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Using limited independent variables in a multivariable regression model
Further to my prior question on multivariable adjustment in regression models, using covariates which are available only for some cases, I have researched in some detail the main methods for limited dependent variables, including Heckman correction or tobit models. However, I fear that they do not apply to my issue, which has more to do with limited independent variables.
In particular, I am giving below an example of the dataset and the possible analysis in R (disregard the overfitting, it's just to make an example, my actual dataset has at least 10,000 cases):
dep <- c(8, 9, 21, -3, 4, 6, 9, 10, 8, 9, 11, 39, 91, 51, 38, 28, 21)
cov1 <- c(68, 58, 42, 19, 39, 49, 29, 38, 25, 22, 19, 36, 39,90, 105, 73, 25)
cov2 <- c(0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0)
cov4 <- c(NA, NA, NA, NA, NA, NA, 56, 33, 45, 44, 56, 49, 36, 39, 40, 41, 59)
cov5 <- c(NA, NA, NA, NA, NA, NA, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0)
mydata <- data.frame(cbind(dep, cov1, cov2, cov3, cov4, cov5))
mydata
reg1 <- lm(dep ~ cov1 + cov2, data = mydata, na.action = na.omit)
anova(reg1)
summary(reg1)
reg2 <- lm(dep ~ cov1 + cov2 + cov3 + cov4 + cov5, data = mydata, na.action = na.omit)
What should I do to best adjust for covariates cov1, cov2, cov3, cov4 and cov5, having dep as dependent variable, given that cov4 and cov5 are available only for patients with cov3 = 1?
Should I discard all cases with cov3 = 0? Should I instead conduct two separate analyses and then pool the regression coefficients according to their standard error? Or is there any other more reasonable approach?
Unfortunately I did not find anything meaningful searching Google, Google Scholar, or PubMed:
https://www.google.it/search?q=limited+independent+variable&nirf=limited+dependent+variable
https://scholar.google.it/scholar?hl=en&q=limited+independent+variable
http://www.ncbi.nlm.nih.gov/pubmed/?term=limited+independent+variable*
To further clarify what is at stake, this is my real problem: I want to create a clinical prediction score (to predict prognosis and future quality of life) for patients undergoing myocardial perfusion imaging (a non-invasive cardiac test used in subjects with or at risk for coronary artery disease). The imaging test follows immediately an exercise stress test in fit patients, and a pharmacologic stress test in those who are not fit. The latter test is worse than the former, and does not provide several important prognostic features (eg maximum heart rate, or workload), so I must include exercise test variables in the multivariable model. But if I do so, I lose more than 1000 patients who only underwent a pharmacologic stress test.
r regression multiple-regression model predictor
Joe_74Joe_74
$\begingroup$ The $x_i$ can have any features, expect they cannot be constant or a linear combination of each other. If there is not much variation in $x_i$ then the standard error will be larger than otherwise. In itself this is not a problem $\endgroup$ – Repmat Apr 1 '16 at 9:19
$\begingroup$ I am not sure I follow you. If I use all the covariates in the model I loose several cases (those with NA). If I only use cov1, cov2, and cov3 I don't use the information in cov4 and cov5... $\endgroup$ – Joe_74 Apr 1 '16 at 9:28
$\begingroup$ You can make some arbitrary assumptions, and do data imputation. But for the sample data posted I dont see the need, you do not loose an entire variable. But yeah sure, you will loose data... $\endgroup$ – Repmat Apr 1 '16 at 9:35
$\begingroup$ The question is not peregrine. Basically, I want to create a clinical prediction score for patients undergoing myocardial perfusion imaging. The imaging test follows an exercise stress test in fit patients, and a pharmacologic stress test in those who are not fit. The latter test is worse than the former, and does not provide several important prognostic features (eg maximum heart rate, or workload), so I must include exercise test variables in the multivariable model. But if I do so, I loose more than 1000 patients who only underwent a pharmacologic stress. I added this also in the question. $\endgroup$ – Joe_74 Apr 1 '16 at 9:42
$\begingroup$ I saw a lot of hits on google.scholar for +"model selection" +"missing covariate", as well as +"model building" +"missing covariate". I suspect that it may be possible - if it is plausible that covariates are simply missing at random - would be to impute them using multiple imputation, do whatever model building you do and combine the results across imputations. I believe there's also models that implicity impute them. However, if covariates will be missing in practice when people are trying to use the prediction score also, that would be an even harder problem. $\endgroup$ – Björn Apr 9 '16 at 5:33
I think in this case it is more appropriate to use Classification and Regression Trees (CART) models.
I found the ctree package very helpful, which is an implementation of rpart. Logistic models (or imputation) in this case do not make much sense to me.
Umberto benedettoUmberto benedetto
$\begingroup$ Note that recursive partitioning requires enormous sample sizes, perhaps 10x larger than regression if single trees are used. $\endgroup$ – Frank Harrell Apr 9 '16 at 12:36
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How to determine which independent variables to add in a multivariable regression model when sample size is small | CommonCrawl |
How exactly to show that s-matrix elements diverges because time-ordering is not well determined?
I was reading the "in-in" formalism (or "closed time path formalism" used in condensed matter physics) in cosmology created by Schwinger in 1961, and there is a saying: "they care about correlation functions instead of S-matrix scattering amplitudes". When I learn QFT, these two things are almost the same thing and are related by LSZ formula. Why they use in-in instead of in-out? what's the difference between correlation functions and S-matrix?
Correlation functions (or Wightman N-point functions) are expectation values of renormalized products of field operators at finite times. The ordering of the operators matters since fields at general arguments do not commute.. The correlation functions need for their nonperturbative definition via a path integral definition the in-in formalism (= closed time path, CTP, Schwinger-Keldysh formalism) where one integrates over a doubled time contour.
The S-matrix elements are computed from the expectations of time-ordered products of field operators (hence independent of the ordering of the operators), which occur in the LSZ formula and in functional derivatives of the standard path integral. They express in-out properties of asymptotic states of scattering experiments. They are obtained in a path integral formulation by integration along a single time path from $t=-\infty$ to $t=+\infty$. As such they also appear inside the CTP formalism.
The information in a time-ordered products is less than in the ordinary product as one can calculate $T(\phi(x)\phi(y))$ from $\phi(x)\phi(y)$ and $\phi(y)\phi(x)$ (away from its singularity at $(x-y)^2=0$), while the converse is not possible.
Correlation functions are important if you want to see the Hilbert space. Therefore the CTP path integral takes a doubled time path, so that it returns to the initial state, which computes expectation values in the initial state. The images of the initial state under products of field operators span a dense set of vectors in the Hilbert space. Therefore, at least in in principle, one can compute inner products of arbitrary state vectors using the CTP formalism. The S-matrix doesn't contain this information.
As a consequence, the in-out description of quantum field theory - though simpler and covered by every textbook on QFT - is incomplete as it only gives the asymptotic properties of a quantum field, while the in-in description - though more involved and only in textbooks treating nonequilibrium statistical mechanics - gives everything - the asymptotics and the finite time behavior. | CommonCrawl |
Dynamics for the damped wave equations on time-dependent domains
DCDS-B Home
Palindromic control and mirror symmetries in finite difference discretizations of 1-D Schrödinger equations
June 2018, 23(4): 1623-1643. doi: 10.3934/dcdsb.2018064
Ion size effects on individual fluxes via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Analysis without electroneutrality boundary conditions
Hong Lu 1, , Ji Li 2, , Joseph Shackelford 3, , Jeremy Vorenberg 3, and Mingji Zhang 3,
School of Mathematics and Statistics, Shandong University, Weihai, Shandong 264209, China
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA
Received May 2017 Revised September 2017 Published June 2018 Early access February 2018
A quasi-one-dimensional steady-state Poisson-Nernst-Planck model with Bikerman's local hard-sphere potential for ionic flows of two oppositely charged ion species through a membrane channel is analyzed. Of particular interest is the qualitative properties of ionic flows in terms of individual fluxes without the assumption of electroneutrality conditions, which is more realistic to study ionic flow properties of interest. This is the novelty of this work. Our result shows that ⅰ) boundary concentrations and relative size of ion species play critical roles in characterizing ion size effects on individual fluxes; ⅱ) the first order approximation $\mathcal{J}_{k1} = D_kJ_{k1}$ in ion volume of individual fluxes $\mathcal{ J}_k = D_kJ_k$ is linear in boundary potential, furthermore, the signs of $\partial_V \mathcal{ J}_{k1}$ and $\partial^2_{Vλ} \mathcal{J}_{k1}$, which play key roles in characterizing ion size effects on ionic flows can be both negative depending further on boundary concentrations while they are always positive and independent of boundary concentrations under electroneutrality conditions (see Corollaries 3.2-3.3, Theorems 3.4-3.5 and Proposition 3.7). Numerical simulations are performed to identify some critical potentials defined in (2). We believe our results will provide useful insights for numerical and even experimental studies of ionic flows through membrane channels.
Keywords: Ion channel, local hard-sphere potential, critical potentials, individual fluxes, electroneutrality conditions.
Mathematics Subject Classification: Primary: 34A26, 34B16, 34D15; Secondary: 37D10, 92C35.
Citation: Hong Lu, Ji Li, Joseph Shackelford, Jeremy Vorenberg, Mingji Zhang. Ion size effects on individual fluxes via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Analysis without electroneutrality boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1623-1643. doi: 10.3934/dcdsb.2018064
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Figure 1. Numerical detection of critical values for $\lambda$ (left graph) and $m$ (right one) in Theorem 3.2.
Figure 2. Numerical detection of critical values for $\sigma$, which corresponds to statement (Ⅰ) in Theorem 3.2. The left graph is for $\lambda<\lambda_1^* = 0.072$, and the right one is for $\lambda>\lambda_2^* = 13.93.$
Figure 3. Numerical detection of critical values $\sigma$ corresponding to statement (Ⅱ) in Theorem 3.2 with $\lambda_1^*<\lambda<\lambda_2^*$. The left graph is for $0<m<m^* = 0.4934$, and the right one is for $m^*<m<\frac{1}{2}.$
Figure 4. Numerical identification of six critical potentials in (15) with $z_1 = -z_2 = 1$. In the left column, the vertical axis actually represents, from top to bottom, ${\mathcal I}(V;\nu, \lambda)-{\mathcal I}_0(V), \ {\mathcal J}_1(V;\nu, \lambda)-{\mathcal J}_{10}(V)$ and ${\mathcal J}_2(V;\nu, \lambda)-{\mathcal J}_{20}(V)$, respectively. In particular, the x-axis for all figures actually represents $\frac{e}{k_BT}V$.
Figure 5. Numerical identification of critical potentials $V_{1c}$ and $ V_1^c$ for individual flux ${\mathcal J}_1$ with $z_1 = -z_2 = 1$ and nonzero permanent charge. The x-axis for all figures actually represents $\frac{e}{k_BT}V$.
Figure 6. Numerical approximations of critical potentials $V_1^c$ for individual flux ${\mathcal J}_1$ with $z_1 = -z_2 = 1$ and nonzero permanent charge as illustrated in Proposition 3.14. The x-axis for all figures actually represents $\frac{e}{k_BT}V$.
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\begin{document}
\title{Formalizing set theory in weak logics, searching for the weakest logic with G\"odel's incompleteness property.} \author{Hajnal Andr\'eka and Istv\'an N\'emeti \thanks{Research supported by the
Hungarian National Foundation for scientific research grants
T81188.}} \date{2011. July} \maketitle
\begin{abstract} We show that first-order logic can be translated into a very simple and weak logic, and thus set theory can be formalized in this weak logic. This weak logical system is equivalent to the equational theory of Boolean algebras with three commuting complemented closure operators, i.e., that of diagonal-free 3-dimensional cylindric algebras ($\mbox{\sf Df\/}_3$'s). Equivalently, set theory can be formulated in propositional logic with 3 commuting S5 modalities (i.e., in the multimodal logic [S5,S5,S5]). There are many consequences, e.g., free finitely generated $\mbox{\sf Df\/}_3$'s are not atomic and [S5,S5,S5] has G\"odel's incompleteness property. The results reported here are strong improvements of the main result of the book: Tarski, A. and Givant, S. R., Formalizing Set Theory without variables, AMS, 1987. \end{abstract}
\section{Introduction}\label{intro-sec}
Tarski in 1953 \cite{T53a, T53b} formalized set theory in the theory of relation algebras. Why did he do this? Because the equational theory of relation algebras ($\mbox{\sf RA\/}$) corresponds to a logic without individual variables, in other words, to a propositional logic. This is why the title of the book \cite{TG} is ``Formalizing set theory without variables". Tarski got the surprising result that a propositional logic can be strong enough to ``express all of mathematics", to be the arena for mathematics. The classical view before this result was that propositional logics in general were weak in expressive power, decidable, uninteresting in a sense. By using the fact that set theory can be built up in it, Tarski proved that the equational theory of $\mbox{\sf RA\/}$ is undecidable. This was the first propositional logic shown to be undecidable.
From the above it is clear that replacing $\mbox{\sf RA\/}$ in Tarski's result with a ``weaker" class of algebras is an improvement of the result and it is worth doing. For more on this see Tarski-Givant~\cite[pp.$89_2-90^4$ and footnote 17 on p.90]{TG}, especially the open problem formulated therein.
A result of J.\ D.\ Monk says that for every finite $n$ there is a 3-variable first-order logic (FOL) formula which is valid but which can be proved (in FOL) with more than $n$ variables only. Intuitively this means that during any proof of this formula there are steps when we have to use $n$ independent data (stored in the $n$ variables as in $n$ machine registers). For example, the associativity of relation composition of binary relations can be expressed with 3 variables but 4 variables are needed for any of its proofs.
Tarski's main idea in \cite{TG} is to use pairing functions to form ordered pairs, and so to store two pieces of data in one register. He used this technique to translate usual infinite-variable first-order logic FOL into the three-variable fragment of it. From then on, he used that any three-variable FOL-formula about binary relations can be expressed by an $\mbox{\sf RA\/}$-equation, \cite[sec 5.3]{HMTII}. He needed two registers for storing the data belonging to a binary relation and he had one more register available for making computations belonging to a proof.
The finite-variable fragment hierarchy of FOL corresponds to cylindric algebras (\mbox{\sf CA\/}'s). The $n$-variable fragment $\mathcal{L}_n$ of FOL consists of all FOL-formulas which use only the first $n$ variables. By Monk's result, $\mathcal{L}_n$ is essentially incomplete for all $n\ge 3$, it cannot have a finite Hilbert-style complete and sound inference system. We get a finite Hilbert style inference system $\tiny{\undrn}$ for $\mathcal{L}_n$ by restricting a usual complete one for infinite-variable FOL to the first $n$ variables (see \cite[sec. 4.3]{HMTII}). This inference system $\tiny{\undrn}$ belonging to $\mathcal{L}_n$ expresses $\mbox{\sf CA\/}_n$, it is sound but not complete: $\tiny{\undrn}$ is much weaker than validity $\models_n$.
Relation algebras are halfway between $\mbox{\sf CA\/}_3$ and $\mbox{\sf CA\/}_4$, the classes of 3-dimen\-sional and 4-dimensional cylindric algebras, respectively. We sometimes jokingly say that $\mbox{\sf RA\/}$ is $\mbox{\sf CA\/}_{3.8}$. Why is $\mbox{\sf RA\/}$ stronger than $\mbox{\sf CA\/}_3$? Because, the so-called relation algebra reduct of a $\mbox{\sf CA\/}_3$ is not necessarily an $\mbox{\sf RA\/}$, e.g., associativity of relation composition can fail in the reduct. See \cite[sec 5.3]{HMTII}, and for more in this line see N\'emeti-Simon~\cite{NeSiIGPL97}. \rmk{The class $\mbox{\sf SA\/}$, defined by weakening the associativity, corresponds closely to $\mbox{\sf CA\/}_3$. Write more, about Maddux's work.??} Why is $\mbox{\sf CA\/}_4$ stronger than $\mbox{\sf RA\/}$? Because not every $\mbox{\sf RA\/}$ can be obtained, up to isomorphism, as the relation algebra reduct of a $\mbox{\sf CA\/}_4$
. However, the same equations are true in $\mbox{\sf RA\/}$ and in the class of all relation algebra reducts of $\mbox{\sf CA\/}_4$'s (
Maddux's result, see \cite[sec 5.3]{HMTII}). Thus Tarski formulated Set Theory, roughly, in $\mbox{\sf CA\/}_4$, i.e., in $\mathcal{L}_4$ with $\tiny{\undrf}$, or in $\mathcal{L}_3$ with validity $\models$.
N\'emeti \cite{NPrep}, \cite{NDis} improved this result by formalizing set theory in $\mbox{\sf CA\/}_3$, i.e., $\mathcal{L}_3$ with $\tiny{\undra}$ in place of validity $\models$.
The main idea for this improvement was using the paring functions to store all data always, during every step of a proof, in one register only and so one got two registers to work with in the proofs. In this approach one represents binary relations as unary ones (of pairs). For the ``execution" of this idea see sections~\ref{qra-sec}-\ref{set-sec} of the present paper.
First-order logic has equality as a built-in relation. One of the uses of equality in FOL is that it can be used to express (simulate) substitutions of variables, thus to ``transfer" content of one variable to the other. The reduct $\mbox{\sf SCA\/}_3$ of $\mbox{\sf CA\/}_3$ ``forgets" equality $\mbox{\sf d\/}_{ij}$ but retains substitution in the form of the term-definable operations $\mbox{\sf s\/}^i_j$. The logic belonging to $\mbox{\sf SCA\/}_3$ is weaker than 3-variable fragment of FOL. Zal\'an Gyenis \cite{Gyenis} improved parts of N\'emeti's result by using $\mbox{\sf SCA\/}_3$ in place of $\mbox{\sf CA\/}_3$.
We get a much weaker logic by forgetting substitutions, too, this is the logic corresponding to $\mbox{\sf Df\/}_3$ in which we formalize Set Theory in the present paper. Without equality or substitutions, if one has only binary relations, one cannot really use the third variable for anything; and it is known that the two-variable fragment of FOL is decidable, so it is already too weak for formalizing set theory. Therefore we need at least one ternary relation symbol (or atomic formula) in order to use the third variable, while in the language of set theory we only have one binary relation symbol, the elementhood-relation $\epsilon$. Therefore, while in formalizing set theory in the three-variable fragment of FOL (in $\mbox{\sf CA\/}_3$) we could do with one binary relation symbol, we did not have to change vocabulary during the formalization, in the present equality- and substitution-free case we have to change vocabulary, and we have to pay attention to this new feature of the translation mapping. A key device of our proofs will be a recursive ``translation mapping" translating FOL into the equational language of $\mbox{\sf Df\/}_3$, or equivalently into the logic $\mbox{$\mathcal Ld$\/}_3$ defined in section~\ref{fmd-sec} below.
$\mbox{\sf Df\/}_3$ is nothing more than Boolean algebras with three commuting complemented closure operators. The only connection between these operators is commutativity. We know that without commutativity the class is too weak for supporting set theory because its equational theory is decidable \cite{NDis}. We know that two commuting such operators do not suffice, for the same reason. We do not know how much complemented-ness of the closure operators is important for supporting set theory.
In section \ref{fmd-sec} we introduce our simple logic $\mbox{$\mathcal Ld$\/}_3$ in several different forms, which reveal its propositional logic character. Then we state three of the main theorems about this logic: it is only seemingly weak, because set theory can be built up in it (Thm.\ref{zfc-t}), and also G\"odel's incompleteness theorem holds for it (Thm.\ref{incomp-t}). In contrast with the fact that $\mbox{$\mathcal Ld$\/}_3$ cannot have a sound and complete Hilbert-style proof system, we state a completeness theorem for $\mbox{$\mathcal Ld$\/}_3$ which comes very close to having a Hilbert-style sound and complete proof system (Thm.\ref{comp-t}). Sections \ref{qra-sec}-\ref{set-sec} contain a full proof for Thm.\ref{zfc-t}, and a proof for a weaker version of Thm.\ref{comp-t}. In section \ref{free-sec} we prove, as a corollary of Thm.\ref{zfc-t}, that the finitely generated $\mbox{\sf Df\/}_3$'s are not atomic. This proof also contains the main ideas for a proof of Thm.\ref{incomp-t}.
We make the paper available in the present form because so many people expressed strong interest in the proofs of two of the main theorems, Thm.\ref{zfc-t} and Thm.\ref{free-t}. We will keep developing the paper and new versions will be found on our home-page, via the link \url{http://www.renyi.hu/~nemeti/FormalizingST.htm}. Via that link one can find more on the history of the problem settled in the present paper, see \cite{shortnote}, and some unpublished works, see \cite{NDis}, \cite{NPrep}.
\section{\label{fmd-sec} A simple logical system: three-variable logic without equality or substitutions} In this section we define the ``target logic" $\mbox{$\mathcal Ld$\/}_3$ of our translation. We give several different forms for it to give a feeling of its expressive power. After this, we formulate three of our main theorems, all stating unexpected properties of this logic.
The language of our system contains three variable symbols, $x,y,z$, one ternary relational symbol $P$, and only one atomic formula, namely $P(x,y,z)$. (We note that, e.g., the formula $P(y,x,z)$ is not available in this language.) The logical connectives are $\lor, \neg, \exists x, \exists y, \exists z$. We denote the set of formulas (of $\mbox{$\mathcal Ld$\/}_3$) by $\mbox{\it Fmd\/}_3$. We will use the derived connectives $\forall, \land, \to, \leftrightarrow$, too, as abbreviations: $\forall v\varphi\de\neg\exists v\neg\varphi$, $\varphi\land\psi\de\neg(\neg\varphi\lor\neg\psi)$, $\varphi\to\psi\de \neg\varphi\lor\psi$, $\varphi\leftrightarrow\psi\de (\varphi\to\psi)\land(\psi\to\varphi)$. Sometimes we will write, e.g., $\exists xy$ or $\forall xyz$ in place of $\exists x\exists y$ or $\forall x\forall y\forall z$, respectively. $\mbox{\it Fmd\/}^1_3$ denotes the set of formulas in $\mbox{\it Fmd\/}_3$ with one free variable $x$, we will often deal with these in section~\ref{qra-sec} on.
The proof system ${\tiny{\undharom}}$ which we will use is a Hilbert style one with the following logical axioms and rules.
The logical axioms are the following. Let $\varphi, \psi \in \mbox{\it Fmd\/}_3$ and $v,w \in \{ x,y,z\}$.\\ ((1))\quad $\varphi$, if $\varphi$ is a propositional tautology.\\ ((2))\quad $\forall v (\varphi \to \psi) \to (\exists v \varphi \to \exists v \psi)$.\\ ((3))\quad $\varphi \to \exists v \varphi$.\\ ((4))\quad $\exists v \exists v \varphi \to \exists v\varphi$.\\ ((5))\quad $\exists v (\varphi\lor\psi)\leftrightarrow (\exists v\varphi\lor\exists v\psi)$.\\ ((6))\quad $\exists v\neg\exists v\varphi\to \neg\exists v\varphi$.\\ ((7))\quad $\exists v\exists w\varphi\to\exists w\exists v\varphi$.\\
The inference rules are Modus Ponens ((MP), or detachment), and Generalization~((G)).
This proof system is a direct translation of the equational axiom system of $\mbox{\sf Df\/}_3$. Axiom ((2)) is needed for ensuring that the equivalence relation defined on the formula algebra by $\varphi\equiv\psi \Leftrightarrow {\tiny{\undharom}}\varphi\!\leftrightarrow\!\psi$ be a congruence with respect to (w.r.t.) the operation $\exists v$. It is congruence w.r.t.\ the Boolean connectives $\lor,\lnot$ by axiom ((1)). Axiom ((1)) expresses that the formula algebra factorized with $\equiv$ is a Boolean algebra, axiom ((5)) expresses that the quantifiers $\exists v$ are operators on this Boolean algebra (i.e., they distribute over $\lor$), axioms ((3)),((4)) express that these quantifiers are closure operations, axiom ((6)) expresses that they are complemented closure operators (i.e., the negation of a closed element is closed again). Together with ((5)) they imply that the closed elements form a Boolean subalgebra, and hence the quantifiers are normal operators (i.e., the Boolean zero is a closed element). Finally, axiom ((7)) expresses that the quantifiers commute with each other.
We define $\mbox{$\mathcal Ld$\/}_3$ as the logic with formulas $\mbox{\it Fmd\/}_3$ and with proof system ${\tiny{\undharom}}$. The logic $\mbox{$\mathcal Ld$\/}_3$ inherits a natural semantics from first-order logic (FOL). The proof system ${\tiny{\undharom}}$ is sound with respect to this semantics, but it is not complete. Moreover, there is no finite Hilbert-style inference system which would be complete and sound at the same time w.r.t.\ this semantics (because the quasi-equational theory of $\mbox{\sf RDf\/}_3$ is not finitely axiomatizable, see \cite{HMTII} and \cite{HbPhL}).
We note that in the above system, axiom ((6)) can be replaced with the following ((8)):\\ ((8))\quad $\exists v(\varphi\land\exists v\psi)\leftrightarrow (\exists v\varphi\land\exists v\psi)$.\\
In the present paper we will use our logic $\mbox{$\mathcal Ld$\/}_3$ as introduced above. However, it has several different but equivalent forms, each of which has advantages and disadvantages. We review some of the different forms below.
Restricted 3-variable FOL is introduced in \cite[Part II, p.157]{HMTII}, with proof system $\tiny{\undre}$. If we restrict this system $\tiny{\undre}$ to formulas not containing the equality $=$ then we get a system equivalent to our $\mbox{$\mathcal Ld$\/}_3$. Lets call this system {\it restricted 3-variable FOL without equality}.
That is, the formulas are those of restricted 3-variable FOL which contain no equality, and we leave out from the axioms of $\tiny{\undre}$ the axioms which contain equality. This way we get a proof system with Modus Ponens and Generalization as deduction rules and with the following axioms:\\
\noindent ((V1))\quad $\varphi$, if $\varphi$ is a propositional tautology.\\ ((V2))\quad $\forall v (\varphi \to \psi) \to (\forall v \varphi \to \forall v \psi)$.\\ ((V3))\quad $\forall v\varphi \to \varphi$.\\ ((V4))\quad $\varphi \to \forall v \varphi$, if $v$ does not occur free in~$\varphi$.\\
Lets call this\footnote{We note that we also omitted ((V9)) of \cite{HMTII} because $\forall v$ abbreviates $\lnot\exists v\lnot$ in our approach, so ((V9)) is not needed.} {\it equality-free $\tiny{\undre}$}. Rule ((V4)) in this system essentially uses individual variables in its using the notion of free variables of a formula. On the other hand, no axiom in ${\tiny{\undharom}}$ needs to use the structure of a formula occurring in a rule, it is essentially variable-free. So, an advantage of ${\tiny{\undharom}}$ over equality-free $\tiny{\undre}$ is that it is more ``algebraic", more like propositional logic. On the other hand, equality-free $\tiny{\undre}$ contains fewer axioms (it contains only ((V1))-((V4)) as axioms).
The logic $\mbox{$\mathcal Ld$\/}_3$ has a neat {\it modal logic form}: three commuting S5 modalities. This is denoted as [S5,S5,S5], see \cite[p.379, lines 15-20]{GKWZ}. We recall this logic in a slightly simplified form. The language contains one propositional variable $p$, the connectives are $\lor, \neg, \Diamond_1, \Diamond_2, \Diamond_3$. We use $\Box_i\de\neg\Diamond_i\neg$, $\to, \leftrightarrow$ as derived connectives as before, and the axioms are the following (where $\varphi, \psi$ are arbitrary formulas of the language and $i,j\in\{ 1,2,3\}$):\\
\noindent ((B))\quad $\varphi$, if $\varphi$ is a propositional tautology,\\ ((K))\quad $\Box_i(\varphi\to\psi)\to(\Box_i\varphi\to\Box_i\psi)$,\\ ((S5))\quad $\Diamond_i\varphi\to\Box_i\Diamond_i\varphi$,\\ ((C1))\quad $\Diamond_i\Diamond_j\varphi\to\Diamond_j\Diamond_i\varphi$,\\ ((C2))\quad $\Diamond_i\Box_j\varphi\to\Box_j\Diamond_i\varphi$.\\
\noindent The rules are Modus Ponens and Generalization (or, in other word, Necessitation, i.e., $\varphi\vdash\Box_i\varphi$). This modal logic is complete w.r.t.\ the frames consisting of three commuting equivalence relations as accessibility relations for the three modalities.
One can present this logic in yet one different form: {\it Equational logic} as the background logic, and the defining axioms of $\mbox{\sf Df\/}_3$ as logical axioms. (Occasionally, we refer to this logic informally as ``the equational theory of $\mbox{\sf Df\/}_3$".) For completeness, we include this form of $\mbox{$\mathcal Ld$\/}_3$ here, too. The language consists of equations $\tau = \sigma$ where $\tau, \sigma$ are terms built up from (arbitrarily many) variables by the use of the function symbols $+,-,f,g,h$ where $+$ is binary and the rest are unary. The axioms are the following, where $x,y,z$ are variables and $F\in\{ f,g,h\}$:\\
\noindent
((B1))\quad $x+y=y+x$, \\ ((B2))\quad $x+(y+z) = (x+y)+z$,\\ ((B3))\quad $-(-(x+y)+-(x+-y))=x$,\\ ((D1))\quad $x+Fx=Fx$, \\ ((D2))\quad $FFx=Fx$, \\ ((D3))\quad $F(x+y)=Fx+Fy$, \\ ((D4))\quad $F(-Fx)=-Fx$, \\ ((D5))\quad $fgx=gfx$,\quad $fhx=hfx$,\quad $ghx=hgx$.\\
\noindent The rules are those of the equational logic:
\noindent Rules of equivalence:\\ $\tau=\tau$,\quad $\tau=\sigma\vdash\sigma=\tau$,\quad $\tau=\sigma, \sigma=\rho\vdash\tau=\rho$,\\ Rules of congruence:\\ $\tau=\sigma, \rho=\delta\quad\vdash\quad -\tau=-\sigma, f\tau=f\sigma, g\tau=g\sigma, h\tau=h\sigma$, $\tau+\rho=\sigma+\delta$,\\ Rule of invariance:\\ $\tau=\sigma\vdash\tau'=\sigma'$ where $\tau',\sigma'$ are obtained from $\tau,\sigma$ by replacing the variables simultaneously with arbitrary terms.
We note that the first three axioms are an axiom system for Boolean algebras, see \cite[Problem 1.1, p.245]{HMTII} (this problem was solved affirmatively by a theorem prover program).
Consider the four ``logics" (or inference systems) ${\tiny{\undharom}}, \tiny{\undre}$, [S5,S5,S5], equational logic with (B1 - D5) introduced so far. We claim that they are equivalent to each other, hence our theorems stated below apply to all of them.
Having formulated our logic $\mbox{$\mathcal Ld$\/}_3$ in several different ways, we now formulate some theorems. The first theorem says that this simple logic $\mbox{$\mathcal Ld$\/}_3$ is strong enough for doing all of mathematics in it. It says that we can do set theory in $\mbox{$\mathcal Ld$\/}_3$ as follows: in place of formulas $\varphi$ of set theory we use their ``translated" versions ${\sf Tr\/}(\varphi)$ in $\mbox{$\mathcal Ld$\/}_3$, and then we use the proof system ${\tiny{\undharom}}$ of $\mbox{$\mathcal Ld$\/}_3$ between the translated formulas in place of the proof system of FOL between the original formulas of set theory. Moreover, for sentences $\varphi$ in the language of set theory, $\varphi$ and ${\sf Tr\/}(\varphi)$ mean the same thing (are equivalent) modulo a ``bridge" $\Delta$ between the two languages. We need this bridge because the language $\mathcal{L}_{\omega}$ of set theory contains only one binary relation symbol $\epsilon$ and equality, and the language $\mbox{$\mathcal Ld$\/}_3$ contains only one ternary relation symbol $P$. When $f:A\to B$ is a function and $X\subseteq A$ then $f(X)\de\{ f(a) : a\in X\}$ denotes the image of $X$ under this function $f$.
\begin{thm}\label{zfc-t}{\sf (Formalizability of set theory in $\mbox{$\mathcal Ld$\/}_3$)\/} There is a recursive translation function ${\sf Tr\/}$ from the language $\mathcal{L}_{\omega}$ of set theory into $\mbox{$\mathcal Ld$\/}_3$ for which the following are true for all sentences $\varphi$ in $\mathcal{L}_{\omega}$: \begin{itemize} \item[(i)] $\mbox{\it ZF\/}\models\varphi$\quad iff\quad ${\sf Tr\/}(\mbox{\it ZF\/}){\tiny{\undharom}}{\sf Tr\/}(\varphi)$. \item[(ii)] $\mbox{\it ZF\/}+\Delta\models \varphi\leftrightarrow{\sf Tr\/}(\varphi)$,\quad where\\ $\Delta\de\forall xyz[(P(x,y,z)\leftrightarrow(x=y=z\,\,\lor \,\,\epsilon(x,y))]$.
\end{itemize} \end{thm}
Theorem~\ref{zfc-t} is proved in section \ref{set-sec}.
\goodbreak The next theorem is a partial completeness theorem for $\mbox{$\mathcal Ld$\/}_3$. It is as good as it can be, see below.
\begin{thm}\label{comp-t}{\sf (Partial completeness theorem for $\mbox{$\mathcal Ld$\/}_3$)\/} Let $\mathcal{L}$ be a FOL-language having countably many relation symbols of each finite arity. There is a recursive subset $K\subseteq\mbox{\it Fmd\/}_3$ and there is a recursive function ${\sf tr\/}$ mapping all $\mathcal{L}$-formulas into $K$ such that the following are true: \begin{itemize} \item[(i)] $\models\varphi$\quad iff\quad ${\tiny{\undharom}}\varphi$ \quad for all $\varphi\in K$. \item[(ii)] $\models\varphi$\quad iff\quad $\models{\sf tr\/}(\varphi)$\quad for all $\mathcal{L}$-sentences $\varphi$. \end{itemize} \end{thm}
According to the above theorem, the proof system ${\tiny{\undharom}}$ is complete within $K$. But is $K$ big enough? Yes, we can prove any valid FOL-formula $\varphi$ by translating it into $K$ and then proving the translated formula by ${\tiny{\undharom}}$. We know that ${\tiny{\undharom}}$ is not strong enough to prove all valid $\mbox{\it Fmd\/}_3$ formulas (i.e., $K$ is necessarily a proper subset of $\mbox{\it Fmd\/}_3$), because as stated in the introduction, no finite Hilbert-style axiom system can be sound and complete at the same time for $\mbox{$\mathcal Ld$\/}_3$. However, we can formulate each sentence in a slightly different form, namely as ${\sf tr\/}(\varphi)$ so that this ``version" of $\varphi$ can now be proved by ${\tiny{\undharom}}$ iff it is valid.
\begin{thm}\label{incomp-t}{\sf (G\"odel style incompleteness theorem for $\mbox{$\mathcal Ld$\/}_3$)\/} There is a formula $\varphi\in\mbox{\it Fmd\/}_3$ such that no consistent recursive extension $T$ of $\varphi$ is complete, and moreover, no recursive extension of $\varphi$ separates the consequences of $\varphi$ from the $\varphi$-refutable sentences. \end{thm}
\begin{discussion}\label{discussion} In Theorems~\ref{zfc-t}-\ref{incomp-t}, at least one at least ternary relation symbol $P$ is needed in the ``target-language" $\mbox{$\mathcal Ld$\/}_3$, the axiom of commutativity ((7)) is needed in the proof system ${\tiny{\undharom}}$ (because omitting ((7)) from ${\tiny{\undharom}}$ results decidability of the so obtained proof system, see \cite{NDis}). We do not know whether complementedness of the closure operators ((6)) is needed or not. Also, two variables do not suffice because the satisfiability problem of the two-variable fragment of FOL is decidable.\qed \end{discussion}
\section{Finding \mbox{\sf QRA\/}-reducts in $\mbox{\sf Df\/}_3$}\label{qra-sec}
In this section we begin the proof of Theorem~\ref{zfc-t}. For the definitions of relation algebras, quasi-projective and representable relation algebras see \cite{TG} or \cite[sec.5.3]{HMTII}. We briefly recall these. Relation algebras, $\mbox{\sf RA\/}$s are Boolean algebras with operators $\langle A,+,-,;,^{\smallsmile},1'\rangle$ such that the operators form an involuted monoid satisfying a further equation. Here, $\langle A,+,-\rangle$ is the Boolean reduct of the $\mbox{\sf RA\/}$ in question, and $;,^{\smallsmile},1'$ stand for relation composition, converse, and identity constant, respectively. The elements $p,q$ in a relation algebra are called {\it quasi-projections} if $p^{\smallsmile};p+q^{\smallsmile};q\le 1'$ and $p^{\smallsmile};q=1'+-1'$, and a relation algebra is called a quasi-projective relation algebra, a $\mbox{\sf QRA\/}$, if there is a pair of quasi-projections in it. We call a relation algebra {\it representable} if its elements are binary relations and the operations are union, complementation (w.r.t.\ the biggest element), relation composition of binary relations, converse of a binary relation, and the identity relation, respectively (more precisely, an $\mbox{\sf RA\/}$ is representable if it is isomorphic to such a concrete algebra). Quasi-projective relation algebras are representable, by a theorem of Tarski.
We show that every $\mbox{\sf Df\/}_3$ contains lots of quasi-projective relation algebras in them. We do this by defining relation algebra type operations in the term language of $\mbox{\sf Df\/}_3$ and proving that these operations form $\mbox{\sf QRA\/}$s in appropriate relativizations. Since $\mbox{\sf QRA\/}$s are representable, this will amount to a ``partial" representation theorem for $\mbox{\sf Df\/}_3$s, and to ``partial" completeness theorem for $\mbox{$\mathcal Ld$\/}_3$ (see Thm.\ref{comp-t}), in the spirit of \cite{HbPhL}. We will work in $\mbox{$\mathcal Ld$\/}_3$ in place of $\mbox{\sf Df\/}_3$.
There will be parameters in the definitions to come. These will be formulas in $\mbox{\it Fmd\/}_3$, namely $\delta_{xy}, \delta_{xz}$ with free variables $\{x,y\}$ and $\{x,z\}$ respectively, together with two other formulas $p_0, p_1$ with free variables $\{x,y\}$. Thus, if you choose $\delta_{xy},\delta_{xz},p_0,p_1$ with the above specified free variables then you will arrive at a \mbox{\sf QRA\/}-reduct of any $\mbox{\sf Df\/}_3$ corresponding to these. We get the \mbox{\sf QRA\/}-reduct by assuming some properties of the meanings of these formulas, this will be expressed by a formula $\mbox{\sf Ax\/}$. In section~\ref{set-sec} then we will choose these parameters so that they fit set theory, which means that the formula $\mbox{\sf Ax\/}$ built up from them is provable in set theory. Intuitively, the formulas $\delta_{xy},\delta_{xz}$ stand for equality $x=y,x=z$ and $p_0,p_1$ will be arbitrary pairing functions.
So, choose formulas $\delta_{xy},\delta_{xz},p_0,p_1$ with the above specified free variables arbitrarily, they will be parameters of the definitions to come. To simplify notation, we will {\it not} indicate these parameters.
We now set ourselves to defining the above relation algebra type operations on $\mbox{\it Fmd\/}_3$. To help readability, we often write just comma in place of conjunction in formulas, especially when they begin with a quantifier. E.g., we write $\exists x(\varphi,\psi)$ in place of $\exists x(\varphi\land\psi)$. Further, $\mbox{\sf True\/}$ denotes a provably true formula, say $\mbox{\sf True\/}\de \delta_{xy}\lor\lnot \delta_{xy}$. First we introduce notation to support the intuitive meaning of the parameters $\delta_{xy}, \delta_{xz}$ as equality.
\begin{defn}\label{veq-d}(Simulating equality between variables)\\ $\mbox{$x\!\doteq\! y$\/} \de \delta_{xy}$,\\ $\mbox{$x\!\doteq\! z$\/} \de \delta_{xz}$,\\ $\mbox{$y\!\doteq\! z$\/} \de \exists x (\mbox{$x\!\doteq\! y$\/}, \mbox{$x\!\doteq\! z$\/})$,\\ $\mbox{$y\!\doteq\! x$\/} \de \mbox{$x\!\doteq\! y$\/}$,\\ $\mbox{$z\!\doteq\! x$\/} \de \mbox{$x\!\doteq\! z$\/}$,\\ $\mbox{$z\!\doteq\! y$\/} \de \mbox{$y\!\doteq\! z$\/}$,\\ $\mbox{$x\!\doteq\! x$\/} \de \mbox{\sf True\/}$,\\ $\mbox{$y\!\doteq\! y$\/} \de \mbox{\sf True\/}$,\\ $\mbox{$z\!\doteq\! z$\/} \de \mbox{\sf True\/}$.\qed\\ \end{defn}
\begin{defn}\label{vsubs-d}(Simulating substitution with (simulated) equality)\\ $\varphi\(x,y\) \de \varphi$,\\ $\varphi\(x,z\) \de \exists y (\mbox{$y\!\doteq\! z$\/} , \varphi)$,\\ $\varphi\(y,z\) \de \exists x (\mbox{$x\!\doteq\! y$\/} , \varphi\(x,z\))$,\\ $\varphi\(y,x\) \de \exists z (\mbox{$x\!\doteq\! z$\/} , \varphi\(y,z\))$,\\ $\varphi\(z,x\) \de \exists y (\mbox{$y\!\doteq\! z$\/} , \varphi\(y,x\))$,\\ $\varphi\(z,y\) \de \exists x (\mbox{$x\!\doteq\! z$\/} , \varphi)$,\\ $\varphi\(x,x\) \de \exists y (\mbox{$x\!\doteq\! y$\/} , \varphi)$,\\ $\varphi\(y,y\) \de \exists x (\mbox{$x\!\doteq\! y$\/} , \varphi)$,\\ $\varphi\(z,z\) \de \exists x (\mbox{$x\!\doteq\! z$\/} , \varphi\(x,x\))$.\qed\\ \end{defn}
\begin{remark}\label{subs-r} In FOL, $\varphi\(u,v\)$ is semantically equivalent with the formula we get from $\varphi$ by replacing $x,y$ with $u,v$ everywhere simultaneously, when $\delta_{xy},\delta_{xz}$ are $x=y, x=z$ respectively. This is Tarski's fabulous trick to simulate substitutions.
\qed \end{remark}
Next we introduce notation supporting intuition about the pairing functions $p_0, p_1$. First we define some auxiliary formulas. We will use the notation $2=\{ 0,1\}$, to make the text shorter. Let
$2^*$ denote the set of all finite sequences of $0,1$ including the empty sequence $\mbox{$\langle\rangle$}$ as well. If $i,j \in 2^*$ then $ij$ denotes their ``concatenation'' usually denoted by $i^\cap \!j $, and $|i|$ denotes the ``length'' of~$i$. Further, if $k \in 2$, then we write $k$ instead of $\langle k\rangle$ for the sequence $\langle k\rangle$ of length~$1$. Accordingly, $00$ denotes the sequence $\langle 0,0\rangle$.
We are going to define $\mbox{\it Fmd\/}_3$-formulas $u_i \!\doteq\! v_j$ for $u,v\in\{ x,y,z\}$ and $i, j \in 2^*$. The intuitive meaning of $u_{i_0 \dots i_n} \!\doteq\! v_{j_0 \dots j_k}$ is that if $p_0, p_1$ are partial functions then $p_{i_n} \dots p_{i_0} u = p_{j_k} \dots p_{j_0} v$. As usual in the partial algebra literature, the equality holds if both sides are defined and are equal. E.g., the intuitive meaning of $x_0\!\doteq\! y_{01}$ is that all of $p_0x, p_0y, p_1p_0y$ exist and $p_0x=p_1p_0y$.
\begin{defn}\label{pid-d}(Simulating projections)\\ Let $\{ u,v,w\}=\{ x,y,z\}$, $i,j\in 2^*$ and $k\in 2$.\\ $(u_{\mbox{$\langle\rangle$}}\!\doteq\! v_{\mbox{$\langle\rangle$}}) \de \mbox{$u\!\doteq\! v$\/}$,\\ $(u_k\!\doteq\! v_{\mbox{$\langle\rangle$}}) \de p_k\(u,v\)$,\\ $(u_{ik}\!\doteq\! v_{\mbox{$\langle\rangle$}}) \de \exists w (u_i\!\doteq\! w_{\mbox{$\langle\rangle$}}, p_k\(w,v\))$\quad if $i\ne \mbox{$\langle\rangle$}$,\\ $(u_i\!\doteq\! v_j) \de \exists w (u_i\!\doteq\! w_{\mbox{$\langle\rangle$}},v_j\!\doteq\! w_{\mbox{$\langle\rangle$}})$\quad if $j\ne \mbox{$\langle\rangle$}$,\\ $(x_i\!\doteq\! x_j) \de \exists y (\mbox{$x\!\doteq\! y$\/}, x_i\!\doteq\! y_j)$,\\ $(y_i\!\doteq\! y_j) \de \exists x (\mbox{$x\!\doteq\! y$\/}, x_i\!\doteq\! y_j)$,\\ $(z_i\!\doteq\! z_j) \de \exists x (z_i\!\doteq\! x_{\mbox{$\langle\rangle$}},z_j\!\doteq\! x_{\mbox{$\langle\rangle$}})$.\qed\\ \end{defn}
We will omit the index $\mbox{$\langle\rangle$}$ in formulas $u_i\!\doteq\! v_j$, i.e., we write $u_i \!\doteq\! v$ and $u \!\doteq\! v_i$ for $u_i \!\doteq\! v_{\mbox{$\langle\rangle$}}$ and $u_{\mbox{$\langle\rangle$}} \!\doteq\! v_i$ respectively if $i \in 2^*$.
So far we did nothing but introduced notation supporting the intuitive meanings of the parameters $\delta_{xy}, \delta_{xz}, p_0, p_1$ as equality and partial pairing functions. Almost any of the concrete formulas supporting this would do, we only had to fix one of them since our proof system ${\tiny{\undharom}}$ is very weak, it would not prove equivalence of most of the semantically equivalent forms. Now we write up a statement $\mbox{\sf Ax\/}$ about the parameters using the just introduced notation. Let $H\de \{ i\in 2^* : |i|\le 3\}$. Notice that $H$ is finite.
\begin{defn}[pairing axiom \mbox{\sf Ax\/}]\label{ax-d} We define $\mbox{\sf Ax\/}\in\mbox{\it Fmd\/}_3$ to be the conjunction of the union of the following finite sets (A1),...,(A4) of formulas: \begin{itemize} \item[(A1)] $\{ u_i\!\doteq\! v_j, v_j\!\doteq\! w_k \to u_i\!\doteq\! w_k : u,v,w\in\{x,y,z\}, i,j,k\in H\}$ \item[(A2)] $\{ u_i\!\doteq\! v_j, u_{ik}\!\doteq\! u_{ik}\to u_{ik}\!\doteq\! v_{jk} : u,v\in\{x,y,z\}, ik,jk\in H, k\in 2\}$ \item[(A3)] $\{ u_i\!\doteq\! u_i, v_j\!\doteq\! v_j\to \exists w(w_0\!\doteq\! u_i, w_1\!\doteq\! v_j) : u,v,w\in\{x,y,z\}, w\notin\{ u,v\}, i,j\in H\}$ \item[(A4)] $\{ \exists w\,u\!\doteq\! w : u,w\in\{ x,y,z\}\}$.\qed \end{itemize} \end{defn}
In the above definition, (A1), (A2), (A4) express usual properties of the equality, while (A3) states the existence of pairs. We say that $x$ is a pair if both $p_0$ and $p_1$ are defined on $x$ and then we think of $x$ as the pair $\langle p_0(x),p_1(x)\rangle$. That $p_i$ is defined on $x$ is expressed by $p_i(x)\!\doteq\! p_i(x)$, i.e., by $x_i\!\doteq\! x_i$ (for $i\in 2$). Following \cite{TG}, we do not require pairs to be unique, i.e., for different $u,v$ it can happen that $u_0=v_0, u_1=v_1$. (This is why $\mbox{\sf QRA\/}$s are called {\it quasi-projective} $\mbox{\sf RA\/}$s and not just projective $\mbox{\sf RA\/}$s in \cite{TG}.) In the next section, just for simplicity, we will use a stronger axiom $\mbox{\sf SAx\/}$ in place of $\mbox{\sf Ax\/}$ in which we require uniqueness of pairs.
We are ready to define our relation-algebra type operations on $\mbox{\it Fmd\/}_3$. They will have the intended meanings on formulas with one free variable $x$, where $x$ denotes a pair. This is expressed by the definition of $\mbox{\sf Dra\/}$, the universe of the algebra defined below. If we assume uniqueness of pairs (as in $\mbox{\sf SAx\/}$ later) then the definition of $\mbox{\sf Dra\/}$ in Def.\ref{qra-d} below can be simplified to be $\mbox{\sf Dra\/}\de\{\varphi\in\mbox{\it Fmd\/}_3^1 :\mbox{\sf Ax\/}{\tiny{\undharom}}\varphi\to\mbox{\sf pair\/}\}$, where $\mbox{\sf pair\/}$ is the formula expressing that $x$ is a pair. Since $\varphi$ has only one free variable $x$ which is a pair, we can think of $\varphi$ as a unary relation of pairs, i.e., as a binary relation. With this intuition, the definitions of the operations $\mbox{{\small{$\odot$}}},^{{\cup\hspace*{-1.43mm} {\boldsymbol \cdot} }},\id$ below in Def.\ref{qra-d} are the natural ones, see Figure~1. For more on the intuition behind Def.\ref{qra-d} see the remark after the definition.
\begin{defn}[relation algebra reduct ${\mathfrak Dra\/}$ of $\mbox{\it Fmd\/}_3$] \label{qra-d} Let $\varphi,\psi\in\mbox{\it Fmd\/}_3$. \begin{itemize} \item[] $\mbox{\sf pair\/} \de \exists y p_0 \land \exists y p_1$. \\ $\varphi u_i \de \exists x(x\!\doteq\! u_i,\varphi)$ \quad if $u\in\{ y,z\}$ and $i\in 2^*$. \item[] $\varphi\mbox{{\small{$\odot$}}}\psi \de \exists y(\varphi y_0, \psi y_1, x_0\!\doteq\! y_{00}, y_{01}\!\doteq\! y_{10}, y_{11}\!\doteq\! x_1)$,\quad see Figure~1,\\
$\varphi^{{\cup\hspace*{-1.43mm} {\boldsymbol \cdot} }} \de \exists y(\varphi y, y_0\!\doteq\! x_1, y_1\!\doteq\! x_0)$,\\
$\id \de x_0\!\doteq\! x_1$,\\
$\div\, \varphi \de \mbox{\sf pair\/}\land\lnot\varphi$,\\
$\varphi+\,\psi \de \varphi\lor\psi$. \item[] $\mbox{\sf Dra\/} \de \{\varphi\in\mbox{\it Fmd\/}_3 : \mbox{\sf Ax\/}{\tiny{\undharom}}\varphi\leftrightarrow \psi\mbox{{\small{$\odot$}}}\id\mbox{
for some }\psi\in\mbox{\it Fmd\/}_3^1\}$,\\
${\mathfrak Dra\/} \de \langle\mbox{\sf Dra\/},+,\,\div,\mbox{{\small{$\odot$}}},^{{\cup\hspace*{-1.43mm} {\boldsymbol \cdot} }},\id\rangle$.\qed \end{itemize} \end{defn}
\begin{figure}\label{comp-fig}
\end{figure}
Let us define $x\sim y\de x_0\!\doteq\! y_0, x_1\!\doteq\! y_1$, and let us call the pairs $x,y$ equivalent if $x\sim y$. Since we do not require pairs to be unique in $\mbox{\sf Ax\/}$, we may have distinct $x,y$ which are equivalent. However, in the above definition, we can see that the result of an operation from $\mbox{{\small{$\odot$}}},^{{\cup\hspace*{-1.43mm} {\boldsymbol \cdot} }},\id$ is always closed under the equivalence relation $\sim$, because, intuitively, the result depends only on $x_0, x_1$, thus if, say, $\varphi\mbox{{\small{$\odot$}}}\psi$ holds at $x$ and $x\sim y$, then $\varphi\mbox{{\small{$\odot$}}}\psi$ holds at $y$, too. (Formally, $\mbox{\sf Ax\/}{\tiny{\undharom}}\varphi\mbox{{\small{$\odot$}}}\psi\land x\sim y\to (\varphi\mbox{{\small{$\odot$}}}\psi)y$.) From this one can see that $\varphi\mbox{{\small{$\odot$}}}\id$ represents the same binary relation as $\varphi$ composed with $\sim$; and thus $\mbox{\sf Dra\/}$ consists of those formulas which do not distinguish equivalent pairs.
With the intuition that $\varphi$ represents a binary relation (coded as a unary relation on pairs), we could have defined, say, $\varphi\mbox{{\small{$\odot$}}}\psi$ as $\exists yz(\varphi y, \psi z, x_0\!\doteq\! y_0, y_1\!\doteq\! z_0, z_1\!\doteq\! x_1)$. This definition would leave us with one register (or free variable) to work with, namely $x$ (because $x_0, x_1$ are recoverable from $y,z$). It is more convenient to code up every relevant data in one register ($y$ in Def.\ref{qra-d}) and so have two registers (namely, $x,z$) to work with. This is how we defined the relation algebraic operations in Def.\ref{qra-d} above.
\goodbreak The next theorem is the heart of formalizing set theory in $\mbox{\it Fmd\/}_3$. Let us define the equivalence relation $\equiv_{T}$ on $\mbox{\it Fmd\/}_3$ by $$ \varphi\equiv_{T}\psi \de T{\tiny{\undharom}}\varphi\leftrightarrow\psi,$$ where $T\subseteq\mbox{\it Fmd\/}_3$. Note that with using this notation we have $$ \mbox{\sf Dra\/} = \bigcup\{ \varphi\mbox{{\small{$\odot$}}}\id\slash\equiv_{\mbox{\sf Ax\/}}\ \ :\ \varphi\in\mbox{\it Fmd\/}^1_3\}.$$
\goodbreak \begin{thm}\label{dra-t} \begin{itemize} \item[(i)] ${\mathfrak Dra\/}$ is an algebra, i.e., the operations $+,\div,\mbox{{\small{$\odot$}}},^{{\cup\hspace*{-1.43mm} {\boldsymbol \cdot} }},\id$ do not lead out of $\mbox{\sf Dra\/}$. \item[(ii)]
$\mbox{\sf Dra\/}\supseteq\{ \varphi\mbox{{\small{$\odot$}}}\psi : \varphi,\psi\in\mbox{\it Fmd\/}_3^1\}$. \item[(iii)] ${\mathfrak Dra\/}\slash\equiv_{\mbox{\sf Ax\/}}$ is a relation algebra. \item[(iv)] The images of the formulas $x_1\!\doteq\! x_{00}$ and $x_1\!\doteq\! x_{01}$ form a quasi-projection pair in ${\mathfrak Dra\/}\slash\equiv_{\mbox{\sf Ax\/}}$. \end{itemize} \end{thm}
\noindent{\bf Proof.} The proof of the analogous theorem in \cite[Thm.9, p.43]{NDis} goes through with some modifications. We indicate here these modifications.
The ``proof explanations" (CA),\dots, (KV) introduced in \cite[p.44]{NDis} can be used in our proof, too, except that we always have to check whether the explanation uses only the Df-part of (CA). If it uses axiom C7 of $\mbox{\sf CA\/}$, then we have to give an alternate proof. Let (Df) denote the part of (CA) which does not use C7. We can use (UV) because it can be derived from $\mbox{\sf Ax\/}$ and (Df). Of course, throughout we have to change $=$ to $\!\doteq\!\,$.
We now go through the proof given in \cite[pp.46-64]{NDis} and indicate the changes needed for our proof. All the statements on pp.48-54 beginning with (A1) and ending with (S13) follow from $\mbox{\sf Ax\/}$ and (Df). In fact, we could just add these statements to $\mbox{\sf Ax\/}$ since they do not contain formula-schemes denoting arbitrary formulas (such as, e.g., (0) on p.54 does), and so they amount to finitely many formulas only. Because of this, we do not indicate the changes needed in the proofs of these items.
We have to avoid statement (0) at the end of p.54 by all means, because it uses axiom C7 of (CA) essentially. Fortunately, we do not really use (0) in the proof, changing $\varphi$ to $\varphi x$ in some steps will suffice for eliminating (0).
The proof of (2\slash a) on p.55 has to be modified, and that can be done as follows. Recall from \cite{NDis} that $\varphi$ has at most one free variable $x$, and $\varphi u\de \exists x(x\!\doteq\! u, \varphi)$ when $u$ is different from $x$ while $\varphi x\de\exists y(y\!\doteq\! x, \varphi y)$.
\noindent Assume $x\notin\{ u,v\}$. The proof for $\varphi u, u\!\doteq\! v{\tiny{\undharom}} \varphi v$ is as on \cite[p.55]{NDis}. Then
\noindent $\varphi x, x\!\doteq\! z {\tiny{\undharom}}\qquad\mbox{by definition of $\varphi x$}\\ \exists y(y\!\doteq\! x, \varphi y), x\!\doteq\! z {\tiny{\undharom}}\\ \exists y(y\!\doteq\! x, \varphi y, x\!\doteq\! z ){\tiny{\undharom}}\qquad\mbox{ by \mbox{\sf Ax\/}}\\ \exists y(y\!\doteq\! z, \varphi y) {\tiny{\undharom}}\qquad\mbox{ by the case when $x\notin\{ u,v\}$ }\\ \exists y(\varphi z) {\tiny{\undharom}}\\ \varphi z.$
\noindent $\varphi x, x\!\doteq\! y {\tiny{\undharom}}\qquad\mbox{ by \mbox{\sf Ax\/}}\\ \exists z(z\!\doteq\! x, \varphi x, x\!\doteq\! y) {\tiny{\undharom}}\qquad\mbox{ by \mbox{\sf Ax\/}}\\ \exists z(z\!\doteq\! y, \varphi x, x\!\doteq\! z) {\tiny{\undharom}}\qquad\mbox{ by the previous case}\\ \exists z(z\!\doteq\! y, \varphi z){\tiny{\undharom}}\qquad\mbox{ by the case when $x\notin\{ u,v\}$ }\\ \varphi y.$
\noindent $\varphi y, y\!\doteq\! x {\tiny{\undharom}}\\ \exists y(y\!\doteq\! x, \varphi y) {\tiny{\undharom}}\quad\mbox{ by definition of $\varphi x$}\\ \varphi x .$
\noindent $\varphi z, z\!\doteq\! x {\tiny{\undharom}}\quad\mbox{ by $\mbox{\sf Ax\/}$}\\ \exists y(y\!\doteq\! z, z\!\doteq\! x, \varphi z) {\tiny{\undharom}}\quad\mbox{ by $\mbox{\sf Ax\/}$}\\ \exists y(y\!\doteq\! x, z\!\doteq\! y, \varphi z) {\tiny{\undharom}} \quad\mbox{ by the case when $x\notin\{ u,v\}$ }\\ \exists y(y\!\doteq\! x, \varphi y) {\tiny{\undharom}}\\ \varphi x.$
On p.58, in the last line of the proof of (9) we have to write $\gamma x$ in place of $\gamma$. Similarly, on p.59, in lines 7 and 8 we have to change $\psi$ to $\psi x$ and then we can cross out reference to (0).
In the last line of the proof of (18) on p.62 we have to use the following, which is practically C7 for formulas of form $\varphi\mbox{{\small{$\odot$}}}\psi$:
\begin{itemize} \item[(C)] $[\lnot(\varphi\mbox{{\small{$\odot$}}}\psi)]z\leftrightarrow \lnot[(\varphi\mbox{{\small{$\odot$}}}\psi)z]\ .$ \end{itemize}
\noindent Recall that $\Delta(x,y)$ denotes the following formula: $x_0\!\doteq\! y_{00}, y_{01}\!\doteq\! y_{10}, x_1\!\doteq\! y_{11}$ (cf., Figure~\ref{comp-fig}). To prove (C) we will prove the following two statements from which it follows immediately.
\begin{itemize} \item[(C1)] $[\lnot(\varphi\mbox{{\small{$\odot$}}}\psi)]z\leftrightarrow \neg\exists y(\Delta(z,y),\varphi y_0,\psi y_1)$ . \item[(C2)] $(\varphi\mbox{{\small{$\odot$}}}\psi)z\leftrightarrow \exists y(\Delta(z,y),\varphi y_0,\psi y_1)$ . \end{itemize}
\noindent {\it Proof of (C2):}\\
\noindent $(\varphi\mbox{{\small{$\odot$}}}\psi)z\leftrightarrow\quad\mbox{ by definition of $\chi z$}\\ \exists x(x\!\doteq\! z, \varphi\mbox{{\small{$\odot$}}}\psi) \leftrightarrow\quad\mbox{ by definition of $\mbox{{\small{$\odot$}}}$}\\ \exists x(x\!\doteq\! z, \exists y(\Delta(x,y),\varphi y_0, \psi y_1))\leftrightarrow\quad\mbox{ by SZV}\\ \exists x\exists y(x\!\doteq\! z,\Delta(x,y),\varphi y_0, \psi y_1)\leftrightarrow\quad\mbox{ by $\mbox{\sf Ax\/}$}\\ \exists x\exists y(x\!\doteq\! z,\Delta(z,y),\varphi y_0, \psi y_1)\leftrightarrow\quad\mbox{ by SZV}\\ \exists x(x\!\doteq\! z,\exists y(\Delta(z,y),\varphi y_0, \psi y_1))\leftrightarrow\quad\mbox{ by SZV}\\ \exists x(x\!\doteq\! z),\exists y(\Delta(z,y),\varphi y_0, \psi y_1)\leftrightarrow\quad\mbox{ by $\mbox{\sf Ax\/}$}\\ \exists y(\Delta(z,y),\varphi y_0, \psi y_1)\quad\mbox{ and we are done.}$
\noindent {\it Proof of (C1):}\\
\noindent $[\neg(\varphi\mbox{{\small{$\odot$}}}\psi)]z\leftrightarrow\quad\mbox{ by definitions of $\chi z$ and $\mbox{{\small{$\odot$}}}$}\\ \exists x(x\!\doteq\! z, \neg\exists y(\Delta(x,y),\varphi y_0, \psi y_1))\leftrightarrow\quad\mbox{ by SZV}\\ \exists x(\forall y(x\!\doteq\! z),\forall y\neg(\Delta(x,y),\varphi y_0, \psi y_1))\leftrightarrow\quad\mbox{ by Df}\\ \exists x\forall y(x\!\doteq\! z, \neg(\Delta(x,y),\varphi y_0, \psi y_1))\leftrightarrow\quad\mbox{ by $\mbox{\sf Ax\/}$ and BA}\\ \exists x\forall y(x\!\doteq\! z, \neg(\Delta(z,y),\varphi y_0, \psi y_1))\leftrightarrow\quad\mbox{ by Df, SZV}\\ \exists x(x\!\doteq\! z,\neg\exists y(\Delta(z,y),\varphi y_0, \psi y_1))\leftrightarrow\quad\mbox{ by SZV}\\ \exists x(x\!\doteq\! z),\neg\exists y(\Delta(z,y),\varphi y_0, \psi y_1)\leftrightarrow\quad\mbox{ by $\mbox{\sf Ax\/}$}\\ \neg\exists y(\Delta(z,y),\varphi y_0, \psi y_1)\quad\mbox{ and we are done}.$
That's all the changes we have to do in the proof given in \cite[pp.46-64]{NDis}! \qquad {\bf QED}
\section{Finding \mbox{\sf CA\/}-reducts in $\mbox{\sf Df\/}_3$}\label{ca-sec}
Simon~\cite{Simon} defines a $\mbox{\sf CA\/}_n$-reduct in every $\mbox{\sf QRA\/}$, for all $n\in\omega$, and also proves that these reducts are representable. We will use, in this paper, the $\mbox{\sf CA\/}_3$-reduct of our $\mbox{\sf QRA\/}$ defined in the previous section, i.e., we will use the $\mbox{\sf CA\/}_3$-reduct of ${\mathfrak Dra\/}\slash\equiv_{\mbox{\sf Ax\/}}$.
We will use the following stronger form of $\mbox{\sf Ax\/}$, just for convenience:
\begin{defn}\label{sax-d}(strong pairing axiom \mbox{\sf SAx\/}.) We define $\mbox{\sf SAx\/}\in\mbox{\it Fmd\/}_3$ to be the conjunction of the union of the finite sets (A1),...,(A4) of Def.\ref{ax-d} together with the following: \begin{itemize} \item[(A5)] $\{\ x_0\!\doteq\! y_0, x_1\!\doteq\! y_1\to x\!\doteq\! y, \ \ x_0\!\doteq\! x_0\leftrightarrow x_1\!\doteq\! x_1\}$.\qed \end{itemize} \end{defn}
The formulas in (A5) express that pairs are unique, and that $p_0$ is defined exactly when $p_1$ is defined. We use (A5) for convenience only, this way formulas will be shorter. We could omit (A5) on the expense that formulas will be longer and more complicated. If we assume $\mbox{\sf SAx\/}$ then $\varphi\land\mbox{\sf pair\/}\in\mbox{\sf Dra\/}$ for all $\varphi\in\mbox{\it Fmd\/}_3^1$.
Every $\mbox{\sf QRA\/}$ has a $\mbox{\sf CA\/}_3$-reduct, which is representable, see Simon \cite{Simon}. The following definition is recalling this $\mbox{\sf CA\/}_3$-reduct from \cite{Simon} in our special case of ${\mathfrak Dra\/}\slash\equiv_{\mbox{\sf SAx\/}}$. The definition below is simpler than in \cite{Simon} because we will assume uniqueness of pairs in $\mbox{\sf SAx\/}$, which is not assumed in \cite{Simon}. We will use the abbreviation \begin{itemize} \item[] $\varphi x_i\de\exists y(y\!\doteq\! x_i,\varphi y)$, when $\varphi\in\mbox{\it Fmd\/}^1_3$ and $i\in 2^*$. \end{itemize} Beware: ${\tiny{\undharom}} \varphi\leftrightarrow\varphi x$ usually is not true for $\varphi\in\mbox{\it Fmd\/}_3^1$. (For the definition of $\varphi y_j$ see Def.\ref{qra-d}.)
The intuitive meaning of Def.\ref{ca-d} below is similar to the one of Def.\ref{qra-d}. The universe of our $\mbox{\sf CA\/}_3$ will consist of those formulas which depend only on $x_1$ such that $x_1$ is a triplet; we will look at such a formula as representing a set of these triplets, i.e., a ternary relation. With this is mind, then in the definition below, $\mbox{\sf d\/}_{ij}$ represents the set of those triplets whose $i$-th and $j$-th components equal, ${\sf T\/}_i$ is the binary relation on triplets which correlates two triplets iff only their $i$-th components may differ.
\begin{defn}\label{ca-d}(cylindric reduct ${\mathfrak Dca\/}$ of $\mbox{\it Fmd\/}_3$.)\\ ${\sf Triplet\/}\de x_{11}\!\doteq\! x_{11}$,\quad see Figure~2,\\ $(0)\de 0,\quad (1)\de 10,\quad (2)\de 11$,\quad and for all $i,j<3$ and $\varphi,\psi\in\mbox{\it Fmd\/}_3^1$\\ ${\sf T\/}_i\de{\sf Triplet\/} x_0\land{\sf Triplet\/} x_1\land\bigwedge\{ x_{0(j)}\!\doteq\! x_{1(j)} : i\ne j<3\}$, \\ $\mbox{\sf c\/}_i\varphi\de\varphi\mbox{{\small{$\odot$}}}{\sf T\/}_i$,\\ ${\sf d\/}_{ij}\de {\sf Triplet\/} x_1\land x_{1(i)}\!\doteq\! x_{1(j)}$,\\ $-\, \varphi \de {\sf Triplet\/} x_1\land\lnot\varphi$,\\ $\varphi+\,\psi \de \varphi\lor\psi$,\\ $\mbox{\sf Dca\/}\de\{ \varphi\in\mbox{\it Fmd\/}_3 : \mbox{\sf SAx\/}{\tiny{\undharom}} \varphi\leftrightarrow (\psi x_1\land{\sf Triplet\/} x_1)\ \ \mbox{ for some } \psi\in\mbox{\it Fmd\/}_3^1\}$,\\
${\mathfrak Dca\/} \de \langle\mbox{\sf Dca\/},+,\,-,\,\mbox{\sf c\/}_i,{\sf d\/}_{ij} : i,j<3\rangle$.\qed \end{defn}
\begin{figure}
\caption{$x$ represents the triplet $\langle x_{(0)},x_{(1)},x_{(2)}\rangle$ when $x_{11}$ is defined}
\label{triplet-fig}
\end{figure}
\goodbreak \begin{thm}\label{cared-t} The set $\mbox{\sf Dca\/}$ is closed under the operations defined in Def.\ref{ca-d} and the algebra ${\mathfrak Dca\/}\slash\equiv_{\mbox{\sf SAx\/}}$ is a representable $\mbox{\sf CA\/}_3$. \end{thm}
\noindent{\bf Proof.} We show that the algebra ${\mathfrak Dca\/}\slash\equiv_{\mbox{\sf SAx\/}}$ is the $\mbox{\sf CA\/}_3$-reduct of the quasi-projective relation algebra ${\mathfrak Dra\/}\slash\equiv_{\mbox{\sf SAx\/}}$, as defined in \cite{Simon}. Let $\mbox{\sf p\/}\de x_1\!\doteq\! x_{00}$, $\mbox{\sf q\/}\de x_1\!\doteq\! x_{01}$. In the following we will omit referring to $\equiv_{\mbox{\sf SAx\/}}$, so we will look at $\mbox{\sf p\/}$ as an element of ${\mathfrak Dra\/}\slash\equiv_{\mbox{\sf SAx\/}}$, while only $\mbox{\sf p\/}\slash\equiv_{\mbox{\sf SAx\/}}$ is such. Recall from Thm.\ref{dra-t}(iv) that $\mbox{\sf p\/},\mbox{\sf q\/}$ form a pair of projections in ${\mathfrak Dca\/}\slash\equiv_{\mbox{\sf SAx\/}}$. See Figure~3.
\begin{figure}
\caption{$\mbox{\sf p\/}$ and $\mbox{\sf q\/}$ represent the pairing functions $p_0, p_1$ coded as unary relations of pairs.}
\end{figure}
\goodbreak Let \begin{itemize} \item[] ${\sf e\/}\de {\sf Triplet\/} x_0\land x_0\!\doteq\! x_1$,\\ $\pi_{(i)}\de{\sf Triplet\/} x_0\land x_1\!\doteq\! x_{0(i)}$,\\ $\chi_{(i)}\de{\sf Triplet\/} x_0\land{\sf Triplet\/} x_1\land x_{0(i)}\!\doteq\! x_{1(i)}$. \end{itemize} \noindent Now, one can show that ${\sf e\/}$ is the same as $\in\!\!^{(3)}$ in \cite[Def.3.1]{Simon}, i.e., \begin{itemize} \item[] $\mbox{\sf SAx\/}{\tiny{\undharom}}{\sf e\/}\leftrightarrow(\mbox{\sf q\/}\mbox{{\small{$\odot$}}}\mbox{\sf q\/}\mbox{{\small{$\odot$}}}\mbox{\sf q\/}^{{\cup\hspace*{-1.43mm} {\boldsymbol \cdot} }}\mbox{{\small{$\odot$}}}\mbox{\sf q\/}^{{\cup\hspace*{-1.43mm} {\boldsymbol \cdot} }}\land\id).$ \end{itemize} Similarly, one can show that $\pi_{(i)}, \chi_{(i)}$ are the same as the ones in \cite{Simon}, and ${\sf Triplet\/}$ is the same as $1^{(3)}$ in \cite[Def.3.1]{Simon}, e.g., \begin{itemize} \item[] $\mbox{\sf SAx\/}{\tiny{\undharom}}\pi_{(1)}\leftrightarrow({\sf e\/}\mbox{{\small{$\odot$}}}\mbox{\sf q\/}\mbox{{\small{$\odot$}}}\mbox{\sf p\/})$,\\ $\mbox{\sf SAx\/}{\tiny{\undharom}}\chi_{(1)}\leftrightarrow(\pi_{(1)}\mbox{{\small{$\odot$}}}\pi_{(1)}^{{\cup\hspace*{-1.43mm} {\boldsymbol \cdot} }})$,\\ $\mbox{\sf SAx\/}{\tiny{\undharom}}{\sf Triplet\/}\leftrightarrow(\mbox{\sf pair\/}\mbox{{\small{$\odot$}}}{\sf e\/})$. \end{itemize} From this one can show, similarly to the above, that \cite{Simon}'s ${\sf t\/}_i, \mbox{\sf d\/}_{ij},{\sf t\/}$ are the same as our $T_i, \mbox{\sf d\/}_{ij}$ and ${\sf e\/}$ in Def.\ref{ca-d} and above. Thus, our reduct ${\mathfrak Dca\/}$ is the same as the 3-reduct defined in \cite{Simon}, and then we can use \cite[Thm.3.2,Thm.5.2]{Simon}.\qquad {\bf QED}
\section{Formalizing set theory in $\mbox{$\mathcal Ld$\/}_3$}\label{set-sec}
The 3-variable restricted fragment $\mathcal{L}_3$ of $\mathcal{L}_\omega$ is defined as follows. Language: three variable symbols, $x,y,z$, one binary relational symbol $\epsilon$, so there is one atomic formula, namely $\epsilon(x,y)$. Logical connectives: $\lor, \neg, \exists x, \exists y, \exists z$ and $u=v$ for $u,v\in\{ x,y,z\}$. We denote the set of formulas by $\mbox{{\it Fm\/}}_3$. Derived connectives are $\forall, \lor, \to, \leftrightarrow$. The proof system $\tiny{\undra}$ of $\mathcal{L}_3$ is a Hilbert style one defined in \cite[Part II, p.157]{HMTII}. The word-algebra of $\mathcal{L}_3$ is denoted as $\mathfrak {Fm}_3$, it is the absolutely free $\mbox{\sf CA\/}_3$-type algebra generated by the formula $\epsilon(x,y)$. $\mbox{{\it Fm\/}}_\omega$ denotes the set of formulas of $\mathcal{L}_\omega$, and if $\mbox{{\it Fm\/}}$ is one of $\mbox{\it Fmd\/}_3, \mbox{{\it Fm\/}}_3, \mbox{{\it Fm\/}}_\omega$, then $\mbox{{\it Fm\/}}^0, \mbox{{\it Fm\/}}^1, \mbox{{\it Fm\/}}^2$ denote the set of formulas in $\mbox{{\it Fm\/}}$ with no free variables, with one free variable $x$, and with two free variables $x,y$, respectively.
In this section we prove Theorem \ref{zfc-t} stated in section \ref{fmd-sec}. As a first step, we define a translation function ${\sf h\/}$ from $\mbox{{\it Fm\/}}_3$ to $\mbox{\it Fmd\/}_3$. This will be analogous to the one defined in \cite{NDis}, but here a novelty is that the vocabulary of $\mbox{{\it Fm\/}}_3$ is different (disjoint) from that of $\mbox{\it Fmd\/}_3$, and we will have to pay attention to this difference. Namely, $\mbox{{\it Fm\/}}_3$ contains one binary relation symbol $\epsilon$ and equality $u=v$ for $u,v\in\{ x,y,z\}$ while $\mbox{\it Fmd\/}_3$ contains one ternary relation symbol $P$ and does not contain equality. The formula $\Delta$ introduced in section \ref{fmd-sec} bridges the difference between the two languages. Namely, $\Delta$ is stated in a language which contains both $\mbox{{\it Fm\/}}_3$ and $\mbox{\it Fmd\/}_3$ and $\Delta$ is a definition of $P$ in $\mbox{{\it Fm\/}}_3$, so it leads from $\mbox{{\it Fm\/}}_3$ to $\mbox{\it Fmd\/}_3$. But $\Delta$ is a two-way bridge, because of the following. Let \begin{align*} \Delta'\de [\epsilon(x,y)\leftrightarrow \forall zP(x,y,z)]\land[x=y=z\leftrightarrow (P(x,y,z)\land\lnot\forall zP(x,y,z))]\ . \end{align*} Then $\Delta'$ provides a definition of equality $=$ and $\epsilon$ in $\mbox{\it Fmd\/}_3$, and thus it is a bridge leading from $\mbox{\it Fmd\/}_3$ to $\mbox{{\it Fm\/}}_3$. Now, the two definitions are equivalent in non-trivial models, namely $$ \exists xy(x\ne y)\models \Delta\leftrightarrow \Delta'\ .$$
As usual, we begin with definitions. To start, we work in $\mbox{{\it Fm\/}}_3$. Recall the concrete definitions of $p_0, p_1$, and $\pi$ from \cite[p.71, p.35]{NDis}. Define
$\pi^+\de \pi\land\\ \phantom{\pi^+\de\quad\ \ }\forall xy[\exists z(\mbox{\sf p\/}_0\mbox{(\!(\/} x,z\mbox{)\!)\/},\mbox{\sf p\/}_0\mbox{(\!(\/} y,z\mbox{)\!)\/}),\exists z(\mbox{\sf p\/}_1\mbox{(\!(\/} x,z\mbox{)\!)\/},\mbox{\sf p\/}_1\mbox{(\!(\/} y,z\mbox{)\!)\/})\to x=y]\land\\ \phantom{\pi^+\de\quad\ \ }\forall x(\exists y\mbox{\sf p\/}_0(x,y)\leftrightarrow \exists y\mbox{\sf p\/}_1(x,y))\land\exists xy(x\ne y).$
\noindent Then $p_0, p_1, \pi, \pi^+$ are formulas of $\mbox{{\it Fm\/}}_3$. We note that the notation $\varphi\mbox{(\!(\/} u,v\mbox{)\!)\/}$ in \cite{NDis} is the same as our $\varphi\(u,v\)$ introduced in Def.\ref{vsubs-d}, except that instead of $x\!\doteq\! y$ etc.\ the notation $\varphi\mbox{(\!(\/} u,v\mbox{)\!)\/}$ uses real equality $x=y$ etc.\ available in $\mbox{{\it Fm\/}}_3$.
Next, we work in $\mbox{\it Fmd\/}_3$ and we get the versions of these formulas that the definition $\Delta$ of $P$ provides. We will write out the details. First we fix the parameters $\delta_{xy}, \delta_{xz}, p_0, p_1$ occurring in the formula $\mbox{\sf SAx\/}$ of $\mbox{\it Fmd\/}_3$.
\begin{defn}\label{E-d}(fixing the parameters $\delta_{xy}, \delta_{xz}$ of $\mbox{\it Fmd\/}_3$)\\ $E\de \forall z\,P(x,y,z)$,\\ $D\de P(x,y,z)\land\lnot E$,\\ $\delta_{xy}\de \exists zD$,\\ $\delta_{xz}\de \exists yD$.\qed \end{defn}
The next definition fixes the parameters $p_0,p_1$ of $\mbox{\it Fmd\/}_3$. To distinguish them from their $\mbox{{\it Fm\/}}_3$-versions, we will denote them by $\mbox{\sf p\/}_0, \mbox{\sf p\/}_1$. The definition below is a repetition of the definition of the corresponding formulas $p_0,p_1$ on p.71 of \cite{NDis} such that we write $E$ and the above defined concrete $\delta_{xy}, \delta_{xz}$ in place of $\epsilon$ and $x=y$, $x=z$. We will use the notation introduced in section \ref{fmd-sec} and we will use notation to support set theoretic intuition. Thus, if we introduce a formula $\varphi$ denoted as, say, $x\!\doteq\!\{ y\}$, then $u\!\doteq\!\{ v\}$ denotes the formula $\varphi\(u,v\)$ (see Definitions~\ref{vsubs-d},\ref{veq-d}). Below, ``op" abbreviates ``ordered pair" (to distinguish it from the formula $\mbox{\sf pair\/}$ defined earlier.).
\begin{defn}\label{p-d}(fixing the parameters $\mbox{\sf p\/}_0,\mbox{\sf p\/}_1$ of $\mbox{\it Fmd\/}_3$)\\ $u{\varepsilon\/} v\de E\(u,v\)$,\quad for $u,v\in\{ x,y,z\}$,\\ $x\!\doteq\!\{ y\}\de \forall z(z{\varepsilon\/} x \leftrightarrow z\!\doteq\! y)$,\\ $\{ x\}{\varepsilon\/} y\de \exists z(z\!\doteq\!\{ x\}, z{\varepsilon\/} y)$,\\ $x\!\doteq\!\{\{ y\}\}\de \exists z(z\!\doteq\!\{ y\}, x\!\doteq\!\{ z\})$,\\ $x{\varepsilon\/}\cup y\de \exists z(x{\varepsilon\/} z, z{\varepsilon\/} y)$,\\ $\mbox{\sf op\/}(x)\de \exists y\forall z(\{ z\}{\varepsilon\/} x\leftrightarrow y\!\doteq\! z)\land$\\ $\phantom{\mbox{\sf op\/}(x)\de} \forall yz[(y{\varepsilon\/}\cup x, \lnot\{ y\}{\varepsilon\/} x, z{\varepsilon\/}\cup x, \lnot\{ z\}{\varepsilon\/} x)\rightarrow y\!\doteq\! z], \forall y\exists z(y{\varepsilon\/} x\rightarrow z{\varepsilon\/} y)$,\\ $\mbox{\sf p\/}_0\de \mbox{\sf op\/}(x)\land \{ y\}{\varepsilon\/} x$,\\ $\mbox{\sf p\/}_1\de \mbox{\sf op\/}(x)\land[x\!\doteq\!\{\{ y\}\}\lor (y{\varepsilon\/}\cup x, \lnot\{ y\}{\varepsilon\/} x)]$.\qed\\ \end{defn}
It is not hard to check that \begin{align}\label{bridge} \Delta,\exists xy(x\ne y)\models\ \pi\!\leftrightarrow\mbox{\sf Ax\/},\ \pi^+\!\!\leftrightarrow\mbox{\sf SAx\/} . \end{align}
We are ready to define our translation mapping ${\sf h\/}$ from $\mbox{{\it Fm\/}}_3$ to $\mbox{\it Fmd\/}_3$. For a formula $\varphi\in\mbox{\it Fmd\/}_3^2$ and $i,j\in 2^*$ we define $$\varphi\(x_i,x_j\)\de\exists yz(y\!\doteq\! x_i, z\!\doteq\! x_j, \varphi\(y,z\)).$$
\begin{defn}\label{h-d}(translation mapping ${\sf h\/}$) \begin{itemize} \item[(i)] ${\sf h\/}': \mbox{{\it Fm\/}}_3\to \mbox{\it Fmd\/}_3^1$ is defined by the following:\\ ${\sf h\/}'$ is a homomorphism from $\mathfrak {Fm}_3$ into ${\mathfrak Dca\/}$ such that\\ ${\sf h\/}'(\epsilon(x,y))\de E\(x_{1(0)},x_{1(1)}\)$. \item[(ii)] $\mbox{\sf SAx\/}^*\de\mbox{\sf SAx\/}\land\forall x({\sf Triplet\/} x_1\to{\sf h\/}'(\pi^+))$. \item[(iii)] We define the mapping ${\sf h\/}:\mbox{{\it Fm\/}}_3\to\mbox{\it Fmd\/}_3^0$ as\\ $${\sf h\/}(\varphi)\de \forall x([\mbox{\sf SAx\/}^*\land{\sf Triplet\/} x_1]\to{\sf h\/}'(\varphi)).\qed$$ \end{itemize} \end{defn}
We say that a translation function $f$ is {\it Boolean preserving w.r.t.\ ${\tiny{\undharom}}$} iff for all sentences $\varphi,\psi\in\mbox{\it Fmd\/}_3$ we have that ${\tiny{\undharom}} f(\varphi\land\psi)\leftrightarrow(f(\varphi)\land f(\psi))$ and ${\tiny{\undharom}} f(\varphi\to\psi)\to(f(\varphi)\to f(\psi)).$
\begin{thm}\label{h-t} Let $\varphi$ be a sentence and $T$ be a set of sentences of $\mbox{{\it Fm\/}}_3$. Then the following (i)-(iii) are true. \begin{itemize} \item[(i)]
$T\cup\{\pi^+\}\models\varphi\ \Leftrightarrow\ {\sf h\/}(T){\tiny{\undharom}}{\sf h\/}\varphi$. \item[(ii)] $\pi^+\land\Delta\models \varphi\leftrightarrow{\sf h\/}(\varphi)$. \item[(iii)] ${\sf h\/}$ is Boolean preserving and $\mbox{\sf SAx\/}^*{\tiny{\undharom}}{\sf h\/}(\lnot\varphi)\to\lnot{\sf h\/}(\varphi)$. \end{itemize} \end{thm}
\noindent{\bf Proof.} \underbar{Proof of (ii):} Let $\mathfrak M=\langle M,\epsilon^{\mathfrak M}, P^{\mathfrak M}\rangle$ be a model of $\pi^+\land\Delta$. Let $V\de\{ x,y,z\}$ and $\mbox{\sf Val\/}\de {}^VM$, the set of evaluations of the variables in $\mathfrak M$. Now, by $\mathfrak M\models\pi^+\land\Delta$ we have that $p_i$ and $\mbox{\sf p\/}_i$ have the same meanings in $\mathfrak M$, and they form pairing functions. Then one can show by induction that \begin{align} \mathfrak M\models (u_i=v_j)[k]\quad\mbox{iff}\quad\mathfrak M\models (u_i\!\doteq\! v_j)[k]\quad\mbox{iff}\quad k(u)_i=k(v)_j \mbox{ in }\mathfrak M. \end{align} We say that $a\in M$ is a {\it triplet} iff $a_{11}$ is defined. Then $a_0, a_{10}$ are also defined by $\mathfrak M\models\pi^+$. If $a$ is a triplet, then we assign an evaluation $\mbox{\sf val\/}(a)\in\mbox{\sf Val\/}$ to $a$ such that $\mbox{\sf val\/}(a)$ assigns to $x,y,z$ the elements $a_{(0)}, a_{(1)}, a_{(2)}$ respectively. One can prove by induction the following statement: For all $\varphi\in\mbox{{\it Fm\/}}_3$ and $k\in\mbox{\sf Val\/}$ we have \begin{align}\label{hval} \mathfrak M\models{\sf h\/}'(\varphi)[k]\quad\mbox{iff}\quad \big(k(x)_1\mbox{ is a triplet and }\mathfrak M\models\varphi[\mbox{\sf val\/}(k(x)_1)]\big). \end{align} From (\ref{hval}) we get the following: \begin{align}\label{hcon} \mathfrak M\models \varphi\leftrightarrow \forall x({\sf Triplet\/} x_1\to{\sf h\/}'\varphi)\quad\mbox{for all }\quad \varphi\in\mbox{{\it Fm\/}}_3^0. \end{align} Indeed, assume $\mathfrak M\models\varphi$, then $\mathfrak M\models\varphi[k]$ for all $k\in \mbox{\sf Val\/}$. We show $\mathfrak M\models\forall x({\sf Triplet\/} x_1\to{\sf h\/}'\varphi)$. Indeed, let $k\in\mbox{\sf Val\/}$ be such that $k(x)_1$ is a triplet. By $\mathfrak M\models\varphi$ then $\mathfrak M\models\varphi[\mbox{\sf val\/}(k(x)_1)]$, by (\ref{hval}) then $\mathfrak M\models{\sf h\/}'\varphi[k]$ and we are done.
Assume next $\mathfrak M\mbox{$\,\not\!\models\,$}\varphi$, then $\mathfrak M\mbox{$\,\not\!\models\,$}\varphi[k]$ for all $k\in\mbox{\sf Val\/}$ because $\varphi$ is a sentence (i.e., does not contain free variables). We show $\mathfrak M\mbox{$\,\not\!\models\,$}\forall x({\sf Triplet\/} x_1\to{\sf h\/}'\varphi)$. Indeed, let $a\in M$ be such that $a_{111}$ is defined. Such an $a\in M$ exists by $\mathfrak M\models\pi$. Let $k\in\mbox{\sf Val\/}$ be such that $k(x)=a$. Then $k(x)_1$ is a triplet and $\mathfrak M\mbox{$\,\not\!\models\,$}\varphi[\mbox{\sf val\/}(k(x)_1]$, so $\mathfrak M\mbox{$\,\not\!\models\,$}{\sf h\/}'\varphi[k]$ by (\ref{hval}), so $\mathfrak M\mbox{$\,\not\!\models\,$}({\sf Triplet\/} x_1\to{\sf h\/}'\varphi)[k]$, so $\mathfrak M\mbox{$\,\not\!\models\,$}\forall x({\sf Triplet\/} x_1\to{\sf h\/}'\varphi)$. This shows that (\ref{hcon}) indeed holds.
From (\ref{hcon}) and $\mathfrak M\models\pi^+\land\Delta$ then we have $\mathfrak M\models\mbox{\sf SAx\/}^*$. This together with (\ref{hcon}) and the definition of ${\sf h\/}$ finishes the proof of (ii).
\noindent\underbar{Proof of (iii):} The proofs of (4) and (6) in \cite[p.73]{NDis}, which prove that $\kappa$ is Boolean-preserving w.r.t.\ $\tiny{\undra}$, work for showing that ${\sf h\/}$ is Boolean preserving w.r.t.\ ${\tiny{\undharom}}$, because ${\sf h\/}$ has the same ``structure" as $\kappa$. Similarly, the proof of (5) in \cite[p.73]{NDis} is good for proving the second statement of the present (iii).
\noindent\underbar{Proof of (i):} First we prove (i) for the special case when $T$ is the empty set. Let $\varphi$ be a sentence of $\mbox{{\it Fm\/}}_3$, we want to prove $\mbox{$\quad\not\!\!\!\!\!\!\vd$}{\sf h\/}(\varphi)$ implies $\pi^+\mbox{$\,\not\!\models\,$}\varphi$. So, assume that $\mbox{$\quad\not\!\!\!\!\!\!\vd$}{\sf h\/}\varphi$,\ \ i.e., $\mbox{$\quad\not\!\!\!\!\!\!\vd$}\forall x(\mbox{\sf SAx\/}^*\land{\sf Triplet\/} x_1\to{\sf h\/}'\varphi)$. Then\ \ $\mbox{\sf SAx\/}\mbox{$\quad\not\!\!\!\!\!\!\vd$}\forall x({\sf Triplet\/} x_1\land{\sf h\/}'(\pi^+)\to{\sf h\/}'(\varphi))$, \ \ i.e.,\ \ \begin{align}\label{notvd} \mbox{\sf SAx\/}\mbox{$\quad\not\!\!\!\!\!\!\vd$}{\sf h\/}'(\pi^+\!\!\to\varphi). \end{align} Recall that ${\sf h\/}'$ is a homomorphism from $\mathfrak {Fm}_3$ to ${\mathfrak Dca\/}\slash\equiv_{\mbox{\sf SAx\/}}$, and the latter is a representable $\mbox{\sf CA\/}_3$. Let $\psi\de\pi^+\!\!\to\varphi$. By (\ref{notvd}) we have that the image of $\psi$ is not $1$ under ${\sf h\/}'$, therefore there is a homomorphism $g$ from ${\mathfrak Dca\/}\slash\equiv_{\mbox{\sf SAx\/}}$ to a cylindric set algebra $\mathfrak C$ such that the image of ${\sf h\/}'\psi$ under $g$ is not $1$. Let $f\de g{\sf h\/}'$, then \begin{align}\label{notmodel} f:\mathfrak {Fm}_3\to\mathfrak C\qquad \mbox{ and }\qquad f(\psi)\neq 1. \end{align} Let $U$ be the base set of $\mathfrak C$, let $R\de \{\langle s_0, s_1\rangle : s\in f(\epsilon(x,y))\}$ and define the model $\mathfrak M$ as $\langle U,R\rangle$. Then for all $\varphi\in\mbox{{\it Fm\/}}_3$ and $s\in U^3$ we have that $\mathfrak M\models\varphi[s]$ iff $s\in f(\varphi)$. Thus $\mathfrak M\mbox{$\,\not\!\models\,$}\psi$ by (\ref{notmodel}), and we are done with showing $\pi^+\mbox{$\,\not\!\models\,$}\varphi$.
In the other direction, we have to show that ${\tiny{\undharom}}{\sf h\/}\varphi$ implies $\pi^+\models\varphi$. By soundness of the proof system ${\tiny{\undharom}}$ we have $\models{\sf h\/}\varphi$, then by (ii) we have $\pi^+\land\Delta\models\varphi$. Since $P$ does not occur in $\varphi$ and in $\pi^+$, this means that $\pi^+\models\varphi$ and we are done.
Next, assume that $T$ is a set of sentences of $\mbox{{\it Fm\/}}_3$. We want to show that $T\cup\{\pi^+\}\models\varphi$ iff ${\sf h\/}(T){\tiny{\undharom}}{\sf h\/}(\varphi)$. Now,
\noindent $T\cup\{\pi^+\}\models\varphi$ iff (by compactness of $\mbox{{\it Fm\/}}_3$)\\ $T_0\cup\{\pi^+\}\models\varphi$ for some finite $T_0\subseteq T$, iff\\ $\pi^+\models\bigwedge T_0\to\varphi$ for some finite $T_0\subseteq T$, iff(by first part of (i))\\ ${\tiny{\undharom}}{\sf h\/}(\bigwedge T_0\to\varphi)$ for some finite $T_0\subseteq T$. Then, by Boolean preserving of ${\sf h\/}$ \\ ${\tiny{\undharom}}\bigwedge{\sf h\/}(T_0)\to{\sf h\/}(\varphi)$, and so by Modus Ponens we get\\ $h(T_0){\tiny{\undharom}}{\sf h\/}(\varphi)$. Conversely, from this we get by the soundness of ${\tiny{\undharom}}$ that\\ \begin{align}\label{mod} h(T_0)\models{\sf h\/}(\varphi). \end{align}
From here on we have to deal with the difference between the two languages.
Let $\mathfrak M$ be an arbitrary model of $\mathcal{L}_3$, so $\mathfrak M$ contains one binary relation, say $\mathfrak M=\langle M,\epsilon^{\mathfrak M}\rangle$. Define $P^{\mathfrak M}$ according to the definition $\Delta$, i.e., $P^{\mathfrak M}\de\{ \langle a,a,a\rangle : a\in M\}\cup\{\langle a,b,c\rangle\in M\times M\times M : \langle a,b\rangle\in\epsilon^{\mathfrak M}\}$. Let $\mathfrak M^+\de\langle M,\epsilon^{\mathfrak M},P^{\mathfrak M}\rangle$ be the expansion of $\mathfrak M$ with this new relation, and let $\mathfrak M^-\de\langle M,P^{\mathfrak M}\rangle$ be the reduct of this expansion to the language $\mbox{$\mathcal Ld$\/}_3$.
Assume now $\mathfrak M\models T_0\cup\{\pi^+\}$. Then \\ $\mathfrak M^+\models T_0\cup\{\pi^+\land\Delta\}$. Thus by (ii) we have that\\ $\mathfrak M^+\models{\sf h\/}(T_0)$, then \\ $\mathfrak M^-\models{\sf h\/}(\varphi)$ by (\ref{mod}). Thus\\ $\mathfrak M^+\models {\sf h\/}\varphi\land\pi^+\land\Delta$, so by (ii)\\ $\mathfrak M^+\models \varphi$, and so\\ $\mathfrak M\models\varphi$ and thus, since $\mathfrak M$ was an arbitrary model of $\mathcal{L}_3$\\ ${\sf T\/}_0\cup\{\pi^+\}\models\varphi$ and we are done.\qquad{\bf QED}
We are almost done with proving Thm.\ref{zfc-t}, all we have to do is to use a connection between $\mathcal{L}_\omega$ and $\mathcal{L}_3$ established in \cite{NDis}, \cite{NPrep}.
\noindent{\bf Proof of Thm.\ref{zfc-t}:} By \cite[Lemma 3.1, p.35]{NDis}, or by \cite[Lemma 2.2, Remark 2.4, p.25, p.30]{NPrep} there is a recursive function $f:\mbox{{\it Fm\/}}_\omega^2\to\mbox{{\it Fm\/}}_3^2$ for which \begin{align}\label{preserving} \pi\models f(\varphi)\leftrightarrow\varphi,\quad f(\lnot\varphi)=\lnot f(\varphi), \quad f(\varphi\lor\psi)=f(\varphi)\lor f(\psi), \end{align} for all sentences $\varphi,\psi$ of $\mathcal{L}_\omega$. Take such an $f'$ and extend it to $\mbox{{\it Fm\/}}_\omega$ by letting $f(\varphi)\de f'(\varphi')$ where $\varphi'$ is the universal closure of $\varphi$ (if $n$ is the smallest number such that the free variables of $\varphi$ are among $v_0,...,v_n$, then $\varphi'$ is $\forall v_0\dots\forall v_n\varphi$). Then $f$ has the same properties as $f'$ and it is defined on the whole of $\mbox{{\it Fm\/}}_\omega$, not only on $\mbox{{\it Fm\/}}_\omega^2$. Define ${\sf Tr\/}:\mbox{{\it Fm\/}}_\omega\to\mbox{\it Fmd\/}_3$ by $$ {\sf Tr\/}(\varphi)\de {\sf h\/}(f(\varphi))$$ for all $\varphi\in\mbox{{\it Fm\/}}_\omega$. We show that this translation function satisfies the requirements of Thm.\ref{zfc-t}. First, ${\sf Tr\/}$ is recursive because both $f$ and ${\sf h\/}$ are such. We defined the parameters $p_0, p_1$ so that \begin{align}\label{pi} \mbox{\it ZF\/}\models\pi^+ \end{align} holds. Thus $\mbox{\it ZF\/}\models\mbox{\it ZF\/}+\pi^+\models \varphi\leftrightarrow f(\varphi)$ by the chosen properties of $f$, and then $\mbox{\it ZF\/}+\Delta\models\mbox{\it ZF\/}+\pi^+\land\Delta\models \varphi\leftrightarrow {\sf h\/} f\varphi$ by Thm.\ref{h-t}(ii), so $\mbox{\it ZF\/}+\Delta\models\varphi\leftrightarrow{\sf Tr\/}\varphi$ for all sentences $\varphi$ of $L_\omega$. This is Thm.\ref{zfc-t}(ii). To prove Thm.\ref{zfc-t}(i), first we show \begin{align}\label{f} \mbox{\it ZF\/}\models\varphi\quad\Leftrightarrow\quad f(\mbox{\it ZF\/})+\pi^+\models f(\varphi). \end{align} Indeed, assume $\mbox{\it ZF\/}\models\varphi$ and let $\mathfrak M\models f(\mbox{\it ZF\/})+\pi^+$. Then $\mathfrak M\models\mbox{\it ZF\/}$ by (\ref{preserving}), so $\mathfrak M\models\varphi+\pi^+$ by our assumption $\mbox{\it ZF\/}\models\varphi$, so $\mathfrak M\models f(\varphi)$ by (\ref{preserving}). This shows $f(\mbox{\it ZF\/})+\pi^+\models f(\varphi)$. Conversely, assume now the latter, and we want to prove $\mbox{\it ZF\/}\models\varphi$. Let $\mathfrak M\models\mbox{\it ZF\/}$, then $\mathfrak M\models\mbox{\it ZF\/}+\pi^+$ by (\ref{pi}), thus $\mathfrak M\models f(\mbox{\it ZF\/})+\pi^+$ by (\ref{preserving}), then $\mathfrak M\models f(\varphi)+\pi^+$ by our assumption, and then $\mathfrak M\models \varphi$ by (\ref{preserving}) again. We have shown that (\ref{f}) holds.
By combining this (\ref{f}) with Thm.\ref{h-t}(i) we get $\mbox{\it ZF\/}\models\varphi$ iff ${\sf Tr\/}(\mbox{\it ZF\/}){\tiny{\undharom}}{\sf Tr\/}\varphi$ which is Thm.\ref{zfc-t}(i).
Later, in section \ref{free-sec}, we will also need the following: \begin{align}\label{bool} {\sf Tr\/}\mbox{ is Boolean preserving}\quad\mbox{ and }\quad \mbox{\sf SAx\/}^*{\tiny{\undharom}}{\sf Tr\/}(\lnot\varphi)\to\lnot{\sf Tr\/}\varphi . \end{align} Indeed, this follows from Thm.\ref{h-t}(iii) and from (\ref{preserving}): Let $\varphi,\psi\in\mbox{{\it Fm\/}}_\omega^0$. Then ${\tiny{\undharom}}{\sf Tr\/}(\varphi\land\psi)$,\ \ iff by the definition of ${\sf Tr\/}$\\ ${\tiny{\undharom}}{\sf h\/} f(\varphi\land\psi)$,\ \ iff by (\ref{preserving})\\ ${\tiny{\undharom}}{\sf h\/}(f\varphi\land f\psi)$,\ \ iff by Thm.\ref{h-t}(iii)\\ ${\tiny{\undharom}}{\sf h\/} f\varphi\land{\sf h\/} f\psi$,\ \ iff by the definition of ${\sf Tr\/}$\\ ${\tiny{\undharom}}{\sf Tr\/}\varphi\land{\sf Tr\/}\psi$.
Similarly,\\ ${\tiny{\undharom}}{\sf Tr\/}(\varphi\to\psi)$,\ \ implies by the definition of ${\sf Tr\/}$\\ ${\tiny{\undharom}}{\sf h\/} f(\varphi\to\psi)$,\ \ implies by (\ref{preserving})\\ ${\tiny{\undharom}}{\sf h\/}(f\varphi\to f\psi)$,\ \ implies by Thm.\ref{h-t}(iii)\\ ${\tiny{\undharom}}{\sf h\/} f\varphi\to{\sf h\/} f\psi$,\ \ implies by the definition of ${\sf Tr\/}$\\ ${\tiny{\undharom}}{\sf Tr\/}\varphi\to{\sf Tr\/}\psi$.
Finally, $\mbox{\sf SAx\/}^*{\tiny{\undharom}}{\sf h\/}(\lnot f\varphi)\to\lnot{\sf h\/}(f\varphi)$, by Thm.\ref{h-t}(iii), so $\mbox{\sf SAx\/}^*{\tiny{\undharom}}{\sf h\/}(f(\lnot \varphi))\to\lnot{\sf h\/}(f\varphi)$ by (\ref{preserving}), and then $\mbox{\sf SAx\/}^*{\tiny{\undharom}}{\sf Tr\/}(\lnot\varphi)\to\lnot{\sf Tr\/}(\varphi)$, as was to be shown. \qquad{\bf QED(Thm.\ref{zfc-t})}
\section{Free algebras}\label{free-sec} In this section we prove that the one-generated free $\mbox{\sf Df\/}_3$ is not atomic. In algebraic logic, in the duality between algebras and logics, atomicity of the Lindenbaum-Tarski algebras correspond to G\"odel incompleteness theorem, see e.g., \cite[sec 1.4]{NPrep}, \cite{Gyenis}.
\begin{thm}\label{free-t} The one-generated free $\mbox{\sf Df\/}_3$ is not atomic. \end{thm}
\noindent{\bf Proof.} It is enough to show that the zero-dimensional part of the free $\mbox{\sf Df\/}_3$ is not atomic by \cite[1.10.3(i)]{HMTII}: \begin{align} \mathfrak {Zd}\mathfrak {Fr}_1\mbox{\sf Df\/}_3\mbox{ is not atomic implies that }\mathfrak {Fr}_1\mbox{\sf Df\/}_3\mbox{ is not atomic.} \end{align} Let us define $\equiv$ as $\equiv_{\emptyset}$, i.e. $$\varphi\equiv\psi\qquad\mbox{ iff }\qquad {\tiny{\undharom}}\varphi\leftrightarrow\psi .$$ Let $\mathfrak {Fmd}_3$ and $\mathfrak {Fmd}_3^0$ denote the word-algebra of $\mbox{\it Fmd\/}_3$ and the word-algebra of sentences of $\mbox{\it Fmd\/}_3$, respectively (the latter under the operations of $\lor, \lnot$). It is easy to show that \begin{align} \mathfrak {Fr}_1\mbox{\sf Df\/}_3\mbox{ is isomorphic to }\mathfrak {Fmd}_3\slash\equiv \mbox{ and }\\ \mathfrak {Zd}\mathfrak {Fr}_1\mbox{\sf Df\/}_3\mbox{ is isomorphic to }\mathfrak {Fmd}_3^0\slash\equiv . \end{align} There is a non-separable formula $\lambda\in\mbox{{\it Fm\/}}_\omega^0$ which is consistent with $\pi^+$ (with our concrete pairing formulas), by \cite[pp.69-71]{NDis} (or equivalently by \cite[Lemma 2.7, p.34]{NPrep}). Define $$\psi\de\mbox{\sf SAx\/}^*\land{\sf Tr\/}\lambda .$$ Then $\psi\in\mbox{\it Fmd\/}_3^0$. We will show that there is no atom below $\psi\slash\equiv\ $\ and the latter is nonzero in $\mbox{\it Fmd\/}_3^0\slash\equiv$. Assume the contrary, i.e., that \begin{align}\label{atom} \mbox{$\delta\slash\equiv$\ \ is an atom below\ \ $\psi\slash\equiv$} \end{align}
and we will derive a contradiction. Let
$$T\de\{\varphi\in\mbox{{\it Fm\/}}_\omega^0 : {\tiny{\undharom}}\delta\to{\sf Tr\/}\varphi\} .$$
Then $T\subseteq\mbox{{\it Fm\/}}_\omega^0$. We will show that $T$ is recursive
and it separates the consequences of $\lambda$ from the sentences
refutable from $\lambda$ which contradicts the choice of $\lambda$,
namely that $\lambda$ is inseparable. From now on let $\varphi\in\mbox{{\it Fm\/}}_\omega^0$ be arbitrary.
Now, $\delta\slash\equiv$ being an atom implies
\begin{align}\label{complement}
\mbox{$\quad\not\!\!\!\!\!\!\vd$}\delta\to{\sf Tr\/}\varphi\quad\Leftrightarrow\quad{\tiny{\undharom}}\delta\to\lnot{\sf Tr\/}\varphi .
\end{align}
From (\ref{complement}) we get that both $T$ and the complement of
$T$ are recursively enumerable, so $T$ is recursive. Next we show
\begin{align}\label{sep1}
\lambda\models\varphi\quad\mbox{implies}\quad\varphi\in T .
\end{align}
Indeed, assume that $\lambda\models\varphi$. Then $\models\lambda\to\varphi$.
Then, in particular, $\pi^+\models\lambda\to\varphi$, and so
${\tiny{\undharom}}{\sf Tr\/}(\lambda\to\varphi)$ by Thm.\ref{zfc-t}(i). Then
${\tiny{\undharom}}{\sf Tr\/}(\lambda)\to{\sf Tr\/}\varphi$ by (\ref{bool}). By Modus Ponens then
$\mbox{\sf SAx\/}^*+{\sf Tr\/}(\lambda){\tiny{\undharom}}{\sf Tr\/}\varphi$, i.e., $\psi{\tiny{\undharom}}{\sf Tr\/}\varphi$. By
(\ref{atom}) we have
\begin{align}\label{delta}
{\tiny{\undharom}}\delta\to\psi
\end{align}
so we have ${\tiny{\undharom}}\delta\to{\sf Tr\/}\varphi$, i.e., $\varphi\in T$ as was desired.
Next we show
\begin{align}
\lambda\models\lnot\varphi\quad\mbox{implies}\quad\varphi\notin T .
\end{align}
Indeed, assume $\lambda\models\lnot\varphi$. Then \\
$(\lnot\varphi)\in T$ by the previous case (\ref{sep1}), and this means\\
${\tiny{\undharom}}\delta\to{\sf Tr\/}(\lnot\varphi)$. Now, by (\ref{delta}) and the definition of $\psi$ we
have\\
${\tiny{\undharom}}\delta\to\mbox{\sf SAx\/}^*$, and thus by ${\tiny{\undharom}}\delta\to{\sf Tr\/}(\lnot\varphi)$ and
(\ref{bool})\\
${\tiny{\undharom}}\delta\to\lnot{\sf Tr\/}(\varphi)$. Thus by (\ref{complement}) we get\\
$\mbox{$\quad\not\!\!\!\!\!\!\vd$}\delta\to{\sf Tr\/}\varphi$, by the definition of $T$ then\\
$\varphi\notin T$.
By the above we have shown that there is no atom below
$\psi\slash\equiv$. It remains to show that the latter is nonzero.
This follows from the fact that $\lambda\land\pi^+$ has a model.
Let $\mathfrak M$ be such that $\mathfrak M\models\lambda\land\pi^+$. Expand this
model with $P^{\mathfrak M}$ so that $\mathfrak M^+\models\Delta$ also. Then
$\mathfrak M^+\models{\sf Tr\/}(\lambda)$ by Thm.\ref{zfc-t}(ii), and also
$\mathfrak M^+\models\mbox{\sf SAx\/}^*$ by $\mathfrak M^+\models\pi^+\land\Delta$. Thus
$\mathfrak M^+\models\psi$, and so $\mathfrak M^-\models\psi$ where $\mathfrak M^-$ is the
reduct of $\mathfrak M^+$ to the language of $\psi$.\qquad{\bf QED}
\noindent Hajnal Andr\'eka\ \ and\ \ Istv\'an N\'emeti\\ R\'enyi Mathematical Research Institute\\ Budapest, Re\'altanoda st.\ 13-15\\ H-1053 Hungary\\ [email protected], [email protected]
\end{document} | arXiv |
A matrix-valued generator $\mathcal{A}$ with strong boundary coupling: A critical subspace of $D((-\mathcal{A})^{\frac{1}{2}})$ and $D((-\mathcal{A}^*)^{\frac{1}{2}})$ and implications
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March 2016, 5(1): 147-184. doi: 10.3934/eect.2016.5.147
Hölder-estimates for non-autonomous parabolic problems with rough data
Hannes Meinlschmidt 1, and Joachim Rehberg 2,
Technische Universität Darmstadt, Fachbereich Mathematik, Dolivostr. 15, D-64293 Darmstadt, Germany
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany
Received March 2015 Revised February 2016 Published March 2016
In this paper we establish Hölder estimates for solutions to nonautonomous parabolic equations on non-smooth domains which are complemented with mixed boundary conditions. The corresponding elliptic operators are of divergence type, the coefficient matrix of which depends only measurably on time. These results are in the tradition of the classical book of Ladyshenskaya et al. [40], which also serves as the starting point for our investigations.
Keywords: second order parabolic equations, Hölder continuity of the solution., non-smooth data, Linear.
Mathematics Subject Classification: 35B65, 35K10, 35K1.
Citation: Hannes Meinlschmidt, Joachim Rehberg. Hölder-estimates for non-autonomous parabolic problems with rough data. Evolution Equations & Control Theory, 2016, 5 (1) : 147-184. doi: 10.3934/eect.2016.5.147
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Hannes Meinlschmidt Joachim Rehberg | CommonCrawl |
\begin{document}
\pagestyle{headings}
\title{Numerical Methods of the Maxwell-Stefan Diffusion Equations and Applications in Plasma and Particle Transport} \author{J\"urgen Geiser} \institute{Ruhr University of Bochum, \\ The Institute of Theoretical Electrical Engineering, \\ Universit\"atsstra"se 150, D-44801 Bochum, Germany \\ \email{[email protected]}} \maketitle
\begin{abstract}
In this paper, we present a model based on a local thermodynamic equilibrium, weakly ionized plasma-mixture model used for medical and technical applications in etching processes. We consider a simplified model based on the Maxwell-Stefan model, which describe multicomponent diffusive fluxes in the gas mixture. Based on additional conditions to the fluxes, we obtain an irreducible and quasi-positive diffusion matrix. Such problems results into nonlinear diffusion equations, which are more delicate to solve as standard diffusion equations with Fickian's approach. We propose explicit time-discretisation methods embedded to iterative solvers for the nonlinearities. Such a combination allows to solve the delicate nonlinear differential equations more effective. We present some first ternary component gaseous mixtures and discuss the numerical methods.
\end{abstract}
{\bf Keywords}: Maxwell-Stefan approach, Plasma model, Multi-component mixture, explicit discretization schemes, iterative schemes. \\
{\bf AMS subject classifications.} 35K25, 35K20, 74S10, 70G65.
\section{Introduction}
We are motivated to understand the gaseous mixtures of a normal pressure and room temperature plasma. The understanding of normal pressure, room temperature plasma applications is important for applications in medical and technical processes. Since many years, the increasing importance of plasma chemistry based on the multi-component plasma is a key factor to understand the gaseous mixture processes, see for low pressure plasma \cite{sene06} and for atmospheric pressure regimes \cite{tanaka2004}.
We consider a simplified Maxwell-Stefan diffusion equation to model the gaseous mixture of multicomponent Plasma. While most classical description of the diffusion goes back to the Fickian's approach, see \cite{fick1855}, we apply the modern description of the multicomponent diffusion based on the Maxwell-Stefan's approach, see \cite{maxwell1855}. The novel approach considers are more detail description of the flux and concentration, which are in deed not only proportional coupled as in the simplified Fickian's approach. Here, we deal with a inter-species force balance, which allows to model cross-effects, e.g., so-called reverse diffusion (up-hill diffusion in direction of the gradients).
Such a more detailed modeling results in irreducible and quasi-positive diffusion-matrices, which can be reduced by transforming with reductions or transforming with Perron-Frobenius theorems to solvable partial differential equations, see \cite{bothe2011}. The obtained system of nonlinear partial differential equations are delicate to solve and numerically, we have taken into account linearisation methods, e.g., iterative fix-point schemes.
The paper is outlined as follows.
In section \ref{modell} we present our mathematical model. A possible reduced model for the further approximations is derived in Section \ref{simply}. In section \ref{method}, we discuss the underlying numerical schemes. The first numerical results are presented in Section \ref{exper}. In the contents, that are given in Section \ref{concl}, we summarize our results.
\section{Mathematical Model} \label{modell}
For the full plasma model, we assumes that the neutral particles can be described as fluid dynamical model, where the elastic collision define the dynamics and few inelastic collisions are, among other reasons, responsible for the chemical reactions.
To describe the individual mass densities, as well as the global momentum and the global energy as the dynamical conservation quantities of the system, corresponding conservation equations are derived from Boltzmann equations.
The individual character of each species is considered by mass-conservation equations and the so-called difference equations.
The extension of the non-mixtured multicomponent transport model, \cite{sene06} is done with respect to the collision integrals related to the right-hind side sources of the conservation laws.
The conservation laws of the neutral elements are given as
\begin{eqnarray*}
\frac{\partial}{\partial t} \rho_s + \frac{\partial}{\partial \vec r} \cdot \rho_s \vec u_s &=& m_s Q_n^{(s)}, \\
\frac{\partial}{\partial t} \rho \vec u + \frac{\partial}{\partial \vec r} \cdot \left(\uu{P}^* + \rho \vec u \vec u \right) &=& - Q_m^{(e)}, \\
\frac{\partial}{\partial t} \mathcal{E}_{\ttt{tot}}^*
+ \frac{\partial}{\partial \vec r} \cdot \left(\mathcal{E}_{\ttt{tot}}^* \vec u + \vec q^* + \uu{P}^* \cdot \vec u \right)
&=& - Q_{\mathcal{E}}^{(e)}. \end{eqnarray*}
where $\rho_s$ : density of species $i$, $\rho = \sum_{i=1}^N \rho_i$, $\vec u$ : velocity, $\mathcal{E}_{\ttt{tot}}^*$ : total energy of the neutral particles. \\
Further the variable $Q_n^{(s)}$ is the collision term of the mass conservation equation, $Q_m^{(e)}$ is the collision term of the momentum conservation equation and $ Q_{\mathcal{E}}^{(e)}$ is the collision term of the energy conservation equation.
We derive the collision term with respect to the Chapmen-Enskog method, see \cite{chapman1990}, and achieve for the first derivatives the following results:
\begin{eqnarray} && m_s Q_n^{(s)} = - \nabla \cdot (\rho_i \sum_{j=0} {\bf V}_i^{j} ), \\ && Q_m^{(e)} = - \sum_{i=1}^{n_s} \rho_i F_i \\ && Q_{\mathcal{E}}^{(e)} = - \sum_{i=1}^{n_s} \rho_i \rho F_i ({\bf u} + \sum_{j=0} V_i^{(j)}) , \end{eqnarray}
where $i = 1, \ldots, n_s$, $F_i$ is an external force per unit mass (see Boltzmann equation), further the diffusion velocity is given as:
\begin{eqnarray} && {\bf V}_i^{0} = 0 \\ && {\bf V}_i^{1} = - \sum_{j=1}^N D_{ij} (d_j + k_{T_j} \frac{\Delta T}{T} ) , \end{eqnarray}
where $\sum_{i=1}^N d_i = 0$,
\begin{eqnarray} d^∗_i = \nabla x_i + x_i \frac{\nabla p}{p} - \frac{\rho_i}{\rho} F_i , \\ d_i = d^∗_i - y_i \sum_{j} d_j^* , \end{eqnarray}
where $x_i = \frac{n_s}{n}$ is the molar fraction of species $i$.
We have an additional constraint based on the mass fraction of each species:
\begin{eqnarray} && \frac{\partial}{\partial t} y_i + \nabla y_i = R_i(y_1, \ldots, y_N) , \end{eqnarray}
where $y_i$ is the mass fraction of species $i$, $R_i$ is the net production rate of species $i$ due to his reactions.
\begin{remark} The full model problem consider a full coupled system of conservation laws and Maxwell-Stefan equations. Each equations are coupled such that the gaseous mixture influences the transport equations and vice verse. In the following, we decouple the equations system and consider only the delicate Maxwell-Stefan equations. \end{remark}
\section{Simplified Model with Maxwell-Stefan Diffusion Equations} \label{simply}
We discuss in the following a multicomponent gaseous mixture with three species (ternary mixture). The model-problem is discussed in the experiments of Duncan and Toor, see \cite{duncan1962}.
Here, they studied an ideal gaseous mixture of the following components:
\begin{itemize} \item Hydrogen ($H_2$, first species), \item Nitrogen ($N_2$, second species), \item Carbon dioxide ($CO_2$, third species). \end{itemize}
The Maxwell-Stefan equations are given for the three species as (see also \cite{boudin2012}):
\begin{eqnarray} \label{part_1_eq_1} && \partial_t \xi_i + \nabla \cdot N_i = 0 , \; 1 \le i \le 3 , \\ \label{cond_1} && \sum_{j=1}^3 N_j = 0 , \\ \label{part_1_2} && \frac{\xi_2 N_1 - \xi_1 N_2}{D_{12}} + \frac{\xi_3 N_1 - \xi_1 N_3}{D_{13}} = - \nabla \xi_1 , \\ \label{part_1_3}
&& \frac{\xi_1 N_2 - \xi_2 N_1}{D_{12}} + \frac{\xi_3 N_2 - \xi_2 N_3}{D_{23}} = - \nabla \xi_2 , \end{eqnarray}
where the domain is given as $\Omega \in {\rm I}\! {\rm R}^d, d \in {I\!\!N}^+$ with $\xi_i \in C^2$.
For such ternary mixture, we can rewrite the three differential equations (\ref{part_1_eq_1}) and (\ref{part_1_2})-(\ref{part_1_3}) with the help of the zero-condition (\ref{cond_1}) into two differential equations, given as:
\begin{eqnarray} \label{part_2} && \partial_t \xi_i + \nabla \cdot N_i = 0 , \; 1 \le i \le 2 , \\ \label{part_2_1} && \frac{1}{D_{13}} N_1 + \alpha N_1 \xi_2 - \alpha N_2 \xi_1 = - \nabla \xi_1 , \\ \label{part_2_2}
&& \frac{1}{D_{23}} N_2 - \beta N_1 \xi_2 + \beta N_2 \xi_1 = - \nabla \xi_2 , \end{eqnarray}
where $\alpha = \left(\frac{1}{D_{12}} - \frac{1}{D_{13}}\right)$, $\beta = \left(\frac{1}{D_{12}} - \frac{1}{D_{23}}\right)$.
Further we have the relations:
\begin{itemize} \item Third mole-fraction: $\xi_3 = 1 - \xi_1 - \xi_2$, \\ \item Third molar flux: $N_3 = - N_1 - N_2$. \end{itemize}
\section{Numerical Methods} \label{method}
In the following, we discuss the numerical methods, which are based on iterative schemes with embedded explicit discretization schemes. We apply the following methods:
\begin{itemize} \item Iterative Scheme in time (Global Linearisation, Matrix Method), \item Iterative Scheme in Time (Local Linearisation with Richardson's Method). \end{itemize}
For the spatial discretization, we apply finite volume or finite difference methods. The underlying time-discretization is based on a first order explicit Euler method.
\subsection{Iterative Scheme in time (Global Linearisation, Matrix Method)}
We solve the iterative scheme:
\begin{eqnarray} \label{ord_0} && \xi_{1}^{n+1} = \xi_1^n - \Delta t \; D_+ N_{1}^n , \\ && \xi_{2}^{n+1} = \xi_2^n - \Delta t \; D_+ N_{2}^n , \\ && \left( \begin{array}{c c} A & B \\ C & D \end{array} \right)
\left( \begin{array}{l} N_1^{n+1} \\ N_2^{n+1} \end{array} \right) =
\left( \begin{array}{l}
- D_- \xi_1^{n+1} \\
- D_- \xi_2^{n+1} \end{array} \right) \end{eqnarray}
for $j = 0, \ldots, J$ , where $\xi_1^n = (\xi_{1,0}^n, \ldots, \xi_{1, J}^n)^T$, $\xi_2^n = (\xi_{2,0}^n, \ldots, \xi_{2, J}^n)^T$ and $I_J \in {\rm I}\! {\rm R}^{J+1} \times {\rm I}\! {\rm R}^{J+1}$,
$N_1^n = (N_{1,0}^n, \ldots, N_{1, J}^n)^T$, $N_2^n = (N_{2,0}^n, \ldots, N_{2, J}^n)^T$ and $I_J \in {\rm I}\! {\rm R}^{J+1} \times {\rm I}\! {\rm R}^{J+1}$, where $n=0,1,2, \ldots, N_{end}$ and $N_{end}$ are the number of time-steps, i.d. $N_{end} = T / \Delta t$.
The matrices are given as:
\begin{eqnarray} && A, B, C, D \in {\rm I}\! {\rm R}^{J+1} \times {\rm I}\! {\rm R}^{J+1}, \\ && A_{j,j} = \frac{1}{D_{13}} + \alpha \xi_{2,j} , \; j = 0 \ldots, J ,\\ && B_{j,j} = - \alpha \xi_{1,j} , \; j = 0 \ldots, J , \\ && C_{j,j} = - \beta \xi_{2, j} , \; j = 0 \ldots, J , \\ && D_{j,j} = \frac{1}{D_{23}} + \beta \xi_{1,j} , \; j = 0 \ldots, J ,\\ && A_{i,j} = B_{i,j} = C_{i,j} = D_{i,j} = 0 , \; i,j = 0 \ldots, J, \; i \neq J , \end{eqnarray}
means the diagonal entries given as for the scale case in equation (\ref{part_2}) and the outer-diagonal entries are zero. \\ The explicit form with the time-discretization is given as:
\begin{algorithm}
1.) Initialisation $n=0$:
\begin{eqnarray} \label{ord_0} && \left( \begin{array}{l} N_1^{0} \\ N_2^{0} \end{array} \right) =
\left( \begin{array}{c c} \tilde{A} & \tilde{B} \\ \tilde{C} & \tilde{D} \end{array} \right)
\left( \begin{array}{l}
- D_- \xi_1^{0} \\
- D_- \xi_2^{0} \end{array} \right) \end{eqnarray}
where $\xi_1^{0} = (\xi_{1,0}^{0}, \ldots, \xi_{1, J}^{0})^T$, $\xi_2^0 = (\xi_{2,0}^0, \ldots, \xi_{2, J}^0)^T$ and $\xi_{1,j}^{0} = \xi_1^{in}(j \Delta x), \; \xi_{2,j}^{0} = \xi_2^{in}(j \Delta x)$, $j = 0, \ldots, J$ and given as for the different intialisations, we have:
\begin{enumerate} \item Uphill example
\begin{eqnarray} \label{init} && \xi_1^{in}(x) = \left\{ \begin{array}{l l} 0.8 & \mbox{if} \; 0 \le x < 0.25 , \\ 1.6 (0.75 - x) & \mbox{if} \; 0.25 \le x < 0.75 , \\ 0.0 & \mbox{if} \; 0.75 \le x \le 1.0 , \end{array} \right. , \\ && \xi_2^{in}(x) = 0.2 , \; \mbox{for all} \; x \in \Omega = [0,1] , \end{eqnarray}
\item Diffusion example (Asymptotic behavior)
\begin{eqnarray} \label{init} && \xi_1^{in}(x) = \left\{ \begin{array}{l l} 0.8 & \mbox{if} \; 0 \le x \in 0.5 , \\ 0.0 & \mbox{else} , \end{array} \right. , \\ && \xi_2^{in}(x) = 0.2 , \; \mbox{for all} \; x \in \Omega = [0,1] , \end{eqnarray} \end{enumerate}
The inverse matrices are given as:
\begin{eqnarray} && \tilde{A}, \tilde{B}, \tilde{C}, \tilde{D} \in {\rm I}\! {\rm R}^{J+1} \times {\rm I}\! {\rm R}^{J+1}, \\ && \tilde{A}_{j,j} = \gamma_j (\frac{1}{D_{23}} + \beta \xi_{1,j}^{0}) , \; j = 0 \ldots, J ,\\ && B_{j,j} = \gamma_j \; \alpha \xi_{1,j}^{0} , \; j = 0 \ldots, J , \\ && C_{j,j} = \gamma_j \; \beta \xi_{2, j}^{0} , \; j = 0 \ldots, J , \\ && D_{j,j} = \gamma_j (\frac{1}{D_{13}} + \alpha \xi_{2,j}^{0}) , \; j = 0 \ldots, J ,\\ && \gamma_j = \frac{D_{13} D_{23}}{1 + \alpha D_{13} \xi_{2,j}^{0} + \beta D_{23} \xi_{1,j}^{0}} , \; j = 0 \ldots, J , \\ && \tilde{A}_{i,j} = \tilde{B}_{i,j} = \tilde{C}_{i,j} = \tilde{D}_{i,j} = 0 , \; i,j = 0 \ldots, J, \; i \neq J , \end{eqnarray}
Further the values of the first and the last grid points of $N$ are zero, means $N_{1,0}^{0} = N_{1,J}^{0} = N_{2,0}^{0} = N_{2,J}^{0} = 0$ (boundary condition).
2.) Next time-steps (till $n = N_{end}$ ): \\
2.1) Computation of $\xi_1^{n+1}$ and $\xi_2^{n+1}$
\begin{eqnarray} \label{ord_0} && \xi_{1}^{n+1} = \xi_1^n - \Delta t \; D_+ N_{1}^n , \\ && \xi_{2}^{n+1} = \xi_2^n - \Delta t \; D_+ N_{2}^n , \end{eqnarray}
2.2) Computation of $N_1^{n+1}$ and $N_2^{n+1}$
\begin{eqnarray} && \left( \begin{array}{l} N_1^{n+1} \\ N_2^{n+1} \end{array} \right) =
\left( \begin{array}{c c} \tilde{A} & \tilde{B} \\ \tilde{C} & \tilde{D} \end{array} \right)
\left( \begin{array}{l}
- D_- \xi_1^{n+1} \\
- D_- \xi_2^{n+1} \end{array} \right) \end{eqnarray}
where $\xi_1^{n} = (\xi_{1,0}^{n}, \ldots, \xi_{1, J}^{n})^T$, $\xi_2^n = (\xi_{2,0}^n, \ldots, \xi_{2, J}^n)^T$ and the inverse matrices are given as:
\begin{eqnarray} && \tilde{A}, \tilde{B}, \tilde{C}, \tilde{D} \in {\rm I}\! {\rm R}^{J+1} \times {\rm I}\! {\rm R}^{J+1}, \\ && \tilde{A}_{j,j} = \gamma_j (\frac{1}{D_{23}} + \beta \xi_{1,j}^{n+1}) , \; j = 0 \ldots, J ,\\ && B_{j,j} = \gamma_j \; \alpha \xi_{1,j}^{n+1} , \; j = 0 \ldots, J , \\ && C_{j,j} = \gamma_j \; \beta \xi_{2, j}^{n+1} , \; j = 0 \ldots, J , \\ && D_{j,j} = \gamma_j (\frac{1}{D_{13}} + \alpha \xi_{2,j}^{n+1}) , \; j = 0 \ldots, J ,\\ && \gamma_j = \frac{D_{13} D_{23}}{1 + \alpha D_{13} \xi_{2,j}^{n+1} + \beta D_{23} \xi_{1,j}^{n+1}} , \; j = 0 \ldots, J , \\ && \tilde{A}_{i,j} = \tilde{B}_{i,j} = \tilde{C}_{i,j} = \tilde{D}_{i,j} = 0 , \; i,j = 0 \ldots, J, \; i \neq J . \end{eqnarray}
Further the values of the first and the last grid points of $N$ are zero, means $N_{1,0}^{n} = N_{1,J}^{n} = N_{2,0}^{n} = N_{2,J}^{n} = 0$ (boundary condition).
3.) Do $n = n+1$ and goto 2.)
\end{algorithm}
\subsection{Iterative Scheme in Time (Local Linearisation with Richardson's Method}
We solve the iterative scheme given in the Richardson iterative scheme:
\begin{eqnarray} \label{ord_0} && \xi_{1}^{n+1, k} = \xi_1^n - \Delta t \; D_+ N_{1}^{n+1} , \\ && \xi_{2}^{n+1, k} = \xi_2^n - \Delta t \; D_+ N_{2}^{n+1} , \\ && \left( \begin{array}{c c} A^{n+1, k-1} & B^{n+1, k-1} \\ C^{n+1, k-1} & D^{n+1, k-1} \end{array} \right)
\left( \begin{array}{l} N_1^{n+1} \\ N_2^{n+1} \end{array} \right) =
\left( \begin{array}{l}
- D_- \xi_1^{n+1, k-1} \\
- D_- \xi_2^{n+1, k-1} \end{array} \right) \end{eqnarray}
for $j = 0, \ldots, J$ , where $\xi_1^n = (\xi_{1,0}^n, \ldots, \xi_{1, J}^n)^T$, $\xi_2^n = (\xi_{2,0}^n, \ldots, \xi_{2, J}^n)^T$ and $I_J \in {\rm I}\! {\rm R}^{J+1} \times {\rm I}\! {\rm R}^{J+1}$,
$N_1^n = (N_{1,0}^n, \ldots, N_{1, J}^n)^T$, $N_2^n = (N_{2,0}^n, \ldots, N_{2, J}^n)^T$ and $I_J \in {\rm I}\! {\rm R}^{J+1} \times {\rm I}\! {\rm R}^{J+1}$, where $n=0,1,2, \ldots, N_{end}$ and $N_{end}$ are the number of time-steps, i.d. $N_{end} = T / \Delta t$.
Further $k = 1, 2, \ldots, K$ is the iteration index with \\
where $\xi_1^{n+1, 0} = (\xi_{1,0}^n, \ldots, \xi_{1, J}^n)^T$, $\xi_2^{n+1, 0} = (\xi_{2,0}^n, \ldots, \xi_{2, J}^n)^T$ and $I_J \in {\rm I}\! {\rm R}^{J+1} \times {\rm I}\! {\rm R}^{J+1}$ is the start solution given with the solution at $t = t^n$.
The matrices are given as:
\begin{eqnarray} && A^{n+1, k-1}, B^{n+1, k-1}, C^{n+1, k-1}, D^{n+1, k-1} \in {\rm I}\! {\rm R}^{J+1} \times {\rm I}\! {\rm R}^{J+1}, \\ && A_{j,j}^{n+1, k-1} = \frac{1}{D_{13}} + \alpha \xi_{2,j}^{n+1, k-1} , \; j = 0 \ldots, J ,\\ && B_{j,j}^{n+1, k-1} = - \alpha \xi_{1,j}^{n+1, k-1} , \; j = 0 \ldots, J , \\ && C_{j,j}^{n+1, k-1} = - \beta \xi_{2, j}^{n+1, k-1} , \; j = 0 \ldots, J , \\ && D_{j,j}^{n+1, k-1} = \frac{1}{D_{23}} + \beta \xi_{1,j}^{n+1, k-1} , \; j = 0 \ldots, J ,\\ && A_{i,j}^{n+1, i-1} = B_{i,j}^{n+1, i-1} = C_{i,j}^{n+1, i-1} = D_{i,j}^{n+1, i-1} = 0 , \; i,j = 0 \ldots, J, \; i \neq J , \end{eqnarray}
means the diagonal entries given as for the scale case in equation (\ref{part_2}) and the outer-diagonal entries are zero. \\ The explicit form with the time-discretization is given as:
\begin{algorithm}
1.) Initialisation $n=0$ with an explicit time-step (CFL condition is given):
\begin{eqnarray} \label{ord_0} && \left( \begin{array}{l} N_1^{0} \\ N_2^{0} \end{array} \right) =
\left( \begin{array}{c c} \tilde{A} & \tilde{B} \\ \tilde{C} & \tilde{D} \end{array} \right)
\left( \begin{array}{l}
- D_- \xi_1^{0} \\
- D_- \xi_2^{0} \end{array} \right) \end{eqnarray}
where $\xi_1^{0} = (\xi_{1,0}^{0}, \ldots, \xi_{1, J}^{0})^T$, $\xi_2^0 = (\xi_{2,0}^0, \ldots, \xi_{2, J}^0)^T$ and $\xi_{1,j}^{0} = \xi_1^{in}(j \Delta x), \; \xi_{2,j}^{0} = \xi_2^{in}(j \Delta x)$, $j = 0, \ldots, J$ and given as for the different intialisations, we have:
\begin{enumerate} \item Uphill example
\begin{eqnarray} \label{init} && \xi_1^{in}(x) = \left\{ \begin{array}{l l} 0.8 & \mbox{if} \; 0 \le x < 0.25 , \\ 1.6 (0.75 - x) & \mbox{if} \; 0.25 \le x < 0.75 , \\ 0.0 & \mbox{if} \; 0.75 \le x \le 1.0 , \end{array} \right. , \\ && \xi_2^{in}(x) = 0.2 , \; \mbox{for all} \; x \in \Omega = [0,1] , \end{eqnarray}
\item Diffusion example (Asymptotic behavior)
\begin{eqnarray} \label{init} && \xi_1^{in}(x) = \left\{ \begin{array}{l l} 0.8 & \mbox{if} \; 0 \le x \in 0.5 , \\ 0.0 & \mbox{else} , \end{array} \right. , \\ && \xi_2^{in}(x) = 0.2 , \; \mbox{for all} \; x \in \Omega = [0,1] , \end{eqnarray} \end{enumerate}
The inverse matrices are given as:
\begin{eqnarray} && \tilde{A}, \tilde{B}, \tilde{C}, \tilde{D} \in {\rm I}\! {\rm R}^{J+1} \times {\rm I}\! {\rm R}^{J+1}, \\ && \tilde{A}_{j,j} = \gamma_j (\frac{1}{D_{23}} + \beta \xi_{1,j}^{0}) , \; j = 0 \ldots, J ,\\ && B_{j,j} = \gamma_j \; \alpha \xi_{1,j}^{0} , \; j = 0 \ldots, J , \\ && C_{j,j} = \gamma_j \; \beta \xi_{2, j}^{0} , \; j = 0 \ldots, J , \\ && D_{j,j} = \gamma_j (\frac{1}{D_{13}} + \alpha \xi_{2,j}^{0}) , \; j = 0 \ldots, J ,\\ && \gamma_j = \frac{D_{13} D_{23}}{1 + \alpha D_{13} \xi_{2,j}^{0} + \beta D_{23} \xi_{1,j}^{0}} , \; j = 0 \ldots, J , \\ && \tilde{A}_{i,j} = \tilde{B}_{i,j} = \tilde{C}_{i,j} = \tilde{D}_{i,j} = 0 , \; i,j = 0 \ldots, J, \; i \neq J , \end{eqnarray}
Further the values of the first and the last grid points of $N$ are zero, means $N_{1,0}^{0} = N_{1,J}^{0} = N_{2,0}^{0} = N_{2,J}^{0} = 0$ (boundary condition).
2.) Next timesteps (till $n = N_{end}$ ) (iterative scheme restricted via the CFL condition based on the previous iterative solutions in the matrices): \\
2.1) Computation of $\xi_1^{n+1, I}$ and $\xi_2^{n+1, I}$
\begin{eqnarray} \label{ord_0} && \xi_{1}^{n+1, k} = \xi_1^n - \Delta t \; D_+ N_{1}^{n+1} , \\ && \xi_{2}^{n+1, k} = \xi_2^n - \Delta t \; D_+ N_{2}^{n+1} , \end{eqnarray}
2.2) Computation of $N_1^{n+1, k-1}$ and $N_2^{n+1, k-1}$
\begin{eqnarray} && \left( \begin{array}{l} N_1^{n+1} \\ N_2^{n+1} \end{array} \right) =
\left( \begin{array}{c c} \tilde{A}^{n+1, k-1} & \tilde{B}^{n+1, k-1} \\ \tilde{C}^{n+1, k-1} & \tilde{D}^{n+1, k-1} \end{array} \right)
\left( \begin{array}{l}
- D_- \xi_1^{n+1, k-1} \\
- D_- \xi_2^{n+1, k-1} \end{array} \right) \end{eqnarray}
where $\xi_1^{n} = (\xi_{1,0}^{n}, \ldots, \xi_{1, J}^{n})^T$, $\xi_2^n = (\xi_{2,0}^n, \ldots, \xi_{2, J}^n)^T$ and the inverse matrices are given as:
\begin{eqnarray} && \tilde{A}^{n+1, k-1}, \tilde{B}^{n+1, k-1}, \tilde{C}^{n+1, k-1}, \tilde{D}^{n+1, k-1} \in {\rm I}\! {\rm R}^{J+1} \times {\rm I}\! {\rm R}^{J+1}, \\ && \tilde{A}_{j,j}^{n+1, k-1} = \gamma_j (\frac{1}{D_{23}} + \beta \xi_{1,j}^{n+1, k-1}) , \; j = 0 \ldots, J ,\\ && B_{j,j}^{n+1, k-1} = \gamma_j \; \alpha \xi_{1,j}^{n+1, k-1} , \; j = 0 \ldots, J , \\ && C_{j,j}^{n+1, k-1} = \gamma_j \; \beta \xi_{2, j}^{n+1, k-1} , \; j = 0 \ldots, J , \\ && D_{j,j}^{n+1, k-1} = \gamma_j (\frac{1}{D_{13}} + \alpha \xi_{2,j}^{n+1, k-1}) , \; j = 0 \ldots, J ,\\ && \gamma_j = \frac{D_{13} D_{23}}{1 + \alpha D_{13} \xi_{2,j}^{n+1, k-1} + \beta D_{23} \xi_{1,j}^{n+1, k-1}} , \; j = 0 \ldots, J , \\ && \tilde{A}_{i,j}^{n+1, k-1} = \tilde{B}_{i,j}^{n+1, k-1} = \tilde{C}_{i,j}^{n+1, k-1} = \tilde{D}_{i,j}^{n+1, k-1} = 0 , \; i,j = 0 \ldots, J, \; i \neq J . \end{eqnarray}
Further the values of the first and the last grid points of $N$ are zero, means $N_{1,0}^{n+1} = N_{1,J}^{n+1} = N_{2,0}^{n+1} = N_{2,J}^{n+1} = 0$ (boundary condition).
Further $k = 1, 2, \ldots, K$ is the iteration index with \\
where $\xi_1^{n+1, 0} = (\xi_{1,0}^n, \ldots, \xi_{1, J}^n)^T$, $\xi_2^{n+1, 0} = (\xi_{2,0}^n, \ldots, \xi_{2, J}^n)^T$ and $I_J \in {\rm I}\! {\rm R}^{J+1} \times {\rm I}\! {\rm R}^{J+1}$ is the start solution given with the solution at $t = t^n$.
3.) Do $n = n+1$ and goto 2.)
\end{algorithm}
\section{Numerical Experiments} \label{exper}
In the following, we concentrate on the following three component system, which is given as:
\begin{eqnarray} \label{ord_0} && \partial_t \xi_i + \partial_x N_i = 0 , \; 1 \le i \le 3 , \\ && \sum_{j=1}^3 N_j = 0 , \\ && \frac{\xi_2 N_1 - \xi_1 N_2}{D_{12}} + \frac{\xi_3 N_1 - \xi_1 N_3}{D_{13}} = - \partial_x \xi_1 , \\
&& \frac{\xi_1 N_2 - \xi_2 N_1}{D_{12}} + \frac{\xi_3 N_2 - \xi_2 N_3}{D_{23}} = - \partial_x \xi_2 , \end{eqnarray}
where the domain is given as $\Omega \in {\rm I}\! {\rm R}^d, d \in {I\!\!N}^+$ with $\xi_i \in C^2$.
The parameters and the initial and boundary conditions are given as:
\begin{itemize} \item $D_{12} = D_{13} = 0.833$ (means $\alpha = 0$) and $D_{23} = 0.168$ (Uphill diffusion, semi-degenerated Duncan and Toor experiment), \item $D_{12} = 0.0833, D_{13} = 0.680$ and $D_{23} = 0.168$ (asymptotic behavior, Duncan and Toor experiment, see \cite{duncan1962}), \item $J = 140$ (spatial grid points), \item The time-step-restriction for the explicit method is given as: \\
$\Delta t \le \frac{(\Delta x)^2}{2 \max\{D_{12}, D_{13}, D_{23}\}}$, \item The spatial domain is $\Omega = [0, 1]$, the time-domain $[0, T] = [0, 1]$, \item The initial conditions are:
\begin{enumerate} \item Uphill example
\begin{eqnarray} \label{init} && \xi_1^{in}(x) = \left\{ \begin{array}{l l} 0.8 & \mbox{if} \; 0 \le x < 0.25 , \\ 1.6 (0.75 - x) & \mbox{if} \; 0.25 \le x < 0.75 , \\ 0.0 & \mbox{if} \; 0.75 \le x \le 1.0 , \end{array} \right. , \\ && \xi_2^{in}(x) = 0.2 , \; \mbox{for all} \; x \in \Omega = [0,1] , \end{eqnarray}
\item Diffusion example (Asymptotic behavior)
\begin{eqnarray} \label{init} && \xi_1^{in}(x) = \left\{ \begin{array}{l l} 0.8 & \mbox{if} \; 0 \le x \in 0.5 , \\ 0.0 & \mbox{else} , \end{array} \right. , \\ && \xi_2^{in}(x) = 0.2 , \; \mbox{for all} \; x \in \Omega = [0,1] . \end{eqnarray} \end{enumerate}
\item The boundary conditions are of no-flux type:
\begin{eqnarray} \label{init} && N_1 = N_2 = N_3 = 0 , \mbox{on} \; \partial \Omega \times [0,1] . \end{eqnarray}
\end{itemize}
We could reduce to a simpler model problem as:
\begin{eqnarray} \label{ord_0} && \partial_t \xi_i + \partial_x \cdot N_i = 0 , \; 1 \le i \le 2 , \\ && \frac{1}{D_{13}} N_1 + \alpha N_1 \xi_2 - \alpha N_2 \xi_1 = - \partial_x \xi_1 , \\
&& \frac{1}{D_{23}} N_2 - \beta N_1 \xi_2 + \beta N_2 \xi_1 = - \partial_x \xi_2 , \end{eqnarray}
where $\alpha = \left(\frac{1}{D_{12}} - \frac{1}{D_{13}}\right)$, $\beta = \left(\frac{1}{D_{12}} - \frac{1}{D_{23}}\right)$.
We rewrite into:
\begin{eqnarray} \label{ord_0} && \partial_t \xi_1 + \partial_x \cdot N_1 = 0 , \\ && \partial_t \xi_2 + \partial_x \cdot N_2 = 0 , \\ && \left( \begin{array}{c c} \frac{1}{D_{13}} + \alpha \xi_2 & - \alpha \xi_1 \\
- \beta \xi_2 & \frac{1}{D_{23}} + \beta \xi_1 \end{array} \right)
\left( \begin{array}{l} N_1 \\ N_2 \end{array} \right) =
\left( \begin{array}{l}
- \partial_x \xi_1 \\
- \partial_x \xi_2 \end{array} \right) \end{eqnarray}
and we have
\begin{eqnarray} \label{part_0} && \partial_t \xi_1 + \partial_x \cdot N_1 = 0 , \\ \label{part_1} && \partial_t \xi_2 + \partial_x \cdot N_2 = 0 , \\ \label{part_2} && \left( \begin{array}{l} N_1 \\ N_2 \end{array} \right) = \frac{D_{13} D_{23}}{1 + \alpha D_{13} \xi_2 + \beta D_{23} \xi_1}
\left( \begin{array}{c c} \frac{1}{D_{23}} + \beta \xi_1 & \alpha \xi_1 \\
\beta \xi_2 & \frac{1}{D_{13}} + \alpha \xi_2 \end{array} \right)
\left( \begin{array}{l}
- \partial_x \xi_1 \\
- \partial_x \xi_2 \end{array} \right) . \end{eqnarray}
The next step is to apply the semi-discretization of the partial differential operator $\frac{\partial}{\partial x}$.
We apply the first differential operator in equation (\ref{part_0}) and (\ref{part_1}) as an forward upwind scheme given as
\begin{eqnarray} \frac{\partial}{\partial x} & = & D_+ = \frac{1}{\Delta x}\cdot \left(\begin{array}{rrrrr}
-1 & 0 & \ldots & ~ & 0 \\
1 & -1 & 0 & \ldots & 0 \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
0 & ~ & 1 & -1 & 0 \\
0 & \ldots & 0 & 1 & -1 \end{array}\right)~\in~{\rm I}\! {\rm R}^{(J+1) \times (J+1)} \end{eqnarray}
and the second differential operator in equation (\ref{part_2}) as an backward upwind scheme given as
\begin{eqnarray} \frac{\partial}{\partial x} & = & D_- = \frac{1}{\Delta x}\cdot \left(\begin{array}{rrrrr}
-1 & 1 & 0 & \ldots & 0 \\
0 & -1 & 1 & 0 & \ldots \\
\vdots & \ddots & \ddots & \ddots & \ddots \\
0 & \ldots & 0 & -1 & 1 \\
0 & ~ & \ldots & 0 & -1 \end{array}\right)~\in~{\rm I}\! {\rm R}^{(J+1) \times (J+1)} . \end{eqnarray}
\subsection{Experiments with the Iterative scheme in time (Global Linearisation)}
In the first experiments, we test the first iterative scheme (iterative scheme in time (Global Linearisation)).
We test the different schemes and obtain the results shown in Figure \ref{multi_1}.
\begin{figure}
\caption{ The figures present the results of the concentration $c_1$, $c_2$ and $c_3$.}
\label{multi_1}
\end{figure}
The concentration and their fluxes are given in Figure \ref{multi_2}.
\begin{figure}
\caption{ The upper figures present the results of the concentration $c_1$ and $- \partial_x \xi_1$. The lower figures presents the results of $c_2$ and $- \partial_x \xi_2$.}
\label{multi_2}
\end{figure}
The full plots in time and space of the concentrations and their fluxes are given in Figure \ref{multi_3}.
\begin{figure}
\caption{ The figures present the results of the 3d plots in time and space. The upper figures present the results of the concentration $c_1$ and $- \partial_x \xi_1$. The lower figures presents the results of $c_2$ and $- \partial_x \xi_2$.}
\label{multi_3}
\end{figure}
The space-time regions where $- N2 \partial_x \xi_2 \ge 0$ for the uphill diffusion and asymptotic diffusion, given in Figure \ref{multi_4}.
\begin{figure}
\caption{ The figures present the asymptotic diffusion (left hand side) and uphill diffusion (right hand side) in the space-time region.}
\label{multi_4}
\end{figure}
\begin{remark} The first method applies a global linearization based on the time-steps. All effects are resolved, by the way, we have taken into account the CFL condition. We achieve better results with finer time-steps, e.g., $\Delta t_{CFl} / 8$, such that the global linearisation, via the time-step is important. \end{remark}
\subsection{Iterative Schemes in time (Local Linearisation)}
In the next series of experiments, we apply the more refined linearization scheme, means the iterative approximation in a single time-step.
We apply the numerical convergence of the schemes with the reference solution of the explicit method by $\Xi_{ref} = (\xi_1, \xi_2)$ where the time-step is $\Delta t_{CFL} / 8$ for this refined solution the error is only marginal.
Based on the reference solution, we deal with the following errors:
\begin{eqnarray} \label{kap7_gleich3}
E_{L_{1, \Delta x}}(t) & = & \int_{\Omega} | \Xi_{method, J, \Delta x}(x, t) - \Xi_{ref}(x, t) | \; dx \nonumber \\
& = & \Delta x \sum_{i=1}^N | \Xi_{method, J, \Delta x}(x_i, t) - \Xi_{ref}(x_i, t) | , \end{eqnarray}
where $method, J$ is the Richardson with $J$ iterative steps and \\ $\Delta t = \Delta t_{CFL}, \Delta t_{CFL}/2, \Delta t_{CFL}/4$. Further $method, expl$ is the explicit method with $\Delta t = \Delta t_{CFL}, \Delta t_{CFL}/2, \Delta t_{CFL}/4$.
We apply the different versions of time-steps and iterative steps, a reference solution is obtain with a fine time-step $\Delta t = \Delta t_{CFL}/4$. We see improvements in Figure \ref{multi_4_1} and the errors in Figure \ref{multi_4_2}.
\begin{figure}
\caption{ The figures present the solutions of the different time-step and iterative step of the Richardson-method}
\label{multi_4_1}
\end{figure}
\begin{figure}
\caption{ The figures present the errors of the different time-step and iterative step solutions.}
\label{multi_4_2}
\end{figure}
\begin{remark} Here, we see the benefit of large time-steps with $N=100$ and $K=800$, means we have only $100$ time-steps and $800$ iterative steps, which are not expensive. Therefore we could gain the same results as with many small time-steps $N=80000$ and only one iterative step $K=1$. Such that the relaxation method benefits with the iterative cycles and we could enlarge the time-steps. \end{remark}
\begin{remark} The second method applies a linear linearization based on the iterative approaches in each single time-step. We have the benefit of a relaxation in each local time-step, such that we see a more accurate solution also with larger time-steps than in the global linearization method. \end{remark}
\section{Conclusions and Discussions } \label{concl}
We present a fluid model based on Maxwell-Stefan diffusion equations. The underlying problems for such a more delicate diffusion matrix is discussed. Based on the nonlinear partial differential equations, we have to apply linearisation approaches. For first test-examples, we achieve more accurate results for a so-called local linearized scheme. In future, we concentrate on the numerical convergence analysis and generalize our results to real-life applications.
\end{document} | arXiv |
Derive phase damping quantum operation
I am reading about the phase damping quantum operation on page 384 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition).
Nielsen & Chuang derived the operation elements from an interaction model of two harmonic oscillators where only the first two levels $|0\rangle$ and $|1\rangle$ are considered. Here's a clipping of the corresponding contents in the book:
Another way to derive the phase damping operation is to consider an interaction between two harmonic oscillators, in a manner similar to how amplitude damping was derived in the last section, but this time with the interaction Hamiltonian \begin{equation} \tag{8.126} H = \chi a^\dagger a\left(b+b^\dagger\right), \end{equation} Letting $U = \exp\left(-iH\Delta t\right)$, considering only the $\left|0\right>$ and $\left|1\right>$ states of the $a$ oscillator as our system, and taking the environment oscillator to initially be $\left|0\right>$, we find that tracing over the environment gives the operation elements $E_k = \left<k_b|U|0_b\right>$, which are \begin{equation} \tag{8.127} E_0 = \begin{bmatrix}1 & 0 \\ 0 & \sqrt{1-\lambda}\end{bmatrix}\end{equation} \begin{equation} \tag{8.128} E_1 = \begin{bmatrix}1 & 0 \\ 0 & \sqrt{\lambda}\end{bmatrix}, \end{equation} where $\lambda = 1-\cos^2\left(\chi\Delta t\right)$
I just could not work out the calculations. Anybody can help me with the $\sqrt{1-\lambda}$ and $\sqrt{\lambda}$ terms?
Actually, when I attempted to derive the operation elements along this way, I got the very different answer:
Firstly, we know that if $[A,[A,B]]=[B,[A,B]]=0$ then $e^{A+B}=e^A e^B e^{-[A,B]/2}$. So we have $$E_0=\langle 0_b| e^{-i\chi\Delta t a^\dagger a(b+b^\dagger)} |0_b\rangle =\langle 0_b| e^{-i\chi\Delta t a^\dagger a b} e^{-i\chi\Delta t a^\dagger a b^\dagger} |0_b\rangle e^{(\chi\Delta t a^\dagger a)^2/2}$$Now using $$e^{-i\chi\Delta t a^\dagger a b^\dagger} |0_b\rangle = \sum_{n=0}^{\infty} \dfrac{(-i\chi\Delta t a^\dagger a)^n}{n!} (b^\dagger)^n |0_b\rangle = \sum_{n=0}^{\infty} \dfrac{(-i\chi\Delta t a^\dagger a)^n}{\sqrt{n!}} |n_b\rangle$$ and $$\langle 0_b| e^{-i\chi\Delta t a^\dagger a b} = \sum_{n=0}^{\infty} \langle 0_b| b^n \dfrac{(-i\chi\Delta t a^\dagger a)^n}{n!} = \sum_{n=0}^{\infty} \langle n_b| \dfrac{(-i\chi\Delta t a^\dagger a)^n}{\sqrt{n!}} $$ we are able to get $$E_0 = \sum_{n=0}^{\infty} \dfrac{(-i\chi\Delta t a^\dagger a)^{2n}}{n!} e^{(\chi\Delta t a^\dagger a)^2/2} = e^{-(\chi\Delta t a^\dagger a)^2/2}$$ Following the same line, using $$\langle 1_b| e^{-i\chi\Delta t a^\dagger a b} = \sum_{n=0}^{\infty} \langle 1_b| b^n \dfrac{(-i\chi\Delta t a^\dagger a)^n}{n!} = \sum_{n=1}^{\infty} \langle n_b| \dfrac{(-i\chi\Delta t a^\dagger a)^{n-1}}{\sqrt{n!}} n$$ we are to obtain $$E_1 = \sum_{n=0}^{\infty} \dfrac{(-i\chi\Delta t a^\dagger a)^{2n+1}}{n!} e^{(\chi\Delta t a^\dagger a)^2/2} = (-i\chi\Delta t a^\dagger a) e^{-(\chi\Delta t a^\dagger a)^2/2}$$
Therefore, my answer will be $E_{0}=\left[\begin{array}{cc}{1} & {0} \\ {0} & {e^{-(\chi\Delta t)^2/2}}\end{array}\right]$ and $E_{1}=\left[\begin{array}{cc}{0} & {0} \\ {0} & {-i\chi\Delta t e^{-(\chi\Delta t)^2/2}}\end{array}\right]$. What is the problem?
nielsen-and-chuang quantum-operation noise
Mithrandir24601♦
Conn-CaoYKConn-CaoYK
I am by no means an expert in this sort of calculation, but I think I (mostly) agree with you.
I divided the calculation up slightly differently, which simplified things notationally. Firstly, I considered the input $|0\rangle_A|0\rangle_B$. Clearly, $H$ acting on this is just 0, so this state doesn't evolve. So, the lop-left element of $E_0$ is 1, and that of $E_1$ is 0.
Then I considered the input $|1\rangle_A|0\rangle_B$. We know that $a^\dagger a$ will always just return $|1\rangle_A$, so we only need to consider the evolution of the second system, i.e. $$ e^{-i\chi\Delta t(b+b^\dagger)}|0\rangle_B. $$ We can then follow your strategy for the calculation (I've only gone through the $E_0$ case) to find the bottom-right element is $e^{-\Delta t^2\chi^2/2}$. (Digression: if I assume the b operators are fermionic, then $b+b^\dagger$ is basically just the Pauli $X$ matrix on a qubit. Then you recover the formula that's given.)
What at first glance seems confusing is why you should only consider $E_0$ and $E_1$. Surely, there are also $E_k$ for all natural numbers $k$? Of course, they will all be of the same form as $E_1$ up to some constant of proportionality. Let's assume $$ E_k=\alpha_k|1\rangle\langle 1| $$ for $k\geq 1$. Then the relevant terms of the Master equation look like $$ \sum_k\frac12 E_k^\dagger E_k\rho+\frac12\rho E_k^\dagger E_k-E_k\rho E_k^\dagger=\frac12 \beta|1\rangle\langle 1|\rho+\frac12 \beta\rho|1\rangle\langle 1|-\beta|1\rangle\langle 1|\rho|1\rangle\langle 1|. $$ This is entirely equivalent to the action of a single operator $E_1'=\sqrt{\beta}|1\rangle\langle 1|$ with $\beta=\sum_k\alpha_k^2$. Moreover, by the fact that the map will be trace preserving, I don't need to bother actually calculating $\beta$. I know that $$ \beta+\langle 1|E_0|1\rangle^2=1. $$ (At least this part is consistent with what N&C is telling us.)
DaftWullieDaftWullie
$\begingroup$ I've gone through the question a couple of times and can't spot anything wrong either. I couldn't find anything in the errata either, so I assumed I was missing something but maybe not... $\endgroup$
– Mithrandir24601 ♦
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5 - The Early Republic (Essentials)
jonloomis
Terms in this set (26)
Essentially an IOU sold by the government. Essentially they are loans people make to the government with a promise of repayment with interest.
Taxes collected on imported goods.
Government policies meant to promote domestic business, either with direct help (subsidies) or by making imports more expensive (tariffs).
Strict Interpretation
A way of reading the Constitution that results in a belief that the government can only do with is expressly written.
Loose Interpretation
A way of reading the Constitution that leads to a belief that the Constitution outlines, but does not express every possible power the government has.
A group of people who work together to affect public policy and support candidates for public office.
The principle that the Supreme Court can rule acts of congress or presidential decision in violation of the Constitution.
Federal Supremacy
The principle that the federal government has power over the states. States cannot ignore or overturn federal law.
First Secretary of the Treasury. He was a Federalist, one of the authors of the Federalist Papers during the debate over ratification of the Constitution. His financial plans included assuming state debts, creating a national bank, and promoting manufacturing. He was killed in a duel with Aaron Burr.
Federalist Party
One of the first two political parties. They supported the Constitution, strong central government, Hamilton's financial plans, and favored Britain over France. Washington and Adams were the only president's from this party.
Democratic-Republican Party
One of the first two political parties. They were the successors to the Anti-Federalists, favored state power fearing tyrannical power concentrated in the federal government. Jefferson and Madison were their leaders.
Third Chief Justice of the Supreme Court. He was critical in the establishment of the court as a co-equal branch of government. He wrote the Marbury v. Madison opinion.
Whiskey Rebellion
An uprising in 1794 in the backcountry of Virginia against a government tax on whiskey production. President Washington responded by leading the army to enforce the tax, establishing the power of the new federal government to enact and collect taxes.
Uprising in France begun in 1789 by the people against the royalty and aristocrats. It was supported in America by the Democratic-Republicans but eventually dissolved into a reign of terror in which many people were accused and beheaded without a fair trial. It eventually ended with the rise of the dictator Napoleon Bonaparte.
Presidential election between President John Adams and Jefferson. It resulted in the first transition of power from one party to another.
XYZ Affair
Political scandal during the John Adams administration when letters from American diplomats in France were made public detailing an effort by France's foreign minister to demand bribes from the Americans.
Quasi-War with France
Open conflict with France from 1798 to 1800 on the high seas due to American neutrality. France was attempting to stop American merchants from doing business with Britain. No declaration of war was ever made.
First Barbary War
Naval conflict in the Mediterranean Sea between America and the Barbary Pirates in 1801 during Thomas Jefferson's presidency.
Second Barbary War
Naval conflict in the Mediterranean Sea between America and the Barbary Pirates in 1807 during James Madison's presidency.
Embargo of 1807
Embargo on imports from Europe established by President Jefferson in an effort to stop French and British ships from attacking American merchants. The plan backfired as it hurt Americans more than Europeans.
War with Britain. It was America's first declared war. It lasted three years and resulted in a military stalemate, but affirmed American independence and provided renewed sense of national identity.
Hartford Convention
Meeting of Federalists in 1814 in which secession of the New England States was discussed. It led to the downfall of the party as a force in national politics.
First Bank of the United States
Bank created by an act of congress as part of Alexander Hamilton's financial plan. He hoped that it would help stabilize the nation's financial system by issuing a stable currency.
Alien and Sedition Acts
Laws passed in 1798 by the Federalist congress and President Adams designed to limit the influence of their political opponents. They severely limited the freedom of speech and were clearly in violation of the First Amendment.
1801 Supreme Court case that resulted from the appointment of the Midnight Justices. The resulting decision by Chief Justice John Marshall established the principle of judicial review.
Washington's Farewell Address
Letter and speech by President Washington at the end of his tenure summarizing his political philosophy and outlining his recommendations for the nation. Most remembered, he warned against entering into foreign alliances.
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Look for a pattern and then write the next three numbers. $$ 8,4,2,1, \ldots $$
Jake and Ness are partners who agree that Jake will receive a $60,000 salary allowance and that any remaining income or loss will be shared equally. If Ness's capital account is credited for$1,000 as his share of the net income in a provided period, how much net income did the partnership earn in that period?
Steve Davis earns a 6% commission on sales of $3,714. What is his commission?
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Tagged: unique solution
Are Coefficient Matrices of the Systems of Linear Equations Nonsingular?
(a) Suppose that a $3\times 3$ system of linear equations is inconsistent. Is the coefficient matrix of the system nonsingular?
(b) Suppose that a $3\times 3$ homogeneous system of linear equations has a solution $x_1=0, x_2=-3, x_3=5$. Is the coefficient matrix of the system nonsingular?
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\[\mathbf{v}=\begin{bmatrix}
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The Possibilities For the Number of Solutions of Systems of Linear Equations that Have More Equations than Unknowns
Determine all possibilities for the number of solutions of each of the system of linear equations described below.
(a) A system of $5$ equations in $3$ unknowns and it has $x_1=0, x_2=-3, x_3=1$ as a solution.
(b) A homogeneous system of $5$ equations in $4$ unknowns and the rank of the system is $4$.
(The Ohio State University, Linear Algebra Midterm Exam Problem)
In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems.
Determine all possibilities for the solution set of the system of linear equations described below.
(a) A homogeneous system of $3$ equations in $5$ unknowns.
(b) A homogeneous system of $5$ equations in $4$ unknowns.
(c) A system of $5$ equations in $4$ unknowns.
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Dispersive Lamb systems
Peter J. Olver , and Natalie E. Sheils
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
* Corresponding author: Peter J. Olver
Received October 2017 Revised March 2018 Published May 2019
Figure(11)
Under periodic boundary conditions, a one-dimensional dispersive medium driven by a Lamb oscillator exhibits a smooth response when the dispersion relation is asymptotically linear or superlinear at large wave numbers, but unusual fractal solution profiles emerge when the dispersion relation is asymptotically sublinear. Strikingly, this is exactly the opposite of the superlinear asymptotic regime required for fractalization and dispersive quantization, also known as the Talbot effect, of the unforced medium induced by discontinuous initial conditions.
Keywords: Lamb system, periodic boundary conditions, Talbot effect, dispersive quantization.
Mathematics Subject Classification: Primary: 35B99.
Citation: Peter J. Olver, Natalie E. Sheils. Dispersive Lamb systems. Journal of Geometric Mechanics, 2019, 11 (2) : 239-254. doi: 10.3934/jgm.2019013
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Figure 1. The Lamb Oscillator on the Line.
Figure 2. The Lamb Oscillator on the Line at Large Time.
Figure 3. The Periodic Lamb Oscillator.
Figure 4. The Dispersive Periodic Lamb Oscillator with $ \omega (k) = k^2 $.
Figure 5. The Dispersive Periodic Lamb Oscillator for the Klein-Gordon Model.
Figure 6. The Dispersive Periodic Lamb Oscillator with $ \omega(k) = \sqrt{\left| k \right|} $.
Figure 7. The Dispersive Periodic Lamb Oscillator for the Regularized Boussinesq Model.
Figure 8. The Unidirectional Periodic Lamb Oscillator for the Transport Model.
Figure 9. The Unidirectional Dispersive Periodic Lamb Oscillator for $ \omega(k) = {k^2} $.
Figure 10. The Unidirectional Dispersive Periodic Lamb Oscillator for $ \omega(k) = \sqrt{k} $.
Figure 11. The Unidirectional Dispersive Periodic Lamb Oscillator for $ \omega(k)=k^{2} /\left(1+\frac{1}{3} k^{2}\right) $
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Peter J. Olver Natalie E. Sheils | CommonCrawl |
Org: Sue Ann Campbell and Xinzhi Liu (University of Waterloo)
JACQUES BÉLAIR, Université de Montréal
An age-structured model with dynamic death rate [PDF]
We consider a general age-structured model for the regulation of red blood cells in which the lifespan of the circulating cells is determined by a general survival function. Using techniques borrowed from demography, an associated death rate is introduced into the modeling equations, and standard methods for the integration of the latter yield a coupled system of differential and integral equations. The stability of the equilibria of this system is analyzed, and Hopf bifurcations are detected. We pay particular attention to the relationship between these stability properties, the survival function specifics and the distribution of lifespans of the erythrocytes.
This is joint work with Frédéric Paquin-Lefebvre.
MONICA COJOCARU, University of Guelph
Linking generalized Nash games and replicator dynamics [PDF]
In this talk I plan to introduce an evolutionary steady state concept related to a generalized Nash game (GN) and a replicator dynamics. Generalized Nash games were introduced in the 50's, and represent models of noncooperative behaviour among players whose both strategy sets and payoff functions depend on strategy choices of other players. <p></p> Evolutionary games consist of populations where individuals play many times, against many different opponents, with each contributing a relatively small contribution to the total reward. Given strategies $\{1,...,n\}$, an individual of $type$ $i$ is one using strategy $i$, and $x_i$ is the frequency of $type\,i$ individuals in the population. Thus the vector $x=(x_1,...,x_n)$ in the unit simplex is the state of the population. Interaction between players of different types can be describes by linear or nonlinear payoffs; assume that a player of $type\,i$ has a payoff $a_i(x)$, and they want to maximize it subject to a choice of strategies. The dynamic evolution of such game is described by the replicator dynamics $\dot{x}_i(t)=x_i(a_i(x)-\sum_i x_i a_i(x)), \forall i$. Assuming shared constraints imposed on the game (i.e. $g(x)\leq B$ where $B$ would be the upper limit of resource for instance) a replicator dynamics has to satisfy the new constraints, becoming a constraint dynamics of the type known in literature as a projected dynamical system. The new dynamics and its relation to the GN needs to be investigated: the main questions we present here refers to highlighting under what conditions Folk's Theorem type results would hold in this new setting.
WENYING FENG, Trent University
Eigenvalue Intervals for Nonlinear Operator Equations in Ordered Banach Spaces [PDF]
We first extend results on existence of positive solutions for differential equations with separated boundary conditions. The results are further generalized to study eigenvalue problems for operator equations in the form of $x=\lambda T(x)$, where $T$ is an nonlinear map defined on an ordered Banach space, and $\lambda$ is a parameter. Some new abstract results are obtained and applied to concrete problems such as Hammerstein integral equations, discrete dynamical systems and boundary-value problems for semi-linear fractional differential equations. Part of the work was joint with Kaly Yanlei Zhang.
MICHAEL LI, University of Alberta
Dynamics of a continuous state-structured model for infectious diseases [PDF]
In this talk, I will describe a state-structured epidemic model for infectious diseases in which the state structure is nonlocal. The state is a measure of infectivity of infected individuals or the intensity of viral replications in infected cells. The model gives rise to a system of nonlinear integro-differential equations with a nonlocal term. I will show the well-posedness and dissipativity of the associated nonlinear semigroup by overcoming a lack of compactness due to the integral form of the equations. By establishing an equivalent principal spectral condition between the linearized operator and the next-generation operator, I will show that the basic reproduction number $R_0$ is a sharp threshold: if $R_0<1$, the disease-free equilibrium is globally asymptotically stable, and if $R_0>1$, the disease-free equilibrium is unstable and a unique endemic equilibrium is globally asymptotically stable. Our proof of the global stability of the endemic equilibrium utilizes a global Lyapunov function whose construction was motivated by the graph-theoretic method for coupled systems on discrete networks developed by Guo-Li-Shuai. This is a joint work with Drs. Zhipeng Qiu and Zhongwei Shen.
KYEONGAH NAH, York University
Stability threshold for linear periodic delay differential equations [PDF]
We consider a single species population growth model with periodically varying recruitment and mortality rates with fixed length of developmental period. The linear stability of the trivial solution is determined by the variational equation, \[ x^{\prime}(t) = -a(t)x(t) + b(t)x(t-1). \] We investigate the stability threshold and demonstrate the theoretical results with numerics motivated by tick population dynamics.
MANUELE SANTOPRETE, Wilfrid Laurier University
Mathematical Models of Radicalization [PDF]
Radicalization is the process by which people come to adopt increasingly extreme political, social, or religious ideologies. In recent years radicalization has become a major concern for national security because it can lead to violent extremism. It is in this context that this talk attempts to describe radicalization mathematically by modelling the spread of extremist ideology as the spread of an infectious disease. This is done by using compartmental epidemiological models. We try to use these models to evaluate the effectiveness of some strategies to counter violent extremism.
GAIL WOLKOWICZ, McMaster University
Pest Control by Generalist Parasitoids: A Bifurcation Approach [PDF]
Magal, Cosner, and Ruan (Math. Med. Biol. 25,1-20; 2008) studied both spatial and non-spatial host-parasitoid models motivated by the need for biological control of horse-chestnut leafminers that have spread through Europe. In the non-spatial model, they considered control by predation of leafminers by a generalist parasitoid population with functional response modeled using a Holling type II (Monod) form. They showed that there can be at most six equilibrium points, and discussed their local stability. We revisit their model in the non-spatial case, and identify cases missed in their investigation and the ramifications for possible pest control strategies. Both the local stability of equilibria and global properties are considered. A bifurcation theoretical approach is used. We provide analytical expressions for fold and Hopf bifurcations. Numerical results show very interesting dynamics, e.g., multiple coexisting limit cycles, homoclinic orbits, codimension one bifurcations including: Hopf, fold, transcritical, cyclic-fold, and homoclinic bifurcations, as well as codimension two bifurcations including: Bautin and Bogdanov-Takens bifurcations.
This research was done in collaboration with Gunog Seo of Colgate University.
JIANHONG WU, Laboratory for Industrial and Applied Mathematics, York University
Semiflows which preserve invariance of cones of high ranks [PDF]
We consider semiflows in general Banach spaces motivated by monotone cyclic feedback systems or differential equations with integer-valued Lyapunov functionals. These semiflows preserve the invariance of cones of high ranks, implying order-related structures on the limit sets of precompact semi-orbits. We show that for a pseudo-ordered precompact semi-orbit the limit set is either ordered, or is contained in the set of equilibria, or possesses a certain ordered homoclinic property. We show that if the omega set contains no equilibrium, then this set is ordered and hence the dynamics of the restricted semiflow is topologically conjugate to a compact flow on a finite dimensional space. Applications to Poincare-Bendixson theorem in infinite dimensional spaces is obtained. This is based on joint work with L. Feng and Y. Wang.
YINGFEI YI, University of Alberta
Reducibility of Quasi-Periodic Linear KdV Equation [PDF]
We consider the following one-dimensional, quasi-periodically forced, linear KdV equations $$u_t+(1+ a_{1}(\omega t,x)) u_{xxx}+ a_{2}(\omega t,x) u_{xx}+ a_{3}(\omega t,x)u_{x} +a_{4}(\omega t,x)u=0$$ under the periodic boundary condition $u(t,x+2\pi)=u(t,x)$, where $\omega$'s are frequency vectors lying in a bounded closed region $\Pi_*\subset R^b$ for some $b>1$, $a_i: T^b\times T\to R$, $i=1,\cdots,4$, are bounded above by a small parameter $\epsilon_*>0$ under a suitable norm, real analytic in $\phi\in T^b$ and sufficiently smooth in $x\in T$, and $a_1,a_3$ are even, $a_2,a_4$ are odd. Under the real analyticity assumption of the coefficients, we show that there exists a Cantor set $\Pi_{\epsilon_*}\subset \Pi_*$ with $|\Pi_*\setminus \Pi_{\epsilon_*}|=O(\epsilon_*^{\frac 1{100}})$ such that for each $\omega\in \Pi_{\epsilon_*}$, the corresponding equation is smoothly reducible to a constant-coefficients one. This problem is closely related to the existence and linear stability of quasi-periodic solutions in a nonlinear KdV equation.
PEI YU, Western University
Study on Slow-Fast Motions in Dynamical Systems [PDF]
In this talk, we present a method to analyze certain slow-fast motions in dynamical systems. For singular perturbed dynamical systems, the well-known Geometric Singular Perturbation Method (GSPM) is usually applied to find the special limit cycles -- slow-fast periodic solutions. However, many practical problems might be not able or very difficult to be put in the form of singular perturbed equations, but they still exhibit slow-fast motions. For such cases, based on dynamical system theory, we developed a method to identify and analyze certain slow-fast motions. We will use several biological examples to illustrate our method, and give a comparison between the GSPM and our method.
YUAN YUAN, Memorial University
A stage-structured mathematical model for fish stock with harvesting [PDF]
We propose a mathematical model for a single species fish stock with three stages structure: juveniles, small adults and large adults with two harvesting strategies for mature classes, maturity and size selectivities. The purpose of the work is to investigate the dynamical behavior of the model and discuss the effect of harvesting. We identify the adult reproduction number $\mathcal{R}_A$ for the model; obtain the local and global stability of the trivial equilibrium when $\mathcal{R}_A<1$; discuss the population persistence and existence of a unique positive equilibrium when $\mathcal{R}_A>1$. Numerical simulations are provided to investigate the influence of harvesting functions, discuss the optimal harvesting rates and explore the effect of periodic coefficients on the dynamical system.
XIAOQIANG ZHAO, Memorial University of Newfoundland
Almost Pulsating Waves in Time and Space Periodic Media [PDF]
In this talk, I will report our recent research on almost pulsating waves for monotone semiflows with monostable structure in time and space periodic media. Our method is a combination of the Poincare maps approach and an evolution viewpoint. The developed theory is then applied to two species competitive reaction-advection-diffusion systems. It turns out that the minimal wave speed exists and coincides with the single spreading speed for such a system no matter whether the spreading speed is linearly determinate. This talk is based on a joint work with Drs. Jian Fang and Xiao Yu.
HUAIPING ZHU, York University
Canard cycles in predator-prey models and bifurcation of degenerate graphics for Hilbert's 16th Problem [PDF]
In this talk, I will start with a classical predator-prey type of system to present the limit cycles and their bifurcations, including canard cycles (fast-slow oscillations) and their cyclicity as well as fast-slow dynamics. Lately, we realize that there are two different types of degenerate limit periodic sets which can generate limit cycles in the predator-prey systems with Holling types of function response. I will then use the simple model with Holling type II functional response to present the two mechanisms for the fast-slow dynamics. In the end, I will connect the finiteness part of Hilbert's 16th problem for quadratic vector fields to explain the difficulties in dealing with the finite cyclicity of a degenerate graphics, the last challenge towards the proof of the finiteness part of Hilbert's 16th problem for quadratic vector fields.
XINGFU ZOU, University of Western Ontario
Coexistence of competing species for intermediate dispersal rates in a reaction-diffusion chemostat model [PDF]
In this talk, I will revisit a diffusive chemostat model with two competing species and one nutrient. We show that for large diffusion rate, both species will be washed out, while for small diffusion rate, competition exclusion will occur. This implies that a stable coexistence can only occurs at intermediate diffusion rate. We present an explicit way of determining parameter range which supports a stable coexistence steady state. This is a joint work with Drs Junping Shi and Yixiang Wu. | CommonCrawl |
Spineless cactus (Opuntia ficus-indica) and saltbush (Atriplex halimus L.) as feed supplements for fattening Awassi male lambs: effect on digestibility, water consumption, blood metabolites, and growth performance
Faysal Alhanafi1,
Yahia Kaysi1,
Muhannad Muna2,
Ashraf Alkhtib ORCID: orcid.org/0000-0002-3381-03043,
Jane Wamatu4 &
Emily Burton3
Tropical Animal Health and Production volume 51, pages 1637–1644 (2019)Cite this article
The effect of replacing 13.6% and 20.3% of a total ration of fattening Awassi lambs by two combinations of fresh saltbush (Atriplex halimus) and fresh spineless cactus (Opuntia ficus-indica) cladodes at a ratio of 1.9:1 (TRT1) and 1.7:1 (TRT2) on water intake, digestibility, blood metabolites, and fattening performance was evaluated. Thirty-six lambs with average initial live weight 34.5 ± 4.18 kg were randomly assigned to three diets (control, TRT1, and TRT2). The control received a diet containing 166 g/kg barley straw and 834 g/kg of commercial concentrate mixture; TRT1 comprised 126 g barley straw, 739 g/kg concentrate mixture, 47 g/kg spineless cactus, and 89 g saltbush; TRT2 comprised 67 g/kg barley straw, 704 g/kg commercial concentrate mixture, 86 g/kg spineless cactus, and 144 g saltbush. A growth trial of 100 days (10 days of adaptation and 90 days of collection) followed by a metabolism trial of 17 days (10 days of adaptation and 7 days of a total feces and urine collection) was carried out. Daily dry matter intake, digestibility of crude protein, ether extract and nutrient detergent fiber, nitrogen balance, average daily gain, feed conversion ratio, and blood metabolites were not significantly affected by the treatment. Water consumption in TRT2 was significantly 16% less compared with the control. A combination of saltbush and spineless cactus at a ratio of 1.7:1 (TRT2) replaced 60% of barley straw and 16% of concentrate mixture without adverse effects on health and growth performance of Awassi male lambs. This represents a potential reduction in feed costs for smallholder farmers.
Syria has a large flock of Awassi sheep estimated at 13.8 million heads that supplies 66% of Syria's red meat (MOA 2016). It has been reported that more than 90% of Awassi sheep flock in Syria are raised in arid and semiarid which receive annual rainfall of less than 300 mm (Salhab and Yasin 2008). The sheep are mainly fed on natural pastures, cereal grains, and agricultural by-products (Alkhateeb 2008). Natural pastures, the basal diet of Awassi sheep in arid and semiarid areas, are continuously deteriorating in productivity and nutritive value due to deforestation (Alkhateeb 2008). Costs of cereal grains and their by-products are increasing due to the decrease in cereal yields as a consequence of drought and global climate change (Ben Salem and Smith 2008). Subsequently, feeding costs increase leading to reduced profitability of livestock production systems. In Syria, the use of alternative, cheaper, and underutilized feed options is encouraged to cope with the increasing demand of livestock feed.
Many nonconventional feeds are available for small ruminant nutrition in tropical areas (Awawdeh 2011). Feeding olive cake replaced 149 g/kg DM of the concentrate mixture without adverse effects on performance and carcass quality of Awassi fattening lambs (Abo Omar et al. 2012). Furthermore, feeding lactating sheep on crude olive cake improved fatty acid profile of milk and cheese (Vargas-Bello-Pérez et al. 2013). Incorporating dry grape pomace in diets of growing fattening sheep did not depress growth performance (Bahrami et al. 2010). Dried sugar pulp, dried citrus pulp, and olive cake can be incorporated into Awassi ewes' diets without negative effect on milk yield and composition (Shdaifat et al. 2013). Pistachio by-products could be introduced to small ruminants' diets at level ranging from 21 to 35%, depending on the by-product type and ruminant species, without negative effects on performance (Alkhtib et al. 2017). Inclusion of coffee pulp in growing sheep diets up to a level of 28% did not have negative effect on fattening performance (Hernández-Bautista et al. 2018).
Spineless cactus and saltbush species are reported to be suitable feed options for sheep in arid and semiarid areas. Smallholder farmers in arid and semiarid areas grow spineless cactus to produce fruits for human consumption, fences for plots and homes, and cladodes for livestock feed (Alary et al. 2007). Dry matter (DM) yield of spineless cactus varies from 3.1 to 47.3 t/ha depending on fertilization and plant density (Dubeux et al. 2006). Cladodes of spineless cactus are high in soluble carbohydrates, calcium, and vitamin A but low in crude protein (CP), fiber, and sodium (Le Houérou 1996). Supplementing straw-based diets with cladodes of spineless cactus improves ruminal digestion in sheep (Ben Salem et al. 1996). Saltbush has a high yield of edible fractions (0.5–12.3 t DM/ha), high content of CP (10–25%), high content of neutral detergent fiber (NDF) (30–45%), and moderate organic matter (OM) digestibility (460–540 g/kg) (Ben Salem et al. 2010). However, feeding sheep predominantly on spineless cactus and saltbush is associated with negative consequences on health and performance. Consuming saltbush in large amounts is associated with consumption of large quantities of water to excrete ingested salt (Ben Salem et al. 2010) whereas availability of drinking water is a critical challenge in arid and semiarid areas. Sheep fed mainly on saltbush are prone to sulfur toxicity, oxalate poisoning, and malabsorption of calcium, magnesium, and phosphorus (Ben Salem et al. 2010). High consumption of spineless cactus is expected to cause diarrhea in ruminants (Gebremariam et al. 2006). High concentration of oxalates was reported in saltbush (van Niekerk et al. 2009) and spineless cactus cladodes (Ben Salem et al. 2002b). D'Mello (1997) reported that the presence of oxalates in sheep diets at a level of 1.1 g oxalates/kg live weight is expected to result in chronic renal failure, calcium oxalate urolithiasis, hypocalcemia, and a decrease in overall performance. However, supplementation of diets based on spineless cactus with fiber-rich feeds like saltbush tends to mitigate such problems (Ben Salem et al. 2002a). As cladodes of spineless cactus contain a high level of moisture (813 to 874 g/kg DM; Batista et al. 2009), they contribute to meeting the extra demand of water resulting from feeding on saltbush. Thus, partial replacement of fattening sheep diets by a combination of fresh spineless cactus and fresh saltbush may raise productivity and decrease feeding costs of sheep in arid and semiarid areas. The current study aimed to evaluate the substitution potential of combinations of spineless cactus and saltbush in typical Syrian fattening diets of Syrian Awassi lambs comprising barley straw and concentrate mixture and their effects on voluntary DM and water intake, digestion of nutrients, nitrogen balance, blood metabolites, and growth performance.
Animals were housed in Karahta Research Station of the General Commission of Scientific Agricultural Research, Damascus, Syria (33○ 4′ N, 36○ 5′ E) at an altitude of 616 m.a.s.l. and average rainfall of 125 mm. This study has been approved by the ethical committee of Damascus University, Syria.
Thirty-Six Awassi male lambs (34.5 ± 4.18 kg live weight and 162 ± 6 days age) were used in this trial. Lambs were housed in individual pens (2 × 1.5 m) in an open-sided barn. Each pen was equipped with a feeder and waterer. Lambs were randomly allocated into three dietary treatments with 12 repetitions. Lambs were drenched with ivermectin at rate of 200 mcg/kg live weight to control common parasites and vaccinated against common diseases of fattening sheep in Syria (anthrax, pasteurellosis, and enterotoxemia) and adapted to pens and diets for 2 weeks before the beginning of the 90-day growth trial.
Dietary treatments
Three rations were designed with different combinations of spineless cactus and saltbush. The experimental diets consisted of a control and two treatment diets (TRT1, TRT2). The control consisted of 166 g/kg barley straw and 834 g/kg concentrate mixture. The concentrate mixture in the trial consisted of 500 g/kg DM whole barley grains, 270 g/kg DM whole corn grains, 170 g/kg DM cotton seed cake, 40 g/kg DM wheat bran, and 20 g/kg DM premix. No further process was applied to the concentrate mixture. This diet is commonly used by Syrian smallholders for sheep fattening. In TRT1, saltbush and spineless cactus cladodes (1.9 to 1) replaced 24% of barley straw and 11% of the concentrate mixture (on a DM basis) of the control group. In TRT2, saltbush and spineless cactus cladodes (1.7 to 1) replaced 60% of barley straw and 16% of the concentrate mixture (on a DM basis) of the control group. All rations were formulated to be isoenergetic and isonitrogenous (Table 1) formulated based on nutritional requirements for growing lambs (NRC 2007).
Table 1 Ingredients and chemical composition (on dry matter basis) of the experimental feeds
Experimental procedures
Forages of 5-year-old saltbush (Atriplex halimus L.) shrubs and a 2-year-old spineless cactus (Opuntia ficus-indica) grown in demonstration fields at a density of 2500 and 5000 plants/ha respectively were used. Fresh leaves and young twigs of saltbush biomass in addition to cladodes of spineless cactus were manually harvested on a daily basis during the trial. Both saltbush and spineless cactus were chopped to a theoretical size of 5 cm and fed fresh.
The lambs received a daily total DM of 4% of their live weight. Concentrate mixture and barley straw were distributed daily at 8:30 h and 17:30 h in two equal portions while saltbush and spineless cactus were offered fresh at 12:30 h. All lambs had ad libitum access to clean drinking water. Feed offered and refusals were recorded daily prior to the morning feeding to obtain daily feed intake for each lamb. Live weight of lambs was measured once every 10 days before the morning feeding to estimate daily weight gain. Blood samples were collected into two tubes on start day then monthly (4 samplings in total) before the morning feeding via the jugular vein: one containing heparin to estimate hematological parameters and the other one without heparin to obtain serum. Serum samples were obtained by centrifuging (1677×g; 20 min; 4 °C) of whole blood. The sera were stored at − 20 °C until being analyzed.
At the end of growth trial, 3 lambs were randomly selected from each treatment group and transferred to individual metabolic crates. After a 14-day adaptation to new conditions and diets, fecal output and urine were collected for 10 consecutive days to measure the digestibility of experimental diets. Representative samples of feed distributed to each lamb and refusals were taken daily. These were dried in a forced air oven at 60 °C for 48 h, ground to pass a 1-mm screen, and stored at room temperature for subsequent analysis. Urine was collected in bottles containing 100 ml of 10% sulfuric acid and stored at − 20 °C until analyzed. Daily fecal was recorded and a representative sample for each lamb taken and frozen at − 20 °C for subsequent analysis.
Feed and blood sample analyses
All samples of feed, leftover feeds, and feces were dried at 105 °C overnight in a forced air oven to determine DM (AOAC 2000; method 934.01). Ash was determined by burning samples in a muffle furnace at 550 °C overnight (AOAC 2000; method 942.05). The nitrogen (N) was determined according to Kjeldahl (AOAC 2000; method 954.01) and ether extract (EE) was determined using the Soxhlet method (AOAC 2000; method 920.39). Crude protein content was calculated as N × 6.26. Neutral detergent fiber (NDF) was determined according to Van Soest et al. (1991). Neutral detergent fiber was assayed without use of an alpha amylase but with sodium sulfite and expressed without residual ash. Specific commercial kits (Katal, Belo Horizonte, MG, Brazil) and a semiautomatic analyzer (Bioplus BIO-2000, Barueri, SP, Brazil) were used to analyze serum urea by the kinetic method with the use of urease (Sampson and Baird 1979), total protein by the biuret method (Tietz 1995), albumin by the boromocresol green method (Dumas et al. 1997), alanine aminotransferase activity by the kinetic method (Huang et al. 2006), aspartate aminotransferase activity by the kinetic method (Huang et al. 2006), and glucose with the use of glucose oxidase (Barham and Trinder 1972), triglyceride (McGowan et al. 1983), cholesterol (Lie et al. 1976), calcium (Leary et al. 1992), and phosphorus (Bartels and Roijers 1975). Automated hematology analyzer (Diatron, Abacus 5, Austria) was used to determine hemoglobin and packed cell volume.
All statistical analyses were carried out using SAS 9.1.3 (SAS 2012). The experimental unit was pen, unless otherwise specified. Probability was set at P ≤ 0.05. Data of the growth trial and blood parameters were analyzed using a repeated measurements design. The MIXED procure of SAS with the following model was used:
$$ {Y}_{ij}=\mu +{\mathrm{TRT}}_i+{M}_j+{\left(\mathrm{TRT}\times M\right)}_i+{\varepsilon}_{ij} $$
where Y is the response variable, TRT is the effect of the treatment is the effect of the measurement, TRT × M is the effect of the interaction between treatment and measurement, and ε is the residual. The subject, the variable on which repeated measurements were taken, was defined as a lamb within a treatment. The type of variance-covariance structure used was set as compound symmetry.
Data of metabolism trial was analyzed according to the following model:
$$ {Y}_{ij}=\mu +{\mathrm{TRT}}_i+{\varepsilon}_{ij} $$
where Y is the response variable, TRT is the effect of the treatment, and ε is the residual.
Least significant difference at 0.05 level of significance was used to separate the treatments in both models.
Metabolism trial
Intake, digestibility, nitrogen balance, and water consumption of Awassi male lambs are shown in Table 2. Replacing diets by saltbush and spineless cactus did not reduce (P > 0.05) dry matter intake of Awassi sheep either in form of g/day nor g/kg0.75 (P > 0.05). Increasing levels of saltbush and spineless cactus improved (P < 0.05) the digestibility of CP and NDF but not DM, OM, and EE. Digestibility of CP in TRT1 and TRT2 was respectively higher than that in the control group by 4.4 points and 6.3 points (P < 0.05). Neutral detergent fiber digestibility in the TRT 1 and the TRT 2 was higher than that in the control by 5.4 points and 9.9 points respectively. Nitrogen intake, fecal N loss, N voided in urine, and N retention of lambs were not significantly different among the treatment groups. The consumption of water by lambs decreased (P < 0.05) by 0.64 L/day in TRT1 and 1.07 L/day in TRT2 compared with the control. Lambs in TRT1 and TRT2 consumed less (P < 0.05) water than those in the control by 0.37 L/kg DM and 0.71 L/kg DM respectively. Consumption of water by lambs decreased (P < 0.05) by 0.02 L/kg0.75 in TRT1 and 0.037 L/kg0.75 in TRT2 compared with the control.
Table 2 Effect of dietary treatments on intake, digestibility, and nitrogen balance of Awassi male lambs
Table 3 presents the effect of treatments on growth performance of Awassi male lambs. The difference in dry matter intake, final weight, weight gain, average daily gain, and feed conversion ratio among experimental treatments was insignificant (P > 0.05). There was no significant effect of the measurement nor treatment×measurement interaction on growth performance parameters (Table 3).
Table 3 Effect of dietary treatments on growth performance of Awassi male lambs
Blood metabolites
Table 4 shows blood metabolites of lambs in the control, TRT1, and TRT2. Levels of all blood metabolites of lambs were not different (P > 0.05) among treatments. All blood parameters related to protein metabolism tended to be higher than that of the control group. Concentration of glucose and triglycerides was only numerically but insignificantly higher in TRT1 and TRT2 compared with the control group. Cholesterol level of TRT1 and TRT2 tended to be less than that of the control group. Calcium and phosphorus levels were numerically higher in TRT1 and TRT2 compared with the control group. Effect of the measurement and the interaction between treatment and measurement on blood metabolites was insignificant (P > 0.05) (Table 4).
Table 4 Effect of dietary treatments on blood parameters in Awassi male lambs
Saltbush and spineless cactus have been reported to negatively impact on sheep performance if they are fed separately. However, simultaneously introducing saltbush and spineless cactus to sheep rations in the current study made no significant difference to growth performance, but improved digestibility of CP and NDF. Saltbush content of non-protein nitrogen was reported to be high (Ben Salem et al. 2010). Therefore, replacing commercial concentrate by saltbush and spineless cactus in TRT1 and TRT2 is expected to increase content of non-protein nitrogen. Cladodes of spineless cactus contain high levels of soluble carbohydrates but low levels of NDF and CP (Ben Salem et al. 2002c). The insignificant change in blood metabolites and nitrogen balance data indicates that spineless cactus in TRT1 and TRT2 supplied ruminal bacteria with sufficient quantity of readily available carbohydrates to improve the capacity of microbial bacteria to fix ammonia released from breaking down saltbush non-protein nitrogen which resulted in observed increase in CP digestibility. The reason behind increased NDF digestibility is that NDF of barley is less digestible compared with NDF of saltbush and spineless cactus as it has less lignin. Dry mater intake of lambs was not affected by the treatments. This indicates that inclusion of a combination of saltbush and spineless cactus cladodes at a ratio of 1.7:1 replaced 60% of barley straw and 16% of concentrate mixture of the control group did not compromise palatability. Previous studies have shown consuming saltbush without concurrent spineless cactus intake by sheep was associated with an increase in water consumption (Ben Salem et al. 2004). As spineless cactus cladodes are rich in water, lambs fed on diets containing cactus cladodes consumed less water compared with the control. Therefore, inclusion of a combination of saltbush and cactus in lambs' diets in replacement of 23% of the total diet could contribute significantly to the daily requirement of water. This is of high importance to smallholder farmers in dry lands.
A high concentration of oxalates was reported in saltbush (van Niekerk et al. 2009) and spineless cactus cladodes (Ben Salem et al. 2002a). D'Mello (1997) reported that the presence of oxalates in sheep diets at a level of 1.1 g oxalates/kg live weight is expected to cause chronic renal failure, calcium oxalate urolithiasis, hypocalcemia, and a decrease in overall performance. However, hemoglobin and packed cell volume levels of Awassi lambs were similar across all experimental treatments. That means oxalates of saltbush and spineless cactus did not depress the metabolism of copper, iron, vitamin B11, and vitamin B12. Levels of albumin, alanine transferase, and aspartate transferase in TRT1 and TRT2 were similar to those in the control. This suggests that oxalates in these treatments did not have adverse effects on liver functions, which agrees with Otal et al. (2010). Concentration of urea in blood of lambs in TRT1 and TRT2 was similar to that in the control which signifies to normal renal function in lambs fed on a mixture of saltbush and spineless cactus. Levels of calcium and phosphorus in TRT1 and TRT2 were normal and not significantly different from those in the control which denotes that oxalates in TRT1 and TRT2 did not affect metabolism of calcium and phosphorus. Blood parameters of metabolism of energy and protein were similar among the experimental treatments. That means levels of oxalates in TRT1 and TRT2 did not affect metabolism of nutrients in Awassi lambs. Moreover, it suggests that all experimental groups supplied similar levels of protein and energy. This result is in line with results of the metabolism trial which showed slight differences (P > 0.05) among experimental treatments in terms of digestibility of nutrients. Overall, this indicates that replacement of 60% barley and 16% concentrate with a combination of saltbush and spineless cactus cladodes (1.7:1) did not raise ration content of oxalates to a toxic level. These results agree with Otal et al. (2010) who fed sheep saltbush ad libitum for 4 weeks without negative effects on blood profile. Similarly, Rekik et al. (2010) reported that feeding Barbarine sheep on 3 kg of spineless cactus per day for 60 days did not alter blood metabolites.
Growth performance of Awassi lambs was not different among control, TRT1, and TRT2. This is in line with the results of digestibility and blood metabolites which indicated similar ingestion of nutrients among treatments. Cereal grains and agro-industrial by-products are the main source of concentrates for livestock feeding in Syria (Alkhateeb 2008). The productivity of crops and, thus, availability of their by-products in developing countries including Syria are decreasing as a result of drought and climate change (Ben Salem and Smith 2008). Furthermore, deforestation is continuously degrading productivity and nutritive value of natural pastures which are the basal diet of sheep. This will not only widen the feed gap in Syria but also lead to an increase in feed costs. Rangelands that receive less than 300 mm of rain annually and are not suitable for cropping constitute 44% of the total area of Syria (MOA 2016). Awassi sheep constitute 90% of the livestock kept by pastoralists in these areas (Salhab and Yasin 2008).
Additionally, these forages grow efficiently in arid and semiarid areas. Thus, crowing saltbush and spineless cactus could be a strategic solution to feed shortage in Syria.
Producing vegetation of saltbush and cactus will be with low cost after pastures are established. Accordingly, replacement of commercial concentrates by saltbush and spineless cactus at an optimum level would decrease sheep fattening costs in Syria.
This study pinpoints that a mixture of saltbush and spineless cactus (1.7 saltbush:1 spineless cactus cladodes) can be introduced to fattening sheep diets replacing 60% of barley straw and 16% of the concentrate mixture which would decrease feeding cost without any adverse effect on health and growth. Moreover, this combination provided 16% of water requirements of fattening Awassi lambs which has special advantage in arid and semiarid areas. Thus, saltbush and spineless cactus, provided that incorporated at an optimum level, might be a sustainable feed option for sheep keepers in Syria.
The high yields of saltbush and spineless cactus (0.7–6.3 t of edible DM/ha in saltbush and 3.1–47.3 t DM/ha in spineless cactus) suggest that the excess biomass should be preserved to facilitate transportation for use by farmers in peri-urban areas of the large cities. Therefore, more studies on the effect of preservation method of saltbush and spineless cactus on nutritive value are required. Awassi sheep is meat-milk-wool breed. Thus, effect of introducing saltbush and spineless cactus cladodes to Awassi sheep diets on milk and wool production and quality should be studied.
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This work was funded by the General Commission of Scientific Agricultural Research (Syria).
Faculty of Agricultural Engineering, Department of Animal Production, University of Damascus, P.O Box 5735, Damascus, Syria
Faysal Alhanafi & Yahia Kaysi
General Commission of Scientific Agricultural Research, P.O Box 113, Damascus, Syria
Muhannad Muna
School of Animal, Rural and Environmental Science, Nottingham Trent University, Brackenhurst Lane, Southwell, Nottinghamshire, NG25 0QF, UK
Ashraf Alkhtib & Emily Burton
International Center for Agricultural Research in the Dry Areas (ICARDA), P.O Box 5689, Addis Ababa, Ethiopia
Jane Wamatu
Faysal Alhanafi
Yahia Kaysi
Ashraf Alkhtib
Emily Burton
Correspondence to Ashraf Alkhtib.
This study has been approved by the ethical committee of Damascus University, Syria.
Alhanafi, F., Kaysi, Y., Muna, M. et al. Spineless cactus (Opuntia ficus-indica) and saltbush (Atriplex halimus L.) as feed supplements for fattening Awassi male lambs: effect on digestibility, water consumption, blood metabolites, and growth performance. Trop Anim Health Prod 51, 1637–1644 (2019). https://doi.org/10.1007/s11250-019-01858-6
Issue Date: 01 July 2019
Cactus cladodes
Saltbush
Fattening
Awassi | CommonCrawl |
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How do we know a quantum state isn't just an unknown classical state?
When an observer causes the wave function of a particle to collapse, how can we know that the wave function was not collapsed already before the measurement?
Suppose we measure the z-component of the spin of an electron. After the measurement, it is entirely aligned along the measured direction, e.g. the +z-direction. Before the measurement we need to assume that a probability distribution proportional to $|\Psi|^2$ of the two allowed directions is present.
If we repeat the measurement with many identically prepared electrons, we should see such a distribution finally. For example, we could measure 40% spin-down and 60% spin-up.
However, it seems we could also assume that all of these particles have a defined spin-direction before we measure them.
What is an intuitive (being aware that quantum phenomena as such are rarely intuitive) explanation for why we cannot simply assume that the spin was already aligned completely in that measured direction?
With regards to the suggestion that this two-year old question is a duplicate of the one asked yesterday, I would like to point out that my question isn't limited to entanglement, but asks about a very fundamental principle in quantum mechanics, and as such is not a duplicate.
quantum-mechanics quantum-spin observers superposition wavefunction-collapse
ahemmetter
asked Oct 7 '16 at 9:49
ahemmetterahemmetter
$\begingroup$ Possible Duplicate: Why is the application of probability in QM fundamentally different from application of probability in other areas?. $\endgroup$ – user36790 Oct 7 '16 at 9:55
$\begingroup$ While not necessarily 'intuative' this wiki page attempts to give an answer $\endgroup$ – By Symmetry Oct 7 '16 at 10:39
$\begingroup$ Perhaps try this Intro To Superposition (youtube.com/watch?v=lZ3bPUKo5zc). Color and Hardness are code for z-spin and x-spin. The box is a Stern-Gerlach device. $\endgroup$ – Bruce Greetham Oct 8 '16 at 20:13
$\begingroup$ Take a look at this page $\endgroup$ – valerio Oct 8 '16 at 21:55
$\begingroup$ Possible duplicate of How do we know that entanglement allows measurement to instantly change the other particle's state? $\endgroup$ – knzhou Dec 14 '18 at 13:24
Quantum mechanics was developed in order to match experimental data. The seemingly very weird idea that some observables do not have a definite value before their measurement is not something physicists have been actively promoting, it is something that theoretical considerations followed by many actual experiments have forced them to admit.
I don't think there is an intuitive explanation for this. It is closely linked to the notion of superposition. The basic idea is that we do indirectly observe the effects of interference between superposed quantum states, but upon actual measurement we never see superposed states, only classical, definite values. If we suppose these values where there all along, then why would we have any interference? The whole framework of QM would be pointless.
In other words, a quantum state is what it is (whatever that is) precisely because it is in contrast to a classical state: crucially, it only describes a probability distribution for observables values, not actual, permanent values for these observables.
A wavefunction that would always be collapsed would just be a classical state. Now why (and does?) a measurement "collapse" anything at all is an open question, the measurement problem.
Stéphane RollandinStéphane Rollandin
$\begingroup$ What would be a simple example of an interference of electrons that contains the spin? $\endgroup$ – ahemmetter Oct 7 '16 at 15:37
Imagine the following set of experimental data:
Every day, you decide whether to put on your sunglasses before looking at the sky to check the weather. Every day I, a thousand miles away, do the same thing.
After we've made our observations, we call each other on the phone to compare. We discover that on days when we've looked without sunglasses, we always see the same thing (sometimes sunny, sometimes cloudy). On days when one of us wears sunglasses and the other doesn't, we still always see the same thing. But on days when we both wear sunglasses, it is invariably the case that one of us sees a sunny sky and the other sees a cloudy sky.
Now suppose every day, one of four things is true: Either the sky above your house is sunny (and looks sunny with or without sunglasses), or it's cloudy (and looks cloudy with or without sunglasses), or it's in a condition that looks sunny without sunglasses but cloudy with them, or it's in a condition that looks sunny with sunglasses but cloudy without. Likewise for the sky above my house. And suppose each sky is unambiguously in one of these states before we look at it.
Question: What pattern could account for the experimental data? Answer: None. If your sky and my sky are always either both sunny or both cloudy, that accounts for what we see on three out of four days but can't account for what we see when we both wear sunglasses. If there's some much more complicated pattern (e.g. 8% of the time our skies are both sunny, 7% they're both cloudy, 19% yours is sunny while mine is in the state that looks sunny only through sunglasses, etc), you still won't be able to account for that experimental data. It's not hard to prove that no matter what percentages you assign to the sixteen possible pairs of states, the experimental data just don't fit your predictions.
Conclusion: You can't use ordinary probability theory to explain the weather.
Now in real life we don't have this problem with weather, because we never see the kind of experimental data I supposed in the first place. But in quantum mechanics, we do see such data (not exactly as I've supposed here, but close enough so that the same issue arises). Thereefore you can't use ordinary probability theory, in the sense you're trying to use it, to explain the observed facts.
WillOWillO
The precise answer is contained within the Kochen-Spekker theorem and Bell's theorem. (I know it's awkward that one of them has the form "the [name] theorem" and the other has the form "[name]'s theorem". That's a long-standing inconsistency in English math and physics usage.)
The key point is the fact that you can measure in different bases. If you have a fixed state $|\psi\rangle$ (whose time-evolution you neglect), and you agree to always measure in the same fixed orthonormal basis $\{|i\rangle \}$ (e.g. the position basis), then the probability distribution $\left \{ p_i = |\langle i | \psi \rangle|^2 \right \}$ is completely classical, and could absolutely simply reflect that the system was in an unknown but definite state before the measurement.
But it turns out that there's no single classical probability distribution (which could simply reflect uncertainty in the system's definite pre-measurement state) that simultaneously reproduces the Born statistics in every basis.
So if you try to understand what's so weird about quantum mechanics while only considering measurements in a single basis, then you'll fail, because the quantum mechanics of a single state measured in a single basis really is just classical probability theory. To see what's really going on, you need to consider measuring in different bases (or equivalently, allowing yourself to act a non-diagonal unitary operator on the state before measuring it).
tparkertparker
For that single particular measurement that is exactly what we assume.
Lets make it simpler : here are measurements of the electrons in the orbitals of the hydrogen atom:
Each dot there is an (x,y) measurement from a single hydrogen atom. Orbitals are the probability loci for the electrons about the proton of the hydrogen.When that electron interacted with the detecting system, it was there.
Quantum mechanics is the theory that models and predicts what the accumulation of all the measurements will show ( the probability distribution)
Edit, to address the new title:
People have been trying from the beginning of the formulation of quantum mechanics, unsuccessfully to find an underlying classical deterministic system from which the probabilities could be calculated classically.
The de Broglie-Bohm theoretical model succeeded to have the wavefunction of non relativistic mechanics as emergent from a deterministic model.
In addition to a wavefunction on the space of all possible configurations, it also postulates an actual configuration that exists even when unobserved. The evolution over time of the configuration (that is, of the positions of all particles or the configuration of all fields) is defined by the wave function via a guiding equation. The evolution of the wave function over time is given by Schrödinger's equation.
The problem is that it cannot be extended to the relativistic regime and equations. In addition the theory violates most physicists intuitions that the simplest mathematically models are preferred over complex ones, Occam's razor:" Among competing hypotheses, the one with the fewest assumptions should be selected" .
There are continuous explorations along the line of seeking a deterministic underpinning for quantum mechanics. G. 't Hooft ( Nobel prize winner)is a member of this site and has written on his efforts towards a deterministic quantum mechanics, see here for example..
The answer is that at present quantum mechanics models successfully all of our experimental observations, whereas there exist no deterministic theory that can show that all the quantum mechanical models emerge from an underlying deterministic frame/level.
anna vanna v
$\begingroup$ So it makes no difference to assume an electron has a pre-defined location which prior to the measurement we simply dont know, or to say the electron is described by a wave function, the square of which gives the probability distribution, since both result in the same distribution after many measurements. Is that roughly the idea behind it? $\endgroup$ – ahemmetter Oct 7 '16 at 10:52
$\begingroup$ Yes. There have been efforts to try to create the probability distributions from classical deterministic equations, unsuccessfully most of the time. Bohm's pilot theory reproduces non relativistic quantum mechanics in a more complicated way, but fails when relativity has to be included. $\endgroup$ – anna v Oct 7 '16 at 12:26
Actually there is a simple experiment to distinguish a quantum superposition from a classical one.
Suppose you have two boxes. There are 10000 particles in the state
$|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$
In one box; and 5000 and 5000 in the states
$|\phi_0\rangle = |0\rangle$ and $|\phi_1\rangle = |1\rangle$
In the other $|0\rangle$ is the state with negative spin in $ z$ and $|1\rangle$ the state with positive spin in $z.$
This boxes represent exactly what you are asking for, namely, is there any difference between a superposition state and one that is already collapsed according to the probabilities of QM?
If you measure spin in $z$ you'll get the same results in both boxes, as you said. $50\%$ up and $50\%$ down. So it seems that quantum superpositions are indeed just like ensembles of collapsed wavefunctions. But... what happens if we measure in $x\,?$
Since the state $|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ is the state $|1\rangle_x$ (the state with positive spin in $x$) every measurement in the superposition box will give spin up. However, since both $|0\rangle$ and $|1\rangle$ are linear superpositions of $|0\rangle_x$ and $|1\rangle_x $ with equal probabilities the second box will give $50\%$ up and $50\%$ down. So there is a notorious difference between boxes.
This is what is usually called interference. When the particles are in a superposition of up and down, wavefunctions interfere and modify the probabilities of measuring eigenvalues of other basis.
P. C. SpanielP. C. Spaniel
Consider 3 such electrons in a state that is a superposition of all 3 spins up in the $z$-direction and all 3 spins down:
$$\left|\psi\right> = \frac{1}{\sqrt{2}} \left[\left |\uparrow\uparrow\uparrow\right> - \left|\downarrow\downarrow\downarrow\right>\right]$$
Consider the 3 observables:
$$\begin{split} A_1 &= \sigma_x^{(1)}\sigma_y^{(2)}\sigma_y^{(3)}\\ A_2 &= \sigma_y^{(1)}\sigma_x^{(2)}\sigma_y^{(3)}\\ A_3 &= \sigma_y^{(1)}\sigma_y^{(2)}\sigma_x^{(3)} \end{split} $$
Here the superscript denotes on which spin the operator acts. So, the observable $A_i$ corresponds to measuring the product of the $x$-component of the $i$th spin and the $y$-components of the two other spins. Using:
$$\begin{split} \sigma_x\left|\uparrow\right> &=&\left|\downarrow\right>\\ \sigma_x\left|\downarrow\right> &=&\left|\uparrow\right>\\ \sigma_y\left|\uparrow\right> &=&i\left|\downarrow\right>\\ \sigma_y\left|\downarrow\right> &=-&i\left|\uparrow\right>\\ \end{split}\tag{1} $$
You then find that:
$$A_i\left|\psi\right> = \left|\psi\right>$$
So, measuring any of the three $A_i$'s will always yield $1$; the product of measuring an $x$ component of one spin and the $y$ component of the two other spins is always equal to 1 despite the individual spin measurements yielding completely random results. If we assume that these results were already determined before they were actually measured, then that means that the counterfactual results of measuring a different spin components are also well defined.
While we cannot tell what the results would have been had we measured different spin components, we do know that all the $A_i$'s yield 1. Whatever the spin measurement results would have been, the product $A_1A_2A_3$ must be equal to 1. But written out in terms of the individual spin measurements, this product equals the product of the result of measuring the three $x$-components and the squares of the three $y$-components. Since these squares are equal to 1, the product of the results of measuring the three $x$-components must yield 1.
Now, it's easy to check using (1) that:
$$\sigma_x^{(1)}\sigma_x^{(2)}\sigma_x^{(3)}\left|\psi\right> =- \left|\psi\right>$$
So, the result of measuring the product of three $x$ components and multiplying the results is always $-1$, and not $1$. This thus proves wrong the assumption that the results of the spin measurements are determined regardless of whether they are actually measured.
Count IblisCount Iblis
$\begingroup$ Just because you pointed me to this answer, I'm trying to understand it, even though this is a more complex example than the one I was asking about. Anyway my doubt here is that you are mixing factual and counterfactual results applying to both of them the same quantum constraints. Counterfactual definiteness means you can include measurements that have not been performed as separate results, but without mixing the observables as if they came from the same measurement. $\endgroup$ – user1892538 Jul 2 '17 at 21:38
$\begingroup$ @user1892538 The QM predictions can just be taken to be what an experimentalist who doesn't know QM will find. So, whenever xxx is measured it's always -1 but xyy, yxy and yyx is always 1 and that is enough for the experimentalist to falsify local hidden variables. $\endgroup$ – Count Iblis Jul 2 '17 at 22:16
$\begingroup$ Ok thanks (in fact your answer makes perfectly sense here) but I still don't see the equivalent of the above with a two spins singlet by measurements only along the axes x,y and z. If you do, please answer my question. Of course I understand how Bell inequality can work with different angles in that case, as per my own answer. $\endgroup$ – user1892538 Jul 3 '17 at 1:04
$\begingroup$ Btw as an experimentalist you also need to make spacelike separated all those 3 times 4 (times n for accuracy) measurements, to close the locality loophole at least. Just out of my curiosity, do you have any reference to this experimental setup (maybe from arxiv)? $\endgroup$ – user1892538 Jul 3 '17 at 1:39
$\begingroup$ Yes, you want spacelike separation. The original argument was presented in this paper. $\endgroup$ – Count Iblis Jul 3 '17 at 23:33
First of all an "observer" is any physical interaction with the system deemed "enough" to produce a measurement. Second, "collapse" is a dangerous term to use. We know that the system isn't only the particle-like component we see because of things like interference patterns and entanglement.
Suppose we measure the z-component of the spin of an electron. After the measurement, it is entirely aligned along the measured direction, e.g. the +z-direction. Before the measurement we need to assume that a probability distribution proportional to |Ψ|2|Ψ|2 of the two allowed directions is present.
You just said they were prepared identically. If they had different intrinsic properties, they wouldn't be identical.
One example is Entanglement, where the correlation of measured spins actually depends on the particles "knowing" what axis their twin is measured in.
Yogi DMTYogi DMT
As long as you have a single electron to be measured it does not make any sense to ask yourself whether it was in a certain spin direction before being measured or if you forced it to assume one.
If we have many electrons which we assume to be "identical", for example because they are produced in the same way (for example being emitted by an heated metal), and we see different outcomes when we measure their spin, it does make sense ask ourselves if a classical description or a quantum description holds.
For example, they may have different speed. But we could formulate a classical description of this phenomenon: they come out of the metal with a given speed, unknown to us -the experimenter - but having a certain value before the measure. The distribution of these speed may follow a certain classical probability distribution.
But sometime a quantum picture is needed. For example, for the spin. In this case, we must assume that before the measurement the spin has no definite value and that the probability distribution which describes the outcomes are written as |psi|^2.
Usually when a quantum picture holds a naive classical one fails, because probabilities - as computed with wave functions - have different properties than naive classical probabilities. Think for example of the interference pattern in the two slit experiment.
However, since quantum is so different from everyday experience, it is natural to ask whether it holds to a classical description of a certain "quantum" phenomena, but that we are not able of finding it because we miss something. This attempts are called "hidden variable" descriptions. In this descriptions we postulate that some hidden degree of freedom exists, which we ignore (hence the name hidden). It is this degree of freedom, which is completely classical - described by a standard probability distribution - which rule that outcomes which we see and which appear to follow a quantum picture.
It is possible to show, however , that a hidden variable picture always fail to reproduce exactly a quantum picture. So from a theoretical Point of view, quantum theory and classical theory are indeed different. Moreover, experiments have been made (Google Alan EPR), which rule out that a hidden variable description holds for certain phenomena. So it is fair to claim that we have evidence of at least one situation in nature which cannot be described by a classical theory, not matter how many "hidden variables" exist which we ignore.
giulio bullsavergiulio bullsaver
(emphasis mine) It seems to me that others are explaining why the measurements can't be predicted by classical probabilistic theory but for the question that is actually asked here, there is a simpler answer.
If you measure in the $x$ direction, you get the result that the spin is either towards the $x$ direction or opposite.
If you measure in the $z$ direction, you get the result that the spin is either towards the $z$ direction or opposite.
Now if the spin already has a well-defined direction before our measurement, it would mean that:
If the spin has well-defined direction $+x$ or $-x$, we will measure in $x$ direction.
If the spin has well-defined direction $+z$ or $-z$, we will measure in the $z$ direction.
But how could the direction of the spin (that we yet haven't measured!) cause us to measure in that direction?
JiKJiK
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ddbar lemma
In complex geometry, the $\partial {\bar {\partial }}$ lemma (pronounced ddbar lemma) is a mathematical lemma about the de Rham cohomology class of a complex differential form. The $\partial {\bar {\partial }}$-lemma is a result of Hodge theory and the Kähler identities on a compact Kähler manifold. Sometimes it is also known as the $dd^{c}$-lemma, due to the use of a related operator $ d^{c}=-{\frac {i}{2}}(\partial -{\bar {\partial }})$, with the relation between the two operators being $i\partial {\bar {\partial }}=dd^{c}$ and so $\alpha =dd^{c}\beta $.[1]: 1.17 [2]: Lem 5.50
Statement
The $\partial {\bar {\partial }}$ lemma asserts that if $(X,\omega )$ is a compact Kähler manifold and $\alpha \in \Omega ^{p,q}(X)$ is a complex differential form of bidegree (p,q) (with $p,q\geq 1$) whose class $[\alpha ]\in H_{dR}^{p+q}(X,\mathbb {C} )$ is zero in de Rham cohomology, then there exists a form $\beta \in \Omega ^{p-1,q-1}(X)$ of bidegree (p-1,q-1) such that
$\alpha =i\partial {\bar {\partial }}\beta ,$
where $\partial $ and ${\bar {\partial }}$ are the Dolbeault operators of the complex manifold $X$.[3]: Ch VI Lem 8.6
ddbar potential
The form $\beta $ is called the $\partial {\bar {\partial }}$-potential of $\alpha $. The inclusion of the factor $i$ ensures that $i\partial {\bar {\partial }}$ is a real differential operator, that is if $\alpha $ is a differential form with real coefficients, then so is $\beta $.
This lemma should be compared to the notion of an exact differential form in de Rham cohomology. In particular if $\alpha \in \Omega ^{k}(X)$ is a closed differential k-form (on any smooth manifold) whose class is zero in de Rham cohomology, then $\alpha =d\gamma $ for some differential (k-1)-form $\gamma $ called the $d$-potential (or just potential) of $\alpha $, where $d$ is the exterior derivative. Indeed, since the Dolbeault operators sum to give the exterior derivative $d=\partial +{\bar {\partial }}$ and square to give zero $\partial ^{2}={\bar {\partial }}^{2}=0$, the $\partial {\bar {\partial }}$-lemma implies that $\gamma ={\bar {\partial }}\beta $, refining the $d$-potential to the $\partial {\bar {\partial }}$-potential in the setting of compact Kähler manifolds.
Proof
The $\partial {\bar {\partial }}$-lemma is a consequence of Hodge theory applied to a compact Kähler manifold.[3][1]: 41–44 [2]: 73–77
The Hodge theorem for an elliptic complex may be applied to any of the operators $d,\partial ,{\bar {\partial }}$ and respectively to their Laplace operators $\Delta _{d},\Delta _{\partial },\Delta _{\bar {\partial }}$. To these operators one can define spaces of harmonic differential forms given by the kernels:
${\begin{aligned}{\mathcal {H}}_{d}^{k}&=\ker \Delta _{d}:\Omega ^{k}(X)\to \Omega ^{k}(X)\\{\mathcal {H}}_{\partial }^{p,q}&=\ker \Delta _{\partial }:\Omega ^{p,q}(X)\to \Omega ^{p,q}(X)\\{\mathcal {H}}_{\bar {\partial }}^{p,q}&=\ker \Delta _{\bar {\partial }}:\Omega ^{p,q}(X)\to \Omega ^{p,q}(X)\\\end{aligned}}$
The Hodge decomposition theorem asserts that there are three orthogonal decompositions associated to these spaces of harmonic forms, given by
${\begin{aligned}\Omega ^{k}(X)&={\mathcal {H}}_{d}^{k}\oplus \operatorname {im} d\oplus \operatorname {im} d^{*}\\\Omega ^{p,q}(X)&={\mathcal {H}}_{\partial }^{p,q}\oplus \operatorname {im} \partial \oplus \operatorname {im} \partial ^{*}\\\Omega ^{p,q}(X)&={\mathcal {H}}_{\bar {\partial }}^{p,q}\oplus \operatorname {im} {\bar {\partial }}\oplus \operatorname {im} {\bar {\partial }}^{*}\end{aligned}}$
where $d^{*},\partial ^{*},{\bar {\partial }}^{*}$ are the formal adjoints of $d,\partial ,{\bar {\partial }}$ with respect to the Riemannian metric of the Kähler manifold, respectively.[4]: Thm. 3.2.8 These decompositions hold separately on any compact complex manifold. The importance of the manifold being Kähler is that there is a relationship between the Laplacians of $d,\partial ,{\bar {\partial }}$ and hence of the orthogonal decompositions above. In particular on a compact Kähler manifold
$\Delta _{d}=2\Delta _{\partial }=2\Delta _{\bar {\partial }}$
which implies an orthogonal decomposition
${\mathcal {H}}_{d}^{k}=\bigoplus _{p+q=k}{\mathcal {H}}_{\partial }^{p,q}=\bigoplus _{p+q=k}{\mathcal {H}}_{\bar {\partial }}^{p,q}$
where there are the further relations ${\mathcal {H}}_{\partial }^{p,q}={\overline {{\mathcal {H}}_{\bar {\partial }}^{q,p}}}$ relating the spaces of $\partial $ and ${\bar {\partial }}$-harmonic forms.[4]: Prop. 3.1.12
As a result of the above decompositions, one can prove the following lemma.
Lemma ($\partial {\bar {\partial }}$-lemma)[3]: 311 — Let $\alpha \in \Omega ^{p,q}(X)$ be a $d$-closed (p,q)-form on a compact Kähler manifold $X$. Then the following are equivalent:
1. $\alpha $ is $d$-exact.
2. $\alpha $ is $\partial $-exact.
3. $\alpha $ is ${\bar {\partial }}$-exact.
4. $\alpha $ is $\partial {\bar {\partial }}$-exact. That is there exists $\beta $ such that $\alpha =i\partial {\bar {\partial }}\beta $.
5. $\alpha $ is orthogonal to ${\mathcal {H}}_{\bar {\partial }}^{p,q}\subset \Omega ^{p,q}(X)$.
The proof is as follows.[4]: Cor. 3.2.10 Let $\alpha \in \Omega ^{p,q}(X)$ be a closed (p,q)-form on a compact Kähler manifold $(X,\omega )$. It follows quickly that (d) implies (a), (b), and (c). Moreover, the orthogonal decompositions above imply that any of (a), (b), or (c) imply (e). Therefore, the main difficulty is to show that (e) implies (d).
To that end, suppose that $\alpha $ is orthogonal to the subspace ${\mathcal {H}}_{\bar {\partial }}^{p,q}\subset \Omega ^{p,q}(X)$. Then $\alpha \in \operatorname {im} {\bar {\partial }}\oplus \operatorname {im} {\bar {\partial }}^{*}$. Since $\alpha $ is $d$-closed and $d=\partial +{\bar {\partial }}$, it is also ${\bar {\partial }}$-closed (that is ${\bar {\partial }}\alpha =0$). If $\alpha =\alpha '+\alpha ''$ where $\alpha '\in \operatorname {im} {\bar {\partial }}$ and $\alpha ''={\bar {\partial }}^{*}\gamma $ is contained in $\operatorname {im} {\bar {\partial }}^{*}$ then since this sum is from an orthogonal decomposition with respect to the inner product $\langle -,-\rangle $ induced by the Riemannian metric,
$\langle \alpha '',\alpha ''\rangle =\langle \alpha ,\alpha ''\rangle =\langle \alpha ,{\bar {\partial }}^{*}\gamma \rangle =\langle {\bar {\partial }}\alpha ,\gamma \rangle =0$
or in other words $\|\alpha ''\|^{2}=0$ and $\alpha ''=0$. Thus it is the case that $\alpha =\alpha '\in \operatorname {im} {\bar {\partial }}$. This allows us to write $\alpha ={\bar {\partial }}\eta $ for some differential form $\eta \in \Omega ^{p,q-1}(X)$. Applying the Hodge decomposition for $\partial $ to $\eta $,
$\eta =\eta _{0}+\partial \eta '+\partial ^{*}\eta ''$
where $\eta _{0}$ is $\Delta _{\partial }$-harmonic, $\eta '\in \Omega ^{p-1,q-1}(X)$ and $\eta ''\in \Omega ^{p+1,q-1}(X)$. The equality $\Delta _{\bar {\partial }}=\Delta _{\partial }$ implies that $\eta _{0}$ is also $\Delta _{\bar {\partial }}$-harmonic and therefore ${\bar {\partial }}\eta _{0}={\bar {\partial }}^{*}\eta _{0}=0$. Thus $\alpha ={\bar {\partial }}\partial \eta '+{\bar {\partial }}\partial ^{*}\eta ''$. However, since $\alpha $ is $d$-closed, it is also $\partial $-closed. Then using a similar trick to above,
$\langle {\bar {\partial }}\partial ^{*}\eta '',{\bar {\partial }}\partial ^{*}\eta ''\rangle =\langle \alpha ,{\bar {\partial }}\partial ^{*}\eta ''\rangle =-\langle \alpha ,\partial ^{*}{\bar {\partial }}\eta ''\rangle =-\langle \partial \alpha ,{\bar {\partial }}\eta ''\rangle =0,$
also applying the Kähler identity that ${\bar {\partial }}\partial ^{*}=-\partial ^{*}{\bar {\partial }}$. Thus $\alpha ={\bar {\partial }}\partial \eta '$ and setting $\beta =i\eta '$ produces the $\partial {\bar {\partial }}$-potential.
Local version
A local version of the $\partial {\bar {\partial }}$-lemma holds and can be proven without the need to appeal to the Hodge decomposition theorem.[4]: Ex 1.3.3, Rmk 3.2.11 It is the analogue of the Poincaré lemma or Dolbeault–Grothendieck lemma for the $\partial {\bar {\partial }}$ operator. The local $\partial {\bar {\partial }}$-lemma holds over any domain on which the aforementioned lemmas hold.
Lemma (Local $\partial {\bar {\partial }}$-lemma) — Let $X$ be a complex manifold and $\alpha \in \Omega ^{p,q}(X)$ be a differential form of bidegree (p,q) for $p,q\geq 1$. Then $\alpha $ is $d$-closed if and only if for every point $p\in X$ there exists an open neighbourhood $U\subset X$ containing $p$ and a differential form $\beta \in \Omega ^{p-1,q-1}(U)$ such that $\alpha =i\partial {\bar {\partial }}\beta $ on $U$.
The proof follows quickly from the aforementioned lemmas. Firstly observe that if $\alpha $ is locally of the form $\alpha =i\partial {\bar {\partial }}\beta $ for some $\beta $ then $d\alpha =d(i\partial {\bar {\partial }}\beta )=i(\partial +{\bar {\partial }})(\partial {\bar {\partial }}\beta )=0$ because $\partial ^{2}=0$, ${\bar {\partial }}^{2}=0$, and $\partial {\bar {\partial }}=-{\bar {\partial }}\partial $. On the other hand, suppose $\alpha $ is $d$-closed. Then by the Poincaré lemma there exists an open neighbourhood $U$ of any point $p\in X$ and a form $\gamma \in \Omega ^{p+q-1}(U)$ such that $\alpha =d\gamma $. Now writing $\gamma =\gamma '+\gamma ''$ for $\gamma '\in \Omega ^{p-1,q}(X)$ and $\gamma ''\in \Omega ^{p,q-1}(X)$ note that $d\alpha =(\partial +{\bar {\partial }})\alpha =0$ and comparing the bidegrees of the forms in $d\alpha $ implies that ${\bar {\partial }}\gamma '=0$ and $\partial \gamma ''=0$ and that $\alpha =\partial \gamma '+{\bar {\partial }}\gamma ''$. After possibly shrinking the size of the open neighbourhood $U$, the Dolbeault–Grothendieck lemma may be applied to $\gamma '$ and ${\overline {\gamma ''}}$ (the latter because ${\overline {\partial \gamma ''}}={\bar {\partial }}({\overline {\gamma ''}})=0$) to obtain local forms $\eta ',\eta ''\in \Omega ^{p-1,q-1}(X)$ such that $\gamma '={\bar {\partial }}\eta '$ and ${\overline {\gamma ''}}={\bar {\partial }}\eta ''$. Noting then that $\gamma ''=\partial {\overline {\eta ''}}$ this completes the proof as $\alpha =\partial {\bar {\partial }}\eta '+{\bar {\partial }}\partial {\overline {\eta ''}}=i\partial {\bar {\partial }}\beta $ where $\beta =-i\eta '+i{\overline {\eta ''}}$.
Bott–Chern cohomology
The Bott–Chern cohomology is a cohomology theory for compact complex manifolds which depends on the operators $\partial $ and ${\bar {\partial }}$, and measures the extent to which the $\partial {\bar {\partial }}$-lemma fails to hold. In particular when a compact complex manifold is a Kähler manifold, the Bott–Chern cohomology is isomorphic to the Dolbeault cohomology, but in general it contains more information.
The Bott–Chern cohomology groups of a compact complex manifold[3] are defined by
$H_{BC}^{p,q}(X)={\frac {\ker(\partial :\Omega ^{p,q}\to \Omega ^{p+1,q})\cap \ker({\bar {\partial }}:\Omega ^{p,q}\to \Omega ^{p,q+1})}{\operatorname {im} (\partial {\bar {\partial }}:\Omega ^{p-1,q-1}\to \Omega ^{p,q})}}.$ :\Omega ^{p,q}\to \Omega ^{p+1,q})\cap \ker({\bar {\partial }}:\Omega ^{p,q}\to \Omega ^{p,q+1})}{\operatorname {im} (\partial {\bar {\partial }}:\Omega ^{p-1,q-1}\to \Omega ^{p,q})}}.}
Since a differential form which is both $\partial $ and ${\bar {\partial }}$-closed is $d$-closed, there is a natural map $H_{BC}^{p,q}(X)\to H_{dR}^{p+q}(X,\mathbb {C} )$ from Bott–Chern cohomology groups to de Rham cohomology groups. There are also maps to the $\partial $ and ${\bar {\partial }}$ Dolbeault cohomology groups $H_{BC}^{p,q}(X)\to H_{\partial }^{p,q}(X),H_{\bar {\partial }}^{p,q}(X)$. When the manifold $X$ satisfies the $\partial {\bar {\partial }}$-lemma, for example if it is a compact Kähler manifold, then the above maps from Bott–Chern cohomology to Dolbeault cohomology are isomorphisms, and furthermore the map from Bott–Chern cohomology to de Rham cohomology is injective.[5] As a consequence, there is an isomorphism
$H_{dR}^{k}(X,\mathbb {C} )=\bigoplus _{p+q=k}H_{BC}^{p,q}(X)$
whenever $X$ satisfies the $\partial {\bar {\partial }}$-lemma. In this way, the kernel of the maps above measure the failure of the manifold $X$ to satisfy the lemma, and in particular measure the failure of $X$ to be a Kähler manifold.
Consequences for bidegree (1,1)
The most significant consequence of the $\partial {\bar {\partial }}$-lemma occurs when the complex differential form has bidegree (1,1). In this case the lemma states that an exact differential form $\alpha \in \Omega ^{1,1}(X)$ has a $\partial {\bar {\partial }}$-potential given by a smooth function $f\in C^{\infty }(X,\mathbb {C} )$:
$\alpha =i\partial {\bar {\partial }}f.$
In particular this occurs in the case where $\alpha =\omega $ is a Kähler form restricted to a small open subset $U\subset X$ of a Kähler manifold (this case follows from the local version of the lemma), where the aforementioned Poincaré lemma ensures that it is an exact differential form. This leads to the notion of a Kähler potential, a locally defined function which completely specifies the Kähler form. Another important case is when $\alpha =\omega -\omega '$ is the difference of two Kähler forms which are in the same de Rham cohomology class $[\omega ]=[\omega ']$. In this case $[\alpha ]=[\omega ]-[\omega ']=0$ in de Rham cohomology so the $\partial {\bar {\partial }}$-lemma applies. By allowing (differences of) Kähler forms to be completely described using a single function, which is automatically a plurisubharmonic function, the study of compact Kähler manifolds can be undertaken using techniques of pluripotential theory, for which many analytical tools are available. For example, the $\partial {\bar {\partial }}$-lemma is used to rephrase the Kähler–Einstein equation in terms of potentials, transforming it into a complex Monge–Ampère equation for the Kähler potential.
ddbar manifolds
Complex manifolds which are not necessarily Kähler but still happen to satisfy the $\partial {\bar {\partial }}$-lemma are known as $\partial {\bar {\partial }}$-manifolds. For example, compact complex manifolds which are Fujiki class C satisfy the $\partial {\bar {\partial }}$-lemma but are not necessarily Kähler.[5]
See also
• Poincaré lemma
• Dolbeault–Grothendieck lemma
References
1. Gauduchon, P. (2010). "Elements of Kähler geometry". Calabi’s extremal Kähler metrics: An elementary introduction. {{cite book}}: |journal= ignored (help)
2. Ballmann, Werner (2006). Lectures on Kähler Manifolds. European mathematical society. doi:10.4171/025. ISBN 978-3-03719-025-8.
3. Demailly, Jean-Pierre (2012). Analytic Methods in Algebraic Geometry. Somerville, MA: International Press. ISBN 9781571462343.
4. Huybrechts, D. (2005). Complex Geometry. Universitext. Berlin: Springer. doi:10.1007/b137952. ISBN 3-540-21290-6.
5. Angella, Daniele; Tomassini, Adriano (2013). "On the $\partial {\bar {\partial }}$-Lemma and Bott-Chern cohomology". Inventiones Mathematicae. 192: 71–81. doi:10.1007/s00222-012-0406-3. S2CID 253747048.
External links
• Jean-Pierre, Demailly. "Personal page at Grenoble, including publications".
| Wikipedia |
Let b be a real number randomly selected from the interval $[-17,17]$. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation $x^4+25b^2=(4b^2-10b)x^2$ has $\textit{at least}$ two distinct real solutions. Find the value of $m+n$.
The equation has quadratic form, so complete the square to solve for x.
\[x^4 - (4b^2 - 10b)x^2 + 25b^2 = 0\]\[x^4 - (4b^2 - 10b)x^2 + (2b^2 - 5b)^2 - 4b^4 + 20b^3 = 0\]\[(x^2 - (2b^2 - 5b))^2 = 4b^4 - 20b^3\]
In order for the equation to have real solutions,
\[16b^4 - 80b^3 \ge 0\]\[b^3(b - 5) \ge 0\]\[b \le 0 \text{ or } b \ge 5\]
Note that $2b^2 - 5b = b(2b-5)$ is greater than or equal to $0$ when $b \le 0$ or $b \ge 5$. Also, if $b = 0$, then expression leads to $x^4 = 0$ and only has one unique solution, so discard $b = 0$ as a solution. The rest of the values leads to $b^2$ equalling some positive value, so these values will lead to two distinct real solutions.
Therefore, in interval notation, $b \in [-17,0) \cup [5,17]$, so the probability that the equation has at least two distinct real solutions when $b$ is randomly picked from interval $[-17,17]$ is $\frac{29}{34}$. This means that $m+n = \boxed{63}$. | Math Dataset |
Investigating the effects of absolute humidity and movement on COVID-19 seasonality in the United States
Quantifying human mobility behaviour changes during the COVID-19 outbreak in the United States
Yixuan Pan, Aref Darzi, … Lei Zhang
Mobility restrictions were associated with reductions in COVID-19 incidence early in the pandemic: evidence from a real-time evaluation in 34 countries
Juhwan Oh, Hwa-Young Lee, … Lawrence O. Gostin
Lockdowns result in changes in human mobility which may impact the epidemiologic dynamics of SARS-CoV-2
Nishant Kishore, Rebecca Kahn, … Caroline O. Buckee
The impact of weather on COVID-19 pandemic
Michael Ganslmeier, Davide Furceri & Jonathan D. Ostry
Non-compulsory measures sufficiently reduced human mobility in Tokyo during the COVID-19 epidemic
Takahiro Yabe, Kota Tsubouchi, … Satish V. Ukkusuri
Heterogeneous interventions reduce the spread of COVID-19 in simulations on real mobility data
Haotian Wang, Abhirup Ghosh, … Jie Gao
COVID-19 outbreak response, a dataset to assess mobility changes in Italy following national lockdown
Emanuele Pepe, Paolo Bajardi, … Michele Tizzoni
On the use of aggregated human mobility data to estimate the reproduction number
Fabio Vanni, David Lambert, … Paolo Grigolini
Impacts of social distancing policies on mobility and COVID-19 case growth in the US
Gregory A. Wellenius, Swapnil Vispute, … Evgeniy Gabrilovich
Gary Lin1,
Alisa Hamilton1,
Oliver Gatalo1,
Fardad Haghpanah1,
Takeru Igusa2,3,4 &
Eili Klein1,5,6
Scientific Reports volume 12, Article number: 16729 (2022) Cite this article
Mounting evidence suggests the primary mode of SARS-CoV-2 transmission is aerosolized transmission from close contact with infected individuals. While transmission is a direct result of human encounters, falling humidity may enhance aerosolized transmission risks similar to other respiratory viruses (e.g., influenza). Using Google COVID-19 Community Mobility Reports, we assessed the relative effects of absolute humidity and changes in individual movement patterns on daily cases while accounting for regional differences in climatological regimes. Our results indicate that increasing humidity was associated with declining cases in the spring and summer of 2020, while decreasing humidity and increase in residential mobility during winter months likely caused increases in COVID-19 cases. The effects of humidity were generally greater in regions with lower humidity levels. Given the possibility that COVID-19 will be endemic, understanding the behavioral and environmental drivers of COVID-19 seasonality in the United States will be paramount as policymakers, healthcare systems, and researchers forecast and plan accordingly.
As of October 14, 2021, the coronavirus disease 2019 (COVID-19) pandemic has claimed over 720,000 lives in the United States alone, with more than 44.7 million confirmed cases1. Current evidence suggests that the primary mode of transmission of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is close contact with infected individuals2,3. Aerosols4,5, which are particulates less than 5 µm in diameter6,7, likely play a significant role in transmission8. After the initial rise of cases in the early winter of 2020, cases remained severe through the spring before dropping in the summer. Given the shelter-in-place order in most states and the rise in humidity, cases generally decreased in May and stayed in lower ranges through the summer until the fall months. In most areas of the northern hemisphere, as fall turns to winter, the weather becomes colder and drier. Lower absolute humidity has been shown to be associated with increased transmission rates of other respiratory viruses (e.g., influenza)9, posing significant concerns regarding potential increases in the number of COVID-19 cases in the fall and winter. The surge in cases through the end of 2020 further supports the seasonal effects of COVID-19.
While several studies have suggested a relationship between climatic factors (e.g., temperature and/or humidity) and COVID-1910,11,12,13,14,15,16,17,18, the exact environmental and biological mechanism behind airborne and droplet transmission and viral survival of SARS-CoV-219 is not yet clear. In influenza, lower atmospheric moisture has been shown to increase the production of aerosol nuclei and viral survival time9, which translates to higher risks of airborne and droplet transmission. Other climatic factors that may impact transmission include temperature and air quality20,21; nevertheless, absolute humidity can still provide a surrogate measure for indoor air moisture and temperature22.
Initial efforts to slow the spread of COVID-19 focused on reducing contacts between individuals through social-distancing measures such as large-scale lockdowns, which were significantly associated with reductions in cases23. However, as the initial lockdowns were lifted and the movement of individuals increased, the correlation between mobility and case growth rates weakened overall24, though upticks in cases were associated with increased mobility during national holidays25. During the months of 2020 and 2021 some counties and states saw increases in cases, while others observed decreases without corresponding increases in movement by any metric. Thus, other factors, including environmental factors, must also be considered as important transmission drivers.
Analyses of the factors influencing COVID-19 have used either climate data21,26,27,28 or human mobility data23, but no study to our knowledge has considered changes in both climate and human mobility on COVID-19 outbreaks in the United States. Preliminary studies have investigated these effects in China but did not consider varying sensitivities to humidity for different climatological regimes, leading to a weaker detection of humidity impacts on transmission risks in areas with higher variations of humidity29. Understanding the potential for climatic factors to increase transmission in the fall and winter is crucial for developing policies to combat the spread of SARS-CoV-2. While the interaction between environmental factors and human encounters is complex, accounting for this relationship is necessary for determining appropriate policies that will be effective at reducing transmissions. Furthermore, indoor gatherings typically increase in frequency and size in the winter and are one of the largest risk factors for transmission7,30. Therefore, greater understanding regarding the added risk of weather changes is needed to aid future decisions on restricting gatherings or implementing mandates for protective face coverings. In this study, we assessed the relative impact of absolute humidity and human mobility in different climatological regimes on reported cases of COVID-19 in the US.
Partitioning climatological regimes
The US is geographically large and encompasses several different climatological regimes with varying absolute humidity trends. We partitioned all 3137 US counties into six exclusive clusters (Fig. 1) ranked by average absolute humidity (AH) using a dynamic time warping (DTW) algorithm which considers both magnitude and functional trends of AH (see "Methods"). The cluster with the lowest average AH was primarily located in the western region of the US, while the region with the highest average AH was located on the southern coast bordering the Gulf of Mexico. Large changes of humidity were seen in clusters High 1 and High 2 which, respectively, includes variances of 26.9 and 30.6 g/m3 (see Fig. S1), while Low 1 and Low 2 humidity clusters had a variance of 4.5 and 14.2 g/m3.
(A) Map of US Counties and their respective absolute humidity clusters. Each county is colored based on their cluster. Counties that are included in the regression analysis are indicated by a darker shade. The clustering analysis was conducted using a partitional algorithm that utilized dynamic time warping (DTW) to measure similarity between absolute humidity profiles of 3137 counties in the United States. Expectantly, the clustering of absolute humidity is related to the geography of the counties which serves as a proxy for regional weather patterns and different climatological regimes. (B) The cross-sectional smoothed mean of human encounter absolute humidity, and new case per 10,000 people trends for each cluster group of the 497 counties analyzed in the regression analysis. Map was generated using the ggplot package31 in R.
Associations between humidity and cases rates
We conducted a regression on counties with more than 50,000 people using a generalized linear model (GLM) and controlling for individual movement and behavior with a metric from mobile phone data of visits to non-essential businesses (see Methods), we found that increases in AH were significantly negatively associated with cases per 100,000 of COVID-19 in all the non-high humidity regions (Table 1). We found that counties that belong to the least humid clusters, Low 1 and Low 2, had a 1 g/m3 increase in AH was associated with an average decrease of 14 percent reduction in cases over the entire duration, while the most humid clusters (High 1 and High 2) had a decrease of 4 percent in cases. The largest associations were seen in counties predominantly in the Rocky Mountains (Low 1; 20% decrease in daily cases), Upper Midwest/Northwest (Mid 1; 12% decrease in daily cases), West Coast/Texas/Northeast (Mid 2; 16% decrease in daily cases), and a region stretching along the western edge of the Midwest down to Texas (Low 2; 8% decrease in daily cases). Small but significant effects were detected in two high humidity clusters, both located in the southern region of the US (High 1 and High 2), with respective reductions of 6% and 1% in daily cases with a 1 g/m3 increase in AH.
Table 1 Untransformed GLM coefficient estimates for the entire study period.
The overall associations between AH and COVID-19 cases were negatively correlated when disaggregated across the time periods (Tables 2 and 3). The regression showed that AH had strong associations in the Mid 2 cluster, located in West Coast/Texas/Northeast, during the spring and summer months of 2020 (Table 2). In the fall of 2020 and spring of 2021, AH associations were generally stronger in counties from Mid 2 and High 1 clusters, which are in the West Coast, Texas, Northeast and Southern regions of the US (Table 3).
Table 2 Untransformed GLM coefficient estimates for the 2020 spring to fall period.
Table 3 Untransformed GLM coefficient estimates for the 2020 winter and 2021 spring seasons.
Associations between movement and case rates
In general, movement effects on daily cases are larger than absolute humidity effects, with visits to retail and recreation positively associated with new COVID-19 cases in most of the clusters (Table 1). Mobility trends for retail & recreation and grocery stores & pharmacies had a larger positive effect during the earlier phase of the pandemic for most clusters (March 10 to September 30, 2020) compared to the later phase spanning from October 1, 2020 to March 1, 2021. The residential mobility trend was associated with a decrease in new cases in most clusters during the earlier phase of the pandemic (Table 2), while having a positive effect on daily cases during the later phase (Table 3).
Detecting multicollinearity between movement and absolute humidity
To understand the collinearity of the combined regressions shown in Tables 1, 2 and 3, we conducted robustness checks with additional regressions that included the AH and the mobility trends separately (See Tables S1–S18). Additionally, we calculated the Generalized Variational Inflation Factor (GVIF) for the regressions in our robustness checks. Workplaces and Residential Mobility Trends were the least collinear with other independent variables (absolute humidity, immunity factor, and previous 14-day caseload) supported by GVIF values less than 2. Mobility trends in Retail and Recreation Areas and Grocery Stores and Pharmacies were mostly non-collinear with few exceptions with GVIF values ranging between with a mean of 1.53 (range: 1.15–2.30) and 1.65 (1.28–2.63). And finally, Transit Stations and Parks demonstrated the most collinearity with mean GVIF values of 2.15 (1.45–3.71) and 2.01 (1.56–2.83).
As the COVID-19 epidemic continues in the US and given the surge of COVID-19 in the winter seasons, there is renewed interest in understanding the relationship between outbreaks and seasonal changes, especially climatological factors related to outdoor and indoor humidity. This is not the first study to investigate humidity impacts on transmission, which been associated with increased transmission of respiratory pathogens (e.g., influenza) and SARS-CoV-2. While SARS-CoV-2 is a novel human virus, other pandemic coronaviruses (e.g., MERS-CoV and SARS-CoV-1)9,32,33,34,35 have also been associated with increased transmission in the winter, thus suggesting similar implications for SARS-CoV-2. Here, we found that the relative effect of absolute humidity on transmissions has so far been significant and was greatest in the Western, upper Midwest, and Northeast regions of the United States, which were clustered into the driest climatological regimes. These results support the hypothesis that falling rates of absolute humidity magnify the transmission risk of SARS-CoV-2, particularly in regions that are more arid and dry36. This effect was less noticeable for more humid regions, such as the coastal and southern counties of the US (Fig. 2).
The average daily new cases per 100,000 people plotted against the average Google Mobility Measure of 497 counties for the entire study duration. The plots are organized by type of movement and cluster group. For each plot, we added a simple linear trend line with shaded standard errors.
The effects of behavior and nonpharmaceutical interventions (NPI) are observed in our analysis when we disaggregate the analysis between the early and later phases of the pandemic. In the early phase of the pandemic, we see that an increase in mobility trends for retail & recreation resulted in an increase in daily cases, which measures visits to restaurants, cafes, shopping centers, theme parks, museums, libraries, and movie theaters. While in the later stages during the fall and winter of 2020, retail & recreation mobility had a lesser effect since many of those establishments were closed due to NPI policies. Furthermore, increases in residential mobility played a larger role in transmission, especially during the winter holidays when travel between residential homes occurred at a higher incidence.
The relationship between humidity and transmission is not fully clear, but several studies have shown that as absolute humidity decreases, survival times for enveloped viruses increase nonlinearly, including other coronaviruses9,22,37,38. Our findings support the hypothesis of a nonlinear relationship since the log-linear effects between humidity and case growth varied between climatological regimes. Our stratified regression and Fig. 2 show that different climatological regimes have different sensitivities to humidity changes. The increased survival of the virus in lower AH may be compounded by increased binding capacity, thereby enhancing the potential infectivity of the virus39. As AH falls, relative humidity indoors also decreases, which may increase susceptibility to airborne diseases40. This association suggests that increased humidification of indoor air in high transmission settings may help decrease the burden of COVID-19.
Given that our results suggest COVID-19 cases will increase significantly during winters, areas where humidity typically falls earlier in the fall (e.g., the upper Midwest) are likely to see cases increase earlier. In contrast, more humid regions (e.g., Gulf Coast areas) will likely observe outbreaks later in the winter. However, the results demonstrate that mobility had a larger and significant impact on cases, particularly when humidity was unchanging in the summer. Consequently, falling temperatures and holiday celebrations are likely to increase the risk of people gathering in indoor spaces for longer durations, resulting in a surge of COVID-19 cases through the winter, given that there are no substantial changes in population immunity and behavior.
The prior influenza pandemic in 2009 is instructive here, as increased contact patterns that occurred in the fall likely combined with falling humidity to drive transmission, which resulted in the peak of infections occurring significantly earlier than other years. Given the uncertainty and nonlinear effects of humidity on transmission, increasing vaccination, proper social distancing, and improving healthcare capacities can potentially reduce the toll of the COVID-19 pandemic. In addition, the uncertainty regarding the role of children in transmission41,42,43, who remain largely unvaccinated, suggests that proper precautions related to opening schools is warranted as the potential for transmission increases. While studies linking schools to outbreaks to date have been limited, few have occurred during the winter when transmission is higher.
We suspected that a relationship between human behavior and climate might exist which can cause variations in encounters. During winter months, the likelihood of being indoors increases especially in colder climates. To investigate this potential interaction, we conducted a collinear analysis. We can interpret this collinear analysis as residential and workplace movement patterns not being collinear with meteorological conditions (absolute humidity) and epidemiological factors (immunity factor and new cases per 100,000 (14-day Lag)). Retail/recreation and grocery/pharmacies are moderately collinear, while transit stations and parks were the most collinearly related to meteorological and epidemiological variables.
One limitation of this study includes changing social distancing dynamics and masking adherence between counties. We attempted to account for county-level heterogeneities using fixed effects for each county, but these are static effects. Furthermore, it is difficult to disentangle the epidemiological dynamics that cause exponential growth of cases. Events related to evacuation in natural disasters or mass-gatherings during the summer of 2020 that were not reflected in the Google Mobility Data44 may bias the analysis. Also, as with many COVID-19 analyses on retrospective data, the differences in testing rates at the county-level will result in varying detection rates of actual cases. Potential variations around vaccination efficacy for variants and within-host changes will impact the magnitude and exact timing of outbreaks45.
Transmission of SARS-CoV-2 will likely increase during the winters in the United States and other temperate regions in the northern hemisphere due in part to falling humidity. Studies of prior viruses and preliminary studies of the SARS-CoV-2 virus underpin the theoretical connection between humidity and transmission of droplet and aerosols. Nevertheless, mobility is still a significant driver of transmission.
The United States is geographically large, and the timing and magnitude of changes in absolute humidity can vary widely across regions. In order to account for regional differences in humidity, we utilized a partitional clustering algorithm with dynamic time warping (DTW) similarity measurements46 to classify the absolute humidity temporal profile for all observed counties into six exclusive clusters that are ranked based on average humidity. The clustering algorithm was implemented using the dtw package in R47. These clusters are ranked from lowest to highest as Low 1, Low 2, Mid 1, Mid 2, High 1, and High 2. Clustering allowed us to designate groups of counties based on temporal, climatological regimes and to stratify different absolute humidity patterns, which reduces group-level effects and enhances the independence of the data points. The DTW clustering of absolute humidity was conducted on a larger set of 3,137 counties. In the regression analysis, we included data from a subset of counties that had more than twenty cumulative confirmed cases and a population of more than 50,000 people. We excluded any days with fewer than 20 cumulative confirmed cases within each county because early transmission dynamics had a high rate of undetected cases48, making the data unreliable for this analysis. The final dataset used in the regression analysis included 497 counties, where separate panel data GLM was conducted on counties in each cluster (NLow1 = 39, NLow 2 = 42, NMid1 = 118, NMid2 = 108, NHigh1 = 78, and NHigh2 = 105). We assessed the results of the model over the entirety of the dataset and two time periods in 2020–2021: (1) the entire duration of the dataset (March 10, 2020 to March 1, 2021), (2) spring and summer when humidity increases (March 10, 2020 to September 30, 2020), and (3) the fall and winter months when humidity decreases to its lowest point (October 1, 2020 to March 1, 2021).
Confirmed case data were extracted from the Johns Hopkins Center for Systems Science and Engineering1 for each county. Population data were obtained from the US Census Bureau49 for 3,137 counties from March 10, 2020 to March 1, 2021. Daily cases were obtained from the confirmed case count by taking a simple difference between the days. Any data incongruencies, such as negative case counts, were omitted in our analysis.
Daily average absolute humidity for each US county, excluding territories, was calculated using temperature and dewpoint data from the National Centers for Environmental Information50 at the National Oceanic and Atmospheric Administration (NOAA). Time series data for the year 2020 from US weather stations were acquired from the NOAA Global Summary of the Day Index51. Weather stations were mapped using latitude and longitude to corresponding counties using the Federal Communications Commission (FCC) Census Block API52. For counties without a weather station, we used data from the nearest station, which was calculated based on distance from the county's spatial centroid using the haversine formula. In cases where counties contained multiple stations, data were averaged across all stations in a county. Absolute humidity was calculated using average daily temperature and average daily dew point (see Alduchov and Eskridge53).
Data on mobility from March 10, 2020 to March 1, 2021 was obtained from the Google COVID-19 Community Mobility Reports54. We specifically utilized the metric that measures visits to grocery stores & pharmacies, parks, transit stations, retail & recreation, residential, and workplaces by comparing the median rate on the county-level to a 5-week period Jan 3–Feb 6, 2020. The measure was calculated as the percent difference from before policy interventions (e.g., shelter-in-place orders) began to impact movement. This temporal measure allowed us to compare movement differences across counties.
For each humidity cluster that was classified using the DTW algorithm, we conducted three multivariate regressions using a generalized linear model (GLM) that assessed the time-weighted association between absolute humidity and non-essential visits with the number of new coronavirus cases (Eqs. 1–3). The GLM regression results in Tables 1, 2 and 3 are described in the following equation,
$$\begin{aligned} \log \left( {Y_{it} } \right) = & \log \left( N \right) + \alpha + \beta_{1} IM_{t} + \beta_{2} y_{{i\left( {t - \delta } \right)}} + \beta_{3} AH_{{i\left( {t - \delta } \right)}} + \beta_{4} RR_{{i\left( {t - \delta } \right)}} + \beta_{5} GP_{{i\left( {t - \delta } \right)}} \\ & + \beta_{6} PK_{{i\left( {t - \delta } \right)}} + \beta_{7} TS_{{i\left( {t - \delta } \right)}} + \beta_{8} WP_{{i\left( {t - \delta } \right)}} + \beta_{9} RD_{{i\left( {t - \delta } \right)}} + \gamma_{i} + \epsilon_{it} \\ \end{aligned}$$
where Yit, is the number of daily COVID-19 cases for county i at time t, log(N) is an offset term to control for population-size, and α is the intercept. In order to account for population immunity and exponential growth dynamics, we added the independent variables cumulative cases per 100,000, IMt, and lagged daily cases per 100,000, yi(t-δ) to the regression models. Absolute humidity, AHi(t-δ) is smoothed using a 7-day moving average and lagged by δ days. Google mobility trends to retail and recreation, RRi(t-δ), grocery and pharmacies, GPi(t-δ), parks, PKi(t-δ), transit stations, TSi(t-δ), workplaces, WPi(t-δ), residential places, RDi(t-δ), are smoothed using a 7-day moving average, lagged by δ days, and rescaled and centered on the mean. Fixed effects γi for each county were added to capture unobserved heterogeneities between counties. For our study, we assumed that the lag length δ was equal to 14 days, which is based on previous studies investigating lagged effects due to the incubation period of COVID-1955. As our outcome variable was daily cases, we modeled the variable as a Poisson distributed random variable with a log-transformed link function. Standard errors were calculated for the estimated linear coefficients β.
We conducted additional regressions on the absolute humidity and mobility measures as predictors individually to test for robustness. Specifically, we fit a GLM with absolute humidity for each humidity cluster and one measure from rescaled Google COVID-19 Community Mobility as linear predictors for new daily cases, as described in Eqs. (2) to (8).
$$\log \left( {Y_{it} } \right) = \log \left( N \right) + \alpha + \beta_{1} IM_{t} + \beta_{2} y_{{i\left( {t - \delta } \right)}} + \beta_{3} AH_{{i\left( {t - \delta } \right)}} + \gamma_{i} + \epsilon_{it}$$
$$\log \left( {Y_{it} } \right) = \log \left( N \right) + \alpha + \beta_{1} IM_{t} + \beta_{2} y_{{i\left( {t - \delta } \right)}} + \beta_{3} AH_{{i\left( {t - \delta } \right)}} + \beta_{4} RR_{{i\left( {t - \delta } \right)}} + \gamma_{i} + \epsilon_{it}$$
$$\log \left( {Y_{it} } \right) = \log \left( N \right) + \alpha + \beta_{1} IM_{t} + \beta_{2} y_{{i\left( {t - \delta } \right)}} + \beta_{3} AH_{{i\left( {t - \delta } \right)}} + \beta_{4} GP_{{i\left( {t - \delta } \right)}} + \gamma_{i} + \epsilon_{it}$$
$$\log \left( {Y_{it} } \right) = \log \left( N \right) + \alpha + \beta_{1} IM_{t} + \beta_{2} y_{{i\left( {t - \delta } \right)}} + \beta_{3} AH_{{i\left( {t - \delta } \right)}} + \beta_{4} PK_{{i\left( {t - \delta } \right)}} + \gamma_{i} + \epsilon_{it}$$
$$\log \left( {Y_{it} } \right) = \log \left( N \right) + \alpha + \beta_{1} IM_{t} + \beta_{2} y_{{i\left( {t - \delta } \right)}} + \beta_{3} AH_{{i\left( {t - \delta } \right)}} + \beta_{4} TS_{{i\left( {t - \delta } \right)}} + \gamma_{i} + \epsilon_{it}$$
$$\log \left( {Y_{it} } \right) = \log \left( N \right) + \alpha + \beta_{1} IM_{t} + \beta_{2} y_{{i\left( {t - \delta } \right)}} + \beta_{3} AH_{{i\left( {t - \delta } \right)}} + \beta_{4} RD_{{i\left( {t - \delta } \right)}} + \gamma_{i} + \epsilon_{it}$$
$$\log \left( {Y_{it} } \right) = \log \left( N \right) + \alpha + \beta_{1} IM_{t} + \beta_{2} y_{{i\left( {t - \delta } \right)}} + \beta_{3} AH_{{i\left( {t - \delta } \right)}} + \beta_{4} WP_{{i\left( {t - \delta } \right)}} + \gamma_{i} + \epsilon_{it}$$
To demonstrate robustness in the coefficient estimates, the coefficients in the combined regression analyses with absolute humidity and all mobility trends (Eq. (1)) were compared to the regression coefficients for absolute humidity and each mobility trend (Eqs. (2)–(8)). The analysis using GLM was conducted using the stats package in R (Version 4.0.2). All untransformed coefficient estimates are located in (Tables 1, 2 and 3). In the main text, we reported the logit-transformed estimates as relative change in cases per unit increase (1 g/m3) of absolute humidity. Given the log-linear relationship in a Poisson regression between the covariates and response variable, we can calculate the percent change in daily cases for a unit increase of a covariate to be equal to exp (β) − 1. For example, if β = − 0.112 for absolute humidity, we would state that there is a 9% (= exp (− 0.112) − 1) reduction for 1 g/m3 increase in absolute humidity. To verify that mulicollinearity is not a major issue, we conducted a collinearity analysis by calculating the Generalized Variational Inflation Factor (GVIF) for all regressions, which are listed in Table S19.
In addition to running a GLM regression, we also discretized the data based on months for each humidity cluster and calculated the Pearson correlation coefficient for absolute humidity and Google Mobility Trends against new cases (Fig. S2). Stationarity was checked for absolute humidity and Google mobility trends using the Levin-Lin-Chu unit-root test for unbalanced panel data for the three periods that were analyzed aforementioned regressions. Results for the stationarity are listed in Table S20 in the supplement.
We tested for robustness and externally validated our regressions by conducting additional analysis using K-folds cross-validation. We validated the coefficient estimation of all the GLMs mentioned previously by showing that the relative effect size for each regression was similar. The analysis was conducted over 100 folds or iterations with separate training and test sets derived from a subset of the county-level data. We used test sets for each fold where the mean square error (MSE) was calculated for each fit and shown in Table S22 in the supplement. In order to minimize overfitting, we also excluded county-level fixed effects in our cross-validation analysis. Additionally, we show the 95% confidence intervals of all parameter estimations using the GLM model that includes all variables in Table S23.
The data that support the findings of this study are openly available through the Johns Hopkins Center for Systems Science and Engineering, Unacast Social Distancing Scorecard, and NOAA National Centers for Environmental Information. Population data can be found through the US Census Bureau Website. All input data and code used to conduct the analysis and generate figures are also available on Github at https://github.com/CDDEP-DC/COVID-Humidity-Mobility-GAM.
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This work was funded by the Centers for Disease Control and Prevention (CDC) MInD-Healthcare Program (Grant Numbers U01CK000589, 1U01CK000536, and contract number 75D30120P07912). The funders had no role in the design, data collection and analysis, decision to publish, or preparation of the manuscript.
Center for Disease Dynamics, Economics & Policy, 962 Wayne Avenue, Suite 530, Silver Spring, MD, 20910-4433, USA
Gary Lin, Alisa Hamilton, Oliver Gatalo, Fardad Haghpanah & Eili Klein
Department of Civil and Systems Engineering, Johns Hopkins University, Baltimore, MD, USA
Takeru Igusa
Department of Earth and Planetary Sciences, Johns Hopkins University, Baltimore, MD, USA
Center for Systems Science and Engineering, Johns Hopkins University, Baltimore, MD, USA
Department of Emergency Medicine, Johns Hopkins University, Baltimore, MD, USA
Eili Klein
Gary Lin
Alisa Hamilton
Oliver Gatalo
Fardad Haghpanah
E.K. conceived the research, G.L. designed the study, A.H. and O.G. collected and processed the data, G.L., E.K., F.H., T.I. analyzed and interpreted the data. All authors contributed to interpretation of results and manuscript writing.
Correspondence to Gary Lin.
Supplementary Information.
Lin, G., Hamilton, A., Gatalo, O. et al. Investigating the effects of absolute humidity and movement on COVID-19 seasonality in the United States. Sci Rep 12, 16729 (2022). https://doi.org/10.1038/s41598-022-19898-8
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Scientific Reports (Sci Rep) ISSN 2045-2322 (online) | CommonCrawl |
Tverberg's theorem
In discrete geometry, Tverberg's theorem, first stated by Helge Tverberg in 1966,[1] is the result that sufficiently many points in d-dimensional Euclidean space can be partitioned into subsets with intersecting convex hulls. Specifically, for any positive integers d, r and any set of
$(d+1)(r-1)+1\ $
points there exists a point x (not necessarily one of the given points) and a partition of the given points into r subsets, such that x belongs to the convex hull of all of the subsets. The partition resulting from this theorem is known as a Tverberg partition.
The special case r = 2 was proved earlier by Radon, and it is known as Radon's theorem.
Examples
The case d = 1 states that any 2r-1 points on the real line can be partitioned into r subsets with intersecting convex hulls. Indeed, if the points are x1 < x2 < ... < x2r < x2r-1, then the partition into Ai = {xi, x2r-i} for i in 1,...,r satisfies this condition (and it is unique).
For r = 2, Tverberg's theorem states that any d + 2 points may be partitioned into two subsets with intersecting convex hulls. This is known as Radon's theorem. In this case, for points in general position, the partition is unique.
The case r = 3 and d = 2 states that any seven points in the plane may be partitioned into three subsets with intersecting convex hulls. The illustration shows an example in which the seven points are the vertices of a regular heptagon. As the example shows, there may be many different Tverberg partitions of the same set of points; these seven points may be partitioned in seven different ways that differ by rotations of each other.
Topological Tverberg Theorem
An equivalent formulation of Tverberg's theorem is:
Let d, r be positive integers, and let N := (d+1)(r-1). If ƒ is any affine function from an N-dimensional simplex ΔN to Rd, then there are r pairwise-disjoint faces of ΔN whose images under ƒ intersect. That is: there exist faces F1,...,Fr of ΔN such that $\forall i,j\in [r]:F_{i}\cap F_{j}=\emptyset $ and $f(F_{1})\cap \cdots \cap f(F_{r})\neq \emptyset $.
They are equivalent because any affine function on a simplex is uniquely determined by the images of its vertices. Formally, let ƒ be an affine function from ΔN to Rd. Let $v_{1},v_{2},\dots ,v_{N+1}$ be the vertices of ΔN, and let $x_{1},x_{2},\dots ,x_{N+1}$ be their images under ƒ. By the original formulation, the $x_{1},x_{2},\dots ,x_{N+1}$ can be partitioned into r disjoint subsets, e.g. ((xi)i in Aj)j in [r] with overlapping convex hull. Because f is affine, the convex hull of (xi)i in Aj is the image of the face spanned by the vertices (vi)i in Aj for all j in [r]. These faces are pairwise-disjoint, and their images under f intersect - as claimed by the new formulation. The topological Tverberg theorem generalizes this formluation. It allows f to be any continuous function - not necessarily affine. But, currently it is proved only for the case where r is a prime power:
Let d be a positive integer, and let r be a power of a prime number. Let N := (d+1)(r-1). If ƒ is any continuous function from an N-dimensional simplex ΔN to Rd, then there are r pairwise-disjoint faces of ΔN whose images under ƒ intersect. That is: there exist faces F1,...,Fr of ΔN such that $\forall i,j\in [r]:F_{i}\cap F_{j}=\emptyset $ and $f(F_{1})\cap \cdots \cap f(F_{r})\neq \emptyset $.
Proofs
The topological Tverberg theorem was proved for prime r by Barany, Shlosman and Szucs.[2] Matousek[3]: 162--163 presents a proof using deleted joins.
The theorem was proved for r a prime-power by Ozaydin,[4] and later by Volovikov[5] and Sarkaria.[6]
See also
• Rota's basis conjecture
• Tverberg-type theorems and the Fractional Helly property.[7]
References
1. Tverberg, H. (1966), "A generalization of Radon's theorem" (PDF), Journal of the London Mathematical Society, 41: 123–128, doi:10.1112/jlms/s1-41.1.123
2. Bárány, I.; Shlosman, S. B.; Szücs, A. (1981-02-01). "On a Topological Generalization of a Theorem of Tverberg". Journal of the London Mathematical Society. s2-23 (1): 158–164. doi:10.1112/jlms/s2-23.1.158.
3. Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3
4. Ozaydin, Murad (1987). "Equivariant Maps for the Symmetric Group". {{cite journal}}: Cite journal requires |journal= (help)
5. Volovikov, A. Yu. (1996-03-01). "On a topological generalization of the Tverberg theorem". Mathematical Notes. 59 (3): 324–326. doi:10.1007/BF02308547. ISSN 1573-8876.
6. Sarkaria, K. S. (2000-11-01). "Tverberg partitions and Borsuk–Ulam theorems". Pacific Journal of Mathematics. 196 (1): 231–241. ISSN 0030-8730.
7. Hell, S. (2006), Tverberg-type theorems and the Fractional Helly property, Dissertation, TU Berlin, doi:10.14279/depositonce-1464
| Wikipedia |
AD0 (1011)
For the manufacture of semi-finished products (sheets, tapes, strips, plates, profiles, panels, rods, tubes, wire, stampings and forgings) by hot or cold deformation, as well as ingots and slabs
CIS, Russia, Ukraine Aluminum, aluminum alloys Technical aluminum
GOST 4784-97 < 0.4 < 0.25 < 0.05 < 0.05 < 0.05 < 0.05 < 0.07 < 99.5
Temperature, °C
$$E\cdot 10^{9}$$, $$MPa$$
$$\alpha\cdot 10^{6}$$, $$K^{-1}$$
$$\varkappa$$, $$\frac{W}{m\cdot K}$$
$$\rho$$, $$\frac{kg}{m^3}$$
$$R\cdot 10^{-6}$$, $$\Omega\cdot m$$
20 71 2710 0.0292
Mechanical properties at 20 °C
Size, mm
$$\sigma _{U}$$, $$MPa$$
$$\epsilon_L$$, %
Trumpet 60 20
Trumpet cold-work 80 4–5
Round GOST 21488-97 60 25
Tape annealing 60 20–30
Tape cold-work 130–145 3–5
Profile 59 20
Sheet 64–78 15–18
Value, HBW
Alloy Cast 20
Alloy deformation 30–35
Alloy annealing 25
Casting and technological parameters
melting point, °C
3.0255 A91060
GOST 4784-97
Description of physical characteristics
$$E\cdot 10^{9}$$ $$MPa$$ Elastic modulus
$$\alpha\cdot 10^{6}$$ $$K^{-1}$$ Coefficient of thermal (linear) expansion (range 20°C–T)
$$\varkappa$$ $$\frac{W}{m\cdot K}$$ Coefficient of thermal conductivity (the heat capacity of the material)
$$\rho$$ $$\frac{kg}{m^3}$$ The density of the material
$$R\cdot 10^{-6}$$ $$\Omega\cdot m$$ Electrical resistivity
Description of mechanical properties
$$\sigma _{U}$$ $$MPa$$ Ultimate tensile strength
$$\epsilon_L$$ % Elongation at break (longitudinal)
Description of the casting and technological parameters
melting point °C The temperature at which solid crystalline body makes the transition to the liquid state and Vice versa | CommonCrawl |
The math behind ANN (ANN- Part 2)
The math behind ANN The math behind ANN Table of contents
Multi-layer perceptron
Perceptron
Feedforward loop
Back propagation of error
Update weights and biases
Derivation of learning rules
Introduction¶
This is second in the series of blogs about neural networks. In this blog, we will discuss the back propagation algorithm. In the previous blog, we have seen how a single perceptron works when the data is linearly separable. In this blog, we will look at the working of a multi-layered perceptron (with theory) and understand the maths behind back propagation.
Multi-layer perceptron¶
An MLP is composed of one input layer, one or more layers of perceptrons called hidden layers, and one final perceptron layer called the output layer. Every layer except the input layer is connected with a bias neuron and is fully connected to the next layer.
Perceptron¶
In the previous blog, we have seen a perceptron with a single TLU. A Perceptron with two inputs and three outputs is shown below. Generally, an extra bias feature is added as input. It represents a particular type of neuron called bias neuron. A bias neuron outputs one all the time. This layer of TLUs is called a perceptron.
In the above perceptron, the inputs are x1 and x2. The outputs are y1, y2 and y3. Θ (or f) is the activation function. In the last blog, the step function is taken as the activation function. There are other activation functions such as:
Sigmoid function: It is S-shaped, continuous and differentiable where the output ranges from 0 to 1.
$$ f(z)=\frac{1}{1+e^{-z}} $$
Hyperbolic Tangent function: It is S-shaped, continuous and differentiable where the output ranges from -1 to 1. $$f\left(z\right)=Tanh\left(z\right) $$
Training¶
Training a network in ANN has three stages, feedforward of input training pattern, back propagation of the error and adjustment of weights. Let us understand it using a simple example.
Consider a simple two-layered perceptron as shown:
Nomenclature¶
\(X_i\) Input neuron \(Y_k\)
\(x_i\) Input value \(y_k\) Output value
\(Z_j\) Hidden neuron \(z_j\) The output of a hidden neuron
\(\delta_k\) The portion of error correction for weight \(w_{jk}\) \(\delta_j\) The portion of error correction for weight \(v_{ij}\)
\(w_{jk}\) Weight of j to k \(v_{ij}\) weight of i to j
\(\alpha\) Learning rate \(t_j\) Actual output
f Activation function -- --
During feedforward, each input unit \(X_i\) receives input and broadcasts the signal to each of the hidden units \(Z_1\ldots Z_j\). Each hidden unit then computes its activation and sends its signal (\(z_j\)) to each output unit. Each output unit \(Y_k\) computes its activation (\(y_k\)) to form the response to the input pattern.
During training, each output unit \(Y_k\) compares its predicted output \(y_k\) with its actual output \(t_k\) to determine the error associated with that unit. Based on this error, \(\delta_k\) is computed. This \(\delta_k\) is used to distribute the error at the output unit back to all input units in the previous layers. Similarly, \(\delta_j\) is computed for all hidden layers \(Z_j\) which is propagated to the input layer.
The \(\delta_k\) and \(\delta_j\) are used to update the weights \(w_{jk}\) and \(v_{ij}\) respectively. The weight adjustment is based on gradient descent and is dependent on error gradient (\(\delta\)), learning rate (\(\alpha\)) and input to the neuron.
Mathematically this means the following:
Feedforward loop¶
Each input unit (\(X_i\)) receives the input \(x_i\) and broadcasts this signal to all units to the hidden layers \(Z_j\).
Hidden layer
Each hidden unit (\(Z_j\)) sums its weighted input signals \(z\_in_j=v_{0j}+\sum_{i} x_i\times v_{ij}\)
The activation function is applied to this weighted sum to get the output. \(z_j=f\left(z\_in_j\right)\) (where f is the activation function).
Each hidden layer sends this signal (\(z_j\)) to the output layers.
Output layer
Each output unit (\(Y_k\)) sums its weighted input signals \(y\_in_k=w_{0k}+\sum_{j}z_j\times w_{jk}\)
The activation function is applied to this weighted sum to get the output. \(y_k=f\left(y\_in_k\right)\) (where f is the activation function).
Back propagation of error¶
The error information term (\(\delta_k\)) is computed at every output unit (\(Y_k\)). $$ \delta_k=\left(t_k-y_k\right)f' y_in_k $$
This error is propagated back to the hidden layer. (later weights will be updated using this \(\delta\))
Each hidden unit (\(Z_j\)) sums its weighted error from the output layer $$ \delta_in_j=\sum_{k}\delta_k\times w_{jk} $$
The derivative of the activation function is multiplied to this weighted sum to get the weighted error information term at the hidden layer. $$ \delta_j=\delta_in_j \times f^\prime\left(z_in_j\right) $$ (where f is the activation function).
This error is propagated back to the initial layer.
Update weights and biases¶
The weights are updated based on the error information terms $$ w_{jk}\left(new\right)=w_{jk}\left(old\right)+\Delta w_{jk} $$ where $ \Delta w_{jk}=\alpha\times\delta_k\times z_j $
$$ v_{ij}\left(new\right)=v_{ij}\left(old\right)+\Delta v_{ij} $$ where $ \Delta v_{ij}=\alpha\times\delta_j\times x_i $
Steps 1 to 12 are done for each training epoch until a stopping criterion is met.
Derivation of learning rules¶
In every loop while training, we are changing the weights (\(v_{ij}\) and \(w_{jk}\)) to find the optimal solution. What we want to do is to find the effect of changing the weights on the error, and minimise the error using gradient descent.
The error gradient that has to be minimised is given by: $$E=\frac{1}{2}\sum_{k}\left(t_k-y_k\right)^2 $$ The effect of changing an outer layer weight (\(w_{jk}\)) on the error is given by: $$\frac{\partial E}{\partial w_{jk}}=\frac{\partial}{\partial w_{jk}}\frac{1}{2}\sum_{k}\left(t_k-y_k\right)^2 $$ $$ =\left(y_k-t_k\right)\frac{\partial}{\partial w_{jk}}f\left(y_in_k\right) $$ $$ =\left(y_k-t_k\right)\times z_j\times f'\left(y_in_k\right) $$ Therefore $$ \Delta w_{jk}=\alpha\frac{\partial E}{\partial w_{jk}}=\alpha\times\left(y_k-t_k\right)\times z_j\times f^\prime\left(y_in_k\right)={\alpha\times\delta}_k\times z_j $$
The effect of changing the weight of a hidden layer weight (\(v_{ij}\)) on the error is given by:
$$ \frac{\partial E}{\partial v_{ij}}=\sum_{k}{\left(y_k-t_k\right)\frac{\partial}{\partial v_{ij}}f\left(y_k\right)} $$ $$ =\sum_{k}\left(y_k-t_k\right)f^\prime\left(y_in_k\right)\frac{\partial}{\partial v_{ij}}f\left(y_k\right) $$ $$ =\sum_{k}\delta_kf^\prime\left(z_in_j\right)\left[x_i\right] =\delta_j\times x_i $$ Therefore $$ \Delta v_{ij}=\alpha\frac{\partial E}{\partial v_{ij}}={\alpha\times\delta}_j\times x_i $$ This way, for any number of layers, we can find the error information terms. Using gradient descent, we can minimise the error and find optimal weights for the ANN. In the next blog, we will implement ANN on the Titanic problem and compare it with logistic regression.
Fausett, L., 1994. Fundamentals of neural networks: architectures, algorithms, and applications. Prentice-Hall, Inc.
Previous Artificial Neural Network (Theory)
Next Linear Programming (R) | CommonCrawl |
Demographic modeling of transient amplifying cell population growth
MBE Home
Designing neural networks for modeling biological data: A statistical perspective
2014, 11(2): 343-361. doi: 10.3934/mbe.2014.11.343
Distributed, layered and reliable computing nets to represent neuronal receptive fields
Arminda Moreno-Díaz 1, , Gabriel de Blasio 2, and Moreno-Díaz Jr. 2,
Facultad de Informática, Universidad Politécnica de Madrid (UPM), Spain
Instituto Universitario de Ciencias y Tecnologías Cibernéticas, Universidad de Las Palmas de Gran Canaria, Spain, Spain
Received September 2012 Revised April 2013 Published October 2013
Receptive fields of retinal and other sensory neurons show a large variety of spatiotemporal linear and non linear types of responses to local stimuli. In visual neurons, these responses present either asymmetric sensitive zones or center-surround organization. In most cases, the nature of the responses suggests the existence of a kind of distributed computation prior to the integration by the final cell which is evidently supported by the anatomy. We describe a new kind of discrete and continuous filters to model the kind of computations taking place in the receptive fields of retinal cells. To show their performance in the analysis of different non-trivial neuron-like structures, we use a computer tool specifically programmed by the authors to that effect. This tool is also extended to study the effect of lesions on the whole performance of our model nets.
Keywords: Hermite functions, Layered and distributed computation, Newton filters, weight profile analysis and synthesis., reliable nets.
Mathematics Subject Classification: Primary: 93B15, 93B51; Secondary: 93A3.
Citation: Arminda Moreno-Díaz, Gabriel de Blasio, Moreno-Díaz Jr.. Distributed, layered and reliable computing nets to represent neuronal receptive fields. Mathematical Biosciences & Engineering, 2014, 11 (2) : 343-361. doi: 10.3934/mbe.2014.11.343
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Arminda Moreno-Díaz Gabriel de Blasio Moreno-Díaz Jr. | CommonCrawl |
\begin{definition}[Definition:Cardinality of Structure]
Let $\AA$ be a first-order structure.
Then the '''cardinality of $\AA$''', denoted $\card \AA$, is defined as:
:$\card \AA := \card A$
where $\card A$ is the cardinality of the underlying set $A$.
\end{definition} | ProofWiki |
Expression of the T regulatory cell transcription factor FoxP3 in peri-implantation phase endometrium in infertile women with endometriosis
Shufang Chen1,
Jian Zhang1,
Changxiao Huang1,
Wen Lu1,
Yan Liang1 &
Xiaoping Wan1
Endometriosis (EM) is highly associated with infertility. The precise mechanism underlying EM-associated infertility remains controversial. This study aimed to investigate the pathogenesis of infertility in women with EM by comparing FoxP3+ T regulatory cells (Tregs) expression in the eutopic endometrium of infertile women with EM and endometrium from healthy fertile women.
As a marker of Tregs, FoxP3 expression was analyzed in eutopic endometrium during the peri-implantation phase in infertile women with mild EM (n = 7), advanced EM (n = 20), and normally fertile women without EM (n = 20). FoxP3 mRNA expression was analyzed by quantitative real-time RT-PCR. FoxP3 protein expression was assessed by immunohistochemistry.
FoxP3 mRNA expression in all infertile patients with EM was significantly higher than the control group (P < 0.05) by non-parametric Mann–Whitney U-test. Further analysis based on the extent of EM revealed that FoxP3 mRNA expression in infertile patients with advanced EM was significantly higher than the mild EM group and the control group (P < 0.05). Immunohistochemistry analysis showed predominant positive staining for FoxP3 protein in the endometrial stroma. In addition, the number of FoxP3+ cells in the eutopic endometrium of infertile women with advanced EM was marginally higher than the mild EM group and the control group, although the differences were not statistically significant (P > 0.05) by two-tailed t-tests.
These findings suggest that FoxP3+ Tregs in the peri-implantation endometrium might participate in the pathogenesis of advanced EM. However, they are not directly involved in the pathogenesis of advanced EM associated with infertility. The differential expression of FoxP3 in infertile women with mild EM and advanced EM implicates that notable differences in the uterine immune status are likely involved in the pathogenesis of mild EM associated with infertility in the peri-implantation endometrium.
Endometriosis (EM) is a common and benign gynecological disorder that is highly associated with infertility. It affects approximately 10% to 15% of women of reproductive age and 25% to 50% of women with infertility. Moreover, 30% to 50% of women with EM are infertile [1]. Although the mechanisms underlying EM-associated infertility include abnormal folliculogenesis, elevated oxidative stress, altered immune function and hormonal milieu in the follicular and peritoneal environments, and reduced endometrial receptivity, the precise mechanism of pathogenesis remains controversial. The combination of these factors leads to poor oocyte quality and impaired fertilization and implantation [2, 3]. Recent studies have demonstrated that endometrial molecular defects during the implantation window might be a cause of EM-associated infertility. Increasing evidence suggests that EM patients have an impaired endometrium and/or an abnormal endometrial environment which make them functionally unfavorable for implantation and pregnancy progression [2, 4].
CD4+CD25+FoxP3+T regulatory cells (Tregs) are a specialized subpopulation of T cells that control and suppress a range of immune responses, including T-cell proliferation and activation, macrophage, B cell, DC and NK cell function, mast cell degranulation, cell proliferation, and cytokine release. Forkhead box protein 3 (FoxP3) is a member of the forkhead-box/winged-helix transcription factor family. It is a unique marker of Tregs. FoxP3 has been reported to be an essential controlling gene for the development and function of naturally occurring Treg populations.
Accumulating evidence from both experimental and clinical studies indicates that a balance between regulation and deletion of responder T cells is an effective strategy to control immune responsiveness after organ or cell transplantation [5]. Furthermore, FoxP3+ Tregs are critical for the maintenance of maternal immune tolerance as well as the prevention of autoimmunity and transplantation rejection. Recent studies have demonstrated links between impaired function or diminished Treg cell populations and complications during pregnancy due to defective implantation or placental insufficiency [6]. In miscarriage, reduced responsiveness to pregnancy associated expansion of Treg cell populations, due to numerically fewer Tregs as well as Treg functional deficiency, may underpin reduced immunosuppressive capability [7]. Compared to women with induced abortion, patients experiencing spontaneous abortion exhibit fewer decidual and peripheral blood CD4+CD25high T cells recovery. Women experiencing repeated miscarriage have been shown to have a reduced number of Tregs within the peripheral blood CD4+ pool and reduced suppressive capacity. Primary unexplained infertility is also associated with reduced expression of FoxP3 mRNA in endometrial tissue during mid-secretory phase of the menstrual cycle, suggesting that impaired differentiation and/or recruitment of uterine Tregs, even prior to conception, might affect patients' ability to establish pregnancy [8].
A recent study demonstrated that eutopic endometrial FoxP3 was up-regulated in women with EM, suggesting that FoxP3 plays a pathogenic role in the formation of EM [9]. However, very little is known about the role of FoxP3+ Tregs in the pathophysiologic mechanism of EM-associated infertility and the changes of FoxP3+ Treg population in different EM stages. In this study, FoxP3 expression in the endometrium during the peri-implantation phase was investigated by comparing infertile women with different stages of EM to normal fertile women. The purpose is to elucidate the pathogenesis of infertility in EM.
Patients and samples
All subjects were patients admitted in the Gynecology Department of the International Peace Maternity and Child Health Hospital, Shanghai Jiaotong University (China) between April 2009 and July 2010. There were 27 primary infertile women with regular menstrual cycles in the study group. All women had visual or biopsy-proven EM. They had undergone endometrial curettage and laparoscopic excision of endometriotic ovarian cysts or endometriotic peritoneal lesions between days 19 and 23 of the cycle. Procedures were performed based on endometrial histological dating [10] and the first day of their last menstrual period (LMP). Infertile women with EM exhibited a minimum 1 year of infertility with a current desire for conception, no chromosomal anomalies in either parent, no uterine structural abnormalities, no thrombophilic disorders, and no contribution of male factor infertility. The control group consisted of 20 women without pelvic EM, confirmed during laparoscopic surgery for para-ovarian cysts or mesosalpinx cysts. All of them had regular menstrual cycles and successful pregnancies. Endometrial biopsies from the control group were taken between days 19 and 23 of the cycle with dating confirmed by microscopic examination and LMP. All samples were histologically examined by a histopathology specialist. The extent of EM was staged according to the revised American Society for Reproductive Medicine classification (rAFS) system [11].
None of the patients in the study group and control group had received any hormone therapy within 6 months of the procedure, had experienced a miscarriage, or had a history of in vitro fertilization treatment. All women abstained from intercourse or used barrier methods of contraception for the period between their last menses and sample collection. Exclusion criteria included: history of autoimmune diseases, pelvic inflammatory disease, genital tract infection, use of intrauterine contraception for at least 6 months prior to surgery, endometrial hyperplasia or endometrial polyps, and concomitant adenomyosis and uterine fibroids. All women in the control group had only one living child and had no history of spontaneous abortion, ectopic pregnancy, or preterm delivery.
This study was approved by the human ethics committee of the International Peace Maternity and Child Health Hospital. A written informed consent form was obtained from each participant prior to their inclusion.
On the day of operation, endometrial tissue was obtained by curettage. Each biopsy was divided into two portions. One was snap-frozen in liquid nitrogen and stored at −80°C until further analysis was performed. The other portion was fixed in 10% neutral buffer formalin for 18 to 24 hours and was embedded in paraffin for further histological dating and immunohistochemical analysis. Plasma progesterone levels were measured to ensure that ovulation had occurred.
Tissue processing
Quantitative real-time PCR (qRT-PCR)
Total RNA was extracted using Trizol method (Invitrogen) from the homogenized tissues, and then quantified on a NanoDrop™ 1000 Spectrophotometer (Thermo scientific Waltham, MA, USA). The OD260nm/280 nm ratio was among 1.9 to 2.0. RNA samples were further assessed by electrophoresis on 1.5% agarose gels, and then visualized under UV light after ethidium bromide staining. RNA samples were stored at −80°C in aliquots until use.
cDNA was synthesized with 1 μg total RNA and 1 μl d(T)18 Oligo using RevertAid First Strand cDNA Synthesis Kit (Fermentas Thermo). Final volume was 20 μl. FoxP3 transcripts were relatively quantified by real-time RT-PCR with SYBR-Green master mix (ABI) on 7500 Real-time PCR System (Applied Biosystems, CA, USA). Reaction mixtures, in a total volume of 10 μl, contained 5 μl SYBR Green, 0.15 μl primer (Table 1), 1 μl cDNA (10 fold diluted) and 3.85 μl RNase-free water. As a negative control, H2O was used instead of cDNA. The PCR was carried out as follows: 95°C for 2 min, 40 cycles of 95°C for 15 s, and 60°C for 1 min. A melting curve was performed to check the specificity of amplification. The relative transcript concentration was calculated by 2-ΔΔCt taking GAPDH as interval standard control. Each sample was analyzed in triplicate.
Table 1 Primer sequences
Endometrial tissues were fixed, cut, mounted, deparaffinized, and rehydrate. After blocking with goat serum, the sections were incubated with the primary murine monoclonal anti-human FoxP3 antibody (Abcam, 236A/E7, Hong Kong) at 4°C overnight (1:50 dilution). After incubation with secondary polyperoxidase-anti-mouse IgG antibody (MR-biotech, Shanghai, China), a horseradish peroxidase detection system was applied. Immunoreactivity was detected using the diaminobenzidine tetrahydrochloride chromogen (MR-biotech, Shanghai, China). Sections were counterstained with hematoxylin, dehydrated and cleared. For a negative control, PBS was substituted for the primary antibody in the above protocols. A sample of lymph node tissue, known to contain FoxP3+ cells, was used as a positive control.
The sections were viewed using a light microscope (Zeiss MIC00958) under × 400 magnification (×40 objective, ×10 ocular). Tregs were characterized by the brown intracellular staining of FoxP3 antibody in the stroma of the endometrium. FoxP3+ cell counting was performed on 10 non-overlapping fields for each sample. Each slide was counted by two different observers blinded to the tissue origin. The average counting from each observer was calculated and a mean was calculated.
Statistical analysis was performed with GraphPad Prism Version 5. Biological parameters were presumed to exist in a normal distribution. Therefore, two-tailed t-tests were used to test significance, and the results were reported as means ± SD. As the results for FoxP3 mRNA expression did not conform to the normal distribution, the differences among groups were also assessed with a non-parametric Mann–Whitney U-test, and results were reported as the median and interquartile range. For all tests, P < 0.05 was considered statistically significant.
Basic clinical characteristics of patients
The staging of EM in the study group followed the rAFS classification. Patients were divided into two subgroups as follows: 7 patients with mild EM (stage I and II), and 20 with advanced EM (stage III and IV). The mean ages of patients in these two groups were 31.0 years (range, 27–40 years) and 29.8 years (range, 26–37 years), respectively. The mean durations of infertility were 2.85 years (range, 1–6 years) and 3.27 years (range, 1–8 years), respectively. The mean age in the control group (normally fertile women without EM) was 30.9 years (range, 25–37 years). No significant differences were noted in age, cycle length (mild EM 29.98 ± 1.32 vs. advanced EM 31.01 ± 1.41 vs. control 30.10 ± 1.20 days) or timing of sampling (mild EM 21.07 ± 1.60 vs. advanced EM 20.97 ± 1.98 vs. control 21.45 ± 1.35 days) among these groups. Furthermore, serum progesterone levels were similar in these groups (mild EM 50.34 ± 2.59 vs. advanced EM 51.67 ± 2.37 vs. control 52.10 ± 3.02 nmol/L). The demographic details of the study group and control group are summarized in Table 2.
