $& $A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}B_{\frac{\alpha-1}2}D_{\frac {\alpha+1}2}T_{\frac{v}2-1}$ & $\frac12(z^2u+e(2z+1))$\\ \cline{1-3} (d) & $ M_{1}^\alpha\oplus M_{5}\oplus(1)^2\oplus(-1)$& $A_\alpha^{\frac\beta2-1} A_{\alpha-1}^{\frac{v}2-\frac\beta2}B_{\frac{\alpha+1}2}D_{\frac {\alpha+1}2}T_{\frac{v}2-1}$ & \\ \cline{1-2} (e) & $ M_{1}^{\alpha}\oplus M_{6}\oplus(1)^2\oplus(-1)$& & \\ \hline (f) & $ M_{1}^{\alpha-1}\oplus M_{7}\oplus(1)^3\oplus(-1)^2 $& $A_{\alpha-1}^{\frac v2-2} A_{\alpha-2}B_{\frac{\alpha+1}2}D_{\frac {\alpha+1}2}T_{\frac{v}2-1}$ & $\left \{\begin{array}{ll}\\
\frac{(z^2u+e(2z+1))}2+2 & \textrm{if} \ e=0\\
\frac{(z^2u+e(2z+1))}2 & \textrm{otherwise}
\end{array}\right.$ \\
\hline (g) & $M_1^\alpha\oplus M_2 \oplus (1)$ & $A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}B_{\frac{\alpha}2}D_{\frac {\alpha}2}T_{\frac{v}2-1}$ & $\frac12(z^2u+e(2z+1))$\\ \cline{1-2} (h) & $M_1^\alpha\oplus M_6\oplus (\omega^{\frac v2-1})\oplus (\omega^{-\left(\frac v2-1\right)}) \oplus (1)$ & &\\ \cline{1-3} (i) & $ M_{1}^\alpha\oplus M_{5}\oplus(1)\oplus(-1)^2$& $A_\alpha^{\frac\beta2-1} A_{\alpha-1}^{\frac{v}2-\frac\beta2}B_{\frac{\alpha}2}D_{\frac {\alpha+2}2}T_{\frac{v}2-1}$ & \\ \cline{1-2} (j) & $ M_{1}^\alpha\oplus M_{6}\oplus(1)\oplus(-1)^2$& &\\ \hline (k) & $ M_{1}^{\alpha-1}\oplus M_{9}\oplus(1)^2\oplus(-1)^3 $& $A_{\alpha-1}^{\frac v2-2} A_{\alpha-2}B_{\frac{\alpha}2}D_{\frac {\alpha+2}2}T_{\frac{v}2-1}$ & $\left \{\begin{array}{ll}\\
\frac{(z^2u+e(2z+1))}2+2 & \textrm{if} \ e=0\\
\frac{(z^2u+e(2z+1))}2 & \textrm{otherwise}
\end{array}\right.$ \\
\hline
\end{tabular}}} \end{sidewaystable}
\begin{sidewaystable}[h] \caption{A semisimple element $y$ of $G=(D_{\ell})_{\rm{s.c}}$ of order $v$ giving an upper bound for $d_u(G)$}\label{t:dsc} \footnotesize{{
\begin{tabular}{| l | l | l | l|} \hline Case & Element $\overline{y}$ & $C_{{\rm SO}_{h+1}(K)}(\overline{y})^0$ & $d_q(C_G(y)^0)$\\ \hline
(a) & $ M_{1}^\alpha\oplus M_{3}\oplus (1)^{2-\epsilon_\beta}$& $A_\alpha^{\lceil\frac\beta2\rceil} A_{\alpha-1}^{\frac{v-1}2-\lceil\frac\beta2\rceil}D_{\lceil \frac {\alpha+1}2\rceil}T_{\frac{v-1}2}$ & $\frac12(z^2u+e(2z+1))+\lceil \frac z2 \rceil \epsilon_u+z+1-\epsilon_z$\\
\hline
(b) & $\left\{\begin{array}{ll}(1)^{\ell}\oplus(-1)^{\ell} & \textrm{if} \ \ell \equiv 0 \ (4) \\
(1)^{\ell+1}\oplus(-1)^{\ell-1} & \textrm{if} \ \ell \equiv 1 \ (4)\\
(1)^{\ell-2}\oplus(-1)^{\ell+2} & \textrm{if} \ \ell \equiv 2 \ (4)\\
(1)^{\ell-1}\oplus(-1)^{\ell+1} & \textrm{if} \ \ell \equiv 3 \ (4)
\end{array}\right.$ & $\left\{\begin{array}{ll}D_{\frac{\ell}2}D_{\frac{\ell}2} & \textrm{if} \ \ell \equiv 0 \ (4)\\
D_{\frac{\ell+1}2}D_{\frac{\ell-1}2} & \textrm{if} \ \epsilon_\ell =1\\ D_{\frac{\ell-2}2}D_{\frac{\ell+2}2} & \textrm{if} \ \ell \equiv 2 \ (4)\\
\end{array}\right.$& $\left \{\begin{array}{ll}\\
\frac{z^2u+e(2z+1)}2+z+1-\epsilon_z+4 & \textrm{if} \ \ell \equiv 2 \ (4)\\ & \textrm{and}\ u =2\\
\frac{z^2u+e(2z+1)}2+z+1-\epsilon_z+2 & \textrm{if} \ \ell \equiv 2 \ (4)\ \textrm{and}\\ & e=u-v\neq 0\\
& \textrm{or} \ e=0\neq u-v\\
\frac{z^2u+e(2z+1)}2+z+1-\epsilon_z & \textrm{otherwise}
\end{array}\right.$ \\
\hline (c) & $M_{1}^\alpha\oplus M_{3}\oplus(1)\oplus(-1) $& $A_\alpha^{\frac\beta2} A_{\alpha-1}^{\frac{v}2-1-\frac\beta2}D_{\frac{\alpha+1}2}D_{\frac {\alpha+1}2}T_{\frac{v}2-1}$ & $\frac{z^2u+e(2z+1)}2+z+1-\epsilon_z$\\ \cline{1-2} (d) & $ M_{1}^\alpha\oplus M_{5}\oplus (\omega^{\frac v2-2})\oplus(\omega^{-(\frac v2-2)}) \oplus(1)\oplus(-1)$& &\\
\cline{1-2} (e) & $ M_{1}^\alpha\oplus M_{5}\oplus (\omega^{\frac v2-1})\oplus(\omega^{-(\frac v2-1)}) \oplus(1)\oplus(-1)$& &\\ \cline{1-2} (f) & $ M_{1}^\alpha\oplus M_{6}\oplus (\omega^{\frac v2-2})\oplus(\omega^{-(\frac v2-2)}) \oplus(1)\oplus(-1)$& &\\ & if $\beta\neq v-4$& & \\ & $ M_{1}^\alpha\oplus M_{8} \oplus(1)\oplus(-1)$& &\\ & if $\beta=v-4$& & \\ \cline{1-2} (g) & $ M_{1}^\alpha\oplus M_{6}\oplus (\omega^{\frac v2-1})\oplus(\omega^{-(\frac v2-1)}) \oplus(1)\oplus(-1)$& &\\ \hline (h) & $M_1^\alpha \oplus M_9 \oplus (1) \oplus (-1)^3 $ & $A_\alpha^{\frac v2-2} A_{\alpha-1} D_{\frac{\alpha+1}2} D_{\frac{\alpha+3}2}T_{\frac v2-1}$ & $\left\{ \begin{array}{ll} \frac{z^2u+e(2z+1)}2 +z+1-\epsilon_z +2 & \textrm{if} \ e=u-2\\ \frac{z^2u+e(2z+1)}2 +z+1-\epsilon_z & \textrm{otherwise}
\end{array}\right.$ \\ \hline
(i) & $M_1^{\alpha-1} \oplus M_7 \oplus (\omega)^2 \oplus (\omega^{-1})^2 \oplus (1)^2 $ & $A_{\alpha}A_{\alpha-1}^{\frac v2-2} D_{\frac{\alpha-1}2} D_{\frac{\alpha+1}2}T_{\frac v2-1}$& $\left\{ \begin{array}{ll} \frac{z^2u+e(2z+1)}2 +z+1-\epsilon_z +2 & \textrm{if} \ e=0\\ \frac{z^2u+e(2z+1)}2 +z+1-\epsilon_z & \textrm{otherwise}
\end{array}\right.$ \\ \hline (j) & $M_1^{\alpha -1}\oplus M_9 \oplus (\omega)\oplus (\omega^{-1}) \oplus (1)^2 \oplus(-1)^2 $ & $A_\alpha A_{\alpha-1}^{\frac v2-3}A_{\alpha-2} D_{\frac{\alpha+1}2}^2 T_{\frac v2-1}$ & $\left\{ \begin{array}{ll} \frac{z^2u+e(2z+1)}2 +z+1-\epsilon_z +2 & \textrm{if} \ e=0\\ \frac{z^2u+e(2z+1)}2 +z+1-\epsilon_z & \textrm{otherwise}
\end{array}\right.$ \\ \hline (k) & $M_1^\alpha\oplus M_3 \oplus (1)^2$ & $A_\alpha^{\frac \beta2}A_{\alpha-1}^{\frac v2-1-\frac \beta2}D_{\frac \alpha2}D_{\frac{\alpha+2}2}T_{\frac v2-1}$ & $\frac{z^2u+e(2z+1)}2+z+1-\epsilon_z$ \\ \cline{1-2} (l) & $ M_{1}^\alpha\oplus M_{5}\oplus (\omega^{\frac v2-1})\oplus(\omega^{-(\frac v2-1)}) \oplus(1)^2$ & & \\ \cline{1-2} (m) & $ M_{1}^\alpha\oplus M_{6} \oplus (\omega^{\frac v2-2})\oplus (\omega^{-(\frac v2-2)})\oplus(1)^2$& &\\ & if $\beta\neq v-4$& & \\ & $ M_{1}^\alpha\oplus M_{8} \oplus(1)^2$& &\\ & if $\beta=v-4$& & \\ \cline{1-3} (n) & $ M_{1}^\alpha\oplus M_{5}\oplus (\omega^{\frac v2-2})\oplus(\omega^{-(\frac v2-2)}) \oplus(1)^2$& $A_\alpha^{\frac \beta2}A_{\alpha-1}^{\frac v2-1-\frac \beta2}D_{\frac \alpha2}D_{\frac{\alpha+2}2}T_{\frac v2-1}$&\\ & if $\beta\neq v-2$& if $\beta\neq v-2$ & \\ & $ M_{1}^\alpha\oplus M_{9} \oplus(1)^2\oplus(-1)^2$& $A_\alpha^{\frac v2-2}A_{\alpha-1}D_{\frac {\alpha+2}2}D_{\frac{\alpha+2}2}T_{\frac v2-1}$ &\\ & if $\beta=v-2$& if $\beta=v-2$& \\ \cline{1-3} (o) & $ M_{1}^\alpha\oplus M_{6}\oplus (\omega^{\frac v2-1})\oplus(\omega^{-(\frac v2-1)}) \oplus(1)^2$& $A_\alpha^{\frac \beta2}A_{\alpha-1}^{\frac v2-1-\frac \beta2}D_{\frac \alpha2}D_{\frac{\alpha+2}2}T_{\frac v2-1}$&\\ & if $\beta\neq v-2$& if $\beta\neq v-2$ & \\ & $ M_{1}^\alpha\oplus M_{9} \oplus(1)^2\oplus(-1)^2$& $A_\alpha^{\frac v2-2}A_{\alpha-1}D_{\frac {\alpha+2}2}D_{\frac{\alpha+2}2}T_{\frac v2-1}$ &\\ & if $\beta=v-2$& if $\beta=v-2$& \\
\cline{1-3}
(p) & $M_1^{\alpha}\oplus(-1)^2$ & $A_{{\alpha-1}}^{\frac v2-1}D_{\frac \alpha2}D_{\frac{\alpha+2}2}T_{\frac v2-1}$ & \\
\hline \end{tabular}}} \end{sidewaystable}
\begin{table}
\begin{tabular}{|l|l|l|} \hline $G$ & Cases & $d_u(G_{s.c})-d_u(G_{a.})$ \\ \hline $A_\ell$ & $p\neq 2$, $u$ even, $h=zu$ and $z$ odd & $\leq 2$\\ \hline $C_\ell$ & $p\neq 2$ and $u$ even & $\leq 2\left\lceil \frac z2\right\rceil$\\ \hline $B_{\ell}$ & $p\neq 2$, $u\equiv 2 \mod 4$, $h=zu$, and $z\equiv u/2 \mod 4$ or $z\equiv 2 \mod 4$ & $\leq 2$\\ \cline{2-2} & $p\neq 2$, $u \equiv 4 \mod 8$, $h=zu$ and $z$ odd & \\ \hline $D_\ell$ & $p\neq 2$, $u=2$, $h=zu$ and $z \equiv u/2 \mod 4$ & $\leq 4$\\ \cline{2-3} & $p\neq 2$, $u \equiv 2 \mod 4$, $u>2$, $h=zu$ and $z \equiv u/2 \mod 4$ & $\leq2$ \\ \cline{2-2} & $p\neq 2$, $u\equiv 2 \mod 4$, $e=u-2\neq 0$ and
$z \equiv 1 \mod 4$& \\ \cline{2-2} & $p\neq 2$, $u \equiv 4 \mod 8$, $z$ is odd, and $h=zu$ & \\ \hline \end{tabular} \caption{The possible exceptions to $d_u(G_{s.c.})=d_u(G_{a.})$ for $G$ of classical type}\label{ta:casestoconsider} \end{table}
\end{document} | arXiv |
\begin{document}
\title{Weak Quasicircles Have Lipschitz Dimension 1}
\author{David M. Freeman} \address{University of Cincinnati Blue Ash College, Department of Math, Physics, and Computer Science, 9555 Plainfield Rd., Cincinnati, OH 45236, United States} \email{[email protected]}
\subjclass[2020]{30L05 (Primary); 54F45, 30C65 (Secondary)}
\date{\today}
\commby{Nageswari Shanmugalingam}
\begin{abstract} We prove that the Lipschitz dimension of any bounded turning Jordan circle or arc is equal to 1. Equivalently, the Lipschitz dimension of any weak quasicircle or arc is equal to 1. \end{abstract}
\maketitle
\section{Introduction}
In \cite{CK13}, Cheeger and Kleiner introduced the concept of Lipschitz dimension and proved deep results about metric spaces of Lipschitz dimension at most 1. Subsequently, in \cite{David19}, David further developed various properties of Lipschitz dimension. While studying the non-invariance of Lipschitz dimension under quasisymmetric mappings in a general metric space setting, David asks in Question 8.7 of \cite{David19} if every quasisymmetric image of the unit interval (that is, a \textit{quasiarc}) has Lipschitz dimension equal to 1. We answer this question in the affirmative, thus demonstrating that the Lipschitz dimension of the unit interval is invariant under quasisymmetric homeomorphisms. In fact, we prove something stronger: the Lipschitz dimension of the unit interval is invariant under weakly quasisymmetric homeomorphisms. This can be derived from the following theorem, which constitutes the main result of this paper (see \rf{S:defs} for definitions).
\begin{theorem}\label{T:main} Bounded turning Jordan circles/arcs have Lipschitz dimension $1$. \end{theorem}
Since bounded turning Jordan arcs need not be metrically doubling, not all bounded turning Jordan arcs are quasiarcs. On the other hand, every quasiarc is bounded turning. Analogous statements hold for Jordan circles. Therefore, in answer to \cite[Question 8.7]{David19}, we obtain the following corollary.
\begin{corollary}\label{C:main} Quasicircles/arcs have Lipschitz dimension $1$. \end{corollary}
We also point out results of Cheeger and Kleiner pertaining to spaces of Lipschitz dimension at most 1. In particular, via \cite[Theorem 1.7]{CK13}, we have the following corollary to \rf{T:main}.
\begin{corollary}\label{C:embed} If $\Gamma$ is a bounded turning Jordan circle or arc, then there exists a measure space $(X,\mu)$ such that $\Gamma$ admits a bi-Lipschitz embedding into $L_1(X,\mu)$. \end{corollary}
Furthermore, via \cite[Theorem 1.11]{CK13}, we also have the following.
\begin{corollary}\label{C:inverse} If $\Gamma$ is a bounded turning Jordan circle or arc, then $\Gamma$ is bi-Lipschitz homeomorphic to an inverse limit of an admissible inverse system of graphs. \end{corollary}
Indeed, given a metric space $X$ and a Lipschitz light map $F:X\to\mathsf{R}$, Cheeger and Kleiner explicitly construct an admissible inverse system of graphs whose inverse limit is bi-Lipschitz homeomorphic to $X$ (see \cite[Section 4]{CK13}).
\rf{T:main} is also relevant to the following result of David.
\begin{theorem}[Theorem 5.9 of \cite{David19}]\label{T:Carnot} Let $\mathbb{G}$ denote a non-abelian Carnot group, and let $K\subset \mathbb{G}$ denote a compact subset of positive measure. Then the Lipschitz dimension of $K$ is equal to $\infty$. \end{theorem}
Since the Lipschitz dimension of a compact space is bounded from below by its topological dimension (see \cite[Observation 1.4]{David19}), given an integer $n\geq1$, we note that the Lipschitz dimension of the product of $n$ bounded turning Jordan arcs is at least $n$. Furthermore, the Lipschitz dimension of the product of $n$ bounded turning Jordan arcs is bounded from above by $n$ (here we use \rf{T:main} and \cite[Proposition 3.1]{David19}). Therefore, the Lipschitz dimension of the product of $n$ bounded turning Jordan arcs is equal to $n$. Since Lipschitz dimension is invariant under bi-Lipschitz homeomorphisms, we arrive at the following corollary.
\begin{corollary}\label{C:nonembed} Let $\mathbb{G}$ denote a non-abelian Carnot group. If $K\subset \mathbb{G}$ is a compact subset of positive measure, then $K$ does not admit a bi-Lipschitz embedding into a product of finitely many bounded turning Jordan arcs. \end{corollary}
In connection with the title of this paper, we also point out the following theorem of Meyer. In particular, all the above results pertaining to bounded turning Jordan circles/arcs can be understood as results about weak quasicircles/arcs.
\begin{theorem}[Theorem 1.1 of \cite{Meyer11}]\label{T:Meyer} A Jordan circle/arc $\Gamma$ is a weak quasicircle/arc if and only if $\Gamma$ is bounded turning. \end{theorem}
Finally, we highlight the following result of Herron and Meyer, as it is foundational to our proof of \rf{T:main}.
\begin{theorem}[\cite{HM12}] If $\Gamma$ is a bounded turning Jordan circle, then $\Gamma$ is bi-Lipschitz homeomorphic to some Jordan circle in $\mathcal{S}_1$. \end{theorem}
Here $\mathcal{S}_1$ is the collection of all Jordan circles given by dyadic diameter functions constructed using the snowflake parameter $\sigma=1$ (see \cite{HM12} for definitions). This result allows us to distort any given bounded turning Jordan circle into a form more amenable to the construction of a Lipschitz light map into $\mathsf{R}$.
The organization of this paper is as follows. In \rf{S:defs} we provide a few key definitions. Then, in \rf{S:prelims}, we investigate various aspects of Jordan circles in $\mathcal{S}_1$. In \rf{S:lip}, we construct a $1$-Lipschitz mapping from any Jordan circle $\Gamma\in\mathcal{S}_1$ into the unit circle. Finally, in \rf{S:light} we prove that this mapping is Lipschitz light via a series of technical lemmas.
\begin{acknowledge} The author thanks Guy C. David for enlightening dialogue regarding the results and theoretical context of both this paper and \cite{David19}. \end{acknowledge}
\section{Basic Definitions}\label{S:defs}
We write $\mathsf{N}$ to denote the set $\{0,1,2,\dots\}$ consisting of non-negative integers, and $\mathsf{R}$ to denote the Euclidean line.
Given two metric spaces $X$ and $Y$ and a constant $L\geq 1$, we say that an embedding $f:X\to Y$ is \textit{$L$-Lipschitz} provided that, for all points $x,y\in X$, we have $d(f(x),f(y))\leq L\,d(x,y)$. Furthermore, an embedding is \textit{$L$-bi-Lipschitz} if it is also true that $d(x,y)\leq L\,d(f(x),f(y))$.
An embedding $f:X\to Y$ is \textit{quasisymmetric} provided there exists a homeomorphism $\eta:[0,\infty)\to[0,\infty)$ such that, for all points $x,y,z\in X$ and $t\in[0,\infty)$, \[d(x,y)\leq t\,d(x,z)\quad\text{implies}\quad d(f(x),f(y))\leq \eta(t)d(f(x),f(z)).\] An embedding $f:X\to Y$ is \textit{weakly quasisymmetric} provided there exists a constant $H\geq 1$ such that, for all $x,y,z\in X$, \[d(x,y)\leq d(x,z)\quad \text{implies}\quad d(f(x),f(y))\leq H d(f(x),f(z)).\] While all quasisymmetries are weak quasisymmetries (with $H:=\eta(1)$), in general, a weak quasisymmetry need not be a quasisymmetry. We refer to the weak quasisymmetric image of the unit circle as a \textit{weak quasicircle}, and such an image of the closed unit interval as a \textit{weak quasiarc}. Thus every quasicircle/arc is a weak quasicircle/arc. Conversely, by \cite[Theorem 4.9]{TV80}, every weak quasicircle/arc that is \textit{metrically doubling} is a quasicircle/arc. Here we say that a space $X$ is metrically doubling if there exists $D\geq 1$ such that that any open ball of radius $2r>0$ can be covered by $D$ open metric balls of radius $r$. For additional information about weak quasicircles and quasiarcs, we refer the reader to \cite{Meyer11} and references therein.
A \textit{Jordan circle} $\Gamma$ is a homeomorphic image of the unit circle. Given two points $x,y\in \Gamma$, we write $\Gamma(x,y)$ to denote a component of $\Gamma\setminus\{x,y\}$ of minimal diameter. We write $\Gamma[x,y]$ to denote the topological closure of $\Gamma(x,y)$; thus $\Gamma[x,y]=\Gamma(x,y)\cup\{x,y\}$. Analogously, a \textit{Jordan arc} $\Gamma$ is a homeomorphic image of the closed unit interval. In this setting, given two points $x,y\in\Gamma$, we write $\Gamma(x,y)$ to denote the connected component of $\Gamma\setminus\{x,y\}$. Again, $\Gamma[x,y]=\Gamma(x,y)\cup\{x,y\}$.
A Jordan circle or arc is said to be \textit{bounded turning} provided that there exists a constant $C\geq 1$ such that, for all pairs of points $x,y\in\Gamma$, we have $\diam(\Gamma[x,y])\leq C\,d(x,y)$. In this case we say that $\Gamma$ is $C$-bounded turning. This property is at times referred to in the literature as \textit{linear connectivity}.
Given a metric space $(X,d)$ and $\delta>0$, a \textit{$\delta$-sequence} in $X$ is a finite sequence of points $\{x_i\}_{i=0}^n$ such that, for each $0\leq i\leq n-1$, we have $d(x_i,x_{i+1})\leq \delta$. A subset $U\subset X$ is said to be $\delta$-connected if every pair of points in $U$ is contained in a $\delta$-sequence in $U$. A \textit{$\delta$-component} of $X$ is a maximal $\delta$-connected subset of $X$.
We say that a map $F:X\to Y$ is \textit{Lipschitz light} provided there exists $C\geq 1$ such that $F$ is $C$-Lipschitz, and, for every $r>0$ and every subset $E\subset Y$ with $\diam(E)\leq r$, the $r$-components of $F^{-1}(E)$ have diameter bounded above by $C\,r$. Here we employ the definition of Lipschitz light used in \cite[Definition 1.2]{David19}. As shown by David in \cite[Section 1.4]{David19}, this definitions is equivalent to \cite[Definition 1.14]{CK13} for maps into $\mathsf{R}^n$ (for $n\geq1$).
A metric space $X$ has \textit{Lipschitz dimension} $n\in\mathsf{N}$ if $n$ is minimal such that there exists a Lipschitz light map $F:X\to\mathsf{R}^n$.
\section{Preliminary Results}\label{S:prelims}
Following \cite{HM12}, we view the unit circle $\mathsf{S}$ as $[0,1]/\{0,1\}$, the closed unit interval whose endpoints are identified. We equip $\mathsf{S}$ with the arc-length metric $\lambda$. That is, for two points $s,t\in\mathsf{S}$ such that $0\leq s\leq t\leq 1$, we have \[\lambda(s,t):=\min\{t-s,1-(t-s)\}.\] The space $\mathsf{S}$ inherits the usual left-to-right orientation on $[0,1]$.
Given $n\in\mathsf{N}$, we write $\mathcal{I}_n$ to denote the collection of $2^n$ closed dyadic intervals in $[0,1]$, each of length $2^{-n}$. We write $\hI_n$ to denote the collection $\cup_{m=0}^n\mathcal{I}_m$. Furthermore, we write $\hat{\mathcal{I}}$ to denote the collection $\cup_{n=0}^\infty\mathcal{I}_n$. Given an interval $I\in\hat{\mathcal{I}}$, we write $l(I)$ to denote the unique index $n\in\mathsf{N}$ such that $I\in \mathcal{I}_n$. For convenience, we use the language of a dyadic tree to describe intervals in $\hat{\mathcal{I}}$. In particular, given any $I\in\hat{\mathcal{I}}$, there are exactly two dyadic \textit{children} contained in $I$, and $I$ is contained in its unique dyadic \textit{parent} interval.
Similarly, we write $\mathcal{D}_n$ to denote the collection of $2^{n}$ dyadic endpoints of intervals in $\mathcal{I}_n$. For example, $\mathcal{D}_0=\{0\}=\{1\}$, $\mathcal{D}_1=\{0,1/2\}=\{1,1/2\}$, etc. Note that, for each $n\in\mathsf{N}$, we have $\mathcal{D}_n\subset\mathcal{D}_{n+1}$. We write $\mathcal{D}$ to denote $\cup_{n=0}^\infty\mathcal{D}_n$.
In order to utilize the catalogue $\mathcal{S}_1$, we rely upon notation and terminology from \cite{HM12}. Given a dyadic diameter function $\Delta$, the distance $d_\Delta$ on $\mathsf{S}$ is defined as \[d_\Delta(x,y):=\inf\sum_{k=1}^N\Delta(J_k),\] where the infimum is taken over all chains $J_1,\dots, J_N$ of intervals from $\hat{\mathcal{I}}$ joining $x$ to $y$. That is, $\{x,y\}$ is contained in the connected set $J_1\cup\dots\cup J_N$. For each $n\in\mathsf{N}$, we define a distance $d_n$ on $\mathsf{S}$ using the \textit{truncated} diameter function $\Delta_n$, defined as follows. For $m\leq n$ and $I\in\mathcal{I}_m$, we define $\Delta_n(I):=\Delta(I)$. For every $m> n$ and $I\in \mathcal{I}_m$, we inductively define $\Delta_n(I)=\frac{1}{2}\Delta_n(\tilde{I})$, where $\tilde{I}\in\mathcal{I}_{m-1}$ denotes the dyadic parent of $I$. In analogy with $d_\Delta$, we then define \[d_n(x,y):=\inf\sum_{k=1}^N\Delta_n(J_k),\] where the infimum is taken over all chains $\{J_k\}_{k=1}^N$ joining $x$ to $y$. We write $\Gamma_n$ to denote the metric space $(\mathsf{S},d_n)$, and $\Gamma$ to denote $(\mathsf{S},d_\Delta)$. For $n\in\mathsf{N}$, we write $\diam_n(E)$ to denote the $d_n$-diameter of a set $E\subset\mathsf{S}$. Furthermore, we write $\diam_\Delta(E)$ to denote the $d_\Delta$-diameter of $E$.
We say that a chain of dyadic intervals $\{I_i\}_{i=1}^N$ is \textit{minimal} provided that it consists of intervals with pairwise disjoint interiors and that no union of at least two distinct intervals from the chain forms an interval in $\hat{\mathcal{I}}$. In particular, if the union of intervals $\cup_{k=1}^MI_{i_k}$ from a minimal chain $\{I_i\}_{i=1}^N$ is equal to some interval $J\in\hat{\mathcal{I}}$, then $M=1$ and $J=I_{i_1}\in\{I_i\}_{i=1}^N$.
\begin{lemma}\label{L:minimal} The definitions of $d_n$ and $d_\Delta$ are unchanged by the assumption that the chains of dyadic intervals utilized in these definitions are minimal. \end{lemma}
\begin{proof} We focus on the definition of $d_n$. The proof for $d_\Delta$ is the same. Suppose that $\{I_i\}_{i=1}^N$ is a chain of dyadic intervals joining $x$ and $y$ in $\mathsf{S}$. Suppose $I_j$ and $I_k$ have non-disjoint interiors. Since both $I_j$ and $I_k$ are dyadic, one must be a subset of the other. Without loss of generality, $I_j\subset I_k$. Therefore, the sum $\sum_{i=1}^N\Delta_n(I_i)$ can be decreased by eliminating the interval $I_j$ from $\{I_i\}_{i=1}^N$. It follows that $d_n$ can be defined only using chains consisting of intervals with pairwise disjoint interiors.
Next, suppose there exists $M\geq2$ and a subcollection $\{I_{i_k}\}_{k=1}^M\subset \{I_i\}_{i=1}^N$ such that $J=\cup_{k=1}^MI_{i_k}\in\hat{\mathcal{I}}$. Since $\Delta_n(J)\leq\sum_{k=1}^M\Delta_n(I_{i_k})$, the sum $\sum_{i=1}^N\Delta_n(I_i)$ can be decreased by replacing the intervals $\{I_{i_k}\}_{k=1}^M$ in $\{I_i\}_{i=1}^N$ with the single interval $J$. Since $N<+\infty$, such a replacement can happen at most finitely many times. It follows that $d_n$ can be defined only using minimal chains. \end{proof}
For use below, we record the following technical lemma.
\begin{lemma}\label{L:minimalchain} Assume $\{I_i\}_{i=1}^N$ is a minimal chain of dyadic intervals indexed such that, for $1\leq i\leq N-1$, the right endpoint of $I_i$ is the left endpoint of $I_{i+1}$. Under this assumption, there is either a unique interval or a unique pair of adjacent intervals from $\{I_i\}_{i=1}^N$ of maximal $d_0$-diameter. Write $i_*$ to denote the index of such a maximal interval. If $i_*>1$, then $l(I_i)$ is strictly decreasing for $i=1,\dots,i_*-1$. If $i_*<N$, then $l(I_i)$ is strictly increasing for $i=i_*+1,\dots,N$. \end{lemma}
\begin{proof} We may assume that $\sigma:=\bigcup_{i=1}^NI_i\not=\mathsf{S}$, else $N=1$ and $I_1=\mathsf{S}$.
Suppose there are two distinct intervals $I_j$ and $I_k$ in $\{I_i\}_{i=1}^N$ of maximal $d_0$-diameter, where $j<k$ and $n:=l(I_j)=l(I_k)$. If these intervals are not adjacent, then the chain $\{I_{i}\}_{i=j+1}^{k-1}$ consisting of intervals from $\{I_i\}_{i=1}^N$ joins the right endpoint of $I_j$ to the left endpoint of $I_k$. Since $I_j$ and $I_k$ are not adjacent, the union $\cup_{i=j}^k I_i\subset \sigma$ contains at least three consecutive intervals from $\mathcal{I}_n$. Such a union must contain some interval $J$ from $\mathcal{I}_{n-1}$. It follows from the assumption that $\{I_i\}_{i=1}^N$ is minimal that the interval $J$ must be an element of $\{I_i\}_{i=1}^N$. However, this violates the assumption that $I_j$ and $I_k$ are of maximal $d_0$-diameter in $\mathcal{I}_n$. Therefore, the intervals $I_j$ and $I_k$ must be adjacent.
To verify the second part of the lemma, suppose $i_*>1$. If $i_*=2$ then the desired conclusion is trivial, so we may assume that $i_*\geq3$. Let $1\leq i\leq i_*-2$, and write $m:=l(I_i)$. Since $I_i$ lies to the left of $I_{i_*}$ in $\sigma$ and $\diam_0(I_{i_*})$ is strictly larger than $\diam_0(I_i)$, the interval $J$ of $\mathcal{I}_m$ immediately to the right of $I_i$ is contained in $\sigma$. By the minimality of $\{I_i\}_{i=1}^N$, no union of at least two intervals from $\{I_i\}_{i=1}^N$ is equal to $J$. Therefore, either $J=I_{i+1}$ or $J$ is strictly contained in $I_{i+1}$. If $J=I_{i+1}$, then (lest we violate the minimality of $\{I_i\}_{i=1}^N$), the midpoint of $I_i\cup I_{i+1}$ is an element of $\mathcal{D}_{m-1}$, and so $I_i\cup I_{i+1}\not\in\mathcal{I}_m$. In this case, since $I_{i+1}$ lies to the left of $I_{i_*}$ and $\diam_0(I_{i_*})>\diam_0(I_{i+1})$, the interior of the interval $K$ in $\mathcal{I}_m$ immediately to the right of $I_{i+1}$ is disjoint from the interior of $I_{i_*}$. It follows from the minimality of $\{I_i\}_{i=1}^N$ that $K$ cannot be strictly contained in any element of $\{I_i\}_{i=1}^N$ (since any dyadic interval strictly containing $K$ must also contain $I_{i+1}$), and so (again using minimality), we conclude that $K=I_{i+2}$. But this nevertheless violates the minimality of $\{I_i\}_{i=1}^N$, since $I_{i+1}\cup I_{i+2}\in\mathcal{I}_{m-1}$. Therefore, $J$ is strictly contained in $I_{i+1}$, and so $l(I_{i+1})<l(I_i)$.
An analogous argument verifies the final assertion of the lemma. \end{proof}
Since, for any $I\in\mathcal{I}$, we have $\Delta_n(I)\leq\Delta(I)$, it follows that, for any $x,y\in\mathsf{S}$, \begin{equation}\label{E:leqn} d_n(x,y)\leq d_\Delta(x,y). \end{equation} Therefore, for any set $E\subset \mathsf{S}$, we have $\diam_n(E)\leq \diam_\Delta(E)$. Furthermore, for any $n\in\mathsf{N}$ and $I\in\hI_n$ with endpoints $a,b$, via \cite[Lemma 3.1]{HM12}, we have \begin{equation}\label{E:intn} d_n(a,b)=\diam_{n}(I)=\Delta_n(I)=\Delta(I)=\diam_\Delta(I)=d_\Delta(a,b). \end{equation}
\begin{lemma}\label{L:distconverge} If $x,y\in \mathsf{S}$ and $n\in\mathsf{N}$, then
\[d_\Delta(x,y)\leq d_n(x,y)+2\max\{\Delta(I)\,|\,I\in\mathcal{I}_n\}.\] In particular, $d_n(x,y)\to d_\Delta(x,y)$. \end{lemma} \begin{proof}
Let $M(n):=\max\{\Delta(I)\,|\,I\in\mathcal{I}_n\}$. Fix $x,y\in \mathsf{S}$, $n\in\mathsf{N}$, and let $0<\varepsilon<M(n)$ be given. Let $\{I_i\}_{i=1}^N$ be a minimal chain of dyadic intervals joining $x$ and $y$, indexed as in the assumptions of \rf{L:minimalchain}, such that $\sum_{i=1}^N\Delta_n(I_i)<d_n(x,y)+\varepsilon$. If $\{I_i\}_{i=1}^N\subset\hI_n$, then we are done, because $\Delta=\Delta_n$ on $\hI_n$. If not, then let $I_{i_*}$ denote an interval from $\{I_i\}_{i=1}^N$ of maximal $d_0$-diameter, and write $m:=l(I_{i_*})$. If $x$ and $y$ are contained in adjacent intervals $J$ and $K$ from $\mathcal{I}_{n}$, then \[d_\Delta(x,y)\leq \Delta(J)+\Delta(K)\leq 2M(n)\leq d_n(x,y)+2M(n).\] Therefore, we can assume that $x$ and $y$ are contained in non-adjacent intervals from $\mathcal{I}_n$. It follows from the minimality of $\{I_i\}_{i=1}^N$ that $m\leq n$. Therefore, either $l(I_1)\leq n$, or, by \rf{L:minimalchain}, there exists a maximal index $i$ such that $1\leq i_1<i_*$ and, if $1\leq i\leq i_1$, then $l(I_i)>n$. Similarly, either $l(I_N)\leq n$, or there exists a minimal index $i$ such that $i_*< i_2\leq N$ and, if $i_2\leq i\leq N$, then $l(I_i)>n$. Assume the existence of such $i_1$ and $i_2$ (else the following argument simplifies). Via \rf{L:minimalchain}, one can verify that the interval $\sigma_1:=\bigcup_{i=1}^{i_1}I_i$ is contained in some interval $J_1\in\mathcal{I}_n$ adjacent (on the left) to $I_{i_1+1}$. Similarly, $\sigma_2:=\bigcup_{i=i_2}^{N}I_i$ is contained in some interval $J_2\in\mathcal{I}_n$ adjacent (on the right) to $I_{i_2-1}$. Thus we have \begin{align*} d_\Delta(x,y)&\leq \Delta(J_1)+\sum_{i=i_1+1}^{i_2-1}\Delta(I_i)+\Delta(J_2)=\Delta(J_1)+\sum_{i=i_1+1}^{i_2-1}\Delta_n(I_i)+\Delta(J_2)\\ &\leq\sum_{i=1}^N\Delta_n(I_i)+2M(n)<d_n(x,y)+\varepsilon+2M(n). \end{align*} Since $\varepsilon>0$ was arbitrary, we are done. \end{proof}
\section{Constructing a $1$-Lipschitz map $F_0:\Gamma\to\Gamma_0$}\label{S:lip}
Let $\Gamma$ denote a bounded turning Jordan circle or arc. Our first step towards the construction of a Lipschitz light map $F:\Gamma\to\mathsf{R}$ is to realize that it is sufficient to find a Lipschitz light map $F:\Gamma\to\Gamma_0$. This is because $\Gamma_0$ is easily seen to admit a Lipschitz light map into $\mathsf{R}$, and one can verify that the composition of a Lipschitz light map from $\Gamma$ to $\Gamma_0$ with a Lipschitz light map from $\Gamma_0$ into $\mathsf{R}$ is Lipschitz light (as noted in \cite[Section 5]{David19}). See also our comments at the outset of \rf{S:light}.
Next, we again recall the following result of Herron and Meyer.
\begin{theorem}[\cite{HM12}] If $\Gamma$ is a bounded turning Jordan circle (or arc), then $\Gamma$ is bi-Lipschitz homeomorphic to some Jordan circle in $\mathcal{S}_1$ (or $\mathcal{S}_1'$). \end{theorem}
Here $\mathcal{S}_1$ is defined as in \cite{HM12}. The collection $\mathcal{S}_1'$ can be analogously defined using dyadic diameter functions on the unit interval $[0,1]$. We remark that the validity of this extension of the main result of \cite{HM12} to Jordan arcs is pointed out by Herron and Meyer on page 605 of \cite{HM12}.
Since Lipschitz dimension is invariant under bi-Lipschitz homeomorphisms, we may work exclusively with Jordan circles in $\mathcal{S}_1$ (or arcs in $\mathcal{S}_1'$). We will only present the details for weak quasicircles; the details for weak quasiarcs are analogous. Thus, given a curve $\Gamma=(\mathsf{S},d_\Delta)\in\mathcal{S}_1$, we construct a Lipschitz light map $F:\Gamma\to\Gamma_0$.
We will need the following map $f$ in order to achieve this goal, which we will refer to as a \textit{folding map}. Given an interval $I\subset \mathsf{S}$, divide $I$ into two dyadic subintervals of equal length with disjoint interiors, and denote these two subintervals by $I^0$ and $I^1$, respectively. We also divide $I$ into four consecutive dyadic subintervals of equal length with disjoint interiors, and denote these four subintervals by $I^{00}$, $I^{01}$, $I^{10}$, and $I^{11}$, respectively. Thus $I^0=I^{00}\cup I^{01}$ and $I^1=I^{10}\cup I^{11}$. Assume these intervals are indexed (in binary) such that adjacent intervals proceed consecutively along the positive orientation in $\mathsf{S}$. Finally, divide each of $I^{01}$ and $I^{10}$ into two dyadic subintervals of equal length with disjoint interiors, and denote these subintervals by $I^{010}$, $I^{011}$, $I^{100}$, and $I^{101}$, respectively. Thus $I^{01}=I^{010}\cup I^{011}$ and $I^{10}=I^{100}\cup I^{101}$. Again we index these intervals such that their order reflects the positive orientation of $\mathsf{S}$. The map $f:I\to I$ is defined by its action on these subintervals. It maps \begin{itemize}
\item{$I^{00}$ linearly onto $I^0$ in an orientation preserving manner,}
\item{$I^{010}$ linearly onto $I^{10}$ in an orientation preserving manner,}
\item{$I^{011}$ linearly onto $I^{10}$ in an orientation reversing manner,}
\item{$I^{100}$ linearly onto $I^{01}$ in an orientation reversing manner,}
\item{$I^{101}$ linearly onto $I^{01}$ in an orientation preserving manner, and}
\item{$I^{11}$ linearly onto $I^1$ in an orientation preserving manner.} \end{itemize}
We note that this definition of a folding map can be scaled linearly and applied to any interval. Thus, for any $n\in\mathsf{N}$, let $I\in\mathcal{I}_n$. If the two dyadic children $I'$ and $I''$ of $I$ satisfy $\Delta(I')=\Delta(I'')=\frac{1}{2}\Delta(I)$, then we define the map $f_n:I\to I$ to be the identity map. Thus $f_n$ is an isometry from $(I,d_{n+1})\to(I,d_n)$. If $\Delta(I')=\Delta(I'')=\Delta(I)$, then we define the map $f_n:I\to I$ to be a folding map. The map $f_n:\Gamma_{n+1}\to\Gamma_n$ is defined in this manner on each interval $I\in\mathcal{I}_n$.
\begin{lemma}\label{L:fold} For each $n\in\mathsf{N}$, the map $f_n:\Gamma_{n+1}\to\Gamma_n$ is $1$-Lipschitz. \end{lemma}
\begin{proof} We examine the image of an interval $I\in\hat{\mathcal{I}}$ under the map $f_n$. If $I\in \hI_n$, then $f_n(I)=I\in\hat{\mathcal{I}}$. If $I\in\hat{\mathcal{I}}\setminus\hI_{n+1}$, then $f_n(I)\in\hat{\mathcal{I}}$ and $\diam_n(f_n(I))\leq \diam_{n+1}(I)$. If $I\in\mathcal{I}_{n+1}$ (the only remaining possibility), then $f_n$ fixes the endpoints of $I$, and $I\subset f_n(I)$. In particular, if $f_n$ is the identity on $\tilde{I}$ (the dyadic parent of $I$), then $f_n(I)=I$. If $f_n$ is a folding map on $\tilde{I}$, then $f_n(I)$ is the union of $I$ with an interval $J\in\mathcal{I}_{n+2}$ that is adjacent to $I$. Moreover, it is straightforward to verify that \begin{align*} \diam_n(f_n(I))&=\diam_n(I\cup J)=\diam_n(I)+\diam_n(J)\\
&=\frac{3}{4}\diam_n(\tilde{I})=\frac{3}{4}\diam_{n+1}(I)<\diam_{n+1}(I). \end{align*} Therefore, given a chain $\{I_i\}_{i=1}^N$ of dyadic intervals, $\{f_n(I_i)\}_{i=1}^N$ can be written as a chain of dyadic intervals $\{I_j'\}_{j=1}^{N'}$ such that $\cup_{i=1}^Nf_n(I_i)=\cup_{j=1}^{N'}I_j'$, and \[\sum_{i=1}^N\diam_n(f_n(I_i))=\sum_{j=1}^{N'}\diam_n(I_j')=\sum_{j=1}^{N'}\Delta_n(I_j').\] Thus, if $\{I_i\}_{i=1}^N$ joins $x$ and $y$, then $\{I_j'\}_{j=1}^{N'}$ joins $f_n(x)$ and $f_n(y)$, and \[d_n(f_n(x),f_n(y))\leq\sum_{j=1}^{N'}\Delta_n(I_j')=\sum_{i=1}^N\diam_n(f_n(I_i))\leq\sum_{i=1}^N\Delta_{n+1}(I_i).\] It follows from the definition of $d_{n+1}(x,y)$ that $d_n(f_n(x),f_n(y))\leq d_{n+1}(x,y)$. \end{proof}
Given any $m\leq n\in\mathsf{N}$, we define $F_{m,n}:=f_m\circ f_{m+1}\circ \dots \circ f_n:\Gamma_{n+1}\to \Gamma_m$. If $m=n$, then we understand $F_{m,m}=F_{n,n}$ to denote $f_n$. As a composition of $1$-Lipschitz maps (\rf{L:fold}), each map $F_{m,n}:\Gamma_{n+1}\to\Gamma_m$ is $1$-Lipschitz. Furthermore, this sequence of maps induces a $1$-Lipschitz map $F_m:(\mathcal{D},d_\Delta)\to(\Gamma_m,d_m)$. To see this, let $x\in \mathcal{D}$, and let $k\in\mathsf{N}$ be the smallest integer such that $x\in \mathcal{D}_k$. For any $n\geq k$, the map $f_n$ fixes the set $\mathcal{D}_k$. Therefore, if $n\geq k$, then $f_n(x)=x$. If $n\geq k>m$, we then observe that $\lim_{n\to+\infty}F_{m,n}(x)=F_{m,k}(x)$. In this case, define $F_m(x):=F_{m,k}(x)$. If $n>m\geq k$, then define $F_m(x):=x$.
To see that $F_m$ thus defined is $1$-Lipschitz on $\mathcal{D}$, let $x,y$ denote any two points in $\mathcal{D}$. Choose $k\in\mathsf{N}$ such that $x,y\in \mathcal{D}_k$ and $k>m$. Via \rf{E:leqn} and the fact that, for all $n\geq m$ the map $F_{m,n}$ is $1$-Lipschitz, we have \[d_m(F_m(x),F_m(y))=d_m(F_{m,k}(x),F_{m,k}(y))\leq d_{k+1}(x,y)\leq d_\Delta(x,y).\] Since $\Gamma$ is complete, $\mathcal{D}$ is dense in $\Gamma$ (cf.\,\cite[Section 3.2]{HM12}), and $F_m$ is Lipschitz on $\mathcal{D}$, it is then straightforward to extend $F_m$ such that $F_m:\Gamma\to\Gamma_m$ is $1$-Lipschitz.
We note that we may also view the maps $F_{m,n}$ as acting on $\Gamma$. Moreover, it follows from \rf{E:leqn} that $F_{m,n}:\Gamma\to\Gamma_m$ is $1$-Lipschitz. With this in mind, we prove the following lemma.
\begin{lemma}\label{L:uniform} For each $m\in\mathsf{N}$, the maps $F_{m,n}:\Gamma\to\Gamma_m$ uniformly converge to $F_m:\Gamma\to\Gamma_m$ as $n\to+\infty$. \end{lemma}
\begin{proof} Fix $m\in\mathsf{N}$ and $\varepsilon>0$. Choose $M\in\mathsf{N}$ such that
\[n\geq M \quad\text{implies} \quad\max\{\Delta(I)\,|\,I\in\mathcal{I}_n\}<\varepsilon/2.\] For any $x\in\Gamma$, there exists a nested sequence of dyadic intervals $I_n\in\mathcal{I}_n$ such that, for every $n\in\mathsf{N}$, we have $I_n\subset I_{n-1}$ and $x\in I_n$. Furthermore, there exists $x_n\in\mathcal{D}_n$ such that $x_n\in I_n$, and so $d_\Delta(x_n,x)\to0$. Assuming $n>\max\{m,M\}$, we have $F_m(x_{n+j})=F_{m,n+j}(x_{n+j})$. If $j=0$, write $w_{n,0}:=x_n$. If $j\geq1$, write $w_{n,j}:=f_{n+1}\circ\dots \circ f_{n+j}(x_{n+j})$. In either case, we note that $F_m(x_{n+j})=F_{m,n+j}(x_{n+j})=F_{m,n}(w_{n,j})$. Since $f_{n+j}(I_n)=I_n$, we have $w_{n,j}\in I_n$. Thus, \[F_m(x)=\lim_{j\to+\infty} F_m(x_{n+j})=\lim_{j\to+\infty}F_{m,n}(w_{n,j})\in F_{m,n}(I_n).\] Combining these observations, we find that, for $n\geq \max\{m,M\}$, we have \begin{align*} d_m(F_{m,n}(x),F_m(x))&\leq d_m(F_{m,n}(x),F_{m,n}(x_{n}))+d_m(F_{m,n}(x_{n}),F_m(x))\\ &\leq d_\Delta(x,x_{n})+\diam_m(F_{m,n}(I_n))\\
&\leq 2\diam_\Delta(I_n)\leq 2\max\{\Delta(I)\,|\,I\in\mathcal{I}_n\}<\varepsilon. \end{align*} It follows that $F_{m,n}:\Gamma\to\Gamma_m$ is uniformly convergent to $F_m:\Gamma\to\Gamma_m$. \end{proof}
\section{Proving that $F_0:\Gamma\to\Gamma_0$ is Lipschitz light}\label{S:light}
Our goal in this section is to verify the existence of a constant $C\geq 1$ such that, for any subset $E\subset \Gamma_0$, the $\diam_0(E)$-components of $F_0^{-1}(E)$ have $d_\Delta$-diameter bounded above by $C\diam_0(E)$. Via the following lemma, this will be sufficient to prove that $F_0:\Gamma\to\Gamma_0$ is Lipschitz light, and thus (via the comments at the outset of \rf{S:lip}) that $\Gamma$ has Lipschitz dimension equal to $1$.
\begin{lemma}\label{L:def} Suppose there exists a constant $C\geq 1$ such that $F:\Gamma\to\Gamma_0$ is $C$-Lipschitz, and, for any subset $E\subset \Gamma_0$ such that $\diam_0(E)>0$, the $\diam_0(E)$-components of $F^{-1}(E)$ have $d_\Delta$-diameter bounded by $C\,\diam_0(E)$. This implies that, for any $r>0$ and any subset $E\subset \Gamma_0$ satisfying $\diam_0(E)\leq r$, the $r$-components of $F^{-1}(E)$ have $d_\Delta$-diameter bounded by $C'r$, for $C':=\max\{C,8\}$. \end{lemma}
\begin{proof} Let $r>0$, and let $E\subset \Gamma_0$ be such that $\diam_0(E)\leq r$. We may assume that $E$ is compact, and that $\diam_0(E)<r$. If $r\geq 1/8$, then we note that $\diam_0(F^{-1}(E))\leq \diam_\Delta(\Gamma)\leq 1\leq 8r$. Thus, we may assume that $r<1/8$.
We claim that $E$ is contained in a subset $E'\subset \Gamma_0$ such that $\diam_0(E')=r$. To see this, we modify the argument employed in \cite[Remark 1.9]{David19}. We first note that $d_0=\lambda$ (the normalized length distance defined in \rf{S:prelims}). Therefore, given $x\in\Gamma_0$, the subset $I(x):=\{y\in \Gamma_0\,|\,d_0(x,y)\leq 1/8\}$ is isometric to some interval in $\mathsf{R}$ of length $1/4$. If $x\in E$, then $E\subset I(x)$. Let $a$ and $b$ denote the first and last points in $E$ along the interval $I(x)$. Thus $d_0(a,b)=\diam_0(E)<r$. Let $c\in I(x)$ be such that $d_0(a,c)=r<1/8$ and $\Gamma_0[a,b]\subset\Gamma_0[a,c]=:E'$. Then $E\subset E'$ and $\diam_0(E')=r$. Clearly, $r$-components of $F^{-1}(E)$ are contained in $r$-components of $F^{-1}(E')$, which (by assumption) have $d_\Delta$-diameter bounded by $Cr$. \end{proof}
With the above lemma in hand, we begin our proof that $F_0:\Gamma\to\Gamma_0$ is Lipschitz light. Fix $E\subset \Gamma_0$ such that $\diam_0(E)>0$, and let $M^*\in\mathsf{N}$ be maximal such that \begin{equation}\label{E:delta_size} 2^{-M^*-1}\leq\diam_0(E)<2^{-M^*}. \end{equation} We may assume that $M^*\geq 3$, else $\diam_\Delta(F_0^{-1}(E))\leq\diam_\Delta(\Gamma)\leq8\diam_0(E)$. By definition of $M^*$, there exist two adjacent dyadic subintervals $I,J\in\mathcal{I}_{M^*}$ such that $E\subset I\cup J$. In fact, $E$ may be contained in a single element of $\mathcal{I}_{M^*}$, but it will do no harm to assume $E$ is contained in the union of two such intervals.
We claim that it is sufficient to examine pre-images of $H:=I\cup J$. Indeed, given any $\delta>0$, the $\delta$-components of $F^{-1}(E)$ are contained in $\delta$-components of $F^{-1}(H)$.
For the remainder of this section, we set $\delta:=\diam_0(E)$.
\begin{lemma}\label{L:projection} Given $n\in\mathsf{N}$ and $U\subset \Gamma_0$, we have $F_{n+1}(F_0^{-1}(U))=F_{0,n}^{-1}(U)$. \end{lemma}
\begin{proof} Suppose $x\in F_{n+1}(F_0^{-1}(U))$, so $x=F_{n+1}(w)$ for some $w\in F_0^{-1}(U)$. Then \[F_{0,n}(x)=F_{0,n}(F_{n+1}(w))=\lim_{m\to\infty}F_{0,n}(F_{n+1,m}(w))=\lim_{m\to\infty}F_{0,m}(w)=F_0(w).\] Since $F_0(w)\in U$, it follows that $F_{n+1}(F_0^{-1}(U))\subset F_{0,n}^{-1}(U)$.
Next, let $x\in F_{0,n}^{-1}(U)$. Write $z_{n+1}:=x$, and choose a point $z_{n+2}\in f_{n+1}^{-1}(z_{n+1})$ so that $f_{n+1}(z_{n+2})=z_{n+1}$. Inductively, for each $k\geq2$, define $z_{n+k}$ such that \[f_{n+k-1}(z_{n+k})=z_{n+k-1}.\] We claim that the sequence $\{z_{n+k}\}_{k=1}^\infty$ is Cauchy with respect to $d_\Delta$, and thus convergent to some point $z\in \Gamma$. Indeed, for any $1\leq i< j$, we note that \[z_{n+i}=f_{n+i}\circ\dots\circ f_{n+j-1}(z_{n+j}).\] Let $I\in\mathcal{I}_{n+i}$ denote an interval containing $z_{n+j}$. For all $k\in\mathsf{N}$, we have $f_{n+i+k}(I)=I$. Therefore, $z_{n+i}\in I$, and so
\[d_\Delta(z_{n+i},z_{n+j})\leq\diam_\Delta(I)\leq\max\{\Delta(J)\,|\,J\in\mathcal{I}_{n+i}\}.\]
Since $\max\{\Delta(J)\,|\,J\in\mathcal{I}_{n+i}\}\to0$ as $i\to\infty$, our claim follows.
Next, we claim that $z=\lim_{k\to+\infty}z_{n+k}\in F_0^{-1}(U)$. Via \rf{L:uniform}, we have \begin{align*} F_0(z)&=\lim_{m\to+\infty}F_{0,n+m-1}(z_{n+m})\\ &=\lim_{m\to+\infty}F_{0,n}(F_{n+1,n+m-1}(z_{n+m}))=\lim_{m\to+\infty}F_{0,n}(z_{n+1})=F_{0,n}(x)\in U. \end{align*} Finally, we claim $F_{n+1}(z)=x$. Again via \rf{L:uniform}, we note that \[F_{n+1}(z)=\lim_{m\to\infty}F_{n+1,m}(z_{m+1})=\lim_{m\to\infty}f_{n+1}\circ\dots\circ f_{m}(z_{m+1})=z_{n+1}=x\] Therefore, $x\in F_{n+1}(F_0^{-1}(U))$, and so $F_{0,n}^{-1}(U)\subset F_{n+1}(F_0^{-1}(U))$. \end{proof}
\begin{lemma}\label{L:component_proj} Given $m\leq n\in\mathsf{N}$ and $x\in \Gamma$, we have $F_{m,n}(F_{n+1}(x))= F_m(x)$. \end{lemma}
\begin{proof} $F_{m,n}(F_{n+1}(w))=\lim_{k\to\infty}F_{m,n}(F_{n+1,k}(w))=\lim_{k\to\infty}F_{m,k}(w)$. \end{proof}
Let $W$ denote any fixed $\delta$-component of $F_0^{-1}(H)$. By \rf{L:projection}, we have $F_{n+1}(W)\subset F_{0,n}^{-1}(H)$. Given any $n\in\mathsf{N}$, via \rf{L:uniform}, the set $F_{n+1}(W)$ is $\delta$-connected in $\Gamma_{n+1}$. In particular, it is contained in a single $\delta$-component of $F_{0,n}^{-1}(H)$ in $\Gamma_{n+1}$. We denote this $\delta$-component by $V_{n+1}$. Thus, for every $n\geq1$, write $V_n$ to denote the $\delta$-component of $F_{0,n-1}^{-1}(H)$ containing $F_{n}(W)$. We also write $V_0:=H$.
\begin{lemma}\label{L:Vs} For $n\in\mathsf{N}$, we have $f_n(V_{n+1})\subset V_n$. Furthermore, the set $V_{n+1}$ is a $\delta$-component of $f_n^{-1}(V_n)$. \end{lemma}
\begin{proof} If $n=0$, then $V_1\subset F_0^{-1}(H)=f_0^{-1}(H)$, and so $f_0(V_1)\subset H=V_0$. We assume $n\geq1$. Via \rf{L:component_proj}, we have $F_n(W)= f_n(F_{n+1}(W))\subset f_n(V_{n+1})\subset F_{0,n-1}^{-1}(H)$. By definition, $F_n(W)\subset V_n\subset F_{0,n-1}^{-1}(H)$. Therefore, the sets $V_n$ and $f_n(V_{n+1})$ are both subsets of $F_{0,n-1}^{-1}(H)$ and have non-trivial intersection. Since $f_n:\Gamma_{n+1}\to\Gamma_n$ is $1$-Lipschitz, the set $f_n(V_{n+1})$ is $\delta$-connected. Since $V_n$ is a maximal $\delta$-connected subset of $F_{0,n-1}^{-1}(H)$, we must have $f_n(V_{n+1})\subset V_n$.
Since $V_{n+1}\subset f_n^{-1}(V_n)$ and $V_{n+1}$ is $\delta$-connected, $V_{n+1}$ is contained in a single $\delta$-component of $f_n^{-1}(V_n)\subset F_{0,n}^{-1}(H)$. Since $V_{n+1}$ is a maximal $\delta$-connected subset of $F_{0,n}^{-1}(H)$, the set $V_{n+1}$ is equal to a single $\delta$-component of $f_n^{-1}(V_n)$. \end{proof}
\begin{lemma}\label{L:goal} There exists $N^*\in\mathsf{N}$ such that, if $n\geq N^*$, then \[\diam_\Delta(W)\leq \diam_n(V_n)+\delta.\] \end{lemma}
\begin{proof}
Choose $N^*\in\mathsf{N}$ such that, for $n\geq N^*$, we have $\max\{\Delta(I)\,|\,I\in\mathcal{I}_n\}<\delta/4$. Let $x,y\in W$. Since, for $n\in\mathsf{N}$, the map $F_n$ fixes elements of $\mathcal{I}_n$, we note that
\[d_\Delta(x,y)\leq d_\Delta(F_n(x),F_n(y))+2\max\{\Delta(I)\,|\,I\in\mathcal{I}_n\}<d_\Delta(F_n(x),F_n(y))+\delta/2.\] Furthermore, via \rf{L:distconverge}, we also have (for $n\geq N^*$) \[d_\Delta(F_n(x),F_n(y))\leq \diam_\Delta(F_n(W))\leq \diam_n(F_n(W))+\delta/2.\] Since $F_n(W)\subset V_n$, we conclude that, for any $n\geq N^*$ and any $x,y\in W$, we have \[d_\Delta(x,y)\leq \diam_n(V_n)+\delta.\] It follows that $\diam_\Delta(W)\leq \diam_n(V_n)+\delta$. \end{proof}
\begin{lemma}\label{L:big} If, for some $n\in\mathsf{N}$, the set $V_n$ is contained in an interval $I_n\in\mathcal{I}_n$ and $\diam_n(V_n)\geq\frac{1}{4}\diam_n(I_n)$, then, for any $k\in\mathsf{N}$, we have $\diam_{n+k}(V_{n+k})\leq4\diam_n(V_n)$. \end{lemma}
\begin{proof} We first note that $V_{n+1}\subset I_n$, since, by \rf{L:Vs}, $f_n(V_{n+1})\subset V_n\subset I_n$, and $f_n(I_n)=I_n$. Via induction, for all $k\in\mathsf{N}$, we have $V_{n+k}\subset I_n$. Therefore, via \rf{E:intn}, we have $\diam_{n+k}(V_{n+k})\leq\diam_{n+k}(I_n)=\diam_n(I_n)\leq 4\diam_n(V_n)$. \end{proof}
\begin{lemma}\label{L:cases} Suppose that, for some $n\in\mathsf{N}$, we have \begin{enumerate}
\item{$\diam_n(V_n)=\diam_0(V_0)$}
\item{$V_n$ is the union of two adjacent intervals from $\mathcal{I}_m$, for some $m\geq M^*$,}
\item{$V_n$ is not symmetric about a point in $\mathcal{D}_n$,}
\item{$V_n$ is contained in a single interval $I_n\in\mathcal{I}_n$, and}
\item{$\diam_n(V_n)\leq\frac{1}{4}\diam_n(I_n)$.} \end{enumerate} Under these assumptions, $\diam_{n+1}(V_{n+1})\leq 2\diam_n(V_n)$. If $\diam_{n+1}(V_{n+1})>\diam_n(V_n)$, then $\diam_{n+1}(V_{n+1})=2\diam_n(V_n)$ and $V_{n+1}$ is symmetric about a point in $\mathcal{D}_{n+3}\setminus\mathcal{D}_{n+1}$. If, on the other hand, $\diam_{n+1}(V_{n+1})<\diam_n(V_n)$, then $\diam_{n+1}(V_{n+1})=0$ and $V_{n+1}$ is a point in $\mathcal{D}_{n+3}\setminus\mathcal{D}_{n+2}$. \end{lemma}
\begin{proof} Note that Assumption (3) follows from Assumption (4) when $n\geq1$; we list Assumption (3) to address the case that $n=0$.
We use binary superscripts to index the four second-generation dyadic sub-intervals $I_n^{00}$, $I_n^{01}$, $I_n^{10}$, and $I_n^{11}$ in $I_n$ such that they proceed consecutively along the positive orientation in $I_n$.
If $f_n$ is the identity on $I_n$, then the lemma is trivial. Therefore, we assume that $f_n$ is a folding map on $I_n$, and we consider the cases below. We preface this case analysis with the reminder that \[\delta<\frac{1}{2^{M^*}}=\frac{1}{2}\diam_0(V_0)=\frac{1}{2}\diam_n(V_n)\leq\frac{1}{8}\diam_n(I_n).\]
Case 1: $V_n\subset I_n^{00}$. In this case, $f_n^{-1}(V_n)$ consists of either one $\delta$-component or (if $V_n$ contains the right endpoint of $I^{00}_n$) it consists of two. If one, then, via \rf{L:Vs}, we have $V_{n+1}=f_n^{-1}(V_n)\subset I_n^{00}$ and $\diam_{n+1}(V_{n+1})=\diam_n(V_n)$. If two, then one $\delta$-component is contained in $I^{00}_n$ and satisfies $\diam_{n+1}(V_{n+1})=\diam_n(V_n)$ while the other is a single point located at the midpoint of $I^{10}_n$.
Case 2: $V_n\subset I_n^{01}$. In this case, there are at most three $\delta$-components of $f_n^{-1}(V_n)$: one in $I_n^{00}$ and either one or two in $I_n^{10}$. The component in $I_n^{00}$ has $d_{n+1}$-diameter equal to $\diam_n(V_n)$. If there are two components in $I_n^{10}$, then they each have $d_{n+1}$-diameter equal to $\diam_n(V_n)$. If there is one component in $I_n^{10}$, then it has $d_{n+1}$-diameter equal to $2\diam_n(V_n)$, and it is symmetric about the midpoint of $I_n^{10}$. Via \rf{L:Vs}, if $\diam_{n+1}(V_{n+1})>\diam_n(V_n)$, then $V_{n+1}$ is symmetric about a point in $\mathcal{D}_{n+3}\setminus\mathcal{D}_{n+1}$ and $\diam_{n+1}(V_{n+1})=2\diam_n(V_n)$.
Case 3: $V_n\subset I_n^{10}$. By symmetry, we can apply an argument parallel to that used in Case 2 to conclude that $\diam_{n+1}(V_{n+1})\leq 2\diam_n(V_n)$. Furthermore, if $\diam_{n+1}(V_{n+1})>\diam_n(V_n)$, then $V_{n+1}$ is symmetric about a point in $\mathcal{D}_{n+3}\setminus\mathcal{D}_{n+1}$ and $\diam_{n+1}(V_{n+1})=2\diam_n(V_n)$.
Case 4. $V_n\subset I_n^{11}$. By symmetry, we can apply an argument parallel to that used in Case 1 to conclude that, either $V_{n+1}\subset I_n^{11}$ has diameter equal to $\diam_n(V_n)$, or $V_{n+1}$ is a single point at the midpoint of $I_n^{01}$.
Case 5: $V_n$ is symmetric about a point in $\mathcal{D}_{n+2}\setminus\mathcal{D}_{n+1}$. In this case, $f_n^{-1}(V_n)$ consists of two $\delta$-components. One is contained in $I_n^{00}$ (or $I_n^{11}$), and the other is contained in $I_n^{10}$ (or $I_n^{01}$). Each component has $d_{n+1}$-diameter equal to $\diam_n(V_n)$. Via \rf{L:Vs}, $\diam_{n+1}(V_{n+1})=\diam_n(V_n)$.
Case 6: $V_n$ is symmetric about a point in $\mathcal{D}_{n+1}\setminus\mathcal{D}_n$. In this case, there are three $\delta$-components of $f_n^{-1}(V_n)$, and each has $d_{n+1}$-diameter equal to $\diam_n(V_n)$. We note that one of these $\delta$-components is symmetric about a point in $\mathcal{D}_{n+1}\setminus\mathcal{D}_n$. In particular, this is the only case in which $V_{n+1}$ might not be contained in a single interval from $\mathcal{I}_{n+1}$. Via \rf{L:Vs}, $\diam_{n+1}(V_{n+1})=\diam_n(V_n)$.
Having exhausted the possible cases, we conclude the proof of the lemma. \end{proof}
\begin{lemma}\label{L:none} If there exists $K\in\mathsf{N}$ such that, for all $k\leq K$, the set $V_k$ is not symmetric about a point in $\mathcal{D}_{k+2}$, then, either there exists $n\geq N^*$ (for $N^*$ as in \rf{L:goal}) such that $\diam_n(V_n)\leq 16\delta$, or, for all $k\leq K$, \begin{enumerate}
\item[$(k.1)$]{$\diam_k(V_k)=\diam_0(V_0)$,}
\item[$(k.2)$]{$V_k$ is the union of two adjacent intervals from $\mathcal{I}_{m}$ for some $m\geq M^*$,}
\item[$(k.3)$]{$V_k$ is contained in a single interval $I_k\in\mathcal{I}_k$, and}
\item[$(k.4)$]{$\diam_k(V_k)\leq \frac{1}{4}\diam_k(I_k)$.} \end{enumerate} \end{lemma}
\begin{proof} Suppose $K\in\mathsf{N}$ is such that, for all $k\leq K$, no set $V_k$ is symmetric about a point in $\mathcal{D}_{k+2}$. In preparation for an inductive argument, we affirm the base case $k=0\leq K$. Indeed, for $V_0=H$, we have \begin{itemize}
\item[(0.1)]{$\diam_0(V_0)=\diam_0(V_0)$,}
\item[(0.2)]{$V_0$ is the union of two adjacent intervals from $\mathcal{I}_{M^*}$, and}
\item[(0.3)]{$V_0$ is contained in a single interval $I_0\in\mathcal{I}_0$.}
\item[(0.4)]{$\diam_0(V_0)\leq \frac{1}{4}\diam_0(I_0)$.} \end{itemize} Here we recall that $M^*\geq 3$. To proceed, we assume that, either there exists $n\geq N^*$ such that $\diam_n(V_n)\leq 16\delta$, or, for all $n\leq k-1\leq K-1$, we have \begin{itemize}
\item[$(n.1)$]{$\diam_n(V_n)=\diam_0(V_0)$,}
\item[$(n.2)$]{$V_n$ is the union of two adjacent intervals from $\mathcal{I}_{m}$, for some $m\geq M^*$,}
\item[$(n.3)$]{$V_n$ is contained in a single interval $I_n\in\mathcal{I}_n$, and}
\item[$(n.4)$]{$\diam_n(V_n)\leq \frac{1}{4}\diam_n(I_n)$.} \end{itemize} Therefore, either there exists $n\geq N^*$ such that $\diam_n(V_n)\leq 16\delta$, or we satisfy the assumptions of \rf{L:cases} for $V_{k-1}$. Since $V_k$ is not symmetric about $\mathcal{D}_{k+2}$, \rf{L:cases} tells us that \begin{itemize}
\item[$(k.1)$]{$\diam_k(V_k)=\diam_{k-1}(V_{k-1})$.} \end{itemize} We note that, if $f_{k-1}$ is the identity on $I_{k-1}$, then $V_k=V_{k-1}$. If $f_{k-1}$ is a folding map on $I_{k-1}$, then $V_k$ is the union of two adjacent intervals in $\mathcal{I}_{m+1}$ (here we are using (k.1)). In either case, \begin{itemize}
\item[$(k.2)$]{$V_k$ is the union of two adjacent intervals in $\mathcal{I}_{m}$, for some $m\geq M^*$.} \end{itemize} Furthermore, since $V_{k-1}$ is not symmetric about a point in $\mathcal{D}_{k}$, Case 6 (in the proof of \rf{L:cases}) cannot occur. It follows that \begin{itemize}
\item[$(k.3)$]{$V_k$ is contained in a single interval $I_k\in\mathcal{I}_k$.} \end{itemize} Furthermore, if $\diam_k(V_k)>\frac{1}{4}\diam_k(I_k)$, then, (since $V_k$ is the union of two adjacent dyadic intervals) we must have either $\diam_k(V_k)=\frac{1}{2}\diam_k(I_k)$ or $\diam_k(V_k)=\diam_k(I_k)$. Since, by assumption, $V_k$ is not symmetric about a point in $\mathcal{D}_{k+2}$, neither case can occur. Therefore, \begin{itemize}
\item[$(k.4)$]{$\diam_k(V_k)\leq\frac{1}{4}\diam_k(I_k)$.} \end{itemize}
Thus we conclude our inductive argument, and the proof of the lemma. \end{proof}
\begin{lemma}\label{L:symm_next} Suppose there exist $n\leq K\in\mathsf{N}$ such that, for all $n\leq k\leq K$, the set $V_{k}$ is not symmetric about a point in $\mathcal{D}_{k+1}\setminus\mathcal{D}_{k}$, and $n\in\mathsf{N}$ is such that \begin{enumerate}
\item[$(n.1)$]{Either $\diam_n(V_n)=0$ or $\diam_0(V_0)\leq\diam_n(V_n)\leq2\diam_0(V_0)$}
\item[$(n.2)$]{$V_n$ is symmetric about a point in $\mathcal{D}_{n+2}\setminus\mathcal{D}_{n+1}$,}
\item[$(n.3)$]{$V_n$ is contained in a single interval $I_n\in\mathcal{I}_n$, and}
\item[$(n.4)$]{$\diam_n(V_n)\leq \frac{1}{4}\diam_n(I_n)$.} \end{enumerate} Under these assumptions on $n$ and $K$, for all $n\leq k\leq K$, it is true that \begin{enumerate}
\item[$(k.1)$]{$\diam_k(V_k)=\diam_n(V_n)$.}
\item[$(k.2)$]{$V_{k}$ is symmetric about a point in $\mathcal{D}_{k+2}\setminus\mathcal{D}_{k+1}$,}
\item[$(k.3)$]{$V_{k}$ is contained in a single interval $I_{k}\in\mathcal{I}_{k}$, and}
\item[$(k.4)$]{$\diam_{k}(V_{k})\leq \frac{1}{4}\diam_n(I_{k})$.} \end{enumerate} \end{lemma}
\begin{proof} By way of induction, we first note that the base case $k=n$ is included in our assumptions. Thus, we assume that $K>n$ and, for all $n\leq j\leq k-1\leq K-1$, \begin{itemize}
\item[$(j.1)$]{$\diam_j(V_j)=\diam_n(V_n)$,}
\item[$(j.2)$]{$V_{j}$ is symmetric about a point in $\mathcal{D}_{j+2}\setminus\mathcal{D}_{j+1}$,}
\item[$(j.3)$]{$V_{j}$ is contained in a single interval $I_{j}\in\mathcal{I}_{j}$, and}
\item[$(j.4)$]{$\diam_{j}(V_{j})\leq \frac{1}{4}\diam_{j}(I_{j})$.} \end{itemize} We prove that the analogous conclusions hold for $V_{k}$. Indeed, if $f_{k-1}$ is the identity on $I_{k-1}$, then $V_{k}=V_{k-1}$ is symmetric about a point in $\mathcal{D}_{k+1}\setminus\mathcal{D}_k$. Since, by assumption, this cannot occur, we only need to consider the case that $f_{k-1}$ is a folding map on $I_{k-1}$. Since $\delta<\frac{1}{2}\diam_0(V_0)\leq\frac{1}{8}\diam_{k-1}(I_{k-1})$, it is clear that \begin{enumerate}
\item[$(k.1)$]{$\diam_k(V_k)=\diam_{k-1}(V_{k-1})$,}
\item[$(k.2)$]{$V_{k}$ is symmetric about a point in $\mathcal{D}_{k+2}\setminus\mathcal{D}_{k+1}$,}
\item[$(k.3)$]{$V_{k}$ is contained in a single interval $I_{k}\in\mathcal{I}_{k}$, and}
\item[$(k.4)$]{$\diam_{k}(V_{k})\leq \frac{1}{4}\diam_{k}(I_{k})$.} \end{enumerate} This completes the inductive argument, and the proof of the lemma. \end{proof}
\begin{lemma}\label{L:not_middle} Suppose there exist $n\leq K\in\mathsf{N}$ such that, for all $n\leq k\leq K$, the set $V_{k}$ is not symmetric about a point in $\mathcal{D}_{k}$, and $n$ is such that \begin{enumerate}
\item[$(n.1)$]{Either $\diam_n(V_n)=0$ or $\diam_0(V_0)\leq\diam_n(V_n)\leq2\diam_0(V_0)$,}
\item[$(n.2)$]{$V_n$ is symmetric about a point in $\mathcal{D}_{n+1}\setminus\mathcal{D}_n$,}
\item[$(n.3)$]{$V_n$ is contained in a single interval $I_n\in\mathcal{I}_n$, and}
\item[$(n.4)$]{$\diam_n(V_n)\leq\frac{1}{4}\diam_n(I_n)$.} \end{enumerate} Under these assumptions, for all $n\leq k\leq K$, \begin{enumerate}
\item[$(k.1)$]{$\diam_k(V_k)=\diam_n(V_n)$.}
\item[$(k.2)$]{$V_k$ is symmetric about a point in $\mathcal{D}_{k+1}$,}
\item[$(k.3)$]{$V_k$ is contained in a single interval $I_k\in\mathcal{I}_k$, and}
\item[$(k.4)$]{$\diam_k(V_k)\leq\frac{1}{4}\diam_n(I_k)$.} \end{enumerate} \end{lemma}
\begin{proof} The proof consists of a straightforward inductive argument similar to that used to prove \rf{L:symm_next}. For the sake of brevity, we omit the details. \end{proof}
We are now ready to prove the following, which, via \rfs{L:goal} and \ref{L:def}, will be sufficient to prove that $F_0:\Gamma\to\Gamma_0$ is Lipschitz light.
\begin{lemma}\label{L:final} There exists $n\in\mathsf{N}$ such that $n\geq N^*$ and $\diam_n(V_n)\leq 128\delta$. \end{lemma}
\begin{proof} If there is no index $n_1$ for which $V_{n_1}$ is symmetric about a point in $\mathcal{D}_{n_1+2}$, then, by \rf{L:none}, we conclude that $\diam_N(V_N)=\diam_0(V_0)\leq 4\delta$. Therefore, we may assume $n_1$ is the minimal such index. We first consider the case that $n_1\geq1$. By the definition of $n_1$ and \rf{L:none}, we may assume that \begin{enumerate}
\item{$\diam_{n_1-1}(V_{n_1-1})=\diam_0(V_0)$,}
\item{$V_{n_1-1}$ is the union of two adjacent intervals from $\mathcal{I}_{m}$, for some $m\geq M^*$,}
\item{$V_{n_1-1}$ is contained in a single interval $I_{n_1-1}\in\mathcal{I}_{n_1-1}$, and}
\item{$\diam_{n_1-1}(V_{n_1-1})\leq\frac{1}{4}\diam_{n_1-1}(I_{n_1-1})$.} \end{enumerate} Via \rf{L:Vs}, it follows from the definition of $f_{n_1-1}$ and the minimality of $n_1$ that $V_{n_1}$ is symmetric about a point in $\mathcal{D}_{n_1+2}\setminus\mathcal{D}_{n_1+1}$, and, either $\diam_{n_1}(V_{n_1})=0$, or \[\diam_0(V_0)\leq\diam_{n_1}(V_{n_1})\leq2\diam_{n_1-1}(V_{n_1-1})=2\diam_0(V_0).\] Furthermore, $V_{n_1}$ is contained in a single interval $I_{n_1}\in\mathcal{I}_{n_1}$. If $\diam_{n_1}(V_{n_1})>\frac{1}{4}\diam_{n_1}(I_{n_1})$, then, via \rf{L:big} and \rf{E:delta_size}, there exists $n\geq N^*$ such that \[\diam_n(V_n)\leq4\diam_{n_1}(V_{n_1})\leq 8\diam_0(V_0)\leq 32\delta.\] Therefore, we assume that $\diam_{n_1}(V_{n_1})\leq \frac{1}{4}\diam_{n_1}(I_{n_1})$.
If there is no index $n>n_1$ such that $V_n$ is symmetric about a point in $\mathcal{D}_{n+1}\setminus\mathcal{D}_{n}$, then, by \rf{L:symm_next}, we conclude that there exists $n\geq N^*$ such that $\diam_n(V_n)\leq 2\diam_0(V_0)\leq8\delta$. Therefore, we may assume that there exists $n_2>n_1$ minimal such that $V_{n_2}$ is symmetric about a point in $\mathcal{D}_{n_2+1}\setminus\mathcal{D}_{n_2}$. Furthermore, the proof of \rf{L:symm_next} makes it clear that $f_{n_2-1}$ is the identity on $I_{n_2-1}$ and so $V_{n_2}=V_{n_2-1}$. Therefore, $V_{n_2}$ is contained in an interval $I_{n_2}\in\mathcal{I}_{n_2}$ and, either $\diam_{n_2}(V_{n_2})=0$, or $\diam_0(V_0)\leq \diam_{n_2}(V_{n_2})\leq2\diam_0(V_0)$. If $\diam_{n_2}(V_{n_2})>\frac{1}{4}\diam_{n_2}(I_{n_2})$, then, via \rf{L:big} and \rf{E:delta_size}, there exists $n\geq N^*$ such that $\diam_n(V_n)\leq 32\delta$. Therefore, we may assume that $\diam_{n_2}(V_{n_2})\leq \frac{1}{4}\diam_{n_2}(I_{n_2})$,
If there is no index $n>n_2$ such that $V_n$ is symmetric about a point in $\mathcal{D}_n$, then, via \rf{L:not_middle} and \rf{E:delta_size}, we conclude that there exists $n\geq N^*$ such that $\diam_n(V_n)\leq 2\diam_0(V_0)\leq 8\delta$. Thus, we assume that there exists $n_3>n_2$ minimal such that $V_{n_3}$ is symmetric about a point in $\mathcal{D}_{n_3}$.
If $V_{n_3}$ is a single point, then we note that, for all $n\geq n_3$, we have $f_n^{-1}(V_{n_3})=V_{n_3}$ (since $f_n$ fixes points in $\mathcal{D}_{n_3}$). Therefore, there exists $n\geq N^*$ such that $\diam_n(V_n)=0<\delta$. Thus we may assume that $\diam_{n_3}(V_{n_3})>0$. In this case, we note that $n_3$ is minimal such that $V_{n_3}$ is not contained in a single interval from $\mathcal{I}_{n_3}$. It is also easy to verify (via \rf{L:not_middle}) that $\diam_{n_3}(V_{n_3})=\diam_{n_2}(V_{n_2})$. Indeed, $V_{n_3}$ is contained in the interior of the union of two adjacent intervals from $\mathcal{I}_{n_3}$ whose union forms $I_{n_3-1}\in\mathcal{I}_{n_3-1}$.
Write $I_{n_3}'$ to denote the left dyadic child of $I_{n_3-1}$, and write $V_{n_3}':=V_{n_3}\cap I_{n_3}'$. Inductively, for each $k\geq1$, write $I_{n_3+k}'$ to denote the right dyadic child of $I_{n_3+k-1}'$. Since $\diam_{n_3}(V_{n_3}')\leq\frac{1}{4}\diam_{n_3}(I_{n_3}')$, we have $V_{n_3}'\subset I'_{n_3+2}$. If $\diam_{n_3}(V_{n_3}')=\frac{1}{4}\diam_{n_3}(I_{n_3}')$, then \[\diam_{n_3-1}(V_{n_3-1})=2\diam_{n_3}(V_{n_3}')=\frac{1}{2}\diam_{n_3}(I_{n_3}')\geq\frac{1}{4}\diam_{n_3-1}(I_{n_3-1}).\] Therefore, by \rf{L:big} and \rf{E:delta_size}, there exists $n\geq N^*$ such that \[\diam_n(V_n)\leq 4\diam_{n_3-1}(V_{n_3-1})\leq8\diam_0(V_0)\leq32\delta.\] Therefore, we may assume that $\diam_{n_3}(V_{n_3}')<\frac{1}{4}\diam_{n_3}(I_{n_3}')$.
For each $n\geq n_3$, we define $V_n':=V_n\cap I_{n_3}'$, and we define the ratio \[R(n):=\frac{\diam_n(V'_n)}{\diam_n(I_n')}.\] Since $R(n_3)<\frac{1}{4}$, we have $\diam_{n_3+1}(V_{n_3+1}')=\diam_{n_3}(V_{n_3}')$. If $f_{n_3}$ is a folding map on $I_{n_3}'$, then $V_{n_3+1}'\subset I_{n_3+3}'$ and $R(n_3+1)=R(n_3)<\frac{1}{4}$. If $f_{n_3}$ is the identity on $I_{n_3}$, then $R(n_3+1)=2R(n_3)$. If $R(n_3+1)\geq\frac{1}{4}$, then define $n_4:=n_3+1$. If not, then we proceed inductively, and assume that, for all $n_3+1\leq j\leq k-1$, we have $V_{n_3+j}'\subset I_{n_3+j+2}'$, $\diam_{n_3+j}(V_{n_3+j}')=\diam_{n_3}(V_{n_3}')$, and $R(n_3+j)<\frac{1}{4}$.
Under this inductive hypothesis, we examine $V_{n_3+k}'$. Either $f_{n_3+k-1}$ is a folding map on $I_{n_3+k-1}'$, and $R(n_3+k)=R(n_3+k-1)<\frac{1}{4}$, or $f_{n_3+k-1}$ is the identity on $I_{n_3+k-1}'$, and $R(n_3+k)=2R(n_3+k-1)$. If $R(n_3+k)\geq\frac{1}{4}$, then write $n_4:=n_3+k$.
Via induction, we are faced with two possibilities: either there exists $n_4>n_3$ minimal such that $V_{n_4}'\subset I_{n_4}'$ and $R(n_4)\geq\frac{1}{4}$, or, for all $n> n_3$, we have $V_n'\subset I_{n+2}'$ and $R(n)< \frac{1}{4}$. We claim this latter case cannot occur. Indeed, we note that, for all $n>n_3$, we have $R(n+1)\geq R(n)$. Moreover, $R(n+1)>R(n)$ if and only if $f_n$ is the identity on $I_{n}'$ and $R(n+1)=2R(n)$. Since $\diam_n(I_n')\to 0$, the map $f_n$ must be the identity on $I_n'$ infinitely often, and thus $R(n+1)=2R(n)$ infinitely often. This would imply that $R(n)\to+\infty$, and this contradiction proves our claim.
Thus we have $V_{n_4}'=V_{n_4}\cap I_{n_3}'\subset I_{n_4}'$ such that $\diam_{n_4}(V_{n_4}')\geq\frac{1}{4}\diam_{n_4}(I_{n_4}')$. Recall that, for any $n\geq n_4$, the map $f_n$ fixes elements of $\mathcal{I}_{n_4}$ and $\mathcal{I}_{n_3}$. Therefore, for any $n\geq n_4$, we have $V_n'\subset I_{n_4}'$, and so \begin{align*} \diam_n(V_n')&\leq \diam_n(I_{n_4}')=\diam_{n_4}(I_{n_4}')\\ &\leq4\diam_{n_4}(V_{n_4}')=8\diam_{n_3}(V_{n_3})\leq 16\diam_0(V_0)\leq64\delta. \end{align*}
An analogous argument applies to the set $V_{n_3}'':=V_{n_3}\cap I_{n_3}''$, where $I_{n_3}''$ denotes the right dyadic child of $I_{n_3-1}$. In particular, there exists $n_5>n_3$ such that, if $n\geq n_5$, then $\diam_n(V_n'')\leq 64\delta$. Therefore, there exists $n\geq N^*$ such that \[\diam_n(V_n)\leq \diam_n(V_n\cap I_{n_3}')+\diam_n(V_n\cap I_{n_3}'')\leq 128\delta.\]
We finish by briefly considering the case that $n_1=0$. If $V_0$ is symmetric about a point in $\mathcal{D}_2\setminus \mathcal{D}_1$, then we argue as in the case that $n_1\geq1$. If $V_0$ is symmetric about a point in $\mathcal{D}_1\setminus\mathcal{D}_0$, then we apply the argument utilized in our above analysis of $V_{n_2}$. If $V_0$ is symmetric about the point in $\mathcal{D}_0$, then we apply (a simple modification of) the argument used in our above analysis of $V_{n_3}$. \end{proof}
\end{document} | arXiv |
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Title/Summary/Keyword: Melanogenesis
Search Result 466, Processing Time 0.108 seconds
Regulation of Melanogenesis as Studied by Chemical Analysis of Melanins
Ito, Shosuke
Journal of the Society of Cosmetic Scientists of Korea
v.26 no.2
- Biochemical studies show that in the process of mixed melanogenesis, cysteinyldopas are produced first which are next oxidized to give pheomelanin. After all of the cysteine is consumed, eumelanin is then deposited on the preformed pheomelanin. - In vitro and in vivo studies show that tyrosinase activity is the most important factor that regulates the switch of melanogenesis, with higher activities increasing melanogenesis, especially eumelanogenesis. - In culturted melanocytes, the tyrosine to cysteine ratio is critical in determining the eumelanin to pheomelanin ratio. - Our HPLC method to analyze eumelanin and pheomelanin has become a useful tool in the study of melanogenesis regulation. There are many problems to be solved before we fully understand the regulation of melanogenesis. Mutations in mouse models are ideal models for studying the genetic and molecular control of melanogenesis. Even in the mouse models, it is not known how cysteine is excluded from being incorporated into melanins in black and other eumelaninc mice, Conversely, it is not known how cysteine is continuously incorporated into pheomelanin in lethal yellow and recessive yellow mice.
Effects of Ethyl Acetate Extract from Ulmus davidiana var. japonica on Melanogenesis (느릅나무의 에틸 아세테이트 추출물에 의한 Melanin생성 효과)
천현자;정승일;김일광
YAKHAK HOEJI
pp.724-729
Melanogenesis is a physiological process resulted in the synthesis of melanin pigments, which has a role in protecting skin front the damaging effect of ultra-violet (UV) radiation. The main aim of the present study was to examine the effect of Ulmus davidiana var. japonica(UL) on Melanogenesis. Cells were cultured in the presence of various concentrations of Ulmus davidiana var. japonica for 48 h, and there were estimated total melanin contents as a final product and activity of tyrosinase, a key enzyme, in Melanogenesis. Among the four solvent extracts tested, EtOAc extract mostly increased tyrosinase activity, And EtOAc extract increased the melanin contents and tyrosinase activity in a dose-dependent manner. Especially It was observed that 100$\mu\textrm{g}$/ml EtOAc extract promotes melanin secretion in B16/F10 melanoma cells by 140% at 48 h treatment and activity of tyrosinase increased by 180% in the presence of same concentration. In conclusion, as for EtOAc extract treatment, there was no effect on the viability of B16/F10 cell, only to stimulate Melanogenesis.
Effect of Ailanthi Radicis Cortex Extracts on Melanogenesis
Cho, Young-Ho
Biomedical Science Letters
Melanogenesis refers to the biosynthesis of melanin pigment in melanocytes. Melanogenesis is controlled by the intra- and extracellular environments. In the present study, to develop a new whitening agent, it was investigated the antioxidant activity and the inhibitory effect of Ailanthi Radicis Cortex extract on tyrosinase activity and on melanogenesis in the B16/F1 melanoma cells. The inhibition ratio of tyrosinase activity of ethylacetate fraction from Ailanthi Radicis Cortex was higher than that of arbutin. The ethylacetate fraction showed scavenging activities of 1,1-diphenyl-2-picrylhydrazyl (DPPH) radicals and superoxide anion radicals in a dose dependent manner. The highest inhibitory activity of melanogenesis was also in ethylacetate fraction ($40.0{\pm}5%$ at the concentration of $400{\mu}g/ml$). This study demonstrates that the Ailanthi Radicis Cortex extract might be used to be a potential agent for skin whitening.
Effect of ginseng and ginsenosides on melanogenesis and their mechanism of action
Kim, Kwangmi
Journal of Ginseng Research
Abnormal changes in skin color induce significant cosmetic problems and affect quality of life. There are two groups of abnormal change in skin color; hyperpigmentation and hypopigmentation. Hyperpigmentation, darkening skin color by excessive pigmentation, is a major concern for Asian people with yellowe-brown skin. A variety of hypopigmenting agents have been used, but treating the hyperpigmented condition is still challenging and the results are often discouraging. Panax ginseng has been used traditionally in eastern Asia to treat various diseases, due to its immunomodulatory, neuroprotective, antioxidative, and antitumor activities. Recently, several reports have shown that extract, powder, or some constituents of ginseng could inhibit melanogenesis in vivo or in vitro. The underlying mechanisms of antimelanogenic properties in ginseng or its components include the direct inhibition of key enzymes of melanogenesis, inhibition of transcription factors or signaling pathways involved in melanogenesis, decreasing production of inducers of melanogenesis, and enhancing production of antimelanogenic factor. Although there still remain some controversial issues surrounding the antimelanogenic activity of ginseng, especially in its effect on production of proinflammatory cytokines and nitric oxide, these recent findings suggest that ginseng and its constituents might be potential candidates for novel skin whitening agents.
https://doi.org/10.1016/j.jgr.2014.10.006 인용 PDF KSCI
Role of NADPH Oxidase-mediated Generation of Reactive Oxygen Species in the Apigenin-induced Melanogenesis in B16 Melanoma Cells (B16 흑색종세포에서 아피제닌에 의한 멜라닌 합성에 미치는 NADPH 산화효소-유래 활성산소종의 역할)
Lee, Yong-Soo
Previously, we have reported that apigenin, a natural flavonoid found in a variety of vegetables and fruits, stimulated melanogenesis through the activation of $K^+-Cl^-$-cotransport (KCC) in B16 melanoma cells. In this study we investigated the possible involvement of reactive oxygen species (ROS) in the mechanism of apigenin-induced melanogenesis in B16 cells. Apigenin elevated intracellular ROS level in a dose-dependent manner. Treatment with various inhibitors of NADPH oxidase, diphenylene iodonium (DPI), apocynin (Apo) and neopterine (NP) significantly inhibited both the generation of ROS and melanogenesis induced by apigenin. In addition these inhibitors profoundly inhibited apigenin-induced $Cl^-$-dependent $K^+$ efflux, a hallmark of KCC activity. However, the apigenin-induced ROS generation was not significantly affected by treatment with a specific KCC inhibitor R-(+)-[(2-n-butyl-6,7-dichloro-2-cyclopentyl-2,3-dihydro-1-oxo-1H-inden-5-yl)oxy]acetic acid (DIOA). These results indicate that the ROS production may be a upstream regulator of the apigenin-induced KCC stimulation, and in turn, melanogenesis in the B16 cells. Taken together, these results suggest that the NADPH oxidase-mediated ROS production may play an important role in the apigenin-induced melanogenesis in B16 cells. These results further suggest that NADPH oxidase may be a good target for the management of hyperpigmentation disorders.
Involvement of Transglutaminase-2 in α-MSH-Induced Melanogenesis in SK-MEL-2 Human Melanoma Cells
Kim, Hyun Ji;Lee, Hye Ja;Park, Mi Kyung;Gang, Kyung Jin;Byun, Hyun Jung;Park, Jeong Ho;Kim, Mi Kyung;Kim, Soo Youl;Lee, Chang Hoon
Biomolecules & Therapeutics
Skin hyperpigmentation is one of the most common skin disorders caused by abnormal melanogenesis. The mechanism and key factors at play are not fully understood. Previous reports have indicated that cystamine (CTM) inhibits melanin synthesis, though its molecular mechanism in melanogenesis remains unclear. In the present study, we investigated the effect of CTM on melanin production using ELISA reader and the expression of proteins involved in melanogenesis by Western blotting, and examined the involvement of transglutaminase-2 (Tgase-2) in SK-MEL-2 human melanoma cells by gene silencing. In the results, CTM dose-dependently suppressed melanin production and dendrite extension in a-MSH-induced melanogenesis of SK-MEL-2 human melanoma cells. CTM also suppressed a-MSH-induced chemotactic migration as well as the expressions of melanogenesis factors TRP-1, TRP-2 and MITF in a-MSH-treated SK-MEL-2 cells. Meanwhile, gene silencing of Tgase-2 suppressed dendrite extension and the expressions of TRP-1 and TRP-2 in a-MSH-treated SK-MEL-2 cells. Overall, these findings suggested that CTM suppresses a-MSH-induced melanogenesis via Tgase-2 inhibition and that therefore, Tgase-2 might be a new target in hyperpigmentation disorder therapy.
https://doi.org/10.4062/biomolther.2014.031 인용 PDF KSCI KPUBS
Melanogenesis Inhibitory Activities of Diarylheptanoids from Alnus hirsuta Turcz in B16 Mouse Melanoma Cell
Cho, Soo-Min;Kwon, Young-Min;Lee, Jae-Hee;Yon, Kyu-Hyeong;Lee, Min-Won
Archives of Pharmacal Research
Four diarylheptanoids, (5R)-1,7-bis (3,4-dihydroxyphenyl)-heptane-5-O-$\beta$-D-glucoside (1), (5R)-1,7-bis (3,4-dihydroxyphenyl)-heptane-5-ol (2), oregonin (3), hirsutanonol (4), were isolated from the bark of Alnus hirsuta Turcz and its inhibitory effects on melanogenesis by measuring the melanin level and tyrosinase activity in B16 melanoma cell were examined. Melanin level and tyrosinase activity were reduced to 75 to 85% by addition of diarylheptanoids to incubation medium of the melanoma cell. On the other hand, melanin level and tyrosinase activity were reduced to 13 to 43% by the addition of diarylheptanoids to incubation medium of the melanoma cell treated with melanogenesis stimulator, $\alpha$-MSH and forskolin. These melanogenesis inhibitory effects were significantly different compared with control.
Inhibitory Effect of Melanogenesis by 5-Pentyl-2-Furaldehyde Isolated from Clitocybe sp.
Kim, Young-Hee;Choo, Soo-Jin;Ryoo, In-Ja;Kim, Bo-Yeon;Ahn, Jong-Seog;Yoo, Ick-Dong
Journal of Microbiology and Biotechnology
In the continued search for melanogenesis inhibitors from microbial metabolites, we found that the culture broth of Clitocybe sp. MKACC 53267 inhibited melanogenesis in B16F10 melanoma cells. The active component was purified by solvent extraction, silica gel chromatography, Sephadex LH-20 column chromatography, and finally by preparative HPLC. Its structure was determined as 5-pentyl-2-furaldehyde on the basis of the UV, NMR, and MS spectroscopic analysis. The 5-pentyl-2-furaldehyde potently inhibited melanogenesis in B16F10 cells with an $IC_{50}$ value of 8.4 ${\mu}g/ml$, without cytotoxicity.
https://doi.org/10.4014/jmb.1101.01053 인용 PDF KSCI
Inhibitory Effect on Melanogenesis of Rhizoma Bletillae (白급이 멜라닌 형성 억제에 미치는 영향)
Yoon, Hwa-Jung;Yoon, Jung-Won;Yoon, So-Won;Ko, Woo-Shin;Woo, Won-Hong
The Journal of Korean Medicine Ophthalmology and Otolaryngology and Dermatology
Recently many efforts were focused to understand the mechanical insights of melanogenesis to develop the agents for hyper-pigmentation and hypo-pigmentation. In the melanin biosynthetic pathway, tyrosinase is the rate limiting enzyme, and ${\alpha}$-melanocyte stimulating hormone(MSH) or cAMP-elevating agents stimulate melanogenesis and enhance the melanin synthesis and the tyrosinase activity. The author has analyzed the effects of Rhizoma Bletillae on the basal melanogenic activities of B16 mouse melanoma cells. Rhizoma Bletillae alone markedly suppressed melanin content and tyrosinase activity in a dose-dependent manner. Pretreatment of the cells with Rhizoma Bletillae. The decrease in the tyrosinase activity was paralled by a decrease in the abundance of tyrosinase protein and tyrosinase promoter activity. These results suggest that Rhizoma Bletillae inhibits melanogenesis of B16 melanoma cells via suppression of tyrosinase activity.
Melanogenesis Inhibitory Activity of Epicatechin-3-O-Gallate Isolated from Polygonum amphibium L.
Lee, Young Kyung;Hwang, Buyng Su;Hwang, Yong;Lee, Seung Young;Oh, Young Taek;Kim, Chul Hwan;Nam, Hyeon Ju;Jeong, Yong Tae
Microbiology and Biotechnology Letters
This study aimed to investigate the melanogenesis inhibitory activity of epicatechin-3-O-gallate (ECG) isolated from Polygonum amphibium L. ECG was isolated from the ethanol extract of P. amphibium L, and its chemical structure was determined using spectroscopic methods such as LC-ESI-MS, 1D-NMR, and UV spectroscopy. ECG inhibited the melanogenesis of B16F10 cells in a dose-dependent manner. Particularly, it decreased the melanin content by 27.4% at 200 µM concentration, compared with the control, in B16F10 cells, without causing cytotoxicity. It is noteworthy that the expression of three key proteins, including tyrosinase, tyrosinase-related protein-1 (TRP-1), TRP-2, and microphthalmia-associated transcription factor (MITF), involved in melanogenesis, is significantly inhibited by ECG. The ECG isolated in this study caused the inhibition of body pigmentation and tyrosinase activity in vivo in the zebrafish model. These results suggest that the ECG isolated from P. amphibium L. is an effective anti-melanogenesis agent.
https://doi.org/10.48022/mbl.2010.10012 인용 PDF
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Hadwiger conjecture (combinatorial geometry)
In combinatorial geometry, the Hadwiger conjecture states that any convex body in n-dimensional Euclidean space can be covered by 2n or fewer smaller bodies homothetic with the original body, and that furthermore, the upper bound of 2n is necessary if and only if the body is a parallelepiped. There also exists an equivalent formulation in terms of the number of floodlights needed to illuminate the body.
See also: Hadwiger conjecture (graph theory)
Unsolved problem in mathematics:
Can every $n$-dimensional convex body be covered by $2^{n}$ smaller copies of itself?
(more unsolved problems in mathematics)
The Hadwiger conjecture is named after Hugo Hadwiger, who included it on a list of unsolved problems in 1957; it was, however, previously studied by Levi (1955) and independently, Gohberg & Markus (1960). Additionally, there is a different Hadwiger conjecture concerning graph coloring—and in some sources the geometric Hadwiger conjecture is also called the Levi–Hadwiger conjecture or the Hadwiger–Levi covering problem.
The conjecture remains unsolved even in three dimensions, though the two dimensional case was resolved by Levi (1955).
Formal statement
Formally, the Hadwiger conjecture is: If K is any bounded convex set in the n-dimensional Euclidean space Rn, then there exists a set of 2n scalars si and a set of 2n translation vectors vi such that all si lie in the range 0 < si < 1, and
$K\subseteq \bigcup _{i=1}^{2^{n}}s_{i}K+v_{i}.$
Furthermore, the upper bound is necessary iff K is a parallelepiped, in which case all 2n of the scalars may be chosen to be equal to 1/2.
Alternate formulation with illumination
As shown by Boltyansky, the problem is equivalent to one of illumination: how many floodlights must be placed outside of an opaque convex body in order to completely illuminate its exterior? For the purposes of this problem, a body is only considered to be illuminated if for each point of the boundary of the body, there is at least one floodlight that is separated from the body by all of the tangent planes intersecting the body on this point; thus, although the faces of a cube may be lit by only two floodlights, the planes tangent to its vertices and edges cause it to need many more lights in order for it to be fully illuminated. For any convex body, the number of floodlights needed to completely illuminate it turns out to equal the number of smaller copies of the body that are needed to cover it.[1]
Examples
As shown in the illustration, a triangle may be covered by three smaller copies of itself, and more generally in any dimension a simplex may be covered by n + 1 copies of itself, scaled by a factor of n/(n + 1). However, covering a square by smaller squares (with parallel sides to the original) requires four smaller squares, as each one can cover only one of the larger square's four corners. In higher dimensions, covering a hypercube or more generally a parallelepiped by smaller homothetic copies of the same shape requires a separate copy for each of the vertices of the original hypercube or parallelepiped; because these shapes have 2n vertices, 2n smaller copies are necessary. This number is also sufficient: a cube or parallelepiped may be covered by 2n copies, scaled by a factor of 1/2. Hadwiger's conjecture is that parallelepipeds are the worst case for this problem, and that any other convex body may be covered by fewer than 2n smaller copies of itself.[1]
Known results
The two-dimensional case was settled by Levi (1955): every two-dimensional bounded convex set may be covered with four smaller copies of itself, with the fourth copy needed only in the case of parallelograms. However, the conjecture remains open in higher dimensions except for some special cases. The best known asymptotic upper bound on the number of smaller copies needed to cover a given body is[2]
$\displaystyle {\binom {2n}{n}}\exp(-c{\sqrt {n}})$
where $c$ is a positive constant. For small $n$ the upper bound of $(n+1)n^{n-1}-(n-1)(n-2)^{n-1}$ established by Lassak (1988) is better than the asymptotic one. In three dimensions it is known that 16 copies always suffice, but this is still far from the conjectured bound of 8 copies.[1]
The conjecture is known to hold for certain special classes of convex bodies, including symmetric polyhedra and bodies of constant width in three dimensions.[1] The number of copies needed to cover any zonotope is at most $(3/4)2^{n}$, while for bodies with a smooth surface (that is, having a single tangent plane per boundary point), at most $n+1$ smaller copies are needed to cover the body, as Levi already proved.[1]
See also
• Borsuk's conjecture on covering convex bodies with sets of smaller diameter
Notes
1. Brass, Moser & Pach (2005).
2. Huang et al. (2022).
References
• Boltjansky, V.; Gohberg, Israel (1985), "11. Hadwiger's Conjecture", Results and Problems in Combinatorial Geometry, Cambridge University Press, pp. 44–46.
• Brass, Peter; Moser, William; Pach, János (2005), "3.3 Levi–Hadwiger Covering Problem and Illumination", Research Problems in Discrete Geometry, Springer-Verlag, pp. 136–142.
• Gohberg, Israel Ts.; Markus, Alexander S. (1960), "A certain problem about the covering of convex sets with homothetic ones", Izvestiya Moldavskogo Filiala Akademii Nauk SSSR (in Russian), 10 (76): 87–90.
• Hadwiger, Hugo (1957), "Ungelöste Probleme Nr. 20", Elemente der Mathematik, 12: 121.
• Huang, Han; Slomka, Boaz A.; Tkocz, Tomasz; Vritsiou, Beatrice-Helen (2022), "Improved bounds for Hadwiger's covering problem via thin-shell estimates", Journal of the European Mathematical Society, 24 (4): 1431–1448, doi:10.4171/jems/1132, ISSN 1435-9855.
• Lassak, Marek (1988), "Covering the boundary of a convex set by tiles", Proceedings of the American Mathematical Society, 104 (1): 269–272, doi:10.1090/s0002-9939-1988-0958081-7, MR 0958081.
• Levi, Friedrich Wilhelm (1955), "Überdeckung eines Eibereiches durch Parallelverschiebungen seines offenen Kerns", Archiv der Mathematik, 6 (5): 369–370, doi:10.1007/BF01900507, S2CID 121459171.
| Wikipedia |
Boundary Value Problems
December 2019 , 2019:125 | Cite as
Existence of nontrivial weak solutions for p-biharmonic Kirchhoff-type equations
Jung-Hyun Bae
Jae-Myoung Kim
Jongrak Lee
Kisoeb Park
We are concerned with the following p-biharmonic equations:
$$ \Delta _{p}^{2} u+M \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u) \,dx \biggr) \operatorname{div}\bigl(\varphi (x,\nabla u)\bigr)+V(x) \vert u \vert ^{p-2}u=\lambda f(x,u) \quad \text{in } \mathbb{R}^{N}, $$
where \(2< 2p<N\), \(\Delta _{p}^{2}u=\Delta (|\Delta u|^{p-2} \Delta u)\), the function \(\varphi (x,v)\) is of type \(\lvert v \rvert ^{p-2}v\), \(\varphi (x,v)=\frac{d}{dv}\varPhi _{0}(x,v)\), the potential function \(V:\mathbb{R}^{N}\to (0,\infty )\) is continuous, and \(f:\mathbb{R} ^{N}\times \mathbb{R} \to \mathbb{R}\) satisfies the Carathéodory condition. We study the existence of weak solutions for the problem above via mountain pass and fountain theorems.
p-biharmonic Kirchhoff type Variational method
35J60 35J92 58E05
where \(2< 2p<N\), \(1< p< p_{*}:=\frac{Np}{N-2p}\), \(\Delta _{p} ^{2}u=\Delta (|\Delta u|^{p-2}\Delta u)\) is a p-biharmonic operator, the function \(\varphi (x,v)\) is of type \(\lvert v \rvert ^{p-2}v\), \(\varphi (x,v)=\frac{d}{dv}\varPhi _{0}(x,v)\), the potential function \(V:\mathbb{R}^{N}\to (0,\infty )\) is continuous, and \(f:\mathbb{R} ^{N}\times \mathbb{R} \to \mathbb{R}\) satisfies the Carathéodory condition.
The fourth-order differential equations arise in the study of deflections of elastic beams on nonlinear elastic foundations. Thus, they become very significant in engineering and physics. Many authors considered this type of equation in recent years, and we refer to [9, 13, 27] and the references therein. For this reason, the existence of solutions of p-biharmonic equations has been studied by several authors; see [6, 8, 12, 15, 21, 24, 30, 31, 34]. To obtain the existence and multiplicity results for the p-Laplace type operators, which generalize the usual p-Laplacian, the authors in [10, 28] considered the following condition:
$$ d \vert v \vert ^{p}\le \varphi (x,v)\cdot v \le p\varPhi _{0}(x,v) $$
for all \(x\in \mathbb{R}^{N}\) and \(v\in \mathbb{R}^{N}\), and for some positive constant d; see also [14]. On the other hand, Kirchhoff in [20] initially proposed the following equation:
$$ \rho \frac{\partial ^{2} u}{\partial t^{2}} - \biggl(\frac{\rho _{0}}{h} + \frac{E}{2L} \int _{0}^{L} \biggl\vert \frac{\partial u}{\partial x} \biggr\vert \,dx \biggr) \frac{\partial ^{2} u}{\partial x^{2}} = 0, $$
which is a generalization of the classical D'Alembert's wave equation. Also, Woinowsky and Krieger [33] in the 1950s considered a stationary analogue of the evolution equation of Kirchhoff type, namely
$$ u_{tt} +\Delta ^{2}u - M\bigl( \Vert \nabla u \Vert ^{2}\bigr)\Delta u = f (x, u), $$
as a model for the deflection of an extensible beam on nonlinear foundations. Here, u denotes the displacement, f is the force that the foundations exert on the beam, and M models the effects of the small changes in the length of the beam (see, e.g., [3, 4, 5, 7] for the physics viewpoint model). In view of mathematics, many researchers have extensively studied the existence of weak solutions for the elliptic problem of Kirchhoff type in recent years (see, e.g., [11, 16, 18]). Based on these references, we consider the generalized elliptic equation (P) involving the p-biharmonic and generalized p-Laplacian of Kirchhoff type.
Since the seminal paper of Ambrosetti and Rabinowitz in [2], the existence of solutions for the elliptic problem has been studied by many researchers. A common feature of these works is that the following condition, which is originally due to Ambrosetti and Rabinowitz, is imposed on the nonlinearity f:
(AR)
There exist positive constants m and ζ such that \(\zeta >p\) and
$$ 0< \zeta F(x,t)\le f(x,t)t \quad \text{for } x\in \varOmega \text{ and } \lvert t \rvert \ge m, $$
where \(F(x,t)=\int _{0}^{t}f(x,s) \,ds\), and Ω is a bounded domain in \(\mathbb{R}^{N}\).
The (AR) condition above is somewhat natural and important to guarantee the boundedness of Palais–Smale sequence of Euler–Lagrange functional for an elliptic equation, however, this condition is very restrictive and eliminates many nonlinearities. Thus, many researchers have tried to drop the (AR) condition for elliptic equations associated with the p-Laplacian; see, e.g., [1, 23, 25, 26, 29].
The purpose of this paper is to study the existence of weak solutions for problem (P) without assuming the (AR) condition, but imposing various assumptions for the divergence part φ and nonlinear term f. In particular, as observed by Remark 1.8 in [23], there are many examples which do not fulfill the condition of the nonlinear term f given in [1, 25, 26]. On the other hand, in case of the whole space \(\mathbb{R}^{N}\), the main difficulty of this problem is the lack of compactness for the Sobolev theorem. In that sense, our study is to pursue two goals. First, we show the existence of nontrivial weak solutions for the problem above using the mountain pass theorem. To be precise, we prove the existence of weak solutions for problem (P) under Cerami condition, as a weak version of the Palais–Smale condition. Also, we try to do analysis using the properties of Kirchhoff function M and function φ. Second, we show the multiplicity of weak solutions to problem (P) via the fountain theorem. To the best of our knowledge, there were no such existence results for our problem in this situation.
2 Preliminaries
In this section, we briefly describe the framework for our problem. We assume that the potential \(V\in C(\mathbb{R}^{N})\) is a continuous function with
\(\inf_{x\in \mathbb{R}^{N}}V(x)>0\), and \(\operatorname{meas} \{x\in \mathbb{R}^{N}:V(x)\le K \}<+\infty \) for all \(K\in \mathbb{R}\).
Also, we set \(D^{p}(\mathbb{R}^{N}) = \{u\in L^{p_{*}}(\mathbb{R}^{N})| \Delta u \in L^{p}(\mathbb{R}^{N})\}\). Thus, we define the function space as follows:
$$ X= \biggl\{ u \in D^{p}\bigl(\mathbb{R}^{N}\bigr) : \int _{\mathbb{R}^{N}} \bigl( \lvert \Delta u \rvert ^{p}+ \vert \nabla u \vert ^{p} +V(x) \lvert u \rvert ^{p} \bigr) \,dx < + \infty \biggr\} $$
equipped with the norm
$$ \Vert u \Vert _{X}^{p}= \Vert \Delta u \Vert ^{p}_{L^{p}(\mathbb{R}^{N})}+ \Vert \nabla u \Vert ^{p}_{L^{p}(\mathbb{R}^{N})}+ \bigl\Vert V^{1/p}u \bigr\Vert ^{p}_{L^{p}( \mathbb{R}^{N})}. $$
For our problem, we first assume that \(M:\mathbb{R}^{+} \to \mathbb{R}^{+}\) satisfies the following conditions:
\(M\in C(\mathbb{R}^{+})\) satisfies \(\inf_{t\in \mathbb{R}^{+}} M(t) \geq m_{0} > 0\), where \(m_{0}\) is a constant.
There exists \(\theta \in [1,\frac{N}{N-p})\) such that \(\theta \mathcal{M}(t)=\theta \int _{0}^{t} M(\tau )\,d \tau \geq M(t)t \) for any \(t\geq 0\).
A typical example for M is given by \(M(t)=b_{0} +b_{1}t^{n}\) with \(n>0\), \(b_{0}>0\), and \(b_{1}\geq 0\).
Next, we assume that \(\varphi : \mathbb{R}^{N}\times \mathbb{R}^{N} \to \mathbb{R}^{N}\) is a continuous function with the continuous derivative with respect to v of the mapping \(\varPhi _{0}:\mathbb{R}^{N}\times \mathbb{R}^{N} \to \mathbb{R}\), \(\varPhi _{0}=\varPhi _{0}(x,v)\), that is, \(\varphi (x,v)=\frac{d}{dv}\varPhi _{0}(x,v)\). Suppose that φ and \(\varPhi _{0}\) satisfy the following assumptions:
(J1)
The equality
$$ \varPhi _{0}(x,\mathbf{0})=0 $$
holds for almost all \(x\in \mathbb{R}^{N}\).
There are a nonnegative function \(a\in L^{p'}(\mathbb{R} ^{N})\) and a nonnegative constant b such that
$$ \bigl\vert \varphi (x,v) \bigr\vert \le a(x)+b \vert v \vert ^{p-1} $$
holds for almost all \(x\in \mathbb{R}^{N}\) and for all \(v\in \mathbb{R}^{N}\). Here, \(p'\) is a conjugate number of p.
The relations
$$ d \vert v \vert ^{p}\le \varphi (x,v)\cdot v\quad \text{and} \quad d \vert v \vert ^{p}\le p \varPhi _{0}(x,v) $$
hold for all \(x\in \mathbb{R}^{N}\) and \(v\in \mathbb{R}^{N}\), where d is a positive constant.
\(\varPhi _{0}(x,\cdot )\) is strictly convex in \(\mathbb{R} ^{N}\) for all \(x\in \mathbb{R}^{N}\).
The relation
$$ p\varPhi _{0}(x,v)-\varphi (x,v)\cdot v \ge 0 $$
holds for all \(x\in \mathbb{R}^{N}\) and all \(v\in \mathbb{R}^{N}\).
Let us define the functional \(\varPhi : X \to \mathbb{R}\) by
$$ \varPhi (u)=\frac{1}{p} \int _{\mathbb{R}^{N}} \lvert \Delta u \rvert ^{p} \,dx+ \mathcal{M} \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u) \,dx \biggr)+ \frac{1}{p} \int _{\mathbb{R}^{N}}V(x) \lvert u \rvert ^{p} \,dx. $$
It is not difficult to prove that the functional \(\varPhi \in C^{1}(X, \mathbb{R})\), and its Fréchet derivative is given by
$$ \begin{aligned} \bigl\langle \varPhi ^{\prime }(u),v\bigr\rangle &= \int _{\mathbb{R}^{N}} \lvert \Delta u \rvert ^{p-2} \Delta u \Delta v \,dx+M \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x, \nabla u) \,dx \biggr) \int _{\mathbb{R}^{N}}\varphi (x,\nabla u)\cdot \nabla v \,dx\\ &\quad {}+ \int _{\mathbb{R}^{N}}V(x) \lvert u \rvert ^{p-2}uv \,dx. \end{aligned} $$
We give some examples satisfying assumptions (J1)–(J5).
Example 2.1
Let us consider the following functions:
$$ \varphi (x,v)= \lvert v \rvert ^{p-2}v \quad \text{and} \quad \varPhi _{0}(x,v)=\frac{ \lvert v \rvert ^{p}}{p} $$
for \(v\in \mathbb{R}^{N}\) and \(x\in \mathbb{R}^{N}\). Then it is obvious that assumptions (J1)–(J5) hold.
Suppose that \(a\in L^{2p'}(\mathbb{R}^{N})\), and there is a positive constant \(a_{0}\) such that \(a(x)\ge a_{0}\) for almost all \(x\in \mathbb{R}^{N}\). We consider
$$ \varphi (x,t)= \bigl(a(x)+t^{2} \bigr)^{\frac{p-2}{2}}t \quad \text{and} \quad \varPhi _{0}(x,t)=\frac{1}{p} \bigl[ \bigl(a(x)+t^{2} \bigr)^{\frac{p}{2}}-a(x)^{\frac{p}{2}} \bigr] $$
for \(t\in \mathbb{R}\), where \(p\geq 2\) for all \(x\in \mathbb{R}^{N}\). Then assumptions (J1)–(J5) hold.
By analogous arguments as in [19, 22], the following lemma is easily checked, and thus we omit the proof. That is, the operator \(\varPhi ^{\prime }\) is a mapping of type \((S_{+})\).
Lemma 2.2
Assume that (V), (M1), (M2), and (J1)–(J4) hold. Then the functional\(\varPhi :X\to \mathbb{R}\)is convex and weakly lower semicontinuous onX. Moreover, the operator\(\varPhi ^{\prime }\)is a mapping of type\((S_{+})\), i.e., if\(u_{n}\rightharpoonup u\)inXand\(\limsup_{n\to \infty } \langle \varPhi ^{\prime }(u_{n})-\varPhi ^{ \prime }(u), u_{n}-u \rangle \le 0\), then\(u_{n}\to u\)inXas\(n\to \infty \).
Denoting \(F(x,t)=\int _{0}^{t}f(x,s) \,ds\), for the number θ given in (M2), we assume that
(F1)
\(f: \mathbb{R}^{N}\times \mathbb{R} \to \mathbb{R}\) satisfies the Carathéodory condition in the sense that \(f(\cdot ,t)\) is measurable for all \(t\in \mathbb{R}\) and \(f(x,\cdot )\) is continuous for almost all \(x\in \mathbb{R}^{N}\).
There exist nonnegative functions \(\rho \in L^{q^{\prime }}(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})\) and \(\sigma \in L^{ \infty }(\mathbb{R}^{N})\) such that
$$ \bigl\lvert f(x,t) \bigr\rvert \le \rho (x)+\sigma (x) \lvert t \rvert ^{q-1},\quad q\in (\theta p,p_{*}) $$
for all \((x,t)\in \mathbb{R}^{N}\times \mathbb{R}\).
There exists \(\delta >0\) such that
$$ F(x,t)\leq 0 \quad \text{for } x\in \mathbb{R}^{N} \text{ and } \lvert t \rvert < \delta . $$
\(\lim_{ \lvert t \rvert \to \infty }{\frac{F(x,t)}{ \lvert t \rvert ^{\theta p}}}=\infty \) uniformly for almost all \(x\in \mathbb{R}^{N}\).
There exist \(c_{0}\ge 0\), \(r_{0}\ge 0\), and \(\kappa > \frac{N}{p}\) such that
$$ \bigl\lvert F(x,t) \bigr\rvert ^{\kappa }\le c_{0} \lvert t \rvert ^{\kappa p}{\mathfrak{F}}(x,t) $$
for all \((x,t)\in \mathbb{R}^{N}\times \mathbb{R}\) and \(\lvert t \rvert \ge r_{0}\), where \({\mathfrak{F}}(x,t)=\frac{1}{\theta p}f(x,t)t-F(x,t) \ge 0\).
There exist \(\mu >\theta p\) and \(\varrho >0\) such that
$$ \mu F(x,t)\leq tf(x,t)+\varrho t^{p} $$
for all \((x,t)\in \mathbb{R}^{N} \times \mathbb{R}\).
Next, we give some examples with respect to assumptions (F1)–(F6).
Since assumption (F5) is weaker than the following assumption, namely that
$$ \frac{f(x,t)}{ \lvert t \rvert ^{\theta -2}t} \quad \text{is increasing for $t> 0$ and decreasing for $t< 0$} $$
for any \(x\in \mathbb{R}^{N}\), we check that the following example satisfies assumption (F5) by applying condition (2.1).
Let us consider
$$ f(x,t)= \lvert t \rvert ^{q-2}t\log { \bigl(1+ \lvert t \rvert \bigr)} $$
for all \(t\in \mathbb{R}\). It is clear that function f satisfies assumptions (F1)–(F4). Since the following ratio, namely
$$ \frac{f(x,t)}{ \lvert t \rvert ^{p-2}t}=\frac{ \lvert t \rvert ^{q-2}t\log { (1+ \lvert t \rvert )}}{ \lvert t \rvert ^{p-2}t}= \lvert t \rvert ^{q-p} \log { \bigl(1+ \lvert t \rvert \bigr)}, $$
is increasing for \(t> 0\) and decreasing for \(t< 0\) if \(q>p=\theta \), it follows that assumption (F5) holds.
The following example can be found in [23] for the case of p-Laplace operator.
Consider the following function:
$$ f(x,t)= \lvert t \rvert ^{p-2}t\bigl(4 \lvert t \rvert ^{3}+2t \sin t-4 \cos t\bigr). $$
Then this function satisfies conditions (F2), (F6), but not the (AR) condition.
Define the functional \(\varPsi :X\to \mathbb{R}\) by
$$ \varPsi (u)= \int _{\mathbb{R}^{N}}F(x,u) \,dx. $$
Then it is easy to check that \(\varPsi \in C^{1}(X,\mathbb{R})\) and its Fréchet derivative is
$$ \bigl\langle \varPsi ^{\prime }(u),v \bigr\rangle = \int _{\mathbb{R}^{N}}f(x,u)v \,dx $$
for any \(u,v \in X\). Next we define the functional \(I_{\lambda }:X \to \mathbb{R}\) by
$$ I_{\lambda }(u)=\varPhi (u)-\lambda \varPsi (u). $$
Then it follows that the functional \(I_{\lambda }\in C^{1}(X,\mathbb{R})\) and its Fréchet derivative is
$$\begin{aligned} \bigl\langle I_{\lambda }^{\prime }(u),v \bigr\rangle &= \int _{\mathbb{R}^{N}} \lvert \Delta u \rvert ^{p-2} \Delta u \Delta v \,dx+M \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u) \,dx \biggr) \int _{\mathbb{R}^{N}}\varphi (x,\nabla u)\cdot \nabla v \,dx \\ &\quad{}+ \int _{\mathbb{R}^{N}}V(x) \lvert u \rvert ^{p-2}uv \,dx- \lambda \int _{\mathbb{R}^{N}}f(x,u)v \,dx \end{aligned}$$
for any \(u,v\in X\).
In our setting, first of all, we need the following lemma. Using a similar argument as in [17, Lemma 3.2], we can see that the functionals Ψ and \(\varPsi ^{\prime }\) are weakly strongly continuous on X. We give a detailed proof for the convenience of the reader.
Assume that (V) and (F1)–(F2) hold. ThenΨand\(\varPsi ^{\prime }\)are weakly strongly continuous onX.
See Appendix. □
3 Existence of weak solutions
Definition 3.1
We say that \(u\in X\) is a weak solution of problem (P) if
$$\begin{aligned} & \int _{\mathbb{R}^{N}} \lvert \Delta u \rvert ^{p-2}\Delta u \cdot \Delta v \,dx+M \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u) \,dx \biggr) \int _{\mathbb{R}^{N}}\varphi (x,\nabla u)\cdot \nabla v \,dx \\ &\quad {}+ \int _{\mathbb{R}^{N}}V(x) \lvert u \rvert ^{p-2}uv \,dx-\lambda \int _{\mathbb{R}^{N}}f(x,u) v \,dx=0 \end{aligned}$$
for any \(v\in X\).
The following result is used to show that the energy functional \(I_{\lambda }\) satisfies the geometric conditions of the mountain pass theorem.
Assume that (V), (M1), (M2), (J1)–(J3), and (F1)–(F4) hold. Then the geometric conditions in the mountain pass theorem hold, i.e.,
\(u=0\)is a strict local minimum for\(I_{\lambda }(u)\),
\(I_{\lambda }(u)\)is unbounded from below onX.
By assumption (F3), \(u=0\) is a strict local minimum for \(I_{\lambda }(u)\). Next we claim that condition (2) holds. Assumption (F4) implies that for any \(K_{0}>0\), there exists a constant \(\delta >0\) such that
$$ F(x,t)\ge K_{0} \lvert t \rvert ^{\theta p} $$
for \(\lvert t \rvert >\delta \) and for almost all \(x \in \mathbb{R}^{N}\). Note that for \(t>1\), we can easily check that \(\mathcal{M}(t)\leq \mathcal{M}(1)t\). For any \(v\in X\setminus \{0 \}\), from assumptions (J2), (J3) and relation (3.1), we have
$$\begin{aligned} I_{\lambda }(tv) &=\frac{1}{p} \int _{\mathbb{R}^{N}} \lvert t\Delta v \rvert ^{p} \,dx+ \mathcal{M} \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,t\nabla v) \,dx \biggr)\\ &\quad {}+ \frac{1}{p} \int _{\mathbb{R}^{N}}V(x) \lvert t \rvert ^{p} \lvert v \rvert ^{p} \,dx-\lambda \int _{\mathbb{R}^{N}}F(x,tv) \,dx \\ &\le \frac{1}{p} \int _{\mathbb{R}^{N}} \lvert t\Delta v \rvert ^{p} \,dx+ \mathcal{M}(1) \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,t\nabla v) \,dx \biggr)^{\theta }\\ &\quad {}+\frac{1}{p} \int _{\mathbb{R}^{N}}V(x) \lvert t \rvert ^{p} \lvert v \rvert ^{p} \,dx-\lambda \int _{\mathbb{R}^{N}}F(x,tv) \,dx \\ &\le \frac{1}{p} \lvert t \rvert ^{p} \Vert v \Vert _{X}^{p}+ \mathcal{M}(1) \biggl( \int _{\mathbb{R}^{N}}a(x) \lvert t\nabla v \rvert + \frac{b}{p} \lvert t \nabla v \rvert ^{p} \,dx \biggr)^{ \theta }-\lambda \int _{\mathbb{R}^{N}}F(x,tv) \,dx \\ &\le \lvert t \rvert ^{\theta p} \biggl(\frac{1}{p} \Vert v \Vert _{X}^{p}+\mathcal{M}(1) \biggl( \int _{\mathbb{R}^{N}}a(x) \lvert \nabla v \rvert + \frac{b}{p} \lvert \nabla v \rvert ^{p} \,dx \biggr)^{ \theta }-\lambda K_{0} \int _{\mathbb{R}^{N}} \lvert v \rvert ^{\theta p} \,dx \biggr) \end{aligned}$$
for sufficiently large \(t>1\). If \(K_{0}\) is large enough, then we assert that \(I_{\lambda }(tv)\to -\infty \) as \(t \to \infty \). Hence we conclude that the functional \(I_{\lambda }\) is unbounded from below. This completes the proof. □
With the aid of Lemmas 2.2 and 2.5, we prove that the energy functional \(I_{\lambda }\) satisfies the Cerami condition \((C)_{c}\) condition, for short, i.e., for \(c\in \mathbb{R}\), any sequence \(\{u_{n} \}\subset X\) such that
$$ I_{\lambda }(u_{n})\to c \quad \text{and} \quad \bigl\Vert I_{\lambda }^{ \prime }(u_{n}) \bigr\Vert _{X^{*}}\bigl(1+ \Vert u_{n} \Vert _{X} \bigr)\to 0 \quad \text{as } n\to \infty $$
has a convergent subsequence. This plays a key role in obtaining the existence of a nontrivial weak solution for the given problem.
Assume that (V), (M1), (M2), (J1)–(J5), and (F1)–(F5) hold. Then the functional\(I_{\lambda }\)satisfies the\((C)_{c}\)condition for any\(\lambda >0\).
For \(c\in \mathbb{R}\), let \(\{u_{n}\}\) be a \((C)_{c}\)-sequence in X, that is,
$$ I_{\lambda }(u_{n})\to c \quad \text{and} \quad \bigl\Vert I_{\lambda }^{ \prime }(u_{n}) \bigr\Vert _{X^{*}}\bigl(1+ \Vert u_{n} \Vert _{X}\bigr)\to 0 \quad \text{as } n\to \infty . $$
This says that
$$ c = I_{\lambda }(u_{n})+o(1)\quad \text{and} \quad \bigl\langle I_{ \lambda }^{\prime }(u_{n}), u_{n} \bigr\rangle =o(1), $$
where \(o(1)\to 0\) as \(n\to \infty \). It follows from Lemmas 2.2 and 2.5 that \(\varPhi ^{\prime }\) and \(\varPsi ^{\prime }\) are mappings of type \((S_{+})\). Since \(I_{\lambda }^{\prime }\) is of type \((S_{+})\) and X is reflexive, it suffices to prove that the sequence \(\{u_{n}\}\) is bounded in X. We argue by contradiction. Suppose that the sequence \(\{u_{n} \}\) is unbounded in X. Then we may assume that \(\|u_{n}\|_{X}>1\) and \(\|u_{n}\|_{X}\to \infty \) as \(n\to \infty \). Define a sequence \(\{w_{n} \}\) by \(w_{n}={u_{n}}/{\|u_{n}\|_{X}}\). It is clear that \(\{w_{n} \} \subset X\) and \(\|w_{n}\|_{X}=1\). Hence, up to a subsequence still denoted by \(\{w_{n} \}\), we obtain \(w_{n}\rightharpoonup w\) in X as \(n\to \infty \) and note that
$$ w_{n}(x) \to w(x)\quad \text{a.e. in } \mathbb{R}^{N} \quad \text{and} \quad w_{n} \to w\quad \text{in }L^{s}\bigl(\mathbb{R}^{N}\bigr) \text{ as } n \to \infty $$
for \(1< s<p_{*}\). According to assumptions (M1), (M2), (J3), and relation (3.3), we obtain that
$$\begin{aligned} c &=I_{\lambda }(u_{n})+o(1) \\ &=\frac{1}{p} \int _{\mathbb{R}^{N}} \lvert \Delta u_{n} \rvert ^{p} \,dx+\mathcal{M} \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx \biggr) \\ &\quad {}+\frac{1}{p} \int _{\mathbb{R}^{N}}V(x) \lvert u_{n} \rvert ^{p} \,dx-\lambda \int _{\mathbb{R}^{N}}{F(x,u_{n})} \,dx+o(1) \\ &\geq \frac{1}{p} \int _{\mathbb{R}^{N}} \lvert \Delta u_{n} \rvert ^{p} \,dx+\frac{1}{\theta }M \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx \biggr) \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx \\ & \quad{}+ \frac{1}{p} \int _{\mathbb{R}^{N}}V(x) \lvert u_{n} \rvert ^{p} \,dx-\lambda \int _{\mathbb{R}^{N}}{F(x,u_{n})} \,dx+o(1) \\ &\geq \frac{1}{p} \int _{\mathbb{R}^{N}} \lvert \Delta u_{n} \rvert ^{p} \,dx+\frac{dm_{0}}{\theta p} \int _{\mathbb{R}^{N}} \lvert \nabla u_{n} \rvert ^{p} \,dx \\ &\quad {}+\frac{1}{p} \int _{\mathbb{R}^{N}}V(x) \lvert u _{n} \rvert ^{p} \,dx-\lambda \int _{\mathbb{R}^{N}}{F(x,u_{n})} \,dx+o(1) \\ &\geq \frac{\min \{1,dm_{0}\}}{\theta p} \Vert u_{n} \Vert _{X}^{p}-\lambda \int _{\mathbb{R}^{N}}{F(x,u_{n})} \,dx+o(1). \end{aligned}$$
Since \(\|u_{n}\|_{X}\to \infty \) as \(n\to \infty \), we have
$$ \int _{\mathbb{R}^{N}}{F(x,u_{n})} \,dx\geq \frac{\min \{1,dm_{0}\}}{ \theta p\lambda } \Vert u_{n} \Vert _{X}^{p}- \frac{c}{\lambda }+\frac{o(1)}{ \lambda } \to \infty \quad \text{as } n\to \infty . $$
In addition, we assert that
$$\begin{aligned} I_{\lambda }(u_{n}) &=\frac{1}{p} \int _{\mathbb{R}^{N}} \lvert \Delta u_{n} \rvert ^{p} \,dx+\mathcal{M} \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx \biggr)\\ &\quad {}+\frac{1}{p} \int _{\mathbb{R}^{N}}V(x) \lvert u_{n} \rvert ^{p} \,dx-\lambda \int _{\mathbb{R}^{N}} {F(x,u_{n})} \,dx \\ &\le \frac{1}{p} \Vert u_{n} \Vert _{X}^{p}+\mathcal{M} \biggl( \int _{\mathbb{R} ^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx \biggr)-\lambda \int _{\mathbb{R}^{N}} {F(x,u_{n})} \,dx. \end{aligned}$$
Combining this with relation (3.3), we obtain that
$$ \frac{1}{p} \Vert u_{n} \Vert _{X}^{p}+\mathcal{M} \biggl( \int _{\mathbb{R}^{N}} \varPhi _{0}(x,\nabla u_{n}) \,dx \biggr)\ge \lambda \int _{\mathbb{R}^{N}} {F(x,u_{n})} \,dx+c-o(1) $$
for sufficiently large n. Assumption (F4) implies that there exists \(t_{0}>1\) such that \({F(x,t)}>{ \lvert t \rvert ^{\theta p}}\) for all \(x\in \mathbb{R}^{N}\) and \(\lvert t \rvert >t _{0}\). From assumptions (F1) and (F2), there exists \(\mathcal{C}>0\) such that \(\lvert F(x,t) \rvert \leq \mathcal{C}\) for all \((x,t)\in \mathbb{R}^{N} \times [-t_{0},t_{0}]\). Therefore we can choose a real number \(\mathcal{C}_{0}\) such that \(F(x,t)\geq \mathcal{C}_{0}\) for all \((x,t)\in \mathbb{R}^{N}\times \mathbb{R}\), and thus
$$ \frac{F(x,u_{n})-\mathcal{C}_{0}}{\frac{1}{p } \Vert u_{n} \Vert _{X}^{p}+ \mathcal{M} (\int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx )} \geq 0, $$
for all \(x\in \mathbb{R}^{N}\) and for all \(n\in \mathbb{N}\). Set \(\varOmega _{1}= \{ x\in \mathbb{R}^{N} : w(x)\neq0 \}\). By the convergence in (3.4), we know that
$$ \bigl\lvert u_{n}(x) \bigr\rvert = \bigl\lvert w_{n}(x) \bigr\rvert \Vert u_{n} \Vert _{X}\to \infty \quad \text{as } n\to \infty $$
for all \(x\in \varOmega _{1}\). So then, it follows from assumptions (M2), (J2), (F4), and Hölder's inequality that, for all \(x\in \varOmega _{1}\), we have
$$\begin{aligned} &\lim_{n\to \infty }{\frac{F(x,u_{n})}{\frac{1}{p } \Vert u_{n} \Vert _{X} ^{p}+\mathcal{M} (\int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx )}} \\ &\quad \geq \lim_{n\to \infty }{\frac{F(x,u_{n})}{\frac{1}{p } \Vert u_{n} \Vert _{X} ^{p}+\mathcal{M}(1) (1+ (\int _{\mathbb{R}^{N}}\varPhi _{0}(x, \nabla u_{n}) \,dx )^{\theta } )}} \\ &\quad \geq \lim_{n\to \infty }{\frac{F(x,u_{n})}{\frac{1}{p } \Vert u_{n} \Vert _{X} ^{p}+\mathcal{M}(1) (1+ (\int _{\mathbb{R}^{N}}a(x) \lvert \nabla u _{n} \rvert +\frac{b}{p} \lvert \nabla u_{n} \rvert ^{p} \,dx )^{\theta } )}} \\ &\quad \geq \lim_{n\to \infty }{\frac{F(x,u_{n})}{\frac{1}{p } \Vert u_{n} \Vert _{X} ^{p}+\mathcal{M}(1) (1+ ( \Vert a \Vert _{L^{p'}(\mathbb{R}^{N})} \Vert \nabla u_{n} \Vert _{L^{p}(\mathbb{R}^{N})}+\frac{b}{p} \Vert \nabla u_{n} \Vert _{L ^{p}(\mathbb{R}^{N})}^{p} )^{\theta } )}} \\ &\quad \geq \lim_{n\to \infty }{\frac{F(x,u_{n})}{ (\frac{1}{p }+ \mathcal{M}(1) (1+ ( \Vert a \Vert _{L^{p'}(\mathbb{R}^{N})}+ \frac{b}{p} )^{\theta } ) ) \Vert u_{n} \Vert _{X}^{ \theta p}}} \\ &\quad = \lim_{n\to \infty }{\frac{F(x,u_{n})}{ (\frac{1}{p }+ \mathcal{M}(1) (1+ ( \Vert a \Vert _{L^{p'}(\mathbb{R}^{N})}+ \frac{b}{p} )^{\theta } ) ) \lvert u_{n}(x) \rvert ^{\theta p}} \bigl\lvert w_{n}(x) \bigr\rvert ^{\theta p}} \\ &\quad = \infty , \end{aligned}$$
where we have used the inequality \(\mathcal{M} (t)\leq \mathcal{M}(1) (1+{t}^{\theta } )\) for all \(t\in \mathbb{R}^{+}\), since if \(0\le t<1\), then \(\mathcal{M} (t)=\int _{0}^{t} M(\tau ) \,d\tau \leq \mathcal{M}(1)\) and if \(t>1\), then \(\mathcal{M} (t)\leq \mathcal{M}(1)t ^{\theta }\). Hence we get that \(\operatorname{meas}(\varOmega _{1})=0\). Indeed, if \(\operatorname{meas}(\varOmega _{1})\neq0\), according to (3.6)–(3.8) and Fatou's lemma, we would obtain
$$\begin{aligned} \frac{1}{\lambda } &={\liminf_{n\to \infty }{ \frac{ {\int _{\mathbb{R}^{N}}{F(x,u_{n})} \,dx}}{ \lambda \int _{\mathbb{R}^{N}}{F(x,u_{n})} \,dx+c-o(1)}}} \\ &\geq {\liminf_{n\to \infty } \int _{\mathbb{R}^{N}}{\frac{F(x,u_{n})}{ \frac{1}{p } \Vert u_{n} \Vert _{X}^{p}+\mathcal{M} (\int _{\mathbb{R}^{N}} \varPhi _{0}(x,\nabla u_{n}) \,dx )}}} \,dx \\ &\geq {\liminf_{n\to \infty } \int _{\varOmega _{1}}{\frac{F(x,u_{n})}{ \frac{1}{p } \Vert u_{n} \Vert _{X}^{p}+\mathcal{M} (\int _{\mathbb{R}^{N}} \varPhi _{0}(x,\nabla u_{n}) \,dx )}}} \,dx \\ &\quad {}- \limsup _{n\to \infty } { \int _{\varOmega _{1}}{\frac{\mathcal{C}_{0}}{\frac{1}{p } \Vert u_{n} \Vert _{X} ^{p}+\mathcal{M} (\int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx )}} \,dx} \\ &= \liminf_{n\to \infty }{ \int _{\varOmega _{1}}{\frac{F(x,u_{n})- \mathcal{C}_{0}}{\frac{1}{p } \Vert u_{n} \Vert _{X}^{p}+\mathcal{M} (\int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx )}} \,dx} \\ &\ge \int _{\varOmega _{1}}{\liminf_{n\to \infty }{ \frac{F(x,u_{n})- \mathcal{C}_{0}}{\frac{1}{p } \Vert u_{n} \Vert _{X}^{p}+\mathcal{M} (\int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx )}}} \,dx \\ &\ge \int _{\varOmega _{1}}{\liminf_{n\to \infty }{ \frac{F(x,u_{n})}{ \frac{1}{p } \Vert u_{n} \Vert _{X}^{p}+\mathcal{M} (\int _{\mathbb{R}^{N}} \varPhi _{0}(x,\nabla u_{n}) \,dx )}}} \,dx \\ &\quad {}-{ \int _{\varOmega _{1}} \limsup_{n\to \infty }{ \frac{\mathcal{C}_{0}}{\frac{1}{p } \Vert u_{n} \Vert _{X}^{p}+\mathcal{M} (\int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u _{n}) \,dx )}} \,dx} \\ &=\infty , \end{aligned}$$
which is a contradiction. Thus \(w(x)=0\) for almost all \(x\in \mathbb{R} ^{N}\). Using assumptions (M1)–(M2) and (J5), we get
$$\begin{aligned} c+1 &\ge I_{\lambda }(u_{n})-\frac{1}{\theta p} \bigl\langle I_{ \lambda }^{\prime }(u_{n}),u_{n} \bigr\rangle \\ &=\frac{1}{p} \int _{\mathbb{R}^{N}} \lvert \Delta u_{n} \rvert ^{p} \,dx+\mathcal{M} \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx \biggr) \\ &\quad {}+\frac{1}{p} \int _{\mathbb{R}^{N}}V(x) \lvert u_{n} \rvert ^{p} \,dx-\lambda \int _{\mathbb{R}^{N}}{F(x,u_{n})} \,dx \\ & \quad -\frac{1}{\theta p} \int _{\mathbb{R}^{N}} \lvert \Delta u_{n} \rvert ^{p} \,dx-\frac{1}{\theta p}M \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x, \nabla u_{n}) \,dx \biggr) \int _{\mathbb{R}^{N}}\varphi (x,\nabla u_{n}) \cdot \nabla u_{n} \,dx \\ & \quad -\frac{1}{\theta p} \int _{\mathbb{R}^{N}}V(x) \lvert u_{n} \rvert ^{p} \,dx +\frac{1}{\theta p}\lambda \int _{\mathbb{R}^{N}}f(x,u_{n})u_{n} \,dx \\ &\ge \frac{1}{\theta }M \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u _{n}) \,dx \biggr) \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx- \lambda \int _{\mathbb{R}^{N}}{F(x,u_{n})} \,dx \\ & \quad -\frac{1}{\theta p}M \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx \biggr) \int _{\mathbb{R}^{N}}\varphi (x,\nabla u_{n})\cdot \nabla u_{n} \,dx+\frac{1}{\theta p}\lambda \int _{\mathbb{R}^{N}}f(x,u _{n})u_{n} \,dx \\ & = \frac{1}{\theta }M \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx \biggr) \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx- \frac{1}{p} \int _{\mathbb{R}^{N}}\varphi (x,\nabla u_{n})\cdot \nabla u _{n} \,dx \biggr) \\ &\quad{}+\lambda \int _{\mathbb{R}^{N}}{\mathfrak{F}}(x,u_{n}) \,dx \\ &\ge \lambda \int _{\mathbb{R}^{N}}{\mathfrak{F}}(x,u_{n}) \,dx \end{aligned}$$
for n large enough. Let us define \(\varOmega _{n}(a,b):=\{ x\in \mathbb{R} ^{N} : a \le \lvert u_{n}(x) \rvert < b \}\) for \(a \geq 0\). The convergence in (3.4) means that
$$ w_{n}\to 0 \quad \text{in } L^{r}\bigl(\mathbb{R}^{N}\bigr)\quad \text{and} \quad w_{n}(x)\to 0 \quad \text{a.e. in } \mathbb{R}^{N} \text{ as } n\to \infty $$
for \(1 < r < p_{*}\). Hence by using (3.5) we get
$$ 0< \frac{\min \{1,dm_{0}\}}{\lambda \theta p} \le \limsup_{n\to \infty } \int _{\mathbb{R}^{N}}{\frac{ \lvert F(x,u_{n}) \rvert }{ \Vert u _{n} \Vert _{X}^{p}}} \,dx. $$
On the other hand, from assumption (F2) and relation (3.11), it follows that
$$\begin{aligned} &\int _{\varOmega _{n}(0,r_{0})}{\frac{F(x,u_{n})}{ \Vert u_{n} \Vert _{X}^{p}}} \,dx \\ &\quad \le \int _{\varOmega _{n}(0,r_{0})}\frac{\rho (x) \lvert u_{n}(x) \rvert + \frac{ \sigma (x)}{q} \lvert u_{n}(x) \rvert ^{q}}{ \Vert u_{n} \Vert _{X} ^{p}} \,dx \\ &\quad \le \frac{C_{1}}{ \Vert u_{n} \Vert _{X}^{p}} \Vert \rho \Vert _{L^{q^{\prime }}(\mathbb{R}^{N})} \Vert u_{n} \Vert _{L^{q}(\mathbb{R}^{N})} \\ &\qquad {} + \frac{ \Vert \sigma \Vert _{L^{\infty }(\mathbb{R}^{N})}}{q} \int _{\varOmega _{n}(0,r_{0})} \bigl\lvert u _{n}(x) \bigr\rvert ^{q-p} \bigl\lvert w_{n}(x) \bigr\rvert ^{p} \,dx \\ &\quad \le \frac{C_{1}}{ \Vert u_{n} \Vert _{X}^{p}} \Vert \rho \Vert _{L^{q^{\prime }}(\mathbb{R}^{N})} \Vert u_{n} \Vert _{L^{q}(\mathbb{R}^{N})} + \frac{ \Vert \sigma \Vert _{L^{\infty }(\mathbb{R}^{N})}}{q}r_{0}^{q-p} \int _{\mathbb{R}^{N}} \bigl\lvert w _{n}(x) \bigr\rvert ^{p} \,dx \\ &\quad \le \frac{C_{2}}{ \Vert u_{n} \Vert _{X}^{p}} \Vert \rho \Vert _{L^{q^{\prime }}(\mathbb{R}^{N})} \Vert u_{n} \Vert _{X} + \frac{ \Vert \sigma \Vert _{L^{\infty }(\mathbb{R}^{N})}}{q}r_{0}^{q-p} \int _{\mathbb{R}^{N}} \bigl\lvert w_{n}(x) \bigr\rvert ^{p} \,dx \\ &\quad \le \frac{C_{3}}{ \Vert u_{n} \Vert _{X}^{p-1}} + \frac{ \Vert \sigma \Vert _{L ^{\infty }(\mathbb{R}^{N})}}{q}r_{0}^{q-p} \int _{\mathbb{R}^{N}} \bigl\lvert w _{n}(x) \bigr\rvert ^{p} \,dx \to 0 \quad \text{as } n\to \infty \end{aligned}$$
for some positive constants \(C_{i}\)\((i=1,2,3)\). Set \(\kappa ^{\prime }=\kappa / (\kappa -1)\). Since \(\kappa > N / p\), we get \(1 < \kappa ^{\prime } p < p_{*}\). Hence, it follows from (F5), (3.10), and (3.11) that
$$\begin{aligned} \int _{\varOmega _{n}(r_{0},\infty )}{\frac{ \lvert F(x,u_{n}) \rvert }{ \Vert u_{n} \Vert _{X}^{p}}} \,dx &\leq \int _{\varOmega _{n}(r_{0},\infty )}{\frac{ \lvert F(x,u_{n}) \rvert }{ \lvert u_{n}(x) \rvert ^{p}} \bigl\lvert w_{n}(x) \bigr\rvert ^{p}} \,dx \\ &\le \biggl\{ \int _{\varOmega _{n}(r_{0},\infty )} \biggl(\frac{ \lvert F(x,u _{n}) \rvert }{ \lvert u_{n}(x) \rvert ^{p}} \biggr) ^{\kappa } \,dx \biggr\} ^{\frac{1}{\kappa }} \biggl\{ \int _{\varOmega (r_{0},\infty )} \bigl\lvert w_{n}(x) \bigr\rvert ^{ \kappa ^{\prime } p} \biggr\} ^{\frac{1}{\kappa ^{\prime }}} \\ &\le c_{0}^{\frac{1}{\kappa }} \biggl\{ \int _{\varOmega _{n}(r_{0},\infty )}{\mathfrak{F}}(x,u_{n}) \,dx \biggr\} ^{\frac{1}{\kappa }} \biggl\{ \int _{\mathbb{R}^{N}} \bigl\lvert w_{n}(x) \bigr\rvert ^{\kappa ^{\prime }p} \biggr\} ^{\frac{1}{\kappa ^{\prime }}} \\ &\le c_{0}^{\frac{1}{\kappa }} \biggl(\frac{c+1}{\lambda } \biggr) ^{\frac{1}{\kappa }} \biggl\{ \int _{\mathbb{R}^{N}} \bigl\lvert w_{n}(x) \bigr\rvert ^{\kappa ^{\prime }p} \biggr\} ^{\frac{1}{\kappa ^{\prime }}} \to 0 \quad \text{as } n\to \infty . \end{aligned}$$
Combining the estimates in (3.13) with (3.14), we have
$$ \int _{\mathbb{R}^{N}}{\frac{ \lvert F(x,u_{n}) \rvert }{ \Vert u _{n} \Vert _{X}^{p}}} \,dx = \int _{\varOmega _{n}(0,r_{0})}{\frac{ \lvert F(x,u _{n}) \rvert }{ \Vert u_{n} \Vert _{X}^{p}}} \,dx+ \int _{\varOmega _{n}(r_{0},\infty )}{\frac{ \lvert F(x,u_{n}) \rvert }{ \Vert u_{n} \Vert _{X}^{p}}} \,dx \to 0 \quad \text{as } n \to \infty , $$
which contradicts (3.12). This completes the proof. □
Using Lemma 3.3, we prove the existence of a nontrivial weak solution for our problem under the considered assumptions.
Theorem 3.4
Assume that (V), (M1), (M2), (J1)–(J5), and (F1)–(F5) hold. Then problem (P) has a nontrivial weak solution for all\(\lambda >0\).
Note that \(I_{\lambda }(0)=0\). In view of Lemma 3.2, the geometric conditions in the mountain pass theorem are fulfilled. And also \(I_{\lambda }\) satisfies the \((C)_{c}\) condition for any \(\lambda >0\) by Lemma 3.3. Hence, problem (P) has a nontrivial weak solution for all \(\lambda >0\). This completes the proof. □
Next, under assumption (F6) instead of (F5), we show that \(I_{\lambda }\) satisfies the Cerami condition.
Assume that (V), (M1), (M2), (J1)–(J5), (F1)–(F4), and (F6) hold. Then the functional\(I_{\lambda }\)satisfies the\((C)_{c}\)condition for any\(\lambda >0\).
For \(c\in \mathbb{R}\), let \(\{u_{n}\}\) be a \((C)_{c}\)-sequence in X satisfying (3.2). Following the proof of Lemma 3.3, we only prove that \(\{u_{n}\}\) is bounded in X. To this end, arguing by contradiction, suppose that \(\|u_{n}\|_{X}\rightarrow \infty \) as \(n\to \infty \). Let \(v_{n}=u_{n}/\|u_{n}\|_{X}\). Then \(\|v_{n}\|_{X}=1\). Passing to a subsequence, we may assume that \(v_{n}\rightharpoonup v\) as \(n\to \infty \) in X. Thus by an embedding theorem, for \(1< s< p_{*}\), we have
$$ v_{n} \rightarrow v \quad \text{in } L^{s}\bigl( \mathbb{R}^{N}\bigr)\quad \text{and} \quad v_{n}(x) \rightarrow v(x) \quad \text{a.e. in } \mathbb{R}^{N} \text{ as } n\to \infty . $$
From (M1), (M2), (J5), and (F6), it follows that
$$\begin{aligned} c+1 &\ge I_{\lambda }(u_{n})-\frac{1}{\mu } \bigl\langle I_{\lambda } ^{\prime }(u_{n}),u_{n} \bigr\rangle \\ &=\frac{1}{p} \int _{\mathbb{R}^{N}} \lvert \Delta u_{n} \rvert ^{p} \,dx+\mathcal{M} \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx \biggr) \\ &\quad {}+\frac{1}{p} \int _{\mathbb{R}^{N}}V(x) \lvert u_{n} \rvert ^{p} \,dx- \lambda \int _{\mathbb{R}^{N}}{F(x,u_{n})} \,dx \\ &\quad{}-\frac{1}{\mu } \int _{\mathbb{R}^{N}} \lvert \Delta u_{n} \rvert ^{p} \,dx-\frac{1}{\mu }M \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx \biggr) \int _{\mathbb{R}^{N}}\varphi (x,\nabla u_{n})\cdot \nabla u_{n} \,dx \\ & \quad{}- \frac{1}{\mu } \int _{\mathbb{R}^{N}}V(x) \lvert u_{n} \rvert ^{p} \,dx+ \frac{\lambda }{\mu } \int _{\mathbb{R}^{N}}{f(x,u_{n})u_{n}} \,dx \\ &\geq \biggl(\frac{1}{p}-\frac{1}{\mu } \biggr) \int _{\mathbb{R}^{N}} \lvert \Delta u _{n} \rvert ^{p} \,dx+m_{0} \biggl(\frac{1}{\theta p}- \frac{1}{ \mu } \biggr) \int _{\mathbb{R}^{N}}\varphi (x,\nabla u_{n})\cdot \nabla u _{n} \,dx \\ & \quad{}+ \biggl(\frac{1}{p}-\frac{1}{\mu } \biggr) \int _{\mathbb{R}^{N}}V(x) \lvert u_{n} \rvert ^{p} \,dx-\frac{\lambda \varrho }{ \mu } \int _{\mathbb{R}^{N}} \lvert u_{n} \rvert ^{p} \,dx \\ &\geq \min \{dm_{0},1\} \biggl(\frac{1}{\theta p}- \frac{1}{\mu } \biggr) \biggl( \int _{\mathbb{R}^{N}} \lvert \Delta u_{n} \rvert ^{p} \,dx+ \int _{\mathbb{R}^{N}} \lvert \nabla u_{n} \rvert ^{p} \,dx + \int _{\mathbb{R}^{N}}V(x) \lvert u_{n} \rvert ^{p} \,dx \biggr) \\ &\quad {}-\frac{ \lambda \varrho }{\mu } \int _{\mathbb{R}^{N}} \lvert u_{n} \rvert ^{p} \,dx \\ &\geq \min \{dm_{0},1\} \biggl(\frac{1}{\theta p}- \frac{1}{\mu } \biggr) \Vert u_{n} \Vert _{X}^{p}-\frac{\lambda \varrho }{\mu } \int _{\mathbb{R}^{N}} \lvert u_{n} \rvert ^{p} \,dx \quad \text{for large } n \in \mathbb{N}. \end{aligned}$$
This implies
$$ 1\leq \frac{\lambda \varrho \theta p}{\min \{dm_{0},1\}(\mu -\theta p)} \limsup_{n\rightarrow \infty } \Vert v_{n} \Vert _{L^{p}(\mathbb{R}^{N})}^{p}= \frac{ \lambda \varrho \theta p}{\min \{dm_{0},1\}(\mu -\theta p)} \Vert v \Vert _{L ^{p}(\mathbb{R}^{N})}^{p}. $$
Hence, due to (3.15), we see that \(v\neq 0\). From the same argument as in Lemma 3.3, we can show that the relations (3.6), (3.7), and (3.8) hold, and hence we conclude that relation (3.9) is true. Therefore we get a contradiction. Thus \(\{u_{n}\}\) is bounded in X. This completes the proof. □
Next, applying the fountain theorem in [32, Theorem 3.6] with the oddity of f, we demonstrate infinitely many weak solutions for problem (P). To do this, let W be a reflexive and separable Banach space. Then there are \(\{e_{n}\}\subseteq W\) and \(\{f_{n}^{*} \}\subseteq W^{*}\) such that
$$ W=\overline{\mathrm{span}\{e_{n}:n=1,2,\dots \}}, \qquad W^{*}=\overline{\mathrm{span}\bigl\{ f_{n}^{*}:n=1,2, \dots \bigr\} }, $$
$$\begin{aligned} \bigl\langle f^{*}_{i},e_{j} \bigr\rangle =\textstyle\begin{cases} 1 &\text{if } i=j, \\ 0 &\text{if } i\ne j. \end{cases}\displaystyle \end{aligned}$$
Let us denote \(W_{n}=\operatorname{span}\{e_{n}\}\), \(Y_{k}=\bigoplus_{n=1} ^{k}W_{n}\), and \(Z_{k}= \overline{\bigoplus_{n=k}^{\infty }W_{n}}\). In order to establish the existence result, we use the following Fountain theorem.
([1, 32])
LetXbe a real reflexive Banach space, \(I \in C^{1}(X,\mathbb{R})\)satisfies the\({(C)_{c}}\)condition for any\(c>0\)andIis even. If for each sufficiently large\(k \in \mathbb{N}\), there exist\(\rho _{k}> \delta _{k}>0\)such that the following conditions hold:
\(b_{k}:=\inf \{ I(u):u\in Z_{k}, \|u\|_{X}= \delta _{k}\}\to \infty\)as\(k\to \infty \);
\(a_{k}:=\max \{ I(u):u\in Y_{k}, \|u\|_{X}=\rho _{k}\} \le 0\).
Then the functionalIhas an unbounded sequence of critical values, i.e., there exists a sequence\(\{u_{n}\}\subset X\)such that\(I^{\prime }(u_{n})=0\)and\(I(u_{n})\to +\infty \)as\(n\to +\infty \).
Assume that (V), (M1), (M2), (J1)–(J5), and (F1)–(F5) hold. If\(\varPhi _{0}(x,-v)=\varPhi _{0}(x,v)\)holds for all\((x,v)\in \mathbb{R}^{N}\times \mathbb{R}^{N}\)and\(f(x,-t)=-f(x,t)\)holds for all\((x,t)\in \mathbb{R}^{N} \times \mathbb{R}\), then for any\(\lambda >0\), problem (P) possesses an unbounded sequence of nontrivial weak solutions\(\{u_{n}\}\)inXsuch that\(I_{\lambda }( u_{n})\to \infty \)as\(n\to \infty \).
It is obvious that \(I_{\lambda }\) is an even functional and satisfies the \((C)_{c}\) condition. It suffices to show that there exist \(\rho _{k}> \delta _{k}>0\) such that
\(b_{k}:=\inf \{I_{\lambda }(u):u\in Z_{k}, \|u\|_{X}= \delta _{k}\}\to \infty \quad \text{as } n\to \infty \);
\(a_{k}:=\max \{I_{\lambda }(u):u\in Y_{k}, \|u\|_{X}= \rho _{k}\}\le 0\),
for k large enough. Denote
$$ \alpha _{k}:=\sup_{u \in Z_{k}, \Vert u \Vert _{X}=1} \Vert u \Vert _{L^{q}( \mathbb{R}^{N})}. $$
Then we have \(\alpha _{k} \to 0\) as \(k \to \infty \). In fact, assume to the contrary that there exist \(\varepsilon _{0}>0\) and a sequence \(\{u_{k}\}\) in \(Z_{k}\) such that
$$ \Vert u_{k} \Vert _{X}=1 \quad \text{and} \quad \Vert u_{k} \Vert _{L^{q}( \mathbb{R}^{N})} \ge \varepsilon _{0} $$
for all \(k \ge k_{0}\). By the boundedness of the sequence \(\{u_{k}\}\) in X, we can find an element \(u \in X\) such that \(u_{k} \rightharpoonup u\) in X as \(n\to \infty \) and
$$ \bigl\langle {f_{j}^{*},u}\bigr\rangle =\lim _{k \to \infty }{\bigl\langle {f_{j}^{*},u _{k}}\bigr\rangle }=0 $$
for \(j=1,2,\dots \). Thus we deduce \(u=0\). However, we see that
$$ \varepsilon _{0} \le \lim_{k \to \infty }{ \Vert u_{k} \Vert _{L^{q}(\mathbb{R} ^{N})}}= \Vert u \Vert _{L^{q}(\mathbb{R}^{N})}=0, $$
which is a contradiction.
For any \(u \in Z_{k}\), we may suppose that \(\|u\|_{X}>1\). According to assumptions (M1), (M2), (J3), and (F2), we obtain that
$$\begin{aligned} I_{\lambda }(u) &=\frac{1}{p} \int _{\mathbb{R}^{N}} \lvert \Delta u \rvert ^{p} \,dx+ \mathcal{M} \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u) \,dx \biggr) \\ &\quad {}+ \frac{1}{p} \int _{\mathbb{R}^{N}}V(x) \lvert u \rvert ^{p} \,dx- \lambda \int _{\mathbb{R}^{N}}{F(x,u)} \,dx \\ &\ge \frac{1}{p} \int _{\mathbb{R}^{N}} \lvert \Delta u \rvert ^{p} \,dx+ \frac{1}{\theta }M \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u) \,dx \biggr) \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u) \,dx \\ &\quad {}+ \frac{1}{p} \int _{\mathbb{R}^{N}}V(x) \lvert u \rvert ^{p} \,dx -\lambda \int _{\mathbb{R}^{N}} \bigl\lvert \rho (x) \bigr\rvert \lvert u \rvert \,dx-\lambda \int _{\mathbb{R}^{N}}\frac{ \lvert \sigma (x) \rvert }{q} \lvert u \rvert ^{q} \,dx \\ &\ge \frac{\min \{1,dm_{0}\}}{\theta p} \biggl( \int _{\mathbb{R}^{N}} \lvert \Delta u \rvert ^{p} \,dx+ \int _{\mathbb{R}^{N}} \lvert \nabla u \rvert ^{p} \,dx + \int _{\mathbb{R}^{N}}V(x) \lvert u \rvert ^{p} \,dx \biggr) \\ &\quad{}-\lambda \int _{\mathbb{R}^{N}} \bigl\lvert \rho (x) \bigr\rvert \lvert u \rvert \,dx-\lambda \int _{\mathbb{R}^{N}}\frac{ \lvert \sigma (x) \rvert }{q} \lvert u \rvert ^{q} \,dx \\ &\ge \frac{\min \{1,dm_{0}\}}{\theta p} \Vert u \Vert ^{p}_{X} -2\lambda \Vert \rho \Vert _{L^{q^{\prime }}(\mathbb{R}^{N})} \Vert u \Vert _{L^{q}(\mathbb{R} ^{N})} -\frac{2\lambda }{q} \Vert \sigma \Vert _{L^{\infty }(\mathbb{R}^{N})} \int _{\mathbb{R}^{N}} \lvert u \rvert ^{q} \,dx \\ &\ge \frac{\min \{1,dm_{0}\}}{\theta p} \Vert u \Vert ^{p}_{X}-2 \lambda C _{4} \Vert u \Vert _{X} - \frac{2\lambda }{q} \alpha _{k}^{q}C_{5} \Vert u \Vert ^{q} _{X}, \end{aligned}$$
where \(C_{4}\) and \(C_{5}\) are positive constants. If we take
$$ \delta _{k}= \biggl(\frac{{2\lambda }C_{5}\alpha _{k}^{q}}{\min \{1,dm _{0} \}} \biggr)^{{1}/ (p-q )}, $$
then \(\delta _{k}\to \infty \) as \(k\to \infty \) because \(\theta p< q\) and \(\alpha _{k}\to 0\) as \(k\to \infty \). Hence, if \(u \in Z_{k}\) and \(\|u\|_{X}=\delta _{k}\), then we conclude that
$$ I_{\lambda }(u)\ge {\min \{1,dm_{0} \}} \biggl( \frac{1}{ \theta p}-\frac{1}{q} \biggr)\delta _{k}^{p}-2 \lambda C_{4} \delta _{k} \to \infty \quad \text{as } k \to \infty . $$
This implies that condition (1) holds.
The proof of condition (2) proceeds analogously as in the proof of [1, Theorem 1.3]. For the reader's convenience, we give the proof. Assume that condition (2) is not true. Then for some k there exists a sequence \(\{u_{n}\}\) in \(Y_{k}\) such that
$$ \Vert u_{n} \Vert _{X}\to \infty \quad \text{as } n\to \infty \quad \text{and}\quad I_{\lambda }(u_{n}) \ge 0. $$
Set \(w_{n}=u_{n}/\|u_{n}\|_{X}\). Note that \(\|w_{n}\|_{X}=1\). Since \(\dim {Y_{k}}<\infty \), there exists \(w\in Y_{k}\setminus \{0\}\) such that, up to a subsequence,
$$ \Vert w_{n}-w \Vert _{X}\to 0 \quad \text{and} \quad w_{n}(x)\to w(x) $$
for almost all \(x\in \mathbb{R}^{N}\) as \(n\to \infty \). If \(w(x)\neq 0\), then \(\lvert u_{n}(x) \rvert \to \infty \) for all \(x\in \mathbb{R}^{N}\) as \(n\to \infty \). Hence we obtain by assumption (F4) that
$$ \lim_{n\to \infty }{\frac{F(x,u_{n}(x))}{ \Vert u_{n} \Vert _{X}^{\theta p}}} = \lim _{n\to \infty }{\frac{F(x,u_{n}(x))}{ \lvert u_{n}(x) \rvert ^{\theta p}} \bigl\lvert w_{n}(x) \bigr\rvert ^{\theta p} } = \infty $$
for all \(x\in \varOmega _{2}:= \{ x\in \mathbb{R}^{N} : w(x)\neq0 \}\). As in the proof of Lemma 3.3, we have
$$ \int _{\varOmega _{2}}{\frac{F(x,u_{n}(x))}{ \Vert u_{n} \Vert _{X}^{\theta p}}} \,dx\to \infty \quad \text{as }n\to \infty . $$
Therefore, we conclude that
$$\begin{aligned} I_{\lambda }(u_{n}) &\le \frac{1}{p} \Vert u_{n} \Vert _{X}^{p}+\mathcal{M} \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u_{n}) \,dx \biggr)-\lambda \int _{\mathbb{R}^{N}}{F(x,u_{n})} \,dx \\ &\leq \frac{1}{p} \Vert u_{n} \Vert _{X}^{\theta p}+\mathcal{M}(1) \biggl(1+ \biggl( \Vert a \Vert _{L^{p'}(\mathbb{R}^{N})}+\frac{b}{p} \biggr)^{\theta } \biggr) \Vert u_{n} \Vert _{X}^{\theta p}-\lambda \int _{\mathbb{R}^{N}}{F(x,u_{n})} \,dx \\ &\leq \Vert u_{n} \Vert _{X}^{\theta p} \biggl(\frac{1}{p}+\mathcal{M}(1) \biggl(1+ \biggl( \Vert a \Vert _{L^{p'}(\mathbb{R}^{N})}+\frac{b}{p} \biggr) ^{\theta } \biggr)-\lambda \int _{\varOmega _{2}}{\frac{F(x,u_{n}(x))}{ \Vert u_{n} \Vert _{X}^{p}}} \,dx \biggr)\\ &\to -\infty \quad \text{as }n\to \infty , \end{aligned}$$
Remark 3.8
Although we replace (F5) with (F6) in the assumptions of Theorem 3.7, we can show that problem (P) possesses an unbounded sequence of nontrivial weak solutions \(\{u_{n}\}\) in X such that \(I_{\lambda }( u_{n})\to \infty \) as \(n\to \infty \).
The authors wish to express their sincere thanks to the anonymous referees and the handling editor for many constructive comments, leading to the improved version of this paper.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
The first author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2017R1D1A1B03031104). J.-M. Kim was supported by National Research Foundation of Korea Grant funded by the Korean Government (NRF-2016R1D1A1B03930422). J. Lee was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07048620) and (2019R1A6A1A11051177). K. Park was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2017R1E1A1A03070225).
Appendix: Proof of Lemma 2.5
In this section, we give a proof of Lemma 2.5 for the reader's convenience. In fact, we consider that it is a well-known fact to researchers in this area.
Let \(\{u_{n}\}\) be a sequence in X such that \(u_{n}\rightharpoonup u\) in X as \(n\to \infty \). Then \(\{u_{n}\}\) is bounded in X, and we know the embeddings \(X\hookrightarrow L^{p}(\mathbb{R}^{N})\) and \(X\hookrightarrow L^{q}(\mathbb{R}^{N})\) are compact for \(p< q< p_{*}\). So we know that
$$ u_{n} \to u \quad \text{in } L^{p}\bigl( \mathbb{R}^{N}\bigr) \quad \text{and} \quad u_{n}\to u \quad \text{in } L^{q}\bigl(\mathbb{R}^{N}\bigr) \text{ as } n\to \infty . $$
First we prove that Ψ is weakly strongly continuous in X. Let \(u_{n}\to u\) in \(L^{p}(\mathbb{R}^{N})\cap L^{q}(\mathbb{R}^{N})\) as \(n\to \infty \). By the convergence principle, there exist a subsequence \(\{u_{n_{k}}\}\) such that \(u_{n_{k}}(x)\to u(x)\) as \(k\to \infty \) for almost all \(x\in \mathbb{R}^{N}\) and a function \(u_{0}\in L^{p}(\mathbb{R} ^{N})\cap L^{q}(\mathbb{R}^{N})\) such that \(|u_{n_{k}}(x)|\leq u_{0}(x)\) for all \(k\in \mathbb{N}\) and for almost all \(x\in \mathbb{R}^{N}\). Therefore from (F2), we deduce
$$\begin{aligned} \int _{\mathbb{R}^{N}} \bigl\vert F(x,u_{n_{k}}) \bigr\vert \,dx &\leq \int _{\mathbb{R}^{N}}\rho (x) \bigl\vert u _{n_{k}}(x) \bigr\vert +\frac{\sigma (x)}{q} \bigl\vert u_{n_{k}}(x) \bigr\vert ^{q} \,dx \\ &\leq \Vert \rho \Vert _{L^{q'}(\mathbb{R}^{N})} \Vert u_{0} \Vert _{L^{q}( \mathbb{R}^{N})}+ \Vert \sigma \Vert _{L^{\infty }(\mathbb{R}^{N})} \Vert u_{0} \Vert _{L ^{q}(\mathbb{R}^{N})}^{q}. \end{aligned}$$
Since function f satisfies the Carathéodory condition by (F1), we obtain that \(F(x,u_{n_{k}})\to F(x,u)\) as \(k\to \infty \) for almost all \(x\in \mathbb{R}^{N}\). Therefore, the Lebesgue convergence theorem tells us that
$$ \int _{\mathbb{R}^{N}} F(x,u_{n_{k}}) \,dx \to \int _{\mathbb{R}^{N}} F(x,u) \,dx $$
as \(k\to \infty \), which says \(\varPsi (u_{n_{k}}) \to \varPsi (u)\) as \(k\to \infty \). Thus Ψ is weakly strongly continuous in X.
Next, we show that \(\varPsi '\) is weakly strongly continuous on X. By (F2) and Hölder's inequality, we obtain
$$\begin{aligned} \int _{\mathbb{R}^{N}} \bigl\lvert f(x,u_{n})-f(x,u) \bigr\rvert ^{q'} \,dx &\leq C_{6} \int _{\mathbb{R}^{N}} \bigl\lvert f(x,u_{n}) \bigr\rvert ^{q'}+ \bigl\lvert f(x,u) \bigr\rvert ^{q'} \,dx \\ &\leq C_{7} \int _{\mathbb{R}^{N}} \bigl\lvert \rho (x) \bigr\rvert ^{q'}+ \bigl\Vert \lvert \sigma \rvert ^{q'} \bigr\Vert _{L^{\infty }(\mathbb{R} ^{N})}\bigl( \lvert u_{n} \rvert ^{q} + \lvert u \rvert ^{q}\bigr) \,dx \end{aligned}$$
for some positive constants \(C_{6}\), \(C_{7}\), which implies that \(\lvert f(x,u_{n})-f(x,u) \rvert ^{q'}\leq g(x)\) for almost all \(x\in \mathbb{R}^{N}\) and for some \(g\in L^{1}(\mathbb{R}^{N})\). Since \(u_{n}\to u\) in \(L^{p}(\mathbb{R}^{N})\cap L^{q}(\mathbb{R}^{N})\) and almost all in \(\mathbb{R}^{N}\), it follows from (A.1) and the convergence principle that \(f(x,u_{n})\to f(x,u)\) for almost all \(x\in \mathbb{R} ^{N}\). Combining this with the Lebesgue convergence theorem, we have
$$\begin{aligned} \bigl\Vert \varPsi '(u_{n})-\varPsi '(u) \bigr\Vert _{X^{*}} &=\sup_{ \Vert v \Vert _{X}\leq 1} \bigl\lvert \bigl\langle \varPsi '(u_{n})-\varPsi '(u),v\bigr\rangle \bigr\rvert \\ &=\sup_{ \Vert v \Vert _{X}\leq 1} \int _{\mathbb{R}^{N}} \bigl\lvert f(x,u_{n})-f(x,u) \bigr\rvert \lvert v \rvert \,dx \\ &\leq \biggl( \int _{\mathbb{R}^{N}} \bigl\lvert f(x,u_{n})-f(x,u) \bigr\rvert ^{q'} \,dx \biggr)^{\frac{1}{q'}} \to 0 \quad \text{as } n\to \infty . \end{aligned}$$
Therefore, we derive that \(\varPsi '(u_{n})\to \varPsi '(u)\) in X. This completes the proof. □
Alves, C.O., Liu, S.B.: On superlinear \(p(x)\)-Laplacian equations in \(\mathbb{R}^{N}\). Nonlinear Anal. 73, 2566–2579 (2010) MathSciNetCrossRefGoogle Scholar
Ambrosetti, A., Rabinowitz, P.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973) MathSciNetCrossRefGoogle Scholar
Arosio, A.: A geometrical nonlinear correction to the Timoshenko beam equation. Nonlinear Anal. 47, 729–740 (2001) MathSciNetCrossRefGoogle Scholar
Ball, J.M.: Initial boundary value problem for an extensible beam. J. Math. Anal. Appl. 42, 61–90 (1973) MathSciNetCrossRefGoogle Scholar
Ball, J.M.: Stability theory for an extensible beam. J. Differ. Equ. 14, 399–418 (1973) MathSciNetCrossRefGoogle Scholar
Barletta, G., Livrea, R.: Infinitely many solutions for a class of differential inclusions involving the p-biharmonic. Differ. Integral Equ. 26, 1157–1167 (2013) MathSciNetzbMATHGoogle Scholar
Berger, H.M.: A new approach to the analysis of large deflections of plates. J. Appl. Mech. 22, 465–472 (1955) MathSciNetzbMATHGoogle Scholar
Bernis, F., Azorero, J.G., Peral, I.: Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order. Adv. Differ. Equ. 1, 219–240 (1996) MathSciNetzbMATHGoogle Scholar
Bonanno, G., Di Bella, B.: A boundary value problem for fourth-order elastic beam equations. J. Math. Anal. Appl. 343, 1166–1176 (2008) MathSciNetCrossRefGoogle Scholar
Boureanu, M.-M., Preda, F.: Infinitely many solutions for elliptic problems with variable exponent and nonlinear boundary conditions. Nonlinear Differ. Equ. Appl. 19, 235–251 (2012) MathSciNetCrossRefGoogle Scholar
Cammaroto, F., Vilasi, L.: Multiple solutions for a Kirchhoff-type problem involving the \(p(x)\)-Laplacian operator. Nonlinear Anal. 74, 1841–1852 (2011) MathSciNetCrossRefGoogle Scholar
Candito, P., Li, L., Livrea, R.: Infinitely many solutions for a perturbed nonlinear Navier boundary value problem involving the p-biharmonic. Nonlinear Anal. 75, 6360–6369 (2012) MathSciNetCrossRefGoogle Scholar
Chabrowski, J., do Ó, J.M.: On some fourth-order semilinear elliptic problems in \(\mathbb{R}^{N}\). Nonlinear Anal. 49, 861–884 (2002) MathSciNetCrossRefGoogle Scholar
De Napoli, P., Mariani, M.: Mountain pass solutions to equations of p-Laplacian type. Nonlinear Anal. 54, 1205–1219 (2003) MathSciNetCrossRefGoogle Scholar
Drabek, P., Otani, M.: Global bifurcation result for the p-biharmonic operator. Electron. J. Differ. Equ. 2001, 48, 1–19 (2001) MathSciNetzbMATHGoogle Scholar
Dreher, M.: The Kirchhoff equation for the p-Laplacian. Rend. Semin. Mat. (Torino) 64, 217–238 (2006) MathSciNetzbMATHGoogle Scholar
Fan, X., Han, X.: Existence and multiplicity of solutions for \(p(x)\)-Laplacian equations in \(\mathbb{R}^{N}\). Nonlinear Anal. 59, 173–188 (2004) MathSciNetzbMATHGoogle Scholar
Guo, Y., Nie, J.: Existence and multiplicity of nontrivial solutions for p-Laplacian Schrödinger–Kirchhoff-type equations. J. Math. Anal. Appl. 428, 1054–1069 (2015) MathSciNetCrossRefGoogle Scholar
Kim, Y.-H., Bae, J.-H., Lee, J.: The existence of infinitely many solutions for nonlinear elliptic equations involving p-Laplace type operators in \(\mathbb{R}^{N}\). J. Nonlinear Sci. Appl. 10, 2144–2161 (2017) MathSciNetCrossRefGoogle Scholar
Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883) zbMATHGoogle Scholar
Li, C., Tang, C.-L.: Three solutions for a Navier boundary value problem involving the p-biharmonic. Nonlinear Anal. 72, 1339–1347 (2010) MathSciNetCrossRefGoogle Scholar
Li, L., Ding, L., Pan, W.-W.: Existence of multiple solutions for a \(p(x)\)-biharmonic equation. Electron. J. Differ. Equ., 2013, 139 (2013) MathSciNetCrossRefGoogle Scholar
Lin, X., Tang, X.H.: Existence of infinitely many solutions for p-Laplacian equations in \(\mathbb{R}^{N}\). Nonlinear Anal. 92, 72–81 (2013) MathSciNetCrossRefGoogle Scholar
Liu, L., Chen, C.: Infinitely many solutions for p-biharmonic equation with general potential and concave-convex nonlinearity in \(\mathbb{R} ^{N}\). Bound. Value Probl. 2016, 6 1–9 (2016) CrossRefGoogle Scholar
Liu, S.B.: On ground states of superlinear p-Laplacian equations in \(\mathbb{R}^{N}\). J. Math. Anal. Appl. 361, 48–58 (2010) MathSciNetCrossRefGoogle Scholar
Liu, S.B., Li, S.J.: Infinitely many solutions for a superlinear elliptic equation. Acta Math. Sinica (Chin. Ser.) 46, 625–630 (2003) MathSciNetzbMATHGoogle Scholar
Micheletti, A.M., Pistoia, A.: Multiplicity results for a fourth-order semilinear elliptic problem. Nonlinear Anal. 31, 895–908 (1998) MathSciNetCrossRefGoogle Scholar
Mihăilescu, M., Rădulescu, V.: A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proc. R. Soc., Math. Phys. Eng. Sci. 462, 2625–2641 (2006) MathSciNetCrossRefGoogle Scholar
Miyagaki, O.H., Souto, M.A.S.: Superlinear problems without Ambrosetti and Rabinowitz growth condition. J. Differ. Equ. 245, 3628–3638 (2008) MathSciNetCrossRefGoogle Scholar
Sun, J., Chu, J., Wu, T.-F.: Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian. J. Differ. Equ. 262, 945–977 (2017) MathSciNetCrossRefGoogle Scholar
Talbi, M., Tsouli, N.: On the spectrum of the weighted p-biharmonic operator with weight. Mediterr. J. Math. 4, 73–86 (2007) MathSciNetCrossRefGoogle Scholar
Willem, M.: Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications, vol. 24, p. x+162 pp. Birkhäuser, Boston (1996) CrossRefGoogle Scholar
Woinowsky-Krieger, S.: The effect of axial force on the vibration of hinged bars. J. Appl. Mech. 17, 35–36 (1950) MathSciNetzbMATHGoogle Scholar
Xiu, Z., Zhao, J., Chen, J., Li, S.: Multiple solutions on a p-biharmonic equation with nonlocal term. Bound. Value Probl. 2016, 154, 1–11 (2016) MathSciNetCrossRefGoogle Scholar
1.Department of MathematicsSungkyunkwan UniversitySuwonRepublic of Korea
2.Department of MathematicsYonsei UniversitySeoulRepublic of Korea
3.Institute of Mathematical SciencesEwha Womans UniversitySeoulRepublic of Korea
4.Department of MathematicsIncheon National UniversityIncheonRepublic of Korea
Bae, JH., Kim, JM., Lee, J. et al. Bound Value Probl (2019) 2019: 125. https://doi.org/10.1186/s13661-019-1237-6 | CommonCrawl |
Absolute continuity
From Encyclopedia of Mathematics
2010 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL] (Absolute continuity of measures)
2010 Mathematics Subject Classification: Primary: 26A46 [MSN][ZBL] (Absolute continuity of functions)
1 Absolute continuity of the Lebesgue integral
2 Absolute continuity of measures
3 Absolute continuity of a function
3.1 Metric setting
Absolute continuity of the Lebesgue integral
Describes a property of absolutely Lebesgue integrable functions. Consider the Lebesgue measure $\lambda$ on the $n$-dimensional euclidean space and let $f\in L^1 (\mathbb R^n, \lambda)$. Then for every $\varepsilon>0$ there is a $\delta>0$ such that \[ \left|\int_A f (x) \rd\lambda (x)\right| < \varepsilon \qquad \mbox{for every measurable set}\, A \mbox{ with } \lambda (A)< \delta\, . \] This property can be generalized to measures $\mu$ on a $\sigma$-algebra $\mathcal{B}$ of subsets of a space $X$ and to functions $f\in L^1 (X, \mu)$ (cp. with Theorem 12.34 of [HS]).
Absolute continuity of measures
A concept in measure theory (see also Absolutely continuous measures). If $\mu$ and $\nu$ are two measures on a σ-algebra $\mathcal{B}$ of subsets of $X$, we say that $\nu$ is absolutely continuous with respect to $\mu$ if $\nu (A) =0$ for any $A\in\mathcal{B}$ such that $\mu (A) =0$ (cp. with Defininition 2.11 of [Ma]). This definition can be generalized to signed measures $\nu$ and even to vector-valued measures $\nu$. Some authors generalize it further to vector-valued $\mu$'s: in that case the absolute continuity of $\nu$ with respect to $\mu$ amounts to the requirement that $\nu (A) = 0$ for any $A\in\mathcal{B}$ such that $|\mu| (A)=0$, where $|\mu|$ is the total variation of $\mu$ (see for instance Section 30 of [Ha]).
The Radon-Nikodym theorem (see Theorem B, Section 31 of [Ha]) characterizes the absolute continuity of $\nu$ with respect to $\mu$ with the existence of a function $f\in L^1 (\mu)$ such that $\nu = f \mu$, i.e. such that \[ \nu (A) = \int_A f\, \rd\mu \qquad \text{for every } A\in\mathcal{B}. \] A corollary of the Radon-Nikodym, the Jordan decomposition Theorem, characterizes signed measures as differences of nonnegative measures. We refer to Signed measure for more on this topic (see also Hahn decomposition).
Absolute continuity of a function
A function $f:I\to \mathbb R$, where $I$ is an interval of the real line, is said absolutely continuous if for every $\varepsilon> 0$ there is $\delta> 0$ such that, for every finite collection of pairwise disjoint intervals $(a_1,b_1), (a_2,b_2), \ldots , (a_n,b_n) \subset I$ with $\sum_i (b_i-a_i) <\delta$, we have \[ \sum_i |f(b_i)-f (a_i)| <\varepsilon \] (see Section 4 in Chapter 5 of [Ro]).
An absolutely continuous function is always continuous. Indeed, if the interval of definition is open, then the absolutely continuous function has a continuous extension to its closure, which is itself absolutely continuous. A continuous function might not be absolutely continuous, even if the interval $I$ is compact. Take for instance the function $f:[0,1]\to \mathbb R$ such that $f(0)=0$ and $f(x) = x \sin x^{-1}$ for $x>0$. The space of absolutely continuous (real-valued) functions is a vector space.
A characterization of absolutely continuous functions on an interval might be given in terms of Sobolev spaces: a continuous function $f:I\to \mathbb R$ is absolutely continuous if and only its distributional derivative is an $L^1$ function, cp. with Theorem 1 in Section 4.9 of [EG] (if $I$ is bounded, this is equivalent to require $f\in W^{1,1} (I)$). Vice versa, for any function with $L^1$ distributional derivative there is an absolutely continuous representative, i.e. an absolutely continuous $\tilde{f}$ such that $\tilde{f} = f$ a.e. (cp. again with [EG]). The latter statement can be proved using the absolute continuity of the Lebesgue integral.
An absolutely continuous function is differentiable almost everywhere and its pointwise derivative coincides with the generalized one. The fundamental theorem of calculus holds for absolutely continuous functions, i.e. if we denote by $f'$ its pointwise derivative, we then have \begin{equation}\label{e:fundamental} f (b)-f(a) = \int_a^b f' (x)\rd x \qquad \forall a<b\in I. \end{equation} In fact this is yet another characterization of absolutely continuous functions (see Theorem 13 and Corollary 11 of Section 4 in Chapter 5 of [Ro]).
The differentiability almost everywhere does not imply the absolute continuity: a notable example is the Cantor ternary function or devil staircase (see Problem 46 in Chapter 2 of [Ro]). Though such function is differentiable almost everywhere, it fails to satisfy \ref{e:fundamental} since the derivative vanishes almost everywhere but the function is not constant, cp. with Problems 11 and 12 of Chapter 5 in [Ro] (indeed the generalized derivative of the Cantor ternary function is a measure which is not absolutely continuous with respect to the Lebesgue measure, see [AFP]).
It follows from \ref{e:fundamental} that an absolutely continuous function maps a set of (Lebesgue) measure zero into a set of measure zero (i.e. it has the Luzin-N-property), and a (Lebesgue) measurable set into a measurable set. Any continuous function of bounded variation which maps each set of measure zero into a set of measure zero is absolutely continuous (this follows, for instance, from the Radon-Nikodym theorem). Any absolutely continuous function can be represented as the difference of two absolutely continuous non-decreasing functions.
Metric setting
This notion can be easily generalized when the target of the function is a metric space $(X,d)$. In that case the function $f:I\to X$ is absolutely continuous if for every positive $\varepsilon$ there is a positive $\delta$ such that for any $a_1<b_1\leq a_2<b_2 \leq \ldots \leq a_n<b_n \in I$ with $\sum_i |a_i -b_i| <\delta$, we have \[ \sum_i d (f (b_i), f(a_i)) <\varepsilon\, . \] The absolute continuity guarantees the uniform continuity. As for real valued functions, there is a characterization through an appropriate notion of derivative.
Theorem 1 A continuous function $f$ is absolutely continuous if and only if there is a function $g\in L^1_{loc} (I, \mathbb R)$ such that \begin{equation}\label{e:metric} d (f(b), f(a))\leq \int_a^b g(t)\, dt \qquad \forall a<b\in I\, \end{equation} (cp. with [AGS]). This theorem motivates the following
Definition 2 If $f:I\to X$ is absolutely continuous and $I$ is a closed interval, the metric derivative of $f$ is the function $g\in L^1$ with the smallest $L^1$ norm such that \ref{e:metric} holds (cp. with [AGS]).
The definition can be easily generalized to more general domains of definition. Observe also that, if $X$ is the standard Euclidean space $\mathbb R^k$, then the metric derivative of $f$ is the norm of the classical derivative.
[AFP] L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[AGS] L. Ambrosio, N. Gigli, G. Savaré, "Gradient flows in metric spaces and in the space of probability measures". Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005. MR2129498 Zbl 1090.35002
[Bi] P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[Bo] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523 Zbl 0635.47001
[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Ha] P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[HS] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202
[KF] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961). MR0085462 MR0118796Zbl 0103.08801
[Ma] P. Mattila, "Geometry of sets and measures in euclidean spaces". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[Ro] H.L. Royden, "Real analysis" , Macmillan (1969) MR0151555 Zbl 0197.03501
[Ru] W. Rudin, "Principles of mathematical analysis", Third edition, McGraw-Hill (1976) MR038502 Zbl 0346.2600
[Ta] A.E. Taylor, "General theory of functions and integration" , Blaisdell (1965) MRMR0178100 Zbl 0135.11301
How to Cite This Entry:
Absolute continuity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolute_continuity&oldid=30857
This article was adapted from an original article by A.P. Terekhin, V.F. Emel'yanov, L.D. Kudryavtsev, V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Absolute_continuity&oldid=30857"
Measure and integration
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Bartolomeo Sovero
Bartolomeo Sovero (1576–1629) was a Swiss mathematician.[1]
Bartolomeo Sovero
BornBarthélemy Souvey
1576
Corbières, Switzerland
Died1629 (aged 52–53)
Padua, Republic of Venice
Alma mater
• Collège Saint-Michel
OccupationMathematician
Works
• Sovero, Bartolomeo (1630). Curvi ac recti proportio. Patavii: Varisco Varisco.
References
1. "Bullettino di Bibliografia e Storia delle Scienze Matematiche e Fisiche".
External links
• Nenci, Elio (2019). "SOVERO, Bartolomeo". Dizionario Biografico degli Italiani, Volume 93: Sisto V–Stammati (in Italian). Rome: Istituto dell'Enciclopedia Italiana. ISBN 978-8-81200032-6.
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• ISNI
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National
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• Portugal
People
• Italian People
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• Historical Dictionary of Switzerland
• IdRef
| Wikipedia |
\begin{document}
\title{The Vectorial Lambda Calculus Revisited\thanks{This paper is based on the first author's master thesis~\cite{Noriega20}.}} \author{Francisco Noriega\inst{1} \and Alejandro D\'iaz-Caro\inst{2,3}}
\authorrunning{F.~Noriega and A.~D\'iaz-Caro}
\institute{DC, FCEyN. Universidad de Buenos Aires, Argentina\\
\email{[email protected]}
\and
DCyT. Universidad Nacional de Quilmes, Argentina
\and
ICC. CONICET--Universidad de Buenos Aires, Argentina
\\
\email{[email protected]}}
\maketitle
\begin{abstract}
We revisit the \ensuremath{\textrm{Vectorial}}\ Lambda Calculus, a typed version of \ensuremath{\textrm{Lineal}}. \ensuremath{\textrm{Vectorial}}\ (as
well as \ensuremath{\textrm{Lineal}}) has been originally designed for quantum computing, as an extension to
System F where linear combinations of lambda terms are also terms and linear
combinations of types are also types.
In its first presentation, \ensuremath{\textrm{Vectorial}}\ only provides a weakened version of the
Subject Reduction property. We prove that our revised \ensuremath{\textrm{Vectorial}}\ Lambda Calculus
supports the standard version of said property, answering a long standing
issue. In addition we also introduce the concept of weight of types and
terms, and prove a relation between the weight of terms and of its types.
\keywords{Lambda calculus \and Type theory \and Quantum computing} \end{abstract}
\section{Introduction}\label{ch:introduction} The ``quantum data, classic control'' paradigm has been proposed for programming languages by Selinger~\cite{selinger_2004}. It presumes that quantum computers will have a specialized device, known as QRAM~\cite{Knill04}, attached to a classical computer, with the latter instructing the former which operations to perform over which qubits. In this scheme, the classical computer is the one that reads the output of measurements performed on the qubits to retrieve the classical bits and continue running the program. Hence, the quantum memory and the allowed operations are only provided as black boxes under this paradigm. The quantum lambda calculus~\cite{selinger_valiron_2006}, as well as several programming languages for describing quantum algorithms, such as Qiskit~\cite{Qiskit}, or the more evolved Quipper~\cite{quipper} and QWIRE~\cite{qwire}, follow this scheme. However, a lambda calculus allowing for programming those black boxes continues to be a long-standing problem. This is what is known as ``quantum data and control''. One of the first attempts for quantum control within the lambda calculus was van Tonder's calculus~\cite{tonder04lambda}, which placed the lambda terms directly inside the quantum memory. A completely different path started with Arrighi and Dowek's work~\cite{ArrighiDowekRTA08,ArrighiDowekLMCS17}, who proposed a new untyped calculus called \ensuremath{\textrm{Lineal}}. In \ensuremath{\textrm{Lineal}}, linear combinations (i.e.~superpositions) of terms are also terms, and they showed how to encode quantum operations with it.
\ensuremath{\textrm{Lineal}}\ is a minimalistic language able to model high-level computation with linear algebra, providing a computational definition of vector spaces and bilinear functions. The first problem addressed by this language was how to model higher-order computable operators over infinite dimensional vector spaces. This serves as a basis for studying wider notions of computability upon abstract vector spaces, whatever the interpretation of the vectors might be (probabilities, number of computational paths leading to one result, quantum states, etc.). Thus, the terms are modelled as said state vectors, and if $\ve{t}$ and $\ve{u}$ are valid terms, then so is the term \( \alpha\cdot\ve{t} + \beta\cdot\ve{u} \), representing the superposition of the state vectors $\ve{t}$ and $\ve{u}$ with some scalars $\alpha$ and $\beta$. However, the downside of this generality in the context of quantum computing, is that the operators are not restricted to being unitary (as required by quantum physics). It was not until several years later~\cite{DiazcaroGuillermoMiquelValironLICS19} that the problem of how to restrict such a language to the quantum realm has been somehow solved using a realizability technique. However, such a technique is based on defining the denotational semantics first, and then extracting a type system from there (maybe with an infinite amount of typing rules) that fits such semantics. The problem on how to extract a finite set of typing rules, which is expressive enough, remains open. In~\cite{MalherbeDiazcaro2020b} there is a first attempt to define such a language, called Lambda-$\mathcal S_1$, which is, however, far from the original \ensuremath{\textrm{Lineal}}. For example, in \ensuremath{\textrm{Lineal}}\ it is possible to define an Oracle $U_f$ implementing the one-bit to one-bit function $f$\footnote{See, for
example, \cite[\S 1.4.2]{NC10} for more information about this Oracle, or
\cite[\S 5.2]{NC10}, for a deeper discussion about oracles in general.} with
a lambda-term abstraction taking the function $f$ as a parameter. This is not
possible in Lambda-$\mathcal S_1$, since to ensure that the produced $U_f$ is
unitary would require to do a test of orthogonality between two open
lambda-terms. Therefore, the realizability technique provides only part of
the solution, but more complex type systems that take into account the scalars
within the types might be needed to solve this problem.
\ensuremath{\textrm{Vectorial}}~\cite{vectorial} is a polymorphic typed version of \ensuremath{\textrm{Lineal}}\ providing a formal account of linear operators and vectors at the level of the type system, including both scalars and sums of types. In \ensuremath{\textrm{Vectorial}}, if $\Gamma \vdash \ve t:T$ and $\Gamma \vdash \ve r:R$ then $\Gamma \vdash~\alpha\cdot\ve t + \beta\cdot\ve r~:~\alpha\cdot T + \beta\cdot R$. In general, if $\ve t$ has type $\sum_i\alpha_i\cdot U_i$, it reduces to a superposition $\sum_i\alpha_i\cdot\ve r_i$, with each $\ve r_i$ of type $U_i$. As in \ensuremath{\textrm{Lineal}}, finite vectors and matrices can be encoded within \ensuremath{\textrm{Vectorial}}. The linear combinations of types typing the encoded expressions give some information on the linear combination of values to be obtained. In particular, $U_f$ is typable in \ensuremath{\textrm{Vectorial}}. In addition, \ensuremath{\textrm{Lineal}}, its untyped version, required some kind of restrictions to avoid non confluent terms issued from the fact that not normalising terms can be considered as a form of infinite, and so the subtraction of any two terms is not always well defined\footnote{An easy
example is a term $Y_{\ve b}$ rewriting to $\ve b+Y_{\ve b}$, so without
further restrictions, $Y_{\ve b}-Y_{\ve b}$ may be rewritten both to $\ve 0$
and to $\ve b+Y_{\ve b}-Y_{\ve b}$ and thus to $\ve b$.}. With type systems
ensuring strong normalisation, such kind of issues
disappear~\cite{ArrighiDiazcaroLMCS12,DiazcaroPetitWoLLIC12,lmcs:927,vectorial}.
\ensuremath{\textrm{Vectorial}}\ has been a step into the quest for a quantum lambda calculus in the quantum data and control paradigm. However, despite its many interesting properties, \ensuremath{\textrm{Vectorial}}\ does not feature the subject reduction property. For example, while $(\lambda x.x)+(\lambda x.x)$ can be typed by $(U\to U)+(V\to V)$ for any $U$ and $V$, $2\cdot(\lambda x.x)$ can only by typed by $2\cdot(U\to U)$ or $2\cdot (V\to V)$. So, even if $U+U$ is equivalent to $2\cdot U$, subject reduction is lost if $\ve t+\ve t$ reduces to $2\cdot\ve t$, as it is the case in \ensuremath{\textrm{Lineal}}. In~\cite{vectorial} only a weakened version of subject reduction has been established. This is the reason why, after defining \ensuremath{\textrm{Vectorial}}, the quest for quantum control in the lambda calculus has taken a turn into simpler type systems~\cite{DiazcaroDowekTPNC17,DiazcaroMalherbeLSFA18,DiazcaroGuillermoMiquelValironLICS19,DiazcaroDowekRinaldiBIO19,DiazcaroMalherbeACS20,DiazcaroDowek2020b}, none of them considered to be complete yet.
By revisiting \ensuremath{\textrm{Vectorial}}, we noticed that it is possible to fix its lack of subject reduction, while preserving many properties of the original system. This is the main contribution of our paper: to provide a non-trivial redefinition of \ensuremath{\textrm{Vectorial}}, featuring subject reduction, while still having the main desirable properties of the original system. We think that this modified version of \ensuremath{\textrm{Vectorial}}\ will provide the needed framework in the quest for the quantum-controlled lambda calculus.
\subsection*{Plan of the paper}\label{sec:introduction:plan} The definition of this revised version of \ensuremath{\textrm{Vectorial}}, which we will call \ensuremath{\lvec_{\textrm{\tiny R}}}\ along this paper to avoid confusion, is given in Section~\ref{ch:vecrev}. We also discuss the design decisions behind the revision in order to regain the standard version of the subject reduction property. In Section~\ref{sec:examples} we bring back key examples from \ensuremath{\textrm{Vectorial}}, showing that they are still valid for \ensuremath{\lvec_{\textrm{\tiny R}}}. We prove subject reduction in Section~\ref{ch:sr}. In Section~\ref{ch:other-properties} we present the proof for other desirable properties of the system: progress, strong normalisation, and weight preservation, that is, the weight of a typed term is equal to the weight of its type.
\section{The calculus}\label{ch:vecrev} \subsection{\texorpdfstring{\ensuremath{\llin_{\textrm{\tiny R}}}}{LinealR}: The untyped setting}\label{sec:vecrev:terms} \ensuremath{\textrm{Lineal}}~\cite{ArrighiDowekRTA08,ArrighiDowekLMCS17} extends the lambda calculus with linear combinations of terms. In our revised version, which we call \ensuremath{\llin_{\textrm{\tiny R}}}, the grammar of terms is given by \[
\ve t ::= x\mid\lambda x.\ve t\mid (\ve t)~\ve t\mid\alpha\cdot\ve t\mid\ve
t+\ve t \] where $\alpha$ belongs to a commutative ring $(\ensuremath{\mathsf{S}},+,\times)$.
This grammar differs from that of \ensuremath{\textrm{Lineal}}\ in the fact that we do not include a term $\ve 0$ representing the null linear combination. Indeed, $0\cdot\ve t$ is a proper term, but it differs from $0\cdot\ve r$ when $\ve t\neq\ve r$. This modification comes from the fact that in a typed calculus, $\ve 0$ would have to be typed with any type. Then, for example, $(\lambda x.x+0\cdot\ve t)~\ve r$ may not have a type, if $\ve t$ is not an arrow type for example, while $(\lambda x.x)~\ve r$ can always be typed. So it becomes crucial not to simplify the term $0\cdot\ve t$, and consequently we do not need a term $\ve 0$. In fact, such linear combinations can be seen as forming a ``weak'' module, differing from a module in the fact that there is no neutral element for the addition. See~\cite[\S II.B]{DiazcaroGuillermoMiquelValironLICS19} for a longer discussion about the weak structure, which has been historically used within the concept of unbounded operators, introduced by von Neumann to give a rigorous mathematical definition to the operators that are used in quantum mechanics. For historical reasons we will continue calling the calculus ``The \emph{Vectorial} Lambda Calculus'', while it could be named ``The \emph{Weak Module} Lambda Calculus''.
Variables and abstractions are called basis terms~\cite{ArrighiDowekRTA08,vectorial} or pure values~\cite{DiazcaroGuillermoMiquelValironLICS19}: \[
\ve b ::= x\mid\lambda x.\ve t \]
The reduction rules, given in Figure~\ref{fig:vecrev:terms}, are split in four groups. The groups E (elementary rules) and F (factorisation rules) deal with the (weak) module axioms. The group B is composed by only one rule, the beta-reduction, following a ``call-by-basis'' strategy~\cite{AssafDiazcaroPerdrixTassonValironLMCS14}, that is, the beta-reduction can occur only when the argument is a basis term. Finally, the group A (application rules) deals with applications in linear combinations: If the left hand side or the right hand side of an application is a linear combination (and so, the conditions for applying the call-by-basis beta-rule are not met), then the application is first distributed over the linear combination.
\begin{figure}\label{fig:vecrev:terms}
\end{figure}
\subsection{\texorpdfstring{\ensuremath{\lvec_{\textrm{\tiny R}}}}{VectorialR}: Typed \texorpdfstring{\ensuremath{\llin_{\textrm{\tiny R}}}}{LinealR}}\label{sec:vecrev:typesystem} The grammar of types~\cite{vectorial} consists in a sort of unit types, that is, types which are not linear combinations of other types, aimed to type base terms, and a sort of general types, which are linear combinations of unit types, or type variables of that sort. \[
\begin{array}[t]{l@{\hspace{1.5cm}}r@{$\ ::=\quad$}l}
\text{\em Types:} & T & U~|~\alpha\cdot T\mid T+T\mid \vara{X}\\
\text{\em Unit types:} & U & \varu{X}\mid U\to T\mid\forall\varu{X}.U\mid\forall \vara{X}.U
\end{array} \] We write $T,R,S$ for general types and $U, V, W$ for unit types. Notice that there are two kinds of variables, distinguished by its typography. Variables $\varu X, \varu Y, \varu Z$ are variables meant to be replaced only by unit types, while $\vara X, \vara Y, \vara Z$ can be replaced by any type. Note, however, that, for example, $\forall\varu X.\varu X$ is a valid type (even if not inhabited), while $\forall\vara X.\vara X$ is not even grammatically correct. In the same way, since arrows have the shape $U\to T$, an $\vara X$ variable can only appear in the body of the arrow. The shape of the arrow accounts for the fact that the calculus is call-by-base, and so only base terms can be passed as arguments.
As with terms, types form a (weak) module. Therefore, we consider the equivalence between types given in Figure~\ref{fig:typeequiv}. \begin{figure}
\caption{Equivalence between types}
\label{fig:typeequiv}
\end{figure}
A typing sequent $\Gamma\vdash\ve t:T$ relates a context $\Gamma$, formed by a set of unit-typed term variables (and, as usual, written as a coma-separated list of variables and types), a term $\ve t$ and a type $T$. The rules to construct valid typing sequents are given in Figure~\ref{fig:vecrev:types}, and they have been modified in relation to the set of rules from \ensuremath{\textrm{Vectorial}}~\cite{vectorial}. We write $X$ when we do not want to specify which kind of variable we refer to ($\varu{X}$ or $\vara{X}$). The notation $T[A/X]$ is a way to abbreviate two rules, one where $A$ is a unit type and $X$ is $\varu X$, and another one with $A$ any type and $X$ is $\vara X$. Similarly, $\forall_I$ (resp.~$\forall_E$) stands for $\forall_\varu{I}$ or $\forall_\vara{I}$ (resp.~$\forall_\varu{E}$ or $\forall_\vara{E}$) depending on which kind of variable is being introduced (resp.~eliminated).
\begin{figure}\label{fig:vecrev:types}
\end{figure}
Since the main focus of this work is to provide a revision of $\ensuremath{\textrm{Vectorial}}$ to recover the subject reduction property, we deemed necessary to revise the typing rules. To make it clear how this new type system solves the problem, we analyse the problem the original system had.
In \ensuremath{\textrm{Vectorial}}, instead of $1_E$ and $S$, there is an arguably more natural rule $\alpha_I$: \[
\prftree[r]{$\alpha_I$}{\Gamma\vdash\ve t:T}{\Gamma\vdash\alpha\cdot\ve
t:\alpha\cdot T} \]
However, consider a term $\ve{t}$ typable both by $T$ and $R\not\equiv T$. The term $\alpha \cdot \ve{t} + \beta \cdot \ve{t}$ can be typed by $\alpha \cdot T + \alpha \cdot R$, both, in \ensuremath{\textrm{Vectorial}}\ and in \ensuremath{\lvec_{\textrm{\tiny R}}}. However, upon reducing this term by rule $\alpha \cdot \ve{t} + \beta \cdot \ve{t} \to (\alpha + \beta) \cdot \ve{t}$ (from Group F), the given term in \ensuremath{\textrm{Vectorial}}\ can only be typed either by $(\alpha+\beta)\cdot T$ or $(\alpha+\beta)\cdot R$, breaking subject reduction. Instead, the added rule $S$ in \ensuremath{\lvec_{\textrm{\tiny R}}}\ allows to type such a term with the correct type $\alpha\cdot T+\beta\cdot R$.
We can generalise the problem, so for any term $\ve{t}$ that can be typed with $T_1,\dots,T_n$, the system should be able to type $(\sui{n} \alpha_i) \cdot \ve{t}$ with $\sui{n} \alpha_i \cdot T_i$. The only condition we must satisfy is that the scalar associated with the term is equal to the sum of the scalars of the type, which in this case is $\sui{n} \alpha_i$.
Rule $S$ has been introduced to solve this problem, and it also served as a replacement for rule $\alpha_I$, which is the particular case with $n=1$.
However, the rule $S$ alone is not enough to solve the problem. Continuing with the example, using the new rule $S$ we have \[
\prftree[r]{$S$}
{\prftree
{\vdots}
{\Gamma \vdash \ve{t}: T}}
{\prftree
{\vdots}
{\Gamma \vdash \ve{t}: R}}
{\Gamma \vdash (\alpha + \beta) \cdot \ve{t}: \alpha \cdot T + \beta \cdot R} \] In the particular case when $\alpha + \beta = 1$, the previous conclusion is $\Gamma \vdash 1 \cdot \ve{t}: \alpha \cdot T + \beta \cdot R$, and so by applying the rewriting rule $1 \cdot \ve{t} \to \ve{t}$ (from Group E), we end up having to derive $\Gamma \vdash \ve{t}: \alpha \cdot T + \beta \cdot R$. Such is the reason for the rule $1_E$.
\section{Interpretation of typing judgements}\label{sec:examples} In the general case the calculus can represent infinite-dimensional linear operators such as $\lambda x.x$, $\lambda x.\lambda y.y$, $\lambda x.\lambda f.(f)\,x$,\dots and their applications. Even for such general terms $\ve t$, the vectorial type system provides much information about the superposition of basis terms $\sum_i\alpha_i\cdot\ve b_i$ to which $\ve t$ is reduced to, as proven by~Theorem~\ref{thm:progress} (Progress). How much information is brought by the type system in the finitary case is the topic of this section.
Next we show how to encode finite-dimensional linear operators, i.e.~matrices, together with their applications to vectors. This encoding slightly differs from that of $\ensuremath{\textrm{Vectorial}}$~\cite[\S 6]{vectorial}.
\subsection{In 2 dimensions} In this section we show how $\ensuremath{\lvec_{\textrm{\tiny R}}}$ handles the Hadamard gate\footnote{The
Hadamard gate is a well known quantum operator sending $|0\rangle$ to $\frac 1{\sqrt 2}|0\rangle+\frac 1{\sqrt 2}|1\rangle$
and $|1\rangle$ to $\frac 1{\sqrt 2}|0\rangle-\frac 1{\sqrt 2}|1\rangle$.}, and how to encode matrices and vectors in general.
With an empty typing context, the booleans $\true=\lambda x.\lambda y.x\,$ and
$\,\false=\lambda x.\lambda y.y$ (or $|0\rangle$ and $|1\rangle$ in Dirac notation) can be respectively typed with the types $\True=\forall \varu{XY}.\varu X\to (\varu Y\to\varu X)\,$ and $\,\False=\forall\varu{XY}.\varu X\to (\varu Y\to\varu Y)$. The superposition has the following type $\vdash\alpha\cdot\true+\beta\cdot\false:\alpha\cdot\True + \beta\cdot\False$. (Note that it can also be typed with $(\alpha+\beta)\cdot \forall\varu X.\varu X\to\varu X\to\varu X$).
The linear map $\ve{U}$ sending $\true$ to $a\cdot\true+b\cdot\false$ and $\false$ to $c\cdot\true+d\cdot\false$
is written as \[
\ve U={\lambda
x.\cocanon{((x)~\canon{a\cdot\true+b\cdot\false})~\canon{c\cdot\true+d\cdot\false}}}. \] where $\canon{\ve t}$ stands for $\lambda x.\ve t$, for a fresh variable $x$, and $\cocanon{\ve t}$ stands for $(\ve t)~\lambda x.x$. This way, $\cocanon{\canon{\ve t}}\to^*\ve t$.
Such an encoding is needed to freeze the distribution of an application with respect to its argument. Indeed, $(\ve t)~(\ve r+\ve s)\to(\ve t)~\ve r+(\ve t)~\ve s$, while $(\ve t)~(\lambda x.\ve s+\ve t)$ does not distribute since the argument is already a base term.
The following sequent is valid: \[
\vdash\ve{U}:\forall \vara{X}.((I\to (a\cdot\True+b\cdot\False))\to(I\to
(c\cdot\True+d\cdot\False))\to I\to \vara{X})\to \vara{X}. \] or, using a similar notation $\canon T$ for $I\to T$, \[
\vdash\ve{U}:\forall
\vara{X}.(\canon{a\cdot\True+b\cdot\False}\to\canon{c\cdot\True+d\cdot\False}\to\canon{\vara
X})\to \vara{X}. \] One can check that $\vdash~({\textbf{U}})~\true~:~a\cdot\True+b\cdot\False$, as expected since it reduces to $a\cdot\true+b\cdot\false$: \begin{align*}
&({\textbf{U}})~\true\\
&= \left(\lambda x.\cocanon{\left((x)\canon{a\cdot\true+b\cdot\false}\right)\canon{c\cdot\true+d\cdot\false}}\right)~\left(\lambda x.\lambda y.x\right)\\
&= \lambda x.\left(\left(\left((x)~\left(\lambda f.a\cdot\true+b\cdot\false\right)\right)~\left(\lambda g.c\cdot\true+d\cdot\false\right)\right)~\left(\lambda x.x\right)\right)~(\lambda x.\lambda y.x)\\
&\to (((\lambda x.\lambda y.x)~(\lambda f.a\cdot\true+b\cdot\false))~(\lambda g.c\cdot\true+d\cdot\false))~(\lambda x.x)\\
&\to ((\lambda y.\lambda f.a\cdot\true+b\cdot\false)~(\lambda g.c\cdot\true+d\cdot\false))~(\lambda x.x)\\
&\to (\lambda f.a\cdot\true+b\cdot\false)~(\lambda x.x)\\
&\to a\cdot\true+b\cdot\false \end{align*}
The Hadamard gate $\textbf{H}$ is the particular case $a=b=c=-d=\nicefrac1{\sqrt2}$. The term $({\textbf{H}})~(\nicefrac1{\sqrt2}\cdot\true+\nicefrac1{\sqrt2}\cdot \false)$ has type $\True+0\cdot\False$, and reduces as follows. \begin{align*}
&({\textbf{H}})~\left(\nicefrac1{\sqrt2}\cdot\true+\nicefrac1{\sqrt2}\cdot \false\right)
\ \to^*\ \left(({\textbf{H}})~\left(\nicefrac1{\sqrt2}\cdot \true\right)\right)+\left(({\textbf{H}})~\left(\nicefrac1{\sqrt2}\cdot \false\right)\right)\\
&\to^*\ \nicefrac1{\sqrt2}\cdot(({\textbf{H}})~\true)+\nicefrac1{\sqrt2}\cdot(({\textbf{H}})~\false)\\
&\to^*\ \nicefrac1{\sqrt2}\cdot\left(\nicefrac1{\sqrt2}\cdot\true+\nicefrac1{\sqrt2}\cdot\false\right)+\nicefrac1{\sqrt2}\cdot\left(\nicefrac1{\sqrt2}\cdot\true-\nicefrac1{\sqrt2}\cdot\false\right)\\
&\to^*\ \nicefrac{1}{2}\cdot\true+\nicefrac{1}{2}\cdot\false + \nicefrac{1}{2}\cdot\true-\nicefrac{1}{2}\cdot\false
\ \to^*\ \true + 0\cdot\false \end{align*}
But we can do more than typing $2$-dimensional vectors or $2\times2$-matrices: using the same technique we can encode vectors and matrices of any size.
\subsection{Vectors in \texorpdfstring{$n$}{n} dimensions}\label{sec:vec} The $2$-dimensional space is represented by the span of $\lambda x_1x_2.x_1$ and $\lambda x_1x_2.x_2$: the $n$-dimensional space is simply represented by the span of all the $\lambda x_1\cdots{}x_n.x_i$, for $i \in \left\{1,\dots,n\right\}$. As for the two dimensional case where \[
\vdash~
\alpha_1\cdot\lambda x_1x_2.x_1 +
\alpha_2\cdot\lambda x_1x_2.x_2
~:~
\alpha_1\cdot\forall \varu{X}_1\varu{X}_2.\varu{X}_1
+
\alpha_2\cdot\forall \varu{X}_1\varu{X}_2.\varu{X}_2, \] an $n$-dimensional vector is typed with \[
\vdash~
\sui{n}\alpha_i\cdot\lambda x_1\cdots{}x_n.x_i
~:~
\sui{n}\alpha_i\cdot\forall \varu{X}_1\cdots{}\varu{X}_n.\varu{X}_i. \] We use the notations \[
{\ve e}_i^n = \lambda x_1\cdots{}x_n.x_i,
\qquad
{\ve E}_i^n = \forall \varu{X}_1\cdots{}\varu{X}_n.\varu{X}_i \] and write \[
\begin{array}{r@{~=~}l@{~=~}l}
\left\llbracket
\begin{array}{c}
\alpha_1 \\
\vdots \\
\alpha_n
\end{array}
\right\rrbracket^{\textrm{term}}_{n}
&
\left(\begin{array}{c}
\alpha_{1}\cdot{\ve e}_1^n\\
+\\
\cdots\\
+\\
\alpha_{n}\cdot{\ve e}_n^n
\end{array}\right)
&
\sum\limits_{i=1}^{n}\alpha_i\cdot {\ve e}_i^n
\\[4em]
\left\llbracket
\begin{array}{c}
\alpha_1 \\
\vdots \\
\alpha_n
\end{array}
\right\rrbracket^{\textrm{type}}_{n}
&
\left(\begin{array}{c}
\alpha_{1}\cdot{\ve E}_1^n \\
+\\
\cdots \\
+\\
\alpha_{n}\cdot{\ve E}_n^n
\end{array}\right)
&
\sum\limits_{i=1}^{n}\alpha_i\cdot {\ve E}_i^n
\end{array} \]
\subsection{\texorpdfstring{$n\times m$}{nxm} matrices}\label{sec:mat} Once the representation of vectors is chosen, it is easy to generalise the representation of $2\times 2$ matrices to the $n\times m$ case. Suppose that the matrix $U$ is of the form \[
U =
\left(
\begin{array}{ccc}
\alpha_{11} & \cdots & \alpha_{1m}
\\
\vdots && \vdots
\\
\alpha_{n1} & \cdots & \alpha_{nm}
\end{array}
\right), \] then its representation is \[
\left\llbracket
U
\right\rrbracket^{\textrm{term}}_{n\times m}
={~~~~}
\lambda x.
\left\{
\left(
\cdots
\left(
(x)
\left[
\begin{array}{c}
\alpha_{11}\cdot{\ve e}_1^n
\\+\\
\cdots
\\+\\
\alpha_{n1}\cdot{\ve e}_n^n
\end{array}
\right]
\right)
\cdots
\left[
\begin{array}{c}
\alpha_{1m}\cdot{\ve e}_1^n
\\+\\
\cdots
\\+\\
\alpha_{nm}\cdot{\ve e}_n^n
\end{array}
\right]
\right)
\right\}\qquad \] and its type is \[
\left\llbracket
U
\right\rrbracket^{\textrm{type}}_{n\times m}
={~~~~}
\forall\vara{X}.
\left(
\left[
\begin{array}{c}
\alpha_{11}\cdot{\ve E}_1^n
\\+\\
\cdots
\\+\\
\alpha_{n1}\cdot{\ve E}_n^n
\end{array}
\right]\to
\cdots
\to
\left[
\begin{array}{c}
\alpha_{1m}\cdot{\ve E}_1^n
\\+\\
\cdots
\\+\\
\alpha_{nm}\cdot{\ve E}_n^n
\end{array}
\right]\to
[~\vara{X}~]
\right)
\to
\vara{X}, \] that is, an almost direct encoding of the matrix $U$.
\section{Subject Reduction}\label{ch:sr} Recovering the Subject Reduction property constitutes the main focus of this work. In the original system, the Group F was the group of rules that required special consideration and did not satisfy the property in full.
The proof of the Subject Reduction theorem requires some intermediate results that we develop in this section. \conf{We give enough details for reproducing all the proofs. The full detailed proofs are given in the 51-pages long arXiv'ed version at~\cite{paper:arxiv}.} \arxiv{The full proofs are given in the Appendix~\ref{app:proofsSR}.}
\arxiv{We use the standard notation for equivalence classes: $[x]$ identifies the class from which $x$ is a representative.} Given a type derivation tree $\pi$, we may refer to it simply by its last sequent, $\pi = \Gamma \vdash \ve{t}: T$, when there is no ambiguity. We also write $size(\pi)$ for the number of sequents present on the tree $\pi$.
The following lemma gives a canonical form for types. \begin{lemma}[Characterisation of types~\cite[Lem.~4.2]{vectorial}]\label{lem:sr:typecharact}
For any type $T$, there exist $n,m\in\mathbb{N}$, $\alpha_1,\dots,\alpha_n$,
$\beta_1,\dots,\beta_m\in\ensuremath{\mathsf{S}}$, distinct unit types $U_1,\dots,U_n$ and
distinct general variables $\vara{X}_1,\dots,\vara{X}_m$ such that \(
T\equiv\sui{n}\alpha_i\cdot U_i+\suj{m}\beta_j\cdot\vara{X}_j \).
\conf{\qed} \end{lemma} \arxiv{\begin{proof}
Structural induction on $T$.
The full details are given in Appendix~\ref{app:proofsSR}. \end{proof}}
\arxiv{Our system admits weakening and strengthening, as stated by the following lemma. \begin{lemma}[Weakening and Strengthening]\label{lem:sr:weakening}
Let $\ve t$ be such that $x\not\in\FV{\ve t}$. Then $\Gamma\vdash\ve t:T$ is
derivable if and only if $\Gamma,x:U\vdash\ve t:T$ is derivable. \end{lemma} \begin{proof}
By a straightforward induction on the type derivation. \end{proof}}
{The following two lemmas present some properties of the equivalence relation. \begin{lemma}[Equivalence between sums of distinct elements (up to
$\equiv$)~\cite[Lem.~4.4]{vectorial}]\label{lem:sr:equivdistinctscalars}
Let $U_1,\dots,U_n$ be a set of distinct (not equivalent) unit types, and let
$V_1,\dots,V_m$ be also a set distinct unit types. If $\sui{n}\alpha_i\cdot
U_i\equiv\suj{m}\beta_j\cdot V_j$, then $m=n$ and there exists a permutation
$p$ of $m$ such that $\forall i$, $\alpha_i=\beta_{p(i)}$ and $U_i\equiv
V_{p(i)}$.
\conf{\qed} \end{lemma} \arxiv{\begin{proof}
The full details are given in Appendix~\ref{app:proofsSR}. \end{proof}}
\begin{lemma}[Equivalences $\forall$~\cite[Lem.~4.5]{vectorial}]\label{lem:sr:equivforall}
Let $U_1,\dots,U_n$ be a set of distinct (not equivalent) unit types and let
$V_1,\dots,V_n$ be also a set of distinct unit types.
\begin{enumerate}
\item\label{ap:it:equivforall1} $\sui{n}\alpha_i\cdot
U_i\equiv\suj{m}\beta_j\cdot V_j$ iff $\sui{n}\alpha_i\cdot\forall
X.U_i\equiv\suj{m}\beta_j\cdot\forall X.V_j$.
\item\label{ap:it:equivforall2} If $\sui{n}\alpha_i\cdot\forall
X.U_i\equiv\suj{m}\beta_j\cdot V_j$ then $\forall V_j,\exists
W_j~/~V_j\equiv\forall X.W_j$.
\item\label{ap:it:equivforall3} If $T\equiv R$ then $T[A/X]\equiv R[A/X]$.
\end{enumerate} \end{lemma} \arxiv{\begin{proof}
The full details are given in Appendix~\ref{app:proofsSR}. \end{proof}} }
We follow Barendregt's proof of subject reduction for System F~\cite{Barendregt92}, with the corrections first presented at~\cite{stackexchange,ArrighiDiazcaroLMCS12}. First, we introduce a relation between types, when these types are valid for the same term in the same context.
\begin{definition}\label{def:order} For any types $T, R$, and any context
$\Gamma$ such that for some term $\ve{t}$, the sequent $\Gamma\vdash\ve t:T$ can be derived from the sequent $\Gamma\vdash\ve t:R$, without extra hypothesis, then
\begin{enumerate}
\item If $X\notin\FV{\Gamma}$, write $R\prec_{X,\Gamma} T$ if either:
\begin{itemize}
\item $R\equiv\sui{n}\alpha_i\cdot U_i$ and $T\equiv\sui{n}\alpha_i\cdot
\forall X.U_i$,\quad or
\item $R\equiv\sui{n}\alpha_i\cdot \forall X.U_i$ and $T\equiv
\sui{n}\alpha_i\cdot U_i[A/X]$.
\end{itemize}
\item If $\V$ is a set of type variables such that
$\V\cap\FV{\Gamma}=\emptyset$, we define $\preceq_{\V,\Gamma}$ inductively:
\begin{itemize}
\item If $R\prec_{X,\Gamma} T$, then $R\preceq_{\V \cup \{X\},\Gamma} T$.
\item If $\V_1,\V_2\subseteq\V$, $S\preceq_{\V_1,\Gamma} R$ and
$R\preceq_{\V_2,\Gamma} T$, then $S\preceq_{\V_1\cup\V_2,\Gamma} T$.
\item If $R \equiv T$, then $R\preceq_{\V,\Gamma} T$.
\end{itemize}
Note that these relations only predicate on the types and the context, thus
they hold for any term $\ve t$.
\end{enumerate} \end{definition}
\arxiv{\begin{example}
Consider the following derivation.
\[
\prftree[r]{$\equiv$}
{
\prftree[r]{$\forall_{\vara{I}}$}
{
\prftree[r]{$\forall_{\varu{E}}$}
{
\prftree[r]{$\forall_{\varu{I}}$}
{
\prftree[r]{$\equiv$}{\Gamma\vdash\ve t:T}
{\prfassumption{T\equiv \sui{n}\alpha_i\cdot U_i}}
{\Gamma\vdash\ve t:\sui{n}\alpha_i\cdot U_i}
}
{\prfassumption{\varu{X}\notin\FV{\Gamma}}}
{\Gamma\vdash\ve t:\sui{n}\alpha_i\cdot \forall\varu{X}.U_i}
}
{\Gamma\vdash\ve t:\sui{n}\alpha_i\cdot U_i[V/\varu{X}]}
}
{\prfassumption{\vara{Y}\notin\FV{\Gamma}}}
{\Gamma\vdash\ve t:\sui{n}\alpha_i\cdot \forall\vara{Y}.U_i[V/\varu{X}]}
}
{\prfassumption{\sui{n}\alpha_i\cdot \forall\vara{Y}.U_i[V/\varu{X}]\equiv R}}
{\Gamma \vdash \ve{t}: R}
\]
Then $R\ssubt_{\{\varu{X},\vara{Y}\},\Gamma} T$. \end{example}} \conf{\begin{example}
If there exists $\ve t$ such that $\Gamma\vdash\ve t:\sui{n}\alpha_i\cdot U_i$, then
$$\sui{n}\alpha_i\cdot U_i\quad\ssubt_{\{\varu{X},\vara{Y}\},\Gamma}\quad\sui{n}\alpha_i\cdot \forall\vara{Y}.U_i[V/\varu{X}]$$ \end{example}}
\begin{lemma}\label{lem:sr:sorderhasnofv}
For any unit type $U \not\equiv \forall X. V$, if $U \ssubt_{\V, \Gamma}
\forall X. V$, then $X \notin \FV{\Gamma}$. \end{lemma} \arxiv{\begin{proof}
By definition of $\ssubt$. \end{proof}}
The following lemma states that if two arrow types are ordered, then they are equivalent up to some substitution.
\begin{lemma}[Arrows comparison]\label{lem:sr:arrowscomp}
$V \to R\ssubt_{\V,\Gamma} \forall\vec X.(U\to T) $, then $U\to T\equiv(V\to
R)[\vec{A}/\vec{Y}]$, with $\vec Y\notin \FV{\Gamma}$. \end{lemma} \arxiv{\begin{proof}
Let $(~\cdot~)^\circ$ be a map from types to types defined as follows,
\begin{align*}
X^\circ &= X \\
(U\to T)^\circ &= U\to T \\
(\forall X.T)^\circ &= T^\circ \\
(\alpha\cdot T)^\circ &=\alpha\cdot T^\circ\\
(T+R)^\circ &=T^\circ+R^\circ
\end{align*}
First we prove that for any types $V, U$, there exists $\vec A$ such that if
$V \ssubt_{\V,\Gamma} \forall\vec X.U$, then $U^\circ\equiv V^\circ[\vec
A/\vec X]$. Therefore, we have $U\to T\equiv(U\to T)^\circ\equiv(V\to
R)^\circ[\vec A/\vec X]=(V\to R)[\vec A/\vec X]$. The full details of the
proof are given in the Appendix~\ref{app:proofsSR}. \end{proof}}
\arxiv{Five generation lemmas are required: two classical ones, for applications (Lemma~\ref{lem:sr:app}) and abstractions (Lemma~\ref{lem:sr:abs}); and three new ones for scalars (Lemma~\ref{lem:sr:scalars}), sums (Lemma~\ref{lem:sr:sums}) and basis terms (Lemma~\ref{lem:sr:basevectors}).
\begin{lemma}[Scalars]\label{lem:sr:scalars}
For any context $\Gamma$, term $\ve t$, type $T$, if $\pi = \Gamma\vdash
\alpha\cdot\ve{t}: T$, there exist $R_1, \dots, R_n$, $\alpha_1, \dots,
\alpha_n$ such that
\begin{itemize}
\item $T \equiv \sui{n}\alpha_i \cdot R_i$.