Table 2 Clinical characteristics of all women in the study
Quantitative real-time PCR analysis of FoxP3 mRNA expression
FoxP3 mRNA expression in the study group (median, 1.09; interquartile range, 0.58-1.43) was significantly higher than the control group (median, 0.56; interquartile range, 0.21-0.72; P < 0.05). Further analysis based on the extent of EM revealed that FoxP3 mRNA expression in infertile patients with advanced EM (median, 1.20; interquartile range, 0.86-1.95) was significantly higher than the mild EM group (median, 0.38; interquartile range, 0.21-0.47) and the control group (P < 0.05). The median level of FoxP3 mRNA in infertile patients with mild EM was marginally lower than in the control group (0.38 vs. 0.56), although no statistically significant difference was detected between these two groups (P > 0.05) (Figure 1).
Quantitative real-time RT-PCR analysis of FoxP3 mRNA expression in peri-implantation phase eutopic endometrium. (a) Relative expression of FoxP3 mRNA determined by RT-PCR in peri-implantation phase eutopic endometrium (cycle days 19–23) from infertile women with endometriosis (n = 27) and normal fertile women without endometriosis (n = 20). *Significant difference compared endometriosis group with control group by the Mann–Whitney U-test (P < 0.05). (b) Relative expression of FoxP3 mRNA determined by RT-PCR in peri-implantation phase eutopic endometrium (cycle days 19–23) from infertile women with mild endometriosis (n = 7), advanced endometriosis (n = 20) and normal fertile women without endometriosis (n = 20). *Significant difference compared advanced endometriosis group with mild endometriosis group and control group by the Mann–Whitney U-test (P < 0.05). All values were expressed as median (range). (EM, infertile women with endometriosis; Mild EM, infertile women with mild endometriosis; Advanced EM, infertile women with advanced endometriosis).
Immunohistochemical staining of FoxP3 protein in human endometrium
Immunohistochemical staining showed that FoxP3 protein was predominantly expressed in the endometrial stroma (Figure 2). Enumeration of FoxP3+ Tregs was expressed as the mean number (± SD) of FoxP3+ cells per square millimeter of endometrium. The number of FoxP3+ cells in eutopic endometrium of infertile women with advanced EM (0.79 ± 0.52) was marginally higher than the mild EM group (0.50 ± 0.29) and the control group (0.51 ± 0.30), although there were no statistically significant differences among these groups (P > 0.05) (Table 3).
Immunohistochemical staining of FoxP3 protein in peri-implantation phase eutopic endometrium. Immunohistochemical staining of FoxP3 protein expression in eutopic endometrium during the peri-implantation phase. Positive immunolabeling for FoxP3 (brown diaminobenzidine chromogen coloration) occurs in the endometrial stroma (arrows). (a) FoxP3 protein expression in the endometrium of infertile patients with advanced endometriosis. (b) FoxP3 protein expression in the endometrium of infertile patients with mild endometriosis. (c) FoxP3 expression in the endometrium of the control group. No significant difference was observed among these groups. (all pictures, original magnification × 200; inset, original magnification × 400).
Table 3 FoxP3 levels in the endometrium
The mechanisms by which EM impairs fertility remain poorly understood. Accumulating evidence indicates that the eutopic endometrium of women with EM differs from that of women without EM [12], which may contribute to failure of implantation. A meta-analysis of in-vitro fertilization and embryo transfer (IVF-ET) trials showed that women with EM have similar ovulation and embryo formation rates compared to patients scheduled for IVF treatment without EM (e.g. blocked fallopian tubes). However, the implantation rates in women with EM are 50% lower than those achieved in patients being treated for other causes of infertility [13]. These results indicate a receptivity defect within the eutopic endometrium in women with EM that affects fertility regardless of other causes of infertility in EM (e.g. adhesions). Gene array studies have established aberrant gene expression in the endometrium of women with EM compared to those without EM during the implantation window [14]. Furthermore, studies indicate that an abnormal inflammatory environment is present, not only in pelvic endometriotic lesions, but also in the eutopic endometrium of patients with EM. Therefore, the decrease in fertility experienced by these women might be caused by inflammatory processes, which in turn, affecting ovulation and implantation.
Successful embryo implantation is a dynamic process, requiring dialog between the blastocyst and a receptive endometrium [15]. Although implantation is primarily regulated by the steroid hormones, a host of local immune cells, cytokines, growth factors and adhesion molecules have been identified that mediate the apposition, adhesion and invasion of the blastocyst [16, 17]. Maintenance of an optimal pro- and anti-inflammatory state at the feto-maternal interface is necessary for successful implantation. The leukocyte population in the endometrial environment at the time of implantation includes uterine natural killer (uNK), macrophages, T cells and B cells [18–20]. Dysregulation in the production of these factors may lead to aberrant implantation. As one of these factors, Tregs play a crucial role in regulation and suppression of local immune response during implantation phase.
Recently, studies in reproductive immunology show that Tregs play an important role in maternal tolerance of the conceptus. Their suppressive actions are exerted even prior to embryo implantation. Tregs are enriched at the fetal-maternal interface, showing a suppressive phenotype. Inadequate numbers of Tregs or their functional deficiency might be linked with miscarriage, pre-eclampsia, infertility and the failure of embryo implantation. Several studies have reported an association between Tregs and implantation failure or recurrent spontaneous miscarriage in humans. Women experiencing repeated miscarriage were shown to have a reduced frequency of Tregs within peripheral blood, and reduced suppressive capacity, compared to normal fertile women [7, 21]. Primary unexplained infertility has also been associated with reduced expression of Foxp3 mRNA in endometrial tissue in the mid-secretory phase of the menstrual cycle [8]. These studies suggest that reduction in the size and functional impairment of the Treg population and/or insufficient migration of Tregs to decidual tissue at the feto–maternal interface induce implantation failure in embryo implantation or recurrent spontaneous abortion in humans.
In contrast to other leukocytes, Tregs play the most crucial roles in controlling, suppressing and modulating a vast variety of immune responses in the development of endometriosis. Endometriosis is an inflammatory condition, associated with highly dysregulated immune response at both uterine and peritoneal levels. Recent evidence suggests that dysregulated immune response in EM is likely to originate within the eutopic endometrium [22]. Berbic et al. found that FoxP3+ cells in the eutopic endometrium of women with EM remained highly up-regulated during the secretory phase of the menstrual cycle, while at this time their expression was significantly down-regulated in women without EM [21]. They propose that FoxP3+ cells in eutopic endometrium in women with EM decrease the ability of newly recruited immune cell populations to effectively recognize and target endometrial antigens shed during menstruation, allowing their survival and ability to implant in ectopic sites [9]. Tregs are likely to be linked to pathogenesis and progression of EM. Basta et al. demonstrated that the disturbance in the immunological equilibrium observed in ectopic endometrium and deciduas would seem to be related to the alteration in the Treg cell population that occurs in these ectopic tissues. Additionally, no differences in the percentage of Tregs within the T lymphocyte subpopulation were observed over the course of the menstrual cycle in the ovarian endometriosis. They hypothesized that the absence of Tregs fluctuation can be linked to an immune defect arising with the development of endometriosis [23]. In another genetic marker research,André GM et al. first evaluated the association between FOXP3 polymorphisms in infertile women with and without EM. They suggest that the FOXP3 polymorphisms can be associated with risk of idiopathic infertility (rs2280883 and rs2232368) and EM (rs3761549) in Brazilian women [24]. In addition, recent studies have implicated Tregs in inducing tolerance to tumours. In Prieto's reviews, the proposed hypothesis predicts that local expansion of Tregs might suppress anti-tumour responses and facilitate the progression of EM to ovarian cancer in susceptible women [25].
Very few human studies of FoxP3 expression in the peri-implantation endometrium have been reported. FoxP3 expression in the endometrium of infertile patients with EM compared to healthy fertile women remains to be elucidated. Therefore, this study aimed to study the difference of FoxP3+ expression in endometrial tissue during the peri-implantation window between patients with EM-associated infertility and healthy fertile women. Furthermore, FoxP3 expression in the endometrium of infertile patients with different stages of EM and the role of Tregs in the etiology of infertility in women with EM were investigated as well. Analysis of FoxP3 mRNA expression by quantitative real-time RT-PCR revealed that infertile women with EM have higher levels of FoxP3 mRNA in eutopic endometrium than the control group. Further analysis based on the extent of EM revealed that infertile women with advanced EM have higher levels of FoxP3 mRNA in eutopic endometrium than women with mild EM and the control group. The results of this study conflict with those reported in 2006 by Jasper et al. [8], which demonstrated an association between unexplained infertility and reduced FoxP3 mRNA expression in endometrial tissue. However, the results of this study are consistent with Berbic's report, which demonstrated upregulation of FoxP3 expression in eutopic endometrium in women with EM in the secretory phase [9]. Therefore, it is hypothesized that FoxP3+ Tregs in the peri-implantation endometrium participate in the pathogenesis of EM while they are not directly involved in the pathogenesis of advanced EM associated infertility.
Additionally, our study showed that the expression of FoxP3 mRNA in the infertile women with mild EM was significant lower than patients with advanced EM, but it was similar to the control group. It is suggested by this result that the uterine immune status in peri-implantation endometrium among infertile patients with mild EM is different from that of the advanced EM. This kind of difference might be involved in the failure of embryo implantation and the pathogenesis of infertility in the mild EM. Although there was no statistically significant difference between the mild EM group and the control group, FoxP3 mRNA expression in the infertile women with mild EM was slightly lower. It is not clear whether the inadequate numbers of FoxP3+ Tregs in the peri-implantation endometrium in sub-fertile women with mild EM is related to the pathogenesis of infertility and unsuccessful embryo implantation. This needs to be examined in future studies with large sample size.
Tregs, which comprise only 5-10% of CD4+T cells in human,are few in eutopic endometrium and are periodically regulated by 17-β-estradiol [21]. Tregs will increase and exert full suppressive function when they expose to alloantigen, such as embryo, sperm, and inflammation [26, 27]. Since our study excluded the effect of alloantigen, the level of FoxP3+ cells by semiquantitative immunohistochemical staining was relatively low. Although there was no statistically significant difference, FoxP3+ expression in the advanced EM group was higher than the mild EM group and the control group. This result was consistent with the findings in the quantitative real-time PCR analysis. Because it is unethical to investigate human embryo implantation process in vivo, future studies may focus on the changes of Tregs during peri-implantation phase in the eutopic endometrium of infertile women with EM in vitro.
From the above, our findings suggest that FoxP3+ Tregs in the peri-implantation endometrium might participate in the pathogenesis of advanced EM. However, they are not directly involved in the pathogenesis of advanced EM associated with infertility. The differential expression of FoxP3 in infertile women with mild EM and advanced EM implicates that notable differences in the uterine immune status are likely involved in the pathogenesis of mild EM associated with infertility in the peri-implantation endometrium.
Tregs:
CD4+CD25+FoxP3+T regulatory cells
FoxP3:
$$ f(x) = {{a}_0} + \sum\nolimits_{{n = 1}}^{{00}} ( {{a}_n}\cos \frac{{n\pi x}}{L} + {{b}_n}\sin \frac{{n\pi x}}{L}) $$Forkhead box protein 3
LMP:
Last menstrual period
rAFS:
Revised American Society for Reproductive Medicine classification system
uNK:
Uterine natural killer
IVF-ET:
In-vitro fertilization and embryo transfer.
Giudice LC, Kao LC: Endometriosis. Lancet. 2004, 364: 1789-1799. 10.1016/S0140-6736(04)17403-5.
Ulukus M, Cakmak H, Arici A: The role of endometrium in endometriosis. J Soc Gynecol Investig. 2006, 13: 467-476.
Simo'n C, Gutie'rrez A, Vidal A, de los Santos MJ, Tari'n JJ, Remohi' J, Pellicer A: Outcome of patients with endometriosis in assisted reproduction: results from in vitro fertilization and oocyte donation. Hum Reprod. 1994, 9: 725-729.
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André GM, Barbosa CP, Teles JS, Vilarino FL, Christofolini DM, Bianco B: Analysis of FOXP3 polymorphisms in infertile women with and without endometriosis. Fertil Steril. 2011, 95: 2223-2227. 10.1016/j.fertnstert.2011.03.033.
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This work was supported by the young scientific research project of Shanghai Municipal Health Bureau, grant 2010Y154.
Department of Gynecology, International Peace Maternity and Child Health Hospital, School of Medicine, Shanghai Jiaotong University, Shanghai, China
Shufang Chen, Jian Zhang, Changxiao Huang, Wen Lu, Yan Liang & Xiaoping Wan
Shufang Chen
Jian Zhang
Changxiao Huang
Wen Lu
Yan Liang
Xiaoping Wan
Correspondence to Xiaoping Wan.
SC and JZ contributed to the design of the study, acquisition of data, analysis and interpretation of data, and writing the manuscript. CH, WL and YL were involved in the experimental work of the study. XW was principal project supervisor and was mainly responsible for the intellectual planning of the project. All authors read and approved the final manuscript.
Shufang Chen, Jian Zhang contributed equally to this work.
Chen, S., Zhang, J., Huang, C. et al. Expression of the T regulatory cell transcription factor FoxP3 in peri-implantation phase endometrium in infertile women with endometriosis. Reprod Biol Endocrinol 10, 34 (2012). https://doi.org/10.1186/1477-7827-10-34
FoxP3
T regulatory cells | CommonCrawl |
\begin{definition}[Definition:Triangle (Geometry)/Height]
The '''height''' of a triangle is the length of a perpendicular from the apex to whichever side has been chosen as its base.
That is, the length of the '''altitude''' so defined.
:400px
Thus the length of the '''altitude''' $h_a$ so constructed is called the '''height''' of $\triangle ABC$.
\end{definition} | ProofWiki |
\begin{document}
\title{Sasaki manifolds, K\"ahler cone manifolds and biharmonic submanifolds}
\title[Biharmonic submanifolds]
{Sasaki manifolds, K\"ahler cone manifolds and biharmonic submanifolds}
\author{Hajime Urakawa}
\address{Division of Mathematics, Graduate School of Information Sciences, Tohoku University, Aoba 6-3-09, Sendai, 980-8579, Japan}
\curraddr{Institute for International Education,
Tohoku University, Kawauchi 41, Sendai 980-8576, Japan}
\email{urakawa@@math.is.tohoku.ac.jp}
\keywords{Legendrian submanifold, Sasaki manifold, Lagrangian submanifold,
harmonic map, biharmonic map}
\subjclass[2000]{primary 58E20, secondary 53C43}
\thanks{
Supported by the Grant-in-Aid for the Scientific Reserch, (C) No. 25400154, Japan Society for the Promotion of Science.
} \maketitle \begin{abstract}
For a Legendrian submanifold $M$ of a Sasaki manifold $N$, we study harmonicity and biharmonicity of the corresponding Lagrangian cone submanifold $C(M)$ of a K\"ahler manifold $C(N)$. We show that, if $C(M)$ is biharmonic in $C(N)$, then it is harmonic; and $M$ is proper biharmonic in $N$ if and only if $C(M)$ has a non-zero eigen-section of the Jacobi operator with the eigenvalue $m=\dim M$.
\end{abstract}
\numberwithin{equation}{section} \theoremstyle{plain} \newtheorem{df}{Definition}[section] \newtheorem{th}[df]{Theorem} \newtheorem{prop}[df]{Proposition} \newtheorem{lem}[df]{Lemma} \newtheorem{cor}[df]{Corollary} \newtheorem{rem}[df]{Remark}
\section{Introduction} Harmonic maps play a central role in geometry;\,they are critical points of the energy functional $E(\varphi)=\frac12\int_M\vert d\varphi\vert^2\,v_g$ for smooth maps $\varphi$ of $(M,g)$ into $(N,h)$. The Euler-Lagrange equations are given by the vanishing of the tension filed $\tau(\varphi)$. In 1983, J. Eells and L. Lemaire \cite{EL1} extended the notion of harmonic map to biharmonic map, which are, by definition, critical points of the bienergy functional \begin{equation} E_2(\varphi)=\frac12\int_M \vert\tau(\varphi)\vert^2\,v_g. \end{equation} After G.Y. Jiang \cite{J} studied the first and second variation formulas of $E_2$, extensive studies in this area have been done (for instance, see \cite{CMP}, \cite{LO2}, \cite{MO1}, \cite{OT2}, \cite{S1}, \cite{IIU2}, \cite{IIU}, \cite{II},
etc.). Notice that harmonic maps are always biharmonic by definition.
We say, for a smooth map $\varphi:\,(M,g)\rightarrow (N,h)$
to be {\em proper biharmonic} if it is biharmonic, but not harmonic. B.Y. Chen raised (\cite{C}) so called B.Y. Chen's conjecture and later, R. Caddeo, S. Montaldo, P. Piu and C. Oniciuc raised (\cite{CMP}) the generalized B.Y. Chen's conjecture. \par \textbf{B.Y. Chen's conjecture:}\par
{\em Every biharmonic submanifold of the Euclidean space ${\mathbb R}^n$ must be harmonic (minimal).} \vskip0.6cm\par \textbf{The generalized B.Y. Chen's conjecture:}\par {\em Every biharmonic submanifold of a Riemannian manifold of non-positive curvature must be harmonic (minimal).} \vskip0.6cm\par For the generalized Chen's conjecture, Ou and Tang gave (\cite{OT}, \cite{OT2}) a counter example in a Riemannian manifold of negative curvature. For the Chen's conjecture, affirmative answers were known for the case of surfaces in the three dimensional Euclidean space (\cite{C}), and the case of hypersurfaces of the four dimensional Euclidean space (\cite{HV}, \cite{D}). Furthermore, Akutagawa and Maeta gave (\cite{AM}) recently a final supporting evidence to the Chen's conjecture:
\begin{th} Any complete regular biharmonic submanifold of the Euclidean space ${\mathbb R}^n$ is harmonic (minimal). \end{th}
\vskip0.6cm\par To the generalized Chen's conjecture, we showed (\cite{NUG}) that
\begin{th} Let $(M,g)$ be a complete Riemannian manifold, and the curvature of $(N,h)$, non-positive. Then, \par $(1)$ every biharmonic map $\varphi:\,(M,g)\rightarrow (N,h)$ with finite energy and finite bienergy must be harmonic. \par $(2)$ In the case ${\rm Vol}(M,g)=\infty$, under the same assumtion, every biharmonic map $\varphi:\,(M,g)\rightarrow (N,h)$ with finite bienergy is harmonic. \end{th}
\vskip0.3cm\par We also obtained (cf. \cite{NU1}, \cite{NU2}, \cite{NUG})
\begin{th} Assume that $(M,g)$ is a complete Riemannian manifold, $\varphi:\,(M,g)\rightarrow (N,h)$ is an isometric immersion, and the sectional curvature of $(N,h)$ is non-positive. If $\varphi:\,(M,g)\rightarrow (N,h)$ is biharmonic and $\int_M\vert{\bf H}\vert^2\,v_g<\infty$, then it is minimal. Here, $\bf H$ is the mean curvature normal vector field of the isometric immersion $\varphi$. \end{th}
\vskip0.3cm\par Theorem 1.3 gives an affirmative answer to the generalized B.Y. Chen's conjecture under the $L^2$-condition and completeness of $(M,g)$. \vskip0.3cm\par In this paper, for every Legendrian submanifold $\varphi:\,(M^m,g)\rightarrow (N^{2m+1},h)$ of a Sasaki manifold $(N^{2m+1},h)$, and the Lagrangian cone submanifold $\overline{\varphi}:\,(C(M),\overline{g})\rightarrow (C(N),\overline{h})$ of a K\"ahler cone manifold $(C(N),\overline{h})$,
we show (Theorems 3.3 and 4.4) that (1)
$\overline{\varphi}:\,(C(M),\overline{g})\rightarrow (C(N),\overline{h})$ is biharmonic if and only if it is harmonic, which is equivalent to that
$\varphi:\,(M,g)\rightarrow (N,h)$ is harmonic.
(2) $\varphi:\,(M,g)\rightarrow (N,h)$ is proper biharmonic if and only if
$\tau(\overline{\varphi})$ is a non-zero eigen-section of the Jacobi operator $J_{\overline{\varphi}}$ with the eigenvalue $m=\dim M$.
The assertion (2) can be regarded as a biharmonic map version of T. Takahashi's theorem (cf. Theorem 4.5) which claims that each coordinate function of the isometric immersion of $(M^m,g)$ into the unit sphere $S^n\hookrightarrow {\mathbb R}^{n+1}$ is the eigenfunction of the Laplacian of $(M,g)$ with the eigenvalue $m=\dim M$.
\vskip0.6cm\par {\bf Acknowledgement.} \quad This work was finished during the stay at the University of Bsilicata, Potenza, Italy, June of 2013. The author would like to express his sincere gratitude to Professors Sorin Dragomir and Elisabetta Barletta for their hospitality and helpful discussions, and also Dr. Shun Maeta for his helpful comments on Sasahara's works. The author also express his gratitude to Professor T. Sasahara who pointed several errors in the first draft. \vskip0.6cm\par
\section{Preliminaries} We first prepare the materials for the first and second variational formulas for the bienergy functional and biharmonic maps. Let us recall the definition of a harmonic map $\varphi:\,(M,g)\rightarrow (N,h)$, of a compact Riemannian manifold $(M,g)$ into another Riemannian manifold $(N,h)$, which is an extremal of the {\em energy functional} defined by $$ E(\varphi)=\int_Me(\varphi)\,v_g, $$ where $e(\varphi):=\frac12\vert d\varphi\vert^2$ is called the energy density of $\varphi$. That is, for any variation $\{\varphi_t\}$ of $\varphi$ with $\varphi_0=\varphi$, \begin{equation} \frac{d}{dt}\bigg\vert_{t=0}E(\varphi_t)=-\int_Mh(\tau(\varphi),V)v_g=0, \end{equation} where $V\in \Gamma(\varphi^{-1}TN)$ is a variation vector field along $\varphi$ which is given by $V(x)=\frac{d}{dt}\big\vert_{t=0}\varphi_t(x)\in T_{\varphi(x)}N$, $(x\in M)$, and the {\em tension field} is given by $\tau(\varphi) =\sum_{i=1}^mB(\varphi)(e_i,e_i)\in \Gamma(\varphi^{-1}TN)$, where $\{e_i\}_{i=1}^m$ is a locally defined orthonormal frame field on $(M,g)$, and $B(\varphi)$ is the second fundamental form of $\varphi$ defined by \begin{align} B(\varphi)(X,Y)&=(\widetilde{\nabla}d\varphi)(X,Y)\nonumber\\ &=(\widetilde{\nabla}_Xd\varphi)(Y)\nonumber\\ &=\overline{\nabla}_X(d\varphi(Y))-d\varphi(\nabla_XY), \end{align} for all vector fields $X, Y\in {\frak X}(M)$. Here, $\nabla$, and $\nabla^N$,
are Levi-Civita connections on $TM$, $TN$ of $(M,g)$, $(N,h)$, respectively, and $\overline{\nabla}$, and $\widetilde{\nabla}$ are the induced ones on $\varphi^{-1}TN$, and $T^{\ast}M\otimes \varphi^{-1}TN$, respectively. By (2.1), $\varphi$ is {\em harmonic} if and only if $\tau(\varphi)=0$. \par The second variation formula is given as follows. Assume that $\varphi$ is harmonic. Then, \begin{equation} \frac{d^2}{dt^2}\bigg\vert_{t=0}E(\varphi_t) =\int_Mh(J(V),V)v_g, \end{equation} where $J$ is an elliptic differential operator, called the {\em Jacobi operator} acting on $\Gamma(\varphi^{-1}TN)$ given by \begin{equation} J(V)=\overline{\Delta}V-{\mathcal R}(V), \end{equation} where $\overline{\Delta}V=\overline{\nabla}^{\ast}\overline{\nabla}V =-\sum_{i=1}^m\{ \overline{\nabla}_{e_i}\overline{\nabla}_{e_i}V-\overline{\nabla}_{\nabla_{e_i}e_i}V \}$ is the {\em rough Laplacian} and ${\mathcal R}$ is a linear operator on $\Gamma(\varphi^{-1}TN)$ given by ${\mathcal R}(V)= \sum_{i=1}^mR^N(V,d\varphi(e_i))d\varphi(e_i)$, and $R^N$ is the curvature tensor of $(N,h)$ given by $R^N(U,V)=\nabla^N{}_U\nabla^N{}_V-\nabla^N{}_V\nabla^N{}_U-\nabla^N{}_{[U,V]}$ for $U,\,V\in {\frak X}(N)$. \par J. Eells and L. Lemaire \cite{EL1} proposed polyharmonic ($k$-harmonic) maps and Jiang \cite{J} studied the first and second variation formulas of biharmonic maps. Let us consider the {\em bienergy functional} defined by \begin{equation} E_2(\varphi)=\frac12\int_M\vert\tau(\varphi)\vert ^2v_g, \end{equation} where $\vert V\vert^2=h(V,V)$, $V\in \Gamma(\varphi^{-1}TN)$. \par The first variation formula of the bienergy functional is given by
\begin{equation} \frac{d}{dt}\bigg\vert_{t=0}E_2(\varphi_t) =-\int_Mh(\tau_2(\varphi),V)v_g. \end{equation} Here, \begin{equation} \tau_2(\varphi) :=J(\tau(\varphi))=\overline{\Delta}(\tau(\varphi))-{\mathcal R}(\tau(\varphi)), \end{equation} which is called the {\em bitension field} of $\varphi$, and $J$ is given in $(2.4)$.
\par A smooth map $\varphi$ of $(M,g)$ into $(N,h)$ is said to be {\em biharmonic} if $\tau_2(\varphi)=0$. By definition, every harmonic map is biharmonic. We say, for an immersion $\varphi:\,(M,g)\rightarrow (N,h)$ to be {\em proper biharmonic} if it is biharmonic but not harmonic (minimal). \vskip0.6cm\par
\section{Legendrian submanifolds and Lagrangian submanifolds} In this section, we first show a correspondence between the set of all Legendrian submanifolds of a Sasakian manifold and the one of all Lagrangian submanifolds of a K\"ahler cone manifold. \par An $n(=2m+1)$ dimensional contact Riemannian manifold $(N,h)$ with a contact form $\eta$ is said to be a {\em contact metric manifold} if there exist a smooth $(1,1)$ tensor field $J$ and a smooth vector field $\xi$ on $N$, called a {\em basic vector field}, satisfying that \begin{align} J^2&=-\mbox{I\!d}+\eta\otimes \xi,\\ \eta(\xi)&=1,\\ J\,\xi&=0,\\ \eta\,\circ\,J&=0,\\ h(JX,JY)&=h(X,Y)-\eta(X)\,\eta(Y),\\ \eta(X)&=h(X,\xi),\\ d\eta(X,Y)&=h(X,JY), \end{align} for all smooth vector fields $X$, $Y$ on $N$. Here, $\mbox{I\!d}$ is the identity transformation of $T_xN$ $(x\in N)$. A contact metric manifold $(N,h,J,\xi,\eta)$ is {\em Sasakian} if $(C(N),\overline{h},I)$ is a K\"ahler manifold. Here, a cone manifold $C(N):=N\times {\mathbb R}^+$ where ${\mathbb R}^+:=\{r\in {\mathbb R}\vert\,r>0\}$, $\overline{h}$ is a cone metric on $C(N)$, $\overline{h}:=dr^2+r^2\,h$, which is a Hermitian metric with respect to an almost complex structure $I$ on $C(N)$ given by \begin{equation} \left\{ \begin{aligned} IY&:=JY+\eta(Y)\,\Psi,\qquad (Y\in {\frak X}(N)),\\ I\Psi&:=-\xi, \end{aligned} \right. \end{equation} where $\Psi:=r\,\frac{\partial}{\partial r}$ is called the {\em Liouville vector field} on $C(N)$. We denote by ${\frak X}(N)$, the set of all smooth vector fields on $N$. A contact metric manifold $(N,h,J,\xi, \eta)$ is Sasakian if and only if \begin{equation} (\nabla^N_XJ)(Y)=h(X,Y)\,\xi-\eta(Y)\,X \quad (X,\,Y\in {\frak X}(N)). \end{equation} \par Let us recall the definition
\begin{df} Let $M^m$ be an $m$-dimensional manifold, an immersion $\varphi:\,M^m\rightarrow N^{2m+1}$. $M^m$ is called to be a {\em Legendrian} submanifold
of an $(2m+1)$-dimensional Sasakian manifold $(N,h,J,\xi,\eta)$ if $\varphi^{\ast}\eta\equiv 0$ which is equivalent to that \begin{align} \varphi_{\ast\,x}(X_x)\in \mbox{\rm Ker}(\eta_{\varphi(x)}) \end{align} for all $X_x\in T_xM$ ($x\in M$). \end{df}
A Legendrian submanifold $M^m$ satisfies the following two conditions: \par (1) $\varphi_{\ast}(T_xM)$ is orthogonal $J(\varphi_{\ast}(T_xM))$ with respect to $h$ for all $x\in M$. This is equivalent to that the normal bundle $T^{\perp}M$ of $\varphi:\,M\rightarrow N$ has the following splitting: $$ T_xM^{\perp}={\mathbb R}\xi_{\varphi(x)}\oplus J\,\varphi_{\ast}T_xM \,\, (x\in M). $$ \par (2) The second fundamental form $B$ of $\varphi(M)\subset N$ has its value at $\mbox{\rm Ker}(\eta)$, that is, $$ B(\varphi_{\ast}X,\varphi_{\ast}Y)=\nabla^N_{X}\varphi_{\ast}Y-\varphi_{\ast}(\nabla_XY)\in \varphi_{\ast}(T_xM)^{\perp}, $$ where $T_xM^{\perp}$ is $\varphi_{\ast}(T_xM)^{\perp}$, which is $$ \{W_{\varphi(x)}\in T_{\varphi(x)}N\vert\, h(W_{\varphi(x)},\varphi_{\ast\,x}X_x)=0\,\, (\forall\,\,X_x\in T_xM)\}. $$ Here, $\nabla$, $\nabla^N$ are Levi-Civita connections of $(M,g)$, $(N,h)$ where $g$ is the induced metric on $M$ by $g:=\varphi^{\ast}h$. \par In the following, we identify $\varphi(M)$ with $M$, itself. The following theorem is well known, but essentially important for us.
\begin{th} Let $M^m$ be an $m$-dimensional submanifold of a Sasakian manifold $(N^{2m+1},h,J,\xi,\eta)$. Then, $M$ is a Legendrian submanifold of a Sasaki manifold $N$ if and only if $C(M)\subset C(N)$ is a Lagrangian submanifold of a K\"ahler cone manifold $(C(N),\overline{h},I)$. \end{th}
{\it Proof} \quad We have the equivalence that $M\subset N$ is Legendrian if and only if \begin{equation} \left\{ \begin{aligned} &\xi_x{}^{\perp}= T_xM\oplus JT_xM,\\ &h(T_xM,J\,T_xM)=\{0\} \end{aligned} \right. \end{equation} for all $x\in M$. That is, $h(\xi,X)=0$ and $h(X,J\,Y)=0$ for all $X$,\,$Y\in {\frak X}(M)$. Then, (3.11) is equivalent to that \begin{align} \Omega(f_1\,\Phi+X,f_2\,\Phi+Y) &=r^2\,\{ f_1\,h(\xi,Y)-f_2\,h(\xi,X)+h(X,JY)\}\nonumber\\ &=0 \end{align} for all smooth functions $f_1$, $f_2$ on $C(M)$ and $X$, $Y\in {\frak X}(M)$. Here, $\Omega$ is the K\"ahler form of $C(N)$ which is given by $\Omega =2\,r\,dr\wedge \eta+r^2\,d\eta$. Finally, (3.12) is equivalent to that $C(M)\subset C(N)$ is Lagrangian. \qed \vskip0.6cm\par
Now our main theorem is as follows: \begin{th} Let $\varphi:\,(M,g)\rightarrow (N,h)$ be a Legendrian submanifold of a Sasakian manifold $(N^n,h,J,\xi,\eta)$ ($n=2m+1$) and $\overline{\varphi}:\,(C(M),\overline{g})\ni(r,x)\mapsto (r,\varphi(x))\in (C(N),\overline{h},I)$, a Lagrangian submanifold of a K\"ahler cone manifold. Here $C(M):=M\times {\mathbb R}^+\subset C(N):=N\times {\mathbb R}^+$, $\overline{g}=dr^2+r^2\, g$, and $\overline{h}=dr^2+r^2\, h$. Then, \par
$(1)$ it holds that \begin{equation} \tau(\overline{\varphi})=\frac{1}{r^2}\,\tau(\varphi). \end{equation} Thus, we have the equivalence that $\varphi:\,(M,g)\rightarrow (N,h)$ is harmonic if and only if $\overline{\varphi}(C(M),\overline{g})\rightarrow (C(N),\overline{h})$ is also harmonic. \par
$(2)$ Secondly, it holds that \begin{equation}\tau_2(\overline{\varphi})=\frac{1}{r^4}\,\tau_2(\varphi) +\frac{m}{r^2}\,\tau(\varphi). \end{equation} Then, we have the equivalence that $\varphi:\,(M,g)\rightarrow (N,h)$ is proper biharmonic if and only if for $\overline{\varphi}:\,(C(M),\overline{g})\rightarrow (C(N),\overline{h})$, the tension field $\tau(\overline{\varphi})$ is a non-zero eigen-section of the Jacobi operator $J_{\overline{\varphi}}$ with the eigenvalue $m=\dim M$. And we have the equivalence that $\overline{\varphi}:\,(C(M),\overline{g})\rightarrow (C(N),\overline{h})$ is biharmonic if and only if it is harmonic, which is equivalent to that $\varphi:\,(M,g)\rightarrow (N,h)$ is harmonic. \par
$(3)$ Thirdly, it holds that \begin{equation} \tau_2(\overline{\varphi})^{\perp}=\frac{1}{r^4}\,\tau_{2}(\varphi)^{\perp} +\frac{m}{r^2}\,\tau(\varphi). \end{equation} Then, we have the equivalence that $\varphi:\,(M,g)\rightarrow (N,h)$ is minimal if and only if $\overline{\varphi}:\,(C(M),\overline{g})\rightarrow (C(N),\overline{h})$ is bi-minimal. \par
$(4)$ Finally, it holds that \begin{equation} \mbox{\rm div}_{\overline{g}}(I\,\tau (\overline{\varphi})) =\frac{1}{r^2}\,\mbox{\rm div}_g(J\,\tau(\varphi)). \end{equation} Then, we have also the equivalence that $\varphi:\,(M,g) \rightarrow (N,h,J,\xi,\eta)$ is Legendrian minimal if and only if\, $\overline{\varphi}:\,(C(M),\overline{g})\rightarrow (C(N),\overline{h},I)$ is also Lagrangian minimal. \end{th}
\vskip0.6cm\par To prove Theorem 3.3, we need the following lemma.
\begin{lem} The Levi-Civia connection $\nabla^{C(M)}$ of the cone manifold $(C(M),\overline{g})$ of a Riemannian manifold $(M,g)$, where the cone metric $\overline{g}=dr^2+r^2\,g$, is given as follows: \begin{equation} \left\{ \begin{aligned} \nabla^{C(M)}_XY&=\nabla_XY-r\,g(X,Y)\,\frac{\partial}{\partial r},\\ \nabla^{C(M)}_X\frac{\partial}{\partial r}&=\frac{1}{r}\,X,\\ \nabla^{C(M)}_{\frac{\partial}{\partial r}}Y&=\frac{1}{r}\,Y,\\ \nabla^{C(M)}_{\frac{\partial}{\partial r}}\frac{\partial}{\partial r}&=0. \end{aligned} \right. \end{equation} Here, $X,\,Y\in {\frak X}(M)$, and $\nabla$ is the Levi-Civita connection of $(M,g)$. \end{lem}
The proof of Lemma 3.4 is a direct computation which is omitted. \vskip0.6cm\par To proceed to give a proof of Theorem 3.3, we first take a locally defined orthonormal frame field $\{e_i\}_{i=1}^m$ on $(M,g)$. Define $\overline{e}_i:=\frac{1}{r}\,e_i$ $(i=1,\ldots,m$), and $\overline{e}_{m+1}:=\frac{\partial}{\partial r}$. Then, $\{\overline{e}_i\}_{i=1}^{m+1}$ is a locally defined orthonormal frame field on the cone manifold $(C(M),\overline{g})$. \par Let $\varphi:\,(M^m,g)\rightarrow (N^n,h)$ ($n=2m+1$) be a Legendrian submanifold of a Sasakian manifold, and $\overline{\varphi}:\,(C(M),\overline{g})\rightarrow (C(N),\overline{h})$, the corresponding cone submanifold of a K\"ahler cone $(C(N),\overline{h})$. We should see a relation between the induced bundles $\varphi^{-1}TN$ and $\overline{\varphi}^{-1}TC(N)$. We denote by $\Gamma(E)$, the space of all smooth sections of the vector bundle $E$. Then, every smooth section $W$ of the induced bundle $\overline{\varphi}^{-1}TC(N)$ can be written as \begin{equation} W=V+B\,\frac{\partial}{\partial r} \end{equation} where $V$ is a smooth section of the induced bundle $\varphi^{-1}TN$ and $B$ is a smooth function on $C(M)=M\times {\mathbb R}^+$. Because, for every point $(x,r)\in C(M)=M\times {\mathbb R}^+$, $\overline{\varphi}(x,r)=(\varphi(x),r)$, and $W_{(x,r)}\in T_{\overline{\varphi}(x,r)}C(N)=T_{(\varphi(x),r)}(N\times {\mathbb R}^+) =T_{\varphi(x)}N\oplus T_r{\mathbb R}^+$, so we can write as $W_{(x,r)}=V_x+B(x,r)\,\frac{\partial}{\partial r}$, where $V_x\in T_{\varphi(x)}N$ and $B(x,r)\in {\mathbb R}$. \par Then, if we denote by $\overline{\nabla}$, and $\overline{\overline{\nabla}}$, the induced connections of the induced bundles $\varphi^{-1}TN$, and $\overline{\varphi}^{-1}TC(N)$ from the connections $\nabla^N$, $\nabla^{C(N)}$ of $(N,h)$ and $(C(N),\overline{h})$, respectively, then we have for every $W\in \Gamma(\overline{\varphi}^{-1}TC(N))$, with $W=V+B\,\frac{\partial}{\partial r}$ and $V\in \Gamma(\varphi^{-1}TN)$ and $B\in C^{\infty}(M\times {\mathbb R}^+)$, \begin{equation} \left\{ \begin{aligned} \overline{\overline{\nabla}}_XW&=\overline{\nabla}_XV+\frac{B}{r}\,X+(XB)\,\frac{\partial}{\partial r},\qquad (X\in {\frak{X}}(M)),\\ \overline{\overline{\nabla}}_{\frac{\partial}{\partial r}}W&=\frac{\partial B}{\partial r}\,\frac{\partial}{\partial r}. \end{aligned} \right. \end{equation}
\vskip0.6cm\par {\it Proof of Theorem 3.3.} \par (1) We have, for $i=1,\ldots,m$, ($m=\dim M$), \begin{align} \overline{\varphi}_{\ast}\nabla^{C(M)}_{\overline{e}_i}\overline{e}_i &=\overline{\varphi}_{\ast}\big(\frac{1}{r^2}\,\nabla^{C(M)}_{e_i}e_i\big)\nonumber\\ &=\frac{1}{r^2}\,\overline{\varphi}_{\ast}\big( \nabla_{e_i}e_i-r\,g(e_i,e_i)\,\frac{\partial}{\partial r}\big) \qquad (\mbox{\rm by Lemma 3.4 (3.17)}) \nonumber\\ &=\frac{1}{r^2}\,\big(\nabla_{e_i}e_i-r\,\frac{\partial}{\partial r}\big) \end{align} since $\overline{\varphi}$ is the inclusion map of $C(M)$ into $C(N)$. For $i=m+1$, we have \begin{align} \overline{\varphi}_{\ast}\big(\nabla^{C(M)}_{\overline{e}_{m+1}}\overline{e}_{m+1}\big)= \overline{\varphi}_{\ast}\big( \nabla^{C(M)}_{\frac{\partial}{\partial r}}\frac{\partial}{\partial r}\big)=0. \end{align} Furthermore, we have, for $i=1,\ldots,m$, \begin{align} \overline{\nabla}_{\overline{e}_{\ast}}\overline{\varphi}_{\ast}\overline{e}_i &=\nabla^{C(N)}_{\frac{1}{r}\,e_i}\frac{1}{r}\,e_i\nonumber\\ &=\frac{1}{r^2}\bigg\{ \nabla^N_{e_i}e_i-r\,h(e_i,e_i)\,\frac{\partial}{\partial r} \bigg\}\nonumber\\ &=\frac{1}{r^2}\bigg\{ \nabla^N_{e_i}e_i-r\,\frac{\partial}{\partial r} \bigg\} \end{align} since $\overline{\varphi}^{\ast}\overline{h}=\overline{g}$ and $\varphi^{\ast}h=g$. For $i=m+1$, we have also \begin{align} \overline{\nabla}_{\overline{e}_{m+1}}\overline{\varphi}_{\ast}\overline{e}_{m+1}=\nabla^{C(N)}_{\frac{\partial}{\partial r}}\frac{\partial}{\partial r}=0. \end{align} \par Thus, we have \begin{align} \tau(\overline{\varphi})&=\sum_{i=1}^{m+1}\bigg\{ \overline{\nabla}_{\overline{e}_i} \overline{\varphi}_{\ast}\overline{e}_i -\overline{\varphi}_{\ast} \big( \nabla^{C(M)}_{\overline{e}_i}\overline{e}_i \big) \bigg\}\nonumber\\ &=\frac{1}{r^2}\sum_{i=1}^m\big\{ \nabla^N_{e_i}e_i-\nabla_{e_i}e_i\big\}\qquad\quad (\mbox{\rm by (3.20), (3.21), (3.22), (3.23)}) \nonumber\\ &=\frac{1}{r^2}\,\tau(\varphi ), \end{align} which is (3.13).
\par For (2), we have to see relations between \begin{align} J_{\varphi}(V)&= \overline{\Delta}_{\varphi}V-\sum_{i=1}^mR^N(V,\varphi_{\ast}e_i)\varphi_{\ast}e_i,\qquad (V\in \Gamma(\varphi^{-1}TN)),\\ J_{\overline{\varphi}}(W)&= \overline{\overline{\Delta}}_{\overline{\varphi}}W -\sum_{i=1}^{m+1}R^{C(N)}(W,\overline{\varphi}_{\ast}\overline{e}_i) \overline{\varphi}_{\ast}\overline{e}_i,\quad(W\in \Gamma(\overline{\varphi}^{-1}TC(N)). \end{align} where \begin{align} \overline{\Delta}_{\varphi}V&:=-\sum_{i=1}^m\{ \overline{\nabla}_{e_i}(\overline{\nabla}_{e_i}V)-\overline{\nabla}_{\nabla_{e_i}e_iV} \},\\ \overline{\overline{\Delta}}_{\overline{\varphi}}W&:= -\sum_{i=1}^{m+1}\{ \overline{\overline{\nabla}}_{\overline{e}_i}( \overline{\overline{\nabla}}_{\overline{e}_i}W) -\overline{\overline{\nabla}}_{\nabla^{C(M)}_{\overline{e}_i}\overline{e}_i}W\}. \end{align} Here, $\overline{\nabla}$, and $\overline{\overline{\nabla}}$ are the induced connections of $\varphi^{-1}TN$ and $\overline{\varphi}^{-1}TC(N)$ from the Levi-Civita connections $\nabla^N$ and $\nabla^{C(N)}$ of $(N,h)$ and $(C(N),\overline{h})$ with $\overline{h}=dr^2+r^2\,h$, respectively.
\par
({\it The first step}) \quad By (3.19), we have
\begin{equation}
\left\{
\begin{aligned}
\overline{\overline{\nabla}}_X(\overline{\overline{\nabla}}_YW)
&=
\overline{\nabla}_X(\overline{\nabla}_YV)+\frac{B}{r}\,\nabla^N_XY+\frac{XB}{r}\,Y+\frac{YB}{r}\,X\\ &\quad+X(YB)\,\frac{\partial}{\partial r}, \qquad (X,\,Y\in {\frak X}(M)),\\ \overline{\overline{\nabla}}_{\frac{\partial}{\partial r}}\big( \overline{\overline{\nabla}}_{\frac{\partial}{\partial r}}W \big)&=\frac{\partial ^2B}{\partial r^2}\,\frac{\partial}{\partial r}, \end{aligned} \right. \end{equation} where we used that $\overline{\overline{\nabla}}_X(\overline{\nabla}_YV)=\overline{\nabla}_X(\overline{\nabla}_YV)$, $\overline{\overline{\nabla}}_XY=\overline{\nabla}_XY=\nabla^N_XY$ and $\overline{\overline{\nabla}}_X\frac{\partial}{\partial r}=\frac{1}{r}\,X$
for every $X$, $Y\in {\frak X}(M)$.
Thus, we obtain,
for $W=V+B\,\frac{\partial}{\partial r}\in \Gamma(\overline{\varphi}^{-1}TC(N))$ with
$V\in \Gamma(\varphi^{-1}TN)$ and $B\in C^{\infty}(M\times {\mathbb R}^+)$,
\begin{align}
\overline{\overline{\Delta}}_{\overline{\varphi}}W&=
\frac{1}{r^2}\,\overline{\Delta}_{\varphi}V-\frac{B}{r^3}\,\tau(\varphi)-\frac{2}{r^3}\,\mbox{\rm grad}_MB\nonumber\\
&\quad+\bigg(
\frac{1}{r^2}\,\Delta_MB-\frac{\partial^2B}{\partial r^2}-\frac{m}{r}\,\frac{\partial B}{\partial r}
\bigg)\,\frac{\partial}{\partial r},
\end{align}
where let us recall
\begin{align}
\overline{\Delta}_{\varphi}V
&=-\sum_{i=1}^m\{
\overline{\nabla}_{e_i}(\overline{\nabla}_{e_i}V)-\overline{\nabla}_{\nabla_{e_i}e_i}V\}
\qquad
(V\in \Gamma(\varphi^{-1}TN)),\nonumber\\
\tau(\varphi)&=\sum_{i=1}^m(\nabla^N_{e_i}e_i-\nabla_{e_i}e_i),
\qquad
\mbox{\rm grad}_MB= \sum_{i=1}^m(e_iB)\,e_i,
\nonumber\\
\Delta_MB&=-\sum_{i=1}^m\{e_i(e_iB)-\nabla_{e_i}e_i\,B\}
\qquad
(B\in C^{\infty}(M\times {\mathbb R}^+)).\nonumber
\end{align}
\par
({\it The second step}) \quad By a direct computation, we have
the curvature tensor field $R^{C(N)}$ of $(C(N),\overline{h})$:
\begin{equation}
\left\{
\begin{aligned}
&R^{C(N)}(X,Y)Z=R^N(X,Y)Z-h(Y,Z)\,X+h(X,Z)Y,\\
&R^{C(N)}\bigg(X,\frac{\partial}{\partial r}\bigg)\frac{\partial}{\partial r}=0,\\
&R^{C(N)}\bigg(\frac{\partial}{\partial r},Y\bigg)Z=0,
\end{aligned}
\right.
\end{equation}
for every $X$, $Y$, $Z\in {\frak X}(M)$.
Therefore, we obtain
\begin{align}
\sum_{i=1}^mR^{C(N)}(W,\overline{\varphi}_{\ast}\overline{e}_i)\overline{\varphi}_{\ast}\overline{e}_i
=\frac{1}{r^2}\sum_{i=1}^mR^N(V,\varphi_{\ast}e_i)\varphi_{\ast}e_i
-\frac{m}{r^2}\,V+\frac{1}{r^2}V^{\rm T},
\end{align}
for $W=V+B\,\frac{\partial}{\partial r}\in \Gamma(\overline{\varphi}^{-1}TC(N))$.
\par
({\it The third step}) \quad
Therefore, we have
\begin{align}
J_{\overline{\varphi}}(W)&=
\overline{\overline{\Delta}}_{\overline{\varphi}}W-\sum_{i=1}^mR^{C(N)}(W,\overline{\varphi}_{\ast}\overline{e}_i)
\overline{\varphi}_{\ast}\overline{e}_i\nonumber\\
&=
\frac{1}{r^2}\bigg(\overline{\Delta}_{\varphi}V-\sum_{i=1}^mR^N(V,\varphi_{\ast}e_i)\varphi_{\ast}e_i
\bigg)+\frac{m}{r^2}\,V-\frac{1}{r^2}\,V^{\rm T}
\nonumber\\
&\quad
-\frac{B}{r^3}\,\tau(\varphi)-\frac{2}{r^3}\,\mbox{\rm grad}_MB\nonumber\\
&\quad
+
\bigg(
\frac{1}{r^2}\,\Delta_MB-\frac{\partial^2B}{\partial r^2}-\frac{m}{r}\,\frac{\partial B}{\partial r}
\bigg)\,\frac{\partial}{\partial r}.
\end{align}
Here, we have already $\tau(\overline{\varphi})=\frac{1}{r^2}\,\tau(\varphi)$
in Thoerem 3.3 (1) (3.13).
For this $W:=\tau(\overline{\varphi})$, we have $V=\frac{1}{r^2}\,\tau(\varphi)$,
$B=0$ and $V^{\rm T}=0$, and we have
\begin{align}
J_{\overline{\varphi}}(\tau(\overline{\varphi}))&=
\frac{1}{r^4}\,\bigg(
\overline{\Delta}_{\varphi}(\tau(\varphi))-\sum_{i=1}^mR^N(\tau(\varphi),\varphi_{\ast}e_i)\varphi_{\ast}e_i)\bigg)+\frac{m}{r^2}\,\tau(\varphi)
\nonumber\\
&=\frac{1}{r^4}\,J_{\varphi}(\tau(\varphi))+\frac{m}{r^2}\,\tau(\varphi).
\end{align}
We have (3.14) in (2).
By (3.34), we have
the equivalence between the bi-harmonicity of $\varphi$ and
that
$\tau(\overline{\varphi})$ is a non-zero eigen-section of the Jacobi
operator $J_{\overline{\varphi}}$ with eigenvalue $m=\dim M$.
Furthermore, $\tau_2(\overline{\varphi})=0$ if and only if
$\tau_2(\varphi)+mr^2\,\tau(\varphi)=0$ for all $r>0$, which is equivalent to that
$\tau(\varphi)=0$.
\par
For (3) in Theorem 3.3, we only observe the following orthogonal decompositions:
\begin{align}
T_xN&=T_xM\oplus T_xM^{\perp},
\quad T_xM^{\perp}=J\,T_xM\oplus {\mathbb R}\,\xi_x,\\
T_{(x,r)}C(N)&=T_xN\oplus T_r{\mathbb R}^+\nonumber\\
&=T_xM\oplus J\,T_xM\oplus{\mathbb R}\,\xi_x\oplus T_r{\mathbb R}^+\nonumber\\
&=T_{(x,r)}C(M)\oplus J\,T_xM\oplus {\mathbb R}\,\xi_x\nonumber\\
&=T_{(x,r)}C(M)\oplus T_xM^{\perp},
\end{align}
for every $x\in M\subset N$.
So let us decompose $\tau_2(\overline{\varphi})=\frac{1}{r^4}\,\tau_2(\varphi)$
following (3.35) and (3.36). Then, we have
\begin{align}
\tau_2(\overline{\varphi})=\tau_2(\overline{\varphi})^{\rm T}+\tau_2(\overline{\varphi})^{\perp}
\end{align}
where $\tau_2(\overline{\varphi})^{\rm T}\in T_{(x,r)}C(M)$ and $\tau_2(\overline{\varphi})^{\perp}\in T_xM^{\perp}$, and also we have
\begin{align}
\frac{1}{r^4}\,\tau_2(\varphi)
+\frac{m}{r^2}\,\tau(\varphi)
=\frac{1}{r^4}\,
\tau_2(\varphi)^{\rm T}+
\frac{1}{r^4}\,\tau_2(\varphi)^{\perp} +\frac{m}{r^2}\,\tau(\varphi),
\end{align}
where $\tau_2(\varphi)^{\rm T}\in T_xM$ and
$\tau_2(\varphi)^{\perp}\in T_xM^{\perp}$.
But, since we have $T_xM\subset T_{(x,r)}C(M)$, we have
\begin{equation}
\left\{
\begin{aligned}
\tau_2(\overline{\varphi})^{\rm T}&=\frac{1}{r^4}\,\tau_2(\varphi)^{\rm T},\\
\tau_2(\overline{\varphi})^{\perp}&=\frac{1}{r^4}\,\tau_2(\varphi)^{\perp}
+\frac{m}{r^2}\,\tau(\varphi).
\end{aligned}
\right.
\end{equation}
Then, we have $\tau_2(\varphi)^{\perp}=0$ if and only if
$\tau_2(\varphi)^{\perp}+mr^2\,\tau(\varphi)=0$ for all $r>0$, which is equivalent to
that $\tau(\varphi)=0$.
\par
For (4),
we first show that
\begin{align}
I\,\tau(\overline{\varphi})&=J\,\tau(\overline{\varphi})+\eta(\tau(\overline{\varphi}))
\,\Psi\nonumber\\
&=\frac{1}{r^2}\,J\,\tau(\varphi)+\frac{1}{r^2}\,\eta(\tau(\varphi))\,\Psi\nonumber\\
&=\frac{1}{r^2}\,J\,\tau(\varphi)
\end{align}
Because for a Legendrian submanifold of a Sasaki manifold, the second fundamental form $B$ takes its value in $\mbox{\rm Ker}(\eta)$,
so $\tau(\varphi)=\mbox{\rm Trace} (B)\subset {\rm Ker}(\eta)$, that is,
\begin{align}
\eta(\tau(\varphi))=0.
\end{align}
Then, we have
\begin{align}
{\rm div}_{\overline{g}}(I\,\tau(\overline{\varphi}))&=
\sum_{i=1}^{m+1}\overline{g}(\overline{e}_i,\nabla^{C(M)}_{\overline{e}_i}(I\,\tau(\overline{\varphi})))\nonumber\\
&=\frac{1}{r^4}\,\sum_{i=1}^m\overline{g}(e_i,\nabla^{C(M)}_{e_i}(J\,\tau(\varphi)))
\nonumber\\
&\qquad\qquad
+\frac{1}{r^2}\,\overline{g}\big(
\frac{\partial}{\partial r},\nabla^{C(M)}_{\frac{\partial}{\partial r}}(J\,\tau(\varphi))
\big).
\end{align}
But, the first term of the right hand side of (3.42) coincides with
\begin{align}
\frac{1}{r^4}&\sum_{i=1}^m\overline{g}\bigg(e_i,\nabla_{e_i}(J\,\tau(\varphi))-
r\,g(e_i,J\,\tau(\varphi))\,\frac{\partial}{\partial r}\bigg)\nonumber\\
&=\frac{1}{r^2}\sum_{i=1}^mg(e_i,\nabla_{e_i}(J\,\tau(\varphi)))\nonumber\\
&=\frac{1}{r^2}\,{\rm div}_g(J\,\tau(\varphi)).
\end{align}
On the other hand, the second term of the right hand side of (3.42) coincides with
\begin{align}
\frac{1}{r^2}\,\overline{g}\big(
\frac{\partial}{\partial r},\nabla^{C(M)}_{\frac{\partial}{\partial r}}(J\,\tau(\varphi))
\big)=
\frac{1}{r^3}\,\overline{g}\big(
\frac{\partial}{\partial r},J\,\tau(\varphi)
\big)=0
\end{align}
because $J\,\tau(\varphi)$ is tangential to $T_xM$ for the Legendrian immersion
$\varphi:\,(M,g)\rightarrow (N,h,J)$. Therefore, we obtain the desired formula:
$$
{\rm div}_{\overline{g}}(I\,\tau(\overline{\varphi}))=\frac{1}{r^2}\,{\rm div}_g(J\,\tau(\varphi)).
$$
We obtain Theorem 3.3. \qed
\vskip0.6cm\par
\begin{rem}
The assertion (4) in Theorem 3.3 was given by I. Castro, H.Z. Li and F. Urbano
$($\cite{CLU}$)$, and H. Iriyeh $($\cite{Ir}$)$, independently in a different manner from ours.
\end{rem}
\vskip0.6cm\par
\section{Biharmonic Lgendrian submanifolds of Sasakian manifolds}
By Theorem 3.3, we turn to review studies of a proper biharmonic Legendrian submanifold of
a Sasaki manifold $(N^n,h,J,\xi,\eta)$ and give Takahashi-type theorem (cf. Theorem 4.4).
First let us recall the equations of biharmonicity of an isometric immersions (cf. \cite{MU}).
\begin{lem}
Let $\varphi:\,(M^m,g)\rightarrow (N^n,h)$ be an isometric immersion.
Then, for $\varphi$ to be biharmonic if and only if
\begin{equation}
\left\{
\begin{aligned}
&\sum_{i=1}^m(\nabla_{e_i}A_{\bf H})(e_i)
+\sum_{i=1}^mA_{\nabla^{\perp}_{e_i}{\bf H}}(e_i)
-\sum_{i=1}^m\big( R^N({\bf H},e_i)e_i
\big)^{\rm T}=0,\\
&\Delta^{\perp}{\bf H}+\sum_{i=1}^mB(A_{\bf H}(e_i),e_i)-\sum_{i=1}^m
\big(R^N({\bf H},e_i)e_i\big)^{\perp}=0,
\end{aligned}
\right.
\end{equation}
where
${\bf H}=\frac{1}{m}\sum_{i=1}^mB(e_i,e_i)$ the mean curvature vector field along
$\varphi$, $B$ is the second fundamental form, and $A$ is the shape operator
for the isometric immersion $\varphi:\,(M,g)\rightarrow (N,h)$
\end{lem}
For an isometric immersion of a Legendrian submanifold into a Sasakian manifold,
we have
\begin{th}
Let $\varphi:\,(M^m,g)\rightarrow (N^n,h,J,\xi,\eta)$ $(n=2m+1)$ be an isometric immersion of a Legendrian submanifold of a Sasakian manifold.
Then, for $\varphi$ to be biharmonic if and only if
\begin{align}
\sum_{i=1}^m&(\nabla_{e_i}A_{\bf H})(e_i)
+\sum_{i=1}^mA_{\nabla^{\perp}_{e_i}{\bf H}}(e_i) \nonumber\\ &-\sum_{i,j=1}^mh( (\nabla^{\perp}_{e_j}B)(e_i,e_i)-(\nabla^{\perp}_{e_i}B)(e_j,e_i),{\bf H})\,e_j\nonumber\\ &\qquad=0,\\
\Delta^{\perp}{\bf H}&+\sum_{i=1}^mB(A_{\bf H}(e_i),e_i)\nonumber\\
&+\sum_{j=1}^m{\rm Ric}^N(J{\bf H},e_j)\,Je_j-\sum_{j=1}^m{\rm Ric}^M(J\,{\bf H},e_j)\,Je_j\nonumber\\
&\qquad-\sum_{i=1}^mJ\,A_{B(J\,{\bf H},e_i)}(e_i)+m\,J\,A_{\bf H}(J\,{\bf H})+{\bf H}\nonumber\\
&\qquad\qquad\qquad=0.
\end{align}
\end{th}
\vskip0.6cm\par
In the case that $(N^{2m+1},h,J,\xi,\eta)$ is a Sasaki space form
$N^{2m+1}(\epsilon)$
of constant $J$-sectional curvature $\epsilon$ whose curvature tensor
$R^N$ is given by
\begin{align}
&R^N(X,Y)Z=\frac{\epsilon+3}{4}
\big\{
h(Y,Z)\,X-h(Z,X)\,Y\big\}\nonumber\\
&+\frac{\epsilon-1}{4}
\big\{
\eta(X)\eta(Z)Y-\eta(Y)\eta(Z)X+h(X,Z)\eta(Y)\xi-h(Y,Z)\eta(X)\xi\nonumber\\
&\qquad+h(Z,J\,Y)\,J\,X-h(Z,J\,X)\,J\,Y+2h(X,J\,Y)\,JZ\big\},
\end{align}
for all $X,\,Y,\, Z\in {\frak X}(N)$,
we have (\cite{I1}, \cite{S3})
\begin{th}
Let $\varphi:\,(M^m,g)\rightarrow N^{2m+1}(\epsilon)$ be a Legendrian submanifold of a Sasaki space form of constant $J$-sectional curvature $\epsilon$. Then, for $\varphi$, to be biharmonic if and only if
\begin{align}
\overline{\Delta}_{\varphi}{\bf H}=\frac{\epsilon(m+3)+3(m-1)}{4}\,{\bf H}
\end{align}
which is equivalent to
\begin{equation}
\left\{
\begin{aligned}
&\sum_{i=1}^m(\nabla_{e_i}A_{\bf H})(e_i)
+\sum_{i=1}^m
A_{{\nabla^{\perp}}_{e_i}{\bf H}}
(e_i)=0,\\
&\Delta^{\perp}{\bf H}+\sum_{i=1}^mB(A_{\bf H}(e_i),e_i)
-\frac{\epsilon(m+3)+3(m-1)}{4}\,{\bf H}=0.
\end{aligned}
\right.
\end{equation}
\end{th}
\vskip0.6cm\par
Now, let us consider a Legendrian submanifold $M^m$ of the $(2m+1)$-dimensional unit sphere $S^{2m+1}(1)$ with the standard metric $ds^2_{\rm std}$ of constant sectional curvature $1$. Then, we have, due to Theorem 3.3, and $J_{\overline{\varphi}}=\overline{\overline{\Delta}}$ which follows from that $R^{C(N)}=0$ because of $(C(N), \overline{h})=({\mathbb C}^{m+1}, ds^2)$:
\begin{th} Let $\varphi:\,(M^m,g)\rightarrow (S^{2m+1}(1),ds^2_{\rm std})$ be a Legendrian submanifold of $(S^{2m+1}(1),ds^2_{\rm std})$, and $\overline{\varphi}:\,(C(M),\overline{g})\rightarrow ({\mathbb C}^{m+1},ds^2)$, the corresponding Lagrangian cone submanifold of the standard complex space $({\mathbb C}^{m+1},ds^2)$. Then, it holds that $\varphi:\,(M^m,g)\rightarrow (S^{2m+1}(1), ds^2_{\rm std})$ is proper biharmonic if and only if $\tau(\overline{\varphi})=\frac{1}{r^2}\,\tau(\varphi)=\frac{1}{r^2}\,\frac{{\bf H}}{m}$ is a non-zero eigen-section of the rough Laplacian $\overline{\overline{\Delta}}_{\overline{\varphi}}$ acting on $\Gamma(\overline{\varphi}^{-1}T{\mathbb C}^{m+1})$ with the eigenvalue $m=\dim M$: $\overline{\overline{\Delta}}_{\overline{\varphi}} \,\tau(\overline{\varphi})=m\,\tau(\overline{\varphi})$. \end{th}
\vskip0.6cm\par This Theorem 4.4 could be regarded as a biharmonic map version of the following T. Takahashi's theorem (\cite{T}). Our theorem is a different type from Theorem 4.3. For Takahashi-type theorem for harmonic maps into Grassmannian manifolds, see pp. 42 and 46 in \cite{Ng}:
\begin{th} (T. Takahashi) Let $(M^m,g)$ be a compact Riemannian manifold, and let $\varphi:\,(M^m,g)\rightarrow (S^n,ds^2_{\rm std})$ be an isometric immersion. We write $\varphi=(\varphi_1,\cdots,\varphi_{n+1})$ where $\varphi_i\in C^{\infty}(M)$ $(1\leq i\leq n+1)$ via the canonical embedding $S^n\hookrightarrow{\mathbb R}^{n+1}$. Then, $\varphi:\,(M,g)\rightarrow (S^n,ds^2_{\rm std})$ is minimal if and only if $\Delta_g\,\varphi_i=m\,\varphi_i$, $(1\leq i\leq n+1)$. Here, $\Delta_g$ is the positive Laplacian acting on $C^{\infty}(M)$. \end{th}
\vskip0.6cm\par Certain classification theorems about proper biharmonic Legendrian immersions into the unit sphere $(S^{2m+1}(1), ds^2_{\rm std})$ were obtained by T. Sasahara (\cite{S1}, \cite{S2}, \cite{S3}).
\vskip1.6cm\par
\end{document} | arXiv |
What is the underlying explanation behind fictitious/pseudo forces?
The popular example of the bus: Lets say you are standing in a bus and the bus is moving with a constant velocity, we can therefore agree that you are in an inertial reference frame and therefore the law of inertia applies in your reference frame.
However, as soon the bus decelerates you feel a "push" forward, moreover the frame is no longer inertial, at this point we agree that the law of inertia doesn't hold for that frame since you changed the state of your motion while no force is acting on you in your frame. Consequently, in order to make up for the "discrepancy" between the law of inertia and such situations we introduce a pseudo force (stating that this is the force that caused us to change states) in order to be able to effectively use Newtonian mechanics in a broader domain.
This is the popular explanation as to why pseudo forces are introduced, however no one really touches on to the underlying principles of the occurrence of a pseudo force, so I'm looking forward for a more in depth explanation into the nature of a pseudo force (i.e why it occurs from a physical perspective?), rather than just saying that we introduce it in non-inertial reference frames, for reasons similar to the above.
If we assume its nothing more than just a human correction used for mathematical and physical analysis, and simultaneously we can't say that inertia is the reason we tend to fall forward in the situation stated above since it is a non-inertial frame, then what would be the explanation exactly to such tendency of changing states of motion in an example like the above
newtonian-mechanics forces reference-frames
Qmechanic♦
$\begingroup$ There is no deeper explanation. If inertial systems with newton's laws exist, then in a frame of reference that is accelerated relative to an inertial one Newton's laws will not be valid, unless you introduce such pseudoforces. There is nothing physical about it, an object that moves at contant speed in one reference frame will look accelerated if seen by a reference frame that is accelerated with respect to it. Wheter or not newton's law exist (if space is galilean) $\endgroup$ – user126422 Apr 21 '17 at 1:56
$\begingroup$ Just to add about inertial frames existing, I thought this might be tangentially related. physics.stackexchange.com/q/78317 $\endgroup$ – SpiralRain Apr 21 '17 at 2:10
$\begingroup$ When the bus decelerates what you actually feel is you pulling yourself backward. $\endgroup$ – immibis Apr 21 '17 at 9:34
$\begingroup$ You might find my earlier answer to a related question (dealing with rotating reference frames) relevant. $\endgroup$ – Ilmari Karonen Apr 21 '17 at 14:37
$\begingroup$ Obligatory XKCD: xkcd.com/123 $\endgroup$ – Mark Apr 21 '17 at 22:41
We actually don't introduce a pseudo-force as much as we introduce an acceleration. It is an acceleration which is experienced by all bodies in that non-inertial frame. From time to time, it can be convenient to think of it as a pseudo-force, but the deeper meaning you're looking for deals with accelerations, not forces.
In your bus example, when the bus starts decelerating, every object acquires an acceleration which corresponds to the effect of the reference frame decelerating. Thus, your human on the bus will accelerate forward unless a force generates an opposing acceleration.
A more interesting case is a rotating frame. A rotating frame is non-inertial, and the equations of motion within that frame include a centrifugal acceleration $a=\frac{v^2}{r}$ away from the center of the rotating frame. If no force pushes on the object, it will accelerate away from the center at that rate. However, in most interesting rotating frame problems, there is a force in the opposite direction as well. In the case of an orbiting body like the ISS, that force is the force of gravity, $F=mg$ towards the center of our rotating frame. This generates an acceleration of $g$, and when the acceleration $g$ from the sum of the forces is equal but opposite of the acceleration from the non-inertial reference frame $\frac{v^2}{r}$ the object appears not to move (in the rotating reference frame).
Likewise, if you are spinning a weight on the end of a string, it's the force of tension on the string which directly opposes the accelerations from the non-inertial reference frame.
The idea of a pseudo-force comes about when it is not intuitive to think about these accelerations. Consider the case where you're on a gravitron, which is the carnival ride that spins really fast and pins everyone up against the wall. In this case, it is not intuitive to think about the difference between the accelerations from your reference frame and accelerations caused by the force of the walls pressed up against your back. Every part of your body feels as though there is a force pushing you outward. In fact, if you run the math, the effect of this "centrifugal force" pushing you outward is identical to the effect of an acceleration caused by the non-inertial frame multiplied by your mass.
This is where the pseudo-force comes from. At a deeper level, its really more meaningful to treat it as an acceleration, but in practice it can be convenient to model this acceleration as a force by multiplying the acceleration by the mass of the object. When we choose to deal with these non-inertial effects as forces, we call them pseudo-forces. In particular, we like to do this when we want to say the sum of the forces on a body (that isn't accelerating) is 0. It's convenient to think in all forces instead of having to mix and match forces and accelerations. It's also convenient to think this way because the intuitive wiring in our brains is typically built to assume inertial frames (even when that isn't actually accurate). But the "meaningful" math behind them is all accelerations.
To truly understand this thing you will have to refine your ideas about physics and coordinate systems and you have to learn Tensor Calculus and differential geometry. The main thing to understand is that Laws of nature must not depend on our coordinate system! It took Einstein a lot of time to understand that when he made General Relativity.
Here I have attempted to explain this in as simple terms as possible, using the math of General Relativity but applying it on Newtonian Spacetime.
Newton's law $$\mathfrak{F}=m\mathfrak{a}$$ is valid always, in all frames of reference. (when $\mathfrak{F}$ and $\mathfrak{a}$ are vectors). Laws of nature must not depend on our coordinate system! However $\mathfrak{a}$ is defined geometrically without refrence to coordinate system as $\mathfrak{a} =\nabla_{v} v$. ($\nabla$ is called the Covariant Derivative). In a coordinate system, the expression for $\mathfrak{a}$ (acceleration of a particle moving on a curve $ x$) is not simply $ \ddot x^{i}$ [$x^i$ are the components of $x$ in a coordinate system] but contains extra terms depending on the curvature of the coordinate system (and curvature of spacetime also) which include $ \Gamma^i_{jk}$ (called as Christoffel Symbols).
$$\mathfrak{a}^i \neq \ddot x^i$$
Now in an inertial frame, all the $ \Gamma$'s are equal to $0$, and we get $$F^i=m\ddot x^{i}\tag{Only when $\Gamma^i_{jk}=0$}.$$ If we are not in an inertial frame, we get $$\frac{F^\alpha}m=\underbrace{\ddot x^\alpha+\Gamma^\alpha_{\gamma\delta}\dot x^\gamma\dot x^\delta +2\Gamma^\alpha_{\gamma0}\dot x^\gamma+ \Gamma^\alpha_{00}}_{\mathfrak{a}^\alpha}.$$ Now this equation is still $\mathfrak{F}=m\mathfrak{a}$ but $\mathfrak{a}$ now has a complicated expression. The real force is still $\mathfrak{F}$.
However if you choose to call $\ddot x$ the "acceleration" (which it is not) then you have to deal with 3 new terms in the equation which we now call "fictitious forces". In the above equation you may be able to recognize the "centrifugal force", "Coriolis Force" etc. But they are really the parts of acceleration!
So in the end fictitious forces arise from calling something the "acceleration" which it is not.
*In fact gravity arises from the same equation. The term $\Gamma^\alpha_{00}$ is "the gravitational force".
KartikKartik
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Fictitious Forces in a car crash | CommonCrawl |
A statistical analysis of Pc1–2 waves at a near-cusp station in Antarctica
M. Regi ORCID: orcid.org/0000-0003-3626-24191,
M. Marzocchetti1,
P. Francia1 &
M. De Lauretis1
Earth, Planets and Space volume 69, Article number: 152 (2017) Cite this article
A statistical study of Pc1–2 waves at southern polar latitudes is presented. Ultra-low frequency geomagnetic field measurements collected at the Italian station Mario Zucchelli (Terra Nova Bay, Antarctica, altitude-adjusted corrected geomagnetic latitude 80°S, MLT = UT − 8) from 2003 to 2010 corresponding with the declining phase of solar cycle 23 and the onset of the solar cycle 24 are used. The long data series allows us to analyze the solar cycle, seasonal and magnetic local time dependence and investigate the possible generation processes related to the solar wind–magnetosphere interaction. We found that during the day, Pc1–2 waves occur around local magnetic noon and midnight. Polarized waves show an almost linear polarization, suggesting a wave propagation along a meridional ionospheric waveguide, from the injection region up to the latitude of Terra Nova Bay. The origin of the waves appears to be due to substorm/stormrelated instabilities and, in the dayside, to solar wind compressions of the magnetopause. Based on these results, we propose a simple model to estimate Pc1–2 power variations depending on geomagnetic activity and solar wind density.
ULF waves in the highest frequency band (0.1–5 Hz, Pc1–2 waves) have been extensively observed in the magnetosphere and on the ground at both low and high latitudes (Bolshakova et al. 1980; Anderson et al. 1992a, b; Menk et al. 1993; Mursula et al. 1994; Dyrud et al. 1997). They are generally believed to be generated by electromagnetic ion cyclotron (EMIC) resonance in the near equatorial magnetospheric regions, for example, in the storm-time ring current (Menk 2011).
The waves propagate toward the ionosphere along geomagnetic field lines as left-handed polarized Alfven waves. In the ionosphere, after a conversion into right-handed polarized isotropic compressional modes, the waves propagate horizontally in the waveguide represented by the ionosphere, becoming linearly polarized far from the injection region; the major axis of the polarization ellipse points along the direction of wave arrival in the waveguide (Tepley and Landshoff 1966; Greifinger and Greifinger 1968; Greifinger 1972a, b; Summers and Fraser 1972; Fujita and Tamao 1988). Because ionospheric boundaries are not perfect reflectors, the waves are attenuated during propagation, while leakage through the lower boundary allows their observation on the ground (Manchester 1966).
Frequency time representation of Pc1–2 waves on the ground reveals that they can be generally characterized as structured and unstructured waves.
Structured waves, called "pearls", are characterized by periodic variations in amplitude and have been observed primarily at low and middle latitudes; they are commonly interpreted in terms of wave packets bouncing between conjugate points (Jacobs and Watanabe 1964), as waves modulated by long period ULF waves (Rasinkangas and Mursula 1998; Mursula et al. 2001), or due to a ion cyclotron resonator (Guglielmi et al. 2000), but the formation mechanism has not yet been definitely identified (Paulson et al. 2014, 2017).
Unstructured waves, which do not show a clear repetitive structure, are predominant at high latitudes, suggesting that their source is in the outer magnetospheric regions (Menk et al. 1993; Mursula et al. 1994). The occurrence of waves is characterized by a diurnal variation, with a maximum around noon, and shows an inverse relation with the solar cycle, being significantly higher during the solar minimum (Menk et al. 1993; Mursula et al. 1994; Kangas et al. 1998).
Ground and satellite observations (Popecki et al. 1993; Anderson et al. 1996) have suggested that ions of the plasma sheet, drifting from the nightside to dayside, can develop a temperature anisotropy capable of generating EMIC waves; therefore, equatorially generated EMIC waves can represent the source of high-latitude waves. Dyrud et al. (1997) examined Pc1–2 waves at cusp (74°) and polar (80°) latitudes; they found that broadband waves dominated at higher latitudes. Their occurrence was confined around local magnetic noon (within 4 h), while narrower bandwidth events are predominant at cusp latitudes and are distributed more widely in local time. The authors suggested that the narrowband waves originated in the subsolar and post-noon equatorial region, while the broadband waves originated in the high-latitude plasma mantle and at the poleward edge of the cusp.
In the outer magnetosphere (L > 9), Engebretson et al. (2002) found that most of the observed Pc1–2 waves were associated with compressions of the magnetosphere, due to enhanced solar wind (SW) densities. Analyzing spacecraft and ground data, Usanova et al. (2008) observed structured dayside Pc1 waves at L = 5–6.5 (63°–66°) related to SW density enhancements and magnetospheric compressions, which may be an important source for lower latitude EMIC waves close to the plasmapause. Similarly, Clausen et al. (2011) found that EMIC waves at geosynchronous orbit were preferentially generated during intervals of large SW density, and Tetrick et al. (2017) provided statistical evidence for the association of substorm injections and SW compressions with the onset of EMIC waves at the plasmapause. Usanova et al. (2012), analyzing THEMIS data, observed that EMIC waves occurred preferentially in the dayside outer magnetosphere and are strongly controlled by SW pressure; moreover, high EMIC occurrence, preferentially at 12–15 MLT, is also associated with high AE.
Kim et al. (2011) examined Pc1–2 events simultaneously observed along a meridional array in Antarctica (62°S to 87°S geomagnetic latitude, spanning 2920 km) during 2007; they found that the waves were observed predominantly during the daytime hours, propagated poleward in the ionospheric waveguide, changed to a linear polarization, and had an attenuation factor of ~ 10 dB/1000 km.
The importance of Pc1–2 waves as responsible for relativistic electron precipitation into the high-latitude ionosphere has been recently shown with increasing observational evidence (Rodger et al. 2008; Clilverd et al. 2010; Blum et al. 2015). Theoretical studies have demonstrated that such waves should be an effective mechanism for loss of > 1 MeV electrons from the radiation belts, through pitch angle scattering by gyro-resonant interaction (Engebretson et al. 2008, and references therein). The electron precipitation could modify, by ionization, the chemistry and electric conductivity of the atmosphere (Mironova et al. 2015), with potential effects on local or even global climate. In this regard, it is worth noting that a significant correlation between Pc1–2 power and atmospheric parameters has been recently observed in statistical analyses at Terra Nova Bay (TNB) (Francia et al. 2015; Regi et al. 2016, 2017).
In the present work, we conducted a statistical study of Pc1–2 waves in the 100–500 mHz frequency range at southern polar latitudes. We used ULF geomagnetic field measurements collected at the Italian station Mario Zucchelli (Terra Nova Bay, Antarctica, TNB, AACGM latitude 80°S, MLT = UT − 8) from 2003 to 2010, corresponding to the declining phase of solar cycle 23 and the onset of solar cycle 24. Although TNB is located in the polar cap, its latitude is such that at magnetic local noon the field line approaches the cusp and closed field lines. The availability of a long time series data allowed us to analyze the solar cycle and seasonal and local time dependence of the observed Pc1–2 waves and investigate their relationship with geomagnetic conditions and SW density.
Data and methods
We used geomagnetic field fluctuation measurements collected from a search coil magnetometer installed at TNB from 2003 to 2010, with a large data gap during 2005. The instrument provides northward H, eastward D, and vertically downward Z components of geomagnetic field variations at a 1-s sampling rate. To reduce aliasing, a low-pass filter was applied so that the response at frequencies > 450 mHz was strongly dampened.
We computed, over 30-min intervals, the power spectral density (PSD) and cross-power spectral density (CPSD), using the Hamming window and averaging 200-s subintervals with no overlap, corresponding to a frequency resolution of 5 mHz and 18 degrees of freedom. The spectra were then converted using an instrument transfer function from mV2/Hz to nT2/Hz.
We estimated the polarization parameters applying the technique for partially polarized waves as proposed by Fowler et al. (1967). In particular, the polarization ratio R, the ratio between the polarized and total intensities of the horizontal signal, ellipticity ε, the ratio between the minor and major axes of the polarization ellipse in the horizontal plane, and azimuth θ, the clockwise measured angle between the major axis of the polarization ellipse and the H direction, were evaluated over each interval. Looking downward in the southern hemisphere, a positive (negative) value of ellipticity indicates left-handed (right-handed) polarized waves; if the ellipticity is close to zero, generally |ε| < 0.2, the waves are linearly polarized.
To characterize the interplanetary and magnetospheric conditions, we used the 1-min SW and interplanetary magnetic field (IMF) data and 1-min geomagnetic activity index AE, respectively, from the OMNI database (http://cdaweb.gsfc.nasa.gov/cdaweb/istppublic/).
Experimental results
The statistical analysis
For each 30-min interval, we computed the power on the horizontal H and D components, P H and P D , integrated in the Pc1–2 (100–450 mHz) frequency range. Because the power and AE index typically show variations in several orders of magnitude, when computing time averages, we used log(P) (Francia et al. 2015; Regi et al. 2015, 2016) and log(AE) (Anqin et al. 2008; Schmitter 2010) that follow a quasi-normal distribution. The power was then obtained reconverting the results using the exponential function.
Figure 1 shows the sunspot number, 30-min averages of the SW density and speed, and 30-min averages of the AE index and total power P (i.e., P H + P D ) through from 2003 to 2010. A large data gap in the ULF power P can be seen during local equinoxes and winter in 2005. The sunspot number clearly shows the long descending phase of solar cycle 23 (2003–2009) and onset of the new activity cycle at the beginning of 2010. The SW was characterized by recurrent fast speed streams and associated high densities, especially for 2006–2008, while the average speed reached the lowest values in 2009. The AE index values show a general decrease through the years and exhibit strong variations that closely corresponding to the SW streams. Both the general decrease with the solar cycle and correspondence with the SW stream structure can be observed in the Pc1–2 power, clearly evidenced by the daily averages shown as a red line. In addition, the power is characterized by an annual modulation, with a maximum during the local summer. Figure 2 shows in detail the period from August 15, 2007–January 1, 2008, when recurring SW fast streams are clearly identified, each stream being preceded by enhanced density, which causes the compression of the dayside magnetopause. As shown, the stream structure is well reflected in both the AE index and daily averages of the Pc power.
Interplanetary and geomagnetic conditions from 2003 to 2010. From the top: sunspot number, 30-min averages of SW speed and density, and 30-min averages of AE index and total power P
Interplanetary and geomagnetic conditions in the time interval August 15, 2007–January 1, 2008. 30-min averages of SW density and speed and 30-min averages of AE index and total power
From Fig. 2, it is clear that the Pc1–2 power undergoes a diurnal variation. The analysis of this variation is presented in Fig. 3, where we plot the average value of P for each 30-min interval for 2003–2010, separately for summer, winter, and equinoctial months. There is a clear power maximum just after the magnetic local noon (20 UT), when TNB was near the polar cusp and closed field lines. The maximum is less pronounced and slightly shifted during equinoxes, and becomes an order of magnitude lower during winter. We evaluated if such a power maximum reflects a general broadband increase in power or indicates a specific Pc1–2 activity. In Fig. 4, we compare the average of all power spectra around magnetic noon (20 UT, red line) and midnight (08 UT, blue line), separately during summer, equinoctial, and winter months. In addition to a general increase in power at noon with respect to midnight, a clear power enhancement between 100 and 300 mHz characterizes the noon spectra; in particular, this occurs during summer and equinoxes, indicating a specific intensification of Pc1–2 activity.
Daily variations in power. The UT dependence on the average P value from 2003 to 2010, separately for local summer (red line), winter (blue line), and equinoctial (green line) months. The number of spectra in each average is indicated
Average power spectra. The average power spectra around noon (red line) and midnight (blue line), separately for summer (left panel), winter (middle panel), and equinoctial (right panel) months
To clarify if the noon intensification of Pc1–2 power corresponds to a higher occurrence rate of events, we performed an additional analysis, identifying the Pc1–2 events on the basis of the technique developed by De Lauretis et al. (2010) and Ponomarenko et al. (2002). If, for a given time interval, S T (f) is the experimental total (signal + noise) power spectrum, S(f) is the power spectrum of the unknown signal, and N(f) is the background noise, we evaluate the signal-to-noise ratio SNR(f) as follows:
$${\text{SNR}} = \frac{S}{N} = \frac{{S_{T} - N}}{N} = \frac{{S_{T} }}{N} - 1$$
where the noise N(f) was estimated by fitting a linear function to the log(S T )–log(f) dependence at the extremes of the 100–450 mHz frequency range:
$${ \log }N = \alpha { \log }\left( f \right) + \beta$$
For each 30-min interval, if the main SNR peak was higher than a given threshold, we assumed that an event occurred at the corresponding frequency; we considered a threshold of six, which corresponds to a noise less than 15% of the experimental signal. Impulsive, broadband signals are not selected by our method because their power spectrum does not emerge significantly from the corresponding noise spectrum. For each event, we also conducted a polarization analysis to estimate the polarization ratio, ellipticity, and azimuthal angle. The total number of selected events was 5119; the number of events per day (we applied a smoothing over 31 days) is plotted in Fig. 5 for different years. The most evident feature is the seasonal variation, which is characterized by a clear winter minimum and a maximum extending during summer/equinoxes months. In addition, focusing on the occurrence summer maxima, we note an increase in events from 2005 to 2010, indicating a negative correlation with the solar activity cycle, while the winter minima do not show such a variation.
Occurrence of ULF events from 2003 to 2010. The number of events per day for different years. Shaded areas represent data gaps
The diurnal distribution of the events, normalized to the available data for each 30-min time interval, is presented in Fig. 6 (black line) for the entire time interval from 2003 to 2010 and separately for summer, equinoctial, and winter months; the error bars in the figure represent the 95% confidence intervals estimated using the Clopper–Pearson method (Clopper and Pearson 1934; Regi et al. 2014). As shown, the overall events show a predominant occurrence maximum around local magnetic noon and a secondary one, less evident, around midnight. If we restrict the analysis to polarized events, in particular with R > 0.85 (red line), the occurrence rate is characterized by a very strong attenuation of the noon maximum with respect to the midnight one. This result is generally independent of the choice of the R value above 0.8; the choice of R = 0.85 reduces by no more than 60% the number of events, still allowing good statistical reliability. The same diurnal distribution is observed for summer and equinoctial events, while winter events are very few and do not clearly show the midnight occurrence maximum.
Daily variation in ULF events. The UT distribution of events, normalized to the available data for each time interval, for the entire time interval, 2003–2010, and separately for summer, equinoctial, and winter months. For each time interval, the number of total events is indicated in parentheses. The vertical lines at approximately 08 UT and 20 UT indicate the magnetic midnight and noon, respectively
We investigated the frequency distribution of events detected, respectively, around midnight (Fig. 7, left panel, black line) and noon (Fig. 7, right panel, black line). While the midnight events predominantly occur at frequencies between 200 and 375 mHz, for the noon events, the distribution is clearly shifted toward lower frequencies, mostly in the 175–275 mHz range. This result suggests that the Pc1–2 power maximum observed around noon is related to the predominantly lower frequency events occurring in the same time interval; this feature is due to the rapid increase in the power spectrum, with decreasing frequency in the 200–400 mHz range (Fig. 4). In contrast, polarized events are almost uniformly distributed between 200 and 375 mHz, both at midnight and noon (Fig. 7, red lines). They are characterized by ellipticity values symmetrically distributed around zero, with ε = 0.04 as the median value, indicating an almost linear polarization (Fig. 8, left panel), and by azimuthal angles peaked between 40° and 70°, with the median value ~ 49° (Fig. 8, right panel), suggesting that the signal power tends to be higher along the east–west component. Similar values of the θ angle were found at TNB in a different frequency range (22–100 mHz) by De Lauretis et al. (2005); they suggested that the polarization characteristics may be affected by ground conductivity anomalies due to the proximity of TNB to the coastline.
Frequencies of selected events. The frequency distribution for events around magnetic local midnight (left panel) and magnetic local noon (right panel). The percentages are computed with respect to the number of events for each time interval. Red lines refer to polarized events
Polarization parameters. The distribution of ellipticity (left panel) and azimuthal angle (right panel) for polarized events. The vertical black lines mark the median values
We also investigated the dependence of events on geomagnetic activity and SW conditions. In particular, we considered the AE index in characterizing the high-latitude geomagnetic activity and SW density n SW, related to magnetospheric compressions, as a SW parameter. Figure 9 shows the distribution of events for disturbed geomagnetic conditions (AE > 300 nT, black line) and high SW densities (n SW > 10 cm−3, red line). The midnight maximum of occurrence clearly becomes prevalent with respect to the noon maximum for high AE values, while it almost disappears for high SW densities. This result indicates that the generation mechanism for Pc1–2 waves observed around midnight is mainly associated with substorm/storm development, while magnetopause compressions cause events almost entirely around noon.
ULF dependence on AE and solar wind density. The UT distribution of the events for disturbed geomagnetic conditions (black line) and high SW densities (red line). The vertical lines at approximately 08 UT and 20 UT indicate the magnetic midnight and noon, respectively
To clarify the relationship between Pc1–2 waves, geomagnetic activity, and SW density, we computed the correlation coefficient between ln(P) and ln(AE) and between ln(P) and ln(n SW) as a function of UT. In this analysis, the total power was integrated in the frequency band where most events were detected, i.e., between 175 and 350 mHz. In Fig. 10, we show the results of the correlation analysis, where the 95% confidence interval is estimated using the null hypothesis. As lustrated, the correlation with the AE index is significant throughout the entire day, with a maximum at magnetic midnight. The correlation with SW density is not significant, except around magnetic noon, when it does not exceed a value of ~ 0.2; we interpret such result as a sporadic correlation, corresponding with high SW density values, which could drive Pc1–2 waves.
Correlation analysis. The diurnal variation in correlation between ln(P) and ln(AE) (upper panel) and between ln(P) and ln(n SW) (bottom panel). The shaded areas indicate the 95% confidence intervals. In each panel the number of selected time intervals is shown, and the vertical lines at approximately 08 UT and 20 UT indicate the magnetic midnight and noon, respectively
Based on the observed results, we developed a simple model that describes ln(P) as a linear function of ln(AE) and ln(n SW):
$${ \ln }\left( P \right) = a_{0} + a_{1} ln\left( {\text{AE}} \right) + a_{2} { \ln }\left( {n_{\text{SW}} } \right)$$
To estimate \(a_{0}\), \(a_{1}\), and \(a_{2}\), we first evaluated the hourly values of the intercept and slope obtained from the linear regression analysis; then, the time series of each parameter a i (i = 0, 1, 2) was fitted using four sinusoidal functions as follows
$$a_{i} = C + \mathop \sum \limits_{k = 1}^{4} A_{k} { \sin }\left( {\omega_{k} t + \varphi_{k} } \right)$$
where time t represents the hour in UT.
To account for seasonal effects, we performed the analysis separately for summer, winter, and equinoctial months from 2003 to 2010. The coefficients of the model are presented in Table 1.
Table 1 Model coefficients
As an example, in Fig. 11 we present the UT dependence of the \(a_{0}\), \(a_{1}\), and \(a_{2}\) coefficients and comparison between the experimental and computed P for March 2008 and January 2016. The comparison is particularly significant for the latter interval, which was not used in computing the model coefficients. We note that \(a_{0}\), the basis of the daily variation, is characterized by a maximum just after magnetic local noon during summers and equinoxes, while it is almost constant during winter, as expected in the absence of solar radiation. The \(a_{1}\) coefficient shows an additional maximum at midnight, associated with substorm processes, while the \(a_{2}\) coefficient only shows a non-negligible contribution around noon; these features are consistent with the observed event distribution during conditions with high AE and n SW (Fig. 9). The time series shows a general correspondence for both the selected months, with a correlation coefficient of ~ 0.8, and for short-term variations. We also computed the correlation coefficients in different time windows, and, as shown in Fig. 12, we found that the correlation is significant, at a confidence level of 95%, for time scales longer than 6–7 h.
Pc1–2 power model. Top panels: the UT dependence on model coefficients. Middle and bottom panels: a comparison between experimental and model P values for March 2008 and January 2016
Correspondence time scales. The correlation coefficients between model and experimental data computed in different time windows. The black and blue lines refer to March 2008 and January 2016, respectively
Here, we show two examples of Pc1–2 events, observed in the noon (Fig. 13) and midnight (Fig. 14) sectors. During the event around local magnetic noon (20 UT = 12 MLT) on October 14, 2010, between 17 and 22 UT, the dynamical Fourier spectrum shows a power enhancement at frequencies of ~ 320–370 mHz, lasting several hours and characterized by a signal-to-noise ratio SNR greater than 10. Between 19:30 and 21:10 UT, the polarization ratio R is very high, reaching ~ 0.90, the ellipticity value is ~ 0, indicating linearly polarized waves, while the azimuthal angle θ decreases from ~ 70° to ~ 45°. The IMF is directed northward and the geomagnetic index AE indicates quiet magnetospheric conditions; the SW density is high (> 10 cm−3) during the entire time interval. This is an example of Pc1–2 waves associated with magnetospheric compression, which persists for several hours. The compression increases ion anisotropy in the layers just inside the magnetopause, more efficiently in the subsolar region, which stimulates EMIC waves (Usanova et al. 2008, 2012 and references therein); the excited waves propagate along the outermost closed field lines toward the high-latitude ionosphere. They are clearly observed at TNB, where they show a linear polarization together with a decrease in the azimuthal component signal, indicating a transmission along the ionospheric waveguide to a large distance from the injection region (Fujita and Tamao 1988).
Noon case study. The dynamical power spectrum, signal-to-noise ratio (white areas correspond to SNR < 0), polarization ratio, ellipticity, azimuthal angle, IMF B z component, SW density, and AE index during the October 14, 2010, event. In addition to the OMNI data (blue line), which show large gaps, the WIND data are also plotted (red line). Black and white contours encircle values corresponding to a polarization ratio R > 0.85
Midnight case study. The dynamical power spectrum, signal-to-noise ratio, polarization ratio, ellipticity, azimuthal angle, IMF B z component, SW density, and AE index during the August 27, 2003, event. Black and white contours encircle values corresponding to a polarization ratio R > 0.85
Figure 14 shows the event observed on August 27, 2003, just after local magnetic midnight (08 UT = 00 MLT). Corresponding to the broadband power enhancements observed in the time interval 09:00–10:00 UT, polarized, short-lived waves in the 300–400 mHz frequency band are observable. They are characterized by an ellipticity of ~ 0° (linear polarization) and a θ value of ~ 40°–50°. These waves are associated with moderate geomagnetic activity, evidenced by IMF southward fluctuations and an AE index higher than 300 nT.
Summary and discussion
We conducted a statistical study of the Pc1–2 waves at TNB, a southern polar cap station. The geomagnetic latitude (~ 80°S) is such that, during the day, the local open field lines approach the magnetopause and closed field lines around noon; it is very close to the latitude of the poleward cusp boundary, particularly during summer and equinoctial months and for northward IMF conditions, i.e., for a quiet magnetosphere (Zhou et al. 1999, 2000).
The long data series, almost continuous in the time interval 2003–2010, allowed us to the study solar cycle and seasonal and daily variations in Pc1–2 power.
We found that the Pc1–2 power seems to follow the solar activity variation, slightly decreasing through the descending phase of the solar cycle (2004–2008) to the solar minimum in 2009. In addition to this weak variation, the Pc1–2 power shows a seasonal modulation, characterized by higher values during the local summer, and a MLT dependence, characterized by a maximum around noon. Both features can be explained, considering that the TNB field lines are at the shortest distances from the outermost closed field lines during summer and around magnetic local noon. Indeed, the latitudinal position of the cusp footprint depends on the dipole tilt angle, which exhibits both an annual variation and daily variation. When the geomagnetic dipole axis tilts more toward the Sun, the cusp moves more poleward; Zhou et al. (1999) examined polar cusp crossings at high altitude, obtained from polar satellite data, and showed that the magnetic latitude of the center cusp moves from ~ 77° to ~ 81°, when the tilt angle changes from − 30° to 30°.
An analysis of Pc1–2 events indicates a seasonal variation, consistent with a similar variation in power. In contrast, examining the MLT dependence, in addition to a main peak of occurrence around noon, also observed in the power analysis, there is a minor peak near midnight. While polarized waves represent only a small fraction of the events observed around noon, midnight events are generally polarized waves. Both noon and midnight polarized waves exhibit a similar frequency distribution. The ellipticity values indicate primarily linearly polarized waves, with the major axis directed at ~ 50° with respect to the H component. Our results suggest that the power peak at noon is due primarily to unpolarized waves, with a minor contribution represented by linearly polarized waves. This observation is interpreted as waves propagated far from the injection region up to the TNB latitude, in the high-latitude ionospheric waveguide along the meridional direction (Kim et al. 2011). The midnight events are very few with respect to the noon events, but are generally consistent with linearly polarized waves.
Considering geomagnetic and SW conditions, we found that the noon peak is associated with time intervals characterized by very high SW densities and associated high AE values (Usanova et al. 2012), while the midnight peak is entirely associated with high AE values. In addition, the Pc1–2 power appears significantly related to AE through the day, particularly at midnight, and to n SW only around noon. Based on these results, we hypothesize that this observation indicates different sources for Pc1–2 waves observed at TNB. In the noon sector, compressions of the outermost closed magnetospheric field lines by the SW are important; in the declining phase of the solar cycle SW pressure variations are generally associated with the fast stream occurrence, which also produces high magnetospheric activity (high AE). We note that the noon maximum of both power and event occurrence is slightly asymmetric, shifted to the post-noon sector, in agreement with the results of previous studies. At geosynchronous orbit, a similar MLT occurrence of EMIC waves associated with SW pressure peaks was documented in Clausen et al. (2011) and Park et al. (2016); on the ground, a post-noon occurrence maximum (12–13 MLT) was observed by Kurazhkovskaya et al. (2007) at the Mirny Observatory (77°S invariant latitude, Antarctica) for magnetic impulses accompanied by Pc1 pulsations. This feature could be due to the impact of SW high density regions leading fast streams (Corotating Interaction Regions), mostly on the post-noon magnetopause, due to the IMF average orientation at the Earth's orbit at ~ 45° with respect to the Sun–Earth direction (Rostoker and Sullivan 1987; Villante et al. 2001). In the midnight sector, Pc1–2 waves are likely caused by substorm/stormrelated ion instability occurring in the plasma sheet. The significant correlation observed between Pc1–2 power and both AE index and SW density allows us to develop a simple model to estimate Pc1–2 power from the two parameters. The model fits the experimental data well, and for time intervals outside the dataset used for the model, up to time scales as short as 6–7 h. Taking into account that the wave power attenuation along the ionospheric waveguide is ~ 10 dB/1000 km (Kim et al. 2011), we believe that the applicability of the model can be extended from auroral latitudes up to latitudes near the geomagnetic pole.
The occurrence of events shows a dependence on the solar cycle, opposite with respect to the power dependence, with an increasing number of events through the years. This feature is consistent with previous results, which showed a strong negative correlation between high-latitude Pc1 events and solar activity, and agrees with the negative correlation between SW density and solar activity (Mursula et al. 1994; Kangas et al. 1998).
We suggest that Pc1–2 events observed around noon at TNB could be due to waves generated just inside the magnetopause, near the equatorial plane, by SW compressions, which was observed in the dayside outer magnetosphere by Engebretson et al. (2002) and Usanova et al. (2012). These waves propagate along the outermost closed field lines into the ionosphere; then, they can propagate along the ionospheric waveguide far from the injection region up to the TNB latitude. For example, at noon, this is over ~ 500–600 km, the average distance of the station from the closed field lines. We hypothesize that due to the closeness of the station to the cusp around noon, TNB is subject to a mixture of different waves and only very strong signals can maintain their properties, such as the polarization degree. For example, the polarized waves of the selected noon event are associated with a long duration magnetospheric compression due to very high SW density and show an ellipticity close to zero, which indicates a linear polarization, as expected for waves propagating far in the waveguide (Greifinger and Greifinger 1968; Greifinger 1972a, b).
Nighttime events are less common, probably because in the dark sector, TNB is embedded in the polar cap, with local field lines far from the magnetospheric regions from which waves originate (as the plasma sheet). The observed waves are generally polarized, likely because additional superimposed signals are absent, as suggested by the lower power content in the nighttime spectra (Fig. 4). Pc1–2 waves are observed at TNB during perturbed geomagnetic conditions, when ion instability increases in the plasma sheet. This result is in agreement with the magnetospheric observations by Usanova et al. 2012, who also detected EMIC waves just before midnight at L = 9–10 during moderate and enhanced substorm activity. The selected nighttime event has a short duration, occurred in correspondence with a geomagnetic substorm, and shows a linear ellipticity. These features are consistent with a wave packet propagated toward the high-latitude ionosphere along magnetotail field lines, from the plasma sheet, and then along the ionospheric waveguide up to TNB.
In conclusion, the main results of our statistical study of Pc1–2 waves at southern polar latitudes are as follows:
Pc1–2 waves are observed around local magnetic noon and midnight, and are, respectively, associated with SW compressions of the magnetopause and substorm/stormrelated instabilities.
Polarized waves, primarily observed around midnight, show an almost linear polarization, suggesting wave propagation along a meridional ionospheric waveguide, from the injection region up to the latitude of Terra Nova Bay.
Based on these results, we propose a simple model to estimate Pc1–2 power variations at auroral and polar latitudes that depend on the geomagnetic activity and SW density.
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MR and MM performed the data analysis. MR and PF drafted the manuscript. PF and MDL participated in the study design and interpretation of results. All authors read and approved the final manuscript.
This research activity was supported by the Italian PNRA (Programma Nazionale di Ricerche in Antartide, PdR2013/B2.09). The authors acknowledge J.H. King and N. Papatashvilli at NASA and CDAWeb for solar wind data (http://cdaweb.gsfc.nasa.gov). Measurements of the magnetic field fluctuations at Terra Nova Bay can be requested from Marcello De Lauretis at the following e-mail address: [email protected]. The authors thank both reviewers for their helpful comments and suggestions.
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Department of Physical and Chemical Sciences, University of L'Aquila, L'Aquila, Italy
M. Regi, M. Marzocchetti, P. Francia & M. De Lauretis
M. Regi
M. Marzocchetti
P. Francia
M. De Lauretis
Correspondence to M. Regi.
Regi, M., Marzocchetti, M., Francia, P. et al. A statistical analysis of Pc1–2 waves at a near-cusp station in Antarctica. Earth Planets Space 69, 152 (2017). https://doi.org/10.1186/s40623-017-0738-8
Local Magnetic Noon
Southern Polar Latitudes
Injection Region
EMIC Waves
Usanov | CommonCrawl |
Cauchy-Schwarz with AM-GM
I want to extend CS from two to three variables. Here's a Cauchy-Schwarz proof with two variables, which is proof 4 from here
Let $A = \sqrt{a_1^2 + a_2^2 + \dots + a_n^2}$ and $B = \sqrt{b_1^2 + b_2^2 + \dots + b_n^2}$. By the arithmetic-geometric means inequality (AGI), we have
$$ \sum_{i=1}^n \frac{a_ib_i}{AB} \leq \sum_{i=1}^n \frac{1}{2} \left( \frac{a_i^2}{A^2} + \frac{b_i^2}{B^2} \right) = 1 $$
$$ \sum_{i=1}^na_ib_i \leq AB =\sqrt{\sum_{i=1}^na_i^2} \sqrt{\sum_{i=1}^n b_i^2} $$
How would I extend this method for three variables, i.e. to get the following? $$ \sum_{i=1}^na_ib_i c_i \leq \sqrt{\sum_{i=1}^na_i^2} \sqrt{\sum_{i=1}^n b_i^2} \sqrt{\sum_{i=1}^n c_i^2} $$
Somehow I don't think it's as trivial as the first method, i.e. simply defining $C$ the same way does not seem to work. Maybe there is a better approach?
linear-algebra inequality proof-writing
vegavega
$$\sum_{i=1}^n(a_ib_i) c_i \leq \sqrt{(\sum_{i=1}^{n}a_i^2b_i^2)(\sum_{i=1}^{n}c_i^2) } \leq \sqrt{\sum_{i=1}^{n}a_i^2} \sqrt{\sum_{i=1}^{n}b_i^2} \sqrt{\sum_{i=1}^{n}c_i^2} $$
First inequality follows using AM $\geq$ GM for two variables( $a_ib_i $'s as one variable, and $c_i$'s as another), and the second one follows as $\sum_{i=1}^{n}a_i^2b_i^2\leq (\sum_{i=1}^{n}a_i^2)(\sum_{i=1}^{n}b_i^2).$
SurajitSurajit
$\begingroup$ What is the name of the second inequality you applied? $\endgroup$ – vega Sep 22 '18 at 15:57
$\begingroup$ @vega Don't know if it has a name. But it's very easy to prove as follows: $(\sum_{i=1}^{n}a_i^2)(\sum_{i=1}^{n}b_i^2)=\sum_{i=1}^{n}a_i^2b_i^2+\sum_{i\neq j}a_i^2b_j^2 \geq \sum_{i=1}^{n}a_i^2b_i^2$ $\endgroup$ – Surajit Sep 22 '18 at 19:34
$\begingroup$ yeah I wrote exactly that down later and figured it out. Thanks a lot! $\endgroup$ – vega Sep 22 '18 at 20:10
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Inequality in non-decreasing sequence
Proof of the Bergström inequality using Cauchy
Prove that $n\sum_{i=1}^n a_ib_i \geq \sum_{i=1}^n a_i \cdot \sum_{i = 1}^n b_i.$
$\frac{1}{2}(\frac{b_1}{a_1}-\frac{b_n}{a_n})^2(\sum_{1}^{n}{a_i^2 }) ^2 \ge (\sum_{1}^{n}{a_i^2 }) (\sum_{1}^{n}{b_i^2 })-(\sum_{1}^{n}{a_ib_i })^2$
Proving Cauchy Inequality from Hölder's inequality?
An inequality involving $a_i, b_i$ such that $\sum_{i=1}^n a_i = \sum_{i=1}^n b_i = 1$
Finding minimum value of $\sum a_ib_i$
Proving Cauchy-Schwarz with Arithmetic Geometric mean
Extending Cauchy-Schwarz
Proof of inequality using Cauchy–Schwarz inequality | CommonCrawl |
\begin{definition}[Definition:Perfect Field/Definition 2]
Let $F$ be a field.
$F$ is a '''perfect field''' {{iff}} one of the following holds:
:$\Char F = 0$
:$\Char F = p$ with $p$ prime and $\Frob$ is an automorphism of $F$
where:
:$\Char F$ denotes the characteristic of $F$
:$\Frob$ denotes the Frobenius endomorphism on $F$
\end{definition} | ProofWiki |
Minimal Length Scale Scenarios for Quantum Gravity
Sabine Hossenfelder1
We review the question of whether the fundamental laws of nature limit our ability to probe arbitrarily short distances. First, we examine what insights can be gained from thought experiments for probes of shortest distances, and summarize what can be learned from different approaches to a theory of quantum gravity. Then we discuss some models that have been developed to implement a minimal length scale in quantum mechanics and quantum field theory. These models have entered the literature as the generalized uncertainty principle or the modified dispersion relation, and have allowed the study of the effects of a minimal length scale in quantum mechanics, quantum electrodynamics, thermodynamics, black-hole physics and cosmology. Finally, we touch upon the question of ways to circumvent the manifestation of a minimal length scale in short-distance physics.
In the 5th century B.C., Democritus postulated the existence of smallest objects that all matter is built from and called them 'atoms'. In Greek, the prefix 'a' means 'not' and the word 'tomos' means 'cut'. Thus, atomos or atom means uncuttable or indivisible. According to Democritus' theory of atomism, "Nothing exists except atoms and empty space, everything else is opinion." Though variable in shape, Democritus' atoms were the hypothetical fundamental constituents of matter, the elementary building blocks of all that exists, the smallest possible entities. They were conjectured to be of finite size, but homogeneous and without substructure. They were the first envisioned end of reductionism.
2500 years later, we know that Democritus was right in that solids and liquids are composed of smaller entities with universal properties that are called atoms in his honor. But these atoms turned out to be divisible. And stripped of its electrons, the atomic nucleus too was found to be a composite of smaller particles, neutrons and protons. Looking closer still, we have found that even neutrons and protons have a substructure of quarks and gluons. At present, the standard model of particle physics with three generations of quarks and fermions and the vector fields associated to the gauge groups are the most fundamental constituents of matter that we know.
Like a Russian doll, reality has so far revealed one after another layer on smaller and smaller scales. This begs the question: Will we continue to look closer into the structure of matter, and possibly find more layers? Or is there a fundamental limit to this search, a limit beyond which we cannot go? And if so, is this a limit in principle or one in practice?
Any answer to this question has to include not only the structure of matter, but the structure of space and time itself, and therefore it has to include gravity. For one, this is because Democritus' search for the most fundamental constituents carries over to space and time too. Are space and time fundamental, or are they just good approximations that emerge from a more fundamental concept in the limits that we have tested so far? Is spacetime made of something else? Are there 'atoms' of space? And second, testing short distances requires focusing large energies in small volumes, and when energy densities increase, one finally cannot neglect anymore the curvature of the background.
In this review we will study this old question of whether there is a fundamental limit to the resolution of structures beyond which we cannot discover anything more. In Section 3, we will summarize different approaches to this question, and how they connect with our search for a theory of quantum gravity. We will see that almost all such approaches lead us to find that the possible resolution of structures is finite or, more graphically, that nature features a minimal length scale — though we will also see that the expression 'minimal length scale' can have different interpretations. While we will not go into many of the details of the presently pursued candidate theories for quantum gravity, we will learn what some of them have to say about the question. After the motivations, we will in Section 4 briefly review some approaches that investigate the consequences of a minimal length scale in quantum mechanics and quantum field theory, models that have flourished into one of the best motivated and best developed areas of the phenomenology of quantum gravity.
In the following, we use the unit convention c = ħ = 1, so that the Planck length lPl is the inverse of the Planck mass mPl = 1/lPl, and Newton's constant \(G = l_{{\rm{P1}}}^2 = 1/m_{{\rm{P1}}}^2\). The signature of the metric is (1, −1, −1, −1). Small Greek indices run from 0 to 3, large Latin indices from 0 to 4, and small Latin indices from 1 to 3, except for Section 3.2, where small Greek indices run from 0 to D, and small Latin indices run from 2 to D. An arrow denotes the spatial component of a vector, for example \(\vec a = ({a_1},{a_2},{a_3})\). Bold-faced quantities are tensors in an index-free notation that will be used in the text for better readability, for example p = (p0, p1, p2, p3). Acronyms and abbreviations can be found in the index.
We begin with a brief historical background.
A Minimal History
Special relativity and quantum mechanics are characterized by two universal constants, the speed of light, c, and Planck's constant, ħ. Yet, from these constants alone one cannot construct either a constant of dimension length or mass. Though, if one had either, they could be converted into each other by use of ħ and c. But in 1899, Max Planck pointed out that adding Newton's constant G to the universal constants c and ħ allows one to construct units of mass, length and time [265]:
$$\begin{array}{*{20}c} {{t_{{\rm{Pl}}}} \approx {{10}^{- 43}}\;{\rm{s}}\quad \quad \quad}\\ {{l_{{\rm{Pl}}}} \approx {{10}^{- 33}}\;{\rm{cm}}\quad \quad}\\ {{m_{{\rm{Pl}}}} \approx 1.2 \times {{10}^{19}}\;{\rm{GeV}}.}\\ \end{array}$$
Today these are known as the Planck time, Planck length and Planck mass, respectively. As we will see later, they mark the scale at which quantum effects of the gravitational interaction are expected to become important. But back in Planck's days their relevance was their universality, because they are constructed entirely from fundamental constants.
The idea of a minimal length was predated by that of the "chronon," a smallest unit of time, proposed by Robert Lévi [200] in 1927 in his "Hyphothèse de l'atome de temps" (hypothesis of time atoms), that was further developed by Pokrowski in the years following Lévi's proposal [266]. But that there might be limits to the divisibility of space and time remained a far-fetched speculation on the fringes of a community rapidly pushing forward the development of general relativity and quantum mechanics. It was not until special relativity and quantum mechanics were joined in the framework of quantum field theory that the possible existence of a minimal length scale rose to the awareness of the community.
With the advent of quantum field theory in the 1930s, it was widely believed that a fundamental length was necessary to cure troublesome divergences. The most commonly used regularization was a cut-off or some other dimensionful quantity to render integrals finite. It seemed natural to think of this pragmatic cut-off as having fundamental significance, an interpretation that however inevitably caused problems with Lorentz invariance, since the cut-off would not be independent of the frame of reference. Heisenberg was among the first to consider a fundamentally-discrete spacetime that would yield a cut-off, laid out in his letters to Bohr and Pauli. The idea of a fundamentally finite length or a maximum frequency was in these years studied by many, including Flint [110], March [219], Möglich [234] and Goudsmit [267], just to mention a few. They all had in common that they considered the fundamental length to be in the realm of subatomic physics on the order of the femtometer (10−15 m).
The one exception was a young Russian, Matvei Bronstein. Today recognized as the first to comprehend the problem of quantizing gravity [138], Bronstein was decades ahead of his time. Already in 1936, he argued that gravity is in one important way fundamentally different from electrodynamics: Gravity does not allow an arbitrarily high concentration of charge in a small region of spacetime, since the gravitational 'charge' is energy and, if concentrated too much, will collapse to a black hole. Using the weak field approximation of gravity, he concluded that this leads to an inevitable limit to the precision of which one can measure the strength of the gravitational field (in terms of the Christoffel symbols).
In his 1936 article "Quantentheorie schwacher Gravitationsfelder" (Quantum theory of weak gravitational fields), Bronstein wrote [138, 70]:
"[T]he gravitational radius of the test-body (GρV/c2) used for the measurements should by no means be larger than its linear dimensions (V1/3); from this one obtains an upper bound for its density (ρ ≲ c2/GV2/3). Thus, the possibilities for measurements in this region are even more restricted than one concludes from the quantum-mechanical commutation relations. Without a profound change of the classical notions it therefore seems hardly possible to extend the quantum theory of gravitation to this region."Footnote 1 ([70], p. 150)Footnote 2
Few people took note of Bronstein's argument and, unfortunately, the history of this promising young physicist ended in a Leningrad prison in February 1938, where Matvei Bronstein was executed at the age of 31.
Heisenberg meanwhile continued in his attempt to make sense of the notion of a fundamental minimal length of nuclear dimensions. In 1938, Heisenberg wrote "Über die in der Theorie der Elementarteilchen auftretende universelle Länge" (On the universal length appearing in the theory of elementary particles) [148], in which he argued that this fundamental length, which he denoted r0, should appear somewhere not too far beyond the classical electron radius (of the order 100 fm).
This idea seems curious today, and has to be put into perspective. Heisenberg was very worried about the non-renormalizability of Fermi's theory of β-decay. He had previously shown [147] that applying Fermi's theory to the high center-of-mass energies of some hundred GeV lead to an 'explosion,' by which he referred to events of very high multiplicity. Heisenberg argued this would explain the observed cosmic ray showers, whose large number of secondary particles we know today are created by cascades (a possibility that was discussed already at the time of Heisenberg's writing, but not agreed upon). We also know today that what Heisenberg actually discovered is that Fermi's theory breaks down at such high energies, and the four-fermion coupling has to be replaced by the exchange of a gauge boson in the electroweak interaction. But in the 1930s neither the strong nor the electroweak force was known. Heisenberg then connected the problem of regularization with the breakdown of the perturbation expansion of Fermi's theory, and argued that the presence of the alleged explosions would prohibit the resolution of finer structures:
"If the explosions actually exist and represent the processes characteristic for the constant r0, then they maybe convey a first, still unclear, understanding of the obscure properties connected with the constant r0. These should certainly express themselves in difficulties of measurements with a precision better than r0… The explosions would have the effect…that measurements of positions are not possible to a precision better than r0."Footnote 3 ([148], p. 31)
In hindsight we know that Heisenberg was, correctly, arguing that the theory of elementary particles known in the 1930s was incomplete. The strong interaction was missing and Fermi's theory indeed non-renormalizable, but not fundamental. Today we also know that the standard model of particle physics is renormalizable and know techniques to deal with divergent integrals that do not necessitate cut-offs, such as dimensional regularization. But lacking that knowledge, it is understandable that Heisenberg argued that taking into account gravity was irrelevant for the existence of a fundamental length:
"The fact that [the Planck length] is significantly smaller than r0 makes it valid to leave aside the obscure properties of the description of nature due to gravity, since they — at least in atomic physics — are totally negligible relative to the much coarser obscure properties that go back to the universal constant r0. For this reason, it seems hardly possible to integrate electric and gravitational phenomena into the rest of physics until the problems connected to the length r0 are solved."Footnote 4 ([148], p. 26)
Heisenberg apparently put great hope in the notion of a fundamental length to move forward the understanding of elementary matter. In 1939 he expressed his belief that a quantum theory with a minimal length scale would be able to account for the discrete mass spectrum of the (then known) elementary particles [149]. However, the theory of quantum electrodynamics was developed to maturity, the 'explosions' were satisfactorily explained and, without being hindered by the appearance of any fundamentally finite resolution, experiments probed shorter and shorter scales. The divergences in quantum field theory became better understood and discrete approaches to space and time remained unappealing due to their problems with Lorentz invariance.
In a 1947 letter to Heisenberg, Pauli commented on the idea of a smallest length that Heisenberg still held dearly and explained his reservations, concluding "Extremely put, I would not be surprised if your 'universal' length turned out to be a mere figment of imagination." [254]. (For more about Heisenberg's historical involvement with the universal length, the interested reader is referred to Kragh's very recommendable article [199].)
In 1930, in a letter to his student Rudolf Peierls [150], Heisenberg mentioned that he was trying to make sense of a minimal length by letting the position operators be non-commuting \([{{\hat x}^v},{{\hat x}^\mu}] \ne 0\). He expressed his hope that Peierls ask Pauli how to proceed with this idea:
"So far, I have not been able to make mathematical sense of such commutation relations… Do you or Pauli have anything to say about the mathematical meaning of such commutation relations?"Footnote 5 ([150], p. 16)
But it took 17 years until Snyder, in 1947, made mathematical sense of Heisenberg's idea.Footnote 6 Snyder, who felt that that the use of a cut-off in momentum space was a "distasteful arbitrary procedure" [288], worked out a modification of the canonical commutation relations of position and momentum operators. In that way, spacetime became Lorentz-covariantly non-commutative, but the modification of commutation relations increased the Heisenberg uncertainty, such that a smallest possible resolution of structures was introduced (a consequence Snyder did not explicitly mention in his paper). Though Snyder's approach was criticized for the difficulties of inclusion of translations [316], it has received a lot of attention as the first to show that a minimal length scale need not be in conflict with Lorentz invariance.
In 1960, Peres and Rosen [262] studied uncertainties in the measurement of the average values of Christoffel symbols due to the impossibility of concentrating a mass to a region smaller than its Schwarzschild radius, and came to the same conclusion as Bronstein already had, in 1936,
"The existence of these quantum uncertainties in the gravitational field is a strong argument for the necessity of quantizing it. It is very likely that a quantum theory of gravitation would then generalize these uncertainty relations to all other Christoffel symbols." ([262], p. 336)
While they considered the limitations for measuring the gravitational field itself, they did not study the limitations these uncertainties induce on the ability to measure distances in general.
It was not until 1964, that Mead pointed out the peculiar role that gravity plays in our attempts to test physics at short distances [222, 223]. He showed, in a series of thought experiments that we will discuss in Section 3.1, that this influence does have the effect of amplifying Heisenberg's measurement uncertainty, making it impossible to measure distances to a precision better than Planck's length. And, since gravity couples universally, this is, though usually negligible, an inescapable influence on all our experiments.
Mead's work did not originally attain a lot of attention. Decades later, he submitted his recollection [224] that "Planck's proposal that the Planck mass, length, and time should form a fundamental system of units…was still considered heretical well into the 1960s," and that his argument for the fundamental relevance of the Planck length met strong resistance:
"At the time, I read many referee reports on my papers and discussed the matter with every theoretical physicist who was willing to listen; nobody that I contacted recognized the connection with the Planck proposal, and few took seriously the idea of [the Planck length] as a possible fundamental length. The view was nearly unanimous, not just that I had failed to prove my result, but that the Planck length could never play a fundamental role in physics. A minority held that there could be no fundamental length at all, but most were then convinced that a [different] fundamental length…, of the order of the proton Compton wavelength, was the wave of the future. Moreover, the people I contacted seemed to treat this much longer fundamental length as established fact, not speculation, despite the lack of actual evidence for it." ([224], p. 15)
But then in the mid 1970s then Hawking's calculation of a black hole's thermodynamical properties [145] introduced the 'transplanckian problem.' Due to the, in principle infinite, blue shift of photons approaching a black-hole horizon, modes with energies exceeding the Planck scale had to be taken into account to calculate the emission rate. A great many physicists have significantly advanced our understanding of black-hole physics and the Planck scale, too many to be named here. However, the prominent role played by John Wheeler, whose contributions, though not directly on the topic of a minimal length, has connected black-hole physics with spacetime foam and the Planckian limit, and by this inspired much of what followed.
Unruh suggested in 1995 [308] that one use a modified dispersion relation to deal with the difficulty of transplanckian modes, so that a smallest possible wavelength takes care of the contributions beyond the Planck scale. A similar problem exists in inflationary cosmology [220] since tracing back in time small frequencies increases the frequency till it eventually might surpass the Planck scale at which point we no longer know how to make sense of general relativity. Thus, this issue of transplanckian modes in cosmology brought up another reason to reconsider the possibility of a minimal length or a maximal frequency, but this time the maximal frequency was at the Planck scale rather than at the nuclear scale. Therefore, it was proposed [180, 144] that this problem too might be cured by implementing a minimum length uncertainty principle into inflationary cosmology.
Almost at the same time, Majid and Ruegg [213] proposed a modification for the commutators of spacetime coordinates, similar to that of Snyder, following from a generalization of the Poincaré algebra to a Hopf algebra, which became known as κ-Poincaré. Kempf et al. [175, 174, 184, 178] developed the mathematical basis of quantum mechanics that took into account a minimal length scale and ventured towards quantum field theory. There are by now many variants of models employing modifications of the canonical commutation relations in order to accommodate a minimal length scale, not all of which make use of the complete κ-Poincaré framework, as will be discussed later in Sections 4.2 and 4.5. Some of these approaches were shown to give rise to a modification of the dispersion relation, though the physical interpretation and relevance, as well as the phenomenological consequences of this relation are still under debate.
In parallel to this, developments in string theory revealed the impossibility of resolving arbitrarily small structures with an object of finite extension. It had already been shown in the late 1980s [140, 10, 9, 11, 310] that string scattering in the super-Planckian regime would result in a generalized uncertainty principle, preventing a localization to better than the string scale (more on this in Section 3.2). In 1996, John Schwarz gave a talk at SLAC about the generalized uncertainty principles resulting from string theory and thereby inspired the 1999 work by Adler and Santiago [3] who almost exactly reproduced Mead's earlier argument, apparently without being aware of Mead's work. This picture was later refined when it became understood that string theory not only contains strings but also higher dimensional objects, known as branes, which will be discussed in Section 3.2.
In the following years, a generalized uncertainty principle and quantum mechanics with the Planck length as a minimal length received an increasing amount of attention as potential cures for the transplanckian problem, a natural UV-regulator, and as possible manifestations of a fundamental property of quantum spacetime. In the late 1990s, it was also noted that it is compatible with string theory to have large or warped extra dimensions that can effectively lower the Planck scale into the TeV range. With this, the fundamental length scale also moved into the reach of collider physics, resulting in a flurry of activity.Footnote 7
Today, how to resolve the apparent disagreements between the quantum field theories of the standard model and general relativity is one of the big open questions in theoretical physics. It is not that we cannot quantize gravity, but that the attempt to do so leads to a perturbatively non-renormalizable and thus fundamentally nonsensical theory. The basic reason is that the coupling constant of gravity, Newton's constant, is dimensionful. This leads to the necessity to introduce an infinite number of counter-terms, eventually rendering the theory incapable of prediction.
But the same is true for Fermi's theory that Heisenberg was so worried about that he argued for a finite resolution where the theory breaks down, and mistakenly so, since he was merely pushing an effective theory beyond its limits. So we have to ask then if we might be making the same mistake as Heisenberg, in that we falsely interpret the failure of general relativity to extend beyond the Planck scale as the occurrence of a fundamentally finite resolution of structures, rather than just the limit beyond which we have to look for a new theory that will allow us to resolve smaller distances still?
If it was only the extension of classical gravity, laid out in many thought experiments that will be discussed in Section 3.1, that had us believing the Planck length is of fundamental importance, then the above historical lesson should caution us we might be on the wrong track. Yet, the situation today is different from the one that Heisenberg faced. Rather than pushing a quantum theory beyond its limits, we are pushing a classical theory and conclude that its short-distance behavior is troublesome, which we hope to resolve with quantizing the theory. And, as we will see, several attempts at a UV-completion of gravity, discussed in Sections 3.2–3.7, suggest that the role of the Planck length as a minimal length carries over into the quantum regime as a dimensionful regulator, though in very different ways, feeding our hopes that we are working on unveiling the last and final Russian doll.
For a more exhaustive coverage of the history of the minimal length, the interested reader is referred to [141].
Thought experiments have played an important role in the history of physics as the poor theoretician's way to test the limits of a theory. This poverty might be an actual one of lacking experimental equipment, or it might be one of practical impossibility. Luckily, technological advances sometimes turn thought experiments into real experiments, as was the case with Einstein, Podolsky and Rosen's 1935 paradox. But even if an experiment is not experimentally realizable in the near future, thought experiments serve two important purposes. First, by allowing the thinker to test ranges of parameter space that are inaccessible to experiment, they may reveal inconsistencies or paradoxes and thereby open doors to an improvement in the fundamentals of the theory. The complete evaporation of a black hole and the question of information loss in that process is a good example for this. Second, thought experiments tie the theory to reality by the necessity to investigate in detail what constitutes a measurable entity. The thought experiments discussed in the following are examples of this.
The Heisenberg microscope with Newtonian gravity
Let us first recall Heisenberg's microscope, that lead to the uncertainty principle [146]. Consider a photon with frequency ω moving in direction x, which scatters on a particle whose position on the x-axis we want to measure. The scattered photons that reach the lens of the microscope have to lie within an angle ϵ to produce an image from which we want to infer the position of the particle (see Figure 1). According to classical optics, the wavelength of the photon sets a limit to the possible resolution Δx
$$\Delta x \underset{\sim}{>} {1 \over {2\pi \omega \sin \epsilon}}.$$
Heisenberg's microscope. A photon moving along the x-axis scatters off a probe within an interaction region of radius R and is detected by a microscope (indicated by a lens and screen) with opening angle ϵ.
But the photon used to measure the position of the particle has a recoil when it scatters and transfers a momentum to the particle. Since one does not know the direction of the photon to better than ϵ, this results in an uncertainty for the momentum of the particle in direction x
$$\Delta {p_x}\underset{\sim}{>} \omega \sin \epsilon.$$
Taken together one obtains Heisenberg's uncertainty (up to a factor of order one)
$$\Delta x\Delta {p_x}\underset{\sim}{>} {1 \over {2\pi}}.$$
We know today that Heisenberg's uncertainty is not just a peculiarity of a measurement method but much more than that — it is a fundamental property of the quantum nature of matter. It does not, strictly speaking, even make sense to consider the position and momentum of the particle at the same time. Consequently, instead of speaking about the photon scattering off the particle as if that would happen in one particular point, we should speak of the photon having a strong interaction with the particle in some region of size R.
Now we will include gravity in the picture, following the treatment of Mead [222]. For any interaction to take place and subsequent measurement to be possible, the time elapsed between the interaction and measurement has to be at least on the order of the time, τ, the photon needs to travel the distance R, so that τ ≳ R. The photon carries an energy that, though in general tiny, exerts a gravitational pull on the particle whose position we wish to measure. The gravitational acceleration acting on the particle is at least on the order of
$$a \approx {{G\omega} \over {{R^2}}},$$
and, assuming that the particle is non-relativistic and much slower than the photon, the acceleration lasts about the duration the photon is in the region of strong interaction. From this, the particle acquires a velocity of v ≈ aR, or
$$v \approx {{G\omega} \over R}.$$
Thus, in the time R, the acquired velocity allows the particle to travel a distance of
$$L \approx G \omega.$$
However, since the direction of the photon was unknown to within the angle ϵ, the direction of the acceleration and the motion of the particle is also unknown. Projection on the x-axis then yields the additional uncertainty of
$$\Delta x \underset{\sim}{>} G\omega \sin \epsilon.$$
Combining (8) with (2), one obtains
$$\Delta x\underset{\sim}{>}\sqrt G = {l_{{\rm{Pl}}}}.$$
One can refine this argument by taking into account that strictly speaking during the measurement, the momentum of the photon, ω, increases by Gmω/R, where m is the mass of the particle. This increases the uncertainty in the particle's momentum (3) to
$$\Delta {p_x}\underset{\sim}{>}\omega \left({1 + {{Gm} \over R}} \right)\sin \epsilon,$$
and, for the time the photon is in the interaction region, translates into a position uncertainty Δx ≈ RΔp/m
$$\Delta x\underset{\sim}{>} \omega \left({{R \over m} + G} \right)\sin \epsilon,$$
which is larger than the previously found uncertainty (8) and thus (9) still follows.
Adler and Santiago [3] offer pretty much the same argument, but add that the particle's momentum uncertainty Δp should be on the order of the photon's momentum ω. Then one finds
$$\Delta x\underset{\sim}{>} G\Delta p.$$
Assuming that the normal uncertainty and the gravitational uncertainties add linearly, one arrives at
$$\Delta x\underset{\sim}{>} {1 \over {\Delta p}} + G\Delta p.$$
Any uncertainty principle with a modification of this or similar form has become known in the literature as 'generalized uncertainty principle' (GUP). Adler and Santiago's work was inspired by the appearance of such an uncertainty principle in string theory, which we will investigate in Section 3.2. Adler and Santiago make the interesting observation that the GUP (13) is invariant under the replacement
$${l_{{\rm{Pl}}}}\Delta p \leftrightarrow {1 \over {{l_{{\rm{Pl}}}}\Delta p}},$$
which relates long to short distances and high to low energies.
These limitations, refinements of which we will discuss in the following Sections 3.1.2–3.1.7, apply to the possible spatial resolution in a microscope-like measurement. At the high energies necessary to reach the Planckian limit, the scattering is unlikely to be elastic, but the same considerations apply to inelastic scattering events. Heisenberg's microscope revealed a fundamental limit that is a consequence of the non-commutativity of position and momentum operators in quantum mechanics. The question that the GUP then raises is what modification of quantum mechanics would give rise to the generalized uncertainty, a question we will return to in Section 4.2.
Another related argument has been put forward by Scardigli [275], who employs the idea that once one arrives at energies of about the Planck mass and concentrates them to within a volume of radius of the Planck length, one creates tiny black holes, which subsequently evaporate. This effects scales in the same way as the one discussed here, and one arrives again at (13).
The general relativistic Heisenberg microscope
The above result makes use of Newtonian gravity, and has to be refined when one takes into account general relativity. Before we look into the details, let us start with a heuristic but instructive argument. One of the most general features of general relativity is the formation of black holes under certain circumstances, roughly speaking when the energy density in some region of spacetime becomes too high. Once matter becomes very dense, its gravitational pull leads to a total collapse that ends in the formation of a horizon.Footnote 8 It is usually assumed that the Hoop conjecture holds [306]: If an amount of energy ω is compacted at any time into a region whose circumference in every direction is R ≤ 4πGω, then the region will eventually develop into a black hole. The Hoop conjecture is unproven, but we know from both analytical and numerical studies that it holds to very good precision [107, 168].
Consider now that we have a particle of energy ω. Its extension R has to be larger than the Compton wavelength associated to the energy, so R ≥ 1/ω. Thus, the larger the energy, the better the particle can be focused. On the other hand, if the extension drops below 4πGE, then a black hole is formed with radius 2ωG. The important point to notice here is that the extension of the black hole grows linearly with the energy, and therefore one can achieve a minimal possible extension, which is on the order of \(R \sim \sqrt G\).
For the more detailed argument, we follow Mead [222] with the general relativistic version of the Heisenberg microscope that was discussed in Section 3.1.1. Again, we have a particle whose position we want to measure by help of a test particle. The test particle has a momentum vector \((\omega, \vec k)\), and for completeness we consider a particle with rest mass μ, though we will see later that the tightest constraints come from the limit μ → 0.
The velocity v of the test particle is
$$v = {k \over {\sqrt {{\mu ^2} + {k^2}}}},$$
where k2 = ω2 − μ2, and \(k = |\vec k|\). As before, the test particle moves in the x direction. The task is now to compute the gravitational field of the test particle and the motion it causes on the measured particle.
To obtain the metric that the test particle creates, we first change into the rest frame of the particle by boosting into x-direction. Denoting the new coordinates with primes, the measured particle moves towards the test particle in direction −x′, and the metric is a Schwarzschild metric. We will only need it on the x-axis where we have y = z = 0, and thus
$${g_{00}^{\prime}} = 1 + 2\phi{\prime}\,,\quad {g_{11}^{\prime}} = - {1 \over {{{g}_{00}^{\prime}}}}\,,\quad {g_{22}^{\prime}} = {g_{33}^{\prime}} = - 1,$$
$$\phi{\prime} = {{G\mu} \over {\vert x{\prime}\vert}},$$
and the remaining components of the metric vanish. Using the transformation law for tensors
$${g_{\mu \nu}} = {{\partial {{(x{\prime})}^\kappa}} \over {\partial {x^\mu}}}{{\partial {{(x{\prime})}^\alpha}} \over {\partial {x^\nu}}}{g_{\kappa \alpha}^{\prime}},$$
with the notation x0 = t, x1 = x, x2 = y, x3 = z, and the same for the primed coordinates, the Lorentz boost from the primed to unprimed coordinates yields in the rest frame of the measured particle
$${g_{00}} = {{1 + 2\phi} \over {1 + 2\phi (1 - {v^2})}} + 2\phi, \qquad {g_{11}} = - {{- 1 + 2\phi {v^2}} \over {1 + 2\phi (1 - {v^2})}} + 2{v^2}\phi,$$
$${g_{01}} = {g_{10}} = - {{2v\phi} \over {1 + \phi (1 - {v^2})}} - 2v\phi, \qquad g_{22}^\prime = g_{33}^\prime = - 1,$$
$$\phi = {{\phi{\prime}} \over {1 - {v^2}}} = - {{G\omega} \over R}.$$
Here, R = vt − x is the mean distance between the test particle and the measured particle. To avoid a horizon in the rest frame, we must have 2Φ′ < 1, and thus from Eq. (21)
$$- 2\phi{\prime} = 2{{G\omega} \over R}(1 - {v^2}) < 1.$$
Because of Eq. (2), Δx ≥ 1/ω but also Δx ≥ R, which is the area in which the particle may scatter, thus
$$\Delta {x^2}\underset{\sim}{>} {R \over \omega} \underset{\sim}{>} 2G(1 - {v^2}).$$
We see from this that, as long as v2 ≪ 1, the previously found lower bound on the spatial resolution Δx can already be read off here, and we turn our attention towards the case where 1 − v2 ≪ 1. From (21) we see that this means we work in the limit where −ϕ ≫ 1.
To proceed, we need to estimate now how much the measured particle moves due to the test particle's vicinity. For this, we note that the world line of the measured particle must be timelike. We denote the velocity in the x-direction with u, then we need
$$d{s^2} = \left({{g_{00}} + 2{g_{10}}u + {g_{11}}{u^2}} \right){\rm{d}}{t^2} \ge 0.$$
Now we insert Eq. (20) and follow Mead [222] by introducing the abbreviation
$$\alpha = 1 + 2\phi (1 - {v^2}).$$
Because of Eq. (22), 0 < α < 1. We simplify the requirement of Eq. (24) by leaving u2 alone on the left side of the inequality, subtracting 1 and dividing by u − 1. Taking into account that Φ ≤ 0 and v ≤ 1, one finds after some algebra
$$u \ge {{1 + 2\phi (1 + \alpha)} \over {1 - 2\phi {v^2}(1 + \alpha)}},$$
$${u \over {1 - u}} \ge - {1 \over 2}\left({1 + 2\phi} \right).$$
One arrives at this estimate with reduced effort if one makes it clear to oneself what we want to estimate. We want to know, as previously, how much the particle, whose position we are trying to measure, will move due to the gravitational attraction of the particle we are using for the measurement. The faster the particles pass by each other, the shorter the interaction time and, all other things being equal, the less the particle we want to measure will move. Thus, if we consider a photon with v = 1, we are dealing with the case with the least influence, and if we find a minimal length in this case, it should be there for all cases. Setting v = 1, one obtains the inequality Eq. (27) with greatly reduced work.
Now we can continue as before in the non-relativistic case. The time τ required for the test particle to move a distance R away from the measured particle is at least τ ≳ R/(1 − u), and during this time the measured particle moves a distance
$$L = u\tau \underset{\sim}{>} R{u \over {1 - u}} \underset{\sim}{>} {R \over 2}\left({- 1 - 2\phi} \right).$$
Since we work in the limit −ϕ ≫ 1, this means
$$L \underset{\sim}{>} G \omega,$$
and projection on the x-axis yields as before (compare to Eq. (8)) for the uncertainty added to the measured particle because the photon's direction was known only to precision ϵ
$$\Delta x \underset{\sim}{>} G\omega \sin\epsilon.$$
This combines with (2), to again give
$$\Delta x\underset{\sim}{>} {l_{{\rm{Pl}}}}.$$
Adler and Santiago [3] found the same result by using the linear approximation of Einstein's field equation for a cylindrical source with length l and radius ρ of comparable size, filled by a radiation field with total energy ω, and moving in the x direction. With cylindrical coordinates x, r, ϕ, the line element takes the form [3]
$${\rm{d}}{s^2} = {\rm{d}}{t^2} - {\rm{d}}{x^2} - {\rm{d}}{y^2} - {\rm{d}}{z^2} + f(r,x,t){({\rm{dt}} - {\rm{d}}x)^2},$$
where the function f is given by
$$f(r,x,t) = {{4G\omega} \over l}g(r)\theta (x - t)\theta (t - x - l)$$
$$g(r) = \left\{{\begin{array}{*{20}c} {{r^2}/{\rho ^2}\quad \quad \;} & {{\rm{for}}} & {r < \rho}\\ {1 + \ln ({r^2}/{\rho ^2})} & {{\rm{for}}} & {r > \rho}\\ \end{array}.} \right.$$
In this background, one can then compute the motion of the measured particle by using the Newtonian limit of the geodesic equation, provided the particle remains non-relativistic. In the longitudinal direction, along the motion of the test particle one finds
$${{{{\rm{d}}^2}x} \over {{\rm{d}}{t^2}}} = {1 \over 2}{{\partial f} \over {\partial x}}.$$
The derivative of f gives two delta-functions at the front and back of the cylinder with equal momentum transfer but of opposite direction. The change in velocity to the measured particle is
$$\Delta \dot x = 2G{\omega \over l}g(r).$$
Near the cylinder g(r) is of order one, and in the time of passage τ ∼ l, the particle thus moves approximately
$$2 G \omega,$$
which is, up to a factor of 2, the same result as Mead's (29). We note that Adler and Santiago's argument does not make use of the requirement that no black hole should be formed, but that the appropriateness of the non-relativistic and weak-field limit is questionable.
Limit to distance measurements
Wigner and Salecker [274] proposed the following thought experiment to show that the precision of length measurements is limited. Consider that we try to measure a length by help of a clock that detects photons, which are reflected by a mirror at distance D and return to the clock. Knowing the speed of light is universal, from the travel-time of the photon we can then extract the distance it has traveled. How precisely can we measure the distance in this way?
Consider that at emission of the photon, we know the position of the (non-relativistic) clock to precision Δx. This means, according to the Heisenberg uncertainty principle, we cannot know its velocity to better than
$$\Delta v = {1 \over {2M\Delta x}},$$
where M is the mass of the clock. During the time T = 2D that the photon needed to travel towards the mirror and back, the clock moves by TΔv, and so acquires an uncertainty in position of
$$\Delta x + {T \over {2M\Delta x}},$$
which bounds the accuracy by which we can determine the distance D. The minimal value that this uncertainty can take is found by varying with respect to Δx and reads
$$\Delta {x_{\min}} = \sqrt {{T \over {2M}}}.$$
Taking into account that our measurement will not be causally connected to the rest of the world if it creates a black hole, we require D > 2MG and thus
$$\Delta {x_{\min}} \underset{\sim}{>} {l_{{\rm{Pl}}}}.$$
Limit to clock synchronization
From Mead's [222] investigation of the limit for the precision of distance measurements due to the gravitational force also follows a limit on the precision by which clocks can be synchronized.
We will consider the clock synchronization to be performed by the passing of light signals from some standard clock to the clock under question. Since the emission of a photon with energy spread Δω by the usual Heisenberg uncertainty is uncertain by ΔT ∼ 1/(2Δω), we have to take into account the same uncertainty for the synchronization.
The new ingredient comes again from the gravitational field of the photon, which interacts with the clock in a region R over a time τ ≳ R. If the clock (or the part of the clock that interacts with the photon) remains stationary, the (proper) time it records stands in relation to τ by \(T = \tau \sqrt {g00}\) with g00 in the rest frame of the clock, given by Eq. (20), thus
$$T = \tau \sqrt {1 - {{4G\omega} \over r}}.$$
Since the metric depends on the energy of the photon and this energy is not known precisely, the error on ω propagates into T by
$${(\Delta T)^2} = {\left({{{\partial T} \over {\partial \omega}}} \right)^2}{(\Delta \omega)^2},$$
$$\Delta T\sim{{2G\tau} \over {r\sqrt {1 - 4G\omega/r}}}\Delta \omega.$$
Since in the interaction region τ ≳ R ≳ r, we can estimate
$$\Delta T \underset{\sim}{>} {{2G} \over {\sqrt {1 - 4G\omega/R}}}\Delta \omega \underset{\sim}{>} 2G\Delta \omega.$$
Multiplication of (45) with the normal uncertainty ΔT ≳ 1/(2Δω) yields
$$\Delta T \underset{\sim}{>} {l_{{\rm{Pl}}}}.$$
So we see that the precision by which clocks can be synchronized is also bound by the Planck scale.
However, strictly speaking the clock does not remain stationary during the interaction, since it moves towards the photon due to the particles' mutual gravitational attraction. If the clock has a velocity u, then the proper time it records is more generally given by
$$T = \int {\rm{d}} s\sim\tau \sqrt {{g_{00}} + 2{g_{01}}u + {g_{11}}{u^2}}.$$
Using (20) and proceeding as before, one estimates the propagation of the error in the frequency by using v = 1 and u ≤ 1
$$\left\vert {{{{\rm{d}}T} \over {{\rm{d}}\omega}}} \right\vert \underset{\sim}{>} \tau {{8G} \over r}{1 \over {\sqrt {1 + 4G\omega/r}}},$$
and so with τ ≳ R≳ r
$$\Delta T \underset{\sim}{>} \tau {G \over R}\Delta \omega \underset{\sim}{>} G\Delta \omega.$$
Therefore, taking into account that the clock does not remain stationary, one still arrives at (46).
Limit to the measurement of the black-hole-horizon area
The above microscope experiment investigates how precisely one can measure the location of a particle, and finds the precision bounded by the inevitable formation of a black hole. However, this position uncertainty is for the location of the measured particle however and not for the size of the black hole or its radius. There is a simple argument why one would expect there to also be a limit to the precision by which the size of a black hole can be measured, first put forward in [91]. When the mass of a black-hole approaches the Planck mass, the horizon radius R ∼ GM associated to the mass becomes comparable to its Compton wavelength λ = 1/M. Then, quantum fluctuations in the position of the black hole should affect the definition of the horizon.
A somewhat more elaborate argument has been studied by Maggiore [208] by a thought experiment that makes use once again of Heisenberg's microscope. However, this time one wants to measure not the position of a particle, but the area of a (non-rotating) charged black hole's horizon. In Boyer-Lindquist coordinates, the horizon is located at the radius
$${R_H} = GM\left[ {1 + {{\left({1 - {{{Q^2}} \over {G{M^2}}}} \right)}^{{1 \over 2}}}} \right],$$
where Q is the charge and M is the mass of the black hole.
To deduce the area of the black hole, we detect the black hole's Hawking radiation and aim at tracing it back to the emission point with the best possible accuracy. For the case of an extremal black hole (Q2 = GM2) the temperature is zero and we perturb the black hole by sending in photons from asymptotic infinity and wait for re-emission.
If the microscope detects a photon of some frequency ω, it is subject to the usual uncertainty (2) arising from the photon's finite wavelength that limits our knowledge about the photon's origin. However, in addition, during the process of emission the mass of the black hole changes from M + ω to M, and the horizon radius, which we want to measure, has to change accordingly. If the energy of the photon is known only up to an uncertainty Δp, then the error propagates into the precision by which we can deduce the radius of the black hole
$$\Delta {R_H}\sim\left\vert {{{\partial {R_H}} \over {\partial M}}} \right\vert \Delta p.$$
With use of (50) and assuming that no naked singularities exist in nature M2G ≤ Q2 one always finds that
$$\Delta {R_H} \underset{\sim}{>} 2G\Delta p.$$
In an argument similar to that of Adler and Santiago discussed in Section 3.1.2, Maggiore then suggests that the two uncertainties, the usual one inversely proportional to the photon's energy and the additional one (52), should be linearly added to
$$\Delta {R_H} \underset{\sim}{>} {1 \over {\Delta p}} + \alpha G\Delta p,$$
where the constant α would have to be fixed by using a specific theory. Minimizing the possible position uncertainty, one thus finds again a minimum error of ≈ αlPl.
It is clear that the uncertainty Maggiore considered is of a different kind than the one considered by Mead, though both have the same origin. Maggiore's uncertainty is due to the impossibility of directly measuring a black hole without it emitting a particle that carries energy and thereby changing the black-hole-horizon area. The smaller the wavelength of the emitted particle, the larger the so-caused distortion. Mead's uncertainty is due to the formation of black holes if one uses probes of too high an energy, which limits the possible precision. But both uncertainties go back to the relation between a black hole's area and its mass.
A device-independent limit for non-relativistic particles
Even though the Heisenberg microscope is a very general instrument and the above considerations carry over to many other experiments, one may wonder if there is not some possibility to overcome the limitation of the Planck length by use of massive test particles that have smaller Compton wavelengths, or interferometers that allow one to improve on the limitations on measurement precisions set by the test particles' wavelengths. To fill in this gap, Calmet, Graesser and Hsu [72, 73] put forward an elegant device-independent argument. They first consider a discrete spacetime with a sub-Planckian spacing and then show that no experiment is able to rule out this possibility. The point of the argument is not the particular spacetime discreteness they consider, but that it cannot be ruled out in principle.
The setting is a position operator \({\hat x}\) with discrete eigenvalues {xi} that have a separation of order lPl or smaller. To exclude the model, one would have to measure position eigenvalues x and x′, for example, of some test particle of mass M, with ∣x − x′∣ ≤ lPl. Assuming the non-relativistic Schrödinger equation without potential, the time-evolution of the position operator is given by \({\rm{d}}\hat x(t)/{\rm{d}}t = i[\hat H,\hat x(t)] = \hat p/M\), and thus
$$\hat x(t) = \hat x(0) + \hat p(0){t \over M}.$$
We want to measure the expectation value of position at two subsequent times in order to attempt to measure a spacing smaller than the Planck length. The spectra of any two Hermitian operators have to fulfill the inequality
$$\Delta A\Delta B \ge {1 \over {2i}}\langle [\hat A,\hat B]\rangle,$$
where Δ denotes, as usual, the variance and 〈·〉 the expectation value of the operator. From (54) one has
$$[\hat x(0),\hat x(t)] = i{t \over M},$$
$$\Delta x(0)\Delta x(t) \ge {t \over {2M}}.$$
Since one needs to measure two positions to determine a distance, the minimal uncertainty to the distance measurement is
$$\Delta x \ge \sqrt {{t \over {2M}}}.$$
This is the same bound as previously discussed in Section 3.1.3 for the measurement of distances by help of a clock, yet we arrived here at this bound without making assumptions about exactly what is measured and how. If we take into account gravity, the argument can be completed similar to Wigner's and still without making assumptions about the type of measurement, as follows.
We use an apparatus of size R. To get the spacing as precise as possible, we would use a test particle of high mass. But then we will run into the, by now familiar, problem of black-hole formation when the mass becomes too large, so we have to require
$$M < 2{R \over G}.$$
Thus, we cannot make the detector arbitrarily small. However, we also cannot make it arbitrarily large, since the components of the detector have to at least be in causal contact with the position we want to measure, and so t > R. Taken together, one finds
$$\Delta x \ge \sqrt {{t \over {2M}}} \ge \sqrt {{R \over {2M}}} \ge \sqrt G,$$
and thus once again the possible precision of a position measurement is limited by the Planck length.
A similar argument was made by Ng and van Dam [238], who also pointed out that with this thought experiment one can obtain a scaling for the uncertainty with the third root of the size of the detector. If one adds the position uncertainty (58) from the non-vanishing commutator to the gravitational one, one finds
$$\Delta x\underset{\sim}{>} \sqrt {{R \over {2M}}} + GM.$$
Optimizing this expression with respect to the mass that yields a minimal uncertainty, one finds \(M \sim {(R/l_{{\rm{P1}}}^4)^{1/3}}\) (up to factors of order one) and, inserting this value of M in (61), thus
$$\Delta x \underset{\sim}{>} {\left({Rl_{{\rm{Pl}}}^2} \right)^{{1 \over 3}}}.$$
Since R too should be larger than the Planck scale this is, of course, consistent with the previously-found minimal uncertainty.
Ng and van Dam further argue that this uncertainty induces a minimum error in measurements of energy and momenta. By noting that the uncertainty Δx of a length R is indistinguishable from an uncertainty of the metric components used to measure the length, Δx2 =R2Δg, the inequality (62) leads to
$$\Delta {g_{\mu \nu}} \underset{\sim}{>} {\left({{{{l_{{\rm{Pl}}}}} \over R}} \right)^{{2 \over 3}}}.$$
But then again the metric couples to the stress-energy tensor Tμν, so this uncertainty for the metric further induces an uncertainty for the entries of Tμν
$$({g_{\mu \nu}} + \Delta {g_{\mu \nu}}){T^{\mu \nu}} = {g_{\mu \nu}}({T^{\mu \nu}} + \Delta {T^{\mu \nu}}){.}$$
Consider now using a test particle of momentum p to probe the physics at scale R, thus p ∼ 1/R. Then its uncertainty would be on the order of
$$\Delta p \underset{\sim}{>} p{\left({{{{l_{{\rm{Pl}}}}} \over R}} \right)^{{2 \over 3}}} = p{\left({{p \over {{m_{{\rm{Pl}}}}}}} \right)^{{2 \over 3}}}.$$
However, note that the scaling found by Ng and van Dam only follows if one works with the masses that minimize the uncertainty (61). Then, even if one uses a detector of the approximate extension of a cm, the corresponding mass of the 'particle' we have to work with would be about a ton. With such a mass one has to worry about very different uncertainties. For particles with masses below the Planck mass on the other hand, the size of the detector would have to be below the Planck length, which makes no sense since its extension too has to be subject to the minimal position uncertainty.
Limits on the measurement of spacetime volumes
The observant reader will have noticed that almost all of the above estimates have explicitly or implicitly made use of spherical symmetry. The one exception is the argument by Adler and Santiago in Section 3.1.2 that employed cylindrical symmetry. However, it was also assumed there that the length and the radius of the cylinder are of comparable size.
In the general case, when the dimensions of the test particle in different directions are very unequal, the Hoop conjecture does not forbid any one direction to be smaller than the Schwarzschild radius to prevent collapse of some matter distribution, as long as at least one other direction is larger than the Schwarzschild radius. The question then arises what limits that rely on black-hole formation can still be derived in the general case.
A heuristic motivation of the following argument can be found in [101], but here we will follow the more detailed argument by Tomassini and Viaggiu [307]. In the absence of spherical symmetry, one may still use Penrose's isoperimetric-type conjecture, according to which the apparent horizon is always smaller than or equal to the event horizon, which in turn is smaller than or equal to 16πG2ω2, where ω is as before the energy of the test particle.
Then, without spherical symmetry the requirement that no black hole ruins our ability to resolve short distances is weakened from the energy distribution having a radius larger than the Schwarzschild radius, to the requirement that the area A, which encloses ω is large enough to prevent Penrose's condition for horizon formation
$$A \ge 16\pi {G^2}{\omega ^2}.$$
The test particle interacts during a time ΔT that, by the normal uncertainty principle, is larger than 1/(2ω). Taking into account this uncertainty on the energy, one has
$$A{(\Delta T)^2} \ge 4\pi {G^2}.$$
Now we have to make some assumption for the geometry of the object, which will inevitably be a crude estimate. While an exact bound will depend on the shape of the matter distribution, we will here just be interested in obtaining a bound that depends on the three different spatial extensions, and is qualitatively correct. To that end, we assume the mass distribution fits into some smallest box with side-lengths Δx1, Δx2, Δx3, which is similar to the limiting area
$$A\sim{{\Delta {x^1}\Delta {x^2} + \Delta {x^1}\Delta {x^3} + \Delta {x^2}\Delta {x^3}} \over {{\alpha ^2}}},$$
where we added some constant α to take into account different possible geometries. A comparison with the spherical case, Δxi = 2R, fixes α2 = 3/π. With Eq. (67) one obtains
$${\left({\Delta t} \right)^2}\left({\Delta {x^1}\Delta {x^2} + \Delta {x^1}\Delta {x^3} + \Delta {x^2}\Delta {x^3}} \right) \ge 12l_{\rm{p}}^4.$$
$${\left({\Delta {x^1} + \Delta {x^2} + \Delta {x^3}} \right)^2} \ge \Delta {x^1}\Delta {x^2} + \Delta {x^1}\Delta {x^3} + \Delta {x^2}\Delta {x^3}$$
one also has
$$\Delta t\left({\Delta {x^1} + \Delta {x^2} + \Delta {x^3}} \right) \ge 12l_{\rm{p}}^2,$$
which confirms the limit obtained earlier by heuristic reasoning in [101].
Thus, as anticipated, taking into account that a black hole must not necessarily form if the spatial extension of a matter distribution is smaller than the Schwarzschild radius in only one direction, the uncertainty we arrive at here depends on the extension in all three directions, rather than applying separately to each of them. Here we have replaced ω by the inverse of ΔT, rather than combining with Eq. (2), but this is just a matter of presentation.
Since the bound on the volumes (71) follows from the bounds on spatial and temporal intervals we found above, the relevant question here is not whether ?? is fulfilled, but whether the bound Δx ≳ lPl can be violated [165].
To address that question, note that the quantities Δxi in the above argument by Tomassini and Viaggiu differ from the ones we derived bounds for in Sections 3.1.1–3.1.6. Previously, the Δx was the precision by which one can measure the position of a particle with help of the test particle. Here, the Δxi are the smallest possible extensions of the test particle (in the rest frame), which with spherical symmetry would just be the Schwarzschild radius. The step in which one studies the motion of the measured particle that is induced by the gravitational field of the test particle is missing in this argument. Thus, while the above estimate correctly points out the relevance of non-spherical symmetries, the argument does not support the conclusion that it is possible to test spatial distances to arbitrary precision.
The main obstacle to completion of this argument is that in the context of quantum field theory we are eventually dealing with particles probing particles. To avoid spherical symmetry, we would need different objects as probes, which would require more information about the fundamental nature of matter. We will come back to this point in Section 3.2.3.
String theory is one of the leading candidates for a theory of quantum gravity. Many textbooks have been dedicated to the topic, and the interested reader can also find excellent resources online [187, 278, 235, 299]. For the following we will not need many details. Most importantly, we need to know that a string is described by a 2-dimensional surface swept out in a higher-dimensional spacetime. The total number of spatial dimensions that supersymmetric string theory requires for consistency is nine, i.e., there are six spatial dimensions in addition to the three we are used to. In the following we will denote the total number of dimensions, both time and space-like, with D. In this Subsection, Greek indices run from 0 to D.
The two-dimensional surface swept out by the string in the D-dimensional spacetime is referred to as the 'worldsheet,' will be denoted by Xν, and will be parameterized by (dimensionless) parameters σ and τ, where τ is its time-like direction, and σ runs conventionally from 0 to 2π. A string has discrete excitations, and its state can be expanded in a series of these excitations plus the motion of the center of mass. Due to conformal invariance, the worldsheet carries a complex structure and thus becomes a Riemann surface, whose complex coordinates we will denote with z and \({\bar z}\). Scattering amplitudes in string theory are a sum over such surfaces.
In the following ls is the string scale, and \({\alpha\prime} = l_{\rm{s}}^2\). The string scale is related to the Planck scale by \({l_{{\rm{P1}}}} = g_{\rm{s}}^{1/4}{l_{\rm{s}}}\), where gs is the string coupling constant. Contrary to what the name suggests, the string coupling constant is not constant, but depends on the value of a scalar field known as the dilaton.
To avoid conflict with observation, the additional spatial dimensions of string theory have to be compactified. The compactification scale is usually thought to be about the Planck length, and far below experimental accessibility. The possibility that the extensions of the extra dimensions (or at least some of them) might be much larger than the Planck length and thus possibly experimentally accessible, has been studied in models with a large compactification volume and lowered Planck scale, see, e.g., [1]. We will not discuss these models here, but mention in passing that they demonstrate the possibility that the 'true' higher-dimensional Planck mass is in fact much smaller than mPl, and correspondingly the 'true' higher-dimensional Planck length, and with it the minimal length, much larger than lPl. That such possibilities exist means, whether or not the model with extra dimensions are realized in nature, that we should, in principle, consider the minimal length a free parameter that has to be constrained by experiment.
String theory is also one of the motivations to look into non-commutative geometries. Non-commutative geometry will be discussed separately in Section 3.6. A section on matrix models will be included in a future update.
Generalized uncertainty
The following argument, put forward by Susskind [297, 298], will provide us with an insightful examination that illustrates how a string is different from a point particle and what consequences this difference has for our ability to resolve structures at shortest distances. We consider a free string in light cone coordinates, \({X_ \pm} = ({X^0} \pm {X^1})/\sqrt 2\) with the parameterization \({X_ +} = 2l_{\rm{s}}^2{P_ +}\tau\), where P+ is the momentum in the direction X+ and constant along the string. In the light-cone gauge, the string has no oscillations in the X+ direction by construction.
The transverse dimensions are the remaining Xi with i > 1. The normal mode decomposition of the transverse coordinates has the form
$${X^i}(\sigma, \tau) = {x^i}(\sigma, \tau) + {\rm{i}}\sqrt {{{\alpha{\prime}} \over 2}} \sum\limits_{n \neq 0} {\left({{{\alpha _n^i} \over n}{e^{{\rm{i}}n(\tau + \sigma)}} + {{\tilde \alpha _n^i} \over n}{e^{{\rm{i}}n(\tau - \sigma)}}} \right)},$$
where xi is the (transverse location of) the center of mass of the string. The coefficients \(\alpha _n^i\) and \(\tilde \alpha _n^i\) are normalized to \([\alpha _n^i,\alpha _m^j] = [\tilde \alpha _n^i,\tilde \alpha _m^j] = - {\rm{i}}m{\delta ^{ij}}{\delta _{m, - n}}\), and \([\tilde \alpha _n^i,\alpha _m^j] = 0\). Since the components Xν are real, the coefficients have to fulfill the relations \({(\alpha _n^i)^*} = \alpha _{- n}^i\) and \({(\tilde \alpha _n^i)^*} = \tilde \alpha _{- n}^i\).
We can then estimate the transverse size ΔX⊥ of the string by
$${(\Delta {X_ \bot})^2} = \langle \sum\limits_{i = 2}^D {{{({X^i} - {x^i})}^2}} \rangle,$$
which, in the ground state, yields an infinite sum
$${(\Delta {X_ \bot})^2}\sim l_{\rm{s}}^2\sum\limits_n {{1 \over n}}.$$
This sum is logarithmically divergent because modes with arbitrarily high frequency are being summed over. To get rid of this unphysical divergence, we note that testing the string with some energy E, which corresponds to some resolution time ΔT = 1/E, allows us to cut off modes with frequency > 1/ΔT or mode number n ∼ lsE. Then, for large n, the sum becomes approximately
$$\Delta X_ \bot ^2 \approx l_{\rm{s}}^2\log ({l_{\rm{s}}}E).$$
Thus, the transverse extension of the string grows with the energy that the string is tested by, though only very slowly so.
To determine the spread in the longitudinal direction X−, one needs to know that in light-cone coordinates the constraint equations on the string have the consequence that X− is related to the transverse directions so that it is given in terms of the light-cone Virasoro generators
$${X_ -}(\sigma, \tau) = {x^ -}(\sigma, \tau) + {{\rm{i}} \over {{P_ +}}}\sum\limits_{n \neq 0} {\left({{{{L_n}} \over n}{e^{{\rm{i}}n(\tau + \sigma)}} + {{{{\tilde L}_n}} \over n}{e^{{\rm{i}}n(\tau - \sigma)}}} \right)},$$
where now Ln and \({{\tilde L}_n}\) fulfill the Virasoro algebra. Therefore, the longitudinal spread in the ground state gains a factor ∝ n2 over the transverse case, and diverges as
$${(\Delta {X_ -})^2}\sim{1 \over {P_ + ^2}}\sum\limits_n n.$$
Again, this result has an unphysical divergence, that we deal with the same way as before by taking into account a finite resolution ΔT, corresponding to the inverse of the energy by which the string is probed. Then one finds for large n approximately
$${(\Delta {X_ -})^2} \approx {\left({{{{l_{\rm{s}}}} \over {{P_ +}}}} \right)^2}{E^2}.$$
Thus, this heuristic argument suggests that the longitudinal spread of the string grows linearly with the energy at which it is probed.
The above heuristic argument is supported by many rigorous calculations. That string scattering leads to a modification of the Heisenberg uncertainty relation has been shown in several studies of string scattering at high energies performed in the late 1980s [140, 310, 228]. Gross and Mende [140] put forward a now well-known analysis of the classic solution for the trajectories of a string worldsheet describing a scattering event with external momenta \(p_i^\nu\). In the lowest tree approximation they found for the extension of the string
$${x^\nu}(z,\bar z) \approx l_{\rm{s}}^2\sum\limits_i {p_i^\nu} \log \vert z - {z_i}\vert,$$
plus terms that are suppressed in energy relative to the first. Here, zi are the positions of the vertex operators on the Riemann surface corresponding to the asymptotic states with momenta \(p_i^\nu\). Thus, as previously, the extension grows linearly with the energy. One also finds that the surface of the string grows with E/N, where N is the genus of the expansion, and that the fixed angle scattering amplitude at high energies falls exponentially with the square of the center-of-mass energy s (times \(l_{\rm{s}}^2\)).
One can interpret this spread of the string in terms of a GUP by taking into account that at high energies the spread grows linearly with the energy. Together with the normal uncertainty, one obtains
$$\Delta {x^\nu}\Delta {p^\nu} \underset{\sim}{>} 1 + {l_{\rm{s}}}E,$$
again the GUP that gives rise to a minimally-possible spatial resolution.
However, the exponential fall-off of the tree amplitude depends on the genus of the expansion, and is dominated by the large N contributions because these decrease slower. The Borel resummation of the series has been calculated in [228] and it was found that the tree level approximation is valid only for an intermediate range of energies, and for \(sl_{\rm{s}}^2 \gg {g_{\rm{s}}}^{- 4/3}\) the amplitude decreases much slower than the tree-level result would lead one to expect. Yoneya [318] has furthermore argued that this behavior does not properly take into account non-perturbative effects, and thus the generalized uncertainty should not be regarded as generally valid in string theory. We will discuss this in Section 3.2.3.
It has been proposed that the resistance of the string to attempts to localize it plays a role in resolving the black-hole information-loss paradox [204]. In fact, one can wonder if the high energy behavior of the string acts against and eventually prevents the formation of black holes in elementary particle collisions. It has been suggested in [10, 9, 11] that string effects might become important at impact parameters far greater than those required to form black holes, opening up the possibility that black holes might not form.
The completely opposite point of view, that high energy scattering is ultimately entirely dominated by black-hole production, has also been put forward [48, 131]. Giddings and Thomas found an indication of how gravity prevents probes of distance shorter than the Planck scale [131] and discussed the 'the end of short-distance physics'; Banks aptly named it 'asymptotic darkness' [47]. A recent study of string scattering at high energies [127] found no evidence that the extendedness of the string interferes with black-hole formation. The subject of string scattering in the trans-Planckian regime is subject of ongoing research, see, e.g., [12, 90, 130] and references therein.
Let us also briefly mention that the spread of the string just discussed should not be confused with the length of the string. (For a schematic illustration see Figure 2.) The length of a string in the transverse direction is
$$L = \int {\rm{d}} \sigma {\left({{\partial _\sigma}{X^i}{\partial _\sigma}{X^i}} \right)^2}\,,$$
where the sum is taken in the transverse direction, and has been studied numerically in [173]. In this study, it has been shown that when one increases the cut-off on the modes, the string becomes space-filling, and fills space densely (i.e., it comes arbitrarily close to any point in space).
The length of a string is not the same as its average extension. The lengths of strings in the groundstate were studied in [173].
Spacetime uncertainty
Yoneya [318] argued that the GUP in string theory is not generally valid. To begin with, it is not clear whether the Borel resummation of the perturbative expansion leads to correct non-perturbative results. And, after the original works on the generalized uncertainty in string theory, it has become understood that string theory gives rise to higher-dimensional membranes that are dynamical objects in their own right. These higher-dimensional membranes significantly change the picture painted by high energy string scattering, as we will see in 3.2.3. However, even if the GUP is not generally valid, there might be a different uncertainty principle that string theory conforms to, that is a spacetime uncertainty of the form
$$\Delta X\Delta T \underset{\sim}{>} l_{\rm{s}}^2.$$
This spacetime uncertainty has been motivated by Yoneya to arise from conformal symmetry [317, 318] as follows.
Suppose we are dealing with a Riemann surface with metric \({\rm{d}}s = \rho (z,\bar z)|{\rm{d}}z|\) that parameterizes the string. In string theory, these surfaces appear in all path integrals and thus amplitudes, and they are thus of central importance for all possible processes. Let us denote with Ω a finite region in that surface, and with Γ the set of all curves in Ω. The length of some curve γ ∈ Γ is then L(γ, ρ) = ∫γ ρ∣dz∣. However, this length that we are used to from differential geometry is not conformally invariant. To find a length that captures only the physically-relevant information, one can use a distance measure known as the 'extremal length' λΩ
$${\lambda _\Omega}(\Lambda) =\sup\limits_\rho {{L{{(\Omega, \rho)}^2}} \over {A(\Omega, \rho)}},$$
$$L(\Gamma, \rho) = \inf\limits_{\gamma \in \Lambda} L(\gamma, \rho)\,,\quad A(\Omega, \rho) = \int\nolimits_\Omega {{\rho ^2}} \,{\rm{d}}z\,{\rm{d}}\bar z.$$
The so-constructed length is dimensionless and conformally invariant. For simplicity, we assume that Ω is a generic polygon with four sides and four corners, with pairs of opposite sides named α, α′ and β, β′. Any more complicated shape can be assembled from such polygons. Let Γ be the set of all curves connecting α with α′ and Γ* the set of all curves connecting β with β′. The extremal lengths λΩ(Γ) and λΩ(Γ*) then fulfill property [317, 318]
$${\lambda _\Omega}({\Gamma ^{\ast}}){\lambda _\Omega}(\Gamma) = 1$$
Conformal invariance allows us to deform the polygon, so instead of a general four-sided polygon, we can consider a rectangle in particular, where the Euclidean length of the sides (α, α′) will be named and that of sides (β, β′) will be named b. With a Minkowski metric, one of these directions would be timelike and one spacelike. Then the extremal lengths are [317, 318]
$${\lambda _\Omega}({\Gamma ^{\ast}}) = {b \over a}\,,\quad {\lambda _\Omega}(\Gamma) = {a \over b}{.}$$
Armed with this length measure, let us consider the Euclidean path integral in the conformal gauge (gμν = ημν) with the action
$${1 \over {l_{\rm{s}}^2}}\int\nolimits_\Omega {\rm{d}} z\,{\rm{d}}\bar z\,{\partial _z}{X^\nu}{\partial _{\bar z}}{X^\nu}.$$
(Equal indices are summed over). As before, X are the target space coordinates of the string worldsheet. We now decompose the coordinate z into its real and imaginary part σ1 = Re(z), σ2 = Im(z), and consider a rectangular piece of the surface with the boundary conditions
$$\begin{array}{*{20}c} {{X^\nu}(0,{\sigma _2}) = {X^\nu}(a,{\sigma _2}) = {\delta ^{\nu 2}}B{{{\sigma _2}} \over b},}\\ {{X^\nu}({\sigma _1},0) = {X^\nu}({\sigma _1},b) = {\delta ^{\nu 1}}A{{{\sigma _1}} \over a}.} \end{array}$$
if one integrates over the rectangular region, the action contains a factor ab((B/b)2 + (A/a)2) and the path integral thus contains a factor of the form
$$\exp \left({- {1 \over {l_{\rm{s}}^2}}\left({{{{A^2}} \over {\lambda (\Gamma)}} + {{{B^2}} \over {\lambda ({\Gamma ^{\ast}})}}} \right)} \right).$$
Thus, the width of these contributions is given by the extremal length times the string scale, which quantifies the variance of A and B by
$$\Delta A\sim{l_{\rm{s}}}\sqrt {\lambda (\Gamma)}, \quad \Delta B\sim{l_{\rm{s}}}\sqrt {\lambda ({\Gamma ^{\ast}})}.$$
In particular the product of both satisfies the condition
$$\Delta A\Delta B\sim l_{\rm{s}}^2.$$
Thus, probing short distances along the spatial and temporal directions simultaneously is not possible to arbitrary precision, lending support to the existence of a spacetime uncertainty of the form (82). Yoneya notes [318] that this argument cannot in this simple fashion be carried over to more complicated shapes. Thus, at present the spacetime uncertainty has the status of a conjecture. However, the power of this argument rests in it only relying on conformal invariance, which makes it plausible that, in contrast to the GUP, it is universally and non-perturbatively valid.
Taking into account Dp-Branes
The endpoints of open strings obey boundary conditions, either of the Neumann type or of the Dirichlet type or a mixture of both. For Dirichlet boundary conditions, the submanifold on which open strings end is called a Dirichlet brane, or Dp-brane for short, where p is an integer denoting the dimension of the submanifold. A D0-brane is a point, sometimes called a D-particle; a D1-brane is a one-dimensional object, also called a D-string; and so on, all the way up to D9-branes.
These higher-dimensional objects that arise in string theory have a dynamics in their own right, and have given rise to a great many insights, especially with respect to dualities between different sectors of the theory, and the study of higher-dimensional black holes [170, 45].
Dp-branes have a tension of \({T_p} = 1/({g_{\rm{S}}}l_{\rm{S}}^{p + 1})\); that is, in the weak coupling limit, they become very rigid. Thus, one might suspect D-particles to show evidence for structure on distances at least down to lsgs.
Taking into account the scattering of Dp-branes indeed changes the conclusions we could draw from the earlier-discussed thought experiments. We have seen that this was already the case for strings, but we can expect that Dp-branes change the picture even more dramatically. At high energies, strings can convert energy into potential energy, thereby increasing their extension and counteracting the attempt to probe small distances. Therefore, strings do not make good candidates to probe small structures, and to probe the structures of Dp-branes, one would best scatter them off each other. As Bachas put it [45], the "small dynamical scale of D-particles cannot be seen by using fundamental-string probes — one cannot probe a needle with a jelly pudding, only with a second needle!"
That with Dp-branes new scaling behaviors enter the physics of shortest distances has been pointed out by Shenker [283], and in particular the D-particle scattering has been studied in great detail by Douglas et al. [103]. It was shown there that indeed slow moving D-particles can probe distances below the (ten-dimensional) Planck scale and even below the string scale. For these D-particles, it has been found that structures exist down to \(g_{\rm{s}}^{1/3}{l_{\rm{s}}}\).
To get a feeling for the scales involved here, let us first reconsider the scaling arguments on black-hole formation, now in a higher-dimensional spacetime. The Newtonian potential ϕ of a higher-dimensional point charge with energy E, or the perturbation of g00 = 1 + 2ϕ, in D dimensions, is qualitatively of the form
$$\phi \sim{{{G_D}E} \over {{{(\Delta x)}^{D - 3}}}},$$
where Δx is the spatial extension, and GD is the D-dimensional Newton's constant, related to the Planck length as \({G_D} = l_{{\rm{P1}}}^{D - 2}\). Thus, the horizon or the zero of g00 is located at
$$\Delta x\sim{\left({{G_D}E} \right)^{{1 \over {D - 3}}}}.$$
With E∼1/Δt, for some time by which we test the geometry, to prevent black-hole formation for D = 10, one thus has to require
$$(\Delta t){(\Delta x)^7} \underset{\sim}{>} {G_{10}} = g_{\rm{s}}^2l_{\rm{s}}^8\,,$$
re-expressed in terms of string coupling and tension. We see that in the weak coupling limit, this lower bound can be small, in particular it can be much below the string scale.
This relation between spatial and temporal resolution can now be contrasted with the spacetime uncertainty (82), that sets the limits below which the classical notion of spacetime ceases to make sense. Both of these limits are shown in Figure 3 for comparison. The curves meet at
$$\Delta {x_{\min}}\sim{l_{\rm{s}}}g_{\rm{s}}^{1/3},\quad \Delta {t_{\min}}\sim{l_{\rm{s}}}g_{\rm{s}}^{- 1/3}.$$
If we were to push our limits along the bound set by the spacetime uncertainty (red, solid line), then the best possible spatial resolution we could reach lies at Δxmin, beyond which black-hole production takes over. Below the spacetime uncertainty limit, it would actually become meaningless to talk about black holes that resemble any classical object.
Spacetime uncertainty (red, solid) vs uncertainty from spherical black holes (blue, dotted) in D = 10 dimensions, for gs < 1 (left) and gs > 1 (right). After [318], Figure 1. Below the bound from spacetime uncertainty yet above the black-hole bound that hides short-distance physics (shaded region), the concept of classical geometry becomes meaningless.
At first sight, this argument seems to suffer from the same problem as the previously examined argument for volumes in Section 3.1.7. Rather than combining Δt with Δx to arrive at a weaker bound than each alone would have to obey, one would have to show that in fact Δx can become arbitrarily small. And, since the argument from black-hole collapse in 10 dimensions is essentially the same as Mead's in 4 dimensions, just with a different r-dependence of ϕ, if one would consider point particles in 10 dimensions, one finds along the same line of reasoning as in Section 3.1.2, that actually Δt ≳ lPl and Δx ≳ lPl.
However, here the situation is very different because fundamentally the objects we are dealing with are not particles but strings, and the interaction between Dp-branes is mediated by strings stretched between them. It is an inherently different behavior than what we can expect from the classical gravitational attraction between point particles. At low string coupling, the coupling of gravity is weak and in this limit then, the backreaction of the branes on the background becomes negligible. For these reasons, the D-particles distort each other less than point particles in a quantum field theory would, and this is what allows one to use them to probe very short distances.
The following estimate from [318] sheds light on the scales that we can test with D-particles in particular. Suppose we use D-particles with velocity v and mass m0 = 1/(lsgs) to probe a distance of size Δx in time Δt. Since vΔt∼ Δx, the uncertainty (94) gives
$${(\Delta x)^8} \underset{\sim}{>} vg_{\rm{s}}^2l_{\rm{s}}^8.$$
thus, to probe very short distances one has to use slow D-particles.
But if the D-particle is slow, then its wavefunction behaves like that of a massive non-relativistic particle, so we have to take into account that the width spreads with time. For this, we can use the earlier-discussed bound Eq. (58)
$$\Delta {x_{{\rm{spread}}}} \underset{\sim}{>} \sqrt {{{\Delta t} \over {2{m_0}}}},$$
$$\Delta {x_{{\rm{spread}}}} \underset{\sim}{>} {{{l_{\rm{s}}}{g_{\rm{s}}}} \over {2v}}.$$
If we add the uncertainties (96) and (98) and minimize the sum with respect to v, we find that the spatial uncertainty is minimal for
$$v\sim g_{\rm{s}}^{2/3}.$$
Thus, the total spatial uncertainty is bounded by
$$\Delta x\underset{\sim}{>} {l_{\rm{s}}}g_{\rm{s}}^{1/3},$$
and with this one also has
$$\Delta t \underset{\sim}{>} {l_{\rm{s}}}g_{\rm{s}}^{- 1/3},$$
which are the scales that we already identified in (95) to be those of the best possible resolution compatible with the spacetime uncertainty. Thus, we see that the D-particles saturate the spacetime uncertainty bound and they can be used to test these short distances.
D-particle scattering has been studied in [103] by use of a quantum mechanical toy model in which the two particles are interacting by (unexcited) open strings stretched between them. The open strings create a linear potential between the branes. At moderate velocities, repeated collisions can take place, since the probability for all the open strings to annihilate between one collision and the next is small. At \(v \sim g_{\rm{S}}^{2/3}\), the time between collisions is on the order of Δt ∼ lsg−1/3, corresponding to a resonance of width \(\Gamma \sim g_{\rm{S}}^{1/3}/{l_{\rm{s}}}\). By considering the conversion of kinetic energy into the potential of the strings, one sees that the particles reach a maximal separation of \(\Delta x \sim {l_{\rm{S}}}{g_{\rm{S}}}^{- 1/3}\), realizing a test of the scales found above.
Douglas et al. [103] offered a useful analogy of the involved scales to atomic physics; see Table (1). The electron in a hydrogen atom moves with velocity determined by the fine-structure constant α, from which it follows the characteristic size of the atom. For the D-particles, this corresponds to the maximal separation in the repeated collisions. The analogy may be carried further than that in that higher-order corrections should lead to energy shifts.
Table 1 Analogy between scales involved in D-particle scattering and the hydrogen atom. After [103].
The possibility to resolve such short distances with D-branes have been studied in many more calculations; for a summary, see, for example, [45] and references therein. For our purposes, this estimate of scales will be sufficient. We take away that D-branes, should they exist, would allow us to probe distances down to \(\Delta x \sim g_{\rm{s}}^{1/3}{l_{\rm{s}}}\).
T-duality
In the presence of compactified spacelike dimensions, a string can acquire an entirely new property: It can wrap around the compactified dimension. The number of times it wraps around, labeled by the integer w, is called the 'winding-number.' For simplicity, let us consider only one additional dimension, compactified on a radius R. Then, in the direction of this coordinate, the string has to obey the boundary condition
$${X^4}(\tau, \sigma + 2\pi) = {X^4}(\tau, \sigma) + 2\pi wR.$$
The momentum in the direction of the additional coordinate is quantized in multiples of 1/R, so the expansion (compare to Eq. (72)) reads
$${X^4}(\tau, \sigma) = x_0^4 + {{nl_{\rm{s}}^2} \over R}\tau + wR\sigma + {\rm{i}}{{\alpha{\prime}} \over 2}\sum\limits_{n \neq 0} {\left({{{\alpha _n^i} \over n}{e^{{\rm{i}}n(\tau + \sigma)}} + {{\tilde \alpha _n^i} \over n}{e^{{\rm{i}}n(\tau - \sigma)}}} \right)},$$
where \(x_0^4\) is some initial value. The momentum \({P^4} = {\partial _\tau}{x^4}/(l_{\rm{s}}^2)\) is then
$${P^4}(\tau, \sigma) = {n \over R}\tau + {{\rm{i}} \over {\sqrt 2 {l_{\rm{s}}}}}\sum\limits_{n \neq 0} {\left({\alpha _n^i{e^{{\rm{i}}n(\tau + \sigma)}} + \tilde \alpha _n^i{e^{{\rm{i}}n(\tau - \sigma)}}} \right)}.$$
The total energy of the quantized string with excitation n and winding number is formally divergent, due to the contribution of all the oscillator's zero point energies, and has to be renormalized. After renormalization, the energy is
$${E^2} = \sum\limits_{\mu = 0}^3 {{P^\mu}} {P_\mu} + {m^2}\quad {\rm{with}}$$
$${m^2} = {{{n^2}} \over {{R^2}}} + {w^2}{{{R^2}} \over {{{\alpha\prime}^2}}} + {2 \over {\alpha{\prime}}}\left({N + \tilde N - {{D - 2} \over {12}}} \right),$$
where μ runs over the non-compactified coordinates, and N and Ñ are the levels of excitations of the left and right moving modes. Level matching requires nw + N − Ñ = 0. In addition to the normal contribution from the linear momentum, the string energy thus has a geometrically-quantized contribution from the momentum into the extra dimension(s), labeled with n, an energy from the winding (more winding stretches the string and thus needs energy), labeled with w, and a renormalized contribution from the Casimir energy. The important thing to note here is that this expression is invariant under the exchange
$$R \leftrightarrow {{l_{\rm{s}}^2} \over R}\,,\quad n \leftrightarrow w,$$
i.e., an exchange of winding modes with excitations leaves mass spectrum invariant.
This symmetry is known as target-space duality, or T-duality for short. It carries over to multiples extra dimensions, and can be shown to hold not only for the free string but also during interactions. This means that for the string a distance below the string scale ∼ ls is meaningless because it corresponds to a distance larger than that; pictorially, a string that is highly excited also has enough energy to stretch and wrap around the extra dimension. We have seen in Section 3.2.3 that Dp-branes overcome limitations of string scattering, but T-duality is a simple yet powerful way to understand why the ability of strings to resolves short distances is limited.
This characteristic property of string theory has motivated a model that incorporates T-duality and compact extra dimensions into an effective path integral approach for a particle-like object that is described by the center-of-mass of the string, yet with a modified Green's function, suggested in [285, 111, 291].
In this approach it is assumed that the elementary constituents of matter are fundamentally strings that propagate in a higher dimensional spacetime with compactified additional dimensions, so that the strings can have excitations and winding numbers. By taking into account the excitations and winding numbers, Fontanini et al. [285, 111, 291] derive a modified Green's function for a scalar field. In the resulting double sum over n and w, the contribution from the n = 0 and w = 0 zero modes is dropped. Note that this discards all massless modes as one sees from Eq. (106). As a result, the Green's function obtained in this way no longer has the usual contribution
$$G(x,y) = - {1 \over {{{(x - y)}^2}}}{.}$$
Instead, one finds in momentum space
$$G(p) = \sum\limits_{N = 0}^\infty {\sum\limits_{w,n = 1}^\infty {{{n{l_0}} \over {\sqrt {{p^2} + {m^2}}}}}} {K_1}\left({n{l_0}\sqrt {{p^2} + {m^2}}} \right),$$
where the mass term is given by Eq. (106) and a function of N, n and w. Here, K1 is the modified Bessel function of the first kind, and l0 = 2πR is the compactification scale of the extra dimensions. For n = w = 1 and N = 0, in the limit where p2 ≫ m2 and the argument of K1 is large compared to 1, \({p^2} \gg 1/l_0^2\), the modified Bessel function can be approximated by
$${K_1} \rightarrow {{4{\pi ^2}} \over {\sqrt {p{l_0}}}}\exp \left({- {l_0}\sqrt {{p^2}}} \right),$$
and, in that limit, the term in the sum (109) of the Green's function takes the form
$$\rightarrow {{\sqrt {{l_0}}} \over {{p^{3/2}}}}\exp \left({- {l_0}\sqrt {{p^2}}} \right).$$
Thus, each term of the modified Green's function falls off exponentially if the energies are large enough. The Fourier transform of this limit of the momentum space propagator is
$$G(x,y) \approx {1 \over {4{\pi ^2}}}{1 \over {{{(x - y)}^2} + l_0^2}},$$
and one thus finds that the spacetime distance in the propagator acquires a finite correction term, which one can interpret as a 'zero point length', at least in the Euclidean case.
It has been argued in [285] that this "captures the leading order correction from string theory". This claim has not been supported by independent studies. However, this argument has been used as one of the motivations for the model with path integral duality that we will discuss in Section 4.7. The interesting thing to note here is that the minimal length that appears in this model is not determined by the Planck length, but by the radius of the compactified dimensions. It is worth emphasizing that this approach is manifestly Lorentz invariant.
Loop Quantum Gravity and Loop Quantum Cosmology
Loop Quantum Gravity (LQG) is a quantization of gravity by help of carefully constructed suitable variables for quantization, variables that have become known as the Ashtekar variables [39]. While LQG theory still lacks experimental confirmation, during the last two decades it has blossomed into an established research area. Here we will only roughly sketch the main idea to see how it entails a minimal length scale. For technical details, the interested reader is referred to the more specialized reviews [42, 304, 305, 229, 118].
Since one wants to work with the Hamiltonian framework, one begins with the familiar 3+1 split of spacetime. That is, one assumes that spacetime has topology ℝ × Σ, i.e., it can be sliced into a set of spacelike 3-dimensional hypersurfaces. Then, the metric can be parameterized with the lapse-function N and the shift vector Ni
$${\rm{d}}{s^2} = ({N^2} - {N_a}{N^a})\,{\rm{d}}{t^2} - 2{N_a}\,{\rm{d}}t\,{\rm{d}}x - {q_{ab}}\,{\rm{d}}{x^a}\,{\rm{d}}{x^b},$$
where qij is the three metric on the slice. The three metric by itself does not suffice to completely describe the four dimensional spacetime. If one wants to stick with quantities that make sense on the three dimensional surfaces, in order to prepare for quantization, one needs in addition the 'extrinsic curvature' that describes how the metric changes along the slicing
$${K_{ab}} = {1 \over {2N}}\left({{\nabla _k}{N_i} + {\nabla _i}{N_k} - {\partial _t}{q_{ij}}} \right),$$
where ∇ is the covariant three-derivative on the slice. So far, one is used to that from general relativity.
Next we introduce the triad or dreibein, \(E_i^a\), which is a set of three vector fields
$${q^{ab}} = E_i^aE_j^b{\delta ^{ij}}.$$
The triad converts the spatial indices a, b (small, Latin, from the beginning of the alphabet) to a locally-flat metric with indices i, j (small, Latin, from the middle of the alphabet). The densitized triad
$$\tilde E_i^a = \sqrt {\det q} E_i^a,$$
is the first set of variables used for quantization. The other set of variables is an su(2) connection \(A_a^i\), which is related to the connection on the manifold and the external curvature by
$$A_a^i = \Gamma _a^i + \beta K_a^i,$$
where the i is the internal index and \(\Gamma _a^i = {\Gamma _{ajk}}{\epsilon ^{jki}}\) is the spin-connection. The dimensionless constant β is the 'Barbero-Immirzi parameter'. Its value can be fixed by requiring the black-hole entropy to match with the semi-classical case, and comes out to be of order one.
From the triads one can reconstruct the internal metric, and from A and the triad, one can reconstruct the extrinsic curvature and thus one has a full description of spacetime. The reason for this somewhat cumbersome reformulation of general relativity is that these variables do not only recast gravity as a gauge theory, but are also canonically conjugated in the classical theory
$$\left\{{A_a^i(x),\tilde E_j^b(y)} \right\} = \beta \delta _a^b\delta _j^i{\delta ^3}(x - y),$$
which makes them good candidates for quantization. And so, under quantization one promotes A and E to operators and Ê and replaces the Poisson bracket with commutators,
$$\left[ {\hat A_b^j(x),\hat \tilde E_i^a(y)} \right] = {\rm{i}}\beta \delta _a^b\delta _j^i{\delta ^3}(x - y).$$
The Lagrangian of general relativity can then be rewritten in terms of the new variables, and the constraint equations can be derived.
In the so-quantized theory one can then work with different representations, like one works in quantum mechanics with the coordinate or momentum representation, just more complicated. One such representation is the loop representation, an expansion of a state in a basis of (traces of) holonomies around all possible closed loops. However, this basis is overcomplete. A more suitable basis are spin networks ψs. Each such spin network is a graph with vertices and edges that carry labels of the respective su(2) representation. In this basis, the states of LQG are then closed graphs, the edges of which are labeled by irreducible su(2) representations and the vertices by su(2) intertwiners.
The details of this approach to quantum gravity are far outside the scope of this review; for our purposes we will just note that with this quantization scheme, one can construct operators for areas and volumes, and with the expansion in the spin-network basis ψs, one can calculate the eigenvalues of these operators, roughly as follows.
Given a two-surface Σ that is parameterized by two coordinates x1, x2 with the third coordinate x3 = 0 on the surface, the area of the surface is
$${A_\Sigma} = \int\nolimits_\Sigma {\rm{d}} {x^1}\,{\rm{d}}{x^2}\sqrt {\det {q^{(2)}}},$$
where det \({q^{(2)}} = {q_{11}}{q_{12}} - q_{12}^2\) is the metric determinant on the surface. In terms of the triad, this can be written as
$${A_\Sigma} = \int\nolimits_\Sigma {\rm{d}} {x^1}\,{\rm{d}}{x^2}\sqrt {\tilde E_i^3{{\tilde E}^{3i}}}.$$
This area can be promoted to an operator, essentially by making the triads operators, though to deal with the square root of a product of these operators one has to average the operators over smearing functions and take the limit of these smearing functions to delta functions. One can then act with the so-constructed operator on the states of the spin network and obtain the eigenvalues
$${\hat A_\Sigma}{\psi _s} = 8\pi l_{{\rm{Pl}}}^2\beta \sum\limits_I {\sqrt {{j_I}({j_I} + 1)}} {\psi _s},$$
where the sum is taken over all edges of the network that pierce the surface Σ, and JI, a positive half-integer, are the representation labels on the edge. This way, one finds that LQG has a minimum area of
$${A_{\min}} = 4\pi \sqrt 3 \beta l_{{\rm{Pl}}}^2.$$
A similar argument can be made for the volume operator, which also has a finite smallest-possible eigenvalue on the order of the cube of the Planck length [271, 303, 41]. These properties then lead to the following interpretation of the spin network: the edges of the graph represent quanta of area with area \(\sim l_{\rm{p}}^2\sqrt {j(j + 1)}\), and the vertices of the graph represent quanta of 3-volume.
Loop Quantum Cosmology (LQC) is a simplified version of LQG, developed to study the time evolution of cosmological, i.e., highly-symmetric, models. The main simplification is that, rather than using the full quantized theory of gravity and then studying models with suitable symmetries, one first reduces the symmetries and then quantizes the few remaining degrees of freedom.
For the quantization of the degrees of freedom one uses techniques similar to those of the full theory. LQC is thus not strictly speaking derived from LQG, but an approximation known as the 'mini-superspace approximation.' For arguments why it is plausible to expect that LQC provides a reasonably good approximation and for a detailed treatment, the reader is referred to [40, 44, 58, 57, 227]. Here we will only pick out one aspect that is particularly interesting for our theme of the minimal length.
In principle, one works in LQC with operators for the triad and the connection, yet the semi-classical treatment captures the most essential features and will be sufficient for our purposes. Let us first briefly recall the normal cosmological Friedmann-Robertson-Walker model coupled to scalar field ϕ in the new variables [118]. The ansatz for the metric is
$${\rm{d}}{s^2} = - {\rm{d}}{t^2} + {a^2}(t)\left({{\rm{d}}{x^2} + {\rm{d}}{y^2} + {\rm{d}}{z^2}} \right),$$
and for the Ashtekar variables
$$A_a^i = c\delta _a^i\,,\quad \tilde E_i^a = p\delta _i^a.$$
The variable p is dimensionless and related to the scale factor as a2 = ∣p∣, and c has dimensions of energy. c and p are canonically conjugate and normalized so that the Poisson brackets are
$$\{c,p\} = {{8\pi} \over 3}\beta.$$
The Hamiltonian constraint for gravity coupled to a (spatially homogeneous) pressureless scalar field with canonically conjugated variables Φ, pΦ is
$$16\pi G{\mathcal H} = - {6 \over {{\beta ^2}}}{c^2}\vert p{\vert ^{1/2}} + 8\pi Gp_\phi ^2\vert p{\vert ^{- 3/2}} = 0.$$
This yields
$${c \over \beta} = 2\sqrt {{\pi \over 3}} {l_{{\rm{Pl}}}}{{{p_\phi}} \over {\vert p\vert}}.$$
Since ϕ itself does not appear in the Hamiltonian, the conjugated momentum pϕ is a constant of motion ṗϕ = 0, where a dot denotes a derivative with respect to t. The equation of motion for ϕ is
$$\dot \phi = {{{p_\phi}} \over {{p^{3/2}}}},$$
so we can identify
$${\rho _\phi} = {{p_\phi ^2} \over {2\vert p{\vert ^3}}}$$
as the energy density of the scalar field. With this, Equation (129) can be written in the more familiar form
$${\dot \rho _\phi} = - {3 \over 2}{{\dot p} \over {{p^4}}}p_\phi ^2 = - 3{{\dot a} \over a}{\rho _\phi}.$$
The equation of motion for p is
$$\dot p = - {{8\pi} \over 3}\beta {{\partial {\mathcal H}} \over {\partial c}} = 2{c \over \beta}\vert p{\vert ^{1/2}}.$$
Inserting (128), this equation can be integrated to get
$${p^{3/2}} = 2\sqrt {3\pi} {l_{{\rm{Pl}}}}{p_\phi}t.$$
One can rewrite this equation by introducing the Hubble parameter H = ȧ/a = ṗ/(2p); then one finds
$${H^2} = {{4\pi} \over 3}l_{{\rm{Pl}}}^2{{p_\phi ^2} \over {{p^3}}} = {{8\pi} \over 3}G{\rho _\phi},$$
which is the familiar first Friedmann equation. Together with the energy conservation (131) this fully determines the time evolution.
Now to find the Hamiltonian of LQC, one considers an elementary cell that is repeated in all spatial directions because space is homogeneous. The holonomy around a loop is then just given by exp(iμc), where c is as above the one degree of freedom in A, and μ is the edge length of the elementary cell. We cannot shrink this length μ to zero because the area it encompasses has a minimum value. That is the central feature of the loop quantization that one tries to capture in LQC; μ has a smallest value on the order of μ0 ∼ lPl. Since one cannot shrink the loop to zero, and thus cannot take the derivative of the holonomy with respect to μ, one cannot use this way to find an expression for c in the so-quantized theory.
With that in mind, one can construct an effective Hamiltonian constraint from the classical Eq. (127) by replacing c with sin(μ0c)/μ0 to capture the periodicity of the network due to the finite size of the elementary loops. This replacement makes sense because the so-introduced operator can be expressed and interpreted in terms of holonomies. (For this, one does not have to use the sinus function in particular; any almost-periodic function would do [40], but the sinus is the easiest to deal with.) This yields
$$16\pi G{H_{{\rm{eff}}}} = - {6 \over {{\beta ^2}}}\vert p{\vert ^{{1 \over 2}}}{{{{\sin}^2}({\mu _0}c)} \over {{\mu _0}}} + 8\pi G{1 \over {\vert p{\vert ^{{3 \over 2}}}}}p_\phi ^2.$$
As before, the Hamiltonian constraint gives
$${{\sin ({\mu _0}c)} \over {{\mu _0}\beta}} = 2\sqrt {{\pi \over 3}} {l_{{\rm{Pl}}}}{{{p_\phi}} \over {\vert p\vert}}.$$
And then the equation of motion in the semiclassical limit is
$$\dot p = \{p,{H_{{\rm{eff}}}}\} = - {{8\pi} \over 3}\beta {{\partial {H_{{\rm{eff}}}}} \over {\partial c}} = {{2\vert p{\vert ^{{1 \over 2}}}} \over {\beta {\mu _0}}}\sin ({\mu _0}c)\cos ({\mu _0}c){.}$$
With the previously found identification of ρΦ, we can bring this into a more familiar form
$${H^2} = {{{{\dot p}^2}} \over {4{p^2}}} = {{8\pi} \over 3}G{\rho _\rho}\left({1 - {{{\rho _\phi}} \over {{\rho _{\rm{c}}}}}} \right),$$
where the critical density is
$${\rho _{\rm{c}}} = {3 \over {8\pi G{\beta ^2}\mu _0^2a}}.$$
The Hubble rate thus goes to zero for a finite a, at
$${a^2} = 4\pi Gp_\phi ^2{\beta ^2}\mu _0^2,$$
at which point the time-evolution bounces without ever running into a singularity. The critical density at which this happens depends on the value of pϕ, which here has been a free constant. It has been argued in [43], that by a more careful treatment the parameter μ0 depends on the canonical variables and then the critical density can be identified to be similar to the Planck density.
The semi-classical limit is clearly inappropriate when energy densities reach the Planckian regime, but the key feature of the bounce and removal of the singularity survives in the quantized case [56, 44, 58, 57]. We take away from here that the canonical quantization of gravity leads to the existence of minimal areas and three-volumes, and that there are strong indications for a Planckian bound on the maximally-possible value of energy density and curvature.
Quantized conformal fluctuations
The following argument for the existence of a minimal length scale has been put forward by Padmanabhan [248, 247] in the context of conformally-quantized gravity. That is, we consider fluctuations of the conformal factor only and quantize them. The metric is of the form
$${g_{\mu \nu}}(x) = {(1 + \phi (x))^2}{\bar g_{\mu \nu}}(x),$$
and the action in terms of ḡ reads
$$S[\bar g,\phi ] = {1 \over {16\pi G}}\int {{{\rm{d}}^4}} x\sqrt {- \bar g} \left({\bar R{{(1 + \phi (x))}^2} - 2\Lambda {{(1 + \phi (x))}^4} - 6{\partial ^\nu}\phi {\partial _\nu}\phi} \right).$$
In flat Minkowski background with ḡνκ = ηνκ and in a vacuum state, we then want to address the question what is the expectation value of spacetime intervals
$$\langle 0\vert {\rm{d}}{s^2}\vert 0\rangle = \langle 0\vert {g_{\mu \nu}}\vert 0\rangle \,{\rm{d}}{x^\mu}\,{\rm{d}}{x^\nu} = (1 + \langle 0\vert \phi {(x)^2}\vert 0\rangle){\eta _{\mu \nu}}\,{\rm{d}}{x^\mu}\,{\rm{d}}{x^\nu}.$$
Since the expectation value of ϕ(x)2 is divergent, instead of multiplying fields at the same point, one has to use covariant point-slitting to two points xν and yν = xν + dxν and then take the limit of the two points approaching each other
$$\langle 0\vert {\rm{d}}{s^2}\vert 0\rangle = \underset{{\rm{d}}x \rightarrow 0}{\lim} (1 + \langle 0\vert \phi (x)\phi (x + {\rm{d}}x)\vert 0\rangle){\eta _{\mu \nu}}\,{\rm{d}}{x^\mu}\,{\rm{d}}{x^\nu}\,.$$
Now for a flat background, the action (142) has the same functional form as a massless scalar field (up to a sign), so we can tell immediately what its Green's function looks like
$$\langle 0\vert \phi (x)\phi (y)\vert 0\rangle = {{l_{\rm{p}}^2} \over {4{\pi ^2}}}{1 \over {{{(x - y)}^2}}}.$$
Thus, one can take the limit dxν → 0
$$\langle 0\vert {\rm{d}}{s^2}\vert 0\rangle = {{l_{\rm{p}}^2} \over {4{\pi ^2}}}\underset{{\rm{d}}x \rightarrow 0}{\lim} {1 \over {{{(x - y)}^2}}}{\eta _{\mu \nu}}\,{\rm{d}}{x^\mu}\,{\rm{d}}{x^\nu} = {{l_{\rm{p}}^2} \over {4{\pi ^2}}}.$$
The two-point function of the scalar fluctuation diverges and thereby counteracts the attempt to obtain a spacetime distance of length zero; instead one has a finite length on the order of the Planck length.
This argument has recently been criticized by Cunliff in [92] on the grounds that the conformal factor is not a dynamical degree of freedom in the pure Einstein-Hilbert gravity that was used in this argument. However, while the classical constraints fix the conformal fluctuations in terms of matter sources, for gravity coupled to quantized matter this does not hold. Cunliff reexamined the argument, and found that the scaling behavior of the Greens function at short distances then depends on the matter content; for normal matter content, the limit (146) still goes to zero.
Asymptotically Safe Gravity
String theory and LQG have in common the aim to provide a fundamental theory for space and time different from general relativity; a theory based on strings or spin networks respectively. Asymptotically Safe Gravity (ASG), on the other hand, is an attempt to make sense of gravity as a quantum field theory by addressing the perturbative non-renormalizability of the Einstein-Hilbert action coupled to matter [300].
In ASG, one considers general relativity merely as an effective theory valid in the low energy regime that has to be suitably extended to high energies in order for the theory to be renormalizable and make physical sense. The Einstein-Hilbert action is then not the fundamental action that can be applied up to arbitrarily-high energy scales, but just a low-energy approximation and its perturbative non-renormalizability need not worry us. What describes gravity at energies close by and beyond the Planck scale (possibly in terms of non-metric degrees of freedom) is instead dictated by the non-perturbatively-defined renormalization flow of the theory.
To see how that works, consider a generic Lagrangian of a local field theory. The terms can be ordered by mass dimension and will come with, generally dimensionful, coupling constants gi. One redefines these to dimensionless quantities \({\tilde g_i} = {\lambda ^{- {d_i}}}{g_i}\), where k is an energy scale. It is a feature of quantum field theory that the couplings will depend on the scale at which one applies the theory; this is described by the Renormalization Group (RG) flow of the theory. To make sense of the theory fundamentally, none of the dimensionless couplings should diverge.
In more detail, one postulates that the RG flow of the theory, described by a vector-field in the infinite dimensional space of all possible functionals of the metric, has a fixed point with finitely many ultra-violet (UV) attractive directions. These attractive directions correspond to "relevant" operators (in perturbation theory, those up to mass dimension 4) and span the tangent space to a finite-dimensional surface called the "UV critical surface". The requirement that the theory holds up to arbitrarily-high energies then implies that the natural world must be described by an RG trajectory lying in this surface, and originating (in the UV) from the immediate vicinity of the fixed point. If the surface has finite dimension d, then d measurements performed at some energy λ are enough to determine all parameters, and then the remaining (infinitely many) coordinates of the trajectory are a prediction of the theory, which can be tested against further experiments.
In ASG the fundamental gravitational interaction is then considered asymptotically safe. This necessitates a modification of general relativity, whose exact nature is so far unknown. Importantly, this scenario does not necessarily imply that the fundamental degrees of freedom remain those of the metric at all energies. Also in ASG, the metric itself might turn out to be emergent from more fundamental degrees of freedom [261]. Various independent works have provided evidence that gravity is asymptotically safe, including studies of gravity in 2 + ϵ dimensions, discrete lattice simulations, and continuum functional renormalization group methods.
It is beyond the scope of this review to discuss how good this evidence for the asymptotic safety of gravity really is. The interested reader is referred to reviews specifically dedicated to the topic, for example [240, 202, 260]. For our purposes, in the following we will just assume that asymptotic safety is realized for general relativity.
To see qualitatively how gravity may become asymptotically safe, let λ denote the RG scale. From a Wilsonian standpoint, we can refer to λ as 'the cutoff'. As is customary in lattice theory, we can take λ as a unit of mass and measure everything else in units of λ. In particular, we define with
$$\tilde G = G{\lambda ^2}$$
the dimensionless number expressing Newton's constant in units of the cutoff. (Here and in the rest of this subsection, a tilde indicates a dimensionless quantity.) The statement that the theory has a fixed point means that \({\tilde G}\), and all other similarly-defined dimensionless coupling constants, go to finite values when λ → ∞.
The general behavior of the running of Newton's constant can be inferred already by dimensional analysis, which suggests that the beta function of 1/G has the form
$$\lambda {d \over {d\lambda}}{1 \over G} = \alpha {\lambda ^2},$$
where α is some constant. This expectation is supported by a number of independent calculations, showing that the leading term in the beta function has this behavior, with α > 0. Then the beta function of \({\tilde G}\) takes the form
$$\lambda {{d\tilde G} \over {d\lambda}} = 2\tilde G - \alpha {\tilde G^2}.$$
This beta function has an IR attractive fixed point at \(\tilde G = 0\) and also a UV attractive nontrivial fixed point at \({{\tilde G}_\ast} = 1/\alpha\). The solution of the RG equation (148) is
$$G{(\lambda)^{- 1}} = G_0^{- 1} + {\alpha \over 2}{\lambda ^2},$$
where G0 is Newton's constant in the low energy limit. Therefore, the Planck length, \(\sqrt G\), becomes energy dependent.
This running of Newton's constant is characterized by the existence of two very different regimes:
If \(0 < \tilde G \ll 1\) we are in the regime of sub-Planckian energies, and the first term on the right side of Eq. (149) dominates. The solution of the flow equation is
$$\tilde G(\lambda) = {\tilde G_0}{\left({{\lambda \over {{\lambda _0}}}} \right)^2},$$
where λ0 is some reference scale and \({{\tilde G}_0} = \tilde G({\lambda _0})\). Thus, the dimensionless Newton's constant is linear in λ2, which implies that the dimensionful Newton's constant \(G(\lambda) = {G_0} = l_{{\rm{P1}}}^2\) is constant. This is the regime that we are all familiar with.
In the fixed point regime, on the other hand, the dimensionless Newton's constant \(\tilde G = {{\tilde G}_\ast}\) is constant, which implies that the dimensionful Newton's constant runs according to its canonical dimension, \(G(\lambda) = {{\tilde G}_\ast}/{\lambda ^2}\), in particular it goes to zero for λ → ∞.
One naturally expects the threshold separating these two regimes to be near the Planck scale. With the running of the RG scale, \({\tilde G}\) must go from its fixed point value at the Planck scale to very nearly zero at macroscopic scales.
At first look it might seem like ASG does not contain a minimal length scale because there is no limit to the energy by which structures can be tested. In addition, towards the fixed point regime, the gravitational interaction becomes weaker, and with it weakening the argument from thought experiments in Section 3.1.2, which relied on the distortion caused by the gravitational attraction of the test particle. It has, in fact, been argued [51, 108] that in ASG the formation of a black-hole horizon must not necessarily occur, and we recall that the formation of a horizon was the main spoiler for increasing the resolution in the earlier-discussed thought experiments.
However, to get the right picture one has to identify physically-meaningful quantities and a procedure to measure them, which leads to the following general argument for the occurrence of a minimal length in ASG [74, 261].
Energies have to be measured in some unit system, otherwise they are physically meaningless. To assign meaning to the limit of λ → ∞ itself, λ too has to be expressed in some unit of energy, for example as \(\lambda \sqrt G\), and that unit in return has to be defined by some measurement process. In general, the unit itself will depend on the scale that is probed in any one particular experiment. The physically-meaningful energy that we can probe distances with in some interaction thus will generally not go to ∞ with λ. In fact, since \(\sqrt G \rightarrow 1/\lambda\), an energy measured in units of \(\sqrt G\) will be bounded by the Planck energy; it will go to one in units of the Planck energy.
One may think that one could just use some system of units other than Planck units to circumvent the conclusion, but if one takes any other dimensionful coupling as a unit, one will arrive at the same conclusion if the theory is asymptotically safe. And if it is not, then it is not a fundamental theory that will break down at some finite value of energy and not allow us to take the limit λ → ∞. As Percacci and Vacca pointed out in [261], it is essentially a tautology that an asymptotically-safe theory comes with this upper bound when measured in appropriate units.
A related argument was offered by Reuter and Schwindt [270] who carefully distinguish measurements of distances or momenta with a fixed metric from measurements with the physically-relevant metric that solves the equations of motion with the couplings evaluated at the scale λ that is being probed in the measurement. In this case, the dependence on λ naturally can be moved into the metric. Though they have studied a special subclass of (Euclidian) manifolds, their finding that the metric components go like 1/λ2 is interesting and possibly of more general significance.
The way such a 1/λ2-dependence of the metric on the scale λ at which it is tested leads to a finite resolution is as follows. Consider a scattering process with in and outgoing particles in a space, which, in the infinite distance from the scattering region, is flat. In this limit, to good precision spacetime has the metric gκν(λ → 0) = ηκ,ν. Therefore, we define the momenta of the in- and outgoing particles, as well as their sums and differences, and from them as usual the Lorentz-invariant Mandelstam variables, to be of the form s = ηκνpκpν. However, since the metric depends on the scale that is being tested, the physically-relevant quantities in the collision region have to be evaluated with the metric \({g_{kv}}(\sqrt s) = {\eta _{kv}}m_{{\rm{P}}1}^2/s\). With that one finds that the effective Mandelstam variables, and thus also the momentum transfer in the collision region, actually go to \({g_{kv}}(\sqrt s){p^v}{p^\mu} = m_{{\rm{P}}1}^2\), and are bounded by the Planck scale.
This behavior can be further illuminated by considering in more detail the scattering process in an asymptotically-flat spacetime [261]. The dynamics of this process is described by some Wilsonian effective action with a suitable momentum scale λ. This action already takes into account the effective contributions of loops with momenta above the scale λ, so one may evaluate scattering at tree level in the effective action to gain insight into the scale-dependence. In particular, we will consider the scattering of two particles, scalars or fermions, by exchange of a graviton.
Since we want to unravel the effects of ASG, we assume the existence of a fixed point, which enters the cross sections of the scattering by virtual graviton exchange through the running of Newton's constant. The tree-level amplitude contains a factor 1/mPl for each vertex. In the s-channel, the squared amplitude for the scattering of two scalars is
$$\vert M_{\rm{s}}^2\vert = {1 \over {4{m_{{\rm{Pl}}}}}}{{{t^2}{u^2}} \over {{s^2}}},$$
and for fermions
$$\vert M_{\rm{f}}^2\vert = {1 \over {128{m_{{\rm{Pl}}}}}}{{{t^4} - 6{t^3}u + 18{t^2}{u^2} - 6t{u^3} + {u^4}} \over {{s^2}}}.$$
As one expects, the cross sections scale with the fourth power of energy over the Planck mass. In particular, if the Planck mass was a constant, the perturbative expansion would break down at energies comparable to the Planck mass. However, we now take into account that in ASG the Planck mass becomes energy dependent. For the annihilation process in the s-channel, it is \(\sqrt s\), the total energy in the center-of-mass system, that encodes what scale can be probed. Thus, we replace mPl with \(1/\sqrt {G(s})\). One proceeds similarly for the other channels.
From the above amplitudes the total cross section is found to be [261]
$${\sigma _{\rm{s}}} = {{sG(s)} \over {1920\pi}}\,,\quad {\sigma _{\rm{f}}} = {{sG(s)} \over {5120\pi}},$$
for the scalars and fermions respectively. Using the running of the gravitational coupling constant (150), one sees that the cross section has a maximum at s = 2G0/α and goes to zero when the center-of-mass energy goes to infinity. For illustration, the cross section for the scalar scattering is depicted in Figure 4 for the case with a constant Planck mass in contrast to the case where the Planck mass is energy dependent.
Cross section for scattering of two scalar particles by graviton exchange with and without running Planck mass, in units of the low-energy Planck mass \(1/\sqrt {{G_0}}\). The dot-dashed (purple) line depicts the case without asymptotic safety; the continuous (blue) and dashed (grey) line take into account the running of the Planck mass, for two different values of the fixed point, \(1/\sqrt {{{\tilde G}_\ast}} = 0.024\) and 0.1 respectively. Figure from [261]; reproduced with permission from IOP.
If we follow our earlier argument and use units of the running Planck mass, then the cross section as well as the physically-relevant energy, in terms of the asymptotic quantities \(G\left({\sqrt s} \right)\sqrt s/{G_0}\), become constant at the Planck scale. These indications for the existence of a minimal length scale in ASG are intriguing, in particular because the dependence of the cross section on the energy offers a clean way to define a minimal length scale from observable quantities, for example through the (square root of the) cross section at its maximum value.
However, it is not obvious how the above argument should be extended to interactions in which no graviton exchange takes place. It has been argued on general grounds in [74], that even in these cases the dependence of the background on the energy of the exchange particle reduces the momentum transfer so that the interaction would not probe distances below the Planck length and cross sections would stagnate once the fixed-point regime has been reached, but the details require more study. Recently, in [30] it has been argued that it is difficult to universally define the running of the gravitational coupling because of the multitude of kinematic factors present at higher order. In the simple example that we discussed here, the dependence of G on the \(\sqrt s\) seems like a reasonable guess, but a cautionary note that this argument might not be possible to generalize is in order.
Non-commutative geometry
Non-commutative geometry is both a modification of quantum mechanics and quantum field theory that arises within certain approaches towards quantum gravity, and a class of theories in its own right. Thus, it could rightfully claim a place both in this section with motivations for a minimal length scale, and in Section 4 with applications. We will discuss the general idea of non-commutative geometries in the motivation because there is a large amount of excellent literature that covers the applications and phenomenology of non-commutative geometry. Thus, our treatment here will be very brief. For details, the interested reader is referred to [104, 151] and the many references therein.
String theory and M-theory are among the motivations to look at non-commutative geometries (see, e.g., the nice summary in [104], section VII) and there have been indications that LQG may give rise to a certain type of non-commutative geometry known as κ-Poincaré. This approach has been very fruitful and will be discussed in more detail later in Section 4.
The basic ingredient to non-commutative geometry is that, upon quantization, spacetime coordinates xν are associated to Hermitian operators \({{\hat x}^v}\) that are non-commuting. The simplest way to do this is of the form
$$[{\hat x^\nu},{\hat x^\mu}] = {\rm{i}}{\theta ^{\mu \nu}}.$$
The real-valued, antisymmetric two-tensor θμν of dimension length squared is the deformation parameter in this modification of quantum theory, known as the Poisson tensor. In the limit θμν → 0 one obtains ordinary spacetime. In this type of non-commutative geometry, the Poisson tensor is not a dynamical field and defines a preferred frame and thereby breaks Lorentz invariance.
The deformation parameter enters here much like ħ in the commutation relation between position and momentum; its physical interpretation is that of a smallest observable area in the μν-plane. The above commutation relation leads to a minimal uncertainty among spacial coordinates of the form
$$\Delta {x_\mu}\Delta {x_\nu} \underset{\sim}{>} {1 \over 2}\vert {\theta ^{\mu \nu}}\vert.$$
One expects the non-zero entries of θμν to be on the order of about the square of the Planck length, though strictly speaking they are free parameters that have to be constrained by experiment.
Quantization under the assumption of a non-commutative geometry can be extended from the coordinates themselves to the algebra of functions f(x) by using Weyl quantization. What one looks for is a procedure W that assigns to each element f(x) in the algebra of functions \({\mathcal A}\) a Hermitian operator \(\hat f = W(f)\) in the algebra of operators \({\hat {\mathcal A}}\). One does that by choosing a suitable basis for elements of each algebra and then identifies them with each other. The most common choiceFootnote 9 is to use a Fourier decomposition of the function f(x)
$$\tilde f(k) = {1 \over {{{(2\pi)}^4}}}\int {{{\rm{d}}^4}} x\,{e^{- {\rm{i}}{k_\nu}{x^\nu}}}f(x),$$
and then doing the inverse transform with the non-commutative operators \({{\hat x}^v}\)
$$\hat f = W(f) = {1 \over {{{(2\pi)}^4}}}\int {{{\rm{d}}^4}} k\,{e^{- {\rm{i}}{k_\nu}{{\hat x}^\nu}}}\tilde f(k).$$
One can extend this isomorphism between the vector spaces to an algebra isomorphism by constructing a new product, denoted *, that respects the map W,
$$W(f \star g)(x) = W(f)\cdot W(g) = \hat f\cdot\hat g,$$
for f, \(g \in {\mathcal A}\) and \({\hat f}\), \(\hat g \in \hat {\mathcal A}\). From Eqs. (158) and (159) one finds the explicit expression
$$W(f \star g) = {1 \over {{{(2\pi)}^4}}}\int {{{\rm{d}}^4}} k\,{{\rm{d}}^4}p\,{e^{{\rm{i}}{k_\nu}{{\hat x}^\nu}}}\,{e^{{\rm{i}}{p_\nu}{{\hat x}^\nu}}}\,\tilde f(k)\tilde g(p).$$
With the Campbell-Baker-Hausdorff formula
$${e^A}{e^B} = {e^{A + B + {1 \over 2}[A,B] + {1 \over {12}}[A,[A,B]] - {1 \over {12}}[B,[A,B]] - {1 \over {24}}[B,[A,[A,B]]] + \ldots}}$$
one has
$${e^{{\rm{i}}{k_\nu}{{\hat x}^\nu}}}{e^{{\rm{i}}{p_\nu}{{\hat x}^\nu}}} = {e^{{\rm{i}}({k_\nu} + {p_\nu}){{\hat x}^\nu} - {{\rm{i}} \over 2}{k_\nu}{\theta ^{\nu \kappa}}{p_\kappa}}},$$
$$W(f \star g) = {1 \over {{{(2\pi)}^4}}}\int {{{\rm{d}}^4}} \,k{{\rm{d}}^4}p\,{e^{{\rm{i}}({k_\nu} + {p_\nu}){{\hat x}^\nu} - {{\rm{i}} \over 2}{k_\nu}{\theta ^{\nu \kappa}}{p_\kappa}}}\tilde f(k)\tilde g(p).$$
This map can be inverted to
$$f(x) \star g(x) = \int {{{{{\rm{d}}^4}p} \over {{{(2\pi)}^4}}}} {{{{\rm{d}}^4}k} \over {{{(2\pi)}^4}}}\,\tilde f(k)\tilde g(p){e^{- {{\rm{i}} \over 2}{k_\kappa}{\theta ^{\kappa \nu}}{p_\nu}}}{e^{- {\rm{i}}({k_\kappa} + {p_\kappa}){x^\kappa}}}.$$
If one rewrites the θ-dependent factor into a differential operator acting on the plane-wave-basis, one can also express this in the form
$${\left. {f(x) \star g(x) = \exp \left({{{\rm{i}} \over 2}{\partial \over {\partial {x^\nu}}}{\theta ^{\nu \kappa}}{\partial \over {\partial {x^\kappa}}}} \right)f(x)g(y)} \right\vert _{x \rightarrow y}},$$
which is known as the Moyal-Weyl product [236].
The star product is a particularly useful way to handle non-commutative geometries, because one can continue to work with ordinary functions, one just has to keep in mind that they obey a modified product rule in the algebra. With that, one can build non-commutative quantum field theories by replacing normal products of fields in the Lagrangian with the star products.
To gain some insight into the way this product modifies the physics, it is useful to compute the star product with a delta function. For that, we rewrite Eq. (164) as
$$\begin{array}{*{20}c} {f(x) \star g(x) = \int {{{{{\rm{d}}^4}p} \over {{{(2\pi)}^4}}}} {{\rm{d}}^4}y\,f(x + {1 \over 2}\theta k)g(x + y){e^{- {\rm{i}}{k_\kappa}{y^\kappa}}}\quad \quad \quad \quad \quad \quad \quad \;}\\ {= {1 \over {{\pi ^4}\left\vert {\det \theta} \right\vert}}\int {{{\rm{d}}^4}} z\,{{\rm{d}}^4}y\,f(x + z)g(x + y){e^{- 2{\rm{i}}{z^\nu}\theta _{\nu \kappa}^{- 1}{y^\kappa}}}.}\\ \end{array}$$
And so, one finds the star product with a delta function to be
$$\delta (x) \star g(x) = {1 \over {{\pi ^4}\vert {\det \theta} \vert}}\int {{{\rm{d}}^4}} y\,{e^{2{\rm{i}}{x^\nu}\theta _{\nu \kappa}^{- 1}{y^\kappa}}}g(y).$$
In contrast to the normal product of functions, this describes a highly non-local operation. This non-locality, which is a characteristic property of the star product, is the most relevant feature of non-commutative geometry.
It is clear that the non-vanishing commutator by itself already introduces some notion of fundamentally-finite resolution, but there is another way to see how a minimal length comes into play in non-commutative geometry. To see that, we look at a Gaussian centered around zero. Gaussian distributions are of interest not only because they are widely used field configurations, but, for example, also because they may describe solitonic solutions in a potential [137].
For simplicity, we will consider only two spatial dimensions and spatial commutativity, so then we have
$$[{\hat x^i},{\hat x^j}] = {\rm{i}}\theta {\epsilon^{ij}},$$
where i, j ∈ {1, 2}, ϵij is the totally antisymmetric tensor, and θ is the one remaining free parameter in the Poisson tensor. This is a greatly simplified scenario, but it will suffice here.
A normalized Gaussian in position space centered around zero with covariance σ
$${\Psi _\sigma}(x) = {1 \over {\pi \sigma}}\exp \left({- {{{x^2}} \over {{\sigma ^2}}}} \right)$$
has the Fourier transform
$${\tilde \Psi _\sigma}(k) = \int {{d^2}} x{e^{ikx}}{\Psi _\sigma}(x) = \exp \left({- {\pi ^2}{k^2}{\sigma ^2}} \right).$$
We can then work out the star product for two Gaussians with two different spreads σ1 and σ2 to
$$\begin{array}{*{20}c} {{{\tilde \Psi}_{{\sigma _1}}} \star {{\tilde \Psi}_{{\sigma _2}}}(k) = \int {{d^2}} k{\Psi _{{\sigma _1}}}(k){\Psi _{{\sigma _2}}}(p - k)\exp \left({{{\rm{i}} \over 2}{k^i}{\epsilon _{ij}}{p^j}} \right)\quad} \\ {= {\pi \over {{{(4\sigma _1^2 + \sigma _2^2)}^2}}}\exp \left({- {{{p^2}\sigma _{12}^2} \over 4}} \right),} \\ \end{array}$$
$$\sigma _{12}^2 = {{\sigma _1^2\sigma _2^2 + {\theta ^2}} \over {\sigma _1^2 + \sigma _2^2}}.$$
Back in position space this yields
$${\Psi _{{\sigma _1}}} \star {\tilde \Psi _{{\sigma _2}}}(x) = {1 \over {\pi {\sigma _{12}}{{(4\sigma _1^2 + \sigma _2^2)}^2}}}\exp \left({- {{{x^2}} \over {\sigma _{12}^2}}} \right).$$
Thus, if we multiply two Gaussians with σ1, σ2 < θ, the width of the product σ12 is larger than θ. In fact, if we insert σ1 = σ2 = σ12 = σ in Eq. (172) and solve for σ, we see that a Gaussian with width σ = θ squares to itself. Thus, since Gaussians with smaller width than θ have the effect to spread, rather than to focus, the product, one can think of the Gaussian with width θ as having a minimum effective size.
In non-commutative quantum mechanics, even in more than one dimension, Gaussians with this property constitute solutions to polynomial potentials with a mass term (for example for a cubic potential this would be of the form V(ϕ) = m2ϕ * ϕ + a2ϕ * ϕ * ϕ * ϕ) [137], because they square to themselves, and so only higher powers continue to reproduce the original function.
Besides the candidate theories for quantum gravity so far discussed, there are also discrete approaches, reviewed, for example, in [203]. For these approaches, no general statement can be made with respect to the notion of a minimal length scale. Though one has lattice parameters that play the role of regulators, the goal is to eventually let the lattice spacing go to zero, leaving open the question of whether observables in this limit allow an arbitrarily good resolution of structures or whether the resolution remains bounded. One example of a discrete approach, where a minimal length appears, is the lattice approach by Greensite [139] (discussed also in Garay [120]), in which the minimal length scale appears for much the same reason as it appears in the case of quantized conformal metric fluctuations discussed in Section 3.4. Even if the lattice spacing does not go to zero, it has been argued on general grounds in [60] that discreteness does not necessarily imply a lower bound on the resolution of spatial distances.
One discrete approach in which a minimal length scale makes itself noticeable in yet another way are Causal Sets [290]. In this approach, one considers as fundamental the causal structure of spacetime, as realized by a partially-ordered, locally-finite set of points. This set, represented by a discrete sprinkling of points, replaces the smooth background manifold of general relativity. The "Hauptvermutung" (main conjecture) of the Causal Sets approach is that a causal set uniquely determines the macroscopic (coarse-grained) spacetime manifold. In full generality, this conjecture is so far unproven, though it has been proven in a limiting case [63]. Intriguingly, the causal sets approach to a discrete spacetime can preserve Lorentz invariance. This can be achieved by using not a regular but a random sprinkling of points; there is thus no meaningful lattice parameter in the ordinary sense. It has been shown in [62], that a Poisson process fulfills the desired property. This sprinkling has a finite density, which is in principle a parameter, but is usually assumed to be on the order of the Planckian density.
Another broad class of approaches to quantum gravity that we have so far not mentioned are emergent gravity scenarios, reviewed in [49, 284]. Also in these cases, there is no general statement that can be made about the existence of a minimal length scale. Since gravity is considered to be emergent (or induced), there has to enter some energy scale at which the true fundamental, non-gravitational, degrees of freedom make themselves noticeable. Yet, from this alone we do not know whether this also prevents a resolution of structures. In fact, in the absence of anything resembling spacetime, this might not even be a meaningful question to ask.
Giddings and Lippert [128, 129, 126] have proposed that the gravitational obstruction to test short distance probes should be translated into a fundamental limitation in quantum gravity distinct from the GUP. Instead of a modification of the uncertainty principle, fundamental limitations should arise due to strong gravitational (or other) dynamics, because the concept of locality is only approximate, giving rise to a 'locality bound' beyond which the notion of locality ceases to be meaningful. When the locality bound is violated, the usual field theory description of matter no longer accurately describes the quantum state and one loses the rationale for the usual Fock space description of the states; instead, one would have to deal with states able to describe a quantum black hole, whose full and proper quantum description is presently unknown.
Finally, we should mention an interesting recent approach by Dvali et al. that takes very seriously the previously-found bounds on the resolution of structures by black-hole formation [106] and is partly related to the locality bound. However, rather than identifying a regime where quantum field theory breaks down and asking what quantum theory of gravity would allow one to consistently deal with strong curvature regimes, in Dvali et al.'s approach of 'classicalization', super-Planckian degrees of freedom cannot exist. On these grounds, it has been argued that classical gravity is in this sense UV-complete exactly because an arbitrarily good resolution of structures is physically impossible [105].
Summary of motivations
In this section we have seen that there are many indications, from thought experiments as well as from different approaches to quantum gravity, that lead us to believe in a fundamental limit to the resolution of structure. But we have also seen that these limits appear in different forms.
The most commonly known form is a lower bound on spatial and temporal resolutions given by the Planck length, often realized by means of a GUP, in which the spatial uncertainty increases with the increase of the energy used to probe the structures. Such an uncertainty has been found in string theory, but we have also seen that this uncertainty does not seem to hold in string theory in general. Instead, in this particular approach to quantum gravity, it is more generally a spacetime uncertainty that still seems to hold. One also has to keep in mind here that this bound is given by the string scale, which may differ from the Planck scale. LQG and the simplified model for LQC give rise to bounds on the eigenvalues of the area and volume operator, and limit the curvature in the early universe to a Planckian value.
Thus, due to these different types of bounds, it is somewhat misleading to speak of a 'minimal length,' since in many cases a bound on the length itself does not exist, but only on the powers of spatio-temporal distances. Therefore, it is preferable to speak more generally of a 'minimal length scale,' and leave open the question of how this scale enters into the measurement of physical quantities.
Models and Applications
In this section we will investigate some models that have been developed to deal with a minimal length or, more often, a maximum energy scale. The models discussed in the following are not in themselves approaches to a fundamental description of spacetime like the ones previously discussed that lead us to seriously consider a finite resolution of structures. Instead, the models discussed in this section are attempts to incorporate the notion of a minimal length into the standard model of particle physics and/or general relativity by means of a modification of quantum mechanics, quantum field theory and Poincaré symmetry. These models are meant to provide an effective description of the possible effects of a minimal length with the intention to make contact with phenomenology and thereby ideally constrain the possible modifications.Footnote 10
As mentioned previously, the non-commutative geometries discussed in Section 3.6 could also have rightfully claimed a place in this section on models and applications.
Before we turn towards the implementation, let us spend some words on the interpretation because the construction of a suitable model depends on the physical picture one aims to realize.
Interpretation of a minimal length scale
It is the central premise of this review that there exists a minimal length scale that plays a fundamental role in the laws of nature. In the discussion in Section 5 we will consider the possibility that this premise is not fulfilled, but for now we try to incorporate a minimal length scale into the physical description of the world. There are then still different ways to think about a minimal length scale or a maximum energy scale.
One perspective that has been put forward in the literature [14, 193, 15, 212] is that for the Planck mass to be observer independent it should be invariant under Lorentz boosts. Since normal Lorentz boosts do not allow this, the observer independence of the Planck mass is taken as a motivation to modify special relativity to what has become known as 'Deformed Special Relativity' (DSR). (See also Section 4.5). In brief, this modification of special relativity allows one to perform a Lorentz boost on momentum space in such a way that an energy of Planck mass remains invariant.
While that is a plausible motivation to look into such departures from special relativity, one has to keep in mind that just because a quantity is of dimension length (mass) it must not necessarily transform under Lorentz boosts as a spatial or time-like component of a spacetime (or momentum) four vector. A constant of dimension length can be invariant under normal Lorentz boosts, for example, if it is a spacetime (proper) distance. This interpretation is an essential ingredient to Padmanabhan's path integral approach (see Section 4.7). Another example is the actual mass of a particle, which is invariant under Lorentz boosts by merit of being a scalar. Thinking back to our historical introduction, we recall that the coupling constant of Fermi's theory for the weak interaction is proportional to the inverse of the W-mass and therefore observer independent in the sense that it does not depend on the rest frame in which we determine it — and that without the need to modify special relativity.
Note also that coupling constants do depend on the energy with which structures are probed, as discussed in Section 3.5 on ASG. There we have yet another interpretation for a minimal length scale, that being the energy range (in terms of normally Lorentz-invariant Mandelstam variables for the asymptotically in/out-going states) where the running Planck mass comes into the fixed-point regime. The center-of-mass energy corresponding to the turning point of the total cross section in graviton scattering, for example, makes it a clean and observer-independent definition of an energy scale, beyond which there is no more new structure to be found.
One should also keep in mind that the outcome of a Lorentz boost is not an observable per se. To actually determine a distance in some reference frame one has to perform a measurement. Thus, for the observer independence of a minimal length, it is sufficient if there is no operational procedure that allows one to resolve structures to a precision better than the Planck length. It has been argued in [154] that this does not necessitate a modification of Lorentz boosts for the momenta of free particles; it is sufficient if the interactions of particles do not allow one to resolve structures beyond the Planck scale. There are different ways this could be realized, for example, by an off-shell modification of the propagator that prevents arbitrarily-high momentum transfer. As previously discussed, there are some indications that ASG might realize such a feature and it can, in a restricted sense, be interpreted as a version of DSR [74].
An entirely different possibility that we mentioned in Section 3.7, has recently been put forward in [106, 105], where it was argued that it is exactly because of the universality of black-hole production that the Planck length already plays a fundamental role in classical gravity and there is no need to complete the theory in the high energy range.
Modified commutation relations
The most widely pursued approach to model the effects of a minimal length scale in quantum mechanics and quantum field theory is to reproduce the GUP starting from a modified commutation relation for position and momentum operators. This modification may or may not come with a modification also of the commutators of these operators with each other, which would mean that the geometry in position or momentum space becomes non-commuting.
The modified commutation relations imply not only a GUP, but also a modified dispersion relation and a modified Poincaré-symmetry in momentum and/or position space. The literature on the subject is vast, but the picture is still incomplete and under construction.
Recovering the minimal length from modified commutation relations
To see the general idea, let us start with a simple example that shows the relation of modified commutation relations to the minimal length scale. Consider variables \({\rm{k}} = (\vec k,\omega)\), where \({\vec k}\) is the three vector, components of which will be labeled with small Latin indices, and \(x = (\vec x,t)\). Under quantization, the associated operators obey the standard commutation relations
$$[{x^\nu},{x^\kappa}] = 0,\quad [{x^\nu},{k_\kappa}] = {\rm{i}}\delta _\kappa ^\nu \quad, \quad \left[ {{k_\nu},{k_\kappa}} \right] = 0.$$
Now we define a new quantity \({\rm{p}} = (\vec p,E) = f({\rm{k}})\), where f is an injective function, so that f−1(p) = k is well defined. We will also use the notation \({p_\mu} = h_\mu ^\alpha ({\rm{k)}}{k_\alpha}\) with the inverse \({k_\alpha} = h_\alpha ^\mu ({\rm{p)}}{p_\mu}\). For the variables x and p one then has the commutation relations
$$[{x^\nu},{x^\kappa}] = 0,\quad [{x^\nu},{p_\kappa}] = {\rm{i}}{{\partial {f_\kappa}} \over {\partial {k_\nu}}}\,,\quad \left[ {{p_\nu},{p_\kappa}} \right] = 0.$$
If one now looks at the uncertainty relation between xi and pi, one finds
$$\Delta {x_i}\Delta {p_i} \ge {1 \over 2}\langle {{\partial {f_i}} \over {\partial {k^i}}}\rangle{.}$$
To be concrete, let us insert some function, for example a generic expansion of the form \(\vec p \approx \vec k(1 + \alpha {k^2}/m_{{\rm{p}}1}^2)\) plus higher orders in k/mPl, so that the inverse relation is \(\vec p \approx \vec p(1 - \alpha {p^2}/m_{{\rm{p}}1}^2)\). Here, α is some dimensionless parameter. One then has
$${{\partial {f_i}} \over {\partial {k^j}}} \approx {\delta _{ij}}\left({1 + \alpha {{{p^2}} \over {m_{{\rm{Pl}}}^2}}} \right) + 2\alpha {{{p_i}{p_j}} \over {m_{{\rm{Pl}}}^2}}.$$
Since this function is convex, we can rewrite the expectation value in (176) to
$$\Delta {x_i}\Delta {p_i} \ge {1 \over 2}\left({1 + \alpha {{\langle {p^2}\rangle} \over {m_{{\rm{Pl}}}^2}} + 2\alpha {{\langle p_i^2\rangle} \over {m_{{\rm{Pl}}}^2}}} \right).$$
And inserting the expression for the variance 〈A2〉 − 〈A〉2 = ΔA2, one obtains
$$\begin{array}{*{20}c} {\Delta {x_i}\Delta {p_i} \ge {1 \over 2}\left({1 + \alpha {{\Delta {p^2} + {{\langle p\rangle}^2}} \over {m_{{\rm{Pl}}}^2}} + 2\alpha {{\Delta p_i^2 + {{\langle {p_i}\rangle}^2}} \over {m_{{\rm{Pl}}}^2}}} \right)}\\ {\ge {1 \over 2}\left({1 + 3\alpha {{\Delta p_i^2} \over {m_{{\rm{Pl}}}^2}}} \right),\quad \quad \quad \quad \quad \;}\\ \end{array}$$
$$\Delta {x_i} \ge {1 \over 2}\left({{1 \over {\Delta {p_i}}} + 3\alpha {{\Delta {p_i}} \over {m_{{\rm{Pl}}}^2}}} \right).$$
Thus, we have reproduced the GUP that we found in the thought experiments in Section 3.1 with a minimal possible uncertainty for the position [175, 26, 207, 209, 25].
However, though the nomenclature here is deliberately suggestive, one has to be careful with interpreting this finding. What the inequality (180) tells us is that we cannot measure the position to arbitrary precision if we do it by varying the uncertainty in p. That has an operational meaning only if we assign to p the meaning of a physical momentum, in particular it should be a Hermitian operator. To distinguish between the physical quantity p, and k that fulfills the canonical commutation relations, the k is sometimes referred to as the 'pseudo-momentum' or, because it is conjugated to x, as 'the wave vector.' One can then physically interpret the non-linear relation between p and k as an energy dependence of Planck's constant [156].
To further clarify this, let us turn towards the question of Lorentz invariance. If we do not make statements in addition to the commutation relations, we do not know anything about the transformation behavior of the quantities. For all we know, they could have an arbitrary transformation behavior and Lorentz invariance could just be broken. If we require Lorentz invariance to be preserved, this opens the question of how it is preserved, what is the geometry of its phase space, and what is its Poisson structure. And, most importantly, how do we identify physically-meaningful coordinates on that space?
At the time of writing, no agreed upon picture has emerged. Normally, phase space is the cotangential bundle of the spacetime manifold. One might generalize this to a bundle of curved momentum spaces, an idea that dates back at least to Max Born in 1938 [64]. In a more radical recent approach, the 'principle of relative locality' [287, 286, 169, 27, 23, 21] phase space is instead considered to be the cotangential bundle of momentum space.
To close this gap in our example, let us add some more structure and assume that the phase space is a trivial bundle \({\mathcal S} = {\mathcal M} \otimes {\mathcal P}\), where \({\mathcal M}\) is spacetime and \({\mathcal P}\) is momentum space. Elements of this space have the form (x, p). If we add that the quantity p is a coordinate on \({\mathcal P}\) and transforms according to normal Lorentz transformations under a change of inertial frames, and k is another coordinate system, then we know right away how k transforms because we can express it by use of the function f. If we do a Lorentz transformation on p, so that p′= Λp, then we have
$${\bf{k{\prime}}} = f({\bf{p{\prime}}}) = f(\Lambda {\bf{p}}) = f(\Lambda {f^{- 1}}({\bf{k}})),$$
which we can use to construct the modified Lorentz transformation as \({{\bf k}\prime} = \tilde \Lambda ({\bf k})\). In particular, one can chose f in such a way that it maps an infinite value of p (in either the spatial or temporal entries, or both) to a finite value of k at the Planck energy. The so-constructed Lorentz transformation on p will then keep the Planck scale invariant, importantly without introducing any preferred frame. This is the basic idea of deformations of special relativity, some explicit examples of which we will meet in Section 4.5.
If one assumes that p transforms as a normal Lorentz vector, one has
$$[{J_{\kappa \nu}},{p_\mu}] = {\rm{i}}({{p_\nu}{\eta _{\kappa \mu}} - {p_\kappa}{\eta _{\nu \mu}}}){.}$$
Since this commutator commutes with p, one readily finds
$$[{J_{\kappa \nu}},{k_\mu}] = {\rm{i}}({{p_\nu}{\eta _{\kappa \alpha}} - {p_\kappa}{\eta _{\nu \alpha}}}){{\partial {k_\mu}} \over {\partial {p_\alpha}}},$$
which gives us the infinitesimal version of (181) by help of the usual expansion
$$k_{\prime \nu} = k_\nu - \frac{\mathrm{i}}{2} \omega^{\alpha \beta} [ J_{\alpha \beta}, k_\nu ] + {\mathcal O}(\omega^2),$$
where ωαβ are the group parameters of Λ.
Now that we know how k transforms, we still need to add information for how the coordinates on \({\mathcal P}\) are supposed to be matched to those on \({\mathcal M}\). One requirement that we can use to select a basis on \({\mathcal M}\) along with that on \({\mathcal M}\) under a Lorentz transformation is that the canonical form of the commutation relations should remain preserved. With this requirement, one then finds for the infinitesimal transformation of x [253]
$$[{J_{\kappa \nu}},{x^\mu}] = {\rm{i}}{x^\alpha}{\partial \over {\partial {k_\mu}}}\left({{p_\kappa}{\eta _{\beta \nu}}{{\partial {k_\alpha}} \over {\partial {p_\beta}}} - {p_\nu}{\eta _{\beta \kappa}}{{\partial {k_\alpha}} \over {\partial {p_\kappa}}}} \right)$$
and the finite transformation
$${x\prime^\nu} = {{\partial p_\alpha^\prime} \over {\partial k_\mu^\prime}}\Lambda _{\;\alpha}^\nu {{\partial {k_\beta}} \over {\partial {p_\nu}}}{x^\kappa}.$$
One finds the latter also directly by noting that this transformation behavior is required to keep the symplectic form ω = dxα ∧ dkα canonical.
A word of caution is in order here because the innocent looking indices on these quantities do now implicitly stand for different transformation behaviors under Lorentz transformations. One can amend this possible confusion by a more complicated notation, but this is for practical purposes usually unnecessary, as long as one keeps in mind that the index itself does not tell the transformation behavior under Lorentz transformations. In particular, the derivative ∂kα/∂pβ has a mixed transformation behavior and, in a Taylor-series expansion, this and higher derivatives yield factors that convert the normal to the modified transformation behavior.
So far this might have seemed like a rewriting, so it is important to stress the following: just writing down the commutation relation leaves the structure under-determined. To completely specify the model, one needs to make an additional assumption about how a Lorentz transformation is defined, how the coordinates in position space ought to be chosen along with those in momentum space under a Lorentz transformation and, most importantly, what the metric on the curved momentum space (and possibly spacetime) is.
Needless to say, the way we have fixed the transformation behavior in this simple example is not the only way to do it. Another widely used choice is to require that k obeys the usual transformation behavior, yet interpret p as the physical momentum, or x as 'pseudo-coordinates' (though this is not a word that has been used in the literature). We will meet an example of this in Section 4.2.2. This variety is the main reason why the literature on the subject of modified commutation relations is confusing.
At this point it should be mentioned that a function f that maps an infinite value of p to an asymptotically-finite value of f(p) cannot be a polynomial of finite order. Its Taylor series expansion necessarily needs to have an infinite number of terms. Now if k(p) becomes constant for large p, then ∂k/∂p goes to zero and ∂p/∂k increases without bound, which is why the uncertainty (176) increases for large p. Depending on the choice of f, this might be the case for the spatial or temporal components or both. If one wants to capture the regularizing properties of the minimal length, then a perturbative expansion in powers of E/mPl, will not work in the high energy limit. In addition, such expansions generically add the complication that any truncation of the series produces for the dispersion relation a polynomial of finite order, which will have additional zeros, necessitating additional initial values [158]. This can be prevented by not truncating the series, but this adds other complications, discussed in Section 4.4.
Since ∂p/∂k is a function of p, the position operator i(∂pα/∂kν)∂/∂pα, is not Hermitian if p is Hermitian. This is unfortunate for a quantity that is supposed to be a physical observable, but we have to keep in mind that the operator itself is not an observable anyway. To obtain an observable, we have to take an expectation value. To ensure that the expectation value produces meaningful results, we change its evaluation so that the condition 〈xΨ∣Φ〉 = 〈Φ∣xΦ〉 which in particular guarantees that expectation values are real, is still fulfilled. That is, we want the operator to be symmetric, rather than Hermitian.Footnote 11 To that end, one changes the measure in momentum space to [175, 184]
$${{\rm{d}}^3}p \rightarrow {{\rm{d}}^3}k = \left\vert {{{\partial k} \over {\partial p}}} \right\vert {{\rm{d}}^3}p,$$
which will exactly cancel the non-Hermitian factor in \(\hat x = {\rm{i}}\partial/(\partial k)\), because now
$$\begin{array}{*{20}c} {\langle \Psi \vert x\Phi \rangle = {\rm{i}}\int {{{\rm{d}}^3}} p\left\vert {{{\partial k} \over {\partial p}}} \right\vert {\Psi ^{\ast}}{\partial \over {\partial k}}\Phi \;\quad \quad \quad \quad \quad \quad \quad \quad \;\;}\\ {= {\rm{i}}\int {{{\rm{d}}^3}} k{\Psi ^{\ast}}{\partial \over {\partial k}}\Phi = {\rm{i}}\int {{{\rm{d}}^3}} k({\partial \over {\partial k}}{\Psi ^{\ast}})\Phi}\\ {= \langle x\Psi \vert \Phi \rangle. \;\;\;\quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ \end{array}$$
Note that this does not work without the additional factor because then the integration measure does not fit to the derivative, which is a consequence of the modified commutation relations. We will see in the next section that there is another way to think of this modified measure.
The mass-shell relation, gμν(p)pμpν − m2 = 0, is invariant under normal Lorentz transformations acting on p, and thus gμν(k)pμ(k)pν(k)2 − m2 = 0 is invariant under the modified Lorentz transformations acting on k. The clumsy notation here has stressed that the metric on momentum space will generally be a non-linear function of the coordinates. The mass-shell relation will yield the Hamiltonian constraint of the theory.
There is a subtlety here; since x is not Hermitian, we can't use the representation if this operator. The way this can be addressed depends on the model. One can in many cases just work the momentum representation. In our example, we would note that the operator \({{\bar x}^v} = (\partial {k_v}/\partial {p_\mu}){x^\mu}\) is Hermitian, and use its representation. In this representation, the Hamiltonian constraint becomes a higher-order operator, and thus delivers a modification of the dispersion relation. However, the interpretation of the dispersion then hinges on the interpretation of the coordinates. Depending on the suitable identification of position space coordinates and the function f, the speed of massless particles in this model may thus become a function of the momentum four-vector p. It should be noted however that this is not the case for all choices of f [155].
Many of the applications of this model that we will meet later only use the first or second-order expansion of f. While these are sufficient for some interesting consequences of the GUP, the Planck energy is then generically not an asymptotic and invariant value. If the complete function f is considered, it is usually referred to as an 'all order GUP.' The expansion to first or second order is often helpful because it allows one to parameterize the possible modifications by only a few dimensionless quantities.
To restrict the form of modifications possible in this approach, sometimes the Jacobi identities are drawn upon. It is true that the Jacobi identities restrict the possible commutation relations, but seen as we did here starting from the standard commutation relations, this statement is somewhat misleading. The Jacobi identities are, as the name says, identities. They say more about the properties of the binary operation they represent than about the quantities this operation acts on. They are trivially fulfilled for the commutators of all new variables f(k) one can define (coordinates one can choose) if the old ones fulfilled the identities. However, if one does not start from such a function, one can draw upon the Jacobi identities as a consistency check.
A requirement that does put a restriction on the possible form of the commutation relation is that of rotational invariance. Assuming that \({f_i}(\vec k) = {k_i}h(k)\) where \(k = \left| {\vec k} \right|\), one has
$${{\partial {f_i}} \over {\partial {k_j}}} = \delta _i^jh(k) + {k_i}{{\partial h(k)} \over {\partial {k_j}}}.$$
The expansion to 3rd order in k is
$$h(k) = 1 + \alpha k + \beta {k^2} + {\mathcal O}({k^3}),\quad {{\partial h} \over {\partial {k_j}}} = \alpha {{{k_j}} \over k} + 2\beta {k_j} + {\mathcal O}({k^2}).$$
We denote the inverse of \({f_i}(\vec k)\) with \(f_i^{- 1}(\vec k) = {p_i}\tilde h(p)\). An expansion of \(\tilde h(p)\) and comparison of coefficients yields to third order
$$\tilde h(p) = 1 - \alpha p - (\beta - 2{\alpha ^2}){p^2}{.}$$
With this, one has then
$$\begin{array}{*{20}c} {[{x_i},{p_j}] = {\delta _{ij}} + \alpha \left({k{\delta _{ij}} + {{{k_i}{k_j}} \over k}} \right) + \beta \left({{k^2}{\delta _{ij}} + 2{k_j}{k_i}} \right) + {\mathcal O}({k^3})\quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ {= {\delta _{ij}} + \alpha \left({p{\delta _{ij}} + {{{p_i}{p_j}} \over p}} \right) + (\beta - {\alpha ^2}){p^2}{\delta _{ij}} + (2\beta - {\alpha ^2}){p_i}{p_j} + {\mathcal O}({p^3}){.}}\\ \end{array}$$
For the dimensions to match, the constant α must have a dimension of length and β a dimension of length squared. One would expect this length to be of the order Planck length and play the role of the fundamental length.
It is often assumed that β = α2, but it should be noted that this does not follow from the above. In particular, α may be zero and the modification be even in k, starting only at second order. Note that an expansion of the commutator in the form [xi, pj] = iδij(1 + βp2) does not fulfill the above requirement.
To summarize this section, we have seen that a GUP that gives rise to a minimal length scale can be realized by modifying the canonical commutation relations. We have seen that this modification alone does not completely specify the physical picture, we have in addition to fix the transformation behavior under Lorentz transformations and the metric on momentum space. In Section 4.2.2 we will see how this can be done.
The Snyder basis
As mentioned in the previous example, p is not canonically conjugate to x, and the wave vector k, which is canonically conjugate, is the quantity that transforms under the modified Lorentz transformations. But that is not necessarily the case for models with modified commutation relations, as we will see in this section.
Let us start again from the normal commutation relations (174) and now define new position coordinates X by \({X_v} = {x_v} - {x^\alpha}{k_\alpha}{k_v}/m_{{\rm{P}}1}^2\), as discussed in [135]. (The Planck mass mPl could enter here with an additional dimensionless factor that one would expect to be of order one, if one describes a modification that has its origin in quantum gravitational effects. In the following we will not carry around such an additional factor. It can easily be inserted at any stage just by replacing mPl with αmPl.) In addition, we now require that k transforms under the normal Lorentz transformations. With this replacement, the k's are then still commutative as usual and the remaining commutation relations have the form
$$[{X_\mu},{X_\nu}] = - {1 \over {{m_{{\rm{Pl}}}}}}{J_{\mu \nu}}\,,\quad [{X_\mu},{X_\nu}] = i\left({{\eta _{\mu \nu}} - {{{k_\mu}{k_\nu}} \over {m_{{\rm{Pl}}}^2}}} \right),$$
where we recognize
$${J_{\mu \nu}} = {x_\mu}{k_\nu} - {x_\nu}{k_\mu} = {X_\mu}{k_\nu} - {X_\nu}{k_\mu},$$
as the generators of Lorentz transformations. This reproduces the commutation relations of Snyder's original proposal [288].
The commutator between X and k leads to a GUP by taking the expectation value in the same way as previously, though the reason here is a different one: If it is the Xν's that are representing physically-meaningful positions in spacetime, then it is their non-commutativity that spoils the resolution of structures at the Planck scale. Note that the transformation from x to X is not canonical exactly for the reason that it does change the commutation relations.
In a commonly-used notation, Ji0 = Ni are the generators of boosts and {J23, J31, J12} are the generators of the rotations {M1, M2, M3} that fulfill the normal Lorentz algebra
$$\begin{array}{*{20}c} {[{N_i},{N_j}] = - i{\epsilon _{ijk}}{N_k}\,,\quad [{M_i},{M_j}] = i{\epsilon _{ijk}}{N_k},}\\ {[{M_i},{N_j}] = i{\epsilon _{ijk}}{N_k}.\quad \quad \quad \quad \quad \quad \quad \quad \quad \;}\\ \end{array}$$
Since we have not done anything to the transformation of the momentum k, in the X, k phase-space coordinates one also has
$$\begin{array}{*{20}c} {[{M_i},{k_j}] = i{\epsilon_{ijk}}{k_k},\quad [{M_i},{k_0}] = 0,}\\ {[{N_i},{k_j}]= i{\delta _{ij}}{k_0},\quad [{N_i},{k_0}] = i{k_i}\,.}\\ \end{array}$$
There are two notable things here. First, as in Section 4.2.2, the Lorentz algebra remains entirely unmodified. Second, the X by construction transforms covariantly under normal Lorentz transformations if the x and k do. However, we see that there is exactly one x for which X does not depend on k, and that is x = 0. If we perform a translation by use of the generator k; the coordinate x will be shifted to some value x′ = x + a. Alternatively, one may try to take a different generator for translations than k, the obvious choice is the operator canonically conjugated to Xν
$${\partial \over {\partial {X_\nu}}} = {\partial \over {\partial {x_\alpha}}}{{\partial {x_\alpha}} \over {\partial {X_\nu}}}.$$
If one contracts \({X_v} = {x_v} + {x^\alpha}{k_\alpha}{k_v}/m_{{\rm{P}}1}^2\) with kν, one can invert X(x) to
$${x_\nu} = {X_\nu} - {{{X_\alpha}{k^\alpha}} \over {m_{{\rm{Pl}}}^2 + {k_\kappa}{k^\kappa}}}{k_\nu}.$$
Then one finds the translation operator
$${\partial \over {\partial {X_\nu}}} = {k^\nu}\left({{1 \over {1 - {k^\alpha}{k_\alpha}/m_{{\rm{Pl}}}^2}}} \right).$$
Therefore, it has been argued [230] that one should understand this type of theory as a modification of translation invariance rather than a modification of Lorentz symmetry. However, this depends on which variables are assigned physical meaning, which is a question that is still under discussion.
We should at this point look at Snyder's original motivation for it is richer than just the commutation relations of position, momenta and generators and adds to it in an important way. Snyder originally considered a 5-dimensional flat space of momenta, in which he looked at a hypersurface with de Sitter geometry. The full metric has the line element
$${\rm{d}}{s^2} = {\eta ^{AB}}\,{\rm{d}}{\eta _A}\,{\rm{d}}{\eta _B},$$
where the coordinates ηA have dimensions of energy and capital Latin indices run from 0 to 4. This flat space is invariant under the action of the group SO(4, 1), which has a total of 10 generators. In that 5-dimensional space, consider a 4-dimensional hyperboloid defined by
$$- m_p^2 = {\eta ^{AB}}{\eta _A}{\eta _B} = {\eta ^{\mu \nu}}{\eta _\mu}{\eta _\nu} - \eta _4^2.$$
This hypersurface is invariant under the SO(3, 1) subgroup of SO(4, 1). It describes a de Sitter space and can be parameterized by four coordinates. Snyder chooses the projective coordinates kν = mp1ην/η4. (These coordinates are nowadays rarely used to parameterize de Sitter space, as the fifth coordinate of the embedding space η4 is not constant on the hyperboloid.) The remaining four generators of SO(4, 1) are then identified with the coordinates
$${J_{4\nu}} = {X_\nu} = i\left({{{{\eta _4}} \over {{m_{{\rm{Pl}}}}}}{\partial \over {\partial {\eta _\nu}}} + {{{\eta _\nu}} \over {{m_{{\rm{Pl}}}}}}{\partial \over {\partial {\eta _4}}}} \right).$$
From this one obtains the same commutation relations (193), (195), and (196) as above [55].
However, the Snyder approach contains additional information: We know that the commutation relations seen previously can be obtained by a variable substitution from the normal ones. In addition, we know that the momentum space is curved. It has a de Sitter geometry, a non-trivial curvature tensor and curvature scalar \(12/m_{{\rm{P}}1}^2\). It has the corresponding parallel transport and a volume measure. In these coordinates, the line element has the form
$${\rm{d}}{s^2} = {{{\eta ^{\mu \nu}}\,{\rm{d}}{k_\mu}\,{\rm{d}}{k_\nu}} \over {1 - {\eta ^{\alpha \kappa}}{k_\alpha}{k_\kappa}/m_{{\rm{Pl}}}^2}}.$$
Thus, we see how the previously found need to adjust the measure in momentum space (187) arises here naturally from the geometry in momentum space. The mass-shell condition is
$${m^2} = {{{\eta ^{\mu \nu}}{k_\mu}{k_\nu}} \over {1 - {\eta ^{\alpha \kappa}}{k_\alpha}{k_\kappa}/m_{{\rm{Pl}}}^2}}.$$
We note that on-shell this amounts to a redefinition of the rest mass.
The (X, k) coordinates on phase space have become known as the Snyder basis.
The choice of basis in phase space
The coordinates η that Snyder chose to parameterize the hyperbolic 4-dimensional submanifold are not unique. There are infinitely many sets of coordinates we can choose on this space; most of them will be non-linear combinations of each other. Such non-linear redefinitions of momenta will change the commutation relations between position and momentum variables. More generally, the question that arises here is what coordinates on phase space should be chosen, since we have seen in the previous Section 4.2.2 that a change of coordinates in phase space that mixes position and momentum operators creates non-commutativity. For example, one could use a transformation that mixes p and x to absorb the unusual factor in the [x, p] commutator in (175) at the expense of creating a non-commutative momentum space.
Besides the above-discussed coordinate systems (x, k), (x, p), (X, η), and (X, k), there are various other choices of coordinates that can be found in the literature. One choice that is very common are coordinates \({\tilde x}\) that are related to the Snyder position variables [197] via
$${\tilde x_0} = {X_0}\,,\quad {\tilde x_i} = {X_i} + {{{N_i}} \over {{m_{\rm{p}}}}}.$$
This leads to the commutation relations
$$[{\tilde x_0},{\tilde x_i}] = - {\rm{i}}{{{x_i}} \over {{m_{{\rm{Pl}}}}}}\,,\quad [{\tilde x_i},{\tilde x_j}] = 0.$$
The non-commutative spacetime described by these coordinates has become known as κ-Minkowski spacetime. The name derives from the common nomenclature in which the constant mpl (that, as we have warned previously, might differ from the actual Planck mass by a factor of order one) is κ.
Another choice of coordinates that can be found in the literature [132, 287] is obtained by the transformation
$${\chi _0} = {x_0} + {x_i}{k^i}/{m_{{\rm{Pl}}}}\,,\quad {\chi _i} = {x_i}$$
on the normal coordinates xν. This leads to the commutation relations
$$\begin{array}{*{20}c} {[{\chi _0},{\chi _i}] = {\rm{i}}{{{\chi _i}} \over {{m_{{\rm{Pl}}}}}}\,,\quad [{\chi _i},{\chi _j}] = 0\,,\quad [{k_i},{\chi _j}] = {\rm{i}}{\delta _{ij}},} \\ {[{\chi _0},{k_0}] = {\rm{i}}\,,\quad [{\chi _0},{k_i}] = - {\rm{i}}{{{k_i}} \over {{m_{{\rm{Pl}}}}}}\,,\quad [{k_0},{\chi _i}] = {\rm{i}}.\quad} \\ \end{array}$$
This is the full κ-Minkowski phase space [205], which is noteworthy because it was shown by Kowalski-Glikman and Nowak [197] that the geometric approach to phase space is equivalent to the algebraic approach that has been pursued by deforming the Poincaré-algebra (the algebra of generators of Poincaré transformations, i.e., boosts, rotations and momenta) to a Hopf algebra [213] with deformation parameter κ, the κ-Poincaré Hopf algebra, giving rise to the above κ-Minkowski phase space.
A Hopf algebra generally consists of two algebras that are related by a dual structure and associated product rules that have to fulfill certain compatibility conditions. Here, the dual to the κ-Poincaré algebra is the κ-Poincaré group, whose elements are identified as Lorentz transformations and position variables. The additional structure that we found in the geometric approach to be the curvature of momentum space is, in the algebraic approach, expressed in the co-products and antipodes of the Hopf algebra. As in the geometrical approach, there is an ambiguity in the choice of coordinates in phase space.
In addition to the various choices of position space coordinates, one can also use different coordinates on momentum space, by choosing different parameterizations of the hypersurface than that of Snyder. One such parameterization is using coordinates πν, that are related to the Snyder basis by
$$\begin{array}{*{20}c} {{\eta _0} = - {m_{{\rm{Pl}}}}\sinh \left({{{{\pi _0}} \over {{m_{{\rm{Pl}}}}}}} \right) - {{{{\vec \pi}^2}} \over {2{m_{{\rm{Pl}}}}}}\exp \left({{{{\pi _0}} \over {{m_{{\rm{Pl}}}}}}} \right),\quad}\\ {{\eta _i} = - {\pi _i}\exp \left({{{{\pi _0}} \over {{m_{{\rm{Pl}}}}}}} \right),\quad \quad \quad \quad \quad \quad \quad \quad}\\ {{\eta _4} = - {m_{{\rm{Pl}}}}\cosh \left({{{{\pi _0}} \over {{m_{{\rm{Pl}}}}}}} \right) - {{{{\vec \pi}^2}} \over {2{m_{{\rm{Pl}}}}}}\exp \left({{{{\pi _0}} \over {{m_{{\rm{Pl}}}}}}} \right).}\\ \end{array}$$
(Recall that η4 is not constant on the hypersurface.) The πν's are the bicrossproduct basis of the Hopf algebra [213], and they make a natural choice for the algebraic approach. With the κ-Minkowski coordinates \({{\tilde x}_\mu}\), one then has the commutators [197]
$$\begin{array}{*{20}c} {[{\pi _0},{{\tilde x}_0}] = {\rm{i}}\,,\quad [{\pi _i},{{\tilde x}_0}] = - {\rm{i}}{{{\pi _i}} \over {{m_{{\rm{Pl}}}}}},}\\ {[{\pi _i},{{\tilde x}_j}] = - {\rm{i}}{\delta _{ij}}\,,\quad [{\pi _0},{{\tilde x}_i}] = 0.\;}\\ \end{array}$$
Another choice of coordinates on momentum space is the Magueijo-Smolin basis \({{\mathcal P}_\mu}\), which is related to the Snyder coordinates by
$${k_\mu} = {{{{\mathcal P}_\mu}} \over {1 - {{\mathcal P}_0}/{m_{{\rm{Pl}}}}}}.$$
From the transformation behavior of the k (196), one can work out the transformation behavior of the other coordinates, and reexpress the mass-shell condition 204 in the new sets of coordinates.
Since there are infinitely many other choices of coordinates, listing them all is beyond the scope of this review. So long as one can identify a new set of coordinates by a coordinate transformation from other coordinates, the commutation relations will fulfill the Jacobi identities automatically. Thus, these coordinate systems are consistent choices. One can also, starting from the transformation of the Snyder coordinates, derive the transformation behavior under Lorentz transformation for all other sets of coordinates. For the above examples the transformation behavior can be found in [132, 197]
In summary, we have seen here that there are many different choices of coordinates on phase space that lead to modified commutation relations. We have met some oft used examples and seen that the most relevant information is in the geometry of momentum space. Whether there are particular choices of coordinates on phase space that lend themselves to easy interpretations and are thus natural in some sense is presently an open question.
Multi-particle states
One important consequence of the modified Lorentz symmetry that has been left out in our discussion so far is the additivity of momenta, which becomes relevant when considering interactions.
In the example from Section 4.2.1, the function f has to be non-linear to allow a maximum value of (some components of) k to remain Lorentz invariant, and consequently the Lorentz transformations \({\tilde \Lambda}\) are non-linear functions of k. But that means that the transformation of a sum of pseudo-momenta k1 + k2 is not the same as the sum of the transformations:
$$\tilde \Lambda ({{\bf{k}}_1} + {{\bf{k}}_2}) \neq \tilde \Lambda ({{\bf{k}}_1}) + \tilde \Lambda ({{\bf{k}}_2}).$$
Now in the case discussed in Section 4.2.1, it is the p that is the physical momentum that is conserved, and since it transforms under normal Lorentz transformations its conservation is independent of the rest frame.
However, if one has chosen the p rather than the k to obey the normal Lorentz transformation, as was the case in Sections 4.2.2 and 4.2.3, then this equation looks exactly the other way round
$$\tilde \Lambda ({{\bf{p}}_1} + {{\bf{p}}_2}) \neq \tilde \Lambda ({{\bf{p}}_1}) + \tilde \Lambda ({{\bf{p}}_2}).$$
And now one has a problem, because the momentum p should be conserved in interactions, and the above sum is supposed to be conserved in an interaction in one rest frame, it would not be conserved generally in all rest frames. (For the free particle, both p and k are conserved since the one is a function of the other.)
The solution to this problem is to define a new, non-linear, addition law ⊕; that has the property that it remains invariant and that can be rightfully interpreted as a conserved quantity. This is straightforward to do if we once again use the quantities k that in this case by assumption transform under the normal Lorentz transformation. To each momentum we have an associated pseudo-momentum k1 = f−1(p1), k2 = f−1 (p2). The sum k1 + k2 is invariant under normal Lorentz transformations, and so we construct the sum of the p′s as
$${{\bf{p}}_1} \oplus {{\bf{p}}_2} = f({{\bf{k}}_1} + {{\bf{k}}_2}) = f({f^{- 1}}({{\bf{p}}_1}) + {f^{- 1}}({{\bf{p}}_2})).$$
It is worthwhile to note that this modified addition law can also be found from the algebraic approach; it is the bicrossproduct of the κ-Poincaré algebra [197].
This new definition for a sum is now observer independent by construction, but we have created a new problem. If the function f (or some of its components) has a maximum of the Planck mass, then the sum of momenta will never exceed this maximal energy. The Planck mass is a large energy as far as particle physics is concerned, but in everyday units it is about 10−5 gram, a value that is easily exceeded by some large molecules. This problem of reproducing a sensible multi-particle limit when one chooses the physical momentum to transform under modified Lorentz transformations has become known as the 'soccer-ball problem.'
The soccer-ball problem is sometimes formulated in a somewhat different form. If one makes an expansion of the function f to include the first correction terms in p/mPl, and from that derives the sum ⊕, then it remains to be shown that the correction terms stay small if one calculates sums over a large number of momenta, whose ordinary sum describes macroscopic objects like, for example, a soccer ball. One expects that the sum then has approximately the form p1 ⊕ p2 ≈ p1 + p2 + p1p2Γ/mPl, where Γ are some coefficients of order one. If one iterates this sum for N terms, the normal sum grows with N but the additional term with ∼ N2, so that it will eventually become problematic.
Note that this problem is primarily about sums of momenta, and not even necessarily about bound states. If one does not symmetrize the new addition rule, the result may also depend on the order in which momenta are added. This means in particular the sum of two momenta can depend on a third term that may describe a completely unrelated (and arbitrarily far away) part of the universe, which has been dubbed the 'spectator problem' [195, 134].
There have been various attempts to address the problem, but so far none has been generally accepted. For example, it has been suggested that with the addition of N particles, the Planck scale that appears in the Lorentz transformation, as well as in the modified addition law, should be rescaled to mPlN [212, 171, 210]. It is presently difficult to see how this ad-hoc solution would follow from the theory. Alternatively, it has been suggested that the scaling of modifications should go with the density [157, 246] rather than with the total momentum or energy respectively. While the energy of macroscopic objects is larger than that of microscopic ones, the energy density decreases instead. This seems a natural solution to the issue but would necessitate a completely different ansatz to implement. A noteworthy recent result is [24, 166], where it has been shown how the problem can be alleviated in a certain model, such that the nonlinear term in the sum does not go with N2 but with N3/2.
One should also note that this problem does not occur in the case in which the modification is present only off-shell, which seems to be suggested in some interpretations. Then, if one identifies the momenta of particles as those of the asymptotically free states, the addition of their momenta is linear as usual. For the same reason, the problem also does not appear in the interpretation of such modifications of conservation laws as being caused by a running Planck's constant, put forward [261, 74], and discussed in Section 3.5. In this case, the relevant energy is the momentum transfer, and for bound states this remains small if the total mass increases.
So we have seen that demanding the physical momentum rather than the pseudo-momentum to transform under modified Lorentz transformations leads to the soccer ball and the spectator problem. This is a disadvantage of this choice. On the other hand, this choice has the advantage that it has a geometric base, which is missing for the case discussed in Section 4.2.1.
We have, in Section 4.2.2 and 4.2.3, seen different examples for modified commutation relations with a curved momentum space. But commutation relations alone don't make for physics. To derive physical meaning, one has to define the dynamics of the system and its observables.
This raises the question of which principles to use for the dynamics and how to construct observables. While there are several approaches to this, some of which we will meet in the following, there exists to date no agreed-upon framework by which to derive observables, and therefore the question of whether there is a physical reason to prefer one basis over another is open.
One finds some statements in the literature that a different choice of coordinates leads to a different physics, but this statement is somewhat misleading. One should more precisely say that a choice of coordinates and the corresponding commutation relations do not in and of themselves determine the physics. For that, one has to specify not only the geometry of the phase space, which is not contained merely in the commutation relations, but also a unique procedure to arrive at equations of motion.
Given the complete geometry (or, equivalently, the operations on the Hopf algebra), the Hamiltonian in some basis can be identified as (a function of) the Casimir operator of the Lorentz group.Footnote 12 It can be expressed in any basis one wishes by substitution. However, if the transformation between one basis of phase space and the other is not canonical, then transforming the Hamiltonian by substitution will not preserve the Hamiltonian equations. In particular, ∂H/∂p and [H, x] will generically not yield the same result, thus the notion of velocity requires careful interpretation, especially when the coordinates in position space are in addition non-commuting. It has been argued by Smolin in [286] that commuting coordinates are the sensible choice. The construction of observables with non-commuting coordinates has been worked towards, e.g., in [287, 286, 169, 27, 23, 21, 272, 302, 136].
The one modification that all of these approaches have in common is a non-trivial measure in momentum space that in the geometric approach results from the volume element of the now curved space. But this raises the question of what determines the geometry. Ideally one would like an axiomatic approach that allows one to derive the geometry from an underlying principle, and then everything else from the geometry. One would, in the general case, if dynamical matter distributions are present, not expect the structure of momentum space to be entirely fixed. A step towards a dynamical momentum space has been made in [86], but clearly the topic requires more investigation.
Another open problem with this class of models is the type of non-locality that arises. If the Planck length acts as a minimal length, there clearly has to be some non-locality. However, it has been shown that for those types of models in which the speed of light becomes energy dependent, the non-locality becomes macroscopically large. Serious conceptual problems arising from this were pointed out in [16, 277, 156], and shown to be incompatible with observation in [159, 160].
A very recent development to address this problem is to abandon an absolute notion of locality, and instead settle for a relative one. This 'principle of relative locality' [287, 286, 169, 27, 23, 21, 79] is a promising development. It remains to be seen how it mitigates the problem of non-local particle interactions. For some discussion, see [162, 163, 161]. It should be stressed that this problem does not occur if the speed of light remains constant for free particles.
Quantum mechanics with a minimal length scale
So we have seen that modified commutation relations necessarily go together with a GUP, a modified measure in momentum space and a modified Lorentz symmetry. These models may or may not give rise to a modified on-shell dispersion relation and thus an energy-dependent speed of light [155], but the modified commutation relations and the generalized uncertainty cannot be treated consistently without taking care of the momentum space integration and the transformation behavior.
The literature on the subject of quantum mechanics with a minimal length scale is partly confusing because many models use only some of the previously-discussed ingredients and do not subscribe to all of the modifications, or at least they are not explicitly stated. Some differ in the interpretation of the quantities; notoriously there is the question of what is a physically-meaningful definition of velocity and what is the observable momentum.
Thus, the topic of a minimal length scale has thus given rise to many related approaches that run under the names 'modified commutation relation,' 'generalized uncertainty,' 'deformed special relativity,' 'minimal length deformed quantum mechanics,' etc. and are based on only some features of the modified phase space discussed previously. It is not clear in all cases whether this is consistently possible or what justifies a particular interpretation. For example, one may argue that in the non-relativistic limit, a modified transformation behavior under boosts, that would only become relevant at relativistic energies, is irrelevant. However, one has to keep in mind that the non-linear transformation behavior of momenta results in a non-linear addition law, which becomes problematic for the treatment of multi-particle states. Thus, even in the non-relativistic case, one has to be careful if one deals with a large number of particles.
The lack of clean, agreed upon, axiomatic approach has inevitably given rise to occasional criticism. It has been argued in [5], for example, that the deformations of special relativity are operationally indistinguishable from special relativity. Such misunderstandings are bound to arise if the model is underspecified. Maybe, the easiest way to see that the minimal length modified quantum theory is not equivalent to the unmodified case is to keep in mind that the momentum space is curved: There is no coordinate transformation that will make the curvature of momentum space go away. The non-trivial metric will also produce an infinite series of higher-order derivatives in the Hamiltonian constraint, a reflection of the non-locality that the existence of a minimal length scale implies.
In the following, we will not advocate one particular approach, but just report what results are presently available. Depending on which quantities are raised to physical importance, the resulting model can have very different properties. The speed of light might be energy dependent [26] or not [301, 99], there might be an upper limit to energies and/or momenta, or not, addition laws and thresholds might be modified, or not, coordinates might be non-commuting, or not, there might be non-localities or not, the modification might only be present off-shell, or not. This is why the physical meaning of different bases in phase space is a problem in need of being addressed in order to arrive at more stringent predictions.
Maximal localization states
The most basic information about the minimal length modified quantum mechanics is in the position operator itself. While, in the momentum representation, there exist eigenvectors of the position operator that correspond to arbitrarily-sharply-peaked wave functions, these do not describe physically-possible configurations. It has been shown in [184] that the sharply-peaked wavefunctions with spread below the minimal position uncertainty carry an infinite energy, and thus do not represent a physically-meaningful basis. Instead, one can construct quasi-localized states that are as sharply focused as physically possible. These states are then no longer exactly orthogonal. In [184], the maximal localization states have been constructed in one spatial dimension for a 2nd-order expansion of the GUP.
The Schrödinger equation with potential
The most straight-forward modification of quantum mechanics that one can construct with the modified commutation relations is leaving the Hamiltonian unmodified. For the harmonic oscillator for example, one then has the familiar expression
$$H = {{{{\bf{p}}^2}} \over {2m}} + m{\omega ^2}{{{{\bf{x}}^2}} \over 2}.$$
However, due to the modified commutation relations, if one inserts the operators, the resulting differential equation becomes higher order. In one dimension, for example, in the momentum space representation, one would have to second order \(\hat x = {\rm{i}}(1 + l_{{\rm{P}}1}^2{p^2})\partial/\partial p\) and thus for the stationary equation
$${{{\partial ^2}} \over {\partial {p^2}}}\psi (p) + {{2{l_{{\rm{Pl}}}}} \over {1 + l_{{\rm{Pl}}}^2{p^2}}}{\partial \over {\partial p}}\Psi (p) + {{2E/(2m{\omega ^2}) - {p^2}/{{(m\omega)}^2}} \over {{{(1 + l_{{\rm{Pl}}}^2{p^2})}^2}}}\psi (p) = 0.$$
The same procedure can be applied to other types of potentials in the Schrödinger equation, and in principle this can be done not only in the small-momentum expansion, but to all orders. In this fashion, in the leading-order approximation, the harmonic oscillator in one dimension has been studied in [184, 167, 8, 125], the harmonic oscillator in arbitrary dimensions in [83, 177, 85, 93], the energy levels of the hydrogen atom in [167, 69, 296, 67, 258], the particle in a box in [7], Landau levels and the tunneling current in [8, 96, 97], the uniform gravitational potential in [244, 83], the inverse square potential in [65, 66], neutrino oscillations in [292], reflection and transmission coefficients of a potential step and potential barrier in [8, 97], the Klein paradox in [124], and corrections to the gyromagnetic moment of the muon in [143, 95]. Note that these leading order expansions do not all use the same form of the GUP.
In order to obtain the effects of the minimal length on the transition rate of ultra cold neutrons in gravitational spectrometers, Pedram et al. calculated the quantization of the energy spectrum of a particle in a linear gravitational field in the GUP leading-order approximation [259] and to all orders [256]. The harmonic oscillator in one dimension with an asymptotic GUP has been considered in [257, 255].
While not, strictly speaking, falling into the realm of quantum mechanics, let us also mention here the Casimir effect, which has been studied in [142, 46, 241, 112, 102], and Casimir-Polder intermolecular forces, which have been looked at in [252].
All these calculations do, in principle, cause corrections to results obtained in standard quantum mechanics. As one expects, the correction terms are unobservably small if one assumes the minimal length scale to be on the order of the Planck length. However, as argued previously, since we have no good explanation as to why the Planck length as the scale at which quantum gravity should become important is so small, the minimal length should, in principle, be regarded as a free parameter and then be bound by experiment. A compilation of bounds from the above calculations is presently not available and unfortunately no useful comparison is possible due to the different parameterizations and assumptions used. One can hope that this might improve in the future if a more standardized approach becomes established, for example, using the parameterization (192).
The Klein-Gordon and Dirac equation
The Klein Gordon equation can be obtained directly from the invariant p(k)2 − m2 = 0. The Dirac equation can be constructed using the same prescription that lead to the Schrödinger equation, except that, to make sure Lorentz invariance is preserved, one should first bring it into a suitable form
$$({\gamma ^\nu}{p_\nu} - m)\Psi = 0,$$
and then replace pν with its operator as discussed in Section 4.2.1. In the position representation, this will generally produce higher-order derivatives not only in the spatial, but also in the temporal, components. In order to obtain the Hamiltonian that generates the time evolution, one then has to invert the temporal part.
The Dirac equation with modified commutation relations has been discussed in [167, 188]. The Klein-Gordon equation and the Dirac particle in a rectangular and spherical box has been examined in [98].
Quantum field theory with a minimal length scale
One can construct a quantum field theory along the lines of the quantum mechanical treatment, starting with the modified commutation relations. If the position-space coordinates are non-commuting with a constant Poisson tensor, this leads to the territory of non-commutative quantum field theory for which the reader is referred to the literature specialized on that topic, for example [104, 151] and the many references therein.
Quantum field theory with the κ-Poincaré algebra on the non-commuting κ-Minkowski space-time coordinates has been pioneered in [174, 178, 183]. In [176] it has been shown that introducing the minimal length uncertainty principle into quantum field theory works as a regulator in the ultraviolet, at least for ϕ4 theory. Recently, there has been a lot of progress on the way towards field quantization [31], by developing the Moyal-Weyl product, the Fock space [75, 34], and the conserved Noether charges [113, 4, 35]. The case of scalar field theory has been investigated in [33, 226]. A different second-order modification of the commutation relation has been investigated for the spinor and Klein-Gordon field in [233] and [232] respectively. [77] studied the situation in which the Hamiltonian remains unmodified and only the equal time commutation relations are modified. There are, as yet, not many applications in the literature that investigate modifications of the standard model of particle physics, but one can expect these to follow soon.
Parallel to this has been the development of quantum field theory in the case where coordinates are commuting, the physical momentum transforms under the normal Lorentz transformation, and the speed of light is constant [167]. This approach has the advantage of being easier to interpret, yet has the disadvantage of delivering more conservative predictions. In this approach, the modifications one is left with are the modified measure in momentum space and the higher-order derivatives that one obtains from the metric in momentum space. The main difficulty in this approach is that, when one takes into account gauge invariance, one does not only obtain an infinite series of higher-derivate corrections to the propagator, one also obtains an infinite number of interaction terms. Whether these models are unitary is an open question.
In order to preserve the super-Planckian limit that is necessary to capture the presence of the UV-regulating properties, it has been suggested in [167] that one expand the Lagrangian in terms of gE/mPl, where g is the coupling constant and E is the energy scale. This means that the corrections to the propagator (which do not contain any g) are kept entirely, but one has only the first vertex of the infinite series of interaction terms. One can then explore the interesting energy range > mPl until mPl/g. The virtue of this expansion, despite its limited range of applicability, is that, by not truncating the power series of the propagator, one does not introduce additional poles. The expansion in terms of E/mPl as looked at in [188].
In this modified quantum field theory, in [152] the running gauge couplings, possibly with additional compactified spatial dimensions, were investigated. In [189], the electro-weak gauge interaction with minimal length was studied. In [231], the top quark phenomenology in the case with a lowered Planck scale was studied, and in [153, 81] it has been argued that if the Planck scale is indeed lowered, then its role as a minimal length would decrease the production of black holes.
One recurring theme in these models is the suppression of phase space at high energies [167], which is a direct consequence of the modified measure in momentum space. This has also been found, for the same reason, in the κ-Poincar é approach [311]. Another noteworthy feature of these quantum field theories with a minimal length is that the commutator between the fields ϕ(x) and their canonical conjugate π(y) are not equal to a delta function [157, 221], which is an expression of the non-locality that the higher-order derivatives bring in.
In [190] it has furthermore been suggested that one apply this modification of the quantization procedure to quantum cosmology, which is a promising idea that might allow one to make contact with phenomenology.
Deformed Special Relativity
Deformed special relativity (DSR) is concerned with the departure from Lorentz symmetry that results from the postulate that the Planck energy transforms like a (component of a) momentum four vector and remains an invariant, maximal energy scale. While the modified commutation relations necessarily give rise to some version of DSR, one can also try to extract information from the deformation of the Lorentz symmetry, or the addition law, directly. This gives rise to what Amelino-Camelia has dubbed 'test theories' [18]: simplified and reduced versions of the quantum theory with a minimal length. Working with these test theories has the advantage that one can make contact with phenomenology without working out the — still not very well understood — second quantization and interaction. It has the disadvantage that it makes the ambiguity in identifying physically-meaningful observables worse.
The literature on the topic is vast, and we can not cover it in totality here. For more details on the DSR phenomenology, the reader is referred to [19, 20]. We will just mention the most relevant properties of these types of models here.
As we have seen earlier, a non-linear relation p(k), where k transforms under a normal Lorentz transformation, generates the deformed Lorentz transformation for p by Eq. (181). Note that in DSR it is the non-linearly transforming p that is considered the physical momentum, while the k that transforms under the normal Lorentz transformation is considered the pseudo-momentum. There is an infinite number of such functions, and thus there is an infinite number of ways to deform special relativity.
We have already met the common choices in the literature; they constitute bases in κ-Minkowski phase space, for example, the coordinates (211) proposed by Magueijo and Smolin in [211]. With this relation, a boost in the z-direction takes the form
$${\mathcal P}_0^\prime= {{\gamma ({{\mathcal P}_0} - v{{\mathcal P}_z})} \over {1 + (\gamma - 1){{\mathcal P}_0}/{m_{{\rm{Pl}}}} - \gamma v{{\mathcal P}_z}/{m_{{\rm{Pl}}}}}}$$
$${\mathcal P}_z^\prime = {{\gamma ({{\mathcal P}_z} - v{{\mathcal P}_0})} \over {1 + (\gamma - 1){{\mathcal P}_0}/{m_{{\rm{Pl}}}} - \gamma v{{\mathcal P}_z}/{m_{{\rm{Pl}}}}}}.$$
This example, which has become known as DSR2, is particularly illustrative because this deformed Lorentz boost transforms (mPl, mPl) → (mPl, mPl), and thus keeps the Planck energy invariant. Note that since \({k_0}/k = {{\mathcal P}_0}/{\mathcal P}\), a so defined speed of light remains constant in this case.
Another example that has entered the literature under the name DSR1 [71] makes use of the bicrossproduct basis (209). Then, the dispersion relation for massless particles takes the form
$$\cosh ({\pi _0}/{m_{{\rm{Pl}}}}) = {1 \over 2}{{{{\vec \pi}^2}} \over {m_{{\rm{Pl}}}^2}}{e^{{\pi _0}/{m_{{\rm{Pl}}}}}}.$$
The relation between the momenta and the pseudo-momenta and their inverse has been worked out in [171]. In this example, the speed of massless particles that one derives from the dispersion relation depends on the energy of the particle. This effect may be observable in high-frequency light reaching Earth from distant sources, for example from γ-ray bursts. This interesting prediction is covered in more detail in [26, 19, 20].
As discussed in Section 4.2.4, the addition law in this type of model has to be modified in order to obtain Lorentz-invariant conserved sums of momenta. This gives rise to the soccer-ball problem and can lead to changes in thresholds of particle interactions [19, 20, 78]. It had originally been argued that this would shift the GZK cut-off [17], but this argument has meanwhile been revised. However, in the 'test theory' one does not actually have a description of the particle interaction, so whether or not the kinematical considerations would be realized is unclear.
In these two DSR theories, it is usually assumed that the position variables conjugated to k are not commutative [99], thereby delivering a particular realization of κ-Minkowski space. In the DSR1 model, the speed of massless particles that one derives from the above dispersion relation (220) is energy dependent. However, the interpretation of that speed hinges on the meaning of the conjugated position-space coordinates, which is why it has also been argued that the physically-meaningful speed is actually constant [301, 99]. Without an identification of observable positions, it is then also difficult to say whether this type of model actually realizes a minimal length. One can expect that recent work on the principle of relative locality [287, 286, 169, 27, 23, 21, 79] will shed light on this question. A forthcoming review [196] will be especially dedicated to the development of relative locality.
Composite systems and statistical mechanics
As mentioned previously, a satisfactory treatment of multi-particle states in those models in which the free particles' momenta are bound by a maximal energy scale is still lacking. Nevertheless, approaches to the description of composite systems or many particle states have been made, based on the modified commutation relations either without subscribing to the deformed Lorentz transformations, and thereby generically breaking Lorentz invariance, or by employing an ad hoc solution by rescaling the bound on the energy with the number of constituents. While these approaches are promising in so far that modified statistical mechanics at Planckian energies would allow one to use the early universe as a laboratory, they should be regarded with some caution, because the connection to the single particle description with deformed Lorentz symmetry is missing, and the case in which Lorentz symmetry is broken is strongly constrained already [191].
That having been said, the statistical mechanics from the κ-Poincaré algebra was investigated in general in [194, 109]. In [268] corrections to the effective Hamiltonian of macroscopic bodies have been studied, and in [264] observational consequences of modified commutation relations for a massive oscillator have been considered. In [172] statistical mechanics with a generalized uncertainty and possible applications for cosmology have been looked at. The partition functions of minimal-length quantized statistical mechanics have been derived in [257], in [6] the consequences of the GUP on the Liouville theorem were investigated, and in [84] the modification of the density of states and the arising consequences for black-hole thermodynamics were studied. In [242], one finds the effects of the GUP on the thermodynamics of ultra-relativistic particles in the early universe, and relativistic thermodynamics in [94]. [312] studied the equation of state for ultra-relativistic Fermi gases in compact stars, the ideal gas was addressed in [82] and photon gas thermodynamics in [320].
Path-integral duality
In Sections 3.2.4 and 3.4 we have discussed two motivations for limits of spacetime distances that manifest themselves in the Green's function. While one may question how convincing these motivations are, the idea is interesting and may be considered as a model on its own right. Such a modification that realizes a finite 'zero point length' of spacetime intervals had been suggested by Padmanabhan [247, 249, 250, 251] as a way to effectively take into account metric fluctuations below the Planck scale (the motivation from string theory discussed in Section 3.2.4 was added after the original proposal). This model has the merit of not requiring a modification of Lorentz invariance.
The starting point is to rewrite the Feynman propagator GF(x, y) as a sum over all paths γ connecting x and y
$${G_F}(x,y) = \sum\limits_\gamma {{e^{- m{D_\gamma}(x,y)}}} = \int d \tau {e^{- m\tau}}K(x,y,\tau),$$
where Dγ(x, y) is the proper length of γ, and m is a constant of dimension mass. Note that the length of the path depends on the background metric, which is why one expects it to be subject to quantum gravitational fluctuations. The path integral kernel is
$$K(x,y,\tau) = \int {\mathcal D} x\exp \left({- {m \over 4}\int\nolimits_0^\tau {\rm{d}} \tau{\prime}{g_{\mu \nu}}{{\dot x}^\mu}{{\dot x}^\nu}} \right),$$
where a dot indicates a derivative with respect to τ′. The relevant difference between the middle and right expressions in (221) is that D(x, y) has a square root in it. The equivalence has been shown using a Euclidean lattice approach in [251]. Once the propagator is brought into that form, one can apply Padmanabhan's postulated "principle of duality" according to which the weight for each path should be invariant under the transformation \(D\gamma (x,y) \rightarrow l_{{\rm{P}}1}^2/D\gamma (x,y)\). This changes the propagator (221) to
$${\tilde G_F}(x,y) = \sum\limits_\gamma {\exp} \left[ {- m\left({{D_\gamma}(x,y) + {{l_{{\rm{Pl}}}^2} \over {{D_\gamma}(x,y)}}} \right)} \right].$$
Interestingly enough, it can be shown [251] that with this modification in the Schwinger representation, the path integral kernel remains unmodified, and one merely obtains a change of the weight
$${\tilde G_F}(x,y) = \int {\rm{d}} \tau \exp \left[ {- m\left({\tau + {{l_{{\rm{Pl}}}^2} \over \tau}} \right)} \right]K(x,y,\tau){.}$$
When one makes the Fourier transformation of this expression, the propagator in momentum space takes the form
$${\tilde G_F}(p) = {{2{l_{{\rm{Pl}}}}} \over {\sqrt {{p^2} + {m^2}}}}{K_1}(2{l_{{\rm{Pl}}}}\sqrt {{p^2} + {m^2}}),$$
where K1 is the modified Bessel function of the first kind. This expression has the limiting values (compare to Eq. (109))
$${\tilde G_F}(p) \rightarrow \left\{{\begin{array}{*{20}c} {{1 \over {{p^2} + {m^2}}}\quad \quad \quad \quad} & {{\rm{for}}\sqrt {{p^2} + {m^2}} \ll {m_{{\rm{Pl}}}}} \\ {{{\exp (- 2{l_{{\rm{Pl}}}}\sqrt {{p^2} + {m^2}})} \over {\sqrt {2{l_{{\rm{Pl}}}}} {{({p^2} + {m^2})}^{3/4}}}}} & {{\rm{for}}{m_{{\rm{Pl}}}} \ll \sqrt {{p^2} + {m^2}}} \\ \end{array}} \right..$$
This postulated duality of the path integral thus suppresses the super-Planckian contributions to amplitudes. As mentioned in Section 3.2.4, in position space, the Feynman propagator differs from the ordinary one by the shift (x − y)2 → (x − y)2 + 2lPl. (This idea is so different not from that of March [219], who in 1936 proposed to replace ordinary spacetime distances with a modified distance \(d\tilde s = ds - \rho\). Though at that time, the 'minimal length' ρ was supposed to be of about the size of the atomic nucleus. March's interpretation was that when the newly defined distance between two points vanishes, the points become indistinguishable.)
Some applications for this model for QED, for example the Casimir effect, have been worked out in [294, 282], and consequences for inflation and cosmological models have been looked at in [295, 192]. For a recent criticism see [76].