\item $\pi_i = \Gamma \vdash \ve{t}: R_i$, with $size(\pi) > size(\pi_i)$, for
$i \in \{1, \dots, n\}$.
\item $\sui{n} \alpha_i = \alpha$.
\end{itemize} \end{lemma} \begin{proof}
By induction on the typing derivation. Full details are given in
Appendix~\ref{app:proofsSR}. \end{proof}
\begin{lemma}[Sums]\label{lem:sr:sums}
If $\Gamma\vdash\ve t+\ve r:S$, there exist $R$, $T$ such that
\begin{itemize}
\item $S \equiv T + R$.
\item $\Gamma\vdash\ve t: T$.
\item $\Gamma\vdash\ve r: R$.
\end{itemize} \end{lemma} \begin{proof}
By induction on the typing derivation. Full details are given in
Appendix~\ref{app:proofsSR}. \end{proof}
\begin{lemma}[Application]\label{lem:sr:app}
If $\Gamma\vdash(\ve t)~\ve r:T$, there exist $R_1, \dots, R_h$, $\mu_1,
\dots, \mu_h$, $\V_1,\dots,\V_h$ such that $T \equiv \suk{h} \mu_k \cdot R_k$,
$\suk{h} \mu_k = 1$ and for all $k \in \{1,\dots,h\}$
\begin{itemize}
\item $\Gamma\vdash\ve t: \sui{n_k}{\alpha_{(k,i)} \cdot\forall\vec{X}.(U\to
T_{(k,i)})}$.
\item $\Gamma\vdash\ve r: \suj{m_k}\beta_{(k,j)}\cdot
U[\vec{A}_{(k,j)}/\vec{X}]$.
\item $\sui{n_k}\suj{m_k} \alpha_{(k,i)}\times\beta_{(k,j)}\cdot
{T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}]} \ssubt_{\V_k,\Gamma} R_k$.
\end{itemize} \end{lemma} \begin{proof}
By induction on the typing derivation. Full details are given in
Appendix~\ref{app:proofsSR}. \end{proof}
\begin{lemma}[Abstractions]\label{lem:sr:abs}
If $\Gamma\vdash\lambda x.\ve t:T$, then there exist $T_1,\dots,T_n$,
$R_1,\dots,R_n$, $U_1,\dots,U_n$, $\alpha_1,\dots,\alpha_n$, $\V_1,\dots,\V_n$
such that $T \equiv \sui{n} \alpha_i \cdot T_i$, $\sui{n} \alpha_i = 1$ and
for all $i \in \{1,\dots,n\}$,
\begin{itemize}
\item $\Gamma,x:U_i\vdash\ve t:R_i$.
\item $U_i \to R_i \ssubt_{\V_i,\Gamma} T_i$.
\end{itemize} \end{lemma} \begin{proof}
By induction on the typing derivation. Full details are given in
Appendix~\ref{app:proofsSR}. \end{proof}
\begin{lemma}[Basis terms]\label{lem:sr:basevectors}
For any context $\Gamma$, type $T$ and basis term $\ve{b}$, if
$\Gamma\vdash\ve{b}: T$ there exist $U_1, \dots, U_n$, $\alpha_1, \dots,
\alpha_n$ such that
\begin{itemize}
\item $T \equiv \sui{n} \alpha_i \cdot U_i$.
\item $\Gamma\vdash\ve{b}: U_i$, for $i \in \{1,\dots,n\}$.
\item $\sui{n} \alpha_i = 1$.
\end{itemize} \end{lemma} \begin{proof}
By induction on the typing derivation. Full details are given in
Appendix~\ref{app:proofsSR}. \end{proof} } \conf{The relation between types just defined is taken into account for the generation lemmas. We left the technical details for the arXiv'ed long version~\cite{paper:arxiv}.}
Substitution lemma is standard.
\begin{lemma}[Substitution lemma]\label{lem:sr:substitution}
For any term ${\ve t}$, basis term $\ve b$, term variable $x$, context
$\Gamma$, types $T$, $U$, type variable $X$ and type $A$, where $A$ is a unit
type if $X$ is a unit variable, otherwise $A$ is a general type, we have,
\begin{enumerate}
\item \label{ap:it:substitutionTypes} if $\Gamma\vdash\ve{t}: T$, then
$\Gamma[A/X]\vdash\ve{t}: T[A/X]$;
\item \label{ap:it:substitutionTerms} if $\Gamma,x:U\vdash\ve t:T$ and
$\Gamma\vdash\ve b:U$, then $\Gamma\vdash\ve t[\ve b/x]: T$.
\end{enumerate} \end{lemma} \arxiv{\begin{proof}
Both items are proven by induction on the typing derivation. Full details are
given in Appendix~\ref{app:proofsSR}. \end{proof}}
We extend the equivalence between types as an equivalence between contexts in a natural way: The equivalence between contexts $\Gamma \equiv \Delta$ is defined by $x:U \in \Gamma$ if and only if there exists $x:V \in \Delta$ such that $U \equiv V$.
\begin{theorem}[Subject Reduction]\label{thm:sr}
For any terms $\ve{t}, \ve{t}'$, any context $\Gamma$ and any type $T$, if
$\ve{t} \to \ve{t}'$ and $\Gamma \vdash \ve{t}: T$, then $\Gamma \vdash
\ve{t}': T$. \end{theorem} \arxiv{\begin{proof}
By induction on the rewrite relation. Full details are given in
Appendix~\ref{app:proofsSR}. \end{proof}}
\section{Other properties}\label{ch:other-properties} In this section we present additional properties that are satisfied by $\ensuremath{\lvec_{\textrm{\tiny R}}}$: progress, strong normalisation, and a characterisation property showing that the sum of all the components of a vector, which we call weight, of a type is the weight of the value obtained after reduction. \conf{We give enough details for reproducing all the proofs. The full detailed proofs are given in the 51-pages long arXiv'ed version at~\cite{paper:arxiv}.} \arxiv{The proofs are given in the Appendix~\ref{app:proofsOP}.}
Let $\mathbb{V} = \left\{\sui{n} \alpha_i \cdot \lambda x_i.\ve{t}_i +
\sum^{m}_{j=n+1} \lambda x_j.\ve{t}_j \mid \forall i, j, \lambda x_i.\ve{t}_i
\neq \lambda x_j.\ve{t}_j\right\}$ be the set of values in our calculus, and we write $\mathsf{NF}$ as the set of terms in normal form (that is, terms that cannot be reduced any further). The following theorem relates those two sets.
\begin{theorem}[Progress]\label{thm:progress}
If $\vdash \ve{t}: T$ and $\ve{t} \in \mathsf{NF}$, then $\ve{t} \in
\mathbb{V}$. \end{theorem} \arxiv{\begin{proof}
By induction on $\ve{t}$. Full details are given in
Appendix~\ref{app:proofsOP}. \end{proof}}
\begin{theorem}[Strong Normalisation]\label{thm:sn} If $\Gamma \vdash \ve{t}: T$ is a valid sequent, then $\ve{t}$ is strongly normalising. \end{theorem} \begin{proof}
The proof is by showing that every typed term in \ensuremath{\lvec_{\textrm{\tiny R}}}\ is also typed in \ensuremath{\textrm{Vectorial}}. The full details are given in the Appendix~\ref{app:proofsOP}. \end{proof}
As previously discussed, the objective of the system is to be able to model vector spaces (or, more precisely, weak modules). In this context, we know that the basis terms represent base vectors, while general terms represent any vector. From here, it follows that if $\ve{v} = \alpha \cdot \ve{b}_1 + \beta \cdot \ve{b}_2$, then $\ve{b}_1$ represents the vector $\left(\begin{smallmatrix}1\\0\end{smallmatrix}\right)$, $\ve{b}_2$ represents the vector $\left(\begin{smallmatrix}0\\1\end{smallmatrix}\right)$, and $\ve{v}$ represents the vector $\left(\begin{smallmatrix}\alpha\\\beta\end{smallmatrix}\right) = \alpha\cdot \left(\begin{smallmatrix}1\\0\end{smallmatrix}\right) + \beta \cdot \left(\begin{smallmatrix}0\\1\end{smallmatrix}\right)$. Therefore, the weight of $\ve{v}$ should be $\alpha + \beta$, since that is effectively the weight of $\left(\begin{smallmatrix}\alpha\\\beta\end{smallmatrix}\right)$.
This is analogous for types: the unit types represent base vectors (which is why they type basis terms), and the general types represent any vector.
We proceed then to formalise the concept of weight of types and terms. First we define the weight of types (Definition~\ref{def:wp:weighttypes}), then the weight of values (Definition~\ref{def:wp:weightterms}), and, finally, we can define the weight of a term as the weight of its type, after proving that if a typed term reduces to a value, then the weight of the value and of the type coincides (Theorem~\ref{thm:wp:weightpreserv}).
\begin{definition}[Weight of types]\label{def:wp:weighttypes}
We define the relation $\tnorm{\bullet}: \text{Type} \to \text{Scalar}$
inductively as follows:
\[
\begin{array}{r@{\,}l@{\hspace{1cm}}r@{\,}l@{\hspace{1cm}}r@{\,}l}
\tnorm{U} &= 1
&
\tnorm{\alpha \cdot T} &= \alpha \cdot \tnorm{T}
&
\tnorm{T + R} &= \tnorm{T} + \tnorm{R}
\end{array}
\] \end{definition}
\begin{example}
Consider the type $\sui{n} \alpha_i \cdot U_i$, then
\[
\tnorm{\sui{n} \alpha_i \cdot U_i} = \sui{n} \alpha_i \cdot \tnorm{U_i}=
\sui{n} \alpha_i
\] \end{example}
\begin{definition}[Weight of values]\label{def:wp:weightterms}
We define the relation $\tnorm{\bullet}: \text{Term} \to \text{Scalar}$
inductively as follows:
\[
\begin{array}{r@{\,}l@{\hspace{1cm}}r@{\,}l@{\hspace{1cm}}r@{\,}l}
\tnorm{\ve{b}} &= 1
&
\tnorm{\alpha \cdot \ve{t}} &= \alpha \cdot \tnorm{\ve{t}}
&
\tnorm{\ve{t} + \ve{r}} &= \tnorm{\ve{t}} + \tnorm{\ve{r}}
\end{array}
\] \end{definition} \begin{example}
Consider the term $\sui{n} \alpha_i \cdot \lambda x_i.\ve{t}_i$, then
\[
\tnorm{\sui{n} \alpha_i \cdot \lambda x_i.\ve{t}_i} = \sui{n} \alpha_i \cdot
\tnorm{\lambda x_i.\ve{t}_i}= \sui{n} \alpha_i
\] \end{example}
\begin{lemma}\label{lem:wp:weightequiv}
If $T \equiv R$, then $\tnorm{T} = \tnorm{R}$. \end{lemma} \arxiv{\begin{proof}
We prove the lemma holds for every definition of $\equiv$. Full details are
given in Appendix~\ref{app:proofsOP}. \end{proof}}
\begin{lemma}\label{lem:wp:weightofvalues}
If $\ve{v}\in \mathbb{V}$, and $\vdash \ve{v}: T$, then $\tnorm{T}
\equiv \tnorm{\ve{v}}$. \end{lemma} \arxiv{\begin{proof}
By induction on $n$. Full details are given in Appendix~\ref{app:proofsOP}. \end{proof}}
Finally, the weight of an arbitrary term can be defined as the weight of its type, thanks to the following theorem.
\begin{theorem}[Weight Preservation]\label{thm:wp:weightpreserv}
If $\vdash \ve{t}: T$ and $\ve{t} \to^{*} \ve{v}$, then $\tnorm{T} =
\tnorm{\ve{v}}$. \end{theorem} \begin{proof}
Since $\ve{t} \to^{*} \ve{v}$, by Theorem~\ref{thm:progress}, $\ve{v} = \sui{n}
\alpha_i \cdot \lambda x_i.\ve{t}_i + \sum^{m}_{j=n+1} \lambda x_j.\ve{t}_j$,
where $\lambda x_i.\ve{t}_i \neq \lambda x_j.\ve{t}_j$ for all $i \in \{1,
\dots, n\}$, $j \in \{n+1, \dots m\}$. Also, by Theorem~\ref{thm:sr}, we know then
that $\vdash \ve{v}: T$. Finally, by Lemma~\ref{lem:wp:weightofvalues}, we know
that $\tnorm{T} = \tnorm{\ve{v}}$. \end{proof}
\section{Conclusion}\label{ch:conclusion} We have revisited $\ensuremath{\textrm{Vectorial}}$ redefining it in a careful way, proving that the modified version satisfies the standard formulation of the Subject Reduction property (Theorem~\ref{thm:sr}). It is worth mentioning that the design choices we made are not necessarily the only possibility. Indeed, one of the first approaches we considered involved keeping most of the typing rules as in the original system, and adding subtyping. In the end, we realized that the property could be satisfied in a simpler and more elegant way by modifying the typing rules. The summary of the changes made to the original system is: \begin{itemize} \item We added the $S$ rule, that deals with superposition of types of a single
term. \item We added the $1_E$ rule, to allow the removal of the scalar if said scalar
is equal to 1. \item We removed the term $\ve{0}$, which proved to be
undesirable~\cite{DiazcaroGuillermoMiquelValironLICS19}. \end{itemize}
In addition, we showed that the obtained calculus is still strongly normalising (Theorem~\ref{thm:sn}), by proving that the typable terms in the modified version, are typable in the original system (which has been proved to be strongly normalising as well~\cite{vectorial}). We also provided a proof of the progress property (Theorem~\ref{thm:progress}), which allowed us to characterise the terms that cannot be reduced any further. This enabled us to formalize the concept of weight of types and terms, and to prove that terms had the same weight as their types (Theorem~\ref{thm:wp:weightpreserv}).
We stand by this modified version of \ensuremath{\textrm{Vectorial}}, which we think provides the right framework in the quest for the quantum-controlled lambda calculus.
\arxiv{
} \appendix
\arxiv{ \section{Omitted proofs in Section~\ref{ch:sr}}\label{app:proofsSR} \arxiv{\xrecap{Lemma}{Characterisation of types~\cite[Lem.~4.2]{vectorial}}{lem:sr:typecharact} {
For any type $T$, there exist $n,m\in\mathbb{N}$, $\alpha_1,\dots,\alpha_n$,
$\beta_1,\dots,\beta_m\in\ensuremath{\mathsf{S}}$, distinct unit types $U_1,\dots,U_n$ and
distinct general variables $\vara{X}_1,\dots,\vara{X}_m$ such that \(
T\equiv\sui{n}\alpha_i\cdot U_i+\suj{m}\beta_j\cdot\vara{X}_j \). } \begin{proof}
Structural induction on $T$.
\begin{itemize}
\item Let $T=U$, then take $\alpha=\beta=1$, $n=1$ and $m=0$, and so
$T\equiv\sui{1}1\cdot U=1\cdot U$.
\item Let $T=\alpha\cdot T'$, then by the induction hypothesis
$T'\equiv\sui{n}\alpha_i\cdot U_i+\suj{m}\beta_j\cdot\vara{X}_j$, so
$T=\alpha\cdot T'\equiv\alpha\cdot (\sui{n}\alpha_i\cdot
U_i+\suj{m}\beta_j\cdot\vara{X}_j)\equiv\sui{n}(\alpha\times\alpha_i)\cdot
U_i+\suj{m}(\alpha\times\beta_j)\cdot\vara{X}_j$.
\item Let $T=R+S$, then by the induction hypothesis
$R\equiv\sui{n}\alpha_i\cdot U_i+\suj{m}\beta_j\cdot\vara{X}_j$ and
$S\equiv\sui{n'}\alpha'_i\cdot U'_i+\suj{m'}\beta'_j\cdot\vara{X'}_j$, so
$T=R+S\equiv\sui{n}\alpha_i\cdot U_i+\sui{n'}\alpha'_i\cdot
U'_i+\suj{m}\beta_j\cdot\vara{X}_j+\suj{m'}\beta'_j\cdot\vara{X'}_j$. If the
$U_i$ and the $U'_i$ are all different each other, we have finished, in
other case, if $U_k=U'_h$, notice that $\alpha_k\cdot U_k+\alpha'_h\cdot
U'_h\equiv (\alpha_k+\alpha'_h)\cdot U_k$.
\item Let $T=\vara X$, then take $\alpha=\beta=1$, $m=1$ and $n=0$, and so
$T\equiv\suj{1} 1\cdot\vara{X}\equiv 1\cdot\vara X$. \qed
\end{itemize} \end{proof}}
\begin{definition}
Let $F$ be an algebraic context with $n$ holes. Let $\vec U = U_1,\ldots,U_n$
be a list of $n$ unit types. If $U$ is a unit type, we write $\bar U$ for the
set of unit types equivalent to $U$:
\[
\bar U := \{ V ~|~ V \textrm{ is unit and } V \equiv U \}.
\]
The {\em context vector} $v_F(\vec U)$ associated with the context $F$ and the
unit types $\vec U$ is partial map from the set $\mathcal S = \{ \bar U \}$ to
scalars. It is inductively defined as follows: $v_{\alpha\cdot F}(\vec U) :=
\alpha v_F(\vec U)$, $v_{F + G}(\vec U) := v_F(\vec U) + v_G(\vec U)$, and
finally $v_{[-_i]}(\vec U) := \{\bar U_i \mapsto 1\}$. The sum is defined on
these partial map as follows:
\[
(f + g)(\vec U) = \left\{
\begin{array}{ll}
f(\vec U) + g(\vec U)& \textrm{if both are defined}
\\
f(\vec U)& \textrm{if $f(\vec U)$ is defined but not $g(\vec U)$}
\\
g(\vec U)& \textrm{if $g(\vec U)$ is defined but not $f(\vec U)$}
\\
\textrm{is not defined} & \textrm{if neither $f(\vec U)$ nor
$g(\vec U)$ is defined.}
\end{array}
\right.
\]
Scalar multiplication is defined as follows:
\[
(\alpha f)(\vec U) = \left\{
\begin{array}{ll}
\alpha (f(\vec U))& \textrm{if $f(\vec U)$ is defined}
\\
\textrm{is not defined} & \textrm{if $f(\vec U)$ is not defined.}
\end{array}
\right.
\] \end{definition}
\begin{lemma}\label{lem:equivdistinctscalarsaux}
Let $F$ and $G$ be two algebraic contexts with respectively $n$ and $m$ holes.
Let $\vec U$ be a list of $n$ unit types, and $\vec V$ be a list of $m$ unit
types. Then $F(\vec U) \equiv G(\vec V)$ implies $v_F(\vec U) = v_G(\vec V)$. \end{lemma}
\begin{proof}
The derivation of $F(\vec U) \equiv F(\vec V)$ essentially consists
in a sequence of the elementary rules (or congruence thereof) in
Figure~\ref{fig:typeequiv} composed with transitivity:
\[
F(\vec U) = W_1\equiv W_2 \equiv \cdots \equiv W_k = G(\vec V).
\]
We prove the result by induction on $k$.
\begin{itemize}
\item Case $k=1$. Then $F(\vec U)$ is syntactically equal to
$G(\vec V)$: we are done.
\item Suppose that the result is true for sequences of size $k$, and
let
\[
F(\vec U) = W_1\equiv W_2 \equiv \cdots \equiv W_k
\equiv W_{k+1} = G(\vec V).
\]
Let us concentrate on the first step $F(\vec U) \equiv W_2$: it is an
elementary step from Figure~\ref{fig:typeequiv}. By structural induction on
the proof of $F(\vec U) \equiv W_2$ (which only uses congruence and
elementary steps, and not transitivity), we can show that $W_2$ is of the
form $F'(\vec U')$ where $v_F(\vec U) = v_{F'}(\vec U')$. We are now in
power of applying the induction hypothesis, because the sequence of
elementary rewrites from $F'(\vec U')$ to $G(\vec V)$ is of size $k$.
Therefore $v_{F'}(\vec U') = v_G(\vec V)$. We can then conclude that
$v_F(\vec U) = v_G(\vec V)$.
\end{itemize}
This conclude the proof of the lemma. \end{proof}
\xrecap{Lemma}{Equivalence between sums of distinct elements (up to
$\equiv$)}{lem:sr:equivdistinctscalars} Let $U_1,\dots,U_n$ be a set of distinct (not equivalent) unit types, and let $V_1,\dots,V_m$ be also a set distinct unit types. If $\sui{n}\alpha_i\cdot U_i\equiv\suj{m}\beta_j\cdot V_j$, then $m=n$ and there exists a permutation $p$ of $n$ such that $\forall i$, $\alpha_i=\beta_{p(i)}$ and $U_i\equiv V_{p(i)}$.
\begin{proof}
Let $S = \sui{n}\alpha_i\cdot U_i$ and $T = \suj{m}\beta_j\cdot V_j$. Both $S$
and $T$ can be respectively written as $F(\vec U)$ and $G(\vec V)$. Using
Lemma~\ref{lem:equivdistinctscalarsaux}, we conclude that $v_F(\vec U) =
v_G(\vec V)$. Since all $U_i$'s are pairwise non-equivalent, the $\bar U_i$'s
are pairwise distinct.
\[
v_F(\vec U) = \{ \bar U_i \mapsto \alpha_i~|~i=1\ldots n\}.
\]
Similarly, the $\bar V_j$'s are pairwise disjoint, and
\[
v_G(\vec G) = \{ \bar V_j \mapsto \beta_j~|~i=1\ldots m\}.
\]
We obtain the desired result because these two partial maps are supposed to be
equal. Indeed, this implies:
\begin{itemize}
\item $m=n$ because the domains are equal (so they should have the same size)
\item Again using the fact that the domains are equal, the sets $\{\bar U_i\}$
and $\{\bar V_j\}$ are equal: this means there exists a permutation $p$ of
$n$ such that $\forall i$, $\bar U_i= \bar V_{p(i)}$, meaning $U_i\equiv
V_{p(i)}$.
\item Because the partial maps are equal, the images of a given element $\bar
U_i = \bar V_{p(i)}$ under $v_F$ and $v_G$ are in fact the same: we
therefore have $\alpha_i=\beta_{p(i)}$.
\end{itemize}
And this closes the proof of the lemma. \end{proof}
\arxiv{\xrecap{Lemma}{Equivalences $\forall$~\cite[Lem.~4.5]{vectorial}}{lem:sr:equivforall} {
Let $U_1,\dots,U_n$ be a set of distinct (not equivalent) unit types and let
$V_1,\dots,V_n$ be also a set of distinct unit types.
\begin{enumerate}
\item $\sui{n}\alpha_i\cdot
U_i\equiv\suj{m}\beta_j\cdot V_j$ iff $\sui{n}\alpha_i\cdot\forall
X.U_i\equiv\suj{m}\beta_j\cdot\forall X.V_j$.
\item If $\sui{n}\alpha_i\cdot\forall
X.U_i\equiv\suj{m}\beta_j\cdot V_j$ then $\forall V_j,\exists
W_j~/~V_j\equiv\forall X.W_j$.
\item If $T\equiv R$ then $T[A/X]\equiv R[A/X]$.
\end{enumerate} } \begin{proof}
Item (1) From Lemma~\ref{lem:sr:equivdistinctscalars}, $m=n$, and without loss
of generality, for all $i$, $\alpha_i=\beta_i$ and $U_i=V_i$ in the
left-to-right direction, $\forall X.U_i=\forall X.V_i$ in the right-to-left
direction. In both cases we easily conclude.
\noindent
Item (2) is similar.
\noindent
Item (3) is a straightforward induction on the equivalence $T\equiv R$. \end{proof} }
\xrecap{Lemma}{Arrows comparison}{lem:sr:arrowscomp} { $V \to
R\ssubt_{\V,\Gamma} \forall\vec X.(U\to T) $, then $U\to T\equiv(V\to
R)[\vec{A}/\vec{Y}]$, with $\vec Y\notin \FV{\Gamma}$. } \begin{proof}
Let $(~\cdot~)^\circ$ be a map from types to types defined as follows,
\begin{align*}
X^\circ &= X \\
(U\to T)^\circ &= U\to T \\
(\forall X.T)^\circ &= T^\circ \\
(\alpha\cdot T)^\circ &=\alpha\cdot T^\circ\\
(T+R)^\circ &=T^\circ+R^\circ
\end{align*}
We need three intermediate results:
\begin{enumerate}
\item If $T\equiv R$, then $T^\circ\equiv R^\circ$.
\item For any types $U, A$, there exists $B$ such that
$(U[A/X])^\circ=U^\circ[B/X]$.
\item For any types $V, U$, there exists $\vec A$ such that if $V
\ssubt_{\V,\Gamma} \forall\vec X.U$, then $U^\circ\equiv V^\circ[\vec A/\vec
X]$.
\end{enumerate}
{\textit{Proofs.}}
\begin{enumerate}
\item Induction on the equivalence rules. We only give the basic cases since
the inductive step, given by the context where the equivalence is applied,
is trivial.
\begin{itemize}
\item $(1\cdot T)^\circ=1\cdot T^\circ\equiv T^\circ$.
\item $(\alpha\cdot(\beta\cdot T))^\circ=\alpha\cdot(\beta\cdot
T^\circ)\equiv(\alpha\times\beta)\cdot T^\circ=((\alpha\times\beta)\cdot
T)^\circ$.
\item $(\alpha\cdot T+\alpha\cdot R)^\circ=\alpha\cdot T^\circ+\alpha\cdot
R^\circ\equiv\alpha\cdot(T^\circ+R^\circ)=(\alpha\cdot(T+R))^\circ$.
\item $(\alpha\cdot T+\beta\cdot T)^\circ=\alpha\cdot T^\circ+\beta\cdot
T^\circ\equiv(\alpha+\beta)\cdot T^\circ=((\alpha+\beta)\cdot T)^\circ$.
\item $(T+R)^\circ=T^\circ+R^\circ\equiv R^\circ+T^\circ=(R+T)^\circ$.
\item $(T+(R+S))^\circ=T^\circ+(R^\circ+S^\circ)\equiv
(T^\circ+R^\circ)+S^\circ=((T+R)+S)^\circ$.
\end{itemize}
\item Structural induction on $U$.
\begin{itemize}
\item $U=\varu X$. Then $(\varu X[V/\varu X])^\circ=V^\circ=\varu
X[V^\circ/\varu X]=\varu X^\circ[V^\circ/\varu X]$.
\item $U=\varu Y$. Then $(\varu Y[A/X])^\circ=\varu Y=\varu Y^\circ[A/X]$.
\item $U=V\to T$. Then $((V\to T)[A/X])^\circ=(V[A/X]\to
T[A/X])^\circ=V[A/X]\to T[A/X]=(V\to T)[A/X]=(V\to T)^\circ[A/X]$.
\item $U=\forall Y.V$. Then $((\forall Y.V)[A/X])^\circ=(\forall
Y.V[A/X])^\circ=(V[A/X])^\circ$, which by the induction hypothesis is
equivalent to $V^\circ[B/X]=(\forall Y.V)^\circ[B/X]$.
\end{itemize}
\item It suffices to show this for $V \prec_{X,\Gamma} \forall\vec X.U$.
Cases:
\begin{itemize}
\item $\forall\vec X.U\equiv\forall Y.V$, then notice that $(\forall\vec
X.U)^\circ \equiv_{(1)}(\forall Y.V)^\circ=V^\circ$.
\item $V\equiv\forall Y.W$ and $\forall\vec X.U\equiv W[A/X]$, then
$(\forall\vec X.U)^\circ\equiv_{(1)}(W[A/X])^\circ\equiv_{(2)}
W^\circ[B/X]=(\forall Y.W)^\circ[B/X]\equiv_{(1)}V^\circ[B/X]$.
\end{itemize}
\end{enumerate}
Proof of the lemma. $U\to T\equiv(U\to T)^\circ$, by the intermediate result
3, this is equivalent to $(V\to R)^\circ[\vec A/\vec X]=(V\to R)[\vec A/\vec
X]$. \end{proof}
\xrecap{Lemma}{Scalars}{lem:sr:scalars}{ For any context $\Gamma$, term $\ve t$,
type $T$, if $\pi = \Gamma\vdash \alpha\cdot\ve{t}: T$, there exist $R_1,
\dots, R_n$, $\alpha_1, \dots, \alpha_n$ such that
\begin{itemize}
\item $T \equiv \sui{n}\alpha_i \cdot R_i$.
\item $\pi_i = \Gamma \vdash \ve{t}: R_i$, with $size(\pi) > size(\pi_i)$, for
$i \in \{1, \dots, n\}$.
\item $\sui{n} \alpha_i = \alpha$.
\end{itemize} } \begin{proof}
By induction on the typing derivation. \inductioncase{Case $S$}
\[
\prftree[r]{$S$} {\Gamma \vdash \ve{t}: T_i} {\prfassumption{\forall i \in
\{1,\dots,n\}}}
{\Gamma \vdash \left(\sui{n} \alpha_i\right) \cdot \ve{t}: \sui{n} \alpha_i
\cdot T_i}
\]
Trivial case. \inductioncase{Case $\equiv$}
\[
\prftree[r]{$\equiv$} {\pi' = \Gamma \vdash \alpha \cdot \ve{t}: T} {T
\equiv R} {\pi = \Gamma \vdash \alpha \cdot \ve{t}: R}
\]
By the induction hypothesis there exist $S_1,\dots,S_n$,
$\alpha_1,\dots,\alpha_n$ such that
\begin{itemize}
\item $T \equiv R \equiv \sui{n} \alpha_i \cdot S_i$.
\item $\pi_i = \Gamma \vdash \ve{t}: S_i$, with $size(\pi') > size(\pi_i)$,
for $i \in \{1,\dots,n\}$.
\item $\sui{n} \alpha_i = \alpha$.
\end{itemize}
It is easy to see that $size(\pi) > size(\pi')$, so the lemma holds.
\inductioncase{Case $1_E$}
\[
\prftree[r]{$1_E$} {\pi = \Gamma \vdash 1\cdot (\alpha \cdot \ve{t}): T}
{\Gamma \vdash \alpha \cdot \ve{t}: T}
\]
By induction hypothesis, there exist $R_1,\dots,R_m$, $\beta_1,\dots,\beta_m$
such that
\begin{itemize}
\item ${T \equiv \suj{m} \beta_j \cdot R_j}$.
\item $\pi_j = \Gamma \vdash \alpha \cdot \ve{t}: R_j$ with $size(\pi) >
size(\pi_j)$ for $j = \{1, \dots, m\}$.
\item $\suj{m} \beta_j = 1$.
\end{itemize}
Since $size(\pi) > size(\pi_j)$, then by applying the induction hypothesis
again for all $j = \{1, \dots, m\}$, we have that there exist
$S_{(j,1)},\dots,S_{(j,n_j)}$, $\alpha_{(j,1)},\dots,\alpha_{(j,n_j)}$ such
that
\begin{itemize}
\item $R_j \equiv \sui{n_j} \alpha_{(j,i)} \cdot S_{(j,i)}$.
\item $\pi_{(j,i)} = \Gamma \vdash \ve{t}: S_{(j,i)}$ with $size(\pi_j) >
size(\pi_{(j,i)})$ for $i \in \{1, \dots, n_j\}$.
\item $\sui{n_j} \alpha_{(j,i)} = \alpha$.
\end{itemize}
Given that $\Gamma \vdash \alpha \cdot \ve{t}: T$, then
\[
T \equiv \suj{m} \beta_j \cdot R_j \equiv \suj{m} \beta_j \cdot \sui{n}
\alpha_{(j,i)} \cdot S_{(j,i)} \equiv \suj{m}\sui{n} (\beta_j \times
\alpha_{(j,i)}) \cdot S_{(j,i)}
\]
Finally, we must prove that $\suj{m}\sui{n} (\beta_j \times \alpha_{(j,i)}) =
\alpha$,
\[
\suj{m}\sui{n} (\beta_j \times \alpha_{(j,i)}) = \suj{m} \beta_j \cdot
\underbrace{\sui{n} \alpha_{(j,i)}}_{=~\alpha} = \suj{m} \beta_j \cdot
\alpha = \alpha \cdot \underbrace{\suj{m} \beta_j}_{=~1} = \alpha
\]
\inductioncase{Case $\forall_I$}
\[
\prftree[r]{$\forall_I$} {\pi = \Gamma \vdash \alpha \cdot \ve{t}:
\sui{n}\alpha_i \cdot U_i} {X \notin \FV{\Gamma}} {\pi' = \Gamma \vdash
\alpha \cdot \ve{t}: \sui{n}\alpha_i \cdot \forall X. U_i}
\]
By the induction hypothesis there exist $R_1,\dots,R_m$, $\mu_1,\dots,\mu_m$
such that
\begin{itemize}
\item $\sui{n}\alpha_i\cdot U_i\equiv \suj{m} \mu_j \cdot R_j$.
\item $\pi_j = \Gamma \vdash \ve{t}: R_j$, with $size(\pi) > size(\pi_j)$, for
$j \in \{1, \dots, m\}$.
\item $\suj{m} \mu_j = \alpha$.
\end{itemize}
By applying Lemma~\ref{lem:sr:typecharact} for all $j \in \{1, \dots, m\}$,
and since $\sui{n}\alpha_i\cdot U_i$ does not have any general variable
$\vara{X}$,
then $R_j \equiv \suk{h_j}\beta_{(j,k)}\cdot V_{(j,k)}$.\\
Hence $\sui{n}\alpha_i\cdot U_i\equiv \suj{m}\mu_j \cdot \suk{h_j}\beta_{(j,k)}\cdot V_{(j,k)}$.\\
Without loss of generality, assuming all unit types are distinct (not
equivalent), then by Lemma~\ref{lem:sr:equivforall},
\[
\sui{n-1}\alpha_i\cdot \forall{X}.U_n \equiv \suj{m}\mu_j \cdot
\underbrace{\suk{h_j}\beta_{(j,k)}\cdot \forall X.V_{(j,k)}}_{\equiv~R'_j}
\]
We must prove that for all $j \in \{1,\dots,m\}$, $\pi'_j = \Gamma \vdash
\ve{t}: R'_j$ and that $size(\pi') > size(\pi'_j)$. By applying the
$\forall_I$ rule, we have
\[
\prftree[r]{$\forall_I$} {\Gamma \vdash \ve{t}: R_j} {\prfassumption{X
\notin \FV{\Gamma}}} {\pi'_j = \Gamma \vdash \ve{t}: R'_j}
\]
And notice that using the $S$ rule, obtain
\[
\prftree[r]{$\equiv$} {\prftree[r]{$S$} {\pi'_j = \Gamma \vdash \ve{t}:
R'_j} {\forall j \in \{1,\dots,m\}} {\Gamma \vdash \alpha \cdot \ve{t}:
\suj{m} \mu_j \cdot R'_j}} {\prfassumption{\sui{n}\alpha_i \cdot \forall
X. U_i \equiv \suj{m} \mu_j \cdot R'_j}} {\pi' = \Gamma \vdash \alpha
\cdot \ve{t}: \sui{n}\alpha_i \cdot \forall X. U_i}
\]
So for all $j \in \{1,\dots,m\}$, $size(\pi') > size(\pi'_j)$.
\inductioncase{Case $\forall_E$}
\[
\prftree[r]{$\forall_E$} {\pi = \Gamma \vdash \alpha \cdot \ve{t}:
\sui{n}\alpha_i \cdot \forall X. U_i} {\pi' = \Gamma \vdash \alpha \cdot
\ve{t}: \sui{n}\alpha_i \cdot U_i[A/X]}
\]
By the induction hypothesis there exist $R_1, \dots, R_m$, $\mu_1, \dots,
\mu_m$ such that
\begin{itemize}
\item $\sui{n} \alpha_i \cdot \forall X. U_i \equiv \suj{m} \mu_j \cdot R_j$.
\item $\pi_j = \Gamma \vdash \ve{t}: R_j$, with $size(\pi) > size(\pi_j)$, for
$j \in \{1, \dots, m\}$.
\item $\suj{m} \mu_j = \alpha$.
\end{itemize}
By applying Lemma~\ref{lem:sr:typecharact} for all $j \in \{1, \dots, m\}$,
and since $\sui{n}\alpha_i \cdot \forall X. U_i$
does not have any general variable $\vara{X}$, then $R_j \equiv \suk{h_j}\beta_{(j,k)}\cdot V_{(j,k)}$.\\
Hence $\sui{n}\alpha_i \cdot \forall X. U_i \equiv \suj{m}\mu_j \cdot \suk{h_j}\beta_{(j,k)}\cdot V_{(j,k)}$.\\
Without loss of generality, we assume that all unit types present at both
sides of the equivalence are distinct, then by
Lemma~\ref{lem:sr:equivdistinctscalars}, for all $j \in \{1, \dots, m\}$, $k
\in \{1,\dots,h_j\}$, there exists $V'_{(j,k)}$ such that $V_{(j,k)} \equiv
\forall X.V'_{(j,k)}$. Then,
\[
\sui{n}\alpha_i\cdot \forall{X}.U_i \equiv \suj{m}\mu_j \cdot
\underbrace{\suk{h_j}\beta_{(j,k)}\cdot \forall X.V'_{(j,k)}}_{\equiv~R_j}
\]
By the same lemma, we have that
\[
\sui{n}\alpha_i\cdot U_i[A/X] \equiv \suj{m}\mu_j \cdot
\underbrace{\suk{h_j}\beta_{(j,k)}\cdot V'_{(j,k)}[A/X]}_{\equiv~R'_j}
\]
We must prove that for all $j \in \{1,\dots,m\}$, $\pi'_j = \Gamma \vdash
\ve{t}: R'_j$ and that $size(\pi') > size(\pi'_j)$. By applying the
$\forall_E$ rule, we have
\[
\prftree[r]{$\forall_E$} {\Gamma \vdash \ve{t}: R_j} {\pi'_j = \Gamma \vdash
\ve{t}: R'_j}
\]
And notice that using the $S$ rule, obtain
\[
\prftree[r]{$\equiv$} {\prftree[r]{$S$} {\pi'_j = \Gamma \vdash \ve{t}:
R'_j} {\forall j \in \{1,\dots,m\}} {\Gamma \vdash \alpha \cdot \ve{t}:
\suj{m} \mu_j \cdot R'_j}} {\prfassumption{\sui{n}\alpha_i \cdot
U_i[A/X] \equiv \suj{m} \mu_j \cdot R'_j}} {\pi' = \Gamma \vdash \alpha
\cdot \ve{t}: \sui{n}\alpha_i \cdot U_i[A/X]}
\]
So for all $j \in \{1,\dots,m\}$, $size(\pi') > size(\pi'_j)$. \end{proof}
\xrecap{Lemma}{Sums}{lem:sr:sums}{ If $\Gamma\vdash\ve t+\ve r:S$, there exist
$R$, $T$ such that
\begin{itemize}
\item $S \equiv T + R$.
\item $\Gamma\vdash\ve t: T$.
\item $\Gamma\vdash\ve r: R$.
\end{itemize} } \begin{proof}
By induction on the typing derivation. \inductioncase{Case $+_I$}
\[
\prftree[r]{$+_I$} {\Gamma \vdash \ve{t}: T} {\Gamma \vdash \ve{r}: R}
{\Gamma \vdash \ve{t} + \ve{r}: T + R}
\]
Trivial. \inductioncase{Case $\equiv$}
\[
\prftree[r]{$\equiv$} {\Gamma \vdash \ve{t} + \ve{r}: P} {S \equiv P}
{\Gamma \vdash \ve{t} + \ve{r}: S}
\]
By the induction hypothesis, ${S \equiv P \equiv T + R}$. \inductioncase{Case
$1_E$}
\[
\prftree[r]{$1_E$} {\pi = \Gamma \vdash 1\cdot(\ve{t} + \ve{r}): T} {\Gamma
\vdash \ve{t} + \ve{r}: T}
\]
By Lemma~\ref{lem:sr:scalars}, there exist $R_1, \dots, R_m$, $\beta_1, \dots,
\beta_m$ such that
\begin{itemize}
\item ${T \equiv \suj{m} \beta_j \cdot R_j}$.
\item $\pi_j = \Gamma \vdash \ve{t} + \ve{r}: R_j$ with $size(\pi) >
size(\pi_j)$ for $j \in \{1, \dots, m\}$.
\item $\suj{m} \beta_j = 1$
\end{itemize}
Since $size(\pi) > size(\pi_j)$, by applying the induction hypothesis for all
$j \in \{1, \dots, m\}$,
\begin{itemize}
\item $R_j \equiv S_{(j,1)} + S_{(j,2)}$.
\item $\Gamma\vdash\ve t: S_{(j,1)}$.
\item $\Gamma\vdash\ve r: S_{(j,2)}$.
\end{itemize}
Then,
\[
T \equiv \suj{m} \beta_j \cdot R_j \equiv \suj{m} \beta_j \cdot (S_{(j,1)} +
S_{(j,2)}) \equiv \suj{m} \beta_j \cdot S_{(j,1)} + \suj{m} \beta_j \cdot
S_{(j,2)}
\]
We can rewrite $T$ as follows:
\[
P_1 = \suj{m} \beta_j \cdot S_{(j,1)}\qquad P_2 = \suj{m} \beta_j \cdot
S_{(j,2)}\qquad T \equiv P_1 + P_2
\]
Finally, we must prove that $\Gamma \vdash \ve{t}: P_1$ and $\Gamma \vdash \ve{r}: P_2$.\\
Since $\Gamma\vdash\ve t: S_{(j,1)}$ and $\Gamma\vdash\ve r: S_{(j,2)}$ for
all $j \in \{1,\dots,m\}$, applying the $S$ rule in both cases we
have
\[
\prftree[r]{$S$} {\Gamma\vdash\ve t: S_{(j,1)}~\forall j \in \{1,\dots,m\}}
{\Gamma \vdash 1\cdot \ve{t}: P_1} \qquad \prftree[r]{$S$} {\Gamma\vdash\ve
t: S_{(j,2)}~\forall j \in \{1,\dots,m\}}
{\Gamma \vdash 1\cdot \ve{r}: P_2}
\]
Applying the $1_E$ rule to both sequents, we have
\[
\Gamma\vdash \ve t: P_1 \qquad \Gamma\vdash \ve r: P_2
\]
Finally, by $\equiv$ rule, $\Gamma \vdash \ve{t} + \ve{r}: T$.
\inductioncase{Case $\forall$}
\[
\prftree[r]{$\forall$} {\Gamma \vdash \ve{t} + \ve{r}: \sui{n}\alpha_i\cdot
U_i} {\Gamma \vdash \ve{t} + \ve{r}: \sui{n}\alpha_i\cdot V_i}
\]
Rules $\forall_I$ and $\forall_E$ both have the same structure as shown above.
In any case, by the induction hypothesis $\Gamma\vdash\ve t:T$ and
$\Gamma\vdash\ve r:R$ with
$T+R\equiv\sui{n}\alpha_i\cdot U_i$.\\
Then, there exist $N,M\subseteq\{1,\dots,n\}$ with $N\cup M=\{1,\dots,n\}$
such that
\begin{align*}
T\equiv\sum_{i\in N\setminus M}\alpha_i\cdot U_i+\sum_{i\in N\cap M}\alpha_i'\cdot U_i&\qquad\textrm{and}\qquad
R\equiv\sum_{i\in M\setminus N}\alpha_i\cdot U_i+\sum_{i\in N\cap M}\alpha_i''\cdot U_i&
\end{align*}
where $\forall i\in N\cap M$, $\alpha_i'+\alpha_i''=\alpha_i$.\\
Therefore, using $\equiv$ (if needed) and the same $\forall$-rule,
\begin{align*}
T\equiv\sum_{i\in N\setminus M}\alpha_i\cdot V_i+\sum_{i\in N\cap M}\alpha_i'\cdot V_i&\qquad\textrm{and}\qquad
R\equiv\sum_{i\in M\setminus N}\alpha_i\cdot V_i+\sum_{i\in N\cap M}\alpha_i''\cdot V_i&
\end{align*} \end{proof}
\xrecap{Lemma}{Application}{lem:sr:app}{ If $\Gamma\vdash(\ve t)~\ve r:T$, there
exist $R_1, \dots, R_h$, $\mu_1, \dots, \mu_h$, $\V_1,\dots,\V_h$ such that $T
\equiv \suk{h} \mu_k \cdot R_k$, $\suk{h} \mu_k = 1$ and for all $k \in
\{1,\dots,h\}$
\begin{itemize}
\item $\Gamma\vdash\ve t: \sui{n_k}{\alpha_{(k,i)} \cdot\forall\vec{X}.(U\to
T_{(k,i)})}$.
\item $\Gamma\vdash\ve r: \suj{m_k}\beta_{(k,j)}\cdot
U[\vec{A}_{(k,j)}/\vec{X}]$.
\item $\sui{n_k}\suj{m_k} \alpha_{(k,i)}\times\beta_{(k,j)}\cdot
{T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}]} \ssubt_{\V_k,\Gamma} R_k$.
\end{itemize} } \begin{proof}
By induction on the typing derivation.
\inductioncase{Case $\to_E$}
\[
\prftree[r]{$\to_E$}
{\Gamma \vdash \ve{t}: \sui{n}{\alpha_i \cdot\forall\vec{X}.(U\to T_i)}}
{\Gamma \vdash \ve{r}: \suj{m}\beta_j\cdot U[\vec{A}_j/\vec{X}]}
{\Gamma \vdash (\ve t)~\ve r: \sui{n}\suj{m} \alpha_i\times\beta_j\cdot {T_i[\vec{A}_j/\vec{X}]}}
\]
Take $\mu_1, \dots, \mu_h$ such that $\suk{h} \mu_k = 1$, then
\[
\sui{n}\suj{m} \alpha_i\times\beta_j\cdot T_i[\vec{A}_j/\vec{X}] \equiv \suk{h} \mu_k \cdot \sui{n}\suj{m} \alpha_i\times\beta_j\cdot T_i[\vec{A}_j/\vec{X}]
\]
So this is the trivial case.
\inductioncase{Case $\equiv$}
\[
\prftree[r]{$\equiv$}
{\Gamma \vdash (\ve t)~\ve r: P}
{S \equiv P}
{\Gamma \vdash (\ve t)~\ve r: S}
\]
By the induction hypothesis, there exist $R_1, \dots, R_h$, $\mu_1, \dots, \mu_h$, $\V_1,\dots,\V_h$ such that $P \equiv S \equiv \suk{h} \mu_k \cdot R_k$, $\suk{h} \mu_k = 1$
and for all $k \in \{1,\dots,h\}$,
\begin{itemize}
\item $\Gamma\vdash\ve t: \sui{n_k}{\alpha_{(k,i)} \cdot\forall\vec{X}.(U\to T_{(k,i)})}$.
\item $\Gamma\vdash\ve r: \suj{m_k}\beta_{(k,j)}\cdot U[\vec{A}_{(k,j)}/\vec{X}]$.
\item $\sui{n_k}\suj{m_k} \alpha_{(k,i)}\times\beta_{(k,j)}\cdot {T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}]} \ssubt_{\V_k,\Gamma} R_k$.
\end{itemize}
So the lemma holds.
\inductioncase{Case $1_E$}
\[
\prftree[r]{$1_E$}
{\pi = \Gamma \vdash 1\cdot(\ve t)~\ve r: T}
{\Gamma \vdash (\ve t)~\ve r: T}
\]
By Lemma~\ref{lem:sr:scalars}, there exist $R_1, \dots, R_h$, $\mu_1, \dots, \mu_h$ such that
\begin{itemize}
\item $T \equiv \suk{h} \mu_k \cdot R_k$.
\item $\pi_k = \Gamma \vdash (\ve t)~\ve r: R_k$, with $size(\pi) > size(\pi_k)$, for $k \in \{1, \dots, h\}$..
\item $\suk{h} \mu_k = 1$.
\end{itemize}
Since $size(\pi) > size(\pi_k)$, we apply the inductive hypothesis for all $k \in \{1, \dots, h\}$ (and omiting the $k$ index for readability),
so there exist $S_{1}, \dots, S_{p}$, $\eta_{1}, \dots, \eta_{p}$, $\V_{1},\dots,\V_{p}$ such that
$R \equiv \sug{q}{p} \eta_{q} \cdot S_{q}$, $\sug{q}{p} \eta_{q} = 1$ and for all $q \in \{1,\dots,p\}$,
\begin{itemize}
\item $\Gamma\vdash\ve t: \sui{n_{q}}{\alpha_{(q,i)} \cdot\forall\vec{X}.(U\to T_{(q,i)})}$.
\item $\Gamma\vdash\ve r: \suj{m_{q}}\beta_{(q,j)}\cdot U[\vec{A}_{(q,j)}/\vec{X}]$.
\item $\sui{n_{q}}\suj{m_{q}} \alpha_{(q,i)}\times\beta_{(q,j)}\cdot {T_{(q,i)}[\vec{A}_{(q,j)}/\vec{X}]} \ssubt_{\V_{q},\Gamma} S_{q}$.
\end{itemize}
Then
\[
T \equiv \suk{h} \mu_k \cdot R_k \equiv \suk{h} \mu_k \cdot \sug{q}{p_k} \eta_{(k,q)} \cdot S_{(k,q)} \equiv \suk{h}\sug{q}{p_k} (\mu_k \times \eta_{(k,q)}) \cdot S_{(k,q)}
\]
Finally, we must prove that $\suk{h}\sug{q}{p_k} (\mu_k \times \eta_{(k,q)}) = 1$,
\[
\suk{h}\sug{q}{p_k} (\mu_k \times \eta_{(k,q)}) = \suk{h} \mu_k \cdot \underbrace{\sug{q}{p_k} \eta_{(k,q)}}_{=~1} = \suk{h} \mu_k = 1
\]
\inductioncase{Case $\forall_I$}
\[
\prftree[r]{$\forall_I$}
{\pi' = \Gamma \vdash (\ve{t})~\ve{r}: \sug{a}{b}\sigma_a \cdot V_a}
{X \notin \FV{\Gamma}}
{\Gamma \vdash (\ve{t})~\ve{r}: \sug{a}{b}\sigma_a \cdot \forall X. V_a}
\]
By the induction hypothesis there exist $R_1,\dots,R_h$, $\mu_1,\dots,\mu_h$, $\V_1,\dots,\V_h$ such that
$\sug{a}{b}\sigma_{a} \cdot V_a \equiv \suk{h} \mu_k \cdot R_k$, $\suk{h} \mu_k = 1$ and for all $k \in \{1,\dots,h\}$,
\begin{itemize}
\item $\Gamma\vdash\ve t: \sui{n_k}{\alpha_{(k,i)} \cdot\forall\vec{X}.(U\to T_{(k,i)})}$.
\item $\Gamma\vdash\ve r: \suj{m_k}\beta_{(k,j)}\cdot U[\vec{A}_{(k,j)}/\vec{X}]$.
\item $\sui{n_k}\suj{m_k} \alpha_{(k,i)}\times\beta_{(k,j)}\cdot {T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}]} \ssubt_{\V_k,\Gamma} R_k$.
\end{itemize}
By Lemma~\ref{lem:sr:typecharact}, and since $\sug{a}{b}\sigma_{a} \cdot V_a$ does not have any general variable,
then for all $k \in \{1,\dots,h\}$, $R_k \equiv \sug{c}{d_k}\eta_{(k,c)}\cdot W_{(k,c)}$.\\
Hence $\sug{a}{b}\sigma_{a} \cdot V_a \equiv \suk{h}\mu_h \cdot \sug{c}{d_k}\eta_{(k,c)}\cdot W_{(k,c)}$.\\
Without loss of generality, assuming all unit types are distinct (not equivalent),
then by Lemma~\ref{lem:sr:equivforall},
\[
\sug{a}{b}\sigma_a \cdot \forall X. V_a \equiv
\suk{h}\mu_k \cdot \underbrace{\sug{c}{d_k}\eta_{(k,c)}\cdot \forall X.W_{(k,c)}}_{R'_k}
\]
Finally, for all $k \in \{1,\dots,h\}$ we must prove that
$\sui{n_k}\suj{m_k} \alpha_{(k,i)}\times\beta_{(k,j)}\cdot {T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}]} \ssubt_{\V'_k,\Gamma} R'_k$.\\
Notice that $R_k \ssubt_{\V_k \cup \{X\},\Gamma} R'_k$, then by definition of $\ssubt$, taking $\V'_k = \V_k \cup \{X\}$,\\
$\sui{n_k}\suj{m_k} \alpha_{(k,i)}\times\beta_{(k,j)}\cdot {T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}]} \ssubt_{\V'_k,\Gamma} R'_k$.
\inductioncase{Case $\forall_E$}
\[
\prftree[r]{$\forall_E$}
{\Gamma \vdash (\ve{t})~\ve{r}: \sug{a}{b}\sigma_a \cdot \forall X.V_a}
{\Gamma \vdash (\ve{t})~\ve{r}: \sug{a}{b-1}\sigma_a \cdot V_a[A/X]}
\]
By the induction hypothesis there exist $R_1,\dots,R_h$, $\mu_1,\dots,\mu_h$, $\V_1,\dots,\V_h$ such that
$\sug{a}{b}\sigma_a \cdot \forall X.V_a \equiv \suk{h} \mu_k \cdot R_k$,
$\suk{h} \mu_k = 1$ and for all $k \in \{1,\dots,h\}$,
\begin{itemize}
\item $\Gamma\vdash\ve t: \sui{n_k}{\alpha_{(k,i)} \cdot\forall\vec{X}.(U\to T_{(k,i)})}$.
\item $\Gamma\vdash\ve r: \suj{m_k}\beta_{(k,j)}\cdot U[\vec{A}_{(k,j)}/\vec{X}]$.
\item $\sui{n_k}\suj{m_k} \alpha_{(k,i)}\times\beta_{(k,j)}\cdot {T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}]} \ssubt_{\V_k,\Gamma} R_k$.
\end{itemize}
By Lemma~\ref{lem:sr:typecharact}, and since $\sug{a}{b}\sigma_a \cdot \forall X.V_a$ does not have any general variable,
$R_k \equiv \sug{c}{d_k}\eta_{(k,c)}\cdot W_{(k,c)}$.\\
Hence $\sug{a}{b}\sigma_a \cdot \forall X.V_a \equiv \suk{h} \mu_k \cdot \sug{c}{d_k}\eta_{(k,c)}\cdot W_{(k,c)}$.\\
Without loss of generality, we assume that all unit types present at both sides of the equivalence are distinct,
then by Lemma~\ref{lem:sr:equivforall}, for all $k \in \{1,\dots,h\}, c \in \{1,\dots,d_k\}$, there exists $W'_{(k,c)}$
such that $W_{(k,c)} \equiv \forall X.W'_{(k,c)}$, so we have
\[
\sug{a}{b}\sigma_a \cdot \forall X. V_a \equiv
\suk{h}\mu_k \cdot \underbrace{\sug{c}{d_k}\eta_{(k,c)}\cdot \forall X.W'_{(k,c)}}_{R_k}
\]
By the same lemma, we have that
\[
\sug{a}{b}\sigma_a \cdot V_a[A/X] \equiv
\suk{h}\mu_k \cdot \underbrace{\sug{c}{d_k}\eta_{(k,c)}\cdot W'_{(k,c)}[A/X]}_{R'_k}
\]
Finally, for all $k \in \{1,\dots,h\}$ we must prove that
$\sui{n_k}\suj{m_k} \alpha_{(k,i)}\times\beta_{(k,j)}\cdot {T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}]} \ssubt_{\V'_k,\Gamma} R'_k$.\\
Notice that $R_k \ssubt_{\V_k \cup \{X\},\Gamma} R'_k$, then by definition of $\ssubt$, taking $\V'_k = \V_k \cup \{X\}$,\\
$\sui{n_k}\suj{m_k} \alpha_{(k,i)}\times\beta_{(k,j)}\cdot {T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}]} \ssubt_{\V'_k,\Gamma} R'_k$.
\end{proof}
\xrecap{Lemma}{Abstractions}{lem:sr:abs}{
If $\Gamma\vdash\lambda x.\ve t:T$, then there exist $T_1,\dots,T_n$, $R_1,\dots,R_n$, $U_1,\dots,U_n$,
$\alpha_1,\dots,\alpha_n$, $\V_1,\dots,\V_n$ such
that $T \equiv \sui{n} \alpha_i \cdot T_i$, $\sui{n} \alpha_i = 1$ and for all $i \in \{1,\dots,n\}$,
\begin{itemize}
\item $\Gamma,x:U_i\vdash\ve t:R_i$.
\item $U_i \to R_i \ssubt_{\V_i,\Gamma} T_i$.
\end{itemize} } \begin{proof}
By induction on the typing derivation
\inductioncase{Case $\to_I$}
\[
\prftree[r]{$\to_I$}
{\Gamma,x:U\vdash\ve t:R}
{\Gamma\vdash\lambda x.\ve t: U \to R}
\]
Trivial.
\inductioncase{Case $\equiv$}
\[
\prftree[r]{$\equiv$}
{\Gamma\vdash\lambda x.\ve t: R}
{R \equiv T}
{\Gamma \vdash \lambda x.\ve t: T}
\]
By the induction hypothesis, there exist $T_1,\dots,T_n$, $R_1,\dots,R_n$, $U_1,\dots,U_n$,
$\alpha_1,\dots,\alpha_n$, $\V_1,\dots,\V_n$ such
that $T \equiv R \equiv \sui{n} \alpha_i \cdot T_i$, $\sui{n} \alpha_i = 1$ and for all $i \in \{1,\dots,n\}$,
\begin{itemize}
\item $\Gamma,x:U_i\vdash\ve t:R_i$.
\item $U_i \to R_i \ssubt_{\V_i,\Gamma} T_i$.
\end{itemize}
So the lemma holds.
\inductioncase{Case $1_E$}
\[
\prftree[r]{$1_E$}
{\pi = \Gamma\vdash 1\cdot(\lambda x.\ve t): T}
{\Gamma\vdash \lambda x.\ve t: T}
\]
By Lemma~\ref{lem:sr:scalars}, there exist $R_1, \dots, R_m$, $\beta_1, \dots, \beta_m$ such that
\begin{itemize}
\item $T \equiv \suj{m}\beta_i \cdot R_j$.
\item $\pi_i = \Gamma \vdash \ve{t}: R_j$, with $size(\pi) > size(\pi_j)$, for $j \in \{1, \dots, n\}$.
\item $\suj{n} \beta_i = 1$.
\end{itemize}
Since $size(\pi) > size(\pi_j)$, by induction hypothesis, for all $j \in \{1, \dots, n\}$
there exist $S_{(j,1)},\dots,S_{(j,n_j)}$, $P_{(j,1)},\dots,P_{(j,n_j)}$, $U_{(j,1)},\dots,U_{(j,n_j)}$,
$\eta_{(j,1)},\dots,\eta_{(j,n_j)}$, $\V_{(j,1)},\dots,\V_{(j,n_j)}$ such
that $R_j \equiv \sui{n_j} \eta_{(j,i)} \cdot S_{(j,i)}$, $\sui{n_j} \eta_{(j,i)} = 1$ and for all $i \in \{1,\dots,n_j\}$,
\begin{itemize}
\item $\Gamma,x:U_{(j,i)}\vdash\ve t:P_{(j,i)}$.
\item $U_{(j,i)} \to P_{(j,i)} \ssubt_{\V_{(j,i)},\Gamma} S_{(j,i)}$.
\end{itemize}
Then we have
\[
T \equiv \suj{m} \beta_j \cdot R_j \equiv \suj{m} \beta_j \cdot \sui{n_j} \eta_{(j,i)} \cdot S_{(j,i)} \equiv \suj{m}\sui{n_j} (\beta_j \times \eta_{(j,i)}) \cdot S_{(j,i)}
\]
Finally, we must prove that $\suj{m}\sui{n_j} (\beta_j \times \eta_{(j,i)}) = 1$:
\[
\suj{m}\sui{n_j} (\beta_j \times \eta_{(j,i)}) = \suj{m} \beta_j \cdot \underbrace{\sui{n_j} \eta_{(j,i)}}_{=~1} = \suj{m} \beta_j = 1
\]
\inductioncase{Case $\forall_I$}
\[
\prftree[r]{$\forall_I$}
{\Gamma \vdash \lambda x.\ve t: \sui{n} \alpha_i \cdot U_i}
{\prfassumption{X \notin \FV{\Gamma}}}
{\Gamma \vdash \lambda x.\ve t: \sui{n} \alpha_i \cdot \forall X.U_i}
\]
By the induction hypothesis, there exist $T_1,\dots,T_m$, $R_1,\dots,R_m$, $V_1,\dots,V_m$,
$\alpha_1,\dots,\alpha_m$, $\V_1,\dots,\V_m$ such
that $\sui{n} \alpha_i \cdot U_i \equiv \suj{m} \mu_j \cdot T_j$, $\suj{m} \mu_i = 1$ and for all $j \in \{1,\dots,m\}$,
\begin{itemize}
\item $\Gamma,x:V_j\vdash\ve t:R_j$.
\item $V_j \to R_j \ssubt_{\V_j,\Gamma} T_j$.
\end{itemize}
By Lemma~\ref{lem:sr:typecharact}, and since $\sui{n} \alpha_i \cdot U_i$ does not have any general variable $\vara{X}$, then
$T_i \equiv \sug{k}{h_j} \beta_{(j,k)} \cdot W_{(j,k)}$.
Hence $\sui{n} \alpha_i \cdot U_i \equiv \suj{m} \mu_i \cdot \sug{k}{h_j} \beta_{(j,k)} \cdot W_{(j,k)}$.
Without loss of generality, assuming all unit types are distinct (not equivalent),
then by Lemma~\ref{lem:sr:equivforall},
\[
\sui{n} \alpha_i \cdot U_i \equiv
\suj{m} \mu_i \cdot \underbrace{\sug{k}{h_j} \beta_{(j,k)} \cdot \forall X.W_{(j,k)}}_{T'_j}
\]
Finally, we must prove that $V_j \to R_j \ssubt_{\V'_j,\Gamma} T'_j$ for some $\V'_j$.
Since $V_j \to R_j \ssubt_{\V_j,\Gamma} T_j$ and $T_j \ssubt_{\V''_j,\Gamma} T'_j$, then by $\ssubt$ and using $\V'_j = \V_j \cup \V''_j$,
we conclude that $V_j \to R_j \ssubt_{\V'_j,\Gamma} T'_j$.
\inductioncase{Case $\forall_E$}
\[
\prftree[r]{$\forall_E$}
{\Gamma \vdash \lambda x.\ve t: \sui{n} \alpha_i \cdot \forall X.U_i}
{\Gamma \vdash \lambda x.\ve t: \sui{n} \alpha_i \cdot U_i[A/X]}
\]
By the induction hypothesis, there exist $T_1,\dots,T_m$, $R_1,\dots,R_m$, $V_1,\dots,V_m$,
$\alpha_1,\dots,\alpha_m$, $\V_1,\dots,\V_m$ such
that $\sui{n} \alpha_i \cdot \forall X.U_i \equiv \suj{m} \mu_j \cdot T_j$, $\suj{m} \mu_i = 1$ and for all $j \in \{1,\dots,m\}$,
\begin{itemize}
\item $\Gamma,x:V_j\vdash\ve t:R_j$.
\item $V_j \to R_j \ssubt_{\V_j,\Gamma} T_j$.
\end{itemize}
By Lemma~\ref{lem:sr:typecharact}, and since $\sui{n} \alpha_i \cdot U_i$ does not have any general variable $\vara{X}$, then
$T_i \equiv \sug{k}{h_j} \beta_{(j,k)} \cdot W_{(j,k)}$.
Hence $\sui{n} \alpha_i \cdot U_i \equiv \suj{m} \mu_i \cdot \sug{k}{h_j} \beta_{(j,k)} \cdot W_{(j,k)}$.
Without loss of generality, assuming all unit types are distinct (not equivalent),
then by Lemma~\ref{lem:sr:equivforall}, for all $j \in \{1, \dots, m\}$, $k \in \{1,\dots,h_j\}$,
there exists $W'_{(j,k)}$ such that $W_{(j,k)} \equiv \forall X.W'_{(j,k)}$.
Then,
\[
\sui{n} \alpha_i \cdot \forall X.U_i \equiv
\suj{m} \mu_i \cdot \underbrace{\sug{k}{h_j} \beta_{(j,k)} \cdot \forall X.W'_{(j,k)}}_{T_j}
\]
By the same lemma, we have that
\[
\sui{n} \alpha_i \cdot U_i[A/X] \equiv
\suj{m} \mu_i \cdot \underbrace{\sug{k}{h_j} \beta_{(j,k)} \cdot W'_{(j,k)}[A/X]}_{T'_j}
\]
Finally, we must prove that $V_j \to R_j \ssubt_{\V'_j,\Gamma} T'_j$ for some $\V'_j$.
Since $V_j \to R_j \ssubt_{\V_j,\Gamma} T_j$ and $T_j \ssubt_{\V''_j,\Gamma} T'_j$, then by $\ssubt$ and using $\V'_j = \V_j \cup \V''_j$,
we conclude that $V_j \to R_j \ssubt_{\V'_j,\Gamma} T'_j$.
\end{proof}
\xrecap{Lemma}{Basis terms}{lem:sr:basevectors}{
For any context $\Gamma$, type $T$ and basis term $\ve{b}$, if
$\Gamma\vdash\ve{b}: T$ there exist $U_1, \dots, U_n$, $\alpha_1, \dots, \alpha_n$ such that
\begin{itemize}
\item $T \equiv \sui{n} \alpha_i \cdot U_i$.
\item $\Gamma\vdash\ve{b}: U_i$, for $i \in \{1,\dots,n\}$.
\item $\sui{n} \alpha_i = 1$.
\end{itemize} } \begin{proof}
By induction on the typing derivation.
\inductioncase{Case $ax$}
\[
\prftree[r]{$ax$}
{}
{\Gamma, x:{U}\vdash x:{U}}
\raisebox{9pt}{\qquad\text{and}\qquad}
\prftree[r]{$\to_I$}
{\Gamma,x:U\vdash\ve t:T}
{\Gamma\vdash\lambda x.\ve t: U \to T}
\]
Trivial cases.
\inductioncase{Case $\equiv$}
\[
\prftree[r]{$\equiv$}
{\Gamma\vdash\ve{b}: R}
{R \equiv T}
{\Gamma \vdash \ve{b}: T}
\]
By the induction hypothesis, there exist $U_1, \dots, U_n$, $\alpha_1, \dots, \alpha_n$ such that
\begin{itemize}
\item $T \equiv R \equiv \sui{n} \alpha_i \cdot U_i$.
\item $\Gamma\vdash\ve{b}: U_i$, for $i \in \{1,\dots,n\}$.
\item $\sui{n} \alpha_i = 1$.
\end{itemize}
So the lemma holds.
\inductioncase{Case $1_E$}
\[
\prftree[r]{$1_E$}
{\pi = \Gamma\vdash 1\cdot\ve{b}: T}
{\Gamma\vdash \ve{b}: T}
\]
By Lemma~\ref{lem:sr:scalars}, there exist $R_1, \dots, R_m$, $\beta_1, \dots, \beta_m$ such that
\begin{itemize}
\item $T \equiv \suj{m}\beta_j \cdot R_j$.
\item $\suj{m} \beta_j = 1$, and $\pi_j = \Gamma \vdash \ve{b}: R_j$ with $size(\pi) > size(\pi_j)$ for $j = \{1, \dots, m\}$.
\item $\suj{m} \beta_j = 1$.
\end{itemize}
Since $size(\pi) > size(\pi_j)$, by induction hypothesis, for all $j = \{1, \dots, m\}$ there exist
$U_{(j,1)},\dots,U_{(j,n_j)}$, $\alpha_{(j,1)},\dots,\alpha_{(j,n_j)}$ such that
\begin{itemize}
\item $R_j \equiv \sui{n_j} \alpha_{(j,i)} \cdot U_{(j,i)}$.
\item $\Gamma\vdash\ve{b}: U_{(j,i)}$, for $i \in \{1,\dots,n_j\}$.
\item $\sui{n_j} \alpha_{(j,i)} = 1$.
\end{itemize}
Then
\[
T \equiv \suj{m} \beta_j \cdot R_j \equiv \suj{m} \beta_j \cdot \sui{n_j} \alpha_{(j,i)} \cdot U_{(j,i)} \equiv \suj{m}\sui{n_j} (\beta_j \times \alpha_{(j,i)}) \cdot U_{(j,i)}
\]
Finally, we must prove that $\suj{m}\sui{n_j} (\beta_j \times \alpha_{(j,i)}) = 1$:
\[
\suj{m}\sui{n_j} (\beta_j \times \alpha_{(j,i)}) = \suj{m} \beta_j \cdot \underbrace{\sui{n_j} \alpha_{(j,i)}}_{=~1} = \suj{m} \beta_j = 1
\]
\inductioncase{Case $\forall$}
\[
\prftree[r]{$\forall$}
{\Gamma \vdash \ve{b}: \sui{n} \alpha_i \cdot U_i}
{\Gamma \vdash \ve{b}: \sui{n} \alpha_i \cdot V_i}
\]
$\forall$-rules ($\forall_I$ and $\forall_E$) both have the same structure as shown above.\\
In both cases, by the induction hypothesis, there exist $W_1, \dots, W_m$, $\beta_1, \dots, \beta_m$ such that
\begin{itemize}
\item $\sui{n} \alpha_i \cdot U_i \equiv \suj{m} \beta_j \cdot W_j$.
\item $\Gamma\vdash\ve{b}: W_j$, for $j \in \{1,\dots,m\}$.
\item $\suj{m} \beta_j = 1$.
\end{itemize}
Without loss of generality, we assume that all unit types present at both sides of the equivalence are distinct,
so by Lemma~\ref{lem:sr:equivdistinctscalars},
then $m = n$ and there exists a permutation $p$ of $m$ such that
for all $i \in \{1,\dots,n\}$, then $U_i = W_{p(i)}$ and $\alpha_i = \beta_{p(i)}$,
which means that $\sui{n} \alpha_i = 1$.
Finally, by applying the corresponding $\forall$ rule for all $i \in \{1,\dots,n\}$, we have
\[
\prftree[r]{$\forall$}
{\Gamma \vdash \ve{b}: U_i}
{\Gamma \vdash \ve{b}: V_i}
\] \end{proof}
\xrecap{Lemma}{Substitution lemma}{lem:sr:substitution}{
For any term ${\ve t}$, basis term $\ve b$, term variable $x$, context $\Gamma$, types $T$, $U$, type variable $X$ and type $A$, where $A$ is a unit type if $X$ is a unit variable, otherwise $A$ is a general type, we have,
\begin{enumerate}
\item If $\Gamma\vdash\ve{t}: T$, then $\Gamma[A/X]\vdash\ve{t}: T[A/X]$;
\item If $\Gamma,x:U\vdash\ve t:T$ and $\Gamma\vdash\ve b:U$, then $\Gamma\vdash\ve t[\ve b/x]: T$.
\end{enumerate} } \begin{proof}~
\textleadbydots{Item (1)}
Induction on the typing derivation.
\inductioncase{Case $ax$}
\[
\prftree[r]{$ax$}
{}
{\Gamma, x: {U}: \vdash x: U}
\]
Notice that ${\Gamma[A/X],x:U[A/X]\vdash x:U[A/X]}$ can also be derived with the same rule.
\inductioncase{Case $\to_I$}
\[
\prftree[r]{$\to_I$}
{\Gamma,x:U\vdash\ve t:T}
{\Gamma\vdash\lambda x.\ve t:U\to T}
\]
By the induction hypothesis $\Gamma[A/X],x:U[A/X]\vdash\ve t:T[A/X]$, so by rule $\to_I$, $\Gamma[A/X]\vdash\lambda x.\ve t:U[A/X]\to T[A/X]=(U\to T)[A/X]$.
\inductioncase{Case $\to_E$}
\[
\prftree[r]{$\to_E$}
{\Gamma\vdash\ve t:\sui{n}\alpha_i\cdot\forall\vec Y.(U\to T_i)}
{\Gamma\vdash\ve r:\suj{m}\beta_j\cdot U[\vec B_j/\vec Y]}
{\Gamma\vdash(\ve t)~\ve r:\sui{n}\suj{m}\alpha_i\times\beta_j\cdot T_i[\vec B_j/\vec Y]}
\]
By the induction hypothesis
$\Gamma[A/X]\vdash\ve t:(\sui{n}\alpha_i\cdot\forall\vec Y.(U\to T_i))[A/X]$ and this type is equal to $\sui{n}\alpha_i\cdot\forall\vec Y.(U[A/X]\to T_i[A/X])$.
Also $\Gamma[A/X]\vdash\ve r:(\suj{m}\beta_j\cdot U[\vec B_j/\vec Y])[A/X]=
\suj{m}\beta_j\cdot U[\vec B_j/\vec Y][A/X]$.
Since $\vec Y$ is bound, we can consider $\vec{Y} \notin \FV{A}$.
Hence $U[\vec B_j/\vec Y][A/X]=U[A/X][\vec B_j[A/X]/\vec Y]$, and so, by rule $\to_E$,
\begin{align*}
\Gamma[A/X]\vdash(\ve t)~\ve r&:\sui{n}\suj{m}\alpha_i\times\beta_j\cdot T_i[A/X][\vec B_j[A/X]/\vec Y]\\
&=\left(\sui{n}\suj{m}\alpha_i\times\beta_j\cdot T_i[\vec B_j/\vec Y]\right)[A/X]
\end{align*}
\inductioncase{Case $\forall_I$}
\[
\prftree[r]{$\forall_I$}
{\Gamma\vdash\ve t:\sui{n}\alpha_i\cdot U_i}
{Y\notin\FV{\Gamma}}
{\Gamma\vdash\ve{t}:\sui{n}\alpha_i\cdot \forall Y.U_i }
\]
By the induction hypothesis,
$\Gamma[A/X]\vdash\ve t:(\sui{n}\alpha_i\cdot
U_i)[A/X]=\sui{n}\alpha_i\cdot U_i[A/X]$.
Then, by
rule $\forall_I$, $\Gamma[A/X]\vdash\ve
t:\sui{n}\alpha_i\cdot \forall Y.U_i[A/X]=(\sui{n}\alpha_i\cdot \forall Y.U_i)[A/X]$.
Since $Y$ is bound, we can consider $Y \notin \FV{A}$.
\inductioncase{Case $\forall_E$}
\[
\prftree[r]{$\forall_E$}
{\Gamma\vdash\ve t:\sui{n}\alpha_i\cdot \forall Y.U_i}
{\Gamma\vdash\ve t:\sui{n}\alpha_i\cdot U_i[B/Y]}
\]
By the induction
hypothesis $\Gamma[A/X]\vdash\ve
t:(\sui{n}\alpha_i\cdot \forall Y.U_i)[A/X]=\sui{n}\alpha_i\cdot \forall Y.U_i[A/X]$.
Since $Y$ is bound, we can
consider $Y \notin \FV{A}$.
Then by rule $\forall_E$,
${\Gamma[A/X]\vdash\ve t:\sui{n}\alpha_i\cdot U_i[A/X][B/Y]}$.
We can consider $X\notin\FV{B}$ (in
other case, just take $B[A/X]$ in the $\forall$-elimination), hence
\[
\sui{n}\alpha_i\cdot U_i[A/X][B/Y] =\sui{n}\alpha_i\cdot U_i[B/Y][A/X] =\left(\sui{n}\alpha_i\cdot U_i[B/Y]\right)[A/X]
\]
\inductioncase{Case $S$}
\[
\prftree[r]{$S$}
{\Gamma\vdash\ve t:T_i~\forall i \in \{1,\dots,n\}}
{\Gamma\vdash \left(\sui{n} \alpha_i\right) \cdot\ve t: \sui{n} \alpha_i \cdot T_i}
\]
By the induction hypothesis, for all $i \in \{1, \dots, n\}$, $\Gamma[A/X]\vdash\ve t:T_i[A/X]$,
so by rule $S$,
$\Gamma[A/X]\vdash \left(\sui{n} \alpha_i\right) \cdot\ve t:\sui{n} \alpha_i \cdot T_i[A/X]={(\sui{n} \alpha_i \cdot T_i)[A/X]}$.
\inductioncase{Case $+_I$}
\[
\prftree[r]{$+_I$}
{\Gamma\vdash\ve t:T}
{\Gamma\vdash\ve r:R}
{\Gamma\vdash\ve t+\ve r:T+R}
\]
By the induction hypothesis $\Gamma[A/X]\vdash\ve t:T[A/X]$ and $\Gamma[A/X]\vdash\ve r:R[A/X]$, so by rule $+_I$, ${\Gamma[A/X]\vdash\ve t+\ve r:T[A/X]+R[A/X]=(T+R)[A/X]}$.
\inductioncase{Case $\equiv$}
\[
\prftree[r]{$\equiv$}
{\Gamma\vdash\ve t:T}
{T\equiv R}
{\Gamma\vdash\ve t:R}
\]
By the induction hypothesis $\Gamma[A/X]\vdash\ve t:T[A/X]$, and since $T\equiv R$, then $T[A/X]\equiv R[A/X]$, so by rule $\equiv$, $\Gamma[A/X]\vdash\ve t:R[A/X]$.
\inductioncase{Case $1_E$}
\[
\prftree[r]{$1_E$}
{\Gamma\vdash1\cdot\ve t:T}
{\Gamma\vdash\ve t: T}
\]
By the induction hypothesis $\Gamma[A/X]\vdash1\cdot\ve t:T[A/X]$.
By rule $1_E$, $\Gamma[A/X]\vdash\ve t:T[A/X]$.
\textleadbydots{Item (2)}
We proceed by induction on the typing derivation of $\Gamma,x:U\vdash\ve t:T$.
\inductioncase{Case $ax$}
\[
\prftree[r]{$ax$}
{\Gamma,x:U\vdash\ve t:T}
\]
Cases:
\begin{itemize}
\item $\ve t=x$, then $T=U$, and so $\Gamma\vdash\ve t[\ve b/x]:T$
and $\Gamma\vdash\ve b:U$ are the same sequent.
\item $\ve t=y$.
Notice that $y[\ve b/x]=y$.
\arxiv{By Lemma~\ref{lem:sr:weakening}}\conf{By weakening} $\Gamma,x:U\vdash y:T$ implies
$\Gamma\vdash y:T$.
\end{itemize}
\inductioncase{Case $\to_I$}
\[
\prftree[r]{$\to_I$}
{\Gamma,x:U,y:V\vdash \ve{r}:R}
{\Gamma,x:U\vdash\lambda x.\lambda y.\ve{r}:V\to R}
\]
Since our system admits weakening \arxiv{(Lemma~\ref{lem:sr:weakening})}, the sequent $\Gamma,y:V\vdash\ve b:U$
is derivable.
Then by the induction hypothesis,
$\Gamma,y:V\vdash\ve r[\ve b/x]:R$, from where, by
rule $\to_I$, we obtain $\Gamma\vdash\lambda y.\ve r[\ve
b/x]:V\to R$.
We conclude, since
$\lambda y.\ve r[\ve b/x]=(\lambda y.\ve r)[\ve b/x]$.
\inductioncase{Case $\to_E$}
\[
\prftree[r]{$\to_E$}
{\Gamma,x:U\vdash\ve r:\sui{n}\alpha_i\cdot\forall\vec Y.(V\to T_i)}
{\Gamma,x:U\vdash\ve u:\suj{m}\beta_j\cdot V[\vec B/\vec Y]}
{\Gamma,x:U\vdash(\ve r)~\ve u:\sui{n}\suj{m}\alpha_i\times\beta_j\cdot R_i[\vec B/\vec Y]}
\]
By the induction hypothesis,
$\Gamma\vdash\ve r[\ve b/x]:\sui{n}\alpha_i\cdot\forall\vec Y.(V\to R_i)$
and
$\Gamma\vdash\ve u[\ve b/x]:\suj{m}\beta_j\cdot V[\vec B/\vec Y]$.
Then, by rule $\to_E$,
$\Gamma
\vdash \ve r[\ve b/x]~\ve u[\ve b/x]:
\sui{n}\suj{m}\alpha_i\times\beta_j\cdot R_i[\vec B/\vec Y]$.
\inductioncase{Case $\forall_I$}
\[
\prftree[r]{$\forall_I$}
{\Gamma,x:U\vdash\ve t:\sui{n}\alpha_i\cdot V_i}
{Y\notin\FV{\Gamma}\cup\FV{U}}
{\Gamma,x:U\vdash\ve{t}:\sui{n-1}\alpha_i\cdot V_i + \alpha_n\cdot\forall Y.V_n}
\]
By the induction hypothesis,
$\Gamma\vdash\ve t[\ve b/x]:\sui{n}\alpha_i\cdot V_i$.
Then by rule
$\forall_I$, $\Gamma\vdash\ve t[\ve
b/x]:\sui{n-1}\alpha_i\cdot V_i + \alpha_n\cdot\forall Y.V_n$.
\inductioncase{Case $\forall_E$}
\[
\prftree[r]{$\forall_E$}
{\Gamma,x:U\vdash\ve t:\sui{n-1}\alpha_i\cdot V_i + \alpha_n \cdot \forall Y.V_n}
{\Gamma,x:U\vdash\ve t:\sui{n-1}\alpha_i\cdot U_i + \alpha_n\cdot U_n[B/Y]}
\]
By the induction hypothesis, $\Gamma\vdash\ve t[\ve
b/x]:\sui{n-1}\alpha_i\cdot V_i + \alpha_n \cdot \forall Y.V_n$.
By rule $\forall_E$, $\Gamma\vdash\ve t[\ve
b/x]:\sui{n-1}\alpha_i\cdot V_i + \alpha_n\cdot V_n[B/Y]$.
\inductioncase{Case $S$}
\[
\prftree[r]{$S$}
{\Gamma,x:U\vdash\ve{t}:T_i~\forall i \in \{1,\dots,n\}}
{\Gamma,x:U\vdash \left(\sui{n} \alpha_i\right) \cdot \ve{t}: \sui{n} \alpha_i \cdot T_i}
\]
By the
induction hypothesis, for all $i \in \{1, \dots, n\}$, $\Gamma\vdash\ve t[\ve b/x]:T_i$.
Then by
rule $S$, $\Gamma\vdash \left(\sui{n} \alpha_i\right) \cdot \ve{t}[\ve b/x]: \sui{n} \alpha_i \cdot T_i$.
Notice that $\left(\sui{n} \alpha_i\right) \cdot \ve{t}[\ve b/x]=(\left(\sui{n} \alpha_i\right) \cdot \ve{t})[\ve b/x]$.
\inductioncase{Case $+_I$}
\[
\prftree[r]{$+_I$}
{\Gamma,x:U\vdash\ve r:R}
{\Gamma,x:U\vdash\ve u:S}
{\Gamma,x:U\vdash\ve r+\ve u:R+S}
\]
By the
induction hypothesis, $\Gamma\vdash\ve r[\ve b/x]:R$
and $\Gamma\vdash\ve u[\ve b/x]:S$.
Then by
rule $+_I$, $\Gamma\vdash\ve r[\ve b/x]+\ve u[\ve b/x]:R+S$.
Notice that $\ve r[\ve b/x]+\ve
u[\ve b/x]=(\ve r+\ve u)[\ve b/x]$.
\inductioncase{Case $\equiv$}
\[
\prftree[r]{$\equiv$}
{\Gamma,x:U\vdash\ve t:T}
{T\equiv R}
{\Gamma,x:U\vdash\ve t:R}
\]
By the induction hypothesis, $\Gamma\vdash\ve t[\ve b/x]:R$.
Hence, by rule $\equiv$, $\Gamma\vdash\ve t[\ve b/x]:T$.
\inductioncase{Case $1_E$}
\[
\prftree[r]{$1_E$}
{\Gamma,x:U\vdash1\cdot\ve t:T}
{\Gamma,x:U\vdash\ve t: T}
\]
By the induction hypothesis, $\Gamma\vdash 1 \cdot \ve t[\ve b/x]:R$.
Hence, by rule $1_E$, $\Gamma\vdash\ve t[\ve b/x]:T$. \end{proof}
\xrecap{Theorem}{Subject Reduction}{thm:sr}{
For any terms $\ve{t}, \ve{t}'$, any context $\Gamma$ and any type $T$, if $\ve{t} \to \ve{t}'$ and $\Gamma \vdash \ve{t}: T$, then $\Gamma \vdash \ve{t}': T$. } \begin{proof}
Let $\ve{t} \to \ve{t}'$ and $\Gamma \vdash \ve{t}: T$, we proceed by induction on the rewrite relation:
\inductioncase{Group E}
\inductioncase{Case $1\cdot \ve{t}\to \ve{t}$}
Consider $\Gamma \vdash 1\cdot \ve{t}: T$, then by $1_E$ rule, then $\Gamma \vdash \ve{t}: T$.
\inductioncase{Case $\alpha\cdot(\beta\cdot\ve{t})\to (\alpha\times\beta)\cdot\ve{t}$}
Consider $\pi = \Gamma \vdash \alpha\cdot(\beta\cdot\ve{t}): T$,
then by applying Lemma~\ref{lem:sr:scalars}, there exist $R_1, \dots, R_n$, $\alpha_1, \dots, \alpha_n$ such that
\begin{itemize}
\item $T \equiv \sui{n}\alpha_i \cdot R_i$.
\item $\pi_i = \Gamma \vdash \beta\cdot\ve{t}: R_i$, with $size(\pi) > size(\pi_i)$, for $i \in \{1, \dots, n\}$.
\item $\sui{n} \alpha_i = \alpha$.
\end{itemize}
By applying Lemma~\ref{lem:sr:scalars} for all $i \in \{1,\dots,n\}$,
there exist $S_{(i,1)}, \dots, S_{(i,m_i)}$, $\beta_{(i,1)}, \dots, \beta_{(i,m_i)}$ such that
\begin{itemize}
\item $R_i \equiv \suj{m_i}\beta_{(i,j)} \cdot S_{(i,j)}$.
\item $\pi_{(i,j)} = \Gamma \vdash \ve{t}: S_{(i,j)}$, with $size(\pi_i) > size(\pi_{(i,j)})$, for $j \in \{1, \dots, m_i\}$.
\item $\suj{m_i} \beta_{(i,j)} = \beta$.
\end{itemize}
Notice that
\[
\sui{n}\alpha_i \cdot \underbrace{\suj{m_i}\beta_{(i,j)}}_{\beta} = \sui{n}\alpha_i \cdot \beta\\
= \beta \cdot \underbrace{\sui{n}\alpha_i}_{\alpha} = \beta \times \alpha = \alpha \times \beta
\]
Then applying the $S$ rule,
\[
\prftree[r]{$S$}
{\Gamma \vdash \ve{t}: S_{(i,j)}~\forall i \in \{1,\dots,n\},~\forall j \in \{1,\dots,m_i\}}
{\Gamma \vdash (\alpha \times \beta) \cdot \ve{t}: \sui{n}\alpha_i \cdot \suj{m_i}\beta_{(i,j)} \cdot S_{(i,j)}}
\]
Since for all $i \in \{1,\dots,n\}$, $\suj{m_i}\beta_{(i,j)} \cdot S_{(i,j)} \equiv R_i$, and since
$\sui{n}\alpha_i \cdot R_i \equiv T$, then by $\equiv$ rule, we conclude that $\Gamma \vdash (\alpha\times\beta)\cdot\ve{t}:T$.
\inductioncase{Case $\alpha\cdot(\ve{t} + \ve{r})\to \alpha\cdot\ve{t} + \alpha\cdot\ve{r}$}
Consider $\Gamma \vdash \alpha\cdot(\ve{t} + \ve{r}): T$,
then by Lemma~\ref{lem:sr:scalars} there exist $R_1, \dots, R_n$, $\alpha_1, \dots, \alpha_n$ such that
\begin{itemize}
\item $T \equiv \sui{n}\alpha_i \cdot R_i$.
\item $\pi_i = \Gamma \vdash \ve{t} + \ve{r}: R_i$, with $size(\pi) > size(\pi_i)$, for $i \in \{1, \dots, n\}$.
\item $\sui{n} \alpha_i = \alpha$.
\end{itemize}
Since $size(\pi) > size(\pi_i)$, then by Lemma~\ref{lem:sr:sums}, for all $i \in \{1,\dots,n\}$,
there exist $S_{i,1}, S_{i,2}$ such that
\begin{itemize}
\item $\Gamma \vdash \ve{t}: S_{(i,1)}$.
\item $\Gamma \vdash \ve{r}: S_{(i,2)}$.
\item $S_{(i,1)} + S_{(i,2)} \equiv R_i$.
\end{itemize}
Then applying the $S$ rule,
\begin{align*}
\prftree[r]{$S$}
{\Gamma \vdash \ve{t}: S_{(i,1)}~\forall i \in \{1,\dots,n\}}
{\Gamma \vdash \alpha \cdot \ve{t}: \sui{n} \alpha_i \cdot S_{(i,1)}}
\qquad
\prftree[r]{$S$}
{\Gamma \vdash \ve{r}: S_{(i,2)}~\forall i \in \{1,\dots,n\}}
{\Gamma \vdash \alpha \cdot \ve{r}: \sui{n} \alpha_i \cdot S_{(i,2)}}
\end{align*}
By applying the $+_I$ rule,
\[
\prftree[r]{$+_I$}
{\Gamma \vdash \alpha\cdot\ve{t}: \sui{n} \alpha_i \cdot S_{(i,1)}}
{\Gamma \vdash \alpha\cdot\ve{r}: \sui{n} \alpha_i \cdot S_{(i,2)}}
{\Gamma \vdash \alpha\cdot\ve{t} + \alpha\cdot\ve{r}: \sui{n} \alpha_i \cdot S_{(i,1)} + \sui{n} \alpha_i \cdot S_{(i,2)}}
\]
Notice that
\[
\sui{n} \alpha_i \cdot S_{(i,1)} + \sui{n} \alpha_i \cdot S_{(i,2)} \equiv \sui{n} \alpha_i \cdot (S_{(i,1)} + S_{(i,2)})
\equiv \sui{n} \alpha_i \cdot R_i \equiv T
\]
Finally, applying the $\equiv$ rule, we conclude that $\Gamma \vdash \alpha\cdot\ve{t} + \alpha\cdot\ve{r}: T$.
\inductioncase{Group F}
\inductioncase{Case $\alpha\cdot\ve{t} + \beta\cdot\ve{t}\to (\alpha + \beta)\cdot \ve{t}$}
Consider ${\Gamma \vdash \alpha\cdot\ve{t} + \beta\cdot\ve{t}: T}$.\\
For simplicity, we rename $\alpha = \mu_1$ and $\beta = \mu_2$, then by Lemma~\ref{lem:sr:sums} there exist $S_1, S_2$ such that
\begin{itemize}
\item ${\pi_1 = \Gamma \vdash \mu_1\cdot\ve{t}: S_{1}}$.
\item ${\pi_2 = \Gamma \vdash \mu_2\cdot\ve{t}: S_{2}}$.
\item ${S_{1} + S_{2} \equiv T}$.
\end{itemize}
And by Lemma~\ref{lem:sr:scalars}, for $k = 1, 2$,
there exist $R_{(k,1)},\dots,R_{(k,n_k)}$, $\gamma_{(k,1)},\dots,\gamma_{(k,n_k)}$ such that
\begin{itemize}
\item $S_k \equiv \sui{n_k}\gamma_{(k,i)} \cdot R_{(k,i)}$.
\item $\pi_{(k,i)} = \Gamma \vdash \ve{t}: R_{(k,i)}$, with $size(\pi_k) > size(\pi_{(k,i)})$, for $i \in \{1, \dots, n_k\}$.
\item $\sui{n_k} \gamma_{(k,i)} = \mu_k$.
\end{itemize}
Notice that
\[
\underbrace{\sui{n_1} \mu_{(1,i)}}_{=~\mu_1} + \underbrace{\sui{n_2} \mu_{(2,i)}}_{=~\mu_2} = \mu_1 + \mu_2 = \alpha + \beta
\]
Then applying the $S$ rule,
\[
\prftree[r]{$S$}
{\Gamma \vdash \ve{t}: R_{(1,i)}~\forall i \in \{1,\dots,n_1\}}
{\Gamma \vdash \ve{t}: R_{(2,i)}~\forall i \in \{1,\dots,n_2\}}
{\Gamma \vdash (\alpha + \beta) \cdot \ve{t}: \sui{n_1} \mu_{(1,i)} \cdot R_{(1,i)} + \sui{n_2} \mu_{(2,i)} \cdot R_{(2,i)}}
\]
We also know that
\[
\sui{n_1} \mu_{(1,i)} \cdot R_{(1,i)} \equiv S_1\qquad
\sui{n_2} \mu_{(2,i)} \cdot R_{(2,i)} \equiv S_2\qquad
S_1 + S_2 \equiv T
\]
Finally, we conclude by $\equiv$ rule that ${\Gamma \vdash (\alpha + \beta)\cdot \ve{t}: T}$.
\inductioncase{Case $\alpha\cdot\ve{t} + \ve{t}\to (\alpha + 1)\cdot \ve{t}$}
Consider ${\Gamma \vdash \alpha\cdot\ve{t} + \ve{t}: T}$, then by Lemma~\ref{lem:sr:sums} there exist $S_1, S_2$ such that
\begin{itemize}
\item ${\pi = \Gamma \vdash \alpha\cdot\ve{t}: S_{1}}$.
\item ${\Gamma \vdash \ve{t}: S_{2}}$.
\item ${S_{1} + S_{2} \equiv T}$.
\end{itemize}
And by Lemma~\ref{lem:sr:scalars}, there exist $R_1, \dots, R_n$, $\alpha_1, \dots, \alpha_n$ such that
\begin{itemize}
\item $S_1 \equiv \sui{n}\alpha_{i} \cdot R_i$.
\item $\pi_{i} = \Gamma \vdash \ve{t}: R_{i}$, with $size(\pi) > size(\pi_{i})$, for $i \in \{1, \dots, n\}$.
\item $\sui{n} \alpha_{i} = \alpha$.
\end{itemize}
Then applying the $S$ rule,
\[
\prftree[r]{$S$}
{\Gamma \vdash \ve{t}: R_i~\forall i \in \{1,\dots,n\}}
{\Gamma \vdash \ve{t}: S_2}
{\Gamma \vdash (\alpha + 1) \cdot \ve{t}: \sui{n}\alpha_{i} \cdot R_i + S_{2}}
\]
We also know that
\[
\sui{n} \mu_{i} \cdot R_{i} \equiv S_1\qquad
S_1 + S_2 \equiv T
\]
Finally, we conclude by $\equiv$ rule that ${\Gamma \vdash (\alpha + 1)\cdot \ve{t}: T}$.
\inductioncase{Case $\ve{t} + \ve{t}\to (1 + 1)\cdot\ve{t}$}
Consider $\Gamma \vdash \ve{t} + \ve{t}: T$, then
by Lemma~\ref{lem:sr:sums} there exist $T_1, T_2$ such that
\begin{itemize}
\item $\Gamma \vdash \ve{t}: T_1$.
\item $\Gamma \vdash \ve{t}: T_2$.
\item $T_1 + T_2 \equiv T$.
\end{itemize}
Then applying the $S$ rule,
\[
\prftree[r]{$S$}
{\Gamma \vdash \ve{t}: T_1}
{\Gamma \vdash \ve{t}: T_2}
{\Gamma \vdash (1 + 1)\cdot\ve{t}: T_1 + T_2}
\]
Finally, by $\equiv$ rule we conclude that ${\Gamma \vdash (1 + 1)\cdot \ve{t}: T}$.
\inductioncase{Group B}
\inductioncase{Case $(\lambda x.\ve{t})~\ve{b}\to\ve{t}\subst{\ve b}{x}$}
Consider $\Gamma \vdash (\lambda x.\ve{t})~\ve{b}: T$, then by Lemma~\ref{lem:sr:app}, , there exist $R_1, \dots, R_h$, $\mu_1, \dots, \mu_h$, $\V_1,\dots,\V_h$ such that $T \equiv \suk{h} \mu_k \cdot R_k$,
$\suk{h} \mu_k = 1$ and for all $k \in \{1,\dots,h\}$,
\begin{itemize}
\item $\Gamma\vdash \lambda x.\ve{t}: \sui{n_k}{\alpha_{(k,i)} \cdot\forall\vec{X}.(U\to T_{(k,i)})}$.
\item $\Gamma\vdash\ve{b}: \suj{m_k}\beta_{(k,j)}\cdot U[\vec{A}_{(k,j)}/\vec{X}]$.
\item $\sui{n_k}\suj{m_k} \alpha_{(k,i)}\times\beta_{(k,j)}\cdot {T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}]} \ssubt_{\V_k,\Gamma} R_k$.
\end{itemize}
For the sake of readability, we will split the proof:
\begin{enumerate}
\item We will prove that $\Gamma, x:U[\vec{A}_{(k,j)}/X] \vdash \ve{t}: T_{(k,i)}[\vec{A}_{(k,j)}/X]$, for all $k \in \{1, \dots, h\}$, $j \in \{1, \dots, m\}$, $i \in \{1, \dots, n\}$.
\item We will prove that $\Gamma \vdash \ve{t}\subst{\ve b}{x}: T_{(k,i)}[\vec{A}_{(k,j)}/X]$, for all $k \in \{1, \dots, h\}$, $j \in \{1, \dots, m\}$, $ i \in \{1, \dots, n\}$.
\item We will prove that $\Gamma \vdash \ve{t}\subst{\ve b}{x}: T$.
\end{enumerate}
\textleadbydots{Item (1)}
\noindent We will prove that $\Gamma, x:U[\vec{A}_{(k,j)}/X] \vdash \ve{t}: T_{(k,i)}[\vec{A}_{(k,j)}/X]$, for all $k \in \{1, \dots, h\}$, $j \in \{1, \dots, m\}$, $i \in \{1, \dots, n\}$.\\
For simplicity, we will omit the $k$ index, which would otherwise be present in all the types, scalars and upper bound of the summations.\\
Considering $\lambda x.\ve{t}$ is a basis term, by Lemma~\ref{lem:sr:basevectors} then
there exist $W_1,\dots,W_b$, $\gamma_1,\dots,\gamma_b$ such that
\begin{itemize}
\item $\sug{a}{b} \gamma_{a} \cdot W_{a} \equiv \sui{n}{\alpha_{i} \cdot\forall\vec{X}.(U\to T_{i})}$.
\item $\Gamma \vdash \lambda x.\ve{t}: W_{a}$, for $a \in \{1, \dots,b\}$.
\item $\sug{a}{b} \gamma_{a} = 1$.
\end{itemize}
Without loss of generality, we assume that all unit types present at both sides of the equivalences are distinct, so
by Lemma~\ref{lem:sr:equivdistinctscalars}, then $b = n$ and there exists a permutation of $n$, $p$, such that
$\forall\vec{X}.(U\to T_{i}) \equiv W_{p(i)}$ and $\alpha_{i} = \gamma_{p(i)}$, for all $i \in \{1, \dots, n\}$.\\
Since for all $i \in \{1, \dots, n\}$ we have $\Gamma \vdash \lambda x.\ve{t}: \forall\vec{X}.(U\to T_{i})$,
then by Lemma~\ref{lem:sr:abs} and Lemma~\ref{lem:sr:equivdistinctscalars},
we know that $\Gamma, x:V_{i} \vdash \ve{t}: S_{i}$, and $V_{i} \to S_{i} \ssubt_{\V_i,\Gamma} \forall\vec{X}.(U\to T_{i})$.\\
By applying Lemma~\ref{lem:sr:arrowscomp}, then $U\equiv V_{i}[\vec{B}/\vec{Y}]$ and $T_{i} \equiv S_{i}[\vec{B}/\vec{Y}]$, with $\vec{Y} \notin \FV{\Gamma}$.\\
Then, by Lemma~\ref{lem:sr:substitution} and $\equiv$ rule, we have that $\Gamma, x:U \vdash \ve{t}: T_{i}$ for all $i \in \{1, \dots, n\}$.\\
By Lemma~\ref{lem:sr:sorderhasnofv}, since $V_{i} \to S_{i} \ssubt_{\V_i,\Gamma} \forall\vec{X}.(U\to T_{i})$ for all $i \in \{1, \dots, n\}$, then we know $\vec{X} \notin \FV{\Gamma}$
and so $\Gamma \equiv \Gamma[\vec{C}/\vec{X}]$, for any $\vec{C}$.\\
Therefore, by applying Lemma~\ref{lem:sr:substitution} multiple times, we have $\Gamma, x:U[\vec{A}_{j}/X] \vdash \ve{t}: T_{i}[\vec{A}_{j}/X]$
for all $j \in \{1, \dots, m\}$, $i \in \{1, \dots, n\}$.\\
Following this procedure for all $k \in \{1, \dots, h\}$, then we proved
that $\Gamma, x:U[\vec{A}_{(k,j)}/X] \vdash \ve{t}: T_{(k,i)}[\vec{A}_{(k,j)}/X]$,
for all $k \in \{1, \dots, h\}$, $j \in \{1, \dots, m\}$, $i \in \{1, \dots, n\}$.
\textleadbydots{Item (2)}
\noindent We will prove that $\Gamma \vdash \ve{t}\subst{\ve b}{x}: T_{(k,i)}[\vec{A}_{(k,j)}/X]$, for all $k \in \{1, \dots, h\}$, $j \in \{1, \dots, m_k\}$, $ i \in \{1, \dots, n_k\}$.\\
For simplicity, we will omit the $k$ index, which would otherwise be present in all the types, scalars and upper bound of the summations.\\
Since $\ve{b}$ is a basis term, by Lemma~\ref{lem:sr:basevectors} there exist $W'_1, \dots, W'_c$, $\eta_1, \dots, \eta_c$ such that
\begin{itemize}
\item $\sug{a}{c} \eta_{a} \cdot W'_{a} \equiv \suj{m}\beta_j\cdot U[\vec{A}_{j}/\vec{X}]$.
\item $\Gamma \vdash \ve{b}: W'_{a}$, for $a \in \{1, \dots,c\}$.
\item $\sug{a}{c} \eta_{a} = 1$.
\end{itemize}
Without loss of generality, we assume that all unit types present at both sides of the equivalences are distinct, so
by Lemma~\ref{lem:sr:equivdistinctscalars}, then $c = m$, and there exists a permutation $q$ of $m$, such that
$U[\vec{A}_{j}/\vec{X}] \equiv W'_{q(j)}$ and $\beta_j = \eta_{q(j)}$, for all $j \in \{1, \dots, m\}$.\\
Then, following Item (1), by applying Lemma~\ref{lem:sr:substitution}, we have that $\Gamma \vdash \ve{t}\subst{\ve b}{x}: T_{i}[\vec{A}_{j}/X]$
for all $j \in \{1, \dots, m\}$, $i \in \{1, \dots, n\}$.
Following this procedure for all $k \in \{1, \dots, h\}$, then we proved
that $\Gamma \vdash \ve{t}\subst{\ve b}{x}: T_{(k,i)}[\vec{A}_{(k,j)}/X]$,
for all $k \in \{1, \dots, h\}$, $j \in \{1, \dots, m\}$, $ i \in \{1, \dots, n\}$.
\textleadbydots{Item (3)}
\noindent Using the results of Item (1) and Item (2), and since in both items we already proved
that for all $k \in \{1, \dots, h\}$, $\sui{n_k} \alpha_i = \suj{m_k} \beta_j = 1$,
then by applying the $S$ rule for all $k \in \{1, \dots, h\}$ (we will omit the $k$ index for simplicity, that
will be present in all types, scalars and upper bound of the summations),
\[
\prftree[r]{$1_E$}
{\prftree[r]{$S$}
{\Gamma \vdash \ve{t}\subst{\ve b}{x}: T_{i}[\vec{A}_{j}/X]~\forall i \in \{1,\dots,n\},~\forall j \in \{1,\dots,m\}}
{\Gamma \vdash 1\cdot \ve{t}\subst{\ve b}{x}: \sui{n}\suj{m}\alpha_i\times\beta_j\cdot T_i[\vec{A}_j/X]}}
{\Gamma \vdash \ve{t}\subst{\ve b}{x}: \sui{n}\suj{m}\alpha_i\times\beta_j\cdot T_i[\vec{A}_j/X]}
\]
Since $\sui{n_k}\suj{m_k}\alpha_{(k,i)}\times\beta_{(k,j)}\cdot T_{(k,i)}[\vec{A}_{(k,j)}/X] \ssubt_{\V,\Gamma} R_k$,
then $\Gamma \vdash \ve{t}\subst{\ve b}{x}: R_k$.\\
Considering that $\suk{h} \mu_k = 1$, then by applying the $S$ and the $1_E$ rule again,
\[
\prftree[r]{$1_E$}
{\prftree[r]{$S$}
{\Gamma \vdash \ve{t}\subst{\ve b}{x}: R_k~\forall k \in \{1,\dots,h\}}
{\Gamma \vdash 1\cdot\ve{t}\subst{\ve b}{x}: \suk{h} \mu_k \cdot R_k}}
{\Gamma \vdash \ve{t}\subst{\ve b}{x}: \suk{h} \mu_k \cdot R_k}
\]
Finally, since $\mu_k \cdot R_k \equiv T$,
we conclude by $\equiv$ rule that $\Gamma \vdash \ve{t}[\ve{b}/x]: T$.
\inductioncase{Group A}
\inductioncase{Case $(\ve{t} + \ve{r})~\ve{u}\to (\ve{t})~\ve{u} + (\ve{r})~\ve{u}$}
Consider $\Gamma \vdash (\ve{t} + \ve{r})~\ve{u}: T$, then by
Lemma~\ref{lem:sr:app}, there exist $R_1, \dots, R_h$, $\mu_1, \dots, \mu_h$, $\V_1,\dots,\V_h$ such that $T \equiv \suk{h} \mu_k \cdot R_k$,
$\suk{h} \mu_k = 1$ and for all $k \in \{1,\dots,h\}$
\begin{itemize}
\item $\Gamma\vdash\ve{t} + \ve{r}: \sui{n_k}{\alpha_{(k,i)} \cdot\forall\vec{X}.(U\to T_{(k,i)})}$.
\item $\Gamma\vdash\ve{u}: \suj{m_k}\beta_{(k,j)}\cdot U[\vec{A}_j/\vec{X}]$.
\item $\sui{n_k}\suj{m_k} \alpha_{(k,i)}\times\beta_{(k,j)}\cdot {T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}]} \ssubt_{\V_k,\Gamma} R_k$.
\end{itemize}
We will simplify the rest of this proof by omitting the $k$ index,
which would otherwise be present in all the types, scalars and upper bound of the summations.
The rest of this proof then should be applied to all $k \in \{1, \dots, h\}$.\\
By Lemma~\ref{lem:sr:sums}, there exist $S_1$, $S_2$ such that
\begin{itemize}
\item $\Gamma \vdash \ve{t}: S_{1}$.
\item $\Gamma \vdash \ve{r}: S_{2}$.
\item $S_{1} + S_{2} \equiv \sui{n}{\alpha_i \cdot\forall\vec{X}.(U\to T_i)}$.
\end{itemize}
Hence, there exist $N_1, N_2\subseteq\{1,\dots,n\}$ with $N_1\cup N_2=\{1,\dots,n\}$ such that
\begin{align*}
S_{1}\equiv\sum\limits_{i\in N_1\setminus N_2}\alpha_{i}\cdot\forall\vec X.(U\to T_{i})+
\sum\limits_{i\in N_1\cap N_2}\eta_{i}\cdot\forall\vec X.(U\to T_{i}) & \mbox{\quad and}\\
S_{2}\equiv\sum\limits_{i\in N_2\setminus N_1}\alpha_{i}\cdot\forall\vec X.(U\to T_{i})+
\sum\limits_{i\in N_1\cap N_2}\eta'_{i}\cdot\forall\vec X.(U\to T_{i}) &
\end{align*}
where for all $i \in N_1\cap N_2$, $\eta_{i}+\eta'_{i}=\alpha_{i}$.
Therefore, using $\equiv$ we get
\begin{align*}
\Gamma\vdash\ve t:\sum\limits_{i\in N_1\setminus N_2}\alpha_{i}\cdot\forall\vec X.(U\to T_{i})+
\sum\limits_{i\in N_1\cap N_2}\eta_{i}\cdot\forall\vec X.(U\to T_{i}) & \mbox{\quad and}\\
\Gamma\vdash\ve r:\sum\limits_{i\in N_2\setminus N_1}\alpha_{i}\cdot\forall\vec X.(U\to T_{i})+
\sum\limits_{i\in N_1\cap N_2}\eta'_{i}\cdot\forall\vec X.(U\to T_{i}) &
\end{align*}
So, using rule $\to_E$, we get
\begin{align*}
\Gamma\vdash(\ve t)~\ve u:\sum\limits_{i\in N_1\setminus N_2}\suj{m}\alpha_{i}\times\beta_{j}\cdot T_{i}[\vec{A}_{j}/\vec X]+
\sum\limits_{i\in N_1\cap N_2}\suj{m}\eta'_{i}\times\beta_{j}\cdot T_{i}[\vec{A}_{j}/\vec X] & \mbox{\quad and}\\
\Gamma\vdash(\ve r)~\ve u:\sum\limits_{i\in N_2\setminus N_1}\suj{m}\alpha_{i}\times\beta_{j}\cdot T_{i}[\vec{A}_{j}/\vec X]+
\sum\limits_{i\in N_1\cap N_2}\suj{m}\eta'_{i}\times\beta_{j}\cdot T_{i}[\vec{A}_{j}/\vec X] &
\end{align*}
By rule $+_I$ we can conclude
\[
\Gamma\vdash(\ve t)~\ve u+(\ve r)~\ve u:\sui{n}\suj{m}\alpha_{i}\times\beta_{j}\cdot T_{i}[\vec{A}_{j}/\vec X]
\]
Since $\sui{n_k}\suj{m_k}\alpha_{(k,i)}\times\beta_{(k,j)}\cdot T_{(k,i)}[\vec{A}_{(k,j)}/\vec X] \ssubt_{\V_k, \Gamma} R_k$
for all $k \in \{1, \dots, h\}$, then by definition of $\ssubt$, we can derive $\Gamma\vdash(\ve t)~\ve u+(\ve r)~\ve u: R_k$.\\
By applying the $S$ and $1_E$ rules, then
\[
\prftree[r]{$1_E$}
{\prftree[r]{$S$}
{\Gamma\vdash(\ve t)~\ve u+(\ve r)~\ve u: R_k~\forall k \in \{1,\dots,k\}}
{\Gamma\vdash 1\cdot((\ve t)~\ve u+(\ve r)~\ve u): \suk{h} \mu_k \cdot R_k}}
{\Gamma\vdash (\ve t)~\ve u+(\ve r)~\ve u: \suk{h} \mu_k \cdot R_k}
\]
Finally, by the $\equiv$ rules, then $\Gamma\vdash (\ve t)~\ve u+(\ve r)~\ve u: T$.
\inductioncase{Case $(\ve{t})~(\ve{r} + \ve{u})\to (\ve{t})~\ve{r} + (\ve{t})~\ve{u}$}
Consider $\Gamma \vdash (\ve{t})~(\ve{r} + \ve{u}): T$, then by
Lemma~\ref{lem:sr:app}, there exist $R_1, \dots, R_h$, $\mu_1, \dots, \mu_h$, $\V_1,\dots,\V_h$ such that $T \equiv \suk{h} \mu_k \cdot R_k$,
$\suk{h} \mu_k = 1$ and for all $k \in \{1,\dots,h\}$
\begin{itemize}
\item $\Gamma\vdash\ve{t}: \sui{n_k}{\alpha_{(k,i)} \cdot\forall\vec{X}.(U\to T_{(k,i)})}$.
\item $\Gamma\vdash \ve{r} + \ve{u}: \suj{m_k}\beta_{(k,j)}\cdot U[\vec{A}_{(k,j)}/\vec{X}]$.
\item $\sui{n_k}\suj{m_k} \alpha_{(k,i)}\times\beta_{(k,j)}\cdot {T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}]} \ssubt_{\V_k,\Gamma} R_k$.
\end{itemize}
We will simplify the rest of this proof by omitting the $k$ index,
which would otherwise be present in all the types, scalars and upper bound of the summations.
The rest of this proof then should be applied to all $k \in \{1, \dots, h\}$.\\
By Lemma~\ref{lem:sr:sums}, there exists $S_1$, $S_2$ such that
\begin{itemize}
\item $\Gamma \vdash \ve{r}: S_{1}$
\item $\Gamma \vdash \ve{u}: S_{2}$
\item $S_{1} + S_{2} \equiv \suj{m}\beta_j\cdot U[\vec{A}_j/\vec{X}]$
\end{itemize}
Hence, there exist $N_1, N_2\subseteq\{1,\dots,m\}$ with $N_1\cup N_2=\{1,\dots,m\}$,
such that
\begin{align*}
S_1\equiv\sum\limits_{j\in N_1\setminus N_2}\beta_j\cdot U[\vec{A}_j/\vec{X}]+
\sum\limits_{j\in N_1\cap N_2}\eta_{j}\cdot U[\vec{A}_j/\vec{X}] & \mbox{\quad and}\\
S_2\equiv\sum\limits_{i\in N_2\setminus N_1}\beta_j\cdot U[\vec{A}_j/\vec{X}]+
\sum\limits_{j\in N_1\cap N_2}\eta'_{kj}\cdot U[\vec{A}_j/\vec{X}] &
\end{align*}
where for all $j\in N_1\cap N_2$, $\eta_{j}+\eta'_{j}=\beta_j$.
Therefore, using $\equiv$ we get
\begin{align*}
\Gamma\vdash\ve r:\sum\limits_{j\in N_1\setminus N_2}\beta_j\cdot U[\vec{A}_j/\vec{X}]+
\sum\limits_{j\in N_1\cap N_2}\eta_{j}\cdot U[\vec{A}_j/\vec{X}] & \mbox{\quad and}\\
\Gamma\vdash\ve u:\sum\limits_{j\in N_2\setminus N_1}\beta_j\cdot U[\vec{A}_j/\vec{X}]+
\sum\limits_{j\in N_1\cap N_2}\eta'_{kj}\cdot U[\vec{A}_j/\vec{X}] &
\end{align*}
So, using rule $\to_E$, we get
\begin{align*}
\Gamma\vdash(\ve t)~\ve r:\sui{n}\sum\limits_{j\in N_1\setminus N_2}\alpha_i\times\beta_j\cdot T_i[\vec{A}_j/\vec X]+
\sui{n}\sum\limits_{j\in N_1\cap N_2}\alpha_i\times\eta_{j}\cdot T_i[\vec{A}_j/\vec X] & \mbox{\quad and}\\
\Gamma\vdash(\ve t)~\ve u:\sui{n}\sum\limits_{j\in N_2\setminus N_1}\alpha_i\times\beta_j\cdot T_i[\vec{A}_j/\vec X]+
\sui{n}\sum\limits_{j\in N_1\cap N_2}\alpha_i\times\eta'_{kj}\cdot T_i[\vec{A}_j/\vec X] &
\end{align*}
By rule $+_I$ we can conclude
\[
\Gamma\vdash(\ve t)~\ve r+(\ve t)~\ve u:\sui{n}\suj{m}\alpha_i\times\beta_j\cdot T_i[\vec{A}_j/\vec X]
\]
Since $\sui{n_k}\suj{m_k}\alpha_{(k,i)}\times\beta_{(k,j)}\cdot T_{(k,i)}[\vec{A}_{(k,j)}/\vec X] \ssubt_{\V_k, \Gamma} R_k$
for all $k \in \{1, \dots, h\}$, then by definition of $\ssubt$, we can derive $\Gamma\vdash(\ve t)~\ve r+(\ve t)~\ve u: R_k$.\\
By applying the $S$ and $1_E$ rules, then
\[
\prftree[r]{$1_E$}
{\prftree[r]{$S$}
{\Gamma\vdash(\ve t)~\ve r+(\ve t)~\ve u: R_k~\forall k \in \{1,\dots,h\}}
{\Gamma\vdash 1\cdot((\ve t)~\ve r+(\ve t)~\ve u): \suk{h} \mu_k \cdot R_k}}
{\Gamma\vdash (\ve t)~\ve r+(\ve t)~\ve u: \suk{h} \mu_k \cdot R_k}
\]
Finally, by the $\equiv$ rules, then $\Gamma\vdash (\ve t)~\ve r+(\ve t)~\ve u: T$.
\inductioncase{Case $(\alpha\cdot\ve{t})~\ve{r}\to \alpha\cdot(\ve{t})~\ve{r}$}
Consider $\Gamma \vdash (\alpha\cdot\ve{t})\ \ve{r}: T$, by
Lemma~\ref{lem:sr:app}, there exist $R_1, \dots, R_h$, $\mu_1, \dots, \mu_h$, $\V_1,\dots,\V_h$ such that $T \equiv \suk{h} \mu_k \cdot R_k$,
$\suk{h} \mu_k = 1$ and for all $k \in \{1,\dots,h\}$
\begin{itemize}
\item $\pi_k = \Gamma\vdash \alpha\cdot\ve{t}: \sui{n_{k}}{\alpha_{(k,i)} \cdot\forall\vec{X}.(U\to T_{(k,i)})}$.
\item $\Gamma\vdash \ve{r}: \suj{m_{k}}\beta_{(k,j)}\cdot U[\vec{A}_{(k,j)}/\vec{X}]$.
\item $\sui{n_{k}}\suj{m_{k}} \alpha_{(k,i)}\times\beta_{(k,j)}\cdot {T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}]} \ssubt_{\V_{k},\Gamma} R_k$.
\end{itemize}
We will simplify the rest of this proof by omitting the $k$ index,
which would otherwise be present in all the types, scalars and upper bound of the summations.
The rest of this proof then should be applied to all $k \in \{1, \dots, h\}$.\\
By Lemma~\ref{lem:sr:scalars}, there exist $S_1, \dots, S_b$, $\eta_1, \dots, \eta_b$ such that
\begin{itemize}
\item $\sui{n}{\alpha_{i} \cdot\forall\vec{X}.(U\to T_{i})} \equiv \sug{a}{b}\eta_{a} \cdot S_{a}$.
\item $\pi_i = \Gamma \vdash \ve{t}: S_{a}$, with $size(\pi) > size(\pi_{a})$, for $a \in \{1, \dots, b\}$.
\item $\sug{a}{b} \eta_{a} = \alpha$.
\end{itemize}
Considering $\sui{n}{\alpha_{i} \cdot\forall\vec{X}.(U\to T_{i})}$ does not have any general variable $\vara{X}$ and
that $\sui{n}{\alpha_{i} \cdot\forall\vec{X}.(U\to T_{i})} \equiv \sug{a}{b}\eta_{a} \cdot S_{a}$,
then by Lemma~\ref{lem:sr:typecharact},
$S_{a} \equiv \sug{c}{d_{a}}\gamma_{(a,c)}\cdot V_{(a,c)}$.\\
Without loss of generality, we assume that all unit types present at both sides of the equivalences are distinct, so
by Lemma~\ref{lem:sr:equivdistinctscalars}, then $n = \sug{a}{b} d_{a}$, and by taking a partition
from $\{1,\dots, \sug{a}{b} d_{a}\}$ (defining an equivalence class)
and the trivial permutation $p$ of $n$ such that $p(i) = i$ (which we will omit for readability), we have
\begin{itemize}
\item $\alpha_i = \eta_{[i]}\times\sigma_i$, where $\sigma_i = \gamma_{\left([i],\frac{i}{[i]}\right)}$.
\item $\forall\vec{X}.(U\to T_i) \equiv V_{\left([i],\frac{i}{[i]}\right)}$.
\end{itemize}
Take $f(a) = \sug{e}{a-1} d_e$, so we rewrite $S_a \equiv \sug{c}{d_{a}}\gamma_{(a,c)}\cdot V_{(a,c)}$ as
\[
S_a \equiv \sum^{f(a) + d_a}_{g = f(a)} \sigma_g\cdot V_{\left([g],\frac{g}{[g]}\right)}
\equiv \sum^{f(a) + d_a}_{g = f(a)} \sigma_g\cdot \forall\vec{X}.(U\to T_g)
\]
Applying $\to_E$ for all $a \in \{1,\dots,b\}$,
\[
\prftree[r]{$\to_E$}
{\Gamma \vdash \ve{t}: \sum^{f(a) + d_a}_{g = f(a)} \sigma_g \cdot \forall\vec{X}.(U\to T_g)}
{\Gamma\vdash \ve{r}: \suj{m}\beta_{j}\cdot U[\vec{A}_{j}/\vec{X}]}
{\Gamma \vdash (\ve{t})~\ve{r}: \sum^{f(a) + d_a}_{g = f(a)}\suj{m} \left(\sigma_g\times\beta_{j}\right)\cdot T_{g}[\vec{A}_{j}/\vec{X}]}
\]
We rewrite $\sum^{f(a) + d_a}_{g = f(a)}\suj{m} \left(\sigma_g\times\beta_{j}\right)\cdot T_{g}[\vec{A}_{j}/\vec{X}] \equiv P_a$,
then by applying the $S$ rule we have
\[
\prftree[r]{$S$}
{\Gamma \vdash (\ve{t})~\ve{r}: P_a~\forall a \in \{1,\dots,b\}}
{\Gamma \vdash \alpha\cdot(\ve{t})~\ve{r}: \sug{a}{b} \eta_a \cdot P_a}
\]
Now we begin to unravel the final result
\begin{align*}
\sug{a}{b} \eta_a \cdot P_a &\equiv \sug{a}{b} \eta_a \cdot \sum^{f(a) + d_a}_{g = f(a)}\suj{m} \left(\sigma_g\times\beta_{j}\right)\cdot T_{g}[\vec{A}_{j}/\vec{X}]\\
&\equiv \sug{a}{b}\sum^{f(a) + d_a}_{g = f(a)}\suj{m} \left(\eta_{[g]} \times \sigma_g\times\beta_{j}\right)\cdot T_{g}[\vec{A}_{j}/\vec{X}]\\
&\equiv \sug{a}{b}\sum^{f(a) + d_a}_{g = f(a)}\suj{m} (\alpha_{g}\times\beta_{j})\cdot T_{g}[\vec{A}_{j}/\vec{X}]\\
&\equiv \sui{n}\suj{m} (\alpha_{i}\times\beta_{j})\cdot T_{i}[\vec{A}_{j}/\vec{X}]\\
\end{align*}
Then,
\[
\Gamma \vdash \alpha\cdot(\ve{t})~\ve{r}: \sui{n}\suj{m} (\alpha_{i}\times\beta_{j})\cdot T_{i}[\vec{A}_{j}/\vec{X}]
\]
Since $\sui{n_k}\suj{m_k} (\alpha_{(k,i)}\times\beta_{(k,j)})\cdot T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}] \ssubt_{\V_k,\Gamma} R_k$,
then for all $k \in \{1,\dots,h\}$, $\Gamma \vdash \alpha\cdot(\ve{t})~\ve{r}: R_k$.\\
By applying the $S$ and $1_E$ rules, then
\[
\prftree[r]{$1_E$}
{\prftree[r]{$S$}
{\Gamma\vdash\alpha \cdot(\ve t)~\ve r: R_k~\forall k \in \{1,\dots,h\}}
{\Gamma\vdash 1\cdot(\alpha \cdot(\ve t)~\ve r): \suk{h} \mu_k \cdot R_k}}
{\Gamma\vdash \alpha \cdot(\ve t)~\ve r: \suk{h} \mu_k \cdot R_k}
\]
Finally, by the $\equiv$ rule, then $\Gamma\vdash \alpha \cdot (\ve t)~\ve r: T$.
\inductioncase{Case $(\ve{t})~(\alpha\cdot\ve{r})\to \alpha\cdot(\ve{t})~\ve{r}$}
Consider $\Gamma \vdash (\ve{t})~(\alpha\cdot\ve{r}): T$, by
Lemma~\ref{lem:sr:app}, there exist $R_1, \dots, R_h$, $\mu_1, \dots, \mu_h$, $\V_1,\dots,\V_h$ such that $T \equiv \suk{h} \mu_k \cdot R_k$,
$\suk{h} \mu_k = 1$ and for all $k \in \{1,\dots,h\}$
\begin{itemize}
\item $\Gamma\vdash \ve{t}: \sui{n_{k}}{\alpha_{(k,i)} \cdot\forall\vec{X}.(U\to T_{(k,i)})}$.
\item $\pi_k = \Gamma\vdash \alpha\cdot\ve{r}: \suj{m_{k}}\beta_{(k,j)}\cdot U[\vec{A}_{(k,j)}/\vec{X}]$.
\item $\sui{n_{k}}\suj{m_{k}} \alpha_{(k,i)}\times\beta_{(k,j)}\cdot {T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}]} \ssubt_{\V_k,\Gamma} R_k$.
\end{itemize}
We will simplify the rest of this proof by omitting the $k$ index,
which would otherwise be present in all the types, scalars and upper bound of the summations.
The rest of this proof then should be applied to all $k \in \{1, \dots, h\}$.\\
By Lemma~\ref{lem:sr:scalars}, there exist $S_1, \dots, S_b$, $\eta_1, \dots, \eta_b$ such that
\begin{itemize}
\item $\suj{m}\beta_{j}\cdot U[\vec{A}_{j}/\vec{X}] \equiv \sug{a}{b}\eta_{a} \cdot S_{a}$.
\item $\pi_i = \Gamma \vdash \ve{r}: S_{a}$, with $size(\pi) > size(\pi_{a})$, for $a \in \{1, \dots, b\}$.
\item $\sug{a}{b} \eta_{a} = \alpha$.
\end{itemize}
Considering $\suj{m}\beta_{j}\cdot U[\vec{A}_{j}/\vec{X}]$ does not have any general variable $\vara{X}$ and
that $\suj{m}\beta_{j}\cdot U[\vec{A}_{j}/\vec{X}] \equiv \sug{a}{b}\eta_{a} \cdot S_{a}$,
then by Lemma~\ref{lem:sr:typecharact},
$S_{a} \equiv \sug{c}{d_{a}}\gamma_{(a,c)}\cdot V_{(a,c)}$.\\
Without loss of generality, we assume that all unit types present at both sides of the equivalences are distinct, so
by Lemma~\ref{lem:sr:equivdistinctscalars}, then $m = \sug{a}{b} d_{a}$, and by taking a partition
from $\{1,\dots, \sug{a}{b} d_{a}\}$ (defining an equivalence class)
and the trivial permutation $p$ of $m$ such that $p(j) = j$ (which we will omit for readability), we have
\begin{itemize}
\item $\beta_j = \eta_{[j]}\times\sigma_j$, where $\sigma_j = \gamma_{\left([j],\frac{j}{[j]}\right)}$.
\item $U[\vec{A}_{j}/\vec{X}] \equiv V_{\left([j],\frac{j}{[j]}\right)}$.
\end{itemize}
Take $f(a) = \sug{e}{a-1} d_e$, so we rewrite $S_a \equiv \sug{c}{d_{a}}\gamma_{(a,c)}\cdot V_{(a,c)}$ as
\[
S_a \equiv \sum^{f(a) + d_a}_{g = f(a)} \sigma_g\cdot V_{\left([g],\frac{g}{[g]}\right)}
\equiv \sum^{f(a) + d_a}_{g = f(a)} \sigma_g\cdot U[\vec{A}_{g}/\vec{X}]
\]
Applying $\to_E$ for all $a \in \{1,\dots,b\}$,
\[
\prftree[r]{$\to_E$}
{\Gamma\vdash \ve{t}: \sui{n}{\alpha_{i} \cdot\forall\vec{X}.(U\to T_{i})}}
{\Gamma \vdash \ve{r}: \sum^{f(a) + d_a}_{g = f(a)} \sigma_g\cdot U[\vec{A}_{g}/\vec{X}]}
{\Gamma \vdash (\ve{t})~\ve{r}: \sui{n}\sum^{f(a) + d_a}_{g = f(a)} \left(\alpha_i \times \sigma_g\right)\cdot T_{i}[\vec{A}_{g}/\vec{X}]}
\]
We rewrite $\sui{n}\sum^{f(a) + d_a}_{g = f(a)} \left(\alpha_i \times \sigma_g\right)\cdot T_{i}[\vec{A}_{g}/\vec{X}] \equiv P_a$,
then by applying the $S$ rule we have
\[
\prftree[r]{$S$}
{\Gamma \vdash (\ve{t})~\ve{r}: P_a~\forall a \in \{1,\dots,b\}}
{\Gamma \vdash \alpha\cdot(\ve{t})~\ve{r}: \sug{a}{b} \eta_a \cdot P_a}
\]
Now we begin to unravel the final result
\begin{align*}
\sug{a}{b} \eta_a \cdot P_a &\equiv \sug{a}{b} \eta_a \cdot \sui{n}\sum^{f(a) + d_a}_{g = f(a)} \left(\alpha_i \times \sigma_g\right)\cdot T_{i}[\vec{A}_{g}/\vec{X}]\\
&\equiv \sug{a}{b}\sum^{f(a) + d_a}_{g = f(a)}\suj{m} \left(\alpha_i \times \eta_{[g]} \times \sigma_g\right)\cdot T_{i}[\vec{A}_{g}/\vec{X}]\\
&\equiv \sug{a}{b}\sum^{f(a) + d_a}_{g = f(a)}\suj{m} (\alpha_i\times\beta_{g})\cdot T_{i}[\vec{A}_{g}/\vec{X}]\\
&\equiv \sui{n}\suj{m} (\alpha_{i}\times\beta_{j})\cdot T_{i}[\vec{A}_{j}/\vec{X}]\\
\end{align*}
Then,
\[
\Gamma \vdash \alpha\cdot(\ve{t})~\ve{r}: \sui{n}\suj{m} (\alpha_{i}\times\beta_{j})\cdot T_{i}[\vec{A}_{j}/\vec{X}]
\]
Since $\sui{n_k}\suj{m_k} (\alpha_{(k,i)}\times\beta_{(k,j)})\cdot T_{(k,i)}[\vec{A}_{(k,j)}/\vec{X}] \ssubt_{\V_k,\Gamma} R_k$,
then for all $k \in \{1,\dots,h\}$, $\Gamma \vdash \alpha\cdot(\ve{t})~\ve{r}: R_k$.\\
By applying the $S$ and $1_E$ rules, then
\[
\prftree[r]{$1_E$}
{\prftree[r]{$S$}
{\Gamma\vdash\alpha \cdot(\ve t)~\ve r: R_k~\forall k \in \{1,\dots,h\}}
{\Gamma\vdash 1\cdot(\alpha \cdot(\ve t)~\ve r): \suk{h} \mu_k \cdot R_k}}
{\Gamma\vdash \alpha \cdot(\ve t)~\ve r: \suk{h} \mu_k \cdot R_k}
\]
Finally, by the $\equiv$ rule, then $\Gamma\vdash \alpha \cdot (\ve t)~\ve r:
T$. \end{proof} }
\arxiv{\section{Omitted proofs in Section~\ref{ch:other-properties}}\label{app:proofsOP}} \conf{\section{Proof of Theorem~\ref{thm:sn}}\label{app:proofsOP}} \arxiv{\xrecap{Theorem}{Progress}{thm:progress}{
Given $\mathbb{V} = \left\{\sui{n} \alpha_i \cdot \lambda x_i.\ve{t}_i + \sum^{m}_{j=n+1} \lambda x_j.\ve{t}_j \mid \forall i, j, \lambda x_i.\ve{t}_i \neq \lambda x_j.\ve{t}_j\right\}$
and $\mathsf{NF}$ the set of terms in normal form (the terms that cannot be reduced any further), then
if $\vdash \ve{t}: T$ and $\ve{t} \in \mathsf{NF}$, it follows that $\ve{t} \in \mathbb{V}$. } \begin{proof}
By induction on $\ve{t}$:
\inductioncase{Case $\ve{t} = \sui{n} \alpha_i \cdot \lambda x_i.\ve{t}_i + \sum^{m}_{j=n+1} \lambda x_j.\ve{t}_j \mid \forall i, j, \lambda x_i.\ve{t}_i \neq \lambda x_j.\ve{t}_j$}
Trivial case.
\inductioncase{Case $\ve{t} = \sui{n} \alpha_i \cdot \lambda x_i.\ve{t}_i + \sum^{m}_{j=n+1} \lambda x_j.\ve{t}_j \mid \exists i, j, \lambda x_i.\ve{t}_i = \lambda x_j.\ve{t}_j$}
$\ve{t} \notin \mathsf{NF}$, since at least one reduction rule from Group F can be applied.
\inductioncase{Case $\ve{t} = (\ve{r})~\ve{s}$}
By induction hypothesis, we know that $\ve{r} = \sui{n} \alpha_i \cdot \lambda x_i.\ve{t}_i + \sum^{m}_{j=n+1} \lambda x_j.\ve{t}_j \in \mathbb{V}$.
We consider the following cases:
\begin{itemize}
\item If $m>n+1$ or $n \neq 0$, then at least one reduction rule from Group A can be applied, hence $(\ve{r})~\ve{s} \notin \mathsf{NF}$.
\item If $m=n+1$ and $n=0$, then $\ve{r} = \ve{b}_{n+1} \in \mathbb{V}$.
Since $FV(\ve{r}) = \emptyset$, then $\ve{r} = \lambda x.\ve{r'}$, which implies $(\ve{r})~\ve{s}$ is a beta-redex
or at least one reduction rule from Group A can be applied, hence $(\ve{r})~\ve{s} \notin \mathsf{NF}$.
\end{itemize}
\inductioncase{Case $\ve{t} = \alpha \cdot \ve{r}$}
By induction hypothesis, we know that $\ve{r} = \sui{n} \alpha_i \cdot \lambda x_i.\ve{t}_i + \sum^{m}_{j=n+1} \lambda x_j.\ve{t}_j \in \mathbb{V}$.
We consider the following cases:
\begin{itemize}
\item If $m \neq n+1$ or $n \neq 0$, then at least one reduction rule from Group E can be applied, hence $(\ve{r})~\ve{s} \notin \mathsf{NF}$.
\item If $m=n+1$, $n=0$ and $\alpha = 1$, then $\ve{r} = \lambda x.\ve{t} \in \mathbb{V}$, but $1 \cdot \ve{r} = 1 \cdot \lambda x.\ve{t} \to \lambda x.\ve{t}$, hence $\alpha \cdot \ve{r} \notin \mathsf{NF}$.
\item If $m=n+1$, $n=0$ and $\alpha \neq 1$, then $\ve{r} = \lambda x.\ve{t} \in \mathbb{V}$ and $\alpha \cdot \ve{r} = \alpha \cdot \ve{b} \in \mathbb{V}$.
\end{itemize}
\inductioncase{Case $\ve{t} = \ve{t}_1 + \ve{t}_2$}
By induction hypothesis, we know that $\ve{t}_{k} = \sui{n^k} \alpha^k_i \cdot (\lambda x_i.\ve{t}_i)^k_i + \sum^{m^k}_{j=n+1} (\lambda x_j.\ve{t}_j)^k_j \in \mathbb{V}$, with $k = 1, 2$.\\
We consider the following cases:
\begin{itemize}
\item $\exists i, j ~/~ (\lambda x_i.\ve{t}_i)^1 = (\lambda x_j.\ve{t}_j)^2$, then at least one reduction rule from Group F can be applied, hence $\ve{t}_1 + \ve{t}_2 \notin \mathsf{NF}$.
\item $\forall i, j ~/~ (\lambda x_i.\ve{t}_i)^1 \neq (\lambda x_j.\ve{t}_j)^2$, then by definition of $\mathbb{V}$, $\ve{t}_1 + \ve{t}_2 \in \mathsf{NF}$.\qed
\end{itemize} \end{proof} }
\xrecap{Theorem}{Strong Normalisation}{thm:sn} {If $\Gamma \vdash \ve{t}: T$ is a valid sequent, then $\ve{t}$ is strongly normalising.}
\begin{proof}
Consider the following derivation tree in \ensuremath{\lvec_{\textrm{\tiny R}}}, where $T \equiv T_1$,
\[
\pi_1 = \left\{\vcenter{
\prftree
{\vdots}
{\Gamma \vdash \ve{t}: T_1}}\right.
\]
Since the only difference between $\ensuremath{\lvec_{\textrm{\tiny R}}}$ and $\ensuremath{\textrm{Vectorial}}$ is the replacement of
the $\alpha_I$ rule for the $S$ and $1_E$ rules, then if $S$ and $1_E$ are not
present in $\pi_1$, we have that $\pi_1$ (and particularly, $\Gamma \vdash
\ve{t}: T$) is also a valid derivation for $\ensuremath{\textrm{Vectorial}}$. Also, notice that up to this
point, neither the term nor the types have scalars associated with them. If a
scalar were to be introduced, then the derivation trees (for $\ensuremath{\lvec_{\textrm{\tiny R}}}$ and
$\ensuremath{\textrm{Vectorial}}$) would be
\[
\begin{array}{c@{\qquad}c}
\text{In }\ensuremath{\lvec_{\textrm{\tiny R}}}
&
\text{In }\ensuremath{\textrm{Vectorial}}
\\[2ex]
\vcenter{\prftree[r]{$S$}
{\pi_1}
{\dots}
{\pi_n}
{\Gamma \vdash \left(\sui{n} \alpha_i\right) \cdot \ve{t}: \sui{n} \alpha_i \cdot T_i}}
&
\vcenter{\prftree[r]{$\alpha_I$}
{\pi_1}
{\Gamma \vdash \left(\sui{n} \alpha_i\right) \cdot \ve{t}: \left(\sui{n} \alpha_i\right) \cdot T}}
\end{array}
\]
Where $\pi_i = \Gamma \vdash \ve{t}: T_i$ are valid sequents for some $T_i$,
with $i \in \{2, \dots, n\}$.\\ Now, notice that by having $\Gamma \vdash
\left(\sui{n} \alpha_i\right) \cdot \ve{t}: \sui{n} \alpha_i \cdot T_i$
(specifically, by having a linear combination of types), we are restricting the
terms we can type. In other words, for every derivation tree in $\ensuremath{\lvec_{\textrm{\tiny R}}}$, there
is a simpler derivation tree in $\ensuremath{\textrm{Vectorial}}$, and thus if a sequent $\Gamma \vdash
\ve{t}: T$ is valid in $\ensuremath{\lvec_{\textrm{\tiny R}}}$, then there is a derivation tree for the same
term in $\ensuremath{\textrm{Vectorial}}$. Finally, since $\ensuremath{\textrm{Vectorial}}$ is strongly
normalising~\cite[Thm.~5.7]{vectorial}, then $\ensuremath{\lvec_{\textrm{\tiny R}}}$ is strongly normalising. \end{proof}
\arxiv{ \recap{Lemma}{lem:wp:weightequiv}{
If $T \equiv R$, then $\tnorm{T} = \tnorm{R}$. } \begin{proof}
We prove the lemma holds for every definition of $\equiv$
\inductioncase{Case $1\cdot T \equiv T$}
Trivial case.
\inductioncase{Case $\alpha\cdot(\beta\cdot T) \equiv (\alpha\times\beta)\cdot T$}
\begin{align*}
\tnorm{\alpha\cdot(\beta\cdot T)} &= \alpha\cdot\tnorm{\beta\cdot T} = (\alpha \times \beta) \cdot \tnorm{T}
= \tnorm{(\alpha \times \beta) \cdot T}
\end{align*}
\inductioncase{Case $\alpha\cdot T+\alpha\cdot R \equiv \alpha\cdot (T+R)$}
\begin{align*}
\tnorm{\alpha\cdot T+\alpha\cdot R} &= \tnorm{\alpha\cdot T} + \tnorm{\alpha\cdot R}\\
&= \alpha \cdot \tnorm{T} + \alpha \cdot \tnorm{R} = \alpha \cdot (\tnorm{T} + \tnorm{R})\\
&= \alpha \cdot (\tnorm{T + R}) = \tnorm{\alpha\cdot (T+R)}\\
\end{align*}
\inductioncase{Case $\alpha\cdot T+\beta\cdot T \equiv (\alpha+\beta)\cdot T$}
\begin{align*}
\tnorm{\alpha\cdot T+\beta\cdot T} &= \tnorm{\alpha \cdot T} + \tnorm{\beta \cdot T} = \alpha \cdot \tnorm{T} + \beta \cdot \tnorm{T}\\
&= (\alpha + \beta) \cdot \tnorm{T} = \tnorm{(\alpha+\beta)\cdot T}
\end{align*}
\inductioncase{Case $T+R \equiv R+T$}
\begin{align*}
\tnorm{T + R} = \tnorm{T} + \tnorm{R} = \tnorm{R} + \tnorm{T} = \tnorm{T + R}
\end{align*}
\inductioncase{Case $T+(R+S) \equiv (T+R)+S$}
\begin{align*}
\tnorm{T + (R+S)} &= \tnorm{T} + \tnorm{R + S} = \tnorm{T} + \tnorm{R} + \tnorm{S}\\
&= \tnorm{T + R} + \tnorm{S} = \tnorm{(T + R) + S} \qed
\end{align*} \end{proof}
\recap{Lemma}{lem:wp:weightofvalues}{
If $\ve{v}\in \mathbb{V}$, and $\vdash \ve{v}: T$, then
$\tnorm{T} \equiv \tnorm{\ve{v}}$. } \begin{proof}
Let $\ve{v} = \sui{k} \alpha_i \cdot \lambda x_i.\ve{t}_i + \sum^{n}_{i = k+1} \lambda x_i.\ve{t}$.
We proceed by induction on $n$.
\inductioncase{Case $n = 1$}
There are two possible escenarios:
\textleadbydots{\textbf{$k = 1$}}
\noindent In this scenario, consider $\pi =~\vdash \alpha_1 \cdot \lambda x_1.\ve{t}_1: T$.
By Lemma~\ref{lem:sr:scalars}, there exist $R_1, \dots, R_m$, $\beta_1, \dots, \beta_m$ such that
\begin{itemize}
\item $T \equiv \suj{m}\beta_j \cdot R_j$.
\item $\pi_i = ~ \vdash \lambda x_1.\ve{t}_1: R_j$, with $size(\pi) > size(\pi_j)$, for $j \in \{1, \dots, m\}$.
\item $\sui{m} \beta_j = \alpha_1$.
\end{itemize}
Considering $\lambda x_1.\ve{t}_1$ is a basis term, then by Lemma~\ref{lem:sr:basevectors},
for each $j \in \{1, \dots, m\}$ (we will omit the $j$ index for readability),
there exist $U_1, \dots, U_h$, $\sigma_1, \dots, \sigma_h$ such that
\begin{itemize}
\item $R \equiv \suk{h} \sigma_k \cdot U_k$.
\item $\vdash \lambda x_1.\ve{t}_1: U_k$, for $k \in \{1,\dots,h\}$.
\item $\suk{h} \sigma_k = 1$.
\end{itemize}
Then,
\[
T \equiv \suj{m} \beta_j \cdot R_j \equiv \suj{m} \beta_j \cdot (\suk{h_j} \sigma_{(j,k)} \cdot U_{(j,k)})
\]
Finally, by definition of $\tnorm{\bullet}$, we have
\begin{align*}
\tnorm{\ve{v}} &= \tnorm{\alpha_1 \cdot \lambda x_1.\ve{t}_1} = \alpha_1 \cdot \tnorm{\lambda x_1.\ve{t}_1}\\
&= \alpha_1 = \sui{m} \beta_j = \sui{m} \beta_j \cdot \underbrace{\left(\suk{h_j} \sigma_{(j,k)}\right)}_{=~1}\\
&= \sui{m} \beta_j \cdot \left(\suk{h_j} \sigma_{(j,k)} \cdot \tnorm{U_{(j,k)}}\right)\\
&= \sui{m} \beta_j \cdot \tnorm{\suk{h_j} \sigma_{(j,k)} \cdot U_{(j,k)}}
= \tnorm{\sui{m} \beta_j \cdot \left(\suk{h_j} \sigma_{(j,k)} \cdot U_{(j,k)}\right)}\\
&= \tnorm{T}
\end{align*}
\textleadbydots{\textbf{$k = 0$}}
\noindent In this scenario, consider $\vdash \lambda x_1.\ve{t}_1: T$.
Considering $\lambda x_1.\ve{t}_1$ is a basis term, then
by Lemma~\ref{lem:sr:basevectors}, there exist $U_1, \dots, U_m$, $\beta_1, \dots, \beta_m$ such that
\begin{itemize}
\item $T \equiv \suj{m} \beta_k \cdot U_j$.
\item $\vdash \lambda x_1.\ve{t}_1: U_j$, for $j \in \{1,\dots,m\}$.
\item $\suj{m} \beta_j = 1$.
\end{itemize}
Finally, by definition of $\tnorm{\bullet}$, we have
\begin{align*}
\tnorm{\ve{v}} &= \tnorm{\lambda x_1.\ve{t}_1} = 1 = \suj{m} \beta_j\\
&= \suj{m} \beta_j \cdot \tnorm{U_j} = \tnorm{\suj{m} \beta_j \cdot U_j}\\
&= \tnorm{T}
\end{align*}
\inductioncase{Induction step}
Consider now that $\vdash \ve{v} = \ve{v}' + \ve{v}'': T$,
where $\ve{v}' = \sui{k} \alpha_i \cdot \lambda x_i.\ve{t}_i + \sum^{n}_{i = k+1} \lambda x_i.\ve{t}_i$
and either $\ve{v}'' = \beta \cdot \lambda x.\ve{t}$, or $\ve{v}'' = \lambda x.\ve{t}$.
By Lemma~\ref{lem:sr:sums}, we know there exists $R$ and $S$ such that
\begin{itemize}
\item $T \equiv R + S$.
\item $\Gamma\vdash\ve{v}': R$.
\item $\Gamma\vdash\ve{v}'': S$.
\end{itemize}
By induction hypothesis, since $\vdash \ve{v}' = \sui{k} \alpha_i \cdot \lambda x_i.\ve{t}_i + \sum^{n}_{i = k+1} \lambda x_i.\ve{t}_i: R$,
then $\tnorm{R} = \tnorm{\ve{v}'}$;
and since either $\ve{v}'' = \beta \cdot \lambda x.\ve{t}$ or $\ve{v}'' = \lambda x.\ve{t}$, in both cases we know that $\tnorm{S} = \tnorm{\ve{v}''}$.
Finally, and considering by Lemma~\ref{lem:wp:weightequiv} that $\tnorm{T} = \tnorm{R} + \tnorm{S}$, we have
\begin{align*}
\tnorm{\ve{v}} &= \tnorm{\ve{v}' + \ve{v}''}\\
&= \tnorm{\ve{v}'} + \tnorm{\ve{v}''}\\
&= \tnorm{R} + \tnorm{S}\\
&= \tnorm{T}
\qed
\end{align*} \end{proof} }
\end{document} | arXiv |
Hermite–Minkowski theorem
In mathematics, especially in algebraic number theory, the Hermite–Minkowski theorem states that for any integer N there are only finitely many number fields, i.e., finite field extensions K of the rational numbers Q, such that the discriminant of K/Q is at most N. The theorem is named after Charles Hermite and Hermann Minkowski.
This theorem is a consequence of the estimate for the discriminant
${\sqrt {|d_{K}|}}\geq {\frac {n^{n}}{n!}}\left({\frac {\pi }{4}}\right)^{n/2}$
where n is the degree of the field extension, together with Stirling's formula for n!. This inequality also shows that the discriminant of any number field strictly bigger than Q is not ±1, which in turn implies that Q has no unramified extensions.
References
Neukirch, Jürgen (1999). Algebraic Number Theory. Springer. Section III.2
| Wikipedia |
Sex-specific patterns of senescence in artificial insect populations varying in sex-ratio to manipulate reproductive effort
Charly Jehan1,
Manon Chogne1,
Thierry Rigaud1 &
Yannick Moret ORCID: orcid.org/0000-0002-2435-84161
BMC Evolutionary Biology volume 20, Article number: 18 (2020) Cite this article
The disposable soma theory of ageing assumes that organisms optimally trade-off limited resources between reproduction and longevity to maximize fitness. Early reproduction should especially trade-off against late reproduction and longevity because of reduced investment into somatic protection, including immunity. Moreover, as optimal reproductive strategies of males and females differ, sexually dimorphic patterns of senescence may evolve. In particular, as males gain fitness through mating success, sexual competition should be a major factor accelerating male senescence. In a single experiment, we examined these possibilities by establishing artificial populations of the mealworm beetle, Tenebrio molitor, in which we manipulated the sex-ratio to generate variable levels of investment into reproductive effort and sexual competition in males and females.
As predicted, variation in sex-ratio affected male and female reproductive efforts, with contrasted sex-specific trade-offs between lifetime reproduction, survival and immunity. High effort of reproduction accelerated mortality in females, without affecting immunity, but high early reproductive success was observed only in balanced sex-ratio condition. Male reproduction was costly on longevity and immunity, mainly because of their investment into copulations rather than in sexual competition.
Our results suggest that T. molitor males, like females, maximize fitness through enhanced longevity, partly explaining their comparable longevity.
Life history theory assumes that organisms are constrained to optimally trade-off limited energetic and time resources between reproduction and life span, to maximize fitness [1, 2]. This principle is at the core of the theory of ageing, which predicts that, as reproduction is resource demanding, current reproduction is traded-off against future reproduction and survival, caused by a reduced investment into somatic protection and maintenance [2,3,4]. However, recent studies have sometimes revealed patterns of actuarial (decline in survival rate with age) and reproductive (decline in reproductive success with age) senescence rather contrasted with this prediction [5,6,7]. Since individuals may differ in both resource acquisition and resource allocation between traits, depending on individual and environmental quality, the cost of reproduction can remain undetected at the population level [8, 9].
Studies that investigated cost of reproduction in terms of senescence mainly focused on females [10, 11]. Those on males often referred to sexual selection theory and therefore on the cost of producing and maintaining sexual traits [12]. In males, cost of reproduction may result from resource demands for courtship, mating, struggling with female resistance, mate guarding, the production of sperm and accessory gland proteins [13,14,15,16]. They may also engage into costly intra-sexual competition for females through pre- and post-copulatory contests with other males [17]. In females, cost of reproduction may result from gamete production, offspring care, harassment by males, mating injuries, sexually transmitted diseases and damaging seminal substances [18,19,20,21]. These differential costs may have contributed to the evolution of sexually dimorphic life-history strategies in many species through which males and females achieve maximal fitness. For instance, while males may maximize fitness by increasing mating success at the expense of longevity, females may maximize fitness through longevity because offspring production, although resource intensive, requires time too. The different reproductive costs may also contribute to different patterns of senescence between males and females, which may vary within and among populations, depending on their relative intensity. Strong investment into reproduction early in life seems to contribute to accelerating reproductive and actuarial senescence [22]. However, our understanding of the impacts of the costs of reproduction on senescence mainly relies on theoretical and correlative studies, whereas experimental investigations are still scarce.
Somatic protection partly depends on the immune system, whose competence may diminish with age. Such an immunosenescence causes enhanced sensitivity to infection and inflammatory diseases, increasing risk of morbidity and mortality with age [23, 24]. Increased reproductive effort was found associated with enhanced susceptibility to parasitism and disease [25] or decreased immune activity [17, 26,27,28,29]. Trade-offs between reproductive and immune functions for limited resources, or negative pleiotropic effects of reproductive hormones on immune defence, have been proposed as proximate causes of the cost of reproduction [30]. However, contradictory results are common as studies also failed to demonstrate such a cost [31,32,33]. If investment into reproduction can induce a progressive decline in somatic functions, strong investment into early reproductive effort may generate accelerated immunosenescence and contribute to actuarial senescence.
Recent correlative evidence suggests that population structure, such as sex-ratio, affects individual reproductive effort with potential sex-specific consequences on senescence [34, 35]. In particular, variation in sex-ratio is predicted to modulate the cost of mating, through the strength of sexual selection in males [36], influencing the putative trade-off between reproductive effort and somatic maintenance [11]. Furthermore, cost of reproduction in females is also predicted to depend on population sex-ratio as it is expected to influence male competition for fertilization [16]. Hence, experimentally varying population sex-ratio appears to be a valuable tool to manipulate males and females reproductive effort and test its impacts on senescence at the population level.
Here, in a single experiment, we investigated the consequences of variable levels of investment in breeding effort on lifetime reproduction, survival and immunity of males and females of the mealworm beetle, Tenebrio molitor, of which we have manipulated the sex-ratio in artificial populations. In this highly promiscuous insect, manipulating the sex-ratio of populations is expected to affect both the average intensity of intra sexual competition or sexual selection, and individual mating rate. In male-biased sex-ratio conditions, males should face fewer mating opportunities, whereas females should show high individual reproductive effort. By contrast, in female-biased sex-ratio condition, males should copulate more frequently, whereas females should have fewer opportunities to mate. This experimental design allowed us to test the cost of different key features of male and female reproduction in terms of senescence at the population level by examining their lifetime changes in survival, fertility, reproductive effort, body condition and immunity. Note, however, that manipulating sex-ratio may only affect the opportunity for sexual selection and not the actual sexual selection [37], and life-history particularities of biological models should be taken into account. For example, common wisdom is that male-biased sex-ratio conditions should accelerate male reduction of survival, reproduction and immunity because of intense pre- and post-copulatory intra sexual competition. However, in T. molitor, mating might be where the largest costs arise in reproduction for males (see below), and accelerated senescence is expected in populations with female-biased sex-ratio because males should produce higher reproductive effort. Indeed, direct observations of the mating behaviour of T. molitor suggest that males do not engage in costly physical contests to access females [38, 39]. Courtship and mating are relatively brief during which males transfer a spermatophore that does not release the sperm before 7–10 min post-copulation [40]. Males may then perform rather passive short post-copulatory mate guarding, consisting on staying within 1 cm of the female for more than.
One minute in the presence of competitors [38, 39]. However, males do not appear to have evolved specific post-copulatory mate-guarding behaviour like those observed in other insects [41]. The spermatophore transferred during copulation contains nutrient-rich substances that constitute a nuptial gift [42], whose cost may prevent males to copulate again for 20 min after the last copulation [41]. Hence, as mating is more costly than pre- and post-copulatory sexual competition, T. molitor males may best achieve fitness through longevity, just like females, which would ultimately prevent the evolution of divergent patterns of actuarial senescence between males and females. Females, for their part, may exhibit strong early reproductive effort in populations with male-biased sex-ratio, they also should exhibit accelerated decline in reproduction, and immunity or earlier immune dysregulation, correlating with reduced survival with age. In populations with female-biased sex-ratio, females should survive, reproduce and maintain immunity at older age, as they might exhibit lower early life reproductive effort.
Mealworm beetles
Mealworm beetles are stored grain product pests that live several months in populations of variable density and at sex-ratio of about 50% (± 20%). T. molitor males and females may initiate reproduction from the fifth day post emergence, although they reach their full sexual maturity from the eighth day post emergence. They can mate many times with several partners within their 2 to 5 months of adult life. Females are continuously receptive to mating during adulthood and may produce up to 30 eggs per day although egg production may decline after 3 weeks [43]. Although able to store sperm in their spermatheca, females need to mate frequently to maintain high egg production [44].
The immune system of T. molitor relies on both constitutive cellular (e.g. haemocytes) and enzymatic (e.g. prophenoloxidase system) components at the core of the inflammatory response [45]. Their activity is cytotoxic [46], causing self-damage [47] and lifespan reduction [48,49,50,51]. They were found to decrease after mating [52] and either decline [53] or increase [54] with age. In addition, the inducible production of antibacterial peptides in the haemolymph [45] is an energetically costly process that may reduce survival [55]. As selection on immune expression and immune regulation might be weaker after reproductive senescence, age-related decline of baseline levels of immunity might be observed and immune activation may occur at old age due to dysregulation [54, 56].
Artificial populations and experimental design
Virgin adult beetles of controlled age (10 ± 2 days post-emergence) were obtained from pupae haphazardly sampled from a stock culture maintained in laboratory conditions (24 ± 2 °C, 70% RH in permanent darkness) at Dijon, France. Prior to the experiments, all these experimental insects were maintained separately in laboratory conditions, and supplied ad libitum with bran flour and water, supplemented by apple.
Fifteen artificial populations of 100 adult beetles were made according to three sex-ratio conditions. Five populations had a balanced sex-ratio, each comprising 50 males and 50 females (thereafter named the 50%_males condition), and were considered as the reference populations. Five populations had a male-biased sex-ratio, each comprising 75 males and 25 females (75%_males). Finally, five populations had a female-biased sex-ratio, each comprising 25 males and 75 females (25%_males). Each population was maintained in a plastic tank (L × 1 x H, 27 × 16.5 × 11.5 cm) containing bran flour, supplied once a week with apple and water. Every 2 weeks, each population was transferred into a clean tank supplied with fresh bran flour, thus avoiding the development of the progeny with the experimental adults.
Age specific reproductive assay
Reproductive capacity of females and males in each population was estimated weekly. To this purpose, 4 females haphazardly picked in each population were individually transferred into a plastic Petri dish (9 cm in diameter), containing bleached flour, a 2 mL centrifuge tube of water and a piece of apple. Each female was allowed to lay eggs in the Petri dish for 3 days, and was then returned to their initial population box. Two weeks later, the number of larvae was counted in each Petri dish to quantify female fertility, which is the number of viable larvae produced per female [57].
Concomitantly, four males haphazardly picked in each population were also individually transferred into Petri dishes, as above. Reproductive success of males was estimated through direct measures of their fertility (number of viable offspring per male [57]) instead of measuring spermatophores or counting the sperm, which are rough surrogates of male reproductive success. Each male was provided with a virgin female aged from 8 to 15 days for 24 h and was then returned in its initial population. Each female was then allowed to lay eggs in the Petri dish for three additional days to estimate, as described above, male fertility. In T. molitor, males may affect female fecundity (number of potential eggs produced by the female) and therefore their fertility, according to the respective quality of spermatophores and sperm transferred during mating. Consequently, male's success was a measure of the potential reproductive effort, not the one realized within its experimental population.
While assayed for their reproduction, focal females and males were replaced by marked individuals of the same age and sex in all the populations, to keep sex-ratio and density constant. Substitutes were from the same cohort as the experimental insects, kept in a separate tank of mixed-sex population. They were marked by clipping a piece of one elytra. When focal insects assayed for their reproduction were returned into their initial population box, substitutes were removed and returned into their tank.
Estimation of male and female reproductive effort at the population level
Survival of the insects was checked weekly, and dead insects were replaced by marked substitutes of the same sex and about the same age to keep the population sex-ratio and density of individuals constant. No measurement was performed on these marked individuals.
As the experimental design does not allow gathering measurements of longevity and fertility for each individual of the population, we estimated male and female reproductive effort (RE) at the population level, from the above age-specific measures of fertility, for each of the five population replicates, within sex-ratio conditions. This estimate was calculated as the total number of viable larvae produced per female or male in each replicate (i.e. the cumulative number of larvae produced during the whole experiment in a given replicate divided by the number of females or males tested for this replicate), divided by their respective average lifespan in the population replicate (i.e. the average lifetime of females or males in each relicate). The equation is given as follow:
$$ {RE}_r=\frac{l}{ML} $$
Where l is the total number of offspring (here viable larvae) produced per assayed females or males in the population replicate r, and ML is the recorded mean lifespan (in weeks) of males and females in the replicate r. RE values (as offspring per individuals and per mean weeks of survival in the population) of each sex and population replicate within each sex-ratio condition were used as data points for comparisons between modalities of sex-ratio.
Note that female RE values are likely representative of both female and male conditions resulting from the experiment, because the female reproductive performance resulted from mating with males from their respective population. By contrast, male RE values are representative of the male condition only, because male reproductive performance was standardized by pairing it with a virgin and age-controlled female that did not experience the experimental conditions. Therefore, male RE must be seen as a surrogate of male reproduction potential.
Body condition and haemolymph collection
At weeks 2, 4, 6 and 12 after the start of the experiment, 4 females and 4 males were picked at random in each population to estimate their body condition and immunity. The first three time points were chosen as being relevant of the time period during which most of the beetle reproduction is achieved and survival is still relatively high [44]. It is also within this period of time that a potential decline in somatic protection, including immunity, is predicted. The last time point corresponds to a period of time when reproduction should have almost ceased and when few beetles remain alive. As immunity measurements was destructive sampling, sampled insects were replaced by marked substitutes as above, to keep sex-ratio and density constant. However, this substitution was definitive, as sampled individuals were not returned to their initial population box after being assayed. The below estimation of the insect body condition and immunity was done as described in [58]. Beetles were first sized by measuring the length of the right elytra with Mitutoyo digital callipers (precision 0.1 mm) and weighed to the nearest mg with an OHAUS balance (discovery series, DU114C). Body condition was then estimated by the residuals of the regression between body size and body mass. Then, beetles were chilled on ice for 10 min before the sampling of 5 μL of haemolymph from a wound made in the beetle's neck and flushed in a microcentrifuge tube containing 25 μL of phosphate-buffered saline (PBS 10 mM, pH 7.4). A 10-μL aliquot was immediately used to measure haemocyte count. Another 5-μL aliquot was kept in an N-phenylthiourea-coated microcentrifuge tube (P7629, Sigma-Aldrich, St Louis, MO, USA) and stored at − 80 °C for later examination of its antibacterial activity. The remaining haemolymph solution (15 μL) was further diluted in 15 μL of PBS and stored at − 80 °C for later measurement of its phenoloxidase activity.
Immune parameters
Haemocyte count was measured using a Neubauer improved haemocytometer under a phase-contrast microscope (magnification × 400).
Antimicrobial activity of the haemolymph was measured using the inhibition zone assay described in [58]. Briefly, an overnight culture of the bacterium Arthrobacter globiformis from the Pasteur Institute (CIP105365) was added to a Broth medium containing 1% agar to achieve a final concentration of 105 cells.mL− 1. Six millilitres of the medium was subsequently poured per Petri dish and, after solidification, 12 wells were made inside the agar plate in which 2 μL of each haemolymph sample was deposited. Plates were then incubated overnight at 28 °C and the diameter of each zone of inhibition was measured.
For each haemolymph sample, both (i) the activity of naturally activated phenoloxidase (PO) enzyme only (hereafter PO activity) and (ii) the activity of PO plus that of proenzymes (proPO) (hereafter Total-PO activity) were measured using the spectrophotometric assay described in [59]. Total-PO activity quantification required the activation of proPO into PO with chymotrypsin, whereas PO activity was measured directly from the sample. Frozen haemolymph samples were thawed on ice and centrifuged (3500 g, 5 min, 4 °C). In a 96-well plate, 5 μL of supernatant were diluted in 20 μL of PBS and were added either 140 μL of distilled water to measure PO activity or 140 μL of 0.07 mg. mL− 1 chymotrypsin solution (Sigma-Aldrich, St Louis, MO, USA, C-7762) to measure Total-PO activity. Subsequently, 20 μL of a 4 mg.mL− 1 L-Dopa solution (Sigma-Aldrich, St Louis, MO, USA, D-9628) were added to each well. The reaction proceeded for 40 min at 30 °C, in a microplate reader (Versamax, Molecular Devices, Sunnyval, CA, USA). Reads were taken every 15 s at 490 nm and analysed using the software SOFT-Max®Pro 4.0 (Molecular Devices, Sunnyvale, CA, USA). Enzymatic activity was measured as the slope (Vmax value: change in absorbance unit per min) of the reaction curve during the linear phase of the reaction and reported to the activity of 1 μL of pure haemolymph.
Cox-regressions with a time-dependent covariate were used to analyse the difference in survival rates with respect to sex-ratio during the time (in weeks) from the start of the experiment and the death of all individuals. Sex-ratio was coded as categorical variables. The effect of sex ratio in the statistical model used the reference survival function generated from the data derived from the females or the males of the 50%-male sex-ratio condition. Time (in weeks) was incremented as a covariate in interaction with sex-ratio in the model as hazard ratios when the survival functions where not constant over time (for more details, see [60]). The analyses of fertility (i.e. the number of larvae produced per female or male at each week) and immune parameters were performed using mixed models, either Linear or Generalized linear depending on the nature of the data (see table legends). Starting models included sex-ratio condition, week (continuous variable for fertility, ordinal variable for immunity), their interaction, body condition and replicates treated as a random factor (REML estimates of variance component). The models presented here are those minimizing the AICc, where ΔAICc > 2 is usually considered to be good support [61], after comparisons of all models including predictors and their interactions, in a stepwise fashion (see Additional file 1: Table S1). The analyses of reproductive effort were made using ANOVA testing the effect of sex-ratio conditions. Analyses were made using IBM® SPSS® Statistics 19, JMP v. 10.0 and R version 3.3.2 (The R Foundation for Statistical Computing, Vienna, Austria, http://www.r-project.org). All the data files are available from the Dryad data base [62].
Demography: survival, fertility and reproductive effort
A first survival analysis comparing males and females of the 50%-male sex-ratio condition, which presumably corresponds to the sex-ratio condition in natural populations of T. molitor, showed no difference in longevity between males and females (Cox regression: Wald statistics = 0.004, d.f. = 1, p = 0.947, see Additional file 1: Figure S1). Survival of females and males was significantly affected by the sex-ratio condition (Table 1, Fig. 1). In the 75%-male sex-ratio condition, females exhibited an accelerated mortality with time by a factor of 13% per week compared to females of the 25 and 50%-male sex-ratio conditions (see odd ratio of Sex-ratio*Time-Cov in Table 1a). There was no significant difference in survival between females in the 25 and 50%-male sex-ratio conditions (Table 1a, Fig. 1a). Contrasting with females, males in the 75%-male sex-ratio condition survived significantly longer than those in the 25 and 50%-male sex-ratio conditions (by 53 and 50%, respectively, see odd ratio in Table 1b, Fig. 1b).
Table 1 Survival of adult females (a) and males (b) of Tenebrio molitor according to sex-ratio condition (Sex-ratio). The "simple" contrast was used for Sex-ratio (survival of males in the 50% of male condition was used as baseline). For females (a), a time-dependent procedure was used to account for the time-dependent effect of Sex-ratio on the risk of mortality (T × Sex-ratio). This procedure was not necessary for males as the effect of Sex-ratio on the risk of mortality was constant over time (b)
Age-specific survival according to sex-ratio condition. a females; b males
Whereas female fertility decreased with time, this pattern was dependent on the sex-ratio condition (Table 2a, Fig. 2a, see Additional file 1: Figure S2). Indeed, female fertility in the 75 and 25%-male sex-ratio conditions was lower than that in the 50%-male sex-ratio condition during the first 2 weeks, and became subsequently higher (Fig. 2a). Male fertility decreased with time in all sex-ratio conditions, with no significant effect of sex-ratio condition (Table 2b, Fig. 2b, see Additional file 1: Figure S2). As expected for both sexes, heavier females produced more larvae than lighter ones (Table 2).
Table 2 Fertility: generalized linear mixt models (GLMM, Poisson distribution, Log link function) analysing the factors influencing the number of larvae produced by females (a), (n = 638) and males (b), (n = 737)
Age-specific fertility of females (a) and males (b) according to sex-ratio condition. a the tested females were those coming from the experimental tanks. b the tested females were virgin females mated with males coming from the experimental tanks. Dots are the means (for variation around the means see Additional file 1: Figure S2) and lines are the predictions of the models
Female's RE differed among sex-ratio conditions (F2, 12 = 8.06 p = 0.006) and was significantly higher in the 75%-male than in the 25%-male sex-ratio condition (Fig. 3a). Female RE in the 50%-male sex-ratio condition showed an intermediate value (Fig. 3a). Male RE significantly differed among sex-ratio conditions (F2, 12 = 4.63 p = 0.032). Male RE in the 75%-male sex-ratio condition was significantly lower than in the 25%-male sex-ratio condition (Fig. 3b). Like for females, male's RE from the balanced sex-ratio condition showed an intermediate value, which was not significantly different from the two other sex-ratio conditions (Fig. 3b).
Estimated mean reproductive effort. Reproductive effort (RE - mean number of viable offspring produced per individual and per week of survival in the population) of females (left panel) and males (right panel) according to sex-ratio condition. Lines are means, dots are values of single replicates. Values surrounded by different letters were significantly different after Tukey-Kramer HSD post-hoc test (α = 0.05)
Body condition and immunity
Male and female body condition, estimated by the residuals of the regression between body size and body mass, exhibited a similar decline with age, which was not affected by the sex-ratio condition (Table 3a, see Additional file 1: Figure S3). In females, immunological parameters were never affected by the sex-ratio condition (Table 3b-f). Both PO activity and Total-PO activity changed with age with lowest values at week 6 (Fig. 4a), and were positively influenced by body condition (Table 3b, c). By contrast, anti-bacterial activity of the haemolymph increased with age (Table 3e, Fig. 4b). Haemocyte counts of females only differed among population replicates (Table 3d, see Additional file 1: Figure S3). As opposed to females, some of the immunological parameters of males were affected by the sex-ratio condition (Table 3). Male PO activity was influenced by the interaction between time and sex-ratio (Table 3b). While PO activity of males in the 50%-male sex-ratio condition decreased during the first 6 weeks, PO activity of males in the other two sex-ratio conditions increased between week 2 and 4. In all sex-ratio conditions, PO activity increased again at week 12 (Fig. 4c). In addition, PO activity of males in the 25%-male sex-ratio condition was overall lower than PO activity of males in the other sex-ratio conditions (Fig. 4c). Total-PO activity only differed between population replicates (Table 3c). As in females, antibacterial activity in the haemolymph of males increased with age (Table 3e, Fig. 4d). However, the size of the zones of inhibition of males exhibiting positive antibacterial activity (reflecting the intensity of this activity) was higher for males in the 50%-male sex-ratio condition than for males in the other sex-ratio conditions (Table 3f, Fig. 4e). Finally, haemocyte counts of males in all sex-ratio conditions varied with time, mainly because of its high value at week 6 (Table 3d, Fig. 4f).
Table 3 Body condition and immune parameters. Mixed linear models or generalized linear model analysing the factors influencing body condition (a), PO activity (b), Total PO activity (c), haemocyte count (d), the proportion of beetles exhibiting antibacterial activity in their haemolymph (e) and the intensity of this antibacterial activity as the size of the zone of inhibition (f) in both females (left) and males (right). Models included sex-ratio condition, Age in weeks (ordinal variable), their interaction, body condition, and replicates as a random factor
Immune parameters in females and males according to individual age and/or sex-ratio condition. a females PO activity; b proportion of females exhibiting antibacterial activity; c male PO activity; d proportion of males exhibiting antibacterial activity; e male intensity of antibacterial activity as the size (in mm) of zones of inhibition; f male haemocyte count. Values are means among replicates ± s. e. m. Number in the bars are sample size
By manipulating the sex-ratio of artificial populations of mealworm beetles, Tenebrio molitor, we successfully affected the reproductive effort of both males and females. Note that for males, our estimations are rather relevant of their reproductive potential effort or maximal reproductive effort because they were tested using young virgin females. As predicted, female biased sex-ratio led females to exhibit a relatively low reproductive effort, whereas males reproductive potential was the highest. By contrast, male biased sex-ratio increased the reproductive effort of females while that of males dropped. Males and females from populations with balanced sex-ratio exhibited intermediate reproductive effort. Interestingly, males and females of the 50% sex-ratio condition had similar longevity. This absence of divergent patterns of actuarial senescence between males and females may result from a relatively strong investment of males into mating activity rather than into sexual competition. Therefore, like in females, longevity appears to be an important criterion to maximize fitness in males of T. molitor.
While varying in their reproductive effort according to sex-ratio condition, females exhibited different patterns of actuarial and reproductive senescence. Females in the 75%-male sex-ratio condition (with the higher reproductive effort) suffered from an increased mortality compared to females in the other conditions. While we did not directly observed the mating behaviour in our experiments, frequent mating events and harassments by males may explain this accelerated mortality and is in line with previous observations from other insect species [34, 63,64,65].
Female fertility with time in the 50% sex-ratio condition contrasted to that of females in the other two sex-ratio conditions. They produced many offspring during the first 2 weeks of their adult life, then fewer to become almost null at 8 weeks onward. As previously reported in both vertebrates and invertebrates [22, 66], intense early reproductive activity is associated to earlier reproductive decline. Females in the other sex-ratio conditions (25 and 75% of males) produced relatively fewer offspring when young adults but kept this reproductive effort when becoming older. In the 25% sex-ratio condition, the low proportion of males may have constrained female access to mating, preventing them to reach their maximal early reproductive potential. Such a low early reproductive investment, possibly accompanied by low male harassment, may have preserved female late reproduction. In the 75% male sex-ratio condition, high proportion of males was expected to increase interactions between males as well as enhancing mate guarding [38]. This may also have prevented young females to have optimal access to mating, while compensating by a higher probability of mating as they aged. However, female reproduction in this male-biased sex-ratio condition stopped earlier than in the others, because their survival had rapidly declined. All together, the data suggest that the reproductive effort of females in the 75% male sex-ratio condition was more costly than that of females in the other sex-ratio conditions, constraining them to trade-off their longevity against their reproduction.
Despite a more costly reproductive effort, females in the 75%-male sex-ratio condition did not exhibit any further functional decline compare to females in the other sex-ratio conditions. While body condition declined with female age, such a decline was similar in all sex-ratio conditions. Similarly, changes in female immune activity were never influenced by the sex-ratio. While haemocyte counts remained constant with female age, antibacterial activity increased. Similar results were reported in the bumblebee, Bombus terrestris [53]. With age, the probability of having been exposed to microbes increases. This may explain the higher proportion of older individuals exhibiting induced antibacterial activity in their haemolymph, as insects can produce prophylactic long lasting antibacterial responses after a single bacterial challenge [67, 68]. PO activity declined at week 6 in females, which is consistent with the beginning of senescence, when reproduction started to end but survival is still relatively high (Fig. 1, see Additional file 1: Figure S2). PO activity increased again at week 12, among the rare surviving individuals (Fig. 1). This higher late PO-activity may have two non-exclusive explanations. On the one hand, it may result from selection where individuals with the best somatic protection, involving high PO-activity, survived longer than those having poorer ones. On the other hand, high levels of PO activity at week 12 could also result from a deregulation of the host inflammatory response [69, 70]. The impact of female reproductive effort on immunity seems limited, at least on the constitutive base levels of the immune parameters we have measured. We cannot exclude that female ability to produce an immune response upon challenge or others non-measured immune parameters could be affected. Nonetheless, our results show that increasing the reproductive effort of T. molitor females affected demographic senescence, mainly through longevity reduction, but with apparently limited effect on immune senescence.
Changes in the reproductive effort (reproductive potential) of males through the manipulation of the sex-ratio also affected their survival. It is often assumed that most of the cost of reproduction in males involves sexual pre-copulatory competition. Thus males in the 75%-male sex-ratio condition could have been expected to engage in strong and costly intra-sexual competition for females, resulting in low reproductive success and shorter longevity compared to the other sex-ratio conditions, as previously shown in vertebrates [11, 71, 72] and invertebrates [34]. However, although male fertility slightly declined with age, it was not affected by the experimental sex-ratio condition, suggesting that males exhibited similar patterns of reproductive senescence, independently of their reproductive effort. Male reproductive senescence might also be revealed by the production of lower quality offspring with age [73], which was not tested in this study. In addition, males from the 75%-sex-ratio condition showed longer longevity. This phenomenon may have two main explanations, consistent with predictions linked to the peculiar mating behaviour of T. molitor. On the one hand, competition for females in that sex-ratio condition was not very strong or costly. Under high risk of sexual competition, male reproductive success may depend on their investment into pre-copulatory (e.g., courtship and aggressive behaviours with other males) and/or post-copulatory (e.g., mate guarding) behaviours to limit sperm competition [74]. So far, T. molitor males were never reported to engage in physical contest either before or after copulation [38] and current evidence suggests that male-male competition is unlikely to bear strong costs [38, 41, 75, 76]. Our experimental design nevertheless did not allowed us to verify these assumptions. On the other hand, each mating event is costly for T. molitor males, because nutrient-rich spermatophores are transferred to females in addition to the sperm [42]. Since, on average, males in the 75%-male sex-ratio condition may have copulated less frequently than males in the other sex-ratio conditions, they may have saved resources that contributed to their longer survival. By contrast, males from the 25%-male sex-ratio condition likely performed more mating events than males in the other sex-ratio conditions, but this was apparently not costly enough to significantly impair their longevity compared to the 50%-male sex-ratio condition.
Our results suggest that males in this 25%-male sex-ratio condition had paid a functional cost for their higher reproductive effort, especially in terms of immunity. Indeed, males in the 25%-male sex-ratio condition showed a reduced immune activity possibly resulting from their higher reproductive effort. First, males in the 25%-male sex-ratio condition had reduced PO activity despite having a similar concentration of total phenoloxidase enzymes in their haemolymph than males of the other sex-ratio conditions. This lower PO activity was constant over time and contrasted with that of males of the two other sex-ratio conditions, for which the temporal pattern of PO activity resembled that of females (high levels in early weeks, decline in week 6, and re-increase in week 12). Since mating activity is known to transiently reduce PO activity in T. molitor [52], such a down regulation of the PO activity in males of the 25%-male sex-ratio condition might be reflecting their higher mating activities. Higher secretion of juvenile hormone might be involved in mediating mating-induced PO activity depression [52], which could contribute to reduce longevity [77]. Juvenile hormone also prevents the release of cytotoxic substances by active PO enzymes that could reduce insect longevity by self-damaging host tissues and organs [47, 49,50,51, 78]. These combined effects may have contributed to the observed absence of difference between the survival of males in the 25% and the 50%-male sex-ratio conditions. Second, as observed for females, the proportion of males exhibiting positive antibacterial activity in their haemolymph increased with age in all the sex-ratio conditions. As stated earlier, this was expected as the probability of having being challenged by opportunistic microbes increases with age. However, the size of the zones of inhibition observed from the haemolymph of males in the 25%-male sex-ratio condition was significantly smaller than that of males in the other sex-ratio conditions, suggesting that males in the 25%-male sex-ratio condition produced less antibacterial peptides than the other males. As mating activity, through the production and transfer of spermatophores, is particularly resource-demanding for males in terms of protein content [42], the higher mating activity of males in the 25%-male sex-ratio condition could have depleted the necessary protein resource to produce as much antibacterial peptides as in the other sex-ratio conditions. Mating may mediate such a trade-off through juvenile hormone secretion, which functions to switch on physiological processes associated with gametogenesis and spermatophore production [79].
Manipulating sex-ratio of artificial populations of T. molitor had important impacts on reproductive effort of females and males, but resulted in contrasting sex-specific trade-offs on demographic and immune traits. Increasing female reproductive effort did not affect immunity but strongly reduced longevity. Not surprisingly, females may then maximize fitness by moderate early investment into reproduction and longevity. While decreasing male reproductive effort enhanced longevity, increasing it impairs immunity. Males may therefore favour reproduction at the expense of their immunity when given the opportunity to increase their reproductive effort. This is in line with the Bateman's principle applied to immunity, where males gain fitness by increasing reproductive effort at the expense of immunity [80]. It is also consistent with the disposable soma theory of ageing, as reproduction compromises somatic protection [3, 4]. Nevertheless, our results also suggest that sexual competition in T. molitor is not a strong modulator of the male reproductive strategy towards early mating opportunities [81]. Basically, like in females, most of the cost of reproduction in males results from multiple copulations. This thus contrasts with the hypothesis that males should gain fitness by increasing mating success by investing in sexual competition at the expense of longevity [82, 83]. Since longevity is a key life history trait for both males and females of T. molitor, sex-specific patterns of actuarial senescence are not expected to evolve in this species. Accordingly, males and females showed similar patterns of survival with age in populations with balanced sex-ratio. Our results may further indirectly suggest that divergent actuarial senescence between males and females should evolve in species in which males strongly invest into sexual competition [11, 22].
All data files will be available from the Dryad Digital database at https://doi.org/10.5061/dryad.cvdncjt11.
AICc:
Corrected Akaike Information Criterion
Number of larvae
Mean Lifespan
PO:
Phenoloxidase
ProPO:
ProPhenoloxidase
r :
Reproductive Effort
REML:
REstricted Maximum Likelihood
Stearns SC. The evolution of life histories. Oxford: Oxford University Press; 1992.
Hughes KA, Reynolds RM. Evolutionary and mechanistic theories of aging. Annu Rev Entomol. 2005;50:421–45.
Williams GC. Natural selection, the costs of reproduction, and a refinement of Lack's principle. Am Nat. 1966;100:687–90.
Kirkwood TB, Rose MR. Evolution of senescence: late survival sacrificed for reproduction. Philos Trans R Soc Lond Ser B Biol Sci. 1991;332:15–24.
Jones OR, Scheuerlein A, Salguero-Gómez R, Camarda CG, Schaible R, Casper BB, et al. Diversity of ageing across the tree of life. Nature. 2014;505:169–73.
Jones OR, Gaillard J-M, Tuljapurkar S, Alho JS, Armitage KB, Becker PH, et al. Senescence rates are determined by ranking on the fast-slow life-history continuum. Ecol Lett. 2008;11:664–73.
Rodríguez-Muñoz R, Boonekamp JJ, Liu XP, Skicko I, Fisher DN, Hopwood P, et al. Testing the effect of early-life reproductive effort on age-related decline in a wild insect. Evolution. 2019;73:317–28.
van Noordwijk AJ, de Jong G. Acquisition and allocation of resources: their influence on variation in life history tactics. Am Nat. 1986;128:137–42.
Bleu J, Gamelon M, Sæther B-E. Reproductive costs in terrestrial male vertebrates: insights from bird studies. Proc R Soc B Biol Sci. 2016;283:20152600.
Hamel S, Gaillard J-M, Yoccoz NG, Loison A, Bonenfant C, Descamps S. Fitness costs of reproduction depend on life speed: empirical evidence from mammalian populations. Ecol Lett. 2010;13:915–35.
Lemaître J-F, Gaillard J-M, Pemberton JM, Clutton-Brock TH, Nussey DH. Early life expenditure in sexual competition is associated with increased reproductive senescence in male red deer. Proc R Soc B Biol Sci. 2014;281:20140792.
Tidière M, Gaillard J-M, Müller DWH, Lackey LB, Gimenez O, Clauss M, et al. Does sexual selection shape sex differences in longevity and senescence patterns across vertebrates? A review and new insights from captive ruminants. Evolution. 2015;69:3123–40.
Walker WF. Sperm utilization strategies in nonsocial insects. Am Nat. 1980;115:780–99.
Dewsbury DA. Ejaculate cost and male choice. Am Nat. 1982;119:601–10.
Andersson MB. Sexual selection. Princeton: Princeton University Press; 1994.
Arnqvist G, Rowe L. Sexual conflict. Princeton: Princeton University Press; 2005.
Hosken DJ. Sex and death: microevolutionary trade-offs between reproductive and immune investment in dung flies. Curr Biol. 2001;11:R379–80.
Stockley P. Sexual conflict resulting from adaptations to sperm competition. Trends Ecol Evol. 1997;12:154–9.
Jennions MD, Petrie M. Why do females mate multiply? A review of the genetic benefits. Biol Rev Camb Philos Soc. 2000;75:21–64.
Harshman LG, Zera AJ. The cost of reproduction: the devil in the details. Trends Ecol Evol. 2007;22:80–6.
Fortin M, Meunier J, Laverré T, Souty-Grosset C, Richard F-J. Joint effects of group sex-ratio and Wolbachia infection on female reproductive success in the terrestrial isopod Armadillidium vulgare. BMC Evol Biol. 2019;19:65.
Lemaître J-F, Berger V, Bonenfant C, Douhard M, Gamelon M, Plard F, et al. Early-late life trade-offs and the evolution of ageing in the wild. Proc R Soc B Biol Sci. 2015;282:20150209.
DeVeale B, Brummel T, Seroude L. Immunity and aging: the enemy within? Aging Cell. 2004;3:195–208.
Shanley DP, Aw D, Manley NR, Palmer DB. An evolutionary perspective on the mechanisms of immunosenescence. Trends Immunol. 2009;30:374–81.
Norris K, Evans MR. Ecological immunology: life history trade-offs and immune defense in birds. Behav Ecol. 2000;11:19–26.
Owens IPF, Wilson K, Owens IPF, Wilson K. Immunocompetence: a neglected life history trait or conspicuous red herring? Trends Ecol Evol. 1999;14:170–2.
Zera AJ, Harshman LG. The physiology of life history trade-offs in animals. Annu Rev Ecol Syst. 2001;32:95–126.
Fedorka KM, Zuk M, Mousseau TA. Immune suppression and the cost of reproduction in the ground cricket, Allonemobius socius. Evolution. 2004;58:2478–85.
Simmons LW, Roberts B. Bacterial immunity traded for sperm viability in male crickets. Science. 2005;309:2031.
French SS, DeNardo DF, Moore MC. Trade-offs between the reproductive and immune systems: facultative responses to resources or obligate responses to reproduction? Am Nat. 2007;170:79–89.
Drummond-Barbosa D, Spradling AC. Stem cells and their progeny respond to nutritional changes during Drosophila oogenesis. Dev Biol. 2001;231:265–78.
Greenman CG, Martin LB, Hau M. Reproductive state, but not testosterone, reduces immune function in male house sparrows (Passer domesticus). Physiol Biochem Zool. 2005;78:60–8.
Schwenke RA, Lazzaro BP, Wolfner MF. Reproduction-immunity trade-offs in insects. Annu Rev Entomol. 2016;61:239–56.
Rodríguez-Muñoz R, Boonekamp JJ, Fisher D, Hopwood P, Tregenza T. Slower senescence in a wild insect population in years with a more female-biased sex ratio. Proc Biol Sci. 2019;286:20190286.
Tompkins EM, Anderson DJ. Sex-specific patterns of senescence in Nazca boobies linked to mating system. J Anim Ecol. 2019;88:986–1000.
Clutton-Brock TH, Parker GA. Potential reproductive rates and the operation of sexual selection. Q Rev Biol. 1992;67:437–56.
Klug H, Heuschele J, Jennions MD, Kokko H. The mismeasurement of sexual selection. J Evol Biol. 2010;23:447–62.
Carazo P, Font E, Alfthan B. Chemosensory assessment of sperm competition levels and the evolution of internal spermatophore guarding. Proc Biol Sci. 2007;274:261–7.
Gage MJG, Baker RR. Ejaculate size varies with socio-sexual situation in an insect. Ecol Entomol. 1991;16:331–7.
Gadzama NM, Happ GM. The structure and evacuation of the spermatophore of Tenebrio molitor L. (Coleoptera: Tenebrionidae). Tissue Cell. 1974;6:95–108.
Drnevich JM. Number of mating males and mating interval affect last-male sperm precedence in Tenebrio molitor L. Anim Behav. 2003;66:349–57.
Carver FJ, Gilman JL, Hurd H. Spermatophore production and spermatheca content in Tenebrio molitor infected with Hymenolepis diminuta. J Insect Physiol. 1999;45:565–9.
Dick J. Oviposition in certain Coleoptera. Ann Appl Biol. 1937;24:762–96.
Drnevich JM, Papke RS, Rauser CL, Rutowski RL. Material benefits from multiple mating in female mealworm beetles (Tenebrio molitor L.). J Insect Behav. 2001;14:215–30.
Vigneron A, Jehan C, Rigaud T, Moret Y. Immune defenses of a beneficial pest: the mealworm beetle, Tenebrio molitor. Front Physiol. 2019;10. https://doi.org/10.3389/fphys.2019.00138.
Nappi AJ, Ottaviani E. Cytotoxicity and cytotoxic molecules in invertebrates. Bioessays. 2000;22:469–80.
Sadd BM, Siva-Jothy MT. Self-harm caused by an insect's innate immunity. Proc Biol Sci. 2006;273:2571–4.
Pursall ER, Rolff J. Immune responses accelerate ageing: proof-of-principle in an insect model. PLoS One. 2011;6:e19972.
Daukšte J, Kivleniece I, Krama T, Rantala MRJ, Krams IA. Senescence in immune priming and attractiveness in a beetle. J Evol Biol. 2012;25:1298–304.
Krams I, Daukšte J, Kivleniece I, Kaasik A, Krama T, Freeberg TM, et al. Trade-off between cellular immunity and life span in mealworm beetles Tenebrio molitor. Curr Zool. 2013;59:340–6.
Khan I, Agashe D, Rolff J. Early-life inflammation, immune response and ageing. Proc R Soc B Biol Sci. 2017;284:20170125.
Rolff J, Siva-Jothy MT. Copulation corrupts immunity: a mechanism for a cost of mating in insects. PNAS. 2002;99:9916–8.
Moret Y, Schmid-Hempel P. Immune responses of bumblebee workers as a function of individual and colony age: senescence versus plastic adjustment of the immune function. Oikos. 2009;118:371–8.
Khan I, Prakash A, Agashe D. Immunosenescence and the ability to survive bacterial infection in the red flour beetle Tribolium castaneum. J Anim Ecol. 2016;85:291–301.
Moret Y, Schmid-Hempel P. Survival for immunity: the price of immune system activation for bumblebee workers. Science. 2000;290:1166–8.
Licastro F, Candore G, Lio D, Porcellini E, Colonna-Romano G, Franceschi C, et al. Innate immunity and inflammation in ageing: a key for understanding age-related diseases. Immun Ageing. 2005. https://doi.org/10.1186/1742-4933-2-8.
Shenk MK. Fertility and fecundity. In: The international encyclopedia of human sexuality. New York City: American Cancer Society; 2015. p. 369–426. https://doi.org/10.1002/9781118896877.wbiehs153.
Moret Y. "Trans-generational immune priming": specific enhancement of the antimicrobial immune response in the mealworm beetle, Tenebrio molitor. Proc Biol Sci. 2006;273:1399–405.
Zanchi C, Troussard J-P, Martinaud G, Moreau J, Moret Y. Differential expression and costs between maternally and paternally derived immune priming for offspring in an insect. J Anim Ecol. 2011;80:1174–83.
Norušis MJ. IBM SPSS Statistics 19 statistical procedures companion. Upper Saddle River: Prentice Hall; 2012. https://trove.nla.gov.au/version/173310980. Accessed 31 Oct 2019
Burnham KP, Anderson DR. Multimodel inference: understanding AIC and BIC in model selection. Sociol Methods Res. 2004;33:261–304.
Jehan C, Chogne M, Rigaud T, Moret Y. Data from: sex-specific patterns of senescence in artificial insect populations varying in sex-ratio to manipulate reproductive effort. Dryad Digital Repository. 2020. https://doi.org/10.5061/dryad.cvdncjt11.
Adler MI, Bonduriansky R. The dissimilar costs of love and war: age-specific mortality as a function of the operational sex ratio. J Evol Biol. 2011;24:1169–77.
Archer CR, Zajitschek F, Sakaluk SK, Royle NJ, Hunt J. Sexual selection affects the evolution of lifespan and ageing in the decorated cricket Gryllodes Sigillatus. Evolution. 2012;66:3088–100.
Tatar M, Carey JR, Vaupel JW. Long-term cost of reproduction with and without accelerated senescence in callosobruchus maculatus: analysis of age-specific mortality. Evolution. 1993;47:1302–12.
Creighton JC, Heflin ND, Belk MC. Cost of reproduction, resource quality, and terminal investment in a burying beetle. Am Nat. 2009;174:673–84.
Haine ER, Pollitt LC, Moret Y, Siva-Jothy MT, Rolff J. Temporal patterns in immune responses to a range of microbial insults (Tenebrio molitor). J Insect Physiol. 2008;54:1090–7.
Dhinaut J, Chogne M, Moret Y. Immune priming specificity within and across generations reveals the range of pathogens affecting evolution of immunity in an insect. J Anim Ecol. 2018;87:448–63.
Scherfer C, Tang H, Kambris Z, Lhocine N, Hashimoto C, Lemaitre B. Drosophila Serpin-28D regulates hemolymph phenoloxidase activity and adult pigmentation. Dev Biol. 2008;323:189–96.
Labaude S, Moret Y, Cézilly F, Reuland C, Rigaud T. Variation in the immune state of Gammarus pulex (Crustacea, Amphipoda) according to temperature: are extreme temperatures a stress? Dev Comp Immunol. 2017;76:25–33.
Promislow DEL. Costs of sexual selection in natural populations of mammals. Proc R Soc Lond Ser B Biol Sci. 1992;247:203–10.
Clutton-Brock TH, Isvaran K. Sex differences in ageing in natural populations of vertebrates. Proc Biol Sci. 2007;274:3097–104.
Carazo P, Molina-Vila P, Font E. Male reproductive senescence as a potential source of sexual conflict in a beetle. Behav Ecol. 2011;22:192–8.
Font E, Desfilis E. Chapter 3 - courtship, mating, and sex pheromones in the mealworm beetle (Tenebrio molitor). In: Ploger BJ, Yasukawa K, editors. Exploring animal behavior in laboratory and field. San Diego: Academic Press; 2003. p. 43–58. https://doi.org/10.1016/B978-012558330-5/50005-4.
Happ GM. Multiple sex pheromones of the mealworm beetle, Tenebrio molitor L. Nature. 1969;222:180.
Drnevich JM, Hayes EF, Rutowski RL. Sperm precedence, mating interval, and a novel mechanism of paternity bias in a beetle (Tenebrio molitor L.). Behav Ecol Sociobiol. 2000;48:447–51.
Herman WS, Tatar M. Juvenile hormone regulation of longevity in the migratory monarch butterfly. Proc Biol Sci. 2001;268:2509–14.
Zhao P, Lu Z, Strand MR, Jiang H. Antiviral, anti-parasitic, and cytotoxic effects of 5,6-dihydroxyindole (DHI), a reactive compound generated by phenoloxidase during insect immune response. Insect Biochem Mol Biol. 2011;41:645–52.
Wigglesworth VB. The principles of insect physiology. Netherlands: Springer; 1972. https://www.springer.com/la/book/9780412246609. Accessed 4 Apr 2019
Rolff J. Bateman's principle and immunity. Proc Biol Sci. 2002;269:867–72.
Bonduriansky R, Maklakov A, Zajitschek F, Brooks R. Sexual selection, sexual conflict and the evolution of ageing and life span. Funct Ecol. 2008;22:443–53.
Vinogradov AE. Male reproductive strategy and decreased longevity. Acta Biotheor. 1998;46:157–60.
Carranza J, Pérez-Barbería FJ. Sexual selection and senescence: male size-dimorphic ungulates evolved relatively smaller molars than females. Am Nat. 2007;170:370–80.
We thank G. Sorci for critical comments on the manuscript, C. Sabarly, A. Salis and M. Teixeira Brandao for technical assistance.
This study was funded by the CNRS and a grant from the ANR (ANR-14-CE02–0009). Funding agencies contributed strictly financially to the performed research.
UMR CNRS 6282 BioGéoSciences, Équipe Écologie Évolutive, Université Bourgogne-Franche Comté, Dijon, France
Charly Jehan, Manon Chogne, Thierry Rigaud & Yannick Moret
Charly Jehan
Manon Chogne
Thierry Rigaud
Yannick Moret
CJ, TR and YM conceived the ideas and designed methodology, CJ, MC and YM collected the data, CJ, TR and YM analysed the data, CJ, TR and YM led the writing of the manuscript. All authors contributed critically to the drafts and gave final approval for publication.
Correspondence to Charly Jehan or Yannick Moret.
Table S1. AICc values for the models presented in Tables 2 and 3. The models in bold are those presented in the tables. Figure S1. Male and female age specific mortality rate according to time for each sex-ratio condition. Arrow indicate the time when the values of 50% of mortality rate is reached for the first time and dashed lines indicate 50% of the population is dead. Values are means among replicates ± s. e. m. Figure S2. Details of variation in fertility in females (A) and males (B). Values are means among replicates ± s. e. m. Figure S3. Physiological parameters: females in bright grey and males in black. Body condition, PO activity according to time, Total-PO activity according to time and sex-ratio condition, Haemocyte count, Proportion of individuals producing antibacterial activity, Diameter of inhibition zone according to time and sex-ratio condition. Values are means among replicates ± s. e. m.
Jehan, C., Chogne, M., Rigaud, T. et al. Sex-specific patterns of senescence in artificial insect populations varying in sex-ratio to manipulate reproductive effort. BMC Evol Biol 20, 18 (2020). https://doi.org/10.1186/s12862-020-1586-x
Disposable soma theory
Immuno-senescence
Tenebrio molitor
Evolutionary ecology and behaviour | CommonCrawl |
An anonymous entropy-based location privacy protection scheme in mobile social networks
Lina Ni1,
Fulong Tian1,
Qinghang Ni2,
Yan Yan1 &
Jinquan Zhang ORCID: orcid.org/0000-0001-5193-06871
EURASIP Journal on Wireless Communications and Networking volume 2019, Article number: 93 (2019) Cite this article
The popularization of mobile communication devices and location technology has spurred the increasing demand for location-based services (LBSs). While enjoying the convenience provided by LBS, users may be confronted with the risk of privacy leakage. It is very crucial to devise a secure scheme to protect the location privacy of users. In this paper, we propose an anonymous entropy-based location privacy protection scheme in mobile social networks (MSN), which includes two algorithms K-DDCA in a densely populated region and K-SDCA in a sparsely populated region to tackle the problem of location privacy leakage. The K-DDCA algorithm employs anonymous entropy method to select user groups and construct anonymous regions which can guarantee the area of the anonymous region formed be moderate and the diversity of the request content. The K-SDCA algorithm generates a set of similar dummy locations which can resist the attack of adversaries with background information. Particularly, we present the anonymous entropy method based on the location distance and request contents. The effectiveness of our scheme is validated through extensive simulations, which show that our scheme can achieve enhanced privacy preservation and better efficiency.
Nowadays, the Internet of Things (IoT) is building a connected world seamlessly and enhancing the quality of our daily life throughout applications coming from consumer, commercial, industrial, and infrastructure spaces [1–3]. It provides more intelligent services and makes them more efficient via accessing to and storage as well as processing of data [4–7]. An increasing number of people would prefer making use of smart mobile devices to access the Internet for social activities; thus, the emerging mobile social networks (MSN) have promoted the development of a new application pattern of location-based services (LBS) which is a location-based value-added service provided by a location service provider and brings great convenience to our lives [8–10].
As we know, LBS can be used in location-based point of interest retrieval service, navigation service, social service, motion detection service, advertisement push service, and so on [11, 12]. When using LBS, users may need to send the personal identification information, the location information, or the request content to the LBS server. The service provider receives the request and processes it to provide the user location-based services, such as Meituan WaiMai, Didi ChuXing, and BaiDu Map. Unfortunately, LBS could explore the preferences and behavior patterns of users by analyzing the users' location information [13, 14]. If these information is abused or resold by the service provider or intercepted by the attacker, the users' location information may be disclosed and the potential threat may be posed to them. Therefore, the location privacy protection is of crucial challenge [15, 16].
Current system architectures for location privacy protection include standalone architecture, distributed peer-to-peer architecture, and central server architecture. At present, most of the location privacy protection methods based on the k-anonymity model adopt the central server architecture. In this paper, we employ this architecture in our scheme as well and assume that the central anonymous server is trusted as in many methods [13, 15, 17]. The central server architecture consists of mobile users, LBS servers, and location anonymous server [17], as depicted in Fig. 1. As the core of the whole architecture, the anonymous server is responsible for processing anonymously the information such as the users' real location, filtering the candidate query result set returned by the service provider and then returning them to the users who send LBS queries.
Central server architecture. It consists of mobile users, LBS servers, and location anonymous server. As the core of the whole architecture, the anonymous server is responsible for processing anonymously the information such as the users' real location, filtering the candidate query result set returned by the service provider and then returning them to the users who send LBS queries
In the real MSN scenario, areas where many mobile users using LBS are divided into densely populated regions and sparsely populated regions. In densely populated regions, numerous mobile users send LBS requests simultaneously, while few mobile users send LBS requests or the content requested by the user is single in sparsely populated regions. As for the location privacy protection of mobile users in these two regions, it is of equal importance. The existing location privacy protection methods generally consider the location privacy protection in sparsely populated regions or in densely populated regions separately or do not explicitly indicate the specific region. In fact, the location privacy protection method is different in these two regions. In this work, we propose the K-DDCA algorithm in densely populated regions and the K-SDCA algorithm in sparsely populated regions respectively to protect the location privacy of mobile users, comprehensively considering the user location distance and request content and combining with the kd-tree algorithm.
In recent years, many techniques have been proposed to solve the problem of location privacy protection, such as location perturbation and obfuscation [18, 19], region anonymization [20–22], and dummy location [23]. The region anonymization technique, which reduces the probability of identifying the real user to 1/k, is an important one for location privacy protection. However, there are some problems with this technique. First, the region anonymization technique may form redundant regions in the process of constructing an anonymous region without considering the diversity of the request contents. Second, the neighbors constructing anonymous region may be in the vicinity of real users, which may expose the real users' location and reduce the user's experience. Meanwhile, this technique fails to solve the issue that the anonymity cannot satisfy the user's demand, resulting in privacy leakage of mobile user locations in sparsely populated regions, which poses a challenge to location privacy protection.
In this paper, we propose an anonymous region constructing algorithm based on kd-trees in densely populated regions (K-DDCA), which employs the kd-tree algorithm [24, 25] to search neighbor users. Compared with other neighbor search algorithms, the kd-tree algorithm can improve the search efficiency. Furthermore, according to our proposed anonymity entropy, it selects the nearest neighbor users that meet the requirements and achieve k-anonymity among the returned candidate sets of neighbor users together with the real users to construct an anonymous region. When constructing an anonymous region, we comprehensively consider the distance between the real requesting users and their neighbor users as well as the difference of the content requested by them. Thus, in a densely populated region like a school, the distance between neighbor users may be small, making it easier for an attacker to infer the user's region. However, since we consider the distance and the content of the user's requests and add randomness when constructing the user groups, it can effectively protect the user's identity information and request contents, and the attacker cannot associate a request with a user.
The dummy location generation technique [26, 27] solves the problem of user location privacy leakage caused by neighbor users in sparsely populated regions without satisfying the requirement of anonymity. This technique does not rely on trusted third-party servers to build an anonymous region with the dummy location data and the real location of the requesting users, reducing the communication overhead. However, this technique does not fully consider the context when generating data. For example, the generated dummy location points may be located in sparsely populated regions such as lakes, rivers, and swamps. If the attacker has mastered certain background knowledge (such as maps and historical query records), the dummy location can be easily filtered out by the adversaries, which cannot satisfy the user's anonymity requirement.
Considering this problem of typical dummy location generation technique, we propose an anonymous region constructing algorithm based on kd-trees in sparsely populated regions (K-SDCA). This algorithm takes into account the temporal and spatial factors and finds out multiple candidate users based on historical access records, thus preventing the selected users from clustering together. K-SDCA utilizes anonymous entropy to select user groups with uniform geographical distribution and large differences in request content. In this way, even if the attacker has mastered a certain background knowledge, the dummy users cannot be easily filtered out, thereby achieving the user's anonymity requirement.
Our main contributions are listed as follows:
We present a more efficient location privacy protection scheme based on anonymous entropy in MSN, which utilizes central server architecture and local area anonymization to protect the location privacy in both dense region and sparse region.
We propose two location privacy protection algorithms K-DDCA in a densely populated region and K-SDCA in a sparsely populated region to construct anonymous regions.
We propose the anonymous entropy method to effectively and securely select user groups based on the location distance and request contents and further construct anonymous regions, which can guarantee that the area of the anonymous region formed be moderate, and ensure the diversity of the request content.
The rest of the paper is organized as follows: Section 2 introduces the problems we have studied, the proposed methods, and the simulation environment. We discuss the related work in Section 3. Section 4 introduces the basic concepts and the kd-tree algorithm. Section 5 introduces the system structure model, together with our algorithm design, and the security analysis. Section 6 shows the evaluation results and discussion. We conclude this paper in Section 7.
Methods/experimental
In this paper, we study the issue of location privacy protection in MSN employing kd-tree as the storage structure. Specially, we put forward an anonymous entropy method based on on the location distance and request contents to tackle the problem of location privacy leakage. Our scheme consists of two algorithms K-DDCA in a densely populated region and K-SDCA in a sparsely populated region.
Our experimental environment is 64 bit Windows 7 system with Intel (R) Core i7-8700k CPU @ 3.70 GHz, and the RAM of 32G. The programming language is Python on PyCharm. We adopt OPNET [28, 29] to generate data and execute simulation experiments. A large number of experiments and security analysis prove that our scheme has high security and better efficiency.
The privacy protection has been well studied in MSN. In this section, we introduce some work on location privacy protection in LBS related to our scheme. At present, the privacy-preserving schemes are categorized into three methods including the spatial cloaking method [30–32], dummy location method [33–35], and cryptography primitive-based method [36–38].
Spatial cloaking method
The spatial cloaking technology [30] forms a hiding area that contains k real users for each user employing LBS, which makes it difficult for service providers to determine the real identities and accurate locations of users from the hiding region.
In [30], Gruteser et al. first introduced k-anonymity into location privacy and proposed an anonymous usage method of LBS, which obscures the exact location of the real user into an anonymous region. It reduces the probability that the attacker accurately infers the actual location of the user to 1/k. In [31], Abul et al. proposed a quad-tree-based anonymity algorithm which adopts a recursive method to continuously divide the space region in which the mobile user resides into four quadrants. During constructing an anonymous region, it starts from the quadrant where the real user requests the service and expands to the parent quadrant until the user's anonymity attains k (i.e., at least k users in the anonymous region). However, the anonymous region generated by this method is too large and there may exist a large amount of redundant space, resulting in a decline in service quality. The reason for the existence of redundant space is that the algorithm does not consider the distribution of users with their adjacent quadrants in the formation of anonymous region. Meanwhile, due to lack of users in sparsely populated regions, the anonymous region will fail to be constructed.
In [32], Mokbel et al. proposed an anonymous algorithm based on the Casper model, which effectively improves the performance of the anonymity algorithm in [30]. When the user's quadrant does not satisfy k-anonymity, it first expands the anonymous region to the adjacent quadrant and then expands to the parent quadrant if k-anonymity has not been reached yet. Unfortunately, when the real user is located in a sparsely populated region, the Casper algorithm will fail to construct the anonymous region due to lack of sufficient dummy locations. Since the Casper algorithm still uses quad-tree to partition the spatial region and fails to consider the distribution of the target user's adjacent users when it merges the quadrants, there exists redundant region in the anonymity region constructed. Meanwhile, the Casper anonymous algorithm will gradually increase the area of the merged quadrant after it is recursively extended to the parent quadrant; thus, the next merge may be incorporated into the redundant region.
Dummy location method
The dummy location approach [33] generates multiple dummy locations and integrates the users' real locations into the dummy ones and sends them to the service provider for privacy protection. Since the service providers cannot distinguish between real locations and dummy ones, they can only provide the required services for each submitted location.
In [33], Niu et al. proposed DLS and enhanced DLS algorithms by selecting several candidate dummy locations with similar query probabilities. In [34], Wu et al. formulated a multi-objective optimization (MOS) algorithm considering both the query probability and the area of the anonymous region to generate k−1 dummy locations and achieve k-anonymity. Although these methods reduce the possibility that some location points are filtered out, they do not take into account the inherent differences between the requests. Moreover, since the procedure of the algorithms takes place on mobile clients, the amount of computing is relatively large and they have high requirements on computing power and storage space of mobile clients.
In [35], Liao et al. proposed the K-DLCA algorithm considering the semantic information and the historical query probability of the locations and using greedy strategy to select each dummy location that can maximize the current entropy. However, since the selected dummy locations are located near the real user, this algorithm may expose the real user's location.
Cryptography primitive-based method
The cryptography primitive-based approach [36–38] mainly employs related cryptography to process the requests sent by users, which can protect users' privacy information and obtain the service data. This type of scenario does not require an anonymous server and just needs to encrypt the user's location information and send it to the service provider. The service provider decrypts and executes the corresponding query, then encrypts and returns the results. Thus, the cryptography primitive-based approach has excellent security. However, the obvious drawback is that the computational overhead is comparatively large. Therefore, the feasibility of such schemes is relatively poor.
In our strategy, the K-DDCA algorithm employs kd-tree to search nearby users in the densely populated region. During the generation of anonymous regions, neighbor users are selected according to the distance and the request content among users without generating redundant regions, reducing the time complexity and improving the quality of service. In sparsely populated regions, considering the historical records and geographical distribution, the K-SDCA algorithm achieves k-anonymity by selecting dummy users with high historical query probability, relatively uniform geographical distribution, and large difference in request contents.
In this section, we first present some concepts and then introduce the idea and procedure of the kd-tree algorithm. Specifically, we devise an anonymous entropy method to construct the anonymous region.
Here, we give some basic concepts to lay a foundation for our scheme.
(k-anonymity [39]) k-anonymity is proposed by Sweeney et al. as a privacy protection mechanism in data publishing. A data object is said to have the k-anonymity property if the information for this target data object cannot be distinguished from at least the other k−1 individuals whose information also appear in the data publishing. Thereafter, k-anonymity is considerablely extended to protect location privacy and query privacy in LBS.
In this paper, we adopt the classical notion of Euclidean distance as our distance measure.
(Distance measure) Assume that the locations of two users ui and uj are \(\left ({x_{i}},{y_{i}}\right)\) and \(\left ({x_{j}},{y_{j}}\right)\), respectively, the distance measure (i.e., Euclidean distance) between ui and uj, denoted by dist, is defined as
$$ dist\left(u_i,u_{j}\right)=\sqrt{\left(x_i-x_{j}\right)^{2}+\left(y_i-y_{j}\right)^{2}}, $$
where xi, yi represent the longitude and latitude of the location of user ui, respectively.
(Distance between user and line) Assume that two users ui and uj form a straight line, denoted by lineij, then the distance between user ul and line lineij, denoted by dul, is defined as
$$ d_{ul}=\frac{|\left(y_j-y_{i}\right){x_l}+\left(x_i-x_{j}\right){y_l}+\left({x_j}{y_i}-{x_i}{y_j}\right)|}{\sqrt{\left(y_j-y_{i}\right)^{2}+\left(x_i-x_{j}\right)^{2}}}, $$
where \(\left ({x_{i}},{y_{i}}\right)\), \(\left ({x_{j}},{y_{j}}\right)\), and \(\left ({x_{l}},{y_{l}}\right)\) are the locations of users ui, uj, and ul, respectively.
(Region partition) Assume that Reg is a region and its radius is r. Based on historical experience, we set thresholds MinU and MinC which represent the minimum number of users and the minimum number of request content types of Reg, respectively. For any ui∈Reg, the following region partitions hold:
All the users in Reg who satisfied the condition whose distance from ui is not greater than r are called r-neighborhood of ui, denoted by \(N_{r}\left (u_{i}\right)\), that is,
$$ N_{r}\left(u_{i}\right)=\{u_j \in Reg | dist\left(u_i,u_{j}\right)\le r\}. $$
All the users in Reg who satisfied the condition whose request contents are different from ui are called θ-neighborhood of ui, denoted by \(N_{\theta }\left (u_{i}\right)\), that is,
$$ N_\theta \left(u_{i}\right)=\{u_{j}\in Reg | boolean\left(u_i,u_{j}\right)=False\}, $$
where boolean() is a boolean value, which represents whether the request contents of the users are same.
(Densely/sparsely populated region) Assume there are at least MinU concurrent users initiating the request at the same time as ui in the r-neighborhood of user ui, that is, \(|N_{r}\left (u_{i}\right)| \ge MinU\), and the θ-neighborhood of ui contains at least MinC kinds of request contents, that is, \(|N_{\theta }\left (u_{i}\right)| \ge MinC\), then we call Reg as a densely populated region; otherwise, the region is called a sparsely populated region.
In Definition 5, MinU and MinC are all based on historical experience.
(Historical query probability) Assume that Reg is divided into n∗n cells. Note that, according to the number of historical queries in the cell, each cell may have its own probability of being queried. Then, the historical query probability of cell i is the ratio of the number of cell i queried versus the total number of queries for all cells in Reg, denoted by pi, that is,
$$ phq_i=\frac{QCell_i}{\sum_{j=1}^{n^{2}}QCell_j}, \qquad \left(i=1,\cdots,n^{2}\right), $$
where QCelli is the number of the ith cell queried and \(\sum _{j=1}^{n^{2}}QCell_{j}\) is the total number of queries in the whole region. Obviously, we have \(\sum _{i=1}^{n^{2}}phq_{i}=1\).
(Entropy[33]) Entropy is used to measure the uncertainty of a set, the bigger the value of the entropy the more uncertain of the set. Formally, it is defined as
$$ H(R)=-{{\sum_{i=1}^{n}p_i \log p_i} }, $$
where pi represents the probability that the user is identified.
Many methods [33] use entropy to evaluate the anonymity of anonymous regions. In this paper, we adopt the same evaluation criteria as well.
Anonymous entropy method
In order to select users who are evenly distributed with real users, it is needed to consider the distance and the request content among users. Therefore, we propose the anonymous entropy method to construct the anonymous region.
Suppose that there are 2k users in Reg, k−1 users are selected randomly to form a user group Ui with real users. The process is repeated m times, and m user groups are formed, where m is defined by the user according to his privacy requirements.
Entropy of distance
On the one hand, the distance among users is considered. The goal of the anonymous entropy method on distance is as follows. If the sum of distances between k−1 users and the real user among m user groups are equal, the user group which is evenly distributed is selected. Otherwise, if the total distances are not equal, the user group with a larger distance is selected. In order to achieve the above goal, the entropy is used to select a user group on distance. Here, the weight of the neighbor user ui in the nth user group is denoted as αni, that is,
$$ \alpha_{ni}=\frac{dist\left(u_{real},u_{i}\right)}{{\sum_{j=1}^{2k}{dist\left(u_{real},u_{j}\right)}}} \qquad \left(i=1,\cdots,k-1\right). $$
Then, the entropy of the nth user group for distance is
$$ H(n)=-\sum_{i=1}^{k-1}{\alpha_{ni} \log \alpha_{ni}}. $$
Figure 2 depicts a scenario that MSN users request LBS service, which is an illustration example of the neighbor user selection. Dots represent the users, where red dot A represents the real user and other seven black dots represent the neighbor users. Eight users with labels {A,B,⋯,G} are deployed in the region. Lines between the dots indicate the spatial neighbor relation between the users, where the digits marked on the lines indicate the distance. Suppose that there are three user groups \(G_{1}=\left \{A,B,C,F\right \}\), \(G_{2}=\left \{A,D,E,F\right \}\), and \(G_{3}=\left \{A,C,D,G\right \}\).
Illustration of the neighbor user selection. It depicts a scenario that MSN users request LBS service. Dots represent the users, where red dot A represents the real user and other seven black dots represent the neighbor users. Eight users with labels {A,B,⋯,G} are deployed in the region. Lines between the dots indicate the spatial neighbor relation between the users, where the digits marked on the lines indicate the distance
The total distance of users in G1 to the real user A is less than those of G2 and G3. The total distance between G2 and G3 is equal. As shown in Fig. 2, G2 is evenly distributed; however, G3 is not uniformly distributed from the real user. According to formula (8), the entropy on distance of each group is obtained shown as \(H\left (G_{1}\right)=1.2174\), \(H\left (G_{2}\right)=1.3415\), and \(H\left (G_{3}\right)=1.3155\). It can be seen that G2 has the largest entropy for distance; thus, it has the greatest degree of anonymity. According to the principle of anonymous entropy method, the user group G2 is chosen here.
Content weights in user groups
On the other hand, the request content among users is considered. A weight is assigned to the request content in the user group based on the number of request types and the distribution in the user group, that is, the more request types in the user group and the more uniform the request type distribution, the greater the content weight of the user group. The weight of the request content is denoted by βn, that is,
$$ \beta_n=\frac{2 boolean\left(u_{i}^{c},u_{j}^{c}\right)}{k\left(k-1\right)}, \quad \left(i,j=1,\cdots k,i\not= j\right), $$
where \(u_{i}^{c}\) and \(u_{j}^{c}\) represent the request contents of ui and uj, respectively.
As shown in Fig. 2, suppose a user group GA contains five users. A is the real user and \(\left \{B,C,D,E\right \}\) is the selected neighbors. Assume that there are three types of request contents in GA, such as the requests of the adjacent hospital, hotel, and shopping mall. Then, we have two forms of request type distribution in user groups, \(G_{4}=\left \{\Diamond,\Box,\Delta \Delta \Delta \right \}\), \(G_{5}=\left \{\Diamond \Diamond,\Box \Box,\Delta \right \}\), where the symbols ♢, \(\Box \), and Δ represent different types of request content from user, and each interval separated by a comma represents the same request type. For example, we can suppose that \(G_{4}=\left \{A,B,CDE\right \}\), \(G_{5}=\left \{AB,CD,E\right \}\). In G4, user A may request the location of the adjacent hospital, user B may request the hotel, while users C,D, and E may request the shopping mall. Or user E may request the location of the adjacent hospital, user D may request the hotel, while users A,B, and C may all request the shopping mall. But they have the same request distribution. According to formula (9), the weight of each form of user group can be calculated as \(\beta _{G_{4}}=7/10\), \(\beta _{G_{5}}=8/10\).
If there are two types of request contents in GA, the distribution of the request types in GA is divided into two forms: \(G_{6}=\left \{\Diamond,\Delta \Delta \Delta \Delta \right \}\), \(G_{7}=\left \{\Diamond \Diamond,\Delta \Delta \Delta \right \}\). For example, suppose that \(G_{6}=\left \{A,BCDE\right \}\), \(G_{7}=\left \{AB,CDE\right \}\). According to formula (9), the weights are calculated to be \(\beta _{G_{6}}=4/10\), \(\beta _{G_{7}}=6/10\), respectively. According to the principle of anonymous entropy method, here the user group is selected who has the most number of request types and request types evenly distributed among users. In this example, the user group G5 is selected who has three request types and even distribution.
Combination of distance and content metric
Taking into account the distance between users in the user group and the distribution of the request content jointly, the anonymous entropy for the nth user group is defined as the sum of the entropy in terms of distance and the difference on request content in the nth user group, denoted by
$$ HA(R)=-{{\sum_{i=1}^{k-1}\alpha_{ni} \log \alpha_{ni}}+\beta_n}. $$
Now, the user group with the largest anonymous entropy is selected in m user groups as the anonymous region, represented as
$$ U_{\text{max}}=arg\max \limits_{n \in\left(1,\cdots, m\right)}\{HA_{n}\}. $$
The above is the anonymous entropy method we proposed. In the following, we will illustrate our motivation for calculating the anonymous entropy based on the difference in distance and the request content.
Illustration of the selection of distance and content metric
Figure 3 shows three real scenarios that users send requests to the LBS server. In each subgraph, the left part represents the user's anonymous region and the right part shows the diversity of content requested by the user respectively. The lines connecting the two parts represent the relationship between the user request and the corresponding LBS service.
LBS request type It shows three real scenarios that users send requests to the LBS server. In each subgraph, the left part represents the user's anonymous region and the right part shows the diversity of content requested by the user. The lines connecting the two parts represent the relationship between the user request and the corresponding LBS service. a Location privacy protected. b Content privacy inferred. c Location privacy inferred
Fortunately, as the figure shows that Fig. 3a represents a perfect scenario where the location privacy of the user is protected because both the area of the anonymous region and the difference of the request content are all larger. Unfortunately, on the one hand, Fig. 3b shows that the user's content privacy information can be inferred according to the user's LBS request, namely, content privacy inferred. On the other hand, Fig. 3c shows that the user's real location can be inferred based on the user's LBS request, namely, location privacy inferred.
Note that Fig. 3b and c indicate two imperfect scenarios. In Fig. 3b, since the request contents are the same with different users, there exists risk of privacy leakage. For example, if the content requested by the four users are all nearby hospital, the probability of the users being sick is the greatest, and the attacker may use this information to propose a targeted strategy to attack those users. In Fig. 3c, although the request content is different, unfortunately, they are issued by the same location (i.e., multiple users are clustered at the same location). Thus, the area of the anonymous region is too small so the user's real location could be easily inferred by the attacker.
In a word, we propose the anonymous entropy method comprehensively considering the distance between the request users and the difference of the request contents, which can effectively protect the privacy of the users.
kd-tree algorithm
kd-tree is a kind of balanced binary tree that divides data points in k-dimensional space, which is mainly applied to the search of key data in multi-dimensional space [24,25]. Essentially, it is a spatial partitioning tree. In this paper, we use the nodes of kd-tree to store the users' locations. The operation of kd-tree is divided into two phases including kd-tree construction and kd-tree search [24].
Phase 1: kd-tree construction
The detailed steps of constructing a kd-tree are as follows:
(1) Construct the root node, which represents the region containing all the users, denoted by U.
(2) Select x axis as the split axis and the median of U's x coordinates as the segmentation point. Divide the corresponding region of the root node into two sub-regions, which is achieved by the line that is through segmentation point and perpendicular to the split axis. Thus,
The left child node of the kd-tree corresponds to the sub-region whose x coordinates are smaller than the segmentation point.
The right child node corresponds to the sub-region whose coordinates are greater than the segmentation point.
The root node stores the points that fall on the split axis.
(3) For each node whose depth is j, calculate d=jmod 2. If d=0, select y axis as the split axis. Otherwise, x axis is selected as the split axis.
The median of the split axis corresponding to the region of the node is used as the segmentation point. Thus, the corresponding region of the node is divided into two sub-regions.
(4) The process is iterated until there is no location data in these two sub-regions. The construction of kd-tree is completed.
Phase 2: kd-tree search
In a kd-tree, the neighbor search is achieve by maintaining a queue of 2k nodes. The detailed steps of searching a kd-tree are as follows:
(1) Find the leaf nodes containing the real user.
Recursively visit the kd-tree downwards from the root node.
If the coordinates of the real user on the current split axis are smaller than those of the segmentation points, then move to the left child node to search real user; otherwise, move to the right child node until the child node is a leaf node.
Now, this leaf node is treated as the current nearest point and is put into the queue.
(2) Recursively roll back upwards the search path and perform the following operations on each node.
If a location point saved by this node that is closer to the real user than the current nearest point is found, the current location point is taken as the current nearest point and placed in the queue. If the queue is full, the head of the queue is dequeued.
Draw a circle around the center of the requesting user with the distance from the requesting user to the nearest point.
Check whether the region corresponding to another child node intersects the circle. If it intersects, there may exist points closer to the request user in the another region. Then, move to another region and recursively search for the nearest neighbor. If they do not intersect, go back upwards.
(3) When going back to the root node, the search ends if it does not intersect another sub-region.
From the above description, it can be seen that the ineffective nearest neighbor search can be greatly reduced after the construction of kd-tree. Since many locations do not intersect with the circle, there is no need to calculate the distance at all, which saves a lot of computing time.
Anonymous entropy-based location privacy protection scheme
In this section, we illustrate the system model of our scheme and elaborate the details of the algorithm design as well as security analysis.
The architecture of our proposed system model consists of the following three components: (1) MSN users, (2) anonymous servers, and (3) LBS providers, as shown in Fig. 4. Our system model is inspired by the classical central server architecture. However, our system has more superiorities on anonymous servers.
Architecture of the proposed system model. It consists of the following three components: (1) MSN user, (2) anonymous servers, and (3) LBS providers. The system model is inspired by the classical central server architecture, and it has more superiorities on anonymous servers
1) MSN users: MSN user sends a LBS request LBSQ to an anonymous server through mobile terminals, formally, LBSQ={id,(x,y),c,k,A}, where id and (x,y) are the identity and location coordinates of the user, respectively; c represents the content of the request services; k represents the privacy degree set by the users, namely, the number of users in the anonymous region; and A denotes the minimum area of the anonymous region.
2) Anonymous servers: Anonymous server mainly consists of two modules, that is anonymous processing module and candidate result filter module.
The anonymous processing module mainly deals with the anonymity of MSN users, which is the core of our scheme. It involves in two algorithms K-DDCA and K-SDCA, which will be elaborated whereafter.
After receiving the LBS requests sent by users, the anonymous server anonymously processes the location privacy, identity, request content, and other information according to the requirements of the users. The procedure is as follows:
Construct the kd-tree based on the region where the requesting user is located.
If the kd-tree is constructed successfully, the nearest neighbor users are searched on the kd-tree, and the anonymity is processed according to the K-DDCA algorithm.
If the kd-tree fails to be constructed or the K-DDCA algorithm fails to process anonymously, the K-SDCA algorithm is executed anonymously, and then send the anonymous request to the LBS provider.
The candidate result filter module filters the result set returned by the LBS provider according to the real location of the requesting user and finally sends the accurate result to the MSN user.
3) LBS providers: According to the received LBS requests, LBS provider executes calculation to find out the candidate result set required by the MSN user and returns it to the anonymous server.
Algorithm design
In this paper, we adopt the central server architecture as our system architecture, which can reduce the requirements of the high computing power and storage space of the mobile terminals. Our scheme is divided into two processing alternatives including both K-DDCA algorithm in the densely populated region and K-SDCA algorithm in the sparsely populated region.
K-DDCA
In the densely populated region, we propose the K-DDCA algorithm to construct anonymous regions. K-DDCA mainly includes the following procedures.
It is notable that we use kd-tree to store user data. After the successful construction of kd-tree, we find the nearest 2k users of real users according to the kd-tree search algorithm [24]. We fully consider the location relationship among the neighbor users to ensure that no redundant regions are generated and time complexity is reduced. Here, the neighbor search is realized by maintaining a queue containing 2k nodes.
After finishing the search of the kd-tree, if 2k neighbors are found, then m user groups are formed according to the requirements of the real users. Each user group includes real users and the randomly selected k−1 users among 2k neighbor users.
According to our anonymous entropy method, the uncertainty of the user groups is evaluated. Then, the user group with the largest entropy is selected.
If the queue is not full after finishing the search of kd-tree or the diversity of request content of the final user group is less than the threshold value MinC, the real user is in a sparsely populated region. Then, the K-SDCA algorithm is employed to generate dummy locations.
The pseudo code of K-DDCA is elaborated in Algorithm 1.
The K-DDCA algorithm is mainly divided into two phases. Firstly, K-DDCA needs to find the nearest 2k users from the real user based on the kd-tree algorithm [24]. Since the average computational complexity of the kd-tree search algorithm is O(logN), the average time complexity of searching for neighbors is also O(logN).
Then, K-DDCA selects k−1 neighbors among 2k users to achieve k-anonymity together with real users. We need to execute m rounds. In each round, K-DDCA forms a user group and calculates the anonymous entropy of the user group. Afterward, K-DDCA selects a user group with the largest entropy among them. Thus, the time complexity of this process is O(m).