Direct applications of the uncertainty principle
Maybe the most direct way to look for effects of the minimal length is to start from the GUP itself. This procedure is limited in its applicability because there are only so many insights one can gain from an inequality for variances of operators. However, cases that can be studied this way are everything that can be concluded from modifications of the Bekenstein argument, and with it corrections to the black-hole entropy that one obtains taking into account the modification of the uncertainty principle and modified dispersion relations.
Most interestingly, in [225, 28] it has been argued that comparing the corrections to the black-hole entropy obtained from the GUP to the corrections obtained in string theory and LQG may be used to restrict the functional form of the GUP.
It has also been argued that taking into account the GUP may give rise to black-hole remnants [2], a possibility that has been explored in many follow-up works, e.g., [87, 243, 315]. Corrections to the thermodynamical properties of a Schwarzschild black hole have been looked at in [22, 321, 100, 214, 217], the Reissner-Nordström black hole has been considered in [319], and black holes in anti-de Sitter space in [280, 281, 61]. Black-hole thermodynamics with a GUP has been studied in [201, 237, 54, 186], the thermodynamics of Kerr-Newman black holes in [315], and the entropy of a charged black hole in f(R) gravity in [273]. In [80] the consequences of the GUP for self-dual black holes found in the mini-superspace approximation of LQC have been analyzed.
The thermodynamics of anti-de Sitter space has been looked at in [309], and the dynamics of the Taub cosmological model with GUP in [53]. The thermodynamics of Friedmann-Robertson-Walker in four-dimensional spacetimes with GUP can be found in [52, 216], and with additional dimensions in [279]. The relations of the GUP to holography in extra dimensions have been considered in [276], the effects of GUP on perfect fluids in cosmology in [215], and the entropy of the bulk scalar field in the Randall-Sundrum model with GUP in [185]. In [119] it has been suggested that there is a relationship between black-hole entropy and the cosmological constant. The relations of the GUP to Verlinde's entropic gravity have been discussed in [245]
While most of the work on modified uncertainty relations has focused on the GUP, the consequences of the spacetime uncertainty that arises in string theory for the spectrum of cosmological perturbations have been studied in [68]. In [269] it has been proposed that it might be possible to test Planck scale modifications of the energy-time uncertainty relation by monitoring tritium decay. It should also be mentioned that the classical mechanics of κ-Poincaré has been worked out in [206], and the kinematics of a classical free relativistic particle with deformed phase space in [132, 123, 32]. The effects of such a deformed phase space on scalar field cosmology have been investigated in [263].
In [179] an interesting consequence of the minimal length was studied, the implication of a finite bandwidth for physical fields. Making this connection allows one to then use theorems from classical information theory, such as Shannon's sampling theorem. It was shown in [179] that fields on a space with minimum length uncertainty can be reconstructed everywhere if known only on a discrete set of points (any set of points), if these points are, on average, spaced densely enough. These continuous fields then have a literally finite information density. In [181], it was shown that this information-theoretic meaning of the minimal length generalizes naturally to curved spacetime and in [182] it was then argued that for this reason spacetime would be simultaneously continuous and discrete in the same way that information can be.
A model for spacetime foam in terms of non-local interactions as a description for quantum gravitational effects, which serves as an origin of a minimal length scale, has been put forward in [121, 122]. This model is interesting because it ties together three avenues towards a phenomenology of quantum gravity: the minimal length scale, decoherence from spacetime foam, and non-locality.
Finally, we mention that a minimum time and length uncertainty in rainbow gravity has been found in [115, 116, 117].
After the explicit examples in Sections 3 and 4, here we will collect some general considerations.
One noteworthy remark for models with a minimal length scale is that discreteness seems neither necessary nor sufficient for the existence of a minimal length scale. String theory is an example that documents that discreteness is not necessary for a limit to the resolution of structures, and [60] offered example in which discreteness does not put a finite limit on the resolution of spatial distances (though the physical interpretation, or the observability of these quantities requires more study).
We have also seen that the minimal length scale is not necessarily the Planck length. In string theory, it is naturally the string scale that comes into play, or a product of the string coupling and the string scale if one takes into account D-branes. Also in ASG, or emergent gravity scenarios, the Planck mass might just appear as a coupling constant in some effective limit, while fundamentally some other constant is relevant. We usually talk about the Planck mass because we know of no higher energy scale that is relevant to the physics we know, so it is the obvious candidate, but not necessarily the right one.
Interrelations
The previously-discussed theories and models are related in various ways. We had already mentioned that the path-integral duality (Section 4.7) is possibly related to T-duality (Section 3.2.4) or conformal fluctuations in quantum gravity (Section 3.4), and that string theory is one of the reasons to study non-commutative geometries. In addition to this, it has also been argued that the coherent-state approach to non-commutative geometries represents another model for minimal length modified quantum mechanics [293]. The physics of black holes in light of the coherent state approach has been reviewed in [239].
DSR has been motivated by LQG, though no rigorous derivation exists to date. However, there are non-rigorous arguments that DSR may emerge from a semiclassical limit of quantum gravity theories in the form of an effective field theory with an energy dependent metric [29], or that DSR (in form of a κ-Poincaré algebra) may result from a version of path integral quantization [198]. In addition, it has been shown that in 2+1 dimensional gravity coupled to matter, the gravitational degrees of freedom can be integrated out, leaving an effective field theory for the matter, which is a quantum field theory on κ-Minkowski spacetime, realizing a particular version of DSR [114]. Recently, it has also been suggested that DSR could arise via LQC [59].
As already mentioned, it has been argued in [74] that ASG may give rise to DSR if one carefully identifies the momentum and the pseudo-momentum. In [133] how the running of the Planck's mass can give rise to a modified dispersion relation was studied.
Observable consequences
The most relevant aspect of any model is to make contact with phenomenology. We have mentioned a few phenomenological consequences that are currently under study, but for completeness we summarize them here.
To begin with, experimental evidence that speaks for any one of the approaches to quantum gravity discussed in Section 3 will also shed light on the nature of a fundamental length scale. Currently, the most promising areas to look for such evidence are cosmology (in particular the polarization of the cosmic microwave background) and miscellaneous signatures of Lorentz-invariance violation. The general experimental possibilities to make headway on a theory of quantum gravity have been reviewed in [19, 164]. One notable recent development, which is especially interesting for the question of a minimal length scale, is the possibility that direct evidence for the discrete nature of spacetime may be found in the emission spectra of primordial black holes, if such black holes exist and can be observed [50].
Signatures directly related to the minimal length proposal are a transplanckian cut-off that would make itself noticeable in the cosmic microwave background in the way that the spectrum of fluctuations would not be exactly scale invariant [220, 68]. Imprints from scalar and tensor perturbations have been studied in [37, 38, 36], and in [88, 89] it has been argued in that observable consequences arise at the level of the CMB bispectrum. Deformations of special relativity can lead to an energy-dependent dispersion, which might be an observable effect for photons reaching Earth from γ-ray bursts at high redshift [26, 19, 20]. Minimal length deformations do, in principle, give rise to computable correction terms to a large number of quantum mechanical phenomena (see Section 4.3.2). This allows one to put bounds on the parameters of the model. These bounds are presently many orders of magnitude away from the regime where one would naturally expect quantum gravitational effects. While it is therefore unlikely that evidence for a minimal length can be found in these experiments, it should be kept in mind that we do not strictly speaking know that the minimal length scale is identical to the Planck scale and not lower, and scientific care demands that every new range of parameter space be scrutinized.
Recently, it was proposed that a massive quantum mechanical oscillator might allow one to test Planck-scale physics [264] in a parameter range close to the Planck scale. This proposal should be regarded with caution because the deformations for composite systems used therein do not actually follow from the ones that were motivated by our considerations in Section 3, because the massive oscillator represents a multi-particle state. If one takes into account the ad-hoc solutions to the soccer-ball problem, that are necessary for consistency of the theory when considering multi-particle states (see Section 4.2.4), then the expected effect is suppressed by a mass many orders of magnitude above the Planck mass. Thus, it is unlikely that the proposed experiment will be sensitive to Planck-scale physics.
It is clearly desireable to be able to study composite systems and ensembles, which would allow us to make use of recent advances in quantum optics and data from the early high-density era of the universe. Thus, solving the soccer-ball problem is of central relevance for making contact between these models and phenomenology.
Is it possible that there is no minimal length?
"The last function of reason is to recognize that there are an infinity of things which surpass it."
After having summarized all the motivations for the existence of a minimal length scale, we have to take care that our desire for harmony does not have us neglecting evidence to the contrary.
We have already discussed that there are various possibilities for a minimal length scale to make itself noticeable, and this does not necessarily mean that it appears as a lower bound on the spatial resolution. We could instead merely have a bound on products of spatial and temporal extensions. So in this sense there might not be a minimal length, just a minimal length scale. Therefore, we answer the question posed in this section's title in the affirmative. Let us then ask if it is possible that there is no minimal length scale.
The case for a minimal length scale seems clear in string theory and LQG, but it is less clear in emergent gravity scenarios. If gravity is emergent, and the Planck mass appears merely as a coupling constant in the effective limit, this raises the question, of there is some way in which the fundamental theory cannot have a limiting value at all.
In ASG, the arguments we have reviewed in Section 3.5 are suggestive but not entirely conclusive. The supporting evidence that we discussed comes from graviton scattering, and from a study of a particular type of Euclidean quantum spacetimes.Footnote 13 Notwithstanding the question of whether general relativity actually has a (physically-meaningful) fixed point, the evidence for a minimal length is counterintuitive even in ASG, because gravity becomes weaker at high energies, so, naively, one would expect its distorting effects to also become weaker.
As Mead carefully pointed out in his article investigating the Heisenberg microscope with gravity:
"We have also neglected the effect of quantum fluctuations in the gravitational field. However, these would be expected to provide an additional source of uncertainty, not remove those already present. Hence, inclusion of this effect would, if anything, strengthen the result." ([222], p. B855)
That is correct, one might add, unless gravity itself weakens and counteracts the effect of the quantum fluctuations. In fact, in [51] the validity of the Hoop conjecture in a thought experiment testing short-distance structures has been re-examined in the context of ASG. It was found that the running of the Planck mass avoids the necessity of forming a trapped surface at the scale of the experiment. However, it was also found that still no information about the local physics can be transmitted to an observer in the asymptotic distance.
As previously mentioned, there is also no obvious reason for the existence of a minimal length scale in discrete approaches where the lattice spacing is taken to zero [13]. To study the question, one needs to investigate the behavior of suitably constructed observables in this limit. We also note the central role of the Hoop conjecture for our arguments, and that it is, for general configurations, an unproven conjecture.
These questions are presently very much under discussion; we mention them to show that the case is not as settled as it might have seemed from Sections 3 and 4.
We have seen in this review that there are many motivations for the existence of a minimal length scale. Various thought experiments suggest there are limits to how well we can resolve structures. String theory and LQG, presently the two most widely pursued approaches to quantum gravity, both bring with them a minimal length scale, if in very different realizations. It has been argued that a minimal length scale also exists in the scenario of ASG, and that non-commutative geometries have a minimal length scale built in already.
With that extensive motivation, many models have been proposed that aim at incorporating a minimal length scale into the quantum field theories of the standard model, rather than waiting for a theory of quantum gravity to be developed and eventually connected to the standard model. We have discussed some of these approaches, and also identified some key open problems. While a lot of work has been done directly studying the implications of modified dispersion relations, deformations of special relativity and a GUP, the underlying framework is not yet entirely understood. Most importantly, there is the question of how to construct physically-meaningful observables. One possibility to address this and some other open questions is to develop an axiomatic approach based on the geometry of phase space.
Exploring the consequences of a minimal length scale is one of the best motivated avenues to make contact with the phenomenology of quantum gravity, and to gain insights about the fundamental structure of space and time.
"[D]er Gravitationsradius des zur Messung dienenden Probekörpers (GρV/c2) soll keineswegs größer als seine linearen Abmessungen (V1/3) sein; daraus entsteht eine obere Grenze für seine Dichte (ρ ≲ c2/GV2/3). Die Messungsmöglichkeiten sind also in dem Gebiet noch mehr beschränkt als es sich aus den quantenmechanischen Vertauschungsrelationen schliessen läßt. Ohne eine tiefgreifende Änderung der klassischen Begriffe, scheint es daher kaum möglich, die Quantentheorie der Gravitation auch auf dieses Gebiet auszudehnen."
Translations from German to English: SH.
"Wenn die Explosionen tatsächlich existieren und die für die Konstante r0 eigentlich charakeristischen Prozesse darstellen, so vermitteln sie vielleicht ein erstes, noch unklares Verständnis der unanschaulichen Züge, die mit der Konstanten r0 verbunden sind. Diese sollten sich ja wohl zunächst darin äußern, daß die Messung einer den Wert r0 unterschreitenden Genauigkeit zu Schwierigkeiten führt… [D]ie Explosionen [würden] dafür sorgen…, daß Ortsmessungen mit einer r0 unterschreitenden Genauigkeit unmöglich sind."
"Der Umstand, daß [die Plancklänge] wesentlich kleiner ist als r0, gibt uns das Recht, von den durch die Gravitation bedingten unanschaulichen Zügen der Naturbeschreibung zunächst abzusehen, da sie — wenigstens in der Atomphysik — völlig untergehen in den viel gröberen unanschaulichen Zügen, die von der universellen Konstanten r0 herrühren. Es dürfte aus diesen Gründen wohl kaum möglich sein, die elektrischen und die Gravitationserscheinungen in die übrige Physik einzuordnen, bevor die mit der Länge r0 zusammenhängenden Probleme gelöst sind."
"Mir ist es bisher nicht gelungen, solchen Vertauschungs-Relationen einen vernünftigen mathematischen Sinn zuzuordnen… Fällt Ihnen oder Pauli nicht vielleicht etwas über den mathematischen Sinn solcher Vertauschungs-Relationen ein?"
The story has been told [313] that Peierls asked Pauli, Pauli passed the question on to his colleague Oppenheimer, who asked his student Hartland Snyder. However, in a 1946 letter to Pauli [289], Snyder encloses his paper without any mention of it being an answer to a question posed to him by others.
Though the hope of a lowered Planck scale pushing quantum gravitational effects into the reach of the Large Hadron Collider seems, at the time of writing, to not have been fulfilled.
In the classical theory, inside the horizon lies a singularity. This singularity is expected to be avoided in quantum gravity, but how that works or doesn't work is not relevant in the following.
An example of a different choice of basis can be found in [314].
The word 'effective' should here not be read as a technical term.
According to the Hellinger-Toeplitz theorem, an everywhere-defined symmetric operator on a Hilbert space is necessarily bounded. Since some operators in quantum mechanics are unbounded, one is required to deal with wave functions that are not square integrable. The same consideration applies here.
The Lorentz group has a second Casimir operator, which is the length of the Pauli-Lubanski pseudovector. It can be identified by it being a function of the angular momentum operator.
Though it has meanwhile been shown that the fixed point behavior can be found also in the Lorentzian case [218].
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Some citation-related characteristics of scientific journals published in individual countries
Keshra Sangwal1
Scientometrics volume 97, pages 719–741 (2013)Cite this article
Relationships between publication language, impact factors and self-citations of journals published in individual countries, eight from Europe and one from South America (Brazil), are analyzed using bibliometric data from Thomson Reuters JCR Science Edition databases of ISI Web of Knowledge. It was found that: (1) English-language journals, as a rule, have higher impact factors than non-English-language journals, (2) all countries investigated in this study have journals with very high self-citations but the proportion of journals with high self-citations with reference to the total number of journals published in different countries varies enormously, (3) there are relatively high percentages of low self-citations in high subject-category journals published in English as well as non-English journals but national-language journals have higher self-citations than English-language journals, and (4) irrespective of the publication language, journals devoted to very specialized scientific disciplines, such as electrical and electronic engineering, metallurgy, environmental engineering, surgery, general and internal medicine, pharmacology and pharmacy, gynecology, entomology and multidisciplinary engineering, have high self-citations.
Different aspects of the scientific publication behaviour of researchers publishing in various national and international journals have been studied and differences between the citations of papers in English and non-English languages on a global level have been recognized. Several studies have shown that citations per paper of non-English journals are lower than those of English journals (Garfield 1978; Gonzalez-Alcaide et al. 2012; Liang et al. 2013; Mueller et al. 2006; Poomkottayil et al. 2011; Sangwal 2012; van Raan et al. 2011). Campbell (1990) found that US and UK researchers have a tendency to cite publications produced in their own countries. There are also evidences that researchers are more likely to cite papers published in national languages when publishing in national journals than in international journals (Garfield 1978; Liang et al. 2013; Lin and Zhang 2007). Language self-citation has been suggested as the primary cause of this biased citation behavior in these journals (Liang et al. 2013). Sangwal (2012) analyzed the publication trends of Polish professors and found that: the citability of papers published by physics, chemistry and technical sciences professors in Poland decreases with increasing fraction of the papers in volumes/issues of journals as proceedings of conferences and in non-English language journals.
The scientific impact of journals is traditionally measured in terms of their impact factors (IFs) calculated from the total number of citations, including self-citations, received by the papers published in them and the ranking of a journal in its scientific discipline is determined by the journal IF. These IFs of journals are usually used by research funding agencies as an evaluation measure of scientific performance of individual researchers, faculties and institutes. For example, the Polish Ministry of Higher Education has introduced a system of funding of research in university faculties and institutes and independent research institutes based on their categories determined from consideration of their scientific research outputs. The categories of the research units are determined according to a standardized evaluation criterion based on the number of points assigned to different publications of their publication output. The list of publications valid for research funding until 2010 was based on somewhat ill-defined criteria but the Ministry has updated and revised the list successively in September and December 2012 and is available on the homepage of the Ministry: http://www.nauka.gov.pl/finansowanie/finansowanie-nauki/dzialalnosc-statutowa/. The list of publications is composed of three parts. Part A includes journals, irrespective of their language, belonging to Thomson Reuters' journal IFs and are found in the Journal Citation Reports (JCR) database. A paper published in these journals has been assigned points lying between 15 and 50 (in steps of 5 points), depending on the journal IF. Journals in Part B are those which do not have IFs, and a paper published in these journals is assigned between 1 and 9 points (in steps of 1 point). Journals in Part C, on the other hand, are from the European Reference Index for the Humanities database, and a paper in these journals is assigned 10, 12 and 14 points. A cursory examination of these lists of journals reveals that several journals from the previous list B have entered the new list A of IF journals and some of them have even IF exceeding unity. This is a result of inclusion of more and more national journals in the Thomson Reuters databases in recent years.
For the evaluation of research performance most funding agencies usually use citation data from journals for disciplines such as Science, Engineering and Medicine from Thomson Reuters' Web of Science (WoS). They do not use citation data for disciplines such as Social Sciences and Humanities from WoS or Scopus databases because these databases do not cover citations in books, book chapters, conference papers or journals not indexed in the WoS. Google Scholar has been reported to represent poor coverage for disciplines such as Chemistry and Physics and has a wide coverage which does not vary much across different fields and often includes nearly 90 % of published outputs including books and reports (Harzing 2013; Mingers and Lipitakis 2010). However, the citations it generates come from many different sources which are often not research related (Mingers and Lipitakis 2010). Mingers and Lipitakis (2010) reported that in the field of business and management WoS is more accurate and rigorous. In a recent study, Harzing (2013) compared of coverages of the publication output of 20 Nobel Prize winners in Chemistry, Economics, Medicine and Physics by Google Scholar and WoS, and found that: (1) Google Scholar might provide a less biased comparison across disciplines than the WoS and (2) the use of Google Scholar might redress the traditionally disadvantaged position of the Social Sciences in citation analysis.
The IF of a journal in a particular year is defined as the ratio of the number of citations received in that year by papers published in the journal in the previous 2 years to the number of papers published in that journal in those 2 years. Since it is a measure of the mean citations per paper over a two-year period, there are a number of problems associated with this measure, which are mainly concerned with the short time window for citation record, the robustness/reliability of data sources, and the coverage of data by the source. These problems of the journal IF have been accentuated over years in the literature. The problems are essentially directed to Thomson Reuters which manages its "World of Science" databases used for the calculation of IF of journals. To address the criticism of two-year impact factors (IF2s) of journals, Thomson Reuters has taken a number of steps. For example, since 2007 World of Science database has started publishing five-year impact factors (IF5s) of journals in addition to their classical two-year impact factors (IF2s), and during the last 5 years Thomson Reuters has successively expanded its databases by including new English, non-English and multilingual journals published in different countries across the World.
According to Zitt (2012) the limitations of IF are not its flaw as a measure but it is the vulnerability of the measure to changes, including manipulation, by issues such as the type and the number of documents fetching citations. For example, impact factors of journals can be increased by including high number of self-citations (Bornmann et al. 2008), because journal self-citations are included in the calculation of impact factors. However, despite recognized deficiencies of impact factors of journals, their adoption as a measure of scientific performance has resulted in an omnipresent pressure on editors to improve the impact factors of their journals and on authors to publish in journals with high impact factors, .
Didegah et al. (2012) compared journal publishing behaviors against journal citing behaviors across the world. These authors found that: (1) most papers in five ranges of percentiles of IF2-based quality, from the top 1 %, followed by 1–10 %, 10–20 % and 20–50 %, to the lowest 50–100 %, of journals come from scientifically and economically advanced countries, (2) less developed countries cite high-quality journals at the same rate as developed countries, and (3) research cooperation between developed and less developed countries positively influences the publishing behavior of the latter as their papers coauthored with developed countries are published more often in top quality journals. The influence of research collaboration between countries on their citation impact is also well known. For a review on this subject the reader is referred to a recent paper by Lancho-Barrantes et al. (2013).
Guerrero-Bote et al. (2007) suggested that the distribution of IF of journals belonging to a particular subject category on the journal rank is related to rates of export and import of knowledge in a subject area, denoted here by EX and IM, respectively, defined by the following relations:
$$ EX = \frac{{L_{\text{total}} - L_{\text{sc}} }}{{L_{\text{total}} }}, $$
$$ IM = \frac{{R_{\text{total}} - L_{\text{sc}} }}{{R_{\text{total}} }}, $$
where L total is the number of all citations received in the year Y by papers published in the year (Y-3), (Y-2) or (Y-1) in a subject category, L sc is the number of subcitations (citations from journals of the same subject area) received by the above papers in the year Y, and R total is the number of references of the category. The concept of export and import rates of knowledge, called the iceberg hypothesis, was also explored in a later paper by Lancho-Barrantes et al. (2010) to describe the rank-order distribution of IF in several other subject categories. According to the present author, the above EX and IM parameters do not give important information on the citation behavior of journals belonging to a subject category. For example, the difference (L total − L sc) is equal to the number of citations from journals not from the same subject area and is directly connected to the "external impact" factor (IFext) defined by the above authors such that IFext < IF. In fact, one observes IF > IFext in the plots of IF of journals belonging to various subject categories against the descending journal rank, reported in the above papers. However, apart from the iceberg hypothesis, various other mathematical functions have been proposed to describe the rank-order distributions of items, including IF of journals in various scientific disciplines. For a brief survey of the literature on this subject the reader is referred to a recent paper by the present author (Sangwal 2013).
There is sparse literature on the study of the comparative behavior of journals published in individual countries in English and national languages. No special attention has also been paid until now to analyze the influence of self-citations of journals published in different countries on their impact factors. The present study is addressed to these issues using Thomson Reuters' JCR databases. The aim of the study is three-fold: (1) to compare citation-related characteristics of journals published in nine individual countries from an analysis of their publication languages, two-year and five-year impact factors and self-citations, (2) to examine the factors which lead to changes in the impact factors, and (3) to analyze self-citation characteristics of journals in terms of their publication languages.
Bibliometric data for analysis
We analyzed the citation data of journals published in the following nine countries: Brazil, Croatia, Czech Republic, Italy, Poland, Romania, Slovakia, Spain and Turkey. The countries were selected from the consideration that English is not the national language of these countries, and a high percentage of journals are published in their national languages in different scientific disciplines. Due to their geographical, political and economic background, they represent different publication cultures and organization of research work. For example, in Czech Republic, Poland, Slovakia and Romania research work is carried out in universities as well as institutes of their national academies of sciences but practically in all of the countries considered in this study there are independent research institutes.
We used JCR of Thomson Reuters ISI Web of Knowledge database covering the period 2002–2011, to collect appropriate bibliometric data about the journals, their publication language, publishers, two-year impact factors with journal self-citations (IF2) and without self-citations (IF2nsc), five-year impact factors with self-citations (IF5), journal subject category quartiles (Q1–Q4) based on quartiles of categories, and journal self-citations from the above selected countries. Some basic information about the journals from the 2008–2012 JCR Science Edition is collected in Tables 1 and 2. Table 2 contains data on the numbers of journals from the investigated countries in the 2008 and 2011 JCR databases in: (1) subject category quartiles Q1, Q2, Q3 and Q4, from the topmost subject category Q1 to the lowest subject category Q4, assigned according to the distribution of their decreasing IF2 in the percentile ranges 100–75 %, 75–50 %, 50–25 % and 0–25 %, respectively, in different scientific areas, (2) journal self-citation quartiles F1, F2, F3 and F4 from the lowest to the highest self-citations, assigned according to the distribution of their increasing self-citations, in the percentile ranges 0–25 %, 25–50 %, 50–75 % and 75–100 %, respectively, and (3) English-language (Engl.), multilingual (ML) and national-language (N) journals. Data on subject category quartiles Qs are given in the JCR databases, whereas ranges of journal self-citation quartiles Fs were calculated from the values of ratio f = IF2nsc/IF2 (see Fig. 7) or from self-citations percentiles. The values of IF2, IF2nsc and percentage self-citation of different journals are given in the JCR databases.
Table 1 Total number N and number N IF5 of journals with IF5 from different countries indexed in JCR of 2008–2011
Table 2 Research categories, publication languages and self-citations of journals published in selected countries
It should be mentioned that all non-English journals published in Czech Republic and Slovakia publish papers both in Czech and Slovak languages in addition to papers in English. Therefore, journals published in Czech Republic and Slovakia are typically either English-language or multilingual.
Two-year versus five-year impact factors of journals
All of the journals indexed in the 2008–2011 JCR databases do not have their two-year impact factors (IF2). This situation is observed, for example, in the case of Spain for 2011 journals, where 2 journals do not have their IF2. However, not all of the journals with IF2 have their five-year impact factors (IF5) and the number N IF2 of journals with IF2 is usually much higher than the number N IF5 of journals with IF5. This difference is due to the inclusion in the successive JCR databases of new journals which did not have citation data covering five-year window.
We examined the influence of duration of citation window on impact factors of journals by investigating the relationship between two-year IFs (IF2) of journals published in different countries and their corresponding five-year IFs (IF5). For this purpose we selected the 2011 JCR database which has indexed the highest number of journals among the four databases analyzed here.
Figure 1 shows the dependence of the values of IF2 of journals published in different countries on their corresponding IF5, whereas the solid linear plots represent a slope of unity when IF2 = IF5 for different journals. The slope of the plots of IF2 against IF5 of different journals published in Spain (Fig. 1a), Poland (Fig. 1b), Croatia, Czech Republic and Slovakia (Fig. 1c) is approximately unity. In contrast to these cases of the slope of unity, for the journals published from Turkey and Brazil the slope is lower than unity whereas that for the journal from Romania exceeds unity, as indicated by the dashed line in Fig. 1b. Since the journal impact factor is computed as the ratio of citations received in a given year by papers published over a citation window, these features of the plots of IF2 against IF5 with slopes of lower than, equal to or higher than unity are related to the general trends of increasing, constant or decreasing number of citations received in successive years by the journals published in these countries, respectively. The values of IF2 higher than those of IF5 for the journals published by a country mean higher values of citations during the 2 years of citations considered in the calculations of IF2, whereas lower values of IF2 imply that the journals received lower citations during the 2 years.
Plots of IF2 of journals published in different countries against their corresponding IF5 according to 2011 JCR. Linear plot represents a slope of unity. For the sake of clarity data are presented in separate figures
It is interesting to confront the above general conclusions drawn from a comparison of the IF2 and IF5 of high-ranked international journals and top-ranked journals published from another country. Raj and Zainab (2012) recently reported, among others, data on the IF2 and IF5 of top ten international journals from Thomson Reuters 2008 JCR database and of top ten national journals from the Malaysian citation database. Examination of the data for Malaysian journals reveals that for most of these journals their IF2 is higher than IF5 and for two journal this increase is even 170 and 260 %. Obviously, the citation behavior of Malaysian journals is somewhat similar to that of Romanian journals. However, in the case of data on the international journals, except in the case of one journal (CA-Cancer J. Clin.) where the IF2 (74.58) has increased substantially from its IF5 (50.77), the values of IF2 of the remaining journals have either remained practically constant or somewhat decreased (by <20 %). The present author also examined the recent data of IF2 against IF5 for the top 20 international journals from 2011 JCR database. It was found that, with the exception of the journal CA-Cancer J. Clin. (IF2 101.78; IF5 67.41) where IF2 differs from IF5 enormously, IF2 has remained comparable with IF5 for most of the journals.
Campanario (2011) compared the values of IF2 with those of IF5 of top 20 international journals from Thomson Reuters 2007–2009 JCR databases and found that IF5 > IF2 for most journals but IF5 < IF2 for about a quarter of them. Similar observations have previously been made by other authors (Rousseau et al. 2001). The increase in IF2 of journals was attributed to the citations of more papers published in the latest 2 years than in the previous years (Campanario 2011). Using the scientific publication output of Norwey, Aksnes and Sivertsen (2004) found that: (1) there are large annual variations in the influence of highly cited papers on the average citation rate of the subfields, and (2) the average citation rates of papers in major subfields are highly determined by one or only a few highly cited papers. The above observations are associated with the highly skewed distribution of citations of papers published in journals. Therefore, IF is increased primarily by the highly cited papers (Vinkler 2012; Moed et al. 2012). In view of this skewness of citation distribution of papers in journals, a huge number of citations received by an individual paper published in a journal can have a dramatic effect on its IF (Moed et al. 2012).
Publishing trends of journals
The growth dynamics of the journals published by different countries may be analyzed from the dependence of the ratio (N IF2 − N IF5)/N IF2 on the number N IF2 of journals with IF2. The interval in the values of (N IF2 − N IF5)/N IF2 in the plots of (N IF2 − N IF5)/N IF2 against N IF2 for different countries is a measure of "established" journals published in different countries. The lower and narrower the interval in the values of (N IF2 − N IF5)/N IF2 for a country, the higher is the number of the established journals published by it. However, the slope of the plot of (N IF2 − N IF5)/N IF2 against N IF2 is a measure of the growth dynamics of the journal published in different countries. The lower the value of the slope of the plot of (N IF2 − N IF5)/N IF2 as a function of N IF2 for a country, the higher is the growth dynamics of the journals published in it.
Figure 2 shows the plots of (N IF2 − N IF5)/N IF2 against N IF2 for different countries. Four linear plots of (N IF2 − N IF5)/N IF2 on N IF2 drawn with slopes of 0.02, 0.01, 0.005 and 0.0025 are also shown in the figure for visual reference. The values of the slope of the plots of (N IF2 − N IF5)/N IF2 against N IF2 for different countries indicate that the highest growth dynamics of journals has occurred in countries like Italy and Turkey, whereas the lowest growth dynamics has been observed by journals published in Croatia, Czech Republic and Slovakia. The growth dynamics of journals published in Spain, Brazil, Poland and Romania lies in between the above two extremes.
Plots of (N IF2 − N IF5)/N IF2 against the total number N IF2 of journals with IF2 published in different countries. Data from Table 1
The total number N of journals published in the countries considered here for 2008 as well as 2011 JCR databases is always lower than the total number N Q of subject categories represented by them. For example, according to the 2011 JCR database, the total number of journals published by the countries analyzed in this study is 610 but they are assigned to 738 subject categories. This is due to the fact that many journals are assigned to more than one JCR category. For example, Opto-Electronic Review (Opto-Electron. Rev.), published in Poland by Versita, a Publisher with publication/distribution arrangements with Springer, and Energy Education Science and Technology (Energy Educ. Sci. Tech.), published in Turkey by Sila Science (University Mah, Trabzon), belong to three categories. The share of more than one JCR category in the journals, defined here as percentage of excess of categories (excess %), lies in a wide range for these countries. This excess share lies between 8.5 and 29.3 % for the journals indexed in the 2011 JCR database. The lower the value of the excess share of categories, the higher is the percentage of one-category journals. The 2011 JCR data reveal that a large proportion of journals published in Romania belongs to one-category journals, a large proportion of journals published in Croatia, Czech Republic, Slovakia and Spain are two-category journals, whereas the journals published in Brazil, Italy, Poland and Turkey belong to two as well as three subject categories.
The behavior of excess subject categories for journals published in the countries analyzed here in different years was compared by introducing the parameter q = (N Q − N IF2)/N IF2, where N Q is the total number of categories and N IF2 is the number of journals with IF2. We used N IF2 values of journals instead of the number N of indexed journals because IF2 of a journal is used to assign a category to it. The values of the parameter q for journals indexed in 2008 and 2011 for different countries are compared in Fig. 3. The number N IF2 of journals with IF2 for these 2 years are given at the top of the two columns for each country.
Dependence of parameter q = (N Q − N IF2)/N IF2 on the number N IF2 of journals published in different countries. N Q is the total number of categories. Data from Table 2. See text for details
It may be seen from the figure that the parameter q is not directly related to the number N IF2 of journals published in a country. However, with increasing number N IF2 of journals published by individual countries, the values of q show enormously different trends. With an increase in N IF2, the value of q for Poland, Spain and Czech Republic remains essentially unchanged, it decreases for Romania, Brazil, Italy, and Slovakia, whereas it increases for Turkey and Croatia. These observations are related to changes in the ratio N Q/N IF2 with an increase in the number N IF2 (i.e. N) of journals published in a country. When more new journals from a country with a smaller number of subject categories than those in previous years are indexed in the JCR database, the value of q decreases in later years. When more new journals from a country with a higher number of subject categories than those in previous years are included in the JCR database, the value of q increases in subsequent years. However, when more new journals from countries with the same number of subject categories as in previous years are included in the JCR database, q remains unchanged in later years.
Journal categories and self-citation
Figure 4 compares the relative percentages of English-language, multilingual and national-language journals published in different countries according to 2008 and 2012 JCR databases. Several features may be noted from this figure:
Histogram of relative participation of English, multi-language and local language journals published in different countries according to a 2008 JCR and b 2011 JCR. Total number of journals from a country is given at the top of corresponding columns. Data from Table 2
In the 2008 JCR data the share of English-language journals is always higher than that of national-language journals. However, the share of national-language journals is relatively high about 30 % in countries like Spain and Romania and is equal to that of multilingual journals. In contrast to this, the share of English-language and multilingual journals is equal and is about 45 % for Brazil and Croatia.
The share of English-language journals has remained practically at the same level in the 2008 and 2011 JCR databases for Poland, Italy and Slovakia. However, in 2011 JCR database the relative share of English-language journals published in Romania has practically doubled with respect to the 2008 JCR database at the expense of Romanian-language and multilingual journals.
In the case of Czech Republic and Slovakia publishing English-language and multilingual journals alone, their relative shares in the 2008 and 2011 JCR databases follow different trends. The relative shares of the English-language and multilingual journals from Slovakia have remained practically unchanged at about 85 and 15 %, respectively, but the share of English-language journals published in Czech Republic has increased significantly in the 2011 JCR database at the expense of multilingual journals.
The total number of national-language journals indexed in the 2011 JCR database for all countries has increased to 161 from mere 24 indexed in the 2008 JCR database. This share has approached 26.4 % of the total number of journals in the 2011 JCR database from 9.4 % of the journals in the 2008 JCR database.
Figure 5 shows the relative distribution of four quartiles of the subject categories of journals published in different countries according to 2008 and 2011 JCR databases. As seen from Fig. 5, with insignificant changes in the order of neighboring categories, the share of journals published in a country increases with lowering of their category in the two databases. Among these insignificant changes are an increase or a decrease in categories Q1 and Q2 for different countries, but one also encounters redistribution of shares of categories Q1 and Q2 for a country in the two databases. Large changes are observed in the case of Italy, Spain, Turkey and Romania. The shares of Q1 and Q2 have increased for Italy and Turkey, whereas the shares of different categories have become steadily increasing for Spain and Romania in the 2011 JCR database in comparison with those in the 2008 database.
Histogram of relative participation of four quartiles in the subject categories of journals published in different countries according to a 2008 JCR and b 2011 JCR. Total number of categories for journals from a country is given at the top of corresponding columns. Data from Table 2
Figure 6 shows the relative distribution of self-citation quartiles F of journals published in different countries according to 2008 and 2011 JCR databases. It may be noted that, with the exception of Romania, the relative distribution of self-citation quartiles of journals published in various countries decreases with increasing journal self-citations. However, there are relatively high shares of self-citations quartiles F3 and F4 in the case Romania.
Histogram of relative participation of four quartiles in the groups of self-citations to journal published in different countries according to a 2008 JCR and b 2011 JCR. Total number of journals from a country is given at the top of corresponding columns. Data from Tables: a 3 and b 4
The effect of self-citations and publication languages of journals published in different countries was examined from the distribution of categories of English and non-English journals corresponding to different journal self-citations quartiles. The relevant data are given in Tables 3 and 4 according to the 2008 and 2012 JCR databases. From these tables the following features may be noted:
Table 3 Structure of quartiles of categories Q of English and non-English journals from 2008 JCR database corresponding to different quartiles of journal self-citations
The number of subject category quartiles Q of English-language journals published in a country is mainly confined to self-citation quartiles F1 and F2. However, a majority of the journals in these self-citation quartiles lies in subject categories Q2, Q3 and Q4. Journals published in Romania are exceptions.
The number of subject category quartiles Q of English-language journals published in a country increases in the case of self-citation quartile F1 of journal, but no specific trend of the number of categories is observed for other self-citation quartiles of journals published in different countries.
In Brazil, Spain and Turkey, where the percentage of non-English-language journals is comparable with or higher than that in the case of English-language journals, a majority of the journals belongs to self-citation quartiles F1 and F2 but most of them lie in subject categories Q3 and Q4. In contrast to these countries, in Romania there are no journals belonging to self-citation quartile F1. However, most of the non-English journals published in all countries belong to subject category Q4.
From the above observations it may be concluded that the subject categories of non-English-language journals published in different countries follow trends different from those in the case of English-language journals. Non-English-language journals mainly belong to the lowest category Q4 in comparison with English-language journals a majority of which belongs to categories Q3 and Q4. In other words, English-language journals have higher impact factors than non-English journals. This inference is consistent with the previous findings on differences in the citations of English- and non-English-language journals (Garfield 1978; Gonzalez-Alcaide et al. 2012; Liang et al. 2013; Mueller et al. 2006; Poomkottayil et al. 2011; Sangwal 2012; van Raan et al. 2011).
There are no non-English-language journals published in Croatia and Romania belonging to self-citation quartile F1, whereas there are comparable but relatively high percentage of category Q4 journals published in these countries belonging to self-citation quartiles F3 and F4. This trend of the percentage of category Q4 journals belonging to self-citation quartiles F3 and F4 is different from that encountered in the case of non-English-language journals published in the other countries, and is associated with relatively high contribution of self-citations in the case of Croatia and Romania.
Examination of subject areas of the journals published in different countries revealed that in practically all countries non-English-language journals cover highly specialized areas like agriculture, horticulture, forestry, agronomy, food sciences and technology, veterinary sciences, fisheries, nursing, surgery, oncology, dermatology, cardiology, pediatrics and general and internal medicine. Similar findings have been reported earlier in the case of Spanish-language journals in the fields of clinical medicine or social sciences and humanities (Gonzalez-Alcaide et al. 2012). The main reason of this trend is associated with the localized nature of the subject matter of the papers published in non-English-language journals. Therefore, these journals are not attractive for a relatively wide range of audience, especially publishing their papers in English-language journals. This results in poor citations of the papers published in non-English-language journals and their low impact factors. Consequently, these journals are expected to belong to relatively low category quartiles in comparison with English-language journals.
Self-citation characteristics of English- and non-English-language journals
From the nine countries selected above, the bibliometric data for the journals published in the seven countries (i.e. Brazil, Croatia, Italy, Poland, Romania, Spain and Turkey), containing papers written in English, in the national language of a country, or in both of these languages, were analyzed in detail for 2008 and 2011. Journals published in the remaining two countries, Czech Republic and Slovakia, which contain papers written in English alone or in both Czech and Slovak languages in addition to papers in English, were not considered for the analysis in view of relatively small data and absence of typically national-language journals.
In order to investigate the influence of publication language of journals, the distribution of English-, national- and multi-language journals published in different countries was analyzed quantitatively from the number N E, N N and N ML of journals, respectively, in self-citation quartiles F1–F4. The numbers N E, N N and N ML of journals in the self-citation quartiles F1–F4 may be counted in two ways: (1) directly from the printouts of datafiles, with additional information recorded manually of the values of percent self-citations or of two-year impact factors without self-citations (IF2nsc), from databases for the journals from different countries or (2) from the plots of self-citation parameter f, calculated from the values of IF2nsc and two-year impact factors with self-citations (IF2) as f = IF2nsc/IF2, as a function of IF2 of journals published in a country. Figure 7 shows typical examples of the plots of journal self-citation parameters f for the English-, national- and multi-language journals published in different countries plotted as a function of the values of their IF2 from the 2011 JCR database.
Typical examples of relationship between parameter f = IF2nsc/IF2 and IF2 of self-citations of English-, national- and multi-language journals published in a Poland and b Italy. Original data from 2011 JCR database. See text for details
It should be mentioned that the definition of the parameter f introduced above is similar to that of the parameter EX of Eq. (1), used by Guerrero-Bote et al. (2007). The definition of f is based on IF2 and IF2nsc which correspond to the citations L total and L nsc normalized with respect to the number N IF2 of papers with IF2 published in a given period. For a given year when N IF2 is constant, f = EX. However, for different years when N IF2 does not remain constant and the citation behavior of journals from a country is also different, f ≠ EX.
In Fig. 7, IF2s of the journals from a country are grouped into four quartiles defined by the parameter g = IF2/IF2max, where IF2max is the highest IF2 for a country. An exception is the IF2 = 31.677 of the top journal from Turkey, where the highest IF2 is taken as 2 which is approximately equal to the second top journal with IF2 = 1.991. The groups G1, G1, G3 and G4 defined in this way on the basis of IF2 quartiles are: (1) 0–0.25, (2) 0.25–0.50, (3) 0.5–0.75 and (4) 0.75–1. This categorization of IF2 into G groups is similar to that of categorization of IF2 into subject category quartiles Q used in Thomson Reuters JCR databases. Each self-citation quartile F and each IF2-based group quartile G were further divided into two subgroups. The numbers of journals located in these different subgroups of self-citation and IF2-based group quartiles are denoted in these figures whereas for different countries the numbers of English-, national- and multi-language journals and their total number published are given in the insets.
The numbers N E, N N and N ML of parameter f corresponding to groups F1–F4 for English-, national- and multi-language journals, counted by following the above procedure, from various countries are given in Tables 5 and 6 for 2008 and 2011 JCR databases, respectively. As noted before from Table 2, these tables also show that most of the journals published in different countries belong to self-citation groups F1 and F2 but there are several exceptions where a high percentage of journals belongs to self-citation groups F3 and F4.
Table 5 Numbers N of journals with different quartiles F of self-citations according to 2008 JCR database
Table 6 Numbers N of journals with different quartiles F of self-citations according to 2011 JCR
The effect of publication language of the journals published in different countries was analyzed from normalized fractions p of journals belonging to the four quartiles F1–F4. The normalized fraction p was calculated from the ratio of the number N E, N N or N ML of English-, national- or multi-language journals to their corresponding total number ΣN E, ΣN N or ΣN ML (for example: p E = N E/ΣN E) in groups F1–F4 of self-citation quartiles. Histograms of the fractions p of journals belonging to the four quartiles of self-citations are presented in Figs. 8 and 9. Figures 8 and 9 show data for countries publishing relatively high and low number of journals classified according to the 2011 JCR database, respectively. We discuss below the general features of self-citations in these two classes of journals.
Histograms of relative participation p of four groups F of self-citation quartiles of journals published in a, b Poland, c, d Italy, e, f Brazil and g, h Spain. Data from a, c, e, g 2008 JCR, and b, d, f, h 2011 JCR
Histograms of relative participation p of four groups F of self-citation quartiles of journals published in a, b Turkey, c, d Romania and e, f Croatia. Data from a, c, e 2008 JCR, and b, d, f 2011 JCR
Figures 8 and 9 show that, with the exception of Romania, the fraction p of English-language journals published in different countries decreases with increasing self-citation quartile. The fractions of the journals in these countries are mainly limited to self-citation quartile F3 in the 2008 JCR database but they have gone down to F4 for most of the countries in the 2011 JCR database. The fraction p of multilingual journals published in Poland, Italy, Brazil and Spain also shows a decreasing trend with increasing self-citations. In the case of the remaining countries (Turkey, Czech Republic and Slovakia), it is difficult to establish any specific trends because of small number of multilingual journals published by them. In contrast to the trends of English-language and multilingual journals, the trends of changes in the fraction p of national-language journals published in the countries studied here with increasing self-citations are enormously different from each other, but a distinct difference in the self-citation behavior of journals indexed in the 2008 and 2011 JCR databases may be noted.
The national-language journals indexed in the 2008 JCR database from different countries is relatively small. Therefore, it is difficult to establish any self-citation trends for the national-language journals from the countries studied here. However, for the journals indexed in the 2011 JCR database one finds the following trends with increasing self-citation quartile: (1) the fraction p of non-English-language journals decreases for Italy (Fig. 8d), Brazil (Fig. 8f) and Turkey (Fig. 8b), (2) it remains distributed more or less uniformly over the first three or over all of self-citation quartiles for a country like Poland (Fig. 8b), and (3) it is distributed in such a way that its value is relatively low in self-citation quartile F1 and then changes nonuniformly in quartiles F2, F3 and F4 for countries like Spain (Fig. 8h) and Croatia (Fig. 9f). As judged from the self-citation quartiles of different countries, a clearly increased tendency of self-citations in national-language journals is seen in the histograms for practically all countries.
It should be noted that the self-citation behavior of English-, multi- and national-language journals published in Romania are completely different from their counterparts in other countries. For example, in contrast to the decreasing fraction p of English-language journals published in most countries with increasing self-citation quartile, the self-citation fraction p is equally distributed over all self-citation quartiles (Fig. 9c, d). In comparison with the trends of the distribution of p of the national-language journals published in other countries, the fraction p of Romanian-language journals mainly lies only in the highest self-citation quartiles F3 and F4.
From the above results it may be concluded that non-English language journals published in various countries usually have higher self-citations than English-language journals. A direct consequence of these high self-citations is to increase the values of their IF2. This tendency of high self-citations of journals, especially non-English-language journals, is associated with the national, regional or local nature of the subject matter of the papers published in them (Gonzalez-Alcaide et al. 2012), reluctance of authors of papers published in them to cite international literature due to language barrier and editorial policies of the journals (Bornmann et al. 2008; Gonzalez-Alcaide et al. 2012).
High self-citation journals in different countries during 2008–2012
Journals with self-citations above 70 % published in the selected countries were compared to assess the trends of self-citations with increasing number of journals published in different countries. For this purpose, the relevant data collected from the 2008 and 2011 JCR databases are listed in Tables 7 and 8, respectively.
Table 7 Journals with high self-citations (SC) published in selected countries according to 2008 JCR
It may be noted from these tables that all countries have journals with very high self-citations, but the number of journals with high self-citations with reference to the total number of journals is relatively low in the 2008 JCR database in comparison with that in the 2011 JCR database. In general, the ratio of high self-citation journals to the total number of journals has increased significantly in the 2011 database from the 2008 database for all countries, but the value of the increase in the ratio of high self-citation journals for different countries varies enormously. Journals indexed in the 2008 JCR database from Romania, Turkey and Brazil show high self-citations but there are no high self-citation journals indexed in the 2008 JCR database from Poland, Italy, Spain and Croatia. However, in the 2011 JCR database, there are 12–15 % high self-citation journals from Poland and Romania, about 2 % from Brazil and Turkey, whereas it is intermediate for the remaining countries.
National-language journals have higher self-citations than English-language journals. However, it is a common observation that, irrespective of the publication language, journals devoted to very specialized scientific disciplines typically have relatively high self-citations. Among these disciplines are, for example, electrical and electronic engineering, metallurgy, environmental engineering, surgery, general and internal medicine, pharmacology and pharmacy, gynecology, entomology and multidisciplinary engineering.
The above observations suggest that, although all countries have highly self-cited journals, the proportion of the highly self-cited journals depends on the citation culture in different countries, the publication language of journals, their scientific discipline and the dissemination of their contents. The difference in the proportion of self-cited journals may also be attributed to the editorial policies of the journals published in these countries and the regional/local character of the contents of papers published in very specialized scientific disciplines of some of the journals.
The following conclusions can be drawn from this study:
Analysis of data of two-year impact factor (IF2) against five-year impact factor (IF5) of different journals published in different countries according to the 2011 JCR database with the widest coverage of journals in the JCR databases revealed that IF2 ≈ IF5 for the journals published in Poland, Czech Republic and Croatia, IF2 > IF5 for the journals published from Turkey and Brazil, whereas IF2 < IF5 for the journals published in Romania. These relationships between IF2 and IF5 are related to the increasing, constant or decreasing number of citations received by the journals published in these countries. The unusual behavior of non-English-language journals published in Romania is mainly associated with the high self-citations of these journals which usually lie in categories Q3 and Q4.
English-language journals, as a rule, have higher impact factors than non-English-language journals.
All countries investigated in this study have journals with very high self-citations but the proportion of journals with high self-citations with reference to the total number of journals published in different countries varies enormously. National-language journals have higher self-citations than English-language journals.
Irrespective of the publication language, journals devoted to very specialized scientific disciplines have relatively high self-citations. Among these disciplines are, for example, electrical and electronic engineering, metallurgy, environmental engineering, surgery, general and internal medicine, pharmacology and pharmacy, gynecology, entomology and multidisciplinary engineering.
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Department of Applied Physics, Lublin University of Technology, ul. Nadbystrzycka 38, 20-618, Lublin, Poland
Keshra Sangwal
Correspondence to Keshra Sangwal.
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Sangwal, K. Some citation-related characteristics of scientific journals published in individual countries. Scientometrics 97, 719–741 (2013). https://doi.org/10.1007/s11192-013-1053-1
Issue Date: December 2013
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\begin{document}
\title{The domination number of plane triangulations}
\author{ Simon \v Spacapan\footnote{ University of Maribor, FME, Smetanova 17, 2000 Maribor, Slovenia. e-mail: simon.spacapan @um.si. }} \date{\today}
\maketitle
\begin{abstract} \noindent We introduce a class of plane graphs called weak near-triangulations, and
prove that this class is closed under certain graph operations. Then we use the properties of weak near-triangulations to prove that every plane triangulation on $n>6$ vertices has a dominating set of size at most $17n/53$. This improves the bound $n/3$ obtained by Matheson and Tarjan. \end{abstract}
\noindent {\bf Key words}: triangulation, dominating set
\noindent {\bf AMS subject classification (2010)}: 05C10, 05C69
\section{Introduction}
A dominating set in a graph $G=(V,E)$ is a set $D\subseteq V$, such that every vertex in $V\setminus D$ is adjacent to a vertex in $D$. The domination number of $G$, denoted as $\gamma(G)$, is the minimum size of a dominating set in $G$. A plane triangulation is a connected simple plane graph $G$, such that every face of $G$ is triangular. A plane graph $G$ is a near-triangulation, if $G$ is 2-connected and every face of $G$ is triangular, except possibly the unbounded face.
Matheson and Tarjan conjectured in \cite{mathesontarjan} that every sufficiently large plane triangulation has a dominating set of cardinality at most $n/4$, where $n$ is the number of vertices of the triangulation. They proved that the domination number of every near-triangulation, and therefore also every triangulation, is at most $n/3$.
\begin{theorem}\cite{mathesontarjan}\label{mathesontarjan} Every near-triangulation has a 3-coloring such that each color class is a dominating set. \end{theorem}
We mention that Theorem \ref{mathesontarjan} provides the only known upper bound for $\gamma(G)/n$ in the class of plane triangulations. Several results for sublasses of plane triangulations are known. For example, the conjecture of Matheson and Tarjan was confirmed in \cite{erika} for plane triangulations with maximum degree 6. In \cite{liu} the authors improve the bound $n/4$ obtained in \cite{erika}, by proving that there exists a constant $c$ such that $\gamma(G)\leq n/6+c$
for every plane triangulation $G$ with maximum degree 6.
Another result is given in \cite{plummer1} where the authors prove that $\gamma(G)\leq \max\{\lceil\frac {2n}{7}\rceil,\lfloor\frac{5n}{16}\rfloor\}$ for every 4-connected plane triangulation $G$. The domination number of outerplanar triangulations is considered in \cite{campos} and \cite{tokunaga}, where the authors independently prove that an outerplaner triangulation with $n$ vertices and $t$ vertices of degree 2 has a dominating set of cardinality at most $(n+t)/4$. This result is further improved in \cite{li}.
Theorem \ref{mathesontarjan} was extended to tringulations on the projective plane, the torus and the Klein bottle in \cite{plummer} and in \cite{honjo}. It is proved that every triangulation on any of these surfaces has a dominating set of cardinality at most $n/3$. These results are further generalized in \cite{furuya}, where it is proved that every triangulation on a closed surface has a dominating set of cardinality at most $n/3$.
We also mention that the domination number of planar graphs with small diameter is studied in \cite{mac}, \cite{henning1} and \cite{henning2}. It is proved that every sufficiently large planar
graph with diameter 3 has domination number at most 6.
We approach the problem of finding the smallest constant $c$, such that $\gamma(G)\leq cn$, for every sufficiently large plane triangulation. In the following section we introduce the class of weak near-triangulations. We define reducibility of a weak near-triangulation, and prove that every weak near-triangulation $G$ is reducible, except if all blocks of $G$ are outerplaner or have exactly 6 vertices. This result is then used to prove that every plane triangulation on $n>6$ vertices has a dominating set of cardinality at most 17n/53.
\section{The domination number of triangulations} We refer the reader to \cite{diestel} and \cite{mt} for a complete overwiev of definitions and terminology that we use. In this article we consider only simple graphs with no multiple edges or loops. Let $G$ be a plane graph. An edge $e$ of $G$ is {\em incident} to a face $F$ of $G$ if $e$ is contained in the boundary of $F$. Similarly we define incidence of a verex $x$ and a face $F$.
Vertices and edges incident to the unbounded face of $G$ are called {\em external vertices} and {\em external edges}, respectively. If a vertex (an edge) is not an external vertex (edge), then
it is called an {\em internal vertex} ({\em internal edge}).
A {\em triangle} is a cycle on three vertices. If $a,b$ and $c$ are vertices of a triangle, we denote this triangle by $abc$, and we say that $a,b$ and $c$ are {\em contained} in the triangle $abc$. A {\em facial triangle} of a plane graph $G$ is a triangle whose interior is a face of $G$. A face bounded by a triangle is called a {\em triangular face}. A connected plane graph $G$ is a {\em triangulation} if all faces of $G$ are triangular, and $G$ is a {\em near-triangulation} if it is 2-connected and every bounded face of $G$ is triangular. A {\em block} of a graph $G$ is a maximal connected subgraph of $G$ without a cutvertex.
\begin{definition} A plane graph $G$ is a weak near-triangulation (WNT) if every bounded face of $G$ is triangular and every vertex of $G$ is contained in a triangle. \end{definition}
\begin{figure}
\caption{A weak near-triangulation $G$. }
\label{sibkaskorajtriangulacija}
\end{figure}
We note that an empty graph is a weak near-triangulation, and that in the above definition there are no assumptions about the connectivity of $G$, hence $G$ may be disconnected. Note also that the definition is equivalent to the following: $G$ is a weak near-triangulation if every bounded face of $G$ is triangular and every vertex of $G$ is contained in a facial triangle.
In \cite{diestel} (see Lemma 4.2.2., p. 91, and Lemma 4.2.6, p. 93) the following two results are given. \begin{lemma}\label{prva} Let $G$ be a plane graph and $e$ an edge of $G$. If $e$ lies on a cycle $C\subseteq G$, then $e$ is incident to exactly one face $F$ of $G$, such that $F$ is contained in the interior of $C$. \end{lemma}
\begin{lemma}\label{druga} In a 2-connected plane graph, every face is bounded by a cycle. \end{lemma}
We use Lemma \ref{prva} and \ref{druga} to prove the following.
\begin{lemma} \label{blok} Every block of a weak near-triangulation is either a near-triangulation or a $K_2$. \end{lemma} \noindent{\bf Proof.\ } Let $B$ be a block of a weak near-triangulation $G$. Since every vertex of $G$ is contained in a triangle of $G$, we find that $B$ has more than one vertex. Suppose that $B$ is a block on more than two vertices. Then $B$ is 2-connected, and so by Lemma \ref{druga} every face of $B$ is bounded by a cycle. Let $F$ be a bounded face of $B$ and $C$ the cycle bounding $F$ ($F$ lies in the interior of $C$).
To prove that $B$ is a near-triangulation, we have to prove that $F$ is triangular.
Let $e=uv$ be any edge of $C$. Since $C$ is a cycle of $G$ we find, by Lemma \ref{prva}, that there is a face $F'$ of $G$ contained in the interior of $C$ incident to $e$. Since $F'$ is a bounded face of $G$, it has to be triangular (by the definition of a weak near-triangulation). If $F'=F$ we are done. Suppose that $F'\neq F$, and suppose that $F'$ is bounded by the triangle $uvx$. Then $x$ is not in $B$, for otherwise $F$ and $F'$ are faces of $B$ contained in the interior of $C$, both incident to $e$ (contradictory to Lemma \ref{prva}). So $x\notin B$ and therefore $B$ is not a maximal connected subgraph of $G$ without a cutvertex (we may add $x$ to $B$), a contradiction.
$\square$
Observe also that every endblock (leaf block) of a weak near-triangulation is a near-triangulation. \begin{lemma} Every weak near-triangulation has a 3-coloring such that each color class is a dominating set. \end{lemma}
\noindent{\bf Proof.\ } Let $G$ be a weak near-triangulation. Delete all bridges of $G$ and call the obtained graph $G'$. Every block of $G'$ is a near-triangulation, and two distinct blocks have at most one common vertex. To obtain the desired 3-coloring of a connected component of $G'$ we use induction on number of blocks. In induction step we delete an endblock (except the cutvertex) and color the obtained graph with 3 colors according to the induction hypothesis. Then use Theorem \ref{mathesontarjan} to obtain a coloring of the deleted endblock, if needed permute the color classes in the endblock (so that the cutvertex gets the color that it already has).
$\square$
A straightforward corollary is the following.
\begin{corollary}\label{n3weak} Every weak near-triangulation on $n$ vertices has domination number at most $n/3$. \end{corollary}
For a plane graph $G$ and $X\subseteq V(G)$ we denote by $G-X$ the graph obtained from $G$ by deleting the vertices in $X$ and all edges incident to a vertex in $X$. If $X=\{u\}$ is a singleton, we write $G-u$ instead of $G-\{u\}$. If $H$ is a subgraph of a plane graph $G$, then we assume that the embedding of $H$ in the plane is that given by $G$. This in particular applies for $H=G-X$. Let $N[x]$ denote
the set of vertices that are either adjacent or equal to $x$, $N[x]$ is called the {\em closed neighborhood} of $x$.
\begin{definition} Let $G$ be a weak near-triangulation. We say that $G$ is reducible if there exists a set $D\subseteq V(G)$ and a vertex $x\in D$ with the following properties: \begin{itemize} \item[(i)] $D\subseteq N[x]$
\item[(ii)] $|D|\geq 4$ \item[(iii)] $G-D$ is a weak near-triangulation.
\end{itemize} \end{definition}
The main result of this paper is the following theorem, the proof is postponed to the last section.
\begin{theorem}\label{reducibilnost} Let $G$ be a weak near-triangulation. If $G$ has a block $B$, such that $B$ is not outerplanar and the order of $B$ is different from 6, then $G$ is reducible.
\end{theorem}
Note that it is not possible to extend the above theorem to weak near-triangulations that have only outerplaner blocks and blocks on 6 vertices. We can see this by observing that
an octahedron embedds in the plane as a triangulation, and it is not dominated by one vertex, and therefore also not reducible. Another example is given in Fig.~\ref{slika} where the graph in case $(a)$ is not reducible.
Observe also that every near-triangulation on $n$ vertices with domaination number $n/3$ is not reducible, examples of such outerplaner near-triangulations are exhibited in \cite{mathesontarjan}. In fact, Theorem \ref{reducibilnost} and Corollary~\ref{n3weak} imply the following.
\begin{corollary} Every near-triangulation that attains the bound $n/3$ from Theorem \ref{mathesontarjan}
is outerplanar or has exactly 6 vertices. \end{corollary}
\noindent{\bf Proof.\ }
By Theorem \ref{reducibilnost}, every near-triangulation $G$ which is not outerplaner and has $n\neq 6$ vertices is reducible. So there exists a set $D$, such that $G-D$ is a weak near-triangulation, and $D$ is dominated by one vertex.
The reduced graph $G-D$ has, according to Corollary \ref{n3weak}, a dominating set containing at most $(n-|D|)/3$ vertices.
The result follows from $|D|\geq 4$.
$\square$
It is easy to prove that every near-triangulation $G$ on 6 vertices has a vertex $x$, such that the closed neighborhood of $x$ contains all internal vertices of $G$. The reader may prove this by a case analysis: the outer cylce of $G$ has 5, 4 or 3 vertices.
\begin{corollary} Every plane triangulation on $n>6$ vertices has a dominating set of size at most 17n/53. \end{corollary}
\noindent{\bf Proof.\ } Let $G$ be a plane triangulation on $n$ vertices. We apply Theorem \ref{reducibilnost} until the obtained graph $G'$ is irreducible. More precisely, $G'=G-(D_1\cup\ldots\cup D_k)$,
where $D_i$ is a set of vertices in $G-(D_1\cup\ldots\cup D_{i-1})$, such that there exists a vertex $x_i\in D_i$ with the property $D_i\subseteq N[x_i]$. Moreover, for all $i\leq k$, $|D_i|\geq 4$. By Theorem \ref{reducibilnost}, $G'$ is a weak near-triangulation, and every block of $G'$ is outerplanar or has exactly 6 vertices. If $G'$ contains all three external vertices of $G$, then $G'$ is a triangulation, and thus has exactly one block. If $G'$ is a block of order 6, then at least one reduction was done to get from $G$ to $G'$. The vertices of $V(G)\setminus V(G')$ are dominated by a dominating set of size at most
$\frac 14 |V(G)\setminus V(G')|$. The block $G'$ of order 6 is dominated by 2 vertices. It follows that $G$ has a dominating set of size at most $3n/10$. If $G'$ is outerplaner, then it is a triangle. In this case $G$ has a dominating set of size at most $2n/7$.
Assume therefore that at least one external vertex of $G$ is not in $G'$. We claim that every external vertex of $G'$ is adjacent to a vertex in $V(G)\setminus V(G')$. Let $x$ be an external vertex of $G'$. If $x$ is also an external vertex of $G$, the claim follows from the assumption. If $x$ is an internal vertex of $G$, then let $xy$ be an external edge of $G'$ incident to $x$. Since $x$ is an internal vertex of $G$, the edge $xy$ is an internal edge of $G$, and so there are two facial triangles in $G$ incident to $xy$, and since $xy$ is an external edge of $G'$, one of them is not a triangle of $G'$. So both vertices $x$ and $y$ are adjacent to a vertex in $V(G)\setminus V(G')$. This proves the claim.
Let $q=|V(G')|$. By Corollary \ref{n3weak} $G$ has a dominating set of size at most $(n-q)/4+q/3$, this is one way to dominate $G$. Since every non-outerplanar block of $G'$ of order 6 has a vertex, that dominates all internal vertices of this block, we can choose a vertex in each block of order 6 to dominate internal vertices of $G'$. External vertices of $G'$ are, according to the claim above, dominated by vertices in $V(G)\setminus V(G')$. So we construct a dominating set by choosing all vertices in $V(G)\setminus V(G')$ and a vertex in each block of order $6$ to dominate internal vertices of $G'$.
Since any two distinct blocks share at most one vertex, it follows that at most $|V(G')|/5$ vertices have been chosen in blocks of order 6. This construction gives a dominating set of size at most $n-q+ q/5$. For all $q$ we have $\min\{(n-q)/4+q/3,n-q+ q/5\}\leq 17n/53$.
$\square$
\section{Operations on weak near-triangulations}
We start with definitions of a problematic and a bad vertex of a plane graph.
\begin{definition} Let $G$ be a plane graph and $u\in V(G)$. We say that $u$ is a {\em problematic vertex} in $G$ if $G-u$ is not a weak near-triangulation.
\end{definition}
\begin{definition}\label{bad} Let $G$ be a plane graph and $u\in V(G)$ a problematic vertex in $G$. We say that $x$ is a {\em $u$-bad vertex} in $G$ if $x$ is not contained in a triangle of $G-u$. \end{definition} If $G$ is a weak near-triangulation, then every $u$-bad vertex in $G$ is an external vertex of $G$, and it is adjacent to $u$. Moreover, if $x$ is a $u$-bad vertex, then there exists a facial triangle $uxy$ in $G$. See Fig.\ref{sibkaskorajtriangulacija}, where $u$ is a problematic vertex in $G$, and there are two $u$-bad vertices in $G$ (the white vertices). We skip the proof of the following lemma, it is similar to the proof of Lemma \ref{blok}.
\begin{lemma}\label{dajsklic} Let $G$ be a weak near-triangulation and $x$ an external vertex of $G$. Every bounded face of $G-x$ is triangular. Moreover, if $y$ is adjacent to $x$ in $G$, then $y$ is an external vertex of $G-x$. \end{lemma} Lemma \ref{dajsklic} has an immediate corollary.
\begin{corollary}\label{trikotnalica} Let $G$ be weak near-triangulation, $D\subseteq V(G)$ a set of vertices such that $D\subseteq N[y]$ for some $y\in D$. If $D$ contains at least one external vertex of $G$, then every bounded face of $G-D$ is triangular. \end{corollary}
\begin{lemma}\label{osnovna} Let $G$ be a weak near-triangulation and $u$ a problematic external vertex in $G$. Let $T$ be the set of all $u$-bad verteces in $G$. Then $G-(\{u\}\cup T)$ is a weak near-triangulation. \end{lemma}
\noindent{\bf Proof.\ } Since every $u$-bad vertex in $G$ is adjacent to $u$, we find (by applying Corollary \ref{trikotnalica}) that every face of $G-(\{u\}\cup T)$ is triangular, except possibly the unbounded face. When removing vertices that are not contained in a triangle of $G-u$ we obtain a graph in which every vertex is contained in a triangle (note that this is possibly an empty graph).
$\square$
\noindent If $u$ is an internal vertex, then the following lemma applies.
\begin{lemma}\label{osnovna1} Let $G$ be a weak near-triangulation and $u$ a problematic internal vertex in $G$. Let $T$ be the set of all $u$-bad verteces in $G$, and suppose that $T\neq \emptyset$. Then $G-(\{u\}\cup T)$ is a weak near-triangulation. \end{lemma}
\noindent{\bf Proof.\ } We note that every $u$-bad vertex in $G$ is an external vertex of $G$. The rest of the proof is the same as the proof of Lemma \ref{osnovna}.
$\square$
\begin{lemma}\label{lepljenje} Let $G$ be a weak near-triangulation and $D\subseteq X\subseteq V(G)$ such that \begin{itemize} \item[(i)] $G-X$ is a weak near-triangulation \item[(ii)] $ D\subseteq N[y]$ for some $y\in D$ \item[(iii)] $D$ contains at least one external vertex of $G$ \end{itemize} then $G-D$ is a weak near-tirangulation if and only if every vertex of $X\setminus D$ is contained in a triangle of $G-D$.
\end{lemma}
\noindent{\bf Proof.\ } If a vertex of $X\setminus D$ is not contained in a triangle of $G-D$, then $G-D$ is not a WNT.
Suppose that every vertex in $X\setminus D$ is contained in a triangle of $G-D$. By (i) $G-X$ is a WNT, so every vertex of $G-X$ is contained in a triangle of $G-X$ and hence also in a triangle of $G-D$. It follows that every vertex of $G-D$ is contained in a triangle of $G-D$. We also see that from (ii) and (iii) together with Corollary \ref{trikotnalica} follows that every face of $G-D$ is triangular, except possibly the unbounded face.
$\square$
\begin{lemma} \label{trikotnik} Let $G$ be a weak near-triangulation, and $X\subseteq V(G)$ a set such that $G-X$ is a weak near-triangulation. Suppose that $uvw$ is a triangle of $G$, and let $Y$ be the set of vertices of $G$ contained in the interior of $uvw$. Suppose that $X\cap Y=\emptyset$, $u\in X$ and $v,w\notin X$.
If $y\in Y$ is a problematic vertex in $G-X$, then every $y$-bad vertex in $G-X$ is adjacent to $u$ in $G$.
\end{lemma}
\noindent{\bf Proof.\ } Let $G,X,Y,y$ and $u,v,w$ be as declared in the lemma. Let $x$ be a $y$-bad vertex in $G-X$. If $x=v$ or $x=w$ there is nothing to prove, because $uvw$ is a triangle in $G$. Assume that $x\notin\{v,w\}$, and note that $x$ is adjacent to $y$,
and therefore lies in the interior of $uvw$ in $G$. Since $x$ is a $y$-bad vertex in $G-X$, we find that there is a facial triangle $xyt$ in $G-X$. The edge $xt$ is an internal edge of $G$ and so there is a facial triangle $xty'$ in $G$, such that $y'\neq y$. However, the triangle $xty'$ is not a triangle in $G-X$, for otherwise $x$ is not a $y$-bad vertex in $G-X$. It follows that $y'$ is not in $G-X$, and therefore $y'=u$ (because $y'$ is adjacent to $x$ which is an internal vertex of $uvw$).
$\square$
\section{Reducibility of weak near-triangulations} \label{miki}
In this section we prove Theorem \ref{reducibilnost}. Let $G$ be a weak near-triangulation and $B$ a block of $G$ that is not outerplanar. Suppose also that $B$ is not a block on 6 vertices. Let $u$ be an internal vertex of $B$ so that $u$ has at least two external neighbors (neighbors incident to the unbounded face of $G$). Let $u_1,\ldots,u_n$ be the external neighbors of $u$, and let $R_1,\ldots,R_n$ be regions in the plane where the region $R_k$ is bounded by the outer cycle of $B$ and the edges $uu_k$ and $uu_{k+1}$ (indices are calculated modulo $n$), if $n>2$ two consecutive regions share an edge and if $n=2$ they share two edges. We may choose $u$ so that at most one of the regions $R_k,k\leq n$ contains an internal vertex of $B$ different from $u$. We note that this choice of $u$ only becomes relevant in subsection \ref{uniproblematicen}.
The proof is divided into four main cases (depending on the number of $u$-bad vertices in $G$ ) and several subcases. For each main case there is a subsection. In the proof we work with weak near-triangulation $G$, and with subgraphs of $G$. When we say "adjacent" we mean "adjacent in $G$" (as opposed to "adjacent in a subgraph of $G$"), and unless otherwise stated, a facial triangle means a facial triangle in $G$.
\subsection{There are at least three $u$-bad vertices in $G$ }
Suppose that $u$ has at least three $u$-bad vertices in $G$. Let $T$ be the set of all $u$-bad vertices in $G$, and define $D=\{u\}\cup T$.
By Lemma \ref{osnovna1}, $G-D$ is a WNT. Since $|D|\geq 4$ and $D\subseteq N[u]$, $G$ is reducible.
\subsection{There are exactly two $u$-bad vertices in $G$ }
Suppose that $u$ has exactly two $u$-bad vertices in $G$, and let $x_1,x_2$ be $u$-bad vertices in $G$. By Lemma \ref{osnovna1}, $G-\{u,x_1,x_2\}$ is a WNT. Observe that $u$ is an internal vertex of $G$, so $ux_1$ and $ux_2$ are internal edges of $G$, and therefore there are two facial triangles $ux_1w_1$ and $ux_1z_1$ containing edge $ux_1$, and two facial triangles $ux_2w_2$ and $ux_2z_2$ containing edge $ux_2$ (note that vertices $w_1,x_1,z_1$ are not necessarily distinct from $w_2,x_2,z_2$). Since $x_1$ and $x_2$ are $u$-bad vertices in $G$ neither $x_1$ nor $x_2$ is contained in a triangle of $G-u$. It follows that $x_1w_1, x_1z_1$ and $x_2w_2,x_2z_2$ are external edges of $G$, and therefore $w_1,w_2,x_1,x_2$ are external vertices of $G$. Note also that if $a\neq u$ is a common neighbor of $x_1$ and $x_2$ in $G$, then $a\in \{w_1,w_2,z_1,z_2\}$, because the facial triangles containing edges $ax_2$ and $ax_1$ also contain $u$ (because $x_1$ and $x_2$ are $u$-bad vertices in $G$).
We distinguish three possibilities. 1. $x_1$ and $x_2$ have a common neighbor different from $u$. 2. $x_1$ and $x_2$ are adjacent, and $u$ is their only common neighbor. 3. $x_1$ and $x_2$ are not adjacent, and $u$ is their only common neighbor.
\subsubsection{$x_1$ and $x_2$ have a common neighbor different from $u$} Suppose that $x_1$ and $x_2$ have a common neighbor $t$ different from $u$. Observe that $t$ is adjacent to $u$, because $t\in \{w_1,w_2,z_1,z_2\}$. If $t$ is not problematic in $G-\{u,x_1,x_2\}$, then let $D=\{u,x_1,x_2,t\}$. Since $G-D$ is a WNT and $D\subseteq N[t]$, $G$ is reducible. If $t$ is problematic in $G-\{u,x_1,x_2\}$, then let $T$ be the set of all $t$-bad vertices in $G-\{u,x_1,x_2\}$, and define $D=\{u,x_1,x_2,t\}\cup T$. By Lemma \ref{osnovna}, $G-D$ is a WNT. Since $D\subseteq N[t]$, $G$ is reducible. \label{skupensosed}
\subsubsection{$x_1$ and $x_2$ are adjacent, and $u$ is their only common neighbor} Observe that $ux_1x_2$ is a facial triangle in this case. Let $ux_1w_1$ and $ux_2w_2$ be facial triangles such that $w_1\neq x_2$ and $w_2\neq x_1$. Since $u$ is the only common neighbor of $x_1$ and $x_2$ we have $w_1\neq w_2$. If $w_1$ is not problematic in $G-\{u,x_1,x_2\}$, then define $D=\{u,x_1,x_2,w_1\}$.
By the definition (of a problematic vertex) $G-D$ is a WNT. Since $|D|\geq 4$ and $ D\subseteq N[u]$, $G$ is reducible. If $w_2$ is not problematic in $G-\{u,x_1,x_2\}$, the reduction is analogous as above, so assume that both $w_1$ and $w_2$ are problematic in $G-\{u,x_1,x_2\}$.
Suppose that $w_1$ is a $w_2$-bad vertex in $G-\{u,x_1,x_2\}$, and that $w_2$ is a $w_1$-bad vertex in $G-\{u,x_1,x_2\}$. In this case $w_1$ and $w_2$ are adjacent, moreover $x_1$ and $x_2$ are in the exterior of triangle $w_1w_2u$, because $x_1$ and $x_2$ are external vertices of $G$. If $w_1w_2x$ is in a facial triangle and $x\neq u$, then $x$ lies either in the interior of triangle $w_1w_2u$ or in the exterior. If $x$ is in the interior of $w_1w_2u$, then the edge $xw_1$ is incident to a triangular face $xw_1t$. If $t\notin \{u,w_2\}$, then $xw_1t$ is a triangle in $G-\{u,x_1,x_2,w_2\}$, and so $w_1$ is not a $w_2$-bad vertex in $G-\{u,x_1,x_2\}$, contrary to the assumption. It follows that $xw_1u$ and $xw_1w_2$ are facial triangles in $G$. Similarly, since $w_2$ is a $w_1$-bad vertex in $G-\{u,x_1,x_2\}$, we find that $xw_2u$ is a facial triangle in $G$. This in particular implies that $x$ is the only vertex of $G$ in the interior of triangle $w_1w_2u$.
\begin{figure}
\caption{Cases $(a)$ to $(d)$.}
\label{slika}
\end{figure}
Suppose now that $w_1w_2y$ is a facial triangle such that $y$ is in the exterior of $w_1w_2u$. Since $w_1,w_2,x_1,x_2$ are external vertices of $G$, $y\neq x_1$ and $y\neq x_2$. If $yw_1$ is not an external edge of $G$, then there is a facial triangle $yw_1t$, with $t\neq x_1,x_2$ (note again that $w_1,w_2,x_1,x_2$ are external vertices of $G$). This contradicts the assumption that $w_1$ is a $w_2$-bad vertex in $G-\{u,x_1,x_2\}$. Therefore $w_1y$ is an external edge of $G$, and similarly $w_2y$ is an external edge of $G$.
Since $yw_1, yw_2, w_1x_1,x_1x_2$ and $x_2w_2$ are external edges of $G$, and also $w_1w_2$ is an external edge if there is no facial triangle $w_1w_2y$ (with $y$ in the exterior of $w_1w_2u$), we find that $B$ is isomorphic to one of the four graphs shown in Fig.~\ref{slika}. $B$ is a block on 6 vertices in cases (a) and (b), which contradicts our assumptions. $B$ is a block on 5 vertices in case (c), and since $G-\{u,x_1,x_2\}$ is a WNT, $w_1$ is contained in a facial triangle $w_1ab$ of $G-\{u,x_1,x_2\}$. Clearly, $a,b\notin B$, and therefore $w_1$ is not a $w_2$-bad vertex in $G-\{u,x_1,x_2\}$ (contrary to the assumption). In case (d), $B$ is a block of order 7. If there is a $w_1$-bad vertex $z\notin\{ x,w_2\}$ in $G-\{u,x_1,x_2\}$, then let $T$ be the set of all $w_1$-bad vertices in $G-\{u,x_1,x_2\}$, and define $D=(\{w_1,x_1\}\cup T)\setminus \{w_2\}$. Note that $x$ is a $w_1$-bad vertex in $G-\{u,x_1,x_2\}$ and so $x,z,x_1,w_1\in D$. It follows that
$|D|\geq 4$, moreover $D\subseteq N[w_1]$. Since $G-\{u,x_1,x_2\}$ is a WNT, Lemma \ref{osnovna} implies that $G-(\{u,x_1,x_2,w_1\}\cup T)$ is a WNT. Since $(\{u,x_1,x_2,w_1\}\cup T)\setminus D=\{u,x_2,w_2\}$ we find, by applying Lemma \ref{lepljenje}, that $G-D$ is a WNT, and therefore $G$ is reducible. (When applying
Lemma \ref{lepljenje} we set $X=\{u,x_1,x_2,w_1\}\cup T$, and we note that $X-D$ induces a triangle.) If $x$ and $w_2$ are the only $w_1$-bad vertices in $G-\{u,x_1,x_2\}$, then define $D=\{u,x_1,x_2,w_1,w_2,x\}$. Apply Lemma \ref{osnovna} to $G-\{u,x_1,x_2\}$ (recall that $G-\{u,x_1,x_2\}$ is a WNT), and its
problematic vertex $w_1$. It follows that $G-D$ is a WNT, because $x$ and $w_2$ are the only $w_1$-bad vertices in $G-\{u,x_1,x_2\}$. Reducibility of $G$ follows from $D\subseteq N[u]$. This proves that $w_1$ is not a $w_2$-bad vertex, or $w_2$ is not a $w_1$-bad vertex in $G-\{u,x_1,x_2\}$.
Assume, without loss of generality, that $w_2$ is not a $w_1$-bad vertex in $G-\{u,x_1,x_2\}$. Let $T$ be the set of all $w_1$-bad vertices in $G-\{u,x_1,x_2\}$,
and suppose first that $|T|\geq 2$. Define $D=\{x_1,w_1\}\cup T$. By Lemma \ref{osnovna}, $G-(\{u,x_1,x_2,w_1\}\cup T)$ is a WNT, and therefore by Lemma \ref{lepljenje}, $G-D$ is a WNT if and only if $u$ and $x_2$ are contained in a triangle of $G-D$. Since $ux_2w_2$ is a triangle in $G-D$, we conclude that $G-D$ is a WNT.
Moreover $D\subseteq N[w_1]$ and $|D|\geq 4$, therefore $G$ is reducible. Suppose now that there is exactly one $w_1$-bad vertex in $G-\{u,x_1,x_2\}$. Call it $z$, and note that $z\neq w_2$, and that $G-\{u,x_1,x_2,w_1,z\}$ is a WNT by Lemma \ref{osnovna}.
The edge $w_1z$ is incident to a facial triangle in $G$ (and in $G-\{u,x_1,x_2\}$). Suppose that there exists a vertex $t\neq w_2$ such that $w_1zt$ is a facial triangle. If $t$ is not problematic in $G-\{u,x_1,x_2,w_1,z\}$ then let $D=\{x_1,w_1,z,t\}$. Since $G-\{u,x_1,x_2,w_1,z,t\}$ is a WNT, and vertices $u$ and $x_2$ are contained in the triangle $ux_2w_2$ of $G-D$, we find by Lemma \ref{lepljenje}, that $G-D$ is a WNT. So $G$ is reducible, because $D\subseteq N[w_1]$. If $t$ is problematic in $G-\{u,x_1,x_2,w_1,z\}$, then let $T$ be the set of all $t$-bad vertices in $G-\{u,x_1,x_2,w_1,z\}$. Define $D=\{w_1,z,t\}\cup T$. By Lemma \ref{osnovna}, $G-\{u,x_1,x_2,w_1,z,t\}\cup T$ is a WNT, and so $G-D$ is a WNT by Lemma \ref{lepljenje}, due to the
fact that vertices $u,x_1,x_2$ are contained in a triangle of $G-D$. Since $D\subseteq N[t]$ and $|D|\geq 4$, we find that $G$ is reducible. In both cases we found that $G$ is reducible, so assume that the only facial triangle incident to $w_1z$ is the triangle $w_1zw_2$. We discuss two possibilities.
First is when $z$ is in the interior of the triangle $w_1w_2u$. Then $w_1z$ is an internal edge of $G$, so it's contained in two facial triangles. If one of them is $w_1zu$, then $z$ is adjacent to $u$ in $G$. In this case define $D=\{u,x_1,x_2,w_1,z\}$ and observe that $G-D$ is a WNT, and that $D\subseteq N[u]$, hence $G$ is reducible. Otherwise, if $w_1zu$ is not a facial triangle, there exists a vertex $t\neq w_2$ such that $w_1zt$ is a facial triangle, contradicting the assumption that $w_1zw_2$ is the only facial triangle containing the edge $w_1z$.
The second possibility is that $z$ is in the exterior of triangle $w_1w_2u$. Since $w_1zw_2$ is the only facial triangle incident to $w_1z$ we find that $w_1z$ is an external edge of $G$. Similarly, since $z$ is a $w_1$-bad vertex in $G-\{u,x_1,x_2\}$, it is not contained in a triangle of $G-\{u,x_1,x_2,w_1\}$, and so $zw_2$ is an external edge of $G$. It follows that edges $w_2x_2,x_2x_1,x_1w_1,w_1z,zw_2$ are external edges of $G$, and so the union of these edges is the boundary of $B$. If there is no vertex in the interior of the triangle $w_1w_2u$, then $B$ is a block of order 6. Otherwise $u$ is adjacent to a vertex $y$ in the interior of $w_1w_2u$.
If $y$ is not problematic in $G-\{u,x_1,x_2\}$, then let $D=\{u,x_1,x_2,y\}$. $G-D$ is a WNT, by the definition of a problematic vertex, and since $D\subseteq N[u]$, $G$ is reducible. Assume therefore that $y$ is problematic in
$G-\{u,x_1,x_2\}$. We apply Lemma \ref{trikotnik}, where $X=\{u,x_1,x_2\}$ and where the triangle $uw_1w_2$ takes the role of $uvw$ in Lemma \ref{trikotnik}. The lemma implies that every $y$-bad vertex in $G-\{u,x_1,x_2\}$ is adjacent to $u$ in $G$. Let $T$ be the set of all $y$-bad vertices in $G-\{u,x_1,x_2\}$. Define $D=\{u,x_1,x_2,y\}\cup T$, and note that $D\subseteq N[u]$.
By Lemma \ref{osnovna}, $G-D$ is a WNT, therefore $G$ is reducible.
\subsubsection{$x_1$ and $x_2$ are not adjacent, and $u$ is their only common neighbor}
Let $w_1,z_1,w_2,z_2$ be pairwise distinct vertices such that
$ux_1w_1$ and $ux_1z_1$ are facial triangles containing $ux_1$, and $ux_2w_2$ and $ux_2z_2$ are facial triangles containing $ux_2$. Note that $x_1$ and $x_2$ are external vertices of $G$, and therefore $w_1$ is not adjacent to $z_1$. Moreover, there is no path in $G$ that avoids $x_1,x_2$ and $u$ between $w_1$ and $z_1$ (again, because $x_1$ and $x_2$ are external vertices). Similarly there is no path in $G$ that avoids $x_1,x_2$ and $u$ between $w_1$ and $w_2$, or there is no path in $G$ that avoids $x_1,x_2$ and $u$ between $w_1$ and $z_2$ (if both paths exist, then $x_1$ or $x_2$ is not an external vertex). Assume, without loss of generality, the latter. This, in particular, implies that $w_1$ is not adjacent to $z_2$ in $G$.
If $w_1$ is not problematic in $G-\{u,x_1,x_2\}$, then we define $D=\{u,x_1,x_2,w_1\}$. Since $D\subseteq N[u]$ and $G-D$ is a WNT, $G$ is reducible. Assume therefore that $w_1$ is problematic in $G-\{u,x_1,x_2\}$. If $w_1$ has at least two $w_1$-bad vertices in $G-\{u,x_1,x_2\}$ then let $T$ be the set of $w_1$-bad vertices in $G-\{u,x_1,x_2\}$ and define $D=\{w_1,x_1\}\cup T$. Since $w_1$ is not adjacent to $z_2$, we find that $z_2\notin T$. It follows that $ux_2z_2$ is a triangle of $G-D$. We apply Lemma \ref{lepljenje} to sets $X=\{u,x_1,x_2,w_1\}\cup T$ and $D$. Since
$G-X$ is a WNT, and every vertex of $X\setminus D$ is contained in a triangle of $G-D$, we find that $G-D$ is a WNT. Since $D\subseteq N[w_1]$ and $|D|\geq 4$, $G$ is reducible. Assume now that there is exactly one $w_1$-bad vertex in $G-\{u,x_1,x_2\}$, call this vertex $z$. Let $w_1zt$ be a facial triangle. If $t$ is not problematic in $G-\{u,x_1,x_2,w_1,z\}$, then let $D=\{x_1,w_1,z,t\}$. Since $z_2$ is not adjacent to $w_1$, we find that both $z$ and $t$ are distinct from $z_2$.
It follows that $u$ and $x_2$ are contained in a triangle of $G-D$, the triangle $ux_2z_2$. By Lemma \ref{lepljenje}, $G-D$ is a WNT, moreover $D\subseteq N[w_1]$. Therefore $G$ is reducible. The last possibility is that $t$ is problematic in $G-\{u,x_1,x_2,w_1,z\}$. In this case let $T$ be the set of all $t$-bad vertices in $G-\{u,x_1,x_2,w_1,z\}$, and define $D=\{t,z,w_1\}\cup T$. Since there is no path between $w_1$ and $z_2$ in $G$ that avoids $x_1,x_2$ and $u$, we find that $z_2\notin T\cup\{z,t\}$. Analogous arguments prove that
$z_1\notin T\cup\{z,t\}$. It follows that vertices $u,x_1,$ and $x_2$ are contained in triangles of $G-D$, these are triangles $ux_1z_1$ and $ux_2z_2$. By Lemma \ref{osnovna}, $G-(\{u,x_1,x_2,w_1,z,t\}\cup T)$ is a WNT. We apply Lemma \ref{lepljenje} to find that $G-D$ is a WNT. Moreover $D\subseteq N[t]$, so $G$ is reducible.
\subsection{There is exactly one $u$-bad vertex in $G$ }
Suppose that there is exactly one $u$-bad vertex in $G$, call it $x_1$. By Lemma \ref{osnovna1}, $G-\{u,x_1\}$ is a WNT. Let $ux_1w_1$ and $ux_1z_1$ be facial triangles containing the edge $ux_1$. Note that $x_1z_1$ and $x_1w_1$ are external edges of $G$, and therefore $w_1$ and $z_1$ are external vertices of $G$. If $w_1$ is problematic in $G-\{u,x_1\}$, then let $T$ be the set of all $w_1$-bad vertices in $G-\{u,x_1\}$. Define $D=\{u,x_1,w_1\}\cup T$. By Lemma \ref{osnovna}, $G-D$ is a WNT. Since $D\subseteq N[w_1]$, $G$ is reducible.
Suppose that $w_1$ is not problematic in $G-\{u,x_1\}$, and suppose additionaly that $z_1$ is not problematic in $G-\{u,x_1,w_1\}$. Then define $D=\{u,x_1,w_1,z_1\}$, and observe that $G-D$ is a WNT. Since $D\subseteq N[u]$, $G$ is a reducible.
It remains to prove that $G$ is reducible if $z_1$ is problematic in $G-\{u,x_1,w_1\}$. If $z_1$ is adjacent to $w_1$ in $G$, then let $T$ be the set of all $z_1$-bad vertices in $G-\{u,x_1,w_1\}$. Define $D=\{u,w_1,x_1,z_1\}\cup T$ and observe that $D\subseteq N[z_1]$. By Lemma \ref{osnovna}, $G-D$ is a WNT, and so $G$ is reducible. From now on we assume that $w_1$ and $z_1$ are not adjacent in $G$.
If $\deg_B(w_1)=3$, then $w_1$ is contained in a triangle of $G-\{u,x_1\}$, say $w_1ab$, such that $a,b\notin B$. In this case let $T$ be the set of all $z_1$-bad vertices in $G-\{u,x_1,w_1\}$, and define $D=\{u,x_1,z_1\}\cup T$. Since $G-(\{u,x_1,w_1,z_1\}\cup T)$ is a WNT, also $G-D$ is a WNT by Lemma \ref{lepljenje} (note that $a,b\notin T$, and so $w_1$ is contained in a triangle of $G-D$). Since $D\subseteq N[z_1]$, we find that $G$ is reducible. Therefore $\deg_B(w_1)\geq 4$.
Now we can prove the following claim: every $z_1$-bad vertex in $G-\{u,x_1,w_1\}$ is adjacent to $w_1$ in $G$. If not, then the set of $z_1$-bad vertices in $G-\{u,x_1,w_1\}$ that are not adjacent to $w_1$ in $G$ is nonempty, call this set $T$. Also, let $X$ be the set that contains $u,x_1,w_1,z_1$ and all $z_1$-bad vertices in $G-\{u,x_1,w_1\}$. By Lemma \ref{osnovna}, $G-X$ is a WNT. Define $D=\{u,x_1,z_1\}\cup T$, and observe that $G-D$ contains all neighbors of $w_1$ in $G$, except $x_1$ and $u$. Moreover, every vertex (distinct from $w_1$) in $X\setminus D$ is adjacent to $w_1$ in $G-D$. Since $\deg_B(w_1)\geq 4$, and $w_1x_1$ is an external edge of $B$, and $ux_1w_1$ is a facial triangle, we find that every vertex in $X\setminus D$ is contained in a triangle of $G-D$. Hence, by Lemma \ref{lepljenje}, $G-D$ is a WNT. Since $D\subseteq N[z_1]$, $G$ is reducible. This proves the claim.
If every $z_1$-bad vertex in $G-\{u,x_1,w_1\}$ is adjacent to $u$, then let $T$ be the set of all $z_1$-bad vertices in $G-\{u,x_1,w_1\}$. Define $D=\{u,x_1,w_1,z_1\}\cup T$. Since $D\subseteq N[u]$, and $G-D$ is a WNT, $G$ is reducible. Assume therefore that a $z_1$-bad vertex is not adjacent to $u$.
Let $y$ be a $z_1$-bad vertex in $G-\{u,x_1,w_1\}$ not adjacent to $u$ in $G$. Since $w_1z_1\notin E(G)$ and $yu\notin E(G)$ there exist exactly one vertex adjacent to $y$ in the interior of 4-cycle $yw_1uz_1$ (for otherwise $y$ is not a $z_1$-bad vertex in $G-\{u,x_1,w_1\}$). Call this vertex $z$ and observe that $zyw_1$ and $zyz_1$ are facial triangles. If $z$ and $u$ are not adjacent then let $w_1zy_1$ be the facial triangle such that $y_1\neq y$. Let $T$ be the set of all $z_1$-bad vertices in $G-\{u,w_1,x_1\}$, and define $D=(\{u,x_1,z_1\}\cup T)\setminus \{z,y_1\}$. Since $G-\{u,w_1,x_1,z_1\}\cup T$ is a WNT, also $G-D$ is a WNT (by Lemma \ref{lepljenje}).
As $y\in D$, we have $|D|\geq 4$ and so $G$ is reducible. It follows that $z$ and $u$ are adjacent. Moreover the triangle $w_1uz$ is a facial triangle (we do the same reduction as above if $w_1uz$ is not a facial triangle).
Suppose that there is a vertex $x$ in the interior of triangle $uzz_1$, such that $x$ is adjacent to $u$ in $G$. If $x$ is not problematic in $G-\{u,x_1,w_1\}$, then let $D=\{u,x_1,w_1,x\}$ and observe that $G-D$ is a WNT. Since $D\subseteq N[u]$, $G$ is reducible. Suppose that $x$ is problematic in $G-\{u,x_1,w_1\}$. By Lemma \ref{trikotnik}, every $x$-bad vertex in $G-\{u,x_1,w_1\}$ is adjacent to $u$.
Let $T$ be the set of all $x$-bad vertices in $G-\{u,x_1,w_1\}$. Define $D= \{u,x_1,w_1,x\}\cup T$ and observe that $D\subseteq N[u]$. By Lemma \ref{osnovna}, $G-D$ is a WNT, and therefore $G$ is reducible. It follows that $uz_1z$ is a facial triangle of $G$, and therefore $z$ is also a $z_1$-bad vertex in $G-\{u,x_1,w_1\}$.
If the set $T$ of all $z_1$-bad vertices in $G-\{u,x_1,w_1\}$ contains more than two vertices, then let $D=(\{u,x_1,z_1\}\cup T)\setminus \{z,y\}$. Since $zyw_1$ is a triangle in $G-D$ we find, by applying Lemma \ref{lepljenje}, that $G-D$ is a WNT. Since $D\subseteq N[z_1]$, $G$ is reducible. Assume therefore that $z$ and $y$ are the only $z_1$-bad vertices in $G-\{u,x_1,w_1\}$, and so $G-\{u,x_1,w_1,z_1,z,y\}$ is a WNT. If $w_1y$ is not an external edge of $G$, then define $D=\{u,x_1,z_1,z\}$. Since $w_1y$ is not an externl edge of $G$, $y$ and $w_1$ are contained in a triangle of $G-D$. Lemma \ref{lepljenje} implies that $G-D$ is a WNT. Reducibility of $G$ follows from $D\subseteq N[z_1]$. Assume therefore that $yw_1$ is an external edge of $G$. If $yz_1$ is not an external edge of $G$, then define $D=\{u,x_1,w_1,z\}$. Since $y$ and $z_1$ are contained in a triangle of $G-D$ (note that there is a facial triangle $yz_1t$, where $t\neq z$, and $t\neq w_1$ because $z_1$ is not adjacent to $w_1$, and $t\neq x_1$ because $z_1$ is an external vertex of $G$), we find that $G-D$ is a WNT. Reducibility follows from $D\subseteq N[u]$. Hence also $z_1y$ is an external edge of $G$, and therefore all four edges $w_1x_1,x_1z_1,z_1y$ and $yw_1$ are external edges of $G$. So $B$ is a block of order 6.
\subsection{There are no $u$-bad vertices in $G$ } \label{uniproblematicen}
Suppose that $u$ has no $u$-bad vertices in $G$. Let $u_1$ be any external neighbor of $u$ in $G$. Since $u$ is not problematic in $G$, $u_1$ is contained in a triangle of $G-u$. Note first that $\deg_B(u_1)>2$, because $u_1$ is adjacent to $u$ in $B$, and $u$ is an internal vertex of $B$.
We claim the following: if $\deg_B(u_1)=3$ then $G$ is reducible or $u_1$ is contained in a triangle $u_1ab$, where $a,b\notin B$. Suppose that $\deg_B(u_1)=3$. Let $z_1, z_2$ and $u$ be neighbors of $u_1$ in $B$, where $z_1$ and $z_2$ are external vertices of $B$. Since $u_1$ is contained in a triangle of $G-u$ we find that $u_1$ is contained in a triangle $u_1ab$, where $a,b\notin B$, or $z_1$ is adjacent to $z_2$. Suppose that $z_1$ is adjacent to $z_2$ and that $u_1$ is not contained in a triangle $u_1ab$, where $a,b\notin B$. Then every block $B'\neq B$ of $G$ containing $u_1$ is a $K_2$, and therefore $u_1z_1z_2$ is the only triangle containing $u_1$ in $G-u$. It follows that $z_1$ and $z_2$ are the only vertices that are potentially not in a triangle of $G-\{u,u_1\}$.
Suppose that $\deg(u)\geq 4$ or $z_1z_2$ is an internal edge of $G$. Then all vertices of $G-\{u,u_1\}$ are contained in a triangle of $G-\{u,u_1\}$, and so (note that by Corollary \ref{trikotnalica} all bounded faces of $G-\{u,u_1\}$ are triangular), $G-\{u,u_1\}$ is a WNT. In this case the reductions are defined as follows. If $z_1$ is problematic in $G-\{u,u_1\}$, then let $T$ be the set of all $z_1$-bad vertices in $G-\{u,u_1\}$, and define $D=\{u,u_1,z_1\}\cup T$. Since $G-D$ is a WNT and $D\subseteq N [z_1]$, $G$ is reducible. If $z_1$ is not problematic in $G-\{u,u_1\}$ and $z_2$ is not problematic in $G-\{u,u_1,z_1\}$, then let $D=\{u,u_1,z_1,z_2\}$. Since $D\subseteq N[u]$, $G$ is reducible. If $z_2$ is problematic in $G-\{u,u_1,z_1\}$, then let $T$ be the set of all $z_2$-bad vertices in $G-\{u,u_1,z_1\}$. Define $D=\{u,u_1,z_1,z_2\}\cup T$. Since $z_1$ is adjacent to $z_2$, $D\subseteq N[z_2]$. Since $G-D$ is a WNT, $G$ is reducible.
It remains to define reductions when $z_1z_2$ is an external edge of $G$ and $\deg(u)=3$. In this case $B$ is a $K_4$ (recall that $\deg_B(u_1)=3$, therefore $z_1u_1$ and $z_2u_1$ are external edges of $B$), and $G-u$ is a WNT.
If there is a $z_1$-bad vertex $t\neq u_1$ in $G-u$, then let $T$ be the set of all $z_1$-bad vertices in $G-u$, and define $D=\{u,z_1\}\cup T$. Since $u_1\in T$ we have $|T|\geq 2$, and
therefore $|D|\geq 4$. Since $G-u$ is a WNT, we find (by applying Lemma \ref{osnovna}) that $G-D$ is a WNT. Reducibility follows from $D\subseteq N[z_1]$. Therefore $u_1$ is the only $z_1$-bad vertex in $G-u$, and similarly $u_1$ is also the only $z_2$-bad vertex in $G-u$. It follows that there exist blocks $B_1$ and $B_2$ of $G$, both different from $B$, and both near-triangulations, such that $z_1\in B_1$ and $z_2\in B_2$.
So we may define $D=\{u,u_1,z_1,z_2\}$. Observe that $G-D$ is a WNT because none of the vertices $u_1,z_1$ and $z_2$ have a bad vertex in $G-u$ which is not contained in $B$. Since $D\subseteq N[u]$, $G$ is reducible. This proves the claim, and therefore it remains to define reductions in case if $\deg_B(u_1)\geq 4$, or $\deg_B(u_1)=3$ and $u_1$ is contained in a triangle $u_1ab$, where $a,b\notin B$. Assume in the sequal that $\deg_B(u_1)\geq 4$, or $\deg_B(u_1)=3$ and $u_1$ is contained in a triangle $u_1ab$, where $a,b\notin B$, moreover assume that this is true for any external neighbor $u_1$ of $u$. This in particular implies that every vertex of $G-u$ is contained in a facial triangle of $G-u$ which is also a facial triangle of $G$ (or equvalently: every vertex of $G-u$ is contained in a facial triangle of $G$ not containing $u$).
\subsubsection{$u_1$ is not problematic in $G-u$}
Suppose that $u_1$ is not problematic in $G-u$, and therefore $G-\{u,u_1\}$ is a WNT. The edge $uu_1$ is incident to two triangular faces of $G$. So there are vertices $x$ and $y$ such that $uu_1x$ and $uu_1y$ are facial triangles in $G$. By the choice of $u$ (see Section \ref{miki}), at least one of $x$ and $y$ is an external vertex of $B$ (and $G$), for otherwise more than one region $R_k, k\leq n$ contains an internal vertex of $B$. Assume that $x$ is an external vertex. If $u_1y$ is an external edge of $G$, then let $u_2=y$, otherwise let $u_2=x$.
We claim that $u_1$ is contained in a triangle of $G-\{u,u_2\}$. If $\deg_B(u_1)=3$ then $u_1$ is contained in a triangle $u_1ab$, where $a,b\notin B$ (in which case the claim is true). Assume now that $\deg_B(u_1)\geq 4$. Now if $u_2=y$ then $u_1u_2$ is an external edge of $B$, and $uu_1u_2$ is a facial triangle. Since $\deg_B(u_1)\geq 4$ and $B$ is a near-triangulation we find that $u_1$ is incident to at least three facial triangles of $B$. When we remove $u$ and $u_1$ from $G$ at least one facial triangle containing $u_1$ remains. If $u_2=x$ and $u_1x$ is an external edge, then the argument is the same as above. So assume that $u_1x$ is also not an external edge (we already know that $u_1y$ is not an external edge). Now in this case $\deg_B(u_1)\geq 5$. Removing vertices $u$ and $u_2$ (which form a facial triangle with $u_1$) keeps at least one facial triangle containing $u_1$. This proves the claim.
If $u_2$ is a problamatic vertex in $G-\{u,u_1\}$, then let $T$ be the set of all $u_2$-bad vertices in $G-\{u,u_1\}$. Since $G-\{u,u_1\}$ is a WNT, it follows from
Lemma \ref{osnovna} that $G-(\{u,u_1,u_2\}\cup T)$ is a WNT. We define $D=\{u,u_1,u_2\}\cup T$. Since $D\subseteq N[u_2]$, $|D|\geq 4$ and $G-D$ is a WNT, we find that $G$ is reducible.
Assume that $u_2$ is not problematic in $G-\{u,u_1\}$ and so $G-\{u,u_1,u_2\}$ is a WNT. Let $t\neq u_1$ be such that $uu_2t$ is a facial triangle. If $t$ is not problematic in $G-\{u,u_1,u_2\}$ then let $D=\{u,u_1,u_2,t\}$. We have $D\subseteq N[u]$ and $G-D$ is a WNT, so $G$ is reducible. So assume that $t$ is problematic in $G-\{u,u_1,u_2\}$.
Let $T$ be the set of all $t$-bad vertices in $G-\{u,u_1,u_2\}$. If $t$ and $u_1$ are adjacent in $G$ then let $D=\{u,u_1,u_2,t\}\cup T$. We find that $G-D$ is a WNT according to Lemma \ref{osnovna}. Moreover $D\subseteq N[t]$, and so $G$ is reducible. Assume therefore that $t$ is not adjacent to $u_1$. If no vertex of $T$ is adjacent to $u_1$ then let $D=\{u,u_2,t\}\cup T$. It follows from Lemma \ref{lepljenje} and the fact that $u_1$ is contained in a triangle of $G-\{u,u_2\}$ (and so also in a triangle of $G-D$) that $G-D$ is a WNT. Since $D\subseteq N[t]$, $G$ is reducible. Assume from now on that a vertex of $T$ is adjacent to $u_1$, and call this vertex $t_0$.
{\bf Case $(a)$.} Suppose that $t_0$ is not adjacent to $u$ in $G$. Then the interior of 4-cycle $t_0u_1ut$ contains exactly one vertex adjacent to $t_0$, call it $z$ (if the interior of $t_0u_1ut$ contains two vertices adjacent to $t_0$, then $t_0$ is contained in a triangle of $G-\{u,u_1,u_2,t\}$, and hence $t_0$ is
not a $t$-bad vertex in $G-\{u,u_1,u_2\}$). If $z$ and $u$ are not adjacent in $G$, then $u_1$ is contained in a triangle $u_1zz'$, where $z'$ lies in the interior of $u_1utz$. Let $T$ be the set of all $t$-bad vertices in $G-\{u,u_1,u_2\}$, and define $D=\{u,u_2,t\}\cup T\setminus\{z,z'\}$. Since $G-(\{u,u_1,u_2,t\}\cup T)$ is a WNT, we find (by applying Lemma \ref{lepljenje}) that
$G-D$ is a WNT. Since $t_0\in D$, we have $|D|\geq 4$. Moreover$D\subseteq N[t]$, and therefore $G$ is reducible.
\begin{figure}
\caption{Case $(a)$: $t_0$ is not adjacent to $u$, Case $(b)$: $t_0$ is adjacent to $u$.}
\label{primeri}
\end{figure}
Assume therefore that $z$ and $u$ are adjacent in $G$ (see Fig.~\ref{primeri}, Case $(a)$). If the interior of the triangle $u_1uz$ contains a vertex, then $u_1$ is contained in a triangle $u_1zz'$, where $z'$ lies in the interior of $u_1uz$. In this case we can do the same reduction as in the previous case, by defining $D=\{u,u_2,t\}\cup T\setminus\{z,z'\}$. We may therefore assume that there is no vertex of $G$ in the interior of $uu_1z$.
Suppose that $y$ is a vertex of $G$ in the interior of $utz$ adjacent to $u$. If
$y$ is not problematic in $G-\{u,u_1,u_2\}$, we define $D=\{u,u_1,u_2,y\}$. Since $G-D$ is a WNT, and $D\subseteq N[u]$, $G$ is reducible. Otherwise, if $y$ is problematic in $G-\{u,u_1,u_2\}$, then by Lemma \ref{trikotnik}, every $y$-bad vertex in $G-\{u,u_1,u_2\}$ is adjacent to $u$ in $G$. In this case let $T$ be the set of all $y$-bad vertices in $G-\{u,u_1,u_2\}$ and define $D=\{u,u_1,u_2,y\}\cup T$. Since $G-D$ is a WNT, and $D\subseteq N[u]$, we find that $G$ is reducible. This proves that there is no vertex of $G$ in the interior of $utz$. Observe that we proved that all triangles on Fig.~\ref{primeri}, Case $(a)$, are facial triangles.
Let $T$ be the set of all $t$-bad vertices in $G-\{u,u_1,u_2\}$, and let $X=\{u,u_1,u_2,t\}\cup T$. If $|T|\geq 3$, then $D=X\setminus \{u_1,z,t_0\}$. Since $G-X$ is a WNT, also $G-D$ is a WNT, according to Lemma \ref{lepljenje}. Since $D\subseteq N[t]$, and $D\geq 4$, $G$ is reducible. Assume therefore that $z$ and $t_0$ are the only $t$-bad vertices in $G-\{u,u_1,u_2\}$, and so $X=\{ u,u_1,u_2,t,z,t_0\}$.
If $u_1t_0$ is not an external edge of $G$, there exists a facial triangle $u_1t_0x$, with $x\neq z$, $x\neq t$ (recall that $u_1$ is not adjacent to $t$) and $x\neq u_2$ (recall that $u_1$ is an external vertex). In this case we define $D=X\setminus \{u_1,t_0\}=\{u_2,u,t,z\}$. Since every vertex of $X-D$ is contained in a triangle of $G-D$ we find, by applying Lemma \ref{lepljenje}, that
$G-D$ is a WNT. Since $D\subseteq N[u]$, $G$ is reducible. It follows that $u_1t_0$ is an external edge. With an analogous argument we prove that $u_1u_2$ is an external edge. If $u_2t$ is not an external edge of $G$, then there is a facial triangle $u_2tx$, with $x\neq u$ and $x\neq u_1$ (recall that $u_1$ is not adjacent to $t$). If $x=t_0$, then $u_2$ and $t_0$ are adjacent in $G$. If $u_2t_0$ is an external edge of $G$, then all three edges $u_1u_2,u_2t_0$ and $t_0u_1$ are external edges of $G$, and therefore $B$ is a block of order 6 in $G$. Otherwise, if $u_2t_0$ is an internal edge, then there exists an $x\notin \{t, u_1\}$ such that $u_2t_0x$ is a facial triangle. In this case we define $D=X\setminus \{u_2,t_0\}$. Since $G-X$ is
a WNT, also $G-D$ is a WNT. Moreover $D\subseteq N[u]$, so $G$ is reducible. If $x\neq t_0$, then define $D=\{u,u_1,z,t_0\}$. Since $u_2$ and $t$ are contained in a triangle of $G-D$, we find that $G-D$ is a WNT. Since $D\subseteq N[u_1]$, we find that $G$ is reducible. Therefore we may assume that $u_2t$ is also an external edge of $G$. Finally we argue that $t_0t$ is an external edge of $G$. If not, then there is a facial triangle $tt_0x$, with $x\neq z$. If $x=u_2$, then $u_2t$ is not an external edge, contradicting the above assumption. Otherwise $x\neq u_2$ and $x\neq u_1$. So we define $D=X\setminus \{t,t_0\}$ and argue that $G-D$ is a WNT by refering to Lemma \ref{lepljenje}. Altogether, we proved that $u_1u_2,u_2t,tt_0,t_0u_1$ are external edges of $G$, therefore $B$ contains $u,u_1,u_2,t,z,t_0$ and no other vertices, hence $B$ is a block of order 6.
{\bf Case $(b)$.} Now we discuss the case when $t_0$ and $u$ are adjacent in $G$ (see Fig.~\ref{primeri}, Case $(b)$). If there is a vertex in the interior of $u_1ut_0$, then there is exactly one such vertex, for otherwise either $t_0$ is contained in a triangle of $G-\{u,u_1,u_2,t\}$ (and so $t_0$ is not a $t$-bad vertex) or there is a triangle $u_1zz'$, where $z$ and $z'$ are in the interior of $u_1ut_0$ (note that in this case $z$ and $z'$ are not adjacent to $t$). If the latter happens, then let $T$ be the set of all $t$-bad vertices in $G-\{u,u_1,u_2\}$ and define $D=(\{u,u_2,t\}\cup T)$. Since $u_1zz'$ is a triangle in $G-D$, we find that $G-D$ is a WNT (by Lemma \ref{lepljenje}). Since $D\subseteq N[t]$, we find that $G$ is reducible. If there is exactly one vertex in the interior of $u_1ut_0$, then this vertex is a $u_1$-bad vertex in $G-u$, and so $u_1$ is problematic in $G-u$. This case is treated in Section \ref{u1jeproblematicen}. Assume therefore that there is no vertex in the interior of $u_1ut_0$, so we have Case $(b)$ of Fig.~\ref{primeri}, where $B$ potentially contains some vertices of $G$.
Suppose that $t_0$ is the only $t$-bad vertex in $G-\{u,u_1,u_2\}$. In this case let $D=\{u,u_1,u_2,t,t_0\}$. By Lemma \ref{osnovna}, $G-D$ is a WNT. Since $D\subseteq N[u]$, $G$ is reducible. Assume therefore that there are at least two $t$-bad vertices in $G-\{u,u_1,u_2\}$.
If $t_0u_1$ is not an external edge of $G$, then there is a facial triangle $t_0u_1x$, with $x\neq u$ and $x\neq t$ (recall that $u_1$ is not adjacent to $t$) and $x\neq u_2$ (recall that $u_1$ is an external vertex). If $x$ is a $t$-bad vertex in $G-\{u,u_1,u_2\}$
then we get case $(a)$ of Fig.~\ref{primeri}. Assume therefore that $x$ is not a $t$-bad vertex in $G-\{u,u_1,u_2\}$. Let $T$ be the set of all
$t$-bad vertices in $G-\{u,u_1,u_2\}$, $X=\{u,u_1,u_2,t\}\cup T$ and $D=X\setminus \{u_1,t_0\}$. By Lemma \ref{osnovna} we find that $G-X$ is a WNT, and therefore, by applying Lemma \ref{lepljenje} we see that $G-D$ is a WNT. Reducibility of $G$ follows from $D\subseteq N[t]$ and $|D|\geq 4$. We conclude that $u_1t_0$ is an external edge of $G$.
If $u_1u_2$ is not an external edge of $G$, then there is a facial triangle $u_1u_2x$, with $x\neq u$, $x\neq t$ and $x\neq t_0$. Moreover $x$ is not adjacent to $t$ because $u_1$ and $u_2$ are external vertices of $G$. In this case we define the reduction by $D=X\setminus \{u_1,u_2\}$. This proves that both edges $t_0u_1$ and $u_1u_2$ are external edges of $G$. It follows that $\deg_B(u_1)=3$, and so $u_1$ is contained in a triangle $u_1ab$, with $a,b\notin B$ (see the assumption at the end of Section \ref{uniproblematicen}). Observe that $a$ and $b$ are not adjacent to $t$ (because $t\in B$).
We define the reduction by $D=X\setminus \{u_1\}$. Since $u_1$ is contained in a triangle of $G-D$ we see that $G-D$ is a WNT, and so the reduction is well defined.
\subsubsection{$u_1$ is problematic in $G-u$.}\label{u1jeproblematicen}
Note that there are no $u$-bad vertices in $G$, and so every vertex of $G-u$ is contained in a triangle of $G-u$.
Suppose that $u_1$ is problematic in $G-u$. If there are at least two $u_1$-bad vertices in $G-u$, then let $T$ be the set of all $u_1$-bad vertices in $G-u$, and define $D=\{u,u_1\}\cup T$. Since every vertex of $G-u$ is contained in a triangle of $G-u$, we find that every $u_1$-bad vertex in $G-u$ is adjacent to $u_1$. By Corollary \ref{trikotnalica} every bounded face of $G-D$ is triangular. Moreover, removing $u_1$ and all $u_1$-bad vertices from $G-u$ produces a graph in which all vertices are contained in a triangle.
Therefore $G-D$ is a WNT, and since $D\subseteq N[u_1]$ and $|D|\geq 4$, $G$ is reducible. Assume therefore that there is exactly one $u_1$-bad vertex $x$ in $G-u$, and observe that the same arguments as above prove that $G-\{u,u_1,x\}$ is a WNT. Since every vertex of $G-u$ (and in particular $x$) is contained in a facial triangle of $G$ not containing $u$ (see the assumption at the end of Section \ref{uniproblematicen}), there exists a facial triangle $u_1xt$ in $G$, where $t\neq u$. (since $x$ is a $u_1$-bad vertex in $G-u$, both vertices $u_1$ and $x$ are contained in the same facial triangle, and removing $u_1$ breaks this facial triangle).
First let us prove that $u$ and $t$ are not adjacent. If they are adjacent, then let $T$ be the (possibly empty) set of all $t$-bad vertices in $G-\{u,u_1,x\}$ and define $D=\{u,u_1,x,t\}\cup T$. Since $D\subseteq N[t]$ and $G-D$ is a WNT we find that $G$ is reducible. This proves that $u$ is not adjacent to $t$.
If $t$ is not problematic in $G-\{u,u_1,x\}$, then let $D=\{u,u_1,x,t\}$. By the defintion (of a problematic vertex) $G-D$ is a WNT, moreover $D\subseteq N[u_1]$, so $G$ is reducible. Assume therefore that $t$ is problematic in $G-\{u,u_1,x\}$. First we will assume that there is a $t$-bad vertex $t_0$ in $G-\{u,u_1,x\}$ adjacent to $u$.
\noindent {\bf Case 1: A $t$-bad vertex is adjacent to $u$.}
\noindent There is at most one vertex in the interior of the 4-cycle $uu_1tt_0$ adjacent to $t_0$. If not, $t_0$ is contained in a triangle of $G-\{u,u_1,x,t\}$, and so $t_0$ is not a $t$-bad vertex in $G-\{u,u_1,x\}$. Similarly, there is at most one vertex in the interior of the 4-cycle $uu_1tt_0$ adjacent to $u$. If not, $u$ is contained in a triangle $uw_1w_2$, where both $w_1$ and $w_2$ are in the interior of
$uu_1tt_0$. In this case let $T$ be the set of all $t$-bad vertices in $G-\{u,u_1,x\}$ and define $D=(\{u_1,x,t\}\cup T)\setminus \{w_1,w_2\}$. Since $\{u_1,x,t,t_0\}\in D$ we have $|D|\geq 4$, and since $G-(\{u,u_1,x,t\}\cup T)$ is a WNT, we find (by applying Lemma \ref{lepljenje}) that $G-D$ is a WNT. Therefore assume that either there is no vertex of $G$ in the interior of the 4-cycle $uu_1xt$ adjacent to $t_0$ and $u$, or there is exactly one vertex in the interior of $uu_1xt$ adjacent to $t_0$ and $u$ (and note that this must be the same vertex). Call this vertex (if it exists) $z$, and note that $u_1zu$, $uzt_0$ and $tzt_0$ are facial triangles in this case.
We claim that $t_0$ is the only $t$-bad vertex in $G-\{u,u_1,x\}$. Let us first exclude the possibility $|T|\geq 3$, where $T$ is the set of all $t$-bad vertices in $G-\{u,u_1,x\}$. The edge $uu_1$ is an internal edge of $G$, so there is a facial triangle $uu_1w$, with $w\neq t$.
If $|T|\geq 3$, let $D=(\{x,t\}\cup T)\setminus\{w\}$. $|D|\geq 4$ and since $G-(\{u,u_1,x,t\}\cup T)$ is a WNT we find (by
Lemma \ref{lepljenje}) that $G-D$ is a WNT. Since $D\subseteq N[t]$, $G$ is reducible. This leaves us with the possibility $|T|=2$, to be excluded next.
Let $t_0$
and $y$ be the only $t$-bad vertices in $G-\{u,u_1,x\}$. If there is no vertex in the interior of $uu_1tt_0$, then $u_1$ and $t_0$ are adjacent (recall that $u$ and $t$ are not adjacent). Let $uu_1w$ be the facial triangle, with $w\neq t_0$. If $w=y$, then let $D=\{u,u_1,x,t,t_0,y\}$. Since $D\subseteq N[u_1]$, $G$ is reducible. If $w\neq y$, the reduction is given by $D=\{x,t,t_0,y\}$. Since $w\neq t$ (recal that $t$ and $u$ are not adjacent), and $w\neq x$ (recall that $u_1$ is an external vertex of $G$) we find that $uu_1w$ is a triangle in $G-D$, and so $G-D$ is a WNT (by Lemma \ref{lepljenje}). Since $D\subseteq N[t]$, $G$ is reducible.
Suppose now that there is a vertex $z$ in the interior of $uu_1tt_0$ adjacent to $t_0, u,u_1$ and $t$. First assume that $z=y$. If $u_1$ and $t_0$ are adjacent, then define $D=\{u,u_1,x,t,t_0,y\}$. We have $D\subseteq N[u_1]$ and $G-D$ is a WNT, so $G$ is reducible. Now assume that $u_1$ and $t_0$ are not adjacent. In this case there is a facial triangle $uu_1w$, with $w\neq t_0$ and $w\neq y=z$. Also $w\neq x$ because $u_1$ is an external vertex of $G$, and $w\neq t$ since $u$ and $t$ are not adjacent. In this case we define $D=\{t,x,y,t_0\}$. Observe that $D\subseteq N[t]$ and that $uu_1w$ is a triangle in $G-D$, so $G-D$ is a WNT (by Lemma \ref{lepljenje}). This proves that
$G$ is reducible. It remains to check the case $z\neq y$. In this case we have
$D=\{x,t,t_0,y\}$, and so $uu_1z$ is a triangle in $G-D$. It follows that $G-D$ is a WNT (by Lemma \ref{lepljenje}). Reducibility of $G$ follows from $D\subseteq N[t]$. This proves that $t_0$ is the only $t$-bad vertex in $G-\{u,u_1,x\}$, as claimed.
If $u_1$ and $t_0$ are adjacent then $D=\{u,u_1,x,t,t_0\}$. Since $D\subseteq N[u_1]$, and $G-D$ is a WNT, $G$ is reducible. Suppose therefore that $u_1$ and $t_0$ are not adjacent. In this case there is a vertex $z$ is the interior of $uu_1tt_0$ adjacent to all four vertices of this 4-cycle. Since $ut_0$ is an internal edge, there is a facial triangle $ut_0w$, with $w\neq z$. We have $w\neq t$ (because $u$ is not adjacent to $t$), and $w\neq u_1$ (because $u_1$ and $t_0$ are not adjacent by the assumption) and $w\neq x$ (for otherwise $x$ is contained in the triangle $xtt_0$ in $G-\{u,u_1\}$, and so $x$ is not a $u_1$-bad vertex in $G-u$). If $z$ is not problematic in $G-\{u,u_1,x,t,t_0\}$ then $G-\{u,u_1,x,t,t_0,z\}$ is a WNT. Define $D=\{t,x,z,u_1\}$ and observe that $D\subseteq N[t]$. Since $ut_0w$ is a triangle in $G-D$ we find (by Lemma \ref{lepljenje}) that $G-D$ is a WNT. It remains to check the case when $z$ is problematic in $G-\{u,u_1,x,t,t_0\}$. Let $T$ be the set of all $z$-bad vertices in $G-\{u,u_1,x,t,t_0\}$. Define $D=\{z,u,t_0\}\cup T$. Since $G-\{u,u_1,x,t,t_0,z\}\cup T$ is a WNT, we find that $G-D$ is a WNT, because $u_1xt$ is a triangle in $G-D$. $G$ is reducible because $D\subseteq N[z]$.
\noindent {\bf Case 2: No $t$-bad vertex is adjacent to $u$.}
\noindent Assume now that no $t$-bad vertex is adjacent to $u$. We will prove that $x$ and $u$ are not adjacent in $G$. For the purpose of a contradiction assume that $x$ and $u$ are adjacent. Then there are two possible drawings of the graph induced by $u,u_1,x,t$: either the triangle $u_1xt$ lies in the interior of the triangle $u_1xu$, or it lies in the exterior. If the triangle $u_1xt$ lies in the interior of the triangle $u_1xu$ then the edge $xt$
is incident to a triangular face contained in the interior of the 4-cycle $u_1txu$. Let $xtw$ be the triangle bounding this triangular face. Since there are no multiple edges in $G$ and $u$ is not adjacent to $t$, we find that $w\notin \{u,u_1\}$. It follows that $x$ is contained in a traingle of $G-\{u,u_1\}$, a contradiction (recall that $x$ is a $u_1$-bad vertex in $G-u$).
Let's consider the case when triangle $u_1xt$ lies in the exterior of the triangle $u_1xu$. We claim that, the interior of $uu_1x$ contains at most one vertex of $G$. To see this let $w$ be a vertex in the interior of $uu_1x$ adjacent to $x$. The edge $xw$ is an internal edge, so it is incident to two facial triangles $xwy$ and $xwy'$. If $y$ or $y'$ is not $u$ or $u_1$, then $x$ is contained in a triangle of $G-\{u,u_1\}$, and so $x$ is not a $u_1$-bad vertex in $G-u$, a contradiction. Therefore $y=u_1$ and $y'=u$. If there is a vertex of $G$ in the interior of the tirangle $uu_1w$, then $u$ is contained in a triangle of $G-(\{u_1,x,t\}\cup T)$, where $T$ is the set of all $t$-bad vertices in $G-\{u,u_1,x\}$. Define $D=\{u_1,x,t\}\cup T$. By Lemma \ref{osnovna}, $G-(\{u,u_1,x,t\}\cup T)$ is a WNT, and therefore $G-D$ is a WNT by Lemma \ref{lepljenje}. This proves that there is no vertex in the interior of $uu_1w$, and so $w$ is the only vertex in the interior of triangle $uu_1x$, as claimed.
We now discuss both cases: there is a vertex in the interior of $uu_1x$, or there is no such vertex. Suppose that $w$ is (the only) vertex in the interior of triangle $uu_1x$. Then $w$ is a $u_1$-bad vertex in $G-u$, this contradicts that $x$ is the only $u_1$-bad vertex in $G-u$ (see assumptions in the second paragraph of Section \ref{u1jeproblematicen}). It remains to check the case when there is no vertex in the interior of $uu_1x$. In this case $uu_1x$ is a facial triangle, and if $\deg_G(u)\geq 4$, then $u$ is contained in a triangle of
$G-(\{u_1,x,t\}\cup T)$, because $t$ is not adjacent to $u$, and no vertex in $T$ is adjacent to $u$ (and because $u$ is an internal vertex of $G$). In this case we define $D=\{u_1,x,t\}\cup T$. Since $G-(\{u,u_1,x,t\}\cup T)$ is a WNT, we find by an application of Lemma \ref{lepljenje} that $G-D$ is a WNT. Since $D\subseteq N[t]$, $G$ is reducible. It follows that $\deg_G(u)=3$, and let $w\neq u_1,x$ be the third neighbor of $u$ in $G$. Clearly, $w$ is adjacent to $x$
and $u_1$, because $u$ is an internal vertex and $G$ is a WNT. Let $T$ be the set of all $w$-bad vertices in $G-\{u,u_1,x\}$ (if any), and define $D=\{u,u_1,x,w\}\cup T$. By Lemma \ref{osnovna}, $G-D$ is a WNT, and since $D\subseteq N[w]$, we conclude that $G$ is reducible. This proves that $u$ and $x$ are not adjacent in $G$.
Since no $t$-bad vertex in $G-\{u,u_1,x\}$ is adjacent to $u$ (and $u$ and $t$ are not adjacnet), we find that $u$ is contained in a triangle of $G-(\{u_1,x,t\}\cup T)$, where $T$ is the set of all $t$-bad vertices in $G-\{u,u_1,x\}$. We define $D=\{u_1,x,t\}\cup T$. Lemma \ref{lepljenje} proves that $G-D$ is a WNT. Since $D\subseteq N[t]$, we find that $G$ is reducible. This completes the proof of Theorem \ref{reducibilnost}.
\noindent {\bf Acknowledgement:} The author greatly appreciates discussions with Uro\v s Milutinovi${\rm \acute c}$ while working on this paper. The author is supported by research grant P1-0297 of Ministry of Education of Slovenia.
\end{document} | arXiv |
Systems of ODEs
1 The predator-prey model
2 Qualitative analysis of the predator-prey model
3 Solving the Lotka–Volterra equations
4 Vector fields and systems of ODEs
5 Discrete systems of ODEs
6 Qualitative analysis of systems of ODEs
7 The vector notation and linear systems
8 Classification of linear systems
9 Classification of linear systems, continued
This is the IVP we have considered so far: $$\frac{\Delta y}{\Delta t}=f(t,y),\ y(t_0)=y_0,$$ and $$y'=f(t,y),\ y(t_0)=y_0.$$ The equations show how the rate of change of $y$ depends on $t$ and $y$.
What if we have two variable quantities dependent on $t$?
The simplest example is as follows:
$x$ is the horizontal location, and
$y$ is the vertical location. $\\$
We have already seen this simplest setting of free fall: $$\begin{cases} \frac{\Delta x}{\Delta t}=v_x,\\ \frac{\Delta y}{\Delta t}=v_y-gt, \end{cases} \text{ and } \begin{cases} x'=v_x,\\ y'=v_y-gt. \end{cases}$$ It is just as simple when arbitrary functions are in the right-hand sides of the equations (the continuous case is solved by integration). Here the rate of change of the location depends on the time $t$ only.
More complex is the situation when the rate of change of the location depends on the location. When the former depends only on its own component of the location, the motion is described by this pair of ODEs: $$\begin{cases} \frac{\Delta x}{\Delta t}=g(x),\\ \frac{\Delta y}{\Delta t}=h(y). \end{cases} \text{ and } \begin{cases} x'=g(x),\\ y'=h(y). \end{cases}$$ The solution consists of two solutions to the two, unrelated, ODEs. We can then apply the methods of Chapter 24.
As an example of quantities that do interact, let's consider the predator-prey model.
$x$ be the number of rabbits and
$y$ be the number of foxes in the forest.
$\Delta t$ be the fixed increment of time.
Even though time $t$ is now discrete, the "number" of rabbits $x$ or foxes $y$ isn't. Those are real numbers in our model. One can think of $.1$ rabbits as if the actual number is unknown but the likelihood that there is one somewhere is $10\%$.
We begin with the rabbits. There are two factors affecting their population.
First, we assume that they have an unlimited food supply and reproduce in a manner described previously -- when there is no predator. In other words, the gain of the rabbit population per unit of time through their natural reproduction is proportional to the size of their current population. Therefore, we have: $$\text{rabbits' gain }=\alpha\cdot x \cdot \Delta t,$$ for some $\alpha>0$.
Second, we assume that the rate of predation upon the rabbits to be proportional to the rate at which the rabbits and the foxes meet, which, in turn, is proportional to the sizes of their current populations, $x$ and $y$. Therefore, we have: $$\text{rabbits' loss }=\beta\cdot x\cdot y \cdot \Delta t,$$ for some $\beta>0$.
Combined, the change of the rabbit population over the period of time of length $\Delta t$ is: $$\Delta x = \alpha x \Delta t - \beta x y \Delta t.$$
We continue with the foxes. There are two factors affecting their population.
First, we assume that the foxes have only a limited food supply, i.e., the rabbits. The foxes die out geometrically in a manner described previously -- when there is no prey. In other words, the loss of the fox population per unit of time through their natural death is proportional to the size of their current population. Therefore, we have: $$\text{foxes' loss }=\gamma\cdot y \cdot \Delta t,$$ for some $\gamma >0$.
Second, we again assume that the rate of reproduction of the foxes is proportional to the rate of their predation upon the rabbits which is, as we know, proportional to the sizes of their current populations, $x$ and $y$. Therefore, we have: $$\text{foxes' gain }=\delta\cdot x\cdot y \cdot \Delta t,$$ for some $\delta>0$.
Combined, the change of the fox population over the period of time of length $\Delta t$ is: $$\Delta y = \delta xy \Delta t - \gamma y \Delta t.$$
Putting these two together gives us a discrete predator-prey model: $$\begin{cases} \Delta x = \left( \alpha x - \beta x y \right) \Delta t,\\ \Delta y = \left( \delta xy - \gamma y \right) \Delta t. \end{cases}$$ Then the spreadsheet formulas are for $x$ and $y$ respectively: $$\texttt{=R[-1]C+R2C3*R[-1]C*R3C1-R2C4*R[-1]C*R[-1]C[1]*R3C1},$$ $$\texttt{=R[-1]C-R2C5*R[-1]C*R3C1+R2C6*R[-1]C*R[-1]C[-1]*R3C1.}$$
Let's take a look at an example of a possible dynamics ($\alpha=0.10$, $\beta=0.50$, $\gamma=0.20$, $\delta=0.20$, $h=0.2$):
This is what we see. Initially, there are many rabbits and, with this much food, the number of foxes was growing: $\uparrow$. This was causing the number of rabbits to decline: $\leftarrow$. Combined, this is the direction of the system: $\nwarrow$. Later, the number of rabbits declines so much that, with so little food, the number of foxes also started to decline: $\downarrow$. At the end, both of the populations seem to have disappeared...
Another experiment shows that they can recover ($\alpha=3$, $\beta=2$, $\gamma=3$, $\delta=1$, $h=0.03$):
In fact, we can see a repeating pattern.
Furthermore, with $\Delta t \to 0$, we have two, related, ODEs, a continuous predator-prey model: $$\begin{cases} \frac{dx}{dt} = \alpha x - \beta x y ,\\ \frac{dy}{dt} = \delta xy - \gamma y . \end{cases}$$ It approximates the discrete model. The equations are known as the Lotka–Volterra equations.
Qualitative analysis of the predator-prey model
To confirm our observations, we will carry out qualitative analysis. It is equally applicable to both the discrete model and the system of ODEs. Indeed, they both have the same right-hand side: $$\begin{cases} \frac{\Delta x}{\Delta t} = \alpha x - \beta x y ,\\ \frac{\Delta y}{\Delta t} = \delta xy - \gamma y , \end{cases}\quad\text{and}\quad \begin{cases} \frac{dx}{dt} = \alpha x - \beta x y ,\\ \frac{dy}{dt} = \delta xy - \gamma y . \end{cases}$$
We will investigate the dynamics at all locations in the first quadrant of the $tx$-plane.
First, we find the locations where $x$ has a zero derivative, i.e., $x'=0$, which is the same as the locations where the discrete model leads to no change in $x$, i.e., $\Delta x=0$. The condition is: $$\alpha x - \beta x y=0.$$ We solve the equation: $$x=0 \text{ or } y=\alpha/\beta.$$ We discover that, first,
$x=0$ is a solution, and, second,
the horizontal line $y=\alpha/\beta$ is crossed vertically by the solutions.
In other words, first, the foxes are dying out with no rabbits and, second, there may be a reversal in the dynamics of the rabbits at a certain number of foxes. To find out, solve the inequality: $$x'>0 \text{ or } \Delta x>0\ \Longrightarrow\ \alpha x - \beta x y>0 \ \Longrightarrow\ y<\alpha/\beta.$$ It follows that, indeed, the number of rabbits increases when the number of foxes is below $\alpha/\beta$, otherwise it decreases.
Second, we find the locations where $y$ has a zero derivative, i.e., $y'=0$, which is the same as the locations where the discrete model leads to no change in $y$, i.e., $\Delta y=0$. The condition is: $$\delta xy - \gamma y=0.$$ We solve the equation: $$y=0 \text{ or } x=\gamma/\delta.$$ We discover that, first,
$y=0$ is a solution, and, second,
the vertical line $x=\gamma/\delta$ is crossed horizontally by the solutions.
In other words, first, the rabbits thrive with no foxes and, second, there may be a reversal in the dynamics of the foxes at a certain number of rabbits. To find out, solve the inequality: $$y'>0 \text{ or } \Delta y>0\ \Longrightarrow\ \delta xy - \gamma y>0 \ \Longrightarrow\ x>\gamma/\delta.$$ It follows that, indeed, the number of foxes increases when the number of rabbits is above $\gamma/\delta$, otherwise it decreases.
Now we put these two parts together. We have four sectors in the first quadrant determined by the four different choices of the signs of $x'$ and $y'$, or $\Delta x$ and $\Delta y$:
In either case, the result is a rough description of the dynamics on the local level: if this is the current state, then this is the direction it is going. It is a vector field!
We visualize this vector field with the same parameters as before:
The arrows aren't meant yet to be connected into curves to produce solutions. The only four distinct solutions we know for sure are the following:
the decline of the foxes in the absence rabbits -- on the $y$-axis;
the explosion of the rabbits in the absence of foxes -- on the $x$-axis;
the freezing of both rabbits and foxes at the special levels -- in the middle; and also
the freezing of both rabbits and foxes at the zero level.
Either of the last two is a constant solution called an equilibrium. The main, non-zero, equilibrium is: $$x(t)=\gamma/\delta,\ y(t)=\alpha/\beta.$$ What about the rest of the solutions?
In order to confirm that the solutions circle the main equilibrium, we need a more precise analysis.
In each of the four sectors, the monotonicity of the solutions has been proven. However, it is still possible that some solutions will stay within the sector when one or both of $x$ and $y$ behave asymptotically: $$x(t)\to a \text{ and/or } y(t)\to b \text{ as } t\to \infty.$$ Since both functions are monotonic, this implies that $$x'(t)\to 0 \text{ and/or } y'(t)\to 0 \text{ as } t\to \infty,$$ and the same for $\Delta x$ and $\Delta y$. We can show that this is impossible. For example, suppose we start in the bottom right sector, i.e., the initial conditions are:
$x(0)=p>\gamma/\delta$;
$y(0)=q<\alpha/\beta$.
Then, for as long as the solution is in this sector, we have
$x'>0 \ \Longrightarrow\ x>p$;
$y'>0 \ \Longrightarrow\ y>q$.
Therefore, $$y'=y(\delta x - \gamma)>q(\delta p-\gamma)>0.$$ This number is a gap between $y'$ and $0$. Therefore, $y'$ cannot diminish to $0$, and the same is true for $\Delta y$. It follow that the solution will reach the line $y=\alpha/\beta$ "in finite time".
Exercise. Prove the analogous facts about the three remaining sectors.
We have demonstrated that a solution will go around the main equilibrium, but when it comes back, will it be closer to the center, farther, or will it come to the same location?
The first option is indicated by our spreadsheet result. Next, we set the discrete model aside and concentrate on solving analytically our system of ODEs to answer the question, is this a cycle or a spiral?
Solving the Lotka–Volterra equations
We would like to eliminate time from the equations ($x>0,\ y>0$): $$\begin{cases} \frac{dx}{dt} = \alpha x - \beta x y ,\\ \frac{dy}{dt} = \delta xy - \gamma y . \end{cases}$$ This step is made possible by the following trick. We interpret these derivatives in terms of the differential forms: $$\begin{array}{llll} dx = (\alpha x - \beta x y )dt&\Longrightarrow & dt=\frac{dx}{\alpha x - \beta x y },\\ dy = (\delta xy - \gamma y)dt&\Longrightarrow & dt=\frac{dy}{\delta xy - \gamma y }. \end{array}$$ Therefore, $$dt=\frac{dx}{\alpha x - \beta x y }=\frac{dy}{\delta xy - \gamma y }.$$ We next separate variables: $$\frac{\delta x - \gamma }{x}dx=\frac{\alpha - \beta y}{y}dy.$$ We integrate: $$\int \left(\delta - \frac{\gamma }{x} \right)dx=\int \left(\frac{\alpha}{y} - \beta \right)dy,$$ and we have: $$\delta x - \gamma \ln x = \alpha \ln y - \beta y +C.$$ The system is solved!.. But what does it mean?
Every solution $x=x(t)$ and $y=y(t)$, when substituted into the function $$G(x,y)=\delta x - \gamma \ln x - \alpha \ln y + \beta y,$$ produces a constant. In other words, this parametric curve is a level curve of $z=G(x,y)$.
Once we have no derivatives left, we declare the system solved even though only implicitly. Even though we don't have explicit formulas for $x=x(t)$ or $y=y(t)$, we can use what we have to further study the qualitative behavior of the system.
For example, the fact that this is a level curve already suggests that the parametric curve is not a spiral. Just try to imagine such a surface that its level curves are spirals:
We turn instead to the actual function. First, we plot it with a spreadsheet ($\alpha=3$, $\beta=2$, $\gamma=3$, $\delta=1$):
The level curves are visible. Some of them are clearly circular and others aren't. The reason is that they aren't shown all the way to the axes because the value of $G$ rises so quickly (in fact, asymptotically).
As expected, the surface seems to have a single minimum point. Let's prove that algebraically: $$\begin{array}{lllll} \frac{\partial G}{\partial x}(x,y)&=\delta - \gamma /x,\\ \frac{\partial G}{\partial y}(x,y)&= - \alpha /y + \beta. \end{array}$$ We next find the extreme points of $G$. We set the two derivatives equal to zero and solve for $x$ and $y$: $$\begin{array}{lllll} \frac{\partial G}{\partial x}(x,y)&=\delta - \gamma /x&=0&\Longrightarrow &x=\frac{\gamma}{\delta},\\ \frac{\partial G}{\partial y}(x,y)&= - \alpha /y + \beta&=0&\Longrightarrow &y=\frac{\alpha}{\beta}. \end{array}$$ This point is indeed our main equilibrium point. The surface here has a horizontal tangent plane. We have also demonstrated that there are no others points like that!
But could this be a maximum point? Just as $y=x^3$ crosses the $x$-axis at $0$ degrees, a surface can cross its tangent plane.
We compute the second derivatives: $$\begin{array}{lllll} \frac{\partial^2 G}{\partial x^2}(x,y)&=\gamma \frac{1}{x^2}>0,\\ \frac{\partial^2 G}{\partial y^2}(x,y)&=\alpha \frac{1}{y^2}>0 . \end{array}$$ Both are positive throughout, therefore, either of the cross-sections of the surface along the axes have a minimum point here and it has to stay on one side of the plane. However, it might cross it at other directions. For example, this still might be a saddle point! We invoke the Second Derivative Test from Chapter 9 to resolve this.
We consider the Hessian matrix (discussed in Chapter 18) of $G$. It is the $2\times 2$ matrix of the four partial derivatives of $G$: $$H(x,y) = \begin{pmatrix}\frac{\partial^2 G}{\partial x^2} &\frac{\partial^2 G}{\partial x \partial y}\\ \frac{\partial^2 G}{\partial y \partial x} &\frac{\partial^2 G}{\partial y^2}\end{pmatrix}=\begin{pmatrix} \gamma \frac{1}{x^2} &0\\ 0& \alpha \frac{1}{y^2}\end{pmatrix}.$$ Here, in addition, we have the mixed second derivatives: $$\frac{\partial^2 G}{\partial x \partial y}(x,y)=\frac{\partial^2 G}{\partial y \partial x}(x,y)=0 .$$ Next, we look at the determinant of the Hessian: $$D(x,y)=\det(H(x,y)) = \left( \gamma \frac{1}{x^2}\right)\cdot \left( \alpha \frac{1}{y^2}\right)= \frac{\alpha \gamma}{x^2y^2}>0.$$ It's positive! Therefore, the point is a minimum.
We conclude that every level curve of $G$, i.e., a solution of the system, goes around the equilibrium and comes back to create a cycle.
An easier, but more ad hoc, way to reach this conclusion is to imagine that a solution starts on, say, the line $y=\alpha/\beta$ at $x=x_0$ to the right of the equilibrium and then comes back to the line at $x=x_1$. Since this is a level curve, we have $G(x_0,\alpha/\beta)=G(x_1,\alpha/\beta)$. According to Rolle's Theorem from Chapter 9, this contradicts our conclusion that $\frac{\partial G}{\partial x}> 0$ along this line.
Plotted with the same parameters, this is what these curves look like:
Exercise. Prove that the predator-prey discrete model, i.e., Euler's method for this system of ODEs, produces solutions that spiral away from the equilibrium. Hint: image below.
Vector fields and systems of ODEs
Numerous processes are modeled by systems of ODEs. For example, if little flags are placed on the lawn, then their directions taken together represent a system of ODEs, while the (invisible) air flow is the solutions of this system.
A similar idea is used to model a fluid flow. The dynamics of each particle is governed by the velocity of the flow, at each location, the same at every moment of time.
To solve such a system would require tracing the path of every particle of the liquid.
Let's review the discrete model of a flow: given a flow on a plane, trace a single particle of this stream.
For both coordinates, $x$ and $y$, the following table is being built. The initial time $t_0$ and the initial location $p_0$ are placed in the first row of the spreadsheet. As we progress in time and space, new numbers are placed in the next row of our spreadsheet: $$t_n,\ v_n,\ p_n,\ n=1,2,3,...$$ The following recursive formulas are used:
$t_{n+1}=t_n+\Delta t$.
$p_{n+1}=p_n+v_{n+1}\cdot \Delta t$.
The result is a growing table of values: $$\begin{array}{c|c|c|c|c|c} &\text{iteration } n&\text{time }t_n&&\text{velocity }v_n&\text{location }p_n\\ \hline \text{initial:}&0&3.5&&--&22\\ &1&3.6&&33&25.3\\ &...&...&&...&...\\ &1000&103.5&&4&336\\ &...&...&&...&...\\ \end{array}$$
So, instead of two (velocity -- location, as before), there will be four main columns when the motion is two-dimensional and six when it is three-dimensional: $$\begin{array}{c|c|cc|cc|c} \text{}&\text{time}&\text{horiz.}&\text{horiz.}&\text{vert.}&\text{vert.}&...\\ \text{} n&\text{}t&\text{vel. }x'&\text{loc. }x&\text{vel. }y'&\text{loc. }y&...\\ \hline 0&3.5&--&22&--&3&...\\ 1&3.6&33&25.3&4&3.5&...\\ ...&...&...&...&...&...&...\\ 1000&103.5&4&336&66&4&...\\ ...&...&...&...&...&...&...\\ \end{array}$$
Example. Recall the examples such flows. If the velocity of the flow is proportional to the location: $$v_{n+1}=.2\cdot p_n,$$ for both horizontal and vertical, the result is particles flying away from the center, faster and faster:
If the horizontal velocity is proportional to the vertical location and the vertical velocity proportional to the negative of the horizontal location, the result resembles rotation:
Now, suppose the velocity comes from explicit formulas as functions of the location: $$u=f(x,y),\ v=g(x,y),$$ are there explicit formulas for the location as a function of time: $$x=x(t),\ y=y(t)?$$
We assume that there is a version of our recursive relation, $$p_{n+1}=p_n+v_{n+1}\cdot \Delta t,$$ for every $\Delta t>0$ small enough. Then our two functions have to satisfy for $x$: $$v_n=f(p_n) \text{ and } p_n=x(t_n),$$ and for $y$: $$v_n=g(p_n) \text{ and } p_n=y(t_n),$$
We substitute these two, as well as $t=t_n$, into the recursive formulas for $p_{n+1}$ for $x$: $$x(t+\Delta t)=x(t)+f(x(t+\Delta t),y(t+\Delta t))\cdot \Delta t,$$ and for $y$: $$y(t+\Delta t)=y(t)+g(x(t+\Delta t),y(t+\Delta t))\cdot \Delta t.$$ Then, we have for $x$: $$\frac{x(t+\Delta t)-x(t)}{\Delta t}=f(x(t+\Delta t),y(t+\Delta t)),$$ and for $y$: $$\frac{y(t+\Delta t)-y(t)}{\Delta t}=g(x(t+\Delta t),y(t+\Delta t)).$$ Taking the limit over $\Delta t\to 0$ gives us the following system: $$\begin{cases} x'(t)&=f(x(t),y(t)),\\ y'(t)&=g(x(t),y(t)), \end{cases}$$ provided $x=x(t),\ y=y(t)$ are differentiable at $t$ and $u=f(x,y),\ v=g(x,y)$ are continuous at $(x(t),y(t))$.
Now, let's recall from Chapter 21 that a vector field supplies a direction to every location, i.e., there is a vector attached to each point of the plane: $${\rm point} \mapsto {\rm vector}.$$ A vector field in dimension $2$ is in fact any function: $$F : {\bf R}^2 \to {\bf R}^2 ,$$ given by two functions of two variables: $$F(x,y)=<f(x,y),g(x,y)>.$$ Then, the vector field defines a system of two (time-independent) ODEs.
Definition. A solution of a system of ODEs is a pair of functions $x=x(t)$ and $y=y(t)$ (a parametric curve) with either one differentiable on an open interval $I$ such that for every $t$ in $I$ we have: $$\begin{cases} x'(t)&=f(x(t),y(t)),\\ y'(t)&=g(x(t),y(t)), \end{cases}$$ or abbreviated: $$\begin{cases} x'=f(x,y),\\ y'=g(x,y). \end{cases}$$
The vector field of this system is the slope field of the ODE.
How do we visualize the solutions of such a system? With a single ODE, $$y'=f(t)\ \Longrightarrow\ y=y(t),$$ we simply plot their graphs, i.e., the collections of points $(t,y(t))$, of some of them on the $ty$-plane. This time, the solutions are parametric curves! Their graphs, i.e., the collections of $(t,x(t),y(t))$ lie in the $3$-dimensional $txy$-space. That is why, we, instead, plot their images, i.e., the collections of points $(x(t),y(t))$ on the $xy$-plane. In the theory of ODEs, they are also known as trajectories, or paths.
Then the vectors of the vector field are tangent to these trajectories.
Since the vector field is independent of $t$, such a representation is often sufficient as explained by the following result.
Theorem. If $x=x(t),\ y=y(t)$ is a solution of the system of ODEs then so is $x=x(t+s),\ y=y(t+s)$ for any real $s$.
It is also important to be aware of the fact that the theory of systems of ODEs "includes" the theory of single ODEs. Recall, first, how the graph of every function can be represented by the trajectory of a parametric curve: $$y=r(x)\ \longrightarrow\ \begin{cases} x=t,\\ y=r(x). \end{cases}$$ Similarly, the solutions of every time-independent ODE can be represented by the trajectories of a system of two ODEs: $$y'=g(x) \ \longrightarrow\ \begin{cases} x'&=1,\\ y'&=g(y). \end{cases}$$
Definition. For a given system of ODEs and a given triple $(t_0,x_0,y_0)$, the initial value problem, or an IVP, is $$\begin{cases} x'&=f(x,y),\\ y'&=g(x,y); \end{cases}\quad \begin{cases} x(t_0)&=x_0,\\ y(t_0)&=y_0; \end{cases}$$ and its solution is a solution of the ODE that satisfies the initial condition above.
By the last theorem, the value of $t_0$ doesn't matter; it can always be chosen to be $0$.
Definition. We say that a system of ODEs satisfies the existence property at a point $(t_0,x_0,y_0)$ when the IVP above has a solution.
If your model of a real-life process doesn't satisfy existence, it reflects limitations of your model. It is as if the process starts but never continues.
Definition. We say that an ODE satisfies the uniqueness property at a point $(t_0,x_0,y_0)$ if every pair of solutions, $(x_1,y_1),\ (x_2,y_2)$, of the IVP above are equal, $$x_1(t)=x_2(t), \ y_1(t)=y_2(t),$$ for every $t$ in some open interval that contains $t_0$.
If your model of a real-life process doesn't satisfy uniqueness, it reflects limitations of your model. It's as if you have all the data but can't predict even the nearest future.
Thus systems of ODEs produce families of curves as the sets of their solutions. Conversely, if a family of curves is given by an equation with a single parameter, we may be able to find a system of ODEs for it.
Example. The family of vertically shifted graphs, $$y=x^2+C,$$ creates an ODE if we differentiate (implicitly) with respect to $t$: $$y'=2xx'.$$ Since these are just functions of $x$, we can choose $x=t$. This is a possible vector field for this family: $$\begin{cases} x'&=1,\\ y'&=2x. \end{cases}$$ $\square$
Example. The family of stretched exponential graphs, $$y=Ce^x,$$ creates an ODEs: $$y'=Ce^xx'.$$ This is a possible vector field for this family: $$\begin{cases} x'&=1,\\ y'&=Ce^x. \end{cases}$$ $\square$
Example. What about these concentric circles?
(In the case when $C=0$, we have the origin.) They are given by $$x^2+y^2=C\ge 0.$$ We differentiate (implicitly) with respect to $t$: $$2xx'+2yy'=0.$$ We choose what $x',\ y'$ might be equal to in order for the two terms to cancel. This is a possible vector field for this family: $$\begin{cases} x'&=-y,\\ y'&=x. \end{cases}$$ $\square$
Example. These hyperbolas are given by these equations: $$xy=C.$$ (In the case when $C=0$, we have the two axes.)
Then, we have an ODE: $$x'y+xy'=0.$$ This is a possible vector field for this family: $$\begin{cases} x'&=x,\\ y'&=-y. \end{cases}$$ $\square$
None of the examples have problems with either existence or uniqueness -- in contrast to the corresponding ODEs.
The proofs of the following important theorems lie outside the scope of this book.
Theorem (Existence). Suppose $(x_0,y_0)$ is a point on the $xy$-plane and suppose
$H$ is an open interval that contains $x_0$, and
$G$ is an open interval that contains $y_0$.
Suppose also that functions $z=f(x,y)$ and $z=g(x,y)$ of two variables are continuous with respect to $x$ and $y$ on $H\times G$. Then the system of ODE, $$\begin{cases} x'&=f(x,y),\\ y'&=g(x,y). \end{cases}$$ satisfies the existence property at $(t_0,x_0,y_0)$ for any $t_0$.
Theorem (Uniqueness). Suppose $(x_0,y_0)$ is a point on the $xy$-plane and suppose
Suppose also that function $z=f(x,y)$ and $z=g(x,y)$ of two variables are differentiable with respect to $x$ and $y$ on $H\times G$. Then the system of ODEs, $$\begin{cases} x'&=f(x,y),\\ y'&=g(x,y). \end{cases}$$ satisfies the uniqueness property at $(t_0,x_0,y_0)$ for any $t_0$.
Discrete systems of ODEs
Discrete ODEs approximate and are approximated by continuous ODEs. The same is true for systems. For example, the discrete system for the predator-prey model produces this almost exactly cyclic path:
In other words, Euler's method is capable of tracing solutions very close the ones of the ODE it came from.
Just as in the $1$-dimensional case, the IVP tells us:
where we are (the initial condition), and
the direction we are going (the ODE).
Just as before, the unknown solution is replaced with its best linear approximation.
Example. Let's consider again these concentric circles:
They are the solutions of the ODEs: $$\begin{cases} x'&=y,\\ y'&=-x; \end{cases}$$ We choose the increment of $t$: $$\Delta t=1.$$
We start with this initial condition: $$t_0=0, \quad x_0=0,\quad y_0=2.$$ We substitute these two numbers into the equations: $$\begin{cases} x'&=2,\\ y'&=0; \end{cases}$$ This is the direction we will follow. The increments are $$\begin{cases} \Delta x=2\cdot \Delta t=2\cdot 1 =2,\\ \Delta y=0\cdot \Delta t=0\cdot 1 =0; \end{cases}$$ Our next location on the $xy$-plane is then: $$\begin{cases} x_1=x_0+\Delta x=0+2=2,\\ y_1=y_0+\Delta y=2+0=2. \end{cases}$$
A new initial condition appears: $$x_0=2,\quad y_0=2.$$ We again substitute these two numbers into the equations: $$\begin{cases} x'&=2,\\ y'&=-2; \end{cases}$$ producing the direction we will follow. The increments are $$\begin{cases} \Delta x=2\cdot \Delta t=2\cdot 1 =2,\\ \Delta y=-2\cdot \Delta t=-2\cdot 1 =-2; \end{cases}$$ Our next location on the $xy$-plane is then: $$\begin{cases} x_2=x_1+\Delta x=2+2=4,\\ y_2=y_1+\Delta y=2+(-2)=0. \end{cases}$$
One more IVP: $$x_2=0,\ y_2=-4.$$ The increments are $$\begin{cases} \Delta x=0\cdot \Delta t=0\cdot 1 =0,\\ \Delta y=-4\cdot \Delta t=-4\cdot 1 =-4; \end{cases}$$ Our next location on the $xy$-plane is then: $$\begin{cases} x_3=x_2+\Delta x=4+0=4,\\ y_3=y_2+\Delta y=0-4=-4. \end{cases}$$
These four points form a very crude approximation of one of our circular solutions:
They are clearly spiraling away from the origin. $\square$
In terms of motion,
at our current location and current time, we examine the ODE to find the velocity and then move accordingly to the next location.
Definition. The Euler solution with increment $\Delta t>0$ of the IVP $$\begin{cases} x'&=f(x,y),\\ y'&=g(x,y); \end{cases}\quad \begin{cases} x(t_0)&=x_0,\\ y(t_0)&=y_0; \end{cases}$$ is the two sequences $\{x_n\}$ and $\{y_n\}$ of real numbers given by: $$\begin{cases} x_{n+1}&=x_n+f(x_n,y_n)\cdot \Delta t,\\ y_{n+1}&=y_n+g(x_n,y_n)\cdot \Delta t; \end{cases}$$ where $t_{n+1}=t_n+\Delta t$.
Once again, if we derived our ODEs from a discrete model (via $\Delta t\to 0$), Euler's method will bring us right back to it: $$\newcommand{\la}[1]{\!\!\!\!\!\xleftarrow{\quad#1\quad}\!\!\!\!\!} \begin{array}{cccccc} &&\text{ODEs}\\ &\nearrow&&\searrow\\ \text{discrete model}&&\la{same!}&&\text{Euler's method} \end{array}$$
Example. Let's now carry out this procedure with a spreadsheet. The formulas for $x_n$ and $y_n$ are respectively: $$\texttt{=R[-1]C+R[-1]C[1]*R3C1},$$ $$\texttt{=R[-1]C-R[-1]C[-1]*R3C1}.$$ These are the results:
In contrast to the case of a single ODE, the approximations do not behave erratically close to the $x$-axis. The reason is that there is no division by $y$ anymore. $\square$
Example. Let's consider again these hyperbolas: $$xy=C.$$ They are the solutions of the system: $$\begin{cases} x'&=x,\\ y'&=-y; \end{cases}$$ An Euler solution is shown below:
However, is this asymptotic convergence toward the $x$-axis or do they merge? $\square$
Qualitative analysis of systems of ODEs
Since Euler's method depends on the value of $h$, the increment of $t$, and, even with smaller and smaller values of $h$, the result remains a mere approximation. Meanwhile, qualitative analysis collects information about the solutions without solving the system -- either analytically or numerically. The result is fully accurate but very broad descriptions of the solutions.
Example. Let's review an example of a single ODE from last section: $$y'=-\tan (y).$$ This is what we conclude about the strip $[-\pi/2,\pi/2]$:
for $-\pi/2<y<0$, we have $y'=-\tan y>0$ and, therefore, $y\nearrow$;
for $y=0$, we have $y'=-\tan y=0$ and, therefore, $y$ is a constant solution;
for $0<y<\pi/2$, we have $y'=-\tan y<0$ and, therefore, $y\searrow$.
In fact, every solution $y$ is decreasing (or increasing) throughout its domain. The conclusions are confirmed with Euler's method:
We can match this ODE with a system: $$\begin{cases} x'=1,\\ y'=-\tan y. \end{cases}$$ Its solutions have these trajectories as solutions. $\square$
The directions of a parametric curves are its tangent vectors. Therefore, the directions of the solutions of the system of ODEs: $$\begin{cases} x'=f(x,y),\\ y'=g(x,y), \end{cases}$$ are derived from the signs of the functions in the right-hand side: $$\begin{array}{c|c|c|c} &&x'=f(x,y)<0&x'=f(x,y)=0&x'=f(x,y)>0\\ \hline &&\leftarrow&\bullet&\to\\ \hline y'=g(x,y)<0&\downarrow&\swarrow&\downarrow&\searrow\\ \hline y'=g(x,y)=0&\bullet&\leftarrow&\bullet&\rightarrow\\ \hline y'=g(x,y)>0&\uparrow&\nwarrow&\uparrow&\nearrow \end{array}$$
Example. Consider next: $$y'=\cos y \cdot \sqrt{|y|}.$$ we demonstrated that the monotonicity of the solutions varies with the cosine:
We can see that all solutions progress forward along the $x$-axis. What if we add variability of the direction of $x$? Consider: $$\begin{cases} x'=\sin y,\\ y'=\cos y . \end{cases}$$ We conduct the "sign analysis" for both functions in the right-hand side: $$\begin{array}{r|c|c|c} y&x'&x=x(t)&\text{path}\\ \hline 2\pi&0&&\\ &-&x\downarrow&\leftarrow\leftarrow\leftarrow\leftarrow\\ \pi&0&&\\ &+&x\uparrow&\to\to\to\to\\ 0&0&&\\ &-&x\downarrow&\leftarrow\leftarrow\leftarrow\leftarrow\\ -\pi&0&\\ \end{array} \quad \begin{array}{r|c|c|c} y&y'&y=y(t)&\text{path}\\ \hline 3\pi/2&0&\\ &-&y\downarrow&\downarrow\downarrow\downarrow\downarrow\\ \pi/2&0&\\ &+&y\uparrow&\uparrow\uparrow\uparrow\uparrow\\ -\pi/2&0&\\ \end{array}$$ Now put them together: $$\begin{array}{cccccc} \to&\to&\to&\to\\ \nearrow&\nearrow&\nearrow&\nearrow\\ \uparrow&\uparrow&\uparrow&\uparrow\\ \nwarrow&\nwarrow&\nwarrow&\nwarrow\\ \leftarrow&\leftarrow&\leftarrow&\leftarrow\\ \end{array}$$ The results are confirmed with Euler's method:
Example. The next one: $$\begin{cases} x'=\sin x,\\ y'=\cos y . \end{cases}$$ We conduct the "sign analysis" of the right-hand side: $$\begin{array}{r|c|cccccc} &x&-\pi&&\pi&&2\pi&&3 \pi&\\ \hline y&y' | x'&0&+&0&-&0&+&0\\ \hline 3\pi/2&0&\bullet&\to&\bullet&\leftarrow&\bullet&\to&\bullet\\ &-&\downarrow&\searrow&\downarrow&\swarrow&\downarrow&\searrow&\downarrow\\ \pi/2&0&\bullet&\to&\bullet&\leftarrow&\bullet&\to&\bullet\\ &+&\uparrow&\nearrow&\uparrow&\nwarrow&\uparrow&\nearrow&\uparrow\\ -\pi/2&0&\bullet&\to&\bullet&\leftarrow&\bullet&\to&\bullet\\ &-&\downarrow&\searrow&\downarrow&\swarrow&\downarrow&\searrow&\downarrow\\ -3\pi/2&0&\bullet&\to&\bullet&\leftarrow&\bullet&\to&\bullet\\ \end{array}$$ The results are confirmed with Euler's method:
Example. The next one is discontinuous: $$x'=x−y,\ y'=[x+y].$$ For Euler's method we use the $\texttt{FLOOR}$ function:
File:X'=x−y, y'=x+y.png
Exercise. Confirm the plot below by analyzing this system: $$x'=y,\ y'=\sin y \cdot y.$$
Suppose the system is time-independent, $$x'=f(x,y),\ y'=g(x,y).$$ Then it is thought of as a flow: liquid in a pond or the air over a surface of the Earth.
Then, the right-hand side is recognized as a two-dimensional vector field. In contrast to the $1$-dimensional case with only three main possibilities, we will see a wide variety of behaviors around an equilibrium when the space of location is a plane.
With such a system, the qualitative analysis is much simpler. In fact, the ones above exhibit most of the possible patterns of local behavior.
We concentrate on what is going on in the vicinity of a given location $(x,y)=(a,b)$.
The first main possibility is $$f(a,b)\ne 0 \text{ or } g(a,b)\ne 0,$$ which is equivalent to $$F(a,b)= <f(a,b),g(a,b)>\ne 0.$$ Then, from the continuity of $f$ and $g$, we conclude that neither changes its sign in some disk $D$ that contains $(a,b)$. Then, the solutions with paths located within $D$ proceed in an about the same direction:
The behavior is "generic".
More interesting behaviors are seen around a zero of $F$:
$F(a,b)=0\ \Longrightarrow\ (x,y)=(a,b) $ is a stationary solution (an equilibrium).
Then the pattern in the vicinity of the point, i.e., an open disk $D$, depends on whether this is a maximum of $f$ or $g$, or a minimum, or neither. Some of the ideas come from dimension $1$.
For example, a stable equilibrium, $$y\to a \text{ as } x\to +\infty,$$ is a sink: flow in only. An unstable equilibrium, $$y\to a \text{ as } x\to -\infty,$$ is a source: flow out only.
An semi-stable equilibrium could mean that some the solutions asymptotically approach the equilibrium and others do not. As you can see there are many more possibilities than in dimension $1$.
The vector notation and linear systems
We can combine the two variables to form a point, a location or a location on the plane: $$X=(x,y).$$ We can also use the column-vector notation: $$X=\left[\begin{array}{cc}x \\ y\end{array}\right].$$ Next, the same happens to their derivatives as vectors: $$\frac{\Delta X}{\Delta t}=\left<\frac{\Delta x}{\Delta t}, \frac{\Delta y}{\Delta t}\right>=\left[\begin{array}{cc}\frac{\Delta x}{\Delta t}\\\frac{\Delta y}{\Delta t}\end{array}\right].$$ Then the setup we have been using for real-valued functions reappears: $$\frac{\Delta X}{\Delta t}=F(X),\ X(t_0)=X_0.$$ In this section, our main concern will be ODEs with respect to derivatives: $$X'=<x',y'>=\left[\begin{array}{cc}x'\\y'\end{array}\right],$$ and $$X'=F(X),\ X(t_0)=X_0.$$ All the plotting, however, is done with the discrete ODEs.
The "phase space" ${\bf R}^2$ is the space of all possible locations. Then the position of a given particle is a function $X:{\bf R}^+\to {\bf R}^2$ of time $t\ge 0$. Meanwhile, the dynamics of the particle is governed by the velocity of the flow, at each location, the same at every moment of time: the velocity of a particle if it happens to be at point $X$ is $F(X)$.
Then either the next position is predicted to be $X+F(X)$ -- that's a discrete model -- or $F(X)$ is just a tangent of the trajectory -- that's an ODE.
A vector field supplies a direction to every location, i.e., there is a vector attached to each point of the plane: $${\rm point} \mapsto {\rm vector}.$$
A vector field in dimension $2$ is any function: $$F : {\bf R}^2 \to {\bf R}^2 .$$ Furthermore, one can think of a vector field as a time-independent ODE on the plane: $$X'=F(X).$$ The corresponding IVP adds an initial condition: $$X(t_0)=X_0.$$
Definition. A solution of a system of ODEs is a function $u$ differentiable on an open interval $I$ such that for every $t$ in $I$ we have: $$X'(t)=F(X(t)).$$
Definition. For a given system of ODEs and a given $(t_0,X_0)$, the initial value problem, or an IVP, is $$X'=F(X),\quad X(t_0)=X_0.$$ and its solution is a solution of the ODE that satisfies the initial condition above.
Definition. The Euler solution with increment $\Delta t>0$ of the IVP: $$X'=F(X),\quad X(t_0)=X_0;$$ is a sequence $\{X_n\}$ of points on the plane given by: $$X_{n+1}=X_n+F(X_n)\cdot \Delta t,$$ where $t_{n+1}=t_n+\Delta t$.
All the definitions above remain valid if we think of $X$ as a location in an $N$-dimensional space ${\bf R}^N$.
We now concentrate on linear systems (that may be acquired from non-linear ones via linearization). All the functions involved are linear and, therefore, differentiable; both existence and uniqueness are satisfied!
This is the case of a linear function $F$. As such, it is given by a matrix and is evaluated via matrix multiplication: $$F(X)=FX=\left[ \begin{array}{ll}a&b\\c&d\end{array}\right] \left[ \begin{array}{ll}x\\y\end{array}\right]=\left[ \begin{array}{ll}ax+by\\cx+dy\end{array}\right].$$ It is written simply as $$X'=FX.$$ The characteristics of this matrix -- the determinant, the trace, and the eigenvalues -- will help us classify such a system.
However, the first observation is very simple: $X=0$ is the equilibrium of the system. In fact what we've learned about systems of linear equations tells us the following.
Theorem. When $\det F\ne 0$, $X=0$ is the only equilibrium of the system $X'=FX$; otherwise, there are infinitely many such points.
In other words, what matter is whether $F$, as a function, is or is not one-to-one.
Example (degenerate). The latter is the "degenerate" case such as the following. Let's consider this very simple system of ODEs: $$\begin{cases}x'&=2x\\ y'&=0\end{cases} \qquad\begin{array}{ll}\Longrightarrow\\ \Longrightarrow\end{array} \begin{cases}x&=Ce^{2t}\\ y&=K\end{cases}.$$ It is easy to solve, one equation at a time. We have exponential growth on the $x$-axis and constant on the $y$-axis.
Example (saddle). Let's consider this system of ODEs: $$\begin{cases}x'&=-x\\ y'&=4y\end{cases} \qquad\begin{array}{ll}\Longrightarrow\\ \Longrightarrow\end{array} \begin{cases}x&=Ce^{-t}\\ y&=Ke^{4t}\end{cases}.$$ We solve it instantly because the two variables are fully separated. We can think of either of these two solutions of the two ODEs as a solution of the whole system that lives entirely within one of the two axes:
We have exponential growth on the $x$-axis and exponential decay on the $y$-axis. The rest of the solutions are seen to tend toward one of these. Since not all of the solutions go toward the origin, it is unstable.
This pattern is called a "saddle" because the curves look like the level curves of a function of two variables around a saddle point. Here, the matrix of $F$ is diagonal: $$F=\left[ \begin{array}{ll}-1&0\\0&4\end{array}\right].$$ Algebraically, we have: $$X=\left[ \begin{array}{ll}x\\y\end{array}\right]=\left[ \begin{array}{ll}Ce^{-t}\\Ke^{4t}\end{array}\right]=Ce^{-t}\left[ \begin{array}{ll}1\\0\end{array}\right]+Ke^{4t}\left[ \begin{array}{ll}0\\1\end{array}\right].$$ We have expressed the general solution as a linear combination of the two basis vectors! $\square$
Example (node). A slightly different system is: $$\begin{cases}x'&=2x\\ y'&=4y\end{cases} \qquad\begin{array}{ll}\Longrightarrow\\ \Longrightarrow\end{array} \begin{cases}x&=Ce^{2t}\\ y&=Ke^{4t}\end{cases}.$$ Here, $$F=\left[ \begin{array}{ll}2&0\\0&4\end{array}\right].$$ Once again, either of these two solutions of the two ODEs is a solution of the whole system that lives entirely within one of the two axes (exponential growth on both of the axes) and the rest of the solutions are seen to tend toward one of these. The slight change to the system produces a very different pattern:
Algebraically, we have: $$X=\left[ \begin{array}{ll}x\\y\end{array}\right]=\left[ \begin{array}{ll}Ce^{2t}\\Ke^{4t}\end{array}\right]=Ce^{2t}\left[ \begin{array}{ll}1\\0\end{array}\right]+Ke^{4t}\left[ \begin{array}{ll}0\\1\end{array}\right],$$ a linear combination of the two basis vectors with time-dependent weights. The equilibrium is unstable but changing $2$ and $4$ to $-2$ and $-4$ will reverse the directions of the curves and make it stable. The exponential growth is faster along the $y$-axis; that is why the solutions appear to be tangent to the $x$-axis. In fact, eliminating $t$ gives us $y=x^2$ and similar graphs. When the two coefficients are equal, the growth is identical and the solutions are simply straight lines:
What if the two variables aren't separated? The insight is that they can be -- along the eigenvectors of the matrix. Indeed, the basis vectors are the eigenvectors of these two diagonal matrices. Now just imagine that the two pictures are skewed:
Let's look at them one at a time. The idea is uncomplicated: the system within the eigenspace is $1$-dimensional. In other words, it is a single ODE and can be solved the usual way. This is how we find solutions. Every solution $X$ that lies within the eigenspace, which is a line, is a $t$-dependent multiple of the eigenvector $V$: $$\begin{array}{lll} X=rV& \Longrightarrow X'=(rV)'=F(rV)=rFV=r\lambda V\\ &\Longrightarrow\ r'V=\lambda rV\\ &\Longrightarrow\ r'=\lambda r\\ &\Longrightarrow\ r=e^{\lambda t}. \end{array}$$
Theorem (Eigenspace solutions). If $\lambda$ is an eigenvalue and $V$ a corresponding eigenvector of a matrix $F$, then $$X=e^{\lambda t}V$$ is a solution of the linear system $X'=FX$.
Proof. To verify, we substitute into the equation and use linearity (of both matrix multiplication and differentiation): $$\begin{array}{lll} X=e^{\lambda t}V& \Longrightarrow \\ \text{left-hand side:}&X'=(e^{\lambda t}V)'=(e^{\lambda t})'V=\lambda e^{\lambda t}V;\\ \text{right-hand side:}&FX=F(e^{\lambda t}V)=e^{\lambda t}FV= e^{\lambda t}\lambda V. \end{array}$$ $\square$
The eigenvalue can be complex!
The second idea is to try to express the solution of the general linear system as a linear combination of two solutions found this way.
Theorem (Representation). Suppose $V_1$ and $V_2$ are two eigenvectors of a matrix $F$ that correspond to two eigenvalues $\lambda_1$ and $\lambda_2$. Suppose also that the eigenvectors aren't multiples of each other. Then all solutions of the linear system $X'=FX$ are given as linear combinations of non-trivial solutions within the eigenspaces: $$X=Ce^{\lambda_1 t}V_1+Ke^{\lambda_2 t}V_2,$$ with real coefficients $C$ and $K$.
Proof. Since these solutions cover the whole plane, the conclusion follows from the uniqueness property. $\blacksquare$
Definition. For a given eigenvector $V$ with eigenvalue $\lambda$, we will call $e^{\lambda t}V$ a characteristic solution.
Exercise. Show that when all eigenvectors are multiples of each other, the formula won't give us all the solutions.
Classification of linear systems
We consider a few systems with non-diagonal matrices. The computations of the eigenvalues and eigenvectors come from Chapter 24.
Example (degenerate). Let's consider a more general system of ODEs: $$\begin{cases}x'&=x&+2y,\\ y'&=2x&+4y,\end{cases} \ \Longrightarrow\ F=\left[ \begin{array}{cc}1&2\\2&4\end{array}\right].$$ Euler's method shows the following solutions:
It appears that the system has (exponential) growth in one direction and constant in another. What are those directions? Linear algebra helps.
First, the determinant is zero: $$\det F=\det\left[ \begin{array}{cc}1&2\\2&4\end{array}\right]=1\cdot 4-2\cdot 2=0.$$ That's why there is a whole line of points $X$ with $FX=0$. These are stationary points. To find them, we solve this equation: $$\begin{cases}x&+2y&=0,\\ 2x&+4y&=0,\end{cases} \ \Longrightarrow\ x=-2y.$$ We have, then, eigenvectors corresponding to the zero eigenvalue $\lambda _1=0$: $$V_1=\left[\begin{array}{cc}2\\-1\end{array}\right]\ \Longrightarrow\ FV_1=0 .$$
So, second, there is only one non-zero eigenvalue: $$\det (F-\lambda I)=\det\left[ \begin{array}{cc}1-\lambda&2\\2&4-\lambda\end{array}\right]=\lambda^2-5\lambda=\lambda(\lambda-5).$$ Let's find the eigenvectors for $\lambda_2=5$. We solve the equation: $$FV=\lambda_2 V,$$ as follows: $$FV=\left[ \begin{array}{cc}1&2\\2&4\end{array}\right]\left[\begin{array}{cc}x\\y\end{array}\right]=5\left[\begin{array}{cc}x\\y\end{array}\right].$$ This gives as the following system of linear equations: $$\begin{cases}x&+2y&=5x,\\ 2x&+4y&=5y,\end{cases} \ \Longrightarrow\ \begin{cases}-4x&+2y&=0,\\ 2x&-y&=0,\end{cases} \ \Longrightarrow\ y=2x.$$ This line is the eigenspace. We choose the eigenvector to be: $$V_2=\left[\begin{array}{cc}1\\2\end{array}\right].$$
Every solution starts off the line $y=-x/2$ and continues along this vector. It is a linear combination of the two eigenvectors: $$X=CV_1+KV_2=C\left[\begin{array}{cc}2\\-1\end{array}\right]+Ke^{5t}\left[\begin{array}{cc}1\\2\end{array}\right].$$ $\square$
Exercise. Find the line of stationary solutions.
Example (saddle). Let's consider this system of ODEs: $$\begin{cases}x'&=x&+2y,\\ y'&=3x&+2y.\end{cases} $$ Here, the matrix of $F$ is not diagonal: $$F=\left[ \begin{array}{cc}1&2\\3&2\end{array}\right].$$ Euler's method shows the following:
The two lines the solutions appear to converge to are the eigenspaces. Let's find them: $$\det (F-\lambda I)=\det \left[ \begin{array}{cc}1-\lambda&2\\3&2-\lambda\end{array}\right]=\lambda^2-3\lambda-4.$$ Therefore, the eigenvalues are $$\lambda_1=-1,\ \lambda_2=4.$$
Now we find the eigenvectors. We solve the two equations: $$FV_k=\lambda_k V_k,\ k=1,2.$$ The first: $$FV_1=\left[ \begin{array}{cc}1&2\\3&2\end{array}\right]\left[\begin{array}{cc}x\\y\end{array}\right]=-1\left[\begin{array}{cc}x\\y\end{array}\right].$$ This gives as the following system of linear equations: $$\begin{cases}x&+2y&=-x,\\ 3x&+2y&=-y,\end{cases} \ \Longrightarrow\ \begin{cases}2x&+2y&=0,\\ 3x&+3y&=0,\end{cases} \ \Longrightarrow\ x=-y.$$ We choose $$V_1=\left[\begin{array}{cc}1\\-1\end{array}\right].$$ Every solution within this eigenspace (the line $y=-x$) is a multiple of this characteristic solution: $$X_1=e^{\lambda_1t}V_1=e^{-t}\left[\begin{array}{cc}1\\-1\end{array}\right].$$
The second eigenvalue: $$FV_2=\left[ \begin{array}{cc}1&2\\3&2\end{array}\right]\left[\begin{array}{cc}x\\y\end{array}\right]=4\left[\begin{array}{cc}x\\y\end{array}\right].$$ We have the following system: $$\begin{cases}x&+2y&=4x,\\ 3x&+2y&=4y,\end{cases} \ \Longrightarrow\ \begin{cases}-3x&+2y&=0,\\ 3x&-2y&=0,\end{cases}\ \Longrightarrow\ x=2y/3.$$ We choose $$V_2=\left[\begin{array}{cc}1\\3/2\end{array}\right].$$ Every solution within this eigenspace (the line $y=3x/2$) is a multiple of this characteristic solution: $$X_2=e^{\lambda_2t}V_2=e^{4t}\left[\begin{array}{cc}1\\3/2\end{array}\right].$$
The two solutions $X_1$ and $X_2$, as well as $-X_1$ and $-X_2$, are shown below:
The general solution is a linear combination of these two basic solutions: $$X=Ce^{\lambda_1t}V_1+Ke^{\lambda_2t}V_2=Ce^{-t}\left[\begin{array}{cc}1\\-1\end{array}\right]+Ke^{4t}\left[\begin{array}{cc}1\\3/2\end{array}\right]=\left[\begin{array}{cc}Ce^{-t}+Ke^{4t}\\-Ce^{-t}+3/2Ke^{4t}\end{array}\right],$$ i.e., $$\begin{cases}x&=Ce^{-t}&+Ke^{4t},\\ y&=-Ce^{-t}&+3/2Ke^{4t}.\end{cases} $$ The equilibrium is unstable. $\square$
Example (node). Let's consider this system of ODEs: $$\begin{cases}x'&=-x&-2y,\\ y'&=x&-4y.\end{cases} $$ Here, the matrix of $F$ is not diagonal: $$F=\left[ \begin{array}{cc}-1&-2\\1&-4\end{array}\right].$$ Euler's method shows the following:
The analysis starts with the characteristic polynomial: $$\det (F-\lambda I)=\det \left[ \begin{array}{cc}-1-\lambda&-2\\1&-4-\lambda\end{array}\right]=\lambda^2-5\lambda+6.$$ Therefore, the eigenvalues are $$\lambda_1=-3,\ \lambda_2=-2.$$
To find the eigenvectors, we solve the two equations: $$FV_k=\lambda_k V_k,\ k=1,2.$$ The first: $$FV_1=\left[ \begin{array}{cc}-1&-2\\1&-4\end{array}\right]\left[\begin{array}{cc}x\\y\end{array}\right]=-1\left[\begin{array}{cc}x\\y\end{array}\right].$$ This gives as the following system of linear equations: $$\begin{cases}-x&-2y&=-3x,\\ x&-4y&=-3y,\end{cases} \ \Longrightarrow\ \begin{cases}2x&-2y&=0,\\ x&-y&=0,\end{cases} \ \Longrightarrow\ x=y.$$ We choose $$V_1=\left[\begin{array}{cc}1\\1\end{array}\right].$$ Every solution within this eigenspace (the line $y=x$) is a multiple of this characteristic solution: $$X_1=e^{\lambda_1t}V_1=e^{-3t}\left[\begin{array}{cc}1\\1\end{array}\right].$$
The second eigenvalue: $$FV_2=\left[ \begin{array}{cc}-1&-2\\1&-4\end{array}\right]\left[\begin{array}{cc}x\\y\end{array}\right]=-2\left[\begin{array}{cc}x\\y\end{array}\right].$$ We have the following system: $$\begin{cases}-x&-2y&=-2x,\\ x&-4y&=-2y,\end{cases} \ \Longrightarrow\ \begin{cases}x&-2y&=0,\\ x&-2y&=0,\end{cases}\ \Longrightarrow\ x=2y.$$ We choose $$V_2=\left[\begin{array}{cc}2\\1\end{array}\right].$$ Every solution within this eigenspace (the line $y=x/2$) is a multiple of this this characteristic solution: $$X_2=e^{\lambda_2t}V_2=e^{-2t}\left[\begin{array}{cc}2\\1\end{array}\right].$$
The general solution is a linear combination of these two basic solutions: $$X=Ce^{\lambda_1t}V_1+Ke^{\lambda_2t}V_2=Ce^{-3t}\left[\begin{array}{cc}1\\1\end{array}\right]+Ke^{-2t}\left[\begin{array}{cc}2\\1\end{array}\right].$$ The equilibrium is stable. $\square$
Definition. For a linear system $X'=FX$, the equilibrium solution $X_0=0$ is called a stable node if every other solution $X$ satisfies: $$X(t)\to 0\text{ as } t\to +\infty \text{ and }||X(t)||\to \infty \text{ as } t\to -\infty ;$$ and an unstable node if $$X(t)\to 0\text{ as } t\to -\infty \text{ and }||X(t)||\to \infty \text{ as } t\to +\infty ;$$ provided no $X$ makes a full rotation around $0$.
Definition. For a linear system $X'=FX$, the equilibrium solution $X_0=0$ is called a saddle if it has solutions $X$ that satisfy: $$||X(t)||\to \infty \text{ as } t\to \pm\infty .$$
Theorem (Classification of linear systems I). Suppose matrix $F$ has two real eigenvalues $\lambda _1$ and $\lambda_2$. Then,
if $\lambda _1$ and $\lambda_2$ have the same sign, the system $X'=FX$ has a node, stable when this sign is negative and unstable when this sign is positive;
if $\lambda _1$ and $\lambda_2$ have the opposite signs, the system $X'=FX$ has a saddle.
Proof. The stability is seen in either of the two characteristic solutions, as $t\to +\infty$: $$||X||=||e^{\lambda t}V||=e^{\lambda t}\cdot ||V||\to\begin{cases}\infty&\text{if }\lambda>0,\\0&\text{if }\lambda<0.\end{cases}$$ According to the last theorem, we have a linear combination of the two characteristic solutions. Then, in the former case, we have one or the other pattern, and in the latter, both. There can be no rotation because no solution can intersect an eigenspace, according to the uniqueness property. $\blacksquare$
Classification of linear systems, continued
What if the eigenvalues are complex?
Recall that the characteristic polynomial of matrix $F$ is $$\chi(\lambda)=\det (F-\lambda I)=\lambda^2-\operatorname{tr} F\cdot\lambda+\det F$$ The discriminant of this quadratic polynomial is $$D=(\operatorname{tr} F)^2-4\det F.$$ When $D>0$, we have two distinct real eigenvalues, the case addressed in the last section. We are faced with complex eigenvalues whenever $D<0$. The transitional case is $D=0$.
Example (improper node). In contrast to the last example, a node may be produced by a matrix with repeated (and, therefore, real) eigenvalues: $$F=\left[ \begin{array}{cc}-1&2\\0&-1\end{array}\right].$$ Euler's method shows the following:
The analysis starts with the characteristic polynomial: $$\det (F-\lambda I)=\det \left[ \begin{array}{cc}-1-\lambda&2\\0&-1-\lambda\end{array}\right]=(-1-\lambda)^2.$$ Therefore, the eigenvalues are $$\lambda_1=\lambda_2=-1.$$ The only eigenvectors are horizontal. The solution is given by $$X=Ce^{-t}\left[\begin{array}{cc}1\\0\end{array}\right]+K\left( te^{-t}\left[\begin{array}{cc}1\\0\end{array}\right]+e^{-t}\left[\begin{array}{cc}?\\1\end{array}\right] \right).$$ $\square$
Exercise. Finish the computation in the example.
When $D<0$, the eigenvalues are complex! Therefore, there are no eigenvectors (not real ones anyway). Does the system $X'=FX$ even have solutions? The theorem about characteristic solutions says yes, they are certain exponential functions...
Example (center). Consider $$\begin{cases}x'&=y,\\ y'&=-x.\end{cases} $$ We already know that the solution is found by substitution: $$x' '=(x')'=y'=-x.$$ Therefore the solutions are the linear combinations of $\sin t$ and $\cos t$. The result is confirmed with Euler's method (with a limited number of step to prevent the approximations to spiral out):
According to the theory above, the solutions are supposed to be exponential rather than trigonometric. But the latter are just exponential functions with imaginary exponents.
Let's make this specific; we have $$F=\left[ \begin{array}{ccc}0&1\\-1&0\end{array}\right],$$ and the characteristic polynomial, $$\chi(\lambda)=\lambda^2+1,$$ has these complex roots: $\lambda_{1,2}=\pm i$.
To find the first eigenvector, we solve: $$FV_1=\left[ \begin{array}{ccc}0&1\\-1&0\end{array}\right]\left[\begin{array}{cc}x\\y\end{array}\right]=i\left[\begin{array}{cc}x\\y\end{array}\right].$$ This gives as the following system of linear equations: $$\begin{cases}&y&=ix\\ -x&&=iy\end{cases} \ \Longrightarrow\ y=ix.$$ We choose a complex eigenvector: $$V_1=\left[\begin{array}{cc}1\\i\end{array}\right],$$ and similarly: $$V_2=\left[\begin{array}{cc}1\\-i\end{array}\right],$$
The general solution is a linear combination -- over the complex numbers -- of these two characteristic solutions: $$X=Ce^{\lambda_1t}V_1+Ke^{\lambda_2t}V_2=Ce^{it}\left[\begin{array}{cc}1\\i\end{array}\right]+Ke^{-it}\left[\begin{array}{cc}1\\-i\end{array}\right].$$ The problem is solved! ...in the complex domain. What is the real part?
Let $K=0$. Then the solution is: $$X=Ce^{it}\left[\begin{array}{cc}1\\i\end{array}\right]=C\left[\begin{array}{cc}e^{it}\\ie^{it}\end{array}\right]=C\left[\begin{array}{cc}\cos t+i\sin t\\i(\cos t+i\sin t)\end{array}\right]=C\left[\begin{array}{cc}\cos t+i\sin t\\ -\sin t+i\cos t\end{array}\right].$$ Its real part is: $$\operatorname{Re} X=C\left[\begin{array}{cc}\cos t\\ -\sin t\end{array}\right].$$ These are all the circles. $\square$
Example (focus). Let's consider a more complex system ODEs: $$\begin{cases}x'&=3x&-13y,\\ y'&=5x&+y.\end{cases} $$ Here, the matrix of $F$ is not diagonal: $$F=\left[ \begin{array}{cc}3&-13\\5&1\end{array}\right].$$ Euler's method shows the following:
The analysis starts with the characteristic polynomial: $$\chi(\lambda)=\det (F-\lambda I)=\det \left[ \begin{array}{cc}3-\lambda&-13\\5&1-\lambda\end{array}\right]=\lambda^2-4\lambda+68.$$ Therefore, the eigenvalues are $$\lambda_{1,2}=2\pm 8i.$$
Now we find the eigenvectors. We solve the two equations: $$FV_k=\lambda_k V_k,\ k=1,2.$$ The first: $$FV_1=\left[ \begin{array}{cc}3&-13\\5&1\end{array}\right]\left[\begin{array}{cc}x\\y\end{array}\right]=(2+8i)\left[\begin{array}{cc}x\\y\end{array}\right].$$ This gives as the following system of linear equations: $$\begin{cases}3x&-13y&=(2+8i)x\\ 5x&+y&=(2+8i)y\end{cases} \ \Longrightarrow\ \begin{cases}(1-8i)x&-13y&=0\\ 5x&+(-1-8i)y&=0\end{cases} \ \Longrightarrow\ x=\frac{1+8i}{5}y.$$ We choose $$V_1=\left[\begin{array}{cc}1+8i\\5\end{array}\right].$$
The second eigenvalue: $$FV_2=\left[ \begin{array}{cc}3&-13\\5&1\end{array}\right]\left[\begin{array}{cc}x\\y\end{array}\right]=(2- 8i)\left[\begin{array}{cc}x\\y\end{array}\right].$$ We have the following system: $$\begin{cases}3x&-13y&=(2-8i)x\\ 5x&+y&=(2-8i)y\end{cases} \ \Longrightarrow\ \begin{cases}(1+8i)x&-13y&=0\\ 5x&+(-1+8i)y&=0\end{cases}\ \Longrightarrow\ x=\frac{1-8i}{5}y.$$ We choose $$V_2=\left[\begin{array}{cc}1-8i\\5\end{array}\right].$$
The general complex solution is a linear combination of the two characteristic solutions: $$Z=Ce^{\lambda_1t}V_1+Ke^{\lambda_2t}V_2=Ce^{(2+8i)t}\left[\begin{array}{cc}1+8i\\5\end{array}\right]+Ke^{(2-8i)t}\left[\begin{array}{cc}1-8i\\5\end{array}\right].$$
Let's now examine a simple real solution. We let $C=1$ and $K=0$: $$\begin{array}{lll} X=\operatorname{Re} Z&= \operatorname{Re}e^{(2+8i)t}\left[\begin{array}{cc}1+8i\\5\end{array}\right]\\ &=e^{2t}\operatorname{Re}e^{8it}\left[\begin{array}{cc}1+8i\\5\end{array}\right]\\ &=e^{2t}\operatorname{Re}(\cos 8t+i\sin 8t)\left[\begin{array}{cc}1+8i\\5\end{array}\right]\\ &=e^{2t}\operatorname{Re}\left[\begin{array}{cc}(\cos 8t+i\sin 8t)(1+8i)\\(\cos 8t+i\sin 8t)5\end{array}\right]\\ &=e^{2t}\operatorname{Re}\left[\begin{array}{cc}\cos 8t+i\sin 8t+8i\cos 8t-8\sin 8t\\5\cos 8t+i5\sin 8t\end{array}\right]\\ &=e^{2t}\left[\begin{array}{cc}\cos 8t-8\sin 8t\\5\cos 8t\end{array}\right]. \end{array}$$ Plotting this parametric curve confirms Euler's method result:
Definition. For a linear system $X'=FX$, the equilibrium solution $X_0=0$ is called a stable focus if every other solution $X$ satisfies: $$X(t)\to 0\text{ as } t\to +\infty \text{ and }||X(t)||\to \infty \text{ as } t\to -\infty ;$$ and an unstable focus if $$X(t)\to 0\text{ as } t\to -\infty \text{ and }||X(t)||\to \infty \text{ as } t\to +\infty ;$$ provided every such $X$ makes a full rotation around $0$.
Definition. For a linear system $X'=FX$, the equilibrium solution $X_0=0$ is called a center if all solutions are cycles.
Theorem (Classification of linear systems II). Suppose matrix $F$ has two complex conjugate eigenvalues $\lambda _1$ and $\lambda_2$. Then,
if the real part of $\lambda _1$ and $\lambda_2$ is non-zero, the system $X'=FX$ has a focus, stable when this sign of this number is negative and unstable when this sign is positive;
if the real part of $\lambda _1$ and $\lambda_2$ is zero, the system $X'=FX$ has a center.
Proof. The stability is seen in either of the two characteristic solutions, as $t\to +\infty$: $$||X||=||e^{\lambda t}V||=||e^{(a+bi) t}V||=e^{at} |\cos bt+i\sin bt|\cdot ||V||=e^{at}||V||\to\begin{cases}\infty&\text{if }a>0,\\0&\text{if }a<0.\end{cases}$$ $\blacksquare$
The combination of the two classification theorems is illustrated below:
Thereby we complete the sequence: elementary algebra $\longrightarrow$ matrix algebra $\longrightarrow$ linear differential equations. To summarize, in order to classify a system of linear ODEs $X'=FX$, where $F$ is a $2\times 2$ matrix and $X$ is a vector on the plane, we classify $F$ according to its eigenvalues and visualize how the locations of these two numbers in the complex plane indicate very different behaviors of the trajectories. (The missing patterns are better illustrated dynamically, as the exact "moments" when one pattern transitions into another.)
Exercise. Point out on the complex plane the locations of the center and the improper node.
Exercise. How likely would a given system fall into each of these five categories? What about the center and the improper node?
Exercise. What parameters determine the clockwise vs. counter-clockwise behavior?
Example (predator-prey). Let's classify the equilibria of the predator-prey model -- via linearization. Our non-linear system is given by the following: $$\begin{cases} x' = \alpha x &- \beta x y ,\\ y' = \delta xy &- \gamma y , \end{cases}$$ with non-negative coefficients $\alpha x,\ \beta,\ \delta,\ \gamma$. In other words, we have $$X'=G(X),\text{ with } G(x,y)=(\alpha x - \beta x y ,\delta xy - \gamma y).$$ The Jacobian of $G$ is $$G'(x,y)= \left[\begin{array}{cc} \frac{\partial}{\partial x}(\alpha x - \beta x y)&\frac{\partial}{\partial y}(\alpha x - \beta x y)\\ \frac{\partial}{\partial x}(\delta xy - \gamma y)&\frac{\partial}{\partial y}(\delta xy - \gamma y) \end{array}\right]=\left[\begin{array}{cc} \alpha - \beta y&-\beta x \\ \delta y &\delta x - \gamma \end{array}\right].$$ The matrix depends on $(x,y)$ because the system is non-linear. By fixing locations $X=A$, we create linear vector ODEs: $$X'=G'(A)X.$$
First, we consider the zero equilibrium, $$x=0,\ y=0.$$ Here, $$F=G'(0,0)=\left[\begin{array}{cc} \alpha - \beta \cdot0&-\beta \cdot0 \\ \delta \cdot0 &\delta\cdot 0 - \gamma \end{array} \right] =\left[\begin{array}{cc} \alpha &0 \\ 0& - \gamma \end{array} \right].$$ The eigenvalues are found by solving the following equation: $$\det \left[\begin{array}{cc} \alpha-\lambda &0 \\ 0& - \gamma-\lambda \end{array} \right]=(\alpha-\lambda)(- \gamma-\lambda)=0.$$ Therefore, $$\lambda_1=\alpha,\ \lambda_2=-\gamma.$$ We have two real eigenvalues of opposite signs. This is a saddle! Indeed, around this point the foxes decline while the rabbits increase in numbers.
The main equilibrium is $$x=\frac{\gamma}{\delta},\ y=\frac{\alpha}{\beta}.$$ Here, $$F=G'\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right)=\left[\begin{array}{cc} \alpha - \beta \cdot \frac{\alpha}{\beta} &-\beta \cdot \frac{\gamma}{\delta} \\ \delta \cdot \frac{\alpha}{\beta} &\delta\cdot \frac{\gamma}{\delta} - \gamma \end{array}\right] =\left[\begin{array}{cc} 0 &- \frac{\beta\gamma}{\delta} \\ \frac{\delta\alpha}{\beta} &0 \end{array}\right] .$$ The eigenvalues are found by solving the following equation: $$\det \left[\begin{array}{cc} -\lambda &- \frac{\beta\gamma}{\delta} \\ \frac{\delta\alpha}{\beta} &-\lambda \end{array}\right]=\lambda^2+ \alpha\gamma=0.$$ Therefore, $$\lambda_1=\sqrt{\alpha\gamma}\,i,\ \lambda_2=-\sqrt{\alpha\gamma}\,i.$$ We have two purely imaginary eigenvalues. This is a center! Indeed, around this point we have a cyclic behavior.
The results match our previous analysis. $\square$
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XIII Mexican School of Particles and Fields
2-11 October 2008
San Carlos, Sonora, México
America/Hermosillo timezone
XIII MSPF Portal
[email protected]
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Displaying 196 contributions out of 196
A new approach to flavor?
Session: Beyond SM
The origin of flavor is one of the biggest mysteries of the Standard Model. Despite more than a half-century of data, we still have essentially no explanation for family replication in the Standard Model. Many theoretical models can incorporate or accommodate this replication, but very few actually explain/predict/require it. In this talk, I will describe some recent work on a possible new approac ... More
Presented by Keith DIENES on 11 Oct 2008 at 12:00
A possible Grand Unification Theory with 331 models.
Session: Posters
Presented by Dr. Ricardo GAITAN on 9 Oct 2008 at 19:00
A review of selected next-generation neutrino-less double-beta decay experiments
Session: Review Talks
The basic particle physics and nuclear physics concepts of 0\nu\beta\beta-decay will be discussed, followed by a discussion of experimental techniques. There are five next generation experiments, all with different technologies, that are under construction or in various stages of advanced research and development. The advantages and disadvantages of each will be discussed, as well as their status ... More
Presented by Dr. Frank AVIGNONE on 7 Oct 2008 at 10:00
A study of confinement and dynamical chiral symmetry breaking in QED3
Presented by Mr. Saúl SÁNCHEZ MADRIGAL on 9 Oct 2008 at 19:00
Acceleration and energy loss in strongly-coupled gauge theories
Session: String Theory
Presented by Dr. Alberto GUIJOSA on 7 Oct 2008 at 18:30
Acquisition electronics for cherenkov detectors
Presented by Mario CASTILLO MALDONADO
An analysis of predictability of recent alpha decay half-life formulae and the alpha partial half-lives of some exotic nuclei
Presented by Nabanita DASGUPTA-SCHUBERT on 9 Oct 2008 at 19:00
An introduction to high energy nuclear collisions I
Presented by Dr. Jamal JALILIAN-MARIAN on 2 Oct 2008 at 11:30
An introduction to high energy nuclear collisions II
An introduction to high energy nuclear collisions III
Analytical description of neutrino oscillations in the earth
Session: Neutrino Physics
We derive an analytical description of neutrino oscillations in matter based on the Magnus exponential representation of the time evolution operator. Our approach is valid in a wide range of the neutrino energies and properly accounts for the modifications that the respective probability transitions suffer when neutrinos originated in different sources traverse the Earth. The present approximation ... More
Presented by Dr. Daniel SUPANITSKY on 7 Oct 2008 at 19:00
Anomalies, Beta Functions and Grand Unification with Higher-dimensional Higgs Representations
Presented by Alma Dolores ROJAS PACHECO on 9 Oct 2008 at 19:00
Aspects of strong coupled non-conformal gauge theories at finite temperature
Presented by Dr. Alexander BUCHEL on 6 Oct 2008 at 12:30
Auger Norte: technical design and scientific motivation
Auger Sur and the AMIGA & HEAT enhancements
Auger at the "Red de Altas Energia"
Auger en el CINVESTAV
Auger en la BUAP
Auger en la UMSNH
Auger en la UNAM
BATATA muon telescope: numerical simulations
Session: Cosmic Rays
Presented by Dr. Federico Andres SANCHEZ
Baryons from quarks in color gauge space of constant positive curvature and deconfinement
Session: Non-Perturbative QFT
Detailed account is given of the fact that the Cornell potential predicted by Lattice QCD and its recently reported exactly solvable extension to a trigonometric quark confinement potential can be placed within the context of a Coulomb-type potential in color gauge space of constant positive curvature, the 3D hypersphere. We make the case that the geometric vision on confinement provides a remarka ... More
Presented by Mariana KIRCHBACH on 10 Oct 2008 at 17:10
Despite extensive searching at LEP and the Tevatron, the Standard Model still provides an extremely accurate description of the fundamental particles and their interactions. Yet the Standard Model raises many questions, and theories of what might lie beyond it are numerous. We will focus on the big questions: what is the nature of the Higgs boson in theories beyond the standard model and what do w ... More
Presented by Dr. John CONWAY on 11 Oct 2008 at 10:00
Bose Einstein condesates as unified cosmology
Session: Cosmology
In this talk we present some ideas how to unify the early Universe, that means the inflationary epoch, with the late universe, when the large scale structure formed. If the inflaton decays into another scalar field with a smaller mass, this new scalar field can work as the dark matter. In a SO(1,1) isospin unification, the scalar field can be the dark energy as well. We present the evolution of th ... More
Presented by Dr. Tonatiuh MATOS on 8 Oct 2008 at 17:00
Bottomonium spectroscopy and the eta_b discovery at BaBar
Session: Hadronic Physics
Thirty years after the discovery of the Upsilon resonances and the bottom quark, BaBar has made the first observation of the bottomonium ground state, the eta_b. I will present details of the observation. In addition I will discuss other recent BaBar Bottomonium analyses.
Presented by Dr. Silke NELSON on 7 Oct 2008 at 19:00
Bs Physics at D0
The results of the Bs meson measurement in the D0 experiment are showed. This include the determination of the deltaGamma between the two states the Light and the Heavy, the determination of the Delta Ms the difference between the two mass states, the measurement of the lifetimes in hadronic and semileptonic modes. The very new measurement of the phases and amplitudes in the transversity basis are ... More
Presented by Dr. Pedro PODESTA on 7 Oct 2008 at 18:30
CP violating effects in the ZZV vertex induced by flavor changing neutral currents
Presented by A MOYOTL ACUAUITL on 9 Oct 2008 at 19:00
Characterizing the medium in heavy ion collisions through jets
Session: RHI
I will discuss the possibilities to characterize the medium created in heavy ion collisions at RHIC and the LHC. I will focus on the opportunities that jets offer for such characterization, and comment on the problems related with background substraction.
Presented by Dr. Nestor ARMESTO on 6 Oct 2008 at 16:50
Charged leptonic and quarks sector in several SM extensions of with discrete symmetries
Presented by Eduardo Jonathan TORRES HERRERA on 9 Oct 2008 at 19:00
Charmless hadronic two-body decays of Bs meson including axial-vector mesons in the final state
Presented by Carlos Eduardo VERA AGUIRRE on 9 Oct 2008 at 19:00
Chiral phase transition in peripheral heavy-ion collisions
It is been recently realized that in peripheral heavy-ion collisions, a sizable magnetic field is produced in the interaction region. Although this field becomes weak at the proper times when the chiral phase transition is believed to occur, it is still significant so as to ask whether it influences such transition. In this talk, we use the linear sigma model to study the chiral phase transition i ... More
Presented by Angel SÁNCHEZ on 7 Oct 2008 at 16:50
Collisional parton energy loss in a finite size QCD medium reexamined: Off-mass-shell effects.
Presented by Dr. Luis Manuel MONTANO ZETINA on 9 Oct 2008 at 19:00
Compactifications with torsion: moving beyond Calabi-Yau
Presented by Dr. Brian WECHT on 8 Oct 2008 at 18:30
Conformal hydrodynamics beyond the supergravity approximation
Conformal symmetry on the lattice
The conformal window in a gauge field theory with N_f light fermions is the range of N_f values such that the theory is asymptotically free and the infrared coupling is governed by a fixed point. In an SU(N) gauge theory with N_f fermions in the fundamental representation, the conformal window ranges from 11N/2 down to some critical value N_c at which a transition is expected to one in which there ... More
Presented by Dr. Thomas APPELQUIST on 10 Oct 2008 at 10:00
Constraining dark energy with neutrino physics
We consider the impact on the properties of the dark energy component, which dominates the current expansion of the universe, due to the effects of neutrino physics. By including a large mass of the neutrinos of the order of $\Sigama_i m_i\simeq 1.8 eV$, as claimed by members of the Heidelberg-Moscow double beta decay experiment, we find that a cosmological constant is ruled out at more than $95 \ ... More
Presented by Axel DE LA MACORRA PETTERSSON on 8 Oct 2008 at 19:00
Construction of a Didactic Cloud Chamber for Particle Tracking
Presented by Dr. Edgar CASIMIRO LINARES, Dr. Edgar CASIMIRO LINARES on 9 Oct 2008 at 19:00
Cosmic antimatter, more details and phenomena
More detailed discussion of antimatter effects in cosmology is presented. Some new phenomena are considered.
Presented by Dr. Alexander DOLGOV on 10 Oct 2008 at 16:00
Cosmic rays and the sun
Cosmic rays composition determination methods
Presented by Dr. Daniel SUPANITSKY
Cosmological baryogenesis. Matter and antimatter in the universe.
Existing mechanisms of creation of cosmological excess of matter over antimatter are reviewed. Particular attention is paid to the models leading to significant amount of antimatter in the universe and its observational signatures.
Presented by Dr. Alexander DOLGOV on 9 Oct 2008 at 09:00
Cosmological simulations: the role of scalar fields
We present numerical N-body simulation studies of large-scale structure formation. The main purpose of these studies is to analyze the several models of dark matter and the role they played in the process of large-scale structure formation. We analyze the standard and more successful case, i.e., the cold dark matter with cosmological constant (LCDM). We compare the results of this model with the c ... More
Presented by Mario Alberto RODRIGUEZ-MEZA on 6 Oct 2008 at 16:00
DAQ System to Detect GRBs at the Sierra Negra High Altitude Observatory Using Water Cerenkok Detectors
Presented by Dr. Humberto SALAZAR on 9 Oct 2008 at 19:00
DPyC Medal Award in Honor of Matias Moreno
Session: DPyC Medal Award Ceremony
on 8 Oct 2008 at 11:50
D^+ → K^{-} p^{+} p^{+} and the elastic K p amplitude
Heavy-meson decays are important sources of information about scalar resonances. Hopefully, they may also shed light into meson-meson scattering amplitudes. In a recent work, our group has studied the low-energy sector of the reaction D+ → K- p+ p+ by means of chiral SU(3)×SU(3) effective lagrangians, which include scalar resonances and allow a consistent treatment of the primary weak vertex. I ... More
Presented by Dr. Manoel Roberto ROBILOTTA on 7 Oct 2008 at 17:00
Detection of nclined cosmic rays
Presented by Alejandra PARRA FLORES
Diffractive processes in electron-proton and proton-proton collisions
Presented by Ms. Agnieszka LUSZCZAK on 9 Oct 2008 at 19:00
Discussion session
on 10 Oct 2008 at 19:00
Educational impact, outreach and connection with the industry
Effective gluon propagator from a trigonometric quark confinemet potential
Presented by Cliffor Benjamin COMPEAN JASSO on 9 Oct 2008 at 19:00
Effective theory approach to portly neutrinos: theory and application
I will discuss the effects of heavy Majorana neutrinos with sub-TeV masses. I will argue that the mere presence of these particles would be a signal of physics beyond the minimal seesaw mechanism. Using an effective Lagrangian approach I will describe the most important interactions of these particles and discuss to what extent these interactions can be probed at the LHC.
Presented by Dr. Jose WUDKA on 8 Oct 2008 at 17:00
Electric dipole moment of the quark top within an effective theory
Presented by Hector NOVALES SANCHEZ on 9 Oct 2008 at 19:00
Electron-positron annihilation into $\\phi a_{0}(980)$
Presented by Carlos VAQUERA on 9 Oct 2008 at 19:00
Electroweak scale neutrinos and Higgses
Based on the fact that the only experimental data we have so far for physics beyond the Standard Model is that of neutrino physics, we impose a constraint for any addition not to introduce new higher scales. We present a model consistent with this restriction that includes electroweak scale right-handed neutrinos and a lepton number violating singlet scalar field. We also discuss some of its inter ... More
Presented by Dr. Alfredo ARANDA on 7 Oct 2008 at 18:30
Electroweak symmetry breaking at the LHC (lecture I)
I will give a pedagogical review of the Standard Model predictions for Higgs boson production and decay at the LHC. The theoretical shortcomings of the single Higgs boson model will be discussed and some alternative mechanisms of electroweak symmetry breaking presented.
Presented by Dr. Sally DAWSON on 3 Oct 2008 at 11:30
Electroweak symmetry breaking at the LHC (lecture II)
Electroweak symmetry breaking at the LHC (lecture III)
Electroweak theory basics I
This series of lectures are oriented mainly to particle physics students, covering, on a pedestrian level, the basics and some general model buildings aspects of the Standard Model of the electroweak interactions.
Presented by Dr. Abdel PÉREZ-LORENZANA on 2 Oct 2008 at 09:00
Electroweak theory basics II
Electroweak theory basics III
Evolution of the CKM matrix for the SM, 2HDM and MSSM models
Presented by Jose Halim MONTES DE OCA YEMHA on 9 Oct 2008 at 19:00
Fermionic amplitudes an alternative approch
Several methods for the computation of the general spin 1/2 amplitude are studied.
Presented by Dr. Matías MORENO on 9 Oct 2008 at 19:00
Field theoretic description of the abelian and non-abelian josephson effect
We formulate the Josephson effect in a field theoretic language which affordsa straightforward generalization to the non-abelian case. Our formalisminterprets Josephson tunneling as the excitation of pseudo-Goldstone bosons. We discuss applicationsto various non-abelian symmetry breaking systems in particle and condensedmatter physics.
Presented by Rohana WIJEWARDHANA on 10 Oct 2008 at 16:20
Financing avenues and prospects in Mexico for inversion and operation costs
Fine Structure Constant and Redshift
Presented by Dr. Le LE DUC THONG on 9 Oct 2008 at 19:00
Finite width effects in decay constants of vector and axial-vector mesons
We calculate decay constants of vector and axial-vector mesons from decays $\tau^-\to V^-\nu_\tau$ and $\tau^-\to A^-\nu_\tau$, respectively, considering the effects of the anstability of vector and axial-vector mesons, and the case of strange axial-mesons the effect of $K_{1A}-K_{1B}$ mixing. Moreover, we calculate the impact in some branching ratios of $B$ decays.
Presented by Dr. German CALDERON on 10 Oct 2008 at 17:30
First physics in ALICE
The ALICE experiment is intensively pursuing preparation for first physics data with the Large hadron Collider. After a brief introduction to the ALICE experiment and its present status the following topics will be addressed:. The wide scope of detector related and reconstruction linked issues like the alignements, reconstruction efficiency, systematic errors etc. The operation of the LHC in ... More
Presented by Dr. Guy PAIC on 7 Oct 2008 at 19:20
Flavor and Higgs Physics in Randall-Sundrum Models
Presented by Alfonso DIAZ FURLONG on 9 Oct 2008 at 19:00
Flavor in SUGRA Backgrounds: Beyond the Probe Approximation
Presented by Dr. Elena CACERES on 8 Oct 2008 at 17:00
Formación de estructuras cósmicas: ¿partículas exóticas frias o tibias?
Se presentará una reseña sobre los principales logros y debilidades del escenario actual de formación y evolución de estructuras cósmicas, principalmente de las galáxias. La reseña estará orientada a las implicaciones que tiene el tipo de materia oscura sobre las propiedades y evolución de las galáxias. Se trataran en particular los casos de partículas exóticas frias y tibias.
Presented by Dr. Vladimir AVILA-REESE
Fundamental an composite scalars from extra dimensions
In this talk we discuss a scenario consisting of an effective 4D theory containing fundamental and composite fields. The strong dynamics sector responsible for the compositeness is assumed to be of extra dimensional origin. In the 4D effective theory the SM fermion and gauge fields are taken as fundamental fields. The scalar sector of the theory resembles a bosonic topcolor in the sense there are ... More
Presented by Dr. Roberto NORIEGA PAPAQUI on 10 Oct 2008 at 18:30
GRID/Computational facilities for astroparticle physics
Gauge Transformations as space-time symmetries
Presented by René ANGELES on 9 Oct 2008 at 19:00
Gauge-Higgs unification in warped geometries
It is well known that the Higgs sector is still a lacking piece of the Standard Model. This sector governs the electroweak symmetry breaking and gives masses of quarks and leptons. Furthermore, the quadratic divergent correction to the Higgs mass strongly suggest the existence of new physics at the TeV scale. For this purpose, a lot of scenarios beyond the Standard Model have been proposed. In thi ... More
Presented by Dr. Linares ROMAN on 10 Oct 2008 at 17:15
General gauge mediation
I will discuss the framework of General Gauge Mediation developed with Seiberg and Shih to investigate predictions of all Gauge Mediated models.
Presented by Patrick MEADE on 11 Oct 2008 at 19:15
Gravitational Modification of Breit Hamiltonian
Presented by Jose Alexander CAICEDO on 9 Oct 2008 at 19:00
HAWC/Lago
Hamiltonian perturbative approach for high order theories
Presented by Ana A. AVILEZ LOPEZ on 9 Oct 2008 at 19:00
Hard probes in heavy-ion collisions at RHIC
I will review the status of hard probes at RHIC. I will start with a brief introduction of the benchmark in pp and dAu collisions. Then I will discuss particle production at high transverse momentum, and photon and quarkonium production in AA collisions, with emphasis on the proposed theoretical explanations. I will end with a briefly outlook at the forthcoming experiments at the LHC.
Heavy Quark Physics and CP Violation I
Presented by Dr. Jeffrey RICHMAN on 3 Oct 2008 at 10:00
Heavy Quark Physics and CP Violation II
Heavy Quark Physics and CP Violation III
Heavy scalar mesons and B decays
Presented by Carlos RAMIREZ on 10 Oct 2008 at 18:30
Higgs Branching Ratios to cc-bar at the ILC
Presented by Yambazi BANDA on 9 Oct 2008 at 19:00
Higgs physics at the terascale
The discovery of a Higgs-like particle at the LHC is highly anticipated. I will review the status of current limits on Higgs boson properties and discuss what we can expect to learn about Higgs couplings at the LHC. The possibility of distinguishing between a Standard Model Higgs boson and other mechanisms for electroweak symmetry breaking will be examined.
Higgs sector of the SM &otimes S3 flavour symmetry
Presented by Ezequiel RODRIGUEZ JAUREGUI on 9 Oct 2008 at 19:00
High energy astrophysical neutrinos
The quest for astrophysical neutrinos with energies above TeV`s is the major goal of several astroparticle physics experiments around the world. The detection of these elusive particles would open a new astronomical window to the universe at high energies, in particular, to very hot and distant regions of the cosmos not accessible with TeV´s and ultra high energy cosmic rays. In this talk, a br ... More
Presented by Dr. Juan Carlos ARTEAGA-VELÁZQUEZ, Dr. Juan Carlos ARTEAGA VELAZQUEZ on 8 Oct 2008 at 17:30
High precision standard model physics
Sometime $14 \times 10^9$ years ago, the Universe started in a Big Bang. In that moment, the huge amount of energy liberated in the explosion coaleced forming equal quantities of matter amd antimatter. However, as the Universe expanded and cooled, its composition changed in such a way that, at about one second after the explosion, all the antimatter just disapeared, leaving matter to form the Univ ... More
Presented by Dr. Javier MAGNIN on 9 Oct 2008 at 11:30
Holographic dark matter
In this talk we discuss the viability of a dark matter candidate, called the lightest holographic particle (LHP), that arises within the context of dual 5D AdS/CFT models, where the Higgs boson is identified as a psudogoldstone boson of a (conformal) strongly interacting sector. Constraints on the model from collider physics and cosmology are discussed.
Presented by Dr. Lorenzo DIAZ-CRUZ on 8 Oct 2008 at 18:30
Induced gravity: dark energy and dark matter
We study the recent cosmic evolution using the induced gravity theory in which a Higgs field non-minimally couples to gravity. After the symmetry breaking the field goes to its true vacuum value, a process that induces the creation of a new Higgs particle that can account as dark matter. On the other hand, the evolution of the Higgs field acts as a quintessence field. Taking into account the exper ... More
Presented by Dr. Jorge CERVANTES on 7 Oct 2008 at 17:00
Inflation-Dark Energy Unification via Quantum Regeneration
Instrumentation of Position Sensitive Photomultiplier Tubes Hamamatsu H8500
Presented by Uvaldo REYES on 9 Oct 2008 at 19:00
Invisible Decays of Supersymmetric Higgs Bosons.
Presented by Maria del Rocio APARICIO MENDEZ, Ms. María del Rocío APARICIO MÉNDEZ on 9 Oct 2008 at 19:00
JEM-EUSO / Super-EUSO
Jet quenching in heavy ion collisions from AdS/CFT
Presented by Dr. José Daniel EDELSTEIN on 7 Oct 2008 at 16:00
Jet reconstruction at the LHC (lecture I)
The main goal of this series of mini-lectures is to provide an inside look at the reconstruction of particle jets in hadronic final states in the proton-proton collisions at the Large Hadron Collider (LHC), which is expected to begin operations for physics end of September 2008 at CERN, the European Center for Particle Physics Research. These jets are produced in basically all collision channels, ... More
Presented by Dr. Peter LOCH on 4 Oct 2008 at 12:30
Jet reconstruction at the LHC (lecture II)
Jet reconstruction at the LHC (lecture III)
KASCADE Grande
Lattice QCD advances in baryon physics
Recent years have seen significant advances in numerical studies of QCD. Particularly in the heavy-quark sector, lattice QCD has matured into a precision tool for confronting experimental measurements of nonperturbative QCD. In the light-quark sector, and especially in the case of baryon properties, the field has not yet reached this level of precision. Upon a brief introduction to the methods of ... More
Presented by Daniel ROSS YOUNG on 11 Oct 2008 at 09:00
Learning from the past with a peek into the future
The ultrarelativistic heavy ion collisions have completed 25 years of continuous efforts in theory and experiments. In this years a jump of a factor of ~20 in the center of mass energy has been achieved, and today we are looking to the probably last enrgy jump of ~25 to happen at the Large Hadron Collider (LHC) in CERN. The wealth of data accumulated is enormous and has open some windows in our k ... More
Lee model and Bogoliubov transformations
Presented by Vladimir CUESTA SÁNCHEZ on 9 Oct 2008 at 19:00
Left-right symmetric model with μ-τ symmetry.
Presented by Juan Carlos GOMEZ IZQUIERDO on 9 Oct 2008 at 19:00
Lepton flavour violating processes in Minimal S_3-invariant extension of the Standard Model
A variety of lepton flavour violating effects related to the recent discovery of neutrino oscillations and mixing is here systematically discussed in terms of an S3 flavour permutational symmetry. After a brief updated review of some relevant results on neutrino masses and mixings, that had been derived in the framework of a minimal S3-invariant extension of the Standard Model, we will give explic ... More
Presented by Prof. Alfonso MONDRAGON BALLESTEROS on 7 Oct 2008 at 17:30
Light-front holography and novel QCD phenomena
The AdS/CFT correspondence between Anti-de Sitter space and conformal gauge theories provides an analytically tractable approximation to QCD in the regime where the QCD coupling is large and constant. "Light-front Holography" is a remarkable feature of AdS/QCD: it allows hadronic amplitudes in the AdS fifth dimension to be mapped to frame-independent light-front wavefunctions of hadrons in phy ... More
Presented by Dr. Stan BRODSKY on 10 Oct 2008 at 09:00
MSSM with explicit CP-Violation in the Higgs sector
Presented by Shoaib MUNIR on 9 Oct 2008 at 19:00
Mass spectrum and unification in a B-L extended standard model
Presented by Roger Jose; HERNANDEZ PINTO on 9 Oct 2008 at 19:00
Masses, mixings and degeneracy of a neutral Higgs mixture H-A
Presented by Melina GOMEZ BOCK on 9 Oct 2008 at 19:00
Miguel Angel Peres Angon: what is that really remains?
Modelin the galactic and extragalactic transition from maximum shower depth and energy spectrum data
Presented by Dr. Cinzia DE DONATO
Neutrino mass seesaw version 3: resent developments
The origin of neutrino mass is usually attributed to a seesaw mechanism, either through a heavy Majorana fermion singlet (version 1) or a heavy scalar triplet (version 2). Recently, the idea of using a heavy Majorana fermion triplet (version 3) has gained some attention. This is a review of the basic idea involved, its U(1) gauge extension, and some recent developments.
Presented by Dr. Ernest MA on 7 Oct 2008 at 16:00
Neutrino physics: present and future
Presented by Dr. Boris KAYSER on 7 Oct 2008 at 09:00
Neutrinos phenomenology in OHT-MSSM
Presented by Rodolfo Enrique BARRADAS PALMEROS on 9 Oct 2008 at 19:00
New physics effects in top quark interactions
The quark top stands out as the heaviest elementary particle known to date. Because its mass is of order of the electroweak symmetry breaking (EWSB)energy scale, this article is expected to play an essential role in the mechanism of EWSB. In particular, since the top quark lives very shortly and almost all the time decays into a b quark and a W boson, as predicted by the standard model (SM), new p ... More
Presented by Miguel Angel PEREZ on 11 Oct 2008 at 13:45
New physics with ultra-high energy cosmic rays
Presented by Dr. Lorenzo DIAZ-CRUZ
Non-perturbative QCD modeling and meson physics
An account will be given of recent results and issues arising from QCD modeling embedded in the ladder-rainbow truncation of the Dyson-Schwinger equations of QCD. The emphasis will be upon meson properties and decays. Specific topics will include dynamical chiral symmetry breaking, light flavor mixing in singlet and non-singlet channels, and extensions to heavy quark mesons.
Presented by Peter TANDY on 9 Oct 2008 at 16:20
Non-standard neutrino interactions and low energy neutrino experiments
Non-standard interactions (NSI) arise naturally in different models of physics beyond the Standard Model, and they can have an important influence in the solar neutrino analysis. In this talk, after a discussion of the solar neutrino data I will concentrate in present and future low-energy experiments and their sensitivity to NSI. The current status of NSI constraints will be introduced and it wil ... More
Presented by Dr. Omar MIRANDA on 6 Oct 2008 at 16:00
Nuclear Dependence of Charm Production
Presented by Ernesto Alejandro BLANCO COVARRUBIAS on 9 Oct 2008 at 19:00
Numerical analysis of quark-gluon vertex in a momentun subtraction scheme
Presented by Xiomara GUTIERREZ GUERRERO on 9 Oct 2008 at 19:00
On the condensed matter analog of baryon chiral perturbation theory
It is shown that baryon chiral perturbation theory, i.e., the low-energy effective theory for pions and nucleons in QCD, has its condensed matter analog: A low-energy effective theory describing magnons as well as holes (or electrons) doped into antiferromagnets. We briefly present a symmetry analysis of the Hubbard and $t-J$-type models, and review the construction of the leading terms in the eff ... More
Presented by Christoph HOFMANN on 11 Oct 2008 at 17:10
Paths beyond the standard theory
The Standard Model of particle physics represents a remarkable achievement on the search for an understanding of the fundamental interactions and constituents of matter. Nevertheless, it has a limited capability to answer some of our more fundamental questions about the nature of physics at shorter distances. Thus, many theorist do believe the Standard Model is but an intermediate stage on a longe ... More
Presented by Dr. Abdel PEREZ-LORENZANA on 10 Oct 2008 at 11:30
Performance of the V0A trigger detector for the ALICE first physics program at the LHC
Presented by Dr. Andrés SANDOVAL on 9 Oct 2008 at 16:50
Perturvative and non-perturbative phenomena in QED: the scattering by a solenoidal magnetic field
The confidence we have now-a-days in the quantum field theory stems, between other important results, from the fact that it reduces to the classical calculations in the appropiate regimes. This fact has been exemplified in more elementary textbooks of the area for the case of the Coulombian scattering: the differential cross section of the scattering in the lowest order in perturbation theory, equ ... More
Presented by Gabriela MURGUÍA on 11 Oct 2008 at 18:30
Phase transition dynamics and gravitational waves
During a first-order phase transition, gravitational radiation is generated either by bubble collisions or by turbulence. For phase transitions which took place at the electroweak scale and beyond, the signal is expected to be in the sensitivity range of interferometers such as LISA or BBO. In this talk we review the generation of gravitational waves and discuss the dependence of the spectrum on t ... More
Presented by Ariel MEGEVAND on 11 Oct 2008 at 19:30
Phi K K production in electron-positron anihilation
Presented by Mr. Selim GÓMEZ on 9 Oct 2008 at 19:00
Physics beyond the standard model and high energy cosmic rays
Presented by Prof. Arnulfo ZEPEDA
Predictions of finite unified theories
Finite Unified Theories (FUTs) are N=1 supersymmetric Grand Unified Theories that can be made all-loop finite. The requirement of all-loop finiteness leads to a severe reduction of the free parameters of the theory and, in turn, to a large number of predictions. We investigate these theories in the context of low-energy phenomenology observables. We present a detailed scanning of the all-loop fini ... More
Presented by Dr. Myriam MONDRAGON CEBALLOS on 10 Oct 2008 at 19:15
Probing saturation physics at RHIC and LHC
I will review the latest developments in the high energy limit of QCD and describe the experimental signatures of this new and novel kinematic regime. It is expected that a hadron or nucleus at high energy becomes a Color Glass Condensate (CGC), a dense system of gluons which controls the high energy limit of hadronic/nuclear collisions. I focus on single inclusive particle production in deuteron ... More
Presented by Dr. Jamal JALILIAN-MARIAN on 10 Oct 2008 at 19:20
Properties of single and double charm Hadrons
We will review experimental results obtained in the last few years on charmed hadrons, both mesons and baryons. The B-Factories discovered several charmonion-like states, the nature of some of them still under discussion. New $D_s$ states where observed by several experiments. In the baryon sector, new studies on known baryons were performed as well as several new (excited) states were found.
Presented by Dr. Jurgen ENGELFRIED on 7 Oct 2008 at 17:30
QCD at the LHC (lecture I)
Lecture 1: Parton description of hadron-hadron collisions Deep inelastic scattering + measurement of parton distributions Altarelli-Parisi evolution Lecture 2: Jets in e+e- Altarelli-Parisi evolution of jets Jets in hadron-hadron collisions Parton showers Lecture 3: A littl ... More
Presented by Dr. Michael PESKIN on 3 Oct 2008 at 09:00
QCD at the LHC (lecture II)
QCD at the LHC (lecture III)
Quantum criticality in AdS/CFT
Many low dimensional condensed matter systems exhibit zero temperature phase transitions driven by quantum fluctuations. These are known as "quantum" phase transitions. The description of such transitions is beyond the usual Landau-Ginzburg-Wilson paradigm of a single order parameter. At criticality, such systems exhibit novel behavior including the appearance of fractional statistics, ga ... More
Presented by Dr. Samuel VAZQUEZ on 6 Oct 2008 at 16:00
Quark Dynamics from AdS/CFT
Presented by Eric Josafat PULIDO PADILLA on 9 Oct 2008 at 19:00
Quark flavor mixing and mass matrices
Some implications of the textures of the mass matrices for the flavor mixing matrix V are reviewed. Constraints on the structure of the mass matrices are given using some of the experimently measured properties of V and the quark masses at 2 GeV and MZ energy scales. In addition, to the Fritzsch and Stech type mass matrices, a new type of mass matrix (designated as "CGS") is considered. The CGS ty ... More
Presented by Dr. Virendra GUPTA on 10 Oct 2008 at 17:00
RHIC physics
The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National laboratory has been in operation for almost a decade. It was designed to create the Quark-Gluon Plasma (QGP), the primordial state of matter which must have existed immediately after the Big Bang. I will give an overview of the physics explored at RHIC, focusing on the pre-collision stage and the early times right after the collisio ... More
Radio stabilization from the vacuum
Volume stabilization in models with flat extra dimension could follow from vacuum energy residing in the bulk when translational invariance is spontaneously broken. We study a simple toy model that exemplifies this mechanism which considers a massive scalar field with non trivial boundary conditions at the end points of the compact space, and includes contributions from brane and bulk cosmological ... More
Presented by Dr. Eli SANTOS on 9 Oct 2008 at 16:30
Recent developments in chiral unitary theory on meson-meson and meson-baryon interactions
I shall give a brief exposition of chiral dynamics and unitarity in coupled channels to deal with hadron interactions. Explicit examples will be given for meson baryon interaction showing how some resonances are dynamically generated, in particular two Lambda(1405) states for which some experimental evidence has been found. Similarly the interaction of vector mesons with pseudoscalars leads to axi ... More
Presented by Dr. Eulogio OSET on 9 Oct 2008 at 12:30
Relativistic modifications to the potential generated by a massive point mass
Presented by Oscar Rene SALCIDO VALLE on 9 Oct 2008 at 19:00
SPolarization in exclusive pp reactions from the FNAL e690 experiment
It is an experimental evidence that all baryons are created polarized from unpolarized p-nucleus collisions. So far, the origin of this polarization remains unexplained in spite of the experimental evidences accumulated in the past thirty years. Up to these days, (( is the most studied baryon for polarization, for it is copiously produced in p-nucleus collisions at the energies of the principal hi ... More
Presented by Dr. Julian FELIX on 8 Oct 2008 at 18:30
Screening and energy loss from AdS/CFT
In this talk, I will present a short introduction to the AdS/CFT correspondence and show why it might be a useful tool to study certain strongly coupled gauge theories. In this context, I will show some results related to screening and energy loss of heavy probes traversing an N=4 super-Yang-Mills plasma, which might be helpful for understanding RHIC physics.
Presented by Dr. Mariano CHERNICOFF on 6 Oct 2008 at 16:00
Search of dark mather with HAWC
Presented by Reyna Xoxocotzi AGUILAR
Searches for new physics in the top quark sector
Fifteen years following the first glimpses of the top quark at Fermilab's Tevatron, physicists are now able to explore top quark physics with substantial precision. With almost 30 times the data of Run 1, we are learning much about the nature of this peculiarly heavy quark, while we are searching for hints of physics beyond the Standard Model in the top quark sector. I will present recent results ... More
Presented by Robin ERBACHER, Robin ERBACHER on 10 Oct 2008 at 16:30
Seeking a needle in a haystack - Precision physics with the BaBar experiment
The BaBar Collaboration submitted more than 90 new results to ICHEP 2008 - this summer's major conference in high energy physics. In my presentation I will highlight many of these measurements and discuss new results in bottomonium spectroscopy, the decays of charm mesons, the physics of B mesons as well as new measurements on heavy lepton decays.
Presented by Dr. Klaus HONSCHEID on 10 Oct 2008 at 12:30
Signal fluctuation and multi-layer shower frots in high energy air showers
Presented by Mr. Benjamin MORALES
Soft physics observables in heavy ion experiments
Soft (i.e. low momentum) hadrons constitute the bulk of the particles produced in relativistic heavy ion collisions and they are therefore essential to characterize the system created in these reactions. For the more central (i.e. small impact parameter) collisions and/or at the higher center-of-mass energies, where the higher temperatures and energy densities are attained, it is expected on the b ... More
Presented by Dr. Francesco PRINO on 8 Oct 2008 at 18:30
Solving the flatness problem using inflation produced by scalar fields.
Presented by Dupret Alberto SANTANA BEJARANO on 9 Oct 2008 at 19:00
Some theoretical issues on hyperon semileptonic decays
We discuss some aspects involved in the study of hyperon semileptonic decays from the theoretical perspective. In order to properly analyze these decays, two major corrections should be taken into account: radiative corrections to integrated observables and effects of flavor SU(3) symmetry breaking to weak form factors. We provide a detailed discussion on the current status of these issues.
Presented by Dr. Ruben FLORES MENDIETA on 8 Oct 2008 at 17:00
Spin 3/2 Multipole Moments in NKR Formalism
Presented by Ernesto German DELGADO ACOSTA on 9 Oct 2008 at 19:00
Steps toward Dyson-Schwinger equations for equal-time correlation functions
Dyson-Schwinger equations as a semi-analytical tool have given access, for the first time, to the deep infrared region of QCD (or Yang-Mills theory) in the Landau gauge. In the so-called ghost dominance approximation, even a very simple analytical solution exists. Recent efforts have gone into repeating this success for QCD in the Coulomb gauge, the reason being that the color-Coulomb potential ap ... More
Presented by Prof. Axel WEBER on 10 Oct 2008 at 19:20
Strangeness production in p-p collisions as seen by the ALICE experiment at the CERN LHC
Presented by Erick ALMARAZ on 9 Oct 2008 at 19:00
String theory basics for non-strig-theory people I
This course will consist of two parts. In the first part I will present some of the basic ideas of string theory, while in the second I will give a brief introduction to what is arguably the most useful tool derived from string theory to date, the so-called AdS/CFT correspondence.
Presented by Dr. Mariano CHERNICOFF, Dr. Mariano CHERNICOFF on 4 Oct 2008 at 15:30
String theory basics for non-strig-theory people II
String theory basics for non-strig-theory people III
Structure of the nucleon
Presented by Dr. Roelof BIJKER on 8 Oct 2008 at 17:30
Study of non perturbatives aspects of field theories in mexico
I shall try to give an overview of the work realized in Mexico in the last years on non perturbative aspects of field theories, in particular on confinement and dynamical mass generation through the study of Schwinger-Dyson equations.
Presented by Adnan BASHIR on 11 Oct 2008 at 19:20
Study of the flavor changing vertex Htc in the H->Wbc decay process.
Presented by Gilberto TETLALMATZI XOLOCOTZI on 9 Oct 2008 at 19:00
Supergroup formulation of Plebanski theory of gravity
Presented by Jose Eduardo ROSALES QUINTERO on 9 Oct 2008 at 19:00
Symmetry Restoration at Finite Temperature in a Weak Magnetic Field
Presented by Prof. Jorge NAVARRO on 9 Oct 2008 at 19:00
TUS project - Montana array
The AdS/CFT correspondence and non-perturbative QCD
In the first half of the lecture I will introduce some aspects of one of the most profound theoretical insights in modern physics: the AdS/CFT correspondence or Maldacena conjecture. I will illustrate its potential to shed light on a number of non-perturbative phenomena in non-Abelian gauge theories like QCD. I will then devote the second half of the lecture to present the extension of this framew ... More
The Angra Neutrino Project and double chooz: precise measuremet of theta_{13} and safeguards applications of neutrino detectors
We present an introduction to the Angra Neutrino Project and to the experiment Double Chooz. These two experimental initiatives address the study of neutrino oscillations and particularly the precise determination of the mixing angle theta_13. Both will as well explore, already in an early stage, the issue of the use of neutrino detectors to monitor the reactor fuel composition, in the context of ... More
The FNAL e938 experiment: the mexican contribution to the MINERvA collaboration
The MINERvA (Main INjector ExpeRiment for ν A) collaboration (http://minerva.fnal.gov//) in a neutrino scattering experiment which uses the NuMI beamline at Fermilab. It seeks to measure low energy neutrino interactions both to support neutrino oscillation experiments and to study the strong dynamics of the nucleon and nucleus that affect these interactions. It is currently in its final prototypi ... More
The Fermilab experimental neutrino physics program
The current and planned neutrino physics experiments at Fermilab constitute a rich program aimed at investigating every aspect of the neutrino. The experimental design and results of the neutrino oscillation experiments MiniBooNE and MINOS, currently running, will be described. The techniques and goals of the next generation oscillation experiment, NOvA will be presented. Finally, two experiments ... More
Presented by Dr. George MORFIN on 7 Oct 2008 at 11:30
The Feynman propagator for the harmonic and anharmonic oscillators: three different roads
Presented by Juan Ernesto ROBLES ACOSTA on 9 Oct 2008 at 19:00
The Feynman propagator within Phi^4 Theory: a diagrammatic approach to the perturbation expansion of the two-point correlation function.
Presented by Luis Juan ORTIZ VALDIVIA on 9 Oct 2008 at 19:00
The JEM-EUSO mission
JEM-EUSO is a science mission to explore extremes of the Universe. It will observe the darkside of the Earth and detects UV photons emitted from the extensive air shower caused by an extreme energy particle (about 1020 eV). Such a particle arrives almost straightly through our Milky Way Galaxy and is expected to allow us to trace the source location by its arrival direction. This nature can open t ... More
Presented by Dr. Toshikazu EBISUZAKI on 7 Oct 2008 at 13:30
The KSS bound and phase transitions
Presented by Antonio DOBADO on 9 Oct 2008 at 19:00
The Origin of Mass and Confinement I
Dynamical chiral symmetry breaking and confinement are two crucial features of Quantum Chromodynamics responsible for the nature of the hadron spectrum. These phenomena, presumably coincidental, can account for 98% of the mass of our visible universe. In this set of lectures, I shall present an introductory review of them in the light of the Schwinger-Dyson equations.
Presented by Dr. Alfredo RAYA on 2 Oct 2008 at 10:00
The Origin of Mass and Confinement II
The Origin of Mass and Confinement III
The astroparticle physics program in the ALICE-CERN experiment
Presented by Dr. Mario Ivan MARTINEZ HERNANDEZ on 7 Oct 2008 at 18:30
The charged Higgs as a possible signal of new physics at the LHC
We study feasible extensions of the Higgs sector beyond SM. Specifically we study the charged Higgs as signal of new physics in the framework of supersymmetry, not supersymmetry and some cases from extra dimensions. We present the boson sector and the Yukawa sector of the scalars models. We research the phenomenology of a model of electroweak scale right-handed neutrino masses, which includes a do ... More
Presented by Dr. Jaime HERNÁNDEZ SÁNCHEZ on 11 Oct 2008 at 17:15
The collider physics cosmic ray physics intercourse
Presented by Carmine PAGLIARONE
The derivation of constrains on the mSUGRA parameter space form the entropy of dark matter halos
We construct an expression for the entropy of a dark matter halo modelled by a Navarro-Frenk-White profile with a core. By comparing this entropy with the entropy at dark-matter freeze-out, we obtain constraints on the allowed parameter space for mSUGRA models. Additionally, by imposing consistency with the current dark matter bounds, we severely reduce the allowed region for low tan-beta and deri ... More
Presented by Dr. Lukas NELLEN on 11 Oct 2008 at 18:30
The effective action in Einstein-Maxwell theory
Considerable work has been done on the one-loop effective action in combined electromagnetic and gravitational fields, particularly as a tool for determining the properties of light propagation in curved space. After a short review of previous work, I will present some recent results obtained using the worldline formalism. In particular, I will discuss various ways of generalizing the QED Euler-He ... More
Presented by Christian SCHUBERT on 10 Oct 2008 at 18:30
The hadronic yield from dynamical quark recombination
We present a dynamical quark recombination model to study the hadron yield through the hadron-quark matter transition. The model is based on a variational approach to the many-body system and a truly many-body potential. Using the single variational parameter of the model as a probability we are able to explain, in a Bjorken scenario that incorporates the proper time evolution of the system, the p ... More
Presented by Dr. Genaro TOLEDO SANCHEZ on 7 Oct 2008 at 16:00
The isolated leading particle approach: a tool to study the medium created in heavy ion collisions
In the study of heavy ion reactions, there are many methods to look at the created matter in the collisions. Jets have been of interest for many years, however, on this front, the techniques developed up to date for the analysis are not enough to understand a large body of signals, due to an unclear definition of jets. In this talk we present the details of the leading particles and the remaining ... More
Presented by Dr. Eleazar CUAUTLE on 10 Oct 2008 at 18:30
The next spectroscopy: New elementary particles at the Large Hadron Collider
A new particle accelerator, the Large Hadron Collider (LHC), is now beginning its operation at CERN in Geneva. Particle physicists expect that this accelerator will open to view the next set of interactions beyond the familiar strong, weak, and electromagnetic forces. In this lecture, I will introduce the LHC physics program for proton-proton collisions. I will review experimental results in parti ... More
The status and physics potential of the LHC
The Large Hadron Collider (LHC) at the European Center for Particle Physics (CERN) in Geneva, Switzerland, is presently commissioned for its first its first proton-proton collisions expected in fall of 2008. The design collision energy is 14 TeV at the center of mass, with a luminosity of 10^34 cm^-2 s^-1, making this machine the highest energy man-made particle accelerator with the highest colli ... More
Thermodynamics of the Schwarzschild Black Hole in Noncommutative Space
Presented by Mr. Sinuhe PEREZ-PAYAN on 9 Oct 2008 at 19:00
Title (Campuzano)
Title (Godina)
Title (Santangelo)
Presented by Prof. Andrea SANTANGELO
Title (Szczepanik)
Top quark decays in extended models
The top quark decays are of particular interest as a means to test the standard model (SM) predictions. These include the dominant t -> bW, the Cabibbo-Kobayashi-Maskawa (CKM)-suppressed process t -> cWW, and the rare decays t -> cZ and t -> c gamma. They are highly suppressed and they become an excellent window to probe the predictions of theories beyond the SM. In this work we evaluate the effec ... More
Presented by Dr. Ricardo GAITAN on 11 Oct 2008 at 17:00
Toroidal Dipole Moment of a Massless Neutrino
Presented by Esteban Alejandro Reyes PEREZ MONTANEZ on 9 Oct 2008 at 19:00
Tree-level matastable SUSY-breaking with D-terms
We present a simple construction in which metastable vacua occur classically and in which supersymmetry-breaking is sourced by both D-terms and F-terms. All the relevant dynamics in the scenario described here is perturbative, hence calculations of vacuum energies and lifetimes can be performed explicitly. (In most other metastable SUSY-breaking models, these quantities are obscured by nonperturba ... More
Presented by Brooks THOMAS on 11 Oct 2008 at 12:45
Tri-bimaximal neutrino mixing and CKM matrix from finite group family symmetry in SU(5) GUT
We propose a model based on $SU(5) \times { }^{(d)}T$ which successfully gives rise to near tri-bimaximal leptonic mixing as well as realistic CKM matrix elements for the quarks. The Georgi-Jarlskog relations for three generations are also obtained. There are only nine operators allowed in the Yukawa sector up to at least mass dimension seven due to an additional $Z_{12} \times Z'_{12}$ symmetry, ... More
Presented by Mu-Chun CHEN on 11 Oct 2008 at 18:30
Unification of inflation and dark matter
We review the conditions for a single field to be responsible for inflation and dark matter in the Universe, and the observational constraints that should be accomplished for that purpose. As an example, we take a minimally coupled scalar field endowed with a quadratic potential; we shall show that this model seems to require a second period of inflation in order to satisfy the observational const ... More
Presented by Dr. Ureña LUIS on 7 Oct 2008 at 19:30
Unified treatment of quarks and leptons
Presented by Felix Francisco GONZALEZ CANALES on 9 Oct 2008 at 19:00
Using identified particles to probe the medium produced at RHIC
I will discuss how the experiments at RHIC have made extensive use of identified particles in determining that the hot and dense medium produced is consistent with a state of liberated quarks and gluons prior to hadronization. Further, we have used these identified hadrons to study how this medium interacts with differently flavoured partons.
Presented by Dr. Helen CAINES on 8 Oct 2008 at 19:20
Very high momentum particle identification detector for ALICE
The anomalies observed at RHIC for the baryon - meson ratios have prompted a number of theoretical works on the nature of the hadrochemistry in the hadronisation stage of the pp collisions and in the evolution of the dense system formed in heavy ion collisions. Although the predictions differ in the theoretical approach, generally a substantial increase in the baryon production is predicted in the ... More
Presented by Dr. Edmundo GARCIA on 9 Oct 2008 at 16:00 | CommonCrawl |
# Database design principles: normalization, data types, and efficient querying
Before diving into indexing and caching strategies, it's important to understand the principles of database design. These principles guide the creation of efficient and scalable databases.
- **Normalization**: This is the process of organizing data into tables and relationships to minimize redundancy and improve data integrity. The main goal of normalization is to reduce data redundancy and improve data integrity. Five normal forms exist: 1NF, 2NF, 3NF, BCNF, and 4NF.
- **Data types**: Choosing the appropriate data types for your columns can significantly impact performance. PostgreSQL provides a variety of data types, including numeric, character, date/time, and boolean types. It's crucial to choose the most efficient data type for your specific use case.
- **Efficient querying**: Writing efficient queries is essential for optimizing database performance. Some best practices include using the `EXPLAIN` command to analyze query execution plans, avoiding the use of `SELECT *`, and minimizing the use of `LIKE` queries with leading wildcards.
## Exercise
Create a table with normalized data.
```sql
CREATE TABLE authors (
id SERIAL PRIMARY KEY,
name VARCHAR(255) NOT NULL,
birthdate DATE
);
CREATE TABLE books (
id SERIAL PRIMARY KEY,
title VARCHAR(255) NOT NULL,
author_id INTEGER REFERENCES authors(id)
);
```
Write a query that retrieves all books by a specific author.
```sql
SELECT b.title
FROM books b
JOIN authors a ON b.author_id = a.id
WHERE a.name = 'John Doe';
```
# Introduction to indexing: how it works and its benefits
Indexing is a technique used to improve the speed of data retrieval operations on a database table. An index works by providing a more efficient way to locate and access data.
- **B-tree indexes**: B-trees are the default index type in PostgreSQL. They are balanced search trees that are useful for equality and range queries on data that can be ordered. PostgreSQL query planner considers using a B-tree if any comparison operator is used in the query.
- **Hash indexes**: Hash indexes are a secondary index structure that accesses a file through hashing a search key. They are most effective when there are constraints on the leading columns. However, their use cases are limited because they only support single-column indexes and cannot check uniqueness or perform range operations.
## Exercise
Create a B-tree index on the `title` column of the `books` table.
```sql
CREATE INDEX books_title_idx ON books(title);
```
# Choosing the right index: B-tree, hash, GiST, and GIN indexes
PostgreSQL provides multiple index types, including B-trees, hash indexes, Generalized Search Tree (GiST) indexes, and Generalized Inverted Index (GIN) indexes. The choice of index type depends on the specific use case and query patterns.
- **B-tree indexes**: As mentioned earlier, B-trees are the default index type in PostgreSQL. They are suitable for equality and range queries on data that can be ordered.
- **Hash indexes**: Hash indexes are a secondary index structure that accesses a file through hashing a search key. They are most effective when there are constraints on the leading columns.
- **GiST indexes**: GiST indexes are a general-purpose index type that can be used with any data type. They are useful for complex queries that involve data types that cannot be indexed using B-tree or hash indexes.
- **GIN indexes**: GIN indexes are a more advanced type of index that can be used with any data type. They are useful for complex queries that involve data types that cannot be indexed using B-tree or hash indexes, and they support more advanced operations, such as containment queries.
## Exercise
Create a GiST index on the `name` column of the `authors` table.
```sql
CREATE INDEX authors_name_gist_idx ON authors USING gist(name gist_trgm_ops);
```
# Implementing indexing in PostgreSQL: CREATE INDEX statement
In PostgreSQL, you can create an index using the `CREATE INDEX` statement. This statement specifies the name of the index, the table it belongs to, and the columns it indexes.
```sql
CREATE INDEX index_name ON table_name(column_name);
```
For example, to create a B-tree index on the `title` column of the `books` table, you would use the following statement:
```sql
CREATE INDEX books_title_idx ON books(title);
```
## Exercise
Create a GIN index on the `tags` column of the `books` table.
```sql
CREATE INDEX books_tags_gin_idx ON books USING gin(tags);
```
# Understanding PostgreSQL's query planner and execution plans
PostgreSQL's query planner is responsible for selecting the most efficient execution plan for a given query. To understand the query planner's decisions, you can use the `EXPLAIN` command.
The `EXPLAIN` command provides information about the query execution plan, including the chosen indexes, the order of operations, and the estimated costs of each operation. This information can help you identify potential performance bottlenecks and optimize your queries.
## Exercise
Use the `EXPLAIN` command to analyze the query execution plan for retrieving all books by a specific author.
```sql
EXPLAIN SELECT b.title
FROM books b
JOIN authors a ON b.author_id = a.id
WHERE a.name = 'John Doe';
```
# Introduction to caching: the concept and its importance in optimizing performance
Caching is a technique used to store frequently accessed data in memory, reducing the need for repeated disk I/O operations. In PostgreSQL, you can implement caching using PL/pgSQL functions and triggers.
- **In-memory caching**: In-memory caching stores data in the server's memory, reducing the need for disk I/O operations. This can significantly improve query performance.
- **Distributed caching**: Distributed caching involves storing data in a separate cache server, which can be accessed by multiple PostgreSQL instances. This can be useful for large-scale applications that require high availability and fault tolerance.
## Exercise
Create a PL/pgSQL function that caches the result of a query.
```sql
CREATE OR REPLACE FUNCTION get_books_by_author(p_author_name VARCHAR)
RETURNS TABLE(title VARCHAR) AS $$
DECLARE
v_result RECORD;
BEGIN
FOR v_result IN
SELECT b.title
FROM books b
JOIN authors a ON b.author_id = a.id
WHERE a.name = p_author_name
LOOP
RETURN NEXT v_result;
END LOOP;
END;
$$ LANGUAGE plpgsql;
```
# Caching strategies: in-memory caching and distributed caching
In-memory caching and distributed caching can be implemented using different strategies in PostgreSQL.
- **In-memory caching**: In PostgreSQL, you can use the `pg_buffercache` extension to monitor and analyze the buffer cache, which is PostgreSQL's in-memory cache.
- **Distributed caching**: Distributed caching can be implemented using third-party caching solutions, such as Redis or Memcached. These caching solutions can be integrated with PostgreSQL using the `pg_redis` or `pg_memcached` extensions.
## Exercise
Install and configure the `pg_redis` extension to enable distributed caching in PostgreSQL.
```sql
CREATE EXTENSION IF NOT EXISTS pg_redis;
```
# Implementing caching in PostgreSQL: PL/pgSQL functions and triggers
Caching can be implemented in PostgreSQL using PL/pgSQL functions and triggers.
- **PL/pgSQL functions**: PL/pgSQL functions can be used to cache the result of a query. This can be done by creating a function that retrieves the data from the cache, if available, or executes the query and stores the result in the cache if not.
- **Triggers**: Triggers can be used to automatically update the cache when the underlying data changes. This can be done by creating a trigger that is fired before or after an INSERT, UPDATE, or DELETE operation and updates the cache accordingly.
## Exercise
Create a PL/pgSQL trigger that updates the cache when a book's title is updated.
```sql
CREATE OR REPLACE FUNCTION update_books_title_cache()
RETURNS TRIGGER AS $$
BEGIN
-- Update the cache with the new title
-- ...
RETURN NEW;
END;
$$ LANGUAGE plpgsql;
CREATE TRIGGER books_title_update_trigger
AFTER UPDATE OF title ON books
FOR EACH ROW
EXECUTE FUNCTION update_books_title_cache();
```
# Monitoring and analyzing performance: using EXPLAIN, VACUUM, and pg_stat_statements
To monitor and analyze the performance of your PostgreSQL database, you can use the `EXPLAIN` command, the `VACUUM` command, and the `pg_stat_statements` extension.
- **EXPLAIN**: As mentioned earlier, the `EXPLAIN` command provides information about the query execution plan, including the chosen indexes, the order of operations, and the estimated costs of each operation.
- **VACUUM**: The `VACUUM` command is used to clean up and optimize the physical layout of the data in a table. It removes dead rows, updates table statistics, and reclaims storage space.
- **pg_stat_statements**: The `pg_stat_statements` extension provides statistics on executed SQL statements. It can be used to identify frequently executed queries and optimize their performance.
## Exercise
Analyze the performance of a specific query using the `EXPLAIN` command.
```sql
EXPLAIN SELECT * FROM books WHERE title = 'The Catcher in the Rye';
```
# Performance tuning in PostgreSQL: configuration settings and best practices
To optimize the performance of your PostgreSQL database, you can follow best practices and fine-tune configuration settings.
- **Shared buffers**: The `shared_buffers` configuration setting determines the amount of memory allocated for the buffer cache, which is used to cache frequently accessed data. Increasing the value of this setting can improve performance for read-heavy workloads.
- **Work_mem**: The `work_mem` configuration setting determines the amount of memory allocated for sorting and hash operations. Increasing the value of this setting can improve performance for complex queries.
- **Maintenance tasks**: Regularly running the `VACUUM` command and analyzing tables with the `ANALYZE` command can help maintain optimal performance.
## Exercise
Tune the `shared_buffers` and `work_mem` configuration settings to improve the performance of your PostgreSQL database.
```sql
ALTER SYSTEM SET shared_buffers = '256MB';
ALTER SYSTEM SET work_mem = '16MB';
```
# Common performance issues and their solutions: locks, deadlocks, and query optimization
PostgreSQL can experience various performance issues, such as locks, deadlocks, and inefficient queries.
- **Locks**: Locks are used to prevent concurrent modifications of the same data. In some cases, excessive lock contention can cause performance issues. To resolve this, you can use advisory locks, which allow for more fine-grained control over locking behavior.
- **Deadlocks**: Deadlocks occur when two or more transactions are waiting for each other to release a lock. To prevent deadlocks, you can use the `FOR UPDATE` clause to lock rows for update, or use the `SKIP LOCKED` clause to skip locked rows.
- **Query optimization**: Inefficient queries can cause performance issues. To optimize queries, you can use the `EXPLAIN` command to analyze query execution plans, avoid using `SELECT *`, and minimize the use of `LIKE` queries with leading wildcards.
## Exercise
Optimize a query that retrieves all books by a specific author.
```sql
SELECT b.title
FROM books b
JOIN authors a ON b.author_id = a.id
WHERE a.name = 'John Doe';
```
This query can be optimized by using a B-tree index on the `name` column of the `authors` table. | Textbooks |
Impact of COVID-19 infection on life expectancy, premature mortality, and DALY in Maharashtra, India
Guru Vasishtha ORCID: orcid.org/0000-0001-5477-18451,
Sanjay K. Mohanty2,
Udaya S. Mishra3,
Manisha Dubey4 &
Umakanta Sahoo1
The COVID-19 infections and deaths have largely been uneven within and between countries. With 17% of the world's population, India has so far had 13% of global COVID-19 infections and 8.5% of deaths. Maharashtra accounting for 9% of India's population, is the worst affected state, with 19% of infections and 33% of total deaths in the country until 23rd December 2020. Though a number of studies have examined the vulnerability to and spread of COVID-19 and its effect on mortality, no attempt has been made to understand its impact on mortality in the states of India.
Using data from multiple sources and under the assumption that COVID-19 deaths are additional deaths in the population, this paper examined the impact of the disease on premature mortality, loss of life expectancy, years of potential life lost (YPLL), and disability-adjusted life years (DALY) in Maharashtra. Descriptive statistics, a set of abridged life tables, YPLL, and DALY were used in the analysis. Estimates of mortality indices were compared pre- and during COVID-19.
COVID-19 attributable deaths account for 5.3% of total deaths in the state and have reduced the life expectancy at birth by 0.8 years, from 73.2 years in the pre-COVID-19 period to 72.4 years by the end of 2020. If COVID-19 attributable deaths increase to 10% of total deaths, life expectancy at birth will likely reduce by 1.4 years. The probability of death in 20–64 years of age (the prime working-age group) has increased from 0.15 to 0.16 due to COVID-19. There has been 1.06 million additional loss of years (YPLL) in the state, and DALY due to COVID-19 has been estimated to be 6 per thousand.
COVID-19 has increased premature mortality, YPLL, and DALY and has reduced life expectancy at every age in Maharashtra.
In a short span of 1 year, COVID-19 has emerged as the largest-ever health crisis of the twenty-first century. With over 78 million infections and 1.7 million deaths attributable to it until 23rd December 2020, COVID-19 attributable deaths account for 2.9% of additional deaths worldwide [1, 2]. The global spread of COVID-19 infection and attributable mortality has been highly uneven among and within countries. With 18.6 million infections and 3,30,824 COVID-19 attributable deaths, the USA accounts for 23.8% of global infections and 19.2% of global deaths [1]. India, with over 10 million infected cases and 1,46,476 COVID-19 deaths, is the second-largest country with respect to the size of infection and is ranked third with respect to COVID-19 attributable deaths [1]. The actual number of infections in many countries, including India, remains underestimated due to the asymptomatic nature of the infection and inadequate testing and surveillance system.
As the COVID-19 infection continues to spread, an increasing number of studies have become available on the extent of infection, the associated risk factors, and the crude fatality ratio (CFR) with and without time lag, projecting deaths and estimating the loss of life expectancy, premature mortality, and YPLL across countries [3,4,5,6,7,8,9]. Findings suggest that the infection rate across populations is largely underestimated, while the CFR shows large variations across countries, geographies, and demographic characteristics. The demographic structure, availability of health care resources, and multimorbid conditions explain COVID-19 attributable deaths to a larger extent [3, 10,11,12,13,14]. In China, fever, dyspnea, and chest pain/discomfort have been the more common symptom among the deceased patients, while fever has the most common symptom among the surviving patients [3, 10]. Older adults, people with comorbidities, and men are more susceptible to COVID-19 fatality [11, 15].
Evidence suggests that people with the COVID-19 infection are more prone to many life-threatening morbidities and fatalities [11, 16]. A study conducted in Italy found that fatigue, dyspnea, joint pain, and chest pain were persistent among the recovered patients [16]. Studies have projected premature mortality and reduction in life expectancy due to the infection across countries [4, 8, 17, 18]. After a certain threshold level of COVID-19 prevalence, life expectancy starts decreasing. In North America, Europe, Latin America, and the Caribbean, life expectancy at birth has been estimated to have reduced by 1 year at 10% prevalence of infection [4]. The COVID-19 attributable mortality has the potential to reduce life expectancy in India, weekly and annual life expectancy at birth in Spain, and seasonal life expectancy in Italy [8, 17, 18]. Besides mortality, many studies are available on the vulnerability to the COVID-19 infection, and mental distress, and loss of livelihood due to the preventive measures for containing the virus [19, 20].
The spread of COVID-19 has been largely uneven across the states of India. With 123 million population (9% of India's population), Maharashtra is the second most populous and urbanized state in the country. It is one of the more developed states and ranks high on the human development index [21]. However, Maharashtra is the worst affected state with respect to COVID-19 infections and mortality. Until 23rd December 2020, it had 1.9 million cases and 48,876 deaths due to COVID-19, accounting for 19% of total infections and 34% of all COVID-19 attributable deaths in the country [22]. The case-fatality ratio in the state is higher than the national average. It has been observed that the rapid community transmission of the virus in a short time has resulted in a higher incidence of the disease and deaths resulting from it and, consequently, has affected the life expectancy [12]. Many states, including Maharashtra, are now experiencing the second and the third waves of the COVID-19 pandemic. With the global literature hinting at the implications of COVID-19 for longevity, it becomes imperative to make a regional assessment of the same owing to the disproportionately high load of infections and deaths due to the pandemic in the region. This assessment involves premature mortality, with its consequential bearing on life expectancy, person-years of life lost, and disability-adjusted life years (DALY). With the age-specific load of the infection and fatalities, person-years of life lost offers an understanding into the skewed share of life lost during the productive years, which has implications not only for a macro assessment, but also for household-level micro assessment. Years of potential life lost (YPLL) is a summary measure of premature mortality that reflects the sum of years lost from a predefined age, such as standard life expectancy. A higher YPLL is indicative of premature mortality and contributes to the compression of life expectancy. DALY measures the disease burden of the population and consists of YPLL and Years Lived with Disability (YLD). DALY serves to understand the implications of differential severity of the disease for individuals conditioned by their age, sex, and any pre-disposed condition. In the context of the COVID-19 pandemic, estimating YPLL and DALY is appropriate as over two-thirds of deaths are under 70 years of age 'a standard age for estimating YPLL' [2]. Patients affected by COVID-19 have long-term health complications and are more likely to be morbid than non-COVID-19 patients [23]. In ultimate terms, the loss of life expectancy in a regional setting reflects the severity of the pandemic with sustained and periodic soaring of infection in the state. In this context, this paper examines the effect of COVID-19 on premature mortality, life expectancy, YPLL, and DALY in one of the worst affected states of India, Maharashtra.
Data and methods
Data for this paper was drawn from multiple sources. These include the Report of the Expert Committee on Population Projections, Sample Registration System (SRS) Statistical Report 2018, and other published sources. The population size and distribution for Maharashtra for the year 2020 were taken from the Report of the Expert Committee on Population Projections [24]. The age-specific death rates for the state for the year 2018 (latest available data) were taken from the SRS Statistical Report and labelled as death rate without the COVID-19 infection [25]. The COVID-19 confirmed cases and deaths by age group were taken from the Times of India reports, dated 7th December 2020 and 21st December 2020 [26, 27]. The total number of confirmed cases and deaths until 20th December 2020 for Maharashtra and India were taken from covid19india.org [22]. We redistributed the total deaths until 20th December 2020, as per the distribution of deaths for which age data was available (7th December 2020). Age-specific case fatality ratio (ASCFR) was computed from the given data.
Descriptive statistics, abridged life tables, and estimates of YPLL and DALY were used in the analysis. It was assumed that COVID-19 attributable deaths are unprecedented additional deaths that could have been averted in the absence of COVID-19. A set of abridged life tables were generated to estimate life expectancy at various ages, premature mortality, and YPLL and DALY were used in the analysis. The probability of death by age 70 and between age 20 and age 64 was computed and termed as premature mortality. Estimates were classified under the five scenarios. Scenario 1 provides estimates of deaths without COVID-19. Scenario 2 considers that COVID-19 deaths accounted for 5.3% of total deaths until 20th December 2020. Scenarios 3, 4, and 5 project the estimates assuming that COVID-19 attributable deaths would increases to 6, 8 and 10% of total deaths respectively. Expected deaths due to COVID-19 were distributed in accordance with the age distribution of COVID-19 as of date. Estimates of YPLL and DALY were made as follows.
Years of potential life lost (YPLL)
YPLL is a summary measure of premature mortality that estimates the average years a person would have lived had he or she not died prematurely [28]. YPLL is estimated as:
$$ YPLL=\sum \limits_{i=0}^{\infty }{d}_i\ast {L}_i $$
Where Li represents the life expectancy at age i and di represents the number of deaths at age i. It is a self-weighted estimate, which gives a higher weight to the deaths occurring at younger ages and a lower weight to the deaths at higher ages [28, 29]. The deaths at each age are weighted by age-specific life expectancy.
Disability adjusted life years (DALY)
DALY is a summary measure of health of a population, combining mortality and non-fatal health outcomes. DALY is commonly used to measure the difference between a current situation of health and an ideal situation, where everyone lives up to the age of the standard life expectancy and in a perfect health [30]. It is estimated by summing the potential life lost due to premature mortality and productive years of life lost due to disability/disease [30]. It is calculated as:
$$ DALY= YLL+ YLD $$
Where YLL denotes the years of life lost due to premature mortality, and YLD denotes the years lived with disability.
YLL and YLD were calculated considering the discounting rate of 3%. Discounting rate shows the social preference of a healthy year now, rather than in the future. The value of a year of life is generally decreased annually by a fixed percentage. The World Bank Disease Control Priorities study and the Global Burden of Disease (GBD) project both used a 3% discount rate, and the US Panel on Cost-Effectiveness in Health and Medicine recently recommended that the economic analyses of health also use a 3% real discount rate to adjust both costs and health outcomes [31, 32].
YLL is estimated as:
$$ YLL=\frac{N}{r}\left(1-{e}^{- rL}\right) $$
Where N is the number of deaths, L is the life expectancy at the age of death, and r is the discount rate.
$$ YLD=\frac{\left(I\ast DW\ast L\ast \left(1-{e}^{- rL}\right)\right)}{r} $$
Where I is the number of incidence/prevalence cases, DW is a disability weight (a weight factor that reflects the severity of the disease on a scale from 0 (equivalent to perfect health) to 1 (equivalent to being dead)), and L is the duration of the disability.
As COVID-19 is a novel disease, its disability weight is not available. Since it is a severely infectious disease having an acute period and is associated with the lower respiratory tract infection [33]. Hence, we have used the disability weight of 0.133, as a proxy of COVID-19 (available elsewhere [34]. The duration of disability of 60 days was used because the patients of COVID-19 have been hospitalized for an average of 30 days, and after discharge, are quarantined for 14–28 days approximately [35, 36].
Table 1 provides the key indicators of the COVID-19 infection and the associated mortality for Maharashtra and India. Since the onset of the pandemic, COVID-19 has infected more than 1.8 million people, of whom 48,746 had died in Maharashtra until 20th December 2020. COVID-19 attributable deaths amount to 5.3% of the total deaths. These additional deaths could have been prevented in the absence of COVID-19. The case-fatality ratio in the state is 2.57, higher than the national average of 1.5. The rate of infection in Maharashtra (15 infected per 1000 people) is more than double compared to the national average (7 infected per 1000 people). In 2018 (the pre-COVID-19 period), the life expectancy at birth was 73.2 years in the state compared to 69.7 years in India as a whole.
Table 1 Summary indicators of population and COVID-19 indicators in Maharashtra and India 2020
Appendix 1 presents the number of confirmed cases and COVID-19 attributable deaths by age group in Maharashtra until 20th December 2020. Figure 1 estimates the age-specific case fatality ratio in the state. The ASCFR shows an increasing pattern with age. It is as low as 0.14 in the age group 10-year and reaches 7% by age 60 and 11% in the age group 80 years and above.
Age-specific case fatality ratio in Maharashtra, India 2020
Figure 2 compares the life table probability of deaths with and without the COVID-19 infection in Maharashtra. Age-specific probability of deaths was estimated by assuming that the age pattern of mortality without COVID-19 would have remained the same as during the period 2014–2018. The gap between the two curves shows the difference in probability of death with and without the COVID-19 pandemic. It is observed that the infection has affected the age pattern of mortality. This pattern suggests that after the age of 44, the probability of dying with COVID-19 is increases with age compared to the probability of dying without COVID-19. The probability of death till the age of 44 is similar both pre- and during the COVID-19 pandemic. The probability of dying with COVID-19, compared to without COVID-19, is higher among those aged 45–75 with COVOD-19 compare to without COVID-19.
Life table probability of death without and with 5.3% COVID-19 attributable deaths in Maharashtra, India 2020
Figure 3a and b depict the premature mortality by age 70 (\( {{}_{70}{}q}_0\Big) \) and in the prime working-age group of 20–64 (\( {{}_{44}{}q}_{20}\Big) \) in pre- and during the COVID-19 pandemic. The premature mortality pre-COVID-19 was 0.34 but increased to 0.36 with COVID-19. Given the current mortality pattern, if the share of deaths attributable to the infection reaches 8 and 10%, the premature mortality (\( {{}_{70}{}q}_0 \)) would increase to 0.37 and 0.38, respectively. In the working-age group (\( {{}_{44}{}q}_{20}\Big) \), the probability of death due to the current rate of infection has increased to 0.16 from 0.15 in the pre-COVID-19 period. Under the assumed 10% COVID-19 attributable death share scenario, the probability of death in the working-age group would increase to 0.17.
Fig 3
a Premature mortality in the age group 0–70 pre- and during COVID-19 pandemic in Maharashtra, India 2020. b Premature mortality in the working-age group (20–64) pre- and during COVID-19 pandemic in Maharashtra, India 2020
Table 2 presents the estimates of life expectancy by age group without COVID-19 and with varying degrees of COVID-19 attributable deaths in Maharashtra. Life expectancy in each age group with and without the COVID-19 infection exhibits the changing age-specific survival pattern. It can be observed that life expectancy has reduced in each scenario of the COVID-19 infection. In the pre-COVID-19 period, life expectancy at birth (age 0) was 73.2 years, which with the current infection, has reduced to 72.4 years. Therefore, life expectancy has reduced by 0.8 years in the state, A disproportionate reduction of life expectancy was observed across age. With an increase in the share of COVID-19 attributable deaths to 10%, life expectancy would reduce to 71.8 years in the state.
Table 2 Life expectancy under various scenarios of COVID-19 attributable mortality in Maharashtra, India 2020
Figure 4 shows the reduction in life expectancy at birth in various scenarios of COVID-19 attributable deaths in Maharashtra. Estimates suggest that the ongoing COVID-19 pandemic has significantly affected the life expectancy in the state. Life expectancy has already shrunk by 0.8 years due to the current level of COVID-19 attributable deaths. in the scenario that COVID-19 attributable deaths would amount to 6, 8, and 10% of total deaths in the state, the life expectancy at birth would reduce by 0.9, 1.1, and 1.4 years respectively.
Reduction in life expectancy at birth (in years) due to COVID-19 attributable deaths in Maharashtra, India 2020
Table 3 presents the estimates of years of potential life lost (YPLL) without COVID-19 and with various level of deaths attributable to COVID-19 by age group in Maharashtra. YPLL was estimated at 17.4 million in the absence of infection. COVID-19 added more than 1.06 million YPLL loss in the state. It can be observed that the age composition of YPLL with infection is substantially different without infection. COVID-19 has significantly affected the working adults aged 45–65 years with their percentage share of YPLL increased to 50% compare to 33% without infection. In contrast, the percentage share of YPLL among infants (0–1 years of age) has decreased to 0.03% with COVID-19 compare to 6.5% without it. In the scenario that COVID-19 attributable deaths increase to 6, 8 and 10%, YPLL would increase by 1.2 million, 1.5 million and 1.9 million respectively.
Table 3 Estimates of years of potential life lost (YPLL) in various scenarios of COVID-19 attributable deaths in Maharashtra, India 2020
Table 4 shows the estimated age-specific DALY in various scenarios of COVID-19 attributable mortality in Maharashtra. With the current share of deaths attributable to the infection, we estimated DALYs for all ages to be 6.1 per thousand population. DALY would change in each scenario of COVID-19 attributable deaths. If COVID-19 accounts for a 6% death share, DALYs would increase to 7 per thousand population. With an increase in the share of deaths attributable to COViD-19 to 8 and 10%, DALYs would increase to 9.2 and 11.5 per thousand population respectively. The age pattern of DALY suggests that the age group 60–64 makes the highest contribution to the overall DALY.
Table 4 Age pattern of years of life lost (YLL), years lived with disability (YLD) and disability adjusted life years (DALY) (per 1000 population) in alternative scenarios of COVID-19 infection in Maharashtra, 2020
Discussion and conclusion
In a short period of 1 year, the COVID-19 pandemic has emerged as the largest-ever health crisis globally, nationally, and locally. Despite several measures to contain the spread of the virus, the infection has intensified the disease burden, increased premature mortality, increased short- and long-term morbidity, raised health care costs, reduced income, increased unemployment, and above all, generated a psychological scare worldwide. The second and the third waves of COVID-19 infection are underway in many of the worst-affected countries. The geographical spread of the COVID-19 infection and the associated mortality has largely been uneven. Its impact varies across and within the countries. India is the second-worst affected country with respect to the number of COVID-19 infections, where the disease remains a constant threat due to the large population, densely populated cities, slums, compromised hygiene practices, inadequate health care facilities, and the lack of other development indicators. The COVID-19 infection rate and fatalities are quite uneven across India, with Maharashtra being the worst affected state, accounting for one-third of the total COVID-19 attributable deaths. In this context, this is the first-ever study to examine the impact of COVID-19 on life expectancy, premature mortality, and disability-adjusted life years in Maharashtra, India. The following are the salient findings.
First, the COVID-19 infection accounted for 5.3% of all causes of death in the state until 20th December 2020. The crude death rate has increased from 7 per thousand in the pre-COVID-19 period to 7.4 per thousand population during COVID-19, which is not a minor shift. A similar finding has been reported in a study from USA [37]. Second, the COVID-19 epidemic has significantly raised the likelihood of dying in the age group 45–70, comprising a large majority of the working adult population. At the current infection rate, the share of premature mortality among working adults (20–64) is 16, and 36% before age 70. In a hypothetical scenario where the level of COVID-19 attributable deaths increase to 10%, the probability of dying prematurely in the age group (0–70) and in the working-age group of 20–64 would rise to 38 and 18% respectively. Similar findings have been reported by various studies across the globe [38, 39]. These observations on the mortality front conveys not only the survival adversity brought about by COVID-19, but also composition of the adversity, which is rather alarming and has long-term implications. Apart from the reversal in the trend of the crude death rate (CDR), the increase in the premature mortality among working adults has resulted in a loss of productivity and put a share of households in distress with the loss of the bread earner. This may lead to a range of adversities like discontinuation of education by children and debt burden among distressed households with gendered derivatives, wherein dependent girls, children, and women will become worse off. Premature mortality, unless otherwise sufficiently protected with insurance and economic protection can have a devastating impact on individual households, which may not be apparent in the macro scene. Third, with the current share of COVID-19 induced mortality, life expectancy has already shrunk by 0.8 years in the state. If the virus continues to spread and mortality reaches 10%, loss in life expectancy is likely to be 1.4 years. Our findings are consistent with literature [4, 8, 18]. A high reduction of life expectancy (2.94 years) has been observed in USA [40]. Fourth, The COVID-19 attributable deaths have caused about 1.06 million YPLL in the state. The majority of the loss in YPLL has been among the working adults aged 45–64 years. This disproportionately high share of person-years life lost in the ages 45–64 has undoubtedly increase the dependency burden at the household level, which calls for micro-monitoring of such households and adoption of appropriate protective measures for the dependents of the adult victims. Lastly, the COVID-19 induced mortality has substantial implications for DALY as well. At the current share of COVID-19 induced mortality, the loss of DALY was estimated at 6.1 per thousand population. With an increase in the deaths share to 8 and 10% of mortality, DALY loss is estimated to increase to 9.2 and 11.5 per thousand population, respectively.
The new strains of COVID-19 at various places across the globe have alarmed the world about the higher transmission probability and higher associated mortality than the existing strains of virus [41]. Therefore, advanced preparedness needs to be in place to tackle the rapid spread of the infection until a substantial share of the population is vaccinated. Since the beginning of the pandemic, the national and states governments in India have made several efforts to contain the spread of the virus with measures like imposing a complete lockdown, promoting hygiene and hand wash practices, executing a phase-wise unlocking, promoting social distancing norms and mask wearing, identifying hot spots, and others. Though the infection rate had slowed down in the country, COVID-19 infection is surging again, whereas Maharashtra continues to be the worst affected state in the country. In recent months, Maharashtra has accounted for about half of the new COVID-19 infections in the country. The likely reasons behind this is the densely populated cities, the presence of a large number of slums, the large inflow and outflow of migrants, the engagement of a substantial labour force in the unorganized sector, along with a number of demographic challenges. The potential for rapid transmission and the implied fatalities may well reverse the survival scene in the state as indicated by this study [12].
It is heartening to note that India has been successful in developing two of the vaccines, namely 'Covaxin' and 'Covishield', and started the vaccination exercise on 16th January 2021. As of 5th March 2021, more than 15 million people have been vaccinated in the country of whom 1.4 million are in the state of Maharashtra [22]. The current vaccination program is limited to health care workers, senior citizens and persons with comorbidities. Giving the fact that half of the newly infected cases are from Maharashtra, it is suggested that the state be given priority in vaccination. The vaccines are being supplied by the central government while the vaccinated are being carried out by the state governments. The state of Maharashtra and the cities in the state should be accorded a high priority in the vaccination program. Also the IEC (Information Education and Communication) on the efficacy of the vaccination should be intensified to reduce the vaccine hesitancy and to eliminate the mistrust among some sections of the population.
Given that the risk of The COVID-19 infection is age conditioned, with the vulnerability being greater among those aged 40 and above, and that the virulence of the disease intensifies with multimorbid conditions and a compromised immunity, the rate of infection has been quite high in the later ages, with an increased risk of fatality as well. In view of the need for out-of-home activities for livelihood and the prevalence of inappropriate working conditions that hardly allow for COVID protocols of SMS (Sanitary Practice, Masking and Safe Social Distance) to be followed, there needs to be a greater focus on this vulnerable population to attain a balance between work and life.
In conclusion, this exercise makes a precise assessment of the survival scenario keeping in mind the continuation of the COVID-19 pandemic. The study emphasize the need for robust protection measures to mitigate the consequences of the disease on victim households and for prioritizing the vaccination program in the state of Maharashtra. The very specific vulnerability to this infection calls for suitable action on a variety of fronts like work environments out of home, means of communication in keeping with the COVID-19 protocols, and adequate sanitary and hygiene amenities in the living environment to restrict the spread of the infection and to bring it under control before prior to the vaccination drive goes into in full swing.
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https://timesofindia.indiatimes.com/city/mumbai/one-in-3-covid-deaths-across-maharashtra-was-of-person-in-61-70-age-group/articleshow/79599564.cms, https://timesofindia.indiatimes.com/city/mumbai/maharashtra-mumbai-see-slight-drop-in-cases-marginal-rise-in-deaths/articleshow/79831787.cms
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International Institute for Population Sciences, Govandi Station Road, Deonar, Mumbai, Maharashtra, 400088, India
Guru Vasishtha & Umakanta Sahoo
Department of Fertility Studies, International Institute for Population Sciences, Mumbai, India
Sanjay K. Mohanty
Centre for Development Studies, Prashant Nagar, Medical College P.O, Ullor Thiruvananthapuram, Kerala, India
Udaya S. Mishra
Centre for Chronic Disease Control, New Delhi, India
Manisha Dubey
Guru Vasishtha
Umakanta Sahoo
Conceptualization and design of study: SKM and GV; analysis and interpretation of data: GV; drafting the manuscript: GV and SKM; critical revision of the manuscript for important intellectual content: GV, SKM, USM, MD and US. The authors read and approved the final manuscript.
Correspondence to Guru Vasishtha.
The authors declare that they do not have any competing interest.
Additional file 1: Appendix 1.
Age-specific COVID-19 confirmed cases, deaths and estimated case fatality ratio (CFR) in Maharashtra, 20th December, 2020. Appendix 2. Estimated deaths without COVID-19 and with varying level of COVID-19 attributable deaths in Maharashtra, India, 20th December, 2020.
Vasishtha, G., Mohanty, S.K., Mishra, U.S. et al. Impact of COVID-19 infection on life expectancy, premature mortality, and DALY in Maharashtra, India. BMC Infect Dis 21, 343 (2021). https://doi.org/10.1186/s12879-021-06026-6
Accepted: 30 March 2021
Premature mortality
YPLL
DALY | CommonCrawl |
\begin{document}
\title[Approximation of general 3-variable Jensen $\rho$-functional inequalities in complex Banach spaces] {Approximation of general 3-variable Jensen $\rho$-functional inequalities in complex Banach spaces} \author[G. Lu]{Gang Lu$^*$} \address{Gang Lu \newline \indent Division of Foundational Teaching, Guangzhou College of Technology and Business, Guangzhou 510850, P.R. China } \email{[email protected]}\vskip 2mm
\author[W. Sun]{Wenlong Sun} \address{Wenlong Sun \newline \indent Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110870, P.R. China} \email{[email protected]}\vskip 2mm
\author[H.Qiao]{Hanyue Qiao} \address{Hanyue Qiao \newline \indent Department of Mathematics, Yanbian University, Yanji 133001, P.R. China} \email{[email protected]}\vskip 2mm
\author[Y. Jin]{Yuanfeng Jin$^*$} \address{Yuanfeng Jin \newline \indent Department of Mathematics, Yanbian University, Yanji 133001, P.R. China} \email{[email protected]}\vskip 2mm
\author[C. Park]{Choonkil Park} \address{Choonkil Park\newline \indent Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea} \email{[email protected]}\vskip 2mm
\begin{abstract} In this paper, we introduce and investigate general 3-variable Jensen $\rho$- functional equation, and
prove the Hyers-Ulam stability of the Jensen functional equations associated with the general 3-variable Jensen $\rho$-functional inequalities in complex Banach spaces. \end{abstract}
\subjclass[2010]{Primary 39B62, 39B52, 46B25}
\keywords{Jensen functional inequaty; Hyers-Ulam stability; complex Banach space.\\ $^*$Corresponding authors: [email protected] (G.Lu), [email protected] (Y. Jin).}
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\newtheorem{theorem}[df]{Theorem}
\newtheorem{corollary}[df]{Corollary}
\newtheorem{proposition}[df]{Proposition} \newtheorem{example}[df]{Example} \setcounter{section}{0}
\maketitle
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\section{Introduction and preliminaries}
The stability problem of functional equations originated from a question of Ulam \cite{Ul} concerning the stability of group homomorphisms.The essence of the problem is, under what condition does there exists a homomorphism near an approximate homomorphism? The study of stability for functional equation arises from the Ulam's problem. In 1941,
Hyers \cite{Hy} gave the first affirmative answer to the question of Ulam for Banach spaces. His method was called the {\it direct method}. Later, Hyers' theorem was generalized by Aoki \cite{A} for additive mappings and by Rassias \cite{R3} for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by G\u avruta \cite{Ga} by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias' approach. The stability problems for several functional equations or inequations have been extensively investigated by a number of authors (see \cite{AD}--\cite{EGRG}, \cite{HIR}--\cite{LP}, \cite{pra}, \cite{R1}--\cite{XRX}).
The function equations \begin{eqnarray}\label{eq11} &&f(x+y+z)+f(x+y-z)-2f(x)-2f(y)=0\\ \label{eq12} && f(x+y+z)-f(x-y-z)-2f(y)-2f(z)=0. \end{eqnarray} are called {\it 3-variable Jensen}. In \cite{LLJX,Park, SJPL}, Lu {\it et al.} investigated the 3-variable functional inequalities
and proved their stability.
In this paper, we consider the following functional equations
\begin{eqnarray}\label{eq11} &&f(x+y+\alpha z)+f(x+y-\alpha z)-2f(x)-2f(y)=0, \\ \label{eq12} && f(x+\beta y+\alpha z)-f(x-\alpha z)-\beta f(y)-2f(\alpha z)=0, \end{eqnarray} where $\beta$ and $\alpha $ are nonzero real numbers. And discuss the Hyers-Ulam stability of general 3-variable Jensen $\rho$-functional equations associated with functional inequalities in complex Banach spaces.
Throughout this paper, assume that $X$ is a complex normed vector space with norm $\|\cdot\|$ and that $Y$ is a complex Banach space.
\section{Hyers-Ulam stability of (\ref{eq11}) }
In this section, we prove the Hyers-Ulam stability of the 3-variable functional inequality \begin{eqnarray}\label{eq21} \begin{split}
\;& \|f(x+y+\alpha z)+f(x+y-\alpha z)-2f(x)-2f(y)\|\\
\;& \leq \left\|\rho_1(f(x+y+\alpha z)-f(x+y)-f(\alpha z))\right\|\\\;&
+\left\|\rho_2(f(x+y-\alpha z)+f(-x)+f(\alpha z-y))\right\| \end{split} \end{eqnarray}
in complex Banach spaces, where $\rho_1$ and $\rho_2$ are fixed complex numbers with $|\rho_1|+3|\rho_2|<2$.
\begin{lemma}\label{lm1} Let $f:X\rightarrow Y$ be a mapping. If it satisfies (\ref{eq21}) for all $x,y,z\in X$, then $f$ is additive. \end{lemma}
\begin{proof} Letting $x=y=z=0$ in (\ref{eq21}), we get
$$2\|f(0)\|\leq (|\rho_1|+3|\rho_2|)\|(f(0)\|$$
and thus $f(0)=0$, $|\rho_1|+3|\rho_2|<2$.
Letting $x=y=0$ in (\ref{eq21}), we get
$$\|f(\alpha z)+f(-\alpha z)\|\leq \|\rho_2(f(-\alpha z)+f(\alpha z))\|$$ and so $f(-x)=-f(x)$ for all $x\in X$.
Letting $z=0$ in (\ref{eq21}), we have \begin{eqnarray*} \begin{split}
\;& \|2f(x+y)-2f(x)-2f(y)\| \leq
\|\rho_2 (f(x+y)-f(x)-f(y))\|\\ \;& \end{split} \end{eqnarray*} and so $f(x+y)=f(x)+f(y)$ for all $x,y\in X$. Hence $f:X\rightarrow Y$ is additive. \end{proof}
\begin{corollary} Let $f: X\rightarrow Y$ be a mapping satisfying \begin{eqnarray} \begin{split}
\;& \|f(x+y+\alpha z)+f(x+y-\alpha z)-2f(x)-2f(y)\|\\
\;& = \left\|\rho_1(f(x+y+\alpha z)-f(x+y)-f(\alpha z))\right\|\\\;&
+\left\|\rho_2(f(x+y-\alpha z)+f(-x)+f(\alpha z-y))\right\| \end{split} \end{eqnarray} for all $x,y,z\in X$. Then $f:X\rightarrow Y$ is additive. \end{corollary}
We prove the Hyers-Ulam stability of the additive functional inequality (\ref{eq21}) in complex Banach spaces.
\begin{theorem}\label{thm23} Let $f:X\rightarrow Y$ be a mapping. If there is a function $\varphi:X^3\rightarrow [0,\infty)$ with $\varphi(0,0,0)=0$ such that \begin{eqnarray}\label{eqn24} \begin{split}
\;& \|f(x+y+\alpha z)+f(x+y-\alpha z)-2f(x)-2f(y)\|\\
\;& \leq \left\|\rho_1(f(x+y+\alpha z)-f(x+y)-f(\alpha z))\right\|\\\;&
+\left\|\rho_2(f(x+y-\alpha z)+f(-x)+f(\alpha z-y))\right\|+\varphi(x,y,z) \end{split} \end{eqnarray} and \begin{eqnarray}\label{eqn25} \lim_{j\rightarrow \infty}\frac{1}{2^{ j}} \varphi\left(2^{j}x, 2^{j}y,2^{j}z\right)=0 \end{eqnarray}
\begin{eqnarray}\label{eqn30}
\widetilde{\varphi}(x):=\sum_{i=0}^{\infty}\frac{1}{2^{i+1}}\frac{1}{(2-|\rho_2|)}\left(\varphi(2^{i}x,2^{i}x,0)
+\frac{2|\rho_2|}{1-|\rho_2|}\varphi(0,0,2^{i}x)\right)<\infty
\end{eqnarray} for all $x,y,z\in X$, then there exists a unique additive mapping $A: X\rightarrow Y$ such that \begin{eqnarray}\label{eqn26}
\|f(x)-A(x)\|
\leq\widetilde{\varphi}(x) \end{eqnarray} for all $x\in X$. \end{theorem}
\begin{proof} Letting $x=y=z=0$ in (\ref{eqn24}), we get \begin{eqnarray}\label{eqn24'}
2\|f(0)\|\leq (|\rho_1|+3|\rho_2|)\|(f(0)\|.\end{eqnarray} So $f(0)=0$. Letting $x=y=0$ in (\ref{eq21}), we get
$$\|f(\alpha z)+f(-\alpha z)\|\leq \|\rho_2(f(-\alpha z)+f(\alpha z))\|+\varphi(0,0,z)$$
and so $$\|f(z)+f(-z)\|\leq \frac{\varphi\left(0,0,\frac{z}{\alpha}\right)}{1-|\rho_2|}$$ for all $z\in X$.
Letting $y=x$ and $z=0$ in (\ref{eqn24}), we get \begin{eqnarray}\label{eqn28}
\|2f(2x)-4f(x)\|
\leq |\rho_2|\|f(2x)-2f(x)\|+2|\rho_2|\|f(x)+f(-x)\|+\varphi(x,x,0) \end{eqnarray} and so
$$\|f(2x)-2f(x)\|\leq \frac{1}{2-|\rho_2|}\left(\varphi(x,x,0)+\frac{2|\rho_2|}{1-|\rho_2|}\varphi\left(0,0,\frac{x}{\alpha}\right)\right)$$ for all $x\in X$.
Thus
\begin{eqnarray*}\left\|f(x)-\frac{f(2x)}{2}\right\| \leq \frac{1}{2}\frac{1}{(2-|\rho_2|)}\left(\varphi(x,x,0)+\frac{2|\rho_2|}{1-|\rho_2|}\varphi\left(0,0,\frac{x}{\alpha}\right)\right) \end{eqnarray*} for all $x\in X$. Hence one may have the following formula for positive integers $m, l$ with $m>l$,
\begin{eqnarray}\label{eqn29}
\begin{split}
\;& \left\| \frac{1}{2^{l}}f\left(2^l x\right)-\frac{1}{2^{m}}f\left(2^m x\right)\right\|\\
\;&\leq \sum_{i=l}^{m-1}\frac{1}{2^{i+1}}\frac{1}{(2-|\rho_2|)}\left(\varphi(2^{i}x,2^{i}x,0)
+\frac{2|\rho_2|}{1-|\rho_2|}\varphi\left(0,0,\frac{2^{i}x}{\alpha}\right)\right) \end{split} \end{eqnarray} for all $x\in X$.
It follows from (\ref{eqn30}) that the sequence $\left\{\frac{f(2^kx)}{2^k} \right\}$ is a Cauchy sequence for all $x\in X $. Since $ Y $ is complete, the sequence $\left\{\frac{f(2^kx)}{2^k} \right\}$ converges. So one may define the mapping $A: X\rightarrow Y$ by $$A(x):=\lim_{k\rightarrow \infty}\left\{\frac{f(2^kx)}{2^k} \right\}, \quad \forall x\in X.$$ Taking $l=0$ and letting $m$ tend to $\infty$ in (\ref{eqn29}), we get (\ref{eqn26}).
It follows from (\ref{eqn24}) that \begin{eqnarray} \begin{split}
\;& \|A(x+y+\alpha z)+A(x+y-\alpha z)-2A(x)-2A(y)\|\\
\;& =\lim_{n\rightarrow \infty}\frac{1}{2^n}\left\|f\left[2^n(x+y+\alpha z)\right]+f\left[2^n(x+y-\alpha z)\right]
-2f\left(2^nx\right)-2f\left(2^ny\right)\right\|\\
\;& \leq \lim_{n\rightarrow \infty}\frac{1}{2^n}\left\|\rho_1\left(f\left[2^n(x+y+\alpha z)\right]-f\left(2^nx+2^ny\right)-
f\left(2^n \alpha z\right)\right)\right\|\\ \;&
+\lim_{n\rightarrow \infty}\frac{1}{2^n}\left\|\rho_2\left(f\left[2^n(x+y-\alpha z)\right]+f\left(-2^nx\right)
+f\left(-2^ny+2^n\alpha z\right)\right)\right\|\\\;&+\lim_{n\rightarrow \infty} \frac{1}{2^n}\varphi\left(2^nx,2^ny,2^n\alpha z\right) \\
\;& =\|\rho_1(A(x+y+\alpha z)-A(x+y)-A(\alpha z))\|\\\;&+\|\rho_2(A(x+y-\alpha z)+A(-x)+A(-y+\alpha z))\| \end{split} \end{eqnarray} for all $x,y,z\in X$. One can see that $A$ satisfies the inequality (\ref{eq21}) and so it is additive by Lemma \ref{lm1}.
Now, we show that the uniqueness of $A$. Let $T: X\rightarrow Y$ be another additive mapping satisfying (\ref{eqn24}). Then one has
\begin{eqnarray*} \begin{split}
\; &\|A(x)-T(x)\|= \left\|\frac{1}{2^k} A\left(2^kx\right)-\frac{1}{2^k}
T\left(2^k x\right)\right\| \\
\;&\leq
\frac{1}{2^{k}}\left(\left\|A\left(2^k x\right)-f\left(2^k x\right)\right\|\right.\\ \; &
\left.+\left\|T\left(2^k x\right)-f\left(2^k x\right)\right\|\right) \\
\;& \leq 2\frac{1}{2^{k}}\widetilde{\varphi}(2^k x)=\sum_{i=k}^{\infty}\frac{1}{2^{i+1}}\frac{1}{(2-|\rho_2|)}\left(\varphi(2^{i}x,2^{i}x,0)
+\frac{2|\rho_2|}{1-|\rho_2|}\varphi\left(0,0,\frac{2^{i}x}{\alpha}\right)\right),
\end{split} \end{eqnarray*} which tends to zero as $k\rightarrow \infty$ for all $x\in X$. So we can conclude that $A(x)=T(x)$ for all $x\in X$. \end{proof}
\begin{corollary} Let $r<1$ and $\theta$ be nonnegative real numbers and $f:X\rightarrow Y$ be a mapping such that \begin{eqnarray}\label{eqn211} \begin{split}
\;& \|f(x+y+\alpha z)+f(x+y-\alpha z)-2f(x)-2f(y)\|\\
\;& =\left\|\rho_1(f(x+y+\alpha z)-f(x+y)-f(\alpha z))\right\|\\\;&
+\left\|\rho_2(f(x+y-\alpha z)+f(-x)+f(\alpha z-y))\right\|+\theta(\|x\|^r+\|y\|^r+\|z\|^r) \end{split} \end{eqnarray} for all $x,y,z\in X$. Then there exists a unique additive mapping $A:X\rightarrow Y$ such that \begin{eqnarray}
\|f(x)-A(x)\|\leq \frac{2\theta}{(2-2^r)}\cdot\frac{1}{(1-|\rho_2|)(2-|\rho_2|)}\|x\|^r \end{eqnarray} for all $x\in X$. \end{corollary}
\begin{theorem}\label{thm25} Let $f:X\rightarrow Y$ be a mapping with $\varphi(0,0,0)=0$. If there is a function $\varphi :X^3\rightarrow [0,\infty)$ satisfying (\ref{eqn24}) such that \begin{eqnarray}
\lim_{j\rightarrow \infty}2^{j} \varphi \left(\frac{x}{2^j},\frac{y}{2^j},\frac{z}{2^j}\right)=0 \end{eqnarray} for all $x,y,z\in X$, then there exists a unique additive mapping $A:X\rightarrow Y$ such that
$$\|f(x)-A(x)\|\leq \widetilde{\varphi}\left(\frac{x}{2}\right):=\sum_{i=0}^{\infty}\frac{1}{2^{i}}\frac{1}{2-|\rho_2|}\left(\varphi(\frac{x}{2^{i+1}},\frac{
x}{2^{i+1}},0)+\frac{2|\rho_2|}{1-|\rho_2|}\varphi(0,0,\frac{x}{\alpha 2^{i+1}})\right)$$ for all $x\in X$. \end{theorem}
\begin{proof} Similar to the proof of Theorem \ref{thm23}, we can get \begin{eqnarray*}
\left\|f(x)-2f\left(\frac{x}{2}\right)\right\|
\leq\frac{1}{2-|\rho_2|}\left(\varphi(\frac{x}{2},\frac{x}{2},0)
+\frac{2|\rho_2|}{1-|\rho_2|}\varphi(0,0,\frac{x}{2\alpha })\right) \end{eqnarray*} for all $x\in X$.
Next, we can prove that the sequence $\{2^nf\left(\frac{x}{2^n}\right)$ is a Cauchy sequence for all $x\in X$, and define a mapping $A: X\rightarrow Y$ by $$A(x):=\lim_{n\rightarrow \infty}2^n f\left(\frac{x}{2^n}\right)$$ for all $x\in X$ .
The rest proof is similar to the corresponding part of the proof of Theorem \ref{thm23}. \end{proof}
\begin{corollary} Let $r>1$ and $\theta$ be nonnegative real numbers and $f:X\rightarrow Y$ ba a mapping such that \begin{eqnarray}\label{eqn211} \begin{split}
\;& \|f(x+y+\alpha z)+f(x+y-\alpha z)-2f(x)-2f(y)\|\\
\;&\leq \left\|\rho_1(f(x+y+\alpha z)-f(x+y)-f(\alpha z))\right\|\\\;&
+\left\|\rho_2(f(x+y-\alpha z)+f(-x)+f(\alpha z-y))\right\|+\theta(\|x\|^r+\|y\|^r+\|z\|^r) \end{split} \end{eqnarray} for all $x,y,z\in X$. Then there exists a unique additive mapping $A:X\rightarrow Y$ such that \begin{eqnarray}
\|f(x)-A(x)\|\leq \frac{2^{1+r}\theta}{2^r-1}\frac{1}{(1-|\rho_{2}|)(2-|\rho_{2}|)}\|x\|^r \end{eqnarray} for all $x\in X$. \end{corollary}
\section{Hyers-Ulam stability of (\ref{eq12}) }
In this section, we prove that the Hyers-Ulam stability of the 3-variable functional inequality \begin{eqnarray}\label{eqn31} \begin{split}
\;& \|f(x+\beta y+\alpha z)-f(x-\alpha z)-\beta f(y)-2f(\alpha z)\|\\
\;& \leq \left\|\rho_1(f(x+\alpha z)-f(x)-f(\alpha z))\right\|\\\;&
+\left\|\rho_2(f(x+\beta y-\alpha z)-f(x)-\beta f(y)+f(\alpha z))\right\| \end{split} \end{eqnarray}
in complex Banach space, where $\rho_1$ and $\rho_2$ are fixed complex numbers with $|\rho_2|<1$ and $|\beta +2|\geq |\rho_1|+|\rho_2(1-\beta )|$.
\begin{lemma}\label{lm31} Let $f:X\rightarrow Y$ be a mapping. If it satisfies (\ref{eqn31}) for all $x,y,z\in X$, then $f$ is additive. \end{lemma}
\begin{proof} Letting $x=y=z=0$ in (\ref{eqn31}) for all $x,y,z\in X$, we get \begin{eqnarray}
\|(\beta +2)f(0)\|\leq (|\rho_1|+|\rho_2||\beta -1|)\| f(0)\|. \end{eqnarray} Thus $f(0)=0$.
Letting $x=y=0$ in (\ref{eqn31}), we get
\begin{eqnarray*}(1-|\rho_2|)\|f(\alpha z)+f(-\alpha z)\|\leq 0\end{eqnarray*} and so $f(-x)=-f(x)$ for all $x\in X$.
Letting $x=0$ in (\ref{eqn31}), we have \begin{eqnarray}\label{eqn33} \begin{split}
\|f(\beta y+\alpha z)-f(\alpha z)-\beta f(y)\| \leq
\|\rho_2 (f(\beta y-\alpha z)-\beta f(y)+f(\alpha z))\| \end{split} \end{eqnarray} for all $y,z\in X$.
Letting $z=-z$ in (\ref{eqn33}), we get \begin{eqnarray}
\|f(\beta y-\alpha z)+f(\alpha z)-\beta f(y)\|\leq |\rho_2|\|f(\beta y+\alpha x)-\beta f(y)-f(\alpha z)\| \end{eqnarray} for all $y,z\in X$. Thus \begin{eqnarray}\label{eqn31'}
\|f(\beta y-\alpha z)-\beta f(y)+f(\alpha z)\|\leq 0 \end{eqnarray} and so
$$\|f(y+z)-f(y)-f(z)\|= 0$$ for all $y,z\in X$. Hence $f:X\rightarrow Y$ is additive. \end{proof}
\begin{corollary} Let $f: X\rightarrow Y$ be a mapping satisfying \begin{eqnarray}\label{eqn36} \begin{split}
\;& \|f(x+\beta y+\alpha z)-f(x-\alpha z)-\beta f(y)-2f(\alpha z)\|\\
\;& =\left\|\rho_1(f(x+\alpha z)-f(x)-f(\alpha z))\right\|\\\;&
+\left\|\rho_2(f(x+\beta y-\alpha z)-f(x)-\beta f(y)+f(\alpha z))\right\| \end{split} \end{eqnarray} for all $x,y,z\in X$. Then $f:X\rightarrow Y$ is additive. \end{corollary}
We prove the Hyers-Ulam stability of the functional inequality (\ref{eqn31}) in complex Banach spaces.
\begin{theorem}\label{thm33} Let $f:X\rightarrow Y$ be a mapping. Assume that there is a function $\varphi:X^3\rightarrow [0,\infty)$ with $\varphi(0,0,0)=0$ such that \begin{eqnarray}\label{eqn34} \begin{split}
\;& \|f(x+\beta y+\alpha z)-f(x-\alpha z)-\beta f(y)-2f(\alpha z)\|\\
\;& \leq\left\|\rho_1(f(x+\alpha z)-f(x)-f(\alpha z))\right\|\\\;&
+\left\|\rho_2(f(x+\beta y-\alpha z)-f(x)-\beta f(y)+f(\alpha z))\right\|+\varphi(x,y,z) \end{split} \end{eqnarray} and \begin{eqnarray}\label{eqn35}
\lim_{j\rightarrow \infty}\frac{1}{|1+\beta|^{ j}} \varphi\left((1+\beta)^{j}x, (1+\beta)^{j}y,(1+\beta)^{j}z\right)=0 \end{eqnarray} for all $x,y,z\in X$. Then there exists a unique additive mapping $A:X\rightarrow Y$ such that \begin{eqnarray}\label{eqn36}
\|f(x)-A(x)\|
\leq \widetilde{\varphi}(x,x,0) \end{eqnarray} for all $x\in X$, where \begin{eqnarray}\label{eqn37}
\widetilde{\varphi}(x,y,z):=\frac{1}{|1+\beta|(1-|\rho_1|)}
\sum_{j=0}^\infty\frac{1}{|1+\beta|^{ j}} \varphi\left((1+\beta)^{j}x, 1+\beta)^{j}y,1+\beta)^{j}z\right)<\infty \end{eqnarray} for all $x, y, z\in X$. \end{theorem}
\begin{proof} Letting $x=y=z=0$ in (\ref{eqn34}), we get \begin{eqnarray}\label{eqn24'}
\|(\beta +2)f(0)\|\leq (|\rho_1|+|(1-\beta)\rho_2|)\| f(0)\|.\end{eqnarray} So $f(0)=0$.
Letting $z=0$ and $y=x$ in (\ref{eqn34}), we get \begin{eqnarray}\label{eqn312}
\|f((1+\beta)x)-(1+\beta)f(x)\|
\leq |\rho_2|\|f((1+\beta)x)-(1+\beta)f(x)\|+\varphi(x,x,0) \end{eqnarray} for all $x\in X$.
Thus \begin{eqnarray*}
\left\|f(x)-\frac{f((1+\beta)x)}{1+\beta}\right\|
\leq \frac{1}{1-|\rho_2|}
\frac{1}{|1+\beta|}\varphi\left(x,x,0\right) \end{eqnarray*} for all $x\in X$ .
Hence one may have the following formula for positive integers $m, l$ with $m>l$, \begin{eqnarray}\label{eqn39} \begin{split}
\;&\left\| \frac{1}{|1+\beta|^{l}}f\left((1+\beta)^l x\right)-\frac{1}{|1+\beta|^{m}}f\left((1+\beta)^m x\right)\right\|\\
\;& \leq \frac{1}{|1+\beta|(1-|\rho_1|)} \sum_{i=l}^{m-1}\frac{1}{|1+\beta|^{i}}\varphi\left(|1+\beta|^i x,|1+\beta|^i x, 0\right), \end{split}\end{eqnarray} for all $x\in X$.
It follows from (\ref{eqn37}) that the sequence $\left\{\frac{f(1+\beta)^kx)}{(1+\beta)^k} \right\}$ is a Cauchy sequence for all $x\in X $. Since $ Y $ is complete, the sequence $\left\{\frac{f((1+\beta)^kx)}{(1+\beta)^k} \right\}$ converges. So one may define the mapping $A: X\rightarrow Y$ by $$A(x):=\lim_{k\rightarrow \infty}\left\{\frac{f((1+\beta)^kx)}{(1+\beta)^k} \right\}, \quad \forall x\in X.$$ Taking $m=0$ and letting $l$ tend to $\infty$ in (\ref{eqn39}), we get (\ref{eqn36}).
It follows from (\ref{eqn34}) that \begin{eqnarray} \begin{split}
\;& \|A(x+\beta y+\alpha z)-A(x-\alpha z)-\beta A(y)-2A(\alpha z)\|\\
\;& =\lim_{n\rightarrow \infty}\frac{1}{|1+\beta|^n}
\left\|f\left[(1+\beta)^n(x+\beta y+\alpha z)\right]+ f\left[(1+\beta)^n(x-\alpha z)\right]\right.\\ \;&\left.
-\beta f\left((1+\beta)^ny\right)-2f\left((1+\beta)^n \alpha z\right)\right\|\\
\;& \leq \lim_{n\rightarrow \infty}\frac{1}{|1+\beta|^n}
\left\|\rho_1\left(f\left[(1+\beta)^n(x+\alpha z)\right]-
f\left[(1+\beta)^n\left(x\right)\right]-f\left((1+\beta)^n\alpha z\right)\right)\right\|\\ \;&
+\lim_{n\rightarrow \infty}\frac{1}{|1+\beta|^n}
\left\|\rho_2\left(f\left[(1+\beta)^n(x+\beta y-\alpha z)\right]-f\left((1+\beta)^nx\right)\right.\right.\\ \;&\left.\left.
-\beta f\left((1+\beta)^ny\right)+f\left((1+\beta)^n \alpha z\right)\right)\right\|\\\;&+\lim_{n\rightarrow \infty}
\frac{1}{|1+\beta|^n}\varphi\left((1+\beta)^nx,(1+\beta)^ny,(1+\beta)^nz\right) \\
\;& =\|\rho_1(A(x+\alpha z)-A(x)-A(\alpha z))\|\\\;&+\|\rho_2(A(x+\beta y-\alpha z)-A(x)-\beta A(y)+A(\alpha z))\| \end{split} \end{eqnarray} for all $x,y,z\in X$. One can see that $A$ satisfies the inequality (\ref{eqn31}) and so it is additive by Lemma \ref{lm31}.
Now, we show that the uniqueness of $A$. Let $T: X\rightarrow Y$ be another additive mapping satisfying (\ref{eqn34}). Then one has
\begin{eqnarray*} \begin{split}
\; &\|A(x)-T(x)\|= \left\|\frac{1}{(1+\beta)^k} A\left((1+\beta)^kx\right)-\frac{1}{(1+\beta)^k}
T\left((1+\beta)^k x\right)\right\| \\
\;&\leq
\frac{1}{|1+\beta|^{k}}\left(\left\|A\left((1+\beta)^k x\right)-f\left((1+\beta)^k x\right)\right\|\right.\\ \; &
\left.+\left\|T\left((1+\beta)^k x\right)-f\left((1+\beta)^k x\right)\right\|\right) \\
\;& \leq 2\frac{1}{|1+\beta|^{k}}\widetilde{\varphi}(x,x,0),
\end{split} \end{eqnarray*} which tends to zero as $k\rightarrow \infty$ for all $x\in X$. So we can conclude that $A(x)=T(x)$ for all $x\in X$. \end{proof}
\begin{corollary} Let $r>1$ and $\theta$ be nonnegative real numbers and $f:X\rightarrow Y$ ba a mapping such that \begin{eqnarray}\label{eqn211} \begin{split}
\;& \|f(x+\beta y+\alpha z)-f(x-\alpha z)-\beta f(y)-2f(\alpha z)\|\\
\;& \leq \left\|\rho_1(f(x+\alpha z)-f(x)-f(\alpha z))\right\|\\\;&
+\left\|\rho_2(f(x+\beta y-\alpha z)-f(x)-\beta f(y)+f(\alpha z))\right\|+\theta(\|x\|^r+\|y\|^r+\|z\|^r) \end{split} \end{eqnarray}
for all $x,y,z\in X$ with $|1+\beta|>1$. Then there exists a unique additive mapping $A:X\rightarrow Y$ such that \begin{eqnarray}
\|f(x)-A(x)\|\leq \frac{2\theta}{|1+\beta|-|1+\beta|^r}\frac{1}{1-|\rho_{2}|}\|x\|^r \end{eqnarray} for all $x\in X$. \end{corollary}
\begin{theorem}\label{thm35} Let $f:X\rightarrow Y$ be a mapping with $f(0)=0$. If there is a function $\varphi :X^3\rightarrow [0,\infty)$ satisfying (\ref{eqn34}) such that \begin{eqnarray}
\widetilde{\varphi}(x,y,z):=\sum_{j=1}^\infty |1+\beta|^{j} \varphi \left(\frac{x}{(1+\beta)^j},\frac{y}{(1+\beta)^j},\frac{z}{(1+\beta)^j}\right)<\infty \end{eqnarray} for all $x,y,z\in X$, then there exists a unique additive mapping $A:X\rightarrow Y$ such that \begin{eqnarray}
\|f(x)-A(x)\|\leq\frac{1}{1-|\rho_2|} \widetilde{\varphi}\left(\frac{x}{1+\beta },\frac{x}{1+\beta},0\right) \end{eqnarray} for all $x\in X$. \end{theorem} \begin{proof} Similar to the proof of Theorem \ref{thm33}, we can get \begin{eqnarray*}
\left\|f(x)-(1+\beta)f\left(\frac{x}{1+\beta}\right)\right\|
\leq \frac{1}{1-|\rho_2|}\varphi\left(\frac{x}{1+\beta},\frac{x}{1+\beta},0\right) \end{eqnarray*} for all $x\in X$.
Next, we can prove that the sequence $\{(1+\beta)^nf\left(\frac{x}{(1+\beta)^n}\right)\} $ is a Cauchy sequence for all $x\in X$ and define a mapping $A: X\rightarrow Y$ by $$A(x):=\lim_{n\rightarrow \infty}(1+\beta)^n f\left(\frac{x}{(1+\beta)^n}\right)$$ for all $x\in X$. The rest of the proof is similar to the corresponding part of the proof of Theorem \ref{thm33}. \end{proof}
\begin{corollary} Let $r>1$ and $\theta$ be nonnegative real numbers and $f:X\rightarrow Y$ ba a mapping such that \begin{eqnarray} \begin{split}
\;& \|f(x+\beta y+\alpha z)-f(x-\alpha z)-\beta f(y)-2f(\alpha z)\|\\
\;& \leq \left\|\rho_1(f(x+\alpha z)-f(x)-f(\alpha z))\right\|\\\;&
+\left\|\rho_2(f(x+\beta y-\alpha z)-f(x)-\beta f(y)+f(\alpha z))\right\|+\theta(\|x\|^r+\|y\|^r+\|z\|^r) \end{split} \end{eqnarray}
for all $x,y,z\in X$ and $|1+\beta|<1$. Then there exists a unique additive mapping $A:X\rightarrow Y$ such that \begin{eqnarray}
\|f(x)-A(x)\|\leq \frac{2\theta}{|1+\beta|^r-|1+\beta|}\frac{1}{1-|\rho_{2}|}\|x\|^r \end{eqnarray} for all $x\in X$. \end{corollary}
\section*{Competing interests}
The author declares that he has no competing interests.
\section*{Authors' contributions}
The author conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
\section*{Funding}
This work was supported by National Natural Science Foundation of China (No. 11761074), the Projection of the Department of Science and Technology of JiLin Province and the Education Department of Jilin Province (No. 20170101052JC) and the scientific research project of Guangzhou College of Technology and Business in 2020(No. KA202032).
\end{document} | arXiv |
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Debaditya Choudhury ORCID: orcid.org/0000-0002-0573-36201,2 na1,
Duncan K. McNicholl ORCID: orcid.org/0000-0003-4511-17191,2 na1,
Audrey Repetti ORCID: orcid.org/0000-0002-6296-69573,4 na1,
Itandehui Gris-Sánchez ORCID: orcid.org/0000-0001-5756-51075 nAff9,
Shuhui Li ORCID: orcid.org/0000-0002-3314-070X6,7,
David B. Phillips7,
Graeme Whyte8,
Tim A. Birks5,
Yves Wiaux ORCID: orcid.org/0000-0002-1658-01213 &
Robert R. Thomson ORCID: orcid.org/0000-0003-4978-14881,2
Nature Communications volume 11, Article number: 5217 (2020) Cite this article
Fibre optics and optical communications
Imaging and sensing
The thin and flexible nature of optical fibres often makes them the ideal technology to view biological processes in-vivo, but current microendoscopic approaches are limited in spatial resolution. Here, we demonstrate a route to high resolution microendoscopy using a multicore fibre (MCF) with an adiabatic multimode-to-single-mode "photonic lantern" transition formed at the distal end by tapering. We show that distinct multimode patterns of light can be projected from the output of the lantern by individually exciting the single-mode MCF cores, and that these patterns are highly stable to fibre movement. This capability is then exploited to demonstrate a form of single-pixel imaging, where a single pixel detector is used to detect the fraction of light transmitted through the object for each multimode pattern. A custom computational imaging algorithm we call SARA-COIL is used to reconstruct the object using only the pre-measured multimode patterns themselves and the detector signals.
Endoscopes that use bundles of optical fibres to transmit light in a spatially-selective manner have had a profound impact on minimally-invasive medical procedures. To reduce the device size and increase imaging resolution, this concept has been extended to individual fibres containing thousands of light-guiding cores. These single-fibre coherent fibre bundles (SF-CFBs) can provide resolutions of a few microns in the visible1. When combined with fluorescent contrast agents, they facilitate observation of disease processes at the cellular level2.
SF-CFBs are not without drawbacks. To maintain spatially-selective transmission of light, the fibre cores must be sufficiently spaced to keep core-to-core crosstalk at an acceptable level, intrinsically limiting imaging resolution and throughput. Recently, it has been shown that a highly scattering material can be placed onto the end of the SF-CFB to convert each single mode into a multimode pattern of light, and that these patterns can be used for compressive fluorescence imaging3. Unfortunately, the use of such a highly scattering material between the object and the distal end of the fibre will dramatically reduce the fluorescence collection efficiency—a key parameter that must be maximised for real-world in-vivo applications. The significant drawbacks of SF-CFBs has led to an explosion of interest in multimode fibre (MMF) imaging, where image information is carried by multiple overlapping spatial modes guided by one multimode core, rather than the many spatially separated cores of the SF-CFB. MMF imaging can deliver an order of magnitude higher spatial resolution, but it is far from trivial to implement because the amplitudes and phases of the MMF modes become scrambled along the fibre. This can be addressed by characterising the MMF's transmission matrix and controlling a spatial light modulator to "undo" the scrambling4,5, but any movement of the fibre changes its transmission matrix, and access to the in vivo distal end is required for recalibration unless the new path is precisely known6.
One technology that could provide of a route to high resolution single-fibre microendoscopy is the multicore fibre (MCF) "photonic-lantern" (PL)7. PLs are guided-wave transitions that efficiently couple light from Ns single mode cores (the MCF) to a multimode waveguide like an MMF. PLs can be made by tapering (heating and stretching in a small flame) a single MCF8, such that the entire reduced-diameter MCF acts as the multimode end of the PL. Np = Ns distinct multimode patterns of light are generated at the multimode output by coupling light into each core at the MCF input, one at a time. If the MCF exhibits negligible crosstalk between the cores along the length of the MCF, such that the light propagates along just one core, these patterns do not change when bending the fibre, unlike those of an ordinary MMF. This is because deformation of the MCF merely changes the overall phase of the output pattern. Unlike the spatially-separated modes of a SF-CFB (but like an ordinary MMF), the PL allows the full area of the fibre end-facet to be sampled, and the size of the patterns can be reduced to the minimum allowed by the numerical aperture (NA) of the multimode end.
Here, we demonstrate the feasibility of PL based microendoscopy by using a PL to implement a form of "single-pixel" imaging9 that we call computational optical imaging using a lantern (COIL). Light patterns generated by the PL are projected onto an object (e.g., tissue). Light returned from the object (e.g., fluorescence) is detected by a single-pixel detector, which for the microendoscopy application could be placed at the proximal end of the MCF. In this case, the single-pixel detector would measure the total returned signal across all MCF cores. Information about the distribution of light across these cores is not needed, and the manner in which the light is detected is not critical—as long as it is unchanged between calibration and measurement, and the proximal-end detection is not selecting part of an interference pattern due to waves from multiple cores (which would be sensitive to phases). The known patterns and measured return signals provide information about the object, from which an image can be formed9. We note that both Mahalati at al.10 and Amitonova et al.11 have demonstrated compressive single MMF imaging with proximal end signal detection, but we highlight that both suffer from the stability issues that MCF PLs address. We show that the quality and detail of the computed image can be greatly improved by exploiting an advanced image formation algorithm, that combines the measurement data with a generic prior postulating that the spatial structure of the image is underpinned by a small number of degrees of freedom. We demonstrate that COIL opens a promising route to efficient and practical high-resolution microendoscopy.
Computational imaging algorithm
The starting point for our image reconstruction algorithm is to approach PL based imaging in the context of the theory of compressive sampling. In this context, one assumes that the image under scrutiny is sparse in some transform domain linearly related to the pixel domain (e.g., the domain of a wavelet transform12), that is to say that its spatial structure is underpinned by a small number of degrees of freedom. The sparsity prior information is leveraged to enable the image recovery from incomplete data. Compressive sampling approaches have been developed in a wide variety of imaging applications ranging from magnetic resonance imaging13,14, and astronomical imaging15,16, to ghost imaging17,18 and speckle imaging19. Optimisation algorithms represent the dominant class to solve inverse problems for image recovery from incomplete data. The image estimate is defined as a minimiser of an objective function, consisting of the sum of a data-fidelity term and a sparsity-promoting prior term. The resulting minimisation problem is solved through iterative algorithms progressively minimising the objective function.
From the perspective of the reconstruction algorithm, we choose to work in a regime where the number of pixels in the reconstruction is considerably higher than the number of patterns projected and their transmission values e.g., for reconstructions formed using 121 projected patterns, our reconstructions are 125 × 125 pixels in size (see Supplementary Note 1 in the Supplementary Information (SI) for a detailed description of the reasoning involved in setting the number of pixels in each reconstruction presented in this manuscript.). As we demonstrate through simulations, this regime is of particular interest for COIL as it can, in the future, allow the reconstruction of high resolution images without unrealistic demands on the number of MCF cores. Due to the high number of pixels in the reconstruction compared to the number of projected patterns, the inverse problem becomes heavily ill-posed and image formation requires strong prior information. With that aim we resort to an advanced "average sparsity" model firstly introduced in astronomical imaging16, where multiple wavelet transforms are introduced simultaneously to promote sparsity and reduce the effective number of degrees of freedom well-below the image size.
To solve the resulting minimisation problem, we rely on modern "proximal splitting" optimisation algorithms20,21 whose main features are a guaranteed fast convergence and low computational complexity. These algorithms have been used in computational imaging in a variety of fields (see ref. 20 and references therein). Building on the "average sparsity" approach we developed a proximal algorithm for COIL, dubbed sparsity averaging reweighted analysis (SARA)–COIL. Details of our optimisation approach are provided in the "Methods" section, together with a description of the associated MATLAB toolbox.
Experimental techniques and results
Figure 1a is a schematic of an MCF (with Ns = 25 for clarity) with a PL at one end. For the work reported here, the PL was fabricated at one end of ~3 m of MCF with Ns = 121 single-mode cores in a 11 × 11 square array (Fig. 1b) with negligible core-to-core crosstalk at 514 nm. The multimode output end of the PL had a core diameter of ~35 μm and an NA of ~0.22 (Fig. 1c). See "Methods" section for fabrication details of the PL. Using computer-controlled alignment, each MCF core could be individually excited using coherent 514 nm laser light, generating Np = Ns = 121 different multimode patterns of light at the output. Each output pattern was highly stable regardless of the conformation of the MCF, Fig. 1d–f. (See the "Methods" section for full details of how the stability of these patterns was quantified, together with Supplementary Fig. 1 for a schematic of the experimental setup used to characterise the pattern stability. Supplementary Figs. 2 and 3 present the patterns generated under different MCF deformation and excitation conditions. Supplementary Table 1 presents quantitative evaluations of the pattern stability under different MCF deformation and excitation conditions). This is due to the short length (~4 cm) of the PL transition itself and the minimal crosstalk between the MCF cores. In contrast, similar bending of an ordinary MMF changes the output pattern (Fig. 1g–i).
Fig. 1: Computational imaging using a photonic lantern.
a Schematic Ns = 25 square-array multicore fibre with a photonic lantern at one end. (in green) Light in one core excites a fixed light pattern at the lantern's output. b Optical micrograph of the facet of the Ns = 121 multicore fibre used in this work. Scale bar: 50 μm. c Optical micrograph of the multimode output of the photonic lantern. Scale bar: 10 μm. d–f Near field intensity patterns at the output of the photonic lantern when one core of the multicore fibre is excited with monochromatic light (λ = 514 nm). The patterns are insensitive to fibre bending as shown by the micrographs obtained for three arbitrary conformations of the fibre. Scale bars: 10 μm. g–i Near field intensity patterns at the output of a 105 μm core multimode fibre when excited with monochromatic light (λ = 514 nm). As shown in the micrographs obtained for three arbitrary conformations of the fibre, the patterns are highly sensitive to bending of the fibre. Scale bars: 20 μm. j Experimental setup used to acquire the data during the photonic lantern imaging experiments. Full details of the data acquisition procedure are given in the "Methods" section.
Our experimental setup for single pixel imaging (Fig. 1j) is similar to the computational ghost imaging system presented in the ref. 18, where a spatial light modulator projected random patterns of light onto a test object and detectors measured the fraction of power transmitted through the object. In our experiment the spatial light modulator was replaced with the PL, allowing Np = Ns = 121 different patterns to be projected onto the object by exciting each core of the MCF individually. The experimental data acquisition procedure consisted of two steps. First, a camera was used to record the patterns generated in the object plane by exciting each of the cores individually. In the second, the object was moved into the beam path and a detector was used to record the magnitude of the light transmitted through the object. These pairs of data (patterns + transmission values) are the data used with the SARA–COIL algorithm (further details of the experimental setup and data acquisition procedure are presented in the "Methods" section). We highlight the fact that the MCF was intentionally moved and deformed significantly between pattern calibration and imaging experiments, to further highlight the stability of the PL approach.
Initially, we used a simple "knife-edge" as the object. As shown in the object images of Fig. 2, the knife-edge was orientated either horizontally (H) or vertically (V) and positioned to block ~25, ~50, or ~75% of the pattern projected onto it. As shown in Fig. 2, COIL successfully reconstructs images of 125 × 125 pixels using only Np = 121 patterns. All reconstructions we report using experimental data represent a 0.9 mm × 0.9 mm field of view at the object plane, where the lantern output is imaged with a magnification of ~26 for the purposes of this demonstration. Since the illumination light originates from the lantern itself, the resolution of a near-field imaging modality without the imaging optics would scale by the inverse of the same magnification.
Fig. 2: COIL imaging of a knife edge using experimental data.
SARA–COIL results obtained using Np = 121 patterns. Micrographs of the objects are shown in the first and third columns, while corresponding SARA–COIL reconstructions are presented in the columns to the right of each set of objects. Hi and Vi respectively denote objects formed by horizontally and vertically overlaying a knife edge over ~25% (i = 1), ~50% (i = 2), and ~75% (i = 3) of the intensity pattern. Each reconstructed image has 125 × 125 pixels, with a field of view in the object plane of 0.9 mm × 0.9 mm.
To confirm that COIL is applicable to more complex objects, we repeated the experiment using the objects shown in Fig. 3: an "off-centre cross" and "4 dots" positioned asymmetrically. SARA–COIL can clearly reconstruct the off-centre cross, further confirming the generality of the approach, but cannot reconstruct the small features in the "4-dots" object. To demonstrate how the imaging quality might improve by using an MCF PL with more cores, we repeated the data acquisition nine times with the object rotated by 40° between each, acquiring transmission data for each object using effectively Np = Ns × 9 = 1089 different patterns. As expected, increasing the number of patterns significantly increases image quality for the off-centre cross (Fig. 3). It also reconstructs some features of the "4 dots" object but falls short of fully resolving them.
Fig. 3: COIL imaging of objects using experimentally measured and simulated data.
SARA–COIL reconstructions are presented using either Np = 121 or Np = 1089 patterns, and either experimentally measured or simulated pattern and overlap data. The objects are an off-centre cross and four asymmetrically-positioned elliptical dots, micrographs of which are presented. The reconstructions are either 125 × 125 pixels in size for the Np = 121 case, or 377 × 377 pixels in size for the Np = 1089 case. Reconstructions using simulated patterns and overlap data for the Np = 121 case used patterns generated from random orthonormal superpositions of the 121 lowest order modes of a circular ideal-mirror waveguide. For reconstructions using Np = 1089, the object was rotated about the optical axis by 320° in steps of 40°, effectively creating a total of 121 × 9 patterns. The field of view of all reconstructions using experimentally measured data is 0.9 mm × 0.9 mm in the object plane.
The reconstructions we have presented using experimentally acquired data are a proof-of-concept of the COIL approach, but the quality of the imaging we have obtained is limited and does not yet demonstrate a compressive advantage. As we discuss later, we believe this is primarily due to limitations with the current experimental setup. To investigate the imaging quality that could be achievable in the future using an optimised experimental system, we have performed detailed end-to-end simulations. To simulate the intensity patterns from an ideal Ns = 121 PL, we first calculated the field distributions of the 121 lowest-order spatial modes of a circular ideal-mirror waveguide. We then generated a set of 121 mutually-orthonormal but otherwise random coherent superpositions of the modes, and formed intensity patterns by taking the square modulus. We highlight that previously reported characterisation results have confirmed the adiabatic nature of our MCF PL transitions22, supporting the relevance of the PL pattern simulations to our experimental system. The imaging experiment was simulated by computing the overlap integral between each intensity pattern and the object. The intensity patterns and overlap data were then processed using SARA–COIL to reconstruct an image. The simulated reconstructions for both objects, using either Np = 121 (not rotated) or Np = 1089 (nine rotations), are shown in Fig. 3 alongside the reconstructions based on experimental data for comparison. As expected, images obtained using both measured and simulated data improved considerably as the number of patterns is increased. Furthermore, if we consider that the multimode port of the PL used in our experiments has a diameter of 35 μm, our Np = 1089 simulations suggest that sub-micron resolution could be achievable using a PL generating only a thousand patterns. (The NA of the port would have to be ~0.3 to support this number of modes, rather than the 0.22 of the PL used here).
To further highlight the potential of COIL for the high-resolution imaging of structures in vivo, we simulated (as above) the results that might be expected using a Ns = 2000 PL to project Np = 2000 patterns. The two objects used for this simulation were an image of the 1951 USAF resolution target and a confocal microscope fluorescence image of fixed calcein-stained adenocarcinomic human alveolar basal epithelial (A549) cells. Our images, shown in Fig. 4, are high-quality reconstructions of both objects. Figure 4 also shows that our image reconstruction technique is robust to the presence of additive Gaussian noise in the overlap data. For example, both contrast and resolution are only minimally affected by the noise, and features such as the horizontal and vertical bars in the top right of the USAF target are still clearly resolvable.
Fig. 4: Simulations of COIL imaging with a high core count photonic lantern.
Simulated reconstruction results (511 × 511 pixels) obtained using Np = 2000 intensity patterns generated from random orthonormal superpositions of the 2000 lowest-order modes of a circular ideal-mirror waveguide. The objects were the 1951 USAF resolution target and a confocal microscope image of fixed calcein stained adenocarcinomic human alveolar basal epithelial (A549) cells. For each object, the reconstructed image with additive Gaussian noise (input signal-to-noise ratio iSNR = 50) is shown alongside that with no added noise. We highlight the fact that there is deliberately no spatial scale for the reconstructions, since the size of a waveguide supporting Np = 2000 modes varies depending on its core-cladding refractive index contrast. The reader is referred to the discussion section for more information. We thank Eckhardt Optics for allowing us to use their image of the USAF 1951 resolution test chart presented in the top left.
For completeness, Fig. 5 compares SARA–COIL to a simpler, more intuitive, reconstruction algorithm used for classical ghost imaging—see Eq. 5 in the ref. 18. This algorithm uses only the fractional transmission of the projected pattern to weight its contribution to the image reconstruction. No attempt is made to optimise this towards a realistic object using a prior. The comparison confirms that SARA–COIL significantly improves both resolution and contrast, revealing features that are otherwise barely or not visible. These results provide a compelling justification for the advanced algorithmic approach we adopted.
Fig. 5: Comparing reconstruction algorithms for COIL applications.
Reconstructions of various objects using experimental or simulated data and either an established ghost imaging algorithm (Eq. 5 in [Sun18]) (middle row) or SARA–COIL (bottom row). a 125 × 125 pixel reconstructions of an off-centre cross for Np = 121 using experimental data. b 377 × 377 pixel reconstructions of an offset cross for Np = 1089 using experimental data. c, d 511 × 511 pixel reconstructions of the A549 cells (c) and the USAF target (d) for Np = 2000 using simulated patterns and overlap data. Note that regions with no available information are treated differently by the two algorithms. As seen in the corners of all images, the ghost imaging algorithm assigns a mid-scale value, whereas SARA–COIL assigns a value of 0. In images reconstructed from experimental data, 1 represents the regions of highest transmission, and in those based on simulated data 1 represents regions of highest intensity. The field of view of all reconstructions using experimental data is 0.9 mm × 0.9 mm in the object plane. We thank Eckhardt Optics for allowing us to use their image of the USAF 1951 resolution test chart presented at the top of column d.
The reconstructions presented in Fig. 3 using 1089 patterns clearly indicate that although our experimental results broadly agree with simulations from ideal data, there is considerable potential for more accurate reconstructions. We highlight that the quality of the reconstructions using experimental data is degraded by the fact that the re-centring of the object onto the pattern after each rotation was only performed by eye, using a thin ring around the object to guide alignment. In fact, both reconstructions in Fig. 3 show hints of resolving this ring. This practical limitation can be readily resolved by adopting MCFs with more cores and not rotating them.
Remarkably, Fig. 4 demonstrates that, even in the presence of noise, a future Ns = Np = 2000 COIL system could be capable of resolving objects separated by just ~1.6% of the multimode core diameter (see the three-bar pattern at the top right of the USAF target). If these objects were point-like objects, this would indicate the potential to resolve ~3000 point source objects across the fibre facet. This is a clear demonstration of the future potential advantage of using COIL in combination with the SARA–COIL algorithm for compressive optical imaging. To put this into a future use-case context, if a COIL system is constructed to operate using 488 nm excitation light and an Ns = 2000 MCF, the multimode output of the PL could have a 63 μm diameter core with an NA of 0.22, assuming established fabrication techniques8 with an F-doped silica cladding—see "Methods" section. Such a system could resolve objects separated by just ~1.25 μm. This is close to the 1.35 μm expected from Rayleigh's criterion (0.61 λ/NA), a strong indication that COIL can deliver at least diffraction-limited imaging across the field of view of the core.
The Ns = Np pattern projection is only the simplest imaging modality one might consider using PLs for. In fact, PLs could enable significantly more advanced and powerful modalities, some driven by compressive sampling principles, but these require the controlled simultaneous excitation of multiple MCF cores to generate coherent combinations of the multimode states at the output. To do this in a controlled manner, the key information to be obtained are the relative phases and amplitudes of the individual basis patterns at the multimode output. We envisage future COIL imaging systems exploiting polarisation maintaining MCFs, where the PL's output is coated to partially reflect some pump or metrology light back along the MCF. Since each multimode pattern generates a specific nonbinary phase and amplitude distribution across the MCF cores after reflection, and since there is negligible crosstalk between the MCF's cores, the distribution of reflected light across the cores at the proximal end will encode the relative phases and amplitudes of the multimode patterns at the output. In principle, this could facilitate the coherent synthesis of arbitrary excitation fields at the output of the lantern for both near-field and far-field spot-scanning modalities. To demonstrate the feasibility of these more advanced MCF PL-based imaging approaches, Supplementary Fig. 4 in the SI presents results of transmission matrix measurements, and coherent light control, to generate spot scanning at the multimode output of an integrated photonic lantern. Also see Supplementary Movie 1 in the SI to view the full transmission matrix characterisation, and Supplementary Movie 2 in the SI for a visualisation of the coherent generation of a scanning spot at the multimode end of the integrated lantern. Further details of these preliminary experiments on coherent mode combination in a photonic lantern are given in Supplementary Note 2 in the SI. Such coherent mode combination approaches could also enable the projection of many more than Ns different known multimode patterns. As detailed by Mahalati et al.10, the number of possible "intensity modes", and therefore the number of resolvable features across the output core, could reach a maximum of 4Ns. For the case of an Ns = 2000 PL with a 63 μm diameter 0.22 NA multimode core operating at 488 nm, such an approach could deliver a resolution of ~626 nm—significantly smaller than the Rayleigh limit and opening a potential route to super-resolution microendoscopy. The NA of the PL's multimode output can also be pushed well beyond 0.22 by exploiting more advanced fibre approaches. For example, we foresee the creation of PL's using a polarisation maintaining MCF with a double-cladding geometry, such as those commonly used in fibre lasers for efficient cladding pumping. In this case, the MCF cores and their glass cladding would be surrounded by an air cladding that could facilitate a PL multimode port at the distal end with an in vivo NA of up to ~0.65 at 488 nm23. This might deliver a spatial resolution of ~212 nm, although stability issues during in vivo exposure will obviously play a role in determining this.
We resorted to a powerful framework of optimisation to develop the SARA-COIL algorithm, but further developments may significantly improve image estimation. Firstly, regularisation priors specifically developed for images of interest in microendoscopy can improve quality over our state-of-the-art "average sparsity" prior. Secondly, parallelised "proximal algorithms"24,25 can improve scalability to high-resolution imaging, ultimately to provide real-time microendoscopic imaging. Finally, approximation in the measurement model can severely affect imaging quality in computational imaging (e.g., the alignment between object and patterns). Joint calibration and imaging algorithms can be defined in the theory of optimisation, that can simultaneously solve for unknown parameters in the measurement model and form the image26,27.
To conclude, we have experimentally demonstrated a form of single-pixel imaging using a multicore fibre and photonic lantern to generate distinct multimode light patterns. We have provided compelling evidence that this, underpinned by the powerful SARA-COIL optimisation algorithm, can deliver at least diffraction-limited imaging across the full area of a multimode fibre core, without sensitivity to bending or any need to control or compensate for modal phases. This meets the world-wide need to develop new fibre-optic imaging techniques to deliver high-resolution images of cellular and molecular mechanisms in vivo. We have also discussed how it opens a route to more complex imaging modalities, such as super-resolution microendoscopy with submicron resolution. We also anticipate that COIL could also be useful in applications that benefit from a reduced number of measurements, such as fibre-optic epifluorescence or confocal microendoscopy, which are vulnerable to detrimental effects such as photobleaching and phototoxicity28.
The multicore fibre
The Ns = 121 square-array multicore fibre was originally fabricated for a study of wavelength-to-time mapping22. The cores were positioned on a square grid with a core-to-core spacing of ~10.53 µm. The mode field diameters of the MCF cores were measured at 514 nm using calibrated near-field imaging. The 1/e2 mode field diameter was ~2.1 ± 0.2 μm.
Photonic lantern fabrication
To fabricate the PL8, the MCF was threaded into a fluorine-doped silica capillary, the refractive index of which is lower than the pure silica cladding of the MCF. The capillary was collapsed, by surface tension, on top of the MCF using an oxybutane flame. Using a similar flame, the cladded structure was then softened and stretched by a tapering rig, forming a biconical fibre-like structure. The multimode port of the PL was finally revealed by cleaving the centre of the tapered waist. The resultant multicore-to-multimode taper was ~4 cm long, with an approximately linear profile. The multimode port's core diameter was ~35 µm and its numerical aperture was 0.22.
Quantifying the stability of the multimode patterns
Supplementary Fig. 1 shows the experimental setup we used to characterise the stability of the patterns generated at the multimode end of an MCF lantern. Light from a 514 nm diode laser was transported to the MCF using a single mode fibre (SM450 from Thorlabs, single mode at 514 nm, NA between 0.1 and 0.14) and coupled into the MCF using direct fibre-to-fibre coupling. By minimising the gap between the single mode fibre and the MCF it was possible to excite any core of the MCF individually. By increasing the gap slightly, it was possible to excite multiple cores simultaneously. Light emerging from the multimode end of the photonic lantern was focused using a lens onto a CMOS camera, and near-field images of the multimode output could be digitally captured using a computer. To investigate the stability of the multimode patterns we captured images of the patterns under the following excitation conditions.
Characterising the effect of excitation polarisation: to investigate the effect of laser polarisation on the multimode patterns, five images of the multimode output were captured. During the data acquisition, the MCF was not disturbed but the single mode fibre delivering light from the laser to the MCF was either left undisturbed or wrapped around a 25 mm diameter circular rod one, two, three, or four times to alter the excitation polarisation in a random manner.
Characterising the effect of MCF deformation: to investigate the effect of deforming the MCF, five images of the multimode pattern were captured. During the data acquisition, the single mode fibre delivering light from the laser to the MCF was not disturbed, but the MCF was either left undisturbed or wrapped around a 25 mm diameter circular rod either one, two, three, or four times. This entire procedure was repeated five times, each time coupling light into a different core individually. (The cores were chosen and numbered arbitrarily.) The procedure was also repeated when coupling light into many cores simultaneously. Multicore excitation was achieved by coupling the excitation fibre to the middle of the MCF core array, and then increasing the gap between the excitation fibre and the MCF to ~500 μm. Given the NA of the excitation fibre (~0.1–0.14), we estimate that the spot size on the input the MCF will be in the region of ~50 μm. Given that the MCF cores are spaced by ~10.53 µm, we estimate that ~40 of the MCF cores were excited.
Characterising the measurement precision: light was coupled into one core and five images of the multimode output were captured in quick succession without moving either the MCF or the single mode fibre delivering laser light from the laser to the MCF. This assesses the limits of the measurement system, since the patterns being captured are identical.
The similarity in the multimode patterns for each situation described above was quantified by calculating the overlap integral for each pair-wise combination of the images in each data set. For our purposes, the overlap integral was defined as:
$${\mathrm{Overlap}}\;{\mathrm{integral}} = \frac{{\left( {{\int} {I_A \cdot I_B} } \right)^2}}{{{\int} {\left( {I_{\mathrm{A}}^2} \right){\int} {\left( {I_{\mathrm{B}}^2} \right)} } }},$$
where IA and IB are the intensity distributions of the two patterns. The denominator normalises the calculation such that the overlap is 1 when the two patterns being compared are identical.
Supplementary Figs. 2, 3 present the results of the pattern stability measurements. Each column presents the five images of the multimode pattern acquired under specific excitations and fibre (SMF or MCF) deformations.
Supplementary Table 1 presents a summary of the pattern stability measurements. The patterns generated when exciting more than one MCF core simultaneously are highly unstable, with an overlap of 0.779 ± 0.02. This is to be expected from the clear visual differences in the relevant images shown in Supplementary Fig. 2a, and is due to core-dependent bending-induced phase shifts across the MCF29. The situation is dramatically different when exciting only one core of the MCF at a time. Under this condition the patterns exhibit an overlap of 0.978 ± 0.009. The fact that this value is very close to 1, and within error of the 0.985 ± 0.003 measurement precision limit of our experimental system, confirms the extremely high degree of stability of the multimode patterns generated when exciting only one MCF core at a time. It is interesting to note that an overlap of 0.958 ± 0.012 was measured when varying the excitation polarisation by deforming the single mode excitation fibre. This value may indicate that changing the polarisation of the excitation light has a very slight effect on the generated patterns but raises the question as to why this is not seen when deforming the MCF. One possibility is that the MCF is more polarisation preserving than the single mode excitation fibre, so that deforming the MCF has a reduced impact on the output polarisation.
Imaging data acquisition
Imaging data using the photonic lantern was acquired using the experimental system shown in Fig. 1j. This consisted of a 514 nm continuous wave laser source, the light from which was transported to the experimental setup using a single mode fibre. The light from the single mode fibre was collimated using a fibre collimation package (L1) and focused onto the proximal facet of the MCF using lens L2. The proximal end of the MCF (the end without the photonic lantern transition) was mounted on a computer controlled NanoMax six-axis translation stage which provided nanometre resolution and ~μm bi-directional repeatability in the positioning of the MCF relative to the excitation laser focus. The end of the MCF with the photonic lantern was mounted on a manual three-axis translation stage and its multimode facet was imaged to the object plane using lens L3. The object plane was then relay imaged onto a CMOS camera using lens L4. A fraction of the light from the photonic lantern was sent to a reference detector before the object, allowing instabilities in the laser and any variations in coupling efficiency to be accounted for during data acquisition.
The experimental procedure consisted of two steps. In Step 1, a CMOS camera was used to record the patterns generated in the object plane (without the object in place) by exciting each of the cores individually. During this step, the NanoMax positions were noted to allow each MCF core to be excited and addressed individually during the next step. In Step 2, the object was moved into the beam path at the object plane, the CMOS camera was replaced with the "Transmission detector" to record the magnitude of the light transmitted through the object. The NanoMax positions recorded in Step 1 were used to address and excite each core individually with ~μm precision. For each core, the throughput was maximised using the ~nm resolution movements of the computer-controlled stages. The magnitudes of the signals from both the reference detector and the transmission detector were then recorded for each of the patterns projected onto the object. In this manner, the overlap of each projected pattern on the object was measured. It took ~1 h to collect the data for 121 cores. This data, in combination with the recorded patterns, are the data used with the SARA–COIL algorithm.
SARA–COIL algorithm
The observed data, denoted by \(y \in {\Bbb R}^{N_{\mathrm{p}}}\) (there is one data point per pattern), consist of a linear transform of the image of interest \(x \in {\Bbb R}^{\mathrm{n}}\) with a linear operator whose lines consist of the projection patterns. The measurement model thus reads:
$$y = {\Phi} x + e,$$
where \({\Phi} \in {\Bbb R}^{N_p \times n}\) represents the measurement operator and \(e \in {\Bbb R}^{N_{\mathrm{p}}}\) the acquisition noise.
The SARA–COIL algorithm results from an adaptation of the "SARA" approach developed by Carrillo et al.16. On the one hand, the minimisation problem solved reads as
$${\mathrm{minimise}}\,\left\| {{\it{{\Omega} {\Psi} }}{\mathrm{x}}} \right\|_1{\mathrm{subject}}\;{\mathrm{to}}\;{\it{x}} \in \left[ {0, + \infty } \right[^n{\mathrm{and}}\left\| {{\it{y}} - {\it{{\Phi} x}}} \right\|_2 \le \epsilon .$$
The first element in this expression is the sparsity-promoting prior term to be minimised. ||.||1 denotes the nondifferentiable ℓ1 norm, traditionally invoked in the context of compressive sampling. \({\mathrm{{\Psi} }} \in {\Bbb R}^{L \times n}\) is the linear operator defining the sparsity transform, built as the concatenation of nine wavelet transforms (L = 9n) as in Carrillo et al.16. \({\mathrm{{\Omega} }} \in {\Bbb R}^{L \times L}\) is a diagonal weighting matrix computed using a reweighting procedure introduced by Candès et al.30. The second element of the expression " \(\in [0, + \infty ]^n\) " is a prior term imposing the physical constraint of positivity of the intensity image to be formed. The third element "\(y - {\Phi} x_2 \le {\it{\epsilon }}\)" is the data-fidelity term imposing that the discrepancy between data and model is bounded by the noise energy ϵ.
To solve this minimisation problem, we developed an iterative algorithm based on the primal-dual forward-backward "proximal algorithm"31,32.
Raw data will be made available through the Heriot-Watt University PURE research data management system. https://doi.org/10.17861/a1bebd55-b44f-4b34-82c0-c0fe925762c6.
Code availability
A MATLAB toolbox gathering the algorithm implementation as well as the data necessary to reproduce our simulations results using the USAF resolution target is available on GitHub at https://basp-group.github.io/SARA-COIL/.
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This work was funded through the "Proteus" Engineering and Physical Sciences Research Council (EPSRC) Interdisciplinary Research Collaboration (IRC) (EP/K03197X/1), by the Science and Technology Facilities Council (STFC) through STFC-CLASP grants ST/K006509/1 and ST/K006460/1, STFC Consortium grants ST/N000625/1 and ST/N000544/1. S.L. acknowledges support from the National Natural Science Foundation of China under Grant no. 61705073. DBP acknowledges support from the Royal Academy of Engineering, and the European Research Council (PhotUntangle, 804626). The authors thank Philip Emanuel for the use of his confocal image of A549 cells and Eckhardt Optics for their image of the USAF 1951 target. The authors sincerely thank the anonymous reviewers of this paper for their detailed and considered feedback which helped us to improve the quality of this paper significantly.
Itandehui Gris-Sánchez
Present address: ITEAM Research Institute, Universitat Politècnica de València, 46022, Valencia, Spain
These authors contributed equally: Debaditya Choudhury, Duncan K. McNicholl, Audrey Repetti.
Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK
Debaditya Choudhury, Duncan K. McNicholl & Robert R. Thomson
EPSRC IRC Hub, MRC Centre for Inflammation Research, Queen's Medical Research Institute (QMRI), University of Edinburgh, Edinburgh, UK
Institute of Sensors, Signals and System, Heriot-Watt University, Edinburgh, EH14 4AS, UK
Audrey Repetti & Yves Wiaux
Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh, EH14 4AS, UK
Audrey Repetti
Department of Physics, University of Bath, Claverton Down, Bath, BA2 7AY, UK
Itandehui Gris-Sánchez & Tim A. Birks
Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, 430074, Wuhan, Hubei, China
Shuhui Li
School of Physics and Astronomy, University of Exeter, Exeter, EX4 4QL, UK
Shuhui Li & David B. Phillips
Institute of Biological Chemistry, Biophysics and Bioengineering, Heriot-Watt University, Edinburgh, EH14 4AS, UK
Graeme Whyte
Debaditya Choudhury
Duncan K. McNicholl
David B. Phillips
Tim A. Birks
Yves Wiaux
Robert R. Thomson
R.R.T., T.A.B., and D.C. designed the experimental setups using the MCF photonic lantern. D.C. and D.K.M. conducted the optical experiments using the MCF photonic lantern—apart from the work described in the "Quantifying the stability of the multimode patterns" section of the Methods, which was conducted by R.R.T. D.K.M., and T.A.B simulated the multimode patterns used in the simulations of the MCF lantern imaging. The MCF and PL were simulated by T.A.B., designed by I.G.-S. and T.A.B. and fabricated by I.G.-S., T.A.B. carried out proof-of-concept simulations of image reconstruction using photonic lanterns. A.R and Y.W designed the SARA–COIL algorithm, A.R. implemented the algorithm and Matlab toolbox and performed the reconstruction using both simulated and real data. D.M. developed the ultrafast laser inscribed photonic lantern used for the coherent mode combination work described in the Supplementary Note 2. GW participated in discussions on the coherent mode combination work presented in the supplementary information. S.L. and D.B.P. performed the photonic lantern transmission matrix characterisation and coherent mode combination work described in the Supplementary Note 2. R.R.T. conceived the future application of photonic lanterns to microendoscopy and single-pixel imaging. R.R.T. and Y.W. supervised and led the collaboration. All authors discussed the results and all authors contributed to the writing of the manuscript.
Correspondence to Robert R. Thomson.
The authors declare the following competing interests: D.C., T.A.B., and R.R.T. have applied for a patent covering aspects of using photonic lanterns for imaging applications (WO 2018/203088 A1). The other authors declare no competing interests.
Peer review information Nature Communications thanks the anonymous reviewers 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 affiliations.
Description of Additional Supplementary Files
Supplementary Movie 1
Choudhury, D., McNicholl, D.K., Repetti, A. et al. Computational optical imaging with a photonic lantern. Nat Commun 11, 5217 (2020). https://doi.org/10.1038/s41467-020-18818-6
Memory effect assisted imaging through multimode optical fibres
Simon A. R. Horsley
Nature Communications (2021)
Editors' Highlights
Nature Communications (Nat Commun) ISSN 2041-1723 (online) | CommonCrawl |
Identification of self-interacting proteins by integrating random projection classifier and finite impulse response filter
Volume 20 Supplement 13
Proceedings of the 2018 International Conference on Intelligent Computing (ICIC 2018) and Intelligent Computing and Biomedical Informatics (ICBI) 2018 conference: genomics
Zhan-Heng Chen1,2,
Zhu-Hong You1,2,
Li-Ping Li1,
Yan-Bin Wang1,
Yu Qiu1,2 &
Peng-Wei Hu3
Identification of protein-protein interactions (PPIs) is crucial for understanding biological processes and investigating the cellular functions of genes. Self-interacting proteins (SIPs) are those in which more than two identical proteins can interact with each other and they are the specific type of PPIs. More and more researchers draw attention to the SIPs detection, and several prediction model have been proposed, but there are still some problems. Hence, there is an urgent need to explore a efficient computational model for SIPs prediction.
In this study, we developed an effective model to predict SIPs, called RP-FIRF, which merges the Random Projection (RP) classifier and Finite Impulse Response Filter (FIRF) together. More specifically, each protein sequence was firstly transformed into the Position Specific Scoring Matrix (PSSM) by exploiting Position Specific Iterated BLAST (PSI-BLAST). Then, to effectively extract the discriminary SIPs feature to improve the performance of SIPs prediction, a FIRF method was used on PSSM. The R'classifier was proposed to execute the classification and predict novel SIPs. We evaluated the performance of the proposed RP-FIRF model and compared it with the state-of-the-art support vector machine (SVM) on human and yeast datasets, respectively. The proposed model can achieve high average accuracies of 97.89 and 97.35% using five-fold cross-validation. To further evaluate the high performance of the proposed method, we also compared it with other six exiting methods, the experimental results demonstrated that the capacity of our model surpass that of the other previous approaches.
Experimental results show that self-interacting proteins are accurately well-predicted by the proposed model on human and yeast datasets, respectively. It fully show that the proposed model can predict the SIPs effectively and sufficiently. Thus, RP-FIRF model is an automatic decision support method which should provide useful insights into the recognition of SIPs.
Protein is a significant component of all cells and tissues of an organism. It is organic macro-molecule or large biological molecule, comprising of many amino acids with different length. It is the basic material of life and the main undertaker of life activity. A number of proteins often associate with their partner or other proteins which is called protein-protein interactions (PPIs) [1]. Self-interacting proteins (SIPs) is a particular type of PPIs, where can interact in terms of duplicate their own genes. SIPs occupy an important role in cellular functions and cellular signal transduction. The majority of chemical reactions occur in living systems which mainly depend on the activity of enzymes. Its essence is a large of protein self-interactions. But it exists a certain difficulty for researchers to discover whether protein can interact with each other or not. The functionality of protein refers to that it could handle the transport of ions and small molecules across cell membranes, depends on their homo-oligomers [2]. In particular, homo-oligomerization can also contribute proteins to compose large structures with increasing error control during synthesis and without increasing genome size [3]. From the past years, many researchers elucidated the overall properties of proteins. Ispolatov et.al discovered that the average homodimers of SIPs is more than double the total amount of non-SIPs in the protein interaction networks (PINs) [4]. It is crucial for clarifying the function of SIPs to further understand the regulation of protein function and comprehend whether protein can interact with each other, so that we can better comprehend the mechanism of disease [5]. Liu et al analyzed the properties of SIPs from various aspects information, and applied a logistic regression framework to develop a SIPs prediction model by integrating multiple features [6]. Hence, SIPs will help to improve the stability and prevent the denaturation of a protein via reducing its surface area [7].
So far, a large number of previous methods on the PPIs detection have been proposed [8,9,10]. For instance, Zhang et al. summarized all sorts of computational methods based on their present knowledge, and proposed an algorithm which integrates structural information with other functional clues [11]. Zou et al. presented a novel fingerprint features and dimensionality reduction strategy for predicting TATA binding proteins, which could improve the prediction accuracy [12]. Hamp et al. introduce a new technique to predict PPIs based on evolutionary profiles and profile-kernel support vector machine [13]. Wan et al. exploited an ensemble multi-label classifier for human protein subcellular location prediction with imbalanced protein source [14]. Song et al. designed a predictor to identify DNA-binding proteins based on unbalanced classification [15]. Sylvain et al. put forward a new PPIs Prediction Engine named PIPE, which is capable of predicting PPIs for any target pair of the yeast Saccharomyces cerevisiae proteins from their original structure and without any additional information [16]. Xia et al. presented a sequence-based multi-classifier system that employed autocorrelation descriptor to code an interaction protein pair and chose rotation forest as classifier to infer PPIs [17]. Li et al. provide a scored human PINs with several-fold more interactions and better functional biological relevance than comparable resources by the means of data integration and quality control [18].
However, these approaches could be applied to detect PPIs well [19], but they are not good enough to predict SIPs. Mainly exist in terms of following points: (1) In essence, they also have certain limitations that take the correlation between protein pairs into account for SIPs detection, for example co-expression, co-localization and co-evolution. Nevertheless, these info are of no use for SIPs. (2) The datasets applied to predict PPIs are different from those of SIPs, the datasets of the former are balanced and those of the latter are unbalanced. (3) Besides, prediction of PPIs datasets have no PPIs between same partners. In virtue of reasons, these computational approaches are not suitable for predicting SIPs. Hence, It is becoming more and more significant to exploit an effective calculation method to predict SIPs.
In this paper, we put forward a random projection (RP) bind with Finite Impulse Response Filter (FIRF) model for predicting SIPs from protein sequence information. Furthermore, the main ideas of our raised method includes the following four aspects: (1) The PSI-BLAST could be exploited to convert each protein sequence to a Position Specific Scoring Matrix (PSSM); (2) Employing Finite Impulse Response Filter (FIRF) method to calculate the eigenvalues from protein sequences on a PSSM; (3) To reduce the dimension of feature values which obtained from WT method by applying the Principal Component Analysis (PCA) technique, and removed the noise features from the data, thus the pattern in the data is discovered; (4) RP classifier is applied to build a training set on which the classifiers will be trained. More specifically as follows: first of all, the PSSM of each protein sequence is converted into a 400-dimensional feature vector by employing FIRF method to extract helpful information; then, to remove the influence of noise, we reduced the dimension from 400 to 300 by applying PCA method; At last, realized classification on yeast and human datasets by relying on RP classifier. The experimental results show that this method outperforms the SVM-based method and other previous methods. It is revealed that the presented method is suitable and perform well for predicting SIPs.
Five-fold cross-validation on human and yeast datasets
The performance of the proposed method is estimated on the human and yeast datasets. Aiming at the fairness and over-fitting problems, we repeated the experiment five times on the two same datasets, termed five-fold cross validation. Further, described it in details, we split the human dataset which was mainly composed of characteristic values into five non-overlapping pieces, and four parts was randomly chosen as training set and selected the remaining characteristic values as independent test set. Then, we can obtain the results by repeating five times to test our model. To illustrate the rationality, toughness and stability of our algorithm, we also implemented the method of RP-FIRF on the yeast dataset.
To guarantee impartiality and objectivity of the test, the parameters for human and yeast datasets should be set in the same way. In our task, we obtained the better result by adjusting the diverse parameters of RP classifier constantly. Thus, we set the number of blocks B1 = 10 for independent projections to classify the training and test sets, the size of each block was carefully chosen as B2 = 30, and then applying the K-Nearest Neighbor (KNN) base classifier and the leave-one-out test error estimate, where k = seq (1, 30, by = 8).
Afterwards, we test our RP-FIRF prediction method on the two mentioned datasets, and got the results of the two datasets based on 5-fold cross-validation are discovered in Tables 1 and 2. From the Table 1, the data is observed that our proposed method exhibited the five outcomes of average Accuracy (Acc), Sensitivity (Sen), Precision (PE), and Matthews correlation coefficient (MCC) of 97.89, 74.46, 100.00, and 85.31% on human dataset and the standard deviations of them of 0.17, 2.18, 0.00, and 1.29%, respectively. Similarly, we can get the results in Table 2 by running experiment on yeast dataset, the average Accuracy is 97.35%, average Sensitivity is 77.03%, average Precision is 99.62%, and average MCC is 86.31% and the standard deviations of them of 0.15, 1.17, 0.52, and 0.79%, respectively.
Table 1 Results measured by RP-FIRF method on human dataset with 5-fold cross-validation
Table 2 Results measured by RP-FIRF method on yeast dataset with 5-fold cross-validation
As mentioned above, It is apparent that our method can receive good effect of SIPs detection because of the appropriate feature extraction and classifier. The presented feature extraction technique plays a critical part in enhancing the calculation accuracy. The specific reasons can be summed up in the following three aspects: (1) PSSM could describe the protein sequence in the form of numerical values. It can be employed to find an amino acid that matches a specific location to give the score in a target protein sequence. Not only can it represents the information of protein sequence, but also it preserves helpful enough information as much as possible. Accordingly, A PSSM contains almost the whole information of one protein sequence for detecting SIPs. (2) Finite impulse response filter (FIRF) feature extraction method of protein sequence can further optimize the performance of our proposed model. (3) To drop the negative influence of noise, PCA was employed to reduce the dimension of data on the condition of the integrity of FIRF feature vector, thus the helpful information in the data will be mined. In a few words, experimental results revealed that our RP-FIRF model is extreme fit for SIPs prediction.
Compare our proposed model with the SVM-based method
Although the RP-FIRF model achieved accuracy more than 90%, It still needs further test and verify the effectiveness of our presented model. From the point of classification, support vector machine (SVM) is a generalized linear classifier. The SVM-based method has been widely known in many fields of scientific research. Therefore, it's necessary to compare the prediction accuracy of our RP-FIRF model with the SVM-based method by using the same eigenvalues based on the two above mentioned datasets. We mainly employed the LIBSVM packet tool [20] to implement classification in the experiment. Our first task was to adjust the main parameters of SVM classifier. A radial basis function (RBF) was chosen as the kernel function, and then the two parameters of RBF were adjusted via a grid search algorithm, which were set c = 0.6 and g = 0.02.
As is shown in Tables 3 and 4, we trained and compared the RP-FIRF model with SVM-based model on yeast and human datasets by employing 5-fold cross-validation respectively. The data from Table 3 can be displayed that the mean of Accuracy, the mean of Sensitivity, the mean of Precision, and the mean of MCC from SVM classifier are 92.32, 32.89, 100.00, and 53.07% on yeast dataset, respectively. However, the RP-FIRF method reached 97.35% average Accuracy, 77.03% average Sensitivity, 99.62% average Precision, and 86.31% average MCC on yeast dataset. Equally, the data from Table 4 can be shown that the average Accuracy, the average Sensitivity, the average Precision, and the average MCC of SVM classifier are 96.21, 54.44, 100.00, and 72.30% on human dataset. Nevertheless, the proposed model achieved 97.89% average Accuracy, 74.46% average Sensitivity, 100.00% average Precision, and 85.31% average MCC on human dataset. Stated thus, it is clear that the overall prediction results of RP classifier are much better than those of SVM classifier.
Table 3 Comparison results of RP and SVM with FIRF feature vectors on yeast dataset
Table 4 Comparison results of RP and SVM with FIRF feature vectors on human dataset
Meanwhile, receiver operating characteristic (ROC) curves was applied to analysis the binary classification system (the outcome results only have two categories), was widely applied in many fields such as bioinformatics [21], forecasting of natural hazards [22], machine learning [23], data mining [24] and so on. Therefore, we also used ROC curves to measure the comprehensive index between sensitivity and specificity continuous variable. The area under curves (AUC) could be shown the discriminating capability of the classifier. The closer the top-left corner of the curve is, the higher the prediction accuracy is. Otherwise, the lower the diagnosis result is. In other words, The larger the AUC, the stronger the capability of discernment.
From Fig. 1, we plotted the ROC curves by making a comparison between RP and SVM on human dataset, it is clearly that the AUC of SVM classifier is 0.7754 and that of RP classifier is 0.8842. Plots of the RP and SVM classifier on yeast dataset in the ROC space are plot in Fig. 2, it is sharply that the AUC of SVM classifier is 0.6631 and that of RP classifier is 0.8896. Anyhow, we demonstrate that the AUC of RP classifier is also significantly larger than that of SVM classifier. So the RP method is an accurate and robust technique for SIPs detection.
Comparison of ROC curves between RP and SVM on human dataset
Comparison of ROC curves between RP and SVM on yeast dataset
Measure our proposed model against other previous methods
In the process of practice, we measured the quality of proposed model named RP-FIRF with other existing methods based on the two above mentioned datasets to further testify that our approach could obtain better results. We listed a clear statement of account in Tables 5 and 6, which are the comparison results on the two datasets. From Table 5, it is obvious that the RP-FIRF model achieved the highest average accuracy of 97.35% than the other six methods (range from 66.28 to 87.46%) on yeast dataset. At the same instant, it is clear to see that the other six methods got lower MCC (range from 15.77 to 28.42%) than our proposed model of 86.31% on the same dataset. In exactly the same way, from Table 6, the overall results of our prediction approach is also outperform the other six methods on human dataset. To make a summary of it, we measured our RP-FIRF model against with the other six approaches on yeast and human datasets respectively, the prediction accuracy of the overall experimental results can be improved. This fully illustrates that a good feature extraction tool and a suitable classifier is very important for predicting model. It is further illustrated that our method is superior to the other six approaches and quite suitable for SIPs preditcion.
Table 5 Performance results between RP-FIRF model and the other methods on yeast dataset
Table 6 Performance results between RP-FIRF model and the other methods on human dataset
In the study, a machine learning model was put forward to predict SIPs which based on protein primary sequence. This model was developed by combining Finite Impulse Response Filter with Random Projection classifier, which was termed RP-FIRF. The mainly improvements for this method are attributable to the following aspects: (1) A reasonable representative method FIRF is used to effectively extract the discriminary features, which can process and analyze protein sequence data well. (2) The RP classifier is strongly suitable for predicting SIPs, and a high recognition accuracy can be obtained. The experimental results measured by the presented model on yeast and human datasets revealed that the performance of RP method is significantly better than that of the SVM-based method and other six previous methods. It fully shows that the integration of FIRF method with RP classifier is able to significantly improve the accuracies of SIPs prediction. Overall, we have predicted a reliable set of SIPs suitable for further computational as well as experimental analyses. For the future research, there will be more and more effective feature extraction methods and machine learning approaches exploited for detecting SIPs.
Materials and methodology
In our study, we constructed the datasets mainly derived from the UniProt database [29] which contains 20,199 curated human protein sequences. There are many different types of resources such as DIP [30], BioGRID [31], IntAct [32], InnateDB [33] and MatrixDB [34], we can get the PPIs related information from them. In relational databases, we mainly set up the datasets for SIPs which embodies two identical interacting protein sequences and whose type of interaction was characterized as "direct interaction". Based on that, we can construct the datasets for the experiment by applying 2994 human self-interacting protein sequences.
For the 2994 human SIPs, we need to single out the datasets for the experiment and assess the performance of the RP-FIRF model, which mainly includes three steps [28]: (1) If the protein sequences which may be fragments, we will remove it and retain the length of protein sequences between 50 residues and 5000 residues from all the human proteome; (2) To build up the positive dataset of human, we formed a high-grade SIPs data which should meet one of the following conditions: (a) the self-interactions were revealed by at least one small-scale experiment or two sorts of large-scale experiments; (b) the protein has been announced as homo-oligomer (containing homodimer and homotrimer) in UniProt; (c) it has been reported by more than two publications for self-interactions; (3) For the human negative dataset, we removed the whole types of SIPs from all the human proteome (contains proteins annotated as 'direct interaction' and more extensive 'physical association') and SIPs detection in UniProt database. To sum it up, we obtained the ultimate human dataset for the experiment which was mainly composed of 1441 SIPs and 15,938 non-SIPs [28].
Just as the construction of human dataset, we also further assess the cross-species ability of the RP-FIRF model by repeating the same strategy mentioned above to generate the yeast dataset. Finally, 710 SIPs was assigned to form the yeast positive dataset and 5511 non-SIPs was allocated to constitute the yeast negative dataset [28].
In the field of machine learning, confusion matrix is always employed in evaluating the classification model, also known as an error matrix [35, 36]. It indicates information about actual and predicted classifications for two class classifier which could be shown as the follow Table 7.
Table 7 Confusion Matrix
In our study, in the interest of size up the steadiness and effectiveness of our present model, we computed the values of 5 parameters: Accuracy (Acc), Sensitivity (Sen), specificity (Sp), Precision (PE) and Matthews's Correlation Coefficient (MCC), respectively. These parameters can be described as follows:
$$ Acc=\frac{TP+ TN}{TP+ FP+ TN+ FN} $$
$$ Sen=\frac{TP}{TP+ FN} $$
$$ Sp=\frac{TN}{FP+ TN} $$
$$ PE=\frac{TP}{FP+ TP} $$
$$ MCC=\frac{\left( TP\times TN\right)-\left( FP\times FN\right)}{\sqrt{\left( TP+ FN\right)\times \left( TN+ FP\right)\times \left( TP+ FP\right)\times \left( TN+ FN\right)}} $$
where, TP (i.e. true positives) is the quantity of true interacting pairs correctly predicted. FP (i.e. false positives) represents the number of true non-interacting pairs falsely predicted. TN (i.e. true negatives) is the count of true non-interacting pairs predicted correctly. FN (i.e. false negatives) represents true interacting pairs falsely predicted to be non-interacting pairs. On the basis of these parameters, a ROC curve was plotted to evaluate the performance of random projection method. And then, we can calculate the area under curve (AUC) to measure the performance of the classifier.
Position specific scoring matrix
In our experiment, Position Specific Scoring Matrix (PSSM) is a helpful technique which was employed to detect distantly related proteins [37]. Accordingly, each protein sequence information was transformed into PSSM by using the PSI-BLAST [38]. And then, a given protein sequence can be converted into an H × 20 PSSM which could be represented as follow:
$$ M=\left\{ M\alpha \kern0.1em \beta \kern0.3em \alpha :1=1\cdots H,\beta =1\cdots 20\right\} $$
where H denotes the length of a protein sequence, and 20 is the number of amino acids due to every sequence was constituted by 20 different amino acids. For the query protein sequence, the score Cαβ indicates that the β-th amino acid in the position of α assigned from a PSSM. Therefore, Cαβ could be described as:
$$ C\alpha \beta ={\sum}_{k=1}^{20}p\left(\alpha, k\right)\times q\left(\beta, k\right) $$
where p(α,k) represents the occurrence frequency of the k-th amino acid at location of α, and q(β,k) is the Dayhoff's mutation matrix value between β-th and k-th amino acids. In addition, diverse scores determine different relative location relationships, a greater degree means a strongly conservative position, and otherwise a weakly conservative position can gain a lower value.
Overall, PSSM has been more and more important in the research of SIPs prediction. In a detailed and exact way, we employed PSI-BLAST to obtain the PSSM from each protein sequence for detecting SIPs. To achieve a better score and a large scale of homologous sequences, the E-value parameter of PSI-BLAST was set to be 0.001 which reported for a given result represents the quantity of two sequences' alignments and selected three iterations in this experiment [39, 40]. Afterwards we can achieve a 20-dimensional matrix which consists of M × 20 elements based on PSSM, where M represents the count of residues of a protein, and 20 denote the 20 types of amino acids.
Finite impulse response filters
In the field of digital signal processing (DSP) [41], finite impulse response filter (FIRF) is one of the most commonly used components, which can perform the function of signal pre-modulation and frequency band selection and filtering. FIRF are widely employed in many fields such as communications [42], image processing [43], pattern recognition [44], wireless sensor network [45] and so on. Many methods of DSP were applied in the fundamental research of cytology, brain neurology, genetics and other fields. In our work, we applied FIRF to process the characteristics of protein sequences, which would be used to predict the SIPs. Therefore, many important features of the problem can be fully highlighted by the FIRF method, and then it could devote to the details of the problem. We design it by using Fourier series method in details as follows.
At first, the corresponding Frequency Response Function of FIRF transfer function can be described as:
$$ H\left({e}^{jw}\right)=\sum \limits_{n=0}^{N-1}h(n){e}^{- jwn} $$
where, h(n) is the available impulse response sequence, and N represents the sample sizes of frequency response H (ejw). Given the frequency response Hd (ejw) of ideal filter, and let H (ejw) approach Hd (ejw) infinitely.
$$ {H}_d\left({e}^{jw}\right)=\sum \limits_{n=-\infty}^{\infty }{h}_d(n){e}^{- jwn} $$
And then, we can achieve the -hd(n) by employing inverse Fourier transform of Hd (ejw). The hd(n) is built as
$$ {h}_d(n)=\frac{1}{2\pi }{\int}_{-\pi}^{\pi }{H}_d\left({e}^{jw}\right){e}^{jw n} dw $$
where hd(n) is a finite length. If hd(n) is an infinite length, we can intercept hd(n) by applying a finite length of the windows function sequence w(n).
$$ h(n)={h}_d(n)w(n) $$
According to the above formula, we can gain the unit sample response for our designed FIR filter. To check the filter whether meet the design requirements by follow formula.
$$ H\left({e}^{jw}\right)= DTFT\left[h(n)\right] $$
The integral square error (ISE) between the frequency response of ideal filter and our designed filter can be defined as follow:
$$ {\varepsilon}^2=\frac{1}{2\pi }{\int}_{-\pi}^{\pi }{\left|{H}_d\left({e}^{jw}\right)-H\left({e}^{jw}\right)\right|}^2 dw $$
In our study, we cannot directly extract the eigenvalues from the protein because of each protein sequence have the different amino acids composition. To prevent the generation of unequal lengths of feature vectors, we multiply the transpose of PSSM by PSSM to achieve 20 × 20 matrix. and then, we employ the FIRF technique to transform the PSSM of each protein sequence into a feature vector which have the same size with 20 × 20 matrix. Afterwards, these feature values could be computed as a 400-dimensional vector. Eventually, every protein sequence from the two above mentioned datasets was transformed into a 400-dimensional vector by employing FIRF approach.
For the sake of remove the influence of noise and improve the result of SIPs prediction, we applied the Principal Component Analysis (PCA) to remove the influence of noisy features on the two above mentioned datasets. So as to we can reduce the dimension of the two datasets from 400 to 300. Accordingly, we could employ a small number of information to represent the whole data and push the complexity into smaller, so as to improve the generalization error.
Random projection classifier
In mathematics and statistics, Random Projection (RP) is a classifier for dimensionality reduction of some points which lie in Euclidean space. RP classifier showed that N points in N dimensional space can almost always be mapped to a space of dimension ClogN with command on the ratio of error and distances [46, 47]. It has been successfully applied in rebuilding of frequency-sparse signals [48], face recognition [49], protein subcellular localization [50] and textual and visual information retrieval [51].
We formally describe the RP classifier as follow in details. At first, let
$$ \varGamma ={\left\{ Ai\right\}}_{i=1}^N, Ai\in {R}^n $$
be the primitive high dimensional space dataset, where n represents the high dimension and N denotes the number of the dataset. The goal of dimensionality reduction is embedding the vectors into a lower dimensional space Rq from a high dimension Rn, where q < <n. The output of data is defined as follow:
$$ \overset{\sim }{\varGamma }={\left\{\overset{\sim }{A_i}\right\}}_{i=1}^N,\overset{\sim }{A_i}\in {R}^q $$
where q is close to the intrinsic dimensionality of Γ. Thus, the vectors of Γ was regarded as embedding vectors.
If we want to reduce the dimension of Γ via random projection method, a random vector set γ = {ri} k i = 1 must be constructed at first, where ri∈Rq. The random basis can be obtained by two common choices as follow [46]:
The vectors {ri} k i = 1 are normally distributed over the q dimensional unit sphere.
The components of the vectors {ri} k i = 1 are chosen Bernoulli + 1/− 1 distribution and the vectors are standardized so that ||ri||l2 = 1 for i = 1, …,n.
Then, the columns of q × n matrix R are consisted of the vectors in γ. The embedding result Ãi of Ai can be got by
$$ \overset{\sim }{A_i}=R\cdot {A}_i $$
In our proposed method, random projection classifier will be trained on a training set. And we enrich the component of the ensemble method based on random projection.
Next, the size of target space was set to a part of around the space where the training members reside. We built a size of n × N matrix G whose columns are made up the column vectors in Γ. The training set Γ have given in Eq.14.
$$ G=\left({A}_1|{A}_2|...|{A}_N\right) $$
Then, we construct k random matrices {Ri} k i = 1 whose size is q × n, q and n are introduced in the above mentioned paragraph, and k is the quantity of classifiers. Here, the columns of matrices are normalized so as to the l2 norm is 1.
And then, in our method, to construct the training sets {Ti} k i = 1 by projecting G onto {Ri} k i = 1 which is the k random matrices. It can be represented as follow:
$$ {T}_i={R}_i\cdot G,\kern0.5em i=1,...,k $$
The training sets are imported into an inducer and the export results are a piece of classifiers {ℓi} k i = 1. How to classify a new dataset I through classifier ℓi. At first, we embed I into the dimensionality reduction space Rq. Then, It can be owned via mapping u to the random matrix Ri as follow:
$$ \overset{\sim }{I}={R}_i\cdot I $$
where Ĩ is the inlaying of u, the classification of Ĩ can be garnered from the classification of I by ℓi. In this ensemble method, the random projection classifier use a data-driven voting threshold which is employed to classification outcomes of the whole classifiers {ℓi} k i = 1 for the Ĩ to decide produce the ultimate classification result of Ĩ.
In this experiment, the random projections were split up non-overlapping blocks where B1 = 10 and each one carefully chosen from a block of size B2 = 30 that achieved the smallest estimate of the test error. We used the k-Nearest Neighbor (KNN) as base classifier and the leave-one-out test error estimate, where k = seq (1, 30, by = 8). The prior probability of interaction pairs in the training sample set was taken as the voting parameter. Our classifier integrates the results of taking advantage of the base classifier on the selected projection, with the data-driven voting threshold to confirm the final mission.
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The authors would like to thank all the guest editors and anonymous reviewers for their constructive advices.
About this supplement
This article has been published as part of BMC Genomics Volume 20 Supplement 13, 2019: Proceedings of the 2018 International Conference on Intelligent Computing (ICIC 2018) and Intelligent Computing and Biomedical Informatics (ICBI) 2018 conference: genomics. The full contents of the supplement are available online at https://bmcgenomics.biomedcentral.com/articles/supplements/volume-20-supplement-13.
This work is supported in part by the National Science Foundation of China, under Grants 61373086, 61572506.
The Xinjiang Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Urumqi, 830011, China
Zhan-Heng Chen, Zhu-Hong You, Li-Ping Li, Yan-Bin Wang & Yu Qiu
University of Chinese Academy of Sciences, Beijing, 100049, China
Zhan-Heng Chen, Zhu-Hong You & Yu Qiu
IBM Research, Beijing, 100049, China
Peng-Wei Hu
Zhan-Heng Chen
Zhu-Hong You
Li-Ping Li
Yan-Bin Wang
Yu Qiu
ZHC and ZHY conceived the algorithm, carried out analyses, prepared the data sets, carried out experiments, and wrote the manuscript; LPL, YBW and YQ designed, performed and analyzed experiments and wrote the manuscript; All authors read and approved the final manuscript.
Correspondence to Zhu-Hong You.
Chen, ZH., You, ZH., Li, LP. et al. Identification of self-interacting proteins by integrating random projection classifier and finite impulse response filter. BMC Genomics 20 (Suppl 13), 928 (2019). https://doi.org/10.1186/s12864-019-6301-1
Self-interacting proteins
PSSM
Random projection
Finite impulse response filter | CommonCrawl |
CoordinatedGeometry MAT
Problem - 4760
In the diagram below, a line is tangent to a unit circle centered at $Q (1, 1)$ and intersects the two axes at $P$ and $R$, respectively. The angle $\angle{OPR}=\theta$. The area bounded by the circle and the $x-$axis is $A(\theta)$ and the are bounded by the circle and the $y-$axis is $B(\theta)$.
Show the coordinates of the point $Q$ is $(1+\sin\theta, 1+\cos\theta)$. Find the equation of line $PQR$ and determine the coordinates of $P$.
Explain why $A(\theta)=B\left(\frac{\pi}{2}-\theta\right)$ always holds and calculates $A\left(\frac{\pi}{2}\right)$.
Show that $A\left(\frac{\pi}{3}\right)=\sqrt{3}-\frac{\pi}{3}$.
The solution for this problem is available for $0.99. You can also purchase a pass for all available solutions for $99.
© 2009 - 2023 Math All Star | CommonCrawl |
\begin{definition}[Definition:Finite Semigroup]
A '''finite semigroup''' is a semigroup of finite order.
That is, a semigroup $\struct {S, \circ}$ is a '''finite semigroup''' {{iff}} its underlying set $G$ is finite.
That is, a '''finite semigroup''' is a semigroup with a finite number of elements.
\end{definition} | ProofWiki |
Multiregional single-cell dissection of tumor and immune cells reveals stable lock-and-key features in liver cancer
Intratumoral heterogeneity and clonal evolution in liver cancer
Bojan Losic, Amanda J. Craig, … Augusto Villanueva
Dissecting spatial heterogeneity and the immune-evasion mechanism of CTCs by single-cell RNA-seq in hepatocellular carcinoma
Yun-Fan Sun, Liang Wu, … Xin-Rong Yang
A single-cell atlas of liver metastases of colorectal cancer reveals reprogramming of the tumor microenvironment in response to preoperative chemotherapy
Li-Heng Che, Jing-Wen Liu, … Jian-Ming Li
Transcriptomic analysis of hepatocellular carcinoma reveals molecular features of disease progression and tumor immune biology
K. Okrah, S. Tarighat, … S. M. Huang
The immune contexture of hepatocellular carcinoma predicts clinical outcome
Friedrich Foerster, Moritz Hess, … Ernesto Bockamp
Functional Genomic Complexity Defines Intratumor Heterogeneity and Tumor Aggressiveness in Liver Cancer
So Mee Kwon, Anuradha Budhu, … Xin Wei Wang
The heterogeneous immune landscape between lung adenocarcinoma and squamous carcinoma revealed by single-cell RNA sequencing
Chengdi Wang, Qiuxiao Yu, … Weimin Li
Single-cell RNA transcriptome reveals the intra-tumoral heterogeneity and regulators underlying tumor progression in metastatic pancreatic ductal adenocarcinoma
Qianhui Xu, Shaohuai Chen, … Wen Huang
Single-cell RNA sequencing reveals distinct tumor microenvironmental patterns in lung adenocarcinoma
Philip Bischoff, Alexandra Trinks, … Frederick Klauschen
Lichun Ma ORCID: orcid.org/0000-0001-9809-775X1,2 na1,
Sophia Heinrich1,3 na1,
Limin Wang1,
Friederike L. Keggenhoff4,
Subreen Khatib1,
Marshonna Forgues1,
Michael Kelly ORCID: orcid.org/0000-0003-0654-27785,
Stephen M. Hewitt ORCID: orcid.org/0000-0001-8283-17886,
Areeba Saif7,
Jonathan M. Hernandez7,
Donna Mabry8,
Roman Kloeckner ORCID: orcid.org/0000-0001-5492-47924,9,
Tim F. Greten ORCID: orcid.org/0000-0002-0806-25358,10,
Jittiporn Chaisaingmongkol ORCID: orcid.org/0000-0001-8019-556211,12,
Mathuros Ruchirawat ORCID: orcid.org/0000-0001-6402-244211,12,
Jens U. Marquardt ORCID: orcid.org/0000-0002-8314-26824,13 &
Xin Wei Wang ORCID: orcid.org/0000-0001-9735-606X1,10
Nature Communications volume 13, Article number: 7533 (2022) Cite this article
Cellular signalling networks
Tumour heterogeneity
Intratumor heterogeneity may result from the evolution of tumor cells and their continuous interactions with the tumor microenvironment which collectively drives tumorigenesis. However, an appearance of cellular and molecular heterogeneity creates a challenge to define molecular features linked to tumor malignancy. Here we perform multiregional single-cell RNA sequencing (scRNA-seq) analysis of seven liver cancer patients (four hepatocellular carcinoma, HCC and three intrahepatic cholangiocarcinoma, iCCA). We identify cellular dynamics of malignant cells and their communication networks with tumor-associated immune cells, which are validated using additional scRNA-seq data of 25 HCC and 12 iCCA patients as a stable fingerprint embedded in a malignant ecosystem representing features of tumor aggressiveness. We further validate the top ligand-receptor interaction pairs (i.e., LGALS9-SLC1A5 and SPP1-PTGER4 between tumor cells and macrophages) associated with unique transcriptome in additional 542 HCC patients. Our study unveils stable molecular networks of malignant ecosystems, which may open a path for therapeutic exploration.
Multi-level analyses of human cancer tissues unveil a vast degree of molecular heterogeneity among cells within individual tumors, a feature known as intratumor heterogeneity (ITH)1. A varying degree of ITH can be found in most, if not all, major solid malignancies and these features are universally associated with patient's prognosis2. Tumor cell evolution may be a main contributor to ITH because each tumor cell or their-derived subclones compete with each other in an adverse milieu of the tumor microenvironment (TME) for survival, resulting in a complex tumor ecosystem, where tumor cells may serve as the architect to orchestrate various cell types in the tumor cell community to facilitate its growth3,4. Because stromal and immune cells in the tumor ecosystem may compete with tumor cells for space and nutrients in addition to potentially activated immune surveillance to restrict tumor growth, tumor cells may develop unique features to reprogram the TME to evade from the competition5. Therefore, like the landscape in nature, the survival fitness of each tumor may be dictated by a unique tumor cell landscape shaped by the intrinsic tumor biology, tumor evolution and the TME, which gives rise to an appearance of a complex tumor ecosystem. Obviously, ITH represents a major barrier for implementing effective therapeutic interventions such as systemic therapies because of the difficulty in identifying stable molecular features linked to malignancy due to evolution-driven moving targets4.
Overwhelming evidence indicates the remarkable heterogeneity of solid malignancies at the phenotypic and genetic levels. For example, somatic mutation analysis revealed that most hepatocellular carcinoma (HCC) cases examined show ITH at the genetic levels6. However, recent studies of lung cancer and leukemia with the application of single-cell technologies revealed that malignant clonal dominance is a cell-intrinsic and heritable property and transcriptome-based cellular state arises largely independently of genetic variation7,8. This raises questions about the accuracy in defining tumor cell biodiversity at the genetic levels due to the presence of passenger mutations and that transcriptomic heterogeneity may represent a good alternative to model cancer evolution1,9. The key question remains as how best to define biologically meaningful tumor ecosystem. Since tumor evolution is accompanied by continuous interactions of tumor cells and the TME, defining cellular features and its underlying molecular communication networks that shape tumor biology and consequently drive tumor evolution may be a key to improve molecular understanding of tumor ecosystems and to develop effective therapeutic approaches for solid tumors. Herein, we postulated that each tumor ecosystem may represent the success in enriching a unique combination of tumor-stromal interaction networks that promote tumor evolution under selective pressure. Defining the interactions of tumor and immune/stromal cells may, thus, represent unique fingerprints stable for its tumor biology, a feature analogous to the lock-and-key model that describes the enzyme-substrate interaction proposed by Emil Fischer over 127 years ago10.
Cells are the smallest structural and functional unit of a tumor lesion. Therefore, tumor biology should be studied at the single-cell level11,12. Single-cell-based transcriptomic analysis has been increasingly used to study tumor and immune cell compositions in normal and diseased tissues such as liver cancer to better capture the tumor ecosystem13,14. As the incidence and mortality of HCC and intrahepatic cholangiocarcinoma (iCCA), the two main histological types of liver cancer, are still on the rise at the global level15, several recent studies have exploited HCC and iCCA ecosystems by single-cell transcriptomic analysis16,17,18,19,20. These studies mainly relied on tumor biopsies obtained from a single region within the tumor to generate tumor cell composition, which have revealed vast ITH. It is unclear whether sampling bias influences the observed tumor ecosystem and the associated ITH within each tumor and consequently data interpretation, or if the tumor composition is relatively stable with a secured communication network of tumor and the TME that supports tumor malignancy.
In this study, we aim to determine the spatial distribution of tumor cells and TME by performing a multiregional single-cell RNA sequencing (scRNA-seq) analysis of HCC and iCCA from seven liver cancer patients and validate the results in single-cell data from an additional 37 HCC and iCCA patients. We find that while ITH is evident, variations of the tumor cell composition and the corresponding communication networks of tumor cells and the TME are smaller within each tumor than between tumors regardless of tumor size and corresponding distance among sampling tissues. We identify a fingerprint consisting of LGALS9-SLC1A5, SPP1-PTGER4 as tumor and macrophage-derived ligand–receptor interaction pairs, linked to tumor aggressiveness. We independently validate the stability of the expression patterns and prognostic value of the LGALS9-SLC1A5 and SPP1-PTGER4 pairs using both bulk transcriptome profiles of 542 HCC samples from three independent cohorts and multiplexed fluorescence in situ hybridization-based profiles of 258 HCC samples from two cohorts. Our multiregional single-cell dissection of tumor and immune cells reveals a stable tumor-macrophage interaction network linked to ITH and HCC prognosis, which may provide the basis for further functional exploration including the development of rationale therapeutics for liver cancer.
Multiregional liver tumor cell transcriptome profiles
To determine the spatial distribution of tumor cells and the TMEs as well as its stability within each tumor, we performed multiregional single-cell transcriptomic profiling of liver tumor specimens with varying tumor sizes from four HCC patients and three iCCA patients who underwent surgical resection (Supplementary Table 1). Specifically, we prepared single cells from five separate regions for each tumor, i.e., three tumor cores (T1, T2, and T3), one tumor border (B) and an adjacent normal tissue (N), followed by droplet-based 5' scRNA-seq of these samples (Fig. 1a). We removed one sample (T3 from case 3C) due to single-cell library failure and thus a total of 34 samples were included in this study. We identified malignant and non-malignant cells by using the same method applied successfully in our previous studies and further used adjacent normal liver tissues as a control (Supplementary Fig. 1)17,20. Samples with >10 malignant cells detected were used for the analysis of malignant cells. With this criterion, six patients in our cohort had detectable malignant cells (Supplementary Fig. 1a). We then determined the similarity of tumor cell composition among multiple regions for each tumor from the six patients. t-distributed stochastic neighbor embedding (t-SNE) analysis revealed that malignant cells formed patient-specific clusters regardless of tumor regions (Fig. 1b and c), suggesting a much smaller interregional heterogeneity than intertumoral heterogeneity. This was also evident from hierarchical clustering analysis of multiple regions, where tumor cells from the same patient tended clustering together in the hierarchical tree (Fig. 1d). This was consistent with our previous bulk transcriptome study of HCC21. In contrast to patient-specific patterns of malignant cells, epithelial cells from adjacent normal tissues of different cases were mixed and separated from malignant cells (Supplementary Fig. 2), indicating shared non-malignant epithelial cell states among patients, which further served as a control to define malignant cells.
Fig. 1: Multiregional single-cell transcriptome profiling of liver cancer.
a Workflow of multiregional tissue collection, processing, scRNA-seq, and data analysis. B, tumor border; T1, T2, and T3, three tumor cores; N, adjacent normal tissue. scRNA-seq, single-cell RNA sequencing. The figure was generated using BioRender. b, c t-SNE plot of malignant cells colored by cases (b) or tumor regions (c). Case ID was named according to the histological subtypes of HCC and iCCA. H, HCC; C, iCCA. B, tumor border; T1, T2, T3, three tumor cores. d Hierarchical clustering of malignant cells from each tumor region across all cases. Samples were named according to the histological subtypes and tumor regions. e Representative magnetic resonance imaging (MRI) of case 4H and histopathology of tumors from border, T1 and T2 of this case. Scale bars, 50 μm. Multiregional imaging pictures from all 7 cases are included as supplementary Figure 3. f The distribution of pair-wise correlations of malignant cells within each tumor region (intraregion), across regions within each individual case (interregion) and across cases (intertumor). Pearson's correlation coefficient was applied. N.D., not detectable. Solid and dashed gray lines indicate the mean and standard deviation of all intraregional correlation values. Source data are provided as a Source data file.
While tumor cells from multiple regions of each individual case were clustered together (Fig. 1c), we observed noticeable differences in tumor histology among different regions of each tumor, regardless of a difference in tumor size among these tumors (Fig. 1e and Supplementary Fig. 3). To quantitively determine the heterogeneity of tumor cells, we calculated intraregional heterogeneity as the distribution of pair-wise correlation of malignant cells within a specific region while interregional heterogeneity as the correlation of malignant cells among multiple regions within a patient. We used intertumoral heterogeneity among different patients as a reference. Noticeably, while some regional differences within each tumor were noted, correlations among intraregional and interregional tumor biopsies within each patient were much greater than that of intertumor among different patients (Fig. 1f). We also evaluated the inferred chromosomal copy number variations (CNVs) and found considerable distinct patterns among patients while much smaller differences among multiple tumor regions within each case (Supplementary Fig. 4a). To determine the impact of tumor size on tumor heterogeneity, we calculated its correlation with tumor heterogeneity among different regions and found no relationship in these samples (Supplementary Fig. 4b–d). These results indicate that while tumor size and histology may vary among patients, there was a much smaller difference in tumor cell transcriptomic activity within a patient than between patients. This observation is consistent with a recent report regarding HCC regional transcriptomic heterogeneity22.
Dynamics of transcriptomic heterogeneity of malignant and non-malignant cells
To evaluate the landscape of cellular dynamics of malignant cells from different regions linked to their transcriptomic heterogeneity, we performed trajectory analysis using RNA velocity23,24, which determines cellular dynamics including developmental lineages and differentiation states based on splicing kinetics of tumor cell transcriptome. We found cellular trajectories of malignant cells again appear similar among different tumor regions within each tumor while heterogeneity in expressions of different stemness genes is noted. For example, the malignant cells from case 1C (7.5 cm in tumor size) followed a similar trajectory regardless of the tumor region sampled (Fig. 2a). When ranking tumor cells of 1C along their latent time estimated using RNA velocity with the expression of cell stemness related marker genes (i.e., EPCAM, KRT19, ICAM1)16 and a tumor progression related gene (i.e., SPP1)20, we found a similar gene expression pattern among sampling regions while varying expression among individual cells (Fig. 2b). Similar results were found in other cases (Supplementary Fig. 5). In addition, we performed fluorescence-activated cell sorting (FACS) analysis of EPCAM+ cells or GPC3+ cells for three HCC samples with available cryopreserved single-cell suspension, because the two markers were known elevated in tumor cells. The proportion of EPCAM+ or GPC3+ cells was relatively stable in the tumor core regions while differences were noted when compared to the tumor border regions, suggesting that the proportion of those cells may vary among tumor regions (Supplementary Fig. 6). We also confirmed that SPP1 expression is elevated in tumor cells but its expression is heterogeneous (Supplementary Fig. 7), which is consistent with previous publications21.
Fig. 2: Multiregional tumor cell trajectory of case 1C.
a RNA velocity of malignant cells from all tumor regions with viable malignant cells. b Expression of EPCAM, KRT19, ICAM1, and SPP1 in malignant cells along cellular latent time determined by RNA velocity method in (a).
We also determined the spatial landscape of non-malignant cells of HCC and iCCA by dimensional reduction using a manifold learning method of uniform manifold approximation and projection (UMAP)25. In contrast to the patient-specific patterns in malignant cells, non-malignant cells were mainly grouped based on their cell lineage, i.e., T cells, B cells, tumor-associated macrophages (TAMs), cancer-associated fibroblasts (CAFs), tumor-associated endothelial cells (TECs), hepatocytes and cholangiocytes, which were determined using lineage-specific marker genes (Fig. 3a, b). We observed similar cellular patterns of the TMEs among multiple tumor regions or among patients while a small difference was noted (Fig. 3c, d). When each cell type was analyzed separately, we found a high correlation of T cells from different sampling regions within each case, suggesting that the T-cell transcriptomic profiles appear stable among different sampling regions (Fig. 3e). We further evaluated the presence of CD3+ T cells using immunohistochemistry analysis, where relatively similar CD3+ T-cell numbers between tumor cores and tumor borders within each case than among cases was revealed with available paraffin blocks (Supplementary Fig. 8). In contrast, we found the correlation of CAFs, TECs, TAMs and B cells varied among different regions with some cases showing a high correlation while others showing a low correlation (Supplementary Fig. 9a). However, variations of immune/stromal cell types among different sampling regions were not correlated with tumor size, similar to the features of tumor cells (Supplementary Fig. 9b).
Fig. 3: Landscape of multiregional non-malignant cells.
a UMAP of non-malignant cells colored by cell types. CAFs, cancer-associated fibroblasts; TAMs. tumor-associated macrophages; TECs, tumor-associated endothelial cells. b Violin plots of cell-type specific marker gene expression in non-malignant cells. c, d UMAP of non-malignant cells colored by tumor regions (c) and case IDs (d). e Correlation of T cells (n = 61,561) between different tumor regions of each individual case. In the box plots, the central rectangles span the first quartile to the third quartile, with the segments inside the rectangle corresponding to the median. Whiskers extend 1.5 times the interquartile range. Source data are provided as a Source data file.
Based on the overall features of all cells in the TME, we observed immune activation in 3C and 3H while immune suppression in 4H and 1H (Supplementary Fig. 9c). For the rest of the cases, lack of the immune activities was observed (Supplementary Fig. 9c). Since T-cell profiles appeared most stable, we further determined T-cell subtypes and their composition in each tumor region. We identified 21 subsets and defined them based on the top differentially expressed genes (Supplementary Fig. 10a and b). We observed similarity of the T-cell state composition in some tumor regions within each case while variations among others (Supplementary Fig. 10c). For example, 4H had the most stable T-cell subset composition among all the tumor regions, which is consistent with the highest correlation of T-cell transcriptomic profiles in this case (Fig. 3e and Supplementary Fig. 10c). In contrast, similarity of T-cell states was found in only two of the biopsied regions in the case of 1C. Thus, T-cell states appeared very dynamic although the difference within a patient seems smaller than between patients. Taken together, transcriptomic heterogeneity of malignant cells and immune/stromal cells appears smaller among sampling regions within each tumor lesion than among tumors from different patients while cellular heterogeneity within each tumor is noted. These results are consistent with recent publications in HCC, non-small-cell lung cancer and renal cell carcinoma using single-cell technologies22,26,27.
Multiregional tumor-immune communication networks
Given the noticeable histological and transcriptomic heterogeneity of a tumor lesion described above and elsewhere, it is imperative to identify stable molecular features linked to tumor biology. As tumor progression is a dynamic process involving continuous interactions of tumor cells and stromal/immune cells, tumor cells may profoundly influence various cell types within the tumor ecosystem to promote their own survival and the dissemination of malignancy4. Meanwhile, the TMEs including tumor-associated matrix suppress tumor growth by effective immune surveillance but may also be educated by tumor to support tumor progression28,29,30. We hypothesized that each tumor ecosystem may contain specific molecular communication networks between tumor cellular activities and the immune cell landscape unique to each patient, as these features may have minimum sampling bias. To identify the communication networks, we searched for ligand–receptor interactions of malignant cells and the TME within each patient using CellPhoneDB31,32 (see "Methods"). We identified patient-specific interaction networks that are consistently well-conserved among different tumor regions from each individual patient (Fig. 4a, b, Supplementary Figs. 11–13), regardless of the direction of the interactions (i.e., tumor-to-TME, ligands from malignant cells and receptors from non-malignant cells; TME-to-tumor, ligands from non-malignant cells and receptors from malignant cells). These results indicate that the molecular communication networks appear relatively stable regardless of sampling locations, which likely reflects the intrinsic tumor biology of each tumor. Noticeably, the interactions are more stable in HCC than iCCA (Fig. 4c). We also determined the interactions between tumor cells and T-cell subsets, where relatively stable interactions were observed within each case (Supplementary Fig. 12b, c). In contrast to the consistency among different tumor regions within a specific patient, the ligand–receptor interaction pairs varied across patients, showing cell-to-cell communication networks unique to each patient (Fig. 4a and d). To determine whether patient-specific ligand–receptor interaction networks were mainly contributed by tumors or by TME, we developed a strategy by performing random shuffle of either tumor or TME (Fig. 4e). Not surprisingly, we found that the strength of the network interaction deteriorates faster for tumor random shuffle than TME random shuffle (Fig. 4f), suggesting the uniqueness is mainly controlled by malignant cells. Taken together, these results indicate that each tumor may contain relatively stable but unique communication networks between tumor and the TME, where malignant cells mainly contribute to the patient-specific patterns.
Fig. 4: Communication of malignant cells and non-malignant cells.
a Ligand–receptor interactions of malignant cells and non-malignant cells in six cases with viable malignant cells. Each column indicates a ligand–receptor pair, with the first and the second gene representing a ligand and a receptor, respectively. Each row represents a non-malignant cell type that interacts with malignant cells. The direction of an interaction is indicated by colored dot. Purple, malignant cells provide ligands and interact with receptors from non-malignant cells in the TME; green, non-malignant cells in the TME provide ligands and interact with receptors from malignant cells. The size of each dot represents the proportion of tumor regions within each case in identifying a specific interaction pair, with 1 indicating occurrence in all tumor regions and 0 indicating occurrence in none of the tumor regions. b Illustration of ligand–receptor interactions of tumor and the TME. Purple dots, ligands from tumor; green dots, ligands from TME. c Stacked bar plot of the percentage of ligand–receptor pairs in each individual case found in certain proportion of tumor regions. One means that a pair was found in all tumor regions within a case while zero means that a pair was found in none of the tumor regions. d Similarity of ligand–receptor interactions among multiple regions of different cases. Zero indicates no overlap of ligand–receptor interactions while 1 means a full overlap of ligand–receptor interactions between samples. e Illustration of switching TME and switching tumor. Switching TME indicates that tumor cells from one case are combined with TMEs from other cases to form distinct tumor ecosystems. Switching tumor indicates that TME from one case are combined with tumors from other cases to form distinct tumor ecosystems. f The proportion of matched ligand–receptor interactions from switching tumor or TME with the original search of using paired tumor and TME from the same case. Student's t-test (two-sided) was applied with p value provided. b and e were generated using BioRender.
The tumor-immune communication networks are associated with patient outcomes
To determine if patient-specific tumor-immune interaction networks are biologically important and are associated with tumor aggressiveness, we extended our search for ligand–receptor interactions in additional 46 tumor samples from 25 HCC and 12 iCCA patients described recently20. We included samples with > 15 detectable malignant cells per tumor for determining interactions of malignant cells and non-malignant cells. We further filtered the identified ligand–receptor pairs by using those found in the multiregional analysis (Fig. 4a, see "Methods"), which were selected using stringent criteria and demonstrated stable communication programs across multiple regions within each patient. We found two main clusters with different interaction networks based on a hierarchical relationship of the ligand–receptor interaction activities (Fig. 5a). Noticeably, patients from the two clusters have significantly different overall survival, suggesting that the tumor-immune interaction networks are biologically distinct between the two clusters linked to tumor biology (Fig. 5b). Because this cohort contains both HCC and iCCA patients, we also analyzed HCC patients separately to avoid potential tumor type bias. We found a consistent trend of survival difference between the two clusters (Fig. 5c). We did not analyze iCCA patients separately as they were all in Cluster 1. To determine the key interactions of each cluster, we evaluated the difference between the proportions of each ligand–receptor interaction in the two clusters. Noticeably, the ligand–receptor interaction networks are polarized between the two clusters (Fig. 5d). For example, the LGALS9-SLC1A5 and SPP1-PTGER4 pairs, mainly contributed by the communication of malignant cells and TAMs, were among the top interaction pairs from Cluster 1. To confirm the ligand–receptor pairs identified by CellPhoneDB, we further applied CellChat33 to determine cellular interactions. We found >85% consistency of the ligand–receptor interactions (i.e., SPP1-PTGER4 and LGALS9-SLC1A5) between tumor cells and TAMs using the two methods, suggesting that the identified pairs are stable (Supplementary Fig. 14a). In addition, we evaluated the impact of the number of malignant cells on ligand–receptor pair identification by randomly sampling malignant cells. We applied this strategy to five cases with the highest number of malignant cells and found no linear relationship between the number of cells and the accuracy of ligand–receptor interaction determination (Supplementary Fig. 14b). However, we did notice a slight drop of its accuracy when the number of malignant cells is less than 20 (i.e., an average accuracy of 80.04% with 20 malignant cells and 71% with 10 malignant cells (Supplementary Fig. 14b).
Fig. 5: Communication of malignant cells and non-malignant cells are associated with patient outcome.
a Hierarchical clustering of the ligand–receptor interaction patterns of malignant cells and non-malignant cells. Each row indicates a ligand–receptor pair, with the first and the second gene representing a ligand and a receptor, respectively. Each column represents a tumor sample. The direction of an interaction is indicated by color. Purple, malignant cells provide ligands and interact with receptors from non-malignant cells in the TME; green, non-malignant cells in the TME provide ligands and interact with receptors from malignant cells. Distinct non-malignant cell types that interact with malignant cells are indicated by colors. Clusters were determined based on the hierarchical relationship. b, c Overall survival of all patients (b) or HCC patients (c) from Cluster 1 and Cluster 2 in (a). Log-rank test was preformed to show the statistical difference of the two groups. d The difference between the proportions of each ligand–receptor interaction in Cluster 1 and Cluster 2. Red, pairs enriched in Cluster 1; Blue, pairs enriched in Cluster 2. The direction of each interaction pair is indicated by color. Purple, malignant cells provide ligands and interact with receptors from non-malignant cells in the TME; green, non-malignant cells in the TME provide ligands and interact with receptors from malignant cells. The non-malignant cell types that interact with malignant cells are indicated in parentheses.
To further validate whether the specific ligand–receptor interaction networks were associated with overall survival of HCC patients, we applied the ligand–receptor pairs found from the single-cell analysis to bulk transcriptomic data of 542 patients from three HCC cohorts (i.e., LCI cohort, TCGA HCC cohort, TIGER-LC HCC cohort). Different from directly evaluating cell-cell communications using single-cell data, we developed a strategy by calculating the mean of the ligand and receptor in a specific interaction pair and then determining the occurrence of this pair using the median expression across patients to mimic ligand–receptor interactions among tumor and TME (see "Methods"). We applied this strategy to all the identified ligand–receptor pairs to generate the interaction map of each patient, based on which hierarchical clustering was then performed. We found that patients could be grouped into two main clusters with distinct interaction patterns in the LCI cohort (Fig. 6a) as well as in the TCGA HCC and TIGER-LC HCC cohorts (Supplementary Fig. 15a). To evaluate the consistency of the communication patterns among the three cohorts, we calculated the proportion of each ligand–receptor pair in Cluster 1 for all three cohorts. We found high concordance among three different cohorts with pairs to be assigned to Cluster 1 or Cluster 2, especially for tumor-TAM-derived pairs in Cluster 1, i.e., LGALS9-SLC1A5 and SPP1-PTGER4 pairs (Fig. 6b). Remarkably, patients from the two clusters had a consistently different overall survival in tumor tissues but not in adjacent non-tumor tissues from all three cohorts (Fig. 6c), suggesting that the survival related ligand–receptor interaction activities are imbedded within a tumor lesion. Consistently, when patients from each cohort were divided into three clusters based on the hierarchical relationship, we found a significant trend linking interaction networks to the overall survival of patients, further indicating that the interaction network is a stable feature of HCC aggressiveness (Supplementary Fig. 15b). Collectively, these results imply that the landscape of each tumor ecosystem may contain a unique combination of tumor-immune/stromal interaction pairs from a successful tumor evolution, which is analogous to lock-and-key feature of the enzyme-substrate interaction. These interactions can be exploited as a classifier of tumor aggressiveness.
Fig. 6: Validation of the tumor and TME interaction patterns for patient stratification using bulk transcriptomic data.
a Hierarchical clustering of ligand–receptor interaction activities in LCI cohort. Each column represents a tumor sample. Each row represents a pair. The direction of each interaction pair is indicated by color. Purple, tumor provides ligands and interacts with receptors from the TME; green, TME provides ligands and interacts with receptors from tumor; light purple, both directions (pairs were identified in both directions from single-cell analysis but can only be modeled once in bulk data). b The proportion of each pair in Cluster 1 of three HCC cohorts. Error bar, mean ± standard error of the mean. c Overall survival of patients from Cluster 1 and Cluster 2 in three HCC cohorts and the corresponding non-tumor cohorts. Cluster 1 and Cluster 2 were determined based on hierarchical clustering of ligand–receptor interaction activities in (a) and Supplementary Fig. 15a. The number of samples in each cohort was provided. Log-rank test was preformed to show the statistical difference of the two groups.
Validation of key ligand–receptor pairs by multiplex in situ hybridization in HCC
To further validate the robustness of the ligand–receptor interactions linked to HCC prognosis, we evaluated the top two pairs, i.e., LGALS9-SLC1A5 and SPP1-PTGER4, representing the tumor-TAM communications for proof-of-principle analysis (Fig. 7a), as validation of all significant pairs would be technical challenging. We determined their expression patterns in tissue microarrays consisting of paraffin blocks of both TIGER-LC cohort (HCC, n = 68) and LCI cohort (HCC, n = 190) using the RNAscope multiplex fluorescent in situ hybridization assay (Fig. 7b). We found a significant correlation of the expression levels of all four target genes in both cohorts between transcriptome-based analysis and RNAscope-based analysis, indicating a good quality of the RNAscope assay of the target genes (Fig. 7c and Supplementary Fig. 16a). Based on the RNAscope signal and the resolved spatial context of each gene, we then determined whether the pair associated genes were more likely to be colocalized than by chance. This was implemented by calculating the Bhattacharyya coefficient (BC) of each tumor, which could measure the amount of overlap of two spatially distributed genes. Specifically, we partitioned each tumor into tiles and calculated the probability of each gene in each tile to calculate the BC for each tumor, with one representing a fully spatial overlap of two genes and zero indicating no overlap (Fig. 7d). In addition, we measured the proportion of the filled tiles (FTs) for each tumor to indicate the spatial preference of the pair-related genes in certain tumor locations (Fig. 7d). We found a high colocalization of the pair-related genes for both pairs with high BC values, suggesting the co-dependence of the pair-related two genes (Fig. 7e and Supplementary Fig. 16b). Moreover, the pair-related genes were colocalized in certain tumor regions rather than the whole tumor space (Fig. 7f and Supplementary Fig. 16c). Consistent with single-cell and bulk transcriptome data, patients with high expression of both pairs had a significant worse overall survival than those of the low expression groups in both HCC cohorts (Fig. 7g and Supplementary Fig. 16d). Therefore, the co-occurrence of the two ligand–receptor pairs (LGALS9-SLC1A5 and SPP1-PTGER4) was stable and the elevated expressions were associated with poor patient outcome in HCC.
Fig. 7: Validation of two interaction pairs between tumor and TAM using RNAscope assay.
a Illustration of ligand–receptor interaction pairs between tumor cell and TAM (generated using BioRender). b A representative image of RNAscope multiplex fluorescent in situ hybridization of four genes of an HCC sample from a total of 258 samples analyzed. (c) Correlation of RNAscope signal and bulk transcriptome gene expression in TIGER-LC cohort. Pearson's correlation coefficient (two-sided) was calculated. Dashed line: p = −log10(0.05). d Evaluation of the colocalization of two spatially distributed genes. BC, Bhattacharyya coefficient: 1, a full colocalization; 0, no colocalization. FTs, filled tiles. e The distribution of BCs of LGALS9 and SLC1A5 (top) as well as SPP1 and PTGER4 (bottom) in HCC samples from the TIGER-LC cohort. Dashed line: mean value. f The distribution of the proportion of filled tiles of LGALS9 and SLC1A5 (top) as well as SPP1 and PTGER4 (bottom) in HCC samples from the TIGER-LC cohort. Ten-times of randomization was used to generate random spread of markers on tissue sections as a reference. Gray line, proportion determined based on the ratio of true signal and each random spread; gold line, mean derived from ten gray lines. Dashed line: mean value. Student's t-test (two-sided) was applied. g Overall survival of HCC patients with low expression and high expression of LGALS9 and SLC1A5 as well as SPP1 and PTGER4 from the TIGER-LC cohort. Tumor samples with expression of the four marker genes in between were grouped into others. Log-rank test and a trend test among the groups were preformed. h t-SNE plot of TAMs from samples with or without the two pairs (i.e., LGALS9 and SLC1A5, SPP1 and PTGER4) in Fig. 5a. i The composition of each TAM cluster. j Differentially expressed genes of TAMs from the samples with or without the two pairs (i.e., LGALS9 and SLC1A5, SPP1 and PTGER4) in Fig. 5a. k Differentially expressed genes of malignant cells from the samples with or without the two pairs (i.e., LGALS9 and SLC1A5, SPP1 and PTGER4) in Fig. 5a. Wilcoxon test with multiple test adjustment was applied in (j) and (k).
While evidence for physical interactions of LGALS9-SLC1A5 or SPP1-PTGER4 as the ligand and receptor pairs has been described in the curated database of CellPhoneDB31, the expression patterns of each gene among different cells as well as the functional consequence of these interactions between tumor cells and TAMs have not been determined. We found that SPP1 and SLC1A5 were more abundantly expressed in malignant cells while PTGER4 and LGALS9 were more abundantly expressed in non-malignant cells (Supplementary Fig. 17a). In addition, the expression of these genes was also significantly higher in tumors with the two pairs than those without the two pairs (Supplementary Fig. 17b). To explore the biological features associated with this unique tumor-macrophage link, we compared single-cell transcriptomic profiles of TAMs from tumors with or without the presence of LGALS9-SLC1A5 and SPP1-PTGER4, and found 4 main subtypes of TAMs, i.e., c1 (proliferative), c2 (inflammatory), c3 (restorative), and c4 (CLEC9A+WDFY4+) (Fig. 7h). We found that c1 and c4 TAMs were enriched in tumors with the pairs while c2 and c3 TAMs were enriched in tumors without the pairs (Fig. 7i). We also searched for differentially expressed genes in TAMs or tumor cells from the cases with or without the specific pairs as the biological surrogates of the unique tumor-macrophage link (Fig. 7j and k). TAMs from tumors with the pairs had unique transcriptome profiles with enriched genes involving in oxidative phosphorylation while tumor cells from cases with the pairs showed unique transcriptome profiles with genes enriched in interferon response pathways (Supplementary Data 1). Consistently, both TAM- and tumor cell-derived surrogate signatures could significantly discriminate Cluster-1 HCC from Cluster-2 HCC in all three cohorts evaluated, indicating the biological importance of the surrogate signatures (Supplementary Fig. 17c, d). In addition, there was a significant correlation between the expression of the two ligand–receptor pair-related four genes and their functional surrogate genes (Supplementary Fig. 17e–g). Collectively, these results indicate the presence of a unique and stable tumor-macrophage signaling activity representing unique signatures downstream of the specific ligand–receptor interactions linked to HCC aggressiveness.
Because of tumor evolution and the consequent ITH, cancer research has been confronted for decades by the dilemma as how best to effectively define key drivers and functional biomarkers representing the hallmarks of cancer as the basis for implementing early diagnosis and precision intervention4. The approaches by the TCGA and ICGC consortia to provide big data analytics especially integrative genomics are exciting and enable a rich data source for driver discovery. These initiatives also promote the idea of targeting tumors based on drivers unique to certain molecular subtypes as the central theme of precision oncology34. While this strategy led to initial success in targeting melanoma with BRAF mutations or lung cancer with EGFR mutations, most tumors eventually relapse and the overall prognosis of those patients remains poor35,36. This is especially challenging for liver cancer research in which a complex etiology-related hepatocarcinogenesis results in a vast heterogeneous cancer genome without dominant driver mutations but plenty of passenger mutations that do not provide any phenotypic consequences37. Tumor evolution may be the main reason for therapeutic failure and consequent poor patient outcomes. Recent advances in single-cell technologies have provided an unprecedented sensitivity to better define a tumor ecosystem12. An important question remains as to what key features represent intrinsic tumor biology and its evolution for each tumor lesion and how much sampling bias contributes to the observed ITH.
The success in establishing a tumor colony should satisfy two parallelly evolving processes, i.e., an acquisition of molecular alterations in somatic cells and an appropriate 'molding' of tumor cell landscape known as the TME to support the survival and fitness of somatically altered cells. Therefore, a perfect fit between tumor cells and their TME may reflect a timestamp for a successful tumor evolution and should be unique to each solid tumor lesion. This is analogous to the lock-and-key model to explain the enzyme-substrate interaction in efficiently achieving a biological process. In this study, we attempted to find evidence of a lock-and-key feature in HCC by incorporating spatial single-cell analysis to examine whether sampling bias may contribute to the appearance of ITH and to search for molecular fingerprints of tumor-stromal interactions unique to tumor biology of each liver tumor lesion. We found that some differences in the tumor cell composition and non-malignant cells in the TME could be observed among biopsies from different sampling regions. However, molecular features representing the specific ligand–receptor interactions among tumor cells and non-malignant cells appeared stable among various cohorts from patients with different ethnicities and etiologies. Specifically, we identified two ligand–receptor pairs, i.e., LGALS9-SLC1A5 and SPP1-PTGER4 that may represent a fingerprint of functional interactions between tumor cells and TAMs. We found that SLC1A5 and SPP1 are mainly produced by tumor cells while LGALS9 and PTGER4 are mainly expressed in TAMs. It should be noted that SPP1 is also expressed in TAMs, but the expression is much lower than tumor cells. In addition, SPP1 encodes osteopontin (OPN), a cytokine known to promote HCC metastasis21. It is known that OPN exists several different isoforms. However, it is difficult to determine the status of OPN isoforms using 10x genomics single-cell data. This would be an interesting subject of future studies. SLC1A5 encodes a neutral amino acid transporter38. In contrast, PTGER4 encodes one of four receptors for prostaglandin E2, which may be involved in T-cell factor signaling. PTGER4-expressing macrophages have been shown to promote intestinal epithelial barrier regeneration upon inflammation39. LGALS9 encodes an S-type lectin involved in cell adhesion, immune escape, angiogenesis and tumor metastasis40. While physical interactions among LGALS9-SLC1A5 and SPP1-PTGER4 pairs have been described in the curated database of CellPhoneDB32, a functional consequence of these interactions between tumor cells and TAM has not been determined. We found a significant correlation between differentially expressed genes in tumor cells and TAMs defined by the presence of the fingerprint pairs and the ligand–receptor expressions themselves, suggesting that the activities of these genes may represent a functional interaction of the unique tumor-macrophage network. Moreover, we found that tumor cells with the fingerprint pairs had a unique transcriptome enriching with interferon response pathways, while tumor cells without the fingerprint pairs enriched cellular signaling involving coagulation. In contrast, TAMs with the fingerprint pairs were much more proliferative and enriched genes were involved in oxidative phosphorylation. It is interesting to note that among the identified TAMs, we found a cluster of CLEC9A+WDFY4+ cells. While CLEC9A is a marker for type 1 dendritic cells, it is also expressed in other cell types including macrophages. We further validated the presence of the fingerprint pairs to be associated with overall survival in two independent cohorts of 258 HCC patients using an in situ hybridization approach. Our results suggest that the transcriptome-based single-cell analysis is a robust tool to define each tumor lesion reflecting its tumor biology. The identification of the unique ligand–receptor fingerprint pairs may help provide the rationale for implementing biopsy-based single-cell analysis for biological understanding of tumors in questions for clinical decision making, which should be evaluated further in clinical trials.
HCC and iCCA are two histological subtypes of liver cancer. Bulk transcriptomic profiling of primary HCC and iCCA indicates both distinct and shared molecular features41. In the clustering analysis of ligand–receptor activities using our single-cell cohort, we found Cluster 1 comprised both HCC and iCCA while Cluster 2 was mainly composed of HCC (Fig. 5a), indicating some HCC shared ligand–receptor communication features with iCCA while others did not. Genomic studies of liver cancer demonstrated trunk and branch mutations along with distinct genetic clones among different regions of a liver tumor. However, we found transcriptomic profiles of malignant cells are similar among different sampling locations of a tumor lesion for most of the cases in our single-cell cohort. A recent single-cell study of lung tumor evolution using both scRNA-seq and single-cell DNA sequencing (scDNA-seq) demonstrated that the clones determined by scDNA-seq are largely independent from clones with similar cellular states derived from transcriptomic landscape determined by scRNA-seq7. These results indicate that genomic alterations may be independent of transcriptomic profiles. This is anticipated as most genomic alterations used for clonality analysis have no functional consequence as most of them are passengers1. Consistently, many recent studies including this study have now shown evidence supporting the idea that cellular states defined by scRNA-seq may be better in representing tumor cell clonality and evolution20.
One limitation of this study is that single-cell data were based on a small cohort, especially for iCCA samples, as well as limited longitudinal biopsies for monitoring identified fingerprint during tumor evolution. Another limitation is that the validation of the LGALS9-SLC1A5 and SPP1-PTGER4 interactions is not at the protein level. It's very challenging to target different proteins with multiple antibodies on the same slide using multi-channel chromogenic detection. For this reason, our effort was mainly relying on the use of RNAscope analysis as a proof-of-principle experiment even though this approach has its limitation and therefore needs to reach interpretation with caution. In addition, the functional consequences of the LGALS9-SLC1A5 and SPP1-PTGER4 interactions have not been tested experimentally using both in vitro and in vivo HCC models. However, we validated the stability of the identified interaction networks using bulk transcriptomic data of an additional 542 HCC patients, a necessary step to strengthen their pathophysiological relevance. There is an urgent need in identifying appropriate preclinical models used to explore functional relevance of research biopsy-based observations, a call for ensuring rigor and reproducibility of the follow-up functional studies42. Moreover, it is worth to determine the changes of the communication networks between tumor and TME during tumor evolution in response to therapy and this knowledge may help understand the mechanism of therapeutic resistance. We continue to enroll HCC and iCCA patients with on-treatment longitudinal biopsies at the NIH Clinical Center as a part of the NCI-CLARITY study to address this question in the future.
Liver cancer remains one of the most difficult to treat solid malignancies with a 5-year survival rate of less than 18% in the U.S. for many decades43. In the single-cell studies of different cancer types including liver cancer, the phenomenon of extensive tumor heterogeneity has been noticed, which creates a major barrier for effective cancer interventions. Sampling bias could be an issue when one uses a single biopsy to determine tumor biology and response to treatment. Thus, in clinical practice, it is important to identify features that are relatively stable and can be used to assess molecular features of a tumor during the course of clinical intervention to avoid sampling bias. Our study indicates that a unique tumor-macrophage link via ligand–receptor interactions from LGALS9-SLC1A5 and/or SPP1-PTGER4 signaling pairs appears a stable molecular feature to define ITH linked to overall survival of patients with HCC. This is consistent with the notion that tumor cells continuously communicate with the tumor microenvironment, defining the molecular map underlining tumor biodiversity may be a key to improve our understanding of tumor heterogeneity and further identifying novel therapeutic targets. We suggest that the identified feature may reflect a successful tumor evolution unique to each tumor. Exploiting methods to disrupt these interactions could constitute a viable therapeutic strategy to target HCC and stop tumor evolution, thereby improving treatment efficacy.
Human sample collection
A total of seven primary liver cancer patients treated at the University Medical Center in Mainz and the NIH Clinical Center in Bethesda, have been enrolled prospectively into this study. Among them, three patients were diagnosed with iCCA and four were diagnosed with HCC. Tumor size for each patient can be found in Supplementary Table 1. All patients received surgical resection. A total of five samples from the tumor core, tumor border, and adjacent non-tumor tissue were collected for each patient. Specifically, we collected three samples from the tumor core that were not adjacent, one sample from the tumor border, and one sample from the adjacent non-tumor tissue that was not locally close to the tumor. Each sample was measured about 5 mm diameter in size before single-cell library preparation. Sample collection was performed with written informed consent from patients. We removed one sample (T3) from patient 3C due to single-cell library failure and thus a total of 34 samples were included in this study. This study was approved by the ethics committee of the University Medical Center in Mainz and the National Institutes of Health.
Single-cell suspension preparation
Resected samples were collected in RPMI 1640 media and were immediately processed to keep ischemic time to a minimum. Samples were minced in petri dishes on ice and transferred into gentleMACS C Tubes. Enzymes of the Miltenyi tumor dissociation kit had been added before according to the kit user guide. The tube was placed into the MACS Dissociator for mechanical dissociation. The tube was incubated for 30 min at 37 °C under continuous shaking. Next, the cell suspension was filtered through a nylon mesh and cells were counted to determine number of cells and viability. The samples were then centrifuged at 300 × g, 4 °C for 5 min and re-suspended in freezing media for cryopreservation in liquid nitrogen. Samples from Mainz, Germany, were shipped cryopreserved to the NIH.
Single-cell library preparation and droplet-based scRNA-seq
Cryopreserved samples were thawed and prepared according to the Single Cell 5' Reagent Kits User Guide. Specifically, the samples were washed and re-suspended in PBS + 0.04% BSA. Cell viability was determined. The cDNAs were obtained after the GEM (Gel Bead-in-emulsion) generation and barcoding, followed by GEM RT (reverse transcription) reaction. Purified cDNA was amplified for 14 cycles. A clean up using SPRIselect beads was performed. cDNA libraries were prepared with 10x Genomics Single Cell 5' library & gel bead kit v1.1. cDNA concentration was determined by Bioanalyzer. Libraries were then pooled and normalized to a final loading concentration. The samples were loaded in the lanes according to the 10x Genomics 5' User Guide and were then sequenced using Illumina NovaSeq platform at Frederick National Laboratory for Cancer Research Sequencing Facility, with sequencing parameters of 28 bp (Read1), 8 bp (Index1), and 98 bp (Read2). The targeted sequencing depth for each sample is 50,000 reads/cell. Base calling was carried out with Real-Time Analysis software (version 3.4.4) on Illumina sequencing systems. Demultiplexing was then performed by using bcl2fastq (version 2.20), with one mismatch allowed in the barcodes. The standard 10x Genomics CellRanger (version 3.1.0) pipeline was used to extract FASTQ files and to perform data processing including alignment, tagging, gene, and transcript counting. Sequenced reads were aligned to human reference sequence (refdata-cellranger-GRCh38-3.0.0) provided by the 10x Genomics.
scRNA-seq data pre-processing
We integrated single-cell profiles from different samples by performing read depth normalization for all the 34 samples using cellranger aggr pipeline from the Cell Ranger (version 3.1.0), which equalized the average read depth per cell between samples based on the confidently mapped reads. R Seurat package (version 3.1.2) was applied for the pre-processing of the aggregated data. We kept genes that were expressed in at least three cells and removed cells with less than 500 genes detected. A total of 112,506 cells passed this initial quality control, with an average of 1067 genes and 3359 unique UMIs detected per cell. We then normalized the total counts in each individual cell to 10,000, followed by log transformation to generate the normalized data.
Separation of malignant cells and non-malignant cells
To separate malignant cells and non-malignant cells, we inferred large-scale chromosomal copy-number variations (CNVs) based on single-cell transcriptome profiles as described in previous published single-cell studies17,20,44,45. Briefly, this method infers CNVs by taking the average expression of a set of genomically adjacent genes along each chromosome to eliminate gene-specific patterns and yield CNV profiles, with the assumption of aberrant karyotypes in malignant cells. Because adjacent non-tumor tissues are available in this study, we used cells derived from those samples as a reference during CNV inference in order to reduce background noise. From the inferred CNVs, gains on chromosomes 1q and 8q of malignant cells were observed (Supplementary Fig. 1a), consistent with the CNV profiles generated from genome sequencing of liver cancer41. In contrast, no obvious CNV patterns were observed in non-malignant cells (Supplementary Fig. 1a). To further confirm the successful separation of malignant cells and non-malignant cells, we evaluated the expression of epithelial- and liver-specific marker genes in the derived malignant cells and non-malignant cells, considering the epithelial origins of malignant cells17. We found consistency of tumor aberrant karyotypes and the marker gene expression, with strikingly higher expression of epithelial- and liver-specific genes in malignant cells, further suggesting a confident separation of malignant cells and non-malignant cells (Supplementary Figs. 1b and 1c). As expected, a high number of genes was expressed in malignant cells, with an average of 3106 genes and 16,658 UMIs detected per malignant cell. In the downstream analysis of malignant cells, we only included the samples with >10 malignant cells detected. With this criterion, we didn't detect enough malignant cells in 2C as well as the samples labeled as N.D. (not detected) in Fig. 1f.
Tumor heterogeneity
To evaluate tumor heterogeneity of malignant cells, we used three independent approaches: (1) Dimensional reduction algorithm. We first applied principal component analysis (PCA) to the top 2000 most variable genes of all malignant cells determined using standardized means and variances. We further performed dimensional reduction on the first 20 principal components (PCs) by employing t-distributed stochastic neighbor embedding (t-SNE) method. Samples with more than 10 malignant cells detected were involved in this analysis. In the two-dimensional t-SNE space, we observed heterogeneous tumor cell populations, with a larger tumor heterogeneity between patients than within a patient. (2) Hierarchical clustering method. We also generated a hierarchical tree of malignant cells from different tumor regions of all the cases. Here, we applied the top 2000 most variable genes as described above and calculated the mean expression of each individual gene in all the malignant cells of each tumor sample. Then we constructed the hierarchical tree of all the tumor samples by using correlation-based distance measurement and ward.D2 data agglomeration method. (3) Pearson's correlation analysis. To quantitively measure the level of intraregional (within a tumor region), interregional (across multiple tumor regions within a case) and intertumoral (across multiple tumors from different cases) heterogeneity, we calculated the correlation of the malignant-cell transcriptomic profiles based on the top 2000 most variable genes as described above. Intraregional tumor heterogeneity was measured as the distribution of the pair-wise correlation of malignant cells within a specific tumor region. Interregional tumor heterogeneity was determined as the density distribution of the correlation coefficients of malignant cells across different tumor regions within a specific case. Intertumoral heterogeneity was calculated as the correlation of malignant cells among the samples from different cases. For all the three levels of heterogeneity, Pearson's correlation coefficient was applied, with 1 indicating a perfect positive linear correlation, −1 representing a perfect negative correlation and 0 standing for no linear correlation.
Cellular dynamics determined by RNA velocity method
To recover the cellular dynamics of malignant cells, we applied RNA velocity method from the scvelo python package. RNA velocity allows for learning of cellular lineage trajectory by considering the spliced and unspliced events in the single cells23,24. Specifically, we used the scv.tl.recover_dynamics function to recover the full dynamics, followed by calculating the velocity of each gene using scv.tl.velocity function based on the splicing kinetics. Then the collection of the velocities of all genes were used to calculate the transition probabilities between cells and to predict the direction and movement of a specific cell in a future state. Finally, the velocities were projected onto a uniform manifold approximation and projection (UMAP) embedding by using the scv.pl.velocity_embedding_stream function for visualization. Based on the transcriptional dynamics, the latent time underlying cellular processes of each individual cell was recovered using scv.tl.latent_time function. Cases with more than 50 viable malignant cells were used for RNA velocity analysis since it failed to recover the full dynamics of malignant cells if the number of cells is too small. To indicate gene expression along cellular latent time determined by RNA velocity analysis, we applied 10 genes (tumor stemness-related genes and tumor evolution-related genes) including EPCAM, KRT19, ICAM1, PROM1, LGR5, CD44, ANPEP, HNF4A, ALDH1A1, and SPP1 in our analysis16,20. Due to varying expression of these genes in the malignant cells of each case, we only included the genes that were expressed in at least 20 malignant cells using the function scv.pp.filter_and_normalize with parameters of min_shared_counts=20 and n_top_genes=2000 in the latent time analysis.
Analysis of non-malignant cells
We first performed PCA of non-malignant cells based on the top 2000 most variable genes determined using standardized means and variances. Then we applied UMAP analysis to the first 20 PCs for dimensional reduction and data visualization. We determined cell lineage of non-malignant cells based on the expression of lineage-specific marker genes to T cells, B cells, TAMs, CAFs, TECs, hepatocytes, and cholangiocytes. To ensure a pure population of T cells, we evaluated the total expression of CD3D, CD3E, and CD3G in each individual cell within the T-cell group, and further calculated the proportion of T cells positive for the three marker genes in each T-cell cluster determined using the Louvain algorithm, where the clusters with a proportion of <80% were removed. To measure the similarity of non-malignant cells from different tumor regions within each case, we calculated the correlation of each cell type between different regions. We first selected most variable genes of each cell type, and then applied the mean expression of the determined most variable genes of each tumor region within a case for correlation analysis (Fig. 3e and Supplementary Fig. 9a). In addition, for each cell type, we calculated the pair-wise correlation of cells within a tumor region to represent the intraregional heterogeneity, and then the average correlation value of each region was applied to measure the ratio of tumor border and tumor core (Supplementary Fig. 9b).
Immune features of the TME
We generated pseudo-bulk data based on the single-cell profiles of each case and applied the immune signature46, cytotoxic and exhaustion signatures47,48 to determine the overall immune scores of the TME by using the average expression of the genes from each signature. In addition, we calculated the ratio between cytotoxic and exhaustion immune scores to reflect the overall immune microenvironment of each case.
Communication of malignant cells and the TMEs
We resolved the communications between malignant cells and the TMEs by identifying ligand–receptor interactions using CellPhoneDB31,32, which predicts the significance of a ligand–receptor pair based on the mean expression of the ligand and receptor in two evaluated cell types while using random permutations as background signals. The original significant ligand–receptor pairs were determined with p value <0.05 to indicate their enrichment in the interacting pairs of cell types (Supplementary Fig. 11). We used all cells within a case for the original search and further evaluated the derived pairs in each tumor region. In this study, we applied more stringent criteria of p value <0.01 and mean expression of a ligand-receptor pair >0.5, in order to find biologically more meaningful pairs. We also removed the ligand-receptor pairs which occurred in the same interacting partners of all patients, given the reason that those pairs represent the common features in all patients and may not contribute to the patient-specific communication networks between the tumor and the TMEs. The derived ligand-receptor pairs were shown in Fig. 4a. Based on the ligand-receptor pairs identified in each individual case, we calculated regional stability. Specifically, we determined the proportion of the pairs that were found in all tumor regions or part of the tumor regions. We also determined the interactions between malignant cells and T-cell subtypes using similar strategies. Because the number of cells varies among cell types, we also evaluated the influence of the number of cells on the ligand-receptor interaction search. We performed random selection of T cells and TAMs, which represent two largest populations of non-malignant cells in our study. We then searched for ligand-receptor interactions based on the randomly selected cells. We repeated the process of random selection and interaction pair search for 10 times and found that an average of 92% ligand-receptor pairs can be matched to the original search, suggesting that the identified ligand-receptor pairs are very stable (Supplementary Fig. 13). To confirm our findings of the ligand-receptor interactions (i.e., SPP1-PTGER4 and LGALS9-SLC1A5) between tumor cells and TAMs, we applied CellChat33 to further identify cellular communications. Since the two interactions pairs are not listed in the CellChat database, we manually curated the two pairs. With the derived significant interaction pairs, we also applied a stringent criterion with a probability ≥0.01 similar to our analysis using CellPhoneDB to identify biologically more meaningful pairs.
Interaction networks of tumor and the TMEs for patient stratification
We used our NIH single-cell cohort to increase the power of studying the communications of malignant cells and the TMEs in liver cancer20. In this cohort, single-cell transcriptomes are available for a total of 46 tumor samples from 37 HCC or iCCA patients. The tumor samples with viable malignant cells were used for modeling. For patients with longitudinal tumor samples available, we applied the first sample with viable malignant cells in our analysis. We first searched for ligand-receptor interaction pairs between tumor cells and stromal/immune cells using CellPhoneDB31,32 and identified the pairs based on the criteria of p value <0.01 and mean expression of ligand-receptor pair >0.5, as used in our multiregional single-cell analysis. We then filtered the derived interaction pairs based on those found in the multiregional analysis, as those pairs have been selected using stringent criteria and demonstrated stable across multiple tumor regions. We also removed the interaction pairs that were occurred in less than three cases or more than ten cases, the same idea of selecting most variable genes for downstream data analysis considering their functional importance. Hierarchical clustering of the interaction network was performed using Pearson's correlation-based distance measurement and ward.D2 data agglomeration method. Two clusters of tumors were derived with different interaction patterns and were associated with distinct patient outcomes. In addition, we evaluated the impact of the number of malignant cells on ligand-receptor pair identification by randomly sampling malignant cells. We selected five cases with the highest number of malignant cells in Fig. 5a and randomly sampled 200, 100, 50, 20, and 10 malignant cells from each case for ligand-receptor pair identification. All the non-malignant cells were used in the analyses. We performed five times of random sampling for each setting and further compared the identified pairs with those in Fig. 5a to determine the accuracy of ligand-receptor pair detection.
To further evaluate the communication networks for patient stratification, we analyzed bulk transcriptomic data from three HCC cohorts including both tumor and paired non-tumor samples, i.e., LCI cohort (tumor, n = 239; non-tumor, n = 229), TCGA HCC cohort (tumor, n = 247, non-tumor, n = 48), and TIGER-LC HCC cohort (tumor, n = 56; non-tumor, n = 53). As we cannot directly measure cell-cell interactions using bulk transcriptome, we developed a strategy to evaluate the communication networks for tumor samples. Specially, we first calculated the mean expression of the ligand and receptor from an interaction pair for each individual tumor sample. Then, based on the median expression of the interaction pair across all tumor samples in a cohort, we determined the occurrence of a pair by comparing the expression value in each patient with that median value. Finally, we performed hierarchical clustering analysis of the communication networks using Pearson's correlation-based distance measurement and ward.D2 data agglomeration method. Clusters were determined using cutree function in R dendextend package. In the analysis of bulk transcriptome data, we used the same set of ligand-receptor pairs as those in the analysis of single-cell data (Fig. 5a). But the same ligand-receptor interaction pair in different pairs of interacting cell types from single-cell analysis was considered as one pair in bulk data, because cell types cannot be resolved in bulk transcriptome. We performed the same analysis for non-tumor samples in the three HCC cohorts as a control.
Single-molecule RNA in situ hybridization on tissue microarrays of HCC
Tissue microarrays (TMAs) were constructed using 1.0 millimeter (mm) cores from formalin fixed, paraffin embedded (FFPE) tissue for LCI and TIGER-LC HCC cohorts41. Tissues were mounted as TMAs with Superfrost PLUS Slides (Thermo Fisher Scientific, Cat # 5951PLUS). RNAScopeTM fluorescent 4-plex in situ hybridization49 was performed, by Advanced Cell Diagnostics (acdbio.com), for four genes on 5\(\,{{{{{\rm{\mu }}}}}}\)M TMA sections. Specially, paired double-Z oligonucleotide probes were designed against target RNA for LGALS9 (Cat# 1039898-C1), PTGER4 (Cat# 406778-C2), SPP1 (Cat# 420108-C3), and SLC1A5 (Cat# 427588-C4). The RNAscope LS Fluorescent Multiplex Reagent Kit (Cat# 322800) (Advanced Cell Diagnostics) was used with modified pretreatment conditions. FFPE human TMAs were incubated with Leica BOND Epitope Retrieval Solution 2 (ER2) at 95 °C for 20 min. Then RNAscope 2.5 LS Protease III was used for 15 min at 40 °C. Pretreatment conditions were optimized with the RNAscope LS 2.5 Hs-4-plex Positive Control Probe (Cat# 321808), specific to the housekeeping genes of POLR2A, PPIB, UBC, and HPRT1. Negative control background was evaluated using the RNAscope 4-plex LS Multiplex Negative Control Probe (Cat# 321838) specific to the bacterial dapB gene. A 3D Histech Panoramic Scan Digital Slide Scanner microscope with a 40x objective was used to generate fluorescent images. Images were analyzed using HALO® Image Analysis Platform (Indica Labs), where single-cell probe copies were quantified with resolved spatial coordinates. Samples with minimum tissues on the TMAs were excluded from downstream data analysis.
Colocalization of genes
To evaluate the spatial colocalization of a ligand–receptor pair-related two genes in each tumor core from the RNAscope assay, we calculated the Bhattacharyya coefficient (BC) of each tumor, which provides a measurement of the amount of overlap for two spatially distributed genes. Specifically, we generated partitions of each tumor core with tiles of 500 × 500 pixels (~70 × 70 μm). Then the probability of each gene in each tile was calculated. With the derived probabilities, we calculated the BC as:
$${BC}({{{{{\bf{p}}}}}}{{{{{\bf{1}}}}}},{{{{{\bf{p}}}}}}{{{{{\bf{2}}}}}})=\mathop{\sum }\limits_{i=1}^{n}\sqrt{p{1}_{i}p{2}_{i}}$$
where p1i and p2i represent the probability of the first and the second gene in the ith tile respectively, and n represents the total number of tiles from the partition of the space of a tumor sample. Because probabilities were applied for BC calculation, the derived BC value is between 0 and 1, with 1 representing a full overlap and 0 indicating no overlap. Samples with >1% cells positive for each of the two markers were used for this analysis. To measure the proportion of maker-positive tiles in a tumor sample, we randomly shuffled gene expression of the cells and counted the total number of filled tiles in a random situation, which was used as the denominator for evaluating the proportion of the original resolved marker-positive tiles. This process was repeated for 10 times for each tumor sample as a way of evaluating the spatial preference of a ligand-receptor pair-related two genes.
Analysis of the interaction pair-related downstream signaling
To evaluate the differences of malignant cells between tumors with the two pairs of LGALS9-SLC1A5 and SPP1-PTGER4 and those without the two pairs, we performed differential gene expression analysis of malignant cells between the two groups of tumors from our NIH single-cell cohort (Fig. 5a). With the derived genes, we performed gene set enrichment analysis using GSEA (version 4.0.3) to search for enriched pathways in each group. We further selected genes that were highly expressed in the with-pair group (gene set 1, average log[fold-change] > 1 and adjusted p value < 0.05) and the without-pair group (gene set 2, average log[fold-change] < −1 and adjusted p value < 0.05). The two sets of genes were then applied for PCA of bulk transcriptome data in LCI, TCGA HCC, and TIGER-LC HCC cohorts to demonstrate their roles as downstream surrogates of the two ligand-receptor interaction pairs. To further indicate the relationship of the two pairs and the two surrogate gene sets, we calculated the correlation between the geometric mean of the four genes (LGALS9, SLC1A5, SPP1, and PTGER4) and the ratio of the average gene expression of gene set 1 and gene set 2 for all the three HCC cohorts. We did similar analyses for TAMs to demonstrate the two interaction pairs-related downstream signaling in these cells.
Flowcytometry
Cryopreserved single cells were washed once and re-suspended in Stain Buffer (BD Pharmingen®, Cat. #554656). Cells were then stained with antibodies. For EpCAM staining, cells were stained using PE-conjugated anti-EpCAM antibody (Miltenyi Biotec, Cat.# 130-113-826) or its isotype control (Miltenyi Biotec, Cat. #130-113-762). For GPC3 staining, cells were stained indirectly with mouse monoclonal anti-GPC3 antibody50 (clone YP7) or its isotype control (BD Pharmingen®, Cat. #555746) together with APC-conjugated goat anti-mouse antibody (BD Pharmingen®, Cat. #550826), respectively. Antibodies were used at 1:100 dilution. Antibody-stained cells were washed with the Stain Buffer and then analyzed with BD FACSCanto II cell analyzer (Becton Dickinson).
Deparaffinization and rehydration were performed using xylene and citrate buffer ph 6.0 followed by distilled aqua. Washing steps with TBS-TX 0.03% and Trtion X-100 0.3% TBS were performed before and after incubation with H2O2. Next, blocking was done using 2.5% horse serum and 0.1% triton over 45 min followed by avidin/biotin blocking (Vectorlabs). Slides were then incubated with mouse anti-CD3 (monoclonal, Santa Cruz, Cat. #sc-59010, dilution 1:80) antibody overnight at 4 degrees Celsius. Next day, several washing steps were performed using TBS-TX, H2O2, and distilled water. Secondary antibodies using anti-mouse-biotin (streptavidin HRP, Rockland/Thermos Scientific, Cat. #N100, dilution 1:1000) were incubated for 90 min at room temperature followed by several washing steps. Streptavidin-HRP and later DAB (Vectorlabs) were each added in a separate step followed by washing steps. Finally, another hematoxylin staining was performed over 20 s. To quantitively determine CD3+ cells in different tumor regions of a patient, we counted the number of CD3+ positive cells per selected field of view (~0.2 mm). Five fields of view were randomly selected for each tumor region. This was performed for patient 1C, 2C, 3C, 2H, and 3H with tumor blocks available. For SPP1 staining, Osteopontin (OSP)/ SPP1 monoclonal antibody [Clone OSP/4589] (Rat, MyBioSource, Cat. #MBS4382252) was used at 1:100 dilution.
Paraffin sections were dried overnight on a heating plate at 40 °C. Deparaffinization and rehydration were performed using xylene and EtOH. Sections were first stained in hematoxylin for 3 min and were rinsed in water for three times. Then sections were stained in eosin for 2 min, followed by several dehydration steps starting with 70% EtOH and eventually 100% EtOH. Finally, sections were incubated in xylene for 5 min and covered with cover slides.
Plot generation
Violin plots, box plots, scatter plots, bar plots, and density plots were generated with ggplot and geom_violin, geom_boxplot, geom_point, geom_bar, geom_density functions in R ggplot2 package (version 3.3.5). Heatmap was generated using Heatmap function in R ComplexHeatmap package (version 2.2.0). Kaplan–Meier survival plots were generated with GraphPad Prism (version 8.4.3).
Reporting summary
Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.
The processed single-cell transcriptomic data generated in this study have been deposited in the Gene Expression Omnibus (GEO) and are available without restriction under accession number GSE189903. However, the NCI raw sequencing data are considered protected information and access to raw data is therefore restricted. The raw sequencing data are available in the NCBI dbGaP archive under accession number phs003117.v1.p1. Access via the NCI's dbGaP can be requested by qualified senior and principal investigators overseeing the research. The NCI's Data Access Committee reviews such requests and will make data available for up to 12 months. The publicly available datasets used in this study include a processed single-cell data of GSE151530, bulk transcriptomic data of GSE14520 and GSE76297, as well as the TCGA database (TCGA-LIHC). Source data are provided as a Source data file. Source data are provided with this paper.
Code availability
Code is available upon request. It should be directed to and will be fulfilled by the Lead Contact, Xin Wei Wang ([email protected]).
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We thank members of the Wang laboratory for critical discussions, the patients, families, and nurses for contribution to this study. We also thank Eytan Ruppin and Snorri Thorgeirsson for their critical evaluation of the manuscript. This work was supported by grants (Z01 BC 010877, Z01 BC 010876, Z01 BC 010313, and ZIA BC 011870) from the intramural research program of the Center for Cancer Research, National Cancer Institute of the United States. J.U.M. received funding from the Volkswagen Foundation (Lichtenberg Program) and the Wilhelm-Sander Foundation (2021.089.1).
Open Access funding provided by the National Institutes of Health (NIH).
These authors contributed equally: Lichun Ma, Sophia Heinrich.
Laboratory of Human Carcinogenesis, Center for Cancer Research, National Cancer Institute, Bethesda, MD, 20892, USA
Lichun Ma, Sophia Heinrich, Limin Wang, Subreen Khatib, Marshonna Forgues & Xin Wei Wang
Cancer Data Science Laboratory, Center for Cancer Research, National Cancer Institute, Bethesda, MD, 20892, USA
Lichun Ma
Clinic for Gastroenterology, Hepatology and Endocrinology, Hanover Medical School, Hanover, Germany
Sophia Heinrich
Department of Medicine I, Lichtenberg Research Group, Johannes Gutenberg University, Mainz, Germany
Friederike L. Keggenhoff, Roman Kloeckner & Jens U. Marquardt
Frederick National Laboratory for Cancer Research, Leidos Biomedical Research, Inc., Frederick, MD, 20701, USA
Laboratory of Pathology, Center for Cancer Research, National Cancer Institute, Bethesda, MD, 20892, USA
Stephen M. Hewitt
Surgical Oncology Program, Center for Cancer Research, National Cancer Institute, Bethesda, MD, 20892, USA
Areeba Saif & Jonathan M. Hernandez
Thoracic and GI Malignancies Branch, Center for Cancer Research, National Cancer Institute, Bethesda, MD, 20892, USA
Donna Mabry & Tim F. Greten
Institute for Interventional Radiology, University of Lübeck, Lübeck, Germany
Roman Kloeckner
Liver Cancer Program, Center for Cancer Research, National Cancer Institute, Bethesda, MD, 20892, USA
Tim F. Greten & Xin Wei Wang
Laboratory of Chemical Carcinogenesis, Chulabhorn Research Institute, Bangkok, 10210, Thailand
Jittiporn Chaisaingmongkol & Mathuros Ruchirawat
Center of Excellence on Environmental Health and Toxicology, Office of Higher Education Commission, Ministry of Education, Bangkok, 10400, Thailand
Department of Medicine I, University Medical Center, Lübeck, Germany
Jens U. Marquardt
Limin Wang
Friederike L. Keggenhoff
Subreen Khatib
Marshonna Forgues
Areeba Saif
Jonathan M. Hernandez
Donna Mabry
Tim F. Greten
Jittiporn Chaisaingmongkol
Mathuros Ruchirawat
Xin Wei Wang
L.M. and X.W.W. developed study concept; L.M. performed data analysis; S.H., L.W., F.K., S.K., M.F., S.M.H., A.S., J.M.H., D.M., R.K., T.F.G, J.C., M.R., and J.M. performed sample collection and processing; S.K. and M.K. performed additional analyses; L.M. and X.W.W. interpreted data; L.M., S.H., and X.W.W. wrote the manuscript. All authors read, edited, and approved the manuscript.
Correspondence to Xin Wei Wang.
Nature Communications thanks Qiang Gao, Manu Setty and the other anonymous reviewer(s) for their contribution to the peer review of this work. Peer review reports are available.
Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Peer review file
Description to Additional Supplementary Information
Supplementary Data1
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as 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. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
Ma, L., Heinrich, S., Wang, L. et al. Multiregional single-cell dissection of tumor and immune cells reveals stable lock-and-key features in liver cancer. Nat Commun 13, 7533 (2022). https://doi.org/10.1038/s41467-022-35291-5
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\begin{document}
\title[Toric configuration spaces]{Toric configuration spaces:\\ the bipermutahedron and beyond} \author{Navid Nabijou}
\begin{abstract} We establish faithful tropicalisation for point configurations on algebraic tori. Building on ideas from enumerative geometry, we introduce tropical scaffolds and use them to construct a system of modular fan structures on the tropical configuration spaces. The corresponding toric varieties provide modular compactifications of the algebraic configuration spaces, with boundary parametrising transverse configurations on tropical expansions. The rubber torus, used to identify equivalent configurations, plays a key role. As an application, we obtain a modular interpretation for the bipermutahedral variety. \end{abstract}
\maketitle \tableofcontents
\section*{Introduction} \noindent Fix a lattice $N\cong\mathbb{Z}^d$ and consider the associated algebraic torus and real vector space \begin{align*} T \colonequals N \otimes \mathbb{G}_{\mathrm{m}}, \qquad V \colonequals N \otimes \mathbb{R}. \end{align*} This paper explores the relationship between two moduli problems: \begin{align*} \text{Algebraic: \quad Labelled points $x_0,\ldots,x_n \in T$ up to simultaneous translation.}\\ \text{Tropical: \quad Labelled points $p_0,\ldots,p_n \in V$ up to simultaneous translation.} \end{align*} Points may coincide. Trivially, the corresponding moduli spaces are products modulo diagonals: \begin{align*} T[n] \colonequals T^{n+1}/T, \qquad V[n] \colonequals V^{n+1}/V. \end{align*} Consider $N[n] \colonequals N^{n+1}/N$. We have $T[n] = N[n]\otimes \mathbb{G}_{\mathrm{m}}$ and $V[n] = N[n] \otimes \mathbb{R}$. Consequently, there is a bijection between toric compactifications of $T[n]$ and complete fan structures on $V[n]$. In this paper we refine this into a correspondence: \begin{equation} \label{eqn: correspondence} \left\{ \text{\parbox{0.185\textwidth}{\center{modular \\ compactifications \\ of $T[n]$}}} \right\} \longleftrightarrow \left\{ \text{\parbox{0.185\textwidth}{\center{modular \\ fan structures \\ on $V[n]$}}} \right\}. \end{equation} This is an instance of faithful tropicalisation of moduli spaces \cite{ACP,CCUW,RanganathanSkeletons1}. A key step is identifying the appropriate class of modular fan structures.
The case $d=1$ recovers the identification between the Losev--Manin moduli space and the permutahedral variety. The cases $d \geq 2$ are new. The case $d=2$ in particular furnishes modular interpretations for the toric varieties associated to the bipermuthedron and the square of the permutahedron.
For simplicity we use the translation action to fix $x_0$ as the identity element $1 \in T$. We refer to this as the \textbf{anchor point}. This produces an identification \[ T[n]=T^n\] consisting of the moduli for the remaining points $x_1,\ldots,x_n$. We retain the anchor point $x_0=1 \in T$ to align with the existing literature.
\subsection{Expansions and scaffolds} Fix a valuation ring $R$ with fraction field $K$ and value group $\mathbb{R}$. Consider a family of point configurations in $T$ \[ \operatorname{Spec} K \to T[n].\]
Coordinatewise valuation produces a point configuration in $V$ \[ (x_1,\ldots,x_n) \in T[n](K) \quad \rightsquigarrow \quad (p_1,\ldots,p_n) \in V[n].\] This intertwines algebraic and tropical moduli. The valuation $p_i \in V$ records the asymptotics of the point $x_i \in T(K)$. Since $x_0=1 \in T$ we have $p_0=0 \in V$.
Insights from logarithmic enumerative geometry \cite{NishinouSiebert,MR20} suggest that as the points $x_1,\ldots,x_n$ approach infinity, the ambient variety $T$ should degenerate to a \textbf{tropical expansion}: a union of toric varieties meeting along toric strata. The points $x_1,\ldots,x_n$ will then limit to interior points on the irreducible components of this expansion.
A tropical expansion is encoded by a polyhedral decomposition of $V$, whose vertices index the irreducible components. To ensure that the point $x_i \in T(K)$ limits to the interior of an irreducible component, its valuation $p_i \in V$ must lie on a vertex of this polyhedral decomposition. The situation is summarised in Figure~\ref{fig: limit}.
Producing a modular compactification of $T[n]$ thus requires the following input data (Section~\ref{sec: scaffold}).
\begin{customdef}{X}[$\approx$ {Definition~\ref{def: scaffold}}] A \textbf{tropical scaffold} is the data of, for every point $(p_1,\ldots,p_n) \in V[n]$, a polyhedral decomposition of $V$ containing the points $p_0,p_1,\ldots,p_n \in V$ as vertices. \end{customdef} This data is required to vary linearly with the point $(p_1,\ldots,p_n)$ and is thus encoded as a complete fan $\Lambda$ on $V[n] \times V$ whose fibre over the point $(p_1,\ldots,p_n) \in V[n]$ gives the corresponding polyhedral decomposition of $V$.
\begin{figure}
\caption{A configuration on $T$ limiting to a transverse configuration on an expansion.}
\label{fig: limit}
\end{figure}
\subsection{Moduli} In Section~\ref{sec: moduli} we fix a tropical scaffold and construct the associated configuration space. Our results are summarised as follows.
\begin{customthm}{Y} To each tropical scaffold $\Lambda$ there is an associated configuration space $P_{\Lambda}$. This is a toric Deligne--Mumford stack compactifying $T[n]$, and its stacky fan is constructed explicitly from $\Lambda$ via universal weak semistable reduction (Definition~\ref{def: main construction}). It supports a universal tropical expansion and transverse point configuration (Section~\ref{sec: universal family}): \[ \begin{tikzcd} \mathcal{Y}_\Lambda \ar[d,"\uppi" left] \\ P_\Lambda. \ar[u,bend right, "{x_0,x_1,\ldots,x_n}" right] \end{tikzcd} \] Each locally-closed stratum of $P_\Lambda$ parametrises transverse point configurations on the associated tropical expansion, with two configurations identified if they differ by the action of the rubber torus (Theorem~\ref{thm: strata}). \end{customthm} The rubber torus originates in enumerative geometry \cite{CarocciNabijouRubber}. It is a crucial ingredient, necessary in order to obtain a separated moduli space. This perhaps explains why the above construction was not discovered earlier.
We build the fan of the configuration space $P_\Lambda$ explicitly from the scaffold $\Lambda$ (Section~\ref{sec: main construction}). Conceptually, it is obtained by stratifying $V[n]$ into regions on which the underlying polyhedral complex of the polyhedral decomposition of $V$ is constant. Practically, the construction $\Lambda \rightsquigarrow P_\Lambda$ can be viewed both as an instance of universal weak semistable reduction and as an example of a Chow quotient (Section~\ref{sec: semistable reduction}). This identifies the correct class of modular fan structures for \eqref{eqn: correspondence}.
While in general $P_\Lambda$ is a toric Deligne--Mumford stack, in most cases of interest it is a vanilla toric variety (but see Example~\ref{ex: stacky example}).
\subsection{Bipermutahedral variety} \label{sec: examples introduction} This project began as an attempt to find a modular interpretation for the bipermutahedral variety. The bipermutahedral fan parametrises labelled points in $\mathbb{R}^2$ up to translation, stratified according to bisequence \cite{ArdilaDenhamHuh,ArdilaBipermutahedron}.
The notion of bisequence strongly suggests a particular choice of tropical scaffold. We identify the associated configuration space with the bipermutahedral variety (Figure~\ref{fig: biperm scaffold introduction} and Section~\ref{sec: bipermutahedron}). A coarsening of this scaffold also produces the square of the permutahedral variety (Figure~\ref{fig: 2 perms scaffold introduction} and Section~\ref{sec: square of permutahedron}).
This produces modular interpretations for these spaces, resembling the identification of the permutahedral variety with the Losev--Manin moduli space. \begin{figure}
\caption{Bipermutahedral variety}
\label{fig: biperm scaffold introduction}
\caption{Square of permutahedral variety}
\label{fig: 2 perms scaffold introduction}
\caption{Scaffolds producing the bipermutahedral variety and the square of the permutahedral variety. The former is obtained from the latter by slicing with the supporting antidiagonal.}
\label{fig: dim 2 scaffolds introduction}
\end{figure}
\subsection{Prospects} The freedom to choose a tropical scaffold affords great flexibility. This is by design: the resulting class of configuration spaces is broad enough to include both the bipermutahedral variety and the square of the permutahedral variety.
Despite this flexibility, there is an open structural question. Tropical scaffolds form an inverse system: a refinement of a scaffold is again a scaffold. For $d=1$ and $n$ fixed, this inverse system has a unique minimal scaffold, whose associated configuration space is the permutahedral variety (Section~\ref{sec: permutahedron}). This singles out permutahedral varieties amongst all toric Deligne--Mumford stacks. For $d \geq 2$ there are multiple minimal scaffolds.
\begin{customquestion}{Z} \label{question: minimal} For $d \geq 2$, characterise the toric stacks arising as configuration spaces associated to minimal tropical scaffolds. \end{customquestion} We conjecture that all such toric stacks are in fact toric varieties. The scaffold giving rise to the square of the permutahedral variety is minimal, while the scaffold giving rise to the bipermutahedral variety is not. We suspect that the latter cannot be obtained from \emph{any} minimal scaffold, though at present we have no avenues for proving this.
\subsection{Proximate moduli}
\subsubsection{Toroidal embeddings} This paper may be adapted to study point configurations on toroidal embeddings $(X|D)$. Here $T$ is replaced by the interior $U \colonequals X \setminus D$ and $V$ is replaced by the tropicalisation $\Sigma \colonequals \Sigma(X|D)$. The universal tropical family is the projection \[ \begin{tikzcd}
\Sigma^n \times \Sigma \ar[d,"\uppi" left] \\
\Sigma^n \ar[u,bend right, "{p_1,\ldots,p_n}" right] \end{tikzcd} \] and there is no anchor point $p_0$ or translation action (compare with \eqref{eqn: universal family on vector spaces}). Tropical scaffolds $\Lambda$ are refinements of $\Sigma^n \times \Sigma$ and universal weak semistable reduction \cite[Section~3.2]{MolchoSS} produces $\Pi_\Lambda$.
This is not quite a generalisation of our setup, as it requires an initial choice of compactification $X$ of $U$. Consequently the configuration space is always a modification of $X^n$. We focus on tori in this paper in order to emphasise the pleasant combinatorics.
\subsubsection{Very affine varieties} This paper may also be adapted to study point configurations on a closed subvariety $U$ of an algebraic torus. Here $T$ is replaced by $U$ and $V$ is replaced by $\operatorname{trop}U \subseteq V$. Each tropical scaffold should produce a stacky fan structure on $(\operatorname{trop}U)^n = \operatorname{trop}(U^n)$. This will require a mild generalisation of the stacky Chow quotient of \cite[Section~3.2]{AscherMolcho}
The set $\operatorname{trop} U$ admits a fan structure \cite[Theorem~2.2.5]{EKL}. If $U$ is sch\"{o}n then a choice of fan structure produces a toroidal embedding $(\overline{U}\ | \ \overline{U} \setminus U)$ which reduces this case to the previous one \cite[Theorem~1.4]{Tevelev}. However there is no minimal choice of fan structure on $\operatorname{trop} U$.
\subsubsection{Di Rocco--Schaffler} While preparing this paper we discovered \cite{DiRoccoSchaffler}. Building on \cite{GerritzenPiwek,KTChow,HKTHyperplane,AscherMolcho,SchafflerTevelev} the authors study toric configuration spaces of a similar flavour to ours: the configuration space is also constructed as a Chow quotient, and the boundary also parametrises configurations on tropical expansions.
The difference lies in the input data. Instead of a tropical scaffold, the authors begin with a complete fan $\Sigma$ on $N$. In our language $\Sigma$ gives rise to a canonical tropical scaffold: for each $(p_1,\ldots,p_n) \in V[n]$ we overlay the translates of $\Sigma$ at the points $p_0,p_1,\ldots,p_n$ and take the minimal common refinement (see \cite[Definition~3.4]{DiRoccoSchaffler}). Under the identification $V[n] \times V = V^n \times V = V^{n+1}$ this gives $\Lambda = \Sigma^{n+1}$ and the associated configuration space is the Chow quotient by the diagonal. The configuration spaces studied in \cite{DiRoccoSchaffler} thus constitute an important special case of our construction, and the authors establish many interesting results in this setting.
Applying their construction with $\Sigma=\Sigma(\mathbb{P}^1 \times \mathbb{P}^1)$ gives the scaffold producing the square of the permutahedral variety (Section~\ref{sec: square of permutahedron}). On the other hand no choice of $\Sigma$ gives the scaffold producing the bipermutahedral variety (Section~\ref{sec: bipermutahedron}).
As far as we can tell, the rubber torus does not appear in \cite{DiRoccoSchaffler}.
\subsubsection{Distinct points} Consider configurations of distinct labelled points on a smooth projective variety $X$. The resulting moduli space is $X^n$ minus all diagonals. Fulton--MacPherson construct a beautiful modular compactification of this space \cite{FM}. As points collide, $X$ is replaced by the degeneration to the normal cone of the collision point. The boundary of the moduli space parametrises transverse point configurations on such degenerations, up to automorphisms covering the identity on $X$. This is reminiscent of tropical expansions and rubber tori, though Fulton--MacPherson's automorphism groups are typically not abelian.
Upcoming work of Mok studies configurations of distinct labelled points in the interior of a simple normal crossings pair. The resulting moduli spaces synthesise Fulton--MacPherson degenerations and tropical expansions.
\subsubsection{Unlabelled points} Another variant would be to consider configurations of unlabelled, possibly coincident points. The results of this paper would need to be modified to incorporate $S_n$-invariant structures. Special care must be taken with automorphisms, and cone stacks will likely play a role \cite{CCUW}. This would produce a logarithmic analogue of the punctual Chow variety.
The natural next step would be to study the Hilbert--Chow morphism to the logarithmic punctual Hilbert scheme. The latter was introduced in \cite{MR20} and studied further in \cite{KennedyHuntQuot}.
\subsection*{Funding} The author was partially supported by the Herchel Smith Fund.
\section{Scaffolds}\label{sec: scaffold} \noindent Recall that $V[n]$ parametrises labelled points $p_0,p_1,\ldots,p_n \in V$ up to simultaneous translation. We view this as a vector space with integral structure $N[n] \subseteq V[n]$. The rigidification $x_0=1 \in T$ corresponds to the rigidification $p_0=0 \in V$ and furnishes an isomorphism $V[n]=V^n$. The moduli space $V[n]$ supports a universal family \begin{equation} \label{eqn: universal family on vector spaces} \begin{tikzcd} V[n] \times V \ar[d,"\uppi" left] \\ V[n] \ar[u,bend right,"{p_0,p_1,\ldots,p_n}" right] \end{tikzcd} \end{equation} where, letting $\uppi_i \colon V[n] = V^n \to V$ denote the $i$th projection, the universal sections $p_i$ are given by \[ p_0 = \Id \times 0,\ p_1 = \Id \times \uppi_1,\ \ldots\ ,\ p_n = \Id \times \uppi_n. \]
\begin{definition} \label{def: scaffold} A \textbf{tropical scaffold} $\Lambda$ is a complete fan on $V[n] \times V$ such that the image $p_i(V[n])$ of every section $p_i$ is a union of cones in $\Lambda$.\end{definition}
For every $p \in V[n]$, intersection with the cones of $\Lambda$ gives a polyhedral decomposition of the fibre $\uppi^{-1}(p)=V$ which we denote $\Lambda_p$. The condition that the image of every section is a union of cones ensures that every point $p_i = p_i(p) \in \uppi^{-1}(p)$ belongs to a vertex of $\Lambda_p$.
\begin{remark} Tropical scaffolds are primitive cousins of the universal tropical expansions constructed for logarithmic Donaldson--Thomas theory \cite[Section~3]{MR20}. In that setting it is profitable to permit non-complete expansions. We insist on completeness in order to apply results on universal weak semistable reduction for toric stacks (Section~\ref{sec: semistable reduction}). A benefit is that the construction of the moduli space from the scaffold is canonical. \end{remark}
\section{Moduli} \label{sec: moduli}
\noindent Given a tropical scaffold $\Lambda$ we construct the associated \textbf{configuration space} $P_\Lambda$. This is a toric Deligne--Mumford stack compactifying $T[n]$. We construct the corresponding stacky fan $\Pi_\Lambda$ on $V[n]$ explicitly from the scaffold $\Lambda$. Conceptually this is achieved by stratifying $V[n]$ into regions over which the polyhedral complex $\Lambda_p$ remains constant.
Practically, the construction $\Lambda \rightsquigarrow \Pi_\Lambda$ can be viewed either as an instance of universal weak semistable reduction \cite{AbramovichKaru,MolchoSS} or as a stacky Chow quotient \cite{KapranovSturmfelsZelevinsky,AscherMolcho}. Indeed, the fan $\Pi_\Lambda$ should be such that the projection $V[n] \times V \to V[n]$ is a map of fans $\Lambda \to \Pi_\Lambda$. The corresponding toric morphism is the universal tropical expansion, so in particular should be equidimensional with reduced fibres. We construct $\Pi_\Lambda$ as the coarsest stacky fan on $V[n]$ satisfying these conditions, with the caveat that we need to allow $\Lambda$ to be replaced by a refinement (Section~\ref{sec: main construction})
After constructing $P_\Lambda$ we establish its key properties: we construct the universal family and point configuration (Section~\ref{sec: universal family}) and verify that boundary strata in $P_\Lambda$ parametrise point configurations, up to the rubber action, on the tropical expansions encoded by the scaffold (Section~\ref{sec: strata}).
While in general $P_\Lambda$ is a Deligne--Mumford stack, in most cases of interest it is a variety. The relaxed reader may therefore ignore the stacky subtleties in what follows.
\subsection{Preliminaries} We recall the basics of stacky fans and semistable reduction.
\subsubsection{Stacky fans} \label{sec: stacky fans} Toric Deligne--Mumford stacks and their associated fans appear in the literature in various incarnations \cite{BorisovChenSmith, FantechiMannNironi, GeraschenkoSatriano, GillamMolchoStackyFans}. For us, a \textbf{stacky fan} consists of an ordinary fan $\Sigma$ on a lattice $N$ together with a collection of finite-index sublattices \[ L_\upsigma \subseteq N_\upsigma \colonequals (\upsigma \otimes \mathbb{R}) \cap N \] for every $\upsigma \in \Sigma$, compatible under face inclusions. Locally, the corresponding toric stack has isotropy group $N_\upsigma/L_\upsigma$ along its deepest stratum. These were introduced in \cite{Tyomkin} and studied in \cite{GillamMolchoStackyFans} where they take the name ``lattice KM fans'' after \cite{KottkeMelrose}. A quick introduction can be found in \cite[Section~2.3]{MolchoSS}.
Stacky fans in this sense correspond to toric Deligne--Mumford stacks, possibly singular but with generically trivial isotropy. These are precisely the toric stacks which arise in our study.
\subsubsection{Weak semistability} A map of stacky fans \[ \uppi \colon (N_1,\Sigma_1,\{L_{\upsigma_1}\}) \to (N_2,\Sigma_2,\{L_{\upsigma_2}\})\] is \textbf{weakly semistable} if for all $\sigma_1 \in \Sigma_1$ we have \[ \uppi(\upsigma_1) \in \Sigma_2 \quad \text{and} \quad \uppi(L_{\upsigma_1}) = L_{\uppi(\upsigma_1)}.\] If the lattice map $\uppi \colon N_1 \to N_2$ is surjective, then $\uppi$ is weakly semistable if and only if the corresponding morphism of toric stacks is equidimensional with reduced fibres; see \cite[Sections~4 and 5]{AbramovichKaru} for the case of toric varieties, and \cite[Proposition~3.1.1]{GillamMolchoStackyFans} for the extension to toric stacks. Given a weakly semistable map of stacky fans, the corresponding morphism of toric stacks is always representable; this follows immediately from the fan criterion for representability \cite[Theorem~3.11.2]{GillamMolchoStackyFans}.
\subsubsection{Stacky Chow quotients and universal weak semistable reduction} \label{sec: semistable reduction} In \cite{AscherMolcho} Ascher--Molcho construct a stacky enhancement of the Chow quotient \cite{KapranovSturmfelsZelevinsky}. We recast this as an instance of universal weak semistable reduction \cite{MolchoSS} for maps to the logarithmic torus. We claim no originality here: we suspect that this interpretation was already known to the authors.
Fix lattices $N_1$ and $N_2$, a complete fan structure $\Sigma_1$ on $N_1$, and a surjective map \[ \uppi \colon N_1 \to N_2.\] Crucially, we do not begin with any fan structure on $N_2$. For the intended application, we will have $N_1=N[n] \times N$ and $N_2 = N[n]$ with $\uppi$ the projection and $\Sigma_1=\Lambda$ a tropical scaffold. It is crucial that $\Sigma_1$ is complete.
\begin{remark} We write the above data as $\uppi \colon (N_1,\Sigma_1) \to N_2$ and interpret the codomain as a ``fan'' consisting of a single cone constituting the entire lattice. If $X_1$ is the toric variety corresponding to $(N_1,\Sigma_1)$ then $\uppi$ corresponds to a logarithmic morphism $X_1 \to N_2 \otimes \mathbb{G}_{\mathrm{log}}$. See \cite{RW,MW} for background on logarithmic and tropical tori, and \cite[Section~1]{KennedyHuntQuot} for a treatment of cone stacks with non-convex charts. \end{remark}
\begin{definition} The category $\mathcal{F}$ (for \textbf{flattenings}) has objects consisting of the following data: \begin{enumerate} \item $(N_2,\Sigma_2^\prime,\{L_{\upsigma_2}^\prime\})$ a stacky fan on $N_2$.
\item $(N_1,\Sigma_1^\prime,\{L_{\upsigma_1}^\prime\})$ a stacky fan refining $(N_1,\Sigma_1)$.\footnote{This means that $\Sigma_1^\prime$ is a subdivision of $\Sigma_1$. There is no condition on the sublattices $L_{\upsigma_1}^\prime$. Geometrically $\Sigma_1^\prime$ implements a toric blowup and the $L_{\upsigma_1}^\prime$ implement root constructions.} \end{enumerate} This data is subject to the following conditions: \begin{enumerate}
\item The map $N_1 \to N_2$ is a map of stacky fans $(N_1,\Sigma_1^\prime,\{L_{\upsigma_1}^\prime\}) \to (N_2,\Sigma_2^\prime,\{L_{\upsigma_2}^\prime\})$.
\item This map is weakly semistable. \end{enumerate} Necessarily, the fan $\Sigma_2^\prime$ is complete. Morphisms in $\mathcal{F}$ consist of fan maps which are the identity on the underlying lattices: \begin{center}\begin{tikzcd} (N_1, \Sigma_1^{\prime\prime}, \{L_{\upsigma_1}^{\prime\prime}\}) \ar[r] \ar[d] & (N_1, \Sigma_1^{\prime}, \{L_{\upsigma_1}^{\prime}\}) \ar[r] \ar[d] & (N_1, \Sigma_1) \ar[d] \\ (N_2, \Sigma_2^{\prime\prime}, \{L_{\upsigma_2}^{\prime\prime}\}) \ar[r] & (N_2, \Sigma_2^{\prime}, \{L_{\upsigma_2}^{\prime}\}) \ar[r] & N_2. \end{tikzcd}\end{center} \end{definition}
\begin{theorem}[\!{\cite[Theorem~3.9]{AscherMolcho}}] \label{thm: semistable reduction} The category $\mathcal{F}$ has a terminal object. \end{theorem}
The terminal object $(N_2,\Sigma_2^\prime,\{L_{\upsigma_2}^\prime\})$ is the stacky Chow quotient \cite[Section~3.2]{AscherMolcho} of $(N_1,\Sigma_1)$ by the saturated sublattice $\operatorname{Ker} \uppi \subseteq N_1$. The fan $\Sigma_2^\prime$ is obtained by overlaying the images $\uppi(\upsigma_1) \subseteq N_{2} \otimes \mathbb{R}$ for every cone $\upsigma_1 \in \Sigma_1$. The lattices $L_{\upsigma_2}^\prime$ are obtained by intersecting the images of the lattices $N_{\upsigma_1}$. Finally, pullback produces the stacky fan $(N_1,\Sigma_1^\prime,\{L_{\upsigma_1}^\prime\})$ refining $(N_1,\Sigma_1)$.
\subsection{Main construction} \label{sec: main construction}
\begin{definition} \label{def: main construction} Given a tropical scaffold $\Lambda$ the associated \textbf{configuration fan} $\Pi_\Lambda$ is the stacky fan obtained by applying Theorem~\ref{thm: semistable reduction} to the projection \[ \uppi \colon (N[n] \times N,\Lambda) \to N[n].\] The associated \textbf{configuration space} $P_\Lambda$ is the proper toric Deligne--Mumford stack corresponding to $\Pi_\Lambda$. This application of Theorem~\ref{thm: semistable reduction} also produces a stacky refinement of $\Lambda$ and from now on we replace $\Lambda$ by this refinement. \end{definition} While in general $P_\Lambda$ is a Deligne--Mumford stack, in cases of interest it is usually a variety (see Sections~\ref{sec: permutahedron} and \ref{sec: dim 2}). The necessity of stacky structures in general is explained in Example~\ref{ex: stacky example}.
\subsection{Universal family} \label{sec: universal family} Let $\mathcal{Y}_\Lambda$ denote the toric stack corresponding to $\Lambda$. The map $\Lambda \to \Pi_\Lambda$ is weakly semistable (Theorem~\ref{thm: semistable reduction}) and hence the corresponding morphism of proper toric stacks \[ \mathcal{Y}_\Lambda \to P_\Lambda \] is representable with equidimensional and reduced fibres. We refer to it as the \textbf{universal tropical expansion}. We now turn to the universal point configuration. Consider the section $p_i \colon V[n] \to V[n] \times V$ defined in Section~\ref{sec: scaffold}.
\begin{lemma} The section $p_i$ is a weakly semistable map of stacky fans $\Pi_\Lambda \to \Lambda$.
\end{lemma} \begin{proof} By assumption the image $p_i(V[n])$ is a union of cones of $\Lambda$ (this does not change when we refine $\Lambda$ in Definition~\ref{def: main construction}). We therefore view $p_i(V[n])$ as a subfan of $\Lambda$. The map $\uppi$ restricts to an isomorphism of vector spaces $\uppi \colon p_i(V[n]) \to V[n]$. Since $\uppi$ is weakly semistable, this restriction is an isomorphism of fans. In particular its inverse $p_i$ is also a map of fans, and maps every cone surjectively (in fact, isomorphically) onto another cone. It is straightforward to check that $p_i$ respects the stacky sublattices. \end{proof}
Consequently, we obtain toric morphisms $x_i \colon P_\Lambda \to \mathcal{Y}_\Lambda$ which are sections of the universal tropical expansion. Together these produce the \textbf{universal point configuration} (compare with \eqref{eqn: universal family on vector spaces}) \begin{equation} \label{eqn: universal family} \begin{tikzcd} \mathcal{Y}_\Lambda \ar[d,"\uppi" left] \\ P_\Lambda. \ar[u,bend right,"{x_0,x_1,\ldots,x_n}" right] \end{tikzcd} \end{equation} Since $p_i$ is weakly semistable, $x_i$ is torically transverse. It is not flat because the underlying lattice map is not surjective.
\subsection{Expansion geometry} \label{sec: expansion geometry} As in Section~\ref{sec: scaffold} we view the map $\Lambda \to \Pi_\Lambda$ as a family of polyhedral decompositions of $V$ parametrised by $p \in |\Pi_\Lambda| = V[n]$. The finite edges of $\Lambda_p$ are metrised by the choice of $p$, and the isomorphism class of the polyhedral complex $\Lambda_p$ is constant on the relative interior of every cone $\uprho \in \Pi_\Lambda$. Specialising to a face of $\uprho$ has the effect of setting certain edge lengths to zero, collapsing $\Lambda_p$ to a simpler polyhedral complex.
Fix a cone $\uprho \in \Pi_\Lambda$ and let $P_{\Lambda, \uprho} \subseteq P_\Lambda$ denote the corresponding locally-closed torus orbit. Let \[ Y_\uprho \subseteq \mathcal{Y}_\Lambda \] denote the fibre of $\uppi$ over the distinguished point of $P_{\Lambda, \uprho}$. The fibre of $\uppi$ over any other point of $P_{\Lambda, \uprho}$ is non-canonically isomorphic to $Y_\uprho$. Consider the polyhedral complex \[ \Lambda_\uprho \colonequals \Lambda_p \] for $p$ any point in the relative interior of $\uprho$. Polyhedra of $\Lambda_\uprho$ correspond to cones $\uplambda \in \Lambda$ with $\uppi(\uplambda) = \uprho$ and so there is an inclusion-reversing correspondence between the polyhedra of $\Lambda_\uprho$ and the strata of $Y_\uprho$. In particular the vertices of $\Lambda_\uprho$ index the irreducible components of $Y_\uprho$. Each such irreducible component is a toric variety with dense torus $T$. Its fan is obtained by zooming in to the corresponding vertex of $\Lambda_\uprho$.
Each marking $p_i$ is supported on a vertex $v_i \in \Lambda_\uprho$. This corresponds to the unique cone of $\Lambda$ which is both contained in $p_i(V[n])$ and mapped isomorphically onto $\uprho$ via $\uppi$. The corresponding irreducible component $Y_{v_i} \subseteq Y_\uprho$ is a toric variety and we denote its dense torus by $T_{v_i}$ so that \[ T_{v_i} \subseteq Y_{v_i} \subseteq Y_\uprho.\] There is a natural identification $T_{v_i}=T$. When the universal section $x_i \colon P_\Lambda \to \mathcal{Y}_\Lambda$ is restricted to $P_{\Lambda, \uprho}$ it factors through $T_{v_i}$. For more on the geometry of tropical expansions (in the general context of toroidal embeddings) see \cite{CarocciNabijouExpansions}.
\subsection{Rubber action and strata} \label{sec: strata} In \cite{CarocciNabijouRubber} a canonical torus action is defined for any tropical expansion over a cone $\uprho$. It is referred to as the rubber action; the terminology comes from enumerative geometry. We recall the rubber action in our context.
Fix a cone $\uprho \in \Pi_\Lambda$ and a vertex $v \in \Lambda_\uprho$ indexing an irreducible component $Y_v \subseteq Y_\uprho$. There is a linear tropical position map \[ \varphi_v \colon \uprho \to V \] which records the position of $v$ in terms of the tropical parameters in $\uprho$. We define the \textbf{rubber torus} \[ T_\uprho \colonequals L_\uprho \otimes \mathbb{G}_{\mathrm{m}} \] where $L_\uprho \subseteq N_\uprho$ is the finite-index sublattice encoded by the stacky fan (Section~\ref{sec: stacky fans}). The position map $\varphi_v$ corresponds to a lattice map $N_\uprho \to N$ which we restrict to $L_\uprho$ and tensor by $\mathbb{G}_{\mathrm{m}}$ to produce a homomorphism $T_\uprho \to T$. Since $Y_v$ is a toric variety with dense torus $T_v=T$ we obtain an action \[ T_\uprho \curvearrowright Y_v.\] These actions glue to a global action $T_\uprho \curvearrowright Y_\uprho$ referred to as the \textbf{rubber action} \cite[Theorem~1.8]{CarocciNabijouRubber}. Examples are given in \cite[Section~4]{CarocciNabijouRubber}.
Recall from Section~\ref{sec: expansion geometry} that on the locally-closed stratum $P_{\Lambda, \uprho}$ the marking $x_i$ factors through the dense torus $T_{v_i} \subseteq Y_{v_i}$. The rubber action arises from a homomorphism $T_\uprho \to T_{v_i}=T$ and hence preserves $T_{v_i}$.
\begin{theorem} \label{thm: strata} The locally-closed stratum $P_{\Lambda, \uprho}$ is isomorphic to the moduli space of point configurations \begin{equation} \label{eqn: point configuration stratum} (x_1,\ldots,x_n) \in T_{v_1} \times \cdots \times T_{v_n} \end{equation} considered up to the diagonal action of the rubber torus $T_{\uprho}$. \end{theorem} \begin{proof} For each $v_i$ the tropical position map $\varphi_{v_i} \colon L_\uprho \to N$ records the position of $p_i$. It is obtained as the composite \[ L_\uprho \hookrightarrow N_\uprho \hookrightarrow N[n] \xrightarrow{p_i} N[n] \times N \to N\] where $p_i \colon N[n] \to N[n] \times N$ is the section considered in Section~\ref{sec: scaffold}. Since $p_i = \Id \times \uppi_i$ we can identify the above composite with \[ L_\uprho \hookrightarrow N_\uprho \hookrightarrow N[n] = N^n \xrightarrow{\uppi_i} N.\] It follows that the product of tropical position maps $\varphi_{v_1} \times \cdots \times \varphi_{v_n}$ recovers the lattice inclusion $L_{\uprho} \subseteq N[n]$. The moduli space of point configurations \eqref{eqn: point configuration stratum} is therefore the stack quotient \[ [T[n]/T_\uprho]. \] This is precisely the description of the torus orbit $P_{\Lambda, \uprho} \subseteq P_\Lambda$ corresponding to the cone $\uprho \in \Pi_\Lambda$. \end{proof}
\begin{remark} The anchor point $x_0= 1 \in T$ remains stationary as the target expands. The corresponding vertex $v_0 \in \Lambda_\uprho$ is the origin in $V$ and $x_0$ is the identity element of the torus $T=T_{v_0} \subseteq Y_{v_0}$. \end{remark}
\begin{remark} Consider the degenerate case $\uprho=0$. The fibre $Y_0$ is the generic fibre of $\mathcal{Y}_\Lambda \to P_\Lambda$. It is the complete toric variety corresponding to the asymptotic fan of $\Lambda$ \cite[Section~3]{NishinouSiebert}. There is no rubber torus. Since a point configuration on the interior of $Y_0$ is nothing more than a point configuration on $T$, we identify the interior of $P_\Lambda$ with the moduli space $T[n]$. \end{remark}
\begin{example}\label{ex: stacky example} We provide an example demonstrating the necessity of stacky structures in general. Set $d=n=1$ and coordinatise the ambient space as $V=\mathbb{R}_x$. The position $a_1$ of the unanchored point $p_1$ coordinatises the moduli space $V[1] = \mathbb{R}_{a_1}$ and the projection is \[ V[1] \times V = \mathbb{R}^2_{a_1 x} \xrightarrow{\uppi} \mathbb{R}_{a_1} = V[1].\] We consider the following scaffold $\Lambda$: \[ \begin{tikzpicture}[scale=1.4]
\filldraw [gray] (0, 3) circle (1pt); \filldraw [gray] (0.5, 3) circle (1pt); \filldraw [gray] (1, 3) circle (1pt); \filldraw [gray] (1.5, 3) circle (1pt); \filldraw [gray] (2., 3) circle (1pt); \filldraw [gray] (2.5, 3) circle (1pt); \filldraw [gray] (3, 3) circle (1pt);
\filldraw [gray] (0, 2.5) circle (1pt); \filldraw [gray] (0.5, 2.5) circle (1pt); \filldraw [gray] (1, 2.5) circle (1pt); \filldraw [gray] (1.5, 2.5) circle (1pt); \filldraw [gray] (2., 2.5) circle (1pt); \filldraw [gray] (2.5, 2.5) circle (1pt); \filldraw [gray] (3, 2.5) circle (1pt);
\filldraw [gray] (0, 2) circle (1pt); \filldraw [gray] (0.5, 2) circle (1pt); \filldraw [gray] (1, 2) circle (1pt); \filldraw [gray] (1.5, 2) circle (1pt); \filldraw [gray] (2, 2) circle (1pt); \filldraw [gray] (2.5, 2) circle (1pt); \filldraw [gray] (3, 2) circle (1pt);
\filldraw [gray] (0, 1.5) circle (1pt); \filldraw [gray] (0.5, 1.5) circle (1pt); \filldraw [gray] (1, 1.5) circle (1pt); \filldraw [gray] (1.5, 1.5) circle (1pt); \filldraw [gray] (2., 1.5) circle (1pt); \filldraw [gray] (2.5, 1.5) circle (1pt); \filldraw [gray] (3, 1.5) circle (1pt);
\filldraw [gray] (0, 1) circle (1pt); \filldraw [gray] (0.5, 1) circle (1pt); \filldraw [gray] (1, 1) circle (1pt); \filldraw [gray] (1.5, 1) circle (1pt); \filldraw [gray] (2., 1) circle (1pt); \filldraw [gray] (2.5, 1) circle (1pt); \filldraw [gray] (3, 1) circle (1pt);
\draw [dashed,gray] (-0.25,3.25) -- (3.25,3.25) -- (3.25,0.75) -- (-0.25,0.75) -- (-0.25,3.25);
\draw [->] (-0.5,0.5) -- (-0.5,1); \draw (-0.5,1) node[above]{\small$x$}; \draw [->] (-0.5,0.5) -- (0,0.5); \draw (0,0.5) node[right]{\small$a_1$};
\draw [thick] (-0.25,2) -- (3.25,2); \draw (3.25,2) node[right]{\small$x=0$};
\draw [thick] (0.25,0.75) --(2.75,3.25); \draw (2.75,3.25) node[above]{\small$x=a_1$};
\draw [thick] (-0.25, 1.125) -- (3.25,2.875); \draw (3.25,2.875) node[right]{\small$x=a_1/2$};
\end{tikzpicture} \] For fixed $a_1$ the corresponding vertical slice of $\Lambda$ gives a polyhedral decomposition of $V=\mathbb{R}_x$. For $a_1 \geq 0$ this is \[ \begin{tikzpicture}[scale=1.6];
\draw[<->,gray] (-2,0) -- (2,0);
\draw (0,0) node[above]{\small$p_0$}; \filldraw[black] (0,0) circle (1pt);
\filldraw[black] (0.5,0) circle (1pt);
\filldraw[black] (1,0) circle (1pt); \draw (1,0) node[above]{\small$p_1$};
\draw (0,-0.1) -- (0,-0.15) -- (0.5,-0.15) -- (0.5,-0.1); \draw (0.25,-0.075) node[below]{\tiny$a_1/2$};
\draw (0,-0.2) -- (0,-0.4) -- (1,-0.4) -- (1,-0.2); \draw (0.5,-0.35) node[below]{\tiny$a_1$};
\end{tikzpicture} \] and similarly for $a_1 \leq 0$. The empty vertex between $p_0$ and $p_1$ corresponds to the ray $x=a_1/2$. The universal weak semistable reduction algorithm with respect to the horizontal projection produces the following stacky fan $\Pi_\Lambda$ on $V[1]=\mathbb{R}_{a_1}$ \[ \begin{tikzpicture}[scale=1.6];
\draw[<->] (-3,0) -- (3,0);
\draw (0,0.05) node[above]{\small$0$}; \filldraw[black] (0,0) circle (1pt); \filldraw[black] (-0.5,0) circle (1pt); \filldraw[black] (-1,0) circle (1pt); \filldraw[black] (-1.5,0) circle (1pt); \filldraw[black] (-2,0) circle (1pt); \filldraw[black] (-2.5,0) circle (1pt); \filldraw[black] (0.5,0) circle (1pt); \filldraw[black] (1,0) circle (1pt); \filldraw[black] (1.5,0) circle (1pt); \filldraw[black] (2,0) circle (1pt); \filldraw[black] (2.5,0) circle (1pt);
\draw (0,0) circle[radius=2pt]; \draw (1,0) circle[radius=2pt]; \draw (2,0) circle[radius=2pt]; \draw (-1,0) circle[radius=2pt]; \draw (-2,0) circle[radius=2pt];
\end{tikzpicture} \] The stacky sublattices $L_\uprho$ are indicated with open circles. The corresponding toric stack $P_\Lambda$ is the square root stack \cite{CadmanRoot} of $\mathbb{P}^1$ at $0$ and $\infty$. It has two points with $\mu_2$ isotropy. Theorem~\ref{thm: strata} provides a modular interpretation for this isotropy. As $x_1$ approaches infinity the target breaks into a chain of three projective lines \[ \begin{tikzpicture}
\draw (0,0) to [out=40,in=180] (1,0.4) to[out=0,in=140] (2,0); \draw[fill=black] (1,0.4) circle[radius=2pt]; \draw (1,0.4) node[above]{\small$x_0$};
\draw (1.6,0) to [out=40,in=180] (2.6,0.4) to[out=0,in=140] (3.6,0);
\draw (3.2,0) to [out=40,in=180] (4.2,0.4) to[out=0,in=140] (5.2,0); \draw[fill=black] (4.2,0.4) circle[radius=2pt]; \draw (4.2,0.4) node[above]{\small$x_1$};
\end{tikzpicture} \] The stacky sublattice is coordinatised by $a_1/2$ and therefore the tropical position map $\mathbb{Z} \to \mathbb{Z}$ for the vertex $v_1$ is multiplication by $2$. It follows that the rubber action on the component containing $x_1$ has generic stabiliser $\mu_2$. This endows the point configuration with a nontrivial automorphism.
The above scaffold is clearly not minimal: see Question~\ref{question: minimal}. \end{example}
\section{Dimension one: permutahedral variety} \label{sec: permutahedron}
\noindent Set $d=1$ so that $V \cong \mathbb{R}$. A point configuration $(p_1,\ldots,p_n) \in V[n]$ determines a canonical polyhedral decomposition of $V$: \[ \begin{tikzpicture}[scale=1.6];
\draw[<->] (-2,0) -- (2,0);
\draw (0,0) node[above]{\small$p_0$}; \filldraw[black] (0,0) circle (1pt);
\filldraw[black] (1,0) circle (1pt); \draw (1,0) node[above]{\small$p_1$};
\filldraw[black] (-1.5,0) circle(1pt); \draw (-1.5,0) node[above]{\small$p_2$};
\filldraw[black] (1.5,0) circle(1pt); \draw (1.5,0) node[above]{\small$p_3$}; \end{tikzpicture} \] Consequently there is a unique minimal tropical scaffold $\Lambda_0$. We show that the associated configuration fan $\Pi_{\Lambda_0}$ is the permutahedral fan. This recovers the identification of the Losev--Manin moduli space with the permutahedral variety \cite[Section~2.6]{LosevManin}.
\subsection{Minimal scaffold} Recall from Section~\ref{sec: scaffold} the universal tropical family \[ \begin{tikzcd} V[n] \times V \ar[d,"\uppi" left] \\ V[n]. \ar[u,bend right,"{p_0,p_1,\ldots,p_n}" right] \end{tikzcd} \] The key point is that for $d=1$, the image of each section $p_i$ is a hyperplane. We write \[ H_i \colonequals p_i(V[n]).\] The identification $V[n]=V^n$ furnishes coordinates $a_1,\ldots,a_n$ on $V[n]$; by convention we also set $a_0=0$. Let $x$ denote the standard coordinate on $V$. The hyperplane $H_i$ is then the vanishing locus of the linear form \[ A_i \colonequals x - a_i.\] Recall that a tropical scaffold is a complete fan on $V[n] \times V$ such that each $H_i$ is a union of cones.
\begin{lemma} \label{lem: unique minimal scaffold} There is a unique minimal tropical scaffold $\Lambda_0$. \end{lemma}
\begin{proof} This is a general fact about fans induced by arrangements of linear hyperplanes. The cones of $\Lambda_0$ are indexed by ordered partitions \[ I = (I_0,I_+,I_-), \qquad I_0 \sqcup I_+ \sqcup I_- = \{0,1,\ldots,n\}\] with corresponding cones \begin{align} \label{eqn: definition of lambdaI} \uplambda_I \colonequals & \bigcap_{i \in I_0} \{A_i=0\} \bigcap_{i \in I_+} \{A_i \geq 0\} \bigcap_{i \in I_-} \{A_i \leq 0\} \nonumber \\ = & \bigcap_{i \in I_0} \{ x = a_i \} \bigcap_{i \in I_+} \{ x \geq a_i \} \bigcap_{i \in I_-} \{ x \leq a_i \}. \end{align} Each $H_i$ is the union of those cones $\uplambda_I$ with $i \in I_0$, and $\Lambda_0$ is clearly minimal with this property.\end{proof}
\subsection{Permutahedral fan} Let $\Sigma_n$ denote the permutahedral fan on $V[n]$. Conceptually this stratifies $V[n]$ into regions over which the functions $a_0,a_1,\ldots,a_n$ have a fixed ordering. Geometrically this corresponds to a fixed ordering of the points $p_0,p_1,\ldots,p_n$.
Formally the cones of $\Sigma_n$ are indexed by total preorders on $\{0,1,\ldots,n\}$. We encode these as ordered partitions \[ J=(J_1, \ldots, J_m), \qquad J_i \neq \emptyset,\ J_1 \sqcup \ldots \sqcup J_m = \{0,1,\ldots,n\}. \] The corresponding cones are \begin{equation} \label{eqn: definition of deltaJ} \upsigma_J \colonequals \bigcap_{\substack{1 \leq i \leq m \\ a_k,a_l \in J_i}} \{ a_k=a_l\} \bigcap_{\substack{1 \leq i < j \leq m \\ a_i \in J_i, a_j \in J_j}} \{ a_i \leq a_j \}. \end{equation} \begin{theorem} \label{thm: get permutahedron} The configuration fan $\Pi_{\Lambda_0}$ is equal to the permutahedral fan $\Sigma_n$. \end{theorem}
\begin{proof} We first show that $\Pi_{\Lambda_0}$ is a refinement of $\Sigma_n$. It suffices to show that every cone of $\Sigma_n$ is an intersection of images of cones of $\Lambda_0$. Fix $\upsigma_J \in \Sigma_n$ corresponding to the total preorder $J=(J_1,\ldots,J_m)$. For each $i \in \{1,\ldots,m\}$ we define the partition $I(i) = (I_0(i),I_+(i),I_-(i))$ by: \begin{align*}
I_0(i) & \colonequals J_i, \\
I_+(i) & \colonequals J_1 \sqcup \ldots \sqcup J_{i-1}, \\
I_-(i) & \colonequals J_{i+1} \sqcup \ldots \sqcup J_m. \end{align*}
By \eqref{eqn: definition of lambdaI} on the corresponding cone $\uplambda_{I(i)} \in \Lambda_0$ we have \begin{align*} x & = a_j \qquad \text{for $j \in J_i$},\\ x & \geq a_j \qquad \text{for $j \in J_1 \sqcup \ldots \sqcup J_{i-1}$}, \\ x & \leq a_j \qquad \text{for $j \in J_{i+1} \sqcup \ldots \sqcup J_m$}. \end{align*} Projecting away from the coordinate $x$ we obtain \[ \uppi(\uplambda_{I(i)}) = \bigcap_{a_k,a_l \in J_i} \{ a_k = a_l \} \bigcap_{\substack{1 \leq j < i\\ a_j \in J_j\\ a_i \in J_i}} \{ a_j \leq a_i \} \bigcap_{\substack{i < j \leq m\\ a_i \in J_i\\ a_j \in J_j}} \{ a_i \leq a_j \}.\] It follows from \eqref{eqn: definition of deltaJ} that $\uppi(\uplambda_{I(1)}) \cap \cdots \cap \uppi(\uplambda_{I(m)}) = \upsigma_J$ (in general there is no single cone $\uplambda \in \Lambda_0$ whose image is $\upsigma_J$). We conclude that there is a refinement \[ \Pi_{\Lambda_0} \to \Sigma_n.\] To prove that this is an equality, we invoke the universal property of $\Pi_{\Lambda_0}$ (Theorem~\ref{thm: semistable reduction}). It suffices to construct a refinement $\Lambda_0^\prime \to \Lambda_0$ such that $\uppi$ is a weakly semistable map of fans $\Lambda_0^\prime \to \Sigma_n$.
The fan $\Lambda_0^\prime$ is constructed as the minimal common refinement of $\Lambda_0$ and the preimage of $\Sigma_n$ under $\uppi$. Since both $\Lambda_0$ and $\Sigma_n$ are defined by hyperplanes, this can be described explicitly. We see from \eqref{eqn: definition of lambdaI} that the hyperplanes defining $\Lambda_0$ impose comparisons between each of $a_0,a_1,\ldots,a_n$ and $x$. The hyperplanes defining $\Sigma_n$ imposes pairwise comparisons amongst $a_0,a_1,\ldots,a_n$, and the same is true of their pullbacks under $\uppi$. The union of these hyperplanes therefore impose pairwise comparisons amongst $a_0,a_1,\ldots,a_n,x$. We conclude that \[ \Lambda_0^\prime = \Sigma_{n+1} \] with the coordinate $x$ playing the role of $a_{n+1}$. It is well-known and not hard to check that projecting away from this coordinate gives a weakly semistable map of fans $\Sigma_{n+1} \to \Sigma_n$. This completes the proof. \end{proof}
The configuration space $P_{\Lambda_0}$ is therefore identified with the Losev--Manin moduli space. The universal tropical expansion and point configuration coincide with the universal curve and marking sections \[ \begin{tikzcd} \mathcal{Y}_{\Lambda_0^\prime} \ar[d,"\uppi" left] \\ P_{\Lambda_0}. \ar[u,bend right,"{x_0,x_1,\ldots,x_n}" right] \end{tikzcd} \] This follows from the identification $\Lambda_0^\prime=\Sigma_{n+1}$ and the fact that the forgetful map $\Sigma_{n+1} \to \Sigma_n$ induces the universal curve over the Losev--Manin moduli space \cite[Section~2.1]{LosevManin}.
The rubber action on the universal expansion coincides with the automorphisms of the universal Losev--Manin curve which fix the two heavy markings and $x_0$. This is illustrated in the following example.
\begin{example} Consider the following tropical point configuration with $n=3$: \[ \begin{tikzpicture}[scale=1.6];
\draw[<->] (-3,0) -- (3,0);
\draw (0,0) node[above]{\small$p_0$}; \filldraw[black] (0,0) circle (1pt);
\filldraw[black] (1,0) circle (1pt); \draw (1,0) node[above]{\small$p_1$}; \draw (1,0.2) node[above]{\small$p_2$};
\filldraw[black] (2,0) circle(1pt); \draw (2,0) node[above]{\small$p_3$};
\end{tikzpicture} \] This defines a total preorder $J = (J_1,J_2,J_3) = (\{0\},\{1,2\},\{3\})$ indexing a two-dimensional cone $\upsigma_J \in \Pi_{\Lambda_0}=\Sigma_3$. This cone is defined by the (in)equalities \[ a_0 \leq a_1 = a_2 \leq a_3.\] It follows that the primitive positive coordinates on the cone $\upsigma_J$ are \[ a_1-a_0, \qquad a_3-a_1\] which appear geometrically as the edge lengths in the polyhedral decomposition \[ \begin{tikzpicture}[scale=1.6];
\draw[<->] (-3,0) -- (3,0);
\draw (0,0) node[above]{\small$p_0$}; \filldraw[black] (0,0) circle (1pt);
\filldraw[black] (1,0) circle (1pt); \draw (1,0) node[above]{\small$p_1$}; \draw (1,0.2) node[above]{\small$p_2$};
\filldraw[black] (2,0) circle(1pt); \draw (2,0) node[above]{\small$p_3$};
\draw (0.05,-0.1) -- (0.05,-0.2) -- (0.95,-0.2) -- (0.95,-0.1); \draw (0.5,-0.175) node[below]{\small$a_1-a_0$};
\draw (1.05,-0.1) -- (1.05,-0.2) -- (1.95,-0.2) -- (1.95,-0.1); \draw (1.5,-0.175) node[below]{\small$a_3-a_1$};
\end{tikzpicture} \] The corresponding topical expansion is a chain of three projective lines \[ \begin{tikzpicture}
\draw (0,0) to [out=40,in=180] (1,0.4) to[out=0,in=140] (2,0); \draw[fill=black] (1,0.4) circle[radius=2pt]; \draw (1,0.4) node[above]{\small$x_0$}; \draw (1,0) node[below]{\small$C_0$};
\draw (1.6,0) to [out=40,in=180] (2.6,0.4) to[out=0,in=140] (3.6,0); \draw (2.6,0) node[below]{\small$C_1$}; \draw[fill=black] (2.2,0.35) circle[radius=2pt]; \draw (2.2,0.35) node[above]{\small$x_1$}; \draw[fill=black] (3,0.35) circle[radius=2pt]; \draw (3,0.35) node[above]{\small$x_2$};
\draw (3.2,0) to [out=40,in=180] (4.2,0.4) to[out=0,in=140] (5.2,0); \draw[fill=black] (4.2,0.4) circle[radius=2pt]; \draw (4.2,0.4) node[above]{\small$x_3$}; \draw (4.2,0) node[below]{\small$C_2$};
\end{tikzpicture} \] The two-dimensional rubber torus $T_{\upsigma_J}=N_{\upsigma_J} \otimes \mathbb{G}_{\mathrm{m}}$ is canonically coordinatised by $e_1 \colonequals a_1-a_0$ and $e_2 \colonequals a_3-a_1$. With respect to these coordinates the action on the components $C_1$ and $C_2$ has weights $e_1$ and $e_1+e_2$ respectively. In the latter case this is because the position of the corresponding vertex is $e_1+e_2=a_3-a_0$.
Changing coordinates from $(e_1,e_2)$ to $(e_1,e_1+e_2)$ produces a split torus whose factors act independently on the components $C_1$ and $C_2$. This coincides with the automorphisms of the curve obtained by forgetting the markings $x_1,x_2,x_3$ and introducing a heavy marking on each end component, as in the Losev--Manin moduli space. The rigidification $x_0=1 \in T$ means that this marking does not contribute to moduli and is not forgotten. This is consistent with the fact that the rubber torus acts trivially on $C_0$. \end{example}
\section{Dimension two: bipermutahedral variety} \label{sec: dim 2}
\noindent For $d \geq 2$ there is no longer a unique minimal choice of scaffold. We introduce two scaffolds of particular interest, and show that the associated configuration spaces are the square of the permutahedral variety (Section~\ref{sec: square of permutahedron}) and the bipermutahedral variety (Section~\ref{sec: bipermutahedron}).
\subsection{Coordinates} Set $d=2$. We coordinatise the ambient space as \[ V = \mathbb{R}^2_{xy} \] which also produces coordinates on the configuration space \[ V[n] = \mathbb{R}^{2n}_{a_1 b_1 \ldots a_n b_n}\] where $(a_i,b_i)$ is the position of $p_i$. The anchor point $p_0$ has coordinates $a_0=b_0=0$.
\subsection{Square of the permutahedral fan} \label{sec: square of permutahedron} Given a point configuration $(p_1,\ldots,p_n) \in V[n]$ we obtain a polyhedral decomposition of $V$ by slicing with the horizontal and vertical lines passing through the points $p_i$ \[ \begin{tikzpicture}[scale=1.2]
\draw (5,1.75) node[left]{\small$V$}; \draw [dashed] (5,-2) -- (5,2) -- (9,2) -- (9,-2) -- (5,-2);
\draw (5,-1) -- (9,-1); \draw(9,-1) node[right]{\small$y=b_0$};
\draw (5,0.8) -- (9,0.8); \draw (9,0.8) node[right]{\small$y=b_1$};
\draw (5,1.4) -- (9,1.4); \draw (9,1.4) node[right]{\small$y=b_2$};
\draw (6,2) -- (6,-2); \draw (6,-2) node[below]{\small$x=a_2$};
\draw (7,2) -- (7,-2); \draw (7,-2) node[below]{\small$x=a_0$};
\draw (8,2) -- (8,-2); \draw (8,-2) node[below]{\small$x=a_1$};
\draw (7.25,-1) node[above]{\small$p_0$}; \draw[fill=black] (7,-1) circle[radius=2pt];
\draw (8.25,0.8) node[above]{\small$p_1$}; \draw[fill=black] (8,0.8) circle[radius=2pt];
\draw (6.25,1.4) node[above]{\small$p_2$}; \draw[fill=black] (6,1.4) circle[radius=2pt];
\end{tikzpicture} \] This defines a tropical scaffold $\Lambda_{S}$ ($S$ for ``square''). The horizontal and vertical lines above correspond to hyperplanes in $V[n] \times V$:
\begin{definition} \label{defn: scaffold square of permutahedron} The scaffold $\Lambda_{S}$ is the complete fan on $V[n] \times V$ induced (as in the proof of Lemma~\ref{lem: unique minimal scaffold}) by the following arrangement of hyperplanes: \begin{equation} \label{eqn: hyperplanes for LambdaS} \{ x = a_0 \} , \ldots, \{ x\, = a_n\}, \{ y = b_0 \}, \ldots, \{ y = b_n\}. \end{equation} \end{definition}
The polyhedral complex $\Lambda_{S,p}$ for $p = (a_1,b_1,\ldots,a_n,b_n) \in V[n]$ undergoes phase transitions whenever the ordering of $(a_0,\ldots,a_n)$ or $(b_0,\ldots,b_n)$ changes. This explains the following.
\begin{proposition}
The configuration fan $\Pi_{\Lambda_{S}}$ is equal to the square $\Sigma_n \times \Sigma_n$ of the permutahedral fan. \end{proposition}
\begin{proof} There is a natural isomorphism \[ \mathbb{R}^{2n}_{a_1 b_1 \ldots a_n b_n} \times \mathbb{R}^2_{xy} = (\mathbb{R}^n_{a_1 \ldots a_n} \times \mathbb{R}_x) \times (\mathbb{R}^n_{b_1 \ldots b_n} \times \mathbb{R}_y).\] Under this isomorphism we have $\Lambda_{S} = \Lambda_0 \times \Lambda_0$ (compare Definition~\ref{defn: scaffold square of permutahedron} and the proof of Lemma~\ref{lem: unique minimal scaffold}). The result follows from Theorem~\ref{thm: get permutahedron}, since universal weak semistable reduction commutes with external products. \end{proof}
\begin{example} \label{example: square of permutahedron} Consider the point configuration and scaffold in Figure~\ref{fig: square scaffold}. The corresponding cone $\uprho \in \Pi_{\Lambda_{S}} = \Sigma_n \times \Sigma_n$ is defined by the inequalities \begin{equation} \label{eqn: inequalities defining square cone example} a_1 \leq a_0 \leq a_2,\qquad b_0 \leq b_2 \leq b_1. \end{equation} The fibre of the universal tropical expansion \eqref{eqn: universal family} over the locally-closed stratum $P_{\Lambda_{S},\uprho} \subseteq P_{\Lambda_S}$ is the patchwork quilt made of copies of $\mathbb{P}^1\!\times\!\mathbb{P}^1$ depicted in Figure~\ref{fig: square expansion}. \begin{figure}
\caption{Scaffold}
\caption{Expansion}
\caption{A stratum of $P_{\Lambda_S}$.}
\label{fig: square scaffold}
\label{fig: square expansion}
\end{figure}
The polyhedral edge lengths labelled in Figure~\ref{fig: square scaffold} give the coordinate system on $\uprho$ dual to the basis of primitive ray generators. Indeed, the inequalities \eqref{eqn: inequalities defining square cone example} produce the primitive non-negative linear functions \[ a_0 - a_1 = e_1, \qquad a_2 - a_0 = e_2, \qquad b_2 - b_0 = f_1, \qquad b_1 - b_2 = f_2.\] We use Theorem~\ref{thm: strata} to give a modular description of $P_{\Lambda_S,\uprho}$. The rubber torus is $T_{\uprho} = (\mathbb{G}_{\mathrm{m}}^4)_{e_1 e_2 f_1 f_2}$ and the rubber action is governed by the tropical position maps. For the vertices $v_i$ supporting the points $p_i$ these are \begin{align*} \varphi_{v_0}(e_1,e_2,f_1,f_2) & = (a_0,b_0) = (0,0), \\ \varphi_{v_1}(e_1,e_2,f_1,f_2) & = (a_1,b_1) = (-e_1,f_1+f_2), \\ \varphi_{v_2}(e_1,e_2,f_1,f_2) & = (a_2,b_2) = (e_2,f_1). \end{align*} The positions of $x_1,x_2$ give $2+2=4$ dimensions of moduli. These are precisely cancelled out by the rubber action, so $P_{\Lambda_S,\uprho}$ is a point. This is consistent with the fact that $\uprho \in \Pi_{\Lambda_S}$ is maximal. \end{example}
\subsection{Bipermutahedral fan} \label{sec: bipermutahedron}
We assume familiarity with the bipermutahedral fan. For an accessible introduction, see \cite[Sections~2.3--2.6]{ArdilaDenhamHuh} and \cite{ArdilaBipermutahedron}.
Ardila--Denham--Huh define the bipermutahedral fan $\Sigma_{n,n}$ as the tropical configuration space $V[n]$ stratified according to bisequence \cite[Section~2.4]{ArdilaDenhamHuh}. From our perspective the definition of bisequence strongly suggests a choice of tropical scaffold, namely the subdivision \[ \Lambda_B \to \Lambda_S \] obtained by slicing each polyhedral decomposition $\Lambda_{S,p}$ with the supporting antidiagonal (compare with \cite[Figure~2]{ArdilaDenhamHuh}): \begin{equation} \label{eqn: bipermutahedral scaffold} \begin{tikzpicture}[scale=1,baseline=(current bounding box.center)]
\draw [dashed] (5,-2) -- (5,2) -- (9,2) -- (9,-2) -- (5,-2);
\draw[blue,thick] (8,-2) -- (5,1); \draw[blue] (8,-2) node[below]{\small$x+y=a_0+b_0$};
\draw (5,-1) -- (9,-1);
\draw (5,0.8) -- (9,0.8);
\draw (5,1.4) -- (9,1.4);
\draw (6,2) -- (6,-2);
\draw (7,2) -- (7,-2);
\draw (8,2) -- (8,-2);
\draw (7.25,-1) node[above]{\small$p_0$}; \draw[fill=black] (7,-1) circle[radius=2pt];
\draw (8.25,0.8) node[above]{\small$p_1$}; \draw[fill=black] (8,0.8) circle[radius=2pt];
\draw (6.25,1.4) node[above]{\small$p_2$}; \draw[fill=black] (6,1.4) circle[radius=2pt];
\end{tikzpicture} \end{equation} Formally this is defined as follows. The vector space $V[n] \times V$ is covered by closed convex cones \[ \mathcal{C}_i \colonequals \left\{ \min_{j}(a_j + b_j) = a_i + b_i \right\} \subseteq V[n] \times V \] for $i \in \{0,\ldots,n\}$. Geometrically $\mathcal{C}_i$ is the locus where $p_i$ lies on the supporting antidiagonal; the latter therefore has equation $x+y=a_i+b_i$.
The restricted fan $\Lambda_{S}|_{\mathcal{C}_i}$ is obtained by intersecting each cone of $\Lambda_S$ with $\mathcal{C}_i$. We construct $\Lambda_B|_{\mathcal{C}_i}$ by slicing $\Lambda_S|_{\mathcal{C}_i}$ with the hyperplane \[ D_i \colonequals \{ x+y = a_i+b_i \} \subseteq V[n] \times V\]
so that $\Lambda_B|_{\mathcal{C}_i}$ is the restriction to $\mathcal{C}_i$ of the fan induced by the arrangement of hyperplanes \eqref{eqn: hyperplanes for LambdaS} together with $D_i$. Since \[ D_i \cap (\mathcal{C}_i \cap \mathcal{C}_j) = D_j \cap (\mathcal{C}_i \cap \mathcal{C}_j) \]
the fans $\Lambda_B|_{\mathcal{C}_i}$ glue to produce a scaffold $\Lambda_B$. Unlike $\Lambda_0$ or $\Lambda_S$, $\Lambda_B$ is not globally induced by a hyperplane arrangement.
The fibres $\Lambda_{B,p}$ are depicted in \eqref{eqn: bipermutahedral scaffold}. This polyhedral complex undergoes phase transitions precisely when the bisequence associated to $p \in V[n]$ changes. This explains the following.
\begin{theorem} \label{thm: get bipermutahedron} The configuration fan $\Pi_{\Lambda_B}$ is equal to the bipermutahedral fan $\Sigma_{n,n}$. \end{theorem}
\begin{proof} We first describe $\Sigma_{n,n}$ in the form we require. The vector space $V[n]$ is covered by closed convex cones \[ C_i \colonequals \left\{ \min_j(a_j+b_j) = a_i + b_i \right\} \subseteq V[n] \]
for $i \in \{0,\ldots,n\}$. A cone $\upsigma \in \Sigma_{n,n}|_{C_i}$ is produced by choosing the following comparisons:\footnote{The comparisons (i), (ii), (iii) contain redundancies. Since $\upsigma \subseteq C_i$ we must have $a_i+b_i \leq a_j+b_j$ for all $j$. Consequently, choosing $a_i+b_i = a_j+b_j$ or $a_i+b_i \geq a_j+b_j$ in (iii) produces the same cone. Similarly if we choose $a_i \leq a_j$ in (i) and $b_i \leq b_k$ in (ii) then $a_i+b_i \leq a_j+b_k$, and so choosing $a_i+b_i = a_j+b_k$ or $a_i+b_i \geq a_j+b_k$ in (iii) produces the same cone. These redundancies do not affect the argument.} \begin{enumerate}[label=(\roman*)]
\item Pairwise comparisons of $a_0,\ldots,a_n$.
\item Pairwise comparisons of $b_0,\ldots,b_n$.
\item A comparison of $a_i+b_i$ against $a_j + b_k$ for all $j,k \in \{0,\ldots,n\}$. \end{enumerate}
Together these determine the bisequence associated to a point $p=(a_1,b_1,\ldots,a_n,b_n) \in C_i$. The cone $\upsigma \subseteq C_i$ is cut out by the corresponding (in)equalities. Each fan $\Sigma_{n,n}|_{C_i}$ is induced by a hyperplane arrangement, and these fans glue to produce $\Sigma_{n,n}$. Comparisons of type (i) and (ii) do not depend on $i$ and show that $\Sigma_{n,n}$ is a refinement of $\Sigma_n \times \Sigma_n$.
To show that $\Pi_{\Lambda_B}=\Sigma_{n,n}$ we follow the proof of Theorem~\ref{thm: get permutahedron}. We first show that $\Pi_{\Lambda_B}$ is a refinement of $\Sigma_{n,n}$, by showing that every cone of $\Sigma_{n,n}$ is an intersection of images of cones of $\Lambda_B$.
Fix $\upsigma \in \Sigma_{n,n}$. We have $\upsigma \in \Sigma_{n,n}|_{C_i}$ for some $i$ (not necessarily unique) and $\upsigma$ is cut out by (in)equalities imposing comparisons of type (i), (ii), (iii). For each of these (in)equalities we find a cone $\uplambda \in \Lambda_B$ such that the given (in)equality holds on $\uppi(\uplambda)$, and such that $\sigma \subseteq \uppi(\uplambda)$. The intersection of all such $\uppi(\uplambda)$ is then equal to $\upsigma$.
Since $\uppi^{-1}(C_i) = \mathcal{C}_i$ we must have $\uplambda \in \Lambda_B|_{\mathcal{C}_i}$. A cone $\uplambda \in \Lambda_B|_{\mathcal{C}_i}$ is produced by choosing the following comparisons: \begin{enumerate}[label=(\Roman*)]
\item A comparison of $x$ against each of $a_0,\ldots,a_n$.
\item A comparison of $y$ against each of $b_0,\ldots,b_n$.
\item A comparison of $a_i+b_i$ against $x+y$. \end{enumerate}
We work through the comparisons (i), (ii), (iii) defining $\upsigma$. First consider a type (i) comparison. If the comparison is $a_j \leq a_k$ then we choose the type (I) comparisons $x=a_j,x \leq a_k$. If the comparison is $a_j=a_k$ then we choose the type (I) comparisons $x=a_j,x=a_k$. We choose the remaining type (I) comparisons compatibly with the type (i) comparisons, as in the proof of Theorem~\ref{thm: get permutahedron}. Similarly we choose the type (II) comparisons compatibly with the type (ii) comparisons. This gives $x=a_j$ and $y=b_l$ for some $l$. We then choose the type (III) comparison of $a_i+b_i$ against $x+y$ to be equal to the type (iii) comparison of $a_i+b_i$ against $a_j+b_l$. This produces a cone $\uplambda \in \Lambda_B|_{\mathcal{C}_i}$ with the desired properties.
The case of a type (ii) comparison is identical. It remains to consider a type (iii) comparison. This compares $a_i+b_i$ against $a_j+b_k$. We choose the type (I) and (II) comparisons \[ x = a_j, \qquad y=b_k, \]
and choose the type (III) comparison of $a_i+b_i$ against $x+y$ to be equal to the type (iii) comparison of $a_i+b_i$ against $a_j+b_k$. We choose the remaining type (I) and (II) comparisons compatibly with the type (i) and (ii) comparisons, as in the proof of Theorem~\ref{thm: get permutahedron}. This produces a cone $\uplambda \in \Lambda_B|_{\mathcal{C}_i}$ with the desired properties.
We conclude that there is a refinement \[ \Pi_{\Lambda_B} \to \Sigma_{n,n}.\] To prove that this is an equality, we invoke the universal property of $\Pi_{\Lambda_B}$ (Theorem~\ref{thm: semistable reduction}). It suffices to construct a refinement $\Lambda_B^\prime \to \Lambda_B$ such that $\uppi$ is a weakly semistable map of fans $\Lambda_B^\prime \to \Sigma_{n,n}$.
We construct $\Lambda_B^\prime$ as the minimal common refinement of $\Lambda_B$ and the preimage of $\Sigma_{n,n}$ under $\uppi$. The condition of being a weakly semistable map of fans is local on the source, so it suffices to show that $\uppi$ is a weakly semistable map of fans
\[ \Lambda_B^\prime|_{\mathcal{C}_i} \to \Sigma_{n,n}|_{C_i}.\]
Both $\Lambda_B|_{\mathcal{C}_i}$ and $\Sigma_{n,n}|_{C_i}$ are induced by explicit hyperplane arrangements. We deduce that a cone $\uplambda^\prime \in \Lambda_B^\prime|_{\mathcal{C}_i}$ is produced by choosing the following comparisons: \begin{enumerate}[label=(\Roman*')] \item Pairwise comparisons of $x,a_0,\ldots,a_n$. \item Pairwise comparisons of $y,b_0,\ldots,b_n$. \item A comparison of $a_i+b_i$ against $a_j+b_k$ for $j,k \in \{0,\ldots,n\}$. \item A comparison of $a_i+b_i$ against $x+y$. \end{enumerate}
These determine type (i), (ii), (iii) comparisons, producing a cone $\upsigma \in \Sigma_{n,n}|_{C_i}$ with $\uppi(\uplambda^\prime) \subseteq \upsigma$. The type (III') comparisons and the type (iii) comparisons are not necessarily identical: the latter are a specialisation of the former obtained by combining the type (I'), (II'), (IV') comparisons and invoking transitivity, with $x$ and $y$ functioning as intermediaries.
Now fix $p=(a_1,b_1,\ldots,a_n,b_n) \in \upsigma$. The type (I'), (II'), (IV') comparisons involving $x$ and $y$ do not impose any restrictions on the $a_j$ and $b_j$ beyond the type (i), (ii), (iii) comparisons. It follows that $\uppi(\uplambda^\prime)=\upsigma$ and this completes the proof. \end{proof}
\begin{example} Consider the cone $\uprho \in \Sigma_n \times \Sigma_n$ from Example~\ref{example: square of permutahedron}. This is cut out by the inequalities \eqref{eqn: inequalities defining square cone example}. These correspond to the type (i) and (ii) comparisons appearing in the proof of Theorem~\ref{thm: get bipermutahedron}.
The refinement $\Sigma_{n,n} \to \Sigma_n \times \Sigma_n$ subdivides $\uprho$ into a union of cones. By Theorem~\ref{thm: get bipermutahedron} each such cone corresponds to a different polyhedral complex $\Lambda_{B,p}$. An example is illustrated in Figure~\ref{fig: biperm scaffold}. \begin{figure}
\caption{Scaffold}
\caption{Expansion}
\caption{A stratum of $P_{\Lambda_B}$.}
\label{fig: biperm scaffold}
\label{fig: biperm expansion}
\end{figure}
\noindent The associated bisequence is $2|0|12|1$ and the corresponding cone $\uptau \in \Sigma_{n,n}$ is the subset of $\uprho$ defined by the additional (in)equalities \begin{align*} a_0 + b_0 & \leq a_1 + b_1, \\ a_0 + b_0 & = a_1 + b_2, \\ a_0 + b_0 & \leq a_2 + b_1, \\ a_0 + b_0 & \leq a_2+b_2. \end{align*} The last two inequalities are redundant: $a_0 \leq a_2$ and $b_0 \leq b_2 \leq b_1$ on $\uprho$ already imply $a_0 + b_0 \leq a_2 + b_1$ and $a_0 + b_0 \leq a_2 + b_2$. The first two (in)equalities are essential.
The polyhedral edge lengths again give a coordinate system on $\uptau$. Notice that now we have $e_1=f_1$ because the antidiagonal is required to pass through the vertex $(a_0-e_1,b_0+f_1)$. This arises from the equality \[ a_0+b_0 = a_1 + b_2 \Leftrightarrow a_0-a_1 = b_2-b_0 \Leftrightarrow e_1=f_1.\] A coordinate system for $\uptau$ is therefore given by $(e_1,e_2,f_2)$. Correspondingly we have $\dim \uptau = 3$ whereas $\dim \uprho = 4$.
The associated tropical expansion is illustrated in Figure~\ref{fig: biperm expansion}. The $2$ hexagonal components are two-dimensional permutahedral varieties and each of the remaining $9$ components is a $\mathbb{P}^1 \times \mathbb{P}^1$. As far as the moduli of the $x_i$ is concerned, this expansion is for all intents and purposes identical to the expansion considered in Example~\ref{example: square of permutahedron}. The rubber torus, however, differs in a crucial way: it is now only three-dimensional \[ T_\uptau = (\mathbb{G}_{\mathrm{m}}^3)_{e_1 e_2 f_2}. \] This is precisely the subtorus $T_\uptau \subseteq T_\uprho$ that acts trivially on the divisor joining the two hexagonal components; see \cite[Example~4.4]{CarocciNabijouRubber} for a similar phenomenon. As usual the rubber action is governed by the tropical position maps \begin{align*} \varphi_{v_0}(e_1,e_2,f_2) & = (a_0,b_0) = (0,0), \\ \varphi_{v_1}(e_1,e_2,f_2) & = (a_1,b_1) = (-e_1,e_1+f_2), \\ \varphi_{v_2}(e_1,e_2,f_2) & = (a_2,b_2) = (e_2,e_1). \end{align*} From Theorem~\ref{thm: strata} we see that there are $(2+2)-3=1$ dimensions of moduli for $P_{\Lambda_B,\uptau} \subseteq P_{\Lambda_B}$. This is consistent with the fact that $\operatorname{codim}\uptau=1$. \end{example}
\subsection{Harmonic fan} A close cousin of the bipermutahedral fan is the harmonic fan $H_{n,n}$, introduced in \cite[Section~2.8]{ArdilaDenhamHuh} and studied in \cite{ArdilaEscobar}. It is non-simplicial and sits in a tower of refinements \[ \Sigma_{n,n} \to H_{n,n} \to \Sigma_n \times \Sigma_n.\] A cone of $H_{n,n}$ is produced by choosing pairwise comparisons of $a_0,\ldots,a_n$ and of $b_0,\ldots,b_n$, along with a subset of $\{0,\ldots,n\}$ indexing those points $p_i$ which lie on the supporting antidiagonal.
\begin{question} Does $H_{n,n}$ arise as the configuration fan $\Pi_\Lambda$ associated to any tropical scaffold $\Lambda$? \end{question}
We are unable to identify a suitable scaffold. If the answer is indeed negative, this provides another example of the bipermutahedral fan enjoying properties which the harmonic fan lacks.
\footnotesize
\noindent Navid Nabijou. Queen Mary University of London. \href{mailto:[email protected]}{[email protected]}
\end{document} | arXiv |
(a) Geometrically, in the $x-y$ plane, $T$ is the reflection about the diagonal $x=y$ and $U$ is a projection onto the $x$-axis.
(b) We have
$(U+T)(x_1,x_2)=(x_2,x_1)+(x_1,0)=(x_1+x_2,x_1)$.
$(UT)(x_1,x_2)=U(x_2,x_1)=(x_2,0)$.
$(TU)(x_1,x_2)=T(x_1,0)=(0,x_1)$.
$T^2(x_1,x_2)=T(x_2,x_1)=(x_1,x_2)$, the identity function.
$U^2(x_1,x_2)=U(x_1,0)=(x_1,0)$. So $U^2=U$.
By Theorem 9 part (v), top of page 82, $T$ is invertible if $\{T\epsilon_1,T\epsilon_2,T\epsilon_3\}$ is a basis of $\mathbb C^3$. Since $\mathbb C^3$ has dimension three, it suffices (by Corollary 1 page 46) to show $T\epsilon_1,T\epsilon_2,T\epsilon_3$ are linearly independent. To do this we row reduce the matrix
$$\left[\begin{array}{ccc}1&0&i\\0&1&1\\i&1&0\end{array}\right]$$to row-reduced echelon form. If it reduces to the identity then its rows are independent, otherwise they are dependent. Row reduction follows:
$$\left[\begin{array}{ccc}1&0&i\\0&1&1\\i&1&0\end{array}\right]
\rightarrow\left[\begin{array}{ccc}1&0&i\\0&1&1\\0&1&1\end{array}\right]
$$This is in row-reduced echelon form not equal to the identity. Thus $T$ is not invertible.
The matrix representation of the transformation is
$$\left[\begin{array}{c}x_1\\x_2\\x_3\end{array}\right] \mapsto \left[\begin{array}{ccc}3&0&0\\1&-1&0\\2&1&1\end{array}\right]\cdot\left[\begin{array}{c}x_1\\x_2\\x_3\end{array}\right]$$where we've identified $\mathbb R^3$ with $\mathbb R^{3\times1}$. $T$ is invertible if the matrix of the transformation is invertible. To determine this we row-reduce the matrix – we row-reduce the augmented matrix to determine the inverse for the second part of the Exercise.
$$\left[\begin{array}{ccc|ccc}3&0&0 & 1&0&0 \\ 1&-1&0 & 0&1&0 \\ 2&1&1 & 0&0&1\end{array}\right]$$$$\rightarrow\left[\begin{array}{ccc|ccc}1&-1&0 & 0&1&0 \\ 3&0&0 & 1&0&0 \\ 2&1&1 & 0&0&1\end{array}\right]$$$$\rightarrow\left[\begin{array}{ccc|ccc}1&-1&0 & 0&1&0 \\ 0&3&0 & 1&-3&0 \\ 0&3&1 & 0&-2&1\end{array}\right]$$$$\rightarrow\left[\begin{array}{ccc|ccc}1&-1&0 & 0&1&0 \\ 0&1&0 & 1/3&-1&0 \\ 0&3&1 & 0&-2&1\end{array}\right]$$$$\rightarrow\left[\begin{array}{ccc|ccc}1&0&0 & 1/3&0&0 \\ 0&1&0 & 1/3&-1&0 \\ 0&0&1 & -1&1&1\end{array}\right]$$Since the left side transformed into the identity, $T$ is invertible. The inverse transformation is given by
$$\left[\begin{array}{c}x_1\\x_2\\x_3\end{array}\right] \mapsto \left[\begin{array}{ccc}1/3&0&0\\1/3&-1&0\\-1&1&1\end{array}\right]\cdot\left[\begin{array}{c}x_1\\x_2\\x_3\end{array}\right]$$So$$T^{-1}(x_1,x_2,x_3)=(x_1/3,\ \ x_1/3-x_2,\ \ -x_1+x_2+x_3).$$
Working with the matrix representation of $T$ we must show
$$(A^2-I)(A-3I)=0$$where
$$A=\left[\begin{array}{ccc}3&0&0\\1&-1&0\\2&1&1\end{array}\right].$$Calculating:
$$A^2=\left[\begin{array}{ccc}3&0&0\\1&-1&0\\2&1&1\end{array}\right]\left[\begin{array}{ccc}3&0&0\\1&-1&0\\2&1&1\end{array}\right]$$$$=\left[\begin{array}{ccc}9&0&0\\2&1&0\\9&0&1\end{array}\right]$$Thus
$$A^2-I=\left[\begin{array}{ccc}8&0&0\\2&0&0\\9&0&0\end{array}\right].$$Also
$$A-3I=\left[\begin{array}{ccc}0&0&0\\1&-4&0\\2&1&-2\end{array}\right]$$Thus
$$(A^2-I)(A-3I)=\left[\begin{array}{ccc}8&0&0\\2&0&0\\9&0&0\end{array}\right]\cdot\left[\begin{array}{ccc}0&0&0\\1&-4&0\\2&1&-2\end{array}\right]$$$$=\left[\begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\end{array}\right].$$
An (ordered) basis for $\mathbb C^{2\times2}$ is given by
$$A_{11}=\left[\begin{array}{cc}1&0\\0&0\end{array}\right],\quad A_{21}=\left[\begin{array}{cc}0&0\\1&0\end{array}\right]$$$$A_{12}=\left[\begin{array}{cc}0&1\\0&0\end{array}\right],\quad A_{22}=\left[\begin{array}{cc}0&0\\0&1\end{array}\right].$$If we identify $\mathbb C^{2\times 2}$ with $\mathbb C^4$ by
$$\left[\begin{array}{cc}a&b\\c&d\end{array}\right]\mapsto (a,b,c,d)$$then since
$$A_{11}\mapsto A_{11}-4A_{21}$$$$A_{21}\mapsto -A_{11}+4A_{21}$$$$A_{12}\mapsto A_{12}-4A_{22}$$$$A_{22}\mapsto -A_{12}+4A_{22}$$the matrix of the transformation is given by
$$\left[\begin{array}{cccc}
1&-4&0&0\\
-1&4&0&0\\
0&0&1&-4\\
0&0&-1&4\end{array}\right].$$To find the rank of $T$ we row-reduce this matrix:
$$\rightarrow\left[\begin{array}{cccc}
0&0&0&0\\
0&0&0&0\end{array}\right].$$It has rank two so the rank, so the rank of $T$ is $2$.
Note that $T^2(A)=T(T(A))=T(BA)=B(BA)=B^2A$. Thus $T^2$ is given by multiplication by a matrix just as $T$ is, but multiplication with $B^2$ instead of $B$. Explicitly
$$B^2=\left[\begin{array}{cc}1&-1\\-4&4\end{array}\right]\left[\begin{array}{cc}1&-1\\-4&4\end{array}\right]$$$$=\left[\begin{array}{cc}5&-5\\-20&20\end{array}\right].$$
Let $\{\alpha_1,\alpha_2,\alpha_3\}$ be a basis for $\Bbb R^3$. Then $T(\alpha_1),T(\alpha_2),T(\alpha_3)$ must be linearly dependent in $\Bbb R^2$, because $\Bbb R^2$ has dimension $2$. So suppose $$b_1T(\alpha_1)+b_2T(\alpha_2)+b_3T(\alpha_3)=0$$ and not all $b_1,b_2,b_3$ are zero. Then
$$b_1\alpha_1+b_2\alpha_2+b_3\alpha_3\not=0$$ and
$$UT(b_1\alpha_1+b_2\alpha_2+b_3\alpha_3)$$$$=U(T(b_1\alpha_1+b_2\alpha_2+b_3\alpha_3))$$$$=U(b_1T(\alpha_1)+b_2T(\alpha_2)+b_3T(\alpha_3)$$$$=U(0)=0.$$Thus (by the definition at the bottom of page 79) $UT$ is not non-singular and thus by Theorem 9, page 81, $UT$ is not invertible.
The obvious generalization is that if $n>m$ and $T:\Bbb R^n\rightarrow\Bbb R^m$ and $U:\Bbb R^m\rightarrow\Bbb R^n$ are linear transformations, then $UT$ is not invertible. The proof is an immediate generalization the proof of the special case above, just replace $\alpha_3$ with $\dots,\alpha_n$.
Identify $\mathbb R^2$ with $\mathbb R^{2\times1}$ and let $T$ and $U$ be given by the matrices
$$A=\left[\begin{array}{cc}1&0\\0&0\end{array}\right],\quad B=\left[\begin{array}{cc}0&1\\0&0\end{array}\right].$$More precisely, for
$$X=\left[\begin{array}{c}x\\y\end{array}\right].$$Let $T$ be given by $X\mapsto AX$ and let $U$ be given by $X\mapsto BX$. Thus $TU$ is given by $X\mapsto ABX$ and $UT$ is given by $X\mapsto BAX$. But $BA=0$ and $AB\not=0$ so we have the desired example.
If $T^2=0$ then the range of $T$ must be contained in the null space of $T$ since if $y$ is in the range of $T$ then $y=Tx$ for some $x$ so $Ty=T(Tx)=T^2x=0$. Thus $y$ is in the null space of $T$.
To give an example of an operator where $T^2=0$ but $T\not=0$, let $V=\mathbb R^{2\times1}$ and let $T$ be given by the matrix
$$A=\left[\begin{array}{cc}0&1\\0&0\end{array}\right].$$Specifically, for
$$X=\left[\begin{array}{c}x\\y\end{array}\right].$$Let $T$ be given by $X\mapsto AX$. Since $A\not=0$, $T\not=0$. Now $T^2$ is given by $X\mapsto A^2X$, but $A^2=0$. Thus $T^2=0$.
By the comments in the Appendix on functions, at the bottom of page 389, we see that simply because $TU=I$ as functions, then necessarily $T$ is onto and $U$ is one-to-one. It then follows immediately from Theorem 9, page 81, that $T$ is invertible. Now $TT^{-1}=I=TU$ and multiplying on the left by $T^{-1}$ we get $T^{-1}TT^{-1}=T^{-1}TU$ which implies $(I)T^{-1}=(I)U$ and thus $U=T^{-1}$.
Let $V$ be the space of polynomial functions in one variable over $\Bbb R$. Let $D$ be the differentiation operator and let $T$ be the operator "multiplication by $x$" (exactly as in Example 11, page 80). As shown in Example 11, $UT=I$ while $TU\not=I$. Thus this example fulfills the requirement.
Exercise 3.2.10
Let $\mathcal B=\{\alpha_1,\dots,\alpha_n\}$ be a basis for $F^{n\times1}$ and let $\mathcal B'=\{\beta_1,\dots,\beta_m\}$ be a basis for $F^{m\times1}$. We can define a linear transformation from $F^{n\times1}$ to $F^{m\times1}$ uniquely by specifying where each member of $\mathcal B$ goes in $F^{m\times1}$. If $m<n$ then we can define a linear transformation that maps at least one member of $\mathcal B$ to each member of $\mathcal B'$ and maps at least two members of $\mathcal B$ to the same member of $\mathcal B'$. Any linear transformation so defined must necessarily be onto without being one-to-one. Similarly, if $m>n$ then we can map each member of $\mathcal B$ to a unique member of $\mathcal B'$ with at least one member of $\mathcal B'$ not mapped to by any member of $\mathcal B$. Any such transformation so defined will necessarily be one-to-one but not onto.
Let $\{\alpha_1,\dots,\alpha_n\}$ be a basis for $V$. Then the rank of $T$ is the number of linearly independent vectors in the set $\{T\alpha_1,\dots,T\alpha_n\}$. Suppose the rank of $T$ equals $k$ and suppose WLOG that $\{T\alpha_1,\dots,T\alpha_k\}$ is a linearly independent set (it might be that $k=1$, pardon the notation). Then $\{T\alpha_1,\dots,T\alpha_k\}$ give a basis for the range of $T$. It follows that $\{T^2\alpha_1,\dots,T^2\alpha_k\}$ span the range of $T^2$ and since the dimension of the range of $T^2$ is also equal to $k$, $\{T^2\alpha_1,\dots,T^2\alpha_k\}$ must be a basis for the range of $T^2$. Now suppose $v$ is in the range of $T$. Then $v=c_1T\alpha_1+\cdots+c_kT\alpha_k$. Suppose $v$ is also in the null space of $T$. Then $$0=T(v)=T(c_1T\alpha_1+\cdots+c_kT\alpha_k)=c_1T^2\alpha_1+\cdots+c_kT^2\alpha_k.$$ But $\{T^2\alpha_1,\dots,T^2\alpha_k\}$ is a basis, so $T^2\alpha_1,\dots,T^2\alpha_k$ are linearly independent, thus it must be that $c_1=\cdots=c_k=0$, which implies $v=0$. Thus we have shown that if $v$ is in both the range of $T$ and the null space of $T$ then $v=0$, as required.
We showed in Exercise 2.3.12, page 49, that the dimension of $V$ is $mn$ and the dimension of $W$ is $pn$. By Theorem 9 page (iv) we know that an invertible linear transformation must take a basis to a basis. Thus if there's an invertible linear transformation between $V$ and $W$ it must be that both spaces have the same dimension. Thus if $T$ is inverible then $pn=mn$ which implies $p=m$. The matrix $B$ is then invertible because the assignment $B\mapsto BX$ is one-to-one (Theorem 9 (ii), page 81) and non-invertible matrices have non-trivial solutions to $BX=0$ (Theorem 13, page 23). Conversely, if $p=n$ and $B$ is invertible, then we can define the inverse transformation $T^{-1}$ by $T^{-1}(A)=B^{-1}A$ and it follows that $T$ is invertible.
From http://greggrant.org | CommonCrawl |
For how many values of $x$ is the expression $\frac{x^2-9}{(x^2+2x-3)(x-3)}$ undefined?
The expression is undefined when the denominator of the fraction is equal to zero, so we are looking for values of $x$ that satisfy $(x^2+2x-3)(x-3)=0$. This polynomial factors further as $(x-1)(x+3)(x-3)=0$, giving us the solutions $x=1$, $x=-3$ and $x=3$. Therefore, there are $\boxed{3}$ values of $x$ for which the expression is undefined. | Math Dataset |
Vanishing capillarity limit of the non-conservative compressible two-fluid model
Feedback controllability for blowup points of semilinear heat equations
June 2017, 22(4): 1393-1423. doi: 10.3934/dcdsb.2017067
Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays
Shangzhi Li and Shangjiang Guo ,
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China
* Corresponding author: S. Guo
Received May 2016 Revised November 2016 Published February 2017
Fund Project: The second author is supported by NSF of China (Grants No. 11671123 & 11271115)
This paper is devoted to a cooperative model composed of two species withstage structure and state-dependent maturation delays. Firstly, positivity and boundedness of solutions are addressed to describe the population survival and the natural restriction of limited resources. It is shown that for a given pair of positive initial functions, the two mature populations are uniformly bounded away from zero and that the two mature populations are bounded above only if the the coupling strength is small enough. Moreover, if the coupling strength is large enough then the two mature populations tend to infinity as the time tends to infinity. In particular, the positivity of the two immature populations has been established under some additional conditions. Secondly, the existence and patterns of equilibria are investigated by means of degree theory and Lyapunov-Schmidt reduction. Thirdly, the local stability of the equilibria is also discussed through a formal linearization. Fourthly, the global behavior of solutions is discussed and the explicit bounds for the eventual behaviors of the two mature populations and two immature populations are obtained. Finally, global asymptotical stability is investigated by using the comparison principle of the state-dependent delay equations.
Keywords: State-dependent delay, cooperation, stage structure, comparison principle, global stability.
Mathematics Subject Classification: Primary:34D23, 34K20;Secondary:92D25.
Citation: Shangzhi Li, Shangjiang Guo. Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1393-1423. doi: 10.3934/dcdsb.2017067
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Figure 1. Simulations of system (3) illustrate that the synchronous equilibrium is globally asymptotically stable, where $\alpha=2,\gamma=0.1,\mu=0.1,\beta=0.365$
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Figure 2. Simulations of system (3) illustrate that the synchronous equilibrium is globally asymptotically stable, where $\alpha=1.5,\gamma=0.2,\mu=0.1,\beta=0.365$
Figure 3. Simulations of system (3) illustrate that every solution of (3) is asymptotically synchronous and tends to infinity as $t$ tends to infinity, where $\alpha=2,\gamma=0.1,\mu=0.4,\beta=0.365$
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Shangzhi Li Shangjiang Guo | CommonCrawl |
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Microfluidics and Nanofluidics
March 2017 , 21:50 | Cite as
Formation of inverse Chladni patterns in liquids at microscale: roles of acoustic radiation and streaming-induced drag forces
Junjun Lei
First Online: 03 March 2017
While Chladni patterns in air over vibrating plates at macroscale have been well studied, inverse Chladni patterns in water at microscale have recently been reported. The underlying physics for the focusing of microparticles on the vibrating interface, however, is still unclear. In this paper, we present a quantitative three-dimensional study on the acoustophoretic motion of microparticles on a clamped vibrating circular plate in contact with water with emphasis on the roles of acoustic radiation and streaming-induced drag forces. The numerical simulations show good comparisons with experimental observations and basic theory. While we provide clear demonstrations of three-dimensional particle size-dependent microparticle trajectories in vibrating plate systems, we show that acoustic radiation forces are crucial for the formation of inverse Chladni patterns in liquids on both out-of-plane and in-plane microparticle movements. For out-of-plane microparticle acoustophoresis, out-of-plane acoustic radiation forces are the main driving force in the near-field, which prevent out-of-plane acoustic streaming vortices from dragging particles away from the vibrating interface. For in-plane acoustophoresis on the vibrating interface, acoustic streaming is not the only mechanism that carries microparticles to the vibrating antinodes forming inverse Chladni patterns: In-plane acoustic radiation forces could have a greater contribution. To facilitate the design of lab-on-a-chip devices for a wide range of applications, the effects of many key parameters, including the plate radius R and thickness h and the fluid viscosity μ, on the microparticle acoustophoresis are discussed, which show that the threshold in-plane and out-of-plane particle sizes balanced from the acoustic radiation and streaming-induced drag forces scale linearly with R and \(\sqrt \mu\), but inversely with \(\sqrt h\).
Chladni patterns Acoustic streaming Acoustic radiation force Acoustofluidics Vibrating plates
The online version of this article (doi: 10.1007/s10404-017-1888-5) contains supplementary material, which is available to authorized users.
Arranging particles and cells into desired patterns for lab-on-a-chip biological applications using ultrasonic fields, i.e. acoustophoresis, by means of bulk and surface acoustic wave techniques, have attracted increasing interest in recent years (Bruus et al. 2011; Friend and Yeo 2011). When an ultrasonic standing/travelling wave is established in a micro-channel containing an aqueous suspension of particles, two main forces act on the particles: the acoustic radiation force and the streaming-induced drag force. In most bulk and surface micro-acoustofluidic manipulation devices, the latter is generally considered to be a disturbance because it places a practical lower limit on the particle size that can be manipulated by the former (Wiklund et al. 2012; Drinkwater 2016). Nevertheless, acoustic streaming flows have been applied to play an active role in the functioning of such systems. (Hammarstrom et al. 2012, 2014; Yazdi and Ardekani 2012; Antfolk et al. 2014; Devendran et al. 2014; Ohlin et al. 2015; Cheung et al. 2014; Huang et al. 2014; Patel et al. 2014; Destgeer et al. 2016; Rogers and Neild 2011; Tang and Hu 2015; Leibacher et al. 2015; Agrawal et al. 2013, 2015).
The ability to use ultrasonic fields for manipulation of particles and fluids has a long history which can date back to many eminent scientists including Chladni (1787), Faraday (1831), Kundt and Lehmann (1874), Rayleigh (1883), King (1934), Gorkov (1962). As early as 1787, the German physicist Chladni (1787) observed that randomly distributed sand particles on a vibrating metal plate could group along the nodal lines forming a wide variety of symmetrical patterns. The various patterns formed at different modes of resonance were called Chladni figures. Chladni also reported that fine particles would move in the opposite direction, to the antinodes, which was further studied by Faraday (1831), who found that it was due to air currents in the vicinity of the plate, i.e. acoustic streaming. The latter phenomenon was revisited by Van Gerner et al. (2010, 2011) who showed that it will always occur when the acceleration of the resonating plate is lower than gravity acceleration. Zhou et al. (2016) recently proposed an approach which is able to control the motion of multiple objects simultaneously and independently on a Chladni plate.
Recently, Vuillermet et al. (2016) demonstrated that it is possible to form two-dimensional inverse Chladni patterns on a vibrating circular plate in water at microscale, which extended an earlier work from Dorrestijn et al. (2007), who showed formation of one-dimensional (1D) Chladni patterns on a vibrating cantilever submerged in water, where microparticles and nanoparticles were found to move to the antinodes and nodes of the vibrating interface, respectively. Both works have depicted the two-dimensional streaming field in the near-field and emphasized the effects of in-plane streaming flow on the collections of particles at vibrating antinodes or nodes. Practical manipulation on vibrating plates, however, is three-dimensional (3D) including out-of-plane and in-plane manipulation, and interestingly, in such systems, little work has been done on the impact of acoustic radiation forces, the main engine for particle and cell manipulation in other acoustofluidic manipulation devices. Unlike microparticle acoustophoresis in bulk and surface standing wave devices that have been well studied (Barnkob et al. 2012; Muller et al. 2012, 2013; Lei et al. 2014; Hahn et al. 2015; Nama et al. 2015; Oberti et al. 2009), the literature is lacking a quantitative analysis of microparticle acoustophoresis over vibrating plate systems.
In this paper, we will show a detailed 3D study on the main forces for the formation of inverse Chladni patterns on a clamped vibrating circular plate in contact with water (see Fig. 1 for the configuration). Both out-of-plane and in-plane microparticle acoustophoresis are discussed and the contributions of main driving forces are compared, which enables a clear presentation of the underlying physics of microparticle manipulation in such systems. The many key parameters, including the plate thickness and radius, the vibration amplitude and the fluid viscosity, on the microparticle acoustophoresis are discussed. We believe that this work could provide an excellent tool on analysing microparticle acoustophoresis in vibrating plate systems and on guiding device designs for the better control of patterning of microparticles at various sizes as well as for single particle and cell manipulation.
Sketch of a clamped vibrating circular plate in contact with water, where \(R\) and \(h\) are the radius and thickness of the circular plate, respectively
2 Numerical method
We use bold and normal-emphasis fonts to represent vector and scalar quantities, respectively. Here, we assume a homogeneous isotropic fluid, in which the continuity and momentum equations for the fluid motion are.
$$\frac{\partial \rho }{\partial t} + \nabla \cdot \left( {\rho \varvec{u}} \right) = 0,$$
$$\rho \left( {\frac{{\partial \varvec{u}}}{\partial t} + \varvec{u} \cdot \nabla \varvec{u}} \right) = - \nabla p + \mu \nabla^{2} \varvec{u} + \left( {\mu_{b} + \frac{1}{3}\mu } \right)\nabla \nabla \cdot \varvec{u},$$
(1b)
where \(\rho\) is the fluid density, t is time, \(\varvec{u}\) is the fluid velocity, p is the pressure and μ and μ b are, respectively, the dynamic and bulk viscosity coefficients of the fluid.
Taking the first and second order into account, we write the perturbation series of fluid density, pressure and velocity: (Bruus 2012)
$$\rho = \rho_{0} + \rho_{1} + \rho_{2} ,$$
$$p = p_{0} + p_{1} + p_{2} ,$$
$$\varvec{u} = \varvec{u}_{1} + \varvec{u}_{2} ,$$
(2c)
where the subscripts 0, 1 and 2 represent the static (absence of sound), first-order and second-order quantities, respectively. Substituting Eq. (2) into Eq. (1) and considering the equations to the first order, Eq. (1) for solving the first-order acoustic velocity take the form,
$$\frac{{\partial \rho_{1} }}{\partial t} + \rho_{0} \nabla \cdot \varvec{u}_{1} = 0,$$
$$\rho_{0} \frac{{\partial \varvec{u}_{{\mathbf{1}}} }}{\partial t} = - \nabla p_{1} + \mu \nabla^{2} \varvec{u}_{{\mathbf{1}}} + \left( {\mu_{b} + \frac{1}{3}\mu } \right)\nabla \nabla \cdot \varvec{u}_{{\mathbf{1}}} .$$
Repeating the above procedure, considering the equations to the second order and taking the time average of Eq. (1) using Eq. (2), the continuity and momentum equations for solving the second-order time-averaged acoustic streaming velocity can be turned into
$$\nabla \cdot \overline{{\rho_{1} \varvec{u}_{{\mathbf{1}}} }} + \rho_{0} \nabla \cdot \overline{{\varvec{u}_{{\mathbf{2}}} }} = 0,$$
$$- \nabla \overline{{p_{2} }} + \mu \nabla^{2} \overline{{\varvec{u}_{{\mathbf{2}}} }} + \left( {\mu_{b} + \frac{1}{3}\mu } \right)\nabla \nabla \cdot \overline{{\varvec{u}_{{\mathbf{2}}} }} + \varvec{F} = 0,$$
$$\varvec{F} = - \rho_{0} \overline{{\varvec{u}_{{\mathbf{1}}} \nabla \cdot \varvec{u}_{{\mathbf{1}}} + \varvec{u}_{{\mathbf{1}}} \cdot \nabla \varvec{u}_{{\mathbf{1}}} }} ,$$
where the upper bar denotes a time-averaged value and \(\varvec{F}\) is the Reynolds stress force (Lighthill 1978). When modelling the steady-state streaming flows in most practical acoustofluidic manipulation devices, the inertial force \(\overline{{\varvec{u}_{{\mathbf{2}}} }} \cdot \nabla \overline{{\varvec{u}_{{\mathbf{2}}} }}\) is generally negligible compared to the viscosity force in such systems, which results in the creeping motion. The divergence-free velocity \(\overline{{\varvec{u}_{{\mathbf{2}}}^{\varvec{M}} }} = \overline{{\varvec{u}_{{\mathbf{2}}} }} + \overline{{\rho_{1} \varvec{u}_{{\mathbf{1}}} }} /\rho_{0}\), derived from Eq. (4a), is the mass transport velocity of the acoustic streaming, which is generally closer to the velocity of tracer particles in a streaming flow than \(\overline{{\varvec{u}_{2} }}\) (Nyborg 1998).
In this work, only the boundary-driven streaming field was solved because an evanescent wave field is established (see below) such that the overall streaming field is dominated by the boundary-driven streaming. Moreover, as the inner streaming vortices are confined only at the thin viscous boundary layer [thickness of \(\delta_{v} \approx 0.6\) µm at 1 MHz in water (Bruus 2012)], for numerical efficiency, we solved only the 3D outer streaming fields using Nyborg's limiting velocity method (Nyborg 1958; Lee and Wang 1989) as those published previously (Lei et al. 2013, 2014, 2016). Although the inner streaming fields were not computed in this work, they can, of course, be known from the limiting velocity field.
3 Numerical model, results and discussion
To validate the numerical results, a clamped circular plate of radius R = 800 µm and thickness h = 5.9 µm was firstly considered, which has a same size to the one used in Vuillermet et al.'s experiments (2016). Our model is slightly different to the device in Vuillermet et al.'s experiments. It can be seen from Fig. 1 that our model shows a vibrating clamped plate in a free space while the side boundaries of Vuillermet et al.'s device have sound reflections, which may result in acoustic pressure antinodes at the plate boundaries. More model parameters are found in Table 1. The model configuration is shown in Fig. 3a, where a cylindrical fluid-channel-only model was considered. Cartesian (\(x, y, z\)) and cylindrical (\(r, \theta , z\)) coordinates were used for the convenience of calculations. The finite element package COMSOL 5.2 (COMSOL Multiphysics 2015) was used to solve all equations. The modelled final particle (radius of 30 µm) positions driven by the main forces including acoustic radiation forces, streaming-induced drag forces and buoyancy forces at two vibrating modes are shown in Fig. 2a. It can be seen that the inverse Chladni patterns the microparticles form compare well with Vuillermet et al.'s (2016) experimental observations. In the following, we will show step by step why microparticles are gathered to the vibrating antinodes forming inverse Chladni patterns and the contributions of various driving forces on the acoustophoretic motion of microparticles at various sizes.
Model parameters
Model domain
\(\pi R^{2} \times h\)
\(\pi \times 0.8^{2} \times 0.725\)
Density of plate
kg m−3
Plate Poisson's ratio
Plate Young's modulus
\(E\)
Sound speed in plate
\(u\)
m s−1
Particle density
\(\rho_{p}\)
Sound speed in particle
\(c_{p}\)
Density of water
\(\rho_{f}\)
Sound speed in water
\(c_{f}\)
(Colour online) Top views of the final positions of microparticles (radius of 30 µm) on a plate at various vibrating modes: a modelled, where spheres are the microparticles and colours show the vibrating displacements (white for maximum and black for zero); and b measured, adapted with permission from Vuillermet et al. (2016) Copyrighted by the American Physical Society. The particle properties used in simulations are included in Table 1
It is noteworthy that we have previously applied a fluid-channel-only model to study the 3D transducer-plane streaming fields in bulk acoustofluidic manipulation devices (Lei 2015), where the excitation of transducer was approximated by a Gaussian distribution of boundary vibration. The fluid-channel-only model applied in this work has more merits because we can easily write down the displacement equation when the circular plate vibrates at a resonant mode (see Eq. (6) below), and thus, there is no need to make an approximation on the boundary vibrations as we did in the previous models (Lei et al. 2013, 2016).
3.1 Resonant frequencies
Resonant frequencies at various modes were firstly modelled, which are shown in Table 2. For comparison, the modelled eigenfrequencies of first eight modes for another two cases, namely no load and load with air, are also presented. It can be seen that the resonant frequencies for vibrations in air and those in vacuum are very close; differences are small enough to be considered as numerical errors, suggesting that omitting the influence of air does not introduce any significant error on the resonant frequencies. The resonant frequencies of vibration in contact with water, however, have been reduced at least by a factor of 3 for all the modes presented, which means that we have to consider the influence of external load introduced by the surrounding water. All the results shown in this paper are for the (4, 1) mode (\(\delta_{v} \approx 1.84\) µm) unless otherwise stated.
The modelled resonant frequencies (Hz) of first eight modes for various loads
The computations were performed on a Lenovo Y50 running Windows 8 (64-bit) equipped with 16 GB RAM and Intel(R) Core(TM) i7-4710HQ processor of clock frequency 2.5 GHz. The mesh constitution was chosen based on the method described in a previous work (Lei et al. 2013), which chooses the mesh size to obtain steady solutions, i.e. ensuring further refining of mesh does not change the solution significantly. This model resulted in 131,521 mesh elements, a peak RAM usage of 4.96 GB (at the acoustic step), and a running time of about 4 h for solving the steps described between Sects. 3.2 and 3.6 below.
3.2 First-order acoustic fields
The first-order acoustic fields were modelled using the COMSOL 'Pressure Acoustics, Frequency Domain' interface, which solves the harmonic, linearized acoustic problem, taking the form,
$$\nabla^{2} p_{1} + \frac{{\omega^{2} }}{{c^{2} }}p_{1} = 0,$$
where ω is the angular frequency and c is the speed of sound in the fluid. The acoustic fields in the model regime were created by a harmonic vibration of the bottom edge (i.e. the plate) coupled with radiation boundary conditions on all other edges. For comparison, we also tried adding perfect matching layers around the cylindrical domain to absorb all outgoing waves and found that the differences on all the modelled quantities between these two methods are within 3%. To give a clear presentation of results, we show here the results modelled form radiation boundary conditions.
For a \(\left( {m, n} \right)\) vibrating mode, the plate displacement amplitude can be written as
$$w = J_{m} \left( {\frac{{\alpha_{mn} }}{R}r} \right)\cos \left( {m\theta } \right),$$
where \(J_{m} \left( \cdot \right)\) is the Bessel function of the first kind of order m and \(\alpha_{mn}\) is the nth zero of \(J_{m} \left( \cdot \right)\). The results presented in this paper were obtained at a vibration amplitude of 0.4 µm unless otherwise stated. The vibration amplitude has a limited effect on the shape of microparticle trajectories as both the acoustic radiation force and streaming-induced drag force scale with the square of the vibration amplitude (more discussions can be found below).
As shown in Fig. 3b, a standing wave field was established on the vibrating interface with acoustic pressure nodes and antinodes locating at plate displacement nodes and antinodes, respectively. The standing wave field is shown more clearly in Fig. 3d, where the in-plane circumferential acoustic pressure magnitudes are plotted. The out-of-plane acoustic pressure magnitudes over a vibrating antinode are plotted in Fig. 3c, which shows that the acoustic pressure magnitudes decay exponentially with the increase in distance from the vibrating interface. The reason is that the plate wave travels at the vibrating interface at a subsonic regime leading to an evanescent wave field: the plate wave velocity at substrate surface \(u = \lambda f_{r} \approx 55\) m/s \(\ll u_{l}\), where λ is the acoustic wavelength, f r is the resonant frequency and u l is the speed of sound in the liquid.
(Colour online) a Geometry of the considered problem, where the bottom edge (\(z = 0\)) vibrates at a (4, 1) mode; b 3D acoustic pressure magnitudes (\(\left| {p_{1} } \right|\), Pa); c out-of-plane \(\left| {p_{1} } \right|\) [arrow in (b)]; and d in-plane \(\left| {p_{1} } \right|\) on \(r = 0.56\) mm at the bottom edge. \(r = \sqrt {x^{2} + y^{2} }\) and \(\theta = { \arctan }\left( {y/x} \right)\). The dashed line and the equation in (c) show the exponential fitting of the modelled acoustic pressure magnitudes
3.3 Acoustic radiation forces
The corresponding 3D acoustic radiation forces were solved from the Gorkov equation (Gorkov 1962),
$$\varvec{F}_{{\varvec{ac}}} = \nabla \left\{ {V_{0} \left[ {\frac{{3\left( {\rho_{p} - \rho_{f} } \right)}}{{2\rho_{p} + \rho_{f} }}\overline{{E_{kin} }} - \left( {1 - \frac{{\beta_{p} }}{{\beta_{f} }}} \right)\overline{{E_{pot} }} } \right]} \right\},$$
where \(\overline{{E_{kin} }}\) and \(\overline{{E_{pot} }}\) are the time-averaged kinematic and potential energy, \(\rho_{p}\) and \(\rho_{f}\) are, respectively, the density of particle and fluid, \(\beta_{p} = 1/\left( {\rho_{p} c_{p}^{2} } \right)\) and \(\beta_{f} = 1/\left( {\rho_{f} c_{f}^{2} } \right)\) are the compressibility of particle and fluid, and \(V_{0}\) is the particle volume (see Table 1 for model properties). Equation (7) is valid for particles that are small compared to the acoustic wavelength λ in the limit \(r_{0} /\lambda \ll 1\) (where r 0 is the radius of the particle) in an inviscid fluid in an arbitrary sound field. (Gorkov 1962) When a particle moves close to the vibrating plate, the acoustic radiation forces may oscillate weakly with a decrease in distance to the plate due to the multiple-scattering interaction and wall interference, while the force magnitudes will not be significantly affected (Wang and Dual 2012).
The modelled acoustic radiation force fields are shown in Fig. 4. As shown in Fig. 4c, the out-of-plane acoustic radiation forces also decrease exponentially with the increase in distance from the vibrating interface. In the near-field, at this vibrating amplitude, the out-of-plane acoustic radiation forces have a greater contribution on the sedimentation of microparticles than the buoyancy forces. With an increase in vibration amplitude, we can expect dominant out-of-plane acoustic radiation forces over buoyancy forces. Interestingly, as shown in Fig. 4b, the in-plane acoustic radiation forces carry microparticles away from the acoustic pressure nodes and converge at antinodes from all directions, in contrast with the conditions usually found in bulk and surface standing wave manipulation devices, where the acoustic radiation forces move most particles and cells of interest to the acoustic pressure nodes (Glynne-Jones et al. 2012). Examining Eq. (7), it can be seen that the acoustic radiation force is a gradient of the force potential, which contains a positive contribution from the kinematic energy (weighted by a function of the fluid and particle densities) and a negative contribution from the potential energy (weighted by a function of the fluid and particle compressibility). Comparing the contributions of these two terms in this model, it was found that the kinetic energy term dominates in the force potential, which drives microparticles to the vibrating antinodes.
(Colour online) a 3D acoustic radiation force magnitudes (\(\left| {F_{ac} } \right|\), N) on a particle with a radius of 30 µm; b in-plane \(\left| {F_{ac} } \right|\); and c out-of-plane \(\left| {F_{ac} } \right|\) [red arrow in (a)], where the inset shows the directions of the plotted forces above a vibrating antinode. \(F_{B}\) and \(F_{G}\) are the buoyancy and gravity, respectively. The dashed line and the equation in (c) show the exponential fitting of the modelled acoustic radiation force
3.4 Acoustic streaming fields
The 3D acoustic streaming field was modelled using Nyborg's limiting velocity method (Nyborg 1958; Lee and Wang 1989). It was shown that if the boundary has a radius of curvature that is much larger than the acoustic boundary layer, then the time-averaged velocity at the extremity of the inner streaming (the 'limiting velocity') can be approximated as a function of the local, first-order linear acoustic field. The outer streaming in the bulk of the fluid can then be predicted by a fluidic model that takes the limiting velocity as a boundary condition. The applicability and viability of the limiting velocity method have been further discussed recently (Lei et al. 2017). In Cartesian coordinates, the limiting velocity field at the driving boundaries (\(z = 0\)) can be written as
$$u_{L} = - \frac{1}{4\omega }\text{Re} \left\{ {q_{x} + u_{1}^{*} \left[ {\left( {2 + i} \right)\nabla \cdot \varvec{u}_{{\mathbf{1}}} - \left( {2 + 3i} \right)\frac{{dw_{1} }}{dz}} \right]} \right\},$$
$$v_{L} = - \frac{1}{4\omega }\text{Re} \left\{ {q_{y} + v_{1}^{*} \left[ {\left( {2 + i} \right)\nabla \cdot \varvec{u}_{{\mathbf{1}}} - \left( {2 + 3i} \right)\frac{{dw_{1} }}{dz}} \right]} \right\},$$
$$q_{x} = u_{1} \frac{{du_{1}^{*} }}{dx} + v_{1} \frac{{du_{1}^{*} }}{dy},$$
$$q_{y} = u_{1} \frac{{dv_{1}^{*} }}{dx} + v_{1} \frac{{dv_{1}^{*} }}{dy},$$
(8d)
where u L and v L are the x- and y-components of the limiting velocity field, u 1, v 1 and w 1 are the x-, y- and z-components of the acoustic velocity vector \(\varvec{u}_{{\mathbf{1}}}\), \(\text{Re} \left\{ \cdot \right\}\) denotes the real part of a complex value and \(*\) is the complex conjugate.
A COMSOL 'Creeping Flow' interface was used to model the acoustic streaming field, which solves
$$\nabla \cdot \overline{{\varvec{u}_{{\mathbf{2}}} }} = 0,$$
$$\nabla p_{2} = \mu \nabla^{2} \overline{{\varvec{u}_{{\mathbf{2}}} }} .$$
As only outer streaming fields are solved in this method, with the assumption of low velocity and incompressible flow, the first term in the left-hand side of Eq. (4a) is zero and thus \(\overline{{\varvec{u}_{{\mathbf{2}}} }} = \overline{{\varvec{u}_{{\mathbf{2}}}^{\varvec{M}} }}\) (Hamilton et al. 2003). Then, as discussed by Lighthill (1978), the Reynolds stress in the bulk of the fluid can set up hydrostatic stresses, but in the absence of attenuation these will not create vortices, hence these terms are not included in Eq. (9b). The 3D outer acoustic streaming fields in the considered model regime were generated by the limiting velocity field on the vibrating interface (see Fig. 5a) along with no-slip boundary conditions (\(\overline{{\varvec{u}_{{\mathbf{2}}} }} = 0\)) on all other edges.
(Colour online) a The limiting velocity field (m/s) on the bottom edge (\(z = 0\)); and b, c front and left views of the 3D acoustic streaming fields, where the colours at the bottom edge in (b, c) show the acoustic pressure magnitudes (red for maximum and blue for zero). To give a clear presentation of the 3D acoustic streaming flows, only those above one acoustic pressure antinode are shown. Arrows in (b, c) show the streaming directions
The limiting velocity field and the 3D acoustic streaming fields are shown in Fig. 5. It can be seen that, similar to the distribution of in-plane acoustic radiation forces, the limiting velocities (i.e. the in-plane acoustic streaming velocity field) converge at the acoustic pressure antinodes from all directions leading to acoustic streaming vortices on out-of-planes perpendicular to the vibrating interface as those plotted in Fig. 5b, c, where, in order to give a clear demonstration of the 3D acoustic streaming fields, only the acoustic streaming vortices above one acoustic pressure antinode are plotted.
3.5 Acoustic streaming-induced drag forces
Based on the acoustic streaming velocity field, we can calculate the acoustic streaming-induced drag forces on microparticles from the stokes drag,
$$\varvec{F}_{\varvec{d}} = 6\mu \pi r_{0} \left( {\overline{{\varvec{u}_{{\mathbf{2}}} }} - \varvec{v}} \right),$$
where \(\varvec{v}\) is the particle velocity. Equation (10) is valid for particles sufficiently far from the channel walls (Happel 1965). Since microparticle acoustophoresis discussed in this work is closely associated with the vibrating plate, it is necessary to take into account the wall effect on the streaming-induced drag forces when a particle moves close to the bottom wall. When a sphere particle moves perpendicularly towards or in parallel to the vibrating plate, the streaming-induced drag force should be corrected by multiplying a wall-effect-correction factor χ or γ, respectively, which can be expressed as (Happel 1965)
$$\chi = \frac{4}{3}\sinh \alpha \mathop \sum \limits_{i = 1}^{\infty } \frac{{i\left( {i + 1} \right)}}{{\left( {2i - 1} \right)\left( {2i + 3} \right)}} \times \left[ {\frac{{2\sinh \left( {2i + 1} \right)\alpha + \left( {2i + 1} \right)\sinh 2\alpha }}{{4\sinh^{2} \left( {i + 1/2} \right)\alpha - \left( {2i + 1} \right)^{2} \sinh^{2} \alpha }} - 1} \right],$$
$$\gamma = \frac{1}{{1 - A\left( {r_{0} /H} \right) + B\left( {r_{0} /H} \right)^{3} - C\left( {r_{0} /H} \right)^{4} - D\left( {r_{0} /H} \right)^{5} }},$$
(11b)
$$\alpha = \cosh^{ - 1} \left( {H/r_{0} } \right),$$
(11c)
where \(H\) is the distance from the centre of the particle to the plate and A = 9/16, B = 1/8, C = 45/256 and D = 1/16.
The 3D acoustic streaming-induced drag forces are shown in Fig. 6, where, for comparison, the buoyancy forces are also plotted. As shown in Fig. 6c, with the increase in distance from the vibrating interface, the out-of-plane streaming-induced drag forces rise rapidly to the maximum value in the near-field and then fall gradually to zero in the far-field. The wall effect can increase the maximum our-of-plane streaming-induced drag force by approximately a factor of 2 in this model. Also, it can be seen that, for a small vibration amplitude of w = 0.4 µm, the maximum out-of-plane streaming-induced drag force is larger than the buoyancy force on a particle with a radius of 30 µm. With an increase in vibration amplitude, we can expect even larger acoustic streaming-induced drag forces while the buoyancy forces remain the same. Therefore, it might be reasonable to say that introducing only the streaming effects is not enough to explain the sedimentation of microparticles, especially for those with r 0 < 30 µm, where the differences between the out-of-plane streaming-induced drag forces and the buoyancy forces are even larger, as plotted in Fig. 6d, because the former and the latter scale with the particle radius and particle volume, respectively.
(Colour online) a 3D streaming-induced drag forces on a particle with a radius of 30 µm (\(\left| {F_{d} } \right|\), N); b in-plane \(\left| {F_{d} } \right|\); c out-of-plane \(\left| {F_{d} } \right|\) [red arrow in (a)]; and d comparisons of maximum out-of-plane \(\left| {F_{d} } \right|\) [peak in (c)] with the buoyancy forces for various particle sizes (radius of \(r_{0}\)). The inset in (c) shows the directions of the plotted forces above a vibrating antinode. \(F_{B}\) and \(F_{G}\) are the buoyancy and gravity, respectively
3.6 Microparticle trajectories
From the acoustic radiation forces and streaming-induced drag forces that have been calculated, together with the buoyancy forces, microparticle (polystyrene beads) trajectories were modelled, following
$$\frac{d}{dt}\left( {m_{p} \varvec{v}} \right) = \varvec{F}_{\varvec{d}} + \varvec{F}_{{\varvec{ac}}} + \varvec{F}_{\varvec{B}} + \varvec{F}_{\varvec{G}} ,$$
$$\varvec{F}_{\varvec{B}} + \varvec{F}_{\varvec{G}} = \frac{4}{3}\pi r_{0}^{3} g\left( {\rho_{f} - \rho_{p} } \right),$$
where m p is the particle mass, \(\varvec{F}_{\varvec{B}}\) is the buoyancy, \(\varvec{F}_{\varvec{G}}\) is the gravity and g is the gravity acceleration. In this work, it is assumed that all the forces, including acoustic radiation, streaming-induced drag and buoyancy forces, act on the centre of spherical particles (otherwise, integration of forces over the particle surface would be needed when the particles are close to the boundaries). It is noteworthy that, in addition to these main driving forces, a particle–particle interaction force was used in this model. The particle–particle interaction force can be expressed as
$$\varvec{F} = - k_{s} \mathop \sum \limits_{i = 1}^{N} \left( {\left| {\varvec{r} - \varvec{r}_{\varvec{i}} } \right| - r_{e} } \right)\frac{{\varvec{r} - \varvec{r}_{\varvec{i}} }}{{\varvec{r} - \varvec{r}_{\varvec{i}} }},$$
where k s is the spring constant, \(\varvec{r}_{\varvec{i}}\) is the position vector of the ith particle, and r e is the equilibrium position between particles. In this model, k s = 2.5 × 10−4 N/m for polystyrene beads (Jensenius and Zocchi 1997) and r e was set as 2r 0 to avoid all particles being concentrated to a single point.
Here, a COMSOL 'Particle Tracing for Fluid Flow' interface was used to solve Eq. (12) to model the particle trajectories. The shape of the trajectories is independent of the pressure amplitude since both the acoustic radiation forces and steaming-induced drag forces scale with the square of pressure; results are presented here for an excitation amplitude of w = 0.4 µm. An array of tracer particles (given the properties of polystyrene beads of radius 30 µm) are seeded at \(t = 0\). Acoustic radiation forces, streaming-induced drag forces and buoyancy forces act on the particles, resulting in the motion shown in Fig. 7. It can be seen that, in the considered model regime, particles with a radius of 30 µm first move towards the vibrating interface driven by the predominant out-of-plane forces and are then carried to their closest acoustic pressure antinodes by in-plane forces, resulting in spider-like trajectories and inverse Chladni patterns on the vibrating interface within seconds. Generally, particles closer to the vibrating interface take less time to settle for stronger driving forces. Particles with smaller sizes take longer to locate at the acoustic pressure antinodes for smaller driving forces and will follow the out-of-plane streaming vortices leading to acoustic streaming-dominated trajectories close to those shown in Fig. 5b, c while r 0 < 6.9 µm (see explanations below and videos in the Supplemental material).
(Colour online) Trajectories of microparticles (radius of 30 µm) at: a \(t = 0\); and b \(t = 3\) s. Spheres are the microparticle, black solid lines show particle trajectories and colours at the bottom edge show the vibrating displacements (white for maximum and black for zero). See video in the Supplemental material
Out-of-plane acoustophoresis. A single particle out-of-plane acoustophoresis is directly acted upon by the acoustic radiation force, the buoyancy force and the acoustic streaming-induced drag force. The equation of motion for a spherical particle of out-of-plane velocity \(\varvec{v}^{{\varvec{out}}}\) above an acoustic pressure antinode is then
$$\varvec{v}^{{\varvec{out}}} = \frac{{\varvec{F}_{\varvec{d}}^{{\varvec{out}}} + \varvec{F}_{{\varvec{ac}}}^{{\varvec{out}}} + \varvec{F}_{\varvec{B}} + \varvec{F}_{\varvec{G}} }}{{6\pi \mu r_{0} }}.$$
As we have seen above, particles are concentrated at the acoustic pressure antinodes, so we take here a particle staying above an acoustic pressure antinode to analyse the contributions of the many forces on the microparticle out-of-plane acoustophoresis. As shown in the inset of Fig. 8a, the streaming-induced drag forces, \(\varvec{F}_{\varvec{d}}^{{\varvec{out}}}\), competes with other forces above an acoustic pressure antinode as the acoustic streaming flow drives particles away from the pressure antinode, while other forces bring particles to the pressure antinode. Based on the fact that
$$\varvec{F}_{\varvec{d}}^{{\varvec{out}}} \propto r_{0}\quad {\text{and }}\quad \varvec{F}_{{\varvec{ac}}}^{{\varvec{out}}} + \varvec{F}_{\varvec{B}} + \varvec{F}_{\varvec{G}} \propto r_{0}^{3} ,$$
there should be a threshold out-of-plane particle size, \(r_{0}^{out}\): for \(r_{0} > r_{0}^{out}\), particles can be easily concentred to the acoustic pressure antinodes; while for \(r_{0} < r_{0}^{out}\), particles will follow the out-of-plane acoustic streaming vortices. We define the threshold particle radius \(r_{0}^{out}\) for crossover from these out-of-plane forces. The out-of-plane forces on particles at various sizes are plotted in Fig. 8a, which shows that, at a small vibration amplitude of w = 0.4 µm, the threshold particle size \(r_{0}^{out} \approx 6.9\) µm. Considering the wall-effect-correction for the streaming-induced drag forces, \(r_{0}^{out} \approx 9.1\) µm. This threshold out-of-plane particle size may slightly vary with the vibration amplitude as \(\varvec{F}_{\varvec{B}} + \varvec{F}_{\varvec{G}}\) are independent of the vibration amplitude, while \(\varvec{F}_{\varvec{d}}^{{\varvec{out}}}\) and \(\varvec{F}_{{\varvec{ac}}}^{{\varvec{out}}}\) scale with the square of the vibration amplitude. As shown in Fig. 4c, the buoyancy force is approximately 1/20 of \(\varvec{F}_{{\varvec{ac}}}^{{\varvec{out}}}\) at \(w = 0.4\) µm on the vibrating interface. With an increase in vibration amplitude, the contribution of buoyancy force will be even smaller on the microparticle acoustophoresis in the near-field. To calculate the limit value of \(r_{0}^{out}\), we can set
$$\varvec{F}_{{\varvec{ac}}}^{{\varvec{out}}} + \varvec{F}_{\varvec{d}}^{{\varvec{out}}} = 0$$
by ignoring the buoyancy forces, which gives
$$r_{0}^{out} = \sqrt {\frac{{\left| {F_{d}^{out} } \right|}}{{\left| {F_{ac}^{out} } \right|}}} r_{0} \approx 7.1 \mu m.$$
Considering the wall-effect-correction for the streaming-induced drag forces, the limit value of \(r_{0}^{out} \approx 9.4\) µm.
(Colour online) Comparisons of magnitudes of a out-of-plane forces and b in-plane forces on particles with various sizes (radius of \(r_{0}\)). The insets show the directions of the plotted forces above a vibrating antinode. \(F_{ac}\), \(F_{d}\), \(F_{B}\) and \(F_{G}\) are the acoustic radiation force, streaming-induced drag force, buoyancy and gravity, respectively. The in-plane forces are the average values over the bottom edge
In-plane microparticle acoustophoresis. For the in-plane microparticle acoustophoresis, it is acted upon by the acoustic radiation force and the streaming-induced drag force. Similar to the analyses above, the equation of motion for a spherical particle of in-plane velocity \(\varvec{v}^{{\varvec{in}}}\) is then
$$\varvec{v}^{{\varvec{in}}} = \frac{{\varvec{F}_{\varvec{d}}^{{\varvec{in}}} + \varvec{F}_{{\varvec{ac}}}^{{\varvec{in}}} }}{{6\pi \mu r_{0} }}.$$
As shown in Figs. 4b and 6b, both the in-plane acoustic radiation force, \(\varvec{F}_{{\varvec{ac}}}^{{\varvec{in}}}\), and the streaming-induced drag force, \(\varvec{F}_{\varvec{d}}^{{\varvec{in}}}\), move microparticles to the acoustic pressure antinodes (see also the inset in Fig. 8b). To evaluate the contributions of these two forces on the in-plane microparticle acoustophoresis, we compare their average values over the plate interface because considering the maximum force only may not be accurate. Since both of these in-plane forces point to the acoustic pressure antinodes, they jointly contribute to the focusing of microparticles to the acoustic pressure antinodes provided that the particle sizes are big enough to avoid being driven away from the vibrating interface by out-of-plane acoustic streaming vortices (as discussed in the previous step), which could provide evidence for the much larger particle velocities measured in experiments when compared with the predicted streaming velocities as shown in Vuillermet et al. (2016) work.
Although there is no threshold in-plane particle size for the reason that both the in-plane acoustic radiation force and streaming-induced drag force drive microparticles to the acoustic pressure antinodes, we can figure out the contribution of each force on the in-plane microparticle acoustophoresis. Again, based on the fact that
$$\varvec{F}_{\varvec{d}}^{{\varvec{in}}} \propto r_{0}\quad {\text{and}}\quad \varvec{F}_{{\varvec{ac}}}^{{\varvec{in}}} \propto r_{0}^{3} ,$$
we can expect a critical in-plane particle size, \(r_{0}^{in}\): for \(r_{0} > r_{0}^{in}\), \(\varvec{F}_{{\varvec{ac}}}^{{\varvec{in}}}\) contribute more to the in-plane acoustophoresis; while for \(r_{0} < r_{0}^{out}\), \(\varvec{F}_{\varvec{d}}^{{\varvec{in}}}\) have a higher contribution. The value of \(r_{0}^{in}\) can be found from setting \(\varvec{F}_{{\varvec{ac}}}^{{\varvec{in}}} = \varvec{F}_{\varvec{d}}^{{\varvec{in}}}\), which gives
$$r_{0}^{in} = \sqrt {\frac{{\left| {\varvec{F}_{\varvec{d}}^{{\varvec{in}}} } \right|}}{{\left| {\varvec{F}_{{\varvec{ac}}}^{{\varvec{in}}} } \right|}}} r_{0} \approx 15.7\,\upmu{\text{m}}.$$
Considering the wall-effect-correction for the streaming-induced drag forces, \(r_{0}^{in} \approx 27.6\) µm. The in-plane forces on particles at various sizes are plotted in Fig. 8b. It is noteworthy that, different to the situation for \(r_{0}^{out}\), \(r_{0}^{in}\) is independent of the vibration amplitude \(w\) because both \(\varvec{F}_{{\varvec{ac}}}^{{\varvec{in}}}\) and \(\varvec{F}_{\varvec{d}}^{{\varvec{in}}}\) scale with the square of \(w\).
Actually, it can be seen from Eqs. (17) and (20) that, ignoring the small effect of buoyancy forces in the near-field, the relationships between the in-plane and out-of-plane threshold particle sizes and the ratios of the corresponding streaming-induced drag force and acoustic radiation force are
$$r_{0}^{in} , r_{0}^{out} = \sqrt {\frac{{\left| {\varvec{F}_{\varvec{d}} } \right|}}{{\left| {\varvec{F}_{{\varvec{ac}}} } \right|}}} r_{0} .$$
4 Effects of key parameters on microparticle acoustophoresis
Having demonstrated the acoustophoresis of microparticles at various sizes for a particular plate (thickness of 5.9 µm and radius of 0.8 mm), in this section, we investigate the effects of many key parameters, including the plate radius and thickness and the fluid viscosity, on the performance of microparticle acoustophoresis in order to facilitate device design for a wide range of applications.
Effects of fluid viscosity. On the one hand, it can be seen from Eq. (8) that the magnitudes of limiting velocities (i.e. the strength of the outer streaming velocities) are independent of the fluid viscosity even though viscosity is the initial cause of acoustic streaming flows. Thus, with a change in fluid viscosity, the streaming-induced drag force, \(\varvec{F}_{\varvec{d}}\), scales linearly with \(\mu\), while \(\varvec{F}_{{\varvec{ac}}}\) will remain the same. From Eq. (21), the following relationships are established,
$$r_{0}^{in} , r_{0}^{out} \propto \sqrt \mu .$$
Therefore, to eliminate the 'side effect' of streaming flows on the microparticle manipulation, and we can conclude that lowering the fluid viscosity is a viable way to augment the weight of acoustic radiation force on microparticle acoustophoresis.
Effects of plate thickness and radius. To investigate the effects of plate thickness (\(h\)) and radius (\(R\)) on the microparticle acoustophoresis, we considered a series of h and R ranging from 2 to 14 µm and 0.3 to 1.4 mm, respectively. When one parameter was studied, the other parameter was kept the same. For each case, following the whole numerical procedure described in the sections above, we calculated the threshold in-plane and out-of-plane particle sizes, which are shown in Fig. 9. It can be seen that these two threshold particle sizes have similar variation tendencies: they grow with the increase in R and fall with the rise of h.