Due to N≫m, the total time complexity of the K-DDCA algorithm is O(logN+m)=O(logN). □
K-SDCA
Motivation of K-SDCA
In a sparsely populated region, we utilize the K-SDCA algorithm to generate the dummy location. As the assumption before, the region where the real user is located is divided into n∗n cells. Anonymous server reads all the historical query probabilities of the locations near the real user in the whole region. Here, we adopt the method of calculating historical query probabilities proposed in [33]. The following is the construction process of anonymous region for the K-SDCA algorithm:
Firstly, considering the locations with similar query probability can be aggregated together to facilitate the search of targets, we sort all the historical query probabilities. After that, we can conveniently select the 3k cells with the similar historical query probability to the real user. The reason for selecting 3k cells is to construct an anonymous region with the appropriate number of candidates.
Secondly, the anonymous server constructs a smaller set of 2k candidate dummy locations from the 3k candidates using the heuristic method in order to exclude some dummy locations gathered together.
Thirdly, the anonymous server employs the anonymous entropy method to effectively and securely select k−1 users among 2k users and further achieve k-anonymity together with the real users.
Procedure of K-SDCA
To clarify the procedure of K-SDCA, let ureal denote the cell where the real user is currently located. We employ the heuristic method to choose \(\left (u_{1},\cdots,u_{2k-1}\right)\) in turn through 2k−1 rounds, namely, u1 is selected in the first round, u2 is selected in the second round, and so on. In each round, each remaining candidate is endowed with a weight and the dummy user of this round is selected with a probability proportional to its weight. Let x denote the number of the remaining candidates in each round, and wi denote the weight of ui. The procedure of K-SDCA is as follows.
First, we calculate the distances among 3k users and ureal, sort them in reverse order, and then store them in list \(\widehat {U}\).
In the odd rounds, each remaining candidate ui in list \(\widehat {U}\) is endowed a weight, which is the ratio of the distance between ui and ureal to total distances in \(\widehat {U}\), denoted by woi, that is
$$ w_{oi}=\frac{dist\left(u_i,u_{real}\right)}{\sum{_{u_{j}\in \widehat{U}}}dist\left(u_j,u_{real}\right)}. $$
In the even rounds, a straight line linereal,p is generated through ureal and the point selected in the previous round. Then, each remaining candidate ui in the list \(\widehat {U}\) is endowed a weight based on its distance from linereal,p, denoted by wei, that is
$$ w_{ei}=\frac{d_{ul}\left(u_i,line_{{real},p}\right)}{\sum{_{u_{i}\in \widehat{U}}}d_{ul}\left(u_i,line_{{real},p}\right)}. $$
In each round, \(u_{i}\left (i=1,\cdots,x\right)\) is selected as a dummy user with probability
$$ \frac{w_i}{\sum_{j=1}^{x}w_j}. $$
The selected user ui is added to the selected user group U and then removed from list \(\widehat {U}\).
The detailed pseudo code of K-SDCA is specified in Algorithm 2.
The K-SDCA algorithm is mainly divided into three phases. Firstly, K-SDCA sorts all the historical query probabilities via Shell sort and selects 3k cells with the similar historical query probabilities to real user. Since the average time complexity of Shell sort is \(O\left (NlogN\right)\), the time complexity of this phase is \(O\left (NlogN\right)\).
Secondly, K-SDCA selects 2k evenly distributed cells from the 3k cells through 2k rounds. The time complexity of selecting 2k dummy locations is O(k), where k represents the number of users in the final anonymous region.
Thirdly, K-SDCA selects the user group with the maximum entropy among the m user groups to achieve k-anonymity after executing m rounds. Thus, the time complexity of this phase is O(m).
Due to N>k>m, the total time complexity of the K-SDCA algorithm is \(O(NlogN+k+m)=O\left (NlogN\right)\). □
K-DDCA and K-SDCA can achieve k-anonymity.
According to the definition of k-anonymity (Definition 1) in Section 4, obviously, the meaning of k-anonymity is to form a cloaking region containing k users for each query user. In this way, the real user will become indistinguishable from other k−1 users.
We apply the K-DDCA algorithm in a densely populated region. Specifically, K-DDCA forms m user groups, where each user group includes the real user and the randomly selected k−1 users among 2k neighbor query users. Then, the user group with the largest entropy is selected by utilizing the anonymous entropy method. According to the principle of anonymous entropy, this user group has the maximum uncertainty. Therefore, the k-anonymity can be achieved in the densely populated region.
Due to the lack of enough neighbor query users in a sparsely populated region, our K-SDCA algorithm selects neighbor users and builds user groups by means of historical records in the anonymous server. In addition, each user group contains a real user and k−1 historical query users. According to the anonymous entropy method, the user group which has the largest uncertainty among user groups is chosen. Thus, our designed K-SDCA algorithm can achieve k-anonymity in the sparsely populated region. □
Example of K-SDCA
The illustration of the dummy location (user) selection is shown in Fig. 5, which depicts a scenario that users send requests to the LBS server in the sparsely populated region. The solid red dot A represents the real user, the solid black dots represent the selected dummy locations (users) by the K-SDCA algorithm, and the numbers indicate the order in which they are selected. The hollow dots represent the unselected dummy locations. In addition, the solid lines between two locations represent the connection between dummy locations selected in the odd round, and the dotted lines are vertical lines representing the distances from the location point to the straight line.
Illustration of dummy location (user) selection. It depicts a scenario that users send requests to the LBS server in a sparsely populated region. The solid red dot A represents the real user, the solid black dots represent the selected dummy locations (users) by the K-SDCA algorithm, and the numbers indicate the order in which they are selected. The hollow dots represent the unselected dummy locations. The solid lines between two locations represent the connection between dummy locations selected in the odd round, and the dotted lines are vertical lines representing the distances from the location point to the straight line
When real user A requests an LBS service, since there are no adequate users that send LBS requests simultaneously in a sparsely populated region, the anonymous servers have to utilize the historical locations where previous users sent the LBS requests to construct the anonymous region.
Anonymous servers first locate 3k locations according to similar query probability with A among these historical records and then store the selected locations to \(\widehat {U}\).
Thus, in the first round, the number of remaining candidates x=3k and the weight o f each candidate uj in \(\widehat {U}\) is
$$w_{oj}=\frac{dist\left(u_j,A\right)}{\sum{_{u_{i}\in \widehat{U}}}dist\left(u_i,A\right)} $$
according to formula (12). In Fig. 5, we can see that the cells far away from the real location A have the higher probability of being selected.
Since it may be easier to deduce the actual location in some cases, here we do not directly select the cell that is the farthest from A. The dummy locations are selected according to formula (14). When the dummy user E is selected, it is queued into U and removed from \(\widehat {U}\), then a line lineA,E between A and E is formed.
In the second round, the number of remaining candidates x=3k−1, and according to the distance between uj and lineA,E, the weight of each remaining candidate uj in \(\widehat {U}\) is
$$w_{ej}=\frac{d_{ul}\left(u_j,line_{A,E}\right)}{\sum{_{u_{i}\in \widehat{U}}}d_{ul}\left(u_i,line_{A,E}\right)}. $$
Then, we calculate the probability of uj being selected by formula (14). Suppose the dummy user D is selected here. We add it to U and remove it from \(\widehat {U}\).
Repeat the above process until 2k dummy locations {A,B,C,D,E,F} in Fig. 5 are found. Subsequently, we use the anonymous entropy method to construct an anonymous region.
Since the query probability is similar, it is not easy to filter out the locations for an attacker. Furthermore, 2k evenly distributed locations are selected from the 3k candidates in order to prevent the selected locations from aggregating together which would expose the real users' location. Finally, k users are selected from 2k users to form m user groups. For each user group, one cell is the real location and the other k−1 cells are dummy locations. In addition, we need to calculate the anonymous entropy of each user group to select the user group with maximum anonymous entropy.
In some scenarios, an attacker may collude with some users to obtain additional information about other users. Furthermore, an attacker may also collude with a LBS service provider to utilize the information obtained by the LBS service provider to infer other sensitive information of legitimate users for profit. Fortunately, our scheme can resist the collusion attacks of adversaries and show stronger security. If the probability of successful deducing the real user's location does not increase with the size of colluding group, the algorithm is colluding-attack resistant.
We give the security analysis of our scheme in the following.
K-DDCA and K-SDCA are colluding-attack resistant.
Our scheme can protect the user's location privacy by the interference of other neighbors when the real user is located in densely populated regions. In sparsely populated regions, our scheme also can protect the user's location privacy by generating dummy locations. Now, we prove our algorithm is colluding-attack resistant.
1) In densely populated regions, when an attacker colludes with user uA, the gained information may include the history query probability, the current query, and the historical query.
On the one hand, 2k locations are selected to randomly form m user groups which have the largest anonymous entropy. On the other hand, the distance to the real users and the request content of those is taken into account. Therefore, other users in the same group can not know the real user's location and only know their user group that possesses the largest anonymous entropy, which signifies that the user can only randomly deduce the location of the requesting user, and the probability of successful deducing is 1/k.
Afterwards, the attacker also colludes with the user uB. Because there is no connection between uA and uB, the probability that the attacker deduces the real user's position is still 1/k. Therefore, collusion attacks can be resisted in densely populated regions.
2) In sparsely populated regions, the attacker intercepts user uA to obtain the related information such as the historical search probability. Firstly, the cells which have the similar query probability with the real user are selected. Then, the cells that evenly distributed around the real user are chosen. Finally, it selects the user group with the highest entropy according to the anonymous entropy method. Since the information obtained by the attacker cannot help speculate the actual location of the real user, the successful deducing possibility is 1/k.
Afterwards, the attacker also colludes with user uB. As there is no connection between uA and uB, the probability for the adversary inferring the real user's location is still 1/k, which means that the probability for obtaining the real user privacy cannot be increased with the size of the colluding group; it can resist collusion attack in sparsely populated regions.
In extreme circumstance, the LBS server may be act as an adversary. At the moment, the LBS service provider owns both the history queries and current queries which include the user's identifier, the mix of real and dummy locations, and the request content.
In the K-DDCA algorithm, 2k neighbor query users are first selected, and then the anonymous entropy method is used to select k−1 query users to construct an anonymous region based on the requested content and distance. The LBS service provider cannot speculate the real location of the query user based on the information already available.
In the K-SDCA algorithm, 3k locations with the similar query probability to the real location are first selected, and then the user group is stochastically constructed according to the distance relationship among 3k locations. On this basis, the anonymous entropy method is utilized to select the anonymous region to ensure the uncertainty. Therefore, even if the LBS service provider has possessed global information, it cannot infer the real location of the user. □
In this section, we evaluate the performance of our scheme via extensive experiments. We give the detailed experimental results and discussion.
Experimental settings
The experimental environment is 64 bit Windows 7 system with Intel (R) Core i7-8700k CPU @ 3.70 GHz and the RAM of 32G. The programming language is Python on PyCharm.
We use OPNET [28,29] to generate the simulation data which can better describe human behavior patterns, construct complex network topologies, and simulate the process of sending/receiving of message.
Assume that we take a 4 km × 4 km region as our simulation region which is divided into 40 × 40 cells where 20 points of interests (POIs) are randomly and uniformly generated. In the densely populated region, the K-DDCA algorithm is compared with the quad-tree algorithm [30] and the Casper algorithm [31]. However, in the sparsely populated region, the quad-tree algorithm and the Casper algorithm cannot complete the construction of the anonymous region. We will compare the K-SDCA algorithm with the following four algorithms, that is the random dummy selection algorithm, DLS algorithm, enhanced-DLS algorithm [33], and MOS algorithm [34].
Comparison of anonymous processes between the Casper algorithm and our algorithm
In Fig. 6, we simulate the anonymization process of the Casper algorithm and K-DDCA algorithm. Figure 6a shows the Casper anonymous process, and Fig. 6b indicates the K-DDCA algorithm anonymous process. The shaded area represents the anonymous region. Assume that there is a LBS request req and k=2.
Comparison of anonymous processes between Casper algorithm and K-DDCA algorithm. It depicts the anonymization process of the Casper algorithm and the K-DDCA algorithm. Subgraph a shows the Casper anonymous process, and subgraph b indicates the K-DDCA algorithm anonymous process. The shaded area represents the anonymous region. Assume that there is a LBS request req and k=2
Figure 6a shows that u1 sends a LBS request. Since the adjacent quadrant does not meet the requirement of k, it expands to the parent quadrant until it meets the requirement of k. In order to satisfy k, it should extend to the entire region. If an attacker finds that users u2 to u5 are in the anonymous region of u1 and their respective anonymous regions do not contain u1, it is possible to speculate that the request is issued by u1 when an attacker finds that the anonymous region contains the entire space, which is a risk of privacy leaks. Fortunately, there is no danger of privacy disclosure in Fig. 6b which is an anonymous region formed by our algorithm.
Relationship between the area of the anonymous regions and k value
Figure 7 indicates the evaluation results in details where we can clearly see that the area of the anonymous region formed by the quad-tree algorithm, Casper algorithm, and K-DDCA algorithm gradually increases with the raise of the k value. Assume that the number of people in the region is 1000 here. With the increase of k, the area of anonymous regions becomes increasingly large. However, since the quad-tree algorithm and Casper algorithm use the quad-tree model for storage, they do not take full account of the location relationships of the neighboring users. In particular, when it extends to a high level, each expansion leads to a larger area increase, generating redundant space. Therefore, the size of the anonymous region is too large which means that the anonymous users are denser accordingly, so that the real user's location could be easily identified by the attacker.
Relationship between the area of anonymous region and k value. It shows that as the k value increases, the area of the anonymous region constructed by the K-DDCA algorithm is appropriate compared to the other two algorithms
It can be seen from the figure that the area of anonymous regions of K-DDCA becomes stable to 2 km2 when K=16. The reason is that the K-DDCA algorithm takes full account of the location relationship between neighboring users and then selects evenly distributed users among neighbors. Therefore, it can guarantee that the area of the anonymous region formed will be moderate, and the area of the anonymous region can be set according to the user's needs. If the area does not meet the requirements of the user's k value, then K-SDCA is used to generate the dummy location, and the quad-tree algorithm and Casper algorithm can only achieve the user's requirements by expanding the area of the anonymous region.
Relationship between the number of users and the area of anonymous region
Figure 8 shows the relationship between the area of the anonymous region and the number of users under the condition of k=10, k=15 from the Casper algorithm and K-DDCA algorithm. It can be seen from the curves that the area of the anonymous region gradually decreases with the increase of user number. After the user density reaches 900, it tends to be stable.
Relationship between the number of users and the area of the anonymous region. It shows the relationship between the area of the anonymous region and the number of users under the condition of k=10, k=15 from the Casper algorithm and K-DDCA algorithm
According to the K-DDCA algorithm, the user can determine the area of the anonymous region (set to 4 km2). When the number of users is less than k at the beginning, the K-SDCA algorithm generates dummy locations and construct an anonymous region. Therefore, when the number of users is large enough, the area of generating anonymous region tends to be stable. However, in order to achieve the k value specified by the user, the Casper algorithm will always enlarge the area of the anonymous region. If the number of users in the entire area is less than k, the Casper algorithm will not complete the construction of the anonymous region.
Relationship between the privacy level and k value in densely populated regions
Figure 9 indicates the privacy level in terms of entropy of different schemes; we can see that the anonymity of the K-DDCA algorithm and Casper algorithm gradually increases with the raise of the k value. Obviously, the performance of the K-DDCA algorithm is better than that of the Casper algorithm. The anonymous region formed by the Casper model contains a large number of redundant regions which mean that there is no user there as shown in Fig. 6a. An attacker can exclude a number of users based on the background information, so the anonymity of the Casper algorithm cannot reach 1/k.
Comparison of anonymity between the K-DDCA algorithm and the Casper algorithm under different k values. It indicates the privacy level in terms of entropy of different schemes, and the anonymity of the K-DDCA algorithm and Casper algorithm gradually increases with the raise of the k value
Relationship between privacy level and k value among various algorithms in sparsely populated regions
In Fig. 10, we can see that the degree of anonymity gradually increases with the raise of k. Compared to other four algorithms, the dummy locations generated by K-SDCA has a better effect. In this figure, the random scheme performs worse than other schemes, since it just generates dummy locations randomly and without considering background information. Compared with K-SDCA, enhanced-DLS, and MOS algorithms, DLS algorithm does not work well, since it only considers the condition that the query probability of dummy locations is similar to that of the real user and does not consider the distance between users and the difference in request content.
Comparison of anonymity among different algorithms under different k values. It shows that the degree of anonymity gradually increases with the raise of k. Compared to other four algorithms, the dummy locations generated by K-SDCA has a better effect
On the basis of similar query probability, the enhanced-DLS and MOS algorithms generated dummy locations which are as far as possible from real users. The MOS algorithm is better than the enhanced-DLS algorithm, but it does not consider the diversity of the request content, so the anonymity is relatively good. The dummy locations chosen by the K-SDCA algorithm satisfy three constraint conditions: (1) the query probability is similar to that of the real user, (2) the geographical distribution is relatively uniform, and (3) the content of the request is different. Afterward, the K-SDCA algorithm selects the optimal user group utilizing anonymous entropy method, which increases the uncertainty.
Relationship between anonymous time and k value in sparsely populated regions
In Fig. 11, we can see that the anonymous time of enhanced-DLS, K-SDCA, and MOS algorithms gradually increases with the raise of k. Under the conditions of the same k value, the K-SDCA algorithm is more efficient than the enhanced-DLS and MOS algorithms. The reason is that the MOS algorithm needs to compute overlapping areas when forming anonymous regions and then expand the total anonymous area as much as possible.
Relationship between anonymous time and k value among different algorithms in sparsely populated regions. It indicates that the anonymous time of enhanced-DLS, K-SDCA, and MOS algorithms gradually increases with the raise of k. Under the conditions of the same k value, the K-SDCA algorithm is more efficient than the enhanced-DLS and MOS algorithms
The similarity between the enhanced-DLS algorithm and the K-SDCA algorithm is to select one dummy user per round. However, the difference between them is that it is necessary for the enhanced-DLS algorithm to calculate the distance between the candidate users and the selected users and then compare the product of the distances in each round. Fortunately, the K-SDCA algorithm does not need to calculate the product of the distances. It only needs to calculate the distance from the candidate set to the straight line in even rounds by formula (2). Therefore, the K-SDCA algorithm is more efficient than the enhanced-DLS algorithm.
This paper takes kd-tree as the storage structure and puts forward an anonymous entropy-based location privacy protection scheme in MSN. Our scheme consists of two algorithms K-DDCA in a densely populated region and K-SDCA in a sparsely populated region to tackle the problem of location privacy leakage. Specifically, an anonymous entropy method is proposed which takes into account the distance between users in the user group and the distribution of the request content jointly. According to the anonymous entropy method, we select the user group with the largest anonymous entropy to guarantee that the user group has the greatest uncertainty. In addition, our scheme effectively reduces the time complexity and provides users with high-quality services.
In the future, we will research on the location privacy protecting scheme in the sparsely populated region where there is a lack of historical records. Since LBS service providers have many limitations for the data usage, we will research how to evaluate these demand and challenge. Furthermore, we plan to cooperate with some LBS service providers to further validate the effectiveness of our scheme.
α ni :
The weight of the user ui on distance
β n :
The weight of the user group on request content
d ul :
The distance between user ul and line
H(R):
The anonymity of a user group R
LBSQ :
The LBS request
MinU :
The minimum threshold for the number of user
MinC :
The minimum threshold for the kind of request contents
\(N_{r}\left (u_{i}\right)\) :
The r-neighborhood of ui
\(N_{\theta }\left (u_{i}\right)\) :
The 𝜃-neighborhood of ui
p h q i :
The historical query probability of cell i
\(dist\left (u_{i},u_{j}\right)\) :
The distance between user ui and user uj
U max :
The user group with the largest anonymous entropy
w ei :
The weight of candidate ui in the even round
w oi :
The weight of candidate ui in the odd round
\(\left ({x_{i}},{y_{i}}\right)\) :
The location of user ui
D. E. Kouicem, A. Bouabdallah, H. Lakhlef, Internet of things security: a top-down survey. Comput. Netw.141(4), 199–221 (2018).
Y. Huo, C. Hu, X. Qi, T. Jing, Lodpd: A location difference-based proximity detection protocol for fog computing. IEEE Internet Things J.4(5), 1117–1124 (2017).
J. Mao, Y. Chen, F. Shi, Y. Jia, Z Liang, Toward exposing timing-based probing attacks in web applications. Sensors. 17(3), 464 (2017).
Y. Liang, Z. Cai, J. Yu, Q. Han, Y. Li, Deep learning based inference of private information using embedded sensors in smart devices. IEEE Netw. Mag.32(4), 8–14 (2018).
Z. Cai, Z. He, X. Guan, Y. Li, Collective data-sanitization for preventing sensitive information inference attacks in social networks. IEEE Trans. Dependable Secure Comput.15(4), 577–590 (2017).
Y. Liang, Q. Han, Y. Li, et al., Location privacy leakage through sensory data. Secur. Commun. Netw.2017(11), 1–12 (2017).
L. Ni, J. Zhang, C. Jiang, et al., Resource allocation strategy in fog computing based on priced timed petri nets. IEEE Internet Things J.4(5), 1216–1228 (2017).
y. Li, M. L. Yiu, Route-saver: leveraging route apis for accurate and efficient query processing at location-based services. IEEE Trans. Knowl. Data Eng.27(1), 235–249 (2015).
M. Xin, M. Lu, W. Li, An adaptive collaboration evaluation model and its algorithm oriented to multi-domain location-based services. Expert Syst. Appl.42(5), 2798–2807 (2015).
R. Al-Dhubhani, J. M. Cazalas, An adaptive geo-indistinguishability mechanism for continuous LBS queries. Wirel. Netw.24(8), 3221–3239 (2018).
Y. Jing, L. Hu, W. S. Ku, C. Shahabi, Authentication of k nearest neighbor query on road networks. IEEE Trans. Knowl. Data Eng.26(6), 1494–1506 (2014).
V. Bindschaedler, R. Shokri, in 2016 IEEE Symposium on Security and Privacy (SP). Synthesizing plausible privacy-preserving location traces (IEEE, 2016), pp. 546–563.
Y. Sun, M. Chen, L. Hu, Y. Qian, Hassan, ASA: Against statistical attacks for privacy-aware users in location based service. Futur. Gener. Comput. Syst.70:, 48–58 (2017).
Z. Cai, X. Zheng, A private and efficient mechanism for data uploading in smart cyber-physical systems. IEEE Trans. Netw. Sci. Eng. (2018). https://doi.org/10.1109/TNSE.2018.2830307.
X. Zheng, G. Luo, L. Tian, et al., Privacy-preserved community discovery in online social networks. Futur. Gener. Comput. Syst. (2018). https://doi.org/10.1016/j.future.2018.04.020.
L. Ni, C. Li, X. Wang, H. Jiang, Yu J., DP-MCDBSCAN: Differential privacy preserving multi-core DBSCAN clustering for network user data. IEEE Access.6:, 21053–21063 (2018).
T. Xu, Y. Cai, in Proceedings of the 15th annual ACM international symposium on Advances in geographic information systems. Vol. 39. Location anonymity in continuous location-based services (ACM, 2007), pp. 300–307.
C. Li, B. Palanisamy, in Proceedings of the 24th ACM International on Conference on Information and Knowledge Management. Reversecloak: protecting multi-level location privacy over road networks (ACM, 2015), pp. 673–682.
J. Shao, R. Lu, X. Lin, in IEEE INFOCOM 2014-IEEE Conference on Computer Communications. FINE: a fine-grained privacy-preserving location-based service framework for mobile devices (IEEE, 2014), pp. 244–252.
X. Li, M. Miao, H. Liu, J. Ma, K. -C. Li, An incentive mechanism for k-anonymity in LBA privacy protection based on credit mechanism. Soft Comput.21(14), 3907–3917 (2017).
B. Ying, D. Makrakis, in 2014 IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS). Protecting location privacy with clustering anonymization in vehicular networks (IEEE, 2014), pp. 305–310.
J. Zhang, Y. Yuan, X. Wang, et al., RPAR: location privacy preserving via repartitioning anonymous region in mobile social network. Secur. Commun. Netw.2018:, 1–10 (2018).
T. Hara, A. Suzuki, M. Iwata, et al., Dummy-based user location anonymization under real-world constraints. IEEE Access.4:, 673–687 (2016).
H. Samet, The design and analysis of spatial data structures. Off. Sci. Tech. Inf. Tech. Rep.50255(4), 1211 (1990).
X. Wang, Y. Luo, Y. Jiang, W. Wu, Q. Yu, Probabilistic optimal projection partition kd-tree k-anonymity for data publishing privacy protection. Intell. Data Anal.22(6), 1415–1437 (2018).
S. Hayashida, D. Amagata, T. Hara, X. Xie, Dummy generation based on user-movement estimation for location privacy protection. IEEE Access.6:, 22958–22969 (2018).
B. Niu, Q. Li, X. Zhu, et al., in 2015 IEEE conference on computer communications (INFOCOM). Enhancing privacy through caching in location-based services (IEEE, 2015), pp. 1017–1025.
Opnet. https://www.opnet.com/. Accessed 25 Jul 2018.
G. Sun, D. Liao, H. Li, H. Yu, V. I. Chang, L2P2: A location-label based approach for privacy preserving in LBS. Future Gener. Comput. Syst.74:, 375–384 (2017).
M. Gruteser, D. Grunwald, in Proceedings of the 1st international conference on Mobile systems, applications and services. Anonymous usage of location-based services through spatial and temporal cloaking (ACM, 2003), pp. 31–42.
O. Abul, F. Bonchi, M. Nanni, in 2008 IEEE 24th International Conference on Data Engineering (ICDE '08), IEEE Computer Society,. Never Walk Alone: Uncertainty for Anonymity in Moving Objects Databases, (2008), pp. 376–385.
M. F. Mokbel, C. -Y. Chow, W. G. Aref, in Proceedings of the 32nd International Conference on Very Large Data Bases. The new casper: Query processing for location services without compromising privacy (VLDB Endowment, 2006), pp. 763–774.
B. Niu, Q. Li, X. Zhu, G. Cao, H. Li, in IEEE INFOCOM 2014-IEEE Conference on Computer Communications. Achieving k-anonymity in privacy-aware location-based services (IEEE, 2014), pp. 754–762.
D. Wu, Y. Zhang, Y. Liu, in 2017 IEEE Trustcom/BigDataSE/ICESS. IEEE. Dummy Location Selection Scheme for K-Anonymity in Location Based Services (IEEE, 2017), pp. 441–448.
D. Liao, X. Huang, V. Anand, et al., in 2016 IEEE International Conference on Communications (ICC). k-DLCA: an efficient approach for location privacy preservation in location-based services (IEEE, 2016), pp. 1–6.
R. JiangM, R. Lu, K. -K. R. Choo, Achieving high performance and privacy-preserving query over encrypted multidimensional big metering data. Future Gener. Comput. Syst.78:, 392–401 (2018).
Z. Xu, Z. Cai, J. Li, G. Hong, in IEEE INFOCOM 2017-IEEE Conference on Computer Communications. Location-privacy-aware review publication mechanism for local business service systems (IEEE, 2017), pp. 1–9.
Z. Xu, Z. Cai, J. Li, H. Gao, Data linkage in smart internet of things systems: a consideration from a privacy perspective. IEEE Commun. Mag.56(9), 55–61 (2018). https://doi.org/10.1109/MCOM.2018.1701245.
L. Sweeney, k-anonymity: a model for protecting privacy. Int. J. Uncertain. Fuzziness Knowl.-Based Syst.10(05), 557–570 (2002).
This work is supported by National Key R & D Programs Project of China under Grant 2017YFC0804406, NSF of China under Grant 61672321, 61771289, 61832012 and 61373027, Training Program of the Major Research Plan of NSF of China under Grant 91746104, Project of Shandong Province Higher Educational Science and Technology Program under Grant J15LN19, Open Project of Tongji University Embedded System and Service Computing of Ministry of Education of China under Grant ESSCKF 2015-02.
The datasets used in this paper is generated by OPNET generator, and the website is https://www.opnet.com/.
College of Computer Science and Engineering and Shandong Province Key Laboratory of Wisdom Mine Information Technology, Shandong University of Science and Technology, Qingdao, Shandong, China
Lina Ni
, Fulong Tian
, Yan Yan
& Jinquan Zhang
College of Foreign Languages, Shandong Agricultural University, Taian, Shandong, China
Qinghang Ni
Search for Lina Ni in:
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The main idea of this paper was proposed by LNN. The system model and algorithm were designed by JQZ and FLT. The architecture was given by JQZ. The simulations were conducted by FLT, YY, and LNN. The writing of the paper was completed by FLT, QHN, YY, and JQZ. All authors read and approved the final manuscript.
Correspondence to Jinquan Zhang.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Ni, L., Tian, F., Ni, Q. et al. An anonymous entropy-based location privacy protection scheme in mobile social networks. J Wireless Com Network 2019, 93 (2019) doi:10.1186/s13638-019-1406-4
Location privacy protection
Location-based service
k-anonymity
Anonymous entropy
kd-tree
Recent Advances in Internet of Things Security and Privacy | CommonCrawl |
# Mathematical foundations of gradient descent
Gradient descent is an optimization algorithm used in machine learning and deep learning to minimize a cost function. It is a first-order optimization method that uses the negative gradient (or approximate gradient) of the cost function to update the parameters iteratively. The goal is to find the minimum of the cost function, which represents the optimal solution to the problem.
The basic idea behind gradient descent is to move in the direction of the steepest decrease of the cost function. This is achieved by computing the gradient of the cost function at the current parameter values and using it to update the parameters. The update is performed using the following formula:
$$
\theta_{t+1} = \theta_t - \eta \nabla f(\theta_t)
$$
where $\theta_t$ represents the current parameter values, $\eta$ is the learning rate, and $\nabla f(\theta_t)$ is the gradient of the cost function at $\theta_t$.
Let's consider a simple example of a linear regression model. The cost function is the mean squared error between the predicted values and the actual values. The gradient of the cost function with respect to the parameters can be computed as follows:
$$
\nabla f(\theta) = \frac{2}{m} \sum_{i=1}^m (y_i - \theta^T x_i) x_i
$$
where $m$ is the number of training examples, $y_i$ is the target value for the $i$-th example, $\theta$ is the parameter vector, and $x_i$ is the feature vector for the $i$-th example.
## Exercise
Compute the gradient of the mean squared error cost function for a linear regression model.
### Solution
$$
\nabla f(\theta) = \frac{2}{m} \sum_{i=1}^m (y_i - \theta^T x_i) x_i
$$
# Implementing gradient descent in Python
To implement gradient descent in Python, you can use the NumPy library for vectorized operations and the Matplotlib library for visualization. Here's an example of how to implement gradient descent for linear regression:
```python
import numpy as np
import matplotlib.pyplot as plt
# Generate data
np.random.seed(0)
m = 100
X = np.random.rand(m, 1)
y = 4 + 3 * X + np.random.randn(m, 1)
# Gradient descent
theta = np.random.randn(2, 1)
learning_rate = 0.01
iterations = 1000
for i in range(iterations):
gradients = 2/m * X.T @ (X @ theta - y)
theta = theta - learning_rate * gradients
# Plot the result
plt.scatter(X, y)
plt.plot(X, X @ theta, 'r-')
plt.show()
```
# Understanding the challenges of gradient descent
Gradient descent has several challenges that need to be addressed:
- Scalability: Gradient descent can be computationally expensive for large datasets and high-dimensional parameter spaces.
- Convergence: Gradient descent may not always converge to the global minimum of the cost function, especially if the learning rate is too small or the cost function has local minima.
- Vanishing and exploding gradients: In deep learning, the gradients of the cost function can vanish or explode when using stochastic gradient descent, leading to slow convergence or unstable updates.
## Exercise
Discuss the challenges of gradient descent and how to address them in practice.
### Solution
Gradient descent has several challenges that need to be addressed:
- Scalability: To address this challenge, we can use parallel and distributed computing techniques, such as mini-batch gradient descent or stochastic gradient descent.
- Convergence: We can use techniques like momentum or adaptive learning rate methods to improve the convergence of gradient descent.
- Vanishing and exploding gradients: We can use techniques like gradient clipping, batch normalization, or weight initialization methods to prevent the gradients from vanishing or exploding.
# Introducing stochastic gradient descent
Stochastic gradient descent (SGD) is an extension of gradient descent that addresses some of its challenges. In SGD, instead of computing the gradient of the cost function for the entire dataset, we compute the gradient for a single training example at each iteration. This allows SGD to scale better to large datasets and high-dimensional parameter spaces.
The update rule for stochastic gradient descent is similar to that of gradient descent:
$$
\theta_{t+1} = \theta_t - \eta \nabla f(\theta_t, x_t, y_t)
$$
where $\theta_t$ is the current parameter values, $\eta$ is the learning rate, and $\nabla f(\theta_t, x_t, y_t)$ is the gradient of the cost function at $\theta_t$ for the $t$-th example.
Here's an example of how to implement stochastic gradient descent for linear regression:
```python
# Stochastic gradient descent
theta = np.random.randn(2, 1)
learning_rate = 0.01
iterations = 1000
for i in range(iterations):
random_index = np.random.randint(m)
xi, yi = X[random_index:random_index+1], y[random_index:random_index+1]
gradients = 2 * (X[random_index:random_index+1] @ theta - y[random_index:random_index+1]) * xi
theta = theta - learning_rate * gradients
```
# Implementing stochastic gradient descent in Python
To implement stochastic gradient descent in Python, you can use the NumPy library for vectorized operations and the Matplotlib library for visualization. Here's an example of how to implement stochastic gradient descent for linear regression:
```python
import numpy as np
import matplotlib.pyplot as plt
# Generate data
np.random.seed(0)
m = 100
X = np.random.rand(m, 1)
y = 4 + 3 * X + np.random.randn(m, 1)
# Stochastic gradient descent
theta = np.random.randn(2, 1)
learning_rate = 0.01
iterations = 1000
for i in range(iterations):
random_index = np.random.randint(m)
xi, yi = X[random_index:random_index+1], y[random_index:random_index+1]
gradients = 2 * (X[random_index:random_index+1] @ theta - y[random_index:random_index+1]) * xi
theta = theta - learning_rate * gradients
# Plot the result
plt.scatter(X, y)
plt.plot(X, X @ theta, 'r-')
plt.show()
```
# Comparing gradient descent and stochastic gradient descent
Gradient descent and stochastic gradient descent have some key differences:
- Scalability: Stochastic gradient descent is more scalable than gradient descent because it computes the gradient for a single training example at each iteration.
- Convergence: Stochastic gradient descent can converge faster than gradient descent because it uses a noisy estimate of the gradient, which can help escape local minima.
- Computational complexity: Gradient descent has a higher computational complexity because it computes the gradient for the entire dataset at each iteration.
## Exercise
Compare gradient descent and stochastic gradient descent in terms of their advantages and disadvantages.
### Solution
Gradient descent and stochastic gradient descent have some key differences:
- Scalability: Stochastic gradient descent is more scalable than gradient descent because it computes the gradient for a single training example at each iteration.
- Convergence: Stochastic gradient descent can converge faster than gradient descent because it uses a noisy estimate of the gradient, which can help escape local minima.
- Computational complexity: Gradient descent has a higher computational complexity because it computes the gradient for the entire dataset at each iteration.
# Applications of optimization algorithms in machine learning
Optimization algorithms are widely used in machine learning for various purposes:
- Model training: Optimization algorithms like gradient descent and stochastic gradient descent are used to train machine learning models by minimizing the cost function.
- Feature selection: Optimization algorithms can be used to select the most important features in a dataset, which can improve the performance of machine learning models.
- Hyperparameter tuning: Optimization algorithms can be used to find the best hyperparameters for a machine learning model, which can lead to better generalization performance.
## Exercise
Discuss the applications of optimization algorithms in machine learning.
### Solution
Optimization algorithms are widely used in machine learning for various purposes:
- Model training: Optimization algorithms like gradient descent and stochastic gradient descent are used to train machine learning models by minimizing the cost function.
- Feature selection: Optimization algorithms can be used to select the most important features in a dataset, which can improve the performance of machine learning models.
- Hyperparameter tuning: Optimization algorithms can be used to find the best hyperparameters for a machine learning model, which can lead to better generalization performance.
# Handling overfitting and regularization
Overfitting is a common problem in machine learning where a model performs well on the training data but poorly on the test data. To prevent overfitting, we can use regularization techniques like L1 or L2 regularization. These techniques add a penalty term to the cost function, which discourages the model from fitting the training data too closely.
In gradient descent and stochastic gradient descent, we can add regularization by modifying the cost function as follows:
$$
\text{Regularized cost function} = \text{Original cost function} + \lambda \sum_{i=1}^n \theta_i^2
$$
where $\lambda$ is the regularization parameter, $\theta_i$ are the model parameters, and $n$ is the number of parameters.
## Exercise
Discuss how to handle overfitting and regularization in gradient descent and stochastic gradient descent.
### Solution
To handle overfitting and regularization in gradient descent and stochastic gradient descent, we can use regularization techniques like L1 or L2 regularization. These techniques add a penalty term to the cost function, which discourages the model from fitting the training data too closely.
In gradient descent and stochastic gradient descent, we can add regularization by modifying the cost function as follows:
$$
\text{Regularized cost function} = \text{Original cost function} + \lambda \sum_{i=1}^n \theta_i^2
$$
where $\lambda$ is the regularization parameter, $\theta_i$ are the model parameters, and $n$ is the number of parameters.
# Using optimization algorithms for deep learning
Optimization algorithms are also widely used in deep learning. In deep learning, optimization algorithms like gradient descent and stochastic gradient descent are used to train neural networks by minimizing the cost function.
In deep learning, the cost function is often computed using backpropagation, which is an efficient algorithm for computing the gradient of the cost function with respect to the network parameters. The backpropagation algorithm uses the chain rule of calculus to compute the gradient by propagating the error back through the network.
## Exercise
Discuss how optimization algorithms are used in deep learning.
### Solution
Optimization algorithms are also widely used in deep learning. In deep learning, optimization algorithms like gradient descent and stochastic gradient descent are used to train neural networks by minimizing the cost function.
In deep learning, the cost function is often computed using backpropagation, which is an efficient algorithm for computing the gradient of the cost function with respect to the network parameters. The backpropagation algorithm uses the chain rule of calculus to compute the gradient by propagating the error back through the network.
# Improving the convergence of optimization algorithms
To improve the convergence of optimization algorithms like gradient descent and stochastic gradient descent, we can use techniques like momentum, adaptive learning rate methods, or gradient clipping. These techniques can help prevent the model from getting stuck in local minima and improve the overall convergence of the optimization process.
## Exercise
Discuss techniques to improve the convergence of optimization algorithms.
### Solution
To improve the convergence of optimization algorithms like gradient descent and stochastic gradient descent, we can use techniques like momentum, adaptive learning rate methods, or gradient clipping. These techniques can help prevent the model from getting stuck in local minima and improve the overall convergence of the optimization process.
# Optimizing loss functions in Python
In Python, we can use libraries like TensorFlow or PyTorch to implement optimization algorithms for training machine learning models. These libraries provide built-in functions for computing the gradient of the cost function, which can be used to update the model parameters iteratively.
Here's an example of how to optimize a loss function using TensorFlow:
```python
import tensorflow as tf
# Define the model
model = tf.keras.Sequential([
tf.keras.layers.Dense(10, activation='relu'),
tf.keras.layers.Dense(1)
])
# Define the loss function and the optimizer
loss_function = tf.keras.losses.MeanSquaredError()
optimizer = tf.keras.optimizers.Adam(learning_rate=0.01)
# Train the model
for epoch in range(1000):
with tf.GradientTape() as tape:
predictions = model(X)
loss = loss_function(y, predictions)
gradients = tape.gradient(loss, model.trainable_variables)
optimizer.apply_gradients(zip(gradients, model.trainable_variables))
```
## Exercise
Discuss how to optimize loss functions in Python using TensorFlow or PyTorch.
### Solution
In Python, we can use libraries like TensorFlow or PyTorch to implement optimization algorithms for training machine learning models. These libraries provide built-in functions for computing the gradient of the cost function, which can be used to update the model parameters iteratively.
Here's an example of how to optimize a loss function using TensorFlow:
```python
import tensorflow as tf
# Define the model
model = tf.keras.Sequential([
tf.keras.layers.Dense(10, activation='relu'),
tf.keras.layers.Dense(1)
])
# Define the loss function and the optimizer
loss_function = tf.keras.losses.MeanSquaredError()
optimizer = tf.keras.optimizers.Adam(learning_rate=0.01)
# Train the model
for epoch in range(1000):
with tf.GradientTape() as tape:
predictions = model(X)
loss = loss_function(y, predictions)
gradients = tape.gradient(loss, model.trainable_variables)
optimizer.apply_gradients(zip(gradients, model.trainable_variables))
```
# Evaluating the performance of optimization algorithms
To evaluate the performance of optimization algorithms like gradient descent and stochastic gradient descent, we can use metrics like mean squared error, accuracy, or F1-score. These metrics can be computed by comparing the predicted values from the model with the actual values from the dataset.
In Python, we can use libraries like Scikit-learn or TensorFlow to compute these metrics. Here's an example of how to compute the mean squared error using Scikit-learn:
```python
from sklearn.metrics import mean_squared_error
# Compute the predicted values
predictions = model.predict(X)
# Compute the mean squared error
mse = mean_squared_error(y, predictions)
print("Mean squared error:", mse)
```
## Exercise
Discuss how to evaluate the performance of optimization algorithms in Python.
### Solution
To evaluate the performance of optimization algorithms like gradient descent and stochastic gradient descent, we can use metrics like mean squared error, accuracy, or F1-score. These metrics can be computed by comparing the predicted values from the model with the actual values from the dataset.
In Python, we can use libraries like Scikit-learn or TensorFlow to compute these metrics. Here's an example of how to compute the mean squared error using Scikit-learn:
```python
from sklearn.metrics import mean_squared_error
# Compute the predicted values
predictions = model.predict(X)
# Compute the mean squared error
mse = mean_squared_error(y, predictions)
print("Mean squared error:", mse)
``` | Textbooks |
\begin{document}
\def\compilefullpaper{}
\title{Fault-tolerant quantum computation with few qubits} \author{Rui Chao} \author{Ben W. Reichardt} \affiliation{University of Southern California}
\begin{abstract} Reliable qubits are difficult to engineer, but standard fault-tolerance schemes use seven or more physical qubits to encode each logical qubit, with still more qubits required for error correction. The large overhead makes it hard to experiment with fault-tolerance schemes with multiple encoded~qubits.
The $15$-qubit Hamming code protects seven encoded qubits to distance three. We give fault-tolerant procedures for applying arbitrary Clifford operations on these encoded qubits, using only two extra qubits, $17$ total. In particular, individual encoded qubits within the code block can be targeted. Fault-tolerant universal computation is possible with four extra qubits, $19$ total. The procedures could enable testing more sophisticated protected circuits in small-scale quantum devices.
Our main technique is to use gadgets to protect gates against correlated faults. We also take advantage of special code symmetries, and use pieceable fault tolerance. \end{abstract}
\maketitle
\section{Introduction}
Quantum computers are faulty, but schemes to tolerate errors incur a large space overhead. For example, one qubit encodes into seven physical qubits using the Steane code~\cite{Steane96css}, or into nine physical qubits using the Bacon-Shor and smallest surface codes~\cite{Bacon05operator, BombinMartindelgado07surfaceoverhead, HorsmanFowlerDevittVanmeter11surfacesurgery}. Error correction uses additional qubits. When more than one level of encoding is required for better protection, the overhead multiplies, so that thousands of physical qubits can be required for each logical qubit~\cite{Suchara13overhead}. This overhead will compound the challenge of building large quantum computers. In the near term, it also makes it more difficult to run fault-tolerance experiments, which are important to test different schemes' performance, validate models and learn better approaches.
Codes storing multiple qubits have higher rates~\cite{CalderbankShor96}, but too large codes tend to tolerate less noise since initializing codewords gets difficult~\cite{Steane03}. A key obstacle for using any code with multiple qubits per code block is that it is complicated and inefficient to address the individual encoded qubits to compute on them~\cite{Gottesman97, SteaneIbinson03multiqubitcodes}. For example, to apply a CNOT gate between two logical qubits in a code block, the optimized method in~\cite{SteaneIbinson03multiqubitcodes} requires a full ancillary code block, with no stored data, into which the target logical qubit is transferred temporarily.
We introduce lower-overhead methods for computing fault tolerantly on multiple data qubits in codes of distance two or three.
\begin{enumerate}[leftmargin=*] \item For even $n$, the ${[\![} n, n-2, 2 {]\!]}$ code encodes $n-2$ logical qubits into $n$ physical qubits, protected to distance two. We show that with two more qubits, encoded CNOT and Hadamard gates can be applied fault tolerantly. For $n \geq 6$, four extra qubits suffice to fault-tolerantly apply an encoded ${\ensuremath{\mathrm{CC}Z}}$ gate, for universality. \item For better, distance-three protection, we encode seven qubits into $15$, and give fault-tolerant circuits for the encoded Clifford group using two more qubits, and for a universal gate set with four extra qubits ($19$ total). \end{enumerate} Combined with the two-qubit fault-tolerant error-detection and error-correction methods in~\cite{ChaoReichardt17errorcorrection}, this means that substantial quantum calculations can be implemented fault tolerantly in a quantum device with fewer than $20$ physical qubits. Figure~\ref{f:computationqubits} summarizes our results.
\begin{figure}
\caption{Summary of our constructions. Using two extra qubits, one can apply fault tolerantly either encoded CNOT and Hadamard gates or the full Clifford group. Four extra qubits are enough for fault-tolerant universal computation.}
\label{f:computationqubits}
\end{figure}
In order to compute on data encoded within a single code block, we need to apply two- or three-qubit gates. The particular circuits use symmetries of the codes or a more general round-robin construction from~\cite{YoderTakagiChuang16pieceableft}. This is not fault tolerant, because a single gate failure can cause a correlated error of weight two or worse, which a distance-three code cannot correct. To fix this, we replace each gate with a gadget involving two to four more qubits. With no gate faults, the gadgets are equivalent to the ideal gates they replace. The gadgets' purpose is to detect correlated errors, so that they can be corrected for later. The gadgets cannot prevent the gates from spreading single-qubit faults into problematic multi-qubit errors. To avoid this problem, we design the circuits carefully, and in some cases intersperse partial error correction procedures between gadgets, an idea from~\cite{KnillLaflammeZurek96} recently applied and extended by~\cite{HillFowlerWangHollenberg11codeconversion, YoderTakagiChuang16pieceableft}. Sometimes error correction even needs to overlap the gadgets.
For the basics of stabilizer algebra, quantum error-correcting codes and fault-tolerant quantum computation, we refer the reader to~\cite{Gottesman09faulttolerance}.
\section{Trick: Flagging a gate for correlated faults}
A main trick we use is to replace ${\ensuremath{\mathrm{C}Z}}$ and ${\ensuremath{\mathrm{CC}Z}}$ gates with small gadgets that can catch correlated faults. The gadgets are reminiscent of one-ancilla-qubit fault-tolerant SWAP gate gadgets~\cite{Gottesman00local}.
\subsection{${\ensuremath{\mathrm{C}Z}}$ gadget} \label{s:czflags}
The controlled-phase gate is a two-qubit diagonal gate ${\ensuremath{\mathrm{C}Z}} = \ensuremath{\boldsymbol{1}} - 2 \ketbra{11}{11}$, represented in circuits as \begin{equation*} \raisebox{-.2cm}{\includegraphics[scale=.769]{images/cz}} \end{equation*} Observe that ${\ensuremath{\mathrm{C}Z}}$ gates commute with $Z$ errors, but copy $X$ (or $Y$) errors on one wire into $Z$ errors on the other: \begin{equation} \raisebox{-.3cm}{ \raisebox{.135cm}{\includegraphics[scale=.769]{images/czzerror}} \quad \includegraphics[scale=.769]{images/czxerror} } \end{equation}
For fault tolerance, it suffices to study gates that fail with Pauli faults after the gate. That is, when a noisy ${\ensuremath{\mathrm{C}Z}}$ gate fails, it applies the ideal ${\ensuremath{\mathrm{C}Z}}$ gate followed by one of the $15$ nontrivial two-qubit Pauli operators.
Consider the circuit of \figref{f:czxflag}, using one extra qubit that at the end is measured in the $\ket 0, \ket 1$ basis ($Z$ eigenbasis). If the gates are perfect, then the measurement returns $0$ and this circuit has the effect of a single ${\ensuremath{\mathrm{C}Z}}$ gate. If the ${\ensuremath{\mathrm{C}Z}}$ gate fails with an $X$ or $Y$ fault on the second qubit, however, then the measurement will return~$1$. Thus certain kinds of faults can be detected. If at most one location fails and the measurement returns~$0$, then the output cannot have an $XX$, $XY$, $YX$ or $YY$ error.
\begin{figure}\label{f:czxflag}
\label{f:czzflag}
\label{f:czflags}
\end{figure}
A similar circuit can catch $Z$ faults. If all gates in \figref{f:czzflag} are perfect, then the $X$ basis ($\ket +, \ket -$) measurement will return $+$ and the effect will be of a single ${\ensuremath{\mathrm{C}Z}}$. If there is at most one fault and the measurement returns~$+$, then the output cannot have a $YX$ or $ZZ$ error. (This fact can be verified by, for example, propagating $ZZI$ and $ZZX$ backward through the circuit, and observing that no single gate failure can create either.)
The gadgets to catch $X$ and $Z$ faults can be combined:
\begin{theorem} \label{t:czflags} With no faults, the circuit of \figref{f:czflags} implements a ${\ensuremath{\mathrm{C}Z}}$ gate, with the measurements outputting $0$ and~$+$. If there is at most one fault and the measurements return $0$ and~$+$, then neither $XX$, $XY$, $YX$, $YY$ nor $ZZ$ errors can occur on the output. \end{theorem}
Thus all single faults are caught except those equivalent to a fault on the ${\ensuremath{\mathrm{C}Z}}$ output or input qubits (namely, $IX, IY, IZ, XI, YI, ZI$ and $XZ, YZ, ZX, ZY$). No ${\ensuremath{\mathrm{C}Z}}$ gadget can catch more errors than this.
\begin{figure}\label{f:czzflagwrongorder}
\label{f:czflagswrongorder}
\label{f:czzflagwrongorderczflagswrongorder}
\end{figure}
The order of the gates matters. Although the last two gates in \figref{f:czzflag} formally commute, switching them changes the faulty circuit so that an undetected $ZZ$ error can occur due to a single gate fault. Similarly, in \figref{f:czflags} it is important that the gadget for catching $Z$ faults go inside that for $X$ faults. With the other order, a single fault can lead to an undetected $XX$ error. See \figref{f:czzflagwrongorderczflagswrongorder}.
In practice, not every ${\ensuremath{\mathrm{C}Z}}$ gate need always be replaced with the full ${\ensuremath{\mathrm{C}Z}}$ gate gadget of \figref{f:czflags}. Multiple gates can sometimes be combined under single flags. We will see examples below.
\subsection{${\ensuremath{\mathrm{CC}Z}}$ gadget}
The three-qubit gate ${\ensuremath{\mathrm{CC}Z}} = \ensuremath{\boldsymbol{1}} - 2 \ketbra{111}{111}$ is denoted \begin{equation*} \raisebox{-.4cm}{\includegraphics[scale=.769]{images/ccz}} \end{equation*} Again, it commutes with $Z$ errors. It copies an $X$ error on the input into a ${\ensuremath{\mathrm{C}Z}} = \tfrac12 (II + IZ + ZI - ZZ)$ error on the output, i.e., into a linear combination of $II$, $IZ$, $ZI$ and $ZZ$ errors: \begin{equation*} \raisebox{-.4cm}{\includegraphics[scale=.769]{images/cczxerror}} \end{equation*}
The following gadget, using four ancilla qubits, implements a ${\ensuremath{\mathrm{CC}Z}}$ on the black data qubits. Furthermore, it satisfies that provided there is at most one failed gate and the measurement results are trivial, $0$ and $+$, then the error on the output is a linear combination of Paulis that could result from a one-qubit fault before or after a perfect ${\ensuremath{\mathrm{CC}Z}}$ gate, i.e., $III, ZII, XII, YII, XZI, YZI, XZZ, YZZ$ and qubit permutations thereof. \begin{equation} \label{e:cczgadget} \raisebox{-1.4cm}{\includegraphics[scale=.769]{images/cczgadget}} \end{equation} That without faults the black and blue gates realize a ${\ensuremath{\mathrm{CC}Z}}$ is a special case of the following claim~\cite{YoderTakagiChuang16pieceableft}, with $r = 3$, $S^Z_1 = \{1\}$, $S^Z_2 = \{2, 3\}$, $S^Z_3 = \{5, 6\}$.
\begin{claim} \label{t:logicalccz} Consider an $n$-qubit CSS code with $k$ encoded qubits given by $\widebar X_j = X_{S^X_j}, \widebar Z_j = Z_{S^Z_j}$ for $S^X_j, S^Z_j \subseteq [n]$. Let $U$ be the product of $\mathrm{C}^{(r-1)}Z = \ensuremath{\boldsymbol{1}} - 2 \ketbra{1^r}{1^r}$ gates applied to every tuple of qubits in $S^Z_1 \times S^Z_2 \times \cdots \times S^Z_r$. (If the $S^Z_j$ sets are not disjoint, then some of the applied gates will be $\mathrm{C}^{(s-1)}Z$ for some $s < r$.)
Then $U$ is a valid logical operation. It implements logical $\mathrm{C}^{(r-1)}Z$ on the first $r$ encoded qubits. \end{claim}
\begin{proof} For the case that the sets $S^Z_j$ are disjoint, the claim is immediate in the case of singleton sets and follows in general by induction on the set size using the identity \begin{equation*} \label{e:cnotcommutecgate} \raisebox{-.5cm}{\includegraphics[scale=.769]{images/cnotcommutecgate}} \end{equation*} for any gate $G$, on one or more qubits, with $G^2 = \ensuremath{\boldsymbol{1}}$. (Figure~\ref{f:czzflag} provides one example.)
More generally, the claim follows by writing out computational basis codewords as superpositions of computational basis states, and computing the effect of~$U$. For $(x, y) \in \{0,1\}^r \times \{0,1\}^{k-r}$, the codeword $\ket{\widebar{x y}}$ is a uniform superposition of computational basis states: \begin{equation*} \ket{\widebar{x y}} \propto \prod_{j=1}^r \widebar X_j^{x_j} \prod_{i=r+1}^k \widebar X_i^{y_{i-r}} \sum_{\text{stabilizers $X_S$}} X_S \ket{0^n} \end{equation*} The relationships $\widebar X_i \widebar Z_j = (-1)^{\delta_{ij}} \widebar Z_j \widebar X_i$ imply that $\abs{S^X_i \cap S^Z_j}$ is odd for $i = j$ and even otherwise; and for any stabilizer $X_S$, $[X_S, \widebar Z_j] = 0$ so $\abs{S \cap S^Z_j}$ is even.
If $x \neq 1^r$, say $x_j = 0$, then for every term $\ket z$ in the above sum, $z$ has an even number of $1$s in $S^Z_j$, implying that $U \ket z = \ket z$. Hence $U \ket{\widebar{x y}} = \ket{\widebar{x y}}$.
If $x = 1^r$, then for any term $\ket z$ in $\ket{\widebar{x y}}$, $z$ has an odd number of $1$s in each $S^Z_j$ and therefore $U \ket z = - \ket z$. Hence $U \ket{\widebar{x y}} = - \ket{\widebar{x y}}$. \end{proof}
\section{Fault-tolerant operations \mbox{for ${[\![} \lowercase{n},\lowercase{n}-2,2 {]\!]}$ codes}}
For even $n$, the ${[\![} n, n-2, 2 {]\!]}$ error-detecting code has stabilizers $X^{\otimes n}$ and $Z^{\otimes n}$, and logical operators $\widebar{X}_j = X_1 X_{j+1}$, $\widebar Z_j = Z_{j+1} Z_n$ for $j = 1, \ldots, n-2$. This code, its symmetries, and methods of computing fault tolerantly on the encoded qubits were studied by Gottesman~\cite{Gottesman97}. However, his techniques require at least $2 n$ extra qubits. For example, to apply a CNOT gate between two logical qubits in the same code block, he teleports them each into separate code blocks, applies transversal CNOT gates between the blocks, and then teleports them back.
We will give a fault-tolerant implementation of encoded CNOT and Hadamard gates on arbitrary logical qubits, using only two extra qubits. Two-qubit fault-tolerant procedures for state preparation, error detection and projective measurement were given in~\cite{ChaoReichardt17errorcorrection}. For $n \geq 6$ (so there are at least three encoded qubits), we will give a four-qubit fault-tolerant implementation of the encoded ${\ensuremath{\mathrm{CC}Z}}$ gate, thereby completing a universal gate set.
\subsection{Permutation symmetries and transversal operations}
Fault tolerance for a distance-two code means that any single fault within an operation should either have no effect or lead to a detectable error. For example, of course the $4^{n-2}$ logical Pauli operators can all be applied fault tolerantly, since the operations do not couple any qubits.
All qubit permutations preserve the two stabilizers and therefore preserve the code space. They are also fault tolerant if implemented either by relabeling the qubits or by moving them past each other (and not by using two-qubit SWAP gates~\cite{Gottesman00local}). For $i, j \in [n-2]$, $i \neq j$, the qubit swap $(i+1,j+1)$ swaps the logical qubits~$i$ and~$j$. The qubit swap $(1, 2)$ implements logical CNOTs from qubits $2$ through $n - 2$ into~$1$, and the qubit swap $(2, n)$ implements logical CNOTs in the opposite direction: \begin{equation*} \raisebox{-.8cm}{\includegraphics[scale=.769]{images/errordetectingswap12}} \raisebox{-.1cm}{\quad\text{and}\qquad} \raisebox{-.8cm}{\includegraphics[scale=.769]{images/errordetectingswap2n}} \end{equation*}
Transversal Hadamard, $H^{\otimes n}$, followed by the qubit swap $(1, n)$, implements logical $H^{\otimes (n-2)}$.
The Clifford reflection $G = \tfrac{i}{\sqrt 2}(X + Y)$ conjugates $X \leftrightarrow Y$ and $Z \rightarrow -Z$. Transversal $G$ is a valid logical operation (up to $Z_n$ to correct the $X^{\otimes n}$ syndrome if $n = 2 \mod 4$). It implements logical ${\ensuremath{\mathrm{C}Z}}$ gates between all encoded qubits: \begin{equation*} \raisebox{-.8cm}{\includegraphics[scale=.769]{images/errordetectingtransversalxy}} \end{equation*}
The operations given so far generate a group much smaller than the full $(n-2)$-qubit Clifford group. The qubit permutations generate $n!$ different logical operations (except for $n = 4$, just $6$ operations). With the transversal application of the six one-qubit Clifford gates, up to Paulis, this gives $6 (n!)$ different logical operations (or $36$ for $n = 4$). \tabref{f:groupsizes} gives the sizes of various interesting subgroups of the Clifford group, for comparison.
\begin{table} \begin{center}
\begin{tabular}{c |@{\quad} c @{\quad} l @{\quad} l@{\quad} l} \hline \hline $k$ & $S_k$ & $\GL(k, 2)$ & $\langle \text{CNOT}, H \rangle$ & $\mathcal C_k / \mathcal P_k$ \\ \hline 1 & 1 & $\times 1$ & $\times 2$ & $\times 3$ \\ 2 & 2 & $\times 3$ & $\times 12$ & $\times 10$ \\ 3 & 6 & $\times 28$ & $\times 240$ & $\times 36$ \\ 4 & 24 & $\times 840$ & $\times 17280$ & $\times 136$ \\ 5 & 120 & $\times 83328$ & $\times 4700160$ & $\times 528$ \\ 6 & 720 & $\times 27998208$ & $\times 4963368960$ & $\times 2080$ \\ 7 & 5040 & $\times 3.2 \cdot 10^{10}$ & $\times 2.1 \cdot 10^{13}$ & $\times 8256$ \\ 8 & 40320 & $\times 1.3 \cdot 10^{14}$ & $\times 3.4 \cdot 10^{17}$ & $\times 32896$ \\ \hline \hline \end{tabular} \end{center} \caption{Sizes of the $k$-qubit Clifford group and subgroups. There are $\abs{S_k} = k!$ permutations of $k$ qubits. CNOT gates generate a group of size $\abs{\GL(k,2)} = \prod_{j=0}^{k-1}(2^k - 2^j)$, adding Hadamard gates generates a larger group, and finally the full Clifford group, up to the $\abs{\mathcal P_k} = 4^k$ Paulis, has size $2^{k^2} \prod_{j=1}^k (4^k - 1)$. The sizes are given as multiples of the previous columns, e.g., $\abs{\mathcal C_2 / \mathcal P_2} = 2 \times 3 \times 12 \times 10 = 720$. \comment{$\abs{\langle \text{CNOT}, H \rangle} = \abs{\mathcal C_k / \mathcal P_k} / (2^{k-1} (2^k + 1))$.} } \label{f:groupsizes} \end{table}
We next give fault-tolerant implementations for a logical Hadamard gate on a single encoded qubit, and for a logical ${\ensuremath{\mathrm{C}Z}}$ gate between two encoded qubits. These generate a large subgroup of the Clifford group, the $\langle \text{CNOT}, H \rangle$ column in \tabref{f:groupsizes}.
\subsection{${\ensuremath{\mathrm{C}Z}}$ gate}
By \claimref{t:logicalccz}, a logical ${\ensuremath{\mathrm{C}Z}}_{1,2}$ gate can be implemented by $Z_n \, {\ensuremath{\mathrm{C}Z}}_{2,3} \, {\ensuremath{\mathrm{C}Z}}_{2,n} \, {\ensuremath{\mathrm{C}Z}}_{3,n}$: \begin{equation} \label{e:cz23ncz12} \text{physical} \raisebox{-.5cm}{\includegraphics[scale=.769]{images/cz23n}} \;\, = \;\, \text{logical} \raisebox{-.3cm}{\includegraphics[scale=.769]{images/cz12}} \end{equation}
However, this implementation is not fault tolerant. Some failures are detectable; for example, if the ${\ensuremath{\mathrm{C}Z}}_{2,3}$ gate fails as $XI$, then the final, detectable error is $X_2 Z_n$. Others are not, e.g., if the ${\ensuremath{\mathrm{C}Z}}_{2,3}$ gate fails as $XX$, then the final $X_2 X_3 = \widebar X_1 \widebar X_2$ error is undetectable.
The bad faults that can cause undetectable logical errors are as follows: \begin{center}
\begin{tabular}{c c | c c | c c} \hline \hline
\multicolumn{2}{c|}{${\ensuremath{\mathrm{C}Z}}_{2,3}$} & \multicolumn{2}{c|}{${\ensuremath{\mathrm{C}Z}}_{2,n}$} & \multicolumn{2}{c}{${\ensuremath{\mathrm{C}Z}}_{3,n}$} \\ Fault & Error & Fault & Error & Fault & Error \\ \hline $ZZ$ & $Z_2 Z_3$ & $ZZ$ & $Z_2 Z_n$ & $ZZ$ & $Z_3 Z_n$ \\ $XX$ & $X_2 X_3$ & $XY$ & $X_2 Z_3 Y_n$ & $XX$ & $X_3 X_n$ \\ $YY$ & $Y_2 Y_3$ & $YX$ & $Y_2 Z_3 X_n$ & $YY$ & $Y_3 Y_n$ \\ \hline \hline \end{tabular} \end{center} In particular, all these bad faults are caught by the ${\ensuremath{\mathrm{C}Z}}$ gadget of \thmref{t:czflags}. Therefore replacing each physical ${\ensuremath{\mathrm{C}Z}}$ gate in~\eqnref{e:cz23ncz12} with that gadget gives a fault-tolerant implementation of a logical ${\ensuremath{\mathrm{C}Z}}_{1,2}$ gate. The circuit uses at most two ancilla qubits at a time.
In fact, one can simplify the resulting circuit by using the same $\ket 0$ ancilla to catch $X$ faults on multiple ${\ensuremath{\mathrm{C}Z}}$ gates. The following circuit is also fault tolerant: \begin{equation*} \raisebox{-1.3cm}{\includegraphics[scale=.769]{images/cz23ncombinedxflags}} \end{equation*}
The gadgets to catch $Z$ faults can be merged, too. The following circuit is fault tolerant, and still requires at most two ancilla qubits at a time: \begin{equation} \raisebox{-1.3cm}{\includegraphics[scale=.769]{images/cz23ncombinedxzflags}} \end{equation} Perhaps further simplifications are possible.
\subsection{Targeted Hadamard gate}
A single encoded Hadamard gate can also be implemented fault tolerantly with two extra qubits. The black portion of the circuit below, with \raisebox{-.1cm}{\includegraphics[scale=.45]{images/dualczlong}}$\;$, implements $\widebar H_1$. The red and blue portions, analogous to Figs.~\ref{f:czxflag} and~\ref{f:czzflag}, respectively, catch problematic faults. $X$ measurements should return~$+$ and $Z$ measurements~$0$. \begin{equation} \raisebox{-1.3cm}{\includegraphics[scale=.769]{images/errordetectinghadamard}} \end{equation} The circuit's fault tolerance can be verified by enumerating all the ways in which single gates can fail.
\subsection{Four-ancilla ${\ensuremath{\mathrm{CC}Z}}$ gate}
For $n \geq 6$, a ${\ensuremath{\mathrm{CC}Z}}$ gate on encoded qubits $1, 2, 3$ can be implemented by round-robin ${\ensuremath{\mathrm{CC}Z}}$ gates on $\{2, n\} \times \{3, n\} \times \{4, n\}$, by \claimref{t:logicalccz}: \begin{equation} \label{e:errordetectingroundrobinccz234n} \raisebox{-.8cm}{\includegraphics[scale=.769]{images/errordetectingroundrobinccz234n}} \end{equation} This circuit uses one $Z$, three ${\ensuremath{\mathrm{C}Z}}$ and four ${\ensuremath{\mathrm{CC}Z}}$ gates. To make it fault tolerant, use the gadget from \figref{f:czflags} for each ${\ensuremath{\mathrm{C}Z}}$ gate, and replace each ${\ensuremath{\mathrm{CC}Z}}$ with the gadget of Eq.~\eqnref{e:cczgadget}. Overall, this requires four ancilla qubits.
Single gate faults are either caught by the gadgets or lead to an error that could also arise from a one-qubit fault between the gates in~\eqnref{e:errordetectingroundrobinccz234n}. A one-qubit $X$ or $Y$ fault will be detectable at the end because it is copied only to linear combinations of $Z$s---the $X$ component of the final error will still have weight one---and a one-qubit $Z$ fault will be detectable because it commutes through the ${\ensuremath{\mathrm{C}Z}}$ and ${\ensuremath{\mathrm{CC}Z}}$ gates. Therefore the procedure is fault tolerant.
\section{Fault-tolerant operations for the ${[\![} 15,7,3 {]\!]}$ \mbox{Hamming code}}
The ${[\![} 15,7,3 {]\!]}$ Hamming code is a self-dual CSS, perfect distance-three code. Packing seven logical qubits into $15$ physical qubits, it is considerably more efficient than more commonly used ${[\![} 7,1,3 {]\!]}$ and ${[\![} 9,1,3 {]\!]}$ CSS codes, although it tolerates less noise.
We first give a presentation of the code and its symmetries following~\cite{Harrington11permutations}. Then we give a two-ancilla-qubit method for fault tolerantly implementing the full Clifford group on the encoded qubits, and, to complete a universal gate set, a four-qubit fault-tolerant encoded ${\ensuremath{\mathrm{CC}Z}}$ gate.
Two-qubit fault-tolerant procedures for state preparation and error correction were given in~\cite{ChaoReichardt17errorcorrection}.
\subsection{${[\![} 15,7,3 {]\!]}$ Hamming code}
The ${[\![} 15,7,3 {]\!]}$ Hamming code has four $X$ and four $Z$ stabilizers each given by the following parity-checks: \begin{equation} \label{e:1573stabilizers} \begin{tabular}{c c c c c c c c c c c c c c c} 0&0&0&0&0&0&0&1&1&1&1&1&1&1&1\\ 0&0&0&1&1&1&1&0&0&0&0&1&1&1&1\\ 0&1&1&0&0&1&1&0&0&1&1&0&0&1&1\\ 1&0&1&0&1&0&1&0&1&0&1&0&1&0&1 \end{tabular} \end{equation} Index the qubits left to right from $1$ to $15$. Observe that the columns are these numbers in binary.
As in~\cite{Harrington11permutations}, we define logical operators based on the following seven weight-five strings: \begin{equation} \label{e:1573logicaloperators} \begin{tabular}{c c c c c c c c c c c c c c c} 1&1&0&1&0&0&0&1&0&0&0&0&0&0&1\\ 1&1&0&0&1&0&0&0&0&1&0&1&0&0&0\\ 1&1&0&0&0&1&0&0&0&0&1&0&0&1&0\\ 1&1&0&0&0&0&1&0&1&0&0&0&1&0&0\\ 1&0&0&1&0&1&0&0&1&1&0&0&0&0&0\\ 1&0&0&1&0&0&1&0&0&0&0&1&0&1&0\\ 1&0&0&0&0&0&0&1&0&1&0&0&1&1&0 \end{tabular} \end{equation} From the first string, $\widebar X_1 = XXIXIIIXIIIIIIX$ and $\widebar Z_1 = ZZIZIIIZIIIIIIZ$. The remaining strings specify the logical operators $\widebar X_2, \widebar Z_2$ through $\widebar X_7, \widebar Z_7$.
\subsection{Transversal operations}
Transversal operations are automatically fault tolerant.
Transversal Pauli operators implement logical transversal Pauli operators. Indeed, transversal $X$, i.e., $X^{\otimes 15}$, preserves the code and implements transversal logical $X$, i.e., $X^{\otimes 7}$, on the code space, and similarly for $Y$ and~$Z$.
In fact, any one-qubit Clifford operator applied transversally preserves the code space and implements the same operator transversally on the encoded qubits. For example, since the logical operators are each self-dual, applying the Hadamard gates $H^{\otimes 15}$ implements logical $H^{\otimes 7}$.
Of course, since the code is CSS, transversal CNOT gates between two code blocks implements transversal logical CNOT gates on the code spaces. Furthermore, \cite{PaetznickReichardt13universal} shows that on three code blocks transversal CCZ can be used to obtain a universal gate set. Here, however, we will consider only single code blocks and Clifford operations.
\subsection{Permutation symmetries}
Permutations of the qubits are also fault tolerant, either by physically moving the qubits or by relabeling them.
The code's permutation automorphism group has order $20,\!160$, and is isomorphic to $A_8$ and $\GL(4,2)$~\cite{Harrington11permutations, Grassl13automorphisms}. It is generated by the following three $15$-qubit permutations: \begin{equation} \label{e:1573permutationautomorphisms} \begin{split} \sigma_1 &= (1,2,3)(4,14,10)(5,12,9)(6,13,11)(7,15,8) \\ \sigma_2 &= (1,10,5,2,12)(3,6,4,8,9)(7,14,13,11,15) \\ \sigma_3 &= (1,10,15,3,8,13)(4,6)(5,12,11)(7,14,9) \end{split}\end{equation} These permutations fix the code space, but act nontrivially within it. The permutations $\sigma_1$ and $\sigma_2$ apply the respective permutations $(1,2,3)$ and $(3,4,5,6,7)$ to the seven logical qubits. Together, these generate the alternating group $A_7$ of even permutations.
The logical effect of $\sigma_3$ is not a permutation. It is equivalent to the following circuit of $24$ CNOT gates, in which gates with the same control wire are condensed: \begin{equation} \raisebox{-1.2cm}{\includegraphics[scale=.577]{images/sigma3circuit}} \end{equation} Thus the first logical qubit is fixed, while for $j \in \{2, \ldots, 7\}$ and $P \in \{X, Y, Z\}$, $P_j$ is mapped to $(\prod_{j=2}^7 P_j) P_{j+1}$, wrapping the indices cyclically. This is a six-qubit generalization of a four-qubit operator studied in~\cite[Sec.~6]{Gottesman97}. (Like permutations, this operation has the property of being a valid transversal operation on any stabilizer code.)
\subsection{${\ensuremath{\mathrm{C}Z}}$ circuits based on permutation symmetries} \label{s:cz89to1011and1213to1415}
Any permutation symmetry of the code can be turned into a ${\ensuremath{\mathrm{C}Z}}$ automorphism (\figref{f:permutationcz}):
\begin{figure}\label{f:permutationcz}
\label{f:czsigma3logical}
\end{figure}
\begin{claim} \label{t:permutationcz} For a self-dual CSS code, if $\sigma$ is a qubit permutation that fixes the code space, then the circuit with a ${\ensuremath{\mathrm{C}Z}}$ gate from $i$ to $\sigma(i)$, for all $i \neq \sigma(i)$, fixes the code space up to Pauli $Z$ corrections. \end{claim}
\begin{proof} $Z$ stabilizers commute with the ${\ensuremath{\mathrm{C}Z}}$ gates, so are preserved. An $X$ stabilizer $X_S = \prod_{i \in S} X_i$ is mapped to $\prod_{i \in S} (X_i Z_{\sigma(i)} Z_{\sigma^{-1}(i)}) = \pm X_S Z_{\sigma(S) \cup \sigma^{-1}(S)}$. Up to sign, this is a stabilizer, since $Z_{\sigma(S)} Z_{\sigma^{-1}(S)}$ is a stabilizer. \end{proof}
For example, the physical circuit in Eq.~\eqnref{e:cz23ncz12} comes from the cyclic permutation $(2,3,n)$ of the ${[\![} n, n-2, 2 {]\!]}$ code.
Applying \claimref{t:permutationcz} to $\sigma_3$ of Eq.~\eqnref{e:1573permutationautomorphisms}, the two ${\ensuremath{\mathrm{C}Z}}$ gates for the cycle $(4, 6)$ cancel out, leaving the gates $({\ensuremath{\mathrm{C}Z}}_{1,10} {\ensuremath{\mathrm{C}Z}}_{10,15} \cdots {\ensuremath{\mathrm{C}Z}}_{13,1}) ({\ensuremath{\mathrm{C}Z}}_{5,12} {\ensuremath{\mathrm{C}Z}}_{12,11} {\ensuremath{\mathrm{C}Z}}_{11,5}) \ldots$. As shown in \figref{f:czsigma3logical}, the effect is that of logical ${\ensuremath{\mathrm{C}Z}}$ gates following the cycle $(2,3,4,5,6,7)$.
Notice that the logical effect necessarily consists of encoded ${\ensuremath{\mathrm{C}Z}}$ gates, because logical $Z$ operators are unchanged and logical $X$ operators pick up $Z$ components. Also, the map from \claimref{t:permutationcz} is not a homomorphism from permutations into unitary circuits.
\subsubsection*{${\ensuremath{\mathrm{C}Z}}$ gates $\{8,9\}$ to $\{10,11\}$ and $\{12,13\}$ to $\{14,15\}$}
The permutation $(6,7)(8,10,9,11)(12,14,13,15)$ fixes the code, and under \claimref{t:permutationcz} corresponds to the eight ${\ensuremath{\mathrm{C}Z}}$ gates of \figref{f:cz89to1011and1213to1415}. Figure~\ref{f:cz89to1011and1213to1415logical} gives their logical effect.
\begin{figure}\label{f:cz89to1011and1213to1415}
\label{f:cz89to1011and1213to1415logical}
\label{f:cz89to1011and1213to1415errorcorrection}
\label{f:cz89to1011and1213to1415physicallogical}
\end{figure}
Using Magma~\cite{Magma97}, we compute that the group generated by this operation, the permutations $\sigma_1, \sigma_2, \sigma_3$, and transversal $H$ has the same size as the $\langle \text{CNOT}, H \rangle$ group on \emph{six} qubits (about $1.001 \cdot 10^{20}$). (Adding transversal $G$ only triples the group size.) This hints that by working in a logical basis in which one qubit has $\widebar X = X^{\otimes 15}$ and $\widebar Z = Z^{\otimes 15}$ (operators fixed by the permutations and by \figref{f:cz89to1011and1213to1415}), perhaps arbitrary combinations of CNOT and $H$ can be applied to the other six qubits. But no, only half the $\langle \text{CNOT}, H \rangle$ group can be reached.
The circuit of \figref{f:cz89to1011and1213to1415} is clearly not fault tolerant. (For example, an $XX$ fault after the ${\ensuremath{\mathrm{C}Z}}_{9,11}$ gate gives the error $X_9 X_{11}$, which is indistinguishable from $X_2$.) We can use the gadgets from \secref{s:czflags} for each ${\ensuremath{\mathrm{C}Z}}$ gate to obtain the circuit of \figref{f:cz89to1011and1213to1415errorcorrection}, shown with the trailing error correction. Two ancilla qubits are needed.
We claim that this compiled circuit is fault tolerant. This means that if the input lies in the code space, the compiled circuit has at most one fault (a two-qubit Pauli fault after a gate, or a one-qubit fault on a resting qubit), and the subsequent error correction is perfect; then the final outputs lie in the code space with no logical errors. To verify fault tolerance, there are two cases to check.
First, consider the case that, with at most one fault, all the gadget measurements give the trivial output ($0$ for a $Z$ measurement, $+$ for $X$). Since the gadgets catch two-qubit gate faults, we need only check possible one-qubit faults between gates. Inequivalent fault locations are marked with stars in \figref{f:cz89to1011and1213to1415errorcorrection}. (Faults at other locations either cause the same errors, or will be caught.) In particular, entering error correction the possible error can be $\ensuremath{\boldsymbol{1}}$, $X_1, Z_1, Y_1, \ldots, X_{15}, Z_{15}, Y_{15}$---from the ${\color{Midnight} \bigstar}$ locations. Or, from ${\color{red} \bigstar}$ locations, it can be $XIZZ, YIZZ, IXZZ, IYZZ, ZZXI, ZZYI, ZZIX$, $ZZIY$, and $IZXI, IZYI, IZIX, IZIY$ from ${\color{Tangerine} \bigstar}$ locations, on qubits $8, 9, 10, 11$---and similarly for qubits $12$ to $15$. This give $70$ different errors total. All $70$ have distinct syndromes, and therefore can be corrected. (This fact can be verified either by computing all the syndromes, or by observing from Eq.~\eqnref{e:1573stabilizers} that $Z_8 Z_9 Z_{10} Z_{11}$, $Z_{12} Z_{13} Z_{14} Z_{15}$, $Z_1 Z_6 Z_7$, $Z_1 Z_8 Z_9$, $Z_1 Z_{12} Z_{13}$ and $Z_1 Z_{14} Z_{15}$ are logical operators. Thus, for example, if you observe the $Z$ syndrome $1000$, for error $X_8$, and the $X$ syndrome $0001$, for error~$Z_1$, you can safely correct $X_8 Z_{10} Z_{11}$. $Z_1 X_8$ cannot occur.)
Note that the error-correction procedure needs to take into account the $X$ and $Z$ stabilizer syndromes together to decide what correction to apply. This can work because the ${[\![} 15,7,3 {]\!]}$ code is not a perfect stabilizer code: there are $2^8$ possible syndromes but only $1 + 15 \cdot 3$ possible trivial or weight-one errors. (It is only perfect as a CSS code, i.e., the $2^4$ $Z$ stabilizer syndromes are exactly enough to correct the $1 + 15$ possible trivial or weight-one $X$ errors, and similarly for $Z$ errors.) This leaves room to correct some errors of weight more than one.
Next, consider the case that, with at most one fault in \figref{f:cz89to1011and1213to1415errorcorrection}, one or more of the gadget measurements gives a nontrivial output. This case is much simpler, because the measurement results localize the fault, leaving only a few possibilities for the error entering error correction. One must verify that in all cases, these possibilities are distinguished by their syndromes.
For example, if the first $Z$ measurement returns $1$ and all other measurements are trivial, the errors from single faults that can occur are, on qubits $8, 9, 10, 11$: \begin{equation*} \begin{array}{r@{,\;} r@{,\;} r@{,\;} r} IIII & ZIII & XIII & YIII, \\ XIIZ &YIIZ & XIZZ & YIZZ, \\ XZIX & XZIY & XZXZ & XZYZ, \\ YZIX & YZIY & YZXZ & YZYZ \end{array} \end{equation*} These $16$ possible errors all have distinct syndromes, so are correctable.
As another example, if the last two measurements, of qubits coupled to qubit~$13$, are nontrivial, then the possible errors from single faults are, on qubits $12, 13, 14, 15$: \begin{equation*} IXII, IYII, IXIZ, IYIZ, IXIX, IXIY \end{equation*} Again, these have distinct syndromes.
Other possible measurement outcomes are similar. We have used a computer to check them all.
\subsection{Round-robin ${\ensuremath{\mathrm{C}Z}}$ circuits to complete the Clifford group}
The above operations do not generate the full seven-qubit logical Clifford group, and we have not been able to find a permutation for which applying \claimref{t:permutationcz} enlarges any further the generated logical group. Instead, we turn to the round-robin construction of \claimref{t:logicalccz}.
\subsubsection*{${\ensuremath{\mathrm{C}Z}}$ gates $4$ to $\{5,6,7\}$, $8$ to $\{9,10,11\}$, $12$ to $\{13,14,15\}$}
Observe that $Z_{\{4,8,12\}}$ and $Z_{\{4,5,6,7\}} \sim Z_{\{8,9,10,11\}} \sim Z_{\{12,13,14,15\}}$ are logical operators, implementing respectively $\widebar Z_{\{2,5,7\}}$ and $\widebar Z_{\{1,2,3,4\}}$. By a minor extension of \claimref{t:logicalccz}, applying $Z_{\{4,8,12\}}$ and nine ${\ensuremath{\mathrm{C}Z}}$ gates from $4$ to each of qubits $\{5,6,7\}$, $8$ to $\{9,10,11\}$, and $12$ to $\{13,14,15\}$ preserves the code space. The logical effect is \begin{equation} \raisebox{-1.3cm}{\includegraphics[scale=.769]{images/cz4to567and8to91011and12to131415logical}} \end{equation}
Together with permutations and transversal operations, this circuit completes the seven-qubit Clifford group, without needing the operation from \secref{s:cz89to1011and1213to1415}. To make the operation fault tolerant, we will transform it in three~steps.
First, consider the circuit below, in which we have wrapped the ${\ensuremath{\mathrm{C}Z}}$ gates leaving qubits $4$, $8$ and $12$ with overlapping gadgets to catch $X$ faults. If at most one fault occurs and one or more of the $Z$ measurements gives~$1$, then the errors that can occur are distinguished by their syndromes. \begin{equation} \label{e:cz4to567and8to91011and12to131415flags} \raisebox{-3.7cm}{\includegraphics[scale=.769]{images/cz4to567and8to91011and12to131415flags}} \end{equation} For example, if the orange, first $Z$ measurement off qubit~$4$ gives~$1$ and all others~$0$, then the possible errors entering error correction are, on qubits $4, 5, 6, 7$: \begin{equation*} \begin{array}{r@{,\;} r@{,\;} r@{,\;} r} IIII & ZIII \\ XIZZ & XXZZ & XYZZ & XZZZ \\ YIZZ & YXZZ & YYZZ &YZZZ \end{array} \end{equation*} These errors all have distinct syndromes.
The circuit in Eq.~\eqnref{e:cz4to567and8to91011and12to131415flags} is not fault tolerant, however, because with at most one fault if all the $Z$ measurements give~$0$, some inequivalent errors will have the same syndrome. We can list the problematic errors. For each of the following sets, the errors within the set are all possible, but have the same syndrome: \begin{equation}\begin{gathered} \label{e:baderrors} \{ Z_1, Z_4 Z_5, Z_8 Z_9, Z_{12} Z_{13} \} \\ \{ Z_2, Z_4 Z_6, Z_8 Z_{10}, Z_{13} Z_{14} \} \\ \{ Z_3, Z_4 Z_7, Z_8 Z_{11}, Z_{12} Z_{15} \} \\ \{ X_4, Y_4 Z_5 Z_6 Z_7 \}, \{ Y_4, X_4 Z_5 Z_6 Z_7 \} \\ \{ X_8, Y_8 Z_9 Z_{10} Z_{11} \}, \{Y_8, X_8 Z_9 Z_{10} Z_{11} \} \\ \{ X_{12}, Y_{12} Z_{13} Z_{14} Z_{15} \}, \{ Y_{12}, X_{12} Z_{13} Z_{14} Z_{15} \} \end{gathered}\end{equation}
Next replace each of the blue ${\ensuremath{\mathrm{C}Z}}$ gates in Eq.~\eqnref{e:cz4to567and8to91011and12to131415flags} with the $ZZ$ fault gadget from \figref{f:czzflag}. This gadget has the property that, with at most one failure, a $ZZ$ fault can only occur if the $X$ measurement returns~$-$. These measurements thus distinguish the errors in the first three sets above.
Yet the new circuit is still not fault tolerant. The gadget measurements cannot distinguish the errors in each of the last six sets in~\eqnref{e:baderrors}. For example, consider an $X_4$ error before the circuit. It propagates to $X_4 Z_5 Z_6 Z_7$. Since $Z_4 Z_5 Z_6 Z_7$ is a logical error, $X_4 Z_5 Z_6 Z_7$ is indistinguishable from a $Y_4$ error after the circuit, and no error-correction rules can correct for the possible logical error.
Gadgets cannot protect against single-qubit faults that occur just before or after the circuit. This circuit is qualitatively different from the one we considered in \secref{s:cz89to1011and1213to1415}, and a new trick is needed to make it fault tolerant.
Consider the following circuit from~\cite{ChaoReichardt17errorcorrection}, ignoring for now the orange portion at top right. \begin{equation} \label{e:456712131415ftsyndrome} \raisebox{-2.4cm}{\includegraphics[scale=.769]{images/456712131415ftsyndrome}} \end{equation} This circuit fault-tolerantly extracts the syndrome of the $Z_{\{4, 5, 6, 7, 12, 13, 14, 15\}}$ stabilizer, in the sense that: \begin{itemize} \item With no gate faults, the $Z$ measurement gives the syndrome, and the $X$ measurement gives~$+$. \item With at most one fault, if the $X$ measurement gives~$+$, then the data error has weight $\leq 1$. \item With at most one fault, if the $X$ measurement gives~$-$, then the $X$ component of the data error has weight $\leq 1$. The $Z$ component can be any of \begin{equation*}\begin{gathered} Z_4, Z_{\{4, 5\}}, Z_{\{4, 5, 6\}}, Z_{\{4, 5, 6, 12\}}, Z_{\{4, 5, 6, 7, 12\}} \sim Z_{\{13, 14, 15\}}, \\ Z_{\{4, 5, 6, 7, 12, 14\}} \sim Z_{\{13, 15\}}, Z_{\{4, 5, 6, 7, 12, 13, 14\}} \sim Z_{15}, \ensuremath{\boldsymbol{1}} \end{gathered}\end{equation*} and these errors all have distinct syndromes. (The order of the CNOT gates ensures this property.) \end{itemize} In~\cite{ChaoReichardt17errorcorrection}, this circuit was used in a two-ancilla-qubit fault-tolerant error-correction procedure.
The circuit in~\eqnref{e:456712131415ftsyndrome} is useful for us now to detect an $X_4$ or $Y_4$ error on the input to~\eqnref{e:cz4to567and8to91011and12to131415flags}. However, it is not enough to measure the $Z$ syndrome, or even to run full error correction, before applying the circuit~\eqnref{e:cz4to567and8to91011and12to131415flags}, because an $X_4$ or $Y_4$ fault could happen after the syndrome measurement completes and before~\eqnref{e:cz4to567and8to91011and12to131415flags}. This problem is solved by the orange portion of~\eqnref{e:456712131415ftsyndrome}, which is meant to continue into~\eqnref{e:cz4to567and8to91011and12to131415flags}, replacing the first $\ket 0$ preparation and CNOT. It gives qubit~$4$ temporary protection, so that an $X_4$ or $Y_4$ fault is caught by either the syndrome measurement or the orange $Z$ measurement, or both.
While the above arguments give intuition for the construction, they leave out the details. Let us now present the full fault-tolerant construction.
1. Start by applying~\eqnref{e:456712131415ftsyndrome} to extract the syndrome for $Z_{\{4,5,6,7,12,13,14,15\}}$. If the $Z$ or $X$ measurement is nontrivial, then decouple the orange qubit with another CNOT, apply error correction, and finish by applying unprotected ${\ensuremath{\mathrm{C}Z}}$ gates $4$ to $\{5,6,7\}$, $8$ to $\{9,10,11\}$ and $12$ to $\{13,14,15\}$. (This is safe because one fault has already been detected.)
2. Next, if the $Z$ and $X$ measurements were trivial, apply the top third of circuit~\eqnref{e:cz4to567and8to91011and12to131415flags}, where the orange qubit wire continues from~\eqnref{e:456712131415ftsyndrome}, to implement protected ${\ensuremath{\mathrm{C}Z}}$ gates $4$ to $\{5, 6, 7\}$. If any measurements are nontrivial, then finish by applying unprotected ${\ensuremath{\mathrm{C}Z}}$ gates $8$ to $\{9,10,11\}$ and $12$ to $\{13,14,15\}$, then error correction. We have argued already that this is fault tolerant; the extended orange ``flag" is enough to catch $X_4$ or $Y_4$ faults between~\eqnref{e:456712131415ftsyndrome} and~\eqnref{e:cz4to567and8to91011and12to131415flags}.
3. If the measurements so far were trivial, then apply a circuit analogous to~\eqnref{e:456712131415ftsyndrome} to extract the $Z_{\{8,9,10,11,12,13,14,15\}}$ syndrome. (Note that this is still a stabilizer, even though the ${\ensuremath{\mathrm{C}Z}}$ gates $4$ to $\{5, 6, 7\}$ have changed the code.) If the $Z$ syndrome or $X$ measurement is nontrivial, then apply error correction---a simple error-correction procedure is to apply ${\ensuremath{\mathrm{C}Z}}$ gates $4$ to $\{5, 6, 7\}$ to move back to the ${[\![} 15, 7, 3 {]\!]}$ code and correct there---before finishing with ${\ensuremath{\mathrm{C}Z}}$ gates $8$ to $\{9, 10, 11\}$ and $12$ to $\{13, 14, 15\}$. If the $Z$ and $X$ measurements were trivial, then apply the middle portion of~\eqnref{e:cz4to567and8to91011and12to131415flags}, where the orange qubit wire extends from qubit~$8$, to implement protected ${\ensuremath{\mathrm{C}Z}}$ gates $8$ to $\{9, 10, 11\}$. If any measurements are nontrivial, then finish by applying unprotected ${\ensuremath{\mathrm{C}Z}}$ gates from $12$ to $\{13,14,15\}$, then error correction.
4. If the measurements so far were trivial, then extract the $Z_{\{8,9,10,11,12,13,14,15\}}$ syndrome using~\eqnref{e:456712131415ftsyndrome} except with the data qubits in order $12, 13, 14, 15, 8, 9, 10, 11$ top to bottom (so that the orange flag attaches to qubit~$12$). If the $Z$ or $X$ measurement is nontrivial, then with at most one fault whatever error there is on the data can be corrected. (The easiest way is to move forward to the ${[\![} 15, 7, 3 {]\!]}$ code using ${\ensuremath{\mathrm{C}Z}}$ gates $12$ to $\{13, 14, 15\}$ and correct there. Note that these ${\ensuremath{\mathrm{C}Z}}$ gates turn the weight-one errors $X_{12}, X_{13}, X_{14}, X_{15}$ into $X_{12} Z_{\{13,14,15\}}, Z_{12} X_{13}, Z_{12} X_{14}, Z_{12} X_{15}$, respectively, but these can still be corrected; e.g., if in $X$ error correction you detect $X_{12}$, apply the correction $X_{12} Z_{\{13, 14, 15\}}$.) If the $Z$ and $X$ measurements were trivial, then apply the bottom portion of~\eqnref{e:cz4to567and8to91011and12to131415flags} to implement protected ${\ensuremath{\mathrm{C}Z}}$ gates from $12$ to $\{13, 14, 15\}$, and correct errors based on the measurement results.
Observe that this procedure requires two ancilla qubits.
\subsection{Four-qubit fault-tolerant ${\ensuremath{\mathrm{CC}Z}}$ for universality} \label{s:1573ccz}
In order to realize a universal set of operations on the seven encoded qubits, we give a four-ancilla-qubit fault-tolerant implementation for an encoded ${\ensuremath{\mathrm{CC}Z}}$ gate. The idea is to start with a circuit of round-robin ${\ensuremath{\mathrm{CC}Z}}$ gates to implement the encoded ${\ensuremath{\mathrm{CC}Z}}$ non-fault-tolerantly (\claimref{t:logicalccz}), then replace each ${\ensuremath{\mathrm{CC}Z}}$ with the gadget of Eq.~\eqnref{e:cczgadget} to catch correlated errors. Finally, we intersperse $X$ error correction procedures to catch $X$ faults before they can spread, much like the pieceable fault-tolerance constructions of~\cite{YoderTakagiChuang16pieceableft} except on a single code block. (A similar approach can also be used to implement encoded ${\ensuremath{\mathrm{C}Z}}$ gates.)
By \claimref{t:logicalccz}, an encoded ${\ensuremath{\mathrm{CC}Z}}$ gate can be implemented by round-robin ${\ensuremath{\mathrm{CC}Z}}$ gates on qubits $\{1, 4, 5\} \times \{1, 6, 7\} \times \{1, 8, 9\}$ (one $Z$, six ${\ensuremath{\mathrm{C}Z}}$ and $21$ ${\ensuremath{\mathrm{CC}Z}}$ gates): \begin{equation} \label{e:1573roundrobinccz1456789} \raisebox{-1.3cm}{\includegraphics[scale=.769]{images/1573roundrobinccz1456789}} \end{equation}
To make this circuit fault tolerant, first replace each ${\ensuremath{\mathrm{C}Z}}$ gate with the gadget from \figref{f:czflags}, and replace each ${\ensuremath{\mathrm{CC}Z}}$ gate with the gadget from Eq.~\eqnref{e:cczgadget}. After each gadget, apply $X$ error correction, and at the end apply both $X$ and $Z$ error correction. (As in~\cite{YoderTakagiChuang16pieceableft}, it might be possible to put multiple ${\ensuremath{\mathrm{CC}Z}}$ gadgets before each $X$ error correction, but we have not tried to optimize this.) Observe that $X$ error correction can be implemented even partially through the round-robin circuit because the code's $Z$ stabilizers are preserved by ${\ensuremath{\mathrm{CC}Z}}$ gates.
There are two cases to consider to demonstrate fault tolerance: either a gadget is ``triggered" with a nontrivial, $1$ or $-$, measurement outcome, or no gadgets are triggered.
\renewcommand{\arabic{paragraph}}{\arabic{paragraph}}
\paragraph{A gadget is triggered.}
If a gadget is triggered, then any Pauli errors can be present on its output data qubits. It is straightforward to check mechanically that for each ${\ensuremath{\mathrm{C}Z}}$ gate in~\eqnref{e:1573roundrobinccz1456789}, all four possible $X$ errors, $II$, $IX$, $XI$ and $XX$, have distinct $Z$ syndromes, and so can be corrected immediately in the subsequent $X$ error correction, before the errors can spread. By symmetry, the four possible $Z$ errors have distinct syndromes. These errors commute through~\eqnref{e:1573roundrobinccz1456789} and are fixed by the final $Z$ error correction.
Similar considerations hold for each ${\ensuremath{\mathrm{CC}Z}}$ gate: the possible $X$ and $Z$ error components have distinct syndromes, so an error's $X$ component can be corrected immediately and the $Z$ component corrected at the end.
\paragraph{No gadgets are triggered.}
If there is a single failure in a ${\ensuremath{\mathrm{C}Z}}$ or ${\ensuremath{\mathrm{CC}Z}}$ gadget, but the gadget is not triggered, then the error leaving the gadget is a linear combination of the same Paulis that could result from a one-qubit $X$, $Y$ or $Z$ fault before or after the gadget.
If the error has no $X$ component, then as a weight-one $Z$ error it commutes to the end of~\eqnref{e:1573roundrobinccz1456789}, at which point $Z$ error correction fixes it.
If the error has $X$ component of weight one, then the $Z$ component can be a permutation of any of $III, IIZ, IZZ, ZZZ$ on the three involved qubits (or of $II, IZ, ZZ$ for a ${\ensuremath{\mathrm{C}Z}}$ gadget). As we have already argued, these $Z$ errors have distinct $X$ syndromes. The $X$ error correction immediately following the gadget will catch and correct the error's $X$ component, keeping it from spreading. The final $Z$ error correction, alerted to the $X$ failure, will correct the error's $Z$ component.
\section{Conclusion}
Space-saving techniques for fault-tolerant quantum computation should be useful both for large-scale quantum computers and for nearer-term fault-tolerance experiments. Our techniques can likely be optimized further, and adapted to experimental model systems---but it might also be useful to relax the space optimization and allow a few more qubits. The techniques can also likely be applied to other codes, especially distance-three CSS codes. The round-robin ${\ensuremath{\mathrm{C}Z}}$ and ${\ensuremath{\mathrm{CC}Z}}$ constructions apply to some non-CSS codes, such as the ${[\![} 8,3,3 {]\!]}$ code, but then are more difficult to make fault tolerant.
We thank Ted Yoder for helpful comments. Research supported by NSF grant CCF-1254119 and ARO grant W911NF-12-1-0541.
\end{document} | arXiv |
sample size calculator proportion
the hypothesis of no difference. The desired precision of the estimate will be half the width of the desired confidence interval (i.e) for an example if you give the desired precision of 5%, you would get the confidence interval width to be about 0.1 (10%. Sample Size Calculator You can use this free sample size calculator to determine the sample size of a given survey per the sample proportion, margin of error, and required confidence level. This calculator is useful for tests concerning whether a proportion, $p$, is equal to a reference value, $p_0$. The hypotheses are, This calculator uses the following formulas to compute sample size and power, respectively: We perform a two-sample test to determine whether the proportion in group A, $p_A$, is different from the proportion in group B, $p_B$. a) This calculator uses the following formula for the sample size n:n = (Zα/2+Zβ)2 * (p1(1-p1)+p2(1-p2)) / (p1-p2)2,where Zα/2 is the critical value of the Normal distribution at α/2 (e.g. sample size from two proportions (r=1), the probability. The desired precision of the estimate will be half the width of the desired confidence interval (i.e) for an example if you give the desired precision of 5%, you would get the confidence interval width to be about 0.1 (10%. Choose which calculation you desire, enter the relevant values (as decimal fractions) for p0 (known value) and p1 (proportion in the population to be sampled) and, if calculating power, a sample size. Hypothesis Testing: One-Sample Inference - One-Sample Inference for a we have two samples. Compare Two Proportions ¡V Casagrande, Pike & Smith. where, © 2013-2020 HyLown Consulting LLC • Atlanta, GA, Test Relative Incidence in Self Controlled Case Series Studies, $$n=p(1-p)\left(\frac{z_{1-\alpha/2}+z_{1-\beta}}{p-p_0}\right)^2$$ Power & Sample Size Calculator. and, if calculating power, the sample sizes needed to detect a difference between two binomial Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the NIH. n_A=\kappa n_B \;\text{ and }\; $$n=p(1-p)\left(\frac{z_{1-\alpha/2}+z_{1-\beta}}{p-p_0}\right)^2$$ Please cite this site wherever used in published work: Kohn MA, Senyak J. of P2, proportion of characteristic present in arm 2, d) Null Hypothesis value (%): the pre-specified proportion (the value to compare the observed proportion to), expressed as a percentage. Instructions: Use this calculator to compute probabilities associated to the sampling distribution of the sample proportion. a sample size. Suppose the two groups are 'A' and 'B', and we collect a sample from both groups -- i.e. probability of rejecting the true null hypothesis. Define be the upper This calculator allows you to evaluate the properties of different statistical designs when planning an experiment (trial, test) utilizing a Null-Hypothesis Statistical Test to make inferences. are usually used: It is Calculate Sample Size Needed to Compare 2 Proportions: 2-Sample, 2-Sided Equality. Suppose the two groups are 'A' and 'B', and we collect a sample from both groups -- i.e. sample size from the first population. Sample Size Calculator Determines the minimum number of subjects for adequate study power ClinCalc.com » Statistics » Sample Size Calculator. sizes, the percentage error which results from using the approximation is no The confidence interval (also called margin of error) is the plus-or-minus figure usually reported in newspaper or television opinion poll results. of £], the probability of type II error, or (1-power) of the test, c) value of P1, proportion of characteristic present It is easier to be sure of extreme answers than of middle-of-the-road ones. probability of type I error (significance level) is the probability of rejecting the true null hypothesis. sample size from two proportions (r=1), the probabilityandare considered sufficiently different to warrant rejecting Given below sample size formula to estimate a proportion with specified precision. ). This project was supported by the National Center for Advancing Translational Sciences, National Institutes of Health, through UCSF-CTSI Grant Numbers UL1 TR000004 and UL1 TR001872. With significance level £\=0.05, equal Calculate power given sample size, alpha, and the minimum detectable effect (MDE, minimum effect of interest). (To use this page, your browser must recognize JavaScript.). With significance level £\=0.05, equal achieve an 80% power (£]=0.2) can be Then the required sample size for two arms to achieve an 80% power (β=0.2) can be determined by.Reference: £\: The You can calculate the sample size in five simple steps: Choose the required confidence level from the dropdown menu To calculate $$ m be the required The uncertainty in a given random sample (namely that is expected that the proportion estimate, p̂, is a good, but not perfect, approximation for the true proportion p) can be summarized by saying that the estimate p̂ is normally distributed with mean p and variance p(1-p)/n. As defined below, confidence level, confidence interval… This utility calculates the sample size required to estimate a proportion (or prevalence) with a specified level of confidence and precision. vs. One study group vs. population. Sample Size Calculators [website]. for p0 (known value) and p1 (proportion in the population to be sampled) You may also modify α (type I error rate) and the power, if relevant. Software utilities developed by Michael Kohn. If 99% of your sample said "Yes" and 1% said "No," the chances of error are remote, irrespective of sample size. Available at https://www.sample-size.net/ [Accessed 27 November 2020].
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Accounting for behavioral responses during a flu epidemic using home television viewing
Michael Springborn1,
Gerardo Chowell2,3,
Matthew MacLachlan4 &
Eli P Fenichel5
BMC Infectious Diseases volume 15, Article number: 21 (2015) Cite this article
103 Altmetric
An Erratum to this article was published on 05 September 2016
Theory suggests that individual behavioral responses impact the spread of flu-like illnesses, but this has been difficult to empirically characterize. Social distancing is an important component of behavioral response, though analyses have been limited by a lack of behavioral data. Our objective is to use media data to characterize social distancing behavior in order to empirically inform explanatory and predictive epidemiological models.
We use data on variation in home television viewing as a proxy for variation in time spent in the home and, by extension, contact. This behavioral proxy is imperfect but appealing since information on a rich and representative sample is collected using consistent techniques across time and most major cities. We study the April-May 2009 outbreak of A/H1N1 in Central Mexico and examine the dynamic behavioral response in aggregate and contrast the observed patterns of various demographic subgroups. We develop and calibrate a dynamic behavioral model of disease transmission informed by the proxy data on daily variation in contact rates and compare it to a standard (non-adaptive) model and a fixed effects model that crudely captures behavior.
We find that after a demonstrable initial behavioral response (consistent with social distancing) at the onset of the outbreak, there was attenuation in the response before the conclusion of the public health intervention. We find substantial differences in the behavioral response across age subgroups and socioeconomic levels. We also find that the dynamic behavioral and fixed effects transmission models better account for variation in new confirmed cases, generate more stable estimates of the baseline rate of transmission over time and predict the number of new cases over a short horizon with substantially less error.
Results suggest that A/H1N1 had an innate transmission potential greater than previously thought but this was masked by behavioral responses. Observed differences in behavioral response across demographic groups indicate a potential benefit from targeting social distancing outreach efforts.
The series of flu-like outbreaks over the past decade illustrates the ongoing need for refinement of strategies to control and mitigate the impact of infectious diseases, including SARS in 2003 [1], the 2009 A/H1N1 (swine) influenza pandemic [2,3] and the emergence of a novel A/H7N9 (avian) influenza virus in 2013 [4]. In parallel to standard vaccination efforts, nonpharmaceutical interventions (NPIs) are a critical part of the management toolkit [5-7]. In particular, NPIs become even more relevant in the context of emerging infectious diseases when the availability of a vaccine may be substantially delayed. Chief among NPIs are strategies for enhancing social distancing, whether privately initiated or policy-directed (e.g., closing of schools, businesses and public events) [8]. While behavioral NPIs appear promising, it is important to evaluate empirically their efficacy since they can be costly [9] and could have unintended consequences, such as leading to a net increase in the long-run number of cases or increasing the total cost of the epidemic and policy response [10,11]. The potential for individual response to disease risk and policy presents a challenge for the measurement of the infectivity of a pathogen and design of policy directed social distancing [12,13]. Ferguson [14] argues that despite the need for a holistic approach, current models essentially ignore the feedback between epidemics and behavior.
Empirical analysis of the effect of social distancing behavior on epidemiological dynamics is of clear interest, but it has proven difficult to obtain representative data on actual behavioral responses to epidemics. Empirical investigation of the influence of behavior on flu-like transmission dynamics has been largely limited to binary proxies for behavior, specifically pre-scheduled [6,15] and epidemiologically driven [16] school closings and patterns of weekdays and weekends [17]. Though policy interventions are often coarse, individuals' responses to policy and their own private decisions about risk are likely more nuanced [8]. Fenichel et al. [18] show that private risk reduction may have changed in subtle ways during the 2009 A/H1N1 epidemic. Caley et al. [19] estimate the change in infectious contact rates in Sydney, Australia from the 1918 influenza pandemic but do so indirectly by inferring changes in contacts based on the estimated reproduction number and proportion susceptible conditional on a given value for the reproduction number, R0.
We use novel data on variation in home television viewing behavior as a proxy for changes in the level of daily social interaction. We find a strong viewing behavior response in Central Mexico associated with the A/H1N1 influenza virus in April and May of 2009. The data reveal that proxy behavioral responses were greatest among children and wealthier socio-economic groups. Furthermore, we couple the behavioral response with an epidemiological model, and show that the A/H1N1 influenza virus was likely more transmissible than previously believed because the transmission potential was masked by behavioral responses.
To leverage the television viewing data for exploring the role of behavior during an epidemic, we extend the binary proxy for time varying infectivity in [20], where behavior can change at only one point in time, to allow for daily variation in behavior. Following [17], we decompose a standard model of the transmission rate into the two components of a contact rate and average transmission rate per contact. To inform changes in the contact rate, we use a daily proxy for changes in time spent by individuals in the home, namely variation in home television viewing. While viewing is an imperfect proxy for social distancing behavior, this data has several appealing attributes. The data are collected consistently prior to, during, and after epidemics in all major media markets worldwide. The sample is representative of the local population (by design) and can be disaggregated into various demographic subgroups. Typically the data is collected automatically and electronically (as in our sample) and do not rely on self-reporting. The viewing data in our application were obtained from IBOPE International net-AGB Nielsen Media Research, the largest private research and audience measurement firm in Latin America.
We contribute to the literature by examining variation in the behavioral response across time and demographic subgroups and by calibrating and analyzing dynamic behavioral disease transmission models. First, we quantify the dynamic nature of the behavioral response to the 2009 A/H1N1 influenza pandemic and public intervention in Central Mexico. We show that the aggregate response is not constant and describe how it varies systematically over time. Next, we unpack the aggregate dynamic into demographic subgroups and show how certain age groups and/or socio-economic groups respond more strongly than others. Turning to the modeling of disease transmission dynamics, we assess whether accounting for daily changes in contacts better accounts for the variation in new cases. We then explore the potential for bias in the standard model from ignoring underlying changes in behavior. Previous simulation analysis has shown that intervention focused on children is particularly effective in reducing the attack rate of influenza [21]. We examine how accounting for heterogeneity between adults and children alters conclusions. In the next section we first show how the basic transmission model can be extended to incorporate dynamic behavior and then describe the data and model estimation approach in detail.
Standard epidemiological model
We model the 2009 A/H1N1 epidemic in Central Mexico using an SEIR epidemiological model [22-24]. We define three different model formulations: one that does not account for any behavioral changes, one that assumes that behavioral change is constant throughout the government-imposed health interventions, and one that assumes that behavioral change can be estimated by daily television viewing data. For each model, individuals in the population, of size N, are classified by health status of individuals in into four states in each period, t: susceptible (St), exposed (infected but not yet infectious), (Et), infectious (It), and recovered (Rt). The transition dynamics between health states are described by a system of difference equations:
$$ \begin{array}{c}{S}_{t+1}-{S}_t=-{\beta}_t{S}_t{I}_t/N\\ {}{E}_{t+1}-{E}_t={\beta}_t{S}_t{I}_t/N-\kappa {E}_t\\ {}{I}_{t+1}-{I}_t=\kappa {E}_t-\gamma {I}_t\\ {}{R}_{t+1}-{R}_t=\gamma {I}_t,\end{array} $$
where β t is the transmission rate, κ is the rate at which incubating individuals progress from the exposed to the infectious health status (or the inverse of the latent period) and γ is the recovery rate (or the inverse of the recovery period).
In the standard (SD) model βt is becomes a constant scalar. This confounds the combined effect of contacts and the probability of transmission from a contact [12]. In the classical transmission model, the behavior governing contacts is assumed to be fixed. Yet for many human diseases, including influenza, behavioral shifts and NPI likely play an important role in the transmission process.
Behavioral epidemiological model
To generalize the classical model we decompose βt into the likelihood of transmission conditional on a contact (ρ0) and the average number of contacts experienced by individuals \( \left(\overline{\mathrm{C}}\right) \):
$$ {\beta}_t^{SD}={\rho}_0\overline{C}: $$
The parameters ρ0 and \( \overline{\mathrm{C}} \) are not uniquely identified since they enter the model as a product. Nevertheless, ρ0 can be estimated following [17] and using population estimates from the literature for \( \overline{C} \).a
Despite differentiating between the likelihood of transmission from a contact and the number of contacts, \( {\beta}_t^{SD} \) is assumed to be constant. We explore two alternatives that relax the assumption of a constant transmission rate. The first extension to facilitate a time-varying transmission rate is to allow for two different, but otherwise constant, levels in β t over time. Following [20], we model the behavioral response as a fixed effect (FE) (i.e. using a dummy variable) for the duration of a time period given by τ, for example during a particular public health intervention,
$$ {\beta}_t^{FE}=\left({\rho}_0+{\mathbf{1}}_{\tau }(t){\rho}_1\right)\overline{C}, $$
where ρ0 is a baseline marginal transmission rate (per contact), ρ1 is a shift in the marginal baseline transmission rate during the window τ, and 1τ(t) is the indicator function, equal to one when t ∈ τ, and zero otherwise.
Second, we propose a flexible response model that allows for daily variation in behavior. Given the availability of an empirical proxy for changes in contact rates, we relax the assumption of fixed contact rates. Let Δt represent the percentage deviation from the average \( \overline{\left(\mathrm{C}\right)} \) for a given period t. A dynamic behavioral (DB) transmission function that is similar in form the Equations (2) and (3) but accounts for variation in the contact rate is:
$$ {\beta}_t^{DB}=\left({\rho}_0+{\rho}_1{\Delta}_t\right)\overline{C}. $$
Relative to the SD model in Equation (2), the DB transmission rate model includes an additional term \( \left({\rho}_1{\Delta}_t\overline{C}\right) \) capturing an additive effect of any behavioral response. The SD model (2) is nested within both the FE model (3) and the DB model (4): \( {\beta}_t^{SD}={\beta}_t^{FE}\left({\rho}_1=0\right)={\beta}_t^{DB}\left({\rho}_1=0\right) \). Under all three models, the subset of the population N in each of the health states changes over time. The only other potentially dynamic component is the transmission rate β t , which is either fixed (SD model), takes one of two constant values over time (FE model), or varies daily (DB model).
To examine the implications of social distancing we focus on the initial outbreak of A/H1N1 in Central Mexico, in the spring of 2009.b We obtained laboratory confirmed pandemic A/H1N1 influenza cases from April 1 to May 20 in Central Mexico from a prospective epidemiological surveillance system that was established in response to the 2009 influenza pandemic by the Mexican Institute for Social Security (IMSS) [25]. These data are presented in Table 3 in Appendix A. IMSS is a tripartite Mexican health system that relies on a network of over 1,000 primary health-care units and 259 hospitals nationwide, and covers ~40% of the Mexican population. Importantly, testing rates for novel A/H1N1 influenza remained stable at ~33% [20]. Chowell et al. [20] show that the age distribution of the population affiliated with IMSS is generally representative of the general population of Mexico, rejecting the hypothesis that the distributions are significantly different. Furthermore they note that the male-to-female ratio among the population affiliated with IMSS (47:53) is similar to that of the general population (49:51).
On April 15th 2009, the Mexico Ministry of Health began receiving informal indications of a severe pneumonia in metropolitan Mexico City [3,26]. The novel influenza A/H1N1 virus was confirmed by U.S. and Canadian labs for multiple Mexican patients from April 22–24. On Friday, April 24th, the federal government announced the closure of public schools for metropolitan Mexico City, and a public awareness campaign was initiated by the Ministry of Health. Further "social distancing measures" involved closing restaurants and entertainment venues and cancelling large public events [26]. After May 9, the infection rate declined dramatically and large public health interventions were lifted [20]. Students resumed school on Monday, May 11. The window τ = {April 24, …, May 10} is used in the FE model for the sub-period over which we might expect to observe an effect due to social distancing. We also considered alternative dates for the start of this window, from April 10th through April 23rd, but none were statistically preferred as explained further in the results. A graphical timeline of events related to the outbreak is provided by Chowell et al. [20] (Table 1).
Table 1 Summary statistics for daily percentage deviation from the long-run mean ATV (Δ t ) for various demographic groups
Ethics Committee approval was not necessary according to local regulations. All the data were de-identified. Data employed in this study are routinely collected for epidemiological surveillance purposes.
We use data on home television viewing in metropolitan Mexico City as a proxy measure for dynamic behavioral response in Central Mexico during the influenza outbreak. The logic of this approach relies on two key assumptions. First, we assume that time spent watching television increases in time spent in the home, and that a linear approximation is sufficient to capture this behavior.c With respect to an individual's daily time allocation, since we are mainly concerned with time spent at home or not at home, an increase in the former subtracts from the latter. Second, we assume that the number of contacts an individual makes is proportional to the time spent outside the home.
Viewership data for Mexico City were obtained from IBOPE International net-AGB Nielsen Media Research, the largest private research and audience measurement firm in Latin America.d The specific measure used was individual daily average time viewed (ATV), which is given by the aggregate number of hours viewed by everyone in the sample divided by the number of individuals in the sample (including those with no viewing in a given period). The data reflect aggregate observations for individuals (not households) in a given demographic group. IBOPE's sample is composed of an ongoing panel of individuals, balanced across demographic characteristics to be representative of the population of Mexico City. Daily data were obtained for the months of April and May in 2009. With respect to data on daily confirmed cases of influenza and average TV viewership, ethics committee review was not relevant since all data were de-identified, aggregated before acquisition and collected under existing conditions (i.e. there were no experimental treatments). Similarly, since the data were gathered through existing mechanisms and not for our study, obtaining written informed consent from participants was not relevant.
We used the percentage deviation in average television viewership (relative to the non-intervention period) as a proxy for the percentage deviation in contacts. We choose this simple form for the proxy since a parameterized model of raw contacts as a function of television viewing is not available. Let \( \overline{ATV} \) represent the baseline (non-intervention period) mean of ATVt over an extended time horizon from both before and after the public response to the outbreak, but not during. The baseline period used to determine \( \overline{ATV} \) is April 1-April 23 and May 10-May 31, which includes April and May of 2009, excluding the period τ. \( \overline{ATV} \) for our sample is 1.7 hours per day (with a minimum and maximum ATVt over the baseline period of (1.5, 1.9)). The time-varying deviation from the baseline mean ATVt is given by \( {\varDelta}_t=\left(AT{V}_t-\overline{ATV}\right)/\overline{ATV} \).
We considered both a single homogenous population and a heterogeneous population divided into two groups: adults (age 18 and above, denoted A) and children (individuals below the age of 18, denoted K). For the heterogeneous population model, the disaggregated viewership data allowed for inference on how the behavior of adults and children varied over time. The extension of the homogenous population transmission model in (1) to the heterogeneous subgroup setting is presented in Appendix B. Information is not available to characterize how changes in contacts made by one group (e.g. adults) might differ between contacts they make within the same group (e.g. adult-adult contact) versus another group (e.g. adult-child contact). Therefore, we make the simplifying assumption that deviation in the contact rate for a member of group i is uniform across the different groups they may come in contact with; we used a single time series to inform deviations in children's contacts with either adults or children (Δt, K → A = Δt, K → K = Δt,K) and another single time series similarly for adults (Δt,A → K = Δt, A → A = Δt,A).
We modeled the age-specific contact rates for school-age children and adults for central Mexico based on survey contact data collected from several European countries [27]:
$$ \mathbf{C}=\left[\begin{array}{cc}\hfill {\overline{C}}_{K\to K}\hfill & \hfill {\overline{C}}_{K\to A}\hfill \\ {}\hfill {\overline{C}}_{A\to K}\hfill & \hfill {\overline{C}}_{A\to A}\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 8.9\hfill & \hfill 5.5\hfill \\ {}\hfill 1.9\hfill & \hfill 9.3\hfill \end{array}\right]. $$
The average contact rate for the homogenously mixing population, \( \overline{C}=6.1 \), is given by the population-weighted average of C.
Model estimation
We set the population of Central Mexico to N = 5.3*107 individuals [28] and follow [17] in setting the mean probability of an infection being laboratory-confirmed A/H1N1 influenza at \( \overline{\upvarphi}=0.0015 \). This estimate of is constructed as the product of the symptomatic rate (65% [29,30]), the hospitalization rate (0.45% [31]), and the probability of an infected, hospitalized individual being identified as having A/H1N1 (50%). We control for observed variation in the rate that hospitalized cases were tested by scaling the mean probability of confirmation by the observed deviation from the mean testing rate: \( {\phi}_{\mathrm{t}}=\overline{\upvarphi}\left({\mathrm{TR}}_{\mathrm{t}}/\overline{\mathrm{TR}}\right) \). Testing rate data were obtained from IMSS (the same source as described above for the case data). We set the fraction initially infected on day 1 of the time period (April 1) at π = 1.9 × 10− 5, such that given the population and the probability of confirmation, one case is confirmed on the first day. Consistent with [5,32,33], the daily rate of progression from latent to infected health status and the recovery rate are set to κ = 0.67 and γ = 0.5, respectively.
The main coefficients of interest for estimation are the parameters of the transmission rate functions for each of the three models. Let ρ represent the vector of marginal transmission rate parameters, given by the scalar [ρ0] for the SD model and the vector [ρ0, ρ1] for the FE and DB models. Model parameters were estimated by maximum likelihood. We assumed that the observed number of confirmed new infections each day, \( {I}_t^c \), follows a Poisson process with a mean arrival rate λ t (ρ) given by the number of new observed infections predicted by the disease model, ϕ t κE t . The log-likelihood function is:
$$ L={\displaystyle \sum_{t=1}^T\left({I}_t^c \ln \left[{\lambda}_t\left(\rho \right)\right]-{\lambda}_t\left(\rho \right)- ln\left({I}_t^c!\right)\right)}. $$
Development of the log-likelihood function is explained in further detail in Appendix C.
Because maximum likelihood estimates can be sensitive to the choice of initial values provided to the numerical optimization algorithm, we used a multiple starting point solver in Matlab (version R2013a) designed to identify the global optimum. For each model, the solver was run for each of M different randomly drawn starting vectors for the unknown parameters in ρ. We set M equal to 50 for the standard model (one parameter) and 100 for the alternative models (two parameters). From this set of local maxima, the solution with the greatest likelihood was selected as the estimate for the global maximum. We estimated 95% confidence intervals for the parameters using the likelihood ratio method [34]. To test for statistically significant differences in performance, when comparing the SD model against the FE and DB models we used a likelihood ratio test, since the SD model is nested within both of the alternatives (FE and DB). Since the FE and DB models are not nested, the standard likelihood ratio test is not feasible. Following [35], we used a Cox non-nested test with a parametric bootstrap (see Appendix D for details).
Dynamic behavioral response
In Figure 1 we present the dynamic behavioral response time series for Δ t (percentage deviation from mean ATV) in Mexico City during April and May 2009 in aggregate (Figure 1A) and for various demographic and time subgroups (Figure 1B-D). The range and mean for this variable over the limited intervention period (τ) is presented in Table 1. A positive deviation (Δ t > 0) indicates that an above average amount of time was spent in home TV viewing and, by inference, in the home. The mean level of Δ t over the period τ is positive and, as shown by a one-sample t-test, significantly different from zero at the 1% level for the aggregate population and each subgroup considered here (see Table 1).
Percentage deviations from mean daily individual average time viewed (Δ t ) for various demographic groups in Mexico City. Panel A shows deviations for the aggregate population. Panel B distinguishes between adults and children. Panel C differentiates by socioeconomic level (SEL). Panel D presents the daytime versus nighttime response. The shaded area in each graph represents the intervention period τ.
The dynamic path of Δ t for the aggregate population is presented in Figure 1A. Outside of the shaded intervention window (τ), this measure has a mean of zero (by construction) and typically falls within a range of +/− 5%. During the period τ, Δ t shifts demonstrably upwards. This behavioral response is strongest in the first week (approximately 20%) before gradually tapering off to near zero by the end of the intervention period. This pattern suggests that the population's capacity for social distancing might be limited in duration; before the public health intervention concluded, there was a substantial decline in the behavioral response relative to the peak in the first week. (Alternatively, it may be that the level of viewing per unit of time spent in the home fell as individuals switched to other in-home activities.) After the NPI concluded there was a period of reduced viewing activity in the home (Δ t < 0). Specifically, Δ t reached its most negative point on May 10th at −10.5%. Outside of the post-intervention dip, Δ t dropped below −10% on only one other day. As further evidence that the dip was likely not a coincident random event, we find that this dip persisted at 5% below the non-NPI period mean for four consecutive days—there are no other instances in the data when Δ t falls below 95% of the mean for more than a single day. While the causal mechanism behind these dynamics is not known with certainty, one possibility is that this multi-day period of suppressed in-home activity compensated for forgone social and commercial activities from earlier in the intervention period. The observation of reallocation of risky activities in time is common in the public health literature. Following the introduction of antiretroviral treatment for HIV/AIDS [28,36] find empirical evidence of increased sexual risk taking. Boyes and Faith [2] show that when alcohol consumption is banned at college football games that total alcohol consumption may rise through substitution effects in periods sandwiching the game. Finally, Graff Zivin and Neidell [37] find that while Southern California residents curtail outdoor activity on days with poor air quality, if the episode is prolonged the behavioral response dissipates rapidly.
The age class breakdown for Δ t presented in Figure 1B shows a substantial difference in response between children and adult subgroups during the intervention period. The mean (23.7%) and the maximum (46.2%) behavioral response of children is more than twice as large as the response observed for adults (see Table 1). The difference in responses is statistically significant at the 1% level as indicated by a two sample t-test.
The data from IBOPE are disaggregated into three socioeconomic levels (SELs) based on a set of household characteristics, including the size and amenities of the home, appliance ownership, automobile ownership, and level of education (Figure 1C). During the intervention period, on average the high SEL group shows a response that is over 50% greater than that of the low SEL group. This difference is significant at the 5% level. The medium SEL class displays an intermediate response (Table 1).
Finally, we consider variation in the response by time of day, specifically daytime (6 am-6 pm) versus nighttime (6 pm-6 am) (Figure 1D). The mean daytime response is approximately twice as strong as the nighttime response (Table 1). This is not surprising given that time spent in the home is lower during the daytime to begin with and thus presents a larger opportunity for adjustment.
The time path for each of the subgroups discussed above follows a path that is qualitatively similar to that of the aggregate population, showing a strong initial positive response that largely or entirely decays before the end of the intervention. For each subgroup comparison considered here, there was a significant difference in the average level of the behavioral response.
Transmission model estimation
The maximum likelihood parameter estimates for each model are based on T = 41 days of observations, stretching from April 1 through the end of the intervention period on May 11 (Table 2). Figures illustrating the log-likelihood profile for each model are presented in Appendix E. The time frame used corresponds to the period of time considered in [20]. After this period, additional cases attenuate substantially as shown in the time series of \( {I}_t^c \) (Figure 2). We focus on this initial 41 day period since the performance of each model (in terms of log-likelihood values and residuals) becomes increasingly poor as more of the post-intervention period is included
Table 2 Maximum likelihood parameter estimates
Time series for the number of new confirmed cases (\( {\mathbf{I}}_{\mathbf{t}}^{\mathbf{c}} \), left axis) and percentage deviation from mean daily individual average time viewed (Δ t , right axis) for 50 days beginning April 1, 2009. The shaded area in each graph represents the intervention period τ.
The degree to which accounting for changes in contacts better accounts for the variation in new cases is one of our core research questions. Results show that the standard model is indeed incomplete—we reject the SD model in favor of both the DB model (p < 0.01) and FE model (p < 0.01). However, we do not find that the DB model outperforms the FE model. In fact we reject the DB model in favor of the FE model (p < 0.01). To see why it might be the case that a simple fixed effect is preferred in this case to the dynamic, data-driven behavioral model, consider the time series for \( {I}_t^c \) and Δ t presented in Figure 2. Consistent with expectations under the DB model, when the social distancing proxy Δ t begins to surge on April 24th (day 24) the number of new confirmed cases plateaus. However, when Δ t declines in early May while infections are still common, the number of new confirmed cases \( \left({I}_t^c\right) \) does not grow in a sustained fashion but rather, after a slight delay, begins to fall. Thus the dynamics of initial and early intervention period of the outbreak are consistent with the DB model but the late intervention period is not.
Given that both the FE and DB models outperform the SD model, we explored the potential for biased estimates of the transmission parameter in the SD model as a potential shortcoming of ignoring behavioral change. Estimates of the baseline transmission rate (ρ0) in Table 2 show that while the DB and FE models are in essential agreement, the SD estimate is 12% lower. To explore whether this difference is idiosyncratic or systematic we re-estimate each of the three models starting with only the first M days of data for M ∈ [15, 41]. We exclude the FE model for M ∈ [15, 24] since this model is not differentiated from the SD model until the intervention begins on April 24th. In Figure 3 we present the resulting estimates of ρ0. We find that estimates are variable but roughly consistent across models through April 24th. This is not surprising given that before the public health intervention began on April 24th our proxy suggests that behavior had yet to shift discernibly. After this point, estimates of ρ0 for the DB and FE models remain roughly stable near 0.064 while the baseline transmission coefficient for the SD model declines monotonically. Thus over the intervention period when behavioral response is strong, the SD estimate of ρ0 falls each day to account for the new factor. In contrast, models that allow for a behavioral shift result in estimates for baseline transmission that are essentially level over time.
Estimates for the baseline rate of transmission for the three models: standard (SD), fixed effect (FE) and dynamic behavioral (DB).
As a practical matter, this bias in the SD model has important implications for public health and forecast error. First, the SD model provides an estimate of ρ0 substantially lower than models with behavior. This suggests that A/H1N1 virus is more infectious, but this infectiousness is masked by behavioral shifts. Second, the SD model results in substantial forecast error, a result shown using simulation in [13] to emerge when human adaptive behavior is important in epidemiological systems.
Forecasting error comparison
In Figure 4 we present forecast error over a four-day horizon for time series of increasing length from M ∈ [15, 41]. The exercise is meant to capture the public health official's problem of estimating the current state of an outbreak based on observed cases to date. We assume that there is a four-day lag between the date of testing and reporting of all confirmed cases, a typical lag for reporting infectious disease outbreaks. Thus forecast error appearing in the figure for day M = 15 represents error made on day 19 conditional on case data that is complete through day 15. We assume that behavioral data (Δ t ) is available across this four day lag. From the raw forecast error in Figure 4A, it is clear that prediction performance for the SD model becomes poor relative to the alternatives shortly after the intervention on day 24. From this point on, the SD model leads to systematic over-prediction of the number of new cases. DB model performance deteriorates next towards the end of the intervention period. Finally, by the time the intervention concludes, all three models systematically over-predict new cases. This suggests that factors absent from the models considered here are important for capturing post intervention dynamics (e.g. personal protective measures to reduce risks per contact).
Error from forecasting new confirmed cases over a four-day horizon conditional on the number of days observed under the standard (SD), fixed effect (FE) and dynamic behavioral (DB) models. Panel A shows daily forecast error and Panel B shows cumulative absolute error starting from day 25.
We estimated the transmission model results above assuming a single homogenous population. However, differences in the behavioral response (Δ t ) for children versus adults presented above motivate exploration of age-class heterogeneity. When we modeled children and adults as separate populations (with separate time series for Δ t in the behavioral model), but constrained transmission parameters to be the same for both populations, estimates were not significantly changed. We further tested an extended model in which transmission parameters (ρ0, ρ1) were free to vary between the two groups. This model was not statistically significantly different for either the SD (p = 0.31), DB model (p = 0.41), or FE model (p = 0.12) at the 10% level. For this FE model, relative to the homogeneous (baseline) case, the coefficients ρ0 and ρ1 were roughly 50% larger in magnitude for children and 90% smaller in magnitude for adults. This evidence is not conclusive, but hints that infections between children and from children to adults might have been a leading driver of disease dynamics—and also most sensitive to intervention. However, this effect is too small and imprecisely estimated to assert with statistical significance.
While we failed to find a significant difference in the transmission coefficients between children and adults, this does not mean that there were not significant differences in these populations. Recall that we controlled for differences between children and adults in the baseline contact rate as specified in the matrix C. When this matrix was replaced with the average \( \left(\overline{C}\right) \) a significant difference emerged between the homogeneous and heterogeneous coefficient specification for both the SD (p < 0.01) and DB models (p = 0.08) but not for the FE model.
We examined the sensitivity of transmission model results to several alternative assumptions. First, given the temporal mismatch between the case and behavioral time series in Figure 2, we explored whether the relative preference for the FE model continued to hold under extensions in the latent period, i.e. the number of days individuals were infected but not infectious. In the baseline model the latent period was set to 1/κ = 1.5. The performance of the DB model relative to the FE model was robust to alternative assumptions on the latent period, including 2, 3 or 4 days. We also considered whether idiosyncratic variation or "noise" in the ATV variable might hinder the DB model. As a simple test we set a +/−5% threshold for the Δt measure—any variation that did not exceed this band was set to zero. This did not qualitatively change results. Qualitative results were also not sensitive to a nonlinear quadratic form for the DB model.
Convergence in the performance of the DB and FE models was found when the number of days included in the estimation was limited. For all time series that included 38 days or less, we failed to reject one model in favor of the other. However, after this time frame the FE model emerges as the preferred model (e.g. p < 0.01 at 39 days).
For the FE model, we also considered alternative dates for the start of the intervention window, from April 10th through our baseline window start date of April 24th. For each of these alternative specifications we found that the associated parameter ρ1 was statistically significantly different from zero. However, we also found that the log-likelihood was greatest for the FE window beginning on April 24th (our baseline specification) illustrating that none of the alternative start dates was statistically preferred.
The final parameter examined in our sensitivity analysis was the mean probability of confirmation. Our baseline level for \( \overline{\upvarphi} \) implies that 1.2% of the population was infected by the end of the spring wave (conditional on the observed number of cases and total population). We examined sensitivity to an alternative scenario in which 10% of the population contracts the disease, which implied a mean probability of confirmation of \( \overline{\upvarphi}=8.1\times {10}^{-5} \). Results from this alternative low probability of confirmation scenario were not qualitatively different.
Counterfactual behavioral response
We explore two alternative scenarios in which the behavioral response to the epidemic is either non-existent or enhanced. We present the path of new confirmed cases under these alternatives, along with fitted curves from the baseline models in Figure 5A. Under the first alternative, to eliminate the behavioral response, we multiply the ρ0 term by zero (0ρ0, thin lines). Under the second alternative, to enhance the behavioral response, we multiply the ρ0 term by two (2ρ0, thick lines). Fitted curves from the unaltered baseline models (1ρ0, medium lines) and \( {I}_t^c \) are provided for comparison. For the baseline models, the fit of the DB and FE models is similar until the final few periods in which the DB fit diverges from the observed path \( \left({I}_t^c\right) \). The importance of the behavioral response is evident. With no behavioral response, the projected path of new cases increased sharply, more than quadrupling (Figure 5B) for both models by day 41. Alternatively, with a doubling of response, attenuation of new cases occurs approximately two weeks earlier and cumulative cases by day 41 are cut in half.
Fitted daily cases (A) and cumulative cases (B) for the unaltered fixed effect (FE) and dynamic behavioral (DB) models (1ρ 0 ) and two alternatives where the behavioral response is either eliminated (0ρ 0 ) or doubled (2ρ 0 ). Observed newly confirmed cases \( \left({I}_t^c\right) \) are also provided for comparison.
We used novel data on variation in home television viewing behavior as a proxy for changes in the level of daily social interaction in Central Mexico during the 2009 A/H1N1 influenza pandemic. Results from both behavioral models (FE and DB) suggested that social distancing was a key factor in constraining the initial wave of A/H1N1 in Central Mexico. In the absence of a behavioral response, the estimated counterfactual path of new cases escalated rapidly in initial weeks rather than stabilizing and eventually falling as was observed. The assumption of fixed behavior in the standard (SD) model led to shortcomings in estimation and prediction. Estimates of the baseline rate of transmission systematically shifted over time. If the baseline rate of transmission is interpreted as a measure of biological infectivity in the standard model, this is likely to lead to an underestimate of this parameter, as in our setting, given confounding effects of behavioral responses. This suggests that A/H1N1 had an innate transmission potential much greater than previously thought but this was masked by behavioral responses. This has implications for management advice including the allocation of resources between pharmaceutical and nonpharmaceutical interventions. Furthermore, the error in near term predictions of new cases through time was also substantially greater under the standard model compared to the behavioral models. This error was also systematic—the standard model consistently led to over-prediction in the number of new cases.
Comparing the behavioral models, we found that that the dynamic behavioral model was not preferred to the simpler fixed effect model. One explanation may be the imperfect nature of variation in viewership as a proxy for changes in public contact rate. For example, it is possible that during the public health intervention the observed increase in ATVt was due to a greater share of home time allocated to TV viewing, rather than an increase in time spent at home. Or it could be the case that viewing per unit of time spent at home may be declining in time spent at home. Another explanation might be the inability at this time to empirically capture changes in behavior outside the home to reduce contacts or transmission (e.g. washing hands, wearing facemasks, and avoidance of coughing into open air). Bell [5] notes that while policies promoting social distancing may be effective against pandemic influenza, other individual behavioral measures should be either routine (e.g. hand and respiratory hygiene and disinfection of contaminated household surfaces) or considered for certain settings and risk levels (e.g. mask use).
We found that the home viewership response was stronger in the high (versus low) socioeconomic level (SEL) subgroup. This finding is suggestive but should be interpreted with care. On the one hand, individuals in the high SEL subgroup are arguably less constrained in adjusting contacts than those in the low SEL subgroup. For example, Kumar et al. [38] suggested that workplace policies can impinge on distancing measures and such workplace policies may be more binding on lower SELs. If this hypothesis were tested and verified, it would suggest the potential for targeting of social distancing polices to facilitate self-protective measures for low SEL individuals. On the other hand, it may be that the difference in response is an artifact of the behavioral proxy which might emerge, for example, if the relationship between home viewership and time spent at home differed systematically between SEL subgroups (e.g., if high SEL individuals respond more strongly because ownership of more televisions provides more opportunities to view).
In addition to varied responses across groups, we also found differences over time, namely attenuation in the behavioral response before the conclusion of the public health intervention. Furthermore, we found evidence of a rebound effect in which, after a prolonged period of elevated in-home activity there appeared to be period of suppressed activity. This is consistent with the historical analysis of Caley et al. [19] who found that as the perceived risk of the 1918 swine flu decreased in Australia, the public appeared to revert to normal behavior. Similarly, Fenichel et al. [18] found that air travelers' adaptive to A/H1N1 dissipated after an initially strong response. Further studies of the 2009 A/H1N1 influenza pandemic in other regions with similar intervention measures (e.g. Hong Kong, [39]) could help to confirm and generalize the insights gleaned here.
While the dynamic behavioral model based on the home viewership proxy did not out-perform the simple fixed effect model, the results represent progress in identifying and unpacking the drivers behind this fixed effect. Going forward, further detailed data on private and public behavior during outbreaks would serve to identify behavioral effects on transmission with greater precision. For example, we did not model the effect of antiviral treatment. Capturing additional behavioral adjustments made outside of the home to reduce effective contacts is likely be important for explicit modeling of the behavior underlying disease transmission. To this end, there is value in allocating resources during an outbreak to consistently gather data on public and private protective actions, such as antiviral use or the use of face masks. Although transitioning from empirical analysis based on fixed effect measures of behavior to fully dynamic responses at finer time scales will require additional investment in data collection, potential benefits include the promise of informing more finely tuned and less costly public health interventions.
Ksiazek TG, Erdman D, Goldsmith CS, Zaki SR, Peret T, Emery S, Tong S, Urbani C, Comer JA, Lim W, Rollin PE, Dowell S, Ling A-E, Humphrey C, Shieh W-J, Guarner J, Paddock CD, Rota P, Fields B, DeRisi J, Yang J-Y, Cox N, Hughes J, LeDuc JW, Bellini W, Anderson LJ, and the SARS Working Group. A novel coronavirus associated with severe acute respiratory syndrome. N Engl J Med. 2003; 348(20):1953–66.
Boyes WJ, Faith RL. Temporal regulation and intertemporal substitution: the effect of banning alcohol at college football games. Public Choice. 1993; 77(3):595–609.
Chowell G, Bertozzi SM, Colchero MA, Lopez-Gatell H, Alpuche-Aranda C, Hernandez M, Miller MA. Severe respiratory disease concurrent with the circulation of H1N1 influenza. N Engl J Med. 2009; 361(7):674–79.
Zhu H, Wang D, Kelvin DJ, Li L, Zheng Z, Yoon S-W, Wong S-S, Farooqui A, Wang J, Banner D, Chen R, Zheng R, Zhou J, Zhang Y, Hong W, Dong W, Cai Q, Roehrl MHA, Huang SSH, Kelvin AA, Yao T, Zhou B, Chen X, Leung GM, Poon LLM, Webster RG, Webby RJ, Peiris JSM, Guan Y, Shu Y. Infectivity, transmission, and pathology of human-isolated H7N9 influenza virus in ferrets and pigs. Science. 2013; 341(6142):183–86.
Bell D, Nicoll A, Fukuda K, Horby P, Monto A, Hayden F, Wylks C, Sanders L, Van Tam J. Non-pharmaceutical interventions for pandemic influenza, international measures. Emerg Infect Dis. 2006; 12(1):81–7.
Cauchemez S, Valleron A-J, Boelle P-Y, Flahault A, Ferguson NM. Estimating the impact of school closure on influenza transmission from Sentinel data. Nature. 2008; 452(7188):750–54.
Stern A, Markel H. What mexico taught the world about pandemic influenza preparedness and community mitigation strategies. JAMA. 2009; 302(11):1221–22.
Aiello AE, Coulborn RM, Aragon TJ, Baker MG, Burrus BB, Cowling BJ, Duncan A, Enanoria W, Fabian MP, Ferng Y-H, Larson EL, Leung GM, Markel H, Milton DK, Monto AS, Morse SS, Navarro JA, Park SY, Priest P, Stebbins S, Stern AM, Uddin M, Wetterhall SF, Vukotich CJ. Research findings from nonpharmaceutical intervention studies for pandemic influenza and current gaps in the research. Am J Infect Control. 2010; 38(4):251–58.
Smith RD, Keogh-Brown MR, Barnett T, Tait J. The economy-wide impact of pandemic influenza on the UK: a computable general equilibrium modelling experiment. BMJ (Clinical research ed). 2009; 339:b4571.
Maharaj S, Kleczkowski A. Controlling epidemic spread by social distancing: Do it well or not at all. BMC Public Health. 2012; 12(1):679.
Fenichel EP. Economic considerations for social distancing and behavioral based policies during an epidemic. J Health Econ. 2013; 32(2):440–51.
Fenichel EP, Castillo-Chavez C, Ceddia MG, Chowell G, Parra PAG, Hickling GJ, Holloway G, Horan R, Morin B, Perrings C, Springborn M, Velazquez L, Villalobos C. Adaptive human behavior in epidemiological models. Proc Natl Acad Sci. 2011; 108(15):6306–11.
Fenichel E, Wang X. The mechanism and phenomena of adaptive human behavior during an epidemic and the role of information. In: Manfredi P, D'Onofrio A, editors. Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases. New York: Springer; 2013: p. 153–68.
Ferguson N. Capturing human behaviour. Nature. 2007; 446(7137):733.
Hens N, Ayele G, Goeyvaerts N, Aerts M, Mossong J, Edmunds J, Beutels P. Estimating the impact of school closure on social mixing behaviour and the transmission of close contact infections in eight European countries. BMC Infect Dis. 2009; 9(1):187.
Copeland DL, Basurto-Davila R, Chung W, Kurian A, Fishbein DB, Szymanowski P, Zipprich J, Lipman H, Cetron MS, Meltzer MI, Averhoff F. Effectiveness of a school district closure for pandemic influenza a (H1N1) on acute respiratory illnesses in the community: a natural experiment. Clin Infect Dis. 2013; 56(4):509–16.
Towers S, Chowell G. Impact of weekday social contact patterns on the modeling of influenza transmission, and determination of the influenza latent period. J Theor Biol. 2012; 312:87–95.
Fenichel EP, Kuminoff NV, Chowell G. Skip the trip: Air Travelers' behavioral responses to pandemic influenza. PLoS One. 2013; 8(3):e58249.
Caley P, Philp DJ, McCracken K. Quantifying social distancing arising from pandemic influenza. J R Soc Interface. 2008; 5(23):631–39.
Chowell G, Echevarría-Zuno S, Viboud C, Simonsen L, Tamerius J, Miller MA, Borja-Aburto VH. Characterizing the epidemiology of the 2009 influenza A/H1N1 pandemic in Mexico. PLoS Med. 2011; 8(5):e1000436.
Glass RJ, Glass LM, Beyeler WE, Min HJ. Targeted social distancing design for pandemic influenza. Emerg Infect Dis. 2006; 12(11):1671–81.
Hethcote HW. Qualitative analyses of communicable disease models. Math Biosci. 1976; 28(3–4):335–56.
Diekmann O, Heesterbeek. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. New York: Wiley \& Sons; 2000.
Anderson RM, May RM. Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press; 1991.
Echevarria-Zuno S, Mejia-Arangure JM, Mar-Obeso AJ, Grajales-Muniz C, Robles-Perez E, Gonzalez-Leon M, Ortega-Alvarez MC, Gonzalez-Bonilla C, Rascon-Pacheco RA, Borja-Aburto VH. Infection and death from influenza A H1N1 virus in Mexico: a retrospective analysis. Lancet. 2009; 374(9707):2072–79.
Córdova J, Hernández M, López-Gatell H, Bojorquez I, Palacios E, Rodríguez G, Rosa B, Ocampo R, Alpuche C, Flores R. Update: novel influenza A (H1N1) virus infection-Mexico, March-May 2009. Morb Mortal Wkly Rep. 2009; 58(21):585–89.
Mossong J, Hens N, Jit M, Beutels P, Auranen K, Mikolajczyk R, Massari M, Salmaso S, Tomba GS, Wallinga J, Heijne J, Sadkowska-Todys M, Rosinska M, Edmunds WJ. Social contacts and mixing patterns relevant to the spread of infectious diseases. PLoS Med. 2008; 5(3):e74.
Lakdawalla D, Sood N, Goldman D. HIV breakthroughs and risky sexual behavior. Q J Econ. 2006; 121(3):1063–102.
Elder AG, O'Donnell B, McCruden EAB, Symington IS, Carman WF. Incidence and recall of influenza in a cohort of Glasgow healthcare workers during the 1993–4 epidemic: results of serum testing and questionnaire. BMJ. 1996; 313(7067):1241–42.
King JC, Haugh CJ, Dupont WD, Thompson JM, Wright PF, Edwards KM. Laboratory and epidemiologic assessment of a recent influenza B outbreak. J Med Virol. 1988; 25(3):361–68.
Reed C, Angulo FJ, Swerdlow DL, Lipsitch M, Meltzer MI, Jernigan D, Finelli L. Estimates of the prevalence of pandemic (H1N1) 2009, United States, April-July 2009. Emerg Infect Dis. 2009; 15(12):2004–07.
Cowling BJ, Chan KH, Fang VJ, Lau LL, So HC, Fung RO, Ma ES, Kwong AS, Chan C-W, Tsui WW. Comparative epidemiology of pandemic and seasonal influenza A in households. N Engl J Med. 2010; 362(23):2175–84.
Cauchemez S, Donnelly CA, Reed C, Ghani AC, Fraser C, Kent CK, Finelli L, Ferguson NM. Household transmission of 2009 pandemic influenza A (H1N1) virus in the United States. N Engl J Med. 2009; 361(27):2619–27.
Davidson R, MacKinnon JG. Econometric Theory and Methods, Volume 21. New York: Oxford University Press; 2004.
Dameus A, Richter FGC, Brorsen BW, Sukhdial KP. AIDS versus the Rotterdam demand system: a Cox test with parametric bootstrap. J Agric Resour Econ. 2002; 27(2):335–47.
Mechoulan S. Risky sexual behavior, testing, and HIV treatments. In: Forum for Health Economics & Policy: 2007. 2007.
Zivin JSG, Neidell MJ. The impact of pollution on worker productivity. In: National Bureau of Economic Research. 2011.
Kumar S, Quinn SC, Kim KH, Daniel LH, Freimuth VS. The impact of workplace policies and other social factors on self-reported influenza-like illness incidence during the 2009 H1N1 pandemic. Am J Public Health. 2011; 102(1):134–40.
Wu JT, Cowling BJ, Lau EHY, Ip DKM, Ho L-M, Tsang T, Chuang S-K, Leung P-Y, Lo S-V, Liu S-H, Riley S. School closure and mitigation of pandemic (H1N1) 2009, Hong Kong. Emerg Infect Dis. 2010; 16(3):538–41.
Coulibaly N, Wade Brorsen B. Monte carlo sampling approach to testing nonnested hypothesis: monte carlo results. Econ Rev. 1999; 18(2):195–209.
This publication was made possible by grant number 1R01GM100471-01 from the National Institute of General Medical Sciences (NIGMS) at the National Institutes of Health. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of NIGMS.
Department of Environmental Science & Policy, University of California, 2104 Wickson Hall, One Shields Ave., Davis, CA, 95616, USA
Michael Springborn
Mathematical, Computational & Modeling Sciences Center, School of Human Evolution and Social Change, Arizona State University, 900 S. Cady Mall, Tempe, 85287-2402, AZ, USA
Gerardo Chowell
Division of International Epidemiology and Population Studies, Fogarty International Center, National Institutes of Health, 31 Center Dr, MSC 2220, Bethesda, 20892-2220, MD, USA
Department of Agricultural & Resource Economics, University of California, 2116 Social Sciences & Humanities, One Shields Ave., Davis, CA, 95616, USA
Matthew MacLachlan
Yale School of Forestry and Environmental Studies, 195 Prospect St., New Haven, CT, 06511, USA
Eli P Fenichel
Correspondence to Michael Springborn.
MS, EF, and GC contributed to concept, design, and model development. MS and MM analyzed the data. All authors contributed to, read and approved the final manuscript.
An erratum to this article is available at http://dx.doi.org/10.1186/s12879-016-1795-5.
Table 3 Laboratory confirmed pandemic A/H1N1 influenza cases from April 1 to May 20, 2009, in Central Mexico for children (under 18) and adults (18 and over)
Multiple age class transmission model
All three base models (SD, FE and DB) can be generalized to allow for age structure within the population. Social interactions may vary across demographic groups, for example children attending school versus working adults. We follow [17] in generalizing the system of differential equations for a homogenously mixing population in (1) to allow for variation in the transmission rate between demographic groups in the set G. Dynamics for each subgroup i ∈ G are given by:
$$ \begin{array}{c}{S}_{i,t+1}-{S}_{i,t}=-{S}_{i,t}{\displaystyle \sum_g^G{\beta}_{t,i\to g}{I}_{g,t}/N}\\ {}{E}_{i,t+1}-{E}_{i,t}={S}_{i,t}{\displaystyle \sum_g^G{\beta}_{t,i\to g}{I}_{g,t}/N}-\kappa {E}_{i,t}\\ {}{I}_{i,t+1}-{I}_{i,t}=\kappa {E}_{i,t}-\gamma {I}_{i,t}\\ {}{R}_{i,t+1}-{R}_{i,t}=\gamma {I}_{i,t}.\end{array} $$
The model in (5) captures heterogeneous mixing within the population model. The group-specific transmission function (βt,i → g) is the same as in the homogenous case, except \( \overline{C} \) and Δ t are replaced by \( {\overline{C}}_{i\to g} \) and Δt,i → g, respectively. The parameter \( {\overline{\mathrm{C}}}_{\mathrm{i}}{{}_{\to}}_{\mathrm{g}} \) reflects the average number of contacts that members of group i experience with members of group g, and Δt,i → g is the percent deviation from that average at time t.
Derivation of the log-likelihood function
We assumed that the observed number of confirmed new infections on any given day, \( {I}_t^c \), follows a Poisson process with a mean arrival rate λ t (ρ):
$$ \Pr \left({I}_t^c\Big|{\lambda}_t\left(\rho \right)\right)=\frac{\lambda_t\left(\rho \right)}{ \exp \left({\lambda}_t\left(\rho \right)\right){I}_t^t!}. $$
The likelihood function for all observations from t = 1, …, T is given by the product:
$$ L={\displaystyle \prod_{t=1}^T\left(\frac{\lambda_t\left(\rho \right)}{ \exp \left({\lambda}_t\left(\rho \right)\right){I}_t^t!}\right)}. $$
Taking the log of this expression provides the log-likelihood function:
$$ L={\displaystyle \sum_{t=1}^T\left({I}_t^c \ln \left[{\lambda}_t\left(\rho \right)\right]-{\lambda}_t\left(\rho \right)- ln\left({I}_t^c!\right)\right)} $$
Finally, to connect the likelihood model with the SEIR transmission model, we assume that the mean Poisson arrival rate of newly confirmed cases is given by the number of new observed infections predicted by the disease model, λ t (ρ) = ϕ t κE t .
Cox non-nested test with a parametric bootstrap
Under a given null model (e.g. either FE or DB), each bootstrapped sample of the data (new infections) was generated by simulating draws from the Poisson process governing arrivals of new infections based on the fitted estimates of the mean arrival rate for new infections, λ t ∀ t = 1, …, 50. This process was repeated to create M = 500 bootstrapped samples. The likelihood estimates from each of the bootstrapped samples were used to construct the following p-value [40] for the test of a given alternative model (a) against the null (0):
$$ p\hbox{-} \mathrm{value}=\frac{numb\left[{L}_0\left({\widehat{\theta}}_{0m},{I}_m^{obs}\right)-{L}_a\left({\widehat{\theta}}_{am},{I}_m^{obs}\right)\le {L}_{0a},\ \forall\ m=1,\dots, M\ \right] + 1\ }{M+1}, $$
where \( {I}_m^{obs} \) is the bootstrapped data sample for each iteration m = 1, …, M; \( {\widehat{\theta}}_{jm} \) represents the ML estimates for model j ∈ {FE, DB} given sample m; Lj is the maximum log-likelihood for the model j; \( {\mathrm{L}}_{0\mathrm{a}}={\mathrm{L}}_0\left({\widehat{\uptheta}}_0\right)-{\mathrm{L}}_{\mathrm{a}}\left({\widehat{\uptheta}}_{\mathrm{a}}\right) \) is the difference between the maximum log-likelihood estimates under H0 and Ha given the original data; and numb counts the number of times the condition is true for each of M iterations. A small sample correction is implemented by adding a 1 to the numerator and denominator. Because the FE and DB models are nonnested, selection of a unique null model is not feasible. Instead, the Cox test is conducted twice, with each of the models serving as the null in turn.
Likelihood profiles
In Figures 6, 7, 8 we present log-likelihood profiles underlying the maximum likelihood estimates in Table 2. In each case the log-likelihood value excludes the additive constant term which is not a function of the parameters to be estimated (i.e. the final term in Equation (10)). For each profile the maximum likelihood estimates from Table 2 are indicated with a triangle.
Log-likelihood profile for the standard (SD) model as a function of the marginal transmission rate (per contact) ρ 0 . The triangle represents the maximum likelihood estimate from Table 2. The star on log-likelihood indicates that the constant term from the log-likelihood has been excluded.
Log-likelihood profile for the fixed effect (FE) model as a function of the marginal transmission rate (per contact) ρ 0 , and the shift in the marginal baseline transmission rate during the intervention window, ρ 1 . The triangle represents the maximum likelihood estimates from Table 2. The star on log-likelihood indicates that the constant term from the log-likelihood has been excluded.
Log-likelihood profile for the dynamic behavioral (DB) model as a function of the marginal transmission rate (per contact) ρ 0 , and the behavioral response, ρ 1 . The triangle represents the maximum likelihood estimates from Table 2. The star on log-likelihood indicates that the constant term from the log-likelihood has been excluded.
aTowers and Chowell [17] allow the number of contacts experienced on weekends and weekdays to differ but these levels are taken from the literature and are otherwise constant. They also allow the transmission rate to vary over time according to a first order harmonic process to capture seasonality over a large portion of the year. We do not explore this structure since our period of interest is two months long.
bCentral Mexico includes the Federal District (Mexico City) and states of Guerrero, Hidalgo, Jalisco, Mexico (includes greater Mexico City), Puebla, San Luis Potosi, and Tlaxcala.
cThis assumption is difficult to test for Mexico. However, data from the American Time Use Survey (http://www.bls.gov/tus/) suggest that Americans do watch more television as they spend more time in the house, though the relationship may be nonlinear [37].
dThe data are collected and stored by the regional division IBOPE AGP Mexico (http://www.agbnielsen.net/whereweare/whereweare.asp).
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
Springborn, M., Chowell, G., MacLachlan, M. et al. Accounting for behavioral responses during a flu epidemic using home television viewing. BMC Infect Dis 15, 21 (2015). https://doi.org/10.1186/s12879-014-0691-0
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\begin{document}
\title{Chromatic Clustering in High Dimensional Space} \author{Hu Ding \hspace{0.25in} Jinhui Xu} \institute{
Department of Computer Science and Engineering\\ State University of New York at Buffalo\\
\email{\tt \{huding, jinhui\}@buffalo.edu}\\ } \maketitle
\thispagestyle{empty}
\begin{abstract} In this paper, we study a new type of clustering problem, called {\em Chromatic Clustering}, in high dimensional space. Chromatic clustering seeks to partition a set of colored points into
groups (or clusters) so that no group contains points with the same color and a certain objective function is optimized. In this paper, we consider two variants of the problem, chromatic $k$-means clustering (denoted as $k$-CMeans) and chromatic $k$-medians clustering (denoted as $k$-CMedians), and investigate their hardness and approximation solutions. For $k$-CMeans, we show that the additional coloring constraint destroys several key properties (such as the locality property) used in existing $k$-means techniques (for ordinary points), and significantly complicates the problem. There is no FPTAS for the chromatic clustering problem, even if $k=2$. To overcome the additional difficulty, we develop a standalone result, called {\em Simplex Lemma}, which enables us to efficiently approximate the mean point of an unknown point set through a fixed dimensional simplex. A nice feature of the simplex is its independence with the dimensionality of the original space, and thus can be used for problems in very high dimensional space. With the simplex lemma, together with several random sampling techniques, we show that a $(1+\epsilon)$-approximation of $k$-CMeans can be achieved in near linear time through a sphere peeling algorithm. For $k$-CMedians, we show that a similar sphere peeling algorithm exists for achieving constant approximation solutions.
\end{abstract}
\pagestyle{plain} \pagenumbering{arabic} \setcounter{page}{1}
\section{Introduction} \label{sec-intro}
Clustering is one of the most fundamental problems in computer science and finds applications in many different areas \cite{AV07,BBC,BHI,AP98,DEF,GG06,HM04,KR99,OSS,BKK,HDX11}. Most existing clustering techniques assume that the to-be-clustered data items are independent from each other.
Thus each data item can ``freely'' determine its membership within the resulting clusters, without paying attention to the clustering of other data items.
In recent years, there are also considerable attentions on clustering dependent data and a number of clustering techniques, such as correlation clustering, point-set clustering, ensemble clustering, and correlation connected clustering, have been developed \cite{BBC,BKK,DEF,GG06,HDX11}.
In this paper, we consider a new type of clustering problems, called {\em Chromatic Clustering}, for dependent data. Roughly speaking, a chromatic clustering problem takes as input a set of colored data items and groups them into clusters, according to certain objective functions, so that no pair of items with the same color are grouped together (such a requirement is called {\em chromatic constraint}). Chromatic clustering captures the mutual exclusiveness relationship among data items and is a rather useful model for various applications. Due to the additional chromatic constraint, chromatic clustering is thus expected to simultaneously solve the ``coloring'' and clustering problems, which significantly complicates the problem. As it will be shown later, the chromatic clustering problem is challenging to solve even for the case that each color is shared only by two data items.
For chromatic clustering, we consider in this paper two variants, {\em Chromatic $k$-means Clustering ($k$-CMeans)} and {\em Chromatic $k$-median Clustering ($k$-CMedians)}, in $\mathbb{R}^{d}$ space, where the dimensionality could be very high and $k$ is a fixed number.
In both variants, the input is a set $\mathcal{G}$ of $n$ point-sets $G_{1}, \cdots, G_{n}$ with each containing a maximum of $k$ points in $d$-dimensional space, and the objective is to partition all points of $\mathcal{G}$ into $k$ different clusters so that the chromatic constraint is satisfied and the total squared distance (i.e., $k$-CMeans) or total distance (i.e., $k$-CMedians) from each point to the center point (i.e., median or mean point) of its cluster is minimized.
\textbf{Motivation:} The chromatic clustering problem is motivated by several interesting applications. One of them is for determining the topological structure of chromosomes in cell biology \cite{HDX11}. In such applications, a set of 3D probing points (e.g., using BAC probes) is extracted from each homolog of the interested chromosome (see Figure \ref{fig-probe} in Appendix), and the objective is to determine, for each chromosome homolog, the common spatial distribution pattern of the probes among a population of cells. For this purpose, the set of probes from each homolog is converted into a high dimensional feature point in the feature space, where each dimension represents the distance between a particular pair of probes. Since each chromosome has two (or more as in cancer cells) homologs, each cell contributes $k$ (i.e., two or more) feature points.
Due to technical limitation, it is impossible to identify the same homolog from all cells.
Thus, the $k$ feature points from each cell form a point-set with the same color (meaning that they are undistinguishable). To solve the problem, one could chromatically cluster all point-sets into $k$ clusters (after normalizing the cell size), with each corresponding to a homolog, and use the mean or median point of each cluster as its common pattern.
\textbf{Related works:} As its generalization, chromatic clustering is naturally related to the traditional clustering problem. Due to the additional chromatic constraint, chromatic clustering could behave quite differently from its counterpart.
For example, the $k$-means algorithms in \cite{BHI,KSS} relies on the fact that all input points in a Voronoi cell of the optimal $k$ mean points belong to the same cluster. However, such a key locality property no longer holds for the $k$-CMeans problem.
Chromatic clustering falls in the umbrella of clustering with constraint. For such type of clustering, several solutions exist for some variants \cite{BD06}. Unfortunately, due to their heuristic nature, none of them can yield quality guaranteed solutions for the chromatic clustering problem. The first quality guaranteed solution for chromatic clustering was obtained recently by Ding and Xu. In \cite{HDX11}, they
considered a special chromatic clustering problem, where every point-set has exactly $k$ points in the first quadrant, and the objective is to cluster points by cones apexed at the origin, and presented the first PTAS for constant $k$. The $k$-CMeans and $k$-CMedians problems considered in this paper are the general cases of the chromatic clustering problem. Very recently, Arkin {\em et al.} \cite{ADH} considered a chromatic 2D $2$-center clustering problem and presented both approximation and exact solutions.
\subsection{Main Results and Techniques}
In this paper, we present three main results, a constant approximation and a $(1+\epsilon)$-approximation for $k$-CMeans and their extensions to $k$-CMedians.
\begin{itemize} \item {\bf Constant approximation:} We show that given any $c$-approximation for $k$-means clustering, it could yield a $(2ck^2+2k-1)$-approximation for $k$-CMeans. This not only provides a way for us to generate an initial constant approximation solution for $k$-CMeans through some $k$-means algorithm, but more importantly reveals the intrinsic connection between the two clustering problems.
\item {\bf $(1+\epsilon)$-approximation:} We show that a near linear time $(1+\epsilon)$-approximation solution for $k$-CMeans can be obtained using an interesting sphere peeling algorithm. Due to the lack of locality property in $k$-CMeans, our sphere peeling algorithm is quite different from the ones used in \cite{KSS,BHI}, which in general do not guarantee a $(1+\epsilon)$-approximation solution for $k$-CMeans as shown by our first result. Our sphere peeling algorithm is based on another standalone result, called {\em Simplex Lemma}. The simplex lemma enables us to obtain an approximate mean point of a set of unknown points through a grid inside a simplex determined by some partial knowledge of the unknown point set. A unique feature of the simplex lemma is that the complexity of the grid is {\em independent of the dimensionality}, and thus can be used to solve problems in high dimensional space. With the simplex lemma, our sphere peeling algorithm iteratively generates the mean points of $k$-CMeans with each iteration building a simplex for the mean point.
\item {\bf Extensions to $k$-CMedians:} We further extend the idea for $k$-CMeans to $k$-CMedians. Particularly, we show that any $c$-approximation for $k$-medians can be used to yield a $((2+\epsilon)ck^2+(2+\epsilon)k+1)$-approximation for $k$-CMedians, where the $\epsilon$ error comes from the difficulty of computing the optimal median point (i.e., Fermat Weber point). With this and a similar sphere peeling technique, we obtain a $(5+\epsilon)$-approximation for $k$-CMedians. Note that although $k\ge 2$ is a constant in this paper, a $(5+\epsilon)$-approximation is still much better than a $((2+\epsilon)ck^2+(2+\epsilon)k+1)$-approximation.
\end{itemize}
Due to space limit, many details of our algorithms, proofs, and figures are put in Appendix.
\section{Preliminaries} \label{sec-pre}
In this section, we introduce some definitions which will be used throughout the paper.
\begin{definition}[Chromatic Partition]
\label{cpart} Let $\mathcal{G}=\{G_{1}, \cdots, G_{n}\}$ be a set of $n$ point-sets with each $G_{i}=\{p^{i}_{1}, \dots, p^{i}_{k_{i}}\}$ consisting of $k_{i}\leq k$ points in $\mathbb{R}^d$ space.
A chromatic partition of $\mathcal{G}$ is a partition of the $\sum_{1\leq i\leq n}k_{i}$ points into $k$ sets, $U_{1}, \cdots, U_{k}$, such that each $U_{i}$ contains no more than one point from each $G_{j}$ for $j=1, 2, \cdots, n$.
\end{definition}
\begin{definition}[Chromatic $k$-means Clustering ($k$-CMeans)] \label{def-kcm} Let $\mathcal{G}=\{G_{1}, \cdots,$ $G_{n}\}$ be a set of $n$ point-sets with each $G_{i}=\{p^{i}_{1}, \dots, p^{i}_{k_{i}}\}$ consisting of $k_{i}\leq k$ points in $\mathbb{R}^d$ space.
The chromatic $k$-means clustering (or $k$-CMeans) of $\mathcal{G}$ is to find $k$ points $\{m_{1}, \cdots, m_{k}\}$ in $\mathbb{R}^d$ space and a chromatic partition $U_{1}, \cdots, U_{k}$ of $\mathcal{G}$ such that $\frac{1}{n}\sum_{j}\sum_{q\in U_{j}}||q-m_{j}||^2$ is minimized. The problem is called full $k$-CMeans if $k_1=k_2=\cdots=k_n=k$. \end{definition}
For both $k$-CMedians and $k$-CMeans, a problem often encountered in our approach is ``How to find the best cluster for each point in $G_{i}$ if the $k$ mean or median points $A=\{m_1, \cdots, m_k\}$ are already known?'' An easy way to solve this problem is to first build a complete bipartite graph $(G_{i}\cup A, E_{i})$ with points in $G_{i}$ and $A$ as the two partites and then compute a minimum weight bipartite matching as the solution, where the edge weight is the Euclidean distance or squared distance of the two corresponding vertices. Clearly, this can be done in a total of $O(k^{3}dn)$ time for all $G_{i}$'s. (We call this procedure as {\bf bipartite matching}.)
\section{Hardness of $k$-CMeans} \label{sec-hard}
It is easy to see that $k$-means is a special case of $k$-CMeans (i.e., each $G_i$ contains exactly one point). As shown by Dasgupta \cite{D08}, $k$-means in high dimensional space is NP-hard even if $k=2$. Thus, we immediately have the following theorem.
\begin{theorem} \label{the-fptas} $k$-CMeans is NP-hard for $k\ge 2$ in high dimensional space. \end{theorem}
\subsection{Is Full $k$-CMeans Easier?}
It is interesting to know whether full $k$-CMeans is easier than general $k$-CMeans, since it is disjoint with $k$-means when $k\geq 2$. The following theorem gives a negative answer to this question.
\begin{theorem} \label{mfptas} Full $k$-CMeans is NP-hard and has no FPTAS for $k\ge 2$ in high dimensional space unless P=NP (see Appendix for the proof). \end{theorem}
The above theorem indicates that the fullness of $k$-CMeans does not reduce the hardness of the problem. However, this does not necessarily mean that full $k$-CMeans is as difficult as general $k$-CMeans to achieve a $(1+\epsilon)$-approximation for fixed $k$. Below we show that a $(1+\epsilon)$-approximation can be relatively easily achieved for full $k$-CMeans through some random sampling technique.
First we introduce a key lemma from \cite{IKI}. Let $S$ be a set of $n$ points in $\mathbb{R}^d$ space, $T$ be a randomly selected subset from $S$ with $t$ points, and $\overline{x}(S)$, $\overline{x}(T)$ be the mean points of $S$ and $T$ respectively.
\begin{lemma}[\cite{IKI}] \label{lem-dis}
With probability $1-\eta$, $||\overline{x}(S)-\overline{x}(T)||^2<\frac{1}{\eta t}Var^{0}(S)$, where $Var^{0}(S)$ $=(\sum_{s\in S}||s-\overline{x}(S)||^2)/n$. \end{lemma}
\begin{lemma} \label{lem-select} Let $S$ be a set of elements, and $S'$ be a subset of $S$ such that
$\frac{|S'|}{|S|}=\alpha$. If randomly select $\frac{t\ln\frac{t}{\eta}}{\ln(1+\alpha)}=O(\frac{t}{\alpha}\ln\frac{t}{\eta})$ elements from $S$, with probability at least $1-\eta$, the sample contains at least $t$ elements from $S'$. \end{lemma}
\begin{proof} If we randomly select $z$ elements from $S$, then it is easy to know that with probability $1-(1-\alpha)^z$, there is at least one element from the sample belonging to $S'$. If we want the probability $1-(1-\alpha)^z$ equal to $1-\eta/t$, $z$ has to be $\frac{\ln\frac{t}{\eta}}{\ln\frac{1}{1-\alpha}}=\frac{\ln\frac{t}{\eta}}{\ln(1+\frac{\alpha}{1-\alpha})}\leq\frac{\ln\frac{t}{\eta}}{\ln(1+\alpha)}=O(\frac{1}{\alpha}\ln\frac{t}{\eta})$ (by Taylor series and $\alpha<1$, $\ln(1+\alpha)=O(\alpha)$). Thus if we perform $t$ rounds of random sampling with each round selecting $O(\frac{1}{\alpha}\ln\frac{t}{\eta})$ elements, we get at least $t$ elements from $S'$ with probability at least $(1-\eta/t)^t\geq 1-\eta$.
\qed \end{proof}
Lemma \ref{lem-dis} tells us that if we want to find an approximate mean point within a distance of $\epsilon Var^{0}(S)$ to the mean point, we just need to take a random sample of size $O(1/\epsilon)$. Lemma \ref{lem-select} suggests that for any set $S$ and its subset $S'\subset S$ of size $\alpha |S|$, we can have a random subset $T$ of $S'$ with size $O(1/\epsilon)$ by randomly sampling directly from $S$ $O(\frac{1}{\epsilon\alpha}\ln \frac{1}{\epsilon})$ points, even if $S'$ is an unknown subset of $S$.
Combining the two lemmas, we can immediately compute an approximation solution for full $k$-CMeans in the following way. First, we note that in full $k$-CMeans, each optimal cluster contains exact $n$ points from the total of $kn$ points in $\mathcal{G}$. This means that each cluster has a fraction of $\frac{1}{k}$ points from $\mathcal{G}$. Then, we can obtain an approximate mean point for each optimal cluster by (1) randomly sampling $O(\frac{k}{\epsilon}\ln\frac{1}{\epsilon})$ points from $\mathcal{G}$, (2) enumerating all possible subsets of size $O(1/\epsilon)$ to find the set $T$ which is a random sample of the unknown optimal cluster, and (3) computing the mean of $T$ as the approximate mean point of the optimal cluster. Finally, we can generate the $k$ chromatic clusters from the $k$ approximate mean points by using the bipartite matching procedure (see Section \ref{sec-pre}).
\begin{theorem} \label{the-cptas} With constant probability, a $(1+\epsilon)$-approximation of full $k$-CMeans can be obtained in $O(2^{poly(\frac{k}{\epsilon})}$ $nd)$ time.
\end{theorem}
With the above theorem, we only need to focus on the general $k$-CMeans problem in the remaining sections. Note that in the general case, some clusters may have a very small fraction (rather than $1/k$) of points, thus we can not use the above method to solve the general $k$-CMeans problem.
\section{Constant Approximation from $k$-means} \label{sec-constant}
In this section, we show that a constant approximation solution for $k$-CMeans can be produced from an approximation solution of $k$-means. Below is the main theorem of this section.
\begin{theorem} \label{the-constant} Let $\mathcal{G}=\{G_1, \cdots, G_n\}$ be an instance of $k$-CMeans, and $\mathcal{C}$ be the $k$ mean points of a constant $c$-approximation solution of $k$-means on the points $\cup^n_{i=1}G_i$.
Then $[\mathcal{C}]^k$ contains at least one $k$-tuple which could induce a $(2ck^2+2k-1)$-approximation of $k$-CMeans on $\mathcal{G}$, where $[\mathcal{C}]^k=\underbrace{\mathcal{C}\times\cdots\times\mathcal{C}}_{k}$. \end{theorem}
To prove Theorem \ref{the-constant}, we first introduce two lemmas.
\begin{lemma} \label{lem-meanshift}
Let $P$ be a set of points in $\mathbb{R}^d$ space, and $m$ be the mean point of $P$. For any point $m'\in \mathbb{R}^d$, $\sum_{p\in P}||p-m'||^2=\sum_{p\in P}||p-m||^2+|P| \times ||m-m'||^2$ (see Appendix for the proof).
\end{lemma}
\begin{lemma} \label{lem-close}
Let $P$ be a set of points in $\mathbb{R}^d$ space, and $P_{1}$ be its subset containing $\alpha |P|$ points for some $0<\alpha\leq 1$. Let $m$ and $m_1$ be the mean points of $P$ and $P_{1}$ respectively. Then $||m_{1}-m||\leq\sqrt{\frac{1-\alpha}{\alpha}}\delta$, where $\delta^2=\frac{1}{|P|}\sum_{p\in P}||p-m||^2$. \end{lemma} \begin{proof} Let $P_2=P\setminus P_1$, and $m_{2}$ be its mean point. By Lemma \ref{lem-meanshift} we first have the following two equalities.
\small{
\begin{eqnarray}
\sum_{p\in P_{1}}||p-m||^2 &= &\sum_{p\in P_{1}}||p-m_1||^2+|P_1| \times ||m_1-m||^2. \label{for-1}\\
\sum_{p\in P_{2}}||p-m||^2 &=& \sum_{p\in P_{2}}||p-m_2||^2+|P_2| \times ||m_2-m||^2. \label{for-2}
\end{eqnarray}
}
\normalsize
Then by the definition of $\delta$, we have $\delta^2=\frac{1}{|P|}(\sum_{p\in P_{1}}||p-m||^2+\sum_{p\in P_{2}}||p-m||^2)$. Let $L=||m_{1}-m_{2}||$. By the definition of mean point, we have $m=\frac{1}{|P|}\sum_{p\in P} p=\frac{1}{|P|}(\sum_{p\in P_1} p+\sum_{p\in P_2} p)=\frac{1}{|P|}(|P_1|m_1+|P_2|m_2)$. Thus the three points $\{m, m_1, m_2\}$ are collinear, and $||m_{1}-m||=(1-\alpha) L$ and $||m_{2}-m||=\alpha L$. Combining (\ref{for-1}) and (\ref{for-2}), we have
\small{ \begin{eqnarray*}
\delta^2= \frac{1}{|P|}(\sum_{p\in P_{1}}||p-m_1||^2+ |P_1| \times ||m_1-m||^2+\sum_{p\in P_{2}}||p-m_2||^2+|P_2| \times ||m_2-m||^2)\\
\geq \frac{1}{|P|}( |P_1| \times ||m_1-m||^2+|P_2| \times ||m_2-m||^2)
=\alpha((1-\alpha)L)^2+(1-\alpha)(\alpha L)^2=\alpha(1-\alpha)L.
\end{eqnarray*}
}
\normalsize
Thus, we have $L\leq\frac{\delta}{\sqrt{\alpha(1-\alpha)}}$, which means that $||m_{1}-m||=(1-\alpha) L\leq\sqrt{\frac{1-\alpha}{\alpha}}\delta$. \qed \end{proof}
\begin{proof}[\textbf{of Theorem \ref{the-constant}}] Let $\{c_1, \cdots, c_k\}$ be the $k$ mean points in $\mathcal{C}$, and $\{S_1, \cdots, S_k\}$ be their corresponding clusters.
Let $\{m_{1}, \cdots, m_{k}\}$ be the $k$ unknown optimal mean points of $k$-CMeans, and $\mathcal{OPT}=\{Opt_{1}, \cdots, Opt_{k}\}$ be the corresponding $k$ optimal chromatic clusters.
Let $\Gamma^i_j=Opt_i\cap S_j$, and $\tau^i_j$ be its mean point for $1\leq i, j\leq k$.
\begin{figure}
\caption{An example illustrating Theorem \ref{the-constant}.}
\label{fig-constant}
\end{figure}
Since $\cup^k_{j=1}\Gamma^i_j=Opt_i$, by pigeonhole principle we know that there must exist some index $1\leq j_i \leq k$ such that $|\Gamma^i_{j_i}|\geq \frac{1}{k}|Opt_i|$. Thus by fixing $j_i$, we have the following about $\sum_{p\in Opt_i}||p-c_{j_i}||^2$ (see Figure \ref{fig-constant})
\small{
\begin{eqnarray}
\sum_{p\in Opt_i}||p-c_{j_i}||^2 &= &\sum_{p\in Opt_i}||p-m_i||^2+|Opt_i| \times ||m_i-c_{j_i}||^2
=\sum_{p\in Opt_i}||p-m_i||^2+|Opt_i| \times ||m_i-\tau^i_{j_i}+\tau^i_{j_i}-c_{j_i}||^2 \nonumber\\
&\leq&\sum_{p\in Opt_i}||p-m_i||^2+|Opt_i| \times (||m_i-\tau^i_{j_i}||+||\tau^i_{j_i}-c_{j_i}||)^2 \nonumber\\
&\leq&\sum_{p\in Opt_i}||p-m_i||^2+|Opt_i| \times 2(||m_i-\tau^i_{j_i}||^2+||\tau^i_{j_i}-c_{j_i}||^2), \label{for-3} \end{eqnarray} } \normalsize where the first equation follows from Lemma \ref{lem-meanshift} (note that $m_i$ is the mean point of $Opt_i$), and the last inequality follows from the fact that $(a+b)^2\leq 2(a^2+b^2)$ for any numbers $a$ and $b$. By Lemma \ref{lem-close}, we have
\small{ \begin{eqnarray}
||\tau^i_{j_i}-m_i||^2\leq \frac{1-\frac{1}{k}}{\frac{1}{k}}(\frac{1}{|Opt_i|}\sum_{p\in Opt_i}||p-m_i ||^2). \label{for-4}\\
||\tau^i_{j_i}-c_{j_i}||^2\leq \frac{1-\frac{|\Gamma^i_{j_i}|}{|S_{j_i}|}}{\frac{|\Gamma^i_{j_i}|}{|S_{j_i}|}}(\frac{1}{|S_{j_i}|}\sum_{p\in S_{j_i}}||p-c_{j_i} ||^2). \label{for-5}
\end{eqnarray}
}
\normalsize
Plugging (\ref{for-4}) and (\ref{for-5}) into inequality (\ref{for-3}), we have
\small{ \begin{eqnarray*}
\sum_{p\in Opt_i}||p-c_{j_i}||^2\leq\sum_{p\in Opt_i}||p-m_i||^2+|Opt_i| \times 2(||m_i-\tau^i_{j_i}||^2+||\tau^i_{j_i}-c_{j_i}||^2)\\
\leq\sum_{p\in Opt_i}||p-m_i||^2+|Opt_i| \times 2(\frac{1-\frac{1}{k}}{\frac{1}{k}}(\frac{1}{|Opt_i|}\sum_{p\in Opt_i}||p-m_i ||^2)+\frac{1-\frac{|\Gamma^i_{j_i}|}{|S_{j_i}|}}{\frac{|\Gamma^i_{j_i}|}{|S_{j_i}|}}(\frac{1}{|S_{j_i}|}\sum_{p\in S_{j_i}}||p-c_{j_i} ||^2))\\
=(2k-1)\sum_{p\in Opt_i}||p-m_i||^2+2\frac{|Opt_i|}{|\Gamma^i_{j_i}|} \times (1-\frac{|\Gamma^i_{j_i}|}{|S_{j_i}|})\sum_{p\in S_{j_i}}||p-c_{j_i} ||^2). \end{eqnarray*} } \normalsize
Since $|\Gamma^i_{j_i}|\geq \frac{1}{k}|Opt_i|$, we have $\frac{|Opt_i|}{|\Gamma^i_{j_i}|} \times (1-\frac{|\Gamma^i_{j_i}|}{|S_{j_i}|})\leq k$. Thus the above inequality becomes \small{ \begin{eqnarray}
\sum_{p\in Opt_i}||p-c_{j_i}||^2\leq (2k-1)\sum_{p\in Opt_i}||p-m_i||^2+2k\sum_{p\in S_{j_i}}||p-c_{j_i} ||^2. \label{for-6} \end{eqnarray} } \normalsize Summing both sides of (\ref{for-6}) over $i$, we have \small{ \begin{eqnarray}
\sum^k_{i=1}\sum_{p\in Opt_i}||p-c_{j_i}||^2\leq (2k-1)\sum^k_{i=1}\sum_{p\in Opt_i}||p-m_i||^2+2k\sum^k_{i=1}\sum_{p\in S_{j_i}}||p-c_{j_i} ||^2 \nonumber\\
\leq (2k-1)\sum^k_{i=1}\sum_{p\in Opt_i}||p-m_i||^2+2k^2 \sum^k_{j=1}\sum_{p\in S_{j}}||p-c_{j} ||^2, \label{for-7} \end{eqnarray} }
\normalsize where the second inequality follows from the inequality $\sum_{p\in S_{j_i}}||p-c_{j_i} ||^2\leq \sum^k_{j=1}\sum_{p\in S_{j}}||p-c_{j} ||^2$, which implies that $2k\sum^k_{i=1}\sum_{p\in S_{j_i}}||p-c_{j_i} ||^2\leq 2k^2 \sum^k_{j=1}\sum_{p\in S_{j}}||p-c_{j} ||^2$.
It is obvious that the optimal objective value of $k$-means is no larger than that of $k$-CMeans on the same set of points in $\mathcal{G}$. Thus, $\sum^k_{j=1}\sum_{p\in S_{j}}||p-c_{j} ||^2\leq c\sum^k_{i=1}\sum_{p\in Opt_i}||p-m_i||^2$. Plugging this inequality into inequality (\ref{for-7}), we have \small{ \begin{eqnarray*}
\sum^k_{i=1}\sum_{p\in Opt_i}||p-c_{j_i}||^2\leq (2ck^2+2k-1)\sum^k_{i=1}\sum_{p\in Opt_i}||p-m_i||^2. \end{eqnarray*} } \normalsize
The above inequality means that if we take the $k$-tuple $(c_{j_1}, \cdots, c_{j_k})$ as the $k$ approximate mean points for $k$-CMeans, we have a $(2ck^2+2k-1)$-approximation solution, where the $k$ chromatic clusters can be obtained by the bipartite matching procedure. Thus, the theorem is proved.
\qed \end{proof}
\noindent \textbf{Running Time:} In the above theorem, the bipartite matching procedure takes $O(k^3 nd)$ time for one $k$-tuple. Since there are in total $O(k^k)$ such $k$-tuples,
the total running time is $O(k^{k+3} nd)$ for computing a $(2ck^2+2k-1)$-approximation of $k$-CMeans from a $c$-approximation of $k$-means. As $k$ is assumed to be a constant in this paper, the running time is linear.
\section{$(1+\epsilon)$-Approximation Algorithm} \label{sec-ptas}
This section presents our $(1+\epsilon)$-approximation solution to the $k$-CMeans problem. We first introduce a standalone result, {\em Simplex Lemma}, and then use it to achieve a $(1+\epsilon)$-approximation for $k$-CMeans. The main idea of the algorithm is to use a sphere peeling technique to generate the chromatic clusters iteratively, where the Simplex Lemma helps to determine a proper peeling region.
\subsection{Simplex Lemma} \label{sec-simplex}
Simplex Lemma is mainly for approximating the mean point of some \textbf{unknown} points set $P$. The only known information about $P$ is a set $S$ of $j$ points with each of them being an approximate mean point of a subset of $P$. The following Simplex lemmas show that it is possible to construct a simplex of $S$ and find the desired approximate mean point of $P$ inside the simplex.
\begin{figure}
\caption{An example for Lemma \ref{lem-simplex} with $j=4$.}
\label{fig-simplex}
\caption{An example for Lemma \ref{lem-shift} with $j=4$.}
\label{fig-simplex2}
\end{figure}
\begin{lemma}[Simplex Lemma \Rmnum{1}] \label{lem-simplex} Let $P$ be a set of points in $\mathbb{R}^d$ with a partition of $P=\cup^j_{l=1} P_l$ and $P_{l_1}\cap P_{l_2}=\emptyset$ for any $l_1\neq l_2$.
Let $o$ be the mean point of $P$, and $o_l$ be the mean point of $P_l$ for $1\leq l\leq j$. Further, let $\delta^2=\frac{1}{|P|}\sum_{p\in P}||p-o||^2$, and $V$ be the simplex determined by $\{o_1, \cdots, o_j\}$.
Then for any $0<\epsilon\leq 1$, it is possible to construct a grid of size $O((8j/\epsilon)^j)$ inside $V$ such that at least one grid point $\tau$ satisfies the inequality $||\tau-o||\leq\sqrt{\epsilon}\delta$.
\end{lemma}
\begin{proof} We will prove this lemma by mathematical induction on $j$.
\noindent\textbf{Base case:} For $j=1$, since $P_1=P$, $o_1=o$. Thus, the simplex $V$ and the grid are all simply the point $o_1$. Clearly $\tau=o_{1}$ satisfies the inequality.
\noindent\textbf{Induction step:} Assume that the lemma holds for any $j\leq j_0$ for some $j_{0} \ge 1$ (i.e., Induction Hypothesis). Now we consider the case of $j=j_0+1$. First, we assume that $\frac{|P_l |}{|P|}\geq \frac{\epsilon}{4j}$ for each $1\leq l\leq j$. Otherwise, we can reduce the problem to the case of smaller $j$ in the following way. Let $I=\{l| 1\leq l\leq j, \frac{|P_l |}{|P|}< \frac{\epsilon}{4j}\}$ be the index set of small subsets. Then, $\frac{\sum_{l\in I}|P_l |}{|P|}<\frac{\epsilon}{4}$, and $\frac{\sum_{l\not\in I}|P_l |}{|P|}\geq 1-\frac{\epsilon}{4}$. By Lemma \ref{lem-close}, we know that $||o'-o||\leq\sqrt{\frac{\epsilon/4}{1-\epsilon/4}}\delta$, where $o'$ is the mean point of $\cup_{l\not\in I}P_l$. Let $(\delta')^2$ be the variance of $\cup_{l\not\in I}P_l$. Then, we have $(\delta')^2\le\frac{|P|}{|\cup_{l\not\in I}P_l|}\delta^2\leq \frac{1}{1-\epsilon/4}\delta^2$. Thus, if we replace $P$ and $\epsilon$ by $\cup_{l\not\in I}P_l$ and $\frac{\epsilon}{16}$ respectively, and find a point $\tau$ such that $||\tau-o'||^2\leq\frac{\epsilon}{16}(\delta')^2\leq \frac{\epsilon/16}{1-\epsilon/4}\delta^2$, we have $||\tau-o||^2\leq(||\tau-o'||+||o'-o||)^2\leq \frac{\frac{9}{16}\epsilon}{1-\epsilon/4}\delta^2\leq \epsilon\delta^2$ (where the last inequality is due to the fact $\epsilon<1$). This means that we can reduce the problem to a problem with point set $\cup_{l\not\in I}P_l$ and a smaller $j$ (i.e., $j-|I|$). By the induction hypothesis, we know that the reduced problem can be solved (note that the simplex would be a subset of $V$ determined by $\{o_l\mid 1\leq l\leq j, l\not\in I\}$), and therefore the induction step holds for this case. Thus, in the following discussion, we can assume that $\frac{|P_l |}{|P|}\geq \frac{\epsilon}{4j}$ for each $1\leq l\leq j$.
For each $1\leq l\leq j$, since $\frac{|P_l |}{|P|}\geq \frac{\epsilon}{4j}$, by Lemma \ref{lem-close}, we know that $||o_l-o||\leq\sqrt{\frac{1- \frac{\epsilon}{4j}}{ \frac{\epsilon}{4j}}}\delta\leq 2\sqrt{\frac{j}{\epsilon}}\delta$. This, together with triangle inequality, implies that for any $1\leq l, l'\leq j$, $||o_l-o_{l'}||\leq ||o_l-o||+||o_{l'}-o||\leq 4\sqrt{\frac{j}{\epsilon}}\delta$. Thus, if we pick any index $l_0$, and draw a ball $\mathcal{B}$ centered at $o_{l_0}$ and with radius $r=\max_{1\leq l\leq j}\{||o_l-o_{l_0}||\}\leq 4\sqrt{\frac{j}{\epsilon}}\delta$, the whole simplex $V$ will be inside $\mathcal{B}$. Note that since $o=\sum^j_{l=1}\frac{|P_j|}{|P|}o_l$, $o$ also locates inside $V$. This indicates that we can construct $\mathcal{B}$ in the $j-1$-dimensional space spanned by $\{o_1, \cdots, o_j\}$, rather than the whole $\mathbb{R}^d$ space. Also, if we build a grid inside $\mathcal{B}$ with grid length $\frac{\epsilon r}{4j}$, the total number of grid points is no more than $O((\frac{8j}{\epsilon})^j)$. With this grid, we know that for any point $q$ inside $V$, there exists a grid point $g$ such that $||g-q||\leq \sqrt{j (\frac{\epsilon r}{4j})^2}=\frac{\epsilon}{4\sqrt{j}}r\leq \sqrt{\epsilon}\delta$. This means that can find a grid point $\tau$ inside $V$, such that $||\tau-o||^2\leq\epsilon\delta^2$. Thus, the induction step holds.
With the above base case and induction steps, the lemma holds for any $j\ge 1$.
\qed \end{proof}
In the above lemma, we assume that the exact positions of $\{o_1, \cdots, o_j\}$ are known (see Fig. \ref{fig-simplex}). However, in some scenario (e.g., the exact partition of $P$ is not given, as is the case in $k$-CMeans),
it is possible that we only know the approximate position of each mean point $o_{i}$ (see Fig. \ref{fig-simplex2}). The following lemma shows that an approximate position of $o$ can still be similarly determined.
\begin{lemma}[Simplex Lemma \Rmnum{2}] \label{lem-shift} Let $P$, $o$, $P_{l}, o_{l}, 1\le l \le j$, and, $\delta$ be defined as in Lemma \ref{lem-simplex}.
Let $\{o'_1, \cdots, o'_j\}$ be $j$ points in $R^{d}$ such that $||o'_l-o_l ||\leq L$ for $1\leq l\leq j$ and $L>0$, and $V'$ be the simplex determined by $\{o'_1, \cdots, o'_j\}$. Then for any $0<\epsilon\leq 1$, it is possible to construct a grid of size $O((8j/\epsilon)^j)$ inside $V'$ such that at least one grid point $\tau$ satisfies the inequality $||\tau-o||\leq\sqrt{\epsilon}\delta+(1+\epsilon)L$.
\end{lemma}
\subsection{Sphere Peeling Algorithm} \label{sec-peeling}
This section presents a sphere peeling algorithm to achieve a $(1+\epsilon)$-approximation for $k$-CMeans.
Let $\mathcal{G}=\{G_1, \cdots, G_n\}$ be an instance of $k$-CMeans with $k$ (unknown) optimal chromatic clusters $\mathcal{OPT}=\{Opt_1, \cdots,$ $Opt_k\}$, and $m_{j}$ be the mean point of the cluster $Opt_j$ for $1\leq j\leq k$.
Without loss of generality, we assume that $|Opt_1|\geq |Opt_2|\geq\cdots\geq |Opt_k |$.
\noindent\textbf{Algorithm overview:} Our algorithm first computes a constant $C$-approximation solution (by Theorem \ref{the-constant}) to determine an upper bound $\Delta$ of the optimal objective value $\delta^{2}_{opt}$, and then search for a good approximation of $\delta^{2}_{opt}$ in the interval of $[\Delta/C, \Delta]$. At each search step, our algorithm performs a sphere peeling procedure to iteratively generate $k$ approximate mean points for the chromatic clusters. Initially, the sphere peeling procedure uses random sampling technique (i.e., Lemma \ref{lem-dis} and \ref{lem-select}) to find an approximate mean point for $Opt_{1}$. At $(j+1)$-th iteration, it already has approximate mean points $\{p_{v_1}, \cdots, p_{v_j}\}$ for $Opt_1, \cdots, Opt_j$ respectively. Then it draws $j$ peeling spheres, $B_{j+1,1}, \cdots, B_{j+1,j}$, centered at the $j$ approximate mean points respectively and with a radius determined by the approximation of $\delta_{opt}$.
Denote the set of unknown points $Opt_{j+1}\setminus (\cup^j_{l=1}B_{j+1,l})$ as $\mathcal{A}$. Our algorithm considers two cases: (a) $|\mathcal{A}|$ is large enough and (b) $|\mathcal{A}|$ is small. For case (a), since $|\mathcal{A}|$ is large enough, we can first use Lemma \ref{lem-select} to find an approximate mean point $m_{\mathcal{A}}$ of $\mathcal{A}$, and then construct a simplex determined by $m_{\mathcal{A}}$ and $\{p_{v_1}, \cdots, p_{v_j}\}$. For case (b), it directly constructs a simplex determined just by $\{p_{v_1}, \cdots, p_{v_j}\}$. For either case, our algorithm builds a grid inside the simplex (i.e., using Lemma \ref{lem-shift}) to find an approximate mean point for $Opt_{j+1}$ (i.e., $p_{v_{j+1}}$). Repeat the sphere peeling procedure $k$ times to generate the $k$ approximate mean points.
\noindent\textbf{Algorithm $k$-CMeans} \newline \textbf{Input:} $\mathcal{G}=\{G_1, \cdots, G_n\}$, $k\geq 2$, and a small positive value $\epsilon$. \newline \textbf{Output:} $(1+\epsilon)$-approximation solution for $k$-CMeans on $\mathcal{G}$.
\begin{enumerate} \item Run the PTAS
of $k$-means in \cite{KSS} on $\mathcal{G}$, and let $\Delta$ be the obtained objective value.
\item For $i=1$ to $\frac{2k}{\epsilon}$ do \begin{enumerate} \item Set $\delta=\frac{\sqrt{\Delta}}{2k}+i\frac{\epsilon}{2k}\sqrt{\Delta}$, and run the Sphere-Peeling-Tree algorithm.
\item Let $\mathcal{T}_i$ be the output tree.
\end{enumerate}
\item For each path of every $\mathcal{T}_i$, use bipartite matching procedure to compute the objective value of $k$-CMeans on $\mathcal{G}$. Output the $k$ points from the path with the smallest objective value.
\end{enumerate}
\noindent\textbf{Algorithm Sphere-Peeling-Tree} \newline \textbf{Input:} $\mathcal{G}$, $k\geq 2$, $\epsilon, \delta>0$. \newline \textbf{Output:} A tree $\mathcal{T}$ of height $k$ with each node $v$ associating with a point $p_v \in \mathbb{R}^d$.
\begin{enumerate}
\item Initialize $\mathcal{T}$ with a single root node $v$ associating with no point. \item Recursively grow each node $v$ in the following way \begin{enumerate} \item If the height of $v$ is already $k$, then it is a leaf. \item Otherwise, let $j$ be the height of $v$. Build the radius candidates set $\mathcal{R}=\cup^{\log(kn)}_{t=0}\{\frac{1+l\frac{\epsilon}{2}}{2(1+\epsilon)}j2^{t/2}\sqrt{\epsilon}\delta\mid 0\le l\le 4+\frac{2}{\epsilon}\}$. For each $r\in\mathcal{R}$, do \begin{enumerate} \item Let $\{p_{v_1}, \cdots, p_{v_j}\}$ be the $j$ points associated with nodes on the root-to-$v$ path.
\item For each $p_{v_l}$, $1\leq l\leq j$, construct a ball $B_{j+1,l}$ centered at $p_{v_l}$ and with radius $r$. \item Take a random sample from $\mathcal{G}\setminus\cup^j_{l=1}B_{j+1,l}$ with size $m=\frac{8k^3}{\epsilon^9}\ln\frac{k^2}{\epsilon^6}$. Compute the mean points of all subset of the sample, and denote them as $\Pi=\{\pi_1, \cdots, \pi_{2^m-1}\}$. \item For each $\pi_i \in \Pi$, construct the simplex determined by $\{p_{v_1}, \cdots, p_{v_j}, \pi_i\}$. Also construct the simplex determined by $\{p_{v_1}, \cdots, p_{v_j}\}$. Build a grid inside each simplex with size $O((\frac{32j}{\epsilon^2})^j)$.
\item In total, there are $2^m (\frac{32j}{\epsilon^2})^j$ grid points inside the $2^m$ simplices. For each grid point, add one child to $v$, and associate it with the grid point.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\begin{theorem} \label{the-ptas} With constant probability, Algorithm $k$-CMeans yields a $(1+\epsilon)$-approximation for $k$-CMeans in $O(2^{poly(\frac{k}{\epsilon})}n(\log n)^{k+1} d )$ time. \end{theorem}
\subsection{Proof of Theorem \ref{the-ptas}} \label{sec-proof}
Let $\beta_j= |Opt_j|/|\cup_{i=1}^{n} G_{i}|$, and
$\delta^2_j=\frac{1}{|Opt_j |}\sum_{p\in Opt_j}||p-m_j||^2$, where $m_j$ is the mean point of $Opt_j$.
Clearly, $\beta_1\geq\cdots\geq\beta_k$ (by assumption) and $\sum^k_{j=1}\beta_j =1$. Let $\delta^2_{opt}=\sum^k_{j=1}\beta_j\delta^2_j$.
We prove Theorem \ref{the-ptas} by mathematical induction. Instead of directly proving it, we consider the following two lemmas which jointly ensure the correctness of Theorem \ref{the-ptas}.
\begin{lemma} \label{lem-induction} Among all the trees generated in Algorithm $k$-CMeans, with constant probability, there exists at least one tree, $\mathcal{T}_i$, which has a root-to-leaf path with each node $v_{j}$ at level $j$, $1\leq j\leq k$, on the path associating a point $p_{v_j}$ and satisfying the inequality
$||p_{v_j}-m_j ||\leq \epsilon\delta_j+(1+\epsilon)j\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt} .$
\end{lemma}
Before proving this lemma, we first show its implication.
\begin{lemma} \label{lem-equal} If Lemma \ref{lem-induction} is true, Algorithm $k$-CMeans yields a $(1+O(k^3)\epsilon)$-approximation for $k$-CMeans. \end{lemma} \begin{proof} We first assume that Lemma \ref{lem-induction} is true. Then for each $1\leq j\leq k$, we have \small{ \begin{eqnarray}
\sum_{p\in Opt_j}||p-p_{v_j}||^2&=&\sum_{p\in Opt_j}||p-m_j||^2+|Opt_j|\times||m_j-p_{v_j}||^2
\leq \sum_{p\in Opt_j}||p-m_j||^2+|Opt_j|\times2(\epsilon^2\delta^2_j+(1+\epsilon)^2 j^2\frac{\epsilon}{\beta_j}\delta^2_{opt}) \nonumber\\
&=&(1+2\epsilon^2)|Opt_j |\delta^2_j+2(1+\epsilon)^2 j^2\epsilon|\mathcal{G}|\delta^2_{opt}, \label{for-10} \end{eqnarray} }
\normalsize where the first equation follows from Lemma \ref{lem-meanshift} (note that $m_j$ is the mean point of $Opt_j$), the second inequality follows from Lemma \ref{lem-induction} and the fact that $(a+b)^2\leq 2(a^2+b^2)$ for any two real numbers $a$ and $b$, and the last equality follows from $\frac{|Opt_j|}{\beta_j}=|\mathcal{G}|$. Summing both sides of (\ref{for-10}) over $j$, we have \small{ \begin{eqnarray}
\sum^k_{j=1}\sum_{p\in Opt_j}||p-p_{v_j}||^2 &\leq &\sum^k_{j=1}((1+2\epsilon^2)|Opt_j |\delta^2_j+2(1+\epsilon)^2 j^2\epsilon|\mathcal{G}|\delta^2_{opt})\nonumber\\
&\leq& (1+2\epsilon^2)\sum^k_{j=1}|Opt_j |\delta^2_j+2(1+\epsilon)^2 k^3\epsilon|\mathcal{G}|\delta^2_{opt}=(1+O(k^3)\epsilon)|\mathcal{G}|\delta^2_{opt}, \label{for-11} \end{eqnarray} } \normalsize
where the last equation follows from the fact that $\sum^k_{j=1}|Opt_j |\delta^2_j=|\mathcal{G}|\delta^2_{opt}$. By (\ref{for-11}), we know that $\{p_{v_1}, \cdots, p_{v_k}\}$ will induce a $(1+O(k^3)\epsilon)$-approximation solution for $k$-CMeans via bipartite matching procedure. Since Algorithm $k$-CMeans outputs the best solution generated in all trees, the resulting solution is clearly a $(1+O(k^3)\epsilon)$-approximation solution. Thus the lemma is true. \qed \end{proof}
The above lemma indicates that if we replace $\epsilon$ by $\frac{\epsilon}{k^3}$ in the input of our algorithm, it will result in a $(1+\epsilon)$-approximation solution. This implies that Lemma \ref{lem-induction} is indeed sufficient to ensure the correctness of Theorem \ref{the-ptas} (except for the time complexity). Now we prove Lemma \ref{lem-induction}.
\begin{proof}[\textbf{of Lemma \ref{lem-induction}}]
Note that $\Delta\le4k^2\delta^2_{opt}$, and we build $\epsilon$-net in $[\frac{\sqrt{\Delta}}{2k},\sqrt{\Delta}]$. Let $\mathcal{T}_i$ be the tree generated by Algorithm Sphere-Peeling-Tree and corresponding to the input $\delta\in [\delta_{opt}, (1+\epsilon)\delta_{opt}]$. We will focus our discussion on $\mathcal{T}_{i}$, and prove the lemma by mathematical induction on $j$.
\noindent\textbf{Base case:} For $j=1$, since $\beta_1=\max\{\beta_j |1\leq j\leq k\}$, we have $\beta_1\geq\frac{1}{k}$. By Lemmas \ref{lem-dis} and \ref{lem-select}, we can find the approximation mean point through random sampling. Let $p_{v_{1}}$ be the approximation mean point. Clearly, $||p_{v_1}-m_1||\leq \epsilon\delta_1\leq \epsilon\delta_1+(1+\epsilon)\sqrt{\frac{\epsilon}{\beta_1}}\delta_{opt}$ (By Lemmas \ref{lem-dis} and \ref{lem-select}).
\noindent\textbf{Induction step:} We assume that there is a path in $\mathcal{T}_i$ from the root to the $j_0$-th level, such that for each $1\leq l\leq j_0$, the level-$l$ node $v_{l}$ on the path is associated with a point $p_{v_{l}}$ satisfying the inequality $||p_{v_l}-m_l ||\leq \epsilon\delta_l+(1+\epsilon)l\sqrt{\frac{\epsilon}{\beta_l}}\delta_{opt} $ (i.e., Induction Hypothesis). Now we consider the case of $j=j_0+1$. Below we will show that there is one child of $v_{j-1}$, i.e., $v_{j}$, such that its associated point $p_{v_{l}}$ satisfies the inequality $||p_{v_j}-m_j ||\leq \epsilon\delta_j+(1+\epsilon)j\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt} $. First, we have the following claim (see Appendix for the proof).
\begin{claim}[\textbf{1}] In the set of radius candidates built in Algorithm Sphere-Peeling-Tree, there exists one value $r_j\in \mathcal{R}$ such that \small{
$$j\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}\leq r_j\leq (1+\frac{\epsilon}{2})j\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}.$$
}\normalsize \end{claim}
Now, we construct the $j-1$ peeling spheres, $\{B_{j,1}, \cdots, B_{j,j-1}\}$ (as in Algorithm Sphere-Peeling-Tree). For each $1\leq l\leq j-1$, $B_{j,l}$ is centered at $p_{v_l}$ and with radius $r_j$. By Markov inequality and induction hypothesis, we have the following claim (see Appendix for the proof).
\begin{claim}[\textbf{2}]
For each $1\leq l\leq j-1$, we have $|Opt_l \setminus (\bigcup^{j-1}_{w=1}B_{j,w})|\le \frac{4\beta_j |\mathcal{G}|}{\epsilon}$.
\end{claim}
Claim \textbf{2} shows that $|Opt_l \setminus (\bigcup^{j-1}_{w=1}B_{j,w})|$ is bounded for $1\leq l\leq j-1$, which helps us to find the approximate mean point of $Opt_j $. Induced by the $j-1$ peeling spheres $\{B_{j,1}, \cdots, B_{j,j-1}\}$, $Opt_j$ is divided into $j$ subsets, $Opt_j\cap B_{j,1}$, $\cdots$, $Opt_j\cap B_{j,j-1}$ and $Opt_j \setminus(\bigcup^{j-1}_{w=1}B_{j,w})$. To simplify our discussion, we let $P_l$ denote $Opt_j\cap B_{j,l}$ for $1\leq l\leq j-1$, $P_j$ denote $Opt_j \setminus(\bigcup^{j-1}_{w=1}B_{j,w})$, and $\tau_{l}$ denote the mean point of $P_l$. Note that the peeling spheres may intersect with each other. For any two intersecting spheres $B_{j,l_1}$ and $B_{j,l_2}$, we let the points set $Opt_j\cap (B_{j,l_1}\cap B_{j,l_2})$ belong to either $P_{l_1}$ or $P_{l_2}$ arbitrarily. Thus, we can assume that $\{P_l\mid 1\leq l\leq j\}$ are pairwise disjoint. Now consider the size of $P_{j}$ (i.e., $|P_j|$). We have the following two cases: (a) $|P_j |\geq \epsilon^3\frac{\beta_{j}}{j}|\mathcal{G}|$ and (b) $|P_j |<\epsilon^3\frac{\beta_{j}}{j}|\mathcal{G}|$. In the following, we show how, in each case, Algorithm Sphere-Peeling-Tree can obtain an approximate mean point for $Opt_{j}$ by using the Simplex Lemma (i.e., Lemma \ref{lem-shift}).
\begin{figure}
\caption{Case (a) for $j=4$.}
\label{fig-case1}
\caption{Case (b) for $j=4$.}
\label{fig-case2}
\end{figure}
For case (a), by Claim \textbf{2}, together with the fact that $\beta_l\leq \beta_{j}$ for $l>j$, we know that \small{
$$\frac{|P_j |}{\sum_{1\leq i\leq k}|Opt_{i}\setminus(\bigcup^{j-1}_{l=1}B_{j,l})|}\geq\frac{\frac{\epsilon^3}{j}\beta_j}{\frac{4(j-1)\beta_j}{\epsilon}+\frac{\epsilon^2}{j}\beta_j+(k-j)\beta_j}>\frac{\epsilon^4}{8kj}\ge\frac{\epsilon^4}{8k^2}.$$
}\normalsize
This means that $P_{j}$ is large enough, comparing to the set of points outside the peeling spheres. Hence, we can use random sampling technique to obtain an approximate mean point $\pi $ for $P_j$ in the following way. First, we set $t=\frac{k}{\epsilon^5}$, $\eta=\frac{\epsilon}{k}$, and take a sample of size $\frac{t\ln(t/\eta)}{\epsilon^4 /8k^2}=\frac{8k^3}{\epsilon^9}\ln\frac{k^2}{\epsilon^6}$. By Lemma \ref{lem-select}, we know that with probability $1-\frac{\epsilon}{k}$, the sample contains $\frac{k}{\epsilon^5}$ points from $P_j$. Then we let $\pi$ be the mean point of the $\frac{k}{\epsilon^5}$ points from $P_j$, and $a^{2}$ be the variance of $P_j$. By Lemma \ref{lem-dis}, we know that with probability $1-\frac{\epsilon}{k}$, $||\pi-\tau_j||^2\leq \epsilon^4 a^2$. Also, since $\frac{|P_j|}{|Opt_j|}\geq\frac{\epsilon^3}{j}$, we have $a^2\le\frac{|Opt_j|}{|P_j|}\delta^2_j\leq\frac{j}{\epsilon^3}\delta^2_j$. Thus, $||\pi-\tau_j||^2\leq \epsilon j\delta^2_j$.
Once obtaining $\pi$, we can now use Lemma \ref{lem-shift} to find a point $p_{v_{j}}$ satisfying the condition of $||p_{v_{j}}-m_j||\leq \epsilon\delta_j+(1+\epsilon)j\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}$. First, we construct a simplex $V'_{(a)}$ determined by $\{p_{v_1}, \cdots, p_{v_{j-1}}\}$ and $\pi$ (see Figure. \ref{fig-case1}). Note that $Opt_j$ is divided by the peeling spheres into $j$ disjoint subsets, $P_1, \cdots, P_j$, which is a partition of $Opt_{j}$. Each $P_l$ ($1\le l\le j-1$) locates inside $B_{j,l}$, which implies that $\tau_l$ is also inside $B_{j,l}$. Further, since $||p_{v_l}-\tau_l||\leq r_j\leq (1+\frac{\epsilon}{2})j\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}$ for $1\leq l\leq j-1$ (by Claim \textbf{1}), and $||\pi-\tau_j||\leq \sqrt{\epsilon j}\delta_j\leq \sqrt{\frac{\epsilon j}{\beta_j}}\delta_{opt}$ (by $\beta_j\delta^2_j\le\delta^2_{opt}$, which implies $\delta_j\le\sqrt{1/\beta_j}\delta_{opt}$), after setting the value of $L$ (in Lemma \ref{lem-shift}) to be $\max\{r_j,||\pi-\tau_j|| \}\le\max\{(1+\frac{\epsilon}{2})j\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}, \sqrt{\frac{\epsilon j}{\beta_j}}\delta_{opt}\}\le(1+\frac{\epsilon}{2})j\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}$ and the value of $\epsilon$ (in Lemma \ref{lem-shift}) to be $\epsilon_0=\epsilon^2/4$, by Lemma \ref{lem-shift} we can construct a grid inside the simplex $V'_{(a)}$ with size $O((\frac{8j}{\epsilon_0})^j)$ which ensures the existence of one grid point $\tau$ satisfying the inequality of $||\tau-m_j||\leq \sqrt{\epsilon_0}\delta_j+(1+\epsilon_0)L\leq\epsilon\delta_j+(1+\epsilon)j\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}$. Hence, we can use $\tau$ as $p_{v_j}$, and the induction step holds for this case.
For case (b), since $P_{j}$ has a small size, we cannot directly perform random sampling on it to find its approximate mean point. To overcome this difficulty, we merge $P_{j}$ with some other large subset $P_{l}$.
Particularly, since $\sum^{j-1}_{l=1}|P_l |= |Opt_j |-|P_j|\geq (\beta_{j}-\epsilon^3\frac{\beta_{j}}{j})|\mathcal{G}|$, by pigeonhole principle, we know that there exists one $l_0$ such that $P_{l_0}$ has size at least $\frac{1}{j-1}(\beta_{j}-\epsilon^3\frac{\beta_{j}}{j})|\mathcal{G}|$. Without loss of generality, we assume $l_0=1$. Then $|P_1 |\geq
\frac{1}{j-1}(\beta_{j}-\epsilon^3\frac{\beta_{j}}{j})|\mathcal{G}|$, and we can view $P_1\cup P_j$ as one large enough subset of $Opt_j$. Let $\tau'$ denote the mean point of $P_1\cup P_j$, then we have the following claim (see Appendix for the proof).
\begin{claim}[\textbf{3}]
$||\tau_1-\tau'||\le\frac{\sqrt{2}\epsilon}{1-\epsilon^3}\sqrt{\frac{j\epsilon}{\beta_j}}\delta_{opt}$. \end{claim}
This means that we can also use Lemma \ref{lem-shift} to find an approximate mean point in a way similar to case (a) (see Figure. \ref{fig-case2}); the difference is that $Opt_j$ is divided into $j-1$ subsets (i.e., $P_1$ and $P_j$ is viewed as one subset $P_1\cup P_j$) and the value of $L$ is set to be $r_j+||\tau_1-\tau'||\le r_j+\frac{\sqrt{2}\epsilon}{1-\epsilon^3}\sqrt{\frac{j\epsilon}{\beta_j}}\delta_{opt}$. We can first construct a simplex $V'_{(b)}$ determined by $\{p_{v_1}, \cdots, p_{v_{j-1}}\}$ (see Figure. \ref{fig-case2}), and then build a grid inside $V'_{(b)}$ with size $O((\frac{8j}{\epsilon_0})^j)$, where $\epsilon_0=\epsilon^2/4$.
By Lemma \ref{lem-shift}, we know that there exists one grid point $\tau$ satisfying the condition of $||\tau-m_j||\leq \sqrt{\epsilon_0}\delta_j+(1+\epsilon_0)L\leq\epsilon\delta_j+(1+\epsilon)j\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}$. Thus the induction step holds for this case.
Since Algorithm Sphere-Peeling-Tree executes every step in our above discussion, the induction step, as well as the lemma, is true.
\qed \end{proof}
\noindent\textbf{Success probability:} From the above analysis, we know that in the $j$-th step/iteration, only case (a) (i.e., $|P_j |\geq \epsilon^3\frac{\beta_{j}}{j}|\mathcal{G}|$) needs to consider success probability, since case (b) (i.e., $|P_j |< \epsilon^3\frac{\beta_{j}}{j}|\mathcal{G}|$) does not need to do sampling. Recall that in case (a), we take a sample of size $\frac{8k^3}{\epsilon^9}\ln\frac{k^2}{\epsilon^6}$. Thus with probability $1-\frac{\epsilon}{k}$, it contains $\frac{k}{\epsilon^5}$ points from $P_j$. Meanwhile, with probability $1-\frac{\epsilon}{k}$, $||\pi-\tau_j||^2\leq \epsilon^4 a^2$. Hence, the success probability in the $j$-th step is $(1-\frac{\epsilon}{k})^2$, which means that the success probability in all $k$ steps is $(1-\frac{\epsilon}{k})^{2k}\geq 1-2\epsilon$.
\noindent\textbf{Running time:} Algorithm $k$-CMeans calls Algorithms Sphere-Peeling-Tree $\frac{2k}{\epsilon}$ times. It is easy to see that each node on the tree returned from Algorithm Sphere-Peeling-Tree has $|\mathcal{R}|2^m (\frac{32j}{\epsilon^2})^j$ children, where $|\mathcal{R}|=O(\frac{\log kn}{\epsilon})$, and $m=\frac{8k^3}{\epsilon^9}\ln\frac{k^2}{\epsilon^6}$. Since the tree has a height of $k$, the complexity of the tree is $O(2^{poly(\frac{k}{\epsilon})}(\log n)^k)$. Further, since each node takes $O(|\mathcal{R}|2^m (\frac{32j}{\epsilon^2})^j nd)$ time, the total time complexity of Algorithm $k$-CMeans is $O(2^{poly(\frac{k}{\epsilon})}n(\log n)^{k+1} d )$.
\section{Extension to Chromatic $k$-Medians Clustering} \label{sec-extension}
We extend our ideas for $k$-CMeans to the Chromatic $k$-Medians Clustering problem ($k$-CMedians). Similar to $k$-CMeans, we first show its relationship with $k$-medians, and then present a $(5+\epsilon)$-approximation algorithm using the sphere peeling technique. Due to the lack of a similar Simplex Lemma for $k$-CMedians, we achieve a constant approximation, instead of a PTAS. See details of the algorithm in Section \ref{sec-kmedian} of the Appendix.
\begin{thebibliography}{7}
\bibitem{ADH} Esther M. Arkin, JosŽ Miguel D’az-B‡–ez, Ferran Hurtado, Piyush Kumar, Joseph S. B. Mitchell, BelŽn Palop, Pablo PŽrez-Lantero, Maria Saumell, Rodrigo I. Silveira: Bichromatic 2-Center of Pairs of Points. LATIN 2012: 25-36
\bibitem{AP98} P. K. Agarwal and C. M. Procopiuc. ``Exact and Approximation Algorithms for Clustering,''{\em Proc. 9th ACM-SIAM Sympos. Discrete Algorithms, pages 658-667,} 1998.
\bibitem{AV07} David Arthur, Sergei Vassilvitskii: ''k-means++: the advantages of careful seeding''. {\em SODA 2007: 1027-1035}
\bibitem{BBC} Nikhil Bansal, Avrim Blum, Shuchi Chawla: ''Correlation Clustering''.{\em Machine Learning 56(1-3)}: 89-113 (2004)
\bibitem{BD06} S.Basu, Ian Davidson: Clustering with Constraints Ð Theory and Practice. {\em ACM KDD 2006}
\bibitem{BHI} M.Badoiu, S.Har-Peled, P.Indyk, ``Approximate clustering via core-sets", {\em Proceedings of the 34th Symposium on Theory of Computing}, pp.~250--257, 2002.
\bibitem{BKK} C.Böhm, K.Kailing, P.Kröger, A.Zimek, "Computing Clusters of Correlation Connected Objects".{\em Proc. ACM SIGMOD International Conference on Management of Data (SIGMOD'04), Paris, France. pp. 455–467. doi:10.1145/1007568.1007620. }
\bibitem{D08} S. Dasgupta, ''The hardness of k-means clustering''. {\em Technical Report}, 2008.
\bibitem{DEF} Erik Demaine, Dotan Emanuel, Amos Fiat, and Nicole Immorlica. ''Correlation clustering in general weighted graphs''. {\em Theor. Comput. Sci., 361(2):172-187}, 2006
\bibitem {HDX11} H.Ding, J.Xu, ''Solving Chromatic Cone Clustering via Minimum Spanning Sphere'', {\em ICALP}, 2011
\bibitem{GG06} Ioannis Giotis and Venkatesan Guruswami. '' Correlation clustering with a fixed number of clusters''. {\em Theory of Computing, 2(1):249-266}, 2006.
\bibitem{HM04} S. Har-Peled and S. Mazumdar, ``Coresets for k-Means and k-Median Clustering and their Applications,'' {\em Proc. 36th ACM Symposium on Theory of Computing, pages 291-300,} 2004.
\bibitem{IKI} Mary Inaba, Naoki Katoh, Hiroshi Imai, '' Applications of Weighted Voronoi Diagrams and Randomization to Variance-Based k-Clustering (Extended Abstract)''. {\em Symposium on Computational Geometry 1994: 332-339}
\bibitem{KR99} S. G. Kolliopoulos and S. Rao, ``A nearly linear-time approximation scheme for the euclidean k-median problem,'' {\em Proc. 7th Annu. European Sympos. Algorithms, pages 378-389,} 1999.
\bibitem{KSS} A. Kumar, Y. Sabharwal, S. Sen, '' Linear-time approximation
schemes for clustering problems in any dimensions''. {\em J. ACM 57(2):2010}
\bibitem{OSS} R.Ostrovsky, Y.Rabani, L.J.Schulman, and C.Swamy. "The Effectiveness of Lloyd-Type Methods for the k-Means Problem". {\em Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06). pp. 165–174.}
\end{thebibliography}
\section{Figure. \ref{fig-probe}}
\begin{figure}
\caption{BAC probes of Chromosome 1 in a WI38 cell with homolog having 6 probes.}
\label{fig-probe}
\end{figure}
\section{Proof for Theorem \ref{mfptas}} \label{sec-mfptas}
\begin{proof} Since it is sufficient to show that the theorem holds for the case of $k=2$, we assume in this proof that $k=2$ and each point-set $G_{i}$ has exactly two points. We make use of a construction by Dasgupta for the NP-hardness proof of the 2-mean clustering problem in high dimensional space \cite{D08}. Their proof reduces from the NAE3SAT problem. For better understanding our ideas, below we sketch their construction.
\begin{enumerate} \item For any instance $\phi$ of NAE3SAT with literal set $\{x_{1}, \cdots, x_{n}\}$ and $m$ clauses, construct a $2n \times 2n$ matrix $D_{\alpha,\beta}$ as follows, where the indices correspond to $\{x_{1}, \cdots, x_{n}\}$ when they are in the range of $[1,n]$, and to $\{\overline{x_{1}}, \cdots, \overline{x_{n}}\}$ when they are in the range of $[n+1,2n]$.\\ $D_{\alpha, \beta} = \left\{ \begin{array}{ll} 0 & \textrm{if $\alpha=\beta$}\\ 1+\Delta & \textrm{if $\alpha=\overline{\beta}$}\\ 1+\delta & \textrm{if $\alpha\sim\beta$}\\ 1 & \textrm{otherwise,} \end{array} \right.$
where $\Delta$, $\delta$ are two constants satisfying inequalities $0<\delta<\Delta<1$ and $4\delta m<\Delta\leq 1-2\delta n$, and $\alpha\sim\beta$ means that both $\alpha$ and $\beta$ or both $\overline{\alpha}$ and $\overline{\beta}$ appear in a clause.
\item $D$ can be embedded into $R^{2n}$, i.e., there exist $2n$ points in $R^{2n}$ with $D$ as their distance matrix.
\item Let $C_{1}$ and $C_{2}$ be the two clusters of the $2$-mean clustering of the $2n$ embedding points. If for any $i$, the points corresponding to
$x_{i}$ and $\overline{x_{i}}$ are separated into different clusters, then $\phi$ is satisfiable if and only if $$\frac{1}{2n}\sum_{i,j\in C_{1}}D_{i,j}+\frac{1}{2n}\sum_{i,j\in C_{2}}D_{i,j}\leq n-1+\frac{2\delta m}{n}.$$
\item Since $\frac{1}{2n}\sum_{i,j\in C_{1}}D_{i,j}+\frac{1}{2n}\sum_{i,j\in C_{2}}D_{i,j}$ is the total cost of the $2$-mean clustering for $C_{1}$ and $C_{2}$, a polynomial time solution to the $2$-mean clustering problem in high dimensional space implies a polynomial time solution to NAE3SAT. Thus the 2-mean clustering is NP-hard in high dimensions. \end{enumerate}
The above reduction can be naturally extended to show the NP-hardness of the full chromatic $2$-mean clustering problem. To show this, we only need to construct $G_{i}$ as the set containing the two points corresponding to $x_{i}$ and $\overline{x_{i}}$ (for simplicity, we write it as $G_{i}=\{x_{i}, \overline{x_{i}}\}$), and the remaining proof follows from the same argument.
Next, we show that full $2$-CMean has no FPTAS in high dimensional space unless P=NP. To see this, we still use the same construction. From the above discussion, we know that $\phi$ is unsatisfiable if and only if for any chromatic partition of $\mathcal{G}$, there exists one clause in $\phi$ such that the three points corresponding to the three literals in this clause are clustered into the same cluster. Hence, the total cost for any chromatic partition is at least $$2\frac{1}{n}({n\choose 2}+(m-1)\delta+3\delta)=n-1+\frac{2}{n}(m+2)\delta .$$
The ratio $\eta$ between the minimum chromatic partition cost of an unsatisfiable instance and the upper bound cost of a satisfiable instance is $$\eta=\frac{n-1+\frac{2}{n}(m+2)\delta}{n-1+\frac{2\delta m}{n}}=1+\frac{\frac{4}{n}\delta}{n-1+\frac{2(m+2)}{n}\delta}.$$ If we let $\delta=\frac{1}{5m+2n}$, then $\eta=1+\frac{\frac{4}{n}\delta}{n-1+\frac{2(m+2)}{n}\delta}=1+\frac{4}{n(5m+2n)(n-1)+2(m+2)}$.
Suppose that there exists an FPTAS for the full chromatic $2$-means clustering problem. Then, if we let $\epsilon <\frac{4}{n(5m+2n)(n-1)+2(m+2)}$, the cost of a $(1+\epsilon)$-approximation of the full $2$-CMeans is less than $n-1+\frac{2}{n}(m+2)\delta$ if and only if $\phi$ is satisfiable. Since the running time of the FPTAS for full $2$-CMeans and $\frac{1}{\epsilon}$ are all polynomial functions of $m$ and $n$, this implies that NAE3SAT can be solved in polynomial time. Obviously this can only happen if P=NP. \qed \end{proof}
\section{Proof of Lemma \ref{lem-meanshift}} \label{sec-meanshift} \begin{proof} In the our following discussion, we use $<a, b>$ to denote the inner product of $a$ and $b$. It is easy to see that
$$\sum_{p\in P}||p-m'||^2=\sum_{p\in P}||p-m+m-m'||^2$$
$$=\sum_{p\in P}(||p-m||^2+2<p-m,m-m'>+||m-m'||^2)$$
$$=\sum_{p\in P}||p-m||^2+2\sum_{p\in P}<p-m,m-m'>+|P| \times ||m-m'||^2$$
$$=\sum_{p\in P}||p-m||^2+2<\sum_{p\in P}(p-m),m-m'>+|P| \times ||m-m'||^2.$$
Since $m$ is the mean point of $P$, $\sum_{p\in P}(p-m)=0$. Thus, the above equality becomes $\sum_{p\in P}||p-m'||^2=\sum_{p\in P}||p-m||^2+|P| \times ||m-m'||^2$. \qed \end{proof}
\section{Proof of Lemma \ref{lem-shift}} \label{sec-shift} \begin{proof} Similar to Lemma \ref{lem-simplex}, we prove this lemma by mathematics induction on $j$.
\textbf{Base case.} For $j=1$, since $o_1=o$, we just need to let $\tau=o'_1$. Then, we have $||\tau-o||=||o'_1-o||=||o'_1-o_1||\leq L\leq\sqrt{\epsilon}\delta+(1+\epsilon)L$. Thus, the base case holds.
\textbf{Induction step.} Assume that the lemma holds for any $j\leq j_0$ for some $j_{0} \ge 1$ (i.e., Induction Hypothesis). Now we consider the case of $j=j_0+1$. Similar to the proof of Lemma \ref{lem-simplex}, we assume that $\frac{|P_l |}{|P|}\geq \frac{\epsilon}{4j}$ for each $1\leq l\leq j$. Otherwise, it can be reduced to a problem with smaller $j$, and solved by the induction hypothesis. Hence, in the following discussion, we assume that $\frac{|P_l |}{|P|}\geq \frac{\epsilon}{4j}$ for each $1\leq l\leq j$.
First, we know that $o=\sum^j_{l=1}\frac{|P_l |}{|P|} o_l$. Let $o'=\sum^j_{l=1}\frac{|P_l |}{|P|} o'_l$. Then, we have \begin{eqnarray}
||o-o'||=||\sum^j_{l=1}\frac{|P_l |}{|P|} o_l-\sum^j_{l=1}\frac{|P_l |}{|P|} o'_l||\leq \sum^j_{l=1}\frac{|P_l |}{|P|}||o_l-o'_l||\leq L. \label{for-8} \end{eqnarray}
Thus, if we can find a grid point $\tau$ within a distance to $o'$ no more than $\sqrt{\epsilon}\delta+\epsilon L$ (i.e., $||\tau-o'||\leq \sqrt{\epsilon}\delta+\epsilon L$), by inequality (\ref{for-8}), we will have $||\tau-o||\leq||\tau-o'||+||o'-o||\leq\sqrt{\epsilon}\delta+(1+\epsilon)L$. This means that we only need to find a grid point close enough to $o'$.
To find such a $\tau$, we first consider the distance from $o'_{l}$ to $o'$. For any $1\leq l\leq j$, we have \begin{eqnarray}
||o'_l-o'||\leq ||o'_l-o_l||+||o_l-o||+||o-o'||\leq 2\sqrt{\frac{j}{\epsilon}}\delta+2L, \label{for-9} \end{eqnarray}
where the first inequality follows from triangle inequality, and the second inequality follows from the facts that $||o'_l-o_l||$ and $||o-o'||$ are both bounded by $L$, and $||o_l-o||\leq 2\sqrt{\frac{j}{\epsilon}}\delta$ (by Lemma \ref{lem-close}).