Effects of plate radius on the threshold a in-plane particle sizes, \(r_{0}^{in}\), and b out-of-plane particle sizes, \(r_{0}^{out}\) (with wall effect). For (a, b), the plate thickness is the same, \(h = 5.9\) µm. For (c, d), the plate radius is the same, \(R = 0.8\) mm
Compare with basic theory. Turning to the theoretical aspect, as seen from Eq. (21), to determine how these two threshold particle sizes change with the many key parameters, we only need to figure out how the force ratio on the right-hand side varies with these parameters. If we define \(\varvec{v}^{{\varvec{rad}}}\) as the contribution of acoustic radiation force on the particle velocity, considering Eqs. (15) and (19), we have
$$\frac{{\left| {\varvec{F}_{\varvec{d}} } \right|}}{{\left| {\varvec{F}_{{\varvec{ac}}} } \right|}} = \frac{{\left| {\overline{{\varvec{u}_{{\mathbf{2}}} }} } \right|}}{{\left| {\varvec{v}^{{\varvec{rad}}} } \right|}}.$$
Examining the acoustic field in the near-field, it can be seen from Fig. 3b that, if expanded in the radial direction, the acoustic pressure field (as plotted in Fig. 3d) can be approximated to a 1D standing wave on all circumferences for \(0 < r \ll R\), in which the right-hand side of Eq. (23) has the following relation (Barnkob et al. 2012)
$$\frac{{\left| {\overline{{\varvec{u}_{{\mathbf{2}}} }} } \right|}}{{\left| {\varvec{v}^{{\varvec{rad}}} } \right|}} = \frac{6\mu }{{\varPhi \rho_{f} \omega r_{0}^{2} }},$$
where \(\varPhi \approx 0.1685\) in this work is the acoustic contrast factor and the thermoviscous effects are not included.
For a clamped circular plate with radius of \(R\) and thickness of h, the angular frequency for an unloaded case for each \(\left( {m, n} \right)\) mode follows (Leissa 1993)
$$\omega = \frac{{\alpha_{mn}^{2} }}{{R^{2} }}\sqrt {\frac{{Eh^{2} }}{{12\rho \left( {1 - \upsilon^{2} } \right)}}} ,$$
where \(E\) is the plate Young's modulus, ρ is the plate density and υ is the plate Poisson's ratio. Considering the surrounding water, for a given \(\left( {m, n} \right)\) mode, the angular frequency is reduced to
$$\omega = \frac{{\alpha_{mn}^{2} }}{{R^{2} }}\sqrt {\frac{{Eh^{2} }}{{12\rho \left( {1 - \upsilon^{2} } \right)}}} \frac{1}{C},$$
$$C = \sqrt {1 + \varGamma_{mn} \frac{{\rho_{f} }}{\rho }\frac{R}{h}} ,$$
where \(\varGamma_{mn}\) is the non-dimensional added virtual mass incremental (NAVMI) factor, values of which can be found in Ref. (Amabili and Kwak 1996), Table 5, in the case of a clamped plate.
Combining Eqs. (21), (23), (24) and (26), the relationships between the threshold in-plane particle sizes and the many key parameters in a 1D standing wave field can be expressed as
$$r_{0}^{in} = Rh^{ - 0.5} \left( {\frac{6\mu C}{{\varPhi \rho_{f} \alpha_{mn}^{2} }}} \right)^{0.5} \left[ {\frac{E}{{12\rho \left( {1 - \upsilon^{2} } \right)}}} \right]^{ - 0.25} .$$
The calculated values of r 0 in using Eq. (27) and those obtained from our model for various \(R\) and \(h\) are shown in Fig. 10. It can be seen that the modelled \(r_{0}^{in}\) compare reasonably well with the calculated values under the 1D standing wave approximation. The differences between the calculated values and those modelled may be attributed to the reason that, compared to an approximated 1D standing wave, the acoustic field in the near-field is a more complex pattern. However, due to the complexity of the problem, the good comparisons between our model and the calculated values indicate that the approximated 1D standing wave may have captured the main features of (4, 1) mode and our model can be applied to study the basic physics of microparticle acoustophoresis on vibrating plate systems for even more complex vibrating modes.
(Colour online) Comparisons on the threshold in-plane particle sizes between the modelling and theory, where the diamonds and squares show the modelled values calculated from the averaged and maximum forces over the bottom surface (with wall effect), respectively, and triangles show the calculated values using Eq. (27). For (a), the plate radius is the same, \(R = 0.8\) mm, and the plate thickness is the same for (b), \(h = 5.9\) µm
5 Mode switching
Eigenfrequency studies show that two orthogonal vibrating patterns for each (\(m,n\)) vibrating mode could be excited at two adjacent frequencies (typically hundreds of Hz difference) provided that the modes are high enough (\(m \ge 1\)). As shown in Fig. 11, the phase angle between two adjacent acoustic pressure antinodes of these two orthogonal patterns is
(Colour online) A schematic representation of the underlying mechanism for the circular manipulation of a single particle by continuous mode switching between two \(\left( {m, n} \right)\) orthogonal modes. To complete a full circle of movement (i.e. \(\theta = \pi /2m\)), 4 \(m\) times of mode switching are required
$$\theta = \frac{\pi }{2m}.$$
For this specific model, both the in-plane acoustic radiation force and streaming-induced drag force diverge from the vibrating nodes and converge at the vibrating antinodes, so when switching from one mode (e.g. mode 1 in Fig. 11) to the other orthogonal mode (e.g. mode 2 in Fig. 11) a particle tends to move from the vibrating antinode of the former to its closest antinode of the latter either clockwise or anticlockwise depending on the initial position of the particle (assuming the initial position of the particle slightly shifts from the vibrating antinode). The potential underlying mechanism for the circular manipulation of a single particle is schematically shown in Fig. 11. It can be seen that, for each mode switching, the particle can move by an angle, \(\theta = \pi /2m\), while its distance to the centre of the circular membrane will remain the same. To complete a full circle of manipulation, 4 \(m\) times of mode switching are required. This method is different from the mode switching proposed by Glynne-Jones et al. (2010) who showed that beads can be brought to any arbitrary point between the half and quarter-wave nodes when rapidly switching back and forth between half and quarter wavelength frequencies in bulk acoustofluidic devices.
We have investigated the 3D acoustophoretic motion of microparticles due to acoustic radiation, acoustic streaming, gravity and buoyancy over a clamped vibrating circular plate in contact with water. The underlying physics of microparticle acoustophoresis over vibrating plates has been studied in detail. Previous predominant analyses have emphasized the in-plane acoustic streaming flows on the formation of inverse Chladni patterns, which, according to this study, may not be complete. For in-plane microparticle acoustophoresis, both the in-plane acoustic radiation forces and the in-plane streaming-induced drag forces were shown to drive microparticles to their closest vibrating antinodes. For out-of-plane microparticle acoustophoresis above vibrating antinodes, in addition to the buoyancy forces, one has to consider the acoustic radiation forces in the near-field, which prevent the out-of-plane streaming vortices from dragging microparticles away from the vibrating interface.
Based on the high efficiency of this numerical model, the threshold in-plane and out-of-plane particle sizes balanced from the acoustic radiation and streaming-induced drag force under all vibrating modes can be readily obtained. An important next step is to achieve a direct experimental verification of numerical modelling. Given a successful experimental verification, this 3D model could be extended to include the thermoviscous effects (Muller and Bruus 2014) to obtain more accurate results, but it would be very computationally expensive. According to a study by Rednikov and Sadhal (2011), the thermoviscous effects can increase the streaming velocities by 18% for water at 20 °C which, thus, will shift the threshold particle sizes.
The good comparisons between our modelling and experiments and basic theories indicate that our numerical model could be used together with high-precision experiments as a better research tool to study the many yet unsolved problems. For example, modelling suggests that mode switching between two adjacent frequencies may be used for circumferential manipulation of a single particle or a pair of particles, which might provide routes for the study of particle–particle and particle–wall interactions in acoustofluidics.
While we have shown here 3D particle size-dependent acoustophoresis over an ultrathin circular plate in water, we believe that this strategy could be applied to analyse 3D acoustophoretic motion of microparticles in other vibrating plate systems regardless of fluid medium and thickness, shape and material of plates. One particular application would be acoustophoretic handling of sub-micrometre particles, such as small cells, bacteria and viruses, whose movements are usually dominated by acoustic streaming flows. From the modelled results and the general scaling law given in Eq. (27), we can conclude that increasing plate thickness, decreasing the plate diameter and lowering the viscosity of the liquid are probably the most viable way to conduct such manipulation.
The above-mentioned applications demonstrate that our numerical model is timely and has a huge potential on studies of basic physical aspects of microparticle acoustophoresis in vibrating plate systems and the design of lab-on-a-chip devices.
This work is supported by the EPSRC/University of Southampton Doctoral Prize Fellowship (EP/N509747/1). The authors gratefully acknowledge helpful discussions with Prof M. Hill and Dr P. Glynne-Jones. Models used to generate the simulation data supporting this study are openly available from the University of Southampton repository at http://dx.doi.org/10.5258/SOTON/404258.
10404_2017_1888_MOESM1_ESM.avi (3.9 mb)
Supplementary material 1 (AVI 3956 kb)
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© The Author(s) 2017
Open AccessThis 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.
1.Faculty of Engineering and the EnvironmentUniversity of SouthamptonSouthamptonUK
Lei, J. Microfluid Nanofluid (2017) 21: 50. https://doi.org/10.1007/s10404-017-1888-5
Received 08 November 2016
Accepted 22 February 2017
First Online 03 March 2017
Publisher Name Springer Berlin Heidelberg | CommonCrawl |
Sherman–Takeda theorem
In mathematics, the Sherman–Takeda theorem states that if A is a C*-algebra then its double dual is a W*-algebra, and is isomorphic to the weak closure of A in the universal representation of A.
The theorem was announced by Sherman (1950) and proved by Takeda (1954). The double dual of A is called the universal enveloping W*-algebra of A.
References
• Sherman, S. (1950), "The second adjoint of a C* algebra", Proceedings of the International Congress of Mathematicians 1950 (PDF), vol. 1, Providence, R.I.: American Mathematical Society, p. 470
• Takeda, Zirô (1954), "Conjugate spaces of operator algebras", Proceedings of the Japan Academy, 30 (2): 90–95, doi:10.3792/pja/1195526177, ISSN 0021-4280, MR 0063578
Functional analysis (topics – glossary)
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Applications
• Hardy space
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Advanced topics
• Approximation property
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Spectral theory and *-algebras
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• Involution/*-algebra
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• Gelfand–Mazur theorem
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| Wikipedia |
How do we calculate the distance to the stars
Already antique astronomers used their curiosity and innovative engineering abilities to determine the large distances in our Solar system. When humans start sailing on the oceans they saw how the airframe disappeared before the mast when a boat was passing the horizon which leads to the speculation that Earth was round. This notation was established by 3rd century BC by Greek astronomy. In 240 B.C the Greek astronomer Eratosthenes that also is known as the father of geography as he introduced the concept of longitude and latitude and draw a map over at that time the known world. He made a very accurate measurement of the circumference of the Earth. In the city of Syene 800 kilometers south of Alexandria (Egypt) there where a famous well. Precisely at summer solstice once a year the Sun's rays shone straight down into the well. At the same time in Alexandria. Eratosthenes measured the length of the shadow from a stick and calculated the angle to:
$$\tan^{-1} \frac{d_{shadow}}{d_{stick}}\approx 7.2^{\circ }$$
7 degree is 1/50th the circumference of a circle and knowing the distance to Syene is 800 kilometers the earth circumference should be 50 times that distance 40 000 kilometers.
Or using trigonometry:
$$2\pi \frac{800000}{\tan 7.2^{\circ }}\approx 40241005$$
Another Greek astronomer Aristarchus of Samos at the same time calculated the distance to the Moon (R). By looking at a lunar eclipse and calculating how long time it took for the Earth shadow to cross over the Moon that takes 3 hours and 40 minutes (t) it will take 29 days for the Moon to orbit an entire revolution around the Earth (T). He estimated the distance to 60 earth radii (r) that is correct.
$$\frac{\pi R}{r}=\frac{T}{t}$$
$$\Rightarrow R\approx 60r$$
He also estimated the distance to the Sun. During a solar eclipse, the Moon covers almost the entire disc. This tells us that the Sun is larger than the Moon and farther away. During half moon he assumed that the Moon forms a right angle with the Sun and Earth he measured that angle to 87 degrees.
$$\frac{R_{\odot}}{60r}=\cos 87^{-1}\approx 20$$
He came to the conclusion that the Sun is 20 times farther away from the Moon. This is wrong the Sun is 400 times farther away from the Moon as the angle is closer to 90.
On a side note: trigonometric functions had not yet been invented the ancient greek used geometrical relations to find proportions.
The first measurement of the distance to a planet was made by Gian Domenico Cassini. In 1672, He used a technique called parallax to measure the distance to Mars. If you hold up your thumb at one arm distance look at with just the left eye and then the other you will see that object farther away is shifting position that is caused by the separations of your eyes. You are watching the object from two different positions. The distance the thumb seems moving is its parallax. If you know the distance between your eyes and the angle by which your thumb moved against the background, you can calculate the length of your arm. By making an observation on two different places at Earth one can calculate the distance to objects far away in the same way.
To measure the distance to a star like Proxima Centauri that is 4.24 light-years away. One could take pictures of the star from two points when Earth is at one side of the Sun and then six months later when Earth is on opposite sides of the Sun and then calculating the parallax angle that more distant stars seem moving. The parallax angle Proxima shifting is 0.77 arc second one arc second is 1/3600 of a degree. A distance to a star was calculated for the first time in 1838 by Friedrich Bessel who measured the parallax of 61 Cygni as 0.314 arc second 11.4 light-years away. To measure large distances to stars the unit parsec (pc) is often used instead of light-years. A parsec is a distance that the parallax angle is 1 arc second that is 3.26 light-years. Parallax can only be used to find distances under 100 parsecs away
To measure the luminosity that is the total amount of energy emitted per time by an astronomical object or the brightness a logarithmic scale are used that is called the absolute magnitude. The sun has a magnitude of -27 and the dimmest objects visible with the naked eye has a magnitude of 6. The apparent magnitude is the magnitude of the object seen at 10 parsecs away. The brightness of a star is inversely proportional to the square of its distance.
$$L\sim \frac{1}{D^{2}}$$
French astronomer Charles Messier cataloged 110 astronomical objects the closest large galaxy was cataloged M31 in 1764. He thought it was a nebula within our galaxy. The object is also known as Andromeda and is visible with the naked eye. When astronomers discovered a variable star called novae in Andromeda in 1917 they noticed that it was 10 times less bright than similar stars in our galaxy. A Cepheid variable star is a very bright star that pulsates in a predictable way.
once the period has been measured its luminosity can be estimated. Then the distance to the object could be calculated in parsec with this formula
$$d=10^{(m-M+5)/5}$$
where m is the apparent magnitude and M the absolute magnitude of the Cepheid. Edwin Hubble in 1925 calculated that the galaxy 1.5 million light-years away. Modern calculations show it is 2.5 million light-years away or 778 000 parsec.
Image credit: NASA/JPL-Caltech
Andromeda galaxy is blueshifted it moving towards the milky way due to gravitational forces but all distant galaxies are redshifted they are moving away because the universe is expanding. The velocity of a galaxy is proportional to its distance from us by the equation
$$v=Hd$$
Where H is the Hubble constant that is estimated to be 70.0 km/sec/Mpc
Objects like quasars that are the ultraluminous nuclei of galaxies are extremely redshifted. For example, the quasar 3C 273 has a redshift of 0.158 which means it moving away at a speed of 44000 km/s (0.158 * speed of light)
using Hubble's law its distance could be calculated to 2 billion light-years or 620 Mpc.
The most distant object GN-z11 has a redshift on 11.09 and is 13.39 billion light-years away (actually it is much further away as space has been expanding during the time it takes the light to reach us).
Posted by Göran Bäcklund 5/6/2019 6:00:46 PM
Andromeda Hubble
New observations suggest that the universe is round
The riddle of the size of the universe has involved scientists ever since the childhood of cosmology. Newton believed that the universe is infinite, while Kepler believed in a finite.
Albert Einstein was the first physicist that gave the concept of a finite universe a sustainable theoretical foundation. The gravity can make space curve so much that the overall structure closes itself in the same way as the surface of a globe. The shape of the universe is depending on how much mass it is in the universe. The density parameter was derived by Alexander Friedmann in 1922 from Einstein's field equations.
$$\Omega =\frac{\rho}{\rho_{c}}$$
where ρ is the actual density of the Universe and ρc is the critical density.
The critical density is according to Friedmann equations
$$\rho_{c} =\frac{3H^{2}}{8\pi G}$$
where G is the gravitational constant 6.674×10−11 m3/(kg⋅s2) and H is the Hubble parameter a function of time that tells us how fast the universe is expanding it may be derived from the same equations as
$$H^{2}= \frac{8\pi G \rho }{3}-\frac{kc^{2}}{a}$$
then the density parameter becomes
$$ \Omega = \frac{H^{2}+\frac{kc^{2}}{a}}{H^{2}}$$
where c is the speed of light in vacuum and k is the curvature constant and a is the scale factor
If the density parameter:
Is bigger than 1 and k equals 1, then the universe is finite and has a spherical shape
Is smaller than 1 and k equals -1, then the universe is infinite or finite and has a hyperbolic shape
Is equal to 1 and k equals 0, then the universe is infinite and is flat
Credit: NASA / WMAP Science Team
The Planck space observatory was a spacecraft operated by the European Space Agency (ESA) from 2009 to 2013 and mapped the cosmic microwave background CMB. CMB is the radiation leftover from the big bang.
Several observations have indicated that the universe is flat and that fits very well with our current theoretical models, but re-analysis of the Planck data shows that we live in a finite spherical universe where the density parameter is bigger than 1
Here is the paper: nature.com
Here is an article about it quantamagazine.org
The density of the universe is according to the article calculated to be about 6 hydrogen atoms per cubic meter of space and the critical density is 5.7 hydrogen atoms per cubic meter of space, which gives the density parameter value of 6/5.7 =1.05.
Posted by Göran Bäcklund 11/17/2019 1:08:34 AM
Universe Hubble | CommonCrawl |
Adaptive Information Assimilation using Convolutional Neural Network for Forecast of Breast Cancer from Electronic Health Records
Senthil Kumar Kumar J, Kamala Devi K, Raja Sekar J
Purpose Data acquired from cancer based Electronic Health Records (EHRs) shows key statistics on cancer affected persons. To estimate the impact of the cancer on those persons, we need to extract vital information from those pathology health records. It is an exhaustive procedure to carry out because of large volume of records and data acquired for a continuous period of time.Methods This research portrays, the investigation of convolutional neural network (CNN) and Support Vector Machine (SVM) techniques for extracting topographic codes from the pathology reports of breast cancer. Investigations are carried out using conventional frequency vector space method and the deep learning techniques such as CNN. The learning experience of those algorithms were absorbed on a set of 730 pathology reports.Results We perceived that the CNN technique reliably outperformed the conventional frequency vector methods. It is also observed that it causes the micro and macro average performance to increase up to 0.119, and 0.101, while considering the populated class labels for the CNN model. Unambiguously, the top performing CNN approach attained a micro-F score of 0.821 over the considered topography codes.Conclusion These promising outcomes reveals the prospective of deep learning approaches, particularly CNN for estimating the impact of the cancer from the pathology reports compared to conventional SVM approach. More advanced and accurate approaches to effectively improve the accuracy in information extraction are needed.
Pathology Reports
Health care organizations have started using the Information Technology service in our contemporary society. It has become a mandate choice for many clinical and administrative activities. Usage of Electronic Health Records (EHR) have started playing a significant role for such tasks. For extracting valuable information from EHR data deep learning techniques can be applied (Benjamin et al. 2017). EHRs are also used for making decisions about affected tissues after thorough examinations of them. Decisions from EHRs, are vital for the patient's present and feature health issues. Pathologist use invasive methods on the patients as one technique for obtaining biopsy from affected tissues of human body. They can also do the review by sneaking through the pathology records in EHRs. Primary view of the symptoms from those EHRs can make the pathologist to take appropriate decisions and give proper directions and medications to the patients. Extracting valuable information on the disease form the EHRs is quite challenging, if the volume of data in the EHRs is larger. Lot of researches have been carried out to manage those data and extract the information for performing accurate clinical decisions by pathologists.
For more accurate prediction of health relevant parameters from human body, large volume of data need to be analyzed. Big data technologies can support the health care industries by processing a large volume of EHR data for extracting vital health parameters from the patients (Marco et al. 2015). It enables to estimate sensitive information that cannot be easily determined with individual patient data. EHR holding the physiological variables for patients of different age group and health conditions are smart enough for analyzing and predicting the diseases. Researchers mostly find it challenging to collect detailed information about patients. So, the publically available SEER cancer data is mostly used for training the developed models and estimating accurate insights from them. It includes health record of almost 28% of population in US (SEER, 2016). Raw data from SEER based EHR cannot be directly utilized for processing. The dataset is obtained by proper signed agreement from the National Cancer Institute's Surveillance, Epidemiology, and End Results (SEER) Program.
Wearable devices with low power consumption are gaining popularity for healthcare applications. Using spectral preprocessing on the health data from the wearable devices, it makes the data ready for processing by deep learning frameworks (Daniele et al. 2017). Advent of Internet of Things (IoT) gave made a significant impact for the wide spread usage of wearable health care devices. Data acquired from those smart wearable health care devices also support to large extent to generate health records.
Deep Neural Networks (DNN) are gaining popular the support of deep learning techniques. Those networks have multiple layers, which are capable of extracting meaningful features and learning from the data. Fig.1 shows the structure of a simple DNN, which has an input and output layers along with multiple hidden layers. Learning occurs with the support of forward and backward propagation of the weights associated with each neuron. Deep learning approaches using the EEG data are employed for reducing sleeping disorders of human. From its results diagnosis of insomnia can be done effectively (Mostafa et al. 2017).
Convolutional Neural Networks are a category of DNN. It has been gaining popular with the usage of medical images for classification and prediction of diseases. It can also be used for extracting deep hierarchical features in medical images. Convolutional Neural Networks are used for cervical cancer screening (Ling et al. 2017), Breast Cancer screening (Moi et al. 2017), for diagnosis of auto immune diseases (Zhimin et al. 2017), risk prediction using EMR (Phuoc et al. 2017), Gait parameter extraction (Julius et al. 2017), Lung tissue classifications (Qiangchang et al. 2018), and extracting primary cites from pathology reports (John et al. 2018) and many more. Fig.2 shows the general Architecture of a simple CNN considering the breast pathology report as input document matrix. The convolution layer observers the features from the pathology reports, the pooling layer eliminates the redundant features and the prediction of the tumor is performed at the final softmax output stage of the network.
Recurrent Neural Networks (Edward et al. 2016), are used for assisting doctors by providing best clinical decision support using EHR. From the large volume of patient records RNN based models predicts the future impacts of the diseases to the patients (Trang et al. 2017). Deep dynamic neural networks are also employed to predict future consequences of patients health from their health records. Long short team memory (LSTM) uses memory of historical records with time stamps for predicting the future risk factors (Truyen et al. 2015).
Medical record data objects are embedded using low dimensional vector space using Restricted Boltzmann Machine (RBM). This mechanism is encouraged for usage in medical records that are mostly discrete in nature. By offering embedded space, most knowledge on the data can be exploited. Hierarchical order of learning from medical data are also carried out, by considering the order of visit and co-occurrence of the medication codes during the visits to health care organizations. It is made viable by considering an architecture with multilayer perceptron (MLP) (Edward et al. 2016).
Breast cancer survival rates of both women and men were analyzed from health records (Paulo et al. 2017). The analysis were performed using descriptive statistics on Cox regression and Kaplan-Meier analysis. Apart from gender, their drinking, smoking and other habits are also observed to predict the overall survival and disease free survival of the patients.
Early stages of recurrences in breast cancer are analyzed for estimating the risks and follow up actions for the cancer detected cases (Vinzenz et al. 2019). From the health data collected from German and Dutch people, analysis are performed by considering biological subtypes, surgery type and age of the patients. Decision making during breast cancer treatment need to consider the biological subtype and the pattern of recurrence at different time intervals (Ignatov et al. 2018). With known biological subtype, analysis were performed with the health care data. It is observed that the tumours that were initially low, remains the same even after 10 years. Observation from the enriched biological subtypes shows the increased pattern tumour.
Precision in cancer prediction can be improved by using an appropriate production model implemented using machine learning techniques. Penalized regression technique is implemented to build a predictive model to observe the resistance of epidermal growth factor receptor tyrosine kinase inhibitors (Young et al. 2018).
The research article starts with presenting a brief introduction to usage of EHR data, and deployment of big data and deep learning techniques for modern health care applications. Section II deals with the health data set and the strategies made for information extraction from EHR. Section III discusses the methodology used for training and testing the cancer pathology reports extracted from SEER dataset using CNN. Section IV summaries the results analyzed after the successful deployment of the CNN for effective information extraction from the health records. Finally, Section V summarizes the work in the conclusion part.
HEALTH DATASET FOR INFORMATION ASSIMILATION
The developed multi study derived model for prediction provides good transferability and generalizability along with perfect accuracy during observations. Incidence of brain metastases is observed from the SEER datasets for production of prognosis (Yi-Jun et al. 2018). From a large set of breast cancer patients details collected from the SEER data, more incidence of brain metastases is observed for HER2 subtype. Also visceral metastases is observed from the patients having TNBC and HER2 subtypes. This analysis contributes to earlier metastases and positively increases the survival rate of the affected breast cancer patients.
From the SEER dataset, patients with stage IV breast cancer were identified and they the clinical value of auxiliary lymph nodes were assessed (San-Gang et al. 2017). The effect of the auxiliary lymph node dissection with the survival rates of the patients were analysed. It was observed that the auxiliary lymph node dissection improves with the survival rate of the patients.
From the health records, it is perceived that dissection of axillary lymph node improves the survival rate of patients diagnosed with breast cancer of stage IV. This observation was made on the patients who received tumor surgery in primary case, especially in liver and bone (Wu SG et al. 2017). Gender based survival of breast cancer analysis was performed with the hospital health records. Overall survival and disease free survival studies on them reports no noteworthy dissimilarity in prediction, but changes in clinical features were founded based on their demographic locations (Thuler et al. 2017). With familiar biological subtype of the patients, the breast cancer recurrence patterns are studied. It is evident that with varying time their subtypes are changed accordingly and they need to be considered while making a decision about tumour (Ignatov et al. 2018). From the SEER dataset prediction of brain metastasis is performed from the breast cancer reports. It is evaluated based on the molecular subtypes and estimated that patients with TNBC and HER2 subtypes possess visceral metastasis (Kim et al. 2018)
Pathology reports from SEER dataset that matches with 730 cases of breast cancer are chosen or analysis from the registry. The topography codes used in the training set of the analysis process includes only the final diagnosis part from the pathology report. This kind of choice is made to avoid variation during the training process and to improve the robustness of the estimations from the reports. Table 1 shows the 9 ICD-O-3 topography codes that includes the primary sites of breast cancer chosen for the analysis. In the preprocessing stage, the text contents of the pathology report are aggregated to carefully utilize the empty sections in the reports.
Extraction of valuable information from the SEER health records can be performed by fragmenting the sequence of EHR data and by performing Multi hot encoding of the sequence. Fig.3 shows the sequence of steps to be carried out for feeding the extracted data from EHR for performing the processing using deep neural networks.
Corpus of data can be encoded using feature vectors based on the count of words. This vector space models are basic tasks of NLP systems for relatively simpler extraction of vital health care information from the data set. Based on the observation similarities, the word embedding techniques can be used for information extraction.
Usage of deep learning techniques to learn the representation of words from the data set, unlike conventional observation methods can provide better accuracy and minimizes the efforts in information retrieval.
Few earlier works on extracting of text data using deep learning techniques uses recurrent networks (Mikolov.T, et. al. 2010). Some of the literatures also extracted the data using feature vectors on the encoded documents (Le. Q. et. al. 2014). These category of information extraction largely depends on the structure and form of the documents used. Even though CNN were developed for vision based tasks in deep learning approaches, it has found its deep rooted impact on NLP, and literatures have utilized its extraordinary performance for information extraction from documents (Zhang. Et. al. 2016). Also, utilization of the convolution filters in CNN and its max pooling techniques for information extraction from documents improves the accuracy when compared to the conventional techniques. It is highly applicable for features with higher dimensions and it can utilize the order of words in the document directly.
In the proposed investigation, we use the word segmentation process and word vector representation to the train the classifier using deep learning technique. The process for training and sequence extraction for tumor prediction is illustrated in the Fig. 4.
The sequence of word vectors is trained to maximize the objective function for a word of context. Trained vector of words after the word segmentation are able to capture different meanings of the words in the context.
Analysis And Results
Analysis of the extracted pathology reports from SEER database are performed to test the effectiveness of the proposed DNN. In this research paper, we study the effectiveness of our proposed framework on SEER EHR data. From the extracted SEER dataset F1-score, precision and accuracy, were used to estimate the efficiency and performance of the proposed CNN framework architecture. For estimating much better performance, recall and precision measures are joint together obtain a better thoughtful understanding of the classifier. They are computed using the following expressions shown from eq(1) to eq(5).
Were is true negative, representing total predicted affected region, is true positive, is false negative and is false positive. Accuracy defined in eq(1) depicts the classification success rate considering both true and false values. The precision shown in eq(2), computes only with respect to the positive outcomes of the classifier. Similarly, the eq(5), computes only with respect to the negative outcomes. Performance evaluation measure are dominant while considering the F1-score of the classifier. Both micro and macro F1-score are computed from the eq(4).
The proposed CNN architecture follows the implementation stages as shown in Fig.5. Initially the random weights are initialized and combined with the patient pathology records. The features are being associated to the input nodes. Followed by this initial stages, forward propagation is carried out by calculating the error function and predicting the design. This process is repeated to activate the neurons in the network based on the updated weights with respect to the error, till reaching the desired result.
Updating of the weights are performed in the backpropagation stage with the support of the backpropagation ID. The weights are repeated to update each data input, till reaching the desired result. The entire task is repeated for the training set and the process is done for multiple epochs. Once the desired accuracy is reached the process is stopped.
Quality of the classifications are evaluated using the confusion matrix. Accuracy in the predictions are observed from the diagonal values of the matrix. Normalized confusion matrices are plotted for SVM and CNN based observations. Analysis are also performed for minimally populated tasks and well-populated tasks.
From the normalized confusion matrix shown in Fig.6 and Fig.7, it is evident that the proposed CNN based classifier outperforms the SVM classifier to a larger extent for the minimally populated tasks. The diagonal elements of the figures represent the true positive classification performed successfully. The vertical elements specify the false positive classification performed. The false negatives are represented in the horizontal axis of the confusion matrix. It is evident that the CNN model classifies better for the breast classes c34.0, c34.1 and c34.9 in the minimally populated classes
Similar form of normalized confusion matrix shown in Fig.8 and Fig.9, is plotted for the well populated tasks. From the observations, it is evident that the proposed CNN based classifier outperforms the SVM classifier with better accuracy. The diagonal elements of the CNN confusion matrix for well populated tasks shows more true positives compared to SVM technique. It is evident that the CNN model classifies better for the breast classes c34.1, c34.3 and c34.9 in the well populated classes.
Table 2 shows the comparison and consolidated results of the SVM and CNN based classifiers of the different pathology reports with breast cancer. It is shown for both minimally and well populated tasks. From the eq(1) to eq(5), the performance measures are calculated and they are tabulated with accuracy, precision, F-score and specificity. From the observations made, the CNN model outperforms the SVM for breast cancer information assimilation for both minimally and well populated tasks.
This kind of classifiers are better choice for information extraction from the electronic health record. Deep learning based CNN model show cases with appropriate well defined strategy with better accuracy than the conventional SVM classifier.
In this proposed research article, we have designed and developed a deep neural network for information extraction from pathology reports. Series of experiments were done using CNN and traditional SVM classifiers on the SEER dataset. The performance of CNN is observed to be superior with better micro-F and macro-F scores of 0.821 and 0.794 respectively. Assimilation of information from the highly populated class of embedded randomized data in the CNN layers leads to better performance than the SVM classifiers.
The authors would like to thank the Department of Computer Science and Engineering, and the Management, Principal of Mepco Schlenk Engineering College, Sivakasi, Tamil Nadu, India for providing us the modern state-of-art facilities to carry out this research work.
Compliance with ethical standards
Conflict of interest: The authors confirm that they have no conflict of interest regarding this research article
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Table 1: Breast ICD-O-3 Topographical Codes with Description and Count
Breast ICD-O-3 Topographical Codes
C50.0
Central Portion of breast
Upper inner quadrant of breast
Lower inner quadrant of breast
Upper outer quadrant of breast
Lower outer quadrant of breast
Axillary tail of breast
Overlapping lesion of breast
Breast NOS
Table 2: Comparison of the SVM with proposed CNN model for minimally and well populated tasks.
Proposed CNN algorithm
Minimally Populated Task
Well Populated Task
Accuracy (%)
Precision/ Sensitivity (%)
Micro F-Score
Macro F-Score
Specificity (%) | CommonCrawl |
Horadam sequences: A survey update and extension.
Larcombe, Peter J. 2017. Horadam sequences: A survey update and extension. Bulletin of the Institute of Combinatorics and its Applications (ICA).
Larcombe, Peter J.
We give an update on work relating to Horadam sequences that are generated by a general linear recurrence formula of order two. This article extends a first ever survey published in early 2013 in this Bulletin, and includes coverage of a new research area opened up in recent times.
Horadam sequence
Bulletin of the Institute of Combinatorics and its Applications (ICA)
The Institute of Combinatorics and its Applications (ICA)
(2017) Horadam Sequences - A Survey Update and Extension.pdf
https://repository.derby.ac.uk/item/94w6x/horadam-sequences-a-survey-update-and-extension
Reflections On What Mathematics Is and Isn't: Halmos, Keyser, and Others
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On 'Two Cultures' and Tackling the 'Writing Versus Mathematics' Dichotomy
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Write, and Write Well - Speak, and Speak Well: The Gospel According to Halmos and Rota
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Mathematicians can also write, right?
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Alwyn Francis Horadam, 1923-2016: A personal tribute to the man and his sequence
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On a scaled balanced-power product recurrence
Larcombe, Peter J. and Fennessey, Eric J. 2016. On a scaled balanced-power product recurrence. Fibonacci Quarterly.
A short monograph on exposition and the emotive nature of research and publishing
Larcombe, Peter J. 2016. A short monograph on exposition and the emotive nature of research and publishing. Mathematics Today.
A polynomial based construction of periodic Horadam sequences
Larcombe, Peter J. and Fennessey, Eric J. 2016. A polynomial based construction of periodic Horadam sequences. Utilitas Mathematica.
A new formulation of a result by McLaughlin for an arbitrary dimension 2 matrix power
Larcombe, Peter J. 2016. A new formulation of a result by McLaughlin for an arbitrary dimension 2 matrix power. Bulletin of the Institute of Combinatorics and its Applications (ICA).
On the jacobsthal, horadam and geometric mean sequences
Larcombe, Peter J. and Rabago, Julius, F. T. 2016. On the jacobsthal, horadam and geometric mean sequences. Bulletin of the Institute of Combinatorics and its Applications (ICA).
On the structure of periodic complex Horadam orbits
Bagdasar, Ovidiu, Larcombe, Peter J. and Anjum, Ashiq 2016. On the structure of periodic complex Horadam orbits. Carpathian Journal of Mathematics..
A note on the invariance of the general $2 \times 2$ matrix anti-diagonals ratio with increasing matrix power: Four proofs
Larcombe, Peter J. 2015. A note on the invariance of the general $2 \times 2$ matrix anti-diagonals ratio with increasing matrix power: Four proofs. Fibonacci Quarterly.
Closed form evaluations of some series comprising sums of exponentiated multiples of two-term and three-term Catalan number linear combinations
Larcombe, Peter J. 2015. Closed form evaluations of some series comprising sums of exponentiated multiples of two-term and three-term Catalan number linear combinations. Fibonacci Quarterly.
Closed form evaluations of some series involving Catalan numbers
Larcombe, Peter J. 2014. Closed form evaluations of some series involving Catalan numbers. Bulletin of the Institute of Combinatorics and its Applications (ICA).
A condition for anti-diagonals product invariance across powers of $2 \times 2$ matrix sets characterizing a particular class of polynomial families
Larcombe, Peter J. and Fennessey, Eric J. 2015. A condition for anti-diagonals product invariance across powers of $2 \times 2$ matrix sets characterizing a particular class of polynomial families. Fibonacci Quarterly.
On the phenomenon of masked periodic Horadam sequences
Larcombe, Peter J. and Fennessey, Eric J. 2015. On the phenomenon of masked periodic Horadam sequences. Utilitas Mathematica.
On horadam sequence periodicity: A new approach
Larcombe, Peter J. and Fennessey, Eric J. 2015. On horadam sequence periodicity: A new approach. Bulletin of the Institute of Combinatorics and its Applications (ICA).
A non-linear identity for a particular class of polynomial families
Larcombe, Peter J. and Fennessey, Eric J. 2014. A non-linear identity for a particular class of polynomial families. Fibonacci Quarterly.
On cyclicity and density of some Catalan polynomial sequences
Larcombe, Peter J. and Fennessey, Eric J. 2014. On cyclicity and density of some Catalan polynomial sequences. Bulletin of the Institute of Combinatorics and its Applications (ICA).
Some factorisation and divisibility properties of Catalan polynomials
Jarvis, Frazer A., Larcombe, Peter J. and Fennessey, Eric J. 2014. Some factorisation and divisibility properties of Catalan polynomials. Bulletin of the Institute of Combinatorics and its Applications (ICA).
Generalised Catalan polynomials and their properties
Larcombe, Peter J., Jarvis, Frazer A. and Fennessey, Eric J. 2014. Generalised Catalan polynomials and their properties. Bulletin of the Institute of Combinatorics and its Applications (ICA).
The Asymptotic Form of the Sum $\sum_{i=0}^{n} i^{p} { n+i \choose i }$: Two Proofs
Larcombe, Peter J., Kirschenhofer, Peter and Fennessey, Eric J. 2014. The Asymptotic Form of the Sum $\sum_{i=0}^{n} i^{p} { n+i \choose i }$: Two Proofs. Utilitas Mathematica. | CommonCrawl |
A continuous data driven translational model to evaluate effectiveness of population-level health interventions: case study, smoking ban in public places on hospital admissions for acute coronary events
Hossein Bonakdari ORCID: orcid.org/0000-0001-6169-36541,2,
Jean-Pierre Pelletier ORCID: orcid.org/0000-0001-9930-64531 &
Johanne Martel-Pelletier ORCID: orcid.org/0000-0003-2618-383X1
An important task in developing accurate public health intervention evaluation methods based on historical interrupted time series (ITS) records is to determine the exact lag time between pre- and post-intervention. We propose a novel continuous transitional data-driven hybrid methodology using a non-linear approach based on a combination of stochastic and artificial intelligence methods that facilitate the evaluation of ITS data without knowledge of lag time. Understanding the influence of implemented intervention on outcome(s) is imperative for decision makers in order to manage health systems accurately and in a timely manner.
To validate a developed hybrid model, we used, as an example, a published dataset based on a real health problem on the effects of the Italian smoking ban in public spaces on hospital admissions for acute coronary events. We employed a continuous methodology based on data preprocessing to identify linear and nonlinear components in which autoregressive moving average and generalized structure group method of data handling were combined to model stochastic and nonlinear components of ITS. We analyzed the rate of admission for acute coronary events from January 2002 to November 2006 using this new data-driven hybrid methodology that allowed for long-term outcome prediction.
Our results showed the Pearson correlation coefficient of the proposed combined transitional data-driven model exhibited an average of 17.74% enhancement from the single stochastic model and 2.05% from the nonlinear model. In addition, data demonstrated that the developed model improved the mean absolute percentage error and correlation coefficient values for which 2.77% and 0.89 were found compared to 4.02% and 0.76, respectively. Importantly, this model does not use any predefined lag time between pre- and post-intervention.
Most of the previous studies employed the linear regression and considered a lag time to interpret the impact of intervention on public health outcome. The proposed hybrid methodology improved ITS prediction from conventional methods and could be used as a reliable alternative in public health intervention evaluation.
Due to advances in technology and improvements in recording reliable data and sharing methods, the time series (TS) concept has emerged in many theoretical and practical studies over the past few decades [1]. This concept allows researchers to access the outcome of any phenomenon or intervention, at any time, with minimum cost and effort, and to plan possible solutions and control measures based on the forecasted data [2]. Therefore, improving knowledge about studying TS, preprocessing, modeling and, if needed, post-processing is imperative [3].
In the domain of public health interventions, the interrupted time series (ITS) concept has been widely employed to evaluate the impact of a new intervention at a known point in time in routinely observed data [4,5,6,7,8,9,10]. ITS is fundamentally a sequence of outcomes over uniformly time-spaced intervals that are affected by an intervention at specific points in time or by change points. The outcome of interest shows a variation from its previous pattern due to the effect of the intervention. The applied intervention splits TS data into pre- and post-intervention periods. Based on this definition, Wagner et al. [11] proposed segmented regression analysis for evaluating intervention impacts on the outcomes of interest in ITS studies. In this approach, the choice of each segment is based on the change point, with the possible additional time lag in some cases, in order for the intervention to have an effect [12,13,14,15,16,17]. In addition, for pre- and post-intervention period segments of a TS, the level and trend values should be determined either by linear [17] or nonlinear [6] approaches. Therefore, accurate values of the change point and time lag parameters are essential in segmented regression analysis.
Affecting an intervention at a change point produces different possible outcome patterns in the post-intervention period for both level and trend parameters. Figure 1 illustrates some possible impacts of an intervention on the post-intervention period. As shown in Fig. 1a–c, a change in level (or intercept) may lead to a change in level after a time lag or a temporary level change after the intervention. Other possible patterns are a change in slope (or trend) with a change in slope after a time lag, or a temporary slope change as shown in Fig. 1d–f, respectively. In some cases, a change in both of these parameters could take place as an immediate change, e.g., a change after a time lag or temporary level and slope changes (Fig. 1g–i).
Possible patterns in interrupted time series post-intervention period data analysis
Regardless of the popularity and consensus on using segmented regression-based methods for solving ITS problems, selecting the most appropriate time lag is a challenging task with an important impact on results in this type of modeling. The reason for the delicacy of this task is that there is no specific rule to define the time lag produced between the pre- and post-intervention periods. In some cases, the outcomes of interventions have an unknown delayed response to the implemented strategies and a lag time may occur long after an intervention. However, in ITS modeling, when segmented regression approaches are used, the exact time lag after an intervention should be taken into consideration to guarantee modeling result accuracy and appropriateness. In addition, an undocumented change point seriously complicates ITS analysis. Applying a continuous nonlinear TS method is considered reliable if the ITS analysis can be released from all these fundamental concerns. Therefore, there is a necessity to introduce potential uses of linear, nonlinear or a combination of both models for solving such problems.
Over the past few years, soft computing methods have been employed across domains and have established reliable tools for modeling complex systems and predicting different phenomena in healthcare [18,19,20,21,22,23,24]. Among soft computing techniques, the Group Method of Data Handling (GMDH) is a common self-organizing heuristic model, which can be used for simulating complicated nonlinear problems. This evolutionary procedure is performed by dividing a complex problem into some smaller and simpler problems. Based on GMDH, this study proposes a novel methodology of the continuous modeling of an ITS based on data preprocessing. An example of the novel ITS modeling uses a linear-based stochastic model, a nonlinear-based model and an integration of a stochastic and a nonlinear model (hybrid). In order to run the models, certain tests and preprocessing methods are initially applied to the TS to prepare the data for stochastic modeling. It is crucial to investigate the structure of the TS being studied prior to modeling. Therefore, the TS undergoes stationarity testing along with normality testing. After surveying the characteristics of the TS, stationarizing methods appropriate to the TS are used. Then, in case of non-normal distribution, a normal transformation is applied to the stationarized TS. For the second TS modeling approach, the dataset is modeled with an artificial intelligence (AI) method which is, in this case, the Generalized Structure Group Method of Data Handling (GS-GMDH). In the third and final step, a hybrid model that combines the linear and nonlinear results is applied. Finally, the results are compared according to various indices and methods. Therefore, using this method facilitates modeling the ITS continuously, i.e. there is no need to identify the change point and intervention lag time.
Barone-Adesi et al. [25] carried out an extensive study on the effect of a smoking ban in public places on hospital admissions for acute coronary events (ACEs). In January 2005, Italy introduced legislation that prohibits smoking in indoor public spaces, the goal of which was the reduction of health issues caused by second-hand smoke [25]. Second-hand smoke consists of smoke exhaled by smokers and from lit cigarettes and causes numerous health problems in non-smokers every year, as well as high treatment costs for both patients and the government. The ban was undertaken on 10 January 2005 to confront the growing trend of ACEs and to control this problem.
Bernal et al. [6] used a subset of ACEs data from subjects in Sicily, Italy, between 2002 and 2006 among those aged 0–69 years. They analysed the ITS data by applying segmented linear regression to the standardized rate of ACEs TS associated with the implementation of a ban on smoking in all indoor public places, to calculate the change in the subsequent outcome levels and trends. Based on Barone-Adesi et al.'s [25] assumption, Bernal et al. [6] considered only a level change in ACEs occurring and there was no lag between the pre- and post-segments in the modeling procedure.
Here, we used the dataset from Bernal et al. (Fig. 4, [6]) as an example to illustrate the proposed method's performance in ITS simulation from real data regarding a health problem; it is not meant to contribute to the substantive evidence on the topic. The dataset employed comprises routine hospital admissions with 600–1100 ACEs. More information about the dataset can be found in the Barone-Adesi et al. article [25].
Time series are data recorded continuously and based on time to institute a sequence of measures, each of which refers to a time. Thus, the ACE data collected monthly from 2002 to 2006 is a TS. Each TS consists of four terms: jump + trend + period + stochastic component. The first three terms, known as deterministic terms, are calculable and removable. The jump term represents the sudden changes that occur in TS. These changes are detectable as steps in TS plots or by numerical tests. The trend term represents the gradual upward or downward changes that take place during a long period of time; this term is denoted in TS as a linear fitted line. The third deterministic term, the period, represents the periodic alternations in TS, which are seen as sinusoidal variations. Therefore, only the remaining stochastic term is required for use in stochastic or nonlinear modeling. This term is achieved while stationarity (absence of a deterministic term) occurs. Numerous tests and methods exist for investigating and omitting deterministic terms and some are presented below.
In stochastic modeling, two conditions must be met: the first is the stationarity (for details see below, Stationarizing methods); second, the distribution of the TS should be normal. Thus, in order to start stochastic-based modeling, the existence of deterministic terms must be checked, and when present, they should be removed. The Mann–Whitney (MW), Fisher, and Mann–Kendall (MK) tests are employed to check the jump, period and trend, respectively; the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test to assess the overall stationarity of the TS; and the Jarque–Bera (JB) test to check the normality of the TS.
A non-parametric test is used to assess the trend term in the studied TS. The MK test was developed to detect the gradual changes in TS, both seasonal and non-seasonal. The test equation is as follows [26]:
$$ U_{MK} \, = \,\left\{ \begin{gathered} \left( {MK - 1} \right){\text{var}} \left( {MK} \right)^{ - 0.5} \,\,\,\quad \,\,MK > 0 \hfill \\ 0\,\,\,\quad\quad\quad \,\,MK = 0 \hfill \\ \left( {MK + 1} \right){\text{var}} \left( {MK} \right)^{ - 0.5} \,\,\,\,\,\quad MK < 0 \hfill \\ \end{gathered} \right. $$
where UMK is the standard Mann–Kendall statistic, MK is the Mann–Kendall statistic, and var(MK) is the variance of MK. MK and var(MK) are defined as:
$$ MK\, = \,\sum\limits_{i = 1}^{N - 1} {\sum\limits_{j = i + 1}^{N} {{\text{sgn}} \left( {x_{j} - x_{i} } \right)} } $$
$$ {\text{var}} \left( {MK} \right)\, = \frac{1}{18}\left[ {N\left( {N\, - \,1} \right)\left( {2N\, + \,5} \right)\, - \,\sum\limits_{g}^{p} {t_{g} \left( {t_{g} \, - \,1} \right)\left( {2t_{g} \, + \,5} \right)} } \right] $$
where p is the number of identical groups, tg is the observation number in the gth group, sgn is the sign function, and N is the number of samples.
The (MK) test equation for a seasonal trend is expressed as follows:
$$ S_{k} \, = \,\sum\limits_{i\, = \,1}^{{N_{k} 1}} {\sum\limits_{j\, = \,i\, + \,1}^{{N_{k} \, - \,1}} {{\text{sgn}} \left( {x_{ki} \, - \,x_{kj} } \right)} } $$
$$ SMK\, = \,\sum\limits_{k = 1}^{\omega } {\left( {S_{k} \, - \,{\text{sgn}} \left( {S_{k} } \right)} \right)} $$
$$ {\text{var}} \left( {SMK} \right)\, = \,\sum\limits_{k}^{\omega } {\frac{{N_{k} \left( {N_{k} \, - \,1} \right)\left( {2N_{k} \, + \,5} \right)}}{18}} \, + \,2\sum\limits_{i = 1}^{\omega - 1} {\sum\limits_{j = i + 1}^{\omega } {\sigma_{ij} } } $$
$$ U_{SMK} \, = \,MK\,{\text{var}} \left( {MK} \right)^{ - 0.5} $$
where ω is the number of seasons in a year and σij is the covariance of the statistic test in seasons i and j.
The trend in the TS is insignificant if \(U_{\alpha /2} < \,U_{MK} \, < \,U_{1 - \alpha /2}\) and \(U_{\alpha /2} \, < \,U_{SMK} \, < \,U_{1 - \alpha /2}\) and Uα/2 and U1 − α/2 are the α/2 and 1 − α/2 quartiles of the normal cumulative probability distribution. A probability corresponding to the test statistic less than 5% means the absence of a significant trend in the TS.
A numerical survey for the jump term in the ACE TS, namely the non-parametric MW test, is employed as follows [27]:
$$ U_{MW} \, = \,\frac{{\sum\limits_{t = 1}^{{N_{1} }} {\left( {R\left( {g\left( t \right)} \right)\, - \,\frac{{n_{1} \left( {n_{1} + n_{2} + 1} \right)}}{2}} \right)} }}{{\sqrt {\frac{{n_{1} n_{2} \left( {n_{1} + n_{2} + 1} \right)}}{12}} }} $$
where g(t) is the ascending ordered ACE series, R(g(t)) is the order of g(t), and N1\(( x_{1}( t )\, = \,\{ {x( 1 ),x( 2 ), ..., x( {N_{1} )} \}} )\) and N2\(( x_{2} ( t )\, = \,\{ x( N_{1} + 1), x( {N_{1} + 2} ),.., x( N ) \} )\) are the numbers of sub-series of the main series, such that the sum of these series is equal to main series. If\(P_{{\left| {U_{MW} } \right|}}\) is larger than the significant level (in this study α = 0.01), then the jump term is insignificant.
The significance of periodicity is investigated with the following statistic [28]:
$$ F^{*} \, = \,\frac{{N\left( {N - 2} \right)\left( {\alpha_{k}^{2} + \beta_{k}^{2} } \right)}}{{4\left( {\sum\limits_{z = 1}^{k} {\left( {x\left( t \right) - \alpha_{z} \cos \left( {\Omega_{z} t} \right) - \beta_{z} \sin \left( {\Omega_{z} t} \right)} \right)} } \right)}} $$
where F* is the Fisher statistic, N is the number of samples, αz and βz are Fourier coefficients, and Ωz is the angular frequency. Αz, βz and Ωz are defined as follows:
$$ \alpha_{z} \, = \,\frac{2}{N}\left( {\sum\limits_{t = 1}^{N} {x\left( t \right)\cos \left( {2\pi f_{z} t} \right)} } \right)\,\,\,\;\,z = 1,2,...,k $$
$$ \beta_{z} \, = \,\frac{2}{N}\left( {\sum\limits_{t = 1}^{N} {x\left( t \right)\sin \left( {2\pi f_{z} t} \right)} } \right)\,\;\,z = 1,2,...,k $$
$$ f_{z} \, = \,\frac{z}{N} $$
$$ \Omega_{z} \, = \,\frac{2\pi z}{N}\;z = 1,2,...,k $$
where fz is the zth harmonic of the base frequency.
The periodicity related to Ωz is significant if the critical value of the F distribution at a significant level (F(2, N-2)) is lower than F*:
$$ F^{*} \, \ge \,F\left( {2,N - 2} \right) $$
For the considerable level of 0.05 (α = 0.05), the critical value of freedom degrees in the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test
The test is named after its authors [29] and is used to assess the overall stationarity of the ACE TS:
$$ S^{2} \left( l \right)\,\, = \,\,\frac{1}{n}\sum\limits_{t = 1}^{n} {e_{t}^{2} \, + \,} \frac{2}{n}\sum\limits_{s = 1}^{l} {w\left( {s,l} \right)} \frac{1}{n}\sum\limits_{t = s1}^{n} {e_{t} e_{t - s} } $$
$$ w\left( {s,l} \right)\,\, = \,1\, - \,s\left( {l\,\, + \,\,1} \right) $$
n is the number of TS, et is the residuals, and St2 is the average square of errors between time 1 and t. The statistic used for the "level" and "trend" stationarity tests is given by:
$$ \eta_{\tau ,\mu } \, = \,\frac{1}{{n^{2} }}\sum\limits_{t = 1}^{n} {\frac{{S_{t}^{2} }}{{S^{2} \left( l \right)}}} $$
Kwiatkowski et al. [29] calculated the symmetric critical values via Monte Carlo simulation. The probability corresponding to a test statistic higher than 5% indicates stationarity.
Jarque–Bera (JB) test
The JB test [30] is applied to measure the the goodness of fit and the test statistic is expressed as follows:
$$ JB\, = \,n\left( {\frac{{S_{k}^{2} }}{6}\, + \,\frac{{\left( {K_{u} \, - \,3} \right)^{2} }}{24}} \right) $$
where Ku is kurtosis, Sk is skewness and JB is a chi-square distribution with two degrees of freedom that can be used to assume the data is normal.
Stationarizing methods
In case a significant trend term exists in the TS as detected in the MK, seasonal Mann–Kendall (SMK) or autocorrelation function (ACF) plot, a trend analysis is the best way to remove or reduce its impact on TS. Then, a linear line is fitted to the TS and is subtracted from the TS values; remaining is a detrended TS.
Differencing
One of the most widely employed methods of stationarizing TS is differencing. This method eliminates correlations in TS. The non-seasonal differencing method, which is the subtraction of each value from the previous one, removes the trend in variances and jumps. The equation is as follows:
$$ {\text{Differenced TS }}\left( t \right)\,\, = \,\,{\text{MED}}\left( t \right)\, - \, {\text{MED}}\left( {t \, - \, {1}} \right) $$
where MED(t) represents a studied TS, in this case ACE, recorded at time t.
Stochastic modeling
The auto-regressive moving average (ARMA) and auto-regressive integrated moving average (ARIMA) models are the two most conventional methods of the stochastic approach. The difference between these models is in the data differencing method of the ARIMA model, which makes it suitable for non-stationary TS. The equation for ARIMA(p, d, q) is as follows [31]:
$$ \varphi \left( I \right)\,\,\left( {1\, - \,I} \right)^{d} MED\left( t \right)\, = \,\theta \left( I \right)\,\varepsilon \left( t \right) $$
$$ \varphi \left( I \right)\, = \,\left( {1\, - \,\varphi_{1} I\, - \,\varphi_{2} I^{2} \, - \,\,\varphi_{3} I^{3} \, - \, \cdots - \varphi_{p} I^{p} } \right) $$
$$ \theta \left( I \right)\, = \,\left( {1\, - \,\theta_{1} I\, - \,\theta_{2} I^{2} \,\, - \,\theta_{3} I^{3} \, - \,\, \cdots \, - - \theta_{p} I^{p} } \right) $$
where φ is the autoregressive (AR) process, θ the moving average (MA) parameter, ε(t) the residual, d the non-seasonal differencing, and p and q the AR and MA orders of the model parameters respectively. The value of these orders is determined through autocorrelation function (ACF) and partial autocorrelation (PACF) diagrams [31], I the differencing operator, and (1 − I)d the dth non-seasonal differencing. In the ARMA model, d is equal to 0 and it does not have the differencing operator.
As it is crucial to investigate the structure of the TS being studied prior to modeling, certain tests and preprocessing methods were initially applied to prepare the data for stochastic modeling. After separation of the dataset into training and testing samples, the existence of deterministic terms in the TS should be examined. For this purpose, MW, MK and Fisher tests are employed to check the existence of Jump, Trend and Period (respectively).
If the results of these tests show no deterministic terms, the stationary TS must be checked. Otherwise, any deterministic terms should be eliminated. The KPSS test is applied to check the stationary TS. If the result of this test does not confirm the stationary TS, Trend analysis and differencing is applied and the KPSS test is applied again to check the stationary TS. After ensuring that the TS is stationary, the TS normality is evaluated using the JB test. After making sure that the TS is stationary and normal, the preprocessing is finished and stochastic modeling is initiated. Initially, depending on the type of problem, it is determined whether the problem is seasonal or not. Then, the range of seasonal and non-seasonal parameters related to auto regressive (AR) and moving average (MA) terms, as well as a constant term, are determined using ACF and PACF diagrams. The ACF and PACF diagrams only determine the most important lags, not the optimum ones.
It may be possible to obtain the optimal model; it does not require the use of all the parameters specified by these two diagrams. The first way to obtain the optimum combination is to examine all the compounds resulting from the defined domains for the stochastic model parameters (i.e. 2p(max)+q(max) − 1 models for an ARMA model). Doing this is very time-consuming as one has to examine all the comparisons and compare them, and the results in many models should be examined as well. Therefore, integrating a stochastic model with the continuous genetic algorithm (CGA) is used in the current study. Indeed, the optimal values of the seasonal MA and AR parameters are determined through an evolutionary process. Then, the residual independence of the proposed model is evaluated using the Ljung-Box test. Finally, the performance of the model is appraised using test data. Considering the maximum number of ARMA, seasonal auto regressive (SAR) and seasonal moving average (SMA) as 5, an example of the optimum achieved solution by ARIMA-CGA is provided in Fig. 2.
An example of the integrated stochastic model with genetic algorithm. AR, auto regressive; SAR, seasonal auto regressive; MA, moving average; SMA, seasonal moving average
The objective function of the CGA is defined, in which all possible combinations are considered and the corrected Akaike information criterion (AICC) (Eq. 23) is employed to find the optimum model in terms of accuracy and simplicity simultaneously. The first term of the AICC indicates the accuracy of the model while the second one considers the complexity of the model.
$$ AICC\, = \,N\, \times \,Ln\left( {MSE} \right)\, + \,2\, \times \,Comp.\, + \,\frac{2 \times Comp.(Comp.\, + \,1)}{{N\, - \,Comp.\, - \,1}} $$
where N is the number of samples, MSE is the mean square error and Comp. is the complexity of the model. The Comp. is the summation of stochastic models (p, q, P, Q) and constant term if it exists. The MSE is calculated as:
$$ MSE\, = \,\,\,\,\frac{{\sum\nolimits_{i = 1}^{N} {(MED_{obs,i} \, - \,MED_{P,i} )^{2} } }}{N} $$
where MEDobs,i and MEDp,i are the ith value of the observed and predicted value (respectively). The flowchart of the preprocessing based stochastic model is presented in Fig. 3.
Flowchart of the preprocessing-based stochastic model. KPSS, Kwiatkowski–Phillips–Schmidt–Shin test; F, Fisher test; AR(p), auto regressive model; SAR(P), seasonal auto regressive model; MA(q), moving average model; SMA(Q), seasonal moving average model
Generalized structure of group method of data handling (GMDH)
GMDH is a self-organized approach that gradually produces more complex models when evaluating the performance of the input and output datasets [32]. In this approach, the relationship between the input and output variables is expressed by the Volterra Series, which is similar to the Kolmogrov–Gabor polynomial:
$$ y\, = \,a_{0} \, + \,\sum\limits_{i = 1}^{N} {a_{i} x_{i} } \,\, + \,\,\sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{N} {a_{ij} x_{i} x_{j} } } \, + \,\sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{N} {\sum\limits_{k = 1}^{N} {a_{ijk} x_{i} x_{j} x_{k} } \, + \, \cdots } } $$
where y is the output variable, A = (a0, a1, …, am) is the weights vector and X = (x1, …, xN) is the input variables vector. The GMDH model has been developed based on heuristic self-organization to overcome the complexities of multidimensional problems. This method first considers different neurons with two input variables and then specifies a threshold value to determine the variables that cannot reach the performance level. This procedure is a self-organizing algorithm.
The main purpose of the GMDH network is to construct a function in a feed-forward network on the basis of a second-degree transfer function. The number of layers and neurons within the hidden layers, the effective input variables and the optimal model structure are automatically determined with this algorithm.
In order to model using the GMDH algorithm, the entire dataset should first be divided into training and testing categories. After segmenting the data, it creates neurons with two inputs. Given that each neuron has only two inputs, all possible combinations for a model with n input vectors are as:
$$ NAPC\, = \,\left( \begin{gathered} n \hfill \\ 2 \hfill \\ \end{gathered} \right)\, = \,\frac{n(n\, - \,1)}{2} $$
where NAPC is the number of all possible combinations and n is the number of input vectors.
According to the quadratic regression polynomial function, all neurons have two inputs and one output with the same structure, and each neuron with five weights (a1, a2, a3, a4, a5) and one bias (a0) executes the processing between the inputs (xi, xj) and output data as follows:
$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{y} \, = \,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f} (x_{i} ,x_{j} )\, = \,a_{0} \, + \,a_{{1}} x_{i} \, + \,a_{{2}} x_{j} \, + \,a_{{3}} x_{i}^{2} \, + \,a_{{4}} x_{j}^{2} + \,a_{{5}} x_{i} x_{j} $$
The unknown coefficients (a0, a1, a2, a3, a4, a5) are obtained by ordinary least squares. The performance of all neural network methods is heavily influenced by the chosen parameters. The unknown coefficients are calculated through a least squares solution as follows:
$$ A_{i} \, = \,(x_{i}^{T} x_{i} )^{ - 1} x_{i} Y $$
where A = {a0, a1, a2, a3, a4, a5} is the unknown coefficients vector, Y = {y1, …, yN}T is the output vector and x is the input variable vector.
The AICC criterion (Eq. 23) is applied to determine the optimal network structure and select the neurons describing the target parameter. The Comp. in this equation for the GMDH model is defined as follows:
$$ Comp.\, = \,NL\, + \,NTN $$
where Comp. is the complexity, NL is the number of layers and NTN is the number of total neurons.
The performance of classical GMDH in the modeling of nonlinear problems has been demonstrated in various studies [33,34,35,36]. However, along with its advantages, it possesses the following limitations: (i) second-order polynomials, (ii) only two inputs for each neuron, (iii) inputs of each neuron can only be selected from the adjacent layer [37, 38]. In complex nonlinear problems, the necessity of using second-order polynomials may impede an acceptable result. In addition, considering only two inputs per neuron and using adjacent layer neurons would result in a significant increase in the number of neurons (NN) [39].
In the current study, a new scheme of GMDH as a GS-GMDH is employed and encoded in the MATLAB environment. The developed model removes all the mentioned disadvantages, so that each neuron can connect to two or three neurons at a time, taken from adjacent or non-adjacent layers. In addition, the order of polynomials can also be two or three. Similar to classical GMDH, the best structure is chosen based on the AICC index. According to the provided description, the developed GS-GMDH can offer four modes: (1) second-order polynomial with two inputs, (2) second-order polynomial with three inputs, (3) third-order polynomial with two inputs, and (4) third-order polynomial with three inputs. The first mode is classical GMDH.
Figure 4 indicates an example of the developed GS-GMDH for a model with five inputs and one output. In this figure, 3 different neurons (x11, x12, x21) are presented to provide an equation to estimate the target parameter (y). The two neurons x11 and x21 have three inputs, which are the inputs of the desired problem. The x21 neuron, which is the output of the problem, has three inputs similar to the two previous neurons (x11 and x21), except that it uses the non-adjacent layer neurons (x13) in addition to the adjacent layer neurons (x11 and x21).
An example of the developed generalized structure group method of data handling (GS-GMDH) for a model with five inputs and one output. × 1, ×2, ×3, ×4, ×5, input parameters; ×11, × 2, neurons in first layer; ×21, neuron in second layer; y, output
The GS-GMDH was used in this study to achieve the most precise results in forecasting the studied TS, which we abbreviated as MED data. GS-GMDH is superior to the former method, GMDH, due to the random structure of neurons that is encoded in the genotype string that results in using all neurons from previous layers in subsequent layers. In addition, GS-GMDH facilitates finding the minimized training and prediction errors separately, preventing model overtraining. The flow chart of the developed GS-GMDH model is presented in Fig. 5.
The flow chart of the developed generalized structure group method of data handling (GS-GMDH). MNI, maximum number of inputs; MNN, maximum number of neurons; IM, inputs more; DNN, decrease number of neurons; PD, polynomial degree; NL, number of layer; NN, number of neuron; n, number of input vectors; AICC, Akaike information criterion
Before starting the modeling using the GS-GMDH method, some parameters must first be determined. The first parameter is the Maximum Number of Inputs (MNI) that determines the maximum number of inputs for individual neurons. It could be two or three. If set to three, both two and three inputs are tried. Inputs More (IM) is the other one that should be determined before starting modeling. It could be zero or one. If set to zero, the inputs of each neuron are considered only for previous layer while if IM is set to one, this results in taking input from the non-adjacent layers also. The Maximum Number of Neurons (MNN) is equal to the number of input variables, while it could be twice that number for complex problems. The polynomial degree (PD) could be considered to be two or three. If set to three, both two and three are allowed.
Combining linear and nonlinear models (data-driven method)
ITS consists of stochastic and deterministic components. Thus, by using appropriate data preprocessing methods, it is possible to reduce the problematic effects of deterministic components in the modeling process. The proposed methodology is based on a continuous modeling process. This data-driven method is based on preprocessing to identify linear and nonlinear components of ITS, verification of the validity of decomposed data, and the decomposed model. In the studied case (6), the ACE TS fluctuates greatly. The outcomes of the single stochastic and neural network modeling approaches are relatively weak. Hence, as a third approach, the ACE TS is modeled with a combined stochastic-neural network model. Stochastic models perform efficiently, while TS are linear and do not contain deterministic terms that are responsible for nonlinearity. AI methods, on the other hand, allow the modeling of TS with nonlinear components. The TS, however, is not purely linear or nonlinear; both components are present simultaneously; the integration of which sometimes produces complex structures in the TS. In such cases, the use of single stochastic or nonlinear methods might be improved by a combined model. Combining stochastic models with AI methods is one of the most effective methods of modeling TS with complex structures. As shown in Fig. 6, the residuals of the stochastic models were used as a new TS in GS-GMDH modeling, such that the features of both modeling approaches were utilized.
Flowchart of interrupted time series modeling through a continuous nonlinear approach
Verification indices to evaluate models
To verify the accuracy of modeling performed in the TS MED forecasting, the correlation coefficient (R), scatter index (SI), mean absolute percentage error (MAPE), root mean squared relative error (RMSRE) and performance index (ρ) are used. In addition to these indices, the corrected AICC and Nash–Sutcliffe model efficiency (EN-S) based on comparing the model's simplicity with the goodness-of-fit and amount of deviation from the mean value [40] are used. The AICC index is used to find the best models in each TS modeling, and the lower the index value is the simpler the model. The EN-S index ranges from -∞ to 1, and the closer the index is to one, the more accurate the model.
$$ R\, = \,\frac{{\sum\limits_{i = 1}^{N} {\left( {MED_{obs,i} \, - \,\overline{MED}_{obs,i} } \right)\left( {MED_{pred,i} \, - \overline{\,MED}_{pred,i} } \right)} }}{{\sqrt {\sum\limits_{i = 1}^{N} {\left( {MED_{obs,i} \, - \,\overline{MED}_{obs,i} } \right)^{2} } \sum\limits_{i = 1}^{N} {\left( {MED_{pred,i} \, - \,\overline{MED}_{pred,i} } \right)^{2} } } }} $$
$$ SI\, = \,\frac{{\sqrt {\frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {MED_{obs,i} \, - \,MED_{pred,i} } \right)^{2} } } }}{{\overline{MED}_{obs} }} $$
$$ MAPE = \,\frac{100}{N}\left( {\frac{{\left| {MED_{obs,i} \, - \,MED_{pred,i} } \right|}}{{MED_{obs,i} }}} \right) $$
$$ RMSE\, = \,\sqrt {\frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {MED_{obs,i} \, - \,MED_{pred,i} } \right)^{2} } } $$
$$ \rho \, = \,\frac{{\left( {\sqrt {\frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {MED_{obs,i} \, - \,MED_{pred,i} } \right)^{2} } } /\sum\limits_{i = 1}^{N} {\left( {MED_{obs,i} } \right)} } \right)}}{1\, + \,R} $$
$$ E_{N - S} \,\,\, = \,\,\left[ {1 - \frac{{\sum\limits_{i = 1}^{N} {\left( {MED_{obs,i} - MED_{pred,i} } \right)^{2} } }}{{\sum\limits_{i = 1}^{N} {\left( {MED_{obs,i} - \overline{MED}_{pred,i} } \right)^{2} } }}} \right] $$
$$ AICC\, = \,\frac{{2kN\, + \,\left( {N\ln \left( {\sigma_{\varepsilon }^{2} } \right)\left( {N\, - \,k\, - \,1} \right)} \right)}}{N\, - \,k\, - \,1} $$
where k is the number of parameters, N is the number of samples, σε2 is the residuals' standard deviation, EN-S is the Nash–Sutcliffe test statistic, and MEDobs,i and MEDpred,i are the ith value of actual data and forecasted MED, respectively.
The Ljung-Box test is used to check the independence of the residuals of the modeled TS [41]. The test statistic is calculated as follows:
$$ Q_{Ljung - Box} \, = \,\left( {N^{2} \, + \,2N} \right)\sum\limits_{h = 1}^{m} {\frac{{r_{h} }}{N\, - \,1}} $$
where N is the number of samples, rh is the residual coefficient of the auto regression (εt) in lag h, and the value of m is equal to ln(N). If the probability corresponding to the Ljung-Box test statistic in the χ2 distribution is higher than the α-level (in this case PQ > α = 0.05), the residual series is white noise and the model is adequate.
Preprocessing tests
The values of the JB test show that the desired TS is distributed normally (pJB = 66.29 > 0.05). Figure 7 indicates the ACF and the PACF of the main TS (standardized rate of ACEs TS), and data showed that there is a correlation up to three non-seasonal lags (the time period). Since the values of ACF are rapidly damped and are within the limit boundaries, there is no significant period or trend in the TS. However, to ensure this, the existence of deterministic terms and stationarity of the main TS was also evaluated using quantitative tests.
Standardized rate of acute coronary events (ACEs) time series: a autocorrelation; b partial autocorrelation. Lag indicates the time period
Table 1 provides the results of the quantitative test to evaluate the existence of deterministic terms, stationarity and normality of the main series, and detrended and differenced TS. The results of the non-seasonal and seasonal MK tests show that the p-values of MK and seasonal MK are 0.02 and 0.24 respectively. Therefore, the ACE TS has a non-seasonal trend (pMK = 0.02 is less than critical value, 0.05). Hence, the trend must be removed. Moreover, the p-value of the Fisher test indicates that the TS has a period (pFisher = 5.85) greater than the critical value 3. According to the Fisher test, the severity of the period is not very high as the value is close to critical, then minor. Moreover, the MW test proves there is no jump in the TS (pMW = 2.78, higher than the acceptable value 0.05). The KPSS test also indicates the TS is non-stationary (pKPSS = 0.19, higher than the acceptable value 0.05), which is because of the trend and period detected in the TS.
Table 1 Evaluation of the presence of deterministic terms; stationarity and normality of the standardized rate of acute coronary events (ACEs) time series; and detrended and differenced time series
To remove the deterministic terms, two scenarios are defined: de-trending and stationarizing the ACE TS by differencing before stochastic modeling. The linear trend line is obtained as follows: trend line = 0.4792 × t + 201.72.
After eliminating the linear trend from the main TS, all of the deterministic factors are removed. Indeed, the detrended TS is stationary with no deterministic term. Similar to the main TS, this has a normal distribution. Consequently, the detrended TS is modeled with the ARMA. To find the parameters of the ARMA model, ACF and PACF diagrams are employed. As shown in Fig. 8, it is obvious there is still a correlation to three non-seasonal lags. Therefore, p and q in ARMA(p,q) are considered as p,q = {0,1,2,3}. For the purpose of determining more accurate and simpler models, the value of these parameters is considered 10.
Detrended time series: a autocorrelation, b partial autocorrelation. Lag indicates the time period
In the second scenario, differencing the main TS is proposed (differenced TS) to remove the deterministic terms. The findings in Table 1 in which MW, MK, SMK and JB increases are higher than 0.05, as well as the Fisher test higher than 3, indicate the differenced TS results in an increasing of the period in the new TS. Although the Fisher test exhibits growth in periodicity, the stationarity of the differenced TS increases considerably; thus, enabling the modeling of the TS. Furthermore, the differencing method has considerable impact on the correlation of the lags and decreases them markedly. Hence, an ARIMA model could be employed with fewer parameters and subsequently less error. The ACF and PACF of the differenced TS (Fig. 9) indicate that the values of p and q in this state are lower than the ARMA model.
Differenced time series: a autocorrelation, b partial autocorrelation. Lag indicates the time period
TS modeling, however, offers numerous combinations of previous lags from which to select the most appropriate TS input combination. Therefore, applying suitable preprocessing should lead to determining and selecting the most effective lag for modeling. According to the ACF plots for both preprocessed TS and test results, a maximum of three parameter orders are required for ARMA modeling and one for ARIMA modeling (Figs. 7, 8, 9). For modeling, the first 50 data were considered for the training stage and the remainder (nine data) for the testing stage. The stochastic-based linear modeling results are presented in Table 2. As the results in this table indicate, both linear models are relatively weak in modeling the ACE TS. The ARMA model outperforms ARIMA and the results are marginally better than ARIMA. The ARMA model with seven non-seasonal auto-regressive parameters and five non-seasonal moving average parameters modeled the ACE TS with R = 77.95%, SI = 3.46%, MAPE = 2.89%, RMSRE = 3.54%, EN-S = 0.66 and AICC = −15.84 in testing. The ARIMA model also performed slightly weaker than ARMA with R = 73.74%, SI = 4.21%, MAPE = 2.89%, RMSRE = 4.37, EN-S = 0.50 and AICC = −5.55 in testing. The Ljung-Box results for ARIMA and ARMA models are provided in Fig. 10. The test is done for the first 47 lags of the training part and the eight lags of the test part separately (n-1 data are considered for testing). It is observed that the residuals of both linear models are independent and the white noise and modeling are adequate and correct. Figure 11 demonstrates scatter plots of both ARMA and ARIMA models in testing and training versus the observed data. According to this figure, the majority of forecasted data are located within 5% intervals.
Table 2 Statistical indices for the stochastic-based linear modeling
Ljung–Box results of the auto-regressive moving average (ARMA) and auto-regressive integrated moving average (ARIMA) models for both training (train) and testing (test) stages. Lag indicates the time period
Scatter plot of the auto-regressive moving average (ARMA) and auto-regressive integrated moving average (ARIMA) model predictions of MED data
Generalized structure group method of data handling (GS-GMDH)
As mentioned earlier, AI methods are widely utilized for data simulation and forecasting. Each TS consists of two parts: linear and nonlinear. The stochastic models, also known as linear models, are able to model the linear part of the TS; hence, the nonlinearity is removed from the TS prior to modeling with pre-preprocessing methods. Conversely, AI models are known for their ability in modeling the nonlinear part. The neural network applied to the ACE TS under study is the GS-GMDH model, which is enhanced by the genetic algorithm. In this GS-GMDH, it is allowed to randomly apply crossover and mutation for the whole length of the chromosome string. Neurons are used for all layers and by calculating the errors separately; both training and testing sets have low errors. The results of this method are presented in Table 3. According to the results, the model was able to forecast the original TS without preprocessing with R = 82.35%, SI = 4.22%, MAPE = 3.16%, RMSRE = 4.25% and ρ = 2.33% for the training data and R = 89.35%, SI = 2.60%, MAPE = 2.10%, RMSRE = 2.66% and ρ = 2.33% for the testing data. As the scatter plot for both training and testing data in Fig. 12 indicates, the majority of forecasted data in the testing period have less than 5% error and are located within the intervals. The AI method employed allowed forecasting of the nonlinearity in the ACE TS very well. Though the GS-GMDH method performed better than the single ARIMA and ARMA models, with a mean growth of 11.63% in correlation, these results are relatively close to the stochastic models. Therefore, a complementary method is required.
Table 3 Statistical indices for the developed hybrid model in acute coronary event (ACE) forecasting
Scatter plot of generalized structure group method of data handling (GS-GMDH) in MED prediction
Combined data-driven modeling
As mentioned in previous sections, each model has certain specifications. Stochastic models perform efficiently, while TS are linear and do not contain deterministic terms that are responsible for nonlinearity. AI methods, on the other hand, allow the modeling of TS with nonlinear components. The TS, however, is not purely linear or nonlinear. Both components are present simultaneously, the integration of which sometimes creates complex structures. In such cases, employing single stochastic or nonlinear methods does not provide acceptable results. Therefore, alternative solutions are required to resolve this problem. Hybridizing stochastic models with AI methods is one of the most viable methods of modeling TS with complex structures. In the studied case, the ACE TS fluctuates greatly. The outcomes of the single stochastic and neural network modeling approaches are relatively weak. Thus, as a third approach, the ACE TS is modeled with a combined stochastic-neural network model. The hybrid model results are provided in Table 4.
It is apparent from the information supplied in Table 4 that the correlation between the modeled and observed data is rising. The R exhibited an average of 17.74% enhancement from the single stochastic model and 2.05% from the single GS-GMDH model. Although the results are slightly better than the single GS-GMDH model, model accuracy improved and in fact, the errors are about half those of the linear model. The ARMA-GS-GMDH model with R = 91.03%, SI = 2.52%, MAPE = 2.13%, RMSRE = 2.47% and ρ = 1.29% outperformed the ARIMA–GS-GMDH model with R = 91.91%, SI = 2.86%, RMSRE = 2.99% and ρ = 1.56% as well as all other models. Figure 13 demonstrates the scatter plot of the hybrid modeling results, where almost all forecasted data are within the ± 5% error interval. Figures 14 and 15 provide a good comparison between the observed MED data and the models. The box plot (Fig. 4a) shows that the hybrid model forecasted the interquartile area, mean and median of the data better than other models. However, the maximum and minimum predictions varied between the models (Fig. 4b). The superiority of the ARIMA–GS-GMDH model is demonstrated by the model's maximum, minimum and interquartile areas, which are much closer to the observed data than all other models, especially the regression model used in the Bernal et al. [6] study.
Scatter plot of auto-regressive moving average-generalized structure group method of data handling (ARMA–GS-GMDH) and auto-regressive integrated moving average–generalized structure group method of data handling (ARIMA–GS-GMDH) in acute coronary event (ACE) prediction
Box plot of all models versus observed data. GS-GMDH, generalized structure group method of data handling; ARMA, auto-regressive moving average; ARIMA, auto-regressive integrated moving average; ARMA–GS-GMDH, auto-regressive moving average–generalized structure group method of data handling; ARIMA–GS-GMDH, auto-regressive integrated moving average–generalized structure group method of data handling; Std rate standardized rate of acute coronary events (ACEs) [6]
Taylor graph for checking the performance of the linear, nonlinear and hybrid models in predicting acute coronary events (ACEs). GS-GMDH, generalized structure group method of data handling; ARMA, auto-regressive moving average; ARIMA, auto-regressive integrated moving average; ARMA–GS-GMDH, auto-regressive moving average–generalized structure group method of data handling; ARIMA–GS-GMDH, auto-regressive integrated moving average–generalized structure group method of data handling [6]
The Taylor diagram [42] investigates the performance of the models using the standard deviation (SD) and R of all the tested models simultaneously. The distance from any point to the observed data in the diagram is equivalent to the centered RMSE and a precise model is one with a coefficient of determination of 1 and SD similar to the observed data. [43, 44] As illustrated in Fig. 16, the sample ITS models, including combined data-driven modeling (ARMA–GS-GMDH and ARIMA–GS-GMDH); showed a superior performance to models in the Bernal et al. study [6]. Both data-driven models were situated closer to the reference (observed) point than the models alone (GS-GMDH, ARMA and ARIMA). ARMA–GS-GMDH has a lower SD and higher R. By applying a combined model, the difference between the model and observed data is decreased and accuracy of predicted results is increased (Table 4, Fig. 13) in both training and testing stages. The R of the proposed combined data-driven model (ARMA–GS-GMDH and ARIMA–GS-GMDH) exhibited an average of 17.74% enhancement from the single stochastic model (ARMA and ARIMA) and 2.05% from the nonlinear model (GS-GMDH). Although the results of combined approaches are slightly better than the single GS-GMDH model, the accuracy is improved and the errors were about half those of the linear model.
Scatter plot of observed acute coronary events (ACEs) and Auto-regressive integrated moving average–generalized structure group method of data handling (ARMA–GS-GMDH) superior hybrid model results compared to Bernal et al.'s model results. Std rate, standardized rate of ACEs [6,25]
As illustrated in Fig. 16, compared to the regression model of the Bernal et al. study [6], the single stochastic ARMA model and ARIMA have almost the same location in the diagram, in addition to showing a relatively higher RMSE than the single GS-GMDH and the combined model. The plot in Fig. 16 showed the superiority of the combined ARMA–GS-GMDH model with the observed ACE data [25] and the regression model [6]. Moreover, by combining the features of both models (ARMA and GS-GMDH), the fluctuations in the ACE TS could be better predicted. The series has severe fluctuations, which is why linear models alone cannot adequately forecast the data (Fig. 16). Hence, data (Table 5) showed that the combined model improved the results of the linear regression. The statistical indices indicate that the linear regression model has lower accuracy (R being 11.83% lower) and higher errors (SI, 1.33%; MAPE, 1.25%; and RMSRE, 1.19%) than the proposed model. Index ρ can be employed for measuring model error in addition to examining the correlation between the model and observational values. This index is lower for the ARMA–GS-GMDH model (ρ = 1.86%) compared to the Bernal et al. [6] model (ρ = 2.66%). Moreover, the Nash–Sutcliffe coefficient (EN–S), which is an index showing a model's weakness in forecasting extreme values, revealed an EN–S = 0.58 for the regression model which is considerably lower than the combined model EN–S = 0.78.
Table 5 Statistical indices for the proposed model, ARMA-GS-GMDH, and segmented regression method by Bernal et al. [6] for the same testing and training periods
This study provides a novel approach on the use of ITS modeling based on the continuous translational data driven approach. To validate the developed model, we assessed the effects of the Italian smoking ban in public areas on hospital admissions for acute coronary events. We propose a hybrid methodology using a continuous translational data-driven approach based on a combination of the stochastic and AI methods that will (i) increase the accuracy of prediction results through a continuous modeling process, and (ii) importantly will solve a challenging issue in ITS modeling regarding the time lag between pre- and post-intervention periods, which limits the application of the segmented regression method in ITS modeling.
The complex dynamic behavior of the ACE can be modeled with a TS approach, which deduces the characteristics of the data generation process by analyzing historical data. In a recent study, Bonakdari et al. [24] showed that future prevalence a complex heath care outcome can be evaluated by historical TS at a specific time. As different dependent parameters can have a serious impact on outcome, relevant information regarding the ACE was extracted based on historical data summarized as internal patterns. In this study, the ACE TS was modeled using linear-based stochastic model (ARMA, ARIMA), nonlinear-based GS-GMDH and an integration of a continuous linear (stochastic) with a nonlinear model (data-driven method). Two fundamental premises for stochastic modeling were stationarity and normal distribution. In order to achieve stationarity, the deterministic terms should be removed from the TS. For this purpose, the structure of the TS was investigated by different tests. Initially, the ACE TS structure was investigated by stationarity and normality tests. Data showed that the TS was normally distributed but was not stationary (Table 1). The deterministic term(s) responsible for non-stationarity (trend, jump and period) terms were performed and trend and jump were found in the series. Detrending the ACE TS by trend analysis was done by stationarizing the data and then by differencing the detrended data. The former surprisingly eliminated all deterministic terms and stationarized the TS very well by 44.55% (Table 1). The latter improved the stationarity and removed the linear trend completely, made some fluctuations in the TS and increased the Fisher statistic parameter. Nonetheless, the preprocessed ACE TS was completely stationary and normal. The ARMA and ARIMA models were the first applied to the series. In order to determine the order of the models, ACF plots were used and a maximum of three parameters were required. For further investigation, ten parameters were considered in modeling. The ARMA model with seven non-seasonal auto-regressive parameters and five non-seasonal moving average parameters in the testing period outperformed the ARIMA model. For the second TS modeling approach, the ACE TS data was modeled by GS-GMDH. The most important feature of these models is their ability to model nonlinearity better than linear stochastic models. The results showed that the single nonlinear model improved the accuracy of GS-GMDH. In the third and final step, a combination of linear and nonlinear models was made. As the results depicted, both ARMA–GS-GMDH and ARIMA–GS-GMDH outperformed the single models. The ARMA–GS-GMDH model enhanced the results by an average of 17.74 and 2.03% compared to the single linear and nonlinear models. As illustrated in the Taylor diagram, combined models have a higher R to observed data, lower RMSE and SD closer to the observed data than other models, thus better fitting the observed data.
The proposed methodology, as well as ITS modeling, can be employed for TS prediction. To verify the performance of the methodology in TS data set modeling, another health care real case was assessed. Bhaskaran et al. [45] used TS modeling in environmental epidemiology. They studied the association between ozone levels and the total number of deaths in the city of London (UK) for a time period of five years from 1 January 2002 to 31 December 2006. In brief, the authors [45] investigated three alternative techniques including time stratified model, periodic functions, and flexible spline functions to shed light on key considerations for modeling long term patterns of studied TS. Their prediction for the total number of deaths as TS outcomes yielded a coefficient of R = 0.71, 0.65 and 0.69 for each method, respectively. When applying the present developed methodology to their dataset, data from the hybrid model (ARMA–GS-GMDH) give more accurate results in which R = 0.75 for total number of deaths. These confirm not only that the proposed hybrid model is able to predict ITS outcomes (no need to identify the implemented intervention on outcomes), but it also can be employed for modeling TS with high accuracy compared to conventional approaches.
As detailed by Bonakdari et al. [46], conventional analysis of ITS in healthcare is based on regression methods that highly depend on intervention lag time which is very often difficult to determine. However, the present methodology can continuously be employed in such cases. As examples, the hybrid model could also be applied to several health conditions and include to analyze the relationship between smoking bans and the incidence of acute myocardial infarction [47]; to analyze the quality improvement strategy on the rate of being up-to-date with pneumococcal vaccination [48]; to assess the impact of health information technology initiatives on the performance of rheumatoid arthritis disease activity measures and outcomes [16], to name a few.
As all studies, there are limitations of the hybrid methodology and mostly associated with stochastic and/or nonlinear models. The most important limitation of such a hybrid method is the minimum length of outcome TS dataset needed in the training stage. In addition, selecting appropriate parameters of stochastic models in some cases requires increasing stationarization steps which could lead to differencing, seasonal standardization, and spectral analysis methods. In turn, selecting the best input combination in nonlinear models could also be a challenging task. Finally, designing AI architecture for a given ITS requires several trial and error steps to find the appropriate parameters.
Our study suggested that the proposed continuous translational data-driven model not only predicts ACEs with high accuracy and improved ITS prediction compared to current regression methods, but importantly, does not require any predefined lag time between pre- and post-intervention. This methodology can therefore be used as a reliable alternative in public health intervention evaluation. Hence, the novel hybrid approach provides a step forward by facilitating the modeling of such assessments in a short time. This is important for decision makers to manage health conditions as complex adaptive systems in a timely manner.
Not relevant.
ACEs:
Acute coronary events
ACF:
Autocorrelation function
AI:
AICC:
Akaike information criterion
AR:
Autoregressive
ARMA:
Auto-regressive moving average
ARIMA:
Auto-regressive integrated moving average
Continuous genetic algorithm
GMDH:
Group method of data handling
GS-GMDH:
Generalized structure group method of data handling
IM:
Inputs more
ITS:
Interrupted time series
MA:
TS:
KPSS:
Kwiatkowski–Phillips–Schmidt–Shin
JB:
Jarque–Bera
MK:
Mann–Kendall
MAPE:
Mean absolute percentage error
MNI:
Maximum number of inputs
MNN:
Maximum number of neurons
Mann–Whitney
NN:
Number of neurons
Polynomial degree
RMSRE:
Root mean squared relative error
Seasonal auto regressive
SMA:
Seasonal moving average
SMK:
Seasonal Mann–Kendall
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The authors would like to thank Santa Fiori for her assistance in the manuscript preparation.
This work was supported in part by the Osteoarthritis Research Unit of the University of Montreal Hospital Research Centre (CRCHUM) and the Chair in Osteoarthritis of the University of Montreal, Montreal, Quebec, Canada. The funding sources had no role in the study design, the collection, analysis and interpretation of data the writing of the report; nor in the decision to submit the article for publication.
Osteoarthritis Research Unit, University of Montreal Hospital Research Centre (CRCHUM), 900 Saint-Denis Street, R11.412, Montreal, QC, H2X 0A9, Canada
Hossein Bonakdari, Jean-Pierre Pelletier & Johanne Martel-Pelletier
Department of Soil and Agri-Food Engineering, Laval University, 2425 rue de l'Agriculture, Québec, QC, G1V 0A6, Canada
Hossein Bonakdari
Johanne Martel-Pelletier
HB designed the study and performed the analysis. HB, JPP and JMP interpreted the results and drafted the article. All authors critically revised the article for intellectual content and approved the submitted version of the article. All authors are accountable for the accuracy and integrity of the work.
Correspondence to Johanne Martel-Pelletier.
Bonakdari, H., Pelletier, JP. & Martel-Pelletier, J. A continuous data driven translational model to evaluate effectiveness of population-level health interventions: case study, smoking ban in public places on hospital admissions for acute coronary events. J Transl Med 18, 466 (2020). https://doi.org/10.1186/s12967-020-02628-x
Transitional model
Lag time
Data-driven Clinical Decision Processes | CommonCrawl |
Ind-scheme
In algebraic geometry, an ind-scheme is a set-valued functor that can be written (represented) as a direct limit (i.e., inductive limit) of closed embedding of schemes.
Examples
• $\mathbb {C} P^{\infty }=\varinjlim \mathbb {C} P^{N}$ is an ind-scheme.
• Perhaps the most famous example of an ind-scheme is an infinite grassmannian (which is a quotient of the loop group of an algebraic group G.)
See also
• formal scheme
References
• A. Beilinson, Vladimir Drinfel'd, Quantization of Hitchin’s integrable system and Hecke eigensheaves on Hitchin system, preliminary version
• V.Drinfeld, Infinite-dimensional vector bundles in algebraic geometry, notes of the talk at the `Unity of Mathematics' conference. Expanded version
• http://ncatlab.org/nlab/show/ind-scheme
| Wikipedia |
Broadband light management in hydrogel glass for energy efficient windows
Jia Fu1 na1,
Chunzao Feng1 na1,
Yutian Liao1,
Mingran Mao1,
Huidong Liu1 &
Kang Liu1,2
Frontiers of Optoelectronics volume 15, Article number: 33 (2022) Cite this article
Windows are critically important components in building envelopes that have a significant effect on the integral energy budget. For energy saving, here we propose a novel design of hydrogel-glass which consists of a layer of hydrogel and a layer of normal glass. Compared with traditional glass, the hydrogel-glass possesses a higher level of visible light transmission, stronger near-infrared light blocking, and higher mid-infrared thermal emittance. With these properties, hydrogel-glass based windows can enhance indoor illumination and reduce the temperature, reducing energy use for both lighting and cooling. Energy savings ranging from 2.37 to 10.45 MJ/m2 per year can be achieved for typical school buildings located in different cities around the world according to our simulations. With broadband light management covering the visible and thermal infrared regions of the spectrum, hydrogel-glass shows great potential for application in energy-saving windows.
Avoid the most common mistakes and prepare your manuscript for journal editors.
Energy use in buildings contributes to over 40% of the world's total energy consumption, of which lighting and space cooling make up a significant proportion [1, 2]. Traditional glass windows are the least-energy-efficient components in buildings [3, 4]. In the summer, the near-infrared (NIR) sunlight transmitted through windows produces into undesired heating [4,5,6]. In turn, the high reflection of mid-infrared (MIR) limits heat rejection from the building. This "greenhouse effect" aggravates cooling energy consumption [7]. In the past decades, a lot of efforts have been devoted to manipulation of the optical properties of windows, to save energy in buildings.
Radiative cooling that passively dumps heat into the cold outer space via MIR electromagnetic waves through the atmosphere window is an attractive strategy for building cooling [8,9,10,11]. Visible transparent coatings with high emissivity in MIR have been proposed to enhance heat dissipation by way of windows [12,13,14,15]. However, they do not block the NIR sunlight. Low-E glass can reflect the NIR sunlight to mitigate internal building heating, but the low emissivity of MIR induces heat accumulation and low visible transmittance leads to higher lighting demand [16, 17]. Electrochromic windows, which can switch the NIR transmittance with an applied electric field, have also been developed to improve building energy efficiency [18]. However, the narrow working spectrum of these approaches distances the windows from optimized efficiency, which requires the manipulation of electromagnetic spectrum covering visible, NIR and MIR [19, 20]. Novel windows with broadband light management capability are still highly desirable in order to improve energy-efficiency.
Hydrogels are formed through the cross-linking of hydrophilic polymer chains within an aqueous environment. The water-rich and transparent nature makes them a potential choice for window engineering. Based on its thermo-responsive property, Zhou and Li et al. used hydrogels to fabricate thermochromic hydrogel glass with visible transparency varying with environmental temperature [21, 22]. However, these works focus only on the visible manipulation and temperature responsiveness. The inherent property of hydrogels in near-infrared and mid-infrared, and the potential benefit of low refraction index, are not investigated.
In this work, we propose a hydrogel-glass (Fig. 1a) with broadband light management capability covering the visible, NIR and MIR spectrum and we investigate its performance in energy-saving windows. First, we introduced the fabrication of the hydrogel glass and the basic optical properties in the visible, NIR and MIR regions of the electromagnetic spectrum. Then, the optical and thermal performances of the hydrogel glass were demonstrated in a solar cell and a hand-made simulated house. Lastly, the energy saving potential of the application of hydrogel glass in windows in a school building was estimated via the simulation with EnergyPlus software.
Structure design and working principle of the hydrogel-glass. a Schematical structure of the hydrogel-glass with the functions of enhanced visible transmission, absorbed NIR light and enhanced MIR emission. b Calculated photon penetration depths for liquid water and traditional silica glass. c Photograph of the fabricated hydrogel-glass. d Thermogravimetric analysis of the hydrogel. Inset is the SEM image of the freeze-dried hydrogel
2.1 Preparation of the hydrogel-glass
The hydrogel-glass was prepared by directly photo-polymerizing the hydrogel precursor solution on the glass surface. First, the glass surface (40 mm × 40 mm × 2 mm) was activated by air plasma (10.5 W) for 10 min, during which hydroxyl groups were bonded onto the glass surface. Then, the substrate was immersed into mixed solution containing 3 mL 3-(trimethoxysilyl) propyl methacrylate, 20 µL acetic acid and 150 mL deionized water for 12 h. During the immersion, the silane coupling agents bonded with the hydroxyl groups, forming a layer of silanol groups on the glass surface [23]. Subsequently, the glass was put into a mold thicker than the glass and the precursor solution, containing 2 mol/L acrylamide, 0.001 mol/L N, N′-methylenebis(acrylamide) and 0.002 mol/L 2-hydroxy-4′-(2-hydroxyethoxy)-2-methylpropiophenone, was poured into the mold. Then the mold was clamped by two pieces of glass substrates and irradiated under ultraviolet light (365 nm, ca. 4 mW/cm2) for 8 h with nitrogen protection, obtaining the polyacrylamide (PAAm) hydrogel bonded glass. Here, the thickness of hydrogel was determined by the thickness of the mold. Finally, the hydrogel-glass was soaked in aqueous LiBr solution (8 mol/L) until it was swollen completely, obtaining the hydrogel-glass. The LiBr was introduced into the hydrogel to prevent dehydration.
2.2 Characterizations
The morphologies of the dry hydrogel were characterized by a scanning electron microscope (TESCAN, MIRA3). Thermogravimetric analyzer (TGA 4000, PerkinElmer) was used to measure the water content in the hydrogel. The tests were conducted in the temperature range from 30 to 800 °C with a scanning rate of 15 °C/min. The bonding strength between the hydrogel layer and the glass layer was measured by a universal mechanical test machine (CMT6350, SANS) at a stretching rate of 15 mm/min. Mass variations of hydrogels were measured by an electronic balance (ML-T, Mettler Toledo). Spectral reflectance (R) and transmittance (T) in 0.3–2.5 μm were measured using a UV–Vis-NIR spectrophotometer (Lambda 1050, Perkin Elmer) equipped with an integrating sphere (Labspher8). Spectral reflectance (R) in the range 2.5–25 μm was measured by a Fourier transform infrared spectrometer (FTIR, INVENIO S, Bruker) with a gold integrating sphere (A562). The spectral absorptance (A) can be calculated by A = 1 − R − T in the solar spectrum (AM1.5), and by A = 1 − R in MIR spectrum (T = 0).
2.3 Performance measurements
In the laboratory environment, the photocurrent density–voltage curves of the solar cells with different thick hydrogel layers were characterized by a Keithley 2400 source meter. Temperature of the cell was measured using a thermocouple (TT-K-30, Omega Company) fixed at the bottom of the cell and recorded by a data logger (TC-08, Pico Technology). The simulated solar light was provided by a solar simulator (SS-100A, Class AAA, Sanyou Corporation). The solar intensity was measured by an optical power meter (CEL-NP2000-2).
To measure the radiation cooling performance, solar cells with normal glass and hydrogel-glass were placed in a chamber, made of the polystyrene foam to minimize the influence of environment effects. Temperatures of the solar cells were measured by two thermocouples fixed in the center of the bottom side. Another thermocouple was put in the chamber to measure the ambient temperature. The relative humidity of the environment was monitored using a thermo-hygrometer (CENTER 310).
To demonstrate the visible and NIR light management of the hydrogel glass, two model houses with the same size of 20 cm × 20 cm × 20 cm were used for outdoor experiments. The walls and floor of the houses were made of an outer layer of 2-cm-thick glued wood and an inner layer of 3-cm-thick polystyrene foam to minimize heat leakage. One of the houses was covered by a hydrogel-glass window (3-mm-thick hydrogel on a 6-mm-thick glass substrate). The other house was covered with a 6-mm-thick glass for comparison. In each house, the indoor illuminance and temperature were measured by a split type lux meter (DELIXI DLY-1801) with a silicon detector (working wavelength range in 400−900 nm) and a thermocouple, respectively. The lux meter was placed on the center of the floor of each house, and the thermocouple was in the air about 3 cm above the floor. The temperature of the hydrogel glass was measured with a thermocouple fixed between hydrogel and glass layers. A thermocouple was fixed on the top of the normal glass via glue to measure the temperature. The outdoor solar intensity was recorded by the optical power meter (CEL-NP2000-2).
2.4 Simulations
The energy-saving of the hydrogel-glass was simulated by the EnergyPlus (version 9.5.1) software. A typical primary school building model from the EnergyPlus database was employed. The total building area was 6871 m2, and the building exterior surface included walls (2473.6 m2), roofs (3518.26 m2), vertical windows (865.76 m2), and horizontal skylights (13.38 m2). The total window opening area was 879.14 m2.
The optical and thermal properties of windows used in the simulations are listed in Additional file 1: Table S1. Normal glass (CLEAR 6MM) was selected from the EnergyPlus database. The parameters of transparent NIR shielding (TNS) glass was obtained from the reference [24]. The optical and thermal parameters of hydrogel-glass were calculated from the measured data. (Details about the calculation can be found in Additional file 1: Note 1 and Note 2). The weather data of each city were obtained from the EnergyPlus database website [25].
In the simulation model, extra heating or cooling was needed to balance the heat transfer between the internal building and the environment, so as to maintain a fixed indoor temperature. The extra power can be calculated:
$${P}_{\mathrm{gen}}^{ }+{P}_{\mathrm{sun}}={P}_{\mathrm{conv}}+{P}_{\mathrm{cond}}+{P}_{\mathrm{rad}}+{P}_{\mathrm{extra}},$$
where Pgen is the heat generated by the internal loads, Psun is the absorbed solar flux by the school building (roofs, walls and windows). The right side of the equation represents the heat exchange between the building and the environment. Here, Pconv, Pcond, and Prad are the net convective, conductive, and radiative heat fluxes, respectively. Pextra is the extra power to maintain the room temperature at a determined value.
The lighting energy consumption was calculated from the EnergyPlus daylighting model using the SplitFlux method [26]. Indoor brightness is maintained at 300 Lux by both the natural and artificial lighting.
The hydrogel-glass consisted of a layer of hydrogel and a layer of normal glass (Fig. 1a). The hydrogel layer was composed of small amount of cross-linked polymer and large amount of liquid water. Hence, it possessed similar optical properties to those of water [27, 28]. As shown in Fig. 1b, traditional glasses have large photon penetration depth in the whole solar spectrum without the capability to filter out NIR from sunlight (Additional file 1: Note 3). As a comparison, the photon penetration depth of water is greater than 1 m for most of the visible spectrum, and rapidly decreases to several millimeters for most of the NIR spectrum. Thus, the hydrogel can let through incoming visible light and block incoming NIR. Moreover, according to the figure, the transmittance of the hydrogel can be tuned by its thickness to obtain a desirable transmittance spectrum. While in the wavelengths corresponding to infrared transparency window of atmosphere, both water and glass have low penetration depth, which means they are nontransparent even with the thickness of 100 μm. Here the penetration depth of the glass has the smallest value of about 0.3 μm around the wavelength of ~ 9 μm, which is caused by the Si–O–Si asymmetric stretching vibrations [29, 30]. The strong lattice variation in the glass also means it has a high refraction index, high surface reflection and low thermal emittance [12]. However, no strong lattice or molecular vibration exist in the water or hydrogel [30]. Therefore, it is possible that the hydrogel-glass can enhance the MIR emission, block most NIR sunlight, and keep high transmission of visible light (Fig. 1c).
Thermogravimetric measurement showed that vaporizable water in the hydrogel accounts for 62.2% of the total weight, an equivalent volume fraction of ~ 95% in the hydrogel (Fig. 1d). SEM image of the freeze-dried hydrogel clearly revealed the network of solid polymer, which helped to confine liquid water inside the hydrogel (Fig. 1d). In addition, the LiBr salt inside reduced the vapor pressure of water inside to balance with the ambient humidity [31, 32]. Hence, the hydrogel retained its weight and structure stability under varying environmental conditions, as shown in Additional file 1: Fig. S1. The hydrogel layer was also firmly bonded with the glass substrate, as illustrated by the stress–strain curve in Additional file 1: Fig. S2.
We further characterized the spectral absorptance of the hydrogel-glass in the wavelength range from 0.3 to 25 μm (Additional file 1: Fig. S3). The absorptance of traditional glass was also presented for comparison. As shown in Fig. 2a, the hydrogel-glass had extremely low absorptance approaching zero in the visible light, which is very similar to traditional glass. In the NIR spectrum, the hydrogel-glass had multiple absorption peaks originating from the O–H stretching vibrations [33]. With increase of the thickness of hydrogel layer, the hydrogel-glass presented higher absorptance, and a hydrogel layer with several millimeter thick could block most NIR sunlight. In the atmosphere window spectrum, the hydrogel-glass had higher absorptance than the glass, which means hydrogel-glass had higher thermal emittance. To further confirm the measured results, we calculated the theoretical spectral absorptance of hydrogel layer on a glass substrate based on the model presented in Additional file 1: Fig. S4. Detailed calculation can be found in Additional file 1: Note 1. As shown in Additional file 1: Fig. S3, the theoretical spectral absorptances were close to the measured results.
Spectral characteristics of the hydrogel-glass with different-thickness hydrogel layers compared with traditional glass. a Measured spectral absorptance in the wavelength range from 0.3 to 25 μm. Here, HG in the figure is abbreviation of hydrogel-glass. b Spectral transmittance in visible light. The solid and dashed lines represent the measured and calculated results, respectively. The inset chart shows the average transmittance (\(\overline{T }\)). c Measured spectral reflectance in the wavelengths of infrared transparency window of atmosphere
We also measured the transmittance of the hydrogel-glass in the visible spectrum from 0.38 to 0.76 μm. As shown in Fig. 2b, the hydrogel-glass had slightly higher transmittance than the glass substrate in most of the visible spectrum. The average transmittances (\(\overline{T }\)) of hydrogel-glass in visible spectrum were calculated based on Additional file 1: Eq. (S6). As shown in the inset chart of Fig. 2b, \(\overline{T }\) of the glass substrate was 92.3%. While the average transmittance of the hydrogel-glass was 92.8% with a 0.15-mm-thick hydrogel layer. Two reasons are responsible for the enhanced visible transmittance: firstly, water has a negligible extinction coefficient in the visible range; secondly, the refractive index of water (1.33) is lower than that of glass (1.5), reducing the surface reflection of incident light. When further increasing the hydrogel thickness to 1.7 and 3.4 mm, \(\overline{T }\) decreased to 92.6% and 92.5%, respectively. Figure 2c presents the spectral reflectance of the hydrogel-glass and normal glass in the wavelengths of infrared transparency window of atmosphere. The glass exhibited a reflection band at about 9 μm due to the strong refraction caused by the Si–O–Si asymmetric stretching vibrations [29], and this led to a low thermal emittance of ~ 0.84. In contrast, the surface reflectance of the hydrogel-glass was lower than 5% in the wavelengths of infrared transparency window of atmosphere. By integrating the spectral absorption over the wavelength with respect to black body radiation (Additional file 1: Eq. (S7)), we obtained a high thermal emittance of ~ 96%, which is higher than that of other transparent radiative cooling windows [12, 34]. The results of transmittance and reflectance of hydrogel-glass in the full region of the electromagnetic spectrum can be found in Additional file 1: Fig. S3b and 3c.
To demonstrate the broadband light modulation capability, we simply put the hydrogel layer on a commercial glass encapsulated Si solar cell, as shown in Fig. 3a and Additional file 1: Fig. S5a. Under solar irradiance of 1 kW/m2 from the sunlight simulator, the solar cells with hydrogel-glass presented higher photocurrent densities and efficiencies than those for the cells encapsulated with normal glasses, as shown in Fig. 3b and Additional file 1: Fig. S5b. The photocurrent density increased from 4.48 to 4.59 mA/cm2 with the thickness of hydrogel decreasing from 3.4 to 0.15 mm. The temperature of the solar cells decreased with the decreasing thickness due to the higher surface reflectance in the solar spectrum (Additional file 1: Figs. S5c and d), indicating that smaller thickness is more suitable for enhancing photovoltaic efficiency. Here it is important to note that, the hydrogel-glass enhances the thermal emittance in the wavelengths of infrared transparency window of atmosphere, and can also induce strong radiative cooling and possibly reduce the temperature of solar cells in the working mode. As shown in Fig. 3c and Additional file 1: Fig. S6, the temperature of a hydrogel glass in an outdoor environment is obviously lower than the temperature of traditional glass and ambient temperature, indicating strong radiative cooling. Thus, it is predicted that the efficiency of the solar cell with hydrogel glass can be even higher through heat dissipation to outer space [35, 36].
Demonstrations of the broadband light management performance of the hydrogel-glass. a Schematic of the solar cell with hydrogel-glass. b Current density of the solar cell with different thickness of hydrogel layers. c Temperatures of the solar cell, solar cell with hydrogel-glass and environment from 20:00 to 23:00 at outdoor (May 21, 2021 in Wuhan, China). The relative humidity was ~ 70%. d Schematic of a house with a piece of hydrogel-glass as the window. Here, the 3-mm-thick hydrogel was used due to its high NIR absorption. e Illuminance inside the house with the hydrogel-glass and common glass in a sunny day (Jan. 1st, 2022). f Indoor temperatures inside the houses with hydrogel-glass and common glass
To further demonstrate the potential of the hydrogel-glass for windows in buildings, we set the hydrogel-glass on a small house model with a size of 20 cm × 20 cm × 20 cm, and measured the illuminance and temperature at different spots (Fig. 3d and Additional file 1: Fig. S7a). Figure 3e shows that the indoor illuminance of the house with hydrogel-glass is slightly higher than that in the house with normal glass in a sunny day. The enhanced illuminance helps to reduce the electricity consumption from lighting. More importantly, the indoor temperature (Tin) of the house with the hydrogel-glass is always lower than that of the house with normal glass as shown in Fig. 3f. The largest temperature reduction reaches 3.5 °C at noon with the highest solar intensity of 58.7 mW/cm2. Hence, the hydrogel-glass can reduce the cooling power consumption of a building in the summer. The reduction of the indoor temperatures is mainly caused by the absorption of NIR sunlight in the hydrogel-glass; the temperatures of the hydrogel-glass (Tout) were always higher than those of the normal glass, as presented in Additional file 1: Fig. S7b.
To evaluate the energy-saving potential of the hydrogel-glass in buildings, we calculated the energy consumption of a typical building, using EnergyPlus, considering the daytime lighting and indoor temperature regulation. As shown in Fig. 4a, the building model was a school with two types of windows including the vertical windows in walls and horizontal skylights in roofs. The windows accounted for 13% of the external surface area. The energy consumptions of normal glass and TNS glass were also calculated for comparison [24]. Detailed optical and thermal parameters of the three types of glass are summarized in Additional file 1: Table S1.
Energy saving evaluation of the hydrogel-glass as windows in buildings. a Schematic of a school building. b Annual energy consumption of the school building based on weather data for Wuhan, China (30.62° N, 114.13° E). Three windows of normal glass, TNS, and hydrogel-glass were used in the simulations. c Annual energy savings and the corresponding percentage of building models with hydrogel-glass in eight cities around the world.
We first evaluated the annual energy consumption per building area for lighting and cooling of the building model, assuming its location to be in Wuhan, which has a subtropical monsoon climate (Additional file 1: Table S2). As shown in Fig. 4b, the normal glass was calculated to have annual cooling and lighting energy consumption of 89.57 and 46.65 MJ/m2, respectively. The use of TNS glass reduced the cooling energy to 84.79 MJ/m2, but increased the lighting energy demand to 47.32 MJ/m2. In contrast, the use of hydrogel-glass reduced both the cooling and lighting energy to 84.04 and 46.62 MJ/m2, respectively. The annual energy saving was calculated to be up to 5.56 MJ/m2 as compared with that for normal glass (Fig. 4c). The annual electricity savings in eight representative cities (Additional file 1: Table S2) with different climatic conditions were also calculated [25]. As shown in Fig. 4c, the calculated energy savings were ranged from 2.37 to 10.45 MJ/m2, accounting for ~ 3% to ~ 8% of annual energy consumption. Among them, the Aswan city in the dry and hot tropical desert climate has the highest potential for energy saving up to 10.45 MJ/m2. In cities with longer annual lighting time such as Moscow, the hydrogel-glass was evaluated to have the highest annual lighting energy saving of 0.05 MJ/m2, as shown in Additional file 1: Table S3. These results suggest the great potential of hydrogel-glass for energy saving in different climate conditions.
In summary, we have developed a novel hydrogel-glass by coating environmentally stable hydrogel onto traditional glass. With the low refraction hydrogel layer, the hydrogel-glass increases the visible transparency of windows and reduces the electricity consumption for building illumination. In addition, it blocks most of the NIR sunlight and rejects the heat into outer space via enhanced mid-infrared emittance, and thus reduces the cooling power demand. Its absorption, transmission and reflection properties at different wavelengths endow the hydrogel-glass with capability of simultaneous solar and thermal management. Based on simulations, the hydrogel-glass can achieve energy savings ranging from 2.37 to 10.45 MJ/m2 per year for building models located at different cities around the world, providing a possible approach for next-generation energy-efficient windows.
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K. L. acknowledges the National Natural Science Foundation of China (Grant No. 51976141) and Open Project Program of Wuhan National Laboratory for Optoelectronics (No. 2018WNLOKF018). H. L. acknowledges the National Natural Science Foundation of China (Grant No. 52002291).
Jia Fu and Chunzao Feng contributed equally to this work.
MOE Key Laboratory of Hydraulic Machinery Transients, School of Power and Mechanical Engineering, Wuhan University, Wuhan, 430072, China
Jia Fu, Chunzao Feng, Yutian Liao, Mingran Mao, Huidong Liu & Kang Liu
Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, 430074, China
Kang Liu
Jia Fu
Chunzao Feng
Yutian Liao
Mingran Mao
Huidong Liu
All authors read and approved the final manuscript.
Correspondence to Huidong Liu or Kang Liu.
Note 1. Theoretical calculations of the reflectance, transmittance and absorptance of the hydrogel-glass. Note 2. Thermal conductivity of the hydrogel-glass. Note 3. Photon penetration depth of water and glass. Figure S1. Mass variation of the hydrogel exposed in ambient environment. The size of the samples is 1cm × 2 cm × 2 mm. The ambient temperature is about 26 °C, and the relative humidity is about 50%. Inset shows the hydrogel after storage in environment for two years and the newly prepared hydrogel. Figure S2. Stress-strain curve of the hydrogel sandwich between two glass plates. Inset is the photograph during the test. Figure S3. Optical spectra of the glass and hydrogel-glasses. (a) Theoretical and measured spectral absorptance in region of 0.3-25 μm. Here, it is noted that the liquid water has no absorption in the UV spectrum according to the calculation. The measured absorption of the hydrogel is caused by the polymer networks in the hydrogel. The solid and dashed line represent the measured and calculated results, respectively. (b-c) Measured spectral transmittance (b) and reflectance (c) in the range of 0.3-2.5 μm. Figure S4. Schematic of the calculations of the transmittance and reflectance of the hydrogelglass. Figure S5. Light management of solar cells with hydrogel-glass with different thick hydrogel layers. (a) Photograph of the solar cells. The ccale bar is 4 cm. (b-c) Photovoltaic efficiencies (b) and working temperatures (c) of the solar cells.(d) Measured spectral reflectance of the solar cells in 0.3-2.5 μm. Figure S6. Experiment setup to test the radiative cooling performance of hydrogel-glass. (a) Schematic of the setup. (b) Photograph of the experiment setup. Figure S7. Performance demonstration of the hydrogel-glass in a house model. (a) Photograph on experiment setup. (b) Temperatures of the hydrogel-glass, normal glass and environment. Table S1 Optical and thermal properties of three windows used in the simulations. Table S2 Climatic classification and geographical location of eight cities. Table S3 Annual electricity consumption and energy saving potential in the eight cities.
Fu, J., Feng, C., Liao, Y. et al. Broadband light management in hydrogel glass for energy efficient windows. Front. Optoelectron. 15, 33 (2022). https://doi.org/10.1007/s12200-022-00033-4 | CommonCrawl |
\begin{document}
\title{Necessary and sufficient conditions for the finiteness of the second moment of the measure of level sets } \author{J-M. Aza\"{i}s\thanks{IMT, Universit\'{e} de Toulouse, Toulouse, France. Email: [email protected]} \qquad J. R. Le\'{o}n\thanks{IMERL, Facultad de Inegenier\'{\i}a Universidad de la Rep\'{u}blica, Montevideo, Uruguay. Email: rlramos\,@fing.edu.uy and Universidad Central de Venezuela. Escuela de Matem\'atica.}} \maketitle \begin{abstract} For a smooth vectorial stationary Gaussian random field $X:\Omega\times\R^d\to\R^d$, we give necessary and sufficient conditions to have a finite second moment for the number of roots of $X(t)-u$. The results are obtained by using a method of proof inspired on the one obtained by D. Geman for stationary Gaussian processes long time ago. Afterwards the same method is applied to the number of critical points of a scalar random field and also to the level set of a vectorial process $X:\Omega\times\R^D\to\R^d$ with $D>d$. \end{abstract} \keywords{Level Sets, Kac-Rice formula, Moments, Random fields}\\ \textup{2000} \textit{Mathematics Subject Classification}: \textup{60G15, 60G60}
\section{Introduction} In the 1940s three articles with apparently different orientations appeared in mathematical literature. Firstly was Mark Kac's paper \cite{Kac:Kac} ``On the average number of real roots of a random algebraic equation" and secondly two papers written by S.O. Rice \cite{Rice1}, \cite{Rice2}``Mathematical analysis of random noise I and II". In the work of Kac and in the second of Rice the zeros of Gaussian random functions were studied. In particular they established with precision a formula, known today as the Kac-Rice formula, which allows to compute the expectation of the number of zeros (or crossings by any level) of a Gaussian random function. In spite of the apparently separated that seem the works, M. Kac in the review of the article affirms that ``All these results (of Rice) can also be derived using the methods introduced by the reviewer (M. Kac)".
After these two works an intense research activity has been developed. In particular, the interest in these subjects had a great impulse after the appearance of the book written by H. Cramer and M. R. Leadbetter \cite{C:L}. In this work, there is not only a general demonstration of the Kac-Rice formula for the number of crossings of a Gaussian processes, but also formulas for the factorial moments of this last random variable. An important fact to notice is that in the book a sufficient condition for the second moment of the number of crossings of zero to be finite is established. Then a little time later D. Geman in \cite{Ge} showed that this condition was also necessary. This condition is now known as `` the Geman condition". This result has been extended to any level at \cite{Kratz:Leon}.
The theme gained a new impulse when appear in the eighties two books, the first one written by R. Adler \cite{Ad1} ``The geometry of random fields" and the second one a Lecture Notes \cite{W1} written by M. Wschebor ``Surfaces al\'eatoires. Mesure g\'eom\'etrique des ensembles de niveau". Both books focus their study on crossings or geometric invariants of the level sets, for random fields having a multidimensional domain and taking scalar or vector values. The problems studied by Cramer \& Leadbetter were extended to this new context. In particular we must point out the Adler \& Hasofer's article \cite {Ad:Ha} in which conditions are established so that the number of stationary points for a Gaussian field of $ X: \R ^ 2 \to \R $ have a second moment. It is important to observe that studying the stationary points of a scalar field leads to study the zeros of its gradient, which is a vector field.
The twenty-first century saw two books appear \cite{Ad:Tay} and \cite{AW} that gave a new impetus to the subject. New fields of application of the formulas appeared in the literature and the area has become a large domain of research. We can point out for instance the applications to the number of roots of random polynomial systems (algebraic or trigonometric) and also to the volume of nodal sets when the systems are rectangular \cite{Peccati}. Also Kac-Rice formulas are a basic tool to study the sets of zeros of random waves and it has been much effort to prove or disprove Berry's conjectures \cite{berry1}, see \cite{Peccati} and the references therein. A field of applications where the formulas have been very useful is in random sea modeling, the Lund's School of probability has been very active in these matters, see for instance the paper \cite{Lindgren} and the references therein. In addition, the processes to which the crossings are studied can have their domain in a manifold of finite dimension see \cite{let1}. A very interesting case of this last situation is the article \cite{Au:Ben} where the domain of the random filed is the sphere in large dimension.
In the present paper we obtain necessary and sufficient conditions to have a finite second moment for the number of roots of $X(t)-u$, for a stationary, mean zero Gaussian field $X:\Omega\times\R^d\to\R^d$. The proofs of the main results are rather simple using the case $d=1$ as inspiration. Our results can be extended to the number of critical points of a stationary mean zero scalar Gaussian field. We must note that recently in \cite{estr:four} a sufficient condition for the critical points of a scalar field has finite second moment was given, however our method is rather different. Finally let us point us that as a bonus our method of proof allows obtaining a very simple result for the volume of level sets for Gaussian fields $X:\Omega\times\R^D\to\R^d$ with $D>d.$ Under condition of stationarity and diferentiability, the second moment is always finite.
Suppose that we have a way to check easily that the measure of the level set of a Gaussian field has finite second moment. Then it is ready to obtain an It\^o-Wiener expansion for this functional. Two consequences of this representation are important to remark: firstly the asymptotic variance of the level functional can be computed and also the speed of the divergence of this quantity can be estimated, secondly the fourth moment theorem can be used to obtain diverse CLT. This has been done some time ago in \cite{KL} and more recently in a lot of papers. We can cite by instance the article \cite{Peccati} where one can also consult some recent references.
The organisation of the
paper is the following: in Section \ref{Geman1} we revisit the results of \cite{Kratz:Leon} in dimension 1. Section \ref{dd} studies the number of points of levels sets for a random field $X: \R^d \to \R^d$, $d>1$. The subsection \ref{criti} is devoted to the study of the number of critical points of a random field $X: \R^d \to \R$. Section \ref{Dd} studies the measure of levels sets for a random field $X: \R^D\to \R^d$, $D>d$. The proof of the different lemmas are given in the appendix. \label{dd}
\section{ Real valued process on the line, Geman's condition}\label{Geman1} The results of this section are contained in the paper \cite{Kratz:Leon}. However, we present a new proof as an introduction to the next section.
\noindent Consider a process $X: \R\to\R$ and assume \begin{itemize} \item It is Gaussian stationary, normalized this is: $$\E(X(0))=0;\quad {\rm Var}(X(t))=1.$$ This last point is without loss of generality. \item The second spectral moment $\lambda_2$ is positive and finite. \end{itemize} Let $N_u([0,T]):=\#\{t\in[0,T]:\, X(t)=u\}$ for a given level $u\in \R$. Moreover we define the covariance $$r(\tau)=\E[X(0)X(\tau)].$$
And let set $$\sigma^2(\tau):={\rm Var}(X'(0)|\; X(0)=X(\tau)=0)=\lambda_2-\frac{(r'(\tau))^2}{1-r^2(\tau)}.$$ In what follows $(Const)$ will denote a generic positive constant, its value can change from one occurence to another.\\ The relation $x \leq (Const) y, y \leq (Const)x$ is denoted $ x\asymp y$.
The object of this section is to prove the next theorem: \begin{theorem} \label{t:1} The following statements are equivalent \begin{enumerate}[label=(\alph*)] \item $\E(N_u([0,T])^2)$ is finite for some $u$ and $ T$. \item $\E(N_u([0,T])^2)$ is finite for all $u$ and all finite $T$. \item The intergral $\int \frac{\sigma^2(\tau)}{\tau}d\tau$ converges at zero. \end{enumerate} \end{theorem}
{\bf Remark: } Integrating by parts in (c) we get the classical Geman's condition by using the following lemma, whose proof (as well as the proofs of all lemmas) is referred to the appendix. \begin{lemma}\label{integrales} There is equivalence between the convergence at zero of the two following integrals $$
\int \frac{\lambda_2+r''(\tau)}{\tau}d\tau \quad \mbox { and } \quad \int \frac{\sigma^2(\tau)}{\tau}d\tau.$$ \end{lemma}
Before the proof of the theorem we need some notation and two lemmas. \begin{lemma} \label{resultados} For $\tau$ sufficiently small we set the following definitions and we have the following relations.
\begin{enumerate}[label=(\alph*)]
\item $\mu_{1,\tau,u}:=\E(X(\tau)|\, X(0)=X(\tau)=u)=\frac{r'(\tau)u}{1+r(\tau)}$.
\item $\mu_{2,\tau,u}:=\E(X(0)|\, X(0)=X(\tau)=u)=-\mu_{1,\tau,u}.$
\item Recall that $\sigma^2(\tau)={\rm Var}(X(0)|\, X(0),\, X(\tau))=\lambda_2-\frac{(r'(\tau))^2}{1-r^2(\tau)}.$ \item $\det({\rm Var}(X(0);X(\tau))=1-r^2(\tau)\asymp \tau^2$
\item if the fourth spectral moment $\lambda_4$ satisfies $ \lambda_2^2 < \lambda_4 \leq + \infty$, then \\ $\frac{|\mu_{1,\tau,u}|}{\sigma(\tau)}\leq (Const) u$ . \end{enumerate} \end{lemma}
\begin{lemma} \label{acotacion}Assume that $|m_1|, |m_2|\le K$ for some constant $K$ and that $(Y_1;Y_2)\stackrel{\mathcal L}= N\big((m_1;m_2),\begin{pmatrix}1&\rho\\\rho&1\end{pmatrix}\big)$. Then
$\E|Y_1Y_2|\asymp 1.$ Where the two constants implied in the symbol $\asymp$ depend on $K$. \end{lemma}
\begin{proof}[Proof of the Theorem] First we have to consider the particular case $\lambda_4 = \lambda_2^2$. This corresponds to the Sine-Cosine process: $X(t) = \xi_1 \sin(wt) + \xi_2 \cos(wt)$ where $ \xi_1,\xi_2$ are independent standard normals. In this case a direct calculation shows that (a)-(c) hold true.
We consider now the other cases assuming that $\lambda_2^2<\lambda_4$. We start from (c): we assume that $$ \int_0^T \frac{\sigma^2(\tau)}{\tau}d\tau <+\infty \quad\mbox{ with }T \mbox{ sufficiently small.} $$ The expectation of the number of crossings is finite because the second spectral moment is, see \cite{C:L}. Thus it is enough to work with the second factorial moment. The Kac-Rice formula for this quantity \cite{C:L} writes \begin{align}
\E &(N_u([0,T])(N_u([0,T])-1))= \notag \\ \
&\frac1\pi \int_0^T(T-\tau)\E[|X'(0)||X'(\tau)|\,|X(0)=X(\tau)=u)\frac{e^{-\frac {u^2}{1+r}}}{\sqrt{1-r^2}} d\tau. \notag \\ \label{KR}
& \leq (Const) \int_0^T \E[| \frac{X'(0)}{\sigma(\tau)}| | \frac{X'(\tau)}{\sigma(\tau)}|\, \Big|X(0)=X(\tau)=u) \frac{ \sigma^2(\tau) }{\tau^2} d\tau, \end{align} using Lemma \ref{resultados} (d). By Lemma \ref{resultados} (e), $\frac{X'(0)}{\sigma(\tau}$ and $\frac{X'(\tau)}{\sigma(\tau)}$ have a bounded conditional mean, then applying now Lemma \ref{acotacion}: \begin{equation} \label{e:jma1} \E (N_u([0,T])(N_u([0,T])-1) \leq (Const) \int_0^T \frac{ \sigma^2(\tau) }{\tau^2} d\tau. \end{equation} This give the finiteness of the second moment form $T$ sufficiently small. By the Minkowsky inequality it is also the case for every $T$ giving (b).
In the other direction we start from (a) with $u=0$ and $T $ sufficiently small (which is weaker than (b)) and we prove (c) .
Again we can consider the second factorial moment and apply the Kac-Rice formula to get that \\ \\
$ \displaystyle
\E (N_u([0,T])(N_u([0,T])-1)$ $$\geq (Const) \int_0^{T/2} \E\Big(| \frac{X'(0)}{\sigma(\tau)}| | \frac{X'(\tau)}{\sigma(\tau)}|\, \Big|X(0)=X(\tau)=u \Big) \frac{ \sigma^2(\tau) }{\tau^2} d\tau. $$ It suffices to apply Lemma \ref{acotacion} in the other direction. \end{proof} \begin{remark} We can also obtain \eqref{e:jma1} with an explicit constant by use of the Cauchy-Schwarz inequality. \end{remark}
\section{Random fields $\R^d \to \R^d$, $d>1$} \label{dd}
\subsection{Position of the problem}
Let us consider a random field $X: \R^d\to\R^d$. We assume $(H_1)$: \begin{itemize} \item The field is Gaussian and stationary and has a continuous derivative. \item The distribution of $X(0)$ (respectively $X'(0)$) is non degenerate (N.D.). \end{itemize} By a rescaling in space we can assume without loss of generality that $$\E[X(t)]=0\quad \mbox{ and } \quad {\rm Var}(X(t))=I_d,$$ where ${\rm Var}$ denotes for us the variance-covariance matrix. We keep the notation ${\rm Cov}$ for the matrix $$ {\rm Cov}(X,Y) := \E\Big( \big(X-\E(X)\big) \big(Y-\E(Y)\big)^\top\Big). $$
We also define the following additional hypothesis
\begin{equation} \label{H2}
\tag{$H_2$}\mbox{The coordinates }X_i \mbox{ of }X \mbox{ are independent and isotrope} \end{equation}
We define \begin{align*}
\sigma^2_{i,\lambda}(r) &:= {\rm Var}\big(X'_{i\lambda}\big |\, X(0), X(\lambda r)\big),\\ \sigma^2_{max} (r) &:= \max_{i =1,\ldots,d}\max_{\lambda\in \mathbb S^{d-1}} \sigma^2_{i,\lambda}(r),
\end{align*}
where $X'_{i\lambda}$ denotes the derivative of $X_i$ in the direction $\lambda \in \mathbb S^{d-1}$.
\subsection{Zero level}\label{zerolevel} We set $N(0,S)$ the number of roots of the field $X(\cdot)$ on some compact set $S$. The following result is new as well as all that follows.
\begin{theorem}\label{t:jma12} Under $(H1)$,
if $$
\int \frac{\sigma^2_{max} (r)}{r} dr \mbox{ converges at } 0,
$$
then for all compact $S \subset \R^d $ : $\E\big((N(0,S))^2\big)$ is finite.
\end{theorem}
The proof of the Theorem uses the following lemma.
\begin{lemma} \label{l:gos}
Let $T,(Z_n)_n$ be in the same Gaussian space. Assume that $Z_n\to Z$ a.s. or in probability or in $\mathbb L^2(\Omega)$ and the random variable $Z$ is (N.D.). Then
\begin{align*}
\forall z,\, &\E(T|\, Z_n=z)\to\E(T|\, Z=z), \\
&{\rm Var}(T|Z_n) \to {\rm Var}(T|Z).
\end{align*}
\end{lemma}
\begin{proof}[Proof of Theorem \ref{t:jma12}]
Set
$$\mathcal C = \{ X(0) = X(t) =0\}, $$ and let $E_\mathcal C $ denotes the expectation conditional to $\mathcal C$.
We consider the following quantity
\begin{equation} \label{atu}
\mathcal A(t,0)=\E_\mathcal C\big(|\det X'(0)\det X'(t)|). \end{equation} By applying the Cauchy-Schwarz inequality and by symmetry of the roles of $0$ and $t$: $$ \mathcal A(t,0) \leq \E_\mathcal C \big( \det((X'(0))^\top X'(0) )\big), $$ We define the Jacobian the matrix $X'(0)$ by $X'_{ij}(0) = \frac{\partial X_i}{ \partial t_j} $.
We perform a change of basis so that $ t = r e_1=|t| e_1$ where $e_1$ is the first vector of the new basis. We denote by $ \bar X$ the expression of $X$ in this basis. Let $ \bar X'_{:j} $ denote the $j$th column of $ \bar X'$. Using Gram representation of the semidefinite positive matrix $M=(M_{ij})$, we know that \begin{equation} \label{zaza}
\det(M) \leq M_{1,1} \ldots M_{d,d}.
\end{equation}
This gives
\begin{equation} \label{som}
\mathcal A(t,0) \leq \E_\mathcal C \big( \| \bar X'_{:1}\|^2 \ldots \| \bar X'_{:d}\|^2\big)
= \sum_{ 1\leq i_1,\ldots,i_d \leq d} \E_\mathcal C( (X'_{i_1,1})^2 \ldots (X'_{i_d,d})^2).
\end{equation}
Because the conditional expectation is contractive, for $j>1$,
\begin{equation} \label{sc1}
\E_\mathcal C \big( (\bar X'_{i_j,j})^2 \big) \leq \E\big( ( \bar X'_{i_j,j})^2 \big) \leq (Const).
\end{equation}
In addition
\begin{equation} \label{sc2}
\E_\mathcal C \big( (\bar X'_{i_1,1})^2 \big) \leq \sigma^2_{max} (r) .
\end{equation}
If we consider a term of \eqref{som}, we can apply Cauchy-Schwarz inequality to get that it is bounded by
$$
\big( E_\mathcal C( (X'_{i_1,1})^4\big)^{1/2} \big( E_\mathcal C( (X'_{i_2,2})^4\ldots (X'_{i_d,d})^4)\big)^{1/2} .
$$
Using \eqref{sc1} and \eqref{sc2}, we get that this term is bounded by
$$
(Const) \sigma^2_{max}(r).
$$
As a consequence we get the same bound for the whole sum.
We now study the joint density $$p_{X(0),X(t)} (0,0) = (Const) \big( \det {\rm Var}(X(0),X(t))\big) ^{-\frac 1 2} .$$
Using the fact that a determinant is invariant by adding to some row (or column) a linear combination of the others rows (or columns) we get
$$
\det ({\rm Var}(X(0),X(t))) = \det ({\rm Var}(X(0),X(t) -X(0))).
$$
Using Lemma \ref{l:gos}.
\begin{equation}\label{density}
p_{X(0),X(t)} (0,0) \simeq (Const) r^{-d} ( \det {\rm Var}(X(0),X_\lambda'(0))\big) ^{-\frac 1 2}
\asymp r^{-d} ,
\end{equation}
where $\lambda:= t/ \|t\|$.
We are now able to apply the Kac-Rice formula see, for example, \cite{AW}, Theorem 6.3. As in the case $d=1$ we can limit our attention to the second factorial moment. We have
\begin{align*}
\E &(N(0,S)(N(0,S)-1) \notag \\ =& \int_{S^2} \mathcal A(t-s,0) P_{X(s),X(t)} (0,0) ds dt
\le (Const) |S| \int_{S} \sigma^2_{max}(t) \|t\|^{-d} dt, \end{align*}
where $|S|$ is the Lebesgue measure of $S$. Passing in polar coordinates, including $S$ in a centered ball with radius $a$, we get that the term above is bounded by
$$ (Const) \int _0^a\ r^{d-1} r^{-d} \sigma^2_{max}(r) d r = (Const) \int _0^a \frac{ \sigma^2_{max}(r)} {r} d r. $$
\end{proof}
\subsection{General level}
In this section we assume $(H_1)$ and $(H_2)$. Note that $ \sigma^2_{i,\lambda} (r)$ depends no more on $\lambda$. We denote by $ \sigma^2_{i} (r)$ its value. We have
$ \sigma^2_{max} (r) = \max_{i =1,\ldots,d}\sigma^2_{i}(r)$.
Our result is the following
\begin{theorem}\label{t:jma} Under the hypotheses above \\
$\bullet$ If $$
\int \frac{ \sigma^2_{max} (r)}{r} dr \mbox{ converges at } 0,
$$
then for all compact $S \subset \R^d $ an all $ u \in \R^d $: $\E\big((N(u,S))^2\big)$ is finite.
$\bullet$
If $\E\big((N(u,S))^2\big)$ is finite for some $u$ and some compact $S$ with non-empty interior, then
$$
\int \frac{ \sigma^2_{max} (r)}{r} dr \mbox{ converges at } 0.
$$ \end{theorem}
Because of stationarity and isotropy we have $$
{\rm Cov}(X_i (s), X_i (t)) = \rho_i(\|s-t\|^2), $$ where $\rho_i$ is some function of class $C^2(\mathbb R)$.
Before the proof of the Theorem we state the following lemmas.
\begin{lemma} \label{Chichi} Let $\mathcal F$ be a family of Gaussian distributions for
$X,Y$ two $d\times d $ Gaussian matrices. Let $Z$ be the $2 d^2$ vector obtained by the elements of $X,Y$ in any order.
$(a)$ Suppose that for all distribution in $\mathcal F$, $\E(Z) \in K_1$ and ${\rm Var} (Z) \in K_2$ where $K_1,K_2$ are two compacts sets .
Then there exists $C$ such that :
$$
\sup_{f \in \mathcal F} \E_f(| \det(X) \det(Y)|) \leq C.$$ The constant $C$ depends only on $K_1$, $K_2$ and $d$.
$(b)$ Suppose in addition that for every $f \in \mathcal F$, $$\P\{ \det(X) =0\}=0, \P\{ \det(Y) =0\}=0$$
then
there exists $c$ such that :
$$
\E (| \det(X) \det(Y)|) \geq c.
$$
The positive constant $c$ depends only on $K_1$, $K_2$ and $d$.
\end{lemma}
For establishing the next lemma let introduce the following definitions
$$\sigma^2_i(r)=-2\rho'_i(0)-\frac{4r^2(\rho'_i(r^2))^2}{1-\rho^2_i(r^2)},$$ $$b_i(r)\sigma_i(r)=\big({-2\rho'_i(r^2)-4r^2\rho_i''(r^2)-\frac{4r^2\rho_i(r^2)(\rho'_i(r^2))^2}{1-\rho^2_i(r^2)}}\big).$$
Then we have the following, denoting ${\rm Var}_\mathcal C$ the variance-covariance matrix conditional to $\mathcal C$.
\begin{lemma}\label{l:cov}[see \cite{AW} p. 336]
$${\small{\rm Var}_\mathcal C \big(X'_i(0); X'_i(r e_1)\big)} ={\tiny\begin{bmatrix}\sigma_i(r)&0&\ldots&0&b_i(r)\sigma_i(r)&0&\ldots&0\\ 0&-2\rho'_i(0)&\ldots&0&0&0&\ldots&0\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ 0&0&\ldots&-2\rho'_i(0)&0&0&\ldots&0\\ b_i(r)\sigma_i(r)&0&\ldots&0&\sigma_i(r)&0&\ldots&0\\ 0&0&\ldots&0&0&-2\rho'_i(0)&\ldots&0\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ 0&0&\ldots&0&0&0&\ldots&-2\rho'_i(0)\end{bmatrix}}.$$ \end{lemma}
\begin{proof}[Proof of the Theorem \ref{t:jma}]
We begin by considering instead of (\ref{atu}) the quantity
\begin{equation} \label{atu}
\mathcal A(t,u)=\E_\mathcal C\big(|\det X'(0)\det X'(t)|), \end{equation} where now $$\mathcal C=\{X(0)=X(t)=u\}.$$
Because of isotropy we can assume, without loss of generality, that $ t = r e_1$. Because of the independence of each coordinates assumed in \eqref{H2} $$
\E_\mathcal C(X'_{i,1}(0)) = \E(X'_{i,1}(0) \big |\, X_i(0) = X_i(r e_1) =u_i). $$
So we have to consider a one dimensional problem as in Section \ref{Geman1}.
In addition the spectral measure of each $X_i $ is invariant by isometry so its projection on the first axis cannot be reduced to one point (or two taking into account symmetry). As a consequence Lemma \ref{resultados} $(e)$ holds true implying that
$$
| \E_\mathcal C(X'_{i,1}(0) | \leq (Const) u_i \sigma_i(r) . $$
Let us consider now $E_\mathcal C(X'_{i,j}) = \E\big(X'_{i,j} \big |\, X_i(0) = u_i, \frac{X_i(t) -X_i(0)} {r} =0)$ for $j \neq 1$. From Lemma \ref{l:gos}
$$E_\mathcal C(X'_{i,j}) \simeq \E\big(X'_{i,j} \big |\, X_i(0) = u_i, X'_{i,1}(0) =0\big). $$ By independence $$
\E_\mathcal C(X'_{i,j}) \simeq \E\big(X'_{i,j} \big |\,X'_{i,1}(0) = 0 ) = 0. $$ Of course we have the same kind of result for $X'(r e_1)$.
So, if we divide the first column of $X'(0)$ and $X'(r e_1)$ by $\sigma_{max}(r)$ to obtain $ \tilde X'(0)$ and $ \tilde X'(r e_1)$, Lemma \ref{l:cov}
implies that
all the terms of the variance-covariance matrix are bounded, the expectation is bounded. Using lemma \ref{Chichi} we get that
\begin{equation}\label{e:majo}
\mathcal A(t,u) \leq (Const) \sigma_{max}^2(r).
\end{equation}
The end of the proof of the first assertion is similar to the one of Theorem \ref{t:jma12}.
We turn now to the proof of the second assertion.
To get the inequality on the other direction we must carefully apply (b) of Lemma \ref{Chichi}. For this purpose we need to describe the compact sets $K_1$ and $K_2$.
We know, from the proof of the first assertion, that all the expectations are bounded so $K_1$ is just $[-a,a]^{2d^2\times2d^2}$ for some $a$. For the domain $K_2$ of the variance-covariance matrix of $\tilde X' (0),\tilde X' (t) $ :
\begin{itemize}
\item First we have an independence between the coordinates $X_i$. Denoting by
$X'_{i,:} (t)$ the $i$ row of $X'$ i.e. the gradient of $X_i$ at $t$, then the variables $ \big(X'_{i,:} (0),X'_{i,:} (t) \big),
i=1,\ldots,d$ are independent.
\item If we consider $ \big(X'_{i,:} (0),X'_{i,:} (t) \big)$ for some fixed $i$, we see from lemma \ref{l:cov} that (i) only
one variance varies : $\sigma_i(r)$, (ii) the only non-zero covariance is between $X'_{i,1} (0)$ and $X'_{i,1} (t)$.
\item After dividing $X'_{i,1} (0)$ and $X'_{i,1} (t)$ by $\sigma_{max}(r)$ to obtain $ \tilde X'_{i,1} (0),\tilde X'_{i,1} (t) $
the variance becomes $$\tilde\sigma_i(r):=\frac{\sigma_i(r)}{\sigma_{\max}(r)}.$$ The domain for these variance, when $i$ varies is $$K'_{2}:=\{\tilde\sigma(r) \in \R^d \,:\,0\le\tilde\sigma_i(r)\le1,\,\mbox{ at least one }\tilde\sigma_i(r)=1\}. $$ That is a compact set.
\item The domain for the covariance between $ \tilde X'_{i,1} (0)$ and $ \tilde X'_{i,1} (t) $
is given by Cauchy-Schwarz inequality : $$ {\rm Cov}\big( \tilde X'_{i,1} (0), \tilde X'_{i,1} (t) \big) \leq \tilde\sigma_i(r),$$ that defines another compact set.
\item The other variables are independent between them and independent of the variables above. Their variance are fixed. \end{itemize}
It remains to prove that for every element of $K_1$ and $K_2$ the Gaussian distribution satisfies $$\mathbb P\{\det (X)=0\}=0\mbox{ and } \mathbb P\{\det (Y)=0\}=0,$$ where $X,Y$ is a representation of the conditional distribution of $\tilde X'(0), \tilde X'(t)$. It is sufficient to study the case of $\det(X)$. Recall that we have proved above that all the coordinates of $X$ are independent. The only difficulty is that the variance of the first column may vanish.
Let us consider the $d\times(d-1)$ matrix $X'_{:,-1}$ that consists of columns $2,\ldots,d$ of $X$. Because all the entries of $X_{:,-1}$ are independent they span a subspace of dimension $d-1$ a.s.
Then the rank of $X'_{:,-1}$ is almost surely $(d-1)$ or $\mbox{Im}(X'_{:,-1})$ is a $(d-1)$ space. The distribution of the random matrix $ Y:=X'_{:,-1}$ has a density that can be written
$$ f_Y[dY]=(Const) e^{-\frac12\mbox{Trace}(Y^\top\Sigma^{-1}Y)}[dY],$$ where as before we have denoted $\Sigma$ the covariance matrix of each column vector, which is diagonal. This density function is translated on the Grassmannian giving a bounded density with respect to the Haar measure.
Recall that we have proved above that all the coordinates of $X$ are independent. Let $X_{:,1}$ be the first column of $X$. Conditioning on $X_{:,-1}$, by independence, the distribution of $X_{:,1}$ remains unchanged. A representation for this random variable is $$X_{:,1}=\mu+\xi,\mbox{ with }\mu=\E(X_{:,1}),$$ where $\xi$ (because some $\tilde \sigma_i$ can vanish) has an absolute continuous distribution on the space $$S_I=(\xi_i=0,\mbox{ for } i\in I),\mbox{ with } I=\{i:\, \tilde\sigma_i=0\}. $$ But since at least one $\tilde \sigma_i=1$, we have $S_I\neq\{1,\ldots,d\}$.
Because of its absolute continuity, almost surely, $ \xi$ can not be included in a given subspace $E$ that does not contain $S_I$. In conclusion, given its absolutely continuous distribution over the Grassmannian, with probability one, $\mbox{Im}(X_{:,-1})$ cannot contain any fixed affine space.
As a consequence we can apply Lemma \ref{Chichi} (b) for getting the inequality in the other direction.
It remains to give a lower bound to the density. $$p_{X(0),X(t)}(u,u)=\prod_{i=1}^d\frac1{2\pi}\frac1{\sqrt{1-\rho^2_i(r^2)}}e^{-\frac{u^2}{1-\rho_i(r^2)}}.$$ Since $\rho_i(r^2)\to1$ as $r\to0$ the term $e^{-\frac{u^2}{1-\rho_i(r^2)}}$ is lower bounded. Then is suffices to use (\ref{density}).
\end{proof}
\subsection{Critical points} \label{criti} Let $ Y : Y(t)$ be a random field from $\R^d \to \R$. Critical points points of $ Y$ are in fact zeros of $X= X'(t)$.
Strictly speaking this process does not satisfies the hypotheses of Theorem \ref{t:jma2} because $X'(t)$ is the Hessian of $Y(t)$ so it is symmetric and its distribution in $\R^{d^2}$ is not N.D. But the result holds true with a very similar proof mutatis mutandis.
\begin{theorem}\label{t:jma2} Suppose that
\begin{itemize}
\item$Y$ is Gaussian stationary, centred and has $ C^2$ sample paths.
\item $ Y'(t)$ is N.D., $Y''(t)$ has a non degenerated distribution in the space of symmetric matrices of dimension $d\times d$.
\end{itemize}
Define
$$
\bar S^2_{max} (r) := \max_{i =1,\ldots,d}\max_{\lambda\in \mathbb S^{d-1}} {\rm Var}\big(Y''_{i\lambda}\big |\, Y'(0), Y''(\lambda r)\big),\\ $$
if $$
\int \frac{ \bar S^2_{max} (r)}{r} dr \mbox{ converges at } 0,
$$
then for all compact $S \subset \R^d $ : the second moment of the number of critical points of $Y$ included in $S$ is finite.
\end{theorem}
\section{Random fields from $\R^D$ to $\R^d$, $d<D$} \label{Dd}
In this section we study the level sets of a random field $\R^D$ to $\R^d$. Of course the case
$d>D$ has no interest because almost surely the level set is empty. The case $d=D$ has been considered in the preceding sections so we assume $d<D$. The result presented here is, in some sense, a by-product of Theorem \ref{t:jma}, but by its simplicity it is the most surprising result and one of the main results of this paper.
\begin{theorem}\label{t:jma3}
Let $X:X(t)$ a stationary random field $\R^D$ to $\R^d$, $d<D$ with $C^1$ paths. By the implicit function theorem, a.s. for every $u$ the level set $C_u$ is a manifold and its $D-d$ dimensional measure $\sigma_{D-d}$ is well defined. Let $C(u,K)$ be the restriction of $C_u$ to a compact set $K \subset \R^D$. Assume that the distributions of $X(t)$ and $X'(t)$ are N.D. \\
Then for every $u$ and $K$
\begin{equation}\label{dD}
\E \big(\sigma^2_{D-d}C(u,K)) < + \infty.
\end{equation}
\end{theorem}
\begin{proof}
The Kac-Rice formula reads \begin{multline*} \E \big(\sigma^2_{D-d}C(u,K)) \\ = \int_{K^2}
\E_\mathcal C\big(\big(\det (X'(s)X'(s)^\top)\det (X'(t)X'(t)^\top)\big)^{\frac12}|\,|\,\big)\\ p_{X(s),X(t)} (u,u) ds dt, \end{multline*}
where again $\E_\mathcal C$ denotes the expectation conditional to $\mathcal C = \{X(0)=X(t)=u\}$. Using the arguments in the proof of Theorem \ref{t:jma}, we have $$
p_{X(0),X(t)} (u,u) \leq (Const) \|t\|^{-d} . $$ By Cauchy-Schwarz inequality and symmetry $$A(t,u):
=\E_\mathcal C\left(\big(\det (X'(0)X'(0)^\top)\det (X'(t)X'(t)^\top)\big)^{\frac12}| \right)$$
$$
\leq \E_\mathcal C\left(\big(\det (X'(0)X'(0)^\top)|\big)\right).
$$
Using \eqref{zaza} we have to bound
$$
\E_\mathcal C \prod_{i=1} ^d \| \nabla X_i(0)\|^2.
$$
Now we borrow results from the proof of Theorem \ref{t:jma}, to get that for every $i$:
\begin{align*}
\E_\mathcal C( X'_{i,1}(0)) & \to 0 \\
\E_\mathcal C( X'_{i,j}(0)) & \mbox{ is bounded } \quad j \neq 1,
\end{align*}
Because of the contracting property of the conditional expectation
$ {\rm Var}_\mathcal C( X'_{i,j}(0))$ is bounded. So, it follows directly that $\mathcal A(t,u)$ is upper-bounded.
The integrability of $t^{-d}$ in $\R^D$ gives the result.
\end{proof}
\section{Appendix}
\begin{proof}[Proof of lemma \ref{integrales}] Let us consider the integral $$ \int_0^\delta\frac{\sigma^2(\tau)}{\tau}d\tau.$$ For $\tau$ small enough $$\frac{\sigma^2(\tau)}{\tau}\sim(\frac1{\lambda_2})^{\frac32}\frac{\lambda_2(1-r^2(\tau))-(r'(\tau))^2}{\tau^3}.$$ Then integrating by parts\\ \\ $\displaystyle \int_0^{\delta}\frac{\lambda_2(1-r^2(\tau))-(r'(\tau))^2}{\tau^3}d\tau$ $$=\frac{\lambda_2(1-r^2(\delta))-(r'(\delta))^2}{2\delta^2}+\int_0^\delta \frac{r'(\tau)}\tau (\frac{-\lambda_2r(\tau)-r''(\tau)}\tau) d\tau.$$Hence we need to consider the second term that is equal to $$-\lambda_2 \int_0^\delta\frac{r'(\tau)}\tau(\frac{r(\tau)-1}{\tau})d\tau-\int_0^\delta\frac{r'(\tau)}\tau(\frac{r''(\tau)+\lambda_2}{\tau})d\tau,$$as the first term is evidently convergent, the above sum is convergent if and only if $$\int_0^{\delta}\frac{r''(\tau)+\lambda_2}{\tau}d\tau<\infty.$$ \end{proof}
\begin{proof}[Proof of lemma \ref{resultados}] For short we will write $r, r', r''$ instead of $r(\tau), r'(\tau), r''(\tau)$. Items $(a)-(d)$ are easy consequences of regression formulas (see \cite{AW} page 100 for example).
To prove (e) we study first the behavior of $\frac{r'}{\sigma(\tau)}$ near to zero. We need consider two cases.
The first one is when the fourth spectral moment $ \lambda_4$ is finite: we have $r(t) = 1-\lambda_2 t^2/2 + \lambda_4 t^4/(4!) + o(t^4)$ . By using a Taylor expansion of fourth order on the numerator and the denominator of the fraction $(\frac{r' u}{(1+r)\sigma(\tau)})^2,$ we obtain $$(\frac{r' u}{(1+r)\sigma(\tau)})^2\to\frac{\lambda_2^2u^2}{\lambda_4-\lambda_2^2}\le (Const)\,u^2, \quad \mbox{ giving {\em (e)}}.$$
Consider now the second case : $\lambda_4=+\infty$. \\ Given that $r''(\tau)-r''(0)=\int_0^\infty\frac{1-\cos(\tau\lambda)}{\tau^2 \lambda^2/2}d\mu(\lambda)$, we have by Fatou's lemma \begin{align} \label{e:jma} \liminf_{\tau\to0}\frac{r''(\tau)-r''(0))}{\tau^2}& \ge\int_0^{+\infty}\liminf_{\tau\to0}\frac{1-\cos(\tau\lambda)}{\tau^2 \lambda^2/2} \lambda^4d\mu(\lambda) \notag \\ &=\int_0^\infty\lambda^4d\mu(\lambda)=+\infty. \end{align} Since $1+r$ tends to $2$, it suffices to study the behaviour of $$\frac{r'^2}{\sigma^2(\tau)}\simeq \frac{\lambda_2^3}{\frac{\lambda_2(1-r^2)-r'^2}{\tau^4}}\, ,$$ Note that $\lambda_2(1-r^2(\tau))-r'^2(\tau)= 2\,\lambda_2(1-r(\tau))-r'^2(\tau)+O(\tau^4)$. Furthermore by using the l'Hospital rule $$\lim_{\tau\to0}\frac{2\,\lambda_2(1-r(\tau))-r'^2(\tau)}{\tau^4}= \lim_{\tau\to0} \left({\frac{-r'(\tau)}{2\tau}}\right)\left({\frac{r"(\tau)-r"(0))}{\tau^2}}\right)=+\infty,$$
because of \eqref{e:jma} and since we know that $\frac{-r'(\tau)}{2\tau}\to \frac{\lambda_2}{2}$. Thus $\frac{r'(\tau)}{\sigma(\tau)}\to 0$. These two results imply that {\em(e)} holds. \end{proof}
\begin{proof}[Proof of the lemma \ref{acotacion}] We can write $$Y_2-m_2=\rho (Y_1-m_1)+\sqrt{1-\rho^2}Z_1, $$ where $Z_1$ is a standard Gaussian independent of $Y_1$. Thus\\ \\ $\displaystyle Y_1Y_2=(m_1m_2+\rho)+(m_2+\rho m_1)(Y_1-m_1)$ $$+m_1\sqrt{1-\rho^2}Z_1+\rho ((Y_1-m_1)^2-1)+\sqrt{1-\rho^2}(Y_1-m_1)Z_1. $$
This formula yields that $\E|Y_1Y_2|$ is a continuous function of $(m_1,m_2,\rho)$ and by compactness is upper bounded.
In the other direction, setting $Y_i= m_i + \bar Y_i$, $i=1,2$ we have $$ Y_1 Y_2 = [m_1 m_2 ]+[ m_1 \bar Y_2 + m_2 \bar Y_1] + [ \bar Y_1\bar Y_2 ], $$ Where the three brackets are in different It\^o-Wiener chaos and thus the joint variance is the sum of the variance of each term. This implies that $$ {\rm Var}(Y_1 Y_2) \geq {\rm Var}( \bar Y_1 \bar Y_2) = \E ( \bar Y_1^2 \bar Y_2^2)=
\E \Big( \big(H_2 (\bar Y_1) +1\big) \big(H_2 (\bar Y_2) +1\big) \Big) = 1 +2 \rho^2, $$ the function $H_2(y) = y^2-1 $ is the second Hermite polynomial and also we have used the Mehler formula: $\E( H_i(\bar Y_1) H_j(\bar Y_2 ) = \delta_{ij} \rho ^i i!$. Note that $H_0(y) = 1 $.
As a consequence
$${\rm Var}(Y_1Y_2)\ge 1.$$ and $Y_1Y_2$ is never a.s. constant. Thus $\E|Y_1Y_2|>0$ and by compactness is lower bounded. \end{proof}
\begin{proof} [Proof of lemma \ref{l:gos}]
Note that for $n$ large enough, the distribution of $ Z_n$ is N.D. As a consequence
\begin{align*}
\E(T|\,Z_n=z)&={\rm Cov}(T,Z_n)({\rm Var}(Z_n))^{-1}z \\
{\rm Var}(T|Z_n) &= {\rm Var}(T)- {\rm Cov}(T,Z_n)({\rm Var}(Z_n))^{-1}{\rm Cov}(Z_n,T)
\end{align*}
but
$${\rm Var}(Z_n)\to{\rm Var}(Z),\mbox{ N.D.}$$This implies
$( {\rm Var}(Z_n) )^{-1}\to({\rm Var}(Z))^{-1}).$ The rest is plain.
\end{proof}
\begin{proof}[Proof of lemma \ref{Chichi}]
Let $\Sigma$ be the variance-covariance matrix of $Z$ and let $\mu $ be its expectation. Both vary in a compact sets $K_1,K_2$. Let $\Sigma^\frac 1 2$ be the square root of $\Sigma$ defined in the spectral way. Using operator norm \cite{Far:Nik}, it is easy to prove that $\Sigma^\frac 1 2$ is a (uniformly) continuous function of $\Sigma$. The random vector $Z$ admits the following représentation $$ Z = \mu + \Sigma^\frac 1 2 \xi \quad \xi \sim N(0,I_{2d^2}). $$
The function $\det(X) \det(Y)$, as a polynomial, is a continuous function of $Z$ and by consequence
$ \E\big(|\det(X) \det(Y)|\big)$ is a continuous function of $ \mu,\Sigma$. The first assertion follows by compactness.
In the other direction we have by additivity
$$\P\{ \det(X)=0\} =0, \quad \P\{ \det(Y)=0\} =0.
$$
This implies that
$$
\E(|\det(X)\det(Y)|) >0.
$$
Again the second inequality is obtained by compactness.
\end{proof}
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309 MYC Inhibition Overcomes IMiD Resistance in Heterogeneous Multiple Myeloma Populations
JCTS 2022 Abstract Collection
Lorraine Davis, Zachary J. Walker, Denis Ohlstrom, Brett M. Stevens, Peter A. Forsberg, Tomer M. Mark, Craig T. Jordan, Daniel W. Sherbenou
Journal: Journal of Clinical and Translational Science / Volume 6 / Issue s1 / April 2022
Published online by Cambridge University Press: 19 April 2022, p. 54
OBJECTIVES/GOALS: Immunomodulatory drugs (IMiDs) are critical to multiple myeloma (MM) disease control. IMiDs act by inducing Cereblon-dependent degradation of IKZF1 and IKZF3, which leads to IRF4 and MYC downregulation (collectively termed the "Ikaros axis"). We therefore hypothesized that IMiD treatment fails to downregulate the Ikaros axis in IMiD resistant MM. METHODS/STUDY POPULATION: To measure IMiD-induced Ikaros axis downregulation, we designed an intracellular flow cytometry assay that measured relative protein levels of IKZF1, IKZF3, IRF4 and MYC in MM cells following ex vivo treatment with the IMiD Pomalidomide (Pom). We established this assay using Pom-sensitive parental and dose-escalated Pom-resistant MM cell lines before assessing Ikaros axis downregulation in CD38+CD138+ MM cells in patient samples (bone marrow aspirates). To assess the Ikaros axis in the context of MM intratumoral heterogeneity, we used a 35-marker mass cytometry panel to simultaneously characterize MM subpopulations in patient samples. Lastly, we determined ex vivo drug sensitivity in patient samples via flow cytometry. RESULTS/ANTICIPATED RESULTS: Our hypothesis was supported in MM cell lines, as resistant lines showed no IMiD-induced decrease in any Ikaros axis proteins. However, when assessed in patient samples, Pom treatment caused a significant decrease in IKZF1, IKZF3 and IRF4 regardless of IMiD sensitivity. Mass cytometry in patient samples revealed that individual Ikaros axis proteins were differentially expressed between subpopulations. When correlating this with ex vivo Pom sensitivity of MM subpopulations, we observed that low IKZF1 and IKZF3 corresponded to Pom resistance. Interestingly, most of these resistant populations still expressed MYC. We therefore assessed whether IMiD resistant MM was MYC dependent by treating with MYCi975. In 88% (7/8) of patient samples tested, IMiD resistant MM cells were sensitive to MYC inhibition. DISCUSSION/SIGNIFICANCE: While our findings did not support our initial hypothesis, our data suggest a mechanism where MYC expression becomes Ikaros axis independent to drive IMiD resistance, and resistant MM is still dependent on MYC. This suggests targeting MYC directly or indirectly via a mechanism to be determined may be an effective strategy to eradicate IMiD resistant MM.
MARINE ORGANIC CARBON AND RADIOCARBON—PRESENT AND FUTURE CHALLENGES
Ellen R M Druffel, Steven R Beaupré, Hendrik Grotheer, Christian B Lewis, Ann P McNichol, Gesine Mollenhauer, Brett D Walker
Journal: Radiocarbon / Volume 64 / Issue 4 / August 2022
Published online by Cambridge University Press: 25 January 2022, pp. 705-721
We discuss present and developing techniques for studying radiocarbon in marine organic carbon (C). Bulk DOC (dissolved organic C) Δ14C measurements reveal information about the cycling time and sources of DOC in the ocean, yet they are time consuming and need to be streamlined. To further elucidate the cycling of DOC, various fractions have been separated from bulk DOC, through solid phase extraction of DOC, and ultrafiltration of high and low molecular weight DOC. Research using 14C of DOC and particulate organic C separated into organic fractions revealed that the acid insoluble fraction is similar in 14C signature to that of the lipid fraction. Plans for utilizing this methodology are described. Studies using compound specific radiocarbon analyses to study the origin of biomarkers in the marine environment are reviewed and plans for the future are outlined. Development of ramped pyrolysis oxidation methods are discussed and scientific questions addressed. A modified elemental analysis (EA) combustion reactor is described that allows high particulate organic C sample throughput by direct coupling with the MIniCArbonDAtingSystem.
Edited by Audrey Walker, Albert Einstein College of Medicine, New York, Steven Schlozman, Jonathan Alpert, Albert Einstein College of Medicine, New York
Book: Introduction to Psychiatry
Print publication: 12 August 2021, pp vii-xvi
Print publication: 12 August 2021, pp iv-iv
Print publication: 12 August 2021, pp v-vi
Print publication: 12 August 2021, pp 497-512
1 - Introduction
By Audrey M. Walker, Steven C. Schlozman, Jonathan E. Alpert
Print publication: 12 August 2021, pp 1-8
The first edition of Introduction to Psychiatry is a textbook designed to reach medical students, house staff, primary care clinicians, and early-career mental health practitioners. It is the editors' hope that this text will enable its readers to understand the neuroscientific basis of psychiatry, best practices in the psychiatric assessment and treatment of the patient, the current understanding of core psychiatric diagnoses, and the important underlying issues of population health, public policy, and workforce recruitment and training that must be tackled to bring these advances to all.
Why create a textbook of psychiatry specifically for clinicians not trained for the mental health field? To answer this question, one must understand the troubling challenges facing the mental health workforce, the changing face of mental health care delivery, the enormous comorbidity between psychiatric illnesses and other health conditions, and the impact on non-psychiatric medical illnesses when a comorbid psychiatric disorder is present.
Introduction to Psychiatry
Preclinical Foundations and Clinical Essentials
Edited by Audrey Walker, Steven Schlozman, Jonathan Alpert
Print publication: 12 August 2021
Buy the print book
The current global crisis in mental health has seen psychiatry assume an increasingly integral role in healthcare. This comprehensive and accessible textbook provides an evidence-based foundation in psychiatry for medical students and serves as an excellent refresher for all mental health professionals. Written by medical school faculty and experts in the field, with comprehensive coverage from neurobiology to population health, this essential textbook is an invaluable guide to the evaluation, treatment and current understanding of the major disorders in psychiatry. The book introduces the basics of clinical assessment and all major modalities of evidence based treatment, along with topics often not covered adequately in textbooks such as gender and sexuality, and global mental health. Chapters are complemented by easy to navigate tables, self-assessment questions, and a short bibliography of recommended reading. An essential resource for medical students, trainees, and other medical professionals seeking a clear and comprehensive introduction to psychiatry.
Making the mundane remarkable: an ethnography of the 'dignity encounter' in community district nursing
Emma Stevens, Elizabeth Price, Elizabeth Walker
Journal: Ageing & Society , First View
Published online by Cambridge University Press: 08 July 2021, pp. 1-23
The concept of dignity is core to community district nursing practice, yet it is profoundly complex with multiple meanings and interpretations. Dignity does not exist absolutely, but, rather, becomes socially (de)constructed through and within social interactions between nurses and older adult patients in relational aspects of care. It is a concept, however, which has, to date, received little attention in the context of the community nursing care of older adults. Previous research into dignity in health care has often focused on care within institutional environments, very little, however, explores the variety of ways in which dignity is operationalised in community settings where district nursing care is conducted 'behind closed doors', largely free from the external gaze. This means dignity (or the lack of it) may go unobserved in community settings. Drawing on observational and interview data, this paper highlights the significance of dignity for older adults receiving nursing care in their own homes. We will demonstrate, in particular, how dignity manifests within the relational aspects of district nursing care delivery and how tasks involving bodywork can be critical to the ways in which dignity is both promoted and undermined. We will further highlight how micro-articulations in caring relationships fundamentally shape the 'dignity encounter' through a consideration of the routine and, arguably, mundane aspects of community district nursing care in the home.
Remnant radio galaxies discovered in a multi-frequency survey
Murchison Widefield Array
GAMA Legacy ATCA Southern Survey
Australian SKA Pathfinder
Benjamin Quici, Natasha Hurley-Walker, Nicholas Seymour, Ross J. Turner, Stanislav S. Shabala, Minh Huynh, H. Andernach, Anna D. Kapińska, Jordan D. Collier, Melanie Johnston-Hollitt, Sarah V. White, Isabella Prandoni, Timothy J. Galvin, Thomas Franzen, C. H. Ishwara-Chandra, Sabine Bellstedt, Steven J. Tingay, Bryan M. Gaensler, Andrew O'Brien, Johnathan Rogers, Kate Chow, Simon Driver, Aaron Robotham
Journal: Publications of the Astronomical Society of Australia / Volume 38 / 2021
Published online by Cambridge University Press: 09 February 2021, e008
The remnant phase of a radio galaxy begins when the jets launched from an active galactic nucleus are switched off. To study the fraction of radio galaxies in a remnant phase, we take advantage of a $8.31$ deg $^2$ subregion of the GAMA 23 field which comprises of surveys covering the frequency range 0.1–9 GHz. We present a sample of 104 radio galaxies compiled from observations conducted by the Murchison Widefield Array (216 MHz), the Australia Square Kilometer Array Pathfinder (887 MHz), and the Australia Telescope Compact Array (5.5 GHz). We adopt an 'absent radio core' criterion to identify 10 radio galaxies showing no evidence for an active nucleus. We classify these as new candidate remnant radio galaxies. Seven of these objects still display compact emitting regions within the lobes at 5.5 GHz; at this frequency the emission is short-lived, implying a recent jet switch off. On the other hand, only three show evidence of aged lobe plasma by the presence of an ultra-steep-spectrum ( $\alpha<-1.2$) and a diffuse, low surface brightness radio morphology. The predominant fraction of young remnants is consistent with a rapid fading during the remnant phase. Within our sample of radio galaxies, our observations constrain the remnant fraction to $4\%\lesssim f_{\mathrm{rem}} \lesssim 10\%$; the lower limit comes from the limiting case in which all remnant candidates with hotspots are simply active radio galaxies with faint, undetected radio cores. Finally, we model the synchrotron spectrum arising from a hotspot to show they can persist for 5–10 Myr at 5.5 GHz after the jets switch of—radio emission arising from such hotspots can therefore be expected in an appreciable fraction of genuine remnants.
2 - On Discourse-Intensive Approaches to Environmental Decision-Making: Applying Social Theory to Practice
from Part I - Methods
By Steven E. Daniels, Gregg B. Walker
Edited by Katharine Legun, Julie C. Keller, University of Rhode Island, Michael Carolan, Colorado State University, Michael M. Bell, University of Wisconsin, Madison
Book: The Cambridge Handbook of Environmental Sociology
Print publication: 03 December 2020, pp 29-46
The rise of multi-party processes in which people with quite different ties to a region, natural resource-related industry, or environmental issue work collaboratively to hammer out mutually acceptable agreements is arguably one of the biggest shifts in environmental management over the past twenty-five years. This chapter engages in some sensemaking around this diverse and evolving phenomenon in two ways. First, an approach to designing collaborative natural resource-related discourse with a particularly strong theoretical foundation (Collaborative Learning) is presented to illustrate how theory is manifest in practice. Second a recent best practices/common features list is examined through the perspectives of four social science theorists: Max Weber, Pierre Bourdieu, Niklas Luhmann, and Muzafer Sherif. The practical recommendations that emerge from this list is largely consistent with the larger social and communicative dynamics articulated by these theorists.
Innovation in Urban Transit at the Start of the Twentieth Century: A Case Study of Metropolitan Street Railway's Stealth Hostile Takeover of Third Avenue Railroad
TIMOTHY A. KRUSE, STEVEN KYLE TODD, MARK D. WALKER
Journal: Enterprise & Society / Volume 23 / Issue 2 / June 2022
Published online by Cambridge University Press: 28 October 2020, pp. 357-407
Print publication: June 2022
In 1900, a syndicate of investors used open market purchases and manipulative trading strategies to exploit an ongoing financial crisis at the Third Avenue Railroad Company and stealthily gain control of the company. The acquisition occurred during the first great merger wave in U.S. history and represented the street railway industry's response to a new technology, namely electrification. The lax regulatory environment of the period allowed operators and insiders to profit handsomely and may have benefited consumers, but possibly harmed some minority shareholders. Our case study illuminates an unusual acquisition, when capital markets were less transparent.
Calibration database for the Murchison Widefield Array All-Sky Virtual Observatory
Marcin Sokolowski, Christopher H. Jordan, Gregory Sleap, Andrew Williams, Randall Bruce Wayth, Mia Walker, David Pallot, Andre Offringa, Natasha Hurley-Walker, Thomas M. O. Franzen, Melanie Johnston-Hollitt, David L. Kaplan, David Kenney, Steven J. Tingay
Published online by Cambridge University Press: 11 June 2020, e021
We present a calibration component for the Murchison Widefield Array All-Sky Virtual Observatory (MWA ASVO) utilising a newly developed PostgreSQL database of calibration solutions. Since its inauguration in 2013, the MWA has recorded over 34 petabytes of data archived at the Pawsey Supercomputing Centre. According to the MWA Data Access policy, data become publicly available 18 months after collection. Therefore, most of the archival data are now available to the public. Access to public data was provided in 2017 via the MWA ASVO interface, which allowed researchers worldwide to download MWA uncalibrated data in standard radio astronomy data formats (CASA measurement sets or UV FITS files). The addition of the MWA ASVO calibration feature opens a new, powerful avenue for researchers without a detailed knowledge of the MWA telescope and data processing to download calibrated visibility data and create images using standard radio astronomy software packages. In order to populate the database with calibration solutions from the last 6 yr we developed fully automated pipelines. A near-real-time pipeline has been used to process new calibration observations as soon as they are collected and upload calibration solutions to the database, which enables monitoring of the interferometric performance of the telescope. Based on this database, we present an analysis of the stability of the MWA calibration solutions over long time intervals.
Effect of cerebral white matter changes on clinical response to cholinesterase inhibitors in dementia
M.E. Devine, J.A. Saez Fonseca, R.W. Walker, T. Sikdar, T. Stevens, Z. Walker
Journal: European Psychiatry / Volume 22 / Issue S1 / March 2007
Published online by Cambridge University Press: 16 April 2020, p. S305
Cerebral white matter changes (WMC) represent cerebrovascular disease (CVD) and are common in dementia. Cholinesterase inhibitors (ChEIs) are effective in Alzheimer's Disease (AD) with or without CVD, and in Dementia with Lewy Bodies/Parkinson's Disease Dementia (DLB/PDD). Predictors of treatment response are controversial.
To investigate the effect of WMC severity on response to ChEIs in dementia.
CT or MRI brain scans were rated for WMC severity in 243 patients taking ChEIs for dementia. Raters were blind to patients' clinical risk factors, dementia subtype and course of illness. Effects of WMC severity on rates of decline in cognition, function and behaviour were analysed for 140 patients treated for nine months or longer. Analysis was performed for this group as a whole and within diagnostic subgroups AD and DLB/PDD. The main outcome measure was rate of change in Mini Mental State Examination (MMSE) score. Secondary measures were rates of change in scores on the Cambridge Cognitive Examination (CAMCOG), Instrumental Activities of Daily Living (IADL) and Clifton Assessment Procedures for the Elderly – Behaviour Rating Scale (CAPE-BRS).
There was no significant correlation between severity of WMC and any specified outcome variable for the cohort as a whole or for patients with AD. In patients with DLB/PDD, higher WMC scores were associated with more rapid cognitive decline.
Increased WMC severity does not predict response to ChEIs in AD, but may weaken response to ChEIs in patients with DLB/PDD.
Outcomes of an electronic medical record (EMR)–driven intensive care unit (ICU)-antimicrobial stewardship (AMS) ward round: Assessing the "Five Moments of Antimicrobial Prescribing"
Misha Devchand, Andrew J. Stewardson, Karen F. Urbancic, Sharmila Khumra, Andrew A. Mahony, Steven Walker, Kent Garrett, M. Lindsay Grayson, Jason A. Trubiano
Journal: Infection Control & Hospital Epidemiology / Volume 40 / Issue 10 / October 2019
Published online by Cambridge University Press: 13 August 2019, pp. 1170-1175
The primary objective of this study was to examine the impact of an electronic medical record (EMR)–driven intensive care unit (ICU) antimicrobial stewardship (AMS) service on clinician compliance with face-to-face AMS recommendations. AMS recommendations were defined by an internally developed "5 Moments of Antimicrobial Prescribing" metric: (1) escalation, (2) de-escalation, (3) discontinuation, (4) switch, and (5) optimization. The secondary objectives included measuring the impact of this service on (1) antibiotic appropriateness, and (2) use of high-priority target antimicrobials.
A prospective review was undertaken of the implementation and compliance with a new ICU-AMS service that utilized EMR data coupled with face-to-face recommendations. Additional patient data were collected when an AMS recommendation was made. The impact of the ICU-AMS round on antimicrobial appropriateness was evaluated using point-prevalence survey data.
For the 202 patients, 412 recommendations were made in accordance with the "5 Moments" metric. The most common recommendation made by the ICU-AMS team was moment 3 (discontinuation), which comprised 173 of 412 recommendations (42.0%), with an acceptance rate of 83.8% (145 of 173). Data collected for point-prevalence surveys showed an increase in prescribing appropriateness from 21 of 45 (46.7%) preintervention (October 2016) to 30 of 39 (76.9%) during the study period (September 2017).
The integration of EMR with an ICU-AMS program allowed us to implement a new AMS service, which was associated with high clinician compliance with recommendations and improved antibiotic appropriateness. Our "5 Moments of Antimicrobial Prescribing" metric provides a framework for measuring AMS recommendation compliance.
UV Photochemical Oxidation and Extraction of Marine Dissolved Organic Carbon at UC Irvine: Status, Surprises, and Methodological Recommendations
Brett D Walker, Steven R Beaupré, Sheila Griffin, Ellen R M Druffel
Journal: Radiocarbon / Volume 61 / Issue 5 / October 2019
Published online by Cambridge University Press: 15 April 2019, pp. 1603-1617
The first ultraviolet photochemical oxidation (UVox) extraction method for marine dissolved organic carbon (DOC) as CO2 gas was established by Armstrong and co-workers in 1966. Subsequent refinement of the UVox technique has co-evolved with the need for high-precision isotopic (Δ14C, δ13C) analysis and smaller sample size requirements for accelerator mass spectrometry radiocarbon (AMS 14C) measurements. The UVox line at UC Irvine was established in 2004 and the system reaction kinetics and efficiency for isolating seawater DOC rigorously tested for quantitative isolation of ∼1 mg C for AMS 14C measurements. Since then, improvements have been made to sampling, storage, and UVox methods to increase overall efficiency. We discuss our progress, and key UVox system parameters for optimizing precision, accuracy, and efficiency, including (1) ocean to reactor: filtration, storage and preparation of DOC samples, (2) cryogenic trap design, efficiency and quantification of CO2 break through, and (3) use of isotopic standards, blanks and small sample graphitization techniques for the correction of DOC concentrations and Fm values with propagated uncertainties. New DOC UVox systems are in use at many institutions. However, rigorous assessment of quantitative UVox DOC yields and blank contributions, DOC concentrations and carbon isotopic values need to be made. We highlight the need for a community-wide inter-comparison study.
The Phase II Murchison Widefield Array: Design overview
Randall B. Wayth, Steven J. Tingay, Cathryn M. Trott, David Emrich, Melanie Johnston-Hollitt, Ben McKinley, B. M. Gaensler, A. P. Beardsley, T. Booler, B. Crosse, T. M. O. Franzen, L. Horsley, D. L. Kaplan, D. Kenney, M. F. Morales, D. Pallot, G. Sleap, K. Steele, M. Walker, A. Williams, C. Wu, Iver. H. Cairns, M. D. Filipovic, S. Johnston, T. Murphy, P. Quinn, L. Staveley-Smith, R. Webster, J. S. B. Wyithe
Published online by Cambridge University Press: 23 November 2018, e033
We describe the motivation and design details of the 'Phase II' upgrade of the Murchison Widefield Array radio telescope. The expansion doubles to 256 the number of antenna tiles deployed in the array. The new antenna tiles enhance the capabilities of the Murchison Widefield Array in several key science areas. Seventy-two of the new tiles are deployed in a regular configuration near the existing array core. These new tiles enhance the surface brightness sensitivity of the array and will improve the ability of the Murchison Widefield Array to estimate the slope of the Epoch of Reionisation power spectrum by a factor of ∼3.5. The remaining 56 tiles are deployed on long baselines, doubling the maximum baseline of the array and improving the array u, v coverage. The improved imaging capabilities will provide an order of magnitude improvement in the noise floor of Murchison Widefield Array continuum images. The upgrade retains all of the features that have underpinned the Murchison Widefield Array's success (large field of view, snapshot image quality, and pointing agility) and boosts the scientific potential with enhanced imaging capabilities and by enabling new calibration strategies.
The Engineering Development Array: A Low Frequency Radio Telescope Utilising SKA Precursor Technology
Randall Wayth, Marcin Sokolowski, Tom Booler, Brian Crosse, David Emrich, Robert Grootjans, Peter J. Hall, Luke Horsley, Budi Juswardy, David Kenney, Kim Steele, Adrian Sutinjo, Steven J. Tingay, Daniel Ung, Mia Walker, Andrew Williams, A. Beardsley, T. M. O. Franzen, M. Johnston-Hollitt, D. L. Kaplan, M. F. Morales, D. Pallot, C. M. Trott, C. Wu
Published online by Cambridge University Press: 17 August 2017, e034
We describe the design and performance of the Engineering Development Array, which is a low-frequency radio telescope comprising 256 dual-polarisation dipole antennas working as a phased array. The Engineering Development Array was conceived of, developed, and deployed in just 18 months via re-use of Square Kilometre Array precursor technology and expertise, specifically from the Murchison Widefield Array radio telescope. Using drift scans and a model for the sky brightness temperature at low frequencies, we have derived the Engineering Development Array's receiver temperature as a function of frequency. The Engineering Development Array is shown to be sky-noise limited over most of the frequency range measured between 60 and 240 MHz. By using the Engineering Development Array in interferometric mode with the Murchison Widefield Array, we used calibrated visibilities to measure the absolute sensitivity of the array. The measured array sensitivity matches very well with a model based on the array layout and measured receiver temperature. The results demonstrate the practicality and feasibility of using Murchison Widefield Array-style precursor technology for Square Kilometre Array-scale stations. The modular architecture of the Engineering Development Array allows upgrades to the array to be rolled out in a staged approach. Future improvements to the Engineering Development Array include replacing the second stage beamformer with a fully digital system, and to transition to using RF-over-fibre for the signal output from first stage beamformers.
Coenobichnus currani (new ichnogenus and ichnospecies): Fossil trackway of a land hermit crab, early Holocene, San Salvador, Bahamas
Sally E. Walker, Steven M. Holland, Lisa Gardiner
Journal: Journal of Paleontology / Volume 77 / Issue 3 / May 2003
Published online by Cambridge University Press: 20 May 2016, pp. 576-582
Land hermit crabs (Coenobitidae) are widespread and abundant in Recent tropical and subtropical coastal environments, yet little is known about their fossil record. A walking trace, attributed to a land hermit crab, is described herein as Coenobichnus currani (new ichnogenus and ichnospecies). This trace fossil occurs in an early Holocene eolianite deposit on the island of San Salvador, Bahamas. The fossil trackway retains the distinctive right and left asymmetry and interior drag trace that are diagnostic of modern land hermit crab walking traces. The overall size, dimensions and shape of the fossil trackway are similar to those produced by the modem land hermit crab, Coenobita clypeatus, which occurs in the tropical western Atlantic region. The trackway was compared to other arthropod traces, but it was found to be distinct among the arthropod traces described from dune or other environments. The new ichnogenus Coenobichnus is proposed to accommodate the asymmetry of the trackway demarcated by left and right tracks. The new ichnospecies Coenobichnus currani is proposed to accommodate the form of the proposed Coenobichnus that has a shell drag trace.
By Mitchell Aboulafia, Frederick Adams, Marilyn McCord Adams, Robert M. Adams, Laird Addis, James W. Allard, David Allison, William P. Alston, Karl Ameriks, C. Anthony Anderson, David Leech Anderson, Lanier Anderson, Roger Ariew, David Armstrong, Denis G. Arnold, E. J. Ashworth, Margaret Atherton, Robin Attfield, Bruce Aune, Edward Wilson Averill, Jody Azzouni, Kent Bach, Andrew Bailey, Lynne Rudder Baker, Thomas R. Baldwin, Jon Barwise, George Bealer, William Bechtel, Lawrence C. Becker, Mark A. Bedau, Ernst Behler, José A. Benardete, Ermanno Bencivenga, Jan Berg, Michael Bergmann, Robert L. Bernasconi, Sven Bernecker, Bernard Berofsky, Rod Bertolet, Charles J. Beyer, Christian Beyer, Joseph Bien, Joseph Bien, Peg Birmingham, Ivan Boh, James Bohman, Daniel Bonevac, Laurence BonJour, William J. Bouwsma, Raymond D. Bradley, Myles Brand, Richard B. Brandt, Michael E. Bratman, Stephen E. Braude, Daniel Breazeale, Angela Breitenbach, Jason Bridges, David O. Brink, Gordon G. Brittan, Justin Broackes, Dan W. Brock, Aaron Bronfman, Jeffrey E. Brower, Bartosz Brozek, Anthony Brueckner, Jeffrey Bub, Lara Buchak, Otavio Bueno, Ann E. Bumpus, Robert W. Burch, John Burgess, Arthur W. Burks, Panayot Butchvarov, Robert E. Butts, Marina Bykova, Patrick Byrne, David Carr, Noël Carroll, Edward S. Casey, Victor Caston, Victor Caston, Albert Casullo, Robert L. Causey, Alan K. L. Chan, Ruth Chang, Deen K. Chatterjee, Andrew Chignell, Roderick M. Chisholm, Kelly J. Clark, E. J. Coffman, Robin Collins, Brian P. Copenhaver, John Corcoran, John Cottingham, Roger Crisp, Frederick J. Crosson, Antonio S. Cua, Phillip D. Cummins, Martin Curd, Adam Cureton, Andrew Cutrofello, Stephen Darwall, Paul Sheldon Davies, Wayne A. Davis, Timothy Joseph Day, Claudio de Almeida, Mario De Caro, Mario De Caro, John Deigh, C. F. Delaney, Daniel C. Dennett, Michael R. DePaul, Michael Detlefsen, Daniel Trent Devereux, Philip E. Devine, John M. Dillon, Martin C. Dillon, Robert DiSalle, Mary Domski, Alan Donagan, Paul Draper, Fred Dretske, Mircea Dumitru, Wilhelm Dupré, Gerald Dworkin, John Earman, Ellery Eells, Catherine Z. Elgin, Berent Enç, Ronald P. Endicott, Edward Erwin, John Etchemendy, C. Stephen Evans, Susan L. Feagin, Solomon Feferman, Richard Feldman, Arthur Fine, Maurice A. Finocchiaro, William FitzPatrick, Richard E. Flathman, Gvozden Flego, Richard Foley, Graeme Forbes, Rainer Forst, Malcolm R. Forster, Daniel Fouke, Patrick Francken, Samuel Freeman, Elizabeth Fricker, Miranda Fricker, Michael Friedman, Michael Fuerstein, Richard A. Fumerton, Alan Gabbey, Pieranna Garavaso, Daniel Garber, Jorge L. A. Garcia, Robert K. Garcia, Don Garrett, Philip Gasper, Gerald Gaus, Berys Gaut, Bernard Gert, Roger F. Gibson, Cody Gilmore, Carl Ginet, Alan H. Goldman, Alvin I. Goldman, Alfonso Gömez-Lobo, Lenn E. Goodman, Robert M. Gordon, Stefan Gosepath, Jorge J. E. Gracia, Daniel W. Graham, George A. Graham, Peter J. Graham, Richard E. Grandy, I. Grattan-Guinness, John Greco, Philip T. Grier, Nicholas Griffin, Nicholas Griffin, David A. Griffiths, Paul J. Griffiths, Stephen R. Grimm, Charles L. Griswold, Charles B. Guignon, Pete A. Y. Gunter, Dimitri Gutas, Gary Gutting, Paul Guyer, Kwame Gyekye, Oscar A. Haac, Raul Hakli, Raul Hakli, Michael Hallett, Edward C. Halper, Jean Hampton, R. James Hankinson, K. R. Hanley, Russell Hardin, Robert M. Harnish, William Harper, David Harrah, Kevin Hart, Ali Hasan, William Hasker, John Haugeland, Roger Hausheer, William Heald, Peter Heath, Richard Heck, John F. Heil, Vincent F. Hendricks, Stephen Hetherington, Francis Heylighen, Kathleen Marie Higgins, Risto Hilpinen, Harold T. Hodes, Joshua Hoffman, Alan Holland, Robert L. Holmes, Richard Holton, Brad W. Hooker, Terence E. Horgan, Tamara Horowitz, Paul Horwich, Vittorio Hösle, Paul Hoβfeld, Daniel Howard-Snyder, Frances Howard-Snyder, Anne Hudson, Deal W. Hudson, Carl A. Huffman, David L. Hull, Patricia Huntington, Thomas Hurka, Paul Hurley, Rosalind Hursthouse, Guillermo Hurtado, Ronald E. Hustwit, Sarah Hutton, Jonathan Jenkins Ichikawa, Harry A. Ide, David Ingram, Philip J. Ivanhoe, Alfred L. Ivry, Frank Jackson, Dale Jacquette, Joseph Jedwab, Richard Jeffrey, David Alan Johnson, Edward Johnson, Mark D. Jordan, Richard Joyce, Hwa Yol Jung, Robert Hillary Kane, Tomis Kapitan, Jacquelyn Ann K. Kegley, James A. Keller, Ralph Kennedy, Sergei Khoruzhii, Jaegwon Kim, Yersu Kim, Nathan L. King, Patricia Kitcher, Peter D. Klein, E. D. Klemke, Virginia Klenk, George L. Kline, Christian Klotz, Simo Knuuttila, Joseph J. Kockelmans, Konstantin Kolenda, Sebastian Tomasz Kołodziejczyk, Isaac Kramnick, Richard Kraut, Fred Kroon, Manfred Kuehn, Steven T. Kuhn, Henry E. Kyburg, John Lachs, Jennifer Lackey, Stephen E. Lahey, Andrea Lavazza, Thomas H. Leahey, Joo Heung Lee, Keith Lehrer, Dorothy Leland, Noah M. Lemos, Ernest LePore, Sarah-Jane Leslie, Isaac Levi, Andrew Levine, Alan E. Lewis, Daniel E. Little, Shu-hsien Liu, Shu-hsien Liu, Alan K. L. Chan, Brian Loar, Lawrence B. Lombard, John Longeway, Dominic McIver Lopes, Michael J. Loux, E. J. Lowe, Steven Luper, Eugene C. Luschei, William G. Lycan, David Lyons, David Macarthur, Danielle Macbeth, Scott MacDonald, Jacob L. Mackey, Louis H. Mackey, Penelope Mackie, Edward H. Madden, Penelope Maddy, G. B. Madison, Bernd Magnus, Pekka Mäkelä, Rudolf A. Makkreel, David Manley, William E. Mann (W.E.M.), Vladimir Marchenkov, Peter Markie, Jean-Pierre Marquis, Ausonio Marras, Mike W. Martin, A. P. Martinich, William L. McBride, David McCabe, Storrs McCall, Hugh J. McCann, Robert N. McCauley, John J. McDermott, Sarah McGrath, Ralph McInerny, Daniel J. McKaughan, Thomas McKay, Michael McKinsey, Brian P. McLaughlin, Ernan McMullin, Anthonie Meijers, Jack W. Meiland, William Jason Melanson, Alfred R. Mele, Joseph R. Mendola, Christopher Menzel, Michael J. Meyer, Christian B. Miller, David W. Miller, Peter Millican, Robert N. Minor, Phillip Mitsis, James A. Montmarquet, Michael S. Moore, Tim Moore, Benjamin Morison, Donald R. Morrison, Stephen J. Morse, Paul K. Moser, Alexander P. D. Mourelatos, Ian Mueller, James Bernard Murphy, Mark C. Murphy, Steven Nadler, Jan Narveson, Alan Nelson, Jerome Neu, Samuel Newlands, Kai Nielsen, Ilkka Niiniluoto, Carlos G. Noreña, Calvin G. Normore, David Fate Norton, Nikolaj Nottelmann, Donald Nute, David S. Oderberg, Steve Odin, Michael O'Rourke, Willard G. Oxtoby, Heinz Paetzold, George S. Pappas, Anthony J. Parel, Lydia Patton, R. P. Peerenboom, Francis Jeffry Pelletier, Adriaan T. Peperzak, Derk Pereboom, Jaroslav Peregrin, Glen Pettigrove, Philip Pettit, Edmund L. Pincoffs, Andrew Pinsent, Robert B. Pippin, Alvin Plantinga, Louis P. Pojman, Richard H. Popkin, John F. Post, Carl J. Posy, William J. Prior, Richard Purtill, Michael Quante, Philip L. Quinn, Philip L. Quinn, Elizabeth S. Radcliffe, Diana Raffman, Gerard Raulet, Stephen L. Read, Andrews Reath, Andrew Reisner, Nicholas Rescher, Henry S. Richardson, Robert C. Richardson, Thomas Ricketts, Wayne D. Riggs, Mark Roberts, Robert C. Roberts, Luke Robinson, Alexander Rosenberg, Gary Rosenkranz, Bernice Glatzer Rosenthal, Adina L. Roskies, William L. Rowe, T. M. Rudavsky, Michael Ruse, Bruce Russell, Lilly-Marlene Russow, Dan Ryder, R. M. Sainsbury, Joseph Salerno, Nathan Salmon, Wesley C. Salmon, Constantine Sandis, David H. Sanford, Marco Santambrogio, David Sapire, Ruth A. Saunders, Geoffrey Sayre-McCord, Charles Sayward, James P. Scanlan, Richard Schacht, Tamar Schapiro, Frederick F. Schmitt, Jerome B. Schneewind, Calvin O. Schrag, Alan D. Schrift, George F. Schumm, Jean-Loup Seban, David N. Sedley, Kenneth Seeskin, Krister Segerberg, Charlene Haddock Seigfried, Dennis M. Senchuk, James F. Sennett, William Lad Sessions, Stewart Shapiro, Tommie Shelby, Donald W. Sherburne, Christopher Shields, Roger A. Shiner, Sydney Shoemaker, Robert K. Shope, Kwong-loi Shun, Wilfried Sieg, A. John Simmons, Robert L. Simon, Marcus G. Singer, Georgette Sinkler, Walter Sinnott-Armstrong, Matti T. Sintonen, Lawrence Sklar, Brian Skyrms, Robert C. Sleigh, Michael Anthony Slote, Hans Sluga, Barry Smith, Michael Smith, Robin Smith, Robert Sokolowski, Robert C. Solomon, Marta Soniewicka, Philip Soper, Ernest Sosa, Nicholas Southwood, Paul Vincent Spade, T. L. S. Sprigge, Eric O. Springsted, George J. Stack, Rebecca Stangl, Jason Stanley, Florian Steinberger, Sören Stenlund, Christopher Stephens, James P. Sterba, Josef Stern, Matthias Steup, M. A. Stewart, Leopold Stubenberg, Edith Dudley Sulla, Frederick Suppe, Jere Paul Surber, David George Sussman, Sigrún Svavarsdóttir, Zeno G. Swijtink, Richard Swinburne, Charles C. Taliaferro, Robert B. Talisse, John Tasioulas, Paul Teller, Larry S. Temkin, Mark Textor, H. S. Thayer, Peter Thielke, Alan Thomas, Amie L. Thomasson, Katherine Thomson-Jones, Joshua C. Thurow, Vzalerie Tiberius, Terrence N. Tice, Paul Tidman, Mark C. Timmons, William Tolhurst, James E. Tomberlin, Rosemarie Tong, Lawrence Torcello, Kelly Trogdon, J. D. Trout, Robert E. Tully, Raimo Tuomela, John Turri, Martin M. Tweedale, Thomas Uebel, Jennifer Uleman, James Van Cleve, Harry van der Linden, Peter van Inwagen, Bryan W. Van Norden, René van Woudenberg, Donald Phillip Verene, Samantha Vice, Thomas Vinci, Donald Wayne Viney, Barbara Von Eckardt, Peter B. M. Vranas, Steven J. Wagner, William J. Wainwright, Paul E. Walker, Robert E. Wall, Craig Walton, Douglas Walton, Eric Watkins, Richard A. Watson, Michael V. Wedin, Rudolph H. Weingartner, Paul Weirich, Paul J. Weithman, Carl Wellman, Howard Wettstein, Samuel C. Wheeler, Stephen A. White, Jennifer Whiting, Edward R. Wierenga, Michael Williams, Fred Wilson, W. Kent Wilson, Kenneth P. Winkler, John F. Wippel, Jan Woleński, Allan B. Wolter, Nicholas P. Wolterstorff, Rega Wood, W. Jay Wood, Paul Woodruff, Alison Wylie, Gideon Yaffe, Takashi Yagisawa, Yutaka Yamamoto, Keith E. Yandell, Xiaomei Yang, Dean Zimmerman, Günter Zoller, Catherine Zuckert, Michael Zuckert, Jack A. Zupko (J.A.Z.)
Edited by Robert Audi, University of Notre Dame, Indiana
Book: The Cambridge Dictionary of Philosophy
Published online: 05 August 2015
Print publication: 27 April 2015, pp ix-xxx | CommonCrawl |
\begin{definition}[Definition:Idempotence/Mapping]
Let $S$ be a set.
Let $f: S \to S$ be a mapping.
Then $f$ is '''idempotent''' {{iff}}:
:$\forall x \in S: \map f {\map f x} = \map f x$
That is, {{iff}} applying the same mapping a second time to an argument gives the same result as applying it once.
And of course, that means the same as applying it as many times as you want.
The condition for '''idempotence''' can also be written:
:$f \circ f = f$
where $\circ$ denotes composition of mappings.
\end{definition} | ProofWiki |
\begin{document}
\title{Simultaneous preparation of two optical cat states based on a nondegenerate optical parametric amplifier}
\author{Dongmei Han$^{1}$, Na Wang$^{1}$, Meihong Wang$^{1,2}$, and Xiaolong Su$^{1,2}$}
\email{[email protected]}
\affiliation{$^{1}$State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan, 030006, China \\ $^{2}$Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi, 030006, China\\ }
\begin{abstract} The optical cat state, known as the superposition of coherent states, has broad applications in quantum computation and quantum metrology. Increasing the number of optical cat states is crucial to implement complex quantum information tasks based on them. Here, we prepare two optical cat states simultaneously based on a nondegenerate optical parametric amplifier. By subtracting one photon from each of two squeezed vacuum states, two odd cat states with orthogonal superposition direction in phase space are prepared simultaneously, which have similar fidelity of 60\% and amplitude of 1.2. Compared with the traditional method to generate two odd optical cat states based on two degenerate optical parametric amplifiers, only one nondegenerate optical parametric amplifier is applied in our experiment, which saves half of the quantum resource of nonlinear cavities. The presented results make a step toward preparing the four-component cat state, which has potential applications in fault-tolerant quantum computation. \end{abstract}
\maketitle
\section{Introduction}
As an important quantum phenomenon, Schr\"{o}dinger cat state~\cite{Schrodinger1935}, which is the superposition of `alive' and `dead' cats, plays an essential role in fundamental physics~\cite{HarocheRMP2013,Arndt2014} and quantum information science~\cite{JeongPRA2002,RalphPRA2003,QEC2022,vanEnkPRA2001,Ulanov2017,Sychev2018,Hacker2019}. It has been experimentally prepared in different systems, such as cavity atomic system~\cite{Leibfried2005,Qin2021}, ion trap~\cite{Monroe1996}, superconducting quantum circuits~\cite{Vlastakis2013,backaction2015,chaosong2019} and optical system~\cite{Ourjoumtsev2006,Neergaard2006,kentaro2007,Takahash2008,Thomas2010,zhang2021,Squeezedcat2022}. The optical cat state, which is defined as the superposition of two coherent states with opposite phases, can be prepared by subtracting photons from a squeezed vacuum state~\cite{Ourjoumtsev2006,Neergaard2006,kentaro2007,Takahash2008,Thomas2010,zhang2021,Squeezedcat2022} and with dissipatively coupled degenerate optical parametric oscillators~\cite{ZYzhou2021,ZYzhou2022}. It is an important quantum resource for quantum communication~\cite{Thomas2010}, quantum computation~\cite{JeongPRA2002,RalphPRA2003,LundPRL2008,Tipsmark2011}, and quantum metrology~\cite{Gilchrist2004,JooPRL2011}.
To realize a quantum algorithm, a series of quantum logic gates need to be implemented, which requires more than one qubit. A large number of physical qubits are the necessary resource in implementing multi-qubit geometric gates~\cite{Chen2022}, realizing nonadiabatic geometric quantum computation~\cite{Kang2022}, and performing quantum error correction algorithms~\cite{Fukui2018}. For example, two odd cat states with orthogonal superposition directions in phase space, $N(| \alpha \rangle - | -\alpha \rangle)$ and $N(| i\alpha \rangle - | -i\alpha \rangle)$, can be encoded as $|0\rangle$ and $|1\rangle$ respectively for the fault-tolerant quantum error correction~\cite{extending2016}, where $N$ is the normalization parameter which is expressed by $1/\sqrt{2(1- e^{-2\left| \alpha \right|^{2}}).}$ Recently, a scheme to generate the optical `four-component cat state', which has potential in fault-tolerant quantum computation, has been proposed \cite{FCCol}. In this scheme, two cat states are coupled on a beam splitter and followed by photon number projection measurement to prepare the four-component cat state. Thus, it is essential to increase the number of cat states and generate cat states with different superposition directions in phase space for scalable and fault-tolerant quantum computation with cat states.
To generate two cat states with different directions in phase space, a direct scheme is to employ two sets of cat state generation systems. In each system, a degenerate optical parametric amplifier (DOPA) is used to produce a squeezed vacuum state~\cite{Sychev2017}. An approximate odd cat state is heralded if a photon is subtracted from the squeezed vacuum state~\cite{Ourjoumtsev2006,Neergaard2006,kentaro2007,Thomas2010,zhang2021,Squeezedcat2022}. However, such a scheme costs two optical cavities. The nondegenerate optical parametric amplifier (NOPA), on the other hand, enables one to prepare two squeezed states simultaneously from one optical cavity instead of two~\cite{su2012,su2020,hao2021}.
Here, we prepare two independent optical cat states simultaneously based on a NOPA, where the two prepared cat states $N(| \alpha \rangle - | -\alpha \rangle)$ and $N(| i\alpha \rangle - | -i\alpha \rangle)$ have similar properties except the superposition direction in the phase space. At first, two squeezed vacuum states with orthogonal squeezed direction in phase space are produced from a NOPA simultaneously in the experiment. Then, one photon is subtracted from each squeezed vacuum state simultaneously. Finally, the Wigner functions of two photon-subtracted states are reconstructed, which confirms that two cat states with fidelities of 61\% and 60\% and amplitudes of 1.19 and 1.21 are prepared simultaneously in our experiment.
\section{The Principle}
\begin{figure}
\caption{The principle of preparation of two optical cat states simultaneously. HWP: half-wave plate, PBS: polarization beam splitter, NOPA: nondegenerate optical parametric amplifier.}
\label{fig:boat1}
\end{figure}
The principle of our experiment is shown in Figure 1. When the NOPA is operated at the parametric de-amplification situation and the half-wave plate after the NOPA is set to 22.5 degrees, an amplitude squeezed state and a phase squeezed state are prepared respectively. Then, photon subtraction on each squeezed state is realized with the detection of a small portion of the squeezed mode with a single photon detector. The clicks of coincidence detection from both single photon detectors herald the generation of two optical cat states simultaneously.
The amplitude quadratures $\hat{X}_{1}=(\hat{a}_{1}+\hat{a}^{\dag}_{1})/\sqrt{2}$, $\hat{X}_{2}=(\hat{a}_{2}+\hat{a}^{\dag}_{2})/\sqrt{2}$ and phase quadratures $\hat{P}_{1}=i(\hat{a}^{\dag}_{1}-\hat{a}_{1})/\sqrt{2}$, $\hat{P}_{2}=i(\hat{a}^{\dag}_{2}-\hat{a}_{2})/\sqrt{2}$ of signal mode $\hat{a}_{1}$ and idler mode $\hat{a}_{2}$ from the output of the NOPA are correlated with each other, where the variance of amplitude sum $V[\hat{X}_{1}+\hat{X}_{2}]=e^{-2r}$ and phase difference $V[\hat{P}_{1}-\hat{P}_{2}]=e^{-2r}$ are lower than vacuum noise if the squeezing parameter $r$ is larger than zero ~\cite{NOPA2002,liuyang2022,haijunpr2022,xiaowei2021}. By coupling those two modes on a beam splitter, the coupled modes are expressed by \begin{equation} \hat{d}_{+}=\frac{\hat{a}_{1}+\hat{a}_{2}}{\sqrt{2}}, \end{equation} \begin{equation} \hat{d}_{-}=\frac{\hat{a}_{1}-\hat{a}_{2}}{\sqrt{2}}. \end{equation}
The amplitude and phase quadratures of the coupled mode $\hat{c}$ are expressed by $\hat{X}_{c}=(\hat{d}_{+}+\hat{d}^{\dag}_{+})/\sqrt{2}=(\hat{X}_{1}+\hat{X}_{2})/\sqrt{2}$ and $\hat{P}_{c}=i(\hat{d}^{\dag}_{+}-\hat{d}_{+})/\sqrt{2}=(\hat{P}_{1}+\hat{P}_{2})/\sqrt{2}$. The amplitude and phase quadratures of the coupled mode $\hat{d}$ are expressed by $\hat{X}_{d}=(\hat{d}_{-}+\hat{d}^{\dag}_{-})/\sqrt{2}=(\hat{X}_{1}-\hat{X}_{2})/\sqrt{2}$ and $\hat{P}_{d}=i(\hat{d}^{\dag}_{-}-\hat{d}_{-})/\sqrt{2}=(\hat{P}_{1}-\hat{P}_{2})/\sqrt{2}$. The variances of the amplitude and phase quadratures of coupled modes $\hat{c}$ and $\hat{d}$ are given by $V[\hat{X}_{c}]=V[\hat{P}_{d}]=e^{-2r}/2$, which are squeezed in amplitude and phase quadratures, respectively ~\cite{meihong2020}. Thus, one can produce two squeezed vacuum states with the orthogonal squeezed direction simultaneously by using a NOPA.
The produced amplitude squeezed state $|\psi \rangle_{c}$ and phase squeezed state $|\psi \rangle_{d}$ from the NOPA in the Fock basis are given by \cite{liangguang} \begin{equation}
|\psi \rangle_{c}=\frac{1}{\sqrt{coshr}}\sum^{\infty }_{m=0}\frac{\sqrt{(2m)!}}{2^{m}m!}(-tanhr)^m|2m\rangle, \end{equation} \begin{equation}
|\psi \rangle_{d}=\frac{1}{\sqrt{coshr}}\sum^{\infty }_{m=0}\frac{\sqrt{(2m)!}}{2^{m}m!}(tanhr)^m|2m\rangle, \end{equation} where $r$ is the squeezing parameter, and $m$ represents the photon number. After subtracting one photon from each squeezed state, the states we obtained are given by \begin{equation}
\hat{a}|\psi \rangle_{c}=coshr^{-\frac{3}{2}}\sum^{\infty }_{m=1}\frac{\sqrt{(2m-1)!}}{2^{m-1}(m-1)!}(-tanhr)^{m-1}|2m-1\rangle, \end{equation} \begin{equation}
\hat{a}|\psi \rangle_{d}=coshr^{-\frac{3}{2}}\sum^{\infty }_{m=1}\frac{\sqrt{(2m-1)!}}{2^{m-1}(m-1)!}(tanhr)^{m-1}|2m-1\rangle. \end{equation}
It has been shown that the state obtained by subtracting a single photon from a squeezed vacuum state approximates an odd cat state when $\alpha \leq 1.2$~\cite{Lund2004}. That is, the states $\hat{a}|\psi \rangle_{c}$ and $\hat{a}|\psi \rangle_{d}$ approximate to the optical cat states $|cat_{x} \rangle$ and $|cat_{p} \rangle$ respectively, which are expressed by
\begin{equation}
|cat_{x} \rangle=\frac{1}{\sqrt{2(1- e^{-2\left| \alpha \right|^{2}})}} (|i\alpha \rangle-|-i\alpha \rangle), \end{equation} \begin{equation}
| cat_{p} \rangle=\frac{1} {\sqrt{2(1- e^{-2\left| \alpha \right|^{2}})}} (|\alpha \rangle-|-\alpha \rangle), \end{equation} respectively.
\section{The Experiment}
\begin{figure}
\caption{Experimental setup. MC: mode cleaner, NOPA: nondegenerate optical parametric amplifier, EOM: electro-optic modulator, AOM: acousto-optic modulator, PBS: polarization beam splitter, HWP: half-wave plate, SNSPD: superconducting nanowire single-photon detector, HD: homodyne detector, LO: local oscillator, IF: interference filter, FC: filter cavity, PD: photodiode, BS: beam splitter.}
\label{fig:boat2}
\end{figure}
As shown in Figure 2, a continuous wave intra-cavity frequency-doubled and frequency-stabilized Nd: YAP/LBO (Nd-doped YAIO3 perovskite lithium triborate) laser generates both the 1080 nm and 540 nm laser beams, which are served as the seed beam and the pump beam of the NOPA. Two mode cleaners (MC) are used to filter the spatial and frequency modes of the seed and pump beams respectively. The NOPA consists of a 10 mm $\alpha $-cut KTP (potassium titanyl phosphate) crystal and a concave mirror with a 50 mm radius. The front face of the KTP crystal is coated for the input coupler and the concave mirror serves as the output coupler. The transmissivities of the front face of the KTP crystal at 540 nm and 1080 nm are 40\% and 0.04\%, respectively. The end face of the KTP crystal is antireflection at both 540 nm and 1080 nm. The transmissivities of the output coupler at 540 nm and 1080 nm are 0.5\% and 12.5\%, respectively. The bandwidth of the NOPA is 50 MHz. The triple resonance condition is achieved by adjusting the frequency of the laser and the temperature of the KTP crystal \cite{liuyang2022,haijunpr2022,xiaowei2021}.
The NOPA is operated in the parametric de-amplification situation, where the relative phase difference of the seed and pump beams is locked to $\pi$. When we set the half-wave plate (HWP) at 22.5 degrees, two squeezed vacuum states are generated from two output ports of the polarization beam splitter (PBS) \cite{liuyang2022,haijunpr2022,xiaowei2021}. We inject around 70 mW pump power into the NOPA and generate squeezed vacuum states with squeezing and anti-squeezing levels of around $-$3.2 dB and $4.2$ dB respectively~\cite{han2022ol,han2022prl}.
To implement photon subtraction, we use a variable beam splitter which consists of an HWP and a PBS to reflect around 4\% of the squeezed mode toward the superconducting nanowire single-photon detectors (SNSPDs). For the reflected mode of the variable beam splitter, an interference filter with a bandwidth of 0.6 nm and a filter cavity (FC) with a bandwidth of 212 MHz are used to select out the degenerate modes of the NOPA, which have around -39 dB rejection ratio for the nondegenerate modes~\cite{han2022ol,han2022prl}. The FC is locked by using the laser beam reflected by the front mirror of the FC. The transmitted photons of the FC is coupled to the SNSPD to realize photon detection.
\begin{figure}
\caption{(a, b) Wigner functions and the corresponding contour plots of $|cat_{p} \rangle$ and $|cat_{x} \rangle$, respectively. (c, d) The real parts of the reconstructed density matrix elements of $|cat_{p} \rangle$ and $|cat_{x} \rangle$, respectively. The reconstructed density matrixes are corrected with 81\% efficiency during the reconstruction process which includes the detection efficiency of 90\% of the HD and the transmission efficiency of 90\% of the signal mode.}
\label{fig:boat3}
\end{figure}
In our experiment, the lock-and-hold technique is applied for NOPA locking~\cite{lockhold2012}, which is performed with the help of two acousto-optic modulators (AOMs). The seed beam of 1080 nm is chopped into a cyclic form with $50$ ms period, which corresponds to each locking and holding period. When AOMs are switched on, the first order of the AOM transmission is injected into the NOPA for cavity locking. Two shutters are closed during the locking period to avoid unwanted trigger events from the SNSPDs. When the AOMs are switched off, the seed beam is chopped off and the NOPA is holding. Two shutters are opened during this period and the tapped-down-converted photons are detected by the SNSPDs. Two squeezed vacuum states are generated and the measurement is performed during the holding period.
The transmitted modes of the variable beam splitters are transmitted through the optical isolators to isolate the backscattered light from the homodyne detectors (HDs). Then, they are detected by two HDs with a bandwidth of around 80 MHz. The quantum noise limit of the HD is 18 dB above the electrical noise when the local oscillator power is around 18 mW, which means that the clearance of the HD is 96\%. The detection efficiency of HD is around 90\% considering the efficiency caused by the clearance of 96\%, the mode matching efficiency of 98\%, and the quantum efficiency of the photodiode of 98\%. The total transmission efficiencies of two signal modes are around 90\% considering the loss caused by the optical isolators and other optical components.
In order to measure the quantum noise of two signal modes, the AC signals from two HDs are filtered by two 60 MHz low-pass filters respectively and recorded simultaneously by a digital storage oscilloscope (LeCroy WaveRunner) which is triggered by the coincident clicks of two single-photon detectors. The DC signals of two HDs, which represent the interference between signal beams and local oscillators, are also recorded for the phase inference~\cite{likelihood2004}. The sample rate of the oscilloscope is 1GS/s and we record 50000 points for each data file. According to the measured quadrature value $X_{\theta}$ and the corresponding phase $\theta$ of two output states, the Wigner functions of the cat states are constructed by using the Maximum likelihood algorithm with the size of the Fock space up to 12, which is enough for the state with average photon numbers of around 1.2 \cite{likelihood2009}. The generation rate of the two cat states is around 20 Hz when the coincidence window of the single photon trigger events is around 100 ns.
\section{Results}
The reconstructed Wigner functions and corresponding contour plots of the prepared two optical cat states are shown in Figures 3a,b, respectively. It is obvious that cat states $|cat_{p} \rangle$ and $|cat_{x} \rangle$ present the superposition of coherent states along the phase and amplitude quadrature directions respectively. The obtained negativities at the origin of the Wigner functions of two cat states are $-$0.12 and $-$0.14 respectively, which show clearly the non-classical feature of the prepared quantum states. The real parts of the density matrix elements of two prepared cat states are shown in Figures 3c,d, respectively. Compared with cat state $|cat_{p} \rangle$, some elements of the density matrix of the cat state $|cat_{x} \rangle$ are negative, which comes from the fact that the phase of coherent state $|i\alpha \rangle$ is different from $|\alpha \rangle$. In the density matrix, the existence of $|0\rangle$ photon number is caused by the system imperfection in our experiment.
To quantify the quality of the prepared cat state, the fidelity is applied to characterize the overlap between the experimentally prepared state $\rho_{exp}$ and the ideal state $\rho_{ideal}$~\cite{fidelity1994}, which is expressed by $F=Tr[\rho_{exp}.\rho_{ideal}]$. In the case of subtracting one photon from a pure squeezed vacuum state, the fidelity of the generated cat state is decreased with the increase of the squeezing parameter of the squeezed vacuum state, while the amplitude is increased with the increase of the squeezing parameter~\cite{Ourjoumtsev2006,Lund2004}. If the squeezed vacuum state is not pure, the fidelity of the generated cat state is decreased with the decrease of the purity of the squeezed vacuum state~\cite{Laghaout2013}. In our experiment, the purity of the squeezed vacuum state is around 0.9, which limits the fidelity of prepared cat states.
Based on the reconstructed density matrix of the prepared state and the density matrix of the ideal cat state $|cat_{ideal} \rangle$ (shown in Equations 7 and 8), we obtain the fidelities of prepared states. The dependence of the fidelities on the amplitudes of two cat states are shown by blue curves in Figures 4a,b respectively. It is obvious that the maximum fidelities of the prepared states are obtained with amplitudes of 1.19 and 1.21, which means that two cat states $|cat_{p} \rangle$ and $|cat_{x} \rangle$ with fidelities of 61\% and 60\% are experimentally prepared simultaneously.
\begin{figure}
\caption{(a, b)Dependence of the fidelity on the amplitude of the prepared cat states $|cat_{p} \rangle$ and $|cat_{x} \rangle$.}
\label{fig:boat4}
\end{figure}
\section{Discussion and Conclusion}
As an application, our prepared cat states can be used as the input states for the preparation of the optical four-component cat state $| \psi_{FCC} \rangle = (| \beta \rangle + (-1)^{k} | -\beta \rangle+ (-i)^{k} | i\beta \rangle +i^{k} | -i\beta \rangle)/N_{k}$, where $N_{k}$ ($k=0,1,2,3$) is a normalization factor and $\beta$ is the coherent state amplitude, which has potential applications in fault-tolerant continuous variable quantum computing~\cite{FCCol}. It has been shown that by coupling two single-photon-subtracted states on a 50:50 beam splitter and projecting one output mode of the beam splitter on the Fock state with photon number larger than 2, the other output mode of the beam splitter can approximate the $| \psi_{FCC} \rangle$. Besides this application, the prepared two cat states can also be applied in quantum error correction with cat codes to correct loss errors \cite{extending2016,error2018} and backaction errors \cite{error2018,backaction2015}.
In summary, we experimentally prepare two optical cat states $|cat_{p} \rangle$ and $|cat_{x} \rangle$ simultaneously by subtracting photons from two squeezed states generated by a NOPA. By reconstructing the Wigner functions of output states, we show that two cat states with fidelities of 61\% and 60\% and the corresponding amplitudes of 1.19 and 1.21 are obtained. Since the advantage of generating two squeezed states from one NOPA is applied in our experiment, we save half of the nonlinear cavity as well as other corresponding instruments compared with the preparation of two cat states based on two DOPAs. Especially, the advantage of our scheme becomes more obvious with the increase of the number of cat states. Besides, we simultaneously prepare two kinds of cat states, i.e. $|cat_{p} \rangle$ and $|cat_{x} \rangle$, with considerable generation rates. Our results demonstrate a new method to prepare cat states with different superposition directions simultaneously and make a step toward fault-tolerant quantum computation with continuous variable error correction code.
\section{ACKNOWLEDGMENTS}
This research was supported by the NSFC (Grants Nos. 11834010 and 62005149), and the Fund for Shanxi \textquotedblleft 1331 Project\textquotedblright\ Key Subjects Construction.
\end{document} | arXiv |
\begin{definition}[Definition:Number Prefixes/Half/Hemi-]
The prefix '''hemi-''' denotes $\dfrac 1 2$.
\end{definition} | ProofWiki |
Common-path dual-wavelength quadrature phase demodulation of EFPI sensors using a broadly tunable MG-Y laser
Qiang Liu, Zhenguo Jing, Ang Li, Yueying Liu, Zhiyuan Huang, Yang Zhang, and Wei Peng
Qiang Liu,1 Zhenguo Jing,1,3 Ang Li,2 Yueying Liu,2 Zhiyuan Huang,2 Yang Zhang,2 and Wei Peng1,4
1School of Physics, Dalian University of Technology, Dalian 116024, China
2School of Optoelectronic Engineering and Instrumentation Science, Dalian University of Technology, Dalian 116024, China
[email protected]
[email protected]
Q Liu
Z Jing
A Li
Y Liu
Z Huang
Y Zhang
W Peng
Qiang Liu, Zhenguo Jing, Ang Li, Yueying Liu, Zhiyuan Huang, Yang Zhang, and Wei Peng, "Common-path dual-wavelength quadrature phase demodulation of EFPI sensors using a broadly tunable MG-Y laser," Opt. Express 27, 27873-27881 (2019)
Three-wavelength passive demodulation technique for the interrogation of EFPI sensors with...
Jingshan Jia, et al.
Opt. Express 27(6) 8890-8899 (2019)
Multiplexing fiber-optic Fabry–Perot acoustic sensors using self-calibrating wavelength...
Qiang Liu, et al.
Symmetrical demodulation method for the phase recovery of extrinsic Fabry–Perot...
Instrumentation, Measurement, and Optical Sensors
Distributed Bragg reflectors
Electrooptical modulators
Optical coherence tomography
Tunable lasers
White light interferometry
Original Manuscript: August 15, 2019
Revised Manuscript: September 6, 2019
Manuscript Accepted: September 9, 2019
Setup and principles
Experimental results and discussion
A common-path dual-wavelength phase demodulation technique for extrinsic Fabry–Perot interferometric (EFPI) sensors is proposed on the basis of a broadly tunable modulated grating Y-branch (MG-Y) laser. It can address the three main concerns of existing dual-wavelength phase interrogation methods: the imbalances and disturbances caused by two optical paths utilizing two lasers or two photodetectors, the restrictions between two operating wavelengths and the cavity length of EFPI, and the difficulty in eliminating the direct current (DC) component of the interferometric fringe. Dual-wavelength phase interrogation is achieved in a common optical path through high-speed wavelength switching. Taking advantage of the MG-Y laser's full spectrum scanning ability (1527 ∼ 1567 nm), initial cavity length and DC component can be directly measured by white light interferometry. Two quadrature wavelengths are then selected to perform high speed phase demodulation scheme. Three polyethylene terephthalate (PET) diaphragm based EFPI acoustic sensors with cavity lengths of 127.954 µm, 148.366 µm and 497.300 µm, are used to demonstrate the effectiveness.
High-performance dynamic sensing of acoustic or vibration signals has been widely used in various applications such as non-destructive testing [1], process control [2] and structural condition monitoring [3,4]. Fiber optic extrinsic Fabry–Perot interferometric (EFPI) sensors have been extensively investigated as a promising high-speed dynamic sensor technology owing to their unique features of compact size, immunity to electromagnetic interference, remote sensing and multiplexing ability [5–7]. The demodulation techniques for extracting target dynamic signals from the output of EFPI sensors is one of the crucial points to be studied. The spectrum based white light interferometry (WLI) demodulation method is generally applied by a broadband light source and an optical spectrum analyzer (OSA). Absolute cavity lengths can be interrogated from the full spectrum by a cross-correlation algorithm or fast Fourier transformation [8–10]. The WLI method usually attains high precision and large dynamic range, but the high system cost and slow measurement speed limit its application. Intensity demodulation methods employ a laser whose wavelength is fixed at the quadrature point (Q-point) to ensure the maximum sensitivity as well as the linear dynamic range. However, the Q-point may experience large, but relatively low-frequency environmental drifts caused by ambient temperature or static background pressure variations, requiring a complex tuning mechanism with feedback control [11,12]. Moreover, the dynamic range is limited owing to the narrow linear range [13]. Phase demodulation methods, such as phase generated carrier (PGC) methods [14] and passive quadrature phase-shifted demodulation methods [15,16] can address these shortcomings of WLI and Q-point intensity demodulation methods. Recently, researchers have shown an increased interest in the dual-wavelength phase interrogation scheme [17–19] for its advantages of high-frequency response, large dynamic range and high stability.
However, most dual-wavelength quadrature phase demodulation methods utilize two lasers or two photodetectors (PD) [16,17,20,21], bringing power imbalances between the two paths. The imbalances maybe induced by the optical path difference variations relating to ambient temperature or pressure fluctuations, and the different responsivity of two PDs. In order to obtain two orthogonal signals, the two wavelengths need to satisfy a quarter of the free space range (FSR), thereby confining the length of EFPI cavity. In recent years, direct current (DC) compensation or estimated algorithms [21–25] have been reported to solve the restrictions between the EFPI cavity length and the two operating wavelengths. EFPIs with different cavity lengths can be applied in the methods utlizing two fixed wavelengths. A novel ellipse fitting algorithm was used. However, it might introduce additional estimation errors or loss of sensitivity at the same time. Xia [22] established a wavelength-switched phase demodulation system to track the phase variations in one optical path. Wavelength switching at a speed of 10 kHz is achieved by electro-optic modulators (EOMs) and a polarizer, increasing the system complexity and instability. Moreover, the polarization switching device limits its frequency response. Therefore, there is still a lack of universal, practical and high-performance demodulation techniques up to now.
Vernier tuned distributed Bragg reflector (VT-DBR) lasers were developed for telecommunications applications. They are one of the most promising wavelength tunable lasers for their characteristics of broad tuning range (>40 nm), high-speed switching (<20 ns) and high side mode suppression ratio (SMSR) (>40 dB) [26,27]. In addition to the telecommunications field, they are also used in optical coherence tomography (OCT) and lidar applications [28–30]. Their high-speed switching, linear scanning and narrow linewidth characteristics are very important in various fiber sensing applications. Among various types of VT-DBR lasers, the sampled grating DBR (SG-DBR) laser and modulated grating Y-branch (MG-Y) laser [31] are widely used. In this paper, we proposed a common-path dual-wavelength quadrature phase demodulation technique utilizing an MG-Y tunable laser. The MG-Y laser is electronically tuned and precisely switched without mechanical movement, resulting in high wavelength stability and repeatability [29]. Thanks to its wide wavelength tuning range (1527 ∼ 1567 nm), initial cavity lengths of EFPI sensors and the DC component can be directly measured through full spectrum scanning. Two wavelengths with accurate quadrature phase difference are then chosen to perform high-speed quadrature phase demodulation of dynamic signals. Wavelength switching frequency is up to 500 kHz under the current configuration. Two orthogonal signals for extracting phase variations are separated and recovered in the time domain. The proposed system is highly flexible and adaptable. It is capable of demodulating the phase variations of polyethylene terephthalate (PET) diaphragm based EFPI acoustic sensors with different cavity lengths in one optical path completely.
2. Setup and principles
Schematic diagram of the EFPI demodulation setup is illustrated in Fig. 1. The light beam emitted from a MG-Y laser is guided into the acoustic sensor probe via a circulator. Then, the backreflected signals are detected by a PD. We use a field programmable gate array (FPGA) for precise wavelength control and simultaneous data acquisition. The collected data is uploaded to a computer for phase demodulation. There are two modulated grating reflectors to extand the tuning range based on vernier effect. Output wavelength is determined by injection currents of three sections: the right reflector section (IRR), the left reflector section (ILR) and the phase section (IPH) [32–34]. Injection current of the semiconductor optical amplifier segment (ISOA) enables fine adjustment of output power. More details about its tuning characteristics are introduced in the experimental section. First of all, full spectrum scanning is carried out to get the interference spectrum of acoustic sensor. The initial cavity length, FSR and the DC component can be measured. Output wavelength is linearly scanned from 1527 nm to 1567 nm with an interval of 8 pm. Two quadrature wavelengths are then chosen to achieve high-speed phase interrogation. The wavelength switched continuously at a rate of 500 kHz. Unlike traditional dual-wavelength passive demodulation methods which utilized two lasers or two PDs, the two orthogonal signals are separated and extracted in the time domain. PET diaphragm based EFPI acoustic sensors with cavity lengths of 127.954 µm, 148.366 µm and 497.300 µm were used to verify the performance of the technique. The thickness and diameter of the PET diaphragm are 6 µm and 3.4 mm, respectively.
Fig. 1. Schematic diagram of the common-path dual-wavelength quadrature phase demodulation system. (Inset) Diagram of the EFPI sensor with a polyethylene terephthalate (PET) diaphragm.
Fiber optic EFPI can be considered as a low-finesse two-beam interferometer. Figure 2 presents the interference spectrum of an EFPI acoustic sensor obtained by wavelength scanning of Mg-Y laser. Cavity length of EFPI was calculated as 148.366 µm by a cross-correlation algorithm [10]. The intensity of reflected interferential light corresponding to λ1 and λ2 can be expressed as
(1)$${I_1} = A + B\cos (\frac{{4n\pi }}{{{\lambda _1}}}L + {\varphi _0})$$
Fig. 2. Interference spectrum of the EFPI acoustic sensor with a cavity length of 148.366 µm.
Where A and B are the DC value and the fringe visibility, respectively. L is the cavity length, φ0 is the initial phase, n is the effective refractive index of the cavity (for air, n = 1). We use L0 to represent the initial cavity length when there is no external vibration signal at the beginning. A, B and L0 can be measured directly from the interference spectrum. The phase difference between the two wavelengths can be defined as β, to get two quadrature signals, let
(3)$$\beta = 4\pi nL(\frac{1}{{{\lambda _1}}} - \frac{1}{{{\lambda _2}}}) \approx 4\pi n{L_0}(\frac{1}{{{\lambda _1}}} - \frac{1}{{{\lambda _2}}}) = \frac{\pi }{2} + k\pi$$
Where k is an integer. To obtain the highest sensitivity and reduce the complexity of wavelength selection [17,35], k = 0, so that β is π/2. The wavelength interval between λ2 and λ1 is
(4)$$\Delta \lambda = {\lambda _2} - {\lambda _1} = \frac{{{\lambda _1}{\lambda _2}}}{{8n{L_0}}} \approx \frac{{{\lambda _1}^2}}{{8n{L_0}}}$$
For convenience, we make λ1 at a fix wavelength, for example, 1541.579 nm. For a given initial cavity length L0, λ2 can be determined according to Eq.u. (4). If L0 = 148.366 µm, λ2 is calculated as 1543.581 nm. Equation (2) can then be expressed as
(5)$${I_2} = A + B\cos (\frac{{4n\pi }}{{{\lambda _1}}}L + {\varphi _0} - \frac{\pi }{2}) = A + B\sin (\frac{{4n\pi }}{{{\lambda _1}}}L + {\varphi _0})$$
(6)$${\varphi _1} = \frac{{4n\pi }}{{{\lambda _1}}}L$$
Equation (1) and (5) can be written as
(7)$${I_1} = A + B\cos ({\varphi _1} + {\varphi _0})$$
(8)$${I_2} = A + B\sin ({\varphi _1} + {\varphi _0})$$
According to Eq. (7) and (8), the phase difference can be calculated by a differential cross multiplication (DCM) method [36] or an arctangent method [18]. In this paper, we utilize an arctangent algorithm to extract the phase signals.
(9)$${\varphi _1} + {\varphi _0} = \arctan (\frac{{{I_2} - A}}{{{I_1} - A}}) + m\pi$$
The φ0 is considered constant, m is a phase compensation integer value used for unwraping the large phase variations. The dynamic phase variations caused by vibration or acoustic signals can be recovered by the calculation of Δφ1. One of the main drawbacks of Q-point intensity demodulation is its limited dynamic range, which appears signal distortion when measuring strong acoustic signals. In this paper, continuous phase variations can be monitored and recovered by phase compensation for high dynamic range measurements [37]. The DC value, A, is measured directly by full spectrum scanning based on the same configuration. Moreover, based on linear wavelength scanning of the MG-Y laser, A and the initial cavity length L0 can be easily re-measured, the two selected output wavelengths can be calibrated to meet the quadrature phase condition when needed, making it suitable for engineering applications in harsh environments.
3. Experimental results and discussion
Linear wavelength sweeping and fast switching is the foundation of common-path quadrature phase demodulation technique. Therefore, precise wavelength output and switching are the prerequisites for this work. The tuning mechanism of MG-Y lasers is much more complicated than that of distributed feedback (DFB) lasers. Output wavelength of the laser is controlled by three injection currents that are applied to IRR, ILR, and IPH [32–34]. In order to determine the initial lookup table of the injected current corresponding to each target wavelength, we have established an automated calibration system based on an optical wavelength meter (Yokogawa, AQ6151) to achieve closed-loop feedback. The calibration system is shown in Fig. 3(a). At the beginning, ISOA was set as a constant value. Tuning paths of three injection currents are shown in Fig. 3(b). With the decreasing of IRR or ILR, the output intensity increased as a result of decreased absorption by injected carriers [28], so that the output intensity will have a saw-tooth like variations. The intensity was then calibrated to a constant value (10.51 dBm) by fine tuning of SOA current ISOA as indicated in Fig. 3(c). After the calibration, the standard deviation (SD) of the output intensity is about 0.028 dBm. It can be considered that the laser output intensity is flat throughout the C-band.
Fig. 3. Automated calibration of MG-Y lasers. (a) Diagram of the laboratory-built automated calibration system. (b) Tuning paths of three injection currents for creating a linear wavelength ramp between 1527 nm and 1567 nm. (c) Output intensity and SOA-injection current ISOA after intensity calibration.
An EFPI acoustic sensor (148.366 µm cavity length) was interrogated by the proposed common-path dual-wavelength phase demodulation method to verify its demodulation performance. WLI interrogation was firstly carried out to get the interference spectrum. In WLI mode, there is no external acoustic disturbance applied to the EFPI probe. The DC component can be measured directly. Then switched to dual-wavelength phase interrogation mode. Two quadrature wavelengths were chosen (λ1 = 1541.579 nm and λ2 = 1543.581 nm). The channel numbers corresponding to the selected wavelengths are sent to the FPGA program by host computer, enabling two wavelengths switching and simultaneous data acquisition. In the current configuration, the wavelength switching frequency is up to 500 kHz. A signal generator and a speaker were used to generate the acoustic signals. Firstly, the speaker was driven by a 15 kHz sinewave. Two orthogonal signals extracted from the time domain are presented in Fig. 4(a). Owing to the deviation from the linear operation range, the extracted waveform corresponding to λ1 has a distortion compared with that for λ2. The vibrational phase signal, Δφ1, was recovered successfully, as presented in Fig. 4(b). Its power spectrum in the frequency domain is displayed in Fig. 4(c). It can be seen that the demodulated phase variation frequency is 15 kHz, which is consistent with the external acoustic signals. The signal to noise ratio (SNR) at 15 kHz is approximately 75 dB. To verify the demodulation stability of the demodulator, the same acoustic signal was measured 100 times. The peak-to-valley amplitude values of demodulated phase variations are plotted in Fig. 4(d). The relative standard deviation (RSD) of the peak-to-valley amplitude values is about 0.49%, which indicates that the common-path quadrature phase demodulation system can work stably.
Fig. 4. Demodulation of an EFPI acoustic sensor with cavity length of 148.366 µm at 15 kHz acoustic signals: (a) Extracted orthogonal signals corresponding to λ1 and λ2. (b) Demodulated phase variations. (c) Power spectrum in the frequency domain. (d) Peak-to-valley amplitude fluctuations of the demodulated phase variations at 15 kHz acoustic signals.
Then the EFPI probe was driven at 1 kHz and 8 kHz acoustic signals respectively. The corresponding time domain phase waveforms and power spectrums are shown in Fig. 5. Indicating that the system can correctly demodulate signals at different applied acoustic frequency. Since the wavelength switching frequency is 500 kHz, dual-wavelength phase demodulation frequency is 250 kHz. In practical applications, the sampling frequency should be 5 times larger than the detected acoustic frequency, so that the maximum detectable acoustic frequency is about 50 kHz in this work.
Fig. 5. Power spectrum of phase variations at 1 kHz and 8 kHz acoustic signals. (Inset) Time domain waveform signals. (a) 1 kHz. (b) 8 kHz.
One of the most concerned shortcomings of conventional dual-wavelength phase demodulation methods is the inability to demodulate another EFPI sensor, usually with a different cavity length. Thanks to the wide tuning range of Mg-Y lasers, this disadvantage can be excluded. For another EFPI sensor, after a simple and fast calibration of full-spectrum scanning, two wavelengths with quadrature phase condition will be redefined. Since we used a WLI method to calculate the initial cavity length of the EFPI sensor, considering the wavelength scanning range of the MG-Y laser, the theoretical minimum cavity length that can be demodulated is about 30 µm. Which demonstrates the advantages of this quadrature phase demodulation system in demodulating short cavity EFPI acoustic sensors. Three EFPI acoustic sensors with different cavity lengths (127.954 µm, 148.366 µm and 497.300 µm) were used to verify the adaptability of our demodulation system. Figure 6 displays their frequency domain power spectrum at 8 kHz acoustic signals. We can see that the acoustic signals applied to all three EFPI sensors can be successfully detected and the three responses are highly consistent.
Fig. 6. Power spectrum of three EFPI acoustic sensors with cavity lengths of 127.954 µm, 148.366 µm and 497.300 µm at 8 kHz acoustic signals.
In fact, the proposed method is an intelligent combination of the traditional WLI demodulation scheme and the two-wavelength quadrature phase demodulation scheme. WLI demodulation is performed to directly measure the DC component of the interferometric fringe and calculate the initial cavity length of EFPI sensors. Dual-wavelength quadrature phase demodulation is performed to achieve high-speed phase interrogation. Taking advantages of the broad wavelength tuning range and high-speed switching capability of MG-Y lasers, these two practical demodulation methods can be implemented in one compact and robust system, addressing the drawbacks of existing WLI and dual-wavelength phase demodulation techniques.
In summary, we have demonstrated a highly flexible common-path dual-wavelength quadrature phase demodulation technique for EFPI sensors. High-speed phase variations can be tracked in a single optical path completely. A widely tunable monolithic MG-Y laser is used to achieve high-speed wavelength switching. Then two orthogonal signals for extracting phase variations are separated in the time domain. Thanks to MG-Y laser's full spectrum scanning capability, initial cavity length of the EFPI sensor and the DC component can be measured by WLI interrogation without additional OSA, which is very promising in practical engineering applications. The proposed technique is capable of demodulating different lengths of EFPI cavity. Two wavelengths with accurate quadrature phase relationship are selected to perform high-speed phase demodulation for each specific EFPI sensor probe. Considering its high applicability and stability in demodulating dynamic signals, further research will focus on its application in photoacoustic spectroscopy and photoacoustic imaging.
National Natural Science Foundation of China (61520106013, 61727816); Dalian University of Technology (DUT18RC016).
1. A. Zaki, H. Chai, D. Aggelis, and N. Alver, "Non-destructive evaluation for corrosion monitoring in concrete: A review and capability of acoustic emission technique," Sensors 15(8), 19069–19101 (2015). [CrossRef]
2. Q. Y. Lu and C. H. Wong, "Additive manufacturing process monitoring and control by non-destructive testing techniques: challenges and in-process monitoring," Virtual Phys. Prototy. 13(2), 39–48 (2018). [CrossRef]
3. C. Hu, Z. Yu, and A. Wang, "An all fiber-optic multi-parameter structure health monitoring system," Opt. Express 24(18), 20287–20296 (2016). [CrossRef]
4. R. Di Sante, "Fibre optic sensors for structural health monitoring of aircraft composite structures: Recent advances and applications," Sensors 15(8), 18666–18713 (2015). [CrossRef]
5. B. Fischer, "Optical microphone hears ultrasound," Nat. Photonics 10(6), 356–358 (2016). [CrossRef]
6. Y.-J. Rao, "Recent progress in fiber-optic extrinsic Fabry–Perot interferometric sensors," Opt. Fiber Technol. 12(3), 227–237 (2006). [CrossRef]
7. J. Yin, T. Liu, J. Jiang, K. Liu, S. Wang, F. Wu, and Z. Ding, "Wavelength-division-multiplexing method of polarized low-coherence interferometry for fiber Fabry–Perot interferometric sensors," Opt. Lett. 38(19), 3751–3753 (2013). [CrossRef]
8. Z. Wang, Y. Jiang, W. Ding, and R. Gao, "A cross-correlation based fiber optic white-light interferometry with wavelet transform denoising," in Fourth Asia Pacific Optical Sensors Conference (International Society for Optics and Photonics2013), p. 89241J.
9. X. Zhou and Q. Yu, "Wide-range displacement sensor based on fiber-optic Fabry–Perot interferometer for subnanometer measurement," IEEE Sens. J. 11(7), 1602–1606 (2011). [CrossRef]
10. J. Zhenguo and Y. Qingxu, "White light optical fiber EFPI sensor based on cross-correlation signal processing method," in Proc. 6th Int. Symp. Test and Measurement (2005), 3509.
11. X. Mao, X. Zhou, and Q. Yu, "Stabilizing operation point technique based on the tunable distributed feedback laser for interferometric sensors," Opt. Commun. 361, 17–20 (2016). [CrossRef]
12. Q. Zhang, Y. Zhu, X. Luo, G. Liu, and M. Han, "Acoustic emission sensor system using a chirped fiber-Bragg-grating Fabry–Perot interferometer and smart feedback control," Opt. Lett. 42(3), 631–634 (2017). [CrossRef]
13. J. Ma, M. Zhao, X. Huang, H. Bae, Y. Chen, and M. Yu, "Low cost, high performance white-light fiber-optic hydrophone system with a trackable working point," Opt. Express 24(17), 19008–19019 (2016). [CrossRef]
14. X. Mao, X. Tian, X. Zhou, and Q. Yu, "Characteristics of a fiber-optical Fabry–Perot interferometric acoustic sensor based on an improved phase-generated carrier-demodulation mechanism," Opt. Eng. 54(4), 046107 (2015). [CrossRef]
15. K. A. Murphy, M. F. Gunther, A. M. Vengsarkar, and R. O. Claus, "Quadrature phase-shifted, extrinsic Fabry–Perot optical fiber sensors," Opt. Lett. 16(4), 273–275 (1991). [CrossRef]
16. E. Lu, Z. Ran, F. Peng, Z. Liu, and F. Xu, "Demodulation of micro fiber-optic Fabry–Perot interferometer using subcarrier and dual-wavelength method," Opt. Commun. 285(6), 1087–1090 (2012). [CrossRef]
17. H. Liao, P. Lu, L. Liu, S. Wang, W. Ni, X. Fu, D. Liu, and J. Zhang, "Phase Demodulation of Short-Cavity Fabry–Perot Interferometric Acoustic Sensors With Two Wavelengths," IEEE Photonics J. 9(2), 1–9 (2017). [CrossRef]
18. A. Pinto, O. Frazão, J. Santos, M. Lopez-Amo, J. Kobelke, and K. Schuster, "Interrogation of a suspended-core Fabry–Perot temperature sensor through a dual wavelength Raman fiber laser," J. Lightwave Technol. 28(21), 3149–3155 (2010). [CrossRef]
19. J. Cheng, D.-f. Lu, R. Gao, and Z.-m. Qi, "Fiber optic microphone with large dynamic range based on bi-fiber Fabry-Perot cavity," in AOPC 2017: Fiber Optic Sensing and Optical Communications (International Society for Optics and Photonics2017), 104642C.
20. J. Jiang, T. Zhang, S. Wang, K. Liu, C. Li, Z. Zhao, and T. Liu, "Noncontact Ultrasonic Detection in Low-Pressure Carbon Dioxide Medium Using High Sensitivity Fiber-Optic Fabry–Perot Sensor System," J. Lightwave Technol. 35(23), 5079–5085 (2017). [CrossRef]
21. J. Jia, Y. Jiang, L. Zhang, H. Gao, S. Wang, and L. Jiang, "Dual-wavelength DC compensation technique for the demodulation of EFPI fiber sensors," IEEE Photonics Technol. Lett. 30(15), 1380–1383 (2018). [CrossRef]
22. J. Xia, S. Xiong, F. Wang, and H. Luo, "Wavelength-switched phase interrogator for extrinsic Fabry–Perot interferometric sensors," Opt. Lett. 41(13), 3082–3085 (2016). [CrossRef]
23. H. Liao, P. Lu, D. Liu, L. Liu, and J. Zhang, "Demodulation of diaphragm based fiber-optic acoustic sensor using with symmetric 3× 3 coupler," in 2017 Opto-Electronics and Communications Conference (OECC) and Photonics Global Conference (PGC) (IEEE, 2017), 1–3.
24. H. Liao, P. Lu, L. Liu, D. Liu, and J. Zhang, "Phase demodulation of Fabry-Perot interferometer-based acoustic sensor utilizing tunable filter with two quadrature wavelengths," in Photonic Instrumentation Engineering IV(International Society for Optics and Photonics), 101101J (2017).
25. J. Jia, Y. Jiang, H. Gao, L. Zhang, and Y. Jiang, "Three-wavelength passive demodulation technique for the interrogation of EFPI sensors with arbitrary cavity length," Opt. Express 27(6), 8890–8899 (2019). [CrossRef]
26. M. P. Minneman, J. Ensher, M. Crawforda, and D. Derickson, "All-semiconductor high-speed akinetic swept-source for OCT," in Asia Communications and Photonics Conference and Exhibition (Optical Society of America, 2011), 831116.
27. J. Zhao, H. Zhou, F. Liu, and Y. Yu, "Numerical analysis of phase noise characteristics of SGDBR lasers," IEEE J. Sel. Top. Quantum Electron. 21(6), 223–231 (2015). [CrossRef]
28. D.-h. Choi, R. Yoshimura, and K. Ohbayashi, "Tuning of successively scanned two monolithic Vernier-tuned lasers and selective data sampling in optical comb swept source optical coherence tomography," Biomed. Opt. Express 4(12), 2962–2987 (2013). [CrossRef]
29. J. Park, E. F. Carbajal, X. Chen, J. S. Oghalai, and B. E. Applegate, "Phase-sensitive optical coherence tomography using an Vernier-tuned distributed Bragg reflector swept laser in the mouse middle ear," Opt. Lett. 39(21), 6233–6236 (2014). [CrossRef]
30. D. MacDougall, J. Farrell, J. Brown, M. Bance, and R. Adamson, "Long-range, wide-field swept-source optical coherence tomography with GPU accelerated digital lock-in Doppler vibrography for real-time, in vivo middle ear diagnostics," Biomed. Opt. Express 7(11), 4621–4635 (2016). [CrossRef]
31. J.-O. Wesström, G. Sarlet, S. Hammerfeldt, L. Lundqvist, P. Szabo, and P.-J. Rigole, "State-of-the-art performance of widely tunable modulated grating Y-branch lasers," in Optical Fiber Communication Conference (Optical Society of America), TuE2 (2004).
32. S. O'Connor, M. A. Bernacil, and D. Derickson, "Generation of high speed, linear wavelength sweeps using sampled grating distributed Bragg reflector lasers," in LEOS 2008-21st Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE 2008), 147–148.
33. B. George and D. Derickson, "High-speed concatenation of frequency ramps using sampled grating distributed Bragg reflector laser diode sources for OCT resolution enhancement," in Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XIV (International Society for Optics and Photonics), 75542O (2010).
34. J. Poliak, H. Heininger, F. Mohr, and O. Wilfert, "Modelling of MG-Y laser tuning characteristics," in Infrared Sensors, Devices, and Applications III (International Society for Optics and Photonics 2013), p. 88680N.
35. Y. Wei and Z. Zhai, "Error analysis of dual wavelength quadrature phase demodulation for low-finesse Fabry–Pérot cavity based fibre optic sensor," Optik 122(14), 1309–1311 (2011). [CrossRef]
36. K. P. Koo, A. B. Tveten, and A. Dandridge, "Passive stabilization scheme for fiber interferometers using (3×3) fiber directional couplers," Appl. Phys. Lett. 41(7), 616–618 (1982). [CrossRef]
37. T. Chang, J. Lang, W. Sun, J. Chen, M. Yu, W. Gao, and H. Cui, "Phase Compensation Scheme for Fiber-Optic Interferometric Vibration Demodulation," IEEE Sens. J. 17(22), 7448–7454 (2017). [CrossRef]
A. Zaki, H. Chai, D. Aggelis, and N. Alver, "Non-destructive evaluation for corrosion monitoring in concrete: A review and capability of acoustic emission technique," Sensors 15(8), 19069–19101 (2015).
Q. Y. Lu and C. H. Wong, "Additive manufacturing process monitoring and control by non-destructive testing techniques: challenges and in-process monitoring," Virtual Phys. Prototy. 13(2), 39–48 (2018).
C. Hu, Z. Yu, and A. Wang, "An all fiber-optic multi-parameter structure health monitoring system," Opt. Express 24(18), 20287–20296 (2016).
R. Di Sante, "Fibre optic sensors for structural health monitoring of aircraft composite structures: Recent advances and applications," Sensors 15(8), 18666–18713 (2015).
B. Fischer, "Optical microphone hears ultrasound," Nat. Photonics 10(6), 356–358 (2016).
Y.-J. Rao, "Recent progress in fiber-optic extrinsic Fabry–Perot interferometric sensors," Opt. Fiber Technol. 12(3), 227–237 (2006).
J. Yin, T. Liu, J. Jiang, K. Liu, S. Wang, F. Wu, and Z. Ding, "Wavelength-division-multiplexing method of polarized low-coherence interferometry for fiber Fabry–Perot interferometric sensors," Opt. Lett. 38(19), 3751–3753 (2013).
Z. Wang, Y. Jiang, W. Ding, and R. Gao, "A cross-correlation based fiber optic white-light interferometry with wavelet transform denoising," in Fourth Asia Pacific Optical Sensors Conference (International Society for Optics and Photonics2013), p. 89241J.
X. Zhou and Q. Yu, "Wide-range displacement sensor based on fiber-optic Fabry–Perot interferometer for subnanometer measurement," IEEE Sens. J. 11(7), 1602–1606 (2011).
J. Zhenguo and Y. Qingxu, "White light optical fiber EFPI sensor based on cross-correlation signal processing method," in Proc. 6th Int. Symp. Test and Measurement (2005), 3509.
X. Mao, X. Zhou, and Q. Yu, "Stabilizing operation point technique based on the tunable distributed feedback laser for interferometric sensors," Opt. Commun. 361, 17–20 (2016).
Q. Zhang, Y. Zhu, X. Luo, G. Liu, and M. Han, "Acoustic emission sensor system using a chirped fiber-Bragg-grating Fabry–Perot interferometer and smart feedback control," Opt. Lett. 42(3), 631–634 (2017).
J. Ma, M. Zhao, X. Huang, H. Bae, Y. Chen, and M. Yu, "Low cost, high performance white-light fiber-optic hydrophone system with a trackable working point," Opt. Express 24(17), 19008–19019 (2016).
X. Mao, X. Tian, X. Zhou, and Q. Yu, "Characteristics of a fiber-optical Fabry–Perot interferometric acoustic sensor based on an improved phase-generated carrier-demodulation mechanism," Opt. Eng. 54(4), 046107 (2015).
K. A. Murphy, M. F. Gunther, A. M. Vengsarkar, and R. O. Claus, "Quadrature phase-shifted, extrinsic Fabry–Perot optical fiber sensors," Opt. Lett. 16(4), 273–275 (1991).
E. Lu, Z. Ran, F. Peng, Z. Liu, and F. Xu, "Demodulation of micro fiber-optic Fabry–Perot interferometer using subcarrier and dual-wavelength method," Opt. Commun. 285(6), 1087–1090 (2012).
H. Liao, P. Lu, L. Liu, S. Wang, W. Ni, X. Fu, D. Liu, and J. Zhang, "Phase Demodulation of Short-Cavity Fabry–Perot Interferometric Acoustic Sensors With Two Wavelengths," IEEE Photonics J. 9(2), 1–9 (2017).
A. Pinto, O. Frazão, J. Santos, M. Lopez-Amo, J. Kobelke, and K. Schuster, "Interrogation of a suspended-core Fabry–Perot temperature sensor through a dual wavelength Raman fiber laser," J. Lightwave Technol. 28(21), 3149–3155 (2010).
J. Cheng, D.-f. Lu, R. Gao, and Z.-m. Qi, "Fiber optic microphone with large dynamic range based on bi-fiber Fabry-Perot cavity," in AOPC 2017: Fiber Optic Sensing and Optical Communications (International Society for Optics and Photonics2017), 104642C.
J. Jiang, T. Zhang, S. Wang, K. Liu, C. Li, Z. Zhao, and T. Liu, "Noncontact Ultrasonic Detection in Low-Pressure Carbon Dioxide Medium Using High Sensitivity Fiber-Optic Fabry–Perot Sensor System," J. Lightwave Technol. 35(23), 5079–5085 (2017).
J. Jia, Y. Jiang, L. Zhang, H. Gao, S. Wang, and L. Jiang, "Dual-wavelength DC compensation technique for the demodulation of EFPI fiber sensors," IEEE Photonics Technol. Lett. 30(15), 1380–1383 (2018).
J. Xia, S. Xiong, F. Wang, and H. Luo, "Wavelength-switched phase interrogator for extrinsic Fabry–Perot interferometric sensors," Opt. Lett. 41(13), 3082–3085 (2016).
H. Liao, P. Lu, D. Liu, L. Liu, and J. Zhang, "Demodulation of diaphragm based fiber-optic acoustic sensor using with symmetric 3× 3 coupler," in 2017 Opto-Electronics and Communications Conference (OECC) and Photonics Global Conference (PGC) (IEEE, 2017), 1–3.
H. Liao, P. Lu, L. Liu, D. Liu, and J. Zhang, "Phase demodulation of Fabry-Perot interferometer-based acoustic sensor utilizing tunable filter with two quadrature wavelengths," in Photonic Instrumentation Engineering IV(International Society for Optics and Photonics), 101101J (2017).
J. Jia, Y. Jiang, H. Gao, L. Zhang, and Y. Jiang, "Three-wavelength passive demodulation technique for the interrogation of EFPI sensors with arbitrary cavity length," Opt. Express 27(6), 8890–8899 (2019).
M. P. Minneman, J. Ensher, M. Crawforda, and D. Derickson, "All-semiconductor high-speed akinetic swept-source for OCT," in Asia Communications and Photonics Conference and Exhibition (Optical Society of America, 2011), 831116.
J. Zhao, H. Zhou, F. Liu, and Y. Yu, "Numerical analysis of phase noise characteristics of SGDBR lasers," IEEE J. Sel. Top. Quantum Electron. 21(6), 223–231 (2015).
D.-h. Choi, R. Yoshimura, and K. Ohbayashi, "Tuning of successively scanned two monolithic Vernier-tuned lasers and selective data sampling in optical comb swept source optical coherence tomography," Biomed. Opt. Express 4(12), 2962–2987 (2013).
J. Park, E. F. Carbajal, X. Chen, J. S. Oghalai, and B. E. Applegate, "Phase-sensitive optical coherence tomography using an Vernier-tuned distributed Bragg reflector swept laser in the mouse middle ear," Opt. Lett. 39(21), 6233–6236 (2014).
D. MacDougall, J. Farrell, J. Brown, M. Bance, and R. Adamson, "Long-range, wide-field swept-source optical coherence tomography with GPU accelerated digital lock-in Doppler vibrography for real-time, in vivo middle ear diagnostics," Biomed. Opt. Express 7(11), 4621–4635 (2016).
J.-O. Wesström, G. Sarlet, S. Hammerfeldt, L. Lundqvist, P. Szabo, and P.-J. Rigole, "State-of-the-art performance of widely tunable modulated grating Y-branch lasers," in Optical Fiber Communication Conference (Optical Society of America), TuE2 (2004).
S. O'Connor, M. A. Bernacil, and D. Derickson, "Generation of high speed, linear wavelength sweeps using sampled grating distributed Bragg reflector lasers," in LEOS 2008-21st Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE 2008), 147–148.
B. George and D. Derickson, "High-speed concatenation of frequency ramps using sampled grating distributed Bragg reflector laser diode sources for OCT resolution enhancement," in Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XIV (International Society for Optics and Photonics), 75542O (2010).
J. Poliak, H. Heininger, F. Mohr, and O. Wilfert, "Modelling of MG-Y laser tuning characteristics," in Infrared Sensors, Devices, and Applications III (International Society for Optics and Photonics 2013), p. 88680N.
Y. Wei and Z. Zhai, "Error analysis of dual wavelength quadrature phase demodulation for low-finesse Fabry–Pérot cavity based fibre optic sensor," Optik 122(14), 1309–1311 (2011).
K. P. Koo, A. B. Tveten, and A. Dandridge, "Passive stabilization scheme for fiber interferometers using (3×3) fiber directional couplers," Appl. Phys. Lett. 41(7), 616–618 (1982).
T. Chang, J. Lang, W. Sun, J. Chen, M. Yu, W. Gao, and H. Cui, "Phase Compensation Scheme for Fiber-Optic Interferometric Vibration Demodulation," IEEE Sens. J. 17(22), 7448–7454 (2017).
Adamson, R.
Aggelis, D.
Alver, N.
Applegate, B. E.
Bae, H.
Bance, M.
Bernacil, M. A.
Brown, J.
Carbajal, E. F.
Chai, H.
Chang, T.
Chen, J.
Chen, X.
Chen, Y.
Cheng, J.
Choi, D.-h.
Claus, R. O.
Crawforda, M.
Cui, H.
Dandridge, A.
Derickson, D.
Di Sante, R.
Ding, W.
Ding, Z.
Ensher, J.
Farrell, J.
Fischer, B.
Frazão, O.
Fu, X.
Gao, H.
Gao, R.
Gao, W.
George, B.
Gunther, M. F.
Hammerfeldt, S.
Han, M.
Heininger, H.
Hu, C.
Huang, X.
Jia, J.
Jiang, J.
Jiang, L.
Jiang, Y.
Kobelke, J.
Koo, K. P.
Lang, J.
Li, C.
Liao, H.
Liu, D.
Liu, F.
Liu, G.
Liu, K.
Liu, L.
Liu, T.
Liu, Z.
Lopez-Amo, M.
Lu, D.-f.
Lu, E.
Lu, P.
Lu, Q. Y.
Lundqvist, L.
Luo, X.
Ma, J.
MacDougall, D.
Mao, X.
Minneman, M. P.
Mohr, F.
Murphy, K. A.
Ni, W.
O'Connor, S.
Oghalai, J. S.
Ohbayashi, K.
Park, J.
Peng, F.
Pinto, A.
Poliak, J.
Qi, Z.-m.
Qingxu, Y.
Ran, Z.
Rao, Y.-J.
Rigole, P.-J.
Santos, J.
Sarlet, G.
Schuster, K.
Sun, W.
Szabo, P.
Tian, X.
Tveten, A. B.
Vengsarkar, A. M.
Wang, A.
Wang, F.
Wang, Z.
Wei, Y.
Wesström, J.-O.
Wilfert, O.
Wong, C. H.
Wu, F.
Xia, J.
Xiong, S.
Xu, F.
Yin, J.
Yoshimura, R.
Yu, M.
Yu, Q.
Yu, Y.
Yu, Z.
Zaki, A.
Zhai, Z.
Zhang, Q.
Zhang, T.
Zhao, J.
Zhao, M.
Zhao, Z.
Zhenguo, J.
Zhou, H.
Zhou, X.
Zhu, Y.
Biomed. Opt. Express (2)
IEEE J. Sel. Top. Quantum Electron. (1)
IEEE Photonics J. (1)
IEEE Photonics Technol. Lett. (1)
IEEE Sens. J. (2)
J. Lightwave Technol. (2)
Opt. Eng. (1)
Opt. Fiber Technol. (1)
Optik (1)
Virtual Phys. Prototy. (1)
(1) I 1 = A + B cos ( 4 n π λ 1 L + φ 0 )
(3) β = 4 π n L ( 1 λ 1 − 1 λ 2 ) ≈ 4 π n L 0 ( 1 λ 1 − 1 λ 2 ) = π 2 + k π
(4) Δ λ = λ 2 − λ 1 = λ 1 λ 2 8 n L 0 ≈ λ 1 2 8 n L 0
(5) I 2 = A + B cos ( 4 n π λ 1 L + φ 0 − π 2 ) = A + B sin ( 4 n π λ 1 L + φ 0 )
(6) φ 1 = 4 n π λ 1 L
(7) I 1 = A + B cos ( φ 1 + φ 0 )
(8) I 2 = A + B sin ( φ 1 + φ 0 )
(9) φ 1 + φ 0 = arctan ( I 2 − A I 1 − A ) + m π | CommonCrawl |
Abstract: We analyse the evolution of a mildly inclined circumbinary disc that orbits an eccentric orbit binary by means of smoother particle hydrodynamic (SPH) simulations and linear theory. We show that the alignment process of an initially misaligned circumbinary disc around an eccentric orbit binary is significantly different than around a circular orbit binary and involves tilt oscillations. The more eccentric the binary, the larger the tilt oscillations and the longer it takes to damp these oscillations. A circumbinary disc that is only mildly inclined may increase its inclination by a factor of a few before it moves towards alignment. The results of the SPH simulations agree well with those of linear theory. We investigate the properties of the circumbinary disc/ring around KH 15D. We determine disc properties based on the observational constraints imposed by the changing binary brightness. We find that the inclination is currently at a local minimum and will increase substantially before setting to coplanarity. In addition, the nodal precession is currently near its most rapid rate. The recent observations that show a reappearance of Star B impose constraints on the thickness of the layer of obscuring material. Our results suggest that disc solids have undergone substantial inward drift and settling towards to disc midplane. For disc masses $\sim 0.001 M_\odot$, our model indicates that the level of disc turbulence is low $\alpha \ll 0.001$. Another possibility is that the disc/ring contains little gas. | CommonCrawl |
\begin{definition}[Definition:Transversal (Geometry)/Alternate Angles]
:400px
'''Alternate angles''' are interior angles of a transversal which are on opposite sides and different lines.
In the above figure, the pairs of '''alternate angles''' with respect to the transversal $EF$ are:
:$\angle AHJ$ and $\angle DJH$
:$\angle CJH$ and $\angle BHJ$
\end{definition} | ProofWiki |
Three-dimensional flow within shallow, narrow cavities
Published online by Cambridge University Press: 28 October 2013
Sarah D. Crook ,
Timothy C. W. Lau and
Richard M. Kelso
Sarah D. Crook
Nova Systems, Edinburgh, SA 5111, Australia
Timothy C. W. Lau
Centre for Energy Technology, School of Mechanical Engineering, The University of Adelaide, SA 5005, Australia
[email protected]
The three-dimensional structure of incompressible flow in a narrow, open rectangular cavity in a flat plate was investigated with a focus on the flow topology of the time-averaged flow. The ratio of cavity length (in the direction of the flow) to width to depth was $l{: }w{: }d= 6{: }2{: }1$ . Experimental surface pressure data (in air) and particle image velocimetry data (in water) were obtained at low speed with free-stream Reynolds numbers of ${\mathit{Re}}_{l} = 3. 4\times 1{0}^{5} $ in air and ${\mathit{Re}}_{l} = 4. 3\times 1{0}^{4} $ in water. The experimental results show that the three-dimensional cavity flow is of the 'open' type, with an overall flow structure that bears some similarity to the structure observed in nominally two-dimensional cavities, but with a high degree of three-dimensionality both in the flow near the walls and in the unsteady behaviour. The defining features of an open-type cavity flow include a shear layer that traverses the entire cavity opening ultimately impinging on the back surface of the cavity, and a large recirculation zone within the cavity itself. Other flow features that have been identified in the current study include two vortices at the back of the cavity, of which one is barely visible, a weak vortex at the front of the cavity, and a pair of counter-rotating streamwise vortices along the sides of the cavity near the cavity opening. These vortices are generally symmetric about the cavity centre-plane. However, the discovery of a single tornado vortex, located near the cavity centreline at the front of the cavity, indicated that the flow within the cavity is asymmetric. It is postulated that the observed asymmetry in the time-averaged flow field is due to the asymmetry in the instantaneous flow field, which switches between two extremes at large time scales.
JFM classification
Aerodynamics: Aerodynamics Aerodynamics: Flow–structure interactions
Journal of Fluid Mechanics , Volume 735 , 25 November 2013 , pp. 587 - 612
DOI: https://doi.org/10.1017/jfm.2013.519[Opens in a new window]
©2013 Cambridge University Press
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Sarah D. Crook (a1), Timothy C. W. Lau (a2) and Richard M. Kelso (a2)
DOI: https://doi.org/10.1017/jfm.2013.519 | CommonCrawl |
\begin{document}
\title{Dependence spaces II}
\AUTHOR {Ewa} {Graczy\'{n}ska}
\email {[email protected]}
\maketitle
\begin{abstract} This is a continuation of my paper \cite{10} and the presentation at the Conference on Universal Algebra and Lattice Theory, Szeged, Hungary, June 21-25, 2012. The Steinitz exchange lemma is a basic theorem in linear algebra used, for example, to show that any two bases for a finite-dimensional vector space have the same number of elements. The EIS property was introduced by A. Hulanicki, E. Marczewski, E. Mycielski in \cite{EM1} (see \S 31 of \cite{GG}). In this note we show that its analogue holds in a dependence space. \end{abstract} \footnote{AMS Mathematical Subject Classification 2010}:
Primary: 05C69.
\section{Basic notions} According to F. G\'{e}cseg, H. J\"{u}rgensen \cite{6} the result which is usually referred to as the "Exchange Lemma" states that for transitive dependence, every independent set can be extended to form a basis. In \cite{10} we discussed some interplay between notions discussed in \cite{1}--\cite{2} and \cite{7}--\cite{6}. Another proof was presented there, of the result of N.J.S. Hughes \cite{1} on Steinitz' exchange theorem for infinite bases in connection with the notions of transitive dependence, independence and dimension as introduced in \cite{7} and \cite{8}. In that proof we assumed Kuratowski-Zorn's Lemma (see \cite{4}), as a requirement pointed in \cite{6}. In this note we extend the results to EIS property known in general algebra as Exchange of Independent Sets Property.
We use a modification of the the notation of \cite{1}, \cite{2} and \cite{6}: \\$a,b,c,...,x,y,z,...$ (with or without suffices) to denote the elements of a space ${\bf S}$ (or {\bf S}') and $A,B,C,...,X,Y,Z,...,$ for subsets of ${\bf S}$ (or {\bf S}'). $\Delta$, ${\it S}$ denote a family of subsets of ${\bf S}$, $n$ is always a positive integer.
$A \cup B$ denotes the union of sets $A$ and $B$, $A+B$ denotes the disjoint union of $A$ and $B$, $A-B$ denotes the difference of $A$ and $B$, i.e. is the set of those elements of $A$ which are not in $B$.
\section{Dependent and independent sets}
The following definitions are due to N.J.S. Hughes, invented in 1962 in \cite{1}:
\begin{definition} A set ${\bf S}$ is called a {\it dependence space} if there is defined a set $\Delta$, whose members are finite subsets of ${\bf S}$, each containing at least 2 elements, and if the Transitivity Axiom is satisfied. \end{definition}
\begin{definition} A set $A$ is called {\it directly dependent} if $A \in \Delta$. \end{definition}
\begin{definition} An element $x$ is called {\it dependent on A} and is denoted by $x \sim \Sigma A$ if either $x \in A$ or if there exist distinct elements $x_{0},x_{1},...,x_{n}$ such that \begin{center} (1) $\{ x_{0},x_{1},...,x_{n} \} \in \Delta$ \end{center} where $x_{0}=x$ and $x_{1},...,x_{n} \in A$
and {\it directly dependent} on $\{ x \}$ or $\{ x_{1},...,x_{n} \}$, respectively. \end{definition}
\begin{definition} A set $A$ is called {\it dependent} (with respect to $\Delta$) if (1) is satisfied for some distinct elements $x_{0},x_{1},...,x_{n} \in A$. Otherwise $A$ is {\it independent}. \end{definition}
\begin{definition} If a set $A$ is {\it independent} and for any $x \in {\bf S}$, $x \sim \Sigma A$, i.e. $x$ is dependent on $A$, then $A$ is called a {\it basis of} ${\bf S}$. \end{definition}
A similar definition of a {\it dependence} $D$ was introduced in \cite{7}. In the paper \cite{6} authors based on the theory of dependence in universal algebras as outlined in \cite {7}. We accept the well known:
\begin{definition} The {\it span} $<X>$ of a subset $X$ of ${\bf S}$ is the set of all elements of ${\bf S}$ which depends on $X$, i.e. $x \in <X>$ iff $x \sim \Sigma X$. \end{definition}
\begin{definition} {\bf TRANSITIVITY AXIOM:} \begin{center} If $x \sim \Sigma A$ and for all $a \in A$, $a \sim \Sigma B$, then $x \sim \Sigma B$. \end{center} \end{definition}
\section{Steinitz' exchange theorem}
In Linear Algebra, Steinitz exchange Lemma states that:
if $a \not \in <A \cup \{ b \}>$ and $b \not \in <A>$, then $b \not \in <A \cup \{ a \}>$.
In particular, if $A$ is independent and $a \not \in <A>$, then:
$A \cup \{ a \}$ is independent.
The following lemma is a generalization of the result of P.M. Cohn \cite{7} (cf. the property (E) of \cite{EM3}, p. 206, called there an axchange of an idependent sets or Theorem 3.8 of \cite{6}, p. 426): \begin{lemma} In a dependence space ${\bf S}$, assume that $a \not \in <A \cup \{ b \}>$ and $b \not \in <A>$. \\Then $b \not \in <A \cup \{ a \}>$. \end{lemma}
{\it Proof}
If $b \in <A \cup \{ a \} > - <A>$, then there exists $a_1,...,a_n \in A$, \\such that $b \sim \{a,a_1,...,a_n \}$, i.e. $\{ a,a_1,...a_n,b \} \in \Delta$.\\ Therefore $a \in < \{ b \} \cup A>$, a contradiction . $\Box$
It is clear, that for an independent set $A$, one gets for each $a \in A$, that $a \not \in <A - \{ a \}>$.
\begin{remark} The Lemma above can be reformulated in a similar way as conditions (1) and (2) in Theorem 3.8 and conditions (1) -- (3) of Lemma 3.9 of \cite{6}. \end{remark}
\section{EIS property} The EIS (exchange of independent sets) property was introduced by A. Hulanicki, E. Marczewski, E. Mycielski in \cite{EM1}. First we recall their original definition of EIS property (see \cite{EM1}, \cite{EM3}, p. 647--659). In their paper they use the terminology and notation of \cite{EM2} (with slight modifications). An {\it abstract algebra} is a (nonempty) set with a family of fundamental finitary operations. For any nonempty set $E \subset A$. ${\it C}(E)$ denotes the subalgebra generated by $E$, ${\it C}(\emptyset)$ is denoting the set of algebraic constants (i.e. the values of the constant algebraic operations). The operation ${\it C}$ has finite character, i.e. ${\it C}(E) = \bigcup {\it C}(F)$, where $F$ runs over the family of all finite subsets of $F$ of $E$.
The following theorem about exchange of independent sets is true for all algebras (see \cite{EM2}, p. 58, theorem 2.4 (ii)):
\begin{theorem} Let $P,Q$ and $R$ be subsets of an algebra. If $P \cup Q$ is independent, $P \cap Q = \emptyset$, $R$ is independent, ${\it C}(R) = {\it C}(Q)$, then $P \cup R$ is independent. \end{theorem} As the authors of \cite{EM1} noticed, it might seem at first glance that the relation ${\it C}(R) = {\it C}(Q)$ could be replaced by a weaker one: $R \subset {\it C}(Q)$. Since, as it can be seen from the results of \cite{EM1}, this is not generally true, the authors say that an algebra satisfies {\it the condition of exchange of independent sets} (EIS) whenever for any subsets $P,Q$ and $R$ of it, the relations: $P \cup Q$ is independent, $P \cap Q = \emptyset$, $R$ is independent and $R \subset {\it C}(Q)$ imply that $P \cup R$ is independent.
We transform the original definition of EIS property from {\it algebras} to {\it dependence spaces} in the natural way: \begin{definition} A dependence space ${\bf S}$ satisfies the EIS property, if for arbitrary subsets $P,Q$ and $R$ of ${\bf S}$ the conditions:
(7) $P \cap Q = \emptyset$;
(8) $P \cup Q$ is an independent set in ${\bf S}$;
(9) $R$ is an independent set in ${\bf S}$, $R \subseteq <Q>$;
altogether imply that:
(10) $P \cup R$ is an independent set.
\end{definition} \begin{theorem} In a dependence space ${\bf S}$, the EIS property holds. \end{theorem}
{\it Proof}
Assume (7) -- (9). \\To show (10) assume a contrario that $P \cup R$ is a dependent set. Therefore there exist (all different) elements $a_1,...,a_n,b_1,...,b_m \in P \cup R$ with $a_1,...,a_n \in P$ and $b_1,...,b_m \in R$ and such that $\{ a_1,...,a_n,b_1,...,b_m \}\in \Delta$. From (7) and (9) it follows that there exists an element $a_1 \in P$ such that $a_1 \sim \Sigma \{a_2,...,a_n,b_1,...,b_m \}$, i.e. $a_1 \sim \Sigma ((P - \{ a_1 \}) \cup R)$. But for very element $b \in R$, $b \sim \Sigma Q$, therefore $b \sim \Sigma ((P - \{ a _1 \}) \cup Q)$. Moreover, $c \sim \Sigma((P \cup Q) - \{a_{1}\})$, for every $c \in ((P - \{a_{1}\}) \cup R)$. Thus, by the transitivity axiom $a_1 \sim \Sigma ((P \cup Q) - \{a_1 \}))$. That contradicts (8), as $a_1 \in P \cup Q$ and it is clear, that for an independent set $A$, one gets for each $a \in A$, that $a \not \in <A - \{ a \}>$. $\Box$
\end{document} | arXiv |
Shannon wavelet
In functional analysis, the Shannon wavelet (or sinc wavelets) is a decomposition that is defined by signal analysis by ideal bandpass filters. Shannon wavelet may be either of real or complex type.
Shannon wavelet is not well-localized(noncompact) in the time domain,but its Fourier transform is band-limited(compact support). Hence Shannon wavelet has poor time localization but has good frequency localization. These characteristics are in stark contrast to those of the Haar wavelet. The Haar and sinc systems are Fourier duals of each other.
Definition
Sinc funcition is the starting point for the definition of the shannon wavelet.
Scaling function
First, we define the scaling function to be the sinc function.
$\phi ^{\text{(Sha)}}(t):={\frac {\sin \pi t}{\pi t}}=\operatorname {sinc} (t).$
And define the dilated and translated intances to be
$\phi _{k}^{n}(t):=2^{n/2}\phi ^{\text{(Sha)}}(2^{n}t-k)$
where the parameter $n,k$ means the dilation and the translation for the wavelet respectively.
Then we can derive the Fourier transform of the scaling function:
$\Phi ^{\text{(Sha)}}(\omega )={\frac {1}{2\pi }}\Pi ({\frac {\omega }{2\pi }})={\begin{cases}{\frac {1}{2\pi }},&{\mbox{if }}{|\omega |\leq \pi },\\0&{\mbox{if }}{\mbox{otherwise}}.\\\end{cases}}$ where the (normalised) gate function is defined by
$\Pi (x):={\begin{cases}1,&{\mbox{if }}{|x|\leq 1/2},\\0&{\mbox{if }}{\mbox{otherwise}}.\\\end{cases}}$ Also for the dilated and translated instances of scaling function: $\Phi _{k}^{n}(\omega )={\frac {2^{-n/2}}{2\pi }}e^{-i\omega (k+1)/2^{n}}\Pi ({\frac {\omega }{2^{n+1}\pi }})$
Mother wavelet
Use $\Phi ^{\text{(Sha)}}$ and multiresolution approximation we can derive the Fourier transform of the Mother wavelet:
$\Psi ^{\text{(Sha)}}(\omega )={\frac {1}{2\pi }}e^{-i\omega }{\bigg (}\Pi ({\frac {\omega }{\pi }}-{\frac {3}{2}})+\Pi ({\frac {\omega }{\pi }}+{\frac {3}{2}}){\bigg )}$
And the dilated and translated instances:
$\Psi _{k}^{n}(\omega )={\frac {2^{-n/2}}{2\pi }}e^{-i\omega (k+1)/2^{n}}{\bigg (}\Pi ({\frac {\omega }{2^{n}\pi }}-{\frac {3}{2}})+\Pi ({\frac {\omega }{2^{n}\pi }}+{\frac {3}{2}}){\bigg )}$
Then the shannon mother wavelet function and the family of dilated and translated instances can be obtained by the inverse Fourier transform:
$\psi ^{\text{(Sha)}}(t)={\frac {\sin \pi (t-(1/2))-\sin 2\pi (t-(1/2))}{\pi (t-1/2)}}=\operatorname {sinc} {\bigg (}t-{\frac {1}{2}}{\bigg )}-2\operatorname {sinc} {\bigg (}2(t-{\frac {1}{2}}){\bigg )}$
$\psi _{k}^{n}(t)=2^{n/2}\psi ^{\text{(Sha)}}(2^{n}t-k)$
Property of mother wavelet and scaling function
• Mother wavelets are orthonormal, namely,
$<\psi _{k}^{n}(t),\psi _{h}^{m}(t)>=\delta ^{nm}\delta _{hk}={\begin{cases}1,&{\text{if }}h=k{\text{ and }}n=m\\0,&{\text{otherwise}}\end{cases}}$
• The translated instances of scaling function at level $n=0$ are orthogonal
$<\phi _{k}^{0}(t),\phi _{h}^{0}(t)>=\delta ^{kh}$
• The translated instances of scaling function at level $n=0$ are orthogonal to the mother wavelets
$<\phi _{k}^{0}(t),\psi _{h}^{m}(t)>=0$
• Shannon wavelets has an infinite number of vanishing moments.
Reconstruction of a Function by Shannon Wavelets
Suppose $f(x)\in L_{2}(\mathbb {R} )$ such that $\operatorname {supp} \operatorname {FT} \{f\}\subset [-\pi ,\pi ]$ and for any dilation and the translation parameter $n,k$,
${\Bigg |}\int _{-\infty }^{\infty }f(t)\phi _{k}^{0}(t)dt{\Bigg |}<\infty $, ${\Bigg |}\int _{-\infty }^{\infty }f(t)\psi _{k}^{n}(t)dt{\Bigg |}<\infty $
Then
$f(t)=\sum _{k=\infty }^{\infty }\alpha _{k}\phi _{k}^{0}(t)$ is uniformly convergent, where $\alpha _{k}=f(k)$
Real Shannon wavelet
The Fourier transform of the Shannon mother wavelet is given by:
$\Psi ^{(\operatorname {Sha} )}(w)=\prod \left({\frac {w-3\pi /2}{\pi }}\right)+\prod \left({\frac {w+3\pi /2}{\pi }}\right).$
where the (normalised) gate function is defined by
$\prod (x):={\begin{cases}1,&{\mbox{if }}{|x|\leq 1/2},\\0&{\mbox{if }}{\mbox{otherwise}}.\\\end{cases}}$
The analytical expression of the real Shannon wavelet can be found by taking the inverse Fourier transform:
$\psi ^{(\operatorname {Sha} )}(t)=\operatorname {sinc} \left({\frac {t}{2}}\right)\cdot \cos \left({\frac {3\pi t}{2}}\right)$
or alternatively as
$\psi ^{(\operatorname {Sha} )}(t)=2\cdot \operatorname {sinc} (2t)-\operatorname {sinc} (t),$
where
$\operatorname {sinc} (t):={\frac {\sin {\pi t}}{\pi t}}$
is the usual sinc function that appears in Shannon sampling theorem.
This wavelet belongs to the $C^{\infty }$-class of differentiability, but it decreases slowly at infinity and has no bounded support, since band-limited signals cannot be time-limited.
The scaling function for the Shannon MRA (or Sinc-MRA) is given by the sample function:
$\phi ^{(Sha)}(t)={\frac {\sin \pi t}{\pi t}}=\operatorname {sinc} (t).$
Complex Shannon wavelet
In the case of complex continuous wavelet, the Shannon wavelet is defined by
$\psi ^{(CSha)}(t)=\operatorname {sinc} (t)\cdot e^{-2\pi it}$,
References
• S.G. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1999, ISBN 0-12-466606-X
• C.S. Burrus, R.A. Gopinath, H. Guo, Introduction to Wavelets and Wavelet Transforms: A Primer, Prentice-Hall, 1988, ISBN 0-13-489600-9.
| Wikipedia |
\begin{document}
\begin{abstract}
Using the definition of colouring of $2$-edge-coloured graphs derived from $2$-edge-coloured graph homomorphism, we extend the definition of chromatic polynomial to $2$-edge-coloured graphs.
We find closed forms for the first three coefficients of this polynomial that generalize the known results for the chromatic polynomial of a graph.
We classify those $2$-edge-coloured graphs that have a chromatic polynomial equal to the chromatic polynomial of the underlying graph, when every vertex is incident to edges of both colours.
Finally, we examine the behaviour of the roots of this polynomial, highlighting behaviours not seen in chromatic polynomials of graphs. \end{abstract}
\title{Chromatic Polynomials of $2$-edge-coloured Graphs}
\section{Introduction and Preliminary Notions}
A \emph{$2$-edge-coloured graph $G$} is a triple $(\Gamma,R_G,B_G)$ where $\Gamma$ is a simple graph, $R_G \subseteq E(\Gamma)$, and $B_G \subseteq E(\Gamma)$ so that $R_G \cap B_G = \emptyset$ and $R_G\cup B_G = E(\Gamma)$. We call $G$ a \emph{$2$-edge-colouring} of $\Gamma$. We let $G[R_G]$ and $G[B_G]$ be the simple graph induced, respectively, by the set of red and blue edges. We call $\Gamma$ the simple graph \emph{underlying} $G$.
We note $\{R_G,B_G\}$ need not be a partition of $E(\Gamma)$; we permit $R_G =\emptyset$ or $B_G =\emptyset$. In the case that $\{R_G,B_G\}$ is a partition of $E(\Gamma)$, we say that $G$ is \emph{bichromatic}. Otherwise we say $G$ is \emph{monochromatic.} When there is no chance for confusion we refer to $R_G$ and $B_G$ as $R$ and $B$, respectively. Herein we assume all graphs are loopless and have no parallel arcs. Thus we drop the descriptor of simple when we refer to simple graphs. In various corners of the literature \cite{moprs08,S14} $2$-edge-coloured graphs are referred to as \emph{signified graphs} to highlight the absence of a switching operation as in \cite{N15}.
For other notation not defined herein, we refer the reader to \cite{bondy}. Throughout we generally use Greek majuscules to refer to graphs and Latin majuscules to refer to $2$-edge-coloured graphs.
Let $G=\left(\Gamma_G,R_G,B_G\right)$ and $H=\left(\Gamma_H,R_H,B_H\right)$ be $2$-edge-coloured graphs. There is a \emph{homomorphism} of $G$ to $H$ when there exists a homomorphism $\phi:\Gamma_G \to \Gamma_H$ so that $\phi: G[R_G] \to H[R_H]$ and $\phi: G[B_G] \to H[B_H]$ are both graph homomorphisms. That is, a homomorphism of $G$ to $H$ is a vertex mapping from $V(\Gamma_G)$ to $V(\Gamma_H)$ that preserves the existence of edges as well as their colours. When $H$ has $k$ vertices we call $\phi$ a \emph{$k$}-colouring of $G$.
Equivalently, one may view a $k$-colouring of a $2$-edge-coloured graph $G= (\Gamma,R,B)$ as a function $c: V(G) \to \{1,2,3,\dots, k\}$ satisfying the following two conditions \begin{enumerate}
\item for all $yz \in E(\Gamma)$, we have $c(y) \neq c(z)$; and
\item for all $ux\in R$ and $vy \in B$, if $c(u) = c(v)$ then $c(x) \neq c(y)$. \end{enumerate}
Such a proper colouring defines a homomorphism to $2$-edge-coloured target $H$ with vertex set $\{1,2,3, \dots, k\}$ where $ij \in R_H$ (respectively, $\in B_H$) if and only if there is an edge $wx \in R_G$ (respectively, $\in B_G$) so that $c(w) = i$ and $c(x) = j$.
The \emph{chromatic number of $G$}, denoted $\chi(G)$, is the least integer $t$ such that $G$ admits a $t$-colouring. Observe that when $G$ is monochromatic, the definitions above are equivalent to those for graph homomorphism, $k$-colouring and chromatic number. Thus the choice of notation, $\chi$, to denote the chromatic number is indeed appropriate.
We note that for a $2$-edge-coloured graph $G$, the chromatic number of $G = \left(\Gamma,R,B\right)$ may differ vastly from that of $\Gamma$. There exist $2$-edge-colourings of bipartite graphs that have chromatic number equal to their number of vertices \cite{S14}.
Homomorphisms of edge-coloured graphs have received increasing attention in the literature in the last twenty-five years. Early work by Alon and Marshall \cite{AM98} gave an upper bound on the chromatic number of a $2$-edge-coloured planar graph. More recent work has bounded the chromatic number of two-edge-coloured graphs from a variety of graph families \cite{moprs08,O17} as well as considered questions of computational complexity \cite{B17,B00}.
Given the definition of colouring given above as proper colouring of the underlying graph satisfying extra constraints, one can enumerate, for fixed positive integer $k$, the number of $k$-colourings of a $2$-edge-coloured graph $G$. This observation gives rise to a reasonable definition of chromatic polynomial for a $2$-edge-coloured graph. As the definition of homomorphism and colouring of $2$-edge-coloured graphs fully captures those for graphs, we notice results for the chromatic polynomial of $2$-edge-coloured graphs necessarily generalize those for graphs.
To study the chromatic polynomial for $2$-edge-coloured graphs, we introduce a chromatic polynomial of a more generalized graph object. A \emph{mixed $2$-edge-coloured graph} is a pair $M = (G,F_M$) where $G$ is $2$-edge-coloured graph where $G = (\Gamma,R_G,B_G)$ and $F_M \subseteq \overline{E(\Gamma)}$. We consider $F_M$ as a set of edges that are contained neither in $R_G$ nor in $B_G$. We denote by $S(M)$ the graph with vertex set $V(\Gamma)$ and edge set $R_G \cup B_G \cup F_M$. When there is a no chance for confusion we refer to $F_M$ as $F$.
At this point it may be tempting to consider a mixed $2$-edge-coloured graph as a $3$-edge-coloured graph. We resist this temptation as the following definition of $k$-colouring of a mixed $2$-edge-coloured graph does not match what one would expect for a $3$-edge-coloured graph.
Let $M = (G,F)$ be a mixed $2$-edge-coloured graph with $G = \left(\Gamma, R, B\right)$ We define a \emph{$k$-colouring} of $M$ to be a function $c:V(G) \to \{1,2,3, \dots , k\}$ so that \begin{enumerate}
\item $c(u) \neq c(v)$ for all $uv \in R \cup B \cup F$; and
\item for all $ux\in R$ and for all $vy \in B$, if $c(u) = c(v)$ then $c(x) \neq c(y)$. \end{enumerate}
Informally, $c$ is a $k$-colouring of the $2$-edge-coloured graph $G$ with the extra condition that vertices at the ends of an element of $F$ receive different colours. Notice when $M = \emptyset$, we have that $c$ is a $k$-colouring of $G$. Further notice when $R = B = \emptyset$ we have that $c$ is a $k$-colouring of the graph with vertex set $V(\Gamma)$ and edge set $F$.
Let $M$ be a mixed $2$-edge-coloured graph and let $P(M,\lambda)$ be the unique interpolating polynomial so that for every non-negative integer $k$, we have that $P(M,k)$ is the number of $k$-colourings of $M$. We refer to $P(M,\lambda)$ as the chromatic polynomial of $M$. Notice when $F_M = \emptyset$, the polynomial $P(M,\lambda)$ enumerates colourings of a $2$-edge-coloured graph and when $R = B = \emptyset$, this polynomial is identically the chromatic polynomial of the graph with vertex set $V(\Gamma)$ and edge set $F$.
Our work proceed as follows. In Section \ref{sec:chromPoly} we study properties of chromatic polynomials of mixed $2$-edge-coloured graphs. We use a recurrence reminiscent of the standard recurrence for the chromatic polynomial of a graph to give closed forms for the first three coefficients of this polynomial. These results generalize known results for chromatic polynomials of graphs. As every colouring of a $2$-edge-coloured graph is a colouring of the underlying graph, we are led naturally to considering those $2$-edge-coloured graphs that have the same chromatic polynomial as their underlying graph. We study this problem in Section \ref{sec:chromInvar}. For $2$-edge-coloured graphs where each vertex is incident with both a red edge and a blue edge, we find those $2$-edge-coloured graphs that have the same chromatic polynomial as their underlying graph to be those $2$-edge-coloured graphs that arise as a $2$-edge-colouring of a join. In Section \ref{sec:roots} we study roots of chromatic polynomials of $2$-edge-coloured graphs. We find that closure of the real roots of the $2$-edge-coloured chromatic polynomials to be $\mathbb{R}$ and that the non-real roots can have arbitrarily large modulus. Finally in Section \ref{sec:discuss} we compare our results for chromatic polynomials with those for graphs and oriented graphs. We conjecture on the properties of the chromatic polynomial of an $(m,n)$-mixed graph.
\section{The Chromatic Polynomial of a mixed $2$-edge-coloured Graph}\label{sec:chromPoly}
Our study of the chromatic polynomial of a mixed $2$-edge-coloured graph begins by examining two cases for (2) in the definition of a $k$-colouring of a mixed $2$-edge-coloured graph.
Consider first the case $u=v$. In this case this condition enforces that vertices at the end of an induced bichromatic path with three vertices receive different colours in an colouring. Let $M = \left(G,F\right)$ be a mixed $2$-edge-coloured graph. An induced path $xuy$ is called a \emph{bichromatic $2$-path} when $xu \in R$ and $uy \in B$. Let $\mathcal{P}_G$ be the set of bichromatic $2$-paths of $G$.
Let $M = (G,F)$ be a mixed $2$-edge-coloured graph with $n$ vertices. If every pair of vertices of $M$ is either adjacent in $M$ or at the ends of a bichromatic $2$-path in $G$, then in any colouring of $M$ each vertex receives a distinct colour. Thus
\begin{equation} \label{math:complete}
P(M,\lambda) = \Pi_{i =0}^{n-1} \left(\lambda-i\right) \end{equation}
As every vertex of $M$ receives a distinct colour in every colouring of $M$, the chromatic polynomial of $M$ is exactly that of $K_n$. We return to this observation in the proofs of Theorems \ref{thm:secondcoeff} and \ref{thm:thirdcoeff}.
Let $x$ and $y$ be a pair of vertices that are neither adjacent in $M$ nor at the ends of a bichromatic $2$-path in $G$. The $k$-colourings of $M$ can be partitioned into those in which $x$ and $y$ receive the same colour and those in which $u$ and $v$ receive different colours. Thus
\begin{equation} \label{math:reduce} P(M,\lambda) = P(M+xy,\lambda) + P(M_{xy},\lambda) \end{equation}
where \begin{itemize}
\item $M+xy$ is the mixed $2$-edge-coloured graph formed from $M$ by adding $xy$ to $F$; and
\item $M_{xy}$ is the mixed $2$-edge-coloured graph formed from identifying vertices $x$ and $y$ and deleting any edge that is parallel with a coloured edge. \end{itemize}
See Figure \ref{fig:exampleFig} for a sample computation. Elements of $R$ and $B$ are denoted respectively by dotted and dashed lines. Elements of $F$ are denoted by solid lines. Here we follow the usual convention for chromatic polynomials of having the picture of the graph stand in for its polynomial.
\begin{figure}
\caption{Computing the chromatic polynomial of a mixed $2$-edge-coloured graph}
\label{fig:exampleFig}
\end{figure}
From Equations (\ref{math:complete}) and (\ref{math:reduce}) we directly obtain the following.
\begin{theorem}\label{thm:basicResults}
Let $M = (G,F)$ be a mixed graph with $n$ vertices.
\begin{enumerate}
\item $P(M,\lambda)$ is a polynomial of degree $n$ in $\lambda$;
\item the coefficient of $\lambda^n$ is $1$; and
\item $P(M,\lambda)$ has no constant term.
\end{enumerate} \end{theorem}
As with chromatic polynomials, the coefficient of $\lambda^{n-1}$ is easily computed.
\begin{theorem}\label{thm:secondcoeff}
For a mixed graph $M = (G,F)$ the coefficient of $\lambda^{n-1}$ is given by \[-\left(|R_G| + |B_G| + |F_M| + |\mathcal{P}_G| \right).\] \end{theorem}
\begin{proof}
We proceed considering the existence of a counterexample.