This implies that we can use the similar idea in Lemma \ref{lem-simplex} to construct a ball $\mathcal{B}$ centered at any $o'_{l_0}$ and with radius $r=\max_{1\leq l\leq j}\{||o'_l-o'_{l_0}||\}$. Note that since $||o'_l-o'_{l_0}||\leq ||o'_l-o'||+||o'-o'_{l_0}||\leq 4\sqrt{\frac{j}{\epsilon}}\delta+4L$ (by inequality (\ref{for-9})), the simplex $V'$ is inside $\mathcal{B}$. Similar to Lemma \ref{lem-simplex}, we can build a grid inside $\mathcal{B}$ with grid length $\frac{\epsilon r}{4j}$ and total grid points $O((8j/\epsilon)^j)$. Clearly in this grid, we can find a grid point $\tau$ such that $||\tau-o'||\leq \frac{\epsilon}{4\sqrt{j}}r\leq \sqrt{\epsilon}\delta+\epsilon L$. Thus, $||\tau-o||\leq\sqrt{\epsilon}\delta+(1+\epsilon)L$, and the induction step, as well as the lemma, holds. \qed \end{proof}
\section{Proof of Claim 1 in Lemma \ref{lem-induction}} \label{sec-claim1}
\begin{proof}
Since $1\geq \beta_j\geq \frac{1}{|\mathcal{G}|}\geq\frac{1}{kn}$, there is one integer $t$ between $1$ and $\log(kn)$, such that $2^{t-1}\leq\frac{1}{\beta_j}\leq 2^t$. Thus $ 2^{t/2-1}\sqrt{\epsilon}\delta_{opt}\leq\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}\leq 2^{t/2}\sqrt{\epsilon}\delta_{opt}$. Together with $\delta\in [\delta_{opt}, (1+\epsilon)\delta_{opt}]$, we have
$$ 2^{t/2-1}\sqrt{\epsilon}\frac{\delta}{1+\epsilon}\leq\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}\leq 2^{t/2}\sqrt{\epsilon}\delta .$$
Thus if set $\hat{r}_j=2^{t/2}\sqrt{\epsilon}\delta$, we have $\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}\leq \hat{r}_j\leq 2(1+\epsilon)\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}$. Let $x=\frac{j\hat{r}_j}{j\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}}$. Then we have $1\le x\le 2(1+\epsilon)$. We build a grid in the interval $[\frac{x}{2(1+\epsilon)}, x]$ with the grid length $\frac{\epsilon}{4(1+\epsilon)}x$, and obtain a grid set (i.e., number set) $\mathcal{N}=\{\frac{1+l\frac{\epsilon}{2}}{2(1+\epsilon)}x\mid 0\le l\le 4+\frac{2}{\epsilon}\}$. We prove that there must exist one number in $\mathcal{N}$ and is between $1$ and $1+\epsilon/2$. First, we know that $\frac{x}{2(1+\epsilon)}\le 1\le x$. If $x\le 1+\epsilon/2$, we find the the desired number $x$ in $\mathcal{N}$. Otherwise, the whole interval $[1, 1+\epsilon/2]$ is inside $[\frac{x}{2(1+\epsilon)}, x]$. Since the grid has grid length $\frac{\epsilon}{4(1+\epsilon)}x\le\frac{\epsilon}{4(1+\epsilon)}2(1+\epsilon)=\epsilon/2$, there must exist one grid point locating inside $[1, 1+\epsilon/2]$. Thus, the desired number exists in $\mathcal{N}$.
Let $\mathcal{R}_j=\{\frac{1+l\frac{\epsilon}{2}}{2(1+\epsilon)}j\hat{r}_j\mid 0\le l\le 4+\frac{2}{\epsilon}\}$. From the above analysis, we know that there exists one value $r_j\in \mathcal{R}_j$ such that
$$j\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}\leq r_j\leq (1+\frac{\epsilon}{2})j\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}.$$
Note that $\mathcal{R}_j\subset \mathcal{R}$, where $\mathcal{R}=\cup^{\log(kn)}_{t=0}\{\frac{1+l\frac{\epsilon}{2}}{2(1+\epsilon)}j2^{t/2}\sqrt{\epsilon}\delta\mid 0\le l\le 4+\frac{2}{\epsilon}\}$. Thus, the Claim is proved.
\qed \end{proof}
\section{Proof of Claim 2 in Lemma \ref{lem-induction}} \label{sec-optl} \begin{proof}
First, for each $1\leq l\leq j-1$, we have $|Opt_l \setminus (\bigcup^{j-1}_{w=1}B_{j,w})|\leq|Opt_l \setminus B_{j,l} | $. Secondly, by Markov inequality, we have
$$|Opt_l \setminus B_{j,l} | \leq \frac{\delta^2_l}{(r_j-||p_{v_l}-m_l ||)^2}|Opt_l |.$$
Note that $\delta^2_{opt}=\sum^k_{j=1}\beta_j\delta^2_j$, and $\beta_j\le \beta_l$ (by $l<j$). Thus, we have $\delta_{l}\leq \sqrt{\frac{1}{\beta_l}}\delta_{opt}\le \sqrt{\frac{1}{\beta_j}}\delta_{opt}$. Together with $j\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}\leq r_j$ and $||p_{v_l}-m_l ||\leq \epsilon\delta_l+(1+\epsilon)l\sqrt{\frac{\epsilon}{\beta_l}}\delta_{opt} $ (by induction hypothesis), we have
\begin{eqnarray*}
r_j-||p_{v_l}-m_l ||&\ge& j\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}-(\epsilon\delta_l+(1+\epsilon)(j-1)\sqrt{\frac{\epsilon}{\beta_l}}\delta_{opt} )\\ & =& (1-(j-1)\epsilon)\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}-\epsilon\delta_l\\ &\geq& (1-(j-1)\epsilon-\sqrt{\epsilon})\sqrt{\frac{\epsilon}{\beta_j}}\delta_{opt}, \end{eqnarray*}
where the last inequality follows from $\delta_{l}\leq \sqrt{\frac{1}{\beta_l}}\delta_{opt}\le \sqrt{\frac{1}{\beta_j}}\delta_{opt}$. Thus, we have
\begin{eqnarray*}
|Opt_l \setminus B_{j,l} | &\leq& \frac{\delta^2_l}{(1-(j-1)\epsilon-\sqrt{\epsilon})^2\frac{\epsilon}{\beta_j}\delta^2_{opt}}|Opt_l |\\
&\leq& \frac{\delta^2_l}{(1-(j-1)\epsilon-\sqrt{\epsilon})^2\frac{\epsilon}{\beta_j}\beta_l \delta^2_l}|Opt_l | \\
&=& \frac{\beta_j}{(1-(j-1)\epsilon-\sqrt{\epsilon})^2\epsilon\beta_l}|Opt_l |\\
&=&\frac{\beta_j |\mathcal{G}|}{(1-(j-1)\epsilon-\sqrt{\epsilon})^2\epsilon}\leq\frac{\beta_j |\mathcal{G}|}{(1-j\sqrt{\epsilon})^2\epsilon}, \end{eqnarray*}
where the second inequality follows from the fact that $\beta_l \delta^2_l\leq \delta^2_{opt}$, and the fourth equation follows from that $\frac{|Opt_l|}{\beta_l}=|\mathcal{G}|$. Note that we can assume $\epsilon$ is small enough such that $\epsilon\leq \frac{1}{4j^2}$, which implies that $ \frac{\beta_j |\mathcal{G}|}{(1-j\sqrt{\epsilon})^2\epsilon}\leq \frac{4\beta_j |\mathcal{G}|}{\epsilon}$. Otherwise, we can just replace $\epsilon$ by $\frac{\epsilon}{4j^2}$ as the input at the beginning of the algorithm. Thus, in total, we have \begin{eqnarray*}
|Opt_l \setminus B_{j,l} | &\leq& \frac{4\beta_j |\mathcal{G}|}{\epsilon}. \end{eqnarray*}
Thus the Claim is proved. \qed \end{proof}
\section{Proof of Claim 3 in Lemma \ref{lem-induction}} \label{sec-claim3}
\begin{proof}
First, we have
$$\frac{|P_1|}{|P_1\cup P_j |}\geq \frac{\frac{1}{j-1}(1-\frac{\epsilon^3}{j})}{\frac{1}{j-1}(1-\frac{\epsilon^3}{j})+\frac{\epsilon^3}{j}}>\frac{1-\epsilon^3}{1+\epsilon^3}.$$
Let $a^{2}$ denote the variance of $P_1\cup P_j$. By Lemma \ref{lem-close}, we know that $||\tau_1-\tau'||\leq\sqrt{\frac{2\epsilon^3}{1-\epsilon^3}}a$. Meanwhile, since $\frac{|P_1\cup P_j |}{|Opt_j|}\geq\frac{ |P_1|}{|Opt_j|}\geq\frac{ \frac{1}{j-1}(\beta_{j}-\epsilon^3\frac{\beta_{j}}{j})|\mathcal{G}|}{\beta_j |\mathcal{G}|}=\frac{1-\frac{\epsilon^3}{j}}{j-1}$, we have $a^2\le \frac{|Opt_j|}{|P_1\cup P_j|}\delta^2_j\leq \frac{j-1}{1-\frac{\epsilon^3}{j}}\delta^2_j$. Then we have
\begin{eqnarray*}
||\tau_1-\tau'||&\leq&\sqrt{\frac{2\epsilon^3}{1-\epsilon^3}}a\le\sqrt{\frac{2\epsilon^3}{1-\epsilon^3}}\sqrt{\frac{j-1}{1-\frac{\epsilon^3}{j}}}\delta_j\\ &\leq&\sqrt{\frac{2j\epsilon^3}{(1-\epsilon^3)(1-\frac{\epsilon^3}{j})\beta_j}}\delta_{opt}\\ &\le&\sqrt{\frac{2j\epsilon^3}{(1-\epsilon^3)(1-\epsilon^3)\beta_j}}\delta_{opt}=\frac{\sqrt{2}\epsilon}{1-\epsilon^3}\sqrt{\frac{j\epsilon}{\beta_j}}\delta_{opt}, \end{eqnarray*}
where the third inequality follows from $\delta_j\le \sqrt{\frac{1}{\beta_j}}\delta_{opt}$. Thus, the claim is true. \qed \end{proof} \section{Chromatic $k$-Medians Clustering} \label{sec-kmedian}
In this section, we extend our ideas for $k$-CMeans to the Chromatic $k$-Medians Clustering problem ($k$-CMedians). Similar to $k$-CMeans, we first show its relationship with $k$-medians (in Section \ref{sec-conmedian}), and then present a $(5+\epsilon)$-approximation algorithm (in Section \ref{sec-ptasmedian}). Due to the lack of a similar Simplex Lemma for $k$-CMedians, we achieve a constant approximation, instead of a PTAS.
\begin{definition}[Chromatic $k$-Median Clustering ($k$-CMedians)] \label{def-kcmedian}
Let $\mathcal{G}=\{G_{1}, \cdots,$ $G_{n}\}$ be a set of $n$ point-sets with each $G_{i}=\{p^{i}_{1}, \dots, p^{i}_{k_{i}}\}$ consisting of $k_{i}\leq k$ points in $\mathbb{R}^d$ space. The chromatic $k$-median clustering (or $k$-CMedians) of $\mathcal{G}$ is to find $k$ points $\{m_{1}, \cdots, m_{k}\}$ in $\mathbb{R}^d$ space and a chromatic partition $U_{1}, \cdots, U_{k}$ of $\mathcal{G}$ such that $\frac{1}{n}\sum_{j}\sum_{q\in U_{j}}||q-m_{j}||$ is minimized. \end{definition}
\subsection{Constant Approximation from $k$-Medians} \label{sec-conmedian}
Given a set of points in $\mathbb{R}^d$, the optimal median point is also called {\em Fermat Weber point} in geometry. Its main difference with mean point is that no explicit formula exists for computing the optimal median point, while the mean point is simply the average of the given points. Consequently, median point is often approximated using some iterative procedure, such as {\em Weiszfeld's algorithm}. Thus in the following discussion, we only assume the availability of a $(1+\epsilon)$-approximation of the median point.
\begin{lemma} \label{lem-closemedian}
Let $P$ be a set of points in $\mathbb{R}^d$ space, and $P_{1}$ be a subset of $P$ containing a fraction of $\alpha\leq 1$ points of $P$. Let $m_{opt}$ and $m$ be the optimal and $(1+\epsilon)$-approximate median point of $P$ respectively, and $m_1$ be the optimal median of $P_{1}$. Then $||m_{1}-m||\leq\frac{2+\epsilon}{\alpha}\mu$, where $\mu=\frac{1}{|P|}\sum_{p\in P}||p-m_{opt}||$. \end{lemma} \begin{proof}
Let $\mu_1=\frac{1}{|P_1|}\sum_{p\in P_1}||p-m_1||$. Since $P_1\subseteq P$, it is easy to know that $\sum_{p\in P}||p-m||\geq \sum_{p\in P_1}||p-m||$, which implies that $(1+\epsilon)\mu\geq\frac{1}{|P|}\sum_{p\in P_1}||p-m||=\alpha\frac{1}{|P_1|}\sum_{p\in P_1}||p-m||$. By triangle inequality, we also have $||p-m||\geq ||m-m_1||-||p-m_1||$. Thus, \begin{eqnarray}
(1+\epsilon)\mu\geq \alpha(||m-m_1||-\mu_1). \label{for-12} \end{eqnarray}
Since $m_1$ is the optimal median of $P_1$, we have $\mu_1=\frac{1}{|P_1|}\sum_{p\in P_1}||p-m_1||\leq \frac{1}{|P_1|}\sum_{p\in P_1}||p-m_{opt}||\leq\frac{1}{|P_1|}\sum_{p\in P}||p-m_{opt}||=\frac{1}{\alpha}\mu$. Plugging this into inequality (\ref{for-12}), we have $||m-m_1||\leq\frac{2+\epsilon}{\alpha}\mu$. \qed \end{proof}
\begin{theorem} \label{the-constantmedian} Let $\mathcal{G}=\{G_1, \cdots, G_n\}$ be an instance of $k$-CMedians, and $\mathcal{C}$ be the $k$ $(1+\epsilon)$-approximate median points of the $k$ clusters generated by a $c$-approximation $k$-medians algorithm on the points $\cup_{i=1}^{n}G_{i}$. Then, $[\mathcal{C}]^k$ contains at least one $k$-tuple whose elements are the $k$ median points of a $((2+\epsilon)ck^2+(2+\epsilon)k+1)$-approximation of $k$-CMedians on
$\mathcal{G}$, where $[\mathcal{C}]^k=\underbrace{\mathcal{C}\times\cdots\times\mathcal{C}}_{k}$. \end{theorem} \begin{proof} Let $\{c_1, \cdots, c_k\}$ be the set of $k$ approximate median points in $\mathcal{C}$, and $\{S_1, \cdots, S_k\}$ be the $k$ clusters returned by the $c$-approximation $k$-medians algorithm.
Thus, $c_j$ is the $(1+\epsilon)$-approximate median point of $S_j$ for $1\leq j\leq k$. Let $\mathcal{OPT}=\{Opt_{1}, \cdots, Opt_{k}\}$ be the \textbf{unknown} optimal solution for $k$-CMedians on $\mathcal{G}$, and $m_{j}$ be the optimal median point of $Opt_j$ for $1\leq j\leq k$. Denote the set $Opt_i\cap S_j$ as $\Gamma^i_j$, and its optimal median point as $\tau^i_j$ for $1\leq i, j\leq k$ .
Since $\cup^k_{j=1}\Gamma^i_j=Opt_i$, there must exist some index $1\leq j_i \leq k$ such that $|\Gamma^i_{j_i}|\geq \frac{1}{k}|Opt_i|$. Fixing $j_i$, we have the following about $\sum_{p\in Opt_i}||p-c_{j_i}||$. \begin{eqnarray*}
\sum_{p\in Opt_i}||p-c_{j_i}|| &=& \sum_{p\in Opt_i}||p-m_i+m_i-c_{j_i}||\\
&\leq& \sum_{p\in Opt_i}(||p-m_i||+||m_i-c_{j_i}||)\\
&=& \sum_{p\in Opt_i}||p-m_i||+|Opt_i| \times ||m_i-c_{j_i}||\\
&=& \sum_{p\in Opt_i}||p-m_i||+|Opt_i| \times ||m_i-\tau^i_{j_i}+\tau^i_{j_i}-c_{j_i}||\\
&\leq& \sum_{p\in Opt_i}||p-m_i||+|Opt_i| \times (||m_i-\tau^i_{j_i}||+||\tau^i_{j_i}-c_{j_i}||)
\end{eqnarray*}
By Lemma \ref{lem-closemedian}, we have \begin{eqnarray*}
||\tau^i_{j_i}-m_i|| &\leq& \frac{2+\epsilon}{\frac{1}{k}}(\frac{1}{|Opt_i|}\sum_{p\in Opt_i}||p-m_i ||);\\
||\tau^i_{j_i}-c_{j_i}|| &\leq& \frac{2+\epsilon}{\frac{|\Gamma^i_{j_i}|}{|S_{j_i}|}}(\frac{1}{|S_{j_i}|}\sum_{p\in S_{j_i}}||p-c_{j_i} ||). \end{eqnarray*}
From the above inequalities, we have
\begin{eqnarray*}
\sum_{p\in Opt_i}||p-c_{j_i}|| &\leq & \sum_{p\in Opt_i}||p-m_i||+|Opt_i| \times (||m_i-\tau^i_{j_i}||+||\tau^i_{j_i}-c_{j_i}||)\\
&\leq&\sum_{p\in Opt_i}||p-m_i||^2+|Opt_i|(\frac{2+\epsilon}{\frac{1}{k}}(\frac{1}{|Opt_i|}\sum_{p\in Opt_i}||p-m_i ||)+\frac{2+\epsilon}{\frac{|\Gamma^i_{j_i}|}{|S_{j_i}|}}(\frac{1}{|S_{j_i}|}\sum_{p\in S_{j_i}}||p-c_{j_i} ||))\\
&=&((2+\epsilon)k+1)\sum_{p\in Opt_i}||p-m_i||+(2+\epsilon)\frac{|Opt_i|}{|\Gamma^i_{j_i}|} \times \sum_{p\in S_{j_i}}||p-c_{j_i} ||). \end{eqnarray*}
Since $|\Gamma^i_{j_i}|\geq \frac{1}{k}|Opt_i|$, we have $\frac{|Opt_i|}{|\Gamma^i_{j_i}|}\leq k$. Thus, \begin{eqnarray*}
\sum_{p\in Opt_i}||p-c_{j_i}||^2\leq ((2+\epsilon)k+1)\sum_{p\in Opt_i}||p-m_i||+(2+\epsilon)k\sum_{p\in S_{j_i}}||p-c_{j_i} ||. \end{eqnarray*} Summing both sides of the above inequality over $i$, we have \begin{eqnarray}
\sum^k_{i=1}\sum_{p\in Opt_i}||p-c_{j_i}||^2 &\leq& ((2+\epsilon)k+1)\sum^k_{i=1}\sum_{p\in Opt_i}||p-m_i||+(2+\epsilon)k\sum^k_{i=1}\sum_{p\in S_{j_i}}||p-c_{j_i} || \nonumber\\
&\leq& ((2+\epsilon)k+1)\sum^k_{i=1}\sum_{p\in Opt_i}||p-m_i||+(2+\epsilon)k^2 \sum^k_{j=1}\sum_{p\in S_{j}}||p-c_{j} ||. \label{for-13} \end{eqnarray}
It is easy to know that the optimal objective value of $k$-medians is no larger than that of $k$-CMedians on the same set of points in $\mathcal{G}$. Thus, $\sum^k_{j=1}\sum_{p\in S_{j}}||p-c_{j} ||\leq c\sum^k_{i=1}\sum_{p\in Opt_i}||p-m_i||$. Plugging this inequality into (\ref{for-13}), we have \begin{eqnarray*}
\sum^k_{i=1}\sum_{p\in Opt_i}||p-c_{j_i}||\leq ((2+\epsilon)ck^2+(2+\epsilon)k+1)\sum^k_{i=1}\sum_{p\in Opt_i}||p-m_i||. \end{eqnarray*} The above inequality means that if we take the $k$-tuple $(c_{j_1}, \cdots, c_{j_k})$ as the $k$ approximate median points for $k$-CMedians, we have a $((2+\epsilon)ck^2+(2+\epsilon)k+1)$-approximation solution for $k$-CMedians. Thus, the theorem is proved.
\qed \end{proof}
\subsection{Peeling Algorithm for $k$-CMedians} \label{sec-ptasmedian}
The following lemma is a key to the peeling algorithm for $k$-CMedians (i.e., play a similar role as
Lemma \ref{lem-simplex} for $k$-CMeans).
\begin{lemma} \label{lem-simplexmedian} Let $P$ to be a set of points in $\mathbb{R}^d$ with a partition $P=\cup^j_{l-1} P_l$, $o$ be its optimal median point, and $o_l$ be the optimal median point of $P_l$ for $1\leq l\leq j$.
Let $\mu=\frac{1}{|P|}\sum_{p\in P}||p-o||$. Then, there exists some $i_{0}$ such that $||o-o_{i_{0}}||\leq 4\mu$.
\end{lemma} \begin{proof}
Since $\mu=\frac{\sum_{p\in P}||p-o||}{|P|}=\sum^{l}_{i=1}(\frac{|P_{i}|}{|P|}\frac{\sum_{p\in P_{i}}||p-o||}{|P_{i}|})$, there must exist some index $i_{0}$ such that $\frac{\sum_{p\in P_{i_{0}}}||p-o||}{|P_{i_{0}}|}$ $\leq\mu$. By Markov inequality, we know that there exists one subset $U$ of $P_{i_0}$ such that $|U|> |P_{i_{0}}|/2$ and $||p-o||\leq 2\mu$ for any $ p\in U$.
Since $o_{i_{0}}$ is the optimal median point of $P_{i_{0}}$, $\frac{\sum_{p\in P_{i_{0}}}||p-o_{i_{0}}||}{|P_{i_{0}}|}\leq\frac{\sum_{p\in P_{i_{0}}}||p-o||}{|P_{i_{0}}|}\leq\mu$. Similarly, by Markov inequality, we know that there exists one subset $V$ of $P_{i_0}$ such that $|V|> |P_{i_{0}}|/2$ and $||p-o_{i_0}||\leq 2\mu$ for any $ p\in V$.
From the inequalities of $|U|> |P_{i_{0}}|/2$ and $|V|> |P_{i_{0}}|/2$ and the fact that $U\cap V \neq\emptyset$, we know that there exists one point $p_{0}\in U\cap V$ such that $||p_{0}-o||\leq 2\mu$ and $||p_{0}-o_{i_{0}}||\leq 2\mu$. Thus $||o_{i_{0}}-o||\leq ||o_{i_{0}}-p_{0}||+||p_{0}-o||\leq 4\mu$. \qed \end{proof}
Before presenting our peeling algorithm, we still need the following lemma proved by Badoiu {\em et al.} in \cite{BHI} for finding an approximate solution for $1$-median.
\begin{theorem}[\cite{BHI}] \label{the-1med} Let $P$ be a normalized set of $n$ points in $\mathbb{R}^d$ space, $1>\epsilon>0$, and $R$ be a random sample of $O(1/\epsilon^3\log1/\epsilon)$ points from $P$. Then one can compute, in $O(d2^{O(1/\epsilon^4)}\log n)$ time, a point-set $S(P,R)$ of cardinality $O(2^{O(1/\epsilon^4)}\log n)$ , such that with constant probability (over the choices of $R$), there is a point $q\in S(P,R)$ such that $cost(q,P)\leq (1+\epsilon)med_{opt}(P,1)$. \end{theorem}
\noindent\textbf{Algorithm $k$-CMedians} \newline \textbf{Input:} $\mathcal{G}=\{G_1, \cdots, G_n\}$, $k\geq 2$ and an small constant $\epsilon>0$. \newline \textbf{Output:} a $(5+\epsilon)$-approximation solution for $k$-CMedians on $\mathcal{G}$. \begin{enumerate} \item Run the $(1+\epsilon)$-approximation $k$-medians algorithm from \cite{KSS} on $\mathcal{G}$, and let $\Omega$ be the obtained objective value.
\item For $i=1$ to $\frac{4k}{\epsilon}$ do \begin{enumerate} \item Set $\delta=\frac{\Omega}{4k}+i\frac{\epsilon}{4k}\Omega$, and run Algorithm Sphere-Peeling-Tree-2.
\item Let $\mathcal{T}_i$ be the returned tree.
\end{enumerate}
\item For each path of every $\mathcal{T}_i$, use bipartite matching procedure to compute the objective value of $k$-CMeans on $\mathcal{G}$. Output the $k$ points from the path with smallest objective value.
\end{enumerate}
\noindent\textbf{Algorithm Sphere-Peeling-Tree-$2$} \newline \textbf{Input:} $\mathcal{G}$, $k\geq 2$, $\epsilon, \delta>0$. \newline \textbf{Output:} A tree $\mathcal{T}$ of height $k$ with each node $v$ associated with a point $p_v \in \mathbb{R}^d$. \begin{enumerate}
\item Initialize $\mathcal{T}$ with a single root node $v$ associating with no point. \item Recursively grow each node $v$ in the following way \begin{enumerate} \item If the height of $v$ is already $k$, then it is a leaf node. \item Otherwise, let $j$ be the height of $v$. Build the set of radius candidates $\mathcal{R}=\cup^{\log(kn)}_{t=0}\{\frac{1+l\frac{\epsilon}{2}}{2(1+\epsilon)}j2^{t/2}\sqrt{\epsilon}\delta\mid 0\le l\le 4+\frac{2}{\epsilon}\}$.
For each radius candidate $r\in\mathcal{R}$ do \begin{enumerate} \item Let $j$ be the height of $v$, and $\{p_{v_1}, \cdots, p_{v_j}\}$ be the $j$ points associated with nodes on the root-to-$v$ path (including $p_v$).
\item For each $p_{v_l}$, $1\leq l\leq j$, construct a ball $B_{j,l}$ centered at $p_{v_l}$ and with radius $r$.
\item Take a random sample from $\mathcal{G}\setminus\cup^j_{l=1}B_{j,l}$ with size $m=\frac{8k^3}{\epsilon^9}\ln\frac{k^2}{\epsilon^6}$. Compute the approximate median points of all subsets of the sample (by Theorem \ref{the-1med}), and denote the set of the approximate median points as $\Pi$. Clearly, $|\Pi|=2^{m+O(1/\epsilon^4)}\log n$.
\item For each point $p$ in $\Pi$, add one child to $v$, and associate it with $p$; add another $j$ children, with each one associating with a different point in $\{p_{v_1}, \cdots, p_{v_j}\}$. \end{enumerate} \end{enumerate}
\end{enumerate}
We can use a similar approach as in Section \ref{sec-proof} to analyze the correctness of Algorithm $k$-CMedians. \\
Let $\mathcal{OPT}=\{Opt_1, \cdots, Opt_k\}$ be the optimal solution of $k$-CMedians on $\mathcal{G}$.
Without loss of generality, we assume that $|Opt_1|\ge |Opt_2|\ge\cdots \ge |Opt_k|$. For each $Opt_j$, $1\leq j\leq k$, let $m_j$ be its median point, $\beta_j$ be its fraction in $\mathcal{G}$ (i.e., $|Opt_{j}|/|\cup_{i=1}^{n} G_{i}|$), and $\mu_j=\frac{1}{|Opt_j |}\sum_{p\in Opt_j}||p-m_j||$. Thus, $\beta_1\geq\cdots\geq\beta_k$ and $\sum^k_{j=1}\beta_j =1$. Also, let $\mu_{opt}=\sum^k_{j=1}\beta_j\mu_j$.
\begin{lemma} \label{lem-inductionmedian} Among all the trees generated in Algorithm $k$-CMedians, there exists one tree $\mathcal{T}_i$, which has a root-to-leaf path with each node $v_{j}$ at level $j$, $1\leq j\leq k$, on the path associating a point
$p_{v_j}$ and satisfying the inequality
$$||p_{v_j}-m_j ||\leq 4\mu_j+(1+\epsilon)j\frac{\epsilon}{\beta_j}\mu_{opt} .$$ \end{lemma}
\begin{lemma} \label{lem-equalmedian} If Lemma \ref{lem-inductionmedian} is true, Algorithm $k$-CMedians yields a $(5+O(k^2)\epsilon)$-approximation solution for $k$-CMedians. \end{lemma} \begin{proof} We first assume that Lemma \ref{lem-inductionmedian} is true. Then for each $1\leq j\leq k$, we have \begin{eqnarray}
\sum_{p\in Opt_j}||p-p_{v_j}|| & \le & \sum_{p\in Opt_j}||p-m_j||+|Opt_j|\times||m_j-p_{v_j}|| \nonumber\\
&\leq& \sum_{p\in Opt_j}||p-m_j||+|Opt_j|\times(4\mu_j+(1+\epsilon)j\frac{\epsilon}{\beta_j}\mu_{opt}) \nonumber\\
&=& 5|Opt_j |\mu_j+(1+\epsilon) j\epsilon|\mathcal{G}|\mu_{opt} \label{for-14} \end{eqnarray}
Summing the both sides of (\ref{for-14}) over $j$, we have \begin{eqnarray}
\sum^k_{j=1}\sum_{p\in Opt_j}||p-p_{v_j}||^2 &\leq & \sum^k_{j=1}(5|Opt_j |\mu_j+(1+\epsilon) j\epsilon|\mathcal{G}|\mu_{opt}) \nonumber\\
&\leq& 5\sum^k_{j=1}|Opt_j |\mu_j+(1+\epsilon) k^2\epsilon|\mathcal{G}|\mu_{opt} \nonumber\\
&=&(5+O(k^2)\epsilon)|\mathcal{G}|\mu_{opt}. \label{for-15} \end{eqnarray}
In the above, the last equation follows from the fact that $\sum^k_{j=1}|Opt_j |\mu_j=|\mathcal{G}|\mu_{opt}$. By (\ref{for-15}), we know that $\{p_{v_1}, \cdots, p_{v_k}\}$ induces a $(5+O(k^2)\epsilon)$-approximation solution for $k$-CMedians.
\qed \end{proof}
By a similar argument given in the proof of Lemma \ref{lem-induction}, we can show the correctness of Lemma \ref{lem-inductionmedian}. Thus, we have the following theorem. \begin{theorem} \label{the-ptasmedian} With constant probability, Algorithm $k$-CMedians yields a $(5+\epsilon)$-approximation for $k$-CMedians in $O(2^{poly(\frac{k}{\epsilon})}n(\log n)^{2k} d )$ time.
\end{theorem}
\end{document} | arXiv |
Tunable metasurfaces for visible and SWIR applications
Chang-Won Lee ORCID: orcid.org/0000-0003-0546-04391,
Hee Jin Choi1 &
Heejeong Jeong2
Nano Convergence volume 7, Article number: 3 (2020) Cite this article
Demand on optical or photonic applications in the visible or short-wavelength infrared (SWIR) spectra, such as vision, virtual or augmented displays, imaging, spectroscopy, remote sensing (LIDAR), chemical reaction sensing, microscopy, and photonic integrated circuits, has envisaged new type of subwavelength-featured materials and devices for controlling electromagnetic waves. The study on metasurfaces, of which the thickness is either comparable to or smaller than the wavelength of the considered incoming electromagnetic wave, has been grown rapidly to embrace the needs of developing sub 100-micron active photonic pixelated devices and their arrayed form. Meta-atoms in metasurfaces are now actively controlled under external stimuli to lead to a large phase shift upon the incident light, which has provided a huge potential for arrayed two-dimensional active optics. This short review summarizes actively tunable or reconfigurable metasurfaces for the visible or SWIR spectra, to account for the physical operating principles and the current issues to overcome.
Electromagnetic or photonic metamaterials are artificial materials made of natural metals or dielectrics so as to be specially engineered to provide new and exotic interactions between incident waves and matter. Metamaterials show phenomena that are not observed in natural or conventional materials, such as the negative refractive index [1,2,3], the perfect absorption [4], subwavelength focusing [5] and hyperbolically-engineered dispersion [6, 7]. Metasurface, which is the two-dimensional cousin of the metamaterial, has the thickness smaller than the wavelength of the incident light that allows control of the optical wavefront over subwavelength thicknesses [8]. Therefore, the interaction between the metasurface and the light has to be enough in order to alter the characteristics of the incident light. The characteristics of a metamaterial or a metasurface are primarily determined by its inner structure called "meta-atom" and the interaction between them. Even though meta-atom is originally defined as the unit cell in a uniformly periodic structure, many non-periodic or non-uniformly engineered meta-atoms are now available for passive metasurface applications such as lenses [9,10,11,12,13,14,15], axicons [16, 17], polarization converters [18, 19], and holograms [20,21,22].
Recent advancement in metasurfaces allows active control of light beyond manipulating the characteristics of the light under stationary platforms. Manufacturing metasurfaces becomes more viable compared to its three-dimensional cousin, because of its planar geometry and the well-established lithographic fabrication processes. The active tuning of light through a three-dimensional tunable metamaterial can be obtained by various external stimuli by electrical [23,24,25,26], mechanical [27,28,29,30], optical [31,32,33], thermal, or magnetic means. For mid-infrared or terahertz radiation spectra, split-ring-based, dielectric resonator-based, phase-change-materials-based, graphene-based, or liquid–crystal-based metamaterials are available, as reviewed in the previous literature [34,35,36,37,38]. Each control requires materials with significant optical characteristics to change accordingly.
Microscopic origin responsible for metasurface properties can be explained in terms of phase shift of incident light. The phase shift alters reflection, transmission, phase, polarization, and frequency states of the incident light. In order to gain full control of transmitted or reflected electromagnetic waves from a metasurface, it is necessary to have a full phase shift up to 2π upon incident waves. For passive metasurface with thickness t, typical phase shift for normal incidence after the transmission is approximated by
$$\phi = 2\pi nt/\lambda ,$$
where \(\lambda\) is the wavelength of the incident wave and n is the effective refractive index of the metasurface. The phase shift of the planar-shaped metasurfaces can be limited if the used natural materials's refractive index is not high enough. It is noteworthy that the phase distribution of modern metasurfaces is independently controlled my meta-atoms and can be either continuous and discontinuous [39]. Even though there are a number of materials showing such a controlled optical phase shifts mid-infrared or longer wavelengths, there have been quite challenging to obtain thin materials or devices for tunable metasurface in visible and SWIR regions. The difficulty comes from that natural materials, including semiconductors and conducting oxides, tend to show decreasing behavior of dielectric permittivities approaching unity, almost inversely proportional to the square of the radiation frequency (or proportional to the square of the wavelengths in a vacuum) [40]. If the dielectric permittivity of the parallel component to the incident light is close to unity, no interesting metasurface characteristics emerge.
Tunable metasurfaces for visible and SWIR spectra can be categorized, based on its controlling methods, as (1) electrically-tunable, (2) electromechanically-tunable, (3) nonlinear-optically-tunable, and (4) thermally-tunable ones, similar to the categories of three-dimensional metamaterials. There are other tuning mechanisms such as magneto-optical tuning methods and however, the workable wavelengths from other mechanisms are not currently available in the visible or SWIR light.
Electrical control on metasurface accompanies external bias potential across the entire metasurface or on part of the metamaterials nearby. The external bias reaches individual meta-atoms or a local inner structure [41]. The biasing geometries can be similar to the cases of two-terminal diodes or three-terminal field-effect transistors. Electromechanically-tunable metamaterials use electromechanical control for compressing, stretching, or pressuring to change the periodicities of meta-atoms fabricated on a flexible substrate. Likewise, nonlinear-optically-tunable metamaterials use optical pumping for dynamic control of optical properties such as optically induced transition or \(\upchi^{\left( 3 \right)}\) nonlinearity. Thermally tunable metasurfaces use phase-change material, which shows dielectric permittivity changes due to its crystalline or electronic phase change according to the temperature.
So far, the most successfully engineered material in the visible or SWIR light manipulation is a liquid crystal being widely used in visual displays. However, the thickness of the liquid crystal layer has to exceed ~ 100 μm in order to gain the full 2π phase shift. The maximum modulation speed and anchoring problem are inherently limited by molecular orientation liquid crystal molecules. However, recent advancements in semiconductors, transparent conducting oxides, phase-change materials, and two-dimensional materials begin to invoke potentials to provide fast modulation tunability bandwidth exceeding 10 GHz in visible and SWIR spectra on the very thin material platform less than 10 μm [42,43,44]. Therefore, the development of tunable metamaterials and metasurfaces in the visible and SWIR spectra provides a great impact on optical and photonic applications with ultra-small form factors for the upcoming 4th industrial revolution. In this review, physical principles and tuning mechanisms of individual meta-atom in tunable metasurfaces are discussed with emphasis on the distinctive characteristics of the applied materials and potential applications.
Electrically-tunable metasurfaces
Electrically-tunable metamaterials and metasurfaces for visible or SWIR spectra use local refractive index change according to charge carrier redistribution upon external perturbation. Here charge carriers can be electrons, holes, or ions. Biasing individual meta-atom results in temporally changing electromagnetic phase distribution determined by dynamically-controlled carrier concentration changes. Electric field-effect based light modulation has distinct advantages over liquid crystal-based modulation because it provides (1) fast response (typically > 1 MHz bandwidth of modulation), (2) relatively low power consumption with devices sizes smaller than submicron sizes, (3) CMOS fabrication compatibility, and (4) capability of high-density integration. Estimating carrier concentration of material is important because the redistributed carrier determines the possible phase shift range upon incident light wavelength, the ratio of polarization, and scattering direction [45].
There have been four proposed physical mechanisms based on tunable dispersion relations to design the electronically-tunable metasurfaces. (1) First, dispersion with epsilon-near-zero (ENZ) conditions for semiconducting or oxide materials has long been investigated. The ENZ condition is fulfilled when the real part of the dielectric permittivity approaches zero at a certain wavelength at the interface with adjacent materials. Therefore, the electric field virtually goes to infinity to satisfy the boundary conditions of Maxwell's equations [46, 47]. The most intensively studied ENZ material in the visible or SWIR spectra is a transparent conducting oxide, such as indium tin oxide (ITO) and aluminum-zinc oxide (AZO). (2) The second candidate for enhancing light-matter interaction is the hyperbolic dispersion condition [6, 48]. Strong anisotropic dispersion in stacked multilayers or pillars of metallic and dielectric materials leads to field enhancement along a direction to the hyperbolic wave vector becoming imaginary. (3) Another proposed mechanism is based on altering Mie resonance condition [24, 49], however, this idea has yet been realized in the visible or SWIR spectra. (4) The most recently physical mechanism is based on dynamic control of quantum-confined confined Stark effect [50].
The ENZ dispersion condition for tunable metasurfaces can be understood qualitatively as in the following manner. The meta-atom with two or more electrodes can be regarded as a capacitor. Therefore, the carrier concentration under ENZ condition voltage can be approximately estimated by matching the driving voltage and the dielectric permittivity. The driving voltage required to reach ENZ is defined as
$${\text{V}} = N_{ENZ} d_{ACL} /{\text{C}}$$
where \(N_{ENZ}\) is the free carrier concentration in the considered material at the ENZ condition and \(d_{ACL}\) is the thickness of the charge accumulation layer. Because the capacitance \({\text{C}}\) is given by
$${\text{C}} = \varepsilon_{0} \varepsilon_{r} /d_{r}$$
where \(\varepsilon_{0}\) is the permittivity of vacuum, \(\varepsilon_{r}\) is the dielectric constant of the material, and \(d_{r}\) is the thickness of the material, the electric field at ENZ condition is given by
$${\text{E}} = \frac{\text{V}}{{d_{r} }}$$
The wavelength and the angle of incidence at ENZ condition can be found by searching poles in the Fresnel formula for transmittance or reflectance. Using the Drude model, it is possible to estimate carrier concentration using this form of the electric field as an external stimulus.
One of the noticeable metasurface structures based on electric field-effect based SWIR light modulation has been demonstrated by Huang et al. [51]. The electrically tunable metasurface consists of an electrically bus-connected one-dimensional gold nanoantenna array patterned on thin Al2O3 and ITO layers, deposited on a gold mirror as shown in Fig. 1a. Al2O3 layer, grown by atomic-layer-deposition (ALD) method, provides good thermal stability and a high breakdown field larger than 10 MV/cm [54]. Identical antenna arrays are connected either to right or left external gold connections and electrodes to permit phase and amplitude modulation by electrical gating. Oxide-based field-effect modulation consists of metal–transparent conducting oxide–metal configuration where transparent conducting oxide acts as a semiconductor. All the optical antennae have identical geometries, but applying different voltages to neighboring antennae controls the phase shift imposed by the different antennae. Each antenna was designed with a shape to impose a phase shift on the incident plane wave, which is pre-defined by the lithographic patterning process.
Tunable meta-atoms with transparent conducting oxides. a An ITO layer under external bias modulates the charge accumulation region near ENZ dispersion condition, resulting in a phase shift for the reflected light in the SWIR region [51]. Reproduced with permission from ©2016, American Chemical Society. b Dual gated ITO layer for enhanced phase shift up to 303° at 1550 nm [52]. Reproduced with permission from ©2018, American Chemical Society. c Ionic-conduction-induced visible tunable meta-atom [40]. Reproduced with permission from ©2017, Wiley–VCH. d Oxide heterostructure for Berreman dispersion mode [53]. Reproduced with permission from ©2018, American Chemical Society
Each unit cell is a metal/insulating oxide/semiconductor (MOS) capacitor where the top antenna and back reflector act as capacitive electrodes. By increasing the bias voltage between the top and back electrodes, charge accumulation or depletion region which can induce the change of carrier density forms in the ITO layer, at the ITO/Al2O3 interface. This results in the modification of the complex permittivity of ITO, and in the phase shift between the reflected and incident lights. Because each antenna can be biased individually, the phase distribution across the metasurface effectively controls the overall wavefront, leading to manipulated reflected light. Also, dynamic gate voltage control enables dynamic control of the phase shifts reflected by the individual antenna, facilitating active beam directionality control or high-speed intensity modulation of the reflected light. The authors designed device structure to obtain as close ENZ condition at the interface between the ITO/Al2O3 interface as possible because ENZ leads to large electric field enhancement that occurs in the accumulation layer for short-wavelength infrared wavelengths [55, 56]. Based on the Drude model, the carrier concentration and the resultant dielectric permittivity could be estimated [45, 51, 57]. Regarding ITO as a semiconductor material with a bandgap of Ebg = 2.8 eV, the electron affinity of \(\upchi\) = 5 eV, and effective electron mass of m* = 0.25 me, the dielectric permittivity can be written as
$$\varepsilon_{ITO} \left(\upomega \right) = \varepsilon_{\infty } - \frac{{\upomega_{p}^{2} }}{{\upomega^{2} + i\upomega\varGamma }},\quad\upomega_{p}^{2} = \frac{{Ne^{2} }}{{\varepsilon_{0} m^{*} m_{e} }},$$
where me is the mass of the free electron, \(\varepsilon_{\infty } = 4.2\), \(\upomega_{p}\) is the plasma frequency, \(\varGamma\) is the damping constant of 1.8 × 1014 rad Hz, e is the electron charge, ε0 is the dielectric permittivity of vacuum, and N is the carrier density. As a result, the estimated background carrier concentration of the fabricated ITO film ranged from No = 8 × 1019 cm−3 to 7 × 1020 cm−3 depending on the growth conditions. The accumulated carrier concentration and the real part of the dielectric permittivity as a function of the position from the Al2O3/ITO interface are shown in Fig. 1a. Under the varied gate bias from 0 V to 5 V, the electron carrier density can be increased or decreased at the Al2O3/ITO interface, resulting in the change of the real part of the permittivities from positive to negative values.
The performance of the device shows the reflectance change, defined as \(\frac{\Delta R}{R} = \frac{{R\left( V \right) - R\left( {V = 0V} \right)}}{{R\left( {V = 0V} \right)}}\), almost reaches ~ 30% at 1600 nm and ~ − 20% at 1500 nm, under an applied bias of 2.5 V. The phase shift, measured by a Michelson interferometer incorporating self-phase referencing method, show increasing phase shift with applied bias. A maximum phase shift was obtained under 2.5 V gate bias around 184 °. The authors also confirmed the modulation bandwidth as high as 10 MHz, mainly limited by large capacitance value owing to the large area of the antenna. The largest bandwidth with antennas of submicron could be estimated to exceed 100 GHz [51].
Shirmanesh et al. introduced a dual-gated geometry to the ITO active material to maximize phase shift, as shown in Fig. 1b [52]. This configuration allows the application of two independent gate biases between the back reflector and ITO, and between top antenna and ITO, enabling larger phase tunability compared to the single gate. For enhanced field between the gate dielectric/ITO interfaces, they also incorporated ENZ condition at the interface between the gate dielectric and ITO layers to design the structure, leading to a 5 nm-thick ITO layer. For the top gate, a fishbone structure was introduced. Another improved feature is ALD-grown 9.5 nm-thick Al2O3/HfO2 nanolaminates (HAOL) as gate dielectrics. This hybrid material improved gate-dielectric material enabled larger DC permittivity up to 22, which is close to the value of HfO2 (~ 25). Two independent gating configurations were tested: (I) The gates are applied with V0 while the ITO layer is kept ground. (II) The top gate is applied with +V0 and the bottom gate is applied with −V0, while the ITO layer is kept ground. In the case (I), the authors could achieve significant reflectance change and phase shifts, from − 212 ° to + 91 °, up to 303 ° phase shift at 1550 °nm. This maximum phase shift was obtained when the injected carrier in the entire ITO layer is unipolar.
Optical modulation in visible spectra was demonstrated by Thyagarajan et al., using ionic conduction change [40]. In this work, an Ag/Al2O3/ITO heterostructure, which allows Ag diffusion into Al2O3 and ITO layer, significantly alters dielectric characteristics, resulting in large phase shifts in the visible. Figure 1c shows the device structure with Ag ion diffusion. Like resistive switching devices, these ions diffuse repetitively in and out in the oxide layers depending on the applied bias. When a positive voltage is applied to the Ag electrode, the Ag ions migrate to the opposite inert electrode (ITO) and form Ag filaments in the Al2O3 layer. The growth of Ag nanoparticles in the ITO with increasing applied bias voltages alters optical reflectance and transmittance. Owing to the large device area up to 1 mm × 1 mm, RC time delay limits the modulation frequency up to 600 Hz with a 20% on–off ratio of normalized reflectance. One of the concerns of the ionic diffusion is the endurance of the device, which often leads to reduced functionality in time, as common as in the resistive random-access memory (ReRAM).
Anopchenko et al. demonstrated electrical tuning of reflectance and perfect absorption from multilayer stacks of ITO layers with a gradient of electron densities near ENZ condition as shown in Fig. 1d [53]. The gate bias up to 5 V was applied, similar to a conventional metal–oxide–semiconductor transistor, to make incident light absorbed by matching Berreman and ENZ modes to perfection absorption condition with an optimized thickness of ITO layer. For better absorption, the authors introduced a 5 nm-thick HfO2 layer, which leads to near unity absorption with less than 16 nm of oxide multilayers.
Howes et al. combined ENZ condition of thin dielectric resonators to realize Huygens-mode tunable metasurface, as shown in Fig. 2a [58]. The field enhancement in an ITO layer sandwiched by Si resonator and solid electrolyte can be largely modulated owing to charge accumulation near ENZ condition. The on-state transmittance was 70% in the SWIR and the modulation depth reaches 31%.
Semiconducting material based tunable metasurfaces. a ITO-Si nanoantenna heterojunction for tunable metasurface. Si nanoantennas support optical resonance and electrical bias simultaneously [58]. Reproduced with permission from ©2018, Optical Society of America. b GaAs-based semiconductor heterostructure for tunable quantum-confined Stark effect [50]. Reproduced with permission from ©2019, Springer Nature. c Bi2Se3-based tunable absorber [59]. Reproduced with permission from ©2015, American Chemical Society
Wu et al. introduced another type of electro-optic effect, based on the quantum-confined Stark effect in a III–V multiple quantum well, as shown in Fig. 2b [50]. The modulated dispersion coupled both with Mie resonance and with guided-mode allows 270% relative reflectance modulation depth and a phase shift up 70 °. At 914 nm, the tunable metasurface works as an electrically-driven beam steering device for remote sensing applications.
Two-dimensional materials, namely 2D materials, which comprises of atomically thin layers, have recently significant attention owing to optically tunable phase transitions [60], and intervalley transitions [61, 62]. Since the discovery of graphene's gate-tunable optical properties has been known, various types of 2D materials, especially transition metal dichalcogenides (TMDs) have been intensively studied. TMDs show strong excitonic properties and distinctive valley-dependent dispersion, which is originated pseudospin splitting induced by the magnetic field from Berry curvature in momentum space. In the lack of inversion symmetry, the carriers in the valleys can be selectively excited or detected by circularly-polarized light (chiral behavior) [63, 64], Because the carriers in 2D materials are strongly affected by electric gating field as well, the dielectric permittivities, especially chirality dependent Kerr permittivities also undergo strong change due to electrostatic gating. A recent study shows TMD materials can also serve optically active materials for photodetectors and photon emitters [63].
Even though 2D materials show tunability for GHz or THz frequencies by electro-optic perturbation or by nonlinear responses [65], there are few demonstrations for the spectral region of visible and SWIR light. 2D materials do not have enough free carrier concentrations to directly gate through dielectric space. Therefore, ionic liquid gating was introduced to increase carrier concentration and Fermi level. Liu et al. demonstrated ionic liquid gating across Bi2Se3 to show a great change in transmittance in the visible, as shown in Fig. 2c [59]. The transmittance varies over 60% in the visible with increasing gate voltage from − 1.5 V to 1.5 V. They figured out the dynamically tunable optical properties of Bi2Se3 is originated from optical bandgap due to the change in free-electron concentration (Burstein-Moss shift). They also performed similar experiments using MoSe2. However, MoSe2 rather show ambipolar tunability, which implies both electron and hole concentration affects optical property due to the applied bias polarity.
One of the problems in the classical ENZ model is the fixed or overestimated value of dielectric permittivity under external stimuli. If carrier concentration is redistributed, the real part of the dielectric permittivity \(\varepsilon_{TCO}\) should either be increased or decreased depending on the sign of the bias voltage. To account for this problem, a quantum mechanical model has been recently suggested. The quantum mechanical model is based on the density gradient current, with the expression for electron current given by
$$\overrightarrow {{J_{n} }} = - \mu k_{B} {\text{T}}\nabla n - \mu n\nabla \left( {\varPhi + \varLambda } \right)$$
Here, \(n\) is the electron or hole concentration, \({\text{T}}\) is the lattice temperature, \(\varPhi\) is the potential including both band edges and electrostatic potential. The last term is the quantum-mechanically-corrected potential given by
$$\varLambda = - \frac{{\gamma \hbar^{2} }}{6m}\frac{{\nabla^{2} \sqrt n }}{\sqrt n },$$
where γ is a fit factor, m is the carrier effective mass, and n represents electron or hole concentration as appropriate [66, 67]. The quantum model can be solved by numerical calculation or commercial software package to extract accumulated charge concentration.
The classical model and the quantum model use different boundary conditions, which leads to a significant difference in tuning range and charge concentration. Depending on the geometry, the quantum model and the classical model may produce two orders of magnitude absorption rate difference in dB/μm scale [67]. The quantum mechanical model tends to produce smaller values of accumulated charge concentration under the same external bias condition. The discrepancy in the classical Drude model and the experimental results lies in the fact that the dielectric constants of the material in the structure are actually altered due to nonequilibrium charge redistribution. Nevertheless, it is undeniable that the classical model is more convenient for the initial design of the tunable meta-atom structure.
One of the main advantages of electrically-tunable metasurfaces is that the shape of meta-atoms do not change.
Electromechanically-tunable metasurfaces
Tunable metamaterials can be made by dynamic structural changes by the external stimuli altering the size, the shape of, and distance between meta-atoms. These mechanical stimuli can be attained by a controllable actuation of the sub-wavelength structure. However, the realization of these structures for visible or SWIR spectra is expensive due to complex submicron fabrication and mechanical endurance has to be ensured to gain popularity like a pixelated micro-electro-mechanical system (MEMS) mirror arrays.
In a practical way, manufacturing arrays of resonators on stretchable elastomeric substrates offer a dynamic tuning of the optical metamaterials. The flexible elastomeric substrate can achieve large tuning ranges and relative ease of fabrication. Because stretching or shrinking elastomeric substrates has restoration and cycling problem, the geometric design has been carefully chosen to keep linear response and restoration under repetitive stimuli.
Ou et al. demonstrated electromechanically-driven metasurface working in SWIR spectra, as shown in Fig. 3a [28]. The metasurface is fabricated by focused ion beam milling on a 50-nm-thick silicon nitride membrane. When a bias ~ 3 V is applied, the strings in the metasurface are exposed to electrostatic force, which leads to string fields in the gaps between them. As a result, the transmittance could be modulated by 5% in SWIR. The modulation depth was changing as a function of the modulation frequency.
Electromechanically tunable metasurfaces. a Tunable microbeam array actuated by electrostatic force [28]. Reproduced with permission from ©2013, Springer Nature. b Strain-induced tunable grating on a PDMS substrate [68]. Reproduced with permission from ©2018, American Chemical Society. c Micro-electro-mechanical-system-based metalens doublet [30]. Reproduced with permission from ©2018, Springer Nature
Chen et al. demonstrated a tunable plasmonic lattice grating patterned on a flexible and stretchable PDMS substrate as shown in Fig. 3b [68]. The tunable plasmonic lattice shows the linear and reversible optical response of metasurface with little hysteresis. It shows the geometrical structure with a pair of tapered microrod at the end of plasmonic lattice grating, realizing tailorable strain amplification. The fabricated metasurface shows linear mechanical response under external strain varying 0% – 10%, while reflectance spectra modulation depth reaches 40% by external strain change from 1.6% to 3.5%. The surface plasmon resonance shifts approximately 80 nm in the visible at ~ 780 nm under the same strain variation.
Arbabi et al. demonstrated a focal-length tunable lens using a pair of metalens as shown in Fig. 3c [30]. The metalens is based on the high-contrast dielectric transmit arrays. One metalens is fabricated on a fixed glass substrate whereas the other metalens is fabricated on a movable SiN membrane. The doublet shifts the focal length up to 30 μm around 800 nm wavelength, only by moving one metalens of 1 μm. Higher bias voltage shifts the focal length up to 80 μm.
Reconfigurable metasurfaces based on optical nonlinearity
The properties of light in metasurfaces can be controlled dynamically by means of optical pumping. An optical tuning can be achieved by inserting an active layer that responds to the pumping light in a metallic or dielectric nanocavity.
Zhu et al. realized plasmon-induced nonlinear tunable transparency using gold meta-atom on top of a thin polycrystalline ITO layer, as shown in Fig.4a [69]. The Kerr nonlinear index of refraction is given by
$${\text{n}} = n_{0} + n_{2} I = n_{0} + \frac{{3{\text{Re}}\left( {\chi^{\left( 3 \right)} } \right)}}{{4{\text{c}}\varepsilon_{0} n_{0}^{2} }}I_{0},$$
where \(n_{0}\) and \(n_{2}\) are linear anthe d nonlinear refractive index of ITO, \({\text{Re}}\left( {\chi^{\left( 3 \right)} } \right)\) is the real part of the third-order nonlinear susceptibility of the ITO, \(\varepsilon_{0}\) is the permittivity of the vacuum, and \({\text{c}}\) is the light velocity in the vacuum. The plasmonic resonances of the meta-atom lead to superradiance and subradiance depending on the pump laser intensity. This effect leads to an optical transparency window shift to the short-wavelength direction in SWIR.
Nonlinear-optically tunable metasurfaces. a Plasmonic-antenna assisted tunable meta-atom by Kerr nonlinearity [69]. Reproduced with permission from ©2013, Springer Nature. b Optical pumping leads to Fabry-Pérot resonance shift owing to the large nonlinearity of a nanocavity [70]. Reproduced with permission from ©2018, American Chemical Society. c Polarization converter based on molecular transformation [71]. Reproduced with permission from ©2017, Springer Nature. d Fifth harmonic generation from a hybrid oxide heterostructure near ENZ condition [72]. Reproduced with permission from ©2019, Springer Nature
Kim et al. demonstrated optically tunable metasurface made of a metal–insulator-metal (Ag-Al2O3-Ag) nanocavity with a 70 nm-thick Ga:ZnO layer as an active layer, as shown in Fig. 4b [70]. Harnessed by fast switching mechanism of optical pumping, sub-picosecond switching modulation speed and 80% of modulation depth was obtained by sub 10 mJ/cm2 pumping fluence level. Under degenerate optical pumping, a 15 nm transmittance redshift of Fabry–Pérot resonance shift was obtained in the SWIR spectrum excited near the ENZ wavelength.
Ren et al. demonstrated a polarization-state-switching metasurface made of a plasmonic structure with isomeric ethyl-red polymers, as shown in Fig. 4c [71]. The structure consists of a 100-nm-thick gold film on a 500-μm-thick fused quartz substrate (SiO2) and an ~ 300 nm ethyl-red polymer layer thereon. The gold film forms a periodic array of L-shaped slots to construct metasurfaces. The polarization tuning is achieved by irradiating the green light (532 nm) that leads to change in the isomeric state of ethyl-red, that is, from trans-state to cis state which can efficiently modify the refractive index of the polymer layer. Hence the coupling between the resonant plasmonic modes and isomeric state of ethyl-red leads to polarization control of metasurfaces. When the control light is present, the refractive index decreases as ethyl-red changes to cis state, resulting in a blue shift of polarization azimuth rotation angle ϕ and ellipticity angle χ. As a result, 80% of modulation depth at 6 Hz. The switching speed is inherently limited by the recovery time from the isomerization process.
Yang et al. demonstrated visible high-harmonic generation from In-doped CdO as shown in Fig. 4d, using epsilon-near-zero condition [72]. Even though the ENZ condition is satisfied at 2250 nm, the strong field enhancement at the boundary between In-doped CdO and MgO boosts the harmonic generation, resulting in visible light generation in the fifth order. The authors attributed the origin of harmonics generation to the photo-induced electronic temperature elevation in the conduction band by comparing a relaxation model with the experimental observation of decay time.
Thermally tunable metasurfaces based on phase change materials
Phase-change materials show near-unity refractive index change as the structural or electronic phase changes across a critical temperature. Since the application of digital versatile disc (DVD), germanium-antimony-tellurium (GST) with a crystallization temperature Tc of ~ 433 K (160 °C) and a melting temperature of ~ 873 K (600 °C) has long been used as a common phase change material. For the SWIR spectra application, vanadium dioxide (VO2) shows a hysteretic structural phase transition from the monoclinic phase to the tetragonal phase when the temperature rises above from 340 K (67 °C). Because the temperature could be risen by electrical current pulse injection, these two materials have been intensively investigated for tunable metasurface applications.
Gholipour et al. demonstrated all-optical bidirectional metasurface based on the GST as shown in Fig. 5a [73]. A 15 nm-thick GST layer is sandwiched between SiO2 and ZnS/SiO2 layers. Plasmonic thin trenches made of 50-nm-thick Au layer support plasmonic resonance and enhanced photoabsorption for a temperature change of GST. As the phase of the GST layer changes from the crystalline phase to the amorphous phase, the transmission at SWIR spectra is enhanced from 20% to 40%. The temperature control of the GST layer is provided by a temporally-modulated light to morph phases between amorphous one and crystalline one.
Phase-change-material-based tunable metasurfaces. a Optically tunable thermal metamaterial based on GST [73]. Reproduced with permission from ©2013, WILEY–VCH. b Meta-atom made of a bow-tie-shaped unit cell that has a small amount of the VO2 [74]. Reproduced with permission from ©2017, American Chemical Society. c Metal-dielectric heterostructure for hyperbolic dispersion under thermal modulation [75]. Reproduced with permission from ©2016, American Chemical Society. d UV light-tunable meta-atom made of GST sandwiched by ZnS/SiO2 layers [76]. Reproduced with permission from ©2019, American Chemical Society
Zhu et al. demonstrated electrically-controllable metasurface based on the phase change of VO2 as shown in Fig. 5b [74]. VO2 is laterally sandwiched by plasmonically structured Au electrodes and the modulated voltage pulse train changes the phase of VO2 between amorphous and crystalline phases. As a result, modulation depth up to 33% at the SWIR spectrum could be obtained with sub 10 ms response time.
Chen et al. demonstrated temperature-dependent tunable metamaterial based on vanadium dioxide(VO2)-titanium dioxide(TiO2) multilayer as shown in Fig. 5c [75]. The VO2–TiO2 multilayer shows dispersion relation change from the elliptic form to hyperbolic form as the temperature increases over the critical temperature of VO2 around 325 K. As a result, the real part of the dielectric permittivity of the composite metamaterial measured by spectroscopic ellipsometry changes at the SWIR spectrum.
Gholipour et al. demonstrated an optically-tunable metasurface, based on a patterned GST heterostrutcture with ZnS/SiO2 (Fig. 5d), showing tuning capability for ultraviolet (UV) and visible spectra [76]. GST shows a significantly low refractive index at the UV spectrum, as low as 1.07 at 245 nm in the crystalline phase. Also, an amorphous composite of zinc sulfide and silica with a 1:9 atomic % ratio shows almost constant refractive index 2.4 in the UV and visible. By optical pumping, the phase changes of the GST layer leads to reflectance modulation depth over 10%.
Conclusion and outlook
We have briefly summarized recent advances in tunable metasurfaces in the visible and SWIR, focusing particularly on available tuning mechanisms. Despite all the advancements made in the past few years, we have witnessed there are quite a number of challenges to overcome. One of the challenges in the material point of view is that the natural materials in the visible do not have enough refractive indices. This limits the minimum thickness of the materials to allow enough phase shifts originated from the electric field, electrical power to change phase or thermal or optical power. From fundamental point-of-view, tunable metasurfaces also have similar challenges as passive metasurfaces such as polarization-control, chromatic aberration, large-deflection angle, high efficiency, and the number of available degree of freedoms to solve these issues in single surface or a specific volume [8].
These problems are also related to the problems in unresolved practical issues to allow high-volume, low-cost metasurface devices. Each mechanism has clear advantages and disadvantages to meet the needs of highly pixelated tunable optical and photonic devices in visible and SWIR spectra. For thorough control of the tunable meta-atoms, individual excess to meta-atom and the required integrated biasing circuit, and simultaneous and locked operation of multiple meta-atoms are required. Also, the endurance of meta-atoms under repetitive biasing or optical stimuli and sufficient modulation depth and bandwidths to meet the full 2π phase shift in the target spectra should be overcome.
Despite the difficulties in realizing practical, real-life tunable metasurface devices, we witness continuous efforts to overcome the challenges. We recently noticed different class of tuning mechanism, based on of time-domain-control of metasurfaces, different from nonlinear-optical tunable metasurfaces [77, 78]. Nonconventional metasurfaces, based on non-Hermitian coupling, topological, non-local, and quantum–mechanical interactions are also actively studied [79]. Not to mention the importance of the new class of tuning mechanisms, we expect new tuning mechanisms could revolutionize optics and photonics beyond conventional diffractive optics and electromagnetics in the tunable metamaterials for the visible or SWIR spectra. Even though developing highly integrated tunable optoelectronic or photonic devices and systems with small form factor is a formidable task, we believe the advancement of technology will be able to bring us a new powerful tool for the development of full-random-accessible meta-atoms in a metasurface platform and to find new entrepreneur applications with tunable optical metamaterials and metasurfaces.
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
SWIR:
short-wavelength infrared
LIDAR:
light detection and ranging
ENZ:
epsilon-near-zero
ITO:
indium tin oxide
AZO:
aluminum-zinc oxide
ALD:
atomic-layer-deposition
MOS:
metal/insulating oxide/semiconductor
HAOL:
Al2O3/HfO2 nanolaminates
TMD:
transition metal dichalcogenide
MEMS:
micro-electro-mechanical system
digital versatile disc
germanium-antimony-tellurium
UV:
R.A. Shelby, D.R. Smith, S. Schultz, Science 292, 77 (2001)
J.B. Pendry, Phys. Rev. Lett. 85, 3966 (2000)
D. Schurig, J.J. Mock, B.J. Justice, S.A. Cummer, J.B. Pendry, A.F. Starr, D.R. Smith, Science 314, 977 (2006)
N.I. Landy, S. Sajuyigbe, J.J. Mock, D.R. Smith, W.J. Padilla, Phys. Rev. Lett. 100, 207402 (2008)
L. Yin, V.K. Vlasko-Vlasov, J. Pearson, J.M. Hiller, J. Hua, U. Welp, D.E. Brown, C.W. Kimball, Nano. Lett. 5, 1399 (2005)
A. Poddubny, I. Iorsh, P. Belov, Y. Kivshar, Nat. Photonics. 7, 958 (2013)
P. Shekhar, J. Atkinson, Z. Jacob, Nano. Converg. 1, 14 (2014)
A. Alu, V. Shalaev, M. Loncar, V.J. Sorger, Nanophotonics 7, 949 (2018)
M. Khorasaninejad, F. Aieta, P. Kanhaiya, M.A. Kats, P. Genevet, D. Rousso, F. Capasso, Nano. Lett. 15, 5358 (2015)
M. Khorasaninejad, W.T. Chen, R.C. Devlin, J. Oh, A.Y. Zhu, F. Capasso, Science 352, 1190 (2016)
N. Yu, F. Capasso, Nat. Mater. 13, 139 (2014)
M. Khorasaninejad, F. Capasso, Science 358, eaam8100 (2017)
A. Arbabi, Y. Horie, A.J. Ball, M. Bagheri, A. Faraon, Nat. Commun. 6, 7069 (2015)
A. Arbabi, R.M. Briggs, Y. Horie, M. Bagheri, A. Faraon, Opt. Express. 23, 33310 (2015)
A. Arbabi, Y. Horie, M. Bagheri, A. Faraon, Nat. Nanotechnol. 10, 937 (2015)
D. Lin, P. Fan, E. Hasman, M.L. Brongersma, Science 345, 298 (2014)
W.T. Chen, M. Khorasaninejad, A.Y. Zhu, J. Oh, R.C. Devlin, A. Zaidi, F. Capasso, Light. Sci. App. 6, e16259 (2017)
F. Ding, Z. Wang, S. He, V.M. Shalaev, A.V. Kildishev, ACS Nano. 9, 4111 (2015)
J. Li, S. Chen, H. Yang, J. Li, P. Yu, H. Cheng, C. Gu, H.T. Chen, J. Tian, Adv. Funct. Mater. 25, 704 (2015)
X. Ni, A.V. Kildishev, V.M. Shalaev, Nat. Commun. 4, 2807 (2013)
L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, S. Zhang, Nat. Commun. 4, 2808 (2013)
G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, S. Zhang, Nat. Nanotechnol. 10, 308 (2015)
J. Guo, Y. Tu, L. Yang, R. Zhang, L. Wang, B. Wang, Sci. Rep. 7, 1 (2017)
P.P. Iyer, M. Pendharkar, J.A. Schuller, Adv. Opt. Mater. 4, 1582 (2016)
V.E. Babicheva, A. Boltasseva, A.V. Lavrinenko, Nanophotonics 4, 165 (2015)
Y. Yao, R. Shankar, M.A. Kats, Y. Song, J. Kong, M. Loncar, F. Capasso, Nano. Lett. 14, 6526 (2014)
A.Q. Liu, W.M. Zhu, D.P. Tsai, N.I. Zheludev, J. Opt. 14, 114009 (2012)
J.Y. Ou, E. Plum, J. Zhang, N.I. Zheludev, Nat. Nanotechnol. 8, 252 (2013)
N.I. Zheludev, E. Plum, Nat. Nanotechnol. 11, 16 (2016)
E. Arbabi, A. Arbabi, S.M. Kamali, Y. Horie, M.S. Faraji-Dana, A. Faraon, Nat. Commun. 9, 812 (2018)
M. Abb, P. Albella, J. Aizpurua, O.L. Muskens, Nano. Lett. 11, 2457 (2011)
Q. Wang, E.T.F. Rogers, B. Gholipour, C.M. Wang, G. Yuan, J. Teng, N.I. Zheludev, Nat. Photonics. 10, 60 (2016)
K.J.A. Ooi, P.C. Leong, L.K. Ang, D.T.H. Tan, Sci. Rep. 7, 1 (2017)
I.B. Vendik, O.G. Vendik, M.A. Odit, D.V. Kholodnyak, S.P. Zubko, M.F. Sitnikova, P.A. Turalchuk, K.N. Zemlyakov, I.V. Munina, D.S. Kozlov, V.M. Turgaliev, A.B. Ustinov, Y. Park, J. Kihm, C.W. Lee, I.E.E.E. Trans, Terahertz. Sci. Technol. 2, 538 (2012)
A. Nemati, Q. Wang, M. Hong, J. Teng, Opto.Electronic. Adv. 1, 18000901 (2018)
T. Cui, B. Bai, H.B. Sun, Adv. Funct. Mater. 29, 1806692 (2019)
Q. He, S. Sun, L. Zhou, Research 2019, 1 (2019)
A.M. Shaltout, V.M. Shalaev, M.L. Brongersma, Science 364, 3100 (2019)
S.M. Kamali, E. Arbabi, A. Arbabi, A. Faraon, Nanophotonics 7, 1041 (2018)
K. Thyagarajan, R. Sokhoyan, L. Zornberg, H.A. Atwater, Adv. Mater. 29, 1 (2017)
I. the near future we expect individual meta-atom in three-dimensional metamaterial can be biased like the way how to bias unit cell in a memory Structure, (n.d.)
M. Akbulut, J. Davila-Rodriguez, I. Ozdur, F. Quinlan, S. Ozharar, N. Hoghooghi, P.J. Delfyett, Opt. Express. 19, 16851 (2011)
N. Andriolli, T. Cassese, M. Chiesa, C. De Dios, G. Contestabile, J. Light. Technol. 36, 5685 (2018)
W. Dong, H. Liu, J.K. Behera, L. Lu, R.J.H. Ng, K.V. Sreekanth, X. Zhou, J.K.W. Yang, R.E. Simpson, Adv. Funct. Mater. 29, 1806181 (2019)
E. Feigenbaum, K. Diest, H.A. Atwater, Nano. Lett. 10, 2111 (2010)
A. Alù, M.G. Silveirinha, A. Salandrino, N. Engheta, Phys. Rev. B 75, 155410 (2007)
X. Niu, X. Hu, S. Chu, Q. Gong, Adv. Opt. Mater. 6, 1701292 (2018)
L. Lu, R.E. Simpson, S.K. Valiyaveedu, J Opt. 20, 103001 (2018)
A. Forouzmand, H. Mosallaei, Opt. Express. 26, 17948 (2018)
P.C. Wu, R.A. Pala, G.K. Shirmanesh, W.H. Cheng, R. Sokhoyan, M. Grajower, M.Z. Alam, D. Lee, H.A. Atwater, Nat. Commun. 10, 3654 (2019)
Y.W. Huang, H.W.H. Lee, R. Sokhoyan, R.A. Pala, K. Thyagarajan, S. Han, D.P. Tsai, H.A. Atwater, Nano. Lett. 16, 5319 (2016)
G.K. Shirmanesh, R. Sokhoyan, R.A. Pala, H.A. Atwater, Nano. Lett. 18, 2957 (2018)
A. Anopchenko, L. Tao, C. Arndt, H.W.H. Lee, ACS. Photonics. 5, 2631 (2018)
P.D. Ye, B. Yang, K.K. Ng, J. Bude, G.D. Wilk, S. Halder, J.C.M. Hwang, Appl. Phys. Lett. 86, 1 (2005)
J. Park, J.H. Kang, X. Liu, M.L. Brongersma, Sci. Rep. 5, 15754 (2015)
H.W. Lee, G. Papadakis, S.P. Burgos, K. Chander, A. Kriesch, R. Pala, U. Peschel, H.A. Atwater, Nano. Lett. 14, 6463 (2014)
K. Shi, R.R. Haque, B. Zhao, R. Zhao, Z. Lu, Opt. Lett. 39, 4978 (2014)
A. Howes, W. Wang, I. Kravchenko, J. Valentine, Optica 5, 787 (2018)
Y. Liu, K. Tom, X. Wang, C. Huang, H. Yuan, H. Ding, C. Ko, J. Suh, L. Pan, K.A. Persson, J. Yao, Nano. Lett. 16, 488 (2016)
H. Yang, S.W. Kim, M. Chhowalla, Y.H. Lee, Nat. Phys. 13, 931 (2017)
S. Manzeli, D. Ovchinnikov, D. Pasquier, O.V. Yazyev, A. Kis, Nat. Rev. Mater. 2, 17033 (2017)
J.R. Schaibley, H. Yu, G. Clark, P. Rivera, J.S. Ross, K.L. Seyler, W. Yao, X. Xu, Nat. Rev. Mater. 1, 16055 (2016)
K.F. Mak, J. Shan, Nat. Photonics. 10, 216 (2016)
K.F. Mak, D. Xiao, J. Shan, Nat. Photonics. 12, 451 (2018)
S. Yu, X. Wu, Y. Wang, X. Guo, L. Tong, Adv. Mater. 29, 1606128 (2017)
A. Wettstein, A. Schenk, W. Fichtner, Electron. Devices. 48, 279 (2001)
Q. Gao, E. Li, A.X. Wang, Opt. Mater. Express. 8, 2850 (2018)
W. Chen, W. Liu, Y. Jiang, M. Zhang, N. Song, N.J. Greybush, J. Guo, A.K. Estep, K.T. Turner, R. Agarwal, C.R. Kagan, ACS Nano. (2018). https://doi.org/10.1021/acsnano.8b04889
Y. Zhu, X. Hu, Y. Fu, H. Yang, Q. Gong, Sci. Rep. 3, 2338 (2013)
J. Kim, E.G. Carnemolla, C. DeVault, A.M. Shaltout, D. Faccio, V.M. Shalaev, A.V. Kildishev, M. Ferrera, A. Boltasseva, Nano. Lett. 18, 740 (2018)
M.X. Ren, W. Wu, W. Cai, B. Pi, X.Z. Zhang, J.J. Xu, Light. Sci. Appl. 6, e16254 (2017)
Y. Yang, J. Lu, A. Manjavacas, T.S. Luk, H. Liu, K. Kelley, J.-P. Maria, E.L. Runnerstrom, M.B. Sinclair, S. Ghimire, I. Brener, Nat. Phys. 15, 1022–1026 (2019)
B. Gholipour, J. Zhang, K.F. MacDonald, D.W. Hewak, N.I. Zheludev, Adv. Maters. 25, 3050 (2013)
Z. Zhu, P.G. Evans, R.F. Haglund, J.G. Valentine, Nano Lett 17, 4881 (2017)
Z. Chen, X. Wang, Y. Qi, S. Yang, J.A.N.T. Soares, B.A. Apgar, R. Gao, R. Xu, Y. Lee, X. Zhang, J. Yao, L.W. Martin, ACS. Nano. 10, 10237 (2016)
B. Gholipour, D. Piccinotti, A. Karvounis, K.F. Macdonald, N.I. Zheludev, Nano. Lett. 19, 1643 (2019)
A. Shaltout, A. Kildishev, V. Shalaev, Opt. Mater. Express. 5, 2459 (2015)
W. Divitt, C. Zhu, H.J. Zhang, A. Lezec, Agrawal. Science 364, 890 (2019)
X. Ren, P.K. Jha, Y. Wang, X. Zhang, Nanophotonics 7, 1233 (2018)
This research was supported by the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (MSIT) through Grant No. NRF-2017R1A2B4007219. H. J. Choi is grateful to the financial support from the NRF and the Center for Women In Science, Engineering and Technology Grant funded by the MSIT through Grant No. WISET-2019-272. HJ is supported by University of Malaya Impact Oriented Interdisciplinary Research Grant (IIRG001-19FNW).
Institute of Advanced Optics and Photonics, Department of Applied Optics, Hanbat National University, Daejeon, 34158, Korea
Chang-Won Lee & Hee Jin Choi
Department of Physics, Faculty of Science, University of Malaya, 50603, Kuala Lumpur, Malaysia
Heejeong Jeong
Chang-Won Lee
Hee Jin Choi
CWL, HJC, and HJ wrote the manuscript. All authors read and approved the final manuscript.
Correspondence to Chang-Won Lee or Heejeong Jeong.
Lee, CW., Choi, H.J. & Jeong, H. Tunable metasurfaces for visible and SWIR applications. Nano Convergence 7, 3 (2020). https://doi.org/10.1186/s40580-019-0213-2
Metasurface
Metamaterial
SWIR | CommonCrawl |
Exploring use of unsupervised clustering to associate signaling profiles of GPCR ligands to clinical response
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The Cancer Surfaceome Atlas integrates genomic, functional and drug response data to identify actionable targets
Zhongyi Hu, Jiao Yuan, … Lin Zhang
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Marina A. Guvakova & Serguei Sokol
The landscape of receptor-mediated precision cancer combination therapy via a single-cell perspective
Saba Ahmadi, Pattara Sukprasert, … Eytan Ruppin
A statistical framework for high-content phenotypic profiling using cellular feature distributions
Yanthe E. Pearson, Stephan Kremb, … Kristin C. Gunsalus
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Adam Byron, Stephan Bernhardt, … Leanne de Koning
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Min Young Lee, Taek-Kyun Kim, … Kai Wang
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Marcin Pilarczyk, Mehdi Fazel-Najafabadi, … Mario Medvedovic
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Ciara Tierney, Despina Bazou, … Peter O'Gorman
Besma Benredjem1,2 na1,
Jonathan Gallion3 na1,
Dennis Pelletier4,
Paul Dallaire1,2,
Johanie Charbonneau2,
Darren Cawkill4 nAff7,
Karim Nagi ORCID: orcid.org/0000-0002-4700-55685,
Mark Gosink4,
Viktoryia Lukasheva6,
Stephen Jenkinson4 nAff8,
Yong Ren4 nAff9,
Christopher Somps4,
Brigitte Murat6,
Emma Van Der Westhuizen ORCID: orcid.org/0000-0001-9165-85266 nAff10,
Christian Le Gouill6,
Olivier Lichtarge3,
Anne Schmidt4,
Michel Bouvier ORCID: orcid.org/0000-0003-1128-01006 &
Graciela Pineyro1,2
Nature Communications volume 10, Article number: 4075 (2019) Cite this article
Computational biology and bioinformatics
An Author Correction to this article was published on 16 April 2020
This article has been updated
Signaling diversity of G protein-coupled (GPCR) ligands provides novel opportunities to develop more effective, better-tolerated therapeutics. Taking advantage of these opportunities requires identifying which effectors should be specifically activated or avoided so as to promote desired clinical responses and avoid side effects. However, identifying signaling profiles that support desired clinical outcomes remains challenging. This study describes signaling diversity of mu opioid receptor (MOR) ligands in terms of logistic and operational parameters for ten different in vitro readouts. It then uses unsupervised clustering of curve parameters to: classify MOR ligands according to similarities in type and magnitude of response, associate resulting ligand categories with frequency of undesired events reported to the pharmacovigilance program of the Food and Drug Administration and associate signals to side effects. The ability of the classification method to associate specific in vitro signaling profiles to clinically relevant responses was corroborated using β2-adrenergic receptor ligands.
G protein-coupled receptors (GPCRs) modulate practically every aspect of human physiology and are the target of ~30% of FDA-approved medicines1. When activated these receptors undergo conformational changes2,3 that determine the type and the magnitude of signals triggered within the cell4. This signaling configuration supports ligand-specific activation of the different pathways4, and provides a theoretical opportunity for directing pharmacological stimulus toward pathways that underlie desired therapeutic responses and away from those responsible for undesired side effects5,6. However, in spite of this promise5 development of therapeutic biased ligands has yet to translate into more effective and/or better-tolerated medicines7,8,9,10,11.
Different challenges have hindered the development of clinically effective biased ligands. Except for limited examples12,13,14, we ignore the signals underlying desired and undesired clinical responses of GPCR ligands. To access this knowledge and apply it to drug discovery, it is necessary to identify signaling preferences and to associate distinct signaling profiles to desired/undesired clinical outcomes15,16. The way in which signaling preferences are currently identified in drug discovery efforts involves calculation of "bias factors", an approach that uses consolidated (Log(τ/KA)) transduction coefficients to measure the extent to which a ligand preferentially activates one pathway over another17,18,19. This type of evaluation compares signals in a pairwise manner, a dichotomous approach that provides a fragmented view of a ligand's signaling preferences across the multiplicity of pathways. Perhaps more troubling for the use of "bias factors" as descriptors of potential clinical responses is the fact that their estimated magnitudes vary with the calculation method used to produce them15. Finally, the same "bias factor" may describe drugs with very different efficacies at the pathways of interest20 further questioning the value of these measures as predictors of desired/undesired in vivo responses. In an effort to circumvent at least some of these limitations, we sought an alternative way to identify signaling preferences and classify GPCR ligands.
One of the most studied examples of how biased signaling may support development of more effective and/or better-tolerated therapeutic agents is that of opioid analgesics. Preclinical models have indicated that β-arrestin2 (βarr2) knockout mitigates constipation and respiratory depression induced by morphine21, pointing to the possibility that mu opioid receptor (MOR) agonists that preferentially activate G protein signaling over βarr2 recruitment could induce less of these side effects12,13,14. Here, we use this prototypical example to establish that clustering MOR ligands according to similarities in pharmacodynamic parameters for multiple responses, captures their signaling differences and preferences. We show that ligands with similar G protein/βarr responses cluster together, and provide evidence that ligands within different categories display distinct frequencies of gastrointestinal and respiratory events reported to the FDA pharmacovigilance program. Moreover, when ligands are clustered according to either G protein or βarr responses both signals directly associate to side effects. The practical value of the classification method proposed is further illustrated by the fact that ligand categories defined by similarity of G protein responses at β2-adrenergic receptor (β2AR) correlate with sympatholytic CV events and bronchoconstriction.
Clustering ligands according to pharmacodynamic similarities
We sought a method to identify and group together ligands with overall similarities in a multiplicity of signaling pathways while simultaneously discerning those with overall differences in features, such as efficacy, potency, and signaling preferences. To test the ability of the method to accomplish this task independent of idiosyncrasies in experimental data sets, we generated a set of 320 virtual compounds as variations of 16 prototypical profiles characterized by a combination of pharmacodynamic features across six different readouts (see the Methods section). Profiles are shown in Supplementary Fig. 1. Criteria to classify ligands according to pharmacodynamic similarities were empirically established by generating matrices, in which each ligand was represented by individual logistic (Emax, pEC50) or operational (Log(τ), pKA, Log(τ/KA)) parameters, as well as their combinations. Matrices were then subject to nonnegative matrix factorization (NNMF)22 to identify essential, nonredundant features, and k-means clustering was subsequently used to classify ligands according to these features23. Iterations were used to incorporate the error associated with each mean parameter value, ensuring its propagation throughout the clustering procedure (see the Methods and Supplementary Fig. 2). The result of this procedure was a ligand × ligand similarity matrix that quantifies how frequently any two compounds clustered together over the iterations. Final similarity matrices were submitted to hierarchical clustering to establish row and column ordering according to similarity, and visualized as heatmaps. Figure 1 shows heatmaps for the progressive associations of parameters leading to identification of Log(τ), Emax, and Log(τ/KA) as a combination faithfully recreating the 16 profiles initially defined. Operational efficacy (τ) by itself was not sufficient to fully distinguish ligands with different profiles (Fig. 1a, d). Introducing measures of signaling capacity (Emax) improved the classification (Fig. 1, e), but discrimination was not optimal unless values for transduction coefficient Log(τ/KA) were also included (Fig. 1c, f). Unlike Log(τ) and Emax values, transduction coefficients incorporate potency information24,25 and thus provide a different dimension on which ligands can be distinguished. In effect, Log(τ/KA) coefficients were correlated with logistic potency estimates (pEC50) (Supplementary Fig. 3a), so the classification afforded by Log(τ)-Emax-pEC50 (Supplementary Fig. 3b, c) was quite similar to the one produced with Log(τ)-Emax-Log(τ/KA). Profiles recreated by classifying ligands according to Log(τ)-Emax-Log(τ/KA), are shown in Supplementary Fig. 4, revealing a minimal number of displaced compounds.
Ligands are classified according to similarities in multidimensional signaling profiles using pharmacodynamic parameters. 320 virtual compounds were defined by logistic and operational parameters to represent 16 distinct signaling profiles describing response at six different readouts. Indicated parameters were then subject to NNMF followed by k-means clustering, to produce corresponding similarity matrices that were represented as heatmaps and hierarchical clustering trees (using the R heatmap function with the metric:ward.D2) (a–c) or t-SNE plots (using the R package tsne with default parameters) (d–f). Ligands were color-coded to highlight how different combinations of parameters differentiated compounds originating from the different profiles originally defined
We then applied the proposed classification strategy on experimental data. Multidimensional signaling profiles for this analysis were generated using ten different BRET-based biosensors that monitor βarr recruitment and G protein signaling. G-protein signaling was monitored through conformational rearrangements within Gαi1-2/oAβ1γ2 heterotrimers26, at the interface of Gβ1γ2/Kir3 channel subunits27, or as changes in cAMP levels28,29. βarr recruitment to the receptor was assessed for βarr1, βarr2, and βarr2 in presence of GRK2, GRK5, or GRK6 to account for possible impact of expression differences between the screening system (HEK 293) and target neurons where GRK levels are higher30. Net BRET values obtained in cells co-expressing human MORs (hMOR) and different biosensors in presence/absence of the endogenous ligand Met-Enkephalin (Met-ENK) are shown for reference in Supplementary Fig. 5. Concentration response curves (CRCs) for 10 known opioids (Fig. 2), and 15 novel compounds (Supplementary Fig. 6) identified in the context of a screening campaign at Pfizer Inc.31, were then generated and analyzed with the logistic equation and the operational32 model. Each ligand was phenotypically described by corresponding τ, EMAX, and Log(τ/KA) values (± SEM) derived from 5 G protein- and 5 βarr-related responses (Supplementary Data 1). These were analyzed with NNMF/k-means clustering as above and represented as heat maps for ligands (Fig. 3a) and for parameters (Fig. 3b).
βarr recruitment and G-protein responses generated by opioid ligands. Responses for prescription opioids and known hMOR ligands were monitored using BRET-based biosensors. The results correspond to mean± SEM of at least three independent experiments, normalized to the maximal effect of Met-ENK, which was tested in all experimental runs (n = 16–29). Curves were fit with the operational model and the logistic equation (the results from the logistic fit are shown). a βarr1 recruitment, (b) βarr2 recruitment, (c) βarr2 recruitment in presence of GRK2, (d) βarr2 recruitment in presence of GRK5, (e) βarr2 recruitment in presence of GRK6, (f) cAMP, (g) Gαi1 activation, (h) Gαi2 activation, (i) GαoA activation, and (j) Kir3.1/3.2 activation. Net BRET values for Met-ENK are shown in Supplementary Fig. 5 and dose–response curves for all novel compounds appear in Supplementary Fig. 6. Operational and logistic parameters provided in Supplementary Data 1. Source data are provided as a source data file
Assignment of hMOR ligands into clusters is primarily driven by βarr responses. Ligand (a) and parameter (b) similarity heatmaps for the complete hMOR data set. Yellow and blue, respectively indicate ligands/parameters that never or always cluster together. Distribution of parameters describing ligands within clusters shown in (a) was compared to their distribution in the whole population using a two-sample Kolmogorov–Smirnov test. Resulting p-values were plotted according to clusters shown in (b) (c) or to parameter type (d), mean ± SD are also shown. Red line: p = 0.05. Similarity matrices corresponding to partial data sets for βarr- or G-protein-mediated responses were compared with the complete, reference hMOR data set. Filled bars: proportion of ligands changing clusters when comparing actual βarr and G-protein data sets to the reference; empty bars: proportions observed by comparing simulations of random clustering to the reference data set. ***p < 0.001; -zscore βarr: −5.375; z-score G protein: −6.092. #p < 0.05; z-score difference: −2.22 (e). Similarity matrices generated for indicated partial responses were compared with the hMOR reference matrix. The results for actual data matrices are shown while results for random simulations were omitted. **p < 0.01; ***p < 0.001 comparing partial data sets to their randomized controls; z-scores for comparisons between actual and randomized data: βarr2-GRK2: −12.724; βarr2-GRK6: −10.583; βarr2-GRK2/GRK6:− 8.835; Gαi2: −7.315; cAMP: 6.297; Gαi2- cAMP: −2.351; four assays −8.541; βarr2-GKR2/6/Gαi2: −7.391. ###p < 0.001; z-score difference: βarr2-GRK2/6 vs. Gαi2/cAMP: −3.308; z-score difference: Gαi2 vs βarr2-GRK2/6-Gαi2: 3.754 (f). Source data provided in Supplementary Data 1 and as a source data file
Ligands within the same cluster share quality and magnitude of response
An essential pharmacological question is to identify the pathways and pharmacodynamic properties primarily responsible for ligand clustering. The overall resemblance among relative magnitudes of operational and logistic parameters from different functional readouts is shown in Fig. 3b delineating three clusters of parameters. To further characterize differences among ligand categories, we investigated whether the magnitude of parameters describing ligand response in each assay was different across the three clusters of ligands. To do so, we used a Kolmogorov–Smirnoff test to compare the distribution of parameter values in each cluster to that of the overall population (detailed in Supplementary Fig. 7). Only certain parameters in each cluster contributed to ligand discrimination, and they did so to different extents (Fig. 3c). Those in cluster A had the most weight, as 29.9% of comparisons identified at least one distribution of parameters significantly different from that of the whole population. Overall, 14.9% of comparisons in cluster B and 3.5% in cluster C also significantly contributed to ligand discrimination. Alternatively, ordering parameters by type (Fig. 3d) revealed that efficacy-related parameters (Emax and τ) had the most weight in the classification (52.0% and 36.0% of the comparisons, respectively, identified distributions different from the whole population) while Log(τ/KA) played a smaller role separating primary compound clusters (11.0%). Finally, Supplementary Table 1 shows assay parameters significantly contributing to ligand clustering. The diversity of signals determining cluster assignment distinguishes this multidimensional classification from dichotomous comparisons underlying bias magnitudes.
Importantly, despite independence from bias magnitudes, the proposed classification strategy still allows to evaluate relative contributions of βarr and G protein signaling to ligand assignment to clusters. To access this information, drugs were re-clustered using subsets of the data corresponding exclusively to G protein (Supplementary Fig. 8a) or to βarr (Supplementary Fig. 8b) assays, and the resulting similarity matrix produced with each partial data set was compared with the matrix generated using the complete set of values. Differences between matrices were quantified as described in the Methods section and Supplementary Fig. 9, and expressed as the proportion of changes in ligand distances that were compatible with a switch in clusters between the two compared matrices. Clusters generated with βarr data differed by only 11.5% from clusters produced with the complete data set (Fig. 3e), underscoring the similarity of drug classes defined by βarr signaling patterns and complete signaling profiles. The partial data set for G protein responses differed by 27.2% (Fig. 3e) from the complete ligand similarity matrix. Thus, although clusters generated with βarr or with G protein data sets resembled clusters produced with the complete data set more than did clusters generated with the corresponding randomized values, initial ligand clustering was more faithfully recreated by βarr responses (Fig. 3e), indicating that that this signal was the one primarily driving classification in the complete matrix. Profiles graphically representing Emax and Log(τ/KA) values for G protein and βarr readouts further highlight how the analysis clustered ligands according to type and magnitude of responses elicited (Fig. 4). Ligands in cluster #3 were full, reasonably balanced agonists characterized by maximal effects at βarr and G protein readouts. Ligands in cluster #2 were partial agonists for G protein-mediated responses with measurable βarr recruitment only in presence of overexpressed GRKs, while ligands in cluster #1 displayed minimal or no βarr recruitment and G protein responses were overall smaller than in cluster #2.
Graphic representation of operational and logistic parameters for hMOR ligands populating different clusters. Operational transduction coefficients (Log(τ/KA)) and logistic Emax values derived from concentration response curves generated by hMORs at ten different biosensors were represented as radial graphs. Each radius corresponds to the magnitude of Log(τ/KA) or Emax values. Transduction coefficients are in logarithmic scale, Emax values were normalized to maximal Met-ENK response, and are presented on linear scale. Source data provided in Supplementary Data 1
While it may be feasible to monitor ten different signaling outcomes for a small group of ligands, it is unlikely that this could be done in high-throughput screening or structure–activity profiling. Hence, once we had identified the signals that contributed the most to drug classification, we determined whether a reduced number of assays could convey similar diversity. To this end, βarr2 + GRK2, βarr2 + GRK6, Gαi2, and cAMP were chosen as respective prototypes of βarr- and G protein responses. Similarity matrices generated from these individual signals or from their combinations were compared with the complete similarity matrix. As above, we compared each partial data set to the complete set of hMOR parameters, and then established if the proportion of ligands switching clusters was less than that observed for the corresponding randomized data set. For cAMP, the proportion of changes were larger than the expected random value (Fig. 3f), indicating minimal contribution of this signal to ligand classification. In contrast, for clusters generated with the other data subsets, the proportion of ligands switching clusters was significantly smaller than the random expectation (Fig. 3f), indicating that each of these signals significantly supported ligand discrimination in the complete data set, albeit to different extents.
The combination of βarr2 + GRK2 and βarr2 + GRK6 data was the best at reproducing clustering obtained with the complete hMOR similarity matrix (91.4%; Fig. 3f). In comparison, the Gαi2 data set either combined with cAMP or in isolation moderately recreated the clusters of the complete matrix (Gαi2 = 74.4%, Gαi2 + cAMP = 69.0% differences). Combining all four assays added little extra precision as compared with βarr2 + GRK2 with βarr2 + GRK6 (Fig. 3f). Thus, taken together, these data indicate that it is possible to first identify the signals that primarily contribute to signaling diversity of a group of compounds at a given receptor, and then use these signals as surrogate readout for screening campaigns over large collections of compounds.
Ligand clusters are informative of possible side effects
Preclinical studies suggest that signaling diversity of MOR agonists provides a means of improving tolerability of opioid analgesics12,13,14. Therefore, it was of interest to determine if the pharmacodynamic clusters just defined could inform us about clinical side effects of ligands in each category. To address this issue, we first used standardized gamma (SD gamma) scores33 to identify adverse events most frequently reported for opioids in the Food and Drug Administration's pharmacovigilance data base (Adverse Effects Report System (AERS)), and then calculated the scores of these events for each of the prescription opioids used in the study. These measures of side effect prevalence were associated to ligands in the different clusters by using the Euclidian distance between ligands in the similarity matrix. Tramadol was set as the arbitrary origin, and distances separating the rest of prescription opioids from tramadol in the Log(τ)-Emax-Log(τ/KA) matrix were consigned as measures of ligand similarity. These measures were then correlated to the SD gamma scores for each ligand's side effects. A complete list of the 80 events considered along with r2 and p-values for each correlation is provided in Supplementary Data 2. Correlations that were significant (p ≤ 0.05) and/or explained at least 60% of the variance (r2 ≥ 0.60) were considered. Applying these criteria, ligand position in the Log(τ)-Emax-Log(τ/KA) matrix was correlated to 6 out of a total of 80 associations considered (7.5%), including gastrointestinal (GI) events, respiratory depression, and somnolence (Table 1), all typically associated to opioid therapy34,35. These correlations confirm that signaling categories defined by unsupervised clustering can be associated to distinct frequency of report of undesired effects of opioid ligands.
Table 1 Pharmacodynamic and structural categories associate with frequency of report of undesired events for clinically available hMOR ligands#
Log(τ) and Emax were the main determinants of ligand position in the matrix constituting 89.0% of parameters effectively grouping ligands into clusters (Fig. 3d). Not surprisingly, if ligands were classified exclusively using these efficacy-related parameters, all side effects previously associated with ligand position in the Log(τ)-Emax-Log(τ/KA) matrix remained correlated with their positions in this efficacy-only matrix (Table 1). Actually, categories driven by efficacy measures associated with more side effects than clusters established by including Log(τ/KA) as an additional classification criterion (Table 1). Thus, even if functional affinity information within transduction coefficient affords better discrimination of ligands, it also acts as a confounder for cluster association to side effects.
Associating side effects to specific signals
Preclinical studies have suggested that MOR agonists that preferentially engage G protein over βarr responses could display less gastrointestinal and respiratory side effects in the clinic12,13,14. Hence, we were interested to find out whether AERS reports for opioids would distinctively correlate with ligand categories defined either by G protein or βarr signaling. There has been considerable debate as to whether biased signaling is best identified using Log(τ/KA) transduction coefficients17,24,25 or efficacy-related measures36,37. Hence, partial matrices in which drugs were classified according to G protein or βarr responses were generated using either Log(τ)-Emax or Log(τ)-Emax-Log(τ/KA) as classification criteria. Supplementary Fig. 10 shows how frequency of faecaloma report correlates to similarity scores in these four partial matrices. Considering the classification based exclusively on Log(τ)-Emax values, categories defined by βarr and G protein responses were both correlated to faecaloma report (Supplementary Fig. 10a), implying no differential association of these signals to the undesired event. In contrast, when Log(τ/KA) coefficients were additionally considered, faecaloma report correlated to βarr, but not G protein responses (Supplementary Fig. 10b). However, it is worth considering how inclusion of Log(τ/KA) values breaks the correlation previously established with efficacy-based categories. BUP has a high transduction coefficient that cannot be distinguished from those of fentanyl (FEN) or loperamide (LOP), causing the partial agonist to move closer to these efficacious ligands in the matrix. Yet BUP's transduction coefficient is driven by its high affinity38,39, and regardless of its position among efficacious agonists its side effects profile remains determined by its partial efficacy, disrupting the correlation.
Opioid modulation of acute ileum contractility is G protein-driven by effectors that hyperpolarize myenteric neurons and inhibit neurotransmitter release by vagal terminals40,41,42,. We used this G protein-mediated response to ascertain that failure to correlate faecaloma report to categories partly defined by Log(τ/KA) was not due to the method itself. In effect, as shown in Supplementary Fig. 10c–e, frequency of faecaloma report correlated with Log(τ) but not Log(τ/KA) values describing ligand inhibition of ileum contraction.
Signaling and structural clusters convey complementary information about side effects
Structural criteria are used to classify, compare, and infer possible commonalities of in vivo responses for drug candidates43. We therefore compared categories of ligands defined by pharmacodynamic and structural criteria. Ligand structure was described using Tanimoto values44 derived from standard fingerprint representations (Supplementary Data 3–5), and these values were then clustered using the same NNMF/k-means method as previously applied on signaling profiles. The resulting clusters are shown in Supplementary Fig. 11a, and representatives of each structural group are provided in Supplementary Fig. 12. Structural and pharmacodynamic similarity matrices were then compared, indicating that 36.0% of changes in ligand distance were compatible with a switch in cluster when the two different criteria were applied. This value was significantly lower when compared with 43.5% switches observed using randomized structural data (z-score = −2.803; p < 0.01), denoting some degree of statistical similarity between signaling and structural categories (Source data provided). However, the degree of similarity was low as schematically represented in Supplementary Fig. 11b. In keeping with this notion, clusters based on chemical structures were correlated with a different set of reported events than those associated with the pharmacodynamic clusters (Table 1). In particular, ligand distances in the matrix generated with chemical structures correlated with 12.5% of reported events, including pruritus, a typical opioid associated complaint45, as well as with reports of withdrawal and fluctuations in response and drug levels (Supplementary Data 6). Since structure determines pharmacokinetic properties, it is not surprising that structural categories associate with fluctuations in drug effects and even withdrawal symptoms46. On the other hand, and in spite of pharmacodynamic properties also being determined by structure, categories based on structural fingerprint representations failed to identify any of the events that associated with signaling categories, emphasizing the value of complementing structural information with a signal-based classification.
Ligand clusters generated with different GPCRs
We next examined whether clustering analysis could reveal pharmacodynamic similarities and differences among ligand responses generated at different opioid receptor subtypes. To do so, we used the same set of biosensors as for hMORs to monitor ligand activity at rat MORs (rMORs), human delta opioid receptor (hDORs) and rat DORs (rDORs). Corresponding input matrices containing logistic and operational parameters for each receptor (Supplementary Data 7–9) were analyzed as before to yield individual similarity matrices and associated heatmaps (Fig. 5a–c). Differences in clustering across receptor subtypes and species were evaluated by comparing similarity matrices for each receptor and are summarized in Fig. 5d. These comparisons revealed that the pattern of signaling diversity of this group of opioid ligands was reasonably conserved within the same receptor from different species. Indeed, in comparisons between rat and human MORs or rat and human DORs, the proportion of changes in ligand distances that were compatible with a switch in cluster was significantly less for actual as compared with randomized data sets (Fig. 5e), confirming congruent patterns across species.
Signaling profiles of opioid receptor ligands are conserved across species but not receptor subtypes. Ligand similarity heatmaps for rat MOR (a), human DOR (b), and rat DOR (c) data sets. Yellow and blue, respectively, indicate ligands/parameters that never or always cluster together. Proportion of ligands changing cluster for indicated comparisons; **p < 0.01, ***p < 0.001 comparing actual to randomized data sets as in Fig. 3 (d). Similarity matrices for the same receptor compared across species. Filled and empty bars: proportion of ligands changing clusters when, respectively, comparing actual data and random clustering simulations to the reference matrix; **p < 0.01, ***p < 0.001; z-score MOR subtypes: −7.153; z-score DOR subtypes: −2.742 (e). Similarity matrices for receptor subtypes within species compared as in (e). Actual and randomized data sets did not differ: z-score hMOR vs. hDOR: −1.502; z-score rMOR vs. rDOR subtypes: −1.567 (f). Similarity matrices for βarr- or G protein-partial data sets were compared with the complete hDOR matrix as in (e); only comparisons for actual data are shown; **p < 0.01, ***p < 0.001; z-scores versus corresponding randomized data: βarr: −3.309; G protein: −2.644; cAMP-Kir3.2: −0.309, Gα proteins: −3.286. ##p < 0.01; z-score difference: cAMP-Kir3.2 vs. Gα proteins: 2.515. p = 0.256; z-score difference βarrs vs. All G prot responses: −0.656 (g). A difference was calculated between every valueij in the complete hDOR matrix and corresponding valueij in indicated partial data sets. Histogram shows the fraction of differences with absolute value above indicated thresholds (h). Calculations described in (h) were repeated for the complete hMOR matrix and partial data sets for βarr or G protein (i). Source data provided in Supplementary Data 7, 8, and 9 and source data files
In contrast, when clusters generated with data sets from MORs and DORs within the same species were compared, their differences were statistically indistinguishable from those obtained by comparing corresponding sets of randomized data (Fig. 5f), indicating that the analysis discerned the distinct pharmacological profiles of the two receptor subtypes. To identify the source of these differences, we compared the relative contribution of βarr and G protein responses in driving ligand clustering this time according to hDOR responses. Clusters generated with each partial data set bore statistical similarity to clustering done using the complete data set, and no statistical difference was revealed between clusters produced with βarr and G protein parameters (Fig. 5g). To more precisely establish the weight of βarr and Gprotein responses to clustering of ligands according to hDOR signaling, we investigated how every value in the similarity matrix changed when considering hDOR clusters produced with the complete data set, and clusters generated with pathway-specific data sets. As shown in Fig. 5h, the variations between the complete and the G protein similarity matrices paralleled the differences between the complete and the βarr similarity matrices, indicating the two types of signals similarly contributed to the classification of ligands according to responses generated at hDORs. In contrast, and consistent with the fact that βarr recruitment was the main determinant in hMOR clustering, the differences between G protein and complete matrices were more frequent and larger than those for the corresponding βarr comparison (Fig. 5i). A graphical representation of hDOR clusters is given in Supplementary Fig. 13.
Finally, we assessed whether clustering according to signaling profiles could be extended to GPCRs that couple to effectors different than those activated by opioid receptors. For this purpose, we considered published47 and novel data generated with β2-adrenergic receptor (β2AR) ligands including: (a) G protein-dependent responses (Gαs activation, cAMP production, Ca2+ mobilization)48, (b) βarr-mediated responses (βarr2 recruitment and receptor internalization), and (c) ERK signaling, a multifaceted response involving both G proteins and βarrs49. Parameters describing concentration response curves for each of the readouts (Supplementary Data 10) were analyzed by NNMF and k-means to reveal four different drug categories (Fig. 6a). Cluster #1, including isoproterenol (ISO) and norepinephrine (NE), was characterized by measurable agonist efficacy at all readouts. Salbutamol (SALB) and salmeterol (SALM) in cluster #2 could be distinguished from the first category because of their minimal responses at βarr-dependent readouts. Carvedilol (CARV) and propranolol (PRO) behaved as agonists only in the ERK pathway (Cluster #3), while ICI118,555 and metoprolol (MET) had no efficacy except for inverse agonism at Gαs and cAMP assays (cluster #4). The complete signaling profiles for ligands in different clusters are provided in Supplementary Fig. 14.
β2ADR ligands cluster according to similarity in G protein and βarr-mediated responses. Ligand similarity heatmaps for hβ2ADRs. Yellow and blue, respectively, indicate ligands/parameters that never or always cluster together. (a). Similarity matrices for partial data sets corresponding to ERK, G protein- (Gαs, cAMP, Ca+2), and βarr2- (recruitment, endocytosis) mediated responses were compared with the reference hβ2ADRs data set. Filled bars: proportion of ligands changing clusters when comparing actual ERK, G protein, and βarr2 data sets to the reference; empty bars: proportions observed comparing the reference data set to corresponding simulations of random clustering for partial matrices. *p < 0.05, **p < 0.01; z-score ERK: 1.523; z-score G protein: −2.446 and z-score βarr: −2.636 (b). Source data provided in Supplementary Data 10 and source data files
As shown in Fig. 6b, partial data sets for Gαs/cAMP/Ca2+ and for βarr-recruitment/endocytosis recreated original clusters better than their corresponding randomized controls, indicating significant contribution of these signals to ligand clustering. To establish whether these categories were also relevant to human pharmacology, we evaluated their association to pharmacovigilance data. For this purpose, undesired cardiovascular and respiratory events most frequently reported for β2AR agonists and antagonists were first identified, and SD gamma scores representing the frequency with which these events were reported for the prescription ligands used in the study was correlated to their signaling similarity (measured as Euclidian distances in the full matrix) (Supplementary Data 11). We found that increasing distance from the agonist ISO was significantly correlated (p < 0.05) with increasing frequency of reports for hypotension, decrease in blood pressure, sinus bradycardia, atrioventricular block, sinus arrest, and need for inhalation therapy (Table 2). Interestingly, the first four events in this list typically correspond to reduced sympathetic tone on cardiovascular function50. Hence, their more frequent association with ligands that clustered furthest apart from ISO is entirely consistent with gradual loss of efficacy at β2AR. Moreover, these four events were also negatively correlated with ligand efficacy to induce Gαs, cAMP, and Ca2+ signaling (Table 2), an overlap that is consistent both with the well-document role of these signals in maintaining heart chrono-, and inotropism51 and with the fact that G protein-dependent signals significantly drove ligand clustering. Gαs, cAMP, and Ca2+ signaling categories also showed increasing reports of deleterious effects on respiratory function, some of them such as asthma, asthmatic crisis, status asthmaticus consistent with bronchoconstriction, as distance from ISO and β2AR antagonism increased. Thus, pharmacodynamic categories defined by as few as eight ligands were sufficiently robust to reveal a well-known association between sympatholytic CV events or manifestations of bronchoconstriction and modulation of G protein activity by β2AR ligands. The observation reinforces the notion that unsupervised clustering of multidimensional signaling profiles allows the association of signals generated in simple cellular models to possible clinical effects of GPCR ligands.
Table 2 Pharmacodynamic categories associate with frequency of report of undesired cardiovascular and respiratory events for clinically available β2ADR ligands#
This study introduced a stepwise analysis in which GPCR ligands were organized into pharmacodynamic categories that could be then associated with clinically relevant responses. Pharmacodynamic parameters that best supported classification of ligands ((Log(τ)-Emax-Log(τ/KA)) were empirically chosen by consecutively applying NNMF and k-means clustering to different combination of parameters informative of efficacies and functional affinities. The procedure was successfully applied to classify groups of ligands ranging from 8–320 in number, and representative of a multiplicity of signaling profiles.
Ligand categories generated with (Log(τ)-Emax-Log(τ/KA)) values from hMOR data were primarily driven by ligand diversity in βarr signaling efficacy, but G protein responses also contributed to the classification. Measured similarity among signaling profiles of prescription opioids present in the different categories was correlated with their corresponding frequencies of AERS reports for typical opioid side effects, indicating that the categories established by applying this exploratory method may allow to establish meaningful associations between in vitro signals and clinically relevant drug actions. This notion was further supported by observations that pharmacodynamic categories defined for β2AR ligands were essentially driven by G protein responses and associated to G protein-driven sympatholytic effects52,53. Hence, by unveiling these well-documented associations, we established that clustering analysis of concentration–response parameters allows to associate multidimensional in vitro signaling profiles to clinical responses. Such use of curve parameters should prove beneficial for identifying signals that support specific responses of interest and for which mechanistic information is unavailable. Pharmacodynamic categories defined by efficacy-related parameters (Log(τ)-Emax) had stronger and more frequent correlations to side effects than those defined by additional inclusion of affinity information provided by transduction coefficients. On the other hand, ligand differentiation was optimal when transduction coefficients were taken into account, calling for a discretionary decision on which parameters to use depending on the goal of the classification.
By classifying opioid ligands according to pathway-specific responses, it was possible to explore whether specific signals were driving typical side effects of opioids. We found that association of faecaloma report to categories defined by G protein responses was influenced by the parameters used in ligand classification. In Log(τ)-Emax matrices, G protein and βarr categories were both directly correlated with reports for this side effect, implying that weaker agonists were associated with lower frequency of reports. In contrast, the correlation with G protein responses was disrupted if Log(τ/KA) values were also considered, suggesting a scenario were G protein signaling would not associate to side effects. This divergence as compared with Log(τ)-Emax matrices is linked to the fact that despite its partial efficacy and a side effects profile consistent with partial agonism, Log(τ/KA) coefficients could not distinguish this low-efficacy–high-affinity ligand from much more efficacious agonists, such as morphine or oxycodone. In contrast to G proteins, βarr categories correlated to faecaloma report independent of the parameters used for classification, as transduction coefficients for βarr responses were consistent with BUP's low efficacy. Log(τ/KA) transduction coefficients are largely used to identify biased ligands17,25, so a word of caution is warranted for bias measures driven by functional affinity since only efficacy parameters are predictive of the magnitude of in vivo responses54.
It is also of interest that maximal responses for βarr and G proteins decreased in parallel across the different clusters, albeit not to the same extent. Indeed, βarr signals gradually disappeared while those of G proteins grew progressively smaller without completely vanishing. Such systematic imbalance between the two types of signals has been previously reported55, and is akin to system bias25 where βarr responses are less well coupled to the receptor than G protein signals. Within this context, absence of βarr responses may simply indicate partial agonism and not signaling bias. As a matter of fact, all novel biased hMOR ligands presented herein as well as those published to date (i.e.: TRV13012; PZM2113 and Scripps compounds14) are partial agonists at G protein responses. This raises the possibility that currently available biased ligands could simply produce less side effects because they are partially effective at stimulating the receptor, and not necessarily because of greater efficacy for activating the G protein over βarr. A miss-interpretation of bias might be a problem for future clinical applications since a partial agonist may also produce a submaximal analgesic response. In this sense, it is of interest that βarr-G protein signaling profiles of the latest hMOR ligands14 resemble those obtained in this study for BUP, a partially effective analgesic56. It is also worth considering that the clinical profile of TRV130, a partial hMOR agonist which was clinically tested as the first biased agonist for MORs, did not significantly differ from morphine's profile at doses with equivalent analgesic effects10. Finally, when PZM21 was independently tested after its initial description as biased agonist, it was shown to behave as a partial agonist in βarr and G protein readouts, and to produce respiratory depression commensurate with partial signaling10.
Structural similarity is another means for inferring common in vivo responses of therapeutic drug candidates early in the discovery process43. Here, when clusters established on the bases of signaling and structural resemblance were compared, they displayed nonrandom but marginal similarity. Different reasons could explain the low degree of similarity between categories established with structural and pharmacodynamic criteria, including incomplete representation of structural diversity of opioid ligands within the sample used, or different discriminatory power of signaling profiles and current descriptors of structural properties. Consistent with their low degree of similarity, structural, and pharmacodynamic categories were associated with different types of undesired events. Indeed, structural similarities were more frequently associated with fluctuations in therapeutic response, which are typically associated with pharmacokinetic properties42. On the other hand, signaling categories specifically correlated with on-target side effects, pointing to the complementarity of both approaches when characterizing a limited number of compounds of interest.
In conclusion, we presented an unsupervised classification method that incorporates distinct and complementary data sources to comprehensively describe signaling diversity of GPCR ligands. The procedure identifies signaling imbalance independent of whether bias in the response co-varies with efficacy, it was applied to a large diversity of signaling profiles and distinguishes subtle differences in signaling preferences.
Materials and reagents
Standard opioids were purchased from Cedarlane (Burlington, Canada) and Sigma-Aldrich (St. Louis, MO, USA). Fifteen novel compounds were provided by Pfizer Inc. (Worldwide Research and Development). (−)-Isoproterenol hydrochloride, (−)-norepinephrine, DL-propranolol hydrochloride, ( ± ) metoprolol ( + )-tartrate salt, carvedilol, and salmeterol xinafoate were purchased from Sigma-Aldrich (St Louis, MO). ICI 118,551 and salbutamol hemisulfate were purchased from Tocris Bioscience (Ellisville, MO). Coelenterazine 400a was purchased from Biotium.
Plasmids and DNA constructs
A cleavable signal sequence of influenza hemagglutinin (MKTIIALSYIFCLVFA) and a Flag tag (MDYKDDDDA) were added to the human MOR1, rat MOR1, human DOR, and rat DOR and, their coding sequence optimized and synthetized as Strings DNA Fragments at GeneART (ThermoFisher Scientific). The DNA Strings were subcloned by Gibson assembly (New England Biolabs Canada) in pLVX-IRES-Puro (Clontech Laboratories, Inc). Untagged versions of the receptors were made by an internal NcoI deletion, removing the coding sequence of the Flag tag. Constructs encoding for GFP10-tagged receptors were made by PCR overlap; the coding sequence of each signal-peptide Flag-receptors was PCR-amplified to remove the stop codon and assembled by PCR overlap with the coding sequence of GFP10. The resulting PCR products were subcloned by Gibson assembly in pLVX-IRES-Puro. Constructs encoding the Epac-based cAMP sensor (GFP10-Epac-RlucII), RlucII-tagged Gα (αi1, αi2, αoA, αs), GFP10-Gγ1, GFP10-Gγ2, β-arrestin1-RlucII, β-arrestin2-RlucII, RlucII-Gγ2, human β2-adrenergic receptor (hβ2AR), and cmyc-hβ2AR-GFP10 were previously described (PMID: 15782186, PMID: 22534132, PMID: 23175530, PMID: 24309376, PMID: 19584306, PMID: 26658454, PMID: 16901982, PMID: 15155738). pCDNA3.1 ( + ) Gβ1 was bought at Missouri University of Science and Technology (cdna.org). Plasmids encoding for the following proteins were generously provided as follows: GRK6 and GRK2 by Dr Antonio De Blasi (Istituto Neurologico Mediterraneo Neuromed, Pozzilli, Italy), GRK5 by Dr Robert Lefkowitz (Duke University, Durham, NC). Kir3.2-GFP10 by Dr Terry Hebert (McGill University, Montréal, Canada). Kir3.1 subunit by Dr. Deborah J. Nelson (University of Chicago, Chicago, IL).
HEK293 cells were a kind gift of Dr. Laporte, McGill University57. They were cultured in 100 mm Petri dishes (Sarstedt, Germany) at 37 °C and 5% CO2 in the Dulbecco's modified Eagle's medium (DMEM) supplemented with 10% fetal bovine serum, 2 mm l-glutamine and 100 unit mL−1 penicillin-streptomycin.
Transient transfections of vectors encoding BRET biosensors in combination with complementary signaling partners were performed in 100 mm Petri dishes (3 × 106 cells) for G protein and Kir3.2 channel activation assays and in 96-wells culture plates coated with polyD-lysine (PerkinElmer, MA, USA) for βarr recruitment assays (32,000 cells/well), using the polyethylenimine transfection reagent (Polysciences, PA, USA) at a PEI/DNA ratio of 3:158. For cAMP production assays, stable cell lines expressing the GFP10-Epac-RlucBRET2-cAMP biosensor59 were plated in six-wells plates (Greiner bio-one, Austria) and stably transfected with 1 μg of either MORs or DORs (human or rat) biosensor using PEI. They were selected respectively using hygromycin (100 µg mL−1) and puromycin (10 mg mL−1).
BRET assays
Ligand preparation: Agonists were dissolved in DMSO and spotted on 96-well white bottom microplates (Greiner bio-one) using the HP D300 Digital Dispenser (Tecan Life Sciences). DMSO concentration was normalized for each point at 0.334%.
Gαi and Gαo-activation assay: HEK 293 were co-transfected with DOR or MOR (human or rat), either of the BRET biosensors pairs: γ2-GFP10/GαoA-99-RlucII (Ratio Receptor/GFP/RlucII: 1:0.6:0.12), γ2-GFP10/Gαi1–91-RlucII (Ratio Receptor/GFP/RlucII: 1:0.6:0.12), or γ2-GFP10/Gαi2-99-RlucII (Ratio Receptor/GFP/RlucII: 1:0.72:0.12) together with untagged Gβ1(Ratio 1: 0.6)26. Forty-eight hours after the transfection, the media was removed and the cells were washed with phosphate-buffered solution (PBS) then re-suspended in PBS + MgCl2 (0.429 mM) at a protein concentration ≥0.6 µg µL−1. Coelenterazine 400a was added to the cells to a final concentration of 5 µM for 3 min, and 100 µL per well of this mix were subsequently distributed into the 96-well-printed plates. Plates were read 5 min after on the Mithras LB 940 microplate reader (Berthold Technologies, Bad Wildbad, Germany), 3 s per well, with filters set at 400 nm (RlucII) and 515 nm (GFP10); BRET ratios were calculated as GFP10/RlucII emissions. Net BRET values were calculated by subtracting background BRET ratio observed in cells expressing donor G biosensors alone.
Gs activation assay: HEK293 cells stably expressing β2AR were transiently transfected with 200 ng Gαs-67-RlucII, 100 ng Gβ1, and 100 ng GFP10-Gγ1. The day of the experiment, cells were washed with Hank's balanced salt solution (HBSS) (137 mM NaCl, 5.4 mM KCl, 0.25 mM NaHPO4, 0.44 mM KH2PO4, 1.8 mM CaCl2, 0.8 mM MgSO4, 4.2 mM NaHCO3, pH 7.4) supplemented with 0.1% glucose and 0.1% BSA. Coelenterazine 400a (Coel-400a, Biotium) was added for 5 min to the wells (2.5 μM), then β-adrenergic compounds were added for 4.5 min. BRET was measured and calculated as described above.
Kir 3.2 channel activation assay: HEK 293 were plated onto 100 mm Petri dish and transfected with DOR or MOR (human or rat), the Kir3.2-GFP10/γ2-LucII BRET biosensor pair together with untagged Kir3 channel and G protein subunits27 at a ratio of 1:1:0.075:1:0.5, respectively. The BRET assay was performed as described above.
β-arrestin recruitment: HEK 293 cells were co-transfected with sp-FLAG-DOR-GFP10 or sp-FLAG -MOR-GFP10 (human or rat) and βarr1/2-RlucII for β-arrestin1/2 recruitment at a ratio receptor/construct of 1:0.06. Recruitment of βarr2-RlucII was also tested in the presence of, GRK2, GRK5, GRK6 (Ratio receptor/GRK DNA: 1:0.1). Forty-eight hours after transfection, cells were washed with PBS then incubated in Tyrode's solution (140 mM NaCl, 2.7 mM KCl, 1 mM CaCl2, 12 mM NaHCO3, 5.6 mM D-glucose, 0.49 mM MgCl2, 0.37 mM NaH2PO4, 25 mM HEPES, pH 7.4) for 30–60 min at 37 °C. Indicated concentrations of agonists, diluted in Tyrode buffer, were added to the wells for 10 min, then cells were incubated for 5 min with Coelenterazine 400a (2.5 µM). BRET2 readings were taken at 37 °C as detailed above. For β2AR β-arrestin2 recruitment, HEK cells were transiently transfected with 50 ng βarr2-RLucII and 300 ng β2AR-GFP10. The day of the experiment, cells were washed with HBSS supplemented with 0.1% glucose and 0.1% BSA. β-adrenergic compounds were added to the wells for 10 min, then coelenterazine 400a was added for 5 min to the wells (2.5 μM). BRET was measured and calculated as described above.
cAMP production assay: Stably-transfected cells expressing the GFP10-Epac-RlucBRET2-cAMP biosensor59 and either MORs or DORs were seeded at a density of 30,000 cells/well in a high glucose medium supplemented with 10% newborn calf serum, and grown on 96-well polylysine-coated plates for 48 h. Cells were later transferred to Tyrode buffer and incubated for 15 min at 37 °C. Coelenterazine 400a was then added to a final concentration of 5 µM. Five min later, forskolin (Bioshop, Canada) was introduced (final concentration: 10 µM for rMOR, 15 µM for rDOR, and 25 µM for hMOR and hDOR) followed, 3.5 min later, by increasing concentrations of ligands. BRET2 readings were taken 5 min after ligands were introduced28.
Guinea pig ileum assays
Male Hartley guinea pigs were anesthetized using isoflurane followed by exsanguination. The myenteric plexus of the ileum was dissected according to the method described by Cowie & al.41. Briefly, a portion of the ileum was removed (10 cm distal to the cecum) into which a glass rod was inserted. The myenteric plexus was removed from the circular longitudinal muscle via gentle scraping with a moist cotton swab and separated from the muscle using forceps. The resulting myenteric tissue was cut into 2.5 mm strips and placed in oxygenated Krebs buffer (37 °C, gassed with 95% O2/5% CO2) and tensioned to a baseline tension of 2000 mg. The tissues were washed, equilibrated for 30 min, and subsequently tested for viability with a maximal concentration of Carbachol (300 nM, three times with 10 min of washing, and 10 min of equilibrating in between additions). The final prime was followed by a 20 min wash period followed by a 20 min equilibration period before the start of the experiment. Tissues were continually stimulated with 0.1 Hertz for 1 ms at 20 volts (producing a stimulation equivalent to 80% of the maximal contractile response). Following a 10 min baseline stimulation period, the kappa opioid antagonist nor-binaltorphimine was added (5 nM final) and incubated for 10 min. Finally, cumulative concentration–response curves were generated to each test ligand or vehicle control (DMSO). Isometric tension data (in mg) were collected.
All procedures performed on these animals were in accordance with regulations and established guidelines and were reviewed and approved by Pfizer Institutional Animal Care and Use Committee.
Concentration response curves describing ligand responses by different receptors (hMOR, hDOR, rMOR, rDOR, and hb2ADR) were analyzed with Graphpad Prism6, using built-in 3 or 4 parameter logistic equations to obtain independent pEC50 and Emax values for each receptor–biosensor pair:
$$y = a + \left( {b - a} \right)/\left( {1 + 10^{\left( {logEC50 - x} \right) \ast c}} \right)$$
(y → measured response; a → minimal asymptote, b → maximal asymptote; b − a → Emax; c → slope).
Concentration response curves were additionally analyzed with the operational model of Black and Leff32. As above, curves representing responses elicited by the same receptor at each of the ten different biosensors were fit independently. Fitting was done using Graphpad Prism6 after introducing a set of equations kindly provided by Dr Christopoulos:
A = 10\(^x\)
$$\begin{array}{l}{\it{operate1}} = \hfill \\ \hfill\left( \left( {1 + A} \right) / \left( { {10^{{\mathrm{Log}}R}} \ast A} \right) \right)^n\hfill \\ \hfill\left( {{\mathrm{used}\, \mathrm{to}\, {\mathrm{fit}}\, {\mathrm{full}}\, {\mathrm{agonists}}}} \right)\end{array}$$
$$\begin{array}{l}{\it{operate2}} = \hfill \\ \hfill\left( {\left( {1 + A/{10^{{\mathrm{Log}}KA}}} \right)/\left( {{10^{{\mathrm{Log}}R}}\ast A} \right)} \right)^n\hfill \\ \hfill\left( {{\mathrm{used}}\;{\mathrm{to}}\;{\mathrm{fit}}\;{\mathrm{partial}}\;{\mathrm{agonists}}} \right)\end{array}$$
$$\begin{array}{l}{\mathrm{Full}}\;{\mathrm{agonist}} = \hfill \\ \hfill{\mathrm{basal}} + \left( {E_{\mathrm{max}} - {\mathrm{basal}}} \right)/\left( {1 + {\it{operate1}}} \right)\hfill\end{array}$$
$$\begin{array}{l}{\mathrm{Partial}}\;{\mathrm{agonist}} = \hfill \\ \hfill{\mathrm{basal}} + \left( {E_{{\mathrm{max}}} - {\mathrm{basal}}} \right)/\left( {1 + {\it{operate2}}} \right)\hfill\end{array}$$
basal → response observed in the absence of agonist; Emax → maximal response of the system; n → slope of the function which links occupancy to response; KA → functional affinity (partial agonists); Log(R) → Log(τ/KA).
When using the logistic model, the fits for three and four parameter curves were compared and the best fit taken. If no fitting was possible without constraints, the minimal asymptote was fixed to zero; if this was unsuccessful, the Hill coefficient was additionally fixed to one (i.e.,: only the three parameter fit was considered). If both these constraints proved unsuccessful, and in curves with no inflection point for maximal effect, the highest experimental value was considered Emax. The latter procedure forced the maximal response of very weak partial agonists within the range of experimental data avoiding aberrant predictions due to extrapolation. If no fitting was possible following these constraints, no fitting (NF) status was consigned. If fitting was possible, we made sure that all curves had a Span > 3x SEM, otherwise they were considered as no response (NR).
As used in this study, the operational model does not yield Log(τ) or pKA values for full agonists25, which were consigned as not available (NA). In these circumstances, Emax values were used to differentiate these compounds from partial agonists, and differences among full agonists were established through their consolidated Log(τ/KA) coefficients. It should also be noted that by independently fitting curves for different biosensors, the model does not contemplate interconversion among distinct receptor states stabilized by different effectors.
Feature reduction, and ligand clustering
Each receptor was represented by a matrix composed of 25 ligands (21 for DORs) × 30 parameters (Emax, Log(τ) and Log(τ/KA) for ten assays). This matrix was created by sampling from the normal distribution around each parameter using the mean and standard deviation thereby incorporating the variance associated with each data point and propagating it through the clustering method. In order to correct for scale differences between parameters, we standardized each column to range between 0 and 1 according to:
$${\mathrm{Standardized}}\;{\mathrm{value}} =\frac{{{X}_{ij}} - {\mathrm{minimum}}_j} {{\mathrm{maximum}}_j - {\mathrm{minimum}}_j}$$
for every ligand i and every parameter j.
Process (1) was repeated 1000 times to create 1000 data matrices each independently put through the following procedure (Supplementary Fig. 2).
Nonnegative matrix factorization (NNMF) was used to reduce dimensionality of the data and create the W (ligand * k) and H (k *parameter) basis vectors thereby removing noise and redundancy. We used sparse NNMF to ignore missing data ("NA", "NF", and "NR" curves). Difference between the original matrix V and the product of W * H was minimized to less than 1e-7.
K-means clustering was performed on the W basis vector, where the number of clusters equals the number of basis vectors from NNMF (K = k), to assign each compound into a cluster. Note: the phenotypic parameter clusters were obtained using the H vector instead of the W one.
Steps 3 and 4 were repeated 250 times to quantify the fraction of times each compound clustered together resulting in a ligand * ligand frequency matrix ranging from 1 (always clustered together) to 0 (never clustered together). This iterative process quantifies both global and local minima/maxima arising from small variances in clustering resulting from the randomized starting vectors for NNMF and k-means.
(a) The entire process including feature reduction and clustering (3–5) was repeated for different values of k (k = 2 to k = 7), providing a frequency matrix for each k. (b) These six frequency matrices were averaged together to quantify ligand similarity independent of the number of features used as each K may extract unique patterns that may be complementary or orthogonal to results from different K's.
Steps 3–6 were independently performed on each of the 1000 sampled data matrices providing 1000 composite similarity matrices.
These 1000 matrices were averaged to create a final frequency matrix quantifying how often ligands clustered together and representing total ligand similarity across all concentration response curves.
We visualized the similarity matrix using a dendrogram and a heat map using Orange, created from the distance between each compound in the similarity matrix using a Pearson Correlation.
Simulation of virtual compounds
We built 16 profiles showing bias and various potencies/efficacies by selecting ranges of KA-τ pairs across six imaginary biosensors. So that our virtual compounds respect these ranges, we invented them by sampling random values of KA and τ within the bounds associated to the imaginary biosensors specified per profile. We used this procedure 20 times for each profile yielding 320 virtual ligands.
As for curve fitting, simulations were conducted under the assumption of independence across biosensors. Using the operational model equation, we generated corresponding concentration response curves (CRC) to which we added 10% noise using the flat distribution. Noisy CRCs were then fitted to both the logistic equation and the operational model equation using the Bayesian inference engine STAN60 to yield values of Emax, pEC50, τ, and KA and their associated distributions (from which we computed a standard error of the mean to use in the NNMF pipeline) (values in Supplementary Data 12). The best estimate for τ/KA ratio and its distribution of draws were also computed directly within the fitting process as a transformed parameter.
Selections of subsets of parameter estimations and associated SEM were used in NNMF/k-means clustering. The resulting matrices of frequency of co-occurrences were used to compute distance metrics, hierarchical clustering trees, and tSNE plots whose leaves and data points were colored by profile (Fig. 1, Supplementary Fig. 3, Supplementary Fig. 4).
Clustering of pharmacological parameters
The 25 (1 per ligand) values for each parameter array (P) were distributed into four smaller arrays corresponding the ligand clusters. We then utilized a two-sample Kolmogorov–Smirnov test to compare each sub-array to the original array (P) to measure if these were randomly sampled from the original array. This provides four p-values for each parameter. A significant p-value indicates ligands in that cluster are biased toward a specific response for that parameter. This process was repeated for each of the 30 parameters. These p-values were then sorted according to (i) the type of parameter considered (e.g., pEC50, Emax, or Log(τ/KA)) or (ii) measurement similarity acquired from the similarity matrix obtained by the NNMF/k-means method detailed above using the H basis vector instead of the W. The procedure is summarized in Supplementary Fig. 7.
Comparing clusters among complete data sets
To compare clustering similarity between two different data sets we implemented two approaches: (a) Directly comparing the two similarity matrices using pairwise differences and (b) quantifying the overall difference between the two matrices using random simulation to obtain a difference threshold and to establish significance.
Direct comparison: We calculated the difference between every paired value in similarity matrix A and B (representing the similarity between compounds i and j):
$${\mathrm{Difference}} = {A_{ij}} - {B_{ij}}$$
The resulting difference matrix is of equal dimensionality to A and B, ranging from 1 (compounds i and j are always clustered together in A but never in B) to −1 (always in B but never in A).
Thresholding and random simulation: We compared the difference in Euclidian distance for every pair of ligands i and j between similarity matrices for data set A and B:
$${\mathrm{Difference}} = \left| {\left| {L_{iA} - L_{jA}} \right|} \right| - \left| {\left| {L_{iB} - L_{jB}} \right|} \right|$$
Where LiA and LjA are row vectors representing the similarity of Li and Lj to all other ligands in matrix A. We then used random clustering replicates (detailed below) to identify a cutoff value to determine which difference values corresponded to a significant variation between A and B. The final comparison between data set A and B was represented as a proportion:
$${\mathrm{Fraction}}\;{\mathrm{change}} = \frac{\# \; significant \;differences}{Total \# \; of\; comparisons}$$
Only ligands tested in both data sets were used (e.g., comparing hMOR to hDOR only used the 21 shared ligands).
Random clustering: For each data set, we created 50 random input data matrices by permuting all mean-standard deviation pairs of data points within the original data matrix. Each random matrix was therefore specific to and equal in size and shape to the original data (e.g., hMOR: (25 * 30); hMOR-βarr: (25 * 15)). We then repeated the entire NNMF/k-means clustering method on each data-shuffled random matrix resulting in 50 random clustering frequency matrices for each data type.
To determine a cutoff value representing significant variation between any two data sets (Difference threshold), we calculated the Euclidian Distance between pairs of compounds in the same cluster and compounds in different clusters for each of the 50 trials. The threshold is the mean value of the overlap between the "same cluster" distribution and the "different cluster" distribution (see Supplementary Fig. 9). Threshold values range between 0.95 and 1.5. Using these thresholds, it was possible to calculate the proportion of significant variation between two matrices. To quantify if this change was significant, we calculated the fraction of significant changes (using thresholds) between the clustering from the 50 randomized data sets (e.g., 50 random hDOR) compared with reference cluster (e.g., hMOR). The resulting distribution of 50 values represented the proportion of random changes from the reference. This distribution was used to calculate a z-score for the difference value of the actual data (hMOR vs hDOR):
$${\mathrm{z}} {{-}} {\mathrm{score}} = \frac{{( {{\mathrm{fraction}}\;{\mathrm{change}}\;{\mathrm{in}}\;{\mathrm{actual}}\;{\mathrm{data}}} ) - ( {{\mathrm{mean}}\;{\mathrm{random}}\;{\mathrm{change}}} )}} {( {{\mathrm{STD}}\;{\mathrm{random}}\;{\mathrm{changes}}} )}$$
Comparing clusters generated with complete data and subsets
In order to calculate whether clustering from data subset I (e.g., hMOR-Barr) changed more than data subset J (e.g., hMOR-G protein) compared with the complete data set clusters (e.g., hMOR), we compared the 50 random similarity matrices to the reference (e.g., hMOR) and calculated the fraction of significant changes using the method detailed above. As a result, we obtained an array of 50 values representing the random change from reference. This array was created for both subsets (e.g., hMOR-βarr and hMOR-G protein). We then iteratively, with replacement, randomly sampled 1 value from each of these arrays and calculated the difference to create a distribution of 1000 values indicating the random expected difference between these two subsets of data. We then calculated a z-score using the mean and standard deviation of this distribution and the actual observed difference.
Clustering ligands according to structural similarities
Each ligand was represented using three standard fingerprint representations: (ECFP-6) Extended-Connectivity Fingerprints (ECFPs) (http://accelrys.com/products/collaborative-science/biovia-pipeline-pilot/), Functional-Class Fingerprints (FCFPs) and MDL MACCS keys. A similarity matrices for each different fingerprint was generated for the 25 ligands in the data set, where each value in the matrix (Si,j) corresponds to the Tanimoto similarity value between compound i and compound j and ranges from a value 0 to 1 (1 being most similar). We combined these three matrices into a single matrix of dimensions (25 compounds × 75 comparisons), and repeated the NNMF/k-means clustering algorithm on the data to yield a structural similarity matrix.
Correlating signaling data to side effect report frequency
A list of all MOR-active compounds was created by searching DrugBank for all approved drugs which activate MOR. The resulting list was intersected with the list of drugs in the FDA's Adverse Event Reporting System data for which a standardized gamma (SD gamma) score could be generated at the preferred term (PT) level according to the method of Johnson et al.33. Briefly, SD gamma scoring is a statistical approach to identify disproportionately high, or low, numbers of drug-event occurrences by normalizing for number of drugs and number of event reports. SD gamma scores for each event were averaged for all resulting MOR compounds, and PT events were sorted by average score to produce a listing of high-scoring events most clinically relevant to opioid therapy (80 highest scores were considered). A similar procedure was completed to find the 80 side effects associated with β2ADR-active compounds.
Individual drug SD gamma scores for frequently reported events were then correlated to Euclidian distances separating prescription opioids (tramadol, buprenorphine, oxycodone, morphine, fentanyl, and loperamide) in hMOR and structural similarity matrices. SD gamma scores were additionally correlated to transduction coefficients for BRET or guinea pig contractility responses respectively normalized to Met-ENK (∆Log(τ/KA)MET) or loperamide (∆Log(τ/KA)LOP). Note that LOP and Met-ENK are balanced ligands that co-cluster in every data set such that differences due to normalization are simply scalar. Individual drug SD gamma scores clinically prescribed β2ADR ligands used in the study (isoproterenol, norepinephrine, salbutamol, salmeterol, pindolol, carvedilol, and metoprolol) were similarly correlated to Euclidian distances separating these ligands in the β2ADR similarity matrix or to transduction coefficients for BRET responses where isoproterenol was the standard.
Correlation analysis: GraphPad Prism6 was used to evaluate correlation between drug distance in cluster and the frequency of reports of adverse events.
All statistical comparisons were two-sided except when contrasting partial and whole similarity matrices.
All data generated or analyzed in this study are included in the article and supplementary materials or provided as source data files.
All clustering and cluster comparisons were conducted using Python 2.7.6. Complete source code is available for download at http://github.com/JonathanGallion/Benredjem-Gallion.
Lafferty-Whyte, K., Mormeneo, D. & Del Fresno Marimon, M. Trial watch: opportunities and challenges of the 2016 target landscape. Nat. Rev. Drug Disco. 16, 10–11 (2017).
Lohse, M. J., Nuber, S. & Hoffmann, C. Fluorescence/bioluminescence resonance energy transfer techniques to study G-protein-coupled receptor activation and signaling. Pharm. Rev. 64, 299–336 (2012).
Huang, W. et al. Structural insights into micro-opioid receptor activation. Nature 524, 315–321 (2015).
ADS CAS PubMed PubMed Central Google Scholar
Kenakin, T. & Miller, L. J. Seven transmembrane receptors as shapeshifting proteins: the impact of allosteric modulation and functional selectivity on new drug discovery. Pharm. Rev. 62, 265–304 (2010).
Kenakin, T. Signaling bias in drug discovery. Expert Opin. Drug Disco. 12, 321–333 (2017).
Wacker, D., Stevens, R. C. & Roth, B. L. How ligands illuminate GPCR molecular pharmacology. Cell 170, 414–427 (2017).
Luttrell, L. M., Maudsley, S. & Gesty-Palmer, D. Translating in vitro ligand bias into in vivo efficacy. Cell Signal 41, 46–55 (2018).
Costa-Neto, C. M., ESLT, Parreiras & Bouvier, M. A pluridimensional View of Biased Agonism. Mol. Pharm. 90, 587–595 (2016).
Pang, P. S. et al. Biased ligand of the angiotensin II type 1 receptor in patients with acute heart failure: a randomized, double-blind, placebo-controlled, phase IIB, dose ranging trial (BLAST-AHF). Eur. Heart J. 38, 2364–2373 (2017).
Hill, R. et al. The novel mu-opioid receptor agonist PZM21 depresses respiration and induces tolerance to antinociception. Br. J. Pharm. 175, 2653–2661 (2018).
Singla N. et al. A randomized, Phase IIb study investigating oliceridine(TRV130), a novel µ-receptor G-protein pathway selective (μ-GPS) modulator, for the management of moderate to severe acute pain following abdominoplasty. J Pain Res. 10, 2413–2424 https://doi.org/10.2147/JPR.S137952 (2017).
DeWire, S. M. et al. A G protein-biased ligand at the mu-opioid receptor is potently analgesic with reduced gastrointestinal and respiratory dysfunction compared with morphine. J. Pharm. Exp. Ther. 344, 708–717 (2013).
Manglik, A. et al. Structure-based discovery of opioid analgesics with reduced side effects. Nature 537, 185–190 (2016).
Schmid, C. L. et al. Bias factor and therapeutic window correlate to predict safer opioid analgesics. Cell 171, 1165–1175 e1113 (2017).
Onaran, H. O. et al. Systematic errors in detecting biased agonism: analysis of current methods and development of a new model-free approach. Sci. Rep. 7, 44247 (2017).
Benredjem, B., Dallaire, P. & Pineyro, G. Analyzing biased responses of GPCR ligands. Curr. Opin. Pharm. 32, 71–76 (2017).
Kenakin, T., Watson, C., Muniz-Medina, V., Christopoulos, A. & Novick, S. A simple method for quantifying functional selectivity and agonist bias. ACS Chem. Neurosci. 3, 193–203 (2012).
Rajagopal, S. et al. Quantifying ligand bias at seven-transmembrane receptors. Mol. Pharm. 80, 367–377 (2011).
Lane, J. R., May, L. T., Parton, R. G., Sexton, P. M. & Christopoulos, A. A kinetic view of GPCR allostery and biased agonism. Nat. Chem. Biol. 13, 929–937 (2017).
Kenakin, T. Quantifying biological activity in chemical terms: a pharmacology primer to describe drug effect. ACS Chem. Biol. 4, 249–260 (2009).
Raehal, K. M., Walker, J. K. & Bohn, L. M. Morphine side effects in beta-arrestin 2 knockout mice. J. Pharm. Exp. Ther. 314, 1195–1201 (2005).
Lee, D. D. & Seung, H. S. Learning the parts of objects by non-negative matrix factorization. Nature 401, 788–791 (1999).
ADS CAS PubMed MATH Google Scholar
Forgy, Edward W. Cluster analysis of multivariate data: efficiency versus interpretability of classifications. Biometrics. 21, 768–769 (1965). JSTOR 2528559.
Kenakin, T. & Christopoulos, A. Measurements of ligand bias and functional affinity. Nat. Rev. Drug Disco. 12, 483 (2013).
Kenakin, T. & Christopoulos, A. Signalling bias in new drug discovery: detection, quantification and therapeutic impact. Nat. Rev. Drug Disco. 12, 205–216 (2013).
Gales, C. et al. Probing the activation-promoted structural rearrangements in preassembled receptor-G protein complexes. Nat. Struct. Mol. Biol. 13, 778–786 (2006).
Richard-Lalonde, M. et al. Conformational dynamics of Kir3.1/Kir3.2 channel activation via delta-opioid receptors. Mol. Pharm. 83, 416–428 (2013).
Tudashki, H. B., Robertson, D. N., Schiller, P. W. & Pineyro, G. Endocytic profiles of delta-opioid receptor ligands determine the duration of rapid but not sustained cAMP responses. Mol. Pharm. 85, 148–161 (2014).
Salahpour, A. et al. BRET biosensors to study GPCR biology, pharmacology, and signal transduction. Front Endocrinol. (Lausanne) 3, 105 (2012).
Charfi, I. et al. Ligand- and cell-dependent determinants of internalization and cAMP modulation by delta opioid receptor (DOR) agonists. Cell Mol. Life Sci. 71, 1529–1546 (2014).
Winpenny, D., Clark, M. & Cawkill, D. Biased ligand quantification in drug discovery: from theory to high throughput screening to identify new biased mu opioid receptor agonists. Br. J. Pharm. 173, 1393–1403 (2016).
Black, J. W. & Leff, P. Operational models of pharmacological agonism. Proc. R. Soc. Lond. B Biol. Sci. 220, 141–162 (1983).
ADS CAS PubMed Google Scholar
Johnson, K., Guo, C., Gosink, M., Wang, V. & Hauben, M. Multinomial modeling and an evaluation of common data-mining algorithms for identifying signals of disproportionate reporting in pharmacovigilance databases. Bioinformatics 28, 3123–3130 (2012).
Kalso, E., Edwards, J. E., Moore, R. A. & McQuay, H. J. Opioids in chronic non-cancer pain: systematic review of efficacy and safety. Pain 112, 372–380 (2004).
Ballantyne, J. C. Opioid analgesia: perspectives on right use and utility. Pain. Physician 10, 479–491 (2007).
Rajagopal, S. Quantifying biased agonism: understanding the links between affinity and efficacy. Nat. Rev. Drug Disco. 12, 483 (2013).
Onaran, H. O., Rajagopal, S. & Costa, T. What is biased efficacy? Defining the relationship between intrinsic efficacy and free energy coupling. Trends Pharm. Sci. 35, 639–647 (2014).
Villiger, J. W. Binding of buprenorphine to opiate receptors. Regul. guanyl nucleotides Met. Ions. Neuropharmacol. 23, 373–375 (1984).
Brown, S. M., Holtzman, M., Kim, T. & Kharasch, E. D. Buprenorphine metabolites, buprenorphine-3-glucuronide and norbuprenorphine-3-glucuronide, are biologically active. Anesthesiology 115, 1251–1260 (2011).
Galligan, J. J. & Sternini, C. Insights into the role of opioid receptors in the GI tract: experimental evidence and therapeutic relevance. Handb. Exp. Pharm. 239, 363–378 (2017).
Cowie, A. L., Kosterlitz, H. W., Lydon, R. J. & Waterfield, A. A. The effects of morphine-like substances and their antagonists on transmission at the neuro-effector junction of the myenteric plexus-longitudinal muscle preparation of the guinea-pig ileum. Br. J. Pharm. 38, 465P–466P (1970).
Kosterlitz, H. W. & Waterfield, A. A. In vitro models in the study of structure-activity relationships of narcotic analgesics. Annu Rev. Pharm. 15, 29–47 (1975).
Yamanishi, Y., Pauwels, E. & Kotera, M. Drug side-effect prediction based on the integration of chemical and biological spaces. J. Chem. Inf. Model 52, 3284–3292 (2012).
Willett P. Similarity-based virtual screening using 2D fingerprints. Drug Discov Today. 11, 1046–1053 (2006)
Els, C. et al. Adverse events associated with medium- and long-term use of opioids for chronic non-cancer pain: an overview of Cochrane Reviews. Cochrane Database Syst. Rev. 10, CD012509 (2017).
Cumming, J. G., Davis, A. M., Muresan, S., Haeberlein, M. & Chen, H. Chemical predictive modelling to improve compound quality. Nat. Rev. Drug Disco. 12, 948–962 (2013).
van der Westhuizen, E. T., Breton, B., Christopoulos, A. & Bouvier, M. Quantification of ligand bias for clinically relevant beta2-adrenergic receptor ligands: implications for drug taxonomy. Mol. Pharm. 85, 492–509 (2014).
Woo, A. Y. & Xiao, R. P. beta-Adrenergic receptor subtype signaling in heart: from bench to bedside. Acta Pharm. Sin. 33, 335–341 (2012).
Luttrell, L. M. & Miller, W. E. Arrestins as regulators of kinases and phosphatases. Prog. Mol. Biol. Transl. Sci. 118, 115–147 (2013).
Charkoudian, N. & Rabbitts, J. A. Sympathetic neural mechanisms in human cardiovascular health and disease. Mayo Clin. Proc. 84, 822–830 (2009).
Hove-Madsen, L., Mery, P. F., Jurevicius, J., Skeberdis, A. V. & Fischmeister, R. Regulation of myocardial calcium channels by cyclic AMP metabolism. Basic Res. Cardiol. 91 Suppl 2, 1–8 (1996).
Finney, P. A. et al. Albuterol-induced downregulation of Gsalpha accounts for pulmonary beta(2)-adrenoceptor desensitization in vivo. J. Clin. Invest. 106, 125–135 (2000).
Sebastian, S. et al. The in vivo regulation of heart rate in the murine sinoatrial node by stimulatory and inhibitory heterotrimeric G proteins. Am. J. Physiol. 305, R435–R442 (2013).
Kenakin, T. Agonists: The Measurement of Affinity and Efficacy in Functional Assays. A Pharmacology Primer, 4th Edition 85–117 (Elsevier, 2014).
Molinari, P. et al. Morphine-like Opiates Selectively Antagonize Receptor-Arrestin Interactions. J. Biol. Chem. 285, 12522–12535 (2010).
Raffa, R. B. et al. The clinical analgesic efficacy of buprenorphine. J. Clin. Pharm. Ther. 39, 577–583 (2014).
Robertson, D. N. et al. Design and construction of conformational biosensors to monitor ion channel activation: a prototype FlAsH/BRET-approach to Kir3 channels. Methods 92, 19–35 (2016).
Boussif, O. et al. A versatile vector for gene and oligonucleotide transfer into cells in culture and in vivo: polyethylenimine. Proc. Natl Acad. Sci. USA 92, 7297–7301 (1995).
Audet, N. et al. Differential association of receptor-G betagamma complexes with beta-arrestin2 determines recycling bias and potential for tolerance of delta opioid receptor agonists. J. Neurosci. 32, 4827–4840 (2012).
Carpenter, B. et al. Stan: a probabilistic programming language. J. Stat. Softw. 76, 1–32 (2017).
This research was supported by a research contract from Pfizer Inc. and grants from the Natural Sciences and Engineering Research Council of Canada (Grant 311997 to G.P.) and the Canadian Institutes of Health Research MOP 324876 (to G.P.), MOP 102630 (to M.B. and O.L.) and Foundation grant (FDN-148431) to MB. MB holds a Canada Research Chair in Signal Transduction and Molecular Pharmacology. Dr Lichtarge's research was supported by National Institutes of Health (NIH 2R01 GM066099; NIH 5R01 GM079656). B.B. was supported by a studentship from Fonds de Recherche en Santé du Québec. P.D. was supported by a MITACS fellowship. This study was supported by the Quebec Consortium on Adverse effects of pain medications, an initiative funded by the Quebec Pain Research Network (QPRN) of the Fonds de recherche du Québec Santé.
Darren Cawkill
Present address: Apollo Therapeutics LLP, Stevenage Bioscience Catalyst, Gunnels Wood Road, Stevenage, SG1, 2FX, UK
Present address: Pfizer Inc, La Jolla, CA, 92121, USA
Yong Ren
Present address: Decibel Therapeutics, 1325 Boylston Street, Boston, MA, 02215, USA
Emma Van Der Westhuizen
Present address: Monash Institute of Pharmaceutical Sciences, Parkville, VIC, 3052, Australia
These authors contributed equally: Besma Benredjem, Jonathan Gallion.
Department of Pharmacology and Physiology, Faculty of Medicine, Université de Montréal, Montréal, QC, H3T 1J4, Canada
Besma Benredjem, Paul Dallaire & Graciela Pineyro
CHU Sainte-Justine research center, Montréal, QC, H3T 1C5, Canada
Besma Benredjem, Paul Dallaire, Johanie Charbonneau & Graciela Pineyro
Baylor College of Medicine, Houston, TX, 77030, USA
Jonathan Gallion & Olivier Lichtarge
Pfizer Inc, Groton, CT, 06340, USA
Dennis Pelletier, Darren Cawkill, Mark Gosink, Stephen Jenkinson, Yong Ren, Christopher Somps & Anne Schmidt
College of Medicine, Member of QU Health, Qatar University, Doha, Qatar
Karim Nagi
Institute for Research in Immunology and Cancer, Department of Biochemistry and Molecular Medicine, Université de Montréal, Montréal, QC, H3T 1J4, Canada
Viktoryia Lukasheva, Brigitte Murat, Emma Van Der Westhuizen, Christian Le Gouill & Michel Bouvier
Besma Benredjem
Jonathan Gallion
Dennis Pelletier
Paul Dallaire
Johanie Charbonneau
Mark Gosink
Viktoryia Lukasheva
Christopher Somps
Brigitte Murat
Christian Le Gouill
Olivier Lichtarge
Anne Schmidt
Michel Bouvier
Graciela Pineyro
G.P., M.B., B.B., J.G., A.S., D.C., C.L., P.D., and O.L. conceived the study. B.B., J.G., G.P., M.B., P.D., and M.G. wrote the paper. B.B., J.C., J.G., K.N., M.G., C.L., S.J., V.L., C.S., D.P., B.M., Y.R., P.D., and E.V.W. performed experiments and/or data analysis.
Correspondence to Michel Bouvier or Graciela Pineyro.
M.B., C.L., and G.P. have a patent on BRET biosensors licensed to Domain Therapeutics North America and Pfizer Inc. The remaining authors declare no competing interests.
Peer review information: Nature Communications thanks Stephen Duffull and other anonymous reviewer(s) for their contribution to the peer review of this work.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Description of Additional Supplementary Files
Supplementary Data 1
Supplementary Data 10
Source data
Benredjem, B., Gallion, J., Pelletier, D. et al. Exploring use of unsupervised clustering to associate signaling profiles of GPCR ligands to clinical response. Nat Commun 10, 4075 (2019). https://doi.org/10.1038/s41467-019-11875-6
Characterization of recent non-fentanyl synthetic opioids via three different in vitro µ-opioid receptor activation assays
Marthe M. Vandeputte
Mattias Persson
Christophe P. Stove
Archives of Toxicology (2022)
Selective activation of Gαob by an adenosine A1 receptor agonist elicits analgesia without cardiorespiratory depression
Mark J. Wall
Emily Hill
Bruno G. Frenguelli
Nature Communications (2022)
Determination of G-protein–coupled receptor oligomerization by molecular brightness analyses in single cells
Ali Işbilir
Robert Serfling
Paolo Annibale
Nature Protocols (2021)
In vitro functional characterization of a panel of non-fentanyl opioid new psychoactive substances
Annelies Cannaert
Editors' Highlights
Nature Communications (Nat Commun) ISSN 2041-1723 (online) | CommonCrawl |
\begin{definition}[Definition:Sequence/Empty Sequence]
An '''empty sequence''' is a (finite) sequence containing no terms.
Thus an '''empty sequence''' is a mapping from $\O$ to $S$, that is, the empty mapping.
Thus by definition an '''empty sequence''' is a (finite) sequence whose length is $0$.
\end{definition} | ProofWiki |
A collaborative EPQ inventory model for a three-echelon supply chain with multiple products considering the effect of marketing effort on demand
$ E $-eigenvalue localization sets for tensors
Caili Sang 1,2, and Zhen Chen 1,,
School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou 550025, China
College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025, China
* Corresponding author: Zhen Chen
Received September 2018 Revised December 2018 Published May 2019
Fund Project: This work is supported by National Natural Science Foundation of China (No. 11501141), Science and Technology Projects of Education Department of Guizhou Province (Grant No. KY[2015]352), and Science and Technology Top-notch Talents Support Project of Education Department of Guizhou Province (Grant No. QJHKYZ [2016]066)
Several existing $Z$-eigenvalue localization sets for tensors are first generalized to $E$-eigenvalue localization sets. And then two tighter $ E$-eigenvalue localization sets for tensors are presented. As applications, a sufficient condition for the positive definiteness of fourth-order real symmetric tensors, a sufficient condition for the positive semi-definiteness of fourth-order real symmetric tensors, and a new upper bound for the $ Z$-spectral radius of weakly symmetric nonnegative tensors are obtained. Finally, numerical examples are given to verify the theoretical results.
Keywords: Z-eigenvalues, E-eigenvalues, localization sets, nonnegative tensors, spectral radius.
Mathematics Subject Classification: Primary: 15A18, 15A69, 15A72, 15A42.
Citation: Caili Sang, Zhen Chen. $ E $-eigenvalue localization sets for tensors. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019042
A. Ammar, F. Chinesta and A. Falcó, On the convergence of a greedy rank-one update algorithm for a class of linear systems, Arch. Comput. Methods Eng., 17 (2010), 473-486. doi: 10.1007/s11831-010-9048-z. Google Scholar
B. D. Anderson, N. K. Bose and E. I. Jury, Output feedback stabilization and related problems-solutions via decision methods, IEEE Trans. Automat. Control, AC20 (1975), 53-66. doi: 10.1109/tac.1975.1100846. Google Scholar
L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, Med. Image Comput. Comput. Assist. Interv., 5241 (2008), 1–8. Available from: https://www.ncbi.nlm.nih.gov/pubmed/18979725. doi: 10.1007/978-3-540-85988-8_1. Google Scholar
N. K. Bose and P. S. Kamt, Algorithm for stability test of multidimensional filters, IEEE Trans. Acoust. Speech Signal Process, 22 (1974), 307-314. doi: 10.1109/TASSP.1974.1162592. Google Scholar
N. K. Bose and R. W. Newcomb, Tellegon's theorem and multivariate realizability theory, Int. J. Electron, 36 (1974), 417-425. doi: 10.1080/00207217408900421. Google Scholar
K. C. Chang, K. J. Pearson and T. Zhang, Some variational principles for $Z$-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438 (2013), 4166-4182. doi: 10.1016/j.laa.2013.02.013. Google Scholar
R. A. Devore and V. N. Temlyakov, Some remarks on greedy algorithms, Adv. comput. Math., 5 (1996), 173-187. doi: 10.1007/BF02124742. Google Scholar
A. Falco and A. Nouy, A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach, J. Math. Anal. Appl., 376 (2011), 469-480. doi: 10.1016/j.jmaa.2010.12.003. Google Scholar
J. He, Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl., 20 (2016), 1290-1301. Google Scholar
J. He, Y. M. Liu, H. Ke, J. K. Tian and X. Li, Bounds for the $Z$-spectral radius of nonnegative tensors, Springerplus, 5 (2016), 1727. doi: 10.1186/s40064-016-3338-3. Google Scholar
J. He and T. Z. Huang, Upper bound for the largest $Z$-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114. doi: 10.1016/j.aml.2014.07.012. Google Scholar
J. C. Hsu and A. U. Meyer, Modern Control Principles and Applications, The McGraw-Hill Series in Advanced Chemistry McGraw-Hill Book Co., Inc., New York-Toronto-London, 1956. Google Scholar
E. Kofidis and P. A. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884. doi: 10.1137/S0895479801387413. Google Scholar
T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), 1095-1124. doi: 10.1137/100801482. Google Scholar
L. D. Lathauwer, B. D. Moor and J. Vandewalle, On the best rank-1 and rank-($R_1,R_2,\ldots ,R_N $) approximation of higer-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342. doi: 10.1137/S0895479898346995. Google Scholar
W. Li, D. Liu and S. W. Vong, $Z$-eigenpair bounds for an irreducible nonnegative tensor, Linear Algebra Appl., 483 (2015), 182-199. doi: 10.1016/j.laa.2015.05.033. Google Scholar
C. Li and Y. Li, An eigenvalue localization set for tensors with applications to determine the positive (semi-)definitenss of tensors, Linear Multilinear Algebra, 64 (2016), 587-601. doi: 10.1080/03081087.2015.1049582. Google Scholar
C. Li, Y. Li and X. Kong, New eigenvalue inclusion sets for tensors, Numer. Linear Algebra Appl., 21 (2014), 39-50. doi: 10.1002/nla.1858. Google Scholar
C. Li, Z. Chen and Y. Li, A new eigenvalue inclusion set for tensors and its applications, Linear Algebra Appl., 481 (2015), 36-53. doi: 10.1016/j.laa.2015.04.023. Google Scholar
C. Li, J. Zhou and Y. Li, A new Brauer-type eigenvalue localization set for tensors, Linear Multiliear Algebra, 64 (2016), 727-736. doi: 10.1080/03081087.2015.1119779. Google Scholar
C. Li, A. Jiao and Y. Li, An $S $-type eigenvalue location set for tensors, Linear Algebra Appl., 493 (2016), 469-483. doi: 10.1016/j.laa.2015.12.018. Google Scholar
L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in CAMSAP'05: Proceeding of the IEEE International Workshop on Computational Advances in MultiSensor Adaptive Processing, 2005,129–132.Google Scholar
Q. Liu and Y. Li, Bounds for the $ Z$-eigenpair of general nonnegative tensors, Open Math., 14 (2016), 181-194. doi: 10.1515/math-2016-0017. Google Scholar
L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007. Google Scholar
L. Qi, G. Yu and E. X. Wu, Higher order positive semidefinite diffusion tensor imaging, SIAM J. Imaging Sciences, 3 (2010), 416-433. doi: 10.1137/090755138. Google Scholar
L. Qi, Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines, J. Symbolic Comput., 41 (2006), 1309-1327. doi: 10.1016/j.jsc.2006.02.011. Google Scholar
L. Qi, The best rank-one approximation ratio of a tensor space, SIAM J. Matrix Anal. Appl., 32 (2011), 430-442. doi: 10.1137/100795802. Google Scholar
C. Sang, A new Brauer-type $ Z$-eigenvalue inclusion set for tensors, Numer. Algor., (2018), 1–14. doi: 10.1007/s11075-018-0506-2. Google Scholar
Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581-1595. doi: 10.1137/130909135. Google Scholar
Y. Wang and G. Wang, Two $ S$-type $ Z$-eigenvalue inclusion sets for tensors, J. Inequal. Appl., 2017 (2017), Paper No. 152, 12 pp. doi: 10.1186/s13660-017-1428-6. Google Scholar
G. Wang, G. L. Zhou and L. Caccetta, $Z$-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst., Ser. B., 22 (2017), 187-198. doi: 10.3934/dcdsb.2017009. Google Scholar
Y. Wang and L. Qi, On the successive supersymmetric rank-1 decomposition of higher-order supersymmetric tensors, Numer. Linear Algebra Appl., 14 (2007), 503-519. doi: 10.1002/nla.537. Google Scholar
T. Zhang and G. H. Golub, Rank-one approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550. doi: 10.1137/S0895479899352045. Google Scholar
J. Zhao, A new $ Z$-eigenvalue localization set for tensors, J. Inequal. Appl., 2017 (2017), Paper No. 85, 9 pp. doi: 10.1186/s13660-017-1363-6. Google Scholar
J. Zhao and C. Sang, Two new eigenvalue localization sets for tensors and theirs applications, Open Math., 15 (2017), 1267-1276. doi: 10.1515/math-2017-0106. Google Scholar
Figure 1. Comparisons of $ \mathcal{K}(\mathcal{A}) $, $ \mathcal{L}(\mathcal{A}) $, $ \Psi(\mathcal{A}) $, $ \Upsilon(\mathcal{A}) $ and $ \Omega(\mathcal{A}) $.
Figure 2. Comparisons of $ \Omega(\mathcal{A}) $ and $ \triangle(\mathcal{A}) $.
Table 1. Upper bounds of $ \varrho(\mathcal{A}) $
Method $ \varrho(\mathcal{A})\leq $
Theorem 3.1, i.e., Corollary 4.5 of [29] 23.0000
Theorem 3.3 of [16] 22.8625
Theorem 3.4 of [35], where $ S=\{3\},\bar{S}=\{1,2\} $ 22.8521
Theorem 3.2, i.e., Theorem 4.5 of [31] 22.8149
Theorem 3.5 of [9] 22.7163
Theorem 6 of [10] 22.6290
Theorem 3.3, i.e., Theorem 5 of [34] 22.5000
Theorem 4 of [30], where $ S=\{3\},\bar{S}=\{1,2\} $ 22.4195
Theorem 3.5 21.2604
Chaoqian Li, Yaqiang Wang, Jieyi Yi, Yaotang Li. Bounds for the spectral radius of nonnegative tensors. Journal of Industrial & Management Optimization, 2016, 12 (3) : 975-990. doi: 10.3934/jimo.2016.12.975
Chen Ling, Liqun Qi. Some results on $l^k$-eigenvalues of tensor and related spectral radius. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 381-388. doi: 10.3934/naco.2011.1.381
Juan Meng, Yisheng Song. Upper bounds for Z$ _1 $-eigenvalues of generalized Hilbert tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-8. doi: 10.3934/jimo.2018184
Vesselin Petkov, Georgi Vodev. Localization of the interior transmission eigenvalues for a ball. Inverse Problems & Imaging, 2017, 11 (2) : 355-372. doi: 10.3934/ipi.2017017
Yining Gu, Wei Wu. New bounds for eigenvalues of strictly diagonally dominant tensors. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 203-210. doi: 10.3934/naco.2018012
Bruno Colbois, Alexandre Girouard. The spectral gap of graphs and Steklov eigenvalues on surfaces. Electronic Research Announcements, 2014, 21: 19-27. doi: 10.3934/era.2014.21.19
Lixing Han. An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 583-599. doi: 10.3934/naco.2013.3.583
Jun Zhang, Xinyue Fan. An efficient spectral method for the Helmholtz transmission eigenvalues in polar geometries. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4799-4813. doi: 10.3934/dcdsb.2019031
Victor Kozyakin. Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 143-158. doi: 10.3934/dcdsb.2010.14.143
Manuel V. C. Vieira. Derivatives of eigenvalues and Jordan frames. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 115-126. doi: 10.3934/naco.2016003
Fioralba Cakoni, Drossos Gintides. New results on transmission eigenvalues. Inverse Problems & Imaging, 2010, 4 (1) : 39-48. doi: 10.3934/ipi.2010.4.39
Andreas Kirsch. On the existence of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (2) : 155-172. doi: 10.3934/ipi.2009.3.155
Yining Gu, Wei Wu. Partially symmetric nonnegative rectangular tensors and copositive rectangular tensors. Journal of Industrial & Management Optimization, 2019, 15 (2) : 775-789. doi: 10.3934/jimo.2018070
Gang Wang, Guanglu Zhou, Louis Caccetta. Z-Eigenvalue Inclusion Theorems for Tensors. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 187-198. doi: 10.3934/dcdsb.2017009
Armin Lechleiter. The factorization method is independent of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (1) : 123-138. doi: 10.3934/ipi.2009.3.123
Gianne Derks, Sara Maad, Björn Sandstede. Perturbations of embedded eigenvalues for the bilaplacian on a cylinder. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 801-821. doi: 10.3934/dcds.2008.21.801
Alexandre Girouard, Iosif Polterovich. Upper bounds for Steklov eigenvalues on surfaces. Electronic Research Announcements, 2012, 19: 77-85. doi: 10.3934/era.2012.19.77
Luc Robbiano. Counting function for interior transmission eigenvalues. Mathematical Control & Related Fields, 2016, 6 (1) : 167-183. doi: 10.3934/mcrf.2016.6.167
Todd Kapitula, Björn Sandstede. Eigenvalues and resonances using the Evans function. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 857-869. doi: 10.3934/dcds.2004.10.857
Caili Sang Zhen Chen
\begin{document}$ E $\end{document}-eigenvalue localization sets for tensors" readonly="readonly"> | CommonCrawl |
miRNA analysis with Prost! reveals evolutionary conservation of organ-enriched expression and post-transcriptional modifications in three-spined stickleback and zebrafish
Thomas Desvignes ORCID: orcid.org/0000-0001-5126-87851 na1,
Peter Batzel1 na1,
Jason Sydes1 na1,
B. Frank Eames2 &
John H. Postlethwait ORCID: orcid.org/0000-0002-5476-21371
Scientific Reports volume 9, Article number: 3913 (2019) Cite this article
Sequence annotation
MicroRNAs (miRNAs) can have organ-specific expression and functions; they can originate from dedicated miRNA genes, from non-canonical miRNA genes, or from mirror-miRNA genes and can also experience post-transcriptional variation. It remains unclear, however, which mechanisms of miRNA production or modification are organ-specific and the extent of their evolutionary conservation. To address these issues, we developed the software Prost! (PRocessing Of Short Transcripts), which, among other features, helps quantify mature miRNAs, accounts for post-transcriptional processing, such as nucleotide editing, and identifies mirror-miRNAs. Here, we applied Prost! to annotate and analyze miRNAs in three-spined stickleback (Gasterosteus aculeatus), a model fish for evolutionary biology reported to have a miRNome larger than most teleost fish. Zebrafish (Danio rerio), a distantly related teleost with a well-known miRNome, served as comparator. Our results provided evidence for the existence of 286 miRNA genes and 382 unique mature miRNAs (excluding mir430 gene duplicates and the vaultRNA-derived mir733), which doesn't represent a miRNAome larger than other teleost miRNomes. In addition, small RNA sequencing data from brain, heart, testis, and ovary in both stickleback and zebrafish identified suites of mature miRNAs that display organ-specific enrichment, many of which are evolutionarily-conserved in the brain and heart in both species. These data also supported the hypothesis that evolutionarily-conserved, organ-specific mechanisms may regulate post-transcriptional variations in miRNA sequence. In both stickleback and zebrafish, miR2188-5p was edited frequently with similar nucleotide changes in the seed sequence with organ specific editing rates, highest in the brain. In summary, Prost! is a new tool to identify and understand small RNAs, to help clarify a species' miRNA biology as shown here for an important model for the evolution of developmental mechanisms, and to provide insight into organ-enriched expression and the evolutionary conservation of miRNA post-transcriptional modifications.
microRNAs (miRNAs) are small non-coding RNA molecules about 20-22 nucleotides long that control gene expression post-transcriptionally by repressing translation or inducing the decay of targeted messenger RNA transcripts (mRNAs)1,2,3. miRNAs participate in virtually all biological processes, including the control of cell specification, cell differentiation, organ development, and organ physiology4,5,6 as well as pathologies in humans and other animals7,8,9,10. miRNA genes also appear to be evolutionarily-conserved in number, sequence, and syntenies across metazoans3,11,12,13,14,15, but the evolutionary conservation of miRNA organ expression patterns remain incompletely understood.
Canonically, miRNA genes are transcribed into a primary transcript (pri-miRNA) that folds into a hairpin, from which the enzyme Drosha cleaves off the free 5′ and 3′ ends, thereby producing the precursor miRNA (pre-miRNA). In the case of clustered miRNAs, the pri-miRNAs can be monocistronic or polycistronic and can fold into the same number of hairpins as the number of miRNAs in the cluster. The pre-miRNA, which assumes a stem-loop hairpin conformation, is then exported into the cytoplasm where a second enzyme, Dicer, trims off the loop and releases a miRNA duplex16,17. One strand of the miRNA duplex is usually degraded, while the other strand loads into the RNA-Induced Silencing Complex (RISC), the effector of the miRNA regulation system. Once incorporated into the RISC, the miRNA drives the association of the enzymatic complex to specific mRNA transcripts by base pairing of the miRNA seed (nucleotides 2–8 from the 5′ end) to the targeted transcript's 3′ UTR. The association of the RISC to the messenger RNA will either induce the decay of the transcript or prevent its translation, depending on pairing strength. Other pathways and other gene types can also produce miRNAs (e.g. miRNAs from Drosha- or Dicer-independent pathways, miRNAs produced by both DNA strands at the same locus (mirror miRNAs), lncRNAs, and snoRNAs18,19,20,21) and the most common alternative miRNA biogenesis pathway is the processing of miRtrons, which are miRNA hairpins originating from spliced introns of protein-coding genes22,23.
Besides originating from a variety of biogenesis pathways and gene types, miRNA sequence variations can arise post-transcriptionally, resulting in variations in size and nucleotide sequence; these variants are called isomiRs2,24,25. The most frequent post-transcriptional modification involves variations in length at the 3′ end of miRNAs. Length modifications at the 5′ end of miRNAs occur less frequently than at the 3′ end, perhaps because they cause a shift in the seed, which can modify the identity of targeted transcripts and thus can drastically change the miRNA's function26,27. miRNA sequence variation can also occur due to post-transcriptional editing, in which ADAR (adenosine deaminase, RNA-specific) enzymes post-transcriptionally modify a nucleotide, usually an adenosine (A), into another base, usually an inosine (I)28,29,30. These post-transcriptional modifications have now been shown to be physiologically relevant30,31,32,33,34, but whether post-transcriptional editing occurs in a directed and regulated, organ-specific manner is still currently unknown.
The diversity of miRNAs, their variations, and the rapid expansion of small RNA sequencing reveal the need for small RNA analysis tools that can encompass the full diversity of gene origins and variations in miRNA sequences. Several bioinformatic tools are currently available to study miRNAs using small RNA sequencing datasets35,36,37,38,39,40,41. To study miRNA expression, some tools compare sequenced reads to annotated RNA sequences without aligning directly to a genome38,42. Many tools start by filtering reads that can readily be annotated as miRNAs and then report their expression, sometimes using a genomic reference. Other tools make use of genomic alignments and specialize in the discovery of novel miRNAs36,43 or the study of isomiRs35,44. These tools often perform well for their respective functions, but in many cases, lack transparency in their filtering and annotating algorithms, have few user-defined parameter choices that might help tune a user's specific application, and/or lack the ability to inspect the entire small RNA dataset and omit sequences not already annotated as a miRNA. With increasing amounts of data and sequence read diversity, a more global approach was required to address sequencing output by analyzing every single read – even if it is not yet annotated – as a type of coding or non-coding RNA. In addition, analysis tools should give attention to read alignments on a genomic reference to differentiate fragments potentially originating from one or multiple loci. While many tools are available to study small RNA sequencing datasets, current tools usually do not provide a comprehensive, genome-based analysis of small RNA datasets, thus limiting the study of the full complexity of an experiment by failing to report some of the post-transcriptional processes affecting the diversity of small RNAs.
To help study the complexity and diversity of miRNA sequences in small RNA-seq data, we generated a new software tool Prost! (PRocessing Of Small Transcripts) that facilitates the identification of miRNAs for annotation, quantifies annotated miRNAs, and details variations (isomiRs) observed in each sample. Prost! is open-source, publicly available software45. Earlier versions of Prost! have been used to annotate zebrafish and spotted gar miRNAs46,47, as well as to identify erythromiRs in white-blooded Antarctic icefish48.
To investigate the evolutionary conservation of miRNAs in teleost fish, we performed small RNA sequencing on four organs (brain, heart, testis, and ovary) in two distantly related teleost laboratory model fishes: the medical genetics model zebrafish Danio rerio and the evolutionary genetics model three-spined stickleback Gasterosteus aculeatus. While zebrafish miRNAs are well annotated46,49,50, stickleback miRNAs aren't, and current predicted annotations provide miRNA gene number estimates ranging from several hundred to well over a thousand genes51,52,53,54, which is more than four times the number of miRNA genes in zebrafish. In addition, no study has so far investigated the potential conservation of miRNA expression patterns across teleost fish species, or studied post-transcriptional modifications in teleost mature miRNAs. Here, we addressed the following questions: (1) Is the stickleback miRNome significantly larger than that in other teleost species as reported? (2) Do zebrafish and stickleback share organ-enriched expression of specific miRNAs? And (3) Do animals regulate post-transcriptional modifications to display organ-specificity and are organ-specific modifications shared by zebrafish and stickleback?
Origin of sampled fish
Four zebrafish individuals (Danio rerio, AB strain, two males and two females) were obtained from the University of Oregon Aquatic Animal Core Facility and four three-spined stickleback (Gasterosteus aculeatus, two males and two females) of a fresh water laboratory strain derived from Boot Lake, Alaska were obtained from Mark Currey in the W. Cresko Laboratory (University of Oregon). To limit biases that might arise from differences in culture, physiological state, and sampling conditions, animals of each species were raised under their respective optimal conditions of temperature (20 °C and 28.5 °C for stickleback and zebrafish, respectively), photoperiod (12/12 h light/dark for both species), and densities (one fish per four liters and 10 fish per liter for stickleback and zebrafish55, respectively). In addition, all animals were sexually mature adults that had been reproductively active for several months. All animals were handled in accordance with good animal practice as approved by the University of Oregon Institutional Animal Care and Use Committee (Animal Welfare Assurance Number A‐3009‐01, IACUC protocol 12‐02RA).
RNA extraction and small RNA library preparation
Immediately following euthanasia by overdose of MS-222, fin clips, brains, heart ventricles, and testes were sampled from two male zebrafish and two male stickleback, and fin clips and ovaries were sampled from two female zebrafish and two female stickleback. All organs were dissected by the same person and extractions were processed identically. DNA was extracted from fin clips. Proteinase K was used to break open the cells, cell debris was then precipitated by centrifugation (10 minutes, 4 °C, 12,000 g), and the DNA extract was washed once with isopropanol and centrifuged (10 minutes, 4 °C, 12,000 g), followed by two 75% ethanol washes and centrifugation steps (10 minutes, 4 °C, 12,000 g), before resuspension of DNA in nuclease-free water. Both small and large RNAs from each individual organ were extracted using Norgen Biotek microRNA purification kit (Norgen Biotek, Thorold, ON, Canada) according to the manufacturer's instructions. Using the small RNA extract fractions, for each male of each species, we prepared three individual libraries (brain, heart ventricle, and testis), and for each female of each species we prepared a single library (ovary). In total, 16 small RNA libraries were then prepared and barcoded using the Bioo Scientific NEXTflexTM small RNA sequencing v1 kit (Bioo Scientific, Austin, TX, USA) with 15 PCR cycles. Libraries were sequenced on the Illumina HiSeq 2500 platform at the University of Oregon Genomics and Cell Characterization Core Facility (GC3F). Raw single-end 50-nt long reads were deposited in the NCBI Short Read Archive under project accession numbers SRP157992 and SRP039502 for stickleback and zebrafish, respectively.
Prost! workflow
Raw reads from all sixteen libraries were pre-processed identically. Reads that did not pass Illumina's chastity filter were discarded. Adapter sequences were trimmed from raw reads using cutadapt56 with parameters:--overlap 10 -a TGGAATTCTCGGGTGCCAAGG --minimum-length 1. Reads were then quality filtered using fastq_quality_filter of the FASTX-Toolkit (http://hannonlab.cshl.edu/fastx_toolkit/commandline.html) (with parameters: −Q33 −q 30 −p 100). Remaining reads were converted from FASTQ format to FASTA format.
Filtered reads were processed using Prost!, which is available online at https://prost.readthedocs.io and https://github.com/uoregon-postlethwait/prost 45. Briefly, Prost! size-selects reads for lengths typical of miRNAs and tracks the number of reads matching any given sequence. For miRNAs, we configured Prost! to select for reads 17 to 25 nucleotides in length. Prost! then aligns the unique set of sequences to a reference genome using bbmapskimmer.sh of the BBMap suite (https://sourceforge.net/projects/bbmap/) (with parameters: mdtag = t scoretag = f inserttag = f stoptag = f maxindel = 0 slow = t outputunmapped = f idtag = f minid = 0.50 ssao = f strictmaxindel = t usemodulo = f cigar = t sssr = 0.25 trimreaddescriptions = t secondary = t ambiguous = all maxsites = 4000000 k = 7 usejni = f maxsites2 = 4000000 idfilter = 0.50). We configured Prost! to use the publicly available genome assemblies for three-spined stickleback (BROAD S1) and zebrafish (GRCz10)54 for the study of stickleback and zebrafish reads, respectively. Prost! then groups small RNA sequences that have overlapping genomic locations on each respective genome assembly. We configured Prost! to retain only sequences with a minimum of five identical reads for the initial annotation pass, and only sequences with a minimum of 30 reads for the differential expression analysis.
Prost! annotates reads grouped according to genomic location by aligning against the mature and hairpin sequences of known miRNAs using bbmap.sh of the BBMap suite, as well as by performing a reverse alignment of known mature sequences against the unique set of reads (with parameters: mdtag = t scoretag = f inserttag = f stoptag = f maxindel = 0 slow = t outputunmapped = f idtag = f minid = 0.50 ssao = f strictmaxindel = t usemodulo = f cigar = t sssr = 0.25 trimreaddescriptions = t secondary = t ambiguous = all maxsites = 4000000 k = 7 usejni = f maxsites2 = 4000000 idfilter = 0.50 nodisk). In the current study, we configured Prost! to use all mature and hairpin sequences for chordates in miRBase Release 2149, as well as the predicted stickleback miRNA annotations51,53,54, the extended zebrafish miRNA annotation46, and the spotted gar miRNA annotation47. All annotation datasets used in this study are available on the Prost! Github page (https://github.com/uoregon-postlethwait/prost). Gene nomenclature follows recent conventions2, including those for zebrafish57. For miRNA genes that didn't display phylogenetic conservation and were only predicted by one study51,53,54 following criteria for confident annotation proposed by previous studies2,49,50, each miRNA was annotated if (1) it originated from a maximum of six loci on the genome (which is the maximum number of locations that members of the largest miRNA families yet known originate from, i.e. let7a-5p and miR9-5p); (2) both strands of the hairpin were present in the sequencing dataset; (3) it displayed consistent 5′-end processing; and (4) it had a minimal expression level of 5 RPM for at least one of the two strands. For each mature miRNA locus, the most expressed isomiR was retrieved and used as the reference in the annotation. The annotation strategy used is detailed in a specific file on the Prost! documentation page (https://prost.readthedocs.io). Supplementary Table 1 provides a description of the sequencing depth and annotation statistics for each library.
Manual miRNA annotation
For miRNA genes known in several teleost species but absent from our sequencing data, we performed a manual search in the stickleback genome assembly BROAD S1. We first retrieved the precursor sequence deposited in miRBase Release 2149 for the gene in the most closely related teleost fish, and aligned the sequence to the stickleback genome assembly using BLASTN with sensitivity set for "short sequences" using Ensembl54. Candidate regions (E-val < 1), were declared to have conserved syntenies by comparing three genes upstream and three genes downstream of the genomic hit in the stickleback genome to the corresponding region in other species in which the miRNA was annotated. If Ensembl called one or more of the six flanking genes as orthologs, synteny was considered to be conserved, the gene was manually annotated in stickleback, and the precursor sequence was extracted from the stickleback genome assembly.
Differential expression analyses
From Prost! output, we used the non-normalized read counts of annotated miRNA reads to perform differential expression analysis between organs by pair-wise comparisons using the DESeq2 package58. For isomiR reads predicted to be variants of two or more miRNAs with equal probability, we partitioned their read counts proportionally based on counts of unambiguously annotated miRNAs that might give rise to the isomiR. In addition, when expression of an annotated miRNA was not detected in an organ, a read count of one was used instead of zero to facilitate the calculation of an adjusted p-value for that miRNA. We selected the "local" type trend line fitting model (FitType) and, to avoid false positives, at the expense of potential false negatives, we used a stringent maximum adjusted p-value of 1% (Benjamini and Hochberg procedure to adjust for multiple testing) to consider miRNAs as differentially expressed between two organs. Each pairwise comparison was subsequently verified for appropriate p-value distributions and compatibility with the negative binomial probability model used by DESeq2 (Supplementary File 1 for stickleback and Supplementary File 1 for zebrafish). Heat maps were generated using the Broad Institute Morpheus webserver55,59 (https://software.broadinstitute.org/morpheus/) using log2-transformed normalized counts from annotated miRNAs that displayed a minimum normalized average expression of 5 Reads-per-Million (RPM) over the entire dataset. Hierarchical clustering on both rows and columns was performed using the "one minus Pearson's correlation" model and the "average" linkage method.
Organ-Enrichment Index
To evaluate organ-specific expression enrichment, we calculated for each miRNA an organ enrichment index (OEI), which is analogous to the tissue specificity index (TSI) 'tau' for mRNAs60 and has been previously used for miRNAs61. The OEI varies from 0 to 1, with OEI close to 0 corresponding to miRNAs expressed in many or most organs at similar levels (i.e. 'housekeeping' miRNAs), and OEI close to 1 corresponding to miRNAs expressed in a specific organ (i.e. organ-enriched miRNAs). The OEI for miRNA j is calculated as:
$$oe{i}_{j}=\frac{{\sum }_{i=1}^{N}\,(1-{x}_{j,i})}{N-1}$$
where N corresponds to the total number of organs studied and xi.j is the expression of miRNA j in organ i normalized by the maximal expression among all organs.
PCR analyses and target predictions
To confirm miRNA editing events, we designed PCR primers to amplify primary miRNAs (pri-miRNAs) both from genomic DNA and from large RNA extracts of each investigated individual. This process allows the verification of putatively edited bases, rules out single nucleotide polymorphisms (SNPs) with respect to the reference genome sequence, and tests whether the transcribed pri-miRNA contains the edited base. Supplementary Table 2 contains primer sequences. PCR reactions were performed as previously described62, and the product of each reaction was cleaned using Diffinity RapidTip (Diffinity Genomics, USA) and sequenced by Genewiz (South Plainfield, NJ, USA). The relative frequency of each base at various positions in each miRNAs was displayed using the WebLogo3 webserver63. Putative miRNA targets were predicted using miRAnda 3.3a64,65 with default parameters (i.e. -sc 140.0 -en 1.0) and the 3′ UTR longer than 24 nucleotides present in Ensembl release 79 genome assemblies (BROAD S1 for stickleback, Zv9 for zebrafish). Stickleback to zebrafish gene orthology was called by taking the ortholog with the lowest accession number as called by Ensembl Biomart.
Prost!, a tool for analyzing small RNA sequencing reads
Prost! differs in three main ways from the majority of other tools developed to investigate small RNA-seq data. First, Prost! aligns reads to a user-defined genomic dataset (e.g. a genome assembly, Fig. 1). This initial alignment permits retention of all sequencing reads that match, perfectly or with a few errors, the "genomic dataset", even if these matches are not yet known to be coding or non-coding RNA fragments. As such, Prost! enables the study of not only miRNAs, but also the identification of other small RNAs, such as piRNAs, t-RNA fragments, or the degradome of other RNA biotypes (e.g. snoRNAs, Y_RNA, vault-RNA). Second, Prost! groups reads based on their potential genomic origin(s), on their seed sequence, and ultimately on their annotation (Fig. 1). This step allows the regrouping of sequence variants that could originate from one locus or from multiple loci. Conversely, this step discriminates reads that could originate only from a limited number of paralogous loci, increasing the understanding of gene expression and locus-specific expression levels. Third, Prost! analyzes in depth the subset of reads that had been annotated based on the user-provided annotation dataset (e.g. miRNA or piRNA) and reports frequencies of individual sequence variations with respect to both the reference genome and the most expressed sequence that aligns perfectly to the genome from a genomic location group or annotation group (Fig. 1). This step ultimately provides a comprehensive report on potential post-transcriptional modifications for each group of sequences.
Prost! data processing flowchart. Flow chart displaying the input, pre-processing, categorization, alignment, and output report steps of Prost!
Prost! was written in Python and takes as input a list of sequencing sample files. Prost! can be configured with a simple and well annotated configuration file and optional command line flags, allowing the user to optimize Prost! for each specific dataset, experimental design, and experimental goal (e.g. annotation or quantification). The output can be retrieved either as an individual report per analysis step as tab-separated value files, or a single combined Excel file with each step provided as an individual tab that contains indexes, similar to primary-foreign keys of relational databases, facilitating the navigation from tab-to-tab to retrace and understand the entire analysis process (Fig. 1). Documentation on the Prost! Github page provides a complete description of the output file (https://prost.readthedocs.io). Supplementary Files 3 and 4 are Prost! output files used for differential expression analysis for stickleback and zebrafish, respectively.
Stickleback miRNome
ZooMir51, Ensembl54, and Rastorguev et al.53 predicted that stickleback has 483, 504, and 595 miRNA genes respectively, and Chaturvedi et al.52 predicted 1486 mature miRNAs. Other well annotated teleost fish have substantially smaller miRNomes, consisting of about 250–350 genes49,50,54. This discrepancy raises the question of whether the stickleback miRNome is comparable to other well-annotated teleost genomes and contains approximately 250 to 300 miRNA genes, or whether the additional predicted stickleback miRNA genes are lineage-specific miRNAs and/or false predictions.
Using Prost! on our small RNA sequencing data of brain, heart, testis and ovary, we annotated 273 miRNA genes in stickleback with a total of 382 unique mature miRNAs (excluding the highly replicated mir430 genes and the vaultRNA-derived mir733) (Table 1). Among these 273 miRNA genes, we were able to annotate both 5p and 3p strands for 221 genes (81%) and only one strand for 52 genes (19%) (Table 1). Among these 273 miRNA genes, three genes (Rastorguev-366, -458, and -44353, see Supplemental Files 5 and Table 3 for sequences) had reads in our sequencing data, but none of them displayed phylogenetic conservation. These three miRNAs are therefore likely to be stickleback-specific miRNAs. The manual annotation of known conserved teleost miRNAs46,49,50 that were not among the 273 stickleback miRNA genes annotated with Prost! identified 13 more miRNA genes (Table 1). For these 13 predicted miRNA genes, however, no mature miRNAs were present in our four-organ sequencing data, so we only annotated the putative pre-miRNAs.
Table 1 Stickleback miRNA annotation statistics.
In partial summary, with Prost! and additional manual annotation, 286 miRNA genes were annotated in stickleback. This collection represents a miRNAome similar in size to other well-annotated teleost species that typically contain approximatively 250 to 300 miRNA genes (excluding the mir430 genes). Supplementary Table 3 displays names, sequences, Ensembl Accession numbers if available, and positions on the stickleback 'BROAD S1' genome assembly for stickleback pre-miRNAs and mature miRNAs. Supplementary Files 5 and 6 provide FASTA format annotation for pre-miRNAs and unique mature miRNAs, respectively.
By comparing all available annotations and by excluding mir430 genes and the vaultRNA-derived mir733 miRNAs that form unique large families of miRNA genes, we found that the stickleback annotation generated using Prost! contained many of the miRNAs found in other stickleback annotations51,53,54, lacked some other genes, and contained two additional genes not present in previous annotations. We did not include in our comparison the Chaturvedi et al.52 annotation because it was generated without strand-specificity. Our annotation included 194 of 419 (46%) miRNA genes in ZoomiR, 242 of 593 miRNA genes (41%) in Rastorguev et al., and 251 of 365 miRNA genes (69%) in Ensembl (Fig. 2), after excluding one, 64, and 139 mir430 genes from Rastorguev et al., ZooMir, and Ensembl, respectively. The vaultRNA-derived mir733 miRNAs were not present in any other annotations. Only three miRNAs were missing in our annotation that were present in at least two of the three other annotations, but all three of these (mir204a, mir705, and mir1788) were among the 13 known evolutionarily-conserved miRNAs that we annotated by orthology and for which no sequencing reads were present in our dataset. In addition, our annotation contained two genes that are absent from all other annotations. These two genes are mir3120 (the mirror-miRNAs of mir214, see following section) and the ortholog of the zebrafish mir723b gene; both genes were previously annotated in zebrafish46. All other miRNAs missing from our annotation were predicted in only one of the other three annotations (Fig. 2) and our expression data couldn't confirm their predictions according to confident annotation criteria defined previously2,50. Prost! thus provides conservative results in the annotation of miRNA sequencing data. This finding suggests that predicted miRNAs not found in our annotation either correspond to false predictions or are stickleback-specific genes that our sequencing data lacks because we explored just four tissues at one developmental stage. Sequencing of a larger diversity of tissues, developmental stages, and/or greater sequencing depth could, however, provide evidence of consistent biogenesis and expression of some of these miRNAs, therefore validating them as miRNA genes.
Overlap of several existing stickleback miRNA annotations. Genomic locations of stickleback pre-miRNAs were retrieved from other stickleback annotations47,49,50 and compared with each other and with those identified by Prost! A pre-miRNA from one annotation was considered the same as a pre-miRNA from another annotation if they shared at least a 25 nucleotide overlap.
Identification of mirror-miRNAs in teleosts
In addition to the annotation of mature miRNAs and miRNA genes, Prost! facilitates the identification of mirror-miRNA candidates by automatically filtering small RNA reads that originate from opposite DNA strands at the same location in the genome21,46,66. Teleost fish have mirror-miRNAs and some are conserved across vertebrate species, including the conserved mirror-miRNA pair mir214/mir3120 in human and zebrafish, and at least two other teleost mirror-miRNA pairs (mir7547/mir7553 and mir7552a/mir7552aos)46. In the list of candidate mirror-miRNAs generated by Prost!, we found the mir214/mir3120 pair in both stickleback and zebrafish, but the two other known zebrafish mirror-miRNA pairs did not appear in our stickleback data. Although miR7552a-5p originating from the gene mir7552a was present in our stickleback sequencing data, sequencing reads from mirror mir7552aos were not; given the limited number of organs we studied, the mirror-miRNA pair mir7552a/mir7552aos might be expressed in other stickleback organs. The mirror-miRNA pair mir7547/mir7553 was not only lacking from our sequencing data, but also was not found in the stickleback genome assembly by sequence homology or conserved synteny, providing no evidence for this miRNA pair in stickleback. Each of these three pairs of mirror-miRNAs, as well as potential new ones, might appear after sequencing a wider array of organs, more developmental stages, or with deeper sequencing. This analysis shows that Prost! readily confirmed the conservation of the mirror-miRNA pair mir214/mir3120 in stickleback, demonstrating the genomic and transcriptomic conservation of these mirror-miRNAs among teleost fish.
Organ-enriched miRNA expression
miRNAs are generally considered to be specialized in function and to display organ- and even cell type-specific enrichment4,11,67. Most of these data, however, are from mammals61,68,69,70, so the extent to which conservation of expression and function is similar among teleosts is unknown. We hypothesized that evolutionarily-conserved miRNAs should display organ-specific enrichment in stickleback and zebrafish.
To investigate organ-specific enrichment of miRNA expression in stickleback, we studied the expression of 267 mature miRNAs (of the 321 that Prost! detected) that displayed an overall expression of at least 5 RPM across the entire dataset. Pairwise differential expression analysis using the DESeq2R package58 showed that (1) the brain displayed the greatest number of differentially expressed (DE) miRNAs among the four studied organs, and (2) the gonads (testis and ovary) displayed the fewest DE miRNAs and showed the largest intra-group variability (Fig. 3A).
Differential expression and conservation of miRNAs in stickleback and zebrafish brain, heart, testis, and ovary. (A) Heat map showing the number of stickleback mature miRNAs over-expressed in each organ compared to each other organ along with a sample identity plot that compares the similarities for each of two samples of each organ to the other seven samples tested. (B) Heat map of the 123 stickleback mature miRNAs (in rows) that were consistently enriched in one organ (in columns) compared to the three other organs, or in gonads compared to brain and heart. (C) Heat map of the number of zebrafish mature miRNAs over-expressed in each organ compared to each other organ along with a sample identity plot. (D) Heat map of the 148 zebrafish mature miRNAs consistently enriched in one organ compared to the three other organs, or in gonads compared to brain and heart. For all heat maps, the deepest blue indicates the lowest level of expression in the row and the most intense red indicates the highest level of expression in the row. (E) Lists of organ-enriched miRNAs that are evolutionarily-conserved between stickleback and zebrafish. Bold lettering denotes that the miRNA has an OEI > 0.85 in both species. Superscripted SB (Stickleback) or ZF (Zebrafish) denotes that this specific miRNA has an OEI > 0.85 in the corresponding species but not in the other.
In the six pairwise DE analyses of the four organs, 66 miRNAs were consistently over-expressed in the stickleback brain compared to each of the three other organs, compared to only 32, 10 and nine for heart, testis, and ovary, respectively (Fig. 3B, Supplementary Table 4). Because testis and ovary showed few organ-enriched DE miRNAs and share some common developmental processes in gametogenesis (e.g. meiosis and proliferation), we looked at miRNAs that were over-expressed in both testis and ovary compared to both heart and brain. Six additional miRNAs were similarly enriched in both testis and ovaries compared to the other organs, bringing the total number of miRNAs that are enriched in one or both gonads to 25 (Supplementary Table 4). Altogether, 123 miRNAs (i.e. 46% of the 267 minimally expressed miRNAs) displayed organ enrichment in either brain, heart, testis, ovary or in both gonads. This result validates the hypothesis that miRNAs in stickleback display organ-specific enrichment. More organ-enriched miRNAs would likely be identified after study of more organs.
To confirm this differential expression result, we analyzed the organ enrichment of each minimally expressed miRNA using the organ enrichment index (OEI), combining the testis and ovary data into a common 'gonad' organ type. Among the studied miRNAs, most (154/267 = 58%) displayed intermediate enrichment – they were predominantly expressed in one or more organs but were expressed significantly in at least one other organ (Fig. 4A). A total of 97 miRNAs (36%) showed an OEI > 0.85, which is considered a threshold for organ-enrichment60,61, and only 16 miRNAs (6%) showed ubiquitous, statistically similar expression levels among the organs studied with an OEI ≤ 0.3 (Fig. 4A). In addition, DE miRNAs tended to have the highest OEI scores (Figs 3B and 4A). Among the DE miRNAs that also had an OEI > 0.85, some displayed clear enrichment in brain (e.g. miR9-5p, miR124-3p, and miR138-5p, Fig. 4C–E), in testis (i.e. miR31a-3p, Fig. 4F), in both gonads (e.g. miR196a-5p and miR202-5p, Fig. 4G,H), or in heart (e.g. miR1-3p, miR133-3p, miR499-5p, Fig. 4I–K). Because we studied only four organs (and combined ovary and testis in the OEI analysis), some miRNAs that we categorized as not-organ-enriched might be enriched in organs that we didn't study. For example, miR122-5p, which is known to be mostly expressed in liver in vertebrates50,61, showed low, non-specific expression in all four organs we investigated with an average of 18 RPM and an OEI score of 0.53.
miRNA organ-enriched expression. (A,B) Frequency plot of OEI (organ enrichment index) values for all stickleback and zebrafish miRNAs expressed at more than 5 RPM across the entire dataset. Grey bars represent miRNAs that were also enriched in brain, heart, testis, ovary, or in both gonads, and white bars represent miRNAs that were not found to be enriched in a specific organ. (C–N) Average organ expression of evolutionarily-conserved, organ-enriched miRNAs that have an OEI > 0.85. Expression levels are given in RPM (Reads per Million) with associated standard deviations for the four organs studied in both stickleback (grey bars) and zebrafish (black bars).
To identify organ-specific enrichment of miRNAs in zebrafish, we studied the expression of 314 mature miRNAs (of the 402 that Prost! detected) that displayed an average expression of at least 5 RPM across all eight zebrafish samples. Similar to stickleback, the brain had the most differently expressed miRNAs, while ovary and testis had the least and showed the largest intra-group variability (Fig. 3C).
In all zebrafish pairwise comparisons, 66 miRNAs were consistently enriched in brain, 34 in heart, nine in testis, 21 in ovary, and an additional 18 miRNAs were equally enriched in both gonads (Fig. 3D, Supplementary Table 5). Altogether, 148 miRNAs (47% of the 314 minimally expressed zebrafish miRNAs) displayed organ-enrichment in brain, heart, testis, ovary, or in both gonads in zebrafish. Similar to the stickleback OEI analysis, of the 314 zebrafish miRNAs studied, most miRNAs (158/314 = 51%) displayed intermediate enrichment, 10 miRNAs (3%) showed overall ubiquitous expression levels among the studied organs, and 146 miRNAs (46%) showed organ-enrichment (OEI > 0.85) (Fig. 4A). Also similar to stickleback, we observed that miRNAs identified as organ-enriched by differential expression analyses are among the miRNAs that have the highest OEI (Figs 3D and 4B).
Taken together, these results demonstrated that a large proportion of miRNAs displayed enrichment in a single organ in both stickleback and zebrafish (46% and 47%, respectively) and organ-enrichment scores above 0.85 (36 and 46% in stickleback and zebrafish, respectively).
Organ-enriched miRNAs are conserved between stickleback and zebrafish
The hypothesis that miRNA functions are conserved predicts that at least some of the organ-enriched miRNAs in stickleback would also be enriched in the same organ in zebrafish. To test this prediction, we compared the list of organ-enriched miRNAs in stickleback and zebrafish and found that 44 miRNAs were brain-enriched in both species (Fig. 3B,D,E), with many of them already known to be brain-associated miRNAs in several fish species71,72,73; for example miR9-5p, miR124-3p, and miR138-5p (Fig. 4C–E), which are also highly expressed in brain and nervous organs in mammals50,61,74,75,76,77,78. These observations suggest a strong evolutionary conservation of function for brain-related miRNAs among vertebrates. The heart also displayed a substantial number of evolutionarily-conserved, organ-enriched miRNAs (13 miRNAs) (Fig. 3B,D,E). Heart-enriched miRNAs included mature products of the well-described vertebrate cardiac myomiR genes mir1, mir133, and mir49961,72,73,79,80,81,82 and erythromiRs mir144 and mir45148. The former group participates in muscle formation and function, and the latter may reflect the presence of red blood cells in the heart ventricle at the time of RNA extraction.
Surprisingly, gonad-enriched miRNAs appeared to be less conserved. Only one miRNA, miR31a-5p, was found to be testis-enriched in both stickleback and zebrafish (Figs 3B,D,E and 4F), while no miRNAs were ovary-enriched in both species (Fig. 3B,D,E). In chicken, Mir31 has been hypothesized to be involved in gonadal sex differentiation because it is expressed significantly higher in testes compared to ovaries at early sexual differentiation stages83. In human, MIR31 was down-regulated in the testis of an infertile adult human patient84. In fish, mir31 has not yet been associated with either gonad differentiation or testicular function, but our data are consistent with a role for mir31 that is conserved in testicular function among various vertebrate lineages. In addition, nine miRNAs were enriched in one or both gonads in both species (Fig. 3B,D,E), potentially reflecting a shared role in reproduction in one or both sexes in both species. Interestingly, among the gonad-associated miRNAs in stickleback and zebrafish, most displayed species-specific organ enrichment. For example, miR429a-3p was enriched in testis in stickleback but enriched in ovary in zebrafish; miR10c-5p was enriched in ovary in stickleback but enriched in testis in zebrafish; miR204-5p was enriched in ovary in zebrafish but enriched in both gonads in stickleback; miR196a-5p was enriched in testis in zebrafish but in both gonads in stickleback (Fig. 4G); and miR19c-3p, miR194a-5p, and miR725-3p were enriched in testis in stickleback but enriched in both gonads in zebrafish. In the case of the well-known gonad-enriched miR202-5p85,86,87, the expression level in the stickleback ovary was significantly higher than in testis; although the trend was the same in zebrafish, the difference was not statistically significant (Fig. 4H). The significance of these sex-specific differences is as yet unknown.
The relatively weak evolutionary conservation of sex-specific gonad enrichment in teleost fish is surprising and suggests reduced selective pressure on their function compared to other organ-enriched miRNAs, and/or that differences in the genetic control of reproduction exist between zebrafish and stickleback. Not enough information is currently available to distinguish between these two non-exclusive hypotheses. The large range of variations in sex-determination mechanisms, reproductive systems, reproductive state, and frequency of reproduction in teleost fish88,89, however, might help explain the weak conservation of stickleback and zebrafish miRNA expression in gonads. Indeed, the miRNA regulation system might be evolving with each species' reproductive biology and its associated genetic regulation. Some ancestral functions of a miRNA could be conserved in one species, could have evolved novel targets and regulatory pathways in another, or may simply be lost in a lineage-specific fashion. For example, a gonad-enriched ancestral miRNA might have specialized in testis in one lineage, while remaining gonad-enriched or becoming ovary-enriched in another lineage, as could have happened with miR10c-5p or miR429a-5p in our data.
Only three other miRNAs displayed organ-enriched expression in different organs in zebrafish and stickleback: miR145-3p, miR375-3p, and miR460-5p were enriched in one organ in one species but in a different organ in the other species. miR145-3p was moderately enriched in heart in zebrafish, but was largely enriched in testes and moderately enriched in ovary in stickleback, while displaying similar levels of heart expression in both species. In zebrafish, miR375-3p and miR460-5p were strongly enriched in testis and in "gonads" (testis and ovary) but both were strongly enriched in brain in stickleback (Fig. 4M,N). The study of more teleost species, and in particular, the inclusion of an outgroup to represent the common ancestor, such as spotted gar, is necessary to test hypotheses regarding the loss-of-function, sub-functionalization, and neo-functionalization of miRNAs in teleost fishes.
Evolutionarily-conserved, brain-enriched post-transcriptional miRNA seed editing
Post-transcriptional modification of miRNAs is frequent and generally originates from variation in biogenesis or enzyme-catalyzed nucleotide modification, generating groups of related sequences called isomiRs2,24,25. To study post-transcriptional modifications in teleost fish and to ask whether modifications are evolutionarily-conserved, we developed a feature in Prost! to calculate and color-code (in the Excel output) each type of post-transcriptional modification at individual genomic loci.
An analysis of overall read diversity and isomiR composition revealed that, in both stickleback and zebrafish, most isomiRs were 21 to 23 nucleotides long (Supplemental File 7A), and that the 10 most expressed miRNAs, including their respective isomiRs, accounted for more than 75% of total reads with most of these miRNAs common to both species (Supplemental File 7B). Furthermore, analysis of post-transcriptional modification types revealed that, in both species, approximately 35% of reads displayed 3′-end length variations (templated or non-templated), while only about 3.5% of reads displayed 5′-end length variations (Supplemental File 7C). This result is consistent with the fact that 5′-end length variations shift the seed sequence, which is likely to drastically alter the function of a miRNA by modifying the pool of its targeted mRNAs; in contrast, 3′-end length variations are less likely to strongly impact target recognition2,26. In addition, we observed that miRNA editing is rare in both species with, on average, 0.91% and 4.40% of reads displaying edition in the seven-nucleotide seed in stickleback and zebrafish respectively, and about 1.28% and 0.94% of reads displaying editing in the 15 nt or so outside of the seed in stickleback and zebrafish, respectively (Supplemental File 7C). Further automated analysis of Prost! seed-edition calculations revealed that in stickleback, miR2188-5p was prone to seed editing, especially in the brain (34%) compared to other organs (9.6%) (Fig. 5A). In zebrafish, similar analyses of post-transcriptional modifications revealed that the same miRNA, miR2188-5p, also displayed a higher rate of seed editing in the brain (12%), compared to other organs (1.0%; Fig. 5A). Prost! output also revealed that the stickleback miR2188-5p isomiR pool was composed of sequences displaying three different seeds: the genome-encoded seed (Gac-a in Fig. 5D; 67% of the sequences), a seed with an adenosine-to-guanosine (A-to-G) substitution at position 8 of the miRNA (Gac-b; 26% of the sequences), and a seed with two A-to-G substitutions, one at nucleotide 2 and one at nucleotide 8 (Gac-c; 7% of the sequences) (Fig. 5B). Examination of seed variations in zebrafish identified two different seeds: the genome-encoded seed (Dre-a in Fig. 5D; 89% of the sequences) and a seed with an A-to-G substitution at the second nucleotide of the mature miRNA (Dre-b; 11% of the sequences) (Fig. 5B). Because inosine bases are replaced by guanosine bases during the cDNA synthesis step of the preparation protocols for sequencing libraries, sequencers report a guanosine where an inosine could have originally been present in the RNA molecule. Therefore, nucleotides sequenced as guanosine in place of a genome-encoded adenosine in our sequencing data may have been due to be post-transcriptionally ADAR-edited inosine nucleotides.
Evolutionarily-conserved brain-enriched seed-editing of miR2188-5p. (A) miR2188-5p seed editing frequency is higher in brain compared to other organs in both stickleback and zebrafish. (B) Frequency of seed variants generated using WebLogo3 webserver. (C) Sanger sequencing of genomic DNA and pri-miRNAs of both stickleback specimens. (C') Sanger sequencing of genomic DNA and pri-miRNAs of both zebrafish specimens. The blue box highlights regions corresponding to mature miRNAs and red boxes highlight editing sites. (D) Mature miR2188-5p sequences used for target prediction in both stickleback and zebrafish. (D') Overlap of the predicted target mRNA sets for each of the three mature miR2188-5p isomiRs in stickleback. (D") Overlap of the predicted target mRNA sets for each of the two mature miR2188-5p isomiRs in zebrafish. (E) Overlap of predicted target mRNA sets for either genetically encoded or edited mature miR2188-5p sequences in both stickleback and zebrafish.
To verify that the A-to-G substitutions we observed are due to post-transcriptional modification instead of potential miRNA allelic variations, we sequenced the mir2188 gene and pri-miR2188 transcript from each stickleback and zebrafish individual used for small RNA sequencing. In both species, the genomes of both individuals contained the same nucleotides at seed editing sites as found in the reference genome (Fig. 5C,C'). In addition, we found that the pri-miR2188 was free of nucleotide substitutions at the second and/or eighth nucleotide of the mature miRNA in all individuals (Fig. 5C,C'). The genome sequencing results show that sequence variants in the miR2188-5p seed were not genetically encoded and the pri-miRNA sequencing result suggests that they occurred post-transcriptionally after the processing of the hairpin. We conclude that post-transcriptional seed editing of miR2188-5p is organ-enriched and evolutionarily-conserved. This finding represents, to our knowledge, the first example of teleost organ-enriched evolutionarily-conserved seed editing.
miRNA seed editing, by changing the seed sequence, can alter the set of targeted transcripts and therefore modify a miRNA's function30,31,32,33,34. To evaluate the potential biological effects of miR2188-5p seed editing, we used miRAnda64,65 and 3′ UTR sequences of mRNAs to predict mRNA targets of both the reference miRNA and the edited miRNAs (Fig. 5D, Supplemental Table 6). For stickleback, the three different miR2188-5p isomiRs had few overlapping predicted target genes; in most cases, putative targets were unique to each isomiR (Fig. 5D'). For zebrafish, the targets predicted for each isomiR were also largely non-overlapping, with less than 15% of predicted targets in common for both isomiRs (Fig. 5D").
To see if seed editing is likely to affect genes conserved between zebrafish and stickleback, we analyzed overlaps among putative mRNA target sets for miR2188-5p isomiRs in both species. Two, not mutually exclusive, hypotheses could explain the function of miRNA editing. Under one hypothesis, editing offers a new set of targets, and under the other hypothesis, miRNA editing removes targets from the list hit by the reference sequence. Among genes targeted by either the non-edited and/or the edited miRNAs in each species (394 and 1421 genes in stickleback and zebrafish, respectively), 309 genes displayed orthology relationship between stickleback and zebrafish based on Ensembl Biomart. Under the first hypothesis (gain of targets), results identified only two of 309 orthologs that were not predicted targets of the genomically encoded isomiR but were predicted targets of the seed-edited isomiRs in both stickleback and zebrafish. One of these two genes was cntnap3 (contactin associated protein like 3, ENSDARG00000067824), a cell adhesion molecule of unknown function in teleost fish. In human and mouse, however, CNTNAP3 is expressed in brain and spinal cord90,91, and its dysregulation in developing mice impairs motor learning and social behavior92,93. In addition, high CNTNAP3 levels have been associated with schizophrenia in humans94. The second conserved target of the seed-edited miR2188-5p is pdha1a (pyruvate dehydrogenase alpha 1, ENSDARG00000012387), which is strongly expressed in the brain of developing zebrafish embryos95. PDHA1 mutations in human cause acid lactic buildup, resulting in impaired psychomotor development and chronic neurologic dysfunction with structural abnormalities in the central nervous system (OMIM 300502, ORPHA:79243).
Under the second hypothesis for the function of seed editing (removal of targets), only five genes were predicted to be targets of the genomically encoded isomiR but were not predicted targets of the seed-edited isomiRs in both stickleback and zebrafish. Among these five conserved targets of the genomically encoded, but not the edited miR2188-5p, four genes (cyth1b, dcn, dcun1d1, and polr3e) don't seem to be involved in neuron or brain function in vertebrates. The fifth gene, snap29 (synaptosome associated protein 29, ENSDARG00000038518), a soluble N-ethylmaleimide-sensitive factor-attachment protein receptor (SNARE), has also not yet been shown to be associated with brain phenotypes in zebrafish57. SNAP29 mutations in human, however, cause a unique constellation of clinical manifestations including microcephaly, severe neurologic impairment, psychomotor retardation, referred as CEDNIK (cerebral dysgenesis, neuropathy, ichthyosis, and keratoderma) syndrome (OMIM 604202, ORPHA:66631).
The predicted conservation of potential targets urges functional analyses to study the effect of native or seed-edited miR2188-5p on brain transcript translation and its subsequent phenotypic consequences. To our knowledge, however, no prior publications have identified miR2188-5p as an edited miRNA nor suggested a role for it in brain function, arguing for more research. The study of additional species is needed to understand the phylogenetic conservation of the brain-enriched seed-editing of miR2188-5p and functional analyses are required to decipher a potential role in vertebrate brain development and physiology.
Results reported here show that the novel software Prost! permitted the annotation of 273 miRNA genes in three-spined stickleback and that subsequent manual annotation annotated 13 additional genes. The stickleback miRNome, with a set of 286 miRNA genes, is thus comparable to the miRNome of other teleost species rather than being greatly enlarged as suggested by previous analyses51,52,53,54. Prost! analysis of small RNA sequencing libraries from more tissues or stages, however, is likely to permit the annotation of additional miRNA genes. In addition, as predicted, the differential expression analysis of miRNAs in four organs revealed significant organ-enrichment in either brain, heart, testis, ovary, or both gonads for about 46% and 47%of minimally expressed miRNAs in adult stickleback and zebrafish. Furthermore, supporting the hypothesis that organ-enriched miRNAs are evolutionarily-conserved, enriched expression of specific miRNAs was found in both brain and heart of both stickleback and zebrafish. In ovary and testis, however, fewer expressed miRNAs were conserved between these two teleosts, although several miRNAs that were enriched in both gonads of one species tended to be enriched in at least one of the two gonad types in the other species. Finally, we demonstrated the conservation of organ-specific miR2188-5p seed editing in the brains of both zebrafish and stickleback, suggesting potential conservation of this organ-specific, post-transcriptional seed editing process among teleosts.
All data generated or analyzed during this study are included in this published article (and its Supplementary Information files) and raw Illumina sequencing reads were deposited in the NCBI Short Read Archive under project accession numbers SRP157992 and SRP039502 for stickleback and zebrafish, respectively.
Carthew, R. W. & Sontheimer, E. J. Origins and Mechanisms of miRNAs and siRNAs. Cell 136, 642–655 (2009).
Desvignes, T. et al. miRNA Nomenclature: A View Incorporating Genetic Origins, Biosynthetic Pathways, and Sequence Variants. Trends Genet. 31, 613–626 (2015).
Bartel, D. P. Metazoan MicroRNAs. Cell 173, 20–51 (2018).
Christodoulou, F. et al. Ancient animal microRNAs and the evolution of tissue identity. Nature 463, 1084–1088 (2010).
ADS CAS PubMed PubMed Central Article Google Scholar
Kosik, K. S. MicroRNAs and Cellular Phenotypy. Cell 143, 21–26 (2010).
Ameres, S. L. & Zamore, P. D. Diversifying microRNA sequence and function. Nat. Rev. Mol. Cell Biol. 14, 475–488 (2013).
Bhaskaran, M. & Mohan, M. MicroRNAs: History, Biogenesis, and Their Evolving Role in Animal Development and Disease. Vet. Pathol. 51, 759–774 (2014).
Leva, G. D., Garofalo, M. & Croce, C. M. MicroRNAs in Cancer. Annu. Rev. Pathol. Mech. Dis. 9, 287–314 (2014).
Ranganathan, K. & Sivasankar, V. MicroRNAs - Biology and clinical applications. J. Oral Maxillofac. Pathol. 18, 229 (2014).
Tüfekci, K. U., Öner, M. G., Meuwissen, R. L. J. & Genç, Ş. The Role of MicroRNAs in Human Diseases. In miRNomics: MicroRNA Biology and Computational Analysis 33–50, https://doi.org/10.1007/978-1-62703-748-8_3 (Humana Press, Totowa, NJ, 2014).
Lee, C.-T., Risom, T. & Strauss, W. M. Evolutionary Conservation of MicroRNA Regulatory Circuits: An Examination of MicroRNA Gene Complexity and Conserved MicroRNA-Target Interactions through Metazoan Phylogeny. DNA Cell Biol. 26, 209–218 (2007).
Grimson, A. et al. Early origins and evolution of microRNAs and Piwi-interacting RNAs in animals. Nature 455, 1193–1197 (2008).
ADS CAS PubMed Article Google Scholar
Wheeler, B. M. et al. The deep evolution of metazoan microRNAs. Evol. Dev. 11, 50–68 (2009).
Loh, Y.-H. E., Yi, S. V. & Streelman, J. T. Evolution of microRNAs and the diversification of species. Genome Biol. Evol. 3, 55–65 (2011).
Tarver, J. E., Donoghue, P. C. J. & Peterson, K. J. Do miRNAs have a deep evolutionary history? BioEssays 34, 857–866 (2012).
Winter, J., Jung, S., Keller, S., Gregory, R. I. & Diederichs, S. Many roads to maturity: microRNA biogenesis pathways and their regulation. Nat. Cell Biol. 11, 228–234 (2009).
Ha, M. & Kim, V. N. Regulation of microRNA biogenesis. Nat. Rev. Mol. Cell Biol. 15, 509–524 (2014).
Cheloufi, S., Dos Santos, C. O., Chong, M. M. W. & Hannon, G. J. A dicer-independent miRNA biogenesis pathway that requires Ago catalysis. Nature 465, 584–589 (2010).
Cifuentes, D. et al. A Novel miRNA Processing Pathway Independent of Dicer Requires Argonaute2 Catalytic Activity. Science 328, 1694–1698 (2010).
Scott, M. S. & Ono, M. From snoRNA to miRNA: Dual function regulatory non-coding RNAs. Biochimie 93, 1987–1992 (2011).
Scott, H. et al. MiR-3120 Is a Mirror MicroRNA That Targets Heat Shock Cognate Protein 70 and Auxilin Messenger RNAs and Regulates Clathrin Vesicle Uncoating. J. Biol. Chem. 287, 14726–14733 (2012).
Berezikov, E., Chung, W.-J., Willis, J., Cuppen, E. & Lai, E. C. Mammalian Mirtron Genes. Mol. Cell 28, 328–336 (2007).
Ruby, J. G., Jan, C. H. & Bartel, D. P. Intronic microRNA precursors that bypass Drosha processing. Nature 448, 83–86 (2007).
Neilsen, C. T., Goodall, G. J. & Bracken, C. P. IsomiRs – the overlooked repertoire in the dynamic microRNAome. Trends Genet. 28, 544–549 (2012).
Guo, L. & Chen, F. A challenge for miRNA: multiple isomiRs in miRNAomics. Gene 544, 1–7 (2014).
Tan, G. C. et al. 5′ isomiR variation is of functional and evolutionary importance. Nucleic Acids Res. 42, 9424–9435 (2014).
Engkvist, M. E. et al. Analysis of the miR-34 family functions in breast cancer reveals annotation error of miR-34b. Sci. Rep. 7, 9655 (2017).
Yang, W. et al. Modulation of microRNA processing and expression through RNA editing by ADAR deaminases. Nat. Struct. Mol. Biol. 13, 13–21 (2006).
Heale, B. S. E. et al. Editing independent effects of ADARs on the miRNA/siRNA pathways. EMBO J. 28, 3145–3156 (2009).
Nishikura, K. A-to-I editing of coding and non-coding RNAs by ADARs. Nat. Rev. Mol. Cell Biol. 17, 83–96 (2016).
Kawahara, Y., Zinshteyn, B., Chendrimada, T. P., Shiekhattar, R. & Nishikura, K. RNA editing of the microRNA-151 precursor blocks cleavage by the Dicer–TRBP complex. EMBO Rep. 8, 763–769 (2007).
Kawahara, Y. et al. Redirection of Silencing Targets by Adenosine-to-Inosine Editing of miRNAs. Science 315, 1137–1140 (2007).
Kume, H., Hino, K., Galipon, J. & Ui-Tei, K. A-to-I editing in the miRNA seed region regulates target mRNA selection and silencing efficiency. Nucleic Acids Res. gku662, https://doi.org/10.1093/nar/gku662 (2014).
Warnefors, M., Liechti, A., Halbert, J., Valloton, D. & Kaessmann, H. Conserved microRNA editing in mammalian evolution, development and disease. Genome Biol. 15, R83 (2014).
Pantano, L., Estivill, X. & Martí, E. SeqBuster, a bioinformatic tool for the processing and analysis of small RNAs datasets, reveals ubiquitous miRNA modifications in human embryonic cells. Nucleic Acids Res. 38, e34–e34 (2010).
PubMed Article CAS Google Scholar
Friedländer, M. R., Mackowiak, S. D., Li, N., Chen, W. & Rajewsky, N. miRDeep2 accurately identifies known and hundreds of novel microRNA genes in seven animal clades. Nucleic Acids Res. 40, 37–52 (2012).
Barturen, G. et al. sRNAbench: profiling of small RNAs and its sequence variants in single or multi-species high-throughput experiments. Methods Gener. Seq. 1 (2014).
Baras, A. S. et al. miRge - A Multiplexed Method of Processing Small RNA-Seq Data to Determine MicroRNA Entropy. PLOS ONE 10, e0143066 (2015).
Rueda, A. et al. sRNAtoolbox: an integrated collection of small RNA research tools. Nucleic Acids Res. 43, W467–W473 (2015).
Lukasik, A., Wójcikowski, M. & Zielenkiewicz, P. Tools4miRs – one place to gather all the tools for miRNA analysis. Bioinformatics 32, 2722–2724 (2016).
Urgese, G., Paciello, G., Acquaviva, A. & Ficarra, E. isomiR-SEA: an RNA-Seq analysis tool for miRNAs/isomiRs expression level profiling and miRNA-mRNA interaction sites evaluation. BMC Bioinformatics 17, 148 (2016).
Vitsios, D. M. & Enright, A. J. Chimira: analysis of small RNA sequencing data and microRNA modifications. Bioinformatics 31, 3365–3367 (2015).
Hansen, T. B., Venø, M. T., Kjems, J. & Damgaard, C. K. miRdentify: high stringency miRNA predictor identifies several novel animal miRNAs. Nucleic Acids Res. 42, e124 (2014).
Guo, L., Yu, J., Liang, T. & Zou, Q. miR-isomiRExp: a web-server for the analysis of expression of miRNA at the miRNA/isomiR levels. Sci. Rep. 6, 23700 (2016).
Batzel, P., Desvignes, T., Postlethwait, J. H., Eames, B. F. & Sydes, J. Prost!, a tool for miRNA annotation and next generation smallRNA sequencing experiment analysis. Zenodo, https://doi.org/10.5281/zenodo.1937101 (2018).
Desvignes, T., Beam, M. J., Batzel, P., Sydes, J. & Postlethwait, J. H. Expanding the annotation of zebrafish microRNAs based on small RNA sequencing. Gene 546, 386–389 (2014).
Braasch, I. et al. The spotted gar genome illuminates vertebrate evolution and facilitates human-teleost comparisons. Nat. Genet. 48, 427–437 (2016).
Desvignes, T., Detrich, H. W. III & Postlethwait, J. H. Genomic conservation of erythropoietic microRNAs (erythromiRs) in white-blooded Antarctic icefish. Mar. Genomics 30, 27–34 (2016).
Kozomara, A. & Griffiths-Jones, S. miRBase: annotating high confidence microRNAs using deep sequencing data. Nucleic Acids Res. 42, D68–D73 (2013).
Fromm, B. et al. A Uniform System for the Annotation of Vertebrate microRNA Genes and the Evolution of the Human microRNAome. Annu. Rev. Genet. 49, 213–242 (2015).
Li, S.-C. et al. Identification of homologous microRNAs in 56 animal genomes. Genomics 96, 1–9 (2010).
Chaturvedi, A., Raeymaekers, J. A. M. & Volckaert, F. A. M. Computational identification of miRNAs, their targets and functions in three-spined stickleback (Gasterosteus aculeatus). Mol. Ecol. Resour. 14, 768–777 (2014).
Rastorguev, S. M. et al. Identification of novel microRNA genes in freshwater and marine ecotypes of the three-spined stickleback (Gasterosteus aculeatus). Mol. Ecol. Resour. 16, 1491–1498 (2016).
Aken, B. L. et al. Ensembl 2017. Nucleic Acids Res. 45, D635–D642 (2017).
Castranova, D. et al. The Effect of Stocking Densities on Reproductive Performance in Laboratory Zebrafish (Danio rerio). Zebrafish 8, 141–146 (2011).
Martin, M. Cutadapt removes adapter sequences from high-throughput sequencing reads. EMBnet.journal 17, 10–12 (2011).
Bradford, Y. et al. ZFIN: enhancements and updates to the zebrafish model organism database. Nucleic Acids Res. 39, D822–D829 (2011).
Love, M. I., Huber, W. & Anders, S. Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2. Genome Biol. 15, 550 (2014).
Morpheus. Available at, https://software.broadinstitute.org/morpheus/.
Yanai, I. et al. Genome-wide midrange transcription profiles reveal expression level relationships in human tissue specification. Bioinformatics 21, 650–659 (2005).
Ludwig, N. et al. Distribution of miRNA expression across human tissues. Nucleic Acids Res. 44, 3865–3877 (2016).
Desvignes, T., Contreras, A. & Postlethwait, J. H. Evolution of the miR199-214 cluster and vertebrate skeletal development. RNA Biol. 11, 281–294 (2014).
Crooks, G. E., Hon, G., Chandonia, J.-M. & Brenner, S. E. WebLogo: A Sequence Logo Generator. Genome Res. 14, 1188–1190 (2004).
Enright, A. J. et al. MicroRNA targets in Drosophila. Genome Biol. 5, R1 (2003).
Betel, D., Wilson, M., Gabow, A., Marks, D. S. & Sander, C. The microRNA.org resource: targets and expression. Nucleic Acids Res. 36, D149–D153 (2008).
Tyler, D. M. et al. Functionally distinct regulatory RNAs generated by bidirectional transcription and processing of microRNA loci. Genes Dev. 22, 26–36 (2008).
Halushka, M. K., Fromm, B., Peterson, K. J. & McCall, M. N. Big Strides in Cellular MicroRNA Expression. Trends Genet. 34, 165–167 (2018).
de Rie, D. et al. An integrated expression atlas of miRNAs and their promoters in human and mouse. Nat. Biotechnol. 35, 872–878 (2017).
Juzenas, S. et al. A comprehensive, cell specific microRNA catalogue of human peripheral blood. Nucleic Acids Res. 45, 9290–9301 (2017).
McCall, M. N. et al. Toward the human cellular microRNAome. Genome Res., https://doi.org/10.1101/gr.222067.117 (2017).
Kitano, J., Yoshida, K. & Suzuki, Y. RNA sequencing reveals small RNAs differentially expressed between incipient Japanese threespine sticklebacks. BMC Genomics 14, 214 (2013).
Vaz, C. et al. Deep sequencing of small RNA facilitates tissue and sex associated microRNA discovery in zebrafish. BMC Genomics 16, 950 (2015).
Andreassen, R. et al. Discovery of miRNAs and Their Corresponding miRNA Genes in Atlantic Cod (Gadus morhua): Use of Stable miRNAs as Reference Genes Reveals Subgroups of miRNAs That Are Highly Expressed in Particular Organs. PLOS ONE 11, e0153324 (2016).
Miska, E. A. et al. Microarray analysis of microRNA expression in the developing mammalian brain. Genome Biol. 5, R68 (2004).
Obernosterer, G., Leuschner, P. J. F., Alenius, M. & Martinez, J. Post-transcriptional regulation of microRNA expression. RNA 12, 1161–1167 (2006).
Makeyev, E. V., Zhang, J., Carrasco, M. A. & Maniatis, T. The MicroRNA miR-124 Promotes Neuronal Differentiation by Triggering Brain-Specific Alternative Pre-mRNA Splicing. Mol. Cell 27, 435–448 (2007).
Cheng, L.-C., Pastrana, E., Tavazoie, M. & Doetsch, F. miR-124 regulates adult neurogenesis in the subventricular zone stem cell niche. Nat. Neurosci. 12, 399–408 (2009).
Jung, H.-J. et al. Regulation of prelamin A but not lamin C by miR-9, a brain-specific microRNA. Proc. Natl. Acad. Sci. 109, E423–E431 (2012).
Chen, J.-F. et al. The role of microRNA-1 and microRNA-133 in skeletal muscle proliferation and differentiation. Nat. Genet. 38, 228–233 (2006).
Bhuiyan, S. S. et al. Evolution of the myosin heavy chain gene MYH14 and its intronic microRNA miR-499: muscle-specific miR-499 expression persists in the absence of the ancestral host gene. BMC Evol. Biol. 13, 142 (2013).
Horak, M., Novak, J. & Bienertova-Vasku, J. Muscle-specific microRNAs in skeletal muscle development. Dev. Biol. 410, 1–13 (2016).
Siddique, B. S., Kinoshita, S., Wongkarangkana, C., Asakawa, S. & Watabe, S. Evolution and Distribution of Teleost myomiRNAs: Functionally Diversified myomiRs in Teleosts. Mar. Biotechnol. 18, 436–447 (2016).
Cutting, A. D. et al. The potential role of microRNAs in regulating gonadal sex differentiation in the chicken embryo. Chromosome Res. 20, 201–213 (2012).
Muñoz, X., Mata, A., Bassas, L. & Larriba, S. Altered miRNA Signature of Developing Germ-cells in Infertile Patients Relates to the Severity of Spermatogenic Failure and Persists in Spermatozoa. Sci. Rep. 5, 17991 (2015).
ADS PubMed PubMed Central Article CAS Google Scholar
Wainwright, E. N. et al. SOX9 Regulates MicroRNA miR-202-5p/3p Expression During Mouse Testis Differentiation. Biol. Reprod. 89 (2013).
Zhang, J. et al. MiR-202-5p is a novel germ plasm-specific microRNA in zebrafish. Sci. Rep. 7, 7055 (2017).
Gay, S. et al. MiR-202 controls female fecundity by regulating medaka oogenesis. PLOS Genet. 14, e1007593 (2018).
Helfman, G., Collette, B. B., Facey, D. E. & Bowen, B. W. The Diversity of Fishes: Biology, Evolution, and Ecology, 2nd Edition (Wiley-Blackwell, 2009).
Martínez, P. et al. Genetic architecture of sex determination in fish: applications to sex ratio control in aquaculture. Front. Genet. 5 (2014).
Spiegel, I., Salomon, D., Erne, B., Schaeren-Wiemers, N. & Peles, E. Caspr3 and Caspr4, Two Novel Members of the Caspr Family Are Expressed in the Nervous System and Interact with PDZ Domains. Mol. Cell. Neurosci. 20, 283–297 (2002).
Hirata, H. et al. Cell adhesion molecule contactin‐associated protein 3 is expressed in the mouse basal ganglia during early postnatal stages. J. Neurosci. Res. 94, 74–89 (2015).
Hirata, H., Takahashi, A., Shimoda, Y. & Koide, T. Caspr3-Deficient Mice Exhibit Low Motor Learning during the Early Phase of the Accelerated Rotarod Task. PLOS ONE 11, e0147887 (2016).
Tong, D. et al. The critical role of ASD-related gene CNTNAP3 in regulating synaptic development and social behavior in mice. bioRxiv 260083, https://doi.org/10.1101/260083 (2018).
Okita, M., Yoshino, Y., Iga, J. & Ueno, S. Elevated mRNA expression of CASPR3 in patients with schizophrenia. Nord. J. Psychiatry 71, 312–314 (2017).
Thisse, B. & Thisse, C. Fast Release Clones: A High Throughput Expression Analysis. ZFIN Direct Data Submiss (2004).
We thank the University of Oregon Aquatics Facility, Trevor Enright and Tim Mason for zebrafish care; Mark Currey and William A. Cresko for the gift of stickleback (grants: NSF DEB 0949053, NSF IOS 102728, and NIH 1R24GM079486-01A1 to W.A.C.); and the UO Genomics and Cell Characterization Core Facility (GC3F). We also thank Teretha Taylor (Albany State University, GA; UO SPUR Program NIH grant 2R25HD070817) and Michael J. Beam who assisted in the early steps of stickleback miRNA annotation, and John Willis, David Clouthier, Kristin Artinger, Julien Bobe, Jérôme Montfort, Joanna Kelley, Igor Babiak, Leonardo M Martin, Teshome T Bizuayehu, Peter Alestrom, and Havard Aanes for constructive comments during the development of Prost!, as well as Brian Bushnell for helpful advice and insight concerning use of BBMap We also thank two anonymous reviewers whose comments helped to improve the manuscript. This work benefited from access to the University of Oregon high performance computers Talapas and ACISS (NSF grant OCI-0960354). This work was funded by the grants NIH U01 DE020076, NIH R24 OD011199, NIH 5R01 OD011116, NIH R01 AG031922, NIH R01 GM085318, NIH P01 HD22486, and NSF PLR-1543383.
Thomas Desvignes, Peter Batzel and Jason Sydes contributed equally.
Institute of Neuroscience, University of Oregon, Eugene, OR, 97403, USA
Thomas Desvignes, Peter Batzel, Jason Sydes & John H. Postlethwait
Department of Anatomy, Physiology, and Pharmacology, University of Saskatchewan, Saskatoon, SK, S7N 5E5, Canada
B. Frank Eames
Thomas Desvignes
Peter Batzel
Jason Sydes
John H. Postlethwait
Study concept and design: T.D., B.F.E., P.B., J.S., J.H.P. Software development: P.B., J.S., T.D., B.F.E., J.H.P. Acquisition of data: T.D. Analysis and interpretation of data: T.D., J.S., P.B. Wrote the manuscript: T.D. Critical revision of the manuscript: T.D., B.F.E., P.B., J.S., J.H.P. Obtained funding: J.H.P., T.D. Study supervision: T.D. and J.H.P.
Corresponding authors
Correspondence to Thomas Desvignes or John H. Postlethwait.
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Supplementary Datasets Description
Supplementary File 3
Desvignes, T., Batzel, P., Sydes, J. et al. miRNA analysis with Prost! reveals evolutionary conservation of organ-enriched expression and post-transcriptional modifications in three-spined stickleback and zebrafish. Sci Rep 9, 3913 (2019). https://doi.org/10.1038/s41598-019-40361-8
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, John H. Postlethwait
& Peter Konstantinidis
Polar Biology (2020)
Encyclopedia of tools for the analysis of miRNA isoforms
Georges Pierre Schmartz
, Fabian Kern
, Tobias Fehlmann
, Viktoria Wagner
, Bastian Fromm
& Andreas Keller
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List of axioms
This is a list of axioms as that term is understood in mathematics. In epistemology, the word axiom is understood differently; see axiom and self-evidence. Individual axioms are almost always part of a larger axiomatic system.
ZF (the Zermelo–Fraenkel axioms without the axiom of choice)
Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory. They can be easily adapted to analogous theories, such as mereology.
• Axiom of extensionality
• Axiom of empty set
• Axiom of pairing
• Axiom of union
• Axiom of infinity
• Axiom schema of replacement
• Axiom of power set
• Axiom of regularity
• Axiom schema of specification
See also Zermelo set theory.
Axiom of choice
With the Zermelo–Fraenkel axioms above, this makes up the system ZFC in which most mathematics is potentially formalisable.
Equivalents of AC
• Hausdorff maximality theorem
• Well-ordering theorem
• Zorn's lemma
Stronger than AC
• Axiom of global choice
Weaker than AC
• Axiom of countable choice
• Axiom of dependent choice
• Boolean prime ideal theorem
• Axiom of uniformization
Alternates incompatible with AC
• Axiom of real determinacy
Other axioms of mathematical logic
• Von Neumann–Bernays–Gödel axioms
• Continuum hypothesis and its generalization
• Freiling's axiom of symmetry
• Axiom of determinacy
• Axiom of projective determinacy
• Martin's axiom
• Axiom of constructibility
• Rank-into-rank
• Kripke–Platek axioms
• Diamond principle
Geometry
• Parallel postulate
• Birkhoff's axioms (4 axioms)
• Hilbert's axioms (20 axioms)
• Tarski's axioms (10 axioms and 1 schema)
Other axioms
• Axiom of Archimedes (real number)
• Axiom of countability (topology)
• Dirac–von Neumann axioms
• Fundamental axiom of analysis (real analysis)
• Gluing axiom (sheaf theory)
• Haag–Kastler axioms (quantum field theory)
• Huzita's axioms (origami)
• Kuratowski closure axioms (topology)
• Peano's axioms (natural numbers)
• Probability axioms
• Separation axiom (topology)
• Wightman axioms (quantum field theory)
• Action axiom (praxeology)
See also
• Axiomatic quantum field theory
• Minimal axioms for Boolean algebra
| Wikipedia |
Li Wang, Shen-long Jiang, Qun Zhang, Yi Luo. Multi-domain High-Resolution Platform for Integrated Spectroscopy and Microscopy Characterizations[J]. Chinese Journal of Chemical Physics , 2020, 33(6): 680-685. doi: 10.1063/1674-0068/cjcp2006093
Citation: Li Wang, Shen-long Jiang, Qun Zhang, Yi Luo. Multi-domain High-Resolution Platform for Integrated Spectroscopy and Microscopy Characterizations[J]. Chinese Journal of Chemical Physics , 2020, 33(6): 680-685. doi: 10.1063/1674-0068/cjcp2006093
doi: 10.1063/1674-0068/cjcp2006093
Li Wang,
Shen-long Jiang , ,
Qun Zhang,
Yi Luo
Hefei National Laboratory for Physical Sciences at the Microscale, Department of Chemical Physics, Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China
Corresponding author: Shen-long Jiang, E-mail: [email protected]
Accepted Date: 2020-07-02
Multi-domain platform,
Spectral/spatial/temporal resolution,
Integrated characterizations,
Microscopy,
[1] L. Liu, J. Han, L. Xu, J. Zhou, C. Zhao, S. Ding, H. Shi, M. Xiao, L. Ding, Z. Ma, C. Jin, Z. Zhang, and L. Peng, Science 368, 850 (2020).
[2] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Nature 556, 43 (2018).
[3] H. T. Chen, R. Kersting, and G. C. Cho, Appl. Phys. Lett. 83, 3009 (2003).
[4] V. J. Rao, M. Matthiesen, K. P. Goetz, C. Huck, C. Yim, R. Siris, J. Han, S. Hahn, U. H. F. Bunz, A. Dreuw, G. S. Duesberg, A. Pucci, and J. Zaumseil, J. Phys. Chem. C 124, 5331 (2020).
[5] B. Li, G. Zhang, Y. Liang, Q. Hao, J. Sun, C. Zhou, Y. Tao, X. Yang, and Z. Ren, Chin. J. Chem. Phys. 32, 399 (2019).
[6] A. Rousse, C. Rischel, S. Fourmaux, I. Uschmann, S. Sebban, G. Grillon, P. Balcou, E. Förster, J. P. Geindre, P. Audebert, J. C. Gauthier, and D. Hulin, Nature 410, 65 (2001).
[7] A. H. Zewail and J. M. Thomas, 4D Electron Microscopy: Imaging in Space and Time, World Scientific, (2010).
[8] J. Müller, W. Ibach, K. Weishaupt, and O. Hollricher, Confocal Raman Microscopy, Springer, (2018).
[9] H. Cai, Q. Meng, H. Zhao, M. Li, Y. Dai, Y. Lin, H. Ding, N. Pan, Y. Tian, Y. Luo, and X. Wang, ACS Appl. Mater. Interfaces 10, 20189 (2018).
[10] M. Yi and Z. Shen, J. Mater. Chem. A 3, 11700 (2015).
[11] A. C. Ferrari and D. M. Basko, Nat. Nanotechnol. 8, 235 (2013).
[12] N. S. Mueller, S. Heeg, M. P. Alvarez, P. Kusch, S. Wasserroth, N. Clark, F. Schedin, J. Parthenios, K. Papagelis, C. Galiotis, M. Kalbáč, A. Vijayaraghavan, U. Huebner, R. Gorbachev, O. Frank, and S. Reich, 2D Mater. 5, 015016 (2018).
[13] M. A. Bissett, M. Tsuji, and H. Ago, Phys. Chem. Chem. Phys. 16, 11124 (2014).
[14] R. W. Havener, H. Zhuang, L. Brown, R. G. Hennig, and J. Park, Nano Lett. 12, 3162 (2012).
[15] X. Lu, X. Luo, J. Zhang, S. Y. Quek, and Q. Xiong, Nano Res. 9, 3559 (2016).
[16] R. Loudon, Adv. Phys. 50, 813 (2001).
[17] X. Liu, X. Zhang, M. Lin, and P. Tan, Chin. Phys. B 26, 067802 (2017).
[18] P. Tan, W. Han, W. Zhao, Z. Wu, K. Chang, H. Wang, Y. Wang, N. Bonini, N. Marzari, N. Pugno, G. Savini, A. Lombardo, and A. C. Ferrari, Nat. Mater. 11, 294 (2012).
[19] P. Ramnani, M. R. Neupane, S. Ge, A. A. Balandin, R. K. Lake, and A. Mulchandani, Carbon 123, 302 (2017).
[20] L. Liang, J. Zhang, B. G. Sumpter, Q. Tan, P. Tan, and V. Meunier, ACS Nano 11, 11777 (2017).
[21] C. Cong and T. Yu, Nat. Commun. 5, 1 (2014).
[22] I. Maity, P. K. Maiti, and M. Jain, Phys. Rev. B 97, 1 (2018).
[23] L. Jiang, R. Liu, R. Su, Y. Yu, H. Xu, Y. Wei, Z. Zhou, and X. Wang, Nanoscale 10, 13565 (2018).
[24] H. Ding, M. Liu, N. Pan, Y. Dong, Y. Lin, T. Li, J. Zhao, Z. Luo, Y. Luo, and X. Wang, J. Phys. Chem. Lett. 10, 1355 (2019).
[25] V. Shcheslavskiy, P. Morozov, A. Divochiy, Y. Vakhtomin, K. Smirnov, and W. Becker, Rev. Sci. Instrum. 87, 053117 (2016).
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Shen-long Jiang, ,
Multi-domain platform /
Spectral/spatial/temporal resolution /
Integrated characterizations /
Spectroscopy /
Microscopy /
Abstract: In recent decades, materials science has experienced rapid development and posed increasingly high requirements for the characterizations of structures, properties, and performances. Herein, we report on our recent establishment of a multi-domain (energy, space, time) high-resolution platform for integrated spectroscopy and microscopy characterizations, offering an unprecedented way to analyze materials in terms of spectral (energy) and spatial mapping as well as temporal evolution. We present several proof-of-principle results collected on this platform, including in-situ Raman imaging (high-resolution Raman, polarization Raman, low-wavenumber Raman), time-resolved photoluminescence imaging, and photoelectrical performance imaging. It can be envisioned that our newly established platform would be very powerful and effective in the multi-domain high-resolution characterizations of various materials of photoelectrochemical importance in the near future.
Ⅰ. INTRODUCTION
In recent decades, a huge advancement in the field of materials science has been witnessed. For instance, the size of semiconductor devices is pushed to a few nanometers [1, 2], the length scale in the measurements of electron transport properties reaches the order of optical wavelength [3, 4], and the time scale in the measurements of charge carrier dynamics enters a few picoseconds or even femtoseconds [5, 6]. The in-situ microscopic and real-time dynamic processes, which usually play key roles in the functionalization of materials, have aroused great attention from both experimentalists and theorists in a wide spectrum of fundamental and application research fields. The development of multi-domain (energy, space, time) high-resolution characterization techniques has become a topical trend in the field of high-precision measurement of material systems [7]. These techniques can lay a solid foundation for the accurate determination of important basic parameters and processes, such as physical and chemical structures, excited-state properties, charge-carrier interactions, and enigmatic quantum effects under external fields. The confocal Raman microscopy is a robust tool for investigating the form and structure of materials [8]. Benefited from some innovations, its spatial resolution has been routinely improved to hundreds of nanometers. Besides, if a pulsed ultrashort laser (picosecond or femtosecond) is used as the excitation source and a delayed synchronized acquisition is arranged, it can also achieve a high temporal resolution. Herein, we describe a multi-domain (energy, space, time) high-resolution platform for integrated spectroscopic and microscopic characterizations of materials. The proof-of-principle results collected on this platform demonstrate its powerful functionalities, promising extensive applications toward multi-domain characterizations of various materials with high spectral, spatial, and temporal resolutions.
Ⅱ. INSTRUMENTATION
FIG. 1 shows the schematic diagram of our newly established multi-domain platform, which is partly based on a WITec alpha300 RAS+ confocal Raman microscopy (WITec GmbH) but with some important modifications.
Figure 1. The Schematic diagram of the multi-domain (energy, space, time) high-resolution platform for integrated spectroscopy and microscopy characterizations.
As for the function of scanning near-field optical microscopy (SNOM), we succeed in improving its spatial resolution from $ \sim $100 nm to $ \sim $60 nm by using an aperture whose pinhole diameter is reduced as much as possible without significantly reducing the amount of light transmission; meanwhile, the sample substrate is modified by an aperture array made of Au nano-gaps [9], as can be clearly visualized from an atomic-force microscopy (AFM) image shown in FIG. 2(a). Such an arrangement turns out to greatly improve the signal intensity and signal-to-noise ratio due to the gap-plasmon effect. As such, the near-field resolution is further improved from $ \sim $60 nm to $ \sim $20 nm, as revealed in FIG. 2 (b) and (c).
Figure 2. (a) AFM image of an Au nano-gaps aperture array. (b) SNOM image corresponding to (a). (c) The variation of AFM height (red line) and SNOM intensity (black line) with distances along the white dashed lines in (a) and (b), respectively.
As for the laser systems used, we achieve a convenient switching among the following three: (ⅰ) a diode-pumped continuous-wave (CW) laser (355 or 532 nm, Cobalt Laser); (ⅱ) a picosecond pulsed diode laser (PDL 800-D, 373 nm, PicoQuant GmbH); (ⅲ) a tunable femtosecond laser, which is delivered by an optical parametric amplifier (TOPAS-Prime, 240$ - $2600 nm, Light Conversion) excited by a Ti:sapphire regenerative amplifier (Astrella, center wavelength 800 nm, pulse duration 35 fs, pulse energy 7 mJ, repetition rate 1 kHz, Coherent).
As for the Raman or photoluminescence (PL) measurements, the polarization of the excitation laser is adjusted by a half-wave plate and a Glan-Taylor prism, and the polarization of the signal is adjusted by another Glan-Taylor prism. The optical delay line, which is used to variably adjust the time delays between the excitation and probe lasers, comprises several mirrors with low group-delay dispersion and a linear DC motor stage (Parker GmbH). The excitation laser is reflected by a dichroic mirror into an objective or a tip with a tiny hole of 60/90/100 nm diameter and then is focused onto the sample. The movement of the sample is controlled by a piezo stage. The diffraction-limited spot size of the CW 532-nm excitation laser is determined to be $ \sim $300 nm (with a 100$ \times $ objective, NA = 0.90 where NA stands for numerical aperture). The probe laser dispersed by an $ f $ = 600 mm spectrometer equipped with 150, 600, or 1800 grooves/mm grating (UHTS 300). The collected signals are detected using a back-illuminated charge-coupled device (CCD) camera (DU970N-BV, Andor) that is thermoelectrically cooled down to $ - $60 $ ^{\circ} $C. It is worth noting here that the dichroic mirror can be replaced by BragGrate notch filters (OptiGrate Corp.) with an optical density of 3 and a spectral bandwidth of $ \sim $5$ - $10 cm$ ^{-1} $ such that the Rayleigh line can be effectively suppressed to aid in the acquisition of low-wavenumber Raman spectra.
As for the micro-zone photoelectrical performance measurements, a source meter (Keithley 2400, Tektronix Inc.) is used as the power supplier to provide the background potential difference. When a photovoltaic device is irradiated, the separation of photogenerated electrons and holes gives rise to a weak photocurrent, which is amplified by a current amplifier (FEMTO Messtechnik GmbH) and then sent to a computer for processing and analysis. Note that all of the following proof-of-principle tests are performed under ambient conditions.
Ⅲ. PROOF-OF-PRINCIPLE TESTS
To examine the spatial and spectral resolutions accomplished on our platform, we perform measurements on an ultrathin graphene sample, which is prepared using a routine mechanical exfoliation method [10]. As shown in FIG. 3(a), the area of Raman imaging (approximately 20 µm$ \times $20 µm) is surrounded by the blue dotted lines. Through a point-by-point scanning, we can readily analyze the spectral profile of each point to obtain the structural information of interest. Displayed in FIG. 3 (b)-(d) are the Raman mapping results for the G band of graphene ($ \sim $1582 cm$ ^{-1} $, in-plane C$ - $C stretching) [11], including the spatial distributions of its intensity, peak frequency ($ \omega_ \rm{G} $), and full width at half maximum (FWHM, $ \Gamma_ \rm{G} $), respectively. From these spatial mappings one can clearly see that the degree of wrinkling-induced stress varies across the large-scale graphene [12, 13]. Moreover, the spectral structures of the 2D band ($ \sim $2677 cm$ ^{-1} $, the first overtone of D mode) [11] in the labelled regions (FIG. 3(e)) exhibit an obvious position dependence with rich evolution information. These results would enable a further understanding of the correlation between energy space and real space.
Figure 3. (a) Optical imaging of the crystal structure showing the tested graphene sample. Spatially resolved Raman mappings of the G-band (b) intensity, (c) peak frequency, and (d) FWHM, all corresponding to the blue rectangle region in (a). (e) Left panel: spatially resolved Raman imaging of the 2D-band intensity distributions corresponding to the green rectangle region in (a), right panel: position-dependent spectral evolution of 2D band. (f) Polar plot of the G-band intensity $ I_ \rm{G} $ as a function of laser polarization angle $ \theta $ for configurations of back-scattering geometry. (g) Low-wavenumber Raman spectra showing Stokes and anti-Stokes branches for the C and ZO$ ' $ modes.
Angle-resolved polarization Raman spectroscopy can be employed to identify the Raman modes (based on crystal symmetry and Raman selection rules) and to determine the crystallographic orientation of anisotropic materials [14, 15]. In our back-scattering geometry ($ Z $-in and $ Z $-out for the light), the electric polarization vectors of the incident $ e_ {\rm{i}} $ and scattered $ e_ {\rm{s}} $ light are in the $ X $-$ Y $ plane. By setting the angle between the $ e_ {\rm{i}} $ polarization and the $ X $ axis as $ \theta $ and fixing the $ e_ {\rm{s}} $ polarization to be parallel with the $ X $ axis, we have
According to the relationship of Raman intensity
where $ \; {R} $ is the Raman tensor [16], we have
As an example, given that the Raman tensors of the G band are $ \tilde{R_1} $ and $ \tilde{R_2} $ [17]:
the calculated $ I_ \rm{G} $ equals to a constant of $ c^2 $, independent of laser polarizations and sample azimuth angles. As such, the G band manifests itself as an isotropic distribution on the basal plane, as shown in FIG. 3(f).
Low vibrational-frequency (typically below 140 cm$ ^{-1} $) micro-Raman spectroscopy has been used to unveil such details as stacking order and interlayer interactions in two-dimensional materials [18-20]. The reflective-volume Bragg-grating-based notch filters can be adopted for facilitating the acquisition of low-wavenumber Raman spectra. Such a notch filter can reflect light with a bandwidth as narrow as 10 cm$ ^{-1} $, not affected by other wavelengths passing through and with an overall transmittance up to 95%. By integrating it into the platform, we achieve simultaneous recording of the Stokes and anti-Stokes Raman spectra featuring low wavenumbers. As a demonstration, FIG. 3(g) shows a representative result on the two important rigid-plane Raman modes in bilayer graphene, i.e., shear mode (or C mode, $ \sim $28 cm$ ^{-1} $) [21] and out-of-plane interlayer optical photon breathing mode (or ZO$ ' $ mode, $ \sim $85 cm$ ^{-1} $) [22], linking directly to the interlayer interactions of interest therein.
The above proof-of-principle characterizations demonstrate that our platform has been of high caliber in the spectral (energy) and spatial domains. To exploit its functionality in the temporal domain, we replace the CW light source with a pulsed laser (pulse duration 40 ps) such that the time-resolved imaging measurements can be executed with the aid of an external synchronization system. FIG. 4(a) shows the optical imaging for the crystal structure of a typical perovskite material of CsPbBr$ _3 $ with square and stick shapes. FIG. 4(b) exhibits a representative PL emission spectrum excited with the 373-nm pulsed laser, showing a strong excitonic emission band peaking at $ \sim $526 nm [23]. Considering that such an excitonic band may consist of contributions from bright and dark exciton states [24], we use 528-nm short/long-wave-pass filters to record the asymmetrically distributed PL emissions, as divided into spectral regions A and B in FIG. 4(b), mainly reflecting the bright and dark excitonic nature, respectively [24]. The time-correlated single photon counting technique [25] is adopted to map out the PL lifetimes ($ \tau_{ \rm{PL}} $) of the two spectral regions, with an instrument response function of $ \sim $100 ps. FIG. 4(c) shows a typical exponential fitting to yield the specific $ \tau_{ \rm{PL}} $ of region A for the square-shaped sample. The resulting $ \tau_{ \rm{PL}} $ mappings for the square- and stick-shaped samples are displayed in FIG. 4 (d)-(f) and FIG. 4 (g)-(i), respectively. Obviously, the two samples exhibit distinctly different patterns of spatially non-uniform $ \tau_{ \rm{PL}} $ distributions. As for the square-shaped sample, one can see from FIG. 4 (d) and (e) that the upper and left edges emit longer-lived PL than the lower and right edges for both regions A and B. Nevertheless, the lifetime differences ($ \Delta $$ \tau $$ _{ \rm{PL}} $) between regions A and B (point-by-point), as plotted in FIG. 4(f), manifest as an interesting nested pattern where the narrow exterior with negatively-valued $ \Delta $$ \tau $$ _{ \rm{PL}} $ (i.e., region-A emissions are shorter-lived than region-B emissions) surrounds the large-area interior with positively-valued $ \Delta $$ \tau_{ \rm{PL}} $ (i.e., region-A emissions are longer-lived than region-B emissions). As for the stick-shaped sample, one can see from FIG. 4 (g) and (h) that longer-lived PL emissions appear at its two ends for both regions A and B. Remarkably, the corresponding $ \Delta $$ \tau $$ _{ \rm{PL}} $ mapping, as plotted in FIG. 4(i), exhibits also a nested pattern but with exactly opposite $ \Delta $$ \tau $$ _{ \rm{PL}} $ signs. These proof-of-principle tests (taking time-resolved PL as an example) highlight the utility of time-domain mapping function integrated in our multi-domain platform. Along this line, we are currently making efforts to further extend to higher temporal resolution by introducing femtosecond light sources, such as the broadband tunable TOPAS-Prime system (see Instrumentation section). By doing so, in conjunction with the already achieved high spectral (energy) and spatial resolutions, we expect that our multi-domain (energy, space, time) platform would be capable of producing a wealth of high-quality data full of new physics to be explored, such as the elusive interaction, correlation, and interplay involved in a variety of exotic material systems.
Figure 4. (a) Optical imaging of the crystal structure of CsPbBr$ _3 $ perovskite material with different shapes. (b) PL emission spectrum (excitation at 373 nm). (c) Time-resolved PL kinetics (excitation at 373 nm). (d) Lifetime mapping of the square-shaped CsPbBr$ _3 $ for region A in (b). (e) Lifetime mapping of the square-shaped CsPbBr$ _3 $ for region B in (b). (f) Lifetime difference pattern derived from (d) and (e). (g) Lifetime mapping of the stick-shaped CsPbBr$ _3 $ for region A in (b). (h) Lifetime mapping of the stick-shaped CsPbBr$ _3 $ for region B in (b). (i) Lifetime difference pattern derived from (g) and (h).
Last but not least, the function of photocurrent imaging is also integrated in our multi-domain high-resolution platform, as schematically illustrated in FIG. 5(a). The spatially resolved photocurrent imaging is operated in a micro-zone scanning mode. As a proof of principle, we conduct an evaluation of photoelectrical performance on a test sample of silicon electrode material (FIG. 5(b)). FIG. 5(c) shows a typical photocurrent response (excitation at 532 nm) under laser on/off conditions. FIG. 5(d) maps out the spatial distribution of photocurrent for a specific zone of the test sample (as marked by the red rectangle in FIG. 5(b)). Such a mapping, in combination with the above-described multi-domain functions, would be very useful for gleaning in-situ information from certain material devices. Hence, the relationship between the photoelectrical performances and the physical structures, properties, and effects can be scrutinized and elucidated, thereby providing instructive guidance for rational design of materials and material devices.
Figure 5. (a) Schematic configuration of the photocurrent mapping setup that is also integrated in our multi-domain platform. (b) Optical imaging of a test sample of silicon electrode material. (c) A representative test recording photocurrent response (excitation at 532 nm). (d) Photocurrent mapping for the red rectangle zone in (b).
Ⅳ. Conclusion
To summarize, we have successfully established a multi-domain high-resolution platform for integrated spectroscopic and microscopic characterizations, in which the best resolution values are about 10 cm$ ^{-1} $ (energy), 20 nm (space), and 100 ps (time). The multiple functions of our platform are carefully tested with a set of proof-of-principle measurements including Raman imaging (high-resolution Raman, polarization Raman, low-wavenumber Raman), time-resolved photoluminescence imaging, and photoelectrical performance imaging, under in-situ and/or real-time operating conditions. Such a multi-domain high-resolution platform, with further modifications (e.g., by introducing femtosecond time-resolved light sources, external fields, and temperature-controlling systems) that are currently underway in our laboratory, would enable more comprehensive, robust, and effective characterizations in a highly integrated fashion. The obtained multi-domain high-resolution data would allow us to explore new physics such as the elusive interaction, correlation, and interplay involved in photoelectrochemical, photonic, and plasmonic material systems as well as the related prototypical devices.
Ⅴ. Acknowledgments
This work is supported by the National Key Research and Development Program of China (No.2016YFA0200602, No.2017YFA0303500, and No.2018YFA0208702), the National Natural Science Foundation of China (No.21573211, No.21633007, No.21803067, and No.91950207), the Anhui Initiative in Quantum Information Technologies (AHY090200), and the USTC-NSRL Joint Funds (UN2018LHJJ). | CommonCrawl |
Total phenolic and flavonoid contents and antioxidant, anti-inflammatory, analgesic, antipyretic and antidiabetic activities of Cordia myxa L. leaves
Enas R. Abdel-Aleem1,
Eman Z. Attia1,
Fatma F. Farag1,
Mamdouh N. Samy ORCID: orcid.org/0000-0002-2128-53891 &
Samar Y. Desoukey1
Many plants of genus Cordia are traditionally used as astringent, anti-inflammatory, anthelminthic, antimalarial, diuretic, febrifuge, appetite suppressant and cough suppressant and to treat urinary infections, lung diseases and leprosy. The aim of the study is to determine the total phenolic and flavonoid contents of total ethanol extract and different fractions of C. myxa L. leaves, in addition to evaluation of some pharmacological activities including antioxidant, anti-inflammatory, analgesic, antipyretic and antidiabetic effects.
Air dried powder of C. myxa leaves were extracted using 95% ethanol and fractionated successively with petroleum ether, dichloromethane, ethyl acetate and finally with n-butanol. The fractions were concentrated and then investigated for their antioxidant, anti-inflammatory, analgesic, antipyretic and antidiabetic activities using phosphomolybidinum and DPPH assays, carrageenan-induced paw edema, hot plate, yeast -induced pyrexia and streptozotocin-induced hyperglycemia methods, respectively.
The ethyl acetate fraction showed the highest antioxidant activity with high phenolic and flavonoid contents (31.03 ± 0.15 mg gallic acid equivalent/g dried weight and 811.91 ± 0.07 mg rutin equivalent/g dried weight, respectively). Dichloromethane and ethyl acetate fractions exhibited higher anti-inflammatory activity with percentages of inhibition 45.16% and 40.26%, respectively, which were quite comparable to that of indomethacin (51.61%). The petroleum ether and dichloromethane fractions showed the highest analgesic activity with reaction time 289.00 ± 3.00 and 288.33 ± 20.82, respectively. Evaluation of antipyretic activity revealed that the total ethanol extract and different fractions showed high antipyretic activities after 2 h, which were very close to that of the standard acetyl salicylic acid with a rapid onset (30 min). The total ethanol extract and the petroleum ether fraction exhibited the most potent hypoglycemic effect with a significant reduction in blood glucose level especially after 3 h to 95.67 ± 5.77 mg/dl and 87.67 ± 10.26 mg/dl, respectively and percentages decrease in blood glucose level were 68.22% and 70.78%, respectively.
Cordia myxa L. extract and fractions exhibited antioxidant, anti-inflammatory analgesic, antipyretic and antidiabetic activities which may be attributed by the presence of active phytoconstituents.
Boraginaceae (borage) family comprises about 2740 species distributed in 148 genera [1]. Various chemical constituents isolated and characterized from Boraginaceous plants, including pyrrolizidine alkaloids, naphthaquinones, flavonoids, terpenoids, triterpenoids and phenols [2]. Cordia is an important and representative genus of this family that could grow as trees, shrubs or sometimes subscandents [1]. The generic name honours a sixteenth century botanist, Valerius Cordus [3]. The genus Cordia originates from tropical and subtropical regions. About 300 species have been identified worldwide, mostly in the warmer regions. Cordia myxa (Syn. Cordia obliqua, Cordia crenata) is a medium sized deciduous tree about 10.5 m [4]. The chemical characteristics of this genus are the presence of quinones which are known as cordiaquinones [1]. Pyrrolizidine alkaloids are generally present as esters. More than 200 pyrrolizidine alkaloids have been isolated from these plants. Although these alkaloids are cytotoxic and cause poisoning, Cordia myxa was reported to contain the nontoxic alkaloid macrophylline [2].
The plant parts like fruits, leaves, stem bark, seeds and roots of most species of this genus, especially Cordia dichotoma, C. myxa, C. verbenacea, C. martinicensis, C. salicifolia, C. spinescens, C. latifolia, C. ulmifolia, among others, have long been used in traditional medicine as astringent, anti-inflammatory, anthelminthic, antimalarial, diuretic, febrifuge, appetite suppressant and cough suppressant and to treat urinary infections, lung diseases and leprosy. Cordia myxa fruits and seeds are also used as expectorants and effective in treating the diseases of the lungs. Raw fruits contain a gum which can be used beneficially in gonorrhea. They can remove pain from the joints and the burning of the throat, effective in treating the diseases of the spleen, and are used as a demulcent in southern Iran [1]. For this reasons the current study evaluate some pharmacological activities of total ethanol extract and different fractions of C. myxa L. leaves including antioxidant, anti-inflammatory, analgesic, antipyretic, antidiabetic and antimalarial effects.
The leaves of C. myxa were collected from May to November 2015 from El-Orman botanical garden, Cairo, Egypt and identified by Prof. Dr. Nasser Barakat, Professor of Botany, Faculty of Science, Minia University. A voucher sample (Mn-Ph-Cog-023) was kept in the Herbarium of Pharmacognosy Department, Faculty of Pharmacy, Minia University, Minia, Egypt.
Preparation of the extract and fractions
The air dried powdered leaves (4.24 Kg) of C. myxa. Were extracted by maceration with 95% ethanol at room temperature with occasional agitation for 7 days till exhaustion, and then concentrated under reduced pressure till dryness. The concentrated ethanol extract (275 g) was suspended in the least amount of distilled water, transferred to a separating funnel and partitioned successively by liquid/liquid extraction with petroleum ether, dichloromethane, ethyl acetate and finally with n-butanol. The fractions were concentrated under reduced pressure at 40 °C to afford petroleum ether (107 g), dichloromethane (13.2 g), ethyl acetate (11.74 g) and n-butanol (49.92 g) fractions.
The animals used in this study include female and male albino rats weighing 200 ± 50 g and mice weighing 30 ± 5 g, obtained from animal house of Faculty of Medicine, Assiut University. They were housed under standardized environmental conditions, and fed with standard diet and water. The study was conducted following approval by the Institutional Animal Ethical Committee of Faculty of Pharmacy, Minia University, Minia, Egypt.
The acute toxicity of the total ethanolic extract of Cordia myxa leaves was evaluated by measuring the lethal dose for 50% of the laboratory animals (LD50) [5]. Different dose levels (1, 2, 2.5, 3 up to 3.5 g/ kg, p.o) of the total ethanolic extract (suspended in 0.5% CMC) were orally administrated to different groups of mice (30 ± 5 g, each containing six mice). The control group received an equivalent dose of the vehicle (0.5% CMC). Both the test and control groups were noticed for 24 h under normal environmental conditions, with free access to food and water.
Evaluation of antioxidant study
Each sample was dissolved in 95% methanol to make a concentration of 40 mg/ml from total ethanolic extract and different fractions except the ethyl acetate fraction which was prepared as 10 mg/ml and then diluted to prepare the series concentrations for antioxidant assays.
Estimation of total phenolic content
The content of total phenolic compounds for total ethanol extract and different fractions of C. myxa leaves were determined by Folin–Ciocalteu method [6]. Analysis was performed by adding 3.5 ml of deionized water, 50 μl of sample extract, 50 μl Folin-Ciocalteu reagent (2 N) and 300 μl of sodium carbonate (10%). The reaction was left for 30 min. and then the absorbance was measured in triplicate at 730 nm. The blank consisted of all reagents excluding the sample extract. A standard curve was made using gallic acid so that total phenolic concentration was expressed as mg of gallic acid equivalents per gram dried fraction.
Estimation of total flavonoid content
Total flavonoid content was determined following a method in literature [6], where 0.3 ml of extracts, 3.4 ml of 30% methanol, 0.15 ml of NaNO2 (0.5 M) and 0.15 ml of AlCl3.6H2O (0.3 M) were mixed. After 5 min, 1 ml of NaOH (1 M) was added. The solution was mixed well and the absorbance was measured in triplicate against the reagent blank at 506 nm. The standard curve for total flavonoids was made using rutin standard solution at different concentration under the same procedure. The total flavonoid content was expressed as milligrams of rutin equivalents per gram of dried fraction.
Phosphomolybdate assay (total antioxidant capacity)
The total antioxidant capacity of the fractions was determined by phosphomolybdate method using ascorbic acid as a standard [6]. An aliquot of 0.3 ml of sample solution was mixed with 3 ml of reagent solution (0.6 M sulphuric acid, 28 mM sodium phosphate and 4 mM ammonium molybdate). The tubes were capped and incubated in a water bath at 95 °C for 90 min. After the samples had cooled to room temperature, the absorbance of the mixture was measured at 695 nm against a blank. A typical blank contained 3 ml of the reagent solution and the appropriate volume of the solvent and incubated under the same conditions. A standard curve was made using ascorbic acid; hence, the antioxidant activity was expressed relative to that of ascorbic acid. All determinations were done in triplicate.
DPPH radical scavenging activity assay
The free radical scavenging activity of the fractions was measured using 1,1- diphenyl-2-picryl-hydrazyl (DPPH) [6]. Briefly, 200 μl of each of the extract or fractions at various concentrations was added to 2 ml of DPPH solution (0.1 mM), The reaction mixture was shaken well and incubated in the dark for 15 min at room temperature. Methanol was used instead of the extract and fractions as a control. Then the absorbance was measured in triplicate at 517 nm. The capability to scavenge the DPPH radical was calculated using the following equation:
DPPH scavenging effect (%) = [(A0 -A1/A0) × 100].
Where A0 was the absorbance of the control reaction and A1 equal the absorbance in the presence of the extract. The extract concentration providing 50% inhibition (IC50) was calculated from the graph of DPPH scavenging effect against extract concentration.
Anti-inflammatory activity
The total ethanol extract and different fractions of C. myxa leaves were evaluated for their anti-inflammatory activity using the carrageenan-induced paw edema method [7]. Female albino rats (200 ± 50 g) were randomly divided into seven groups (six animals per group). The specified dose of extract, fractions, and standard drug were suspended in 0.5% CMC solution. The –ve control group administered the vehicle (0.5% CMC solution), while the standard drug indomethacin (+ve control) was given orally at a dose level of 8 mg/kg. The total ethanol extract and different fractions were administered orally at a dose of 350 mg/kg through 2 h after carrageenan injection 0.1 ml, 1% w/v in normal saline, s.c.) into the sub-plantar tissue of the right hind paw. The paw thickness (mm) was measured using a vernier calliper at 0, 0.5, 1, 2, 3, 4 and 5 h after administration of the tested extract, fractions and standard drug. The percentage inhibition of the rat paw edema was calculated as follows [8]:
$$ \%\mathrm{Inhibition}=\frac{\left(\mathrm{Control}\ \mathrm{mean}-\mathrm{treated}\ \mathrm{mean}\right)}{\mathrm{Control}\ \mathrm{mean}}\mathrm{X}\ 100 $$
Analgesic activity
The analgesic activity of the total ethanol extract and different fractions of C. myxa L. leaves was evaluated using hot plate method [9]. Mice (30 ± 5 g) were grouped into seven groups (six animals each). The –ve control group administered the vehicle (0.5% CMC solution) and the standard drug paracetamol 100 mg/kg, p.o. (+ve control). The tested extract and different fractions were suspended in 0.5% CMC solution and were administered orally at a dose level of 350 mg/kg. The animals were placed on a hot plate and the temperature of the metal surface was maintained at 54 °C. The time (sec) of the response produced by the animal as tail withdrawn, licking paws or jumping due to radient heat is noted and recorded at 0, 0.5, 1, 2, 3, 4 and 5 h after the administration of the tested extract, fractions and the standard drug.
The percentage of protection against thermal stimulus was calculated as follows [10]:
$$ \%\mathrm{protection}\ \mathrm{against}\ \mathrm{thermal}\ \mathrm{stimulus}=\frac{\mathrm{Test}\ \mathrm{mean}\left(\mathrm{Ta}\right)\hbox{-} \mathrm{Control}\ \mathrm{mean}\left(\mathrm{Tb}\right)}{\mathrm{Control}\ \mathrm{mean}\left(\mathrm{Tb}\right)}\mathrm{X}\ 100 $$
Antipyretic activity
The total ethanol extract and different fractions of C. myxa L. leaves were evaluated for their antipyretic activity using yeast-induced pyrexia method [11, 12]. The test was performed on female albino rats (200 ± 50 g) by subcutaneous injection (in the back, below the nape of the neck) of 20% aqueous suspension of yeast in a dose of 10 ml/kg to induce pyrexia. The pyretic animals were grouped into seven groups (six animals each). The –ve control group orally administered the vehicle (0.5% CMC solution), while the +ve control was given the reference drug acetylsalicylic acid at a dose level of 330 mg/kg, p.o). The tested extract and different fractions were suspended in 0.5% CMC solution and were administered orally at a dose level of 350 mg/kg through 2 h after yeast injection. The rectal temperature of each animal was recorded by inserting a thermometer 2 cm into the rectum at 0, 0.5, 1, 2, 3, 4 and 5 h following the administration of the tested extract, fractions and the reference drug.
Anti-diabetic activity
The total ethanol extract and different fractions of C. myxa L. leaves were evaluated for their anti-diabetic effects using streptozotocin-induced hyperglycemia method [12]. The test was performed on adult male albino rats (200 ± 50 g) by intraperitoneal injection of streptozotocin (80 mg/kg). Blood glucose level was measured after 3 days up to 1 week for assessment of hyperglycemia. Rats with blood glucose level above (200 mg/dl) were considered to be diabetic and were used in this study. The diabetic rats were divided into seven groups (six rats each). The control group was administered the vehicle (0.5% CMC solution) and the standard drug vildagliptin 50 mg/kg p.o. (+ve control). The tested extract and different fractions were suspended in 0.5% CMC solution and were orally administered at a dose level of 350 mg/kg. Blood glucose levels were measured at intervals of 0 (fasting), 0.5, 1, 2, 3, 4 and 5 h by collecting blood samples from the tail vein (caudal vein). The percentage of change in blood glucose level was calculated by the following formula [12, 13]:
$$ \%\mathrm{lowering}\ \mathrm{of}\ \mathrm{blood}\ \mathrm{glucose}\ \mathrm{level}=\frac{\mathrm{Wc}\left(\mathrm{fasting}\right)-\mathrm{Wt}\left(\mathrm{test}\right)}{\mathrm{Wc}\left(\mathrm{fasting}\right)}\mathrm{X}\ 100 $$
Results of all biological studies were expressed as means ± S.E.M. One-way analysis of variance (ANOVA) followedby Dunnett's test was used to determine significance when compared to the control group. p values less than 0.05, 0.01, and 0.001 were considered significant (*p < 0.05, **p < 0.01,*** p < 0.001). Graph Pad Prism 5 was used for statistical calculations (Graph pad Software, San Diego California, USA).
Concerning the safety of C. myxa L. plant, a preliminary toxicity study showed no mortality up to a dose level of 2500 mg/Kg and this was in accordance with published data which reported that both the leaves and the fruits are safe and edible in India and other countries [14, 15].
The content of total phenolic compounds of the total ethanol extract and the different fractions of C. myxa L. leaves was determined by Folin– Ciocalteu method. The results were shown in Table 1 and Additional file 1.
Table 1 Results representing the total phenolic content of the total ethanol extract and the different fractions of C. myxa L. leaves using Folin–Ciocalteu method
Results were varied widely, ranging from 4.54 ± 0.04 to 31.03 ± 0.15 mg gallic acid equivalent/g dried weight. The highest amount of phenols was found in the ethyl acetate fraction (31.03 ± 0.15), followed by the dichloromethane, total extract, petroleum ether and n-butanol fractions (8.81 ± 0.04), (5.76 ± 0.23), (4.75 ± 0.22) and (4.54 ± 0.04), respectively.
The total flavonoid content of the total ethanol extract and different fractions of C. myxa L. leaves was determined and the results were shown in Table 2 and Additional file 2, where they varied widely, ranging from 13.09 ± 0.11 to 811.91 ± 0.07 mg rutin equivalent/g dried weight. The highest amounts of flavonoids was.
Table 2 Results representing the total flavonoid content of the total ethanol extract and the different fractions of C. myxa L. leaves
found in the ethyl acetate fraction (811.91 ± 0.07) followed by the dichloromethane fraction (295.45 ± 0.13), the total extract (192.24 ± 0.33), the n-butanol fraction (84.29 ± 0.017) and lastly the petroleum ether fraction which contained remarkably lower amounts of these compounds (13.09 ± 0.11).
The total ethanol extract and the different fractions of C. myxa L. leaves showed varied antioxidant capacity. The highest antioxidant capacity was displayed by the ethyl acetate fraction (103.40 ± 0.05) mg of ascorbic acid equivalent/g dried weight. The other fractions and the total ethanol extract showed mild antioxidant capacities, where the butanol fraction showed the lowest capacity (12.58 ± 0.60) mg of ascorbic acid equivalent/g dried fraction. The results of phosphomolybdate assay were listed in Table 3 and illustrated in Additional file 3.
Table 3 Results of the total antioxidant capacities of the total ethanol extract and the different fractions of C. myxa L. leaves using phosphomolybdate assay
By analyzing the antioxidant activity of the total ethanol extract and the different fractions of C. myxa L. leaves, it was demonstrated that all showed dose-dependent antioxidant activity. Results were listed in Table 4 and illustrated in Additional file 4. The ethyl acetate fraction was the most effective DPPH free radical scavenger with IC50 with a value of 0.03 ± 0.01 mg/ml. The other fractions and the total ethanol extract showed higher IC50, thus lower DPPH radical scavenging activity, where the total ethanol extract exhibited the lowest DPPH radical scavenging activity with IC50 value of 1.33 ± 0.03 mg/ml as compared with BHT (0.96 ± 0.03 μg/ml) as a positive control.
Table 4 Results of the free radical scavenging activities of the total ethanol extract and the different fractions of C. myxa L. leaves using DPPH assay
It is obvious from Tables 5 and 6 and Additional file 5 that the percentages of edema inhibition varied within the different fractions and the total extract. A gradual increase was observed for the first 4 h, then, the activity started to decline again. The highest anti-inflammatory activity was observed after 3 h in the total extract and after 4 h for the rest of the fractions, while the standard showed this activity after both the third and fourth hour from administration. After 3 h, the dichloromethane fraction showed moderate anti-inflammatory activity with a percentage of inhibition (40.26%), while, after 4 h, both the dichloromethane and the ethyl acetate fractions showed higher anti-inflammatory activity with percentages of inhibition (45.16%) and (40.26%), respectively, which were quite comparable to that of indomethacin (51.61%). The total extract and the butanol fraction showed mild inhibition of the inflammation that was increased by time to be 19.35% and 25.81%, respectively, after 4 h. On the other hand, the least activity was shown by the petroleum ether fraction throughout the 5 h of the experiment.
Table 5 Results of the anti-inflammatory activity of the total ethanol extract and the different fractions of C. myxa L. leaves using carrageenan-induced paw edema method
Table 6 Results representing the percentages of edema inhibition of the total ethanol extract and the different fractions of C. myxa L. leaves
The results of the analgesic activity of the total ethanol extract and different fractions of C. myxa L. leaves revealed that the total ethanol extract and all fractions exhibited potent analgesic activity compared to the control but less than the used standard paracetamol (Tables 7 and 8 and Additional file 6). One hour from the beginning of the experiment, the total ethanol extract, petroleum ether and dichloromethane fractions significantly increased the reaction time (106.33 ± 3.06***), (191.00 ± 31.08**) and (170.33 ± 21.39**), respectively. After 3 h, the petroleum ether and dichloromethane fractions showed the highest analgesic activity (289.00 ± 3.00***) and (288.33 ± 20.82***), respectively. They increased the reaction time more than three times as that of the –ve control. The total extract showed moderate analgesic activity. It increased the reaction time double that of the control. The ethyl acetate and butanol fractions showed slight analgesic activities compared to other fractions.
Table 7 Results of the analgesic activity of the total ethanol extract and the different fractions of C. myxa L. leaves in mice using the hot plate method using paracetamol as a standard
Table 8 Results representing the percentage of protection against external stimulus of the total ethanol extract and the different fractions of C. myxa L. leaves using paracetamol as a standard
The results of the antipyretic activity of the total ethanol extract and different fractions of C. myxa L. leaves revealed that the total ethanol extract and most of the fractions exhibited a significant (p < 0.001) antipyretic activity up to 4 h (Table 9 and Additional file 7). After 2 h, the total ethanol extract, the petroleum ether, the dichloromethane and the ethyl acetate fractions showed high antipyretic activities which were very close to that of the standard acetyl salicylic acid with a rapid onset (30 min). The antipyretic activity lasted up to 4 h. The butanol fraction showed antipyretic activity with a slower onset after 2 h which also lasted for 4 h.
Table 9 Results of the anti-pyretic activity of the total ethanol extract and the different fractions of C. myxa L. leaves using yeast-induced pyrexia method
From the mentioned results in (Tables 10 and 11 and Additional file 8) the total ethanol extract and petroleum ether fraction showed potent anti-diabetic effect with a significant decrease in blood glucose level with a rapid onset (30 min). Their effects were increased gradually until reaching the highest level after 3 h, and then a gradual decline was demonstrated for the rest of experiment. The total ethanol extract and the petroleum ether fraction were the most potent. They showed a significant reduction in blood glucose level especially after 3 h 95.67 ± 5.77***mg/dl and 87.67 ± 10.26***mg/dl, respectively. Also the percentage decrease in blood glucose level was (68.22%) and (70.78%), respectively, which demonstrates their close resemblance to the standard vildagliptin (78.69%). The dichloromethane fraction also showed a moderate gradual reduction in blood glucose level up to 121.67 ± 3.06*** mg/dl after 2 h with a percentage decrease of (42.79%) while both ethyl acetate and butanol fractions showed no anti-diabetic activity throughout the experiment time.
Table 10 Results of the antidiabetic activity of the total ethanol extract and the different fractions of C. myxa L. leaves using streptozotocin-induced hyperglycemia method and vildagliptin as a standard
Table 11 Results representing the percentage of decrease in of blood glucose level of the total ethanol extract and the different fractions of C. myxa L. leaves using streptozotocin-induced hyperglycemia method
Safety of C. myxa was evaluated and results confirmed the previous published data [14, 15]. The ethyl acetate fraction showed the highest antioxidant activity compared to the total ethanol extract and other fractions. This activity may be attributed to its high content of flavonoids and other phenolic compounds as phenyl propanoids. The total ethanol extract and dichloromethane fraction exhibited highest anti-inflammatory activities compared to indomethacin. Many mechanisms were proved for anti-inflammatory activity of flavonoids, such as: inhibition of cyclooxygenase and 5- lipoxygenase pathways, and inhibition of eicosanoid biosynthesis, together with their ability to inhibit neutrophil degradation [16]. The potent antipyretic activity of the total extract and different fractions of C. myxa L. may be attributed to its content of flavonoids and sterols [17]. Analgesic activity may be attributed to their content of sterols which were reported to have analgesic activity [18]. Finally; the hypoglycaemic effect is in accordance with reported literature [19] for the total ethanol extract which proved that the hypoglycemic mechanism is throughout the inhibition of α-glucosidase enzyme. Thus it can decrease blood glucose level or inhibit glucose absorption.
The present study confirmed that total ethanol extract and different fractions of C. myxa leaves showed significant antioxidant, antiinflammatory, analgesic, antipyretic and antidiabetic activities in different in vitro and in vivo experimental models, which are in accordance with folk medicine of many Cordia plants. The presence of high content of flavonoids and other phenolic compounds in the ethyl acetate fraction might have some role in the observed pharmacological activities. In addition, the acute toxicity does not show any symptoms, changes in behavior or mortality at 2.5 g/kg doses that indicate a therapeutic safety for the doses pharmacologically active. The further detailed investigation is ongoing to determine the exact bioactive phytoconstituents that are responsible for those actions of this plant as traditional medicine. Moreover, it could be a potential source for discovery of antioxidant, antiinflammatory, analgesic, antipyretic and antidiabetic drug development.
C. myxa :
Cordia myxa
CMC:
DPPH:
2,2-diphenyl-1-picrylhydrazyl
Inhibitory concentration 50
p.o.:
Taken orally
s.c:
Subcutaneous
Yadav R, Yadav SK. Evaluation of antimicrobial activity of seeds and leaves of Cordia obliqua wild against some oral pathogens. Indo Am J Pharm Res. 2013;3:6035–43.
Sharma RA, Singh B, Singh D, Chandrawat P. Ethnomedicinal, pharmacological properties and chemistry of some medicinal plants of Boraginaceae in India. J Med Plant Res. 2009;3:1153–75.
Hussain N, Kakoti BB. Review on ethnobotany and phytopharmacology of Cordia dichotoma. J Drug Deliver Therap. 2013;3:110–3.
Abdel El-Aleem ER, Seddik FE-Z, Samy MN, Desoukey SY. Botanical studies of the leaf of Cordia myxa L. J Pharmacogn Phytochem. 2017;6:2086–91.
Akhila JS, Deepa S, Alwar MC. Acute toxicity studies and determination of median lethal dose. Curr Sci. 2007;93:917–20.
Abdel-Wahab NM, El-kashef DF, Attia EZ, Desoukey SY, Elkhayat ES, Fouad MA, Kamel MS. Total phenol content and antioxidant activities of the fungus Ulocladium botrytis. J Adv Biomed Pharm Sci. 2018;1:17–9.
Winter CA, Risley GA, Nuss GW. Carrageenan induced edema in hind paw of the rat as an assay for inflammatory drugs. Proc Soc Exp Biol Med. 1962;111:544–7.
Sudjarwo ASS. The potency of piperine as anti-inflammatory and analgesic in rats and mice. Folia Med Ind. 2005;41:190–4.
Hosseinzadeh H, Ramezani M, Salmani G. Antinociceptive, anti-inflammatory and acute toxicity effects of Zataria multiflora Boiss extracts in mice and rats. J Ethanopharmacol. 2000;73:379–85.
Wang Y, Chen Y, Xu H, Luo H, Jiang R. Analgesic effects of glycoproteins from Panax ginseng root in mice. J Ethanopharmacol. 2013;148:946–50.
Panthong A, Kanjanapoth D, Taesotikul T, Wongcome T, Reutrakul V. Anti-inflammatory and antipyretic properties of Clerodendrum petasites S.Moore. J Ethanopharmacol. 2003;85:151–6.
Gomaa AA-R, Samy MN, Desoukey SY, Kamel MS. Antiinflammatory, analgesic, antipyretic and anti-diabetic activities of Abutilon hirtum (Lam.) Sweet. Clin Phytosci. 2018;4:11.
Peungvicha P, Thirawarapan SS, Temsiririrkkul R, Watanabe H, Prasain JK, Kadota S. Hypoglycemic effect of the water extract of Piper sarmentosum in rats. J Ethanopharmacol. 1998;60:27–32.
Chauhan D, Shrivastava AK, Patra S. Diversity of leafy vegetables used by tribal peoples of Chhattisgarh, India. Int J Curr Microbiol App Sci. 2014;3:611–22.
Gupta R, Das GG. Toxicity assessment and evaluation of analgesic, antipyretic and anti-inflammatory activities on Cordia obliqua leaf methanol extract. Pharm J. 2017;9:856–61.
Nijveldt RJ, Nood EV, Hoorn DE, Boelens PG, Norren KV, Paul AL. Flavonoids: a review of probable mechanisms of action and potential applications. Am J Clin Nutr. 2001;74:418–25.
Niazi J, Gupta V, Chakarborty P, Kumar P. Anti-inflammatory and antipyretic activity of Aleuritis moluccana leaves. Asian J Pharm Clin Res. 2010;3:35–7.
Owoyele BV, Oguntoye SO, Dare K, Ogun biyi BA, Aruboula EA, Soladoye AO. Analgesic, anti-inflammatory and antipyretic activities from flavonoid fractions of Chromolaena odorata. J Med Plant Res. 2008;2:219–25.
Malik A, Ahmad AR. Antidiabetic effect of standardized extract of Indonesian Kanunang leaves (Cordia myxa L.). Int J Pharm Tech Res. 2016;9(8):268–75.
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Department of Pharmacognosy, Faculty of Pharmacy, Minia University, Minia, Egypt
Enas R. Abdel-Aleem, Eman Z. Attia, Fatma F. Farag, Mamdouh N. Samy & Samar Y. Desoukey
Enas R. Abdel-Aleem
Eman Z. Attia
Fatma F. Farag
Mamdouh N. Samy
Samar Y. Desoukey
ERA performed all of the experiments in the laboratory and data collection, analysis, graphical representation and interpretation. EZA did critical statistical analysis and experiments. FFF did experiment design and overall monitoring. Critical revision of the article was done by MNS. Conception, experiment design, overall monitoring and final approval of the article was done by SYD. All authors read and approved the final manuscript.
Correspondence to Mamdouh N. Samy.
The study was approved by the Institutional Animal Ethical Committee of Faculty of Pharmacy, Minia University, Minia, Egypt.
The authors declare that they have no competing interest.
Total phenolic content of the total ethanol extract and the different fractions of C. myxa L. leaves. (DOCX 26 kb)
Total flavonoid content of the total ethanol extract and the different fractions of C. myxa L. leaves. (DOCX 27 kb)
Antioxidant capacities of the total ethanol extract and the different fractions of C. myxa L. leaves using phosphomolybdate assay. (DOCX 22 kb)
DPPH radical scavenging activity of the total ethanol extract and the different fractions of C. myxa L. leaves. (DOCX 150 kb)
Anti-inflammatory effects of the total ethanol extract and the different fractions of C. myxa L. leaves using carrageenan – induced paw edema method. (DOCX 94 kb)
Analgesic effects of the total ethanol extract and the different fractions of C. myxa L. leaves using hot plate method. (DOCX 93 kb)
Antipyretic effects of the total ethanol extract and the different fractions of C. myxa L. leaves using Yeast– induced pyrexia method. (DOCX 94 kb)
Antidiabetic effects of the total ethanol extract and the different fractions of C. myxa L. leaves using streptozotocin-induced hyperglycemia method. (DOCX 96 kb)
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Abdel-Aleem, E.R., Attia, E.Z., Farag, F.F. et al. Total phenolic and flavonoid contents and antioxidant, anti-inflammatory, analgesic, antipyretic and antidiabetic activities of Cordia myxa L. leaves. Clin Phytosci 5, 29 (2019). https://doi.org/10.1186/s40816-019-0125-z
Total phenolics
Total flavonoids
Antipyretic | CommonCrawl |
\begin{document}
\title{K\"ahler-Einstein metrics and algebraic geometry} \author{Simon Donaldson} \date{\today} \maketitle
\begin{abstract} This paper is a survey of some recent developments in the area described by the title, and follows the lines of the author's lecture in the 2015 Harvard Current Developments in Mathematics meeting. The main focus of the paper is on the Yau conjecture relating the existence of K\"ahler-Einstein metrics on Fano manifolds to K-stability. We discuss four different proofs of this, by different authors, which have appeared over the past few years. These involve an interesting variety of approaches and draw on techniques from different fields.
\end{abstract}
\newtheorem{thm}{Theorem} \newtheorem{defn}{Definition} \newcommand{\partial\overline{\partial}}{\partial\overline{\partial}} \newcommand{{\bf C}}{{\bf C}} \newcommand{{\bf P}}{{\bf P}} \newcommand{\overline{\partial}}{\overline{\partial}} \newcommand{{\bf R}}{{\bf R}} \newcommand{\beta_{n}}{\beta_{n}} \newcommand{{\rm Ricci}}{{\rm Ricci}} \newcommand{\Omega_{{\rm Euc}}}{\Omega_{{\rm Euc}}} \newcommand{{\rm reg}}{{\rm reg}} \newcommand{{\cal KS}(n,\kappa)}{{\cal KS}(n,\kappa)} \section{Introduction}
General existence questions involving the Ricci curvature of compact K\"ahler manifolds go back at least to work of Calabi in the 1950's \cite{kn:Cal1}, \cite{kn:Cal2}. We begin by recalling some very basic notions in K\"ahler geometry.
\begin{itemize}\item All the K\"ahler metrics in a given cohomology class can be described in terms of some fixed reference metric $\omega_{0}$ and a potential function, that is \begin{equation}\omega= \omega_{0}+ i \partial\overline{\partial} \phi. \end{equation} \item A hermitian holomorphic line bundle over a complex manifold has a unique {\it Chern connection} compatible with both structures. A Hermitian metric on the anticanonical line bundle $K_{X}^{-1}=\Lambda^{n}TX$ is the same as a volume form on the manifold. When this volume form is derived from a K\"ahler metric the curvature of the Chern connection can be identified with the Ricci tensor. (In general in this article we will not distinguish between metrics and Ricci tensors regarded as symmetric tensors or $(1,1)$-forms.) \item If the K\"ahler class is a $2\pi$ times an integral class a metric can be regarded as the curvature of the Chern connection on a holomorphic line bundle. The K\"ahler potential parametrising metrics in (1) has a geometrical meaning as a change in the Hermitian metric on the line bundle: $\vert \ \vert =e^{\phi}\vert\ \vert_{0}$.
\end{itemize} One of the questions initiated by Calabi was that of prescribing the volume form of a K\"ahler metric in a fixed cohomology class. By the $\partial\overline{\partial}$-lemma this is the same as prescribing the Ricci tensor (as a closed $(1,1)$-form in the class $c_{1}(X)$). This Calabi conjecture was established by Yau in 1976 \cite{kn:Y}. In particular when $c_{1}(X)=0$ this gives the existence of {\it Calabi-Yau metrics}, with vanishing Ricci curvature. Another question raised by Calabi involved {\it K\"ahler-Einstein metrics} with \lq\lq cosmological constant'' $\lambda$, so $ {\rm Ricci}=\lambda \omega$. The case $\lambda=0$ is the Calabi-Yau case as above and and if $\lambda$ is non-zero we may assume that it is $\pm 1$. The K\"ahler-Einstein condition can then be expressed as saying that we require that the Hermitian metric on the holomorphic line bundle $K_{X}^{-1}$ given by the volume form of $\omega$ realises $\pm \omega$ as the curvature form of its Chern connection. Explicitly, in complex dimension $n$ the equation to be solved for a K\"ahler potential $\phi$ is \begin{equation} (\omega_{0}+ i \partial\overline{\partial}\phi)^{n}= \omega_{0}^{n} \exp(\pm \phi +h_{0})\end{equation} where $h_{0}$ is the solution of $\partial\overline{\partial} h_{0}+\omega_{0}=\pm {\rm Ricci}(\omega_{0}) $ given by the $\partial\overline{\partial}$-lemma.
In the case when $\lambda=-1$ there is a straightforward existence theorem, established by Aubin and Yau. In fact the proof is significantly simpler than that when $\lambda=0$. The point is that the nonlinearity from the exponential in (2) occurs with a favourable sign. The condition on the K\"ahler class means that the manifold $X$ is of general type and the metrics are generalisations of metrics of constant curvature $-1$ on Riemann surfaces of genus 2 or more. The harder case is when $\lambda=1$. Then the condition on the K\"ahler class means that $X$ is a {\it Fano manifold}. A result of Matsushima \cite{kn:Mat}, also from the 1950's, shows that existence can fail in this case. Matsushima showed that if there is a solution the holomorphic automorphism group of $X$ is reductive (the complexification of a compact Lie group). This means that Fano manifolds, such as the projective plane blown up in one or two points, with non-reductive automorphism groups cannot support such a metric. The same Calabi-Aubin-Yau scheme which proved existence in the cases $\lambda\leq 0$--the \lq\lq continity method'' (see 4.2 below)--- can be set up in the positive case, but must break down for these manifolds. More precisely, the difference arises because of the absence of a $C^{0}$-estimate for the K\"ahler potential due to the unfavourable sign in the equation (2). On the other hand there are many cases where the existence of a solution has been established, using arguments exploiting detailed geometric features of the manifolds and the theory of \lq\lq log canonical thresholds''. The question which arises is to characterise in terms of the complex geometry of the manifold $X$ exactly when a solution exists. In the early 1990's Yau conjectured that the appropriate criterion should be in terms of the {\it stability} of the manifold $X$ and, after two decades of work by many mathematicians, this is now known to the case. The precise formulation is in terms of a algebro-geometric notion of {\it K-stability} and the statement is that a Fano manifold admits a K\"ahler-Einstein metric if and only if it is K-stable.
One of the attractive features of this problem is the range of techniques which can be brought to bear. By its nature, the statement involves an interaction between algebraic geometry and complex differential geometry and, as we shall see, there are important connections with global Riemannian geometry, with pluripotential theory in complex analysis and with nonlinear PDE. Four different proofs of the main result have appeared up to the time of writing. \begin{enumerate} \item Deformation of cone singularities. (Chen, Donaldson and Sun\cite{kn:CDS}); \item The continuity method (Datar, Szekelyhidi \cite{kn:DaS}); \item Proof via Kahler-Ricci flow (Chen, Sun, Wang \cite{kn:CSW}); \item Proof by the variational method (Berman, Boucksom, Jonnson \cite{kn:BBJ}-the statement proved here is slightly different). \end{enumerate}
The purpose of this article is to survey these recent developments.
\section{K-stability}
The notion of \lq\lq stability'', in this context, arose from the study of moduli problems in algebraic geometry and geometric invariant theory. It is not usually possible to give a good structure to a set of all isomorphism classes of algebro-geometric objects. For example, generic 4-tuples of points in the projective line up to the action of projective transformations are classified by the cross-ratio, but there is no satisfactory way to define the cross-ratio if three or more points coincide. The idea is that the isomorphism classes of a suitable restricted class of stable objects do form a good space. There is a general circle of ideas relating the algebraic approach to these questions to metric structures and differential geometry involving the notion of a moment map and the \lq\lq equality of symplectic and complex quotients''. In particular there is a large literature, going back to the early 1980's, developing these notions in the framework of gauge theories and results such as the existence of Hermitian Yang-Mills connections on stable vector bundles \cite{kn:UY}. But the author recently wrote a survey which emphasised this side of the story \cite{kn:D2}, so we will not go into it here beyond noting that the K\"ahler-Einstein problem fits naturally into wider questions of the existence of constant scalar curvature and extremal K\"ahler metrics; questions which remain largely open.
Typically, stability is defined by a {\it numerical criterion} on {\it degenerations} of the objects in question. In our situation, we consider polarised varieties $(X,L)$, so $L$ is an ample line bundle over $X$ and the sections of $L$ embed $X$ as a projective variety in some projective space. The relevant degenerations are {\it test configurations} which are defined as follows. For an integer $m>0$ a test configuration of exponent $m$ for $(X,L)$ is a flat family of schemes $\pi:{\cal X}\rightarrow {\bf C}$, with a relatively ample line bundle ${\cal L}\rightarrow {\cal X}$ and a ${\bf C}^{*}$-action on ${\cal X},{\cal L}$ covering the standard action on ${\bf C}$. For $t\in {\bf C}$ we write $X_{t}$ for the scheme-theoretic fibre $\pi^{-1}(t)$ and $L_{t}$ for the restriction of ${\cal L}$ to $X_{t}$. We require that for all non-zero $t$ the pair $(X_{t}, L_{t})$ is isomorphic to $(X,L^{m})$. (Note that, due to the ${\bf C}^{*}$-action, it suffices to know this for {\it some} non-zero $t$.) For technical reasons we also suppose that the total space ${\cal X}$ is normal.
The numerical criterion, in our situation, is provided by the Futaki invariant. In its general form, \cite{kn:D0}, this is defined for any $n$-dimensional projective scheme $Z$ with ${\bf C}^{*}$-action and ${\bf C}^{*}$-equivariant ample line bundle $\Lambda\rightarrow Z$, as follows. For each integer $k\geq 0$ we have a vector space $H^{0}(Z;\Lambda^{k})$ with an induced ${\bf C}^{*}$ action. Write $d(k)$ for the dimension of the space and $w(k)$ for the sum of the weights of the action. For large $k$, $d(k)$ is given by a Hilbert polynomial which has degree exactly $n$ (since $\Lambda$ is ample), while $w(k)$ is given by a polynomial of degree at most $n+1$. Thus $d(k)/kw(k)$ has an expansion, for large $k$: $$ \frac{d(k)}{k w(k)} = F_{0} + k^{-1} F_{1} + \dots, $$ and the Futaki invariant is defined to be the co-efficient $F_{1}$. In our situation, we define the Futaki invariant ${\rm Fut}({\cal X})$ of a test configuration of exponent $m$ to be $m^{-1}$ times the Futaki invariant of the central fibre, with line bundle $L_{0}$ and the induced ${\bf C}^{*}$-action.
With these definitions in place we can state the main definition of this section. \begin{defn}
A polarised variety $(X,L)$ is K-semistable if for any test configuration ${\cal X}$ we have ${\rm Fut}({\cal X})\geq 0$. It is $K$-stable if equality holds only when ${\cal X}$ is a product $X\times {\bf C}$. \end{defn} Note that in the last clause we allow the ${\bf C}^{*}$-action on $X\times {\bf C}$ to be induced from a non-trivial action on $X$. What we have called K-stability is often called K-polystability in the literature. The precise statement of the result mentioned in the previous section, verifying Yau's conjecture, is \begin{thm} A Fano manifold $X$ admits a Kahler-Einstein metric if and only if $(X,K_{X}^{-1})$ is K-stable. \end{thm} Here the \lq\lq only if''is usually regarded as the easier direction and is due to Berman \cite{kn:Berm}, following related results of various authors The uniqueness of the metric, modulo holomorphic automorphisms, is a relatively old result of Bando and Mabuchi \cite{kn:BM}. We will not say more about these results here but focus on the \lq\lq if'' direction.
To give some background to the technical aspects of the proofs sketched in Section 4 below we will now try to explain why Theorem 1 is plausible. First we go back to the definition of the Futaki invariant of $(Z,\Lambda)$ in the case when $Z$ is a manifold, which was in fact the original context for Futaki's definition \cite{kn:Fut}. Choose a K\"ahler metric $\omega$ on $Z$ in the class $c_{1}(\Lambda)$ preserved by the action of $S^{1}\subset {\bf C}^{*}$. Viewing $\omega$ as a symplectic structure, this action is generated by a Hamiltonian function $H$ on $Z$. Then the Futaki invariant can be given by a differential geometric formula \begin{equation} \int_{Z} (R-\hat{R}) H \frac{\omega^{n}}{n!}, \end{equation} where $R$ is the scalar curvature of $\omega$ and $\hat{R}$ is the average value of $R$ over $Z$. This formula can be derived from the equivariant Riemann-Roch theorem and can also be understood in terms of the asymptotic geometry of sections of $L^{k}$ as $k\rightarrow \infty$, in the vein of quasi-classical asymptotics in quantisation theory. What this formula shows immediately is that if $\omega$ can be chosen to have constant scalar curvature---in particular if it is a K\"ahler-Einstein metric---then the Futaki invariant vanishes. This given another way, different from the Matshusima theorem, of ruling out K\"ahler-Einstein metrics on 1 or 2 point blow-ups of ${\bf C}{\bf P}^{2}$. The definition of K-stability employs the Futaki invariant in a more subtle way; it is not just the automorphisms of $X$ which need to be considered but of the degenerations. The {\it Mabuchi functional} gives a way to understand this phenomenon. This is a functional ${\cal F}$ on the space ${\cal H}$ of K\"ahler metrics in a given cohomology class on a manifold $X$ defined via its first variation \begin{equation} \delta {\cal F} = \int_{X} (R-\hat{R}) \delta \phi \frac{\omega_{\phi}^{n}}{n!}. \end{equation} Here $\delta \phi$ is an infinitesimal variation in the K\"ahler potential and one shows that such a functional ${\cal F}$ is well-defined, up to the addition of an arbitrary constant. By construction a critical point of ${\cal F}$ is exactly a constant scalar curvature metrics which, in the setting of Theorem 1 can be shown to be K\"ahler-Einstein. (We mention here that there is another functional, the {\it Ding functional} which has many similar properties to the Mabuchi functional and plays an important part in many developments. This is only defined for manifolds polarised by $K^{\pm 1}$.)
There are three possibilities: \begin{itemize} \item ${\cal F}$ is bounded below on ${\cal H}$ and attains its infimum; \item ${\cal F}$ is bounded below but does not attain its infimum; \item ${\cal F}$ is not bounded below. \end{itemize} An extension of Theorem 1 is the statement that these three possibilities correspond to $X$ being respectively $K$-stable, $K$-semistable (but not $K$-stable) and not $K$-semistable.
Now suppose that ${\cal X}$ is a test configuration for $(X,K_{K}^{-1})$ and, for simplicity, that the total space is smooth. Choose a K\"ahler metric on this total space, invariant under $S^{1}\subset {\bf C}^{*}$. Pulling back by the ${\bf C}^{*}$-action the restrictions of this metric to the fibres $X_{t}$ for non-zero $t$ can be regarded as a family of metrics $\omega(t)$ on the fixed manifold $X$ parametrised by $t\in {\bf C}^{*} $ but these metrics have no limit, among metrics on $X$, as $t\rightarrow 0$. It is natural to think of this limit as a \lq\lq point at infinity'' in the space ${\cal H}$ of K\"ahler metrics on $X$. As we discuss further in Section 4.4 below, the role of the Futaki invariant is to determine the asymptotic behaviour as $t\rightarrow 0$ of the Mabuchi functional in such families obtained from test configurations.
Theorem 1 can be understood roughly as saying that if there is no minumum of ${\cal F}$ this can be detected by studying the asymptotics at points at infinity of this kind (derived from algebro-geometric data).
Apart from its intrinsic interest---in giving an algebro-geometric criterion for the solubility of a PDE---Theorem 1 result also has implications for the construction of compactified moduli spaces of Fano manifolds, see work of Odaka \cite{kn:O2}, Spotti, Sun and Yao \cite{kn:SSY} and Li, Wang and Xu \cite{kn:LWX0}, \cite{kn:LWX}. For moduli questions, the notion of K-stability is also relevant in the negative case of varieties with ample canonical bundle. In this case Odaka \cite{kn:O1} showed that a variety is K-stable if and only if has semi log canonical singularities, which is equivalent to stability in the sense of Alexeev, K\'ollar and Shepherd-Barron. Berman and Guenancia showed that for such varieties this is equivalent to the existence of a K\"ahler-Einstein metric \cite{kn:BG} (with a suitable definition in the singular case). Compactifying moduli spaces of manifolds involves adding points corresponding to singular varieties and it is interesting to relate the behaviour of the K\"ahler-Einstein metrics to the algebraic geometry of the singularities. There is much recent progress in this direction, see \cite{kn:HS} for example.
\section{Riemannian convergence theory and projective embeddings}
In this section we will discuss some ideas which play an important role in three of the four proofs considered in Section 4 below. The general context can be explained as follows. In solving a PDE problem {\it compactness}---the ability to take limits in some kind of approximating scheme---is usually crucial. On the other hand in our problem we need to exhibit the obstruction to solving the problem (the existence of a K\"ahler-Einstein metric) as an algebro-geometric object (a test configuration with non-positive Futaki invariant). In the framework of Ricci curvature in Riemannian geometry there is a well-developed convergence theory of {\it Gromov-Hausdorff limits}; thus, in the K\"ahler situation, we would like to relate these limits to algebraic geometry and that is the topic of this section.
We begin by recalling some of the main results from the Riemannian theory of manifolds with a lower bound on the Ricci curvature. The foundation of the theory is the link between the Ricci curvature and volume expressed by the Bishop comparison theorem. For simplicity we just consider the case of an $m$-dimensional Riemannian manifold $M$ with ${\rm Ricci}\geq 0$. Then Bishop's theorem states that for each $p\in M$ the volume ratio \begin{equation} v_{p}(r) = \frac {{\rm Vol}(B_{p}(r))}{\Omega_{m} r^{m}},\end{equation} is a weakly decreasing function of $r$. Here $B_{p}(r)\subset M$ is the metric ball of radius $r$, and we introduce the normalising constant $\Omega_{m}$---the volume of the unit ball in ${\bf R}^{m}$---so that $v_{p}(r)$ tends to $1$ as $r$ tends to $0$. If $M$ is compact with total volume $V$ and diameter $\leq D$ it follows that \begin{equation} {\rm Vol}(B_{p}(r)) \geq \kappa r^{m}, \end{equation} with $\kappa= V/D^{m}$. Recall that the {\it Gromov-Hausdorff distance} between two compact metric spaces $A,B$ is defined as the infimum of the numbers $\delta$ such that there is a metric on the disjoint union $A\sqcup B$ which extends the given metrics on $A,B$ and such that both $A,B$ are $\delta$-dense in $A\sqcup B$. If $(M_{i}, g_{i})$ is a sequence of compact Riemannian $m$-manifolds with ${\rm Ricci}\geq 0$, ${\rm Vol}(M_{i})=V$, ${\rm diam}(M_{i})\leq D$ then Gromov's compactness theorem asserts that there is a subsequence which converges in the sense of this Gromov-Hausdorff distance to some limiting metric space $(Z, d_{Z})$. (More generally, the same result applies if we have any fixed lower bound on the Ricci curvatures.) The proof is an elementary argument based on ball-packing considerations and the lower bound (6). It is sometimes convenient to express this Gromov-Hausdorff convergence in terms of a natural topology on the disjoint union
$$ {\cal M}= Z\cup \bigsqcup_{i}M_{i} . $$
Thus for $q\in Z$ it makes sense to talk about points $p_{i}\in M_{i}$ which are close to $q$.
Results of Anderson \cite{kn:And}, Cheeger-Colding \cite{kn:CC} and Cheeger-Colding-Tian\cite{kn:CCT} give finer information about such \lq\lq non-collapsed'' Gromov-Hausdorff limits. (Here the non-collapsing refers to the volume lower bound, which rules out the collapse of the sequence of $m$-dimensional manifolds to some lower dimensional space.) The notion of Gromov-Hausdorff convergence can be extended to sequences of spaces with base points: the metric balls of any fixed radius centred at the base points are required to converge as above. For each point $q\in Z$ and sequence of real numbers $\lambda_{j}\rightarrow \infty$ we consider the sequence of based metric spaces $(q, Z, \lambda_{i} d_{Z})$. After perhaps passing to a subsequence we have a based Gromov-Hausdorff limit which is a metric cone, a {\it tangent cone} of $Z$ at $q$. The regular set $R\subset Z$ is defined to be the subset where some tangent cone is ${\bf R}^{m}$ and the complement $Z\setminus R$ is the singular set $S$. If the manifolds $M_{i}$ satisfy a fixed bound on the Ricci curvature $\vert {\rm Ricci}\vert \leq \Lambda$ then more can be said. Anderson showed that there are fixed $\delta_{m},\kappa_{m}$, depending only the dimension, such that if $p\in M_{i}$ and $r\leq \Lambda^{-1/2}$ then if the if the volume ratio $v_{p}(r)$ is greater than $1-\delta_{m}$ there are harmonic co-ordinates on the sub-ball $B_{p}(\kappa r)$ in which the metric satisfies $C^{1,\nu}$ estimates. Together with results of Cheeger and Colding this shows that the regular set is open in $Z$ and the Riemannian metrics converge to a $C^{1,\nu}$-Riemannian metric $g_{\infty}$ on $R$. ( This means that if $U$ is a pre-compact open set in $R$ there are $C^{2,\nu}$ diffeomorphisms $\chi_{i}:U\rightarrow M_{i}$ which, regarded as maps into ${\cal M}$, converge to the inclusion $U\rightarrow Z$ and such that the $\chi^{*}_{i}(g_{i})$ converge in $C^{1,\nu}$ to $g_{\infty}$.) The singular set $S$ has Hausdorff codimension at least $4$. (In the general Riemannian context, this is a recent result of Cheeger and Naber \cite{kn:CN}, but in the K\"ahler case which will be our concern it goes back to Cheeger, Colding and Tian \cite{kn:CCT}.) The corresponding statements apply to tangent cones: each has a smooth Ricci-flat metric outside a closed singular set of codimension at least 4.
We want to relate these ideas to algebraic geometry and in this section we will focus on the case considered in \cite{kn:DS}. Thus we suppose that $(X_{i}, \omega_{i})$ are K\"ahler manifolds of complex dimension $n$ with fixed volumes $V$, diameters $\leq D$ and $\vert {\rm Ricci}\vert\leq \Lambda$ and with a Gromov-Hausdorff limit $Z$, as above. We suppose that that these are polarised manifolds, so that $\omega_{i}$ is the curvature of a Hermitian holomorphic line bundle $L_{i}\rightarrow X_{i}$. The main result is that $Z$ can be endowed with the structure of an algebraic variety. More precisely, we allow passage to a subsequence and form the disjoint union ${\cal M}$ as above. Then there is a continuous map $I:{\cal M}\rightarrow {\bf C}{\bf P}^{N}$ with the two properties. \begin{itemize}\item There is some fixed $k$ such that for sufficiently large $i$, the restriction of $I$ to $X_{i}$ is an embedding defined by the holomorphic sections of $L_{i}^{k}\rightarrow X_{i}$; \item The restriction of $I$ to $Z$ is a homeomorphism to its image, which is a normal projective variety in ${\bf C}{\bf P}^{N}$. \end{itemize}
This result can be seen as an extension of the Kodaira embedding theorem to singular limit spaces and the proof extends some the ideas in one approach to the Kodaira theorem (an approach which seems to be well-known to experts but does not feature in standard textbooks). Suppose that $L\rightarrow X$ is a holomorphic line bundle over a compact complex manifold and $\sigma_{0}$ is a holomorphic section of $L$ over an open subset $V\subset X$. Let $\beta$ be a cut-off function with compact support in $V$, extended by $0$ over $X$ and with $\beta=1$ on some interior region $V_{0}\subset V$. Then we can regard $\sigma=\beta \sigma_{0}$ as a smooth section of $L$ over $X$ in an obvious way. This will not be a holomorphic section because we have a term coming from the cut-off function $$ \overline{\partial} \sigma = (\overline{\partial} \beta) \sigma_{0}, $$
but if we have Hermitian metrics on $L$ and $X$ we can project $\sigma$ to the space of holomorphic sections using the $L^{2}$ inner product on sections of $L$, arriving at a holomorphic section $s$. Of course in this generality the construction need not be useful---the projection $s$ could be $0$. The idea is that under suitable hypotheses we can arrange that $s$ is not zero and is very close to the original section $\sigma_{0}$ over $V_{0}$. The key is to estimate the error term $\eta=\sigma-s$, which is given by a Hodge Theory formula $$ \eta= \overline{\partial}^{*} \Delta^{-1} \overline{\partial} \sigma, $$ where $\Delta$ is the $\overline{\partial}$-Laplacian on the $L$-valued $(0,1)$ forms. Of course this formula only makes sense if $\Delta$ is invertible, i.e. if the cohomology group $H^{1}(X;L)$ is zero. Then we have $$ \Vert \eta\Vert^{2}_{L^{2}}= \langle \overline{\partial}^{*} \Delta^{-1} \overline{\partial} \sigma, \overline{\partial}^{*}\Delta^{-1} \overline{\partial} \sigma\rangle= \langle \overline{\partial}\overline{\partial}^{*}\Delta^{-1}\overline{\partial} \sigma, \Delta^{-1}\overline{\partial}\sigma\rangle=\langle \overline{\partial}\sigma, \Delta^{-1}\overline{\partial}\sigma\rangle, $$ and so $$ \Vert \eta\Vert_{L^{2}}^{2}\leq \Vert \Delta^{-1}\Vert \ \Vert \overline{\partial}\sigma\Vert^{2}, $$ where $\Vert \Delta^{-1}\Vert$ is the $L^{2}$-operator norm. Now suppose that $L$ is a positive line bundle and the metric on $X$ is the K\"ahler metric $\omega$ given by the curvature of $L$. There is then a formula of Weitzenb\"ock type
\begin{equation} \Delta = P^{*} P + {\rm Ric} + 1 , \end{equation}
where $P$ is the $(0,1)$-component of the covariant derivative on $L$-valued $(0,1)$ forms. But all we use is that $P^{*}P$ is a non-negative operator, so if ${\rm Ric} \geq -1/2$ (say) then $\Delta\geq 1/2$ and $\Delta^{-1}$ is defined and has $L^{2}$-operator norm at most $2$. So we get $$\Vert \eta\Vert_{L^{2}}\leq \sqrt{2} \Vert \overline{\partial} \sigma \Vert_{L^{2}}.$$
So if we can arrange that $\overline{\partial}\sigma$ is small, in $L^{2}$ norm compared with $\sigma$, then we get a non-trivial holomorphic section $s$. By construction, $\eta$ is holomorphic over $V_{0}$ and the $L^{2}$ norm of $\eta$ controls all derivatives there, by the usual elliptic estimates, and we can hope to show that $s$ is a small perturbation of $\sigma_{0}$ in any $C^{r}$ norm. The whole approach extends easily to a case when the original section $\sigma_{0}$ is not exactly holomorphic but approximately so, measured in terms of bounds on the $L^{2}$ norm of $\overline{\partial} \sigma_{0}$.
To prove the usual Kodaira embedding theorem in this way we first make the simple observation that changing $L$ to $L^{k}$ for some $k>1$ corresponds to rescaling the metric by a factor $k$---i.e. scaling lengths of paths by a factor $\sqrt{k}$. Under this rescaling the Ricci curvature is multiplied by $k^{-1}$ so if we start with any positive line bundle and take a suitable power we can arrange that the condition ${\rm Ric}\geq - 1/2$ holds after rescaling. We take $U\subset X$ to be a small co-ordinate ball centred on a point $p\in X$. The flat model is given by the trivial line bundle $\Lambda$ over ${\bf C}^{n}$ with a Hermitian structure corresponding to the Euclidean metric on ${\bf C}^{n}$. In this structure the trivialising section $\tau$ of $\Lambda$ has Gaussian norm
$$ \vert \tau(z) \vert = e^{-\vert z\vert^{2}/4}, $$
which decays very rapidly at infinity. After rescaling, the geometry of the manifold $X$ in the small ball $U$ is close to the flat model and we get an approximately holomorphic section $\sigma_{0}$ modelled on $\tau$. By suitable choice of parameters one arranges a cut-off function $\beta$ with the support of $\nabla \beta$ contained in the region where $\sigma_{0}$ is very small, so that $\Vert \overline{\partial} \sigma\Vert$ is also very small. Here the \lq\lq suitable choice of parameters'' involves choosing $k$ large. The upshot is that for a suitable $k$ and for each $p\in X$ we construct a holomorphic section of $L^{k}$ \lq\lq peaked'' around $p$ and in particular not vanishing at $p$. Thus the sections of $L^{k}$ define a holomorphic map of $X$ to a projective space and one can go further to show that this map is an embedding, once $k$ is sufficiently large. (Note that the formula (7) is the same as that used in the proof of the Kodaira-Nakano vanishing theorem, which is invoked in the usual proof of the embedding theorem via blow-ups.)
Returning to our main discussion, we do not want to do analysis directly on the Gromov-Hausdorff limit $Z$ but instead establish uniform estimates on the converging sequence $X_{i}$. Consider any polarised manifold $(X,L)$ with K\"ahler metric given by the curvature of $L$. We endow $H^{0}(X,L)$ with the standard $L^{2}$ metric and for $x$ in $X$ we consider the evaluation map $${\rm ev}_{x}: H^{0}(X,L)\rightarrow L_{x}.$$ We define $\rho_{L}(x)$ to be square of the norm of this map so the statement that $\rho_{L}(x)>0$ for all $x\in X$ is the same as saying that the sections of $L$ define a map $\tau: X\rightarrow {\bf P}(H^{0}(X,L)^{*})={\bf P}$. More generally, a lower bound on the $\rho_{L}(x)$ gives metric control of this map. The operator norm of
$$ d\tau_{x}: TX_{x}\rightarrow T{\bf P}_{\tau(x)}$$ is
$\rho_{L}(x)^{-1/2} \max (\vert (\nabla s)_{x}\vert) $, where the maximum is taken over holomorphic sections $s$ of $L$ with $L^{2}$ norm $1$ vanishing at $x$. In our situation, with Ricci curvature and diameter bounds, there is a well-known upper bound on $\vert \nabla s\vert$, so a strictly positive lower bound on the $\rho_{L}(x)$ gives a Lipschitz bound on the map $\tau$. Replacing $L$ by $L^{k}$ we see that the crucial point is to find some $k$ and $b>0$ so that for all $i$ and for all $x$ in $X_{i}$ we have \begin{equation} \rho_{L^{k}}(x)\geq b. \end{equation} (Such a bound is sometimes referred to as a {\it partial $C^{0}$-estimate}.) It is straightforward to reduce to the case when the dimension of $H^{0}(X_{i},L_{i}^{k})$ is independent of $i$, so that we can regard $\tau_{i}:X_{i}\rightarrow {\bf P}$ as mapping into a fixed projective space. If the bound (8) holds then, after possibly taking a subsequence, we can pass to the Gromov-Hausdorff limit and define a continuous map $\tau_{\infty}:Z\rightarrow {\bf P}$ with image a projective variety. Further arguments then show that after perhaps increasing $k$ this map $\tau_{\infty}$ is an homeomorphism from $Z$ to a normal projective variety.
The central issue then is to establish the lower bound (8), as has been emphasised in many places by Tian. Let $q$ be a point in $Z$ and $C(Y)$ be some tangent cone to $Z$ at $q$. If $q$ is a smooth point then $C(Y)$ is ${\bf C}^{n}$ and we have the model Gaussian section as discussed above. In general we always have a Hermitian line bundle $\Lambda$ over the regular part of $C(Y)$ with a holomorphic section $\tau$ satisfying exactly the same Gaussian decay with respect to the distance to the vertex of the cone. We choose a suitable open subset $U$ of the regular part of the cone. It follows from the definitions that for large $i$ there are diffeomorphisms $\chi_{i}:U\rightarrow X_{i}$ which are approximately holomorphic isometries with respect to rescalings of the metrics $k\omega_{i}$ for some suitable large $k$. (Here the approximation can be made as close as we like by taking $k$ large.) Then are then two main technical points to address. \begin{itemize}
\item We want to have lifts $\tilde{\chi}_{i}:\Lambda \rightarrow \chi^{*}_{i}(L^{k})$ to approximate isomorphisms of Hermitian line bundles over $U$. \item We want a suitable cut-off function $\beta$ on $U$ with $\vert \overline{\partial} \beta \tau\vert$ small in $L^{2}$. \end{itemize} Given these, we can transport the section $\beta \tau$ to an approximately holomorphic section of $L_{i}^{k}$ over $\chi_{i}(U)$and follow the projection procedure to get a holomorphic section $s$ of $L_{i}$ modelled on $\tau$. The derivative bounds on $s$ give a lower bound on $\vert s\vert$ over all points in $X_{i}$ close to $q$ and an elementary covering argument establishes the bound (8).
The first technical point involves considerations of the holonomy of the connection on $\Lambda^{*}\otimes \chi_{i}^{*}(L_{i}^{k})$, which has very small curvature by construction--- this is straightforward if $U$ is simply connected. The second technical point involves the singular set in $C(Y)$, and in particular the fact that this has Hausdorff codimension strictly greater than $2$ (see 4.1 below).
\section{Four proofs} We now come to the core of this survey in which we discuss four different proofs of the equivalence between stability and K\"ahler-Einstein metrics. In total these proofs run to many hundreds of pages so it is impossible to give any kind of thorough account of them here. All we can do is to explain general strategies and some salient points in the arguments.
\subsection{ The proof by cone singularities}
(Note that the announcement \cite{kn:CDS0} contains an outline of this proof.) Given a Fano manifold $X$ we fix some suitable $m\geq 1$ and a smooth divisor $D$ in $\vert -m K_{X}\vert$. For $0<\beta\leq 1$ we can define a class of K\"ahler metrics on $X$ with cone singularity of angle $2\pi \beta$ along $D$ and extend the whole theory to this case. (When $\beta= r^{-1}$ for an integer $r$ these are orbifold metrics, and hence well-established. There are also close analogies with the theory of parabolic structures and singular Hermitian Yang-Mills connections as developed in \cite{kn:Biq} for example.) We can define a modified Mabuchi functional \begin{equation}\delta {\cal F}_{\beta}= \int_{X} (R-\hat{R}) \delta \phi + (1-\beta) \int_{D} \delta \phi - c \int_{X}\delta \phi \end{equation} where $c= (m c_{1}(X)^{n})^{-1}$ is the ratio of the volume of D to the volume of X, so that the right hand side vanishes when $\delta \phi$ is a constant. Roughly speaking, we have a family of functionals ${\cal F}_{\beta}$ with critical points the K\"ahler-Einstein metrics with cone angle $\beta$ and the strategy of the proof is to follow a family of such critical points as $\beta$ increases. We want to show that either the family continues up to $\beta=1$, which gives our desired K\"ahler-Einstein metric on $X$, or that the critical point moves off to infinity and that this yields a test configuration violating the K-stability condition.
To begin we need to show that a solution exists for {\it some} $\beta$. Take $m>1$ and $\beta= r^{-1}$ so that we are in the orbifold case. If $r>m/m-1$ then characteristic class arguments show that the we are in the situation of negative Ricci curvature and the desired solution follows from a straightforward orbifold extension of the standard Aubin-Yau theory. Next one shows that the set of $\beta$ for which a solution exists is {\it open}. This can be achieved using a suitable linear elliptic theory on manifolds with cone singularities \cite{kn:D1}.
Suppose then that there is some $\beta_{0}\in (0,1]$ such that a solution $\omega_{\beta}$ exists for $\beta<\beta_{0}$ but that there is no solution for $\beta=\beta_{0}$. We need to extend the theory sketched in Section 3, for smooth metrics with bounded Ricci curvature, to K\"ahler-Einstein metrics with cone singularities. To begin we show that a metric with cone singularity can be approximated in the Gromov-Hausdorff sense by smooth metrics with Ricci curvature bounded below (\cite{kn:CDS}, Part I). Then the Cheeger-Colding theory implies that there is a subsequence $\beta_{i}$ increasing to $\beta_{\infty}$ and a Gromov-Hausdorff limit $Z$ of the $(X,\omega_{\beta_{i}})$ and $Z$ has metric tangent cones at each point. We call a tangent cone $C(Y)$ \lq\lq good'' if the regular set $ C(Y_{{\rm reg}})$ is open, the metric is induced by a smooth K\"ahler metric there and and for each compact subset $K\subset Y_{ {\rm reg}}$ and each $\eta>0$ there is a cut-off function $\gamma$ of compact support in $Y_{{\rm reg}}$, equal to $1$ on $K$ and with \begin{equation} \Vert \nabla \gamma\Vert_{L^{2}}\leq \eta. \end{equation} The main technical result is that all tangent cones to $Z$ are good. Given this, an extension of the arguments outlined in Section 3 above show that $Z$ is naturally a normal projective variety, carrying a singular K\"ahler-Einstein metric $\omega_{\infty}$. Moreover if we write $X_{i}$ for the metric space $(X,\omega_{\beta_{i}})$ and $D_{i}\subset X_{i}$ for the divisor $D$ then there is a divisor $\Delta\subset Z$ such that the pairs $(X_{i}, D_{i})$ converge to $(Z,\Delta)$.
The new feature in this case, which leads to the difficulty in proving that tangent cones are \lq\lq good'', is the possibility of codimension 2 singular sets. This is the critical dimension with respect to the cut-off control (10). If $\psi$ is a compactly-supported function on ${\bf R}^{m}$ and if $\psi_{\lambda}$ is the rescaled function $\psi_{\lambda}(x)=\psi(\lambda^{-1} x)$ then $$ \Vert \nabla \psi_{\lambda}\Vert_{L^{2}} = O(\lambda^{(m-2)/2})$$ which tends to $0$ with $\lambda$ if $m>2$. This allows one to construct cut-off functions with derivative arbitrarily small in $L^{2}$ adapted to a compact set $A$ of Hausdorff codimension strictly greater than $2$. In the codimension 2 case one needs appropriate control of the volume of the $\lambda$-neighbourhood $N_{\lambda}(A)$: $$ {\rm Vol}\ N_{\lambda}(A))\leq C \lambda^{2}. $$ This is equivalent to the notion of {\it Minkoswki codimension}$\geq 2$ i.e. for any $r$ the set $A$ can be covered by $O(r^{-2})$ balls of radius $r$.
To complete the proof of the Theorem one wants to show that if indeed the family of solutions breaks down at some $\beta_{\infty}$ as considered above then there is test configuration with central fibre $Z$ and with non-positive Futaki invariant, so the original manifold $X$ is not K-stable. To this end one can extend the whole theory of stability and test configurations to pairs consisting of a variety and divisor, with a real parameter $\beta$. There is a modified Futaki invariant ${\rm Fut}_{\beta}$ which compares with the usual formula (3) (in the smooth case) just as the modified Mabuchi functional (9) compares with (4). One wants to construct a test configuration $({\cal X}, {\cal D})$ for the pair $(X,D)$ with central fibre $(Z,\Delta)$ and show that \begin{equation} {\rm Fut}_{\beta_{\infty}}({\cal X}, {\cal D})= 0. \end{equation} The Futaki invariant ${\rm Fut}_{\beta}$ depends linearly on $\beta$ so the fact that $(X,D)$ is stable for small $\beta$ implies that ${\rm Fut}({\cal X})={\rm Fut}_{1}({\cal X}, {\cal D})\leq 0$.
Let $G$ be the automorphism group of the pair $(Z,\Delta)$--a complex Lie group. The existence of this test configuration $({\cal X}, {\cal D})$ follows from general principles once it is established that $G$ is {\it reductive}; the complexification of a compact subgroup $K\subset G$. Using projective embeddings in some ${\bf C}{\bf P}^{N}$, the pairs correspond to points $[X,D]$, $[Z,\Delta]$ in a suitable Hilbert scheme ${\bf S}$ which in turn is embedded in some large projective space ${\bf P}$. The group $PGL(N+1,{\bf C})$ acts on ${\bf S}$ and ${\bf P}$ and what we know from the convergence discussion above is that $[Z,\Delta]$ is in the closure of the orbit of $[X,D]$. The group $G$ can be identified with the stabiliser of $[Z,\Delta]$ in $PGL(N+1,{\bf C})$. If $G$ is reductive a version of the Luna slice theorem gives a slice for the action of $PGL(N+1,{\bf C})$ at $[Z,\Delta]$ (one takes a $G$-invariant complement to the action of $G$ on the tangent space of ${\bf P}$ at $[Z,\Delta]$). A well-known result of Hilbert and Mumford, applied to the $G$ action on this slice, shows that there is a $1$-parameter subgroup ${\bf C}^{*}\subset G$ such that $[Z,\Delta]$ is in the closure of the ${\bf C}^{*}$-orbit of $[X,D]$ and this is equivalent to the desired test configuration.
The reductivity of $G$ is an extension of the Matsushima result to the case of pairs and singular varieties. Likewise the vanishing of the Futaki invariant (11) is an extension of the simple case considered in Section 2 above, for manifolds of constant scalar curvature. For the proofs one has to work with the singular metric $\omega_{\infty}$ using techniques from pluripotential theory.
\subsection{The proof by the continuity method}
In the continuity method one fixes a positive $(1,1)$ form $\alpha$ representing $c_{1}(X)$ and tries to solve the family of equations for $\omega_{s}$, with parameter $s\in [0,1]$:
\begin{equation} {\rm Ricci}(\omega_{s})= (1-s) \alpha + s \omega_{s}. \end{equation}
Yau's solution of the Calabi conjecture shows that there is a solution for $s=0$ and it is well-known that the set of parameters for which a solution exists is open; if the solution can be continued up to $s=1$ we have a K\"ahler Einstien metric, so the problem is to prove closedness (under the stability hypothesis). Thus we suppose that for some $S<1$ there are solutions for $s<S$ but none for $s=S$. The set-up is very similar to that of cone singularities above, indeed the latter can be regarded as a variant of the continuity method, replacing the smooth form $\alpha$ with the current defined by a divisor.
If one knew that the fixed form $\alpha$ was bounded with respect to $\omega_{s}$ then the $\omega_{s}$ would have bounded Ricci curvature and the results discussed in Section 3 above would apply immediately to give a limiting metric on a normal projective variety. So the major difficulty is that we do not have such a bound. The Ricci curvature of the $\omega_{s}$ is positive so the fundamentals of the Cheeger-Colding theory apply and we obtain a Gromov-Hausdorff limit $Z$ (more precisely, a limit of some sequence $(X, \omega_{s_{i}})$ with $s_{i}$ increasing to $S$). But this theory does not ensure that the regular set in $Z$ is open, or give the good convergence properties over the regular set exploited in the argument of Section 3. This is one of the main problems overcome by Sz\'ekelyhidi in \cite{kn:Gabor2}.
To explain some of Sz\'ekelyhidi's arguments, we restrict attention to the case when $S<1$. Consider first a unit ball $B$ centred at a point $p$ in a K\"ahler manifold, with metric $\omega$, and a vector-valued holomorphic function $ f:B\rightarrow {\bf C}^{m}$. Suppose that the Ricci curvature is bounded in $B$, say $\vert {\rm Ric}\vert \leq 4$, and that the pair $(\omega, f)$ satisfy the equation \begin{equation} {\rm Ric}(\omega) = \sigma \omega + f^{*}(\Omega_{{\rm Euc}}), \end{equation} for some $\sigma\in [0,1]$ where $\Omega_{{\rm Euc}}$ is the standard Euclidean K\"ahler form on ${\bf C}^{m}$. We claim that for any $\epsilon>0$ we can find a $\delta$ (independent of $\sigma$) such that for any such $\omega$, if the volume ratio $v_{p}(1)$ exceeds $(1-\delta)$ then in fact $$\vert {\rm Ric}(p)\vert \leq \epsilon. $$
First, the Ricci curvature is non-negative so by the Bishop inequality we can pass to a smaller ball with centre $p$ (rescaled) and preserve the volume bound. By the results of Anderson we may as well assume that the metric on $B$ is $C^{1,\nu}$-close to the Euclidean metric in harmonic co-ordinates. By a suitable version of the Newlander-Nirenberg integrability theorem we can also suppose that these co-ordinates are actually holomorphic. The bound on the Ricci curvature and the equation (13) means that $\vert \nabla f\vert^{2} \leq 2n $ and since $f$ is holomorphic we get interior bounds on all higher derivatives of $f$ and hence on the Ricci tensor, in these holomorphic co-ordinates. Thus if $\vert{\rm Ricci}(p)\vert>\epsilon$ we will have $\vert {\rm Ricci}\vert>\epsilon/2$ over a ball of definite size centred at $p$. The $C^{1,\nu}$ bound on the metric tensor gives $C^{,\nu}$ bounds on the Christoffel symbols. From this Sz\'ekelyhidi shows that there is some unit tangent vector $v$ at $p$ such that, in geodesic polar-coordinates centred at $p$, there is a definite lower bound on
${\rm Ric} (\frac{\partial}{\partial r}, \frac{\partial}{\partial r})$ at all points close to $p$ and along geodesics starting from $p$ at a sufficiently small angle to $v$. Then the proof of the Bishop inequality shows that this Ricci curvature {\it reduces} the volume of $B$ by a definite amount (compared with the Euclidean ball) determined by $\epsilon$. So ${\rm Vol}(B)\leq (1-\delta(\epsilon))\Omega_{2n}$, say. Choosing $\delta=\delta(\epsilon)$ we get a contradiction to the hypothesis that $\vert {\rm Ric}(p)\vert\geq \epsilon$ and the claim is established.
Clearly the result extends (with a suitable $\delta(\epsilon)$) to the case when $\Omega_{{\rm Euc}}$ is replaced by any smooth positive $(1,1)$-form $A$ defined over a suitable neighbourhood of the image of $f$. In the case at hand one can cover $X$ by a finite number of holomorphic co-ordinate charts. Working near a given point in $X$ such a chart yields the holomorphic map $f$ above (with $m=n$) and we take the form $A$ corresponding to $(1-s)\alpha$ in this chart. If $s\leq S<1$ we get a $\delta(\epsilon)$ such that the discussion above applies to any rescaling of a small ball in $(X,\omega_{s})$.
Now let $q$ be a point in the regular set of the Gromov-Hausorff limit $Z$ and let $p_{i}\in (X,\omega_{s_{i}})$ be a sequence converging to $q$ in the sense of the Gromov-Hausdorff convergence. Let $B_{i}$ be the unit ball obtained by rescaling a small ball (of fixed radius $\rho$, independent of $i$) about $p_{i}$. For any given $\delta>0$ we can suppose that for all subballs $\tilde{B}\subset B_{i}$ (not necessarily centred at $p_{i}$) the \lq\lq volume defect'' of $\tilde{B}$ is less than $\delta$. We choose $\delta$ as above, for some $\epsilon<1$. Now let $$M={\rm max}_{x\in B_{i}} \left(\vert {{\rm Ricci}}(x)\vert\ d(x,\partial B_{i})^{2}\right). $$ A standard line of argument shows that in fact $M\leq 4$. For if not
let $\tilde{p}\in B_{i}$ be a point where the maximum is attained and $\tilde{B}$ be the ball of radius $\tilde{d}/\sqrt{M}$ centred at $\tilde{p}$, where $\tilde{d}$ is the distance from $\tilde{p}$ to the boundary of $B_{i}$. Rescaling $\tilde{B}$ to unit size we get a ball to which the previous results apply. After rescaling those results give $\vert {\rm Ricci}(\tilde{p})\leq \epsilon d_{0}^{-2} M< d_{0}^{-2}M$ which is a contradiction to the choice of $\tilde{x}$. Thus $M\leq 4$ and in particular $\vert {\rm Ricci}(p_{i})\vert \leq 4 \rho^{-2}$. The conclusion is that Sz\'ekelyhidi is able to show that, when $S<1$ the Ricci curvature is bounded near points in the regular set in $Z$. It follows that the regular set is open and carries a $C^{1,\nu}$ K\"ahler metric. Going further, he extends the discussion to tangent cones and shows that these are all \lq\lq good'' in the sense discussed in 4.1 above. Note that in this situation the singular set can have real codimension 2, different from the simpler situation considered in Section 3. (In fact results of C. Li \cite{kn:Li} in the toric case suggest that the limit as $s\rightarrow S$ will develop cone singularities along a divisor.)
In \cite{kn:Gabor2}, Sz\'ekelyhidi established the partial $C^{0}$ estimate along the continiuty method and used this to show that another notion of stability, introduced by S. Paul, implies the existence of a K\"ahler-Einstein metric. The corresponding result for $K$-stability was established in the subsequent paper \cite{kn:DaS} of Datar and Sz\'ekelyhidi. The output from the convergence theory is that if the continuity method breaks down at $S\in (0,1]$ there is a limiting projective variety $Z$, a singular K\"ahler metric $\omega_{S}$ and a closed non-negative $(1,1)$ current $\alpha_{\Psi}$ on $Z$ satisfying the equation $$ {{\rm Ricci}}(\omega_{S})= (1-S)\omega_{S}+ S \alpha_{\Psi}. $$ The possible presence of singularities means that that this equation needs to be interpreted. The current $\alpha_{\Psi}$ is locally written as $i\partial\overline{\partial}\psi$ where $ \psi$ is an $L^{1}$ purisubharmonic function. Globally, $\Psi$ is a singular Hermitian metric on the anticanonical bundle. Datar and Sz\'ekelyhidi set up a theory for pairs consisting of a variety and a $(1,1)$-current, analogous to the theory for pairs (variety, divisor) discussed above. The new feature is that their space of pairs is infinite dimensional. They are able to carry through a similar strategy to that outlined in 4.1 above by approximating $(1,1)$-currents by those defined by divisors. The form $\alpha$ on $X$ can be taken to be the restriction of the Fubini-Study metric under an embedding $X\subset {\bf P}$, then there is a integral geometry formula $$ \alpha= \int_{{\bf P}^{*}} [H\cap X] d\mu(H), $$ where ${\bf P}^{*}$ is the dual projective space parametrising hyperplanes $H\subset {\bf P}$, $\mu$ is the standard measure on ${\bf P}^{*}$ and $[H\cap X]$ is the current of the divisor $H\cap X$ in $X$. Replacing the integral by a finite sum gives the approximation procedure which is the starting point for these arguments.
The results of Datar and Sz\'ekelyhidi go further than the statement of Theorem 1 in two directions. First they prove an analogous result for solutions of the {\it K\"ahler-Ricci soliton} equation. Recall that this equation is \begin{equation} {\rm Ricci}(\omega) - \omega= L_{v}\omega, \end{equation} where $v$ is a holomorphic vector field and $L_{v}$ is the Lie derivative. Such metrics are the appropriate analogues of K\"ahler-Einstein metrics on Fano manifolds with non-vanishing Futaki invariant and represent fixed points of the K\"ahler-Ricci flow (modulo holomorphic diffeomorphisms), which we discuss further below.
In another direction, Datar and Sz\'ekelyhidi's proof is compatible with group actions so they prove that to test K-stability of a Fano manifold $X$ it suffices to consider test configurations with an additional compatible action of ${\rm Aut}(X)$. This is important because an outstanding defect of the general theory is that it is very hard to verify K-stability of a polarised variety. The problem becomes more tractable for manifolds with large symmetry groups. Toric manifolds, with a complex torus action having an dense orbit, can be described in terms of polytopes. Both K\"ahler metrics invariant under the action of the corresponding real torus and toric test configurations can be described by convex functions on the polytope and the stability condition is relatively explicit. However in the toric Fano case the existence problem for K\"ahler-Einstein metrics and K\"ahler-Ricci solitons was completely settled by Wang and Zhu in 2004 \cite{kn:WZ}, and no interesting phenomena arise, from the point of view of stability. Ilten and S\"uss \cite{kn:IS} consider $n$-dimensional varieties with an action of an $(n-1)$-dimensional complex torus and develop a combinatorial description of these. In this way they are able to produce new examples, of manifolds which are K-stable, and the theorem of Datar and Szekelyhidi gives corresponding explicit new results about the existence of K\"ahler-Einstein metrics. In a similar vein, Delcroix studied group compactifications \cite{kn:Delcroix1}; that is he considered a manifold $X$ which contains a complex reductive Lie group $G$ as a dense open subset and such that both left and right translations on $G$ extend to $X$. These can be described by polytopes in the Lie algebra of a maximal compact real torus in $G$ and Delcroix extends the arguments of Wang and Zhu to find an explicit condition for the existence of a K\"ahler-Einstein metric. This work of Delcroix was essentially self-contained and did not invoke general existence results such as Theorem 1. A subsequent paper \cite{kn:Delcroix2} later showed that his condition emerges from an analysis of equivariant degenerations of $X$ and extended the results to the larger class of spherical varieties, using the theorem of Datar and Sz\'ekelyhidi.
\subsection {The proof by Ricci flow}
If $X$ is a Fano mainfold the relevant version of the Ricci flow is the evolution equation \begin{equation} \frac{\partial \omega_{t}}{\partial t} = \omega_{t} -{\rm Ricci}(\omega_{t}), \end{equation} for a one-parameter family of metrics $\omega_{t}$ in the class $c_{1}(X)$. This can be expressed in terms of the K\"ahler potential. For each $t$ there is a unique $h_{t}$ such that
$$ \omega_{t}-{\rm Ricci}(\omega_{t}) = i\partial\overline{\partial} h_{t}, $$
normalised so that $\max_{X} h_{t}=0$ and we can write $\omega_{t}=\omega_{0}+ i \partial\overline{\partial} \phi_{t}$ where $\phi_{t}$ evolves by: \begin{equation} \frac{\partial\phi_{t}}{\partial t} = h_{t}. \end{equation}
It has been known for many years that this equation has a solution for all $t\in[0,\infty)$, starting with any initial condition. The main result of the paper of Chen, Sun and Wang \cite{kn:CSW} is that this flow converges as $t\rightarrow \infty$ to a \lq\lq weak'' Kahler-Ricci soliton metric $\omega_{\infty}$ on a normal projective variety $X_{\infty}$ (in fact a \lq\lq Q-Fano variety''). That is to say there is an algebraic torus action on $X_{\infty}$ an element $\xi$ in the Lie algebra of this torus and a metric on the regular part of $X_{\infty}$ (locally, in $X_{\infty}$, given by bounded potential) and which satisfies the equation (12) with respect to the holomorphic vector field generated by $\xi$.
In the case when the limiting metric is weak K\"ahler-Einstein but $X_{\infty}$ is not isomorphic to $X$ the same arguments as in 4.1 above, using the reductivity of the automorphism group, show that there is a test configuration for $X$ with central fibre $X_{\infty}$ and Futaki invariant zero. When the limit is a genuine Kahler-Ricci soliton the statment of \cite{kn:CSW} is slightly more subtle. They show that there is a destabilising test configuration for $X$, with central fibre $\overline{X}$ and strictly negative Futaki invariant, and a further degeneration (which might be trivial) of $\overline{X}$ to $X_{\infty}$. The upshot then is a trichotomy: \begin{itemize} \item $X$ is K-stable: the Ricci flow converges to a Kahler-Einstein metric on $X$; \item $X$ is K-semistable but not $K$-stable, the limit is K\"ahler-Einstein (possibly singular) but $X_{\infty}$ is not isomorphic to $X$; \item $X$ is not K-semistable: the limit is a genuine Ricci soliton and $X_{\infty}$ is not isomorphic to $X$. \end{itemize}
The foundation for these results is the subsequential convergence of the flow established by Chen and Wang in the earlier paper \cite{kn:CW}. That is, they prove that for any sequence $t_{i}\rightarrow \infty$ there is a subsequence $i'$ such that $(X, \omega_{t_{i'}})$ converges (in the same sense as in the previous subsections) to a weak K\"ahler-Ricci soliton metric on a Q-Fano variety. A fundamental difficulty in proving this is that it is not known that the Ricci curvature is bounded, either above or below, along the flow. This prevents a direct application of the Cheeger-Colding convergence theory. Results from the previous literature on K\"ahler-Ricci flow, including those of Perelman elaborated and extended by Sesum and Tian \cite{kn:ST}, yield three important pieces of information. \begin{enumerate}\item The scalar curvature $R_{t}$ is bounded along the flow: $\vert R_{t}\vert\leq C_{1}$ \item The potential $h_{t}$ is bounded along the flow: $\vert h_{t}\vert\leq C_{2}$ \item There is a uniform Sobolev inequality \cite{kn:Ye}, \cite{kn:Zhang}: $$ \Vert f \Vert_{L^{2n/n-1}}\leq C_{3}\left( \Vert \nabla f \Vert_{L^{2}} + \Vert f \Vert_{L^{2}}\right) . $$
\end{enumerate}
The general strategy of Chen and Wang is to establish a compactness theorem for segments of the flow over a fixed time interval, say $(T-1\leq t\leq T+1)$. That is, they show that if $T_{i}$ is a sequence tending to infinity then, after passing to a subsequence, these segments of the flow converge (to a possibly singular limit). They show that the limit is \lq\lq stationary'', in that it is the solution of the K\"ahler-Ricci flow given by a K\"ahler-Ricci soliton, evolving by the action of holomorphic automorphisms. The proofs of Chen and Wang are based on a blow-up argument and the comparison with suitable \lq\lq canonical neighbourhoods''. For $\kappa>0$, Chen and Wang define a class ${\cal KS}(n,\kappa)$ of non-compact length spaces. A space $W$ in ${\cal KS}(n,\kappa)$ is a smooth Ricci-flat K\"ahler $n$-manifold outside a closed singular set of codimension $>3$, and with asymptotic volume ratio $\geq \kappa$, that is $$ {\rm liminf}_{r\rightarrow \infty} v_{p}(r)\geq \kappa . $$
In their application $\kappa=\kappa(C_{3})$ is determined by the constant $C_{3}$ in the Sobolev bound. They prove from their definition that spaces in ${\cal KS}(n,\kappa)$ have many of the properties of the limit spaces treated by the Cheeger-Colding theory. In particular, they adapt that theory to show the existence of metric tangent cones. They also establish a compactness property of ${\cal KS}(n,\kappa)$, with respect to based convergence over bounded sets. This compactness means that spaces in ${\cal KS}(n,\kappa)$ satisfy certain uniform estimates.
Chen and Wang's blow-up argument is governed by a canonical radius $cr(p)\in (0,\infty]$ which they define for any point $p$ in a Riemannian manifold $M$. This notion is in the same order of ideas as others in the literature such as the harmonic radius and curvature scale, but Chen and Wang's definition is tailored to the particular case at hand. The general idea is that $cr(p)\geq r$ if the $r$-ball centred at $p$, scaled to unit size, satisfies various definite estimates. The parameters in these estimates are chosen in line with the uniform estimates established in ${\cal KS}(n,\kappa)$, so that roughly speaking $cr(p)=\infty$ for a point in a space $W$ in ${\cal KS}(n,\kappa)$ (or more precisely $cr(p)$ is arbitarily large for a point in a Riemannian manifold which is sufficiently close to some $W\in {\cal KS}(n,\kappa)$). Now Chen and Wang establish a lower bound $cr(p)\geq \epsilon>0$ for any $p$ in a manifold $(X,\omega_{t})$ along the Ricci flow. In outline, the argument is to suppose not, so there is a sequence of times $t_{i}$ and points $p_{i}\in (X,\omega_{t_{i}})$ such that $cr(p_{i})\rightarrow 0$. Rescaling by $r_{i}^{-1}$ they arrive at a sequence of based manifolds $(p_{i}, M_{i})$ with $cr$ bounded below by $1$ and with $cr(p_{i})=1$. They show that these manifolds converge to a space in ${\cal KS}(n,\kappa)$ and derive a contradiction from the fact that $cr=\infty$ on ${\cal KS}(n,\kappa)$.
In this argument the bound $\vert R\vert\leq C_{1}$ on the scalar curvature enters in the following way. Under the Ricci flow the scalar curvature $R$ evolves by $$ \frac{\partial R}{\partial t}= \Delta R + \vert {\rm Ric}\vert^{2} - n $$
After rescaling a portion of the Ricci flow, by a large factor $r_{i}^{-1}$ in the space direction and by $r_{i}^{-1/2}$ in the time direction, the scalar curvature $R'$ satisfies
$$ \frac{\partial R'}{\partial t}= \Delta R' + \vert {\rm Ric'}\vert^{2} - n r_{i}^{2} $$ and $\vert R'\vert \leq C_{1} r_{i}^{2}$. Thus on any region where the rescaled flows, with a sequence of scalings $r_{i}^{-1}\rightarrow \infty$, converge in $C^{\infty}$ the limit must be a stationary Ricci-flat manifold. This is the fundamental mechanism which leads to the singular Ricci-flat spaces in ${\cal KS}(n,\kappa)$. The parameter $\kappa=\kappa(C_{3})$ is determined by a standard relation between the Sobolev constant and volume ratio. (If a space has small asymptotic volume ratio one can write down a compactly supported function $f$ with $\Vert\nabla f\Vert_{L^{2}}$ small compared with $\Vert f \Vert_{L^{2n/n-1}}$. If the volume ratio is less than $\kappa(C_{3})$, such a space cannot arise as a blow-up limit of manifolds with Sobolev bound $C_{3}$.)
The $L^{2}$ construction of holomorphic sections features in two ways in Chen and Wang's arguments. One is global, to produce a projective embedding of the limit space. The tangent cone information from the blow-up limit in ${\cal KS}(n,\kappa)$ is transferred to the manifolds in the limiting sequence and the techniques outlined in Section 3 apply. The other is local, to produce local holomorphic co-ordinates as ratios of suitable holomorphic sections. The bound on the potential $\vert h_{t}\vert \leq C_{2}$ is important here. First, since there is no lower bound on the Ricci curvature the argument based on the formula (7)does not immediately apply. Changing the metric on the line bundle by a factor $e^{h_{t}}$ introduces an extra term which precisely cancels the Ricci curvature contribution in (7) and the bound on the $h_{t}$ means that this change does not substantially effect the estimates. Second, the evolution equation (13) gives control of the change in the metric on the line bundle in time, and Chen and Wang are able to use this to obtain local holomorphic co-ordinates that are adapted to the metrics $\omega_{t}$ over definite time intervals. Alongside these complex geometry arguments they also use the the Ricci flow techniques of Perelman.
The existence of a limit of the K\"ahler-Ricci flow, as established by Chen, Sun and Wang in \cite{kn:CSW} introduces further new ideas. The Chen and Wang result leaves open the possibility that different sequences of times $t_{i}\rightarrow \infty$ could lead to different limits. Let ${\cal C}$ be the set of all limits that arise. By general principles this set is connected. One major step is to show that all $X_{\infty}$ in ${\cal C}$ can be embedded in projective space in such a way that the soliton vector fields are generated by the same fixed 1-parameter subgroup. This uses an algebro-geometric characterisation of the vector field of a Ricci soliton, via a generalisation of the Futaki invariant theory, which leads to a rigidity property.
\subsection{ Proof by variational method}
The result proved by Berman, Boucksom and Jonsson in \cite{kn:BBJ} involves a notion of \lq\lq uniform K-stability''. Let ${\cal X}$ be a test configuration for a polarised manifold $(X,L)$, so we have a ${\bf C}^{*}$-action on the central fibre $X_{0}$. Suppose first that $X_{0}$ is smooth and we fix an $S^{1}$-invariant K\"ahler metric in the class $c_{1}({\cal L}_{0})$ which yields a symplectic structure. The $S^{1}$-action is generated by a Hamiltonian function $H$ which we can normalise to have maximum value $0$. Then we define $\Vert {\cal X}\Vert$ to be the $L^{1}$ norm of $H$. This is a quantity which is independent of the choice of metric in the cohomology class. The definition can be extended to any scheme, using the asymptotics of the trace of the action on sections of ${\cal L}^{k}$ as $k\rightarrow\infty$, similar to the definition of the Futaki invariant. Then $(X,L)$ is said to be {\it uniformly K-stable} if there is some $\epsilon>0$ such that \begin{equation} {\rm Fut}({\cal X})\geq \epsilon \Vert {\cal X}\Vert \end{equation}for all non-trivial test configurations ${\cal X}$. The main result of \cite{kn:BBJ} is that for a Fano manifold $X$, polarised by $K_{X}^{-1}$ and with finite automorphism group, the existence of a K\"ahler-Einstein metric is equivalent to uniform $K$-stability. Note that {\it a priori} uniform $K$-stability is a stronger condition than $K$-stability although {\it a posteriori} they are equivalent, for Fano manifolds with finite automorphism group. One can consider many other norms on test configurations and the general notion of uniform stability goes back to the thesis of Szekelyhidi. It is also related to another variant of $K$-stability developed by Szekelyhidi which considers filtrations of the co-ordinate ring $\bigoplus H^{0}(X,L^{k})$ \cite{kn:Gabor1}. A definition of uniform stability which turns out to be equivalent to that in \cite{kn:BBJ} was given by Dervan \cite{kn:Dervan}.
To give some indication of the proof in \cite{kn:BBJ} we begin by considering an analogous situation in finite dimensions. Let $V$ be a complete (finite-dimensional) Riemannian manifold with the property that the each two points can be joined by a unique geodesic segment---for example a simply connected manifold of non-positive curvature. Let $v_{0}\in V$ be a base point and let $F$ be function on $V$ which is convex along geodesics. If $\gamma:[0,\infty)\rightarrow V$ is a geodesic ray emanating from $v_{0}$, parametrised by arc length, the ratio $\gamma(t)/t$ is increasing with $t$ and we can define the {\it asymptotic slope} $ S_{\gamma}\in [0,\infty]$ to be the limit as $t\rightarrow \infty$. If $S_{\gamma}>0$ then for any $\delta< S_{\gamma}$ there is a $C_{\gamma}$ such that \begin{equation} F(\gamma(s)) \geq \delta s -C_{\gamma}. \end{equation}
An elementary argument, hinging on the compactness of the set of geodesics through $v_{0}$, shows that if there is some $\delta>0$ such that if $S_{\gamma}\geq \delta$ for all such rays $\gamma$ then given some $\delta'<\delta$ we can find $C$ such \begin{equation} F(v)\geq \delta' d(v,v_{0}) - C \end{equation} for all $v\in V$. It also follows easily that $F$ attains a minimum in $V$.
The relevance of this to the K\"ahler-Einstein problem on a Fano manifold $X$ is that, as we have discussed in Section 2, a K\"ahler-Einstein metric can be seen as a critical point of the Mabuchi functional ${\cal F}$ on the space of K\"ahler metrics ${\cal H}$. The pair $({\cal H}, {\cal F})$ has many properties analogous to $(V,F)$ above. There is a Mabuchi metric which makes ${\cal H}$ formally a symmetric space of non-positive curvature and ${\cal F}$ is convex along geodesics. The programme roughly speaking, is to extend the arguments above to this infinite-dimensional setting and to relate the asymptotics of the Mabuchi functional, analogous to the asymptotic slope $S_{\gamma}$, to the condition (17) on test configurations.
A point $\omega$ in ${\cal H}$ defines a volume form on $X$ and the tangent space to ${\cal H}$ at $\omega$ can be identified with the functions $\delta\phi$ of integral $0$. For each $p\geq1$ the $L^{p}$ norm defines a Finsler structure on ${\cal H}$. The case $p=2$ gives the infinite dimensional Riemannian structure first considered by Mabuchi \cite{kn:Mab1} but the case $p=1$ is also important, as shown by Darvas \cite{kn:Darvas}. The completion, in the metric defined by this Finsler structure, is a space ${\cal E}^{1}$ of currents defined by \lq\lq finite energy'' potentials. Geodesics in ${\cal H}$ also have a good geometric meaning. Smooth geodesics segments parametrised by an interval $[a,b]$ correspond to $S^{1}$-invariant closed $(1,1)$-forms $\Omega$ on the product $X\times S^{1}\times [a, b]$ which satisfy $\Omega^{n+1}=0$ and which restrict to a metric in ${\cal H}$ on each copy of $X$ in the product. (The different Finsler structures share the same geodesics.) If we stay in the smooth category it is not true that any two points in ${\cal H}$ can be joined by a geodesic \cite{kn:Lempert} but this is the case if one relaxes the definitions to allow singularities---in fact forms with $C^{1,1}$ potentials---as shown by Chen \cite{kn:XC}. It is an elementary calculation that the Mabuchi functional is convex along smooth geodesics. The convexity in the general case is a deep recent result of Berman and Berndtsson \cite{kn:BB}.
Now fix a base point $\omega_{0}$ in ${\cal H}$---analogous to $v_{0}\in V$. According to Darvas, the distance to $\omega_{0}$ in the $L^{1}$-Finsler structure is equivalent to functional $J$ which is well-known in the literature and which can be characterised by the property that $J(\omega_{0})=0$ and $$ \delta J = \int_{X} \delta \phi (\omega^{n}-\omega_{0}^{n}) . $$ Thus the analogue of (19) is an inequality. \begin{equation} {\cal F}(\omega)\geq \delta' J(\omega)-C\end{equation} for some fixed $\delta'>0, C$ and $\omega$. This is sometimes referred to as the \lq\lq properness'' of the Mabuchi functional. Tian showed \cite{kn:Ti} that if this inequality (20) holds then a K\"ahler-Einstein metric exists. The Mabuchi functional is decreasing along the continuity path, as discussed in 4.2 above, and so ${\cal F}(\omega_{s})$ controls $J(\omega_{s})$ and Tian showed that this allows the continuity path to be continued to $s=1$. Darvas and Rubinstein have given another proof of this result, and generalisations, recently \cite{kn:DR}.
There is a large circle of results relating geodesic rays in ${\cal H}$ to test configurations, see \cite{kn:PS} for example. Using the conformal equivalence between $S^{1}\times [0,\infty)$ and the punctured disc $\Delta^{*}$, geodesic rays correspond to $S^{1}$-invariant closed $(1,1)$-forms $\Omega$ on the product $X\times \Delta^{*}$ with $\Omega^{n+1}=0$ and which are positive on each $X\times \{t\}$. For certain purposes one can work with {\it subgeodesics} which correspond to positive definite forms $\Omega$. In particular if ${\cal X}$ is a test configuration we can consider a \lq\lq smooth'' metric $\Omega$ on ${\cal X}$. For example we can embed ${\cal X}$ in some $\Delta\times {\bf C}{\bf P}^{N}$ and take the restriction of a metric on the ambient manifold. The ${\bf C}^{*}$-action on ${\cal X}$ gives an open embedding $X\times \Delta^{*}$ and we get a subgeodesic ray $\omega_{s}$ in ${\cal H}$. Boucksom, Hisamoto and Jonsson \cite{kn:BHJ} prove that (if the central fibre is reduced)
$$ J(\omega_{s}) \sim \Vert {\cal X}\Vert s\ \ \ ;\ \ \ \ {\cal F}(\omega_{s})\sim {\rm Fut}({\cal X}) s , $$ as $s\rightarrow \infty$. (There are many earlier results of this kind in the literature, under various hypotheses.) Thus the uniform stability condition is equivalent to the statement that there is a $\delta>0$ such that for any such subgeodesic ray, arising from a test configuration, we have \begin{equation} {\cal F}(\omega_{s})\geq \delta J(\omega_{s})- C, \end{equation} where $C$ depends on the ray.
There are now two main aspects to the proof. (The exposition in \cite{kn:BBJ} involves some sophisticated techniques which go well beyond this writers knowledge, and indeed \cite{kn:BBJ} is described by the authors as an outline to be followed by a more detailed version. What we write below is extremely sketchy.) \begin{itemize} \item
To pass from the subgeodesic rays arising from test configurations to general geodesic rays and establish an inequality (21) along any geoedesic ray.
A sub-geodesic ray comes with a family of K\"ahler potentials which can be viewed as a metric on the pull-back of $L$ to $X\times \Delta^{*}$ or as a singular metric on the pull-back to $X\times \Delta$. This metric defines a multiplier ideal sheaf: the local holomorphic sections which are in $L^{2}$ with respect to the metric. Fundamental results of Nadel show that this is a coherent sheaf and so one can construct corresponding blow-ups of $X\times \Delta$ along the powers of this ideal sheaf, which yield test configurations. Bermann, Boucksom and Jonnson use these to approximate the original ray by those arising from test configurations and eventually to pass from the algebro-geometric uniform K-stability hypothesis to the estimate (18) on general geodesic rays. For these purposes they also work with the Ding functional (which we mentioned briefly in Section 2). They also use ideas and results from non-Archimedean geometry.
\item To carry the elementary arguments from the finite dimensional model over to the infinite dimensional situation. Results of Berman, Boucksom, Eyssidieux, Guedj and Zeriahi \cite{kn:BBEGZ} are used here to give the relevant compactness property in ${\cal E}^{1}$ for geodesics segments with bounded Mabuchi functional. \end{itemize}
These variational techniques based on convex geometry in the space ${\cal H}$ of K\"ahler metrics have been used by Darvas and Rubinstein \cite{kn:DR} and Berman, Darvas and Lu \cite{kn:BDL} to produce interesting results in the more general framework of constant scalar curvature metrics. The outstanding problem is to prove the regularity of weak solutions produced by minimising the Mabuchi functional.
\end{document} | arXiv |
Ackermann–Teubner Memorial Award
The Alfred Ackermann–Teubner Memorial Award for the Promotion of Mathematical Sciences recognized work in mathematical analysis. It was established in 1912 by engineer Alfred Ackermann-Teubner (1857–1941),[1] and was an endowment of the University of Leipzig.[2]
It was awarded 14 times between 1914 and 1941.[3] Subsequent awards were to be made every other year until a surplus of 60,000 marks was accumulated within the endowment, at which time, the prize was to be awarded annually. The subjects included:[4]
• History, philosophy, teaching
• Mathematics, especially arithmetic and algebra
• Mechanics
• Mathematical physics
• Mathematics, especially analysis
• Astronomy and theory of errors
• Mathematics, especially geometry
• Applied mathematics, especially geodesy and geophysics.
Honorees
The fifteen honorees between 1914 and 1941 are:[5]
• 1914: Felix Klein[3]
• 1916: Ernst Zermelo, prize of 1,000 marks[6]
• 1918: Ludwig Prandtl[7]
• 1920: Gustav Mie[8]
• 1922: Paul Koebe[9]
• 1924: Arnold Kohlschütter[10]
• 1926: Wilhelm Blaschke[11]
• 1928: Albert Defant[12]
• 1930: Johannes Tropfke
• 1932: Emmy Noether and Emil Artin, co-honorees[13]
• 1934: Erich Trefftz(de)[14]
• 1937: Pascual Jordan[15]
• 1938: Erich Hecke[16]
• 1941: Paul ten Bruggencate[17]
Jurists
In 1937, Constantin Carathéodory and Erhard Schmidt were invited to jury the award.[18] Along with Wilhelm Blaschke, Carathéodory was invited again in 1944 by the German Union of Mathematicians.[19]
See also
• List of mathematics awards
References
1. "Ackermann, Gustav Alfred Benedictus". personen-wiki.slub-dresden.de. Retrieved 2008-09-04.
2. Georgiadou, Maria (2004). Constantin Carathéodory: Mathematics and Politics in Turbulent Times. New York: Springer. p. 348. ISBN 3-540-20352-4.
3. "Notes". Bulletin of the American Mathematical Society. Providence, Rhode Island: American Mathematical Society. 21 (8): 419. May 1915. doi:10.1090/S0002-9904-1915-02671-6.
4. "Notes and News". The American Mathematical Monthly. Mathematical Association of America. 19 (8/9): 157. August–September 1912. JSTOR 2972758.
5. Ackermann-Teubner memorial prize (Leipzig 1914-1941)
6. "Notes". Bulletin of the American Mathematical Society. Providence, Rhode Island: American Mathematical Society. 23 (7): 336. April 1917. doi:10.1090/S0002-9904-1917-02963-1.
7. Society, American Mathematical (July 1919). "Notes". Bulletin of the American Mathematical Society. Providence, Rhode Island: American Mathematical Society. 25: 477. doi:10.1090/S0002-9904-1919-03240-6.
8. "Notes". Bull. Amer. Math. Soc. 27 (5): 237–241. 1921. doi:10.1090/s0002-9904-1921-03418-5.
9. "Notes". Bulletin of the American Mathematical Society. Providence, Rhode Island: American Mathematical Society. 29 (5): 235. May 1923. doi:10.1090/S0002-9904-1923-03715-4.
10. "Notes". Bull. Amer. Math. Soc. 31 (7): 375–378. 1925. doi:10.1090/s0002-9904-1925-04073-2..
11. "Notes". Bull. Amer. Math. Soc. 33 (3): 373. 1927. doi:10.1090/s0002-9904-1927-04389-0..
12. "Notes". Bull. Amer. Math. Soc. 35 (5): 741. 1929. doi:10.1090/S0002-9904-1929-04836-5..
13. Felder, D.G.; Rosen, D. (2005-02-01). Fifty Jewish women who changed the world. New York: Citadel Press. p. 100. ISBN 0-8065-2656-4.
14. "Notes" (PDF). Bulletin of the American Mathematical Society. Providence, Rhode Island: American Mathematical Society. 41: 178. May 1935. doi:10.1090/S0002-9904-1935-06071-9.
15. Teubner, ed. (1937). "Ackermann-Teubnerpreis 1937". Jahresbericht der Deutschen Mathematiker-Vereinigung. 47: 76..
16. "Ackermann-Teubnerpreis 1938". Mathematische Annalen. Springer-Verlag. 117 (1): 140. 1940. doi:10.1007/BF01450014..
17. "Alfred Ackermann-Teubner Gedächnispreis". Mathematische Annalen. 118: 440. 1941. doi:10.1007/bf01487379.
18. Georgiadou, p. 348
19. Georgiadou, p. 399
| Wikipedia |
\begin{document}
\begin{abstract}
In this paper we study an extension of the Bernstein Theorem for minimal spacelike surfaces of the four dimensional Minkowski vector space form and we obtain the class of those surfaces which are also graphics and have non-zero Gauss curvature. That is the class of entire solutions of a system of two elliptic non-linear equations that is an extension of the equation of minimal graphic of $\mathbb R^3$. Therefore, we prove that the so-called Bernstein property does not hold in general for the case of graphic spacelike surfaces in $\mathbb R^4_1$. In addition, we also obtain explicitly the conjugated minimal spacelike surface, and identify the necessary conditions to extend continuously a local solution of the generalized Cauchy-Riemann equations. \end{abstract}
\title{ Minimal Spacelike Surfaces and the Graphic Equations in $\mink$} {\bf Keywords}: Minimal spacelike surface, Bernstein Theorem, Weierstrass representation
{\bf MSC}: 53C42; 53C50
\section{Introduction}
One relevant classic result in the context of the global geometry of spacelike surfaces it is the Bernstein Theorem, which assures that if a minimal surface in the Euclidean $3$-dimensional space $\mathbb E^3$ is an entire graphic of a function $f:\Omega \subset \mathbb R^2 \to \mathbb R$, then it is a plane. Or equivalently, for $S$ being a regular surface of ${\mathbb{E}}^{3}$ and for a fixed direction $\rm{span \{\partial_{3}}\}$ and a system of coordinates $(O,x,y,z)$, such that in those coordinates
$\partial_{3} = (0,0,1)$, the Bernstein Theorem (\cite{1}) assures that {\em If $S$ is a minimal surface and the orthogonal projection in the coordinate plane $(O,x,y)$ is 1-1 and onto, then the surface is a plane.}
In the context lorentzian, it is well known the Cabali-Bernstein Theorem which establishes that in the $3$-dimensional Minkowski space $\mathbb R^3_1$ the only entire minimal graphic $\{(f(x,y), x, y)| (x,y) \in \Omega \subset \mathbb R^2\}$ are the spacelike planes. One can see the E. Calabi work in \cite{2} as a transposition of the Bernstein Theorem for $\mathbb R^3_1$, where the fixed direction is a timelike unit vector.
After the Bernstein and Calabi-Bernstein results, several authors have shown interest in these global results, and hence in the literature are found several works proving the Bernstein property from different viewpoints, providing diverse extensions or new proofs of those theorems.
Although in codimension one the Bernstein property is hold, it is worth pointing out that the property may be not hold in codimension bigger that one. That is the case in codimension two, where the Kommerell work (\cite{5}) considers minimal surface in the Euclidean $4$-dimensional space $\mathbb R^4$, and proves that graphic of entire holomorphic function gives minimal surfaces such that its projection in the plane $(O,x,y)$ is 1-1, onto and its Gauss curvature is not zero.
Motived by the results above and on the influence of the works of J. C. C. Nitsche (\cite{7}) and of T. Rad\'o (\cite{9}), we show through of this paper that the Bernstein property does not hold for spacelike surfaces in the 4-dimensional Minkowski space $\mathbb R^4_1$. More than it, in this paper we also provide answers to the question whether it is possible to establish some extension of the Bernstein Theorem for those kind of surfaces. Since the inner product used in this case is undefined, we need to consider two cases: fixing a timelike plane or a spacelike plane. So, explicitly, we work on answering the following question, which is a generalization type of the Bernstein and Kommerell theorems:
{\it Are there complex functions, not necessarily holomorphic, defined in the whole plane, whose graphic spacelike surface associated to orthogonal projection on a timelike plane or on a spacelike plane in ${\real}_{1}^{4}$, is onto and with non-zero Gauss curvature?}
Through of this paper we answer the previous question. In fact, we obtain two classes of minimal entire graphic spacelike surfaces in $\mathbb R^4_1$ of type $(A(x,y), x, y, B(x,y))$ and $(x, A(x,y), B(x,y), y)$, for $A(x,y), B(x,y)$ being smooth functions to real-valued, for which there exist points with non-zero Gauss curvature. We call the graphics above, as the first and second type, respectively. Our technique involves the use of a Weierstrass representation involving three holomorphic functions a complex-valued $a(w), b(w)$ and $\mu(w)$. That representation allows us to show that for getting the graphic minimal spacelike surfaces the holomorphic functions $a$ and $b$ have to be proportional complexes if the graphic is of first type, or inversely proportional complexes with the imaginary part different from zero if the graphic is the second type. Moreover, if the functions $a$ and $b$ are assumed to be defined in whole the complex plane $\mathbb C$, we find classes of graphic surfaces of first and second type which are entire and minimal with Gauss curvature different from zero. Therefore our theorems \ref{326} and \ref{500} provide explicit examples which prove that the Bernstein property does not hold in general for spacelike surfaces in $\mathbb R^4_1$.
Carrying out our study of the spacelike surfaces in $\mathbb R^4_1$, we also obtain explicitly the conjugate minimal spacelike surface using the Weierstrass representation. In addition, we identify under what conditions we can guarantee that a non-isothermic neighborhood can be extended to the entire complex plane. That is done using the generalized Cauchy-Riemman equations on neighborhood in non-isothermic coordinates. So, our work can be seen as an extension of the program developed by T. Rad\'o in \cite{9}.
In this paper we also give several examples of graphic minimal spacelike surfaces in $\mathbb R^4_1$ with Gauss curvature non-zero, and we find conditions to construct graphic minimal spacelike surfaces which have a new type of singularities, it called lightlike singularities, as defined by Kobayashi in \cite{4}. Those singularities are points where the tangent plane of the surface is also tangent to the lightcone of ${\real}_{1}^{4}$.
Finally, in the last section of this paper, we construct a $\theta$-family of minimal spacelike surfaces in $\mathbb R^4_1$ which transports a minimal first type graphic surface in $\mathbb E^3$ to a associated minimal first type graphic surface in $\mathbb L^3$. That allows us to conclude, as expected, that the Bernstein Theorem holds if and only if the Calabi Theorem holds.
For obtaining our results, we use the integral representation of the spacelike surfaces in $\mathbb R^4_1$, and of the adaptation of \cite{3} to the Minkowski space ${\real}_{1}^{4}$. The details of this adaptation can be found in the article of authors M.P. Dussan, A. P. Franco Filho and P. Sim\~oes (\cite{DPS}). Moreover, we pay attention to the Kobayashi work in \cite{4}, where he used the technique of Weierstrass representation to find several examples of minimal spacelike surfaces $\mathbb R^3_1$ and to find new type of singularities for these surfaces. Those singularities are points where as manifold these surfaces are defined but where the metric vanishes. That means in those points the tangent plane of $S$ is also tangent to the lightcone of $\mathbb R^3_1$. The Helicoid is a beautiful example that we can find in \cite{4}.
\section{Basic Facts and Notations} The Minkowski space ${\real}_{1}^{4}$ will be the $4$-dimensional real space ${\mathbb R}^{4}$ equipped with the bilinear form called of Lorentzian product, which is given by $$\lpr{(a,b,c,d)}{(t,x,y,z)} = -a t + b x + c y + d z.$$
A spacelike plane $V \subset {\real}_{1}^{4}$ is a $2$-dimensional vector subspace where $\lpr{v}{v} > 0$ for each $v \neq 0$ of the plane $V$. A timelike plane $T \subset {\real}_{1}^{4}$ is a $2$-dimensional vector subspace where there exists a timelike vector $t \in T$, that means that $\lpr{t}{t} < 0$, and an other spacelike vector $n \in T$ such that $\lpr{n}{n} > 0$ with $\lpr{t}{n} = 0$.
We say that timelike plane $T$ is the orthogonal complement of the spacelike plane $V$, denoted by the symbols $V = T^{\perp}$ and $T = V^{\perp}$, if $$\lpr{x}{y} = 0 \; \; \mbox{ for all } \; \; x \in T \; \; \mbox{ and } \; \; y \in V.$$
The following proposition is very useful throughout this work, it establishes a special orthonormal basis for each timelike plane. We denote by $\partial_0$ the vector $(1,0,0,0)$.
\begin{prop} For each spacelike plane $V \not\subset {\mathbb{E}}^{3}$ equipped with a orthonormal basis $\{e_{1},e_{2}\}$, the (unique) timelike plane $T = V^{\perp}$ has an orthonormal basis $\{\tau,\nu\}$ satisfying the following conditions:
1. \ $\lpr{\tau}{\tau} = -1$ and $\lpr{\tau}{\partial_{0}} < 0$. That means $\tau$ is timelike, future directed unit vector of $T$.
2.\ $\lpr{\nu}{\nu} = 1$ with $\lpr{\nu}{\partial_{0}} = 0$. That means that $\nu$ is a vector into the $3$-dimensional subspace $\{0\} \times {\mathbb R}^{3} \subset {\real}_{1}^{4}$, which we will identify with the Euclidean $3$-dimensional vector space ${\mathbb{E}}^{3}$.
3. $\lpr{\tau}{\nu} = 0$ and for all other orthonormal basis $\{t,n\}$ of $T$ we have that $\tau^{0} \leq \vert t^{0} \vert$.
4. The orthonormal basis $\{\tau, e_{1}, e_{2}, \nu\}$, in this order, is positive relative to the Minkowski referential $\{\partial_{0}, \partial_{1}, \partial_{2}, \partial_{3}\}$ given by the canonical basis of ${\mathbb R}^{4}$. \end{prop}
\begin{proof} We need to define the vector $\tau$, therefore all the statements of the proposition follow immediately. In fact, we take $\tau$ as being \begin{equation} \tau = \frac{1}{\sqrt{1 + (e^{0}_{1})^{2} + (e^{0}_{2})^{2} \;}}(\partial_{0} + e^{0}_{1} e_{1} + e^{0}_{2} e_{2}), \end{equation} where $e^{0}_{i} = - \lpr{\partial_{0}}{e_{i}}$ for $i = 1,2$. It is trivial to see that $\lpr{\tau}{\tau} = -1$, $\lpr{\tau}{e_{i}} = 0$ for $i = 1,2$, and that $\tau$ is directed future. Since by the assumption $V \not\subset {\mathbb{E}}^{3}$, we have the timelike plane generated by $\{\partial_{0}, \tau\}$. Then we take $\nu$ to be the unique unit vector of the line $\mathrm{span}\{\partial_{0}, \tau\} \cap {\mathbb{E}}^{3}$ such that $\{\tau, e_{1}, e_{2}, \nu\}$ is a positive basis.
Now, assuming that we have other orthonormal basis $\{t,n\}$ for $T$ we can take the Lorentz transformation given by $$t = \cosh \varphi \; \tau + \sinh \varphi \; \nu \; \; \; \mbox{ and } \; \; \; n = \sinh \varphi \; \tau + \cosh \varphi \; \nu,$$ assumed that $t^{0} > 0$. Since $-\lpr{t}{\partial_{0}} = - \cosh \varphi \lpr{\tau}{\partial_{0}}$ it follows then that $t^{0} > \tau^{0}$. \end{proof}
\subsection{A Semi-rigid frame} Let ${\real}_{1}^{4} = E \oplus T$ be given by the directed sum of a spacelike plane $E$ and its orthogonal complement $T$, which is a timelike plane.
\begin{dfn}\label{def} A semi-rigid referential of the Minkowski space ${\real}_{1}^{4}$ associated to a directed sum ${\real}_{1}^{4} = E \oplus T$, is a positive basis $\{l_{0},e_{1},e_{2},l_{3}\}$ of ${\real}_{1}^{4}$ satisfying the following conditions:
1. \ $E = \mathrm{span}\{e_{1},e_{2}\}$ and \ $T = \mathrm{span}\{l_{0},l_{3}\}$.
2. \ $\{e_{1},e_{2}\}$ is an orthonormal basis for $E$.
3. \ $\{l_{0},l_{3}\}$ is a null basis for $T$ such that $l_{0}^{0} = 1 = l_{3}^{0}$. \end{dfn}
\begin{prop} If we have two semi-rigid referential $\{l_{0},e_{1},e_{2},l_{3}\}$ and $\{\tilde{l}_{0},\tilde{e}_{1},\tilde{e}_{2},\tilde{l}_{3}\}$, associated to the directed sum ${\real}_{1}^{4} = E \oplus T$ with $T = E^{\perp}$, then $l_{0} = \tilde{l}_{0} \; \; \mbox{ and } \; \; l_{3} = \tilde{l}_{3}.$
Therefore the complex numbers given by $$a(l_{3}) = \frac{l_{3}^{1} + i l_{3}^{2}}{1 - l_{3}^{3}} \; \; \; \mbox{ and } \; \; \; b(l_{0}) = \frac{l_{0}^{1} + i l_{0}^{2}}{1 + l_{0}^{3}}$$ are univocally determined by the directed sum ${\real}_{1}^{4} = E \oplus T$. \end{prop} \begin{proof} In the Lorentz plane $T$ with induced orientation by $\partial_{0}$, there exists only two independent lightlike directions $L_{0}$ and $L_{3}$. Therefore adding the condition $$\lpr{L_{0}}{\partial_{0}} = -1 = \lpr{L_{3}}{\partial_{0}},$$ we obtain the unique basis $\{l_{0},l_{3}\}$ for $T$ given by 3. of the Definition \ref{def}. \end{proof}
\begin{corol} The matrix associated to the set of the semi-rigid referentials of the directed sum ${\real}_{1}^{4} = E \oplus T$, is given by $$\M(\vartheta) = \left[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos \vartheta & \sin \vartheta & 0 \\ 0 & - \sin \vartheta & \cos \vartheta & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right] \ \ \ \ {\rm for} \ \; \; \vartheta \in {\mathbb R}.$$ \end{corol}
Moreover, $\M(\vartheta)$ is a $1$-parameter sub-group of the Minkowski group of isometry of ${\real}_{1}^{4}$, and all geometric facts that we will see in this work, are invariant by this sub-group. Indeed, we will see that the complex functions $a(l_{3})$ and $b(l_{0})$ determine the geometric properties of minimal spacelike surfaces of ${\real}_{1}^{4}$.
\begin{prop} The frame associated to the vector subspace $E$ can be taken in terms of $a(p)$ and $b(p)$, namely, $$e_{1}(p) = \frac{W(p) + \overline{W(p)}}{2 \vert 1 - a(p) \overline{b(p)} \vert} \; \; \mbox{ and } \; \; e_{2}(p) = \frac{W(p) - \overline{W(p)}}{2 i \vert \vert 1 - a(p) \overline{b(p)} \vert \vert},$$ where $$W(p) = (a(p) + b(p), 1 + a(p)b(p), i(1 - a(p) b(p)), a(p) - b(p)) $$ with $\lpr{W(p)}{\overline{W(p)}} = 2\vert 1 - a(p) \overline{b(p)} \vert^{2}$ \end{prop}
\subsection{Spacelike Surfaces in ${\real}_{1}^{4}$}
\begin{dfn} A spacelike surface $S \subset {\real}_{1}^{4}$ is a smooth $2$-dimensional sub-manifold of the topological real vector space ${\mathbb R}^{4}$ that at each point $p \in S$ its tangent plane $T_{p}S$ relative to the lorentz product of ${\real}_{1}^{4}$ is a spacelike plane.
A spacelike parametric surface of ${\real}_{1}^{4}$ is a two parameters map $(U,X)$ from a connected open subset $U \subset {\mathbb R}^{2}$ into ${\real}_{1}^{4}$, such that the topological subspace $X(U)$ is a spacelike surface.
Henceforward we will assume that $(X(U),X^{-1})$ is a chart of a complete atlas for a spacelike surface $S$ of ${\real}_{1}^{4}$. \end{dfn}
Let $((x,y),U)$ be a connected and simply connected open subset of the Euclidean plane ${\mathbb R}^{2}$. If $X(x,y) = (X^{0}(x,y), X^{1}(x,y), X^{2}(x,y), X^{3}(x,y))$ is a spacelike parametric surface of ${\real}_{1}^{4}$ then, we have a metric tensor induced by the lorentzian semi-metric of ${\real}_{1}^{4}$ given by $$\mathbf{g} = \sum_{i,j} \lpr{D_{i} X}{D_{j} X} dx^{i} \otimes dx^{j},$$ and the second quadratic form of $S = X(U)$ is a quadratic symmetric $2$-form $$B = \sum_{i,j} \Psi_{ij} dx^{i} \otimes dx^{j},$$ that is given by covariant partial derivative by the formula $$D_{ij} X - \sum_k \Gamma_{ij}^{k} D_{k} X = \Psi_{ij}.$$
From the definition of the Christoffel symbols $\Gamma_{ij}^{k}$ it follows that $\lpr{\Psi_{ij}}{D_{k} X} \equiv 0$. Setting a pair of pointwise orthonormal vectors for the normal bundle $NS$ given by $\tau(x,y)$ and $\nu(x,y)$, where $\tau(x,y)$ is a future directed timelike unit vector and $\nu(x,y)$ is a spacelike unit vector, we can assume that $\lpr{\nu(x,y)}{(1,0,0,0)} \equiv 0$. Therefore we have $$\Psi_{ij} = h_{ij} \tau + n_{ij} \nu$$ where by definition $$h_{ij} = - \lpr{D_{ij} X}{\tau} \; \; \mbox{ and } \; \; n_{ij} = \lpr{D_{ij} X}{\nu}.$$
Since $\dim (N_{p}S) = 2$ we need to define the normal connection for $S$, which is given by a covariant vector $\gamma = \sum \gamma_{k} dx^{k}$ where $$\gamma_{k} = \lpr{D_{k} \tau}{\nu} = \lpr{D_{k} \nu}{\tau}.$$
Next we will display this set of structural equations for $S = X(U)$, equation (2) being the Gauss equation, (3) and (4) corresponding to Weingarten equations for $S$. Namely, \begin{align} D_{ij} X - \sum_k \Gamma_{ij}^{k} D_{k} X = h_{ij} \tau + n_{ij} \nu \\ D_{k} \tau = \sum_m h_{m}^{k} D_{m} X + \gamma_{k} \nu \\ D_{k} \nu = - \sum_m n_{m}^{k} D_{m} X + \gamma_{k} \tau. \end{align}
\begin{dfn}\label{1} We say that the surface $S = X(U)$ is a minimal surface if and only if $$H_{S} = \frac{1}{2}\sum \Psi_{ij} g^{ij} = 0.$$ The vector field $H_{S}$ is called the mean curvature vector of $S$.
It follows from equations (2) that an equivalent definition for minimal surfaces is the condition $$2 H_{S} = \sum g^{ij} (D_{ij} X - \sum \Gamma_{ij}^{k} D_{k} X) = (\Delta_{\mathbf{g}} X^{0}, \Delta_{\mathbf{g}} X^{1}, \Delta_{\mathbf{g}} X^{2}, \Delta_{\mathbf{g}} X^{3}) = 0,$$ where $\Delta_{\mathbf{g}}$ is the Laplace-Beltrami operator over $S = (U, {\bf g})$. \end{dfn}
Next we observe that one can associate a Riemann surface to $S$. In fact, from the well know theorem which assures that any spacelike surface admits an isothermic coordinate atlas, that means, there is a parametrization $$f(w) = (f^{0}(w), f^{1}(w), f^{2}(w), f^{3}(w)), \; \; \; w = u + i v \in U' \subset \mathbb{C},$$ such that $f(U') \subset S = X(U)$ and the induced metric tensor is $\mathbf{g} = \lambda^{2} dw d\overline{w}$, or more explicitly $$\lpr{f_{u}}{f_{u}} = \lambda^{2} = \lpr{f_{v}}{f_{v}} \; \; \mbox{ and } \; \; \lpr{f_{u}}{f_{v}} = 0.$$
Since $f_{w} = \frac{1}{2} (f_{u} - i f_{v})$, we extend the bilinear form of ${\real}_{1}^{4}$ to a complex bilinear form over $\mathbb{C}^{4} \equiv {\mathbb R}^{4} + i {\mathbb R}^{4}$, namely, $$\lpr{X + i Y}{A + i B} = \lpr{X}{A} - \lpr{Y}{B} + i(\lpr{X}{B} + \lpr{Y}{A}.$$ Hence it implies that \begin{equation}\label{11} \lpr{f_{w}}{f_{w}} = 0 \; \; \mbox{ and } \; \; \lpr{f_{w}}{\overline{f_{w}}} = \lpr{f_{w}}{f_{\overline{w}}} = \lambda^{2}/2. \end{equation}
Now, if we have two isothermic charts $(U',f)$ and $(V,h)$ for $S$ then, when makes sense, the overlapping map is a holomorphic function, and so we can see $M = (S,\A)$ as a Riemann surface equipped with the conformal atlas $\A$, and such that the induced metric tensor $ds^{2} = \lambda^{2}(w) \vert dw \vert^{2}$ is a compatible metric for the Riemann surface $M$.
Finally, we note that does not exist compact spacelike surfaces in ${\real}_{1}^{4}$, so from now $M$ will be either the disk $$D = \{z \in \mathbb{C} : z \overline z < 1\} \; \; \mbox{ that is a hyperbolic Riemann surface},$$ or the complex plane $\mathbb{C}$ which is a parabolic Riemann surface, since we are assuming that $M$ is a connected and simply connected Riemann surface. Moreover, if $h(z(w)) = f(w)$ then from chain rule it follows $$f_{w}(w) = h_{z}(z(w)) \frac{dz}{dw}(w) \; \; \mbox{ and } \; \; \lpr{f_{w}}{f_{\overline{w}}} = \lpr{h_{z}}{h_{\overline{z}}} \left\vert \frac{dz}{dw} \right\vert^{2}.$$
\subsection{A solution for the equations (\ref{11})} Expanding in its coordinates we have that the equation (\ref{11}) becomes $$-(f^{0}_w)^{2} + (f^{1}_w)^{2} + (f^{2}_w)^{2} + (f^{3}_w)^{2} = 0.$$ Denoting the complex derivate of the components $f^i_w$ by $Z^i$ and assuming that $Z^{1} - i Z^{2} \neq 0$, we have $$\frac{Z^{0} - Z^{3}}{Z^{1} - i Z^{2}} \; \frac{Z^{0} + Z^{3}}{Z^{1} - i Z^{2}} = \frac{Z^{1} + i Z^{2}}{Z^{1} - i Z^{2}}.$$
Defining by $$a = \frac{Z^{0} + Z^{3}}{Z^{1} - i Z^{2}}, \; \; \; \; b = \frac{Z^{0} - Z^{3}}{Z^{1} - i Z^{2}} \; \; \; \; \mbox{ and } \; \; \; \; \mu = \frac{Z^{1} - i Z^{2}}{2},$$ we obtain that the derivate $f_w$ can be represented by $$f_{w} = \mu W(a,b) \; \; \mbox{ where } \; \; W(a,b) = (a + b, 1 + ab, i(1 - ab), a - b),$$ from $(a,b) \in \F(M,\mathbb{C}) \times \F(M,\mathbb{C})$ in $\mathbb{C}^{4}$.
Moreover, we have that $\lambda^{2} = 2 \lpr{f_{w}}{\overline{\; f_{w}}\;} = 4 \mu \overline{\mu} (1 - a \overline{b})(1 - \overline{a} b).$ Therefore, $\mu \neq 0$ and $1 - a \overline{b} \neq 0$ are the conditions to obtain a surface without singularities in its metric.
Now, since we can write $W(a,b) = (a,1,i,a)+ b(1,a,-ia,-1)$ we obtain the cases where happens $(Z^{1} + i Z^{2})(Z^{1} - iZ^{2}) = 0$ through of the expressions $f_{w} = \eta (a,1,i,a)$ and $f_{w} = \xi (1,a,-ia,-1)$. Moreover when $Z^{0} = 0 = Z^{3}$ we obtain $f_{w} = \eta(0,1,i,0)$ which can be identified with the plane $\{0\} \times {\mathbb R}^{2} \times \{0\}$.
The following lemma is an extension to ${\real}_{1}^{4}$ of a theorem obtained by Monge: \begin{lemma}\label{2} For a $\lambda$-isothermic spacelike parametric surface $(U,f)$ the following statement are equivalent:
(i) The surface $f(U)$ is minimal, $H_{f}(w) \equiv 0$.
(ii) The maps $\mu,a,b$ are holomorphic functions from $U$ into $\mathbb{C}$. \end{lemma}
\begin{proof} It follows from the Laplace-Beltrami operator that $\Delta_{M} f^{i}(w) = \frac{2}{\lambda^{2}} (f^{i}(w))_{w \overline{w}} = 0$ for $i = 0,1,2,3.$ \end{proof}
\subsection{An integral representation} Let $(U,X)$ be a spacelike parametric surface of ${\real}_{1}^{4}$ where $$X(x,y) = (X^{0}(x,y), X^{1}(x,y), X^{2}(x,y), X^{3}(x,y))$$ and $U \subset {\mathbb R}^{2}$ is a simply connected domain. Then, the vector $1$-form given by $$dX = \frac{\partial X}{\partial x} dx + \frac{\partial X}{\partial y} dy$$ is exact and therefore closed. So, the integral equation associated to $(U,X)$ is \begin{equation}\label{3} X(x,y) = X(x_{0},y_{0}) + \int_{(x_{0},y_{0})}^{(x,y)} \frac{\partial X}{\partial x} dx + \frac{\partial X}{\partial y} dy. \end{equation}
Moreover, each solution of equation (\ref{3}) is a spacelike parametric surface $(U,X)$ if it holds $$E = \lpr{X_{x}}{X_{x}} > 0, \; \; G = \lpr{X_{y}}{X_{y}} > 0, \; \; F = \lpr{X_{x}}{X_{y}} \; \mbox{ and } \; \; EG - F^{2} > 0.$$
From Definition \ref{1} and Lemma \ref{2}, we obtain:
\begin{corol} Let $U \subset {\mathbb R}^{2}$ a simply connected domain. If $(U,X)$ is a minimal spacelike parametric surface which is solution of the integral equation (\ref{3}), then each coordinate function of $X(x,y)$ is a harmonic real-valued function on $U$. \end{corol} \begin{proof} Indeed, the Laplace-Beltrami operator $\Delta_{M}$ is a tensorial operator defined by contraction of the Gauss equation (2), as follows in Definition \ref{1}. \end{proof}
We note that with isothermic local coordinates the integral representation (\ref{3}) is usually called of Weierstrass integral equation, namely, $$f(w) = p_{0} + 2 \Re \int_{w_{0}}^{w} \mu(\xi) W(a(\xi),b(\xi)) d \xi,$$ where $f_w(w)$ is the solution of equation (\ref{11}) in Subsection 2.2.
\subsection{The structural equations with isothermic parameters} Let $(U,f)$ be a parametric sub-surface of $X(M)$ given with isothermic parameters $w = u + iv$ such that $\lpr{f_{w}}{f_{w}} = 0$ and $\lpr{f_{w}}{f_{\overline{w}}} = \lambda^{2}/2$. In this case we have the following version of structural equations (2), (3) and (4) for minimal surfaces. \begin{lemma} Let $(U,f)$ be a $\lambda^{2}$-isothermic coordinates system for a minimal surface $(M,X)$ of ${\real}_{1}^{4}$. We have the following structural equations in $w = u + i v \in U$: \begin{align} \tau_{w} = \sigma f_{\overline{w}} + \Gamma \; \nu \; \; \; \; \; \; \mbox{ and } \; \; \; \; \nu_{w} = \chi f_{\overline{w}} + \Gamma \; \tau \\ f_{ww} = 2 \frac{\lambda_{w}}{\lambda} f_{w} + \frac{\sigma \lambda^{2}}{2} \; \tau - \frac{\chi \lambda^{2}}{2} \; \nu \; \; \; \; \; \; and \; \; \; \; f_{w\overline{w}} = 0 \\ \Gamma(w) = \lpr{\tau_{w}}{\nu} = - \lpr{\nu_{w}}{\tau} = \frac{\gamma_{1}(w) - i \gamma_{2}(w)}{2} \end{align} \end{lemma} \begin{proof} We start showing equation (8). For that we take $f_{ww} = A f_{w} + B f_{\overline{w}} + C \tau + D \nu$, and assume that equations (7) and (9) are the definition of the functions associated to the normal connection for $(U,f)$.
From $\lpr{f_{w}}{f_{w}} = 0$ it follows $\lpr{f_{ww}}{f_{w}} = 0$, therefore $B = 0$. From $\lpr{f_{w}}{f_{\overline{w}}} = \lambda^{2}/2$ it follows $\lpr{f_{ww}}{f_{\overline{w}}} + \lpr{f_{w}}{f_{w\overline{w}}} = \lambda_{w} \lambda$, and, since $f_{w\overline{w}} = 0$ we obtain $A = 2 \frac{\lambda_{w}}{\lambda}.$
Now, from $\lpr{f_{w}}{\tau} = 0$ we have that $\lpr{f_{ww}}{\tau} + \lpr{f_{w}}{\tau_{w}} = 0$, therefore we obtain $ C = \sigma \frac{\lambda^{2}}{2}$. Analogously one has $D =- \chi \frac{\lambda^{2}}{2}.$ So we have showed equation (8).
The definition of the functions $\sigma$ and $\chi$ is obtained by equations (7), that from $\lpr{f_{\overline{w}}}{\tau} = 0$ and from the minimal condition for $(M,f)$ it follows that $\lpr{\tau_{w}}{f_{\overline{w}}} + \lpr{\tau}{f_{w\overline{w}}} = 0$. Thus the tangential component of $\tau_{w}$ is $\sigma f_{\overline{w}}$. Then, we take the equations (7) as a definition of the functions associated to the normal connection of $(M,f)$. Equation (9) defines the function $\Gamma$. \end{proof}
\section{Two types of Graphics for Minimal Surfaces of ${\real}_{1}^{4}$}
First, let us recall that ${\real}_{1}^{4}$ has topological structure and differential structure of the Euclidean space ${\mathbb R}^{4}$.
If $R(u,v) = (\varphi(u,v),\psi(u,v))$ is a function from $U \subset {\mathbb R}^{2}$ in ${\mathbb R}^{2}$, we can see as a graphic of $R$ the set of point of ${\mathbb R}^{4}$ such that $$\mbox{graphic(R)} = \{((u,v),(\varphi(u,v),\psi(u,v))) \in {\mathbb R}^{4}: (u,v) \in U \subset {\mathbb R}^{2}\}.$$ Since we can choose four equivalent positions for the timelike axis in ${\real}_{1}^{4}$, we only need to pick
two of those positions to get all the possibilities of graphic surfaces. In fact:
Fixing the signature of ${\real}_{1}^{4}$ by $(-1,+1,+1,+1)$ we take by definition:
(1) The first type of graphic surfaces as given by $$X(x,y) = (A(x,y), x, y, B(x,y)) \; \mbox{ where } \; (x,y) \in U \subset {\mathbb R}^{2}.$$
(2) The second type of graphic surfaces as given by $$X(x,y) = (x, A(x,y), B(x,y), y) \; \mbox{ where } \; (x,y) \in U \subset {\mathbb R}^{2}.$$
We will always assume that the functions $A$ and $B$ are $\C^{\infty}(U)$, $U$ is a connected and simply connected open subset of ${\mathbb R}^{2}$ and that $X(U)$ is a spacelike surface of ${\real}_{1}^{4}$.
\begin{prop}\label{14} A minimal graphic surface (first or second type) of ${\real}_{1}^{4}$ satisfies the following system of equations \begin{equation}\label{10} \left\{\begin{matrix} g_{22} D_{11} A - 2 g_{12} D_{12} A + g_{11} D_{22} A = 0 \\ g_{22} D_{11} B - 2 g_{12} D_{12} B + g_{11} D_{22} B = 0 \end{matrix} \right. \end{equation} where $\mathbf{g} = \sum_{ij} g_{ij} du^{i} du^{j}$ is the positive defined metric tensor associated to the surface $S = X(U)$.
The system of equations (\ref{10}) only says that $A$ and $B$ are harmonic functions of the Riemann surface $(U,X)$. \end{prop} \begin{proof} Taking the matrix representation of metric tensor and its inverse tensor $$[g_{ij}] = \left[\begin{matrix} E & F \\ F & G \end{matrix} \right], \; \; \; \; \; \; \; [g^{ij}] = \frac{1}{EG - F^{2}} \left[\begin{matrix} G & -F \\ -F & E \end{matrix} \right],$$ one has, from Definition \ref{1}, that the mean curvature vector is given by $$2 H_{X} = \frac{1}{EG - F^{2}} (G \Psi_{11} - 2F \Psi_{12} + E \Psi_{22}).$$
Now, for each type of surface we take a pointwise basis $\{N_{1},N_{2}\}$ for its normal bundle, as follows.
If $X(x,y) = (A(x,y), x, y, B(x,y))$ we take the orthogonal vectors $$N_{1} = (1, A_{x}, A_{y}, 0) \; \; \mbox{ and } \; \; N_{2} = (0, B_{x}, B_{y}, -1),$$ and so in this case, $D_{ij}X = (D_{ij} A, 0 ,0, D_{ij} B)$.
If $X(x,y) = (x, A(x,y), B(x,y), y)$ we take the orthogonal vectors $$N_{1} = (A_{x},1,0,-A_{y}) \; \; \mbox{ and } \; \; N_{2} = (B_{x},0,1,-B_{y}).$$ Then in this case $D_{ij}X = (0,D_{ij} A, D_{ij} B, 0)$. Therefore, the system (\ref{10}) follows immediately. \end{proof}
Our first example corresponds to minimal spacelike surfaces, which are graphic surfaces of the first type
defined in the whole plane ${\mathbb R}^{2}$.
\begin{example}\label{323} For each harmonic function $\theta : {\mathbb R}^{2} \longrightarrow {\mathbb R}$ the maps $$X(x,y) = (\theta(x,y), x, y, \theta(x,y)) \; \; \mbox{ or } \; \; X(x,y) = (\theta(x,y), x, y, -\theta(x,y))$$ are both minimal spacelike parametric surfaces, locally isometric to the Euclidean plane ${\mathbb R}^{2}$, and therefore flat surfaces.
In fact, it assumes the first expression of $X(x,y)$. Since $X_{x} = (\theta_{x}, 1, 0, \theta_{x})$ and $X_{y} = (\theta_{y}, 0, 1, \theta_{y})$ it follows $\lpr{X_{x}}{X_{x}} = 1 = \lpr{X_{y}}{X_{y}}$ with $\lpr{X_{x}}{X_{y}} = 0$. Now, by assumption $\Delta \theta = \theta_{xx} + \theta_{yy} = 0$, it follows that $H_{X}(x,y) = (0,0,0,0)$.
We also observe that, according the notation of Subsection 2.3, this class of surfaces corresponds to when $Z^1 + i Z^2 =0,$ with $Z^0 - Z^3 =0$ and $Z^0 \ne 0$, where $Z^i$ are the components in the representation $X_w(w) = (\theta_w, \frac{1}{2}, \frac{i}{2}, \theta_w).$
Moreover we can write these parametric surfaces as follows: For $A = B = \theta(x,y)$ we have that $X(x,y) = (0, x, y, 0) + \theta(x,y) (\partial_{0} + \partial_{3})$, therefore $X({\mathbb R}^{2})$ is a subset of a degenerated hyperplane, and this shows that its normal curvature vanishes identically. \end{example}
The Example \ref{323} shows that we need a formula of the second quadratic form in terms of functions $\mu$, $a$ and $b$. That formula was already obtained in Theorem 3.3 from \cite{DPS}, so we rewrite next. \begin{lemma}\label{2.99} Let $f_{w} = \mu W(a,b)$, where $a$ and $b$ are holomorphic functions from $M$ into $\mathbb{C}$. The second quadratic form in complex notation is given by \begin{equation}\label{2.9} (f_{ww})^{\perp} = \frac{\mu a_{w}}{1 - a \overline{b}} L_{0}(b) + \frac{\mu b_{w}}{1 - b \overline{a}} L_{3}(a), \end{equation} where $L_0(b)$ and $L_3(a)$ are future directed lightlike vectors given by $$ L_0(b) = (1+ b\overline b, b + \overline b, -i(b-\overline b), 1- b \overline b)
\ \ \ {\rm and} \ \ \ L_3(a) = (1+ a\overline a, a + \overline a, -i(a-\overline a), - 1+ a \overline a). $$ \end{lemma}
It follows from Lemma \ref{2.99} the next corollary.
\begin{corol} The second quadratic form of a minimal spacelike surface $(U,f)$ is lightlike type if and only if $a_{w} = 0$ or $b_{w} = 0$. Therefore in this case, the Gauss curvature $K(f) = 0$ and the surface is contained in a degenerated hyperplane.
Reciprocally, if the Gauss curvature $K(f) = 0$ then the second quadratic form is lightlike type or it is zero, $(f_{ww})^{\perp} = 0$. \end{corol}
Now, we apply equations (\ref{10}) for graphic minimal surfaces in ${\mathbb{E}}^{3}$ and $\mathbb{L}^{3}$. We give the explicit equation for each case.
For the first type:
(1) When $A(x,y) \equiv 0$ we obtain the graphics in ${\mathbb{E}}^{3}$ given by an unique function $B(x,y)$: $$f(x,y) = (0,x,y,B(x,y)) \in {\mathbb{E}}^{3},$$ with the induced metric tensor over $f(U)$ as a spacelike surface of ${\real}_{1}^{4}$. Then system (\ref{10}) becomes to the equation \begin{equation}\label{4} (1 + B_{y}^{2}) B_{xx} - 2 B_{x} B_{y} B_{xy} + (1 + B_{x}^{2}) B_{yy} = 0, \end{equation} which is called the equation of minimal graphic for smooth surface of the Euclidean space ${\mathbb R}^{3}\equiv \mathbb E^3$. In this case Bernstein showed that if $U = {\mathbb R}^{2}$ then the solution of equation (\ref{4}) is a plane.
(2) When $B(x,y) \equiv 0$ we obtain the graphics in ${\mathbb{E}}^{3}$ given by an unique function $A(x,y)$: $$f(x,y) = (A(x,y),x,y,0)) \in \mathbb{L}^{3},$$ with the induced metric tensor over $f(U)$ as a spacelike surface of ${\real}_{1}^{4}$. System (\ref{10}) becomes to the equation \begin{equation}\label{5} (1 - A_{y}^{2}) A_{xx} + 2 A_{x} A_{y} A_{xy} + (1 - A_{x}^{2}) A_{yy} = 0 \; \; \; \mbox{ with } \; A_{x}^{2} < 1 \; \mbox{ and } \; A_{x}^{2} + A_{y}^{2} < 1, \end{equation} which is called the equation of minimal graphic for smooth surface of the Lorentzian space $\mathbb{L}^{3}$. For this case, Calabi showed that if $U = {\mathbb R}^{2}$ then the solution of equation (\ref{5}) is a plane.
Now we turn our attention for graphic minimal spacelike surfaces of the second type, given by the representation $f(x,y) = (x, A(x,y),B(x,y), y)$. In this case,
(3) When $B(x,y) \equiv 0$ we obtain the graphics given by an unique function $A(x,y)$: $$f(x,y) = (x,A(x,y),0, y)) \in \mathbb{L}^{3},$$ with the induced metric tensor over $f(U)$ as a spacelike surface of ${\real}_{1}^{4}$. Then system (\ref{10}) becomes to the equation \begin{equation} (1 + A_{y}^{2}) A_{xx} - 2 A_{x} A_{y} A_{xy} + (-1 + A_{x}^{2}) A_{yy} = 0 \; \; \; \mbox{ with } \; A_{x}^{2} > A_{y}^{2} + 1, \end{equation} and, we will say that this equation is the equation for graphic of second type of minimal smooth surface of $\mathbb{L}^{3}$.
\section{About the Extension of Local Solutions of the Graphic Equations} In this section we study whether it is possible to extend to whole the complex plane $\mathbb C$ the local solutions for the graphic equations given in system (\ref{10}).
We start identifying a formula for the Gauss curvature of the surface. In fact, for $f_{w} = \mu W(a,b)$ where $(U,f)$ is a minimal spacelike surface of ${\real}_{1}^{4}$, with holomorphic functions $a(w)$, $b(w)$, $\mu(w)$, we know that the expression for the Gauss curvature is given by $$K(f) = - \frac{ \Delta \ln \lambda^{2}}{2\lambda^{2}} = - \frac{1}{\lambda^{2}} \Delta \ln \lambda.$$
Now, since $\lambda^{2} = 4 \mu \overline{\mu} (1 - a \overline{b})(1 - \overline{a}b)$ and $\Delta = 4 \partial_{w \overline{w}},$ we obtain $$K(f) = - \frac{(\ln (1 - a \overline{b})(1 - \overline{a}b))_{w\overline{w}}}{2\mu \overline{\mu} (1 - a \overline{b})(1 - \overline{a}b)}.$$ Since $$(\ln(1 - a \overline{b})(1 - \overline{a}b))_{w\overline{w}} = -a_w\left(\frac{\overline{b}}{1 - a \overline{b}}\right)_{\overline{w}} - b_w\left(\frac{\overline{a}}{1 - b \overline{a}}\right)_{\overline{w}},$$ it follows that \begin{equation} K(f) = \frac{\Re(a_{w}\overline{b}_{\overline{w}}(1 - \overline{a}b)^{2})} {\mu \overline{\mu} (1 - a \overline{b})^{3}(1 - \overline{a}b)^{3}}. \end{equation}
{\bf First case}. We will focus our attention to find surfaces given by $$X(x,y) = (A(x,y), x, y, B(x,y)) \; \; \mbox{ for all } \; \; (x,y) \in {\mathbb R}^{2},$$ satisfying the equations (10), which means that $X({\mathbb R}^{2}) = S$ is a minimal surface of ${\real}_{1}^{4}$.
So a question arises: {\em Is there a non-flat solution to this problem}?
For answering that question we proceed as follows. First, we construct a pointwise basis for the normal bundle.
In fact, it takes the vector fields $N_1$ and $N_2$, along $S = X({\mathbb R}^{2})$, used in the proof of Proposition \ref{14}, namely, $$N_{1} = (1, A_{x}, A_{y}, 0) \; \; \mbox{ and } \; \; N_{2} = (0, -B_{x}, -B_{y}, 1).$$
Then we have the following proposition.
\begin{prop} The spacelike Gauss map $\nu(x,y)$ for the minimal surface $S \subset {\real}_{1}^{4}$ is given by $$\nu(x,y) = \frac{1}{\sqrt{1 + (B_{x})^{2} + (B_{y})^{2}}} (0, - B_{x}, - B_{y}, 1).$$ \end{prop} \begin{proof} We only need to see if the orientation of $\{N_{1},N_{2}\}$ and the orientation of $\{\partial_{0},\partial_{3}\}$ are compatible each other. The compatibly orientations follow from the projected vectors $(N_{1}^{0},0,0,N_{1}^{3}) = \partial_{0}$ and $(N_{2}^{0},0,0,N_{2}^{3}) = \partial_{3}$. \end{proof}
\begin{corol} The Gauss map $\nu : S \longrightarrow \mathbb{S}^{2} \subset {\mathbb{E}}^{3}$ is such that $$\nu^{3} = \frac{1}{\sqrt{1 + (B_{x})^{2} + (B_{y})^{2}}} > 0.$$
In other words, $\nu(S)$ is the (open) north hemisphere of the Riemann sphere $\mathbb{S}^{2}$. \end{corol}
Now we assume that we have a local representation $(U,f)$ such that $f(U) \subset S$ and $$f_{w} = \mu(a + b, 1 + ab, i(1 - ab), a - b),$$ where $a,b,\mu$ are holomorphic functions from $U$ into $\mathbb{C}$, and $U$ is a connected and simply connected open subset of $\mathbb{C}$. Then the normal bundle has a pointwise basis
of lightlike vectors $\{L_{3}(a), L_{0}(b)\}$ like in Lemma 3.2, which allows, in easier form, to compute the fourth component of the spacelike Gauss map $\nu(a,b)$, as follows.
\begin{lemma}\label{111} For an isothermic local representation $(U,f)$ such that $f(U) \subset S$ we have \begin{equation}\label{3.9.9} \nu^{3}(a,b) = \frac{1}{\vert 1 - \overline{a}b \vert \sqrt{1 + \vert a \vert^{2}} \; \sqrt{1 + \vert b \vert^{2}}} (1 - \vert ab \vert^{2}). \end{equation}
Moreover, the maximal extension of holomorphic functions $a,b$, is conditioned by the inequalities: \begin{equation}\label{2.8.8} \vert 1 - \overline{a}b \vert \neq 0 \; \; \mbox{ and } \; \; \vert ab \vert^{2} \neq 1. \end{equation} \end{lemma} \begin{proof} Taking the normalization of the vector $N_3$ given by $$N_{3}(a,b) = \frac{1}{1 + b \overline{b}} L_{0}(b) - \frac{1}{1 + a \overline{a}} L_{3}(a)$$ one gets $\nu(a,b)$ since $N_{3}^{0}(a,b) = 0$. Therefore, we obtain the component $\nu^{3}(a,b)$ given in (\ref{3.9.9}) and the inequalities (\ref{2.8.8}). \end{proof}
Now we observe that the first inequality in (\ref{2.8.8}) is the functional area $\sqrt{EG - F^{2}} = \vert \mu \vert \; \vert 1 - \overline{a}b \vert$. Then for our purposes, we will find a necessary and sufficient condition to obtain a maximal extension of the function $\sqrt{EG - F^{2}}$. Hence if we assume the integrating factor being constant $\mu =1$, we need just to consider the maximal extension of $\vert 1 - \overline{a}b \vert$.
For achieving that goal we give the next corollary, which follows from Liouville Theorem and Theorem \ref{111}, since for $a(w) b(w)$ being an entire bounded function, it must be constant.
\begin{corol}\label{27} If $a(w)$ and $ b(w)$ can be extended for whole the plane $\mathbb{C}$, then there exists a constant $c \in \mathbb{C}$ such that $a(w) b(w) = c$. \end{corol}
Hence it follows as direct consequence of Corollary \ref{27}, that if $a(w) = b(w)$ or $a(w) = - b(w)$ for all $w \in \mathbb{C}$, then $a(w) = \sqrt{c}$. That means that $(\mathbb{C},f)$ is a spacelike plane of ${\real}_{1}^{4}$.
Moreover from the Corollary \ref{27}, we can also construct an example of a minimal surface $(\mathbb{C},f)$, which is a graphic with Gauss curvature $K(f) \neq 0$. This means a set of points $p$ of the surface such that the condition $K(p) = 0$ is not satisfied on the entire plane $\mathbb{C}$. Even more, now we are abled to prove our next result which provides a general class of examples of entire graphic minimal surfaces of first type such that the Gauss curvature $K(f) \neq 0$.
\begin{theor}\label{326} Let $a = a(w)$ be a holomorphic function defined in whole the plane $\mathbb{C}$ such that $a(w) \neq 0$ for each $w \in \mathbb{C}$. Let $c = \alpha + i \beta \in \mathbb{C} \setminus \{0,1,-1\}$ such that $\alpha^{2} + \beta^{2} \neq 1$, and it takes the holomorphic function $b(w) = \frac{c}{ a(w)}$ from $\mathbb{C}$ in $\mathbb{C}$. Then the surface given by \begin{equation}\label{29} f(w) = X_{0} + 2 \Re \int_{0}^{w} \left(a(\xi) + \frac{c}{a(\xi)}, 1 + c, i(1 - c), a(\xi) - \frac{c}{a(\xi)}\right) d \xi, \end{equation} is a minimal surface of ${\real}_{1}^{4}$, which is a graphic surface of type $X(x,y) = (A(x,y), x, y, B(x,y))$ through of the transformation of coordinates given by $x_{w} = (1 + c)$ and $y_{w} = i(1 - c)$.
Moreover, assuming that $a(w)$ is not a constant function then, there exists a point $p \in S$ such that $K(p) \neq 0$. Hence the surface can not be contained in hyperplanes of ${\real}_{1}^{4}$. \end{theor} \begin{proof} Taking $x(u,v) = 2[(1 + \alpha)u - \beta v]$ and $y(u,v) = 2[ \beta u + ( \alpha -1) v]$ we get the equation of the coordinates change, namely, \begin{equation}\label{30} \left[\begin{matrix} u \\ v \end{matrix} \right] = \frac{1}{2[ \alpha^{2} + \beta^{2} -1]} \left[\begin{matrix} \alpha -1 & \beta \\ - \beta & 1 + \alpha \end{matrix} \right] \; \left[\begin{matrix} x \\ y \end{matrix} \right]. \end{equation}
Therefore, since $a(w)$ and $b(w)$ are holomorphic functions and $\alpha^{2} + \beta^{2} \neq 1$, we obtain that equation (\ref{29}) represents a graphic minimal surfaces of first type.
Since the metric is given by $\lambda^{2} = 4 \vert 1 - \overline{a} c/a \vert^{2}$ follows that $\Delta \ln \lambda \neq 0$ in points where $a_w(w) \neq 0$. Then, since $K(f) = - \frac{1}{\lambda^2}\Delta \ln \lambda$, it follows that in those points happen $K(f) \neq 0$.
Next, by integration we can obtain the components functions $A(w) = f^{0}(w)$ and $B(w) = f^{3}(w)$, and through of the coordinate transformation given by the equation (\ref{30}) we obtain the explicit representation as graphic surface.
To finish, we see the real spacial property of surface $S$. In fact, it supposes that there is a vector $v = (v^{0},v^{1},v^{2},v^{3}) \in {\real}_{1}^{4}$ such that $\lpr{v}{f_{w}} = 0$. Then from the equality $-v^{0}(a + b) + v^{1}(1 + ab) + iv^{2} (1 - ab) + v^{3}(a - b) = 0$, we obtain $$(v^{3} - v^{0}) a - (v^{3} + v^{0}) b + (v^{1} + i v^{2}) + ab (v^{1} - i v^{2}) = 0.$$ It defines $T = v^{3} - v^{0}$, $S = v^{3} + v^{0}$, $Z = v^{1} + i v^{2}$, then we obtain $(T a + Z) + b(a \overline{Z} - S) = 0,$ which implies that $$b = \frac{ Ta + Z}{S - a \overline{Z}} = \frac{c}{a} \; \; \mbox{ if and only if } \; \; T = 0 = S \; \; \mbox{ and } \; \; c = - \frac{Z}{\overline Z}.$$
Thus, from $\frac{Z}{\overline Z} = -c$ and $v^{0} - v^{3} = 0 = v^{0} + v^{3}$, it follows that $v \not\in {\real}_{1}^{4}$. Contradiction. \end{proof}
So from Theorem \ref{326} we can construct a classe of minimal graphic surfaces of first type, whose Gauss curvature is not null in some points of the surface. That means the classic Bernstein theorem does not hold in this case. Next we give some particular examples of that fact.
\begin{example} For a simple example, we take $a = e^{w}$ and $c = 2$. Then according to Theorem \ref{326} we can take $b = \frac{c}{a}= 2 e^{-w}$ and $X_0 = 2(-1,0,0,3)$, to have the parametrization $$ f(w) = 2((e^u - 2 e^{-u}) \cos v, 3u, v, (e^u + 2 e^{-u}) \cos v). $$ Therefore taking the coordinates transformation given by $ x = 6u$ and $y = 2v$, we get the graphic parametrization given by $$ X(x,y) = (2 (e^{\frac{x}{6}} - 2 e^{- \frac{x}{6}}) \cos (\frac{y}{2}), \; x, \; y, \; 2 (e^{\frac{x}{6}} + 2 e^{- \frac{x}{6}}) \cos (\frac{y}{2})), $$ for which there are points such that the Gaussian curvature is not zero. In fact, it is just to take $\alpha$ and $\beta$ such that $\alpha \ne \cos (2y)$ and $\beta \ne \sin(2y)$, that means, $c \ne e^{2iy}$. \end{example}
\begin{example}\label{90} In this example we use Theorem \ref{326} to construct minimal graphic surfaces of first type. We start assuming $a(w) = e^{w}$ and $b(w) =\frac{2 e^{i\theta}}{a(w)}$ for $\theta \in (0,\pi)$. Since $\vert c \vert = \vert 2 e^{i\theta} \vert = 2$, the condition $\alpha^2 + \beta^2 \ne 1$ is hold. Then $W(a,b)$ is given by $$W(a,b) = (e^{w} + 2 e^{i\theta} e^{-w}, 1 + 2 e^{i\theta}, i(1 - 2 e^{i\theta}), e^{w} - 2 e^{i\theta}e^{-w}).$$ Now we take the factor of integration $\mu = 1$, to obtain the integral representation (\ref{29}) given by $$f(w) = 2 \Re \int_{0}^{w}(e^{\xi} + 2 e^{i\theta} e^{-\xi}, 1 + 2 e^{i\theta}, i(1 - 2 e^{i\theta}), e^{\xi} - 2 e^{i\theta}e^{-\xi}) d \xi,$$ more explicitly \begin{equation}\label{800} f(u,v) = 2(e^u \cos v - 2 e^{-u} (\cos v \cos \theta + \sin v \sin \theta), (1+ 2\cos \theta) u - 2 v \sin \theta, \end{equation} $$ (-1+ 2 \cos \theta) v + 2 u \sin \theta, e^u \cos v + 2 e^{-u}( \cos v \cos \theta + \sin v \sin \theta)). $$ Hence making the coordinates transformation $x_w = 1+ 2 e^{i\theta}$ and $y_w = i(1-2 e^{i\theta})$, we get $$x = 2[ (1+2 \cos \theta) u - 2 v \sin \theta] \ \ \ \ \ {\rm and} \ \ \ \ \ y = 2[ (-1+ 2 \cos \theta) v + 2 u \sin \theta]. $$ Thus the minimal graphic surface is given by $X(x,y) = (A(x,y), x, y, B(x,y))$, where the functions $A(x,y), B(x,y)$ are given by the first and fourth component of formula (\ref{800}) with $$ u= \frac{1}{6}((2 \cos \theta -1) x + 2 y \sin \theta) \ \ \ \ \ {\rm and} \ \ \ \ \ v = \frac{1}{6}(-2 x \sin \theta + (1+ 2 \cos \theta)y). $$ We observe that since $a_w = e^w$ never vanishes, all the points of the graphic surface are such that $K(p) \ne 0$.
\end{example}
{\bf Second case}. We will focus our attention to find surfaces given by $$X(x,y) = (x, A(x,y), B(x,y), y) \; \; \mbox{ for all } \; \; (x,y) \in {\mathbb R}^{2},$$ satisfying the equations (10). That means that $X({\mathbb R}^{2}) = S$ is a graphic minimal surface of ${\real}_{1}^{4}$ of second type.
So a question arises: {\em Is there a non-flat solution to this problem}?
For answering that question we proceed as before, constructing first a pointwise basis for the normal bundle.
Let us take the attitude matrix of $dX$: $$[dX]^{t} = \left[\begin{matrix} 1 & A_{x} & B_{x} & 0 \\ 0 & A_{y} & B_{y} & 1 \end{matrix}\right].$$
The unit spacelike Gauss map $\nu = \nu(x,y)$ is given by $$\nu(x,y) = \frac{1}{\sqrt{J^{2} + (B_{x})^{2} + (A_{x})^{2}}}(0, B_{x}, -A_{x}, J) \; $$ for $J = \frac{\partial(A,B)}{\partial(x,y)} = A_{x}B_{y} - A_{y} B_{x}.$
Since we can not control the functions $\nu^{i}$ for $i = 1,2,3$, we will work with the Weierstrass form $$f_{w} = \mu (a + b, 1 + ab, i(1 - ab), a - b)$$ and the transformation of coordinates \begin{equation}\label{299} x_{w} = \mu (a + b) \; \mbox{ and } \; y_{w} = \mu( a - b), \; \mbox{ where } \; x_{w}y_{\overline{w}} - x_{\overline{w}}y_{w} = 2 \vert \mu \vert^{2} \; (\overline{a} b - a \overline{b}). \end{equation}
\begin{lemma}\label{200} It considers the transformation of coordinates given by equations (\ref{299}). Then Jacobian function $x_{w}y_{\overline{w}} - x_{\overline{w}}y_{w} = 2 \vert \mu \vert^{2} \; (\overline{a} b - a \overline{b})$ does not vanish in a domain $U \subset M$ if and only if, for each $w \in U$, \begin{equation}\label{277} a(w) \ne 0 \ne b(w) \ \ \ \ {\rm and} \ \ \ \ \ \ \Im(\frac{a(w)}{b(w)}) \neq 0. \end{equation}
A maximal extension of holomorphic functions $a,b$ is conditioned by the inequalities (\ref{277}) and by $\vert 1 - \overline{a}b \vert \neq 0$. \end{lemma} \begin{proof} First we observe that $a(w) \neq 0 \neq b(w)$ is a necessary condition. Moreover, for each $w \in U$, $$-2i \Im(\frac{a(w)}{b(w)} )= \frac{\overline{a(w)}}{\overline{b(w)}} - \frac{a(w)}{b(w)} = \frac{\overline{a(w)} b(w) - a(w) \overline{b(w)}}{b(w) \overline{b(w)}}. $$ Hence, since the Jacobian function does not vanish, it follows that $ \Im(\frac{a(w)}{b(w)} ) \ne 0$. The conversely follows
immediately. \end{proof}
From Lemma \ref{200} and from Little Picard Theorem, it follows the next corollary.
\begin{corol}\label{300} It assumes that the holomorphic functions $a(w)$ and $b(w)$ can be extended for whole the plane $\mathbb{C}$. Then
there exists a constant $c \in \mathbb{C} \setminus \{0,1,-1\}$ such that $b(w) = c a(w)$.
Moreover, as consequence, if $f_w$ is such that $f_{w} = \mu (a(1 + c), 1 + ca^{2}, i(1 - ca^{2}), a(1 - c))$ then $$x(w) = 2 \Re \left((1 + c) \int_{0}^{w} \mu(\xi) a(\xi) d \xi\right) \; \; \mbox{ and } \; \; y(w) = 2 \Re \left((1 - c) \int_{0}^{w} \mu(\xi) a(\xi) d \xi\right).$$
Taking $P(w) + i Q(w) = \int_{0}^{w} \mu(\xi) a(\xi) d \xi$ and $c = \alpha + i \beta$ we obtain $$x(u,v) = 2 [(1 + \alpha) P(w) - \beta Q(w)] \; \; \mbox{ and } \; \; y(w) = 2[(1-\alpha) P(w) + \beta Q(w)].$$ \end{corol} \begin{proof} Since for $a(w) \ne 0 \ne b(w)$, with $\frac{a(w)}{b(w)}$ entire and such that $\Im(\frac{a(w)}{b(w)}) \neq 0$ (Lemma \ref{200}), the map $\frac{a(w)}{b(w)}$ does not cover whole the complex plane, then from Little Picard Theorem, it follows that $\frac{a(w)}{b(w)}$ is constant. Under the hypotheses that constant can not be 0, 1 neither -1. \end{proof}
\begin{rema} We observe that Corollary \ref{300} has a weakness because while in Theorem \ref{32} the equation (\ref{30}) gives us the inversion function which is linear, and which we can use to construct the graphic over whole the complex plane $\mathbb{C}$,
Corollary 4.7 can not guarantee that we have a graphic over all complex plane, since it could exist ramifications. For instance, taking $a(w) = e^{w}$ and $\mu =1$, we obtain $P(u,v) = e^{u} \cos v$ and $Q(u,v) = e^{u} \sin v$. So, $x(u,v) = 2[ (1 + \alpha) e^{u} \cos v - \beta e^u \sin v]$ and $y(u,v) = 2 [(1-\alpha) e^u \cos v + \beta e^u \sin v]$, which are periodic functions in the variable $v$. \end{rema}
In the next theorem we answer the question whether there exist a non-flat solution which is entire graphic surface of second type. In fact, we argue that if $a = a(w)$ is a given holomorphic function defined in whole $\mathbb{C}$ and such that $a(w) \neq 0$, then we can take the holomorphic function $\mu(w) = \frac{1}{a(w)}$ and take also $f_{w} = \mu W(a(w),c a(w))$, with constant $c \in \mathbb{C} \setminus \{0,1,-1\}$. Then next we will show that in this case, it can exist points in the surface such that the
Gauss curvature is not zero.
\begin{theor}\label{500} Let $a = a(w)$ be a holomorphic function defined in whole the plane $\mathbb{C}$ such that $a(w) \neq 0$ for each $w \in \mathbb{C}$. For $c = \alpha + i \beta \in \mathbb{C} \setminus {\mathbb R}$ we take $b(w) = c a(w)$ and $\mu(w) = \frac{1}{a(w)}$. Then the surfaces given by \begin{equation}\label{400} f(w) = X_{0} + 2 \Re \int_{0}^{w} \left(1 + c, \frac{1}{a(\xi)} + c a(\xi), i \left(\frac{1}{a(\xi)} - c a(\xi)\right), 1 - c\right) d \xi, \end{equation} are minimal surfaces of ${\real}_{1}^{4}$, which represent graphic of type $X(x,y) = (x, A(x,y), B(x,y), y)$, where the transformation of coordinates is given by $x_{w} = (1 + c)$ and $y_{w} = (1 - c)$.
Moreover, in this case, the Gauss curvature $K(f)(w) = 0$ if and only if $a_w(w) = 0$. Therefore, assuming that $a = a(w)$ is not a constant function, there exists $p \in S$ such that $K(p) \neq 0$. Again, there is not a hyperplane containing the surface $S$. \end{theor} \begin{proof} By integration we obtain $x = 2 \Re(((1 + \alpha) + i\beta)(u + iv)) = 2 [(1 + \alpha)u - \beta v]$ and $y = 2[(1 - \alpha)u + \beta v]$. That means, \begin{equation} \left[\begin{matrix} u \\ v \end{matrix} \right] = \frac{1}{4 \beta} \left[\begin{matrix} \beta & \beta \\ \alpha - 1 & \alpha + 1 \end{matrix} \right] \; \left[\begin{matrix} x \\ y \end{matrix} \right]. \end{equation}
Since $a, b$ and $\mu$ are holomorphic functions, formula (\ref{400}) in the $(x,y)$-coordinates, represents a graphic minimal surface of second type. Moreover, since the Gauss curvature is given by $K(f) = - \frac{1}{\lambda^2}\Delta \ln \lambda$ where
$\lambda^{2} = 4 \vert \frac{1}{a \overline a} - c \vert^{2}$, it follows that $\Delta \ln \lambda \neq 0$ in points where $a_w(w) \neq 0$. Hence in those points $K(f) \neq 0$.
Next, by integration we can obtain the components functions $A(w) = f^{1}(w)$ and $B(w) = f^{2}(w)$, and through of the transformation of coordinate we get the explicit representation as graphic surface.
Finally we note that it is needed to assume $c \not\in {\mathbb R}$, since we can not have $x_{w} y_{\overline{w}} - x_{\overline{w}} y_{w} = 0$. It is also impossible to obtain a timelike vector $v \in {\real}_{1}^{4}$ such that $\lpr{v}{f_{w}} = 0$, so we have the real spacial property of the surface in $\mathbb R^4_1$. \end{proof}
So from Theorem \ref{500} one can construct a classe of minimal graphic surfaces of second type, whose Gauss curvature is not null in any point of the surface. That means the property of Bernstein does not hold in this case. The following explicit example illustrates this fact.
\begin{example} We use Theorem \ref{500} to construct second type of minimal graphic surfaces. Let $a = e^{w}$ and \ $b = e^{i\theta} a$ \ for $\theta \in (0,\pi)$. Then the expression of $W(a,b)$ is $$W(a,b) = ((1 + e^{i\theta})e^{w}, 1 + e^{i\theta} e^{2w},i(1 - e^{i\theta} e^{2w}), (1 - e^{i\theta})e^{w}).$$ Taking the factor $\mu(w) = e^{-w}$, the integral representation (\ref{400}) is given by $$f(w) = 2 \Re \int_{0}^{w} \left(1 + e^{i\theta}, e^{-\xi} + e^{i\theta} e^{\xi}, i(e^{-\xi} - e^{i\theta} e^{\xi}), 1 - e^{i\theta}\right) d \xi,$$ or more explicitly \begin{equation}\label{600} f(u,v) = 2 ((1+\cos \theta) u - v \sin \theta, -e^{-u} \cos v + e^u (\cos v \cos \theta - \sin v \sin \theta), \end{equation} $$ - e^{-u} \sin v + e^u(\sin v \cos \theta + \cos v \sin \theta), (1- \cos \theta) u + v \sin \theta)). $$ Now making the transformation of coordinates $x_w = 1+ e^{i\theta}$ and $y_w = 1- e^{i\theta}$, we get $$x = 2[(1+\cos \theta) u - v \sin \theta] \ \ \ \ {\rm and } \ \ \ \ \ y = 2[(1- \cos \theta) u + v \sin \theta],$$ and hence the graphic minimal surface is given by $ X(x,y) = (x, A(x,y), B(x,y), y)$ where the functions $A(x, y)$ and $B(x,y)$ are given by the second and third component of formula (\ref{600}) with $$ u = \frac{x+y}{4} \ \ \ \ \ \ { \rm and } \ \ \ \ v = \frac{1}{4 \sin \theta}[(-1 + \cos \theta)x + (1 + \cos \theta)y]. $$ Finally we observe that since $a_w = e^w$ never vanishes, for all the points of the surface one gets that $K(p) \ne 0$. \end{example}
\section{The construction of the conjugated surface $(M,Y)$}
We dedicate this section for looking the explicit expression of the conjugated surface to a minimal spacelike surface $(M,X)$ of ${\real}_{1}^{4}$, using the Weierstrass notation. We start defining a special operator on tangent bundle $TS$ to a surface, as follows.
\begin{dfn} Let $(M,X)$ be a spacelike surface with line element $ds^{2}(X) = E dx^{2} +2 F dx dy + G dy^{2}$, and $TS$ be its tangent bundle, where, pointwise, $\{X_{x}(p),X_{y}(p)\}$ is a base of $T_{p}S$. Let $J : TS \longrightarrow TS$ be the function given by \begin{equation}\label{obrigada} J(V) = \frac{1}{\sqrt{EG - F^{2}}}\left(\lpr{X_{x}}{V} X_{y} - \lpr{X_{y}}{V} X_{x}\right). \end{equation} \end{dfn}
\begin{prop} Let $J : TS \longrightarrow TS$ be the function given by the equation (\ref{obrigada}). Then $\forall V \in TS,$ the following equations are satisfied: $$\lpr{V}{J(V)} = 0, \; \; \; \; \lpr{J(V)}{J(V)} = \lpr{V}{V} \; \; \mathrm{ and } \; \; J(J(V)) = - V.$$ \end{prop} \begin{proof} The first equation follows from $\sqrt{EG - F^{2}} \; \lpr{V}{J(V)} = \lpr{X_{x}}{V} \lpr{X_{y}}{V} - \lpr{X_{y}}{V} \lpr{X_{x}}{V} = 0$. For getting second equation we take the values of $J$ in the basis, namely, \begin{equation}\label{912} J(X_{x}) = \frac{1}{\sqrt{EG - F^{2}}}\left(E X_{y} - F X_{x}\right) \; \mbox{ and } \; J(X_{y}) = \frac{1}{\sqrt{EG - F^{2}}}\left(F X_{y} - G X_{x}\right). \end{equation} Then $$\lpr{J(X_{x})}{J(X_{x})} = E, \ \ \ \ \ \ \lpr{J(X_{y})}{J(X_{y})} = G \ \ \ \ {\rm and} \ \ \lpr{J(X_{x})}{J(X_{y})} = F.$$ Now we note that from the pointwise bi-linearity of $<,>$, it follows the pointwise linearity of $J$. Therefore if $V= a X_x + bX_y$ the second equation of the proposition holds.
The third equation follows directly from the linearity and from the facts $J(J(X_x)) = - X_x$ and $J(J(X_y)) = - X_y.$ \end{proof}
We observe that if $S = (M,X)$ be a spacelike surface of ${\real}_{1}^{4}$, the vector $1$-form associated to $S$ is given by $\beta = X_{x} dx + X_{y} dy$. Therefore, by definition $J(\beta)$ is the $1$-form given by \begin{equation}\label{911} J(\beta) = J(X_{x}) dx + J(X_{y}) dy. \end{equation}
Next we related the operator $J$ with the special normal frame $\{\tau, \nu\}$ in $\mathbb R^4_1$.
Let $l = \mathfrak{X}(v_{1}, v_{2}, v_{3})$ be the exterior product in ${\real}_{1}^{4}$ of a set of vectors $\{v_1, v_2, v_3\}$. By definition, since $\Omega({\real}_{1}^{4}) = (-dx^{0}) \wedge dx^{1} \wedge dx^{2} \wedge dx^{3}$ is the volume form, then $l$ is defined by $$\lpr{l}{w} = \Omega({\real}_{1}^{4})(v_{1}, v_{2}, v_{3}, w), \ \ \ \ \forall w \in {\real}_{1}^{4}.$$ Then the $J$ operator is equivalent to $J(V) = \mathfrak{X}(\tau,\nu,V)$.
\begin{theor}\label{211} Let $S = (M,X)$ be a spacelike surface of ${\real}_{1}^{4}$ and let $\beta = X_{x} dx + X_{y} dy$ be the vector $1$-form associated to $S$. Then \begin{equation}\label{900} J(\beta) = \frac{-F dx - G dy}{\sqrt{EG - F^{2}}} \; X_{x} + \frac{E dx + F dy}{\sqrt{EG - F^{2}}} \; X_{y}. \end{equation} The $1$-form $J(\beta)$ is closed if and only if $(M,X)$ is a minimal spacelike surface. \end{theor} \begin{proof} The equation (\ref{900}) follows from equations (\ref{912}) and (\ref{911}).
For the second statement, we use the representation of the operator $J$ as an exterior product, to obtain $$J(\beta) = \mathfrak{X}(\tau,\nu, X_x dx + X_y dy) = \tau \times \nu \times \beta.$$
Now, since $d\beta = 0$, we get the exterior derivative $d J(\beta) = ((d \tau) \times \nu \times \beta) + (\tau \times (d \nu) \times \beta)$.
Next we will calculate explicitly $dJ(\beta)$. For that, we use $d \tau = \tau_{x} dx + \tau_{y} dy$,
$d \nu = \nu_{x} dx + \nu_{y} dy$, the Weingarten formulas (3), (4) and the anti-commutative properties of the exterior product in ${\real}_{1}^{4}$ and of the exterior product of $1$-forms, to obtain $$ d(J \beta) = (h^1_1 + h^2_2) (X_x \times \nu \times X_y) dx \wedge dy + (n^1_1 + n^2_2) (\tau \times X_x \times X_y) dx \wedge dy. $$ Since $X_{x} \times \nu \times X_{y} = - \sqrt{EG - F^{2} \;} \tau$ and $\tau \times X_{x} \times X_{y} = \sqrt{EG - F^{2} \;} \nu$ one gets \begin{equation}\label{100} d J(\beta) = - 2 H_{X} \sqrt{EG - F^{2} \;} dx \wedge dy. \end{equation} Hence it follows from equation (\ref{100}) that, $d J(\beta) = 0$ if and only if $(M,X)$ is minimal. \end{proof}
Theorem \ref{211} allows us to establish the next corollary which shows the explicit expression of the minimal conjugate spacelike surface $(M,Y)$ in ${\real}_{1}^{4}$. It comes from the fact that since $J(\beta)$ is a closed 1-form in a connected simply-connected open subset of $\mathbb C$ then it is exact. \begin{corol} Let $M$ be a connected and simply connected open subset of the plane $\mathbb{C}$, and let $(M,X)$ be a solution of the minimal graphic equations (\ref{10}). The integral representation (\ref{3}) can be extended to $Z = X + i Y \in \mathbb{C}^{4}$ by \begin{equation} Z(x,y) = Z(x_{0},y_{0}) + \int_{z_{0}}^{z} \beta + i J(\beta), \end{equation} where \begin{equation}\label{7} Y(x,y) = Y(x_{0},y_{0}) + \int_{z_{0}}^{z} \frac{-F dx - G dy}{\sqrt{EG - F^{2}}} \; X_{x} + \frac{E dx + F dy}{\sqrt{EG - F^{2}}} \; X_{y}. \end{equation} Moreover, $Y$ gives us the parametrization of the conjugated minimal spacelike surface $(M,Y)$ of ${\real}_{1}^{4}$. \end{corol}
\begin{proof} Since $J(dY) = J(J(dX) = - dX = - (X_xdx + X_y dy)$ is a closed vector 1-form, from Theorem \ref{211} it follows that $H_Y(p) =0$ for each $p \in M$. \end{proof}
\begin{example} Let $X(x,y) = (0,x \cos y, x \sin y, y)$ be a parametrizated Helicoid of ${\mathbb{E}}^{3}$. The conjugated minimal spacelike surface, given by equation (\ref{7}) with $Y(0,0) = (0, 0, 1, 0)$, is the Catenoid given in coordinates by $$Y(x,y) = (0, - \sqrt{1 + x^{2}} \sin y, \ \sqrt{1 + x^{2}} \cos y, \ \ln(x + \sqrt{1 + x^{2}})).$$ In fact, from $X_{x} = (0,\cos y, \sin y, 0)$ and $X_{y} = (0, -x \sin y, x \cos y, 1)$ it follows that $E = 1$, $F = 0$ and $G = 1 + x^{2}$. Now from the integral equation (\ref{7}) we obtain $$d Y = \frac{1}{\sqrt{1 + x^{2}}}(0,-x \sin y, x \cos y, 1) dx - \sqrt{1 + x^{2}}(0, \cos y, \sin y, 0) dy.$$ Hence by integrating $Y_x = \frac{1}{\sqrt{1 + x^{2}}}(0,-x \sin y, x \cos y, 1)$ and $Y_y = - \sqrt{1 + x^{2}}(0, \cos y, \sin y, 0)$, we get the Catenoid surface $({\mathbb R}^{2}, Y(x,y))$.
Moreover, if $x \geq 0$ we have the part corresponding to $Y^{3} \geq 0$ and, if $x \leq 0$ we have the part corresponding to $Y^{3} \leq 0$. Both surfaces $({\mathbb R}^{2},X(x,y))$ and its conjugated $({\mathbb R}^{2}, Y(x,y))$ are ramified.
Finally, if we make $x = \sinh u$ and $y = v$, we obtain $$\tilde{X}(u,v) = (0, \sinh u \cos v, \sinh u \sin v, v) \; \ \mbox{ and } \; \ \tilde{Y}(u,v) = (0, - \cosh u \sin v, \cosh u \cos v, u),$$ in the isothermic coordinates $(u,v)$. As it is expected it follows $\tilde X_u = - \tilde Y_v$ and $\tilde X_v = \tilde Y_u$. \end{example}
Next example shows an applicability of the $J$ operator.
\begin{example} Let $X(x,y) = (x \cosh y, x \sinh y, f(x), 0)$ be a graphic type of hyperbolic rotation in $\mathbb R^3_1$ in hyperbolic polar coordinates. Since $X_x = (\cosh y, \sinh y, f'(x), 0)$ and $X_y =(x \sinh y, x \cosh y, 0, 0)$, we get $$ E(x,y) = -1 + (f'(x))^2 > 0, \ \ F(x,y) =0, \ \ G(x,y)= x^2 >0, \ \ W = x\sqrt{(f')^2 -1}. $$
Hence, a needed condition for obtaining a minimal spacelike surface $Y$, in terms of the operator $J$, is $$dJ(\beta) = dJ(dX) = 0.$$ Then the $Y^2$-coordinate gives us the equation $\frac{x f'}{\sqrt{-1 + (f')^2}} = k$ or more specifically $$
(k^2 -x^2)(f')^2 = k^2 \ \ \ {\rm with} \ k \in \mathbb R - \{0\}, \ \ |x| < |k|. $$ Now integrating, one obtains $$ f(x) = b + (\pm k) arcsin(x/k). $$ Assuming $k>0$ and $b=0$, we get the parametric surface $$X(x,y) = (x \cosh y, x \sinh y, k. arcsin(x/k), 0).$$ If we take $x = k \sin u$ and $y= v$, we get the correspondent minimal parametric surface with isothermic parameters given by $$ f(u,v) = k(\sin u \cosh v, \sin u \sinh v, u, 0), $$ where $E(u,v) = k^2 \sin^2 u = G(u,v)$ and with lightlike singularities for $f_u(u,v)$ when $u = n \pi$, $n \in \mathbb Z$.
In similar way we get the surface given by $$ g(u,v) = k (\cos u \cosh v, \cos u \sinh v, v, 0), $$ which is a minimal ruled surface with the same metric tensor, it is a type of hyperbolic helicoid surface of $\mathbb R^4_1$. \end{example}
\subsection{Generalized Cauchy-Riemann equations over $(M,X)$} In this subsection we continue studying the local geometry of the spacelike surfaces in $\mathbb R^4_1$. In particular in this first part, we identify the generalized Cauchy-Riemann type equations over the surface $(M,X)$ when the parameters are not isothermic, and then we obtain the needed conditions to extend in continua way any local solution of those equations.
For starting, we observe that if we have a sub-surface $f(U) \subset X(M)$ with isothermic parameters $w = (u,v) \in U$ such that $X(x,y) = f(u(x,y),v(x,y))$, then $$\frac{\partial X}{\partial x} = \frac{\partial u}{\partial x} \; f_{u} + \frac{\partial v}{\partial x} \; f_{v} \ \ \ \; \; \mathrm{ and } \; \; \ \ \ \frac{\partial X}{\partial y} = \frac{\partial u}{\partial y} \; f_{u} + \frac{\partial v}{\partial y} \; f_{v}.$$
\begin{lemma} For each local solution of the equations \begin{equation}\label{222} \frac{\partial w}{\partial y} = \alpha(x,y) \; \frac{\partial w}{\partial x} \; \mbox{ where } \; \alpha(x,y) = \frac{F(x,y) + i \sqrt{E(x,y) G(x,y) - F^{2}(x,y)}}{E(x,y)}, \end{equation}
in a neighborhood $U \subset M$ of a point $p \in M$, there exists a parametric isothermic sub-surface $(U,f)$ of $(M,X)$ such that
$X(x,y) = f(u(x,y),v(x,y))$. Moreover, $\alpha \overline{\alpha} = \frac{G}{E}$. \end{lemma} \begin{proof} Let $W = \sqrt{EG - F^{2}}$ be the area function in coordinates $z = x + i y \in U$. Taking the operator $J$, since
$J(f_{u}) = f_{v}, \ J(f_v) = - f_u$, it follows $$J(X_{x}) = u_x J(f_u) + v_x J(f_v) = u_x f_v - v_x f_u. $$ Hence by equation (\ref{912}) one gets $$u_x f_v - v_x f_u = \frac{E}{W}(u_{y} f_{u} + v_{y} f_{v}) - \frac{F}{W}(u_{x} f_{u} + v_{x} f_{v}).$$
From this last equation, we obtain the following equations with matrix representation \begin{equation}\label{36} \left[\begin{matrix} u_{y} \\ v_{y} \end{matrix}\right] = \left[\begin{matrix} F/E & -W/E \\ W/E & F/E \end{matrix}\right] \; \left[\begin{matrix} u_{x} \\ v_{x} \end{matrix}\right] \; \; \mbox{ and } \; \; \left[\begin{matrix} u_{x} \\ v_{x} \end{matrix}\right] = \frac{E}{G} \left[\begin{matrix} F/E & W/E \\ -W/E & F/E \end{matrix}\right] \; \left[\begin{matrix} u_{y} \\ v_{y} \end{matrix}\right]. \end{equation} Now we observe that the square matrices of order $2\times 2$ of these equations are the matrix representation of a complex number. Therefore we can write $$u_{y} + i v_{y} = \frac{F + i W}{E} \; (u_{x} + i v_{x}),$$ that is equation (\ref{222}). \end{proof}
We note that if the $(x,y)$-coordinates are already isothermic coordinates then $\alpha = i$ and so equations (\ref{222}) for $(M, X)$ becomes to the classic expression of the Cauchy Riemann equations, namely, $u_y = - v_x$ and $u_x = v_y$. So we will call equations (\ref{36}) or (\ref{222}) as the generalized Cauchy-Riemann equations for $(M, X)$.
Then as expected we have the following definition-corollary.
\begin{corol} A smooth function $h = \varphi + i \psi : S \longrightarrow \mathbb{C}$ is generalized holomorphic over the Riemann surface $S = X(M)$ if and only if \begin{equation}\label{certa} \left[\begin{matrix} \varphi_{y} \\ \psi_{y} \end{matrix}\right] = \left[\begin{matrix} F/E & -W/E \\ W/E & F/E \end{matrix}\right] \; \left[\begin{matrix} \varphi_{x} \\ \psi_{x} \end{matrix}\right] \; \; \; {\rm or } \; \; \; \frac{\partial h}{\partial y} = \alpha \frac{\partial h}{\partial x}, \end{equation} where $W = \sqrt{EG - F^{2}}$. \end{corol} \begin{proof} Since in an isothermic neighborhood $(U,\tilde h)$ the function $\tilde{h}(u,v)$ is holomorphic in the sense of complex variable if and only if $h(x,y)$ is holomorphic over $S$, we have $$\frac{\partial h}{\partial x} = \frac{\partial \tilde{h}}{\partial u} (u_{x} + i v_{x}) \; \; \mbox{ and } \; \; \frac{\partial h}{\partial y} = \frac{\partial \tilde{h}}{\partial u} (u_{y} + i v_{y}),$$ because $i \tilde{h}_{u} = \tilde{h}_{v}$ holds for $\mathbb{C}$-holomorphic functions. Therefore $h_{y} = \alpha h_{x}$ follows from the definition of the function $\alpha(x,y)$. The conversely is immediate. \end{proof}
We note that for smooth function $h = \varphi + i \psi : U \subset S \longrightarrow \mathbb{C}$ is a generalized holomorphic if and only if in isothermic coordinates $(u,v)$ the harmonic functions $\varphi, \psi$ are conjugated harmonic functions satisfying the usual Cauchy-Riemann equations.
If we use the operator $J$, we can also give an equivalent definition, namely: $h$ is a generalized holomorphic function if and only if $$ dJ(\varphi(x,y)) = d \psi(x,y) \ \ \ \ \ \ \ \ {\rm and} \ \ \ \ \ \ dJ(\psi(x,y)) =- d \varphi(x,y). $$ They are generalized harmonic functions conjugated each other.
Next we are interested in relating the isothermic neighborhood $(U, f)$ with the Weierstrass datas $a(w)$ and $b(w)$ for graphic spacelike surfaces in $\mathbb R^4_1$.
In fact, fixing the semi-rigid referential associated to $(M,X)$ given by $$\M_{0} = \{l_{0}(b(p)), e_{1}(p),e_{2}(p),l_{3}(a(p))\}$$ where $$e_{1}(p) = \frac{1}{\sqrt{E}} \frac{\partial X}{\partial x} \; \; \mbox{ and } \; \; e_{2} = J(e_{1}) = \frac{1}{\sqrt{E}} J(X_{x}),$$ we obtain the next result.
\begin{prop}\label{32} Let $S = (M,X)$ be a solution of the minimal graphic equation in ${\real}_{1}^{4}$ and $(U,f)$ be a given locally isothermic sub-surface of $S$. Let $r(u,v)$ be a real-valued function and $\M(\vartheta) = \{l_{0}(b), e_{1},e_{2},l_{3}(a)\}_{(u,v)}$ be the semi-rigid referential associated to $f_w(w) = \mu(w) W(a(w),b(w)) = r(w)(\hat{e}_{1}(w) - i \hat{e}_{2}(w))$. Then the following relation is hold: $$\hat{e}_{1}(w) - i \hat{e}_{2}(w) = (\cos \vartheta \; e_{1} + \sin \vartheta \; e_{2}) - i(-\sin \vartheta \; e_{1} + \cos \vartheta \; e_{2})) = e^{i\vartheta}(e_{1} - i e_{2}).$$ \end{prop}
From Proposition \ref{32} it follows that if the coordinates $(M,X)$, $(U,f)$ and $(\tilde{U},\tilde{f})$ around a point $p \in f(U) \cap \tilde{f}(\tilde{U})$ are related by the equations $$X(x,y) = f(u(x,y),v(x,y)) = f \circ \Phi(x,y), \; \; X(x,y) = \tilde{f}(\tilde{u}(x,y),\tilde{v}(x,y)) = \tilde{f} \circ \tilde{\Phi}(x,y),$$ then the transition functions are given by \begin{equation} f \circ \Phi(x,y) = \tilde{f} \circ \tilde{\Phi} \; \; \mbox{ therefore } \; \; \Psi = \tilde{\Phi} \circ \Phi^{-1} = \tilde{f}^{-1} \circ f. \end{equation}
Now, applying the Proposition \ref{32}, we obtain: $$\frac{1}{\hat{r}} f_{w} = e^{i \hat{\phi}}(\hat{e}_{1} - i \hat{e}_{2}) \; \; \mbox{ with } \; \; \frac{1}{\tilde{r}} \tilde{f}_{\tilde{w}} = e^{i \tilde{\phi}}(\tilde{e}_{1} - i \tilde{e}_{2}),$$ which imply that the angle functions are related each other by the equation: \begin{equation} \label{34} \hat{\phi}(u,v) - \tilde{\phi} \circ \Psi(u,v) = \hat{\vartheta}(u,v) - \tilde{\vartheta} \circ \Psi(u,v). \end{equation}
Now we have the following facts, which come from equation (\ref{34}).
(1) {\em If two holomorphic functions agree with each other along a Jordan arc, then they agree with each other along all connected component of this arc}.
From (1) we obtain.
(2) {\em If $(U,f)$ and $(\tilde{U},\tilde{f})$ agree with each other along an Jordan arc in $S$, they agree with each other along the open subset $f(U) \cap \tilde{f}(\tilde{U})$}.
(3) {\em The overlapping or transition map between two isothermic coordinates system for a spacelike surface of ${\real}_{1}^{4}$ are holomorphic function in sense of complex analysis}.
(4) {\em Each holomorphic function $h : U \subset \mathbb{C} \longrightarrow V \subset \mathbb{C}$ can be seen as a pointwise $\mathbb{C}$-linear transformation $dh_{z_{0}} : T_{z_{0}} \mathbb{C} \longrightarrow T_{h(z_{0})} \mathbb{C}$ that preserves oriented angles}.
\begin{lemma}\label{35} The angle function $\tilde{\vartheta} - \vartheta$ determines the transition map of $(U,f)$ and $(\tilde{U},\tilde{f})$ for two isothermic parametrizations of the neighborhood $f(U) \cap \tilde{f}(\tilde{U}) \subset S$, around $p \in S$. \end{lemma}
From Lemma \ref{35} it follows the next result about the extension of local solutions of equation (\ref{222}).
\begin{prop}\label{1020} Let $w, \tilde w$ two local solutions of equation (\ref{222}), around a point $p \in S$, with $w_{y} = \alpha w_{x}$ and $\tilde{w}_{y} = \alpha \tilde{w}_{x}$. If $w_{x} = \tilde{w}_{x}$ then $w_{y} = \tilde{w}_{y}$.
Therefore, all local solution of the equation (\ref{222}) can be continuously extended whenever $E(x,y) > 0$ and $\sqrt{EG -
F^{2}}(x,y) > 0$. \end{prop} \begin{proof} The conclusions are immediate. \end{proof}
Next we prove that the solutions $w = (u,v)$ of the generalized Cauchy-Riemann equations (\ref{36}) or (\ref{222}), are of the form of Nitsche type functions (equation (8), page 23 of \cite{7}).
\begin{theor}\label{1010} The solution for equations (\ref{36}) are given by Nitsche type functions, that means \begin{equation} \begin{matrix}\label{390} u = u(x,y) = x + \int_{z_{0}}^{z} \frac{E dx + F dy}{W} \\ v = v(x,y) = y + \int_{z_{0}}^{z} \frac{F dx + G dy}{W}. \end{matrix} \end{equation} Moreover, from equations (\ref{390}), it is possible to obtain global isothermic coordinates $(U,f)$ for the surface $S = X(M)$. \end{theor} \begin{proof} In fact, since $$\frac{\partial u}{\partial x} = \frac{W + E}{W}, \; \; \; \; \frac{\partial u}{\partial y} = \frac{F}{W}, \; \; \; \; \frac{\partial v}{\partial x} = \frac{F}{W}, \; \; \; \; \frac{\partial v}{\partial y} = \frac{W + G}{W}$$ the matrix equation (\ref{36}) is satisfied. In fact, remembering that $W^{2} + F^{2} = EG$, we obtain $$\left[\begin{matrix} F/W \\ (W + G)/W \end{matrix} \right] = \left[\begin{matrix} F/E & -W/E \\ W/E & F/E \end{matrix}\right] \; \left[\begin{matrix} (E + W)/W \\ F/W \end{matrix} \right].$$ Now, since the solution for equations (\ref{36}) are in the form (\ref{390}), we obtain that the local isothermic parameters $(u,v)$ can be extended globally for the surface $S$ since the conditions of Proposition \ref{1020} are hold. \end{proof}
We highlight in this moment that our generalized Cauchy-Riemann equations (\ref{36}) and its solutions in (\ref{390}) can be applicated when we want to construct the conjugate minimal spacelike surface $(M,Y)$ (\ref{7}), since the solutions
(\ref{390}) involve terms of the local parametrization of $(M,Y)$.
Finally we have the following corollary for equations of minimal graphic surfaces in $\mathbb R^4_1$. \begin{corol}\label{9080} If $S = ({\mathbb R}^{2},X)$ is a solution of the minimal graphic equation (\ref{10}) then, for all $p \in S$, the functions $a(w)$ and $b(w)$ satisfy either $b(p) = c a(p)$ with $c \notin \{-1,1\}$ or $a(p) b(p) = c$ with $\Im(c) \neq 0$ for some constant $c \in \mathbb{C}.$
The Bernstein Theorem and the Calabi Theorem follows from that $c \neq 1$ and $c \neq -1$ and for the second type of surfaces from $\Im(c) \neq 0$.