Let $n$ be the least integer so that there exists a mixed $2$-edge-coloured graph with $n$ vertices so that the statement of the theorem is false.
Among all counterexamples on $n$ vertices, let $M = (G,F)$ be a counterexample that maximizes the number of edges in $S(M)$.
If $\mathcal{P}_G \neq \emptyset$, then there exists vertices $u$ and $v$ which are the ends of an induced bichromatic path in $M$.
Let $M^\prime$ be the mixed $2$-edge-coloured graph formed from $M$ by adding $uv$ to $F$.
We observe that every colouring of $M$ is a colouring of $M^\prime$ and also that every colouring of $M^\prime$ is a colouring of $M$.
Thus $M$ and $M^\prime$ have the same chromatic polynomial.
This contradicts our choice of $M$ as a counterexample on $n$ vertices with the maximum number of edges in $S(M)$.
Thus we may assume $\mathcal{P}_G = \emptyset$.
We claim there exists a pair of vertices $u$ and $v$ so that $u$ and $v$ are not adjacent in $S(M)$.
If $u$ and $v$ do not exist, then $S(M) \cong K_n$ and so $P(M,\lambda) = P(K_n,\lambda)$.
The coefficient of $\lambda^{n-1}$ in $P(K_n,\lambda)$ is given by $-{n \choose 2}$ \cite{R68}.
Recall $\mathcal{P}_G = \emptyset$.
For $M$ we observe
\[
-\left(|R_G| + |B_G| + |F_M| + |\mathcal{P}_G| \right) = -{n \choose 2}
\]
Thus we consider $u,v \in V(M)$ so that $u$ and $v$ are not adjacent in $S(M)$.
Let $c_{n-1}$ be the coefficient of $\lambda^{n-1}$ in $P(M,\lambda)$.
By Theorem \ref{thm:basicResults}, our choice of $M$, and Equation (\ref{math:reduce}), we have
\begin{align*}
c_{n-1} &= -\left(|R_{G}| + |B_{G}| + |F_{M+uv}| + |\mathcal{P}_{G}| \right) +1
\end{align*}
We observe $|F_{M+uv}| = |F_{M}| + 1$.
Simplifying yields
\begin{align*}
c_{n-2} = -\left(|R_G| + |B_G| + |F_M| + |\mathcal{P}_G| \right)
\end{align*} Thus $M$ is not a counterexample. And so by choice of $M$, no counterexample exists. \end{proof}
\begin{corollary}
For a $2$-edge-coloured graph $G$, the coefficient of $\lambda^{n-1}$ in $P(G,\lambda)$ is given by $-\left(|R_G| + |B_G| + |\mathcal{P}_G|\right)$ \end{corollary}
We return now to our examination of cases for (2) in the definition of $k$-colouring of a mixed $2$-edge-coloured graph $M = (G,F)$. Consider now the case where $ux$ and $vy$ induce a bichromatic copy of $2K_2$ in $G$. For $c$, a $k$-colouring of $M$, we note that $\left|\{c(u),c(x),c(v),c(z)\}\right| \in \{3,4\}$. As the subgraph induced by $\{u,v,y,z\}$ has chromatic number $2$ in $S(M)$, such pairs of edges, in a sense, obstruct a colouring of $S(M)$ from being a colouring of $M$.
Let $M = (G,F)$ be a mixed $2$-edge-coloured graph. Let $\Lambda$ be the graph formed from $M$ by adding to $F$ the edge between any pair of vertices which are the ends of an induced bichromatic $2$-path in $M$ and considering coloured edges in $G$ as edges in $\Lambda$. (See Figure \ref{fig:MMstar} for an example)
\begin{figure}
\caption{$\Lambda$ (right) constructed from $M$ (left)}
\label{fig:MMstar}
\end{figure}
For $ux \in R$ and $vy \in B$ we say that $ux$ and $vy$ are \emph{obstructing edges} when $\chi(\Lambda[u,x,v,y]) = 2$. Let $\mathcal{O}_M$ be the set of pairs of obstructing edges in $M$. For a $2$-edge-coloured graph $G$, we let $\mathcal{O}_G$ be the set of obstructing edges in $G$.
Using the notion of obstructing edges, we find a closed form for the coefficient of $\lambda^{n-2}$ of a chromatic polynomial of a mixed $2$-edge-coloured graph.
\begin{theorem}\label{thm:thirdcoeff}
For $M = (G,F)$ a mixed $2$-edge-coloured graph, the coefficient of $\lambda^{n-2}$ in $P(M,\lambda)$ is given by
\[
{ {|R_G| + |B_G| + |\mathcal{P}_G|} + |F_M|\choose 2}
- |T_{S(M)}| - |\mathcal{P}_G| - |\mathcal{O}_M|,
\]
where $T_{S(M)}$ is the set of induced subgraphs of $S(M)$ isomorphic to $K_3$. \end{theorem}
\begin{proof}
We proceed considering the existence of a counterexample.
Let $n$ be the least integer so that there exists a mixed $2$-edge-coloured graph with $n$ vertices so that the statement of the theorem is false.
Among all counterexamples on $n$ vertices, let $M = (G,F)$ be a counterexample that maximizes the number of edges in $S(M)$.
If $\mathcal{P}_G \neq \emptyset$, then there exists vertices $u$ and $v$ which are the ends of an induced bichromatic path in $M$.
Let $M^\prime$ be the mixed $2$-edge-coloured graph formed from $M$ by adding $uv$ to $F$.
We observe that every colouring of $M$ is a colouring of $M^\prime$ and also that every colouring of $M^\prime$ is a colouring of $M$.
Thus $M$ and $M^\prime$ have the same chromatic polynomial.
This contradicts our choice of $M$.
Thus we may assume $\mathcal{P}_G = \emptyset$.
We claim there exists a pair of vertices $u$ and $v$ so that $u$ and $v$ are not adjacent in $S(M)$.
If $u$ and $v$ do not exist, then $S(M) \cong K_n$ and so $P(M,\lambda) = P(K_n,\lambda)$.
The coefficient of $\lambda^{n-2}$ in $P(K_n,\lambda)$ is given by$ {{n \choose 2} \choose 2} - {n \choose 3}$ \cite{R68}.
For $M$ we observe
\begin{align*}
{{n \choose 2} \choose 2} - {{n \choose 3}} &= {|R_G| + |B_G| + |\mathcal{P}_G| + |F_M|\choose 2} - |T_{S(M)}| \\
&= {|R_G| + |B_G| + |\mathcal{P}_G| + |F_M|\choose 2} - |T_{S(M)}| - |\mathcal{P}_G| - |\mathcal{O}_M|
\end{align*}
This last equality follows by observing $\mathcal{P}_G=\emptyset$ (by hypothesis) and $\mathcal{O}_M = \emptyset$ when $M$ is complete. This equality contradicts our choice of $M$, and so we conclude that such vertices $u$ and $v$ exist.
Thus we consider $u,v \in V(M)$ so that $u$ and $v$ are not adjacent in $S(M)$.
Let $c_{n-2}$ be the coefficient of $\lambda^{n-2}$ in $P(M,\lambda)$.
By Theorem \ref{thm:secondcoeff}, our choice of $M$ and Equation (\ref{math:reduce}) we have
\begin{align*}
c_{n-2} &={|R_{G}| + |B_{G}| + |\mathcal{P}_{G}| + |F_{M+uv}| \choose 2}
- \left(|T_{S(M+uv)}| + |\mathcal{P}_{G}| + |\mathcal{O}_{M+uv}|\right)\\
&-\left(|R_{G_{uv}}| + |B_{G_{uv}}| + |F_{M_{uv}}| + |\mathcal{P}_{G_{uv}}| \right)
\end{align*}
Observe $|F_{M+uv}| = |F_{M}| + 1$. Let $C$ be the set of common neighbours of $u$ and $v$ in $S(M)$. We observe $|T_{S(M+uv)}| = |T_{S(M)}| + |C|$ and $|R_{G_{uv}}| + |B_{G_{uv}}| + |F_{M_{uv}}| = |R_{G}| + |B_{G}| + |F_{M}| - |C|$. Thus
\begin{align*}
c_{n-2} &={|R_{G}| + |B_{G}| + |\mathcal{P}_{G}| + |F_{M}| + 1 \choose 2}
- \left(|T_{S(M)}| + |\mathcal{P}_{G}| + |\mathcal{O}_{M+uv}|\right)\\
&-\left(|R_{G}| + |B_{G}| + |F_{M}| + |\mathcal{P}_{G_{uv}}| \right) \end{align*}
Since $\mathcal{P}_{G} = \emptyset$, a pair of obstructing edges, $\left(ux, vy\right)$ in $M$, is not obstructing in $M+uv$ if and only if $xy \notin F$ and one of $uy$ or $vx$ is contained in $F$. Let $\mathcal{O}^{uv}_{M}$ be the set of such obstructing edges. Therefore $|\mathcal{O}_{M+uv}| = |\mathcal{O}_{M}| - |\mathcal{O}^{uv}_{M}|$. Notice now that every element of $\mathcal{O}^{uv}_{M}$ contributes an element of $\mathcal{P}_{G_{uv}}$ that was not an element of $\mathcal{P}_{G}$. Thus $|\mathcal{P}_{G_uv}| = |\mathcal{P}_{G}| +|\mathcal{O}^{uv}_{M}|$.
Substituting yields
\begin{align*}
c_{n-2} &={|R_{G}| + |B_{G}| + |\mathcal{P}_{G}| + |F_{M}| + 1 \choose 2}
- \left(|T_{S(M)}| + |\mathcal{P}_{G}| + \mathcal{O}_{M} - |\mathcal{O}^{uv}_{M}|\right)\\
&-\left(|R_{G}| + |B_{G}| + |F_{M}| + |\mathcal{P}_{G}| +|\mathcal{O}^{uv}_{M}| \right)\\
& ={|R_{G}| + |B_{G}| + |\mathcal{P}_{G}| + |F_{M}| + 1 \choose 2} -\left(|R_{G}| + |B_{G}| + |F_{M}| + |\mathcal{P}_{G}| \right) - \left(|T_{S(M)}| + |\mathcal{P}_{G}| + |\mathcal{O}_{M}|\right)\\
& ={|R_{G}| + |B_{G}| + |\mathcal{P}_{G}| + |F_{M}|\choose 2} - \left(|T_{S(M)}| + |\mathcal{P}_{G}| + |\mathcal{O}_{M}|\right) \end{align*}
Thus $M$ is not a counterexample. And so by choice of $M$, no counterexample exists. \end{proof}
\begin{corollary}
For a $2$-edge-coloured graph $G=\left(\Gamma,R,B\right)$, the coefficient of $\lambda^{n-2}$ in $P(G,\lambda)$ is given by
\[
{|R_G| + |B_G| + |\mathcal{P}_G|\choose 2}
- |T_{\Gamma}| - |\mathcal{P}_G| - |\mathcal{O}_G| \] \end{corollary}
The results in Theorems \ref{thm:secondcoeff} and \ref{thm:thirdcoeff} relied on known results about the second and third coefficients of the chromatic polynomial of a complete graph. Such results can be proved independently for mixed $2$-edge-coloured complete graphs, forgoing the need to reference prior work on the chromatic polynomial of a graph. In this case we notice that the results for the second and third coefficients of the chromatic polynomial may be obtained as corollaries of Theorems \ref{thm:secondcoeff} and \ref{thm:thirdcoeff} by considering mixed $2$-edge-coloured graphs with $R = B = \emptyset$. We contextualize our results within the current state-of-the-art in Section \ref{sec:discuss}.
\section{Chromatically Invariant $2$-edge-coloured Graphs}\label{sec:chromInvar}
Consider a $2$-edge-coloured graph $G = \left(\Gamma,R,B\right)$. By definition every colouring of $G$ is necessarily a colouring of $\Gamma$. Thus for each integer $k \geq 1$ we have $P(\Gamma,k) \geq P(G,k)$. In this section we study the structure of $2$-edge-coloured graphs $G$ for which $P(\Gamma,\lambda) = P(G,\lambda)$. We refer to such $2$-edge-coloured graphs as \emph{chromatically invariant}. Trivially, every $2$-edge-coloured graph in which $R = \emptyset$ (or $B = \emptyset$) is chromatically invariant. We refer to those chromatically invariant $2$-edge-coloured graphs with $R,B \neq \emptyset$ \emph{non-trivially chromatically invariant}.
\begin{lemma}\label{lem:doubleEmpty}
Let $G$ be a $2$-edge-coloured graph. If $\mathcal{P}_G = \mathcal{O}_G = \emptyset$, then $G$ is chromatically invariant. \end{lemma}
\begin{proof}
Let $G=\left(\Gamma,R,B\right)$ be a $2$-edge-coloured graph so that $\mathcal{P}_G = \mathcal{O}_G = \emptyset$.
For each $k \geq 1$, let $C_{G,k}$ be the set of $k$-colourings of $G$.
Similarly, let $C_{\Gamma,k}$ be the set of $k$-colourings of $\Gamma$.
Recalling the definition of $k$-colouring of a $2$-edge-coloured graph, it follows directly that $C_{G,k} \subseteq C_{\Gamma,k}$.
To complete the proof it suffices to show $C_{\Gamma,k} \subseteq C_{G,k}$.
Let $c$ be a $k$-colouring of $\Gamma$.
Consider $ux\in R$ and $vy \in B$ so that $c(u) = c(v)$.
If $u = v$, then $xy \in E(\Gamma)$ as $\mathcal{P}_G = \emptyset$.
Thus $c(x) \neq c(y)$.
Consider now the case where $u \neq v$.
Since $\mathcal{O}_G = \emptyset$ it follows that $\left|\{c(u),c(x),c(v), c(y)\}\right| \in \{3,4\}$.
Thus $c(x) \neq c(y)$.
Therefore $c \in C_{G,k}$ and so it follows $C_{\Gamma,k} \subseteq C_{G,k}$. \end{proof}
\begin{theorem}\label{thm:firstCharact}
A $2$-edge-coloured graph $G = \left(\Gamma ,R , B\right)$ is chromatically invariant if and only if $G$ has no induced bichromatic $2$-path and $G$ contains no induced bichromatic copy of $2K_2$. \end{theorem}
\begin{proof}
Let $G=\left(\Gamma,R,B\right)$ be a $2$-edge-coloured graph.
Begin by assuming $G$ is chromatically invariant.
For a contradiction, we first assume $\mathcal{P} \neq \emptyset$.
Consider $uvw \in \mathcal{P}$.
Let $k$ be the least integer so that there is a $k$-colouring $c$ of $\Gamma$ for which $c(u) = c(v)$.
Let $C_{\Gamma,k}$ be the set of $k$-colourings of $\Gamma$.
Let $C_{G,k}$ be the set of $k$-colourings of $G$.
By construction, $c \in C_{\Gamma,k}$ but $c \notin C_{G,k}$.
By the argument in the proof of Lemma \ref{lem:doubleEmpty}, we have $C_{G,k} \subseteq C_{\Gamma,k}$.
Therefore $|C_{G,k}| < |C_{\Gamma,k}|$.
Thus $P(G,k) < P(\Gamma, k)$, which implies $P(G,\lambda) \neq P(\Gamma, \lambda)$.
Assume now $G$ contains an induced bichromatic copy of $2K_2$.
Let $ux \in R$ and $vy \in B$ so that $G[u,x,v,y]$ is a bichromatic copy of $2K_2$.
Let $k$ be the least integer so that there is a $k$ colouring $c$ of $\Gamma$ for which $c(u) = c(v)$ and $c(x) = c(y)$.
A contradiction follows as in the previous paragraph.
Assume $\mathcal{P} = \emptyset$ and $G$ contains no induced bichromatic copy of $2K_2$.
By Lemma \ref{lem:doubleEmpty} it suffices to show $\mathcal{O}_G = \emptyset$.
Consider $ux \in R$ and $vy \in B$ so that $u \neq v,y$ and $x \neq v,y$.
Since $G$ contains no induced bichromatic copy of $2K_2$, there exists an edge with an end in $\{u,x\}$ and an end in $\{v,y\}$.
Without loss of generality, assume $uv \in R$.
Since $\mathcal{P} = \emptyset$ and $vy \in B$ it follows that $uy \in E(\Gamma)$.
Therefore $\Gamma[u,v,x,y]$ contains a copy of $K_3$.
Thus $\left(ux,vy\right)$ is not a pair of obstructing edges.
Therefore $\mathcal{O}_G = \emptyset$.
The result follows by Lemma \ref{lem:doubleEmpty}. \end{proof}
Theorem \ref{thm:firstCharact} gives a full characterization of chromatically invariant $2$-edge-coloured graphs. This characterization allows us to further characterize these $2$-edge-coloured graphs by way of pairs of independent sets.
\begin{theorem}\label{thm:I1I2}
A $2$-edge-coloured graph $G=\left(\Gamma,R,B\right)$ is chromatically invariant if and only if for every disjoint pair of non-empty independent sets $I_1$ and $I_2$ in $\Gamma$, the $2$-edge-coloured subgraph induced by $I_1$ and $I_2$ is monochromatic. \end{theorem}
\begin{proof}
Let $G=\left(\Gamma,R,B\right)$ be a $2$-edge-coloured graph and let $I_1$ and $I_2$ be disjoint non-empty independent sets of $\Gamma$.
Assume $G$ is chromatically invariant.
Thus by Theorem \ref{thm:firstCharact}, it follows that $G$ has no induced bichromatic $2$-path and no induced bichromatic copy of $2K_2$.
If $G[I_1 \cup I_2]$ has at most one edge, then the result holds -- necessarily this subgraph is monochromatic.
Otherwise, assume $e = u_1u_2$ and $f = v_1v_2$ are edges of $G[I_1 \cup I_2]$ ($u_1,v_1 \in I_1$ and $u_2,v_2 \in I_2$).
Assume, without loss of generality, that $e \in R$ and $f \in B$.
We first show that $e$ and $f$ have a common end point.
Recall $G$ contains no induced bichromatic copy of $2K_2$
Thus if $e$ and $f$ do not have a common end point, then, without loss of generality, we have $u_1v_2 \in E(\Gamma)$.
If $u_1v_2 \in R$ then $u_1v_2v_1 \in \mathcal{P}$.
Similarly, if $u_1v_2 \in B$, then $v_2u_1v_1 \in \mathcal{P}$.
However, we have $\mathcal{P} = \emptyset$.
Therefore $e$ and $f$ have a common end point.
If $e$ and $f$ have a common endpoint, then $ef \in \mathcal{P}$.
However, we have $\mathcal{P} = \emptyset$.
Therefore $e$ and $f$ do not have a common end point, a contradiction.
Therefore the $2$-edge-coloured subgraph induced by $I_1$ and $I_2$ is monochromatic.
Assume $G$ is not chromatically invariant.
By Theorem \ref{thm:firstCharact} we have that $P \neq \emptyset$ or $G$ contains an induced bichromatic copy of $2K_2$.
In either case we find a pair of disjoint independent sets $I_1,I_2$ so that $G[I_1 \cup I_2]$ is not monochromatic. \end{proof}
\begin{corollary}\label{cor:subgraphs}
If $G$ is a $2$-edge-coloured chromatically invariant graph, then every induced subgraph of $G$ is chromatically invariant. \end{corollary}
We turn now to the problem of classifying those graphs which admit a chromatically invariant $2$-edge-coloured colouring. We focus our attention on those graphs which admit a chromatically invariant $2$-edge-colouring in which every vertex is incident with both a red and blue edge.
Recall that a graph $\Gamma$ is a \emph{join} when $V(\Gamma)$ has a partition $\{X,Y\}$ so that $xy \in E(\Gamma)$ for all $x\in X$ and $y \in Y$. For $\Gamma_1 = \Gamma[X]$ and $\Gamma_2 = \Gamma[Y]$ we say that \emph{$\Gamma$ is the join of $\Gamma_1$ and $\Gamma_2$} and we write $\Gamma = \Gamma_1 \vee \Gamma_2$. We call an edge $uv \in V(\Gamma_1 \vee \Gamma_2)$ a \emph{joining edge} when $u \in V(\Gamma_1)$ and $v \in V(\Gamma_2)$. Notice that if $\Gamma$ is a join, then $V(\Gamma)$ admits a partition: $\{X_1,X_2,\dots, X_k\}$ such that $\Gamma[X_i]$ is not a join for each $1 \leq i \leq k$ and for each $1 \leq i < j \leq k$ we have $\Gamma[X_i \cup X_j] = \Gamma[X_i] \vee \Gamma[X_j]$. We denote such a decomposition as $\Gamma= \bigvee_{1 \leq i \leq k} \Gamma[X_i]$.
\begin{lemma}\label{lem:joinsWork}
Let $\Gamma_1$ and $\Gamma_2$ be graphs such that each of $\Gamma_1$ and $\Gamma_2$ have no isolated vertices.
The graph $\Gamma_1 \vee \Gamma_2$ admits a non-trivial chromatically invariant $2$-edge colouring in which every vertex is incident with at least one red edge and one blue edge. \end{lemma}
\begin{proof}
Let $J$ be the set of joining edges of $\Gamma_1 \vee \Gamma_2$.
Let $R = E(\Gamma_1) \cup E(\Gamma_2)$ and $B = J$.
Since each of $\Gamma_1$ and $\Gamma_2$ have no isolated vertices, each vertex of $\Gamma_1 \vee \Gamma_2$ is incident with at least one red edge and one blue edge.
For any pair of disjoint independent sets $I_1,I_2 \subset V(\Gamma)$ we have $I_1,I_2 \subset V(\Gamma_1)$ or $I_1,I_2 \subset V(\Gamma_2)$.
The result follows by observing that the $2$-edge-coloured graph $(\Gamma,R,B)$ satisfies the hypothesis of Theorem \ref{thm:I1I2} \end{proof}
\begin{lemma}\label{lem:monochromeJoin}
Let $G=\left(\Gamma,R,B\right)$ be non-trivial chromatically invariant $2$-edge-coloured graph. If there exists graphs $\Gamma_1$ and $\Gamma_2$ so that $\Gamma = \Gamma_1 \vee \Gamma_2$ and neither of $\Gamma_1$ or $\Gamma_2$ is a join, then all of the joining edges of $\Gamma_1 \vee \Gamma_2$ have the same colour in $G$. \end{lemma}
\begin{proof}
Let $\Gamma_1$ and $\Gamma_2$ be graphs that are not joins.
Let $\Gamma = \Gamma_1 \vee \Gamma_2$.
Let $G=\left(\Gamma,R,B\right)$ be non-trivial chromatically invariant $2$-edge-coloured graph.
We first show that for any pair $y_1,y_2 \in V(\Gamma_2)$ there is a sequence of independent sets $I_1,I_2,\dots, I_\ell$ so that $y_1 \in I_1$, $y_2 \in I_\ell$ and $I_i \cap I_{i+1} \neq \emptyset$ for all $1 \leq i \leq \ell-1$.
Since $\Gamma_2$ is not a join, its complement, $\overline{\Gamma_2}$, is connected.
Therefore there is a path from $y_1$ to $y_2$ in $\overline{\Gamma_2}$.
The edges of such a path form a sequence of independent sets in $\Gamma_2$: $I_1,I_2,\dots, I_\ell$ so that $y_1 \in I_1$, $y_2 \in I_\ell$ and $I_i \cap I_{i+1} \neq \emptyset$ for all $1 \leq i \leq \ell$.
Consider $v \in V(\Gamma_1)$ and $y_1,y_2 \in V(\Gamma_2)$.
Since $\Gamma = \Gamma_1 \vee \Gamma_2$, we have $vy_1,vy_2 \in E(\Gamma)$.
Let $I_1,I_2,\dots, I_\ell$ be a sequence of independent sets in so that $y_1 \in I_1$, $y_2 \in I_\ell$ and $I_i \cap I_{i+1} \neq \emptyset$ for all $1 \leq i \leq\ell-1$.
By Lemma \ref{thm:I1I2} edges between $I_i$ and $v$ all have the same colour.
Since $I_i \cap I_{i+1} \neq \emptyset$, all of the edges between $I_{i+1}$ and $v$ all have that same colour.
Since $v$ is adjacent to every vertex in $\Gamma_2$ it follows that the edges between $I_{1}$ and $v$ have the same colour as those between $I_{\ell}$ and $v$.
Therefore every joining edge with an end at $v$ has the same colour.
Similarly, for any $u \in V(\Gamma_2)$, every joining edge with an end at $u$ has the same colour in $G$.
Thus it follows that all of the joining edges of $\Gamma_1 \vee \Gamma_2$ have the same colour in $G$. \end{proof}
\begin{lemma}\label{lem:isaJoin}
If $G = \left(\Gamma,R,B\right)$ is a chromatically invariant $2$-edge-coloured graph in which every vertex is incident with a red edge and a blue edge, then $\Gamma$ is a join. \end{lemma}
\begin{proof}
Let $G = (\Gamma,R,B)$ be a minimum counterexample with respect to number of vertices.
We first show that there exists a vertex $v \in V(\Gamma)$ so that every vertex of $G-v$ is incident with both a red edge and a blue edge.
If no such vertex exists, then for every $x \in V(\Gamma)$ there exists $y \in V(\Gamma)$ so that $xy \in E(\Gamma)$ and the edge $xy$ is the only one of its colour incident with $y$.
We proceed in cases based on the existence of a pair $u,v \in V(\Gamma)$ so that $uv$ is the only one of its colour incident with $u$ and the only one of its colour incident with $v$.
If such a pair exists, then, without loss of generality, let $uv$ be red.
Since $G$ has no induced bichromatic $2$-path, $N_{G[B]}(u) = N_{G[B]}(v)$.
If $V(\Gamma) = \{u,v\} \cup N_{G[B]}(u)$, then $\Gamma$ is a join.
As such the set $Q = V(\Gamma) \setminus \left(\{u,v\} \cup N_{G[B]}(u)\right)$ is non-empty.
Notice that as $G$ has no induced bichromatic $2$-path, all edges between $Q$ and $N_{G[B]}(u)$ are blue.
Since every vertex of $G$ is incident with both a red and a blue edge, it follows that every vertex of $Q$ is incident with a red edge in the $2$-edge-coloured graph $G[Q]$.
As $\Gamma$ is not a join, there exists $q \in Q$ and $x \in N_{G[B]}(u)$ so that $qx \notin E(\Gamma)$.
Let $rq$ be a red edge in $G[Q]$.
Notice $rx \notin E(\Gamma)$, as otherwise such an edge is blue in $G$ and so $xrq$ is an induced bichromatic $2$-path in $G$.
Therefore the subgraph induced by $\{u,x,q,r\}$ is a bichromatic copy of $2K_2$.
This is a contradiction as $G$ is chromatically invariant.
Therefore no such pair $u,v$ exists.
Since no such pair $u,v$ exists, there is a maximal sequence of vertices of $\Gamma$: $u_1,u_2,\dots u_k$ so that $u_iu_{i+1} \in E(\Gamma)$ and the edge $u_iu_{i+1}$ is the only one of its colour incident with $u_{i+1}$ for all $1 \leq i \leq k-1$.
We further note that, without loss of generality, vertices with an even index are adjacent with a single red edge and vertices with an odd index (other that $u_1$) are incident with a single blue edge.
Thus $u_1,u_2,\dots,u_k$ is an path whose edges alternate being red and blue.
If $k \geq 4$, then since $G$ has no induced bichromatic $2$-path, we have $u_2u_4 \in E(\Gamma)$.
However, this edge is either a second red edge incident with $u_2$ or a second blue edge incident with $u_4$. This is a contradiction, and so $k = 3$.
Since this path was chosen to be maximal, it follows that the edge between $u_1$ and $u_3$ is the only one of its colour incident with $u_1$.
Since $u_1u_2$ is red, it follows that $u_1u_3$ is blue.
But then $u_3$ is incident with two blue edges: $u_2u_3$ and $u_1u_3$.
This is a contradiction.
And so there exists a vertex $v \in V(\Gamma)$ so that every vertex of $G-v$ is incident with both a red edge and a blue edge.
Consider $v \in V(\Gamma)$ so that every vertex of $G-v$ is incident with both a red edge and a blue edge.
By Lemma \ref{lem:monochromeJoin}, $G-v$ is a chromatically invariant $2$-edge-coloured graph.
By the minimality of $G$, we have that $\left(\Gamma - v \right)$ is a join.
Therefore $V(\Gamma-v)$ admits a partition $\{X_1,X_2,\dots, X_k\}$ such that $\left(\Gamma - v \right)[X_i]$ is not a join for each $1 \leq i \leq k$ and for each $1 \leq i < j \leq k$ we have $\left(\Gamma - v \right)[X_i \cup X_j] = \left(\Gamma - v \right)[X_i] \vee \left(\Gamma - v \right)[X_j]$.
Notice for any $1 \leq i \leq k$ that if $v$ is adjacent to every vertex of $X_i$, then $\Gamma$ is necessarily a join.
Thus, for every for every $1 \leq i \leq k$ vertex $v$ is not adjacent to at least one vertex of $X_i$.
By hypothesis, $v$ in incident with a red edge and a blue edge.
Let $vr$ and $vb$ be such edges for some $r,b \in V(\Gamma)$.
We proceed in cases based on the location of $r$ and $b$ within the partition $\{X_1,X_2,\dots, X_k\}$ of $V(\Gamma-v)$.
Consider, without loss of generality, $r,b \in X_1$.
Notice that by Theorem \ref{cor:subgraphs} and Lemma \ref{lem:monochromeJoin}, for every $1 \leq i < j \leq k$, the joining edges of $\left(\Gamma - v \right)[X_i \cup X_j] = \left(\Gamma - v \right)[X_i] \vee \left(\Gamma - v \right)[X_j]$ are the same colour.
Since each of $r$ and $b$ are adjacent to every vertex of $X_2$, then either $vry$ or $vby$ is an induced bichromatic $2$-path for every $y \in X_2$.
(Whether or not $vry$ or $vby$ is an induced bichromatic $2$-path depends on the colour of the joining edges between $X_1$ and $X_2$ ).
Since $G$ is chromatically invariant, by Theorem \ref{thm:firstCharact} no such path can be exist
And so $v$ is adjacent to every vertex in $X_2$, a contradiction.
Consider, without loss of generality, $r \in X_1$ and $b \in X_2$.
Further assume without loss of generality, that all of the joining edges of $\left(\Gamma - v \right)[X_1 \cup X_2] = \left(\Gamma - v \right)[X_1] \vee \left(\Gamma - v \right)[X_2]$ are red.
Since $\Gamma$ is not a join, there is at least one vertex of $X_1$, say $x_1$, that is not adjacent to $v$.
However, $x_1bv$ is an induced bichromatic $2$-path.
Since $G$ is chromatically invariant, by Theorem \ref{thm:firstCharact} no such path can be exist.
This is a contradiction. \end{proof}
Together Lemmas \ref{lem:joinsWork} and \ref{lem:isaJoin} imply the following characterization of those graphs which admit non-trivial chromatically invariant $2$-edge-colourings in which every vertex is incident with at least one edge of both colours.
\begin{theorem}\label{thm:fullJoin}
A graph $\Gamma$ admits a non-trivial chromatically invariant $2$-edge-colouring in which every vertex is incident with both a red edge and a blue edge if and only if $\Gamma$ is the join of two graphs, each of which has no isolated vertices. \end{theorem}
In \cite{C19} the authors fully characterize those oriented graphs that admit a chromatically invariant orientation. Theorem \ref{thm:fullJoin} presents a partial analogue for $2$-edge-coloured graphs. It remains unknown which graphs admit a non-trivial chromatically invariant $2$-edge-colouring when we allow for vertices to be incident with edges of a single colour.
\section{Roots of Chromatic Polynomials of $2$-edge-coloured Graphs}\label{sec:roots}
As with many graph polynomials it is natural to study the location of the roots of the chromatic polynomial of $2$-edge-coloured graphs. The chromatic polynomial of graphs is well studied, and one topic of interest is the location of the roots. A root of a chromatic polynomial of a graph is called a \emph{chromatic root}. See Figure~\ref{figchromroots6} for the chromatic roots of all connected graphs on 6 vertices obtained by computer search. We call a root of a chromatic polynomial of a $2$-edge-coloured graph a \emph{monochromatic root} if the graph is monochromatic and a \emph{bichromatic root} if the graph is bichromatic. See Figure~\ref{figroots6} for the bichromatic roots of all connected $2$-edge-coloured graphs on 6 vertices obtained by computer search. Recall the chromatic polynomial of a monochromatic graph is simply the chromatic polynomial of the underlying graph. Therefore the collection of all chromatic roots is exactly the collection of all monochromatic roots. In this section we provide results on bichromatic roots.
\begin{figure}
\caption{Bichromatic roots of all connected $2$-edge-coloured graphs on 6 vertices}
\caption{Chromatic roots of all connected graphs on 6 vertices}
\label{figroots}
\label{figroots6}
\label{figchromroots6}
\end{figure}
We begin with a study of the real roots. The real chromatic roots are always positive \cite{R68} as the coefficients of the chromatic polynomial of a graph alternate in sign and there are no real roots in $(0,1)\cup (1,\frac{32}{27}]$. In contrast we will show the bichromtic roots are dense in $\mathbb{R}$ and the collection of all rational bichromatic roots is $\mathbb{Z}$. For $n > 1$, let $K_n^r$ and $K_n^b$ denote monochromatic copies of $K_n$ with red and blue edges, respectively. For a pair of $2$-edge-coloured graphs $G= \left(\Gamma_G,R_G, B_G\right)$ and $H = \left(\Gamma_H, R_H, B_H\right)$, let $G \cup H$ denote the disjoint union of $G$ and $H$.
\begin{theorem}
\label{thm:GK2}
Let $G=(\Gamma,R,B)$ be a $2$-edge-coloured graph on $n$ vertices so that $\chi(G)=n$.
We have
\[P(G \cup K_2^r, \lambda)=\lambda(\lambda-1) \cdots (\lambda-n+1)(\lambda^2-\lambda-2|B|).\] \end{theorem}
\begin{proof}
Let $G=(\Gamma,R,B)$ be a $2$-edge-coloured graph on $n$ vertices so that $\chi(G)=n$.
For fixed $k > 0$, we construct a $k$-colouring of $G \cup K_2^r$ by first colouring vertices $G$ and then those of $K_2^r$.
As $\chi(G)=n$, any $k$-colouring assigns each vertex of $G$ a unique colour.
There are $k(k-1) \cdots (k-|V|+1)$ such colourings.
We can then colour vertices of $K_2^r$ with any two different colours unless there exists $b \in B$ whose ends have been assigned those two colours.
Thus each $b \in B$ prohibits two possible colourings of the $K_2^r$.
Therefore given any $k$-colouring of $G$ there are $k^2-k-2|B|$ such colourings of $K_2^r$.
And so \[P(G \cup K_2^r, \lambda)=\lambda(\lambda-1) \cdots (\lambda-n+1)(\lambda^2-\lambda-2|B|).\] \end{proof}
\begin{corollary}
\label{cor:introot}
Let $n > 1$ be an integer.
We have
\[P(K_n^b \cup K_2^r, \lambda)=\lambda(\lambda-1) \cdots (\lambda-n)(\lambda+n-1).\] \end{corollary}
\begin{theorem}
The closure of the rational bichromatic roots is $\mathbb{Z}$. \end{theorem}
\begin{proof}
By Theorem \ref{thm:basicResults}, for any $2$-edge-coloured graph $G$, the leading coefficient of $P(G,\lambda)$ is 1.
Therefore by the rational root theorem, any rational root of an oriented chromatic polynomial must be an integer.
Let $m > 0$ be an integer
By Corollary \ref{cor:introot}, we have \[ P(K_{m+1}^b \cup K_2^r,\lambda)= \lambda(\lambda-1) \cdots (\lambda-(m+1))(\lambda+m).\]
We observe $P(K_{m+1}^b \cup K_r^2,\lambda)$ has roots at $\lambda = -m,0,1,\ldots, m,m+1$. \end{proof}
\begin{theorem}
The closure of the real bichromatic roots is $\mathbb{R}$. \end{theorem}
\begin{proof}
For any $r \in \mathbb{R}$ let $d(r)=r-\lfloor r \rfloor$.
Furthermore let
\[A = \left\{ \frac{1-\sqrt{1+8m}}{2}: m \in \mathbb{N} \right\}.\]
Let $G$ be a $2$-edge-colouring of a complete graph so that $|B|=m$.
By Theorem \ref{thm:GK2} each element of $A$ is a real root of $P(G \cup K_2^r, \lambda)$.
We first show that for any $r \in \mathbb{R}$ and $\epsilon >0$, there exists an $a \in A$ such that $|d(r)-d(a)|< \epsilon$.
Let $f(m)= \frac{1-\sqrt{1+8m}}{2}$ and $M=\frac{2}{\epsilon^2}$.
Note that $|f(m+1)-f(m)| < \epsilon$ for $m \geq M$.
Furthermore $f(m) \rightarrow -\infty$ as $m\rightarrow \infty$.
Thus for any $s \leq f(M)$, there exists an $m \geq M$ such that $f(m+1) \leq s \leq f(m) \leq r$.
This implies $|s-f(m)| < \epsilon$.
Furthermore $|d(s)-d(f(m))| < \epsilon$.
By choosing $s \leq f(M)$ such that $d(s)=d(r)$ and let $a=f(m) \in A$.
It then follows that $|d(r)-d(a)|< \epsilon$.
Let $G_a$ be a $2$-edge-coloured graph so that $P(G_a,\lambda)$ has a real root at $a$.
Let $H_a$ be the $2$-edge-coloured graph formed the join of $G_a$ and any $2$-edge-coloured $K_n$, where all of the joining edges are red.
We have
\[P(H_a,\lambda)=\lambda(\lambda-1) \cdots (\lambda-n+1)P(G_a,\lambda-n).\]
Consider $n=\lfloor r \rfloor- \lfloor a \rfloor$.
As $r=d(r)+\lfloor r \rfloor$ and $a+n=d(a)+\lfloor a \rfloor+n = d(a)+\lfloor r \rfloor$ we have
\[|r-(a+n)| = |d(r)+\lfloor r \rfloor-(d(a)+\lfloor r \rfloor)|=|d(r)-d(a)|< \epsilon.\]
Thus $H_a$ has a root at $a+n$.
\end{proof}
We turn now to study complex roots of chromatic polynomials of $2$-edge-coloured graphs. We show they may have arbitrarily large modulus. To do this we study the limit of the complex roots of a $2$-edge-coloured complete bipartite graph.
Let $p_n(z)=\sum_{j=1}^k\alpha_j(z)\lambda_j(z)^n$. Beraha, Kahne and Weiss studied the limits of the complex roots of such functions (as arising in recurrences). They fully classified those values that occur as limits of roots of a family of polynomials. See \cite{W78} for a full statement of the Bereha-Kahane-Weiss Theorem.
A limit of roots of a family of polynomials ${P_n}$ is a complex number, $z$, for which there are sequences of integers $(n_k)$ and complex numbers $(z_k)$ such that $z_k$ is a zero of $P_{n_k}$, and $z_k \to z$ as $k \to \infty$. The Bereha-Kahane-Weiss Theorem requires non-degeneracy conditions: no $\alpha_i$ is identically $0$, and $\lambda_i\not=\omega\lambda_k$ for any $i\not = k$ and any root of unity $\omega$. The Bereha-Kahane-Weiss Theorem implies that the limit of roots of $P_n(z)$ are precisely those complex numbers $z$ such that one of the following hold:
\begin{itemize}
\item one of the $|\lambda_i(z)|$ exceeds all others and $\alpha_i(z)=0$; or
\item $|\lambda_1(z)|=|\lambda_2(z)|=...=\lambda_{\ell}(z)>|\lambda_j(z)|$ for $\ell + 1\leq j \leq k$ for some $\ell \geq 2$. \end{itemize}
\begin{theorem}~\label{Thrm:complexroots}
Non-real bichromatic roots can have arbitrarily large modulus. \end{theorem}
\begin{proof}
Let $n \geq 5$ be an integer.
Consider $K_{2,n-2}$ with partition $\{X,Y\}$ with $X = \{u,v\}$.
Let $G = \left(K_{2,n-2},R,B\right)$ so that three of the edges incident with $v$ are blue and all other edges are red.
Let $x,y,z\in Y$ be the vertices of $G$ that are adjacent to $v$ by a blue edge.
In any $k$-colouring $c$, we have $c(u)\neq c(v)$.
Further, for each $w \in Y \setminus\{x,y,z\}$ we must have $c(w) \neq c(x),c(y),c(z)$.
We proceed to count the number of $k$-colourings of $G$ based on the cardinality of $\left|\left\{ c(x),c(y),c(z)\right\}\right|$.
When $\left|\left\{ c(x),c(y),c(z)\right\}\right| = 3$, there are
\[
k(k-1)(k-2)(k-3)(k-4)(k-5)^{n-5}
\]
$k$-colourings of $G$.
When $\left|\left\{ c(x),c(y),c(z)\right\}\right| = 2$, there are
\[
3k(k-1)(k-2)(k-3)(k-4)^{n-5}
\]
$k$-colourings of $G$.
Finally, when $\left|\left\{ c(x),c(y),c(z)\right\}\right| = 1$, there are
\[
k(k-1)(k-2)(k-3)^{n-5}
\]
$k$-colourings of $G$.
Thus
\begin{align*}
P(G,\lambda) &= (\lambda-2)(\lambda-3)(\lambda-4)(\lambda-5)^{n-5} + 3(\lambda-2)(\lambda-3)(\lambda-4)^{n-5} + \lambda(\lambda-1)(\lambda-2)(\lambda-3)^{n-5}\\
&= \lambda(\lambda-1)(\lambda-2)(\lambda-3)\left((\lambda-3)^{n-6}+3(\lambda-4)^{n-5}+(\lambda-4)(\lambda-5)^{n-5}\right)
\end{align*}
Consider the polynomial $g(n,\lambda) = (\lambda-3)^{n-6}+3(\lambda-4)^{n-5}+(\lambda-4)(\lambda-5)^{n-5}$.
We may express this polynomial as:
\[
p(n,z)=\alpha_1(z)(\lambda_1(z))^{n-6}+\alpha_2(z)(\lambda_2(z))^{n-5}+\alpha_3(z)(\lambda_3(z))^{n-5}.
\]
Here the non-degeneracy conditions hold for $p(n,z)$.
Applying the Bereha-Kahne-Weiss Theorem and setting $|\lambda_1(z)|=|\lambda_3(z)|>|\lambda_4(z)|$ we solve for $z=a+bi$ such that $|z-3|=|z-5|>|z-4|$.
One can verify when $a=4$ we have $|z-3|=|z-5|$ and $|z-5|>|z-4|$ for all values of $b$.
Thus the curve $z=4+bi$ is a limit of the roots for $2$-edge-coloured chromatic polynomial of $K_{2,n-2}$.
As there are no restrictions on $b$, it then follows that $P(G,\lambda)$ can have complex roots of arbitrarily large modulus. \end{proof}
Consider our above $K_{2,\ell-2}$ with partition $\{X,Y\}$ with $X = \{u,v\}$ and the 2-edge-coloured graph $G_{\ell} = \left(K_{2,\ell-2},R,B\right)$ with three of the edges incident with $v$ are blue and all other edges are red. Let $H_{n,\ell}$ be the $2$-edge-coloured graph formed by $G_{\ell}$ by joining $G_{\ell}$ with a copy of $K^r_n$ so that all joining edges are blue. Every vertex of $K^r_n$ requires a distinct colour and the joining edges are all blue. Therefore no vertex of $G$ can be assigned any of the $n$ colours appearing on the vertices of $K^r_n$. Thus
\[ P(H_{n,\ell},\lambda)=\lambda(\lambda-1) \cdots (\lambda-(n-1))P(G_{\ell},\lambda -n)\]
Taken with Theorem~\ref{Thrm:complexroots}, this implies that the curve $f(n,b)=4+n+bi$ is also limit of the roots for $n\geq 1$. See Figures~\ref{figrootscomplex1} and \ref{figrootscomplex2} for a plot of the roots of these polynomials.
\begin{figure}
\caption{Complex Chromatic Roots of the 2-edge coloured $K_{2,n-2}$}
\label{figrootscomplex1}
\caption{Complex chromatic roots of $H_{n,\ell}$ for $\ell=6,...,18$, $n=1,...,18$}
\label{figrootscomplex2}
\end{figure}
From the plots in Figure~\ref{figrootscomplex1} and Figure~\ref{figrootscomplex2} one can see that the closure of the roots contain an infinite number of vertical curves crossing the real axis at integer values of at least $4$. The real and complex chromatic roots (and hence monochromaitc roots) are dense in the complex plane \cite{S04}. It remains to be seen if bichromatic roots that dense in the complex plane.
\section{Further Remarks} \label{sec:discuss} That the results and methods in Section \ref{sec:chromPoly} closely resemble results and methods for the chromatic polynomial comes as no surprise to these authors -- Equations \ref{math:complete} and \ref{math:reduce} both hold for the chromatic polynomial of a graph. We note, however, that the standard \emph{delete and contract} technique for the chromatic polynomial of a graph does not apply, in general, for $2$-edge-coloured graphs. Deleting an edge, in some sense, forgets the colour of the adjacency between a pair of vertices -- important information for a vertex colouring a $2$-edge-coloured graph. This implies that the $2$-edge-coloured chromatic polynomial is not an evaluation of the Tutte polynomial.
The results and methods in Section \ref{sec:chromPoly} closely mirror those for the oriented chromatic polynomial in \cite{C19}. Such a phenomenon has been observed in the study of the chromatic number oriented graphs and $2$-edge-coloured graphs. In \cite{RASO94} Raspaud and Sopena give an upper bound for the chromatic number of an orientation of a planar graph. And in \cite{AM98} Alon and Marshall use the same techniques to derive the same upper bound for the chromatic number of a $2$-edge-coloured planar graph. In this latter work, Alon and Marshall profess the similarity of their techniques to those appearing in \cite{RASO94}, yet see no way to derive one set of results from the other. In the following years Ne\v{s}et\v{r}il and Raspaud \cite{NR00} showed these results were in fact special cases of a more general result for $(m,n)$-mixed graphs -- graphs in which there are $m$ different arc colours and $n$ different edge colours. Oriented graphs are graphs are $(0,1)$-mixed graphs, oriented graphs are $(1,0)$-mixed graphs and $2$-edge-coloured graphs $(0,2)$-mixed graphs.
By way of homomorphism, one can define, for each $(m,n) \neq (0,0)$, colouring for $(m,n)$-mixed graphs that generalizes graph colouring, oriented graph colouring and $2$-edge-coloured graph colouring.
As our results in Section \ref{sec:chromPoly} closely mirror those in \cite{C19} for oriented graphs we expect that the results in Section \ref{sec:chromPoly} in fact special cases of a more general result for the, to be defined, chromatic polynomial of an $(m,n)$-mixed graph. Showing such a result would require successfully generalizing the notions of obstructing arcs/edge as well as the notions of $2$-dipath and bichromatic $2$-path. This latter problem is considered in \cite{BDS17}.
\end{document} | arXiv |
1PE3PE5PE7PE9PE11PE13PE15PE17PE19PE21PE23PE25PE27PE29PE31PE33PE35PE37PE39PE41PE43PE45PE47PE49PE1AP3AP5AP7AP9AP11AP
(a) The ideal size (most efficient) for a broadcast antenna with one end on the ground is one-fourth the wavelength ($\dfrac{\lambda}{4}$) of the electromagnetic radiation being sent out. If a new radio station has such an antenna that is 50.0 m high, what frequency does it broadcast most efficiently? Is this in the AM or FM band? (b) Discuss the analogy of the fundamental resonant mode of an air column closed at one end to the resonance of currents on an antenna that is one- fourth their wavelength.
$1.50 \times 10^6 \textrm{ Hz}$. This is in the AM band.
Resonance for an electromagnetic wave at $\lambda_1 = \dfrac{h}{4}$ is analogous to sound resonance in a closed end tube which also has $\lambda_1 = \dfrac{L}{4}$. $h$ is analogous to $L$. Further resonances for the antenna might occur at $f_n = \dfrac{nc}{4h}, n=1, 3, 5, ...$
This is College Physics Answers with Shaun Dychko. A broadcast antenna has a height which is one quarter of the wavelength of the electromagnetic radiation its most efficiently emitting and so it will be built such that whatever wavelength it's meant to broadcast is going to be… umm… four times its height or its height would be one quarter of the times of wavelength whichever way you want to look at it. So, yeah, wavelength is four times the height. So, that's four times 50 meters and that makes 200 meters and then we can calculate its frequency using the wave equation which says the speed of the wave equals its frequency times the wavelength and we will solve for f by dividing both sides by lambda. So, f is the speed of light divided by lambda which is 3 times 10 to the 8 meters per second over 200 meters and that makes a frequency of 1.50 times 10 to the 6 hertz. This frequency is 1500 kilo-hertz which is in the AM band and what we are seeing here with this height of the broadcast antenna being related to the resonant wavelength that it is emitting is analogous to tube full of air closed at one end and in this case the length will equal the first resonant frequency over four in which case lambda one equals four times L and then we can solve for the resonant frequency by using this wave equation where I have v instead of C because we are not dealing with light anymore, we are dealing with sound in this particular picture and the first frequency is gonna be v over lambda where lambda is four times L and this would be an L here. So, that's the first resonant frequency. The third harmonic is going to be three v over four L because in this case we have a node here and anti-node here and the next wavelength that fits in this tube such that there is an anti-node in the opening and node at the closed end is… has a… the length of the tube is gonna be three quarters of the wavelength in which case lambda is four L over three and so that's why we get this equation for the third harmonic frequency and in general, the resonant frequencies for a tube closed at one end then is this integer n times the wave speed over four L where n is some odd integer, one, three, five or so on. So, resonance for this electromagnetic wave from the antenna having a resonant wavelength of the height over four is analogous to sound resonance in a closed end tube which has a resonance wavelength of the same form, a length divided by 4. So, the length in this case is length of the tube and the length for the antenna is the height of the antenna. So, the height is analogous to length and further resonance for the antenna might occur at, umm… at some frequencies where n is multiplied by the wave speed which in the case of antenna is C since its emitting light divided by four times the height of the antenna h . | CommonCrawl |
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\Big | 015017\Big |.\Big |
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\Big |
\Big |bibitem\Big |{Zhangxu1\Big |}\Big | \Big |{\Big |sc\Big | X\Big |.\Big |~Zhang\Big |}\Big |,\Big | \Big |{\Big |em\Big | Explicit\Big | observability\Big | estimate\Big | for\Big | the\Big | wave\Big | equation\Big | with\Big | potential\Big | and\Big | its\Big | application\Big |}\Big |,\Big | \Big | R\Big |.\Big | Soc\Big |.\Big | Lond\Big |.\Big | Proc\Big |.\Big | Ser\Big |.\Big | A\Big | Math\Big |.\Big | Phys\Big |.\Big | Eng\Big |.\Big | Sci\Big |.\Big |,\Big | \Big |{\Big |bf\Big | 456\Big |}\Big |(2000\Big |)\Big |,\Big | pp\Big |.\Big |~1101\Big |-\Big |-1115\Big |.\Big |
\Big |
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\Big |
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\Big |
\Big |bibitem\Big |{Zhangxu2\Big |}\Big | \Big |{\Big |sc\Big | X\Big |.\Big |~Zhang\Big |}\Big |,\Big | \Big |{\Big |em\Big | Unique\Big | continuation\Big | for\Big | stochastic\Big | parabolic\Big | equations\Big |}\Big |,\Big | \Big | Diff\Big |.\Big | Int\Big |.\Big | Eqs\Big |.\Big |,\Big | \Big |{\Big |bf\Big | 21\Big |}\Big |(2008\Big |)\Big |,\Big | pp\Big |.\Big |~81\Big |-\Big |-93\Big |.\Big |
\Big |
\Big |bibitem\Big |{Zhangxu3\Big |}\Big | \Big |{\Big |sc\Big | X\Big |.\Big |~Zhang\Big |}\Big |,\Big | \Big |{\Big |em\Big | Carleman\Big | and\Big | observability\Big | estimates\Big | for\Big | stochastic\Big | wave\Big | equations\Big |}\Big |,\Big | Math\Big |.\Big | Ann\Big |.\Big |,\Big | \Big |{\Big |bf\Big | 325\Big |}\Big |(2003\Big |)\Big |,\Big | pp\Big |.\Big |~543\Big |-\Big |-582\Big |.\Big |
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\Big |bibitem\Big |{Zuazua\Big |}\Big | \Big |{\Big |sc\Big | E\Big |.\Big |~Zuazua\Big |}\Big |,\Big | \Big |{\Big |em\Big | Remarks\Big | on\Big | the\Big | controllability\Big | of\Big | the\Big | Schr\Big |\Big |"\Big |{o\Big |}dinger\Big | equation\Big |}\Big |,\Big | \Big | Quantum\Big | control\Big |:\Big | mathematical\Big | and\Big | numerical\Big | challenges\Big |,\Big | \Big | CRM\Big | Proc\Big |.\Big | Lecture\Big | Notes\Big |,\Big | 33\Big |,\Big | Amer\Big |.\Big | Math\Big |.\Big | Soc\Big |.\Big |,\Big | Providence\Big |,\Big | RI\Big |,\Big | 2003\Big |,\Big | pp\Big |.\Big |~193\Big |-\Big |-211\Big |.\Big |
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\Big |end\Big |{document\Big |}\Big | | arXiv |
International Journal of Behavioral Nutrition and Physical Activity
Teaching approaches and strategies that promote healthy eating in primary school children: a systematic review and meta-analysis
Dean A Dudley1,
Wayne G Cotton2 &
Louisa R Peralta2
International Journal of Behavioral Nutrition and Physical Activity volume 12, Article number: 28 (2015) Cite this article
Healthy eating by primary school-aged children is important for good health and development. Schools can play an important role in the education and promotion of healthy eating among children. The aim of this review was to: 1) perform a systematic review of randomised controlled, quasi-experimental and cluster controlled trials examining the school-based teaching interventions that improve the eating habits of primary school children; and 2) perform a meta-analysis to determine the effect of those interventions.
The systematic review was limited to four healthy eating outcomes: reduced food consumption or energy intake; increased fruit and vegetable consumption or preference; reduced sugar consumption or preference (not from whole fruit); increased nutritional knowledge. In March 2014, we searched seven electronic databases using predefined keywords for intervention studies that were conducted in primary schools which focused on the four healthy eating outcomes. Targeted internet searching using Google Scholar was also used. In excess of 200,000 possible citations were identified. Abstracts and full text of articles of potentially relevant papers were screened to determine eligibility. Data pertaining to teaching strategies that reported on healthy eating outcomes for primary school children was extracted from the 49 eligible papers.
Experiential learning strategies were associated with the largest effects across the reduced food consumption or energy intake; increased fruit and vegetable consumption or preference; and increased nutritional knowledge outcomes. Reducing sugar consumption and preference was most influenced by cross-curricular approaches embedded in the interventions.
As with most educational interventions, most of the teaching strategies extracted from the intervention studies led to positive changes in primary school children's healthy eating behaviours. However, given the finite resources, increased overcrowding of school curriculum and capacity of teachers in primary schools, a meta-analysis of this scope is able to provide stakeholders with the best evidence of where these resources should be focused.
The Australian National Health and Medical Research Council (2013) [1] states that optimum nutrition is essential for the healthy growth and development of children. Healthy eating contributes to achieving and maintaining a healthy weight, and provides protection against chronic disease and premature mortality. Conversely, unhealthy eating early in life, in particular the over-consumption of energy-dense, nutrient-poor foods and drinks, as well as physical inactivity and a sedentary lifestyle, are predictors of overweight and obesity [2,3]. There is good evidence that many other non-communicable diseases (such as diabetes, osteoporosis, and hypertension) are also related to unhealthy eating habits and patterns formed during childhood [4]. As such, it is important to establish healthy eating behaviours early, as evidence shows that eating habits and patterns track into adulthood [5,6]. Therefore, childhood is a period where education about healthy eating is essential for establishing healthy eating practices in later years. Schools have been a popular setting for the implementation of health promotion and prevention interventions, as they offer continuous, intensive contact with children and that lifelong health and wellbeing begins with promoting healthy behaviours early in life [7]. School infrastructure, physical environment, policies, curricula, teaching and learning, and staff all have the potential to positively influence child health. Whilst schools have remained a popular infrastructure for health promotion initiatives, teachers will remain the key agent of promoting health and nutrition within schools post-2015 [8]. No systematic review or meta-analysis have been undertaken to date which ascertain the strategies teachers should employ in order to yield maximum effect from their teaching interventions when it comes to fostering healthy eating behaviours in primary school-aged children.
Our aim was to systematically review the evidence related to interventions designed to improve healthy eating habits and patterns of primary school students. Our objectives were to: 1) describe the nature of the interventions that had been conducted (i.e., theories and teaching strategies and approaches); and 2) conduct meta-analyses to determine the effectiveness of these interventions.
This systematic review and meta-analyses report on data extracted and synthesised in 2014 as part of a review project undertaken for the New South Wales (NSW) Department of Education and Communities and the NSW Ministry of Health. The PRISMA (Preferred Reporting Items for Systematic Review and Meta-Analysis) Statement [9] was followed to ensure the transparent reporting of the study.
Interventions types
We included teaching and school-based elementary school interventions delivered by teachers or teacher substitutes that sought to bring about positive nutritional consumption, preference or knowledge change in elementary school children. The following types of teaching and school-based interventions included:
Curriculum initiatives or evaluations
Nutrition-friendly school initiatives
Community programs linked to curricula or delivered by schools (e.g. community gardens)
Health/nutrition education programs related to improving dietary habits
Environmental school change strategies implemented by classroom teachers
Environmental interventions/industry partnerships focused on point-of-purchase consumption linked through classroom based education; this might include campaigns to draw attention to healthier products in school canteens or school lunch choices
Social marketing campaigns
Policies that seek to improve dietary habits of elementary school children (i.e. school board level, provincial/national level).
Acceptable designs for this review included randomised, quasi-experimental and cluster controlled studies conducted in elementary schools (Grades K-6) whereby the primary change agent in the intervention was the classroom teacher (or their teaching substitute). Relevant clusters within studies included individual students, classrooms, schools or communities as the unit of analysis.
Intervention locations had to include elementary schools and/or their immediate community settings. We excluded programs or strategies delivered solely through homes, religious institutions other than schools, non-governmental organisations, primary health care settings, universities, hospitals, outpatient clinics located within hospital settings, commercial programs and metabolic or weight loss clinics.
Outcomes of interest (Healthy eating behaviours)
Our primary outcomes included student consumption, preference and knowledge of nutrient dense foods. Evidence of intervention effects included measures at individual, family, school or community levels. They also included measures of food consumption, preference or knowledge and change in food environments, food disappearance, and food sales (in school cafeterias). Measures of consumption included: diet and food intake records, self-reported and/or reported by parents, teachers or both; food frequency questionnaires/balance sheets; food wastage and plate waste; and micronutrient measures (i.e., biomarkers of exposure to food). Measures of preference included: questionnaires, surveys or self-report instruments that included Likert scales, pairing activities, or self-reported preferences. Measures of knowledge included: questionnaires or tests on food-related knowledge (i.e., Recommended Dietary Intakes, ingredients, nutritional knowledge).
These primary outcomes were then grouped into four dominant healthy eating outcomes that the authors determined aligned with the National Health and Medical Research Council (NHMRC) and their Healthy Eating for Children [10] Guidelines. Our outcomes were therefore;
Food Consumption and Energy Intake–NHMRC Guideline 1 (Limiting energy intake to meet energy needs)
Fruit and Vegetable (FV) Consumption or Preference–NHMRC Guideline 2 (Enjoy a wide variety of nutritious foods)
Reduced Sugar Consumption or Preference (Not from whole fruit)–NHMRC Guideline 3 (Limit intake of foods containing added sugar)
Nutritional Knowledge–NHMRC Guideline 5 (Care for food)
Note: The instruments used and the number studies included in the review and meta-analysis did not allow for segregation of consumption and preference of fruit and vegetables or sugar. We acknowledge that preference for certain food types may have a greater affect on long-term consumption habits.
Outcomes of interest (Teaching strategies)
The primary outcomes of interest included any recognised teaching strategy or articulated approach to teaching that has a known effect on student learning and behaviour. The categorisation for these teaching strategies and approaches to curricula were largely derived from (but not limited to) those articulated in Hattie's synthesis of meta-analysis relating to teaching, learning and student achievement [11].
Our search strategy included: electronic bibliographic databases; grey literature databases; reference lists of key articles; targeted internet searching via Google Scholar; and targeted internet searching of key organisation websites.
We searched the following databases, adapting search terms according to the requirements of individual databases in terms of subject heading terminology and syntax: PUBMED; MEDLINE; the Cochrane Central Register of Controlled Trials (CENTRAL); PsycINFO; ERIC; ScienceDirect; and A + Education. These search terms were based on; 1) participants (e.g. child* OR young people OR youth OR pediatric OR paediatric OR primary school-age* OR elementary school-age* OR primary student* OR elementary student* OR primary school* OR elementary school*); 2) delivery (e.g. teach* OR class* OR health* ed* teach* OR learn* OR teach* polic* OR nutrition ed* OR health* eat*); 3) strategies (e.g. phys* edu* OR health* edu* OR curricul* OR outdoor* OR cook* OR food* OR fruit* OR veg* OR know* OR test*); 4) design (e.g. RCT OR randomi* OR control* OR trial* OR evaluat* OR quasi-exper* OR cluster*). The dates range for search were from database inception to 31st May, 2014.
The search results were then refined to include the full text copies retrieved from these databases and Google Scholar that were published after 1970. These citations were then cross-referenced electronically with 15 reference lists from scoping and systematic review papers in the field of nutrition, education, and health promotion published between 1997 and 2012. A final database and internet search was then conducted to identify studies published between January 2010 (year prior to publication of most recent systematic review) and May 2014.
Screening of citations
Initially duplicate citations were removed from the search by the lead author. The abstract of each citation was then reviewed by a single researcher (DAD) to determine whether it would be included in the systematic review. The full-text articles of all potentially relevant citations were obtained and saved as Adobe-PDF files. Whenever it was uncertain as to whether a citation was appropriate, the full-text copy was obtained. The lead author then screened the citation list. Citations that were deemed ineligible were reviewed by the remaining two authors (WGC, LRP) to determine if any potentially relevant citations were missed, and full-text copies of these citations were then obtained.
Study selection
Following the screening process, full-text articles were then reviewed by the three researchers against the inclusion criteria; if uncertain as whether or not to include an article, the article in question was reviewed again until a final decision was made by majority consensus.
Data was initially extracted from the included studies by the lead researcher from full-text articles and placed in tabulated form (see Table 1). This data included:
Table 1 Studies examining the teaching strategies/approaches used to promote healthy eating to primary school students
Study authors;
Year of publication;
Country (s) of study;
Funding agency of study;
Study design;
Dominant Theoretical Framework used to inform study design
Study sample (Size, Grade, Mean age of participants);
Intervention length;
Whether the intervention was coupled with a physical activity or specially resourced teacher;
Relevant outcome categories
Statistical significance (p value/95%CI)
The effect size of different teaching strategies on each outcome (Cohen's d). Note: If these were not reported in the study and Mean and Standard Deviations could be extracted either directly or indirectly, the Cohen's d was calculated by the lead researcher and verified by the co-authors.
These data were tabulated by the lead author and shared with co-authors for feedback and review. Changes to these interpretations were decided by majority consensus by all three researchers.
The three researchers then reviewed each of the articles independently and each identified the teaching approaches employed in the intervention phase of the studies. Researchers met and cross-referenced their identification of each teaching approach and decided though consensus how each approach would be classified as a wider teaching strategy (if appropriate) that would allow for comparison between studies.
Assessment of methodological quality
Included articles were also assessed for methodological quality using a 10-item quality assessment scale derived from van Sluijs and colleagues [12] (see Table 2). For each included article, three reviewers independently assessed whether the assessed item was present or if the assessed item was absent. Where an item was insufficiently described it was allocated an absent score. Agreement between reviewers for each article was set a priori at 80% [12]. That is, for each article, reviewers were required to agree that the items were either present or absent for 8 of the 10 items. In the case of less than 80% agreement, consensus was reached by further discussion. Results for the assessment of methodological quality are reported in Table 3.
Table 2 Methodological quality assessment items (Adapted from van Sluijs et al. 2007) [12 ]
Table 3 Methodological quality and risk of bias assessment
Effect sizes are the preferred metric for estimating the magnitude of effect of an intervention because they make possible between study as well as within study comparisons [13]. Cohen's d, the effect-size metric constituting the focus of this meta-analysis, is one of the most widely used measures of magnitude of effect and commonly used in educational meta-analyses [11]. The formula for calculating Cohen's d is:
$$ d=\left({M}_1-{M}_2\right)/S{D}_p, $$
where M 1 is the mean of one group of study participants, M 2 is the mean of a second group of study participants, and SD P is the pooled standard deviation for both groups of study participants.
In instances where the groups have been given different learning experiences (e.g. an intervention), d is a measure of the magnitude of effect of the experience on the group receiving the enhanced teaching and learning experience. In cases where SD was not reported but SE (Standard Error) was, SE was converted to SD using the following formula:
$$ SD=SE\ \mathrm{x}\ \sqrt{\mathrm{N}} $$
As Cohen's d accounts for sample size, mean effect sizes for the purposes of the meta-analysis were calculated as follows:
$$ {M}_d={\displaystyle \sum d/{\mathrm{N}}_{\mathrm{s}}} $$
M d is the mean Cohen's d calculated by the sum of all d values and divided by the number of studies (N s ) from which a d value could be extracted for that outcome.
Data pertaining to each study were initially collated and described in a narrative summary (see Table 1). To facilitate comparison between the effect of teaching strategies/approaches, studies were divided according to their outcome measure as follows: Decreased food consumption/energy intake, increased FV consumption/preference, decreased sugar consumption/preference, and increased nutritional knowledge. Meta-analyses were conducted using the standardised mean difference approach (Cohen's d) regardless of their statistical significance where at least two studies existed for a particular outcome measure and sufficient statistical data were reported to allow such synthesis to occur.
Studies incorporated into the meta-analyses included a comparison between teaching strategies/approaches and reported post-test/follow-up values or change scores along with measures of distribution (i.e. mean and standard deviation). For studies that included post-test and follow-up assessments, the assessments completed at the end of the study period (i.e., follow-up) were included in the meta-analyses. The standardized effect sizes were interpreted as minimal (<.02), small (0.2), medium (0.5), and large (0.8) [14]. Analyses also considered whether they represented an effective investment in education given the average effect size of most educational interventions is d = 0.4 [11].
The study selection process is shown in Figure 1. It initially retrieved in excess of 200,000 possible citations. We refined searches to include only full text copies available online and published after 1970 in each of the databases and in Google Scholar reducing this to 18,100 possible citations. These citations were then cross-referenced electronically with reference lists from scoping and systematic review papers published in the field of nutrition, education, and health promotion (n = 15) [15-29] published between 1997 and 2012 that yielded 454 likely studies. A final database and internet search was then conducted to identify studies published between January 2010 (year prior to publication of most recent systematic review) and May 2014. This revealed an additional 23 possible citations totalling 487 publications that were considered for review.
Flowchart of study selection.
These 487 publications were then reviewed based on abstract and excluded if they were not conducted in primary schools or on primary school-aged children. This reduced the number of studies to 233. Studies were then excluded if they were not: a) randomised controlled trials; b) quasi-experimental studies; or c) cluster-controlled trials. This left 55 studies. On review of the full-text paper, another 6 studies were excluded for not meeting the inclusion criteria (i.e. conducted in a laboratory setting) or being a duplicate study. The final 49 studies were all in the form of peer-reviewed journal publications.
To ensure a complete review of the relevant literature is given, all 49 of the included articles are presented in Table 1. Specifically, the table outlines the details of the studies, including author(s), title, year, location, design and stated dominant theoretical framework, target population, and types of outcomes measured. The year of publication for included articles ranged from 1973 to 2011.
Study and intervention characteristics
The final 49 studies included one randomised controlled trial, 13 quasi-experiential studies and 35 cluster-controlled trials. These studies captured data from 38001 primary school children in 13 different countries. Data capable for inclusion in the meta-analyses came from 20234 (53%) participants. All but one country (Trinidad and Tobago) included in these studies were member nations of the Organization for Economic Co-operation and Development (OECD). Only 27 of the 49 studies reported the theoretical frameworks used to inform their intervention design. Whilst some studies reported multiple theoretical approaches (see Table 1), Social Cognitive Theory was the most frequently used theoretical framework and was reported in 16 of 27 studies.
Teaching strategies/approaches
There were eight dominant teaching strategies or approaches to teaching exhibited across the 49 studies that addressed the pre-determined areas of healthy eating for primary school students (i.e. food consumption/energy intake, fruit and vegetable consumption or preference, sugar consumption or preference, and nutritional knowledge). Some studies included more than one of these teaching strategies/approaches in their intervention group. The dominant teaching strategies/approaches were: 1) Enhanced curriculum approaches (i.e. speciality nutrition education programs beyond existing health curricula delivered by teachers or specialists) (n = 29); 2) cross-curricular approaches (i.e. nutrition education programs that were delivered across two or more traditional primary school subjects) (n = 11); 3) parental involvement (i.e. programs requiring active participation or assistance from a parent within or outside the school environment) (n = 10); 4) experiential learning approaches (i.e. school/community garden, cooking and food preparation activities) (n = 10); 5) contingent reinforcement approaches (i.e. rewards or incentives given to students in response to desired behaviours) (n = 7); 6) literary abstraction approaches (i.e. literature read by/to children whereby a character promotes/exemplifies positive behaviours) (n = 3); 7) games-based approaches (i.e. board/card games played by students at school designed to promote positive behaviour and learning of new knowledge) (n = 2); and 8) web-based approaches (i.e. internet-based resources or feedback mechanisms that could be accessed by students at home or school) (n = 2).
The results of the systematic review indicate that several dominant evidence-based approaches to teaching healthy eating in the randomised controlled trial, quasi-experimental and cluster controlled trial literature. In order to determine the strength of the evidence for these approaches, they are analysed against each of the major outcomes used to determine healthy eating and if the study achieved p-values of p < .05 for 50% of the studies, the magnitude of M d (i.e. minimal, small, medium, large) and/or if M d > .40. The decision to use an effect size of M d > .40 is based on Hattie's Zone of Desired Effects reside above this hinge point [11] and therefore have the greatest influence and represent the best investment for improving educational outcomes.
Food consumption and energy intake
Eleven studies reported on outcomes of food consumption and the overall energy intake of primary school-aged children. Curriculum-based approaches were the most popular (seven studies) and reported achieving statistical significance of p < .05 or better across nine studies reducing food consumption or energy intake outcomes. However, researchers were able to calculate effect sizes across six of the reported outcomes and found that four showed minimal or no effect, one had a negative effect and one reported a small effect size. The mean effect size of curriculum-based approaches is minimal (M d = 0.12) and would suggest that curriculum-based approaches alone are not the best influence on reducing food consumption or energy intake.
Three studies utilising experiential learning approaches (i.e., school/community gardens, cooking lessons and food preparation) reported on outcomes associated with reducing food consumption and energy intake. Two of these studies reported achieving statistical significance of p < .05 or better for at least one food consumption or energy intake variable. Effect sizes could be calculated on three of the reported outcomes from two studies. Two large effect sizes were recorded and the other showed no effect. Whilst there were only a small number of effect sizes that were able to be calculated based on the reporting method in these studies, the mean effect size was M d = 1.31 and within the Zone of Desired Effects. These approaches warrant greater investigation to reduce the amount of variance in the calculated effect but show promise in their ability to reduce food consumption and energy intake.
Fruit and vegetable (FV) consumption or preference
In terms of FV consumption or preference, curriculum-based approaches were again the most popular. 60% of curriculum-based approaches found statistically significant (p < .05) improvements in FV consumption or preference among primary school-aged children. However, it is important to note that many of the studies that used curriculum-based approaches (especially those with stronger p values) also coupled their interventions with secondary approaches (e.g., experiential-learning, parental-involvement). Given the way in which data was reported in these studies, it is difficult at this stage to determine the degree to which curriculum-based approaches alone contributed to statistical significance.
Of the 30 effect sizes that were calculated by the researchers, 33% had a medium to large effect and a further 23% had a small effect size. The mean effect size for curriculum-based approaches was M d = 0.45 indicating that having a nutrition curriculum delivered in primary schools makes an important investment in improving FV consumption or preference based on the educational hinge-point of effect sizes described by Hattie [11]. All but one study that was included in the analysis appeared to be based on behavioural, mastery, or didactic approaches and curricula models. The study driven by a socio-cultural perspective of health [30] had only 33 participants and effect sizes ranging from-0.26 to 1.04 for a range of different FV consumption or preference behaviours.
Experiential-learning approaches were used in eight studies to improve FV consumption or preference in primary school children and proved to be very effective with 75% of these types of studies yielding statistical significance at p < .05 or better. Of the 11 effect sizes that were calculated by the researchers, 45% had a large effect and the remaining 55% had a minimal effect size. However, the mean effect size for experiential-learning approaches that included school/community gardens, cooking skills, or food preparation was M d = 0.68, indicating experiential-learning approaches were within the Zone of Desired Effects [11] for improving FV consumption or preference in primary school children.
Cross-curricular approaches (i.e., learning experiences taught across two or more learning areas/subjects) to improving FV consumption or preference in primary school children also proved to very effective. Of the 10 studies using cross-curricular approaches, 90% of these yield statistical significance at p < .05 or better and of the 6 effect sizes calculated by the researchers, 50% had large effect sizes and the remaining 50% had a small or medium effect size. Whilst there were only a small number of effect sizes that were able to be calculated based on the reporting method in these studies, the mean effect size was M d = 0.63, which was within the Zone of Desired Effects.
Four studies used a contingent reinforcement (i.e., reward for behaviour) approach in promotion of FV consumption or preference among primary school children. All four (100%) of these studies reported achieving statistical significance of at least p < .05. There were six effect sizes reported across only two studies [31,32] and four of these effect sizes (67%) were considered large and two (33%) were considered minimal. Based on these two studies, the average effect size for contingent reinforcement in promoting FV consumption or preference is M d = 1.34. More studies are needed in order to ascertain an average effect size with less variance, however, based on available data this approach is well above M d = 0.4 with strong statistical significance in every study indicates it is a worthwhile investment strategy in improving FV consumption or preference among primary school children.
Parental involvement was incorporated into 10 studies that reported against 23 FV consumption or preference outcomes in primary school children. 91% of the outcomes reported against were statistically significant at the p < .05 level. The researchers were able to calculate 14 effect sizes in five of the studies. The results were varied with three large, two medium, three small, two minimal and four negative effect sizes being calculated. The mean effect was M d = 0.39 that was just below the Zone of Desired effects however it is worthwhile noting that no parent involvement approach was ever 'stand-alone'. They all included elements of enhanced curriculum, cross-curricular, experiential learning or web-based support.
Sugar consumption or preference (not from whole fruit)
Enhanced curriculum approaches (mainly based on behavioural or social cognitive theories) in primary schools provided 10 studies for reducing sugar consumption or preference in students however only three yielded statistical significance of p < .05 or better for reducing any sugar-laden beverage (SLB), fruit juice or carbohydrate consumption. Six effect sizes were calculated from these studies that showed one large, one small and four minimal effect sizes. The mean effect size of curriculum approaches for reducing sugar consumption however was only M d = 0.28 suggesting that greater investment beyond curriculum is required to make a substantial difference in reducing the sugar consumption of primary school children.
Cross-curricular approaches were reported in two studies [33,34] in reducing SLB or fruit juice consumption. Both studies reported statistically significant reductions in both SLB and fruit juice consumption at p < .05 or better. Taylor et al. [34] reported two minimal effect sizes whilst James et al. [35] reported a large effect size. The mean effect size for cross-curricular approaches at reducing SLB or fruit juice consumption was M d = 0.42. This was within the Zone of Desired Effects [11], but more investigation may be required given the small number of studies included in the analysis.
Nutritional knowledge
There were 12 studies that adopted enhanced curricula approaches to improving the nutritional knowledge of primary school children. There were 13 nutritional knowledge outcomes that achieved a statistically significant improvement of p < .05 or better. In fact, 8 of the 13 studies reported statistical significance of p < .001. Researchers were able to calculate 7 effect sizes (3 × large, 1 × medium, 3 × minimal) with the mean effect size being M d = 0.75. This indicates that quality curriculum interventions (largely based on behavioural or social cognitive learning theory) are capable of achieving improvements in student nutritional knowledge with the Zone of Desired Effects [11].
An experiential learning-approach was adopted in four studies and all reported achieving statistical significance of p < .05 across seven nutritional knowledge-related outcomes. The researchers were able to calculate effect sizes for six of them and found five large and one minimal effect size. The mean effect size for the experiential learning approaches to nutritional knowledge was M d = 1.35 indicating this approach is a particularly strong evidence-based strategy for improving the nutritional knowledge of primary school-aged children
This meta-analysis of school-based teaching interventions that have focused on improving the eating habits of primary school children found that experiential learning approaches had the greatest effect on reducing the food consumption, energy intake and nutritional knowledge of primary school children, and a smaller effect on primary school children's FV consumption or preference. The other strategies that had a smaller effect on improving primary school children's nutritional knowledge and reducing sugar consumption or preferences were cross-curricular approaches and quality curriculum interventions, respectively. In regards with improving primary school children's FV consumption or preferences, both cross-curricular and quality curriculum interventions were effective.
In light of these findings, it is important to note that the high levels of heterogeneity among the included primary school healthy eating programs, does not make it possible to make firm conclusions. However, the findings have been supported in other literature, with experiential learning strategies, such as garden-enhanced learning strategies, positively influencing vegetable preferences and consumption among primary school children, which has been found to be the strongest predictor of future consumption [36-39]. Similar to this review, Langellotto & Gupta [39], who used meta-analytic techniques, found that school gardens and associated teaching strategies increased vegetable consumption in children, whereas the impacts of nutrition education programs were marginal or non-significant. There are two possible reasons for these findings: 1) school gardens increase access to vegetables; and 2) gardening decreases children's reluctance to try new foods. Birch and colleagues [38] have also stated that in order to improve primary school children's healthy food preferences, experiences and strategies need to increase availability and accessibility to increase exposure to those foods that will then affect their willingness to taste.
Whilst some studies report FV consumption or preference independently of each other, this tends to be the exception rather than the rule of reporting FV consumption or preference in primary school-based studies. Future studies should seek to promote, analyse and report vegetable consumption independent of fruit consumption to ascertain what physiological and behavioural effects this may have on students and findings of the study. This is because excessive consumption of fruit-based sugars (i.e. consuming fructose >50 g/d) may be one of the underlying aetiologies of Metabolic Syndrome and Type 2 Diabetes [35].
This study has some important considerations with regard to its generalizability. The target population were the students attending primary schools from any country around the world but all the studies bar one [40] were conducted in nations of the OECD. As such, they represent some of the most developed and advanced economies on the planet and should be taken into serious consideration when seeking to generalise these findings. Of the 49 studies analysed, more than half (n = 28) were conducted in the United States followed by the United Kingdom (n = 7). This may be attributed of the growing percentage of children in the USA and UK with non-communicable diseases attributed to diet-related factors [4,41]. It may also be indicative of the capacity of advanced economies, such as the USA and UK, to conduct empirically robust studies in primary school settings [42].
There are several strengths of this systematic review and meta-analysis. First, this is the first known paper to systematically extract specific teaching strategies and approaches that facilitate the healthy eating of primary school children. As such, we conducted a systematic review using broad search terms to increase the probability of identifying all eligible publications, which yielded a well-sized (k = 49) evidence base. Second, the method of meta-analysis allowed for these strategies to be considered against other nutritional as well as the educational meta-analytic literature. Third, teaching strategies and approaches were reliably coded using schema of existing evidence of 'what works' in educational settings [11].
There were a few limitations associated with this review. The heterogeneity of primary school healthy eating interventions is large. This fact alone limited our ability to measure the effectiveness of each teaching strategy in the multi-faceted nutrition education programs. Moreover, it is possible that some strategies are commonly clustered with others, thus our findings should be considered carefully in terms of these strategies having similar effects when implemented on their own. Given that all the articles were identified from the peer-reviewed literature, there is some possibility of publication bias on the nature of evidence available to inform the review. Publication bias by particular journals, or more specifically the inability and discouragement of publishing articles that report negative results, may distort conclusions reached. Further, due to all but one study were conducted in OECD countries, findings from this systematic review and meta-analyses should be limited to informing decision making of stakeholders in those of similar nations.
Most teaching strategies extracted from intervention studies lead to positive changes in primary school children's nutritional knowledge and behaviours. However, the most effective strategies for facilitating healthy eating in primary school children are enhanced curricula, cross-curricula and experiential learning approaches. Other strategies that showed some promising effect, but need to be further investigated include contingent reinforcement and parental involvement approaches.
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This study was brokered by the Sax Institute for the NSW Department of Education and Communities and the NSW Ministry of Health. The funding agencies instigated the research questions but had no influence (either directly or indirectly) over the search strategy, included studies, data analysis or findings.
School of Education, Faculty of Human Sciences, Macquarie University, Sydney, NSW, Australia
Dean A Dudley
Faculty of Education and Social Work, University of Sydney, Sydney, NSW, Australia
Wayne G Cotton & Louisa R Peralta
Wayne G Cotton
Louisa R Peralta
Correspondence to Dean A Dudley.
DAD, WGC and LRP conceptualized and designed the study. DAD, WGC and LRP collected the data. DAD conducted the statistical analyses. DAD, WGC and LRP contributed to the writing of the manuscript. All authors read and approved the final manuscript.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Dudley, D.A., Cotton, W.G. & Peralta, L.R. Teaching approaches and strategies that promote healthy eating in primary school children: a systematic review and meta-analysis. Int J Behav Nutr Phys Act 12, 28 (2015). https://doi.org/10.1186/s12966-015-0182-8
Energy intake | CommonCrawl |
\begin{document}
\title{On sets in ${\mathbb R}^d$ with DC distance function} \author{Du\v san Pokorn\'y}\author{Lud\v ek Zaj\'i\v cek}
\thanks{The research was supported by GA\v CR~18-11058S}
\begin{abstract}
We study closed sets $F \subset {\mathbb R}^d$ whose distance function $d_F\coloneqq {\rm dist}\,(\cdot,F)$ is DC
(i.e., is the difference of two convex functions on ${\mathbb R}^d$). Our main result asserts that if $F \subset {\mathbb R}^2$
is a graph of a DC function $g:{\mathbb R}\to {\mathbb R}$, then $F$ has the above property. If $d>1$, the same holds
if $g:{\mathbb R}^{d-1}\to {\mathbb R}$ is semiconcave, however the case of a general DC function $g$ remains open. \end{abstract}
\email{[email protected]} \email{[email protected]}
\keywords{DC function, Distance function, Set of positive reach, Semiconcave function} \subjclass[2010]{26B25} \date{\today} \maketitle
\section{Introduction}
Let $F \neq \varnothing$ be a closed subset of ${\mathbb R}^d$ and let $d_F\coloneqq {\rm dist}\,(\cdot,F)$ be its distance
function. Recall that a function on ${\mathbb R}^d$ is called DC, if it is the difference of two convex functions. It is well-known (see, e.g., \cite[p. 976]{BB}) that
\begin{equation}\label{odist}
\text{ the function $(d_F)^2$ is DC but $d_F$ need not be DC.} \end{equation}
However, the distance function of some interesting special $F\subset {\mathbb R}^d$
is DC; it is true for example for $F$ from Federer's class of sets with positive reach, see \eqref{prdd}.
Our article was motivated by \cite{BB} and by the following question which naturally arises
in the theory of WDC sets (see \cite[Question 2, p. 829]{FPR} and \cite[10.4.3]{Fu2}).
{\bf Question.} \
Is $d_F$ a DC function if $F$ is a graph of a $DC$ function $g: {\mathbb R}^{d-1} \to {\mathbb R}$?
Note that WDC sets form a substantial generalization of sets
with positive reach and still admit the definition of curvature measures (see \cite{PR} or \cite{Fu2}) and $F$ as in Question is a natural example of a WDC set in ${\mathbb R}^d$.
Our main result (Theorem \ref{hlav}) gives the affirmative answer to Question in the case $d=2$;
the case $d>2$ remains open. However, known results relatively easily imply that the answer
is positive if $g$ in Question is semiconcave (Corollary \ref{grsemi}).
In \cite{PZ} we show that our main result has some interesting consequences for WDC subsets of ${\mathbb R}^2$, in particular that these sets have DC distance functions.
In Section 2 we recall some notation and needed facts about DC functions. In Section 3 we prove
our main result (Theorem \ref{hlav}). In last Section 4, we prove a number of further results on the system of sets in ${\mathbb R}^d$ which have DC distance function, including Corollary \ref{grsemi} mentioned above.
We were not able to prove a satisfactory complete characterisation of sets $F\subset \mathbb R^2$ with DC distance function, but we believe that our methods and results should lead to such a characterisation. However, in our opinion, the case of $F\subset \mathbb R^d$, $d\geq 3$, needs some new ideas.
\section{Preliminaries}
In any vector space $V$, we use the symbol $0$ for the zero element. We denote by $B(x,r)$ ($U(x,r)$) the closed (open) ball with centre $x$ and radius $r$.
The boundary and the interior of a set $M$ are denoted by $\partial M$ and $\mathrm{int} M$, respectively. A mapping is called $K$-Lipschitz if it is Lipschitz with a (not necessarily minimal) constant $K\geq 0$.
In the Euclidean space ${\mathbb R}^d$, the norm is denoted by $|\cdot|$ and the scalar product by $\langle \cdot,\cdot\rangle$. By $S^{d-1}$ we denote the unit sphere in ${\mathbb R}^d$.
If $x,y\in{\mathbb R}^d$, the symbol $[x,y]$ denotes the closed segment (possibly degenerate). If also
$x \neq y$, then $l(x,y)$ denotes the line
joining $x$ and $y$.
The distance function from a set $A\subset {\mathbb R}^d$ is $d_A\coloneqq {\rm dist}\,(\cdot,A)$ and
the metric projection of $z\in {\mathbb R}^d$ to $A$ is
$\Pi_A(z)\coloneqq \{ a\in A:\, {\rm dist}\,(z,A)=|z-a|\}$.
If $f$ is defined in ${\mathbb R}^d$, we use the notation $f'_+(x,v)$ for the one sided directional derivative of $f$ at $x$ in direction $v$.
Let $f$ be a real function defined on an open convex set $C \subset {\mathbb R}^d$. Then we say
that $f$ is a {\it DC function}, if it is the difference of two convex functions. Special DC
functions are semiconvex and semiconcave functions. Namely, $f$ is a {\it semiconvex} (resp.
{\it semiconcave}) function, if there exist $a>0$ and a convex function $g$ on $C$ such that
$$ f(x)= g(x)- a \|x\|^2\ \ \ (\text{resp.}\ \ f(x)= a \|x\|^2 - g(x)),\ \ \ x \in C.$$
We will use the following well-known properties of DC functions.
\begin{lem}\label{vldc} Let $C$ be an open convex subset of ${\mathbb R}^d$. Then the following assertions hold. \begin{enumerate} \item[(i)]
If $f: C\to {\mathbb R}$ and $g: C\to {\mathbb R}$ are DC, then (for each $a\in {\mathbb R}$, $b\in {\mathbb R}$) the functions $|f|$, $af + bg$,
$\max(f,g)$ and $\min(f,g)$ are DC.
\item[(ii)]
Each locally DC function $f:C \to {\mathbb R}$ is DC.
\item[(iii)] Each DC function $f:C \to {\mathbb R}$ is
Lipschitz on each compact convex set $Z\subset C$.
\item[(iv)] Let $f_i: C \to {\mathbb R}$, $i=1,\dots,m$, be DC functions. Let $f: C \to {\mathbb R}$ be a continuous function
such that $f(x) \in \{f_1(x),\dots,f_m(x)\}$ for each $x \in C$. Then $f$ is DC on $C$.
\item[(v)] Each $\mathcal C^2$ function $f:C\to\mathbb R$ is DC. \end{enumerate} \end{lem} \begin{proof} Property (i) follows easily from definitions, see e.g. \cite[p. 84]{Tuy}. Property (ii) was proved in \cite{H}.
Property (iii) easily follows from the local Lipschitzness of convex functions. Assertion (iv) is a special case of \cite[Lemma 4.8.]{VeZa} (``Mixing lemma'').
To prove (v) observe that (e.g. by \cite[Prposition~1.1.3~(d)]{CS}) each $C^2$ function is locally semiconcave and therefore locally DC, hence, DC by (ii). \end{proof}
By well-known properties of convex and concave functions, we easily obtain that each locally DC function $f$ on an open set $U \subset {\mathbb R}^d$ has all one-sided directional derivatives finite and \begin{equation}\label{zlose}
g_+'(x,v) + g_+'(x,-v) \leq 0,\ \ x \in U, v\in {\mathbb R}^d,\ \ \ \text{if}\ \ g\ \ \text{is locally semiconcave on}\ \ U. \end{equation}
Recall that if $\varnothing \neq A\subset {\mathbb R}^d$ is closed, then $d_A$ need not be DC; however (see, e.g., \cite[Proposition 2.2.2]{CS}), \begin{equation}\label{loksem} \text{$d_A $ is locally semiconcave (and so locally DC) on ${\mathbb R}^d \setminus A$.} \end{equation}
\section{Main result}\label{graf}
In the proof of Theorem \ref{hlav} below we will use the following simple ``concave mixing lemma''. \begin{lem}\label{comix} Let $U \subset {\mathbb R}^d$ be an open convex set and let $\gamma: U \to {\mathbb R}$ have finite one-sided directional
derivatives $\gamma_+'(x,v)$, ($x \in U, \ v\in {\mathbb R}^d$). Suppose that \begin{equation}\label{nezlo}
\gamma_+'(x,v) + \gamma_+'(x,-v) \leq 0,\ \ x \in U, v\in {\mathbb R}^d, \end{equation}
and that \begin{multline}\label{pokrc} \text{ $\operatorname{graph} \gamma$ is covered by graphs of a finite number}\\ \text{of concave functions defined on $U$.}
\end{multline}
Then $\gamma$ is a concave function. \end{lem} \begin{proof} Since $\gamma$ is clearly concave if each function $t\mapsto \gamma(a + tv),\ (a\in C, v \in S^{d-1})$
is concave on its domain, it is sufficient to prove the case $d=1$, $C=(a,b)$. Set $h(x)\coloneqq -\gamma(x),\ x \in (a,b)$; we need to prove that $h$ is convex. Observe that
\eqref{nezlo} easily implies the condition \begin{equation}\label{prle} h'_-(x)\leq h'_+(x),\ \ \ x \in (a,b). \end{equation}
and \eqref{pokrc} implies that there exists a finite set $\{h_{\alpha}:\ \alpha\in A\}$ of convex functions on
$(a,b)$ such that $\operatorname{graph} h \subset \bigcup\{\operatorname{graph} h_{\alpha}:\ \alpha \in A\}$. To prove the convexity of $h$, it is sufficient to show that the function $h'_+$ is nondecreasing on
$(a,b)$ (see e.g. \cite[Chap. 5, Prop. 18, p. 114]{Roy}); equivalently (it follows e.g. from \cite[Chap. IX, \S7, Lemma 1, p. 266]{Na}) to prove that \begin{multline}\label{movbo} \forall x_0 \in (a,b)\ \exists \delta>0\ \forall x: \ (x \in (x_0,x_0+\delta) \Rightarrow h'_+(x)\geq h'_+(x_0)) \\ \wedge\ \ (x \in (x_0-\delta,x_0) \Rightarrow h'_+(x)\leq h'_+(x_0)). \end{multline} So suppose, to the contrary, that \eqref{movbo} does not hold; then there exists a sequence
$x_n \to x_0$ such that either \begin{equation}\label{nale} x_n < x_0\ \ \text{and}\ \ h'_+(x_n) > h'_+(x_0)\ \ \text{for each}\ \ n \in \mathbb N \end{equation}
or \begin{equation}\label{napr} x_n > x_0\ \ \text{and}\ \ h'_+(x_n) < h'_+(x_0)\ \ \text{for each}\ \ n \in \mathbb N. \end{equation} Since $h$ is clearly continuous, each set $F_{\alpha}\coloneqq \{x \in (a,b):\ h_{\alpha}(x)=h(x)\}$,
$\alpha \in A$, is closed in $(a,b)$. Since $A$ is finite, it is easy to see that for each $n \in \mathbb N$ there exists $\alpha(n) \in A$ such that $x_n \in F_{\alpha(n)}$ and $x_n$ is a right accumulation
point of $ F_{\alpha(n)}$. Using finiteness of $A$ again, we can suppose that there exists $\alpha \in A$
such that $\alpha(n)= \alpha$, $n \in \mathbb N$ (otherwise we could consider a subsequence of $(x_n)$).
Now suppose that \eqref{nale} holds. Since $x_n \in F_{\alpha},\ n=0,1,\dots,$ we obtain
that $h'_+(x_n)= (h_{\alpha})_+'(x_n),\ n \in \mathbb N$, and $h'_-(x_0) = (h_{\alpha})_-'(x_0)$. Using also the convexity of $h_{\alpha}$ and \eqref{prle}, we obtain $$ h'_+(x_n)= (h_{\alpha})_+'(x_n) \leq (h_{\alpha})_-'(x_0) = h'_-(x_0) \leq h'_+(x_0),$$
which contradicts \eqref{nale}. Since the case when \eqref{napr} holds is quite analogous,
neither \eqref{nale} nor \eqref{napr} is possible and so we are done. \end{proof} We will need also the following easy lemma.
\begin{lem}\label{kofu}
Let $V$ be a closed angle in ${\mathbb R}^2$ with vertex $v$ and measure $0<\alpha< \pi$.
Then there exist an affine function $A$ on ${\mathbb R}^2$ and a concave function $\psi$ on ${\mathbb R}^2$ which is Lipschitz with constant
$\sqrt{2}\tan (\alpha/2)$ such that $|z-v| + \psi(z)= A(z),\ z \in V$.
\end{lem}
\begin{proof}
We can suppose without any loss of generality that $v= (0,0)$ and
$$ V= \{(x,y):\ x \geq 0, |y| \leq x\, \tan(\alpha/2)\}.$$
Then $|z-v| = \sqrt{x^2+y^2}$ for $z=(x,y)$. Define the convex function
$$\varphi(x,y)\coloneqq \sqrt{x^2+y^2} - x,\ (x,y)\in V.$$
We will show that
\begin{equation}\label{jelip}
\varphi\ \ \text{is Lipschitz with constant}\ \ \sqrt{2}\tan (\alpha/2).
\end{equation}
To this end estimate, for $(x,y) \in \mathrm{int}\, V$,
$$ \left|\frac{\partial \varphi}{\partial x}(x,y)\right| = \left| \frac{x}{\sqrt{x^2+y^2}}-1\right|
= \frac{y^2}{ ( x + \sqrt{x^2+y^2}) \sqrt{x^2+y^2} } \leq \frac{|y|}{x} \leq \tan (\alpha/2), $$
$$ \left|\frac{\partial \varphi}{\partial y}(x,y)\right| =
\frac{|y|}{ \sqrt{x^2+y^2} } \leq \frac{|y|}{x} \leq \tan (\alpha/2). $$
Thus $|\operatorname{grad} \varphi (x,y)| \leq \sqrt{2}\tan (\alpha/2)$ for $(x,y) \in \mathrm{int}\, V$
and \eqref{jelip} follows. So $\varphi$ has a convex extension $\tilde \varphi$ to ${\mathbb R}^2$
which is also Lipschitz with constant $\sqrt{2}\tan (\alpha/2)$ (see, e.g., \cite[Theorem 1]{CM}).
Now we can put $\psi\coloneqq - \tilde \varphi$, since $\sqrt{x^2+y^2} + \psi(x,y) =x=: A(x,y),\ (x,y) \in V$. \end{proof}
\begin{thm}\label{hlav}
Let $f: {\mathbb R} \to {\mathbb R}$ be a DC function. Then the distance function $d\coloneqq {\rm dist}\,(\cdot, \operatorname{graph} f)$
is DC on ${\mathbb R}^2$.
\end{thm}
\begin{proof}
By \eqref{loksem}, $d$ is locally DC on
${\mathbb R}^2 \setminus \operatorname{graph} f$. So, by Lemma \ref{vldc} (ii),
it is sufficient to prove that, for each $z \in \operatorname{graph} f$, the distance function $d$ is DC
on a convex neighbourhood of $z$. Since we can clearly suppose that $z= (0, f(0))$,
it is sufficient to prove that
\begin{equation}\label{suff}
\text{$d$ is DC on $U\coloneqq U((0,f(0)), 1/10)$.}
\end{equation}
Write $f= g - h$, where $g$, $h$ are convex functions on ${\mathbb R}$. For each $n \in \mathbb N$, consider
the equidistant partition $D_n= \{x^n_0= -1 < x^n_1< \dots
< x^n_n=1\}$ of $[-1,1]$. Let $g_n$, $h_n$ be the piece-wise linear function on $[-1,1]$
such that $g_n(x^n_i) = g(x^n_i)$, $h_n(x^n_i) = h(x^n_i)$ ($0\leq i \leq n$) and
$g_n$, $h_n$ are affine on each interval $[x_{i-1}, x_i]$ ($1\leq i \leq n$).
Put $f_n\coloneqq g_n - h_n$ and $d_n\coloneqq {\rm dist}\,(\cdot, \operatorname{graph} f_n)$. Choose $L > 0$ such that both
$g\restriction_{[-1,1]}$ and $h\restriction_{[-1,1]}$ are $(L/2)$-Lipschitz and observe
that all $g_n$, $h_n$, $f_n$
are $L$-Lipschitz. Since $f_n$ uniformly converge to $f$ on $[-1,1]$, we easily
see that $d_n \to d$ on $\overline{U}$.
Choose an integer $n_0$ such that
\begin{equation}\label{nnula}
n_0 \geq 6 \ \ \text{and}\ \ |f_n(0)- f(0)| < \frac{1}{10}\ \ \text{for each}\ \ n \geq n_0.
\end{equation}
We will prove that there exist $L^* >0$ and concave functions $c_n$ ($n\geq n_0$) on
$\overline{U}$
such that
\begin{equation}\label{lcn}
\text{each}\ \ c_n\ \ \text{is Lipschitz with constant}\ \ L^*\ \ \text{and}
\end{equation}
\begin{equation}\label{dnpcn}
c_n^*\coloneqq d_n + c_n\ \ \text{is concave on}\ \ \overline{U}.
\end{equation}
Then we will done, since \eqref{lcn} and \eqref{dnpcn} easily imply \eqref{suff}. Indeed,
we can suppose that $c_n((0,f(0)))=0$ and, using Arzel\`a-Ascoli theorem, we obtain that
there exists an increasing sequence of indices $(n_k)$ such that $c_{n_k} \to c$, where
$c$ is a continuous concave function on $\overline{U}$. So
$d_{n_k} + c_{n_k} \to d+c=:c^*$ on $\overline{U}$. Using \eqref{dnpcn}, we obtain that $c^*$ is concave and thus $d=c^*-c$ is DC
on $U$.
To prove the existence of $L^*$ and $(c_n)$, fix an arbitrary $n\geq n_0$. For brevity
denote $\Pi\coloneqq \Pi_{\operatorname{graph} f_n}$ and
put $x_i\coloneqq x^n_i$,\
$z_i\coloneqq (x_i, f_n(x_i)),\ i=0,\dots,n$. For $i=1,\dots,n-1$, let $0 \leq \alpha_i < \pi$
be the angle between the vectors $z_i - z_{i-1}$ and $z_{i+1}- z_i$.
Denote
$$s_i: = \frac{f_n(x_{i+1})- f_n(x_i)}{x_{i+1}-x_i}\ \text{and}\ \beta_i\coloneqq \arctan s_i,\ \
i=0,\dots,n-1.$$
Then clearly $\alpha_i= |\beta_i - \beta_{i-1}|$. One of the main ingredients of the present proof is the easy fact
that
\begin{equation}\label{ssi}
\sum_{i=1}^{n-1} |s_i- s_{i-1}| \leq 4L.
\end{equation}
It immediately follows from the well-known estimate of (the ``convexity'') $K_a^b(f_n)$ (see \cite[p. 24, line 5]{RoVa}).
To give, for completeness, a direct proof, denote
$$ \tilde s_i: = \frac{g_n(x_{i+1})- h_n(x_i)}{x_{i+1}-x_i},\ \ s^*_i: = \frac{h_n(x_{i+1})- h_n(x_i)}{x_{i+1}-x_i},
\ \ i=0,\dots,n-1,$$
and observe that the finite sequences $(\tilde s_i)$, $(s^*_i)$ are nondecreasing. Consequently
$$ \sum_{i=1}^{n-1} |\tilde s_i- \tilde s_{i-1}| = \tilde s_n - \tilde s_1 \leq 2L \ \ \text{and}\ \
\sum_{i=1}^{n-1} | s^*_i- s^*_{i-1}| = s^*_n - s^*_1 \leq 2L.
$$
Since $s_i= \tilde s_i - \tilde s^*_i$, \eqref{ssi} easily follows.
Since
$$ \alpha_i = |\beta_i - \beta_{i-1}| \leq |\tan(\beta_i) - \tan(\beta_{i-1})|= |s_i- s_{i-1}|,$$
we obtain
\begin{equation}\label{odsa}
\sum_{i=1}^{n-1} \alpha_i \leq 4L.
\end{equation}
Since $|\beta_i| \leq \arctan L$, we have $\alpha_i/2 \leq \arctan L$. Further, since
the function $\tan$ is convex on $[0, \pi/2)$, the function $s(x)= \tan x/x$ is increasing on
$(0, \pi/2)$. These facts easily imply
$$ \tan \left(\frac{\alpha_i}{2}\right) \leq \frac{\alpha_i}{2}\cdot \frac{L}{\arctan L}.$$
Thus we obtain by \eqref{odsa} \begin{equation}\label{otap} \sum_{i=1}^{n-1} \sqrt{2} \tan \left(\frac{\alpha_i}{2}\right) \leq \frac{2\sqrt{2}L^2}{\arctan L} =: M. \end{equation}
Further observe that each $d_n$ is DC on ${\mathbb R}^2$ and consequently
\begin{equation}\label{exdir}
(d_n)_{+}'(x,v) \in {\mathbb R}\ \ \ \text{exists for every}\ \ \ x,v \in {\mathbb R}^2.
\end{equation}
Indeed, since each segment $[z_{i-1},z_i]$ is a convex set, by the well known fact the distance functions ${\rm dist}\,(\cdot, [z_{i-1}, z_i])$, $i=1,\dots,n,$ are convex and consequently $d_n$ is DC by \eqref{sjedno} below.
If $\alpha_i \neq 0$, set
$$V_i\coloneqq \{ z \in {\mathbb R}^2:\ \langle z-z_i, z_{i+1}-z_i\rangle \leq 0,\ \langle z-z_i, z_{i-1}-z_i\rangle \leq 0 \},$$
which is clearly a closed angle with vertex $z_i$
and measure $\alpha_i$. Let $\psi_i$ and $A_i$ be the (concave and affine) functions on
${\mathbb R}^2$ which correspond to $V_i$ by Lemma \ref{kofu}. If $\alpha_i =0$, put $\psi_i(z)\coloneqq 0,\ z \in {\mathbb R}^2$.
Now set $$ \eta_n: = \sum_{i=1}^{n-1} \psi_i.$$ Then $\eta_n$ is a concave function on ${\mathbb R}^2$ and Lemma \ref{kofu} with \eqref{otap} imply that \begin{equation}\label{lipeta}
\eta_n\ \ \text{ is Lipschitz with constant}\ \ M,
\end{equation}
and, if $\alpha_i \neq 0$,
\begin{equation}\label{kompvi}
|z - z_i| + \psi_i(z) = A_i(z),\ z\in V_i. \end{equation}
The concave function $c_n$ with properties \eqref{lcn}, \eqref{dnpcn} will be defined
as $c_n(x)\coloneqq \eta_n(x) + \xi_n(x),\ x \in \overline{U}$, where the concave function $\xi_n$ on $A\coloneqq (-1,1) \times {\mathbb R}$ will be defined to
``compensate the non-concave behaviour of $d_n$ at points of $\operatorname{graph} f_n$'' in the sense
that, for each point $z \in A \cap \operatorname{graph} f_n$, \begin{equation}\label{kompgr}
(d_n + \xi_n)_+'(z,v) + (d_n + \xi_n)_+'(z,-v) \leq 0\ \ \ \text{whenever}\ \ \ v \in {\mathbb R}^2. \end{equation} We set, for $(x,y) \in A$,
$$ \xi_n(x,y)\coloneqq - \max(2g_n(x)-y, 2h_n(x)+y)\ \ \ \text{and}\ \ \ p_n(x,y)\coloneqq |f_n(x)-y|.$$ Obviously, \begin{equation}\label{lipxi}
\text{$\xi_n$ is concave and Lipschitz with constant $2L+1$.}
\end{equation} Further, for $(x,y) \in A$, \begin{multline*}
p_n(x,y) = \max(g_n(x)-h_n(x)-y, h_n(x)-g_n(x)+y)\\ = \max(2g_n(x)-y, 2 h_n(x)+y)
- h_n(x) - g_n(x),
\end{multline*} which shows that $p_n$ is a DC function and $p_n + \xi_n$ is concave. Consequently, for
each $z\in A$ and $v \in {\mathbb R}^2$,
\begin{equation}\label{pxi}
(p_n)'_+(z,v) + (\xi_n)'_+(z,v) + (p_n)'_+(z,-v) + (\xi_n)'_+(z,-v) \leq 0.
\end{equation}
Since, for each point
$z \in \operatorname{graph} f_n \cap A$, we have $d_n(z)= p_n(z)=0$ and for each $(x,y) \in A$ clearly
$d_n(x,y) \leq |(x,y) - (x,f_n(x))| = p_n(x,y)$, we easily obtain (for each $v \in {\mathbb R}^2$) \begin{equation}\label{troj}
(p_n)'_+(z,v) + (p_n)'_+(z,-v) \geq (d_n)'_+(z,v) + (d_n)'_+(z,-v), \end{equation}
which, together with \eqref{pxi},
implies \eqref{kompgr}.
Now set $$c_n(x)\coloneqq \eta_n(x) + \xi_n(x),\ x \in \overline{U}.$$ By \eqref{lipeta} and \eqref{lipxi} we obtain that \eqref{lcn} holds with $L^*\coloneqq M + 2L+1$.
To prove \eqref{dnpcn}, it is clearly sufficient to show that $\gamma=c_n^*: = d_n + c_n$
is concave on $U$; we will prove it by Lemma \ref{comix}.
First we verify the validity
of \eqref{nezlo} for each $z \in U$. If $z \notin \operatorname{graph} f_n$, then \eqref{nezlo} holds by \eqref{zlose}, since
$\gamma = d_n + \eta_n + \xi_n$ on $U$, $d_n$ is locally semiconcave on ${\mathbb R}^2 \setminus \operatorname{graph} f_n$ and $\eta_n+ \xi_n $
is concave on $U$. If $z \in \operatorname{graph} f_n$, then \eqref{nezlo} follows by \eqref{kompgr} and the concavity of $\eta_n$ on $U$.
So it is sufficient to verify \eqref{pokrc}. To this end, first define on $U$ the functions
$$ \omega_i\coloneqq {\rm dist}\,(\cdot, l(z_i,z_{i+1}))\ \ \text{and}\ \ \mu_i\coloneqq \omega_i+ \eta_n + \xi_n,\ \ \ \ i=1,\dots,n-1. $$
Since each $\operatorname{graph} \omega_i$ is covered by graphs of two affine functions, we see
that
\begin{equation}\label{podvco}
\text{$\operatorname{graph} \mu_i$ is covered by graphs of two concave functions.}
\end{equation}
Now consider an arbitrary $z\in U$
and choose a point $z^* \in \Pi(z)$.
Since $d_n(z) \leq 1/5$ by \eqref{nnula} and $n \geq n_0 \geq 6$,
we obtain $z^* \in \bigcup_1^{n-2} [z_i,z_{i+1}]$.
If $z^*= z_i$ for some $1\leq i \leq n-1$
with $\alpha_i \neq 0$, then we easily see that $z\in V_i$ and $d_n(z)= |z-z_i|$, and consequently
$$ \gamma(z)= (|z-z_i| + \psi_i(z)) + \sum_{1\leq j\leq n-1, j\neq i} \psi_j(z) +\xi_n(z) = \nu_i(z),$$ where $$\nu_i(z)\coloneqq A_i(z) + \sum_{1\leq j\leq n-1, j\neq i} \psi_j(z) +\xi_n(z) ,\ \ z \in U,$$
is concave on $U$.
If $z^*= z_i$ and $\alpha_i=0$, or $z^*\in [z_i,z_{i+1}]\setminus\{z_i, z_{i+1}\}$ for some $1\leq i \leq n-1$, then
clearly $d_n(z) = \omega_i(z)$ and so $\gamma(z) = \mu_i(z)$.
So we have proved that the graph of $\gamma=c^*_n$ is covered by graphs of functions $\nu_i,\ 1\leq i \leq n-1$, $\alpha_i \neq 0$, and functions $\mu_i,\ 1\leq i \leq n-1$. Using \eqref{podvco}, we obtain \eqref{pokrc}
and Lemma \ref{comix} implies that $\gamma=c^*_n$ is concave.
\end{proof}
\section{Other results}\label{other} We finish the article with a number of additional results on the systems $$ \mathcal D_d\coloneqq \{\varnothing\} \cup \{\varnothing \neq A \subset {\mathbb R}^d: \ A\ \ \text{is closed}\ \ \text{and}\ \ d_A\ \ \text{is DC}\},\ \ \ d=1,2,\dots. $$ First we observe that the description of $\mathcal D_1$ is very simple since \begin{equation}\label{cald1} \text{$A \subset {\mathbb R}$ belongs to $\mathcal D_1$ iff the system of all components of $A$
is locally finite.} \end{equation} Indeed, if the system of all components of $\varnothing \neq A\subset {\mathbb R}$ is locally finite,
then Lemma \ref{vldc} (ii) easily implies that $d_A$ is DC.
If the system of all components of $A$ is not locally finite, then there exists a sequence $(c_n)$ of centres of components of
${\mathbb R}\setminus A$ converging to a point $a\in A$. Therefore $d_A$ is not one-sidedly strictly
differentiable at $a$, since $(d_A)'_{\pm}(c_n) = \mp1$. Consequently
$d_A$ is not DC, since each DC function on ${\mathbb R}$ is one-sidedly strictly
differentiable at each point (see \cite[Note 3.2]{VeZa} or \cite[Proposition 3.4(i) together
with Remark 3.2]{VZ2}).
From this characterisation easily follows that $\mathcal D_1$ is closed with respect to finite unions and intersections and that, for a closed set $M\subset \mathbb R$, \begin{equation}\label{eq:boundaryEquivalence} M\in \mathcal D_1\iff \partial M\in\mathcal D_1. \end{equation}
Concerning $d\geq 2$ further observe that \begin{equation}\label{sjedno} \mathcal D_d\ \ \text{is closed with respect to finite unions}. \end{equation} Indeed, if $\varnothing \neq A \in \mathcal D_d$ and $\varnothing \neq B \in \mathcal D_d$, then $d_{A\cup B} = \min (d_A, d_B)$ and so $d_{A\cup B}$ is DC by Lemma \ref{vldc} (i).
Example~\ref{ex:boundaryAndIntersection} below shows that already $\mathcal D_2$ is not closed with respect to finite intersections. Equivalence \eqref{eq:boundaryEquivalence} does not generalize already to dimension $2$ either (see again Example~\ref{ex:boundaryAndIntersection}), however, one can see that, for a closed set $M\subset \mathbb R^d$, \begin{equation}\label{eq:boundaryEquivalenceGeneral} \partial M \in \mathcal D_d\iff (M \in \mathcal D_d \quad\text{and}\quad \overline{{\mathbb R}^d \setminus M}\in \mathcal D_d),\quad d\in\mathbb N. \end{equation}
To prove one implication suppose $\partial M\in \mathcal D_d$. If $x\not\in M$ then clearly $\Pi_M(x)\in\partial M$ and so $d_{M}(x)=d_{\partial M}(x)$. Consequently, for each $x\in\mathbb R^d$, $d_{M}(x)\in\{0,d_{\partial M}(x)\}$ and so $M\in\mathcal D_d$ by Lemma~\ref{vldc}~(iv). Similarly, if $x\not\in \overline{{\mathbb R}^d \setminus M}$ then $\Pi_{\overline{{\mathbb R}^d \setminus M}}(x)\in\partial (\overline{{\mathbb R}^d \setminus M})\subset\partial M$ so again $d_{\overline{{\mathbb R}^d \setminus M}}(x)\in\{0,d_{\partial M}(x)\}$ and $\overline{{\mathbb R}^d \setminus M}\in \mathcal D_d$ follows.
To prove the opposite implication it is enough to show that $d_{\partial M}=\max(d_M,d_{\overline{{\mathbb R}^d \setminus M}})$ if $\partial M\not=\varnothing$. Clearly $d_{\partial M}\geq \max(d_M,d_{\overline{{\mathbb R}^d \setminus M}})$, since $\partial M= M\cap\overline{{\mathbb R}^d \setminus M}$. To prove the opposite inequality suppose to the contrary that $$r\coloneqq d_{\partial M}(x)> \max(d_M(x),d_{\overline{{\mathbb R}^d \setminus M}}(x))$$ for some $x\in\mathbb R^d$. Consequently $U(x,r)\cap M\not=\varnothing$ and $U(x,r)\cap\overline{{\mathbb R}^d \setminus M}\not=\varnothing$. Then also $U(x,r)\cap{\mathbb R}^d \setminus M\not=\varnothing$ and thus $U(x,r)\cap\partial M\not=\varnothing$ which is a contradiction.
Before presenting the following example we first observe that the function $g(x)=x^5\cos\frac{\pi}{x}$, $x\not=0$, $g(0)=0$, is $\mathcal C^2$ on $\mathbb R$ and therefore DC by Lemma~\ref{vldc}~(v). Indeed, a direct computation shows that $g''(x)=x\left(8\pi x\sin\frac{\pi}{x}-(\pi^2-20x^2)\cos\frac{\pi}{x}\right)$, $x\not =0$, and $g''(0)=0$.
\begin{ex}\label{ex:boundaryAndIntersection}
There are sets $A,B\in\mathcal D_2$ such that $A\cap B\not\in\mathcal D_2$. Further,
there is a set $K\in {\mathcal D}_2$ such that $\partial K\not\in{\mathcal D}_2$ and
$\overline{\mathbb R^2\setminus K}\not\in{\mathcal D}_2$.
\end{ex} \begin{proof}
Define $g(x)=x^5\cos\frac{\pi}{x}$, $x\not=0$, $g(0)=0$, and $f(x)=0$, $x\in \mathbb R$. Put
$$
A=\{ (x,y): y\geq f(x) \},\;
B=\{ (x,y): y\leq g(x) \},\; H=\{x: f(x)\leq g(x) \}.
$$
Since both $f$ and $g$ are DC, we obtain that $A,B\in \mathcal D_2$ by Theorem~\ref{hlav} and \eqref{eq:boundaryEquivalenceGeneral}.
Put $M=A\cap B$ and $K=\overline{\mathbb R^2\setminus M}$.
Clearly also $M=\overline{\mathbb R^2\setminus K}$.
First note that $M\not\in\mathcal D_2$ since the function $x\mapsto d_M(x,0)$ is equal to $d_H$, but clearly $H\not\in \mathcal D_1$ by\eqref{cald1}.
We obtain that $K\in\mathcal D_2$ by \eqref{sjedno}, since $K=C\cup D$, where $C=\{ (x,y): y\leq f(x) \}$ and $D=\{ (x,y): y\geq g(x) \}$, and $C,D\in \mathcal D_2$ by Theorem~\ref{hlav} and \eqref{eq:boundaryEquivalenceGeneral}.
Finally, $\partial K\not\in\mathcal D_2$ by \eqref{eq:boundaryEquivalenceGeneral} applied to $K$. \end{proof}
Now we will show that equivalence \eqref{eq:boundaryEquivalence} holds for sets $M$ of positive reach (cf. \eqref{prdd}). We first recall their definition.
If $A\subset {\mathbb R}^d$ and $a\in A$, we define $${\rm reach}\,(A,a)\coloneqq \sup\{r\geq 0:\, \Pi_A(z)\ \text{is a singleton for each}\ \ z \in U(a,r)\}$$ and the reach of $A$ as $${\rm reach}\, A\coloneqq \inf_{a\in A}{\rm reach}\, (A,a).$$ Note that each set with positive reach is clearly closed.
As mentioned in Introduction, it is essentially well-known that \begin{equation}\label{prdd} \text{if $A \subset {\mathbb R}^d$ has positive reach, then $A \in \mathcal D_d.$} \end{equation} Indeed, for each $a\in A$ \cite[Proposition 5.2]{CH} implies that $d_A$ is semiconvex on $U(a, {\rm reach}\, A/2)$, which with \eqref{loksem} and Lemma \ref{vldc} (ii) implies that $d_A$ is DC.
\begin{prop}\label{doplpr}
Let $\varnothing \neq A \subset {\mathbb R}^d$ be a set with positive reach and $B\coloneqq \overline{{\mathbb R}^d \setminus A}$. Then both $B$ and $\partial A$ belong to $\mathcal D_d$. \end{prop} \begin{proof}
By \eqref{prdd} and \eqref{eq:boundaryEquivalenceGeneral} it is sufficient to prove that $B\in\mathcal D_d$.
Since $d_B$ is locally DC on ${\mathbb R}^d\setminus B$ (see \eqref{loksem}) and on $\mathrm{int} B$
(trivially), by Lemma \ref{vldc} (ii) it is sufficient
to prove that
\begin{equation}\label{lokhr}
\text{for each $a \in \partial B$ there exists $\rho>0$ such that $d_B$ is DC on $U(a,\rho)$.}
\end{equation}
To prove \eqref{lokhr}, choose $0<r< {\rm reach}\, A$ and denote $A_r\coloneqq \{x:\ {\rm dist}\,(x,A)=r\}$.
We will first prove that
\begin{equation}\label{soucet}
{\rm dist}\,(x,B) +r = {\rm dist}\,(x, A_r),\ \ \text{whenever}\ \ x \in {\mathbb R}^d \setminus B = \mathrm{int} A.
\end{equation}
To this end, choose an arbitrary $x \in \mathrm{int} A$. Obviously, there exists $y\in \partial B \subset \partial A$ such that ${\rm dist}\,(x,B) = |x-y|$. Since $A$ has positive reach and $y \in \partial A$,
there exists $z \in A_r$ such that $|y-z| = r$ (It follows, e.g., from \cite[Proposition 3.1 (v),(vi)]{RZ}). Therefore
$$ {\rm dist}\,(x, A_r) \leq |x-z| \leq |x-y| + |y-z| = {\rm dist}\,(x,B) +r.$$
To prove the opposite inequality, choose a point $z^* \in A_r$ such that ${\rm dist}\,(x, A_r) = |x-z^*|$.
Obviously, on the segment $[x,z^*]$ there exists a point $y^* \in \partial A \subset B$. Then
$$ {\rm dist}\,(x, A_r) = |x-z^*| = |x-y^*| + |y^* - z^*| \geq {\rm dist}\,(x,B) +r,$$
and \eqref{soucet} is proved.
Now let $a \in \partial B \subset \partial A$ be given. Then $a \notin A_r$ and so by \eqref{loksem}
there exists $\rho>0$ such that ${\rm dist}\,(\cdot, A_r)$ is DC on $U(a,\rho)$. For $x \in U(a,\rho)$,
$d_B(x) = {\rm dist}\,(x, A_r)-r$ if $x \in \mathrm{int} A$ (by \eqref{soucet}) and $d_B(x)=0$ if $x \notin \mathrm{int} A$. Thus Lemma \ref{vldc} (iv) implies that $d_B$ is DC on $U(a,\rho)$, which proves \eqref{lokhr}. \end{proof}
Further recall that our main result (Theorem \ref{hlav}) asserts that \begin{equation}\label{grdd} \operatorname{graph} g \in \mathcal D_2\ \ \ \text{whenever}\ \ \ g:{\mathbb R} \to {\mathbb R}\ \ \text{is}\ \ DC. \end{equation}
Motivated by a natural question, for which non DC functions $g$ \eqref{grdd} holds,
we present the following result, whose proof is implicitly contained in the proof
of \cite[Proposition 6.6]{PRZ}; see Remark \ref{mani} below. \begin{prop}\label{lidc} If $g: {\mathbb R}^{d-1} \to {\mathbb R}$ ($d\geq 2$) is locally Lipschitz and $A\coloneqq \operatorname{graph} g \in \mathcal D_d$,
then $g$ is DC. \end{prop} \begin{remark}\label{mani} One implication of \cite[Proposition 6.6]{PRZ} gives that if $A$ is as in Proposition \ref{lidc}
(or, more generally, $A$ is a Lipschitz manifold of dimension $0<k<d$; see \cite[Definition 2.4]{PRZ} for this notion) and $A$ is WDC, then $g$ is DC (or is a DC manifold of dimension $0<k<d$, respectively). The proof of this implication works with an aura $f=f_M$ of a set $M$, but
under the assumption that $A \in \mathcal D_d$, the proof clearly also works, if we use the distance function $d_A$ instead of $f$. So we obtain not only Proposition \ref{lidc}, but also
the following more general result.
{\it If $A \subset {\mathbb R}^d$ is a Lipschitz manifold of dimension $0<k<d$ and $A \in \mathcal D_d$,
then $A$ is a DC manifold of dimension $k$.} \end{remark}
Recall that it is an open question, whether $\operatorname{graph} g\in \mathcal D_d$, whenever
$g:\mathbb R^{d-1}\to\mathbb R$ is a DC function. However, using Proposition \ref{doplpr},
we easily obtain:
\begin{cor}\label{grsemi}
If $g:\mathbb R^{d-1}\to\mathbb R$ is a semiconcave function then $\operatorname{graph} g \in \mathcal D_d$.
\end{cor} \begin{proof} The set $S\coloneqq\{ (a,b)\in\mathbb R^{d-1}\times\mathbb R: b\leq g(a) \}$ has positive reach by \cite[Theorem~2.3]{Fu} and consequently $d_{\operatorname{graph} g}=d_{\partial S}$ is DC by Proposition~\ref{doplpr}. \end{proof}
\begin{rem}\label{hrsemi}
Let $M\subset\mathbb R^d$ be a closed set whose boundary can be locally expressed as a graph of a semiconvex function (i.e., for each $a \in \partial M$ there exist
a semiconcave function $g:\mathbb R^{d-1}\to\mathbb R$, $\delta>0$ and an isometry $\varphi: {\mathbb R}^d \to {\mathbb R}^d$
such that $\partial M \cap U(a,\delta) = \varphi(\operatorname{graph} g) \cap U(a,\delta)$).
Then $d_{\partial M}$ is locally DC (and therefore DC) by Corollary~\ref{grsemi} and \eqref{loksem} and so $M\in\mathcal D_d$ by Lemma~\ref{vldc} (iv) and \eqref{eq:boundaryEquivalenceGeneral}. \end{rem}
Before the next results, we present the following definitions:
we say that a set $A\subset \mathbb R^d$ is a DC hypersurface, if there exist a vector $v\in S^{d-1}$ and a DC function (i.e. the difference of two convex functions) $g$ on $W\coloneqq (\operatorname{span} v)^\bot$ such that $A=\{w+g(w)v: w\in W\}$. A set $P\subset\mathbb R^2$ will be called a DC graph if it is a rotated copy of $\operatorname{graph}(f|_{I})$ for a DC function $f:\mathbb R\to\mathbb R$ and some compact (possibly degenerated) interval $\varnothing\not=I\subset\mathbb R$. Note that $P$ is a DC graph if and only if it is a nonempty connected compact subset of a DC hypersurface in $\mathbb R^2$.
\begin{prop}\label{pokr}
Let $d\geq 2$ and $F \in \mathcal D_d$. Then each bounded set $C \subset \partial F$ can be covered by finitely many DC hypersurfaces. \end{prop} \begin{proof}
By our assumptions, $f\coloneqq {\rm dist}\,(\cdot, F)$ is a DC function on ${\mathbb R}^d$ and $f(x)=0$
for every $x \in C$. So, by \cite[Crollary 5.4]{PRZ} it is sufficient to prove that
for each $x \in C$ there exists $y^* \in \partial f(x)$ with $|y^*|> \varepsilon\coloneqq 1/4$, where
$\partial f(x)$ is the Clarke generalized gradient of $f$ at $x$ (see \cite[p. 27]{C}).
To this end, suppose to the contrary that $x \in C$ and $\partial f(x)\subset B(0,1/4)$.
Since the mapping $x \mapsto \partial f(x)$ is upper semicontinuous (see \cite[Proposition 2.1.5 (d)]{C}), there exists $\delta>0$ such that $\partial f(u) \subset U(0, 1/2)$ for each $u \in U(x, \delta)$.
Since $x \in \partial F$, we can choose $z \in U(x,\delta/2) \setminus F$ and $p \in \Pi_F(z)$.
Then $p \in U(x, \delta)$, $f(z)- f(p) = |z-p|$ and Lebourg's mean-value theorem
(see \cite[Theorem 2.3.7]{C}) implies that there exist $u \in U(x,\delta)$ and $\alpha \in
\partial f(u)$ such that
$$ \langle \alpha, z-p\rangle = f(z)- f(p) = |z-p|.$$
Therefore $|\alpha| \geq 1$, which is a contradiction. \end{proof}
The above proposition easily implies the following fact. \begin{cor}
If $F\in \mathcal D_2$ then $\partial F$ is a subset of the union of a locally finite system of DC graphs. \end{cor}
Using Theorem~\ref{hlav} we obtain the following easy result. \begin{prop}\label{prop:unionOfGraphs}
If $A\subset\mathbb R^2$ is the union of a locally finite system of DC graphs then $A\in \mathcal D_2$.
\end{prop}
\begin{proof} First note that it is enough to prove that any DC graph $P$ belongs to $\mathcal D_2$. Indeed, if $M$ is a locally finite system of DC graphs and each DC graph belongs to $\mathcal D_2$, then $d_M$ is locally DC by \eqref{sjedno} (and so DC) and $M\in\mathcal D_2$.
So assume that $A$ is a DC graph. Without any loss of generality we may assume that $A=\operatorname{graph} f|_{[0,p]}$ for some DC function $f:\mathbb R\to\mathbb R$. If $p=0$ then $d_A=|\cdot|$ is even convex, so assume that $p>0$. We may also assume that $f(0)=0$.
First note that (by Theorem~\ref{hlav} and \eqref{loksem}) $d_A$ is locally DC on $\mathbb R^2\setminus\{(0,0),(p,f(p))\}$. It remains to prove that $d_A$ is DC on some neighbourhood of $(0,0)$ and $(p,f(p))$. We will prove only the case of the point $(0,0)$, the other case can be proved quite analogously. By Lemma \ref{vldc} (iii) we can choose $L>0$ such that $f$ is $L$-Lipschitz on $[0,p]$. Define \begin{equation*} f_\pm(x)\coloneqq \begin{cases} f(x)& \text{if}\quad 0\leq x \leq p,\\ f(p)& \text{if}\quad p<x,\\ \pm 2Ly& \text{if}\quad x<0. \end{cases} \end{equation*}
It is easy to see that both $f_+$ and $f_-$ are continuous and so they are DC by Lemma \ref{vldc} (iv).
Put \begin{equation*} M_0\coloneqq \left\{ (u,v)\in\mathbb R^2: u\geq0, \; v=f_+(u)\right\}, \end{equation*} \begin{equation*} M_1\coloneqq \left\{ (u,v)\in\mathbb R^2: u\geq0, \; f_+(u)< v\right\}\cup \left\{ (u,v)\in\mathbb R^2: u<0, \; -\frac{u}{2L}< v\right\}, \end{equation*} \begin{equation*} M_2\coloneqq \left\{ (u,v)\in\mathbb R^2: u\geq0, \; \tilde f_-(u)> v\right\}\cup \left\{ (u,v)\in\mathbb R^2: u<0, \; \frac{u}{2L}> v\right\} \end{equation*} and \begin{equation*} M_3\coloneqq \left\{ (u,v)\in\mathbb R^2:\frac uL< v< -\frac uL\right\}. \end{equation*} Clearly $\mathbb R^2= M_0\cup M_1\cup M_2\cup M_3$ and $M_1$, $M_2$, $M_3$ are open.
Set $\tilde d\coloneqq{\rm dist}\,(\cdot, M_0)$ and, for each $y \in {\mathbb R}^2$, define $$d_0(y)=0,\ \ d_1(y)\coloneqq {\rm dist}\,(y,\operatorname{graph} f_+),\ \ d_2(y)\coloneqq {\rm dist}\,(y,\operatorname{graph} f_-),
\ \ d_3(y)\coloneqq |y|.$$ Functions $d_1$ and $d_2$ are DC on $\mathbb R^2$ by Theorem~\ref{hlav}, $d_0$ and $d_3$ are even convex on $\mathbb R^2$.
Using (for $K= 1/L, -1/L, 1/(2L), -1/(2L)$) the facts that the lines with the slopes $K$ and $-1/K$ are orthogonal and $M_0\subset \{(u,v): u\geq 0,\; -Lu\leq v \leq Lu \}$, easy geometrical observations show that \begin{equation}\label{mjdt} \tilde d(y)= d_i(y)\ \ \text{ if}\ \ y \in M_i,\ 0\leq i \leq 3, \end{equation} and so Lemma \ref{vldc} (iv) implies that $\tilde d$ is DC. To finish the proof it is enough to observe that $d_A=\tilde d$ on $U(0,\frac{p}{2})$. \end{proof}
However, the following example shows that the opposite implication does not hold even for nowhere dense sets $A$.
\begin{ex}
There is a nowhere dense set $A\in {\mathcal D}_2$ which is not the union of a locally finite system of DC graphs. \end{ex} \begin{proof}
Define
$$\text{$f(x)= \max(x^5,0)$, $x \in {\mathbb R}$, $g(x)=x^5\cos\frac{\pi}{x}$, $x \in {\mathbb R}$, and
$g_k\coloneqq g\restriction_{[\frac{1}{2k+1},\frac{1}{2k}]}$, $k \in \mathbb N $.}$$
Put $A^{\pm}\coloneqq \operatorname{graph} (\pm f)$, $A_k\coloneqq \operatorname{graph} g_k$, $k\in\mathbb N$, and
\begin{equation*}
A\coloneqq A^+\cup A^-\cup \bigcup_{k\in \mathbb N} A_k.
\end{equation*}
$A$ is clearly closed and nowhere dense, and it
is not the union of a locally finite system of DC graphs since every DC graph $B\subset A$ can intersect at most one of the sets $A_i$.
It remains to prove that $A\in {\mathcal D}_2$.
First we will describe all components of ${\mathbb R}^2 \setminus A$. To this end, for each $k \in \mathbb N$, define
\begin{equation*}
U_0(x)=
\begin{cases}
g(x), &\; x\in [1/3,1/2],\\
f(x), &\; x\in [1/2,\infty)
,
\end{cases},
\quad
U_k(x)=
\begin{cases}
g(x), &\, x\in \left[\frac{1}{2k+3},\frac{1}{2k+2}\right],\\
f(x), &\, x\in \left[\frac{1}{2k+2},\frac{1}{2k}\right],
\end{cases}
\end{equation*}
\begin{equation*}
L_0(x)=
-f(x), x\in [1/3,\infty),
\quad
L_k(x)=
\begin{cases}
-f(x), &\, x\in \left[\frac{1}{2k+3},\frac{1}{2k+1}\right],\\
g(x), &\, x\in \left[\frac{1}{2k+1},\frac{1}{2k}\right].
\end{cases}
\end{equation*}
Set $G_k\coloneqq \{(x,y):\ L_k(x) < y <U_k(x)\}$, $ k=0,1,2,\dots$. Then
$$ G^+\coloneqq \{(x,y):\ f(x) < y \}, G^-\coloneqq \{(x,y):\ y< -f(x) \}\ \text{and}\ G_0,G_1,\dots$$
are clearly all components of ${\mathbb R}^2 \setminus A$.
Recall that both $U_k$ and $L_k$ is defined on $D_k$, where $D_k= [\frac{1}{2k+3}, \frac{1}{2k}]$
for $k\in \mathbb N$ and $D_0= [1/3, \infty)$. Using the facts that $D_k$ and $D_{k+2}$ are disjoint
($k=0,1,\dots$),
$$ U_k\left(\frac{1}{2k+3}\right)= L_k\left(\frac{1}{2k+3}\right) = g\left(\frac{1}{2k+3}\right),\
U_k\left(\frac{1}{2k}\right)= L_k\left(\frac{1}{2k}\right) = g\left(\frac{1}{2k}\right)$$
and $U_0(1/3)= L(1/3)=g(1/3)$, it is easy to see that there exist unique functions
$U$, $\tilde U$ which are continuous on ${\mathbb R}$, $U$ (resp. $\tilde U$) extends all
$U_k$, $k=0,2,4,\dots$ (resp. $k=1,3,5,\dots$) and $U$ (resp. $\tilde U$) equals to $g$
at all points at which no $U_k$, $k=0,2,4,\dots$ (resp. $k=1,3,5,\dots$) is defined.
Quite analogously a continuous function $L$ (resp. $\tilde L$) extending all
$L_k$, $k=0,2,4,\dots$ (resp. $k=1,3,5,\dots$) is defined. Since the functions
$g$, $f$, $-f$ are DC,
Lemma \ref{vldc} (iv) implies that
the functions $U$, $\tilde U$, $L$, $\tilde L$ are DC. So Theorem \ref{hlav} implies that
the distance functions
\begin{equation}\label{vzdal}
d_{A^+},\ d_{A^-},\ d_{\operatorname{graph} U},\ d_{\operatorname{graph} \tilde U},\ d_{\operatorname{graph} L},\ d_{\operatorname{graph} \tilde L}
\end{equation}
are DC.
Obviously $d_A(x)=0$ for $x\in A$, $d_A(x)=d_{A^+}(x)$ for $x\in G^+$ and
$d_A(x)=d_{A^-}(x)$ for $x\in G^-$. Further, if $x \in G_k$ with $k=2,4,6,\dots$, then
$$d_A(x) \in \{ d_{\operatorname{graph} U}(x), d_{\operatorname{graph} L}(x)\},$$
which easily follows from the facts that
$$ \partial G_k \subset (\operatorname{graph} U \cup \operatorname{graph} L)\ \ \text{ and}\ \
(\operatorname{graph} U \cup \operatorname{graph} L) \cap G_k = \varnothing.$$
Similarly we obtain that, if $x \in G_k$ with $k=1,3,5,\dots$, then
$$d_A(x) \in \{ d_{\operatorname{graph} \tilde U}(x), d_{\operatorname{graph} \tilde L}(x)\}.$$
Thus, using \eqref{vzdal} and Lemma~\ref{vldc} (iv), we obtain that
$d_A$ is DC. \end{proof}
It seems that there does not exist an essentially simpler example. Iterating the construction of the example we can obtain nowhere dense sets in $\mathcal D_2$ of quite complicated topological structure.
In our opinion, using Proposition~\ref{pokr} and Theorem~\ref{hlav} it is possible to give an optimal complete characterisation of sets in $\mathcal D_2$, but it appears to be a rather hard task. We believe that we succeeded to find some characterisation, however, it is not quite satisfactory and our current proof is very technical. We aim to find a better characterisation, hopefully with a simpler proof.
\end{document} | arXiv |
Generation and filtering of gene expression noise by the bacterial cell cycle
Noreen Walker1,
Philippe Nghe1 &
Sander J. Tans1
Gene expression within cells is known to fluctuate stochastically in time. However, the origins of gene expression noise remain incompletely understood. The bacterial cell cycle has been suggested as one source, involving chromosome replication, exponential volume growth, and various other changes in cellular composition. Elucidating how these factors give rise to expression variations is important to models of cellular homeostasis, fidelity of signal transmission, and cell-fate decisions.
Using single-cell time-lapse microscopy, we measured cellular growth as well as fluctuations in the expression rate of a fluorescent protein and its concentration. We found that, within the population, the mean expression rate doubles throughout the cell cycle with a characteristic cell cycle phase dependent shape which is different for slow and fast growth rates. At low growth rate, we find the mean expression rate was initially flat, and then rose approximately linearly by a factor two until the end of the cell cycle. The mean concentration fluctuated at low amplitude with sinusoidal-like dependence on cell cycle phase. Traces of individual cells were consistent with a sudden two-fold increase in expression rate, together with other non-cell cycle noise. A model was used to relate the findings and to explain the cell cycle-induced variations for different chromosomal positions.
We found that the bacterial cell cycle contribution to expression noise consists of two parts: a deterministic oscillation in synchrony with the cell cycle and a stochastic component caused by variable timing of gene replication. Together, they cause half of the expression rate noise. Concentration fluctuations are partially suppressed by a noise cancelling mechanism that involves the exponential growth of cellular volume. A model explains how the functional form of the concentration oscillations depends on chromosome position.
Single-cell experiments have shown gene expression to fluctuate randomly under constant conditions [1–7], which can have key consequences for the fidelity of signal propagation [8], cell fate decisions [9, 10], and fitness [4, 11–16]. Noise in gene expression is often quantified by the observed cell-to-cell variability in the production rate or concentration of a protein when observing many cells in an isogenic population [1, 17]. Fluctuations in gene expression can be caused by many local and global factors such as random binding events of RNA polymerase [18], fluctuating concentration of ribosomes, or availability of amino acids [4, 19]. The cell cycle has been suggested as a general source of gene expression noise [17, 19], meaning that, in a snapshot of a population, two cells can differ in protein production rate or concentration because they are in different phases of their cell cycle. Alternatively, two cells at the same cell cycle phase can differ because of cell cycle-independent effects. The key aim of this study is to quantify and disentangle these effects in Escherichia coli, and to mechanistically understand cell cycle contributions.
Eukaryotes exhibit distinct cell cycle phases that display different levels of growth activity and of DNA replication, which in turn can result in varying expression levels as the cell cycle progresses. Single-cell investigations of Saccharomyces cerevisiae have indeed shown quasi-periodic fluctuations of protein expression rates [20] and concentrations [21] in synchrony with the cell cycle. The prokaryotic cell cycle does not display such distinct replication and growth phases. E. coli, for instance, grows and replicates DNA continuously throughout its cell cycle, though for slow growth there are periods without replication activity [22, 23]. Expression activity can be dependent on the cell cycle nonetheless, for example because the replication of a gene may double the transcriptional activity at a specific moment in time, as suggested by recent single-cell studies [17, 24, 25]. That doubling would then in turn affect enzyme concentration and could cause quasi-periodic fluctuations. However, at the same time, cells may exploit specific regulatory mechanisms to filter such perturbations [26, 27]. Direct experimental investigations of the impact of the bacterial cell cycle on expression variability are lacking. Elucidating this question is important to understand the origins of gene expression noise, modeling of genetic circuits, and resulting impact on growth variability [28] as well as other forms of cellular heterogeneity [10].
To address these questions, we followed a single-cell approach. We imaged E. coli cells as they grew into micro-colonies and measured gene expression as the fluorescence signal of chromosomally encoded fluorescent proteins (Additional file 1: movie S1). As shown herein, understanding the temporal dynamics requires detailed information on cellular volume increases in time, as protein concentrations are affected both by time-dependent expression and dilution. Thus, we accurately determined protein expression and cell size at sub-cell cycle resolution. We further developed a model to predict the cell cycle dependence and amplitude of these quasi-periodic fluctuations in expression rate and concentration. The model predicted their dependence on chromosomal position, which we tested with genetic constructs.
The protein production rate fluctuates quasi-periodically
To measure the effect of the cell cycle on protein expression, we first determined protein production rate as quantified by the time derivative of the total cellular fluorescence (Methods). Taking the data for all cells with a completed cell cycle (n = 393) over all cell cycle phases, the protein expression rate displayed a total noise intensity (defined as standard deviation divided by the mean) of 0.48 [17]. When plotting the production rate versus cell cycle phase ϕ (where 0 is cell birth and 1 is cell division) and averaging over all cells (Fig. 1a), it displayed the following trend: it was approximately constant in the first half, after which it rose to about two-fold at the end of the cycle (Fig. 1b, Additional file 2: Figure S1). An initially constant rate and two-fold increase is consistent with the known chromosome replication pattern for the observed mean growth rate (0.6 dbl/h): a single chromosome copy in the first period of the cell cycle, after which replication occurs in the second period that produces two copies [29]. Each chromosome copy then yields a fixed expression rate. This is not unreasonable, as other components required for expression, such as RNA polymerases and ribosomes, also double throughout the cell cycle. At faster growth, replication occurs throughout the cell cycle for multiple nested chromosome copies [30]. Consistently, we found that the production rate was not initially flat, but instead rose continuously throughout the cell cycle when growing on a different medium that supported a higher mean growth rate of 1.8 dbl/h (Additional file 2: Figure S2). The total increase remained two-fold, in agreement with an expected doubling of the number of gene copies. Overall, these data indicate that the mean protein expression rate is likely proportional to the gene copy number and hence doubles during chromosome replication. This variation is more continuous at high growth rate because of the nested replication and overall higher gene copy numbers.
Dependence of protein production rate (a, b), protein concentration (c, d) and cell length (e, f) on cell cycle phase. Observables are normalized by the respective population average and therefore unitless. (a, c, e) Data for 393 cells (gray) with three example traces and the binned colony average (thick black line). Histograms display the total frequency of production rate or concentration values summed over all phases. To convey the differences in noise intensity, a bar of size 0.2 times the population mean is displayed. (b, d, f) Phase-dependence of the binned data. In (f) an exponential function (black dashed line) is fitted to the averaged cell length. Error bars are obtained by bootstrapping. For cell length, error bars are plotted but are smaller than the line thickness. Growth was on minimal medium supporting a growth rate of 0.6 dbl/h
Deterministic cell cycle variations contribute to expression noise
To quantify the contribution of the mean cell cycle fluctuations (Fig. 1b) to protein production noise we split the single-cell production rate (which is distinct from the protein concentration) p(ϕ, x) into the population averaged rate \( \overline{p_c}\left(\phi \right) \) and individual deviations δp(ϕ, x), which together capture all cell-to-cell variability (Fig. 1a,b): \( p\left(\phi, \mathbf{x}\right)=\overline{p_c}\left(\phi \right)+\delta p\left(\phi, \mathbf{x}\right) \). Here, ϕ denotes the cell cycle phase and x all other causes of cell-to-cell variability; c refers to cell cycle dependence, which here is redundant because it is implied by the ϕ dependence but used for notation consistency. \( \overline{p_c}\left(\phi \right) \) can be estimated by the curve in Fig. 1b, and subtracted from individual traces to obtain an estimate for δp(ϕ, x). The noise intensity caused by the deterministic cell cycle fluctuation \( \overline{p_c}\left(\phi \right) \) is 0.26, which was obtained by considering the phase ϕ as a random variable and then calculating the variance of the trace. Noise of the individual expression traces δp(ϕ, x), averaged over all cells and ϕ, was 0.42 (Additional file 2: Figure S3a). These values are consistent with a scenario in which population mean trace \( \overline{p_c}\left(\phi \right) \) and deviation traces δp(ϕ, x) are independent and thus their variances (squared noise) can be added up: 0.482 ≈ 0.262 + 0.422. This population-average cell cycle contribution towards production rate noise does not include cell cycle stochasticity of individual cells and we will consider that below.
Concentration fluctuations are buffered by dilution
Fluctuating production rates can cause noise in the protein concentration. To determine the latter, we quantified the mean fluorescence per unit area of the cell. The noise intensity of 0.15 (0.10 for fast growth), which was obtained by taking the data of all cells and at all cell cycle phases, was consistent with previous reports [1]. After ordering by cell cycle phase and averaging (Fig. 1c), the concentration also showed systematic variations (Fig. 1d , Additional file 2: Figure S1): it increased slightly right after cell birth, then decreased and finally rose again. The amplitude of these variations was 4 % of the mean. This low value (Additional file 2: Figure S3b) and the initial increase seemed inconsistent with the large amplitude of variations in production rate caused by the cell cycle, as well as with the initially constant value of production rate (Fig. 1b) [25].
To get a more intuitive understanding of these differences, we formulated a minimal cell cycle model based on the measured cell cycle dependency of production rate (Fig. 1b). The concentration cannot be determined by simply integrating the production rate, as this would ignore dilution due to volume growth. To quantify the volume growth, we determined for each cell its length and its dependence on the cell cycle phase (Fig. 1e, Methods) [28]. The population mean cell length \( \overline{L}\left(\phi \right) \) was well described by an exponential function (Fig. 1f) [31–33], and not by bi-linear or linear functions (Additional file 2: Figure S4), as suggested previously [34–37]. Therefore, an exponential function for cell size was used as input for the minimal model (Fig. 2b). With a mean protein production \( \overline{p}\left(\phi \right) \) at phase ϕ (Fig. 2a), the concentration \( \overline{E}\left(\phi \right) \) can then be written as: \( \overline{E}\left(\phi \right)=\left(\overline{F_0}+{\displaystyle {\int}_0^{\phi }}\overline{p}\left({\phi}^{\hbox{'}}\right)d{\phi}^{\hbox{'}}\right)/\overline{L}\left(\phi \right) \), where \( \overline{F_0} \) is the total amount of protein at cell birth.
Model for cell cycle dependence of protein concentration. a Average protein production rate normalized by the mean. b Exponential length increase normalized to a mean of one (black). Population average protein production rate integrated in time, or the population average total fluorescence (green). c Determined cellular protein concentration, given by the green signal divided by the black line in panel b
By design, \( \overline{E}\left(\phi \right) \) (Fig. 2c) reproduced the measured data (Fig. 1d) and provided an explanation for the observed functional form, including the counterintuitive increase in concentration at the beginning of the cell cycle, before duplication occurs. As \( \overline{E}\left(\phi \right) \) is periodic, we know that increases (dilution rate smaller than expression rate) are balanced by decreases (dilution rate larger than expression rate). In cases where duplication occurs late, the expression rate is essentially constant because there is only one gene copy, while the dilution rate changes. Thus, dilution is then comparatively weak at the beginning of the cell cycle, resulting in an increasing concentration, while dilution is comparatively strong further into the cell cycle, resulting in decreasing concentrations. This rationale also explains why concentration fluctuations are small: the functional form of the total fluorescence (as a function of the cell cycle phase) is almost identical to that of the volume (Fig. 2b).
Stochastic replication timing contributes to expression noise
The single cell data also suggested that stochasticity in replication timing is a source of protein production noise, which is supported by previous studies [23, 38] (Fig. 1a, thin lines). In other words, δp(ϕ, x) would be the sum of fluctuations caused by cell cycle stochasticity δp c (ϕ, ν) and of fluctuations δp nc (x) unrelated to the cell cycle (Fig. 3a). Here, v is the cell cycle phase at which the gene of interest is replicated and v varies from cell to cell. Thus, the sum of δp c (ϕ, ν) and the population-average \( \overline{p_c}\left(\phi \right) \) yield all the fluctuations p c (ϕ, ν) caused by the cell cycle. To determine the stochastic contribution of the cell cycle to the expression noise, one needs to quantify δp c (ϕ, ν). However, it is not trivial to distinguish δp c (ϕ, ν) from the other stochastic, non-cell cycle variations in the experimental single-cell traces.
Production rates of single cells. a Description of variables used for noise decomposition. The protein production rate p(ϕ, x) (red line) is the sum of three contributions: (1) the population-average cell cycle fluctuations \( \overline{p_c}\left(\phi \right) \) (black line), (2) the contribution due to stochastic replication timing (difference between blue and black line, δp c (ϕ, ν)), and (3) stochasticity resulting from other, unknown, noise sources (difference between red and blue line, δp nc (x)). The sum of δp c (ϕ, ν) and δp nc (x) represents all of the stochastic contributions δp(ϕ, x). The phase at which replication occurs is denoted by v. b Experimental traces of three different cells (thick lines) and fitted step functions (thin lines). See Additional file 2 for definition of step function. Initial value was set to 1 and data is slightly vertically shifted for clarity. c Histogram of v. Data is from 53 cells in which a step-function could be discerned from the rest of the noise (13.5 % of the traces). d Comparison of experimental average production rate curve (gray line) and mean of ideal step functions (orange line). Experimental curve is the same as in Fig. 1b, which was obtained from binning the data according to cell cycle phase and averaging over n = 393 cells
To overcome this problem, we started with p c (ϕ, ν) and followed a variance decomposition approach using the law of total variance [39, 40]. The variance of the full cell cycle fluctuations can be decomposed as follows:
$$ Var\left({p}_c\left(\phi, \nu \right)\right)=\left\langle Var\left({p}_c\left(\phi, \nu \right)\Big|\phi \right)\right\rangle +Var\left(\left\langle {p}_c\left(\phi, \nu \right)\Big|\phi \right\rangle \right) $$
Here, angular brackets denote averaging, and the notations Var(… |ϕ) and 〈 … |ϕ〉 indicate, respectively, the variance and the average for a given phase ϕ (conditioned on ϕ). In the second term, the brackets thus indicate an averaging over the stochastic variable v, which yields \( \overline{p_c}\left(\phi \right) \). Next, the variance is taken. This variance was in fact calculated previously, and found to be (0.26)2 (Fig. 1b). Thus, the second term indicates the deterministic contribution to the cell cycle induced noise.
In the first term, the variance of p c (ϕ, ν) is determined conditionally on ϕ, and then averaged. This term thus denotes the stochastic contribution to the cell cycle-induced noise. The data does not directly provide an estimate of this variance, because the cell cycle-induced noise and noise from other sources are confounded in the measured single-cell traces of the production rate (Fig. 1a). Indeed, in these traces, other noise sources, such as metabolism [28] and fluctuating transcription factors [1], are substantial and can mask the quick two-fold increase expected from gene replication events. However, in a subset of traces, the two-fold increase was clear (Fig. 3b,c, Methods). Fitting each of these traces with a step-function (Additional file 2: Figure S5) provided a distribution of the step-moment, v. We obtained a wide distribution for v with a mean 0.64 and a standard deviation of 0.17 (Fig. 3c). To check whether this distribution was consistent with the full dataset, we compared the average of the fitted step-functions to the average of all measured traces (\( \overline{p_c}\left(\phi \right) \), Fig. 1b), and found that they were similar (Fig. 3d). These findings suggested that gene duplication events with stochastic timing in individual cells underlie the smooth shape of the population average production rate (Fig. 1b).
The distribution of v (Fig. 3c) now allowed us to estimate the first term in eq. (1), by first determining the variance of the step-functions at fixed phase, and then averaging over all phases (Additional file 2: Figure S6a). We obtained a value of (0.23)2 for this stochastic contribution of the cell cycle to expression noise, which is comparable in magnitude to the deterministic contribution denoted by the second term ((0.26)2, Additional file 2: Table S1). Thus, variability in initiation timing contributes substantially to the cell cycle-induced noise. The deterministic and stochastic contributions together (p c (ϕ, ν)) thus caused a variance of (0.23)2 + (0.26)2 = (0.35)2, which is about half (52 %) of the protein production variance (Fig. 5b, Additional file 2: Table S1).
To estimate how the protein concentration noise is affected by the cell cycle, we computed the concentration traces resulting from the step-like production rate functions (Additional file 2: Figure S6a). For each p c (ϕ, ν) of the set (Fig. 3c) the corresponding concentration curve was computed, considering that proteins are diluted due to volume growth (Additional file 2: Figure S6b). We found that the quasi-periodic concentration fluctuations caused by the cell cycle (which includes deterministic and stochastic components) contributed less than 1.5 % to the variance in protein concentration (Additional file 2: Figure S6b and Fig. 5b). Note that one can distinguish contributions from the population average trend (Fig. 1d) and the stochastic deviations around it due to variability in replication timing (less than 1 % contribution each, Additional file 2: Table S1).
Location on the chromosome affects expression fluctuations
Chromosome replication is initiated at the origin of replication (oriC) from which two replication forks then progress simultaneously and bi-directionally along the two strands of DNA [41]. This raises two expectations: first, genes located at opposite sides but at the same distance from oriC should be duplicated at the same time and thus show the same cell cycle dependence of protein production and concentration. Second, if one gene is located upstream of the other, the increase in protein production should occur earlier. To test the first prediction, we investigated a cfp gene positioned symmetrical to the yfp gene studied so far, at the opposite strand at the same distance from oriC (Methods, Fig. 4a inset). We indeed found that both reporters displayed a similar dependence of production rate and concentration on cell cycle phase (Fig. 4ab, Additional file 2: Figure S1).
Influence of chromosomal position. Production rate (a) and concentration (b) for genes at equidistant and symmetric positions with respect to the origin of replication. Observables were normalized by the population mean. (Inset) Locations of fluorescent genes and origin of replication oriC on the DNA. Different replication times (c, light to dark gray) and their predicted influence on the concentration (d). Production rates are slightly vertically shifted for clarity. (Normalized) production rate (e) and concentration (f) of GFP (strain ASC636, green line) compared to YFP (strain MG22, gray line). GFP data is from 296 cells with complete cell cycle that have on average seven data points/cell cycle. (Inset) Location of gfp compared to other fluorescent genes. Error bars are obtained by bootstrapping
To change the position we studied a gfp gene under P lac control closer to oriC than yfp or cfp (Methods, Fig. 4e). As expected from the earlier replication, the GFP production rate indeed increased earlier than the previous YFP signal (Fig. 4e). It started comparatively low, then increased more than two-fold and subsequently decreased again to end at twice the initial rate (Fig. 4e). The cause of the high fold-change and decrease is unknown, but changes in chromosome structure or transient improvement in competition for RNA polymerases for this promoter (two binding sites at the two replicated genes) could play a role. As predicted by the model (Fig. 4c,d), the dip in GFP concentration occurred earlier and the initial increase disappeared (Fig. 4f). The magnitude of fluctuations remained at around 4 %. Overall, these data show that gene position on the chromosome affects cell cycle-related noise.
In summary, we found that the cell cycle can be a major causal factor of observed noise in the rate of gene expression (52 %), with the rest coming from other sources such as metabolism [28, 42, 43], transcription factors [8], or expression machinery [18] (Fig. 5a). Within the cell cycle contribution, the data suggests two components: a deterministic mean determined by the cell phase (29 %), and a stochastic contribution caused by variability in the timing of replication (23 %) (Fig. 5b). The initially flat production rate suggested gene copy number is the main factor in cell cycle-induced expression rate variations, though alternative factors, such as cell cycle-induced variations in transcription factor concentrations, could also contribute.
Summary of observed contributions to gene expression noise. a The cell cycle causes fluctuations in the protein production rate, through deterministic and stochastic contributions. Other non-cell cycle-related sources contribute as well. The fluctuations in protein concentration are determined by protein production noise and dilution due to growth. b Contributions of the different noise sources, as described in panel a, as fractions of the total observed variance in gene expression (Additional file 2: Table S1)
The analysis indicated a noise-cancelling mechanism: even sudden two-fold production rate increases caused by replication of the gene are effectively compensated for by an acceleration of dilution due to exponential growth [26, 27] (Fig. 5b). The observed minor effect of the cell cycle on the protein concentration is thus due to a passive homeostasis mechanism that exploits the balance between synthesis and dilution. When proteins are actively degraded, this noise cancelling mechanism would be less efficient. We note that a similar, but likely active, balancing between synthesis and dilution was observed in mammalian cells where transcription rate is adjusted to cell size [44, 45]. The homeostatic mechanism we observed does not necessarily act on noise from other sources, such as fluctuations in RNA polymerase availability [18] or transcription factors [2], if they are not synchronized with exponential volume growth. Indeed, concentrations do display significant noise intensities (0.15 for slow growth, 0.10 for fast growth). We note that other cancelling mechanisms can act on non-cell cycle expression noise. For instance, metabolic noise that causes expression noise is partially compensated for by increased growth [28].
Our findings provide insight into how elementary processes, such as gene replication events and volume growth, can cause and filter noise in bacterial cells. Elucidating the sources of gene expression noise is important to obtaining a bottom-up understanding of cellular heterogeneity, cellular homeostasis, and cell cycle regulation, and to providing input for mathematical models of gene expression networks. Our results confirm previous demonstrations that variance decomposition can be a useful tool in disentangling different noise sources within cells.
Strains, growth media, and cell growth
E. coli strain MG22 [1] was used for all experiments unless noted otherwise. This strain is a derivative of MG1655 that contains yfp and cfp under control of a lac promoter, which were inserted into the chromosome at the intC and galK locus. These two loci are equidistant from the origin of replication, on opposite halves of the circular chromosome. Additionally, we used strain ASC636 in which gene lacA of the lac operon was replaced by gfp (constructed by A. Böhm).
For microscopy experiments we used either M9 minimal medium (main text figures) or rich defined medium (MOPS EZ rich defined medium from Teknova, Additional file 2: Figure S2). M9 was supplemented with 200 μM uracil and 0.1 % maltose was added as a carbon source, yielding a growth rate of 0.6 dbl/h. To the rich medium we added 0.2 % glycerol as a carbon source, yielding a growth rate of 1.8 dbl/h. In all experiments we also added 200 μM IPTG to fully induce the lac promoters.
Cells were inoculated in the morning from −80 °C glycerol stock into tryptone yeast (TY) medium and grown for 7 h at 37 °C. Then, cells were diluted highly (~1/10,000 to ~1/100,000) to three different final concentrations into the defined experimental medium (see above). TY was thereby diluted to <0.05 vol% and remaining TY was consumed by the cells. Cells were grown overnight and the next morning a falcon tube with still exponentially growing cells (OD600 < 0.2) was chosen. Cells were diluted again to OD600 = 0.005 and then used for microscopy.
Sample preparation and microscopy
The sample preparation for microscopy was done in the warm room at 37 °C to minimize temperature stress. We placed a polyacrylamide gel into a glass chamber, pipetted 1 μL of cells onto the gel and closed it with a coverslip. The sample was sealed with a metal clamp (Additional file 2: Figure S7a). This setup provides an oxygen reservoir for the cells but avoids gel dehydration.
To produce polyacrylamide gels, we mixed 1.25 mL 40 % acrylamide, 3.7 mL water, 50 μL fresh 10 % ammonium persulfate, and 5 μL TEMED, poured it into a glass chamber and let it polymerize [28, 46]. Then, the gel was cut into pieces of ~1 cm2 and stored in a falcon tube with water (Milli-Q). Before the experiment a gel pad was placed into a falcon tube with 5 mL of the respective growth medium and the medium was exchanged several times.
Cells were imaged with a Nikon TE2000-E inverted microscope using a 100× oil objective (Nikon, Plan Fluor NA 1.3) and additionally 1.5× intermediate magnification. It was equipped with a cooled CCD camera (Photometrix, Cool-Snap HQ), a Xenon arc lamp with liquid light guide (Sutter, Lambda LS) for fluorescence illumination, computer controlled shutters (Sutter, Lambda 10–3 with SmartShutter), and an automated stage (Märzhäuser). The microscope was located in an incubation chamber (Solent) to keep the temperature at 37 °C. Fluorescence filters were obtained from Chroma and we used #49003 (YFP), #49001 (CFP), and #41017 (GFP). Microscope and image acquisition were computer controlled with MetaMorph software (Molecular Devices).
We searched the pad for isolated cells and manually saved the positions of 4–6 cells. Then, an automated script was looped repeatedly (for up to 20 h) over all positions and images of the growing microcolonies were acquired. Loop times were adjusted to cell doubling times and we used 100 sec/loop for μ = 0.6 dbl/h (main text figures) and 45 sec/loop for μ = 1.8 dbl/h (Additional file 2: Figure S2). Each loop, the routine refocused based on image contrast (Brenner algorithm) and acquired phase contrast images at three different heights (±0.2 μm off focus and in focus). Fluorescence images were acquired approximately every 7 loops to reduce photodamage and bleaching. This resulted in 55 phase contrast and 8 fluorescence images per cell cycle, on average. Fluorescence images were 2 × 2 binned to increase signal-to-noise ratio and illumination was kept as short as possible (YFP: 25 ms, CFP: 30 ms, GFP: 50 ms). Growth rates under the microscope were very similar to bulk growth rates in a plate reader.
Movies were obtained until cells formed a second layer or outgrew the field of view, which happened usually after 8–9 generations and a colony size of several hundred cells. Each experiment was performed at least twice (Additional file 2: Figure S1). Data of YFP and CFP (which are encoded in the same strain) that is displayed in a single figure was obtained from the same microcolony in the same experiment (e.g. Fig. 4ab, Additional file 2: Figure S1 and Figure S2). The replicate displayed in Additional file 2: Figure S1 is a different experiment than Fig. 1. Experiments at different growth rates (e.g. Fig. 4ab versus Additional file 2: Figure S2) are also independent experiments. GFP is encoded in a different strain, and hence the GFP data was measured in different experiments than the YFP and CFP data.
Images were analyzed with custom MATLAB software (MathWorks) based on Schnitzcells [47]. An automatic script determined cell outlines by applying a Laplacian of Gaussian filter to the averaged phase contrast image and then cut accidently connected groups of cells based on the concavity of the cell outline (Additional file 2: Figure S7b). Segmentation was checked manually and corrected if necessary. Cells were then tracked from image to image, resulting in a branch-structured lineage tree, and tracking was checked manually. For all cells with an observed complete cell cycle each measured time point was associated with a cell cycle phase, with 0 being cell birth and 1 cell division.
Cellular length was used as measure for cell size because the rod shaped E. coli only grow along their long axis and cell width is independent of cell cycle phase (Additional file 2: Figure S8, ref. [48], contrary to results in ref. [49]). We fitted a third degree polynomial through the silhouette of the segmented cell [28]. The cell length was then obtained by integrating the curve between both cell poles.
Fluorescence images were corrected for alignment offset, background (camera noise), uneven illumination, and were deconvolved to suppress blur (Additional file 2: Figure S7c). To obtain autofluorescence intensity we measured a non-fluorescent strain (MG1655) with the same illumination settings as standard experiments. Measured signal intensity was 2.5 % (CFP) resp. 0.4 % (YFP) of the actual concentration signal from fluorescent proteins. Fluorescence signals were not corrected for autofluorescence because this was small compared to the signal and only introduced a constant offset with no effect on the results.
The total fluorescence of each cell was determined by summing up the pixel intensities within the cellular outline. The protein production rate p(t) was determined as the slope of a linear fit of three consecutive total fluorescence data points centered around t. For the first and last measurement of the cell cycle, fluorescence information of parent and daughter cells, respectively, was used to determine the slope. Protein concentrations were calculated by dividing total fluorescence by the segmented cellular area.
We analyzed cell cycles within a time window of the experiment that showed a constant population-mean growth rate and mean protein concentration. Population mean growth and concentration were considered constant when they fluctuated less than 5 % around the long-term average (for GFP in strain ASC636 10 % was used as cutoff criterion). Cells that stopped growing or were filamented were removed (less than 15 cells per dataset). The fraction of analyzed cells relative to all cells observed with a complete cell cycle was over 86 % for MG22 datasets and over 50 % for ASC636 datasets. The main conclusions were robust to taking the complete data set of growing non-filamentous cells with complete cell cycles. Datasets contained between 215 selected cells (large cells in rich medium) to 435 selected cells (minimal medium). If one dataset was used for multiple plots (e.g. Figs. 1 and 4a,b), the same cells were analyzed.
Traces of production rate (Fig. 3) for individual cells were considered 'step-traces' when they deviated from a fitted step-trace (see also Additional file 2) less than a fixed threshold. To be considered a step-trace, the mean squared deviation of a data point on the trace from the fitted value had to be below 2 % of the squared trace-average. Figures and percentages in the main text are determined from one microcolony per strain and growth condition. Results for the repeat experiment are shown in Additional file 2: Figure S1.
To determine the average dependence of a signal (e.g. concentration) on cell cycle phase we binned the signal according to phase and averaged it within each bin (Fig. 1b,d,f). Error bars are standard errors of the mean from a resampled distribution of the signal, obtained by bootstrapping from the experimental data for each bin.
The contribution of a specific noise source X (for example, deterministic cell cycle variations) to total protein production noise was calculated by using the additivity of variances for independent variables. The production rate p was, for example, written as sum p = X + Y with Y being the unknown, unmeasured, fluctuations (such as δp(ϕ, x), see also main text). Then, Var(p) = Var(X) + Var(Y), and the fraction of the variance in p, which is caused by X, is Var(X)/Var(p). We normalized variables such as p by their mean so that the squared noise is identical to the variance. For example, the contribution of \( \overline{p_c}\left(\phi \right) \) to p(ϕ, x) is 0.262/0.482. For protein concentration, the calculation is identical.
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This work is part of the research program of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organization for Scientific Research (NWO).
FOM Institute AMOLF, Amsterdam, 1098 XG, The Netherlands
Noreen Walker, Philippe Nghe & Sander J. Tans
Noreen Walker
Philippe Nghe
Sander J. Tans
Correspondence to Sander J. Tans.
NW and SJT designed the research. NW performed the experiments. NW, PN, and SJT analyzed the data. NW and PN developed the model. NW, PN, and SJT wrote the paper. All authors read and approved the final manuscript.
Movie S1. Time lapse movie of growing cells. Phase contrast and yfp fluorescence channel for MG22 cells grown on M9 + 0.1 % maltose + 200 μM Iptg (for example Fig. 1). The movie covers a time of 12 h 40 min. (AVI 863 kb)
Additional file 2: Figures S1-S8, Table S1.
Supplementary methods. (DOCX 1071 kb)
Walker, N., Nghe, P. & Tans, S.J. Generation and filtering of gene expression noise by the bacterial cell cycle. BMC Biol 14, 11 (2016). https://doi.org/10.1186/s12915-016-0231-z
Single-cell microscopy
Beyond Mendel: modeling in biology | CommonCrawl |
Burgers' equation
Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation[1] occurring in various areas of applied mathematics, such as fluid mechanics,[2] nonlinear acoustics,[3] gas dynamics, and traffic flow.[4] The equation was first introduced by Harry Bateman in 1915[5][6] and later studied by Johannes Martinus Burgers in 1948.[7]
For a given field $u(x,t)$ and diffusion coefficient (or kinematic viscosity, as in the original fluid mechanical context) $\nu $, the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system:
${\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}=\nu {\frac {\partial ^{2}u}{\partial x^{2}}}.$
When the diffusion term is absent (i.e. $\nu =0$), Burgers' equation becomes the inviscid Burgers' equation:
${\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}=0,$
which is a prototype for conservation equations that can develop discontinuities (shock waves). The previous equation is the advective form of the Burgers' equation. The conservative form is found to be more useful in numerical integration
${\frac {\partial u}{\partial t}}+{\frac {1}{2}}{\frac {\partial (u^{2})}{\partial x}}=0.$
Terms
There are 4 parameters in Burgers' equation: $u,x,t$ and $\nu $. In a system consisting of a moving viscous fluid with one spatial ($x$) and one temporal ($t$) dimension, e.g. a thin ideal pipe with fluid running through it, Burgers' equation describes the speed of the fluid at each location along the pipe as time progresses. The terms of the equation represent the following quantities:[8]
• $x$: spatial coordinate
• $t$: temporal coordinate
• $u(x,t)$: speed of fluid at the indicated spatial and temporal coordinates
• $\nu $: viscosity of fluid
The viscosity is a constant physical property of the fluid, and the other parameters represent the dynamics contingent on that viscosity.
Inviscid Burgers' equation
The inviscid Burgers' equation is a conservation equation, more generally a first order quasilinear hyperbolic equation. The solution to the equation and along with the initial condition
${\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}=0,\quad u(x,0)=f(x)$
can be constructed by the method of characteristics. The characteristic equations are
${\frac {dx}{dt}}=u,\quad {\frac {du}{dt}}=0.$
Integration of the second equation tells us that $u$ is constant along the characteristic and integration of the first equation shows that the characteristics are straight lines, i.e.,
$u=c,\quad x=ut+\xi $
where $\xi $ is the point (or parameter) on the x-axis (t = 0) of the x-t plane from which the characteristic curve is drawn. Since $u$ at $x$-axis is known from the initial condition and the fact that $u$ is unchanged as we move along the characteristic emanating from each point $x=\xi $, we write $u=c=f(\xi )$ on each characteristic. Therefore, the family of trajectories of characteristics parametrized by $\xi $ is
$x=f(\xi )t+\xi .$
Thus, the solution is given by
$u(x,t)=f(\xi )=f(x-ut),\quad \xi =x-f(\xi )t.$
This is an implicit relation that determines the solution of the inviscid Burgers' equation provided characteristics don't intersect. If the characteristics do intersect, then a classical solution to the PDE does not exist and leads to the formation of a shock wave. Whether characteristics can intersect or not depends on the initial condition. In fact, the breaking time before a shock wave can be formed is given by[9]
$t_{b}=\inf _{x}\left({\frac {-1}{f'(x)}}\right)$
Inviscid Burgers' equation for linear initial condition
Subrahmanyan Chandrasekhar provided the explicit solution in 1943 when the initial condition is linear, i.e., $f(x)=ax+b$, where a and b are constants.[10] The explicit solution is
$u(x,t)={\frac {ax+b}{at+1}}.$
This solution is also the complete integral of the inviscid Burgers' equation because it contains as many arbitrary constants as the number of independent variables appearing in the equation.[11] Using this complete integral, Chandrasekhar obtained the general solution described for arbitrary initial conditions from the envelope of the complete integral.
Viscous Burgers' equation
The viscous Burgers' equation can be converted to a linear equation by the Cole–Hopf transformation,[12][13][14]
$u=-2\nu {\frac {1}{\phi }}{\frac {\partial \phi }{\partial x}},$
which turns it into the equation
${\frac {\partial }{\partial x}}\left({\frac {1}{\phi }}{\frac {\partial \phi }{\partial t}}\right)=\nu {\frac {\partial }{\partial x}}\left({\frac {1}{\phi }}{\frac {\partial ^{2}\phi }{\partial x^{2}}}\right)$
which can be integrated with respect to $x$ to obtain
${\frac {\partial \phi }{\partial t}}=\nu {\frac {\partial ^{2}\phi }{\partial x^{2}}}+g(t)\phi $
where $g(t)$ is a function that depends on boundary conditions. If $g(t)=0$ identically (e.g. if the problem is to be solved on a periodic domain), then we get the diffusion equation
${\frac {\partial \phi }{\partial t}}=\nu {\frac {\partial ^{2}\phi }{\partial x^{2}}}.$
The diffusion equation can be solved, and the Cole–Hopf transformation inverted, to obtain the solution to the Burgers' equation:
$u(x,t)=-2\nu {\frac {\partial }{\partial x}}\ln \left\{(4\pi \nu t)^{-1/2}\int _{-\infty }^{\infty }\exp \left[-{\frac {(x-x')^{2}}{4\nu t}}-{\frac {1}{2\nu }}\int _{0}^{x'}f(x'')dx''\right]dx'\right\}.$
Other forms
Generalized Burgers' equation
The generalized Burgers' equation extends the quasilinear convective to more generalized form, i.e.,
${\frac {\partial u}{\partial t}}+c(u){\frac {\partial u}{\partial x}}=\nu {\frac {\partial ^{2}u}{\partial x^{2}}}.$
where $c(u)$ is any arbitrary function of u. The inviscid $\nu =0$ equation is still a quasilinear hyperbolic equation for $c(u)>0$ and its solution can be constructed using method of characteristics as before.[15]
Stochastic Burgers' equation
Added space-time noise $\eta (x,t)={\dot {W}}(x,t)$, where $W$ is an $L^{2}(\mathbb {R} )$ Wiener process, forms a stochastic Burgers' equation[16]
${\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}=\nu {\frac {\partial ^{2}u}{\partial x^{2}}}-\lambda {\frac {\partial \eta }{\partial x}}.$
This stochastic PDE is the one-dimensional version of Kardar–Parisi–Zhang equation in a field $h(x,t)$ upon substituting $u(x,t)=-\lambda \partial h/\partial x$.
See also
• Euler–Tricomi equation
• Chaplygin's equation
• Conservation equation
• Fokker–Planck equation
References
1. Misra, Souren; Raghurama Rao, S. V.; Bobba, Manoj Kumar (2010-09-01). "Relaxation system based sub-grid scale modelling for large eddy simulation of Burgers' equation". International Journal of Computational Fluid Dynamics. 24 (8): 303–315. Bibcode:2010IJCFD..24..303M. doi:10.1080/10618562.2010.523518. ISSN 1061-8562. S2CID 123001189.
2. It relates to the Navier–Stokes momentum equation with the pressure term removed Burgers Equation (PDF): here the variable is the flow speed y=u
3. It arises from Westervelt equation with an assumption of strictly forward propagating waves and the use of a coordinate transformation to a retarded time frame: here the variable is the pressure
4. Musha, Toshimitsu; Higuchi, Hideyo (1978-05-01). "Traffic Current Fluctuation and the Burgers Equation". Japanese Journal of Applied Physics. 17 (5): 811. Bibcode:1978JaJAP..17..811M. doi:10.1143/JJAP.17.811. ISSN 1347-4065. S2CID 121252757.
5. Bateman, H. (1915). "Some recent researches on the motion of fluids". Monthly Weather Review. 43 (4): 163–170. Bibcode:1915MWRv...43..163B. doi:10.1175/1520-0493(1915)43<163:SRROTM>2.0.CO;2.
6. Whitham, G. B. (2011). Linear and nonlinear waves (Vol. 42). John Wiley & Sons.
7. Burgers, J. M. (1948). "A Mathematical Model Illustrating the Theory of Turbulence". Advances in Applied Mechanics. 1: 171–199. doi:10.1016/S0065-2156(08)70100-5. ISBN 9780123745798.
8. Cameron, Maria. "Notes on Burgers's Equation" (PDF).
9. Olver, Peter J. (2013). Introduction to Partial Differential Equations. Undergraduate Texts in Mathematics. Online: Springer. p. 37. doi:10.1007/978-3-319-02099-0. ISBN 978-3-319-02098-3. S2CID 220617008.
10. Chandrasekhar, S. (1943). On the decay of plane shock waves (Report). Ballistic Research Laboratories. Report No. 423.
11. Forsyth, A. R. (1903). A Treatise on Differential Equations. London: Macmillan.
12. Cole, Julian (1951). "On a quasi-linear parabolic equation occurring in aerodynamics". Quarterly of Applied Mathematics. 9 (3): 225–236. doi:10.1090/qam/42889. JSTOR 43633894.
13. Eberhard Hopf (September 1950). "The partial differential equation ut + uux = μuxx". Communications on Pure and Applied Mathematics. 3 (3): 201–230. doi:10.1002/cpa.3160030302.
14. Kevorkian, J. (1990). Partial Differential Equations: Analytical Solution Techniques. Belmont: Wadsworth. pp. 31–35. ISBN 0-534-12216-7.
15. Courant, R., & Hilbert, D. Methods of Mathematical Physics. Vol. II.
16. Wang, W.; Roberts, A. J. (2015). "Diffusion Approximation for Self-similarity of Stochastic Advection in Burgers' Equation". Communications in Mathematical Physics. 333 (3): 1287–1316. arXiv:1203.0463. Bibcode:2015CMaPh.333.1287W. doi:10.1007/s00220-014-2117-7. S2CID 119650369.
External links
• Burgers' Equation at EqWorld: The World of Mathematical Equations.
• Burgers' Equation at NEQwiki, the nonlinear equations encyclopedia.
| Wikipedia |
Upwind scheme
In computational physics, the term upwind scheme (sometimes advection scheme) typically refers to a class of numerical discretization methods for solving hyperbolic partial differential equations, in which so-called upstream variables are used to calculate the derivatives in a flow field. That is, derivatives are estimated using a set of data points biased to be more "upwind" of the query point, with respect to the direction of the flow. Historically, the origin of upwind methods can be traced back to the work of Courant, Isaacson, and Rees who proposed the CIR method.[1]
Model equation
To illustrate the method, consider the following one-dimensional linear advection equation
${\frac {\partial u}{\partial t}}+a{\frac {\partial u}{\partial x}}=0$
which describes a wave propagating along the $x$-axis with a velocity $a$. This equation is also a mathematical model for one-dimensional linear advection. Consider a typical grid point $i$ in the domain. In a one-dimensional domain, there are only two directions associated with point $i$ – left (towards negative infinity) and right (towards positive infinity). If $a$ is positive, the traveling wave solution of the equation above propagates towards the right, the left side of $i$ is called upwind side and the right side is the downwind side. Similarly, if $a$ is negative the traveling wave solution propagates towards the left, the left side is called downwind side and right side is the upwind side. If the finite difference scheme for the spatial derivative, $\partial u/\partial x$ contains more points in the upwind side, the scheme is called an upwind-biased or simply an upwind scheme.
First-order upwind scheme
The simplest upwind scheme possible is the first-order upwind scheme. It is given by[2]
${\frac {u_{i}^{n+1}-u_{i}^{n}}{\Delta t}}+a{\frac {u_{i}^{n}-u_{i-1}^{n}}{\Delta x}}=0\quad {\text{for}}\quad a>0$
(1)
${\frac {u_{i}^{n+1}-u_{i}^{n}}{\Delta t}}+a{\frac {u_{i+1}^{n}-u_{i}^{n}}{\Delta x}}=0\quad {\text{for}}\quad a<0$
(2)
where $n$ refers to the $t$ dimension and $i$ refers to the $x$ dimension. (By comparison, a central difference scheme in this scenario would look like
${\frac {u_{i}^{n+1}-u_{i}^{n}}{\Delta t}}+a{\frac {u_{i+1}^{n}-u_{i-1}^{n}}{2\Delta x}}=0,$
regardless of the sign of $a$.)
Compact form
Defining
$a^{+}={\text{max}}(a,0)\,,\qquad a^{-}={\text{min}}(a,0)$
and
$u_{x}^{-}={\frac {u_{i}^{n}-u_{i-1}^{n}}{\Delta x}}\,,\qquad u_{x}^{+}={\frac {u_{i+1}^{n}-u_{i}^{n}}{\Delta x}}$
the two conditional equations (1) and (2) can be combined and written in a compact form as
$u_{i}^{n+1}=u_{i}^{n}-\Delta t\left[a^{+}u_{x}^{-}+a^{-}u_{x}^{+}\right]$
(3)
Equation (3) is a general way of writing any upwind-type schemes.
Stability
The upwind scheme is stable if the following Courant–Friedrichs–Lewy condition (CFL) is satisfied.[3]
$c=\left|{\frac {a\Delta t}{\Delta x}}\right|\leq 1$ and $0\leq a$.
A Taylor series analysis of the upwind scheme discussed above will show that it is first-order accurate in space and time. Modified wavenumber analysis shows that the first-order upwind scheme introduces severe numerical diffusion/dissipation in the solution where large gradients exist due to necessity of high wavenumbers to represent sharp gradients.
Second-order upwind scheme
The spatial accuracy of the first-order upwind scheme can be improved by including 3 data points instead of just 2, which offers a more accurate finite difference stencil for the approximation of spatial derivative. For the second-order upwind scheme, $u_{x}^{-}$ becomes the 3-point backward difference in equation (3) and is defined as
$u_{x}^{-}={\frac {3u_{i}^{n}-4u_{i-1}^{n}+u_{i-2}^{n}}{2\Delta x}}$
and $u_{x}^{+}$ is the 3-point forward difference, defined as
$u_{x}^{+}={\frac {-u_{i+2}^{n}+4u_{i+1}^{n}-3u_{i}^{n}}{2\Delta x}}$
This scheme is less diffusive compared to the first-order accurate scheme and is called linear upwind differencing (LUD) scheme.
See also
• Finite difference method
• Upwind differencing scheme for convection
• Godunov's scheme
References
1. Courant, Richard; Isaacson, E; Rees, M. (1952). "On the Solution of Nonlinear Hyperbolic Differential Equations by Finite Differences". Comm. Pure Appl. Math. 5 (3): 243..255. doi:10.1002/cpa.3160050303.
2. Patankar, S. V. (1980). Numerical Heat Transfer and Fluid Flow. Taylor & Francis. ISBN 978-0-89116-522-4.
3. Hirsch, C. (1990). Numerical Computation of Internal and External Flows. John Wiley & Sons. ISBN 978-0-471-92452-4.
Numerical methods for partial differential equations
Finite difference
Parabolic
• Forward-time central-space (FTCS)
• Crank–Nicolson
Hyperbolic
• Lax–Friedrichs
• Lax–Wendroff
• MacCormack
• Upwind
• Method of characteristics
Others
• Alternating direction-implicit (ADI)
• Finite-difference time-domain (FDTD)
Finite volume
• Godunov
• High-resolution
• Monotonic upstream-centered (MUSCL)
• Advection upstream-splitting (AUSM)
• Riemann solver
• Essentially non-oscillatory (ENO)
• Weighted essentially non-oscillatory (WENO)
Finite element
• hp-FEM
• Extended (XFEM)
• Discontinuous Galerkin (DG)
• Spectral element (SEM)
• Mortar
• Gradient discretisation (GDM)
• Loubignac iteration
• Smoothed (S-FEM)
Meshless/Meshfree
• Smoothed-particle hydrodynamics (SPH)
• Peridynamics (PD)
• Moving particle semi-implicit method (MPS)
• Material point method (MPM)
• Particle-in-cell (PIC)
Domain decomposition
• Schur complement
• Fictitious domain
• Schwarz alternating
• additive
• abstract additive
• Neumann–Dirichlet
• Neumann–Neumann
• Poincaré–Steklov operator
• Balancing (BDD)
• Balancing by constraints (BDDC)
• Tearing and interconnect (FETI)
• FETI-DP
Others
• Spectral
• Pseudospectral (DVR)
• Method of lines
• Multigrid
• Collocation
• Level-set
• Boundary element
• Method of moments
• Immersed boundary
• Analytic element
• Isogeometric analysis
• Infinite difference method
• Infinite element method
• Galerkin method
• Petrov–Galerkin method
• Validated numerics
• Computer-assisted proof
• Integrable algorithm
• Method of fundamental solutions
| Wikipedia |
\begin{document}
\title[Motion of a Rigid body in a Compressible Fluid with Navier-slip boundary condition]{Motion of a Rigid body in a Compressible Fluid with Navier-slip boundary condition}
\date{\today}
\author{\v S. Ne\u{c}asov\'a} \address{Institute of Mathematics, Czech Academy of Sciences, \v Zitn\' a 25, 11567 Praha 1, Czech Republic} \email{[email protected]}
\author{M. Ramaswamy} \address{NASI Senior Scientist, ICTS-TIFR, Survey No. 151, Sivakote, Bangalore, 560089, India} \email{[email protected]}
\author{A. Roy} \address{Institute of Mathematics, Czech Academy of Sciences, \v Zitn\' a 25, 11567 Praha 1, Czech Republic} \email{[email protected]}
\author{A. Schl\"{o}merkemper} \address{Institute of Mathematics, University of W\"urzburg, Emil-Fischer-Str.~40, 97074 W\"urzburg, Germany} \email {[email protected]}
\begin{abstract} In this work, we study the motion of a rigid body in a bounded domain which is filled with a compressible isentropic fluid. We consider the Navier-slip boundary condition at the interface as well as at the boundary of the domain. This is the first mathematical analysis of a compressible fluid-rigid body system where Navier-slip boundary conditions are considered. We prove existence of a weak solution of the fluid-structure system up to collision. \end{abstract}
\maketitle
\tableofcontents
\section{Introduction}\label{sec_intro} Let $\Omega \subset \mathbb{R}^3$ be a bounded smooth domain occupied by a fluid and a rigid body. Let the rigid body $\mathcal{S}(t)$ be a regular, bounded domain and moving inside $\Omega$. The motion of the rigid body is governed by the balance equations for linear and angular momentum. We assume that the fluid domain $\mathcal{F}(t)=\Omega \setminus \overline{\mathcal{S}(t)}$ is filled with a viscous isentropic compressible fluid. We also assume the slip boundary conditions at the interface of the interaction of the fluid and the rigid body as well as at $\partial \Omega$. The evolution of this fluid-structure system can be described by the following equations \begin{align} \frac{\partial {\rho}_{\mathcal{F}}}{\partial t} + \operatorname{div}({\rho_{\mathcal{F}}} u_{\mathcal{F}}) =0, \quad &\forall \ (t,x)\in (0,T)\times\mathcal{F}(t),\label{mass:comfluid}\\ \frac{\partial ({\rho_{\mathcal{F}}} u_{\mathcal{F}})}{\partial t}+ \operatorname{div}({\rho_{\mathcal{F}}} u_{\mathcal{F}}\otimes u_{\mathcal{F}})- \operatorname{div} \mathbb{T}(u_{\mathcal{F}})+\nabla p_{\mathcal{F}} =\rho_{\mathcal{F}}g_{\mathcal{F}},\quad &\forall \ (t,x)\in (0,T)\times \mathcal{F}(t),\label{momentum:comfluid}\\ mh''(t)= -\int\limits_{\partial \mathcal{S}(t)} \left(\mathbb{T}(u_{\mathcal{F}}\right) - p_{\mathcal{F}}\mathbb{I}) \nu\, d\Gamma + \int\limits_{\mathcal{S}(t)} \rho_{\mathcal{S}}g_{\mathcal{S}}\, dx,\quad &\forall \ t\in (0,T), \label{linear momentumcomp:body}\\ (J\omega)'(t) = -\int\limits_{\partial \mathcal{S}(t)} (x-h(t)) \times (\mathbb{T}(u_{\mathcal{F}}) - p_{\mathcal{F}}\mathbb{I}) \nu\, d\Gamma + \int\limits_{\mathcal{S}(t)} (x-h(t)) \times \rho_{\mathcal{S}}g_{\mathcal{S}}\, dx,\quad &\forall \ t\in (0,T), \label{angular momentumcomp:body} \end{align} the boundary conditions \begin{align} u_{\mathcal{F}}\cdot \nu = u_{\mathcal{S}} \cdot \nu, \quad &\forall \ (t,x) \in (0,T)\times \partial \mathcal{S}(t), \label{boundarycomp-1}\\ (\mathbb{T}(u_{\mathcal{F}}) \nu)\times \nu = -\alpha (u_{\mathcal{F}}-u_{\mathcal{S}})\times \nu, \quad &\forall \ (t,x) \in (0,T)\times\partial \mathcal{S}(t), \label{boundarycomp-2}\\ u_{\mathcal{F}}\cdot \nu = 0, \quad &\forall \ (t,x) \in (0,T)\times \partial \Omega, \label{boundarycomp-3}\\ (\mathbb{T}(u_{\mathcal{F}}) \nu)\times \nu = -\alpha (u_{\mathcal{F}}\times \nu), \quad &\forall \ (t,x) \in (0,T)\times \partial \Omega, \label{boundarycomp-4} \end{align}
and the initial conditions \begin{align} {\rho_{\mathcal{F}}}(0,x)=\rho_{\mathcal{F}_{0}}(x),\quad ({\rho_{\mathcal{F}}}u_{\mathcal{F}})(0,x)=q_{\mathcal{F}_0}(x), & \quad \forall \ x\in \mathcal{F}_0,\label{initial cond}\\
h(0)=0,\quad h'(0)=\ell_{0},\quad \omega(0)=\omega_{0}.\label{initial cond:comp} \end{align}
The fluid occupies, at $t=0$, the domain $\mathcal{F}_0=\Omega \setminus \mathcal{S}_0$, where the initial position of the rigid body is $\mathcal{S}_0$. In equations \eqref{linear momentumcomp:body}--\eqref{boundarycomp-4}, $\nu(t , x )$ is the unit normal to $\partial\mathcal{S}(t)$ at the point $x \in \partial\mathcal{S}(t)$, directed to the interior of the body. In \eqref{momentum:comfluid} and \eqref{linear momentumcomp:body}--\eqref{angular momentumcomp:body}, $g_{\mathcal{F}}$ and $g_{\mathcal{S}}$ are the densities of the volume forces on the fluid and on the rigid body, respectively. Moreover, $\alpha >0$ is a coefficient of friction.
Here, the notation $u \otimes v$ is the tensor product for two vectors $u,v \in \mathbb{R}^3$ and it is defined as $u \otimes v=(u_{i}v_{j})_{1\leqslant i,j \leqslant 3}$. In the above equations, $\rho_{\mathcal{F}}$ and $u_{\mathcal{F}}$ represent respectively the mass density and the velocity of the fluid, and the pressure of the fluid is denoted by $p_{\mathcal{F}}$.
We assume that the flow is in the barotropic regime and we focus on the isentropic case where the relation between $p_{\mathcal{F}}$ and $\rho_{\mathcal{F}}$ is given by the constitutive law: \begin{equation} \label{const-law} p_{\mathcal{F}}= a_{\mathcal{F}}\rho_{\mathcal{F}}^{\gamma}, \end{equation} with $a_{\mathcal{F}}>0$ and the adiabatic constant $\gamma > \frac{3}{2}$, which is a necessary assumption for the existence of a weak solution of compressible fluids (see for example \cite{EF70}).
As is common, we set $$\mathbb{T}(u_{\mathcal{F}})= 2\mu_{\mathcal{F}} \mathbb{D}(u_{\mathcal{F}}) + \lambda_{\mathcal{F}}\operatorname{div}u_{\mathcal{F}}\mathbb{I},$$ where $\mathbb{D}(u_{\mathcal{F}})=\frac{1}{2}\left(\nabla u_{\mathcal{F}} + \nabla u_{\mathcal{F}}^{\top}\right)$ denotes the symmetric part of the velocity gradient, $\nabla u_{\mathcal{F}}^{\top}$ is the transpose of the matrix $\nabla u_{\mathcal{F}}$, and $\lambda_{\mathcal{F}},\mu_{\mathcal{F}}$ are the viscosity coefficients satisfying \begin{equation*} \mu_{\mathcal{F}} > 0, \quad 3\lambda_{\mathcal{F}} + 2\mu_{\mathcal{F}} \geqslant 0. \end{equation*}
The Eulerian velocity $u_{\mathcal{S}}(t,x)$ at each point $x\in \mathcal{S}(t)$ of the rigid body is given by \begin{equation}\label{Svel1} u_{\mathcal{S}}(t,x)= h'(t)+ \omega(t) \times (x-h(t)), \end{equation} where $h(t)$ is the centre of mass and $h'(t)$, $\omega(t)$ are the linear and angular velocities of the rigid body. We remark that $\mathcal{S}(t)$ is uniquely defined by $h(t)$, $\omega(t)$ and $\mathcal{S}_0$. Similarly, the knowledge of $\mathcal{S}(t)$ and $\mathcal{S}_0$ yields $h(t)$ and $\omega(t)$. The initial velocity of the rigid body is given by \begin{equation}\label{Svel2} u_{\mathcal{S}}(0,x)= u_{\mathcal{S}_0}:= \ell_0+ \omega_0 \times x,\quad x\in \mathcal{S}_0. \end{equation} Here the mass density of the body $\rho_{\mathcal{S}}$ satisfies the following continuity equation \begin{equation}\label{eq:vrBeq}
\frac{\partial \rho_{\mathcal{S}}}{\partial t} + u_\mathcal{S}\cdot\nabla \rho_{\mathcal{S}} = 0, \quad \forall \ (t,x)\in (0,T)\times\mathcal{S}(t),\quad \rho_{\mathcal{S}}(0,x)=\rho_{\mathcal{S}_0}(x),\quad \forall \ x\in \mathcal{S}_0. \end{equation} Moreover, $m$ is the mass and $J(t)$ is the moment of inertia matrix of the solid. We express $h(t)$, $m$ and $J(t)$ in the following way: \begin{align} m &= \int\limits_{\mathcal{S}(t)} \rho_{\mathcal{S}} \ dx, \label{def:m} \\
h(t) &= \frac{1}{m} \int\limits_{\mathcal{S}(t)} \rho_{\mathcal{S}} \ x \ dx, \\
J(t) &= \int\limits_{\mathcal{S}(t)} \rho_{\mathcal{S}} \big[ |x-h(t)|^2\mathbb{I} - (x-h(t)) \otimes (x-h(t)) \big] \ dx. \label{def:J} \end{align} In the remainder of this introduction, we present the weak formulation of the system, discuss our main result regarding the existence of weak solutions and put it in a larger perspective.
\subsection{Weak formulation}\label{S2} We derive a weak formulation with the help of multiplication by appropriate test functions and integration by parts by taking care of the boundary conditions. Due to the presence of the Navier-slip boundary condition, the test functions will be discontinuous across the fluid-solid interface. We introduce the set of rigid velocity fields: \begin{equation} \label{defR} \mathcal{R}=\left\{ \zeta : \Omega \to \mathbb{R}^3 \mid \mbox{There exist }V, r, a \in \mathbb{R}^3 \mbox{ such that }\zeta(x)=V+ r \times \left(x-a\right)\mbox{ for any } x\in\Omega\right\}. \end{equation} For any $T>0$, we define the test function space $V_{T}$ as follows: \begin{equation}\label{def:test} V_{T}= \left\{\!\begin{aligned} &\phi \in C([0,T]; L^2(\Omega))\mbox{ such that there exist }\phi_{\mathcal{F}}\in \mathcal{D}([0,T); \mathcal{D}(\overline{\Omega})),\, \phi_{\mathcal{S}}\in \mathcal{D}([0,T); \mathcal{R})\\ &\mbox{satisfying }\phi(t,\cdot)=\phi_{\mathcal{F}}(t,\cdot)\mbox{ on }\mathcal{F}(t),\quad \phi(t,\cdot)=\phi_{\mathcal{S}}(t,\cdot)\mbox{ on }\mathcal{S}(t)\mbox{ with }\\ &\phi_{\mathcal{F}}(t,\cdot)\cdot \nu = \phi_{\mathcal{S}}(t,\cdot)\cdot \nu \mbox{ on }\partial\mathcal{S}(t),\ \phi_{\mathcal{F}}(t,\cdot)\cdot \nu=0 \mbox{ on }\partial\Omega\mbox{ for all }t\in [0,T] \end{aligned}\right\}, \end{equation} where $\mathcal{D}$ denotes the sets of all infinitely differentiable functions that have compact support. We multiply equation \eqref{momentum:comfluid} by a test function $\phi\in V_T$ and integrate over $\mathcal{F}(t)$ to obtain \begin{multline}\label{express1} \frac{d}{dt} \int\limits_{\mathcal{F}(t)} \rho_{\mathcal{F}}u_{\mathcal{F}}\cdot \phi_{\mathcal{F}} - \int\limits_{\mathcal{F}(t)} \rho_{\mathcal{F}}u_{\mathcal{F}}\cdot \frac{\partial}{\partial t}\phi_{\mathcal{F}} - \int\limits_{\mathcal{F}(t)} (\rho_{\mathcal{F}}u_{\mathcal{F}} \otimes u_{\mathcal{F}}) : \nabla \phi_{\mathcal{F}} + \int\limits_{\mathcal{F}(t)} (\mathbb{T}(u_{\mathcal{F}}) - p_{\mathcal{F}}\mathbb{I}) : \mathbb{D}(\phi_{\mathcal{F}}) \\ = \int\limits_{\partial \Omega} (\mathbb{T}(u_{\mathcal{F}}) - p_{\mathcal{F}}\mathbb{I}) \nu\cdot \phi_{\mathcal{F}} + \int\limits_{\partial \mathcal{S}(t)} (\mathbb{T}(u_{\mathcal{F}}) - p_{\mathcal{F}}\mathbb{I}) \nu\cdot \phi_{\mathcal{F}} +\int\limits_{\mathcal{F}(t)}\rho_{\mathcal{F}}g_{\mathcal{F}}\cdot \phi_{\mathcal{F}}. \end{multline} We use the identity $(A\times B)\cdot (C\times D)=(A\cdot C)(B\cdot D)-(B\cdot C)(A\cdot D)$ to have \begin{equation*} \mathbb{T}(u_{\mathcal{F}}) \nu\cdot \phi_{\mathcal{F}}= [\mathbb{T}(u_{\mathcal{F}}) \nu \cdot \nu](\phi_{\mathcal{F}}\cdot \nu) + [\mathbb{T}(u_{\mathcal{F}}) \nu \times \nu]\cdot (\phi_{\mathcal{F}}\times \nu), \end{equation*} \begin{equation*} \mathbb{T}(u_{\mathcal{F}}) \nu\cdot \phi_{\mathcal{S}}= [\mathbb{T}(u_{\mathcal{F}}) \nu \cdot \nu](\phi_{\mathcal{S}}\cdot \nu) + [\mathbb{T}(u_{\mathcal{F}}) \nu \times \nu]\cdot (\phi_{\mathcal{S}}\times \nu). \end{equation*} Now by using the definition of $V_T$ and the boundary conditions \eqref{boundarycomp-1}--\eqref{boundarycomp-4}, we get \begin{equation}\label{express2} \int\limits_{\partial \Omega} (\mathbb{T}(u_{\mathcal{F}}) - p_{\mathcal{F}}\mathbb{I}) \nu\cdot \phi_{\mathcal{F}} = -\alpha \int\limits_{\partial \Omega} (u_{\mathcal{F}}\times \nu)\cdot (\phi_{\mathcal{F}}\times \nu), \end{equation} \begin{equation}\label{express3} \int\limits_{\partial \mathcal{S}(t)} (\mathbb{T}(u_{\mathcal{F}}) - p_{\mathcal{F}}\mathbb{I}) \nu\cdot \phi_{\mathcal{F}} = -\alpha \int\limits_{\partial \mathcal{S}(t)} [(u_{\mathcal{F}}-u_{\mathcal{S}})\times \nu]\cdot [(\phi_{\mathcal{F}}-\phi_{\mathcal{S}})\times \nu] + \int\limits_{\partial \mathcal{S}(t)} (\mathbb{T}(u_{\mathcal{F}}) - p_{\mathcal{F}}\mathbb{I}) \nu\cdot \phi_{\mathcal{S}}. \end{equation} Using the rigid body equations \eqref{linear momentumcomp:body}--\eqref{angular momentumcomp:body} and some calculations, we obtain \begin{equation}\label{express4} \int\limits_{\partial \mathcal{S}(t)} (\mathbb{T}(u_{\mathcal{F}}) - p_{\mathcal{F}}\mathbb{I}) \nu\cdot \phi_{\mathcal{S}} = -\frac{d}{dt}\int\limits_{\mathcal{S}(t)} \rho_{\mathcal{S}}u_{\mathcal{S}}\cdot \phi_{\mathcal{S}} + \int\limits_{\mathcal{S}(t)} \rho_{\mathcal{S}}u_{\mathcal{S}}\cdot \frac{\partial}{\partial t} \phi_{\mathcal{S}} + \int\limits_{\mathcal{S}(t)} \rho_{\mathcal{S}}g_{\mathcal{S}}\cdot \phi_{\mathcal{S}}. \end{equation} Thus by combining the above relations \eqref{express1}--\eqref{express4} and then integrating from $0$ to $T$, we have \begin{multline}\label{weak-momentum} - \int\limits_0^T\int\limits_{\mathcal{F}(t)} \rho_{\mathcal{F}}u_{\mathcal{F}}\cdot \frac{\partial}{\partial t}\phi_{\mathcal{F}} - \int\limits_0^T\int\limits_{\mathcal{S}(t)} \rho_{\mathcal{S}}u_{\mathcal{S}}\cdot \frac{\partial}{\partial t}\phi_{\mathcal{S}} - \int\limits_0^T\int\limits_{\mathcal{F}(t)} (\rho_{\mathcal{F}}u_{\mathcal{F}} \otimes u_{\mathcal{F}}) : \nabla \phi_{\mathcal{F}} + \int\limits_0^T\int\limits_{\mathcal{F}(t)} (\mathbb{T}(u_{\mathcal{F}}) - p_{\mathcal{F}}\mathbb{I}) : \mathbb{D}(\phi_{\mathcal{F}}) \\
+ \alpha \int\limits_0^T\int\limits_{\partial \Omega} (u_{\mathcal{F}}\times \nu)\cdot (\phi_{\mathcal{F}}\times \nu) + \alpha \int\limits_0^T\int\limits_{\partial \mathcal{S}(t)} [(u_{\mathcal{F}}-u_{\mathcal{S}})\times \nu]\cdot [(\phi_{\mathcal{F}}-\phi_{\mathcal{S}})\times \nu] \\
= \int\limits_0^T\int\limits_{\mathcal{F}(t)}\rho_{\mathcal{F}}g_{\mathcal{F}}\cdot \phi_{\mathcal{F}} + \int\limits_0^T\int\limits_{\mathcal{S}(t)} \rho_{\mathcal{S}}g_{\mathcal{S}}\cdot \phi_{\mathcal{S}} + \int\limits_{\mathcal{F}(0)} (\rho_{\mathcal{F}}u_{\mathcal{F}}\cdot \phi_{\mathcal{F}})(0) + \int\limits_{\mathcal{S}(0)} (\rho_{\mathcal{S}}u_{\mathcal{S}}\cdot \phi_{\mathcal{S}})(0). \end{multline} \begin{remark} We stress that in the definition of the set $U_T$ (in Definition~\ref{weaksolution-main} below) the function $u_{\mathcal{F}}$ on $\Omega$ is a regular extension of the velocity field $u_{\mathcal{F}}$ from $\mathcal{F}(t)$ to $\Omega$, see \eqref{Eu1}--\eqref{Eu2}. Correspondingly, $u_{\mathcal{S}}\in \mathcal{R}$ denotes a rigid extension from $\mathcal{S}(t)$ to $\Omega$ as in \eqref{Svel1}. Moreover, by the density $\rho _{\mathcal{F}}$ in \eqref{NO2}, we mean an extended fluid density $\rho _{\mathcal{F}}$
from $\mathcal{F}(t)$ to $\Omega$ by zero, see \eqref{Erho}--\eqref{15:21}. Correspondingly, $\rho_{\mathcal{S}}$ refers to an extended solid density from $\mathcal{S}(t)$ to $\Omega$ by zero. \end{remark} \begin{remark}\label{u-initial}
In \eqref{NO2}, the initial fluid density $\rho _{\mathcal{F}_0}$ on $\Omega$ represents a zero extension of $\rho _{\mathcal{F}_0}$ (defined in \eqref{initial cond})
from $\mathcal{F}_0$ to $\Omega$. Correspondingly, $\rho_{\mathcal{S}_0}$ in equation \eqref{NO5} stands for an extended initial solid density (defined in \eqref{eq:vrBeq}) from $\mathcal{S}_0$ to $\Omega$ by zero. Obviously, $q_{\mathcal{F}_0}$ refers to an extended initial momentum from $\mathcal{F}_0$ to $\Omega$ by zero and $u_{\mathcal{S}_0}\in \mathcal{R}$ denotes a rigid extension from $\mathcal{S}_0$ to $\Omega$ as in \eqref{Svel2}. \end{remark} \begin{defin}\label{weaksolution-main} Let $T> 0$, and let $\Omega$ and $\mathcal{S}_0 \Subset \Omega$ be two regular bounded domains of $\mathbb{R}^3$. A triplet $(\mathcal{S},\rho,u)$ is a finite energy weak solution to system \eqref{mass:comfluid}--\eqref{initial cond:comp} if the following holds: \begin{itemize} \item $\mathcal{S}(t) \Subset \Omega$ is a bounded domain of $\mathbb{R}^3$ for all $t\in [0,T)$ such that \begin{equation}\label{NO1} \chi_{\mathcal{S}}(t,x) = \mathds{1}_{\mathcal{S}(t)}(x) \in L^{\infty}((0,T) \times \Omega). \end{equation} \item $u$ belongs to the following space
\begin{equation*} U_{T}= \left\{\!\begin{aligned} &u \in L^{2}(0,T; L^2(\Omega)) \mbox{ such that there exist }u_{\mathcal{F}}\in L^2(0,T; H^1(\Omega)),\, u_{\mathcal{S}}\in L^{2}(0,T; \mathcal{R})\\ &\mbox{satisfying }u(t,\cdot)=u_{\mathcal{F}}(t,\cdot)\mbox{ on }\mathcal{F}(t),\quad u(t,\cdot)=u_{\mathcal{S}}(t,\cdot)\mbox{ on }\mathcal{S}(t)\mbox{ with }\\ &u_{\mathcal{F}}(t,\cdot)\cdot \nu = u_{\mathcal{S}}(t,\cdot)\cdot \nu \mbox{ on }\partial\mathcal{S}(t),\ u_{\mathcal{F}}\cdot \nu=0 \mbox{ on }\partial\Omega\mbox{ for a.e }t\in [0,T] \end{aligned}\right\}. \end{equation*}
\item $\rho \geqslant 0$, $\rho \in L^{\infty}(0,T; L^{\gamma}(\Omega))$ with $\gamma>3/2$, $\rho|u|^2 \in L^{\infty}(0,T; L^1(\Omega))$, where \begin{equation*} \rho= (1-\mathds{1}_{\mathcal{S}})\rho_{\mathcal{F}} + \mathds{1}_{\mathcal{S}}\rho_{\mathcal{S}},\quad u= (1-\mathds{1}_{\mathcal{S}})u_{\mathcal{F}} + \mathds{1}_{\mathcal{S}}u_{\mathcal{S}}. \end{equation*} \item The continuity equation is satisfied in the weak sense, i.e.\ \begin{equation}\label{NO2} \frac{\partial {\rho_{\mathcal{F}}}}{\partial t} + \operatorname{div}({\rho}_{\mathcal{F}} u_{\mathcal{F}}) =0 \mbox{ in }\, \mathcal{D}'([0,T)\times {\Omega}),\quad \rho_{\mathcal{F}}(0,x)=\rho_{\mathcal{F}_0}(x),\ x\in \Omega. \end{equation} Also, a renormalized continuity equation holds in a weak sense, i.e.\ \begin{equation}\label{NO3} \partial_t b(\rho_{\mathcal{F}}) + \operatorname{div}(b(\rho_{\mathcal{F}})u_{\mathcal{F}}) + (b'(\rho_{\mathcal{F}})-b(\rho_{\mathcal{F}}))\operatorname{div}u_{\mathcal{F}}=0 \mbox{ in }\, \mathcal{D}'([0,T)\times {\Omega}) , \end{equation} for any $b\in C([0,\infty)) \cap C^1((0,\infty))$ satisfying \begin{equation}\label{eq:b}
|b'(z)|\leqslant cz^{-\kappa_0},\, z\in (0,1],\ \kappa_0 <1, \qquad |b'(z)|\leqslant cz^{\kappa_1},\, z\geqslant 1,\ -1<\kappa_1 <\infty. \end{equation} \item The transport of $\mathcal{S}$ by the rigid vector field $u_{\mathcal{S}}$ holds (in the weak sense) \begin{equation}\label{NO4} \frac{\partial {\chi}_{\mathcal{S}}}{\partial t} + \operatorname{div}(u_{\mathcal{S}}\chi_{\mathcal{S}}) =0 \, \mbox{ in }(0,T)\times {\Omega},\quad \chi_{\mathcal{S}}(0,x)=\mathds{1}_{\mathcal{S}_0}(x),\ x\in \Omega. \end{equation} \item The density $\rho_{\mathcal{S}}$ of the rigid body $\mathcal{S}$ satisfies (in the weak sense) \begin{equation}\label{NO5} \frac{\partial {\rho}_{\mathcal{S}}}{\partial t} + \operatorname{div}(u_{\mathcal{S}}\rho_{\mathcal{S}}) =0 \, \mbox{ in }(0,T)\times {\Omega},\quad \rho_{\mathcal{S}}(0,x)=\rho_{\mathcal{S}_0}(x),\ x\in \Omega. \end{equation} \item Balance of linear momentum holds in a weak sense, i.e.\ for all $\phi \in V_{T}$ the relation \eqref{weak-momentum} holds. \item The following energy inequality holds for almost every $t\in (0,T)$: \begin{multline}\label{energy}
\int\limits_{\mathcal{F}(t)}\frac{1}{2} \rho_{\mathcal{F}}|u_{\mathcal{F}}(t,\cdot)|^2 + \int\limits_{\mathcal{S}(t)} \frac{1}{2} \rho_{\mathcal{S}}|u_{\mathcal{S}}(t,\cdot)|^2 + \int\limits_{\mathcal{F}(t)} \frac{a_{\mathcal{F}}}{\gamma-1}\rho_{\mathcal{F}}^{\gamma}+ \int\limits_0^t\int\limits_{\mathcal{F}(\tau)} \Big(2\mu_{\mathcal{F}} |\mathbb{D}(u_{\mathcal{F}})|^2 + \lambda_{\mathcal{F}} |\operatorname{div} u_{\mathcal{F}}|^2\Big) \\
+ \alpha \int\limits_0^t\int\limits_{\partial \Omega} |u_{\mathcal{F}}\times \nu|^2
+ \alpha \int\limits_0^t\int\limits_{\partial \mathcal{S}(\tau)} |(u_{\mathcal{F}}-u_{\mathcal{S}})\times \nu|^2\\ \leqslant \int\limits_0^t\int\limits_{\mathcal{F}(\tau)}\rho_{\mathcal{F}}g_{\mathcal{F}}\cdot u_{\mathcal{F}} + \int\limits_0^t\int\limits_{\mathcal{S}(\tau)} \rho_{\mathcal{S}}g_{\mathcal{S}}\cdot u_{\mathcal{S}} + \int\limits_{\mathcal{F}_0}\frac{1}{2} \frac{|q_{\mathcal{F}_0}|^2}{\rho_{\mathcal{F}_0}}
+ \int\limits_{\mathcal{S}_0} \frac{1}{2}\rho_{\mathcal{S}_0}|u_{\mathcal{S}_0}|^2 + \int\limits_{\mathcal{F}_0} \frac{a_{\mathcal{F}}}{\gamma-1}\rho_{\mathcal{F}_0}^{\gamma}. \end{multline} \end{itemize}
\end{defin}
\begin{remark}\label{16:37}
We note that our continuity equation \eqref{NO2} is different from the corresponding one in \cite{F4}. We have to work with $u_{\mathcal{F}}$ instead of $u$ because of the Navier boundary condition. The reason is that we need the $H^{1}(\Omega)$ regularity of the velocity in order to achieve the validity of the continuity equation in $\Omega$. Observe that $u \in L^{2}(0,T; L^2(\Omega))$ but the extended fluid velocity has better regularity, in particular, $u_{\mathcal{F}}\in L^2(0,T; H^1(\Omega))$, see \eqref{Eu1}--\eqref{Eu2}.
\end{remark}
\begin{remark}
In the weak formulation \eqref {weak-momentum}, we need to distinguish between the fluid velocity $u_{\mathcal{F}}$ and the solid velocity $u_{\mathcal{S}}$. Due to the presence of the discontinuities in the tangential components of $u$ and $\phi$, neither $\partial_t \phi$ nor $\mathbb{D}(u)$, $\mathbb{D}(\phi)$ belong to $L^2(\Omega)$. That's why it is not possible to write \eqref {weak-momentum} in a global and condensed form (i.e.\ integrals over $\Omega$).
\end{remark}
\begin{remark}
Let us mention that in the whole paper we assume the regularity of domains $\Omega$ and $\mathcal{S}_0$ as $C^{2+\kappa}$, $\kappa >0$. However, we expect that our assumption on the regularity of the domain can be relaxed to a less regular domain like in the work of Kuku\v cka \cite{kuku}.
\end{remark}
\subsection{Discussion and main result} The mathematical analysis of systems describing the motion of a rigid body in a viscous \textit{incompressible} fluid is nowadays well developed. The proof of existence of weak solutions until a first collision can be found in several papers, see \cite{CST,DEES1,GLSE,HOST,SER3}. Later, the possibility of collisions in the case of a weak solution was included, see \cite{F3,SST}. Moreover, it was shown that under Dirichlet boundary conditions collisions cannot occur, which is paradoxical with respect to real situations; for details see \cite{HES, HIL, HT}. Neustupa and Penel showed that under a prescribed motion of the rigid body and under Navier type of boundary conditions collision can occur \cite{NP}. After that G\'erard-Varet and Hillairet showed that to construct collisions one needs to assume less regularity of the domain or different boundary conditions, see e.g.\ \cite{GH,MR3272367,GHC}. In the case of very high viscosity, under the assumption that rigid bodies are not touching each other or not touching the boundary at the initial time, it was shown that collisions cannot occur in finite time, see \cite{FHN}. For an introduction we refer to the problem of a fluid coupled with a rigid body in the work by Galdi, see \cite{G2}. Let us also mention results on strong solutions, see e.g.\ \cite{GGH13,T, Wa}.
A few results are available on the motion of a rigid structure in a \textit{compressible} fluid with Dirichlet boundary conditions. The existence of strong solutions in the $L^2$ framework for small data up to a collision was shown in \cite{BG,roy2019stabilization}. The existence of strong solutions in the $L^p$ setting based on $\mathcal{R}$-bounded operators was applied in the barotropic case \cite{HiMu} and in the full system \cite{HMTT}.
The existence of a weak solution, also up to a collision but without smallness assumptions, was shown in \cite{DEES2}. Generalization of this result allowing collisions was given in \cite{F4}. The weak-strong uniqueness of a compressible fluid with a rigid body can be found in \cite{KrNePi2}. Existence of weak solutions in the case of Navier boundary conditions is not available; we explore it in this article.
For many years, the \emph{no-slip boundary condition} has been the most widely used given its success in reproducing the standard velocity profiles for incompressible/compressible viscous fluids. Although the no-slip hypothesis seems to be in good agreement with experiments, it leads to certain rather surprising conclusions. As we have already mentioned before, the most striking one being the absence of collisions of rigid objects immersed in a linearly viscous fluid \cite{HES,HIL}.
The so-called \emph{Navier boundary conditions}, which allow for slip, offer more freedom and are likely to provide a physically acceptable solution at least to some of the paradoxical phenomena resulting from the no-slip boundary condition, see, e.g.\ Moffat \cite{MOF}. Mathematically, the behavior of the tangential component $[{u}]_{tan}$ is a delicate issue.
The main result of our paper (Theorem~\ref{exist:upto collision}) asserts local-in-time existence of a weak solution for the system involving the motion of a rigid body in a compressible fluid in the case of Navier boundary conditions at the interface and at the outer boundary. It is the first result in the context of rigid body-compressible fluid interaction in the case of Navier type of boundary conditions. Let us mention that the main difficulty which arises in our problem is the jump in the velocity through the interface boundary between the rigid body and the compressible fluid. This difficulty cannot be resolved by the approach introduced in the work of Desjardins, Esteban \cite{DEES2}, or Feireisl \cite{F4} since they consider the velocity field continuous through the interface. Moreover, G\'erard-Varet, Hillairet \cite{MR3272367} and Chemetov, Ne\v casov\' a \cite{CN} cannot be used directly as they are in the incompressible framework. Our weak solutions have to satisfy the jump of the velocity field through the interface boundary.
Our idea is to introduce a new approximate scheme which combines the theory of compressible fluids introduced by P. L. Lions \cite{LI4} and then developed by Feireisl \cite{EF70} to get the strong convergence of the density (renormalized continuity equations, effective viscous flux, artificial pressure) together with ideas from G\'erard-Varet, Hillairet \cite{MR3272367} and Chemetov, Ne\v casov\' a \cite{CN} concerning a penalization of the jump. We remark that such type of difficulties do not arise for the existence of weak solutions of compressible fluids without a rigid body neither for Dirichlet nor for Navier type of boundary conditions.
Let us mention the main issues that arise in the analysis of our system and the methodology that we adapt to deal with it:
\begin{itemize}
\item It is not possible to define a uniform velocity field as in \cite{DEES2, F4}
due to the presence of a discontinuity through the interface of interaction. This is the reason why we introduce the regularized fluid velocity $u_{\mathcal{F}}$ and the solid velocity $u_{\mathcal{S}}$ and why we treat them separately.
\item We introduce approximate problems and recover the original problem as a limit of the approximate ones. In fact, we consider several levels of approximations; in each level we ensure that our solution and the test function do not show a jump across the interface so that we can use several available techniques of compressible fluids (without body). In the limit, however, the discontinuity at the interface is recovered. The particular construction of the test functions is a delicate and crucial issue in our proof, which we outline in \cref{approx:test}.
\item Recovering the velocity fields for the solid and fluid parts separately is also a challenging issue. We introduce a penalization in such a way that, in the last stage of the limiting process, this term allows us to recover the rigid velocity of the solid, see \eqref{12:04}--\eqref{re:solidvel}.
The introduction of an appropriate extension operator helps us to recover the fluid velocity, see \eqref{ext:fluid}--\eqref{re:fluidvel}.
\item Since we consider the compressible case, our penalization with parameter $\delta>0$, see \eqref{approx3}, is different from the penalization for the incompressible fluid in \cite{MR3272367}.
\item Due to the Navier-slip boundary condition, no $H^1$ bound on the velocity on the whole domain is possible. We can only obtain the $H^1$ regularity of the extended velocities of the fluid and solid parts separately. We have introduced an artificial viscosity that vanishes asymptotically on the solid part so that we can capture the $H^1$ regularity for the fluid part (see step 1 of the proof of \cref{exist:upto collision} in Section \ref{S4}).
\item We have already mentioned that the main difference with \cite{MR3272367} is that we consider compressible fluid whereas they have considered an incompressible fluid. We have encountered several issues that are present due to the compressible nature of the fluid (vanishing viscosity in the continuity equation, recovering the renormalized continuity equation, identification of the pressure). One important point is to see that passing to the limit as $\delta$ tends to zero in the transport for the rigid body is not obvious because our velocity field does not have regularity $L^{\infty}(0,T,L^2(\Omega))$ as in the incompressible case see e.g.\ \cite{MR3272367} but $L^2(0,T,L^2(\Omega))$ (here we have $\sqrt{\rho}u \in L^{\infty}(0,T,L^2(\Omega))$ only). To handle this problem, we apply \cref{sequential2} in the $\delta$-level, see \cref{S4}. \end{itemize}
Next we present the main result of our paper.
\begin{theorem}\label{exist:upto collision}
Let $\Omega$ and $\mathcal{S}_0 \Subset \Omega$ be two regular bounded domains of $\mathbb{R}^3$. Assume that for some $\sigma > 0$
\begin{equation*}
\operatorname{dist}(\mathcal{S}_0,\partial\Omega) > 2\sigma.
\end{equation*}
Let $g_{\mathcal{F}}$, $g_{\mathcal{S}} \in L^{\infty}((0,T)\times \Omega)$ and the pressure $p_{\mathcal{F}}$ be determined by \eqref{const-law} with $\gamma >3/2$. Assume that the initial data (defined in the sense of \cref{u-initial}) satisfy
\begin{align} \label{init}
\rho_{\mathcal{F}_{0}} \in L^{\gamma}(\Omega),\quad \rho_{\mathcal{F}_{0}} \geqslant 0 &\mbox{ a.e. in }\Omega,\quad \rho_{\mathcal{S}_0}\in L^{\infty}(\Omega),\quad \rho_{\mathcal{S}_0}>0\mbox{ a.e. in }\mathcal{S}_0,\\
\label{init1}
q_{\mathcal{F}_{0}} \in L^{\frac{2\gamma}{\gamma+1}}(\Omega), \quad q_{\mathcal{F}_{0}}\mathds{1}_{\{\rho_{\mathcal{F}_0}=0\}}=0 &\mbox{ a.e. in }\Omega,\quad \dfrac{|q_{\mathcal{F}_{0}}|^2}{\rho_{\mathcal{F}_0}}\mathds{1}_{\{\rho_{\mathcal{F}_0}>0\}}\in L^1(\Omega),\\
\label{init2}
u_{\mathcal{S}_0}= \ell_0+ \omega_0 \times x& \quad\forall\ x \in \Omega \mbox{ with }\ell_0,\ \omega_0 \in \mathbb{R}^3.
\end{align}
Then there exists $T > 0$ (depending only on $\rho_{\mathcal{F}_0}$, $\rho_{\mathcal{S}_0}$, $q_{\mathcal{F}_0}$, $u_{\mathcal{S}_0}$, $g_{\mathcal{F}}$, $g_{\mathcal{S}}$, $\operatorname{dist}(\mathcal{S}_0,\partial\Omega)$) such that a finite energy weak solution to \eqref{mass:comfluid}--\eqref{initial cond:comp} exists on $[0,T)$. Moreover, \begin{equation*}
\mathcal{S}(t) \Subset \Omega,\quad\operatorname{dist}(\mathcal{S}(t),\partial\Omega) \geqslant \frac{3\sigma}{2},\quad \forall \ t\in [0,T].
\end{equation*}
\end{theorem}
The outline of the paper is as follows. We introduce three levels of approximation schemes in \cref{F1351}. In \cref{S5}, we describe some results on the transport equation, which are needed in all the levels of approximation. The existence results of approximate solutions have been proved in \cref{S3}. \cref{sec:Galerkin} and \cref{14:14} are dedicated to the construction and convergence analysis of the Faedo-Galerkin scheme associated to the finite dimensional approximation level. We discuss the limiting system associated to the vanishing viscosity in \cref{14:18}. \cref{S4} is devoted to the main part: we derive the limit as the parameter $\delta$ tends to zero.
\section{Approximate Solutions}\label{F1351}
In this section, we present the approximate problems by combining the penalization method, introduced in \cite{MR3272367}, and the approximation scheme developed in \cite{MR1867887} along with a careful treatment of the boundary terms of the rigid body to solve the original problem \eqref{mass:comfluid}--\eqref{initial cond:comp}. There are three levels of approximations with the parameters $N,\varepsilon,\delta$. Let us briefly explain these approximations: \begin{itemize}
\item The parameter $N$ is connected with solving the momentum equation using the Faedo-Galerkin approximation.
\item The parameter $\varepsilon > 0$ is connected with a new diffusion term $\varepsilon \Delta \rho $ in the continuity equation together with a term $\varepsilon \nabla \rho \nabla u$ in the momentum equation.
\item The parameter $\delta > 0$ is connected with the approximation in the viscosities (see \eqref{approx-viscosity}) together with a penalization of the boundary of the rigid body to get smoothness through the interface (see \eqref{approx3}) and together with the artificial pressure containing the approximate coefficient, see \eqref{approx-press}. \end{itemize} At first, we state the existence results for the different levels of approximation schemes and then we will prove these later on. We start with the $\delta$-level of approximation via an artificial pressure. We are going to consider the following approximate problem: Let $\delta>0$. Find a triplet $(S^{\delta}, \rho^{\delta}, u^{\delta})$ such that \begin{itemize} \item $\mathcal{S}^{\delta}(t) \Subset \Omega$ is a bounded, regular domain for all $t \in [0,T]$ with \begin{equation}\label{approx1} \chi^{\delta}_{\mathcal{S}}(t,x)= \mathds{1}_{\mathcal{S}^{\delta}(t)}(x) \in L^{\infty}((0,T)\times \Omega) \cap C([0,T];L^p(\Omega)), \, \forall \, 1 \leqslant p < \infty. \end{equation}
\item The velocity field $u^{\delta} \in L^2(0,T; H^1(\Omega))$, and the density function $\rho^{\delta} \in L^{\infty}(0,T; L^{\beta}(\Omega))$, $\rho^{\delta}\geqslant 0$ satisfy \begin{equation}\label{approx2} \frac{\partial {\rho}^{\delta}}{\partial t} + \operatorname{div}({\rho}^{\delta} u^{\delta}) =0 \mbox{ in } \mathcal{D}'([0,T)\times {\Omega}). \end{equation} \item
For all $\phi \in H^1(0,T; L^{2}(\Omega)) \cap L^r(0,T; W^{1,{r}}(\Omega))$, where $r=\max\left\{\beta+1, \frac{\beta+\theta}{\theta}\right\}$, $\beta \geqslant \max\{8,\gamma\}$ and $\theta=\frac{2}{3}\gamma -1$ with $\phi\cdot\nu=0$ on $\partial\Omega$ and $\phi|_{t=T}=0$, the following holds: \begin{multline}\label{approx3} - \int\limits_0^T\int\limits_{\Omega} \rho^{\delta} \left(u^{\delta}\cdot \frac{\partial}{\partial t}\phi + u^{\delta} \otimes u^{\delta} : \nabla \phi\right) + \int\limits_0^T\int\limits_{\Omega} \Big(2\mu^{\delta}\mathbb{D}(u^{\delta}):\mathbb{D}(\phi) + \lambda^{\delta}\operatorname{div}u^{\delta}\mathbb{I} : \mathbb{D}(\phi)- p^{\delta}(\rho^{\delta})\mathbb{I} : \mathbb{D}(\phi)\Big) \\
+ \alpha \int\limits_0^T\int\limits_{\partial \Omega} (u^{\delta} \times \nu)\cdot (\phi \times \nu) + \alpha \int\limits_0^T\int\limits_{\partial \mathcal{S}^{\delta}(t)} [(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta})\times \nu]\cdot [(\phi-P^{\delta}_{\mathcal{S}}\phi)\times \nu]
+ \frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi^{\delta}_{\mathcal{S}}(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta})\cdot (\phi-P^{\delta}_{\mathcal{S}}\phi)\\ = \int\limits_0^T\int\limits_{\Omega}\rho^{\delta} g^{\delta} \cdot \phi
+ \int\limits_{\Omega} (\rho^{\delta} u^{\delta} \cdot \phi)(0), \end{multline} where $\mathcal{P}_{\mathcal{S}}^{\delta}$ is defined in \eqref{approx:projection} below. \item ${\chi}^{\delta}_{\mathcal{S}}(t,x)$ satisfies (in the weak sense) \begin{equation}\label{approx4}
\frac{\partial {\chi}^{\delta}_{\mathcal{S}}}{\partial t} + P^{\delta}_{\mathcal{S}}u^{\delta} \cdot \nabla \chi^{\delta}_{\mathcal{S}} =0\, \mbox{ in }(0,T)\times {\Omega},\quad \chi^{\delta}_{\mathcal{S}}|_{t=0}= \mathds{1}_{\mathcal{S}_0}\, \mbox{ in } {\Omega}. \end{equation} \item $\rho^{\delta}{\chi}^{\delta}_{\mathcal{S}}(t,x)$ satisfies (in the weak sense) \begin{equation}\label{approx5}
\frac{\partial }{\partial t}(\rho^{\delta}{\chi}^{\delta}_{\mathcal{S}}) + P^{\delta}_{\mathcal{S}}u^{\delta} \cdot \nabla (\rho^{\delta}{\chi}^{\delta}_{\mathcal{S}})=0\, \mbox{ in }(0,T)\times {\Omega},\quad (\rho^{\delta}{\chi}^{\delta}_{\mathcal{S}})|_{t=0}= \rho_0^{\delta}\mathds{1}_{\mathcal{S}_0}\, \mbox{ in } {\Omega}. \end{equation} \item Initial data are given by \begin{equation}\label{approx:initial} \rho^{\delta}(0,x)=\rho_0^{\delta}(x),\quad \rho^{\delta} u^{\delta}(0,x) = q_0^{\delta}(x),\quad x\in \Omega . \end{equation} \end{itemize} Above we have used the following quantities: \begin{itemize} \item The density of the volume force is defined as \begin{equation*} g^{\delta}=(1-\chi^{\delta}_{\mathcal{S}})g_{\mathcal{F}} + \chi^{\delta}_{\mathcal{S}}g_{\mathcal{S}}. \end{equation*} \item The artificial pressure is given by \begin{equation}\label{approx-press} p^{\delta}(\rho)= a^{\delta}\rho^{\gamma} + {\delta} \rho^{\beta},\quad\mbox{ with }\quad a^{\delta} = a_{\mathcal{F}} (1-\chi^{\delta}_{\mathcal{S}}), \end{equation} where $a_{\mathcal{F}}>0$ and the exponents $\gamma$ and $\beta$ satisfy $\gamma > 3/2, \ \beta \geqslant \max\{8,\gamma\}$. \item The viscosity coefficients are given by \begin{equation}\label{approx-viscosity}
\mu^{\delta} = (1-\chi^{\delta}_{\mathcal{S}})\mu_{\mathcal{F}} + {\delta}^2\chi^{\delta}_{\mathcal{S}},\quad \lambda^{\delta} = (1-\chi^{\delta}_{\mathcal{S}})\lambda_{\mathcal{F}} + {\delta}^2\chi^{\delta}_{\mathcal{S}}\quad\mbox{ so that }\quad\mu^{\delta} >0,\ 2\mu^{\delta}+ 3\lambda^{\delta} \geqslant 0. \end{equation} \item The orthogonal projection to rigid fields, $P^{\delta}_{\mathcal{S}}:L^2(\Omega)\rightarrow L^2(\mathcal{S}^{\delta}(t))$, is such that, for all $t\in [0,T]$ and $u\in L^2(\Omega)$, we have $P^{\delta}_{\mathcal{S}}u \in \mathcal{R}$ and it is given by \begin{equation}\label{approx:projection} P^{\delta}_{\mathcal{S}}u(t,x)= \frac{1}{m^{\delta}} \int\limits_{\Omega} \rho^{\delta}\chi_{\mathcal{S}}^{\delta} u + \left((J^{\delta})^{-1} \int\limits_{\Omega}\rho^{\delta}\chi_{\mathcal{S}}^{\delta}((y-h^{\delta}(t)) \times u)\ dy \right)\times (x-h^{\delta}(t)), \quad \forall x\in \Omega, \end{equation} where $h^{\delta}$, $m^{\delta}$ and $J^{\delta}$ are defined as \begin{align*}
h^{\delta}(t) = \frac{1}{m^{\delta}} \int\limits_{\mathbb{R}^3} \rho^{\delta}\chi_{\mathcal{S}}^{\delta} x \ dx,\quad m^{\delta} = \int\limits_{\mathbb{R}^3} \rho^{\delta}\chi_{\mathcal{S}}^{\delta} \ dx, \\
J^{\delta}(t) = \int\limits_{\mathbb{R}^3} \rho^{\delta}\chi_{\mathcal{S}}^{\delta} \left[ |x-h^{\delta}(t)|^2\mathbb{I} - (x-h^{\delta}(t)) \otimes (x-h^{\delta}(t))\right] \ dx. \end{align*} \end{itemize} \begin{remark} The penalization which we apply in our case is different from that in \cite{F4}. We do not use the high viscosity limit but our penalization contains an $L^2$ penalization (see \eqref{approx3}), which is necessary because of the discontinuity of the velocity field through the fluid-structure interface. Moreover, we consider a penalization of the viscosity coefficients \eqref{approx-viscosity} together with the additional regularity of the pressure, see \eqref{approx-press}. This approach is completely new. \end{remark} A weak solution of problem \eqref{mass:comfluid}--\eqref{initial cond:comp} in the sense of \cref{weaksolution-main} will be obtained as a weak limit of the solution $(\mathcal{S}^{\delta},\rho^{\delta},u^{\delta})$ of system \eqref{approx1}--\eqref{approx:initial} as $\delta \rightarrow 0$. The existence result of the approximate system reads: \begin{proposition}\label{thm:approxn-delta} Let $\Omega$ and $\mathcal{S}_0 \Subset \Omega$ be two regular bounded domains of $\mathbb{R}^3$. Assume that for some $\sigma>0$
\begin{equation*}
\operatorname{dist}(\mathcal{S}_0,\partial\Omega) > 2\sigma.
\end{equation*}
Let $g^{\delta}=(1-\chi^{\delta}_{\mathcal{S}})g_{\mathcal{F}} + \chi^{\delta}_{\mathcal{S}}g_{\mathcal{S}} \in L^{\infty}((0,T)\times \Omega)$ and \begin{equation}\label{cc:dbg} {\delta} >0,\ \gamma > 3/2, \ \beta \geqslant \max\{8,\gamma\}. \end{equation} Further, let the pressure $p^{\delta}$ be determined by \eqref{approx-press} and the viscosity coefficients $\mu^{\delta}$, $\lambda^{\delta}$ be given by \eqref{approx-viscosity}. Assume that the initial conditions satisfy \begin{align} \rho_{0}^{\delta} &\in L^{\beta}(\Omega), \quad \rho_0^{\delta}\geqslant 0 \mbox{ a.e. in }\Omega,\quad \rho_0^{\delta}\mathds{1}_{\mathcal{S}_0}\in L^{\infty}(\Omega),\quad \rho_0^{\delta}\mathds{1}_{\mathcal{S}_0}>0\mbox{ a.e. in }\mathcal{S}_0,\label{rhonot}\\
&q_0^{\delta} \in L^{\frac{2\beta}{\beta+1}}(\Omega), \quad q_0^{\delta}\mathds{1}_{\{\rho_0^{\delta}=0\}}=0 \mbox{ a.e. in }\Omega,\quad \dfrac{|q_0^{\delta}|^2}{\rho_0^{\delta}}\mathds{1}_{\{\rho_0>0\}}\in L^1(\Omega)\label{qnot}. \end{align} Let the initial energy $$
E^{\delta}[\rho_0^{\delta},q_0^{\delta}] = \int\limits_{\Omega}\Bigg(\frac{1}{2} \frac{|q_0^{\delta}|^2}{\rho_0^{\delta}}\mathds{1}_{\{\rho_0^{\delta}>0\}} + \frac{a^{\delta}(0)}{\gamma-1}(\rho_0^{\delta})^{\gamma} + \frac{\delta}{\beta-1}(\rho_0^{\delta})^{\beta} \Bigg):=E^{\delta}_0$$ be uniformly bounded with respect to $\delta$. Then there exists $T > 0$ (depending only on $E^{\delta}_0$, $g_{\mathcal{F}}$, $g_{\mathcal{S}}$, $\operatorname{dist}(\mathcal{S}_0,\partial\Omega)$) such that system \eqref{approx1}--\eqref{approx:initial} admits a finite energy weak solution $(S^{\delta},\rho^{\delta},u^{\delta})$, which satisfies the following energy inequality: \begin{multline}\label{10:45}
E^{\delta}[\rho ^{\delta}, q^{\delta}] + \int\limits_0^T\int\limits_{\Omega} \Big(2\mu^{\delta}|\mathbb{D}(u^{\delta})|^2 + \lambda^{\delta}|\operatorname{div}u^{\delta}|^2\Big) + \alpha \int\limits_0^T\int\limits_{\partial \Omega} |u^{\delta} \times \nu|^2
+ \alpha \int\limits_0^T\int\limits_{\partial \mathcal{S}^{\delta}(t)} |(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta})\times \nu|^2 \\+ \frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi^{\delta}_{\mathcal{S}}|u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta}|^2 \leqslant \int\limits_0^T\int\limits_{\Omega}\rho^{\delta}
g^{\delta} \cdot u^{\delta}
+ E_0^{\delta}. \end{multline} Moreover, \begin{equation*}
\operatorname{dist}(\mathcal{S}^{\delta}(t),\partial\Omega) \geqslant {2\sigma},\quad \forall \ t\in [0,T],
\end{equation*} and the solution satisfies the following properties: \begin{enumerate} \item For $\theta=\frac{2}{3}\gamma-1$, $s=\gamma+\theta$, \begin{equation}\label{rho:improved}
\|(a^{\delta})^{1/s}\rho^{\delta}\|_{L^{s}((0,T)\times\Omega)} + \delta^{\frac{1}{\beta+\theta}}\|\rho^{\delta}\|_{L^{\beta+\theta}((0,T)\times\Omega)} \leqslant c. \end{equation} \item The couple $(\rho^{\delta},u^{\delta})$ satisfies the identity \begin{equation}\label{rho:renorm1} \partial_t b(\rho^{\delta}) + \operatorname{div}(b(\rho^{\delta})u^{\delta})+[b'(\rho^{\delta})\rho^{\delta} - b(\rho^{\delta})]\operatorname{div}u^{\delta}=0, \end{equation} a.e.\ in $(0,T)\times\Omega$ for any $b\in C([0,\infty)) \cap C^1((0,\infty))$ satisfying \eqref{eq:b}. \end{enumerate} \end{proposition}
In order to prove \cref{thm:approxn-delta}, we consider a problem with another level of approximation: the $\varepsilon$-level approximation is obtained via the continuity equation with dissipation accompanied by the artificial pressure in the momentum equation. We want to find a triplet $(S^{\varepsilon}, \rho^{\varepsilon}, u^{\varepsilon})$ such that we can obtain a weak solution $(\mathcal{S}^{\delta},\rho^{\delta},u^{\delta})$ of the system \eqref{approx1}--\eqref{approx:initial} as a weak limit of the sequence $(\mathcal{S}^{\varepsilon},\rho^{\varepsilon},u^{\varepsilon})$ as $\varepsilon \rightarrow 0$. For $\varepsilon >0$, the triplet is supposed to satisfy: \begin{itemize} \item $\mathcal{S}^{\varepsilon}(t) \Subset \Omega$ is a bounded, regular domain for all $t \in [0,T]$ with \begin{equation}\label{varepsilon:approx1} \chi^{\varepsilon}_{\mathcal{S}}(t,x)= \mathds{1}_{\mathcal{S}^{\varepsilon}(t)}(x) \in L^{\infty}((0,T)\times \Omega) \cap C([0,T];L^p(\Omega)), \, \forall \, 1 \leqslant p < \infty. \end{equation} \item The velocity field $u^{\varepsilon} \in L^2(0,T; H^1(\Omega))$ and the density function $\rho^{\varepsilon} \in L^{\infty}(0,T; L^{\beta}(\Omega)) \cap L^2(0,T;H^1(\Omega))$, $\rho^{\varepsilon}\geqslant 0$ satisfy \begin{equation}\label{varepsilon:approx2} \frac{\partial {\rho}^{\varepsilon}}{\partial t} + \operatorname{div}({\rho}^{\varepsilon} u^{\varepsilon}) =\varepsilon \Delta\rho^{\varepsilon} \mbox{ in }\, (0,T)\times \Omega, \quad \frac{\partial \rho^{\varepsilon}}{\partial \nu}=0 \mbox{ on }\, \partial\Omega. \end{equation}
\item For all $\phi \in H^1(0,T; L^{2}(\Omega)) \cap L^{\beta+1}(0,T; W^{1,{\beta+1}}(\Omega))$ with $\phi\cdot \nu=0$ on $\partial\Omega$, $\phi|_{t=T}=0$, where $\beta \geqslant \max\{8,\gamma\}$, the following holds: \begin{multline}\label{varepsilon:approx3} - \int\limits_0^T\int\limits_{\Omega} \rho^{\varepsilon} \left(u^{\varepsilon}\cdot \frac{\partial}{\partial t}\phi + u^{\varepsilon} \otimes u^{\varepsilon} : \nabla \phi\right) + \int\limits_0^T\int\limits_{\Omega} \Big(2\mu^{{\varepsilon}}\mathbb{D}(u^{\varepsilon}):\mathbb{D}(\phi) + \lambda^{{\varepsilon}}\operatorname{div}u^{\varepsilon}\mathbb{I} : \mathbb{D}(\phi) - p^{\varepsilon}(\rho^{\varepsilon})\mathbb{I} : \mathbb{D}(\phi)\Big) \\ +\int\limits_0^T\int\limits_{\Omega} \varepsilon \nabla u^{\varepsilon} \nabla \rho^{\varepsilon} \cdot \phi
+ \alpha \int\limits_0^T\int\limits_{\partial \Omega} (u^{\varepsilon} \times \nu)\cdot (\phi \times \nu) + \alpha \int\limits_0^T\int\limits_{\partial \mathcal{S}^{\varepsilon}(t)} [(u^{\varepsilon}-P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon})\times \nu]\cdot [(\phi-P^{\varepsilon}_{\mathcal{S}}\phi)\times \nu] \\
+ \frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi^{\varepsilon}_{\mathcal{S}}(u^{\varepsilon}-P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon})\cdot (\phi-P^{\varepsilon}_{\mathcal{S}}\phi) = \int\limits_0^T\int\limits_{\Omega}\rho^{\varepsilon} g^{\varepsilon} \cdot \phi
+ \int\limits_{\Omega} (\rho^{\varepsilon} u^{\varepsilon} \cdot \phi)(0). \end{multline} \item ${\chi}^{\varepsilon}_{\mathcal{S}}(t,x)$ satisfies (in the weak sense) \begin{equation}\label{varepsilon:approx4}
\frac{\partial {\chi}^{\varepsilon}_{\mathcal{S}}}{\partial t} + P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon} \cdot \nabla \chi^{\varepsilon}_{\mathcal{S}} =0\mbox{ in }(0,T)\times {\Omega},\quad \chi^{\varepsilon}_{\mathcal{S}}|_{t=0}= \mathds{1}_{\mathcal{S}_0}\mbox{ in } {\Omega}. \end{equation} \item $\rho^{\varepsilon}{\chi}^{\varepsilon}_{\mathcal{S}}(t,x)$ satisfies (in the weak sense) \begin{equation}\label{varepsilon:approx5}
\frac{\partial }{\partial t}(\rho^{\varepsilon}{\chi}^{\varepsilon}_{\mathcal{S}}) + P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon} \cdot \nabla (\rho^{\varepsilon}{\chi}^{\varepsilon}_{\mathcal{S}})=0\mbox{ in }(0,T)\times {\Omega},\quad (\rho^{\varepsilon}{\chi}^{\varepsilon}_{\mathcal{S}})|_{t=0}= \rho^{\varepsilon}_0\mathds{1}_{\mathcal{S}_0}\mbox{ in }{\Omega}. \end{equation} \item The initial data are given by \begin{equation}\label{varepsilon:initial}
\rho^{\varepsilon}(0,x)=\rho_0^{\varepsilon}(x), \quad \rho^{\varepsilon} u^{\varepsilon}(0,x) = q_0^{\varepsilon}(x)\quad\mbox{in }\Omega,\quad \frac {\partial \rho_0^{\varepsilon }}{\partial \nu}\big |_{\partial \Omega} =0. \end{equation} \end{itemize} Above we have used the following quantities: \begin{itemize} \item The density of the volume force is defined as \begin{equation}\label{gepsilon} g^{\varepsilon}=(1-\chi^{\varepsilon}_{\mathcal{S}})g_{\mathcal{F}} + \chi^{\varepsilon}_{\mathcal{S}}g_{\mathcal{S}}. \end{equation} \item The artificial pressure is given by \begin{equation}\label{p1} p^{\varepsilon}(\rho)= a^{\varepsilon}\rho^{\gamma} + {\delta} \rho^{\beta},\quad\mbox{ with }\quad a^{\varepsilon} = a_{\mathcal{F}} (1-\chi^{\varepsilon}_{\mathcal{S}}), \end{equation} where $a_{\mathcal{F}},{\delta} >0$, and the exponents $\gamma$ and $\beta$ satisfy $\gamma > 3/2, \ \beta \geqslant \max\{8,\gamma\}$. \item The viscosity coefficients are given by \begin{equation}\label{vis1}
\mu^{\varepsilon} = (1-\chi^{\varepsilon}_{\mathcal{S}})\mu_{\mathcal{F}} + {\delta}^2\chi^{\varepsilon}_{\mathcal{S}},\quad \lambda^{\varepsilon} = (1-\chi^{\varepsilon}_{\mathcal{S}})\lambda_{\mathcal{F}} + {\delta}^2\chi^{\varepsilon}_{\mathcal{S}}\quad\mbox{ so that }\quad\mu^{\varepsilon} >0,\ 2\mu^{\varepsilon}+3\lambda^{\varepsilon} \geqslant 0. \end{equation} \item $P^{\varepsilon}_{\mathcal{S}}:L^2(\Omega)\rightarrow L^2(\mathcal{S}^{\varepsilon}(t))$ is the orthogonal projection to rigid fields; it is defined as in \eqref{approx:projection} with $\chi^{\delta}_{\mathcal{S}}$ is replaced by $\chi^{\varepsilon}_{\mathcal{S}}$. \end{itemize} \begin{remark} Above, the triplet $\left(\mathcal{S}^{\varepsilon}, \rho^{\varepsilon}, u^{\varepsilon}\right)$ should actually be denoted by $\left(\mathcal{S}^{\delta,\varepsilon}, \rho^{\delta,\varepsilon}, u^{\delta,\varepsilon}\right)$. The dependence on $\delta$ is due to the penalization term $\Big(\tfrac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi^{\varepsilon}_{\mathcal{S}}(u^{\varepsilon}-P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon})\cdot (\phi-P^{\varepsilon}_{\mathcal{S}}\phi)\Big)$ in \eqref{varepsilon:approx3} and in the viscosity coefficients $\mu^{\varepsilon}$, $\lambda^{\varepsilon}$ in \eqref{vis1}. To simplify the notation, we omit $\delta$ here. \end{remark}
In \cref{14:14} we will prove the following existence result of the approximate system \eqref{varepsilon:approx1}--\eqref{varepsilon:initial}: \begin{proposition}\label{thm:approxn} Let $\Omega$ and $\mathcal{S}_0 \Subset \Omega$ be two regular bounded domains of $\mathbb{R}^3$. Assume that for some $\sigma >0$,
\begin{equation*}
\operatorname{dist}(\mathcal{S}_0,\partial\Omega) > 2\sigma.
\end{equation*}
Let $g^{\varepsilon}=(1-\chi^{\varepsilon}_{\mathcal{S}})g_{\mathcal{F}} + \chi^{\varepsilon}_{\mathcal{S}}g_{\mathcal{S}} \in L^{\infty}((0,T)\times \Omega)$ and $\beta$, $\delta$, $\gamma$ be given as in \eqref{cc:dbg}. Further, let the pressure $p^{\varepsilon}$ be determined by \eqref{p1} and the viscosity coefficients $\mu^{\varepsilon}$, $\lambda^{\varepsilon}$ be given by \eqref{vis1}. The initial conditions satisfy, for some $\underline{\rho}$, $\overline{\rho}>0$, \begin{equation}\label{inteps} 0<\underline{\rho}\leqslant \rho_0^{\varepsilon} \leqslant \overline{\rho}\ \mbox{ in }\ \Omega, \quad \rho_0^{\varepsilon} \in W^{1,\infty}(\Omega),\quad q_0^{\varepsilon}\in L^2(\Omega).
\end{equation}
Let the initial energy $$E^{\varepsilon}[\rho _0 ^{\varepsilon},q_0^{\varepsilon}] =\int\limits_{\Omega}\Bigg(\frac{1}{2} \frac{|q_0^{\varepsilon}|^2}{\rho^{\varepsilon}_0}\mathds{1}_{\{\rho^{\varepsilon}_0>0\}} + \frac{a^{\varepsilon}(0)}{\gamma-1}(\rho_0^{\varepsilon})^{\gamma} + \frac{\delta}{\beta-1}(\rho_0^{\varepsilon})^{\beta} \Bigg):= E_0^{\varepsilon}$$ be uniformly bounded with respect to $\delta$ and $ \varepsilon$.
Then there exists $T > 0$ (depending only on $E^{\varepsilon}_0$, $g_{\mathcal{F}}$, $g_{\mathcal{S}}$, $\operatorname{dist}(\mathcal{S}_0,\partial\Omega)$) such that system \eqref{varepsilon:approx1}--\eqref{varepsilon:initial} admits a weak solution $(S^{\varepsilon},\rho^{\varepsilon},u^{\varepsilon})$, which satisfies the following energy inequality: \begin{multline}\label{energy-varepsilon}
E^{\varepsilon}[\rho ^{\varepsilon},q^{\varepsilon}]+ \int\limits_0^T\int\limits_{\Omega} \Big(2\mu^{\varepsilon}|\mathbb{D}(u^{\varepsilon})|^2 + \lambda^{\varepsilon}|\operatorname{div}u^{\varepsilon}|^2\Big) + \delta\varepsilon \beta\int\limits_0^T\int\limits_{\Omega} (\rho^{\varepsilon})^{\beta-2}|\nabla \rho^{\varepsilon}|^2
+ \alpha \int\limits_0^T\int\limits_{\partial \Omega} |u^{\varepsilon} \times \nu|^2 \\
+ \alpha \int\limits_0^T\int\limits_{\partial \mathcal{S}^{\varepsilon}(t)} |(u^{\varepsilon}-P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon})\times \nu|^2
+ \frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi^{\varepsilon}_{\mathcal{S}}|u^{\varepsilon}-P^{\delta}_{\mathcal{S}}u^{\varepsilon}|^2 \leqslant \int\limits_0^T\int\limits_{\Omega}\rho^{\varepsilon}
g^{\varepsilon} \cdot u^{\varepsilon}
+ E_0^{\varepsilon}. \end{multline}
Moreover, \begin{equation*}
\operatorname{dist}(\mathcal{S}^{\varepsilon}(t),\partial\Omega) \geqslant {2\sigma},\quad \forall \ t\in [0,T],
\end{equation*} and the solution satisfies \begin{equation*} \partial_t\rho^{\varepsilon},\ \Delta \rho^{\varepsilon}\in {L^{\frac{5\beta-3}{4\beta}}((0,T)\times\Omega)}, \end{equation*} \begin{equation}\label{est:indofepsilon}
\sqrt{\varepsilon} \|\nabla \rho^{\varepsilon}\|_{L^2((0,T)\times\Omega)} + \|\rho^{\varepsilon}\|_{L^{\beta+1}((0,T)\times\Omega)} + \|(a^{\varepsilon})^{\frac{1}{\gamma+1}}\rho^{\varepsilon}\|_{L^{\gamma+1}((0,T)\times\Omega)} \leqslant c, \end{equation} where $c$ is a positive constant depending on $\delta$ but which is independent of $\varepsilon$. \end{proposition} To solve the problem \eqref{varepsilon:approx1}--\eqref{varepsilon:initial}, we need yet another level of approximation. The $N$-level approximation is obtained via a Faedo-Galerkin approximation scheme.
Suppose that $\{e_k\}_{k\geqslant 1} \subset \mathcal{D}(\overline{\Omega})$ with $e_k\cdot\nu=0$ on $\partial\Omega$ is a basis of $ L^2(\Omega)$. We set \begin{equation*} X_N = \mbox{ span}(e_1,\ldots,e_N). \end{equation*} $X_N$ is a finite dimensional space with scalar product induced by the scalar product in $L^2(\Omega)$. As $X_N$ is finite dimensional, norms on $X_N$ induced by $W^{k,p}$ norms, $k\in \mathbb{N},\ 1\leqslant p\leqslant \infty$ are equivalent. We also assume that \begin{equation*} \bigcup_{N}X_N \mbox{ is dense in }\left\{v\in W^{1,p}(\Omega) \mid v\cdot \nu=0\mbox{ on }\partial\Omega\right\},\mbox{ for any }1\leqslant p < \infty. \end{equation*} Such a choice of $X_N$ has been constructed in \cite[Theorem 11.19, page 460]{MR3729430}.
The task is to find a triplet $(S^N, \rho^N, u^N)$ satisfying: \begin{itemize} \item $\mathcal{S}^N(t) \Subset \Omega$ is a bounded, regular domain for all $t \in [0,T]$ with \begin{equation}\label{galerkin-approx1} \chi^N_{\mathcal{S}}(t,x)= \mathds{1}_{\mathcal{S}^N(t)}(x) \in L^{\infty}((0,T)\times \Omega) \cap C([0,T];L^p(\Omega)), \, \forall \, 1 \leqslant p < \infty. \end{equation}
\item The velocity field $u^N (t,\cdot)=\sum\limits_{k=1}^N \alpha_k(t)e_k$ with $(\alpha_1,\alpha_2,\ldots,\alpha_N)\in C([0,T])^N$ and the density function $\rho^{N} \in L^2(0,T;H^{2}(\Omega)) \cap H^{1}(0,T;L^{2}(\Omega))$, $\rho^{N} > 0$ satisfies \begin{equation}\label{galerkin-approx2} \frac{\partial {\rho}^N}{\partial t} + \operatorname{div}({\rho}^N u^N) =\varepsilon \Delta\rho^N \mbox{ in }\, (0,T)\times \Omega, \quad \frac{\partial \rho^N}{\partial \nu}=0 \mbox{ on }\, \partial\Omega. \end{equation} \item For all $\phi \in \mathcal{D}([0,T); X_N)$ with $\phi\cdot \nu=0$ on $\partial\Omega$, the following holds: \begin{multline}\label{galerkin-approx3} - \int\limits_0^T\int\limits_{\Omega} \rho^N \left(u^N\cdot \frac{\partial}{\partial t}\phi + u^N \otimes u^N : \nabla \phi\right) + \int\limits_0^T\int\limits_{\Omega} \Big(2\mu^N\mathbb{D}(u^N):\mathbb{D}(\phi) + \lambda^N\operatorname{div}u^N\mathbb{I} : \mathbb{D}(\phi) - p^{N}(\rho^N)\mathbb{I}:\mathbb{D}(\phi)\Big) \\ \int\limits_0^T\int\limits_{\Omega} \varepsilon \nabla u^N \nabla \rho^N \cdot \phi
+ \alpha \int\limits_0^T\int\limits_{\partial \Omega} (u^N \times \nu)\cdot (\phi \times \nu) + \alpha \int\limits_0^T\int\limits_{\partial \mathcal{S}^N(t)} [(u^N-P^N_{\mathcal{S}}u^N)\times \nu]\cdot [(\phi-P^N_{\mathcal{S}}\phi)\times \nu] \\
+ \frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi^N_{\mathcal{S}}(u^N-P^N_{\mathcal{S}}u^N)\cdot (\phi-P^N_{\mathcal{S}}\phi) = \int\limits_0^T\int\limits_{\Omega} \rho^N g^N \cdot \phi
+ \int\limits_{\Omega} (\rho^N u^N \cdot \phi)(0). \end{multline} \item ${\chi}^N_{\mathcal{S}}(t,x)$ satisfies (in the weak sense) \begin{equation}\label{galerkin-approx4}
\frac{\partial {\chi}^N_{\mathcal{S}}}{\partial t} + P^N_{\mathcal{S}}u^N \cdot \nabla \chi^N_{\mathcal{S}} =0\mbox{ in }(0,T)\times {\Omega},\quad \chi^N_{\mathcal{S}}|_{t=0}= \mathds{1}_{\mathcal{S}_0}\mbox{ in } {\Omega}. \end{equation} \item $\rho^{N}{\chi}^{N}_{\mathcal{S}}(t,x)$ satisfies (in the weak sense) \begin{equation}\label{N:approx5}
\frac{\partial }{\partial t}(\rho^{N}{\chi}^{N}_{\mathcal{S}}) + P^{N}_{\mathcal{S}}u^{N} \cdot \nabla (\rho^{N}{\chi}^{N}_{\mathcal{S}})=0\mbox{ in }(0,T)\times {\Omega},\quad (\rho^{N}{\chi}^{N}_{\mathcal{S}})|_{t=0}= \rho_{0}^N\mathds{1}_{\mathcal{S}_0}\mbox{ in } {\Omega}. \end{equation} \item The initial data are given by \begin{equation}\label{galerkin-initial}
\rho^N(0)=\rho_0^N, \quad u^N(0) = u_0^N \quad\mbox{ in }\Omega,\quad \frac {\partial \rho_0^{N }}{\partial \nu}\Big |_{\partial \Omega} =0. \end{equation} \end{itemize} Above we have used the following quantities: \begin{itemize} \item The density of the volume force is defined as \begin{equation}\label{gN} g^N=(1-\chi^N_{\mathcal{S}})g_{\mathcal{F}} + \chi^N_{\mathcal{S}}g_{\mathcal{S}}. \end{equation} \item The artificial pressure is given by \begin{equation}\label{p2} p^{N}(\rho)= a^{N}\rho^{\gamma} + {\delta} \rho^{\beta},\quad\mbox{ with }\quad a^{N} = a_{\mathcal{F}} (1-\chi^{N}_{\mathcal{S}}), \end{equation} where $a_{\mathcal{F}},{\delta} >0$ and the exponents $\gamma$ and $\beta$ satisfy $\gamma > 3/2, \ \beta \geqslant \max\{8,\gamma\}$. \item The viscosity coefficients are given by \begin{equation}\label{vis2}
\mu^N = (1-\chi^N_{\mathcal{S}})\mu_{\mathcal{F}} + {\delta}^2\chi^N_{\mathcal{S}},\quad \lambda^N = (1-\chi^N_{\mathcal{S}})\lambda_{\mathcal{F}} + {\delta}^2\chi^N_{\mathcal{S}}\quad\mbox{ so that }\quad\mu^N >0,\ 2\mu^N+3\lambda^N \geqslant 0.
\end{equation}
\item $P^{N}_{\mathcal{S}}:L^2(\Omega)\rightarrow L^2(\mathcal{S}^{N}(t))$ is the orthogonal projection to rigid fields; it is defined as in \eqref{approx:projection} with $\chi^{\delta}_{\mathcal{S}}$ replaced by $\chi^{N}_{\mathcal{S}}$. \end{itemize} \begin{remark} Actually the triplet $\left(\mathcal{S}^N, \rho^N, u^N\right)$ above should be denoted by $\left(\mathcal{S}^{\delta,\varepsilon,N}, \rho^{\delta,\varepsilon,N}, u^{\delta,\varepsilon,N}\right)$. The dependence on $\delta$ and $\varepsilon$ is due to the penalization term $\left(\tfrac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi^N_{\mathcal{S}}(u^N-P^N_{\mathcal{S}}u^N)\cdot (\phi-P^N_{\mathcal{S}}\phi)\right)$, the viscosity coefficients $\mu^{N}$, $\lambda^{N}$ and the artificial dissipative term $(\varepsilon\Delta\rho)$. To simplify the notation, we omit $\delta$ and $\varepsilon$ here. \end{remark}
A weak solution $(S^{\varepsilon},\rho^{\varepsilon},u^{\varepsilon})$ to the system \eqref{varepsilon:approx1}--\eqref{varepsilon:initial} is obtained through the limit of $(S^N, \rho^N, u^N)$ as $N\rightarrow \infty$. The existence result of the approximate solution of the Faedo-Galerkin scheme reads: \begin{proposition}\label{fa} Let $\Omega$ and $\mathcal{S}_0 \Subset \Omega$ be two regular bounded domains of $\mathbb{R}^3$. Assume that for some $\sigma>0$,
\begin{equation*}
\operatorname{dist}(\mathcal{S}_0,\partial\Omega) > 2\sigma.
\end{equation*}
Let $g^N=(1-\chi^{N}_{\mathcal{S}})g_{\mathcal{F}} + \chi^{N}_{\mathcal{S}}g_{\mathcal{S}} \in L^{\infty}((0,T)\times \Omega)$ and $\beta$, $\delta$, $\gamma$ be given by \eqref{cc:dbg}. Further, let the pressure $p^{N}$ be determined by \eqref{p2} and the viscosity coefficients $\mu^N$, $\lambda^N$ be given by \eqref{vis2}. The initial conditions are assumed to satisfy \begin{equation}\label{initialcond} 0<\underline{\rho}\leqslant \rho_0^N \leqslant \overline{\rho}\ \mbox{ in }\ \Omega, \quad \rho_0^N \in W^{1,\infty}(\Omega),\quad u_0^N\in X_N.
\end{equation}
Let the initial energy $$
E^N(\rho_0^N,q_0^N) =\int\limits_{\Omega} \left( \frac{1}{\rho_0^N}|q_0^N|^2\mathds{1}_{\{\rho_0>0\}} + \frac{a^N(0)}{\gamma-1}(\rho_0^N)^{\gamma} + \frac{\delta}{\beta-1}(\rho_0^N)^{\beta} \right):= E_0^N$$ be uniformly bounded with respect to $N,\varepsilon,\delta$. Then there exists $T>0$ (depending only on $E^{N}_0$, $g_{\mathcal{F}}$, $g_{\mathcal{S}}$, $\overline{\rho}$, $\underline{\rho}$, $\operatorname{dist}(\mathcal{S}_0,\partial\Omega)$) such that the problem \eqref{galerkin-approx1}--\eqref{galerkin-initial} admits a solution $(\mathcal{S}^N,\rho^N,u^N)$ and it satisfies the energy inequality: \begin{multline*}
E^N[\rho ^N, q^N] + \int\limits_0^T\int\limits_{\Omega} \Big(2\mu^N|\mathbb{D}(u^N)|^2 + \lambda^N |\operatorname{div}u^N|^2\Big) + \delta\varepsilon \beta\int\limits_0^T\int\limits_{\Omega} (\rho^N)^{\beta-2}|\nabla \rho^N|^2
+ \alpha \int\limits_0^T\int\limits_{\partial \Omega} |u^N \times \nu|^2 \\
+ \alpha \int\limits_0^T\int\limits_{\partial \mathcal{S}^N(t)} |(u^N-P^N_{\mathcal{S}}u^N)\times \nu|^2
+ \frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi^N_{\mathcal{S}}|u^N-P^N_{\mathcal{S}}u^N|^2 \leqslant \int\limits_0^T\int\limits_{\Omega}\rho^N g^N \cdot u^N
+ E^N_0.
\end{multline*}
Moreover, \begin{equation*}
\operatorname{dist}(\mathcal{S}^{N}(t),\partial\Omega) \geqslant {2\sigma},\quad \forall \ t\in [0,T].
\end{equation*} \end{proposition}
We prove the above proposition in \cref{sec:Galerkin}.
\section{Isometric propagators and the motion of the body} \label{S5} In this section, we state and prove some results regarding the transport equation that we use in our analysis. We mainly concentrate on the following equation: \begin{equation}\label{transport1}
\frac{\partial {\chi}_{\mathcal{S}}}{\partial t} + \operatorname{div}(P_{\mathcal{S}}u\chi_{\mathcal{S}}) =0 \mbox{ in }(0,T)\times\mathbb{R}^3,\quad {\chi}_{\mathcal{S}}|_{t=0}=\mathds{1}_{\mathcal{S}_0}\mbox{ in }\mathbb{R}^3, \end{equation} where $P_{\mathcal{S}}u \in \mathcal{R}$. Note that here $\mathcal{R}$ is referring to the set of rigid fields on $\mathbb{R}^3$ in the spirit of \eqref{defR}. It is given by \begin{equation}\label{projection:P} P_{\mathcal{S}}u (t,x)= \frac{1}{m} \int\limits_{\Omega} \rho\chi_{\mathcal{S}} u + \left(J^{-1} \int\limits_{\Omega}\rho\chi_{\mathcal{S}}((y-h(t)) \times u)\ dy \right)\times (x-h(t)),\quad \forall \ (t,x)\in (0,T)\times\mathbb{R}^3. \end{equation}
In \cite[Proposition 3.1]{MR3272367}, the existence of a solution to \eqref{transport1} and the characterization of the transport of the rigid body have been established with constant $\rho$ in the expression \eqref{projection:P} of $P_{\mathcal{S}}u$. Here we deal with the case when $\rho$ is evolving. We start with some existence results when the velocity field and the density satisfy certain regularity assumptions. \begin{proposition}\label{reg:chiS} Let $u \in C([0,T];\mathcal{D}(\overline{\Omega}))$ and $\rho \in L^2(0,T;H^2(\Omega)) \cap C([0,T];H^1(\Omega))$. Then the following holds true: \begin{enumerate} \item There is a unique solution $\chi_{\mathcal{S}} \in L^{\infty}((0,T)\times \mathbb{R}^3) \cap C([0,T];L^p(\mathbb{R}^3))$ $\forall \ 1\leqslant p<\infty$ to \eqref{transport1}. More precisely, $$\chi_{\mathcal{S}}(t,x)= \mathds{1}_{\mathcal{S}(t)}(x),\quad\forall \ t\geqslant 0,\ \forall \ x\in \mathbb{R}^3.$$ Moreover, $\mathcal{S}(t)=\eta_{t,0}(\mathcal{S}_0)$ for the isometric propagator $\eta_{t,s}$ associated to $P_{\mathcal{S}}u$: \begin{equation*} (t,s)\mapsto \eta_{t,s} \in C^1([0,T]^2; C^{\infty}_{loc}(\mathbb{R}^3)), \end{equation*} where
$\eta_{t,s}$ is defined by \begin{equation}\label{ODE-propagator} \frac{\partial \eta_{t,s}}{\partial t}(y)=P_{\mathcal{S}}u (t,\eta_{t,s}(y)),\quad \forall\ (t,s,y)\in (0,T)^2 \times \mathbb{R}^3, \quad \eta_{s,s}(y)=y,\quad \forall\ y\in \mathbb{R}^3. \end{equation} \item
Let $\rho_{0}\mathds{1}_{\mathcal{S}_0} \in L^{\infty}(\mathbb{R}^3)$. Then there is a unique solution $\rho\chi_{\mathcal{S}} \in L^{\infty}((0,T)\times \mathbb{R}^3) \cap C([0,T];L^p(\mathbb{R}^3))$, $\forall \ 1\leqslant p<\infty$ to the following equation: \begin{equation}\label{re:transport1}
\frac{\partial (\rho {\chi}_{\mathcal{S}})}{\partial t} + \operatorname{div}((\rho\chi_{\mathcal{S}})P_{\mathcal{S}}u) =0 \mbox{ in }(0,T)\times\mathbb{R}^3,\quad \rho{\chi}_{\mathcal{S}}|_{t=0}=\rho_{0}\mathds{1}_{\mathcal{S}_0}\mbox{ in }\mathbb{R}^3. \end{equation} \end{enumerate} \end{proposition} \begin{proof} Following \cite[Proposition 3.1]{MR3272367}, we observe that proving existence of solution to \eqref{transport1} is equivalent to establishing the well-posedness of the ordinary differential equation \begin{equation}\label{ODE-cauchy} \frac{d}{dt}\eta_{t,0}=U_{\mathcal{S}}(t,\eta_{t,0}),\quad \eta_{0,0}=\mathbb{I}, \end{equation} where $U_{\mathcal{S}}\in \mathcal{R}$ is given by \begin{equation*} U_{\mathcal{S}}(t,\eta_{t,0})= \frac{1}{m} \int\limits_{\mathcal{S}_0} \rho(t,\eta_{t,0}(y))\mathds{1}_{\Omega} u(t,\eta_{t,0}(y)) + \left(J^{-1} \int\limits_{\mathcal{S}_0}\rho(t,\eta_{t,0}(y))\mathds{1}_{\Omega}((\eta_{t,0}(y)-h(t)) \times u(t,\eta_{t,0}(y)))\ dy \right)\times (x-h(t)). \end{equation*}
According to the Cauchy-Lipschitz theorem, equation \eqref{ODE-cauchy} admits the unique $C^1$ solution if $U_{\mathcal{S}}$ is continuous in $(t,\eta)$ and uniformly Lipschitz in $\eta$. Thus, it is enough to establish the following result analogous to \cite[Lemma 3.2]{MR3272367}: Let $u \in C([0,T];\mathcal{D}(\overline{\Omega}))$ and $\rho \in L^2(0,T;H^2(\Omega)) \cap C([0,T];H^1(\Omega))$. Then the function
\begin{equation*}
\mathcal{M}:[0,T]\times \operatorname{Isom}(\mathbb{R}^3)\mapsto \mathbb{R},\quad \mathcal{M}(t,\eta)=\int\limits_{\mathcal{S}_0} \rho(t,\eta(y))\mathds{1}_{\Omega}(\eta(y)) u(t,\eta(y))
\end{equation*}
is continuous in $(t,\eta)$ and uniformly Lipschitz in $\eta$ over $[0,T]$.
Observe that the continuity in the $t$-variable is obvious. Moreover, for two isometries $\eta_1$ and $\eta_2$, we have
\begin{align*}
\mathcal{M}(t,\eta_1)-&\mathcal{M}(t,\eta_2)\\=& \int\limits_{\mathcal{S}_0} \rho(t,\eta_1(y))\mathds{1}_{\Omega}(\eta_1(y)) (u(t,\eta_1(y))-u(t,\eta_2(y))) + \int\limits_{\mathcal{S}_0} \rho(t,\eta_1(y))(\mathds{1}_{\Omega}(\eta_1(y))-\mathds{1}_{\Omega}(\eta_2(y))) u(t,\eta_2(y)) \\&+ \int\limits_{\mathcal{S}_0} (\rho(t,\eta_1(y))-\rho(t,\eta_2(y)))\mathds{1}_{\Omega}(\eta_2(y)) u(t,\eta_2(y)):=M_1 +M_2 + M_3.
\end{align*}
As $\rho \in L^2(0,T;H^2(\Omega)) \cap C([0,T];H^1(\Omega))$, the estimates of the terms $M_1$ and $M_2$ are similar to \cite[Lemma 3.2]{MR3272367}. The term $M_3$ can be estimated in the following way:
\begin{equation*}
|M_3|\leqslant C\|\rho\|_{L^{\infty}(0,T;H^1(\Omega))}\|u\|_{L^{\infty}(0,T;L^2(\Omega))}\|\eta_1 -\eta_2\|_{\infty}.
\end{equation*}
This finishes the proof of the first part of \cref{reg:chiS}. The second part of this Proposition is similar and we skip it here. \end{proof} Next we prove the analogous result of \cite[Proposition 3.3, Proposition 3.4]{MR3272367} on strong and weak sequential continuity which are essential to establish the existence result of the Galerkin approximation scheme in \cref{sec:Galerkin}. The result obtained in the next proposition is used to establish the continuity of the fixed point map in the proof of the existence of Galerkin approximation. \begin{proposition}\label{sequential1} Let $\rho^N_0 \in W^{1,\infty}(\Omega)$, let $\rho^k \in L^2(0,T;H^{2}(\Omega)) \cap C([0,T]; H^{1}(\Omega)) \cap H^{1}(0,T;L^{2}(\Omega))$ be the solution to \begin{equation}\label{eq:rhoN}
\frac{\partial {\rho^k}}{\partial t} + \operatorname{div}({\rho}^k u^k) = \Delta\rho^k \mbox{ in }\, (0,T)\times \Omega, \quad \frac{\partial \rho^k}{\partial \nu}=0 \mbox{ on }\, \partial\Omega, \quad\rho^k(0,x)=\rho_0^N(x)\quad\mbox{in }\Omega,\quad\frac {\partial \rho_0^{k}}{\partial \nu}\big |_{\partial \Omega} =0. \end{equation}
\begin{equation*} u^k \rightarrow u \mbox{ strongly in }C([0,T];\mathcal{D}(\overline{\Omega})),\quad \chi_{\mathcal{S}}^k \mbox{ is bounded in }L^{\infty}((0,T)\times \mathbb{R}^3)\mbox{ satisfying } \end{equation*} \begin{equation}\label{n:transport}
\frac{\partial {\chi}^k_{\mathcal{S}}}{\partial t} + \operatorname{div}(P^k_{\mathcal{S}}u^k\chi^k_{\mathcal{S}}) =0 \mbox{ in }(0,T)\times\mathbb{R}^3,\quad {\chi}^k_{\mathcal{S}}|_{t=0}=\mathds{1}_{\mathcal{S}_0}\mbox{ in }\mathbb{R}^3, \end{equation} and let $\{\rho^{k}\chi_{\mathcal{S}}^{k}\}$ be a bounded sequence in $L^{\infty}((0,T)\times\mathbb{R}^3)$ satisfying \begin{equation}\label{N:rhotrans}
\frac{\partial}{\partial t}(\rho^{k}{\chi}^{k}_{\mathcal{S}}) + \operatorname{div}(P^{k}_{\mathcal{S}}u^{k}(\rho^{k}\chi^{k}_{\mathcal{S}})) =0 \mbox{ in }(0,T)\times\mathbb{R}^3,\quad \rho^{k}{\chi}^{k}_{\mathcal{S}}|_{t=0}=\rho^N_0\mathds{1}_{\mathcal{S}_0}\mbox{ in }\mathbb{R}^3, \end{equation}
where $P^{k}_{\mathcal{S}}:L^2(\Omega)\rightarrow L^2(\mathcal{S}^{k}(t))$ is the orthogonal projection to rigid fields with $\mathcal{S}^{k}(t) \Subset \Omega$ being a bounded, regular domain for all $t \in [0,T]$. Then \begin{align*} & \chi_{\mathcal{S}}^k \rightarrow \chi_{\mathcal{S}} \mbox{ weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) \mbox{ and}\mbox{ strongly} \mbox{ in }C([0,T]; L^p_{loc}(\mathbb{R}^3)), \ \forall \ 1\leqslant p<\infty,\\ &\rho^k\chi_{\mathcal{S}}^k \rightarrow \rho \chi_{\mathcal{S}} \mbox{ weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) \mbox{ and}\mbox{ strongly} \mbox{ in }C([0,T]; L^p_{loc}(\mathbb{R}^3)), \ \forall \ 1\leqslant p<\infty, \end{align*} where $\chi_{\mathcal{S}}$ and $\rho\chi_{\mathcal{S}}$ are satisfying \eqref{transport1} and \eqref{re:transport1} with initial data $\mathds{1}_{\mathcal{S}_0}$ and $\rho^{N}_{0}\mathds{1}_{\mathcal{S}_0}$, respectively. Moreover, \begin{align*} & P_{\mathcal{S}}^k u^k \rightarrow P_{\mathcal{S}} u \mbox{ strongly} \mbox{ in }C([0,T]; C^{\infty}_{loc}(\mathbb{R}^3)), \\ &\eta_{t,s}^k \rightarrow \eta_{t,s} \mbox{ strongly} \mbox{ in }C^{1}([0,T]^2; C^{\infty}_{loc}(\mathbb{R}^3)). \end{align*} \end{proposition}
\begin{proof}
As $\{u^{k}\}$ converges strongly in $C([0,T];\mathcal{D}(\overline{\Omega}))$
and $\{ \rho^{k} \chi_{\mathcal{S}}^{k}\} \mbox{ is bounded in }L^{\infty}((0,T)\times \mathbb{R}^3)$,
we obtain that $P^{k}_{\mathcal{S}}u^{k}$ is bounded in $L^2(0,T;\mathcal{R})$. Thus, up to a subsequence,
\begin{equation}\label{PN:weak}
P^{k}_{\mathcal{S}}u^{k} \rightarrow \overline{u_{\mathcal{S}}} \mbox{ weakly in }L^2(0,T;\mathcal{R}).
\end{equation}
Here, obviously $P^{k}_{\mathcal{S}}u^{k} \in L^1(0,T; L^{\infty}_{loc}(\mathbb{R}^3))$, $\operatorname{div}(P^{k}_{\mathcal{S}}u^{k})=0$ and $\overline{u_{\mathcal{S}}} \in L^{1}(0,T;W^{1,1}_{loc}(\mathbb{R}^3))$ satisfies \begin{equation*}
\frac{\overline{u_{\mathcal{S}}}}{1+|x|} \in L^1(0,T;L^1(\mathbb{R}^3)). \end{equation*} Moreover, $\{\chi_{\mathcal{S}}^{k}\} \mbox{ is bounded in }L^{\infty}((0,T)\times \mathbb{R}^3)$, $\chi_{\mathcal{S}}^{k}$ satisfies \eqref{n:transport} and $\{\rho^{N}\chi_{\mathcal{S}}^{k}\}$ is bounded in $L^{\infty}((0,T)\times\mathbb{R}^3)$, $\rho^{k}\chi_{\mathcal{S}}^{k}$ satisfies \eqref {N:rhotrans}. As we have verified all the required conditions, we can apply \cite[Theorem II.4, Page 521]{DiPerna1989} to obtain $$\chi_{\mathcal{S}}^{k} \mbox{ converges weakly-}*\mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3),\mbox{ strongly in }C([0,T]; L^p_{loc}(\mathbb{R}^3)) \ (1\leqslant p<\infty),$$ $$\rho^{k}\chi_{\mathcal{S}}^{k} \mbox{ converges weakly-}*\mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3),\mbox{ strongly in }C([0,T]; L^p_{loc}(\mathbb{R}^3)) \ (1\leqslant p<\infty).$$ Let the limit of $\chi_{\mathcal{S}}^{k}$ be denoted by ${\chi_{\mathcal{S}}}$, it satisfies \begin{equation*}
\frac{\partial {\chi_{\mathcal{S}}}}{\partial t} + \operatorname{div}(\overline{u_{\mathcal{S}}}\ {\chi_{\mathcal{S}}}) =0 \mbox{ in }(0,T)\times \mathbb{R}^3,\quad {\chi_{\mathcal{S}}}|_{t=0}=\mathds{1}_{\mathcal{S}_0}\mbox{ in }\mathbb{R}^3. \end{equation*} Let the weak limit of $\rho^{k}\chi_{\mathcal{S}}^{k}$ be denoted by $\overline{\rho\chi_{\mathcal{S}}}$; it satisfies \begin{equation*}
\frac{\partial (\overline{\rho\chi_{\mathcal{S}}})}{\partial t} + \operatorname{div}(\overline{u_{\mathcal{S}}}\ \overline{\rho\chi_{\mathcal{S}}}) =0 \mbox{ in }(0,T)\times\mathbb{R}^3,\quad \overline{\rho\chi_{\mathcal{S}}}|_{t=0}=\rho^N_0\mathds{1}_{\mathcal{S}_0}\mbox{ in }\mathbb{R}^3. \end{equation*} We follow the similar analysis as for the fluid case explained in \cite[Section 7.8.1, Page 362]{MR2084891} to conclude that \begin{equation*}
\rho^k \rightarrow \rho\mbox{ strongly in } L^p((0,T)\times \Omega), \quad\forall \ 1\leqslant p< \frac{4}{3}\beta\mbox{ with } \ \beta \geqslant \max\{8,\gamma\},\ \gamma > 3/2. \end{equation*} The strong convergences of $\rho^{k}$ and $\chi_{\mathcal{S}}^{k}$ help us to identify the limit: \begin{equation*} \overline{\rho\chi_{\mathcal{S}}}= \rho\chi_{\mathcal{S}}. \end{equation*} Using the convergences of $\rho^{k}\chi_{\mathcal{S}}^{k}$ and $u^{k}$ in the equation \begin{equation*} P_{\mathcal{S}}^{k}u^{k}(t,x)= \frac{1}{m^{k}} \int\limits_{\Omega} \rho^{k}\chi_{\mathcal{S}}^{k} u^{k} + \left((J^k)^{-1} \int\limits_{\Omega}\rho^{k}\chi_{\mathcal{S}}^{k}((y-h^{k}(t)) \times u^{k})\ dy \right)\times (x-h^{k}(t)), \end{equation*} and the convergence in \eqref{PN:weak}, we conclude that \begin{equation*} \overline{u_{\mathcal{S}}}={P_{\mathcal{S}}}u. \end{equation*}
The convergence of the isometric propagator $\eta_{t,s}^{k}$ follows from the convergence of $P_{\mathcal{S}}^{k} u^{k}$ and equation \eqref{ODE-propagator}.
\end{proof}
We need the next result on weak sequential continuity to analyze the limiting system of Faedo-Galerkin as $N\rightarrow\infty$ in \cref{14:14}. The proof is similar to that of \cref{sequential1} and we skip it here. \begin{proposition}\label{sequential11}
Let us assume that $\rho^N_0 \in W^{1,\infty}(\Omega)$ with $\rho^N_0 \rightarrow \rho_0$ in $W^{1,\infty}(\Omega)$,
$\rho^N$ satisfies \eqref{eq:rhoN} and \begin{equation*} \rho^N \rightarrow \rho\mbox{ strongly in }L^p((0,T)\times \Omega),\ 1\leqslant p< \frac{4}{3}\beta\mbox{ with } \ \beta \geqslant \max\{8,\gamma\},\ \gamma > 3/2. \end{equation*} Let $\{u^N,\chi_{\mathcal{S}}^N\}$ be a bounded sequence in $L^{\infty}(0,T; L^2(\Omega)) \times L^{\infty}((0,T)\times \mathbb{R}^3)$ satisfying \eqref{n:transport}. Let $\{\rho^{N}\chi_{\mathcal{S}}^{N}\}$ be a bounded sequence in $L^{\infty}((0,T)\times\mathbb{R}^3)$ satisfying \eqref{N:rhotrans}. Then, up to a subsequence, we have \begin{align*} & u^N \rightarrow u \mbox{ weakly-}* \mbox{ in }L^{\infty}(0,T; L^{2}(\Omega)),\\ & \chi_{\mathcal{S}}^N \rightarrow \chi_{\mathcal{S}} \mbox{ weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) \mbox{ and}\mbox{ strongly} \mbox{ in }C([0,T]; L^p_{loc}(\mathbb{R}^3)), \ \forall\ 1\leqslant p<\infty,\\ &\rho^N\chi_{\mathcal{S}}^N \rightarrow \rho\chi_{\mathcal{S}} \mbox{ weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) \mbox{ and}\mbox{ strongly} \mbox{ in }C([0,T]; L^p_{loc}(\mathbb{R}^3)), \ \forall \ 1\leqslant p<\infty, \end{align*} where $\chi_{\mathcal{S}}$ and $\rho\chi_{\mathcal{S}}$ satisfying \eqref{transport1} and \eqref{re:transport1}, respectively. Moreover, \begin{align*} & P_{\mathcal{S}}^N u^N \rightarrow P_{\mathcal{S}} u \mbox{ weakly-}* \mbox{ in }L^{\infty}(0,T; C^{\infty}_{loc}(\mathbb{R}^3)),\\ &\eta_{t,s}^N \rightarrow \eta_{t,s} \mbox{ weakly-}* \mbox{ in }W^{1,\infty}((0,T)^2; C^{\infty}_{loc}(\mathbb{R}^3)). \end{align*}
\end{proposition}
At the level of the Galerkin approximation, we have boundedness of $\sqrt{\rho^N}u^N$ in $L^{\infty}(0,T;L^2(\Omega))$ and $\rho^N$ is strictly positive, which means that we get the boundedness of $u^N$ in $L^{\infty}(0,T; L^2(\Omega))$. So, we can use \cref{sequential11} in the convergence analysis of the Galerkin scheme. In the case of the $\varepsilon$-level for the compressible fluid, we have boundedness of $\sqrt{\rho ^{\varepsilon}}u^{\varepsilon}$ in $L^{\infty}(0,T;L^2(\Omega))$ but $\rho^{\varepsilon}$ is only non-negative. On the other hand, we establish boundedness of $u^{\varepsilon}$ in $L^{2}(0,T;H^1(\Omega))$. We need the following result for the convergence analysis of the vanishing viscosity limit in \cref{14:18}. \begin{proposition}\label{sequential-varepsilon} Let $\rho^{\varepsilon}_0 \in W^{1,\infty}(\Omega)$ with $\rho^{\varepsilon}_0 \rightarrow \rho_0$ in $L^{\beta}(\Omega)$,
$\rho^{\varepsilon}$ satisfies
\begin{equation*}
\frac{\partial {\rho^{\varepsilon}}}{\partial t} + \operatorname{div}({\rho}^{\varepsilon} u^{\varepsilon}) = \Delta\rho^{\varepsilon} \mbox{ in }\, (0,T)\times \Omega, \quad \frac{\partial \rho^{\varepsilon}}{\partial \nu}=0 \mbox{ on }\, \partial\Omega, \quad\rho^{\varepsilon}(0,x)=\rho_0^{\varepsilon}(x)\mbox{ in }\ \Omega,\quad\frac {\partial \rho_0^{\varepsilon}}{\partial \nu}\big |_{\partial \Omega} =0., \end{equation*} and \begin{equation}\label{epsilon-rhoweak} \rho^{\varepsilon}\rightarrow \rho\mbox{ weakly in }L^{\beta+1}((0,T)\times \Omega),\mbox{ with } \ \beta \geqslant \max\{8,\gamma\},\ \gamma > 3/2. \end{equation} Let $\{u^{\varepsilon},\chi_{\mathcal{S}}^{\varepsilon}\}$ be a bounded sequence in $L^{2}(0,T; H^1(\Omega)) \times L^{\infty}((0,T)\times \mathbb{R}^3)$ satisfying
\begin{equation}\label{epsilon:transport}
\frac{\partial {\chi}^{\varepsilon}_{\mathcal{S}}}{\partial t} + \operatorname{div}(P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon}\chi^{\varepsilon}_{\mathcal{S}}) =0 \mbox{ in }(0,T)\times\mathbb{R}^3,\quad {\chi}^{\varepsilon}_{\mathcal{S}}|_{t=0}=\mathds{1}_{\mathcal{S}_0}\mbox{ in }\mathbb{R}^3, \end{equation} and let $\{\rho^{\varepsilon}\chi_{\mathcal{S}}^{\varepsilon}\}$ be a bounded sequence in $L^{\infty}((0,T)\times\mathbb{R}^3)$ satisfying \begin{equation}\label{epsilon:rhotrans}
\frac{\partial}{\partial t}(\rho^{\varepsilon}{\chi}^{\varepsilon}_{\mathcal{S}}) + \operatorname{div}(P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon}(\rho^{\varepsilon}\chi^{\varepsilon}_{\mathcal{S}})) =0 \mbox{ in }(0,T)\times\mathbb{R}^3,\quad \rho^{\varepsilon}{\chi}^{\varepsilon}_{\mathcal{S}}|_{t=0}=\rho^{\varepsilon}_0\mathds{1}_{\mathcal{S}_0}\mbox{ in }\mathbb{R}^3, \end{equation} where $P^{\varepsilon}_{\mathcal{S}}:L^2(\Omega)\rightarrow L^2(\mathcal{S}^{\varepsilon}(t))$ is the orthogonal projection onto rigid fields with $\mathcal{S}^{\varepsilon}(t) \Subset \Omega$ being a bounded, regular domain for all $t \in [0,T]$. Then up to a subsequence, we have \begin{align*} & u^{\varepsilon} \rightarrow u \mbox{ weakly } \mbox{ in }L^{2}(0,T; H^{1}(\Omega)),\\ & \chi_{\mathcal{S}}^{\varepsilon} \rightarrow \chi_{\mathcal{S}} \mbox{ weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) \mbox{ and }\mbox{ strongly } \mbox{ in }C([0,T]; L^p_{loc}(\mathbb{R}^3)) \ (1\leqslant p<\infty),\\ &\rho^{\varepsilon}\chi_{\mathcal{S}}^{\varepsilon} \rightarrow \rho\chi_{\mathcal{S}} \mbox{ weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) \mbox{ and }\mbox{ strongly } \mbox{ in }C([0,T]; L^p_{loc}(\mathbb{R}^3)) \ (1\leqslant p<\infty), \end{align*} with $\chi_{\mathcal{S}}$ and $\rho\chi_{\mathcal{S}}$ satisfying \eqref{transport1} and \eqref{re:transport1} respectively. Moreover, \begin{align*} & P_{\mathcal{S}}^{\varepsilon} u^{\varepsilon} \rightarrow P_{\mathcal{S}} u \mbox{ weakly } \mbox{ in }L^{2}(0,T; C^{\infty}_{loc}(\mathbb{R}^3)),\\ &\eta_{t,s}^{\varepsilon} \rightarrow \eta_{t,s} \mbox{ weakly } \mbox{ in }H^{1}((0,T)^2; C^{\infty}_{loc}(\mathbb{R}^3)). \end{align*}
\end{proposition}
\begin{proof}
As $\{u^{\varepsilon}\}$ is a bounded sequence in $L^2(0,T;H^1(\Omega))$
and $\{ \rho^{\varepsilon} \chi_{\mathcal{S}}^{\varepsilon}\} \mbox{ is bounded in }L^{\infty}((0,T)\times \mathbb{R}^3)$,
we obtain that $\{P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon}\}$ is bounded in $L^2(0,T;\mathcal{R})$. Thus, up to a subsequence,
\begin{equation}\label{Pepsilon:weak}
P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon} \rightarrow \overline{u_{\mathcal{S}}} \mbox{ weakly in }L^2(0,T;\mathcal{R}).