Finally, if as a submanifold of the topological vector space ${\mathbb R}^{4}$ there exists $S = ({\mathbb R}^{2}, X)$ such that with the induced metric of ${\real}_{1}^{4}$, is a spacelike graphic solution in connected and simply connected open subset $M \subset \mathbb{C}$, with the condition that in some point $p \in S$ the following statement fails:
\lq \lq {\em either $b(p) = c a(p)$ with $c \notin \{-1,1\}$ \ or \ $a(p) b(p) = c$ \ with \ $\Im(c) \neq 0$ and for some constant $c \in \mathbb{C}$}",
then the points $X(x,y)$ where $E G - F^{2} = 0$, are points such that the tangent planes of $X({\mathbb R}^{2})$ are tangent to the lightcone of ${\real}_{1}^{4}$. \end{corol}
So from the second part of Corollary \ref{9080}, we have found conditions to create graphic minimal spacelike surfaces which have new type of singularities, it called lightlike singularities, as defined by Kobayashi in \cite{4}. Those singularities are points where the tangent plane of the surface is also tangent to the lightcone of ${\real}_{1}^{4}$.
\section{A Particular Family of Minimal Surfaces of ${\real}_{1}^{4}$} In this section we construct examples of minimal spacelike surfaces in ${\real}_{1}^{4}$ which are very close related to surfaces in ${\mathbb{E}}^{3}$ and $\mathbb{L}^{3}$.
For the representation $f_{w} = \mu(a + b, 1 + a b, i(1 - a b),a - b)$ with $\mu, a, b$ holomorphic functions from $M$ into $\mathbb{C}$, with $M$ being connected and simply connected open subset of the complex plane, we assume the relation \ $b = a e^{i\theta}$ for a parameter $\theta \in {\mathbb R}$. \begin{dfn} A $\theta$-family is a set of minimal surfaces defined on a connected and simply connected domain $M \subset \mathbb{C}$, linking each other by a parameter $\theta \in {\mathbb R}$, given by the following equation \begin{equation} F(\theta;w) = P_{0} + 2 \Re \int_{w_{0}}^{w} \mu(\xi)\left((1 + e^{i\theta})a(\xi), 1 + e^{i\theta} a^{2}(\xi), i(1 - e^{i\theta} a^{2}(\xi)), (1 - e^{i\theta})a(\xi)\right) d \xi. \end{equation}
When $\theta = 0$ we say that the surface of $\mathbb{L}^{3}$, given by $X(w) = F(0;w)$, is the initial surface of the family, and when $\theta = \pi$ we say that the surface of ${\mathbb{E}}^{3}$, given by $Y(w) = F(\pi;w)$, is the associated surface of the initial surface of the family. \end{dfn}
\begin{lemma} For a $\theta$-family $(M,F(\theta;w))$ of minimal spacelike isothermic parametric surfaces in ${\real}_{1}^{4}$ the equations that related the initial surface $(M,X)$ and the associated surface $(M,Y)$, are given by: \begin{equation}\label{141} \frac{\partial Y^{3}}{\partial w} = \frac{\partial X^{0}}{\partial w}, \; \; \; \; \frac{\partial Y^{1}}{\partial w} = -i \frac{\partial X^{2}}{\partial w} \; \; \; {\rm and} \ \ \ \ \frac{\partial Y^{2}}{\partial w} = i \frac{\partial X^{1}}{\partial w}. \end{equation} \end{lemma} \begin{proof} The equations of lemma follows from $X_{w} = \mu(2a, 1 + a^{2}, i(1 - a^{2}), 0)$ and $Y_{w} = \mu (0, 1 - a^{2},i(1 + a^{2}), 2a)$. \end{proof}
Now we construct an example for these equations:
\begin{example}\label{12} Let $(M,X)$ be the minimal spacelike surface of $\mathbb{L}^{3}$ given, in isothermic parameters, by $$X(u,v) = (u, \sinh u \cos v, \sinh u \sin v,0).$$
Since $X_{u} = (1, \cosh u \cos v, \cosh u \sin v,0)$ and $X_{v} = (0,-\sinh u \sin v, \sinh u \cos v,0)$ we obtain $\lambda^{2}(X) = \sinh^{2} u$. We assume that $(u,v) \in M$ for $u > 0$.
Therefore, it follows $X_{w} = \frac{1}{2}(1, \cosh w, -i \sinh w,0).$ To obtain the associated surface we find the functions $a(w)$ and $\mu(w)$. In fact, since $$2 \mu a = \frac{1}{2}, \; \; \; \; \mu (1 + a^{2}) = \frac{\cosh w}{2}, \; \; \; \; i \mu (1 - a^{2}) = -i \frac{\sinh w}{2},$$ it follows that $4 \mu(w) = e^{- w}$ and $a(w) = e^{w}$.
For obtaining the associated surface $(M,Y)$, we use $Y_{w} = \mu (0, 1 - a^{2},i(1 + a^{2}), 2a)$, and so the surface is such $Y_{w} = \frac{1}{2}(0, - \sinh w, i \cosh w, 1)$. Hence the holomorphic integral curve is given by $$\tilde{Y}(w) = \frac{1}{2} (0, -\cosh w, i\sinh w ,w).$$
Thus, the real part of $\tilde Y$ gives us a Catenoid of ${\mathbb{E}}^{3}$ parametrized by $$Y(u,v) = (0, -\cosh u \cos v, - \cosh u \sin v, u) \; \; \mbox{ com } \; \; \lambda^{2}(Y) = \cosh^{2} u.$$
Now, we look for the representation of those two associated surfaces as graphics of first type. In fact, for $(M,X)$ and the representation $P(x,y) = (A(x,y), x, y, 0)$: It takes $$x = \sinh u \cos v \ \ \ \ \ and \ \ \ \ y = \sinh u \sin v.$$ Therefore $\sinh u = \sqrt{x^{2} + y^{2}}$. For $(M,X)$, we obtain that function $A$ in the graphic representation is given by $$A(x,y) = \ln(\sqrt{x^{2} + y^{2}} + \sqrt{x^{2} + y^{2} + 1}).$$
For $(M,Y)$ and the representation $Q(p,q) = (0, p, q, B(p,q))$: It takes $$p = -\cosh u \cos v \ \ \ \ \ and \ \ \ \ q = -\cosh u \sin v.$$ So, $\cosh u = \sqrt{p^{2} + q^{2}}$. For $(M,Y)$ we obtain the function $B$ as given by $$B(p,q) = \ln(\sqrt{p^{2} + q^{2}} + \sqrt{p^{2} + q^{2} - 1}).$$ \end{example}
Example \ref{12}, and equations linking the initial surface $(M,X)$ and its associated surface $(M,Y)$ in the $\theta$-family, suggest the following result.
\begin{lemma}\label{151} For the associated surfaces of the $\theta$-family given by $$X(w) = (A(x(w),y(w)), x(w), y(w), 0) \; \; \mbox{ and } \; \; Y(w) = (0, p(w), q(w), B(p(w),q(w)),$$ the Jacobian functions of the transformation of coordinates, are related by $$\frac{\partial (x,y)}{\partial (u,v)} = \frac{\partial (p,q)}{\partial (u,v)}.$$ \end{lemma} \begin{proof} From equations of associated surfaces (\ref{141}) it follows that $p_{w} = -i y_{w}$ and $q_{w} = i x_{w}$. Then $p_{w} q_{\overline{w}} - p_{\overline{w}} q_{w} = x_{w} y_{\overline{w}} - x_{\overline{w}} y_{w},$ which implies the relation $\frac{i}{2} [p_u q_v - p_v q_u] = \frac{i}{2} [y_v x_u - y_u x_v].$ \end{proof}
Finally, from Lemma \ref{151} and from our version of the Nitsche equations for transformation of coordinates (\ref{390}), we obtain the following result.
\begin{theor} The $\theta$-family transports minimal first type graphic solutions $P(x,y) = (A(x,y), x, y, 0)$ to minimal associated graphic solutions $Q(p,q) = (0, p, q, B(p,q))$ preserving the domain $dom (A) = dom (B) = M$.
If $M = \mathbb{C}$ then $P(\mathbb{C})$ and $Q(\mathbb{C})$ are spacelike planes of ${\real}_{1}^{4}$.
We can say that \lq \lq the Bernstein theorem holds if and only if the Calabi theorem holds". \end{theor}
{\bf Acknowledgments} \ The first author's research was supported by Projeto Tem\'atico Fapesp n. 2016/23746-6. S\~ao Paulo. Brazil. This paper is part of the Ph.D. thesis of R.S. Santos \cite{6}, which was presented in Universidade de S\~ao Paulo, Brazil, in February 2021.
\end{document} | arXiv |
Chevalley restriction theorem
In the mathematical theory of Lie groups, the Chevalley restriction theorem describes functions on a Lie algebra which are invariant under the action of a Lie group in terms of functions on a Cartan subalgebra.
Statement
Chevalley's theorem requires the following notation:
assumptionexample
Gcomplex connected semisimple Lie groupSLn, the special linear group
${\mathfrak {g}}$the Lie algebra of G${\mathfrak {sl}}_{n}$, the Lie algebra of matrices with trace zero
$\mathbb {C} [{\mathfrak {g}}]^{G}$the polynomial functions on ${\mathfrak {g}}$ which are invariant under the adjoint G-action
${\mathfrak {h}}$a Cartan subalgebra of ${\mathfrak {g}}$the subalgebra of diagonal matrices with trace 0
Wthe Weyl group of Gthe symmetric group Sn
$\mathbb {C} [{\mathfrak {h}}]^{W}$the polynomial functions on ${\mathfrak {h}}$ which are invariant under the natural action of Wpolynomials f on the space $\{x_{1},\dots ,x_{n},\sum x_{i}=0\}$ which are invariant under all permutations of the xi
Chevalley's theorem asserts that the restriction of polynomial functions induces an isomorphism
$\mathbb {C} [{\mathfrak {g}}]^{G}\cong \mathbb {C} [{\mathfrak {h}}]^{W}$.
Proofs
Humphreys (1980) gives a proof using properties of representations of highest weight. Chriss & Ginzburg (2010) give a proof of Chevalley's theorem exploiting the geometric properties of the map ${\widetilde {\mathfrak {g}}}:=G\times _{B}{\mathfrak {b}}\to {\mathfrak {g}}$.
References
• Chriss, Neil; Ginzburg, Victor (2010), Representation theory and complex geometry., Birkhäuser, doi:10.1007/978-0-8176-4938-8, ISBN 978-0-8176-4937-1, S2CID 14890248, Zbl 1185.22001
• Humphreys, James E. (1980), Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer, Zbl 0447.17002
| Wikipedia |
What's the difference between a realization, a representation and an implementation in metrology?
In a recent answer, a metrologist casually used the terms 'realization' and 'implementation' of an SI unit as if they were different, which looks very strange to the untrained eye. Some further digging (example) also throws up uses of 'representation' of an SI unit as a technical term with its own distinct meaning.
What is the precise meaning of these terms in metrology, and what are the precise differences between them? What are good examples of currently implemented realizations and representations vs implementations?
terminology units si-units metrology
Emilio PisantyEmilio Pisanty
The reference document for metrological terms is the International Vocabulary of Metrology (VIM). Definitions there are carefully crafted, but frequently they might seem a bit obscure to non metrologists and further remarks might be needed.
For what concerns realization and reproduction (representation is also found in the literature for reproduction), their meaning is found under the term measurement standard:
Realization of the definition of a given quantity, with stated quantity value and associated measurement uncertainty, used as a reference.
In particular, the related notes 1 and 3 say:
NOTE 1 A "realization of the definition of a given quantity" can be provided by a measuring system, a material measure, or a reference material.
NOTE 3 The term "realization" is used here in the most general meaning. It denotes three procedures of "realization". The first one consists in the physical realization of the measurement unit from its definition and is realization sensu stricto. The second, termed "reproduction", consists not in realizing the measurement unit from its definition but in setting up a highly reproducible measurement standard based on a physical phenomenon, as it happens, e.g. in case of use of frequency-stabilized lasers to establish a measurement standard for the metre, of the Josephson effect for the volt or of the quantum Hall effect for the ohm. The third procedure consists in adopting a material measure as a measurement standard. It occurs in the case of the measurement standard of 1 kg.
Therefore, the terms realization and reproduction denote an object or an experiment with specific properties.
To illustrate the difference between a strict realization and a reproduction, let's take the example of a specific quantity, the unit ohm (note that a unit is a quantity, albeit a specially chosen one).
First, we have to define what this quantity is: this can be done in words, possibly with the help of mathematical relationships involving other quantities, and by adding specifications on influence quantities.
The ohm in the SI is defined as follows [CIPM, 1946: Resolution 2]:
The ohm is the electric resistance between two points of a conductor when a constant potential difference of 1 volt, applied to these points, produces in the conductor a current of 1 ampere, the conductor not being the seat of any electromotive force.
So far, so good, or, at least, it seems. Actually we are a bit stuck because we can realize the ampere and the volt respectively through current and voltage balances, but the reproducibility of the ohm realized in this way would be low, roughly at the $10^{-6}$ level. And the procedure would be rather complex. We are saved in 1956 by Thompson and Lampard who discovered a new theorem in electrostatics [1], which is the electrostatic dual of the van der Pauw theorem [2,3]. This theorem essentially says that you can build a standard of capacitance (that is, realize the farad or one of its submultiples), whose capacitance can be accurately calculated (something you cannot do with a parallel plate capacitor, for instance). If we have a standard of capacitance, through the relationships $Y = \mathrm{j}\omega C$ and $Z = 1/Y$, we have the standards of admittance and impedance, that is, we have the siemens and the ohm, however in the AC regime.
Thus, the strict SI realization of the ohm, as standard of resistance, is roughly the following:
You build a calculable capacitor (and ten years of your life are gone). Typically a 1 m long calculable capacitor has a capacitance of around 1 pF, which at kHz frequency correspond to a quite high impedance (for a short bibliography on the calculable capacitor, see this page).
By means of impedance bridges, you scale the capacitance to higher values (e.g., 1 nF).
By means of a quadrature impedance bridge, you compare the impedance value of a standard resistor with calculable AC-DC behaviour to that of the scaled capacitance.
You calculate the DC value of the resistance.
You scale down the resistance to 1 ohm by means of a resistance bridge.
Once you have all the experiments working (after many years), the realization of the ohm through the above chain of experiments can take more than one month, but the most important issue is that the reproducibility of the ohm realized in this way, though better than that obtainable through the realizations of the volt and the ampere, is just at the $10^{-7}$-$10^{-8}$ level.
Then it arrives the quantum Hall effect (QHE). A QHE element under conditions of low temperature and high magnetic field, realizes a four terminal resistance (or transresistance) with resistance value $R_\mathrm{H} = R_\mathrm{K}/i$, where $R_\mathrm{K}$ is a constant, the von Klitzing constant, and $i$ is an integer, called plateau index (typically we use the plateau corresponding to $i=2$). By the end of the 1980s it was clear that QHE elements could provide resistance standards with much better reproducibility than the other methods described above: at the time, of the order of $10^{-8}$-$10^{-9}$; nowadays, of the order of $10^{-10}$-$10^{-11}$ (two-three order of magnitudes better than that obtainable with a calculable capacitor). It turns out, also, that the von Klitzing constant is linked to two fundamental constants, the Planck constant and the elementary charge, $R_\mathrm{K} = h/e^2$.
The situation in the late 1980s is thus the following:
A QHE experiment is much easier to implement than that of a calculable capacitor (and much less expensive).
The resistance realized by a QHE experiment has a much better reproducibility than that realizable by a calculable capacitor experiment.
The accuracy of the von Klitzing constant, however, is only at the level of the SI ohm realization, that is, of about $10^{-7}$, and the relationship $R_\mathrm{H} = R_\mathrm{K}/i = h/(e^2 i)$ has not yet enough sound theoretical foundation to be exploited.
The first two points suggest the adoption of a conventional unit of resistance, by defining a conventional value of the von Klitzing constant [CIPM, 1988: Recommendation 2]. This conventional value of the von Klitzing constant is denoted by $R_{\mathrm{K}-90}$ (because it was adopted in 1990) and has value
$$R_\mathrm{K-90} = 25\,812.807\,\Omega\quad \text{(exact)}.$$
The conventional unit of resistance is the $\mathit{\Omega}_{90}$, defined as 1
$$\mathit{\Omega}_{90} = \frac{R_\mathrm{K}}{\{R_\mathrm{K-90}\}} = \frac{R_\mathrm{K}}{25\,812.807}.$$
At present, virtually all national resistance scales are traceable to this conventional unit.
It is now worth pointing out that the quantity $\mathit{\Omega}_{90}$ has no links to the SI ohm: it's close to (the relative discrepancy is of the order of $10^{-8}$), but quite not the same thing. Thus, the $\mathit{\Omega}_{90}$ is called a reproduction (or a representation) of the ohm, because it realizes somehow the ohm, but not according to its definition.
At present, this is not the only reproduced unit: the volt is currently reproduced by means of the Josephson effect through a conventional value of the Josephson constant, and the thermodynamic temperature scale is reproduced through two conventional temperature scales, the International Temperature Scale of 1990 (ITS-90) and the Provisional Low Temperature Scale of 2000 (PLTS-2000).
Instead, with the forthcoming revision of the International System of Units, the so called "new SI", the quantum Hall effect and the Josephson effect will really provide SI realizations of the ohm and the volt (see this draft of the mise en pratique of the electrical units).
Finally, for what concerns the term implementation, as far as I know, it has no specific technical meaning within the community of metrologists, and it is used in the common English meaning (whereas realization has a somehow different connotation). Thus, for instance, we can speak of two different implementations of a quantum Hall resistance experiment (because some details might be different).
1 A note on notation: the quantity $\mathit{\Omega}_{90}$ is typeset in italics because it's not an SI unit; braces denote the numerical value of a quantity, according to the notation $Q = \{Q\}[Q]$ [4,5, and this question].
[1] A. M. Thompson and D. G. Lampard (1956), "A New Theorem in Electrostatics and its Application to Calculable Standards of Capacitance", Nature, 177, 888.
[2] L.J. van der Pauw (1958), "A method of measuring specific resistivity and Hall effect of discs of arbitrary shape", Philips Research Reports, 13, 1–9.
[3] L.J. van der Pauw (1958), "A method of measuring the resistivity and Hall coefficient on lamellae of arbitrary shape", Philips Technical Review, 20, 220–224.
[4] E. R. Cohen et al. (2008), Quantities, Units and Symbols in Physical Chemistry, IUPAC Green Book, 3rd Edition, 2nd Printing, IUPAC & RSC Publishing, Cambridge [Online]
[5] E R Cohen and P. Giacomo (1987), Symbols, Units, Nomenclature and Fundamental Constants in Physics, IUPAP SUNAMCO Red Book, 1987 revision, IUPAP & SUNAMCO, Netherlands [Online]
Massimo OrtolanoMassimo Ortolano
$\begingroup$ You know, I was just about to start a bounty on this one, but I realized that it's likely that you're the only one here who can answer this, so I was wondering whether to start it right away or to tell flippiefanus to bug you, to see whether you were back ;-). So, here goes, but I still want to know about that QHE stuff. $\endgroup$ – Emilio Pisanty Aug 25 '16 at 13:06
$\begingroup$ I also want to quiz you on the quantum metrological triangle, but it'll take a bit of thinking to phrase a good question. $\endgroup$ – Emilio Pisanty Aug 25 '16 at 13:08
$\begingroup$ @EmilioPisanty I went back just yesterday ;-) QHE will arrive (historically it's not the first reproduction of the ohm, because of course we somehow had the ohm before the calculable capacitor, but I wanted to start from the strict realization). However, I'm a slow writer, I also want to complete the other answer: keep your other questions for later ;-) $\endgroup$ – Massimo Ortolano Aug 25 '16 at 13:11
$\begingroup$ @EmilioPisanty Added the QHE part: let me know if there is something unclear or some part that you would like to see expanded (in reference to the original question). $\endgroup$ – Massimo Ortolano Aug 26 '16 at 12:52
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Conformal Geometry and Dynamics
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The existence of quasimeromorphic mappings in dimension 3
by Emil Saucan PDF
Conform. Geom. Dyn. 10 (2006), 21-40 Request permission
We prove that a Kleinian group $G$ acting on $\mathbb {H}^{3}$ admits a non-constant $G$-automorphic function, even if it has torsion elements, provided that the orders of the elliptic elements are uniformly bounded. This is accomplished by developing a method for meshing distinct fat triangulations which is fatness preserving. We further show how to adapt the proof to higher dimensions.
[Ab]ab W. Abikoff, Kleinian Groups, lecture notes, The Technion—Israel Institute of Technology, Haifa, Israel, 1996–1997. [Al]al J.W. Alexander, Note on Riemmann spaces, Bull. Amer. Math. Soc. 26 (1920), 370–372.
B. N. Apanasov, Kleinian groups in space, Sibirsk. Mat. Ž. 16 (1975), no. 5, 891–898, 1129 (Russian). MR 0404474
Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995. Corrected reprint of the 1983 original. MR 1393195
Marcel Berger, Geometry. I, Universitext, Springer-Verlag, Berlin, 1987. Translated from the French by M. Cole and S. Levy. MR 882541, DOI 10.1007/978-3-540-93815-6
B. H. Bowditch and G. Mess, A $4$-dimensional Kleinian group, Trans. Amer. Math. Soc. 344 (1994), no. 1, 391–405. MR 1240944, DOI 10.1090/S0002-9947-1994-1240944-6
Robert Brooks and J. Peter Matelski, Collars in Kleinian groups, Duke Math. J. 49 (1982), no. 1, 163–182. MR 650375
Stewart S. Cairns, On the triangulation of regular loci, Ann. of Math. (2) 35 (1934), no. 3, 579–587. MR 1503181, DOI 10.2307/1968752
Stewart S. Cairns, Polyhedral approximations to regular loci, Ann. of Math. (2) 37 (1936), no. 2, 409–415. MR 1503287, DOI 10.2307/1968452
Stewart S. Cairns, A simple triangulation method for smooth manifolds, Bull. Amer. Math. Soc. 67 (1961), 389–390. MR 149491, DOI 10.1090/S0002-9904-1961-10631-9
Jeff Cheeger, Werner Müller, and Robert Schrader, On the curvature of piecewise flat spaces, Comm. Math. Phys. 92 (1984), no. 3, 405–454. MR 734226, DOI 10.1007/BF01210729
H. S. M. Coxeter, Regular polytopes, 2nd ed., The Macmillan Company, New York; Collier Macmillan Ltd., London, 1963. MR 0151873
D. A. Derevnin and A. D. Mednykh, Geometric properties of discrete groups acting with fixed points in a Lobachevskiĭ space, Dokl. Akad. Nauk SSSR 300 (1988), no. 1, 27–30 (Russian); English transl., Soviet Math. Dokl. 37 (1988), no. 3, 614–617. MR 948799
Mark Feighn and Geoffrey Mess, Conjugacy classes of finite subgroups of Kleinian groups, Amer. J. Math. 113 (1991), no. 1, 179–188. MR 1087807, DOI 10.2307/2374827
F. W. Gehring and G. J. Martin, Commutators, collars and the geometry of Möbius groups, J. Anal. Math. 63 (1994), 175–219. MR 1269219, DOI 10.1007/BF03008423
F. W. Gehring and G. J. Martin, On the Margulis constant for Kleinian groups. I, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 2, 439–462. MR 1404096
F. W. Gehring, C. Maclachlan, G. J. Martin, and A. W. Reid, Arithmeticity, discreteness and volume, Trans. Amer. Math. Soc. 349 (1997), no. 9, 3611–3643. MR 1433117, DOI 10.1090/S0002-9947-97-01989-2
Emily Hamilton, Geometrical finiteness for hyperbolic orbifolds, Topology 37 (1998), no. 3, 635–657. MR 1604903, DOI 10.1016/S0040-9383(97)00043-8
J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees. MR 0248844
Troels Jørgensen, On discrete groups of Möbius transformations, Amer. J. Math. 98 (1976), no. 3, 739–749. MR 427627, DOI 10.2307/2373814
M. È. Kapovich and L. D. Potyagaĭlo, On the absence of finiteness theorems of Ahlfors and Sullivan for Kleinian groups in higher dimensions, Sibirsk. Mat. Zh. 32 (1991), no. 2, 61–73, 212 (Russian); English transl., Siberian Math. J. 32 (1991), no. 2, 227–237. MR 1138441, DOI 10.1007/BF00972769
Michael Kapovich and Leonid Potyagailo, On the absence of Ahlfors' finiteness theorem for Kleinian groups in dimension three, Topology Appl. 40 (1991), no. 1, 83–91. MR 1114093, DOI 10.1016/0166-8641(91)90060-Y
[Med]med A.D. Mednikh, Automorphism groups of the three-dimensional hyperbolic manifolds, Soviet Math. Dokl. 32(3) (1985), 633–636. [Mor]mor J.W. Morgan, On Thurston's Uniformization Theorem for Three-Dimensional Manifolds, in The Smith Conjecture (Morgan, J.W. and Bass, H. ed.), Academic Press, NY, 1984, 37–126.
Olli Martio and Uri Srebro, Automorphic quasimeromorphic mappings in $R^{n}$, Acta Math. 135 (1975), no. 3-4, 221–247. MR 435388, DOI 10.1007/BF02392020
O. Martio and U. Srebro, On the existence of automorphic quasimeromorphic mappings in $\textbf {R}^{n}$, Ann. Acad. Sci. Fenn. Ser. A I Math. 3 (1977), no. 1, 123–130. MR 0585312, DOI 10.5186/aasfm.1977.0317
Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
Volker Mayer, Uniformly quasiregular mappings of Lattès type, Conform. Geom. Dyn. 1 (1997), 104–111. MR 1482944, DOI 10.1090/S1088-4173-97-00013-1
James R. Munkres, Elementary differential topology, Revised edition, Annals of Mathematics Studies, No. 54, Princeton University Press, Princeton, N.J., 1966. Lectures given at Massachusetts Institute of Technology, Fall, 1961. MR 0198479
Peter J. Nicholls and Peter L. Waterman, The boundary of convex fundamental domains of Fuchsian groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), no. 1, 11–25. MR 1050778, DOI 10.5186/aasfm.1990.15
Kirsi Peltonen, On the existence of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 85 (1992), 48. MR 1165363
John G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR 1299730, DOI 10.1007/978-1-4757-4013-4
[S1]s1 E. Saucan, The Existence of Quasimeromorphic Mappings, Ann. Acad. Sci. Fenn., Ser A I Math, 31, (2006), 131–142. [S2]s2 E. Saucan, Note on a theorem of Munkres, Mediterr. j. math 2(2) (2005), 215–229. [S3]s3 E. Saucan, in preparation.
D. M. Y. Sommerville, An introduction to the geometry of $n$ dimensions, Dover Publications, Inc., New York, 1958. MR 0100239
Michael Spivak, A comprehensive introduction to differential geometry. Vol. IV, Publish or Perish, Inc., Boston, Mass., 1975. MR 0394452
Uri Srebro, Non-existence of quasimeromorphic automorphic mappings, Analysis and topology, World Sci. Publ., River Edge, NJ, 1998, pp. 647–651. MR 1667838
[SA]sa E. Saucan and E. Apleboim, Quasiconformal Fold Elimination for Seaming and Tomography, in preparation.
William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR 1435975, DOI 10.1515/9781400865321
Pekka Tukia, Automorphic quasimeromorphic mappings for torsionless hyperbolic groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 545–560. MR 802519, DOI 10.5186/aasfm.1985.1061
Jussi Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009, DOI 10.1007/BFb0061216
J. H. C. Whitehead, On $C^1$-complexes, Ann. of Math. (2) 41 (1940), 809–824. MR 2545, DOI 10.2307/1968861
Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 30C65, 57R05, 57M60
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Emil Saucan
Affiliation: Departments of Mathematics and Electrical Engineering, Technion, Haifa, Israel
Email: [email protected], [email protected]
Received by editor(s): December 1, 2003
Published electronically: March 1, 2006
Dedicated: For Meir, who insisted
Journal: Conform. Geom. Dyn. 10 (2006), 21-40
MSC (2000): Primary 30C65, 57R05, 57M60 | CommonCrawl |
\begin{document}
\title{Fair Division through Information Withholding}
\begin{abstract} Envy-freeness up to one good (\EF{1}) is a well-studied fairness notion for indivisible goods that addresses pairwise envy by the removal of at most one good. In the worst case, each pair of agents might require the (hypothetical) removal of a different good, resulting in a weak \emph{aggregate} guarantee. We study allocations that are nearly envy-free in aggregate, and define a novel fairness notion based on \emph{information withholding}. Under this notion, an agent can withhold (or hide) some of the goods in its bundle and reveal the remaining goods to the other agents. We observe that in practice, envy-freeness can be achieved by withholding only a small number of goods overall. We show that finding allocations that withhold an optimal number of goods is computationally hard even for highly restricted classes of valuations. In contrast to the worst-case results, our experiments on synthetic and real-world preference data show that existing algorithms for finding \EF{1} allocations withhold close-to-optimal number of goods. \end{abstract}
\section{Introduction} \label{sec:Introduction}
When dividing discrete objects, one often strives for a fairness notion called \emph{envy-freeness} \citep{F67resource}, under which no agent prefers the allocation of another agent to its own. Envy-free outcomes might not exist in general (even with only two agents and a single indivisible good), motivating the need for approximations. Among the many approximations of envy-freeness proposed in the literature \citep{LMM+04approximately,B11combinatorial,NR14minimizing,CKM+16unreasonable}, the notion called \emph{envy-freeness up to one good} (\EF{1}) has received significant attention recently. \EF{1} requires that pairwise envy can be eliminated by the removal of some good in the envied bundle. It is known that an \EF{1} allocation always exists and can be computed in polynomial time~\citep{LMM+04approximately}.
On closer scrutiny, however, we find that \EF{1} is not as strong as one might think. Indeed, an \EF{1} allocation could entail the (hypothetical) removal of \emph{many} goods, because the elimination of envy for different pairs of agents may require the removal of distinct goods. To see this, consider an instance with six goods $g_1,\dots,g_6$ and three agents $a_1,a_2,a_3$ whose (additive) valuations are as shown below:
\begin{table}[ht]
\label{tab:Motivating_HEF}
\centering
\small
\begin{tabular}{ c|cccccc }
& $g_1$ & $g_2$ & $g_3$ & $g_4$ & $g_5$ & $g_6$\\ \hline
$a_1$ & $\circled{1}$ & $\circled{1}$ & $\underline{4}$ & $1$ & $1$ & $\underline{4}$\\
$a_2$ & $1$ & $\underline{4}$ & $\circled{1}$ & $\circled{1}$ & $\underline{4}$ & $1$\\
$a_3$ & $\underline{4}$ & $1$ & $1$ & $\underline{4}$ & $\circled{1}$ & $\circled{1}$\\
\end{tabular}
\end{table}
Observe that the allocation shown via circled goods is \EF{1}, since any pairwise envy can be addressed by removing an underlined good. However, each pair of agents requires the removal of a \emph{different} good (e.g., $a_1$'s envy towards $a_2$ is addressed by removing $g_3$ whereas $a_3$'s envy towards $a_2$ is addressed by removing $g_4$, and so on), resulting in a weak approximation overall (since all goods need to be removed over all pairs of agents).
The above example shows that \EF{1}, on its own, is too \emph{coarse} to distinguish between allocations that remove a \emph{large} number of goods (such as the one with circled entries) and those that remove only a \emph{few} (such as the one with underlined entries, which, in fact, is envy-free). This limitation highlights the need for a fairness notion that (a) can distinguish between allocations in terms of their \emph{aggregate} approximation, and (b) retains the ``up to one good'' style approximation of \EF{1} that has proven to be practically useful~\citep{GP15spliddit}. Our work aims to fill this important gap.
We propose a new fairness notion called \emph{envy-freeness up to $k$ hidden goods} (\HEF{k}) defined as follows: Say there are $n$ agents, $m$ goods, and an allocation $A = (A_1,\dots,A_n)$. Suppose there is a set $S$ of $k$ goods (called the \emph{hidden} set) such that each agent $i$ withholds the goods in $A_i \cap S$ (i.e., the hidden goods owned by $i$) and only discloses the goods in $A_i \setminus S$ to the other agents. Any other agent $h \neq i$ only observes the goods disclosed by $i$ (i.e., those in $A_i \setminus S$), and its valuation for $i$'s bundle is therefore $v_h(A_i \setminus S)$ instead of $v_h(A_i)$. Additionally, agent $h$'s valuation for its own bundle is $v_h(A_h)$ (and not $v_h(A_h \setminus S)$) because it can observe its own hidden goods. If, under the disclosed allocation, no agent prefers the bundle of any other agent (i.e., if $v_h(A_h) \geq v_h(A_i \setminus S)$ for every pair of agents $i,h$), then we say that $A$ is \emph{envy-free up to $k$ hidden goods} (\HEF{k}). In other words, by withholding the information about $S$, allocation $A$ can be made free of envy.
Notice how \HEF{k} addresses the limitations associated with \EF{1}: Like \EF{1}, \HEF{k} is a relaxation of envy-freeness that is defined in terms of the \emph{number of goods}. However, unlike \EF{1}, \HEF{k} offers a \emph{precise quantification} of the extent of information that must be withheld in order to achieve envy-freeness.
Clearly, any allocation can be made envy-free by hiding all the goods (i.e., if $k = m$). The real strength of \HEF{k} lies in $k$ being \emph{small}; indeed, an \HEF{0} allocation is envy-free. As we will demonstrate below, there are natural settings that admit \HEF{k} allocations with a small $k$ (i.e., hide only a small number of goods) even when (exact) envy-freeness is unlikely.
\subsection*{Information Withholding is Meaningful in \mbox{Practice}.}
To understand the usefulness of \HEF{k}, we generated a synthetic dataset where we varied the number of agents $n$ from $5$ to $10$, and the number of goods $m$ from $5$ to $20$ (we ignore the cases where $m < n$). For every fixed $n$ and $m$, we generated $100$ instances with \emph{binary} valuations. Specifically, for every agent $i$ and every good $j$, the valuation $v_{i,j}$ is drawn i.i.d. from $\textup{Bernoulli}(0.7)$. \Cref{subfig:HEFk_motivation_0.7_NOTEF} shows the heatmap of the number of instances out of $100$ that \emph{do not} admit envy-free outcomes. (Thus, a `hot' cell indicated by red color is one where \emph{none} of the $100$ instances admits an envy-free allocation.) \Cref{subfig:HEFk_motivation_0.7_maxk} shows the heatmap of the number of goods that must be hidden in the worst-case. That is, the color of each cell denotes the smallest $k$ such that each of the corresponding $100$ instances admits some \HEF{k} allocation.
\begin{figure}
\caption{Heatmap of the fraction of instances that are not envy-free.}
\label{subfig:HEFk_motivation_0.7_NOTEF}
\caption{Heatmap of the number of goods that must be hidden.}
\label{subfig:HEFk_motivation_0.7_maxk}
\caption{In both figures, each cell corresponds to $100$ instances with binary valuations for a fixed number of goods $m$ (on X-axis) and a fixed number of agents $n$ (on Y-axis).}
\label{fig:HEFk_motivation_0.7}
\end{figure}
It is evident from \Cref{fig:HEFk_motivation_0.7} that even in the regime where envy-free outcomes are unlikely (in particular, the red-colored cells in \Cref{subfig:HEFk_motivation_0.7_NOTEF}), there exist \HEF{k} allocations with $k \leq 3$ (the light blue-colored cells in \Cref{subfig:HEFk_motivation_0.7_maxk}). This observation, along with the foregoing discussion, suggests that fairness through information withholding is a well-motivated approach towards approximate envy-freeness that could provide promising results in practice.
\paragraph{Our Contributions} We make contributions on three fronts. \begin{itemize}
\item On the \emph{conceptual} side, we propose a novel fairness notion called envy-freeness up to $k$ hidden goods (\HEF{k}) as a fine-grained generalization of envy-freeness in terms of aggregate approximation.
\item Our \emph{theoretical} results (\Cref{sec:Theoretical_Results}) show that computing \HEF{k} allocations is computationally hard even for highly restricted classes of valuations (\Cref{thm:HEFk_Existence_NPcomplete_IdenticalVals,cor:HEFk_Existence_NPcomplete_BinaryVals}). We show a similar result when \HEF{k} is required alongside Pareto optimality (\Cref{prop:HEFkPOExistence_NPcomplete_BinaryVals}). A related technical contribution is an alternative proof of \textrm{\textup{NP-complete}}{}ness of determining the existence of an envy-free allocation for binary valuations (\Cref{prop:EF_Existence_NPcomplete_BinaryValuations}).
\item Our \emph{experiments} show that \HEF{k} allocations with a small $k$ often exist, even when (exact) envy-free allocations do not (\Cref{fig:HEFk_motivation_0.7}). We also compare several known algorithms for computing \EF{1} allocations on synthetic and real-world preference data, and find that the round-robin algorithm and an algorithm of \citet{BKV18Finding} withhold close-to-optimal number of goods, often hiding no more than three items (\Cref{sec:Experiments}). \end{itemize}
\section{Related Work} \label{sec:RelatedWork} An emerging line of work in the fair division literature considers relaxations of envy-freeness by limiting the information available to the agents. Notably, \citet{ABC+18knowledge} consider a setting where each agent is aware only of its own bundle and has no knowledge about the allocations of the other agents. They propose the notion of \emph{epistemic envy-freeness} (\textrm{\textup{EEF}}{}) under which each agent believes that an envy-free allocation of the remaining goods among the other agents is possible. Note that in \textrm{\textup{EEF}}{}, each agent might consider a different hypothetical assignment of the remaining goods, and each of these could be significantly different from the \emph{actual} underlying allocation. By contrast, under \HEF{k}, each agent evaluates its valuation with respect to the same (underlying) allocation. \citet{CS17ignorance} study a related model where agents have probabilistic beliefs about the allocations of the other agents, and envy is defined in expectation. \citet{CCL+19maximin} study a setting similar to \citet{ABC+18knowledge} wherein each agent is unaware of the allocations of the other agents, with the guarantee that it does not get the worst bundle.
Another related line of work considers settings where the agents constitute a social network and can only observe the allocations of their neighbors \citep{AKP17fair,BQZ17networked,CEM17distributed,ABC+18knowledge,BCG+18local,BKN18envy}. These works place an informational constraint on the \emph{set of agents}, whereas our model restricts the \emph{set of revealed goods} per agent.
Several other forms of fairness approximations have been proposed recently, such as introducing side payments~\citep{HS19fair}, permitting sharing of some goods~\citep{SE19fair}, or donating a small fraction of goods~\citep{CGH19envy,CKM+20little}.
\section{Preliminaries} \label{sec:Preliminaries} \paragraph{Problem instance} An \emph{instance} $\mathcal{I} = \langle [n], [m], \mathcal{V} \rangle$ of the fair division problem is defined by a set of $n \in {\mathbb{N}}$ \emph{agents} $[n] = \{1,2,\dots,n\}$, a set of $m \in {\mathbb{N}}$ \emph{goods} $[m] = \{1,2,\dots,m\}$, and a \emph{valuation profile} $\mathcal{V} = \{v_1,v_2,\dots,v_n\}$ that specifies the preferences of every agent $i \in [n]$ over each subset of the goods in $[m]$ via a \emph{valuation function} $v_i: 2^{[m]} \rightarrow {\mathbb{N}} \cup \{0\}$. Notice that each agent's valuation for any subset of goods is assumed to be a non-negative integer. We will assume that the valuation functions are \emph{additive}, i.e., for any $i \in [n]$ and $G \subseteq [m]$, $v_i(G) \coloneqq \sum_{j \in G} v_i(\{j\})$, where $v_i(\emptyset) = 0$. We will write $v_{i,j}$ instead of $v_i(\{j\})$ for a singleton good $j \in [m]$. We say that an instance has \emph{binary valuations} if for every $i \in [n]$ and every $j \in [m]$, $v_{i,j} \in \{0,1\}$.
\paragraph{Allocation} An \emph{allocation} $A \coloneqq (A_1,\dots,A_n)$ refers to an $n$-partition of the set of goods $[m]$, where $A_i \subseteq [m]$ is the \emph{bundle} allocated to agent $i$. Given an allocation $A$, the utility of agent $i \in [n]$ for the bundle $A_i$ is $v_i(A_i) = \sum_{j \in A_i} v_{i,j}$.
\begin{definition}[\textbf{Envy-freeness}]
An allocation $A$ is \emph{envy-free} (\EF{}) if for every pair of agents $i,h \in [n]$, $v_i(A_i) \geq v_i(A_h)$. An allocation $A$ is \emph{envy-free up to one good} (\EF{1}) if for every pair of agents $i,h \in [n]$ such that $A_h \neq \emptyset$, there exists some good $j \in A_h$ such that $v_i(A_i) \geq v_i(A_h \setminus \{j\})$. An allocation $A$ is \emph{strongly envy-free up to one good} (\sEF{1}) if for every agent $h \in [n]$ such that $A_h \neq \emptyset$, there exists a good $g_h \in A_h$ such that for all $i \in [n]$, $v_i(A_i) \geq v_i(A_h \setminus \{g_h\})$. The notions of \EF{}, \EF{1}, and \sEF{1} are due to \citet{F67resource}, \citet{B11combinatorial}, and \citet{CFS+19group}, respectively.\footnote{A slightly weaker notion than \EF{1} was previously studied by \citet{LMM+04approximately}. However, their algorithm can be shown to compute an \EF{1} allocation.} \end{definition}
\begin{definition}[\textbf{Envy-freeness with hidden goods}]
An allocation $A$ is said to be \emph{envy-free up to $k$ hidden goods} (\HEF{k}) if there exists a set $S \subseteq [m]$ of at most $k$ goods such that for every pair of agents $i,h \in [n]$, we have $v_i(A_i) \geq v_i(A_h \setminus S)$. An allocation $A$ is \emph{envy-free up to $k$ uniformly hidden goods} (\uHEF{k}) if there exists a set $S \subseteq [m]$ of at most $k$ goods satisfying $|S \cap A_i| \leq 1$ for every $i \in [n]$ such that for every pair of agents $i,h \in [n]$, we have $v_i(A_i) \geq v_i(A_h \setminus S)$. We say that allocation $A$ \emph{hides} the goods in $S$ and \emph{reveals} the remaining goods. Notice that a \uHEF{k} allocation is also \HEF{k} but the converse is not necessarily true. Indeed, in \Cref{prop:HEFk_vs_uHEF}, we will present an instance that, for some $k \in {\mathbb{N}}$, admits an \HEF{k} allocation but no \uHEF{k} allocation.
\label{defn:HEF-k} \end{definition}
\begin{remark}
It follows from the definitions that \HEF{0} $\Rightarrow$ \HEF{1} $\Rightarrow$ \HEF{2} $\dots$, and that an allocation satisfies \HEF{0} if and only if it satisfies \EF{}. It is also easy to verify that an allocation is \sEF{1} if and only if it is \uHEF{n}. This is because the unique hidden good for every agent is also the one that is (hypothetically) removed under \sEF{1}. Additionally, as discussed in \Cref{sec:Introduction}, an \EF{1} allocation might not be \uHEF{k} for any $k \leq n$.
\label{rem:EF_HEF_Relationship} \end{remark}
We say that allocation $A$ is \emph{\HEF{} with respect to set $S$} if $A$ becomes envy-free after hiding the goods in $S$, i.e., for every pair of agents $i,h \in [n]$, we have $v_i(A_i) \geq v_i(A_h \setminus S)$. We say that $k$ goods \emph{must be hidden} under $A$ if $A$ is \HEF{} with respect to some set $S$ such that $|S|=k$, and there is no set $S'$ with $|S'| < k$ such that $A$ is \HEF{} with respect to $S'$.
\begin{definition}[\textbf{Pareto optimality}]
An allocation $A$ is Pareto dominated by another allocation $B$ if $v_i(B_i) \geq v_i(A_i)$ for every agent $i \in [n]$ with at least one of the inequalities being strict. A \emph{Pareto optimal} (\textrm{\textup{PO}}{}) allocation is one that is not Pareto dominated by any other allocation. \end{definition}
\begin{definition}[\textbf{\EF{1} algorithms}]
\label{defn:EF1_algorithms}
We will now describe four known algorithms for finding \EF{1} allocations that are relevant to our work.
\paragraph{Round-robin algorithm (\textup{\texttt{RoundRobin}}{}):} Fix a permutation $\sigma$ of the agents. The \textup{\texttt{RoundRobin}}{} algorithm cycles through the agents according to $\sigma$. In each round, an agent gets its favorite good from the pool of remaining goods.
\paragraph{Envy-graph algorithm (\textup{\texttt{EnvyGraph}}{}):} This algorithm, proposed by \citet{LMM+04approximately}, works as follows: In each step, one of the remaining goods is assigned to an agent that is not envied by any other agent. The existence of such an agent is guaranteed by resolving cyclic envy relations (if any exists) in a combinatorial structure called the \emph{envy-graph} of an allocation.
\paragraph{Fisher market-based algorithm (\textup{\texttt{Alg-EF1+PO}}{}):} This algorithm, due to \citet{BKV18Finding}, uses local search and price-rise subroutines in a Fisher market associated with the fair division instance, and returns an \EF{1} and \textrm{\textup{PO}}{} allocation. The bound on running time of this algorithm is pseudopolynomial, i.e., has a polynomial dependence on $v_{i,j}$ instead of $\log v_{i,j}$.
\paragraph{Maximum Nash Welfare solution (\textup{\texttt{MNW}}{}):} The \emph{Nash social welfare} of an allocation $A$ is defined as $\textrm{\textup{NSW}}(A) \coloneqq \left( \prod_{i \in [n]} v_i(A_i) \right)^{1/n}$. The \textup{\texttt{MNW}}{} algorithm computes an allocation with the highest Nash social welfare (called a \emph{Nash optimal} allocation). It is known that a Nash optimal allocation is both \EF{1} and \textrm{\textup{PO}}{}~\citep{CKM+16unreasonable}. \end{definition}
\begin{remark}
\citet{CFS+19group} observed that \textup{\texttt{RoundRobin}}{}, \textup{\texttt{Alg-EF1+PO}}{}, and \textup{\texttt{MNW}}{} algorithms all satisfy \sEF{1}. It is easy to see that \textup{\texttt{EnvyGraph}}{} algorithm is also \sEF{1}. Among these four algorithms, only \textup{\texttt{MNW}}{} and \textup{\texttt{Alg-EF1+PO}}{} are provably also \textrm{\textup{PO}}{}.\footnote{It is also known that \textup{\texttt{RoundRobin}}{} and \textup{\texttt{EnvyGraph}}{} fail to satisfy \textrm{\textup{PO}}{}; see, e.g., \citep{CFS17fair}.} The allocations computed by all four algorithms have the property that there exists some agent that is not envied by any other agent. Indeed, \textup{\texttt{MNW}}{} and \textup{\texttt{Alg-EF1+PO}}{} are both \textrm{\textup{PO}}{} and therefore cannot have cyclic envy relations, and \textup{\texttt{RoundRobin}}{} and \textup{\texttt{EnvyGraph}}{} algorithms have this property by design. For such an agent (not necessarily the same agent for all four algorithms), no good needs to be removed under \sEF{1}. Therefore, from \Cref{rem:EF_HEF_Relationship}, all these algorithms are also envy-free up to $n-1$ uniformly hidden goods, or $\uHEF{(n-1)}$.
\label{rem:EF1_algorithms_uHEF_n-1} \end{remark}
\begin{restatable}{prop}{uHEFExistence}
\label{prop:uHEF(n-1)}
Given an instance with additive valuations, a $\uHEF{(n-1)}$ allocation always exists and can be computed in polynomial time, and a $\uHEF{(n-1)}+\textrm{\textup{PO}}{}$ allocation always exists and can be computed in pseudopolynomial time. \end{restatable}
\begin{remark}
Note that for any $k < n-1$, an \HEF{k} allocation might fail to exist. Indeed, with $n$ agents that have identical and positive valuations for $m = n-1$ goods, some agent will surely miss out and force the allocation to hide all $n-1$ (i.e., $k+1$ or more) goods. Therefore, the bound in \Cref{prop:uHEF(n-1)} for \uHEF{k} (and hence, for \HEF{k}) is tight in terms of $k$.
\label{rem:HEFk_Tight_Bound_For_k} \end{remark}
\subsection{Relevant Computational Problems} \label{subsec:Computational_Problems}
\Cref{defn:HEFkExistence} formalizes the decision problem of checking whether a given instance admits a fair (i.e., \HEF{k}) allocation.
\begin{definition}[\textbf{\textup{\textsc{\HEF{k}-Existence}}}]
Given an instance $\mathcal{I}$, does there exist an allocation $A$ and a set $S \subseteq [m]$ of at most $k$ goods such that $A$ is \HEF{} with respect to $S$?
\label{defn:HEFkExistence} \end{definition}
Notice that a certificate for \textup{\textsc{\HEF{k}-Existence}}{} consists of an allocation $A$ as well as a set $S$ of at most $k$ hidden goods.
Another relevant computational question involves checking whether a given allocation $A$ is \HEF{} with respect to some set $S \subseteq [m]$ of at most $k$ goods.
\begin{definition}[\textbf{\textup{\textsc{\HEF{k}-Verification}}}]
Given an instance $\mathcal{I}$ and an allocation $A$, does there exist a set $S \subseteq [m]$ of $k$ goods such that $A$ is \HEF{} with respect to $S$?
\label{defn:HEFkVerification} \end{definition}
For additive valuations, both \textup{\textsc{\HEF{k}-Existence}}{} and \textup{\textsc{\HEF{k}-Verification}}{} are in \textrm{\textup{NP}}{}. The next problem pertains to the existence of envy-free allocations.
\begin{definition}[\textbf{\textup{\textsc{\EF{}-Existence}}}]
Given an instance $\mathcal{I}$, does there exist an envy-free allocation for $\mathcal{I}$?
\label{defn:EFExistence} \end{definition}
\textup{\textsc{\EF{}-Existence}}{} is known to be \textrm{\textup{NP-complete}}{}~\citep{LMM+04approximately}. From \Cref{rem:EF_HEF_Relationship}, it follows that \textup{\textsc{\HEF{k}-Existence}}{} is \textrm{\textup{NP-complete}}{} when $k=0$ for additive valuations.
\section{Theoretical Results} \label{sec:Theoretical_Results}
This section presents our theoretical results concerning the existence and computation of \HEF{k} and \uHEF{k} allocations. We will first show that \uHEF{k} is a strictly more demanding notion than \HEF{k} (\Cref{prop:HEFk_vs_uHEF}).
\begin{restatable}{prop}{HEFvsuHEF}
\label{prop:HEFk_vs_uHEF}
There exists an instance $\mathcal{I}$ that, for some fixed $k \in {\mathbb{N}}$, admits an \HEF{k} allocation but no \uHEF{k} allocation. \end{restatable} \begin{proof}
Consider the fair division instance $\mathcal{I}$ with five agents $a_1,\dots,a_5$ and six goods $g_1,\dots,g_6$ shown in \Cref{tab:HEFk_vs_uHEFk}. Observe that the allocation $A = (A_1,\dots,A_5)$ with $A_1 = \{g_1,g_2\}$, $A_2 = \{g_3\}$, $A_3 = \{g_4\}$, $A_4 = \{g_5\}$, $A_5 = \{g_6\}$ satisfies \HEF{2} with respect to the set $S = \{g_1,g_2\}$.
\begin{table}[h]
\centering
\small
\begin{tabular}{ c|cccccc }
& $g_1$ & $g_2$ & $g_3$ & $g_4$ & $g_5$ & $g_6$\\ \hline
$a_1$ & $1$ & $1$ & $2$ & $0$ & $0$ & $0$ \\
$a_2$ & $1$ & $1$ & $2$ & $0$ & $0$ & $0$ \\
$a_3$ & $10$ & $10$ & $1$ & $1$ & $1$ & $1$ \\
$a_4$ & $10$ & $10$ & $1$ & $1$ & $1$ & $1$ \\
$a_5$ & $10$ & $10$ & $1$ & $1$ & $1$ & $1$ \\
\end{tabular}
\caption{The instance used in the proof of \Cref{prop:HEFk_vs_uHEF}.}
\label{tab:HEFk_vs_uHEFk}
\end{table}
We will show that $\mathcal{I}$ does not admit a \uHEF{2} allocation. Suppose, for contradiction, that there exists an allocation $B$ satisfying \uHEF{2}. Then, $B$ must hide $g_1$ and $g_2$ (otherwise, at least one of $a_3$, $a_4$ or $a_5$ will envy the owner(s) of these goods). Thus, in particular, the good $g_3$ must be revealed by $B$. Assume, without loss of generality, that $g_3$ is \emph{not} assigned to $a_1$ in $B$ (otherwise, a similar argument can be carried out for $a_2$). Then, $B$ must assign both $g_1$ and $g_2$ to $a_1$ (so that $a_1$ does not envy the owner of $g_3$). However, this violates the one-hidden-good-per-agent property of \uHEF{k}, which is a contradiction. \end{proof}
Recall from \Cref{subsec:Computational_Problems} that \textup{\textsc{\HEF{k}-Existence}}{} is \textrm{\textup{NP-complete}}{} when $k=0$. This still leaves open the question whether \textup{\textsc{\HEF{k}-Existence}}{} is \textrm{\textup{NP-complete}}{} for \emph{any} fixed $k \in {\mathbb{N}}$. Our next result (\Cref{thm:HEFk_Existence_NPcomplete_IdenticalVals}) shows that this is indeed the case, even under the restricted setting of \emph{identical} valuations (i.e., for every $j \in [m]$, $v_{i,j}=v_{h,j}$ for every $i,h \in [n]$).
\begin{restatable}[\textbf{Hardness of \textup{\textsc{\HEF{k}-Existence}}{}}]{theorem}{HEFkExistenceNPcompleteIdenticalVals}
\label{thm:HEFk_Existence_NPcomplete_IdenticalVals}
For any fixed $k \in {\mathbb{N}}$, \textup{\textsc{\HEF{k}-Existence}}{} is \textrm{\textup{NP-complete}}{} even for identical valuations. \end{restatable} \begin{proof}
We will show a reduction from \textrm{\textsc{Partition}}{}, which is known to be \textrm{\textup{NP-complete}}{}~
\citep{GJ79computers}. An instance of \textrm{\textsc{Partition}}{} consists of a multiset $X = \{x_1,x_2,\dots,x_n\}$ with $x_i \in {\mathbb{N}}$ for all $i \in [n]$. The goal is to determine whether there exists $Y \subset X$ such that $\sum_{x_i \in Y} x_i = \sum_{x_i \in X \setminus Y} x_i = T$, where $T \coloneqq \frac{1}{2} \sum_{x_i \in X} x_i$.
We will construct a fair division instance with $k+3$ agents $a_1,\dots,a_{k+3}$ and $n+k+1$ goods. The goods are classified into $n+1$ \emph{main goods} $g_1,\dots,g_{n+1}$ and $k$ \emph{dummy goods} $d_1,\dots,d_k$. The (identical) valuations are defined as follows: Every agent values the goods $g_1,\dots,g_n$ at $x_1,\dots,x_n$ respectively; the good $g_{n+1}$ at $T$, and each dummy good at $4T$.
($\Rightarrow$) Suppose $Y$ is a solution of \textrm{\textsc{Partition}}{}. Then, an \HEF{k} allocation can be constructed as follows: Assign the main goods corresponding to the set $Y$ to agent $a_1$ and those corresponding to $X \setminus Y$ to agent $a_2$. The good $g_{n+1}$ is assigned to agent $a_3$. Each of the remaining $k$ agents is assigned a unique dummy good. Note that every agent in the set $\{a_1,a_2,a_3\}$ envies every agent in the set $\{a_4,\dots,a_{k+3}\}$, and these are the only pairs of agents with non-zero envy. Therefore, the allocation can be made envy-free by hiding the $k$ dummy goods, i.e., the allocation is \HEF{} with respect to the set $\{d_1,\dots,d_k\}$.
($\Leftarrow$) Now suppose there exists an \HEF{k} allocation $A$. Since there are $k$ dummy goods and $k+3$ agents, there must exist at least three agents that do not receive any dummy good in $A$. Without loss of generality, let these agents be $a_1$, $a_2$ and $a_3$ (otherwise, we can reindex). We claim that all dummy goods must be hidden under $A$. Indeed, agent $a_1$ does not receive any dummy good, and therefore its maximum possible valuation can be $v(g_1 \cup \dots \cup g_{n+1}) = 3T < v(d_j)$ for any dummy good $d_j$. If some dummy good $d_j$ is not hidden, then $a_1$ will envy the owner of $d_j$, contradicting \HEF{k}. Therefore, all dummy goods must be hidden, and since there are $k$ such goods, these are the only ones that can be hidden.
The above observation implies that the good $g_{n+1}$ must be revealed by $A$. Furthermore, $g_{n+1}$ must be assigned to one of $a_1$, $a_2$ or $a_3$ (otherwise, by pigeonhole principle, one of these agents will have valuation at most $\frac{2T}{3}$ and will envy the owner of $g_{n+1}$). If $g_{n+1}$ is assigned to $a_3$, then the remaining main goods $g_1,\dots,g_n$ must be divided between $a_1$ and $a_2$ such that $v(A_1) \geq T$ and $v(A_2) \geq T$. This gives a partition of the set $X$. \end{proof}
Another commonly used preference restriction is that of \emph{binary} valuations (i.e., for every $i \in [n]$ and $j \in [m]$, $v_{i,j} \in \{0,1\}$). We note that even under this restriction, \textup{\textsc{\HEF{k}-Existence}}{} remains \textrm{\textup{NP-complete}}{} when $k=0$ (\Cref{cor:HEFk_Existence_NPcomplete_BinaryVals}). This observation follows from a result of \citet{AGM+15fair}, who showed that determining the existence of an envy-free allocation is \textrm{\textup{NP-complete}}{} even for binary valuations (\Cref{prop:EF_Existence_NPcomplete_BinaryValuations}). We provide an alternative proof of this statement in \Cref{subsec:Proof_EF_Existence_NPcomplete_BinaryValuations} in the appendix.
\begin{restatable}[\citealp{AGM+15fair}; Theorem 11]{prop}{EFExistenceNPcompleteBinaryVals}
\label{prop:EF_Existence_NPcomplete_BinaryValuations}
\textup{\textsc{\EF{}-Existence}}{} is \textrm{\textup{NP-complete}}{} even for binary valuations. \end{restatable}
\begin{restatable}{corollary}{HEFkExistenceNPcompleteBinaryVals}
\label{cor:HEFk_Existence_NPcomplete_BinaryVals}
For $k=0$, \textup{\textsc{\HEF{k}-Existence}}{} is \textrm{\textup{NP-complete}}{} even for binary valuations. \end{restatable}
\Cref{prop:EF_Existence_NPcomplete_BinaryValuations} is also useful in establishing the computational hardness of finding an \HEF{k}+\textrm{\textup{PO}}{} allocation. Note that unlike \Cref{cor:HEFk_Existence_NPcomplete_BinaryVals}, \Cref{prop:HEFkPOExistence_NPcomplete_BinaryVals} holds for every fixed $k \in {\mathbb{N}}$.
\begin{restatable}[\textbf{Hardness of \HEF{k}+\textrm{\textup{PO}}{}}]{theorem}{HEFkPOExistenceNPcompleteBinaryVals}
\label{prop:HEFkPOExistence_NPcomplete_BinaryVals}
Given any instance $\mathcal{I}$ with binary valuations and any fixed $k \in {\mathbb{N}} \cup \{0\}$, it is \textrm{\textup{NP-hard}}{} to determine if $\mathcal{I}$ admits an allocation that is envy-free up to $k$ hidden goods $(\HEF{k})$ and Pareto optimal $(\textrm{\textup{PO}}{})$. \end{restatable} \begin{proof} (Sketch)
Starting from any instance of \textup{\textsc{\EF{}-Existence}}{} with binary valuations (\Cref{prop:EF_Existence_NPcomplete_BinaryValuations}), we add to it $k$ new goods and $k+1$ new agents such that all new goods are approved by all new agents (and no one else). Also, the new agents have zero value for the existing goods. In the forward direction, an arbitrary allocation of new goods among the new agents works. In the reverse direction, \textrm{\textup{PO}}{} forces each new (respectively, existing) good to be assigned among new (respectively, existing) agents only. The imbalance between new agents and new goods means that all (and only) the new goods must be hidden. Then, the restriction of the \HEF{k} allocation to the existing agents/goods gives the desired \EF{} allocation. \end{proof}
We will now proceed to analyzing the computational complexity of \textup{\textsc{\HEF{k}-Verification}}{}. Here, we show a hardness-of-approximation result (\Cref{prop:HEFk_Verification_Hardness_Of_Approximation_BinaryVals}). The inapproximability factor is stated in terms of the aggregate envy, defined as follows: Given any allocation $A$, the \emph{aggregate envy} in $A$ is the sum of all pairwise envy values, i.e., \begin{align*}
E \coloneqq \sum_{h \in [n]} \sum_{i \neq h} \max\{0, v_i(A_h) - v_i(A_i)\}. \end{align*}
Note that \textup{\textsc{\HEF{k}-Verification}}{} is stated as a decision problem (\Cref{defn:HEFkVerification}). However, one can consider an approximation version of this problem as follows: A $c$-approximation algorithm for \textup{\textsc{\HEF{k}-Verification}}{} takes as input a fair division instance and an allocation, and computes a set of goods of size at most $c \cdot k^{\textup{opt}}$, where $k^{\textup{opt}}$ is the size of the smallest hidden set for the given allocation. Under this definition, \Cref{prop:HEFk_Verification_Hardness_Of_Approximation_BinaryVals} can be interpreted as follows: Given any ${\varepsilon}>0$, there is no polynomial-time $(1-{\varepsilon}).\ln E$-approximation algorithm for \textup{\textsc{\HEF{k}-Verification}}{}, unless P=NP.
\begin{restatable}[\textbf{\textup{\textsc{\HEF{k}-Verification}}{} inapproximability}]{theorem}{HEFkVerificationHardnessOfApproximationBinaryVals}
\label{prop:HEFk_Verification_Hardness_Of_Approximation_BinaryVals}
Given any ${\varepsilon} > 0$, it is \text{NP-hard}{} to approximate \textup{\textsc{\HEF{k}-Verification}}{} to within $(1-{\varepsilon}) \cdot \ln E$ even for binary valuations, where $E$ is the aggregate envy in the given allocation. \end{restatable} \begin{proof}
We will show a reduction from \textup{\textsc{Hitting Set}}{}. An instance of \textup{\textsc{Hitting Set}}{} consists of a finite set $X = \{x_1,\dots,x_p\}$, a collection $\mathcal{F} = \{F_1,\dots,F_q\}$ of subsets of $X$, and some $k \in {\mathbb{N}}$. The goal is to determine whether there exists $Y \subseteq X$, $|Y| \leq k$ that intersects every member of $\mathcal{F}$ (i.e., for every $F \in \mathcal{F}$, $Y \cap F \neq \emptyset$). It is known that given any ${\varepsilon} > 0$, it is \text{NP-hard}{} to approximate \textup{\textsc{Hitting Set}}{} to within a factor $(1-{\varepsilon}) \cdot \ln |\mathcal{F}|$ \citep{DS14analytical}.
We will construct a fair division instance with $n = q+1$ agents and $m = p + \sum_{i=1}^q (|F_i|-1)$ goods. The agents are classified into $q$ \emph{dummy agents} $a_1,\dots,a_q$ and one \emph{main agent} $a_{q+1}$. The goods are classified into $p$ \emph{main goods} $g_1,\dots,g_p$ and $q$ distinct sets of dummy goods, where the $i^\text{th}$ set consists of the goods $f^i_{1},\dots,f^i_{|F_i|-1}$.
The valuations are as follows: The main agent approves all the main goods, i.e., for all $j \in [p]$, $v_{q+1}(\{g_j\}) = 1$. Each dummy agent $a_i$ approves the dummy goods in the $i^\text{th}$ set as well as those main goods that intersect with $F_i$, i.e., for every $i \in [q]$, $v_i(\{f^i_j\}) = 1$ for all $j \in [|F_i|-1]$, and $v_i(\{g_j\}) = 1$ whenever $x_j \in F_i$. All other valuations are set to $0$.
The input allocation $A = (A_1,\dots,A_{q+1})$ is defined as follows: The main agent $a_{q+1}$ is assigned all the main goods, i.e., $A_{q+1} \coloneqq \{g_1,\dots,g_p\}$. For every $i \in [q]$, the dummy agent $a_i$ is assigned the $|F_i|-1$ dummy goods in the $i^\text{th}$ set, i.e., $A_i \coloneqq \{f^i_{1},\dots,f^i_{|F_i|-1}\}$. Note that in the allocation $A$, each dummy agent envies the main agent by one approved good, and these are the only pairs of agents with envy.
($\Rightarrow$) Suppose $Y \subseteq X$, $|Y| \leq k$ is solution of the \textup{\textsc{Hitting Set}}{} instance. We claim that the allocation $A$ is \HEF{} with respect to the set $S \coloneqq \{g_j : x_j \in Y\}$ with $|S| \leq k$. Indeed, since $S$ is induced by a hitting set, each dummy agent approves at least one good in $S$. Therefore, by hiding the goods in $S$, the envy from the dummy agents can be eliminated.
($\Leftarrow$) Now suppose there exists $S \subseteq [m]$, $|S| \leq k$ such that $A$ is \HEF{} with respect to $S$. Then, for every $i \in [q]$, the set $S$ must contain at least one good that is approved by the dummy agent $a_i$ (otherwise $A$ will not be envy-free after hiding the goods in $S$). It is easy to see that the set $Y \coloneqq \{x_j : g_j \in S\}$ constitutes the desired hitting set of cardinality at most $k$.
Finally, to show the hardness-of-approximation, notice that the aggregate envy in $A$ is $q$ because each dummy agent envies the main agent by one unit of utility. The claim now follows by substituting $|\mathcal{F}| = q = E$ in the inapproximability result of \textup{\textsc{Hitting Set}}{} stated above. \end{proof}
Our next result (\Cref{thm:HEFk_Verification_ApproxAlgo}) provides an approximation algorithm that (nearly) matches the hardness-of-approximation result in \Cref{prop:HEFk_Verification_Hardness_Of_Approximation_BinaryVals}. We remark that the algorithm in \Cref{thm:HEFk_Verification_ApproxAlgo} applies to \emph{any} instance with additive and possibly non-binary valuations.
\begin{restatable}[\textbf{Approximation algorithm}]{theorem}{HEFkVerificationApproxAlgo}
\label{thm:HEFk_Verification_ApproxAlgo}
There is a polynomial-time algorithm that, given as input any instance of \textup{\textsc{\HEF{k}-Verification}}{}, finds a set $S \subseteq [m]$ with $|S| \leq k^{\textup{opt}} \cdot \ln E + 1$ such that the given allocation is \HEF{} with respect to $S$. Here, $E$ and $k^{\textup{opt}}$ denote the aggregate envy and the number of goods that must be hidden under the given allocation, respectively. \end{restatable}
The proof of \Cref{thm:HEFk_Verification_ApproxAlgo} is deferred to \Cref{subsec:Proof_HEFk_Verification_ApproxAlgo} in the appendix but a brief idea is as follows: For any set $S \subseteq [m]$, define the \emph{residual envy function} $f : 2^{[m]} \rightarrow \mathbb{R}$ so that $f(S)$ is the aggregate envy in allocation $A$ after hiding the goods in $S$. That is, \begin{align*} f(S) \coloneqq \sum_{h \in [n]} \sum_{i \neq h} \max\{0, v_i(A_h \setminus S) - v_i(A_i)\}. \end{align*} The relevant observation is that $f$ is \emph{supermodular}. Given this observation, the approximation guarantee in \Cref{thm:HEFk_Verification_ApproxAlgo} can be obtained by the standard greedy algorithm for submodular maximization, or, equivalently, supermodular minimization~ \citep{NWF78analysis}; see Algorithm~\ref{alg:Greedy_HEFk_ApproxAlgo} in \Cref{subsec:Proof_HEFk_Verification_ApproxAlgo}.
\section{Experimental Results} \label{sec:Experiments}
\begin{table*}
\centering
\begin{tabular}{|cccc|}
\multicolumn{4}{c}{\textbf{Normalized average-case regret}}\\
\hline
\footnotesize{\textup{\texttt{Alg-EF1+PO}}{}} & \footnotesize{\textup{\texttt{RoundRobin}}{}} & \footnotesize{\textup{\texttt{MNW}}{}} & \footnotesize{\textup{\texttt{EnvyGraph}}{}}\\
\includegraphics[width=0.22\textwidth]{market_regret_7_binary_norm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{rr_regret_7_binary_norm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{mnw_binary_regret_7_binary_norm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{envy_graph_regret_7_binary_norm-eps-converted-to} \\
\hline
\multicolumn{4}{c}{}\\
\multicolumn{4}{c}{\textbf{Number of goods that must be hidden on average} (averaged over non-\EF{} instances only)}\\
\hline
\footnotesize{\textup{\texttt{Alg-EF1+PO}}{}} & \footnotesize{\textup{\texttt{RoundRobin}}{}} & \footnotesize{\textup{\texttt{MNW}}{}} & \footnotesize{\textup{\texttt{EnvyGraph}}{}}\\
\includegraphics[width=0.22\textwidth]{market_k_7_binary_notnorm-eps-converted-to} & \includegraphics[width=0.22\textwidth]{rr_k_7_binary_notnorm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{mnw_binary_k_7_binary_notnorm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{envy_graph_k_7_binary_notnorm-eps-converted-to} \\
\hline
\end{tabular}
\caption{Results for synthetic data.}
\label{tab:Expt_BinaryVals_bias_0.7} \end{table*}
We have seen that the worst-case computational results for \HEF{k}, even in highly restricted settings, are largely negative (\Cref{sec:Theoretical_Results}). In this section, we will examine whether the known algorithms for computing approximately envy-free allocations---in particular, the four \EF{1} algorithms described in \Cref{defn:EF1_algorithms} in \Cref{sec:Preliminaries}---can provide meaningful approximations to \HEF{k} in practice. Recall from \Cref{rem:EF1_algorithms_uHEF_n-1} that all four discussed algorithms---\textup{\texttt{RoundRobin}}{}, \textup{\texttt{MNW}}{}, \textup{\texttt{Alg-EF1+PO}}{}, and \textup{\texttt{EnvyGraph}}{}---satisfy $\uHEF{(n-1)}$.
We evaluate each algorithm in terms of (a) its \emph{regret} (defined below), and (b) the \emph{number of goods that the algorithm must hide}. Given an instance $\mathcal{I}$ and an allocation $A$, let $\kappa(A,\mathcal{I})$ denote the number of goods that must be hidden under $A$. The \emph{regret} of allocation $A$ is the number of extra goods that must be hidden under $A$ compared to the optimal. That is, $\texttt{\textup{reg}}(A,\mathcal{I}) \coloneqq \kappa(A,\mathcal{I}) - \min_{B} \kappa(B,\mathcal{I})$. Similarly, given an algorithm \textup{\textsc{Alg}}{}, the regret of \textup{\textsc{Alg}}{} is given by $\texttt{\textup{reg}}(\textup{\textsc{Alg}}(\mathcal{I}),\mathcal{I})$, where $\textup{\textsc{Alg}}(\mathcal{I})$ is the allocation returned by \textup{\textsc{Alg}}{} for the input instance $\mathcal{I}$. Note that the regret can be large due to the suboptimality of an algorithm, but also due to the size of the instance. To negate the effect of the latter, we normalize the regret value by $n-1$, which is the worst-case upper bound on the number of hidden goods for all four algorithms of interest.
\subsection{Experiments on Synthetic Data} \label{subsec:Expt_Synthetic} The setup for synthetic experiments is similar to that used in \Cref{fig:HEFk_motivation_0.7}. Specifically, the number of agents, $n$, is varied from $5$ to $10$, and the number of goods, $m$, is varied from $5$ to $20$ (we ignore the cases where $m < n$). For every fixed $n$ and $m$, we generated $100$ instances with \emph{binary} valuations drawn i.i.d. from Bernoulli distribution with parameter $0.7$ (i.e., $v_{i,j} \sim \textrm{\textup{Ber}}(0.7)$). \Cref{tab:Expt_BinaryVals_bias_0.7} shows the heatmaps of the normalized regret (averaged over $100$ instances) and the number of goods that must be hidden (averaged over non-\EF{} instances, i.e., whenever $k \geq 1$) for all four algorithms.\footnote{Additional results for $v_{i,j} \sim \textrm{\textup{Ber}}(0.7)$, and $v_{i,j} \sim \textrm{\textup{Ber}}(0.5)$ can be found in \Cref{subsec:Additional_Experiments} in the appendix.}
It is clear that \textup{\texttt{Alg-EF1+PO}}{} and \textup{\texttt{RoundRobin}}{} algorithms have a superior performance than \textup{\texttt{MNW}}{} and \textup{\texttt{EnvyGraph}}{}. In particular, both \textup{\texttt{Alg-EF1+PO}}{} and \textup{\texttt{RoundRobin}}{} have small normalized regret, suggesting that they hide close-to-optimal number of goods. Additionally, the number of hidden goods itself is small for these algorithms (in most cases, no more than \emph{three} goods need to be hidden), suggesting that the worst-case bound of $n-1$ is unlikely to arise in practice. Overall, our experiments suggest that \textup{\texttt{Alg-EF1+PO}}{} and \textup{\texttt{RoundRobin}}{} can achieve useful approximations to \HEF{k} in practice, especially in comparison to \textup{\texttt{MNW}}{} and \textup{\texttt{EnvyGraph}}{}.\footnote{In \Cref{subsec:MNW_Large_Regret} in the appendix, we provide two families of instances where the normalized worst-case regret of \textup{\texttt{MNW}}{} is large.}
\subsection{Experiments on Real-World Data} \label{subsec:Expt_Spliddit} For experiments with real-world data, we use the data from the popular fair division website \emph{Spliddit} \citep{GP15spliddit}. The Spliddit data has $2212$ instances in total, where the number of agents $n$ varies between $3$ and $10$, and the number of goods $m \geq n$ varies between $3$ and $93$. Unlike the synthetic data, the distribution of instances here is rather uneven (see \Cref{fig:Spliddit_data_distribution} in \Cref{subsec:Additional_Experiments} in the appendix); in fact, $1821$ of the $2212$ instances have $n=3$ agents and $m=6$ goods. Therefore, instead of using heatmaps, we compare the algorithms in terms of their normalized regret (averaged over the entire dataset) and the cumulative distribution function of the hidden goods (see \Cref{fig:Results_Spliddit}).
\Cref{fig:Results_Spliddit} presents an interesting twist: \textup{\texttt{MNW}}{} is now the best performing algorithm, closely followed by \textup{\texttt{RoundRobin}}{} and \textup{\texttt{Alg-EF1+PO}}{}. For any fixed $k$, the fraction of instances for which these three algorithms compute an \HEF{k} allocation is also nearly identical. As can be observed, these algorithms almost never need to hide more than \emph{three} goods. By contrast, \textup{\texttt{EnvyGraph}}{} has the largest regret and significantly worse cumulative performance. Therefore, once again, \textup{\texttt{Alg-EF1+PO}}{} and \textup{\texttt{RoundRobin}}{} algorithms perform competitively with the optimal solution, making them attractive options for achieving fair outcomes without withholding too much information.
\begin{figure}
\caption{Results for Spliddit data.}
\label{fig:Results_Spliddit}
\end{figure}
\section{Future Work} Analyzing the asymptotic behavior of \HEF{k} allocations, as has been done for envy-free allocations~\citep{DGK+14computational,MS19when}, is an interesting direction for future work. It would also be interesting to explore the connections with other recently proposed relaxations that involve discarding goods~\citep{CGH19envy,CKM+20little} or sharing a small subset of goods~\citep{SE19fair}.
\section*{Acknowledgments} We thank the anonymous conference reviewers for their helpful comments. We are grateful to Ariel Procaccia and Nisarg Shah for sharing with us the data from Spliddit, and to Haris Aziz for bringing to our attention the proof of \textup{\textsc{\EF{}-Existence}}{} for binary valuations in \citep{AGM+15fair}. RV thanks Rupert Freeman, Nick Gravin, and Neeldhara Misra for very helpful discussions and several useful suggestions for improving the presentation of the paper. Thanks also to Erel Segal-Halevi for many helpful comments on \Cref{subsec:Proof_EF_Existence_NPcomplete_BinaryValuations}. LX acknowledges NSF \#1453542 and \#1716333, and HH acknowledges NSF \#1850076 for support.
\section{Appendix}
\label{sec:Appendix}
\subsection{Proof of Proposition~\ref{prop:EF_Existence_NPcomplete_BinaryValuations}} \label{subsec:Proof_EF_Existence_NPcomplete_BinaryValuations}
Recall the statement of \Cref{prop:EF_Existence_NPcomplete_BinaryValuations}. \EFExistenceNPcompleteBinaryVals*
Our proof of \Cref{prop:EF_Existence_NPcomplete_BinaryValuations} uses a reduction from \textup{\textsc{Equitable Coloring}}{}, which is defined below.
\begin{definition}[\textup{\textsc{Equitable Coloring}}] Given a graph $G$ and a number $\ell \in {\mathbb{N}}$, does there exist a proper $\ell$-coloring of $G$ such that all color classes are of equal size? \label{defn:EquitableColoring} \end{definition}
The standard definition of \textup{\textsc{Equitable Coloring}}{} requires the color classes to differ in size by at most one. We overload the term to refer to the version where all color classes are of the same size. \textup{\textsc{Equitable Coloring}}{} can be shown to be \textrm{\textup{NP-complete}}{} by a straightforward reduction from \textup{\textsc{Graph $k$-Colorability}}{}~\citep{GJ79computers}. In addition, we can assume that $\ell \geq 3$ without loss of generality.
\begin{proof} (of \Cref{prop:EF_Existence_NPcomplete_BinaryValuations})
We will show a reduction from \textup{\textsc{Equitable Coloring}}{}. Recall from \Cref{defn:EquitableColoring} that an instance of \textup{\textsc{Equitable Coloring}}{} consists of a graph $G = (V,E)$ and a number $\ell \in {\mathbb{N}}$. The goal is to determine if $G$ admits a proper $\ell$-coloring wherein the color classes are of the same size. For simplicity, we will write $n \coloneqq |V|$ and $m \coloneqq |E|$.\footnote{Not be confused with the number of agents, $n$, and the number of goods, $m$, as defined in \Cref{sec:Preliminaries}.} Note that we can assume, without loss of generality, that $G$ is connected.\footnote{Given any graph $G$, we can construct another connected graph $G' = (V',E')$ as follows: Let $V' \coloneqq V \cup \{x_1,\dots,x_\ell\} \cup \{y_1,\dots,y_{\frac{n}{\ell}+1}\}$. There is an edge between every pair of vertices in $\{x_1,\dots,x_\ell\}$ so as to induce an $\ell$-clique. In addition, each $y_i$ is connected to every vertex in $\{x_1,\dots,x_\ell\}$ as well as to every vertex in $V$. It is easy to see that $G$ admits an equitable $\ell$-coloring if and only if $G'$ admits an equitable $(\ell+1)$-coloring. Indeed, the $x_i$'s consume $\ell$ colors, and, as a result, all $y_i$'s must have the same color. This, in turn, leaves exactly $\ell$ colors for the vertices in $V$. Furthermore, there are $\frac{n}{\ell}+1$ vertices in each color class, implying that the coloring is equitable.} Since a connected graph with $n$ vertices has at least $n-1$ edges, we have that \begin{align} m \geq n-1. \label{eqn:Connected_Graph_Inequality} \end{align}
In addition, we will also assume that each vertex in $G$ has degree at least two. Indeed, for any vertex $v$ with degree at most one, we can add $\ell$ new vertices $v_1, v_2,\dots,v_{\ell}$ that are connected as follows: The vertices $v_1,\dots,v_{\ell}$ constitute an $\ell$-clique (that is, for every pair of distinct $i,j \in [\ell]$, $v_i$ is connected to $v_j$), and $v$ is connected to each vertex in $\{v_2,\dots,v_{\ell-1}\}$ but not $v_1$. Call the new graph $G'$. It is easy to see that $G$ has an equitable $\ell$-coloring if and only if $G'$ does.
We will construct a fair division instance with $m+n$ goods and $m+\ell$ agents. The agents are classified into $m$ \emph{edge} agents $a_1, \dots, a_m$ and $\ell$ \emph{dummy} agents $d_1,\dots,d_\ell$. The goods are classified into $n$ \emph{vertex} goods $v_1,\dots,v_n$ and $m$ \emph{edge} goods $e_1,\dots,e_m$. Note that we use the same notation for the vertices (edges) and the corresponding vertex (edge) goods.
The preferences of the agents are defined as follows: For every edge $e = (v_i,v_j)$, an edge agent $a_e$ approves all the edge goods and exactly two vertex goods $v_i$ and $v_j$. Each dummy agent approves all the vertex goods and has zero value for the edge goods.
($\Rightarrow$) Suppose $G$ admits an equitable coloring with each color class of size $\frac{n}{\ell}$. Then, an envy-free allocation $A$ can be constructed as follows: Assign each edge good $e$ to the edge agent $a_e$ and each vertex good $v$ to the dummy agent $d_i$ if vertex $v$ has color $i$. Notice that all goods are allocated under $A$. Also note that no two edge agents envy each other since each of them gets exactly one edge good. Furthermore, due to the proper coloring condition, for any edge $e = (v_i,v_j)$ in $G$, the corresponding vertex goods $v_i$ and $v_j$ are assigned to distinct dummy agents. Hence, no edge agent envies a dummy agent. The dummy agents have zero value for the edge goods and therefore do not envy the edge agents. Finally, since all color classes are of the same size, each dummy agent gets exactly $\frac{n}{\ell}$ approved goods, and therefore does not envy any other dummy agent. Overall, the allocation is envy-free.
($\Leftarrow$) Now suppose there exists an envy-free allocation $A$. We will show that $A$ satisfies \Cref{property:Edge_agent_cannot_get_two_edge_goods,property:Dummy_agent_gets_at_least_one_vertex_good,property:Dummy_agents_cannot_get_any_edge_good,property:Edge_agent_gets_exactly_one_edge_good,property:Edge_agents_cannot_get_any_vertex_good,property:Dummy_agents_cannot_get_adjacent_vertices} that will help us infer an equitable coloring of $G$.
\begin{property} No edge agent can get two or more edge goods under $A$. \label{property:Edge_agent_cannot_get_two_edge_goods} \end{property} \begin{proof} (of \Cref{property:Edge_agent_cannot_get_two_edge_goods}) Suppose, for contradiction, that some edge agent $a_e$ gets two or more edge goods. Then, any other edge agent $a_{e'}$ has a utility of at least $2$ for the bundle of $a_e$. For $A$ to be envy-free, $a_{e'}$ must have a utility of at least $2$ for its own bundle. For binary valuations, this means that $a_{e'}$ must be assigned two or more goods that it approves. Therefore, we need at least $2m$ goods to satisfy the edge agents. The total number of available goods is $m + n$, which, using \Cref{eqn:Connected_Graph_Inequality}, evaluates to at most $2m+1$. This leaves at most one good to be allocated among $\ell$ dummy agents. Since $\ell \geq 3$, some dummy agent is bound to be envious, contradicting the envy-freeness of $A$. \end{proof}
\begin{property} Every dummy agent gets at least one vertex good under $A$. \label{property:Dummy_agent_gets_at_least_one_vertex_good} \end{property} \begin{proof} (of \Cref{property:Dummy_agent_gets_at_least_one_vertex_good}) Fix a vertex good $v$ and a dummy agent $d$. Then, either $v$ is assigned to $d$, or $d$ gets some other (approved) good to prevent it from envying the owner of $v$. Since the only goods approved by the dummy agents are the vertex goods, the claim follows. \end{proof}
\begin{property} No dummy agent can get an edge good under $A$.\label{property:Dummy_agents_cannot_get_any_edge_good} \end{property} \begin{proof} (of \Cref{property:Dummy_agents_cannot_get_any_edge_good}) Suppose, for contradiction, that a dummy agent $d$ gets an edge good $e$ under $A$. From \Cref{property:Dummy_agent_gets_at_least_one_vertex_good}, we know that $d$ also gets some vertex good, say $v_0$. By assumption, the graph $G$ has minimum degree two, so there must exist some edge $e_1 = (v_0,v_1)$ adjacent to the vertex $v_0$. Notice that the edge agent $a_{e_1}$ has a utility of (at least) $2$ for the bundle of $d$. Therefore, for $A$ to be envy-free, $a_{e_1}$ must get at least two goods that it approves. \Cref{property:Edge_agent_cannot_get_two_edge_goods} limits the number of edge goods assigned to any edge agent to at most one. Therefore, in addition to some edge good, $a_{e_1}$ must also get the vertex good $v_1$. Once again using the bound on minimum degree of $G$, we get that there must exist some edge $e_2 = (v_1,v_2)$ adjacent to the vertex $v_1$. A similar argument shows that the vertex good $v_2$ must be assigned to the edge agent $a_{e_2}$. Continuing in this manner, we will eventually encounter an edge $e_i = (v_{i-1},v_i)$ such that $v_{i-1}$ is already assigned to $a_{e_{i-1}}$ and $v_{i}$ is already assigned to either $d$ or some other edge agent. This would imply that $a_{e_i}$ is envious of some other agent under $A$---a contradiction. \end{proof}
\begin{property} Every edge agent gets exactly one edge good under $A$. \label{property:Edge_agent_gets_exactly_one_edge_good} \end{property} \begin{proof} (of \Cref{property:Edge_agent_gets_exactly_one_edge_good}) Follows from \Cref{property:Dummy_agents_cannot_get_any_edge_good,property:Edge_agent_cannot_get_two_edge_goods}. \end{proof}
\begin{property} No edge agent can get a vertex good under $A$. \label{property:Edge_agents_cannot_get_any_vertex_good} \end{property} \begin{proof} (of \Cref{property:Edge_agents_cannot_get_any_vertex_good}) Suppose, for contradiction, that some edge agent $a_{e_0}$ is assigned a vertex good $v_0$.
Let $e_1 = (v_0,v_1)$ be an edge incident to the vertex $v_0$ in $G$ (such an edge must exist due to the bound on minimum degree). From \Cref{property:Edge_agent_gets_exactly_one_edge_good}, we know that each edge agent gets exactly one edge good. Thus, the edge agent $a_{e_1}$ has a utility of (at least) $2$ for the bundle of the agent $a_{e_0}$. For $A$ to be envy-free, $a_{e_1}$ must receive two or more goods that it approves, only one of which can be an edge good. Therefore, agent $a_{e_1}$ must also receive the vertex good $v_1$. Now let $e_2 = (v_1,v_2)$ be another edge incident to the vertex $v_1$ in $G$ (again, such an edge exists because $v_1$ has degree at least two). A similar argument implies that the vertex good $v_2$ must be assigned to the edge agent $a_{e_2}$. Continuing in this manner, let $e_i = (v_{i-1},v_i)$ denote the first edge in the sequence for which one of the following mutually exclusive conditions is true:
\begin{enumerate}
\item Either, the vertex good $v_i$ is assigned to an agent different from $a_{e_i}$, or
\item the vertex good $v_i$ is assigned to $a_{e_i}$ and $v_i = v_0$ (thus $a_{e_i}=a_{e_0}$). \end{enumerate}
Notice that due to the finiteness of the graph $G$, there must exist an edge $e_i$ satisfying one of the aforementioned conditions. We will now argue that each of these conditions leads to a contradiction.
\begin{enumerate}
\item First, suppose that the vertex good $v_i$ is assigned to an agent different from $a_{e_i}$. Then, the edge agent $a_{e_i}$ has a utility of (at least) $2$ for the bundle of edge agent $a_{e_{i-1}}$ and a utility of $1$ for its own bundle, contradicting the envy-freeness of $A$.
\item Next, suppose that $v_i = v_0$. That is, there exists a cycle $C = \{(v_0,v_1),(v_1,v_2),\dots,(v_{i-1},v_0)\}$ in the graph $G$ such that for every $j \in \{0,1,\dots,i-1\}$, the vertex good $v_j$ is assigned to the edge agent $a_{e_j}$.
Recall from \Cref{property:Dummy_agent_gets_at_least_one_vertex_good} that each dummy agent gets at least one vertex good. Since all vertex goods corresponding to the vertices in $C$ are assigned to the edge agents, there must exist at least one vertex outside the cycle $C$. Furthermore, since $G$ is connected, there must exist a path from this vertex to a vertex in $C$, say $v_1$. Thus, there must exist a vertex $v'_1 \in V$ such that $(v_1,v'_1) \in E$ and $v'_1 \notin C$. Let $e'_1 \coloneqq (v_1,v'_1)$. Then, the edge agent $a_{e'_1}$ has a utility of $2$ for the bundle of agent $a_{e_1}$ (recall that $a_{e_1}$ receives an edge good and the vertex good $v_1$), and must therefore be assigned the vertex good $v'_1$. Since the vertex $v'_1$ has degree at least two, there must exist another edge $e'_2 = (v'_1,v'_2)$ in $G$. By a similar argument as before, the edge agent $a_{e'_2}$ must be assigned the vertex good $v'_2$. Continuing in this manner, we will encounter an edge, say $e'_i = (v'_{i-1},v'_i)$ such that the vertex good $v'_i$ is assigned to an agent different from $a_{e'_i}$. This means that the edge agent $a_{e'_i}$ has a utility of (at least) $2$ for the bundle of the edge agent $a_{e'_{i-1}}$ and a utility of $1$ for its own bundle, contradicting the envy-freeness of $A$. \end{enumerate}
This completes the proof of \Cref{property:Edge_agents_cannot_get_any_vertex_good}. \end{proof}
\begin{property} For any edge $e = (v_i,v_j)$, no dummy agent is assigned both vertex goods $v_i$ and $v_j$ under $A$. \label{property:Dummy_agents_cannot_get_adjacent_vertices} \end{property} \begin{proof} (of \Cref{property:Dummy_agents_cannot_get_adjacent_vertices}) Suppose, for contradiction, that for some edge $e = (v_i,v_j)$, a dummy agent $d$ is assigned both $v_i$ and $v_j$. \Cref{property:Edge_agent_gets_exactly_one_edge_good} implies that the utility of $a_e$ for its own bundle is exactly $1$. However, the utility of $a_e$ for the bundle of $d$ is $2$, contradicting the envy-freeness of $A$. \end{proof}
It follows from \Cref{property:Edge_agents_cannot_get_any_vertex_good} that all vertex goods must be allocated among the dummy agents. Now consider the following coloring of the graph $G$: For each vertex $v$, the color of $v$ is the index of the dummy agent that gets the vertex good $v$. \Cref{property:Dummy_agents_cannot_get_adjacent_vertices} implies that the coloring is proper. Furthermore, due to envy-freeness of $A$, agents with identical valuations must have equal utilities. Therefore, each dummy agent gets the same number of vertex goods, implying that the coloring is equitable. This completes the proof of \Cref{prop:EF_Existence_NPcomplete_BinaryValuations}. \end{proof}
\subsection{Proof of Theorem~\ref{thm:HEFk_Verification_ApproxAlgo}} \label{subsec:Proof_HEFk_Verification_ApproxAlgo}
Recall the statement of \Cref{thm:HEFk_Verification_ApproxAlgo}.
\HEFkVerificationApproxAlgo*
Recall from \Cref{sec:Theoretical_Results} that given any allocation $A$, the \emph{residual envy function} $f : 2^{[m]} \rightarrow \mathbb{R}$ is defined as follows: \begin{align*}
f(S) \coloneqq \sum_{h \in [n]} \sum_{i \neq h} \max\{0, v_i(A_h \setminus S) - v_i(A_i)\}. \end{align*}
Here, $f(S)$ is the aggregate envy in $A$ after hiding the goods in $S \subseteq [m]$. We will show in \Cref{lem:Supermodularity} that $f$ is \emph{supermodular}, i.e., for any pair of sets $S,T \subseteq [m]$ such that $S \subseteq T$ and any good $j \notin T$, $f(S) - f(S \cup \{j\}) \geq f(T) - f(T \cup \{j\})$. The proof of \Cref{thm:HEFk_Verification_ApproxAlgo} will then follow from the standard greedy algorithm for submodular maximization, or, equivalently, supermodular minimization~ \citep{NWF78analysis}.
\begin{restatable}{lemma}{Supermodularity}
\label{lem:Supermodularity}
The residual envy function $f$ is supermodular. \end{restatable} \begin{proof} We will start with the necessary notation. For any agent $h \in [n]$ and any other agent $i \in [n] \setminus \{h\}$, define $f_{h,i}(S) \coloneqq \max\{0, v_i(A_h \setminus S) - v_i(A_i)\}$ as the envy of $i$ towards $h$ after hiding the goods in $S$. Also, let $f_h(S) \coloneqq \sum_{i \neq h} f_{h,i}(S)$ denote the total (aggregate) envy towards $h$. We therefore have $f(S) = \sum_{h \in [n]} f_h(S) = \sum_{h \in [n]} \sum_{i \neq h} f_{h,i}(S)$.
Notice that $f_{h,i}$ is a monotone non-increasing set function, i.e., for any $S \subseteq T$, we have $f_{h,i}(S) \geq f_{h,i}(T)$. Also notice that for any $T \subseteq [m]$ and any $j \in [m] \setminus T$, we have that $f_{h,i}(T) - f_{h,i}(T \cup \{j\}) \leq v_{i,j}$.
For any set of goods $S \subseteq [m]$ and any agent $h \in [n]$, define $E_h(S) \coloneqq \{i \in [n] : f_{h,i}(S) > 0\}$ as the set of agents that envy agent $h$ even after the goods in $S$ are hidden. Notice that if $S \subseteq T$, then $E_h(T) \subseteq E_h(S)$. Thus, if for some agent $i$ we have that $i \notin E_h(S)$, then $i \notin E_h(T)$, and therefore $f_{h,i}(S) = f_{h,i}(T) = 0$.
Define $N_j \coloneqq \{i \in [n] : v_{i,j} > 0\}$ as the set of agents that have a strictly positive valuation for the good $j$.
We will now prove that $f$ is supermodular, i.e., for any $S \subseteq T$ and any good $j \notin T$, $f(S) - f(S \cup \{j\}) \geq f(T) - f(T \cup \{j\})$. Let $r \in [n]$ be the owner of good $j$ under $A$, i.e., $j \in A_r$. Notice that if $i \notin N_j$ (i.e., $v_{i,j}=0$), then additivity of valuations implies $v_i(A_r \setminus S) = v_i(A_r \setminus S \cup \{j\})$. Thus, \begin{align}
f(S) - f(S \cup \{j\}) & = f_r(S) - f_r(S \cup \{j\}) \nonumber \\
& = \sum_{i \neq r} f_{r,i}(S) - f_{r,i}(S \cup \{j\}) \nonumber \\
& = \sum_{i \in E_r(S)} f_{r,i}(S) - f_{r,i}(S \cup \{j\}) \nonumber \\
& = \sum_{i \in E_r(S) \cap N_j} f_{r,i}(S) - f_{r,i}(S \cup \{j\}),\label{eqn:supermodular_temp1} \end{align} where the first equality uses the fact that for any $h \neq r$, we have $f_h(S) = f_h(S \cup \{j\})$, the third equality uses the fact that if $i \notin E_r(S)$, then $f_{r,i}(S) = f_{r,i}(S \cup \{j\}) = 0$, and the fourth equality uses the fact that $v_i(A_r \setminus S) = v_i(A_r \setminus S \cup \{j\})$ whenever $i \notin N_j$. By a similar reasoning for the set $T$, we get that \begin{align} f(T) &- f(T \cup \{j\}) = \sum_{i \in E_r(T) \cap N_j} f_{r,i}(T) - f_{r,i}(T \cup \{j\}). \label{eqn:supermodular_temp2} \end{align}
Recall that $E_r(T) \subseteq E_r(S)$. Therefore, \Cref{eqn:supermodular_temp1} can be rewritten as \begin{align}
f(S) - f(S \cup \{j\}) & = \sum_{i \in E_r(T) \cap N_j} f_{r,i}(S) - f_{r,i}(S \cup \{j\}) + \nonumber \\
& \qquad \sum_{i \in E_r(S) \setminus E_r(T) \cap N_j} f_{r,i}(S) - f_{r,i}(S \cup \{j\}) \nonumber \\
& \geq \sum_{i \in E_r(T) \cap N_j} f_{r,i}(S) - f_{r,i}(S \cup \{j\}), \label{eqn:supermodular_temp3} \end{align} where the inequality follows from the use of the monotonicity of $f_{r,i}$ for all $i \in E_r(S) \setminus E_r(T) \cap N_j$.
Therefore, from \Cref{eqn:supermodular_temp2,eqn:supermodular_temp3}, it suffices to show that for every $i \in E_r(T) \cap N_j$, $f_{r,i}(S) - f_{r,i}(S \cup \{j\}) \geq f_{r,i}(T) - f_{r,i}(T \cup \{j\})$. We will prove this by contradiction.
Suppose, for contradiction, that for some $i \in E_r(T) \cap N_j$, we have $f_{r,i}(S) - f_{r,i}(S \cup \{j\}) < f_{r,i}(T) - f_{r,i}(T \cup \{j\})$. Then, we must have $i \in E_r(S \cup \{j\})$, since otherwise we get $i \notin E_r(T \cup \{j\})$ and therefore $f_{r,i}(S \cup \{j\}) = f_{r,i}(T \cup \{j\}) = 0$. This would imply that $f_{r,i}(S) < f_{r,i}(T)$, which contradicts the monotonicity of $f_{r,i}$. Hence, for any $i \in E_r(T) \cap N_j$, we also have that $i \in E_r(S \cup \{j\})$.
Notice that for any $i \in E_r(S \cup \{j\}) \cap N_j$, we have $f_{r,i}(S) - f_{r,i}(S \cup \{j\}) = v_{i,j}$ by the additivity of valuations. However, this would require that $f_{r,i}(T) - f_{r,i}(T \cup \{j\}) > v_{i,j}$, which is a contradiction. Therefore, the function $f$ must be supermodular. \end{proof}
We are now ready to prove \Cref{thm:HEFk_Verification_ApproxAlgo}.
\begin{proof} (of \Cref{thm:HEFk_Verification_ApproxAlgo})
Note that allocation $A$ is \HEF{} with respect to a set $S$ if and only if $f(S) \leq 0$. For integral valuations, $f(S) \leq 0$ if and only if $f(S) < 1$. Therefore, it suffices to compute a set $S$ in polynomial time such that $|S| \leq k^{\textup{opt}} \cdot \ln E + 1$ and $f(S) < 1$.
Consider the greedy algorithm described in Algorithm~\ref{alg:Greedy_HEFk_ApproxAlgo}. \begin{algorithm}[t]
\DontPrintSemicolon
\KwIn{An instance $\langle [n], [m], \mathcal{V} \rangle$ and an allocation $A$.}
\KwOut{A set $S \subseteq [m]$.}
\BlankLine
Initialize $S = \emptyset$.\;
\While{$f(S) \geq 1$}{
Set $j' \leftarrow \arg\max_{j \in [m] \setminus S} f(S) - f(S \cup \{j\})$\;
\Comment*[r]{tiebreak lexicographically}
Update $S \leftarrow S \cup \{j'\}$
}
\KwRet $S$
\caption{Greedy Approximation Algorithm for \textup{\textsc{\HEF{k}-Verification}}{}} \label{alg:Greedy_HEFk_ApproxAlgo} \end{algorithm}
At each step, the algorithm adds to the current set the good that provides the largest reduction in the residual envy. This process is continued as long as $f(S) \geq 1$. Since there are $m$ goods, it is clear that the algorithm terminates in at most $m$ steps. Furthermore, from the above observation, it follows that the allocation $A$ is \HEF{} with respect to the set $S$ returned by the algorithm. Therefore, all that remains to be shown is a bound on $|S|$.
Observe that $f(\emptyset) = E$. Recall from the proof of \Cref{lem:Supermodularity} that $f$ is a sum of monotone non-increasing set functions, and is therefore itself monotone non-increasing. Define another set function $g: 2^{[m]} \rightarrow \mathbb{R}$ as follows: \begin{align*}
g(S) \coloneqq E - f(S). \end{align*}
Notice that $g$ is a non-negative, monotone non-decreasing, and integer-valued submodular function with $g(\emptyset) = 0$. Therefore, our goal is to find a set $S$ such that $g(S) > E - 1$.
We will now use the result of \citet{NWF78analysis} for submodular maximization stated below as \Cref{prop:Submodular_Greedy_Approx}. In particular, let $p \coloneqq k^{\textup{opt}}$ be the size of the optimal hidden set (i.e., the number of goods that must be hidden under $A$). Then,
$$\max\limits_{S : |S|\leq p} g(S) = E - \min\limits_{S : |S|\leq k^{\textup{opt}}} f(S) = E.$$
From the bound in \Cref{prop:Submodular_Greedy_Approx}, we have that \begin{alignat*}{2}
& \qquad & (1 - e^{-q/p}) \max\limits_{S : |S|\leq p} g(S) & > E - 1 \\
& \Longleftrightarrow & (1 - e^{-q/k^{\textup{opt}}}) \cdot E & > E - 1 \\
& \Longleftrightarrow & 1 - e^{-q/k^{\textup{opt}}} & > 1 - 1/E \\
& \Longleftrightarrow & \ln \frac{1}{E} & > -q/k^{\textup{opt}} \\
& \Longleftrightarrow & q & > k^{\textup{opt}} \ln E. \end{alignat*}
Thus, after $q > k^{\textup{opt}} \ln E$ steps, any set $S$ constructed by the algorithm satisfies $g(S) > E - 1$, or, equivalently, $f(S) < 1$, giving us the desired bound $|S| \leq k^{\textup{opt}} \ln E + 1$. This completes the proof of \Cref{thm:HEFk_Verification_ApproxAlgo}. \end{proof}
\begin{restatable}[\citet{NWF78analysis}, \citet{KG14submodular}]{prop}{SubmodularGreedyApprox}
\label{prop:Submodular_Greedy_Approx}
Let $g: 2^{[m]} \rightarrow \mathbb{R}_{\geq 0}$ be a monotone non-decreasing submodular function, and let $\{S_i\}_{i \geq 0}$ be the sequence of sets constructed in Algorithm~\ref{alg:Greedy_HEFk_ApproxAlgo}. Then, for any positive integers $p$ and $q$, we have that
\begin{align*}
g(S_q) \geq (1 - e^{-q/p}) \max\limits_{S : |S|\leq p} g(S).
\end{align*} \end{restatable}
\subsection{\textup{\texttt{MNW}}{} can have large regret in the worst-case} \label{subsec:MNW_Large_Regret}
This section presents two results concerning the worst-case regret of \textup{\texttt{MNW}}{} solution. In \Cref{prop:NSW_vs_HEFk}, we will provide a family of instances for which the normalized regret of \textup{\texttt{MNW}}{} approaches $1$ (i.e., the maximum possible value). In \Cref{prop:NSW_vs_HEFk_Binary_Vals}, we will show a slightly weaker limit ($\nicefrac{1}{2}$ instead of $1$) that holds even for the restricted domain of binary valuations. We will use $\kappa^{\textup{\texttt{opt}}}(\mathcal{I}) \coloneqq \min_{A} \kappa(A,\mathcal{I})$ to denote the smallest number of goods that must be hidden under any allocation in the instance $\mathcal{I}$.
\begin{restatable}{prop}{NSWRegret}
\label{prop:NSW_vs_HEFk} There exists a family of instances for which the normalized regret of any Nash optimal allocation approaches $1$ in the limit. \end{restatable} \begin{proof} Consider the fair division instance $\mathcal{I}$ with five agents $a_1,\dots,a_5$ and five goods $g_1,\dots,g_5$ shown in \Cref{tab:Instance_NSW_vs_HEFk}. \begin{table} \centering
\begin{tabular}{ c|cccccc }
& $g_1$ & $g_2$ & $g_3$ & $g_4$ & $g_5$\\ \hline
$a_1$ & $1$ & $0$ & $0$ & $0$ & $0$\\
$a_2$ & $10$ & $1$ & $0$ & $0$ & $0$\\
$a_3$ & $0$ & $10$ & $1$ & $0$ & $0$\\
$a_4$ & $0$ & $0$ & $10$ & $1$ & $0$\\
$a_5$ & $0$ & $0$ & $0$ & $10$ & $1$\\
\end{tabular}
\caption{The instance used in proof of \Cref{prop:NSW_vs_HEFk}.}
\label{tab:Instance_NSW_vs_HEFk} \end{table} The unique Nash optimal allocation (say $A$) for this instance assigns $g_i$ to $a_i$ for every $i \in [5]$. Thus, the goods $g_1$, $g_2$, $g_3$, $g_4$ must be hidden under $A$, i.e., $\kappa(A,\mathcal{I}) = 4$. On the other hand, an allocation (say $B$) that assigns $g_5$ to $a_1$, and $g_{i-1}$ to $a_i$ for every $i \in \{2,\dots,5\}$ only needs to hide the good $g_1$. Indeed, $\kappa^{\textup{\texttt{opt}}}(\mathcal{I}) = 1$ since any allocation must hide $g_1$ to avoid envy from $a_1$ or $a_2$. The desired family of instances is the natural extension of the above example to $n$ agents and $n$ goods. In the limit, the normalized regret of the Nash optimal allocation is $\lim_{n \rightarrow \infty} \frac{(n-1) - 1}{n-1} = 1$. \end{proof}
\begin{restatable}{prop}{NSWvsHEFBinary}
\label{prop:NSW_vs_HEFk_Binary_Vals} There exists a family of instances with binary valuations for which the normalized regret of any Nash optimal allocation approaches $\nicefrac{1}{2}$ in the limit. \end{restatable} \begin{proof} Fix some $t \in {\mathbb{N}}$. Consider an instance $\mathcal{I}_n$ with $2t+1$ agents, consisting of $t$ groups of \emph{ordinary agents} $\{a_{i},b_{i}\}_{i \in [t]}$ and one \emph{special agent} $s$. The goods are also classified into $t$ groups, with group $i$ comprising of five goods $g_{i,1},\dots,g_{i,5}$. For each $i \in [t]$, both $a_i$ and $b_i$ approve all five goods in group $i$ and have zero value for all the other goods. The special agent $s$ approves all the goods.
The above instance admits an envy-free allocation $A$ in which $s$ gets one good from each group, and the other goods are allocated evenly among the group members. That is, for each $i \in [t]$, $a_{i}$ gets $\{g_{i,1},g_{i,2}\}$, $b_{i}$ gets $\{g_{i,3},g_{i,4}\}$, and $s$ gets $g_{i,5}$. Thus, $\kappa^{\textup{\texttt{opt}}}(\mathcal{I}_n) = 0$.
Let $B$ denote any Nash optimal allocation. It is easy to see that $B$ is of one of the following two canonical forms: \begin{itemize}
\item Either $s$ gets two goods from two different groups and the rest of the goods are assigned `evenly,' i.e., for each $i \in [t-2]$, $a_i$ gets $\{g_{i,1},g_{i,2},g_{i,3}\}$ and $b_i$ gets $\{g_{i,4},g_{i,5}\}$, and for $i \in \{t-1,t\}$, $a_i$ gets $\{g_{i,1},g_{i,2}\}$, $b_i$ gets $\{g_{i,3},g_{i,4}\}$ and $s$ gets $g_{i,5}$,
\item or, $s$ gets three goods from three different groups and the other goods are assigned `evenly,' i.e., for each $i \in [t-3]$, $a_i$ gets $\{g_{i,1},g_{i,2},g_{i,3}\}$ and $b_i$ gets $\{g_{i,4},g_{i,5}\}$, and for $i \in \{t-2,t-1,t\}$, $a_i$ gets $\{g_{i,1},g_{i,2}\}$, $b_i$ gets $\{g_{i,3},g_{i,4}\}$ and $s$ gets $g_{i,5}$. \end{itemize}
Either way, $B$ must hide at least $t-3$ goods (one good in each of the groups $1,\dots,t-3$ to avoid envy from $b_i$). Thus, $\texttt{\textup{reg}}(B,\mathcal{I}_n) = \kappa(B,\mathcal{I}_n) = t-3$.
The desired family of instances can be obtained by choosing an arbitrarily large $t$. In the limit, the normalized regret of $B$ is $\lim_{t \rightarrow \infty} \frac{t-3}{2t} = \frac{1}{2}$. \end{proof}
\subsection{Additional Experimental Results} \label{subsec:Additional_Experiments}
\Cref{tab:Expt_BinaryVals_bias_0.7_Part2} presents additional results for the synthetic data used in \Cref{subsec:Expt_Synthetic} (i.e., binary valuations with $v_{i,j} \sim \textrm{\textup{Ber}}(0.7)$ i.i.d.). This time, we compare the algorithms in terms of their (a) normalized worst-case regret (over the $100$ instances), (b) the frequency with which the algorithms compute envy-free outcomes, and (c) the worst-case number of goods that must be hidden by each algorithm. The trend is similar to that in \Cref{subsec:Expt_Synthetic}, with \textup{\texttt{Alg-EF1+PO}}{} and \textup{\texttt{RoundRobin}}{} outperforming \textup{\texttt{MNW}}{} and \textup{\texttt{EnvyGraph}}{}. \Cref{tab:Expt_BinaryVals_bias_0.5} presents similar results for Bernoulli parameter $0.5$. Finally, \Cref{fig:Spliddit_data_distribution} illustrates the distribution of the Spliddit data. As can be seen, a large fraction of instances have between $3$ and $6$ agents and between $3$ and $15$ goods, with a sharp spike at $n=3$ and $m=6$.
\begin{table*} \centering
\begin{tabular}{|cccc|}
\multicolumn{4}{c}{}\\
\multicolumn{4}{c}{\textbf{Normalized worst-case regret}}\\
\hline
\footnotesize{\textup{\texttt{Alg-EF1+PO}}{}} & \footnotesize{\textup{\texttt{RoundRobin}}{}} & \footnotesize{\textup{\texttt{MNW}}{}} & \footnotesize{\textup{\texttt{EnvyGraph}}{}}\\
\includegraphics[width=0.22\textwidth]{market_maxregret_7_binary_norm-eps-converted-to} & \includegraphics[width=0.22\textwidth]{rr_maxregret_7_binary_norm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{mnw_binary_maxregret_7_binary_norm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{envy_graph_maxregret_7_binary_norm-eps-converted-to} \\
\hline
\multicolumn{4}{c}{}\\
\multicolumn{4}{c}{\textbf{Frequency of envy-freeness}}\\
\hline
\footnotesize{\textup{\texttt{Alg-EF1+PO}}{}} & \footnotesize{\textup{\texttt{RoundRobin}}{}} & \footnotesize{\textup{\texttt{MNW}}{}} & \footnotesize{\textup{\texttt{EnvyGraph}}{}}\\
\includegraphics[width=0.22\textwidth]{market_notef_7_binary_notnorm-eps-converted-to} & \includegraphics[width=0.22\textwidth]{rr_notef_7_binary_notnorm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{mnw_binary_notef_7_binary_notnorm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{envy_graph_notef_7_binary_notnorm-eps-converted-to} \\
\hline
\multicolumn{4}{c}{}\\
\multicolumn{4}{c}{\textbf{Number of goods that must be hidden in the worst-case} (max over all $100$ instances)}\\
\hline
\footnotesize{\textup{\texttt{Alg-EF1+PO}}{}} & \footnotesize{\textup{\texttt{RoundRobin}}{}} & \footnotesize{\textup{\texttt{MNW}}{}} & \footnotesize{\textup{\texttt{EnvyGraph}}{}}\\
\includegraphics[width=0.22\textwidth]{market_maxk_7_binary_notnorm-eps-converted-to} & \includegraphics[width=0.22\textwidth]{rr_maxk_7_binary_notnorm-eps-converted-to} &
\includegraphics[width=0.223\textwidth]{mnw_binary_maxk_7_binary_notnorm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{envy_graph_maxk_7_binary_notnorm-eps-converted-to} \\
\hline
\end{tabular}
\caption{Comparing various \EF{1} algorithms over synthetically generated binary instances with $v_{i,j} \sim \textrm{\textup{Ber}}(0.7)$ i.i.d.}
\label{tab:Expt_BinaryVals_bias_0.7_Part2} \end{table*}
\begin{figure*}
\caption{Distribution of the Spliddit data. The color of each cell denotes the number of instances in the dataset with the corresponding number of goods, $m$, on the X axis, and number of agents, $n$, on the Y axis.}
\label{fig:Spliddit_data_distribution}
\end{figure*}
\begin{table*} \centering
\begin{tabular}{|cccc|}
\multicolumn{4}{c}{\textbf{Normalized average-case regret}}\\
\hline
\footnotesize{\textup{\texttt{Alg-EF1+PO}}{}} & \footnotesize{\textup{\texttt{RoundRobin}}{}} & \footnotesize{\textup{\texttt{MNW}}{}} & \footnotesize{\textup{\texttt{EnvyGraph}}{}}\\
\includegraphics[width=0.22\textwidth]{market_regret_5_binary_norm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{rr_regret_5_binary_norm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{mnw_binary_regret_5_binary_norm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{envy_graph_regret_5_binary_norm-eps-converted-to} \\
\hline
\multicolumn{4}{c}{}\\
\multicolumn{4}{c}{\textbf{Normalized worst-case regret}}\\
\hline
\footnotesize{\textup{\texttt{Alg-EF1+PO}}{}} & \footnotesize{\textup{\texttt{RoundRobin}}{}} & \footnotesize{\textup{\texttt{MNW}}{}} & \footnotesize{\textup{\texttt{EnvyGraph}}{}}\\
\includegraphics[width=0.22\textwidth]{market_maxregret_5_binary_norm-eps-converted-to} & \includegraphics[width=0.22\textwidth]{rr_maxregret_5_binary_norm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{mnw_binary_maxregret_5_binary_norm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{envy_graph_maxregret_5_binary_norm-eps-converted-to} \\
\hline
\multicolumn{4}{c}{}\\
\multicolumn{4}{c}{\textbf{Frequency of envy-freeness}}\\
\hline
\footnotesize{\textup{\texttt{Alg-EF1+PO}}{}} & \footnotesize{\textup{\texttt{RoundRobin}}{}} & \footnotesize{\textup{\texttt{MNW}}{}} & \footnotesize{\textup{\texttt{EnvyGraph}}{}}\\
\includegraphics[width=0.22\textwidth]{market_notef_5_binary_notnorm-eps-converted-to} & \includegraphics[width=0.22\textwidth]{rr_notef_5_binary_notnorm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{mnw_binary_notef_5_binary_notnorm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{envy_graph_notef_5_binary_notnorm-eps-converted-to} \\
\hline
\multicolumn{4}{c}{}\\
\multicolumn{4}{c}{\textbf{Number of goods that must be hidden on average} (averaged over non-\EF{} instances only)}\\
\hline
\footnotesize{\textup{\texttt{Alg-EF1+PO}}{}} & \footnotesize{\textup{\texttt{RoundRobin}}{}} & \footnotesize{\textup{\texttt{MNW}}{}} & \footnotesize{\textup{\texttt{EnvyGraph}}{}}\\
\includegraphics[width=0.22\textwidth]{market_k_5_binary_notnorm-eps-converted-to} & \includegraphics[width=0.22\textwidth]{rr_k_5_binary_notnorm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{mnw_binary_k_5_binary_notnorm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{envy_graph_k_5_binary_notnorm-eps-converted-to} \\
\hline
\multicolumn{4}{c}{}\\
\multicolumn{4}{c}{\textbf{Number of goods that must be hidden in the worst-case} (max over all $100$ instances)}\\
\hline
\footnotesize{\textup{\texttt{Alg-EF1+PO}}{}} & \footnotesize{\textup{\texttt{RoundRobin}}{}} & \footnotesize{\textup{\texttt{MNW}}{}} & \footnotesize{\textup{\texttt{EnvyGraph}}{}}\\
\includegraphics[width=0.22\textwidth]{market_maxk_5_binary_notnorm-eps-converted-to} & \includegraphics[width=0.22\textwidth]{rr_maxk_5_binary_notnorm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{mnw_binary_maxk_5_binary_notnorm-eps-converted-to} &
\includegraphics[width=0.22\textwidth]{envy_graph_maxk_5_binary_notnorm-eps-converted-to} \\
\hline
\end{tabular}
\caption{Comparing various \EF{1} algorithms over synthetically generated binary instances with $v_{i,j} \sim \textrm{\textup{Ber}}(0.5)$ i.i.d.}
\label{tab:Expt_BinaryVals_bias_0.5} \end{table*}
\end{document} | arXiv |
\begin{document}
\title{Toward $L_\infty$-recovery of Nonlinear Functions: A Polynomial Sample Complexity Bound for Gaussian Random Fields}
\begin{abstract}
Many machine learning applications require learning a function with a small worst-case error over the entire input domain, that is, the $L_\infty$-error, whereas most existing theoretical works only guarantee recovery in average errors such as the $L_2$-error.
$L_\infty$-recovery from polynomial samples is even impossible for seemingly simple function classes such as constant-norm infinite-width two-layer neural nets. This paper makes some initial steps beyond the impossibility results by leveraging the randomness in the ground-truth functions. We prove a polynomial sample complexity bound for random ground-truth functions drawn from Gaussian random fields. Our key technical novelty is to prove that the degree-$k$ spherical harmonics components of a function from Gaussian random field cannot be spiky in that their $L_\infty$/$L_2$ ratios are upperbounded by $O(d \sqrt{\ln k})$ with high probability. In contrast, the worst-case $L_\infty$/$L_2$ ratio for degree-$k$ spherical harmonics is on the order of $\Omega(\min\{d^{k/2},k^{d/2}\})$.
\end{abstract}
\section{Introduction}\label{sec:intro}
Classical statistical learning theory primarily concerns with recovering functions from examples with small errors \textit{averaged} over a distribution of inputs, e.g., the mean-squared error (that is, the $L_2$-error with respect to the test distribution). However, the worst-case error over the entire input domain, that is, the $L_\infty$-error, is crucial for many applications, and also challenging to achieve. For example, an $L_\infty$-error recovery guarantee will make the learned function more robust to adversarial examples, while standard training is vulnerable~\citep{goodfellow2015explaining,madry2017towards}. The $L_\infty$-recovery is also necessary for many applications where the recovered models will be further used in a downstream decision making process, such as model-based bandits~\citep{huang2021optimal}, reinforcement learning~\citep{huang2021going,sutton2018reinforcement}, and physics informed neural networks \citep{raissi2019physics,wang20222}. In particular, recent theoretical works on deep reinforcement learning heavily rely on the $L_\infty$-recovery of the $Q$-function to prevent the actions from misusing a small worst-case region of inputs where the error is much larger than the average error~\citep{huang2021going}. (See Section~\ref{sec:relatedwork} for more discussions on the applications.)
This paper focuses on $L_\infty$-error recovery of nonlinear functions from polynomial samples. For a compact domain $D$, we aim to learn a function that is \emph{pointwise} close to the ground-truth over the entire domain $D$. Formally, given polynomial random input-output pairs from a ground-truth function $f$, our goal is to learn a function $g$ with small $L_\infty$-distance/error to $f$, defined by $$\|f-g\|_\infty\defeq\sup_{x\in D}|f(x)-g(x)|.$$
For linear function class, we can straightforwardly $L_\infty$-recovery guarantees by relating the $L_\infty$-error to the $L_2$-error on the test distribution, which in turn can be bounded standard tools such as uniform convergence~(e.g., \citet{bartlett2002rademacher,koltchinskii2002empirical,wei2019improved}). Concretely, suppose $P$ is the training/test distribution on domain $D$ and the covariance matrix $\Sigma =\mathbb{E}_{x\sim P}[xx^\top]$ is full-rank, we have for any linear functions $f$ and $g$, \begin{align}
\textstyle{\|f-g\|_\infty\le \left(\sup_{x\in D}\|x\|_2 \right) \cdot \lambda_{\min}(\Sigma)^{-1/2} \cdot \|f-g\|_{L_2(P)} \,.} \label{eqn:2} \end{align}
where $\lambda_{\min}(\Sigma)$ is the minimum eigenvalue of $\Sigma$ and $\|f-g\|_{L_2(P)} = \left(\mathbb{E}_{x\sim P} (f(x)-g(x))^2\right)^{1/2}$ is the squared error on the distribution $P$. Therefore, we can reduce the $L_\infty$-recovery to $L_2$-recovery, and the inequality above is tight for most scenarios.
In contrast, $L_\infty$-recovery of nonlinear functions is much more challenging. When the model's parameters are identifiable and can be recovered, e.g., for finite-width two-layer neural nets (without biases)~\citep{zhong2017recovery,zhou2021local} or low-degree polynomials \citep{huang2021going}, $L_\infty$-recovery of the functions follows straightforwardly from parameter recovery. Parameter recovery fundamentally requires the sample size to be larger than parameter dimension, and therefore does not apply to the over-parameterized settings that are ubiquitous in modern machine learning (\citet{zagoruyko2016wide,du2018gradient,allen2019learning,zhang2021understanding} and references therein) or infinite dimensional features in the kernel method settings. Existing $L_\infty$-recovery algorithms require quasi-polynomial or exponential in dimension samples for two-layer neural networks~\citep{mhaskar2006weighted,mhaskar2019function} or general very smooth functions (with decaying higher-order derivatives)~\citep{vybiral2014weak,krieg2019uniform} .
In fact, $L_\infty$-recovery from polynomial samples is impossible for even seemingly simple function classes, such as two-layer single-neuron neural nets with bias~\citep{dong2021provable,li2021eluder} or constant-norm infinite-width two-layer neural nets without bias (Theorem~\ref{thm:relu-lb} of this paper). The fundamental challenge is that these function classes contain many \emph{spiky} functions $f$ such that $\|f\|_2 \ll \|f\|_\infty$, which means that inequalities analogous to Eq.~\eqref{eqn:2} cannot hold. Moreover, these functions may mostly have tiny values except a spike on an exponentially-small region. Likely, none of the polynomial number of examples falls into the spiky region. As a result, the spike cannot be identified, and $L_\infty$-recovery cannot be achieved.
Interestingly, $L_\infty$-recovery of functions in reproducing kernel Hilbert space (RKHS) with polynomial samples is still a challenging open question, even though $L_2$-recovery with polynomial samples and time has been well established \citep{bartlett2002rademacher,hofmann2008kernel}. Even though they are essentially linear functions with an infinite dimensional features, analysis analogous to linear models (e.g., Eq~\eqref{eqn:2}) is vacuous because the covariance of the features, that is the kernel function, typically has a sequence of eigenvalues that decays to zero. In fact, $L_\infty$-recovery of functions with constant RKHS norm for various kernels (e.g., the radial basis function (RBF) kernels kernel) requires exponential number of samples~\citep{scarlett2017lower,kuo2008multivariate}. Intuitively, this is because RKHS still contains spiky functions, e.g., the $k$-th eigenfunctions of the RBF kernel (or any inner product kernel) on the unit sphere can be spiky for relatively large $k$.
Towards going beyond these intractability results and achieve polynomial sample complexity bounds, we make additional randomized and smooth assumptions on the ground-truth functions that we aim to recover. We essentially assume that the ground-truth function has decaying and random high-frequency components.
Concretely, we work with random ground-truth functions $f$ drawn from a Gaussian random field (GRF, also known as Gaussian process) on the unit sphere \citep{seeger2004gaussian,lang2015isotropic}. We assume that the covariance (or kernel) function, denoted by $K:\mathbb{S}^{d-1}\times\mathbb{S}^{d-1}\to\mathbb{R}$, is an inner product function given by $K(x,x')=\kappa(x^\top x')$ for some function $\kappa:[-1,1]\to\mathbb{R}$, which means that the GRF is isotropic.
All the inner product kernels $\kappa(x^\top x')$ on the unit sphere share the same eigenfunctions called spherical harmonics \citep{atkinson2012spherical}. This brings opportunities for us to use the spherical harmonics tools to analyze the problem. Spherical harmonics form a complete set of basis for the square integrable functions over the sphere (in analog to the Fourier basis in $\mathbb{R}^d$). Intuitively, fast decay of the spherical harmonics components of the function implies the function is smoother. Moreover, some higher-degree spherical harmonics can be more spiky and challenging to recover in $L_\infty$-error.
Our main result is an $L_\infty$-recovery algorithm (Alg.~\ref{alg:main}) with polynomial sample complexity for a random ground-truth $f$ drawn from Gaussian random fields, given that the $k$-th eigenvalues of the covariance function $K$ decays at a rate $O(k^{-(1+\alpha) d})$ for any universal constant $\alpha>0$ (Theorem~\ref{thm:mainsc}). This decay rate is equivalent to that the degree-$k$ spherical harmonics component of $f$ is on the order $O(k^{-\alpha d/2})$. We note that the randomness from the Gaussian random field is the key for us to work with this decay (that is, $\alpha > 0$), because a worst-case function with $O(k^{-d/2})$ decay in the spherical harmonics components is impossible to recover with polynomial samples (Lemma~\ref{lem:relu-lb}). This suggest that the randomness property in the ground-truth function makes the $L_\infty$-recovery much easier. Moreover, for comparison, $L_2$-recovery with polynomial sample is only possible when $\alpha \ge 0$.\footnote{The $\alpha = 0$ case is subtle for both $L_2$- and $L_\infty$-recovery and we leave it as an open question for future work. } In other words, the randomness assumption qualitatively makes $L_\infty$-recovery as easy as $ L_2$-recovery.
Our main technique is to prove a much tighter upper bound for the $L_\infty$-norm of the high-degree components of the ground-truth $f$ using the randomness. Lemma~\ref{lem:GRF} shows that the high-degree components of $f$ drawn from Gaussian random processes are \emph{not} spiky: their $L_\infty$/$L_2$ ratios are upper bounded by $O(d\sqrt{\ln k})$ with high probability, whereas the worst-case ratio is $\Omega(\min\{d^{k/2},k^{d/2}\})$. This lemma might be of independent interest as it extends \citet[Theorem 1]{burq2014probabilistic} to the high-dimensional case with a precise bound on the dependency on $d$. It was not known that the dependency on $d$ is polynomial. Our proof is also surprisingly much simpler than that in~\citet{burq2014probabilistic}. Hence, when the eigenvalues of kernel $K$ decays, we can truncate $f$ at degree $\widetilde{O}(1)$ and get a low-degree polynomial approximation with small $L_\infty$-error (Lemma~\ref{lem:main}).
The rest of this paper is organized as follows. Section~\ref{sec:relatedwork} discusses additional related works and the applications of $L_\infty$-recovery to bandits and reinforcement learning problems. In Section~\ref{sec:preliminary} we give a concise overview of the spherical harmonics, the important tools in this paper. Section~\ref{sec:main-results} states our algorithm and proves a polynomial sample complexity bound for recovering random functions from a Gaussian random field. \Cref{sec:lowerbounds} proves that $L_\infty$-recovery is impossible for two-layer neural nets without bias (Theorem~\ref{thm:relu-lb}), which may be of independent interest.
\paragraph{Additional notations.} Let $\mathbb{S}^{d-1}$ be the $(d-1)$-dimensional unit sphere. We assume that the training distribution is uniform over the $\mathbb{S}^{d-1}$.
That is, the data $x_i$ is sampled independently and uniformly from the sphere, and $y_i=f(x_i)+{\mathcal{N}}(0,1)$ where $y$ is the ground-truth. With slight abuse of notations, we also use $\mathbb{S}^{d-1}$ to denote the uniform distribution over the unit sphere. For a function $g:\mathbb{S}^{d-1}\to\mathbb{R}$, let $\|g\|_p\defeq \mathbb{E}_{x\sim \mathbb{S}^{d-1}}[g(x)^p]^{1/p}$ be its $L_p$-norm with respect to the uniform distribution. For two functions $g,h:\mathbb{S}^{d-1}\to\mathbb{R}$, $\dotp{g}{h}\defeq \mathbb{E}_{x\sim \mathbb{S}^{d-1}}[h(x)g(x)]$ denotes their inner product. For a function $h:\mathbb{R}\to\mathbb{R}$, we use $h^{(k)}$ to denote its $k$-th derivative.
In the following, for two non-negative sequences $a_{k},b_{k}$, we write $a_k=O(b_k)$ or $a_k\lesssim b_k$ if there exists an \emph{absolute} constant $c$ such that $a_k\le c b_k$ for every $k\ge 0$. We write $a_k=\widetilde{O}(1)$ if $a_k=O(\mathrm{polylog}(k))$, and $a_k=\Theta(b_k)$ if $a_k\lesssim b_k$ and $b_k\lesssim a_k.$
\section{Related Works}\label{sec:relatedwork}
\sloppy The classical uniform convergence framework (e.g., \citet{bartlett2002rademacher,koltchinskii2002empirical,kakade2009complexity,bartlett2017spectrally,wei2019improved}) does not directly solve the $L_\infty$-recovery problem. This is because for any $p>0$, approximating the $L_p$-error (defined by $\|f-g\|_p\defeq\mathbb{E}_{x\sim D}[|f(x)-g(x)|^p]^{1/p}$) with $\epsilon$ precision requires $\mathrm{poly}(\epsilon^{-p})$ samples. Hence, as $p\to\infty$, we cannot avhieve uniform convergence using polynomial samples, meaning that bounding the $L_\infty$-error of the learned function requires novel analysis.
\paragraph{Gaussian process bandits and kernelized bandits.} A closely related line of research is the Gaussian process bandits. Instead of learning a function with small $L_\infty$-error, Gaussian process bandit algorithms aim to find a $x$ that maximizes the function $f(x)$ when $f$ is drawn from a Gaussian process \citep{grunewalder2010regret}. Most of the existing results focuses on radial basis function kernel and Mat\'ern kernels, and the regret is exponential in the ambient dimension $d$ \citep{srinivas2009gaussian,krause2011contextual,shekhar2018gaussian,vakili2021scalable}. For Gaussian processes with non-isotropic kernels, \citet{grunewalder2010regret} prove exponential regret upper and lower bounds. For general kernels, \citet{lederer2019uniform,lederer2021uniform} proves a $L_\infty$-error bounds for Gaussian process regression with no assumption on the spectrum of the covariance, and the sample complexity is also exponential.
Another line of research focuses on kernelized bandits and assumes that ground-truth $f$ has a small RKHS norm (see \citet{valko2013finite,wang2014theoretical,chowdhury2017kernelized,vakili2021information,zhang2021neural} and references therein). For the RBF and Mat\'ern kernels, their regret bounds are exponential in $d$ and \citet{scarlett2017lower} prove that no algorithm can achieve polynomial sample complexities. Since a small RKHS norm does not exclude spiky functions in general, the results in this setting requires a stronger assumption on ground-truth $f$. In fact, a function drawn from a Gaussian process has a infinite RKHS norm (defined by the same kernel) almost surely \citep{wahba1990spline}.
\paragraph{Neural nets recovery.} The parameters of finite-width two-layer neural networks can be recoverd with additional assumptions on the correlation between neurons \citep{zhong2017recovery,fu2020guaranteed,zhou2021local}, or the condition number of the first-layer weights \citep{zhang2019learning}. For two-layer neural networks with unbiased ReLU activation, \citet{bakshi2019learning} design algorithms whose sample complexity scales exponentially in the number of hidden neurons. In addition, \citet{milli2019model} proves a recovery guarantee of two-layer ReLU neural networks when the algorithm can query the gradient of the ground-truth. These methods cannot be applied to infinite-width neural networks, and $L_\infty$-recovery of a infinite-width neural network requires exponential samples (Theorem~\ref{thm:relu-lb}).
\paragraph{Applications to bandits, reinforcement learning, and PINN.} The best-arm identification problem in nonlinear bandits can be reduced to an $L_\infty$-recovery problem. If we can learn a function $g$ that approximates the true reward $f$ with $\|f-g\|_\infty\le \epsilon/2$, the action $\hat{x}\defeq \argmax_{x\in D}g(x)$ is $\epsilon$-optimal. Similarly, if the $Q$-function can be learned with a small $L_\infty$-error for finite horizon reinforcement learning, we can guarantee the optimality of the learned policy \citep{huang2021going}.
For physics informed neural networks \citep{raissi2019physics}, minimizing the $L_2$ loss may not be satisfactory \citep{wang20222,krishnapriyan2021characterizing,wang2021understanding}. When learning the Hamilton-Jacobi-Bellman equations (an analog of Bellman equations for continuous time), \citet{wang20222} proves that a small $L_\infty$-error can guarantee a good final performance, while a small $L_2$-error cannot.
\paragraph{$L_\infty$-recovery for other nonlinear functions.} Several other related works study the $L_\infty$-recovery with different assumptions on the ground-truth function. \citet{bertin2004minimax,korostelev1994asymptotically,tsybakov1998pointwise,golubev2000adaptive} study the minimax rate for $L_\infty$-recovering for one-dimensional smooth functions (e.g., functions in H\"older, Sobolev, or Besov classes). For general functions with bounded or decaying high-order derivatives, \citet{vybiral2014weak,krieg2019uniform} design estimators with quasi-polynomial or exponential in dimension samples. \citet{ibragimov1984asymptotic,stone1982optimal,nyssbaum1987nonparametric,bertin2004asymptotically} determine the asymptotically optimal rate for $L_p$-recovery of H\"older smooth functions for general $p\in [1,\infty]$ in high dimensions.
When the ground-truth function lies in the reproducing kernel Hilbert space, \citet{kuo2009power} prove some sufficient conditions for $L_\infty$-recovery with polynomial sample complexity. In general, $L_\infty$-recovery with polynomial samples is impossible unless the eigenvalues of the kernel decay very fast \citep{long2023reinforcement, kuo2008multivariate}. We refer the readers to \citet{ebert2021tractability} for a comprehensive survey in this direction.
Another line of research focuses on learning a nonlinear function with respect to the Sobolev norm \citep{fischer2020sobolev,steinwart2009optimal}. While their analysis can lead to $L_\infty$-recovery bounds, they require stronger smoothness assumptions to exclude the worst-case hard instances shown in Lemma~\ref{lem:relu-lb}. In contrast, our algorithms achieve $L_\infty$-recovery in the average case using much weaker smoothness assumptions.
\section{Preliminaries on Spherical Harmonics}\label{sec:preliminary} Now we give a brief overview of spherical harmonics, the essential tools in this paper, based on \citet[Section 2]{atkinson2012spherical}. Spherical harmonics are the eigenfunctions of the Laplacian operator on the sphere. The eigenfunctions corresponding to the $k$-th eigenvalue are degree-$k$ polynomials, and form a Hilbert space denoted by $\mathbb{Y}_{k,d}.$ The dimension of $\mathbb{Y}_{k,d}$ is $N_{k,d}\defeq \binom{d+k-1}{d-1} - \binom{d+k-3}{d-1}.$ When $k\to\infty$, $N_{k,d}=\Theta(d^k)$ and when $d\to\infty$, $N_{k,d}=\Theta(k^d).$ Spherical harmonics with different degrees are orthogonal to each other, and their linear combinations can represent all square integrable functions over the sphere.
We use $\Pi_k$ to denote the projection operator to the degree-$k$ spherical harmonics space $\mathbb{Y}_{k,d}$. We use $\mathbb{Y}_{\le k,d}$ to denote the space of spherical harmonics up to degree $k$, and $\Pi_{\le k}\defeq \sum_{l=0}^{k}\Pi_l$ the projection operator to $\mathbb{Y}_{\le k,d}$.
Spherical harmonics are closely related to Legendre polynomials. The degree-$k$ Legendre polynomial $P_{k,d}:\mathbb{R}\to\mathbb{R}$ is defined by the following recursive relationship \begin{align}
&P_{0,d}(t)=1,\quad P_{1,d}(t)=t,\\
&P_{k,d}(t)=\frac{2k+d-4}{k+d-3}tP_{k-1,d}(t)-\frac{k-1}{k+d-3}P_{k-2,d}(t),\quad \forall k\ge 2. \end{align} Let $\bar{P}_{k,d}(t)\defeq \sqrt{N_{k,d}}P_{k,d}(t)$ be the normalized Legendre polynomial. Normalized Legendre polynomial is a set of complete orthonormal basis for square-integrable functions over $[-1,1]$ with respect to the measure $\mu_d(t)\defeq (1-t^2)^{\frac{d-3}{2}}\frac{\Gamma(d/2)}{\Gamma((d-1)/2)}\frac{1}{\sqrt{\pi}}$, which equals to the density of $x_1$ when $x=(x_1,\cdots,x_d)$ is uniformly drawn from sphere $\mathbb{S}^{d-1}$. In other words, $\dotp{\bar{P}_{k,d}}{\bar{P}_{k',d}}_{{\mu_d}}\defeq \int_{-1}^{1}\bar{P}_{k,d}(t)\bar{P}_{k',d}(t)\mu_d(t)\mathrm{d} t=\ind{k=k'}.$
\paragraph{Properties of spherical harmonics and Legendre polynomials.} Our proof heavily replies on the following properties of spherical harmonics and Legendre polynomials.
Let $\{Y_{k,j}\}_{j=1}^{N_{k,d}}$ be an orthonormal basis of $\mathbb{Y}_{k,d}$. Then for any function $f:\mathbb{S}^{d-1}\to\mathbb{R}$ with $\|f\|_2<\infty$, there is a unique decomposition $f(\cdot)=\sum_{k\ge 0}\sum_{j=1}^{N_{k,d}}a_{k,j}Y_{k,j}(\cdot)$ with coefficients $\{a_{k,j}\}_{k\ge 0,1\le j\le N_{k,d}}$ that satisfies $\|f\|_2^2=\sum_{k\ge 0}\sum_{j=1}^{N_{k,d}}a_{k,j}^2$.
Spherical harmonics are the eigenfunctions of any inner-product kernels on the sphere, summarized by the following theorem \citep[Theorem 2.22]{atkinson2012spherical}. \begin{theorem}[Funk-Hecke formula]\label{thm:funk-hecke}
\sloppy Let $\sigma:[-1,1]\to\mathbb{R}$ be any one-dimensional function with $\int_{-1}^{1}|\sigma(t)|{\mu_d}(t)\mathrm{d} t<\infty$, and $\lambda_k=N_{k,d}^{-1/2}\dotp{\sigma}{\bar{P}_{k,d}}_{\mu_d}$. Then for any function $Y_k\in \mathbb{Y}_{k,d}$,
\begin{align}
\textstyle{\forall x\in\mathbb{S}^{d-1}, \quad \mathbb{E}_{z\sim \mathbb{S}^{d-1}}[\sigma(x^\top z)Y_k(z)]=\lambda_k Y_k(x).}
\end{align}
In other words, $\mathbb{Y}_{k,d}$ is the space of eigenfunctions of the inner product kernel $K(x,z)\defeq \sigma(x^\top z)$ corresponding to the eigenvalue $\lambda_k.$ \end{theorem}
We can construct spherical harmonics using Legendre polynomials. For any degree $k\ge 0$, let $g_u:\mathbb{S}^{d-1}\to\mathbb{R}$ be the function $g_u(x)=\bar{P}_{k,d}(\dotp{x}{u})$. Then for any $u\in\mathbb{S}^{d-1}$, $g_u\in \mathbb{Y}_{k,d}$ and $\|g_u\|_2=1$.
In the worst-case, high-order spherical harmonics can be very spiky because their $L_\infty/L_2$ ratio is very large: \begin{fact}\label{fact:infty-2-SH}
For every fixed $k\ge 0, g\in \mathbb{Y}_{k,d}$ we have $\|g\|_\infty\le \sqrt{N_{k,d}}\|g\|_2$, and the equality is achieved by $g_u(\cdot)=\bar{P}_{k,d}(\dotp{\cdot}{u})$ for any $u\in\mathbb{S}^{d-1}$. \end{fact}
\section{Main Results}\label{sec:main-results} In this section, we will first design a $L_\infty$-recovery algorithm that achieves polynomial sample complexity when the ground-truth function $f$ satisfies two conditions (Conditions~\ref{cond:decay} and \ref{cond:random}). We then establish these two conditions when $f$ is drawn from an isotropic Gaussian random fields (Lemma~\ref{lem:GRF}).
The first condition states that the spherical harmonics decomposition of the ground-truth $f$ decays at a proper rate. \begin{cond}\label{cond:decay}
The ground-truth function $f$ satisfies $\|\Pi_k f\|_2\le c_1N_{k,d}^{-\alpha/2},\forall k\ge 0$ for some $c_1>0$ and $\alpha>0$. \end{cond} We treat $\alpha$ as a constant that doesn't depend on the ambient dimension $d$. The parameter $\alpha>0$ is intuitively a notion of smoothness of the function $f$. This is because the derivatives of higher-degree spherical harmonics are larger. Hence, qualitatively speaking, functions with a faster decay (larger $\alpha$) is smoother.
Condition~\ref{cond:decay} holds for a wide range of functions. For example, any function of the form $g(\cdot)=h(\dotp{\cdot}{u})$, where $u\in\mathbb{S}^{d-1}$ and $h:[-1,1]\to \mathbb{R}$ with $\sup_{t\in[-1,1]} |h^{(k)}(t)|\le 1,\forall k\ge 0$, satisfies Condition~\ref{cond:decay} with parameter $\alpha=1$ and $c_1=1$ (Proposition~\ref{prop:smooth-inner-product-decay}). In addition, if two functions $g,h$ satisfy Condition~\ref{cond:decay}, so do their convex combinations $\theta g+(1-\theta)h,\forall \theta\in[0,1]$. Hence Condition~\ref{cond:decay} holds for \emph{any} two-layer NNs with bounded $L_1$ norm and infinitely smooth activation $h$ (e.g., exponential activation).
The following condition states that $f$ is not spiky when projected to the degree-$k$ spherical harmonics space. This condition is central to our analysis because it excludes the hard instances in the lower bounds (e.g., spiky functions constructed in Lemma~\ref{lem:relu-lb}). \begin{cond}\label{cond:random}
The ground-truth function $f$ satisfies $\|\Pi_k f\|_\infty\le c_2\sqrt{\ln (k+1)}\|\Pi_k f\|_2$,$\forall k\ge 0$ for some $c_2>0$. \end{cond}
Condition~\ref{cond:random} requires that the $L_\infty$/$L_2$ ratio of $\Pi_k f$ is bounded by $c_2 \sqrt{\ln (k+1)}$, whereas the worst case ratio is $\sqrt{N_{k,d}}$ (Fact~\ref{fact:infty-2-SH}). As we will show later, Condition~\ref{cond:random} holds with high probability for random functions drawn from degree-$k$ spherical harmonics space $\mathbb{Y}_{k,d}$ (Lemma~\ref{lem:random-sh}) and functions drawn from isotropic Gaussian random fields (Lemma~\ref{lem:GRF}).
With Conditions~\ref{cond:decay} and \ref{cond:random}, our main theorem states that there exists an algorithm (described later in Alg.~\ref{alg:main}) that achieves $L_\infty$-recovery using only polynomial samples drawn from the uniform distribution over the sphere $\mathbb{S}^{d-1}.$ \begin{theorem}\label{thm:mainsc}
Suppose the ground-truth function $f$ satisfies Conditions~\ref{cond:decay} and \ref{cond:random} for some fixed $\alpha\in (0,1]$ and $c_1,c_2>0$. If $d\ge 10\alpha^{-1}2^{5/\alpha}+2$, then for any $\epsilon>0,\delta>0$, with probability at least $1-\delta$ over the randomness of the data, Alg.~\ref{alg:main} outputs a function $g$ such that $\|f-g\|_\infty\le \epsilon$ using $O(\mathrm{poly}(c_1c_2,d,1/\epsilon,\ln 1/\delta)^{1/\alpha})$ samples. \end{theorem}
In comparison, the classical kernel methods assume $f$ has a bounded RKHS norm, which is equivalent to assuming that $\|\Pi_k f\|_2$ decays at a rate determined by the choice of kernel. For any function $f$ with decay parameter $\alpha\in(0,1/2)$, Proposition~\ref{prop:rkhs-norm-of-f} shows that the RKHS norm of $f$ is infinite with respect to \emph{any} bounded inner product kernel (e.g., the RBF kernel), and thus violates the assumption of kernel methods. In contrast, Theorem~\ref{thm:mainsc} still implies polynomial sample complexity for $\alpha\in(0,1/2)$ thanks to the additional randomness condition (Condition~\ref{cond:random})
Our algorithm is stated in Alg.~\ref{alg:main}. On a high level, given any desired error $\epsilon>0$, the algorithm selects a truncation threshold $k\ge 0$ (Line~\ref{line:truncate}), and uses empirical risk minimization to find the best degree-$k$ polynomial approximation to the ground-truth $f$.
Instead of directly learning a degree-$k$ polynomial, we can also use two-layer neural networks with polynomial activation to approximate the function $f$. The algorithm and discussion are deferred to Appendix~\ref{app:nns}. \begin{algorithm}[ht]
\caption{$L_\infty$-learning via Low-degree Polynomial Approximation}
\label{alg:main}
\hspace*{\algorithmicindent} \textbf{Parameters:} $\alpha,c_1,c_2>0$, desired error $\epsilon>0$, and failure probability $\delta>0.$
\hspace*{\algorithmicindent} \textbf{Input:} Dataset ${\mathcal{D}}=\{(x_i,y_i)\}_{i=1}^{n}$ where $x_i\sim \mathbb{S}^{d-1}$ are i.i.d.~samples from the unit sphere, and $y_i=f(x_i)+{\mathcal{N}}(0,1)$.
\begin{algorithmic}[1]
\State Set the truncation threshold $k\gets\inf_{l\ge 0}\{2c_1c_2(l+1)^{3/2}(N_{l+1,d})^{-\alpha/2}\le \epsilon/2\}.$\label{line:truncate}
\State Define the function class
\begin{align}
\textstyle{{\mathcal{F}}_k\gets\left\{g\in\mathbb{Y}_{\le k,d}: \|\Pi_l g\|_2\le c_1,\forall l\in[0,k]\right\}}.\label{equ:function-class}
\end{align}
\State Run empirical risk minimization:
$
g\gets \textstyle{\argmin_{h\in{\mathcal{F}}_k} \sum_{i=1}^{n}(h(x_i)-y_i)^2.}
$
\State \textbf{Return} $g$.
\end{algorithmic} \end{algorithm}
We present a proof sketch of Theorem~\ref{thm:mainsc} in Section~\ref{sec:proofsketch-mainsc}, and defer the full proof to Appendix~\ref{app:pf-mainsc}.
\subsection{Instantiation of Theorem~\ref{thm:mainsc} on Gaussian Random Fields} In this section, we instantiate Theorem~\ref{thm:mainsc} on isotropic Gaussian random fields.
Given any positive semi-definite covariance function $K:\mathbb{S}^{d-1}\times\mathbb{S}^{d-1}\to\mathbb{R}$, the mean-zero Gaussian random field is a collection of random variables $\{h(x)\}_{x\in \mathbb{S}^{d-1}}$ such that the distribution of any finite subset $(h(x_1),\cdots,h(x_n))$ is a Gaussian vector with covariance $\Sigma_{ij}=K(x_i, x_j).$ When the distribution is rotationally invariant, i.e., the distribution of $h(x_1),\cdots,h(x_n)$ equals to the distribution of $h(Rx_1),\cdots,h(Rx_n)$ for any rotation matrix $R\in\mathbb{R}^{d\times d}$, the covariance $K(x,x')$ only depends on the inner product $x^\top x'$ and can be written as $K(x,x')=\kappa(x^\top x')$ for some $\kappa:[-1,1]\to \mathbb{R}$. The corresponding GRF is called isotropic.
We focus on the case where the eigenvalues of the covariance (or equivalently, the Legendre polynomial decomposition of $\kappa$, by the Funk-Hecke formula) decays with a proper rate. Concretely, we assume $\kappa$ has the decomposition $\kappa(t)=\sum_{k\ge 0}\hat\kappa_k\bar{P}_{k,d}(t)$ where $\hat\kappa_k\le O(N_{k,d}^{-1/2-\alpha})$ for some $\alpha>0$. Later we will show that a function $f$ drawn from GRF with covariance $K(x,x')=\kappa(x^\top x')$ satisfies $\|\Pi_k f\|_2\le O(N_{k,d}^{-\alpha/2})$ and this inequality is tight. The decay rate $O(N_{k,d}^{-1/2-\alpha})$ is slightly faster than the decay of RBF kernels (given by $\kappa(t)=\exp(t)$), which is $\approx N_{k,d}^{-1/2}$ when $k$ is small \citep{minh2006mercer}.
The following theorem proves that Alg.~\ref{alg:main} can achieve $L_\infty$-recovery for function drawn from Gaussian random fields. \begin{theorem}\label{thm:GRF-main}
Let $f:\mathbb{S}^{d-1}\to \mathbb{R}$ be a function drawn from a Gaussian random field with covariance $K(x,x')=\kappa(x^\top x)$. Suppose for all $k\ge 0$, $\dotp{\kappa}{\bar{P}_{k,d}}_{{\mu_d}}\le c^2 N_{k,d}^{-1/2-\alpha}$ for some $c>0,\alpha>0$.
Given any $\epsilon>0,\delta>0$, with probability at least $1-\delta$ over the randomness of $f$ and the dataset, Alg.~\ref{alg:main} outputs a function $g:\mathbb{S}^{d-1}\to\mathbb{R}$ such that $\|g-f\|_\infty\le \epsilon$ using $O(\mathrm{poly}(c,\epsilon^{-1},d,\ln 1/\delta)^{1/\alpha})$ samples. \end{theorem} To the best of our knowledge, Theorem~\ref{thm:GRF-main} is the first result that achieves a $L_\infty$-error guarantee for isotropic Gaussian processes using only polynomial samples drawn uniformly from the unit sphere.
A closely related line of research to Theorem~\ref{thm:GRF-main} is the Gaussian process bandit problem, where the algorithm can adaptively query any data point and the goal is to maximize the function $f$ drawn from a Gaussian random field \citep{srinivas2010gaussian}. We can modify the GP-UCB algorithm in \citet{srinivas2010gaussian} to a $L_\infty$-recovery algorithm with \emph{adaptive} samples, and this modification, together with the analysis in \citet{vakili2021information}, lead to a polynomial sample complexity with the same condition as Theorem~\ref{thm:GRF-main}.\footnote{On a high level, at every iteration $t\ge 1$ the original GP-UCB algorithm selects the query $x_t$ that maximizes the upper confidence bound of $f$ \citep{srinivas2010gaussian}. \citet{srinivas2010gaussian} construct the upper confidence bound by analytically compute the posterior mean and variance of $f$ given any data points, assuming that the ground-truth $f$ is drawn from a Gaussian process prior. To get an algorithm for $L_\infty$-recovery, we can choose $x_t$ that maximizes the posterior variance of $f(\cdot)$. In this case, the analysis in \citep{srinivas2010gaussian} implies that with high probability after $n$ iterations, the $L_\infty$-error of the posterior mean is upper bounded by the maximum information gain, denoted by $\sqrt{\gamma_n/n}$. Combining with the refined analysis in \citet{vakili2021information}, we can upper bound the information gain $\gamma_n$ using the spectrum decay of $\kappa$, which leads to a polynomial sample complexity in our setting. } In comparison, our algorithm only requires samples from the uniform distribution while GP-UCB must be adaptive. In addition, Theorem~\ref{thm:mainsc} holds for general functions with Condition~\ref{cond:decay} and \ref{cond:random} while the analysis of \citet{srinivas2010gaussian} is specialized to Gaussian processes.
We prove Theorem~\ref{thm:GRF-main} by establishing Conditions~\ref{cond:decay} and \ref{cond:random} for functions drawn from isotropic Gaussian random fields using the following lemma, and then directly invoking Theorem~\ref{thm:mainsc}. \begin{lemma}\label{lem:GRF}
In the setting of Theorem~\ref{thm:GRF-main}, with probability at least $1-\delta$ we have
\begin{align}\label{equ:GRF-1}
\forall k\ge 0,&\quad \|\Pi_k f\|_\infty\le 5\sqrt{2\ln(6/\delta)+2(d^2+1)\ln(k+1)}\|\Pi_k f\|_2,
\end{align}
and
\begin{align}
\forall k\ge 0,&\quad \|\Pi_k f\|_2\le 3c\sqrt{\ln(2/\delta)}N_{k,d}^{-\alpha/2}.\label{equ:GRF-2}
\end{align} \end{lemma}
We the proof of Lemma~\ref{lem:GRF} is deferred to Section~\ref{sec:proofsketch-GRF}.
\subsection{Proof Sketch of Theorem~\ref{thm:mainsc}}\label{sec:proofsketch-mainsc} In this section, we present the proof sketches of Theorem~\ref{thm:mainsc}. On a high level, we prove that (a) the ground-truth $f$ can be approximated by a low-degree polynomial with a small $L_\infty$-error, and (b) learning a low-degree polynomial in $L_\infty$-error only requires polynomial samples.
\begin{proof}[Proof sketch of Theorem~\ref{thm:mainsc}]
For better exposition, in the following we present the proof sketch for the case $\alpha=1/2$, and the general case is proved similarly.
For any fixed threshold $k\ge 0$, we first upper bound the $L_\infty$-distance between the ground-truth $f$ and its low-degree components $\Pi_{\le k}f$. Concretely,
\begin{align}
\textstyle{\|f-\Pi_{\le k}f\|_\infty=\|\sum_{l>k}\Pi_l f\|_\infty\le \sum_{l>k}\|\Pi_l f\|_\infty.}
\end{align}
Under Conditions~\ref{cond:decay} and \ref{cond:random}, the term $\|\Pi_l f\|_\infty$ decays at rate $\sqrt{\ln (l+1)}N_{l,d}^{-1/2}$. Since $N_{l,d}^{-1/2}\approx \min\{l^d,d^l\}^{-1/2}$ decays very fast, we get
\begin{align}\label{equ:pf-thm-mainsc-1}
\textstyle{\|f-\Pi_{\le k}f\|_\infty\le \sum_{l>k}\sqrt{\ln (l+1)}N_{l,d}^{-1/2}\lesssim \sqrt{\ln (k+1)}N_{k,d}^{-1/2}}.
\end{align}
Next we show that the low-degree components $\Pi_{\le k}f$ can be learned w.r.t. $L_\infty$-error using polynomial samples because the $L_\infty$-error of a low-degree polynomial is upper bounded by its $L_2$-error. Indeed, Fact~\ref{fact:infty-2-SH} states that $\|h\|_\infty\le \sqrt{N_{k,d}}\|h\|_2,\forall h\in\mathbb{Y}_{k,d}$. Then for any low-degree polynomial $g\in\mathbb{Y}_{\le k,d}$,
\begin{align}\label{equ:pf-thm-mainsc-2}
\textstyle{\|g-\Pi_{\le k}f\|_\infty=\|\Pi_{\le k}(g-f)\|_\infty\le \sum_{l=0}^{k} \|\Pi_{l}(g-f)\|_\infty \le \sum_{l=0}^{k} \|\Pi_{l}(g-f)\|_2 N_{l,d}^{1/2}.}
\end{align}
When $g\in\mathbb{Y}_{\le k,d}$, we have $\|g-\Pi_{\le k}f\|_2^2=\|\Pi_{\le k}(g-f)\|_2^2=\sum_{l=0}^{k}\|\Pi_{l}(g-f)\|_2^2$. Continuing Eq.~\eqref{equ:pf-thm-mainsc-2} by applying Cauchy-Schwarz, we get
\begin{align}\label{equ:pf-thm-mainsc-3}
\textstyle{\|g-\Pi_{\le k}f\|_\infty\le N_{k,d}^{1/2}\sqrt{k+1}\|\Pi_{\le k}(g-f)\|_2=N_{k,d}^{1/2}\sqrt{k+1}\|g-\Pi_{\le k}f\|_2.}
\end{align}
Now we can choose an threshold $k\ge 0$ to balance the two terms in Eq.~\eqref{equ:pf-thm-mainsc-1} and Eq.~\eqref{equ:pf-thm-mainsc-3}. For any desired error level $\epsilon>0$, we can choose an $k$ such that $\sqrt{\ln (k+1)}N_{k,d}^{-1/2}=\Theta(\epsilon/2)$ and get
\begin{align}\label{equ:pf-thm-mainsc-4}
\textstyle{\|g-f\|_\infty\le \|g-\Pi_{\le k}f\|_\infty+\|f-\Pi_{\le k}f\|_\infty\lesssim \mathrm{poly}(1/\epsilon)\|g-\Pi_{\le k}f\|_2+\epsilon/2.}
\end{align}
Finally, for any truncation threshold $k>0$, $\Pi_{\le k}f$ is a low-degree polynomial and belongs to the family ${\mathcal{F}}_k$ defined in Eq.~\eqref{equ:function-class}. Therefore classic statistical learning theory implies that empirical risk minimization outputs a function $g$ with $\|g-\Pi_{\le k}f\|_2\le \mathrm{poly}(\epsilon)$ using only $\mathrm{poly}(1/\epsilon)$ samples (Lemma~\ref{lem:ERM-l2-kernel}), which completes the proof. \end{proof}
\subsection{Proof of Lemma~\ref{lem:GRF}}\label{sec:proofsketch-GRF} To prove Lemma~\ref{lem:GRF}, we first characterize an isotropic Gaussian random field in the spherical harmonics expansion.
Let $f:\mathbb{S}^{d-1}\to\mathbb{R}$ be a function drawn from an isotropic Gaussian random field with covariance $\kappa:[0,1]\to\mathbb{R}$, and $\{Y_{k,j}\}_{k\ge 0,1\le j\le N_{k,d}}$ a set of orthonormal spherical harmonics basis. We will show that the projection of $f$ to the degree-$k$ spherical harmonics space is isotropic. In other words, $\{\dotp{f}{Y_{k,j}}\}_{1\le j\le N_{k,d}}$ are i.i.d. random variables.
Indeed, by \citet[Theorem 5.5]{lang2015isotropic}, $f$ admits the following spherical harmonics decomposition \begin{align}\label{equ:pfs-GRF-1}
\textstyle{f(\cdot)\stackrel{d}{=}\sum_{k\ge 0}\(\hat\kappa_k^{1/2}N_{k,d}^{-1/4}\sum_{j=1}^{N_{k,d}}a_{k,j}Y_{k,j}(\cdot)\),} \end{align} where $a_{k,j}$ are i.i.d.~unit Gaussian random variables. Hence, to prove Lemma~\ref{lem:GRF} we only need to examine the property of a random function drawn from the spherical harmonics space $\mathbb{Y}_{k,d}$, which is a $N_{k,d}$-dimensional Hilbert space.
The following lemma shows that a random spherical harmonics is not spiky because its $L_\infty$/$L_2$ ratio is upperbounded by $O(d\sqrt{\ln k})$ with high probability, whereas the worst case ratio is $\sqrt{N_{k,d}}=\Omega(\min\{d^{k/2},k^{d/2}\}).$ \begin{lemma}\label{lem:random-sh}
For any fixed $k\ge 0$, let $\{Y_{k,j}\}_{j=1}^{N_{k,d}}$ be any set of orthonormal basis for degree-$k$ spherical harmonics $\mathbb{Y}_{k,d}$.
Let $g=\sum_{j=1}^{N_{k,d}}a_{j}Y_{k,j}$ be a random spherical harmonics where $a_j\sim {\mathcal{N}}(0,1)$ are independent unit Gaussian random variables. For any $\delta>0$ we have, with probability at least $1-\delta$,
\begin{align}\label{equ:random-sh}
\|g\|_\infty\le 5\sqrt{\ln(3/\delta)+2d^2\ln(k+1)}\|g\|_2.
\end{align} \end{lemma} Lemma~\ref{lem:random-sh} is a high-dimensional version of \citet[Theorem 2]{burq2014probabilistic}. The proof of \citet{burq2014probabilistic} relies on the Sobolev embedding theorem, which treats the dimension $d$ as a constant. In contrast, we compute the exact dependency on the dimension $d$ by instantiating the Riesz representation theorem on the space of spherical harmonics and then applying a uniform convergence argument.
In the following, we present a proof sketch of Lemma~\ref{lem:random-sh}. The full proof is deferred to Appendix~\ref{app:random-sh}.
\begin{proof}[Proof Sketch of Lemma~\ref{lem:random-sh}]
To prove the $L_\infty$/$L_2$ norm ratio of $g$, we first invoke Lemma~\ref{lem:riesz-SH} which states that
\begin{align}
\forall x\in\mathbb{S}^{d-1},\quad g(x)=\sqrt{N_{k,d}}\dotp{g}{\bar{P}_{k,d}(\dotp{x}{\cdot})}.
\end{align}
Since $\bar{P}_{k,d}(\dotp{x}{\cdot})\in \mathbb{Y}_{k,d}$, Lemma~\ref{lem:riesz-SH} is an instantiation of the Riesz representation theorem on the space $\mathbb{Y}_{k,d}$. The Riesz representation theorem states that for a Hilbert space, every continuous linear functional (in this case, the evaluation functional ${\rm{ev}}_x:g\to g(x)$) can be represented by the inner product with an element in the space (in this case, $\sqrt{N_{k,d}}\bar{P}_{k,d}(\dotp{x}{\cdot})$).
For any fixed $x\in\mathbb{S}^{d-1}$, because $g=\Pi_k f$ is a Gaussian vector in the $N_{k,d}$-dimensional space $\mathbb{Y}_{k,d}$ and $\bar{P}_{k,d}(\dotp{x}{\cdot})\in\mathbb{Y}_{k,d}$ is a fixed vector, the function value $g(x)$ has a Gaussian distribution.
Formally speaking, we can write $\bar{P}_{k,d}(\dotp{x}{\cdot})=\sum_{j=1}^{N_{k,d}}u_{k,j}Y_{k,j}(\cdot)$ for some fixed parameters $u_{k,j}.$ Let $\vec{a}_k=[a_{k,j}]_{1\le j\le N_{k,d}}$ and $\vec{u}_k=[u_{k,j}]_{1\le j\le N_{k,d}}$, then we get
\begin{align}
g(x)=\sqrt{N_{k,d}}\dotp{g}{\bar{P}_{k,d}(\dotp{x}{\cdot})}=\sqrt{N_{k,d}}\dotp{\vec{a}_k}{\vec{u}_k}\sim \sqrt{N_{k,d}}{\mathcal{N}}(0,\|\vec{u}_k\|_2^2).
\end{align}
Since $\|\vec{u}_k\|_2=\|\bar{P}_{k,d}(\dotp{x}{\cdot})\|_2=1$ and $\|g\|_2=\|\vec{a}_k\|_2\approx \sqrt{N_{k,d}}$, by concentration inequality of Gaussian vectors (Lemma~\ref{lem:gaussian-proj-concentration}) we get for any fixed $x\in\mathbb{S}^{d-1}$, with high probability
$
|g(x)|\lesssim \|\vec{a}_k\|_2=\|g\|_2.
$
Finally, we can use a covering number argument to prove a uniform convergence of all $x\in\mathbb{S}^{d-1}$. Hence, we prove that with high probability, $\forall x\in\mathbb{S}^{d-1}, |g(x)|\le \widetilde{O}(d\sqrt{\ln k})\|g\|_2$, which implies Eq.~\eqref{equ:random-sh}. \end{proof}
With Lemma~\ref{lem:random-sh}, we can now prove Lemma~\ref{lem:GRF}. \begin{proof}[Proof of Lemma~\ref{lem:GRF}]
Recall that \citet[Theorem 5.5]{lang2015isotropic} gives the following spherical harmonics decomposition
\begin{align}
\textstyle{f(x)\stackrel{d}{=}\sum_{k\ge 0}\(\hat\kappa_k^{1/2}N_{k,d}^{-1/4}\sum_{j=1}^{N_{k,d}}a_{k,j}Y_{k,j}(x)\)}
\end{align}
where $a_{k,j}\sim {\mathcal{N}}(0,1)$ are independent Gaussian random variables. By Lemma~\ref{lem:random-sh}, for any fixed $k\ge 0$, with probability at least $1-\delta/(2(k+1)^2)$ we have
\begin{align}
\|\Pi_k f\|_\infty&\le 5\sqrt{\ln(6(k+1)^2/\delta)+2d^2\ln(k+1)}\|\Pi_k f\|_2\\
&\le 5\sqrt{2\ln(6/\delta)+2(d^2+1)\ln(k+1)}\|\Pi_k f\|_2.
\end{align}
By union bound over $k$, with probability at least $1-\delta$ we get
\begin{align}
\forall k\ge 0, \quad \|\Pi_k f\|_\infty\le 5\sqrt{2\ln(6/\delta)+2(d^2+1)\ln(k+1)}\|\Pi_k f\|_2,
\end{align}
which proves Eq.~\eqref{equ:GRF-1}.
Now we prove the second part of lemma. Since $\{Y_{k,j}\}_{j=1}^{N_{k,d}}$ forms an orthonormal basis of $\mathbb{Y}_{k,d}$, we get
\begin{align}\label{equ:pf-GRF-1}
\textstyle{\|\Pi_k f\|_2^2=\hat{\kappa}_kN_{k,d}^{-1/2}\sum_{j=1}^{N_{k,d}}a_{k,j}^2\le c^2N_{k,d}^{-1-\alpha}\sum_{j=1}^{N_{k,d}}a_{k,j}^2.}
\end{align}
For any fixed $k\ge 0$, since $a_{k,j}$ are i.i.d. unit Gaussian random variables, by the concentration of the norm of Gaussian vectors \citep[Lemma 1]{laurent2000adaptive}, we have
\begin{align}
\textstyle{\forall t>0, \quad \mathop{\rm Pr}\nolimits\(\sum_{j=1}^{N_{k,d}}a_{k,j}^2\ge N_{k,d}+2\sqrt{N_{k,d}}\sqrt{t}+2t\)\le \exp(-t).}
\end{align}
Take $t=\ln (2(k+1)^2/\delta)$. Note that $N_{k,d}\ge k\ge \ln(k+1)$. As a result, for all $k\ge 0$ we get
\begin{align}
N_{k,d}+2\sqrt{N_{k,d}}\sqrt{t}+2t\le 9N_{k,d}\ln(2/\delta).
\end{align}
Consequently,
\begin{align}
\textstyle{\mathop{\rm Pr}\nolimits\(\sum_{j=1}^{N_{k,d}}a_{k,j}^2\ge 9N_{k,d} \ln(2/\delta)\)\le (k+1)^{-2}\delta/2.}
\end{align}
Combining with Eq.~\eqref{equ:pf-GRF-1} and union bound over $k$, with probability at least $1-\delta$ we get
\begin{align}
\textstyle{\forall k\ge 0,\quad \|\Pi_k f\|_2\le cN_{k,d}^{-1/2-\alpha/2}\(\sum_{j=1}^{N_{k,d}}a_{k,j}^2\)^{1/2} \le 3c\sqrt{\ln(2/\delta)}N_{k,d}^{-\alpha/2},}
\end{align}
which proves Eq.~\eqref{equ:GRF-2}. \end{proof}
\section{Lower Bounds}\label{sec:lowerbounds} In this section, we present two lower bounds to motivate our Condition~\ref{cond:random}. Both lower bounds hold for any algorithm that can \emph{adaptively} choose its data point $x_i$ and observes a noisy signal $f(x_i)+{\mathcal{N}}(0,1)$, where $f$ denotes the ground-truth function. Our lower bounds may be of independent interest.
\paragraph{Lower bounds for functions with decay rate $N_{k,d}^{-1/2}$.} The following lemma proves that, in the worst case, $L_\infty$-recovery is hard even when the function's spherical harmonics decomposition decays at a rate of $N_{k,d}^{-1/2}.$ \begin{lemma}\label{lem:relu-lb}
For a fixed integer $k\ge 4$ and $\beta_k\in (0,1)$, define ${\mathcal{F}}_k=\{\beta_kP_{k,d}(\dotp{\cdot}{u}):u\in\mathbb{S}^{d-1}\}$ be the hypothesis class.
For any fixed algorithm, let $E_{f,n}$ be the probability that the algorithm outputs $\hat{f}$ such that $\|\hat{f}-f\|_\infty\le \beta_k/4$ using $n$ samples when the ground-truth function is $f$. Then if $n<N_{k,d}\beta_k^{-2}$,
$\min_{f\in {\mathcal{F}}_k}E_{f,n}\le 1/2.$ \end{lemma}
Since $\|P_{k,d}(\dotp{\cdot}{u})\|_2=N_{k,d}^{-1/2}$, the function class ${\mathcal{F}}_k$ (when $\beta_k=1$) is a subset of functions that satisfies Condition~\ref{cond:decay} with $\alpha=1$. Therefore, no algorithm can achieve polynomial sample complexity for $L_\infty$-recovery with only the smoothness condition (Condition~\ref{cond:decay}).
Lemma~\ref{lem:relu-lb} is proved by showing that no algorithm can distinguish all the functions $f\in{\mathcal{F}}_k$ using $o(N_{k,d})$ samples because the average signal-to-noise ratio of any data point is roughly $N_{k,d}^{-1/2}$. Hence, the worst-case sample complexity is at least $\Omega(N_{k,d})$. The proof is deferred to Appendix~\ref{app:pf-lem-relu-lb}.
\paragraph{Lower bounds for two-layer ReLU neural networks.} We first formally define the class of two-layer neural networks used in this paper. Let $\mathrm{NN}\text{-}\mathrm{ReLU}(L_p)$ be the family of two layer neural networks (NNs) with $L_p$-norm bounds. Formally speaking, \begin{align}
\mathrm{NN}\text{-}\mathrm{ReLU}(L_p)=\{g(x)\defeq\mathbb{E}_{\xi\sim \mathbb{S}^{d-1}}[\sigma(x^\top \xi)c(\xi)]:\|c\|_p\le 1\}, \end{align} where $\sigma$ is the ReLU activation and $c:\mathbb{S}^{d-1}\to\mathbb{R}$ is the weight of the NN. Classical finite width neural networks belong to $\mathrm{NN}\text{-}\mathrm{ReLU}(L_1)$ because their weights $c$ can be represented by the mixtures of Dirac measures.
The following theorem shows that learning two-layer neural networks with ReLU activation is statistically hard even when the NN has a constant norm. The lower bound holds for $\mathrm{NN}\text{-}\mathrm{ReLU}(L_2)$, which is a subset of $\mathrm{NN}\text{-}\mathrm{ReLU}(L_1)$. \begin{theorem}\label{thm:relu-lb}
Given the hypothesis class $\mathrm{NN}\text{-}\mathrm{ReLU}(L_2)$. If an algorithm, when running on every possible instance $f\in\mathrm{NN}\text{-}\mathrm{ReLU}(L_2)$, takes in $n$ data points uniformly sampled from the sphere $\mathbb{S}^{d-1}$ and outputs a function $g$ such that $\|f-g\|_\infty\le \epsilon$ with probability at least $1/2$, then $n\ge \Omega\(\(0.002\epsilon^{-1}d^{-7/4}\)^{d/2}\)$. As a corollary, the minimax sample complexity of learning $\mathrm{NN}\text{-}\mathrm{ReLU}(L_2)$ with $L_\infty$-error $\epsilon=O(d^{-7/4})$ requires at least $2^{d}$ samples. \end{theorem} Theorem~\ref{thm:relu-lb} does not contradict with existing results on the recovery of two-layer neural networks \citep{zhong2017recovery,zhou2021local} because they focus on the finite-width case while our lower bound holds for infinite-width neural networks. Compared with the lower bound in \citet{dong2021provable}, Theorem~\ref{thm:relu-lb} does not rely on the bias term in the ReLU activation to kill the signal. Instead, we invoke the Funk-Hecke formula (Theorem~\ref{thm:funk-hecke}) to show that two-layer ReLU NNs can represent spiky functions with constant norm.
Theorem~\ref{thm:relu-lb} is proved by showing that ${\mathcal{F}}_k$ defined in Lemma~\ref{lem:relu-lb} is a subset of $\mathrm{NN}\text{-}\mathrm{ReLU}(L_2)$ if we take $\beta_k\approx k^{-2}$. The proof is deferred to Appendix~\ref{app:pf-thm-relu-lb}.
\section{Conclusion}\label{sec:conclusion} In this paper, we make some initial steps toward $L_\infty$-recovery for nonlinear models by proving a polynomial sample complexity bound for random function drawn from Gaussian random fields. We also prove a $\exp(d)$ sample complexity lower bound for recovering the worst-case infinite-width two-layer neural nets with unbiased ReLU activation, which may be of independent interest.
For future works, we raise the following open questions: \begin{enumerate}
\item To instantiate Condition~\ref{cond:random}, this paper focuses on functions $f$ drawn from Gaussian random fields because they have \emph{independent} components in the spherical harmonics space. However, Condition~\ref{cond:random} also holds when $f$ has correlated components. For example, when $\Pi_k f=\sum_{j=1}^{N_{k,d}}a_{k,j}Y_{k,j}$ where $[a_{k,j}]_{j=1}^{N_{k,d}}$ lies on the $(N_{k,d}-1)$-dimensional sphere. Is it possible to prove Condition~\ref{cond:random} for functions drawn from other distribution?
\item A two-layer single-neuron neural nets with exponential activation, i.e., functions of the form $g(\cdot)=\exp(\dotp{\cdot}{u})$ for some $u\in\mathbb{S}^{d-1}$, does not satisfy Condition~\ref{cond:random}. In fact, $\Pi_k g$ is the most spiky function in $\mathbb{Y}_{k,d}$ because $\Pi_k g=\lambda_k \bar{P}_{k,d}(\dotp{\cdot}{u})$. Can we find a natural (random) subset of two-layer neural networks that satisfy Condition~\ref{cond:random}? \end{enumerate}
\subsection*{Acknowledgment}
The authors would like to thank Ruixiang Zhang, Yakun Xi, Yuhao Zhou, Jason D. Lee for helpful discussions. The authors would also like to thank anonymous reviewers for the references to additional related works. The authors would like to thank the support of NSF CIF 2212263.
\appendix
\section*{List of Appendices}
\startcontents[sections]
\printcontents[sections]{l}{1}{\setcounter{tocdepth}{2}}
\section{Missing Proofs}
\subsection{Proof of Theorem~\ref{thm:mainsc}}\label{app:pf-mainsc} In the following, we first state two lemmas that are critical to the proof of Theorem~\ref{thm:mainsc}.
The next lemma proves that the empirical risk minimization step used in Alg.~\ref{alg:main} outputs a function with small $L_2$ loss, whose proof is deferred to Appendix~\ref{app:pf-ERM-l2-kernel}. \begin{lemma}\label{lem:ERM-l2-kernel}
Suppose the function $f:\mathbb{S}^{d-1}\to \mathbb{R}$ satisfies Condition~\ref{cond:decay} for some fixed $\alpha\in (0,1],c_1,c_2>0$. For any $\epsilon>0$, let $k=\inf_{l\ge 0}\{2c_1c_2(l+1)^{3/2}(N_{l+1,d})^{-\alpha/2}\le \epsilon/2\}$.
Let ${\mathcal{F}}_k\gets\left\{g\in\mathbb{Y}_{\le k,d}: \|\Pi_l g\|_2\le c_1,\forall l\in[0,k]\right\}$ be the function class defined in Alg.~\ref{alg:main}.
For a given dataset $\{(x_i,y_i)\}_{i=1}^{n}$, let $\hat{\mathcal{L}}(h)\defeq \frac{1}{n}\sum_{i=1}^{n}(h(x_i)-y_i)^2$ be the empirical $L_2$ loss, and $g=\argmin_{h\in{\mathcal{F}}_k}\hat{\mathcal{L}}(h).$
For any $\delta>0,\epsilon_1>0$, when $d\ge \max\{2e,4/\alpha\}$ and the number of samples $n\ge \Omega(\mathrm{poly}(c_1c_2,1/\epsilon)^{1/\alpha}\mathrm{poly}(1/\epsilon_1,\ln(1/\delta)))$, with probability at least $1-\delta$,
\begin{align}
\|\Pi_{\le k} (f-g)\|_2=\|\Pi_{\le k} f-g\|_2\le \epsilon_1.
\end{align} \end{lemma}
The following lemma proves that with Conditions~\ref{cond:decay} and \ref{cond:random}, $\|f-\Pi_{\le k} g\|_\infty$ can be upper bounded by $\|\Pi_{\le k}(f-g)\|_2$ for properly chosen $k$. \begin{lemma}\label{lem:main}
Suppose the function $f:\mathbb{S}^{d-1}\to \mathbb{R}$ satisfies Conditions~\ref{cond:decay} and \ref{cond:random} for some fixed $\alpha\in (0,1],c_1,c_2>0$, and $d\ge 10\alpha^{-1}2^{5/\alpha}+2$. For any $\epsilon>0$, define $k=\inf_{l\ge 0}\{2c_1c_2(l+1)^{3/2}(N_{l+1,d})^{-\alpha/2}\le \epsilon/2\}.$
Then for any function $g:\mathbb{S}^{d-1}\to \mathbb{R}$ with $\|\Pi_{\le k} (f-g)\|_2\le \frac{1}{4}\epsilon^{3/\alpha+1}(4c_1c_2)^{-3/\alpha}d^{-4/\alpha}$, we have
$
\|f-\Pi_{\le k} g\|_\infty\le \epsilon.
$ \end{lemma} Proof of Lemma~\ref{lem:main} is deferred to Appendix~\ref{app:pf-lem-main}.
Now we are ready to prove Theorem~\ref{thm:mainsc}. \begin{proof}[Proof of Theorem~\ref{thm:mainsc}]
Let $\epsilon_1=\frac{1}{4}\epsilon^{3/\alpha+1}(4c_1c_2)^{-3/\alpha}d^{-4/\alpha}$. We prove Theorem~\ref{thm:mainsc} in the following two steps.
\paragraph{Step 1: upper bound the population $L_2$ loss.} In this step, we use classic statistical learning tools to show that the ERM step (i.e., $g=\argmin_{h\in{\mathcal{F}}_k} \sum_{i=1}^{n}(h(x_i)-y_i)^2$) returns a function $g$ with small $L_2$ loss. In particular, by Lemma~\ref{lem:ERM-l2-kernel} we get
$
\|\Pi_{\le k} (f-g)\|_2\le \epsilon_1.
$
\paragraph{Step 2: upper bound the $L_\infty$-error via truncation.} In this step we show that with Conditions~\ref{cond:decay} and \ref{cond:random} on the ground-truth function, any function $g$ with a small $L_2$-error will also have a small $L_\infty$-error when projected to the low-degree spherical harmonics space. Formally speaking, invoking Lemma~\ref{lem:main} we get
$
\|\Pi_{\le k} (f-g)\|_2\le \epsilon_1 \implies \|f-\Pi_k g\|_\infty\le \epsilon.
$
Finally, since $g\in{\mathcal{F}}_k\subset \mathbb{Y}_{\le k,d}$, we get $g=\Pi_k g$. Hence, combining these two steps we prove the desired result. \end{proof}
\subsection{Proof of Lemma~\ref{lem:ERM-l2-kernel}}\label{app:pf-ERM-l2-kernel} In the following we prove Lemma~\ref{lem:ERM-l2-kernel}. \begin{proof}[Proof of Lemma~\ref{lem:ERM-l2-kernel}]
We prove Lemma~\ref{lem:ERM-l2-kernel} in two steps.
\paragraph{Step 1: expressivity.} In this step, we prove that $\Pi_{\le k}f\in{\mathcal{F}}_k.$ Indeed, by Condition~\ref{cond:decay} we get
\begin{align}
\|\Pi_k f\|_2\le c_1N_{k,d}^{-\alpha/2}\le c_1,
\end{align}
meaning that $\Pi_{\le k}f\in{\mathcal{F}}_k.$
Consequently, using the definition $g=\argmin_{h\in{\mathcal{F}}_k} \hat{{\mathcal{L}}}(g)$ we have $\hat{{\mathcal{L}}}(g)\le \hat{{\mathcal{L}}}(\Pi_{\le k} f).$
\paragraph{Step 2: uniform convergence.} In this step, we prove that using $$n=\Omega(\mathrm{poly}(c_1,N_{k,d},\ln(1/\delta),1/\epsilon_1))$$ samples, Alg.~\ref{alg:main} outputs a function $g\in{\mathcal{F}}$ such that
\begin{align}\label{equ:ERM-l2-kernel-1}
\|g-\Pi_{\le k}f\|_2\le \epsilon_1.
\end{align}
To this end,
by the uniform convergence of ${\mathcal{F}}_k$ (Lemma~\ref{lem:uniform-convergence-kernel}), when $$n= \Omega(\mathrm{poly}(c_1,N_{k,d},\ln(1/\delta),1/\epsilon_1)),$$ with probability at least $1-\delta$,
\begin{align}\label{equ:ERM-l2-kernel-0}
&\|g-f\|_2^2\le \hat{\mathcal{L}}(g)+\epsilon_1^2/2\le \hat{\mathcal{L}}(\Pi_{\le k}f)+\epsilon_1^2/2\le\|\Pi_{\le k}f-f\|_2^2+\epsilon_1^2.
\end{align}
Since ${\mathcal{F}}_k\subseteq \mathbb{Y}_{\le k,d}$, by the Parseval's identity we get
\begin{align}
\forall h\in{\mathcal{F}}_k, \quad \|h-f\|_2^2&=\|\Pi_{\le k}(h-f)\|_2^2+\|\Pi_{>k}(h-f)\|_2^2\\
&=\|\Pi_{\le k}(h-f)\|_2^2+\|\Pi_{>k}f\|_2^2=\|h-\Pi_{\le k}f\|_2^2+\|f-\Pi_{\le k}f\|_2^2.
\end{align}
Note that $g\in{\mathcal{F}}_k$ and $\Pi_{\le k}f\in{\mathcal{F}}_k$. Combining with Eq.~\eqref{equ:ERM-l2-kernel-0} we get
\begin{align}
\|g-\Pi_{\le k} f\|_2^2\le \epsilon_1^2.
\end{align}
Finally, by the choice of $k$ and Proposition~\ref{prop:truncation-upperbound}, $N_{k,d}=\mathrm{poly}(c_1c_2,1/\epsilon)^{1/\alpha}$, which means that
\begin{align}
n= \Omega(\mathrm{poly}(c_1,N_{k,d},\ln(1/\delta),1/\epsilon_1))=\Omega(\mathrm{poly}(c_1c_2,1/\epsilon)^{1/\alpha}\mathrm{poly}(\ln(1/\delta),1/\epsilon_1)).
\end{align} \end{proof}
The following lemma proves uniform convergence results for the function class ${\mathcal{F}}_k.$ \begin{lemma}\label{lem:uniform-convergence-kernel}
In the setting of Lemma~\ref{lem:ERM-l2-kernel}, for any $\delta>0,\epsilon_1>0$ and $n\ge \Omega(\mathrm{poly}(c_1,N_{k,d},\ln(1/\delta),1/\epsilon_1)),$ with probability at least $1-\delta$ we have
\begin{align}
\sup_{g\in{\mathcal{F}}_k}|\|g-f\|_2^2-\hat{\mathcal{L}}(g)|\le \epsilon_1.
\end{align} \end{lemma} \begin{proof}
We prove this lemma using the Rademacher complexity of kernel methods \citep{bartlett2002rademacher}. First we upper bound the Rademacher complexity of ${\mathcal{F}}_k$. Let $x_1,\cdots,x_n$ be a set of data points and $\hat{R}_n({\mathcal{F}}_k)$ the empirical Rademacher complexity of ${\mathcal{F}}_k$, defined by
\begin{align}
\hat{R}_n({\mathcal{F}}_k)=\frac{1}{n}\mathbb{E}_{\sigma_1,\cdots,\sigma_n\sim \{-1,1\}^n}\[\sup_{g\in {\mathcal{F}}_k}\abs{\sum_{i=1}^{n}\sigma_i g(x_i)}\right].
\end{align}
Recall that $\{Y_{k,j}\}_{j=1}^{N_{k,d}}$ is an orthonormal basis of $\mathbb{Y}_{k,d}$, and any function $g\in{\mathcal{F}}_k$ can be written as $g(x)=\sum_{l=0}^{k}\sum_{j=1}^{N_{l,d}}a_{l,j}Y_{l,j}(x)$ where $\sum_{j=1}^{N_{k,d}}a_{l,j}^2\le c_1^2,\forall l\in[0,k]$. Hence, after defining $\phi_k(x)\defeq [Y_{l,j}(x)]_{l\in[0,k],j\in[N_{l,d}]}$ as the feature vector, and $\vec{a}\defeq [a_{l,j}]_{l\in[0,k],j\in[N_{l,d}]}$, we have $g(x)=\dotp{\phi_k(x)}{\vec{a}}$ and $\|\vec{a}\|_2\le \sqrt{k+1}c_1$.
Let $k(x,x')=\dotp{\phi_k(x)}{\phi_k(x')}$ be the kernel function. Then by the fact that $\sum_{j=1}^{N_{l,d}}Y_{l,j}(x)^2=N_{l,d},\forall l\ge 0$ \citep[Theorem 2.9]{atkinson2012spherical}, we have
\begin{align}
k(x,x)=\sum_{l=0}^{k}\sum_{j=1}^{N_{l,d}}Y_{l,j}(x)^2=\sum_{l=0}^{k}N_{k,d}\le (k+1)N_{k,d}.
\end{align}
By \citet[Lemma 22]{bartlett2002rademacher} we get $\hat{R}_n({\mathcal{F}}_k)\le \frac{2(k+1)\sqrt{N_{k,d}}}{\sqrt{n}}.$
Since for any $x$, we get $g(x)\le \|\phi_k(x)\|_2\|a\|_2=c_1(k+1)\sqrt{N_{k,d}}$, the $L_2$ loss is $(2c_1(k+1)\sqrt{N_{k,d}})$-Lipschitz. As a result, \citet[Theorem 3]{kakade2008complexity} implies that with probability at least $1-\delta$, $\forall g\in{\mathcal{F}}_k$
\begin{align}
|\|g-f\|_2^2-\hat{\mathcal{L}}(g)|=|\mathbb{E}[\hat{\mathcal{L}}(g)]-\hat{\mathcal{L}}(g)|\lesssim \frac{c_1(k+1)^2N_{k,d}}{\sqrt{n}}+c_1(k+1)^2N_{k,d}\sqrt{\frac{\ln(1/\delta)}{n}}.
\end{align}
Note that $N_{k,d}\ge k$. As a result, when $n\ge \Omega(\mathrm{poly}(c_1,N_{k,d},\ln(1/\delta),1/\epsilon_1))$, we get
\begin{align}
\forall g\in{\mathcal{F}}_k,\quad |\|g-f\|_2^2-\hat{\mathcal{L}}(g)|\le \epsilon_1.
\end{align}
which proves the desired result. \end{proof}
\subsection{Proof of Lemma~\ref{lem:truncation}} In the following we present and prove Lemma~\ref{lem:truncation}, which is used to prove Lemma~\ref{lem:main}. \begin{lemma}\label{lem:truncation}
Suppose the function $f:\mathbb{S}^{d-1}\to \mathbb{R}$ satisfies Conditions~\ref{cond:decay} and \ref{cond:random} for some fixed $\alpha\in (0,1]$ and $c_1,c_2>0$.
When $d\ge 10\alpha^{-1}2^{5/\alpha}+2$, we have
\begin{align}
\|f-\Pi_{\le k-1}f\|_\infty\le 2c_1c_2k^{3/2}(N_{k,d})^{-\alpha/2},\quad\forall k\ge 1.
\end{align} \end{lemma} \begin{proof}[Proof of Lemma~\ref{lem:truncation}]
Let $c=c_1c_2$. By basic algebra we get
\begin{align}
\|f-\Pi_{\le k-1}f\|_\infty=\norm{\sum_{l\ge 0}\Pi_{l}f-\Pi_{\le k-1}f}_\infty=\norm{\sum_{l\ge k}\Pi_{l}f}_\infty\le \sum_{l\ge k}c\sqrt{\ln (l+1)}(N_{l,d})^{-\alpha/2}.
\end{align}
Therefore we only need to prove
\begin{align}\label{equ:truncation-1}
\sum_{l\ge k}c\sqrt{\ln (l+1)}(N_{l,d})^{-\alpha/2}\le 2ck^{3/2}(N_{k,d})^{-\alpha/2}.
\end{align}
Recall that $N_{l,d}=\frac{2l+d-2}{l+d-2}\frac{\Gamma(l+d-1)}{\Gamma(l+1)\Gamma(d-1)}$. It follows that
\begin{align}\label{equ:truncation-4}
&\sum_{l\ge k}c\sqrt{\ln (l+1)}(N_{l,d})^{-\alpha/2}\le \;\sum_{l\ge k}c\sqrt{ l}\(\frac{\Gamma(l+d-1)}{\Gamma(l+1)\Gamma(d-1)}\)^{-\alpha/2}.
\end{align}
Let $a_l\defeq \(\frac{\Gamma(l+d-1)}{\Gamma(l+1)\Gamma(d-1)}\)^{-\alpha/2}\sqrt{l}.$ We first prove that when $d\ge \frac{10}{\alpha}2^{5/\alpha}+2$, $\frac{a_{l+1}}{a_l}\le \(\frac{l}{l+1}\)^2,\forall l\ge 1$.
By basic algebra we get
\begin{align}
\frac{a_{l+1}}{a_l}=\sqrt{\frac{l+1}{l}}\(\frac{l+1}{l+d-1}\)^{\alpha/2}.
\end{align}
Let $\kappa=2^{5/\alpha+1}$. We first focus on the case when $l\ge \frac{d}{\kappa-1}$. Since $\alpha(d-2)/5\ge \kappa$, we have
\begin{align}
\(\frac{l+1}{l+d-1}\)^{\alpha/5}=\(1-\frac{d-2}{l+d-1}\)^{\alpha/5}\le 1-\frac{\alpha(d-2)/5}{l+d-1}\le 1-\frac{\kappa}{l+d-1}.
\end{align}
When $l\ge \frac{d}{\kappa-1}$ we have $\frac{\kappa}{l+d-1}\ge \frac{1}{l+1}.$ As a result, $\(\frac{l+1}{l+d-1}\)^{\alpha/5}\le 1-\frac{1}{l+1}=\frac{l}{l+1}.$ Equivalently, we get
\begin{align}
\sqrt{\frac{l+1}{l}}\(\frac{l+1}{l+d-1}\)^{\alpha/2}\le \(\frac{l}{l+1}\)^{2}.\label{equ:truncation-2}
\end{align}
Now we focus on the case when $l<\frac{d}{\kappa-1}.$ In this case we have
\begin{align}
&\(\frac{l+1}{l+d-1}\)^{\alpha/2}< \(\frac{\frac{d}{\kappa-1}+1}{\frac{d}{\kappa-1}+d-1}\)^{\alpha/2}\le \(\frac{\frac{d}{\kappa-1}+2}{\frac{d}{\kappa-1}+d}\)^{\alpha/2}.
\end{align}
Since $\frac{d}{\kappa-1}\ge 2$, we have
\begin{align}
\(\frac{\frac{d}{\kappa-1}+2}{\frac{d}{\kappa-1}+d}\)^{\alpha/2}\le \(\frac{2\frac{d}{\kappa-1}}{\frac{d}{\kappa-1}+d}\)^{\alpha/2}=\(\frac{2}{\kappa}\)^{\alpha/2}\le 2^{5/2}\le \(\frac{l}{l+1}\)^{5/2}.
\end{align}
Consequently,
\begin{align}
\(\frac{l+1}{l+d-1}\)^{\alpha/2}\(\frac{l+1}{l}\)^{1/2}\le \(\frac{l}{l+1}\)^{2}. \label{equ:truncation-3}
\end{align}
Combining Eq.~\eqref{equ:truncation-2} and Eq.~\eqref{equ:truncation-3}, in both cases we have
\begin{align}
\sqrt{\frac{l+1}{l}}\(\frac{l+1}{l+d-1}\)^{\alpha/2}\le \(\frac{l}{l+1}\)^{2}.
\end{align}
Now continue Eq.~\eqref{equ:truncation-4} we get,
\begin{align}
&\sum_{l\ge k}c\sqrt{l}\(\frac{\Gamma(l+d-1)}{\Gamma(l+1)\Gamma(d-1)}\)^{-\alpha/2}= ca_k\sum_{l\ge k}\frac{a_l}{a_k}=ca_k\sum_{l\ge k}\prod_{l'=k}^{l-1}\frac{a_{l'+1}}{a_{l'}}\\
\le\;&ca_k\sum_{l\ge k}\frac{k^2}{l^2}\le cka_k=ck^{3/2}\(\frac{\Gamma(k+d-1)}{\Gamma(k+1)\Gamma(d-1)}\)^{-\alpha/2}\le ck^{3/2}2^{\alpha/2}(N_{k,d})^{-\alpha/2}\\
\le\;& 2ck^{3/2}(N_{k,d})^{-\alpha/2}.
\end{align} \end{proof}
\subsection{Proof of Lemma~\ref{lem:main}}\label{app:pf-lem-main} In this section we prove Lemma~\ref{lem:main}.
\begin{proof}[Proof of Lemma~\ref{lem:main}]
Let $c=c_1c_2$. Recall that $k=\inf_{l\ge 0}\{2c(l+1)^{3/2}(N_{l+1,d})^{-\alpha/2}\le \epsilon/2\}.$ By Lemma~\ref{lem:truncation} we get
\begin{align}\label{equ:thm-main-1}
\|f-\Pi_{\le k}f\|_\infty\le \epsilon/2.
\end{align}
Hence, we only need to prove $\|\Pi_{\le k}g-\Pi_{\le k}f\|_\infty\le \epsilon/2$ and the desired result follows directly from triangle inequality.
Since $\Pi_{\le k}(g-f)$ has degree at most $k$, applying Fact~\ref{fact:infty-2-SH} we get
\begin{align}
\|\Pi_{\le k}g-\Pi_{\le k}f\|_\infty\le \sum_{l=0}^{k}\|\Pi_{l}(g-f)\|_\infty\le \sqrt{N_{k,d}}\sum_{l=0}^{k}\|\Pi_{l}(g-f)\|_2.
\end{align}
By Cauchy-Schwarz and Parseval's theorem we have
\begin{align}
\sum_{l=0}^{k}\|\Pi_{l}(g-f)\|_2\le \((k+1)\sum_{l=0}^{k}\|\Pi_{l}(g-f)\|_2^2\)^{1/2}\le \sqrt{k+1}\|\Pi_{\le k}(g-f)\|_2.
\end{align} As a result,
\begin{align}\label{equ:thm-main-2}
\|\Pi_{\le k}g-\Pi_{\le k}f\|_\infty\le \sqrt{k+1}\sqrt{N_{k,d}}\|\Pi_{\le k}(g-f)\|_2.
\end{align}
In the following, we show that
\begin{align}
\sqrt{k+1}\sqrt{N_{k,d}}\le 2(4c/\epsilon)^{3/\alpha}d^{4/\alpha}.
\end{align}
By the definition of $k$ we have $2ck^{3/2}(N_{k,d})^{-\alpha/2}> \epsilon/2.$
Hence,
\begin{align}\label{equ:thm-main-3}
\sqrt{N_{k,d}}\le \(\frac{4c}{\epsilon}k^{3/2}\)^{1/\alpha}.
\end{align}
To upper bound $k$, note that $N_{k,d}\ge (k/d)^{d-2}$. Therefore,
\begin{align}
\epsilon<4ck^{3/2}(N_{k,d})^{-\alpha/2}\le 4ck^{3/2} \(\frac{d}{k}\)^{-(d-2)\alpha/2}.
\end{align}
Solving for $k$ we get
$
k\le \(4c/\epsilon\)^{\frac{2}{d\alpha-5}}d^{\frac{d\alpha-2}{d\alpha-5}}.
$ Combining with Eq.~\eqref{equ:thm-main-3} and using the assumption that $d\ge \frac{10}{\alpha}2^{5/\alpha}+2$, we get
\begin{align}
&\sqrt{k+1}\sqrt{N_{k,d}}\le 2 \(4c/\epsilon\)^{\frac{1}{\alpha}}k^{\frac{3}{2\alpha}+\frac{1}{2}}\\
\le\;& 2\(4c/\epsilon\)^{\frac{1}{\alpha}}(4c/\epsilon)^{\frac{2}{d\alpha-5}\(\frac{3}{2\alpha}+\frac{1}{2}\)}d^{\frac{d\alpha-2}{d\alpha-5}\(\frac{3}{2\alpha}+\frac{1}{2}\)}\le 2(4c/\epsilon)^{3/\alpha}d^{4/\alpha}.\label{equ:thm-main-4}
\end{align}
Finally, combining Eq.~\eqref{equ:thm-main-4}, Eq.~\eqref{equ:thm-main-2} and the assumption $\|\Pi_{\le k}(g-f)\|_2\le \frac{1}{4}\epsilon(4c/\epsilon)^{-3/\alpha}d^{-4/\alpha}$ we get
\begin{align}
\|\Pi_{\le k}g-\Pi_{\le k}f\|_\infty\le \epsilon/2.
\end{align}
By triangle inequality and Eq.~\eqref{equ:thm-main-1}, we prove the desired result:
\begin{align}
\|f-\Pi_{\le k}g\|_\infty\le \|f-\Pi_{\le k}f\|_\infty+ \|\Pi_{\le k}g-\Pi_{\le k}f\|_\infty\le \epsilon.
\end{align} \end{proof}
\subsection{Proof of Lemma~\ref{lem:relu-lb}}\label{app:pf-lem-relu-lb} In this section we prove Lemma~\ref{lem:relu-lb}. \begin{proof}[Proof of Lemma~\ref{lem:relu-lb}]
In the following, we prove that for any $T<N_{k,d}$, there exists $f\in{\mathcal{F}}_k$ such that $\mathop{\rm Pr}\nolimits_{f,n}(\|\hat{f}_T-f\|_\infty<\beta_k/4)< 1/2$, and the desired result follows directly.
Suppose at round $i$ the algorithm query $x_i\in \mathbb{S}^{d-1}$ and receive $y_i=f(x_i)+{\mathcal{N}}(0,1)$ where $f$ is the ground-truth. At round $T$, the algorithm outputs $\hat{f}_T$. Let $\mathop{\rm Pr}\nolimits_{u,n}(\cdot)$ be the probability space of $(x_1,y_1,\cdots,x_T,y_T)$ when the ground-truth is $f=\beta_kP_{k,d}(\dotp{\cdot}{u}),$ and $\mathop{\rm Pr}\nolimits_{0,n}(\cdot)$ the space when the ground-truth is $f=0$. We use $\mathbb{E}_{u,n}$ and $\mathbb{E}_{0,n}$ to denote the corresponding expectation, respectively. Let ${\mathcal{H}}_i$ be the $\sigma$-field of random variable $(x_1,y_1,\cdots,x_{i-1},y_{i-1},x_i).$
For every $u\in\mathbb{S}^{d-1}$, let $E_{u,n}\defeq \ind{\|\hat{f}_T-\beta_kP_{k,d}(\dotp{\cdot}{u})\|_\infty<\beta_k/4}$ be the event that $\hat{f}_T$ is close to $\beta_kP_{k,d}(\dotp{\cdot}{u})$. By Pinsker's inequality and chain rule of KL divergence, we have
\begin{align}
\mathbb{E}_{u,n}[E_{u,n}]&\le \mathbb{E}_{0,n}[E_{u,n}]+D_{\rm TV}(\mathop{\rm Pr}\nolimits_{0,n}\|\mathop{\rm Pr}\nolimits_{u,n})\\
&\le \mathbb{E}_{0,n}[E_{u,n}]+\sqrt{\frac{1}{2}D_{\mathrm{KL}}(\mathop{\rm Pr}\nolimits_{0,n}\|\mathop{\rm Pr}\nolimits_{u,n})}\\
&= \mathbb{E}_{0,n}[E_{u,n}]+\sqrt{\frac{1}{2}\mathbb{E}_{0,n}\[\sum_{i=1}^{n}D_{\mathrm{KL}}(\mathop{\rm Pr}\nolimits_{0,n}(y_i\mid {\mathcal{H}}_i)\| \mathop{\rm Pr}\nolimits_{u,n}(y_i\mid {\mathcal{H}}_i))\right]}\\
&= \mathbb{E}_{0,n}[E_{u,n}]+\sqrt{\frac{\beta_k^2}{4}\mathbb{E}_{0,n}\[\sum_{i=1}^{n}P_{k,d}(x_i^\top u)^2\right]}.
\end{align}
Consequently,
\begin{align}
\mathbb{E}_{u\sim \mathbb{S}^{d-1}}[\mathbb{E}_{u,n}[E_{u,n}]]&\le \mathbb{E}_{u\sim \mathbb{S}^{d-1}}\[\mathbb{E}_{0,n}[E_{u,n}]+\sqrt{\frac{\beta_k^2}{4}\mathbb{E}_{0,n}\[\sum_{i=1}^{n}P_{k,d}(x_i^\top u)^2\right]}\right]\\
&\le \mathbb{E}_{u\sim \mathbb{S}^{d-1}}\[\mathbb{E}_{0,n}[E_{u,n}]\right]+\sqrt{\frac{\beta_k^2}{4}\mathbb{E}_{u\sim \mathbb{S}^{d-1}}\[\mathbb{E}_{0,n}\[\sum_{i=1}^{n}P_{k,d}(x_i^\top u)^2\right]\right]}\\
&= \mathbb{E}_{u\sim \mathbb{S}^{d-1}}\[\mathbb{E}_{0,n}[E_{u,n}]\right]+\sqrt{\frac{\beta_k^2}{4}\mathbb{E}_{0,n}\[\sum_{i=1}^{n}\mathbb{E}_{u\sim \mathbb{S}^{d-1}}\[P_{k,d}(x_i^\top u)^2\right]\right]}\\
&= \mathbb{E}_{u\sim \mathbb{S}^{d-1}}\[\mathbb{E}_{0,n}[E_{u,n}]\right]+\sqrt{\frac{\beta_k^2}{4}\mathbb{E}_{0,n}\[\frac{n}{N_{k,d}}\right]}\\
&= \mathbb{E}_{u\sim \mathbb{S}^{d-1}}\[\mathbb{E}_{0,n}[E_{u,n}]\right]+\sqrt{\frac{\beta_k^2}{4}\frac{n}{N_{k,d}}}.\label{equ:relu-lb-1}
\end{align}
Now we upper bound the first term in Eq.~\eqref{equ:relu-lb-1}. Let $\hat{u}_T=\min_{u\in\mathbb{S}^{d-1}}\|\hat{f}_T-P_{k,d}(\dotp{\cdot}{u})\|_\infty$. Consider the event $E_{u,n}'=\ind{\|P_{k,d}(\dotp{\cdot}{u})-P_{k,d}(\dotp{\cdot}{\hat{u}_n})\|_\infty <\beta_k/2}$. In the following we prove that $\neg E_{u,n}'\implies \neg E_{u,n}$. Indeed, when $\|P_{k,d}(\dotp{\cdot}{u})-P_{k,d}(\dotp{\cdot}{\hat{u}_n})\|_\infty \ge \beta_k/2$ we get
\begin{align}
\|\hat{f}_T-P_{k,d}(\dotp{\cdot}{u})\|_\infty&\ge \frac{1}{2}\(\|\hat{f}_T-P_{k,d}(\dotp{\cdot}{u})\|_\infty+\|\hat{f}_T-P_{k,d}(\dotp{\cdot}{\hat{u}_n})\|_\infty\)\tag{By the optimality of $\hat{u}_T$}\\
&\ge\frac{1}{2}\|P_{k,d}(\dotp{\cdot}{u})-P_{k,d}(\dotp{\cdot}{\hat{u}_n})\|_\infty\ge \beta_k/4.\tag{Triangle inequality}
\end{align}
Therefore, we get
\begin{align}
&\mathbb{E}_{u\sim \mathbb{S}^{d-1}}\[\mathbb{E}_{0,n}[E_{u,n}]\right]\le \mathbb{E}_{u\sim \mathbb{S}^{d-1}}\[\mathbb{E}_{0,n}[E_{u,n}']\right]\\
=\;&\mathbb{E}_{0,n}\[\mathbb{E}_{u\sim \mathbb{S}^{d-1}}\[\ind{\|P_{k,d}(\dotp{\cdot}{u})-P_{k,d}(\dotp{\cdot}{\hat{u}_n})\|_\infty\le \beta_k/4}\right]\right]\\
\le \;&\mathbb{E}_{0,n}\[\mathbb{E}_{u\sim \mathbb{S}^{d-1}}\[\ind{|P_{k,d}(\dotp{u}{u})-P_{k,d}(\dotp{u}{\hat{u}_n})|\le \beta_k/4}\right]\right]\\
= \;&\mathbb{E}_{0,n}\[\mathbb{E}_{u\sim \mathbb{S}^{d-1}}\[\ind{|1-P_{k,d}(\dotp{u}{\hat{u}_n})|\le \beta_k/4}\right]\right]\\
\le \;&\mathbb{E}_{0,n}\[\mathop{\rm Pr}\nolimits_{u\sim \mathbb{S}^{d-1}}\({P_{k,d}(\dotp{u}{\hat{u}_n})\ge 1-\beta_k/4}\)\right]\\
\le \;&\frac{16}{9N_{k,d}}\le \frac{1}{4}\tag{Proposition~\ref{prop:pnd-tail}}.
\end{align}
Finally, when $T<N_{k,d} \beta_k^{-2}$ we have
\begin{align}
\min_{f\in{\mathcal{F}}_k}\mathop{\rm Pr}\nolimits_{f,n}(\|\hat{f}_T-f\|_\infty<\beta_k/4)=\min_{u\in\mathbb{S}^{d-1}} \mathbb{E}_{u,n}[E_{u,n}]\le \mathbb{E}_{u\in\mathbb{S}^{d-1}}\mathbb{E}_{u,n}[E_{u,n}]<\frac{1}{2}.
\end{align} \end{proof}
\subsection{Proof of Theorem~\ref{thm:relu-lb}}\label{app:pf-thm-relu-lb} In this section we present the proof of Theorem~\ref{thm:relu-lb}. \begin{proof}[Proof of Theorem~\ref{thm:relu-lb}]
When $\epsilon>d^{-7/4}$ the lower bound is trivial. Hence we focus on the regime $\epsilon<d^{-7/4}$.
Let $k$ be the largest even number smaller than $\frac{d^{1/8}}{\sqrt{480\epsilon}}$ and $\tau_k=\dotp{\mathrm{ReLU}}{\bar{P}_{k,d}}_{\mu_d}$. First we prove that the set ${\mathcal{F}}_k\defeq \{\tau_kP_{k,d}(\dotp{\cdot}{u}):u\in\mathbb{S}^{d-1}\}$ belongs to $\mathrm{NN}\text{-}\mathrm{ReLU}(L_2).$
To this end, we prove that for every $f\in{\mathcal{F}}_k$, we can construct $c:\mathbb{S}^{d-1}\to\mathbb{R}$ such that $\|c\|_2\le 1$ and $f(x)=\mathbb{E}_{\xi\in\mathbb{S}^{d-1}}[\mathrm{ReLU}(\xi^\top x)c(\xi)]$ for every $x\in\mathbb{S}^{d-1}.$ For every $f=\tau_kP_{k,d}(\dotp{\cdot}{u})\in{\mathcal{F}}_k$, by Funk-Hecke formula (Theorem~\ref{thm:funk-hecke}) we have
\begin{align}
f(x)=\tau_kP_{k,d}(\dotp{\cdot}{u})=\sqrt{N_{k,d}}\mathbb{E}_{\xi\in\mathbb{S}^{d-1}}[\mathrm{ReLU}(\xi^\top x)P_{k,d}(\dotp{\cdot}{u})]=\mathbb{E}_{\xi\in\mathbb{S}^{d-1}}[\mathrm{ReLU}(\xi^\top x)\bar{P}_{k,d}(\dotp{\cdot}{u})].
\end{align}
Since $\|\bar{P}_{k,d}(\dotp{\cdot}{u})\|_2=1$, we get $f\in \mathrm{NN}\text{-}\mathrm{ReLU}(L_2).$
In the following we prove the desired result by invoking Lemma~\ref{lem:relu-lb} with the hypothesis ${\mathcal{F}}_k.$ First of all, by Lemma~\ref{lem:relu-eigen-approx} and the definition of $k$ we get
\begin{align}
\tau_k/4>\frac{d^{1/4}}{480k^{5/4}(k+d)^{3/4}}\ge \frac{d^{1/4}}{480k^2}\ge \epsilon.
\end{align}
Therefore, Lemma~\ref{lem:relu-lb} implies that the minimax sample complexity is at least $N_{k,d}\tau_k^{-2}.$ By basic Lemma~\ref{lem:relu-eigen-approx} and algebra we have
\begin{align}
&N_{k,d}\tau_k^{-2}\ge \binom{k+d-2}{d-2}\frac{k^{5/2}(k+d)^{3/2}}{1200 d^{1/2}}\gtrsim \(\frac{k}{d-2}+1\)^{d-2}\frac{k^{5/2}(k+d)^{3/2}}{ d^{1/2}}\\
\ge&\(\frac{k}{d}\)^{d}=(0.002\epsilon^{-1}d^{-7/4})^{d/2},
\end{align}
which proves the desired result. \end{proof}
\subsection{Missing Propositions} In this section we state and prove the missing propositions in Section~\ref{sec:main-results}.
\begin{proposition}\label{prop:smooth-inner-product-decay}
Let $h:[-1,1]\to \mathbb{R}$ be a one-dimensional function satisfies $\sup_{t\in[-1,1]}|h^{(k)}(t)|\le 1,\forall k\ge 0.$ Then
\begin{align}
\|\Pi_k h(\dotp{\cdot}{u})\|_2\le 2N_{k,d}^{-1/2}.
\end{align} \end{proposition} \begin{proof}
For a fixed $u\in \mathbb{S}^{d-1}$, by the completeness of the Legendre polynomial basis, we have
\begin{align}
h(\dotp{\cdot}{u})=\sum_{k\ge 0} \tau_k \bar{P}_{k,d}(\dotp{\cdot}{u}),
\end{align}
where $\tau_k\defeq \dotp{h}{\bar{P}_{k,d}}_{{\mu_d}}$. Since $\bar{P}_{k,d}(\dotp{\cdot}{u})\in\mathbb{Y}_{k,d}$, it follows that
\begin{align}
\|\Pi_k \exp(\dotp{\cdot}{u})\|_2=\tau_k \|\bar{P}_{k,d}(\dotp{\cdot}{u})\|_2=\tau_k.
\end{align}
As a result, we only need to prove that
\begin{align}\label{equ:prop-sipd-1}
\tau_k\le N_{k,d}^{-1/2},\quad \forall k\ge 0.
\end{align}
By Rodrigues formula \citet[Proposition 2.26]{atkinson2012spherical} we get
\begin{align}
\tau_k=\;&\int_{-1}^{1}h(t)\bar{P}_{k,d}(t){\mu_d}(t)\mathrm{d} t=\frac{\sqrt{N_{k,d}}\Gamma\(\frac{d}{2}\)}{\sqrt{\pi}\Gamma\(\frac{d-1}{2}\)}\int_{-1}^{1}h(t)P_{k,d}(t)(1-t^2)^{\frac{d-3}{2}}\mathrm{d} t\\
=\;&\frac{\sqrt{N_{k,d}}\Gamma\(\frac{d}{2}\)}{\sqrt{\pi}\Gamma\(\frac{d-1}{2}\)}\frac{\Gamma\(\frac{d-1}{2}\)}{2^k\Gamma\(k+\frac{d-1}{2}\)}\int_{-1}^{1}h^{(k)}(t)(1-t^2)^{k+\frac{d-3}{2}}\mathrm{d} t\\
\le \;&\frac{\sqrt{N_{k,d}}\Gamma\(\frac{d}{2}\)}{\sqrt{\pi}2^k\Gamma\(k+\frac{d-1}{2}\)}\int_{-1}^{1}|h^{(k)}(t)|(1-t^2)^{k+\frac{d-3}{2}}\mathrm{d} t\\
\le \;&\frac{\sqrt{N_{k,d}}\Gamma\(\frac{d}{2}\)}{\sqrt{\pi}2^k\Gamma\(k+\frac{d-1}{2}\)}\int_{-1}^{1}(1-t^2)^{k+\frac{d-3}{2}}\mathrm{d} t\\
\le \;&\frac{\sqrt{N_{k,d}}\Gamma\(\frac{d}{2}\)}{2^k\Gamma\(k+\frac{d-1}{2}\)}\frac{\Gamma\(k+\frac{d-1}{2}\)}{\Gamma\(k+\frac{d}{2}\)}=\frac{\sqrt{N_{k,d}}\Gamma\(\frac{d}{2}\)}{2^k\Gamma\(k+\frac{d}{2}\)}.
\end{align}
As a result, we only need to prove $\frac{\Gamma\(\frac{d}{2}\)}{2^k\Gamma\(k+\frac{d}{2}\)}\le 2N_{k,d}^{-1}$ and then Eq.~\eqref{equ:prop-sipd-1} follows directly.
By the recursive formula of $\Gamma$ function we get
\begin{align}
\frac{2^k\Gamma\(k+\frac{d}{2}\)}{\Gamma\(\frac{d}{2}\)}=2^k\prod_{l=1}^{k}\(k+\frac{d}{2}-l\)=\prod_{l=1}^{k}\(2k+d-2l\).
\end{align}
By the definition of $N_{k,d}$ we have
\begin{align}
N_{k,d}=\frac{2k+d-2}{k+d-2}\binom{k+d-2}{k}\le \frac{2\prod_{l=1}^{k}(k+d-1-l)}{k!}.
\end{align}
Observe that for any $l\in[1,k]$, $k+d-1-l\le 2k+d-2l.$ Consequently,
\begin{align}
N_{k,d}\le \frac{2\prod_{l=1}^{k}(k+d-1-l)}{k!}\le 2\prod_{l=1}^{k}\(2k+d-2l\)=2\frac{2^k\Gamma\(k+\frac{d}{2}\)}{\Gamma\(\frac{d}{2}\)}.
\end{align}
Equivalently,
\begin{align}
\frac{\Gamma\(\frac{d}{2}\)}{2^k\Gamma\(k+\frac{d}{2}\)}\le 2N_{k,d}^{-1}.
\end{align} \end{proof}
\begin{proposition}\label{prop:nn-exp-decay}
Let $\mathrm{NN}\text{-}\mathrm{Exp}(L_p)$ be the family of two layer NNs with activation $\exp(\cdot)$ and $L_p$ norm bounds. Then any function $f\in\mathrm{NN}\text{-}\mathrm{Exp}(L_1)$ satisfies $\|\Pi_k f\|_2\le 2eN_{k,d}^{-1/2}.$ \end{proposition} \begin{proof}
Recall that if two functions $f,g$ satisfies $\|\Pi_k f\|_2\le N_{k,d}^{-1/2}$ and $\|\Pi_k g\|_2\le N_{k,d}^{-1/2}$, their convex combinations $h=\theta f+(1-\theta)g$ also satisfies $\|\Pi_k g\|_2\le N_{k,d}^{-1/2}$. Since any function in $\mathrm{NN}\text{-}\mathrm{Exp}(L_1)$ can be written as a convex combination of functions $\{\pm \exp(\dotp{\cdot}{u}):u\in\mathbb{S}^{d-1}\}$, we only need to prove that
$\|\Pi_k \exp(\dotp{\cdot}{u})\|_2\le N_{k,d}^{-1/2}$ for every $u\in \mathbb{S}^{d-1}$.
Let $h(t)=e^{-1}\exp(t)$. Then we have $\sup_{t\in[-1,1]}|h^{(k)}(t)|\le 1.$ Invoking Proposition~\ref{prop:smooth-inner-product-decay} we get
\begin{align}
\|\Pi_k \exp(\dotp{\cdot}{u})\|_2=e\|\Pi_k h(\dotp{\cdot}{u})\|_2\le 2e N_{k,d}^{-1/2}.
\end{align} \end{proof}
\begin{proposition}\label{prop:rkhs-norm-of-f}
Suppose the function $f$ satisfies $\|\Pi_k f\|_2= \Omega(1)N_{k,d}^{-\alpha/2},\forall k\ge 0$ for some constant $\alpha>0$. For any inner product kernel $K(x,x')$ on the sphere where $\sup_{x,x'\in\mathbb{S}^{d-1}}|K(x,x')|\le 1$, $f$ has a infinite RKHS norm induced by $K$ when $\alpha<1/2$. \end{proposition} \begin{proof}
Since $K(x,x')$ is a bounded inner product kernel, we can write $K(x,x')=h(\dotp{x}{x'})$ for some one-dimensional function $h:[-1,1]\to[-1,1].$ Let $\lambda_k$ be the eigenvalues of kernel $K$. By the Funk-Hecke formula (Theorem~\ref{thm:funk-hecke}) we get
\begin{align}
\lambda_k=N_{k,d}^{-1/2}\dotp{h}{\bar{P}_{k,d}}_{{\mu_d}}\le N_{k,d}^{-1/2}\|h\|_{\mu_d}\|\bar{P}_{k,d}\|_{\mu_d}\le N_{k,d}^{-1/2}\|h\|_\infty\|\bar{P}_{k,d}\|_{\mu_d}\le N_{k,d}^{-1/2}.
\end{align}
Since $\mathbb{Y}_{k,d}$ is the space of eigenfunctions of kernel $K$ corresponding to the eigenvalue $\lambda_k$, the RKHS norm of $f$ is defined by
\begin{align}
\|f\|_K^2=\sum_{k\ge 0}\frac{\|\Pi_k f\|_2^2}{\lambda_k}\ge \sum_{k\ge 0}\|\Pi_k f\|_2^2N_{k,d}^{1/2}.
\end{align}
As a result, when $\alpha<1/2$ we get
\begin{align}
\|f\|_K^2\gtrsim \sum_{k\ge 0}N_{k,d}^{1/2-\alpha}=\infty.
\end{align} \end{proof}
\section{Learning with Two-layer Finite-width Neural Networks}\label{app:nns} In this section, we show that using a finite-width two-layer neural network with polynomial activation can also achieve a small $L_\infty$-error bound. \begin{algorithm}[ht]
\caption{$L_\infty$-learning via Two-layer NNs with Polynomial Activation}
\label{alg:NN}
\hspace*{\algorithmicindent} \textbf{Input:} parameters $\alpha,c_1,c_2>0$, desired error level $\epsilon>0$, and failure probability $\delta>0.$
\hspace*{\algorithmicindent} \textbf{Input:} Dataset ${\mathcal{D}}=\{(x_i,y_i)\}_{i=1}^{n}$ where $x_i\sim \mathbb{S}^{d-1}$ are independent and uniformly sampled from the unit sphere $\mathbb{S}^{d-1}$, and $y_i=f(x_i)+{\mathcal{N}}(0,1)$.
\begin{algorithmic}[1]
\State Set the truncation threshold $k\gets\inf_{l\ge 0}\{2c_1c_2(l+1)^{3/2}(N_{l+1,d})^{-\alpha/2}\le \epsilon/2\}.$\label{nn-line:3}
\State Set the parameters for the neural network: norm bound $B=35 c_1\sqrt{d}\(\frac{4c_1c_2}{\epsilon}\)^{3+4/\alpha}$, and width $m\gets 256B^2\epsilon^{-6/\alpha-2}(4c_1c_2)^{6/\alpha}d^{8/\alpha}$.
\State Define the family of two-layer NNs with polynomial activation $\sigma_k$ defined in Eq.~\eqref{equ:activation}:
$$\textstyle{{\mathcal{F}}_k=\left\{g(x)=\sum_{j=1}^{m}a_j\sigma_k(w_j^\top x): w_j\in \mathbb{S}^{d-1},\sum_{j=1}^{m}|a_j|\le B\right\}}.$$
\State Run empirical risk minimization and get
$
g=\textstyle{\argmin_{f\in{\mathcal{F}}_k} \sum_{i=1}^{n}(f(x_i)-y_i)^2.}
$
\State \textbf{Return} $g$.
\end{algorithmic} \end{algorithm}
Our algorithm is presented as Alg.~\ref{alg:NN}. On a high level, given any desired error level $\epsilon>0$, the algorithm selects a truncation threshold $k\ge 0$ (Line~\ref{nn-line:3}), and use empirical risk minimization to find two-layer neural network $g$ with polynomial activation that fits the ground-truth $f$ the best. The activation $\sigma_k:[-1,1]\to\mathbb{R}$ is the degree-$k$ approximation of the ReLU activation in the Legendre polynomial space, given by \begin{align}\label{equ:activation}
\textstyle{\sigma_k(t)\defeq \sum_{l=0}^{k}\dotp{\mathrm{ReLU}}{\bar{P}_{l,d}}_{\mu_d} \bar{P}_{l,d}(t).} \end{align} Since $\bar{P}_{k,d}(\dotp{w}{\cdot})\in \mathbb{Y}_{k,d}$ for every $k\ge 0,w\in\mathbb{S}^{d-1}$, any two-layer NN with activation $\sigma_k$ is a degree-$k$ polynomial (more precisely, it is the projection of a two-layer ReLU network with the same parameters to the space $\mathbb{Y}_{\le k,d}$). Hence, our algorithm essentially aims to find the best low-degree approximation of the ground-truth $f$ using noisy data.
The following theorem states the sample complexity of Alg.~\ref{alg:NN} \begin{theorem}\label{thm:nn-sc}
Suppose the ground-truth function satisfies Conditions~\ref{cond:decay} and \ref{cond:random} for some fixed $\alpha\in (0,1]$ and $c_1,c_2>0$. If $d\ge 10\alpha^{-1}2^{5/\alpha}+2$, then for any $\epsilon>0,\delta>0$, with probability at least $1-\delta$ over the randomness of the data, Alg.~\ref{alg:NN} outputs a function $g$ such that $\|f-g\|_\infty\le \epsilon$ using $O(\mathrm{poly}(c_1c_2,d,1/\epsilon,\ln 1/\delta)^{1/\alpha})$ samples. \end{theorem}
\subsection{Proof of Theorem~\ref{thm:nn-sc}} \begin{proof}[Proof of Theorem~\ref{thm:nn-sc}]
Let $\epsilon_1=\frac{1}{4}\epsilon^{3/\alpha+1}(4c_1c_2)^{-3/\alpha}d^{-4/\alpha}$. We prove Theorem~\ref{thm:mainsc} in the following two steps.
\paragraph{Step 1: upper bound the population $L_2$ loss.} In this step, we use classic statistical learning tools to show that the ERM step (i.e., $g=\argmin_{h\in{\mathcal{F}}_k} \sum_{i=1}^{n}(h(x_i)-y_i)^2$) returns a function $g$ with small $L_2$ loss. In particular, by Lemma~\ref{lem:ERM-l2-NN} we get
$
\|\Pi_{\le k} (f-g)\|_2\le \epsilon_1.
$
\paragraph{Step 2: upper bound the $L_\infty$-error via truncation.} This step is exactly the same as in the proof of Theorem~\ref{thm:mainsc}.
Combining these two steps we prove the desired result. \end{proof}
\begin{lemma}\label{lem:ERM-l2-NN}
Suppose the function $f:\mathbb{S}^{d-1}\to \mathbb{R}$ satisfies Conditions~\ref{cond:decay} and \ref{cond:random} for some fixed $\alpha\in (0,1],c_1,c_2>0$. For any $\epsilon>0$, let $k=\inf_{l\ge 0}\{2c_1c_2(l+1)^{3/2}(N_{l+1,d})^{-\alpha/2}\le \epsilon/2\}$.
For any $\epsilon_1>0$, let $B=35 c_1\sqrt{d}\(\frac{4c_1c_2}{\epsilon}\)^{3+4/\alpha}$, $\sigma_k(t)=\sum_{l=0}^{k}\dotp{\mathrm{ReLU}}{\bar{P}_{l,d}}_{\mu_d} \bar{P}_{l,d}(t)$, and $m=16B^2/\epsilon_1^2$, define the function class
\begin{align}
{\mathcal{F}}_k=\left\{h(x)=\sum_{j=1}^{m}a_j\sigma_k(w_j^\top x): w_j\in \mathbb{S}^{d-1},\sum_{j=1}^{m}|a_j|\le B\right\}.
\end{align}
For a given dataset $\{(x_i,y_i)\}_{i=1}^{n}$, let $\hat{\mathcal{L}}(h)\defeq \frac{1}{N}\sum_{i=1}^{n}(h(x_i)-y_i)^2$ be the empirical $L_2$ loss, and $g=\argmin_{h\in{\mathcal{F}}}\hat{\mathcal{L}}(h).$
For any $\delta>0$, when $d\ge \max\{2e,4/\alpha\}$ and $n\ge \Omega(\mathrm{poly}(d,(c_1c_2)^{1/\alpha},\epsilon^{-1/\alpha},\ln(1/\delta),1/\epsilon_1))$, with probability at least $1-\delta$,
\begin{align}
\|\Pi_{\le k} (f-g)\|_2=\|\Pi_{\le k} f-g\|_2\le \epsilon_1.
\end{align} \end{lemma}
\subsection{Proof of Lemma~\ref{lem:ERM-l2-NN}}\label{app:pf-ERM-l2-NN} In this section, we prove Lemma~\ref{lem:ERM-l2-NN}. \begin{proof}[Proof of Lemma~\ref{lem:ERM-l2-NN}]
First we prove that there exists $\hat{f}\in{\mathcal{F}}$ such that the population loss is small. Since ${\mathcal{F}}\subseteq \mathbb{Y}_{\le k,d}$, we get
\begin{align}\label{equ:ERM-l2-0}
\forall h\in{\mathcal{F}}, \quad \|h-f\|_2^2=\|h-\Pi_{\le k}f\|_2^2+\|f-\Pi_{\le k}f\|_2^2.
\end{align}
By Lemma~\ref{lem:truncated-f-l1-norm}, $\Pi_{\le k}f$ can be represented by a infinite-width two-layer ReLU neural network with weight $c$ such that $\|c\|_1\le 35 c_1\sqrt{d}\(\frac{4c_1c_2}{\epsilon}\)^{3+4/\alpha}=B$. By Lemma~\ref{lem:nn-sample}, when $m>16B^2/\epsilon_1^2$ there exists a finite-width approximation $\hat{f}\in{\mathcal{F}}$ such that $\|\hat{f}-\Pi_{\le k}f\|_2\le \epsilon_1/2.$
In the following we show that ERM outputs a function $g\in{\mathcal{F}}$ such that
\begin{align}\label{equ:ERM-l2-1}
\|g-f\|_2^2\le \|\hat{f}-f\|_2^2+\epsilon_1^2/2.
\end{align}
By the uniform convergence of two-layer neural networks (Lemma~\ref{lem:uniform-convergence}), when $$n\ge \Omega(\mathrm{poly}(d,(c_1c_2)^{1/\alpha},\epsilon^{-1/\alpha},\ln(1/\delta),1/\epsilon_1))$$ we have
\begin{align}
&\|g-f\|_2^2\le \hat{\mathcal{L}}(g)+\epsilon_1^2/4\le \hat{\mathcal{L}}(\hat{f})+\epsilon_1^2/4\le\|\hat{f}-f\|_2^2+\epsilon_1^2/2.
\end{align}
Combining with Eq.~\eqref{equ:ERM-l2-0} we get
\begin{align}
\|g-\Pi_{\le k} f\|_2^2\le \|\hat{f}-\Pi_{\le k} f\|_2^2+\epsilon_1^2/2< \epsilon_1^2.
\end{align} \end{proof}
The following lemma proves the uniform convergence result for the function class used in Lemma~\ref{lem:ERM-l2-NN}. \begin{lemma}\label{lem:uniform-convergence}
In the setting of Lemma~\ref{lem:ERM-l2-NN}, when $n\ge \Omega(\mathrm{poly}(B,N_{k,d},\ln(1/\delta),1/\epsilon_1))$, for any $\delta>0$, with probability at least $1-\delta$ we have
\begin{align}
\sup_{g\in{\mathcal{F}}}|\|g-f\|_2^2-\hat{\mathcal{L}}(g)|\le \epsilon_1.
\end{align} \end{lemma} \begin{proof}
The proof is essentially the same as the proof of Lemma~\ref{lem:uniform-convergence-kernel}, with the only difference that here we use the Rademacher complexity upper bound for two-layer neural networks~\citep[Theorem 18]{bartlett2002rademacher}. \end{proof}
The following lemma proves the realizability result for the function class used in Lemma~\ref{lem:ERM-l2-NN}. \begin{lemma}\label{lem:truncated-f-l1-norm}
In the setting of Lemma~\ref{lem:ERM-l2-NN}, $\Pi_{\le k}f$ can be represented by an infinite-width two-layer ReLU neural network with weight $c:\mathbb{S}^{d-1}\to\mathbb{R}$ such that $\|c\|_1\le 35 c_1\sqrt{d}\(\frac{4c_1c_2}{\epsilon}\)^{3+4/\alpha}$. \end{lemma} \begin{proof}
Recall that we can write $\Pi_{\le k} f(x)=\sum_{l=0}^{k} \sum_{j=1}^{N_{k,d}}a_{l,j}Y_{l,j}(\cdot)$.
Let \begin{align}
\lambda_l=N_{l,d}^{-1/2}\dotp{\mathrm{ReLU}}{\bar{P}_{l,d}}_{\mu_d}
\end{align} and
define the weight $c:\mathbb{S}^{d-1}\to\mathbb{R}$ by $c(x)=\sum_{l=0}^{k}\lambda_l^{-1}\sum_{j=1}^{N_{l,d}}a_{l,j}Y_{l,j}(\cdot).$ Then by the Funk-Hecke formula (Theorem~\ref{thm:funk-hecke}) we get
\begin{align}
\Pi_{\le k} f(x)=\mathbb{E}_{w\sim\mathbb{S}^{d-1}}[\sigma(x^\top w)c(w)],\quad\forall x\in\mathbb{S}^{d-1}.
\end{align}
Hence, we only need to upper bound $\|c\|_2,$ and then the desired result is proved by the fact that $\|c\|_1\le \|c\|_2.$
Let $\beta_l^2=\|\Pi_l f\|_2^2=\sum_{j=1}^{N_{l,d}}a_{l,j}^2$ for all $l\in[0,k]$. Then we have
\begin{align}
&\|c\|_2^2=\sum_{l=0}^{k}\lambda_l^{-2}\beta_l^2\le \sum_{l=0}^{k}\lambda_l^{-2}c_1N_{l,d}^{-\alpha}
\le1200c_1^2\sum_{l=0}^{k}N_{l,d}^{1-\alpha}l(l+d) \tag{By Lemma~\ref{lem:relu-eigen-approx}}\\
\le\;&1200c_1^2N_{k,d} k^2(k+d).
\end{align}
By the definition of $k$ we have
\begin{align}
2c_1c_2k^{3/2}(N_{k,d})^{-\alpha/2}> \epsilon/2.
\end{align}
Consequently,
\begin{align}
N_{k,d}< \(\frac{4c_1c_2}{\epsilon}\)^{2/\alpha}k^{3/\alpha}.
\end{align}
Applying Proposition~\ref{prop:truncation-upperbound} we get $k\le \(\frac{4c_1c_2}{\epsilon}\)^{\frac{2}{d\alpha-3}}$. Using the assumption that $d>4/\alpha$ we get
\begin{align}
c_1^2N_{k,d} k^2(k+d)\le dc_1^2\(\frac{4c_1c_2}{\epsilon}\)^{2/\alpha}k^{3+3/\alpha}
\end{align}
As a result,
\begin{align}
\|c\|_1^2\le \|c\|_2^2\le 1200 dc_1^2\(\frac{4c_1c_2}{\epsilon}\)^{6+8/\alpha}.
\end{align} \end{proof}
\section{Decomposition of ReLU in the Legendre Polynomial Space} The following lemma analytically computes the spherical harmonics decomposition of ReLU activation (see also \citet{bach2017breaking,mhaskar2006weighted,bourgain1988projection,schneider1967problem}). \begin{lemma}\label{lem:relu-eigen}
Let $\tau_k=\dotp{\mathrm{ReLU}}{\bar{P}_{k,d}}_{\mu_d}$ be the projection of ReLU function to degree-$k$ Legendre polynomial. Then we have
\begin{align}
\tau_k=\begin{cases}
(-1)^{\frac{k-2}{2}}\sqrt{N_{k,d}}\frac{1}{2^k\sqrt{\pi}}\frac{\Gamma(d/2)\Gamma(k-1)}{\Gamma(k/2)\Gamma((k+d+1)/2)},&\text{when $k$ is even},\\
\frac{1}{2\sqrt{d}},&\text{when $k=1$},\\
0,&\text{when $k>1$ and $k$ is odd}.
\end{cases}
\end{align} \end{lemma} \begin{proof}
Recall that by definition,
\begin{align}\label{equ:tau-def}
\tau_k=\int_{-1}^{1}\mathrm{ReLU}(t)\bar{P}_{k,d}(t)\mu_d(t)\mathrm{d} t=\sqrt{N_{k,d}}\int_{0}^{1}t P_{k,d}(t)\mu_d(t)\mathrm{d} t.
\end{align}
When $k$ is odd we have $P_{k,d}(-t)=-P_{k,d}(t).$ As a result,
\begin{align}
\int_{0}^{1}t P_{k,d}(t)\mu_d(t)\mathrm{d} t=\frac{1}{2}\int_{-1}^{1}t P_{k,d}(t)\mu_d(t)\mathrm{d} t.
\end{align}
Recall that $P_{1,d}(t)=t$, and we have $$\int_{-1}^{1}t P_{k,d}(t)\mu_d(t)\mathrm{d} t=\int_{-1}^{1}P_{1,d}(t) P_{k,d}(t)\mu_d(t)\mathrm{d} t=\frac{1}{N_{k,d}}\ind{k=1}.$$ It follows directly that (1) $\tau_k=0$ if $k>1$ and $k$ is odd, and (2) $\tau_1=\frac{1}{2\sqrt{N_{1,d}}}=\frac{1}{2\sqrt{d}}.$
Now we focus on the case when $k$ is even. By the Rodrigues representation formula \citep[Theorem 2.23]{atkinson2012spherical} we get
\begin{align}
P_{k,d}(t)=(-1)^k\frac{\Gamma(\frac{d-1}{2})}{2^k\Gamma(k+\frac{d-1}{2})}(1-t^2)^{-\frac{d-3}{2}}\(\frac{\mathrm{d}}{\mathrm{d} t}\)^k(1-t^2)^{k+\frac{d-3}{2}}.
\end{align}
As a result,
\begin{align}
&\int_{0}^{1}t P_{k,d}(t)\mu_d(t)\mathrm{d} t\\
=&(-1)^k\frac{\Gamma(\frac{d-1}{2})}{2^k\Gamma(k+\frac{d-1}{2})}\frac{\Gamma(d/2)}{\Gamma((d-1)/2)}\frac{1}{\sqrt{\pi}}\int_{0}^{1}t\(\frac{\mathrm{d}}{\mathrm{d} t}\)^k(1-t^2)^{k+\frac{d-3}{2}}\mathrm{d} t\\
=&(-1)^{k+1}\frac{\Gamma(\frac{d-1}{2})}{2^k\Gamma(k+\frac{d-1}{2})}\frac{\Gamma(d/2)}{\Gamma((d-1)/2)}\frac{1}{\sqrt{\pi}}\int_{0}^{1}\(\frac{\mathrm{d}}{\mathrm{d} t}\)^{k-1}(1-t^2)^{k+\frac{d-3}{2}}\mathrm{d} t\tag{integration by parts}\\
=&(-1)^{k+1}\frac{\Gamma(\frac{d-1}{2})}{2^k\Gamma(k+\frac{d-1}{2})}\frac{\Gamma(d/2)}{\Gamma((d-1)/2)}\frac{1}{\sqrt{\pi}}\(\frac{\mathrm{d}}{\mathrm{d} t}\)^{k-2}(1-t^2)^{k+\frac{d-3}{2}}\bigg\vert_{0}^{1}\\
=&(-1)^{k}\frac{\Gamma(\frac{d-1}{2})}{2^k\Gamma(k+\frac{d-1}{2})}\frac{\Gamma(d/2)}{\Gamma((d-1)/2)}\frac{1}{\sqrt{\pi}}\(\frac{\mathrm{d}}{\mathrm{d} t}\)^{k-2}(1-t^2)^{k+\frac{d-3}{2}}\bigg\vert_{t=0}.\label{equ:relu-pf-1}
\end{align}
By binomial theorem, we have
\begin{align}
&\(\frac{\mathrm{d}}{\mathrm{d} t}\)^{k-2}(1-t^2)^{k+\frac{d-3}{2}}\bigg\vert_{t=0}=\(\frac{\mathrm{d}}{\mathrm{d} t}\)^{k-2} \sum_{j=0}^{k+\frac{d-3}{2}}{{k+\frac{d-3}{2}}\choose j}(-1)^jt^{2j}\bigg\vert_{t=0}\\
=&(-1)^{\frac{k-2}{2}}(k-2)!{{k+\frac{d-3}{2}}\choose \frac{k-2}{2}}=(-1)^{\frac{k-2}{2}}\frac{\Gamma(k-1)\Gamma(k+\frac{d-1}{2})}{\Gamma(k/2)\Gamma(\frac{k+d+1}{2})}.\label{equ:relu-pf-2}
\end{align}
Combining Eq.~\eqref{equ:relu-pf-1} and Eq.~\eqref{equ:relu-pf-2} we get
\begin{align}
&\int_{0}^{1}t P_{k,d}(t)\mu_d(t)\mathrm{d} t\\
=&(-1)^{k}\frac{\Gamma(\frac{d-1}{2})}{2^k\Gamma(k+\frac{d-1}{2})}\frac{\Gamma(d/2)}{\Gamma((d-1)/2)}\frac{1}{\sqrt{\pi}}(-1)^{\frac{k-2}{2}}\frac{\Gamma(k-1)\Gamma(k+\frac{d-1}{2})}{\Gamma(k/2)\Gamma(\frac{k+d+1}{2})}\\
=&(-1)^{\frac{k-2}{2}}\frac{\Gamma(\frac{d}{2})\Gamma(k-1)}{2^k\Gamma(\frac{k}{2})\Gamma(\frac{k+d+1}{2}))}\frac{1}{\sqrt{\pi}}.
\end{align}
Finally, combining with Eq.~\eqref{equ:tau-def} we prove the desired result. \end{proof}
\begin{lemma}\label{lem:relu-eigen-approx}
Let $\tau_k=\dotp{\mathrm{ReLU}}{\bar{P}_{k,d}}_{\mu_d}$ be the projection of ReLU to degree-$k$ Legendre polynomial. Then we have $\abs{\tau_k}=\Theta(d^{1/4}k^{-5/4}(k+d)^{-3/4}).$ In particular, for all dimension $d\ge 3$ and even degree $k\ge 4$ the following upper and lower bounds hold:
\begin{align}
\frac{2^{5/4}\pi^{3/4}}{\exp(13/2)} d^{1/4}k^{-5/4}(k+d)^{-3/4}\le \abs{\tau_k}\le \frac{\exp(13/2)}{2\pi^2} d^{1/4}k^{-5/4}(k+d)^{-3/4}.
\end{align} \end{lemma} \begin{proof}
Recall that Stirling's formula states
\begin{align}
\sqrt{2\pi}k^{k+1/2}e^{-k}\le \Gamma(k+1)\le e k^{k+1/2}e^{-k}.
\end{align}
We first prove the upper bound. By Lemma~\ref{lem:relu-eigen}, when $k$ is even we have
\begin{align}
\abs{\tau_k}=&\;\sqrt{N_{k,d}}\frac{1}{2^k\sqrt{\pi}}\frac{\Gamma(d/2)\Gamma(k-1)}{\Gamma(k/2)\Gamma((k+d+1)/2)}\\
=&\;\sqrt{\frac{2k+d-2}{k+d-2}\frac{\Gamma(k+d-1)}{\Gamma(k+1)\Gamma(d-1)}}\frac{1}{2^k\sqrt{\pi}}\frac{\Gamma(d/2)\Gamma(k-1)}{\Gamma(k/2)\Gamma((k+d+1)/2)}\\
\le &\;\frac{\sqrt{2}}{\sqrt{\pi}}\(\sqrt{\frac{\Gamma(k+d-1)}{\Gamma(k+1)\Gamma(d-1)}}\frac{1}{2^k}\frac{\Gamma(d/2)\Gamma(k-1)}{\Gamma(k/2)\Gamma((k+d+1)/2)}\)\\
\le &\;\frac{\sqrt{2}}{\sqrt{\pi}}\(\sqrt{\frac{e}{2\pi}\frac{(k+d-2)^{k+d-\frac{3}{2}}}{k^{k+\frac{1}{2}}(d-2)^{d-\frac{3}{2}}}}\frac{1}{2^k}\frac{\exp(7/2)}{2\pi}\frac{(\frac{d}{2}-1)^{\frac{d-1}{2}}(k-2)^{k-\frac{3}{2}}}{(\frac{k}{2}-1)^{\frac{k-1}{2}}(\frac{k+d-1}{2})^{\frac{k+d}{2}}}\)\\
\le &\;\frac{\exp(4)}{2\pi^2}\(\exp(5/2)\sqrt{\frac{(k+d)^{k+d-\frac{3}{2}}}{k^{k+\frac{1}{2}}d^{d-\frac{3}{2}}}}\frac{1}{2^k}\frac{(\frac{d}{2})^{\frac{d-1}{2}}k^{k-\frac{3}{2}}}{(\frac{k}{2})^{\frac{k-1}{2}}(\frac{k+d}{2})^{\frac{k+d}{2}}}\)\tag{Since $(1-1/t)^t=\Theta(1)$}\\
\le &\;\frac{\exp(13/2)}{2\pi^2}\(\sqrt{\frac{(k+d)^{k+d-\frac{3}{2}}}{k^{k+\frac{1}{2}}d^{d-\frac{3}{2}}}}\frac{d^{\frac{d-1}{2}}k^{k-\frac{3}{2}}}{k^{\frac{k-1}{2}}(k+d)^{\frac{k+d}{2}}}\)\\
\le&\;\frac{\exp(13/2)}{2\pi^2}\((k+d)^{-3/4}k^{-5/4}d^{1/4}\).
\end{align}
Now we prove the lower bound. Similarly,
\begin{align}
\abs{\tau_k}=&\;\sqrt{N_{k,d}}\frac{1}{2^k\sqrt{\pi}}\frac{\Gamma(d/2)\Gamma(k-1)}{\Gamma(k/2)\Gamma((k+d+1)/2)}\\
=&\;\sqrt{\frac{2k+d-2}{k+d-2}\frac{\Gamma(k+d-1)}{\Gamma(k+1)\Gamma(d-1)}}\frac{1}{2^k\sqrt{\pi}}\frac{\Gamma(d/2)\Gamma(k-1)}{\Gamma(k/2)\Gamma((k+d+1)/2)}\\
\ge &\;\frac{1}{\sqrt{\pi}}\(\sqrt{\frac{\Gamma(k+d-1)}{\Gamma(k+1)\Gamma(d-1)}}\frac{1}{2^k}\frac{\Gamma(d/2)\Gamma(k-1)}{\Gamma(k/2)\Gamma((k+d+1)/2)}\)\\
\ge &\;\frac{1}{\sqrt{\pi}}\(\sqrt{\frac{\sqrt{2\pi}}{e^2}\frac{(k+d-2)^{k+d-\frac{3}{2}}}{k^{k+\frac{1}{2}}(d-2)^{d-\frac{3}{2}}}}\frac{1}{2^k}\frac{2\pi}{\exp(1/2)}\frac{(\frac{d}{2}-1)^{\frac{d-1}{2}}(k-2)^{k-\frac{3}{2}}}{(\frac{k}{2}-1)^{\frac{k-1}{2}}(\frac{k+d-1}{2})^{\frac{k+d}{2}}}\)\\
\ge &\;\frac{2^{5/4}\pi^{3/4}}{\exp(3/2)}\(\exp(-5)\sqrt{\frac{(k+d)^{k+d-\frac{3}{2}}}{k^{k+\frac{1}{2}}d^{d-\frac{3}{2}}}}\frac{1}{2^k}\frac{(\frac{d}{2})^{\frac{d-1}{2}}k^{k-\frac{3}{2}}}{(\frac{k}{2})^{\frac{k-1}{2}}(\frac{k+d}{2})^{\frac{k+d}{2}}}\)\tag{Since $(1-1/t)^t=\Theta(1)$}\\
\ge &\;\frac{2^{5/4}\pi^{3/4}}{\exp(13/2)}\(\sqrt{\frac{(k+d)^{k+d-\frac{3}{2}}}{k^{k+\frac{1}{2}}d^{d-\frac{3}{2}}}}\frac{d^{\frac{d-1}{2}}k^{k-\frac{3}{2}}}{k^{\frac{k-1}{2}}(k+d)^{\frac{k+d}{2}}}\)\\
\ge&\;\frac{2^{5/4}\pi^{3/4}}{\exp(13/2)}\((k+d)^{-3/4}k^{-5/4}d^{1/4}\).
\end{align} \end{proof}
\section{Random Spherical Harmonics}\label{app:random-sh} In this section, we prove the $L_\infty$-norm bound for random spherical harmonics. \citet{burq2014probabilistic} prove a similar result without explicitly computes the $d$-dependency in Eq.~\eqref{equ:random-sh}. \begin{proof}[Proof of Lemma~\ref{lem:random-sh}]
For any fixed $x\in\mathbb{S}^{d-1}$, by Lemma~\ref{lem:riesz-SH} we get
\begin{align}\label{equ:random-sh-1}
g(x)=\sqrt{N_{k,d}}\mathbb{E}_{\xi\sim \mathbb{S}^{d-1}}[g(\xi)\bar{P}_{k,d}(x^\top \xi)].
\end{align}
Since $\{Y_{k,j}\}_{j=1}^{N_{k,d}}$ is a set of orthonormal basis, there exists weights $\{u_j\}_{j=1}^{N_{k,d}}$ (that depends on $x$) such that
\begin{align}
\bar{P}_{k,d}(x^\top \xi)=\sum_{j=1}^{N_{k,d}}u_jY_{k,j}(\xi),\quad \forall \xi\in\mathbb{S}^{d-1},
\end{align}
and $\sum_{j=1}^{N_{k,d}}u_j^2=\mathbb{E}_{\xi\sim \mathbb{S}^{d-1}}[\bar{P}_{k,d}(x^\top \xi)^2]=1.$
Define $a=[a_1,\cdots,a_{N_{k,d}}]\in\mathbb{R}^{N_{k,d}}$ and $u=[u_1,\cdots,u_{N_{k,d}}]\in\mathbb{R}^{N_{k,d}}$. Then we have
\begin{align}
&g(x)=\sqrt{N_{k,d}}\mathbb{E}_{\xi\sim \mathbb{S}^{d-1}}[g(\xi)\bar{P}_{k,d}(x^\top \xi)]=\sqrt{N_{k,d}}\mathbb{E}_{\xi\sim \mathbb{S}^{d-1}}\[\(\sum_{j=1}^{N_{k,d}}a_{j}Y_{k,j}(\xi)\)\(\sum_{j=1}^{N_{k,d}}u_jY_{k,j}(\xi)\)\right]\nonumber\\
=\;&\sqrt{N_{k,d}}\sum_{j=1}^{N_{k,d}}a_{j}u_j=\sqrt{N_{k,d}}a^\top u.
\end{align}
In addition, $\|g\|_2^2=\sum_{j=1}^{N_{k,d}}a_j^2=\|a\|_2^2$. Hence, by Lemma~\ref{lem:gaussian-proj-concentration} we get
\begin{align}
\forall t>0,\mathop{\rm Pr}\nolimits\(\frac{|a^\top u|}{\|a\|_2}\ge \frac{2t}{\sqrt{N_{k,d}}}\)\le 3\exp(-t^2/2).
\end{align} Equivalently,
\begin{align}\label{equ:random-sh-3}
\forall x\in\mathbb{S}^{d-1},t>0,\quad \mathop{\rm Pr}\nolimits\(|g(x)|\ge 2t\|g\|_2 \)\le 3\exp(-t^2/2).
\end{align}
In the following, we upper bound $|g(x)|/\|g\|_2$ uniformly over all $x\in\mathbb{S}^{d-1}$ by the covering number argument and uniform concentration.
Let $h(x)=g(x)/\|g\|_2$. First we prove that $h(x)$ is Lipschitz on $\mathbb{S}^{d-1}$ with respect to the great-circle distance $d(x,y)\defeq \arccos(x^\top y).$ To this end, we only need to upper bound the manifold gradient $\nabla^\star_x h(x)$ on the sphere. By Eq.~\eqref{equ:random-sh-1} we get,
\begin{align}
&\|\nabla^\star_x h(x)\|_2=\frac{1}{\|g\|_2}\sqrt{N_{k,d}}\|\mathbb{E}_{\xi\sim \mathbb{S}^{d-1}}[g(\xi)\nabla^\star_x \bar{P}_{k,d}(x^\top \xi)]\|_2\\
\le\;&\frac{1}{\|g\|_2}\sqrt{N_{k,d}}\mathbb{E}_{\xi\sim \mathbb{S}^{d-1}}[|g(\xi)| \|\nabla^\star_x \bar{P}_{k,d}(x^\top \xi)\|_2]\\
\le\;&\frac{1}{\|g\|_2}\sqrt{N_{k,d}}\(\mathbb{E}_{\xi\sim \mathbb{S}^{d-1}}[g(\xi)^2] \mathbb{E}_{\xi\sim \mathbb{S}^{d-1}}[\|\nabla^\star_x \bar{P}_{k,d}(x^\top \xi)\|_2^2]\)^{1/2}\tag{Cauchy-Schwarz inequality}\\
\le\;&\sqrt{N_{k,d}}\sqrt{k(k+d-2)}.\tag{\citet[Proposition 3.6]{atkinson2012spherical}}
\end{align}
which implies that $h(x)$ is $(\sqrt{k(k+d-2)N_{k,d}})$-Lipschitz.
Let $\epsilon=(2\sqrt{k(k+d-2)N_{k,d}})^{-1}$ and ${\mathcal{C}}$ an $\epsilon$-covering of $\mathbb{S}^{d-1}$ with respect to the great-circle distance. By Proposition~\ref{prop:covering-sphere} we get $|{\mathcal{C}}|\le (3/\epsilon)^d$. In addition, for every $x\in\mathbb{S}^{d-1}$ there exists $\hat{x}\in {\mathcal{C}}$ such that
\begin{align}\label{equ:random-sh-4}
|h(x)-h(\hat{x})|\le \sqrt{k(k+d-2)N_{k,d}}\epsilon=\frac{1}{2}.
\end{align}
By union bound and Eq.~\eqref{equ:random-sh-3}, with probability at least $1-\delta$ we get,
\begin{align}
\forall x\in{\mathcal{C}}, \quad |h(x)|&\le 4\sqrt{\ln \frac{3|{\mathcal{C}}|}{\delta}}\le 4\sqrt{\ln (3/\delta)+d\ln (3/\epsilon)}\\
&\le 4\sqrt{\ln(3/\delta)+2d^2\ln(k+1)}.
\end{align}
Combining with Eq.~\eqref{equ:random-sh-4} we get, with probability at least $1-\delta$,
\begin{align}
\forall x\in\mathbb{S}^{d-1},\quad \frac{|g(x)|}{\|g\|_2}\le 5\sqrt{\ln(3/\delta)+2d^2\ln(k+1)},
\end{align}
which proves the desired result. \end{proof}
The following lemma is an realization of Riesz representation theorem. \begin{lemma}\label{lem:riesz-SH}
For any fixed $k\ge 0$ and $f\in\mathbb{Y}_{k,d}$, we have
\begin{align}
f(x)=\sqrt{N_{k,d}}\mathbb{E}_{\xi \sim \mathbb{S}^{d-1}}[f(\xi)\bar{P}_{k,d}(x^\top \xi)],\quad \forall x\in \mathbb{S}^{d-1}.
\end{align} \end{lemma} \begin{proof}
let $\{Y_{k,j}\}_{j=1}^{N_{k,d}}$ be any set of orthonormal basis for degree $k$ spherical harmonics $\mathbb{Y}_{k,d}$. By addition theorem \citet[Theorem 2.9]{atkinson2012spherical}, for any $\xi\in\mathbb{S}^{d-1}$ we get
\begin{align}
\sqrt{N_{k,d}}\bar{P}_{k,d}(x^\top \xi)=\sum_{j=1}^{N_{k,d}}Y_{k,j}(x)Y_{k,j}(\xi).
\end{align}
Since $\{Y_{k,j}\}_{j=1}^{N_{k,d}}$ is a set of orthonormal basis, there exists unique coefficients $\{a_j\}_{j=1}^{N_{k,d}}$ such that $
f(x)=\sum_{j=1}^{N_{k,d}}a_{j}Y_{k,j}(x),\quad \forall x\in\mathbb{S}^{d-1}.
$ As a result,
\begin{align}
&\sqrt{N_{k,d}}\mathbb{E}_{\xi \sim \mathbb{S}^{d-1}}[f(\xi)\bar{P}_{k,d}(x^\top \xi)]=\mathbb{E}_{\xi \sim \mathbb{S}^{d-1}}\[\(\sum_{j=1}^{N_{k,d}}a_{j}Y_{k,j}(\xi)\)\(\sum_{j=1}^{N_{k,d}}Y_{k,j}(x)Y_{k,j}(\xi)\)\right]\notag\\
=\;&\sum_{j=1}^{N_{k,d}}a_{j}Y_{k,j}(x)=f(x).
\end{align} \end{proof}
\section{Helper Lemmas} In this section, we present some low-level helper lemmas.
\begin{proposition}\label{prop:covering-sphere}
Let $N(\epsilon)$ be the $\epsilon$-covering number of $\mathbb{S}^{d-1}$ with respect to the great-circle distance $d(x,y)\defeq \arccos(x^\top y).$ Then for any $\epsilon\in(0,1)$ we have
\begin{align}
N(\epsilon)\le (3/\epsilon)^d.
\end{align} \end{proposition} \begin{proof}
Note that any $(\epsilon/2)$-cover of the unit ball $B^{d}$ (w.r.t. the Euclidean distance) induces an $\epsilon$-covering of the unit sphere $\mathbb{S}^{d-1}$ with the same size. As a result,
\begin{align}
N(\epsilon)\le \frac{(1+\epsilon/2)^d}{(\epsilon/2)^d}\le \(\frac{3}{\epsilon}\)^d.
\end{align} \end{proof}
\begin{proposition}\label{prop:bessel}
Let $I_\nu(z)\defeq\sum_{j=0}^{\infty}\frac{1}{j!\Gamma(\nu+j+1)}\(\frac{z}{2}\)^{\nu+2j}$ be the modified Bessel function of the first kind. Then for every $\nu>1$ we get
\begin{align}
\sqrt{2}e^{-1}\frac{e^{\nu}}{(2\nu)^{\nu+1/2}} \le I_{\nu}(1)\le \frac{e^{1/4}}{\sqrt{\pi}}\frac{e^{\nu}}{(2\nu)^{\nu+1/2}}.
\end{align} \end{proposition} \begin{proof}
By algebraic manipulation we get,
\begin{align}
I_\nu(1)=\sum_{j=0}^{\infty}\frac{1}{j!\Gamma(\nu+j+1)}\(\frac{1}{2}\)^{\nu+2j}
\le \frac{(1/2)^\nu}{\Gamma(\nu+1)}\sum_{j=0}^{\infty}\frac{(1/2)^{2j}}{j!}= \frac{(1/2)^\nu}{\Gamma(\nu+1)}e^{1/4}.
\end{align}
By Stirling's formula, we get
\begin{align}
\frac{(1/2)^\nu}{\Gamma(\nu+1)}e^{1/4}\le \frac{(1/2)^\nu e^\nu}{\sqrt{2\pi}\nu^{\nu+1/2}}e^{1/4}=\frac{e^{1/4}}{\sqrt{\pi}}\frac{e^{\nu}}{(2\nu)^{\nu+1/2}}.
\end{align}
Similarly, we have
\begin{align}
I_\nu(1)=\sum_{j=0}^{\infty}\frac{1}{j!\Gamma(\nu+j+1)}\(\frac{1}{2}\)^{\nu+2j}\ge \frac{(1/2)^\nu}{\Gamma(\nu+1)}.
\end{align}
Using Stirling's formula again we have,
\begin{align}
\frac{(1/2)^\nu}{\Gamma(\nu+1)}\ge \frac{(1/2)^\nu e^\nu}{e\nu^{\nu+1/2}}=\sqrt{2}e^{-1}\frac{e^{\nu}}{(2\nu)^{\nu+1/2}}.
\end{align} \end{proof}
\begin{proposition}\label{prop:pnd-tail}
For any fixed $k\ge 0$, $u\in\mathbb{S}^{d-1}$, and $t>0$ we have
\begin{align}
\mathop{\rm Pr}\nolimits_{x\sim \mathbb{S}^{d-1}}(|P_{k,d}(x^\top u)|>t)\le \frac{1}{t^2N_{k,d}}.
\end{align} \end{proposition} \begin{proof}
By Markov ineqaulity we have
\begin{align}
&\mathop{\rm Pr}\nolimits_{x\sim \mathbb{S}^{d-1}}(|P_{k,d}(x^\top u)|>t)=\mathop{\rm Pr}\nolimits_{x\sim \mathbb{S}^{d-1}}(P_{k,d}(x^\top u)^2>t^2)\\
\le\; &t^{-2}\mathbb{E}_{x\sim \mathbb{S}^{d-1}}[P_{k,d}(x^\top u)^2]=t^{-2}N_{k,d}^{-1},
\end{align}
which proves the desired result. \end{proof}
\begin{lemma}[Lemma 1 of \citet{laurent2000adaptive}]\label{lem:laurent-massart}
Let $a_1, \cdots, a_d$ be i.i.d. ${\mathcal{N}}(0,1)$ Gaussian variables. Then for any $t>0$,
\begin{align}
\mathop{\rm Pr}\nolimits\(\sum_{i=1}^{d}a_i^2\le d-2\sqrt{dt}\)\le \exp(-t).
\end{align} \end{lemma} \begin{lemma}\label{lem:gaussian-proj-concentration}
Let $a=(a_1, \cdots, a_d)\sim {\mathcal{N}}(0,I)$ be a Gaussian vector and $u\in\mathbb{R}^{d}$ a fixed vector with unit norm. Then for any $t>0$,
\begin{align}
\mathop{\rm Pr}\nolimits\(\frac{|\dotp{a}{u}|}{\|a\|_2}\ge \frac{2t}{\sqrt{d}}\)\le 3\exp(-t^2/2).
\end{align} \end{lemma} \begin{proof}
Since $a\sim {\mathcal{N}}(0,I)$ is a Gaussian vector, we have $\dotp{a}{u}\sim {\mathcal{N}}(0,1).$ Hence,
\begin{align}
\mathop{\rm Pr}\nolimits\(|\dotp{a}{u}|>t\)\le 2\exp(-t^2/2).
\end{align}
By Lemma~\ref{lem:laurent-massart} with $t=\frac{9d}{64}$, we also have
\begin{align}
\mathop{\rm Pr}\nolimits\(\|a\|_2\le \sqrt{d}/2\)=\mathop{\rm Pr}\nolimits\(\|a\|_2^2\le d/4\)\le \exp\(-\frac{9}{64}d\)\le \exp\(-\frac{1}{8}d\).
\end{align}
By union bound we have
\begin{align}
&\mathop{\rm Pr}\nolimits\(\frac{|\dotp{a}{u}|}{\|a\|_2}\ge \frac{2t}{\sqrt{d}}\)\le \mathop{\rm Pr}\nolimits\(|\dotp{a}{u}|>t\)+\mathop{\rm Pr}\nolimits\(\|a\|_2\le \sqrt{d}/2\)\\
\le\;&2\exp(-t^2/2)+\exp\(-\frac{1}{8}d\).
\end{align}
Note that when $t>\sqrt{d}/2$, the desired result is trivial because $|\dotp{a}{u}|\le \|a\|_2$ with probability 1. Therefore, when $t\le \sqrt{d}/2$ we have
\begin{align}
\mathop{\rm Pr}\nolimits\(\frac{|\dotp{a}{u}|}{\|a\|_2}\ge \frac{2t}{\sqrt{d}}\)\le 2\exp(-t^2/2)+\exp\(-\frac{1}{8}d\)\le
3\exp(-t^2/2).
\end{align} \end{proof}
\begin{lemma}\label{lem:nn-sample}
Let $f$ be a infinite-width two-layer neural network with activation $\sigma:[-1,1]\to [-1,1]$, defined by
\begin{align}
f(x)=\mathbb{E}_{w\sim \mathbb{S}^{d-1}}[\sigma(x^\top w)c(w)]
\end{align}
for some weight $c:\mathbb{S}^{d-1}\to\mathbb{R}$ with $\|c\|_1<\infty.$ For any $\epsilon>0$, there exists a neural network $\hat{f}$ with $m=4\|c\|_1^2/\epsilon^2$ neurons, defined by $\hat{f}(x)=\sum_{j=1}^{m}a_i\sigma(w_i^\top x)$, such that
$\|\hat{f}-f\|_2\le \epsilon$, and $\sum_{j=1}^{m}|a_i|\le \|c\|_1.$ \end{lemma} \begin{proof}
We prove this theorem by probabilistic method. Let $p:\mathbb{S}^{d-1}\to\mathbb{R}_+$ be a function given by $p(w)=|c(w)|/\|c\|_1$. Then $p$ is a probability density function. Let $m=4\|c\|_1^2/\epsilon^2$. We sample $w_1,\cdots,w_m$ independently from $p$ and let $a_i=\sign(c(w_i))\frac{\|c\|_1}{m}.$ Define the two-layer neural network $\hat{f}$ by $\hat{f}(x)\defeq \sum_{j=1}^{m}a_i\sigma(w_i^\top x).$ In the following we prove that $\mathbb{E}[\|\hat{f}-f\|_2^2]\le \epsilon^2$ where the expectation is taken over the random variables $w_1\cdots,w_m$.
For any fixed $x\in\mathbb{S}^{d-1}$, we have
\begin{align}
&\hat{f}(x)-f(x)=\sum_{j=1}^{m}a_i\sigma(w_i^\top x)-f(x)
=\frac{\|c\|_1}{m}\sum_{j=1}^{m}\(\sign(c(w_i))\sigma(w_i^\top x)-\frac{f(x)}{\|c\|_1}\).
\end{align}
Let $X_i\defeq \sign(c(w_i))\sigma(w_i^\top x)-\frac{f(x)}{\|c\|_1}.$ By basic algebra we have
\begin{align}
\mathbb{E}_{w_i}[X_i]&=\int_{\mathbb{S}^{d-1}} p(w_i)\sign(c(w_i))\sigma(w_i^\top x)\mathrm{d} w_i - \frac{f(x)}{\|c\|_1}\\
&=\frac{1}{\|c\|_1}\(\int_{\mathbb{S}^{d-1}} |c(w_i)|\sign(c(w_i))\sigma(w_i^\top x)\mathrm{d} w_i - f(x)\)\\
&=\frac{1}{\|c\|_1}\(\int_{\mathbb{S}^{d-1}} c(w_i)\sigma(w_i^\top x)\mathrm{d} w_i - f(x)\)=0.
\end{align}
In addition, $|X_i|\le |\sign(c(w_i))\sigma(w_i^\top x)|+|\frac{f(x)}{\|c\|_1}|\le 2.$ It follows that
\begin{align}
\mathbb{E}_{\hat{f}}[(\hat{f}(x)-f(x))^2]&=\frac{\|c\|_1^2}{m^2}\(\sum_{j=1}^{m}X_i\)^2=\frac{\|c\|_1^2}{m^2}\sum_{j=1}^{m}X_i^2\le \frac{4\|c\|_1^2}{m}.
\end{align}
Consequently,
\begin{align}
\mathbb{E}_{\hat{f}}[\|\hat{f}-f\|_2^2]=\mathbb{E}_{\hat{f}}[\mathbb{E}_{x\in\mathbb{S}^{d-1}}[\|\hat{f}-f\|_2^2]]\le \frac{4\|c\|_1^2}{m}\le \epsilon^2.
\end{align}
By the probabilistic method, there exists $\hat{f}$ such that $\|\hat{f}-f\|_2^2\le \epsilon^2$, which proves the first part of the lemma.
By construction, we also have
\begin{align}
\sum_{j=1}^{m}|a_j|=\sum_{j=1}^{m}\frac{\|c\|_1}{m}\le \|c\|_1
\end{align}
almost surely, which proves the second part of the lemma. \end{proof}
\begin{proposition}\label{prop:truncation-upperbound}
For any fixed $\alpha\in (0,1],c_1,c_2>0,\epsilon>0$, let $$k=\inf_{l\ge 0}\{2c_1c_2(l+1)^{3/2}(N_{l+1,d})^{-\alpha/2}\le \epsilon/2\}.$$ When $d> \max\{2e,4/\alpha\}$ we have $k\le \max\{2e,(4c_1c_2/\epsilon)^{\frac{2}{d\alpha-3}}\}$ and $N_{k,d}\le \(\frac{4c_1c_2}{\epsilon}\)^{8/\alpha}$. \end{proposition} \begin{proof}
Let $c=c_1c_2$. By the definition $k$ we get
\begin{align}\label{equ:tu-1}
2c(k+1)^{3/2}N_{k+1,d}^{-\alpha/2}\le \epsilon/2.
\end{align}
Consequently, by the fact that $N_{k+1,d}={{d+k}\choose {d-1}} - {{d+k-2}\choose {d-1}}\le {d+k+1 \choose d}\le \(\frac{e(d+k+1)}{d}\)^{d}$ we get
\begin{align}
4c(k+1)^{3/2}\le \epsilon N_{k+1,d}^{\alpha/2}\le \epsilon \(\frac{e(k+d+1)}{d}\)^{d\alpha/2}.
\end{align}
When $d\ge 2e$ and $k\ge 2e$, we get $\frac{e(k+d+1)}{d}\le k+1.$
As a result,
\begin{align}
4c(k+1)^{3/2}\le \epsilon (k+1)^{d\alpha/2},
\end{align}
which implies that
\begin{align}
k\le \(\frac{\epsilon}{4c_1c_2}\)^{\frac{2}{3-d\alpha}}=\(\frac{4c_1c_2}{\epsilon}\)^{\frac{2}{d\alpha-3}}.
\end{align}
By the definition $k$ we also have
\begin{align}
2ck^{3/2}N_{k,d}^{-\alpha/2}> \epsilon/2.
\end{align}
Hence,
\begin{align}
N_{k,d}\le \(\frac{4c}{\epsilon}\)^{2/\alpha}k^{3/\alpha}\le \(\frac{4c}{\epsilon}\)^{8/\alpha}
\end{align} \end{proof}
\begin{proposition}\label{prop:lipschitzness-poly}
For any $k\ge 0$, let $\sigma_k(t)=\sum_{l=0}^{k}\dotp{\mathrm{ReLU}}{\bar{P}_{l,d}}_{\mu_d} \bar{P}_{l,d}(t)$. Then for all $k\ge 0$ we have
\begin{align}
&\sup_{t\in[-1,1]}|\sigma_k(t)|\le 1200\sqrt{N_{k,d}},\\
&\sup_{t\in[-1,1]}|\sigma_k'(t)|\le 1200k\sqrt{N_{k,d}}.
\end{align} \end{proposition} \begin{proof}
Let $\tau_l=\dotp{\mathrm{ReLU}}{\bar{P}_{l,d}}_{\mu_d}.$ Then we have
\begin{align}
\sup_{t\in[-1,1]}|\sigma_k(t)|\le \sum_{l=0}^{k}\tau_l \sup_{t\in[-1,1]}|\bar{P}_{l,d}(t)|=\sum_{l=0}^{k}\tau_l\sqrt{N_{l,d}}\le k\tau_k\sqrt{N_{k,d}}.
\end{align}
By Lemma~\ref{lem:relu-eigen-approx} we get $l\tau_l\le 1200.$ As a result,
\begin{align}
\sup_{t\in[-1,1]}|\sigma_l(t)|\le 1200\sqrt{N_{l,d}}.
\end{align}
By \citet[Eq. (2.89)]{atkinson2012spherical} we have $\sup_{t\in[-1,1]}|\bar{P}_{l,d}'(t)|\le \frac{l(l+d-2)}{d-1}.$ As a result,
\begin{align}
&\sup_{t\in[-1,1]}|\sigma_l'(t)|\le \sum_{l=0}^{k}\tau_l \sup_{t\in[-1,1]}|\bar{P}_{l,d}'(t)|=\sum_{l=0}^{k}\tau_l\frac{l(l+d-2)}{d-1}\sqrt{N_{l,d}}\\
\le\;& \tau_k\frac{k^2(k+d-2)}{d-1}\sqrt{N_{k,d}}\le 1200k\sqrt{N_{k,d}}.
\end{align} \end{proof}
\end{document} | arXiv |
Ideal norm
In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an ideal in a less complicated ring. When the less complicated number ring is taken to be the ring of integers, Z, then the norm of a nonzero ideal I of a number ring R is simply the size of the finite quotient ring R/I.
Relative norm
Let A be a Dedekind domain with field of fractions K and integral closure of B in a finite separable extension L of K. (this implies that B is also a Dedekind domain.) Let ${\mathcal {I}}_{A}$ and ${\mathcal {I}}_{B}$ be the ideal groups of A and B, respectively (i.e., the sets of nonzero fractional ideals.) Following the technique developed by Jean-Pierre Serre, the norm map
$N_{B/A}\colon {\mathcal {I}}_{B}\to {\mathcal {I}}_{A}$
is the unique group homomorphism that satisfies
$N_{B/A}({\mathfrak {q}})={\mathfrak {p}}^{[B/{\mathfrak {q}}:A/{\mathfrak {p}}]}$
for all nonzero prime ideals ${\mathfrak {q}}$ of B, where ${\mathfrak {p}}={\mathfrak {q}}\cap A$ is the prime ideal of A lying below ${\mathfrak {q}}$.
Alternatively, for any ${\mathfrak {b}}\in {\mathcal {I}}_{B}$ one can equivalently define $N_{B/A}({\mathfrak {b}})$ to be the fractional ideal of A generated by the set $\{N_{L/K}(x)|x\in {\mathfrak {b}}\}$ of field norms of elements of B.[1]
For ${\mathfrak {a}}\in {\mathcal {I}}_{A}$, one has $N_{B/A}({\mathfrak {a}}B)={\mathfrak {a}}^{n}$, where $n=[L:K]$.
The ideal norm of a principal ideal is thus compatible with the field norm of an element:
$N_{B/A}(xB)=N_{L/K}(x)A.$[2]
Let $L/K$ be a Galois extension of number fields with rings of integers ${\mathcal {O}}_{K}\subset {\mathcal {O}}_{L}$.
Then the preceding applies with $A={\mathcal {O}}_{K},B={\mathcal {O}}_{L}$, and for any ${\mathfrak {b}}\in {\mathcal {I}}_{{\mathcal {O}}_{L}}$ we have
$N_{{\mathcal {O}}_{L}/{\mathcal {O}}_{K}}({\mathfrak {b}})=K\cap \prod _{\sigma \in \operatorname {Gal} (L/K)}\sigma ({\mathfrak {b}}),$
which is an element of ${\mathcal {I}}_{{\mathcal {O}}_{K}}$.
The notation $N_{{\mathcal {O}}_{L}/{\mathcal {O}}_{K}}$ is sometimes shortened to $N_{L/K}$, an abuse of notation that is compatible with also writing $N_{L/K}$ for the field norm, as noted above.
In the case $K=\mathbb {Q} $, it is reasonable to use positive rational numbers as the range for $N_{{\mathcal {O}}_{L}/\mathbb {Z} }\,$ since $\mathbb {Z} $ has trivial ideal class group and unit group $\{\pm 1\}$, thus each nonzero fractional ideal of $\mathbb {Z} $ is generated by a uniquely determined positive rational number. Under this convention the relative norm from $L$ down to $K=\mathbb {Q} $ coincides with the absolute norm defined below.
Absolute norm
Let $L$ be a number field with ring of integers ${\mathcal {O}}_{L}$, and ${\mathfrak {a}}$ a nonzero (integral) ideal of ${\mathcal {O}}_{L}$.
The absolute norm of ${\mathfrak {a}}$ is
$N({\mathfrak {a}}):=\left[{\mathcal {O}}_{L}:{\mathfrak {a}}\right]=\left|{\mathcal {O}}_{L}/{\mathfrak {a}}\right|.\,$
By convention, the norm of the zero ideal is taken to be zero.
If ${\mathfrak {a}}=(a)$ is a principal ideal, then
$N({\mathfrak {a}})=\left|N_{L/\mathbb {Q} }(a)\right|$.[3]
The norm is completely multiplicative: if ${\mathfrak {a}}$ and ${\mathfrak {b}}$ are ideals of ${\mathcal {O}}_{L}$, then
$N({\mathfrak {a}}\cdot {\mathfrak {b}})=N({\mathfrak {a}})N({\mathfrak {b}})$.[3]
Thus the absolute norm extends uniquely to a group homomorphism
$N\colon {\mathcal {I}}_{{\mathcal {O}}_{L}}\to \mathbb {Q} _{>0}^{\times },$
defined for all nonzero fractional ideals of ${\mathcal {O}}_{L}$.
The norm of an ideal ${\mathfrak {a}}$ can be used to give an upper bound on the field norm of the smallest nonzero element it contains:
there always exists a nonzero $a\in {\mathfrak {a}}$ for which
$\left|N_{L/\mathbb {Q} }(a)\right|\leq \left({\frac {2}{\pi }}\right)^{s}{\sqrt {\left|\Delta _{L}\right|}}N({\mathfrak {a}}),$
where
• $\Delta _{L}$ is the discriminant of $L$ and
• $s$ is the number of pairs of (non-real) complex embeddings of L into $\mathbb {C} $ (the number of complex places of L).[4]
See also
• Field norm
• Dedekind zeta function
References
1. Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics, vol. 7 (second ed.), Providence, Rhode Island: American Mathematical Society, Proposition I.8.2, ISBN 0-8218-0429-4, MR 1362545
2. Serre, Jean-Pierre (1979), Local Fields, Graduate Texts in Mathematics, vol. 67, translated by Greenberg, Marvin Jay, New York: Springer-Verlag, 1.5, Proposition 14, ISBN 0-387-90424-7, MR 0554237
3. Marcus, Daniel A. (1977), Number fields, Universitext, New York: Springer-Verlag, Theorem 22c, ISBN 0-387-90279-1, MR 0457396
4. Neukirch, Jürgen (1999), Algebraic number theory, Grundlehren der mathematischen Wissenschaften, vol. 322, Berlin: Springer-Verlag, Lemma 6.2, doi:10.1007/978-3-662-03983-0, ISBN 3-540-65399-6, MR 1697859
| Wikipedia |
\begin{definition}[Definition:Complete Bipartite Graph]
A '''complete bipartite graph''' is a bipartite graph $G = \struct {A \mid B, E}$ in which every vertex in $A$ is adjacent to every vertex in $B$.
The complete bipartite graph where $A$ has $m$ vertices and $B$ has $n$ vertices is denoted $K_{m, n}$.
Note that $K_{m, n}$ is the same as $K_{n, m}$.
\end{definition} | ProofWiki |
\begin{definition}[Definition:Sample Statistic/Discrete]
Data which can be described with a discrete variable are known as '''discrete data'''.
\end{definition} | ProofWiki |
Zero-symmetric graph
In the mathematical field of graph theory, a zero-symmetric graph is a connected graph in which each vertex has exactly three incident edges and, for each two vertices, there is a unique symmetry taking one vertex to the other. Such a graph is a vertex-transitive graph but cannot be an edge-transitive graph: the number of symmetries equals the number of vertices, too few to take every edge to every other edge.[1]
The smallest zero-symmetric graph, with 18 vertices and 27 edges
The truncated cuboctahedron, a zero-symmetric polyhedron
Graph families defined by their automorphisms
distance-transitive → distance-regular ← strongly regular
↓
symmetric (arc-transitive) ← t-transitive, t ≥ 2 skew-symmetric
↓
(if connected)
vertex- and edge-transitive
→ edge-transitive and regular → edge-transitive
↓ ↓ ↓
vertex-transitive → regular → (if bipartite)
biregular
↑
Cayley graph ← zero-symmetric asymmetric
The name for this class of graphs was coined by R. M. Foster in a 1966 letter to H. S. M. Coxeter.[2] In the context of group theory, zero-symmetric graphs are also called graphical regular representations of their symmetry groups.[3]
Examples
The smallest zero-symmetric graph is a nonplanar graph with 18 vertices.[4] Its LCF notation is [5,−5]9.
Among planar graphs, the truncated cuboctahedral and truncated icosidodecahedral graphs are also zero-symmetric.[5]
These examples are all bipartite graphs. However, there exist larger examples of zero-symmetric graphs that are not bipartite.[6]
These examples also have three different symmetry classes (orbits) of edges. However, there exist zero-symmetric graphs with only two orbits of edges. The smallest such graph has 20 vertices, with LCF notation [6,6,-6,-6]5.[7]
Properties
Every finite zero-symmetric graph is a Cayley graph, a property that does not always hold for cubic vertex-transitive graphs more generally and that helps in the solution of combinatorial enumeration tasks concerning zero-symmetric graphs. There are 97687 zero-symmetric graphs on up to 1280 vertices. These graphs form 89% of the cubic Cayley graphs and 88% of all connected vertex-transitive cubic graphs on the same number of vertices.[8]
Unsolved problem in mathematics:
Does every finite zero-symmetric graph contain a Hamiltonian cycle?
(more unsolved problems in mathematics)
All known finite connected zero-symmetric graphs contain a Hamiltonian cycle, but it is unknown whether every finite connected zero-symmetric graph is necessarily Hamiltonian.[9] This is a special case of the Lovász conjecture that (with five known exceptions, none of which is zero-symmetric) every finite connected vertex-transitive graph and every finite Cayley graph is Hamiltonian.
See also
• Semi-symmetric graph, graphs that have symmetries between every two edges but not between every two vertices (reversing the roles of edges and vertices in the definition of zero-symmetric graphs)
References
1. Coxeter, Harold Scott MacDonald; Frucht, Roberto; Powers, David L. (1981), Zero-symmetric graphs, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, ISBN 0-12-194580-4, MR 0658666
2. Coxeter, Frucht & Powers (1981), p. ix.
3. Lauri, Josef; Scapellato, Raffaele (2003), Topics in Graph Automorphisms and Reconstruction, London Mathematical Society Student Texts, Cambridge University Press, p. 66, ISBN 9780521529037.
4. Coxeter, Frucht & Powers (1981), Figure 1.1, p. 5.
5. Coxeter, Frucht & Powers (1981), pp. 75 and 80.
6. Coxeter, Frucht & Powers (1981), p. 55.
7. Conder, Marston D. E.; Pisanski, Tomaž; Žitnik, Arjana (2017), "Vertex-transitive graphs and their arc-types", Ars Mathematica Contemporanea, 12 (2): 383–413, arXiv:1505.02029, doi:10.26493/1855-3974.1146.f96, MR 3646702
8. Potočnik, Primož; Spiga, Pablo; Verret, Gabriel (2013), "Cubic vertex-transitive graphs on up to 1280 vertices", Journal of Symbolic Computation, 50: 465–477, arXiv:1201.5317, doi:10.1016/j.jsc.2012.09.002, MR 2996891.
9. Coxeter, Frucht & Powers (1981), p. 10.
| Wikipedia |
# Discrete Fourier Transform and its applications
The Discrete Fourier Transform (DFT) is a mathematical algorithm that transforms a sequence of values in the time domain into a sequence of values in the frequency domain. It is widely used in signal processing and data analysis for various applications, such as audio and image processing, communication systems, and data compression.
The DFT can be defined as a sum of complex exponential functions, where each function corresponds to a specific frequency component in the input signal. The frequency components are determined by the index of the summation, which is denoted as $k$. The DFT is given by:
$$X_k = \sum_{n=0}^{N-1} x_n e^{-j2\pi kn/N}$$
where $X_k$ is the $k$-th frequency component of the transformed signal, $x_n$ is the $n$-th sample of the input signal, $N$ is the number of samples in the input signal, and $j$ is the imaginary unit.
The inverse DFT, also known as the Discrete Inverse Fourier Transform (IDFT), is the process of transforming the frequency domain representation back to the time domain. The IDFT is given by:
$$x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k e^{j2\pi kn/N}$$
where $x_n$ is the $n$-th sample of the reconstructed signal, $X_k$ is the $k$-th frequency component of the input signal, and $N$ is the number of samples in the input signal.
The DFT and IDFT are the fundamental building blocks for various signal processing and data analysis applications. They provide a way to analyze and manipulate signals in the frequency domain, which can be more intuitive and easier to work with than the time domain.
## Exercise
Calculate the DFT of the following input signal:
$$x[n] = \begin{cases}
1, & n = 0 \\
0, & n \neq 0
\end{cases}$$
# Fast Fourier Transform algorithm and its advantages
The Fast Fourier Transform (FFT) algorithm is an efficient method for computing the DFT. It reduces the computational complexity of the DFT from $O(N^2)$ to $O(N\log N)$, making it practical for large input signals.
The FFT algorithm is based on the divide-and-conquer approach, where the input signal is divided into smaller overlapping sub-signals, and the DFT is computed for each sub-signal recursively. The FFT algorithm uses a specific sequence of twiddle factors, which are complex exponential functions that facilitate the computation of the DFT.
The FFT algorithm has several advantages over the direct implementation of the DFT:
- Lower computational complexity: The FFT algorithm reduces the computational complexity of the DFT from $O(N^2)$ to $O(N\log N)$, making it practical for large input signals.
- Faster convergence: The FFT algorithm converges faster than the direct implementation of the DFT, which can be beneficial for solving differential equations and other applications that require a high level of accuracy.
- Efficient use of resources: The FFT algorithm is more efficient in terms of memory usage and computational resources, as it avoids redundant calculations and uses memory more effectively.
## Exercise
Implement the FFT algorithm to compute the DFT of the input signal from the previous exercise.
# Inverse Fast Fourier Transform
The Inverse Fast Fourier Transform (IFFT) is the process of transforming the frequency domain representation back to the time domain. It is the inverse operation of the FFT, and it is used to reconstruct the original signal from its frequency domain representation.
The IFFT is given by:
$$x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k e^{j2\pi kn/N}$$
where $x_n$ is the $n$-th sample of the reconstructed signal, $X_k$ is the $k$-th frequency component of the input signal, and $N$ is the number of samples in the input signal.
The IFFT is important for various signal processing and data analysis applications, as it allows us to reconstruct the original signal from its frequency domain representation.
## Exercise
Calculate the IFFT of the following frequency domain representation:
$$X_k = \begin{cases}
1, & k = 0 \\
0, & k \neq 0
\end{cases}$$
# Window functions and their importance in signal processing
Window functions are mathematical functions that are applied to a signal before performing the DFT. They are used to reduce the effects of spectral leakage and improve the accuracy of the DFT.
There are several types of window functions, including rectangular windows, triangular windows, and various other specialized windows. The choice of the window function depends on the specific application and the desired trade-off between accuracy and computational complexity.
Window functions are important in signal processing and data analysis, as they help to reduce the effects of spectral leakage and improve the accuracy of the DFT. They are widely used in various applications, such as spectral analysis, spectral estimation, and spectral density estimation.
## Exercise
Apply a rectangular window function to the input signal from the first exercise.
# Applications of the Fast Fourier Transform algorithm in signal processing and data analysis
The FFT algorithm has a wide range of applications in signal processing and data analysis. Some of the most common applications include:
- Spectral analysis: The FFT is used to analyze the frequency content of a signal, which is important for applications such as audio and image processing, communication systems, and radar systems.
- Spectral estimation: The FFT is used to estimate the spectral density of a signal, which is important for applications such as noise reduction, equalization, and filtering.
- Spectral density estimation: The FFT is used to estimate the spectral density of a signal, which is important for applications such as noise reduction, equalization, and filtering.
- Data compression: The FFT is used to compress and decompress data, which is important for applications such as image and video compression.
- Digital signal processing: The FFT is used to process digital signals, which is important for applications such as filtering, modulation, and demodulation.
## Exercise
Perform spectral analysis on the input signal from the first exercise using the FFT algorithm.
# Real-world examples and case studies
The FFT algorithm is widely used in various real-world applications. Some examples of these applications include:
- Audio signal processing: The FFT is used to analyze and process audio signals, which is important for applications such as music and speech analysis, noise reduction, and equalization.
- Image processing: The FFT is used to analyze and process image signals, which is important for applications such as image compression, filtering, and feature extraction.
- Communication systems: The FFT is used to analyze and process communication signals, which is important for applications such as wireless communication, radar systems, and satellite communication.
- Data analysis: The FFT is used to analyze and process data signals, which is important for applications such as data compression, encryption, and machine learning.
These examples demonstrate the versatility and importance of the FFT algorithm in signal processing and data analysis.
## Exercise
Design a communication system that uses the FFT algorithm for signal processing and data analysis.
# Implementation and optimization of the Fast Fourier Transform algorithm
The FFT algorithm can be implemented in various programming languages and libraries. Some popular implementations include:
- FFTW: The FFTW library is a widely used C/C++ library for performing the FFT. It provides efficient and highly optimized implementations of the FFT for various data types and sizes.
- MATLAB: MATLAB provides a built-in function called `fft` for computing the FFT. It is based on the Cooley-Tukey algorithm and is highly optimized for various data types and sizes.
- Python: The `numpy` library in Python provides a function called `fft.fft` for computing the FFT. It is based on the Cooley-Tukey algorithm and is highly optimized for various data types and sizes.
The FFT algorithm can be optimized in various ways, such as using efficient algorithms, parallelization, and hardware acceleration. Some common optimization techniques include:
- Algorithm optimization: The FFT algorithm can be optimized by using more efficient algorithms, such as the Bluestein algorithm or the Chirp-Z transform.
- Parallelization: The FFT algorithm can be parallelized by using multicore processors or GPUs to perform the computations in parallel.
- Hardware acceleration: The FFT algorithm can be accelerated by using dedicated hardware, such as FPGA or ASIC, to perform the computations more efficiently.
## Exercise
Implement the FFT algorithm using the FFTW library in C/C++.
# Conclusion and future directions
The Fast Fourier Transform algorithm is a powerful and efficient method for computing the Discrete Fourier Transform. It has a wide range of applications in signal processing and data analysis, and it continues to be a fundamental tool in various fields, such as communication systems, image processing, and machine learning.
Future research directions in the field of FFT algorithms include:
- Developing more efficient algorithms for the FFT, such as the Bluestein algorithm or the Chirp-Z transform.
- Investigating the use of advanced hardware architectures, such as FPGA or ASIC, for accelerating the computation of the FFT.
- Exploring the use of machine learning techniques, such as deep learning or reinforcement learning, for optimizing the FFT algorithm.
## Exercise
Research and compare the performance of different FFT algorithms for a given input signal. | Textbooks |
\begin{document}
\title[European Apportionment via the Cambridge Compromise]{European Apportionment via the{\\}Cambridge Compromise} \author{Geoffrey R.\ Grimmett} \address{Statistical Laboratory,\\Centre for Mathematical Sciences, Cambridge University,\\Wilberforce Road, Cambridge CB3 0WB, UK} \email{[email protected]} \urladdr{http://www.statslab.cam.ac.uk/$\sim$grg/}
\begin{abstract} Seven mathematicians and one political scientist met at the Cambridge Apportionment Meeting in January 2011. They agreed a unanimous recommendation to the
European Parliament for its future apportionments between the EU Member States. This is a short factual account of the reasons that led to the Meeting, of its debates and report, and of some of the ensuing Parliamentary debate. \end{abstract}
\date{Revised August 20, 2011} \renewcommand{\subjclassname}{\textup{2010} Mathematics Subject Classification}
\keywords{Apportionment problem, European Parliament, degressive proportionality, \bp\ method, D'Hondt method} \subjclass[2010]{91B12}
\maketitle \section{Background and Brief}
\subsection{Background} As the European Union has grown and its population has developed, so has the constitution and structure of the European Parliament. In recognition of the need for an orderly allocation of Parliamentary seats between the EU Member States, its Committee on Contitutional Affairs (AFCO) commissioned a Symposium of Mathematicians to \lq\lq identify a mathematical formula for the distribution of seats which will be durable, transparent and impartial to politics''. The purposes of the reform were described thus in \cite{duff}: \begin{dashlist} \item The aim of the symposium is to discuss and, if possible, to propose to the Committee on Constitutional Affairs a mathematical formula for the redistribution of the 751 seats in the European Parliament. The formula should be as transparent as possible and capable of being sustained from one Parliamentary mandate to the next. \item The purpose of the Symposium is to eliminate the political bartering which has characterised the distribution of seats so far by enabling a smooth reallocation of seats once every five years which takes account of migration, demographic shifts and the accession of new Member States. \end{dashlist}
The current note is more a record of the events surrounding the Cambridge Apportionment Meeting than it is a critical analysis of the politics. An account of the history of the current apportionment of Parliament, and of the associated \lq\lq political bartering", may be found in \cite{duff2}.
\subsection{Cambridge Apportionment Meeting (CAM)} The Symposium took place in the Centre for Mathematical Sciences, Cambridge University, on 28--29 January 2011. The following participated: \begin{dashlist} \item Geoffrey Grimmett (University of Cambridge), Director, \item Friedrich Pukelsheim (University of Augsburg), co-Director, \item Jean-Fran\c cois Laslier (\'Ecole Polytechnique, Paris), \item Victoriano Ram\'\i rez Gonz\'alez (University of Granada), \item Richard Rose (University of Aberdeen; European University Institute, Florence), \item Wojciech S\l omczy\'nski (Jagiellonian University, Krak\'ow), \item Martin Zachariasen (University of Copenhagen), \item Karol \.{Z}yczkowski (Jagiellonian University, Krak\'ow), \end{dashlist} \noi advised by \begin{dashlist} \item Andrew Duff MEP (AFCO Rapporteur), \item Rafa\l\ Trzaskowski MEP (AFCO Vice-President), \item Guy Deregnaucourt (AFCO Administrator), \item Wolfgang Leonhardt (AFCO Administrator), \item Kevin Wilkins (Assistant to Andrew Duff), \end{dashlist} \noi in the presence of \begin{dashlist} \item Thomas Kellermann (College of Europe, Natolin, Warsaw), \item Kai-Friederike Oelbermann (University of Augsburg). \end{dashlist}
The formal Report of the Cambridge Apportionment Meeting to the Congressional Affairs Committee may be found at \cite{camcom}. The discussions and recommendations of CAM are summarized in the current article, together with an account of some of the subsequent debate within the Committee. Opinions expressed here are those of the author alone.
\subsection{The constraints}\label{sec:const} Seat allocations are currently required to adhere to the terms of the Treaty of Lisbon. \begin{dashlist} \item Each Member State is to receive a minimum of 6 seats, \item and a maximum of 96 seats, \item Parliament is constrained to have no more than 751 seats in total (including that of the President), \item allocations are required to satisfy a condition of \lq\lq degressive proportionality". \end{dashlist}
CAM was advised by the AFCO representatives that the first three constraints are indeed \emph{inequalities} rather than \emph{equalities}, but nevertheless there existed a general expectation in Parliament that its total size should not be less than 751, and that the smallest States should receive an allocation not greater than 6 seats. The issue of \lq\lq degressive proportionality" is formulated in more detail in Section \ref{sec:dp0}. In reaching its conclusions, the Symposium took into account the following additional observations concerning the general structure of the European Parliament: \begin{dashlist} \item the EU has currently 27 Member States, \item the smallest population (as published officially by Eurostat\footnote{\url{http://epp.eurostat.ec.europa.eu/}} is currently 412,970, and the largest 81,802,257, \item future accessions may include a number of States with a spread of sizes, \item there will be migration and demographic changes, \item Member States' population figures will be used as input to the formula. \end{dashlist}
\subsection{The criteria}\label{sec:crit}
Participants were sensitive in discussions to the three descriptors
provided by the AFCO Committee, namely that the \lq \lq formula'' was required to be \emph{durable}, \emph{transparent} and \emph{impartial to politics}.
\noi \emph{Durable}: A formula that adapts naturally to possible structural changes in the architecture of the EU, arising for example through accessions by new States, through migration, or through demographic shifts.
\noi \emph{Transparent}: An apportionment method that is capable of simple and reasonable explanation to EU citizens, irrespective of their backgrounds.
\noi \emph{Impartial to politics}: A principled and fresh approach, unprejudiced with respect to particular Member States or Political Groups, and free of influence from historical positions beyond the constraints of Section \ref{sec:const}.
\subsection{Summary}
A discussion of degressive proportionality is to be found in Section \ref{sec:dp0}. Section \ref{sec:cc} contains a discussion of the main recommendations of the Cambridge Apportionment Meeting, which are listed explicitly in Section \ref{sec:recom}. A brief account of the subsequent debate and resolutions of the Committee on Constitutional Affairs is presented in Section \ref{sec:afco}. This chapter in the story of European Apportionment ends with the shelving of the mathematical approach.
\section{Degressive Proportionality}\label{sec:dp0}
\subsection{Lamassoure--Severin definition}\label{sec:dp}
\emph{Degressive proportionality} has been defined in Paragraph 6 of the Lamassoure--Severin (2007) Motion of \cite{lamsev} as follows. \begin{numlist} \addtocounter{mycount}{5} \item
{[The European Parliament]} \lq\lq Considers that the principle of degressive proportionality means that the ratio between the population and the number of seats of each Member State must vary in relation to their respective populations in such a way that each Member from a more populous Member State represents more citizens than each Member from a less populous Member State and conversely, but also that no less populous Member State has more seats than a more populous Member State.'' \end{numlist}
The principle of degressive proportionality attracted significant debate and a major recommendation at CAM.
\subsection{CAM recommendation}\label{sec:dp2}
It was noted that \emph{degressive proportionality} comprises two requirements: \begin{numlist} \item no smaller State shall receive more seats than a larger State, \item the ratio population/seats shall increase as population increases. \end{numlist} Condition 1 is easy to accept. Condition 2, on the other hand, poses a serious practical difficulty, and has in addition been violated in recent Parliamentary apportionments. As noted in \cite{ma-rg08,puk10,RG10,rg-pal-m06} and elsewhere, there are hypothetical instances of apportionment for which there exists no solution satisfying both Condition 1 and Condition 2. There was an extensive discussion of this issue at CAM, centred on the following two Options. \begin{Letlist} \item Adopt a method whose outcomes invariably satisfy Condition 2 but with a possibly reduced Parliament-size. \item Propose a change to the Lamassoure--Severin definition of degressive proportionality
lying within existing law and allowing greater flexibility and transparency. \end{Letlist} A method satisfying Option A was presented at CAM (and is summarized in \cite[Sect.\ 6.2]{camcom}). However, CAM preferred Option B on the grounds of transparency of method, and the desirability of achieving a given Parliament-size.
The recommendation of CAM was to amend Paragraph 6 of the Lamassoure--Severin Motion \cite{lamsev} through the addition of the italicized phrase as follows. \begin{numlist} \addtocounter{mycount}{5} \item {[The European Parliament]} Considers that the principle of degressive proportionality means that the ratio between the population and the number of seats of each Member State \emph{before rounding to whole numbers} must vary in relation to their respective populations in such a way that each Member from a more populous Member State represents more citizens than each Member from a less populous Member State and conversely, but also that no less populous Member State has more seats than a more populous Member State. \end{numlist}
\section{Cambridge Compromise} \label{sec:cc}
\subsection{Base+prop method}\label{sec:ccbp}
The `Cambridge Compromise' recommendation\footnote{The Cambridge Compromise proposal is named in harmony with the so-called Jagiellonian Compromise proposal of \cite{slomz10,slomz04} for voting within the European Commission.} to the European Parliament is to adopt a \bp\ system, formulated in \cite{puk10} as follows.
The \emph{\bp} method proceeds in two stages. At the first stage, a fixed \emph{base} number of seats is allocated to each Member State. At the second stage, the remaining seats are allocated to States in \emph{prop}ortion to their population-sizes (subject to rounding, and capping at the maximum). In order to achieve the given Parliament-size, one introduces a further ingredient called the \emph{divisor}.
For given \emph{base} $b$, \emph{maximum} $M$, and \emph{divisor} $d$, define the associated \emph{allocation function} $A_d: [0,\infty) \to [0,\infty)$ by $$ A_d(p) = \min\bigl\{b+p/d,M\bigr\}, $$ The \bp\ method is formulated as follows in mathematical terms. \begin{numlist} \item Assign to a Member State with population $p$ the \emph{seat share} $A_d(p)$, \item perform a rounding of the seat share $A_d(p)$ into an integer \emph{seat number} $[A_d(p)]$, \item adjust the divisor $d$ in such a way that the sum of the seat numbers of all Member States equals the given Parliament-size. \end{numlist}
The total house-size with divisor $d$ is $$ T(d) = \sum_i [A_d(p_i)], $$ where the summation is over all Member States. The value of $d$ is chosen in such a way that $T(d)$ equals the prescribed total\footnote{There is normally an interval of such $d$-values, and there are standard approaches to the question of so-called \emph{ties}. See \cite{baly}, for example, and also Section \ref{sec:dhondt}.}.
The CAM recommendation is to use the base $b= 5$, and to use \emph{rounding upwards}. Outcomes of the Cambridge Compromise are presented in Tables \ref{table1} and \ref{table2}, with 2011 population figures taken from the Eurostat website, and with 27, 28, and 29 Member States.
It was through principled discussion that this recommendation was reached; CAM was instructed to overlook historical apportionments, including the \emph{status quo} as listed in Table \ref{table2}. Participants recognised the challenges that can be presented by change, and these challenges proved formidable for the AFCO Committee (see Section \ref{sec:afco}).
\subsection{Why \bp?}
The CAM participants considered a variety of apportionment schemes based around several different linear and non-linear apportionment functions\footnote{Note that every non-decreasing concave apportionment function leads invariably to allocations satisfying the revised form of degressive proportionality of Section \ref{sec:dp2}.}. Linear functions were preferred over non-linear functions on grounds of transparency and greater potential for proportionality. The dual constraints of maximum and house-size are obstacles to the search for a \emph{smooth} linear apportionment function (that is, a function that is continuously differentiable, say).
Non-linear apportionment functions (following a power\footnote{A power-weighted variant of the Cambridge Compromise is analysed in \cite{gop}.} or parabolic law, for example) can accommodate numerical constraints in a smoother manner. They can be used to fit curves to plots of data points distributed along (possibly concave) lines of trends, such as the current allocations to Member States. On the other hand, they suffer from arbitrariness, and from lack of transparency. The exercise confronting CAM was not one of fitting a curve to historic data, but rather to form a fresh view of apportionment that is impartial to yesterday's politics.
From amongst linear systems, the \emph{\bp} method stands out for its transparency. It is degressively proportional in an active way, since the base operates to the profit of Member States at the lower end of the population table. CAM considered that its simplicity outweighed the discontinuity in the first derivative that arises currently through the maximum cap of 96 seats. We noted that this discontinuity will diminish as the EU changes its shape through accessions. The recommendation to adopt the \bp\ method was reached through consideration of durability, transparency, impartiality, and degressive proportionality.
CAM noted in passing that the \bp\ method can be interpreted as one in which the base is an allocation to Member States, and the remaining seats (prop) are proportional to population (subject to capping at the maximum). This resonates with the founding principles of the EU, enshrined in the Treaty, that the Union is made up both of Member States (enjoying equality in international law) and of citizens (enjoying democratic equality).
\subsection{Choice of base and rounding method}\label{sec:choice}
The choices of base and rounding methods are intertwined. A smaller base tends to favour larger States; rounding \emph{upwards} is usually viewed as tending to favour smaller States. These choices are informed by the existence of a minimum number $m$ of seats per State, and by degressive proportionality.
Let us write $b+$R to denote the system with base $b$ and rounding method R, where R may denote one of: \begin{itemize} \item [{U:}] upwards rounding, \item [{S:}] standard rounding to the nearest integer, \item [{D:}] downwards rounding. \end{itemize} We say that the roundings of a real number $x$ are \emph{well defined} if $x$ is not an integer multiple of $\frac12$. It was considered preferable, in the interests of transparency, that the base be an integer.
Recall that $m=6$, and there is an expectation that the smallest States will indeed receive 6 seats. It was therefore natural to concentrate on the two possibilities: \begin{list}{}{\setlength{\labelwidth}{1cm} \setlength{\leftmargin}{2cm}} \item [{6+S:}] base $b=6$, \emph{standard rounding} (S), \item [{5+U:}] base $b=5$, \emph{upwards rounding} (U). \end{list} Each of these two systems allocates at least 6 seats to every State. The minimum allocation is however fragile under the first system (6+S), as illustrated in \cite[Sect.\ 5.3]{camcom} as follows. The currently smallest Member State is Malta, with a population of 412,970, and it receives an allocation of 6 seats under both the above systems. If, however, its population were to increase by only 8,000 (other populations remaining unchanged), its allocation under 6+S rises to 7. This was considered unacceptable, and for this reason CAM recommended 5+U.
There is an explicit trade-off between base and rounding method (see \cite{sj11,sj11b,zach}). Let $x$ be a real number, and let $\lfloor\cdot\rfloor$ (\resp, $\lceil\cdot\rceil$, $[\cdot]$) denote
rounding downwards (\resp, upwards, and to the nearest integer). For any `base' $b$, we have $$ \lceil b+x\rceil = [b+\tfrac12+x] = \lfloor b+1+x\rfloor, $$ whenever the roundings are well defined. Subject to the last assumption, the three systems $b+$U, $(b+\frac12)+$S, $(b+1)+$D result in the same allocations. In this sense, the systems 5+U, $5\frac12$+S, 6+D are equivalent.
\begin{table}[htbp]
\centering \begin{tabular*}{\textwidth}{@{\extracolsep{\fill}} cc r c @{}r r} \toprule &\multirow{3}{*}{\it Member State}&\multirow{3}{*}{\it Population}&\multirow{3}{*}{\it Seats}&\multicolumn{1}{c}{\it Popn/seats}&\multicolumn{1}{c}{\it Popn/seats}\\ &&&&\multicolumn{1}{c}{\it before}&\multicolumn{1}{c}{\it after}\\ &&&&\multicolumn{1}{c}{\it rounding}&\multicolumn{1}{c}{\it rounding}\\ \midrule 1&Germany&81,802,257&96& 852,106.8 &852,106.8\\ 2&France&64,714,074&85&770,259.3&\emph{761,342.0}\\ 3&UK&62,008,048&81&768,264.0&765,531.5\\ 4&Italy&60,340,328&79&766,950.8&763,801.6\\ 5&Spain&45,989,016&62&752,036.4&741,758.3\\ \seprule 6&Poland&38,167,329&52&739,643.2&733,987.1\\ 7&Romania&21,462,186&32&687,772.5&670,693.3\\ 8&Netherlands&16,574,989&26&656,745.2&637,499.6\\ 9&Greece&11,305,118&19&601,222.1&595,006.2\\ 10&Belgium&10,839,905&19&594,438.5&\emph{570,521.3}\\ \seprule 11&Portugal&10,637,713&18&591,356.6&590,984.1\\ 12&Czech Rep.&10,506,813&18&589,315.9&583,711.8\\ 13&Hungary&10,014,324&18&581,298.7&556,351.3\\ 14&Sweden&9,340,682&17&569,380.7&549,451.9\\ 15&Austria&8,375,290&16&550,056.4&523,455.6\\ \seprule 16&Bulgaria&7,563,710&15&531,334.8&504,247.3\\ 17&Denmark&5,534,738&12&470,724.2&461,228.2\\ 18&Slovakia&5,424,925&12&466,706.8&452,077.1\\ 19&Finland&5,351,427&12&463,965.8&445,952.2\\ 20&Ireland&4,467,854&11&427,330.9&406,168.5\\ \seprule 21&Lithuania&3,329,039&10&367,250.6&332,903.9\\ 22&Latvia&2,248,374&8&290,290.0&281,046.8\\ 23&Slovenia&2,046,976&8&272,953.4&255,872.0\\ 24&Estonia&1,340,127&7&201,939.0&191,446.7\\ 25&Cyprus&803,147&6&134,291.1&133,857.8\\ \seprule 26&Luxembourg&502,066&6&89,446.6&83,677.7\\ 27&Malta&412,970&6&75,027.7&68,828.3\\ [1ex] \midrule &{\it Total}&501,103,425&751& &\\ \bottomrule \end{tabular*}
\small\caption{Each State receives one non-base seat for every 819,000 citizens or part thereof. Population/seat ratios are strictly decreasing before rounding, but there are two violations after rounding, namely Belgium and France when reading for the bottom. Data in this and the next table are taken from the Eurostat website \url{http://epp.eurostat.ec.europa.eu/}.} \label{table1} \end{table}
\begin{table}[htbp]
\centering \begin{tabular*}{\textwidth}{@{\extracolsep{\fill}} cc r c c c c} \toprule &\multirow{2}{*}{\it Member State}&\multirow{2}{*}{\it Population}&\multirow{2}{*}{\it Now}&{\it Seats}&{\it Seats}&{\it Seats}\\ &&&&{\it 27 States}&{\it 28 States}&{\it 29 States}\\ \midrule 1&Germany&81,802,257&99&96&96&96\\ 2&France&64,714,074&74&85&83&82\\ 3&UK&62,008,048&73&81&80&79\\ 4&Italy&60,340,328&73&79&78&77\\ 5&Spain&45,989,016&54&62&61&60\\ \seprule 6&Poland&38,167,329&51&52&51&51\\ 7&Romania&21,462,186&33&32&31&31\\ 8&Netherlands&16,574,989&26&26&25&25\\ 9&Greece&11,305,118&22&19&19&19\\ 10&Belgium&10,839,905&22&19&18&18\\ \seprule 11&Portugal&10,637,713&22&18&18&18\\ 12&Czech Rep.&10,506,813&22&18&18&18\\ 13&Hungary&10,014,324&22&18&17&17\\ 14&Sweden&9,340,682&20&17&17&17\\ 15&Austria&8,375,290&19&16&16&15\\ \seprule 16&Bulgaria&7,563,710&18&15&15&14\\ 17&Denmark&5,534,738&13&12&12&12\\ 18&Slovakia&5,424,925&13&12&12&12\\ 19&Finland&5,351,427&13&12&12&12\\ 20&Ireland&4,467,854&12&11&11&11\\ \seprule 21&Croatia&4,425,747&--&--&11&11\\ 22&Lithuania&3,329,039&12&10&9&9\\ 23&Latvia&2,248,374&9&8&8&8\\ 24&Slovenia&2,046,976&8&8&8&8\\ 25&Estonia&1,340,127&6&7&7&7\\ \seprule 26&Cyprus&803,147&6&6&6&6\\ 27&Luxembourg&502,066&6&6&6&6\\ 28&Malta&412,970&6&6&6&6\\ 29&Iceland&317,630&--&--&--&6\\ \midrule &{\it Total}&505,529,172&751&751 &751&754\\ \bottomrule \end{tabular*}
\small\caption{The column labelled `27 States' is the Cambridge Compromise with the present European Union. The next two columns include Croatia and Iceland in that order. The divisors are 819,000 (27 States), 835,000 (28 States), 844,000 (29 States).} \label{table2} \end{table}
\subsection{Divisors or D'Hondt?} \label{sec:dhondt}
Democracies have extensive experience of voting systems, and a variety of nomenclature has evolved. The following trans-Atlantic translation chart is included here.
{\centering\begin{tabular*}{0.6\textwidth}{@{\extracolsep{\fill}} ccc} \emph{rounding} & \emph{Europe} & \emph{USA}\\ \midrule downwards & D'Hondt & Jefferson\\ standard & Sainte-Lagu\"e & Webster\\ upwards & & Adams\\ \end{tabular*}
}
The Cambridge Compromise may be reformulated as a system of any of these three types, and we illustrate this with the case of D'Hondt's method. Allocate to every State the minimum $m$ seats (currently $m=6$). The remaining seats are allocated according to D'Hondt's method subject to the condition that, when any State attains a total of 96 seats, then it receives no further seats. The ensuing allocation is identical to that of the Cambridge Compromise.
The better to aid the reader, we give a brief explanation of the relevant D'Hondt method in the presence of an integral base and maximum. Write $B$ (\resp, $M$) for the base (\resp, maximum) allocation, and $H$ for the house-size. Let the population-sizes be $p_1,p_2,\dots,p_n$. \begin{numlist} \item At stage 0, allocate $B$ seats to every State. The remaining $R=H-nB$ seats will be allocated sequentially as follows, until none remain. \item Suppose, at some stage, that State $i$ has been allocated $a_i$ seats in all. Find a State $j$ such that $p_j/(a_j-B+1)$ is a maximum, and allocate the next seat to this State. \item Repeat the previous step until no seats remain, subject to the condition that any State achieving the maximum number $M$ of seats is removed from the process. \end{numlist} It may be checked that the outcome agrees with the system $B+$D, which was shown in Section \ref{sec:choice} to be equivalent to the Cambridge Compromise with base $b=B-1$. Similar algorithms are of course valid for the Sainte-Lagu\"e and Adams methods.
Ties can occur in the above algorithm, and these correspond to the non-existence of a divisor for some house-size in the formulation of Section \ref{sec:ccbp}. There are standard ways of breaking ties by casting lots. However, ties are very unlikely to occur in instances of the EU apportionment problem since populations are large and varied. Indeed, subject to a reasonable probabilistic model for population-sizes, the probability of a tie may estimated rigorously.
For further reading, see \cite{sj11}, or perhaps \cite[p.\ 99]{baly},
\subsection{Choosing the minimum and maximum}
The better to understand the role of the minimum, CAM discussed how the minimum and base could be reduced as further States accede to the Union. No final recommendation was reached but two Schemes emerged.
In Scheme A, a cap is introduced on the proportion of seats allocated via the minimum, and the value of the minimum is taken as large as possible subject to this cap. For example, there are currently $27\times 6 = 162$ seats allocated thus, a proportion of approximately 22\%. If, for example, one caps this at 25\%, the minimum remains at 6 for a larger Union of 27--31 States, and is reduced to 5 for 32--37 States, and so on. The base $b$ might either be one fewer than the minimum (with rounding upwards), or might follow a rule of the type: $b$ is the smallest fraction such that the smallest State receives exactly the minimum number of seats (with rounding upwards, say).
In Scheme B, one determines the base as a function of the number $n$ of States, and current practice indicates a formula of the type $b=135/n$. This has the advantage of decreasing steadily as $n$ increases. However, the associated \emph{minimum} decreases in a manner that is sensitive to the smallest population.
Since each State receives by necessity an integral number of seats, one effect of the allocation of seats to new States is a notable lumpiness at the upper end of the population chart. With the minimum held constant, the seats granted to an acceding State are taken from other States in proportion to their populations, and thus mostly from the larger States. Conversely, any adjustment downwards in the minimum allocation releases seats for proportional distribution between the States, of which the largest States gain most.
CAM recommended that consideration be given to the manner in which the minimum allocation should vary in the light of changes to the European Union, and also that the functioning of the maximum allocation be reviewed prior to future apportionments.
\subsection{Population statistics}
Census data is key to the allocation of seats in the European Parliament. Such population data is usually collected only once a decade. Both the year of the census and the manner of updating can vary between countries. In addition, there can be national variation in the definition of a resident. CAM's final recommendation was that the European Commission be encouraged to ensure that Eurostat review the methods used across the Union.
\section{Summary of Recommendations} \label{sec:recom}
\noi \emph{Principal recommendations} \begin{numlist} \item Adopt the revised definition of degressive proportionality proposed in Section \ref{sec:dp2} above. \item For future apportionments of the European Parliament, the method \emph{\bp} should be employed. \item The base should be 5, and fractions should be rounded upwards. \end{numlist}
\noi \emph{Further recommendations} \begin{Letlist} \item Due consideration should be given to the manner in which the minimum, currently 6, and base should vary in the light of future changes in the number of Member States in the European Union. \item The European Parliament should review the manner of functioning of the maximum constraint on number of seats, currently 96, prior to future apportionments. \item The Commission should be encouraged to ensure that Eurostat reviews the methods used by Member States in calculating their current populations, in order to ensure accuracy and consistency. \end{Letlist}
\section{Debate in the AFCO Committee} \label{sec:afco}
The timetable of discussion in Brussels was as follows. In advance of completion of the final CAM Report, the author was invited (as Director of CAM) to deliver a preview to the Committee on Constitutional Affairs (AFCO) in Brussels on 7 February 2011. There was a Committee discussion on 15 March. The Rapporteur, Andrew Duff, tabled a proposal \lq\lq for a modification of the Act concerning the election of the Members of the European Parliament by direct universal suffrage of 20 September 1976'', and this was the subject of amendments by Committee members, leading in turn to a set of so-called \lq\lq Compromise Amendments'' from the Rapporteur\footnote{Video recordings of the two meetings may be found at \url{http://tinyurl.com/5s63d8r}. Versions of the proposals and amendments may be consulted at \url{http://tinyurl.com/6bzedza}.}. A vote was taken on 19 April 2011.
The initial responses of Committee members to the CAM recommendation varied between curiosity verging on support, a desire for clarification, simple misunderstanding, and downright opposition. Several members expressed dismay at the \lq\lq political'' challenges of such a reorganization, and everyone was doubtless sensitive to the needs of Member States, Political Groups, and individual Members of the European Parliament. Amongst the issues that stimulated some MEPs were the changes in allocations to Member States with populations in the 7--11 million range, and the claim by one MEP of unfair treatment of the largest Member State.
The five week intermission between the two Committee meetings permitted a period of reflection and analysis, and contributions at the second meeting were generally more refined. There was some agreement in principle on the desirability of a formulaic approach to apportionment, but only one speaker (apart from the Rapporteur) spoke in support of the Cambridge Compromise. Representatives of several medium-sized countries were particularly implacable.
Committee members tabled 138 amendments to the Rapporteur's Proposal for a modification of the relevant Act. The final three were proposals to employ, \resp, the Cambridge Compromise, a parabolic method, and a power method. These three amendments were not destined to survive the vote, presumably as the consequences of formulaic approaches became clearer to some members of the Committee and of Parliament.
Two of the Rapporteur's twelve \lq\lq Compromise Amendments'' were directly relevant to the Cambridge Compromise. Amendment B proposed a formal definition of degressive proportionality along the lines of Section \ref{sec:dp2}, while withdrawing the proposal to adopt a specific mathematical approach. Amendment F compressed the discussion of a \lq\lq mathematical formula'' as follows: \begin{quotation} [The European Parliament] \lq\lq Proposes to enter into a dialogue with the European Council to explore the possibility of reaching agreement on a \emph{durable}\footnote{Italics by the current author. Recall the \emph{three} criteria of Section \ref{sec:crit}; the criterion of \lq\lq impartiality" has been omitted.} and \emph{transparent} mathematical formula for the apportionment of seats in the Parliament respecting the criteria laid down in the Treaties and the principles of plurality between political parties and solidarity among States.'' \end{quotation} These Compromise Amendments were agreed by the Committee on 19 April 2011, and the amended Proposal was duly carried.
It is not the purpose of this paper to speculate about the reasons for the unenthusiastic response of the AFCO Committee to this proposal in particular, and to formulaic approaches in general. Change can be tricky to manage and to explain to electorates, especially fundamental change requiring unanimity across EU Member States and affecting the livelihoods and ambitions of individual MEPs. The current allocations give preferential treatment to citizens of medium-sized States at the expense of those of larger States. The tentacles of the Political Groups entangle the EU, and alliances harness power and can frustrate change.
There is also the problem of the largest State. According to the Treaty of Lisbon, no State shall receive more than 96 seats, whereas an uncapped allocation would currently give a greater number to Germany. This feature of Parliamentary structure is illuminated baldly by the Cambridge Compromise using current population figures (the prominence of this cap will fade as the EU is enlarged).
It was argued by some MEPs that, in preferring a linear system, CAM had misunderstood the meaning of \lq \lq degressive proportionality". Such critics considered that CAM should have designed a formula to reproduce the current profile of Parliament. Not only is this contrary to the terms of reference received from the AFCO Committee, but also the author believes that mathematics is best not used as a tool to legitimize blatantly political deals.
The argument provides, however, a clue as to why formulaic approaches were disfavoured in the vote. Calculations indicate that, as the number of Member States increases, the allocations of many formulaic systems approach the simple linearity of the Cambridge Compromise. For example, with 29 States (including Croatia and Iceland) the allocations of both the parabolic and power methods differ only very slightly from that of the Cambridge formula. It seems that the mid-range bulge can be preserved only through \lq \lq political bartering", and that the discussion of this paper will resurface in the future.
\end{document} | arXiv |
CS: Fault Identification via Non-parametric Belief Propagation, Recovery of Functions of many Variables via Compressive Sensing, Sparse Signal Recovery and Dynamic Update of the Underdetermined System, A Nonlinear Approach to Dimension Reduction, Heat Source Identification Based on L1 Constrained Minimization
Has Compressive Sensing peaked ? This cannot be the impression you get from reading today's contributions.
Dror let me know of his new paper: Fault Identification via Non-parametric Belief Propagation by Danny Bickson, Dror Baron, Alexander Ihler, Harel Avissar, Danny Dolev. The abstract reads:
We consider the problem of identifying a pattern of faults from a set of noisy linear measurements. Unfortunately, maximum a posteriori probability estimation of the fault pattern is computationally intractable. To solve the fault identification problem, we propose a non-parametric belief propagation approach. We show empirically that our belief propagation solver is more accurate than recent state-of-the-art algorithms including interior point methods and semidefinite programming. Our superior performance is explained by the fact that we take into account both the binary nature of the individual faults and the sparsity of the fault pattern arising from their rarity.
The NBP solver is here.
Recovery of Functions of many Variables via Compressive Sensing by Albert Cohen, Ronald A. DeVore, Simon Foucart, Holger Rauhut. The abstract reads:
Recovery of functions of many variables from sample values usually suffers the curse of dimensionality: The number of required samples scales exponentially with the spatial dimension. In order to avoid this severe bottleneck, one needs to impose further structural properties of the function to be recovered apart from smoothness. Here, we build on ideas from compressive sensing and introduce a function model that involves "sparsity with respect to dimensions" in the Fourier domain. Using recent estimates on the restricted isometry constants of measurement matrices associated to randomly sampled trigonometric systems, we show that the number of required samples scales only logarithmically in the spatial dimension provided the function to be recovered follows the newly introduced highdimensional function model.
Sparse Signal Recovery and Dynamic Update of the Underdetermined System by M. Salman Asif and Justin Romberg. The abstract reads:
Sparse signal priors help in a variety of modern signal processing tasks. In many cases, a sparse signal needs to be recovered from an underdetermined system of equations. For instance, sparse approximation of a signal with an overcomplete dictionary or reconstruction of a sparse signal from a small number of linear measurements. The reconstruction problem typically requires solving an `1 norm minimization problem. In this paper we present homotopy based algorithms to update the solution of some `1 problems when the system is updated by adding new rows or columns to the underlying system matrix. We also discuss a case where these ideas can be extended to accommodate for more general changes in the system matrix.
A Nonlinear Approach to Dimension Reduction by Lee-Ad Gottlieb, Robert Krauthgamer. The abstract reads:
The ℓ2 flattening lemma of Johnson and Lindenstrauss [JL84] is a powerful tool for dimension reduction. It has been conjectured that the target dimension bounds can be refined and bounded in terms of the intrinsic dimensionality of the data set (for example, the doubling dimension). One such problem was proposed by Lang and Plaut [LP01] (see also [GKL03, Mat02, ABN08, CGT10]), and is still open. We prove another result in this line of work: The snowflake metric d1/2 of a doubling set S ⊂ ℓ2 can be embedded with arbitrarily low distortion into ℓD2, for dimension D that depends solely on the doubling constant of the metric. In fact, the target dimension is polylogarithmic in the doubling constant. Our techniques are robust and
extend to the more difficult spaces ℓ1 and ℓ∞, although the dimension bounds here are quantitatively inferior than those for ℓ2.
Heat Source Identification Based on L1 Constrained Minimization by Yingying Li, Stanley Osher, Richard Tsai. The abstract reads:
The inverse problem of finding sparse initial data from the solutions to the heat equation is considered. The initial data is assumed to be a sum of an unknown but nite number of Dirac delta functions at unknown locations. Point-wise values of the heat solution at only a few locations are used in an L1 constrained optimization to find such sparse initial data. A concept of domain of effective sensing is introduced to speed up the already fast Bregman iterative algorithm for L1 optimization. Furthermore, an algorithm which successively adds new measurements at intelligent locations is introduced. By comparing the solutions of the inverse problem that are obtained from different number of measurements, the algorithm decides where to add new measurements in order to improve the reconstruction of the sparse initial data.
I note from the paper:
In compressed sensing [7], we can solve an L0 problems by solving its L1 relaxation when the associated matrix has the restricted isometry property (RIP) [6]. The heat operator does not satisfy RIP, but we can adopt the idea of substituting L0 with L1 for sparse optimization. We will show numerical results which indicate the effectiveness of this strategy. Our approach to this problem is to solve a L1 minimization problem with constraints. We apply the Bregman iterative method [9, 13] to solve the constrained problem as a sequence of unconstrained subproblems. To solve these subproblems, we use the greedy coordinate descent method developed in [11] for solving the L1 unconstrained problem, which was shown to be very efficient for sparse recovery.
It is nice to see this fact acknowledged.
An LANL/UCSD newsletter features some hardware performing some compressive sensing:
Embedded computing and sensing are entrenched in many facets of daily life. Embedded devices must be able to collect large amounts of data from multiple sources, and then present the user with an "executive summary" of the observations. A user can then use this distilled information to quickly plan a course of action. It is therefore imperative that new methods are explored for distilling data to a form that is suitable for a user. Furthermore, the prevalence of wireless communications demands that relevant information be preextracted from high-dimension data in order to reduce bandwidth, memory, and energy requirements. Applications of interest to the EI such as structural health monitoring (SHM) and treaty verification typically require the collection of data from large arrays of wireless sensor nodes at high data rates. Data from different sensors must be combined in order to draw inferences about the state of the system under observation. The sensor nodes used to collect these data typically have severe data storage and energy constraints. Wireless transmission of data must be executed in a thoughtful manner and only the most relevant data should be committed to memory. Recently, compressed sensing has presented itself as a candidate solution for directly collecting relevant information from sparse, highdimensional measurements. The main idea behind compressed sensing is that, by directly collecting a relatively small number of coefficients, it is possible to reconstruct the original measurement. The coefficients are derived from linear combinations of measurements. Conveniently, most signals found in nature are indeed sparse with the notable exception of noise. The findings of the compressed sensing community hold great potential for changing the way data are collected. EI and ISR researchers (David Mascarenas, Chuck Farrar, Don Hush, James Thieler) has begun exploring the possibility of incorporating compressed sensing principles into SHM and treaty verification wireless sensor nodes. Advantages offered by compressed sensing include lower energy use for data collection and transmission, as well as reduced memory requirements. Compressed coefficients also have the interesting property that they are democratic, in the sense that no individual coefficient has any more information than any other. In this way they exhibit some robustness against data corruption. It is also worth noting that compressed sensing versions of conventional statistical signal processing techniques have been adopted by EI based on the use of smashed filters. The extension of statistical signal processing to the compressed domain helps facilitate the transition of the SHM strategies to take advantage of compressed sensing techniques. Currently, a digital version of the compressed sensor onboard a microcontroller is developed. (shown in Figure). This compressed sensor node is being tested for a SHM application requiring acceleration measurements, as well as a CO2 climate treaty verification application. The prototype compressed sensor is capable of collecting compressed coefficients from measurements and sending them to an off-board processor for reconstruction. In addition, the smashed filter has also been implemented onboard the embedded compressed sensor node. Preliminary results have shown that the smashed filter successfully distinguishes between the damaged and undamaged states using only 1/32 the number of measurements used in the conventional matched filter. EI plans to extend the compressive sensing to the mobile-host wireless sensing network architectures that has been studied in the past, as compressed sensing holds great promise for distilling data collected from wireless sensor networks.
By Igor at 1/31/2011 12:01:00 AM No comments:
labels: compressed sensing, compressive sampling, compressive sensing, CS
Compressive Sensing Landscape version 0.2
Upon a commenter's remark on the obvious need for mentioning statistics, I just updated the mental picture of the communities involved in contributing to compressive sensing. This is version 0.2, anybody has a better one?
By Igor at 1/30/2011 03:09:00 PM 6 comments:
Landscape - version 0.1
Here is a mental picture of the communities more or less involved in contributing to compressive sensing. This is version 0.1, anybody has a better one ?
By Igor at 1/30/2011 11:17:00 AM 2 comments:
CS: Op-ed, some videos, Compressive Sensing Using the Entropy Functional, A Message-Passing Receiver for BICM-OFDM
Every so often, I get an email asking me to link back to certain sites, here is the latest instance, NB is listed in the Rest of the Best category, whatever that means, here is what struck me: the definition:
Math blogs are a great way for professors, scientists, and researchers to convey the purpose of their work to a broader audience. Some bloggers offer an introduction to mathematical concepts and current research, while others post updates on their research and interests. The following 50 blogs are the cream of the crop: entertaining and informative posts that have something to offer for math geeks of all stripes.
All I know is that Nuit Blanche ain't:
about Math
about conveying my work most of the time.
there to offer an introduction to current research most of the time.
there to post updates on my interests.
It's a chronicle... and, from my point of view, a good way to broker important and honest discussions.
One of you just contacted me and wondered if this paper was an instance of compressive sensing. I'll have to read the paper first. In the meantime, we have several videos, two papers and two announcements for talks. Enjoy.
First, the videos. The last one isn't about CS but rather how to use a Kinect sensor to perform some SLAM computations, wow.
Compressive Estimation for Signal Integration in Rendering
Robustness of Compressed Sensing Parallel MRI in the Presence of Inaccurate Sensitivity Estimates
6D SLAM with RGB-D Data
The papers:
Compressive Sensing Using the Entropy Functional by Kivanc Kose, Osman Gunay, A. Enis Cetin. The abstract reads:
In most compressive sensing problems l1 norm is used during the signal reconstruction process. In this article the use of entropy functional is proposed to approximate the l1 norm. A modified version of the entropy functional is continuous, differentiable and convex. Therefore, it is possible to construct globally convergent iterative algorithms using Bregman's row action D-projection method for compressive sensing applications. Simulation examples are presented.
I wonder when a solver will be available.
A Message-Passing Receiver for BICM-OFDM over Unknown Clustered-Sparse Channels by Philip Schniter. The abstract reads:
We propose a factor-graph-based approach to joint channel-estimation-and-decoding (JCED) of bit- interleaved coded orthogonal frequency division multiplexing (BICM-OFDM). In contrast to existing designs, ours is capable of exploiting not only sparsity in sampled channel taps but also clustering among the large taps, behaviors which are known to manifest at larger communication bandwidths. In order to exploit these channel-tap structures, we adopt a two-state Gaussian mixture prior in conjunction with a Markov model on the hidden state. For loopy belief propagation, we exploit a "generalized approximate message passing" (GAMP) algorithm recently developed in the context of compressed sensing, and show that it can be successfully coupled with soft-input soft-output decoding, as well as hidden Markov inference, through the standard sum-product framework. For N subcarriers and M bits per subcarrier (and any channel length L \lt N), the resulting JCED-GAMP scheme has a computational complexity of only O(N log2 N+N 2^M). Numerical experiments show that our scheme yields BER performance within 1 dB of the known-channel bound and 4 dB better than decoupled channel-estimation-and-decoding via LASSO.
At UT, there is going to be talk from somebody in industry on his use of compressed sensing and related techniques:
Detection of Patterns in Networks
ECE Seminar Series
ENS 637
Dr. Randy C. Paffenroth
Discuss the development and application of a mathematical and computational framework for detecting and classifying weak, distributed patterns in sensor networks. The work being done at Numerica demonstrates the effectiveness of space-time inference on graphs, robust matrix completion and second order analysis in the detection and classification of distributed patterns that are not discernible at the level of individual nodes. Our focus is on cyber security scenarios where computer nodes (such as terminals, routers and servers) are sensors that provide measurements of packet rates, user activity, central processing unit usage, etc. When viewed independently, they cannot provide a definitive determination of the underlying pattern, but when fused with data from across the network – both spatially and temporally – the relevant patterns emerge. The clear underlying suggestion is that only detectors and classifiers that use a rigorous mathematical analysis of temporal measurements at many spatially distributed points in the network can identify network attacks. This research builds upon work in compressed sensing and robust matrix completion and is an excellent example of industry-academic collaboration.
whereas at University of Illinois, there will be:
GE/IE 590 Seminar-Large-Scale Optimization with Applications in Compressed Sensing
Speaker Professor Fatma Kilinc-Karzan
Date Feb 3, 2011
Time 4:00 pm
Location 101 Transportation Building
Cost Free
Sponsor ISE
Contact Holly Tipsword
E-Mail [email protected]
In this talk, we will cover some of the recent developments in large-scale optimization motivated by the compressed sensing paradigm. Sparsity plays a key role in dealing with high-dimensional data sets. Exploiting this fact, compressed sensing suggests a new paradigm by directly acquiring and storing only a few linear measurements (possibly corrupted with noise) of high-dimensional signals, and then using efficient -recovery procedures for reconstruction. Successful applications of this theory range from MRI image processing to statistics and machine learning. This talk will have two main parts. In the first part, after presenting the necessary background from compressed sensing, we will show that prior results can be generalized to utilize a priori information given in the form of sign restrictions on the signal. We will investigate the underlying conditions allowing successful -recovery of sparse signals and show that these conditions although difficult to evaluate lead to sufficient conditions that can be efficiently verified via linear or semidefinite programming. We will analyze the properties of these conditions, and describe their limits of performance. In the second part, we will develop efficient first-order methods with both deterministic and stochastic oracles for solving large-scale well-structured convex optimization problems. As an application of this theory, we will show how large-scale problems originating from compressed sensing fall into this framework. We will conclude with numerical results demonstrating the effectiveness of our algorithms.
labels: AMP, compressed sensing, compressive sampling, compressive sensing, CS
Around the blogs in 80 hours.
I have changed the underlying template of the blog in order to have a larger width for every entries. One can now also re-tweet entries seamlessly.. Let me know if this improves the reading of the blog.
Here is a small selection of blog entries mixing both hardware and theoretical elements you have seen mentioned on Nuit Blanche.
Greg: MIT IAP '11 radar course SAR example, imaging with coffee cans, wood, and the audio input from your laptop
Gustavo (Greg's student): Cleaner Ranging and Multiple Targets
Anand: Privacy and entropy (needs improvement)
Vladimir (ISW): Albert Theuwissen Reports from EI 2011 - Part 1 and 3D Sensing Forum at ISSCC 2011
ALENEX: Experiments with Johnson-Lindenstrauss
ALENEX/ANALCO II
ALENEX/ANALCO
Jordan: Compressed sensing, compressed MDS, compressed clustering, and my talk tomorrow
Bob: Paper of the Day (Po'D): Performance Limits of Matching Pursuit Algorithms Edition
Terry: An introduction to measure theory
Arxiv blog: The Nuclear Camera Designed to Spot Hidden Radiation Sources
This last entry pointed to this paper on arxiv that featured a position sensitive radiation camera. I noted in this paper the following interesting graphs:
as they point to the ability to readily perform spectrum unfolding. More on this later.
Credit: NASA/JPL/University of Arizona (view from Mars Orbiter and view from the ground of the same scene)
Postdoc Position at EPFL on Compressed Sensing for Interferometric Imaging
Yves Wiaux just sent me the following:
POSTDOCTORAL POSITION at EPFL ON COMPRESSED SENSING FOR INTERFEROMETRIC IMAGING
Announcement: January 25 2011
Application Deadline: March 1 2011
Starting date: as soon as June 1 2011
A postdoctoral position on "Compressed sensing imaging techniques for radio interferometry" is available, as soon as in June 2011, at the Signal Processing Laboratories (SP Lab, splabs.epfl.ch) of EPFL. The opening relies on new funding obtained by Dr Yves Wiaux and Pierre Vandergheynst at the Swiss National Science Foundation (SNSF, www.snf.ch), in the aim of strengthening the activities of the BASP (lts2www.epfl.ch/~ywiaux/index-baspnode) research node in astrophysical signal processing, in collaboration with the Signal Processing Laboratory 2 (LTS2, lts2www.epfl.ch).
The position is opened to any dynamic and highly qualified candidate, Ph.D. in Electrical Engineering, Physics or equivalent and with a strong background in signal processing, specifically on compressive sampling. Competence in programming (MATLAB, C) is required. Knowledge of signal processing on the sphere, or radio-interferometric imaging is a plus. The successful candidate will be in charge of the collaborations between the BASP node and the international radio astronomy community. In addition to his/her main activities, he/she will also be welcome to collaborate with other researchers of the BASP node and of the SP Lab on signal processing for magnetic resonance imaging.
The appointment is initially for one-year, with the possibility of renewal, based upon performance. Note that EPFL offers very attractive salaries.
Requests for further information and applications (including cover letter, CV, and three reference letters), should be sent to Dr Y. Wiaux, directly by email (yves.wiaux[at]epfl.ch), ideally by March 1 2011 (though applications will be considered until the position is filled).
Thanks Yves..... and Pierre. It is now listed on the compressive sensing jobs page.
Image Credit: NASA/JPL/Space Science Institute
W00066496.jpg was taken on January 21, 2011 and received on Earth January 23, 2011. The camera was pointing toward SATURN at approximately 2,594,465 kilometers away, and the image was taken using the CL1 and BL1 filters.
CS: The really long post of the week: Lost in Translation, lots of papers and summer jobs
Today we have a really long post but first, I wanted to mention the following:
* This looks to me like structured illumination for repairing the brain ?
* Bob's entry on Path-finding back to Copenhagen where he talks about the A* algorithm and greedy algorithm.
* Amit Agrawal released some of the data used in the heterodyning computatioal photography work
Dataset B: Coded Exposure Images using Canon Camera and Ferro-Electric Shutter
(Download complete matlab code and all four input files)
Indoor Toy Train
(118 pixel blur)
Outdoor Green Taxi
Outdoor White Taxi
(66 pixel blur)
Outdoor Car License Plate
* Three of you kindly pointed out the french newsletter of Matlab came out with a Cleve's corner article on compressed sensing. We mentioned it before but what got my attention is the title "La note de Cleve - une reconstruction « magique » : la détection compressée". I am not happy with the translation because before you do some detection, you really need to acquire a signal, a concept that seems to have been lost on Mathworks France. Translation mishaps are the reason I had asked Emmanuel for a translation of the term before it got to be maimed back into French. It's really Acquisition Comprimee. I have written to the PR firm and to Cleve, Moler the founder of Matworks. Let's see how much time it takes to remove this abomination. In the meantime, Cleve kindly responded with
Thanks. I agree that Candes is particularly well qualified to make this decision.
-- Cleve
But the newsletter still has the wrong translation, muuuhhh. I like Cleve and his software, but maybe we should call their software Mathlabs or something.
* While we are on the subject of language, I think I should be able to learn Spanish by just reading this introduction to compressed sensing (the whole document is Algoritmos y aplicaciones de Compressive Sensing by María José Díaz Sánchez) Though, I really need to hear someone speak it. It is fascinating how a domain expertise really allows one to hop back and forth between languages. Maybe we should have a Rosetta stone explaining Compressed Sensing in many languages, including Indonesian.
* Stephane Chretien updated his page to include a code to his recent paper. From his page:
Sparse recovery with unknown variance: a LASSO-type approach. (with Sebastien Darses) PDF. Submitted. An illustration using NESTAv1.1 of Lemma 5.3 about the support of the LASSO solution can be found here. The results can be checked using the modification of S. Becker and J. Bobin's Matlab code available here.
* Yesterday I mentioned a presentation by Vivek Goyal, Alyson Fletcher, Sundeep Rangan entitled The Optimistic Bayesian: Replica Method Analysis of Compressed Sensing
Now let's get on with the new papers that appeared on various webpages and on arxiv:
Reconciling Compressive Sampling Systems for Spectrally-sparse Continuous-time Signals by Michael A. Lexa, Mike E. Davies and John S. Thompson.The abstract reads:
The Random Demodulator (RD) and the ModulatedWideband Converter (MWC) are two recently proposed compressed sensing (CS) techniques for the acquisition of continuous-time spectrally-sparse signals. They extend the standard CS paradigm from sampling discrete, finite dimensional signals to sampling continuous and possibly infinite dimensional ones, and thus establish the ability to capture these signals at sub-Nyquist sampling rates. The RD and the MWC have remarkably similar structures (similar block diagrams), but their reconstruction algorithms and signal models strongly differ. To date, few results exist that compare these systems, and owing to the potential impacts they could have on spectral estimation in applications like electromagnetic scanning and cognitive radio, we more fully investigate their relationship in this paper. Specifically, we show that the RD and the MWC are both based on the general concept of random filtering, but that the sampling functions characterising the systems differ significantly. We next demonstrate a previously unreported model sensitivity the MWC has to short duration signals that resembles the known sensitivity the RD has to nonharmonic tones. We also show that block convolution is a fundamental aspect of the MWC, allowing it to successfully sample and reconstruct block-sparse (multiband) signals. This aspect is lacking in the RD, but we use it to propose a new CS based acquisition system for continuous-time signals whose amplitudes are block sparse. The paper includes detailed time and frequency domain analyses of the RD and the MWC that differ, sometimes substantially, from published results.
Compressive Power Spectral Density Estimation by Michael A. Lexa, Mike E. Davies, John S. Thompson, Janosch Nikolic. The abstract reads:
In this paper, we consider power spectral density estimation of bandlimited, wide-sense stationary signals from sub-Nyquist sampled data. This problem has recently received attention from within the emerging field of cognitive radio for example, and solutions have been proposed that use ideas from compressed sensing and the theory of digital alias-free signal processing. Here we develop a compressed sensing based technique that employs multi-coset sampling and produces multi-resolution power spectral estimates at arbitrarily low average sampling rates. The technique applies to spectrally sparse and nonsparse signals alike, but we show that when the widesense stationary signal is spectrally sparse, compressed sensing is able to enhance the estimator. The estimator does not require signal reconstruction and can be directly obtained from a straightforward application of nonnegative least squares.
Sparse Interactions: Identifying High-Dimensional Multilinear Systems via Compressed Sensing by Bobak Nazer and Robert D. Nowak. The abstract reads:
This paper investigates the problem of identifying sparse multilinear systems. Such systems are characterized by multiplicative interactions between the input variables with sparsity meaning that relatively few of all conceivable interactions are present. This problem is motivated by the study of interactions among genes and proteins in living cells. The goal is to develop a sampling/sensing scheme to identify sparse multilinear systems using as few measurements as possible. We derive bounds on the number of measurements required for perfect reconstruction as a function of the sparsity level. Our results extend the notion of compressed sensing from the traditional notion of (linear) sparsity to more refined notions of sparsity encountered in nonlinear systems. In contrast to the linear sparsity models, in the multilinear case the pattern of sparsity may play a role in the sensing requirements.
Structured sublinear compressive sensing via dense belief propagation by Wei Dai, Olgica Milenkovic, Hoa Vin Pham. The abstract reads:
Compressive sensing (CS) is a sampling technique designed for reducing the complexity of sparse data acquisition. One of the major obstacles for practical deployment of CS techniques is the signal reconstruction time and the high storage cost of random sensing matrices. We propose a new structured compressive sensing scheme, based on codes of graphs, that allows for a joint design of structured sensing matrices and logarithmic-complexity reconstruction algorithms. The compressive sensing matrices can be shown to offer asymptotically optimal performance when used in combination with Orthogonal Matching Pursuit (OMP) methods. For more elaborate greedy reconstruction schemes, we propose a new family of dense list decoding belief propagation algorithms, as well as reinforced- and multiple-basis belief propagation algorithms. Our simulation results indicate that reinforced BP CS schemes offer very good complexity-performance tradeoffs for very sparse signal vectors.
RIP-Based Near-Oracle Performance Guarantees for SP, CoSaMP, and IHT by Raja Giryes and Michael Elad. The abstract reads:
This paper presents an average case denoising performance analysis for SP, CoSaMP and IHT algorithms. This analysis considers the recovery of a noisy signal, with the assumptions that it is corrupted by an additive random white Gaussian noise and has a K-sparse representation with respect to a known dictionary D. The proposed analysis is based on the RIP, establishing a near-oracle performance guarantee for each of these algorithms. Similar RIP-based analysis was carried out previously for Dantzig Selector (DS) and Basis Pursuit (BP). Past work also considered a mutual-coherence-based analysis of OMP and thresholding. The performance guarantees developed in this work resemble those obtained for the relaxation-based methods (DS and BP), suggesting that the performance is independent of the sparse representation entries contrast and magnitude which is not true for OMP and thresholding.
Sparse Recovery, Kashin Decomposition and Conic Programming by Alexandre d'Aspremont. The abstract reads:
We produce relaxation bounds on the diameter of arbitrary sections of the l1 ball in R^n. We use these results to test conditions for sparse recovery.
CSSF MIMO RADAR: Low-Complexity Compressive Sensing Based MIMO Radar That Uses Step Frequency by Yao Yu, Athina P. Petropulu, H. Vincent Poor. The abstract reads:
A new approach is proposed, namely CSSF MIMO radar, which applies the technique of step frequency (SF) to compressive sensing (CS) based multi-input multi-output (MIMO) radar. The proposed approach enables high resolution range, angle and Doppler estimation, while transmitting narrowband pulses. The problem of joint angle-Doppler-range estimation is first formulated to fit the CS framework, i.e., as an L1 optimization problem. Direct solution of this problem entails high complexity as it employs a basis matrix whose construction requires discretization of the angle-Doppler-range space. Since high resolution requires fine space discretization, the complexity of joint range, angle and Doppler estimation can be prohibitively high. For the case of slowly moving targets, a technique is proposed that achieves significant complexity reduction by successively estimating angle-range and Doppler in a decoupled fashion and by employing initial estimates obtained via matched filtering to further reduce the space that needs to be digitized. Numerical results show that the combination of CS and SF results in a MIMO radar system that has superior resolution and requires far less data as compared to a system that uses a matched filter with SF.
A Novel Approach for Fast Detection of Multiple Change Points in Linear Models by Xiaoping Shi, Yuehua Wu, Baisuo Jin. The abstract reads:
A change point problem occurs in many statistical applications. If there exist change points in a model, it is harmful to make a statistical analysis without any consideration of the existence of the change points and the results derived from such an analysis may be misleading. There are rich literatures on change point detection. Although many methods have been proposed for detecting multiple change points, using these methods to find multiple change points in a large sample seems not feasible. In this article, a connection between multiple change point detection and variable selection through a proper segmentation of data sequence is established, and a novel approach is proposed to tackle multiple change point detection problem via the following two key steps: (1) apply the recent advances in consistent variable selection methods such as SCAD, adaptive LASSO and MCP to detect change points; (2) employ a refine procedure to improve the accuracy of change point estimation. Five algorithms are hence proposed, which can detect change points with much less time and more accuracy compared to those in literature. In addition, an optimal segmentation algorithm based on residual sum of squares is given. Our simulation study shows that the proposed algorithms are computationally efficient with improved change point estimation accuracy. The new approach is readily generalized to detect multiple change points in other models such as generalized linear models and nonparametric models.
Peak Reduction and Clipping Mitigation by Compressive Sensing by Ebrahim B. Al-Safadi, Tareq Y. Al-Naffouri. The abstract reads:
This work establishes the design, analysis, and fine-tuning of a Peak-to-Average-Power-Ratio (PAPR) reducing system, based on compressed sensing at the receiver of a peak-reducing sparse clipper applied to an OFDM signal at the transmitter. By exploiting the sparsity of the OFDM signal in the time domain relative to a pre-defined clipping threshold, the method depends on partially observing the frequency content of extremely simple sparse clippers to recover the locations, magnitudes, and phases of the clipped coefficients of the peak-reduced signal. We claim that in the absence of optimization algorithms at the transmitter that confine the frequency support of clippers to a predefined set of reserved-tones, no other tone-reservation method can reliably recover the original OFDM signal with such low complexity.
Afterwards we focus on designing different clipping signals that can embed a priori information regarding the support and phase of the peak-reducing signal to the receiver, followed by modified compressive sensing techniques for enhanced recovery. This includes data-based weighted {\ell} 1 minimization for enhanced support recovery and phase-augmention for homogeneous clippers followed by Bayesian techniques.
We show that using such techniques for a typical OFDM signal of 256 subcarriers and 20% reserved tones, the PAPR can be reduced by approximately 4.5 dB with a significant increase in capacity compared to a system which uses all its tones for data transmission and clips to such levels. The design is hence appealing from both capacity and PAPR reduction aspects.
Remarks on the Restricted Isometry Property in Orthogonal Matching Pursuit algorithm by Qun Mo, Yi Shen. The abstract reads:
This paper demonstrates theoretically that if the restricted isometry constant $\delta_K$ of the compressed sensing matrix satisfies $$ \delta_{K+1} < \frac{1}{\sqrt{K}+1}, $$ then a greedy algorithm called Orthogonal Matching Pursuit (OMP) can recover a signal with $K$ nonzero entries in $K$ iterations. In contrast, matrices are also constructed with restricted isometry constant $$ \delta_{K+1} = \frac{1}{\sqrt{K}} $$ such that OMP can not recover $K$-sparse $x$ in $K$ iterations. This result shows that the conjecture given by Dai and Milenkovic is ture.
I am sure they meant True, not ture.
Distributed Representation of Geometrically Correlated Images with Compressed Linear Measurements by Vijayaraghavan Thirumalai ; Pascal Frossard. The abstract reads:
The distributed representation of correlated images is an important challenge in applications such as multi-view imaging in camera networks or low complexity video coding. This paper addresses the problem of distributed coding of images whose correlation is driven by the motion of objects or the positioning of the vision sensors. It concentrates on the problem where images are encoded with compressed linear measurements, which are used for estimation of the correlation between images at decoder. We propose a geometry-based correlation model in order to describe the correlation between pairs of images. We assume that the constitutive components of natural images can be captured by visual features that undergo local transformations (e.g., translation) in different images. These prominent visual features are estimated with a sparse approximation of a reference image by a dictionary of geometric basis functions. The corresponding features in the other images are then identified from the compressed measurements. The correlation model is given by the relative geometric transformations between corresponding features. We thus formulate a regularized optimization problem for the estimation of correspondences where the local transformations between images form a consistent motion or disparity map. Then, we propose an efficient joint reconstruction algorithm that decodes the compressed images such that they stay consistent with the quantized measurements and the correlation model. Experimental results show that the proposed algorithm effectively estimates the correlation between images in video sequences or multi-view data. In addition, the proposed reconstruction strategy provides effective decoding performance that compares advantageously to distributed coding schemes based on disparity or motion learning and to independent coding solution based on JPEG-2000.
Efficient Image Reconstruction Under Sparsity Constraints with Application to MRI and Bioluminescence Tomography by Matthieu Guerquin-Kern, J.-C. Baritaux, Michael Unser. The abstract reads:
Most bioimaging modalities rely on indirect measurements of the quantity under investigation. The image is obtained as the result of an optimization problem involving a physical model of the measurement system. Due to the ill-posedness of the above problem, the impact of the noise on the reconstructed images must be controlled. The recent emphasis in biomedical image reconstruction is on regularization schemes that favor sparse solutions, which renders the optimization problem non-smooth. In this work, we show how step-size adaptation can be used to speed up the most recent multi-step algorithms (e.g. FISTA) employed in sparse image recovery. We present experiments in MRI and Fluorescence Molecular Tomography with specifically tailored step-adaptation strategies. Our results demonstrate the possibility of an order-of-magnitude speed enhancement over state-of-the-art algorithms.
Analyzing Weighted `1 Minimization for Sparse Recovery with Nonuniform Sparse Models by M. Amin Khajehnejad, Weiyu Xu, A. Salman Avestimehr and Babak Hassibi. The abstract reads:
In this paper we introduce a nonuniform sparsity model and analyze the performance of an optimized weighted `1 minimization over that sparsity model. In particular, we focus on a model where the entries of the unknown vector fall into two sets, with entries of each set having a specific probability of being nonzero. We propose a weighted `1 minimization recovery algorithm and analyze its performance using a Grassmann angle approach. We compute explicitly the relationship between the system parameters-the weights, the number of measurements, the size of the two sets, the probabilities of being nonzero- so that when i.i.d. random Gaussian measurement matrices are used, the weighted `1 minimization recovers a randomly selected signal drawn from the considered sparsity model with overwhelming probability as the problem dimension increases. This allows us to compute the optimal weights. We demonstrate through rigorous analysis and simulations that for the case when the support of the signal can be divided into two different subclasses with unequal sparsity fractions, the weighted `1 minimization outperforms the regular `1 minimization substantially. We also generalize our results to signal vectors with an arbitrary number of subclasses for sparsity.
Should penalized least squares regression be interpreted as Maximum A Posteriori estimation? by Rémi Gribonval. The abstract reads::
Penalized least squares regression is often used for signal denoising and inverse problems, and is commonly interpreted in a Bayesian framework as a Maximum A Posteriori (MAP) estimator, the penalty function being the negative logarithm of the prior. For example, the widely used quadratic program (with an $\ell^1$ penalty) associated to the LASSO / Basis Pursuit Denoising is very often considered as MAP estimation under a Laplacian prior in the context of additive white Gaussian noise (AWGN) reduction. This paper highlights the fact that, while this is {\em one} possible Bayesian interpretation, there can be other equally acceptable Bayesian interpretations. Therefore, solving a penalized least squares regression problem with penalty $\phi(x)$ need not be interpreted as assuming a prior $C\cdot \exp(-\phi(x))$ and using the MAP estimator. In particular, it is shown that for {\em any} prior $P_X$, the minimum mean square error (MMSE) estimator is the solution of a penalized least square problem with some penalty $\phi(x)$, which can be interpreted as the MAP estimator with the prior $C \cdot \exp(-\phi(x))$. Vice-versa, for {\em certain} penalties $\phi(x)$, the solution of the penalized least squares problem is indeed the MMSE estimator, with a certain prior $P_X$. In general $dP_X(x) \neq C \cdot \exp(-\phi(x))dx$.
Sparsity Equivalence of Anisotropic Decompositions by Gitta Kutyniok. The abstract reads;
Anisotropic decompositions using representation systems such as curvelets, contourlet, or shearlets have recently attracted significantly increased attention due to the fact that they were shown to provide optimally sparse approximations of functions exhibiting singularities on lower dimensional embedded manifolds. The literature now contains various direct proofs of this fact and of related sparse approximation results. However, it seems quite cumbersome to prove such a canon of results for each system separately, while many of the systems exhibit certain similarities. In this paper, with the introduction of the concept of sparsity equivalence, we aim to provide a framework which allows categorization of the ability for sparse approximations of representation systems. This framework, in particular, enables transferring results on sparse approximations from one system to another. We demonstrate this concept for the example of curvelets and shearlets, and discuss how this viewpoint immediately leads to novel results for both systems.
Behind a paywall:
Automatic Container Code Recognition Using Compressed Sensing Method by Chien-Cheng Tseng and Su-Ling Lee. The abstract reads:
In this paper, an automatic container code recognition method is presented by using compressed sensing (CS). First, the compressed sensing approach which uses the constrained L1 minimization method is reviewed. Then, a general pattern recognition framework based on CS theory is described. Next, the CS recognition method is applied to construct an automatic container code recognition system. Finally, the real-life images provided by trading port of Kaohsiung are used to evaluate the performance of the proposed method.
Here are a list of papers that caught my attention as they werre directly or indirectly related to compressive sensing:
Computational Cameras: Approaches, Benefits and Limits by Shree K. Nayar. The abstract reads:
A computational camera uses a combination of optics and software to produce images that cannot be taken with traditional cameras. In the last decade, computational imaging has emerged as a vibrant field of research. A wide variety of computational cameras have been demonstrated - some designed to achieve new imaging functionalities, and others to reduce the complexity of traditional imaging. In this article, we describe how computational cameras have evolved and present a taxonomy for the technical approaches they use. We explore the benefits and limits of computational imaging, and discuss how it is related to the adjacent and overlapping fields of digital imaging, computational photography and computational image sensors.
Efficient algorithms for chemical threshold testing problems by Annalisa De Bonis, Luisa Gargano and Ugo Vaccaro. The abstract reads:
We consider a generalization of the classical group testing problem. Let us be given a sample contaminated with a chemical substance. We want to estimate the unknown concentration c of this substance in the sample. There is a threshold indicator which can detect whether the concentration is at least a known threshold. We consider both the case when the threshold indicator does not affect the tested units and the more difficult case when the threshold indicator destroys the tested units. For both cases, we present a family of efficient algorithms each of which achieves a good approximation of c using a small number of tests and of auxiliary resources. Each member of the family provides a different tradeoff between the number of tests and the use of other resources involved by the algorithm. Previously known algorithms for this problem use more tests than most of our algorithms do.
Identification and Classification Problems on Pooling Designs for Inhibitor Models by Huilan Chang, Hong-Bin Chen, Hung-Lin Fu. The abstract reads:
Pooling designs are common tools to efficiently distinguish positive clones from negative clones in clone library screening. In some applications, there is a third type of clones called \inhibitors" whose effect is in a sense to obscure the positive clones in pools. Various inhibitor models have been proposed in the literature. We address the inhibitor problems of designing efficient nonadaptive procedures for both identification and classification problems, and improve previous results in three aspects:
* The algorithm that is used to identify the positive clones works on a more general inhibitor model and has a polynomial-time decoding procedure that recovers the set of positives from the knowledge of the outcomes.
* The algorithm that is used to classify all clones works in one-stage, i.e., all tests are arranged in advance without knowing the outcomes of other tests, along with a polynomial-time decoding procedure.
We extend our results to pooling designs on complexes where the property to be screened is defined on subsets of biological objects, instead of on individual ones.
An Almost Optimal Algorithm for Generalized Threshold Group Testing with Inhibitors. by Chen HB, Annalisa De Bonis.. The abstract reads:
Group testing is a search paradigm where one is given a population [Formula: see text] of n elements and an unknown subset [Formula: see text] of defective elements and the goal is to determine [Formula: see text] by performing tests on subsets of [Formula: see text]. In classical group testing a test on a subset [Formula: see text] receives a YES response if [Formula: see text], and a NO response otherwise. In group testing with inhibitors (GTI), identifying the defective items is more difficult due to the presence of elements called inhibitors that interfere with the queries so that the answer to a query is YES if and only if the queried group contains at least one defective item and no inhibitor. In the present article, we consider a new generalization of the GTI model in which there are two unknown thresholds h and g and the response to a test is YES both in the case when the queried subset contains at least one defective item and less than h inhibitors, and in the case when the queried subset contains at least g defective items. Moreover, our search model assumes that no knowledge on the number [Formula: see text] of defective items is given. We derive lower bounds on the minimum number of tests required to determine the defective items under this model and present an algorithm that uses an almost optimal number of tests.
On Parsimonious Explanations for 2-D Tree- and Linearly-Ordered Data by Howard Karloff, Flip Korn, Konstantin Makarychev, Yuval Rabani. The abstract reads:
This paper studies the "explanation problem" for tree- and linearly-ordered array data, a problem motivated by database applications and recently solved for the one-dimensional tree-ordered case. In this paper, one is given a matrix A whose rows and columns have semantics: special subsets of the rows and special subsets of the columns are meaningful, others are not. A submatrix in A is said to be meaningful if and only if it is the cross product of a meaningful row subset and a meaningful column subset, in which case we call it an "allowed rectangle." The goal is to "explain" A as a sparse sum of weighted allowed rectangles. Specifically, we wish to find as few weighted allowed rectangles as possible such that, for all i,j, a_{ij} equals the sum of the weights of all rectangles which include cell (i,j).
In this paper we consider the natural cases in which the matrix dimensions are tree-ordered or linearly-ordered. In the tree-ordered case, we are given a rooted tree T1 whose leaves are the rows of A and another, T2, whose leaves are the columns. Nodes of the trees correspond in an obvious way to the sets of their leaf descendants. In the linearly-ordered case, a set of rows or columns is meaningful if and only if it is contiguous.
For tree-ordered data, we prove the explanation problem NP-Hard and give a randomized 2-approximation algorithm for it. For linearly-ordered data, we prove the explanation problem NP-Hard and give a 2.56-approximation algorithm. To our knowledge, these are the first results for the problem of sparsely and exactly representing matrices by weighted rectangles.
Asynchronous Code-Division Random Access Using Convex Optimization by Lorne Applebaum, Waheed U. Bajwa, Marco F. Duarte, Robert Calderbank. The abstract reads:
Many applications in cellular systems and sensor networks involve a random subset of a large number of users asynchronously reporting activity to a base station. This paper examines the problem of multiuser detection (MUD) in random access channels for such applications. Traditional orthogonal signaling ignores the random nature of user activity in this problem and limits the total number of users to be on the order of the number of signal space dimensions. Contention-based schemes, on the other hand, suffer from delays caused by colliding transmissions and the hidden node problem. In contrast, this paper presents a novel asynchronous (non-orthogonal) code-division random access scheme along with a convex optimization-based MUD algorithm that overcomes the issues associated with orthogonal signaling and contention-based methods. Two key distinguishing features of the proposed algorithm are that it does not require knowledge of the delay or channel state information of every user and it has polynomial-time computational complexity. The main analytical contribution of this paper is the relationship between the performance of the proposed MUD algorithm and two simple metrics of the set of user codewords. The study of these metrics is then focused on two specific sets of codewords, random binary codewords and specially constructed algebraic codewords, for asynchronous random access. The ensuing analysis confirms that the proposed scheme together with either of these two codeword sets significantly outperforms the orthogonal signaling-based random access in terms of the total number of users in the system.
Following are three announcement for three talks, some have passed but what is important is the abstract of the talks:
iCME Colloquium, Resolution Analysis for Compressive Imaging
* Wednesday
* 4:15 pm - 5:05 pm
* Bldg. 380, Rm. 380/380X
Albert Fannjiang (University of California, Davis)
In this talk we will focus on the issue of resolution when compressed sensing techniques are applied to imaging and inverse problems. The physical process of measurement consists of illumination/emission, wave propagation and reception/sampling, giving rise to measurement matrices more structured than those typically studied in the compressed sensing literature. We will show how to achieve the diffraction limit in diffraction tomography, inverse scattering and radio interferometry by using random sampling. We will also discuss the superresolution effect resulting from near-field measurements, random illumination and multiple measurements in conjunction with random sampling.
Fast dimensionality reduction: improved bounds and implications for compressed sensing
Event on 2011-01-20 16:10:00
Fast dimensionality reduction: improved bounds and implications for compressed sensing Colloquium | January 20 | 4:10-5 p.m. | 60 Evans Hall
Speaker: Rachel Ward, Courant Institute, New York University Sponsor: Mathematics, Department of
Embedding high-dimensional data sets into subspaces of much lower dimension is important for reducing storage cost and speeding up computation in several applications, including numerical linear algebra, manifold learning, and computer science. The relatively new field of compressed sensing is based on the observation that if the high-dimensional data are sparse in a known basis, they can be embedded into a lower-dimensional space in a manner that permits their efficient recovery through l1-minimization. We first give a brief overview of compressed sensing, and discuss how certain statistical procedures like cross validation can be naturally incorporated into this set-up. The latter part of the talk will focus on a near equivalence of two fundamental concepts in compressed sensing: the Restricted Isometry Property and the Johnson-Lindenstrauss Lemma; as a consequence of this result, we can improve on the best-known bounds for dimensionality reduction using structured, or fast linear embeddings. Finally, we discuss the Restricted Isometry Property for structured measurement matrices formed by subsampling orthonormal polynomial systems, and high-dimensional function approximation from a few samples.
Event Contact: 510-642-6550
at University of California, Berkeley
Berkeley, United States
Speaker: Qiyu Sun, Approximation Property in Compressive Sampling
3076 Duncan Hall
Houston, Texas, USA
The null space property and the restricted isometry property for a measurement matrix are two basic properties in compressive sampling, and are closely related to the sparse approximation. In this talk, we introduce the sparse approximation property for a measurement matrix, a weaker version of the restricted isometry property and a stronger version of the null space property. We show that the sparse approximation property for a measurement matrix could be an appropriate condition to consider stable recovery of any compressible signal from its noisy measurements.
Finally, some summer jobs:
Internship Opportunity (2011 summer) at Mitsubishi Electric Research Labs (MERL), Cambridge, MA, USA (http://www.merl.com/)
Posting Date: Jan 23, 2011
Internship Positions: Multiple
MERL is the North American corporate research and development organization of the Mitsubishi Electric Corporation (MELCO). Based in Cambridge, MA, MERL offers a challenging and stimulating research environment and opportunity to pursue independent projects and receive authorship on publications. The imaging group at MERL does research concerning information capture/extraction/analysis and display including computer vision, computational photography, multi-projector systems, image processing, machine learning and 3D graphics & modeling.
MERL invites applications for summer internship in 2011 in the imaging group. Graduate students pursuing a Phd degree in EE, CS or a related discipline are encouraged to apply. Publications at the flagship conferences (ICCV/CVPR/SIGGRAPH) as a result of the internship are highly encouraged. Development will be a mix of prototyping in Matlab and/or C/C++ and strong programming skills are required.
Specific research topics of interest include sparse/dense 3D reconstruction, stereo and multi-view stereo, SLAM, structure light scanning, wide-angle imaging using catadioptric sensors and 1D signal processing on DSP's. Applicants are expected to be familiar with the fundamentals of computer vision and image/video processing, along with sufficient knowledge in these areas. Prior publication in these areas is a plus.
Please submit resumes to Amit Agrawal. For more information about past projects, please visit www.amitkagrawal.com
labels: compressed sensing, compressive sampling, compressive sensing, CS, QuantCS
"...I found this idea of CS sketchy,..."
Yesterday I stated the following:
"...Anyway, in seismology it was known since at least 1973 that l_1 minimization was promoting sparsity but nobody really knew why and what to make of it in the sense that none of the sensors were modified as a result of this finding. What the papers of Tao, Candes, Romberg [1] and Donoho [2] did was give a sense of the kind of acquisition that would be acceptable ( measurement matrix satisfying RIP, KGG....) which is the reason we can contemplate building hardware..."
I added the emphasis after a commenter stated that
Dear Igor
"but nobody really knew why and what to make of it in the sense that none of the sensors were modified as a result of this finding"
I respectfully disagree
1) L1 in seismic was for deconvolution, nothing to do with CS.
2) The "Golden Oldies" section of Donoho's publications is worth reading, in particular "Superresolution via Sparsity Constraints", "Uncertainty Principles and Signal Recovery". It is quite clear that even in 1990 Donoho's understanding of L1 (and other sparsity promoting functional like entropy) and his relation with L0 was almost perfect.
To what I responded:
You seem to be arguing about the first part of the sentence, but are you sure you are arguing about the second part of that sentence (which is really the point I was making) ? Namely:
"..in the sense that none of the sensors were modified as a result of this finding.."
I realize that most of you in the signal processing community will feel strongly about what I am about to say but this is mostly because you have had a very raw deal all these years. Other people take data and then tell you : please clean it up I'll come back in the morning to pick it up when you're done. Would you want your kids to have that kind of a job ? It reminds me of being a thermal engineer in the space business, they would always lament being the last one to add anything in the design as it generally amounts to put wrinkled cloth on top of something. At dinner parties, you have a dry martini in one hand and you look sharp, and /or gorgeous while people laugh at your cynical view of what you are doing, you are so funny being the center of attention but deep down you feel dirty just explaining it.
Since this is the second time I see this type of blockage (the first one was here), let me be forthright about what I think on the matter:
Wavelets alone are not important
l_1 solvers promoting sparsity alone are not important
However, these unimportant pieces of the puzzle crystallize into something important because of these two papers [1] [2]. I realize that l_1 promoting sparsity was discovered before empirically, I realize that wavelets were developed in the late 1980's and I realize that they both made the work of signal processing people tremendously easier, but none of these two findings amount to much unless you can change the sensors and the way we (literally) see the world. I could go further:
Fractals alone are not important
Structured sparsity solvers alone are not important
[1] E. J. Candès, J. Romberg and T. Tao. Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math., 59 1207-1223. (pdf)
[2] D. Donoho, Compressed Sensing
Islands of Knowledge
[Yesterday's poll is at the end of this entry.] [Update: I have highlighted the most important part of a sentence]
On the NIPS videos entry, I said:
"...Of particular interest is a question asked to Francis Bach at the end of his presentation where it looks like known bounds are O(p^6) whereas is own empirical experience seems to show an O(p^2) bound, mmmuh, a little bit like what happened in the 1970s' when the seismic folks thought the l_1 norm was a good proxy for l_0........"
To what a youngster commented:
"...I wonder if you'd be kind enough to elucidate what happened in the 70s for those of us not old enough to remember, and not well-versed enough in seismology?.."
I don't think I qualify for either of these statements, but I'll take them to mean that I know everything, muaaaahahahahahahah... Anyway, in seismology it was known since at least 1973 that l_1 minimization was promoting sparsity but nobody really knew why and what to make of it in the sense that none of the sensors were modified as a result of this finding. What the papers of Tao, Candes, Romberg [1] and Donoho [2] did was give a sense of the kind of acquisition that would be acceptable ( measurement matrix satisfying RIP, KGG....) which is the reason we can contemplate building hardware. From [3]:
[3] Vivek Goyal, Alyson Fletcher, Sundeep Rangan, The Optimistic Bayesian: Replica Method Analysis of Compressed Sensing
Question of the Day; Does this patent cover a hardware based computational photography solution ?
In Is Turbulence Ripe for Compressive Imaging ?, I suggested that an imager could be used to detect microbursts on airplanes. Boeing seems to have filed a patent for a similar idea (Boeing invention lets planes spot dangerous turbulence) on July 17, 2009.
Now here is my generic question (you can answer in the poll below), is this claim (a good idea by the way) too tied to a specific technology / or would a specific computational photography or random lens imager avoid fitting the description of the system in the claims ? (I am specifically thinking of item 3)
"......1. A clear air turbulence detection system, comprising:an image capturing element;a lens having a focal length adapted to focus a scene onto said image capturing element such that a combination of said lens and said image capturing element are adapted to optically resolve a visual distortion of a feature in said scene due to turbulence; and a processor adapted to process an image of said scene from said image capturing element, said processor adapted to compare a plurality of said images to detect a change in refraction of light received from said feature due to turbulence and produce an indication of an area of turbulence.
2. The clear air turbulence detection system of claim 1, wherein said image capturing element is a CCD camera.
3. The clear air turbulence detection system of claim 1, wherein said lens is a telephoto lens.
4. The clear air turbulence detection system of claim 1, wherein said combination of said lens and said image capturing element is adapted to have a minimum resolving capability of between approximately 10 microradians of angle and approximately 100 microradians of angle.
5. The clear air turbulence detection system of claim 1, wherein the clear air turbulence system is mounted on a moving platform and wherein said processor is further adapted to transform a scene in a first image relative to a scene in a second image such that a feature in said first image approximates a position, a size and an orientation of said feature in said second image.
6. The clear air turbulence detection system of claim 5, wherein said processor is adapted to transform said scene in said first image using a transformation selected from the group consisting of translating said scene, scaling said scene, rotating said scene, and projecting said scene.
7. The clear air turbulence detection system of claim 5, further comprising:a inertial navigation device providing an orientation data of said feature; anda global positioning system data providing a position of said moving platform, and wherein said processor is adapted to process said global positioning system data and said orientation data to query a geographic information system source for an image data of said feature, and wherein said processor is adapted to compare an image of said feature captured by said image capturing element with said image data of said feature from said geographic information system source to detect a change in spatial information content of said feature due to an area of turbulence.
8. The clear air turbulence detection system of claim 1, wherein said processor detects said change in refraction of said feature due to an area of turbulence by performing a power spectral density operation selected from the group consisting of measuring a decrease in a high spatial frequency content of a measured power spectral density of a feature of an image compared with a calculated power spectral density of said feature of said image, and an increase in a high frequency content of a measured temporal power spectral density of a feature over a plurality of images compared with a calculated power spectral density of said feature over said plurality of images.
9. The clear air turbulence detection system of claim 1, wherein said processor detects turbulence by detecting angular blurring in an image.
10. The clear air turbulence detection system of claim 1, wherein said processor detects turbulence by detecting temporal flickering between images.
11. The clear air turbulence detection system of claim 1, wherein said processor detects a range to said turbulence by tracking a size of an area of scintillation through a plurality of images to determine a rate of change in an angular extent of said area of scintillation.
12. The clear air turbulence detection system of claim 1, wherein said turbulence indication is selected from the group consisting of an audible alarm, a graphical display of said turbulence, a graphical overlay superimposed on a real-time image, an input to an autopilot navigation system, and a broadcasted electronic message.
13. A method of detecting clear air turbulence, comprising:capturing a first image of a scene;capturing a second image consisting essentially of said scene;selecting a feature present in said first image and said second image;registering said first image to said second image such that said feature in said first image has approximately a same position, scale, and orientation as said feature in said second image;comparing said feature in said first image with said feature in said second image to determine a change to a portion of said feature between said first image and said second image; anddisplaying a turbulence indication based upon said change to said feature.
14. The method of detecting clear air turbulence of claim 13, wherein said first image and second image are adapted to resolve a minimum angular resolution of approximately 10 microradians to approximately 100 microradians.
15. The method of detecting clear air turbulence of claim 13, further comprising:removing a change to a portion of said feature caused by moving objects.
16. The method of detecting clear air turbulence of claim 13, further comprising:detecting an angular blurring in a image; anddisplaying a turbulence indication base upon said angular blurring.
17. The method of detecting clear air turbulence of claim 13, further comprising:detecting a temporal flickering in a set of images; anddisplaying a turbulence indication base upon said temporal flickering.
18. The method of detecting clear air turbulence of claim 13, wherein said change to a portion of said feature is an increase in a high temporal frequency content of a power spectrum distribution of said feature.
19. The method of detecting clear air turbulence of claim 13, further comprising:computing a power spectral density of said feature of said second image;computing a power spectral density of said feature of said first image; andcomparing said power spectral density of a said first image and said second image to determine said turbulence indication.
20. The method of detecting clear air turbulence of claim 13, further comprising:receiving an orientation data of said orientation of said scene of said first image relative to said feature;receiving a position data of a distance to said feature of said first image; andtransforming said scene of said second image to have approximately a same position, scale, and orientation as said feature in said first image.
21. The method of detecting clear air turbulence of claim 20, further comprising:querying a geographic information service for a data correlating to said feature; andcomputing said feature of said second image from the data of said geographic information service.
22. The method of detecting clear air turbulence of claim 13, wherein said displaying a turbulence indication is selected from the group consisting of playing an audible alarm, displaying a graphical display of said turbulence, superimposing a graphical overlay on a real-time image, sending a message to an autopilot navigation system, and broadcasting an electronic message.
23. An aircraft with a turbulence detection system, comprising:a turbulence detection system, further comprising a camera system and an image processing system;an aircraft, said aircraft adapted to mount said camera system; anda turbulence alerting system displaced in a cockpit of said aircraft, said turbulence alerting system in electronic communication with said image processing system.
24. The aircraft of claim 23, wherein said camera system further comprises:a CCD camera having a pixel size and focal length adapted to resolve visual distortions in a scene imaged by said CCD camera that are caused by turbulent air.
25. The aircraft of claim 24, wherein said image processing system is adapted to receive a plurality of images of said scene from said camera system, process said plurality of images, and produce an indication of turbulence.
26. The aircraft of claim 25, wherein said image processing system is adapted to select a selected image from said plurality of images, to select a reference image with which to compare said selected image, to perform a geometric image transformation selected from the group consisting of translating a scene, scaling a scene, rotating a scene, and projecting a scene, said geometric image transformation registering scenes from two different perspectives with one another, and to compare said selected image to said reference image to produce said indication of turbulence.
27. The aircraft of claim 26, wherein said reference image is selected from the group consisting of a second selected image from said plurality of images, a previously stored image of the scene, an image of the scene from a geographic information system, and an image of a feature in the scene from a geographic information system.
28. The aircraft of claim 25, wherein said image processing system detects a change selected from the group consisting of an angular blurring of a feature in an image, a temporal flickering of a feature in a set of images, a change in size of an area of scintillation of a feature in an image, a change in high temporal frequency content of a feature in an image, and a change in power spectral density of a feature in an image.
29. The aircraft of claim 23, wherein said turbulence alerting system receives said indication of turbulence from said image processing system, and said turbulence alerting system is selected from the group consisting of an audible alarm, a graphical display of said turbulence, a graphical overlay superimposed on a real-time image, an input to an autopilot navigation system, and a broadcasted electronic message...."
&amp;lt;br /&amp;gt; &amp;lt;a href="http://polldaddy.com/poll/4429413/"&amp;gt;Do you think a random lens imager fit into the description of this patent ?&amp;lt;/a&amp;gt;&amp;lt;span style="font-size:9px;"&amp;gt;&amp;lt;a href="http://polldaddy.com/features-surveys/"&amp;gt;customer surveys&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
nathalie senglet
snapshot from Martinique² [jatp] -
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Download Lectures on Biostatistics (1971). Corrected and searchable version of Google books edition
Download review of Lectures on Biostatistics (THES, 1973).
"An anti-EU movement can't also be anti-US, not without looking as if it hates everyone" by @Freedland www.theguardian.com/commentisf…
Boris: "less a lovable maverick than a rather unpleasant oddball." by @Freedland www.theguardian.com/commentisf…
RT @MartinShovel: My cartoon - is our #NHS safe in #Tory hands? #efficiencysavings #weaselwords pic.twitter.com/muYCcCPCZL
@CaulfieldTim sure! But it's 1 am. here -good night
@CaulfieldTim sounds good. But the problems mostly lie with academics. Self-inflicted wounds
@CaulfieldTim Trudeau looks great. Can we have him please?
@CaulfieldTim Here is an early example www.dcscience.net/2007/12/05/w… and recent ones www.dcscience.net/2014/11/02/t…
RT @taslimanasreen: Had meeting with Elmar Brok, Chair of the Committee on Foreign Affairs at the European Parliament today. pic.twitter.com/sb9lYfqKD6
RT @taslimanasreen: I spoke at the European Parliament today about how freethinkers getting killed,& govt remains silent in Bangladesh. pic.twitter.com/rW0XW5KBw9
@CaulfieldTim no, it isn't fair to blame reporting. They hypsecomes from jml and university PR people, endorsed by authors
Exactly. Bur publishers make cash publishing junk, and senior academics push you to publish every 10 min. Disaster twitter.com/CaulfieldTim/statu…
@CaulfieldTim harms science because every time the latest diet/exercise advice appears in the media, people just jeer (often rightly)
@CaulfieldTim do you think there's a case for stopping obs epidemiology? It so often gives wrong answer, harms people & harms science
How does such dubious junk science get published. I often wonder twitter.com/CaulfieldTim/statu…
If there was ever a case for reverse causality, that's it twitter.com/CaulfieldTim/statu…
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Statistics and the law: the prosecutor's fallacy
This post arose from a recent meeting at the Royal Society. It was organised by Julie Maxton to discuss the application of statistical methods to legal problems. I found myself sitting next to an Appeal Court Judge who wanted more explanation of the ideas. Here it is.
Some preliminaries
The papers that I wrote recently were about the problems associated with the interpretation of screening tests and tests of significance. They don't allude to legal problems explicitly, though the problems are the same in principle. They are all open access. The first appeared in 2014:
http://rsos.royalsocietypublishing.org/content/1/3/140216
Since the first version of this post, March 2016, I've written two more papers and some popular pieces on the same topic. There's a list of them at http://www.onemol.org.uk/?page_id=456.
I also made a video for YouTube of a recent talk.
In these papers I was interested in the false positive risk (also known as the false discovery rate) in tests of significance. It turned out to be alarmingly large. That has serious consequences for the credibility of the scientific literature. In legal terms, the false positive risk means the proportion of cases in which, on the basis of the evidence, a suspect is found guilty when in fact they are innocent. That has even more serious consequences.
Although most of what I want to say can be said without much algebra, it would perhaps be worth getting two things clear before we start.
The rules of probability.
(1) To get any understanding, it's essential to understand the rules of probabilities, and, in particular, the idea of conditional probabilities. One source would be my old book, Lectures on Biostatistics (now free), The account on pages 19 to 24 give a pretty simple (I hope) description of what's needed. Briefly, a vertical line is read as "given", so Prob(evidence | not guilty) means the probability that the evidence would be observed given that the suspect was not guilty.
(2) Another potential confusion in this area is the relationship between odds and probability. The relationship between the probability of an event occurring, and the odds on the event can be illustrated by an example. If the probability of being right-handed is 0.9, then the probability of being not being right-handed is 0.1. That means that 9 people out of 10 are right-handed, and one person in 10 is not. In other words for every person who is not right-handed there are 9 who are right-handed. Thus the odds that a randomly-selected person is right-handed are 9 to 1. In symbols this can be written
\[ \mathrm{probability=\frac{odds}{1 + odds}} \]
In the example, the odds on being right-handed are 9 to 1, so the probability of being right-handed is 9 / (1+9) = 0.9.
Conversely,
\[ \mathrm{odds =\frac{probability}{1 – probability}} \]
In the example, the probability of being right-handed is 0.9, so the odds of being right-handed are 0.9 / (1 – 0.9) = 0.9 / 0.1 = 9 (to 1).
With these preliminaries out of the way, we can proceed to the problem.
The legal problem
The first problem lies in the fact that the answer depends on Bayes' theorem. Although that was published in 1763, statisticians are still arguing about how it should be used to this day. In fact whenever it's mentioned, statisticians tend to revert to internecine warfare, and forget about the user.
Bayes' theorem can be stated in words as follows
\[ \mathrm{\text{posterior odds ratio} = \text{prior odds ratio} \times \text{likelihood ratio}} \]
"Posterior odds ratio" means the odds that the person is guilty, relative to the odds that they are innocent, in the light of the evidence, and that's clearly what one wants to know. The "prior odds" are the odds that the person was guilty before any evidence was produced, and that is the really contentious bit.
Sometimes the need to specify the prior odds has been circumvented by using the likelihood ratio alone, but, as shown below, that isn't a good solution.
The analogy with the use of screening tests to detect disease is illuminating.
A particularly straightforward application of Bayes' theorem is in screening people to see whether or not they have a disease. It turns out, in many cases, that screening gives a lot more wrong results (false positives) than right ones. That's especially true when the condition is rare (the prior odds that an individual suffers from the condition is small). The process of screening for disease has a lot in common with the screening of suspects for guilt. It matters because false positives in court are disastrous.
The screening problem is dealt with in sections 1 and 2 of my paper. or on this blog (and here). A bit of animation helps the slides, so you may prefer the Youtube version.
The rest of my paper applies similar ideas to tests of significance. In that case the prior probability is the probability that there is in fact a real effect, or, in the legal case, the probability that the suspect is guilty before any evidence has been presented. This is the slippery bit of the problem both conceptually, and because it's hard to put a number on it.
But the examples below show that to ignore it, and to use the likelihood ratio alone, could result in many miscarriages of justice.
In the discussion of tests of significance, I took the view that it is not legitimate (in the absence of good data to the contrary) to assume any prior probability greater than 0.5. To do so would presume you know the answer before any evidence was presented. In the legal case a prior probability of 0.5 would mean assuming that there was a 50:50 chance that the suspect was guilty before any evidence was presented. A 50:50 probability of guilt before the evidence is known corresponds to a prior odds ratio of 1 (to 1) If that were true, the likelihood ratio would be a good way to represent the evidence, because the posterior odds ratio would be equal to the likelihood ratio.
It could be argued that 50:50 represents some sort of equipoise, but in the example below it is clearly too high, and if it is less that 50:50, use of the likelihood ratio runs a real risk of convicting an innocent person.
The following example is modified slightly from section 3 of a book chapter by Mortera and Dawid (2008). Philip Dawid is an eminent statistician who has written a lot about probability and the law, and he's a member of the legal group of the Royal Statistical Society.
My version of the example removes most of the algebra, and uses different numbers.
Example: The island problem
The "island problem" (Eggleston 1983, Appendix 3) is an imaginary example that provides a good illustration of the uses and misuses of statistical logic in forensic identification.
A murder has been committed on an island, cut off from the outside world, on which 1001 (= N + 1) inhabitants remain. The forensic evidence at the scene consists of a measurement, x, on a "crime trace" characteristic, which can be assumed to come from the criminal. It might, for example, be a bit of the DNA sequence from the crime scene.
Say, for the sake of example, that the probability of a random member of the population having characteristic x is P = 0.004 (i.e. 0.4% ), so the probability that a random member of the population does not have the characteristic is 1 – P = 0.996. The mainland police arrive and arrest a random islander, Jack. It is found that Jack matches the crime trace. There is no other relevant evidence.
How should this match evidence be used to assess the claim that Jack is the murderer? We shall consider three arguments that have been used to address this question. The first is wrong. The second and third are right. (For illustration, we have taken N = 1000, P = 0.004.)
(1) Prosecutor's fallacy
Prosecuting counsel, arguing according to his favourite fallacy, asserts that the probability that Jack is guilty is 1 – P , or 0.996, and that this proves guilt "beyond a reasonable doubt".
The probability that Jack would show characteristic x if he were not guilty would be 0.4% i.e. Prob(Jack has x | not guilty) = 0.004. Therefore the probability of the evidence, given that Jack is guilty, Prob(Jack has x | Jack is guilty), is one 1 – 0.004 = 0.996.
But this is Prob(evidence | guilty) which is not what we want. What we need is the probability that Jack is guilty, given the evidence, P(Jack is guilty | Jack has characteristic x).
To mistake the latter for the former is the prosecutor's fallacy, or the error of the transposed conditional.
Dawid gives an example that makes the distinction clear.
"As an analogy to help clarify and escape this common and seductive confusion, consider the difference between "the probability of having spots, if you have measles" -which is close to 1 and "the probability of having measles, if you have spots" -which, in the light of the many alternative possible explanations for spots, is much smaller."
(2) Defence counter-argument
Counsel for the defence points out that, while the guilty party must have characteristic x, he isn't the only person on the island to have this characteristic. Among the remaining N = 1000 innocent islanders, 0.4% have characteristic x, so the number who have it will be NP = 1000 x 0.004 = 4 . Hence the total number of islanders that have this characteristic must be 1 + NP = 5 . The match evidence means that Jack must be one of these 5 people, but does not otherwise distinguish him from any of the other members of it. Since just one of these is guilty, the probability that this is Jack is thus 1/5, or 0.2— very far from being "beyond all reasonable doubt".
(3) Bayesian argument
The probability of the having characteristic x (the evidence) would be Prob(evidence | guilty) = 1 if Jack were guilty, but if Jack were not guilty it would be 0.4%, i.e. Prob(evidence | not guilty) = P. Hence the likelihood ratio in favour of guilt, on the basis of the evidence, is
\[ LR=\frac{\text{Prob(evidence } | \text{ guilty})}{\text{Prob(evidence }|\text{ not guilty})} = \frac{1}{P}=250 \]
In words, the evidence would be 250 times more probable if Jack were guilty than if he were innocent. While this seems strong evidence in favour of guilt, it still does not tell us what we want to know, namely the probability that Jack is guilty in the light of the evidence: Prob(guilty | evidence), or, equivalently, the odds ratio -the odds of guilt relative to odds of innocence, given the evidence,
To get that we must multiply the likelihood ratio by the prior odds on guilt, i.e. the odds on guilt before any evidence is presented. It's often hard to get a numerical value for this. But in our artificial example, it is possible. We can argue that, in the absence of any other evidence, Jack is no more nor less likely to be the culprit than any other islander, so that the prior probability of guilt is 1/(N + 1), corresponding to prior odds on guilt of 1/N.
We can now apply Bayes's theorem to obtain the posterior odds on guilt:
\[ \text {posterior odds} = \text{prior odds} \times LR = \left ( \frac{1}{N}\right ) \times \left ( \frac{1}{P} \right )= 0.25 \]
Thus the odds of guilt in the light of the evidence are 4 to 1 against. The corresponding posterior probability of guilt is
\[ Prob( \text{guilty } | \text{ evidence})= \frac{1}{1+NP}= \frac{1}{1+4}=0.2 \]
This is quite small –certainly no basis for a conviction.
This result is exactly the same as that given by the Defence Counter-argument', (see above). That argument was simpler than the Bayesian argument. It didn't explicitly use Bayes' theorem, though it was implicit in the argument. The advantage of using the former is that it looks simpler. The advantage of the explicitly Bayesian argument is that it makes the assumptions more clear.
In summary The prosecutor's fallacy suggested, quite wrongly, that the probability that Jack was guilty was 0.996. The likelihood ratio was 250, which also seems to suggest guilt, but it doesn't give us the probability that we need. In stark contrast, the defence counsel's argument, and equivalently, the Bayesian argument, suggested that the probability of Jack's guilt as 0.2. or odds of 4 to 1 against guilt. The potential for wrong conviction is obvious.
Although this argument uses an artificial example that is simpler than most real cases, it illustrates some important principles.
(1) The likelihood ratio is not a good way to evaluate evidence, unless there is good reason to believe that there is a 50:50 chance that the suspect is guilty before any evidence is presented.
(2) In order to calculate what we need, Prob(guilty | evidence), you need to give numerical values of how common the possession of characteristic x (the evidence) is the whole population of possible suspects (a reasonable value might be estimated in the case of DNA evidence), We also need to know the size of the population. In the case of the island example, this was 1000, but in general, that would be hard to answer and any answer might well be contested by an advocate who understood the problem.
These arguments lead to four conclusions.
(1) If a lawyer uses the prosecutor's fallacy, (s)he should be told that it's nonsense.
(2) If a lawyer advocates conviction on the basis of likelihood ratio alone, s(he) should be asked to justify the implicit assumption that there was a 50:50 chance that the suspect was guilty before any evidence was presented.
(3) If a lawyer uses Defence counter-argument, or, equivalently, the version of Bayesian argument given here, (s)he should be asked to justify the estimates of the numerical value given to the prevalence of x in the population (P) and the numerical value of the size of this population (N). A range of values of P and N could be used, to provide a range of possible values of the final result, the probability that the suspect is guilty in the light of the evidence.
(4) The example that was used is the simplest possible case. For more complex cases it would be advisable to ask a professional statistician. Some reliable people can be found at the Royal Statistical Society's section on Statistics and the Law.
If you do ask a professional statistician, and they present you with a lot of mathematics, you should still ask these questions about precisely what assumptions were made, and ask for an estimate of the range of uncertainty in the value of Prob(guilty | evidence) which they produce.
Postscript: real cases
Another paper by Philip Dawid, Statistics and the Law, is interesting because it discusses some recent real cases: for example the wrongful conviction of Sally Clark because of the wrong calculation of the statistics for Sudden Infant Death Syndrome.
On Monday 21 March, 2016, Dr Waney Squier was struck off the medical register by the General Medical Council because they claimed that she misrepresented the evidence in cases of Shaken Baby Syndrome (SBS).
This verdict was questioned by many lawyers, including Michael Mansfield QC and Clive Stafford Smith, in a letter. "General Medical Council behaving like a modern inquisition"
The latter has already written "This shaken baby syndrome case is a dark day for science – and for justice"..
The evidence for SBS is based on the existence of a triad of signs (retinal bleeding, subdural bleeding and encephalopathy). It seems likely that these signs will be present if a baby has been shake, i.e Prob(triad | shaken) is high. But this is irrelevant to the question of guilt. For that we need Prob(shaken | triad). As far as I know, the data to calculate what matters are just not available.
It seem that the GMC may have fallen for the prosecutor's fallacy. Or perhaps the establishment won't tolerate arguments. One is reminded, once again, of the definition of clinical experience: "Making the same mistakes with increasing confidence over an impressive number of years." (from A Sceptic's Medical Dictionary by Michael O'Donnell. A Sceptic's Medical Dictionary BMJ publishing, 1997).
Appendix (for nerds). Two forms of Bayes' theorem
The form of Bayes' theorem given at the start is expressed in terms of odds ratios. The same rule can be written in terms of probabilities. (This was the form used in the appendix of my paper.) For those interested in the details, it may help to define explicitly these two forms.
In terms of probabilities, the probability of guilt in the light of the evidence (what we want) is
\[ \text{Prob(guilty } | \text{ evidence}) = \text{Prob(evidence } | \text{ guilty}) \frac{\text{Prob(guilty })}{\text{Prob(evidence })} \]
In terms of odds ratios, the odds ratio on guilt, given the evidence (which is what we want) is
\[ \frac{ \text{Prob(guilty } | \text{ evidence})} {\text{Prob(not guilty } | \text{ evidence}} =
\left ( \frac{ \text{Prob(guilty)}} {\text {Prob((not guilty)}} \right )
\left ( \frac{ \text{Prob(evidence } | \text{ guilty})} {\text{Prob(evidence } | \text{ not guilty}} \right ) \]
or, in words,
\[ \text{posterior odds of guilt } =\text{prior odds of guilt} \times \text{likelihood ratio} \]
This is the precise form of the equation that was given in words at the beginning.
A derivation of the equivalence of these two forms is sketched in a document which you can download.
It's worth pointing out the following connection between the legal argument (above) and tests of significance.
(1) The likelihood ratio works only when there is a 50:50 chance that the suspect is guilty before any evidence is presented (so the prior probability of guilt is 0.5, or, equivalently, the prior odds ratio is 1).
(2) The false positive rate in signiifcance testing is close to the P value only when the prior probability of a real effect is 0.5, as shown in section 6 of the P value paper.
However there is another twist in the significance testing argument. The statement above is right if we take as a positive result any P < 0.05. If we want to interpret a value of P = 0.047 in a single test, then, as explained in section 10 of the P value paper, we should restrict attention to only those tests that give P close to 0.047. When that is done the false positive rate is 26% even when the prior is 0.5 (and much bigger than 30% if the prior is smaller –see extra Figure), That justifies the assertion that if you claim to have discovered something because you have observed P = 0.047 in a single test then there is a chance of at least 30% that you'll be wrong. Is there, I wonder, any legal equivalent of this argument?
Tagged Clive Stafford Smith, false conviction, false discovery rate, False positive risk, false positives, FPR, Law, lawyers, Michael Mansfield, Philip Dawid, Squier, statistics, Waney Squier | 10 Comments
Most alternative medicine is illegal
Jump to follow-up
I'm perfectly happy to think of alternative medicine as being a voluntary, self-imposed tax on the gullible (to paraphrase Goldacre again). But only as long as its practitioners do no harm and only as long as they obey the law of the land. Only too often, though, they do neither.
When I talk about law, I don't mean lawsuits for defamation. Defamation suits are what homeopaths and chiropractors like to use to silence critics. heaven knows, I've becomes accustomed to being defamed by people who are, in my view. fraudsters, but lawsuits are not the way to deal with it.
I'm talking about the Trading Standards laws Everyone has to obey them, and in May 2008 the law changed in a way that puts the whole health fraud industry in jeopardy.
The gist of the matter is that it is now illegal to claim that a product will benefit your health if you can't produce evidence to justify the claim.
I'm not a lawyer, but with the help of two lawyers and a trading standards officer I've attempted a summary. The machinery for enforcing the law does not yet work well, but when it does, there should be some very interesting cases.
The obvious targets are homeopaths who claim to cure malaria and AIDS, and traditional Chinese Medicine people who claim to cure cancer.
But there are some less obvious targets for prosecution too. Here is a selection of possibilities to savour..
Universities such as Westminster, Central Lancashire and the rest, which promote the spreading of false health claims
Hospitals, like the Royal London Homeopathic Hospital, that treat patients with mistletoe and marigold paste. Can they produce any real evidence that they work?
Edexcel, which sets examinations in alternative medicine (and charges for them)
Ofsted and the QCA which validate these exams
Skills for Health and a whole maze of other unelected and unaccountable quangos which offer "national occupational standards" in everything from distant healing to hot stone therapy, thereby giving official sanction to all manner of treatments for which no plausible evidence can be offered.
The Prince of Wales Foundation for Integrated Health, which notoriously offers health advice for which it cannot produce good evidence
Perhaps even the Department of Health itself, which notoriously referred to "psychic surgery" as a profession, and which has consistently refused to refer dubious therapies to NICE for assessment.
The law, insofar as I've understood it, is probably such that only the first three or four of these have sufficient commercial elements for there to be any chance of a successful prosecution. That is something that will eventually have to be argued in court.
But lecanardnoir points out in his comment below that The Prince of Wales is intending to sell herbal concoctions, so perhaps he could end up in court too.
The laws
We are talking about The Consumer Protection from Unfair Trading Regulations 2008. The regulations came into force on 26 May 2008. The full regulations can be seen here, or download pdf file. They can be seen also on the UK Statute Law Database.
The Office of Fair Trading, and Department for Business, Enterprise & Regulatory Reform (BERR) published Guidance on the Consumer Protection from Unfair Trading Regulations 2008 (pdf file),
Statement of consumer protection enforcement principles (pdf file), and
The Consumer Protection from Unfair Trading Regulations: a basic guide for business (pdf file).
Has The UK Quietly Outlawed "Alternative" Medicine?
On 26 September 2008, Mondaq Business Briefing published this article by a Glasgow lawyer, Douglas McLachlan. (Oddly enough, this article was reproduced on the National Center for Homeopathy web site.)
"Proponents of the myriad of forms of alternative medicine argue that it is in some way "outside science" or that "science doesn't understand why it works". Critical thinking scientists disagree. The best available scientific data shows that alternative medicine simply doesn't work, they say: studies repeatedly show that the effect of some of these alternative medical therapies is indistinguishable from the well documented, but very strange "placebo effect" "
"Enter The Consumer Protection from Unfair Trading Regulations 2008(the "Regulations"). The Regulations came into force on 26 May 2008 to surprisingly little fanfare, despite the fact they represent the most extensive modernisation and simplification of the consumer protection framework for 20 years."
The Regulations prohibit unfair commercial practices between traders and consumers through five prohibitions:-
General Prohibition on Unfair Commercial
Practices (Regulation 3)
Prohibition on Misleading Actions (Regulations 5)
Prohibition on Misleading Omissions (Regulation 6)
Prohibition on Aggressive Commercial Practices (Regulation 7)
Prohibition on 31 Specific Commercial Practices that are in all Circumstances Unfair (Schedule 1). One of the 31 commercial practices which are in all circumstances considered unfair is "falsely claiming that a product is able to cure illnesses, dysfunction or malformations". The definition of "product" in the Regulations includes services, so it does appear that all forms medical products and treatments will be covered.
Just look at that!
One of the 31 commercial practices which are in all circumstances considered unfair is "falsely claiming that a product is able to cure illnesses, dysfunction or malformations"
Section 5 is equally powerful, and also does not contain the contentious word "cure" (see note below)
Misleading actions
5.—(1) A commercial practice is a misleading action if it satisfies the conditions in either paragraph (2) or paragraph (3).
(2) A commercial practice satisfies the conditions of this paragraph—
(a) if it contains false information and is therefore untruthful in relation to any of the matters in paragraph (4) or if it or its overall presentation in any way deceives or is likely to deceive the average consumer in relation to any of the matters in that paragraph, even if the information is factually correct; and
(b) it causes or is likely to cause the average consumer to take a transactional decision he would not have taken otherwise.
These laws are very powerful in principle, But there are two complications in practice.
One complication concerns the extent to which the onus has been moved on to the seller to prove the claims are true, rather than the accuser having to prove they are false. That is a lot more favourable to the accuser than before, but it's complicated.
The other complication concerns enforcement of the new laws, and at the moment that is bad.
Who has to prove what?
That is still not entirely clear. McLachlan says
"If we accept that mainstream evidence based medicine is in some way accepted by mainstream science, and alternative medicine bears the "alternative" qualifier simply because it is not supported by mainstream science, then where does that leave a trader who seeks to refute any allegation that his claim is false?
Of course it is always open to the trader to show that his the alternative therapy actually works, but the weight of scientific evidence is likely to be against him."
On the other hand, I'm advised by a Trading Standards Officer that "He doesn't have to refute anything! The prosecution have to prove the claims are false". This has been confirmed by another Trading Standards Officer who said
"It is not clear (though it seems to be) what difference is implied between "cure" and "treat", or what evidence is required to demonstrate that such a cure is false "beyond reasonable doubt" in court. The regulations do not provide that the maker of claims must show that the claims are true, or set a standard indicating how such a proof may be shown."
The main defence against prosecution seems to be the "Due diligence defence", in paragraph 17.
Due diligence defence
17. —(1) In any proceedings against a person for an offence under regulation 9, 10, 11 or 12 it is a defence for that person to prove—
(a) that the commission of the offence was due to—
(i) a mistake;
(ii) reliance on information supplied to him by another person;
(iii) the act or default of another person;
(iv) an accident; or
(v) another cause beyond his control; and
(b) that he took all reasonable precautions and exercised all due diligence to avoid the commission of such an offence by himself or any person under his control.
If "taking all reasonable precautions" includes being aware of the lack of any good evidence that what you are selling is effective, then this defence should not be much use for most quacks.
Douglas McLachlan has clarified, below, this difficult question
False claims for health benefits of foods
A separate bit of legislation, European regulation on nutrition and health claims made on food, ref 1924/2006, in Article 6, seems clearer in specifying that the seller has to prove any claims they make.
Scientific substantiation for claims
1. Nutrition and health claims shall be based on and substantiated by generally accepted scientific evidence.
2. A food business operator making a nutrition or health claim shall justify the use of the claim.
3. The competent authorities of the Member States may request a food business operator or a person placing a product on the market to produce all relevant elements and data establishing compliance with this Regulation.
That clearly places the onus on the seller to provide evidence for claims that are made, rather than the complainant having to 'prove' that the claims are false.
On the problem of "health foods" the two bits of legislation seem to overlap. Both have been discussed in "Trading regulations and health foods", an editorial in the BMJ by M. E. J. Lean (Professor of Human Nutrition in Glasgow).
"It is already illegal under food labelling regulations (1996) to claim that food products can treat or prevent disease. However, huge numbers of such claims are still made, particularly for obesity "
"The new regulations provide good legislation to protect vulnerable consumers from misleading "health food" claims. They now need to be enforced proactively to help direct doctors and consumers towards safe, cost effective, and evidence based management of diseases."
In fact the European Food Standards Agency (EFSA) seems to be doing a rather good job at imposing the rules. This, predictably, provoked howls of anguish from the food industry There is a synopsis here.
"Of eight assessed claims, EFSA's Panel on Dietetic Products, Nutrition and Allergies (NDA) rejected seven for failing to demonstrate causality between consumption of specific nutrients or foods and intended health benefits. EFSA has subsequently issued opinions on about 30 claims with seven drawing positive opinions."
". . . EFSA in disgust threw out 120 dossiers supposedly in support of nutrients seeking addition to the FSD's positive list.
If EFSA was bewildered by the lack of data in the dossiers, it needn't hav been as industry freely admitted it had in many cases submitted such hollow documents to temporarily keep nutrients on-market."
Or, on another industry site, "EFSA's harsh health claim regime"
"By setting an unworkably high standard for claims substantiation, EFSA is threatening R&D not to mention health claims that have long been officially approved in many jurisdictions."
Here, of course,"unworkably high standard" just means real genuine evidence. How dare they ask for that!
Enforcement of the law
Article 19 of the Unfair Trading regulations says
19. —(1) It shall be the duty of every enforcement authority to enforce these Regulations.
(2) Where the enforcement authority is a local weights and measures authority the duty referred to in paragraph (1) shall apply to the enforcement of these Regulations within the authority's area.
Nevertheless, enforcement is undoubtedly a weak point at the moment. The UK is obliged to enforce these laws, but at the moment it is not doing so effectively.
A letter in the BMJ from Rose & Garrow describes two complaints under the legislation in which it appears that a Trading Standards office failed to enforce the law. They comment
" . . . member states are obliged not only to enact it as national legislation but to enforce it. The evidence that the government has provided adequate resources for enforcement, in the form of staff and their proper training, is not convincing. The media, and especially the internet, are replete with false claims about health care, and sick people need protection. All EU citizens have the right to complain to the EU Commission if their government fails to provide that protection."
This is not a good start. A lawyer has pointed out to me
"that it can sometimes be very difficult to get Trading Standards or the OFT to take an interest in something that they don't fully understand. I think that if it doesn't immediately leap out at them as being false (e.g "these pills cure all forms of cancer") then it's going to be extremely difficult. To be fair, neither Trading Standards nor the OFT were ever intended to be medical regulators and they have limited resources available to them. The new Regulations are a useful new weapon in the fight against quackery, but they are no substitute for proper regulation."
Trading Standards originated in Weights and Measures. It was their job to check that your pint of beer was really a pint. Now they are being expected to judge medical controversies. Either they will need more people and more training, or responsibility for enforcement of the law should be transferred to some more appropriate agency (though one hesitates to suggest the MHRA after their recent pathetic performance in this area).
Who can be prosecuted?
Any "trader", a person or a company. There is no need to have actually bought anything, and no need to have suffered actual harm. In fact there is no need for there to be a complainant at all. Trading standards officers can act on their own. But there must be a commercial element. It's unlikely that simply preaching nonsense would be sufficient to get you prosecuted, so the Prince of Wales is, sadly, probably safe.
Universities who teach that "Amethysts emit high Yin energy" make an interesting case. They charge fees and in return they are "falsely claiming that a product is able to cure illnesses".
In my view they are behaving illegally, but we shan't know until a university is taken to court. Watch this space.
The fact remains that the UK is obliged to enforce the law and presumably it will do so eventually. When it does, alternative medicine will have to change very radically. If it were prevented from making false claims, there would be very little of it left apart from tea and sympathy
New Zealand must have similar laws.
Just as I was about to post this I found that in New Zealand a
"couple who sold homeopathic remedies claiming to cure bird flu, herpes and Sars (severe acute respiratory syndrome) have been convicted of breaching the Fair Trading Act."
They were ordered to pay fines and court costs totalling $23,400.
A clarification form Douglas McLachlan
On the difficult question of who must prove what, Douglas McLachlan, who wrote Has The UK Quietly Outlawed "Alternative" Medicine?, has kindly sent the following clarification.
"I would agree that it is still for the prosecution to prove that the trader committed the offence beyond a reasonable doubt, and that burden of proof is always on the prosecution at the outset, but I think if a trader makes a claim regarding his product and best scientific evidence available indicates that that claim is false, then it will be on the trader to substantiate the claim in order to defend himself. How will the trader do so? Perhaps the trader might call witness after witness in court to provide anecdotal evidence of their experiences, or "experts" that support their claim – in which case it will be for the prosecution to explain the scientific method to the Judge and to convince the Judge that its Study evidence is to be preferred.
Unfortunately, once human personalities get involved things could get clouded – I could imagine a small time seller of snake oil having serious difficulty, but a well funded homeopathy company engaging smart lawyers to quote flawed studies and lead anecdotal evidence to muddy the waters just enough for a Judge to give the trader the benefit of the doubt. That seems to be what happens in the wider public debate, so it's easy to envisage it happening a courtroom."
The "average consumer".
The regulations state
(3) A commercial practice is unfair if—
(a) it contravenes the requirements of professional diligence; and
(b) it materially distorts or is likely to materially distort the economic behaviour of the average consumer with regard to the product.
It seems,therefore, that what matters is whether the "average consumer" would infer from what is said that a claim was being made to cure a disease. The legal view cited by Mojo (comment #2, below) is that expressions such as "can be used to treat" or "can help with" would be considered by the average consumer as implying successful treatment or cure.
The drugstore detox delusion. A nice analysis "detox" at .Science-based Pharmacy
Tagged Academia, alternative medicine, Anti-science, antiscience, CAM, Central Lancashire, chiropractic, Fair trading, herbalism, homeopathy, Law, New Zealand, nutribollocks, nutrition, nutritional therapy, Prince's Foundation, Trading Standards, Unfair Trading, Universities, Westminster university | 30 Comments
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communications medicine
A stakeholder group assessment of interactions between child health and the sustainable development goals in Cambodia
Decision-making fitness of methods to understand Sustainable Development Goal interactions
Lorenzo Di Lucia, Raphael Slade & Jamil Khan
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Daniel Helldén ORCID: orcid.org/0000-0001-8969-21941,
Thy Chea2,
Serey Sok3,
Linn Järnberg4,
Helena Nordenstedt1,
Göran Tomson5,6,
Måns Nilsson4,7 &
Tobias Alfvén1,8
Communications Medicine volume 2, Article number: 68 (2022) Cite this article
With the implementation of the Sustainable Development Goals, a systematic assessment of how the goals influence child health and vice versa has been lacking. We aimed to contribute to such an assessment by investigating the interactions between child health and the Sustainable Development Goals in Cambodia.
Based on the SDG Synergies approach, 272 interactions between 16 Cambodian Sustainable Development Goals and child health were evaluated by an interdisciplinary Cambodian stakeholder group. From this a cross-impact matrix was derived and network analysis applied to determine first and second-order effects of the interactions with a focus on child health.
We show that with the exception of Cambodian Sustainable Development Goal 15 (life on land) the interactions are perceived to be synergistic between the child health and the Cambodian Sustainable Development Goals, and progress on Cambodian Sustainable Development Goal 16 (peace, justice and strong institutions) could have the largest potential to contribute to the achievement of the Cambodian Sustainable Development Goals, both when it comes to first and second-order interactions.
In this stakeholder assessment, our findings provide novel insights on how complex relationships play out at the country level and highlight important synergies and trade-offs, vital for accelerating the work toward the betterment of child health and achieving the Sustainable Development Goals.
Plain language summary
The Sustainable Development Goals (SDGs) are a set of 17 global goals set by the United Nations to guide the world toward development that meets the needs of the present without compromising the ability of future generations to meet their own needs. The efforts to achieve the different SDGs are interconnected. To better understand in what way, a group with different expertize and perspectives was assembled in Cambodia to score the linkages between the SDGs and child health. This identified that most goals promote better child health and that advancements in child health also help achieve the SDGs in Cambodia. Our study provides useful knowledge and a practical approach for policy makers trying to accelerate the work toward better child health in Cambodia.
The world has experienced an impressive decline in global child mortality over the last decades, however still 5.2 million children die before they reach their 5th birthday1. The 17 Sustainable Development Goals (SDGs) represent the global community's most comprehensive and people-centered set of universal targets to date that have been endorsed by governments2. The health and well-being of children stand to benefit, stagnate, or regress depending on progress in other sectors of society toward the attainment of the SDGs. It is nearly impossible to untangle the health of children from their social, natural and economic environments3. For example, it has been demonstrated that approximately half of the reduction in under-five mortality between 1990 and 2010 can be attributed to investments outside of the health sector4.
Without losing sight of the unfinished progress on reducing global child mortality, the global strategy for women´s, children´s and adolescents' health implore researchers and decision makers to aspire beyond a world in which all children not only survive but thrive in order to realize their potential to transform communities5. Social, economic, political, environmental, and cultural determinants have important effects on child health6,7,8 while the survival, health and well-being of children are crucial to reach multiple sustainable development outcomes9,10.
The SDGs are presented in the 2030 Agenda as integrated, indivisible and interdependent and can be seen as a large system of goals that interact and affect each other directly and indirectly. However, the 2030 Agenda does not attempt to identify or characterize the interactions. A field of SDG interactions studies has emerged where a range of mostly quantitative methods have been applied to try to distinguish these interactions and subsequent network effects11. One such method, the SDG Synergies, a semi-quantitative participatory approach originally developed by the International Science Council and the Stockholm Environment Institute12,13,14 can be used for untangling the direct and indirect effects of interactions between the SDGs. Through the scoring of relevant interactions by a multidisciplinary stakeholder group, the method allows for context-specific analysis of interactions since these vary in position and nature depending on the context within which the interaction occurs15. Furthermore, the framework can serve as a basis for more complex analysis and visualization of the interactions through network analysis14,16. The approach has previously been applied in a variety of policy contexts, ranging from global policy issues such as climate change to interactions within a specific country12,14,16,17,18,19. Using this approach, Blomstedt et al.20 showed that several SDGs, including SDG 1 (no poverty), 2 (zero hunger), 4 (quality education), 5 (gender equality), 8 (decent work and economic growth and 17 (partnership for the goals) have strong and reciprocal links with child health. Their theoretical analysis also suggested that multisectoral collaboration on some targets are essential for sustainable progress on child health, while it found few negative interactions indicating the limited number of trade-offs with health. The method quantifies expert opinions through the scoring of the interactions, and although the subjectivity of the SDG Synergies approach can be in contrast to the classical paradigm of rational and data driven decision making21,22, real world prioritization processes are influenced by many different factors and biases16,23,24. To some extent, the SDG Synergies approach integrates real world human behavior into prioritization and decision making models which is necessary for understanding complex context dependent systems24,25, forming a bridge across sectors and promoting evidence informed policy, particularly given the absence of quality quantitative data to assess the SDGs26.
Cambodia was among the few low- and middle-income countries that achieved the Millennium Development Goal 4 and reduced the under-five mortality from 116 to 27 deaths per 1000 live births between 1990 and 20191,27. However, an estimated 12,000 children still die from preventable causes every year and mortality rates among low income, less educated and more rural populations have not declined as much1,28. Investments outside of the health sector in education, nutrition, water and sanitation, and poverty reduction measures together with multisectoral planning and collaborative initiatives between non-health sectors have been key to the betterment of child health in Cambodia29,30. However, multidimensional poverty and non-monetary deprivation such as overcrowded housing, suboptimal water and sanitation facilities and lack of school attendance are still prevalent with almost half of all children under 18 years of age experienced three or more deprivations in 201831. The development and adoption of the Cambodia Sustainable Development Goals (CSDGs) with its 18 goals and 88 targets offers a comprehensive framework for sustainable development localized to the country context and holds the promise of delivering for children in Cambodia32. The country has improved the health and well-being of children, however the role of different sectors in this achievement has not been systematically assessed. Furthermore, the interactions between the SDGs and child health have not been examined at a country level before. The aim of this study was therefore to contribute to such an assessment by determining the strength, position and nature of interactions between the SDGs and child health in Cambodia. We show that with the exception of CSDG 15 (life on land) the interactions are perceived to be synergistic between the child health and the CSDG, and progress on CSDG 16 (peace, justice and strong institutions) could have the largest potential to contribute to the achievement of the CSDGs, both when it comes to first and second-order interactions.
The semi-quantitative SDG Synergies approach14, applied to the Cambodia national-level context and with a primary focus on child health was utilized in this study. In brief, the SDG Synergies approach follows three overarching stages that enable the investigation of the strength, position and nature of interactions in a network, as outlined below. Further, we provide some additional analysis to ground the results in the country context.
Identification of goals
Between the 169 targets of the SDGs there are almost 300,000 possible pairwise interactions, hence the first step is to limit the scope of the analysis and select the goals or targets of interest. Through matching SDG priorities with national developmental goals, ministry consultations and investigations into possible data sources, the Royal Government of Cambodia has put forward the CSDGs as 18 nationalized goals and a localized set of 88 targets from the 2030 Agenda. On a goal level, the CSDGs include one additional goal (number 18) on the ending of the negative impact of Mine/Explosive remnants of war (ERW) and promote victim assistance, while the targets for each goal are fewer but designed so that data indicators can be obtained to measure the progress toward the targets32. Guided by CSDGs32, the analysis done by Blomstedt et al. 20 and the relevant SDG targets identified by UNICEF33 as well as in-depth discussions within the research team and with local partners to ensure relevancy to the Cambodian context, it was considered most adequate to include all CSDGs with the exception of CSDG 17 (partnerships for the goals) since the goal was deemed too broad for meaningful assessment. It was further decided to limit CSDG 3 (good health and well-being) to only representing child health, which we defined as a state of complete physical, mental and social well-being and not merely the absence of disease or infirmity among human beings below 18 years. The list of CSDGs and their definitions are detailed in Table 1. The selection led to a total of 17 goals, translating into 272 interactions.
Table 1 List of included Cambodia sustainable development goals and their definitions.
Assessing the interactions
Over a 2-day workshop on the 24–25th of August 2020, taking place in Phnom Penh, 29 participants representing a range of governmental and non-governmental stakeholders (see Supplementary Table 1) assessed the interactions between the selected goals, taking advantage of the breadth of country expertize. The participants were purposively selected based on predefined criteria of having either expertize in child health in Cambodia, or being from a non-health sector (for example water and sanitation, agriculture, infrastructure etc.) reflecting the social, economic, political, environmental, and cultural determinants of health and working in a capacity that includes multisectoral collaboration in the country.
Based on the SDG Synergies approach, groups of 5–6 people discussed direct interactions between pairs of goals, by answering a guiding question: "In the Cambodia context, if progress is made on Goal X, how does this influence progress on Goal Y?". The group arrived at a score according to the Weimer-Jehle seven-point scale34, which ranges from strongly restricting (−3) to strongly promoting (+3). The participants also recorded a 1–2 sentence motivation for the score. The exercise was held in Khmer, official published Khmer CSDG descriptions of goals and targets were used and all documents were translated and back-translated for validity. As a basis for scoring, the participants used their expert and working knowledge, as well as a fact sheet for each goal with descriptions of the associated targets and key statistics derived from the latest Cambodia Sustainable Development Report35. It was emphasized that the participants should think about child health in a broad perspective, in line with the definition in Table 1, and not only on child mortality. After the first scoring of interactions, the groups double-checked their own scoring and also verified a set of interactions originally scored by another group. All identified discrepancies and differences were discussed in plenary session, where final scores were arrived at in consensus.
Cross-impact matrix and network analysis
All scores were directly entered into a tailor-made digital software36 developed by the Stockholm Environment Institute, which also includes the statistical analysis features outlined below. From the final scoring of all interactions, a cross-impact matrix was developed, which served to illustrate the results and was the basis for applying network analysis. By utilizing a cross-impact matrix and keeping the analysis at the goal level, a whole of 2030 Agenda approach to child health and SDG interactions in Cambodia could be achieved. While the data presented in the cross-impact matrix provides information on the frequency of different types of interactions and how different goals influence the overall agenda, network analysis methods can be used to assert more systemic properties of the interactions. By using network analysis, where a goal is considered a node (N) and the interaction is considered a link (L) and the subsequent network can be described as G = (N, L), the network can be visualized, clusters of more related goals highlighted, and the impact of certain goals and/or interactions more clearly assessed37. Moving beyond the direct interactions that are evident from the cross-impact matrix, analysis of the second-order interactions shows the net influence of a certain goal on the network as a whole as well as on other individual goals. Following the method described by Weitz et al. 14, the net influence (I) of a goal (g) on the network as a whole including the second-order interactions was calculated according to [Eq. 1]
$${I}_{g}^{Total}={I}_{g}^{1st}+\sum {I}^{2nd}=\,{D}_{g}^{Out}+\mathop{\sum }\limits_{j\ne g}{I}_{gj}{D}_{j}^{Out}$$
where \({I}_{g}^{1st}\) is the influence of goal g on its closest neighbors, I2nd is the influence from g's neighbor's on their neighbors, \({D}_{g}^{Out}\) is the out-degree of goal g, Ig,j is the strength of link from goal g to goal j, and \({D}_{j}^{Out}\) is the out-degree of goal j. Similarly, the aggregated second-order influence of a goal A on another goal D is estimated by
$${I}_{A\to D}^{2nd}=\mathop{\sum }\limits_{i}{w}_{Ai}{w}_{iD}$$
where I runs over all goals connecting A and D, and wij is the weight on the link between goal I and goal j. A more detailed explanation of the concepts outlined above is available in the Supplementary Methods.
Situating of results
Situating the results from the cross-impact matrix and network analysis is relevant to ground the analysis in the country context. Due to the lack of data on the CSDGs an overview of relevant indicators for the SDGs are provided in Supplementary Fig. 1 and Supplementary Data 1, which form the basis for a Pearson paired-observational correlation analysis to assess the trends provided in Supplementary Fig. 2. Notably, included variables were re-coded to showcase progress toward the CSDGs similar to other correlation based assessments of SDG interactions38,39. An overview of key developmental and child health policies are further provided in the Supplementary Fig. 3 while the annual budget expenses for each ministry between 2000-2013 is also provided in Supplementary Fig. 4 and Supplementary Fig. 5. All available data on the indicators of the CSDGs and their SDG counterpart as well as the annual budget expenses has been compiled and can be found in the Supplementary Data 1.
The study received ethical approval from the National Ethics Committee for Health Research in Cambodia (NECHR-023) and written informed consent was obtained from all participants.
Cross-impact matrix and first and second-order interactions of the SDGs in Cambodia
The interactions between 17 CSDGs as defined in Table 1 were scored on a seven-point scale from strongly restricting (−3) to strongly promoting (+3) by an interdisciplinary stakeholder group leading to the cross-impact matrix with 272 interactions illustrated in Fig. 1. There is a high frequency of perceived positive interactions (n = 212, 78%) versus negative (n = 12, 4%) and a substantial amount deemed to have no direct influence (n = 48, 18%). The row sum implies the net first order influence of the goal on the network, and the column sum shows how much the goal is directly influenced by all other goals in the network. It stands clear that CSDG 16 (peace, justice and strong institutions) has the largest first order positive influence on the network, with CSDG 11 (sustainable cities and communities) and CSDG 6 (clean water and sanitation) having the second largest direct positive influence. CSDG 1 (no poverty) has the least positive influence on the network, with negative impacts on CSDG 11 (sustainable cities and communities), 12 (responsible consumption and production), 14 (life below water) and 15 (life on land). Conversely, CSDG 1 (no poverty) together with CSDG 8 (decent work and economic growth) and CSDG 3 (child health) is promoted the most by progress on other goals, whereas CSDG 15 (life on land) is promoted the least by progress on other CSDGs. Importantly, neither the row or column sum details whether the perceived influence results from strong influence by a few targets or weak influence by many, or the distribution between positive and negative interactions.
Fig. 1: Cross-impact matrix of the 17 Cambodian Sustainable Development Goals.
Color according to scale. The row sum implies the net influence of the goal on the network, and the column net sum show the how much the goal is influenced by all other goals in the network. Cambodia Sustainable Development Goals 1 no poverty, 2 zero hunger, 3 child health, 4 quality education, 5 gender equality, 6 clean water and sanitation, 7 affordable and clean energy, 8 decent work and economic growth, 9 industry, innovation and infrastructure, 10 reduced inequalities, 11 sustainable cities and communities, 12 responsible consumption and production, 13 climate change, 14 life below water, 15 life on land, 16 peace, justice and strong institutions, and 18 mine/ERW free. The underlying data can be found in Supplementary Data 2.
Expanding the network from only direct first order interactions to second-order interactions, the ranks of the row sums of the goals change as illustrated in Table 2. CSDG 16 is even more clearly perceived as the most positively influencing goal of the network, while CSDG 6 (clean water and sanitation) falls from 2nd to 4th rank and CSDG 5 (gender equality) jumps from 5th to 3rd. A similar movement is made by CSDG 7 (affordable and clean energy) from 12th to 10th, while notably the bottom five goals and child health remain in their rank. The net influence in absolute terms between the ranks is however close.
Table 2 Rank of goals influencing the network based on first and second-order interactions.
The goals and their interactions can be visualized as a network, seen in Fig. 2. Although no clear clusters can be identified, it is yet again emphasized that the goals are closely interlinked but that some goals such as CSDG 7 (affordable and clean energy), 14 (life below water) and 18 (mine/ERW free) are relatively more distant from other goals in the network.
Fig. 2: Illustration of the full network of 17 goals and 272 linkages based on the cross-impact matrix.
Cambodia Sustainable Development Goals 1 no poverty, 2 zero hunger, 3 child health, 4 quality education, 5 gender equality, 6 clean water and sanitation, 7 affordable and clean energy, 8 decent work and economic growth, 9 industry, innovation and infrastructure, 10 reduced inequalities, 11 sustainable cities and communities, 12 responsible consumption and production, 13 climate change, 14 life below water, 15 life on land, 16 peace, justice and strong institutions, and 18 mine/ERW free.
Child health within the network
The CSDGs in general were perceived to have a positive influence on child health in Cambodia and child health directly and positively influences many of the other CSDGs. Specifically, progress on child health was assessed to strongly promote the achievement of CSDG 1 (no poverty), 4 (quality education) and 8 (decent work and economic growth), moderately promote CSDG 10 (reduced inequalities) and 12 (responsible consumption and production) and weakly promote progress toward CSDG 9 (industry, innovation and infrastructure), 11 (sustainable cities and communities) and 16 (peace, justice and strong institutions) (Fig. 3a). The participants assessed that progress on child health does not have any direct influence on the achievement of CSDG 2 (zero hunger), 5 (gender equality), 6 (clean water and sanitation), 7 (affordable and clean energy), 13 (climate change), 14 (life below water), 15 (life on land) and 18 (mine/ERW free) in Cambodia. They acknowledged, however, that there are many second-order influences from child health on the aforementioned goals. On the other hand, child health is deemed to be strongly influenced by CSDG 6 (clean water and sanitation) and 11 (sustainable cities and communities), moderately influenced by CSDG 2 (zero hunger), 4 (quality education), 5 (gender equality), 7 (affordable and clean energy), 8 (decent work and economic growth), 9 (industry, innovation and infrastructure), 10 (reduced inequalities), 12 (responsible consumption and production), 16 (peace, justice and strong institutions) and lastly weakly influenced by CSDG 1 (no poverty), 13 (climate change), 14 (life below water), 15 (life on land) and 18 (Mine/ERW free) in Cambodia (Fig. 3b). Importantly, there does not seem to be any directly restricting interactions.
Fig. 3: The Cambodia Sustainable Development Goals from the perspective of child health.
a, b: (a) Illustrate the first-order influence of child health on the CSDGs and (b) vice versa. c, d: (c) Illustrate the second-order influence of child health on the CSDGs and (d) vice versa. Color according to scale. Cambodia Sustainable Development Goals 1 no poverty, 2 zero hunger, 3 child health, 4 quality education, 5 gender equality, 6 clean water and sanitation, 7 affordable and clean energy, 8 decent work and economic growth, 9 industry, innovation and infrastructure, 10 reduced inequalities, 11 sustainable cities and communities, 12 responsible consumption and production, 13 climate change, 14 life below water, 15 life on land, 16 peace, justice and strong institutions, and 18 mine/ERW free.
The aggregated second-order interactions found in Fig. 3c and d provide some additional insights. First, all goals, in particular CSDG 16 (peace, justice and strong institutions), have a net positive influence on child health when second-order interactions are considered. Secondly, there seems to be a potentially important positive feedback-loop, whereby improving child health in itself lead to the promotion of child health through promoting interactions of other CSDGs. Thirdly, although the second-order interactions are generally positive, they show a negative influence of child health on CSDG 15 (life on land) not showcased before. This implies an important trade-off that must be handled, which could have been overlooked if researchers and policy makers only focus on direct interactions.
This study constitutes the first attempt at an empirical investigation at a national level into the interactions between child health and the SDGs. In general, the interactions were perceived to be synergistic between the child health and the CSDGs, and progress on CSDG 16 (peace, justice and strong institutions) could have the largest potential to contribute to the achievement of the CSDGs, both when it comes to first and second-order interactions in Cambodia. All goals were deemed to positively influence child health in some way and child health was thought to have a promoting influence on the achievement of other goals except for CSDG 15 (life on land). These findings are in line with similar assessment noting the overall positive influence of good health and well-being on the possibility of achieving other sustainable development outcomes12,17,20.
The SDGs and the locally adapted CSDGs offer an overarching framework that encompasses many of the determinants of child health. Within the Cambodian context, our analysis suggests that in line with the literature on the advancements in child mortality, stakeholders perceive that child health is heavily dependent on progress in other sectors. Further, comparing with data on the key indicators that exist (Table 3) showcase that progress on child health has been positively correlated with a number of CSDGs in Cambodia, but that progress has not coincided with progress toward CSDG 12, 13, 14, 15, and 16 (See Supplementary Fig. 2). Interestingly, there were no restricting interactions found at the first or second-order analysis on child health from making progress on any of the other goals in our analysis, which might suggest the fact that child health in general is closely related to the social determinants of health which the other goals reflect. When considering second-order interactions, CSDG 16 (peace, justice and strong institutions) was perceived to have the largest net positive influence on child health. An example of how effective institutions in Cambodia positively influence child health is the success of the community based poverty identification system ID-Poor which has served as a platform for multisectoral collaboration on various health and non-health issues targeting the poor in Cambodia and strengthening institutional frameworks29. However, a continuous high rate of out of pocket spending have impeded the advancement and coverage of health services40 and ID-Poor beneficiaries who are disabled do not yet have full access to health care services and other social protection schemes41.During the same time the roles and engagement of civil society in health service delivery and social support have decreased42,43,44. Expanding social protection systems and strengthening local institutions together with increased collaboration with civil society might help to accelerate gains in child health and well-being even further. Conversely, progress on child health is deemed to be strongly promoting progress directly on a few key CSDGs, including CSDG 1 (no poverty), CSDG 4 (quality education) and CSDG 8 (decent work and economic growth). The relationships between child health and these policy areas have been characterized as positive and important in multiple studies4,45,46 and is in line with historic trends seen in Table 3, as such our results add empirical country level evidence to the knowledge base. When deciphering second-order interactions, our findings show a positive feedback-loop regarding child health in Cambodia, where progress on child health and well-being in itself through the promoting effect on other goals, leads to further progress in child health. Further, child health has a net promoting second-order influence on all other goals except CSDG 15 (life on land) on which it has a relatively small negative net influence. This is derived from the fact that child health is perceived to promote CSDGs that in total have a net restricting influence on CSDG 15 (life on land), primarily through CSDG 1 (no poverty), 8 (decent work and economic growth), 9 (industry, innovation and infrastructure) and 11 (sustainable cities and communities) in our analysis. These interactions might be explained by the apparent trade-offs between land conservation efforts and progress on other development goals such as reducing poverty and increasing agricultural productivity in Cambodia47,48. When examining the trends in the historic data provided in Table 3 combined with (i) the targets set for 2030 of restoring forests to around 50% of the total land (CSDG 15.1.1) and (ii) simultaneously keeping the 7% annual growth rate of real GDP per capita (CSDG 8.1.1) and (iii) almost eradicating extreme poverty32 it becomes evident that there might be some cause for this concern by the participants especially given the decreasing annual spending of the Ministry of Agriculture, Forestry and Fishery (Supplementary Fig. 5). Importantly, stakeholders considered child health to not have any direct influence on the possibility to make progress on CSDG 15 (life on land), and it is clear that the two goals could have synergistic potential49. The indivisible and complex relationships between sustainable development outcomes showcase that trade-offs that are not apparent at first glance might have implications for the overall achievement of the agenda.
Table 3 Overview of key Cambodian sustainable development indicators.
When applying novel methods for answering research questions adequate reflections on the merits of the analysis are due. The SDG Synergies approach hinges crucially on the selection of goals or targets to analyze, the group of participants that are tasked to make the scoring and the quality of the scoring process. Moving from the goal level to the target level within the SDG framework would probably alter the results. Further, the goals are broadly defined and can be interpreted in different ways when making assessments. A different set of in-country experts, and including private sector representatives, might therefore have judged the interactions in another way. Nevertheless, by clearly framing the goals and utilizing local stakeholders' expert judgment through a double-scoring process leading up to a consensus choice ensured that relevant and relatively unbiased scores were identified. The scoring of the interactions, however, does not rely on any pure quantitative assessment and should therefore be interpreted with caution. While grounding the interactions found in the country context and in the available data allows for a deeper understanding of the relationships found, it is not possible to derive a definitive causal direction for each individual interaction or the network as a whole. The primary results from this study, which are focused on systemic patterns from the perceptions of stakeholders, are only a small contribution to the knowledge base and would benefit from being complemented with research focusing on more specific goals, zooming in on a smaller regional or district geographical area and perhaps include a more formal assessment of how the interactions noted by the stakeholders corresponds to the policies formed and implemented to achieve better child health in Cambodia. Crucially, the advantages of contextualization and applicability must be weighed against the desired generalizability when using the SDG Synergies approach, as findings become harder to generalize across political, economic, geographical, and social settings14,20. Altogether, the strengths and limitations of the method of the results reflect the complexity of the 2030 Agenda itself.
An integrated analysis that transcend sectoral boundaries is necessary to form a bridge between science and policy making for sustainable development in general50 and child health specifically20,51. As our findings illustrate, progress on several CSDGs are important for child health and well-being, while child health in itself promotes progress for sustainable development in Cambodia. In particular, policy makers should consider direct and indirect interactions between child health and CSDG 16 (peace, justice and strong institutions) given the strong net positive effect on child health and the worryingly negative trend over the last decade, while being observant of the net negative effect of progress on child health on CSDG 15 (life on land).
Beside the findings in themselves, the participatory approach of the SDG Synergies approach was greatly appreciated by the included stakeholders and served as an opportunity to meet and discuss multisectoral issues and potential partnerships, framing the discussions around sustainable development, synergies and trade-offs in a common language. Framing key common determinants and prioritizing multisectoral efforts are vital to ending preventable child deaths9. With the risk of competing priorities and limited funding to reach the SDGs, continuous assessment and dialogue of potential synergies and trade-offs are essential to overcome bottlenecks and promote policy relevance. The SDG Synergies approach can serve as one tool for better governance on these issues, allowing also for comparison of interactions found with actual policy priorities and investments52. Overall, a participatory approach such as the SDG Synergies which allows for a systematic assessment of the interactions surrounding the SDGs and child health can provide novel insights on how complex relationships play out on a country level. With the need to further place the child in the centre of the SDGs51 and given the multifaceted challenges facing global child health49 this understanding will be vital for informing policy coherence and exploring innovative multisectoral partnerships that can accelerate the work toward achieving the 2030 Agenda in general and the betterment of global child health in particular.
Source data are included in this published article in Supplementary Data 1 and Supplementary Data 2.
UN Inter-agency Group for Child Mortality Estimation. Levels & Trend in Child Mortality: Report 2020. https://www.unicef.org/media/79371/file/UN-IGME-child-mortality-report-2020.pdf.pdf (2020).
United Nations. Transforming our world: the 2030 Agenda for Sustainable Development. Resolution adopted by the General Assembly on 25 September 2015. A/RES/70/1. https://www.un.org/en/development/desa/population/migration/generalassembly/docs/globalcompact/A_RES_70_1_E.pdf (2015).
Marmot, M. Achieving health equity: from root causes to fair outcomes. Lancet 370, 1153–1163 (2007).
Kuruvilla, S. et al. Policy and Practice: Success factors for reducing maternal and child mortality. Bull World Heal. Organ 92, 533–544 (2014).
Every Woman Every Child. The global strategy for women´s children´s and adolescents' health (2016-2030). https://www.everywomaneverychild.org/wp-content/uploads/2016/11/EWEC_globalstrategyreport_200915_FINAL_WEB.pdf (2015).
Marmot, M., Friel, S., Bell, R., Houweling, T. A. J. & Taylor, S. Closing the gap in a generation: health equity through action on the social determinants of health. Lancet 372, 1661–1669 (2008).
Ottersen, O. P. et al. The political origins of health inequity: Prospects for change. Lancet 383, 630–667 (2014).
Napier, A. D. et al. Culture and health. Lancet 384, 1607–1639 (2014).
Rasanathan, K. et al. Ensuring multisectoral action on the determinants of reproductive, maternal, newborn, child, and adolescent health in the post-2015 era. BMJ 351, h4213 (2015).
Alfvén, T. et al. Placing children at the center of the Sustainable Development Report: A SIGHT – Swedish Society of Medicine Road Map on Global Child Health. https://sight.nu/wp-content/uploads/2021/02/roadmaponglobalchildhealth_2.pdf (2019).
Bennich, T., Weitz, N. & Carlsen, H. Deciphering the scientific literature on SDG interactions: a review and reading guide. Sci. Total Environ. 728, 138405 (2020).
International Council for Science. A guide to SDG interactions: from science to implementation. (2017).
Nilsson, M., Griggs, D. & Visbeck, M. Policy: Map the interactions between Sustainable Development Goals. Nature 534, 320–322 (2016).
Weitz, N., Carlsen, H., Nilsson, M. & Skånberg, K. Towards systemic and contextual priority setting for implementing the 2030 Agenda. Sustain. Sci 13, 531–548 (2018).
International Council for Science. A draft framework for understanding SDG interactions. https://council.science/wp-content/uploads/2017/05/SDG-interactions-working-paper.pdf (2016).
Barquet, K., Järnberg, L., Alva, I. L. & Weitz, N. Exploring mechanisms for systemic thinking in decision-making through three country applications of SDG Synergies. Sustain. Sci. (2021) https://doi.org/10.1007/s11625-021-01045-3.
Nilsson, M. Important interactions among the Sustainable Development Goals under review at the High-Level Political Forum 2017. Stockholm Environ. Institute. https://mediamanager.sei.org/documents/Publications/SEI-WP-2017-06-Nilsson-SDG-interact-HLPF2017.pdf (2017).
Weitz, N. et al. SDGs and the environment in the EU: A systems view to improve coherence. Stockholm Environment Institute. https://www.sei.org/wp-content/uploads/2019/10/sei-2019-pr-weitz-sdg-synergies-eu-env.pdf (2019).
Whitmee, S. et al. Safeguarding human health in the Anthropocene epoch: report of The Rockefeller Foundation-Lancet Commission on planetary health. Lancet 386, 1973–2028 (2015).
Blomstedt, Y. et al. Partnerships for child health: capitalising on links between the sustainable development goals. BMJ 360, k125 (2018).
Wilkins, H. The need for subjectivity in EIA: Discourse as a tool for sustainable development. Environ. Impact Assess. Rev. 23, 401–414 (2003).
Bell, D., Raiffa, H. & Tversky, A. Decision Making, Descriptive, Normative and Prescriptive Interactions. (Cambridge University Press, 1999).
Ajzen, I. The theory of planned behavior. Organ. Behav. Hum. Decis. Process. 50, 179–211 (1991).
Schlüter, M. et al. A framework for mapping and comparing behavioural theories in models of social-ecological systems. Ecol. Econ. 131, 21–35 (2017).
World Bank. World Development Report 2015: Mind, Society, and Behaviour. http://documents1.worldbank.org/curated/en/645741468339541646/pdf/928630WDR0978100Box385358B00PUBLIC0.pdf (2014).
Espey, J. Sustainable development will falter without data. Nature 571, 299 (2019).
Ahmed, S. M. et al. Cross-country analysis of strategies for achieving progress towards global goals for women's and children's health. Bull. World Health Organ 94, 351–361 (2016).
Ministry of Health Cambodia, PMNCH, WHO, World Bank & AHPSR and participants in the Cambodia multistakeholder policy review. Success factors for women's and children's health: Cambodia. https://apps.who.int/iris/bitstream/handle/10665/254481/9789241509008-eng.pdf?sequence=1&isAllowed=y (2014).
Kaba, M. W. et al. IDPoor: a poverty identification programme that enables collaboration across sectors for maternal and child health in Cambodia. BMJ 363, k4698 (2018).
UNICEF. Reducing Stunting in Children Under Five Years of Age: A Comprehensive Evaluation of UNICEF's Strategies and Programme Performance – Cambodia Country Case Study. https://www.unicef.org/cambodia/media/1141/file/Reducing%20Stunting%20Full%20Report.pdf (2017).
UNICEF. Child Poverty in Cambodia Summary Report. https://www.unicef.org/cambodia/media/1496/file/Child%20poverty%20report%20in%20Cambodia_Full%20Report_Eng.pdf (2018).
Royal Government of Cambodia. Cambodian Sustainable Development Goals (CSDGs) Framework (2016-2030). https://data.opendevelopmentmekong.net/dataset/3aacd312-3b1e-429c-ac1e-33b90949607d/resource/d340c835-e705-40a4-8fb3-66f957670072/download/csdg_framework_2016-2030_english_last_final-1.pdf (2018).
UNICEF. Progress for Every Child in the SDG Era. https://www.unicef.org/media/48066/file/Progress_for_Every_Child_in_the_SDG_Era.pdf (2018).
Weimer-Jehle, T. W. Cross-impact balances: a system-theoretical approach to cross-impact analysis. Tecnol. Forecast. Soc. Chang. 73, 334–361 (2006).
Royal Government of Cambodia. Cambodia Sustainable Development Goals progress report 2019. https://sustainabledevelopment.un.org/content/documents/23603Cambodia_VNR_SDPM_Approved.pdf (2019).
Stockholm Environment Institute. SDG Synergies. https://www.sdgsynergies.org/.
Newman, M. Networks: An introduction. (Oxford University Press, 2010).
Pradhan, P. Antagonists to meeting the 2030 Agenda. Nat. Sustain. 2, 171–172 (2019).
Kroll, C., Warchold, A. & Pradhan, P. Sustainable Development Goals (SDGs): Are we successful in turning trade-offs into synergies? Palgrave Commun. 5, 1–11 (2019).
Ensora, T., Chhun, C., Kimsun, T., McPake, B. & Edokad, I. Impact of health financing policies in Cambodia: a 20 year experience. Soc. Sci. Med. 117, 118–126 (2017).
Harder, J. L. & Wendt, C. Access barriers to health and social services in Cambodia: experiences in healthcare use of people with disabilities. South East Asia Res. 29, 469–485 (2021).
Un, K. Cambodia: Moving away from democracy? Artic. Int. Polit. Sci. Rev. 32, 546–562 (2011).
Frewer, T. Doing NGO Work: the politics of being 'civil society' and promoting 'good governance' in Cambodia. Aust. Geogr. 44, 97–114 (2013).
World Bank. Worldwide Governance Indicators. https://info.worldbank.org/governance/wgi/Home/Reports.
Mensch, B. S., Chuang, E. K., Melnikas, A. J. & Psaki, S. R. Systematic Review Evidence for causal links between education and maternal and child health: systematic review. Trop. Med. Int. Heal 24, 504–522 (2019).
Rees, N., Chai, J. & Anthony, D. Right in principle and in practice: a review of the social and economic returns to investing in children. www.unicef.org/socialpolicy/index_59577.html (2012).
Beauchamp, E., Clements, T. & Milner-Gulland, E. J. Exploring trade-offs between development and conservation outcomes in Northern Cambodia. Land Use policy 71, 431–444 (2018).
Royal Government of Cambodia. The Report of Land and Human Development in Cambodia. https://www.un.org/esa/agenda21/natlinfo/countr/cambodia/land.pdf (2007).
Clark, H. et al. The Lancet Commissions A future for the world' s children? A WHO – UNICEF – Lancet Commission. Lancet 395, 605–658 (2020).
United Nations. The Future is Now – Science for Achieving Sustainable Development. Global Sustainable Development Report 2019. https://sustainabledevelopment.un.org/content/documents/24797GSDR_report_2019.pdf (2019).
Alfvén, T. et al. Placing children and adolescents at the centre of the Sustainable Development Goals will deliver for current and future generations. Glob. Health Action 12, 1670015 (2019).
Nilsson, M. & Weitz, N. Governing Trade-Offs and Building Coherence in Policy-Making for the 2030 Agenda. Polit. Gov. 7, 254–263 (2019).
The authors would like to express our thankfulness to all the participants that took part in the study. We are further grateful for the assistance from Mr. Tim Vora, Mr. Sou Veasna, Mr. Bunhoeuth Thou in helping to lead the workshop, Mr. Roy Nijhof for unwavering technical support and Mr. Mark Debackere for graciously lending time and effort making this study possible despite the COVID-19 pandemic.
Open access funding provided by Karolinska Institute.
Department of Global Public Health, Karolinska Institutet, Tomtebodavägen 18A, SE-171 77, Stockholm, Sweden
Daniel Helldén, Helena Nordenstedt & Tobias Alfvén
Malaria Consortium, Phnom Penh, Cambodia
Thy Chea
Research Office, Royal University of Phnom Penh, Phnom Penh, Cambodia
Serey Sok
Stockholm Environment Institute, Stockholm, Sweden
Linn Järnberg & Måns Nilsson
Presidents Office, Karolinska Institutet, Stockholm, Sweden
Göran Tomson
Swedish Institute for Global Health Transformation (SIGHT), Royal Swedish Academy of Sciences, Stockholm, Sweden
Department of Sustainable Development, Environmental Science and Engineering, KTH Royal Institute of Technology, Stockholm, Sweden
Måns Nilsson
Sachs' Children and Youth Hospital, Stockholm, Sweden
Tobias Alfvén
Daniel Helldén
Linn Järnberg
Helena Nordenstedt
D.H. was responsible for study design, data analysis, interpretation of data and for the writing process including writing a first draft of the paper and approval of the final draft. T.C. and S.S. contributed to data collection and revised the paper and approved the final draft. L.J. contributed to study design, data analysis, revised the paper and approved the final draft. H.N. and G.T. contributed to interpretation of data, revised the paper and approved the final draft. M.N. contributed to study design, paper revision and approved the final draft. T.A. contributed to the study design, data interpretation, paper revision and approved the final draft.
Correspondence to Daniel Helldén.
Communications Medicine thanks Zulfiqar A. Bhutta, Stefanos (A) Nastis and Nicholas Kassebaum for their contribution to the peer review of this work. Peer reviewer reports are available.
Description of Additional Supplementary Files
Supplementary Data 1.
Helldén, D., Chea, T., Sok, S. et al. A stakeholder group assessment of interactions between child health and the sustainable development goals in Cambodia. Commun Med 2, 68 (2022). https://doi.org/10.1038/s43856-022-00135-2
Communications Medicine (Commun Med) ISSN 2730-664X (online) | CommonCrawl |
Two sides of scalene $\bigtriangleup ABC$ measure $3$ centimeters and $5$ centimeters. How many different whole centimeter lengths are possible for the third side?
Using the Triangle Inequality, we see that the third side of a triangle with sides $3\text{ cm}$ and $5\text{ cm}$ must be larger than $2\text{ cm}$ but smaller than $8\text{ cm}.$ If the third side must be a whole centimeter length, and the triangle is scalene, that means that the only possible lengths for the third side are: $4\text{ cm},$ $6\text{ cm},$ and $7\text{ cm}.$ That makes $\boxed{3}$ possible lengths for the third side. | Math Dataset |
3-step group
In mathematics, a 3-step group is a special sort of group of Fitting length at most 3, that is used in the classification of CN groups and in the Feit–Thompson theorem. The definition of a 3-step group in these two cases is slightly different.
CN groups
In the theory of CN groups, a 3-step group (for some prime p) is a group such that:
• G = Op,p',p(G)
• Op,p′(G) is a Frobenius group with kernel Op(G)
• G/Op(G) is a Frobenius group with kernel Op,p′(G)/Op(G)
Any 3-step group is a solvable CN-group, and conversely any solvable CN-group is either nilpotent, or a Frobenius group, or a 3-step group.
Example: the symmetric group S4 is a 3-step group for the prime p = 2.
Odd order groups
Feit & Thompson (1963, p.780) defined a three-step group to be a group G satisfying the following conditions:
• The derived group of G is a Hall subgroup with a cyclic complement Q.
• If H is the maximal normal nilpotent Hall subgroup of G, then G′′⊆HCG(H)⊆G′ and HCG is nilpotent and H is noncyclic.
• For q∈Q nontrivial, CG(q) is cyclic and non-trivial and independent of q.
References
• Feit, Walter; Thompson, John G. (1963), "Solvability of groups of odd order", Pacific Journal of Mathematics, 13: 775–1029, doi:10.2140/pjm.1963.13.775, ISSN 0030-8730, MR 0166261
• Feit, Walter; Thompson, John G.; Hall, Marshall, Jr. (1960), "Finite groups in which the centralizer of any non-identity element is nilpotent", Mathematische Zeitschrift, 74: 1–17, doi:10.1007/BF01180468, ISSN 0025-5874, MR 0114856{{citation}}: CS1 maint: multiple names: authors list (link)
• Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR 0569209
| Wikipedia |
Nor am I sure how important the results are - partway through, I haven't noticed anything bad, at least, from taking Noopept. And any effect is going to be subtle: people seem to think that 10mg is too small for an ingested rather than sublingual dose and I should be taking twice as much, and Noopept's claimed to be a chronic gradual sort of thing, with less of an acute effect. If the effect size is positive, regardless of statistical-significance, I'll probably think about doing a bigger real self-experiment (more days blocked into weeks or months & 20mg dose)
Board-certified neuropsychologist Brian Lebowitz, PhD and associate clinical professor of neurology at Stony Brook University, explains to MensHealth.com that the term "encompasses so many things," including prescription medications. Brain enhancers fall into two different categories: naturally occurring substances like Ginkgo biloba, creatine and phenibut; and manmade prescription drugs, like Adderall, and over-the-counter supplements such as Noopept.
But there would also be significant downsides. Amphetamines are structurally similar to crystal meth – a potent, highly addictive recreational drug which has ruined countless lives and can be fatal. Both Adderall and Ritalin are known to be addictive, and there are already numerous reports of workers who struggled to give them up. There are also side effects, such as nervousness, anxiety, insomnia, stomach pains, and even hair loss, among others.
Yet some researchers point out these drugs may not be enhancing cognition directly, but simply improving the user's state of mind – making work more pleasurable and enhancing focus. "I'm just not seeing the evidence that indicates these are clear cognition enhancers," says Martin Sarter, a professor at the University of Michigan, who thinks they may be achieving their effects by relieving tiredness and boredom. "What most of these are actually doing is enabling the person who's taking them to focus," says Steven Rose, emeritus professor of life sciences at the Open University. "It's peripheral to the learning process itself."
If you're suffering from blurred or distorted vision or you've noticed a sudden and unexplained decline in the clarity of your vision, do not try to self-medicate. It is one thing to promote better eyesight from an existing and long-held baseline, but if you are noticing problems with your eyes, then you should see an optician and a doctor to rule out underlying medical conditions.
Herbal supplements have been used for centuries to treat a wide range of medical conditions. Studies have shown that certain herbs may improve memory and cognition, and they can be used to help fight the effects of dementia and Alzheimer's disease. These herbs are considered safe when taken in normal doses, but care should be taken as they may interfere with other medications.
Blinding stymied me for a few months since the nasty taste was unmistakable and I couldn't think of any gums with a similar flavor to serve as placebo. (The nasty taste does not seem to be due to the nicotine despite what one might expect; Vaniver plausibly suggested the bad taste might be intended to prevent over-consumption, but nothing in the Habitrol ingredient list seemed to be noted for its bad taste, and a number of ingredients were sweetening sugars of various sorts. So I couldn't simply flavor some gum.)
As I am not any of the latter, I didn't really expect a mental benefit. As it happens, I observed nothing. What surprised me was something I had forgotten about: its physical benefits. My performance in Taekwondo classes suddenly improved - specifically, my endurance increased substantially. Before, classes had left me nearly prostrate at the end, but after, I was weary yet fairly alert and happy. (I have done Taekwondo since I was 7, and I have a pretty good sense of what is and is not normal performance for my body. This was not anything as simple as failing to notice increasing fitness or something.) This was driven home to me one day when in a flurry before class, I prepared my customary tea with piracetam, choline & creatine; by the middle of the class, I was feeling faint & tired, had to take a break, and suddenly, thunderstruck, realized that I had absentmindedly forgot to actually drink it! This made me a believer.
Smart pills have revolutionized the diagnosis of gastrointestinal disorders and could replace conventional diagnostic techniques such as endoscopy. Traditionally, an endoscopy probe is inserted into a patient's esophagus, and subsequently the upper and lower gastrointestinal tract, for diagnostic purposes. There is a risk of perforation or tearing of the esophageal lining, and the patient faces discomfort during and after the procedure. A smart pill or wireless capsule endoscopy (WCE), however, can easily be swallowed and maneuvered to capture images, and requires minimal patient preparation, such as sedation. The built-in sensors allow the measurement of all fluids and gases in the gut, giving the physician a multidimensional picture of the human body.
Armodafinil is sort of a purified modafinil which Cephalon sells under the brand-name Nuvigil (and Sun under Waklert20). Armodafinil acts much the same way (see the ADS Drug Profile) but the modafinil variant filtered out are the faster-acting molecules21. Hence, it is supposed to last longer. as studies like Pharmacodynamic effects on alertness of single doses of armodafinil in healthy subjects during a nocturnal period of acute sleep loss seem to bear out; anecdotally, it's also more powerful, with Cephalon offering pills with doses as low as 50mg. (To be technical, modafinil is racemic: it comes in two forms which are rotations, mirror-images of each other. The rotation usually doesn't matter, but sometimes it matters tremendously - for example, one form of thalidomide stops morning sickness, and the other rotation causes hideous birth defects.)
Took full pill at 10:21 PM when I started feeling a bit tired. Around 11:30, I noticed my head feeling fuzzy but my reading seemed to still be up to snuff. I would eventually finish the science book around 9 AM the next day, taking some very long breaks to walk the dog, write some poems, write a program, do Mnemosyne review (memory performance: subjectively below average, but not as bad as I would have expected from staying up all night), and some other things. Around 4 AM, I reflected that I felt much as I had during my nightwatch job at the same hour of the day - except I had switched sleep schedules for the job. The tiredness continued to build and my willpower weakened so the morning wasn't as productive as it could have been - but my actual performance when I could be bothered was still pretty normal. That struck me as kind of interesting that I can feel very tired and not act tired, in line with the anecdotes.
A week later: Golden Sumatran, 3 spoonfuls, a more yellowish powder. (I combined it with some tea dregs to hopefully cut the flavor a bit.) Had a paper to review that night. No (subjectively noticeable) effect on energy or productivity. I tried 4 spoonfuls at noon the next day; nothing except a little mental tension, for lack of a better word. I think that was just the harbinger of what my runny nose that day and the day before was, a head cold that laid me low during the evening.
That is, perhaps light of the right wavelength can indeed save the brain some energy by making it easier to generate ATP. Would 15 minutes of LLLT create enough ATP to make any meaningful difference, which could possibly cause the claimed benefits? The problem here is like that of the famous blood-glucose theory of willpower - while the brain does indeed use up more glucose while active, high activity uses up very small quantities of glucose/energy which doesn't seem like enough to justify a mental mechanism like weak willpower.↩
The miniaturization of electronic components has been crucial to smart pill design. As cloud computing and wireless communication platforms are integrated into the health care system, the use of smart pills for monitoring vital signs and medication compliance is likely to increase. In the long term, smart pills are expected to be an integral component of remote patient monitoring and telemedicine. As the call for noninvasive point-of-care testing increases, smart pills will become mainstream devices.
But, if we find in 10 or 20 years that the drugs don't do damage, what are the benefits? These are stimulants that help with concentration. College students take such drugs to pass tests; graduates take them to gain professional licenses. They are akin to using a calculator to solve an equation. Do you really want a doctor who passed his boards as a result of taking speed — and continues to depend on that for his practice?
More recently, the drug modafinil (brand name: Provigil) has become the brain-booster of choice for a growing number of Americans. According to the FDA, modafinil is intended to bolster "wakefulness" in people with narcolepsy, obstructive sleep apnea or shift work disorder. But when people without those conditions take it, it has been linked with improvements in alertness, energy, focus and decision-making. A 2017 study found evidence that modafinil may enhance some aspects of brain connectivity, which could explain these benefits.
"I have a bachelors degree in Nutrition Science. Cavin's Balaster's How to Feed a Brain is one the best written health nutrition books that I have ever read. It is evident that through his personal journey with a TBI and many years of research Cavin has gained a great depth of understanding on the biomechanics of nutrition has how it relates to the structure of the brain and nervous system, as well as how all of the body systems intercommunicate with one another. He then takes this complicated knowledge and breaks it down into a concise and comprehensive book. If you or your loved one is suffering from ANY neurological disorder or TBI please read this book."
One symptom of Alzheimer's disease is a reduced brain level of the neurotransmitter called acetylcholine. It is thought that an effective treatment for Alzheimer's disease might be to increase brain levels of acetylcholine. Another possible treatment would be to slow the death of neurons that contain acetylcholine. Two drugs, Tacrine and Donepezil, are both inhibitors of the enzyme (acetylcholinesterase) that breaks down acetylcholine. These drugs are approved in the US for treatment of Alzheimer's disease.
ATTENTION CANADIAN CUSTOMERS: Due to delays caused by it's union's ongoing rotating strikes, Canada Post has suspended its delivery standard guarantees for parcel services. This may cause a delay in the delivery of your shipment unless you select DHL Express or UPS Express as your shipping service. For more information or further assistance, please visit the Canada Post website. Thank you.
In sum, the evidence concerning stimulant effects of working memory is mixed, with some findings of enhancement and some null results, although no findings of overall performance impairment. A few studies showed greater enhancement for less able participants, including two studies reporting overall null results. When significant effects have been found, their sizes vary from small to large, as shown in Table 4. Taken together, these results suggest that stimulants probably do enhance working memory, at least for some individuals in some task contexts, although the effects are not so large or reliable as to be observable in all or even most working memory studies.