\end{equation}
Here, obviously $P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon} \in L^1(0,T; L^{\infty}_{loc}(\mathbb{R}^3))$, $\operatorname{div}(P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon})=0$ and $\overline{u_{\mathcal{S}}} \in L^{1}(0,T;W^{1,1}_{loc}(\mathbb{R}^3))$ satisfies \begin{equation*}
\frac{\overline{u_{\mathcal{S}}}}{1+|x|} \in L^1(0,T;L^1(\mathbb{R}^3)). \end{equation*} Moreover, $\{\chi_{\mathcal{S}}^{\varepsilon}\} \mbox{ is bounded in }L^{\infty}((0,T)\times \mathbb{R}^3)$, $\chi_{\mathcal{S}}^{\varepsilon}$ satisfies \eqref{epsilon:transport} and $\{\rho^{\varepsilon}\chi_{\mathcal{S}}^{\varepsilon}\}$ is bounded in $L^{\infty}((0,T)\times\mathbb{R}^3)$, $\rho^{\varepsilon}\chi_{\mathcal{S}}^{\varepsilon}$ satisfies \eqref {epsilon:rhotrans}. As we have verified all the required conditions, we can apply \cite[Theorem II.4, Page 521]{DiPerna1989} to obtain $$\chi_{\mathcal{S}}^{\varepsilon} \mbox{ converges weakly-}*\mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3),\mbox{ strongly in }C([0,T]; L^p_{loc}(\mathbb{R}^3)) \ (1\leqslant p<\infty),$$ $$\rho^{\varepsilon}\chi_{\mathcal{S}}^{\varepsilon} \mbox{ converges weakly-}*\mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3),\mbox{ strongly in }C([0,T]; L^p_{loc}(\mathbb{R}^3)) \ (1\leqslant p<\infty).$$ Let the limit of $\chi_{\mathcal{S}}^{\varepsilon}$ be denoted by ${\chi_{\mathcal{S}}}$; it satisfies \begin{equation*}
\frac{\partial {\chi_{\mathcal{S}}}}{\partial t} + \operatorname{div}(\overline{u_{\mathcal{S}}}\ {\chi_{\mathcal{S}}}) =0 \mbox{ in }\mathbb{R}^3,\quad {\chi_{\mathcal{S}}}|_{t=0}=\mathds{1}_{\mathcal{S}_0}\mbox{ in }\mathbb{R}^3. \end{equation*} Let the limit of $\rho^{\varepsilon}\chi_{\mathcal{S}}^{\varepsilon}$ be denoted by $\overline{\rho\chi_{\mathcal{S}}}$; it satisfies \begin{equation*}
\frac{\partial (\overline{\rho\chi_{\mathcal{S}}})}{\partial t} + \operatorname{div}(\overline{u_{\mathcal{S}}}\ \overline{\rho\chi_{\mathcal{S}}}) =0 \mbox{ in }(0,T)\times\mathbb{R}^3,\quad \overline{\rho\chi_{\mathcal{S}}}|_{t=0}=\rho_0\mathds{1}_{\mathcal{S}_0}\mbox{ in }\mathbb{R}^3. \end{equation*} The weak convergence of $\rho^{\varepsilon}$ and strong convergence of $\chi_{\mathcal{S}}^{\varepsilon}$ help us to identify the limit: \begin{equation*} \overline{\rho\chi_{\mathcal{S}}}= \rho\chi_{\mathcal{S}}. \end{equation*} Using the convergences of $\rho^{\varepsilon}\chi_{\mathcal{S}}^{\varepsilon}$ and $u^{\varepsilon}$ in the equation \begin{equation*} P_{\mathcal{S}}^{\varepsilon}u^{\varepsilon}(t,x)= \frac{1}{m^{\varepsilon}} \int\limits_{\Omega} \rho^{\varepsilon}\chi_{\mathcal{S}}^{\varepsilon} u^{\varepsilon} + \left((J^{\varepsilon})^{-1} \int\limits_{\Omega}\rho^{\varepsilon}\chi_{\mathcal{S}}^{\varepsilon}((y-h^{\varepsilon}(t)) \times u^{\varepsilon})\ dy \right)\times (x-h^{\varepsilon}(t)), \end{equation*} and the convergence in \eqref{Pepsilon:weak}, we conclude that \begin{equation*} \overline{u_{\mathcal{S}}}={P_{\mathcal{S}}}u. \end{equation*}
The convergence of the isometric propagators $\eta_{t,s}^{\varepsilon}$ follows from the convergence of $P_{\mathcal{S}}^{\varepsilon} u^{\varepsilon}$ and equation \eqref{ODE-propagator}.
\end{proof}
In the limit of $u^{\delta}$, we can expect the boundedness of the limit only in $L^{2}(0,T;L^2(\Omega))$ but not in $L^{2}(0,T;H^1(\Omega))$.
That is why we need a different sequential continuity result, which we use in \cref{S4}. \begin{proposition}\label{sequential2} Let $\rho^{\delta}_0 \in L^{\beta}(\Omega)$ with $\rho^{\delta}_0 \rightarrow \rho_0$ in $L^{\gamma}(\Omega)$, let
$\rho^{\delta}$ satisfy
\begin{equation*} \frac{\partial {\rho^{\delta}}}{\partial t} + \operatorname{div}({\rho}^{\delta} u^{\delta}) = 0 \mbox{ in }\, (0,T)\times \Omega, \quad\rho^{\delta}(0,x)=\rho_0^{\delta}(x)\mbox{ in }\ \Omega, \end{equation*} and \begin{equation}\label{delta-rhoweak} \rho^{\delta}\rightarrow \rho\mbox{ weakly in }L^{\gamma+\theta}((0,T)\times \Omega),\mbox{ with }\gamma>3/2,\, \theta=\frac{2}{3}\gamma-1. \end{equation}
Let $\{u^{\delta},\chi_{\mathcal{S}}^{\delta}\}$ be a bounded sequence in $L^{2}(0,T; L^2(\Omega)) \times L^{\infty}((0,T)\times \mathbb{R}^3)$ satisfying
\begin{equation}\label{delta:transport}
\frac{\partial {\chi}^{\delta}_{\mathcal{S}}}{\partial t} + \operatorname{div}(P^{\delta}_{\mathcal{S}}u^{\delta}\chi^{\delta}_{\mathcal{S}}) =0 \mbox{ in }(0,T)\times\mathbb{R}^3,\quad {\chi}^{\delta}_{\mathcal{S}}|_{t=0}=\mathds{1}_{\mathcal{S}_0}\mbox{ in }\mathbb{R}^3, \end{equation} and let $\{\rho^{\delta}\chi_{\mathcal{S}}^{\delta}\}$ be a bounded sequence in $L^{\infty}((0,T)\times\mathbb{R}^3)$ satisfying \begin{equation}\label{delta:rhotrans}
\frac{\partial}{\partial t}(\rho^{\delta}{\chi}^{\delta}_{\mathcal{S}}) + \operatorname{div}(P^{\delta}_{\mathcal{S}}u^{\delta}(\rho^{\delta}\chi^{\delta}_{\mathcal{S}})) =0 \mbox{ in }(0,T)\times\mathbb{R}^3,\quad \rho^{\delta}{\chi}^{\delta}_{\mathcal{S}}|_{t=0}=\rho^{\delta}_0\mathds{1}_{\mathcal{S}_0}\mbox{ in }\mathbb{R}^3, \end{equation} where $P^{\delta}_{\mathcal{S}}:L^2(\Omega)\rightarrow L^2(\mathcal{S}^{\delta}(t))$ is the orthogonal projection onto rigid fields with $\mathcal{S}^{\delta}(t) \Subset \Omega$ being a bounded, regular domain for all $t \in [0,T]$. Then, up to a subsequence, we have \begin{align*} & u^{\delta} \rightarrow u \mbox{ weakly } \mbox{ in }L^{2}(0,T; L^{2}(\Omega)),\\ & \chi_{\mathcal{S}}^{\delta} \rightarrow \chi_{\mathcal{S}} \mbox{ weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) \mbox{ and }\mbox{ strongly } \mbox{ in }C([0,T]; L^p_{loc}(\mathbb{R}^3)) \ (1\leqslant p<\infty),\\ & \rho^{\delta}\chi_{\mathcal{S}}^{\delta} \rightarrow \rho\chi_{\mathcal{S}} \mbox{ weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) \mbox{ and }\mbox{ strongly } \mbox{ in }C([0,T]; L^p_{loc}(\mathbb{R}^3)) \ (1\leqslant p<\infty), \end{align*} with $\chi_{\mathcal{S}}$ and $\rho\chi_{\mathcal{S}}$ satisfying \eqref{transport1} and \eqref{re:transport1} respectively. Moreover, \begin{align*} & P_{\mathcal{S}}^{\delta} u^{\delta} \rightarrow P_{\mathcal{S}} u \mbox{ weakly } \mbox{ in }L^{2}(0,T; C^{\infty}_{loc}(\mathbb{R}^3)),\\ & \eta_{t,s}^{\delta} \rightarrow \eta_{t,s} \mbox{ weakly } \mbox{ in }H^{1}((0,T)^2; C^{\infty}_{loc}(\mathbb{R}^3)). \end{align*}
\end{proposition}
\begin{proof}
As $\{u^{\delta}\}$ is a bounded sequence in $L^2(0,T;L^2(\Omega))$ and $\{ \rho^{\delta} \chi_{\mathcal{S}}^{\delta}\} \mbox{ is bounded in }L^{\infty}((0,T)\times \mathbb{R}^3)$, we obtain that $\{P^{\delta}_{\mathcal{S}}u^{\delta}\}$ is bounded in $L^2(0,T;\mathcal{R})$. Thus, up to a subsequence,
\begin{equation}\label{P:weak}
P^{\delta}_{\mathcal{S}}u^{\delta} \rightarrow \overline{u_{\mathcal{S}}} \mbox{ weakly in }L^2(0,T;\mathcal{R}).
\end{equation}
Here, obviously $P^{\delta}_{\mathcal{S}}u^{\delta} \in L^1(0,T; L^{\infty}_{loc}(\mathbb{R}^3))$, $\operatorname{div}(P^{\delta}_{\mathcal{S}}u^{\delta})=0$ and $\overline{u_{\mathcal{S}}} \in L^{1}(0,T;W^{1,1}_{loc}(\mathbb{R}^3))$ satisfies \begin{equation*}
\frac{\overline{u_{\mathcal{S}}}}{1+|x|} \in L^1(0,T;L^1(\mathbb{R}^3)). \end{equation*} Moreover, $\{\chi_{\mathcal{S}}^{\delta}\} \mbox{ is bounded in }L^{\infty}((0,T)\times \mathbb{R}^3)$, $\chi_{\mathcal{S}}^{\delta}$ satisfies \eqref{delta:transport} and $\{\rho^{\delta}\chi_{\mathcal{S}}^{\delta}\}$ is bounded in $L^{\infty}((0,T)\times\mathbb{R}^3)$, $\rho^{\delta}\chi_{\mathcal{S}}^{\delta}$ satisfies \eqref {delta:rhotrans}. Now we can apply \cite[Theorem II.4, Page 521]{DiPerna1989} to obtain $$\chi_{\mathcal{S}}^{\delta} \mbox{ converges weakly-}*\mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3),\mbox{ and strongly in }C([0,T]; L^p_{loc}(\mathbb{R}^3)) \ (1\leqslant p<\infty),$$ $$\rho^{\delta}\chi_{\mathcal{S}}^{\delta} \mbox{ converges weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) \mbox{ and }\mbox{ strongly } \mbox{ in }C([0,T]; L^p_{loc}(\mathbb{R}^3)) \ (1\leqslant p<\infty).$$ Let the weak limit of $\chi_{\mathcal{S}}^{\delta}$ be denoted by ${\chi_{\mathcal{S}}}$. Then it satisfies \begin{equation*}
\frac{\partial {\chi_{\mathcal{S}}}}{\partial t} + \operatorname{div}(\overline{u_{\mathcal{S}}}\ {\chi_{\mathcal{S}}}) =0 \mbox{ in }(0,T)\times\mathbb{R}^3,\quad {\chi_{\mathcal{S}}}|_{t=0}=\mathds{1}_{\mathcal{S}_0}\mbox{ in }\mathbb{R}^3, \end{equation*} Let the limit of $\rho^{\delta}\chi_{\mathcal{S}}^{\delta}$ be denoted by $\overline{\rho\chi_{\mathcal{S}}}$; it satisfies \begin{equation*}
\frac{\partial (\overline{\rho\chi_{\mathcal{S}}})}{\partial t} + \operatorname{div}(\overline{u_{\mathcal{S}}}\ \overline{\rho\chi_{\mathcal{S}}}) =0 \mbox{ in }(0,T)\times\mathbb{R}^3,\quad \overline{\rho\chi_{\mathcal{S}}}|_{t=0}=\rho_0\mathds{1}_{\mathcal{S}_0}\mbox{ in }\mathbb{R}^3. \end{equation*} From \eqref{delta-rhoweak}, we know that\begin{equation*} \rho^{\delta}\rightarrow \rho\mbox{ weakly in }L^{\gamma+\theta}((0,T)\times \Omega),\mbox{ with }\gamma>3/2,\, \theta=\frac{2}{3}\gamma-1. \end{equation*} The weak convergence of $\rho^{\delta}$ to $\rho$ and strong convergence of $\chi_{\mathcal{S}}^{\delta}$ to $\chi_{\mathcal{S}}$ help us to identify the limit: \begin{equation*} \overline{\rho\chi_{\mathcal{S}}}= \rho\chi_{\mathcal{S}}, \end{equation*} Using the convergences of $\rho^{\delta}\chi_{\mathcal{S}}^{\delta}$ and $u^{\delta}$ in the equation \begin{equation*} P_{\mathcal{S}}^{\delta}u^{\delta}(t,x)= \frac{1}{m^{\delta}} \int\limits_{\Omega} \rho^{\delta}\chi_{\mathcal{S}}^{\delta} u^{\delta} + \left((J^{\delta})^{-1} \int\limits_{\Omega}\rho^{\delta}\chi_{\mathcal{S}}^{\delta}((y-h^{\delta}(t)) \times u^{\delta})\ dy \right)\times (x-h^{\delta}(t)), \end{equation*} and the convergence in \eqref{P:weak}, we conclude that \begin{equation*} \overline{u_{\mathcal{S}}}={P_{\mathcal{S}}}u. \end{equation*}
The convergence of the isometric propagator $\eta_{t,s}^{\delta}$ follows from the convergence of $P_{\mathcal{S}}^{\delta} u^{\delta}$ and equation \eqref{ODE-propagator}.
\end{proof}
\section{Existence proofs of Approximate solutions}\label{S3} In this section, we present the proofs of the existence results of the three approximation levels. We start with the $N$-level approximation in \cref{sec:Galerkin} and the limit as $N\to\infty$ in \cref{14:14}, which yields existence at the $\varepsilon$-level. The convergence of $\varepsilon\to 0$, considered in \cref{14:18}, then shows existence of solutions at the $\delta$-level. The final limit problem as $\delta\to 0$ is the topic of \cref{S4}.
\subsection{Existence of the Faedo-Galerkin approximation}\label{sec:Galerkin} In this subsection, we construct a solution $(\mathcal{S}^N,\rho^N,u^{N})$ to the problem \eqref{galerkin-approx1}--\eqref{galerkin-initial}. First we recall a known maximal regularity result for the parabolic problem \eqref{galerkin-approx2}: \begin{proposition}\cite[Proposition 7.39, Page 345]{MR2084891}\label{parabolic} Suppose that $\Omega$ is a regular bounded domain and assume $\rho_0 \in W^{1,\infty}(\Omega)$, $\underline{\rho} \leqslant \rho_0 \leqslant \overline{\rho}$, $u \in L^{\infty}(0,T;W^{1,\infty}(\Omega))$. Then the parabolic problem \eqref{galerkin-approx2} admits a unique solution in the solution space \begin{equation*} \rho \in L^2(0,T;H^{2}(\Omega)) \cap C([0,T]; H^{1}(\Omega)) \cap H^{1}(0,T;L^{2}(\Omega)) \end{equation*} and it satisfies \begin{equation} \label{bounds-on-rho}
\underline{\rho}\exp \left(-\int\limits_0^{\tau} \|\operatorname{div}u(s)\|_{L^{\infty}(\Omega)}\ ds\right)\leqslant \rho(\tau,x)\leqslant \overline{\rho}\exp \left(\int\limits_0^{\tau} \|\operatorname{div}u(s)\|_{L^{\infty}(\Omega)}\ ds\right) \end{equation} for any $\tau \in [0,T]$. \end{proposition}
\begin{proof}[Proof of \cref{fa}] The idea is to view our Galerkin approximation as a fixed point problem and then apply Schauder's fixed point theorem to it. We set \begin{equation*}
B_{R,T}=\{u\in C([0,T]; X_N),\ \|u\|_{L^{\infty}(0,T;L^2(\Omega))}\leqslant R\}, \end{equation*} for $R$ and $T$ positive which will be fixed in Step 3.
\underline{Step 1: Continuity equation and transport of the body.} Given $u \in B_{R,T}$, let $\rho$ be the solution to \begin{equation}\label{eq:rho} \frac{\partial {\rho}}{\partial t} + \operatorname{div}({\rho} u) =\varepsilon \Delta\rho \mbox{ in }\, (0,T)\times \Omega, \quad \frac{\partial \rho}{\partial \nu}=0 \mbox{ on }\, \partial\Omega, \quad\rho(0)=\rho_0^N,\quad 0<\underline{\rho}\leqslant \rho_0^N \leqslant \overline{\rho}, \end{equation} and let ${\chi}_{\mathcal{S}}$ satisfy \begin{equation}\label{eq:chinew}
\frac{\partial {\chi}_{\mathcal{S}}}{\partial t} + P_{\mathcal{S}}u \cdot \nabla \chi_{\mathcal{S}} =0,\quad \chi_{\mathcal{S}}|_{t=0}= \mathds{1}_{\mathcal{S}_0}, \end{equation} and \begin{equation}\label{eq:rhochinew}
\frac{\partial }{\partial t}(\rho{\chi}_{\mathcal{S}}) + P_{\mathcal{S}}u \cdot \nabla (\rho{\chi}_{\mathcal{S}})=0,\quad (\rho{\chi}_{\mathcal{S}})|_{t=0}= \rho_0^{N}\mathds{1}_{\mathcal{S}_0}, \end{equation} where $P_{\mathcal{S}}u \in \mathcal{R}$ and it is given by \eqref{projection:P}.
Since $\rho_0^N \in W^{1,\infty}(\Omega)$, $u\in B_{R,T}$ in \eqref{eq:rho}, we can apply \cref{parabolic} to conclude that $\rho >0$ and \begin{equation*} \rho \in L^2(0,T;H^{2}(\Omega)) \cap C([0,T]; H^{1}(\Omega)) \cap H^{1}(0,T;L^{2}(\Omega)). \end{equation*} Moreover, by \cref{reg:chiS} we obtain \begin{align*} &\chi_{\mathcal{S}} \in L^{\infty}((0,T)\times \Omega) \cap C([0,T];L^p(\Omega)), \, \forall \, 1 \leqslant p < \infty,\\ &\rho\chi_{\mathcal{S}} \in L^{\infty}((0,T)\times \Omega) \cap C([0,T];L^p(\Omega)), \, \forall \, 1 \leqslant p < \infty. \end{align*} Consequently, we define \begin{align*} & \mu = (1-\chi_{\mathcal{S}})\mu_{\mathcal{F}} + \delta^2\chi_{\mathcal{S}},\quad \lambda = (1-\chi_{\mathcal{S}})\lambda_{\mathcal{F}} + \delta^2\chi_{\mathcal{S}}\mbox{ so that }\mu >0,\ 2\mu+3\lambda \geqslant 0, \\ & g=(1-\chi_{\mathcal{S}})g_{\mathcal{F}} + \chi_{\mathcal{S}}g_{\mathcal{S}},\quad p(\rho)= a\rho^{\gamma} + {\delta} \rho^{\beta}\quad \mbox{ with } \quad a = a_{\mathcal{F}} (1-\chi_{\mathcal{S}}). \end{align*} \underline{Step 2: Momentum equation.} Given $u\in B_{R,T}$, let us consider the following equation satisfied by $\widetilde{u}: [0,T]\mapsto X_N$: \begin{multline}\label{tilde-momentum} - \int\limits_0^T\int\limits_{\Omega} \rho \Big(\widetilde{u}'(t)\cdot e_j + (u \cdot \nabla e_j)\cdot \widetilde{u} \Big) + \int\limits_0^T\int\limits_{\Omega} \Big(2\mu\mathbb{D}(\widetilde{u}):\mathbb{D}(e_j) + \lambda\operatorname{div}\widetilde{u}\mathbb{I} : \mathbb{D}(e_j) - p(\rho)\mathbb{I}:\mathbb{D}(e_j)\Big) \\ +\int\limits_0^T\int\limits_{\Omega} \varepsilon \nabla e_j \nabla \rho \cdot \widetilde{u}
+ \alpha \int\limits_0^T\int\limits_{\partial \Omega} (\widetilde{u} \times \nu)\cdot (e_j \times \nu) + \alpha \int\limits_0^T\int\limits_{\partial \mathcal{S}^N(t)} [(\widetilde{u}-P_{\mathcal{S}}\widetilde{u})\times \nu]\cdot [(e_j-P_{\mathcal{S}}e_j)\times \nu] \\
+ \frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi_{\mathcal{S}}(\widetilde{u}-P_{\mathcal{S}}\widetilde{u})\cdot (e_j-P_{\mathcal{S}}e_j) = \int\limits_0^T\int\limits_{\Omega}\rho g \cdot e_j, \end{multline} where $\rho$, $\chi_{\mathcal{S}}$ are defined as in Step 1. We can write \begin{equation*} \widetilde{u}(t,\cdot)= \sum\limits_{i=1}^N g_{i}(t) e_i, \quad \widetilde{u}(0)=u_{0}^N= \sum\limits_{i=1}^N \left(\int\limits_{\Omega} u_{0} \cdot e_i\right)e_i.
\end{equation*} Thus, we can identify the function $\widetilde{u}$ with its coefficients $\{g_{i}\}$ which satisfy the ordinary differential equation, \begin{equation}\label{tildeu-ODE} \sum\limits_{i=1}^N a_{i,j}g'_{i}(t) + \sum\limits_{i=1}^N b_{i,j}g_{i}(t) = f_j(t),\quad g_{i}(0)= \int\limits_{\Omega} u_{0}^N \cdot e_i, \end{equation} where $a_{i,j}$, $b_{i,j}$ and $f_j$
are given by \begin{align*} a_{i,j} &= \int\limits_0^T\int\limits_{\Omega} \rho e_i e_j, \\ b_{i,j} &= \int\limits_0^T\int\limits_{\Omega} \rho (u\cdot \nabla e_j)\cdot e_i + \int\limits_0^T\int\limits_{\Omega} \Big(2\mu\mathbb{D}(e_i):\mathbb{D}(e_j) + \lambda\operatorname{div}e_i\mathbb{I} : \mathbb{D}(e_j) \Big) + \int\limits_0^T\int\limits_{\Omega} \varepsilon \nabla e_j \nabla \rho \cdot e_i \\
&+ \alpha \int\limits_0^T\int\limits_{\partial \Omega} (e_i \times \nu)\cdot (e_j \times \nu) + \alpha \int\limits_0^T\int\limits_{\partial \mathcal{S}(t)} [(e_i-P_{\mathcal{S}}e_i)\times \nu]\cdot [(e_j-P_{\mathcal{S}}e_j)\times \nu] + \frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi_{\mathcal{S}}(e_i-P_{\mathcal{S}}e_i)\cdot (e_j-P_{\mathcal{S}}e_j),\\
f_j &= \int\limits_0^T\int\limits_{\Omega} \rho g\cdot e_j + \int\limits_0^T\int\limits_{\Omega}p(\rho)\mathbb{I}:\mathbb{D}(e_j). \end{align*} Observe that the positive lower bound of $\rho$ in \cref{parabolic} guarantees the invertibility of the matrix $(a_{i,j}(t))_{1\leqslant i,j\leqslant N}$. We use the regularity of $\rho$ (\cref{parabolic}), of $\chi_{\mathcal{S}}$ and of the propagator associated to $P_{\mathcal{S}}u$ (\cref{reg:chiS}) to conclude the continuity of $(a_{i,j}(t))_{1\leqslant i,j\leqslant N}$, $(b_{i,j}(t))_{1\leqslant i,j\leqslant N}$, $(f_{i}(t))_{1\leqslant i \leqslant N}$. The existence and uniqueness theorem for ordinary differential equations gives that system \eqref{tildeu-ODE} has a unique solution defined on $[0,T]$
and therefore equation \eqref{tilde-momentum} has a unique solution \begin{equation*} \widetilde{u} \in C([0,T]; X_N). \end{equation*}
\underline{Step 3: Well-definedness of $\mathcal{N}$.} Let us define a map \begin{align*} \mathcal{N}: B_{R,T} &\rightarrow C([0,T],X_N) \\
u &\mapsto \widetilde{u}, \end{align*} where $\widetilde{u}$ satisfies \eqref{tilde-momentum}. Since we know the existence of $\widetilde{u} \in C([0,T]; X_N)$ to the problem \eqref{tilde-momentum}, we have that $\mathcal{N}$ is well-defined from $B_{R,T}$ to $C([0,T]; X_N)$. Now we establish the fact that $\mathcal{N}$ maps $B_{R,T}$ to itself for suitable $R$ and $T$.
We fix \begin{equation*} 0< \sigma < \frac{1}{2}\operatorname{dist}(\mathcal{S}_0,\partial \Omega). \end{equation*}
Given $u\in B_{R,T}$, we want to estimate $\|\widetilde{u}\|_{L^{\infty}(0,T;L^2(\Omega))}$. We have the following identities via a simple integration by parts: \begin{align}\label{id:1}
\int\limits_0^t\int\limits_{\Omega} \rho \widetilde{u}'\cdot \widetilde{u} &=-\frac{1}{2}\int\limits_0^t\int\limits_{\Omega}\frac{\partial \rho}{\partial t}|\widetilde{u}|^2 + \frac{1}{2}(\rho|\widetilde{u}|^2)(t)-\frac{1}{2}\rho_0|u_0|^2, \\ \label{id:2}
\int\limits_0^T\int\limits_{\Omega} \rho (u\cdot\nabla)\widetilde{u}\cdot \widetilde{u} &= - \frac{1}{2}\int\limits_0^T\int\limits_{\Omega} \operatorname{div}(\rho u)|\widetilde{u}|^2,\\ \begin{split}\label{id:3} \int\limits_{\Omega} \nabla (\rho^{\gamma})\cdot \widetilde{u} &= \frac{\gamma}{\gamma-1} \int\limits_{\Omega} \nabla (\rho^{\gamma-1})\cdot \rho \widetilde{u} = -\frac{\gamma}{\gamma-1}\int\limits_{\Omega}\rho^{\gamma-1} \operatorname{div}(\rho \widetilde{u})=\frac{1}{\gamma-1} \frac{d}{dt}\int\limits_{\Omega} \rho^{\gamma} - \frac{\varepsilon\gamma}{\gamma-1}\int\limits_{\Omega} \rho^{\gamma-1}\Delta\rho \\
&= \frac{1}{\gamma-1} \frac{d}{dt}\int\limits_{\Omega} \rho^{\gamma} + \varepsilon \gamma\int\limits_{\Omega} \rho^{\gamma-2}|\nabla \rho|^2 \geqslant \frac{1}{\gamma-1} \frac{d}{dt}\int\limits_{\Omega} \rho^{\gamma}. \end{split} \end{align} Similarly, \begin{equation}\label{nid:4}
\int\limits_{\Omega} \nabla (\rho^{\beta})\cdot \widetilde{u} = \frac{1}{\beta-1} \frac{d}{dt}\int\limits_{\Omega} \rho^{\beta} + \varepsilon \beta\int\limits_{\Omega} \rho^{\beta-2}|\nabla \rho|^2. \end{equation} We multiply equation \eqref{tilde-momentum} by $g_{j}$, add these equations for $j=1,2,...,N$, use the relations \eqref{id:1}--\eqref{nid:4} and the continuity equation \eqref{eq:rho} to obtain the following energy estimate: \begin{multline}\label{energy:tildeu}
\int\limits_{\Omega}\Big(\frac{1}{2} \rho |\widetilde{u}|^2 + \frac{a}{\gamma-1}\rho^{\gamma} + \frac{\delta}{\beta-1}\rho^{\beta}\Big) + \int\limits_0^T\int\limits_{\Omega} \Big(2\mu|\mathbb{D}(\widetilde{u})|^2 + \lambda |\operatorname{div}\widetilde{u}|^2\Big) + \delta\varepsilon \beta\int\limits_0^T\int\limits_{\Omega} \rho^{\beta-2}|\nabla \rho|^2
+ \alpha \int\limits_0^T\int\limits_{\partial \Omega} |\widetilde{u} \times \nu|^2 \\
+ \alpha \int\limits_0^T\int\limits_{\partial \mathcal{S}(t)} |(\widetilde{u}-P_{\mathcal{S}}\widetilde{u})\times \nu|^2
+ \frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi_{\mathcal{S}}|\widetilde{u}-P_{\mathcal{S}}\widetilde{u}|^2 \leqslant \int\limits_0^T\int\limits_{\Omega}\rho g \cdot \widetilde{u}
+ \int\limits_{\Omega} \Bigg( \frac{1}{2}\frac{\rho_0^N}{|q_0^N|^2}\mathds{1}_{\{\rho_0>0\}} + \frac{a}{\gamma-1}(\rho_0^N)^{\gamma} + \frac{\delta}{\beta-1}(\rho_0^N)^{\beta} \Bigg)\\
\leqslant \sqrt{\overline{\rho}}T\left(\frac{1}{2\widetilde{\varepsilon}}\|g\|^2_{L^{\infty}(0,T;L^2(\Omega))} + \frac{\widetilde{\varepsilon}}{2}\|\sqrt{\rho}\widetilde{u}\|^2_{L^{\infty}(0,T;L^2(\Omega))}\right) + \int\limits_{\Omega} \Bigg( \frac{1}{2}\frac{\rho_0^N}{|q_0^N|^2}\mathds{1}_{\{\rho_0>0\}} + \frac{a}{\gamma-1}(\rho_0^N)^{\gamma} + \frac{\delta}{\beta-1}(\rho_0^N)^{\beta} \Bigg). \end{multline}
An appropriate choice of $\widetilde{\varepsilon}$ in \eqref{energy:tildeu} gives us
\begin{equation*}
\|\widetilde{u}\|^2_{L^{\infty}(0,T;L^2(\Omega))} \leqslant \frac{4{\overline{\rho}}}{\underline{\rho}}T^2\|g\|^2_{L^{\infty}(0,T;L^2(\Omega))} + \frac{4}{\underline{\rho}}E_0^N, \end{equation*}
where $\overline{\rho}$ and $\underline{\rho}$ are the upper and lower bounds of $\rho$. In order to get $\|\widetilde{u}\|_{L^{\infty}(0,T;L^2(\Omega))} \leqslant R$,
we need
\begin{equation}\label{choice-R}
R^2 \geqslant \frac{4{\overline{\rho}}}{\underline{\rho}}T^2\|g\|^2_{L^{\infty}(0,T;L^2(\Omega))} + \frac{4}{\underline{\rho}}E_0^N.
\end{equation} We also need to verify that for $T$ small enough and for any $u\in B_{R,T}$, \begin{equation}\label{no-collision} \inf_{u\in B_{R,T}} \operatorname{dist}(\mathcal{S}(t),\partial \Omega) \geqslant 2\sigma> 0 \end{equation} holds. We follow \cite[Proposition 4.6, Step 2]{MR3272367} and write $\mathcal{S}(t)=\eta_{t,0}(\mathcal{S}_0)$ with the isometric propagator $\eta_{t,s}$ associated to the rigid field $P_{\mathcal{S}}u=h'(t) + \omega(t)\times (y-h(t))$. Then, proving \eqref{no-collision} is equivalent to establishing the following bound: \begin{equation}\label{equivalent-T}
\sup_{t\in [0,T]}|\partial_t \eta_{t,0}(t,y)| < \frac{ \operatorname{dist}(\mathcal{S}_0,\partial\Omega) - 2\sigma}{T},\quad t\in [0,T],\, y\in \mathcal{S}_0. \end{equation} We have \begin{equation*}
|\partial_t \eta_{t,0}(t,y)|=|P_{\mathcal{S}}u(t, \eta_{t,0}(t,y))| \leqslant |h'(t)| + |\omega(t)||y-h(t)|. \end{equation*} Furthermore, if $\overline{\rho}$ is the upper bound of $\rho$, then for $u\in B_{R,T}$ \begin{equation}\label{18:37}
|h'(t)|^2 + J(t)\omega(t)\cdot \omega(t)= \int\limits_{\mathcal{S}(t)} \rho|P_{\mathcal{S}}u(t,\cdot)|^2 \leqslant \int\limits_{\Omega} \rho|u(t,\cdot)|^2 \leqslant \overline{\rho}R^2 \end{equation}
for any $R$ and $t\in (0,T)$. As $J(t)$ is congruent to $J(0)$, they have the same eigenvalues and we have
\begin{equation*}
\lambda_0|\omega(t)|^2 \leqslant J(t)\omega(t)\cdot \omega(t),
\end{equation*}
where $\lambda_0$ is the smallest eigenvalue of $J(0)$. Observe that for $t\in [0,T],\, y\in \mathcal{S}_0$,
\begin{align}
\begin{split}\label{18:39}
|h'(t)| + |\omega(t)||y-h(t)|&\leqslant \sqrt{2}(|h'(t)|^2 + |\omega(t)|^2|y-h(t)|^2)^{1/2} \leqslant \sqrt{2}\max\{1,|y-h(t)|\}(|h'(t)|^2 + |\omega(t)|^2)^{1/2}
\\ &\leqslant C_0\left(|h'(t)|^2 + J(t)\omega(t)\cdot \omega(t)\right)^{1/2},
\end{split}
\end{align}
where $C_0=\sqrt{2}\frac{\max\{1,|y-h(t)|\}}{\min\{1,\lambda_0\}
^{1/2}}$.
Thus, with the help of \eqref{18:37}--\eqref{18:39} and the relation of $R$ in \eqref{choice-R}, we can conclude that any
\begin{equation}\label{choice-T}
T < \frac{ \operatorname{dist}(\mathcal{S}_0,\partial\Omega) - 2\sigma}{C_0 |\overline{\rho}|^{1/2}[\frac{4{\overline{\rho}}}{\underline{\rho}}T^2\|g\|^2_{L^{\infty}(0,T;L^2(\Omega))} + \frac{4}{\underline{\rho}}E_0^N]^{1/2}},
\end{equation}
satisfies the relation \eqref{no-collision}.
Thus, we choose $T$ satisfying \eqref{choice-T} and fix it. Then we choose $R$ as in \eqref{choice-R} to conclude that $\mathcal{N}$ maps $B_{R,T}$ to itself.
\underline{Step 4: Continuity of $\mathcal{N}$.} We show that if a sequence $\{u^k\} \subset B_{R,T}$ is such that $u^k \rightarrow u$ in $B_{R,T}$, then $\mathcal{N}(u^k) \rightarrow \mathcal{N}(u)$ in $B_{R,T}$. As $\mbox{span}(e_1,e_2,...,e_N)$ is a finite dimensional subspace of $\mathcal{D}(\overline{\Omega})$, we have $u^k \rightarrow u$ in $C([0,T];\mathcal{D}(\overline{\Omega}))$. Given $\{u^k\} \subset B_{R,T}$, we have that $\rho^k \in L^2(0,T;H^{2}(\Omega)) \cap C([0,T]; H^{1}(\Omega)) \cap H^{1}(0,T;L^{2}(\Omega))$ is the solution to \eqref{eq:rho}, $\chi_{\mathcal{S}}^k \mbox{ is bounded in }L^{\infty}((0,T)\times \mathbb{R}^3)\mbox{ satisfying }$ \eqref{eq:chinew} and $\{\rho^{k}\chi_{\mathcal{S}}^{k}\}$ is a bounded sequence in $L^{\infty}((0,T)\times\mathbb{R}^3)$ satisfying \eqref{eq:rhochinew}. We apply \cref{sequential1} to obtain \begin{align*} & \chi_{\mathcal{S}}^k \rightarrow \chi_{\mathcal{S}} \mbox{ weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) \mbox{ and }\mbox{ strongly } \mbox{ in }C([0,T]; L^p_{loc}(\mathbb{R}^3)), \, \forall \, 1 \leqslant p < \infty,\\ & P_{\mathcal{S}}^k u^k \rightarrow P_{\mathcal{S}} u \mbox{ strongly } \mbox{ in }C([0,T]; C^{\infty}_{loc}(\mathbb{R}^3)),\\ & \eta_{t,s}^k \rightarrow \eta_{t,s} \mbox{ strongly } \mbox{ in }C^{1}([0,T]^2; C^{\infty}_{loc}(\mathbb{R}^3)). \end{align*} We use the continuity argument as in Step 2 to conclude \begin{equation*} a^k_{i,j}\rightarrow a_{i,j}, \quad b^k_{i,j}\rightarrow b_{i,j}, \quad f^k_j \rightarrow f_j \mbox{ strongly in }C([0,T]), \end{equation*} and so we obtain \begin{equation*} \mathcal{N}(u^k)=\widetilde{u}^k \rightarrow \widetilde{u}=\mathcal{N}(u)\mbox{ strongly in }C([0,T]; X_N). \end{equation*}
\underline{Step 5: Compactness of $\mathcal{N}$.} If $\widetilde{u}(t)=\sum\limits_{i=1}^N g_i(t) e_i$, we can view \eqref{tildeu-ODE} as \begin{equation*} A(t)G'(t) + B(t)G(t) = F(t), \end{equation*} where $A(t)=(a_{i,j}(t))_{1\leqslant i,j\leqslant N},\quad B(t)=(b_{i,j}(t))_{1\leqslant i,j\leqslant N},\quad F(t)=(f_{i}(t))_{1\leqslant i \leqslant N},\quad G(t)=(g_i(t))_{1\leqslant i\leqslant N}$. We deduce \begin{equation*}
|g'_i(t)| \leqslant R|A^{-1}(t)||B(t)| + |A^{-1}(t)||F(t)|. \end{equation*} Thus, we have \begin{equation*}
\sup_{t\in [0,T]} \Big(|g_i(t)| + |g'_i(t)|\Big) \leqslant C. \end{equation*} This also implies \begin{equation*}
\sup_{u\in B_{R,T}} \|\mathcal{N}(u)\|_{C^1([0,T]; X_N)} \leqslant C. \end{equation*} The $C^1([0,T]; X_N)$-boundedness of $\mathcal{N}(u)$ allows us to apply the Arzela-Ascoli theorem to obtain compactness of $\mathcal{N}$ in $B_{R,T}$.
Now we are in a position to apply Schauder's fixed point theorem to $\mathcal{N}$ to conclude the existence of a fixed point $u^N \in B_{R,T}$. Then we define $\rho^N$ satisfying the continuity equation \eqref{galerkin-approx2} on $(0,T)\times \Omega$, and $\chi_{\mathcal{S}}^N=\mathds{1}_{\mathcal{S}^N}$ is the corresponding solution to the transport equation \eqref{galerkin-approx4} on $(0,T)\times \mathbb{R}^3$. It only remains to justify the momentum equation \eqref{galerkin-approx3}. We multiply equation \eqref{tilde-momentum} by $\psi\in \mathcal{D}([0,T))$ to obtain: \begin{multline}\label{22:49} - \int\limits_0^T\int\limits_{\Omega} \rho^N \Big((u^N)'(t)\cdot \psi(t)e_j + (u^N \cdot \nabla (\psi(t)e_j))\cdot {u}^N \Big)+\int\limits_0^T\int\limits_{\Omega} \varepsilon \nabla (\psi(t)e_j) \nabla \rho^N \cdot u^N
+ \alpha \int\limits_0^T\int\limits_{\partial \Omega} (u^N \times \nu)\cdot (\psi(t)e_j \times \nu)\\ + \int\limits_0^T\int\limits_{\Omega} \Big(2\mu^N\mathbb{D}({u}^N):\mathbb{D}(\psi(t)e_j) + \lambda^N\operatorname{div}{u}^N\mathbb{I} : \mathbb{D}(\psi(t)e_j) - p^{N}(\rho^N)\mathbb{I}:\mathbb{D}(\psi (t)e_j)\Big) \\
+ \alpha \int\limits_0^T\int\limits_{\partial \mathcal{S}^N(t)} [({u}^N-P^N_{\mathcal{S}}{u}^N)\times \nu]\cdot [(\psi(t)e_j-P^N_{\mathcal{S}}\psi(t)e_j)\times \nu]
+ \frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi_{\mathcal{S}}({u}^N-P^N_{\mathcal{S}}{u}^N)\cdot (\psi(t)e_j-P^N_{\mathcal{S}}\psi(t)e_j)\\ = \int\limits_0^T\int\limits_{\Omega}\rho^N g^N \cdot \psi(t)e_j, \end{multline} We have the following identities via integration by parts: \begin{equation}\label{id:4} \int\limits_0^T \rho^N (u^N)'(t)\cdot \psi(t)e_j = -\int\limits_0^T (\rho^N)' u^N\cdot \psi(t)e_j - \int\limits_0^T \rho^N u^N\cdot \psi'(t)e_j - (\rho^N u^N\cdot \psi e_j)(0), \end{equation} \begin{equation}\label{id:5} \int\limits_{\Omega} \rho^N (u^N \cdot \nabla (\psi(t)e_j))\cdot {u}^N = -\int\limits_{\Omega} \operatorname{div}(\rho^N u^N) (\psi(t)e_j \cdot {u}^N) - \int\limits_{\Omega}{ \rho^N (u^N \cdot \nabla ) {u}^N\cdot \psi(t)e_j.} \end{equation} Thus we can use the relations \eqref{id:4}--\eqref{id:5} and continuity equation \eqref{galerkin-approx2} in the identity \eqref{22:49} to obtain equation \eqref{galerkin-approx3} for all $\phi \in \mathcal{D}([0,T); X_N)$. \end{proof} \subsection{Convergence of the Faedo-Galerkin scheme and the limiting system}\label{14:14} In \cref{fa}, we have already constructed a solution $(\mathcal{S}^N,\rho^N,u^{N})$ to the problem \eqref{galerkin-approx1}--\eqref{galerkin-initial}. In this section, we establish \cref{thm:approxn} by passing to the limit in \eqref{galerkin-approx1}--\eqref{galerkin-initial} as $N\rightarrow\infty$ to recover the solution of \eqref{varepsilon:approx1}--\eqref{varepsilon:initial}, i.e.\ of the $\varepsilon$-level approximation. \begin{proof} [Proof of \cref{thm:approxn}]
If we multiply \eqref{galerkin-approx3} by $u^N$, then as in \eqref{energy:tildeu}, we derive \begin{multline}\label{energy:uN}
E^N[\rho ^N, q^N] + \int\limits_0^T\int\limits_{\Omega} \Big(2\mu^N|\mathbb{D}(u^N)|^2 + \lambda^N |\operatorname{div}u^N|^2\Big) + \delta\varepsilon \beta\int\limits_0^T\int\limits_{\Omega} (\rho^N)^{\beta-2}|\nabla \rho^N|^2
+ \alpha \int\limits_0^T\int\limits_{\partial \Omega} |u^N \times \nu|^2
\\+ \alpha \int\limits_0^T\int\limits_{\partial \mathcal{S}^N(t)} |(u^N-P^N_{\mathcal{S}}u^N)\times \nu|^2
+ \frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi^N_{\mathcal{S}}|u^N-P^N_{\mathcal{S}}u^N|^2 \leqslant \int\limits_0^T\int\limits_{\Omega}\rho^N g^N \cdot u^N
+ E^N_0, \end{multline} where
$$E^N[\rho ^N, q^N] = \int\limits_{\Omega}\Big(\frac{1}{2} \rho^N |u^N|^2 + \frac{a^N}{\gamma-1}(\rho^N)^{\gamma} + \frac{\delta}{\beta-1}(\rho^N)^{\beta}\Big).$$ Following the idea of the footnote in \cite[Page 368]{MR2084891}, the initial data $(\rho_0^N, u_0^N)$ is constructed in such a way that \begin{equation*} \rho_0^N \rightarrow \rho_0^{\varepsilon} \mbox{ in }W^{1,\infty}(\Omega),\quad \rho_0^N u_0^N \rightarrow q_0^{\varepsilon} \mbox{ in }L^{2}(\Omega) \end{equation*} and \begin{equation} \label{lim}
\int\limits_{\Omega}\Bigg( \frac{1}{2}{\rho_0^N}|u_0^N|^2\mathds{1}_{\{\rho_0^N>0\}} + \frac{a^N}{\gamma-1}(\rho_0^N)^{\gamma} + \frac{\delta}{\beta-1}(\rho_0^N)^{\beta} \Bigg) \rightarrow \int\limits_{\Omega}\Bigg(\frac{1}{2} \frac{|q_0^{\varepsilon}|^2}{\rho_0^{\varepsilon}}\mathds{1}_{\{\rho_0^{\varepsilon}>0\}} + \frac{a^{\varepsilon}}{\gamma-1}(\rho_0^{\varepsilon})^{\gamma} + \frac{\delta}{\beta-1}(\rho_0^{\varepsilon})^{\beta} \Bigg)\mbox{ as }N\rightarrow \infty. \end{equation}
Precisely, we approximate $q_0^{\varepsilon}$ by a sequence $q_0^N$ satisfying \eqref{initialcond} and such that \eqref{lim} is valid. It is sufficient to take $ u_{0}^N = P_N(\frac{q_0^{\varepsilon}}{\rho_0^{\varepsilon}})$, where by $P_N$ we denote the orthogonal projection of $L^2(\Omega) \mbox { onto } X_N$. \cref{fa} is valid with these new initial data. Therefore we can apply the arguments which we will explain below to get Proposition \ref{thm:approxn}.
The construction of $\rho^N$ and \eqref{bounds-on-rho} imply that $\rho^N >0$. Thus the energy estimate \eqref {energy:uN} yields that up to a subsequence \begin{enumerate} \item $u^N\rightarrow u^{\varepsilon}$ weakly-$*$ in $L^{\infty}(0,T;L^2(\Omega))$ and weakly in $L^2(0,T;H^1(\Omega))$, \item $\rho^N \rightarrow \rho^{\varepsilon}$ weakly-$*$ in $L^{\infty}(0,T; L^{\beta}(\Omega))$, \item $\nabla\rho^N \rightarrow \nabla\rho^{\varepsilon}$ weakly in $L^{2}((0,T)\times\Omega)$. \end{enumerate} We follow the similar analysis as for the fluid case explained in \cite[Section 7.8.1, Page 362]{MR2084891} to conclude that \begin{itemize} \item $\rho^N \rightarrow \rho^{\varepsilon}$ in $C([0,T]; L^{\beta}_{weak}(\Omega))$ and $\rho^N \rightarrow \rho^{\varepsilon}$ strongly in $L^p((0,T)\times \Omega)$, $\forall \ 1\leqslant p< \frac{4}{3}\beta$, \item $\rho^N u^N \rightarrow \rho^{\varepsilon} u^{\varepsilon}$ weakly in $L^2(0,T; L^{\frac{6\beta}{\beta+6}})$ and weakly-$*$ in $L^{\infty}(0,T; L^{\frac{2\beta}{\beta+1}})$. \end{itemize} We also know that $\chi_{\mathcal{S}}^N$ is a bounded sequence in $ L^{\infty}((0,T)\times \mathbb{R}^3)$ satisfying \eqref{galerkin-approx4} and $\{\rho^{N}\chi_{\mathcal{S}}^{N}\}$ is a bounded sequence in $L^{\infty}((0,T)\times\mathbb{R}^3)$ satisfying \eqref{N:approx5}. We use \cref{sequential11} to conclude \begin{align}\label{xi} \chi_{\mathcal{S}}^N \rightarrow \chi_{\mathcal{S}}^{\varepsilon} \mbox{ weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) &\mbox{ and }\mbox{ strongly } \mbox{ in } C([0,T]; L^p_{loc}(\mathbb{R}^3)),\ \forall \ 1 \leqslant p <\infty, \end{align} with $\chi_{\mathcal{S}}^{\varepsilon}$ satisfying \eqref{varepsilon:approx4} along with \eqref{varepsilon:approx1}. Thus, we have recovered the transport equation for the body \eqref{varepsilon:approx4}. From \eqref{xi} and the definitions of $g^N$ and $g^{\varepsilon}$ in \eqref{gN} and \eqref{gepsilon}, it follows that \begin{equation}\label{g} g^N \rightarrow g^{\varepsilon} \mbox{ weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) \mbox{ and }\mbox{ strongly } \mbox{ in }C([0,T]; L^p_{loc}(\mathbb{R}^3)) \ \forall \ 1 \leqslant p <\infty. \end{equation} These convergence results make it possible to pass to the limit $N\rightarrow \infty$ in \eqref{galerkin-approx2} to achieve \eqref{varepsilon:approx2}. Now we concentrate on the limit of the momentum equation \eqref{galerkin-approx3}. The four most difficult terms are: \begin{align*} A^N(t,e_k)&= \int\limits_{\partial \mathcal{S}^N(t)} [(u^N-P^N_{\mathcal{S}}u^N)\times \nu]\cdot [(e_k-P^N_{\mathcal{S}}e_k)\times \nu],\quad B^N(t,e_k)= \int\limits_{\Omega} \rho^N u^N \otimes u^N : \nabla e_k,\\ C^N(t,e_k)&= \int\limits_{\Omega} \varepsilon \nabla u^N \nabla \rho^N \cdot e_k,\quad D^N(t,e_k)= \int\limits_{\Omega} (\rho^N)^{\beta}\mathbb{I}: \mathbb{D}(e_k),\quad 1\leqslant k\leqslant N. \end{align*}
To analyze the term $A^N(t,e_k)$, we do a change of variables to rewrite it in a fixed domain and use the convergence results from \cref{sequential1} for the projection and the isometric propagator: \begin{align*} &P_{\mathcal{S}}^N u^N \rightarrow P_{\mathcal{S}}^{\varepsilon} u^{\varepsilon} \mbox{ weakly-}* \mbox{ in }L^{\infty}(0,T; C^{\infty}_{loc}(\mathbb{R}^3)),\\ &\eta_{t,s}^N \rightarrow \eta_{t,s}^{\varepsilon} \mbox{ weakly-}* \mbox{ in }W^{1,\infty}((0,T)^2; C^{\infty}_{loc}(\mathbb{R}^3)). \end{align*} We follow a similar analysis as in \cite[Page 2047--2048]{MR3272367} to conclude that $A^N$ converges weakly in $L^1(0,T)$ to \begin{equation*} A(t,e_k)= \int\limits_{\partial \mathcal{S}^{\varepsilon}(t)} [(u^{\varepsilon}-P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon})\times \nu]\cdot [(e_k-P^{\varepsilon}_{\mathcal{S}}e_{k})\times \nu]. \end{equation*} We proceed as explained in the fluid case \cite[Section 7.8.2, Page 363--365]{MR2084891} to analyze the limiting process for the other terms $B^N(t,e_k)$, $C^N(t,e_k)$, $D^N(t,e_k)$. The limit of $B^N(t,e_k)$ follows from the fact \cite[Equation (7.8.22), Page 364]{MR2084891} that \begin{equation}\label{conv:convective} \rho^N u^N \otimes u^N \rightarrow \rho^{\varepsilon} u^{\varepsilon} \otimes u^{\varepsilon} \mbox{ weakly in }L^2(0,T; L^{\frac{6\beta}{4\beta +3}}(\Omega)). \end{equation} To get the limit of $C^N(t,e_k)$, we use \cite[Equation (7.8.26), Page 365]{MR2084891}: \begin{equation*} \varepsilon\nabla u^N \nabla \rho^N \rightarrow \varepsilon\nabla u^{\varepsilon} \nabla \rho^{\varepsilon} \mbox{ weakly in }L^2(0,T; L^{\frac{5\beta-3}{4\beta}}(\Omega)), \end{equation*} and the limit of $D^N(t,e_k)$ is obtained by using \cite[Equation (7.8.8), Page 362]{MR2084891}: \begin{equation}\label{conv:rho} \rho^N \rightarrow \rho^{\varepsilon} \mbox{ strongly in }L^p(0,T; \Omega),\quad 1\leqslant p < \frac{4}{3}\beta. \end{equation} Thus, using the above convergence results for $B^N$, $C^N$, $D^N$ and the fact that \begin{equation*} \bigcup_{N}X_N\mbox{ is dense in }\left\{v\in W^{1,p}(\Omega) \mid v\cdot \nu=0\mbox{ on }\partial\Omega\right\}\mbox{ for any }p\in [1,\infty), \end{equation*} we conclude the following weak convergences in $L^1(0,T)$: \begin{equation*} B^N(t,\phi^N)\rightarrow B(t,\phi^{\varepsilon})= \int\limits_{\Omega} \rho^{\varepsilon} u^{\varepsilon} \otimes u^{\varepsilon} : \nabla \phi^{\varepsilon}, \end{equation*} \begin{equation*} C^N(t,\phi^N)\rightarrow C(t,\phi^{\varepsilon})=\int\limits_{\Omega} \varepsilon \nabla u^{\varepsilon} \nabla \rho^{\varepsilon} \cdot \phi^{\varepsilon}, \end{equation*} \begin{equation*} D^N(t,\phi^N)\rightarrow D(t,\phi^{\varepsilon})= \int\limits_{\Omega} (\rho^{\varepsilon})^{\beta}\mathbb{I}: \mathbb{D}(\phi^{\varepsilon}). \end{equation*} Thus we have achieved \eqref{varepsilon:approx2} as a limit of equation \eqref{galerkin-approx3} as $N\rightarrow \infty$. Hence, we have established the existence of a solution $(\mathcal{S}^{\varepsilon},\rho^{\varepsilon},u^{\varepsilon})$ to system \eqref{varepsilon:approx1}--\eqref{varepsilon:initial}. Now we establish energy inequality \eqref{energy-varepsilon} and estimates independent of $\varepsilon$: \begin{itemize}
\item Notice that the solution $(\rho^N,u^N)$ of the Galerkin scheme satisfies \eqref{energy:uN} uniformly in $N$. The convergence of $\rho^N|u^N|^2$ in \eqref{conv:convective} and $\rho^N$ in \eqref{conv:rho} ensures that, up to the extraction of a subsequence, \begin{equation*}
\int\limits_{\Omega}\Big(\frac{1}{2} \rho^N |u^N|^2 + \frac{a^N}{\gamma-1}(\rho^N)^{\gamma} + \frac{\delta}{\beta-1}(\rho^N)^{\beta}\Big) \rightarrow \int\limits_{\Omega}\Big(\frac{1}{2} \rho^{\varepsilon} |u^{\varepsilon}|^2 + \frac{a^{\varepsilon}}{\gamma-1}(\rho^{\varepsilon})^{\gamma} + \frac{\delta}{\beta-1}(\rho^{\varepsilon})^{\beta}\Big) \mbox{ as }N\rightarrow\infty. \end{equation*} \item Due to the weak lower semicontinuity of convex functionals, the weak convergence of $u^N$ in $L^2(0,T;H^1(\Omega))$, the strong convergence of $\chi_{\mathcal{S}}^N$ in $C([0,T];L^p(\Omega))$ and the strong convergence of $P_{\mathcal{S}}^N$ in $C([0,T]; C^{\infty}_{loc}(\mathbb{R}^3))$, we obtain \begin{equation}\label{N1}
\int\limits_0^T\int\limits_{\Omega} \Big(2\mu^{\varepsilon}|\mathbb{D}(u^{\varepsilon})|^2 + \lambda^{\varepsilon}|\operatorname{div}u^{\varepsilon}|^2\Big) \leqslant\liminf_{N\rightarrow \infty}\int\limits_0^T\int\limits_{\Omega} \Big(2\mu^N|\mathbb{D}(u^N)|^2 + \lambda^N |\operatorname{div}u^N|^2\Big), \end{equation} \begin{equation}\label{N2}
\int\limits_0^T\int\limits_{\Omega} \chi^{\varepsilon}_{\mathcal{S}}|u^{\varepsilon}-P^{\delta}_{\mathcal{S}}u^{\varepsilon}|^2\leqslant\liminf_{N\rightarrow \infty}\int\limits_0^T\int\limits_{\Omega} \chi^N_{\mathcal{S}}|u^N-P_{\mathcal{S}}u^N|^2. \end{equation} \item Using the fact that $\nabla\rho^N\rightarrow\nabla\rho$ strongly in $L^2((0,T)\times\Omega)$ (by \cite[Equation (7.8.25), Page 365]{MR2084891}), strong convergence of $\rho^N$ in \eqref{conv:rho} and Fatou's lemma, we have \begin{equation}\label{N3}
\int\limits_0^T\int\limits_{\Omega} (\rho^{\varepsilon})^{\beta-2}|\nabla \rho^{\varepsilon}|^2 \leqslant \liminf_{N\rightarrow \infty}\int\limits_0^T\int\limits_{\Omega} (\rho^N)^{\beta-2}|\nabla \rho^N|^2. \end{equation} \item For passing to the limit in the boundary terms, we follow the idea of \cite{MR3272367}. Define the extended velocities $U^N$, $U_{\mathcal{S}}^N$ to whole $\mathbb{R}^3$ associated with $u^N$, $P_{\mathcal{S}}u^N$ respectively. According to \cite[Lemma A.2]{MR3272367}, we have the weak convergences of $U^N$, $U_{\mathcal{S}}^N$ to $U^{\varepsilon}$, $U_{\mathcal{S}}^{\varepsilon}$ in $L^2(0,T;H^1_{loc}(\mathbb{R}^3))$. These facts along with the lower semicontinuity of the $L^2$-norm yield \begin{align}
\int\limits_0^T\int\limits_{\partial \mathcal{S}^{\varepsilon}(t)} |(u^{\varepsilon}-P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon})\times \nu|^2 &=\int\limits_0^T\int\limits_{\partial\mathcal{S}_0} |(U^{\varepsilon}-U_{\mathcal{S}}^{\varepsilon})\times \nu|^2\notag\\ &\leqslant \liminf_{N\rightarrow\infty} \int\limits_0^T\int\limits_{\partial\mathcal{S}_0} |(U^{N}-U_{\mathcal{S}}^{N})\times \nu|^2 \leqslant \liminf_{N\rightarrow\infty} \int\limits_0^T\int\limits_{\partial \mathcal{S}^N(t)} |(u^N-P_{\mathcal{S}}u^N)\times \nu|^2\label{N4}. \end{align} Similar arguments also help us to obtain \begin{equation}\label{N5}
\int\limits_0^T\int\limits_{\partial \Omega} |u^{\varepsilon} \times \nu|^2 \leqslant \liminf_{N\rightarrow\infty}\int\limits_0^T\int\limits_{\partial \Omega} |u^N \times \nu|^2. \end{equation} \item Regarding the term on the right hand side of \eqref{energy:uN}, the weak convergence of $u^N$ in $L^2(0,T;H^1(\Omega))$, the strong convergence of $\rho^N$ in \eqref{conv:rho} and the strong convergence of $g^N$ in \eqref{g} yield \begin{equation}\label{N6} \int\limits_0^T\int\limits_{\Omega} \rho^N g^N \cdot u^N \rightarrow\int\limits_0^T\int\limits_{\Omega} \rho^{\varepsilon} g^{\varepsilon} \cdot u^{\varepsilon}, \; \mbox{ as }N\rightarrow\infty. \end{equation} \end{itemize} Thus, we have established energy inequality \eqref{energy-varepsilon}: \begin{multline}\label{re:epsilon-energy}
E^{\varepsilon}[\rho ^{\varepsilon},q^{\varepsilon}]+ \int\limits_0^T\int\limits_{\Omega} \Big(2\mu^{\varepsilon}|\mathbb{D}(u^{\varepsilon})|^2 + \lambda^{\varepsilon}|\operatorname{div}u^{\varepsilon}|^2\Big) + \delta\varepsilon \beta\int\limits_0^T\int\limits_{\Omega} (\rho^{\varepsilon})^{\beta-2}|\nabla \rho^{\varepsilon}|^2 \\
+ \alpha \int\limits_0^T\int\limits_{\partial \Omega} |u^{\varepsilon} \times \nu|^2
+ \alpha \int\limits_0^T\int\limits_{\partial \mathcal{S}^{\varepsilon}(t)} |(u^{\varepsilon}-P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon})\times \nu|^2
+ \frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi^{\varepsilon}_{\mathcal{S}}|u^{\varepsilon}-P^{\delta}_{\mathcal{S}}u^{\varepsilon}|^2 \leqslant \int\limits_0^T\int\limits_{\Omega}\rho^{\varepsilon}
g^{\varepsilon} \cdot u^{\varepsilon}
+ E^{\varepsilon}_0, \end{multline} where
$$E^{\varepsilon}[\rho^{\varepsilon} ,q^{\varepsilon}] =\int\limits_{\Omega}\left(\frac{1}{2}\frac{|q^{\varepsilon}|^2}{\rho^{\varepsilon}} + \frac{a^{\varepsilon}}{\gamma-1}(\rho^{\varepsilon})^{\gamma} + \frac{\delta}{\beta-1}(\rho^{\varepsilon})^{\beta}\right).$$ We obtain as in \cite[Equation (7.8.14), Page 363]{MR2084891}: \begin{equation*} \partial_t\rho^{\varepsilon},\ \Delta \rho^{\varepsilon}\in {L^{\frac{5\beta-3}{4\beta}}((0,T)\times\Omega)}. \end{equation*}
Regarding the $\sqrt{\varepsilon} \|\nabla \rho^{\varepsilon}\|_{L^2((0,T)\times\Omega)}$ estimate in \eqref{est:indofepsilon}, we have to multiply \eqref{varepsilon:approx2} by $\rho^{\varepsilon}$ and integrate by parts to obtain \begin{equation*}
\frac{1}{2}\int\limits_{\Omega} |\rho^{\varepsilon}(t)|^2 + \varepsilon\int\limits_0^T\int\limits_{\Omega} |\nabla\rho^{\varepsilon}(t)|^2 = \frac{1}{2}\int\limits_{\Omega} |\rho_0^{\varepsilon}|^2 - \frac{1}{2}\int\limits_0^T\int\limits_{\Omega} |\rho^{\varepsilon}|^2\operatorname{div} u^{\varepsilon} \leqslant \frac{1}{2}\int\limits_{\Omega} |\rho_0^{\varepsilon}|^2 + \sqrt{T}\||\rho^{\varepsilon}\|^2_{L^{\infty}(0,T;L^4(\Omega))}\|\operatorname{div}u^{\varepsilon}\|_{L^2(0,T;L^2(\Omega))}. \end{equation*}
Now, the pressure estimates $\|\rho^{\varepsilon}\|_{L^{\beta+1}((0,T)\times\Omega)}$ and $\|\rho^{\varepsilon}\|_{L^{\gamma+1}((0,T)\times\Omega)}$ in \eqref{est:indofepsilon} can be derived by means of the test function $\phi(t,x) = \psi(t)\Phi(t,x)$ with $\Phi(t,x)=\mathcal{B}[ \rho^{\varepsilon}-\overline{m}]$ in \eqref{varepsilon:approx3}, where \begin{equation*}
\psi \in \mathcal{D}(0,T),\quad \overline{m}=|\Omega|^{-1}\int\limits_{\Omega} \rho^{\varepsilon}, \end{equation*} and $\mathcal{B}$ is the Bogovskii operator related to $\Omega$ (for details about $\mathcal{B}$, see \cite[Section 3.3, Page 165]{MR2084891}). After taking this special test function and integrating by parts, we obtain \begin{multline}\label{bogovski:mom} \int\limits_0^T \psi\int\limits_{\Omega}\Big(a^{\varepsilon}(\rho^{\varepsilon})^{\gamma} + {\delta} (\rho^{\varepsilon})^{\beta}\Big) \rho^{\varepsilon}= \int\limits_0^T \psi\int\limits_{\Omega}\Big(a^{\varepsilon}(\rho^{\varepsilon})^{\gamma} + {\delta} (\rho^{\varepsilon})^{\beta}\Big) \overline{m} + \int\limits_0^T 2\psi\int\limits_{\Omega} \mu^{{\varepsilon}}\mathbb{D}(u^{\varepsilon}):\mathbb{D}(\Phi) + \int\limits_0^T \psi\int\limits_{\Omega}\lambda^{\varepsilon}\rho^{\varepsilon}\operatorname{div}u^{\varepsilon}\\- \overline{m}\int\limits_0^T \psi\int\limits_{\Omega}\lambda^{\varepsilon}\operatorname{div}u^{\varepsilon} + \int\limits_0^T \psi\int\limits_{\Omega} \varepsilon \nabla u^{\varepsilon} \nabla \rho^{\varepsilon} \cdot \Phi + \alpha \int\limits_0^T \psi\int\limits_{\partial \mathcal{S}^{\varepsilon}(t)} [(u^{\varepsilon}-P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon})\times \nu]\cdot [(\Phi-P^{\varepsilon}_{\mathcal{S}}\Phi)\times \nu] \\ + \frac{1}{\delta}\int\limits_0^T \psi\int\limits_{\Omega} \chi^{\varepsilon}_{\mathcal{S}}(u^{\varepsilon}-P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon})\cdot (\Phi-P^{\varepsilon}_{\mathcal{S}}\Phi) + \int\limits_0^T \psi\int\limits_{\Omega} \rho^{\varepsilon} g^{\varepsilon} \cdot \Phi. \end{multline} We see that all the terms can be estimated as in \cite[Section 7.8.4, Pages 366--368]{MR2084891} except the penalization term. Using H\"{o}lder's inequality and bounds from energy estimate \eqref{energy-varepsilon}, the penalization term can be dealt with in the following way \begin{equation}\label{bogovski:extra}
\int\limits_0^T \psi\int\limits_{\Omega} \chi^{\varepsilon}_{\mathcal{S}}(u^{\varepsilon}-P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon})\cdot (\Phi-P^{\varepsilon}_{\mathcal{S}}\Phi) \leqslant |\psi|_{C[0,T]} \left(\int\limits_0^T\int\limits_{\Omega} \chi^{\varepsilon}_{\mathcal{S}}|(u^{\varepsilon}-P^{\varepsilon}_{\mathcal{S}}u^{\varepsilon})|^2\right)^{1/2}\|\Phi\|_{L^2((0,T)\times\Omega)}\leqslant C |\psi|_{C[0,T]}, \end{equation}
where in the last inequality we have used $\|\Phi\|_{L^2(\Omega)}\leqslant c\|\rho^{\varepsilon}\|_{L^2(\Omega)}$ and the energy inequality \eqref{energy-varepsilon}. Thus, we have an improved regularity of the density and we have established the required estimates of \eqref{est:indofepsilon}.
The only remaining thing is to check the following fact: there exists $T$ small enough such that if $\operatorname{dist}(\mathcal{S}_0,\partial \Omega) > 2\sigma$, then \begin{equation}\label{epsilon-collision}
\operatorname{dist}(\mathcal{S}^{\varepsilon}(t),\partial \Omega) \geqslant 2\sigma> 0 \quad \forall \ t\in [0,T]. \end{equation} It is equivalent to establishing the following bound: \begin{equation}\label{equivalent-Tag}
\sup_{t\in [0,T]}|\partial_t \eta_{t,0}(t,y)| < \frac{ \operatorname{dist}(\mathcal{S}_0,\partial\Omega) - 2\sigma}{T},\quad y\in \mathcal{S}_0. \end{equation} We show as in Step 3 of the proof of \cref{fa} that (see \eqref{no-collision}--\eqref{18:39}): \begin{equation}\label{00:32}
|\partial_t \eta^{\varepsilon}_{t,0}(t,y)| \leqslant |(h{^\varepsilon})'(t)| + |\omega^{\varepsilon}(t)||y-h^{\varepsilon}(t)|\leqslant C_0\left(\int\limits_{\Omega} \rho^{\varepsilon} |u^{\varepsilon}(t)|^2\right)^{1/2}, \end{equation}
where $C_0=\sqrt{2}\frac{\max\{1,|y-h(t)|\}}{\min\{1,\lambda_0\}
^{1/2}}$. Moreover, the energy estimate \eqref{re:epsilon-energy} yields
\begin{equation*}
\frac{d}{dt}E^{\varepsilon}[\rho ^{\varepsilon},q^{\varepsilon}]+ \int\limits_{\Omega} \Big(2\mu^{\varepsilon}|\mathbb{D}(u^{\varepsilon})|^2 + \lambda^{\varepsilon}|\operatorname{div}u^{\varepsilon}|^2\Big) \leqslant \int\limits_{\Omega}\rho^{\varepsilon}
g^{\varepsilon} \cdot u^{\varepsilon}\\
\leqslant E^{\varepsilon}[\rho ^{\varepsilon},q^{\varepsilon}] + \frac{1}{2\gamma_1}\left(\frac{\gamma -1}{2\gamma}\right)^{\gamma_1/\gamma}\|g^{\varepsilon}\|^{2\gamma_1}_{L^{\frac{2\gamma}{\gamma -1}}(\Omega)},
\end{equation*}
with $\gamma_1=1-\frac{1}{\gamma}$, which implies
\begin{equation}\label{00:33}
E^{\varepsilon}[\rho ^{\varepsilon},q^{\varepsilon}] \leqslant e^{{T}}E^{\varepsilon}_0 + C{T} \|g^{\varepsilon}\|^{2\gamma_1}_{L^{\infty}((0,T)\times\Omega)}.
\end{equation}
Thus, with the help of \eqref{equivalent-Tag} and \eqref{00:32}--\eqref{00:33}, we can conclude that for any $T$ satisfying
\begin{equation*}
T < \frac{ \operatorname{dist}(\mathcal{S}_0,\partial\Omega) - 2\sigma}{C_0 \left[e^{{T}}E^{\varepsilon}_0 + C{T} \|g^{\varepsilon}\|^{2\gamma_1}_{L^{\infty}((0,T)\times\Omega)}\right]^{1/2}},
\end{equation*}
the relation \eqref{epsilon-collision} holds. This completes the proof of \cref{thm:approxn}. \end{proof} \subsection{Vanishing dissipation in the continuity equation and the limiting system}\label{14:18} In this section, we prove \cref{thm:approxn-delta} by taking $\varepsilon\rightarrow 0$ in the system \eqref{varepsilon:approx1}--\eqref{varepsilon:initial}. In order to do so, we have to deal with the problem of identifying the pressure corresponding to the limiting density. First of all, following the idea of the footnote in \cite[Page 381]{MR2084891}, the initial data $(\rho_0^{\varepsilon}, q_0^{\varepsilon})$ is constructed in such a way that \begin{equation*} \rho_0^{\varepsilon}>0,\quad \rho_0^{\varepsilon} \in W^{1,\infty}(\Omega),\quad \rho_0^{\varepsilon} \rightarrow \rho_0^{\delta} \mbox{ in }L^{\beta}(\Omega),\quad q_0^{\varepsilon} \rightarrow q_0^{\delta} \mbox{ in }L^{\frac{2\beta}{\beta + 1}}(\Omega) \end{equation*} and \begin{equation*}
\int\limits_{\Omega}\Bigg( \frac{|q_0^{\varepsilon}|^2}{\rho_0^{\varepsilon}}\mathds{1}_{\{\rho_0^{\varepsilon}>0\}} + \frac{a}{\gamma-1}(\rho_0^{\varepsilon})^{\gamma} + \frac{\delta}{\beta-1}(\rho_0^{\varepsilon})^{\beta} \Bigg) \rightarrow \int\limits_{\Omega}\Bigg( \frac{|q_0^{\delta}|^2}{\rho_0^{\delta}}\mathds{1}_{\{\rho_0^{\delta}>0\}} + \frac{a}{\gamma-1}(\rho_0^{\delta})^{\gamma} + \frac{\delta}{\beta-1}(\rho_0^{\delta})^{\beta} \Bigg)\mbox{ as }{\varepsilon}\rightarrow 0. \end{equation*} More precisely, let $(\rho^{\delta}_0,q^{\delta}_0)$ satisfy \eqref{rhonot}--\eqref{qnot}; then, following \cite[Section 7.10.7, Page 392]{MR2084891}, we can find $\rho^{\varepsilon}_{0} \in W^{1,\infty}({\Omega)}$, $\rho^{\varepsilon}_0 > 0$ by defining \begin{equation*}
\rho^{\varepsilon}_{0}= \mathcal{K}_{\varepsilon}(\rho^{\delta}_{0}) + \varepsilon,
\end{equation*}
where $\mathcal{K}_{\varepsilon}$ is the standard regularizing operator in the space variable.
Then our initial density satisfies
\begin{equation*}
\begin{array}{l}
\rho^{\varepsilon}_{0} \to \rho^{\delta}_{0} \mbox{ strongly in } L^{\beta}(\Omega) .
\end{array} \end{equation*} We define \begin{align*} \overline{{q}^{\varepsilon}_0}= \begin{cases} q_{0}^{\delta}\sqrt{\frac{\rho^{\varepsilon}_{0}}{\rho_{0}^{\delta}}} &\mbox { if } \rho^{\delta}_{0} >0,\\ 0 \quad \quad \quad \quad \quad &\mbox { if } \rho^{\delta}_{0} =0. \end{cases} \end{align*} From \eqref{qnot}, we know that \begin{equation*}
\frac{|\overline{{q}^{\varepsilon}_0}|}{\sqrt{\rho^{\varepsilon}_{0}}} \in { L^2(\Omega)}. \end{equation*} Due to a density argument, there exists
$h^{\varepsilon} \in W^{1,\infty}({\Omega})$ such that \begin{equation*}
\left\|\frac{q^{\varepsilon}_0}{\sqrt{\rho^{\varepsilon}_{0}}} -h^{\varepsilon} \right\|_{L^2(\Omega)}< \varepsilon. \end{equation*} Now, we set $ q^{\varepsilon}_0= h^{\varepsilon}\sqrt{\rho^{\varepsilon}_{0}}$, which implies that
\begin{equation*}
q^{\varepsilon}_0 \to q_{0}^{\delta} \mbox { in } L^{\frac{2\beta}{\beta +1}}(\Omega), \end{equation*} and \begin{equation*}
E^{\varepsilon}_0 \to E^{\delta}_0. \end{equation*}
\begin{proof} [Proof of \cref{thm:approxn-delta}] The estimates \eqref{energy-varepsilon} and \eqref{est:indofepsilon} help us to conclude that, up to an extraction of a subsequence, we have \begin{align}
& u^{\varepsilon}\rightarrow u^{\delta}\mbox{ weakly in }L^2(0,T; H^1(\Omega)),\label{conv1}\\
&\rho^{\varepsilon}\rightarrow \rho^{\delta}\mbox{ weakly in }L^{\beta+1}((0,T)\times \Omega),\mbox{ weakly-}*\mbox{ in } L^{\infty}(0,T;L^{\beta}(\Omega)),\label{conv2}\\
& (\rho^{\varepsilon})^{\gamma}\rightarrow \overline{ (\rho^{\delta})^{\gamma}}\mbox{ weakly in }L^{\frac{\beta+1}{\gamma}}((0,T)\times\Omega),\label{conv3}\\
& (\rho^{\varepsilon})^{\beta}\rightarrow \overline{ (\rho^{\delta})^{\beta}}\mbox{ weakly in } L^{\frac{\beta+1}{\beta}}((0,T)\times\Omega),\label{conv4}\\
& \varepsilon\nabla\rho^{\varepsilon}\rightarrow 0 \mbox{ strongly in }L^2((0,T)\times \Omega)\label{conv5} \end{align} as $\varepsilon\to 0$. Below, we denote by $\left(\rho^{\delta},u^{\delta}, \overline{ (\rho^{\delta})^{\gamma}},\overline{ (\rho^{\delta})^{\beta}}\right)$ also the extended version of the corresponding quantities in $(0,T)\times \mathbb{R}^3$.
\underline{Step 1: Limit of the transport equation.} We obtain from \cref{thm:approxn} that $\rho^{\varepsilon}$ satisfies \eqref{varepsilon:approx2}, $\{u^{\varepsilon},\chi_{\mathcal{S}}^{\varepsilon}\}$ is a bounded sequence in $L^{2}(0,T; H^1(\Omega)) \times L^{\infty}((0,T)\times \mathbb{R}^3)$ satisfying \eqref{varepsilon:approx4} and $\{\rho^{\varepsilon}\chi_{\mathcal{S}}^{\varepsilon}\}$ is a bounded sequence in $L^{\infty}((0,T)\times\mathbb{R}^3)$ satisfying \eqref{varepsilon:approx5}. Thus, we can use \cref{sequential-varepsilon} to conclude that up to a subsequence: \begin{align}\label{13:02} &\chi_{\mathcal{S}}^{\varepsilon} \rightarrow \chi_{\mathcal{S}}^{\delta} \mbox{ weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) \mbox{ and }\mbox{ strongly } \mbox{ in }C([0,T]; L^p_{loc}(\mathbb{R}^3)) \ (1 \leqslant p < \infty), \\ \label{18:11} & \rho^{\varepsilon}\chi_{\mathcal{S}}^{\varepsilon} \rightarrow \rho^{\delta}\chi_{\mathcal{S}}^{\delta} \mbox{ weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) \mbox{ and }\mbox{ strongly } \mbox{ in }C([0,T]; L^p_{loc}(\mathbb{R}^3)) \ (1 \leqslant p < \infty), \end{align} with $\chi_{\mathcal{S}}^{\delta}$ and $\rho^{\delta}\chi_{\mathcal{S}}^{\delta}$ satisfying \eqref{approx4} and \eqref{approx5} respectively. Moreover, \begin{equation}\label{13:01} P_{\mathcal{S}}^{\varepsilon} u^{\varepsilon} \rightarrow P_{\mathcal{S}}^{\delta} u^{\delta} \mbox{ weakly } \mbox{in }L^{2}(0,T; C^{\infty}_{loc}(\mathbb{R}^3)). \end{equation} Hence, we have recovered the regularity of $\chi_{\mathcal{S}}^{\delta}$ in \eqref{approx1} and the transport equations \eqref{approx4} and \eqref{approx5} as $\varepsilon\rightarrow 0$.
\underline{Step 2: Limit of the continuity and the momentum equation.} We follow the ideas of \cite[Auxiliary lemma 7.49]{MR2084891} to conclude: if $\rho^{\delta}, u^{\delta}, \overline{ (\rho^{\delta})^{\gamma}}, \overline{ (\rho^{\delta})^{\beta}}$ are defined by \eqref{conv1}--\eqref{conv4}, we have \begin{itemize} \item $(\rho^{\delta},u^{\delta})$ satisfies: \begin{equation}\label{rho:delta} \frac{\partial {\rho}^{\delta}}{\partial t} + \operatorname{div}({\rho}^{\delta} u^{\delta}) =0 \mbox{ in }\mathcal{D}'([0,T)\times \mathbb{R}^3). \end{equation}
\item For all $\phi \in H^1(0,T; L^{2}(\Omega)) \cap L^r(0,T; W^{1,{r}}(\Omega))$, where $r=\max\left\{\beta+1, \frac{\beta+\theta}{\theta}\right\}$, $\beta \geqslant \max\{8,\gamma\}$ and $\theta=\frac{2}{3}\gamma -1$ with $\phi\cdot\nu=0$ on $\partial\Omega$ and $\phi|_{t=T}=0$, the following holds: \begin{multline}\label{mom:delta} - \int\limits_0^T\int\limits_{\Omega} \rho^{\delta} \left(u^{\delta}\cdot \frac{\partial}{\partial t}\phi + u^{\delta} \otimes u^{\delta} : \nabla \phi\right) + \int\limits_0^T\int\limits_{\Omega} \Big(2\mu^{\delta}\mathbb{D}(u^{\delta}):\mathbb{D}(\phi) + \lambda^{\delta}\operatorname{div}u^{\delta}\mathbb{I} : \mathbb{D}(\phi) - \left(a^{\delta}\overline{ (\rho^{\delta})^{\gamma}}+\delta \overline{ (\rho^{\delta})^{\beta}}\right)\mathbb{I}: \mathbb{D}(\phi)\Big) \\
+ \alpha \int\limits_0^T\int\limits_{\partial \Omega} (u^{\delta} \times \nu)\cdot (\phi \times \nu) + \alpha \int\limits_0^T\int\limits_{\partial \mathcal{S}^{\delta}(t)} \left[(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta})\times \nu\right]\cdot \left[(\phi-P^{\delta}_{\mathcal{S}}\phi)\times \nu\right] \\
+ \frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi^{\delta}_{\mathcal{S}}(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta})\cdot (\phi-P^{\delta}_{\mathcal{S}}\phi) = \int\limits_0^T\int\limits_{\Omega}\rho^{\delta} g^{\delta} \cdot \phi
+ \int\limits_{\Omega} (\rho^{\delta} u^{\delta} \cdot \phi)(0). \end{multline} \item The couple $(\rho^{\delta},u^{\delta})$ satisfies the identity \begin{equation}\label{renorm:delta} \partial_t b(\rho^{\delta}) + \operatorname{div}(b(\rho^{\delta})u^{\delta})+[b'(\rho^{\delta})\rho^{\delta} - b(\rho^{\delta})]\operatorname{div}u^{\delta}=0 \mbox{ in }\mathcal{D}'([0,T)\times \mathbb{R}^3), \end{equation}
with any $b\in C([0,\infty)) \cap C^1((0,\infty))$ satisfying \eqref{eq:b}. \item $\rho^{\delta} \in C([0,T];L^p(\Omega))$, $1\leqslant p< \beta$. \end{itemize} We outline the main lines of the proof of the above mentioned result. We prove \eqref{rho:delta} by passing to the limit $\varepsilon\rightarrow 0$ in equation \eqref{varepsilon:approx2} with the help of the convergence of the density in \eqref{conv2}, \eqref{conv5} and the convergence of the momentum \cite[Section 7.9.1, page 370]{MR2084891} \begin{align}\label{product1} \rho^{\varepsilon}u^{\varepsilon}\rightarrow \rho^\delta u^\delta \mbox{ weakly-}* \mbox{ in }L^{\infty}(0,T;L^{\frac{2\beta}{\beta+1}}(\Omega)),\mbox{ weakly in }L^2(0,T;L^{\frac{6\beta}{\beta+6}}(\Omega)). \end{align} We obtain identity \eqref{mom:delta}, corresponding to the momentum equation, by passing to the limit in \eqref{varepsilon:approx3}. To pass to the limit, we use the convergences of the density and the velocity \eqref{conv1}--\eqref{conv4} and of the transport part \eqref{13:02}--\eqref{13:01} along with the convergence of the product of the density and the velocity \eqref{product1} and the convergence of the following terms \cite[Section 7.9.1, page 371]{MR2084891}: \begin{align*} &\rho^{\varepsilon}u^{\varepsilon}_i u^{\varepsilon}_j \rightarrow \rho^\delta u^\delta_i u^\delta_j\mbox{ weakly in }L^2(0,T;L^{\frac{6\beta}{4\beta + 3}}(\Omega)), \quad i,j=1,2,3,\\ &\varepsilon (\nabla \rho^{\varepsilon}\cdot \nabla)u^{\varepsilon}\rightarrow 0 \mbox{ weakly in }L^{\frac{5\beta-3}{4\beta}}((0,T)\times\Omega). \end{align*} Since, we have already established the continuity equation \eqref{rho:delta} and the function $b\in C([0,\infty)) \cap C^1((0,\infty))$ satisfies \eqref{eq:b}, the renormalized continuity equation \eqref{renorm:delta} follows from the application of \cite[Lemma 6.9, page 307]{MR2084891}. Moreover, the regularity of the density $\rho^{\delta} \in C([0,T];L^p(\Omega))$, $1\leqslant p< \beta$ follows from \cite[Lemma 6.15, page 310]{MR2084891} via the appropriate choice of the renormalization function $b$ in \eqref{renorm:delta} and with the help of the regularities of $\rho^\delta\in L^{\infty}(0,T; L^{\beta}_{loc}(\mathbb{R} ^3)) \cap C([0,T];L^{\beta}_{loc}(\Omega))$, $u^\delta\in L^2(0,T; H^1_{loc}(\mathbb{R}^3))$. Hence we have established the continuity equation \eqref{approx2} and the renormalized one \eqref{rho:renorm1}.
\underline{Step 3: Limit of the pressure term.}
In this step, our aim is to identify the term $\left(\overline{ (\rho^{\delta})^{\gamma}}+\delta \overline{ (\rho^{\delta})^{\beta}}\right)$ by showing that $\overline{ (\rho^{\delta})^{\gamma}}=(\rho^{\delta})^{\gamma}$ and $\overline{ (\rho^{\delta})^{\beta}}=(\rho^{\delta})^{\beta}$. To prove this, we need some compactness of $\rho^{\varepsilon}$, which is not available. However, the quantity $(\rho^{\varepsilon})^{\gamma}+\delta (\rho^{\varepsilon})^{\beta}-(2\mu+\lambda)\rho ^{\varepsilon}\mathrm{div}u^{\varepsilon}$, called ``effective viscous flux", possesses a convergence property that helps us to identify the limit of our required quantity. We have the following weak and weak-$*$ convergences from the boundedness of their corresponding norms \cite[Section 7.9.2, page 373]{MR2084891}:
\begin{align}
\rho ^{\varepsilon}\mathrm{div}u^{\varepsilon}\rightarrow \overline{\rho^{\delta}\mathrm{div}u^{\delta}}&\mbox{ weakly in }L^2(0,T;L^{\frac{2\beta}{2+\beta}}(\Omega)),\label{conv6}\\ (\rho^{\varepsilon})^{\gamma+1}\rightarrow \overline{ (\rho^{\delta})^{\gamma+1}}&\mbox{ weakly-}*\mbox{ in } [C((0,T)\times\Omega)]',\label{conv7}\\ (\rho^{\varepsilon})^{\beta+1}\rightarrow \overline{ (\rho^{\delta})^{\beta+1}}&\mbox{ weakly-}*\mbox{ in } [C((0,T)\times\Omega)]'\label{conv8}.
\end{align}
We apply the following result regarding the "effective viscous flux" from \cite[Lemma 7.50, page 373]{MR2084891}:
Let $u^{\delta}$, $\rho^{\delta}$, $\overline{ (\rho^{\delta})^{\gamma}}$, $\overline{ (\rho^{\delta})^{\beta}}$, $\overline{ (\rho^{\delta})^{\gamma+1}}$, $\overline{ (\rho^{\delta})^{\beta+1}}$, $\overline{\rho^{\delta}\mathrm{div}u^{\delta}}$ be defined in \eqref{conv1}--\eqref{conv4}, \eqref{conv6}--\eqref{conv8}. Then we have
\begin{align}
&\overline{ (\rho^{\delta})^{\gamma+1}}\in L^{\frac{\beta+1}{\gamma+1}}((0,T)\times\Omega),\quad \overline{ (\rho^{\delta})^{\beta+1}} \in L^1((0,T)\times \Omega),\label{effective1}\\
&\overline{ (\rho^{\delta})^{\gamma+1}} + \delta\overline{ (\rho^{\delta})^{\beta+1}} - (2\mu+\lambda)\overline{\rho^{\delta}\mathrm{div}u^{\delta}}=\overline{ (\rho^{\delta})^{\gamma}}\rho^{\delta} + \delta\overline{ (\rho^{\delta})^{\beta}}\rho^{\delta} - (2\mu+\lambda)\rho^{\delta}\mathrm{div}u^{\delta}\mbox{ a.e. in }(0,T)\times\Omega \label{effective2}.
\end{align}
Using the above relation \eqref{effective1} and an appropriate choice of the renormalization function in \eqref{renorm:delta}, we deduce the strong convergence of the density as in \cite[Lemma 7.51, page 375]{MR2084891}: Let $\rho^{\delta}$, $\overline{ (\rho^{\delta})^{\gamma}}$, $\overline{ (\rho^{\delta})^{\beta}}$, $\overline{ (\rho^{\delta})^{\gamma+1}}$, $\overline{ (\rho^{\delta})^{\beta+1}}$ be defined in \eqref{conv2}--\eqref{conv4}, \eqref{conv6}--\eqref{conv7}. Then we have
\begin{equation*}
\overline{ (\rho^{\delta})^{\gamma}}= (\rho^{\delta})^{\gamma},\quad \overline{ (\rho^{\delta})^{\beta}}= (\rho^{\delta})^{\beta} \mbox{ a.e. in }(0,T)\times\Omega.
\end{equation*}
In particular,
\begin{equation}\label{strong:rhoepsilon}
\rho^{\varepsilon}\rightarrow \rho^{\delta}\mbox{ strongly in }L^p((0,T)\times\Omega),\ 1\leqslant p < \beta+1.
\end{equation}
Thus, we have identified the pressure term in equation \eqref{mom:delta}. Hence, we have recovered the momentum equation \eqref{approx3} and we have proved the existence of a weak solution $(\mathcal{S}^{\delta},\rho^{\delta},u^{\delta})$ to system \eqref{approx1}--\eqref{approx:initial}. It remains to prove the energy inequality \eqref{10:45} and the improved regularity for the density \eqref{rho:improved}.
\underline{Step 4: Energy inequality and improved regularity of the density.} Due to the convergences
\begin{align*} &\rho^{\varepsilon}u^{\varepsilon}_i u^{\varepsilon}_j \rightarrow \rho^\delta u^{\delta}_i u^{\delta}_j \mbox{ weakly in }L^2(0,T;L^{\frac{6\beta}{4\beta + 3}}(\Omega)), \quad i,j=1,2,3,\\ &\rho^{\varepsilon}\rightarrow \rho^{\delta}\mbox{ strongly in }L^p((0,T)\times\Omega),\ 1\leqslant p < \beta+1, \end{align*} we have \begin{align*}
&\int\limits_{\Omega} \rho^{\varepsilon}|u^{\varepsilon}|^2 \rightarrow \int\limits_{\Omega} \rho^{\delta}|u^{\delta}|^2 \mbox{ weakly in }L^2(0,T),\\ &\int\limits_{\Omega} \left((\rho^{\varepsilon})^{\gamma} +\delta(\rho^{\varepsilon})^{\beta}\right) \rightarrow \int\limits_{\Omega} \left((\rho^{\delta})^{\gamma} +\delta(\rho^{\delta})^{\beta}\right) \color{black}\mbox{ weakly in }L^{\frac{\beta+1}{\beta}}(0,T). \end{align*} In particular, \begin{equation*}
\int\limits_{\Omega}\Big( \rho^{\varepsilon} |u^{\varepsilon}|^2 + \frac{a^{\varepsilon}}{\gamma-1}(\rho^{\varepsilon})^{\gamma} + \frac{\delta}{\beta-1}(\rho^{\varepsilon})^{\beta}\Big) \rightarrow \int\limits_{\Omega}\Big( \rho^{\delta} |u^{\delta}|^2 + \frac{a^{\delta}}{\gamma-1}(\rho^{\delta})^{\gamma} + \frac{\delta}{\beta-1}(\rho^{\delta})^{\beta}\Big) \mbox{ as }\varepsilon\rightarrow 0. \end{equation*} Due to the weak lower semicontinuity of the corresponding $L^2$ norms, the weak convergence of $u^{\varepsilon}$ in $L^2(0,T;H^1(\Omega))$, the strong convergence of $\rho^{\varepsilon}$ in $L^p((0,T)\times\Omega),\ 1\leqslant p < \beta+1$, the strong convergence of $\chi_{\mathcal{S}}^{\varepsilon}$ in $C([0,T];L^p(\Omega))$ and the strong convergence of $P_{\mathcal{S}}^{\varepsilon}$ in $C([0,T]; C^{\infty}_{loc}(\mathbb{R}^3))$, we follow the idea explained in \eqref{N1}--\eqref{N6} to pass to the limit as $\varepsilon \rightarrow 0$ in the other terms of inequality \eqref{energy-varepsilon} to establish the energy inequality \eqref{10:45}.
To establish the regularity \eqref{rho:improved}, we use an appropriate test function of the type $$\mathcal{B}\left((\rho^{\delta})^{\theta} - |\Omega|^{-1}\int\limits_{\Omega}(\rho^{\delta})^{\theta}\right)$$ in the momentum equation \eqref{approx3}, where $\mathcal{B}$ is the Bogovskii operator. The detailed proof is in the lines of \cite[Section 7.9.5, pages 376-381]{MR2084891} and the extra terms can be treated as we have already explained in \eqref{bogovski:mom}--\eqref{bogovski:extra}. Moreover, we follow the same idea as in the proof of \cref{thm:approxn} (precisely, the calculations in \eqref{epsilon-collision}--\eqref{00:33}) to conclude that there exists $T$ small enough such that if $\operatorname{dist}(\mathcal{S}_0,\partial \Omega) > 2\sigma$, then \begin{equation}\label{delta-collision}
\operatorname{dist}(\mathcal{S}^{\delta}(t),\partial \Omega) \geqslant 2\sigma> 0 \quad \forall \ t\in [0,T]. \end{equation} This settles the proof of \cref{thm:approxn-delta}. \end{proof}
\section{Proof of the main result}\label{S4} We have already established the existence of a weak solution $(\mathcal{S}^{\delta},\rho^{\delta},u^{\delta})$ to system \eqref{approx1}--\eqref{approx:initial} in \cref{thm:approxn-delta}. In this section, we study the convergence analysis and the limiting behaviour of the solution as $\delta\rightarrow 0$ and recover a weak solution to system \eqref{mass:comfluid}--\eqref{initial cond:comp}, i.e., we show \cref{{exist:upto collision}}. \begin{proof} [Proof of \cref{exist:upto collision}]
\underline{Step 0: Initial data.}
We consider initial data $\rho_{\mathcal{F}_0}$, $q_{\mathcal{F}_0}$, $\rho_{\mathcal{S}_0}$, $q_{\mathcal{S}_0}$ satisfying the conditions \eqref{init}--\eqref{init2}. In this step we present the construction of the approximate initial data $(\rho^{\delta}_0,q^{\delta}_0)$ satisfying \eqref{rhonot}--\eqref{qnot} so that, in the limit $\delta\rightarrow 0$, we can recover the initial data $\rho_{\mathcal{F}_0}$ and $q_{\mathcal{F}_0}$ on $\mathcal{F}_0$.
We set
\begin{equation*}
\rho_0 = \rho_{\mathcal{F}_0}(1-\mathds{1}_{\mathcal{S}_0}) + \rho_{\mathcal{S}_0}\mathds{1}_{\mathcal{S}_0},
\end{equation*}
\begin{equation*}
q_0 = q_{\mathcal{F}_0}(1-\mathds{1}_{\mathcal{S}_0}) + \rho_{\mathcal{S}_0}u_{\mathcal{S}_0}\mathds{1}_{\mathcal{S}_0}.
\end{equation*}
Similarly as in \cite[Section 7.10.7, Page 392]{MR2084891}, we can find $\rho^{\delta}_{0} \in L^{\beta}({\Omega)}$ by defining \begin{equation}\label{init-apprx1}
\rho^{\delta}_{0}= \mathcal{K}_{\delta}(\rho_{0}) + \delta,
\end{equation}
where $\mathcal{K}_{\delta}$ is the standard regularizing operator in the space variable.
Then our initial density satisfies
\begin{equation}\label{ini-rho}
\begin{array}{l}
\rho^{\delta}_{0} \to \rho_{0} \mbox{ strongly in } L^{\gamma }(\Omega) .
\end{array} \end{equation} We define \begin{align} \label{init-apprx2} \overline{{q}^{\delta}_0}= \begin{cases} q_{0}\sqrt{\frac{\rho^{\delta}_{0}}{\rho_{0}}} &\mbox { if } \rho _{0} >0,\\ 0 \quad \quad \quad \quad \quad &\mbox { if } \rho _{0} =0. \end{cases} \end{align} From \eqref{init1}, we know that \begin{equation*}
\frac{|\overline{{q}^{\delta}_0}|}{\sqrt{\rho^{\delta}_{0}}} \in { L^2(\Omega)}. \end{equation*} Due to a density argument, there exists
$h^{\delta} \in W^{1,\infty}({\Omega})$ such that \begin{equation*}
\left\|\frac{q^{\delta}_0}{\sqrt{\rho^{\delta}_{0}}} -h^{\delta} \right\|_{L^2(\Omega)}< \delta. \end{equation*} Now, we set $ q^{\delta}_0= h^{\delta}\sqrt{\rho^{\delta}_{0}}$, which implies that
\begin{equation*}
q^{\delta}_0 \to q_{0} \mbox { in } L^{\frac{2\gamma}{\gamma +1}}(\Omega) \end{equation*} and \begin{equation*}
E^{\delta} [\rho ^{\delta}_0, q^{\delta}_0] \to E[\rho_0,q_0]. \end{equation*}
Next we start with the sequence of approximate solutions $\rho^{\delta},u^{\delta}$ of the system \eqref{approx1}--\eqref{approx:initial} (\cref{thm:approxn-delta}). Since the energy $E^{\delta}[\rho ^{\delta}_0,q^{\delta}_0]$ is uniformly bounded with respect to $\delta$, we have from inequality \eqref{10:45} that \begin{multline}\label{again-10:45}
\|\sqrt{\rho^{\delta}} u^{\delta}\|_{L^{\infty}(0,T;L^2(\Omega))}^2 + \|\rho^{\delta}\|_{L^{\infty}(0,T;L^{\gamma}(\Omega))}^2 + \|\sqrt{2\mu^{\delta}}\mathbb{D}(u^{\delta})\|_{L^2((0,T)\times\Omega)}^2 + \|\sqrt{\lambda^{\delta}}\operatorname{div}u^{\delta}\|_{L^2((0,T)\times\Omega)}^2 \\
+ \frac{1}{\delta}\|\sqrt{ \chi^{\delta}_{\mathcal{S}}}\left(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta}\right)\|^2_{L^2((0,T)\times \Omega)} \leqslant C,
\end{multline}
with $C$ independent of $\delta$.
\underline{Step 1: Recovery of the transport equation for body.}
Since $\{u^{\delta},\chi_{\mathcal{S}}^{\delta}\}$ is a bounded sequence in $L^{2}(0,T; L^2(\Omega)) \times L^{\infty}((0,T)\times \mathbb{R}^3)$ satisfying \eqref{approx4}, we can apply \cref{sequential2} to conclude that: up to a subsequence, we have \begin{align}
& u^{\delta} \rightarrow u \mbox{ weakly } \mbox{ in }L^{2}(0,T; L^{2}(\Omega)),\notag\\ & \chi_{\mathcal{S}}^{\delta} \rightarrow \chi_{\mathcal{S}} \mbox{ weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) \mbox{ and }\mbox{ strongly } \mbox{ in }C([0,T]; L^p_{loc}(\mathbb{R}^3)) \ (1\leqslant p<\infty),\label{19:30} \end{align} with \begin{equation*} \chi_{\mathcal{S}}(t,x)=\mathds{1}_{\mathcal{S}(t)}(x),\quad \mathcal{S}(t)=\eta_{t,0}(\mathcal{S}_0), \end{equation*} where $\eta_{t,s}\in H^{1}((0,T)^2; C^{\infty}_{loc}(\mathbb{R}^3))$ is the isometric propagator. Moreover, \begin{align} & P_{\mathcal{S}}^{\delta} u^{\delta} \rightarrow P_{\mathcal{S}} u \mbox{ weakly } \mbox{ in }L^{2}(0,T; C^{\infty}_{loc}(\mathbb{R}^3)),\label{prop1}\\ & \eta_{t,s}^{\delta} \rightarrow \eta_{t,s} \mbox{ weakly } \mbox{ in }H^{1}((0,T)^2; C^{\infty}_{loc}(\mathbb{R}^3)).\notag \end{align} Also, we obtain that $\chi_{\mathcal{S}}$ satisfies \begin{equation*} \frac{\partial {\chi}_{\mathcal{S}}}{\partial t} + \operatorname{div}(P_{\mathcal{S}}u\chi_{\mathcal{S}}) =0 \, \mbox{ in }{\Omega},\quad \chi_{\mathcal{S}}(t,x)=\mathds{1}_{\mathcal{S}(t)}(x). \end{equation*} Now we set \begin{equation}\label{uS} u_{\mathcal{S}}=P_{\mathcal{S}}u \end{equation} to recover the transport equation \eqref{NO4}. Note that we have already recovered the regularity of $\chi_{\mathcal{S}}$ in \eqref{NO1}.
Observe that the fifth term of inequality \eqref{again-10:45} yields \begin{equation}\label{12:04} \sqrt{ \chi^{\delta}_{\mathcal{S}}}\left(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta}\right) \rightarrow 0 \mbox{ strongly in } L^2((0,T)\times \Omega). \end{equation}
The strong convergence of $\chi_{\mathcal{S}}^{\delta}$ and the weak convergence of $u^{\delta}$ and $P^{\delta}_{\mathcal{S}}u^{\delta}$ imply that \begin{equation}\label{re:solidvel}
\chi_{\mathcal{S}}\left(u-u_{\mathcal{S}}\right)=0.
\end{equation}
To analyze the behaviour of the velocity field in the fluid part, we introduce the following continuous extension operator:
\begin{equation}\label{Eu1}
\mathcal{E}_u^{\delta}(t): \left\{ u\in H^{1}(\mathcal{F}^{\delta}(t)),\ u\cdot \nu=0\mbox{ on }\partial\Omega\right\} \rightarrow H^{1}(\Omega).
\end{equation}
Let us set
\begin{equation}\label{Eu2}
u^{\delta}_{\mathcal{F}}(t,\cdot)=\mathcal{E}_u^{\delta}(t)\left[u^{\delta}(t,\cdot)|_{\mathcal{F}^{\delta}}\right].
\end{equation}
We have
\begin{equation}\label{ext:fluid}
\{u^{\delta}_{\mathcal{F}}\} \mbox{ is bounded in }L^2(0,T;H^1(\Omega)),\quad u^{\delta}_{\mathcal{F}}=u^{\delta} \mbox{ on } \mathcal{F}^{\delta},\mbox{ i.e.\ } (1-\chi_{\mathcal{S}}^{\delta})(u^{\delta}-u^{\delta}_{\mathcal{F}})=0.
\end{equation}
Thus, the strong convergence of $\chi_{\mathcal{S}}^{\delta}$ and the weak convergence of $u^{\delta}_{\mathcal{F}}\rightarrow u_{\mathcal{F}}$ in $L^2(0,T;H^1(\Omega))$ yield that
\begin{equation}\label{re:fluidvel}
(1-\chi_{\mathcal{S}})\left(u-u_{\mathcal{F}}\right)=0.
\end{equation}
By combining the relations \eqref{re:solidvel}--\eqref{re:fluidvel}, we conclude that the limit $u$ of $u^{\delta}$ satisfies $u\in L^{2}(0,T; L^2(\Omega))$ and there exists $u_{\mathcal{F}}\in L^2(0,T; H^1(\Omega))$, $u_{\mathcal{S}}\in L^{2}(0,T; \mathcal{R})$ such that $u(t,\cdot)=u_{\mathcal{F}}(t,\cdot)$ on $\mathcal{F}(t)$ and $u(t,\cdot)=u_{\mathcal{S}}(t,\cdot)$ on $\mathcal{S}(t)$.
\underline{Step 2: Recovery of the continuity equations.} We recall that $\rho^{\delta}{\chi}^{\delta}_{\mathcal{S}}(t,x)$ satisfies \eqref{approx5}, i.e. \begin{equation*}
\frac{\partial }{\partial t}(\rho^{\delta}{\chi}^{\delta}_{\mathcal{S}}) + P^{\delta}_{\mathcal{S}}u^{\delta} \cdot \nabla (\rho^{\delta}{\chi}^{\delta}_{\mathcal{S}})=0,\quad (\rho^{\delta}{\chi}^{\delta}_{\mathcal{S}})|_{t=0}=\rho_0^{\delta}\mathds{1}_{\mathcal{S}_0}. \end{equation*} We proceed as in \cref{sequential2} to conclude that \begin{align}\label{19:31} \rho^{\delta}\chi_{\mathcal{S}}^{\delta} \rightarrow \rho\chi_{\mathcal{S}} \mbox{ weakly-}* \mbox{ in }L^{\infty}((0,T)\times \mathbb{R}^3) &\mbox{ and }\mbox{ strongly } \mbox{ in }C([0,T]; L^p_{loc}(\mathbb{R}^3)) \ (1\leqslant p<\infty), \end{align} and $\rho\chi_{\mathcal{S}}$ satisfies \begin{equation*}
\frac{\partial }{\partial t}(\rho{\chi}_{\mathcal{S}}) + P_{\mathcal{S}}u \cdot \nabla (\rho{\chi}_{\mathcal{S}})=0,\quad (\rho{\chi}_{\mathcal{S}})|_{t=0}= \rho_{\mathcal{S}_0}\mathds{1}_{\mathcal{S}_0}. \end{equation*} We set \begin{equation}\label{rhoS} \rho_{\mathcal{S}}= \rho\chi_{\mathcal{S}} \end{equation}
and use the definition of $u_{\mathcal{S}}$ in \eqref{uS} to conclude that $\rho_{\mathcal{S}}$ satisfies: \begin{equation*} \frac{\partial {\rho}_{\mathcal{S}}}{\partial t} + \operatorname{div}(u_{\mathcal{S}}\rho_{\mathcal{S}}) =0 \, \mbox{ in }(0,T)\times{\Omega},\quad \rho_{\mathcal{S}}(0,x)=\rho_{\mathcal{S}_0}(x)\mathds{1}_{\mathcal{S}_0}\mbox{ in }\Omega. \end{equation*} Thus, we recover the equation of continuity \eqref{NO5} for the density of the rigid body.
We introduce the following extension operator: \begin{equation*}
\mathcal{E}_{\rho}^{\delta}(t): \left\{ \rho\in L^{\gamma+\theta}(\mathcal{F}^{\delta}(t))\right\} \rightarrow L^{\gamma+\theta}(\Omega),
\end{equation*}
given by
\begin{equation}\label{Erho}
\mathcal{E}_{\rho}^{\delta}(t)\left[\rho^{\delta}(t,\cdot)|_{\mathcal{F}^{\delta}}\right]=
\begin{cases}
\rho^{\delta}(t,\cdot)|_{\mathcal{F}^{\delta}}&\mbox{ in }\mathcal{F}^{\delta}(t),\\
0 &\mbox{ in }\Omega\setminus \mathcal{F}^{\delta}(t).
\end{cases}
\end{equation}
Let us set
\begin{equation}\label{15:21}
\rho^{\delta}_{\mathcal{F}}(t,\cdot)=E_{\rho}^{\delta}(t)\left[\rho^{\delta}(t,\cdot)|_{\mathcal{F}^{\delta}}\right].
\end{equation} From estimates \eqref{10:45}, \eqref{rho:improved}, \eqref{ext:fluid} and the definition of $\rho^{\delta}_{\mathcal{F}}$ in \eqref{15:21}, we obtain that \begin{align}
u^{\delta}_{\mathcal{F}}\rightarrow u_{\mathcal{F}}&\mbox{ weakly in }L^2(0,T; H^1(\Omega)),\label{ag:conv1}\\
\rho^{\delta}_{\mathcal{F}}\rightarrow \rho_{\mathcal{F}}\mbox{ weakly in }L^{\gamma+\theta}((0,T)&\times \Omega),\ \theta=\frac{2}{3}\gamma-1\mbox{ and weakly-}*\mbox{ in } L^{\infty}(0,T;L^{\beta}(\Omega)),\label{ag:conv2}\\
(\rho^{\delta}_{\mathcal{F}})^{\gamma}\rightarrow \overline{ \rho^{\gamma}_{\mathcal{F}}}&\mbox{ weakly in }L^{\frac{\gamma+\theta}{\gamma}}((0,T)\times\Omega),\label{ag:conv3}\\
\delta(\rho^{\delta}_{\mathcal{F}})^{\beta}\rightarrow 0&\mbox{ weakly in }L^{\frac{\beta+\theta}{\beta}}((0,T)\times \Omega)\label{ag:conv4}. \end{align} Next, we follow the ideas of \cite[Auxiliary lemma 7.53, Page 384]{MR2084891} to assert: \color{black} if $u_{\mathcal{F}}, \rho_{\mathcal{F}}, \overline{ \rho_{\mathcal{F}}^{\gamma}}$ are defined by \eqref{ag:conv1}--\eqref{ag:conv3}, we have \begin{itemize} \item $(\rho_{\mathcal{F}},u_{\mathcal{F}})$ satisfies: \begin{equation}\label{agrho:delta} \frac{\partial {\rho_{\mathcal{F}}}}{\partial t} + \operatorname{div}({\rho}_{\mathcal{F}} u_{\mathcal{F}}) =0 \mbox{ in }\mathcal{D}'([0,T)\times \mathbb{R}^3). \end{equation}
\item The couple $(\rho_{\mathcal{F}},u_{\mathcal{F}})$ satisfies the identity \begin{equation}\label{agrenorm:delta} \partial_t \overline{b(\rho_{\mathcal{F}})} + \operatorname{div}(\overline{b(\rho_{\mathcal{F}})}u_{\mathcal{F}})+\overline{[b'(\rho_{\mathcal{F}})\rho_{\mathcal{F}} - b(\rho_{\mathcal{F}})]\operatorname{div}u_{\mathcal{F}}}=0 \mbox{ in }\mathcal{D}'([0,T)\times \mathbb{R}^3), \end{equation}
for any $b\in C([0,\infty)) \cap C^1((0,\infty))$ satisfying \eqref{eq:b} and the weak limits $\overline{b(\rho_{\mathcal{F}})}$ and $\overline{[b'(\rho_{\mathcal{F}})\rho_{\mathcal{F}} - b(\rho_{\mathcal{F}})]\operatorname{div}u_{\mathcal{F}}}$ being defined in the following sense: \begin{align*} & b(\rho^{\delta}_{\mathcal{F}}) \rightarrow \overline{b(\rho_{\mathcal{F}})} \mbox{ weakly-}*\mbox{ in }L^{\infty}(0,T; L^{\frac{\gamma}{1+\kappa_1}}(\mathbb{R}^3)),\\ & [b'(\rho^{\delta}_{\mathcal{F}})\rho^{\delta}_{\mathcal{F}} - b(\rho^{\delta}_{\mathcal{F}})]\operatorname{div}u^{\delta}_{\mathcal{F}} \rightarrow \overline{[b'(\rho_{\mathcal{F}})\rho_{\mathcal{F}} - b(\rho_{\mathcal{F}})]\operatorname{div}u_{\mathcal{F}}} \mbox{ weakly in }L^2(0,T; L^{\frac{2\gamma}{2+2\kappa_1+\gamma}}(\mathbb{R}^3)). \end{align*} \end{itemize} We outline the main idea of the proof of the asserted result. We derive \eqref{agrho:delta} by letting $\delta\rightarrow 0$ in equation \eqref{approx2} with the help of the convergence of the density in \eqref{ag:conv2} and the convergence of the momentum \cite[Section 7.10.1, page 383]{MR2084891} \begin{align}\label{ag:product1} \rho^{\delta}_{\mathcal{F}}u^{\delta}_{\mathcal{F}}\rightarrow \rho_{\mathcal{F}} u_{\mathcal{F}} \mbox{ weakly-}* \mbox{ in }L^{\infty}(0,T;L^{\frac{2\gamma}{\gamma+1}}(\mathbb{R}^3)),\mbox{ weakly in }L^2(0,T;L^{\frac{6\gamma}{\gamma+6}}(\mathbb{R}^3)). \end{align}
Recall that when we pass to the limit $\varepsilon\rightarrow 0$, we do have $\rho_{\mathcal{F}}^{\delta} \in L^2((0,T)\times \Omega)$. But in this step, we do not have $\rho_{\mathcal{F}} \in L^2((0,T)\times \Omega)$. So, it is not straightforward to obtain the renormalized continuity equation. Observe that this difficulty is not present in the case of $\gamma\geqslant \frac{9}{5}$ as in that case, $\rho_{\mathcal{F}}\in L^{\gamma+\theta}((0,T)\times\Omega)\subset L^2((0,T)\times\Omega)$ since $\gamma+\theta=\frac{5}{3}\gamma-1\geqslant 2$ for $\gamma\geqslant \frac{9}{5}$.
We use equation \eqref{rho:renorm1} and \cite[Lemma 1.7]{MR2084891} to establish that $\{b(\rho^{\delta}_{\mathcal{F}})\}$ is uniformly continuous in $W^{-1,s}(\Omega)$ with $s=\min\left\{\frac{6\gamma}{6\kappa_1+6+\gamma},2\right\}$, where the function $b\in C([0,\infty)) \cap C^1((0,\infty))$ satisfies \eqref{eq:b}. We apply \cite[Lemma 6.2, Lemma 6.4]{MR2084891} to get \begin{align*} b(\rho^{\delta}_{\mathcal{F}}) \rightarrow \overline{b(\rho_{\mathcal{F}})} &\mbox{ in }C([0,T]; L^{\frac{\gamma}{1+\kappa_1}}(\Omega)),\\ b(\rho^{\delta}_{\mathcal{F}}) \rightarrow \overline{b(\rho_{\mathcal{F}})}& \mbox{ strongly in }L^p(0,T; W^{-1,2}(\Omega)), \quad 1\leqslant p < \infty. \end{align*} The above mentioned limits together with \eqref{ag:conv1} help us to conclude \begin{equation*} b(\rho^{\delta}_{\mathcal{F}})u^{\delta}_{\mathcal{F}}\rightarrow \overline{b(\rho_{\mathcal{F}})}u_{\mathcal{F}} \mbox{ weakly in }L^2\left(0,T; L^{\frac{6\gamma}{6\kappa_1+6+\gamma}}(\Omega)\right). \end{equation*} Eventually, we obtain \eqref{agrenorm:delta} by taking the limit $\delta\rightarrow 0$ in \eqref{rho:renorm1}.
\underline{Step 3: Recovery of the renormalized continuity equation.} The method of an effective viscous flux with an appropriate choice of functions \cite[Lemma 7.55, page 386]{MR2084891} helps us to establish boundedness of oscillations of the density sequence and we have an estimate for the amplitude of oscillations \cite[Lemma 7.56, page 386]{MR2084891}:
\begin{equation*} \limsup_{\delta\rightarrow 0} \int\limits_{0}^T\int\limits_{\Omega} [T_k(\rho^{\delta}_{\mathcal{F}})-T_k(\rho_{\mathcal{F}})]^{\gamma+1} \leqslant \int\limits_{0}^T\int\limits_{\Omega} \left[\overline{\rho_{\mathcal{F}}^{\gamma}T_k(\rho_{\mathcal{F}})} - \overline{\rho^{\gamma}_{\mathcal{F}}}\overline{T_k(\rho_{\mathcal{F}})}\right], \end{equation*} where $T_k(\rho_{\mathcal{F}})=\min\{\rho_{\mathcal{F}},k\}$, $k >0$, are cut-off operators and $\overline{T_k(\rho_{\mathcal{F}})}$, $\overline{\rho_{\mathcal{F}}^{\gamma}T_k(\rho_{\mathcal{F}})}$ stand for the weak limits of $T_k(\rho_{\mathcal{F}}^{\delta})$, $(\rho_{\mathcal{F}}^{\delta})^{\gamma}T_k(\rho_{\mathcal{F}}^{\delta})$. This result allows us to estimate the quantities \begin{align*}
& \sup_{\delta> 0} \|\rho_{\mathcal{F}}^{\delta}\mathds{1}_{\{\rho_{\mathcal{F}}^{\delta}\geqslant k\}}\|_{L^p((0,T)\times\Omega)}, \quad \sup_{\delta> 0} \|T_k(\rho_{\mathcal{F}}^{\delta})-\rho_{\mathcal{F}}^{\delta}\|_{L^p((0,T)\times\Omega)},\\ & \|\overline{T_k(\rho_{\mathcal{F}})}-\rho_{\mathcal{F}}\|_{L^p((0,T)\times\Omega)},\quad \|T_k(\rho_{\mathcal{F}})-\rho_{\mathcal{F}}\|_{L^p((0,T)\times\Omega)} \mbox{ with }k>0, \ 1\leqslant p< \gamma+\theta. \end{align*} Using the above estimate and taking the renormalized function $b=T_k$ in \eqref{agrenorm:delta}, after several computations we obtain \cite[Lemma 7.57, page 388]{MR2084891}: Let $b\in C([0,\infty)) \cap C^1((0,\infty))$ satisfy \eqref{eq:b} with $\kappa_1+1\leqslant \frac{\gamma+\theta}{2}$ and let $u_{\mathcal{F}}$, $\rho_{\mathcal{F}}$ be defined by \eqref{ag:conv1}--\eqref{ag:conv2}. Then we obtain the renormalized continuity equation \eqref{NO3}: \begin{equation*} \partial_t b(\rho_{\mathcal{F}}) + \operatorname{div}(b(\rho_{\mathcal{F}})u_{\mathcal{F}}) + (b'(\rho_{\mathcal{F}})-b(\rho_{\mathcal{F}}))\operatorname{div}u_{\mathcal{F}}=0 \mbox{ in }\, \mathcal{D}'([0,T)\times {\Omega}) . \end{equation*} So far, we have recovered the transport equation of the body \eqref{NO4}, the continuity equation \eqref{NO2} and the renormalized one \eqref{NO3}. It remains to prove the momentum equation \eqref{weak-momentum} and establish the energy inequality \eqref{energy}.
\underline{Step 4: Recovery of the momentum equation.} Notice that the test functions in the weak formulation of momentum equation \eqref{weak-momentum} belong to the space $V_T$ (the space is defined in \eqref{def:test}), which is a space of discontinuous functions. Precisely, \begin{equation*} \phi=(1-\chi_{\mathcal{S}})\phi_{\mathcal{F}} + \chi_{\mathcal{S}}\phi_{\mathcal{S}}\mbox{ with }\phi_{\mathcal{F}}\in \mathcal{D}([0,T);\mathcal{D}(\overline{\Omega})),\quad \phi_{\mathcal{S}}\in \mathcal{D}([0,T);\mathcal{R}), \end{equation*} satisfying \begin{equation*} \phi_{\mathcal{F}}\cdot \nu=0 \mbox{ on }\partial\Omega,\quad \phi_{\mathcal{F}}\cdot \nu= \phi_{\mathcal{S}}\cdot \nu\mbox{ on }\partial\mathcal{S}(t). \end{equation*} Whereas, if we look at the test functions in momentum equation \eqref{approx3} in the $\delta$-approximation, we see that it involves an $L^p(0,T; W^{1,p}(\Omega))$-regularity. Hence we approximate this discontinuous test function by a sequence of test functions that belong to $L^p(0,T; W^{1,p}(\Omega))$. The idea is to construct an approximation $\phi^{\delta}_{\mathcal{S}}$ of $\phi$ without jumps at the interface such that \begin{equation}\label{cond1-good} \phi^{\delta}_{\mathcal{S}}(t,x)=\phi_{\mathcal{F}}(t,x) \quad \forall\ t\in (0,T),\ x\in \partial\mathcal{S}^{\delta}(t), \end{equation} and \begin{equation}\label{cond2-good} \phi^{\delta}_{\mathcal{S}}(t,\cdot)\approx \phi_{\mathcal{S}}(t,\cdot)\mbox{ in }\mathcal{S}^{\delta}(t)\mbox{ away from a }\delta^{\vartheta}\mbox{ neighborhood of }\partial\mathcal{S}^{\delta}(t)\mbox{ with }\vartheta >0. \end{equation} In the spirit of \cite[Proposition 5.1]{MR3272367}, at first, we give the precise result regarding this construction and then we will continue the proof of \cref{exist:upto collision}. \begin{proposition}\label{approx:test} Let $\phi \in V_T$ and $\vartheta >0$. Then there exists a sequence
$$\phi^{\delta} \in H^1(0,T; L^{2}(\Omega)) \cap L^r(0,T; W^{1,{r}}(\Omega)),\mbox{ where } r=\max\left\{\beta+1, \frac{\beta+\theta}{\theta}\right\},\ \beta \geqslant \max\{8,\gamma\}\mbox{ and }\theta=\frac{2}{3}\gamma -1 $$ of the form \begin{equation}\label{form:phi} \phi^{\delta}=(1-\chi^{\delta}_{\mathcal{S}})\phi_{\mathcal{F}} + \chi^{\delta}_{\mathcal{S}}\phi^{\delta}_{\mathcal{S}} \end{equation} that satisfies for all $p\in [1,\infty)$: \begin{enumerate}
\item $\|\chi^{\delta}_{\mathcal{S}}(\phi^{\delta}_{\mathcal{S}}-\phi_{\mathcal{S}})\|_{L^p((0,T)\times \Omega))}=\mathcal{O}(\delta^{\vartheta/p})$, \item $\phi^{\delta} \rightarrow \phi$ strongly in $L^p((0,T)\times \Omega)$,
\item $\|\phi^{\delta}\|_{L^p(0,T;W^{1,p}(\Omega))}=\mathcal{O}(\delta^{-\vartheta(1-1/p)})$,
\item $\|\chi^{\delta}_{\mathcal{S}}(\partial_t + P^{\delta}_{\mathcal{S}}u^{\delta}\cdot\nabla)(\phi^{\delta}-\phi_{\mathcal{S}})\|_{L^2(0,T;L^p(\Omega))}=\mathcal{O}(\delta^{\vartheta/p})$, \item $(\partial_t + P^{\delta}_{\mathcal{S}}u^{\delta}\cdot\nabla)\phi^{\delta}\rightarrow (\partial_t + P_{\mathcal{S}}u\cdot \nabla)\phi$ weakly in $L^2(0,T;L^p(\Omega))$. \end{enumerate} \end{proposition}
We give the proof of \cref{approx:test} at the end of this section. Next we continue the proof of \cref{exist:upto collision}.
\underline{Step 4.1: Linear terms of the momentum equation.} We use $\phi^{\delta}$ (constructed in \cref{approx:test}) as the test function in \eqref{approx3}. Then we take the limit $\delta\rightarrow 0$ in \eqref{approx3} to recover equation \eqref{weak-momentum}. Let us analyze the passage to the limit in the linear terms. To begin with, we recall the following convergences of the velocities of the fluid part and the solid part, cf.\ \eqref{ag:conv1} and \eqref{prop1}: \begin{align*} & (1-\chi_{\mathcal{S}}^{\delta})u^{\delta}_{\mathcal{F}}= (1-\chi^{\delta}_{\mathcal{S}})u^{\delta},\quad u^{\delta}_{\mathcal{F}}\rightarrow u_{\mathcal{F}}\mbox{ weakly in }L^2(0,T;H^1(\Omega)),\\ & u^{\delta}_{\mathcal{S}}=P^{\delta}_{\mathcal{S}}u^{\delta},\ u_{\mathcal{S}}=P_{\mathcal{S}}u,\quad u^{\delta}_{\mathcal{S}}\rightarrow u_{\mathcal{S}}\mbox{ weakly in }L^2(0,T; C^{\infty}_{loc}(\mathbb{R}^3)). \end{align*} Let us start with the diffusion term $2\mu^{\delta}\mathbb{D}(u^{\delta}):\mathbb{D}(\phi^{\delta}) + \lambda^{\delta}\operatorname{div}u^{\delta}\mathbb{I} : \mathbb{D}(\phi^{\delta})$ in \eqref{approx3}. We write \begin{align*} \int\limits_0^T\int\limits_{\Omega} 2\mu^{\delta}\mathbb{D}(u^{\delta}):\mathbb{D}(\phi^{\delta}) &= \int\limits_0^T\int\limits_{\Omega} \Big(2\mu_{\mathcal{F}}(1-\chi^{\delta}_{\mathcal{S}})\mathbb{D}(u_{\mathcal{F}}^{\delta}) + \delta^2 \chi_{\mathcal{S}}^{\delta}\mathbb{D}(u^{\delta})\Big): \mathbb{D}(\phi^{\delta})\\ &=\int\limits_0^T\int\limits_{\Omega} 2\mu_{\mathcal{F}}(1-\chi^{\delta}_{\mathcal{S}})\mathbb{D}(u_{\mathcal{F}}^{\delta}):\mathbb{D}(\phi_{\mathcal{F}}) + \delta^2\int\limits_0^T\int\limits_{\Omega} \chi_{\mathcal{S}}^{\delta}\mathbb{D}(u^{\delta}): \mathbb{D}(\phi^{\delta}). \end{align*} The strong convergence of $\chi^{\delta}_{\mathcal{S}}$ to $\chi_{\mathcal{S}}$ and the weak convergence of $u_{\mathcal{F}}^{\delta}$ to $u_{\mathcal{F}}$ imply that \begin{equation*} \int\limits_0^T\int\limits_{\Omega} 2\mu_{\mathcal{F}}(1-\chi^{\delta}_{\mathcal{S}})\mathbb{D}(u_{\mathcal{F}}^{\delta}):\mathbb{D}(\phi_{\mathcal{F}}) \rightarrow \int\limits_0^T\int\limits_{\Omega} 2\mu_{\mathcal{F}}(1-\chi_{\mathcal{S}})\mathbb{D}(u_{\mathcal{F}}):\mathbb{D}(\phi_{\mathcal{F}}). \end{equation*} We know from \eqref{again-10:45}, definition of $\mu^{\delta}$ in \eqref{approx-viscosity} and \cref{approx:test} (with $p=2$ case) that \begin{equation*}
\|\delta\chi^{\delta}_{\mathcal{S}}\mathbb{D}(u^{\delta})\|_{L^2((0,T)\times\Omega)} \leqslant C,\quad \|\phi^{\delta}\|_{L^2(0,T;H^1(\Omega))}=\mathcal{O}({\delta}^{-\vartheta/2}). \end{equation*} These estimates yield that \begin{equation} \label{alpha2}
\left|\delta^2\int\limits_0^T\int\limits_{\Omega} \chi_{\mathcal{S}}^{\delta}\mathbb{D}(u^{\delta}): \mathbb{D}(\phi^{\delta})\right|\leqslant \delta\|\delta\chi^{\delta}_{\mathcal{S}}\mathbb{D}(u^{\delta})\|_{L^2((0,T)\times\Omega)}\|\mathbb{D}(\phi^{\delta})\|_{L^2(0,T;L^2(\Omega))}\leqslant C\delta^{1-\vartheta/2}. \end{equation} If we consider $\vartheta<2$ and $\delta\rightarrow 0$, we have \begin{equation*} \delta^2\int\limits_0^T\int\limits_{\Omega} \chi_{\mathcal{S}}^{\delta}\mathbb{D}(u^{\delta}): \mathbb{D}(\phi^{\delta}) \rightarrow 0. \end{equation*} Hence, \begin{equation*} \int\limits_0^T\int\limits_{\Omega} \Big(2\mu^{\delta}\mathbb{D}(u^{\delta}):\mathbb{D}(\phi^{\delta}) + \lambda_{\mathcal{F}}\operatorname{div}u_{\mathcal{F}}\mathbb{I} : \mathbb{D}(\phi^{\delta})\Big)\rightarrow \int\limits_0^T\int\limits_{\mathcal{F}(t)} \Big(2\mu_{\mathcal{F}}\mathbb{D}(u_{\mathcal{F}}) + \lambda_{\mathcal{F}}\operatorname{div}u_{\mathcal{F}}\mathbb{I}\Big):\mathbb{D}(\phi_{\mathcal{F}}) \end{equation*} as $\delta\rightarrow 0$. Next we consider the boundary term on $\partial\Omega$ in \eqref{approx3}. The weak convergence of $u_{\mathcal{F}}^{\delta}$ to $u_{\mathcal{F}}$ in $L^2(0,T;H^1(\Omega))$ yields \begin{equation*} \int\limits_0^T\int\limits_{\partial \Omega} (u^{\delta} \times \nu)\cdot (\phi^{\delta} \times \nu)=\int\limits_0^T\int\limits_{\partial \Omega} (u_{\mathcal{F}}^{\delta}\times \nu)\cdot (\phi_{\mathcal{F}}\times \nu) \rightarrow\int\limits_0^T\int\limits_{\partial \Omega} (u_{\mathcal{F}}\times \nu)\cdot (\phi_{\mathcal{F}}\times \nu) \mbox{ as }\delta\rightarrow 0. \end{equation*} To deal with the boundary term on $\partial \mathcal{S}^{\delta}(t)$ we do a change of variables such that this term becomes an integral on the fixed boundary $\partial \mathcal{S}_0$. Then we pass to the limit as $\delta\rightarrow 0$ and afterwards transform back to the moving domain. Next, we introduce the notation $r^{\delta}_{\mathcal{S}}=P^{\delta}_{\mathcal{S}}\phi^{\delta}$ to write the following: \begin{align*} \int\limits_0^T\int\limits_{\partial \mathcal{S}^{\delta}(t)} [(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta})\times \nu]\cdot [(\phi^{\delta}-P^{\delta}_{\mathcal{S}}\phi^{\delta})\times \nu]=& \int\limits_0^T\int\limits_{\partial \mathcal{S}^{\delta}(t)} [(u^{\delta}_{\mathcal{F}}-u^{\delta}_{\mathcal{S}})\times \nu]\cdot [(\phi^{\delta}_{\mathcal{F}}-r^{\delta}_{\mathcal{S}})\times \nu]\\=& \int\limits_0^T\int\limits_{\partial \mathcal{S}_0} [(U^{\delta}_{\mathcal{F}}-U^{\delta}_{\mathcal{S}})\times \nu]\cdot [(\Phi^{\delta}_{\mathcal{F}}-R^{\delta}_{\mathcal{S}})\times \nu] ,\end{align*} where we denote by capital letters the corresponding velocity fields and test functions in the fixed domain. By \cref{approx:test} we have that $\phi^{\delta} \rightarrow \phi$ strongly in $L^2(0,T;L^6(\Omega))$. Hence we obtain, as in \cref{sequential2}, that \begin{equation*} r^{\delta}_{\mathcal{S}} \rightarrow r_{\mathcal{S}}=P_{\mathcal{S}}\phi \mbox{ strongly in } L^2(0,T; C^{\infty}_{loc}(\mathbb{R}^3)). \end{equation*} Now using \cite[Lemma A.2]{MR3272367}, we obtain the convergence in the fixed domain \begin{equation*} R^{\delta}_{\mathcal{S}} \rightarrow R_{\mathcal{S}} \mbox{ strongly in } L^2(0,T; H^{1/2}(\partial\mathcal{S}_0)). \end{equation*} Similarly, the convergences of $u^{\delta}_{\mathcal{F}}$ and $u^{\delta}_{\mathcal{S}}$ with \cite[Lemma A.2]{MR3272367} imply \begin{equation*} U^{\delta}_{\mathcal{F}}\rightarrow U_{\mathcal{F}},\ U^{\delta}_{\mathcal{S}}\rightarrow U_{\mathcal{S}}\quad\mbox{weakly in }L^2(0,T;H^1(\Omega)). \end{equation*} These convergence results and going back to the moving domain gives \begin{multline*} \int\limits_0^T\int\limits_{\partial \mathcal{S}^{\delta}(t)} [(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta})\times \nu]\cdot [(\phi^{\delta}-P^{\delta}_{\mathcal{S}}\phi^{\delta})\times \nu]= \int\limits_0^T\int\limits_{\partial \mathcal{S}_0} [(U^{\delta}_{\mathcal{F}}-U^{\delta}_{\mathcal{S}})\times \nu]\cdot [(\Phi^{\delta}_{\mathcal{F}}-R^{\delta}_{\mathcal{S}})\times \nu]\\ \rightarrow \int\limits_0^T\int\limits_{\partial \mathcal{S}_0} [(U_{\mathcal{F}}-U_{\mathcal{S}})\times \nu]\cdot [(\Phi_{\mathcal{F}}-R_{\mathcal{S}})\times \nu]= \int\limits_0^T\int\limits_{\partial \mathcal{S}(t)} [(u_{\mathcal{F}}-u_{\mathcal{S}})\times \nu]\cdot [(\phi_{\mathcal{F}}-\phi_{\mathcal{S}})\times \nu]. \end{multline*} The penalization term can be estimated in the following way: \begin{align}
\left|\frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi^{\delta}_{\mathcal{S}}(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta})\cdot (\phi^{\delta}-P^{\delta}_{\mathcal{S}}\phi^{\delta})\right|=&\left|\frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi^{\delta}_{\mathcal{S}}(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta})\cdot ((\phi^{\delta}-\phi_{\mathcal{S}})-P^{\delta}_{\mathcal{S}}(\phi^{\delta}-\phi_{\mathcal{S}}))\right|\notag\\=\left|\frac{1}{\delta}\int\limits_0^T\int\limits_{\Omega} \chi^{\delta}_{\mathcal{S}}(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta})\cdot (\phi^{\delta}_{\mathcal{S}}-\phi_{\mathcal{S}})\right| &\leqslant \delta^{-1/2}\frac{1}{\delta^{1/2}}\left\|\sqrt{ \chi^{\delta}_{\mathcal{S}}}\left(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta}\right)\right\|_{L^2((0,T)\times \Omega)}\left\|\sqrt{\chi^{\delta}_{\mathcal{S}}}(\phi^{\delta}_{\mathcal{S}}-\phi_{\mathcal{S}})\right\|_{L^2(0,T;L^2(\Omega)))}\notag\\& \leqslant C\delta^{-1/2+\vartheta/2},\label{12:27} \end{align} where we have used the estimates obtained from \eqref{again-10:45} and \cref{approx:test}. By choosing $\vartheta>1$ and taking $\delta\rightarrow 0$, the penalization term vanishes. Note that we also need $\vartheta <2$ in view of \eqref{alpha2}.
\underline{Step 4.2: Nonlinear terms of the momentum equation.} In this step, we analyze the following terms: \begin{multline}\label{convection} \int\limits_0^T\int\limits_{\Omega} \rho^{\delta} \left(u^{\delta}\cdot \frac{\partial}{\partial t}\phi + u^{\delta} \otimes u^{\delta} : \nabla \phi^{\delta}\right) = \int\limits_0^T\int\limits_{\Omega} \rho^{\delta}_{\mathcal{F}}(1-\chi_{\mathcal{S}}^{\delta}) u^{\delta}_{\mathcal{F}}\cdot \frac{\partial}{\partial t}\phi_{\mathcal{F}} + \int\limits_0^T\int\limits_{\Omega} \rho^{\delta}_{\mathcal{F}}(1-\chi_{\mathcal{S}}^{\delta}) u^{\delta}_{\mathcal{F}} \otimes u^{\delta}_{\mathcal{F}} : \nabla \phi_{\mathcal{F}} \\+ \int\limits_0^T\int\limits_{\Omega}\rho^{\delta}\chi^{\delta}_{\mathcal{S}}(\partial_t+u^{\delta}_{\mathcal{S}}\cdot\nabla)\phi^{\delta}\cdot u^{\delta} \end{multline} The strong convergence of $\chi^{\delta}_{\mathcal{S}}$ to $\chi_{\mathcal{S}}$
and the weak convergence of $\rho^{\delta}u^{\delta}_{\mathcal{F}}$ to $\rho_{\mathcal{F}}u_{\mathcal{F}}$ (see \eqref {ag:product1}) imply \begin{equation}\label{A1} \int\limits_0^T\int\limits_{\Omega} \rho^{\delta}_{\mathcal{F}}(1-\chi_{\mathcal{S}}^{\delta}) u^{\delta}_{\mathcal{F}}\cdot \frac{\partial}{\partial t}\phi_{\mathcal{F}} \rightarrow \int\limits_0^T\int\limits_{\Omega} \rho_{\mathcal{F}}(1-\chi_{\mathcal{S}}) u_{\mathcal{F}}\cdot \frac{\partial}{\partial t}\phi_{\mathcal{F}} \quad\mbox{ as }\delta\rightarrow 0. \end{equation}
We use the convergence result for the convective term from \cite[Section 7.10.1, page 384]{MR2084891} \begin{equation*} \rho^{\delta}_{\mathcal{F}}(u^{\delta}_{\mathcal{F}})_i (u^{\delta}_{\mathcal{F}})_j \rightarrow \rho_{\mathcal{F}} (u_{\mathcal{F}})_i (u_{\mathcal{F}})_j\mbox{ weakly in }L^2(0,T;L^{\frac{6\gamma}{4\gamma + 3}}(\Omega)), \quad \forall i,j\in \{1,2,3\}, \end{equation*} to pass to the limit in the second term of the right-hand side of \eqref{convection}: \begin{equation}\label{A2} \int\limits_0^T\int\limits_{\Omega} \rho^{\delta}_{\mathcal{F}}(1-\chi_{\mathcal{S}}^{\delta}) u^{\delta}_{\mathcal{F}} \otimes u^{\delta}_{\mathcal{F}} : \nabla \phi_{\mathcal{F}} \rightarrow \int\limits_0^T\int\limits_{\Omega} \rho_{\mathcal{F}}(1-\chi_{\mathcal{S}}) u_{\mathcal{F}} \otimes u_{\mathcal{F}} : \nabla \phi_{\mathcal{F}}. \end{equation} Next we consider the third term on the right-hand side of \eqref{convection}: \begin{multline*} \int\limits_0^T\int\limits_{\Omega}\rho^{\delta}\chi^{\delta}_{\mathcal{S}}(\partial_t+u^{\delta}_{\mathcal{S}}\cdot\nabla)\phi^{\delta}\cdot u^{\delta}=\int\limits_0^T\int\limits_{\Omega}\rho^{\delta}\chi_{\mathcal{S}}^{\delta}(\partial_t+u^{\delta}_{\mathcal{S}}\cdot\nabla)(\phi^{\delta}-\phi_{\mathcal{S}})\cdot u^{\delta} + \int\limits_0^T\int\limits_{\Omega}\rho^{\delta}\chi_{\mathcal{S}}^{\delta}\partial_t\phi_{\mathcal{S}}\cdot u^{\delta}\\ + \int\limits_0^T\int\limits_{\Omega}\rho^{\delta}\chi_{\mathcal{S}}^{\delta}(u^{\delta}_{\mathcal{S}}\cdot\nabla)\phi_{\mathcal{S}}\cdot u^{\delta}=: T_1^{\delta} + T_2^{\delta}+ T_3^{\delta}. \end{multline*}
We write \begin{equation*} T_1^{\delta} = \int\limits_0^T\int\limits_{\Omega}\rho^{\delta}\chi_{\mathcal{S}}^{\delta}(\partial_t+u^{\delta}_{\mathcal{S}}\cdot\nabla)(\phi^{\delta}-\phi_{\mathcal{S}})\cdot (u^{\delta}-P_{\mathcal{S}}^{\delta}u^{\delta}) + \int\limits_0^T\int\limits_{\Omega}\rho^{\delta}\chi_{\mathcal{S}}^{\delta}(\partial_t+u^{\delta}_{\mathcal{S}}\cdot\nabla)(\phi^{\delta}-\phi_{\mathcal{S}})\cdot P_{\mathcal{S}}^{\delta}u^{\delta}. \end{equation*} We estimate these terms in the following way: \begin{multline*}
\left|\int\limits_0^T\int\limits_{\Omega}\rho^{\delta}\chi_{\mathcal{S}}^{\delta}(\partial_t+u^{\delta}_{\mathcal{S}}\cdot\nabla)(\phi^{\delta}-\phi_{\mathcal{S}})\cdot (u^{\delta}-P_{\mathcal{S}}^{\delta}u^{\delta})\right| \\ \leqslant \|\rho^{\delta}\chi_{\mathcal{S}}^{\delta}\|_{L^{\infty}((0,T)\times \Omega)}\|\chi^{\delta}_{\mathcal{S}}(\partial_t + P^{\delta}_{\mathcal{S}}u^{\delta}\cdot\nabla)(\phi^{\delta}-\phi_{\mathcal{S}})\|_{L^2(0,T;L^6(\Omega))}\frac{1}{\delta^{1/2}}\left\|\sqrt{ \chi^{\delta}_{\mathcal{S}}}\left(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta}\right)\right\|_{L^2((0,T)\times \Omega)}\delta^{1/2}, \end{multline*} \begin{multline*}
\left|\int\limits_0^T\int\limits_{\Omega}\rho^{\delta}\chi_{\mathcal{S}}^{\delta}(\partial_t+u^{\delta}_{\mathcal{S}}\cdot\nabla)(\phi^{\delta}-\phi_{\mathcal{S}})\cdot P_{\mathcal{S}}^{\delta}u^{\delta}\right|\\ \leqslant \|\rho^{\delta}\chi_{\mathcal{S}}^{\delta}\|_{L^{\infty}((0,T)\times \Omega)}\|\chi^{\delta}_{\mathcal{S}}(\partial_t + P^{\delta}_{\mathcal{S}}u^{\delta}\cdot\nabla)(\phi^{\delta}-\phi_{\mathcal{S}})\|_{L^2(0,T;L^6(\Omega))}\|P^{\delta}_{\mathcal{S}}u^{\delta}\|_{L^2((0,T)\times \Omega)}, \end{multline*}
where we have used $\rho^{\delta}\chi_{\mathcal{S}}^{\delta} \in L^{\infty}((0,T)\times \Omega)$ as it is a solution to \eqref{approx5}. Moreover, by \cref{approx:test} (with the case $p=6$), we know that for $\vartheta > 0$$$\|\chi^{\delta}_{\mathcal{S}}(\partial_t + P^{\delta}_{\mathcal{S}}u^{\delta}\cdot\nabla)(\phi^{\delta}-\phi_{\mathcal{S}})\|_{L^2(0,T;L^6(\Omega))}=\mathcal{O}(\delta^{\vartheta/6}).$$ Hence, \begin{equation}\label{A3} T_1^{\delta} \rightarrow 0\mbox{ as } \delta\rightarrow 0. \end{equation} Observe that \begin{equation*} T_2^{\delta}= \int\limits_0^T\int\limits_{\Omega}\rho^{\delta}\chi_{\mathcal{S}}^{\delta}\partial_t\phi_{\mathcal{S}}\cdot (u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta}) + \int\limits_0^T\int\limits_{\Omega}\rho^{\delta}\chi_{\mathcal{S}}^{\delta}\partial_t\phi_{\mathcal{S}}\cdot P^{\delta}_{\mathcal{S}}u^{\delta}. \end{equation*} Now we use the following convergences: \begin{itemize} \item the strong convergence of $\sqrt{ \chi^{\delta}_{\mathcal{S}}}\left(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta}\right) \rightarrow 0$ in $ L^2((0,T)\times \Omega)$ (see the fifth term of inequality \eqref{again-10:45}), \item the strong convergence of $\chi^{\delta}_{\mathcal{S}}$ to $\chi_{\mathcal{S}}$ (see the convergence in \eqref{19:30}), \item the weak convergence of $\rho^{\delta}\chi_{\mathcal{S}}^{\delta}P^{\delta}_{\mathcal{S}}u^{\delta}\mbox{ to }\rho \chi_{\mathcal{S}}P_{\mathcal{S}}u$ (see the convergences in \eqref{19:31} and \eqref{prop1}), \end{itemize} to deduce \begin{equation*} T_2^{\delta} \rightarrow \int\limits_0^T\int\limits_{\mathcal{S}(t)}\rho \chi_{\mathcal{S}}\partial_t \phi_{\mathcal{S}}\cdot P_{\mathcal{S}}u\quad \mbox{ as }\quad\delta\rightarrow 0. \end{equation*} Recall the definition of $u_{\mathcal{S}}$ in \eqref{uS} and the definition of $\rho_{\mathcal{S}}$ in \eqref{rhoS}
to conclude \begin{equation}\label{A4} T_2^{\delta} \rightarrow \int\limits_0^T\int\limits_{\mathcal{S}(t)}\rho_{\mathcal{S}}\partial_t \phi_{\mathcal{S}}\cdot u_{\mathcal{S}}\quad \mbox{ as }\quad\delta\rightarrow 0. \end{equation} Notice that \begin{equation}\label{A5} T_3^{\delta}= \int\limits_0^T\int\limits_{\Omega}\rho^{\delta}\chi^{\delta}_{\mathcal{S}}(u^{\delta}_{\mathcal{S}}\cdot\nabla)\phi_{\mathcal{S}}\cdot u^{\delta} = \int\limits_0^T\int\limits_{\Omega}\rho^{\delta}\chi^{\delta}_{\mathcal{S}}(u^{\delta}_{\mathcal{S}}\otimes u^{\delta}_{\mathcal{S}}): \nabla\phi_{\mathcal{S}}= \int\limits_0^T\int\limits_{\Omega}\rho^{\delta}\chi^{\delta}_{\mathcal{S}}(u^{\delta}_{\mathcal{S}}\otimes u^{\delta}_{\mathcal{S}}): \mathbb{D}(\phi_{\mathcal{S}})=0. \end{equation} Eventually, combining the results \eqref{A1}--\eqref{A5}, we obtain \begin{equation*} \int\limits_0^T\int\limits_{\Omega} \rho^{\delta} \left(u^{\delta}\cdot \frac{\partial}{\partial t}\phi + u^{\delta} \otimes u^{\delta} : \nabla \phi\right) \rightarrow \int\limits_0^T\int\limits_{\mathcal{F}(t)} \rho_{\mathcal{F}}u_{\mathcal{F}}\cdot \frac{\partial}{\partial t}\phi_{\mathcal{F}} + \int\limits_0^T\int\limits_{\mathcal{S}(t)} \rho_{\mathcal{S}}u_{\mathcal{S}}\cdot \frac{\partial}{\partial t}\phi_{\mathcal{S}} + \int\limits_0^T\int\limits_{\mathcal{F}(t)} (\rho_{\mathcal{F}}u_{\mathcal{F}} \otimes u_{\mathcal{F}}) : \nabla \phi_{\mathcal{F}}. \end{equation*}
\underline{Step 4.3: Pressure term of the momentum equation.} We use the definition of $\phi^{\delta}$ \begin{equation*} \phi^{\delta}=(1-\chi^{\delta}_{\mathcal{S}})\phi_{\mathcal{F}} + \chi^{\delta}_{\mathcal{S}}\phi^{\delta}_{\mathcal{S}}, \end{equation*} to write
\begin{equation*} \int\limits_0^T\int\limits_{\Omega} \left(a^{\delta}(\rho^{\delta})^{\gamma} + {\delta} (\rho^{\delta})^{\beta} \right)\mathbb{I} : \mathbb{D}(\phi^{\delta}) = \int\limits_0^T\int\limits_{\Omega} \left[a_{\mathcal{F}} (1-\chi^{\delta}_{\mathcal{S}})(\rho^{\delta}_{\mathcal{F}})^{\gamma}+ {\delta}(1-\chi^{\delta}_{\mathcal{S}}) (\rho_{\mathcal{F}}^{\delta})^{\beta}\right]\mathbb{I} : \mathbb{D}(\phi_{\mathcal{F}}),
\end{equation*}
where we have used the fact that $\operatorname{div}\phi_{\mathcal{S}}^{\delta}=0$.
Due to the strong convergence of $\chi^{\delta}_{\mathcal{S}}$ to $\chi_{\mathcal{S}}$ and the weak convergence of $(\rho_{\mathcal{F}}^{\delta})^{\gamma}$, $(\rho_{\mathcal{F}}^{\delta})^{\beta}$ in \eqref{ag:conv3}, \eqref{ag:conv4} we obtain \begin{equation*}
\int\limits_0^T\int\limits_{\Omega} a_{\mathcal{F}} (1-\chi^{\delta}_{\mathcal{S}})(\rho^{\delta}_{\mathcal{F}})^{\gamma}\mathbb{I} : \mathbb{D}(\phi_{\mathcal{F}}) \rightarrow \int\limits_0^T\int\limits_{\Omega} a_{\mathcal{F}} (1-\chi_{\mathcal{S}})\overline{(\rho_{\mathcal{F}})^{\gamma}}\mathbb{I} : \mathbb{D}(\phi_{\mathcal{F}}) \mbox{ as }\delta \to 0,
\end{equation*}
and
\begin{equation*}
\int\limits_0^T\int\limits_{\Omega} {\delta}(1-\chi^{\delta}_{\mathcal{S}}) (\rho_{\mathcal{F}}^{\delta})^{\beta}\mathbb{I} : \mathbb{D}(\phi_{\mathcal{F}}) \rightarrow 0 \mbox{ as }\delta \to 0.
\end{equation*}
In order to establish \eqref{weak-momentum}, it only remains to show that $\overline{\rho^{\gamma}_{\mathcal{F}}}=\rho^{\gamma}_{\mathcal{F}}$. This is equivalent to establishing some strong convergence result of the sequence $\rho^{\delta}_{\mathcal{F}}$. We follow the idea of \cite[Lemma 7.60, page 391]{MR2084891} to prove: Let $\{\rho^{\delta}_{\mathcal{F}}\}$ be the sequence and $\rho_{\mathcal{F}}$ be its weak limit from \eqref{ag:conv2}. Then, at least for a chosen subsequence,
\begin{equation*}
\rho^{\delta}_{\mathcal{F}}\rightarrow \rho_{\mathcal{F}} \mbox{ in }L^p((0,T)\times\Omega),\quad 1\leqslant p < \gamma+\theta.
\end{equation*}
This immediately yields $\overline{\rho^{\gamma}_{\mathcal{F}}}=\rho^{\gamma}_{\mathcal{F}}$. Thus, we have recovered the weak form of the momentum equation.
\underline{Step 5: Recovery of the energy estimate.} We derive from \eqref{10:45} that \begin{multline*}
\int\limits_{\Omega}\Big( \rho^{\delta} |u^{\delta}|^2 + \frac{a_{\mathcal{F}}}{\gamma-1}(1-\chi^{\delta}_{\mathcal{S}})(\rho^{\delta}_{\mathcal{F}})^{\gamma} + \frac{\delta}{\beta-1}(\rho^{\delta}_{\mathcal{F}})^{\beta}\Big) + \int\limits_0^T\int\limits_{\Omega} \Big(2\mu_{\mathcal{F}}(1-\chi^{\delta}_{\mathcal{S}})|\mathbb{D}(u_{\mathcal{F}}^{\delta})|^2 + \lambda_{\mathcal{F}}(1-\chi^{\delta}_{\mathcal{S}})|\operatorname{div}u^{\delta}_{\mathcal{F}}|^2\Big)
+ \alpha \int\limits_0^T\int\limits_{\partial \Omega} |u^{\delta} \times \nu|^2 \\
+ \alpha \int\limits_0^T\int\limits_{\partial \mathcal{S}^{\delta}(t)} |(u^{\delta}-P^{\delta}_{\mathcal{S}}u^{\delta})\times \nu|^2
\leqslant \int\limits_0^T\int\limits_{\Omega}\rho^{\delta} g^{\delta} \cdot u^{\delta}
+ \int\limits_{\Omega}\Bigg( \frac{|q_0^{\delta}|^2}{\rho_0^{\delta}}\mathds{1}_{\{\rho_0^{\delta}>0\}} + \frac{a}{\gamma-1}(\rho_0^{\delta})^{\gamma} + \frac{\delta}{\beta-1}(\rho_0^{\delta})^{\beta} \Bigg). \end{multline*} To see the limiting behaviour of the above inequality as $\delta$ tends to zero, we observe that the limit as $\delta\to 0$ is similar to the limits $\varepsilon\rightarrow 0$ or $N\rightarrow \infty$ limit. Hence we obtain the energy inequality \eqref{energy}.
\underline{Step 6: Rigid body is away from boundary.} It remains to check that there exists $T$ small enough such that if $\operatorname{dist}(\mathcal{S}_0,\partial \Omega) > 2\sigma$, then \begin{equation}\label{final-collision}
\operatorname{dist}(\mathcal{S}(t),\partial \Omega) \geqslant \frac{3\sigma}{2}> 0 \quad \forall \ t\in [0,T]. \end{equation} Let us introduce the following notation: \begin{equation*} (\mathcal{U})_{\sigma} = \left\{x\in \mathbb{R}^3\mid \operatorname{dist}(x,\mathcal{U})<\sigma\right\}, \end{equation*} for an open set $\mathcal{U}$ and $\sigma > 0$. We recall the following result \cite[Lemma 5.4]{MR3272367}: Let $\sigma > 0$. There exists $\delta_0 >0$ such that for all $0< \delta \leqslant \delta_0$, \begin{equation}\label{cond:col1} \mathcal{S}^{\delta}(t) \subset (\mathcal{S}(t))_{\sigma/4} \subset (\mathcal{S}^{\delta}(t))_{\sigma/2},\quad \forall\ t\in [0,T]. \end{equation} Note that condition \eqref{cond:col1} and the relation \eqref{delta-collision}, i.e., $\operatorname{dist}(\mathcal{S}^{\delta}(t),\partial \Omega) \geqslant 2\sigma> 0$ for all $t\in [0,T]$ imply our required estimate \eqref{final-collision}. Thus, we conclude the proof of \cref{exist:upto collision}. \end{proof}
It remains to prove \cref{approx:test}. The main difference between \cref{approx:test} and \cite[Proposition 5.1]{MR3272367} is the time regularity of the approximate test functions.
Since here we only have weak convergence of $u^{\delta}$ in $L^2(0,T;L^2(\Omega))$, according to \cref{sequential2} we have convergence of $\eta^{\delta}_{t,s}$ in $H^{1}((0,T)^2; C^{\infty}_{loc}(\mathbb{R}^3))$. In \cite[Proposition 5.1]{MR3272367}, they have weak convergence of $u^{\delta}$ in $L^{\infty}(0,T;L^2(\Omega))$, which yields higher time regularity of the propagator $\eta^{\delta}_{t,s}$ in $W^{1,\infty}((0,T)^2; C^{\infty}_{loc}(\mathbb{R}^3))$. Still, the construction of the approximate function is essentially similar, which is why we skip the details and only present the main ideas of the proof here.
\begin{proof}[Proof of \cref{approx:test}] The proof relies on the construction of the approximation $\phi^{\delta}_{\mathcal{S}}$ of $\phi_{\mathcal{S}}$ so that we can avoid the jumps at the interface for the test functions such that \eqref{cond1-good}--\eqref{cond2-good} holds.
The idea is to write the test functions in Lagrangian coordinates through the isometric propagator $\eta^{\delta}_{t,s}$ so that we can work on the fixed domain. Let $\Phi_{\mathcal{F}}$, $\Phi_{\mathcal{S}}$ and $\Phi^{\delta}_{\mathcal{S}}$ be the transformed quantities in the fixed domain related to $\phi_{\mathcal{F}}$, $\phi_{\mathcal{S}}$ and $\phi^{\delta}_{\mathcal{S}}$ respectively: \begin{equation}\label{chofv}
\phi_{\mathcal{S}}(t,\eta^{\delta}_{t,0}(y)) = J_{\eta_{t,0}^{\delta}}\Big|_{y} (\Phi_{\mathcal{S}}(t,y)),\quad \phi_{\mathcal{F}}(t,\eta^{\delta}_{t,0}(y)) = J_{\eta_{t,0}^{\delta}}\Big|_{y} \Phi_{\mathcal{F}}(t,y)\quad\mbox{ and }\quad\phi^{\delta}_{\mathcal{S}}(t,\eta^{\delta}_{t,0}(y)) = J_{\eta_{t,0}^{\delta}}\Big|_{y} \Phi^{\delta}_{\mathcal{S}}(t,y), \end{equation} where $J_{\eta_{t,0}^{\delta}}$ is the Jacobian matrix of $\eta_{t,0}^{\delta}$. Note that if we define \begin{equation*} \Phi^{\delta}(t,y)=(1-\chi^{\delta}_{\mathcal{S}})\Phi_{\mathcal{F}} + \chi^{\delta}_{\mathcal{S}}\Phi^{\delta}_{\mathcal{S}}, \end{equation*} then the definition of $\phi^{\delta}$ in \eqref{form:phi} gives \begin{equation}\label{chofv1}
\phi^{\delta}(t,\eta^{\delta}_{t,0}(y)) = J_{\eta_{t,0}^{\delta}}\Big|_{y} (\Phi^{\delta}(t,y)). \end{equation}
Thus, the construction of the approximation $\phi^{\delta}_{\mathcal{S}}$ satisfying \eqref{cond1-good}--\eqref{cond2-good} is equivalent to building the approximation $\Phi^{\delta}_{\mathcal{S}}$ so that there is no jump for the function $\Phi^{\delta}$ at the interface and the following holds: \begin{equation*} \Phi^{\delta}_{\mathcal{S}}(t,x)=\Phi_{\mathcal{F}}(t,x) \quad \forall\ t\in (0,T),\ x\in \partial\mathcal{S}_0, \end{equation*} and \begin{equation*} \Phi^{\delta}_{\mathcal{S}}(t,\cdot)\approx \Phi_{\mathcal{S}}(t,\cdot)\mbox{ in }\mathcal{S}_0\mbox{ away from a }\delta^{\vartheta}\mbox{ neighborhood of }\partial\mathcal{S}_0\mbox{ with }\vartheta >0. \end{equation*} Explicitly, we set (for details, see \cite[Pages 2055-2058]{MR3272367}): \begin{equation}\label{decomposePhi} \Phi^{\delta}_{\mathcal{S}} = \Phi^{\delta}_{\mathcal{S},1} + \Phi^{\delta}_{\mathcal{S},2}, \end{equation} with \begin{equation}\label{phis1} \Phi^{\delta}_{\mathcal{S},1}= \Phi_{\mathcal{S}} + \chi (\delta^{-\vartheta}z) \left[(\Phi_{\mathcal{F}}-\Phi_{\mathcal{S}}) - ((\Phi_{\mathcal{F}}-\Phi_{\mathcal{S}})\cdot e_z) e_z\right], \end{equation}
where $\chi : \mathbb{R} \rightarrow [0,1]$ is a smooth truncation function which is equal to 1 in a neighborhood of 0 and $z$ is a coordinate transverse to the boundary $\partial \mathcal{S}_0 = \{z=0\}$. Moreover, to make $\Phi^{\delta}_{\mathcal{S}}$ divergence-free in $\mathcal{S}_0$, we need to take $\Phi^{\delta}_{\mathcal{S},2}$ such that
\begin{equation*}
\operatorname{div}\Phi^{\delta}_{\mathcal{S},2}=-\operatorname{div}\Phi^{\delta}_{\mathcal{S},1}\quad\mbox{in }\mathcal{S}_0,\quad \Phi^{\delta}_{\mathcal{S},2}=0\quad\mbox{on }\partial\mathcal{S}_0.
\end{equation*} Observe that, the explicit form \eqref{phis1} of $\Phi^{\delta}_{\mathcal{S},1}$ yields \begin{equation}\label{divphis1} \operatorname{div}\Phi^{\delta}_{\mathcal{S},2}=-\operatorname{div}\Phi^{\delta}_{\mathcal{S},1} = -\chi(\delta^{-\vartheta}z)\operatorname{div}\left[(\Phi_{\mathcal{F}}-\Phi_{\mathcal{S}}) - ((\Phi_{\mathcal{F}}-\Phi_{\mathcal{S}})\cdot e_z) e_z\right]. \end{equation} Thus, the expressions \eqref{phis1}--\eqref{divphis1} give us: for all $p<\infty$, \begin{align}\label{Phi11}
\|\Phi^{\delta}_{\mathcal{S},1}-\Phi_{\mathcal{S}}\|_{H^1(0,T; L^p(\mathcal{S}_0))} &\leqslant C\delta^{\vartheta/p}, \\ \label{Phi12}
\|\Phi^{\delta}_{\mathcal{S},1}-\Phi_{\mathcal{S}}\|_{H^1(0,T; W^{1,p}(\mathcal{S}_0))} &\leqslant C\delta^{-\vartheta(1-1/p)}, \end{align} and \begin{equation}\label{Phi2}
\|\Phi^{\delta}_{\mathcal{S},2}\|_{H^1(0,T; W^{1,p}(\mathcal{S}_0))}\leqslant C\|\chi({\delta}^{-\vartheta}z)\operatorname{div}\left[(\Phi_{\mathcal{F}}-\Phi_{\mathcal{S}}) - ((\Phi_{\mathcal{F}}-\Phi_{\mathcal{S}})\cdot e_z) e_z\right]\|_{H^1(0,T; L^{p}(\mathcal{S}_0))} \leqslant C\delta^{\vartheta/p}. \end{equation} Using the decomposition \eqref{decomposePhi} of $\Phi_{\mathcal{S}}^{\delta}$ and the estimates \eqref{Phi11}--\eqref{Phi12}, \eqref{Phi2}, we obtain \begin{align*}
\|\Phi^{\delta}_{\mathcal{S}}-\Phi_{\mathcal{S}}\|_{H^1(0,T; L^p(\mathcal{S}_0))} &\leqslant C\delta^{\vartheta/p},\\
\|\Phi^{\delta}_{\mathcal{S}}-\Phi_{\mathcal{S}}\|_{H^1(0,T; W^{1,p}(\mathcal{S}_0))} &\leqslant C\delta^{-\vartheta(1-1/p)}. \end{align*} Furthermore, we combine the above estimates with the uniform bound of the propagator $\eta^{\delta}_{t,0}$ in $H^1(0,T; C^{\infty}(\Omega))$ to obtain \begin{align}\label{est:Phi}
\left\|J_{\eta_{t,0}^{\delta}}|_{y}(\Phi^{\delta}_{\mathcal{S}}-\Phi_{\mathcal{S}})\right\|_{H^1(0,T; L^p(\mathcal{S}_0))} &\leqslant C\delta^{\vartheta/p},\\ \label{est:dPhi}
\left\|J_{\eta_{t,0}^{\delta}}|_{y}(\Phi^{\delta}_{\mathcal{S}}-\Phi_{\mathcal{S}})\right\|_{H^1(0,T; W^{1,p}(\mathcal{S}_0))} &\leqslant C\delta^{-\vartheta(1-1/p)}. \end{align} Observe that due to the change of variables \eqref{chofv} and estimate \eqref{est:Phi}: \begin{equation}\label{est1}
\|\chi^{\delta}_{\mathcal{S}}(\phi^{\delta}_{\mathcal{S}}-\phi_{\mathcal{S}})\|_{L^p((0,T)\times \Omega))}\leqslant C\|J_{\eta_{t,0}^{\delta}}|_{y}(\Phi^{\delta}_{\mathcal{S}}-\Phi_{\mathcal{S}})\|_{L^p((0,T)\times \mathcal{S}_0)}\leqslant C \delta^{\vartheta/p}. \end{equation} Since \begin{equation*}
\|\phi^{\delta}-\phi\|_{L^p((0,T)\times \Omega))}\leqslant \|(\chi^{\delta}_{\mathcal{S}}-\chi_{\mathcal{S}})\phi_{\mathcal{F}}\|_{L^p((0,T)\times \Omega))} + \|\chi^{\delta}_{\mathcal{S}}(\phi_{\mathcal{S}}^{\delta}-\phi_{\mathcal{S}})\|_{L^p((0,T)\times \Omega))} + \|(\chi^{\delta}_{\mathcal{S}}-\chi_{\mathcal{S}})\phi_{\mathcal{S}}\|_{L^p((0,T)\times \Omega))}, \end{equation*} using the strong convergence of $\chi^{\delta}_{\mathcal{S}}$ and the estimate \eqref{est1}, we conclude that \begin{equation*} \phi^{\delta} \rightarrow \phi\mbox{ strongly in }L^p((0,T)\times \Omega). \end{equation*} We use estimate \eqref{Phi12} and the relation \eqref{chofv1} to obtain \begin{equation*}
\|\phi^{\delta}\|_{L^p(0,T;W^{1,p}(\Omega))}\leqslant \delta^{-\vartheta(1-1/p)}. \end{equation*} Moreover, the change of variables \eqref{chofv} and estimate \eqref{est:Phi} give \begin{align}\label{timechi}
\begin{split} \|\chi^{\delta}_{\mathcal{S}}(\partial_t + P^{\delta}_{\mathcal{S}}u^{\delta}\cdot\nabla)(\phi^{\delta}-\phi_{\mathcal{S}})\|_{L^2(0,T;L^p(\Omega))} &\leqslant C\left\|\frac{d}{dt}\left(J_{\eta_{t,0}^{\delta}}\Big|_{y}(\Phi^{\delta}_{\mathcal{S}}-\Phi_{\mathcal{S}})\right)\right\|_{L^2(0,T;L^p(\mathcal{S}_0))}\\
& \leqslant C\left\|J_{\eta_{t,0}^{\delta}}|_{y}(\Phi^{\delta}_{\mathcal{S}}-\Phi_{\mathcal{S}})\right\|_{H^1(0,T;L^p(\mathcal{S}_0))}\leqslant C\delta^{\vartheta/p}. \end{split} \end{align} The above estimate \eqref{timechi}, strong convergence of $\chi_{\mathcal{S}}^{\delta}$ to $\chi_{\mathcal{S}}$ in $C(0,T;L^p(\Omega))$ and weak convergence of $P^{\delta}_{\mathcal{S}}u^{\delta}$ to $P_{\mathcal{S}} u \mbox{ weakly } \mbox{ in }L^{2}(0,T; C^{\infty}_{loc}(\mathbb{R}^3))$, give us \begin{equation*} (\partial_t + P^{\delta}_{\mathcal{S}}u^{\delta}\cdot\nabla)\phi^{\delta}\rightarrow (\partial_t + P_{\mathcal{S}}u\cdot \nabla)\phi \mbox{ weakly in } L^2(0,T;L^p(\Omega)), \end{equation*} where \begin{equation*} \phi^{\delta}=(1-\chi^{\delta}_{\mathcal{S}})\phi_{\mathcal{F}} + \chi^{\delta}_{\mathcal{S}}\phi^{\delta}_{\mathcal{S}}\quad\mbox{ and }\quad \phi=(1-\chi_{\mathcal{S}})\phi_{\mathcal{F}} + \chi_{\mathcal{S}}\phi_{\mathcal{S}}. \end{equation*}
\end{proof}
\section*{Acknowledgment} {\it \v S. N. and A. R. have been supported by the Czech Science Foundation (GA\v CR) project GA19-04243S. The Institute of Mathematics, CAS is supported by RVO:67985840.}
\section*{Compliance with Ethical Standards} \section*{Conflict of interest} The authors declare that there are no conflicts of interest.
\end{document} | arXiv |
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Based on the idea and the provided source code of Andrej Karpathy (arxiv-sanity)
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Fernando Becerra, Thomas H. Greif, Volker Springel, Lars Hernquist
Dec. 4, 2014 astro-ph.CO, astro-ph.GA
We present the highest-resolution three-dimensional simulation to date of the collapse of an atomic cooling halo in the early Universe. We use the moving-mesh code arepo with the primordial chemistry module introduced in Greif (2014), which evolves the chemical and thermal rate equations for over more than 20 orders of magnitude in density. Molecular hydrogen cooling is suppressed by a strong Lyman-Werner background, which facilitates the near-isothermal collapse of the gas at a temperature of about $10^4\,$K. Once the central gas cloud becomes optically thick to continuum emission, it settles into a Keplerian disc around the primary protostar. The initial mass of the protostar is about $0.1\,{\rm M}_\odot$, which is an order of magnitude higher than in minihaloes that cool via molecular hydrogen. The high accretion rate and efficient cooling of the gas catalyse the fragmentation of the disc into a small protostellar system with 5-10 members. After about 12 yr, strong gravitational interactions disrupt the disc and temporarily eject the primary protostar from the centre of the cloud. By the end of the simulation, a secondary clump has collapsed at a distance of $\simeq 150\,$au from the primary clump. If this clump undergoes a similar evolution as the first, the central gas cloud may evolve into a wide binary system. High accretion rates of both the primary and secondary clumps suggest that fragmentation is not a significant barrier for forming at least one massive black hole seed.
The Interstellar Medium and Star Formation in Local Galaxies: Variations of the Star Formation Law in Simulations (1401.5082)
Fernando Becerra, Andres Escala
April 23, 2014 astro-ph.CO, astro-ph.GA
We use the Adaptive Mesh Refinement code Enzo to model the interstellar medium in isolated local disk galaxies. The simulation includes a treatment for star formation and stellar feedback. We get a highly supersonic turbulent disk, which is fragmented at multiple scales and characterized by a multi-phase interstellar medium. We show that a Kennicutt-Schmidt (KS) relation only holds when averaging over large scales. However, values of star formation rates and gas surface densities lie close in the plot for any averaging size. This suggests an intrinsic relation between stars and gas at cell-size scales, which dominates over the global dynamical evolution. To investigate this effect, we develop a method to simulate the creation of stars based on the density field from the snapshots, without running the code again. We also investigate how the star formation law is affected by the characteristic star formation timescale, the density threshold and the efficiency considered in the recipe. We find that the slope of the law might vary from ~1.4 for a free-fall timescale, to ~1.0 for a constant depletion timescale. We further demonstrate that a power-law is recovered just by assuming that the mass of the new stars is a fraction of the mass of the cell $m_\star=\epsilon\rho_{\rm gas}\Delta x^3$, with no other physical criteria required. We show that both efficiency and density threshold do not affect the slope, but the right combination of them can adjust the normalization of the relation, which in turn could explain a possible bi-modality in the law.
Gravitational Fragmentation in Galaxy Mergers: A Stability Criteria (1202.1283)
Andres Escala, Fernando Becerra, Luciano del Valle, Esteban Castillo
Oct. 4, 2012 astro-ph.CO
We study the gravitational stability of gaseous streams in the complex environment of a galaxy merger, because mergers are known to be places of ongoing massive cluster formation and bursts of star formation. We find an analytic stability parameter for case of gaseous streams orbiting around the merger remnant. We test our stability criteria using hydrodynamical simulations of galaxy mergers, obtaining satisfactory results. We find that our criteria successfully predicts the streams that will be gravitationally unstable to fragment into clumps. | CommonCrawl |
Second-order differential operators with interior singularity
Kadriye Aydemir1 &
Oktay S Mukhtarov1,2
The purpose of this study is to investigate a new class of boundary value transmission problems (BVTPs) for a Sturm-Liouville equation on two separate intervals. We introduce a modified inner product in the direct sum space \(L_{2}[a,c)\oplus L_{2}(c,b]\oplus C^{2}\) and define a symmetric linear operator in it in such a way that the considered problem can be interpreted as an eigenvalue problem of this operator. Then, by suggesting own approaches, we construct the Green's function for the BVTP under consideration and find the resolvent function for the corresponding inhomogeneous problem.
Many interesting applications of Sturm-Liouville theory arise in quantum mechanics. For instance, for a single quantum-mechanical particle of mass m moving in one space dimension in a potential \(V (x)\), the time-dependent Schrödinger equation is
$$i\hbar\psi_{t}=-\frac{\hbar^{2}}{2m}\psi_{xx}+V(x)\psi. $$
Looking for separable solutions \(\psi(x,t)=\varphi(x)e^{-iEt/\hbar}\), we find that \(\varphi(x)\) satisfies the differential equation
$$-\frac{\hbar^{2}}{2m}\varphi''+V(x)\varphi=E\varphi. $$
That is a Sturm-Liouville equation of the form
$$-y^{\prime\prime}+ q y=\lambda y. $$
The coefficient q is proportional to the potential V, and the eigenvalue parameter λ is proportional to the energy E. Physical problems such as this and those involving sound, surface waves, heat conduction, electromagnetic waves, and gravitational waves, for example, can be solved using the mathematical theory of boundary value problems. Boundary value problems can be investigated also through the methods of Green's function and eigenfunction expansion. The main tool for solvability analysis of such problems is the concept of Green's function. The concept of Green's function is very close to physical intuition (see [1]). If one knows the Green's function of a problem, one can write down its solution in a closed form as linear combinations of integrals involving the Green's function and the functions appearing in the inhomogeneities. Green's functions can often be found in an explicit way, and in these cases it is very efficient to solve the problem in this way. Determination of Green's functions is also possible using Sturm-Liouville theory. This leads to a series representation of Green's functions (see [2]).
In this study we shall investigate a new class of BVPs which consist of the Sturm-Liouville equation
$$ \ell(y):=-p(x)y^{\prime\prime}(x)+ q(x)y(x)=\lambda y(x) $$
to hold in a finite interval \([a, b]\) except at one inner point \(c \in (a, b)\), where discontinuities in y and \(y'\) are prescribed by the transmission conditions at the interior point \(x=c\),
$$ V_{j}(y):=\beta^{-}_{j1}y'(c-)+ \beta^{-}_{j0}y(c-)+\beta ^{+}_{j1}y'(c+)+ \beta^{+}_{j0}y(c+)=0,\quad j=1,2, $$
together with eigenparameter-dependent boundary conditions at end points \(x=a, b\),
$$\begin{aligned}& U_{1}(y):=\alpha_{10}y(a)- \alpha_{11}y'(a)-\lambda\bigl(\alpha '_{10}y(a)- \alpha'_{11}y'(a)\bigr)=0, \end{aligned}$$
$$\begin{aligned}& U_{2}(y):=\alpha_{20}y(b)- \alpha_{21}y'(b)+\lambda\bigl(\alpha '_{20}y(b)- \alpha'_{21}y'(b)\bigr)=0, \end{aligned}$$
where \(p(x)=p^{-}>0\) for \(x \in[a, c) \), \(p(x)=p^{+}>0\) for \(x \in(c, b]\), the potential \(q(x)\) is a real-valued function continuous in each of the intervals \([a, c)\) and \((c, b]\), and it has a finite limit \(q( c\mp0)\); λ is a complex spectral parameter, \(\alpha_{ij}\), \(\beta^{\pm }_{ij}\), \(\alpha'_{ij}\) (\(i=1,2\) and \(j=0,1\)) are real numbers. We want to emphasize that the boundary value problem studied here differs from standard boundary value problems in that it contains transmission conditions and the eigenvalue parameter appears not only in the differential equation, but also in the boundary conditions. Moreover, the coefficient functions may have discontinuity at one interior point. Naturally, eigenfunctions of this problem may have discontinuity at the one inner point of the considered interval. The problems with transmission conditions have become an important area of research in recent years because of the needs of modern technology, engineering and physics. Many of the mathematical problems encountered in the study of boundary-value-transmission problem cannot be treated with the usual techniques within the standard framework of a boundary value problem (see [3]). Note that some special cases of this problem arise after an application of the method of separation of variables to a varied assortment of physical problems. For example, some boundary value problems with transmission conditions arise in heat and mass transfer problems [4], in vibrating string problems when the string is loaded additionally with point masses [5], in diffraction problems [6]. Such properties as isomorphism, coerciveness with respect to the spectral parameter, completeness and Abel bases of a system of root functions of similar boundary value problems with transmission conditions and its applications to the corresponding initial boundary value problems for parabolic equations were investigated in [7–10]. Also some problems with transmission conditions which arise in mechanics (thermal conduction problems for a thin laminated plate) were studied in [11]. Boundary value problems with transmission conditions were investigated extensively in the recent years (see, for example, [3, 7–9, 12–21]).
The 'basic' solutions and a characteristic function
With a view to constructing the characteristic function \(\omega(\lambda)\), we shall define two basic solutions \(\varphi^{-}(x,\lambda)\) and \(\psi^{-}(x,\lambda )\) on the left interval \([a,c)\) and two basic solutions \(\varphi^{+}(x,\lambda)\) and \(\psi^{+}(x,\lambda)\) on the right interval \((c,b]\) by the following procedure. Let \(\varphi^{-}(x,\lambda)\) and \(\psi^{+}(x,\lambda )\) be the solutions of equation (1) on \([a,c)\) and \((c,b]\) satisfying the initial conditions
$$\begin{aligned}& \varphi^{-}(a,\lambda)=\alpha_{11}-\lambda \alpha' _{11},\qquad \frac{\partial \varphi^{-}(a,\lambda)}{\partial x}=\alpha _{10}-\lambda\alpha' _{10}, \end{aligned}$$
$$\begin{aligned}& \psi^{+}(b,\lambda)=\alpha_{21}+\lambda \alpha' _{21},\qquad \frac{\partial \psi^{+}(b,\lambda)}{\partial x}=\alpha _{20}+\lambda\alpha' _{20}, \end{aligned}$$
respectively. In terms of these solutions, we shall define the other solutions \(\varphi^{+}(x,\lambda)\) and \(\psi^{-}(x,\lambda)\) by the initial conditions
$$\begin{aligned}& \varphi^{+}(c+,\lambda) =\frac{1}{\Delta_{12}}\biggl( \Delta_{23}\varphi^{-}(c-,\lambda )+\Delta_{24} \frac{\partial\varphi^{-}(c-,\lambda)}{\partial x}\biggr), \end{aligned}$$
$$\begin{aligned}& \frac{\partial\varphi^{+}(c+,\lambda)}{\partial x} =\frac{-1}{\Delta_{12}}\biggl(\Delta_{13} \varphi^{-}(c-,\lambda )+\Delta_{14}\frac{\partial\varphi^{-}(c-,\lambda)}{\partial x}\biggr) \end{aligned}$$
$$\begin{aligned}& \psi^{-}(c-,\lambda) =\frac{-1}{\Delta_{34}}\biggl( \Delta_{14}\psi^{+}(c+,\lambda )+\Delta_{24} \frac{\partial\psi^{+}(c+,\lambda)}{\partial x}\biggr), \end{aligned}$$
$$\begin{aligned}& \frac{\partial\psi^{-}(c-,\lambda)}{\partial x} =\frac{1}{\Delta_{34}}\biggl( \Delta_{13}\psi^{+}(c+,\lambda )+\Delta_{23} \frac{\partial\psi^{+}(c+,\lambda)}{\partial x}\biggr), \end{aligned}$$
respectively, where \(\Delta_{ij}\) (\(1\leq i< j \leq4\)) denotes the determinant of the ith and jth columns of the matrix
$$T= \left [\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \beta^{+}_{10} & \beta^{+}_{11} & \beta^{-}_{10} & \beta^{-}_{11} \\ \beta^{+}_{20} & \beta^{+}_{21} & \beta^{-}_{20} & \beta^{-}_{21} \end{array} \right ]. $$
The existence and uniqueness of these solutions follow from the well-known existence and uniqueness theorem of ordinary differential equation theory. Moreover, by applying the method of [12], we can prove that each of these solutions is an entire function of the parameter \(\lambda\in\mathbb{C}\) for each fixed x. Taking into account (7)-(10) and the fact that the Wronskians \(\omega^{\pm}(\lambda):=W[\varphi^{\pm}(x,\lambda),\psi^{\pm }(x,\lambda)]\) are independent of variable x, we have
$$\begin{aligned} \omega^{+}(\lambda) =&\varphi^{+}(c+,\lambda) \frac{\partial\psi ^{+}(c+,\lambda)}{\partial x}-\frac{\partial\varphi ^{+}(c+,\lambda)}{\partial x}\psi^{+}(c+,\lambda) \\ =&\frac{\Delta_{34}}{\Delta_{12}}\biggl(\varphi^{-}(c-,\lambda)\frac {\partial\psi ^{-}(c-,\lambda)}{\partial x}- \frac{\partial\varphi ^{-}(c-,\lambda)}{\partial x}\psi^{-}(c-,\lambda)\biggr) \\ =&\frac{\Delta_{34}}{\Delta_{12}} \omega^{-}(\lambda). \end{aligned}$$
It is convenient to define the characteristic function \(\omega(\lambda )\) as
$$\omega(\lambda):= \Delta_{34} \omega^{-}(\lambda) = \Delta_{12} \omega ^{+}(\lambda). $$
Obviously, \(\omega(\lambda)\) is an entire function. By applying the technique of [12], we can prove that there are infinitely many eigenvalues \(\lambda_{n}\), \(n=1,2,\ldots\) , of problem (1)-(4) which coincide with zeros of the characteristic function \(\omega(\lambda)\).
Operator treatment in a modified Hilbert space
To analyze the spectrum of BVTP (1)-(4), we shall construct an adequate Hilbert space and define a symmetric linear operator in it in such a way that the considered problem can be interpreted as the eigenvalue problem of this operator. For this we assume that
$$\begin{aligned}& \Delta_{12}>0, \qquad \Delta_{34}>0, \\& \theta_{1}= \left [\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \alpha_{11} & \alpha_{10} \\ \alpha'_{11} & \alpha'_{10} \end{array} \right ]>0, \qquad \theta_{2}= \left [\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \alpha_{21} & \alpha_{20} \\ \alpha'_{21} & \alpha'_{20} \end{array} \right ]>0 \end{aligned}$$
and introduce modified inner products on the direct sum spaces \(\mathcal{H}_{1}= L_{2}[a,c)\oplus L_{2}(c,b]\) and \(\mathcal {H}=\mathcal{H}_{1}\oplus C^{2}\) by
$$ [f,g]_{\mathcal{H}_{1}}:=\frac{\Delta_{34}}{p^{-}} \int_{a}^{c-} f(x)\overline{g(x)}\, dx + \frac{\Delta_{12}}{p^{+}} \int_{c+}^{b} f(x)\overline{g(x)}\, dx $$
$$ [F,G]_{\mathcal{H}}:=[f,g]_{\mathcal{H}_{1}} + \frac{\Delta_{34}}{\theta_{1}}f_{1}\overline{g}_{1}+ \frac{\Delta _{12}}{\theta_{2}}f_{2}\overline{g}_{2} $$
for \(F= ( f(x), f_{1}, f_{2} ), G= ( g(x), g_{1}, g_{2} )\in\mathcal{H}\), respectively. Obviously, each of these inner products is equivalent to the standard inner products of the Hilbert spaces \(L_{2}[a,c)\oplus L_{2}(c,b]\) and \(L_{2}[a,c)\oplus L_{2}(c,b]\oplus C^{2}\), respectively, so \((\mathcal{H}, [\cdot,\cdot]_{\mathcal{H}})\) and \((\mathcal{H}_{1}, [\cdot,\cdot]_{\mathcal{H}_{1}})\) are also Hilbert spaces. Let us now define the boundary functionals
$$\begin{aligned}& B_{a}[f]:= \alpha_{10}f(a)-\alpha_{11}f'(a), \qquad B'_{a}[f]:= \alpha'_{10}f(a)- \alpha'_{11}f'(a), \\& B_{b}[f]:= \alpha_{20}f(b)-\alpha_{21}f'(b), \qquad B'_{b}[f]:= \alpha'_{20}f(b)- \alpha'_{21}f'(b) \end{aligned}$$
and construct the operator \(\mathcal{L}:\mathcal{H}\rightarrow \mathcal{H}\) with the domain
$$\begin{aligned} \operatorname{dom}(\mathcal{L}) :=& \bigl\{ F=\bigl(f(x), f_{1}, f_{2}\bigr):f(x), f'(x) \in AC_{\mathrm{loc}}(a, c)\cap AC_{\mathrm{loc}}(c, b), \\ &\mbox{and has a finite limits } f(c\mp0)\mbox{ and }f'(c\mp0), \ell(f) \in L_{2}[a,b], \\ &V_{1}(f)=V_{2}(f)=0, f_{1}= B'_{a}[f], f_{2}=-B'_{b}[f] \bigr\} \end{aligned}$$
and action low
$$\mathcal{L}\bigl(f(x),B'_{a}[f], -B'_{b}[f] \bigr)=\bigl(\ell(f), B_{a}[f], B_{b}[f]\bigr). $$
Then problem (1)-(4) can be written in the operator equation form as follows:
$$\mathcal{L}F=\lambda F,\quad F=\bigl(f(x), B'_{a}[f], -B'_{b}[f]\bigr) \in \operatorname{dom}(\mathcal{L}) $$
in the Hilbert space ℋ.
Theorem 1
The linear operator ℒ is symmetric.
By applying the method of [12] it is not difficult to show that \(\operatorname{dom}(\mathcal{L})\) is dense in the Hilbert space ℋ. Now, let \(F=(f(x),B'_{a}[f], -B'_{b}[f]),G=(g(x),B'_{a}[g], -B'_{b}[g]) \in \operatorname{dom}(\mathcal{L})\). By partial integration we have
$$\begin{aligned}{} [\mathcal{L}F,G]_{\mathcal{H}}-[F,\mathcal{L}G]_{\mathcal{H}} =& \Delta_{34} W(f, \overline{g};c-) - \Delta_{34} W(f, \overline{g};a) \\ &{}+\Delta_{12} W(f,\overline{g};b) - \Delta_{12} W(f, \overline{g};c+) \\ &{}+\frac{\Delta_{34}}{\theta_{1}}\bigl(B_{a}[f]\overline {B'_{a}[g]}-B'_{a}[f] \overline{B_{a}[g]}\bigr) \\ &{}+ \frac{\Delta_{12}}{\theta_{2}}\bigl(B'_{b}[f]\overline {B_{b}[g]}-B_{b}[f]\overline{B'_{b}[g]} \bigr), \end{aligned}$$
where, as usual, \(W(f, \overline{g};x)\) denotes the Wronskians of the functions f and \(\overline{g}\). From the definitions of the boundary functionals \(B_{a}\) and \(B_{b}\) we get that
$$\begin{aligned}& B_{a}[f]\overline {B'_{a}[g]}-B'_{a}[f] \overline{B_{a}[g]}=\theta_{1} W(f,\overline{g};a), \end{aligned}$$
$$\begin{aligned}& B'_{b}[f]\overline{B_{b}[g]}-B_{b}[f] \overline{B'_{b}[g]}=-\theta_{2} W(f, \overline{g};b). \end{aligned}$$
Further, taking in view the definition of ℒ and applying the initial conditions (5)-(10), we derive that
$$ W(f, \overline{g};c-) =\frac{ \Delta _{12}}{\Delta_{34}} W(f, \overline{g};c+). $$
Finally, substituting (14), (15) and (16) in (13), we have
$$[\mathcal{L}F,G]_{\mathcal{H}}=[F,\mathcal{L}G]_{\mathcal{H}} \quad \text {for every } F,G \in \operatorname{dom}(\mathcal{L}), $$
so the operator ℒ is symmetric in ℋ. The proof is complete. □
Corollary 1
All the eigenvalues of problem (1)-(4) are real.
If \(f(x)\) and \(g(x)\) are eigenfunctions corresponding to distinct eigenvalues, then they are 'orthogonal' in the sense of the equality
$$ [f,g]_{\mathcal{H}_{1}} + \frac{\Delta_{34}}{\theta_{1}}{}B'_{a}[f]B'_{a}[g] +\frac{\Delta_{12}}{\theta_{2}}{}B'_{b}[f]B'_{b}[g] =0, $$
where \(F=(f(x),B'_{a}[f], -B'_{b}[f]),G=(g(x),B'_{a}[g], -B'_{b}[g]) \in \operatorname{dom}(\mathcal{L})\).
Solvability of the corresponding inhomogeneous problem
Now, let \(\lambda\in C\) not be an eigenvalue of ℒ. Consider the operator equation
$$ (\lambda I-\mathcal{L})Y=U $$
for arbitrary \(U=(u(x),u_{1},u_{2}) \in\mathcal{H}\). This operator equation is equivalent to the following inhomogeneous BVTP:
$$\begin{aligned}& (\lambda-\ell)y(x)= u(x),\quad x \in[a,c)\cup(c,b], \end{aligned}$$
$$\begin{aligned}& V_{3}(y)=V_{4}(y)=0,\qquad \lambda B'_{a}[y]-B_{a}[y]= u_{1},\qquad - \lambda B'_{b}[y]-B_{b}[y]=u_{2}. \end{aligned}$$
We shall search the resolvent function of this BVTP in the form
$$ Y(x,\lambda)=\left \{ \begin{array}{l@{\quad}l} d_{11}(x,\lambda)\varphi^{-}(x,\lambda)+d_{12}(x,\lambda)\psi ^{-}(x,\lambda)& \mbox{for }x\in [ a,c ), \\ d_{21}(x,\lambda)\varphi^{+}(x,\lambda)+d_{22}(x,\lambda)\psi ^{+}(x,\lambda)& \mbox{for }x\in ( c,b ],\end{array} \right . $$
where the functions \(d_{11}(x,\lambda)\), \(d_{12}(x,\lambda)\) are the solutions of the system of equations
$$ \left \{ \begin{array}{l} \frac{\partial d_{11}(x,\lambda)}{\partial x}\varphi^{-}(x,\lambda)+\frac{\partial d_{12}(x,\lambda )}{\partial x}\psi^{-}(x,\lambda)=0, \\ \frac{\partial d_{11}(x,\lambda)}{\partial x}\frac{\partial\varphi^{-}(x,\lambda)}{\partial x}+\frac{\partial d_{12}(x,\lambda )}{\partial x}\frac{\partial\psi^{-}(x,\lambda)}{\partial x}=\frac {u(x)}{p^{-}}\end{array} \right . $$
for \(x\in[ a,c ) \), and \(d_{21}(x,\lambda)\), \(d_{22}(x,\lambda)\) are the solutions of the system of equations
$$ \left \{ \begin{array}{l} \frac{\partial d_{21}(x,\lambda)}{\partial x}\varphi^{+}(x,\lambda)+\frac{\partial d_{22}(x,\lambda)}{\partial x}\psi^{+}(x,\lambda)=0, \\ \frac{\partial d_{21}(x,\lambda)}{\partial x}\frac{\partial\varphi^{+}(x,\lambda)}{\partial x}+\frac{\partial d_{22}(x,\lambda)}{\partial x}\frac{\partial\psi^{+}(x,\lambda)}{\partial x}=\frac{u(x)}{p^{+}}\end{array} \right . $$
for \(x\in ( c,b ] \). Since λ is not an eigenvalue, we have \(\omega^{\pm}(\lambda) \neq0 \). Hence, from (22) and (23) it follows that
$$\begin{aligned}& d_{11}(x,\lambda)=\frac{\Delta_{34}}{p^{-}\omega(\lambda )}\int_{x}^{c-}u(y) \psi^{-}(y,\lambda)\, dy+h_{11}(\lambda), \quad x\in [ a,c ), \\& d_{12}(x,\lambda )=\frac{\Delta_{34}}{p^{-}\omega(\lambda )}\int_{a}^{x}u(y) \varphi^{-}(y,\lambda)\, dy+h_{12}(\lambda),\quad x\in [ a,c ), \\& d_{21}(x,\lambda)=\frac{\Delta_{12}}{p^{+}\omega(\lambda )}\int_{x}^{b}u(y) \psi^{+}(y,\lambda)\, dy+h_{21}(\lambda), \quad x\in( c,b ], \\& d_{22}(x,\lambda )=\frac{\Delta_{12}}{p^{+}\omega(\lambda )}\int_{c+}^{x}u(y) \varphi^{+}(y,\lambda)\, dy+h_{22}(\lambda),\quad x\in ( c,b], \end{aligned}$$
where \(h_{ij}(\lambda)\) (\(i,j=1,2\)) are arbitrary functions of the parameter λ. Substituting this into (21) gives
$$ Y(x,\lambda)=\left \{ \begin{array}{l} \frac{\Delta_{34}\psi^{-}(x,\lambda)}{p^{-}\omega(\lambda)}\int_{a}^{x}\varphi^{-}(y,\lambda)u(y)\, dy + \frac{\Delta_{34}\varphi^{-}(x,\lambda)}{p^{-}\omega(\lambda)}\int_{x}^{c-}\psi^{-}(y,\lambda)u(y)\, dy \\ \quad {}+h_{11}(\lambda)\varphi ^{-}(x,\lambda)+h_{12}(\lambda)\psi^{-}(x,\lambda) \quad \mbox{for }x \in[a,c), \\ \frac{\Delta_{12}\psi^{+}(x,\lambda)}{p^{+}\omega(\lambda)}\int_{c+}^{x}\varphi_{2}(y,\lambda)u(y)\, dy + \frac{\Delta_{12}\varphi^{+}(x,\lambda)}{p^{+}\omega(\lambda)}\int_{x}^{b}\psi^{+}(y,\lambda)u(y)\, dy \\ \quad {}+h_{21}(\lambda)\varphi ^{+}(x,\lambda)+h_{22}(\lambda)\psi^{+}(x,\lambda) \quad \mbox{for }x \in(c,b]. \end{array} \right . $$
By differentiating we have
$$ \frac{\partial Y(x,\lambda)}{\partial x}=\left \{ \begin{array}{l} \frac{\Delta_{34}}{p^{-}\omega(\lambda)} \frac{\partial\psi ^{-}(x,\lambda)}{\partial x}\int_{a}^{x}\varphi^{-}(y,\lambda)u(y)\, dy + \frac{\Delta_{34}}{p^{-}\omega(\lambda)} \frac{\partial\varphi ^{-}(x,\lambda)}{\partial x}\int_{x}^{c-}\psi^{-}(y,\lambda)u(y)\, dy \\ \quad {}+h_{11}(\lambda)\frac{\partial\varphi^{-}(x,\lambda)}{\partial x}+h_{12}(\lambda)\frac{\partial\psi^{-}(x,\lambda)}{\partial x} \quad \mbox{for }x \in[a,c), \\ \frac{\Delta_{12}}{p^{+}\omega(\lambda)} \frac{\partial\psi ^{+}(x,\lambda)}{\partial x}\int_{c+}^{x}\varphi^{+}(y,\lambda)u(y)\, dy + \frac{\Delta_{12}}{p^{+}\omega(\lambda)} \frac{\partial\varphi ^{+}(x,\lambda)}{\partial x}\int_{x}^{b}\psi^{+}(y,\lambda)u(y)\, dy \\ \quad {}+h_{21}(\lambda)\frac{\partial\varphi^{+}(x,\lambda)}{\partial x}+h_{22}(\lambda)\frac{\partial\psi^{+}(x,\lambda)}{\partial x} \quad \mbox{for }x \in(c,b]. \end{array} \right . $$
By using equalities (24), (25) and boundary conditions (20), we can derive that
$$\begin{aligned}& h_{12}(\lambda)=\frac{u_{1}}{\omega ^{-}(\lambda)},\qquad h_{21}( \lambda)=\frac{u_{2}}{\omega^{+}(\lambda)}, \\& h_{11}(\lambda)=\frac{1}{p^{+}\omega^{+}(\lambda)}\int _{c+}^{b}\psi ^{+}(y,\lambda)u(y)\, dy+ \frac{u_{2}}{\omega^{+}(\lambda)} \end{aligned}$$
$$ h_{22}(\lambda)=\frac{1}{p^{-}\omega^{-}(\lambda)}\int_{a}^{c-} \varphi ^{-}(y,\lambda)u(y)\, dy+\frac{u_{1}}{\omega^{-}(\lambda)}. $$
Putting in (24) gives
$$ Y(x,\lambda)=\left \{ \begin{array}{l@{\quad}l} \frac{\Delta_{34}\psi^{-}(x,\lambda)}{p^{-}\omega(\lambda)}\int_{a}^{x}\varphi^{-}(y,\lambda)u(y)\, dy + \frac{\Delta_{34}\varphi^{-}(x,\lambda)}{p^{-}\omega(\lambda)}\int_{x}^{c-}\psi^{-}(y,\lambda)u(y)\, dy \\ \quad {}+\frac{\Delta_{12}\varphi^{-}(x,\lambda)}{\omega(\lambda)}(\frac {1}{p^{+}}\int_{c+}^{b}\psi^{+}(y,\lambda)u(y)\, dy+u_{2}) \\ \quad {}+\frac{\Delta_{34}u_{1} \psi^{-}(x,\lambda)}{\omega(\lambda)}\quad \mbox{for }x \in[a,c), \\ \frac{\Delta_{12}\psi^{+}(x,\lambda)}{p^{+}\omega(\lambda)}\int_{c+}^{x}\varphi^{+}(y,\lambda)u(y)\, dy +\frac{\Delta_{12}\varphi^{+}(x,\lambda)}{p^{+}\omega(\lambda)}\int_{x}^{b}\psi^{+}(y,\lambda)u(y)\, dy \\ \quad {}+ \frac{\Delta_{34}\psi^{+}(x,\lambda)}{\omega(\lambda)}(\frac {1}{p^{-}}\int_{a}^{c-}\varphi^{-}(y,\lambda)u(y)\, dy+u_{1}) \\ \quad {}+\frac{\Delta _{12}u_{2} \varphi^{+}(x,\lambda)}{\omega(\lambda)}\quad \mbox{for }x \in(c,b]. \end{array} \right . $$
Let us introduce the Green's function as
$$ G_{1}(x,y;\lambda)=\left \{ \begin{array}{l@{\quad}l} \frac{\varphi^{-}(x,\lambda)\psi^{-}(y,\lambda)}{\Delta_{34}p^{-}\omega ^{-}(\lambda)}& \mbox{if }x \in[a,c), y \in[a,x), \\ \frac{\psi^{-}(x,\lambda)\varphi^{-}(y,\lambda)}{\Delta_{34}p^{-}\omega ^{-}(\lambda)} &\mbox{if }x \in[a,c), y \in[x,c), \\ \frac{\psi^{-}(x,\lambda)\varphi^{+}(y,\lambda)}{\Delta_{34}p^{-}\omega ^{-}(\lambda)} &\mbox{if }x \in[a,c), y \in(c,b], \\ \frac{\varphi^{+}(x,\lambda)\psi^{-}(y,\lambda)}{\Delta_{12}p^{+}\omega ^{+}(\lambda)} &\mbox{if }x \in(c,b], y \in[a,c), \\ \frac{\varphi^{+}(x,\lambda)\psi^{+}(y,\lambda)}{\Delta_{12}p^{+}\omega ^{+}(\lambda)} &\mbox{if }x \in(c,b], y \in(c,x], \\ \frac{\psi^{+}(x,\lambda) \varphi^{+}(y,\lambda)}{\Delta_{12}p^{+}\omega ^{+}(\lambda)} &\mbox{if }x \in(c,b], y \in[x,b]. \end{array} \right . $$
Then from (26) and (27) it follows that the considered problem (19), (20) has a unique solution given by
$$\begin{aligned} Y(x,\lambda) =& \frac{\Delta_{34}}{p^{-}}\int_{a}^{c-} G_{1}(x,y;\lambda)u(y)\, dy+ \frac{\Delta_{12}}{p^{+}}\int _{c+}^{b} G_{1}(x,y;\lambda)u(y)\, dy \\ &{}+ \Delta_{34}u_{1}\frac{\psi(x,\lambda)}{\omega(\lambda)}+\Delta _{12}u_{2}\frac{\varphi(x,\lambda)}{\omega(\lambda)}. \end{aligned}$$
In fact, we have proved the following theorem.
The resolvent operator can be represented as
$$ (\lambda I-\mathcal{L})^{-1}U(x)=\left ( \begin{array}{@{}c@{}} \int_{a}^{b}G(x,y;\lambda)u(y)\, dy+\Delta_{34}u_{1}\frac{\psi(x,\lambda )}{\omega(\lambda)}+\Delta_{12}u_{2}\frac{\varphi(x,\lambda)}{\omega (\lambda)} \\ B'_{a}[u] \\ -B'_{b}[u] \end{array} \right ) , $$
$$ G(x,y;\lambda)=\left \{ \begin{array}{l@{\quad}l} \frac{\Delta_{34}}{p^{-}}G_{1}(x,y;\lambda) &\textit{if }a < y < c, \\ \frac{\Delta_{12}}{p^{+}}G_{1}(x,y;\lambda)& \textit{if }c < y < b. \end{array} \right . $$
Although the Green's function looks as simple as that of standard Sturm-Liouville problems, it is rather complicated because of the transmission conditions. To illustrate this situation, let us give the following example.
Consider the following simple case of BVTP's (1)-(4) on \([-1,1]\) with \(c=0\):
$$\begin{aligned}& -y^{\prime\prime}(x)=\lambda y(x), \end{aligned}$$
$$\begin{aligned}& y(-1)+\lambda y'(-1)=0, \end{aligned}$$
$$\begin{aligned}& \lambda y(1)+y'(1)=0, \end{aligned}$$
$$\begin{aligned}& \begin{aligned} &y'(0-)=y(0+), \\ &y'(0-)=2y'(0+), \end{aligned} \end{aligned}$$
where λ is a complex spectral parameter. Putting \(\lambda=\mu ^{2}\) we find easily that
$$\begin{aligned}& \varphi^{-}(x,\mu)=\mu^{2}\cos\bigl[\mu(x+1) \bigr]-\frac{1}{\mu}\sin\bigl[\mu(x+1)\bigr], \\& \varphi^{+}(x,\mu)=\biggl(\mu^{2} \cos\mu-\frac{1}{\mu} \sin\mu\biggr)\cos(\mu x) \\& \hphantom{\varphi^{+}(x,\mu)=}{}- \frac{1}{2}\biggl(\mu^{2}\sin\mu+ \frac{1}{\mu}\cos\mu\biggr)\sin(\mu x), \\& \psi^{-}(x,\mu)=(\cos\mu-\mu\sin\mu)\cos(\mu x) \\& \hphantom{\psi^{-}(x,\mu)=}{}+ 2(\sin\mu+ \mu\cos\mu)\sin(\mu x), \\& \psi^{+}(x,\mu)= \cos\bigl[\mu(1-x)\bigr]-\mu\sin\bigl[\mu(1-x)\bigr]. \end{aligned}$$
Using these formulas, we have
$$\begin{aligned} w(\mu) =&\bigl(2\mu^{4}+1\bigr)\cos^{2}\mu-\bigl( \mu^{4}+2\bigr)\sin^{2}\mu \\ &{}+3\bigl(\mu^{3}-\mu \bigr)\sin \mu\cos\mu. \end{aligned}$$
Consequently, the Green's function has the following form:
$$\begin{aligned} G(x,y,\mu) =&\bigl\{ \bigl(2\mu^{4}+1\bigr) \cos^{2}\mu-\bigl(\mu^{4}+2\bigr)\sin^{2}\mu +3 \bigl(\mu^{3}-\mu\bigr)\sin\mu\cos \mu\bigr\} \\ &{}\times \left \{ \begin{array}{l} \{ \mu^{2}\cos[\mu(x+1)]-\frac{1}{\mu}\sin[\mu(x+1)]\}\times \{(\cos\mu-\mu\sin\mu)\cos(\mu y) \\ \quad {}+ 2(\sin\mu+\mu\cos\mu)\sin(\mu y)\},\quad -1\leq x\leq y<0, \\ \{(\cos\mu-\mu\sin\mu)\cos(\mu x)+ 2(\sin\mu+\mu\cos\mu)\sin(\mu x)\} \\ \quad {}\times\{\mu^{2}\cos[\mu(y+1)]-\frac{1}{\mu}\sin[\mu(y+1)]\},\quad -1\leq y\leq x<0, \\ \{(\cos\mu-\mu\sin\mu)\cos(\mu x)+ 2(\sin\mu+\mu\cos\mu)\sin(\mu x)\} \\ \quad {}\times\{(\mu^{2} \cos\mu-\frac{1}{\mu}\sin\mu)\cos(\mu y)- \frac{1}{2}(\mu^{2}\sin\mu+\frac{1}{\mu}\cos\mu)\sin(\mu y)\}, \\ \quad -1 \leq y<0, 0< x\leq1, \\ \{(\mu^{2} \cos\mu-\frac{1}{\mu}\sin\mu)\cos(\mu x)- \frac{1}{2}(\mu^{2}\sin\mu+\frac{1}{\mu}\cos\mu)\sin(\mu x)\} \\ \quad {}\times\{(\cos\mu-\mu\sin\mu)\cos(\mu y)+ 2(\sin\mu+\mu\cos\mu)\sin(\mu y)\}, \\ \quad -1 \leq x<0, 0< y\leq1, \\ \{(\mu^{2} \cos\mu-\frac{1}{\mu}\sin\mu)\cos(\mu x)- \frac{1}{2}(\mu^{2}\sin\mu+\frac{1}{\mu}\cos\mu)\sin(\mu x)\} \\ \quad {}\times\{\cos[\mu(1-y)]-\mu\sin[\mu(1-y)]\},\quad 0< y\leq x\leq1, \\ \{\cos[\mu(1-x)]-\mu\sin[\mu(1-x)]\}\times\{(\mu^{2} \cos\mu-\frac{1}{\mu}\sin\mu)\cos(\mu y) \\ \quad {}- \frac{1}{2}(\mu^{2}\sin\mu+\frac{1}{\mu}\cos\mu)\sin(\mu y)\},\quad 0< x\leq y\leq1. \end{array} \right . \end{aligned}$$
The graph of the Green's function is displayed in Figure 1 and Figure 2 for two different values of the spectral parameter.
The graph of the Green's function \(\pmb{G(x,t,\mu)}\) for \(\pmb{\mu=3}\) .
The graph of the Green's function \(\pmb{G(x,t,\mu)}\) for \(\pmb{\mu=15}\) .
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The authors would like to thank the referees for their valuable comments. This work was supported by the research fund of Gaziosmanpaşa University under Grant No. 2012/126.
Department of Mathematics, Faculty of Arts and Science, Gaziosmanpaşa University, Tokat, 60250, Turkey
Kadriye Aydemir & Oktay S Mukhtarov
Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan
Oktay S Mukhtarov
Kadriye Aydemir
Correspondence to Kadriye Aydemir.
The authors contributed equally to this work. The authors read and approved the final manuscript.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Aydemir, K., Mukhtarov, O.S. Second-order differential operators with interior singularity. Adv Differ Equ 2015, 26 (2015). https://doi.org/10.1186/s13662-015-0360-7
Accepted: 06 January 2015
Sturm-Liouville problems
Green's function
transmission conditions
resolvent operator
3rd International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2014) | CommonCrawl |
Find the expected count and the contribution to the chi-square statistic for the (Group 1, No)
Lewis Harvey 2021-01-16 Answered
Find the expected count and the contribution to the chi-square statistic for the (Group 1, No) cell in the two-way table below.
\(\begin{array}{|c|c|c|}\hline&\text{Yes}&\text{No}&\text{Total}\\\hline\text{Group 1} &56 & 42 & 98\\ \hline \ \text{Group 2}&135&67&202 \\ \hline \text{Group 3}&66&23&89 \\ \hline \text{Total}&257&132&389 \\ \hline \end{array}\)
Round your answer for the excepted count to one decimal place, and your answer for the contribution to the chi-square statistic to three decimal places.
Expected count =?
contribution to the chi-square statistic = ?
Want to know more about Two-way tables?
joshyoung05M
Observed values with marginal totals:
Expected values:
Expected values are the product of the column and row total divided by the table total. \(\begin{array}{|c|c|c|}\hline&\text{Yes}&\text{No}\\\hline\text{Group 1} &\frac{257\times98}{389}=64.7 & \frac{132\times98}{389}=33.2\\ \hline \text{Group 2}&\frac{257\times202}{389}=133.5&\frac{132\times202}{389}=68.5 \\ \hline \text{Group 3}&\frac{257\times89}{389}=58.8&\frac{132\times89}{389}=30.2 \\ \hline \end{array}\)
The expected count for (Group 1, No) is as follows:
\(\text{Expected count}=\frac{132 \times 98}{389}\)
\(=33.2545\)
\(\approx33.3\) The contribution to test statistic is as follows: \(\text{contribution to chi-square}=\frac{(O-E)^2}{E}\)
\(=\frac{(42-33.2545)^2}{33.2545}\)
\(=2.29995\)
\(\approx2.300\)
The General Social Survey (GSS) asked a random sample of adults their opinion about whether astrology is very scientific, sort of scientific, or not at all scientific. Here is a two-way table of counts for people in the sample who had three levels of higher education:
\(\begin{array}{l|c|c|c|c} & \text { Associate's } & \text { Bachelor's } & \text { Master's } & \text { Total } \\ \hline \begin{array}{l} \text { Not al all } \\ \text { scientific } \end{array} & 169 & 256 & 114 & 539 \\ \hline \begin{array}{l} \text { Very or sort } \\ \text { of scientific } \end{array} & 65 & 65 & 18 & 148 \\ \hline \text { Total } & 234 & 321 & 132 & 687 \end{array}\)
State appropriate hypotheses for performing a chi-square test for independence in this setting.
A group of students were polled to find out how many were planning to major in a scientific field of study in college. The results of the poll are shown in the two-way table. Which of the following statements is true? A. Three hundred sixty students were polled in all. B. A student in the senior class is more likely to be planning on a scientific major than a nonscientific major. C. A student planning on a scientific major is more likely to be a junior than a senior. D. More seniors than juniors plan to enter a scientific field of study. \($$\begin{matrix} \text{Majoring in a science field}\ \quad & \quad & \text{Yes} & \text{No}\ \text{Class} & \text{Junior} & \text{150} & \text{210}\ \quad & \text{Senior} & \text{112} & \text{200}\ \end{matrix}$$\)
A group of children and adults were polled about whether they watch a particular TV show. The survey results, showing the joint relative frequencies and marginal relative frequencies, are shown in the two-way table.
\(\begin{array}{c|c|c| c} & Yes & No & Total \\ \hline Children & 0.3 & 0.4 & 0.7 \\ Adults & 0.25& x & 0.3 \\ \hline Total & 0.55 & 0.45 & 1 \end{array}\)
What is the value of x?
Use the two-way table of data from another student survey to answer the following question.
\(\begin{array}{|c|cc|c|} \hline &Like\ Aerobic&Exercise\\ \hline Like\ Weight\ Lifting & Yes&No&Total \\ \hline Yes& 7&14&21\\ \hline No& 12&7&19\\ \hline Total& 29&21&40\\ \hline \end{array}\)
Find the conditional relative frequency that a student likes to lift weights, given that the student likes aerobics.
Find the expected count and the contribution to the chi-square statistic for the (Group 1, Yes) cell in the two-way table below.
\(\begin{array}{|c|c|c|}\hline&\text{Yes}&\text{No}&\text{Total}\\\hline\text{Group 1} &710 & 277 & 987\\ \hline\text{Group 2}& 1175 & 323&1498\\\hline \ \text{Total}&1885&600&2485 \\ \hline \end{array}\)
Expected count=?
contribution to the chi-square statistic=?
A group of children and adults were polled about whether they watch a particular TV show. The survey results, showing the joint relative frequencies and marginal relative frequencies, are shown in the two-way table. What is the value of x? \(\begin{matrix} & \text{Yes} & \text{No} & \text{Total}\\ \text{Children} & \text{0.3} & \text{0.4} & \text{0.7}\\ \text{Adults} & \text{0.25} & \text{x} & \text{0.3}\\ \text{Total} & \text{0.55} & \text{0.45} & \text{1}\\ \end{matrix}\)
Raul is conducting a survey for the school news blog. He surveys 200 senior-class students and finds that 78 students have access to a car on weekends, 54 students have regular chores assigned at home, and 80 students neither have access to a car, nor have regular chores to do. Raul is having a hard time putting the data into a two-way table. Car No car Chores No chores a. Copy and complete the two-way table to help Raul figure out the number of students in each situation. b. Two-way tables often include row and column totals. Add row and column totals to your two-way table. Is there an association between car privileges and having regular chores for these students? Explain your answer in the context of the problem.
Data distributions | CommonCrawl |
Cantor–Dedekind axiom
In mathematical logic, the Cantor–Dedekind axiom is the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one-to-one correspondence between real numbers and points on a line.
This axiom is the cornerstone of analytic geometry. The Cartesian coordinate system developed by René Descartes implicitly assumes this axiom by blending the distinct concepts of real number system with the geometric line or plane into a conceptual metaphor. This is sometimes referred to as the real number line blend.[1]
A consequence of this axiom is that Alfred Tarski's proof of the decidability of first-order theories of the real numbers could be seen as an algorithm to solve any first-order problem in Euclidean geometry.
However, with the development of axiom systems for synthetic geometry that filled in the axioms that Euclid implicitly assumed, and the development of modern notions of the real numbers, both the Euclidean line and the Reals are complete Archimedean fields, thus canonically isomorphic, and the Cantor–Dedekind "axiom" is actually a theorem.
Notes
1. George Lakoff and Rafael E. Núñez (2000). Where Mathematics Comes From: How the embodied mind brings mathematics into being. Basic Books. ISBN 0-465-03770-4.
References
• Ehrlich, P. (1994). "General introduction". Real Numbers, Generalizations of the Reals, and Theories of Continua, vi–xxxii. Edited by P. Ehrlich, Kluwer Academic Publishers, Dordrecht
• Bruce E. Meserve (1953) Fundamental Concepts of Algebra, p. 32, at Google Books
• B.E. Meserve (1955) Fundamental Concepts of Geometry, p. 86, at Google Books
Real numbers
• 0.999...
• Absolute difference
• Cantor set
• Cantor–Dedekind axiom
• Completeness
• Construction
• Decidability of first-order theories
• Extended real number line
• Gregory number
• Irrational number
• Normal number
• Rational number
• Rational zeta series
• Real coordinate space
• Real line
• Tarski axiomatization
• Vitali set
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16/01/2020 - 12:00 Soft Constraints for Data Management Batya Kenig
Speaker: Batya Kenig
Date : 16/01/2020 - 12:00
Title : Soft Constraints for Data Management
Integrity constraints such as functional dependencies (FD), and multivalued dependencies (MVD)
are fundamental in database schema design, query optimization, and for enforcing data integrity.
Current data intensive applications such as ML algorithms process observational data that is often
unnormalized, inconsistent, erroneous and noisy. In these applications, quite often the constraints
need to be inferred from the data, and are not required to hold exactly, but it suffices if they hold
only to a certain degree.
In this work, we use information theory to quantify the degree of satisfaction of a constraint,
giving rise to two major challenges that I will cover in this talk: the implication problem for soft constraints, and discovering soft constraints in data. The implication problem for soft constraints asks whether a set of constraints (antecedents) that hold in the data to a large degree imply a high degree of satisfaction of another constraint (consequent). The implication problem has been investigated in both the Database and AI literature, but only under the assumption that all constraints hold exactly; our work extends this to the case of soft constraints.
Next, we address the problem of mining soft constraints from data, and present an algorithm for discovering complete schemas from data. The algorithm employs pruning techniques that take advantage of the properties of the information-theoretic measures associated with the constraints, and allow it to scale to datasets with up to 1M tuples, and up to 30 attributes.
Based on joint work with Dan Suciu, Pranay Mundra, Guna Prasad, and Babak Salimi (to be presented at ICDT 2020 and SIGMOD 2020)
09/01/2020 - 12:00 VBNets: Learning Entity Representations via Variational Bayesian Networks Oren Barkan
Speaker: Oren Barkan
Title : VBNets: Learning Entity Representations via Variational Bayesian Networks
In this talk, we present Variational Bayesian Networks (VBNets) - A novel scalable Bayesian hierarchical model that utilizes both implicit and explicit relations for learning entity representations. VBNets are designed for Microsoft Store and Xbox services that handle around a billion users worldwide. Different from point estimate solutions that map entities to vectors and are usually over confident, VBNets map entities to densities and hence model uncertainty. VBNets are based on analytical approximations of the intractable entities' posterior and the posterior predictive distribution of the data. We demonstrate the effectiveness of VBNets on linguistic, recommendations, and medical informatics tasks, where it is shown to outperform other alternative methods that facilitate Bayesian modeling with or without semantic priors. In addition, we show that VBNets produce superior representations for rare words and cold items.
If time permits, we will give a brief overview of several recent deep learning works in the domains of deep neural attention mechanisms, multiview representation learning and inverse problems with applications for natural language understanding, recommender systems, computer vision, sound synthesis and biometrics.
08/01/2020 - 13:00 Verification of distributed protocols using decidable logics ODED PADON
Speaker: ODED PADON
Title : Verification of distributed protocols using decidable logics
Formal verification of infinite-state systems, and distributed systems in particular, is a long standing research goal. I will describe a series of works that develop a methodology for verifying distributed algorithms and systems using decidable logics, employing decomposition, abstraction, and user interaction. This methodology is implemented in an open-source tool, and has resulted in the first mechanized proofs of several important distributed protocols. I will also describe a novel approach to the problem of invariant inference based on a newly formalized duality between reachability and mathematical induction. The duality leads to a primal-dual search algorithm, and a prototype implementation already handles challenging examples that other state-of-the-art techniques cannot handle. I will briefly describe several other works, including a new optimization technique for deep learning computations that achieves significant speedups relative to existing deep learning frameworks.
02/01/2020 - 12:00 Recovering nonlinear dynamics via Koopman Theory OMRI AZENCOT
Speaker: OMRI AZENCOT
Title : Recovering nonlinear dynamics via Koopman Theory
In this talk, we will present the main building blocks that allow for an interpretative analysis and process of dynamical systems data. The overarching theme of our work is based on the theory of Bernard Koopman (1931). Key to our approach is the interplay between the underlying dynamics and its embedding onto an infinite-dimensional space of scalar functions. This embedding encodes a potentially nonlinear system via a linear object known as the Koopman Operator. Koopman's perspective is advantageous since it involves the manipulation of linear operators which can be done efficiently. Moreover, algebraic properties of the Koopman matrix are directly associated with dynamical features of the system, which in turn are linked to high-level questions. Overall, the combination of Koopman Theory with novel dimensionality reduction techniques and data science approaches leads to a highly powerful framework.
To demonstrate the effectiveness of our approach, we consider several challenging problems in various fields. In geometry processing, we show how optimizing for a spectral basis and a Koopman operator (a functional map) leads to improved shape matching results. In fluid mechanics, we develop a provably convergent ADMM scheme for computing Koopman operators that admits state-of-the-art results on data with high levels of sensor noise. In image processing, our methodology generates a discrete transform of a nonlinear flow as faster as two orders of magnitude when compared to existing approaches. Finally, we construct a novel framework for recovering dynamics from questionnaire data that arise in the social sciences.
26/12/2019 - 12:00 Optimal Euclidean metric compression TAL WAGNER
Speaker: TAL WAGNER
Title : Optimal Euclidean metric compression
In the metric compression problem, we are given n points in a metric space, and the goal is to construct a compact representation (sketch) of the points, such that the distance between every pair can be approximately recovered from the sketch, up to a small distortion of (1 +/- epsilon). Such sketches are widely used for fast nearest neighbor search in high-dimensional Euclidean spaces.
We give a new algorithm for sketching Euclidean metric spaces. It provably achieves the optimal compression bound, improving over the classical dimension reduction theorem of Johnson and Lindenstrauss. In particular, while the latter theorem represents each point by log(n)/epsilon^2 *coordinates* (each containing a multi-bit number), we show that log(n)/epsilon^2 *bits* are both sufficient and necessary. Empirically, our algorithm either matches or improves over state-of-the-art heuristics.
Based on joint works with Piotr Indyk and Ilya Razenshteyn.
19/12/2019 - 12:00 Beyond Worst Case In Machine Learning: The Oracle Model ALON GONEN
Speaker: ALON GONEN
Title : Beyond Worst Case In Machine Learning: The Oracle Model
In recent years there has been an increasing gap between the success of machine learning algorithms and our ability to explain their success theoretically.
Namely, many of the problems that are solved to a satisfactory degree of precision are computationally hard in the worst case. Fortunately, there are often reasonable assumptions which help us to get around these worst-case impediments and allow us to rigorously analyze heuristics that are used in practice.
In this talk I will advocate a complementary approach, where instead of explicitly characterizing some desired "niceness" properties of the data, we assume access to an optimization oracle that solves a relatively simpler task. This allows us to identify the sources of hardness and extend our theoretical understanding to new domains. Furthermore we will show that seemingly innocents (and arguably justifiable) modifications to the oracle can lead to tractable reductions and even to bypass hardness results.
We demonstrate these ideas using the following results: i) An efficient algorithm for non-convex online learning using an optimization oracle. b) A faster boosting algorithm using a "simple" weak learner. iii) An efficient reduction from online to private learning.
Joint works with Naman Agarwal, Noga Alon, Elad Hazan, and Shay Moran.
12/12/2019 - 12:00 Some new approaches to the heavy hitters problem Jelani Nelson
Speaker: Jelani Nelson
Title : Some new approaches to the heavy hitters problem
In the 'frequent items' problem one sees a sequence of items in a stream (e.g. a stream of words coming into a search query engine like Google) and wants to report a small list of items containing all frequent items. In the 'change detection' problem one sees two streams, say one from yesterday and one from today, and wants to report a small list of items containing all those whose frequencies changed significantly. For both of these problems, we would like algorithms that use memory substantially sublinear in the length of the stream.
We describe new state-of-the-art solutions to both problems. For the former, we make use of chaining methods to control the suprema of Rademacher processes to develop an algorithm BPTree with provably near-optimal memory consumption. For the latter, we develop the first
algorithm to simultaneously achieve asymptotically optimal space, fast query and update time, and high success probability (over the random coins flipped by the algorithm) by reducing the problem to a certain graph partitioning problem, which we then give a new algorithm for.
Based on joint works with Vladimir Braverman, Stephen Chestnut, Nikita Ivkin, Kasper Green Larsen, Huy Le Nguyen, Mikkel Thorup, Zhengyu Wang, and David P. Woodruff.
05/12/2019 - 12:00 Breaking (and Fixing) Real World Crypto Eyal Ronen
Speaker: Eyal Ronen
Title : Breaking (and Fixing) Real World Crypto
In recent years, new forms of communication between people and devices have revolutionized our daily lives. The Internet has become the leading platform for human interaction (e.g., social networks), commerce, information, and also control of physical devices (e.g., Internet of Things). This new connectivity creates new security and privacy risks for individual users and organizations. It also increases the complexity and diversity of the different security and cryptographic solutions we need to protect against increasingly sophisticated and motivated attackers. Designing and implementing a secure system is a very elusive process. One needs to clearly identify the security targets (e.g., maintaining the confidentiality of the messages or preventing access from non-authorized entities) as well as the adversarial capabilities.
In this talk, I will show how we can combine cryptanalytic techniques with various side-channels to break the security guarantees of real-world implementations of cryptographic protocols, and how novel solutions can help mitigate the root causes of these vulnerabilities.
28/11/2019 - 12:00 Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling Przemek Uznański
Speaker: Przemek Uznański
Title : Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling
A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. An important class of distance labeling schemes is that of hub labelings, where a node v∈G stores its distance to the so-called hubs S(v)⊆V, chosen so that for any u,v∈V there is w∈S(u)∩S(v) belonging to some shortest uv path. Notice that for most existing graph classes, the best distance labelling constructions existing use at some point a hub labeling scheme at least as a key building block. Our interest lies in hub labelings of sparse graphs, i.e., those with |E(G)|=O(n), for which we show a lowerbound of n/2^O(√logn) for the average size of the hubsets. Additionally, we show a hub-labeling construction for sparse graphs of average size O(n/RS(n)^c) for some 0<c<1, where RS(n) is the so-called Ruzsa-Szemerédi function, linked to structure of induced matchings in dense graphs. This implies that further improving the lower bound on hub labeling size to n/2^(logn)^o(1) would require a breakthrough in the study of lower bounds on RS(n), which have resisted substantial improvement in the last 70 years. For general distance labeling of sparse graphs, we show a lowerbound of 1/2O(√logn) * SumIndex(n), where SumIndex(n) is the communication complexity of the Sum-Index problem over Z_n. Our results suggest that the best achievable hub-label size and distance-label size in sparse graphs may be Θ(n/2^(logn)^c) for some 0<c<1.
Joint work with Adrian Kosowski and Laurent Viennot (presented at PODC 2019).
21/11/2019 - 12:00 Benefits and Challenges of combining Deep Learning and Search EDO LIBERTY
Speaker: EDO LIBERTY
Title : Benefits and Challenges of combining Deep Learning and Search
Deep Learning has fundamentally changed Information Retrieval. Traditional search retrieves text documents based on text queries. Today, however, companies want to search for images, videos, customers, jobs, shopping catalog items, friends, places, and many more. For such searches, Deep Learning models provide the most accurate results. Unfortunately, deploying and serving machine learning models at massive scale is still a herculean effort even for highly tech-heavy companies. This talk will survey some of those challenges. We will also introduce a new cloud-managed service by HyperCube which serves such real-time workloads.
12/11/2019 - 09:45 Sliding window property testing for regular languages Tatiana Starikovskaya
Speaker: Tatiana Starikovskaya
Title : Sliding window property testing for regular languages
In this talk, we discuss the problem of recognizing regular languages in a variant of the streaming model of computation, called the sliding window model. In this model, we are given a size of the sliding window n and a stream of symbols. At each time instant, we must decide whether the suffix of length n of the current stream ("the active window") belongs to a given regular language.
Recent works showed that the space complexity of an optimal deterministic sliding window algorithm for this problem is either constant, logarithmic, or linear in the window size n; and either constant, double logarithmic, logarithmic, or linear in the randomized setting.
In this work, we make an important step forward and combine the sliding window model with the property testing setting, which results in ultra-efficient algorithms for all regular languages. Informally, a sliding window property tester must accept the active window if it belongs to the language and reject if it is far from the language. We consider deterministic and randomized sliding window property testers with one-sided and two-sided errors. In particular, we show that for any regular language, there is a deterministic sliding window property tester that uses logarithmic space and a randomized sliding window property tester with two-sided error that uses constant space.
The talk is based on a joint work with M. Ganardi, D. Hucke, and M. Lohrey accepted to ISAAC 2019.
07/11/2019 - 12:00 בדיקות בזמן ריצה (Run time verification) Prof. Doron Peled
Speaker: Prof. Doron Peled
Title : בדיקות בזמן ריצה (Run time verification)
הנדסת תכנה כוללת שיטות לפיתוח מערכות המאפשרות חלוקת עבודה, יעילות והקטנת מספר התקלות. כתת-תחום של הנדסת תוכנה, "שיטות פורמליות" עוסקת בטכניקות לבדיקת מערכות וניפוי טעויות. לקראת סוף שנות ה- 60 פותחו שיטות להוכחת נכונות של תכנה, כשהמוטיבציה הראשונית נבעה, בין היתר, מפרויקטי החלל, שדרשו אמינות חסרת תקדים למערכות. נושא זה הלך והתפתח עם השנים, כאשר מערכות מחשב חודרות לתחומים רבים כמו רפואה, תחבורה וכולי. השיטה הישנה של בדיקות תכנה שרדה אף היא, ולמרות שהאמינות שהיא מבטיחה הנה פחותה מזאת של הוכחת נכונות, הרי שהיא עדיין יותר ישימה מבחינת כוח חישוב. בשנים האחרונות עולה הפופולריות של בדיקות בזמן ריצה: האפשרות לעקוב אחרי ביצוע המערכת ולהתריע או אף למנוע תקלות, תוך כדי ריצה. זוהי טכניקה שנמצאת בין בדיקות תכנה, שכן אינה מבטיחה מראש הוכחה מלאה של נכונות התכנה, ובין בדיקות תכנה, שכן היא בודקת את המערכת, בזמן ריצה, מול מפרט מורכב.
בהרצאה זאת אתאר את התחום של בדיקות בזמן ריצה, את המפרט הניתן לתכנה ואת הכלים המשמשים לבדיקות כאלו. אתאר גם כלי לבדיקות בזמן ריצה שפותח בשיתוף פעולה עם NASA ומשמש אף לאנליזה של נתונים המגיעים מגששית כוכב מארס של סוכנות החלל.
20/06/2019 - 09:25 Approximate Similarity Search Under Edit Distance Using Locality-Sensitive Hashing Samuel McCauley
Speaker: Samuel McCauley
Title : Approximate Similarity Search Under Edit Distance Using Locality-Sensitive Hashing
13/06/2019 - 12:00 The Online Event-Detection Problem Shikha Singh
Speaker: Shikha Singh
Title : The Online Event-Detection Problem
Given a stream of N elements, a f-heavy hitter is an item that occurs at least fN times in S. The problem of finding heavy-hitters has been extensively studied in the streaming literature. In this talk, I will present a related problem. We say that there is a f-event at time t if an element occurs exactly fN times in the stream of elements till time t. Thus, for each f-heavy hitter there is a single f-event which occurs when its count reaches the reporting threshold fN. We define the online event-detection problem (OEDP) as: given f and a stream S, report all f-events as soon as they occur.
Many real-world monitoring systems demand event detection where all events must be reported (no false negatives), in a timely manner, with no non-events reported (no false positives), and a low reporting threshold. As a result, the OEDP requires a large amount of space (Omega(N) words) and is not solvable in the streaming model or via standard sampling-based approaches. I will focus on cache-efficient algorithms in the external-memory model and present provide algorithms for the OEDP that are within a log factor of optimal.
06/06/2019 - 12:00 Network Coding Gaps for Completion Times of Multiple Unicasts David Wajc
Speaker: David Wajc
Title : Network Coding Gaps for Completion Times of Multiple Unicasts
Arguably the most common network communication problem is multiple-unicasts: Distinct packets at different nodes in a network need to be delivered to a destination specific to each packet, as fast as possible.
The famous multiple-unicast conjecture posits that, for this natural problem, there is no performance gap between routing and network coding, at least in terms of throughput. We study the same network coding gap, but in terms of completion time.
While throughput corresponds to the completion time for asymptotically-large transmissions, we look at completion times of multiple unicasts for arbitrary amounts of data. We develop nearly-matching upper and lower bounds. In particular, we prove that the network coding gap for the completion time of k unicasts is at most polylogarithmic in k, and there exist instances of k unicasts for which this coding gap is polylogarithmic in k.
This talk is based on joint work with Bernhard Haeupler and Goran Zuzic.
30/05/2019 - 12:00 Osprey: Weak Supervision of Imbalanced Extraction Problems without Code Eran Bringer
Speaker: Eran Bringer
Title : Osprey: Weak Supervision of Imbalanced Extraction Problems without Code
Supervised methods are commonly used for machine-learning based applications but require expensive labeled dataset creation and maintenance. Increasingly, practitioners employ weak supervision approaches, where training labels are programmatically generated in higher-level but noisier ways. However, these approaches require domain experts with programming skills. Additionally, highly imbalanced data is often a significant practical challenge for these approaches. In this work, we propose Osprey, a weak-supervision system suited for highly-imbalanced data, built on top of the Snorkel framework. In order to support non-coders, the programmatic labeling is decoupled into a code layer and a configuration one. This decoupling enables a rapid development of end-to-end systems by encoding the business logic into the configuration layer. We apply the resulting system on highly-imbalanced (0.05% positive) social-media data using a synthetic data rebalancing and augmentation approach, and a novel technique of ensembling a generative model over the legacy rules with a learned discriminative model. We demonstrate how an existing rule-based model can be transformed easily into a weakly-supervised one. For 3 relation extraction applications based on real-world deployments at Intel, we show that with a fraction of the cost, we achieve gains of 18.5 precision points and 28.5 coverage points over prior traditionally supervised and rules-based approaches.
23/05/2019 - 12:00 Truly Sub-quadratic Time Constant Factor Approximation of Edit Distance Diptarka Chakraborty
Speaker: Diptarka Chakraborty
Title : Truly Sub-quadratic Time Constant Factor Approximation of Edit Distance
Edit distance is a measure of similarity of two strings based on the minimum number of character insertions, deletions, and substitutions required to transform one string into the other. This distance measure finds applications in fields such as computational biology, pattern recognition, text processing, and information retrieval. The edit distance can be computed exactly using a dynamic programming algorithm that runs in quadratic time, which is also known to be almost optimal under SETH assumption [Backurs, Indyk 2015]. Andoni, Krauthgamer and Onak (2010) gave a nearly linear time (randomized) algorithm that approximates edit distance within poly-log(n) approximation factor. In this talk we discuss a randomized algorithm with running time $\tilde{O}(n^{2-2/7})$ that approximates the edit distance within a constant factor.
(based on a joint work with Das, Goldenberg, Koucky and Saks, appeared in FOCS'18)
16/05/2019 - 12:00 String Synchronizing Sets: Sublinear-Time BWT Construction and Optimal LCE Data Structure Tomasz Kociumaka
Speaker: Tomasz Kociumaka
Title : String Synchronizing Sets: Sublinear-Time BWT Construction and Optimal LCE Data Structure
In this talk, I will present the first algorithm that breaks the O(n)-time barrier for BWT construction. Given a binary string of length n, it builds the Burrows–Wheeler transform in O(n / √log n) time and O(n / log n) space. Any further progress in the time complexity of BWT construction would yield faster algorithms for the problem of counting inversions: it would improve upon the state-of-the-art O(m √log m)-time solution by Chan and Pǎtraşcu (SODA 2010).
The new algorithm is based on a concept of string synchronizing sets, originally designed for answering Longest Common Extension (LCE) queries, which is of independent interest. I will also show that how this technique yields a data structure of the optimal size O(n / log n) that answers LCE queries in O(1) time and, furthermore, can be deterministically constructed in the optimal O(n / log n) time.
29/04/2019 - 11:00 Near-optimal Sample Complexity Bounds for Robust Learning of Gaussians Mixtures via Compression Schemes SHAI BEN DAVID
Speaker: SHAI BEN DAVID
Title : Near-optimal Sample Complexity Bounds for Robust Learning of Gaussians Mixtures via Compression Schemes
We prove that Θ(kd2 /ε^2 ) samples are necessary and sufficient for learning a mixture of k Gaussians in R^d , up to error ε in total variation distance. This improves both the known upper bounds and lower bounds for this problem. For mixtures of axis-aligned Gaussians, we show that O(kd/ε^2 ) samples suffice, matching a known lower bound. Moreover, these results hold in the agnostic-learning/robust-estimation setting as well, where the target distribution is only approximately a mixture of Gaussians. The upper bound is shown using a novel technique for distribution learning based on a notion of compression. Any class of distributions that allows such a compression scheme can also be learned with few samples. Moreover, if a class of distributions has such a compression scheme, then so do the classes of products and mixtures of those distributions. The core of our main result is showing that the class of Gaussians in R^d admits a small-sized compression scheme.
(Joint work with Hassan Ashtiani, Nicholas J. A. Harvey, Christopher Liaw, Abbas Mehrabian and Yaniv Plan)
04/04/2019 - 12:00 Unsupervised Learning for Translation across Languages and Images Yedid Hoshen
Speaker: Yedid Hoshen
Title : Unsupervised Learning for Translation across Languages and Images
This talk will describe my past and ongoing work on translating images and words between very different datasets without supervision. Although Humans often do not require supervision to make connections between very different sources of data, this is still difficult for machines. Recently great progress was made by using adversarial training – a powerful yet tricky method. Although adversarial methods have had great success, they have critical failings which significantly limit their breadth of applicability and motivate research into alternative non-adversarial methods. In this talk, I will describe novel non-adversarial methods for unsupervised word translation and for translating images between very different datasets (with Lior Wolf). As image generation models are an important component in our method, I will present a non-adversarial image generation approach, which is often better than current adversarial approaches (with Jitendra Malik).
28/03/2019 - 12:00 Homomorphic Secret Sharing Yuval Ishai
Speaker: Yuval Ishai
Title : Homomorphic Secret Sharing
A homomorphic secret-sharing scheme is a secret-sharing scheme that allows locally mapping shares of a secret x to compact shares of a function of x. The talk will survey the current state of the art on homomorphic secret sharing, covering applications in cryptography and complexity theory, efficient constructions, and open questions.
14/03/2019 - 12:00 Information Theory of Deep Learning - The computational benefits of the hidden layers Naftali Tishby
Speaker: Naftali Tishby
Title : Information Theory of Deep Learning - The computational benefits of the hidden layers
The surprising success of machine learning with deep neural networks poses two fundamental challenges. One is understanding why these multilayer networks work so well on many different artificial intelligence tasks, in some cases close or better than human performance. The second is: what can this success tell us about human intelligence and our biological brain.
Our recent Information Bottleneck theory of Deep Learning provides new insights and answers to the first question. It shows that the layers of deep neural networks achieve the optimal information theoretic tradeoff between training sample size and accuracy, and that this optimality is achieved through the noisy process of stochastic gradient decent. Moreover, it shows that the benefit of the multilayers structure is mainly computational - it exponentially boosts the time of convergence to these optimal representations. In this talk I will address the relevance of of these findings to the emergence of hierarchies in deep neural networks and in biological brains.
07/03/2019 - 12:00 Machine Learning, Computer Vision, and Drones at the Supermarket DAN FELDMAN - HAIFA UNIVERSITY
Speaker: DAN FELDMAN - HAIFA UNIVERSITY
Title : Machine Learning, Computer Vision, and Drones at the Supermarket
Machine Learning, Computer Vision, and Drones at the Supermarket
We will show that the following problems are surprisingly related, together with
provable approximations, and cool videos that demonstrate their real-world real-time performance.
1) Have a group of lawful (<200 grams) autonomous toy-drones that carry personalized ads inside the supermarket "Rami-Levi Hashikma LTD" near the university.
This is related to SLAM (simultaneous localization and mapping).
2) Given n lines and a n points, compute a rotation, translation and 1-to-1 matching that will minimize the sum of distances between every point to its matched line.
This is a special case of a fundamental problem in computer vision, called PnP (Perspective-n-Point).
3) Minimize ||Ax-b||+\lambda ||x||-\lambda over every \lambda>0 and x in R^d.
This is called Ridge regression in machine learning, if lambda is given. We suggest the first provable approximation without tuning lambda .
4) Headache caused by current Augmented Reality glasses.
Partially based on papers in IEEE International Conference on Robotics and Automation (ICRA'19) and IEEE Robotics and Automation Letters (RA-L'19).
Joint work with Ibrahim Jubran, David Cohn, Ariel Hutterer, and Daniel Jeryes from the lab.
17/01/2019 - 12:00 Towards intent-based self-driving networks KIRILL KOGAN
Speaker: KIRILL KOGAN
Title : Towards intent-based self-driving networks
Growing complexity of network operations is the subject of many debates. In general there are two major factors impacting complexity of network operations: the size and structure of a manageable state and frequency of its changes. Networks should be autonomous as possible, ideally, completely excluding operators from the operational loop. However, this can significantly increase the size of a manageable state and complexity of network infrastructure. Currently, there is no way to systematically quantify the implemented efficiency versus complexity of network operations by potential design principles. In this talk we will consider this fundamental tradeoff. Our goal is to understand design principles and show their effectiveness allowing networks to become self-driving.
13/01/2019 - 12:00 Challenges and directions Ethan Fetaya
Speaker: Ethan Fetaya
Title : Challenges and directions
Deep learning has become one of the leading methods of machine learning, with significant breakthroughs in the the domains of computer vision and speech recognition, among others. In this talk, I will describe some major obstacles, such as computation time and adversarial attacks, and present a number of possible directions we are taking to overcome these challenges.
10/01/2019 - 12:00 Practical Reliability of Systems DANA DRACHSLER COHEN
Speaker: DANA DRACHSLER COHEN
Title : Practical Reliability of Systems
Recent years have shown adoption of new software trends such as machine learning, programmable computer networks, and blockchain frameworks. Several software bugs and attacks have demonstrated the importance of bringing formal guarantees to such systems. Techniques from verification, automated reasoning and program synthesis were shown successful in providing such guarantees for software systems. My research extends these techniques to these new software trends.
In this talk, I will focus on deep learning and will present two novel, complementary methods, leveraging abstract interpretation and logical constraints, for making deep learning models more reliable and interpretable.
03/01/2019 - 12:00 Cloud Computing and Graph Coloring ILAN COHEN
Speaker: ILAN COHEN
Title : Cloud Computing and Graph Coloring
Cloud computing opens a new chapter in information technology, by enabling global access to shared pools of resources such as services, data, servers, and computer networks. It drives new digital businesses across enterprises. In the last few years, an unprecedented amount of data center capacity has been built to support cloud computing services' growth. Therefore, optimizing the energy budget of data centers, without harming service level agreements, would result in massive savings for their operators, and significantly contribute to greater environmental sustainability. A key challenge in optimizing cloud computing services is their online nature. That is, they require immediate and irrevocable decisions to be made, based on incomplete input.
In this talk, I will discuss my work on two major online optimization problems for cloud services: switch routing and virtual machine placement. I will show how these problems relate to graph coloring, one of the most well-known, popular and extensively-researched areas in the field of graph theory. First, I will present tight bounds for online edge coloring in bipartite graphs, which leads to an optimal switch routing scheduler. I will then discuss vector balancing problems, a well-studied model for virtual machine placement in cloud services. In these problems, jobs have vector loads and the goal is to balance the load on all dimensions simultaneously. For this purpose, I will first consider two natural relaxations of the vertex coloring problem, and show new online lower bounds for them. I will then show how to port these bounds back to vector balancing, and prove that the bounds are tight by presenting matching upper bounds. Finally, for practical applications, these bounds are unsatisfactory, so I will also discuss how to improve the upper bounds by adding restricting, yet practical, assumptions.
30/12/2018 - 10:00 When Existing Techniques Preserve Differential Privacy OR SHEFFET
Speaker: OR SHEFFET
Title : When Existing Techniques Preserve Differential Privacy
It is no secret that online companies, hospitals, credit-card companies and governments hold massive datasets composed of our sensitive personal details. Information from such datasets is often released using some privacy preserving heuristics, which have been repeatedly shown to fail. That is why in recent years the notion of differential privacy has been gaining much attention, as an approach for conducting data-analysis that adheres to a strong and mathematically rigorous notion of privacy. Indeed, many differentially private analogs of existing data-analysis techniques have already been devised. These are, however, new algorithms, that require the use of additional random noise on top of existing techniques.
In this talk we will demonstrate how existing techniques, that were developed independently of any privacy consideration, preserve differential privacy by themselves --- when parameters are properly set. The main focus of the talk will be the Johnson-Lindenstrauss Transform, which preserves differential privacy provided the input satisfies some ``well spread'' properties. We will discuss applications of this algorithm in approximating multiple linear regressions and in statistical inference. Moreover, focusing on linear regression, we will exhibit additional techniques that preserve privacy: regularization, addition of random datapoints and Bayesian sampling.
(Time permitting, we will survey a very different technique, de-noising neural networks, that also aligns with the definition of differential privacy in the local model.)
The talk is self-contained and no prior knowledge is assumed.
27/12/2018 - 12:00 Robust and Efficient Visual Matching SIMON KORMAN
Speaker: SIMON KORMAN
Title : Robust and Efficient Visual Matching
In a majority of computer vision applications, establishing visual correspondence is a fundamental computational stage, since it provides a geometrical understanding of the data, that is crucial for high level decision making. It includes a wide range of alignment, registration or matching tasks, between color images, range scans, 3D models and more, which can provide information like the calibration of a camera, its position in space and the 3D structure and motion of objects in a scene.
In such tasks, real world conditions, due to complex object or camera motion, scene geometry and illumination, are difficult to model and hence algorithms are typically challenged with high levels of noise and outlier data.
In this talk, for a range of matching problems, I will present efforts in designing rich models along with robust and efficient optimization algorithms that extend the performance limits of existing methods. I will present a very general and efficient template matcher as well as some new approaches to robust estimation that excel in the regime of high noise and outlier rates. Lastly, I will discuss the state of visual matching in the era of deep learning, including some current and future research directions.
20/12/2018 - 12:00 On Optimization and Expressiveness in Deep Learning NADAV COHEN
Speaker: NADAV COHEN
Title : On Optimization and Expressiveness in Deep Learning
Understanding deep learning calls for addressing three fundamental questions: expressiveness, optimization and generalization. Expressiveness refers to the ability of compactly sized deep neural networks to represent functions capable of solving real-world problems. Optimization concerns the effectiveness of simple gradient-based algorithms in solving non-convex neural network training programs. Generalization treats the phenomenon of deep learning models not overfitting despite having much more parameters than examples to learn from. This talk will describe a series of works aimed at unraveling some of the mysteries behind optimization and expressiveness. I will begin by discussing recent analyses of optimization for deep linear neural networks. By studying the trajectories of gradient descent, we will derive the most general guarantee to date for efficient convergence to global minimum of a gradient-based algorithm training a deep network. Moreover, in stark contrast to conventional wisdom, we will see that, sometimes, gradient descent can train a deep linear network faster than a classic linear model. In other words, depth can accelerate optimization, even without any gain in expressiveness, and despite introducing non-convexity to a formerly convex problem. In the second (shorter) part of the talk, I will present an equivalence between convolutional and recurrent networks --- the most successful deep learning architectures to date --- and hierarchical tensor decompositions. The equivalence brings forth answers to various questions concerning expressiveness, resulting in new theoretically-backed tools for deep network design.
Optimization works covered in this talk were in collaboration with Sanjeev Arora, Elad Hazan, Noah Golowich and Wei Hu. Expressiveness works were with Amnon Shashua, Or Sharir, Yoav Levine, Ronen Tamari and David Yakira.
13/12/2018 - 12:00 Towards the Next Generation of Proof Assistants: Enhancing the Proofs–as–Programs Paradigm LIRON COHEN
Speaker: LIRON COHEN
Title : Towards the Next Generation of Proof Assistants: Enhancing the Proofs–as–Programs Paradigm
As software has grown increasingly critical to our society's infrastructure, mechanically-verified software has grown increasingly important, feasible, and prevalent. Proof assistants have seen tremendous growth in recent years because of their success in the mechanical verification of high-value applications in many areas, including cyber security, cyber-physical systems, operating systems, compilers, and microkernels. These proof assistants are built on top of constructive type theory whose computational interpretation is given by the proofs-as-programs paradigm, which establishes a correspondence between formal proofs and computer programs. However, while both proof theory and programming languages have evolved significantly over the past years, the cross-fertilization of the independent new developments in each of these fields has yet to be explored in the context of this paradigm. This naturally gives rise to the following questions: how can modern notions of computation influence and contribute to formal foundations, and how can modern reasoning techniques improve the way we design and reason about programs?
In this talk I first demonstrate how using programming principles that go beyond the standard lambda-calculus, namely state and non-determinism, promotes the specification and verification of modern systems, e.g. distributed systems. I then illustrate the surprising fragility of proof assistants in the presence of such new computational capabilities, and outline my ongoing efforts to develop a more robust foundation. For the converse direction, I show how incorporating modern proof-theoretic techniques offers a more congenial framework for reasoning about hard programming problems and hence facilitates the verification effort.
29/11/2018 - 12:00 Optimal Short-Circuit Resilient Formulas RAN GELLES
Speaker: RAN GELLES
Title : Optimal Short-Circuit Resilient Formulas
We consider fault-tolerant boolean formulas in which the output of a faulty gate is short- circuited to one of the gate's inputs. A recent result by Kalai et al. [FOCS 2012] converts any boolean formula into a resilient formula of polynomial size that works correctly if less than a fraction 1/6 of the gates (on every input-to-output path) are faulty. We improve the result of Kalai et al., and show how to efficiently fortify any boolean formula against a fraction 1/5 of short-circuit gates per path, with only a polynomial blowup in size. We additionally show that it is impossible to obtain formulas with higher resilience and sub-exponential growth in size.
Towards our results, we consider interactive coding schemes when noiseless feedback is present; these produce resilient boolean formulas via a Karchmer-Wigderson relation. We develop a coding scheme that resists up to a fraction 1/5 of corrupted transmissions in each direction of the interactive channel. We further show that such a level of noise is maximal for coding schemes with sub-exponential blowup in communication. Our coding scheme takes a surprising inspiration from Blockchain technology.
Joint work with Mark Braverman, Klim Efremenko, and Michael A. Yitayew
22/11/2018 - 12:00 New Paradigms for Cryptographic Hashing Ilan Komargodski
Speaker: Ilan Komargodski
Title : New Paradigms for Cryptographic Hashing
Cryptographic hash functions are the basis of many important and far reaching results in cryptography, complexity theory, and beyond. In particular, hash functions are the primary building block of fundamental applications like digital signatures and verifiable computation, and they are the tool underlying the proofs-of-work which drive blockchains.
Because of the central role of cryptographic hash functions in both theory and practice, it is crucial to understand their security guarantees, toward basing applications on the minimal possible notion of security. Indeed, there are many ways to formalize the security requirement of a hash function; each way is sufficient for different applications. Also, there are many candidate hash functions, offering various trade-offs between security, efficiency, and other desirable properties.
In this talk, I will present an application [Komargodski-Naor-Yogev, FOCS 2017] of a relatively weak notion of hashing (collision resistance) that goes well beyond cryptography into a fundamental problem in the intersection of complexity theory and combinatorics -- the Ramsey problem. This will lead us to new emerging aspects of hash functions, including relaxed security notions and their applications, addressing a recent attack on SHA-1. I will conclude with several exciting open problems and challenges.
15/11/2018 - 12:00 Beyond SGD: Data Adaptive Methods for Machine Learning Kfir Levy
Speaker: Kfir Levy
Title : Beyond SGD: Data Adaptive Methods for Machine Learning
The tremendous success of the Machine Learning paradigm heavily relies on the development of powerful optimization methods. The canonical algorithm for training learning models is SGD (Stochastic Gradient Descent), yet this method has its limitations. It is often unable to exploit useful statistical/geometric structure, it might degrade upon encountering prevalent non-convex phenomena, and it is hard to parallelize. In this talk I will discuss an ongoing line of research where we develop alternative methods that resolve some of SGD's limitations. The methods that I describe are as efficient as SGD, and implicitly adapt to the underlying structure of the problem in a data dependent manner.
In the first part of the talk, I will discuss a method that is able to take advantage of hard/easy training samples. In the second part, I will discuss a method that enables an efficient parallelization of SGD. Finally, I will briefly describe a method that implicitly adapts to the smoothness and noise properties of the learning objective.
08/11/2018 - 12:00 Universal hitting sets and minimizers Ron Shamir
Speaker: Ron Shamir
Title : Universal hitting sets and minimizers
Handling the flood of deep DNA sequencing data calls for better algorithms. Minimizers are a central recent paradigm that has improved many sequence analysis tasks. We present an alternative paradigm that leads to substantial further improvement in such tasks. For integers k and L>k, we say that a set of k-mers is a universal hitting set (UHS) if every possible L-long sequence must contain a k-mer from the set. We present a heuristic called DOCKS to find a compact UHS.
We show that DOCKS works well in practice and produces UHSs that are very close to a theoretical lower bound. We present results for various values of k and L and by applying them to real genomes show that UHSs indeed offers substantial saving compared to minimizers. In particular, DOCKS uses less than 30% of the 10-mers needed to span the human genome compared to minimizers.
Joint work with Yaron Orenstein (BGU), David Pellow (TAU), Guillaume Marcais, Daniel Bork and Carl Kingsford (CMU)
25/10/2018 - 12:00 Coding For Interactive Communication over Networks Klim Efremenko - BGU
Speaker: Klim Efremenko - BGU
Title : Coding For Interactive Communication over Networks
The modern era is an era of interactive communication between many parties, where many parties are actively sending messages based on the information they received. However, the errors may ruin the communication. In this talk, I will describe how one can convert multi-party protocol into error resilient as well I will show the limits of such schemes.
18/10/2018 - 12:00 Verifying the Correctness of Deep Neural Networks Guy Katz - Hebrew Univ.
Speaker: Guy Katz - Hebrew Univ.
Title : Verifying the Correctness of Deep Neural Networks
Deep neural networks have emerged as an effective means for tackling complex, real-world problems. However, a major obstacle in applying them to safety-critical systems is the great difficulty in providing formal guarantees about their behavior. We present an efficient technique for verifying properties of deep neural networks (or providing counter-examples). The technique can also be used to measure a network's robustness to "adversarial inputs" - slight perturbations to a network's input that cause it to err. Our approach is based on the simplex method, extended to handle piecewise-linear activation functions, which are a crucial ingredient in many modern neural networks. The verification procedure tackles neural networks as a whole, without making any simplifying assumptions. We evaluated our technique on a deep neural network implementation of the next-generation Airborne Collision Avoidance System for unmanned aircraft (ACAS Xu), proving various properties about them and in one case identifying incorrect behavior.
[Based on joint work with Clark Barrett, David Dill, Kyle Julian and Mykel Kochenderfer]
Guy Katz is a senior lecturer at the Hebrew University of Jerusalem, Israel. He received his Ph.D. at the Weizmann Institute of Science in 2015. His research interests lie at the intersection between Formal Methods and Software Engineering, and in particular in the application of formal methods to software systems with components generated via machine learning.
14/06/2018 - 12:00 Axioms and Algorithms for Participatory Budgeting Nimrod Talmon
Speaker: Nimrod Talmon
Title : Axioms and Algorithms for Participatory Budgeting
Participatory budgeting is a recent, exciting development in deliberative grassroots democracy in which, e.g., residents of a city specify their preferences on how to construct their city budget, and an algorithm is then applied to aggregate their preferences and decide upon the budget. Even though it is gaining increasing popularity and is used to decide upon many millions of dollars every year, the foundations of participatory budgeting are not yet well understood. I will discuss some recent developments, specifically novel, efficient aggregation algorithms satisfying certain axiomatic properties desired in participatory budgeting scenarios.
31/05/2018 - 12:00 On Odor Reproduction, and How to Test For It David Harel
Speaker: David Harel
Title : On Odor Reproduction, and How to Test For It
In this talk I will tell you how analyzing economic markets where agents have cognitive biases has lead to a better understanding of the communication complexity of local search procedures.
We begin the talk with studying combinatorial auctions with bidders that exhibit endowment effect. In most of the previous work on cognitive biases in algorithmic game theory (e.g., [Kleinberg and Oren, EC'14] and its follow-ups) the focus was on analyzing the implications and mitigating their negative consequences. In contrast, we show how cognitive biases can sometimes be harnessed to improve the outcome.
24/05/2018 - 12:00 From Cognitive Biases to the Communication Complexity of Local Search Shahar Dobzinski
Speaker: Shahar Dobzinski
Title : From Cognitive Biases to the Communication Complexity of Local Search
03/05/2018 - 12:00 A Sublinear Tester for Outerplanarity Reut Levi
Speaker: Reut Levi
Title : A Sublinear Tester for Outerplanarity
We consider one-sided error property testing of $\calF$-minor freeness in bounded-degree graphs for any finite family of graphs $\calF$ that contains a minor of $K_{2,k}$, the $k$-circus graph, or the $(k\times 2)$-grid for any $k\in\mathbb{N}$.
This includes, for instance, testing whether a graph is outerplanar or a cactus graph.
This is joint work with Hendrik Fichtenberger, Yadu Vasudev and Maximilian Wötzel.
26/04/2018 - 12:00 Near-Optimal Compression for the Planar Graph Metric SHAY MOZES
Speaker: SHAY MOZES
Title : Near-Optimal Compression for the Planar Graph Metric
The Planar Graph Metric Compression Problem is to compactly encode the distances among $k$ nodes in a planar graph of size $n$. Two naive solutions are to store the graph using $O(n)$ bits, or to explicitly store the distance matrix with $O(k^2 \log{n})$ bits. The only lower bounds are from the seminal work of Gavoille, Peleg, Pérennes, and Raz [SODA'01], who rule out compressions into a polynomially smaller number of bits, for \emph{weighted} planar graphs, but leave a large gap for unweighted planar graphs. For example, when $k=\sqrt{n}$, the upper bound is $O(n)$ and their constructions imply an $\Omega(n^{3/4})$ lower bound. This gap is directly related to other major open questions in labeling schemes, dynamic algorithms, and compact routing.
12/04/2018 - 12:00 Taking Turing to the Theater %: On Imitation Algorithms Zvi Lotker
Speaker: Zvi Lotker
Title : Taking Turing to the Theater %: On Imitation Algorithms
Computer science has grown out of the seed of imitation. From von Neumann's machine to the famous Turing test, which sparked the field of AI, algorithms have always tried to imitate humans and nature. Examples of such ``imitation algorithms'' are simulated annealing which imitates thermodynamics, genetic algorithms which imitate biology, or deep learning which imitates human learning.
In this talk, I describe an algorithm which imitates human psychology. Specifically, I discuss $M$ algorithms, which serve as a simple example of psychology-based imitation algorithms. The $M$ algorithm is one of the simplest natural language processing (NLP) algorithms.
Respecting the long tradition of imitation algorithms, the $M$ algorithm is both extremely simple and extremely powerful. Like other imitation algorithms, the $M$ algorithm is able to solve extraordinarily difficult problems efficiently. The $M$ algorithm efficiently pinpoints critical events in films, theater productions, and other scripts, revealing the rhythm of the texts.
At first glance, when trying to design an algorithm which pinpoints critical events of a text, it seems necessary for the algorithm to understand the complete text. Additionally, it would be expected that all layers of the narrative, background information, etc., would also be necessary. In short, it would be expected that the algorithm would imitate the human process of comprehending a text.
Surprisingly, the $M$ algorithm utilizes the structure of the complete text itself without understanding even a \emph{single} word, sentence, or character in order to discover critical events. The content of the narrative is not necessary for the algorithm to work. Other than an awareness of the illusion of time, borrowed from psychology, the $M$ algorithm circumvents the human process of reading.
This talk is based on a book (in process).
22/03/2018 - 12:00 Brain computer interfaces for human motor- cognitive enhancement MIRIAM REINER
Speaker: MIRIAM REINER
Title : Brain computer interfaces for human motor- cognitive enhancement
I discuss two models of BCi that correlate to neural error detection and to enhanced motor memory consolidation, then apply to a three tier model of applied BCI for enhanced human relearning of motor skills such as after stroke, accidents or in general a BCI system for motor learning.
Three components are embedded in relearning of motor skills: brain signals of error detection, training to reduce errors, and memory consolidation of the improved motor execution. I integrate here three components that I have previously studied separately, into a triple-stage paradigm that includes EEG markers for error detection, BCI-based motor correction and BCI for memory consolidation of the corrected motor motion through neurofeedback processes. In my earlier work I show that error potentials are uniquely associated with error characteristics, suggesting a potential BCI system for error-tailored correction, rather than generic BCI. The third component relates to neurofeedback. My earlier studies of neurofeedback theta after motor learning, showed a significant effect of enhanced motor memory consolidation in the experimental group compared to the control groups. Here I describe a preliminary three stage protocol of motor rehabilitation: error detection, BCI for motor correction and then a neurofeedback for consolidation of the corrected motion.
15/03/2018 - 12:00 Improved *Deterministic* Algorithms for Partially Dynamic Shortest Paths. Shiri Chechik
Speaker: Shiri Chechik
Title : Improved *Deterministic* Algorithms for Partially Dynamic Shortest Paths.
Computing shortest paths is one of the fundamental problems of graph algorithms.
The goal of *dynamic* single source shortest paths (SSSP) is to maintain a shortest path tree from a fixed source s as the edges of the graph change over time.
The most general case is the fully dynamic one, where each adversarial update inserts or deletes an arbitrary edge. The trivial algorithm is to recompute SSSP after every update in O(m) time. For the fully dynamic case, no non-trivial algorithm is known.
We can, however, improve upon the trivial algorithm by restricting the update sequence to be partially dynamic: only insertions (referred to as incremental), or only deletions (referred to as decremental).
08/03/2018 - 12:00 Nearly Work-Efficient Parallel Algorithm for Digraph Reachability Jeremy Fineman
Speaker: Jeremy Fineman
Title : Nearly Work-Efficient Parallel Algorithm for Digraph Reachability
One of the simplest problems on directed graphs is that of identifying the set of vertices reachable from a designated source vertex. This problem can be solved easily sequentially by performing a graph search, but efficient parallel algorithms have eluded researchers for decades. For sparse high-diameter graphs in particular, there is no previously known parallel algorithm with both nearly linear work and nontrivial parallelism. This talk presents the first such algorithm. Specifically, this talk presents a randomized parallel algorithm for digraph reachability with work O(m*polylog(n)) and span O(n^{2/3}*polylog(n)), with high probability, where n is the number of vertices and m is the number of arcs.
The main technical contribution is an efficient Monte Carlo algorithm that, through the addition of O(n*polylog(n)) shortcuts, reduces the diameter of the graph to O(n^{2/3}*polylog(n)) with high probability. While there are existing algorithms that achieve similar guarantees, they are not computationally efficient; the sequential algorithms have running times much worse than linear. This talk presents a surprisingly simple sequential algorithm with running time O(m log^2(n)) that achieves the stated diameter reduction. Parallelizing that algorithm yields the main result, but doing so involves overcoming several additional challenges.
25/01/2018 הרצאת קולוקוויום ביום חמישי 25.1.2018 מבוטלת .
Speaker: .
Title : הרצאת קולוקוויום ביום חמישי 25.1.2018 מבוטלת
הרצאת קולוקוויום ביום חמישי 25.1.2018 מבוטלת
18/01/2018 - 12:00 Conditional hardness of string similarity and tree similarity Oren Weimann
Speaker: Oren Weimann
Title : Conditional hardness of string similarity and tree similarity
The theory of NP-hardness allows computer scientists to classify nearly all problems as polynomial-time solvable (in P) or NP-hard. Recently, there has been a similar development in classifying problems within the class P. These are known as conditional lower bounds, that is, they depend on the conjectured hardness of certain archetypal problems such as CNF Satisfiability and all-pairs shortest paths (APSP). Proving conditional lower bounds for fundamental problems in P is now a flourishing research area with the goal of better understanding the landscape of P. Most of the problems for which conditional lower bounds were recently shown are Stringology problems.
In my talk I will focus on the problems of computing the edit distance of strings and the edit distance of trees. I will first show that these problems are very similar in terms of their upper bounds by showing that their state of the art algorithm is in fact the same algorithm. Its running time for strings is O(n^2) and for trees is O(n^3). Then, I will show that in terms of lower bounds the problems are in fact very different: String edit distance exhibits the hardness of the strong exponential time hypothesis (asserting that CNF Satisfiability requires O(2^n) time) while tree edit distance exhibits the hardness of the ASPS hypothesis (asserting that APSP requires O(n^3) time).
11/01/2018 - 12:00 Oblivious Routing via Random Walks Gal Shahaf
Speaker: Gal Shahaf
Title : Oblivious Routing via Random Walks
We present novel oblivious routing algorithms for both splittable and unsplittable multicommodity flow. Our algorithm for minimizing congestion for unsplittable multicommodity flow is the first oblivious routing algorithm for this setting. As an intermediate step towards this algorithm, we present a novel generalization of Valiant's classical load balancing scheme for packet-switched networks to arbitrary graphs, which is of independent interest. Our algorithm for minimizing congestion for splittable multicommodity flow improves upon the state-of-theart, in terms of both running time and performance, for graphs that exhibit good expansion guarantees. Our algorithms rely on diffusing traffic via iterative applications of the random walk operator. Consequently, the performance guarantees of our algorithms are derived from the convergence of the random walk operator to the stationary distribution and are expressed in terms of the spectral gap of the graph (which dominates the mixing time).
Joint work with Michael Schapira
04/01/2018 - 12:00 Secure Computation in the Real World Daniel Genkin
Speaker: Daniel Genkin
Title : Secure Computation in the Real World
The security of any system is only as good as its weakest link. Even if the system's security is theoretically proven under some set of assumptions, when faced with real-word adversaries, many of these assumptions become flaky, inaccurate and often completely incorrect.
In this talk I will present two cases for bringing this gap between security theory and security practice:
* Utilizing unintentional and abstraction-defying side-channel leakage from physical computing devices in order to extract secret cryptographic keys and the relation of these attacks to leakage resilient cryptography.
* Constructing and deploying secure computation schemes for arbitrary C programs.
The talk will discuss cryptographic techniques and will include live demonstrations.
01/01/2018 - 10:00 Dynamic graph matching and related problems Shay Solomon
Speaker: Shay Solomon
Title : Dynamic graph matching and related problems
Graph matching is one of the most well-studied problems in combinatorial optimization, with applications ranging from scheduling and object recognition to numerical analysis and computational chemistry.
Nevertheless, until recently very little was unknown about this problem in real-life **dynamic networks**, which aim to model the constantly changing physical world.
In the first part of the talk we'll discuss our work on dynamic graph matching, and in the second part we'll highlight our work on a few related problems.
Shay Solomon is currently a Herman Goldstine Postdoctoral Fellow at IBM T. J. Watson Research Center.
Prior to joining IBM, he was a Rothschild and Fulbright Postdoctoral Fellow at Stanford University, hosted by Prof. Moses Charikar and Prof. Virginia Vassilevska Williams.
Solomon received a Ph.D. degree in Computer Science from the Ben-Gurion University under the guidance of Prof. Michael Elkin.
Solomon's Ph.D. dissertation investigates several longstanding graph compression problems, and has received numerous awards, including a best student paper award for his single-authored SODA'11 paper.
His postdoctoral work focuses on fundamental computational challenges that arise when dealing with dynamic networks.
21/12/2017 - 12:00 (How) should we use domain knowledge in the era of deep learning? (A perspective from speech processing) Karen Livescu
Speaker: Karen Livescu
Title : (How) should we use domain knowledge in the era of deep learning? (A perspective from speech processing)
Deep neural networks are the new default machine learning approach in many domains, such as computer vision, speech processing, and natural language processing. Given sufficient data for a target task, end-to-end models can be learned with fairly simple, almost universal algorithms. Such models learn their own internal representations, which in many cases appear to be similar to human-engineered ones. This may lead us to wonder whether domain-specific techniques or domain knowledge are needed at all.
This talk will provide a perspective on these issues from the domain of speech processing. It will discuss when and how domain knowledge can be helpful, and describe two lines of work attempting to take advantage of such knowledge without compromising the benefits of deep learning. The main application will be speech recognition, but the techniques discussed are general.
14/12/2017 - 12:00 When Codes for Storage Systems Meet Storage Systems Gala Yadgar
Speaker: Gala Yadgar
Title : When Codes for Storage Systems Meet Storage Systems
Large-scale storage systems lie at the heart of the big data revolution. As these systems grow in scale and capacity, their complexity grows accordingly, building on new storage media, hybrid memory hierarchies, and distributed architectures. Numerous layers of abstraction hide this complexity from the applications, but also hide valuable information that could improve the system's performance considerably.
I will demonstrate how to bridge this semantic gap in the context of erasure codes, which are used to guarantee data availability and durability. Current theoretical research efforts focus on codes that will reduce the storage, network, and compute overheads of the systems that use them, without sacrificing their reliability. However, the semantic gap makes it difficult to observe the theoretical benefit of the resulting codes in real implementations. I will follow the example of regeneration and locally recoverable codes, showing the key challenges in applying optimal erasure codes to real systems, and how they can be addressed. This part is based on joint work with Matan Liram, Oleg Kolosov, Eitan Yaakobi, Itzhak Tamo and Alexander Barg.
I will then briefly describe the challenges introduced by the semantic gap in other layers of the "storage stack", and my experience in addressing them. I will refer to the memory hierarchy, flash-based solid-state drives, workload analysis, and aspects of data security.
07/12/2017 - 12:00 Interactive verification of distributed protocols Oded Padon
Title : Interactive verification of distributed protocols
Distributed protocols such as Paxos play an important role in many computer systems.
Therefore, a bug in a distributed protocol may have tremendous effects.
Accordingly, a lot of effort has been invested in verifying such protocols.
However, due to the infinite state space (e.g., unbounded number of nodes, messages) and protocols complexity, verification is both undecidable and hard in practice.
I will describe a deductive approach for verification of distributed protocols, based on first-order logic, inductive invariants and user interaction.
The use of first-order logic and a decidable fragment of universally quantified invariants allows to completely automate some verification tasks.
Tasks that remain undecidable (e.g. finding inductive invariants) are solved with user interaction, based on graphically displayed counterexamples.
I will also describe the application of these techniques to verify safety of several variants of Paxos, and a way to extend the approach to verify liveness and temporal properties.
Oded Padon is a fourth year PhD student in Tel Aviv University, under the supervision of Prof. Mooly Sagiv. His research focuses on verification of distributed protocols using first-order logic.
He is a recipient of the 2017 Google PhD fellowship in programming languages.
30/11/2017 - 12:00 On Notions of Distortion and an Almost Minimum Spanning Tree with constant Average Distortion Arnold Filtser
Speaker: Arnold Filtser
Title : On Notions of Distortion and an Almost Minimum Spanning Tree with constant Average Distortion
Minimum Spanning Trees of weighted graphs are fundamental objects in numerous applications. In particular in distributed networks, the minimum spanning tree of the network is often used to route messages between network nodes. Unfortunately, while being most efficient in the total cost of connecting all nodes, minimum spanning trees fail miserably in the desired property of approximately preserving distances between pairs. While known lower bounds exclude the possibility of the worst case distortion of a tree being small, Abraham et al showed that there exists a spanning tree with constant average distortion. Yet, the weight of such a tree may be significantly larger than that of the MST. In this paper, we show that any weighted undirected graph admits a spanning tree whose weight is at most (1+\rho) times that of the MST, providing constant averagedistortion O(1/\rho).
The constant average distortion bound is implied by a stronger property of scaling distortion, i.e., improved distortion for smaller fractions of the pairs. The result is achieved by first showing the existence of a low weight spanner with small prioritized distortion, a property allowing to prioritize the nodes whose associated distortions will be improved. We show that prioritized distortion is essentially equivalent to coarse scaling distortion via a general transformation, which has further implications and may be of independent interest. In particular, we obtain an embedding for arbitrary metrics into Euclidean space with optimal prioritized distortion.
Joint work with Yair Bartal and Ofer Neiman.
23/11/2017 - 12:00 Deterministic Random Matrices Ilya Soloveychik
Speaker: Ilya Soloveychik
Title : Deterministic Random Matrices
Random matrices have become a very active area of research in the recent years and have found enormous applications in modern mathematics, physics, engineering, biological modeling, and other fields. In this work, we focus on symmetric sign (+/-1) matrices (SSMs) that were originally utilized by Wigner to model the nuclei of heavy atoms in mid-50s. Assuming the entries of the upper triangular part to be independent +/-1 with equal probabilities, Wigner showed in his pioneering works that when the sizes of matrices grow, their empirical spectra converge to a non-random measure having a semicircular shape. Later, this fundamental result was improved and substantially extended to more general families of matrices and finer spectral properties. In many physical phenomena, however, the entries of matrices exhibit significant correlations. At the same time, almost all available analytical tools heavily rely on the independence condition making the study of matrices with structure (dependencies) very challenging. The few existing works in this direction consider very specific setups and are limited by particular techniques, lacking a unified framework and tight information-theoretic bounds that would quantify the exact amount of structure that matrices may possess without affecting the limiting semicircular form of their spectra.
From a different perspective, in many applications one needs to simulate random objects. Generation of large random matrices requires very powerful sources of randomness due to the independence condition, the experiments are impossible to reproduce, and atypical or non-random looking outcomes may appear with positive probability. Reliable deterministic construction of SSMs with random-looking spectra and low algorithmic and computational complexity is of particular interest due to the natural correspondence of SSMs and undirected graphs, since the latter are extensively used in combinatorial and CS applications e.g. for the purposes of derandomization. Unfortunately, most of the existing constructions of pseudo-random graphs focus on the extreme eigenvalues and do not provide guaranties on the whole spectrum. In this work, using binary Golomb sequences, we propose a simple completely deterministic construction of circulant SSMs with spectra converging to the semicircular law with the same rate as in the original Wigner ensemble. We show that this construction has close to lowest possible algorithmic complexity and is very explicit. Essentially, the algorithm requires at most 2log(n) bits implying that the real amount of randomness conveyed by the semicircular property is quite small.
16/11/2017 - 12:00 Let a thousand filters Bloom Martin Farach-Colton
Speaker: Martin Farach-Colton
Title : Let a thousand filters Bloom
Bloom filters and other approximate membership query data structures (AMQs), are one of the great successes of theoretical computer science, with uses throughout databases, file systems, networks and beyond. An AMQ maintains a set under insertions, sometimes deletions, and queries with one-sided error: if a queried element is in the set, the AMQ returns \texttt{present}, if it is not, the AMQ returns \texttt{present} with probability at most $\epsilon$. An optimal AMQ uses $n\log\epsilon^{-1}$ bits of space when the represented set has $n$ elements.
Yet there is a gap between their theoretical bounds provided by AMQs and their requirements in the field.
In this talk, I will describe how AMQs are used in practice and how this changes (for the better) our theoretical understanding of these data structures.
Martin Farach-Colton is a Professor of Computer Science at Rutgers University, New Brunswick, New Jersey. His research focuses on both the theory and practice of external memory and storage systems. He was a pioneer in the theory of cache oblivious analysis. His current research focuses on the use of write optimization to improve performance in both read- and write-intensive big data systems. He has also worked on the algorithmics of strings and metric spaces, with applications to bioinformatics. In addition to his academic work, Professor Farach-Colton has extensive industrial experience. He was CTO and co-founder of Tokutek, a database company that was founded to commercialize his research and acquired in 2015. During 2000–2002, he was a Senior Research Scientist at Google.
09/11/2017 - 12:00 Speaker separation in the wild, and the industry's view Raphael Cohen
Speaker: Raphael Cohen
Title : Speaker separation in the wild, and the industry's view
Audio recordings are a data source of great value used for analyzing conversations and enabling digital assistants. An important aspect of analyzing single-channel audio conversations is identifying who said what, a task known as speaker diarization. The task is further complicated when the number of speakers is a priori unknown. In this talk we'll review the motivation for speaker diarization and verification, outline previous approaches to diarization and their relation to "real" diarization needs (as in audio applications such as Chorus.ai's Conversation Analytics platform), and present the pipeline required for integrating these solutions, as well as recent SOTA end-to-end deep learning approaches to this problem.
02/11/2017 - 12:00 Efficient Logging in Non-Volatile Memory by Exploiting Coherency Protocols Nachshon Cohen
Speaker: Nachshon Cohen
Title : Efficient Logging in Non-Volatile Memory by Exploiting Coherency Protocols
Non-volatile memory (NVM) technologies such as PCM, ReRAM and STT-RAM allow processors to directly write values to persistent storage at speeds that are significantly faster than previous durable media such as hard drives or SSDs.
Many applications of NVM are constructed on a logging subsystem, which enables operations to appear to execute atomically and facilitates recovery from failures.
Writes to NVM, however, pass through a processor's memory system, which can delay and reorder them and can impair the correctness and cost of logging algorithms.
Reordering arises because of out-of-order execution in a CPU and the inter-processor cache coherence protocol.
By carefully considering the properties of these reorderings, this paper develops a logging protocol that requires only one round trip to non-volatile memory while avoiding expensive computations.
We show how to extend the logging protocol to building a persistent set (hash map) that also requires only a single round trip to non-volatile memory for insertion, updating, or deletion.
26/10/2017 - 12:00 Efficient admission policies for cache management and heavy hitter detection GIL EINZIGER
Speaker: GIL EINZIGER
Title : Efficient admission policies for cache management and heavy hitter detection
In this talk, I introduce the often overlooked concept of cache admission policies. Once the cache is full, in order to admit a new item, we need to evict one of the stored items. An admission policy decides if the new item should be admitted to the cache. I will present two admission policies that enhance performance in different fields. The TinyLFU cache admission policy uses past frequency to estimate the benefit of data items. This policy is implemented in the Caffeine high-performance Java library and is used by many open source projects such as Apache Cassandra, Apache Accumulo, JBoss Infinispan, Druid, Neo4J, Spring, to name a few. It is also used in industry by companies such as Google, Netflix, and Linkedin. Alternatively, the RAP admission policy is a randomized streaming algorithm for finding the most frequent items. In RAP, we use a coin flip to decide if a new item should be admitted to the cache. This approach improves the space to accuracy ratio of state of the art algorithms by up to x32 on real traces and up to x128 on synthetic ones.
29/06/2017 - 12:00 Distributed construction of graph spanners Ami Paz - Technion
Speaker: Ami Paz - Technion
Title : Distributed construction of graph spanners
A spanner of a given graph is a sparse subgraph that approximately preserves distances. Since their introduction in the late 1980's, spanners have found numerous applications in synchronization problems, information dissemination, routing schemes and more.
Many applications of spanners are in computer networks, where the network needs to find a spanner for its own communication graph. We present distributed algorithms for constructing additive spanners in networks of bounded message size, namely in the CONGEST model. In addition, we present an innovative technique for showing lower bounds for constructing spanners in this setting, a technique that can be useful for other distributed graph problems.
Based on joint works with Keren Censor-Hillel, Telikepalli Kavitha, Noam Ravid and Amir Yehudayoff
28/06/2017 - 12:00 Mechanism Design for Constrained Matching + Facility Location on Discrete Cycles and Grids Makoto Yokoo + Taiki Todo
Speaker: Makoto Yokoo + Taiki Todo
Title : Mechanism Design for Constrained Matching + Facility Location on Discrete Cycles and Grids
The theory of two-sided matching (e.g., assigning residents to hospitals, students to schools) has been extensively developed, and it has been applied to design clearinghouse mechanisms in various markets in practice. As the theory has been applied to increasingly diverse types of environments, however, researchers and practitioners have encountered various forms of distributional constraints. As these features have been precluded from consideration until recently, they pose new challenges for market designers. One example of such distributional constraints is a minimum quota, e.g., school districts may need at least a certain number of students in each school in order for the school to operate. In this talk, I present an overview of research on designing mechanisms that work under distributional
In this work we study a facility location problem from a perspective of mechanism design. We focus on the problem of locating a facility deterministically on discrete graphs with cycles, including grids and hypercubes. We further assume that mechanisms are defined for any number
of participating agents, so that we can discuss consistency properties of mechanisms' behavior with respect to the number of agents. One of such well-studied properties is false-name-proofness, requiring that no agent can benefit by adding fake agents into the market. Our contribution is two fold. First, for cycle graphs, we show that there exists a false-name-proof and Pareto efficient mechanism if and only if the underlying graph has at most five vertices. Second, for l*m-grid
graphs (where l,m >= 2), we show that such a mechanism exists if and only if either l=2 or m=2 holds. We finally show that for any n(>2)-dimensional binary-cube graphs, there is no such mechanisms.
22/06/2017 - 12:00 Benchmarking Machine-Learning Performance Amitai Armon
Speaker: Amitai Armon
Title : Benchmarking Machine-Learning Performance
Machine-learning attracts a lot of interest in recent years, more than ever before, as it utilizes the increasing amounts of data for many smart applications. This creates a flood of innovative ideas in this domain, constantly introducing new algorithmic and modeling techniques. The rapid pace of innovation poses a challenge in designing hardware that would best fit the needs of this domain several years from now. The goal of this work was to help coping with this challenge, by constructing a benchmark that represents the variety of fundamental compute requirements relevant to this field. The talk describes the analysis method used for building this benchmark. While no one can predict the future of this domain, interesting insights were deduced from its past, and helped hardware and software experts assess several possibilities for the future.
15/06/2017 - 12:00 Reliable cognitive solutions EITAN FARCHI
Speaker: EITAN FARCHI
Title : Reliable cognitive solutions
Traditional software is built using hiding and the decomposition process. The system is broken into sub-systems which are then broken into components and so on until the lowest object in the hierarchy is defined. With the introduction of components implemented using machine learning and dependent on data a host of new challenges to the system readability are introduced. Challenges include a good match of business objectives, machine learning optimization objectives and relevant data, stability of the quality of the solution due to changes in data, artistic choices of machine learning parameters as well as correctness of components in probability. In this talk I present an end-to-end view of the challenges and possible ways to address them as well as some resulting interesting research directions.
08/06/2017 - 12:00 The Password Reset MitM Attack NETHANEL GELERNTER
Speaker: NETHANEL GELERNTER
Title : The Password Reset MitM Attack
Joint work with Senia Kalma, Bar Magnezi and Hen Porcilan (The College of Management Academic Studies)
Was presented in IEEE Security & Privacy 2017.
We present the password reset MitM (PRMitM) attack and show how it can be used to take over user accounts. The PRMitM attack exploits the similarity of the registration and password reset processes to launch a man in the middle (MitM) attack at the application level. The attacker initiates a password reset process with a website and forwards every challenge to the victim who either wishes to register in the attacking site or to access a particular resource on it. The attack has several variants, including exploitation of a password reset process that relies on the victim's mobile phone, using either SMS or phone call. We evaluated the PRMitM attacks on Google and Facebook users in several experiments, and found that their password reset process is vulnerable to the PRMitM attack. Other websites and some popular mobile applications are vulnerable as well. Although solutions seem trivial in some cases, our experiments show that the straightforward solutions are not as effective as expected. We designed and evaluated two secure password reset processes and evaluated them on users of Google and Facebook. Our results indicate a significant improvement in the security. Since millions of accounts are currently vulnerable to the PRMitM attack, we also present a list of recommendations for implementing and auditing the password reset process.
My affiliation: (Cyberpion & College of Management Academic Studies)
25/05/2017 - 12:00 Neural Networks, Graphical Models and Image Restoration Prof. Yair Weiss
Speaker: Prof. Yair Weiss
Title : Neural Networks, Graphical Models and Image Restoration
This talk discusses some of the history of graphical models and neural networks and speculates on the future of both fields with examples from the particular problem of image restoration.
18/05/2017 - 12:00 Labeling schemes for trees and planar graphs Pawel Gawrychowski
Speaker: Pawel Gawrychowski
Title : Labeling schemes for trees and planar graphs
Labeling schemes seek to assign a label to each node in a network, so that a function on two nodes can be computed by examining their labels alone. The goal is to minimize the maximum length of a label and (as a secondary goal) the time to evaluate the function. As a prime example, we might want to compute the distance between two nodes of a network using only their labels. We consider this question for two natural classes of networks: trees and planar graphs. For trees on n nodes, we
design labels consisting of 1/4log^2(n) bits (up to lower order terms), thus matching a recent lower bound of Alstrup et al. [ICALP 2016]. For planar graphs, the situation is much more complex. A major open problem is to close the gap between an upper bound of O(n^{1/2}log(n))-bits and a lower bound of O(n^{1/3})-bits for unweighted planar graphs. We show that, for undirected unweighted planar graphs, there is no hope to achieve a higher lower bound using the known techniques. This is done by designing a centralized structure of size ~O(min(k^2,(kn)^{1/2})) that can calculate the distance between any pair of designated k terminal nodes. We show that this size it tight, up to polylogarithmic terms, for such centralized structures. This is complemented by an improved upper bound of O(n^{1/2}) for labeling nodes of an undirected unweighted planar graph for calculating the distances.
11/05/2017 - 12:00 Sequence to Sequence Learning with Neural Networks: from Characters to Words and Beyond Roee Aharoni
Speaker: Roee Aharoni
Title : Sequence to Sequence Learning with Neural Networks: from Characters to Words and Beyond
Sequence-to-Sequence learning (Sutskever et. al, 2014) equipped with the attention mechanism (Bahdanau et. al, 2015) became the state-of-the-art approach to machine translation, automatic summarization and many other natural language processing tasks, while being much simpler than the previously dominant methods. In this talk we will overview sequence-to-sequence learning with neural networks, followed by two of our recent contributions: the Hard Attention mechanism for linear time inference, and the String-to-Tree model for syntax-aware neural machine translation.
20/04/2017 - 12:00 Secure Centrality Computation Over Multiple Networks TAMIR TASSA
Speaker: TAMIR TASSA
Title : Secure Centrality Computation Over Multiple Networks
Consider a multi-layered graph, where the different layers correspond to different proprietary social networks on the same ground set of users. Suppose that the owners of the different networks (called hosts) are mutually non-trusting parties: how can they compute a centrality score for each of the users using all the layers, but without disclosing information about their private graphs? Under this setting we study a suite of three centrality measures whose algebraic structure allows performing that computation with provable security and efficiency. The first measure counts the nodes reachable from a node within a given radius. The second measure extends the first one by counting the number of paths between any two nodes. The final one is a generalization to the multi-layered graph case: not only the number of paths is counted, but also the multiplicity of these paths in the different layers is considered.
We devise a suite of multiparty protocols to compute those centrality measures, which are all provably secure in the information-theoretic sense. One typical challenge and limitation of secure multiparty computation protocols is their scalability. We tackle this problem and devise a protocol which is highly scalable and still provably secure.
30/03/2017 - 12:00 Exponentially vanishing sub-optimal local minima in multilayer neural networks Daniel Soudry - Columbia Univ.
Speaker: Daniel Soudry - Columbia Univ.
Title : Exponentially vanishing sub-optimal local minima in multilayer neural networks
Multilayer neural networks, trained with simple variants of stochastic gradient descent (SGD), have achieved state-of-the-art performances in many areas of machine learning. It has long been a mystery why does SGD work so well – rather than converging to sub-optimal local minima with high training error (and therefore, high test error).
We examine a neural network with a single hidden layer, quadratic loss, and piecewise linear units, trained in a binary classification task on a standard normal input. We prove that the volume of differentiable regions of the empiric loss containing sub-optimal differentiable local minima is exponentially vanishing in comparison with the same volume of global minima, given "mild" (polylogarithmic) over-parameterization. This suggests why SGD tends to converge to global minima in such networks.
23/03/2017 - 12:00 Toward Natural Human-Robot Collaboration Henny Admoni - CMU
Speaker: Henny Admoni - CMU
Title : Toward Natural Human-Robot Collaboration
As robots become integrated into human environments, they increasingly interact directly with people. This is particularly true for assistive robots, which help people through social interactions (like tutoring) or physical interactions (like preparing a meal). Developing effective human-robot interactions in these cases requires a multidisciplinary approach involving both fundamental algorithms from robotics and insights from cognitive science. My research brings together these two areas to extend the science of human-robot interaction, with a particular focus on assistive robotics. In the process of developing cognitively-inspired algorithms for robot behavior, I seek to answer fundamental questions about human-robot interaction: what makes a robot appear intelligent? How can robots communicate their internal states to human partners to improve their ability to collaborate? Vice versa, how can robots "read" human behaviors that reveal people's goals, intentions, and difficulties, to identify where assistance is required?
In this talk, I describe my vision for robots that collaborate with humans on complex tasks by leveraging natural, intuitive human behaviors. I explain how models of human attention, drawn from cognitive science, can help select robot behaviors that improve human performance on a collaborative task. I detail my work on algorithms that predict people's mental states based on their eye gaze and provide assistance in response to those predictions. And I show how breaking the seamlessness of an interaction can make robots appear smarter. Throughout the talk, I will describe how techniques and knowledge from cognitive science help us develop robot algorithms that lead to more effective interactions between people and their robot partners.
19/03/2017 - 12:00 On Context for Language-Endowed Intelligent Agents Sergei Nirenburg
Speaker: Sergei Nirenburg
Title : On Context for Language-Endowed Intelligent Agents
That context plays an important – possibly, crucial – role in language processing and decision making is a truism. But what counts as context is different for different tasks and different people. I will discuss several facets of context used as major sources of heuristic knowledge required of an artificial intelligent agent emulating human functioning. I will concentrate on tasks related to language understanding and generation as well as decision making in a goal-oriented task environment where the agent is a member of a team whose members can be other agents or humans.
I will be happy to talk to whoever would want to talk with me!
But, of course, the main reason for my visit is being able to talk with you about potential collaborations.
02/02/2017 - 12:00 computational aspects of communication Elad Haramaty - Harvard University
Speaker: Elad Haramaty - Harvard University
Title : computational aspects of communication
Communication has played a growing role in our lives over the past decade. It often becomes a bottleneck in the performance of our daily tasks. This motivates the pursuit for more efficient communication. However, efficiency is becoming more challenging from the computational aspect, due to several of its characteristics in modern communications.
One such characteristic is the interactivity of the protocols in today's noisy communication environments. One natural approach to overcome such a challenge is called Interactive Coding. Interactive coding is an efficient black-box mechanism to convert any interactive protocol that performs well under noiseless environment into one that is also resilient to errors, while preserving the efficiency of the protocol.
Another characteristic which challenges today's communications is the dynamic and interactive nature of modern protocols. This may lead to desynchronization between the communicating parties. However, until now, almost all designed systems assume that both parties are perfectly synchronized and all context is shared perfectly by the communicating agents. Thus, any violation of those protocols leads to a breakdown in communication.
This talk will address both of the aforementioned challenges.The first part of the talk will be devoted to interactive coding and in the second part, we will focus on designing protocols under uncertainty of the shared context.
23/01/2017 - 12:00 Graph Algorithms for Distributed Networks Merav Parter - MIT
Speaker: Merav Parter - MIT
Title : Graph Algorithms for Distributed Networks
I will describe two branches of my work related to algorithms for distributed networks.
The main focus will be devoted for Fault-Tolerant (FT) Network Structures.
The undisrupted operation of structures and services is a crucial requirement in modern day communication networks. As the vertices and edges of the network may occasionally fail or malfunction, it is desirable to make those structures robust against failures.
FT Network Structures are low cost highly resilient structures, constructed on top of a given network, that satisfy certain desirable performance requirements
concerning, e.g., connectivity, distance or capacity. We will overview some results on fault tolerant graph structures with a special focus on FT Breadth-First-Search.
The second part of the talk will discuss distributed models and algorithms for large-scale networks. Towards the end, we will see some connections between distributed computing and other areas such as EE and Biology.
22/01/2017 - 13:30 Voting on restricted preference domains: a survey EDITH ELKIND
Speaker: EDITH ELKIND
Title : Voting on restricted preference domains: a survey
Arrow's famous impossibility theorem (1951) states that there is no perfect voting rule: for three or more candidates, no voting rule can satisfy a small set of very appealing axioms. However, this is no longer the case if we assume that voters' preferences satisfy certain restrictions, such as being single-peaked or single-crossing. In this talk, we discuss single-peaked and single-crossing elections, as well as some other closely related restricted preference domains, and provide an overview of recent algorithmic results for these domains.
19/01/2017 - 12:00 Robust and Simple Market Design Inbal TALGAM-COHEN
Speaker: Inbal TALGAM-COHEN
Title : Robust and Simple Market Design
Algorithms and the Internet are revolutionizing how society allocates its resources. Examples range from wireless spectrum and electricity to online advertising and carpooling opportunities. A fundamental question is how to allocate such resources efficiently by designing robust computational markets.
In this talk I will demonstrate recent progress on this question by considering a problem crucial for major industry players like Google: how to design revenue-maximizing allocation mechanisms. Most existing designs hinge on "getting the price right" – selling goods to buyers at prices low enough to encourage a sale, but high enough to garner non-trivial revenue. This approach is difficult to implement when the seller has little or no a priori information about buyers' valuations, or when the setting is sufficiently complex, as in the case of markets with heterogeneous goods.
I will show a robust approach to designing auctions for revenue, which "lets the market do the work" by allowing prices to emerge from enhanced competition for scarce goods. This work provides guidelines for a seller in choosing among data acquisition and sophisticated pricing, and investment in drawing additional buyers.