Cocoa flavanols (CF) positively influence physiological processes in ways which suggest that their consumption may improve aspects of cognitive function. This study investigated the acute cognitive and subjective effects of CF consumption during sustained mental demand. In this randomized, controlled, double-blinded, balanced, three period crossover trial 30 healthy adults consumed drinks containing 520 mg, 994 mg CF and a matched control, with a 3-day washout between drinks. Assessments included the state anxiety inventory and repeated 10-min cycles of a Cognitive Demand Battery comprising of two serial subtraction tasks (Serial Threes and Serial Sevens), a Rapid Visual Information Processing (RVIP) task and a mental fatigue scale, over the course of 1 h. Consumption of both 520 mg and 994 mg CF significantly improved Serial Threes performance. The 994 mg CF beverage significantly speeded RVIP responses but also resulted in more errors during Serial Sevens. Increases in self-reported mental fatigue were significantly attenuated by the consumption of the 520 mg CF beverage only. This is the first report of acute cognitive improvements following CF consumption in healthy adults. While the mechanisms underlying the effects are unknown they may be related to known effects of CF on endothelial function and blood flow.
The abuse liability of caffeine has been evaluated.147,148 Tolerance development to the subjective effects of caffeine was shown in a study in which caffeine was administered at 300 mg twice each day for 18 days.148 Tolerance to the daytime alerting effects of caffeine, as measured by the MSLT, was shown over 2 days on which 250 g of caffeine was given twice each day48 and to the sleep-disruptive effects (but not REM percentage) over 7 days of 400 mg of caffeine given 3 times each day.7 In humans, placebo-controlled caffeine-discontinuation studies have shown physical dependence on caffeine, as evidenced by a withdrawal syndrome.147 The most frequently observed withdrawal symptom is headache, but daytime sleepiness and fatigue are also often reported. The withdrawal-syndrome severity is a function of the dose and duration of prior caffeine use…At higher doses, negative effects such as dysphoria, anxiety, and nervousness are experienced. The subjective-effect profile of caffeine is similar to that of amphetamine,147 with the exception that dysphoria/anxiety is more likely to occur with higher caffeine doses than with higher amphetamine doses. Caffeine can be discriminated from placebo by the majority of participants, and correct caffeine identification increases with dose.147 Caffeine is self-administered by about 50% of normal subjects who report moderate to heavy caffeine use. In post-hoc analyses of the subjective effects reported by caffeine choosers versus nonchoosers, the choosers report positive effects and the nonchoosers report negative effects. Interestingly, choosers also report negative effects such as headache and fatigue with placebo, and this suggests that caffeine-withdrawal syndrome, secondary to placebo choice, contributes to the likelihood of caffeine self-administration. This implies that physical dependence potentiates behavioral dependence to caffeine.
Oxiracetam is one of the 3 most popular -racetams; less popular than piracetam but seems to be more popular than aniracetam. Prices have come down substantially since the early 2000s, and stand at around 1.2g/$ or roughly 50 cents a dose, which was low enough to experiment with; key question, does it stack with piracetam or is it redundant for me? (Oxiracetam can't compete on price with my piracetam pile stockpile: the latter is now a sunk cost and hence free.)
At small effects like d=0.07, a nontrivial chance of negative effects, and an unknown level of placebo effects (this was non-blinded, which could account for any residual effects), this strongly implies that LLLT is not doing anything for me worth bothering with. I was pretty skeptical of LLLT in the first place, and if 167 days can't turn up anything noticeable, I don't think I'll be continuing with LLLT usage and will be giving away my LED set. (Should any experimental studies of LLLT for cognitive enhancement in healthy people surface with large quantitative effects - as opposed to a handful of qualitative case studies about brain-damaged people - and I decide to give LLLT another try, I can always just buy another set of LEDs: it's only ~$15, after all.)
When it comes to coping with exam stress or meeting that looming deadline, the prospect of a "smart drug" that could help you focus, learn and think faster is very seductive. At least this is what current trends on university campuses suggest. Just as you might drink a cup of coffee to help you stay alert, an increasing number of students and academics are turning to prescription drugs to boost academic performance.
I almost resigned myself to buying patches to cut (and let the nicotine evaporate) and hope they would still stick on well enough afterwards to be indistinguishable from a fresh patch, when late one sleepless night I realized that a piece of nicotine gum hanging around on my desktop for a week proved useless when I tried it, and that was the answer: if nicotine evaporates from patches, then it must evaporate from gum as well, and if gum does evaporate, then to make a perfect placebo all I had to do was cut some gum into proper sizes and let the pieces sit out for a while. (A while later, I lost a piece of gum overnight and consumed the full 4mg to no subjective effect.) Google searches led to nothing indicating I might be fooling myself, and suggested that evaporation started within minutes in patches and a patch was useless within a day. Just a day is pushing it (who knows how much is left in a useless patch?), so I decided to build in a very large safety factor and let the gum sit for around a month rather than a single day.
Thursday: 3g piracetam/4g choline bitartrate at 1; 1 200mg modafinil at 2:20; noticed a leveling of fatigue by 3:30; dry eyes? no bad after taste or anything. a little light-headed by 4:30, but mentally clear and focused. wonder if light-headedness is due simply to missing lunch and not modafinil. 5:43: noticed my foot jiggling - doesn't usually jiggle while in piracetam/choline. 7:30: starting feeling a bit jittery & manic - not much or to a problematic level but definitely noticeable; but then, that often happens when I miss lunch & dinner. 12:30: bedtime. Can't sleep even with 3mg of melatonin! Subjectively, I toss & turn (in part thanks to my cat) until 4:30, when I really wake up. I hang around bed for another hour & then give up & get up. After a shower, I feel fairly normal, strangely, though not as good as if I had truly slept 8 hours. The lesson here is to pay attention to wikipedia when it says the half-life is 12-15 hours! About 6AM I take 200mg; all the way up to 2pm I feel increasingly less energetic and unfocused, though when I do apply myself I think as well as ever. Not fixed by food or tea or piracetam/choline. I want to be up until midnight, so I take half a pill of 100mg and chew it (since I'm not planning on staying up all night and I want it to work relatively soon). From 4-12PM, I notice that today as well my heart rate is elevated; I measure it a few times and it seems to average to ~70BPM, which is higher than normal, but not high enough to concern me. I stay up to midnight fine, take 3mg of melatonin at 12:30, and have no trouble sleeping; I think I fall asleep around 1. Alarm goes off at 6, I get up at 7:15 and take the other 100mg. Only 100mg/half-a-pill because I don't want to leave the half laying around in the open, and I'm curious whether 100mg + ~5 hours of sleep will be enough after the last 2 days. Maybe next weekend I'll just go without sleep entirely to see what my limits are.
But notice that most of the cost imbalance is coming from the estimate of the benefit of IQ - if it quadrupled to a defensible $8000, that would be close to the experiment cost! So in a way, what this VoI calculation tells us is that what is most valuable right now is not that iodine might possibly increase IQ, but getting a better grip on how much any IQ intervention is worth.
Accordingly, we searched the literature for studies in which MPH or d-AMP was administered orally to nonelderly adults in a placebo-controlled design. Some of the studies compared the effects of multiple drugs, in which case we report only the results of stimulant–placebo comparisons; some of the studies compared the effects of stimulants on a patient group and on normal control subjects, in which case we report only the results for control subjects. The studies varied in many other ways, including the types of tasks used, the specific drug used, the way in which dosage was determined (fixed dose or weight-dependent dose), sample size, and subject characteristics (e.g., age, college sample or not, gender). Our approach to the classic splitting versus lumping dilemma has been to take a moderate lumping approach. We group studies according to the general type of cognitive process studied and, within that grouping, the type of task. The drug and dose are reported, as well as sample characteristics, but in the absence of pronounced effects of these factors, we do not attempt to make generalizations about them.
There is no official data on their usage, but nootropics as well as other smart drugs appear popular in the Silicon Valley. "I would say that most tech companies will have at least one person on something," says Noehr. It is a hotbed of interest because it is a mentally competitive environment, says Jesse Lawler, a LA based software developer and nootropics enthusiast who produces the podcast Smart Drug Smarts. "They really see this as translating into dollars." But Silicon Valley types also do care about safely enhancing their most prized asset – their brains – which can give nootropics an added appeal, he says.
Looking at the prices, the overwhelming expense is for modafinil. It's a powerful stimulant - possibly the single most effective ingredient in the list - but dang expensive. Worse, there's anecdotal evidence that one can develop tolerance to modafinil, so we might be wasting a great deal of money on it. (And for me, modafinil isn't even very useful in the daytime: I can't even notice it.) If we drop it, the cost drops by a full $800 from $1761 to $961 (almost halving) and to $0.96 per day. A remarkable difference, and if one were genetically insensitive to modafinil, one would definitely want to remove it.
Nootropics are a responsible way of using smart drugs to enhance productivity. As defined by Giurgea in the 1960's, nootropics should have little to no side-effects. With nootropics, there should be no dependency. And maybe the effects of nootropics are smaller than for instance Adderall, you still improve your productivity without risking your life. This is what separates nootropics from other drugs.
Long-term use is different, and research-backed efficacy is another question altogether. The nootropic market is not regulated, so a company can make claims without getting in trouble for making those claims because they're not technically selling a drug. This is why it's important to look for well-known brands and standardized nootropic herbs where it's easier to calculate the suggested dose and be fairly confident about what you're taking.
Burke says he definitely got the glow. "The first time I took it, I was working on a business plan. I had to juggle multiple contingencies in my head, and for some reason a tree with branches jumped into my head. I was able to place each contingency on a branch, retract and go back to the trunk, and in this visual way I was able to juggle more information."
This calculation - reaping only \frac{7}{9} of the naive expectation - gives one pause. How serious is the sleep rebound? In another article, I point to a mice study that sleep deficits can take 28 days to repay. What if the gain from modafinil is entirely wiped out by repayment and all it did was defer sleep? Would that render modafinil a waste of money? Perhaps. Thinking on it, I believe deferring sleep is of some value, but I cannot decide whether it is a net profit.
So is there a future in smart drugs? Some scientists are more optimistic than others. Gary Lynch, a professor in the School of Medicine at the University of California, Irvine argues that recent advances in neuroscience have opened the way for the smart design of drugs, configured for specific biological targets in the brain. "Memory enhancement is not very far off," he says, although the prospects for other kinds of mental enhancement are "very difficult to know… To me, there's an inevitability to the thing, but a timeline is difficult."
Neuroplasticity, or the brain's ability to change and reorganize itself in response to intrinsic and extrinsic factors, indicates great potential for us to enhance brain function by medical or other interventions. Psychotherapy has been shown to induce structural changes in the brain. Other interventions that positively influence neuroplasticity include meditation, mindfulness , and compassion.
Zach was on his way to being a doctor when a personal health crisis changed all of that. He decided that he wanted to create wellness instead of fight illness. He lost over a 100 lbs through functional nutrition and other natural healing protocols. He has since been sharing his knowledge of nutrition and functional medicine for the last 12 years as a health coach and health educator.
The stimulant now most popular in news articles as a legitimate "smart drug" is Modafinil, which came to market as an anti-narcolepsy drug, but gained a following within the military, doctors on long shifts, and college students pulling all-nighters who needed a drug to improve alertness without the "wired" feeling associated with caffeine. Modafinil is a relatively new smart drug, having gained widespread use only in the past 15 years. More research is needed before scientists understand this drug's function within the brain – but the increase in alertness it provides is uncontested.
When I spoke with Jesse Lawler, who hosts the podcast Smart Drugs Smarts, about breakthroughs in brain health and neuroscience, he was unsurprised to hear of my disappointing experience. Many nootropics are supposed to take time to build up in the body before users begin to feel their impact. But even then, says Barry Gordon, a neurology professor at the Johns Hopkins Medical Center, positive results wouldn't necessarily constitute evidence of a pharmacological benefit.
In 2011, as part of the Silk Road research, I ordered 10x100mg Modalert (5btc) from a seller. I also asked him about his sourcing, since if it was bad, it'd be valuable to me to know whether it was sourced from one of the vendors listed in my table. He replied, more or less, I get them from a large Far Eastern pharmaceuticals wholesaler. I think they're probably the supplier for a number of the online pharmacies. 100mg seems likely to be too low, so I treated this shipment as 5 doses:
The above information relates to studies of specific individual essential oil ingredients, some of which are used in the essential oil blends for various MONQ diffusers. Please note, however, that while individual ingredients may have been shown to exhibit certain independent effects when used alone, the specific blends of ingredients contained in MONQ diffusers have not been tested. No specific claims are being made that use of any MONQ diffusers will lead to any of the effects discussed above. Additionally, please note that MONQ diffusers have not been reviewed or approved by the U.S. Food and Drug Administration. MONQ diffusers are not intended to be used in the diagnosis, cure, mitigation, prevention, or treatment of any disease or medical condition. If you have a health condition or concern, please consult a physician or your alternative health care provider prior to using MONQ diffusers.
The smart pill that FDA approved is called Abilify MyCite. This tiny pill has a drug and an ingestible sensor. The sensor gets activated when it comes into contact with stomach fluid to detect when the pill has been taken. The data is then transmitted to a wearable patch that eventually conveys the information to a paired smartphone app. Doctors and caregivers, with the patient's consent, can then access the data via a web portal.
Phenotropil is an over-the-counter supplement similar in structure to Piracetam (and Noopept). This synthetic smart drug has been used to treat stroke, epilepsy and trauma recovery. A 2005 research paper also demonstrated that patients diagnosed with natural lesions or brain tumours see improvements in cognition. Phenylpiracetam intake can also result in minimised feelings of anxiety and depression. This is one of the more powerful unscheduled Nootropics available.
I took the pill at 11 PM the evening of (technically, the day before); that day was a little low on sleep than usual, since I had woken up an hour or half-hour early. I didn't yawn at all during the movie (merely mediocre to my eyes with some questionable parts)22. It worked much the same as it did the previous time - as I walked around at 5 AM or so, I felt perfectly alert. I made good use of the hours and wrote up my memories of ICON 2011.
During the 1920s, Amphetamine was being researched as an asthma medication when its cognitive benefits were accidentally discovered. In many years that followed, this enhancer was exploited in a number of medical and nonmedical applications, for instance, to enhance alertness in military personnel, treat depression, improve athletic performance, etc.
While the primary effect of the drug is massive muscle growth the psychological side effects actually improved his sanity by an absurd degree. He went from barely functional to highly productive. When one observes that the decision to not attempt to fulfill one's CEV at a given moment is a bad decision it follows that all else being equal improved motivation is improved sanity.
Null results are generally less likely to be published. Consistent with the operation of such a bias in the present literature, the null results found in our survey were invariably included in articles reporting the results of multiple tasks or multiple measures of a single task; published single-task studies with exclusively behavioral measures all found enhancement. This suggests that some single-task studies with null results have gone unreported. The present mixed results are consistent with those of other recent reviews that included data from normal subjects, using more limited sets of tasks or medications (Advokat, 2010; Chamberlain et al., 2010; Repantis, Schlattmann, Laisney, & Heuser, 2010).
The intradimensional– extradimensional shift task from the CANTAB battery was used in two studies of MPH and measures the ability to shift the response criterion from one dimension to another, as in the WCST, as well as to measure other abilities, including reversal learning, measured by performance in the trials following an intradimensional shift. With an intradimensional shift, the learned association between values of a given stimulus dimension and reward versus no reward is reversed, and participants must learn to reverse their responses accordingly. Elliott et al. (1997) reported finding no effects of the drug on ability to shift among dimensions in the extradimensional shift condition and did not describe performance on the intradimensional shift. Rogers et al. (1999) found that accuracy improved but responses slowed with MPH on trials requiring a shift from one dimension to another, which leaves open the question of whether the drug produced net enhancement, interference, or neither on these trials once the tradeoff between speed and accuracy is taken into account. For intradimensional shifts, which require reversal learning, these authors found drug-induced impairment: significantly slower responding accompanied by a borderline-significant impairment of accuracy.
Take at 11 AM; distractions ensue and the Christmas tree-cutting also takes up much of the day. By 7 PM, I am exhausted and in a bad mood. While I don't expect day-time modafinil to buoy me up, I do expect it to at least buffer me against being tired, and so I conclude placebo this time, and with more confidence than yesterday (65%). I check before bed, and it was placebo.
Going back to the 1960s, although it was a Romanian chemist who is credited with discovering nootropics, a substantial amount of research on racetams was conducted in the Soviet Union. This resulted in the birth of another category of substances entirely: adaptogens, which, in addition to benefiting cognitive function were thought to allow the body to better adapt to stress. | CommonCrawl |
This circle passes through the points $(-1, 2)$, $(3,0)$ and $(9,0)$. The center of the circle is at $(h,k)$. What is the value of $h+k$?
The center of the circle must lie on the perpendicular bisector of the points $(3,0)$ and $(9,0),$ which is the line $x = 6,$ so $h = 6.$ Thus, the center of the circle is $(6,k).$
This point must be equidistant to $(-1,2)$ and $(3,0),$ so
\[7^2 + (k - 2)^2 = 9 + k^2.\]This gives us $k = 11.$ Hence, $h + k = 6 + 11 = \boxed{17}.$ | Math Dataset |
\begin{document}
\title[Analyticity for periodic elliptic equations]{Large-scale analyticity and unique continuation for periodic elliptic equations}
\begin{abstract} We prove that a solution of an elliptic operator with periodic coefficients behaves on large scales like an analytic function, in the sense of approximation by polynomials with periodic corrections. Equivalently, the constants in the large-scale~$C^{k,1}$ estimate scale exponentially in~$k$, just as for the classical estimate for harmonic functions.
As a consequence, we characterize entire solutions of periodic, uniformly elliptic equations which exhibit growth like~$O(\exp(\delta|x|))$ for small~$\delta>0$. The large-scale analyticity also implies quantitative unique continuation results, namely a three-ball theorem with an optimal error term as well as a proof of the nonexistence of~$L^2$ eigenfunctions at the bottom of the spectrum. \end{abstract}
\author[S. Armstrong]{Scott Armstrong} \address[S. Armstrong]{Courant Institute of Mathematical Sciences, New York University, USA} \email{[email protected]}
\author[T. Kuusi]{Tuomo Kuusi} \address[T. Kuusi]{Department of Mathematics and Statistics, P.O. Box 68 (Gustaf H\"allstr\"omin katu 2), FI-00014 University of Helsinki, Finland}
\email{[email protected]}
\author[C. Smart]{Charles Smart} \address[C. Smart]{Department of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637.} \email{[email protected]}
\keywords{} \subjclass[2010]{35B10, 35B27, 35P05} \date{\today}
\maketitle
\setcounter{tocdepth}{1}
\section{Introduction}
\subsection*{Motivation and informal summary of results} We consider the linear, divergence-form, uniformly elliptic equation \begin{equation} \label{e.pde} -\nabla\cdot \mathbf{a} \nabla u = 0 \quad \mbox{in} \ U \subseteq{\mathbb{R}^d}, \end{equation} where $\mathbf{a}(\cdot)$ is a measurable, $\mathbb{Z}^d$--periodic map from ${\mathbb{R}^d}$ into the set of~$d$-by-$d$ matrices satisfying a uniform ellipticity assumption. Precisely, we assume there exists a constant $\Lambda\in [1,\infty)$ such that $\mathbf{a} \in L^\infty\left({\mathbb{R}^d}; \mathbb{R}^{d\times d} \right)$ satisfies \begin{equation} \label{e.a.ue}
\left| \xi \right|^2 \leq \xi \cdot \mathbf{a}(x)\xi \quad \mbox{and} \quad
\left| \eta \cdot \mathbf{a}(x)\xi \right|
\leq \Lambda \left| \eta \right| \left| \xi \right|, \quad \forall x,\xi,\eta \in{\mathbb{R}^d}, \end{equation} and \begin{equation} \label{e.periodicity} \mathbf{a}(\cdot + z) = \mathbf{a}, \quad \forall z\in\mathbb{Z}^d. \end{equation} In particular, we make no assumption of regularity on the coefficient field~$\mathbf{a}(\cdot)$ beyond measurability.
This paper concerns the behavior of solutions of~\eqref{e.pde} on length scales much larger than the unit scale on which the coefficients oscillate. The particular focus is on regularity and quantitative unique continuation properties of solutions. Our main result (see Theorem~\ref{t.analyticity} below) states that, on large scales, solutions of~\eqref{e.pde} possess higher-order regularity properties analogous to the classical pointwise estimates for the derivatives of harmonic functions which imply analyticity. This roughly means that a solution of~\eqref{e.pde} can be locally approximated by a certain family of ``heterogeneous polynomials'' (polynomials with periodic corrections) with the same precision as the approximation of a harmonic function by its Taylor polynomial. We call this a ``large-scale analyticity'' estimate.
Large-scale regularity estimates, which play a fundamental role in the theory of homogenization, originated in the work of Avellaneda and Lin~\cite{AL1,AL2}. A qualitative version of large-scale $C^{k,1}$ estimates were proved in~\cite{AL2} for every~$k\in\mathbb{N}$ in the form of a Liouville theorem characterizing all entire solutions with at most polynomial growth. The large-scale analyticity estimate we prove can be considered to be a quantification of the dependence in~$k$ of the prefactor constant in the~$C^{k,1}$ estimate (it is exponential in~$k$) and of the ``minimal scale'' on which it is valid (it is linear in $k$). Moreover, each of these estimates are optimal for~$k\gg1$.
We also present three consequences of the large-scale analyticity estimate. The first is a Liouville-type result (see Theorem~\ref{t.Liouville} below) which characterizes, for a small exponent~$\delta>0$, the set of solutions of~\eqref{e.pde} in the whole space ${\mathbb{R}^d}$ which grow at most like~$\exp(\delta|x|)$ as~$|x|\to \infty$. We show that any such solution can be written as an infinite series of heterogeneous polynomials, analogous to a power series representation. This can be seen as a qualitative version of large-scale analyticity.
The second application is a quantitative unique continuation result in the form of a so-called ``three-ball theorem'' (see Theorem~\ref{t.quc}). It states roughly that given, three balls $B_r\subseteq B_s\subseteq B_R$ with $R/s = s/r$ and a solution $u$ of~\eqref{e.pde} in~$B_R$, the ratio $\| u \|_{L^2(B_R)} / \| u \|_{L^2(B_s)}$ is controlled by $\| u \|_{L^2(B_s)} / \| u \|_{L^2(B_r)}$, plus an error term which depends on the scale~$r$. Recall that the strong unique continuation property is well known to hold for elliptic equations with Lipschitz coefficients~\cite{AKS,GL}, but is false, in general, even for equations with coefficients belonging to~$C^{0,\alpha}$ for every $\alpha\in (0,1)$: see~\cite{P,Mi,Fil}. Therefore quantitative unique continuation estimates necessarily depend on the Lipschitz norm of the coefficients and degenerate on large length scales. Here we prove the opposite: a quantitative unique continuation result which becomes stronger as the length scale becomes larger and is completely useless below the unit scale.
One of the main motivations for studying unique continuation properties lies in their implications for the spectrum of the operator. It is a long-standing conjecture that the spectrum of the elliptic operator~$-\nabla \cdot \mathbf{a} \nabla $ with periodic \emph{and sufficiently smooth} coefficients~$\mathbf{a}(\cdot)$ must be absolutely continuous. For a complete discussion of the history and recent progress on this problem, which is still wide-open, we refer to the survey of Kuchment~\cite[Section 6]{Ku}. We mention only that the only progress made on this question for equations with variable coefficients is the work of Friedlander~\cite{F}, who proved the conjecture under some symmetry assumptions on the coefficients which are unfortunately rather restrictive. In the negative direction, Filonov~\cite{Fil} constructed an explicit example of a coefficient field~$\mathbf{a}(\cdot)$ belonging to~$C^{0,\alpha}$ for every $\alpha<1$ such that the operator $-\nabla \cdot \mathbf{a}\nabla$ admits a nontrivial and compactly-supported eigenfunction (note that compact support allows the example to be periodized).
The third consequence of the large-scale analyticity estimate we present asserts that the operator~$-\nabla \cdot\mathbf{a}\nabla$ has no $L^2$ eigenfunctions near the bottom of the spectrum (see Theorem~\ref{t.bottom} below). We emphasize that this result is valid without regularity assumptions on~$\mathbf{a}(\cdot)$ and therefore, in view of the counterexample of~\cite{Fil} mentioned above, cannot be improved. Indeed, the result of~\cite{Fil} shows the optimality of each of the four theorems presented below.
\subsection*{Statement of the main results} Before presenting the main results, we must introduce some notation. For further notation, including the notation for multiindices and tensors, see Section~\ref{ss.prelim}. As we find it convenient to work with cubes rather than balls, we denote, for every $r>0$, \begin{equation} Q_r:= \left( -\frac12r,\frac12 r \right)^d. \end{equation}
The tensor of~$k$th order partial derivatives of a function~$u$ is denoted~$\nabla^k u$. For $u \in L^p(U)$ with $p\in[1,\infty)$ and $0<|U|<\infty$, we denote the volume-normalized $L^p(U)$ norm by \begin{equation} \label{e.L2norm}
\left\| u \right\|_{\underline{L}^p(U)} := \left( \strokedint_{U} |u(x)|^p \, dx \right)^{\frac1p} := \left( \frac{1}{|U|}\int_{U} |u(x)|^p \, dx \right)^{\frac1p} . \end{equation} For each $m\in\mathbb{N}$, we denote the linear space of real-valued, homogeneous polynomials of order~$m$ on ${\mathbb{R}^d}$ by \begin{equation} \mathbb{P}_m^*:= \left\{
w \, : \, w =\sum_{|\alpha|=m} c_\alpha x^\alpha, \ c_\alpha\in\mathbb{R} \right\} \end{equation} and the linear of all real-valued polynomials of order at most $m$ by \begin{equation} \mathbb{P}_m:= \left\{ w \, : \, w = \sum_{n=0}^m w_n, \ w_n \in \mathbb{P}_n^* \right\}. \end{equation} For convenience we also denote $\mathbb{P}_{-1} := \mathbb{P}_{-2} := \{ 0 \}$.
In order to present the statement of the large-scale analyticity result, we briefly describe the finite-dimensional vector space~$\mathbb{A}_m$ of ``heterogeneous polynomials of degree at most $m\in\mathbb{N}$'' and refer the reader to Section~\ref{s.correctors} for more details. The result of~\cite{AL2} states that, for every $m\in\mathbb{N}$, $p\in \mathbb{P}_{m-2}$ and $u\in H^1_{\mathrm{loc}}({\mathbb{R}^d})$ satisfying the equation \begin{equation}
-\nabla \cdot \mathbf{a}\nabla u = p \quad \mbox{in} \ {\mathbb{R}^d} \end{equation} and the growth condition \begin{equation}
\limsup_{r\to \infty} r^{-(m+1)}
\left\| u \right\|_{\underline{L}^2(Q_r)} = 0, \end{equation} one can find a polynomial $q \in \mathbb{P}_{m}$ such that \begin{equation} \label{e.uformq} u(x) = \sum_{k=0}^m \nabla^kq(x) : \phi^{(k)}(x), \end{equation} where $\phi^{(k)}$ is the \emph{tensor of $k$th order correctors}, which are constructed in Section~\ref{s.correctors}, and the colon denotes the tensor contraction, see~\eqref{e.colon}. We think of such solutions as the correct notion of ``intrinsic polynomials of degree at most~$m$'' and so we define, for each~$m\in\mathbb{N}$, \begin{equation}
\mathbb{A}_m:= \left\{ \psi = \sum_{k=0}^m \left( \nabla^k q : \phi^{(k)} \right) \,:\, q\in \mathbb{P}_m \right\}. \end{equation} Given~$u$ of the form~\eqref{e.uformq}, we have that $-\nabla \cdot \mathbf{a}\nabla u = \mathscr{A}q$, where $\mathscr{A}$ is the higher-order homogenized operator. It has the form \begin{equation*} \mathscr{A} u := \sum_{n=2}^\infty \overline{\mathbf{a}}^{(n)} : \nabla^n u, \end{equation*} where the tensors $\overline{\mathbf{a}}^{(n)}$ are the higher-order homogenized tensors which arise in the construction of the higher-order correctors (see Section~\ref{s.correctors}). We note that $\overline{\mathbf{a}}^{(2)}=:\overline{\mathbf{a}}$ is the familiar homogenized matrix arising in classical homogenization. It is the equation $-\mathscr{A} u = 0$ which should be considered as the ``true'' homogenized equation, rather than $-\nabla\cdot\mathbf{a}\nabla u =0$, which is merely the leading order approximation of the former. We also denote the subset of~$\mathbb{A}_m$ consisting of~$\mathbf{a}(\cdot)$--harmonic functions by \begin{equation} \label{e.Am0.intro} \mathbb{A}^0_m:= \left\{ u = \sum_{k=0}^m \left( \nabla^k q : \phi^{(k)} \right) \,:\, q\in \mathbb{P}_m, \ \mathscr{A}q = 0 \right\} . \end{equation}
The classical pointwise estimate for harmonic functions states that there exists~$C(d)<\infty$ such that, for every harmonic function~$u$ in~$Q_R$ and~$m\in\mathbb{N}$, \begin{equation} \label{e.harm1}
\left| \nabla^m u (0) \right| \leq
\left( \frac{Cm}{R} \right)^m \left\| u \right\|_{\underline{L}^1(Q_R)}. \end{equation} We can restate this in terms of polynomial approximation as follows: there exists~$C(d)<\infty$ such that, for every harmonic function~$u$ in~$B_R$ and~$m\in\mathbb{N}$, there exists $p\in\mathbb{P}_m$ such that, for every $r\in (0,R)$, \begin{equation} \label{e.harm2}
\left\| u - p \right\|_{L^\infty(Q_r)} \leq
\left( \frac{Cr}{R} \right)^m \left\| u \right\|_{\underline{L}^1(Q_R)}. \end{equation} Indeed, one may take~$p$ to be the $m$th order Taylor polynomial of~$u$ at the origin (which we note is also harmonic) and combine~\eqref{e.harm1} with Taylor's theorem.
The main result of the paper is the following generalization of~\eqref{e.harm2} to equations with variable, periodic coefficients.
\begin{theorem}[Large-scale analyticity] \label{t.analyticity} There exists~$C(d,\Lambda)<\infty$ such that, for every $m\in\mathbb{N}$ with $m\geq 0$, $R\in [2Cm,\infty)$ and solution $u\in H^1(Q_R)$ of \begin{equation} -\nabla \cdot \mathbf{a}\nabla u = 0 \quad \mbox{in} \ Q_R, \end{equation} there exists $\psi \in \mathbb{A}_{m}^0$ such that, for every $r\in \left[Cm, R\right]$, \begin{equation} \label{e.Cm1}
\left\| u - \psi \right\|_{\underline{L}^2(Q_r)} \leq
\left( \frac{Cr}{R} \right)^{m+1} \left\| u \right\|_{\underline{L}^2(Q_R)}. \end{equation} \end{theorem}
The main novelty in the statement of Theorem~\ref{t.analyticity} is the fact that the constant~$C$ on the right side of~\eqref{e.Cm1} depends only on $(d,\Lambda)$ and, in particular, does not depend on~$m$. It is well-known result, essentially due to~\cite{AL2} (see for instance~\cite[Theorem 3.6]{AKMbook} for the complete statement), that~\eqref{e.Cm1} is valid if we replace the right side by \begin{equation}
C(m,d,\Lambda) \left( \frac{r}{R} \right)^{m+1}
\left\| u \right\|_{\underline{L}^2(Q_R)}. \end{equation} Since this estimate is qualitative in its dependence in~$m$, it unfortunately gives no information regarding the ratio $r/R$ of length scales on which it is useful. By carefully quantifying the dependence on~$m$ from previous arguments, one at best obtains $C(m,d,\Lambda) = \exp\left( C(d,\Lambda){m^2} \right)$, which is much worse than what is given by Theorem~\ref{t.analyticity}, namely $C(m,d,\Lambda) = \exp(C(d,\Lambda)m)$. Since the latter is the same as observed in the classical estimate~\eqref{e.harm2} for harmonic functions, it is therefore optimal.
The reason we use the modifier ``large-scale'' in describing Theorem~\ref{t.analyticity} is due to the restriction~$r \geq Cm$ in~\eqref{e.Cm1}, which does not appear in~\eqref{e.harm2}. This restriction is necessary and sharp in the sense that, without at least an assumption of Lipschitz regularity for~$\mathbf{a}(x)$, the statement of the theorem would be false if the restriction~$r\geq Cm$ is replaced by~$r \geq \varepsilon m$ for sufficiently small~$\varepsilon$. This can be seen\footnote{One can add a dummy variable $x_{d+1}$ and multiply by $\exp(\lambda^{\frac12}x_{d+1})$ to make an eigenfunction with eigenvalue $\lambda$ into a solution with zero right-hand side in one more dimension. } from the example of the compactly supported eigenfunction constructed in~\cite{Fil}.
Perhaps a better way to make the motivation behind the terminology clear is to present the following rephrasing of Theorem~\ref{t.analyticity}.
\begin{corollary} \label{c.analyticity.exp} There exists~$c(d,\Lambda)\in (0,1]$ such that, for every~$R\in \left[c^{-1},\infty\right)$ and solution $u\in H^1(Q_R)$ of \begin{equation} -\nabla \cdot \mathbf{a}\nabla u = 0 \quad \mbox{in} \ Q_R, \end{equation} there exists $\psi \in \mathbb{A}_{\lfloor R \rfloor }^0$ such that, for every $r\in \left[1, cR\right]$, \begin{equation} \label{e.Cm1exp}
\left\| u - \psi \right\|_{\underline{L}^2(Q_r)} \leq \exp(-cR)
\left\| u \right\|_{\underline{L}^2(Q_R)}. \end{equation} \end{corollary}
We see from~\eqref{e.Cm1exp} that our large-scale analyticity does not yield ``literal'' analyticity in the sense that an arbitrary solution can be expressed locally as a ``power series,'' interpreted as a limit of elements of $\mathbb{A}_m$ with $m\to \infty$. What Corollary~\ref{c.analyticity.exp} says however is that this is valid up to an error which is exponentially small in the length scale. The example of~\cite{Fil} implies moreover that this is optimal.
As a first application of Theorem~\ref{t.analyticity}, we obtain a characterization of entire solutions of~\eqref{e.pde} which have growth at most like a ``slow'' exponential. That is, we will show that a solution in the whole space which does not grow faster than $\exp(c|x|)$ can indeed be written as a ``power series''---a result which is not very surprising in view of Corollary~\ref{c.analyticity.exp}. For each~$\delta>0$, let us denote \begin{equation*} \mathbb{P}_\infty(\delta) := \left\{ w \, : \, w = \sum_{n=0}^\infty w_n, \ w_n \in \mathbb{P}_n^*, \ \sum_{n=0}^\infty
\delta^{-n} \left| \nabla^n w_n(0) \right| < \infty \right\}. \end{equation*} Using bounds on the $k$th order correctors and homogenized tensors which (are proved in Section~\ref{s.correctors} and) state that these grow at most like $C^k$, it follows that, for every~$\delta>0$ sufficiently small and $q \in \mathbb{P}_\infty(\delta)$, the series \begin{equation} u := \sum_{k=0}^\infty \nabla^kq:\phi^{(k)} \end{equation} is absolutely convergent (locally uniformly in~${\mathbb{R}^d}$) and the expression~$\mathscr{A}q$ is well-defined. For such~$\delta$, we may define the space \begin{equation}
\mathbb{A}_\infty(\delta):= \left\{ \psi = \sum_{k=0}^\infty \left( \nabla^k q:\phi^{(k)} \right) \, :\, q \in \mathbb{P}_\infty(\delta) \right\} \end{equation} as well as $\mathbb{A}_\infty^0(\delta)$ in analogy with~\eqref{e.Am0.intro}. The following theorem asserts that this space contains all solutions of~\eqref{e.pde} in~${\mathbb{R}^d}$ which grow like a slow exponential.
\begin{theorem}[Liouville theorem] \label{t.Liouville} There exists~$C(d,\Lambda)<\infty$ and~$\delta_0(d,\Lambda)>0$ such that, for every $u\in H^1_{\mathrm{loc}}({\mathbb{R}^d})$ and $\delta \in (0,\delta_0]$ satisfying \begin{equation} -\nabla \cdot \mathbf{a}\nabla u = 0 \quad \mbox{in}\ {\mathbb{R}^d} \end{equation} and \begin{equation}
\limsup_{r\to \infty}\ \exp\left( - \delta r \right) \left\| u \right\|_{\underline{L}^2(Q_r)} < \infty, \end{equation} we have that $u \in \mathbb{A}_\infty(C\delta)$. \end{theorem}
We next present a quantitative unique continuation result, which comes in the form of a three-ball theorem. To our knowledge, the first quantitative unique continuation result for periodic coefficients appeared in the recent work of Lin and Shen~\cite{LS}, who proved a large-scale doubling condition for solutions of~\eqref{e.pde}. Subsequently, a statement very similar to our Theorem~\ref{t.quc} was obtained in a paper of Kenig and Zhu~\cite{KZ}. Their result does not have the restriction that the ratio of the radii be small, but their error term has a factor of~$R^{-\beta}$, for an exponent~$\beta \in (0,1)$, rather than the sharp exponential factor $\exp(-cr)$ in~\eqref{e.quc}.
\begin{theorem}[Three-ball theorem] \label{t.quc} For each $\alpha \in \left( 0,\tfrac12 \right)$, there exist~$c(d,\Lambda)>0$ and $\theta(\alpha, d,\Lambda) \in \left(0,\tfrac12\right]$ such that, for every $R,r,s\in [2,\infty)$ with \begin{equation} \frac{r}{s} = \frac{s}{R} \in (0,\theta] \end{equation} and $u \in H^1(B_R)$ satisfying \begin{equation}
-\nabla \cdot \mathbf{a}\nabla u = 0\quad \mbox{in} \ B_R, \end{equation} we have the estimate \begin{equation} \label{e.quc}
\left\| u \right\|_{\underline{L}^2(B_s)} \leq
\left\| u \right\|_{\underline{L}^2(B_r)}^{\alpha}
\left\| u \right\|_{\underline{L}^2(B_R)}^{1-\alpha} +
\exp(-cr) \left\| u \right\|_{\underline{L}^2(B_R)}. \end{equation} \end{theorem}
The error term in the estimate~\eqref{e.quc} is sharp in the sense that, without at least an assumption of Lipschitz regularity on the coefficients~$\mathbf{a}(x)$, the multiplicative factor must at least~$\exp( -Cr )$. This is due to same example of~\cite{Fil} mentioned several times above. It remains a very interesting open question whether, under suitable smoothness assumptions on the coefficient~$\mathbf{a}(\cdot)$, the~$c$ inside the exponential in~\eqref{e.quc} may be replaced by any large constant~$A>1$. (If this could be affirmatively resolved, then the restriction~$\lambda \leq \lambda_0$ in Theorem~\ref{t.bottom} below could be also be removed.) We remark that, as can be observed from the proof of Theorem~\ref{t.quc} in Section~\ref{s.conseq}, the size of~$c$ is closely related to the constant~$C$ in the restriction $r\geq Cm$ in Theorem~\ref{t.analyticity}. The optimal exponent~$\alpha$ in~\eqref{e.quc} should be $\frac12$, thus the restriction $\alpha\in (0,\tfrac12)$ is slightly suboptimal.
Our last main result, another direct application of Theorem~\ref{t.analyticity}, asserts without regularity assumptions on the coefficient field~$\mathbf{a}(\cdot)$ beyond measurability that the operator~$-\nabla\cdot\mathbf{a}\nabla$ has no~$L^2$ eigenfunctions near the bottom of its spectrum. Note that the exponential decay condition~\eqref{e.superfluous} below is superfluous and can be replaced by the weaker assumption that~$u\in L^2$ by a result of Kuchment~\cite[Theorem 6.15]{Ku}.
\begin{theorem}[Absence of embedded eigenvalues, bottom of the spectrum] \label{t.bottom} \emph{}\\ There exist constants $\lambda_0(d,\Lambda)>0$ and~$C(d,\Lambda)<\infty$ such that, for every pair \begin{equation*} (\lambda,u) \in [0,\lambda_0]\times H^1_{\mathrm{loc}}({\mathbb{R}^d}) \end{equation*} satisfying \begin{equation} -\nabla \cdot \mathbf{a}\nabla u = \lambda u \quad \mbox{in} \ {\mathbb{R}^d} \end{equation} and the growth condition \begin{equation} \label{e.superfluous} \limsup_{r \to \infty} \, \exp(Cr)
\left\| u \right\|_{\underline{L}^2(Q_r)} = 0, \end{equation} we have that $u\equiv 0$ in ${\mathbb{R}^d}$. \end{theorem}
While we are unaware of Theorem~\ref{t.bottom} appearing previously in the literature, it seems to have attained a kind of folklore status. Indeed, Jonathan Goodman demonstrated to us in a private communication that Theorem~\ref{t.bottom} can also be proved using a Bloch wave perturbation argument. It seems that, like the proof of Theorem~\ref{t.bottom}, this argument is also restricted to the bottom of the spectrum.
None of the results stated in this introduction can be improved without at least an assumption of Lipschitz continuity of the coefficients (with the possible exception of improving $\alpha$ to $\alpha=\frac12$ in the statement of Theorem~\ref{t.quc}). This however begs the question of whether a smoothness assumption on~$\mathbf{a}(\cdot)$ could be used to augment the proof of Theorem~\ref{t.analyticity} by relaxing the restriction~$r\geq Cm$. Unfortunately, we do not expect this to be the case.
In the next section, we introduce the heterogeneous polynomials, the higher-order correctors and homogenized equation and prove some auxiliary estimates. The proof of Theorem~\ref{t.analyticity} is the focus of Section~\ref{s.analyticity}. In Section~\ref{s.conseq} we demonstrate that the other four results stated above are corollaries of Theorem~\ref{t.analyticity}.
\section{Higher-order correctors and homogenized tensors} \label{s.correctors}
\subsection{Preliminaries} \label{ss.prelim}
The standard basis for~${\mathbb{R}^d}$ is denoted~$\{ e_1,\ldots,e_d\}$. We denote the set of multi-indices by~$\mathbb{N}^d$. For each $\alpha\in\mathbb{N}^d$, we define the \emph{order of $\alpha$} by~$\left| \alpha \right| = \sum_{i=1}^d \alpha_i$. The factorial of a multiindex~$\alpha$ is~$\alpha!:= \prod_{i=1}^d \alpha_i!$. For each~$\alpha\in\mathbb{N}^d$, we define the monomial $x^\alpha$ and partial derivative operator~$\partial^\alpha$ by \begin{equation} \label{e.pars} x^\alpha:= \prod_{i=1}^d x_i^{\alpha_i} \quad \mbox{and} \quad \partial^\alpha = \prod_{i=1}^d \partial_{x_i}^{\alpha_i}. \end{equation}
We denote, for $|\alpha|=m\in\mathbb{N}$, the multinomial coefficient \begin{equation} \binom{m}{\alpha} := \frac{m!}{\alpha!}. \end{equation} By a \emph{symmetric tensor of order $m\in\mathbb{N}$}, we mean a mapping \begin{equation}
T := \{ \alpha \in\mathbb{N}^d\,:\, |\alpha|=m \} \to \mathbb{R}. \end{equation} We may write $T =
\{T_\alpha\}_{|\alpha|=m}$. We let $\mathbb{T}_m$ denote the set of symmetric tensors of order~$m$.
The gradient~$\nabla u = (\partial_{x_1}u,\ldots,\partial_{x_d}u)$ of a function~$u$ is a tensor of order~one which we may regard as an element of~${\mathbb{R}^d}$. If $u\in C^m$, then we may regard $\nabla^m u = ( \partial^\alpha u)_{|\alpha|=m}$ as a symmetric tensor of order~$m$. For each $m\in\mathbb{N}$ and $x\in{\mathbb{R}^d}$, we also let $x^{\otimes m} \in \mathbb{T}_m$ be the tensor with coordinates \begin{equation}
(x^{\otimes m})_\alpha:= x^\alpha, \quad |\alpha| = m. \end{equation} We denote the contraction~$S:T$ of~$S,T\in\mathbb{T}_m$ by \begin{equation} \label{e.colon} (S:T) :=
\sum_{|\alpha|=m} \binom{m}{\alpha} S_\alpha T_\alpha. \end{equation} The multinomial coefficient appears in order to properly count the multiplicities of the multiindices. It makes writing Taylor expansions convenient, as a polynomial~$p\in\mathbb{P}_m$ of degree~$m\in\mathbb{N}$ can be expressed as \begin{equation}
p(x) = \sum_{k=0}^m \frac{1}{k!} \left( \nabla^kp(0) : x^{\otimes k} \right). \end{equation} For each direction $i\in \{1,\ldots,d\}$, we define the forward finite difference operator \begin{equation} D_i u(x) := u(x+e_i) - u(x). \end{equation} In analogy with~\eqref{e.pars}, for each $n\in\mathbb{N}$, we let $D^nu(x) \in \mathbb{T}_n$ be defined by \begin{equation} (D^\alpha u)(x) = \left( D_{1}^{\alpha_1} D_{2}^{\alpha_2} \cdots D_{d}^{\alpha_d} u \right)(x). \end{equation} Note that the forward finite difference operators commute and hence $D^\alpha u$ is indeed a symmetric tensor.
We next discuss the norms we put on the linear space~$\mathbb{T}_m$ of $m$th order tensors. Even though~$\mathbb{T}_m$ is finite dimensional, since we will work with~$m$ very large and need to keep track of dependence in~$m$, this requires some care. We let $| \cdot |$ denote the Euclidean norm \begin{equation}
\left| T \right| := \left( T:T \right)^{\frac12} =
\left( \sum_{|\alpha|=m} \binom{m}{\alpha}
\left| T_\alpha\right|^2 \right)^{\frac12}, \quad T \in \mathbb{T}_m. \end{equation} Observe in particular that, for every $x\in{\mathbb{R}^d}$ and $m\in\mathbb{N}$, \begin{equation} \label{e.xotimesm.size}
\left| x^{\otimes m} \right| = |x|^m. \end{equation}
\subsection{Higher-order correctors} \label{ss.higherorder}
In order to work with the space~$\mathbb{A}_n$, we need to introduce the higher-order periodic correctors which are classical in the theory of periodic homogenization (see for instance~\cite{ABV,BG}) and represent the ``periodic wiggles'' in the heterogeneous polynomials. Our construction of the correctors is minimalistic because here we are interested only in building the space~$\mathbb{A}_m$ (and not, for instance, in quantifying homogenization errors). In particular, we have no interest in the flux correctors or in the nonsymmetric part of the homogenized tensors.
The higher-order correctors and homogenized tensors can be defined in various different ways, and different choices amount to different parametrizations of the space~$\mathbb{A}_n$ (which is what is actually intrinsic). Here we take the correctors to be a family~$\{\phi^{(k)}\}_{k\in\mathbb{N}}$ of $\mathbb{Z}^d$-periodic, mean-zero functions with~$\phi^{(k)}$ valued in the set~$\mathbb{T}_k$ of~$k$th order tensors. They are constructed simultaneously with a sequence~$\{\overline{\mathbf{a}}^{(k)}\}_{k\in\mathbb{N}}$ of constant tensors, with $\overline{\mathbf{a}}^{(k)}\in \mathbb{T}_k$, such that, for every~$n\in\mathbb{N}$ and~$p\in\mathbb{P}_n$, \begin{equation} \label{e.correq_n} \nabla \cdot \left( \mathbf{a} \nabla \left( \sum_{k=0}^{n} \phi^{(k)} \, \colon \nabla^k p \right) \right) = \sum_{k=1}^{n} \overline{\mathbf{a}}^{(k)} \, \colon \nabla^{k} p . \end{equation} Here we use the abbreviation $\phi^{(0)}=1$ and $\overline{\mathbf{a}}^{(1)} = 0$. In the case~$n=1$, this is the familiar first-order corrector equation which states that, for every affine~$p\in\mathbb{P}_1$, \begin{equation} \label{e.correq_1} \nabla \cdot \left( \mathbf{a} \left( \nabla p + \nabla \left( \phi^{(1)} \, \colon \nabla p \right) \right) \right) = 0. \end{equation} The tensor $\overline{\mathbf{a}}^{(2)}$, defined by \begin{equation} \label{e.ahom7} \overline{\mathbf{a}}^{(2)} = \overline{\mathbf{a}} := \left\langle \mathbf{a} \left(I_d + \nabla \phi^{(1)} \right) \right\rangle, \end{equation} is the homogenized matrix, which satisfies the ellipticity condition \begin{equation} \label{e.ahom.ue}
\left| \xi \right|^2 \leq \xi \cdot \overline{\mathbf{a}} \xi \quad \mbox{and} \quad
\left| \eta \cdot \overline{\mathbf{a}} \xi \right|
\leq \Lambda \left| \eta \right| \left| \xi \right|, \quad \forall \xi,\eta \in{\mathbb{R}^d}. \end{equation} We remark that, since we have made no assumption of symmetry of the coefficient field~$\mathbf{a}(\cdot)$, the homogenized tensor defined in~\eqref{e.ahom7}, as well as those appearing below in the proof of Lemma~\ref{l.correctorsexist}, are not in general symmetric. However, since in this paper we only use the tensors to define~$\mathscr{A}$ (i.e, $\overline{\mathbf{a}}^{(k)}$ appears only in expressions of the form $\overline{\mathbf{a}}^{(k)}:\nabla^k u$), there is no need to distinguish between these tensors and their symmetric part. We will therefore abuse notation by assuming that all tensors are symmetric (in other words, one should take the symmetric part of any formula defining a tensor).
The next lemma gives us the existence of correctors of arbitrary degree.
\begin{lemma} \label{l.correctorsexist} For each $k \in \mathbb{N}$, there exist correctors $\phi^{(k)} \in H^1(\mathbb{R}^d / \mathbb{Z}^d; \mathbb{T}_k )$ with zero mean in $Q_1$ and tensors $\overline{\mathbf{a}}^{(k)} \in \mathbb{T}_k$ such that, for every~$n\in \mathbb{N}$ and~$p \in \mathbb{P}_n$, \begin{equation} \label{e.corr.eq.n}
-\nabla \cdot \left( \mathbf{a} \nabla \left( \sum_{k=0}^{n} \phi^{(k)} \, \colon \nabla^k p \right) \right) = - \sum_{k=2}^{n} \overline{\mathbf{a}}^{(k)} \, \colon \nabla^{k} p \quad \mbox{in} \ {\mathbb{R}^d}. \end{equation} Moreover, there exists a constant $C(d,\Lambda)<\infty$ such that, for every $k \in \mathbb{N}$, \begin{equation} \label{e.corrector.bounds}
\left\| \nabla \phi^{(k)} \right\|_{L^2(Q_1)}
+ \left| \overline{\mathbf{a}}^{(k)} \right| \leq C^k . \end{equation} \end{lemma}
\begin{proof} We proceed inductively. Let us assume that we have $\phi^{(k)} \in H^1(\mathbb{R}^d / \mathbb{Z}^d; \mathbb{T}_k)$ and $\overline{\mathbf{a}}^{(k)} \in \mathbb{T}_k$ for $k \in \{1,\ldots,m-1\}$, such that~\eqref{e.corr.eq.n} is valid for $n \in \{1,\ldots,m-1\}$. We will construct $\phi^{(m)} \in H^1(\mathbb{R}^d / \mathbb{Z}^d; \mathbb{T}_m )$ and $\overline{\mathbf{a}}^{(m)} \in \mathbb{T}_m$ such that~\eqref{e.corr.eq.n} is valid for $n=m$. The base case~$m=1$ for the induction is valid since $\overline{\mathbf{a}}^{(1)} = 0$ and $\phi^{(0)}=1$ by the equation of $\phi^{(1)}$ in~\eqref{e.correq_1}. Let us denote, for $n\in\{1,\ldots, m-1\}$, \begin{equation}
L_n[p] := \nabla \cdot \left( \mathbf{a} \nabla \left( \sum_{k=0}^{n} \phi^{(k)} \, \colon \nabla^k p \right) \right) - \sum_{k=2}^{n} \overline{\mathbf{a}}^{(k)} \, \colon \nabla^{k} p. \end{equation} Let $p \in \mathbb{P}_m$. Note that, for every $z\in{\mathbb{R}^d}$, we have that~$\widetilde p := p - p(\cdot-z) \in \mathbb{P}_{m-1}$. We set $p_m := \frac1{m!} \nabla^m p(0) : x^{\otimes m}$ and observe, by the induction assumption and the linearity of $L_{n}$, that $L_{m-1}[p] = L_{m-1}[p_m]$ and \begin{align} \notag
L_{m-1}[p] - L_{m-1}[p(\cdot-z)] = L_{m-1}[p - p(\cdot-z)] = 0. \end{align} Taking $z\in \mathbb{Z}^d$ above and using the periodicity of~$\mathbf{a}(\cdot)$, we deduce that $L_{m-1}[p_m]$ is a periodic distribution. We then define the tensor $\overline{\mathbf{a}}^{(m)}\in \mathbb{T}_m$ by \begin{equation*} \left( \overline{\mathbf{a}}^{(m)} \, \colon \nabla^m p \right) := \left\langle L_{m-1} [p_m] \right\rangle, \quad p\in\mathbb{P}_m. \end{equation*} It follows that $L_{m-1}[p] - \overline{\mathbf{a}}^{(m)} \, \colon \nabla^m p$ is periodic, belongs to $H^{-1}(\mathbb{R}^d / \mathbb{Z}^d )$, and has zero mean. Therefore, by Lax-Milgram lemma, the equation \begin{equation*}
\nabla \cdot \left( \mathbf{a} \nabla u_p \right) = \overline{\mathbf{a}}^{(m)} \, \colon \nabla^m p_m - L_{m-1}[p_m] \end{equation*} has a solution $u_p\in H^1(\mathbb{R}^d / \mathbb{Z}^d; \mathbb{T}_m )$ with zero mean and satisfying the bound \begin{equation*}
\left\| \nabla u_p \right\|_{L^2(Q_1)} \leq C \left\| \overline{\mathbf{a}}^{(m)} \, \colon \nabla^m p_m - L_{m-1}[p_m] \right\|_{H^{-1}(\mathbb{R}^d / \mathbb{Z}^d )}. \end{equation*} By the linearity of the map $p\mapsto u_p$ we may write this in tensor form, that is, $u_p = \phi^{(m)} \colon \nabla^m p $ for all $p \in \mathbb{P}_m$. This proves the induction step and completes the proof of the first statement.
The bound~\eqref{e.corrector.bounds} is straightforward to obtain by induction. \end{proof}
\subsection{Heterogeneous polynomials: the space $\mathbb{A}_m$}
We define~$\mathscr{A}$ to be the ``full'' higher-order macroscopic (or homogenized) operator \begin{equation} \label{e.def.scrA} \mathscr{A} u := \sum_{n=2}^\infty \left( \overline{\mathbf{a}}^{(n)} : \nabla^n u \right). \end{equation} This operator is a more precise large-scale approximation of the heterogenous operator~$\nabla \cdot \mathbf{a} \nabla$ than the usual homogenized operator~$\nabla \cdot \overline{\mathbf{a}}\nabla$, which is nothing other than the first summand on the right of~\eqref{e.def.scrA}. Unfortunately,~$\mathscr{A}$ is not elliptic. However, as we will see, in many situations it is dominated by its \emph{lowest}-order coefficients (at least for sufficiently regular data) and thus, in a certain sense,~$\nabla \cdot \overline{\mathbf{a}}\nabla$ is the leading order approximation of~$\mathscr{A}$. Therefore~$\mathscr{A}$ is relatively well-behaved if its domain is restricted to very regular functions.
For each $m\in\mathbb{N}$, define \begin{equation}
\mathbb{A}_m:= \left\{ \psi = \sum_{k=0}^m \left( \nabla^k q : \phi^{(k)} \right) \,:\, q\in \mathbb{P}_m \right\}. \end{equation} By Lemma~\ref{l.correctorsexist}, $\nabla \cdot \mathbf{a}\nabla \psi \in \mathbb{P}_{m-2}$ for every $\psi\in \mathbb{A}_{m}$. Indeed, for every $q\in \mathbb{P}_m$, \begin{equation} \label{e.poly.consistency} -\nabla \cdot \mathbf{a}\nabla \psi = -\mathscr{A}q \quad \mbox{where} \quad \psi = \sum_{k=0}^m \nabla^k q: \phi^{(k)}. \end{equation} In this sense, the approximation of~$\nabla \cdot \mathbf{a}\nabla$ by~$\mathscr{A}$ is exact, for elements of~$\psi$. The result of Avellaneda and Lin~\cite{AL2} tells us that the set~$\mathbb{A}_n$ precisely characterizes the set of solutions~$u$ of the equation \begin{equation} \label{e.pde.encour} -\nabla \cdot \mathbf{a}\nabla u = p \end{equation}
where~$p\in \mathbb{A}_{m-2}$ and $|u(x)|$ grows at most like~$o(|x|^{m+1})$ at infinity. In other words, for every $m\in\mathbb{N}$, \begin{align} \label{e.Liouville.poly} \left\{ u\in H^1_{\mathrm{loc}} ({\mathbb{R}^d}) \, :\,
-\nabla \cdot \mathbf{a}\nabla u \in\mathbb{P}_{m-2}, \
\limsup_{r\to \infty} r^{-(m+1)} \left\| u \right\|_{\underline{L}^2(B_r)} = 0 \right\}
= \mathbb{A}_m. \end{align} In particular, the set of~$\mathbf{a}(x)$--harmonic functions (solutions of~\eqref{e.pde.encour} with $p=0$) with at-most polynomial growth is characterized by \begin{align} \label{e.Liouville.poly.0} & \left\{ u\in H^1_{\mathrm{loc}} ({\mathbb{R}^d}) \, :\,
-\nabla \cdot \mathbf{a}\nabla u = 0, \
\limsup_{r\to \infty} r^{-(m+1)} \left\| u \right\|_{\underline{L}^2(B_r)} = 0 \right\} \\ & \notag \qquad\qquad\qquad\qquad = \left\{ u = \sum_{k=0}^m \left( \nabla^k q : \phi^{(k)} \right) \,:\, q\in \mathbb{P}_m, \ \mathscr{A}q = 0 \right\} = : \mathbb{A}^0_m. \end{align} This Liouville-type theorem for solutions with polynomial growth is a qualitative version of the $C^{m,1}$ estimate~\eqref{e.Cm1}. It is an immediate consequence of the weaker version of~\eqref{e.Cm1} in which the constant~$C$ on the right side is allowed to depend on~$m$. Due to~\eqref{e.Liouville.poly}, we informally refer to elements of~$\mathbb{A}_m$ as ``heterogeneous polynomials.'' We will give another proof of~\eqref{e.Liouville.poly} in this paper, as the argument for Theorem~\ref{t.analyticity} does not depend on it.
We remark that coefficients in the operator~$\mathscr{A}$ are not universal. Indeed, if one normalized the correctors differently or defined them with respect to the cube $Q_2$ instead of $Q_1$, then all of the homogenized coefficients would in general be different, with the exception of~$\overline{\mathbf{a}}^{(2)}$. The universal objects are the tensor~$\overline{\mathbf{a}}^{(2)}$ (the homogenized matrix) and the spaces~$\mathbb{A}_n$.
We note that there exists $C(d,\Lambda)<\infty$ such that, for every $m\in\mathbb{N}$, $\psi\in \mathbb{A}_m$ and $r\geq 1$, \begin{equation} \label{e.psi.easybound}
\left\| \psi \right\|_{\underline{L}^2(Q_r)} \leq \sum_{k=0}^m
\left( \frac{Cr}{k+1} \right)^k \left| \nabla^kq(0) \right|, \quad \mbox{where} \quad \psi = \sum_{k=0}^m \nabla^k q:\phi^{(k)},\quad q\in\mathbb{P}_m. \end{equation} This is an immediate consequence of~\eqref{e.corrector.bounds}.
In the rest of this section, we establish some properties of the space~$\mathbb{A}_m$ and macroscopic operator~$\mathscr{A}$. We first give a basic result concerning the Laplacian operator restricted to polynomials.
\begin{lemma} \label{l.polyharmonic} There exists $C(d,\Lambda)<\infty$ such that, for every~$m\in\mathbb{N}$ and~$p\in \mathbb{P}_m^*$, there exists $q\in \mathbb{P}_{m+2}^*$ satisfying \begin{equation} -\nabla \cdot \overline{\mathbf{a}}\nabla q = p \quad \mbox{in} \ {\mathbb{R}^d} \end{equation} and \begin{equation} \label{e.polyharmonic}
\left| \nabla^{m+2}q \right| \leq
C^m \left| \nabla^m p \right|. \end{equation}
\end{lemma} \begin{proof} By performing an affine change of coordinates, we may suppose that~$\overline{\mathbf{a}}=\mathbf{I}$, in other words,~$\nabla \cdot \overline{\mathbf{a}}\nabla = \Delta$. We claim that the polynomial $q\in \mathbb{P}_{m+2}^*$ defined by \begin{align} \notag
q(x) := - \sum_{j=0}^\infty a_{j,m} |x|^{2(j+1)} (-\Delta)^{j} p(x) \end{align} satisfies the equation \begin{equation} \label{e.Deltapisq} - \Delta q = p \quad \mbox{in} \ {\mathbb{R}^d}, \end{equation} where the coefficients $\{ a_{j,m}\}$ are defined, recursively by \begin{equation*} a_{-1,m} = 1, \quad a_{j,m} = \frac{1}{2(j+1)( 2(m-j) + d)} a_{j-1,m} , \quad j \in \mathbb{N}. \end{equation*} Notice that we have \begin{equation*} a_{j,m} = \prod_{i=0}^j \frac{1}{2(i+1)( 2(m-i) + d)} .
\end{equation*} To prove~\eqref{e.Deltapisq}, a direct computation gives that \begin{align} \notag \lefteqn{
\Delta \left( |x|^{2(j+1)} (-\Delta)^j p(x) \right) } \quad & \\ \notag &
= 2(j+1)|x|^{2j} \left( (2j+d)+ 2 x\cdot \nabla \right) (-\Delta)^j p(x) - |x|^{2(j+1)} (-\Delta)^{j+1} p(x) \\ \notag &
= 2(j+1) \left( 2(m - j) + d \right) |x|^{2j} (-\Delta)^j p(x) - |x|^{2(j+1)} (-\Delta)^{j+1} p(x) . \end{align} By the definition of $a_{j,m}$, we have that \begin{equation*} 2(j+1) \left( 2(m - j) + d\right) a_{j,m} = a_{j-1,m} \end{equation*} and, therefore, \begin{align} \label{e.Lap.telescoping}
\Delta \left( a_{j,m} |x|^{2(j+1)}(-\Delta)^{j} p(x) \right) & =
a_{j-1,m} |x|^{2j} (-\Delta)^{j} p(x) \\ \notag & \quad
- a_{j,m} |x|^{2(j+1)} (-\Delta)^{j+1} p(x) . \end{align} Thus,~\eqref{e.Deltapisq} follows by telescoping since $a_{-1,m} = 1$.
Next, taking gradients, we see that \begin{align*} \nabla^{m+2} q &
= \sum_{j=0}^\infty a_{j,m} \sum_{k=0}^{m+2} \binom{m+2}{k} \left( \nabla^{k} |x|^{2(j+1)} \otimes \nabla^{m+2-k} (-\Delta)^{j} p(x) \right)(0) \\ & = \sum_{j=0}^{\lfloor \frac m2 \rfloor} a_{j,m} \binom{m+2}{2(j+1)} (2(j+1))! I_d^{\otimes (j+1)} \otimes \nabla^{m - 2j} (-\Delta)^{j} p \\ & = \sum_{j=0}^{\lfloor \frac m2 \rfloor} (-1)^j a_{j,m} \frac{(m+2)!}{(m-2j)!} I_d^{\otimes (j+1)} \otimes \nabla^{m - 2j} (\Delta)^{j} p . \end{align*} The tensor $\nabla^{m - 2j} (-\Delta)^{j}$ can be estimated as \begin{equation*}
\left| I_d^{\otimes (j+1)} \otimes \nabla^{m - 2j} (-\Delta)^{j} p \right| \leq d^{\frac{j}2} |\nabla^m p| , \end{equation*}
and it follows that \begin{equation*}
\left| \nabla^{m+2} q \right| \leq |\nabla^m p| \sum_{j=0}^{\lfloor \frac m2 \rfloor} a_{j,m} C^j \frac{(m+2)!}{(m-2j)!} . \end{equation*} Now, we estimate \begin{equation*} \sum_{j=0}^{\lfloor \frac m2 \rfloor} a_{j,m}C^j \frac{(m+2)!}{(m-2j)!} = \sum_{j=0}^{\lfloor \frac m2 \rfloor} a_{j,m} (Cj)^j \binom{m+2}{2j} \leq C^m, \end{equation*} using \begin{equation*} (2j)! a_{j,m} = \frac1{2j(2m+d)} \prod_{i=0}^{j-1} \frac{ 2(i+1) (2i + 1)}{2(i+1)( 2(m-j) + 2 i + d)} \leq \frac1{2j(2m+d)} . \end{equation*} This proves~\eqref{e.polyharmonic}. \end{proof}
The following lemma asserts that the macroscopic operator~$\mathscr{A}$ may be inverted on the space of polynomials.
\begin{lemma} \label{l.twist} There exists $C(d,\Lambda)<\infty$ such that, for every~$m\in\mathbb{N}$ and~$p\in \mathbb{P}_m$, there exists $q\in \mathbb{P}_{m+2}$ satisfying \begin{equation} -\mathscr{A} q = p \quad \mbox{in} \ {\mathbb{R}^d} \end{equation} such that $q(0)=0$, $\nabla q(0)=0$ and, for every $n\in\{0,\ldots,m\}$, \begin{equation} \label{e.twist.bounds}
\left| \nabla^{n+2}q(0) \right| \leq \sum_{k=n}^m
C^{k} \left| \nabla^k p(0) \right|. \end{equation}
\end{lemma} \begin{proof} By Lemma~\ref{l.polyharmonic}, there exists $q_0\in \mathbb{P}_{m+2}$ satisfying \begin{equation} \left\{ \begin{aligned} & -\nabla \cdot \overline{\mathbf{a}} \nabla q_0 = p \\ & q_0(0)= 0, \ \nabla q_0(0)=0, \\ &
\left| \nabla^{n} q_0(0) \right|
\leq C^n \left|\nabla^{n-2} p(0) \right|, \ \ \forall n\in\{2,\ldots,m+2\}. \end{aligned} \right. \end{equation} Arguing inductively using Lemma~\ref{l.polyharmonic} and~\eqref{e.scrA.forward}, there exist~$q_k \in \mathbb{P}_{m+2-k}$ satisfying, for every $k\in\{ 1,\ldots,m+1\}$, \begin{equation} \left\{ \begin{aligned} & -\nabla \cdot \overline{\mathbf{a}} \nabla q_{k} = \mathscr{A}q_{k-1} - \nabla \cdot \overline{\mathbf{a}} \nabla q_{k-1}, \\ & q_k(0)= 0, \ \nabla q_k(0)=0, \\ &
\left| \nabla^{n} q_0(0) \right|
\leq C^{n+k} \left|\nabla^{n-2+k} p(0) \right|, \quad \forall n\in\{2,\ldots,m+2-k\}. \end{aligned} \right. \end{equation} Note that $q_{m+1}=0$. Thus, setting $q:= \sum_{k=0}^{m} q_k$, we obtain the lemma.
\end{proof}
We next compare the kernel of~$\mathscr{A}$ to that of~$\nabla \cdot \overline{\mathbf{a}}\nabla$ and show that they agree at leading order.
\begin{lemma} \label{l.Akernel} There exists $C(d,\Lambda)<\infty$ such that, for every $m\in\mathbb{N}$ and $p\in \mathbb{P}_m$ satisfying~$-\nabla \cdot \overline{\mathbf{a}}\nabla p = 0$, there exists $p' \in \mathbb{P}_{m}$ satisfying \begin{equation} -\mathscr{A}p' = 0, \end{equation} and \begin{equation} p'(0)=p(0), \quad \nabla p'(0) = \nabla p(0) \quad \mbox{and} \quad \nabla^m p' = \nabla^m p \end{equation} such that, for every $n\in\{2,\ldots,m-1\}$, \begin{equation} \label{e.twist.it}
\left| \nabla^{n}(p-p')(0) \right| \leq
\sum_{k=n+1}^{m} C^{k} \left| \nabla^k p(0) \right|. \end{equation} In particular, for every $r > Cm$, \begin{equation} \label{e.twist.it.far}
\left\| p - p' \right\|_{L^\infty(Q_r)} \leq \sum_{k=3}^{m}
\left( \frac{Cr}{k} \right)^{k-1} \left| \nabla^k p(0) \right|. \end{equation} \end{lemma} \begin{proof} We may suppose $m\geq 3$, since $\mathscr{A}$ and $\nabla \cdot \mathbf{a}\nabla$ coincide on~$\mathbb{P}_{2}$. Let~$p\in \mathbb{P}_m$ be $\overline{\mathbf{a}}$-harmonic. Then $\mathscr{A} p \in \mathbb{P}_{m-3}$ and thus, by applying Lemma~\ref{l.twist} and using~\eqref{e.scrA.forward}, we can find $q \in \mathbb{P}_{m-1}^*$ satisfying \begin{equation} -\mathscr{A}q = -\mathscr{A}p \quad \mbox{in} \ {\mathbb{R}^d} \end{equation} such that $q(0)=0$, $\nabla q(0)= 0$ and, for every $n\in\{3,\ldots,m\}$, \begin{align} \label{e.twisties}
\left| \nabla^{n-1}q(0) \right| & \leq \sum_{k=n-3}^{m-3}
C^{k} \left| \nabla^k \mathscr{A} p(0) \right| \\ & \notag \leq \sum_{k=n-3}^{m-3} C^{k}
\sum_{j=k+3}^m C^{j} \left| \nabla^j p(0) \right| \\ & \notag
\leq \sum_{j=n}^{m} \sum_{k=n-3}^{j-3}
C^j \left| \nabla^j p(0) \right| \leq
\sum_{j=n}^{m} C^{j} \left| \nabla^j p(0) \right|. \end{align} Setting $p':= p-q$ yields~\eqref{e.twist.it}.
To get~\eqref{e.twist.it.far}, we estimate, for $|x|>Cm$, \begin{align*}
\left| p(x) - p'(x) \right| & =
\left| \sum_{n=2}^{m-1} \nabla^n(p-p')(0) : \frac{x^{\otimes n}}{n!}
\right| \\ & \leq \sum_{n=2}^{m-1}
\left( \frac{C |x|}{n} \right)^n
\left| \nabla^n(p-p')(0) \right| \\ & \leq \sum_{n=2}^{m-1}
\left( \frac{C |x|}{n} \right)^n
\sum_{j=n+1}^{m} C^{j} \left| \nabla^j p(0) \right| \\ & \leq \sum_{j=3}^{m} \sum_{n=2}^{j-1}
\left( \frac{C |x|}{n} \right)^n C^{j} \left| \nabla^j p(0) \right| \\ & \leq \sum_{j=3}^{m}
\left( \frac{C|x|}{j} \right)^{j-1} \left| \nabla^j p(0) \right|. \end{align*} This completes the proof. \end{proof}
\begin{lemma} \label{l.polyhit.bound} There exists $C(d,\Lambda)<\infty$ such that, for every $m\in\mathbb{N}$ and~$p\in\mathbb{P}_m$, there exists $\psi \in \mathbb{A}_{m+2}$ satisfying \begin{equation} \label{e.polyhit} \left\{ \begin{aligned} & -\nabla \cdot \mathbf{a}\nabla \psi = p \quad \mbox{in} \ {\mathbb{R}^d} \\ & \widehat{\psi}(0) = 0, \ D\widehat{\psi}(0) = 0, \end{aligned} \right. \end{equation} and such that, for every $r\in [m,\infty)$, \begin{equation} \label{e.polyhit.bounds}
\left\| \psi \right\|_{\underline{L}^2(Q_r)} \leq
\sum_{n=0}^m \left( \frac{Cr}{n+1} \right)^{n+2} \left| \nabla^np(0) \right|. \end{equation} \end{lemma} \begin{proof} By Lemma~\ref{l.twist}, we can find~$q\in \mathbb{P}_{m+2}$ satisfying $-\mathscr{A}q = p$, $\widehat{q}(0)=0$, $D\widehat{q}(0)=0$ and, for every $n\in\{0,\ldots,m\}$, \begin{equation} \label{e.tw2}
\left| \nabla^{k+2}q(0) \right| \leq \sum_{n=k}^m
C^{n} \left| \nabla^n p(0) \right|. \end{equation} Define $\psi:= \sum_{k=0}^{m+2} \nabla^k q:\phi^{(k)}$. It is clear from~\eqref{e.poly.consistency} that~$\psi$ satisfies~\eqref{e.polyhit}. By~\eqref{e.psi.easybound} and~\eqref{e.tw2}, for every $r\in [m,\infty)$, \begin{align*}
\left\| \psi \right\|_{\underline{L}^2(Q_r)} & \leq \sum_{k=2}^{m+2}
\left( \frac{Cr}{k+1} \right)^k \left| \nabla^kq(0) \right| \\ & \leq \sum_{k=0}^m \sum_{n=k}^m \left( \frac{Cr}{k+1} \right)^{k+2}
C^{n-k} \left| \nabla^n p(0) \right| \\ & = \sum_{n=0}^m \sum_{k=0}^n \left( \frac{Cr}{k+1} \right)^{k+2}
C^{n-k} \left| \nabla^n p(0) \right| \\ & \leq \sum_{n=0}^m \left( \frac{Cr}{n+1} \right)^{n+2}
\left| \nabla^n p(0) \right|. \end{align*} The proof is complete. \end{proof}
\begin{remark} Notice that we can also put the right hand side of the estimate~\eqref{e.polyhit.bounds} in terms of the differences~$D^np(0)$ rather than the pointwise derivatives~$\nabla^np(0)$. That is, we can replace~\eqref{e.polyhit.bounds} by \begin{equation} \label{e.polyhit.bounds.D}
\left\| \psi
\right\|_{\underline{L}^2(Q_r)}
\leq C \sum_{n=0}^m \left( \frac{Cr}{n+1} \right)^{n+2} \left| D^np(0) \right|. \end{equation} To see this, we first notice that, for any polynomial $p\in\mathbb{P}_m$, \begin{align} \label{e.poly.nab.D}
\left| \nabla^n p(0) \right| & \leq \sum_{k=n}^m
\frac{C^{k-n}}{(k-n)!} \left|D^{k} p(0) \right| k^{k-n}
\leq \sum_{k=n}^m
\left( \frac{Ck}{k-n} \right)^{k-n}\left|D^{k} p(0) \right|. \end{align} Thus, for every $\psi$ and $p$ as in the statement of Lemma~\ref{l.polyhit.bound} and~$r\in [Cm,\infty)$, we may combine~\eqref{e.polyhit.bounds} and~\eqref{e.poly.nab.D} to obtain \begin{align} \label{e.polyhit.bounds2}
\left\| \psi
\right\|_{\underline{L}^2(Q_r)} &
\leq C \sum_{n=0}^m \left( \frac{Cr}{n+1} \right)^{n+2} \left| \nabla^np(0) \right| \\ & \notag \leq C \sum_{n=0}^m \sum_{k=n}^m \left( \frac{Cr}{n+1} \right)^{n+2}
\left( \frac{Ck}{k-n} \right)^{k-n}\left|D^{k} p(0) \right| \\ & \notag = C \sum_{k=0}^m \left( \frac{Cr}{k+1} \right)^{k+2}
\left|D^{k} p(0) \right| \sum_{n=0}^k \left( \frac{Ckn}{r(k-n)} \right)^{k-n} \\ & \notag \leq C \sum_{k=0}^m \left( \frac{Cr}{k+1} \right)^{k+2}
\left|D^{k} p(0) \right|. \end{align} This is~\eqref{e.polyhit.bounds.D}. \end{remark}
For $u \in L^1(U)$ and $z\in \mathbb{Z}^d$ such that $z+Q_1\subseteq U\subseteq{\mathbb{R}^d}$, we define \begin{equation} \widehat{u}(x) : = \int_{z+Q_1} u(x)\,dx \quad \mbox{for } \, x\in z +Q_1. \end{equation} We next estimate the growth of an element~$\psi\in \mathbb{A}_m$ by its \emph{intrinsic} differences at the origin, namely $\{ D^k\widehat{\psi}(0)\}_{k=0,\ldots,m}$, rather than the differences (or derivatives) of the polynomial~$q$ in its representation, as for instance in~\eqref{e.psi.easybound}.
\begin{lemma} \label{l.corr.growth} There exists $C(d,\Lambda)<\infty$ such that, for every $m,n\in\mathbb{N}$ with $n\leq m$, $q\in \mathbb{P}_m$ and $\psi\in \mathbb{A}_m$ satisfying \begin{equation} \label{e.qpsi.relate} \psi = \sum_{k=0}^m \nabla^kq:\phi^{(k)}, \end{equation} we have the estimate \begin{equation} \label{e.corr.growth.q} \left( \frac{m}{n+1} \right)^n
\left| \nabla^nq(0) \right| \leq \sum_{k=n}^m \left( \frac{Cm}{k+1} \right)^{k}
\left| D^k \widehat{\psi} (0) \right| \end{equation} and, consequently, for every $r\geq m$, \begin{equation} \label{e.corr.growth}
\left\| \psi \right\|_{\underline{L}^2(Q_r)} \leq \sum_{k=0}^m \left( \frac{Cr}{k+1} \right)^k
\left| D^k \widehat{\psi} (0) \right|. \end{equation} Moreover, for symmetric tensors $M^{(0)},\ldots,M^{(m)}$ with $M^{(k)} \in ({\mathbb{R}^d})^{\otimes k}$, there exists a unique $\psi\in \mathbb{A}_m$ such that \begin{equation} \label{e.slap.ya} D^k \widehat{\psi} (0) = M^{(k)}, \qquad \forall k\in\{0,\ldots,m\}. \end{equation} \end{lemma} \begin{proof} We proceed by first proving the estimate~\eqref{e.corr.growth.q} by induction in~$m$. The statement is clearly true for $m\in\{0,1\}$. Suppose, for some $m\in\mathbb{N}$ with $m\geq 1$, that~\eqref{e.corr.growth} is valid for~$m-1$ in place of~$m$, that is, there is $\mathsf{C}(d,\Lambda)<\infty$ such that, for every $\psi \in \mathbb{A}_{m-1}$ and~$q\in \mathbb{P}_{m-1}$ satisfying~\eqref{e.qpsi.relate} and $n\in\{0,\ldots,m-1\}$, we have \begin{equation} \label{e.corr.growth.ind.q} \left( \frac{m-1}{n+1} \right)^n
\left| \nabla^nq(0) \right| \leq \sum_{k=n}^{m-1} \left( \frac{ \mathsf{C} (m-1)}{k+1} \right)^{k}
\left| D^k \widehat{\psi} (0) \right|. \end{equation} Let $\psi \in \mathbb{A}_{m}$ and~$q\in \mathbb{P}_{m}$ be as in the lemma. Let $q_m$ be the unique polynomial of degree~$m$ satisfying \begin{equation} D^mq_m = D^m\psi \quad \mbox{and} \quad D^kq_m(0) = 0, \ \ \forall k\in\{0,\ldots,m-1\}, \end{equation} and let~$\zeta \in \mathbb{A}_{m}$ be defined by \begin{equation} \zeta := \sum_{k=0}^{m} \nabla^k q_m : \phi^{(k)}. \end{equation} Then, for every $k,n\in\{0,\ldots,m\}$ with $k+n\leq m$ and $x\in{\mathbb{R}^d}$, \begin{equation} \label{e.nabkDn.bound}
\left| \nabla^k D^n q_m(x) \right| \leq
\frac{C^{m-n-k}}{(m-n-k)!} \left|D^{m} \psi \right| (k+|x|)^{m-n-k}. \end{equation} Hence, by~\eqref{e.corrector.bounds} and~\eqref{e.psi.easybound}, we get, for every $r\geq m$ and~$n\in\{0,\ldots,m\}$,
\begin{align} \label{e.zeta.bound}
\left| D^n \widehat\zeta(0) \right| & \leq
\left\| D^n \zeta \right\|_{\underline{L}^2(Q_1)} \\ & \notag \leq
\sum_{k=0}^{m-n} \left\| \nabla^k D^nq_m \, \colon \phi^{(k)} \right\|_{\underline{L}^2(Q_1)} \\ & \notag \leq
\left|D^{m}\psi \right| \sum_{k=0}^{m-n} C^k \left( \frac{C(k+1)}{m-n-k} \right)^{m-n-k} \\ & \notag \leq C^{m-n}
\left|D^{m} \psi \right|. \end{align}
Since $\psi - \zeta \in \mathbb{A}_{m-1}$, by the induction hypothesis~\eqref{e.corr.growth.ind.q},~\eqref{e.nabkDn.bound} and~\eqref{e.zeta.bound}, we obtain, for every~$n\in\{0,\ldots,m-1\}$, \begin{align*} \lefteqn{
\left( \frac{m}{n+1} \right)^n\left| \nabla^n q(0) \right| } \quad & \\ & \leq
\left( \frac{m}{n+1} \right)^n\left| \nabla^n (q-q_m)(0) \right|
+ \left( \frac{m}{n+1} \right)^n\left| \nabla^n q_m(0) \right| \\ & \leq \left( \frac{m}{m-1} \right)^n\, \sum_{k=n}^{m-1} \left( \frac{ \mathsf{C} (m-1)}{k+1} \right)^{k} \left(
\left| D^k \widehat{\psi}(0) \right| + \left| D^k \widehat{\zeta}(0) \right|\right) + \left( \frac{Cn}{m-n} \right)^{m-n}
\left| D^m \psi \right| \\ & \leq \sum_{k=n}^{m-1} \left( \frac{ \mathsf{C}m}{k+1} \right)^{k}
\left| D^k \widehat{\psi}(0) \right| +
\left| D^m \psi \right| \sum_{k=n}^{m-1} C^{m-k} \left( \frac{ \mathsf{C}m}{k+1} \right)^{k} +
C^m \left| D^m \psi \right|. \end{align*} If~$\mathsf{C}$ is sufficiently large ($\mathsf{C} \geq 8 e C$ suffices\footnote{Details can be found in a commented-out portion of the latex source of this paper, which can be downloaded from the arXiv.}), then the second term on the right side is bounded by \begin{align*} \sum_{k=n}^{m-1} C^{m-k} \left( \frac{ \mathsf{C}m}{k+1} \right)^{k} \leq \frac12\left( \frac{\mathsf{C}m}{m+1} \right)^{m}. \end{align*}
Combining the above and requiring $\mathsf{C}$ to be sufficiently large once again, we obtain that, for every~$n\in\{0,\ldots,m-1\}$, \begin{equation} \label{e.indystepcomplete} \left( \frac{m}{n+1} \right)^n
\left| \nabla^n q(0) \right| \leq \sum_{k=n}^{m} \left( \frac{ \mathsf{C} m}{k+1} \right)^{k}
\left| D^k \widehat{\psi} (0) \right|. \end{equation} Note that the bound~\eqref{e.indystepcomplete} for $n=m$ was proved already in~\eqref{e.zeta.bound}. This completes the induction step and thus the proof of~\eqref{e.corr.growth.q}.
The second estimate~\eqref{e.corr.growth} is an immediate consequence of~\eqref{e.psi.easybound} and~\eqref{e.corr.growth.q}. Indeed, combining these yields, for every $r\geq m$, \begin{align*}
\left\| \psi \right\|_{\underline{L}^2(Q_r)} & \leq \sum_{k=0}^m
\left( \frac{Cr}{k+1} \right)^k \left| \nabla^k q(0) \right| \\ & \leq \sum_{k=0}^m \sum_{n=k}^m \left( \frac{Cr}{m} \right)^k \left( \frac{Cm}{n+1} \right)^n
\left| D^n \widehat{\psi}(0) \right| \\ & = \sum_{n=0}^m \sum_{k=0}^n \left( \frac{Cr}{m} \right)^k \left( \frac{Cm}{n+1} \right)^n
\left| D^n \widehat{\psi}(0) \right| \\ & \leq \sum_{n=0}^m \left( \frac{Cr}{m} \right)^n \left( \frac{Cm}{n+1} \right)^n
\left| D^n \widehat{\psi}(0) \right| = \sum_{n=0}^m \left( \frac{Cr}{n+1} \right)^n
\left| D^n \widehat{\psi}(0) \right| . \end{align*}
Turning to the proof of the last statement, we note that uniqueness is clear from the fact that $\psi \in \mathbb{A}_{m}$ and $D^m\widehat{\psi}(0)=0$ implies that $\psi \in \mathbb{A}_{m-1}$. We prove the existence part of the second statement by induction. Its validity is clear for~$m\in\{0,1\}.$ We therefore suppose, for some~$m\in\mathbb{N}$ with~$m\geq 1$, the statement is valid for~$m-1$ in place of~$m$. Let $M^{(0)},\ldots,M^{(m)}$ be as in the statement and define~$q\in \mathbb{P}_{m}^*$ and~$\zeta \in \mathbb{A}_{m}$ by \begin{equation}
q(x):= \frac1{m!} M^{(m)} : x^{\otimes m} \quad \mbox{and} \quad \zeta(x):= \sum_{k=0}^{m} \nabla^k q(x) : \phi^{(k)}(x). \end{equation} It is clear that~$D^m\zeta = M^{(m)}$. By the induction hypothesis, there exists~$\xi\in \mathbb{A}_{m-1}$ such that \begin{equation}
D^k \widehat{\xi}(0) = M^{(k)} - D^k \widehat{\zeta}(0), \quad \forall k\in\{0,\ldots,m-1\}. \end{equation} Defining $\psi:= \zeta + \xi \in \mathbb{A}_{m}$, we obtain~\eqref{e.slap.ya}. The proof of the second statement, and thus of the lemma, is now complete. \end{proof}
\subsection{Entire solutions with slow exponential growth: the space $\mathbb{A}_\infty(\delta)$}
Recall that, for each $\delta>0$, we denote \begin{equation} \label{e.Pinftydelta} \mathbb{P}_\infty(\delta) := \left\{ w \, : \, w = \sum_{n=0}^\infty w_n, \ w_n \in \mathbb{P}_n^*, \ \sum_{n=0}^\infty
\delta^{-n} \left| \nabla^n w_n(0) \right| < \infty \right\}. \end{equation} The linear space $\mathbb{P}_\infty(\delta)$ is a Banach space with respect to the norm \begin{equation}
\left\| u \right\|_{\mathbb{P}_\infty(\delta)} := \sum_{n=0}^\infty
\delta^{-n} \left| \nabla^n u(0) \right|. \end{equation} By~\eqref{e.corrector.bounds}, there exists $\delta_0(d,\Lambda)>0$ such that~$\mathscr{A}u$ is well-defined for every $u\in \mathbb{P}_\infty(\delta_0)$ and moreover maps~$\mathbb{P}_\infty(\delta)$ to itself for every $\delta \in (0,\delta_0]$. Indeed, if $\delta$ is sufficiently small, then, for every $u\in \mathbb{P}_{\infty}(\delta)$ and $k\in\mathbb{N}$, \begin{align} \label{e.scrA.forward}
\left\| \mathscr{A} u \right\|_{\mathbb{P}_\infty(\delta)} & = \sum_{n=0}^\infty
\delta^{-n} \left| \nabla^n \mathscr{A} u(0) \right| \\ & \notag \leq \sum_{n=0}^\infty \sum_{k=n+2}^\infty
\delta^{-n} C^{k} \left| \nabla^k u(0) \right| \\ & \notag \leq \sum_{k=2}^\infty \sum_{n=0}^{k-2} (\delta C)^{k}
\delta^{-k} \left| \nabla^k u(0) \right|
\leq C \delta^{2} \left\| u \right\|_{\mathbb{P}_\infty(\delta)} . \end{align} We say that $u\in \mathbb{P}_\infty(\delta)$ is \emph{$\mathscr{A}$--harmonic} if $\mathscr{A} u = 0$.
For $\delta \in (0,\delta_0]$, we define the space \begin{equation}
\mathbb{A}_\infty(\delta):= \left\{ \psi = \sum_{k=0}^\infty \nabla^k u:\phi^{(k)} \, :\, u \in \mathbb{P}_\infty(\delta) \right\}, \end{equation} where the $\delta_0(d,\Lambda)>0$ is taken small enough that the sum is absolutely convergent, locally uniformly in~${\mathbb{R}^d}$. Indeed, observe that if~$\delta_0(d,\Lambda)>0$ is sufficiently small and $\delta\in (0,\delta_0]$, then, for every $u\in \mathbb{P}_\infty(\delta)$, \begin{align*}
\left\| \sum_{k=0}^\infty \nabla^k u:\phi^{(k)}
\right\|_{\underline{L}^2(Q_r)} & \leq \sum_{k=0}^\infty C^k
\left\| \nabla^k u \right\|_{L^\infty(Q_r)} \\ & \leq \sum_{k=0}^\infty C^k \sum_{n=k}^\infty
\left( \frac{Cr}{n-k+1} \right)^{n-k} \left| \nabla^nu(0) \right| \\ & \leq
\left\| u \right\|_{\mathbb{P}_\infty(\delta)} \sum_{k=0}^\infty (C\delta)^k \sum_{n=k}^\infty \left( \frac{C\delta r}{n-k+1} \right)^{n-k} \\ & \leq
\left\| u \right\|_{\mathbb{P}_\infty(\delta)} \exp\left( C\delta r \right) \sum_{k=0}^\infty (C\delta)^k \\ & \leq
\left\| u \right\|_{\mathbb{P}_\infty(\delta)} \exp\left( C\delta r \right). \end{align*} In particular, an element $\psi\in \mathbb{A}_\infty(\delta)$ grows at most like a slow exponential: \begin{equation} \limsup_{r\to \infty} \, \exp(-C\delta r)
\left\| \psi \right\|_{\underline{L}^2(Q_r)} < \infty . \end{equation}
\section{Large-scale analyticity} \label{s.analyticity}
In this section we prove the main result of the paper, namely the quantitative, large-scale analyticity of solutions (Theorem~\ref{t.analyticity}). The first step is to \emph{control the low frequencies} by obtaining regularity of solutions restricted to the lattice~$\mathbb{Z}^d$. This is accomplished in Lemma~\ref{l.diffbound}, below, the proof of which is based on the simple idea of Moser and Struwe~\cite{MS} of exploiting the periodic structure of the equation to obtain estimates on integer finite differences of solutions which are analogous to the classical pointwise bounds for the derivatives of harmonic functions. The second step of obtaining control the \emph{high frequencies} is more involved and the focus of most of the section.
\begin{lemma}[Bounds for iterated differences] \label{l.diffbound} There exists $C(d,\Lambda) <\infty$ such that, for every $m\in\mathbb{N}$, $R \geq m+2$ and $u \in H^1(Q_R)$ satisfying \begin{equation} \label{e.pdediffs} -\nabla \cdot \mathbf{a}\nabla u = 0\quad \mbox{in} \ Q_R, \end{equation} we have the estimate \begin{equation} \label{e.diffbound}
\left\| D^m u \right\|_{L^2(Q_1)} \leq \left( \frac{Cm}{R} \right)^m
\left\| u \right\|_{\underline{L}^2(Q_R)}. \end{equation} \end{lemma} \begin{proof} The argument is essentially identical to the one for harmonic functions, as presented for instance in~\cite[Section 2.2.3.c]{Evans}. We first prove the case $m=1$.
Periodicity implies that~$Du$ is also a solution of~\eqref{e.pdediffs}, and therefore we may apply the Jensen and Caccioppoli inequalities to get, for $R>2$ and $r \in [1, R-2)$, \begin{equation} \label{e.caccforDu}
\left\| Du \right\|_{\underline{L}^2(Q_{r})} \leq
\left\| \nabla u \right\|_{\underline{L}^2(Q_{r+2})} \leq \frac{C}{R-r-2}
\left\| u \right\|_{\underline{L}^2(Q_{R})}, \end{equation} and then the De Giorgi-Nash $L^\infty$--$L^2$ estimate to obtain \begin{equation*}
\left\| Du \right\|_{L^2(Q_{1})} \leq
\left\| Du \right\|_{L^\infty(Q_{R/2})}
\leq C \left\| Du \right\|_{\underline{L}^2(Q_{3R/4})} \leq \frac{C}{R}
\left\| u \right\|_{\underline{L}^2(Q_{R})}. \end{equation*} This gives the result in the case $m=1$.
We argue by induction to obtain the result for general~$m$. Without loss of generality, we may assume that $R \geq 4m$ since otherwise the result follows simply by the triangle inequality. Assuming the statement is true for~$m$ with constant $C_0$ and again using the fact that $Du$ is a solution, we apply the Caccioppoli inequality~\eqref{e.caccforDu} to obtain, for every~$R \geq 4m$, \begin{align*}
\left\| D^{m+1} u \right\|_{L^2(Q_{1})} & \leq \left( \frac{C_0 (m+2)}{R} \right)^m
\left\| D u \right\|_{\underline{L}^2(Q_{(1-\frac{2}{m+2})R})} \\ & \leq \left( \frac{C_0(m+2)}{R} \right)^m
\frac{C(m+2)}{R} \left\| u \right\|_{\underline{L}^2(Q_{R})} \\ & \leq \left( \frac{C^{\frac1{m+1}} C_0^{\frac{m}{m+1}} (m+1)}{R} \right)^{m+1}
\left\| u \right\|_{\underline{L}^2(Q_{R})}. \end{align*} If $C_0 \geq C$ is sufficiently large, we obtain the statement for $m+1$. \end{proof}
We next present a simple but useful lemma which estimates the $L^2$ norm of a function in a cube~$Q_r$ by the values of $\widehat{u}$ on the lattice $Q_r\cap \mathbb{Z}^d$ and the $L^2$ norm of its gradient. This ``small-scale Poincar\'e inequality'' is the basic tool we use to control the high frequencies of the solutions.
\begin{lemma} \label{l.jumpdown} There exists $C(d)<\infty$ such that, for every~$r\in\mathbb{N}$ with $r\geq 1$ and~$u\in H^1(Q_r)$, \begin{equation} \label{e.jumpdown}
\left\| u \right\|_{\underline{L}^2(Q_r)} \leq
\left( \frac{1}{|Q_r|} \sum_{z\in \mathbb{Z}^d\cap Q_r}
\left| \widehat{u}(z) \right|^2 \right)^{\frac12} + C
\left\| \nabla u\right\|_{\underline{L}^2(Q_r)}. \end{equation} \end{lemma} \begin{proof} Using the triangle inequality and the Poincar\'e inequality, we compute \begin{align} \label{e.jumpdown.p}
\left\| u \right\|_{\underline{L}^2(Q_r)} & =
\left( \frac{1}{|Q_r|} \sum_{z\in \mathbb{Z}^d\cap Q_r}
\left\| u \right\|_{{L}^2(z+Q_1)}^2 \right)^{\frac12} \\ & \notag \leq
\left( \frac{1}{|Q_r|} \sum_{z\in \mathbb{Z}^d\cap Q_r}
\left( \left| \widehat{u}(z) \right|^2 +
\left\| u - \widehat{u}(z) \right\|_{{L}^2(z+Q_1)}^2\right) \right)^{\frac12} \\ & \notag \leq
\left( \frac{1}{|Q_r|} \sum_{z\in \mathbb{Z}^d\cap Q_r}
\left|\widehat{u}(z) \right|^2 \right)^{\frac12} + C \left(
\frac{1}{|Q_r|} \sum_{z\in \mathbb{Z}^d\cap Q_r}
\left\| \nabla u \right\|_{{L}^2(z+Q_1)}^2 \right)^{\frac12} \\ & \notag =
\left( \frac{1}{|Q_r|} \sum_{z\in \mathbb{Z}^d\cap Q_r}
\left| \widehat{u}(z) \right|^2 \right)^{\frac12} + C
\left\| \nabla u \right\|_{\underline{L}^2(Q_r)}. \qedhere \end{align} \end{proof}
\begin{remark} While the De Giorgi-Nash estimate was used in the proof of Lemma~\ref{l.diffbound}, one can avoid this (and thus have an argument which works for systems) by instead relying on Lemma~\ref{l.jumpdown}. One proceeds by first iterating~\eqref{e.caccforDu} many times (depending on $d$) and then applying a version of the Sobolev inequality on~$\mathbb{Z}^d$ to get a uniform estimate on~$D\widehat{u}$. Then Lemma~\ref{l.jumpdown} can be invoked to take care of the small scales. \end{remark}
For expository purposes, and as a warm-up to the proof of Theorem~\ref{t.analyticity}, we next give a simple new proof of the large-scale~$C^{0,1}$ and $C^{1,1}$ estimates of Avellaneda-Lin~\cite{AL1}. All previous proofs rely on $\overline{\mathbf{a}}$--harmonic approximation via homogenization, either via a compactness argument~\cite{AL1} or a quantitative homogenization argument~\cite{AS}. Here we give a more direct and elementary argument which makes no (explicit) use of homogenization or $\overline{\mathbf{a}}$--harmonic approximation, and yet is completely quantitative. The proof uses only the previous two lemmas and the existence of first-order correctors.
\begin{lemma}[Large-scale $C^{0,1}$ and $C^{1,1}$ estimates] \label{l.C01C11} There exists $C(d,\Lambda)<\infty$ such that, for every~$R\in [2,\infty)$ and solution $u\in H^1(B_R)$ of the equation \begin{equation} \label{e.pde3} -\nabla \cdot \mathbf{a}\nabla u = 0\quad \mbox{in} \ Q_R, \end{equation} we have, for every $r\in [1,R]$ \begin{equation} \label{e.C01}
\left\| u - \widehat{u}(0) \right\|_{\underline{L}^2(Q_r)} \leq
\frac{Cr}{R} \left\| u \right\|_{\underline{L}^2(Q_R)} \end{equation} and, with $\psi \in \mathbb{A}_{1}$ such that $\widehat{\psi}(0) = \widehat{u}(0)$ and $D\widehat \psi(0) = D\widehat{u}(0)$, \begin{equation} \label{e.C11}
\left\| u - \psi \right\|_{\underline{L}^2(Q_r)} \leq
\frac{Cr^2}{R^2} \left\| u \right\|_{\underline{L}^2(Q_R)}. \end{equation} \end{lemma} \begin{proof} Assume that~$R\geq 4$ and~$u\in H^1(B_R)$ is a solution of~\eqref{e.pde3}.
\emph{Step 1.} The proof of~\eqref{e.C01}. By~\eqref{e.diffbound} for $m=1$, \begin{align} \label{e.applydiffbounds}
\left\| Du \right\|_{L^\infty(Q_{3R/4})} \leq
\frac{C}{R} \left\| u \right\|_{\underline{L}^2(Q_R)}. \end{align} This implies that, for every $r\in [1,\tfrac12R]$, \begin{equation} \sup_{z\in Q_r}
\left| \widehat{u}(z) - \widehat{u}(0) \right| \leq
Cr \sup_{z\in Q_{r+1}} \left| D\widehat{u}(z) \right| \leq
\frac{Cr}{R} \left\| u \right\|_{\underline{L}^2(Q_R)}. \end{equation} Combining this with~\eqref{e.jumpdown}, we obtain, for every~$r\in \left[1,\tfrac12 R\right] \cap \mathbb{N}$, \begin{align} \label{e.iterready.0}
\left\| u - \widehat{u}(0) \right\|_{\underline{L}^2(Q_r)} \leq
\frac{Cr}{R} \left\| u \right\|_{\underline{L}^2(Q_R)} +
C \left\| \nabla u \right\|_{\underline{L}^2(Q_r)}. \end{align}
Testing~\eqref{e.pde3} with $\phi^2u$, where~$\phi\in C^\infty_c({\mathbb{R}^d})$ is chosen to satisfy $\mathds{1}_{Q_r} \leq \phi \leq \mathds{1}_{Q_{r+s}}$ and $\| \nabla \phi \|_{L^\infty({\mathbb{R}^d})} \leq Cs^{-1}$, we obtain, for every $r\in \left[1,\tfrac12 R \wedge (R-s)\right]$, \begin{equation} \label{e.cacc.your.u}
\left\| \nabla u \right\|_{\underline{L}^2(Q_r)}^2 \leq \frac{C}{s}
\left\| u \right\|_{\underline{L}^2(Q_{r+s})}^2. \end{equation} Requiring $s=s(d,\Lambda)<\infty$ to be large enough that $Cs^{-1}\leq \frac12$ and combining the result with~\eqref{e.iterready.0}, we obtain, for every $r\in \left[1,\tfrac12 R \wedge(R-s)\right]$, \begin{equation} \label{e.iterready}
\left\| u - \widehat{u}(0) \right\|_{\underline{L}^2(Q_r)} \leq
\frac{Cr}{R} \left\| u \right\|_{\underline{L}^2(Q_R)} +
\frac12 \left\| u - \widehat{u}(0) \right\|_{\underline{L}^2(Q_{r+s})}. \end{equation} An iteration of~\eqref{e.iterready} implies that~\eqref{e.C01} is valid for every $r\in \left[1,\frac12 R \wedge (R-s)\right]$, which also requires that $R \geq s+1$. Since $s\leq C$, these restrictions may be removed by enlarging the constant~$C$ in~\eqref{e.C01}.
\emph{Step 2.} The proof of~\eqref{e.C11}. We suppose that~$R\geq 4+s$.
Applying~\eqref{e.diffbound} for $m=2$, we have, for every~$r\in\left[1,\tfrac12 R\right]$, \begin{align} \label{e.applydiffbounds.11}
\left\| D^2u \right\|_{L^\infty(Q_{3R/4})} \leq
\frac{C}{R^2} \left\| u \right\|_{\underline{L}^2(Q_R)}. \end{align} By~$(\widehat{u}-\widehat{\psi})(0)=0$, $D(\widehat{u}-\widehat{\psi})(0)=0$ and $D^2\psi = 0$, for every $r\in [1,R-2]$, \begin{equation} \sup_{z\in Q_r}
\left| \widehat{u}(z)-\widehat{\psi}(z) \right| \leq
Cr^2 \sup_{z\in Q_{r+2}} \left| D^2\widehat{u}(z) \right| \leq
\frac{Cr^2}{R^2} \left\| u \right\|_{\underline{L}^2(Q_R)}. \end{equation} Applying~\eqref{e.jumpdown} and using~\eqref{e.cacc.your.u} again (with $u-\psi$ in place of $u$), we deduce that, for every $r \in [1,R-2-s]$, \begin{align}
\left\| u -\psi \right\|_{\underline{L}^2(Q_r)} & \leq
\frac{Cr^2}{R^2} \left\| u \right\|_{\underline{L}^2(Q_R)} +
\frac12 \left\| u -\psi \right\|_{\underline{L}^2(Q_{r+s})}. \end{align} An iteration of this inequality implies, for every $r \in [1,R-2-s]$, \begin{equation} \label{e.C11a}
\left\| u -\psi \right\|_{\underline{L}^2(Q_r)} \leq \frac{Cr^2}{R^2}
\left\| u \right\|_{\underline{L}^2(Q_R)}. \end{equation} The conditions~$r \leq R-2-s$ and~$R\geq s+2$ can be removed by enlarging the constant~$C$ on the right side of~\eqref{e.C11a}. This completes the proof. \end{proof}
The proof of Theorem~\ref{t.analyticity} for $m\geq 2$ follows along the same lines as the one of Lemma~\ref{l.C01C11}, but is more involved. We take a heterogeneous polynomial~$\psi\in \mathbb{A}_m$ satisfying $D^k \widehat{\psi} (0) = D^k \widehat{u}(0)$ for every $k\in\{0,\ldots,m\}$, and we endeavor to show that~\eqref{e.Cm1} holds. Compared to the case $m\in\{0,1\}$, the additional difficulty in the case $m\geq 2$ is that the polynomial $p:= -\nabla \cdot \mathbf{a}\nabla \psi$ does not vanish, in general, and must be estimated. This turns out to be the most difficult part of the argument and is the content of Lemma~\ref{l.gaussian.cacc}, below.
Dealing with this difficulty forces us to make some modifications to our strategy of a more technical nature. In particular, we need to work with heat kernels rather than balls or cubes and use some properties of Hermite polynomials. We denote \begin{equation*}
\Phi_{t,y}(x) := (4 \pi t)^{-\frac d2} \exp \left( - \frac{|x-y|^2}{4t} \right). \end{equation*} We begin the proof of Theorem~\ref{t.analyticity} for~$m \geq 2$ by presenting a variant of Lemma~\ref{l.jumpdown} for heat kernels (rather than cubes).
\begin{lemma} \label{l.jumpdown2} Let $r \in \mathbb{N}$ and $u \in H^1(Q_r)$. There exists a constant $C(d)<\infty$ such that, for every $y \in \mathbb{R}^d$ and $t \in [1,\infty)$, \begin{equation} \label{e.jumpdown2} \int_{Q_r} u^2 \Phi_{t,y} \leq 2 \int_{Q_r} \widehat{u}^2 \Phi_{t,y} +
C \int_{Q_r} \left| \nabla u \right|^2 \Phi_{t + 1 ,y} \end{equation} \end{lemma} \begin{proof}
Observe that, for every $t \geq 1$, $x,x',y\in{\mathbb{R}^d}$ with $|x-x'| \leq \sqrt{d}$, we have \begin{align*} \frac{\Phi_{t,y} (x')}{\Phi_{t+1,y}(x)} =
\exp\left( -\frac{|x'-y|^2}{4t(t+1)} + \frac{(x-x')\cdot(x+x'-2y)}{4(t+1)}
\right) \leq C. \end{align*} Thus, by the Poincar\'e inequality, \begin{align*}
\int_{z + Q_1} \left| u(x) - \widehat u(z) \right|^2 \Phi_{t,y}(x)\,dx & \leq C \sup_{x\in z+Q_1} \Phi_{t,y}(x)
\int_{z + Q_1} \left| \nabla u \right|^2 \\ & \leq
C\int_{z + Q_1} \left| \nabla u(x) \right|^2 \Phi_{t+1,y} (x)\,dx. \end{align*} Summing over $z\in \mathbb{Z}^d\cap Q_r$ yields \begin{align*}
\int_{Q_r} \left| u(x) - \widehat u(z) \right|^2 \Phi_{t,y}(x)\,dx \leq
C\int_{Q_r} \left| \nabla u(x) \right|^2 \Phi_{t+1,y} (x)\,dx \end{align*} Using the triangle inequality, we get~\eqref{e.jumpdown2}. \end{proof}
The next lemma gives two simple estimates for polynomials which are needed in the proof of Lemma~\ref{l.gaussian.cacc}.
\begin{lemma}[Polynomial estimates] \label{l.polynomial} For every $y \in {\mathbb{R}^d}$, $t \in (0,\infty)$, $m\in \mathbb{N}$ and polynomial $p \in \mathbb{P}_m$, we have \begin{equation} \label{e.polystupid1}
\int_{\mathbb{R}^d} \left( \frac{|\nabla ( p \Phi_{t,y})|}{ \Phi_{t,y}} \right)^2 \Phi_{t,y} \leq \frac{2d(m+1)}{t} \int_{\mathbb{R}^d} p^2 \Phi_{t,y} \end{equation} and \begin{equation} \label{e.polystupid2}
\int_{\mathbb{R}^d} p^2 \Phi_{t,y} \leq 2 \int_{ Q_{ 4 \sqrt{m t} }(y)} p^2 \Phi_{t,y} . \end{equation} \end{lemma} \begin{proof} It suffices by Fubini to prove the result in dimension~$d=1$. By scaling and translation we may assume that $t = 1/4$ and $y= 0$. We denote $\Phi(x) := \Phi_{1/4,0}(x) = \pi^{-\frac12} \exp(-x^2)$ and represent~$p$ using Hermite polynomials as \begin{equation*} p(x) = \sum_{k = 0}^m c_k h_k(x), \end{equation*} where $h_k$ is the $k$th (physicist) Hermite polynomial, defined by \begin{equation}
h_k(x):= (-1)^k \exp\left(x^2\right) \left( \frac{d}{dx}\right)^n \exp\left(-x^2\right). \end{equation} We note that these Hermite polynomials satisfy \begin{equation} \label{e.hk.one} ( h_n \Phi)' = ( h_n' - 2 x h_n) \Phi = - h_{n+1} \Phi \end{equation} and \begin{equation} \label{e.hk.two} \left\{ \begin{aligned} & h_{n+1}(x) = 2x h_n(x) - 2nh_{n-1}(x), \\ & h_0 = 1, \ h_1(x) = 2x. \end{aligned} \right. \end{equation} They are orthogonal with respect to~$\Phi(x)\,dx$ and satisfy, for every $m,n\in\mathbb{N}$, \begin{equation} \label{e.Hermite.orth} \int_{\mathbb{R}} h_m(x) h_n(x) \Phi(x)\,dx = \left\{ \begin{aligned} & 2^m m! & \mbox{if} \ n=m, \\ & 0 & \mbox{if} \ n\neq m. \end{aligned} \right. \end{equation}
Using~\eqref{e.hk.one} and~\eqref{e.Hermite.orth}, we see that \begin{align*} \int_{\mathbb{R}} \left( \frac{(p\Phi)'}{\Phi}\right)^2 \Phi & = \int_{\mathbb{R}} \left( \sum_{k=0}^m c_k \frac{( h_k \Phi)'}{\Phi} \right)^2 \Phi \\& = \sum_{k=0}^m c_k^2 \int_{\mathbb{R}} h_{k+1}^2 \Phi \\& = \sum_{k=0}^m c_k^2 2(k+1) \int_{\mathbb{R}} h_{k}^2 \Phi \leq 2(m+1) \int_{\mathbb{R}} p^2\Phi. \end{align*} This yields~\eqref{e.polystupid1}.
We turn to the proof of~\eqref{e.polystupid2}. We first observe that, for every $k\in\mathbb{N}$, \begin{equation} \label{e.hk.tailbound}
\left| h_k(x) \right| \leq 4^k|x|^k \quad \forall |x| \geq k^{\frac12}. \end{equation} We can prove~\eqref{e.hk.tailbound} by induction and~\eqref{e.hk.two}: if it holds for $k\in \{n-1,n\} $, then \begin{align*}
\left| h_{n+1}(x) \right| & \leq
2|x| \left| h_n(x) \right| + 2n \left| h_{n-1}(x) \right| \\ & \leq
2|x|\cdot 4^n |x|^n +
2n\cdot 4^{n-1}|x|^{n-1} \\ & \leq
4^{n+1} |x|^{n+1}. \end{align*} It clearly holds for $k\in\{0,1\}$, so it is valid for all $k$. We deduce from~\eqref{e.hk.tailbound} that, for fixed $\alpha\geq 1$ and every $k\in\{0,\ldots,m\}$, \begin{align*}
\int_{|x| > \sqrt{\alpha m}} h_k^2 \Phi \leq 2^{4k+1} \int_{\sqrt{\alpha m}}^\infty x^{2k} \exp\left(-x^2\right) \,dx \leq 2^{6k+2} k! \exp\left( -\tfrac12 \alpha m \right) \end{align*} Thus, using also~\eqref{e.Hermite.orth}, \begin{align*}
\int_{|x|>\sqrt{\alpha m}} p^2 \Phi & \leq (m+1) \sum_{k=0}^m c_k^2
\int_{|x|>\sqrt{\alpha m}} h_k^2 \Phi \\ & \leq (m+1) \exp\left( -\tfrac12 \alpha m \right) \sum_{k=0}^m c_k^2 2^{6k+2} k! \\ & = (m+1) \exp\left( -\tfrac12 \alpha m \right) \sum_{k=0}^m c_k^2 2^{5k+2} \int_{\mathbb{R}} h_k^2 \Phi \\ & \leq (m+1)2^{5m+2} \exp\left( -\tfrac12 \alpha m \right) \int_{\mathbb{R}} p^2\Phi . \end{align*} Taking $\alpha=16$ so that $(m+1)2^{5m+2} \exp\left( -\tfrac12 \alpha m \right) \leq \frac14$ for all $m\geq 1$, we then obtain~\eqref{e.polystupid2} by the triangle inequality. \end{proof}
The next lemma contains the key additional step needed to adapt the argument of Lemma~\ref{l.C01C11} to~$m\geq 2$. The interesting point is that the polynomial~$p$ does not appear on the right side of the estimate~\eqref{e.gaussian.cacc}.
\begin{lemma} \label{l.gaussian.cacc} For each $\delta\in \left(0,1\right]$, there exist $C(\delta,d,\Lambda)<\infty$ and $c(d,\Lambda)>0$ such that, for every $m\in\mathbb{N}$, $R \in [Cm,\infty)$, $t\in [Cm,R]$, $y\in Q_{R/2}$, $p\in\mathbb{P}_m$ and solution~$u\in H^1(Q_R)$ of the equation \begin{equation} \label{e.pde.withp} -\nabla \cdot \mathbf{a} \nabla u = p \quad \mbox{in} \ Q_R, \end{equation} we have the estimate \begin{align} \label{e.gaussian.cacc} \int_{Q_{3R/4}}\left(
\left| \nabla u \right|^2 + p^2 \right) \Phi_{t,y} & \leq \delta \int_{Q_{3R/4}} u^2 \Phi_{t+C,y} + \exp\left( -c R \right) \int_{Q_{R}} u^2. \end{align} \end{lemma} \begin{proof} Let $\delta \in \left( 0,1\right]$ and~$\psi \in W_0^{1,\infty} (Q_R)$ be a cutoff function satisfying \begin{equation} \label{e.cutoff.gc} \mathds{1}_{Q_{3R/4}} \leq \psi \leq \mathds{1}_{Q_{R}} \quad \mbox{and} \quad
|\nabla \psi| \leq \frac{8}{R}. \end{equation} We first test the equation~\eqref{e.pde.withp} with $u \psi^2 \Phi_{t,y}$ to obtain \begin{align*} \int_{Q_R} p u \psi^2 \Phi_{t,y} & = \int_{Q_R} \left( \psi^2 \Phi_{t,y} \nabla u\cdot \mathbf{a} \nabla u + u \nabla ( \psi^2 \Phi_{t,y}) \cdot \mathbf{a} \nabla u \right) \\ & \notag \geq \frac12 \int_{Q_R}
\left| \nabla u \right|^2 \psi^2 \Phi_{t,y}
-C \int_{Q_R} u^2 \Phi_{t,y} \left( \psi^2 \frac{\left| \nabla \Phi_{t,y} \right|^2 }{\Phi_{t,y}^2 } + | \nabla \psi|^2 \right) . \end{align*} Rerranging and using Young's inequality, we get \begin{align} \label{e.testwithu} \lefteqn{ \int_{Q_R}
| \nabla u|^2 \psi^2 \Phi_{t,y} } \qquad & \\ & \notag \leq \!\!\int_{Q_R} u^2 \,\bigg( \frac{\delta}{8}\psi^2 + C
\psi^2 \frac{\left| \nabla \Phi_{t,y} \right|^2 }{\Phi_{t,y}^2 } + C | \nabla \psi|^2 \bigg)\,\Phi_{t,y} + \frac{2}{\delta} \int_{Q_R} p^2 \psi^2 \Phi_{t,y}. \end{align} Testing~\eqref{e.pde.withp} with $p\psi^2 \Phi_{t,y}$ and using Young's inequality, we get \begin{align} \label{e.testwithp} \int_{Q_R} \! p^2 \psi^2 \Phi_{t,y} & = \int_{Q_R} \mathbf{a} \nabla u \cdot \left( \psi^2 \nabla ( p \Phi_{t,y}) + 2\Phi_{t,y} p \psi \nabla\psi \right) \\ & \notag \leq \frac \delta{16} \! \int_{Q_R} \!
| \nabla u|^2 \psi^2 \Phi_{t,y} + \frac{C}{\delta} \! \int_{Q_R} \!\!
\Phi_{t,y}\bigg( \frac{ \left| \nabla ( p \Phi_{t,y} ) \right|^2 }{\Phi_{t,y}^2 } \psi^2 + p^2 |\nabla \psi|^2 \bigg). \end{align} Combining the two previous displays, we obtain \begin{align} \label{e.tricky.combine} \int_{Q_R} \left(
\left| \nabla u \right|^2 + \frac1\delta p^2 \right) \psi^2 \Phi_{t,y} & \leq
\int_{Q_R} u^2 \left( \frac{\delta}{4}+ C \frac{\left| \nabla \Phi_{t,y} \right|^2 }{\Phi_{t,y}^2 } \right) \psi^2 \Phi_{t,y} \\ & \qquad \notag + \frac{C}{\delta^{2}} \int_{Q_R}
\left( \frac{ \left| \nabla ( p \Phi_{t,y} ) \right|^2 }{\Phi_{t,y}^2 } \right) \psi^2 \Phi_{t,y} \\ & \qquad \notag + C \int_{Q_R} \left(u^2 + \frac1{\delta^{2}} p^2 \right)
| \nabla \psi|^2 \Phi_{t,y}. \end{align} We next estimate the three terms on the right.
Observe that, for every $s \in (0,t]$, we have \begin{equation*}
\frac{ \left| \nabla\Phi_{t,y}(x) \right|^2 }{\Phi_{t,y}^2(x) } \frac{\Phi_{t,y}(x)}{\Phi_{t+s,y}(x)} \leq
\frac{|x-y|^2}{4t^2} \frac{\Phi_{t,y}(x)}{\Phi_{t+s,y}(x)} \leq
\frac{2}{s} \left( \frac{s |x-y|^2}{8t^2} \exp \left( - \frac{s|x-y|^2}{8t^2} \right) \right) \leq \frac{1}{s}. \end{equation*} Therefore, we can find~$C(\delta,d,\Lambda)<\infty$ sufficiently large that, for every~$y \in Q_{R/2}$ and $t \in \left[ C, R\right]$, we can estimate the first term on the right side of~\eqref{e.tricky.combine} by \begin{align} \label{e.fatten.up}
\int_{Q_{3R/4}} u^2 \left( \frac{\delta}{4}+ C \frac{\left| \nabla \Phi_{t,y} \right|^2 }{\Phi_{t,y}^2 } \right) \psi^2 \Phi_{t,y} & \leq \frac{\delta}{2} \int_{Q_{3R/4}} u^2 \Phi_{t+C,y} \end{align} Since $y \in Q_{R/2}$ and the fact that $\nabla \psi$ is supported in $Q_R \setminus Q_{3R/4}$, we estimate \begin{multline} \label{e.glotten.up} \int_{Q_R \setminus Q_{3R/4}}
u^2 \left( \frac{\delta}{2}+ C \frac{\left| \nabla \Phi_{t,y} \right|^2 }{\Phi_{t,y}^2 } \right) \psi^2 \Phi_{t,y} + \int_{Q_R}
(u^2 + \delta^{-2}p^2) |\nabla \psi|^2 \Phi_{t,y}
\\ \leq C \exp\left( - \frac{R^2}{Ct} \right) \int_{Q_R} (u^2 + p^2) . \end{multline} Turning to the second term on the right side of~\eqref{e.tricky.combine}, we use Lemma~\ref{l.polynomial} to obtain, for every $y \in Q_{R/2}$ and $t \in \left( 0 , (64 m)^{-1} R^2 \right]$ (note that $Q_{4 \sqrt{mt}}(y) \subseteq Q_{3R/4}$), \begin{align} \label{e.polyineq} \int_{\mathbb{R}^d}
\frac{|\nabla (p\Phi_{t,y}) |^2}{\Phi_{t,y}^2} \Phi_{t,y} \leq \frac{4dm}{t} \int_{{\mathbb{R}^d}} p^2 \Phi_{t,y} \leq \frac{8dm}{t} \int_{Q_{3R/4}} p^2 \Phi_{t,y}. \end{align} Combining the above inequalities~\eqref{e.tricky.combine},~\eqref{e.fatten.up},~\eqref{e.glotten.up} and~\eqref{e.polyineq}, we obtain \begin{align*} \lefteqn{ \int_{Q_{3R/4}} \Phi_{t,y} \left(
\left| \nabla u \right|^2 + \frac1\delta p^2 \right) } \qquad & \\ & \notag \leq \frac{\delta}{2} \int_{Q_{3R/4}} u^2 \Phi_{t+C,y} + C \exp\left(-\frac{R^2}{Ct} \right) \int_{Q_{R}} \left( u^2 +p^2 \right) + \frac{Cm}{\delta^2 t} \int_{Q_{3R/4}} \Phi_{t,y} p^2 . \end{align*} If $t \in [ 2C\delta^{-1} m,R]$, then the last term on the right may be reabsorbed on the left side, giving us the estimate \begin{equation} \label{e.tricky.approach} \int_{Q_{3R/4}} \!\! \Phi_{t,y} \left(
\left| \nabla u \right|^2 + p^2 \right) \leq \delta \int_{Q_{3R/4}} u^2 \Phi_{t+C,y} +
C \exp\left(-cR \right) \int_{Q_{R}} \left( u^2 +p^2 \right). \end{equation} It remains to estimate the second term on the right side of~\eqref{e.tricky.approach}.
We proceed as above, this time testing the equation~\eqref{e.pde.withp} with $u \psi^2$ and $p \psi^2$ (but without the factor of $\Phi_{t,y}$). We obtain, respectively, for every $\theta \in (0,\infty)$, \begin{equation*}
\int_{Q_{R}} |\nabla u|^2 \psi^2 \leq \frac{\theta R^2}{2} \int_{Q_{R}} p^2 \psi^2 + C \int_{Q_R} \left(\frac1{\theta R^2} + |\nabla \psi|^2 \right) u^2 \end{equation*} and \begin{equation*} \int_{Q_{R}} p^2 \psi^2 \leq
\frac1{\theta R^2} \int_{Q_{R}} |\nabla u|^2 \psi^2 +
C \theta R^2 \int_{Q_{R}} |\nabla p|^2 \psi^2 + C\theta \int_{Q_{R}} |\nabla \psi|^2 p^2. \end{equation*} Combining these and using the properties of~$\psi$ in~\eqref{e.cutoff.gc}, we obtain \begin{equation*} \int_{Q_{3R/4}} p^2 \leq
C \theta R^2 \int_{Q_{R}} |\nabla p|^2 + C\theta \int_{Q_{R}} p^2 + \frac{C}{\theta^2 R^4}\int_{Q_R} u^2 . \end{equation*} Choosing $\theta := C^{-m}$ for $C(d)<\infty$, we may reabsorb the first two terms on the right side of the previous inequality to obtain \begin{equation} \label{e.poly.yomama}
\left\| p \right\|_{L^\infty(Q_R)}^2 \leq C^m \int_{Q_{3R/4}} p^2 \leq \frac{C^{m}}{R^4} \int_{Q_R} u^2 . \end{equation} Therefore, if $R \geq Cm$ for $C$ sufficiently large, we obtain \begin{align} \label{e.crude.pbound}
\exp\left(-cR \right) \int_{Q_{R}} \left( u^2 +p^2 \right) & \leq \left( 1 + C^{m} \right) \exp(-cR ) \int_{Q_{R}} u^2 \\ & \notag \leq \exp\left(-\tfrac 12c R \right) \int_{Q_R} u^2. \end{align} Combining~\eqref{e.tricky.approach} and~\eqref{e.crude.pbound} and shrinking~$c$ yields the lemma. \end{proof}
We are now ready to give the proof of Theorem~\ref{t.analyticity}.
\begin{proof}[{Proof of Theorem~\ref{t.analyticity}}] We proved the statement of the theorem for $m\in\{0,1\}$ already in Lemma~\ref{l.C01C11}, so we may suppose $m\geq 2$. By Lemmas~\ref{l.corr.growth} and~\ref{l.diffbound}, there exists~$\psi \in \mathbb{A}_m$ satisfying \begin{equation} \label{e.yourotherass} D^k\widehat{\psi}(0) = D^k\widehat{u}(0) \quad \mbox{for every} \ k\in\{0,\ldots,m\} \end{equation} and, for every $r\geq m$, \begin{align} \label{e.psi.control}
\left\| \psi \right\|_{\underline{L}^2(Q_r)} & \leq \sum_{k=0}^m \left( \frac{Cr}{k+1} \right)^k
\left| D^k\widehat{u}(0) \right| \\ & \notag \leq \sum_{k=0}^m \left( \frac{Cr}{k+1} \right)^k
\left( \frac{C(k+1)}{r} \right)^k \left\| u \right\|_{\underline{L}^2(Q_r)}
\leq C^m \left\| u \right\|_{\underline{L}^2(Q_r)}. \end{align} For convenience, we denote $v:= u - \psi$. Using~\eqref{e.yourotherass} and that $D^{m+1}\psi = 0$, we find that, for every $r\in [2m,\tfrac 12R]$, \begin{align} \label{e.grid.hammer}
\sup_{z\in Q_r}
\left| \widehat{v} (z) \right|
\leq
\frac{Cr^{m+1}}{(m+1)!} \sup_{z\in Q_{r+m+1}} \left| D^{m+1}\widehat{u}(z) \right| \leq
\left( \frac{Cr}{R} \right)^{m+1} \left\| u \right\|_{\underline{L}^2(Q_R)}. \end{align} By Lemma~\ref{l.jumpdown2}, for every $r\in [Cm,R]$ and $y\in Q_{R/2}$, \begin{align} \label{e.jumpdown.applied} \int_{Q_{3R/4}} v^2 \Phi_{r,y} & \leq 2 \int_{Q_{3R/4}}\widehat{v}^2\Phi_{r,y} +
C \int_{Q_{3R/4}} \left| \nabla v \right|^2 \Phi_{r + 1,y}. \end{align} We may estimate the first term on the right side of~\eqref{e.jumpdown.applied}, using~\eqref{e.grid.hammer}, to get, for each $r\in [Cm,\tfrac1{8}R]$ and $y\in Q_r$, \begin{align} \label{e.layercaking} \lefteqn{ \int_{Q_{3R/4}} \widehat{v}^2 \Phi_{r,y} \leq \int_{Q_{7R/8}} \widehat{v}(\cdot-y)^2 \Phi_{r} } \qquad & \\ & \notag = \int_0^{2r} \int_{\partial Q_s} \widehat{v}(\cdot-y)^2 \Phi_r\,ds + \int_{2r}^{7R/8} \int_{\partial Q_s} \widehat{v}(\cdot-y)^2 \Phi_r\,ds \\ & \notag \leq \sup_{Q_{3r}} \widehat{v}^2 \int_{Q_r} \Phi_r + \int_{2r}^{7R/8} \sup_{Q_{s+r}} \widehat{v}^2 \exp\left( -\frac{cs^2}{r} \right) \,ds \\ & \notag \leq
\left\| u \right\|_{\underline{L}^2(Q_R)}^2 \left( \left( \frac{Cr}{R} \right)^{2(m+1)} + \int_{2r}^{7R/8} \left( \frac{Cs}{R} \right)^{2(m+1)} \exp\left( -\frac{cs^2}{r} \right) \,ds \right) \\ & \notag \leq \left( \frac{Cr}{R} \right)^{2(m+1)}
\left\| u \right\|_{\underline{L}^2(Q_R)}^2. \end{align} Indeed, the second integral on the fourth line of the display above can be estimated straightforwardly by changing variables as follows: \begin{align*} \int_{2r}^{7R/8} \left( \frac{Cs}{R} \right)^{2(m+1)} \exp\left( -\frac{cs^2}{r} \right) \,ds & \leq \left( \frac{Cr}{R} \right)^{2(m+1)} \int_1^\infty s^{2(m+1)} \exp\left( -crs^2 \right)\,ds \\ & \leq \left( \frac{Cr}{R} \right)^{2(m+1)} \int_1^\infty s^{2(m+1)} \exp\left( -m s^2 \right)\,ds \\ & \leq \left( \frac{Cr}{R} \right)^{2(m+1)} \int_m^\infty \frac1{2m} \left( \frac{s}{m} \right)^{m+\frac12} \exp(-s)\,ds \\ & \leq \left( \frac{Cr}{R} \right)^{2(m+1)}. \end{align*} For second term on the right side~\eqref{e.jumpdown.applied}, we fix $\delta>0$ to be selected below and apply Lemma~\ref{l.gaussian.cacc}, noting that~$-\nabla\cdot \mathbf{a} \nabla v = -\nabla \cdot\mathbf{a}\nabla \psi \in \mathbb{P}_{m-2}$ in $B_R$, to obtain, under the condition that~$Cm\leq r\leq R$ for $C=C(\delta,d,\Lambda)$, the estimate \begin{equation} \label{e.applybighammer} \int_{Q_{3R/4}} \left(
\left| \nabla v \right|^2 + p^2 \right) \Phi_{r + 1 ,y} \leq \delta \int_{Q_{3R/4}} v^2 \Phi_{r + 1 + C ,y} + \exp\left( - c R \right)
\left\| v \right\|_{\underline{L}^2(Q_R)}^2. \end{equation} Combining~\eqref{e.jumpdown.applied},~\eqref{e.layercaking} and~\eqref{e.applybighammer} and taking $\delta(d,\Lambda)>0$ sufficiently small, we get, for every $r\in [Cm,\tfrac1{8}R]$ and $y\in Q_r$, \begin{align*} \lefteqn{ \int_{Q_{3R/4}} v^2 \Phi_{r,y} } \qquad & \\ & \leq \frac12 \int_{Q_{3R/4}} v^2 \Phi_{r + C ,y}
+ \left( \frac{Cr}{R} \right)^{2(m+1)}
\left\| u \right\|_{\underline{L}^2(Q_R)}^2
+ C \exp\left( - c R \right) \left\| v \right\|_{\underline{L}^2(Q_R)}^2. \end{align*}
Integrating this over $y\in Q_r$ and using that $|Q_{r+C}| \leq \frac 32|Q_r|$ for $r\geq C$, we obtain, for every $r\in [Cm,\tfrac1{8}R]$, \begin{align*}
\int_{Q_{3R/4}} v^2 \left( \frac{\mathds{1}_{Q_r}}{|Q_r|} \ast \Phi_{r} \right) & \leq \frac34
\int_{Q_{3R/4}} v^2 \left(\frac{\mathds{1}_{Q_{r+C}}}{|Q_{r+C}|} \ast \Phi_{r+C} \right) \\ & \qquad +
\left( \frac{Cr}{R} \right)^{2(m+1)} \left\| u \right\|_{\underline{L}^2(Q_R)}^2
+ C \exp\left( - c R \right) \left\| v \right\|_{\underline{L}^2(Q_R)}^2. \end{align*} An iteration now yields, for every $r\in [Cm,\tfrac1{8}R]$, \begin{equation}
\int_{Q_{3R/4}} v^2 \left( \frac{\mathds{1}_{Q_r}}{|Q_r|} \ast \Phi_{r} \right) \leq
\left( \frac{Cr}{R} \right)^{2(m+1)} \left\| u \right\|_{\underline{L}^2(Q_R)}^2
+ C \exp\left( - c R \right) \left\| v \right\|_{\underline{L}^2(Q_R)}^2. \end{equation} We deduce that, for every $r\in [Cm,\tfrac1{8}R]$, \begin{equation} \label{e.vboundsesxp}
\left\| v \right\|_{\underline{L}^2(Q_r)} \leq
\left( \frac{Cr}{R} \right)^{m+1} \left\| u \right\|_{\underline{L}^2(Q_R)} +
C \exp\left( - c R \right) \left\| v \right\|_{\underline{L}^2(Q_R)}. \end{equation} By~\eqref{e.psi.control} and $R\geq Cm$, we have that \begin{equation*}
\left\| v \right\|_{\underline{L}^2(Q_R)} \leq
\left\| u \right\|_{\underline{L}^2(Q_R)} +
\left\| \psi \right\|_{\underline{L}^2(Q_R)} \leq
C^m \left\| u \right\|_{\underline{L}^2(Q_R)}. \end{equation*} We therefore obtain, for every~$R\geq Cm$ and $r \in [Cm,R]$, \begin{equation}
\exp\left( - c R \right) \left\| v \right\|_{\underline{L}^2(Q_R)} \leq
\exp(-cR) \left\| u \right\|_{\underline{L}^2(Q_R)} \leq
\left( \frac{Cr}{R} \right)^{m+1} \left\| u \right\|_{\underline{L}^2(Q_R)}. \end{equation} Combining this with~\eqref{e.vboundsesxp} and substituting $v=u-\psi$, we finally obtain, for every $r\in \left[Cm,\tfrac18 R \right]$, \begin{equation} \label{e.upsi.yodama}
\left\| u - \psi \right\|_{\underline{L}^2(Q_r)} \leq
\left( \frac{Cr}{R} \right)^{m+1} \left\| u \right\|_{\underline{L}^2(Q_R)}. \end{equation} This is~\eqref{e.Cm1}, although the heterogeneous polynomial $\psi$ is not necessary~$\mathbf{a}(x)$--harmonic as in the statement of the theorem.
To complete the proof, we must therefore replace $\psi$ by an element of~$\mathbb{A}_m$ which is $\mathbf{a}(x)$--harmonic. To do this, we must estimate~$p$. By~\eqref{e.poly.yomama} applied to~$u-\psi$ instead of~$u$, we have, for every $r \in \left[Cm,\tfrac18R\right]$, \begin{equation}
\left\| p \right\|_{L^\infty(Q_r)} \leq
\frac{C^m}{r^2} \left\| u - \psi \right\|_{\underline{L}^2(Q_r)} \leq
\frac1{r^2} \left( \frac{Cr}{R} \right)^{m+1} \left\| u \right\|_{\underline{L}^2(Q_R)}. \end{equation} Likewise, for every~$k\in\{1,\ldots,m\}$ and~$r\in\left[Cm,\tfrac1{16}R\right]$, we have \begin{align*}
\left\| D^kp \right\|_{L^\infty(Q_r)} & \leq
\frac{C^{m-k}}{r^2} \left\| D^k(u - \psi) \right\|_{\underline{L}^2(Q_r)} \\ & \notag \leq \frac{C^{m-k}}{r^2} \left( \frac{Cr}{R} \right)^{m+1-k}
\left\| D^ku \right\|_{\underline{L}^2\left(Q_{R/2} \right)} \\ & \notag \leq \frac{C^{m-k}}{r^2} \left( \frac{Cr}{R} \right)^{m+1-k} \left( \frac{Ck}{R} \right)^k
\left\| u \right\|_{\underline{L}^2\left(Q_{R} \right)} \\ & \notag \leq \frac1{r^2} \left( \frac{k}{r} \right)^k \left( \frac{Cr}{R} \right)^{m+1}
\left\| u \right\|_{\underline{L}^2\left(Q_{R} \right)}. \end{align*} The first line above is obtained by applying~\eqref{e.poly.yomama} with $D^k(u-\psi)$ and $D^kp$ in place of $u$ and $p$, while the second line above is obtained by applying~\eqref{e.upsi.yodama} to $D^ku$ and $D^k\psi$ in place of $u$ and $\psi$ and the third line is an application of Lemma~\ref{l.diffbound}. By Lemma~\ref{l.polyhit.bound}, the bound~\eqref{e.polyhit.bounds.D}, and the previous two displays, we can find $\zeta \in \mathbb{A}_m$ such that $-\nabla \cdot \mathbf{a}\nabla \zeta = p$ and, for every $r \in \left[Cm,\tfrac 1{16}R\right]$, \begin{equation*}
\left\| \zeta \right\|_{\underline{L}^2(Q_r)}
\leq Cr^2 \sum_{n=0}^m \left( \frac{Cr}{n+1} \right)^{n} \left| D^n\widehat{p}(0) \right|
\leq
\left( \frac{Cr}{R} \right)^{m+1} \left\| u \right\|_{\underline{L}^2(Q_R)}. \end{equation*} By the triangle inequality, we therefore obtain, for every~$r\in [Cm,\tfrac1{16}R]$, \begin{equation*}
\left\| u - \psi -\zeta \right\|_{\underline{L}^2(Q_r)} \leq
\left\| u - \psi \right\|_{\underline{L}^2(Q_r)} +
\left\| \zeta \right\|_{\underline{L}^2(Q_r)} \leq
\left( \frac{Cr}{R} \right)^{m+1} \left\| u \right\|_{\underline{L}^2(Q_R)}. \end{equation*} Enlarging $C$, we obtain this inequality for every~$r\in [Cm,R]$. Renaming $\psi + \zeta$ to be $\psi$, we obtain the theorem. \end{proof}
\begin{remark} \label{r.pick.psi} Observe that, in addition to proving the statement of Theorem~\ref{t.analyticity}, we also proved the estimate~\eqref{e.Cm1} for the unique $\psi\in \mathbb{A}_m$ which satisfies \begin{equation} D^k\widehat{\psi}(0) = D^k\widehat{u}(0), \quad \forall k\in\{0,\ldots,m\}. \end{equation} Of course, this $\psi$ is not necessarily $\mathbf{a}(x)$--harmonic, unlike the one in the statement of the theorem. \end{remark}
\section{Consequences} \label{s.conseq}
We conclude the paper by showing that Corollary~\ref{c.analyticity.exp} and Theorems~\ref{t.Liouville},~\ref{t.quc} and~\ref{t.bottom} are easy consequences of Theorem~\ref{t.analyticity} and some of the lemmas which were used to prove it.
\begin{proof}[{Proof of Corollary~\ref{c.analyticity.exp}}] Applying Theorem~\ref{t.analyticity} with $m=\left\lfloor \delta R \right\rfloor$, we obtain $\psi\in \mathbb{A}^0_m\subseteq \mathbb{A}_{\lfloor R \rfloor }^0$ such that, for every $s\in [C\delta R, R]$, \begin{equation}
\left\| u - \psi \right\|_{\underline{L}^2(Q_s)} \leq \left( \frac{Cs}{R} \right)^{m+1}
\left\| u \right\|_{\underline{L}^2(Q_R)} . \end{equation} Taking $\delta(d,\Lambda)>0$ sufficiently small and setting $s: = C\delta R$, we obtain \begin{equation}
\left\| u - \psi \right\|_{\underline{L}^2(Q_s)} \leq
\left( \frac12 \right)^{m+1} \left\| u \right\|_{\underline{L}^2(Q_R)} \leq
\exp(-cR) \left\| u \right\|_{\underline{L}^2(Q_R)} . \end{equation} By the $C^{0,1}$ estimate (Lemma~\ref{l.C01C11}), we have that, for every $r\in [1,s]$, \begin{equation}
\left\| u - \psi \right\|_{\underline{L}^2(Q_r)}
\leq C \left\| u - \psi \right\|_{\underline{L}^2(Q_s)}. \end{equation} This completes the proof. \end{proof}
We begin with the proof Theorem~\ref{t.Liouville}.
\begin{proof}[Proof of Theorem~\ref{t.Liouville}] Assume that $u\in H^1_{\mathrm{loc}}({\mathbb{R}^d})$ satisfies \begin{equation} \label{e.pde4} -\nabla\cdot \mathbf{a}\nabla u = 0 \quad\mbox{in} \ {\mathbb{R}^d} \end{equation} and, for some $\delta\in (0,1]$, \begin{equation} \label{e.slowexp.growth}
\limsup_{r\to \infty}\ \exp\left( - \delta r \right) \left\| u \right\|_{\underline{L}^2(Q_r)} \leq 1. \end{equation} We will show that there exist $\delta_0(d,\Lambda)>0$ and $C(d,\Lambda)<\infty$ such that $\delta \leq \delta_0(d,\Lambda)$ implies that $u \in \mathbb{A}_\infty(C\delta)$. Let $C_1$ be the constant $C$ in the statement of Theorem~\ref{t.analyticity}.
By Lemma~\ref{l.diffbound}, for every $m\in\mathbb{N}$ and $r>m+2$, we have \begin{equation}
\left| D^m\widehat{u}(0) \right| \leq \left( \frac{Cm}{r} \right)^m
\left\| u \right\|_{\underline{L}^2(Q_r)}. \end{equation} If $\delta \in (0,\delta_0]$ with $\delta_0(d,\Lambda)>0$ sufficiently small, then we may take $r:=C\delta^{-1} m$ in the previous inequality and use~\eqref{e.slowexp.growth} to obtain, for~$m\in\mathbb{N}$ sufficiently large, \begin{equation} \label{e.Dmhatu.bound}
\left| D^m\widehat{u}(0) \right| \leq \delta^m \exp\left( Cm \right) \leq \left( C\delta \right)^m. \end{equation} By Lemma~\ref{l.corr.growth}, for each $m\in\mathbb{N}$, there exists a unique element~$\psi_m$ of~$\mathbb{A}_m$ satisfying \begin{equation*}
D^k \widehat{\psi}_m (0) = D^k \widehat{u}(0) \quad \forall k\in\{ 0,\ldots,m\}. \end{equation*} Let $q_m\in\mathbb{P}_m$ be such that $\psi_m=\sum_{k=0}^m \nabla^k q_m:\phi^{(k)}$. By~\eqref{e.corr.growth.q} and~\eqref{e.Dmhatu.bound}, we have that, for every $n,m\in\mathbb{N}$ with $n\leq m$, \begin{align*}
\left| \nabla^n(q_m-q_{m+1}) (0) \right| & \leq
\left( \frac{n+1}{m} \right)^{n} C^m \left| D^{m+1} \widehat{\psi}_m (0) \right| \\ & \leq \left( \frac{n+1}{m} \right)^{n} C^m (C\delta)^{m+1}
\leq (C\delta)^{m+1} \end{align*} and hence, if $\delta_0(d,\Lambda)>0$ is sufficiently small and $\delta \in (0,\delta_0]$, \begin{equation}
\left| \nabla^n q_m(0) \right| \leq
\left| D^n\widehat{u}(0) \right| + \sum_{k=n}^{m-1}
\left| \nabla^n(q_k-q_{k+1}) (0) \right| \leq \sum_{k=n}^{m} \left( C\delta \right)^k \leq (C\delta)^n. \end{equation} Thus for $\delta \in (0,\delta_0]$, we have that~$\nabla^nq_m(0)$ is uniformly bounded in~$m\geq n$ by $(C\delta)^n$ and converges as $m\to \infty$. In particular, there exists~$q\in \mathbb{P}_\infty(C\delta)$.
such that, for every $m,n\in\mathbb{N}$ with $n\leq m$, \begin{equation} \label{e.vermicious.knid}
\left| \nabla^n q_m (0) - \nabla^n q(0)\right| \leq (C\delta)^m \longrightarrow 0 \quad \mbox{as} \ m\to \infty. \end{equation} Define~$\psi\in \mathbb{A}_\infty(C\delta)$ by $\psi:=\sum_{k=0}^\infty \nabla^k q:\phi^{(k)}$. By~\eqref{e.psi.easybound} and~\eqref{e.vermicious.knid}, \begin{equation} \label{e.psi.to.psim}
\left\| \psi_m - \psi \right\|_{\underline{L}^2(Q_m)} \leq (C\delta)^m. \end{equation} We next apply Theorem~\ref{t.analyticity} to get, for $m\in\mathbb{N}$, $R:= 3C_1^2m$ and $r\in \left[C_1m, R \right]$, \begin{align*}
\left\| u - \psi_m \right\|_{\underline{L}^2(Q_r)} \leq
\left( \frac {C_1r}{3C_1^2m} \right)^{m+1} \left\| u \right\|_{\underline{L}^2(B_R)}. \end{align*} Taking $r :=C_1m$ and applying~\eqref{e.slowexp.growth} yields, for all $m\in\mathbb{N}$ sufficiently large, \begin{equation}
\left\| u - \psi_m \right\|_{\underline{L}^2(Q_{C_1m} )} \leq \left( \frac13 \right)^{m+1} \exp\left( 2\delta R \right) = \exp\left( -m + 4C_1^2m \delta \right). \end{equation} Thus if~$\delta_0(d,\Lambda)>0$ is small enough, we obtain, for all $m$ sufficiently large, \begin{equation}
\left\| u - \psi_m \right\|_{\underline{L}^2(Q_{Cm})} \leq \exp\left(-\frac12 m\right). \end{equation} Combining this with~\eqref{e.psi.to.psim} and sending $m \to \infty$ yields $u=\psi$. \end{proof}
We turn next to the proof of Theorem~\ref{t.quc}.
\begin{proof}[{Proof of Theorem~\ref{t.quc}}] We may suppose without loss of generality that \begin{equation} \label{e.your.ball.ass}
\left\| u \right\|_{\underline{L}^2(Q_r)} \leq
\left\| u \right\|_{\underline{L}^2(Q_R)}, \end{equation}
otherwise the result is immediate from the inequality $\left\| u \right\|_{\underline{L}^2(Q_s)} \leq
C\left\| u \right\|_{\underline{L}^2(Q_R)}$.
Applying Theorem~\ref{t.analyticity} gives, for every $Cm\leq s \leq cR$, \begin{equation}
\left\| u - \psi_m \right\|_{\underline{L}^2(Q_s)} \leq
\left( \frac{Cs}{R} \right)^{m+1} \left\| u \right\|_{\underline{L}^2(Q_R)}. \end{equation} By Lemmas~\ref{l.corr.growth} and~\ref{l.diffbound}, for every $Cm<r<cs$, \begin{align} \label{e.psi.gb}
\left\| \psi_m \right\|_{\underline{L}^2(Q_s)} & \leq \sum_{k=0}^m \left( \frac{Cs}{k+1} \right)^k
\left| D^k \widehat{u}(0) \right| \\ & \notag \leq \sum_{k=0}^m \left( \frac{Cs}{r} \right)^k
\left\| u \right\|_{\underline{L}^2(Q_r)}
\leq \left( \frac{Cs}{r} \right)^{m}
\left\| u \right\|_{\underline{L}^2(Q_r)}. \end{align} Thus, by the triangle inequality, for every $Cm\leq r \leq cs \leq c^2R$, \begin{align} \label{e.trapped}
\left\| u \right\|_{\underline{L}^2(Q_s)} & \leq
\left\| u - \psi_m \right\|_{\underline{L}^2(Q_s)} +
\left\| \psi_m \right\|_{\underline{L}^2(Q_s)} \\ & \notag \leq \left( \frac{Cs}{r} \right)^{m}
\left\| u \right\|_{\underline{L}^2(Q_r)} + \left( \frac{Cs}{R} \right)^{m+1}
\left\| u \right\|_{\underline{L}^2(Q_R)}. \end{align} Fix $\theta \in (0,c]$ small enough that, if~$C$ is the largest of the constants in~\eqref{e.trapped}, then~$C\theta \leq \tfrac12$. Applying~\eqref{e.trapped} with $r = \theta s = \theta^2R$ and with $m\in\mathbb{N}$ chosen to be the largest integer such that we have $Cm \leq r$ (ensuring that~\eqref{e.trapped} is valid) and \begin{equation} \label{e.cond2} \left( \frac{C}{\theta} \right)^{2m} \leq
\frac{\left\| u \right\|_{\underline{L}^2(Q_R)}}{4\left\| u \right\|_{\underline{L}^2(Q_r)}}. \end{equation} With these choices, we can bound the first term on the right side of~\eqref{e.trapped} by \begin{equation*} \left( \frac{Cs}{r} \right)^{m}
\left\| u \right\|_{\underline{L}^2(Q_r)} = \left( \frac{C}{\theta} \right)^m
\left\| u \right\|_{\underline{L}^2(Q_r)} \leq \frac12
\left\| u \right\|_{\underline{L}^2(Q_r)}^{\frac12}
\left\| u \right\|_{\underline{L}^2(Q_R)}^{\frac12}. \end{equation*} For the second term, we break into two cases: either $C(m+1)>r$ or~\eqref{e.cond2} is false for $m+1$. If the former holds, then \begin{align*}
\left( \frac{Cs}{R} \right)^{m+1}
\left\| u \right\|_{\underline{L}^2(Q_R)} & = \left( C\theta \right)^{m+1}
\left\| u \right\|_{\underline{L}^2(Q_R)} \\ & \leq \left( \frac12 \right)^{m+1}
\left\| u \right\|_{\underline{L}^2(Q_R)} \leq \exp(-cr)
\left\| u \right\|_{\underline{L}^2(Q_R)}, \end{align*} while if the latter holds, then for each $\alpha \in (0,1)$, \begin{align*}
\left( \frac{Cs}{R} \right)^{m+1}
\left\| u \right\|_{\underline{L}^2(Q_R)} & = \left( C\theta \right)^{m+1}
\left\| u \right\|_{\underline{L}^2(Q_R)} \\ & \notag
\leq
\left( C\theta \right)^{m+1} \left\| u \right\|_{\underline{L}^2(Q_R)}^{1-\alpha} \left( \frac{C}{\theta} \right)^{2 \alpha m} \left\| u \right\|_{\underline{L}^2(Q_r)}^\alpha \\ & = C\theta \left( C^{1+2\alpha} \theta^{1-2\alpha} \right)^{m}
\left\| u \right\|_{\underline{L}^2(Q_R)}^{1-\alpha}
\left\| u \right\|_{\underline{L}^2(Q_r)}^\alpha. \end{align*} Using that \begin{equation*}
C^2 \theta^{\frac12-\alpha} \leq 1 \quad \implies \quad C\theta \left( C^{1+2\alpha} \theta^{1-2\alpha} \right)^{m} \leq \frac12, \end{equation*} taking $\theta(\alpha,d,\Lambda)\in (0,1)$ small enough and recalling~\eqref{e.your.ball.ass}, we may combine the above to obtain \begin{align*}
\left\| u \right\|_{\underline{L}^2(Q_s)} \leq
\left\| u \right\|_{\underline{L}^2(Q_r)}^{\alpha}
\left\| u \right\|_{\underline{L}^2(Q_R)}^{1-\alpha} +
\exp(-cr) \left\| u \right\|_{\underline{L}^2(Q_R)}. \end{align*} This completes the proof. \end{proof}
We conclude with proof of Theorem~\ref{t.bottom}.
\begin{proof}[{Proof of Theorem~\ref{t.bottom}}] Suppose that $A\in [4,\infty)$ and $\lambda \in \left( 0,\delta_0^2\right]$, with $\delta_0\leq 1$ the constant in Theorem~\ref{t.Liouville} in $d+1$ dimensions, and suppose that $u\in H^1_{\mathrm{loc}}({\mathbb{R}^d})$ satisfies \begin{equation} -\nabla \cdot \mathbf{a}\nabla u = \lambda u \quad \mbox{in} {\mathbb{R}^d} \end{equation} and \begin{equation} \limsup_{r\to\infty} \, \exp( Ar )
\left\| u \right\|_{\underline{L}^2(Q_r)} = 0. \end{equation} We will show that if $\lambda$ is sufficiently small and $A$ is sufficiently large, each depending only on $(d,\Lambda)$, then $u\equiv 0$. We may assume~$u \leq 1$.
Add a dummy variable to~$u$ by defining \begin{equation}
v(x,x_{d+1}) = \exp\left(\lambda^{\frac12} x_{d+1}\right) u(x) \end{equation} we observe that $v$ is a solution of the equation \begin{equation}
-\nabla \cdot \widetilde\mathbf{a} \nabla v = 0 \quad \mbox{in} \ \mathbb{R}^{d+1}, \end{equation} where $\widetilde{\mathbf{a}}$ is the $(d+1)\times (d+1)$ matrix defined by \begin{equation}
\widetilde\mathbf{a}(x) := \begin{pmatrix} \mathbf{a}(x) & 0 \\ 0 & 1 \end{pmatrix}. \end{equation} Select $R\geq 10$ satisfying \begin{equation} \label{e.bare.ball}
\left\| v \right\|_{\underline{L}^2(B_R(2Re_1))} \leq \exp\left( - AR + 2\lambda^{\frac12}R \right) \leq \exp\left( -\tfrac12 AR\right). \end{equation} Let $c_1 \in \left(0,\tfrac12\right]$ and $C_1 \in [1,\infty)$ be constants to be selected below such that $c_1C_1<1$. Set $m:= \left\lfloor c_1 R \right\rfloor$ and note that $C_1 m < R$. If $c_1$ is sufficiently small, depending only on $(C_1,d,\Lambda)$, then we may apply Theorem~\ref{t.analyticity} to obtain $\psi\in \mathbb{A}_m^0$ satisfying, for every $S \geq R$ and $r\in \left[ C_1 m, S\right]$, \begin{equation} \label{e.reg.app}
\left\| v - \psi \right\|_{\underline{L}^2(B_r)} \leq \left( \frac{Cr}{S} \right)^{m+1}
\left\| v \right\|_{\underline{L}^2(B_{S})}. \end{equation} Here~$\mathbb{A}_m$ is understood to be defined with respect to the coefficients~$\widetilde{\mathbf{a}}$. By taking $r:= C_1 m$ and $S:= R$ we obtain, for $c_1(C_1,d,\Lambda)>0$ sufficiently small, \begin{equation*}
\left\| v - \psi \right\|_{\underline{L}^2(B_{C_1m}(2Re_1))} \leq \left( \frac{CC_1c_1R}{R} \right)^{m+1}
\left\| v \right\|_{\underline{L}^2(B_R(2Re_1))} \leq \frac12
\left\| v \right\|_{\underline{L}^2(B_R(2Re_1))}. \end{equation*} This implies by the triangle inequality and~\eqref{e.bare.ball} that \begin{equation}
\left\| \psi \right\|_{\underline{L}^2(B_{C_1m}(2Re_1))} \leq
2\left( \frac{R}{C_1 m} \right)^{d/2} \left\| v \right\|_{\underline{L}^2(B_R(2Re_1))} \leq \exp\left( -\tfrac 14AR \right) \end{equation} provided that $A$ is large enough. If $C_1(d,\Lambda)$ is taken sufficiently large, we deduce, by Lemmas~\ref{l.corr.growth} and~\ref{l.diffbound}, for every $r>C C_1m$, \begin{align} \label{e.psi.nothing}
\left\| \psi \right\|_{\underline{L}^2(B_r(2Re_1))} & \leq \sum_{k=0}^m \left( \frac{Cr}{k+1} \right)^k
\left| D^k \widehat{\psi}(2Re_1) \right| \\ & \notag \leq \sum_{k=0}^m \left( \frac{Cr}{k+1} \right)^k \left( \frac{Ck}{C_1m} \right)^k
\left\| \psi \right\|_{\underline{L}^2(B_{C_1m}(2Re_1))} \\ & \notag \leq \left( \frac{Cr}{m} \right)^m \exp\left( -\tfrac14 AR \right). \end{align} We now, once and for all, fix $C_1$ so that~\eqref{e.psi.nothing} holds and then fix~$c_1$ as above. Taking $r:=3R$ and $S:=C_2R$ in~\eqref{e.reg.app} for $C_2>10$, combining the resulting estimate with~\eqref{e.psi.nothing} and then taking $C_2$ to be a sufficiently large constant, depending only on $(d,\Lambda)$, we deduce that \begin{align*}
\left\| v \right\|_{\underline{L}^2(B_{R})} &
\leq C \left\| v \right\|_{\underline{L}^2(B_{3R}(2Re_1))} \\ & \leq
C \left\| v - \psi \right\|_{\underline{L}^2(B_{3R}(2Re_1))} +
C \left\| \psi \right\|_{\underline{L}^2(B_{3R}(2Re_1))} \\ & \leq \left( \frac{CR}{C_2R} \right)^{m+1}
\left\| v \right\|_{\underline{L}^2(B_{C_2R}(2Re_1))} + \left( \frac{CR}{m} \right)^m \exp\left( -\tfrac14AR \right) \\ & \leq \exp\left( -cR + C\lambda^{\frac12} R \right) + \exp\left( C R - \tfrac14AR \right). \end{align*} Taking $\lambda>0$ sufficiently small and $A>1$ sufficiently large, each depending only on~$(d,\Lambda)$, we obtain \begin{equation}
\left\| v \right\|_{\underline{L}^2(B_R)} \leq 2 \exp\left( -cR \right). \end{equation} Sending $R\to \infty$ yields $v \equiv 0$ and thus $u\equiv 0$, completing the argument. \end{proof}
\subsection*{Acknowledgments} We thank Jonathan Goodman for showing us a Bloch wave perturbation proof of Theorem~\ref{t.bottom} and Yulia Meshkova for pointing us to the paper of Filonov~\cite{Fil}. C.S.~would like to thank Carlos Kenig and Dana Mendelson for many inspiring discussions and for alerting him to the interest of quantitative unique continuation for periodic operators. S.A.~was partially supported by the NSF award DMS-1700329. T.K.~was supported by the Academy of Finland and the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 818437). C.S.~was partially supported by NSF award DMS-1712841.
\end{document} | arXiv |
Avoider-Enforcer game
An Avoider-Enforcer game[1]: 43–60 (also called Avoider-Forcer game[2] or Antimaker-Antibreaker game[3]: sec.5 ) is a kind of positional game. Like most positional games, it is described by a set of positions/points/elements ($X$) and a family of subsets (${\mathcal {F}}$), which are called here the losing-sets. It is played by two players, called Avoider and Enforcer, who take turns picking elements until all elements are taken. Avoider wins if he manages to avoid taking a losing set; Enforcer wins if he manages to make Avoider take a losing set.
A classic example of such a game is Sim. There, the positions are all the edges of the complete graph on 6 vertices. Players take turns to shade a line in their color, and lose when they form a full triangle of their own color: the losing sets are all the triangles.
Comparison to Maker-Breaker games
The winning condition of an Avoider-Enforcer game is exactly the opposite of the winning condition of the Maker-Breaker game on the same ${\mathcal {F}}$. Thus, the Avoider-Enforcer game is the Misère game variant of the Maker-Breaker game. However, there are counter-intuitive differences between these game-types.
For example, consider the biased version of the games, in which the first player takes p elements each turn and the second player takes q elements each turn (in the standard version p=1 and q=1). Maker-Breaker games are bias-monotonic: taking more elements is always an advantage. Formally, if Maker wins the (p:q) Maker-Breaker game, then he also wins the (p+1:q) game and the (p:q-1) game. Avoider-Enforcer games are not bias-monotonic: taking more elements is not always a disadvantage. For example, consider a very simple Avoider-Enforcer game where the losing sets are {w,x} and {y,z}. Then, Avoider wins the (1:1) game, Enforcer wins the (1:2) game and Avoider wins the (2:2) game.
There is a monotone variant of the (p:q) Avoider-Enforcer game-rules, in which Avoider has to pick at least p elements each turn and Enforcer has to pick at least q elements each turn; this variant is bias-monotonic.[1]: 45–46
Partial avoidance
Similarly to Maker-Breaker games, Avoider-Enforcer games too have fractional generalizations.
Suppose Avoider needs to avoid taking at least a fraction t of the elements in any winning-set (i.e., take at most 1-t of the elements in any set), and Enforcer needs to prevent this, i.e., Enforcer needs to take less than a fraction t of the elements in some winning-set. Define the constant: $c_{t}:=(2t)^{t}\cdot (2-2t)^{1-t}=2\cdot t^{t}\cdot (1-t)^{1-t}$(in the standard variant, $t=1,c_{t}\to 2$).
• If $\sum _{E\in {\mathcal {F}}}{c_{t}}^{-|E|}<1$, and the total number of elements is even, then Avoider has a winning strategy when playing first.[3]: thm A5
See also
Biased positional game#A winning condition for Avoider
References
1. Hefetz, Dan; Krivelevich, Michael; Stojaković, Miloš; Szabó, Tibor (2014). Positional Games. Oberwolfach Seminars. Vol. 44. Basel: Birkhäuser Verlag GmbH. ISBN 978-3-0348-0824-8.
2. Bednarska-Bzdęga, Małgorzata (2014-01-12). "Avoider-Forcer Games on Hypergraphs with Small Rank". The Electronic Journal of Combinatorics. 21 (1): 1–2. ISSN 1077-8926.
3. Lu, Xiaoyun (1991-11-29). "A matching game". Discrete Mathematics. 94 (3): 199–207. doi:10.1016/0012-365X(91)90025-W. ISSN 0012-365X.
| Wikipedia |
Molecular model of dynamic social network based on e-mail communication
Marcin Budka1,
Krzysztof Juszczyszyn2,
Katarzyna Musial3 &
Anna Musial2
Social Network Analysis and Mining volume 3, pages543–563(2013)Cite this article
In this work we consider an application of physically inspired sociodynamical model to the modelling of the evolution of email-based social network. Contrary to the standard approach of sociodynamics, which assumes expressing of system dynamics with heuristically defined simple rules, we postulate the inference of these rules from the real data and their application within a dynamic molecular model. We present how to embed the n-dimensional social space in Euclidean one. Then, inspired by the Lennard-Jones potential, we define a data-driven social potential function and apply the resultant force to a real e-mail communication network in a course of a molecular simulation, with network nodes taking on the role of interacting particles. We discuss all steps of the modelling process, from data preparation, through embedding and the molecular simulation itself, to transformation from the embedding space back to a graph structure. The conclusions, drawn from examining the resultant networks in stable, minimum-energy states, emphasize the role of the embedding process projecting the non–metric social graph into the Euclidean space, the significance of the unavoidable loss of information connected with this procedure and the resultant preservation of global rather than local properties of the initial network. We also argue applicability of our method to some classes of problems, while also signalling the areas which require further research in order to expand this applicability domain.
The emergence of complex behaviour in a system composed of many interacting elements is one of the most fascinating phenomena and recently also a prominent area of research. There are many types of complex networked systems, which can be classified in many different ways. One of the approaches distinguishes infrastructural (Internet, WWW, energy and transportation networks) and natural complex systems (biological networks, social systems and ecosystems) (Barrat et al. 2008). Another classification divides complex networks into technological, social, biological and information networks (Kolaczyk 2009). There is no commonly accepted definition of a complex networked system but there is an agreement that such structure consists of multiple interacting components whose global behaviour cannot be simply inferred from the behaviour of the individual components (Holland 1996; Barrat et al. 2008). The elements of the network are not independent but are rather connected via relationships and in consequence they influence each other. The number of nodes in these networks can differ from hundreds to millions (Watts and Strogatz 1998). One of the challenges is to identify which component influences the behaviour of other components, which is directly connected with the dynamics of such structures.
Complex systems that are investigated in this paper are social networks where nodes represent people (but can also be other social entities such as departments or even whole organisations), connected by different types of social relationships (e.g. friendship, co-working, family) (Garton et al. 1997; Wasserman et al. 1994). Although the general concept of social networks seems to be simple, the fact that the underlying structure is a network implies a set of characteristics, which are typical to all complex systems, i.e. the sum of the interactions between the users does not allow to draw conclusions about the behaviour of the social system as a whole. The consequence of this is that tracking changes in social networks is a very challenging and resource-consuming task, especially that the number of edges of the graphs representing social networks that are nowadays at our disposal can be counted in millions.
Due to the scale and complexity of such systems, computer simulations became an increasingly popular tool for investigating the dynamics of complex systems including social networks. Simulations supplement traditional approaches—formal theories and empirical studies and serve as analytical models enabling making certain predictions about the future behaviour of complex systems. In this research, we focus on the predicting the changes in the network structure. This is especially important as the network structure affects the functions of the network (Strogatz 2001). We also face a typical trade-off between simulations that take into account the detailed, microscopic description of the system (an approach, which in theory assures the most accurate predictions, often with an unacceptable computational overhead) and the minimal set of rules that allows to model the evolution of the system (Schweitzer et al. 2003).
It should be emphasized that many properties of complex systems are hardly definable in terms of any analytical model. Therefore, computer simulations seem to be the only way to gain insight into global system dynamics (Schweitzer et al. 2003; Boccaletti et al. 2006). So far, physics has provided several methodological approaches to tackle this issue. We hence argue that the spatial mobility and concentration of interacting particles can be modelled by employing the molecular dynamics paradigm, leading to many interesting extensions of standard approaches, based on the reinterpretation of potentials and distance in a given space (Weidlich 1991). One proposition of such modification is described in detail in this work. Another family of approaches successfully applied to physics, biology, evolutionary biology and social sciences are cellular automata (Wolfram 1986), starting from the famous Game of Life artificial life model of Conway. One of the first researchers who applied the particle–based approach to social dynamics was Dirk Helbing, who in (Helbing 2010) proposed a fundamental dynamic model which includes many established models as special cases, (e.g. logistic equation, gravity model, some diffusion models, the evolutionary game theory and the social field theory) and also implies numerous new results.
However, in this work we argue that the rapid development of social portals and social media gives us an unique opportunity of the investigation of social systems on the basis of real data. When we consider inferring social relations from the records gathered from systems providing communication and recommendation services, the relations may be quantified and directly measured. On the other hand, a standard approach of sociodynamics assumes a global (and relatively simple) definition of social potential (which reflects the character of "social force" driving the changes in the relations between the system components) which is used to simulate and analyse the collective behaviour of system components (Epstein 2008). This approach has been proved useful for many classes of social systems and the modelling of opinion dynamics (Malarz et al. 2011).
Taking the above into account, we propose to infer the character of social potential from the real-life social system data (using an email-based social network as an example) and to verify the possibility of using it to determine the evolution of the system. This requires embedding n-dimensional social space in Euclidean space to apply the physically inspired methods. According to the best of our knowledge no computational models for assessing the evolutionary schemes of real-world internet-based social structures, in which the edges can not only be formed but can also fade, were developed so far. Hence, in this paper, we propose application of molecular dynamics to modelling the evolution of email-based social network. We focus on the equilibrium state of a network, i.e. the state after the molecular simulation has converged and discuss various issues and challenges encountered during this research. Moreover, we argue that, in the presence of the data coming from real system, the verification of such a model should be done by means of checking if it is possible to recreate the social network from simulation results and compare it with the real network structures which have evolved in the period of time covered by the simulation.
The rest of the paper is structured as follows: in Sect. 2 related work in the fields of social networks dynamics, graph embedding and dynamic molecular modelling are presented. In Sect. 3 methodology followed in this paper is outlined and Sect. 4 explains the experimental set-up. Section 5 is devoted to the molecular simulation and its outcomes. Section 6 aims at presenting the concept of social network recreation from the simulation results and Sect. 7 includes the analysis of the retrieved social networks. Finally, in Sect. 8 results arising from the conducted research are summed up and the future work is presented.
Dynamics of social networks
In the past few years the problem of predicting the future interactions between users in social networks has become an important research challenge. Due to the availability of datasets of online activities and communication between people, scientists try to describe both structure and evolution of such networks. Most of the approaches addressing the complex networks growth take into consideration a limited set of global characteristics of the networks and develop models that reproduce only these characteristics, e.g. node degree distribution (Barabasi 2003), clustering coefficient (Watts 2002) or network diameter (Bollobas 1985).
There are some approaches that aim at developing specific models for online social networks and take into consideration some information characteristic to such networks (Kumar et al. 2006; Lescovec et al. 2008; Bringmann et al. 2010; Braha and Bar-Yam 2006; Liben-Nowell and Kleinberg 2007; Davis et al. 2012; Kashoob et al. 2012). Different models propose different methods of network growth. In (Kumar et al. 2006), on the basis of the analysis of real-world networks such as Flickr and Yahoo 360!, the users have been divided into three different types: passive, linkers and inviters. The members of the first group (passive users) join the network out of curiosity or because of being invited by a friend. These users, as their name suggests, never engage in any significant activities within the network and do not interact with other users. Inviters on the other hand, are interested in migrating the group that they have in the real world into a virtual world; thus their actions focus on inviting their friends to participate in an online social network. Linkers actively connect themselves to other members within the online social network. Based on the analysis of datasets the authors define a rule-based system that follows specific rules used for evolution of the social network. The method that describes the network growth can be defined as the set of steps: (1) at each time step, a node arrives, and one of the statuses: passive, linker or inviter is randomly assigned to it; (2) during the same time step, x edges arrive and the source of each of the edges is chosen at random from the existing inviters and linkers in the network using preferential attachment. Depending on the chosen type of the source node (inviter or linker) different actions are performed. If the source is an inviter, then it invites a non-member to join the network, and so the destination is a new node. If the source is a linker, then the destination is chosen from among the existing linkers and inviters, again using preferential attachment (Kumar et al. 2006). This model represents the growth of a network, i.e. it takes into account adding new nodes and edges. However, the problem of link prediction covers not only the creation of new links but also fading of existing relations.
In (Bringmann et al. 2010) the authors have presented another approach that defines a set of rules regarding how the network evolves. They focus on discovering patterns of interactions between users and their evolution over time. The authors propose to create a single graph that represents social network, which is supplemented with additional information—a time-stamp, added to each relation when it appears in the network for the first time. The experiments were performed on the DBLP database (Bringmann et al. 2010). Similarly to the previous presented study, also this one assumes that the users and the relations between them can only be added to the system and will never be deleted. Moreover, both of approaches presented so far allow to investigate the creation of new edges but do not allow to follow the dynamics of the relationships strengths between users, which is one of their limitations.
Yet another framework for the network growth was developed in (Lescovec et al. 2008) where the authors studied four online social networks: Flickr, Delicious, Answers and LinkedIn. They proposed to apply the maximum-likelihood estimation principle to compare a family of parameterised models in terms of their probability of generating the observed data and as a result to select the model that reflects the data in the best possible way. The task in this framework was to predict which nodes will a new edge connect. For every edge arriving to the network the likelihood that it will connect two given nodes under some model is assessed. The product of these values over all edges gives the likelihood of the model and the model with the highest likelihood is chosen. Similarly to the previously presented methods this one also does not consider the strength of the relation as well as the fact that an edge can disappear from the network.
A survey of other link prediction methods can be found in (Liben-Nowell and Kleinberg 2007), where the approaches like common neighborurs, Jaccard's coefficient and Adamic/Adar method, preferential attachment, Katz method, PageRank and its variants, low-rank approximation, unseen bigriams and clustering are discussed.
A set of approaches that take into consideration the fact that links can disappear from the network have been proposed in (Hill et al. 2010; Braha and Bar-Yam 2006) where the authors have detected a dramatic time dependance in network centrality and the role of nodes, something not apparent from static analysis of node connectivity and network topology. Their experiments studied large-scale email networks consisting of 57,000 users based on data gathered over a period of 113 days. They found that although the daily networks were scale-free, the well-connected nodes in these networks changed from day to day.
A recent method also accounting for the disappearing links has been proposed and investigated in (Juszczyszyn et al. 2011a, b, c), where based on the changes in the local structure, a 1st order probabilistic model of transitions between various triad types has been derived. The model results from an observation that there exist distinctive patterns which drive the evolution of connections between nodes. Node disappearance has also been addressed in (Sarr et al. 2012), but in a somewhat different the context of disruption of the information flow.
Our approach differs from these presented above as we do not propose a model for network growth per se but we investigate the limitations of sociodynamic model verified on data coming from real system. Our proposition takes into account both creation and vanishing of the relationships. Additionally, the network investigated in this work is a structure where strength of the relationships changes over time, which is an important factor in social networks due to the cognitive limitations of people (Hill and Dunbar 2002). In our approach, we do not assign roles to users as this may be misleading. We rather assess, based on the current interactions between users, how the relations strength and the structure of the network may look like in the future.
Distance preserving graph embedding
Following the in-depth discussion presented in (Watts 2002) we cannot expect the social space to be metric, i.e. the triangle inequality between any three nodes does not hold. On the other hand, as it was mentioned above, molecular modelling assumes the interaction between the particles embedded in the Euclidean space. For this reason, to apply molecular modelling we must first perform embedding of the social network graph in a metric, Euclidean space. Numerous embedding methods exist whose overview is presented below.
The Big Bang embedding algorithm (BBE) presented in (Shavitt and Tankel 2004) simulates an explosion of particles that represent network users under a force field that is derived from the embedding error. Each particle is the geometric image of a vertex. The force field reduces the potential energy of the particles which is related to the total embedding error of all particle pairs. In the Big Bang Simulation (BBS) all particles are initially placed in the same location in space. The whole process is performed in an iterative manner and each iteration moves the particles in discrete time intervals. Every iteration begins with calculation of the field force on each particle at the current particles' positions (for the first iteration forces are chosen randomly). As it was mentioned, the forces are derived from the potential energy. In the next step, the positions and velocities at the next time step are calculated. The final step of each iteration is to evaluate the new potential energy. This method allows to embed the network into a freely selected number of dimensions.
Another method that can be used to embed a graph in Euclidean space is called the Multidimensional Scaling (MDS) (Torgerson 1965). MDS defines a suite of methods often used in information visualization and exploration of similarities or dissimilarities in data. There are two variations of MDS, i.e. classical multidimensional scaling (CMDS) algorithm and standard MDS (Bronstein and Kimmel 2006; Kruskal et al. 1978). Classical metric MDS develops the metric as a symmetric bilinear form and calculates the leading d eigenvalues of the corresponding matrix (Torgerson 1965). An MDS algorithm starts with a matrix of similarities between objects (similarity relation does not have to be symmetrical) and then assigns a location of each item in a low-dimensional space. It hence estimates the coordinates of a set of objects in a space of specified dimensionality on the basis of measuring the distances (which, however do not have to be metric) between pairs of objects. A variety of models can be used that include different ways of computing distances and various functions relating the distances to the actual data. Both methods allow to embed graph into different numbers of dimensions. However, the problem that we faced during our experiments with MDS was that the computational overhead was very high and we were not able to obtain results within reasonable time.
In High–Dimensional Embedding (Harel and Koren 2004), which is a fast method for creating 2D representations of large graphs, the graph is first embedded into a very high dimensional space—usually associated with the number of nodes—and then projected into a 2D plane using Principal Components Analysis. This method is used for undirected graphs. It will not be useful from the perspective of our experiments as one of the goals of this study is to embed the graph into different dimensions and verify which number of dimensions helps to achieve best results from the link prediction perspective.
Minimum Volume Embedding (MVE) presented in (Shaw and Jebara 2007, 2009) is an algorithm for non–linear dimensionality reduction that uses semi-definite programming (SDP) and matrix factorization to find a low-dimensional embedding that preserves local distances between points while representing the dataset in fewer dimensions. Authors of MVE emphasise that in all cases MVE in comparison with Semi-definite Embedding and Kernel Principal Component Analysis is able to capture more of the variance of the data in the first two eigenvectors, providing a more accurate 2-dimensional embedding (Shaw and Jebara 2007, 2009). The main features of the minimum volume embedding approach are (1) MVE for a given dataset returns always the same set of coordinates, (2) isolated nodes are neglected in the embedding process and (3) MVE is stable, i.e. adding one node with very weak connections does not influence significantly the positions of the remaining nodes. Enumerated characteristics of MVE means that the graph can be embedded only into 2D space which is not enough from the perspective of the proposed experiments in this paper as one of the goals is to find out what is the best number of dimensions to which the graph should be embedded. Moreover, in the case of not connected graphs the algorithm does not work.
Dynamic molecular modelling and simulation
Dynamic molecular modelling is one of the simulation methods applicable to large ensemble of interacting objects. It was historically used to model physical systems with huge number of particles. In its most classical version the particles are identical and indistinguishable and interact with each other through two–particle mutual symmetrical potential, which is identical for every pair of interacting particles and only distance-dependent. This model can be further extended and modified and it has already proven its applicability to more complex systems. The exact form of the interaction potential can differ depending on the details of the modelled system. In some cases it can be obtained experimentally if two–particle interaction can be separated, extracted and the dependence on their basic properties (e.g. mass, charge etc.) and inter–particle distance can be determined or is known from theoretical considerations. Unfortunately, it is not always the case. Usually the microscopic details of interaction potential are not directly accessible experimentally and only the macroscopic characteristics (which can be described as the statistical mean values) of the whole particle ensemble are known (e.g. temperature, energy, entropy, etc.). Although the behaviour of each particle on a microscopic scale is fully deterministic due to the inter-particle interaction being driven and governed by the second Newton's dynamics principle, it is only possible in very limited number of cases to deduce the form of interaction potential from the macroscopic behaviour of the particle ensemble, i.e. if sub-ensembles, characteristic clustering effects or short-range ordering can be observed. In most cases one has to assume a form of interacting potential (basing on the boundary behaviour of analysed system or some descriptive characteristics that can be deduced from macroscopic observations or general features of two-particle behaviour), perform the simulations of the system of interest and check if the behaviour of macroscopic observables can be reproduced. The problem of finding the interaction potential, in the case when the trajectory/time-dependence of particle position is known, is solvable by a number of differentiation and integration steps. Although this procedure is well defined mathematically, it cannot be conducted in the case of many interacting particles as the trajectory is not a simple analytical function but rather seems random (similar to Brownian motion) due to the complexity of analysed system, in which every particle responds to a force originating from all other particles. Based on the potential, the force acting between particles can be calculated using the following formula:
$$ {{\mathbf{F}}}=-\nabla U $$
where U is the interaction potential.
If the force is known, the time evolution of the system can be obtained by solving for each particle separately the classical equation of motion (2nd Newton's principle of dynamics):
$$ {{\mathbf{F}}}=\frac{{{\mathbf{dp}}}}{{\hbox{d}}t} $$
where p denotes the momentum of the particle and F is a vector sum of forces from all other particles in the system. For objects with constant mass this formula takes the following familiar form:
$$ {{\mathbf{F}}}=m {{\mathbf{a}}}=m\frac{{{\mathbf{d^{2}}}s}}{{\hbox{d}}t^{2}} $$
by using the definition of the momentum and acceleration a, defined as a second derivative of the position vector s. The above equation must be solved for every particle in every simulation step. In order to start the algorithm, the initial positions of all particles, the formula for the force which is identical for all particle pairs and the interacting potential need to be specified. One of the standard potential functions used to describe the many-particle problems is Lennard–Jones potential which is given by
$$ V(r)=4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right] $$
where r denotes the distance between particles. The Lennard–Jones potential, which has been depicted in Fig. 1, is fully defined by two parameters: the depth of the potential ε-responsible for the strength of interactions between particles and σ-related to the minimum distance between two particles. As it can be seen the potential has a global minimum equal to ε for r min = 21/6, σ = 1.12σ. An important characteristic of this potential is that the nature of interaction between two particles depends on their distance. Namely, for distances bigger than σ the particles attract each other, while for distances smaller then σ the character of the potential changes to strongly repulsive.
Lennard–Jones potential
Having the analytical formula for the interaction potential one can easily obtain the formula for the force by simple differentiation, which should be performed analytically to avoid accumulation of numerical approximations. As a first approximation all particles are assumed to have unit mass. Knowing the forces, the Verlet algorithm may be used to obtain the position and the velocity of each particle in consecutive time steps (Juszczyszyn et al. 2009).
The concept of molecular modelling and simulation is used in this study to model the dynamics of a social network. The users who are the nodes of the network become particles in Euclidean space and the distance between particles will be determined based on the relationship strength between the users. The changes in the distances between particles over time will be the basis for inferring the potential in a purely data-driven way and in consequence for determining the force between particles.
The approach proposed in this paper is to reformulate the problem of time evolution in social networks and interpersonal relations into the language of multiple-particle interacting system. This is achieved by assigning the position of the node in the social network graph to the position of the particle in a metric space so that the inter-particle distance reflects the strength of the relation.
In our previous attempts the form of the interacting potential was assumed to reflect the tendency of two particles (nodes) to change their distance in social space. The experimental data were used to extract some characteristic features of the interaction and a modified Lennard–Jones potential was used to reproduce the time evolution at the macroscale. Another possibility to gain an insight into the character of social interaction is to examine in details the distance between each pair of nodes in consecutive time windows. The experimental data, i.e. the positions of each particle (node) in each time window are sufficient to obtain the dependence of the variation of the distance between two particles on their distance (see Sect. 4.2 where the distance matrix is created from the adjacency matrix).
Using only this dependence it is possible to simulate the behaviour of interacting particle ensemble in the following way. Knowing the initial distances between all pairs of particles the change of two-particle distance can be read from the experimental curve. The change of the distance between each two particles can be easily transformed into the displacement vector. The displacement vector has its beginning at initial position of the particle and its end in its final position (it is defined as a difference between the initial and final position vector). Its direction coincides with the direction of a line connecting two particles under consideration and it is pointing into the direction of the centre of mass when the distance between two particles is decreasing, and in opposite direction if the distance is increasing. Because of the equal masses of both interacting particles each of them changes its position by the half of the calculated distance change between them. This procedure allows to define the displacement vector for a considered particle and one of all the other particles from the ensemble. Such an operation should be repeated for all other particles to obtain all displacement vectors for a given particle. Since the displacement vectors calculated in this way represent the forces exerted by other particles, their superposition determines the direction in which the particle under consideration should be moved. We ignore the magnitude of the total force, as moving any particle by this value, which is the length of a negative gradient of the field potential, would most likely result in overshooting and lack of convergence. Instead, we optimize our system in an iterative manner, shifting all particles by a small, fixed step at a time until it reaches a steady state. In that way we are able to simulate the time dependence of the position of each particle knowing only the initial positions of all particles and experimentally obtained relation between the change of the distance between two particles and their distance.
The methodology followed in this study is summarised in Fig. 2. The consecutive stages of this research are presented below.
Methodology followed in this paper
Data preparation As the real-world evolving network is investigated in this paper, the first step is to prepare data in a way that they can be used in further parts of the experiments. This includes extraction of the interactions and time stamps of their occurrence from email logs dataset and dividing this set into time windows of a given size. From each time frame a single social network is created. Note, that in the case of email communication the underlying social network is directed and weighted. However, adjacency matrix fed to the embedding process has to be symmetrical. Thus the directed social network is transformed into undirected one by aggregating the communication between every two nodes.
Distance matrix creation: Creating a distance matrix for each social network snapshot, in which the distances between nodes reflect the intensity of communication between them, is the next step. The distance needs to be calculated for all pairs of nodes, including the pairs which are not connected directly or at all. In our approach the distance between two particles reflects the length of the shortest weighted path linking the two nodes in question.
Embedding distance matrices in Euclidean space The goal of this step is to project the created distance matrices into the Euclidean space in a way that the distances between nodes are reflected in the best possible way. After a review of existing embedding methods, Big Bang Simulation and Classic Multidimensional Scaling were chosen. These two methods facilitate embedding of non-metric spaces into almost arbitrary number of dimensions, limited by characteristics of the network under consideration, with moderate computational requirements.
Definition of molecular model In this step the potential field describing the evolution of interactions between nodes is defined. On this basis the whole molecular model of email-based social network is created. The potential function is determined from the changes in distances between two consecutive network snapshots. As the shape of the potential function depends on past data, the force governing the molecular simulation is different for each dataset.
Molecular simulation This part of the experiments utilizes the outputs of the previous steps: the embedded social network windows and the potential force which is used to move particles in the Euclidean space. The simulation terminates when the set of particles achieve a stable state.
Recreation of network snapshots and analysis of the results The study aims at assessing the characteristics of a network in a stable state, which is an outcome of the molecular simulation. In order to do that, the reverse process to the embedding has to be performed. For each time window the results of simulation, which are the set of nodes' positions in Euclidean space, are taken and the network graph is created based on the distances between nodes (particles) after simulation. This process is straightforward: if the distance between two particles is lower than a given threshold value, the link between these particles in social network is created. The experiments were performed for different values of the distance threshold. Finally, the properties of these retrieved networks are investigated. Two main properties were taken into account: node degree distribution and clustering coefficient.
Experiment setup
Data preparation—creating Email-based social network
The network that has been chosen for experiments was extracted from the email logs of the Wroclaw University of Technology (WrUT). The experimental data were collected during the period of 21 months (February 2006–October 2007). The network was created in the course of the data cleansing process and removing fake and external email addresses. The employees of WrUT are the nodes of the network, whereas email messages exchanged between them were used to infer their relationships (edges in the network). Although every single email message provides information about the sender's activity, it can simultaneously be sent to many recipients. An email sent to only one person reflects strong attention of the sender directed to this recipient, while the same email sent to 20 people does not. For that reason, the intensity of email communication I(x, y) between email user x and y has been defined as
$$ I(x,y)=\displaystyle{\sum_{i=1}^{card({\rm EM}(x,y))}\frac{1}{n_i(x,y)}} $$
where EM(x, y) is the set of all email messages sent between x and y and n i (x, y) denotes the number of all recipients of the ith email sent between x and y (Kazienko et al. 2009).
In consequence, every email with more than one recipient is treated as 1/n of a regular one (n is the number of its recipients). Although 'to-list' recipients are likely to be of much greater message-network importance than the 'cc-list' recipients, both groups are treated in the same way, i.e. the total number of the recipients of an email is always taken into account. Such approach results from the fact that the obtained data do not include information if the recipient of the email is on the 'to-list' or 'cc-list'.
The resulting social network \(SN = \langle N, I\rangle\) is defined as a tuple consisting of a set of network nodes N and a set of relationships that are described by their mean intensity \({{I : N\times N \rightarrow\ \mathbb{R}^+ \cup \{0\},}}\) given by Eq. 5. Note that the resulting structure is a non-directed graph with intensity I as a label assigned to the relationships.
It should be emphasized that the social network derived from the email logs does not have a static structure. The existence of any link in such a graph (i.e. relationship) is a result of a series of discrete events (email messages) which occur in certain time instants and usually with changing frequency. We may also think of the computed relationships' intensity as of the social distance between network members (nodes). Greater I reflects smaller distance in the social space. In order to track changes in relationship strength we have used a sliding window approach.
For the experiments the data from a period of 84 days were selected and divided into frames covering 7 days each. This allowed to create 12 social network graphs \({\hbox{SN}}(t_0), {\hbox{SN}}(t_1), \ldots {\hbox{SN}}(t_n)\) where \(t_0, t_1,\ldots, t_n\) are discrete instants of time. Each network is created according to the procedure defined above on the basis of 7-day period starting in \(t_0, t_1, \ldots, t_n.\) The networks \({\hbox{SN}}(t_0), {\hbox{SN}}(t_1), \ldots {\hbox{SN}}(t_n)\) are temporal images of evolving social structure which was built on the basis of email communication. In addition, only users who were active in all time windows were taken into account as they constitute the core of the network.
Distance matrix creation
The distance between two nodes should reflect their proximity. The most obvious choice—graph distance expressed as the length of the shortest path between the nodes, does not really fit the problem of modelling the dynamics of an email network, especially if the graph is weighted. For example, suppose that the shortest path between nodes x and y has a total weight of 0.7, but it passes through two intermediate nodes. At the same time the shortest path between v and w has a total weight of 0.3, but there are no intermediate nodes at all. In the context of an email network it means that v and w communicate directly, but not very often. On the other hand, x and y do not communicate directly with each other, but their nearest neighbours do it frequently, and x and y communicate with the neighbours frequently too. In practice it means that x and y may not even know each other, while v and w certainly do. Hence in this case the standard graph distance is misleading and for our experiments we propose an alternative definition of social distance. Denoting by \(D_{\hbox{EC}}(x \leftrightarrow y)\) the number of edges in the shortest undirected path between nodes x and y and by \(D_{\hbox{EW}}(x \leftrightarrow y)\) the sum of weights along the same path, normalized to the (0,1) range, the total distance between nodes x and y is given by the following formula:
$$D(x \leftrightarrow y) = \left\{ \begin{array}{ll}D_{\hbox{EC}} (x \leftrightarrow y) + D_{\hbox{EW}} (x \leftrightarrow y) \; if\; \exists (x \leftrightarrow y)\\ \max (D_{\hbox{EC}})+\max (D_{\hbox{EW}})+1 \; if\; \nexists (x \leftrightarrow y)\end{array} \right. $$
As a result the distance will always fall into the (1,2) interval for directly connected nodes, (2,3) if there is one intermediate node, etc. Note, that in this setting the number of edges in the shortest path contributes the most, while the additional information given by the edge weights is also taken advantage of. Equation 6 also assigns some finite distance value to all pairs of nodes not connected by any path, as one of the requirements imposed by the embedding algorithm we have used was that the distance should be defined for every pair of nodes.
Embedding networks in the Euclidean space
With the distance matrices in place the graphs \({\hbox{SN}}(t_0), {\hbox{SN}}(t_1),\ldots,{\hbox{SN}}(t_n)\) can be embedded in the Euclidean space (two or more dimensional), where each node is represented by a point with given coordinates. The resulting sets of points \({\hbox{SN}}_0, {\hbox{SN}}_1,\ldots,{\hbox{SN}}_n\) represent the temporal network images.
An important issue, which should be discussed here, is the dimensionality of the embedding space. Most embedding algorithms have been designed for the purpose of graph visualization. This naturally implies a two- or three-dimensional embedding. However, the higher the dimensionality of the embedding, the more accurately the social distances are mapped into the Euclidean space. Figure 3 depicts the average distance distortionFootnote 1 of the embedding as a function of dimensionality for the WrUT email network and for (a) BBS, (b) CMDS methods. As expected, in both cases the accuracy of the embedding grows with dimensionality. Please note that the scales on the vertical axes in the Fig. 3 are different. It should be emphasized that the pace of accuracy growth with increasing dimensionality is much faster in the case of BBS than CMDS. Intuitively we should choose the number of dimensions to be as high as possible. There is a limit, however, which results from the so called 'curse of dimensionality' (Bishop et al. 1995), and especially the 'distance concentration' phenomenon, which as demonstrated in (Budka et al. 2011) is particularly relevant in the context of dynamic molecular simulation of potential fields in the Euclidean space.
Distance distortion as a function of dimensionality
It has been observed that as the number of dimensions grows, the Euclidean distance loses its discriminative power, regardless of the characteristics of the dataset (Aggarwal et al. 1973; Francois et al. 2005). The reason for this is that under a broad set of conditions the mean value of the L 2-norm distribution grows with data dimensionality while the variance remains approximately constant (Fig. 4) (Francois et al. 2005). As a result, the nearest and furthest neighbours of any molecule appear to be at approximately the same distance, which makes the ratio of distances to the nearest and farthest neighbour tend to converge to 1. As argued in (Beyer et al. 1999), it can occur even for sets with as few as ten dimensions and the decrease in the ratio between the farthest and nearest neighbour distance is steepest in the first 20 dimensions. The effect is additionally magnified by the limited precision of calculations a computer can handle and often leads to the molecular simulation failing to converge (Budka et al. 2011). Hence in practice the embedding dimensionality needs to be a compromise between the distance distortion and negative effects of high dimensionality. For this reason we have decided to embed each graph into \(2,3,\ldots,20\) dimensions to investigate the mapping between graph distances and distances in embedded graph.
Distance concentration for the Euclidean norm, for a random vector drawn from a unit hypercube (solid line denotes the mean value, shaded region denotes the mean ± 2 standard deviations)
Embedding algorithm has to assure that the Euclidean distances between points (nodes) fit in the best possible way the distances in a social space (relation strengths in original graphs). As a result one obtains the representation of social system in which the network is seen as an assembly of N particles, representing the nodes of a social network.
After reviewing several embedding methods, it has been decided that two sets of experiments will be performed: (1) the Big-Bang Simulation and (2) CMDS as these methods enable to embed graph into an arbitrary number of dimensions. Additionally, BBS models the network nodes as a set of particles, which is consistent with the next part of the experiments where molecular modelling approach is used to determine the dynamics of a social network.
Embedding was performed on 12 previously extracted social networks. Each of the networks was embedded into \(2,3,\ldots,20\) dimensions using BBS. CMDS inherently selects the best number of dimensions (in excess of 400 in our case), so in this case the parameter was not set during the experiments, but only first \(2,3,\ldots,20\) dimensions produced by CMDS have been used in our simulations.
For each of the dimensionalities given above we have analysed how well the distances between particles from the social networks (graph) are reflected in the embedded space. To avoid negative effects of high dimensionality we decided to select the lowest number of dimensions that allowed to embed the graph in a way that the mean values of the distances after embedding, which correspond to the graph distances in the ranges <1; 2), <2; 3), <3; 4) etc. were well separated. This has been achieved for 12 dimensions, where for both BBS (Fig. 5) and CMDS (Fig. 6) the distributions of distances in the embedding space are approximately unimodal and their expected values are in the required range.
Mapping between graph distance and distances after embedding graph into 12 dimensions using BBS algorithm
Mapping between graph distance and distances after embedding graph into 12 dimensions using CMDS algorithm
Due to the aforementioned, the actual molecular simulation has been performed in 12-dimensional space. However, for the visualisation purposes, where appropriate and to present general idea, the figures were presented for the two-dimensional embedding and molecular simulation.
After selecting the number of dimensions, the next stage of the experiments was to embed the created social networks snapshots into Euclidean space. As discussed in Sect. 3, embedded graphs serve as an input to the molecular simulation process.
Setting up the dynamic molecular model
Because the sets of network nodes in \({\hbox{SN}}(t_0), {\hbox{SN}}(t_1),\ldots,{\hbox{SN}}(t_n)\) are equal, each point (node) is represented in any of the sets \({\hbox{SN}}_0, {\hbox{SN}}_1,\ldots,{\hbox{SN}}_n\) and is active in each of the windows. We may think of these points as of particles moving as a result of interactions (email communication) between them. At this point we use the formalism of molecular dynamics to associate a potential U with every particle (network node). The actual characteristic of this potential depends on the behaviour of the particles changing their positions in time instants \(t_0, t_1,\ldots, t_n.\)
First experiments were performed using standard Lennard–Jones potential function (Juszczyszyn et al. 2009; Musial et al. 2010). The analysis of server logs has revealed some features of the dynamics of email communication—the growing intensity of communication is always followed by the periods of less frequent email activity. This resembles the repelling force emerging between particles when their distance becomes less than some minimum. We noticed that intense email communication (which results in very small distances in social graph) is never sustained for a longer period of time. On the other hand, fading communication is (in most cases) followed by frequent message exchanges.
It should be stressed that the Lennard–Jones potential was used only for the first experiments and did not accurately fit the underlying data. In the experiments presented in this paper the social network-specific potential function on the basis of available data was developed. In order to do that, first the distance transition probability defined as the probability that a given distance in one window will change into another given distance in the next time frame, was calculated. For the 12 time windows, 11 transition probability matrices were obtained. The matrices were then averaged. The resulting final matrix is presented in Fig. 7a. The force that governs the changes of the location of the particles in the Euclidean space is proportional to the distance change. Hence the third-degree polynomial presented in Fig. 7b, inferred from the distance transition probability, describes the force used in the molecular simulation. Please note that the force will be different for different datasets.
Graph distance transition probability and expected graph distance change for the WrUT email network
The presented force allows to simulate the changes between communication patterns in consecutive time instants. The potentials associated with the nodes reflect their abilities and tendency to establish future connections with their neighbours—the nodes which are close in terms of social space (thus changing the distances in social space which is analogous to the behaviour of particles moving under influence of electrical/gravitational forces).
After the network graphs have been embedded in the Euclidean space and the force function has been established the molecular simulation can be performed. The goal of the simulation is to obtain the network that is in a stable state, i.e. does not change from one simulation step to another. In practice it means that the particles oscillate around the point of equilibrium. The whole process is performed iteratively until oscillation is detected by checking if the mean displacement of all particles between two non-consecutive steps of the simulation is below a threshold value (0.0005 in our experiments). Figure 8 presents the mean displacement changes during the simulation of the first time window, embedded using CMDS. This is just one example of the obtained during the experiments; for each time window and for each embedding algorithm this function will have a different shape and the stop condition will be met for different number of iterations. The common feature of all simulations is that mean displacement converges to 0.
Mean displacement of all particles between two non-consecutive steps of the simulation as a function of number of simulation steps for social network from Window 1 embedded using CMDS method
Figures 9 and 10 present the results of the molecular simulation in 2D space for windows 2 and 4, respectively (the windows have been selected for illustration purposes). For each of these windows the BBS and CMDS embedding as well as the result of the simulation are presented. Although the embedded graphs look differently for CMDS and BBS, the final outcomes of simulations are similar. These two windows were chosen to present how different shapes of embedded graphs behave during the simulation process.
Reconstruction of social networks
The result of the molecular simulation is a set of particle collections in their stable states. In order to investigate the characteristics of obtained structures a reverse-embedding process needs to be performed. During this phase the social non-metric graph is created from the particles embedded in Euclidean space whose positions were determined during the molecular simulation. The graph is recreated using the pairwise Euclidean distances between the particles.
First, the distance between each pair of particles is calculated. After that a threshold for the distance is set and a link is created between pairs of nodes for which the distance is below this threshold. Each social network was reconstructed using 100 different threshold values. First, the difference between maximum and minimum distance in a given time window was calculated and then this number was divided in 99 equal parts. Different values of distance threshold influence the number of links in the recreated network. Figure 11 shows the number of links in the 1st window of social network (embedded using CMDS) as a function of distance threshold. Note that most of the distances is in the range (0;8). For each time window and embedding method, 100 networks with different distance threshold were created and these networks and their characteristics are investigated in the next section.
Window 2 before and after molecular simulation
Number of links in the 1st window of social network (embedded using CMDS) as a function of distance threshold
Analysis of the experimental results
The recreated networks have been examined with respect to node degree distribution, clustering coefficient and shortest paths. Although there were 1,100 reconstructed networks (100 threshold values times 11 windows) and all three characteristics were calculated for each of them, only a subset of networks was selected for subsequent analysis. As it was pointed out, during the reconstruction process, the value of the distance threshold directly influences the number of links in the recreated network. The general and intuitive rule is that the higher the threshold the denser the reconstructed network. Because the structures analysed in this study are social networks, and according to Dunbar's number every human being can on average maintain 150 meaningful social relations (Hill and Dunbar 2002), in further analysis we have selected a threshold for each embedding method and for each time window for the average node degree to be as close to 150 as possible. Please note that the average node degree is meaningful in this case as the node degree distributions follow Poisson distribution. In Fig. 12 the node degree distributions for Window 2 for all 100 distance thresholds are presented, with different colours denoting distributions for different distance thresholds. It is clearly visible that the average node degree increases with the growth of the threshold. On the left side of the plot, for threshold equal to 1 the recreated network is empty (probability that a node has 0 edges is 1) and on the other hand for threshold 100 the network is represented by a fully connected graph (probability that a node is connected to all other nodes in a network is 1). For all intermediate thresholds the node degrees oscillate insignificantly around a given number which grows together with the distance threshold. Hence the value of the average node degree can be used to select networks for further analysis.
Node degree distributions for reconstructed social networks from Window 2 for all 100 distance thresholds (different colours represent node degree distributions for different thresholds; distance threshold increases from 1 to 100 looking from left to right)
The distance thresholds chosen for further analysis are presented in Table 1. For all of them, for a given embedding method and for each time window the mean value of node is closest to 150.
Table 1 Distance thresholds for each time window in which average node degree is closest to 150 in comparison to other thresholds
The mean node degree values for each threshold together with their standard deviation are depicted in Fig. 13 for both BBS and CMDS. There is not much variation for the BBS embedding. In the case of CMDS, windows 1 and 10 can be perceived as outliers, which follow a different node degree distribution than the remaining windows. This difference is also visible in Table 1 where in the case of these two windows, the threshold values which result in the average node degree of 150 are considerably smaller then for other windows. The reason for this is that CMDS, due to its dependence on eigendecomposition of the dissimilarity matrix, is very sensitive to outliers, i.e. nodes that are at the periphery of the social network. The result of the embedding and molecular simulation in such a case is that most of the nodes are very close to each other, while a few are very far away. This explains the trend in node degree distributions for Windows 1 and 10 when CMDS was used.
Average node degree distribution for recreated social networks after the BBS and CMDS embedding and the molecular simulation as a function of distance threshold. Results for Time Window 2
Node degree distributions for the reconstructed networks for previously selected thresholds for each window are presented in Fig. 14 (BBS) and in Fig. 15 (CMDS). All of the networks (except Windows 1 and 10 for CMDS discussed before) follow Poisson distribution of the node degree. It means that the reconstructed networks that are in the stable state, in terms of node degree distribution, behave like random or small-world network (see Table 4) and not scale-free networks that have power-law node degree distribution. There are no hubs in the the reconstructed networks and the standard deviation of the node degree is low as it does not exceed 8 in the case of BBS and 50 in the case of CMDS (neglecting Windows 1 and 10 where standard deviations peeks at 200). This indicates that the active core of the organisational email-based social network in its equilibrium states resembles a community where everybody has similar number of connections.
Node degree distribution for recreated social networks after BBS embedding and the molecular simulation for distance thresholds from Table 1 in different time windows (x axis node degree; y axis probability)
Node degree distribution for recreated social networks after the CMDS embedding and the molecular simulation for distance thresholds from Table 1 in different time windows (x axis node degree; y axis probability)
Next, we investigate the clustering coefficient (CC). Suppose a node v has neighbours \(\mathcal{N}(v),\) with \(|\mathcal{N}(v)|=k_v.\) At most k v (k v − 1)/2 edges can exist between them (this occurs when v is part of a k v -clique). The clustering coefficient of a vertex, CC v , is defined as the fraction of these edges that actually exist. The clustering coefficient of the graph is defined as the average clustering coefficient of all the vertices in the graph. The distributions of the clustering coefficient for the selected networks are presented in Fig. 16 (BBS) and Fig. 17 (CMDS). Similarly to the node degree distributions, this one also follows Poisson distribution, i.e. most of the users have similar clustering coefficient (at the level of 0.35 for both BBS and CMDS—see Table 2). Moreover, the standard deviation of the clustering coefficient is low – 0.01 for BBS for all windows; for CMDS it mostly varies between 0.01 and 0.08 and reaches its maximum—0.16—for Windows 1 and 10. The clustering coefficient at this level is characteristic for real-world social network. Comparing these results with random and ordered networks of the same size (Table 4), it is clear that all of the recreated networks share the features of both these types of networks: their clustering coefficient is larger than the one for random network, which is 0.18 and smaller than that in the ordered network −0.74. We can conclude by saying that the analysed networks follow a small-world network model in terms of clustering coefficient.
Clustering coefficient distribution for recreated social networks after the BBS embedding and the molecular simulation for distance thresholds from Table 1 in different time windows (x axis node degree; y axis probability)
Clustering coefficient distribution for recreated social networks after the CMDS embedding and the molecular simulation for distance thresholds from Table 1 in different time windows (x axis node degree; y axis probability)
Table 2 Average clustering coefficient (ACC) and its standard deviation for the reconstructed networks
Last analysed characteristic that describes the reconstructed networks in a comprehensive manner is the length of the shortest path. The experiments were performed in the same way as in the case of clustering coefficient and the histograms are presented in Fig. 18 (BBS) and Fig. 19 (CMDS). The results indicate that the average path lengths (APL) are short for both BBS and CMDS as they are in the range [1.82; 1.84] for BBS and [1.81; 2.06] for CMDS. Also their standard deviation is rather modest: 0.41 for BBS and in the range [0.41; 0.72] for CMDS (see Table 3). This low value of average path length indicates that small-world phenomena, where two people are separated just by few intermediates, is present. Similarly to the clustering coefficient, average path length puts the analysed networks somewhere in between order and randomness. APL is longer than in the case of random network (1.34) and shorter than in an ordered network (2.75), which means that also in regards to average path length the recreated networks are in fact small-worlds (see Table 4).
Shortest path distribution for recreated social networks after the BBS embedding and the molecular simulation for distance thresholds from Table 1 in different time windows (x axis node degree; y axis probability)
Shortest paths distribution for recreated social networks after the CMDS embedding and the molecular simulation for distance thresholds from Table 1 in different time windows (x axis node degree; y axis probability)
Table 3 Average path length and its standard deviation for the reconstructed networks
Table 4 Network models characteristics; N = 825 – number of nodes; k = 150 – average node degree
The performed analyses revealed that the networks recreated after the molecular simulation are small-world networks. They follow Poisson node degree distribution, have big clustering coefficient and small average path length. Molecular simulation terminates when the system achieves stable state. We showed that networks reconstructed after the simulations feature the three, enumerated above characteristics of social networks. The fact that the final networks are small-world ones and resemble typical characteristics of real-world social networks can be an indication that molecular simulation can be a new way of generating this type of networks and may be effectively applied in sociodynamical analysis.
We have proposed to model the dynamics of a complex social system using molecular simulation, where the interactions between the individuals are determined from the data in a form of a social force, which corresponds to the particle interaction force used in the simulation. In our case the social relation was defined on the basis of communication events (message exchange) recorded in the computer system (email server). This allowed to define a social distance as inversely proportional to the number of messages exchanged between users and to estimate the character of the social force determining the changes of social distance. It was also shown that the global dynamics of such system may be modelled by treating the users as interacting particles embedded in an Euclidean social space. The movement of particles is determined by the social force and their trajectories are determined by their initial positions, derived from the email server logs and allowing to create the social network.
To the best of our knowledge this is the first attempt to apply a molecular modelling approach to the problem of social network dynamics. It has hence required careful verification, especially with respect to representation of the network evolutionary processes and chosen network structural properties, commonly used in network analysis. The experiments have shown that the proposed approach allows to reason about structural properties of evolving social network, while benefitting from the algorithmic simplicity of molecular modelling.
In this study we have presented the whole process of building and using a molecular model, while identifying the following key points:
The embedding procedure projecting the non-metric social graph into the Euclidean space should be chosen with care, taking into account the inherent trade-off between preserving the distances from social graph with the required accuracy and limiting the dimensionality of the Euclidean space. This has proven to be especially difficult for network hubs, regardless of the embedding method used.
The character of social force leading to changes in social distances can be generalized; however, this process is inherently connected with the loss of information in the case of individuals who behave statistically differently from the mean pattern (typical behaviour) derived from the whole network data.
The molecular model of social dynamics allows to reconstruct the social network from positions of the users (moving particles) in an Euclidean social space. While the reconstructed network preserves some of the global characteristics, local properties at the level of individual nodes usually cannot be recovered.
The reconstructed social network follows the small–world network model with large clustering coefficient and small average path length.
Our work aimed to show and demonstrate the possibilities and limitations of constructing an evolving sociodynamic model which is inherently data-driven and shows explanatory power. The key challenge was to establish a link between various approaches used in sociodynamics and network mining techniques which are based on the data acquired directly from contemporary computer-based social systems. We conclude that, using a modified molecular dynamic method, it is possible to create the evolutionary model of complex computer-based social network, but its applicability is restricted only to certain network properties measured in social network analysis. Taking into account that there are few works dealing with the predictive modelling of complex social networks, we find these results promising and forming a basis for the future experiments and development of data-driven evolutionary network models. The obedience of Internet-based social networks provides a huge amount of data for the analysis, changing the paradigms for the description of behavioural changes based on computer-supported social interaction processes.
The distance distortion is defined for each pair as the maximum of the ratio between the original and Euclidean distance and its inverse.
Aggarwal C, Hinneburg A, Keim D (2001) On the surprising behavior of distance metrics in high dimensional space. In: Bussche J, Vianu V (eds) Database Theory—ICDT 2001, Lecture Notes in Computer Science, vol. 1973, Springer, Berlin, pp 420–435
Barabasi AL (2003) Linked: how everything is connected to everything else and what it means. Plume, Newyork
Barrat A, Barthelemy M, Vespignani A (2008) Dynamical processes on complex networks. Cambridge University Press, Cambridge
Beyer K, Goldstein J, Ramakrishnan R, Shaft U (1999) When is "nearest neighbor" meaningful. In: Beeri C, Buneman P (eds) Databases Theory—ICDT 1999, Lecture Notes in Computer Science, vol 1540, Springer, Berlin, pp 217–235
Bishop C (1995) Neural networks for pattern recognition. Oxford University Press, New York
Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU (2006) Complex networks: structure and dynamics. Phys Rep 424(4–5):175–308
Bollobas B (1995) Random graphs. Academic, London
Braha D, Bar-Yam Y (2006) From centrality to temporary fame: dynamic centrality in complex networks. Complexity 12:59–63
Bringmann B, Berlingero M, Bonch F, Gionis A (2010) Learning and predicting the evolution of social networks. IEEE Intell Syst 25(4):26–35
Bronstein M, Kimmel R (2006) Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching. Proc Natl Acad Sci 103(5):1168–1172
Budka M, Gabrys B (2011) Electrostatic field framework for supervised and semi-supervised learning from incomplete data. Natural Comput 10:921–945. doi:10.1007/s11047-010-9182-4
Davis D, Lichtenwalter R, Chawla N (2012) Supervised methods for multi-relational link prediction. Social Netw Anal Min. 1–15. doi:10.1007/s13278-012-0068-6
Epstein J (2008) Why model? J Artif Soc Soc Simul 11(4). http://jasss.soc.surrey.ac.uk/11/4/12.html
Francois D, Wertz V, Verleysen M (2005) Non–Euclidean metrics for similarity search in noisy datasets. In: Proceedings of the European symposium on artificial neural networks, d–side publications, pp 339–334
Garton L, Haythornthwaite C, Wellman B (1997) Studying online social networks. J Comput Mediat Commun 3(1). http://jcmc.indiana.edu/vol3/issue1/garton.html
Harel D, Koren Y (2004) Graph drawing by high-dimensional embedding. J Graph Algorithms Appl 8(2):195–214
Helbing D (2010) Quantitative sociodynamics: stochastic methods and models of social interaction processes. Springer, Berlin
Hill RA, Dunbar RIM (2002) Social network size in humans. Human Nat 14(1):53–72
Hill S, Braha D (2010) Dynamic model of time-dependent complex networks. Phys Rev E. 82 (arXiv:0901.4407v2)
Holland J (1996) Hidden order: how adaptation builds complexity. Basic Books, Newyork
Juszczyszyn K, Musial A, Musial K, Brodka P (2009) Molecular dynamics modelling of the temporal changes in complex networks. In: IEEE Congress on Evolutionary Computing, Trondheim, Sweden. IEEE Computer Society Press, Newyork, pp 553–559
Juszczyszyn K, Budka M, Musial K (2011a) The dynamic structural patterns of social networks based on triad transitions. In: 2011 International Conference on Advances in social networks analysis and mining (ASONAM), pp 581–586. doi:10.1109/ASONAM.2011.50.http://dl.acm.org/citation.cfm?id=2055729
Juszczyszyn K, Musial K, Budka M (2011b) Link prediction based on subgraph evolution in dynamic social networks. In: The Third IEEE international conference on social computing (SocialCom 2011), pp 27–34 (2011) doi:10.1109/PASSAT/SocialCom.2011.15. http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=06113091
Juszczyszyn K, Musial K, Budka M (2011c) On analysis of complex network dynamics changes in local topology. In: The fifth workshop on social network mining and analysis co-located with the 17th ACM SIGKDD international conference on knowledge discovery and data mining (SNA-KDD)
Kashoob S, Caverlee J (2012) Temporal dynamics of communities in social bookmarking systems. Social Netw Anal Min 2:387–404. doi: 10.1007/s13278-012-0054-z
Kazienko P, Musial K, Zgrzywa A (2009) Evaluation of node position based on email communication. Control Cybern 38(1):67–86
Kolaczyk E (2009) Statistical analysis of network data. Springer, Berlin
Kruskal JB, Wish M (1978) Multidimensional scaling, Sage University Paper series on Quantitative Application in the Social Sciences. Sage Publications, Thousand Oaks
Kumar R, Novak J, Tomkins A (2006) Microscopic evolution of social network. In: The 14th ACM SIGKDD international conference on knowledge discovery and data mining. ACM Press, Newyork
Lescovec J, Backstrom L, Kumar R, Tomkins A (2008) Microscopic evolution of social networks. In: ACM SIGKDD international conference on knowledge discovery and data mining (KDD)
Liben-Nowell D, Kleinberg J (2007) The link-prediction problem for social networks. J Am Soc Info Sci Technol 58(7):1019–1031
Malarz K, Gronek P, Kulakowski K (2011) Zaller-deffuant model of mass opinion. J Artif Soc Soc Simul 14(1):1–20
Musial A, Juszczyszyn K, Musial K, Brodka P (2010) Utilizing dynamic molecular modelling technique for predicting changes in complex social networks. In: IEEE/WIC/ACM Joint International Conference on Web Intelligence and Intelligent Agent Technology. IEEE Press, Newyork, pp 1–4
Sarr I, Missaoui R (2012) Managing node disappearance based on information flow in social networks. Soc Netw Anal Min. 1–13. doi:10.1007/s13278-012-0071-y
Schweitzer F (2003) Brownian agents and active particles—collective dynamics in the natural and social sciences. Springer Series in Synergetics. Springer, Berlin
Shavitt Y, Tankel T (2004) Big-bang simulation for embedding network distances in euclidean space. IEEE/ACM Trans Netw 12(6):993–1006
Shaw B, Jebara T (2007) Minimum volume embedding. In: Proceedings of the eleventh international conference on artificial intelligence and statistics
Shaw B, Jebara T (2009) Structure preserving embedding. In: Proceedings of the 26th international conference on machine learning
Strogatz SH (2001) Exploring complex networks. Nature 410(6825):268–276
Torgerson W (1965) Multidimensional scaling of similarity. Psychometrika 30(4):379–393
Wasserman S, Faust K (1994) Social network analysis: methods and applications. Cambridge University Press, New York
Watts D (2002) Small worlds: dynamic of networks between order and randomness. Princeton University Press, Princeton
Watts D, Strogatz S (1998) Collective dynamics of 'small-world' networks. Nature 393(6684):440–444
Weidlich W (1991) Physics and social science—the approach of synergetics. Phys Rep 1(204):1–163
Wolfram S (1986) Theory and applications of cellular automata. World Scientific, Singapore
The research presented in this paper has been partially supported by the European Union within the European Regional Development Fund Program No. POIG.01.03.01-00-008/08. The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No. 251617.
Bournemouth University, Poole House, Fern Barrow, Poole, BH12 5BB, UK
Marcin Budka
Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-370, Wroclaw, Poland
Krzysztof Juszczyszyn
& Anna Musial
Kings College London, Strand Campus, London, WC2R 2LS, UK
Katarzyna Musial
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Correspondence to Krzysztof Juszczyszyn.
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Budka, M., Juszczyszyn, K., Musial, K. et al. Molecular model of dynamic social network based on e-mail communication. Soc. Netw. Anal. Min. 3, 543–563 (2013). https://doi.org/10.1007/s13278-013-0101-4
Revised: 12 January 2013
Issue Date: September 2013
Link prediction
Molecular modelling
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History of combinatorics
The mathematical field of combinatorics was studied to varying degrees in numerous ancient societies. Its study in Europe dates to the work of Leonardo Fibonacci in the 13th century AD, which introduced Arabian and Indian ideas to the continent. It has continued to be studied in the modern era.
Earliest records
The earliest recorded use of combinatorial techniques comes from problem 79 of the Rhind papyrus, which dates to the 16th century BCE. The problem concerns a certain geometric series, and has similarities to Fibonacci's problem of counting the number of compositions of 1s and 2s that sum to a given total.[1]
In Greece, Plutarch wrote that Xenocrates of Chalcedon (396–314 BC) discovered the number of different syllables possible in the Greek language. This would have been the first attempt on record to solve a difficult problem in permutations and combinations.[2] The claim, however, is implausible: this is one of the few mentions of combinatorics in Greece, and the number they found, 1.002 × 10 12, seems too round to be more than a guess.[3][4]
Later, an argument between Chrysippus (3rd century BCE) and Hipparchus (2nd century BCE) of a rather delicate enumerative problem, which was later shown to be related to Schröder–Hipparchus numbers, is mentioned.[5][6] There is also evidence that in the Ostomachion, Archimedes (3rd century BCE) considered the configurations of a tiling puzzle,[7] while some combinatorial interests may have been present in lost works of Apollonius.[8][9]
In India, the Bhagavati Sutra had the first mention of a combinatorics problem; the problem asked how many possible combinations of tastes were possible from selecting tastes in ones, twos, threes, etc. from a selection of six different tastes (sweet, pungent, astringent, sour, salt, and bitter). The Bhagavati is also the first text to mention the choose function.[10] In the second century BC, Pingala included an enumeration problem in the Chanda Sutra (also Chandahsutra) which asked how many ways a six-syllable meter could be made from short and long notes.[11][12] Pingala found the number of meters that had $n$ long notes and $k$ short notes; this is equivalent to finding the binomial coefficients.
The ideas of the Bhagavati were generalized by the Indian mathematician Mahavira in 850 AD, and Pingala's work on prosody was expanded by Bhāskara II[10][13] and Hemacandra in 1100 AD. Bhaskara was the first known person to find the generalised choice function, although Brahmagupta may have known earlier.[1] Hemacandra asked how many meters existed of a certain length if a long note was considered to be twice as long as a short note, which is equivalent to finding the Fibonacci numbers.[11]
The ancient Chinese book of divination I Ching describes a hexagram as a permutation with repetitions of six lines where each line can be one of two states: solid or dashed. In describing hexagrams in this fashion they determine that there are $2^{6}=64$ possible hexagrams. A Chinese monk also may have counted the number of configurations to a game similar to Go around 700 AD.[3] Although China had relatively few advancements in enumerative combinatorics, around 100 AD they solved the Lo Shu Square which is the combinatorial design problem of the normal magic square of order three.[1][14] Magic squares remained an interest of China, and they began to generalize their original $3\times 3$ square between 900 and 1300 AD. China corresponded with the Middle East about this problem in the 13th century.[1] The Middle East also learned about binomial coefficients from Indian work and found the connection to polynomial expansion.[15] The work of Hindus influenced Arabs as seen in the work of al-Khalil ibn Ahmad who considered the possible arrangements of letters to form syllables. His calculations show an understanding of permutations and combinations. In a passage from the work of Arab mathematician Umar al-Khayyami that dates to around 1100, it is corroborated that the Hindus had knowledge of binomial coefficients, but also that their methods reached the middle east.
Abū Bakr ibn Muḥammad ibn al Ḥusayn Al-Karaji (c.953-1029) wrote on the binomial theorem and Pascal's triangle. In a now lost work known only from subsequent quotation by al-Samaw'al, Al-Karaji introduced the idea of argument by mathematical induction.
The philosopher and astronomer Rabbi Abraham ibn Ezra (c. 1140) counted the permutations with repetitions in vocalization of Divine Name.[16] He also established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist and mathematician Levi ben Gerson (better known as Gersonides), in 1321.[17] The arithmetical triangle— a graphical diagram showing relationships among the binomial coefficients— was presented by mathematicians in treatises dating as far back as the 10th century, and would eventually become known as Pascal's triangle. Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations.[18]
Combinatorics in the West
Combinatorics came to Europe in the 13th century through mathematicians Leonardo Fibonacci and Jordanus de Nemore. Fibonacci's Liber Abaci introduced many of the Arabian and Indian ideas to Europe, including that of the Fibonacci numbers.[19] Jordanus was the first person to arrange the binomial coefficients in a triangle, as he did in proposition 70 of De Arithmetica. This was also done in the Middle East in 1265, and China around 1300.[1] Today, this triangle is known as Pascal's triangle.
Pascal's contribution to the triangle that bears his name comes from his work on formal proofs about it, and the connections he made between Pascal's triangle and probability.[1] From a letter Leibniz sent to Daniel Bernoulli we learn that Leibniz was formally studying the mathematical theory of partitions in the 17th century, although no formal work was published. Together with Leibniz, Pascal published De Arte Combinatoria in 1666 which was reprinted later.[20] Pascal and Leibniz are considered the founders of modern combinatorics.[21]
Both Pascal and Leibniz understood that the binomial expansion was equivalent to the choice function. The notion that algebra and combinatorics corresponded was expanded by De Moivre, who found the expansion of a multinomial.[22] De Moivre also found the formula for derangements using the principle of principle of inclusion-exclusion, a method different from Nikolaus Bernoulli, who had found it previously.[1] De Moivre also managed to approximate the binomial coefficients and factorial, and found a closed form for the Fibonacci numbers by inventing generating functions.[23][24]
In the 18th century, Euler worked on problems of combinatorics, and several problems of probability which are linked to combinatorics. Problems Euler worked on include the Knights tour, Graeco-Latin square, Eulerian numbers, and others. To solve the Seven Bridges of Königsberg problem he invented graph theory, which also led to the formation of topology. Finally, he broke ground with partitions by the use of generating functions.[25]
Contemporary combinatorics
In the 19th century, the subject of partially ordered sets and lattice theory originated in the work of Dedekind, Peirce, and Schröder. However, it was Garrett Birkhoff's seminal work in his book Lattice Theory published in 1967,[26] and the work of John von Neumann that truly established the subjects.[27] In the 1930s, Hall (1936) and Weisner (1935) independently stated the general Möbius inversion formula.[28] In 1964, Gian-Carlo Rota's On the Foundations of Combinatorial Theory I. Theory of Möbius Functions introduced poset and lattice theory as theories in Combinatorics.[27] Richard P. Stanley has had a big impact in contemporary combinatorics for his work in matroid theory,[29] for introducing Zeta polynomials,[30] for explicitly defining Eulerian posets,[31] developing the theory of binomial posets along with Rota and Peter Doubilet,[32] and more. Paul Erdős made seminal contributions to combinatorics throughout the century, winning the Wolf prize in-part for these contributions.[33]
Notes
1. Biggs, Norman; Keith Lloyd; Robin Wilson (1995). "44". In Ronald Graham; Martin Grötschel; László Lovász (eds.). Handbook of Combinatorics (Google book). MIT Press. pp. 2163–2188. ISBN 0-262-57172-2. Retrieved 2008-03-08.
2. Heath, Sir Thomas (1981). A history of Greek mathematics (Reprod. en fac-sim. ed.). New York: Dover. ISBN 0486240738.
3. Dieudonné, J. "The Rhind/Ahmes Papyrus - Mathematics and the Liberal Arts". Historia Math. Truman State University. Archived from the original on 2012-12-12. Retrieved 2008-03-06.
4. Gow, James (1968). A Short History of Greek Mathematics. AMS Bookstore. p. 71. ISBN 0-8284-0218-3.
5. Acerbi, F. (2003). "On the shoulders of Hipparchus". Archive for History of Exact Sciences. 57 (6): 465–502. doi:10.1007/s00407-003-0067-0. S2CID 122758966.
6. Stanley, Richard P. (2018-04-10). "Hipparchus, Plutarch, Schröder, and Hough". The American Mathematical Monthly. 104 (4): 344–350. doi:10.2307/2974582. ISSN 0002-9890. JSTOR 2974582.
7. Netz, R.; Acerbi, F.; Wilson, N. "Towards a reconstruction of Archimedes' Stomachion". Sciamvs. 5: 67–99.
8. Hogendijk, Jan P. (1986). "Arabic Traces of Lost Works of Apollonius". Archive for History of Exact Sciences. 35 (3): 187–253. doi:10.1007/BF00357307. ISSN 0003-9519. JSTOR 41133783. S2CID 121613986.
9. Huxley, G. (1967). "Okytokion". Greek, Roman, and Byzantine Studies. 8 (3): 203–204.
10. "India". Archived from the original on 2007-11-14. Retrieved 2008-03-05.
11. Hall, Rachel (2005-02-16). "Math for Poets and Drummers-The Mathematics of Meter" (PDF). Retrieved 2008-03-05. {{cite journal}}: Cite journal requires |journal= (help)
12. Kulkarni, Amba (2007). "Recursion and Combinatorial Mathematics in Chandashāstra". arXiv:math/0703658. Bibcode:2007math......3658K. {{cite journal}}: Cite journal requires |journal= (help)
13. Bhaskara. "The Lilavati of Bhaskara". Brown University. Archived from the original on 2008-03-25. Retrieved 2008-03-06.
14. Swaney, Mark. "Mark Swaney on the History of Magic Squares". Archived from the original on 2004-08-07.
15. "Middle East". Archived from the original on 2007-11-14. Retrieved 2008-03-08.
16. The short commentary on Exodus 3:13
17. History of Combinatorics, chapter in a textbook.
18. Arthur T. White, ”Ringing the Cosets,” Amer. Math. Monthly 94 (1987), no. 8, 721-746; Arthur T. White, ”Fabian Stedman: The First Group Theorist?,” Amer. Math. Monthly 103 (1996), no. 9, 771-778.
19. Devlin, Keith (October 2002). "The 800th birthday of the book that brought numbers to the west". Devlin's Angle. Retrieved 2008-03-08.
20. Leibniz's habilitation thesis De Arte Combinatoria was published as a book in 1666 and reprinted later
21. Dickson, Leonard (2005) [1919]. "Chapter III". Diophantine Analysis. History of the Theory of Numbers. Mineola, New York: Dover Publications, Inc. p. 101. ISBN 0-486-44233-0.
22. Hodgson, James; William Derham; Richard Mead (1708). Miscellanea Curiosa (Google book). Volume II. pp. 183–191. Retrieved 2008-03-08.
23. O'Connor, John; Edmund Robertson (June 2004). "Abraham de Moivre". The MacTutor History of Mathematics archive. Retrieved 2008-03-09.
24. Pang, Jong-Shi; Olvi Mangasarian (1999). "10.6 Generating Function". In Jong-Shi Pang (ed.). Computational Optimisation (Google book). Volume 1. Netherlands: Kluwer Academic Publishers. pp. 182–183. ISBN 0-7923-8480-6. Retrieved 2008-03-09.
25. "Combinatorics and probability". Retrieved 2008-03-08.
26. Birkhoff, Garrett (1984). Lattice theory (3d ed., reprinted with corrections. ed.). Providence, R.I.: American Mathematical Society. ISBN 978-0821810255.
27. Stanley, Richard P. (2012). Enumerative combinatorics (2nd. ed.). Cambridge: Cambridge University Press. pp. 391–393. ISBN 978-1107602625.
28. Bender, Edward A.; Goldman, J. R. (1975). "On the applications of Möbius inversion in combinatorial analysis". Amer. Math. Monthly. 82 (8): 789–803. doi:10.2307/2319793. JSTOR 2319793.
29. Stanley, Richard (2007). "An introduction to hyperplane arrangements". Geometric Combinatorics. IAS/Park City Mathematics Series. Vol. 13. pp. 389–496. doi:10.1090/pcms/013/08. ISBN 9780821837368.
30. Stanley, Richard (1974). "Combinatorial reciprocity theorems". Advances in Mathematics. 14 (2): 194–253. doi:10.1016/0001-8708(74)90030-9.
31. Stanley, Richard (1982). "Some aspects of groups acting on finite posets". Journal of Combinatorial Theory. Ser. A 32 (2): 132–161. doi:10.1016/0097-3165(82)90017-6.
32. Stanley, Richard (1976). "Binomial posets, M¨obius inversion, and permutation enumeration". Journal of Combinatorial Theory. Ser. A 20 (3): 336–356. doi:10.1016/0097-3165(76)90028-5.
33. "Wolf Foundation Mathematics Prize Page". Wolffund.org.il. Archived from the original on 2008-04-10. Retrieved 2010-05-29.
References
• N.L. Biggs, The roots of combinatorics, Historia Mathematica 6 (1979), 109–136.
• Katz, Victor J. (1998). A History of Mathematics: An Introduction, 2nd Edition. Addison-Wesley Education Publishers. ISBN 0-321-01618-1.
• O'Connor, John J. and Robertson, Edmund F. (1999–2004). MacTutor History of Mathematics archive. St Andrews University.
• Rashed, R. (1994). The development of Arabic mathematics: between arithmetic and algebra. London.
• Wilson, R. and Watkins, J. (2013). Combinatorics: Ancient & Modern. Oxford.
• Stanley, Richard (2012). Enumerative combinatorics (2nd ed. ed.), 2nd Edition. Cambridge University Press. ISBN 1107602629.
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