Bio: Inbal Talgam-Cohen is a Marie Curie postdoctoral researcher at HUJI and a visiting postdoctoral researcher at TAU. She holds a PhD from Stanford supervised by Tim Roughgarden, an MSc from Weizmann and a BSc from TAU in computer science, as well as a law LLB. Her research is in algorithmic game theory, including computational and data aspects of market design and applications to Internet economics. Her awards include the 2015 Best Doctoral Dissertation Award of ACM SIGecom, the Stanford Interdisciplinary Graduate Fellowship, and the Best Student Paper Award at EC'15.
05/01/2017 - 12:00 A House Divided Against Itself? Dividing Crowds to Improve Analysis & Design of Crowd-Based Activities Omer Lev - University of Toronto
Speaker: Omer Lev - University of Toronto
Title : A House Divided Against Itself? Dividing Crowds to Improve Analysis & Design of Crowd-Based Activities
In the past several years, crowd-activities have expanded dramatically in both their scope and their size – more diverse activities are involving online crowds, and the number of people (and money) involved in such activities is growing fast. I will focus in this talk on 2 topics, which share as a central feature the partitioning of the crowd into sub-groups.
In peer-selection, agents try to select the top k agents among themselves. We present several algorithms that try to make this process impartial, i.e., agents will not benefit from lying about their peers, while still choosing "good" agents. We will show algorithms that provably reach impartiality using the division of the agents into subgroups, and then show a set of simulations that give us further insight into which techniques work better in scenarios resembling real-world settings.
From there, we will move to district selection: decision making in settings where each partition (a company sub-unit, an electoral district, etc.) reaches a decision, and the decisions from all partitions are amalgamated to a final choice. We explore how this effects representability, and in the case of geographical manipulations ("gerrymandering"), present both computational hardness results and experiments on real-world data.
29/12/2016 - 12:00 Parameterized Complexity as a New Computational Lens Meirav Zehavi
Speaker: Meirav Zehavi
Title : Parameterized Complexity as a New Computational Lens
Parameterized Complexity was introduced in the 1990s by Downey and Fellows. Nowadays, various aspects of this field are ubiquitous, vibrant areas of research. In a nutshell, Parameterized Complexity aims to reduce the running times of algorithms for NP-hard problems by confining the combinatorial explosion to a parameter k. More precisely, a problem is fixed-parameter tractable (FPT) with respect to a parameter k if it can be solved by an algorithm, called a parameterized algorithm, whose time complexity is bounded by f(k)n^{O(1)} for some function f that depends only on k. By careful examination of real datasets, one can choose a parameter k that will often be significantly smaller than the input size n. Indeed, parameterized problems arise in a variety of areas outside of pure theoretical computer science and math.
In this talk, I give a brief overview of some of my recent contributions in Parameterized Complexity. In the first part of my talk, I present results in core research areas such as Kernelization and Parameterized Algorithms. Here, problems are primarily concerned with graphs. However, in practice, inputs are not necessarily graphs. In the second part of my talk, I address this issue. Here, I describe my foray into input domains that consist of polygons, preferences and matrices.
22/12/2016 - 12:00 Shared-memory concurrency in programming languages: Ori Lahav
Speaker: Ori Lahav
Title : Shared-memory concurrency in programming languages:
With the proliferation of multi-core processors, shared-memory concurrent programming has become increasingly important. Nevertheless, despite decades of research, there are still no adequate answers to the following fundamental questions:
1. What is the right semantics for concurrent programs in higher-level languages?
2. Which reasoning principles apply to realistic shared-memory concurrency?
Concerning the first question, the challenge lies in simultaneously allowing efficient implementation on modern hardware with compiler optimizations, while exposing a well-behaved programming model. Both the Java and C/C++ standards tried to address this question, but their
solutions were discovered to be flawed. They either allow spurious "out-of-thin-air" program behaviors or incur a prohibitive runtime cost. In the main part of the talk, I will present a solution to this problem based on a novel idea of promises, which supports efficient implementations and useful programming guarantees.
As for the second question, I will outline my ongoing efforts to identify (i) logical laws for deductive reasoning about concurrent programs; and (ii) programming disciplines that enable the application of well-established verification techniques.
Ori Lahav is a postdoctoral researcher at Max Planck Institute for Software Systems (MPI-SWS). His research interests include programming languages, formal methods and verification, semantics and logic. Previously, Ori was a postdoctoral researcher at the Programming Languages Group at Tel Aviv University. He obtained his Ph.D. from Tel Aviv University in 2014 under the supervision of Arnon Avron. Ori received a Dan David Prize scholarship for young researchers in 2014, the Kleene award for the best student paper at LICS 2013, and a Wolf Prize for excellent graduate students in 2012.
20/12/2016 - 12:00 Acoustic word embeddings and neural segmental models Karen Livescu
Title : Acoustic word embeddings and neural segmental models
This talk covers two closely related lines of work.
For a number of speech tasks, it can be useful to represent speech segments of arbitrary length by fixed-dimensional vectors, or embeddings. In particular, vectors representing word segments -- acoustic word embeddings -- can be used in query-by-example tasks, example-based speech recognition, or spoken term discovery. *Textual* word embeddings have been common in natural language processing for a number of years now; the acoustic analogue is only recently starting to be explored. This talk will present our work on acoustic word embeddings, including a variety of models in unsupervised, weakly supervised, and supervised settings.
15/12/2016 - 12:00 Intelligent Information Sharing to Support Loosely-Coupled Teamwork Ofra Amir - Harvard University
Speaker: Ofra Amir - Harvard University
Title : Intelligent Information Sharing to Support Loosely-Coupled Teamwork
Teamwork is a core human activity, essential to progress in many areas. A vast body of research in the social sciences and in computer science has studied teamwork and developed tools to support teamwork. Although the technologies resulting from this work have enabled teams to work together more effectively in many settings, they have proved inadequate for supporting the coordination of distributed teams that operate in a loosely-coupled manner. In this talk, I will present three integrated research efforts towards developing intelligent systems that reduce coordination overhead in such teams: an in depth formative study of complex healthcare teams, the design of new computational methods for personalizing the information shared with team members about collaborators' activities, and an evaluation of those methods in a realistic teamwork setting, demonstrating that personalized information sharing resulted in higher productivity and lower perceived workload of team members.
08/12/2016 - 12:00 Privacy-Preserving Computations in the Digital Era המרצים: פרופ' יהודה לינדל Privacy-Preserving Computations in the Digital Era המרצים: פרופ' יהודה לינדל
Speaker: Privacy-Preserving Computations in the Digital Era המרצים: פרופ' יהודה לינדל
Title : Privacy-Preserving Computations in the Digital Era המרצים: פרופ' יהודה לינדל
אנו שמחים להתכבד בנוכחותו של
מר רוג'ר גראס – Mr. Roger Grass
תורם הבניין למדעי המחשב
Privacy-Preserving Computations in the Digital Era
המרצים: פרופ' יהודה לינדל
Natural Language Processing with Neural Networks
דוקטורנט רועי אהרוני
Automated Agents that Interact Proficiently with People
פרופ' שרית קראוס
08/12/2016 - 12:00 אנו שמחים להתכבד בנוכחותו של Privacy-Preserving Computations in the Digital Era המרצים: פרופ' יהודה לינדל
Title : אנו שמחים להתכבד בנוכחותו של
01/12/2016 - 12:00 Privacy, Economy – and Cryptography? Amir Herzberg - BIU
Speaker: Amir Herzberg - BIU
Title : Privacy, Economy – and Cryptography?
There is a lot of concern, by legislators, experts and the general public, about the loss of privacy. In the computer science research community, extensive efforts are dedicated to develop Privacy Enhancing Technologies, mostly based on cryptography; much of these efforts are designed to provide privacy even against (cloud) providers.
However, the reality is that users explicitly agree to share their private data with few providers (Google, FB,...), although these providers explicitly state that they may make commercial use of this private data, and in spite of numerous incidents where such data was exposed to unauthorized third parties, and papers showing design-failures allowing such unauthorized disclosure of private data. This is known as the privacy paradox.
We present simple economical analysis of privacy, explaining the privacy paradox, and showing potential market failure, harming social welfare. We discuss possible solutions, including regulatory and cryptographic controls.
27/11/2016 - 18:00 "Can Johnny Finally Encrypt? Evaluating E2E-Encryption in Popular IM Applications" Hemi Leibowitz - Bar-lan University
Speaker: Hemi Leibowitz - Bar-lan University
Title : "Can Johnny Finally Encrypt? Evaluating E2E-Encryption in Popular IM Applications"
Recently, many popular Instant-Messaging (IM) applications announced support for end-to-end encryption, claiming confidentiality even against a rogue operator. Is this, finally, a positive answer to the basic challenge of usable-security presented in the seminal paper, 'Why Johnny Can't Encrypt'?
Our work evaluates the implementation of end-to-end encryption in popular IM applications: WhatsApp, Viber, Telegram, and Signal, against established usable-security principles, and in quantitative and qualitative usability experiments. Unfortunately, although participants expressed interest in confidentiality, even against a rogue operator, our results show that current mechanisms are impractical to use, leaving users with only the illusion of security.
Hope is not lost. We conclude with directions which may allow usable End-to-End encryption for IM applications.
24/11/2016 - 12:00 Faster algorithms for maximal 2-connected subgraphs in directed graphs Veronika Loitzenbauer
Speaker: Veronika Loitzenbauer
Title : Faster algorithms for maximal 2-connected subgraphs in directed graphs
Connectivity related concepts are of fundamental interest in graph theory. The area has received extensive attention over four decades, but many problems remain unsettled, especially for directed graphs. A strongly connected directed graph is 2-edge-connected (resp.,2-vertex-connected) if the removal of any edge (resp., vertex) leaves the graph strongly connected. In this talk we present improved algorithms for computing the maximal 2-edge- and 2-vertex-connected induced subgraphs of a given directed graph. These problems were first studied more than 35 years ago, with O(mn) time algorithms for graphs with m edges and n vertices being known since the late 1980s. In contrast, the same problems for undirected graphs are known to be solvable in linear time. For the directed case, we showed 1) O(n^2) time algorithms in joint work with Monika Henzinger and Sebastian Krinninger (ICALP'15) and 2) O(m^1.5) time algorithms in joint work with Shiri Chechik, Thomas D. Hansen, Giuseppe F. Italiano, and Nikos Parotsidis (SODA'17).
17/11/2016 - 12:00 From Programming Languages to Programming Systems – Software Development by Refinement Shachar Itzhaky - MIT
Speaker: Shachar Itzhaky - MIT
Title : From Programming Languages to Programming Systems – Software Development by Refinement
Everyone wants to program with "high-level concepts", rather than meddle with the fine details of the implementation, such as pointers, network packets, and asynchronous callbacks. This is usually achieved by introducing layers of abstraction – but every layer incurs some overhead, and when they accumulate, this overhand becomes significant and sometimes prohibitive. Optimizing the program often requires to break abstractions, which leads to suboptimal design, higher maintenance costs, and subtle hard-to-trace bugs.
I will present two recent projects that attempt to address this difficulty. TransCal is a rewrite-based interactive synthesis engine that helps a developer mechanically transform a high-level program into a tractable, semantically equivalent version. TransCal attempts to encode and reuse practices that programmers usually reason about informally. The programs targeted are general functional-style programs with operations on collections.
Bellmania is a specialized tool for accelerating dynamic-programming algorithms by generating recursive divide-and-conquer implementations of them. Recursive divide-and-conquer is an algorithmic technique that was developed to obtain better memory locality properties. Bellmania includes a high-level language for specifying dynamic programming algorithms and a calculus that facilitates gradual transformation of these specifications into efficient implementations. These transformations formalize the divide-and-conquer technique; a visualization interface helps users to interactively guide the process, while an SMT-based back-end verifies each step and takes care of low-level reasoning required for parallelism.
10/11/2016 - 12:00 Faster Projection-free Machine Learning and Optimization Dan Garber - Toyota Technological Institute at Chicago
Speaker: Dan Garber - Toyota Technological Institute at Chicago
Title : Faster Projection-free Machine Learning and Optimization
Projected gradient descent (PGD), and its close variants, are often considered the methods of choice for solving a large variety of machine learning optimization problems, including empirical risk minimization, statistical learning, and online convex optimization. This is not surprising, since PGD is often optimal in a very appealing information-theoretic sense. However, for many problems PGD is infeasible both in theory and practice since each step requires to compute an orthogonal projection onto the feasible set. In many important cases, such as when the feasible set is a non-trivial polytope, or a convex surrogate for a low-rank structure, computing the projection is computationally inefficient in high-dimensional settings. An alternative is the conditional gradient method (CG), aka Frank-Wolfe algorithm, that replaces the expensive projection step with a linear optimization step over the feasible set. Indeed in many problems of interest, the linear optimization step admits much more efficient algorithms than the projection step, which is the reason to the substantial regained interest in this method in the past decade. On the downside, the convergence rates of the CG method often fall behind that of PGD and its variants.
In this talk I will survey an ongoing effort to design CG variants that on one hand enjoy the cheap iteration complexity of the original method, and on the other hand converge provably faster, and are applicable to a wider variety of machine learning settings. In particular I will focus on the cases in which the feasible set is either a polytope or a convex surrogate for low-rank matrices. Results will be demonstrated on applications including: LASSO, video co-localization, optical character recognition, matrix completion, and multi-class classification.
23/06/2016 - 12:00 הרצאת קולוקוויום ביום חמישי 23.6.16 מבוטלת. הרצאת קולוקוויום ביום חמישי 23.6.16 מבוטלת.
Speaker: הרצאת קולוקוויום ביום חמישי 23.6.16 מבוטלת.
Title : הרצאת קולוקוויום ביום חמישי 23.6.16 מבוטלת.
הרצאת קולוקוויום ביום חמישי 23.6.16 מבוטלת.
16/06/2016 - 12:00 Low-Rank Matrix Recovery from Row-and-Column Affine Measurements Or Zuk - the Hebrew University
Speaker: Or Zuk - the Hebrew University
Title : Low-Rank Matrix Recovery from Row-and-Column Affine Measurements
We propose a new measurement scheme for low-rank matrix recovery, where each measurement
is a linear combination of elements in one row or one column of the unknown matrix.
This setting arises naturally in applications but current algorithms, developed for standard matrix recovery problems, do not perform well in this case, hence the need for developing new algorithms and theory.
We propose a simple algorithm for the problem based on Singular Value Decomposition (SVD) and least-squares (LS), which we term SVLS. We prove favourable theoretical guarantees for our algorithm
for the noiseless and noisy case, compared to standard matrix completion measurement schemes
and algorithms. Simulations show improved speed and accuracy, including for the problem of unknown rank estimation.
Our results suggest that the proposed row-and-column measurements scheme, together with our recovery algorithm, may provide a powerful framework for affine matrix recovery.
Time permitting, I will describe also progress on other data analysis projects for sparse and structured-sparse data, including a new group-sparse clustering algorithm
09/06/2016 - 12:00 Innovations and start ups Eyal Niv - Pitango Venture Capital
Speaker: Eyal Niv - Pitango Venture Capital
Title : Innovations and start ups
02/06/2016 - 12:00 Face recognition: robust transfer learning using the multiverse loss Lior Wold - Tel Aviv University
Speaker: Lior Wold - Tel Aviv University
Title : Face recognition: robust transfer learning using the multiverse loss
Transfer learning plays a major role in the recent success of deep face recognition methods. Deep networks are trained to solve the multiclass classification problem using a cross entropy loss and the learned representations are then transferred to a different domain. Moreover, the task changes post-transfer to face verification (same/not-same). In the talk, I will point to a few research questions, including: What is the ideal source dataset? How to train in the target domain? How to estimate the certainty of the identification?
One of the observations we make is that the transferred representations support only a few modes of separation and much of their dimensionality is underutilized. To alleviate this, we suggest to learn, in the source domain, multiple orthogonal classifiers. We prove that this leads to a reduced rank representation, which, however, supports more discriminative directions. For example, we obtain the 2nd best result on LFW for a single network. This is achieved using a training set that is a few orders of magnitude smaller than that of the leading literature network, and using a very compact representation of only 51 dimensions.
Given time, I will describe recent achievements in other computer vision domains: optical flow, action recognition, image annotation, and more.
26/05/2016 - 12:00 An Exponential Separation Between Randomized and Deterministic Complexity in the LOCAL Model Tsvi Kopelowitz - University of Michigan
Speaker: Tsvi Kopelowitz - University of Michigan
Title : An Exponential Separation Between Randomized and Deterministic Complexity in the LOCAL Model
Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the distributed LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this talk we show that these exponential gaps are necessary and establish connections between the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message:
1. Fast Delta-coloring of trees requires random bits.
2. Randomized lower bounds imply higher deterministic lower bounds.
3. Deterministic lower bounds imply randomized lower bounds.
Joint work with Yi-Jun Chang and Seth Pettie
14/04/2016 - 12:00 New Technologies in Google Matthieu Mayran - Google
Speaker: Matthieu Mayran - Google
Title : New Technologies in Google
We will be giving an overview of technologies built in Google for Google and how we externalize some of them to the world either as a service or through OOS or both. We will talk about several products including:
Google Cloud Platform: A review of the products that are part of it and when it makes sense to use the Cloud
Big Data tools and services: Best practices and demo showing services that make big data manageable
Machine Learning: Tensorflow open-sourcing and CloudML with some preview of the Vision API and other libraries.
I believe lecture will be one hour to an hour and a half and will be held in english.
The presenter will be Matthieu Mayran, Cloud Solutions Architect
07/04/2016 - 12:00 PonderThis, IBM research monthly challenge Oded Margalit - IBM
Speaker: Oded Margalit - IBM
Title : PonderThis, IBM research monthly challenge
The research division of IBM has been publishing a monthly challenge since May 1998. In the talk we will walk through some of these challenges, give some examples of the solvers and their solutions and some insights on the 'behind the scenes' of authoring PonderThis.
22/03/2016 - 14:00 Recent developments in fair division for strategic agents Simina Brânzei - The Hebrew University
Speaker: Simina Brânzei - The Hebrew University
Title : Recent developments in fair division for strategic agents
I will talk about fair division for strategic agents and will start with the cake cutting problem, which models the division of a heterogeneous divisible good among players with different preferences. Here I will talk about an impossibility result (dictatorship theorem), as well as an
algorithmic framework for strategic agents, where fair outcomes can be implemented in the equilibria of protocols.
If enough time I will also mention the problem of allocating multiple divisible goods for the fairness solution concept of Nash social welfare. When the preferences are known, this solution is achieved in the Fisher market equilibrium outcomes. However, with strategic agents the strong fairness guarantees of market equilibria can be destroyed. We find that for
additive valuations, the Fisher market approximates the NSW within a factor of 2, but for Leontief utilities the approximation degrades linearly with the number of players. Surprisingly, a mechanism known as Trading Post not only retains the constant 2-approximation of Fisher for additive utilities, but approximately implements the NSW objective for Leontief valuations: its price of anarchy is 1 + epsilon for every epsilon > 0.
This is joint work with Peter Bro Miltersen, Ioannis Caragiannis, David Kurokawa, Ariel Procaccia, Vasilis Gkatzelis, and Ruta Mehta.
17/03/2016 - 12:00 Randomized Algorithms for Matrix Decomposition Yariv Aizenbud - Tel-Aviv University
Speaker: Yariv Aizenbud - Tel-Aviv University
Title : Randomized Algorithms for Matrix Decomposition
Matrix decompositions, and especially SVD, are very important tools in data analysis. When big data is processed, the computation of matrix decompositions becomes expensive and impractical. In recent years, several algorithms, which approximate matrix decomposition, have been developed. These algorithms are based on metric conservation features for linear spaces of random projections. We present a randomized method based on sparse matrix distribution that achieves a fast approximation with bounded error for low rank matrix decomposition.
10/03/2016 - 12:00 Artificial Intelligence for Software Engineering Meir Kalech - Ben-Gurion University
Speaker: Meir Kalech - Ben-Gurion University
Title : Artificial Intelligence for Software Engineering
In this talk I will present our recent successful integration of various techniques from the Artificial Intelligence (AI) literature into the software debugging and testing process. First, we show how data that is already stored by industry standard software engineering tools can be used to learn a fault prediction model able to predict accurate the software components that are likely to contain bugs. This allows focusing testing efforts on such error-prone components. Then, we show how this learned fault prediction model can be used to augment existing software diagnosis algorithms, providing a better understanding of which software components need to be replaced to correct an observed bug. Moreover, for the case where further tests are needed to identify the faulty software component, we present a test-planning algorithm based on Markov Decision Processes (MDP).
Importantly, the presented approach for considering both a fault prediction model, learned from past failures, and a diagnosis algorithm that is model-based, is general, and can be applied to other fields, beyond software troubleshooting.
03/03/2016 - 12:00 Pricing Online Decisions Amos Fiat
Speaker: Amos Fiat
Title : Pricing Online Decisions
Will describe how dynamic pricing can be used to solve parking related problems. Will also discuss dynamic pricing for k-server problems, task systems, unit demand bidders, and other problems.
21/01/2016 - 12:00 Mechanical Challenges in the Design of Field Robots David Zarrouk - Ben-Gurion University
Speaker: David Zarrouk - Ben-Gurion University
Title : Mechanical Challenges in the Design of Field Robots
Miniature crawling robots, on the order of centimeter scale and below, have a variety of applications including medical procedures, search and rescue operations, maintenance, reconnaissance, environmental monitoring, assisted agriculture, cave exploration, mapping, studying biological organisms, security, defense, exploration of hazardous environments. Their small size allows them to navigate in remote areas otherwise inaccessible to wheeled or larger robots such as collapsed building and caves and biological vessels, while their low cost makes them disposable and allows their use in large quantities and swarm formations. As miniature crawling robots at this scale are under-actuated are achieving stability at high velocities over varying terrain, reducing the cost of transport, overcoming obstacles, controlling jumping and landing, adhering quickly to different surfaces, transitioning between horizontal to vertical motion, and climbing.
This talk will address some of the modeling and actuation challenges of crawling robots outdoors and inside biological vessels, while taking into account contact compliance and sliding.
Biography: David Zarrouk is currently a Senior Lecturer at the ME depart of BGU. He received his M.S. (2007) and Ph.D (2011) degrees from the faculty of mechanical engineering at the Technion. Between Aug. 2011 and Oct. 2013, he was postdoctoral scholar at the Biomimetics and Millisystems Lab. in the EECS dep. of U.C. Berkeley, working on miniature crawling robots. His research interests are in the fields of biomimetics, millisystems, miniature robotics, flexible and slippery interactions, underactuated mechanisms and theoretical kinematics. He received many prizes for excellence in research and teaching, which include a Fulbright postdoctoral Fellowship, Fulbright-Ilan Ramon postdoctoral Fellowship, Hershel Rich Innovation award, Aharon and Ovadia Barazani prize for excellent Ph.D. thesis, and Alfred and Yehuda Weisman prize for consistent excellence in teaching. a
14/01/2016 - 12:00 Supporting Information Sharing in Loosely-Coupled Human Teamwork Ofra Amir - Harvard University
Title : Supporting Information Sharing in Loosely-Coupled Human Teamwork
People collaborate in carrying out such complex activities as treating patients, co-authoring documents and developing software. Technologies such as Google Drive, Dropbox and Github enable teams to share work artifacts remotely and asynchronously. The coordination of team members' activities remains a challenge, however, because these technologies do not have capabilities for focusing people's attention on the actions taken by others that matter most to their own activities. In this talk, I will present our work towards developing intelligent systems for supporting information sharing in distributed teams. Based on a study of complex health care teams, we formalize the problem of information sharing in loosely-coupled extended-duration teamwork. We develop a new representation, Mutual Influence Potential Networks, that implicitly learns collaboration patterns and dependencies among activities from team members' interactions, and an algorithm that uses this representation to determine the information that is most relevant to each team member. Analysis of Wikipedia revision history data and an evaluation in a simulation environment demonstrate the ability of the proposed approach to identify relevant information to share with team members.
Joint work with Barbara Grosz and Krzysztof Gajos.a
07/01/2016 - 12:00 Towards Synthesis in Real Life Dana Fisman - University of Pennsylvania
Speaker: Dana Fisman - University of Pennsylvania
Title : Towards Synthesis in Real Life
System synthesis refers to the task of automatically generating an executable component of a system (e.g. a software or hardware component) from a specification of the component's behavior. The traditional formalization of the problem assumes the specification is given by a logical formalism. Recent trends to synthesis relax the problem definition and consider a variety of inputs including logical requirements, incomplete programs, and examples behaviors. In this talk I will describe some of the challenges on the road to usable synthesis, a variety of current approaches for coping with them, and some success stories.
Dana Fisman is a research scientist at the University of Pennsylvania, the Associate Director of the NSF expedition ExCAPE about system synthesis, and a visiting fellow at Yale University. She did her PhD in Weizmann under the supervision of Amir Pnueli, and worked many years in the industry in IBM Haifa Research Labs, and in Synopsys Inc. Dana's research interests are in the area of formal methods in system design. She is mostly known for her work on PSL, the IEEE standard for property specification language, on which she received numerous awards from IEEE, IBM and Synopsys.
05/01/2016 - 11:00 Scalable Machine Learning for structured high-dimensional outputs Ofer Meshi - Toyota Technological Institute at Chicago
Speaker: Ofer Meshi - Toyota Technological Institute at Chicago
Title : Scalable Machine Learning for structured high-dimensional outputs
In recent years, machine learning has emerged as an important and influential discipline in computer science and engineering. Modern applications of machine learning involve reasoning about complex objects like images, videos, and large documents. Treatment of such high-dimensional data requires the development of new tools, since traditional methods in machine learning no longer apply. In this talk I will present two recent works in this direction. The first work introduces a family of novel and efficient methods for inference and learning in structured output spaces. This framework is based on applying principles from convex optimization while exploiting the special structure of these problems to obtain efficient algorithms. The second work studies the success of a certain type of approximate inference methods based on linear programming relaxations. In particular, it has been observed that such relaxations are often tight in real applications, and I will present a theoretical explanation for this interesting phenomenon.
Ofer Meshi is a Research Assistant Professor at the Toyota Technological Institute at Chicago. Prior to that he obtained his Ph.D. and M.Sc. in Computer Science from the Hebrew University of Jerusalem. His B.Sc. in Computer Science and Philosophy is from Tel Aviv University. Ofer's research focuses on machine learning, with an emphasis on efficient optimization methods for inference and learning with high-dimensional structured outputs. During his doctoral studies Ofer was a recipient of Google's European Fellowship in Machine Learning.
31/12/2015 - 12:00 Nothing but net? content and network factors in information diffusion Oren Tsur - Harvard's School for Engineering and Applied Science (SEAS), Network Science Institute at Northeastern University and Harvard's Institute for Quantitative Social Science (IQSS).
Speaker: Oren Tsur - Harvard's School for Engineering and Applied Science (SEAS), Network Science Institute at Northeastern University and Harvard's Institute for Quantitative Social Science (IQSS).
Title : Nothing but net? content and network factors in information diffusion
Information diffusion is the process in which nuggets of information spread in a network (typically a social network). This is a complex process that depends on the network topology, the social structures and the information itself (content/language). In this talk I will discuss information diffusion from these different yet complementary perspectives. In the first part of the talk I will focus on the features of the diffusing information. I will present a gradient boosted trees algorithm modified for learning user preferences of Twitter hashtags. In the second part of my talk I will focus on the network structure. I will use exponential random graph models (ERGM) in order to learn what latent factors contribute to network formation and I will show how network structure and social roles contribute to the information spread. Specifically, I will present promising results obtained on political networks in the American political systems and analysis of the partisan use of political hashtags.
In both parts I put emphasis on interpretable models that go beyond accurate prediction and facilitate a better understanding of complex social processes.
Bio sketch:
Oren is a postdoctoral fellow at the Harvard's School for Engineering and Applied Science (SEAS) jointly with the Network Science Institute at Northeastern University and is also affiliated with Harvard's Institute for Quantitative Social Science (IQSS). He received his Ph.D. in Computer Science from the Hebrew University at 2013. In his work he combines natural language processing and network analysis in order to model and predict complex social dynamics. His work was published in both the NLP and the web/data science communities. Oren co-organized the workshop on NLP and Social Dynamics at ACL 2014, is co-organizing the EMNLP 2016 workshop on NLP and Computational Social Science and will be giving the tutorial on Understanding Offline Political Systems by Mining Online Political Data at WSDM 2016. His research on sarcasm detection (with Dmitry Davidov and Ari Rappoport) was listed in Time Magazine special issue as one of the 50 best inventions of the year (2010).
Homepage: http://people.seas.harvard.edu/~orentsur, Oren http://people.seas.harvard.edu/~orentsur/
29/12/2015 - 11:00 Coding for Distributed Storage Systems Natalia Silberstein - Technion
Speaker: Natalia Silberstein - Technion
Title : Coding for Distributed Storage Systems
In distributed storage systems (DSS) data is stored over a large number of storage nodes in such a way that a user can always retrieve the stored data, even if some storage nodes fail. To achieve such resilience against node failures, DSS introduce data redundancy based on different coding techniques. When a single node fails, the system performs node repair, i.e., reconstructs the data stored in the failed node in order to maintain the required level of redundancy.
In this talk we introduce a new approach to constructing different families of optimal codes for DSS via rank-metric codes. In the first part of the talk we deal with two important goals that guide the design of codes for DSS: reducing the repair bandwidth, i.e., the amount of data downloaded from the nodes during node repair, and achieving locality, i.e., reducing the number of nodes participating in a node repair process. First, we present a construction of a new family of optimal locally repairable codes, which enable repair of a failed node while using only few other nodes. Second, we introduce a new class of codes which minimize repair bandwidth for given locality parameters.
In the second part of the talk we consider the problem of making DSS resilient to adversarial attacks. In such attacks, an adversary may gain read-only access to information stored in DSS (passive eavesdropper) or may modify it (active adversary). We present a novel construction of codes for DSS which provide perfect secrecy, i.e., codes which allow to store data without revealing any information to an eavesdropper. This construction is also based on rank-metric codes.
Natalia Silberstein received her B.A. degree in Computer Science (cum laude), B.A. degree in Mathematics (cum laude), M.Sc. degree in Applied Mathematics (summa cum laude), and Ph.D. degree in Computer Science from the Technion, in 2004, 2007, and 2011, respectively. She then spent two years as a postdoctoral fellow in the Wireless Networking and Communications Group, at the Department of Electrical and Computer Engineering at the University of Texas at Austin, USA. She is currently a postdoctoral fellow at the Computer Science Department at the Technion.
Her research focuses on coding for distributed storage systems, network coding, algebraic error-correcting codes, applications of combinatorial designs and graphs to coding theory. | CommonCrawl |
KRM Home
One-species Vlasov-Poisson-Landau system near Maxwellians in the whole space
September 2014, 7(3): 591-604. doi: 10.3934/krm.2014.7.591
On a regularized system of self-gravitating particles
René Pinnau 1, and Oliver Tse 1,
Department of Technomathematics, University of Kaiserslautern, 67663 Kaiserslautern, Germany
Received April 2014 Revised May 2014 Published July 2014
We consider a regularized macroscopic model describing a system of self-gravitating particles. We study the existence and uniqueness of nonnegative stationary solutions and allude the differences to results obtained from classical gravitational models. The system is analyzed on a convex, bounded domain up to three spatial dimensions, subject to Neumann boundary conditions for the particle density, and Dirichlet boundary condition for the self-interacting potential. Finally, we show numerical simulations underlining our analytical results.
Keywords: macroscopic quantum models., self-gravitating particles, Bohm potential, Second order elliptic systems.
Mathematics Subject Classification: Primary: 35D30, 35J57; Secondary: 76Y0.
Citation: René Pinnau, Oliver Tse. On a regularized system of self-gravitating particles. Kinetic & Related Models, 2014, 7 (3) : 591-604. doi: 10.3934/krm.2014.7.591
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2018 Impact Factor: 1.38
René Pinnau Oliver Tse | CommonCrawl |
Why isn't general relativity the obvious thing to try after special relativity?
To preface my question, I ask this as a mathematics student, so I don't have a very good sense of how physicists think.
Here is the historical context I'm imagining (in particular taking into account the development of differential geometry in the 19th century):
Classical mechanics is about Lagrangians of matter fields on $\mathbb{R}^3$ (with the flat metric)
Sometime in the 1820s Gauss speculated about replacing the flat metric on $\mathbb{R}^3$ by one with non-vanishing curvature
Special relativity is about Lagrangians of matter fields on $\mathbb{R}^{3,1}$ (with the flat metric). I'm taking this jump for granted since as I understand it, it was arrived at from experimental results on electromagnetism at the end of the 19th century.
Now in line with Gauss, it seems like it would be extremely natural to speculate about replacing the flat metric on $\mathbb{R}^{3,1}$ by one with non-vanishing curvature (and in the same spirit also considering more exotic topologies for the underlying manifold).
Given that then we would have to ask exactly which metric we are looking for, it seems natural to say that there should be a Lagrangian term corresponding to the metric. The Einstein-Hilbert functional is probably the simplest one to try. And so we get the Einstein equations.
Alternatively (as I heard from someone is the actual history) you could observe that the energy-momentum tensor is of course a 2-tensor, and so for an Euler-Lagrange equation the most natural metric-dependent expression would be $\operatorname{Ric}=T$. Since $T$ is always divergence free it would be natural to replace $\operatorname{Ric}$ by $\operatorname{Ric}-\frac{1}{2}Rg$ just from taking the contracted second Bianchi identity into account. And so again we get the Einstein equations.
I've often heard it said that if Einstein had not come up with special relativity, someone else probably would have in the next five or ten years. However if he had not come up with general relativity, it would have taken much longer to discover. Why is this? I feel that I must be missing something here.
general-relativity special-relativity differential-geometry history
youleryouler
$\begingroup$ " However if he had not come up with general relativity, it would have taken much longer to discover." Hilbert came up with it at basically the exact same time. There was also Nordstrom Gravity, which worked off of the trace rather than the full tensor. It really wouldn't have taken nearly as long as people think; the ideas were ripe for the taking at the time they were found $\endgroup$ – Robert Mastragostino Mar 20 '14 at 1:24
$\begingroup$ The curved spacetime approach only works because of the equivalence principle. The deep insight required to be able to come up with general relativity was seeing that the equivalence principle was the main content of Newtonian gravity, and that this led naturally to a metric theory. Only once you have that can you start thinking about field equations and manifold theory. $\endgroup$ – Jerry Schirmer Mar 20 '14 at 2:00
$\begingroup$ @RobertMastragostino: they certainly knew about each others' work, but they were definitely working independently. A bit of a trivia point is that Hilbert published his paper first, but admitted that Einstein deserved the credit for general relativity. $\endgroup$ – Jerry Schirmer Mar 20 '14 at 2:00
$\begingroup$ @youler: it was considered to be a coincidence of the form of the Newtonian force law for a long time. The insight was figuring out that it was the essential feature of gravity from which other things flow. $\endgroup$ – Jerry Schirmer Mar 20 '14 at 2:27
$\begingroup$ I should also add that a modern development of special relativity is very much informed by the subsequent development of general relativity. While the concept of hyperbolic rotation was certainly present in 1905 with Minkowski, it certainly wasn't as developed as it is now, nor was the real significance of hte negative signature metric known. $\endgroup$ – Jerry Schirmer Mar 20 '14 at 2:59
I do think Jerry Schirmer answered the question in the comments, but I'll try to expand just to make clear how he explained everything.
Let us consider given that special relativity is correctly described by physics in Minkowski spacetime. Then we can ask ourselves how to include gravity without violating causality, which is mandatory by the finite velocity of light.
The idea is to consider Einstein's elevator. Namely that there is no local experiment which can be done that can differentiate between bodies in free fall in a constant gravitational field and the same bodies uniformly accelerated. That's because gravity affects everything the same way. A somewhat formalization of this is called Einstein's equivalence principle (in contrast with Galileo's, that say about coordinate transformation by constant velocities).
Note first that this is not the case for eletromagnetism. One can always use test charges to determine the electromagnetic fields, and it is impossible to do away with them using accelerated frames. Also, the equivalence principles is strictly local. If you look at extend regions gravity will appear through tidal forces.
So, if you think that special relativity is a particular case of general relativity (because it's just the same without gravity) the question is: what looks locally like special relativity but not globally? The answer is curved lorentzian manifolds, that locally are Minkowski.
But, as Jerry stressed, if you think in curved manifolds as generalization of flat ones, that does not, in principle, say anything about gravity. Only by noticing it is a force unlike any other, and formalizing it through the equivalence principle, one can justify the physics behind it, that is the use of curved manifolds. For instance, you suggest it is natural to generalize the situation by allowing curved spaces, but from the mathematical point of view one could just as well argue that there are other forms of generalization, e.g. we could instead try to projectify Minkowski. This is indeed usefull in other contexts, but it has nothing to do with gravity. So for a physicist is important we have "conceptual insights" to guide the process of "generalization for comprehension", or in other words we need principles with physical content.
I'm really unsure about what Gauss could be thinking regarding the metric. He did try to formulate classical mechanics in a differential geometrical way (Lanczos "Vartiational principles of classical mechanics" discusses it), but if that's what you're referring to, then it had nothing to do specifically with gravity.
EDIT: Oh boy, that last sentence is very misleading, I'm sorry. I had a look at Lanczos' book and realized that while Gauss pushed for a different formulation of classical mechanics, it's called Principle of Least Constraint, page 106 in Lanczos, it was only after some time that Hertz gave the principle the geometrical interpretation. So really not relevant to you question. I won't erase the paragraph though, in case anyone is interested.
Also, the equivalence principle argument says nothing about the field equations, and would be true even if the correct equations were different. As a matter of fact, a lot of general relativity independs of Einstein Field Equations, like the causal structure and (to some extend) the singularity theorems. This is why the equivalence principle was formulated as early as 1907 but the field equations came only in 1915.
I'm not a big fan of "what if" questions in history, majorly because they don't seem to have answers, but while Poincaré had the Lorentz trasnformations and a lot of understanding of special relativity, I never heard of anyone who anticipated the equivalence principle. So I hope this makes plausible that while others could have done SR, it did not seem likely that GR was coming, because first it was needed to understand what gravity is. Nordstrom's theory is an extension of ideas of eletromagnetism and was bound to failure. Hilbert indeed got the field equations right on his own, but would not get there without the motivation of curved spacetimes
cesarulianacesaruliana
I'm not sure if your main interest lies on the question on the title of this thread or in the question you pose near the end of your text. I'll try to answer both, in spite of not being an expert neither on GR nor on the education on physics.
Why is it, if it is true, that GR would have taken many more years to discover, had Einstein not discovered it?
I agree with you and the comments to your question in saying that GR would most likely have emerged, in the following years (though "how many years?" is a question which I don't think anybody can answer) as a consequence of the equations given by Hilbert in the paper he published at almost the same time. A detailed account on this subject can be found on the following Physics SE question: Did Hilbert publish GR before Einstein?, and I can quote Pais' biography of Einstein on this subject (for a beautiful account of this particular episode of the development of GR, check chapters 11-14 of "Subtle is the Lord":
Let us come back to Einstein's paper of November 18. It was written at a time in which (by his own admission) he was beside himself about his perihelion discovery (formally announced that same day), very tired, unwell, and still at work on the November 25 paper [the paper called "The Field Equations of Gravitation"]. It seems most implausible to me that he would have been in a frame of mind to absorb the content of the technically difficult paper Hilbert had sent him on November 18. More than a year later, Felix Klein wrote that he found the equations in that paper so complicated that he had not checked them. [...]
I rather subscribe to Klein's opinion that the two men 'talked past each other, which is not rare among simultaneously productive mathematicians'[...] I again agree with Klein 'that there can be no question of priority, since both authors pursued entirely different trains of thought to such an extent that the compatibility of the results did not at once seem assured'. I do believe that Einstein was the sole creator of the physical theory of general relativity and that both he and Hilbert should be credited for the discovery of the fundamental equation.
I am not sure that the two protagonists would have agreed [included as a funny note].
Subtle is the Lord, Chapter 14, page 260. I think this supports my previous statement quite well.
I think this question can only be answered if we are talking about the teaching of the subject to undergrads in college. If we were talking about one teaching him-/herself GR after learning SR, I believe that the answer would be "well, mostly because one doesn't want to/couldn't do it; in fact, one could have done it before", or something along those lines, which doesn't quite help.
When talking about teaching SR to undergrads, you've got to understand this: there isn't a standard for teaching the theories of Relativity to undergrads (written as an as-of-yet undergrad). We mostly get taught versions of the theory with not so much emphasis on the mathematical structures lying below the subject, and get mostly taught to do calculations and think of the dynamics. So we, in these cases, don't always get to know the concepts of metric, of flat spacetime (which we actually use, but almost never think about it for the course), of curved spacetime, among others.
Add to that the fact that, in many universities (such as mine and the one's of many of my acquaintances), we don't get to learn much math (besides Calculus, some Linear Algebra and Complex Variables) before plunging into the vast world of physics subjects, so we are not in a position to say "oh, of course, this concept that I saw when learning SR can be, by analogy with these things which I haven't learned, extended to a more general (and complicated) one, which would link to some variational principle in order to get some new and complicated tensor equations, which we can't solve".
So well, if you see it from a mathematician's point of view (yours), the next step from SR is quite obvious because of the mathematics you know, but if you are a physics undergrad, who has (most likely) not had an intense training in mathematics, you are certainly not going to be able to take that step. Though I may be generalizing horribly, but all that I've said is true, in my experience.
DrarpDrarp
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Why was pseudo-Euclidean geometry not enough for general relativity? | CommonCrawl |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers.
Algebraic structure → Ring theory
Ring theory
Basic concepts
Rings
• Subrings
• Ideal
• Quotient ring
• Fractional ideal
• Total ring of fractions
• Product of rings
• Free product of associative algebras
• Tensor product of algebras
Ring homomorphisms
• Kernel
• Inner automorphism
• Frobenius endomorphism
Algebraic structures
• Module
• Associative algebra
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• Initial ring $\mathbb {Z} $
• Terminal ring $0=\mathbb {Z} _{1}$
Related structures
• Field
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• Semiring
• Semifield
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Algebraic number theory
• Algebraic number field
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• Algebraic independence
• Transcendental number theory
• Transcendence degree
p-adic number theory and decimals
• Direct limit/Inverse limit
• Zero ring $\mathbb {Z} _{1}$
• Integers modulo pn $\mathbb {Z} /p^{n}\mathbb {Z} $
• Prüfer p-ring $\mathbb {Z} (p^{\infty })$
• Base-p circle ring $\mathbb {T} $
• Base-p integers $\mathbb {Z} $
• p-adic rationals $\mathbb {Z} [1/p]$
• Base-p real numbers $\mathbb {R} $
• p-adic integers $\mathbb {Z} _{p}$
• p-adic numbers $\mathbb {Q} _{p}$
• p-adic solenoid $\mathbb {T} _{p}$
Algebraic geometry
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Algebraic structures
Group-like
• Group
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Ring-like
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• Rng
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Lattice-like
• Lattice
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Module-like
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Algebra-like
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Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication.
Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology.
Introduction and definition
Motivation
In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. In commutative algebra, both ideals and quotient rings are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In non-commutative algebra, the distinction between left ideals, ideals, and modules becomes more pronounced, though some ring-theoretic conditions can be expressed either about left ideals or left modules.
Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible to the realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis, and even those that do, free modules, need not have a unique rank if the underlying ring does not satisfy the invariant basis number condition, unlike vector spaces, which always have a (possibly infinite) basis whose cardinality is then unique. (These last two assertions require the axiom of choice in general, but not in the case of finite-dimensional spaces, or certain well-behaved infinite-dimensional spaces such as Lp spaces.)
Formal definition
Suppose that R is a ring, and 1 is its multiplicative identity. A left R-module M consists of an abelian group (M, +) and an operation · : R × M → M such that for all r, s in R and x, y in M, we have
1. $r\cdot (x+y)=r\cdot x+r\cdot y$
2. $(r+s)\cdot x=r\cdot x+s\cdot x$
3. $(rs)\cdot x=r\cdot (s\cdot x)$
4. $1\cdot x=x.$
The operation · is called scalar multiplication. Often the symbol · is omitted, but in this article we use it and reserve juxtaposition for multiplication in R. One may write RM to emphasize that M is a left R-module. A right R-module MR is defined similarly in terms of an operation · : M × R → M.
Authors who do not require rings to be unital omit condition 4 in the definition above; they would call the structures defined above "unital left R-modules". In this article, consistent with the glossary of ring theory, all rings and modules are assumed to be unital.[1]
An (R,S)-bimodule is an abelian group together with both a left scalar multiplication · by elements of R and a right scalar multiplication ∗ by elements of S, making it simultaneously a left R-module and a right S-module, satisfying the additional condition (r · x) ∗ s = r ⋅ (x ∗ s) for all r in R, x in M, and s in S.
If R is commutative, then left R-modules are the same as right R-modules and are simply called R-modules.
Examples
• If K is a field, then K-vector spaces (vector spaces over K) and K-modules are identical.
• If K is a field, and K[x] a univariate polynomial ring, then a K[x]-module M is a K-module with an additional action of x on M that commutes with the action of K on M. In other words, a K[x]-module is a K-vector space M combined with a linear map from M to M. Applying the structure theorem for finitely generated modules over a principal ideal domain to this example shows the existence of the rational and Jordan canonical forms.
• The concept of a Z-module agrees with the notion of an abelian group. That is, every abelian group is a module over the ring of integers Z in a unique way. For n > 0, let n ⋅ x = x + x + ... + x (n summands), 0 ⋅ x = 0, and (−n) ⋅ x = −(n ⋅ x). Such a module need not have a basis—groups containing torsion elements do not. (For example, in the group of integers modulo 3, one cannot find even one element which satisfies the definition of a linearly independent set since when an integer such as 3 or 6 multiplies an element, the result is 0. However, if a finite field is considered as a module over the same finite field taken as a ring, it is a vector space and does have a basis.)
• The decimal fractions (including negative ones) form a module over the integers. Only singletons are linearly independent sets, but there is no singleton that can serve as a basis, so the module has no basis and no rank.
• If R is any ring and n a natural number, then the cartesian product Rn is both a left and right R-module over R if we use the component-wise operations. Hence when n = 1, R is an R-module, where the scalar multiplication is just ring multiplication. The case n = 0 yields the trivial R-module {0} consisting only of its identity element. Modules of this type are called free and if R has invariant basis number (e.g. any commutative ring or field) the number n is then the rank of the free module.
• If Mn(R) is the ring of n × n matrices over a ring R, M is an Mn(R)-module, and ei is the n × n matrix with 1 in the (i, i)-entry (and zeros elsewhere), then eiM is an R-module, since reim = eirm ∈ eiM. So M breaks up as the direct sum of R-modules, M = e1M ⊕ ... ⊕ enM. Conversely, given an R-module M0, then M0⊕n is an Mn(R)-module. In fact, the category of R-modules and the category of Mn(R)-modules are equivalent. The special case is that the module M is just R as a module over itself, then Rn is an Mn(R)-module.
• If S is a nonempty set, M is a left R-module, and MS is the collection of all functions f : S → M, then with addition and scalar multiplication in MS defined pointwise by (f + g)(s) = f(s) + g(s) and (rf)(s) = rf(s), MS is a left R-module. The right R-module case is analogous. In particular, if R is commutative then the collection of R-module homomorphisms h : M → N (see below) is an R-module (and in fact a submodule of NM).
• If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring C∞(X). The set of all smooth vector fields defined on X form a module over C∞(X), and so do the tensor fields and the differential forms on X. More generally, the sections of any vector bundle form a projective module over C∞(X), and by Swan's theorem, every projective module is isomorphic to the module of sections of some bundle; the category of C∞(X)-modules and the category of vector bundles over X are equivalent.
• If R is any ring and I is any left ideal in R, then I is a left R-module, and analogously right ideals in R are right R-modules.
• If R is a ring, we can define the opposite ring Rop which has the same underlying set and the same addition operation, but the opposite multiplication: if ab = c in R, then ba = c in Rop. Any left R-module M can then be seen to be a right module over Rop, and any right module over R can be considered a left module over Rop.
• Modules over a Lie algebra are (associative algebra) modules over its universal enveloping algebra.
• If R and S are rings with a ring homomorphism φ : R → S, then every S-module M is an R-module by defining rm = φ(r)m. In particular, S itself is such an R-module.
Submodules and homomorphisms
Suppose M is a left R-module and N is a subgroup of M. Then N is a submodule (or more explicitly an R-submodule) if for any n in N and any r in R, the product r ⋅ n (or n ⋅ r for a right R-module) is in N.
If X is any subset of an R-module M, then the submodule spanned by X is defined to be $ \langle X\rangle =\,\bigcap _{N\supseteq X}N$ where N runs over the submodules of M which contain X, or explicitly $ \left\{\sum _{i=1}^{k}r_{i}x_{i}\mid r_{i}\in R,x_{i}\in X\right\}$, which is important in the definition of tensor products.[2]
The set of submodules of a given module M, together with the two binary operations + and ∩, forms a lattice which satisfies the modular law: Given submodules U, N1, N2 of M such that N1 ⊂ N2, then the following two submodules are equal: (N1 + U) ∩ N2 = N1 + (U ∩ N2).
If M and N are left R-modules, then a map f : M → N is a homomorphism of R-modules if for any m, n in M and r, s in R,
$f(r\cdot m+s\cdot n)=r\cdot f(m)+s\cdot f(n)$.
This, like any homomorphism of mathematical objects, is just a mapping which preserves the structure of the objects. Another name for a homomorphism of R-modules is an R-linear map.
A bijective module homomorphism f : M → N is called a module isomorphism, and the two modules M and N are called isomorphic. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements.
The kernel of a module homomorphism f : M → N is the submodule of M consisting of all elements that are sent to zero by f, and the image of f is the submodule of N consisting of values f(m) for all elements m of M.[3] The isomorphism theorems familiar from groups and vector spaces are also valid for R-modules.
Given a ring R, the set of all left R-modules together with their module homomorphisms forms an abelian category, denoted by R-Mod (see category of modules).
Types of modules
Finitely generated
An R-module M is finitely generated if there exist finitely many elements x1, ..., xn in M such that every element of M is a linear combination of those elements with coefficients from the ring R.
Cyclic
A module is called a cyclic module if it is generated by one element.
Free
A free R-module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the ring R. These are the modules that behave very much like vector spaces.
Projective
Projective modules are direct summands of free modules and share many of their desirable properties.
Injective
Injective modules are defined dually to projective modules.
Flat
A module is called flat if taking the tensor product of it with any exact sequence of R-modules preserves exactness.
Torsionless
A module is called torsionless if it embeds into its algebraic dual.
Simple
A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible.[4]
Semisimple
A semisimple module is a direct sum (finite or not) of simple modules. Historically these modules are also called completely reducible.
Indecomposable
An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules which are not simple (e.g. uniform modules).
Faithful
A faithful module M is one where the action of each r ≠ 0 in R on M is nontrivial (i.e. r ⋅ x ≠ 0 for some x in M). Equivalently, the annihilator of M is the zero ideal.
Torsion-free
A torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring, equivalently rm = 0 implies r = 0 or m = 0.
Noetherian
A Noetherian module is a module which satisfies the ascending chain condition on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated.
Artinian
An Artinian module is a module which satisfies the descending chain condition on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.
Graded
A graded module is a module with a decomposition as a direct sum M = ⨁x Mx over a graded ring R = ⨁x Rx such that RxMy ⊂ Mx+y for all x and y.
Uniform
A uniform module is a module in which all pairs of nonzero submodules have nonzero intersection.
Further notions
Relation to representation theory
A representation of a group G over a field k is a module over the group ring k[G].
If M is a left R-module, then the action of an element r in R is defined to be the map M → M that sends each x to rx (or xr in the case of a right module), and is necessarily a group endomorphism of the abelian group (M, +). The set of all group endomorphisms of M is denoted EndZ(M) and forms a ring under addition and composition, and sending a ring element r of R to its action actually defines a ring homomorphism from R to EndZ(M).
Such a ring homomorphism R → EndZ(M) is called a representation of R over the abelian group M; an alternative and equivalent way of defining left R-modules is to say that a left R-module is an abelian group M together with a representation of R over it. Such a representation R → EndZ(M) may also be called a ring action of R on M.
A representation is called faithful if and only if the map R → EndZ(M) is injective. In terms of modules, this means that if r is an element of R such that rx = 0 for all x in M, then r = 0. Every abelian group is a faithful module over the integers or over some ring of integers modulo n, Z/nZ.
Generalizations
A ring R corresponds to a preadditive category R with a single object. With this understanding, a left R-module is just a covariant additive functor from R to the category Ab of abelian groups, and right R-modules are contravariant additive functors. This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab should be considered a generalized left module over C. These functors form a functor category C-Mod which is the natural generalization of the module category R-Mod.
Modules over commutative rings can be generalized in a different direction: take a ringed space (X, OX) and consider the sheaves of OX-modules (see sheaf of modules). These form a category OX-Mod, and play an important role in modern algebraic geometry. If X has only a single point, then this is a module category in the old sense over the commutative ring OX(X).
One can also consider modules over a semiring. Modules over rings are abelian groups, but modules over semirings are only commutative monoids. Most applications of modules are still possible. In particular, for any semiring S, the matrices over S form a semiring over which the tuples of elements from S are a module (in this generalized sense only). This allows a further generalization of the concept of vector space incorporating the semirings from theoretical computer science.
Over near-rings, one can consider near-ring modules, a nonabelian generalization of modules.
See also
• Group ring
• Algebra (ring theory)
• Module (model theory)
• Module spectrum
• Annihilator
Notes
1. Dummit, David S. & Foote, Richard M. (2004). Abstract Algebra. Hoboken, NJ: John Wiley & Sons, Inc. ISBN 978-0-471-43334-7.
2. Mcgerty, Kevin (2016). "ALGEBRA II: RINGS AND MODULES" (PDF).
3. Ash, Robert. "Module Fundamentals" (PDF). Abstract Algebra: The Basic Graduate Year.
4. Jacobson (1964), p. 4, Def. 1
References
• F.W. Anderson and K.R. Fuller: Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, New York, 1992, ISBN 0-387-97845-3, ISBN 3-540-97845-3
• Nathan Jacobson. Structure of rings. Colloquium publications, Vol. 37, 2nd Ed., AMS Bookstore, 1964, ISBN 978-0-8218-1037-8
External links
• "Module", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• module at the nLab
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| Wikipedia |
Sendov's conjecture
In mathematics, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex variable. It is named after Blagovest Sendov.
The conjecture states that for a polynomial
$f(z)=(z-r_{1})\cdots (z-r_{n}),\qquad (n\geq 2)$
with all roots r1, ..., rn inside the closed unit disk |z| ≤ 1, each of the n roots is at a distance no more than 1 from at least one critical point.
The Gauss–Lucas theorem says that all of the critical points lie within the convex hull of the roots. It follows that the critical points must be within the unit disk, since the roots are.
The conjecture has been proven for n < 9 by Brown-Xiang and for n sufficiently large by Tao.[1][2]
History
The conjecture was first proposed by Blagovest Sendov in 1959; he described the conjecture to his colleague Nikola Obreshkov. In 1967 the conjecture was misattributed[3] to Ljubomir Iliev by Walter Hayman.[4] In 1969 Meir and Sharma proved the conjecture for polynomials with n < 6. In 1991 Brown proved the conjecture for n < 7. Borcea extended the proof to n < 8 in 1996. Brown and Xiang[5] proved the conjecture for n < 9 in 1999. Terence Tao proved the conjecture for sufficiently large n in 2020.
References
1. Terence Tao (2020). "Sendov's conjecture for sufficiently high degree polynomials". arXiv:2012.04125 [math.CV].
2. Terence Tao (9 December 2020). "Sendov's conjecture for sufficiently high degree polynomials". What's new.
3. Marden, Morris. Conjectures on the Critical Points of a Polynomial. The American Mathematical Monthly 90 (1983), no. 4, 267-276.
4. Problem 4.5, W. K. Hayman, Research Problems in Function Theory. Althlone Press, London, 1967.
5. Brown, Johnny E.; Xiang, Guangping Proof of the Sendov conjecture for polynomials of degree at most eight. Journal of Mathematical Analysis and Applications 232 (1999), no. 2, 272–292.
• G. Schmeisser, "The Conjectures of Sendov and Smale," Approximation Theory: A Volume Dedicated to Blagovest Sendov (B. Bojoanov, ed.), Sofia: DARBA, 2002 pp. 353–369.
External links
• Sendov's Conjecture by Bruce Torrence with contributions from Paul Abbott at The Wolfram Demonstrations Project
| Wikipedia |
Finding out the area of a triangle if the coordinates of the three vertices are given
What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane?
One approach is to find the length of each side from the coordinates given and then apply Heron's formula. Is this the best way possible?
Is it possible to compare the area of triangles with their coordinates provided without actually calculating side lengths?
geometry triangles coordinate-systems area
nbro
TSP1993TSP1993
What you are looking for is called the shoelace formula:
\begin{align*} \text{Area} &= \frac12 \big| (x_A - x_C) (y_B - y_A) - (x_A - x_B) (y_C - y_A) \big|\\ &= \frac12 \big| x_A y_B + x_B y_C + x_C y_A - x_A y_C - x_C y_B - x_B y_A \big|\\ &= \frac12 \Big|\det \begin{bmatrix} x_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1 \end{bmatrix}\Big| \end{align*}
The last version tells you how to generalize the formula to higher dimensions.
PS. Another generalization of the formula is obtained by noting that it follows from a discrete version of the Green's theorem:
$$ \text{Area} = \iint_{\text{domain}}dx\,dy = \frac12\oint_{\text{boundary}}x\,dy - y\,dx $$
Thus the signed (oriented) area of a polygon with $n$ vertexes $(x_i,y_i)$ is
$$ \frac12\sum_{i=0}^{n-1} x_i y_{i+1} - x_{i+1} y_i $$
where indexes are added modulo $n$.
sdssds
You know that AB x AC is a vector perpendicular to the plane ABC such that |AB x AC|= Area of the parallelogram ABA'C. Thus this area is equal to ½ |AB x AC|.
From AB= $(x_2 -x_1, y_2-y_1)$; AC= $(x_3-x_1, y_3-y_1)$, we deduce then
Area of $\Delta ABC$ = $\frac12$$[(x_2-x_1)(y_3-y_1)- (x_3-x_1)(y_2-y_1)]$
J. W. Tanner
PiquitoPiquito
$\begingroup$ This is the best answer, as it covers the 6005 and the Jailcu answers. Vector cross product. $\endgroup$ – richard1941 Aug 24 '16 at 21:09
$\begingroup$ By this way can area come as negative value? I am trying to understand this problem. geeksforgeeks.org/orientation-3-ordered-points $\endgroup$ – SRIDHARAN Dec 19 '17 at 13:39
$\begingroup$ What has been taken is the absolute value of the cross product. $\endgroup$ – Piquito Dec 20 '17 at 14:31
$\begingroup$ Yes it can be negative. The sign depends on the orientation of the three vertices. $\endgroup$ – steven gregory Feb 9 '18 at 7:03
$\begingroup$ Looking at the parallelogram, why not drop a perpendicular from C to AB. Now there are two right triangles whose area is easily found (You KNOW some coordinates.) No need to divide by 2 b/c the two right triangles cover the triangle under observation. $\endgroup$ – David Dyer Jul 12 '19 at 18:29
Heron's formula is inefficient; there is in fact a direct formula. If the triangle has one vertex at the origin, and the other two vertices are $(a,b)$ and $(c,d)$, the formula for its area is $$ A = \frac{\left| ad - bc \right|}{2} $$
To get a formula where the vertices can be anywhere, just subtract the coordinates of the third vertex from the coordinates of the other two (translating the triangle) and then use the above formula.
The simplest way to remember how to calculate is by taking $\frac{1}{2}$ the value of the determinant of the matrix $$ \begin{bmatrix} 1 & 1 & 1 \\ x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \end{bmatrix} $$
BunsOfWrath
JailcuJailcu
$\begingroup$ See also: Show that the area of a triangle is given by this determinant $\endgroup$ – Zaz Aug 17 '16 at 16:21
if $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ are the vertices of a triangle then its area is given by:-
$$\frac{ 1}{2}|(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2))|$$
ShobhitShobhit
The area of a triangle $P(x_1, y_1)$, $Q(x_2, y_2)$ and $R(x_3, y_3)$ is given by $$\triangle= \left|\frac{1}{2}(x_{1} (y_{2} – y_{3}) + x_{2} (y_{3} – y_{1}) + x_{3} (y_{1} – y_{2}))\right| $$ If the area of triangle is zero, it means the points are collinear.
If we code this in Python3, it will look like
def triangle_area(x1, y1, x2, y2, x3, y3):
return abs(0.5*(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2)))
If we code in js, it will look like this
function triangle_area(x1, y1, x2, y2, x3, y3){
return Math.abs(0.5*(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2)))
Nadir Laskar
aryamanaryaman
7111 silver badge11 bronze badge
Now here's a solution which works in any vector space with an inner product: take the half of the root of the Gram-Determinant of two sides of the triangle, that is $$\frac12\sqrt{\det\begin{pmatrix}\langle b-a,b-a\rangle& \langle b-a,c-a\rangle\\ \langle b-a,c-a\rangle & \langle c-a,c-a\rangle \end{pmatrix}}.$$
Michael HoppeMichael Hoppe
For fun, I'll just throw out the really long way that I learned in 3rd grade, only because it hasn't been submitted. I don't endorse this, the Shoelace/Surveyor's formula is way better.
Determine the distance between two of the three points, say $ \big( x_{1}, y_{1} \big) $ and $ \big( x_{2}, y_{2} \big) $. $d = \sqrt{ \big(x_{2} - x_{1}\big) ^{2} + \big(y_{2} - y_{1}\big) ^{2} } $
Determine the slope $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} $ and y-intercept, $a = y_{1} - \big(m \times x_{1}\big)$, of the line between $ \big( x_{1}, y_{1} \big) $ and $ \big( x_{2}, y_{2} \big) $.
Determine the slope of the a line perpendicular the line from $ \big( x_{1}, y_{1} \big) $ and $ \big( x_{2}, y_{2} \big) $, which is the negative reciprocal of the first slope. $n = \frac{-1}{m} $
Determine the equation of the line parallel to this second line, that passes through the third point $ \big( x_{3}, y_{3} \big)$, by finding the y-intercept in $y = n*x+b$, since you already have the slope and a point on the line. $ y_{3} - \big(n \times x_{3}\big)=b$.
Determine where this new line intersects the line between $ \big( x_{1}, y_{1} \big) $ and $ \big( x_{2}, y_{2} \big) $, by solving the system of equations of the new line and the original line: $y = m*x+b$ and $y = n*x+b$. Call this point $ \big( x_{4}, y_{4} \big) $. I won't write this out, I'll leave it as an "exercise for the reader".
Determine the distance between $ \big( x_{3}, y_{3} \big) $ and the new point$ \big( x_{4}, y_{4} \big) $. $c = \sqrt{ \big(x_{4} - x_{3}\big) ^{2} + \big(y_{4} - y_{3}\big) ^{2} } $
If $d$ is the base of the triangle, and $c$ the height, the area is $A = \frac{1}{2} c*d$.
Realize you've spent several minutes solving a trivial problem... cry silently.
iwantmyphdiwantmyphd
$\begingroup$ You learnt this in third grade?! $\endgroup$ – YiFan Jan 12 '19 at 0:34
The area $A$ of the triangle two of whose vertices lie on the axes, with coordinates $(a, 0)$, $(0, b)$, and a third vertex $(c, d)$ is obtained from previous formula by a mere horizontal axis shift of -a units as $$A = \frac{|-ad + b(a - c)|}{2}$$
Kamil Jarosz
Beedassy LekrajBeedassy Lekraj
$\begingroup$ What "previous formula" do you mean? $\endgroup$ – Rory Daulton May 25 '15 at 11:41
$\begingroup$ Sorry to be vague about that ! I meant the formula A = |ad - bc|/2 for triangle with coordinates (a, b) , (c, d) and origin. $\endgroup$ – Beedassy Lekraj May 27 '15 at 4:56
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\begin{document}
\title{Constructions for the optimal pebbling of grids}
\author[1,2]{Ervin Gy\H{o}ri\thanks{[email protected]}} \author[3,4]{Gyula Y. Katona\thanks{[email protected]}} \author[3]{L\'aszl\'o F. Papp\thanks{[email protected]}} \affil[1]{Alfr\'ed R\'enyi Institute of Mathematics, Budapest, Hungary} \affil[2]{Department of Mathematics, Central European University, Budapest, Hungary} \affil[3]{Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Hungary} \affil[4]{MTA-ELTE Numerical Analysis and Large Networks Research Group, Hungary}
\date{Received: date / Accepted: date}
\maketitle
\begin{abstract}
In [\textit{Xue, Yerger: \emph{Optimal Pebbling on Grids}, Graphs and Combinatorics \textbf{(32)} no.~3}] the authors conjecture that if every vertex of an
infinite square grid is reachable from a pebble distribution, then
the covering ratio of this distribution is at most $3.25$. First we
present such a distribution with covering ratio $3.5$, disproving
the conjecture. The authors in the above paper also claim to prove that
the covering ratio of any pebble distribution is at most
$6.75$. The proof contains some errors. We present a
few interesting pebble distributions that this proof does not seem
to cover and highlight some other difficulties of this topic.
\end{abstract}
\keywords{optimal pebbling, pebbling, grid graph} \section{Introduction} \label{intro} Graph pebbling has its origin in number theory. It is a model for the transportation of resources. Starting with a pebble distribution on the vertices of a simple connected graph, a \emph{pebbling move} removes two pebbles from a vertex and adds one pebble at an adjacent vertex. We can think of the pebbles as fuel containers. Then the loss of the pebble during a move is the cost of transportation. A vertex is called \emph{reachable} if a pebble can be moved to that vertex using pebbling moves. There are several questions we can ask about pebbling. One of them is: How can we place the smallest number of pebbles such that every vertex is reachable? The minimum number of pebbles in such a pebble distribution is called the \emph{optimal pebbling
number} of the graph. The \emph{optimal covering ratio of a graph} is the number of vertices of the graph divided by the optimal
pebbling number. Moreover, the \emph{covering ratio of an arbitrary distribution} is the number of vertices reachable from the distribution divided by the number of pebbles in the distribution.
For a
comprehensive list of references for the extensive literature see the survey papers \cite{Hurlbert_survey1,Hurlbert_survey2,Hurlbert_survey3}.
In Section~\ref{sec:1} we show a pebble distribution which disproves a conjecture of Xue and Yerger. Section~\ref{sec2} and \ref{sec3} contain some interesting counterexamples for some lemmas stated in \cite{XY}. We also mention some phenomenons why we think that the proof of Theorem 8 of \cite{XY} can not be corrected, and a proper proof requires a different approach.
In the last section we introduce a new problem, called \emph{optimal integer fractional covering ratio}, where the tools introduced in \cite{XY} can be used. We give a lower and an upper bound on the optimal integer fractional covering ratio of large grids.
\section{Definitions} In this section we summarize the definitions which the paper uses. We start with basic ones, which are well known in the area of pebbling, then continue with more complicated ones, which were introduced in \cite{XY}.
\subsection{Traditional pebbling}
A \emph {pebbling distribution} $D$ is a $V(G)\rightarrow \mathbb{N}$ function. If $D(v)\geq 2$ and $u$ and $v$ are adjacent vertices, then we can apply a $(v\rightarrow u)$ \emph{pebbling move}. It decreases $D(v)$ by two and increases $D(u)$ by one. A vertex $v$ is \emph{reachable} under $D$ if either $D(v)\geq 1$ or we can apply a sequence of pebbling moves such that the last one is an $(u\rightarrow v)$ move.
A distribution is \emph{solvable} if each vertex is reachable under it. We use $|D|$ for the \emph{size of distribution} $D$, which is the total number of pebbles placed on the graph. A distribution is \emph{optimal} on graph $G$ if its size is minimal among all solvable distributions of $G$. \emph{The optimal pebbling number} is the size of an optimal distribution and it is denoted by $\pi_{\opt}(G)$.
The \emph{coverage} of distribution $D$ is the set of reachable vertices. We denote the size of this set by $\Cov(D)$.
The \emph{covering ratio} of $D$ is defined as $\frac{\Cov(D)}{|D|}$.
\subsection{Infinite graphs} The infinite square grid is denoted by $G_{inf}$. In \cite{XY} the authors talk about $G_{inf}$, they do not provide a proper definition for the covering ratio in the case when the distribution is infinite. In their reasoning they assume that the number of pebbles is finite, therefore we will assume the same. When solvability comes into play, we think about arbitrary large, but finite square grids, whose border's size is marginal compared to their total size. Therefore their covering ratio is well defined. We define the optimal covering ratio of $G_{inf}$ as the limit of larger and larger square grids's optimal covering ratios.
\begin{claim} This limit exists. \end{claim}
\begin{proof} We prove, that the reciprocal of the series is convergent. We denote the $n\times n$ grid by $G_{n\times n}$.
Let $\epsilon$ be an arbitrary positive real number and $A=\liminf_{n\rightarrow \infty} \frac{\pi_{\opt}(G_{n\times n})}{n^2}$. Choose $m$ large enough such that $\frac{6}{m}<\epsilon$, $\frac{\pi_{\opt}(G_{m\times m})}{m^2}-A\leq \frac{\epsilon}{2}$ and for every $n\geq m$ the inequality $A-\frac{\pi_{\opt}(G_{n\times n})}{n^2}<\epsilon$ holds. We show that $n\geq m^2$ implies $\left|A-\frac{\pi_{\opt}(G_{n\times n})}{n^2}\right|<\epsilon$.
Write $n$ as $km+r$ and partition $G_{n\times n}$ to $k^2$ piece of disjoint $G_{m\times m}$ and $r^2+2rkm$ remaining vertices. We use its optimal distribution on each $G_{m\times m}$ and place one pebble to each remaining vertex. In such a way we obtain a solvable distribution of $G_{n\times n}$. Therefore:
$$-\epsilon<\frac{\pi_{\opt}(G_{n\times n})}{n^2}-A\leq \frac{k^2\pi_{\opt}(G_{m\times m})+r^2+2rkm}{k^2m^2+r^2+2rkm}-A\leq \frac{\pi_{\opt}(G_{m\times m})}{m^2}-A+\frac{3}{m}<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon$$
\end{proof}
We note, that there are several ways to define pebbling parameters for infinite graphs by considering infinite distributions. Nevertheless, it is beyond the scope of this paper.
\subsection{Combining distributions}
Assume that we have two distributions $D$ and $D'$. We say that these two distributions \emph{interact} at vertex $v$ if $v$ is reachable under both. Vertex $v$ is a \emph{boundary vertex} of $D$ if $v$ is reachable under $D$ but one of its neighbours is not.
It is a natural idea to unify two distributions $D$ and $D^*$ by placing $D(v)+D^*(v)$ pebbles at $v$, to create a bigger one. $D$ and $D^*$ are stronger together in the sense, that some vertices are not reachable under $D$ nor $D^*$, but they are reachable under the $D'$ which we get by combining them. This phenomenon requires the presence of interacting vertices.
For example, if an interaction vertex is boundary in both $D$ and $D^*$ and one of its neighbours is not reachable under $D$ and $D^*$, then this neighbour is reachable under $D'$.
A \emph{unit} is a vertex having at least one pebble. A \emph{unit distribution} contains only one unit.
Using the combination method we can build any distribution from unit distributions. It often happens, that the coverage of a unit is disjoint from coverage of the rest of the distribution. So the unit and the rest do not share an interaction vertex and they can be handled separately. We say that these units are \emph{lonely}.
We can also ask that what is the difference between the covering ratios of $D$ and $D'$. Of course, it depends on $D^*$, but we would like to measure it. This motivates the definition of \emph{marginal covering ratio}, which is the following:
$$\frac{\Cov(D')-\Cov(D)}{|D'|-|D|}.$$
We can not compute the covering ratio of $D'$ if we know the covering ratio of $D$ and the marginal covering ratio. On the other hand, we can state upper bounds, which we are interested in.
\subsection{Fractional pebbling} A variation of the pebbling problem if we allow fractional pebbles. This leads to the area of fractional pebbling, which is well studied in \cite{frac}.
A \emph{continuous distribution} on $G$ is a $V(G)\rightarrow \mathbb{R}^+\cup\{0\}$ function. A \emph{continuous pebbling move} removes $t$ pebbles from a vertex and place $t/2$ pebbles to an adjacent vertex, where $t$ can be any positive real number.
The \emph{optimal fractional pebbling} number is the size of the smallest solvable continuous distribution. It can be calculated by solving a linear program and it is a lower bound on the optimal pebbling number.
Let $D$ be a continuous distribution. The \emph{weight function} of $D$, which is defined on the vertex set of $G$, is defined as: $$W_D(u)=\sum_{v\in V(G)}D(v)2^{-d(u,v)}.$$ It tell us that how many pebbles can be moved to a vertex under $D$ by continuous pebbling moves. It is a very useful tool, widely used by several authors in this topic.
In a solvable distribution the weight of each vertex is at least one. However, if we are considering optimal pebbling distributions, then several vertices have more weight than one. We can consider this extra weight as an excess, and try to calculate it. The sum of these values gives an estimate on the difference between the fractional and the traditional optimal pebbling numbers.
So we define the \emph{excess weight} function as: $$\widehat{W}_D(u)=\begin{cases}W_D(u)-1 &\text{if } W_D(u)>1,\\ W_D(u) &\text{if } W_D(u)\leq 1. \end{cases}$$
With the help of this function, we can give a fractional generalization of covering ratio. To construct the numerator we count the number of reachable vertices and also add the weight of not reachable vertices. The \emph{covering ratio ceiling} of $D$ is:
$$\frac{\sum_{v\in V(G)}W_D(v)-\sum_{v\in V(G)}\widehat{W}_D(v)}{\sum_{v\in V(G)}D(v)}.$$
Like the marginal covering ratio we also define a quantity which measures in some way the change of covering ratio ceiling in case of adding some extra pebbles. So let $D$ and $D'$ be distributions, such that $D(v)\leq D'(v)$ for each vertex. The \emph{marginal covering ratio ceiling} of these distributions is defined by
$$\frac{\left(\sum_{v\in V(G)}W_D'(v)-\sum_{v\in V(G)}\widehat{W}_D'(v)\right)-\left(\sum_{v\in V(G)}W_D(v)-\sum_{v\in V(G)}\widehat{W}_D(v)\right)} {\sum_{v\in V(G)}D'(v)-\sum_{v\in V(G)}D(v)}.$$
\section{Lower bound for the optimal covering ratio of the grid} \label{sec:1}
Conjecture 2 in \cite{XY} states that if every vertex of an infinite square grid is reachable from a pebble distribution, then the covering ratio of this distribution is at most $3.25$.
We present a sequence of distributions on big grids whose covering ratios converge to $3.5$, disproving the conjecture. Repeating periodically the optimal distribution of $G_{n\times n}$ results a solvable distribution on the infinite grid. Therefore considering real infinite graphs and distributions can not decrease the optimal covering ratio, if it is defined intuitively.
A distribution is shown on Figure~\ref{fig:27distr}, units consisting of four pebbles are placed to every other vertex on every 7th diagonal. It is easy to calculate that the covering ratios of such distributions tends to $7/2=3.5$. The arrows and the shaded areas on the figure indicate how to reach all vertices of the grid from this distribution. We conjecture that this is best possible.
\begin{figure}
\caption{Pebble distribution of the grid with covering ratio $3.5$.}
\label{fig:27distr}
\end{figure}
\begin{conjecture} If every vertex of an infinite square grid is reachable from a pebble distribution, then the covering ratio of this distribution is at most $3.5$. \end{conjecture}
\section{Comments on the upper bound for the optimal covering ratio
of the grid}\label{sec2}
In \cite[Section 6]{XY} the authors claim to prove that the optimal covering ratio of the grid is at most $6.75$ (see \cite[Theorem 8]{XY}). The proof contains some errors. Although some of these errors may be corrected somehow, in our opinion some others cannot be corrected. We do not see how to complete the proof, but we believe that the statement is true. In forthcoming paper \cite{GKPT} we are to prove a better bound: the covering ratio is at most $6.5$. In the rest of this note we point out some errors in the proof and show some interesting pebble distributions that highlight the difficulties of this problem.
First we summarize the above mentioned proof. It is an inductive proof on the number of units contained in the distribution. First, as the base case, it is shown that the theorem holds for one unit. Now assume that it also holds for any distribution containing $n$ units. Consider a distribution of $n+1$ units, remove an arbitrary unit, apply the inductive hypothesis for the remaining units. Depending on the position of the removed unit and the remaining distribution apply \cite[Lemma 12]{XY} or \cite[Lemma 14]{XY} to complete the proof. (In the original paper in Section 6 all reference to the lemmas are shifted by one, so any reference to Lemma $i$ should be to Lemma $i-1$.)
\subsection{Comments on \cite[Lemma 12]{XY}} \label{sec:2.1}
The lemma states: \textit{For initial distribution $D$ and unit $U$,
if only the boundary vertices of $U$ are reachable via pebbles from
$D$, then the marginal covering ratio of a unit on $G_{inf}$ is at
most $4.25$.}
We present several counterexamples to this lemma. \begin{figure}
\caption{Distribution with large marginal covering ratio.}
\label{fig:ell1}
\end{figure}
In the first example (see Fig. \ref{fig:ell1}) let $D$ be the
distribution consisting units of size 1 in a horizontal row. The
covering ratio of $D$ is clearly $1$. Now let $U$ be a unit of size
$2$ placed at the end of this row. Only a boundary vertex of $U$ is
reachable from $D$ (i.e. a unit of $D$ is on the boundary of $U$),
so the conditions of the lemma are satisfied. However after adding
$U$, all vertices in the shaded area become reachable. Since the size
of $U$ is constant (it is $2$), the marginal covering ratio depends
on the size of $D$. It can be arbitrary large even if we consider
finite $D$ distributions, and it can be infinite if $D$ is made
infinitely large.
\begin{figure}
\caption{Distribution with large marginal covering ratio.}
\label{fig:ell2}
\end{figure}
Another example is shown on Fig. \ref{fig:ell2}, where a similar
problem appears. Adding a unit of size $1$ can increase the number
of newly reachable vertices by an arbitrary large amount.
The proof presented in \cite{XY} works only for the following weaker statement: \begin{lemma}\label{newclaim}
Let $D$ be a distribution and $U$ be a unit such that only the
boundary vertices of $D$ and $U$ interact. Assume, moreover, that if
$D$ has a unit of size one, then $U$ also has size one. Then, the
marginal covering ratio of $U$ on $G_{{inf}}$ is at most $4.25$. \end{lemma}
\subsection{Comments on the inductive step}
In the inductive step, we assume that there is a distribution $D$ with covering ratio at most $6.75$. Now we add a unit $U$. The authors do not explain this step in detail, but we assume that their intention is to apply \cite[Lemma 12]{XY} if \textit{``only the
boundary vertices of $U$ are reachable via pebbles from $D$''}, and apply \cite[Lemma 14]{XY} if \textit{``the unit interacts not only on
the boundary vertices''}.
The examples in the previous subsection show that in some cases, when some units of size 1 are involved, \cite[Lemma 12]{XY} cannot be used. We think that the authors intended to handle this problem with the following sentence in the proof of \cite[Lemma 12]{XY}: \textit{``We assume that $D$ does not contain lonely units with one
pebble because if $D$ contains those units, we can remove them first
and add them after $U$ has been added.''} This suggests that in the inductive step one should be more careful how to select the unit to be removed, remove the lonely units first. The above example suggests that this does not work. On the other hand, it looks promising to always remove a unit which is on the ``boundary'' of $D$, but now the boundary is understood differently, something like the ``convex hull''. However, this approach does not look easy.
Now let us consider the case when \cite[Lemma 14]{XY} is applied. (The lemma in fact states, that the marginal covering ratio ceiling in this case is at most $6$, however, the proof gives $6.75$, so this is clearly a typo). We give an example, when this fails to prove the inductive step: Two units of size $2$ on adjacent vertices of the grid. In this case one of the units is $D$ the other is $U$, the interaction happens not only on the boundaries. The covering ratio of $D$ is $2.5<6.75$, so the inductive hypothesis holds. Now \cite[Lemma 14]{XY} implies that the marginal covering ratio ceiling of $U$ is at most $6.75$. These facts do not imply that the covering ratio of $D\cup U$ is at most $6.75$. (Of course, the covering ratio is in fact $2<6.75$, just the proof does not imply this.)
The covering ratio is $\Cov(D)/|D|$. However, in the definition of the marginal covering ratio ceiling neither $\Cov(D)$ nor, more importantly, $\Cov(D\cup U)$ appear, so it seems impossible that these two inequalities would imply anything useful for $\Cov(D\cup U)$. We suspect that the intention of the authors was to say that if the marginal covering ratio \textit{ceiling} of $D$ is at most $6.75$ and the marginal covering ratio ceiling of the pair $(D,U)$ is at most $6.75$, then the covering ratio \textit{ceiling} of $D\cup U$ is at most $6.75$. This implication is correct, but then we do not see how to obtain the desired bound for the covering ratio. Of course, since the covering ratio ceiling is an upper bound for the covering ratio, it would be enough to prove with the previous argument that the covering ratio ceiling is at most $6.75$. The above argument does not give this. The covering ratio ceiling of $D\cup U$ is in fact $7.25$, implying only that the covering ratio is at most $7.25$. The basic problem is that the covering ratio ceiling of the only unit of size 2 in $D$ is $8.5$.
\section{Connection between the covering ratio and the covering ratio ceiling} \label{sec3}
The covering ratio ceiling is an upper bound for the covering ratio. It is also clear that the marginal covering ratio ceiling cannot be large if we add a new unit to a distribution. Does this imply something about the change in the covering ratio?
\begin{theorem} The distribution given in Fig.~\ref{fig:ell3} shows that the covering ratio can increase by more than one while the covering ratio ceiling decreases. \end{theorem}
\begin{figure}
\caption{Pebble distribution with increasing covering ratio.}
\label{fig:ell3}
\end{figure}
\begin{proof}
Let us consider the distribution in Figure \ref{fig:ell3}. A unit of size $3$ on every second
vertex in a row of length $2n+1$ ($n+1$ such units in a row), repeated in
every fifth row (see
Fig.~\ref{fig:ell3}) in $5m+1$ rows. The marginal covering ratio ceiling of a unit
is $\frac{\Delta W-\Delta\widehat{W}}{\Delta |D|}$, where
$\Delta W$ is the sum of the changes of the weights which all pebbles
contributes to a vertex, $\Delta\widehat{W}$ is the same for the
weight ceiling function, and $\Delta |D|$ is the number of added
pebbles.
Let us first see the weights at all vertices in this
distribution. For example the total weight for every vertex in row
$0$ is clearly at least $3$. Thus the total weight at every vertex in
row $1$ is at least $3/2$. The weight for every vertex in row 2 is
at least $3/4+3/8=9/8$, since the pebbles in row $0$ contribute at least $3/4$
and pebbles in row $5$ contribute at least $3/8$ to its weight. The same
is true for all other rows, therefore it is clear that the weight of
every vertex is at least $1$.
If the weight of a vertex is at least $1$ then any positive change in the
total weight will result in the same amount of positive change of
$\widehat{W}$. So these vertices contribute $0$ to $\Delta
W-\Delta\widehat{W}$. This implies that if further units are added
to this distribution, then their marginal covering ratio ceiling
will always be $0$. This calculation also shows that the covering
ratio ceiling is $$\frac{(5m+1)(2n+1)}{3(n+1)(m+1)}<\frac{10}{3}.$$
Let us calculate the covering ratio of the distribution. It is easy
to see that each vertex in rows $0,1,4,5,6,9,10,$ $11,\ldots$ is
reachable, but no vertex in rows $2,3,7,8,\ldots$ is
reachable. Therefore, the covering ratio is
$$\frac{(3m+1)(2n+1)}{3(n+1)(m+1)}<2.$$
Now we add units of size $2$ near both ends of the row containing
pebbles (the lighter, smaller pebbles on Fig.~\ref{fig:ell3}) one by
one. The above argument shows that the marginal covering ratio
ceiling is $0$ in every step, and the covering ratio ceiling becomes
$$\frac{(5m+1)(2n+1)}{3(n+1)(m+1)+4m}<\frac{(5m+1)(2n+1)}{3(n+1)(m+1)}<\frac{10}{3},$$
so it is decreasing. On the other hand, it is easy to see that one
can move $4$ pebbles to any vertex of row $5k$, thus every vertex of
the grid becomes reachable. Therefore the covering ratio is also
$$\frac{(5m+1)(2n+1)}{3(n+1)(m+1)+4m}$$ which is close to
$\frac{10}{3}$ if $n$ and $m$ are large enough. So while the
covering ratio ceiling is decreasing, the covering ratio is
increasing from $2$ to $\frac{10}{3}$. \end{proof}
\section{Optimal fractional covering ratio of integer distributions}
The above arguments lead to an interesting question. What is the best upper bound we can hope for using \cite[Lemma 14]{XY}? The idea behind this lemma is that the fractional covering ratio is an upper bound on the covering ratio, and if the starting pebble distribution has only integer number of pebbles, then there must be a certain amount of excess weight. It is easy to prove that the optimal fractional covering ratio on the grid is $9$. The optimal distribution is obtained by placing $1/9$ pebbles at every vertex. How does this change if we only consider integer distributions? Let us call this the optimal \emph{integer fractional covering ratio}, and denote it by $\ifcov(G)$. We give upper and lower bounds for this ratio in case of the $n\times n$ grids .
\begin{theorem}\label{tetel} For any $\varepsilon>0$ there exists
$n(\varepsilon)$ such that if $n>n(\varepsilon)$ then \[ 7-\varepsilon \le {\ifcov}(G_{n\times n}) \le \frac{213}{25}=8.52. \] \end{theorem}
\begin{proof} Consider the distribution given in
Fig.~\ref{fig:frac}. It is easy to calculate, that the number of
pebbles is $n^2/7+O(n)$, therefore if $n$ is large enough, then
$\ifcov(G_{n\times n}) \ge 7-\varepsilon$. We need to show, that
every vertex of the grid is covered by this distribution in the
fractional sense, i.e. the weight of each vertex is at least
one. This is clearly true for the vertices, where a pebble is
placed. By the structure of the distribution, it is clear that it is
enough to show this for the vertices marked with $A,B,C$. For $A$:
there is $1$ pebble at distance $1$, $1$ at distance $2$, $1$ at
distance $3$ and $3$ at distance $4$. Thus the weight at $A$ is at least $
\frac12+\frac14+\frac18+\frac3{16}>1$. Similar calculations show
that the weight at $B$ is at least $
\frac24+\frac38+\frac1{16}+\frac{2}{32}=1$, and the weight at $C$ is
at least $ \frac{1}{2}+\frac14+\frac18+\frac3{16}>1$. For vertices near
the border, the extra pebbles placed on the border guarantees the
required weight. Hence the lower bound is proven.
To prove the upper bound, the following Lemma is needed. \begin{lemma}\label{lem2} If a distribution covers all vertices of the
grid in the fractional sense and the vertex $v$ contains a unit of
size $k$ then the excess weight at this vertex is at least
$\frac{12}{25}k$. \end{lemma}
\begin{proof}
If $k>1$ then the excess weight is at least $k-1>\frac{12}{25}k$, so
the claim clearly holds. Therefore we only have to deal with the
$k=1$ case. The contribution of this pebble to the weight of its
neighbours is $\frac12$, thus the contribution of other pebbles must
also be at least $\frac12$. Similarly, the contribution of this
pebble is $\frac14$ to the vertices at distance 2 from $v$, thus the
contribution of other pebbles must also be at least $\frac34$. So
consider the distance 2 neighborhood of $v$ and partition all
vertices of the grid according to Fig.~\ref{fig:frac-also}. For
simplicity denote by $x_i$ ($y_i$) the total weight contribution of
all pebbles in region $X_i$ ($Y_i$) to the corresponding
vertex. Consider now for example vertex $x_1$. The weight from
pebbles in $X_1$ is $x_1$, the weight from pebbles in $Y_1$ is
clearly $\frac12 y_1$, since the distance of any pebble in $Y_1$ and
$x_1$ is one more than the distance to $y_1$. Similarly, the weight
from $X_2$ to $x_1$ is $\frac14x_2$, from $Y_2$ to $x_1$ is
$\frac18y_2$, etc. Since the total weight at $x_1$ must be at least
$1$ and the contribution of $v$ is $\frac12$, \begin{equation*} 1\le \frac12+x_1+\frac14x_2+\frac14x_3+\frac14x_4+\frac12y_1+\frac18y_2+\frac18y_3+\frac12y_4 \end{equation*} must hold. For all $x_i$ and $y_i$ we can obtain similar inequalities: \begin{align*} 1\le &\frac12+x_1+\frac14x_2+\frac14x_3+\frac14x_4+\frac12y_1+\frac18y_2+\frac18y_3+\frac12y_4\\ 1\le &\frac12+\frac14x_1+x_2+\frac14x_3+\frac14x_4+\frac12y_1+\frac12y_2+\frac18y_3+\frac18y_4\\ 1\le &\frac12+\frac14x_1+\frac14x_2+x_3+\frac14x_4+\frac18y_1+\frac12y_2+\frac12y_3+\frac18y_4\\ 1\le &\frac12+\frac14x_1+\frac14x_2+\frac14x_3+x_4+\frac18y_1+\frac18y_2+\frac12y_3+\frac12y_4\\ 1\le &\frac14+\frac12x_1+\frac12x_2+\frac18x_3+\frac18x_4+y_1+\frac1{4}y_2+\frac1{16}y_3+\frac1{8}y_4\\ 1\le &\frac14+\frac18x_1+\frac12x_2+\frac12x_3+\frac18x_4+\frac1{4}y_1+y_2+\frac1{4}y_3+\frac1{16}y_4\\ 1\le &\frac14+\frac18x_1+\frac18x_2+\frac12x_3+\frac12x_4+\frac1{16}y_1+\frac1{4}y_2+y_3+\frac1{4}y_4\\ 1\le &\frac14+\frac12x_1+\frac18x_2+\frac18x_3+\frac12x_4+\frac1{4}y_1+\frac1{16}y_2+\frac1{4}y_3+y_4 \end{align*} The vertex $v$ contains $1$ pebble, so the weights coming from other vertices will give excess weight at $v$. This excess weight is $\frac12(x_1+x_2+x_3+x_4)+\frac14(y_1+y_2+y_3+y_4)$. To determine the minimum value of the excess weight, so that the above inequalities are satisfied, a linear program can be solved. Using duality, one can easily verify that this minimum is $\frac{12}{25}$. (The minimum is taken when $x_i=0$ and $y_i=\frac{12}{25}$.) \end{proof}
\begin{figure}
\caption{Pebble distribution with $7-\varepsilon \le \ifcov(G_{n\times n})$ }
\label{fig:frac}
\end{figure}
Let $D$ be any pebble distribution. An easy calculation shows
that the sum of the weight contributions of a single pebble to all
other vertices on the grid is $9$, therefore the sum of weights for
every vertex on the grid cannot exceed $9|D|$. If $D$ covers every
vertex on the grid then the weight at every vertex is at least $1$,
however, if the vertex contains $k$ pebbles then the excess weight
on this vertex is at least $\frac{12}{25}k$ by
Lemma~\ref{lem2}. Hence $9|D|\ge n^2+\frac{12}{25}|D|$ holds. This
implies that $\frac{n^2}{|D|}\le
9-\frac{12}{25}=\frac{213}{25}=8.52$, proving our claim. \end{proof}
\begin{figure}
\caption{Proof of the upper bound}
\label{fig:frac-also}
\end{figure}
Theorem~\ref{tetel} implies that the best upper bound for the
(integer) covering ratio we can hope using the approach of integer fractional covering is $7$.
\end{document} | arXiv |
\begin{document}
\title{\textbf{Liouville theorems on the upper half space}} \author {Lei Wang and Meijun Zhu} \address{ Lei Wang, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190, P.R. China, and Department of Mathematics, The University of Oklahoma, Norman, OK 73019, USA}
\email{[email protected]}
\address{ Meijun Zhu, Department of Mathematics, The University of Oklahoma, Norman, OK 73019, USA}
\email{[email protected]}
\maketitle
\noindent \begin{abs} In this paper we shall establish some Liouville theorems for solutions bounded from below to certain linear elliptic equations on the upper half space. In particular, we show that for $a \in (0, 1)$ constants are the only $C^1$ up to the boundary positive solutions to $div(x_n^a \nabla u)=0$ on the upper half space.
\end{abs}\\
\section{\textbf{Introduction}\label{Section 1}}
In this paper we shall establish some Liouville theorems for solutions bounded from below to certain linear elliptic equations on the upper half space. These results imply the uniqueness property to various extension operators on the upper half space. They also provide us a new view point on how to obtain positive kernels for the extension operators. The elliptic properties and estimates, as well as the geometric applications of these extension operators were widely studied recently, see, for example, Caffarelli and Silvestre \cite{CS07}, Hang, Wang and Yang \cite{HWY08}, Chen \cite{Chen14}, Dou and Zhu \cite{DZ15}, Dou, Guo and Zhu \cite{DGZ17}, Gluck \cite{G18}, and references therein.
\subsection{Main results} Denote $\mathbb{R}^n_+=\{x=(x', x_n) \in \mathbb{R}^n \ : \ x_n>0\}$ as the upper half space.
We shall prove
\begin{theorem}\label{main-1}
For $n\geq2$ and $a\in\mathbb{R}$, let $u(x) \in C^2(\mathbb{R}^n_{+})\cap C^0(\overline {\mathbb{R}^n_{+}})$ be a solution to
\begin{equation}\label{equ}
\begin{cases}
div(x_{n}^{a} \nabla u)=0, & \quad u>-C \quad \text{in}\quad \mathbb{R}^n_+,\\
u=0, &\quad\text{on}\quad \partial\mathbb{R}^n_+.
\end{cases}
\end{equation} Then $u=C_*x_n^{1-a}$ for some nonnegative constant $C_*$;
\end{theorem}
For Neumann boundary condition, we have
\begin{theorem}\label{main-2}
Assume $n\geq2$ and $\max\{-1,2-n\}< a<1$. Suppose $u(x) \in C^2(\mathbb{R}^n_{+})\cap C^1(\overline {\mathbb{R}^n_{+}})$ satisfies
\begin{equation}\label{equ-1}
\begin{cases}
div(x_{n}^{a} \nabla u)=0, &\quad u>0, \quad \text{in}\quad \mathbb{R}^n_+,\\
x_n^a\frac {\partial u}{\partial x_n}=0&\quad\text{on}\quad \partial\mathbb{R}^n_+.
\end{cases}
\end{equation}
Then $u=C$ for some positive constant $C$. \end{theorem}
The boundary condition in \eqref{equ-1} holds in the following sense: \begin{equation}\label{bdry-1} \lim_{x_n \to 0^+} x_n^a\frac {\partial u}{\partial x_n}=0. \end{equation}
Note that for $a>0$, if $u(x) \in C^1(\overline {\mathbb{R}^n_{+}})$, it automatically satisfies \eqref{bdry-1}. We immediately have the following result. \begin{corollary}\label{str-1} Assume $n\ge2$ and $0<a<1$. Suppose $u(x) \in C^2(\mathbb{R}^n_{+})\cap C^1(\overline {\mathbb{R}^n_{+}})$ satisfies
\begin{equation}\label{equ-1-11}
div(x_{n}^{a} \nabla u)=0, \quad u>0, \quad \text{in}\quad \mathbb{R}^n_+.
\end{equation} Then $u=C$ for some positive constant $C$.
\end{corollary}
Corollary \ref{str-1} is quite striking: there is {\it no assumption} on the boundary value of $u(x)$. It is worth pointing out that the result in Corollary \ref{str-1} does not hold for $a=0$. And the condition $u(x) \in C^1(\overline {\mathbb{R}^n_{+}})$ can not be weakened since $u(x)=x_n^{1-a}$ does satisfy equation \eqref{equ-1-11} and is positive on the upper half space.
Combining Corollary \ref{str-1} with the classical Liouville Theorem for positive harmonic functions in the whole space, we have the following generalized Liouville Theorem.
\begin{corollary}\label{str-2} Assume $n\ge2$ and $0\le a<1$. Any positive $C^2(\mathbb{R}^n) $ solution to
\begin{equation}\label{equ-1-12}
div(|x_{n}|^{a} \nabla u)=0, \quad \text{in}\quad \mathbb{R}^n
\end{equation} must be a constant function.
\end{corollary}
We illustrate some motivations for our work below.
\subsection{Unique solution to the extension operators} In \cite{CS07}, Caffarelli and Silvestre study the following extension problem for $a \in (-1, 1)$: \begin{equation}\label{equ1.2-1}
\begin{cases}
div(x_{n}^{a} \nabla u)=0, & \quad \text{in}\quad \mathbb{R}^n_+,\\
u(x', 0)=f(x'), & \quad\text{on}\quad \partial\mathbb{R}^n_+.
\end{cases}
\end{equation} Besides many interesting properties were obtained, their study provides a nice ``pointwise'' view on a global defined fractional Laplacian operator: $$(-\Delta)^{\frac{1-a}2}f(x')=- \lim_{x_n \to 0^+} x_n^a\frac {\partial u}{\partial x_n}(x', x_n).$$
For $f(x')$ in a good space (for example, Fourier transform can be applied on $f(x')$), solution $u(x', x_n)$ to \eqref{equ1.2-1} can be represented, up to a constant multiplier, by \begin{equation}\label{equ1.2-2}
u(x', x_n)= \int_{\partial \mathbb{R}_+^n} \frac{ x_n^{{1-a}} f(y)}{(|x'-y|^2+x_n^2)^{\frac{n-a}2}}dy. \end{equation} One can also view $u(x', x_n)$ as an extension of $f(x')$ via operator ${\mathcal P}_a$: $$ u(x', x_n)= {\mathcal P}_a(f) :=\int_{\partial \mathbb{R}_+^n} P_a(x'-y, x_n) f(y)dy, $$ whose positive kernel is \begin{equation}\label{equ1.2-3}
P_a(x', x_n)=\frac{ x_n^{{1-a}}}{(|x'|^2+x_n^2)^{\frac{n-a}2}}. \end{equation}
Hang, Wang and Yan \cite{HWY08} obtain the sharp $L^p$ estimates on ${\mathcal P}_0$ for $n \ge 3$ (the standard harmonic extension with Poisson kernel). Their results were generalized by Chen \cite{Chen14} for general $a > 2-n$. Note that for $n=2$, from Chen's result one can obtain a different proof of two dimensional analytic isoperimetric inequality for simply connected domains due to Carleman \cite{Cal}.
Quite naturally, one may ask: are there other solutions to \eqref{equ1.2-1} besides the function given in \eqref{equ1.2-2}? Generally, the answer is yes, since there are many sign-changing solutions to \eqref{equ}. However, if one only considers bounded solutions, our Theorem \ref{main-1} indicates that the function given in \eqref{equ1.2-2} is the only one.
To extend the classical Hardy-Littlewood-Sobolev inequality on the upper half space, Dou and Zhu \cite{DZ15} studied the following extension operator for $\alpha \in (1, n):$ \begin{equation}\label{equ1.2-4}
u(x', x_n)= {\mathcal E}_\alpha (f) :=\int_{\partial \mathbb{R}^n_+} E_\alpha (x'-y, x_n) f(y)dy:= \int_{\partial \mathbb{R}^n_+}\frac{f(y)}{(|x'-y|^2+x_n^2)^{\frac{n-\alpha}2}}dy. \end{equation} The sharp $L^p$ estimates were obtained in \cite{DZ15}. Later, more general extension operators on the upper half space were studied by Dou, Guo and Zhu \cite{DGZ17} and Gluck \cite{G18}.
Direct computation shows that $u(x', x_n)= {\mathcal E}_{2-a}f$, up to some constant multiplier, satisfies \begin{equation}\label{equ1.2-5}
\begin{cases}
div(x_{n}^{a} \nabla u)=0, & \quad \text{in}\quad \mathbb{R}^n_+,\\
x_n^a \frac{\partial u}{\partial x_n}=f(x')&\quad\text{on}\quad \partial\mathbb{R}^n_+.
\end{cases}
\end{equation} Theorem \ref{main-2} indicates that for $a \in (\max\{2-n,-1\}, 1)$ the bounded solution to \eqref{equ1.2-5} is unique.
\subsection{New view point on the positive kernels} The classical way to find the fundamental solution to Laplacian operator on $\mathbb{R}^n$ is to solve an ordinary differential equation, by assuming that the solution is radially symmetric. This approach certainly fails if the domain is the upper half space.
The other view point to find the fundamental solution could be like this. First, the constant solution $u=C$ is the only positive harmonic solutions in the whole space $\mathbb{R}^n$ (for simplicity, let us just consider $n \ge 3$). Its kelvin transformation: $v(x)=\frac 1{|x|^{n-2}}u(\frac x{|x|^2})$, which is a positive harmonic function on $\mathbb{R}^n \setminus \{0\}$, will yield the fundamental solution (up to a constant multiplier).
To find the positive kernel for the equation \eqref{equ1.2-1} and \eqref{equ1.2-5}, we first have the following observation, which will be proved in next section.
\noindent{\bf Lemma \ref{invariant} } If $u(x)\in C^2(\mathbb{R}_+^n)$ satisfies the equation $div(x_{n}^{a} \nabla u)=0$ on $\mathbb{R}_+^n$, then $v(x)=\frac 1{|x|^{n-2+a}}u(\frac x{|x|^2})$ satisfies the same equation.
Combining Lemma \ref{invariant} with Theorem \ref{main-1} and \ref{main-2}, we know that the kernel for the equation \eqref{equ1.2-1} and \eqref{equ1.2-5} for $a\ne 2-n$, up to a constant multiplier, are given by $$
\Gamma_d=\frac {x_n^{1-a}}{|x|^{n-a}}, \quad \text{and} \quad \ \Gamma_n=\frac 1{|x|^{n-2+a}} $$ respectively.
In \cite{DGZ17} Dou, Guo and Zhu studied a general extension operator using a kernel obtained by taking a partial derivative of Riesz kernel along $x_n$ direction. Later, Gluck \cite{G18} studied a more general extension operator ${\mathcal E}_{\alpha, \beta}$ with the positive kernel $$
E_{\alpha, \beta}(x', x_n)= \frac{ x_n^{{\beta}}}{(|x'|^2+x_n^2)^{\frac{n-\alpha}2}}$$ for $\beta\ge0, \ 0<\alpha+\beta<n-\beta.$ Notice that $$x_n^b E_{a, 1-a-b}(x', x_n)=\Gamma_d(x', x_n).$$ So all these in \cite{DGZ17} and \cite{G18} are really not ``new'' positive kernels.
\subsection{Discussion} We point out that: for $a=0$, Theorem \ref{main-1} seems to be a folklore for nonnegative harmonic functions. We do not know the original proof for this fact. One way to prove it is to adapt the approach by Gidas and Spuck in \cite{GS82}. Unfortunately, It seems to us that their approach only works for nonnegative functions and for $a=0$. Here, we use the method of moving sphere, introduced by Li and Zhu in \cite{LZ95}. Note that we only assume that $u(x)$ is bounded from below in Theorem \ref{main-1}.
For $a=0$, Theorem \ref{main-2} (after we make an even reflection of the solutions) follows from the classical Liouville theorem in the whole space: the only positive harmonic functions in $\mathbb{R}^n$ are positive constants. It seems that Theorem \ref{main-2} is still true for $a \notin (-1, 1)$. But our method does not work.
It is also interesting to extend Corollary 1.3 to other unbounded domains.
\section{Invariance}
For any fixed $x\in\partial\mathbb{R}^n_+$ and $\lambda>0$, we define
$$y^{x,\lambda}=x+\frac{\lambda^2(y-x)}{|y-x|^2}, \quad \forall y\in \overline{ \mathbb{R}^n_+},$$ and
$$u_{x,\lambda}(y)=\frac{\lambda^{n-2+a}}{|y-x|^{n-2+a}}u(y^{x, \lambda}), \quad \forall y \in \overline{\mathbb{R}^n_+}.$$
We have the following invariant property. \begin{lemma}\label{invariant}
If $u(y)\in C^2(\mathbb{R}_+^n)$ satisfies equation $div(y_{n}^{a} \nabla u)=0$ on $\mathbb{R}_+^n$, then for any $x\in \partial \mathbb{R}^n_+$ and $\lambda>0$, $u_{x, \lambda}(y)$ satisfies the same equation.
\end{lemma}
\begin{proof}
By a direct computation, we have for $i=1,2,\cdots,n-1,$
\begin{equation}\label{gi}
\begin{split}
\partial_i u_{x,\lambda}(y)=&-\frac{(n-2+a)\lambda^{n-2+a}(y_i-x_i)}{|y-x|^{n+a}}u(y^{x,\lambda})
+\frac{\lambda^{n+a}}{|y-x|^{n+a}}\partial_i u(y^{x,\lambda})\\
&-\frac{2\lambda^{n+a}(y_i-x_i)}{|y-x|^{n+2+a}}\nabla u(y^{x,\lambda})\cdot(y-x),
\end{split}
\end{equation} for $i=n$, \begin{equation}\label{n} \begin{split}
\partial_n u_{x,\lambda}(y)=&-\frac{(n-2+a)\lambda^{n-2+a}y_n}{|y-x|^{n+a}}u(y^{x,\lambda})
+\frac{\lambda^{n+a}}{|y-x|^{n+a}}\partial_n u(y^{x,\lambda})\\
&-\frac{2\lambda^{n+a}y_n}{|y-x|^{n+2+a}}\nabla u(y^{x,\lambda})\cdot(y-x), \end{split} \end{equation} and \begin{equation*}
\begin{split}
\Delta u_{x,\lambda}(y)=&\frac{\lambda^{n+2+a}}{|y-x|^{n+2+a}}(\Delta u)(y^{x,\lambda})
+\frac{2a\lambda^{n+a}}{|y-x|^{n+a+2}}\nabla u(y^{x,\lambda})\cdot(y-x)\\
&+\frac{a(n-2+a)\lambda^{n-2+a}}{|y-x|^{n+a}}u(y^{x,\lambda}). \end{split} \end{equation*}
Then \begin{equation}\label{tran}
\begin{split}
div(y_n^a\nabla u_{x,\lambda})(y)&=y_n^a\Delta u_{x,\lambda}(y)+ay_n^{a-1}\partial_n u_{x,\lambda}(y)\\
&=\frac{\lambda^{n+2+a}y_n^a}{|y-x|^{n+2+a}}(\Delta u)(y^{x,\lambda})+a\frac{\lambda^{n+a}y_n^{a-1}}{|y-x|^{n+a}}\partial_n u(y^{x,\lambda})\\
&=\frac{\lambda^{n+2-a}}{|y-x|^{n+2-a}}div (y_n^a\nabla u)(y^{x,\lambda})\\
&=0.
\end{split} \end{equation}
\end{proof}
It will be interesting to further explore the geometric implication of the above invariance.
\section{Dirichlet condition} We present the proof for Theorem \ref{main-1} in this section. Noting the specialty of $a=2-n$ in Lemma \ref{invariant}, we divide the proof into three cases: $a>2-n$, $a< 2-n$ and $a=2-n$. We shall prove the results using the method of moving sphere.
\noindent \textbf{Case {1}.} $a>2-n$.
Due to technical difficulties in dealing with the zero boundary condition, we shall classify all solutions bounded from below plus a positive constant instead. It is sufficient to prove \begin{theorem}\label{main-1-1} Assume $n\geq2$. Suppose $u(y) \in C^2(\mathbb{R}^n_{+})\cap C^0(\overline {\mathbb{R}^n_{+}})$ satisfies
\begin{equation}\label{1}
\begin{cases}
div(y_{n}^{a} \nabla u)=0, &\quad u>\frac 12 , \quad \text{in}\quad \mathbb{R}^n_+,\\
u=1&\quad\text{on}\quad \partial\mathbb{R}^n_+.
\end{cases} \end{equation}
If $a> 2-n$, then $u=C_*y_n^{1-a}+1$ for some nonnegative constant $C_*$. In particular, for $a \ge 1$, $u=1$. \end{theorem}
From now on to the end of this section, we always assume solution $u(x) \in C^2(\mathbb{R}^n_{+})\cap C^0(\overline {\mathbb{R}^n_{+}})$. W first have
\begin{lemma}\label{lem1} Assume that $a> 2-n$ and $u$ satisfies conditions in Theorem \ref{main-1-1}.
For any $x\in\partial \mathbb{R}^n_+$, and $\lambda>0$, we have that
$$u_{x,\lambda}(y)\leq u(y), \ \ \ \ \ \forall y\in\mathbb{R}^n_+\backslash B_\lambda(x).$$
\end{lemma}
\begin{proof}
For any fixed $x\in\partial\mathbb{R}^n_+$ and $\lambda>0$, define $$w_{x,\lambda}(y)=u(y)-u_{x,\lambda}(y).$$
Since $n-2+a>0$, we have that
\begin{equation}\label{infinite-1}
\varliminf_{|y|\rightarrow\infty}w_{x,\lambda}(y)=\varliminf_{|y|\rightarrow\infty}u(y)-\lim_{|y|\rightarrow\infty}\frac{\lambda^{n-2+a}}{|y-x|^{n-2+a}}u(x+\frac{\lambda^2(y-x)}{|y-x|^2})\geq \frac{1}{2},
\end{equation} and \begin{equation}\label{bd}
w_{x,\lambda}(y)=1-\frac{\lambda^{n-2+a}}{|y-x|^{n-2+a}}>0\quad\textrm{on }\partial\mathbb{R}^n_+\backslash \overline{B_\lambda(x)}. \end{equation} By \eqref{infinite-1}, we know that there is an $N=N(x,\lambda)>0$ large enough, such that $w_{x,\lambda}\geq C>0$ in $\mathbb{R}^n_+\backslash \overline{B_N(x)}$. Define $\Omega=B^+_N(x)\backslash \overline{B_\lambda^+(x)}$, then we have
$$
\begin{cases}
div(y_{n}^{a} \nabla w_{x,\lambda})=0, & \quad \text{in}\quad \Omega \\
w_{x,\lambda} \ge 0 &\quad\text{on}\quad \partial \Omega.
\end{cases}
$$ By the maximum principle, we know $w_{x,\lambda}\geq 0$ in $\Omega$. Therefore, $w_{x,\lambda}\geq 0$ in $\mathbb{R}^n_+\backslash B_{\lambda}(x)$. \end{proof}
To conclude our proof, we need the following key lemma for the method of moving sphere. See, for example, the proof in Dou and Zhu \cite{DZ15}.
\begin{lemma}\label{lem5}
Assume $f(y) \in C^0(\overline {\mathbb{R}^n_+})$, $n\geq2$, and $\tau>0$. If
$$(\frac{\lambda}{|y-x|})^{\tau}f(x+\frac{\lambda^2(y-x)}{|y-x|^2})\leq f(y), \quad \forall \lambda>0,\;x\in\partial\mathbb{R}^n_+,\;|y-x|\geq\lambda,\, y\in \mathbb{R}^n_+,$$
then $$f(y)=f(y',y_n)=f(0',y_n),\quad \forall y=(y', y_n)\in\mathbb{R}^n_+.$$ \end{lemma}
From Lemma \ref{lem1} and Lemma \ref{lem5}, we know that $u(y', y_n)=u(y_n).$ Then by solving the corresponding ODE, we obtain $u=C_1 y_n^{1-a}+C_2$. From the boundary condition, we know: for $2-n<a<1$, $C_2=1$; And for $a\geq 1$, $C_1$ must be $0$ and $C_2$ must be $1$. We thus complete the proof of Theorem \ref{main-1-1}.
\noindent \textbf{Case {2}.} $a < 2-n$.
For $a <2-n$, it is easy to check that $1+y_n^{1-a}$ does not satisfy the monotonic property in Lemma \ref{lem1}, but $y_n^{1-a}-1$ does. It is sufficient to prove
\begin{theorem}\label{main-5-1} Assume $n\geq2$, and $a <2-n$. If $u(y) \in C^2(\mathbb{R}^n_{+})\cap C^0(\overline {\mathbb{R}^n_{+}})$ satisfies
\begin{equation}\label{equ-main-5-1}
\begin{cases}
div(y_{n}^{a} \nabla u)=0, &\quad u\ge -2, \quad \text{in}\quad \mathbb{R}^n_+,\\
u=-1 &\quad\text{on}\quad \partial\mathbb{R}^n_+,
\end{cases}
\end{equation}
then $u=C_*y_n^{1-a}-1$ for some nonnegative constant $C_*$.
\end{theorem}
First, we have
\begin{lemma}\label{lem1-1} Assume that $a<2-n$ and $u$ satisfies conditions in Theorem \ref{main-5-1}.
For any $x\in\partial \mathbb{R}^n_+$, and $\lambda>0$, we have that
$$u_{x,\lambda}(y)\leq u(y), \ \ \ \ \ \forall y\in\mathbb{R}^n_+\backslash B_\lambda(x).$$
\end{lemma}
\begin{proof} Similar to the proof of Lemma \ref{lem1}, for any fixed $x\in\partial\mathbb{R}^n_+$ and $\lambda>0$, we define $$w_{x,\lambda}(y)=u(y)-u_{x,\lambda}(y).$$
Noting $\lim_{|y| \to 0} u(x+ y)=-1$, and $n-2+a<0$, we have that
\begin{equation}\label{infinite-2}
\begin{split}
\varliminf_{|y|\rightarrow\infty}w_{x,\lambda}(y)&=\varliminf_{|y|\rightarrow\infty}u(y)-\lim_{|y|\rightarrow\infty}\frac{\lambda^{n-2+a}}{|y-x|^{n-2+a}} u (x+\frac{\lambda^2(y-x)}{|y-x|^2}) \\
& \ge -2+\lim_{|y|\rightarrow\infty} \frac{ \lambda^{n-2+a}}{2|y-x|^{n-2+a}}\\
&=+\infty,
\end{split}
\end{equation} and \begin{equation}\label{bd-1}
w_{x,\lambda}(y)=-1+\frac{\lambda^{n-2+a}}{|y-x|^{n-2+a}}>0\quad\textrm{on }\partial\mathbb{R}^n_+\backslash \overline{B_\lambda(x)}. \end{equation} By \eqref{infinite-2}, we know that there is an $N=N(x,\lambda)>0$ large enough, such that $w_{x,\lambda}\geq C>0$ in $\mathbb{R}^n_+\backslash \overline{B_N(x)}$. Define $\Omega=B^+_N(x)\backslash \overline{B_\lambda^+(x)}$, then we have
$$
\begin{cases}
div(y_{n}^{a} \nabla w_{x,\lambda})=0, & \quad \text{in}\quad \Omega \\
w_{x,\lambda} \ge 0 &\quad\text{on}\quad \partial \Omega.
\end{cases}
$$
By the maximum principle, we know that $w_{x,\lambda}\geq 0$ in $\mathbb{R}^n_+\backslash B_{\lambda}(x)$. \end{proof}
Similarly, from Lemma \ref{lem1-1} and Lemma \ref{lem5}, we obtain $u=C_* y_n^{1-a}-1$ for some nonnegative constant $C_*$, thus complete the proof of Theorem \ref{main-5-1}.
\noindent \textbf{Case {3}.} $a =2-n$.
We modify the $u_{x,\lambda}$ to be
$$u_{x,\lambda}(y)=u(y^{x,\lambda})+\ln \frac{\lambda}{|y-x|}.$$ Then $u_{x,\lambda}$ satisfies the same equation $$div (y_n^{2-n}\nabla u)=0.$$
\begin{lemma}\label{lem1-2} Assume that $u$ satisfies conditions in Theorem \ref{main-1} with $a=2-n$.
For any $x\in\partial \mathbb{R}^n_+$, and $\lambda>0$, we have that
$$u_{x,\lambda}(y)\leq u(y), \ \ \ \ \ \forall y\in\mathbb{R}^n_+\backslash B_\lambda(x).$$
\end{lemma}
\begin{proof} For any fixed $x\in\partial\mathbb{R}^n_+$ and $\lambda>0$, we define $$w_{x,\lambda}(y)=u(y)-u_{x,\lambda}(y).$$
Noting $\lim_{|y| \to 0}u(x+y)=0,$ we have that
\begin{equation}\label{infinite-4}
\begin{split}
\varliminf_{|y|\rightarrow\infty}w_{x,\lambda}(y)&=\varliminf_{|y|\rightarrow\infty}u(y)-\lim_{|y|\rightarrow\infty}u (x+\frac{\lambda^2(y-x)}{|y-x|^2})-\lim_{|y|\rightarrow\infty}\ln\frac{\lambda}{|y-x|} \\
& \ge -C-(-\infty)\\
&=+\infty,
\end{split}
\end{equation} and \begin{equation}\label{bd-4}
w_{x,\lambda}(y)=\ln\frac{|y-x|}{\lambda}>0\quad\textrm{on }\partial\mathbb{R}^n_+\backslash \overline{B_\lambda(x)}. \end{equation} Similar to the proof of Lemma \ref{lem1-1}, we have $u_{x,\lambda}(y)\leq u(y)$.
\end{proof}
We thus can conclude our proof from the following lemma. See, for example, the proof of Lemma 3.3 in Li and Zhu \cite{LZ95}.
\begin{lemma}\label{m}
Suppose that $f\in C^1(\mathbb{R}^n_+)$ satisfies, for all $x\in\partial\mathbb{R}^n_+$ and $\lambda>0$,
$$f(y)\geq f(x+\frac{\lambda^2(y-x)}{|y-x|^2})+\ln\frac{\lambda}{|y-x|},\quad \forall y\in\mathbb{R}^n_+.$$
Then $$f(y)=f(y',y_n)=f(0',y_n),\quad \forall y=(y',y_n)\in\mathbb{R}^n_+.$$
\end{lemma}
\section{Neumann boundary condition}
To prove Theorem \ref{main-2}, we also consider $u>1$ by replacing $u$ with $u+1$. First, we need to verify the invariance of the boundary condition under the M\"obius transformation.
\begin{proposition}\label{lem4-1} Assume $a>-1$, and $u(x) \in C^2(\mathbb{R}^n_{+})\cap C^1(\overline {\mathbb{R}^n_{+}})$. If $\lim_{y_n\rightarrow 0^+}y_n^a\partial_n u(y)=0$, then for all $x \in \partial \mathbb{R}^n_+$ and $\lambda>0$, $\lim_{y_n\rightarrow 0^+}y_n^a\partial_n u_{x,\lambda}(y)=0$ if $\lim_{y_n \rightarrow 0^+} y \ne x$. \end{proposition}
\begin{proof} By \eqref{n}, we have \begin{equation*} \begin{split} &\lim_{y_n\rightarrow 0^+}y_n^a\partial_n u_{x,\lambda}(y)\\
=&-\lim_{y_n\rightarrow 0^+}y_n^{1+a}\Big(\frac{(n-2+a)\lambda^{n-2+a}}{|y-x|^{n+a}}u(y^{x,\lambda})+\frac{2\lambda^{n+a}}{|y-x|^{n+2+a}}\nabla u(y^{x,\lambda})\cdot(y-x)\Big)\\
&+\lim_{y_n\rightarrow 0^+}\frac{\lambda^{n+a}y_n^{a}}{|y-x|^{n+a}}\partial_n u(y^{x,\lambda}). \end{split} \end{equation*} For $a>-1$ and $\lim_{y_n \rightarrow 0^+} y \ne x$, we have
$$\lim_{y_n\rightarrow 0^+}y_n^a\partial_n u_{x,\lambda}(y)=0+\lim_{y_n\rightarrow 0^+}\frac{\lambda^{n-a}}{|y-x|^{n-a}}(y_n^a\partial_{n}u)(y^{x,\lambda})=0.$$ \end{proof}
We will use the method of moving sphere again to prove Theorem \ref{main-2}.
\begin{lemma}\label{1.2-1}
Assume that $n\ge 2$ and $\max\{2-n,-1\}<a<1$. Suppose that $u(y) \in C^2(\mathbb{R}_+^n) \cap C^1(\overline{\mathbb{R}_+^n}) $ satisfies
$$
\begin{cases}
div(y_{n}^{a} \nabla u)=0, &\quad u>1, \quad \text{in}\quad \mathbb{R}^n_+,\\
y_n^a\frac {\partial u}{\partial y_n}=0&\quad\text{on}\quad \partial\mathbb{R}^n_+.
\end{cases}
$$
Then for any $x\in\partial \mathbb{R}^n_+$, and $\lambda>0$, we have that
$$u_{x,\lambda}(y)\leq u(y), \ \ \ \ \ \forall y\in\mathbb{R}^n_+\backslash B_\lambda(x).$$ \end{lemma}
\begin{proof}
For any fixed $x\in\partial\mathbb{R}^n_+$ and $\lambda>0$, define $$w_{x,\lambda}(y)=u(y)-u_{x,\lambda}(y).$$
Then
\begin{equation*}
\varliminf_{|y|\rightarrow\infty}w_{x,\lambda}(y)=\varliminf_{|y|\rightarrow\infty}u(y)-\lim_{|y|\rightarrow\infty}\frac{\lambda^{n-2+a}}{|y-x|^{n-2+a}}u(x+\frac{\lambda^2(y-x)}{|y-x|^2})\geq 1,
\end{equation*}
Thus there is an $N=N(x,\lambda)>0$ large enough, such that $w_{x,\lambda}\geq C>0$ in $\mathbb{R}^n_+\backslash \overline{B_N(x)}$. Define $\Omega=B^+_N(x)\backslash \overline{B_\lambda^+(x)}$, then we have
\begin{equation}\label{w-equ-1}
\begin{cases}
div(y_{n}^{a} \nabla w_{x,\lambda})=0, & \quad \text{in}\quad \Omega \\
w_{x,\lambda} \ge 0 &\quad\text{on}\quad \partial \Omega\cap\mathbb{R}^n_+,\\
y_n^a\frac{\partial w_{x,\lambda}}{\partial y_n}=0&\quad \text{on}\quad \partial \Omega\cap\partial\mathbb{R}^n_+.
\end{cases}
\end{equation}
We claim that $w_{x,\lambda}\geq 0$ in $\Omega$. Otherwise $m=\min_{\overline{\Omega}}w_{x,\lambda}<0$. By the maximum principle and boudary condition of $w_{x,\lambda}$, we know that $m=\min_{\partial \Omega\cap\partial\mathbb{R}^n_+}w_{x,\lambda}$.
For $\epsilon>0$, we consider $$A_\epsilon(y):= w_{x,\lambda}(y)-\epsilon g(y),$$ where $g(y)=y_n^{1-a}$. Easy to see that $ div(y_{n}^{a} \nabla A_\epsilon)=0$ in $ \Omega$. Thus there is a positive $\epsilon_0<\frac{-m}{N^{1-a}}$, such that, for $\epsilon \in (0, \epsilon_0)$, \begin{equation}\label{min}
\min_{\overline{\Omega} }A_\epsilon(y)=\min_{\partial \Omega \cap \partial \mathbb{R}_+^n} A_\epsilon(y).
\end{equation}
In fact, for $\epsilon\in(0,\epsilon_0)$, we have
$$A_\epsilon|_{\partial\Omega\cap\mathbb{R}^n_+}\geq -\epsilon g|_{\partial\Omega\cap\mathbb{R}^n_+}\geq -\epsilon N^{1-a}>m=\min_{\partial\Omega\cap\partial\mathbb{R}^n_+}w_{x,\lambda}=\min_{\partial\Omega\cap\partial\mathbb{R}^n_+}A_\epsilon.$$
We thus obtain \eqref{min} by the maximum principle.
Let $y^*\in \partial \Omega\cap\partial\mathbb{R}^n_+$ be one minimal point with $A_\epsilon(y^*)=\min_{\overline{\Omega} }A_\epsilon(y)<0$. It follows that
\begin{equation}\label{neum-1}
\lim_{y \to y^*}\big(y_n^a \frac{\partial A_\epsilon }{\partial y_n}\big)(y)\ge 0.
\end{equation}
Thus
$$\lim_{y \to y^*}\big(y_n^a\frac{\partial w_{{x, \lambda}}}{\partial y_n}\big) (y)\geq \lim_{y_n\rightarrow 0^+}\epsilon\big(y_n^a\frac{\partial g}{\partial y_n}\big) (y)=\epsilon(1-a)>0.$$
It is in contradiction to the boundary condition in the equation \eqref{w-equ-1}. \end{proof}
Now, we use Lemma \ref{lem5} to conclude that $u$ only depends on $y_n$. By solving the corresponding ODE we get Theorem \ref{main-2}.
\noindent {\bf Acknowledgements}\\
\noindent L. Wang is supported by the China Scholarship Council for her study/research at the University of Oklahoma. L. Wang would like to thank Department of Mathematics at the University of Oklahoma for its hospitality, where this work has been done.
\small
\end{document} | arXiv |
\begin{document}
\title{Improving Graph Representation Learning by Contrastive Regularization}
\author{Kaili Ma, Haochen Yang, Han Yang, Tatiana Jin, Pengfei Chen, Yongqiang Chen, Barakeel Fanseu Kamhoua, James Cheng} \email{{klma, hyang, tjin, pfchen, yqchen, kamhoua, jcheng}@cse.cuhk.edu.hk,[email protected]}
\affiliation{
\institution{Department of Computer Science and Engineering, The Chinese University of Hong Kong} }
\renewcommand{Trovato and Tobin, et al.}{Trovato and Tobin, et al.}
\newcommand*{\ldblbrace}{\{\mskip-5mu\{} \newcommand*{\rdblbrace}{\}\mskip-5mu\}} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \makeatletter \DeclareRobustCommand{\abs}{\@ifstar\star@abs\normal@abs}
\newcommand{\star@abs}[1]{\left|#1\right|}
\newcommand{\normal@abs}[2][]{\mathopen{#1|}#2\mathclose{#1|}} \newcommand{theorem}{theorem}
\newcommand{\MakeUppercase{\result}}{\MakeUppercase{theorem}} \theoremstyle{acmplain} \newtheorem{claim}{Claim}[section] \newtheorem{remark}{Remark} \newtheorem*{remark*}{Remark} \newcommand{ours}{ours} \newcommand{LC}{LC} \newcommand{ML}{ML} \newcommand{CO}{CO} \newcommand{\PM}[1]{\scriptsize{\(\pm{#1}\)}} \newcommand{\mathrm{Var}}{\mathrm{Var}} \newcommand{Contrast-Reg}{Contrast-Reg} \makeatother
\begin{abstract}
Graph representation learning is an important task with applications in various areas such as online social networks, e-commerce networks, WWW and semantic webs. For unsupervised graph representation learning, many algorithms such as Node2Vec and GraphSAGE make use of ``negative sampling'' and/or noise contrastive estimation loss. This bears similar ideas to contrastive learning, which ``contrasts'' the node representation similarities of semantically similar (positive) pairs against those of negative pairs. However, despite the success of contrastive learning, we found that directly applying this technique to graph representation learning models (e.g., graph convolutional networks) does not always work. We theoretically analyze the generalization performance and propose a light-weight regularization term that avoids the high scales of node representations' norms and the high variance among them to improve the generalization performance. Our experimental results further validate that this regularization term significantly improves the representation quality across different node similarity definitions and outperforms the state-of-the-art methods.
\end{abstract}
\keywords{Graph representation, Contrastive learning, Regularization}
\maketitle
\definecolor{darkgreen}{rgb}{0.29, 0.67, 0.29}
\section{Introduction}
Graph is widely used to capture rich information (e.g., hierarchical structures, communities) in data from various domains such as social networks, e-commerce networks, knowledge graphs, WWW and semantic webs. By incorporating graph topology and node/edge features into machine learning models, \textit{graph representation learning} has achieved great success in many important applications such as node classification, link prediction, and graph clustering.
A large number of graph representation learning algorithms~\cite{DBLP:DGI,DBLP:GMI,DBLP:LINE,DBLP:DeepWalk,DBLP:node2vec,ahmed2013distributed, cao2015grarep,DBLP:NEMF,harp,kipf2017semi,hamilton2017inductive,velickovic2018graph, xu2018how,qu2019gmnn} have been proposed. Among them, many~\cite{DBLP:LINE,DBLP:DeepWalk,DBLP:node2vec,hamilton2017inductive} are designed in an unsupervised manner and make use of ``\textit{negative sampling}'' to learn node representations. This design shares similar ideas as \textit{contrastive learning}~\cite{DBLP:MOCO,DBLP:goodviews,DBLP:multiviews,DBLP:CPC,DBLP:CPCv2,DBLP:MINE,DBLP:deep_infomax,DBLP:instance_discrimination,DBLP:word2vec,DBLP:simultaneous_clustering_and_representation_learning, DBLP:DeepCluster,DBLP:SimCLR}, which ``contrasts'' the similarities of the representations of similar (or positive) node pairs against those of negative pairs. These algorithms adopt \textit{noise contrastive estimation loss} (\textit{NCEloss}), while they differ in the definition of node similarity (hence the design of contrastive pairs) and encoder design.
Existing graph representation learning algorithms mainly fall into three categories: \textit{adjacency matrix factorization based models}~\cite{ahmed2013distributed, cao2015grarep, DBLP:NEMF}, \textit{skip-gram based models}~\cite{DBLP:DeepWalk,DBLP:node2vec,harp}, and \textit{graph neural networks (GNNs)}~\cite{kipf2017semi,hamilton2017inductive,velickovic2018graph, xu2018how,qu2019gmnn}. We focus on GNNs as GNN models can not only capture graph topology information as the skip-gram and factorization based models, but also incorporate node/edge features. Specifically, we formulate \textbf{a contrastive GNN framework} with four components: \textit{a similarity definition, a GNN encoder, a contrastive loss function, and possibly a downstream classification task}.
While graph representation algorithms using the ``negative sampling'' approach are shown to achieve good performance empirically, there is a lack of theoretical analysis on the generalization performance. In addition, we also found that directly applying contrastive pairs and NCEloss to existing GNN models, e.g., graph convolutional networks (GCNs)~\cite{kipf2017semi}, does not always work well (as shown in Section~\ref{section:exp}). In order to understand the generalization performance of the algorithms and find out when the direct application of contrastive pairs and NCEloss does not work, we derive a generalization bound for our contrastive GNN framework using the theoretical framework proposed in~\cite{DBLP:Theoretical_analysis}. Our generalization bound reveals that \textit{the high scales of node representations' norms and the high variance among them are two main factors that hurt the generalization performance}.
To solve the problems caused by the two factors, we propose a novel regularization method, \textbf{Contrast-Reg}. Contrast-Reg\ uses a regularization vector $r$, which is a random vector with each element in the range $(0,1]$. We learn a graph representation model by forcing the representations of all nodes to be similar to $r$ and all the contrastive representations calculated by shuffling node features to be dissimilar to $r$. We show from the geometric perspective that Contrast-Reg\ stabilizes the scales of norms and reduces their variance. We also validate by experiments that Contrast-Reg\ significantly improves the quality of node representations for a popular GNN model using different similarity definitions.
\noindent \textbf{Outline.} Section~\ref{section:related} discusses related work. Section~\ref{section: theory} gives some preliminaries and Section~\ref{section:theoretial analyses} analyzes the generalization bound. Section~\ref{section:contrast_reg} proposes Contrast-Reg\ and Section~\ref{section:model} presents the contrastive GNN framework. Section~\ref{section:exp} reports the experimental results.
\section{Related work}\label{section:related}
\noindent \textbf{Graph representation learning.} Many graph representation learning models have been proposed. Factorization based models~\cite{ahmed2013distributed, cao2015grarep, DBLP:NEMF} factorize an adjacency matrix to obtain node representations. Random walk based models such as DeepWalk~\cite{DBLP:DeepWalk} sample node sequences as the input to skip-gram models to compute the representation for each node. Node2vec~\cite{DBLP:node2vec} balances depth-first and breadth-first random walk when it samples node sequences. HARP~\cite{harp} compresses nodes into super-nodes to obtain a hierarchical graph to provide hierarchical information to random walk. GNN models~\cite{kipf2017semi,hamilton2017inductive,velickovic2018graph, xu2018how,qu2019gmnn} have shown great capability in capturing both graph topology and node/edge feature information.
Most GNN models follow a neighborhood aggregation schema, in which each node receives and aggregates the information from its neighbors in each GNN layer,
i.e., for the \(k\)-th layer, \(\tilde{h}^{k}_i \!=\! aggregate(h_j^{k-1}, j\!\in\! neighborhood(i))\), and \(h^{k}_i\! =\! combine(\tilde{h}^{k}_i,\! h^{k-1}_i)\). This work proposes a regularization method, Contrast-Reg, for GNN models and improves their performance for downstream tasks.
\noindent \textbf{Contrastive learning.} Contrastive learning is a self-supervised learning method that learns representations by contrasting positive pairs against negative pairs. Contrastive pairs can be constructed in different ways for different types of data and tasks, such as multi-view~\cite{DBLP:goodviews,DBLP:multiviews}, target-to-noise~\cite{DBLP:CPC,DBLP:CPCv2}, mutual information~\cite{DBLP:MINE,DBLP:deep_infomax}, instance discrimination~\cite{DBLP:instance_discrimination}, context co-occurrence~\cite{DBLP:word2vec}, clustering~\cite{DBLP:simultaneous_clustering_and_representation_learning, DBLP:DeepCluster}, and multiple data augmentation~\cite{DBLP:SimCLR}. In addition to the above unsupervised learning settings, \cite{DBLP:multiviews,DBLP:SimCLR,DBLP:CPC} also show great success in capturing information that can be transferred to new tasks from different domains.
Contrastive learning has been successfully applied in many graph representation learning models such as~\cite{DBLP:DeepWalk,DBLP:node2vec,DBLP:LINE,DBLP:DGI,DBLP:GMI,hamilton2017inductive}. In this work, we apply contrastive learning in GNNs and propose a contrastive GNN framework. The state-of-the-art models such as~\cite{DBLP:DGI,DBLP:GMI}, which both use the GCN encoder, can be seen as special instances of this algorithmic framework. We show that with the same GCN encoder and similar contrastive pair designs, our models can significantly outperform~\cite{DBLP:DGI,DBLP:GMI} by adopting Contrast-Reg.
\noindent \textbf{Noise Contrastive Estimation loss.} NCEloss was originally proposed to reduce the computation cost of estimating the parameters of a probabilistic model using logistic regression to discriminate between the observed data and artificially generated noises~\cite{DBLP:NCE_of_Unnormalized_Statistical_Models,DBLP:notes_on_nce_and_negative_sampling,DBLP:NCE}. It has been successfully applied to contrastive learning~\cite{DBLP:goodviews,DBLP:multiviews,DBLP:CPC,DBLP:CPCv2,DBLP:MINE,DBLP:deep_infomax,DBLP:instance_discrimination,DBLP:SimCLR}. There are works aiming to explain the success of NCEloss. \citet{DBLP:NCE_of_Unnormalized_Statistical_Models} proved that when NCEloss serves as the objective function of the parametric density estimation problem, the estimation of the parameters converges in probability to the optimal estimation. \citet{DBLP:Understanding_negative_sampling} showed that when NCEloss is applied in graph representation learning, the mean squared error of the similarity between two nodes is related to the negative sampling strategy. However, the definition of optimal representations or optimal parameters does not consider downstream tasks, but is based on pre-defined structures. In this paper, we adopt the theoretical settings from the contrastive learning framework proposed by \citet{DBLP:Theoretical_analysis} to analyze the generalizability of NCEloss in downstream tasks, i.e., linear classification tasks. To the best of our knowledge, we are the first to theoretically analyze NCEloss under the contrastive learning setting~\cite{DBLP:Theoretical_analysis}.
\noindent \textbf{Node-level similarity.} NCEloss has been adopted in many graph representation learning models to capture different types of node-level similarity. We characterize them as follows: \begin{itemize}
\item \textbf{Structural similarity}: We may capture structural similarity from different angles. From the graph theory perspective, GraphWave~\cite{DBLP:GraphWave} leverages the diffusion of spectral graph wavelets to capture structural similarity, and struc2vec~\cite{DBLP:stru2vec} uses a hierarchy to measure node similarity at different scales. From the induced subgraph perspective, GCC~\cite{DBLP:GCC} treats the induced subgraphs of the same ego network as similar pairs and those from different ego networks as dissimilar pairs. To capture the community structure, vGraph~\cite{DBLP:vgraph} utilizes the high correlation of community detection and node representations to make node representations contain more community structure information. To capture the global-local structure, DGI~\cite{DBLP:DGI} maximizes the mutual information between node representations and graph representations to allow node representations to contain more global information.
\item \textbf{Attribute similarity}: Nodes with similar attributes are likely to have similar representations. GMI~\cite{DBLP:GMI} maximizes the mutual information between node attributes and high-level representations, and \citet{DBLP:Pretrain_GNN} applies attribute masking to help capture domain-specific knowledge.
\item \textbf{Proximity similarity}: Most random walk based models such as DeepWalk~\cite{DBLP:DeepWalk}, Node2vec~\cite{DBLP:node2vec}, and LINE~\cite{DBLP:LINE} share an assumption that nodes with more proximity have higher probability to share the same label. \end{itemize} We will show that Contrast-Reg\ facilitates the contrastive training of graph representation learning models regardless of the different designs of contrastive pairs being used, and thus is helpful in capturing all types of similarities.
\noindent\textbf{Regularization for graph representation learning.} In addition to the general regularization terms used in machine learning such as L1/L2 regularization, there are regularizers proposed for graph representation learning models. GraphAT~\cite{feng2019graphat} and BVAT~\cite{deng2019bvat} add adversarial perturbation \(\frac{\partial f}{\partial x}\) to the input data \(x\) as a regularizer to obtain more robust models. GraphSGAN~\cite{ding2018graphsgan} generates fake input data in low density region by taking a generative adversarial network as regularizer. P-reg~\cite{yang2020rethinking} makes use of the smoothness property in real-world graphs to improve GNN models. Most of the above regularizers are designed for supervised tasks and perform well only in supervised settings, while Contrast-Reg\ is the first regularizer designed for contrastive learning and achieves excellent performance in unsupervised settings. We will also show its advantages over traditional regularizers, e.g., weight decay (L2 regularization of the model parameters)~\cite{DBLP:weight_decay} in Section~\ref{section:contrast_reg}.
\section{Preliminaries}\label{section: theory}
We briefly discuss the theoretical framework~\cite{DBLP:Theoretical_analysis} for contrastive learning and NCEloss, which is the foundation of Section~\ref{section:theoretial analyses}.
\subsection{Concepts in Contrastive Learning} Consider the feature space $\mathcal{X}$, the goal of contrastive learning is to train an encoder $f\in\mathcal{F}:\mathcal{X}\to\mathbb{R}^d$ for all input data points $x\in\mathcal{X}$ by constructing positive pairs $(x,x^+)$ and negative pairs $(x,x^-_1,\cdots,x^-_K)$. To formally analyze the behavior of contrastive learning, \citet{DBLP:Theoretical_analysis} introduce the following concepts.
\begin{itemize}
\item \textit{Latent classes}: Data are considered as drawn from latent classes $\mathcal{C}$ with distribution $\rho$. Further, distribution $\mathcal{D}_c$ is defined over feature space $\mathcal{X}$ that is associated with a class $c\in\mathcal{C}$ to measure the relevance between $x$ and $c$.
\item \textit{Semantic similarity}: Positive samples are drawn from the same latent classes, with distribution
\begin{equation}\label{eq:D_sim}
\mathcal{D}_{sim}(x,x^+)=\mathbb{E}_{c\in \rho}\left[ \mathcal{D}_c(x)\mathcal{D}_c(x^+)\right],
\end{equation}
while negative samples are drawn randomly from all possible data points, i.e., the marginal of \(\mathcal{D}_{sim}\), as
\begin{equation}\label{eq:D_neg}
\mathcal{D}_{neg}(x^-)=\mathbb{E}_{c\in \rho}\left[ \mathcal{D}_c(x^-)\right]
\end{equation}
\item \textit{Supervised tasks}: Denote $K$ as the number of negative samples. The object of the supervised task, i.e., feature-label pair \((x,c)\), is sampled from
\begin{equation*}
\mathcal{D}_{\mathcal{T}}(x,c)=\mathcal{D}_c(x)\mathcal{D}_{\mathcal{T}}(c),
\end{equation*}
where $\mathcal{D}_{\mathcal{T}}(c)=\rho(c|c\in\mathcal{T})$, and $\mathcal{T}\subseteq\mathcal{C}$ with $\abs{\mathcal{T}}=K+1$.\\
Mean classifier $W^\mu$ is naturally imposed to bridge the gap between the representation learning performance and linear separability of learn representations, as
\begin{equation*}
W^\mu_c\coloneqq \mu_c = \mathbb{E}_{x\sim\mathcal{D}_c}[f(x)].
\end{equation*}
\item \textit{Empirical Rademacher complexity}: Suppose $\mathcal{F}:\mathcal{X}\to\left[1, 0\right]$. Given a sample $\mathcal{S}$, {\small
\begin{equation*}
\mathcal{R}_{\mathcal{S}}(\mathcal{F})=\mathbb{E}_{\vec{e}}\left[\sup_{f\in\mathcal{F}} \vec{e}^Tf(\mathcal{S})\right],
\end{equation*}}
where $\vec{e}=\left(e_1,\cdots,e_m\right)^T$, with $e_i$ are independent random variables taking values uniformly from $\left\{-1,+1\right\}$. \end{itemize}
In addition, the theoretical framework in~\cite{DBLP:Theoretical_analysis} makes an assumption: encoder $f$ is bounded, i.e., $\max_{x\in\mathcal{X}}{\norm{f(x)}}\le R^2$, \(R \in \mathbb{R}\).
\subsection{Contrastive Learning with NCEloss} The contrastive loss defined by \citet{DBLP:Theoretical_analysis} is {\small \begin{equation*}
\mathcal{L}_{un}\coloneqq \mathop{\mathbb{E}}_{\substack{(x,x^+)\sim\mathcal{D}_{sim},\\(x^-_1,\cdots,x^-_K)\sim\mathcal{D}_{neg}}}
\left[\ell(\{f(x)^T(f(x^+)-f(x^-_i))\}_{i=1}^K)\right], \end{equation*}} where \(\ell\) can be the hinge loss as $\ell(\mathbf{v})=\max{\left\{0, 1+\max_i{\left\{-\mathbf{v}_i\right\}}\right\}}$ or the logistic loss as $\ell(\mathbf{v})=\log_2{\left(1+\sum_{i}{\exp(-\mathbf{v}_i)}\right)}$. And its supervised counterpart is defined as {\small \begin{equation*}
\mathcal{L}_{sup}^\mu \coloneqq \mathop{\mathbb{E}}_{(x,c)\sim \mathcal{D}_{\mathcal{T}(x,c)}}\left[\ell\left(\left\{f(x)^T\mu_c-f(x)^T\mu_{c^\prime}\right\}_{c^\prime\ne c}\right)\right]. \end{equation*}} A more powerful loss function, NCEloss, used in \cite{DBLP:DGI,DBLP:Understanding_negative_sampling,DBLP:NCE,DBLP:notes_on_nce_and_negative_sampling}, can be framed as {\small \begin{equation}\label{eq: loss} \begin{split}
&\mathcal{L}_{nce} \coloneqq \\
&- \! \mathop{\mathbb{E}}_{\substack{(x,x^+)\sim\mathcal{D}_{sim},\\(x^-_1,\cdots,x^-_K)\sim\mathcal{D}_{neg}}} \! \left[
\!\log\!\sigma(f(x)^{T} f(x^+))\!+\!\sum_{k=1}^K\!\log\!\sigma(-f(x)^{T} f(x^-_k))\!
\right]\!, \end{split} \end{equation}} and its empirical counterpart with $M$ samples $\left(\!x_i^{},\!x_i^+,\!x_{i1}^-,\!\cdots,\!x_{iK}^- \!\right)_{i=1}^M$ is given as{\small \begin{equation}
\hat{\mathcal{L}}_{nce}\!\coloneqq\!-\frac{1}{M}\!\sum_{i=1}^{M}\! \left[
\log\sigma(f(x_i)^T f(x^+_i))\!+\!\sum_{k=1}^{K}\log\sigma(-f(x_i)^Tf(x^-_{ij}))
\right]\!, \end{equation}} where $\sigma(\cdot)$ is the sigmoid function.
For its supervised counterpart, it is exactly the cross entropy loss for the $(K+1)$-way multi-class classification task:{\small \begin{equation}
\mathcal{L}_{sup}^\mu \!\coloneqq \!-\!\mathop{\mathbb{E}}_{(x,c)\sim \mathcal{D}_\mathcal{T}(x,c)}\left[\log{\sigma(f(x)^T\mu_c})\!+\!\log{\sigma(-f(x)^T\mathbf{\mu}_{c^\prime})}\! \mid \! c^\prime \!\neq\! c\right]\!. \end{equation}}
\section{Theoretical Analyses on NCEloss}\label{section:theoretial analyses} In this section, we first give the upper bound of supervised loss when training a model \(f\) using NCEloss. Then we discuss the generalization bound of NCEloss along with the generalization bounds of hinge loss and logistic loss, and show their limitations.
\subsection{The Generalization Bound of NCEloss} We give the generalization error of function class $\mathcal{F}$ on the unsupervised loss function $\mathcal{L}_{nce}$ in Theorem~\ref{generalization}. Since we focus the regularization in contrastive learning, we give the result based on a single negative sample, i.e., $K=1$.
Let $c,c^\prime$ be two classes sampled independently from latent classes $\mathcal{C}$ with distribution $\rho$. Let $\tau=\mathbb{E}_{c,c^\prime\sim\rho^2}\mathbb{I}\left\{c=c^\prime\right\}$ be the probability that $c$ and $c^\prime$ come from the same class. And $\mathcal{L}_{nce}^=(f)$ and $\mathcal{L}_{nce}^\ne(f)$ are NCEloss when negative samples come from the same and different class, respectively. We have the following theorem. \begin{theorem}\label{generalization}
$\forall f\in\mathcal{F}$, with probability at least $1-\delta$,
\begin{equation}\label{eq:main}
\mathcal{L}_{sup}^\mu(\hat{f})\le \mathcal{L}^{\ne}_{nce}(f)+\beta s(f)+\eta Gen_M +\alpha ,
\end{equation}
where $\beta=\frac{\tau}{1-\tau}$, $\eta=\frac{1}{1-\tau}$, $\alpha=\frac{1}{1-\tau}(2\log c^\prime-1)$, $c^\prime\in\left[1+e^{-R^2},2\right]$, \(
s(f)=4\sqrt{\mathbb{E}_{(x_i,x_j)\sim\mathcal{D}_{sim}(x_i,x_j)}\left[(f(x_i)^Tf(x_j))^2\right]}\), and {\small
\begin{equation*}
\begin{split}
Gen_M=&\frac{8R\mathcal{R}_\mathcal{S}(\mathcal{F})}{M}-8\log(\sigma(-R^2))\sqrt{\frac{\log\frac{4}{\delta}}{2M}}\\
=&O\left(R\frac{\mathcal{R}_{\mathcal{S}}(\mathcal{F})}{M}+R^2\sqrt{\frac{\log\frac{1}{\delta}}{M}}\right).
\end{split}
\end{equation*}} \end{theorem}
\begin{remark*}
The above theorem tells us if contrastive learning algorithms with NCEloss could make $\beta s(f)+\eta Gen_M +\alpha$ converge to 0 as $M$ increases, the picked encoder $\hat{f}=\arg\min_{f\in\mathcal{F}}\hat{\mathcal{L}}_{nce}$ will have good performance in downtream tasks. In other words, we could guarantee contrastive learning algorithms to obtain high quality representations by minimizing $\hat{\mathcal{L}}_{nce}$ under the condition that $\beta s(f)+\eta Gen_M +\alpha$ will converge to 0 with a large amount of data. \end{remark*}
To prove Theorem~\ref{generalization}, we first list some key lemmas.
\begin{lemma}\label{lemma:supervised}
For all $f\in\mathcal{F}$,
\begin{equation}
\mathcal{L}_{sup}^\mu(f)\le\frac{1}{1-\tau}(\mathcal{L}_{nce}(f)-\tau).
\end{equation} \end{lemma} This bound connects contrastive representation learning algorithms and its supervised counterpart. This lemma is achieved by Jensen's inequality. The details are given in Appendix~\ref{appendix:convex}. \begin{lemma}\label{lemma:generalization}
With probability at least $1-\delta$ over the set $\mathcal{S}$, for all $f\in\mathcal{F}$,
\begin{equation}
\mathcal{L}_{nce}(\hat{f})\le\mathcal{L}_{nce}(f)+Gen_M .
\end{equation} \end{lemma} This bound guarantees that the chosen $\hat{f}=\arg\min_{f\in\mathcal{F}}\mathcal{L}_{nce}^\mu$ cannot be too much worse than $f^*=\arg\min_{f\in\mathcal{F}}\mathcal{L}_{nce}$. The proof applies Rademacher complexity of the function class~\cite{foundations_of_machine_learning} and vector-contraction inequality~\cite{DBLP:vector-contraction}. More details are given in Appendix~\ref{appendix:generalization_bound}. \begin{lemma}\label{lemma:class_collosion}
$\mathcal{L}_{nce}^=(f)\le 4s(f)+2\log c^\prime$. \end{lemma} This bound is derived by the loss caused by both positive and negative pairs that come from the same class, i.e., class collision. The proof uses Bernoulli's inequality (details in Appendix~\ref{apendix: class_collision}). \begin{proof}[Proof to Theorem~\ref{generalization}]
Combining Lemma~\ref{lemma:supervised} and Lemma~\ref{lemma:generalization}, we obtain
with probability at least $1-\delta$ over the set $\mathcal{S}$, for all $f\in\mathcal{F}$,
\begin{equation}\label{eq:connection}
\mathcal{L}_{sup}^\mu(\hat{f})\le \frac{1}{1-\tau}\left(\mathcal{L}_{nce}(f)+Gen_M-\tau\right)
\end{equation}
Then, we decompose $\mathcal{L}_{nce}=\tau\mathcal{L}_{nce}^=(f)+(1-\tau)\mathcal{L}_{nce}^\ne(f)$, apply Lemma~\ref{lemma:class_collosion} to Eq.~(\ref{eq:connection}), and obtain the result of Theorem~\ref{generalization} \end{proof}
\subsection{Discussion on the Generalization Bound}\label{subsection:relation}
We now discuss the generalization error of NCEloss in the contrastive learning setting.
\subsubsection{Discussion on $Gen_M$ and $s(f)$} $Gen_M$ in Eq.~(\ref{eq:main}) is the generalization error in terms of \textit{Rademacher complexity}. It shows that when the encoder function is bounded and the number of samples $M$ is large enough, $\hat{f}$ obtained by minimizing $\hat{\mathcal{L}}_{nce}$ provides performance guarantee. Note that when satisfying $\ell$ is a bounded Lipschitz continuous function and encoder $f$ is bounded, the generalization error of different contrastive loss terms will only differ in the contraction rate, i.e., Lipschitz continuous constant.
$s(f)$ in Eq.~(\ref{eq:main}) can be further rewritten as \begin{equation*}\label{eq:sf} \begin{split}
s(f)=&4\sqrt{\mathbb{E}_{(x_i,x_j)\sim\mathcal{D}_{sim}(x_i,x_j)}\left[f(x_i)^Tf(x_j)f(x_j)^Tf(x_i)\right]}\\
=&4\sqrt{\mathbb{E}_{c\sim\rho}\left[\mathbb{E}_{x_i\sim\mathcal{D}_c}\left[f(x_i)^T\mathbb{E}_{x_j\sim\mathcal{D}_c}\left[f(x_j)f(x_j)^T\right]f(x_i)\right]\right]}\\
\le &4\sqrt{\mathbb{E}_{{c}\sim\rho}\left[\max_{x\sim\mathcal{D}_c} \norm{f(x)}^2\times \norm{M(f,c)}_2\right]}, \end{split} \end{equation*} where $M(f,c)\coloneqq \mathbb{E}_{x\sim\mathcal{D}_c}\left[f(x)f(x)^T\right]$. It shows that NCEloss is prone to be disturbed by large representation norms.
\subsubsection{Cases that make contrastive learning suboptimal} There are two cases where contrastive learning algorithms cannot guarantee that $\hat{f}$ works in downstream tasks as pointed out in \cite{DBLP:Theoretical_analysis}, which also applies for NCELoss. \textit{Case~1}.\
The optimal $f$ for the downstream task can have large $\mathcal{L}_{un}^\ne$ ($\mathcal{L}_{nce}^\ne$) and thus failure of the algorithm, because of large spurious components in the representations that are orthogonal to the separation plain in the downstream task.
\textit{Case~2}. High intraclass deviation makes $\mathcal{L}_{un}^=$ ($\mathcal{L}_{nce}^=$) large even if both of its supervised counterpart losses \(\mathcal{L}_{sup}^\mu\) and $\mathcal{L}_{un}^\ne$ ($\mathcal{L}_{nce}^\ne$) are small, resulting in failure of the algorithm.
There is an additional case for NCEloss (\textit{Case 3}). The optimal $f$ for the downstream task can have large $\mathbb{E}[\norm{f(x)}]$ and $\mathrm{Var}(\norm{f(x)})$, which lead to large $\mathcal{L}_{nce}^\neq$ and large $s(f)$, even if $f$ gives low intraclass deviation.
\noindent \textit{Example.} \ Figure~\ref{fig:case1} depicts an example with $\mathcal{F}=\left\{f_1,f_2\right\}$, $\mathcal{C}=\left\{c_1,c_2\right\}$, $\mathcal{X}=\left\{x_1,x_2\right\}$, and $\mathcal{D}_{c_1}(x_1)=1$ and $\mathcal{D}_{c_2}(x_2)=1$. In this example, the linear separability of $f_1$ is better than $f_2$ in both Figure~\ref{fig:case1_1} and~\ref{fig:case1_2}, while $\norm{f_1(x_1)}\gg\norm{f_1(x_2)}$. In the case of $f_1(x_1)^Tf_1(x_2)>0$ (case 1 and 3), the contrastive learning algorithm using NCEloss will converge to pick $f_2$ since $\mathcal{L}_{nce}^\neq(f_1) \gg \mathcal{L}_{nce}^\neq(f_2)$ and $s(f_1) \gg s(f_2)$. When $f_1(x_1)^Tf_1(x_2)<0$, $f_2$ will be chosen, since $s(f_1) \gg s(f_2)$ in Eq.(\ref{eq:main}) (case 3).
We remark that once we avoid the problems of case 3, the problems of case 1 and case 2 cannot be serious. For case 1, since when both $\mathrm{Var}(\norm{f(x)})$ and $\mathbb{E}\left[\norm{f(x)}\right]$ are not large, the length of the orthogonal project is comparable to the separation plane so as to avoid destroying the contrast ability. For case 2, mild scale $\mathrm{Var}(\norm{f(x)})$ and $\mathbb{E}\left[\norm{f(x)}\right]$ avoids large intra-class deviation caused by large variance of the representation norm.
\begin{figure}
\caption{$f_1(x_1)^Tf_1(x_2)>0$}
\label{fig:case1_1}
\caption{$f_1(x_1)^Tf_1(x_2)<0$}
\label{fig:case1_2}
\caption{Cases of contrastive learning being suboptimal}
\label{fig:case1}
\end{figure}
We further show that case 3 is not an artificial case, but exists in practice. We use the training status of only using one contrastive loss computed by structural similarity on the Cora dataset~\cite{yang2016revisiting} (Section~\ref{section:model}) to demonstrate the issues with high expectation and high variance of representation norms. Denote \begin{equation*} \begin{split}
\mu^+&\coloneqq\mathbb{E}_{c\sim\rho}\left[\mathbb{E}_{(x_i,x_j)\sim\mathcal{D}_{sim}}\left[\abs{f(x_i)^Tf(x_j)}\right]\right]\\
\mu^-&\coloneqq\mathbb{E}_{c1\ne c2, c1,c2\sim\rho}\left[\mathbb{E}_{x_i\sim\mathcal{D}_{c1},x_j\sim\mathcal{D}_{c2}}\left[\abs{f(x_i)^Tf(x_j)}\right]\right] \end{split} \end{equation*} Figure~\ref{fig:norm_problem} shows the variance and the mean of representation norms (left top), the ratio of $\mu^+$ to $\mu^-$ (right top), $\mu^+$ and $\mu^-$ (left bottom), and the testing accuracy (in a node classification task) during training for 300 epochs. The variance and mean value of representation norms increase with the progression of epochs. This increases $\mu^+$ and $\mu^-$ significantly, while the ratio $\frac{\mu^+}{\mu^-}$ and representation quality~(indicated by the test accuracy) decrease. In the following sections, we use the ratio $\frac{\mu^+}{\mu^-}$ to measure the contrasting ability of models, i.e., the ability to contrast different classes.
\begin{figure}
\caption{Norm Problem.}
\label{fig:norm_problem}
\end{figure}
\begin{figure}
\caption{Random initialization}
\label{fig:initial_tsne}
\caption{Optimized}
\label{fig:optimised_tsne}
\caption{t-SNE visualization of $f(x)^TW$ and $f(\hat{x})^TW$}
\label{fig:tsne}
\end{figure}
\section{Contrastive Regularization}\label{section:contrast_reg} The theoretical analysis in Section~\ref{section:theoretial analyses} shows that a good contrastive representation learning algorithm should satisfy the following conditions: 1. avoiding large representation norm $\mathbb{E}\left[\norm{f(x)}\right]$; 2. avoiding large norm variance $\mathrm{Var}(\norm{f(x)})$; 3. preserving contrast.
We remark that the norm variance $\mathrm{Var}(\norm{f(x)})$ measures how far the norms of node representations are from their average value. It is different from intra-class variance $\norm{\Sigma(f,c)}_2$~\cite{DBLP:Theoretical_analysis}, which is the largest eigenvalue of covariance matrix $\Sigma(f,c)$. This is also the reason why case 2 is different from case 3 in the previous example.
In order to satisfy the above conditions, we propose a contrastive regularization term, \textbf{Contrast-Reg}: \begin{equation}\label{eq:reg} \mathcal{L}_{reg}=-\mathop{\mathbb{E}}_{x,\tilde{x}}\left[\log\sigma(f(x)^TW\mathbf{r})+\log\sigma(-f(\tilde{x})^TW\mathbf{r})\right], \end{equation} where $\mathbf{r}$ is a random vector uniformly sampled from $\left(0,1\right]$, $W$ is the trainable parameter, and $\tilde{x}$ is the noisy features. Different data augmentation techniques such as~\cite{DBLP:SimCLR, DBLP:DGI} can be applied to generate the noisy features. In Section~\ref{section:model}, we will discuss how we calculate the noisy features in the GNN setting.
We give the motivation of Contrast-Reg's design as follows. Consider an artificial downstream task that is to learn a classifier to discriminate the representations of regular data and noisy data. $\mathcal{L}_{reg}$ in Eq.~(\ref{eq:reg}) can be viewed as the classification loss and $W$ can be viewed as the parameter of the bi-linear classifier. The classifier prefers the encoder that can make the representations of intra-class data points more condensed and the inter-class representations more separable. We use the GCN model on the Cora dataset~\cite{yang2016revisiting} as an example. Figure~\ref{fig:initial_tsne} and Figure~\ref{fig:optimised_tsne} show the t-SNE visualization of $f(x)^TW$ and $f(\tilde{x})^TW$ before and after the optimization on $\mathcal{L}_{neg}$, respectively.
We can observe that the learned representations are closer to each other (i.e., the range of representations in Figure~\ref{fig:initial_tsne} are smaller than that in Figure~\ref{fig:optimised_tsne}), while preserving the separability among the representations (i.e., the points with the same label share the same color).
\subsection{Theoretical Guarantees for Contrast-Reg} Before stating the theorem, we give the following lemma to show that {$\mathrm{Var}(\norm{f})$ can be effectively reduced when $\norm{f(x)}$ is large by adding Contrast-Reg.}
\begin{lemma}\label{lemma:variance_reduction}
For a random variable $X\in\left[1.5,+\infty\right)$, a constant $\tau\in\left(0,1\right]$ and a constant $c^2$, we have
\begin{equation}
\mathrm{Var}\left(\sqrt{(X+\frac{\tau}{1+e^X})^2+c^2}\right)< \mathrm{Var}\left(\sqrt{X^2+c^2}\right).
\end{equation} \end{lemma} \begin{proof}\label{proof:variance}
First, we consider
\begin{equation*} \small
h(x)=\sqrt{(x+\frac{\tau}{1+e^x})^2+c^2}-\sqrt{x^2+c^2},
\end{equation*}
where $h(x)$ is strictly decreasing in $\left[x_0,+\infty\right)$ and strictly increasing in $\left(-\infty,x_0\right]$, and $x_0$ is the solution of $h^\prime(x)=\frac{\mathrm{d}h(x)}{\mathrm{d} x} = 0$. Thus, we can approximate the range of $x_0 \in (0,1.5)$ by the fact that $h^\prime(0)h^\prime(1.5)<0$ for all $\tau$ and $c^2$.
Thus, for $x > y \ge 1.5$,
\begin{equation*} \small
\sqrt{(x+\frac{\tau}{1+e^x})^2+c^2}-\sqrt{x^2+c^2}<\sqrt{(y+\frac{\tau}{1+e^y})^2+c^2}-\sqrt{y^2+c^2}
\end{equation*}
and since $(x+\frac{\tau}{1+e^x})$ is monotonically increasing, we get
\begin{equation*} \small
0 < \sqrt{(x+\frac{\tau}{1+e^x})^2+c^2}-\sqrt{(y+\frac{\tau}{1+e^y})^2+c^2}<\sqrt{x^2+c^2}-\sqrt{y^2+c^2}.
\end{equation*}
When $y > x \ge 1.5$,
\begin{equation*} \small
\sqrt{x^2+c^2}-\sqrt{y^2+c^2} < \sqrt{(x+\frac{\tau}{1+e^x})^2+c^2}-\sqrt{(y+\frac{\tau}{1+e^y})^2+c^2}<0.
\end{equation*}
Further, we assume that $X$ and $Y$ are i.i.d. random variables sampled from $\left[1.5,+\infty\right)$,
\begin{equation*} \small
\begin{split}
&\mathrm{Var}\left(\sqrt{(X+\frac{\tau}{1+e^X})^2+c^2}\right)\\
=&\frac{1}{2}\times\mathbb{E}_{X,Y} \left[\left(\sqrt{(X+\frac{\tau}{1+e^X})^2+c^2}-\sqrt{(Y+\frac{\tau}{1+e^Y})^2+c^2}\right)^2\right]\\
=&\frac{1}{2}\times\int\left(\sqrt{(x+\frac{\tau}{1+e^x})^2+c^2}-\sqrt{(y+\frac{\tau}{1+e^y})^2+c^2}\right)^2p(x)p(y)\mathrm{d}x\mathrm{d}y\\
<&\frac{1}{2}\times\int\left(\sqrt{x^2+c^2}-\sqrt{y^2+c^2}\right)^2p(x)p(y)\mathrm{d}x\mathrm{d}y\\
=&\mathrm{Var}(\sqrt{X^2+c^2})
\end{split}
\end{equation*} \end{proof}
\begin{theorem}\label{theorem:variance}
Minimizing Eq.~(\ref{eq:reg}) induces the decrease in $\mathrm{Var}(\norm{f(x)})$ when $f(x)^TW\mathbf{r} \in [1.5,+\infty)$.
\end{theorem} \begin{proof}
We minimize $\mathcal{L}_{neg}$ by gradient descent with learning rate $\beta$ .
\begin{equation*} \small
\begin{split}
\frac{\partial}{\partial f(x)}\mathcal{L}_{reg}=-\sigma(-f(x)^TW\mathbf{r})W\mathbf{r}
\end{split}
\end{equation*}
\begin{equation}\label{eq:geometric} \small
\begin{split}
f(x)\leftarrow f(x)+\beta\left(\sigma(-f(x)^TW\mathbf{r})W\mathbf{r}\right)
\end{split}
\end{equation}
Eq.~(\ref{eq:geometric}) shows that in every optimization step, $f(x)$ extends by $\beta\sigma(-f(x)^TW\mathbf{r})\norm{W\mathbf{r}}$ along $\mathbf{r}_0\coloneqq\frac{W\mathbf{r}}{\norm{W\mathbf{r}}}$. If we do orthogonal decomposition for $f(x)$ along $\mathbf{r}_0$ and its unit orthogonal hyperplain $\Pi(\mathbf{r}_0)$, $f(x)=\left(f(x)^T\mathbf{r}_0\right)\mathbf{r}_0+\left(f(x)^T\Pi(\mathbf{r}_0)\right)\Pi(\mathbf{r}_0)$. Thus we have
\begin{equation}\small \label{eq:norm_expansion}
\norm{f(x)}=\sqrt{\left(f(x)^T\mathbf{r}_0\right)^2+\left(f(x)^T\Pi(\mathbf{r}_0)\right)^2}.
\end{equation}
The projection of $f(x)$ along $\mathbf{r}_0$ is $
f(x)^T\mathbf{r}_0= \frac{f(x)^TW\mathbf{r}}{\norm{W\mathbf{r}}}, $ while the projection of $f(x)$ plus the Contrast-Reg\ update along $\mathbf{r}_0$ is {\small\[
\left(f(x)^T\mathbf{r}_0\right)_{reg}= \frac{f(x)^TW\mathbf{r}}{\norm{W\mathbf{r}}}+ \frac{\beta}{1+e^{f(x)^TW\mathbf{r}}}\norm{W\mathbf{r}}. \]} Note that $\left(f(x)^T\Pi(\mathbf{r}_0)\right)_{reg}=f(x)^T\Pi(\mathbf{r}_0)$.
Based on Lemma~\ref{lemma:variance_reduction} and Eq.~(\ref{eq:norm_expansion}),
when $\beta\norm{W\mathbf{r}}^2\le 1$ and $f(x)^TW\mathbf{r}>1.5$, we have
\begin{equation}\label{eq:variance1}
\mathrm{Var}\left(\norm{\left(f(x)\right)_{reg}}\right)
<
\mathrm{Var}\left(\norm{f(x)}\right).
\end{equation}
\end{proof}
\begin{remark}
$\beta\norm{W\mathbf{r}}^2\le 1$, which is the condition of Eq.~(\ref{eq:variance1})
, is not difficult to satisfy, since the magnitude of $\mathbf{r}$ could be tuned. In practice, $\mathbf{r}\in\left(0,1\right]$ can fit in all our experiments. \end{remark}
\begin{remark}
The range of $f(x)^Tw\mathbf{r}$ in Theorem~\ref{theorem:variance} is not a tight bound for $x_0$ in Lemma~\ref{lemma:variance_reduction}. Since when Eq.~(\ref{eq:reg}) converges, $f(x)^TW\mathbf{r}$ is much larger than 1.5 for almost all the samples empirically, we prove the case for $f(x)^Tw\mathbf{r} \in [1.5,+\infty)$. \end{remark}
\subsection{Understanding the effects of Contrast-Reg}
\begin{figure}
\caption{Geometric interpretation}
\label{fig:geometric}
\end{figure}
Theorem~\ref{theorem:variance} shows that Contrast-Reg\ can reduce $\mathrm{Var}(\left[\norm{f}\right])$ when $\norm{f(x)}$ is large, which is proved from the geometric perspective. Figure~\ref{fig:geometric} visualizes the geometric interpretation of Contrast-Reg. In one gradient descent step, $f(x)$ and $f^\prime(x)$ are the representation before and after the gradient descent update of Contrast-Reg. For any data point pair $x_1$ and $x_2$, we decompose $f(x_1)$ and $f(x_2)$ along $W\mathbf{r}$ and its orthogonal direction. Minimizing Eq.~(\ref{eq:reg}) is consequently extending $f(x)$ in each step along $W\mathbf{r}$ while preserving the length in its orthogonal direction. When we compare $f(x_1)$ and $f(x_2)$, together with $f^\prime(x_1)$ and $f^\prime(x_2)$, we conclude that $\mathrm{Var}\left(f^\prime(x)\right)<\mathrm{Var}\left(f(x)\right)$.
Note that the mean and the variance are positively correlated. When we compare $\mathbb{E}\left[\norm{\lambda f}\right]$ with $\mathbb{E}\left[\norm{f}\right]$, the former has higher norm and variance for $\abs{\lambda}>1$. Theorem~\ref{theorem:variance} shows that our Contrast-Reg can reduce $\mathrm{Var}(\left[\norm{f}\right])$ when $\norm{f(x)}$ is large, and it should also prefer lower mean because the variance will increase when the norm is scaled to a larger value. Figure~\ref{fig:norm} shows that the mean and variance are reduced when we apply Contrast-Reg~ compared to only using one contrastive loss computed by structural similarity. Also, the representation quality is improved significantly.
\begin{figure}
\caption{The influence of Contrast-Reg\ on $\norm{f}$}
\label{fig:norm}
\end{figure}
We also discuss two questions regarding the effects of Contrast-Reg. \textbf{\textit{Does Contrast-Reg\ degrade the contrastive learning algorithm to a trivial solution, i.e., all representations converge to one point, even to the origin?}} We highlight that Theorem~\ref{theorem:variance} is to reduce $\mathrm{Var}(\norm{f(x)})$ rather than $\mathrm{Var}(f(x))$, and Contrast-Reg\ does not force all the representations into one point. From Figure~\ref{fig:geometric}, we know that adding Contrast-Reg~only reduces the variance in the representations along the direction of $W\mathbf{r}$, while preserving the difference along its orthogonal direction. Therefore, Contrast-Reg\ not only reduces $\mathrm{Var}(\norm{f(x)})$, but also preserves the contrasting ability of the contrastive learning algorithm, as shown in {Figure~\ref{fig:optimised_tsne}}. Furthermore, as we randomize $\mathbf{r}$ in each training step, the variance reduction on the representation norm is conducted on various directions. Thus, the representations will not have the same dominant direction, and Contrast-Reg\ does not make the representations converge to one point.\\
\textbf{\textit{Can other regularization/normalization methods, e.g., weight decay or final representation normalization, solve the norm problem?}} Other regularization and normalization techniques may help stabilize the mean of the representation norms and reduce their variance, but they cannot replace Contrast-Reg\ as Contrast-Reg\ leads to more stable changes in the representation norms and preserves high contrast between positive and negative samples.
We compare Contrast-Reg\ with weight decay and $\ell_2-$normalization and show their performance during a 300-epoch training process. Figure~\ref{fig:weight_decay} shows the performance of weight decay and Contrast-Reg. From the left-top figure, we can see that Contrast-Reg\ gives smaller and more stable representation norms than weight delay. Specifically, although a large weight decay rate can be applied to obtain smaller representation norms, it leads to fluctuation. The fluctuation in the representation norms impairs the training process, as reflected by the fluctuation in training losses (left-bottom figure). In addition, from the ratio $\frac{\mu^+}{\mu^-}$, we know that Contrast-Reg~preserves better contrasting ability.
\citet{DBLP:SimCLR} show that adding $\ell_2-$normalization~(i.e., using cosine similarity rather than inner product) with a temperature parameter improves the representation quality empirically. Figure~\ref{fig:cosine} compares Contrast-Reg\ with $\ell_2-$normalization. $\ell_2-$normalization gives much smaller $\frac{\mu^+}{\mu^-}$ than Contrast-Reg, meaning that the separation between contrastive pairs provided by $\ell_2-$normalization is not as clear as that provided by Contrast-Reg. This is because $\ell_2-$normalization not only minimizes the variance in the representation norms, but also reduces the differences among the representations, rendering smaller contrast among the representations of data points in different classes. Thus, $\ell_2-$normalization gives less improvement in the representation quality than Contrast-Reg.
\begin{figure}
\caption{Comparing Contrast-Reg\ with weight decay}
\label{fig:weight_decay}
\end{figure} \begin{figure}
\caption{Comparing Contrast-Reg\ with $\ell_2-$normalization}
\label{fig:cosine}
\end{figure}
\section{A Contrastive GNN Framework}\label{section:model}
We present our contrastive GNN framework in Algorithm~\ref{alg:framework}. Given a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ and node attributes $\mathcal{X}$, we train a GNN model $f$ for $e$ epochs. Node representations can be obtained through $f$ and used as the input to downstream tasks. We adopt NCEloss as the contrastive loss in our framework. For each training epoch, we first select a seed node set $\mathcal{C}$ for computing NCEloss by $SeedSelect$ (Line~3). Then Line~4 invokes $Constrast$ to construct a positive sample and a negative sample for each node in $\mathcal{C}$, where $Constrast$ returns a set $\mathcal{P}$ of 3-tuples consisting of the representations for the seed nodes, the positive samples and the negative samples, respectively. After that, Line~5 computes the training loss by adding NCEloss on $\mathcal{P}$ and the regularization loss calculated by Contrast-Reg, and Line~6 updates $f$ by back-propagation.
As mentioned in Section~\ref{section:contrast_reg}, Contrast-Reg\ requires noisy features for contrastive regularization. In our contrastive GNN framework, we generate the noisy features by simply shuffling node features among nodes following the corruption function in~\cite{DBLP:DGI}.
For different node similarity definitions, different $SeedSelect$ functions are designed to select seed nodes that bring good training effects, while different $Contrast$ functions are designed to generate suitable contrastive pairs for the seed nodes. In the following, we demonstrate by examples the designs of three contrastive GNN models for structure, attribute, and proximity similarity, respectively, which are also used in our experimental evaluation in Section~\ref{section:exp}.
\begin{algorithm}[t] \small
\SetAlgoLined
\KwIn{Graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, node attributes $\mathcal{X}$, a GNN model \(f: \mathcal{V} \rightarrow \mathbb{R}^H\), the number of epochs $e$;}
\KwOut{A trained GNN model $f$;}
Initialize training parameters;\\
\For{$epoch \gets 1$ \KwTo $e$}{
$\mathcal{C}$=SeedSelect($\mathcal{G}$, $\mathcal{X}$, $f$, $epoch$); \\
$\mathcal{P}$=Contrast($\mathcal{C}$, $\mathcal{G}$, $\mathcal{X}$, $f$);\\
loss = NCEloss($\mathcal{P}$) + Contrast-Reg($\mathcal{G}$, $\mathcal{X}$, $f$);\\
Back-propagation and update \(f\);
}
\caption{Contrastive GNN Framework}
\label{alg:framework} \end{algorithm}
\subsection{Structure Similarity}
We give an example model~($LC$) that captures the community structure inherent in graph data~\cite{community}. As clustering is a common and effective method for detecting communities in a graph, we conduct clustering in the node representation space \(f(\mathcal{X})\) to capture community structures. LC borrows the design from~\cite{DBLP:AND} and implements local clustering by $Contrast$ and curriculum learning by $SeedSelect$. We remark that other methods such as global clustering~\cite{DBLP:DeepCluster} and instance discrimination~\cite{DBLP:instance_discrimination} can also be adapted into our contrastive GNN framework by different implementations of $Contrast$ and $SeedSelect$.
\begin{algorithm}[!t] \small
\SetAlgoLined
\SetKwFunction{seed}{SeedSelect}
\SetKwFunction{contrast}{Contrast}
\SetKwInput{hyperparameter}{Hyperparameter}
\hyperparameter{$R$: curriculum update epochs; $k$: the number of candidate positive samples for each seed node;}
\SetKwProg{Pn}{Function}{:}{end}
\Pn{\contrast{$\mathcal{C}$, $\mathcal{G}$, $\mathcal{X}$, $f$}}{
For $x_i\in \mathcal{C}$, let $P_i$ be the set of $k$ nodes in $\{x_j \in \text{Neighbor}(x_i)\}$ with largest $f(x_i)^Tf(x_j)$; \\
Randomly pick one positive node $x^+_i$ from $P_i$ for each $x_i\in \mathcal{C}$;\\
Randomly pick one negative node $x^-_i$ from $\mathcal{V}$ for each $x_i\in \mathcal{C}$;\\
\KwRet $\left\{(f(x_i),f(x^+_i),f(x^-_i))\right\}_{x_i\in\mathcal{C}}$;
}
\SetKwProg{Pn}{Function}{:}{end}
\Pn{\seed{$\mathcal{G}$, $\mathcal{X}$, $f$, $epoch$}}{
\If{$epoch$ \% $R$ $\neq$ 1}{\KwRet the same set of seed nodes $C$ as in the last epoch ;}
$p_{i,j} \leftarrow \dfrac{f(i)^Tf(j)}{\sum_{k\in\mathcal{V}}f(i)^Tf(k)}$ for $i,j\in\mathcal{V}$;\\
$H(i)\leftarrow -\sum_{j\in\mathcal{V}}(p_{i,j}\log\left(p_{i,j}\right))$ for $i\in\mathcal{V}$;\\
\KwRet $(\lfloor \frac{epoch}{R} \rfloor+1)\frac{R}{e}\abs{\mathcal{G}}$ nodes with smallest $H$;
}
\caption{LC}
\label{alg:LC} \end{algorithm}
\begin{algorithm}[!t] \small
\SetAlgoLined
\SetKwFunction{seed}{SeedSelect}
\SetKwFunction{contrast}{Contrast}
\SetKwInput{parameter}{Parameter}
\parameter{Parameters of an (additional) GNN layer $g$.}
\SetKwProg{Pn}{Function}{:}{end}
\Pn{\contrast{$\mathcal{C}$, $\mathcal{G}$, $\mathcal{X}$, $f$}}{
Let $g(x_i)$ be the representation of $x_i$ by stacking $g$ upon $f$;\\
Randomly pick a negative node $x^-_i$ from $\mathcal{V}$ for each $x_i \in \mathcal{C}$;\\
\KwRet $\left\{(g(x_i),f(x_i),f(x^-_i))\right\}_{x_i\in\mathcal{G}}$;
}
\SetKwProg{Pn}{Function}{:}{end}
\Pn{\seed{$\mathcal{G}$, $\mathcal{X}$, $f$, $epoch$}}{
\KwRet $\mathcal{V}$;
}
\caption{ML}
\label{alg:ML} \end{algorithm}
Algorithm~\ref{alg:LC} shows the implementation of $Contrast$ and $SeedSelect$ in LC. For each seed node $x_i$, $Contrast$ generates a positive node $x_i^+$ from the nodes that have the highest similarity scores with $x_i$, and a negative node $x_i^-$ randomly sampled from $V$ (Lines~2-4). $SeedSelect$ selects nodes with the smallest entropy to avoid high randomness and uncertainty at the start of the training. For every $R$ epochs, $SeedSelect$ gradually adds more nodes with larger entropy to be computed in the contrastive loss with the progression of epochs (Lines~11-13).
\subsection{Attribute Similarity}
Models adopting attribute similarity assume that nodes with similar attributes are expected to have similar representations, so that the attribute information should be preserved. \citet{DBLP:deep_infomax, DBLP:GMI} proposed contrastive pair designs to maximize the mutual information between low-level representations (the input features) and high-level representations (the learned representations). Algorithm~\ref{alg:ML} presents our model, $ML$, which adapts their multi-level representation design into our contrastive GNN framework.
In Algorithm~\ref{alg:ML}, $SeedSelect$ selects all nodes in a graph as seeds. $Contrast$ uses node $x_i$ itself as the positive node for each seed node $x_i$, but in the returned 3-tuple, the representation of $x_i$ as the seed node is different from the representation of $x_i$ as the positive node. The second element in the 3-tuple is $x_i$'s representation $f(x_i)$, while the first element is calculated by stacking an additional GNN layer upon $f$. For negative nodes, $Contrast$ randomly samples a node in $V$ for each seed node.
\subsection{Proximity Similarity} The assumption behind proximity similarity is that nodes are expected to have similar representation when they have high proximity (i.e., they are near neighbors). To capture proximity information among nodes, we implement $SeedSelect$ and $Contrast$ following the setting in unsupervised GraphSAGE~\cite{kipf2017semi}. Adjacent nodes are selected to be positive pairs, while negative pairs are sampled from non-adjacent nodes.
\begin{table}[t]
\caption{Datasets} \label{tab:summary}
{\small
\begin{tabular}{crrrr}
\toprule
Dataset & Node \# & Edge \# & Feature \# & Class \# \\
\midrule
Cora & 2,708& 5,429& 1,433&7\\
Citeseer & 3,327& 4,732& 3,703&6\\
Pubmed & 19,717& 44,338& 500&3\\
ogbn-arxiv & 169,343& 1,166,243& 128&40\\
Wiki & 2,405& 17,981& 4,973&3\\
Computers & 13,381& 245,778& 767&10\\
Photo & 7,487& 119,043& 745&8\\
ogbn-products & 2,449,029& 61,859,140& 100&47\\
Reddit & 232,965& 114,615,892& 602&41\\
\bottomrule
\end{tabular}} \end{table}
\begin{table}[t]
\caption{Contrastive learning with and w/o Contrast-Reg}
\label{tab:ablation}
{\small
\begin{tabular}{ccccc}
\toprule
Algorithm & Cora & Wiki & Computers & Reddit\\
\midrule
ML & 73.22\PM{0.77} & 58.70\PM{1.51}&77.08\PM{2.48}&94.33\PM{0.07} \\
ML+reg & \textbf{82.65}\PM{0.57}& \textbf{67.20}\PM{0.96}&\textbf{80.30}\PM{1.84}&\textbf{94.38}\PM{0.04} \\
\midrule
LC & 79.73\PM{0.75}& 68.96\PM{0.56}&79.80\PM{1.49}&94.42\PM{0.03} \\
LC+reg & \textbf{82.33}\PM{0.41} & \textbf{69.19}\PM{1.13}&\textbf{80.89}\PM{1.64}&\textbf{94.43}\PM{0.03} \\
\midrule
CO & 75.49\PM{0.81}& 68.52\PM{1.07}& 81.02\PM{1.54}& 93.85\PM{0.03} \\
CO+reg & \textbf{83.63}\PM{0.62}& \textbf{70.05}\PM{0.98}&\textbf{81.37}\PM{1.58}& \textbf{93.92}\PM{0.05}\\
\bottomrule
\end{tabular}} \end{table}
\begin{table*}[t]
\caption{Downstream task: node classification}
\label{tab:node_classification}
{\small
\begin{tabular}{cccccccccc}
\toprule
Algorithm & Cora & Citeseer & Pubmed & ogbn-arxiv & Wiki & Computers & Photo & ogbn-products & Reddit \\
\midrule
GCN & 81.54\PM{0.68}& 71.25\PM{0.67}& 79.26\PM{0.38}& \textbf{71.74}\PM{0.29}& \textbf{72.40}\PM{0.95}& 79.82\PM{2.04}& \textbf{88.75}\PM{1.99}& 75.64\PM{0.21}&94.02\PM{0.05} \\
\midrule
node2vec & 71.07\PM{0.91}& 47.37\PM{0.95}& 66.34\PM{1.40}& \textbf{70.07}\PM{0.13}& 58.76\PM{1.48}& 75.37\PM{1.52}&83.63\PM{1.53} & 72.49\PM{0.10}&93.26\PM{0.04} \\
DGI & 81.90\PM{0.84}& 71.85\PM{0.37}& 76.89\PM{0.53}& 69.66\PM{0.18}& 63.70\PM{1.43}& 64.92\PM{1.93}& 77.19\PM{2.60}& \textbf{77.00}\PM{0.21}& 94.14\PM{0.03}\\
GMI & 80.95\PM{0.65}& 71.11\PM{0.15}& 77.97\PM{1.04}& 68.36\PM{0.19}& 63.35\PM{1.03}& 79.27\PM{1.64}&87.08\PM{1.23} & 75.55\PM{0.39}& 94.19\PM{0.04}\\
ours~(LC) & \textbf{82.33}\PM{0.41}& \textbf{72.88}\PM{0.39}& \textbf{79.33}\PM{0.59}& 69.94\PM{0.11}& \textbf{69.19}\PM{1.13}& \textbf{80.89}\PM{1.64}& \textbf{87.59}\PM{1.50}& \textbf{76.96}\PM{0.34}& \textbf{94.43}\PM{0.03}\\
ours~(ML) & \textbf{82.65}\PM{0.57}&\textbf{72.98}\PM{0.41} &\textbf{80.10}\PM{1.04} &\textbf{70.05}\PM{0.09} &67.20\PM{0.96} &\textbf{80.30}\PM{1.84} &86.78\PM{1.70} & 76.27\PM{0.20}& \textbf{94.38}\PM{0.04}\\
\bottomrule
\end{tabular}} \end{table*} \begin{table*}[t]
\caption{Downstream task: graph clustering}
\label{tab:graph_clustering}
{\small
\begin{tabular}{cccccccccc}
\toprule
Algorithm &&Cora& &&Citeseer& &&Wiki& \\
\cmidrule(lr){2-4}
\cmidrule(lr){5-7}
\cmidrule(lr){8-10}
& Acc & NMI & F1 & Acc & NMI & F1 & Acc & NMI & F1\\
\midrule
node2vec & 61.78\PM{0.30}& 44.47\PM{0.21}& 62.65\PM{0.26}& 39.58\PM{0.37}& 24.23\PM{0.27}& 37.54\PM{0.39}& 43.29\PM{0.58}& 37.39\PM{0.52}& 36.35\PM{0.51}\\
DGI &\textbf{71.81}\PM{1.01} & 54.90\PM{0.66}& \textbf{69.88}\PM{0.90}& 68.60\PM{0.47}& 43.75\PM{0.50}& 64.64\PM{0.41}& 44.37\PM{0.92}& 42.20\PM{0.90}&40.16\PM{0.72} \\
AGC & 68.93\PM{0.02}& 53.72\PM{0.04}& 65.62\PM{0.01}& 68.37\PM{0.02}& 42.44\PM{0.03}& 63.73\PM{0.02}& 49.54\PM{0.07}& 47.02\PM{0.09}& 42.16\PM{0.11} \\
GMI &63.44\PM{3.18}& 50.33\PM{1.48}& 62.21\PM{3.46}& 63.75\PM{1.05}& 38.14\PM{0.84}& 60.23\PM{0.79}& 42.81\PM{0.40}& 41.53\PM{0.20}&38.52\PM{0.22}\\
ours~(LC) & 70.04\PM{2.04}& 55.08\PM{0.75}& 67.36\PM{2.17} & 67.90\PM{0.74}& 43.63\PM{0.57}& 64.21\PM{0.60} & 50.12\PM{0.96}& 49.70\PM{0.49}& 43.74\PM{0.97}\\
ours~(ML) & 71.59\PM{1.07}& \textbf{56.01}\PM{0.64}& 68.11\PM{1.32} & \textbf{69.17}\PM{0.43}& \textbf{44.47}\PM{0.46}& \textbf{64.74}\PM{0.41} & \textbf{53.13}\PM{1.01}& \textbf{51.81}\PM{0.57}& \textbf{46.11}\PM{0.93}\\
\bottomrule
\end{tabular}} \end{table*}
\section{Experimental Results}\label{section:exp}
In this section, we first show that Contrast-Reg\ can be generally used for various designs of contrastive pairs. Then we evaluate the performance of various models trained with Contrast-Reg\ in both graph representation learning and pretraining settings, where contrastive learning is successfully applied.
\noindent \textbf{Datasets.}
The datasets we used include citation networks such as \textit{Cora, Citeseer, Pubmed}~\cite{yang2016revisiting} and \textit{ogbn-arxiv}~\cite{ogb}, web graphs such as \textit{Wiki}~\cite{DBLP:wiki}, co-purchase networks such as \textit{Computers, Photo}~\cite{DBLP:pitfalls} and \textit{ogbn-products}~\cite{ogb}, and social networks such as \text{Reddit}~\cite{hamilton2017inductive}. Some statistics of the datasets are given in Table~\ref{tab:summary}.
\noindent \textbf{Models.} We denote Algorithm~\ref{alg:LC} (local clustering) capturing \textit{structure similarity} as \textbf{ours~(LC)}, Algorithm~\ref{alg:ML} (multi-level representations) capturing \textit{attribute similarity} as \textbf{ours~(ML)} and the algorithm (co-occurrence) capturing \textit{proximity similarity} as \textbf{ours~(CO)}.
\noindent \textbf{Unsupervised training procedure.} We used full batch training for Cora, Citeseer, Pubmed, ogbn-arxiv, Wiki, Computers and Photo, while we used stochastic mini-batch training for Reddit and ogbn-products. For Cora, Citeseer, Pubmed, ogbn-arxiv, ogbn-products and Reddit, we used the standard split provided in the datasets and fixed the randoms seeds from 0 to 9 for 10 different runs.
For Computers, Photo and Wiki, we randomly split the train/validation/test as 20 nodes/30 nodes/all the remaining nodes per class, as recommended in~\cite{DBLP:pitfalls}. The performance was measured on 25 \((5\times5)\) different runs, with 5 random splits and 5 fixed-seed runs (from 0 to 4) for each random split. For Wiki, we removed the edge attributes for all models for fair comparison. The additional special designs for link prediction task and pretraining setting are given in their respective subsections.
\subsection{Generalizability of Contrast-Reg}\label{sec:wo_reg}
To evaluate the performance gain by Contrast-Reg, we tested the model performance (on node classification accuracy) with and without Contrast-Reg\ on four networks from four different domains. GCN encoder was used on Cora, Wiki and Computers. Graph-Sage with GCN-aggregation encoder was used on Reddit.
Table~\ref{tab:ablation} shows that Contrast-Reg\ can help better capture different types of similarity, i.e., \textbf{ML} for attribute similarity, \textbf{LC} for structure similarity, and \textbf{CO} for proximity similarity, and improve the performance of the models in all cases. In the following experiments, we omit CO since its contrast loss is computed by sampled edges and thus the computation cost is larger than the other two contrast designs.
\subsection{Graph Representation Learning}
Next we show that high quality representations can be learned by our method. High quality means using these representations, simple models (e.g., linear classifier for classification and $k$-means for clustering) can easily achieve high performance on various downstream tasks. To show that, we evaluated the learned representations on three downstream tasks that are fundamental for graph analytics: node classification, graph clustering, and link prediction.
\subsubsection{Node Classification}\label{sec:exp_nc} We evaluated the performance of node classification on all datasets, using both full batch training and stochastic mini-batch training. We compared our methods with DGI~\cite{DBLP:DGI}, GMI~\cite{DBLP:GMI}, node2vec~\cite{DBLP:node2vec}, and supervised GCN~\cite{kipf2017semi}. DGI and GMI are the state-of-the-art algorithms in unsupervised graph representation learning. Node2vec is the representative algorithm for random walk based graph representation algorithms~\cite{DBLP:node2vec, DBLP:LINE, DBLP:DeepWalk}. GCN is a classic supervised GNN model. We report ours~(LC) and ours~(ML), both using Contrast-Reg, where their GNN encoder is GCN for full batch training and GraphSage~\cite{hamilton2017inductive} with GCN-aggregation for stochastic training, respectively. The encoder settings are the same as in DGI and GMI. Our framework can also adopt other encoder such as GAT and similar performance improvements over GAT can also be obtained. We omit the detailed results due to the page limit.
Table~\ref{tab:node_classification} reports node classification accuracy with standard deviation. The results show that our algorithms achieve better performance in the majority of the cases, for both full batch training (on Cora, Citeseer, Pubmed, Computers, Photo and Wiki) and stochastic training (on Reddit and Ogbn-products). Our unsupervised algorithms can even outperform the supervised GCN. Compared with DGI and GMI, our model is similar to the DGI model with a properly designed contrast pair and to the GMI model with the Contrast-Reg term, and thus we can achieve better performance in most cases. If we compare Table~\ref{tab:ablation} and Table~\ref{tab:node_classification}, it shows that the performance gain is from Contrast-Reg\ rather than a more proper contrast design.
\subsubsection{Graph Clustering}\label{sec:exp_gc}
Following the work of~\citet{DBLP:clustering_metric}, we used three metrics to evaluate the clustering performance: accuracy~(Acc), normalized mutual information~(NMI), and F1-macro~(F1). For all these three metrics, a higher value indicates better clustering performance. We compared our methods with DGI, node2vec, GMI, and AGC~\cite{DBLP:AGC} on Cora, Citeseer and Wiki. AGC~\cite{DBLP:AGC} is a state-of-the-art graph clustering algorithm, which exploits high-order graph convolution to do attribute graph clustering. For all models and all datasets, we used $k$-means to cluster both the labels and representations of nodes. The clustering results of labels are taken as the ground truth. Since high dimension is harmful to clustering~\cite{Chen2009}, we applied the PCA algorithm to the representations to reduce the dimensionality before using $k$-means.
The random seed setting for model training was the same as that in the node classification task. And to reduce the randomness caused by $k$-means, we set the random seed of clustering from 0 to 4, and took the average result for each learned representations. For each cell in Table~\ref{tab:graph_clustering}, we report the better result with PCA and without PCA. The results show that our algorithms, especially ours~(ML), achieve better performance in most cases, which again demonstrates the effectiveness of Contrast-Reg. Note that graph clustering is applied on attribute graphs, the fact that the results of ours~(ML) are better than ours~(LC) tells us that attributes play an important role in clustering.
\begin{table}[t]
\caption{Downstream task: link prediction}
\label{tab:link_prediction}
{\small
\begin{tabular}{ccccc}
\toprule
Algorithm & Cora & Citeseer & Pubmed & Wiki \\
\midrule
GCN--neg & 92.40\PM{0.51}& 92.27\PM{0.90}& 97.24\PM{0.19}&93.27\PM{0.31}\\
node2vec & 86.33\PM{0.87}& 79.60\PM{1.58}& 81.74\PM{0.57}& 92.41\PM{0.35}\\
DGI & 93.62\PM{0.98}& 95.03\PM{1.73}& 97.24\PM{0.13}&95.55\PM{0.35}\\
GMI & 91.31\PM{0.88}& 92.23\PM{0.80}& 95.14\PM{0.25}& 95.30\PM{0.29} \\
ours~(LC) & \textbf{94.61}\PM{0.64}& \textbf{95.63}\PM{0.88} & \textbf{97.26}\PM{0.15}& \textbf{96.28}\PM{0.21}\\
\bottomrule
\end{tabular}} \end{table}
\subsubsection{Link Prediction} The representations learned in Sections~\ref{sec:exp_nc} and \ref{sec:exp_gc} should not be directly used in the link prediction task, because the encoder already has access to all edges in an input graph when we train it using contrastive learning, which leads to the data linkage issue (i.e., the edges used in the prediction task being accessible in the training process). Thus, for link prediction, an inductive setting of graph representation learning was adopted. We extracted random induced subgraphs (85\% of the edges)
from each origin graph to train the representation learning model and the link predictor. The remaining edges were used to validate and test the link prediction results (10\% of the edges as the test edge set, 5\% as the validation edge set). The performance was evaluated on 25 (5x5) different runs, with 5 different induced subgraphs (fixed-seed random split scheme) and 5 fixed-seed runs (from 0 to 4). we compared our model with DGI, GMI, node2vec and unsupervised GCN (i.e., GCN-neg in Table~\ref{tab:link_prediction}) on Cora, Citeseer, Pubmed and Wiki. The results in Table~\ref{tab:link_prediction} show that our algorithms achieve better performance than the state-of-the-art methods. We did not conduct the \textit{ML} model in this experiment because \textit{ML} pays more attention to the node attributes.
\begin{table}[t]
\caption{Pretraining}
\label{tab:pretraining}
{\small
\begin{tabular}{ccc}
\toprule
Algorithm & Reddit & ogbn-products\\
\midrule
No pretraining & 90.44\PM{1.62}& 84.69\PM{0.79}\\
DGI & 92.09\PM{1.05}& 86.37\PM{0.19}\\
GMI & 92.13\PM{1.16}& 86.14\PM{0.16}\\
ours~(ML) & 92.18\PM{0.97}& 86.28\PM{0.20}\\
ours~(LC) & \textbf{92.52}\PM{0.55}& \textbf{86.45}\PM{0.13}\\
\bottomrule
\end{tabular}} \end{table}
\subsection{Pretraining}
We also evaluated the performance of Contrast-Reg\ for pretraining. For the Reddit dataset, as it can be naturally split by time, we pretrained the models using the first 20 days (by generating an induced subgraph based on the pretraining nodes) from the dataset, the remaining data was split into three parts: the first part generated a new subgraph for fine-tuning the pretrained model and training the classifier, and the second and third parts were used for validation and test. For the ogbn-products dataset, we split the dataset based on node id, i.e., pretraining the models using a subgraph generated by the first 70\% nodes, where the data splitting scheme for the remaining data is the same as the Reddit dataset. The baseline experiments were conducted on DGI and GMI with the same Graph-Sage with GCN-aggregation encoder as in our model. Table~\ref{tab:pretraining} shows that pretraining the model helps the model converge to a better representation model with low variance, and Contrast-Reg\ can improve the transferability of pretraining a model.
\section{Conclusions}
We showed that the high scales of node representations' norms and high variance among them could make contrastive learning algorithms fail. We then proposed Contrast-Reg\ to avoid the cases that are harmful to the representation quality and showed from the geometric perspective that Contrast-Reg\ stabilizes the scales of norms and reduces their variance. Our experiments validated that Contrast-Reg\ improves the representation quality.
\appendix
\section{Proofs} \subsection{Proof of Lemma~\ref*{lemma:supervised}}\label{appendix:convex} First, we prove that {\small $l\ell(f(x^+),\{f(x^-_i)\})=-(\log\sigma(f(x)^T f(x^+))+$} \\ {\small $\sum_{i=1}^{K}\log(\sigma(f(x)^Tf(x^-)))$} is convex w.r.t. {\small $f(x^+),f(x^-_i),\cdots,f(x^-_K)$}. Consider that {\small $\ell_1(z)=-\log\sigma(z)$ and $\ell_2(z)=-\log\sigma(-z)$} are both convex functions since {\small $\ell_1^{\prime\prime}>0$ and $\ell_2^{\prime\prime}>0$ for $z\in\mathbb{R}$}. Given {\small $f(x)\in\mathbb{R^d}$, $z^+=f(x)^Tf(x^+)$ and $z^-=f(x)^Tf(x^+)$} are affine transformation w.r.t. {\small $f(x^+)$ and $f(x^-)$}. Thus, when {\small $f(x)$} is fixed, {\small $\ell_1(f(x^+))=-\log\sigma(f(x)^Tf(x^+))$ and $\ell_2(f(x^-))=-\log\sigma(-f(x)^Tf(x^-))$} are convex functions. As {\small $\ell_1>0$ and $\ell_2>0$}, we obtain {\small $\ell(f(x^+),\{f(x^-_i)\})=-(\log\sigma(f(x)^T f(x^+))+\sum_{i=1}^{K}\log(\sigma(f(x)^Tf(x^-))))$} \ is convex since non-negative weighted sums preserve convexity~\cite{DBLP:convex}. By the definition of convexity, \begin{equation*} {\small
\begin{split}
\mathcal{L}_{nce}(f)&=\mathbb{E}_{\substack{c^+,c^-\sim \rho^2;\\x\in\mathcal{D}_{c^+}}}\mathbb{E}_{\substack{x^+\sim\mathcal{D}_{c^+};\\x^-\sim\mathcal{D}_{c^-}}}\left[\ell(f(x^+),\{f(x^-_i)\})\right]\\
&\ge \mathbb{E}_{c^+,c^-\sim \rho^2}\mathbb{E}_{x\sim\mathcal{D}_{c^+}}\left[\ell(f(x)^T(\mu_{c^+}-\mu_{c^-}))\right]\\
&=(1-\tau)\mathcal{L}_{sup}^\mu(f)+\tau
\end{split} } \end{equation*}
\subsection{Generalization bound}\label{appendix:generalization_bound}
Denote \begin{equation*}
{\small \begin{split} \tilde{\mathcal{F}}= &\left\{ \tilde{f} \left( x_i,x^+_i,x^-_{i1},\cdots,x^-_{iK}\right) = \right.\\
& \qquad\left. \left( f(x_i),f(x_i^+),f(x^-_{i1}),\cdots,f(x^-_{iK})\right) |f\in\mathcal{F} \right\}.\\ \end{split}} \end{equation*} Let \begin{math} {\small q_{\tilde{f}}=h\circ \tilde{f}} \end{math}, and its function class, \begin{equation*}
{\small \mathcal{Q}=\left\{q=h\circ \tilde{f}|\tilde{f}\in\tilde{\mathcal{F}}\right\}.} \end{equation*} Denote \begin{math} {\small z_i=\left(x_i,x^+_i,x^-_{i1},\cdots,x^-_{iK}\right)} \end{math}, suppose $\ell$ is bounded by $B$, then we can decompose $h=\frac{1}{B}\ell\circ \phi$. Then we have $q_{\tilde{f}}(z_i)=\frac{1}{B}\ell(\phi(\tilde{f}(z_i)))$, where
\begin{equation}\label{eq:decomposition} {\small \begin{split} \phi(\tilde{f}(z_i))&=\left(\sum_{t=1}^{d}f(x_i)_t f(x_{i0}^+)_t,\sum_{t=1}^{d}f(x_i)_t f(x_{i1}^-)_t,\cdots,\right.\\ &\left.\qquad\qquad\qquad\sum_{t=1}^{d}f(x_i)_t f(x_{iK}^-)_t\right)\\ \ell(\mathbf{x})&=-\left(\log \sigma(x_0)+\sum_{i=1}^{K}\log \sigma(-x_i))\right). \end{split}} \end{equation} From Eq.~(\ref{eq:decomposition}), we know that $\phi:\mathbb{R}^{(K+2)d}\to \mathbb{R}^{K+1}$.\\ Then we will prove that $h$ is $L$-Lipschitz by proving that $\phi$ and $\ell$ are both Lipschitz continuity. First,
{\small \begin{align*} &\frac{\partial \phi(\tilde{f}(z_i))}{\partial f(x_i)_t}=f(x_{ik})_{t}= \begin{cases} f(x_{i0}^+)_{t}, & k=0\\ f(x_{ik}^-)_{t}, & k=1,\cdots,K \end{cases}\\ &\frac{\partial \phi(\tilde{f}(z_i))}{f(x_{i0}^+)_t}=f(x_i)_{t},\quad \frac{\partial \phi(\tilde{f}(z_i))}{f(x_{ik}^-)_{t}}=f(x_i)_{t}.\\ \end{align*}} If we assume {\small $\sum_{t=1}^d{f(x_{ik})_t^2}\le R^2$ and $\sum_{t=1}^d{f(x_i)_{t}^2}\le R^2$, \begin{equation*} \begin{split} \norm{J}_F&=\sqrt{\sum_{t=1}^{d}{f(x_{i0}^+)_t^2}+\sum_{k=1}^{K}\sum_{t=1}^{d}{f(x_{ik}^-)_t^2}+(K+1)\sum_{t=1}^{d}{f(x_i)_t^2}}\\
&\le \sqrt{2(K+1)R^2}=\sqrt{2(K+1)}R \end{split} \end{equation*}} Combining $\norm{J}_2\le\norm{J}_F$, we obtain that $\phi$ is $\sqrt{2(K+1)}R$-Lipschitz. Similarly, $\ell$ is $\sqrt{K+1}$-Lipschitz. Since we assume that the inner product of embedding is no more than $R^2$. Thus, $l$ is bounded by $B=-(K+1)\log(\sigma(-R^2))$. Above all, $h$ is $L$-Lipschitz with $L=\frac{\sqrt{2}(K+1)R}{B}$. Applying vector-contraction inequality\cite{DBLP:vector-contraction}. We have {\small \begin{equation*}
\mathbb{E}_{\sigma\sim\{\pm 1\}^{M}}[\sup_{\tilde{f}\in\tilde{\mathcal{F}}} \langle \sigma,(h\circ\tilde{f})_{|\mathcal{S}}\rangle]\le \sqrt{2}L\mathbb{E}_{\sigma\sim\{\pm 1\}^{(K+1)dM}}[\sup_{\tilde{f}\in\tilde{\mathcal{F}}} \langle \sigma,\tilde{f}_{|\mathcal{S}}\rangle]. \end{equation*}} If we write it in Rademacher complexity manner, we have {\small \begin{equation*} \mathcal{R}_{\mathcal{S}}(\mathcal{Q})\le \frac{2(K+1)R}{B}\mathcal{R}_{\mathcal{S}}(\mathcal{F}). \end{equation*}} Applying generalization bounds based on Rademacher complexity \cite{foundations_of_machine_learning} to $q\in\mathcal{Q}$. For any $\delta>0$, with the probability of at least $1-\frac{\delta}{2}$, {\small \begin{equation*} \begin{split} \mathbb{E}[q(\mathbf{z})]\le &\frac{1}{M}\sum_{i=1}^{M}q(\mathbf{z}_i)+\frac{2\mathcal{R}_{\mathcal{S}}(\mathcal{Q})}{M}+3\sqrt{\frac{\log \frac{4}{\delta}}{2M}}\\ \le & \frac{1}{M}\sum_{i=1}^{M}q(\mathbf{z}_i)+\frac{4(K+1)R\mathcal{R}_{\mathcal{S}}(\mathcal{F})}{BM}+3\sqrt{\frac{\log \frac{4}{\delta}}{2M}}. \end{split} \end{equation*}} Thus for any $f$, {\small \begin{equation}\label{eq:Rademacher} \mathcal{L}_{nce}(f)\le \tilde{\mathcal{L}}_{nce}(f)+\frac{4(K+1)R\mathcal{R}_{\mathcal{S}}(\mathcal{F})}{M}+3B\sqrt{\frac{\log \frac{4}{\delta}}{2M}}. \end{equation}} Let $\hat{f}=\arg\min_{f\in\mathcal{F}}\tilde{\mathcal{L}}_{nce}(f)$ and $f^*=\arg\min_{f\in\mathcal{F}}\mathcal{L}_{nce}(f)$. By Hoeffding's inequality, with probability of $1-\frac{\delta}{2}$, {\small \begin{equation} \tilde{\mathcal{L}}_{nce}(f^*)\le \mathcal{L}_{nce}(f^*)+B\sqrt{\frac{\log \frac{2}{\delta}}{2M}} \end{equation}} Substituting $\hat{f}$ into Eq.~(\ref{eq:Rademacher}), combining {\small \begin{math} \tilde{\mathcal{L}}_{nce}(\hat{f})\le \mathcal{L}_{nce}(f^*) \end{math}} and applying union bound, with probability of at most $\delta$ {\small \begin{equation}\label{eq:nce_bound} \begin{split} \mathcal{L}_{nce}(\hat{f})\le &\tilde{\mathcal{L}}_{nce}(\hat{f})+\frac{4(K+1)R\mathcal{R}_{\mathcal{S}}(\mathcal{F})}{M}+3B\sqrt{\frac{\log \frac{4}{\delta}}{2M}}+B\sqrt{\frac{\log \frac{2}{\delta}}{2M}}\\ &\le\mathcal{L}_{nce}(f^*)+\frac{4(K+1)R\mathcal{R}_{\mathcal{S}}(\mathcal{F})}{M}+4B\sqrt{\frac{\log \frac{4}{\delta}}{2M}}\\ &\le\mathcal{L}_{nce}(f)+\frac{4(K+1)R\mathcal{R}_{\mathcal{S}}(\mathcal{F})}{M}-4(K+1)\log(\sigma(-R^2))\sqrt{\frac{\log \frac{4}{\delta}}{2M}}\\
\end{split} \end{equation}} fails. Thus, with probability of at least $1-\delta$, Eq.~(\ref{eq:nce_bound}) holds.\\
\subsection{Class collision loss}\label{apendix: class_collision} Let $p_i=\abs{f(x)^Tf(x_i)}$ and $p=\max_{i\in\{0,1,\cdots,K\}}p_i$. Considering {\small \begin{equation} \begin{split} \mathcal{L}_{nce}^=(f)=& -\mathbb{E}\left[\log \sigma(f(x)^Tf(x^+_0))+\sum_{i=1}^{K}\log \sigma(-f(x)^Tf(x_i^-)))\right]\\ &=\mathbb{E}\left[\log(1+e^{-f(x)^Tf(x^+_0)})+\sum_{i=1}^{K}\log (1+e^{f(x)^Tf(x_i^-)})\right]\\ &\le (K+1)\mathbb{E}\left[\log(1+e^p)\right]\\ &\le (K+1)\log c^\prime+(K+1)\mathbb{E}\left[p\right]\\
\end{split} \end{equation}} where $c^\prime\in\left[1+e^{-R^2},2\right]$.\\ Since $x,x^+_0,x^-_1,\cdots,x^-_{K}$ are sampled i.i.d. from the same class, {\small \begin{equation} \mathbb{E}[p]=\int P[p\ge x]dx=\int (1-(1-P[p_0\ge x])^{K+1}dx. \end{equation}} Applying Bernoulli's inequality, we have {\small \begin{equation}\label{eq:same_class} \begin{split} \mathbb{E}[p]&\le \int (1-(1-(K+1)P[p_0\ge x]))dx\\ &= \int (K+1)P[p_0\ge x]dx\\ &=(K+1)\mathbb{E}[p_0]\\ &=(K+1)\mathbb{E}[\abs{f(x)^Tf(x^+_0)}]\\
&\le (K+1)\sqrt{\mathbb{E}[(f(x)^Tf(x^+_0))^2]}.\\ \end{split} \end{equation} {\small Therefore, {\small \begin{equation}
\mathcal{L}_{nce}^=(f)\le (K+1)\log c^\prime+(K+1)^2s(f) \end{equation}}
\section{Experiment details} \noindent \textbf{Hardware Configuration:} The experiments are conducted on Linux servers installed with an Intel(R) Xeon(R) Silver 4114 CPU @ 2.20GHz, 256GB RAM and 8 NVIDIA 2080Ti GPUs.\\ \noindent \textbf{Software Configuration:} Our models, as well as the DGI, GMI and GCN baselines, were implemented in PyTorch Geometric~\cite{pyg} version 1.4.3, DGL~\cite{dgl} version 0.5.1 with CUDA version 10.2, scikit-learn version 0.23.1 and Python 3.6. Our codes and datasets will be made available.\\ \noindent \textbf{Hyper-parameters:} For full batch training, we used 1-layer GCN as the encoder with prelu activation, for mini-batch training, we used a 3-layer GCN with prelu activation. We conducted grid search of different learning rate (from 1e-2, 5e-3, 3e-3, 1e-3, 5e-4, 3e-4, 1e-4) and curriculum settings (including learning rate decay and curriculum rounds) on the fullbatch version. We used 1e-3 or 5e-4 as the learning rate; 10,10,15 or 10,10,25 as the fanouts and 1024 or 512 as the batch size for mini-batch training.
\end{document} | arXiv |
\begin{document}
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\title[]{Left (right) pseudospectrum and left (right) condition pseudospectrum of bounded linear operators on ultrametric Banach spaces} \author[Jawad Ettayb$^1$]{Jawad Ettayb$^1$} \address{$^{1}$ Department of mathematics and computer science, Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz, F\`es, Morocco.} \email{\textcolor[rgb]{0.00,0.00,0.84}{[email protected]}} \subjclass[2010]{47A10; 47S10}
\keywords{Ultrametric Banach spaces, bounded linear operators, left and right pseudospectrum.} \begin{abstract}
In this note, we introduce and study the left (right) pseudospectrum and left (right) condition pseudospectrum of bounded linear operators on ultrametric Banach spaces. We prove some results about them. \end{abstract} \maketitle
\section{Introduction and Preliminaries }
Throughout this paper, $E$ is an ultrametric infinite-dimensional Banach space over a (n.a) non trivially complete valued field $\mathbb{K}$ with valuation $|\cdot|,$ $\mathcal{L}(E)$ denotes the set of all bounded linear operators on $E.$ Recall that $\mathbb{K}$ is called spherically complete if each decreasing sequence of balls in $\mathbb{K}$ has a non-empty intersection. For more details, see \cite{d} and \cite{r}. We begin with some preliminaries. \begin{definition}\cite{d}\label{a}
Let $E$ be a vector space over $\mathbb{K}$. A non-negative real valued function $\|\cdot\|:E\rightarrow \mathbb{R}_{+}$ is called an ultrametric norm if:
\begin{itemize}
\item[(i)] For all $x\in E,$ $\|x\|=0$ if and only if $x=0,$
\item[(ii)] For any $x\in E$ and $\lambda\in\mathbb{K},$ $\|\lambda x\|=|\lambda|\|x\|,$
\item[(iii)] For any $x,y\in E,$ $\|x+y\|\leq\max(\|x\|,\| y\|).$
\end{itemize} \end{definition} Property $(iii)$ of Definition \ref{a} is referred to as the ultrametric or strong triangle inequality. \begin{definition}\cite{d}
An ultrametric Banach space is a vector space endowed with an ultrametric norm which is complete. \end{definition} \begin{exs}\cite{d}
Let $c_{0}\left(\mathbb{K}\right) $ denote the set of all sequences $ \left( x_{i}\right) _{i\in \mathbb{N}}$ in $\mathbb{K}$ such that
$\displaystyle \lim_{i\rightarrow\infty} x_{i} = 0.$
Then $c_{0}\left( \mathbb{K}\right) $ is a vector space over $\mathbb{K}$ and
$$\| \left( x _{i}\right) _{i\in\mathbb{N}} \|= \sup_{i\in\mathbb{N}}|x_{i}| $$
is an ultrametric norm for which $\left( c_{0}\left(\mathbb{K}\right),\|\cdot\|\right)$ is an ultrametric Banach space. \end{exs}
\begin{theorem}\cite{r}\label{lp} Let $E$ be an ultrametric Banach space over a spherically complete field $\mathbb{K}.$ For each $x\in E\backslash\{0\},$ there exists $x^{\ast}\in E^{\ast}$ such that $x^{\ast}(x)=1$ and $\|x^{\ast}\|=\|x\|^{-1}.$ \end{theorem} \section{left (right) pseudospectum and left (right) condition pseudospectum of bounded linear operators on ultrametric Banach spaces} We introduce the following definitions. \begin{definition} Let $E$ be an ultrametric Banach space over $\mathbb{K}$ and let $A\in\mathcal{L}(E).$ \begin{itemize}
\item[(i)] $A$ is said to be left invertible if there exists $B\in\mathcal{L}(E)$ such that $BA=I.$
\item[(ii)] $A$ is said to be right invertible if there exists $C\in\mathcal{L}(E)$ such that $AC=I.$ \end{itemize} \end{definition} \begin{definition} Let $E$ be an ultrametric Banach space over $\mathbb{K}.$ Let $A\in\mathcal{L}(E),$ the left spectrum $\sigma^{l}(A)$ of $A$ is defined by \begin{equation*} \sigma^{l}(A)=\{\lambda\in\mathbb{K}:\;A-\lambda I\;\text{is not left invertible in}\; \mathcal{L}(E)\}.\\ \end{equation*} \end{definition}
\begin{definition}
Let $E$ be an ultrametric Banach space over $\mathbb{K}.$ Let $A\in\mathcal{L}(E),$ the right spectrum $\sigma^{r}(A)$ of $A$ is defined by
\begin{equation*}
\sigma^{r}(A)=\{\lambda\in\mathbb{K}:\;A-\lambda I\;\text{is not right invertible in}\; \mathcal{L}(E)\}.\\
\end{equation*} \end{definition} \begin{definition}\label{d1}
Let $E$ be an ultrametric Banach space over $\mathbb{K},$ let $A\in\mathcal{L}(E)$ and $\varepsilon>0,$ the left spectrum $\sigma^{l}_{\varepsilon}(A)$ of $A$ is defined by \begin{equation*}
\sigma^{l}_{\varepsilon}(A)=\sigma^{l}(A)\cup\{\lambda\in\mathbb{K}:\inf\{\|C_{l}\|:C_{l}\; \text{a left inverse of}\; A-\lambda I\}>\frac{1}{\varepsilon}\},\\ \end{equation*}
by convention $\inf\{\|C_{l}\|:C_{l}\; \text{a left inverse of}\; A-\lambda I\}=\infty$ if $A-\lambda I$ is not left invertible. \end{definition} \begin{definition}\label{d2}
Let $E$ be an ultrametric Banach space over $\mathbb{K},$ let $A\in\mathcal{L}(E)$ and $\varepsilon>0,$ the right spectrum $\sigma^{r}_{\varepsilon}(A)$ of $A$ is defined by
\begin{equation*}
\sigma^{r}_{\varepsilon}(A)=\sigma^{r}(A)\cup\{\lambda\in\mathbb{K}:\inf\{\|C_{r}\|:C_{r}\; \text{a right inverse of}\; A-\lambda I\}>\frac{1}{\varepsilon}\},\\
\end{equation*}
by convention $\inf\{\|C_{r}\|:C_{r}\; \text{a right inverse of}\; A-\lambda I\}=\infty$ if $A-\lambda I$ is not right invertible. \end{definition} We have the following results. \begin{remark} From Definition \ref{d1} and Definition \ref{d2}, we get \begin{equation*} \sigma^{l}(A)\subset\sigma^{l}_{\varepsilon}(A)\subset\sigma_{\varepsilon}(A) \end{equation*} and \begin{equation*} \sigma^{r}(A)\subset\sigma^{r}_{\varepsilon}(A)\subset\sigma_{\varepsilon}(A). \end{equation*}
\end{remark}
\begin{proposition} Let $E$ be an ultrametric Banach space over $\mathbb{K},$ let $A\in\mathcal{L}(E)$ and $\varepsilon>0$, we have \begin{itemize}
\item[(i)] $\sigma^{l}(A)=\displaystyle
\bigcap_{\varepsilon>0}\sigma^{l}_{\varepsilon}(A)$ and $\sigma^{r}(A)=\displaystyle
\bigcap_{\varepsilon>0}\sigma^{r}_{\varepsilon}(A).$
\item[(ii)] For all $\varepsilon_{1}$ and $\varepsilon_{2}$ such that $0<\varepsilon_{1}<\varepsilon_{2},$ $\sigma^{l}(A)\subset\sigma^{l}_{\varepsilon_{1}}(A)\subset\sigma^{l}_{\varepsilon_{2}}(A)$ and $\sigma^{r}(A)\subset\sigma^{r}_{\varepsilon_{1}}(A)\subset\sigma^{r}_{\varepsilon_{2}}(A).$ \end{itemize} \end{proposition} \begin{proof} \begin{itemize} \item[(i)] From Definition \ref{d1}, for any $\varepsilon>0$,\; $\sigma^{l}(A)\subset\sigma^{l}_{\varepsilon}(A).$ Conversely, if $\lambda\in\displaystyle\bigcap_{\varepsilon>0}\sigma^{l}_{\varepsilon}(A),$ hence for all $\varepsilon>0,\;\lambda\in\sigma^{l}_{\varepsilon}(A).$
If $\lambda\not\in\sigma^{l}(A)$, then $\lambda\in\{\lambda\in\mathbb{K}:\inf\{\|C_{l}\|:C_{l}\; \text{a left inverse of}\; A-\lambda I\}>\varepsilon^{-1}\}$, taking limits as $\varepsilon\rightarrow 0^{+},$ we get $\inf\{\|C_{l}\|:C_{l}\; \text{a left inverse of}\; A-\lambda I\}=\infty.$ Thus $\lambda\in\sigma^{l}(A).$ Similarly, we obtain $\sigma^{r}(A)=\displaystyle \bigcap_{\varepsilon>0}\sigma^{r}_{\varepsilon}(A).$
\item[(ii)] For $\varepsilon_{1}$ and $\varepsilon_{2}$ such that $0<\varepsilon_{1}<\varepsilon_{2}.$ Let $\lambda\in\sigma^{l}_{\varepsilon_{1}}(A)$, then $\inf\{\|C_{l}\|:C_{l}\; \text{a left inverse of}\; A-\lambda I\}>\varepsilon_{1}^{-1}>\varepsilon_{2}^{-1},$ hence $\lambda\in\sigma^{l}_{\varepsilon_{2}}(A).$ Similarly, we have $\sigma^{r}(A)\subset\sigma^{r}_{\varepsilon_{1}}(A)\subset\sigma^{r}_{\varepsilon_{2}}(A).$ \end{itemize} \end{proof} \begin{proposition}\label{prop1} Let $E$ be an ultrametric Banach space over $\mathbb{K},$ let $A\in\mathcal{L}(E)$ and $\varepsilon>0.$ Then \begin{equation*}
\bigcup_{C\in\mathcal{L}(E):\|C\|<\varepsilon}\sigma^{l}(A+C)\subset\sigma^{l}_{\varepsilon}(A). \end{equation*} \end{proposition} \begin{proof}
If $\lambda\in\displaystyle\bigcup_{C\in\mathcal{L}(E):\|C\|<\varepsilon}\sigma^{l}(A+C)$. We argue by contradiction. Suppose that $\lambda\not\in\sigma^{l}_{\varepsilon}(A),$ hence $\lambda\not\in\sigma^{l}(A)$ and $\inf\{\|C_{l}\|:C_{l}\; \text{a left inverse of}\; A-\lambda I\}\leq\frac{1}{\varepsilon},$ thus $\|CC_{l}\|<1.$ Let $D$ defined on $E$ by \begin{align*}
D=\sum_{n=0}^{\infty}C_{l}(-CC_{l})^{n}. \end{align*} One can see that $D$ is well-defined and $D=C_{l}(I+CC_{l})^{-1}.$ Hence for all $y\in E,$ $D(I+CC_{l})y=C_{l}y.$ Set $y=(A-\lambda I)x,$ we have for all $x\in E,$ \begin{align*}
x=&D(I+CC_{l})(A-\lambda I)x=&D(A-\lambda I+CC_{l}(A-\lambda I))x=&D(A-\lambda I+C)x. \end{align*}
Hence $A+C-\lambda I$ is left invertible which is contradiction with $\lambda\in\displaystyle\bigcup_{C\in\mathcal{L}(E):\|C\|<\varepsilon}\sigma^{l}(A+C).$ Consequently \begin{equation*}
\bigcup_{C\in\mathcal{L}(E):\|C\|<\varepsilon}\sigma^{l}(A+C)\subset\sigma^{l}_{\varepsilon}(A). \end{equation*} \end{proof} \begin{theorem}\label{th2}
Let $E$ be an ultrametric Banach space over a spherically complete field $\mathbb{K}$ such that $\|E\|\subseteq|\mathbb{K}|,$ let $A\in\mathcal{L}(E)$ and $\varepsilon>0.$ Then, \begin{equation*}
\sigma^{l}_{\varepsilon}(A)=\bigcup_{C\in\mathcal{L}(E):\|C\|<\varepsilon}\sigma^{l}(A+C). \end{equation*} \end{theorem} \begin{proof}
From Proposition \ref{prop1}, we have $\displaystyle\bigcup_{C\in\mathcal{L}(E):\|C\|<\varepsilon}\sigma^{l}(A+C)\subset\sigma^{l}_{\varepsilon}(A).$
Conversely, let $A\in\mathcal{L}(E)$ and $\varepsilon>0,$ suppose that $\lambda\in\sigma^{l}_{\varepsilon}(A).$ We discuss two cases. \item[First case:] If $\lambda\in\sigma^{l}(A),$ we may set $C=0.$
\item[Second case:] Assume that $\lambda\in\sigma^{l}_{\varepsilon}(A)$ and $\lambda\not\in\sigma^{l}(A),$ then for all $C_{l}$ a left inverse of $A-\lambda I,$ we have $\|C_{l}\|>\frac{1}{\varepsilon}.$ Hence, there exists $y\in E\backslash\{0\}$ such that \begin{equation} \label{e1}
\frac{\|C_{l}y\|}{\|y\|}>\frac{1}{\varepsilon}.
\end{equation} Set $y=(A-\lambda I)x,$ then $C_{l}y=x.$ From \eqref{e1}, we have $\|(A-\lambda I)x\|<\varepsilon\|x\|.$
Since $\|E\|\subseteq|\mathbb{K}|$, then there exists $c\in\mathbb{K}\backslash\{0\}$ such that $|c|=\|x\|.$ Putting $z=c^{-1}x$, then $\|z\|=1,$ hence $\|(A-\lambda I)z\|<\varepsilon.$
By Theorem \ref{lp}, there exists $\phi\in E^{\ast}$ such that $\phi(z)=1$ and $\|\phi\|=\|z\|^{-1}=1.$ Define \begin{center} for all $y\in E,\;Cy=-\phi(y)(A-\lambda I)z.$ \end{center}
Then $C\in\mathcal{L}(E)$ and $\|C\|<\varepsilon,$ since for all $y\in E,$ \begin{eqnarray*}
\|Cy\|&=&\|\phi(y)\|\|(A-\lambda I)z\|\\
&<&\varepsilon\|y\|.
\end{eqnarray*} Furthermore, we have $(A-\lambda I+C)z=0.$ Thus $A-\lambda I+C$ is not left invertible. Consequently, $\lambda\in\displaystyle\bigcup_{C\in\mathcal{L}(E):\|C\|<\varepsilon}\sigma^{l}(A+C).$ \end{proof} We continue with the following definitions. \begin{definition}\label{d3} Let $E$ be an ultrametric Banach space over $\mathbb{K},$ let $A\in\mathcal{L}(E)$ and $\varepsilon>0,$ the left condition pseudospectrum $\Lambda^{l}_{\varepsilon}(A)$ of $A$ is defined by \begin{equation*}
\Lambda^{l}_{\varepsilon}(A)=\sigma^{l}(A)\cup\{\lambda\in\mathbb{K}:\inf\{\|(A-\lambda I)\|\|D_{l}\|:D_{l}\; \text{a left inverse of}\; A-\lambda I\}>\frac{1}{\varepsilon}\},\\ \end{equation*}
by convention $\inf\{\|(A-\lambda I)\|\|D_{l}\|:D_{l}\; \text{a left inverse of}\; A-\lambda I\}=\infty$ if $A-\lambda I$ is not left invertible. \end{definition} \begin{definition}\label{d4} Let $E$ be an ultrametric Banach space over $\mathbb{K},$ let $A\in\mathcal{L}(E)$ and $\varepsilon>0,$ the right condition pseudospectrum $\Lambda^{r}_{\varepsilon}(A)$ of $A$ is defined by \begin{equation*}
\Lambda^{r}_{\varepsilon}(A)=\sigma^{r}(A)\cup\{\lambda\in\mathbb{K}:\inf\{\|A-\lambda I\|\|D_{r}\|:D_{r}\; \text{a right inverse of}\; A-\lambda I\}>\frac{1}{\varepsilon}\},\\ \end{equation*}
by convention $\inf\{\|A-\lambda I\|\|D_{r}\|:D_{r}\; \text{a right inverse of}\; A-\lambda I\}=\infty$ if $A-\lambda I$ is not right invertible. \end{definition} We have the following results. \begin{remark} From Definition \ref{d3} and Definition \ref{d4}, we get \begin{equation*} \sigma^{l}(A)\subset\Lambda^{l}_{\varepsilon}(A)\subset\Lambda_{\varepsilon}(A) \end{equation*} and \begin{equation*} \sigma^{r}(A)\subset\Lambda^{r}_{\varepsilon}(A)\subset\Lambda_{\varepsilon}(A). \end{equation*} \end{remark}
\begin{proposition} Let $E$ be an ultrametric Banach space over $\mathbb{K},$ let $A\in\mathcal{L}(E)$ and $\varepsilon>0$, we have \begin{itemize}
\item[(i)] $\sigma^{l}(A)=\displaystyle
\bigcap_{\varepsilon>0}\Lambda^{l}_{\varepsilon}(A)$ and $\sigma^{r}(A)=\displaystyle
\bigcap_{\varepsilon>0}\Lambda^{r}_{\varepsilon}(A).$
\item[(ii)] For all $\varepsilon_{1}$ and $\varepsilon_{2}$ such that $0<\varepsilon_{1}<\varepsilon_{2},$ $\sigma^{l}(A)\subset\Lambda^{l}_{\varepsilon_{1}}(A)\subset\Lambda^{l}_{\varepsilon_{2}}(A)$ and $\sigma^{r}(A)\subset\Lambda^{r}_{\varepsilon_{1}}(A)\subset\Lambda^{r}_{\varepsilon_{2}}(A).$ \end{itemize} \end{proposition} \begin{proof} \begin{itemize}
\item[(i)] From Definition \ref{d3}, for any $\varepsilon>0$,\; $\sigma^{l}(A)\subset\Lambda^{l}_{\varepsilon}(A).$ Conversely, if $\lambda\in\displaystyle\bigcap_{\varepsilon>0}\Lambda^{l}_{\varepsilon}(A),$ hence for all $\varepsilon>0,\;\lambda\in\Lambda^{l}_{\varepsilon}(A).$
If $\lambda\not\in\sigma^{l}(A)$, then $\lambda\in\{\lambda\in\mathbb{K}:\inf\{\|A-\lambda I\|\|D_{l}\|:D_{l}\; \text{a left inverse of}\; A-\lambda I\}>\varepsilon^{-1}\}$, taking limits as $\varepsilon\rightarrow 0^{+},$ we get $\inf\{\|A-\lambda I\|\|D_{r}\|:D_{l}\; \text{a left inverse of}\; A-\lambda I\}=\infty.$ Hence $\lambda\in\sigma^{l}(A).$ Similarly, we obtain $\sigma^{r}(A)=\displaystyle
\bigcap_{\varepsilon>0}\Lambda^{r}_{\varepsilon}(A).$
\item[(ii)] For $\varepsilon_{1}$ and $\varepsilon_{2}$ such that $0<\varepsilon_{1}<\varepsilon_{2}.$ Let $\lambda\in\Lambda^{l}_{\varepsilon_{1}}(A)$, then $\inf\{\|A-\lambda I\|\|D_{l}\|:D_{l}\; \text{a left inverse of}\; A-\lambda I\}>\varepsilon_{1}^{-1}>\varepsilon_{2}^{-1},$ hence $\lambda\in\Lambda^{l}_{\varepsilon_{2}}(A).$ Similarly, we have $\sigma^{r}(A)\subset\Lambda^{r}_{\varepsilon_{1}}(A)\subset\Lambda^{r}_{\varepsilon_{2}}(A).$ \end{itemize} \end{proof} \begin{proposition}
Let $E$ be an ultrametric Banach space over $\mathbb{K}$ and let $A\in\mathcal{L}(E)$ and for every $\varepsilon>0$ and $\|A-\lambda I\|\neq0.$ Then,
\begin{itemize}
\item[(i)] $\lambda\in\Lambda^{l}_{\varepsilon}(A)$ if, and only if, $\lambda\in\sigma^{l}_{\varepsilon\|A-\lambda I\|}(A).$
\item[(ii)] $\lambda\in\sigma^{l}_{\varepsilon}(A)$ if and only if $\lambda \in\Lambda^{l}_{\frac{\varepsilon}{\|A-\lambda I\|}}(A).$
\end{itemize} \end{proposition} \begin{proof}
\begin{itemize}
\item[(i)] Let $\lambda\in\Lambda^{l}_{\varepsilon}(A),$ then $\lambda\in\sigma^{l}(A)$ or $$\inf\{\|(A-\lambda I)\|\|C_{l}\|:C_{l}\; \text{a left inverse of}\; A-\lambda I\}>\varepsilon^{-1}.$$ Hence
$\lambda\in\sigma^{l}(A)$\;or\;for all $C_{l}$ a left invertible of $A-\lambda I,$ $\|C_{l}\|>\frac{1}{\varepsilon\|(A-\lambda I)\|}.$
Consequently, $\lambda\in\sigma^{l}_{\varepsilon\|A-\lambda I\|}(A).$
The converse is similar.
\item[(ii)] Let $\lambda\in\sigma^{l}_{\varepsilon}(A),$ then, $\lambda\in\sigma^{l}(A)$ or for all $C_{l}$ a left inverse of $A-\lambda I,$ $\|C_{l}\|>\varepsilon^{-1}.$
Thus
\begin{equation*}
\lambda\in\sigma^{l}(A)\;\text{or for all $C_{l}$ a left inverse of $A-\lambda I,$}\;\|(A-\lambda I)\|\|C_{l}\|>\frac{\|(A-\lambda I)\|}{\varepsilon}.
\end{equation*}
Then, $\lambda\in\Lambda^{l}_{\frac{\varepsilon}{\|A-\lambda I\|}}(A).$
The converse is similar.
\end{itemize} \end{proof} One can see the following corollary. \begin{corollary}
Let $E$ be an ultrametric Banach space over $\mathbb{K},$ let $A\in\mathcal{L}(E)$ and $\varepsilon>0.$ If $\alpha,\beta\in\mathbb{K}$ with $\beta\neq 0,$ then $\Lambda^{l}_{\varepsilon}(\beta A+\alpha I)=\alpha +\beta\Lambda^{l}_{\varepsilon}(A).$ \end{corollary} \begin{proposition}
Let $E$ be an ultrametric Banach space over $\mathbb{K},$ let $A\in\mathcal{L}(E)$ such that $A\neq\lambda I$ and $C_{A}=\inf\{\|A-\lambda I\|:\lambda\in\mathbb{K}\}$ and $\varepsilon>0.$ Then $\sigma^{l}_{\varepsilon}(A)\subset\Lambda^{l}_{\frac{\varepsilon}{C_{A}}}(A).$ \end{proposition} \begin{proof}
Let $\mu\in\sigma^{l}_{\varepsilon}(A),$ then $\mu\in\sigma^{l}(A)$ or for all $C_{l}$ a left inverse of $A-\mu I,\;\|C_{l}\|>\frac{1}{\varepsilon}.$
Since $\|A-\mu I\|\geq C_{A}>0.$
Then $\mu\in\sigma^{l}(A)$ or for all $C_{l}$ a left inverse of $A-\mu I, \|A-\mu I\|\|C_{l}\|>\frac{C_{A}}{\varepsilon}.$ Hence $\lambda\in\Lambda^{l}_{\frac{\varepsilon}{C_{A}}}(A).$ \end{proof} \begin{lem}\label{lm1}
Let $E$ be an ultrametric Banach space over $\mathbb{K},$ let $A\in\mathcal{L}(E)$ and $\varepsilon>0.$ If $\lambda\in\Lambda^{l}(A)\backslash\sigma^{l}(A).$ Then there exists $x\in E\backslash\{0\}$ such that $\|(A-\lambda I)x\|<\varepsilon\|A-\lambda I\|\|x\|.$ \end{lem} \begin{proof} If $\lambda\in\Lambda^{l}(A)\backslash\sigma^{l}(A),$ then for all $C_{l}$ a left inverse of $A-\lambda I,$ we have \begin{equation}
\|A-\lambda I\|\|C_{l}\|>\frac{1}{\varepsilon}. \end{equation} Thus \begin{equation}
\|C_{l}\|>\frac{1}{\varepsilon\|A-\lambda I\|}. \end{equation} Then there exists $y\in E\backslash\{0\}$ such that \begin{equation}\label{ee}
\frac{\|C_{l}y\|}{\|y\|}>\frac{1}{\varepsilon\|A-\lambda I\|}. \end{equation}
Set $y=(A-\lambda I)x,$ then $C_{l}y=x.$ From \eqref{ee}, we have $\|(A-\lambda I)x\|<\varepsilon\|A-\lambda I\|\|x\|.$ \end{proof} \begin{theorem}\label{ttp}
Let $E$ be an ultrametric Banach space over $\mathbb{K},$ let $A\in\mathcal{L}(E),$ $\lambda\in\mathbb{K}$ and $\varepsilon>0.$ If there exists $C\in \mathcal{L}(E)$ with $\|C\|<\varepsilon\|A-\lambda I\|$ and $\lambda\in\sigma^{l}(A+C)$. Then, $\lambda\in\Lambda^{l}_{\varepsilon}(A).$ \end{theorem} \begin{proof}
Assume that there exists $C\in\mathcal{L}(E)$ such that $\|C\|<\varepsilon\|A-\lambda I\|$ and $\lambda\in\sigma^{l}(A+C).$ If $\lambda\not\in\Lambda^{l}_{\varepsilon}(A),$ hence $\lambda\not\in\sigma^{l}(A)$ and for each $C_{l}$ a left inverse of $A-\lambda I,\;\|A-\lambda I\|\|C_{l}\|\leq\varepsilon^{-1}.$\\ Consider $D$ defined on $E$ by
\begin{eqnarray}
D=\sum_{n=0}^{\infty}C_{l}(-CC_{l})^{n}.
\end{eqnarray}
Consequently $D=C_{l}(I+CC_{l})^{-1}.$ Hence for all $y\in E,D(I+CC_{l})y=C_{l}y.$ Put $y=(A-\lambda I)x,$ then
\begin{equation*}
(\forall x\in E)\;D(A-\lambda I+C)x=x \end{equation*}
Then $A-\lambda I+C$ is a left invertible which is a contradiction. Thus $\lambda\in\Lambda^{l}_{\varepsilon}(A).$ \end{proof}
Set $\mathcal{C}_{\varepsilon}(E)=\{C\in\mathcal{L}(E):\|C\|<\varepsilon\|A-\lambda I\|\},$ we have. \begin{theorem}\label{th2}
Let $E$ be an ultrametric Banach space over a spherically complete field $\mathbb{K}$ such that $\|E\|\subseteq |\mathbb{K}|,$ let $A\in\mathcal{L}(E)$ and $\varepsilon>0.$ Then,
\begin{equation*}
\Lambda^{l}_{\varepsilon}(A)=\bigcup_{C\in\mathcal{C}_{\varepsilon}(E)}\sigma^{l}(A+C).
\end{equation*} \end{theorem} \begin{proof} By Theorem \ref{ttp}, we have $\displaystyle\bigcup_{C\in\mathcal{C}_{\varepsilon}(E)}\sigma^{l}(A+C)\subset\Lambda^{l}_{\varepsilon}(A).$
Conversely, assume that $\lambda\in\Lambda^{l}_{\varepsilon}(A).$ If $\lambda\in\sigma^{l}(A)$, we may put $C=0.$ If $\lambda\in\Lambda^{l}_{\varepsilon}(A)$ and $\lambda\not\in\sigma^{l}(A)$. By Lemma \ref{lm1} and $\|E\|\subseteq |\mathbb{K}|$, there exists $x\in E\backslash\{0\}$ such that $\|x\|=1$ and
$\|(A-\lambda I)x\|<\varepsilon\|A-\lambda I\|.$\\
By Theorem \ref{lp}, there is a $\varphi\in E^{\ast}$ such that $\varphi(x)=1$ and $\|\varphi\|=\|x\|^{-1}=1.$
Consider $C$ on $E$ defined by for all $y\in X,\;Cy=-\phi(y)(A-\lambda I)x.$ Hence, $\|C\|<\varepsilon\|A-\lambda I\|$ and $D(C)=E$. Moreover, for $x\in E\backslash\{0\},\;(A-\lambda I+C)x=0.$ Then, $(A-\lambda I+C)$ is not left invertible. Consequently,
$\lambda\in\displaystyle\bigcup_{C\in\mathcal{C}_{\varepsilon}(E)}\sigma^{l}(A+C).$ \end{proof}
\end{document} | arXiv |
\begin{document}
\@dblarg\CPM@title[Milnor numbers of projective hypersurfaces]{Milnor numbers of projective hypersurfaces with isolated singularities} \author{June Huh} \email{[email protected]} \address{Department of Mathematics, University of Michigan\\ Ann Arbor, MI 48109\\ USA} \classification{14B05, 14N99.} \keywords{Milnor number, projective hypersurface, homaloidal polynomial.} \thanks{The author was partially supported by NSF grant DMS-0943832.}
\begin{abstract}
Let $V$ be a projective hypersurface of fixed degree and dimension which has only isolated singular points. We show that, if the sum of the Milnor numbers at the singular points of $V$ is large, then $V$ cannot have a point of large multiplicity, unless $V$ is a cone. As an application, we give an affirmative answer to a conjecture of Dimca and Papadima.
\end{abstract}
\title{\@dblarg\CPM@title}
\section{Main results}
\subsection{}
Let $h$ be a nonzero homogeneous polynomial of degree $d \ge 1$ in the polynomial ring $\mathbb{C}[z_0,\ldots,z_n]$. In order to avoid trivialities, we assume throughout that $n \ge 2$. We write $V(h)$ for the projective hypersurface $\{h=0\} \subseteq \mathbb{P}^n$. Associated to $h$ is the \emph{gradient map} obtained from the partial derivatives \[ \text{grad}(h): \mathbb{P}^n \dashrightarrow \mathbb{P}^n, \qquad z \longmapsto \Bigg(\frac{\partial h}{\partial z_0}: \cdots : \frac{\partial h}{\partial z_n}\Bigg). \] The \emph{polar degree} of $h$ is the degree of the gradient map of $h$. The polar degree of $h$ depends only on the set $\{h=0\}$ \cite{Dimca-Papadima}. If $V(h)$ has only isolated singular points, then the polar degree is given by the formula \[ \text{deg}\big(\text{grad}(h)\big) =(d-1)^n - \sum_{p \in V(h)} \mu^{(n)}(p), \] where $\mu^{(n)}(p)$ is the Milnor number of $V(h)$ at $p$. See \cite[Section 3]{Dimca-Papadima}, and also \cite{Fassarella-Medeiros,Huh}.
\begin{theorem}\label{main1} Suppose $V(h)$ has only isolated singular points, and let $m$ be the multiplicity of $V(h)$ at one of its points $x$. Then \[ \text{deg}\big(\text{grad}(h)\big) \ge (m-1)^{n-1}, \] unless $V(h)$ is a cone with the apex $x$. \end{theorem}
Equivalently, we have \[ (d-1)^n \ge (m-1)^{n-1} +\sum_{p \in V(h)} \mu^{(n)}(p), \] unless $V(h)$ is a cone with the apex at $x$. Therefore, for projective hypersurfaces of given degree and dimension, the existence of a point of large multiplicity should be compensated by a smaller sum of the Milnor numbers at the singular points, unless the hypersurface is a cone.
It is interesting to observe how badly the inequality fails when $V(h)$ is a cone over a smooth hypersurface in $\mathbb{P}^{n-1} \subseteq \mathbb{P}^n$. In this case, the polar degree is zero, but the apex of the cone has multiplicity $d$. Both the multiplicity and the sum of the Milnor numbers are simultaneously as large as possible with respect to the degree and the dimension.
The inequality also crucially depends on the assumption that $V(h)$ has only isolated singular points. This can be most clearly seen from Gordan-Noether counterexamples to Hesse's claim that the polar degree is zero if and only if the hypersurface is a cone \cite{Hesse1,Hesse2,Gordan-Noether}. For example, consider the threefold in $\mathbb{P}^4$ defined by \[ h=z_3^{d-1} z_0+ z_3^{d-2}z_4z_1+z_4^{d-1} z_2, \qquad d \ge 3. \] In this case, the degree of the gradient map of $h$ is zero, $V(h)$ has a point of multiplicity of $d-1$, but $V(h)$ is not a cone.
\subsection{}
Let $(V, \mathbf{0})$ be the germ of an isolated hypersurface singularity at the origin of $\mathbb{C}^n$. Associated to the germ are the \emph{sectional Milnor numbers} introduced by Teissier \cite{Teissier1}. The $i$-th sectional Milnor number of the germ, denoted $\mu^{(i)}$, is the Milnor number of the intersection of $V$ with a general $i$-dimensional plane passing through $\mathbf{0}$.
We obtain Theorem \ref{main1} from the following refinement.
\begin{theorem}\label{main2} Suppose $V(h)$ has only isolated singular points, and let $\mu^{(n-1)}$ be the $(n-1)$-th sectional Milnor number of $V(h)$ at one of its points $x$. Then \[ \text{deg}\big(\text{grad}(h)\big) \ge \mu^{(n-1)}, \] unless $V(h)$ is a cone with the apex $x$. \end{theorem}
The Minkowski inequality for mixed multiplicities says that the sectional Milnor numbers always form a log-convex sequence \cite{Teissier2}. In other words, we have \[ \frac{\mu^{(n)}}{\mu^{(n-1)}} \ge \frac{\mu^{(n-1)}}{\mu^{(n-2)}} \ge \cdots \ge \frac{\mu^{(i)}}{\mu^{(i-1)}} \ge \cdots \ge \frac{\mu^{(1)}}{\mu^{(0)}}. \] Since $\mu^{(0)}$ is one and $\mu^{(1)}$ is one less than the multiplicity at the point, we see that Theorem \ref{main2} indeed implies Theorem \ref{main1}.
The inequality of Theorem \ref{main2} is tight relative to the degree and the dimension. In Proposition \ref{EqualityCase} we show that, for each $d \ge 3$ and $n \ge 2$, there is a degree $d$ hypersurface in $\mathbb{P}^n$ with one singular point, for which the equality holds in Theorem \ref{main2}.
\subsection{}\label{homaloidal}
Our interest in Theorems \ref{main1} and \ref{main2} arose from the study of homalodial polynomials. A \emph{homaloidal polynomial} is a homogeneous polynomial whose gradient map is a birational transformation of $\mathbb{P}^n$. A \emph{homaloidal hypersurface} is the projective hypersurface defined by a homaloidal polynomial. See \cite{CRS} for a motivated introduction.
Dolgachev showed in \cite{Dolgachev} that there are exactly three homaloidal plane curves, up to a linear change of homogeneous coordinates:
\begin{itemize} \item a nonsingular conic $h=z_0^2+z_1^2+z_2^2=0$. \item the union of three nonconcurrent lines $h=z_0z_1z_2=0$. \item the union of a conic and one of its tangent $h=z_0(z_1^2+z_0z_2)=0$. \end{itemize}
In contrast, there are abundant examples of homaloidal hypersurfaces in $\mathbb{P}^n$ when $n \ge 3$.
\begin{itemize} \item Any relative invariant of a regular prehomogeneous space is homalodal \cite{Ein-Shepherd-Barron,EKP}. \item Projective duals of certain scroll surfaces are homaloidal \cite{CRS}. \item Determinants of generic sub-Hankel matrices are homaloidal \cite{CRS}. \item The union of a cone over a homaloidal hypersurface in $\mathbb{P}^{n-1} \subseteq \mathbb{P}^n$ and a general hyperplane is homaloidal \cite{Fassarella-Medeiros}. \item There are infinitely many polytopes such that almost all polynomials having any one of them as the Newton polytope are homaloidal \cite{HuhML}. \end{itemize}
In particular, the last construction shows that there are irreducible homaloidal hypersurfaces of any given degree $d \ge 3$ in the projective space of dimension $n \ge 3$. Dimca and Papadima conjectured in \cite[Section 3]{Dimca-Papadima} that none of them has only isolated singular points.
\begin{conjecture}\label{DPConjecture} There are no homaloidal hypersurfaces of degree $d \ge 3$ with only isolated singular points in the projective space of dimension $n\ge 3$. \end{conjecture}
We use Theorem \ref{main2} to give an affirmative answer to this conjecture.
\begin{theorem}\label{Classification1} A projective hypersurface with only isolated singular points has polar degree $1$ if and only if it is one of the following, after a linear change of homogeneous coordinates: \begin{itemize} \item $(n \ge 2, d=2)$ a smooth quadric \begin{flalign*} \hspace{50mm}&h=z_0^2+\cdots+z_n^2=0.& \end{flalign*} \item $(n=2,d=3)$ the union of three nonconcurrent lines \begin{flalign*} \hspace{50mm}&h=z_0z_1z_2=0.& \end{flalign*} \item $(n=2,d=3)$ the union of a smooth conic and one of its tangent \begin{flalign*} \hspace{50mm}&h=z_0(z_1^2+z_0z_2)=0.& \end{flalign*} \end{itemize} \end{theorem}
Theorem \ref{Classification1} shows that, for projective hypersurfaces of given degree and dimension, the sequence of possible values for the sum of the Milnor numbers necessarily contains a gap, except for quadric hypersurfaces and cubic plane curves.
Similar, but stronger results concerning the sum of Tjurina numbers can be found in the works of du Plessis and Wall \cite{duPlessis-Wall1,duPlessis-Wall2}. Other important evidence in support of Conjecture \ref{DPConjecture} was provided by \cite{Dimca,CRS,Ahmed}.
We close this introduction by posing the problem of finding other forbidden values for the sum of the Milnor numbers at the singular points of a degree $d$ hypersurface in $\mathbb{P}^n$, for general $d$ and $n$. See Conjecture \ref{Conjecture}.
\subsection{} We now provide a brief overview of the paper.
In Section \ref{SectionLefschetz}, we formulate and prove a Lefschetz hyperplane theorem with an assigned base point for projective hypersurface complements. The main argument involves \begin{itemize} \item a pencil of hyperplane sections which has only isolated singular points with respect to a Whitney stratification \cite{Tibar1,Tibar2}, and \item a generalized Zariski theorem on the fundamental groups of plane curve complements \cite[Section 4.3]{DimcaBook}. \end{itemize}
In Section \ref{ProofoftheInequality}, we prove Theorem \ref{main2}, using the Lefschetz hyperplane theorem of the previous section. We provide an example showing that the inequality of Theorem \ref{main2} is sharp.
In Section \ref{ProofoftheConjecture}, we use Theorem \ref{main2} to show that all the singularities of a homaloidal hypersurface with only isolated singular points are necessarily simple of type $A$. The proof of Conjecture \ref{DPConjecture} is then obtained from the results of du Plessis and Wall, and Dimca \cite{duPlessis-Wall2,Dimca}. We close with a brief discussion of projective hypersurfaces with polar degree $2$.
\section{Lefschetz theorem with an assigned base point}\label{SectionLefschetz}
Let $D(h)$ be the hypersurface complement $\{h \neq 0\} \subseteq \mathbb{P}^n$. Hamm's Lefschetz theory shows that, if $H$ is a general hyperplane in $\mathbb{P}^n$, then \[ \pi_i\big(D(h),D(h)\cap H\big)=0 \quad \text{for}\quad i<n. \] See \cite{Hamm,Hamm-Le1}. The purpose of this section is to refine this result by allowing hyperplanes to have an assigned base point.
\begin{theorem}\label{attaching} If $H_x$ is a general hyperplane passing through a point $x$ in $\mathbb{P}^n$, then \[ \pi_i\big(D(h),D(h)\cap H_x\big)=0 \quad \text{for}\quad i<n, \] unless \begin{enumerate} \item one of the components of $V(h)$ is a cone with the apex $x$, or \item the singular locus of $V(h)$ contains a line passing through $x$. \end{enumerate} \end{theorem}
Since $D(h)$ and $D(h) \cap H_x$ are homotopic to CW-complexes of dimensions $n$ and $n-1$ respectively, the vanishing of the homotopy groups implies \[ H_i\big(D(h),D(h)\cap H_x\big)=0 \quad \text{for}\quad i\neq n. \]
\begin{example} Let $V(h)$ be the plane curve consisting of a nonsingular conic containing $x$, the tangent line to the conic at $x$, and a general line passing through $x$. Then \[ H_1\big(D(h),D(h) \cap H_x\big) \simeq H_1\big(S^1 \times S^1, S^1\big) \simeq \mathbb{Z}. \]
\end{example}
\begin{example} Let $V(h)$ be the cone over a smooth hypersurface of degree $d$ in $\mathbb{P}^{n-1} \subseteq \mathbb{P}^n$ with the apex $x$. Then \[ H_{n-1}\big(D(h),D(h)\cap H_x\big) \simeq H_{n-1}\big(D(h) \cap \mathbb{P}^{n-1},D(h)\cap H_x \cap \mathbb{P}^{n-1}\big) \simeq \mathbb{Z}^{(d-1)^{n-1}}.
\]
\end{example}
It seems reasonable to expect that the condition on the singular locus of $V(h)$ is also necessary. However, the author does not know this.
The rest of this section is devoted to the proof of Theorem \ref{attaching}.
\subsection{}
We start with a characterization of the apex of irreducible cones in $\mathbb{P}^n$.
\begin{lemma}\label{Cones} Let $V$ be a subvariety of positive dimension $k+1$ in $\mathbb{P}^n$. Then the following conditions are equivalent for a point $x$ in $\mathbb{P}^n$. \begin{enumerate} \item $V$ is a cone with the apex $x$.
\item For any point $y$ of $V$ different from $x$, the line joining $x$ and $y$ is contained in $V$.
\item If $E_x$ is a general codimension $k$ linear subspace in $\mathbb{P}^n$ containing $x$, then every irreducible component of $V \cap E_x$ is a line containing $x$. \item If $E_x$ is a general codimension $k$ linear subspace in $\mathbb{P}^n$ containing $x$, then some irreducible component of $V \cap E_x$ is a line containing $x$. \end{enumerate} \end{lemma}
The irreducibility assumption is clearly necessary in order to deduce (iii) from (iv). \begin{proof} The equivalence of the first three conditions is standard, and (iii) implies (iv). We show that (iv) implies (ii), using a pointed version of the Fano variety of lines in $V$.
Consider the Grassmannians \begin{eqnarray*}
\text{G}_1&:=&\big\{L_x \mid \text{$L_x$ is a line in $\mathbb{P}^n$ containing $x$}\big\} \simeq \text{Gr}(1,n),\\
\text{G}_2&:=&\big\{E_x \mid \text{$E_x$ is a codimension $k$ linear subspace in $\mathbb{P}^n$ containing $x$}\big\} \simeq \text{Gr}(n-k,n), \end{eqnarray*} and the incidence correspondence \[ \mathscr{I}:=\big\{(L_x,E_x) \mid L_x \subseteq V \cap E_x\big\}\subseteq \text{G}_1 \times \text{G}_2. \] Let $\text{pr}_1,\text{pr}_2$ be the projections from $\mathscr{I}$ \[ \xymatrix{ &\mathscr{I} \ar[dl]_{\text{pr}_1} \ar[dr]^{\text{pr}_2}&\\ \text{G}_1 & &\text{G}_2. } \]
We compute the dimension of the image of $\text{pr}_1$, the variety of lines through $x$ contained in $V$. Our assumption (iv) says that $\text{pr}_2$ is generically surjective. Since $\text{pr}_2$ is generically finite in general, it follows that the dimension of $\mathscr{I}$ is equal to that of $\text{G}_2$.
Note also that
\[ \text{pr}_1^{-1}(L_x) \simeq \left\{\begin{array}{cc} \text{Gr}(n-k-1,n-1) & \text{if $L_x \subseteq V $}, \\ \varnothing & \text{if $L_x \nsubseteq V $}.\end{array}\right. \] Therefore \[ \dim \text{Im}(\text{pr}_1)=\dim \text{Gr}(n-k,n)-\dim \text{Gr}(n-k-1,n-1) =k. \]
Next, consider the incidence correspondence \[ \xymatrix{ \mathcal{I}:=\big\{(L_x,p) \mid p \in L_x\big\} \subseteq \text{Im}(\text{pr}_1) \times V, } \] and the associated projections \[ \xymatrix{ &\mathcal{I} \ar[dl]_{\pi_1} \ar[dr]^{\pi_2}&\\ \text{Im}(\text{pr}_1)& & V. } \] $\pi_1$ is a bundle of projective lines, and therefore the dimension of $\mathcal{I}$ is $k+1$. $\pi_2$ is injective over the open subset $\{p\neq x\}$, because there is at most one line containing $p$ and $x$ which is contained in $V$. Since $V$ is assumed to be irreducible, the previous two sentences imply that $\pi_2$ is surjective. In other words, for any point $y$ of $V$ different from $x$, the line joining $x$ and $y$ is contained in $V$.
\end{proof}
\subsection{}\label{TibarLefschetz}
Let $X$ be a smooth projective variety of dimension $n$, and let $A$ be a general codimension $2$ linear subspace of a fixed ambient projective space of $X$. One of the conclusions of the classical Lefschetz theory is the isomorphism \[ H_{i+1}(X,X_c) \simeq H_{i-1}(X_c,X_c \cap A), \qquad i < n-1, \] where $X_c$ is a general member of the pencil of hyperplane sections of $X$ associated to $A$ \cite[Section 3.6]{Lamotke}. By induction, one has the vanishing \[ H_i(X,X_c) = 0, \qquad i < n. \]
The aim of this subsection is to state a generalization due to Tib\u ar \cite{Tibar1,Tibar2}. We state these results in the generality that we need, not necessarily in the generality of the original papers. See also \cite[Section 10.1]{Tibar3}.
We work in the following setting: \begin{itemize} \item $V$ is a closed subset of a projective variety $Y$. \item $X$ is the quasi-projective variety $Y \setminus V$. \item $\mathscr{W}$ is a Whitney stratification of $Y$ such that $V$ is a union of strata. \item $A$ is a codimension $2$ linear subspace of a fixed ambient projective space of $Y$.
\item $\mathscr{W}|_{Y \setminus A}$ is the Whitney stratification of $Y \setminus A$ obtained by restricting $\mathscr{W}$. \item $\mathscr{P}_A$ is the pencil of hyperplanes containing the axis $A$. We write \[ \pi: Y \setminus A \longrightarrow \mathscr{P}_A \] for the map sending $p$ to the member of $\mathscr{P}_A$ containing $p$.
\item $\mathbb{Y}$ is the blow-up of $Y$ along $Y \cap A$. We write \[ p: \mathbb{Y} \longrightarrow \mathscr{P}_A \] for the map which agrees with $\pi$ on $Y \setminus A$.
\item $\mathscr{S}$ is a Whitney stratification of $\mathbb{Y}$ which extends $\mathscr{W}|_{Y \setminus A}$.
\end{itemize}
By a Whitney stratification we mean a complex analytic partition which satisfies the Whitney regularity conditions and the frontier condition. For generalities on Whitney stratifications we refer to \cite{GWdPL,LeTeissier} and references therein.
\begin{definition}\label{DefIsoSing} The \emph{singular locus} of $p$ with respect to $\mathscr{S}$ is the following closed subset of $\mathbb{Y}$: \[
\text{Sing}_\mathscr{S} \hspace{1mm} p:= \bigcup_{\mathcal{S} \in \mathscr{S}} \text{Sing} \hspace{1mm} p |_{\mathcal{S}}. \] We say that $\mathscr{P}_A$ has only \emph{only isolated singular points} with respect to $\mathscr{S}$ if $\dim \text{Sing}_\mathscr{S} \hspace{1mm} p \le 0$. \end{definition}
The singular locus of $p$ is a closed subset of $\mathbb{Y}$ because $\mathscr{S}$ is a Whitney stratification. The notion of isolated singular points in this generalized sense has proved its value, for example, in the works of L\^e \cite{Le1,Le2}.
We are now ready to introduce the theorem of Tib\u ar. We maintain the notations introduced above.
\begin{theorem}[{\cite[Theorem 1.1]{Tibar1}}]\label{NongenericLefschetz} Let $X_c$ be a general member of the pencil on $X$. Suppose that \begin{enumerate} \item the axis $A$ is not contained in $V$, \item the rectified homotopical depth of $X$ is $\ge n$ for some $n \ge 2$, \item the pencil $\mathscr{P}_A$ has only isolated singular points with respect to $\mathscr{S}$, and \item the pair $(X_c,X_c \cap A)$ is $(n-2)$-connected.
\end{enumerate} Then the pair $(X,X_c)$ is $(n-1)$-connected. \end{theorem}
The \emph{rectified homotopical depth} of $X$ is an integer which measures the local connectedness $X$ \cite[Definition1.1]{Hamm-Le2}. If $X$ is locally a complete intersection variety, it is equal to the complex dimension of $X$ \cite[Corollary 3.2.2]{Hamm-Le2}.
\subsection{}
Let $S$ be a smooth and irreducible algebraic subset of $\mathbb{P}^n$, and let $A$ be a codimension $2$ linear subspace of $\mathbb{P}^n$. We write $\mathscr{P}_A$ for the pencil of hyperplanes containing $A$, and \[ \pi_A: S \setminus A \longrightarrow \mathscr{P}_A \] for the map sending $p$ to the member of $\mathscr{P}_A$ containing $p$.
\begin{lemma}\label{Projection} If $A_x$ is a general codimension $2$ linear subspace passing through a point $x$ in $\mathbb{P}^n$, then \[ \pi_{A_x}: S \setminus A_x \longrightarrow \mathscr{P}_{A_x} \]
has only isolated singular points, unless the closure of $S$ in $\mathbb{P}^n$ is a cone with the apex $x$. \end{lemma}
Note that $\pi_{A_x}$ necessarily has nonisolated singularities if, for example, the closure of $S$ in $\mathbb{P}^n$ is the cone over a smooth hypersurface in $\mathbb{P}^{n-1} \subseteq \mathbb{P}^n$ with the apex $x$.
\begin{proof} Let $V$ be the closure of $S$ in $\mathbb{P}^n$, and let $A$ be a codimension $2$ linear subspace of $\mathbb{P}^n$ containing $x$. Denote the conormal variety of $V$ by $I$, the dual variety of $V$ by $\check V$. Consider the projections $\text{pr}_1$, $\text{pr}_2$ from $I$ to $V$, $\check V$ respectively: \[ \xymatrix{ &I \ar[dl]_{\text{pr}_1} \ar[dr]^{\text{pr}_2}&&&&\\ V && \check V \ar[r] & \check{\mathbb{P}}^n & \ar[l] \mathscr{P}_{A} } \] Choosing a line in $\mathbb{P}^n$ disjoint from $A$ identifies $\pi_A$ with the projection from $A$ to the chosen line. Therefore the singular points of $\pi_{A}$ are precisely those points at which the projective tangent space of $S$ is contained in a member of the pencil $\mathscr{P}_A$. In other words, \[ \big\{\text{singular points of $\pi_A$}\big\} = \text{pr}_1\big( \text{pr}_2^{-1}(\mathscr{P}_{A} \cap \check V) \big) \cap S. \]
Suppose from now on that $V$ is not a cone with the apex $x$. Equivalently, we assume that $\check V$ is not contained in $\check x$, where $\check x$ is a hyperplane in $\check{\mathbb{P}}^n$ corresponding to $x$ \cite[Proposition 4.4]{GKZ}.
First consider the case when $\check V$ is not a hypersurface in $\check{\mathbb{P}}^n$. In this case, since $\check V$ is irreducible and not contained in $\check x$, \[ \dim \Big(\check V \cap \check x \Big)\le n-3. \] Therefore a general line contained in $\check x$ is disjoint from $\check V$. In other words, $\pi_{A}$ has no singular points for a general $A$ containing $x$.
Next consider the case when $\check V$ is a hypersurface in $\check{\mathbb{P}}^n$. In this case, the biduality theorem shows that $\text{pr}_2$ is generically a projective bundle with zero dimensional fibers \cite[Theorem 1.1]{GKZ}. Since $I$ is irreducible, the previous sentence implies that \[ \dim \Bigg(D:= \big\{ y \in \check V \mid \text{$\text{pr}_2$ has positive dimensional fiber over $y$} \big\} \Bigg)\le n-3. \] Therefore a general line contained in $\check x$ is disjoint from $D$. In other words, $\pi_{A}$ has only isolated singular points for a general $A$ containing $x$. \end{proof}
\subsection{}
Let $S$ be a smooth and irreducible algebraic subset of $\mathbb{P}^n$, and let $k$ be a positive integer.
\begin{lemma}\label{Bertini}
If $E_x$ is a general linear subspace of codimension $k$ passing through a point $x$ in $\mathbb{P}^n$, then $E_x$ intersects $S \setminus \{x\}$ transversely in $\mathbb{P}^n$. \end{lemma}
\begin{proof} Repeated application of Bertini's theorem shows that $E_x \cap S$ is smooth outside $x$ \cite[Theorem 4.1]{Kleiman}. In other words, for any $p$ in $E_x \cap S$ different from $x$, \[ \text{codim}\big(\text{T}_p E_x \cap \text{T}_pS \subseteq \text{T}_pS\big) = \text{codim}\big(\text{T}_p E_x \subseteq \text{T}_p \mathbb{P}^n\big) = k. \] The conclusion follows from the isomorphism \[ \text{T}_p S / \big( \text{T}_p E_x \cap \text{T}_pS \big) \simeq \big(\text{T}_p E_x + \text{T}_pS\big)/\text{T}_p E_x. \]
\end{proof}
\subsection{}\label{TheStratificationS}
We employ the notation introduced in \ref{TibarLefschetz}. Set $Y=\mathbb{P}^n$, $V=V(h)$, $X=D(h)$, and suppose that \begin{itemize}
\item no component of $V$ is a cone over a smooth variety with the apex $x$, and \item the singular locus of $V$ does not contain a line passing through $x$. \end{itemize} Then we can find a Whitney stratification $\mathscr{W}$ of $Y$ such that \begin{itemize} \item $\{x\}$ is a stratum of $\mathscr{W}$, \item $V$ is a union of strata of $\mathscr{W}$, and \item the closure of a stratum of $\mathscr{W} \setminus \big\{\{x\}\big\}$ is not a cone with the apex $x$.
\end{itemize}
Let $A$ be a codimension $2$ linear subspace of $Y$ containing $x$, and let $\mathbb{Y} \subseteq Y \times \mathbb{P}^1$ be the blow-up of $Y$ along $A$. The projection from $\mathbb{Y}$ onto the $\mathbb{P}^1$ can be identified with the map \[ p: \mathbb{Y} \longrightarrow \mathscr{P}_A. \] The statement below follows, and can be replaced by, the proof of \cite[Proposition 2.4]{Tibar1}.
\begin{lemma}\label{KeyLemma} Let $\mathscr{S}$ be the stratification of $\mathbb{Y}$ with strata \begin{enumerate}[(1)] \item $\big( S \times \mathbb{P}^1 \big) \cap \big( \mathbb{Y} \setminus A \times \mathbb{P}^1 \big)$ for $S \in \mathscr{W} \setminus \big\{\{x\} \big\}$, \item $\big( S\times \mathbb{P}^1 \big) \cap \big( A \times \mathbb{P}^1 \big)$ for $S \in \mathscr{W} \setminus \big\{\{x\} \big\}$, \item $ \{x\} \times \mathbb{P}^1 \setminus E$ and $E$,
\end{enumerate} where $E$ is the set of points at which one of the strata from (1) and (2) fails to be Whitney regular over $\{x\} \times \mathbb{P}^1$. If Lemma \ref{Projection} and Lemma \ref{Bertini} holds for $A$ and each stratum of $\mathscr{W}$, then \begin{enumerate} \item $\mathscr{S}$ is a Whitney stratification, and \item $\mathscr{P}_A$ has only isolated singular points with respect to $\mathscr{S}$. \end{enumerate} \end{lemma}
\begin{proof}
Let $\mathscr{S}_1$ be the Whitney stratification of $\mathbb{Y} \setminus \{x\} \times \mathbb{P}^1$ with strata \[ \mathbb{Y} \setminus A \times \mathbb{P}^1 \ \mathit{and} \ A \times \mathbb{P}^1 \setminus \{x\} \times \mathbb{P}^1, \] and let $\mathscr{S}_2$ be the product Whitney stratification of $Y \times \mathbb{P}^1 \setminus \{x\} \times \mathbb{P}^1$ with strata \[ S \times \mathbb{P}^1 \ \mathit{for} \ S \in \mathscr{W} \setminus \big\{\{x\} \big\}. \] Lemma \ref{Bertini} shows that any pair of strata from $\mathscr{S}_1$ and $\mathscr{S}_2$ intersect transversely in $Y \times \mathbb{P}^1$. It follows that $\mathscr{S}_1 \cap \mathscr{S}_2$ is a Whitney stratification of $\mathbb{Y} \setminus \{x\} \times \mathbb{P}^1$ \cite[1.1.3]{GWdPL}. Now Whitney's fundamental lemma says that $E$ is finite, and all the strata of $\mathscr{S}_1 \cap \mathscr{S}_2$ are Whitney regular over $E$ \cite[Lemma 19.3]{Whitney}. This proves that $\mathscr{S}$ is a Whitney stratification of $\mathbb{Y}$.
For a stratum $\mathcal{S} \in \mathscr{S}$ of the first type,
$p|_\mathcal{S}$ has only isolated singular points by Lemma \ref{Projection}. For a stratum $\mathcal{S} \in \mathscr{S}$ of the second type,
$p|_\mathcal{S}$ is clearly a submersion and has no singular points, and the same is true for the stratum $\{x\} \times \mathbb{P}^1 \setminus E$. Therefore $\mathscr{P}_A$ has only isolated singular points with respect to $\mathscr{S}$.
\end{proof}
\subsection{}
As a final preparation for the proof of Theorem \ref{attaching}, we recall a Zariski theorem on the fundamental group of plane curve complements \cite[Section 4.3]{Dimca}.
Let $x$ be a point in $\mathbb{P}^2$, and let $C$ be a curve in $\mathbb{P}^2$. We say that a line $L_x$ passing through $x$ is \emph{exceptional} with respect to $C$ if \begin{itemize} \item $L_x$ is tangent to the curve $C$, or \item $L_x$ passes through a singular point of the curve $C$ different from $x$. \end{itemize}
\begin{theorem}[{\cite[Corollary 4.3.6]{Dimca}}]\label{Zariski} Suppose that no line containing $x$ is a component of the curve $C$. Then for any line $L_x$ passing through $x$ which is not exceptional, there is an epimorphism \[ \pi_1\big(L_x \setminus C\big) \longrightarrow \pi_1\big(\mathbb{P}^2 \setminus C\big) \] induced by the inclusion. \end{theorem}
Theorem \ref{Zariski} is the base case of the induction for Theorem \ref{attaching}.
\subsection{}
\begin{proof}[Proof of Theorem \ref{attaching}]
We prove by induction on $n$, the base case being Theorem \ref{Zariski}. Suppose that \begin{enumerate}[(a)]
\item no component of $V(h)$ is a cone with the apex $x$, and \item the singular locus of $V(h)$ does not contain a line passing through $x$. \end{enumerate} For the induction step we check that the two conditions on $V(h)$ are also satisfied by $V(h) \cap H_x$. For condition (a), this is the content of Lemma \ref{Cones}. Condition (b) follows from Bertini's theorem that \[ \text{Sing}\big(V(h) \cap H_x\big) \setminus \{x\}= \Big( \text{Sing}\big(V(h)\big) \cap H_x \Big) \setminus \{x\}. \]
Now consider the Whitney stratifications $\mathscr{W}$ and $\mathscr{S}$ of Section \ref{TheStratificationS}. We also employ other notations introduced in that section. When $n \ge 3$, we choose linear subspaces $A \subseteq H$ containing $x$, of codimension $2$ and $1$ respectively, sufficiently general so that \begin{enumerate} \item $A$ is not contained in $V$, \item (a) and (b) are satisfied by $V \cap H$, \item Lemma \ref{Projection} and Lemma \ref{Bertini} holds for $A$ and each stratum of $\mathscr{W}$, and \item the induction hypothesis applies to the pair $(X \cap H, X \cap A)$. \end{enumerate} Then, by Lemma \ref{KeyLemma}, all the assumptions of Theorem \ref{NongenericLefschetz} are satisfied. Therefore the pair $(X,X \cap H)$ is $(n-1)$-connected.
\end{proof}
\subsection{}
For the interested reader we record here a version of what we proved in the generality of Section \ref{TibarLefschetz}. Let $Y$ be a projective variety of dimension $n \ge 2$ in $\mathbb{P}^N$, and let $V$ be a closed algebraic subset of $Y$.
\begin{theorem} Let $E_x \subseteq F_x$ be a general pair of linear subspaces containing a point $x$ in $\mathbb{P}^N$, of codimensions $n-1$ and $n-2$ respectively. Suppose that \begin{enumerate} \item the quasi-projective variety $X:=Y \setminus V$ is locally a complete intersection, \item no component of $V$ (and of $Y$) is a cone with the apex $x$, \item the singular locus of $V$ (and of $Y$) does not contain a line passing through $x$, and \item there is an epimorphism induced by the inclusion \[ \pi_1\big(X \cap E_x\big) \longrightarrow \pi_1\big(X \cap F_x\big). \] \end{enumerate} Then, for a sufficiently general hyperplane $H_x$ passing through $x$, \[ \pi_i\big(X,X\cap H\big)=0 \quad \text{for} \quad i<n. \]
\end{theorem}
\section{Proof of Theorem \ref{main2}}\label{ProofoftheInequality}
\subsection{}
We deduce Theorem \ref{main2} from Theorem \ref{attaching} when $n \ge 3$. A separate argument will be given for plane curves.
\begin{proof}[Proof of Theorem \ref{main2} when $n \ge 3$]
We know from \cite{Dimca-Papadima} that \[ \chi\big(D(h)\big)= (-1)^n\text{deg}\big(\text{grad}(h)\big)+\sum_{i=0}^{n-1}(-1)^{i}(d-1)^{i}. \]
If $V(h)$ is not a cone with the apex $x$, then there is a hyperplane $H_x$ containing $x$ such that
\begin{enumerate} \item Theorem \ref{attaching} applies to $H_x$, \item $V(h) \cap H_x$ is smooth outside $x$, and \item the Milnor number of $V(h) \cap H_x$ at $x$ is the sectional Milnor number $\mu^{(n-1)}$ of $V(h)$ at $x$. \end{enumerate}
It follows from (ii) and (iii) that \[ \chi\big(D(h) \cap H_x\big) = (-1)^{n-1}\Big((d-1)^{n-1}-\mu^{(n-1)}\Big)+\sum_{i=0}^{n-2}(-1)^{i}(d-1)^{i}. \]
Therefore
\[ \text{rank} \ H_n\big(D(h),D(h) \cap H_x \big) =(-1)^n\Big( \chi\big(D(h)\big)-\chi\big(D(h) \cap H_x\big) \Big) = \text{deg}\big(\text{grad}(h)\big) - \mu^{(n-1)} \ge 0. \]
\end{proof}
\subsection{}
We prove Theorem \ref{main2} for plane curves. The main ingredient in this case is Milnor's formula for the double point number \[ 2\delta_x=\mu_x+r_x-1, \] where $\mu_x$ is the Milnor number at $x$, and $r_x$ is the number of branches at $x$ \cite[Theorem 10.5]{Milnor}.
\begin{lemma}\label{PlaneCurves} \ \begin{enumerate} \item Suppose $V(h)$ is a reduced and irreducible plane curve of degree $d$ containing $x$. Then \[ \text{deg}\big(\text{grad}(h)\big) \ge (d-1)+(r_x-1), \] where $r_x$ is the number of branches of $V(h)$ at $x$.
\item Suppose $V(h_1)$ and $V(h_2)$ are plane curves with no common components. Then \[ \text{deg}\big(\text{grad}(h_1h_2)\big) = \text{deg}\big(\text{grad}(h_1)\big) + \text{deg}\big(\text{grad}(h_2)\big) +\#\big( V(h_1) \cap V(h_2) \big)-1. \] \end{enumerate} \end{lemma}
\begin{proof} The first inequality is obtained from \cite[Theorem 10.5]{Milnor} and the fact that $g \ge 0$, where $g$ is the genus of the normalization of $V(h)$.
The second assertion is equivalent to the inclusion-exclusion formula for the topological Euler characteristic.
We refer to \cite[Section 3]{Dolgachev} and \cite[Theorem 3.1]{Fassarella-Medeiros} for details. \end{proof}
\begin{proof}[Proof of Theorem \ref{main2} when $n =2$] Let $V$ be a reduced plane curve of degree $d$ which is not a cone with the apex $x$. Lemma \ref{PlaneCurves} proves the assertion when $V$ is irreducible. We divide the remaining problem into two cases:
\begin{enumerate}[(1)]
\item $V$ has at least two components which are not cones with the apex with $x$. \item $V$ has exactly one component which is not a cone with the apex with $x$. \end{enumerate}
In case (1), we induct on the number of irreducible components. Write $V=V_1 \cup V_2$, where $V_1$ is not a cone with the apex $x$, $V_2$ is irreducible and not a cone with the apex $x$, and $V_1 \cap V_2$ is finite. Let $h_1,h_2$ be reduced equations of degree $d_1,d_2$ defining $V_1,V_2$ respectively. Then, by Lemma \ref{PlaneCurves} and the induction hypothesis, \begin{eqnarray*} \text{deg}\big(\text{grad}(h_1h_2)\big) &=& \text{deg}\big(\text{grad}(h_1)\big) + \text{deg}\big(\text{grad}(h_2)\big) +\Big(\#( V_1 \cap V_2 )-1\Big) \\
&\ge & \Big(m_x(V_1)-1\Big)+\Big(d_2-1\Big)+\Big(\#( V_1 \cap V_2 )-1\Big). \end{eqnarray*} In other words, \[ \text{deg}\big(\text{grad}(h_1h_2)\big) \ge \Big(m_x(V)-1\Big)+\Big(d_2-m_x(V_2)-1\Big)+\Big(\#( V_1 \cap V_2 )-1\Big). \] The second term in the last expression is nonnegative because $V_2$ is not a cone with the apex $x$. The third term is also nonnegative, and this gives the desired inequality.
In case (2), we write $V=V_1 \cup V_2$, where $V_1$ is a cone with the apex $x$, and $V_2$ is irreducible. Then, by Lemma \ref{PlaneCurves}, \begin{eqnarray*} \text{deg}\big(\text{grad}(h_1h_2)\big) &=& \text{deg}\big(\text{grad}(h_2)\big) +\Big(\#( V_1\cap V_2 )-1\Big) \\ &\ge& \Big(d_2-1\Big)+\Big(r_x(V_2)-1\Big)+\Big(\#( V_1 \cap V_2 )-1\Big).
\end{eqnarray*} In other words, \[ \text{deg}\big(\text{grad}(h_1h_2)\big) \ge \Big(m_x(V)-1\Big)+ \Big(d_2-m_x(V_2)-1\Big)+\Big(r_x(V_2)-m_x(V_1)+\#( V_1 \cap V_2 )-1\Big). \] The second term in the last expression is nonnegative because $V_2$ is not a cone with the apex $x$. We claim that the third term is also nonnegative.
Let $t_x(V_2)$ be the number of lines in the tangent cone of $V_2$ at $x$. It follows from Hensel's lemma that \[ r_x(V_2) \ge t_x(V_2). \] See for example \cite[Corollary 2.2.6]{Casas-Alvero}. A local computation shows that a line containing $x$ intersects $V_2$ in at least one point other than $x$, unless the line is contained in the tangent cone of $V_2$ at $x$. Since there are at least $m_x(V_1)-t_x(V_1)$ lines in $V_1$ not contained in the tangent cone of $V_2$ at $x$, we have \[
t_x(V_2)-m_x(V_1)+\#\big( V(h_1) \cap V(h_2) \big)-1 \ge 0. \] This completes the proof. \end{proof}
\subsection{}
We show that, for each $d \ge 3$ and $n \ge 2$, there is a degree $d$ hypersurface in $\mathbb{P}^n$ with one singular point, at which the equality holds in Theorem \ref{main2}.
\begin{proposition}\label{EqualityCase} Let $V(h)$ be the degree $d$ hypersurface in $\mathbb{P}^n$ defined by the equation \[ h=z_0z_1^{d-1}+z_1z_2^{d-1}+(z_3^d+\cdots+z_n^d), \qquad d\ge 3. \] Then the unique singular point $x$ of $V(h)$ satisfies \[ \mu^{(n)}=(d-1)^n-(d-1)^{n-1}+(d-1)^{n-2} \quad \text{and} \quad \mu^{(n-1)}=(d-1)^{n-1}-(d-1)^{n-2}. \]
\end{proposition}
\begin{proof}[Proof of Proposition \ref{EqualityCase} when $n \ge 3$] Locally at $x$, the hypersurface is defined by \[ f=x_1^{d-1}+x_1x_2^{d-1}+x_3^d+\cdots+x_n^d. \] Note that $f$ is weighted homogeneous with weights $w_i$, where \[ \frac{d-1}{w_1}=\frac{1}{w_1}+\frac{d-1}{w_2}=\frac{d}{w_3}=\cdots=\frac{d}{w_n}=1. \]
It follows from \cite[Theorem 1]{Milnor-Orlik} that \[ \mu^{(n)}=\prod_{i=1}^n (w_i-1)=(d-1)^n-(d-1)^{n-1}+(d-1)^{n-2}. \]
Now consider the hyperplane $H$ passing through $x$ defined by \[ x_n=c_1x_1+\cdots +c_{n-1}x_{n-1}, \qquad c = (c_1,\ldots,c_{n-1})\in \big(\mathbb{C}^*\big)^{n-1}. \] Locally at $x$, $V(h) \cap H$ is isomorphic to the hypersurface defined by \[ g = x_1^{d-1}+x_1x_2^{d-1}+x_3^d+\cdots+x_{n-1}^d+(c_1x_1+\cdots +c_{n-1}x_{n-1})^d. \] The principal part of $g$ with respect to the Newton diagram is \[ g_0 = x_1^{d-1}+x_3^d+\cdots+x_{n-1}^d+(c_2x_2+\cdots +c_{n-1}x_{n-1})^d. \] Since $g_0$ defines an isolated singular point at the origin, $g$ is semiquasihomogeneous.
This shows that the singular points defined by $g$ and $g_0$ have the same Milnor number \[ \mu^{(n-1)} = (d-1)^{n-1}-(d-1)^{n-2}. \] See \cite[Chapter 12]{AGV}. \end{proof}
Proposition \ref{EqualityCase} remains valid for $n=2$.
\section{Projective hypersurfaces with small polar degree}\label{ProofoftheConjecture}
\subsection{}
A reduced homogeneous polynomial $h$ is homaloidal if and only if $n$ general first polar hypersurfaces of $V(h)$ intersect at exactly one point outside the singular locus of $V(h)$. However, Conjecture \ref{DPConjecture} cannot be proved by a Cayley-Bacharach type theorem alone.
\begin{example} For each $d \ge 2$ and $n \ge 2$, there is a birational transformation \[ \varphi: \mathbb{P}^n \dashrightarrow \mathbb{P}^n, \qquad z \longmapsto (h_0:\cdots:h_n), \] where $h_0,\ldots,h_1$ are homogeneous polynomials of the same degree $d-1$, with a zero-dimensional base locus. For example, one may take \[ h_n=z_0^{d-1}, \qquad h_{n-1}=2z_0^{d-2}z_1, \qquad h_{n-i}=2z_0^{d-2}z_i+z_{i-1}^{d-1}, \qquad i=2,\ldots,n. \] When $n=2$ and $d=3$, this is the gradient map of the homaloidal polynomial \[ h=z_0(z_1^2+z_0z_2). \] Conjecture \ref{DPConjecture} asserts that no birational transformation with a zero-dimensional base locus is a gradient map when $d \ge 3$ and $n \ge 3$.
\end{example}
\subsection{}\label{NormalForm}
A plane curve singularity of multiplicity $2$ is locally defined by an equation of the form \[ f=x_1^2+x_2^{k+1}, \] where $k$ is the Milnor number at the singular point \cite[Exercise 1.5.14]{Hartshorne}. Similar statements remain valid in all dimensions.
\begin{lemma}\label{TypeA} Let $(V,\mathbf{0})$ be the germ of an isolated hypersurface singularity at the origin of $\mathbb{C}^n$. \begin{enumerate} \item If $\mu^{(n-1)}$ of the germ is equal to $1$, then the singularity is of type $A_k$ for some $k \ge 1$. \item If $\mu^{(n-1)}$ of the germ is equal to $2$, then the singularity is of type $E_{6r}$, $E_{6r+1}$, $E_{6r+2}$ or $J_{r,i}$, for some $r \ge 1$ and $i \ge 0$.
\end{enumerate} \end{lemma}
For the normal form of the above singularities, we refer to \cite[Chapter 15]{AGV}. We set $J_{1,i}=D_{4+i}$ in the second case so that the singularity is simple if and only if $r=1$.
\begin{proof}
We prove the first statement. The second statement can be justified in the same way.
Choose any hyperplane $H$ passing through the origin such that $(V \cap H,\mathbf{0})$ has the Milnor number $1$. Let $f$ be an equation defining $V$, and let $y_n$ be an equation defining $H$. Morse lemma shows that there is a local coordinate system $y_1,\ldots,y_{n-1}$ of $(H,\mathbf{0})$ such that \[
f|_H= y_1^2+\cdots+y_{n-2}^2+y_{n-1}^2. \]
Therefore, we may write \[
f=y_1^2+\cdots+y_{n-1}^2+y_n \big(c_1y_1+\cdots+c_ny_n\big) + \big(\text{terms of degree $\ge 3$ in $y_1,\ldots,y_n$}\big) \] for some $c_1,\ldots,c_n \in \mathbb{C}$.
This shows that the Hessian of $f$ at $\mathbf{0}$ relative to $y_1,\ldots,y_n$ is \[ H(f)=\left(\begin{array}{ccccc} 1 & 0 & \cdots & 0 & c_1 \\ 0 & 1 & \cdots & 0 & c_2 \\ \vdots &\vdots&&\vdots&\vdots \\ 0 & 0 & \cdots & 1 & c_{n-1}\\ c_1&c_2&\cdots &c_{n-1}& c_n \end{array}\right). \] In particular, the corank of the Hessian is at most $1$. Now the classification of corank $\le 1$ isolated singularities says that there is a local coordinate system $x_1,\ldots,x_n$ of $(\mathbb{C}^n,\mathbf{0})$ with \[ f=x_1^2+\cdots+x_{n-1}^2+x_n^{k+1}, \] where $k$ is the Milnor number of $V$ at the origin. See \cite[Chapter 11]{AGV}.
\end{proof}
\subsection{}
Dimca proves Conjecture \ref{DPConjecture} in \cite[Theorem 9]{Dimca} when all the singular points of $V(h)$ are weighted homogeneous. The proof is based on the work of du Plessis and Wall \cite{duPlessis-Wall2}. We use Theorem \ref{main2} to reduce the problem to this case.
\begin{proof}[Proof of Theorem \ref{Classification1}] Let $V(h)$ be a homaloidal hypersurface of degree $d$ in $\mathbb{P}^n$ with only isolated singular points. Theorem \ref{main2} implies that all the singular points of $V(h)$ has the sectional Milnor number $\mu^{(n-1)}=1$. It follows from Lemma \ref{TypeA} that all the singular points of $V(h)$ are locally defined by an equation of the form \[ f=x_1^2+\cdots+x_{n-1}^2+x_n^{k+1}, \] where $k$ is the Milnor number of $V(h)$ at the singular point. In particular, all the singular points of $V(h)$ are weighted homogeneous. Therefore, by \cite[Theorem 9]{Dimca}, either $n \le 2$ or $d \le 2$. If $d \le 2$, the hypersurface should be smooth quadric, and any smooth quadric is defined by \[ h=z_0^2+\cdots+z_n^2, \] after a linear change of coordinates. When $n \le 2$, the assertion is \cite[Theorem 4]{Dolgachev}. \end{proof}
\subsection{}
What can be said about projective hypersurfaces which has polar degree $2$ and only isolated singular points? We propose the following conjecture.
\begin{conjecture}\label{Conjecture}
A projective hypersurface with only isolated singular points has polar degree $2$ if and only if it is one of the following, after a linear change of homogeneous coordinates:
\begin{itemize} \item $(n=3,d=3)$ a normal cubic surface containing a single line \begin{flalign*} \hspace{30mm} h=z_0z_1^2+z_1z_2^2+z_1z_3^2+z_2^3=0, & \qquad (E_6).& \end{flalign*} \item $(n=3,d=3)$ a normal cubic surface containing two lines \begin{flalign*} \hspace{30mm} h=z_0z_1z_2+z_0z_3^2+z_1^3=0, & \qquad (A_5,A_1). & \end{flalign*} \item $(n=3,d=3)$ a normal cubic surface containing three lines and three binodes \begin{flalign*} \hspace{30mm} h=z_0z_1z_2+z_3^3=0, & \qquad (A_2,A_2,A_2). & \end{flalign*} \item $(n=2,d=5)$ two smooth conics meeting at a single point and the common tangent \begin{flalign*} \hspace{30mm} h=z_0(z_1^2+z_0z_2)(z_1^2+z_0z_2+z_0^2)=0, &\qquad (J_{2,4}). & \end{flalign*} \item $(n=2,d=4)$ two smooth conics meeting at a single point \begin{flalign*} \hspace{30mm} h=(z_1^2+z_0z_2)(z_1^2+z_0z_2+z_0^2)=0, &\qquad (A_7).& \end{flalign*} \item $(n=2,d=4)$ a smooth conic, a tangent, and a line passing through the tangency point \begin{flalign*} \hspace{30mm} h=z_0(z_0+z_1)(z_1^2+z_0z_2)=0, &\qquad (D_6,A_1).& \end{flalign*} \item $(n=2,d=4)$ a smooth conic and two tangent lines \begin{flalign*} \hspace{30mm} h=z_0z_2(z_1^2+z_0z_2)=0, &\qquad (A_1,A_3,A_3).& \end{flalign*} \item $(n=2,d=4)$ three concurrent lines and a line not meeting the center point \begin{flalign*} \hspace{30mm} h=z_0z_1z_2(z_0+z_1)=0, &\qquad (D_4,A_1,A_1,A_1).& \end{flalign*} \item $(n=2,d=4)$ a cuspidal cubic and its tangent at the cusp \begin{flalign*} \hspace{30mm} h=z_0(z_1^3+z_0^2z_2)=0, &\qquad (E_7).& \end{flalign*} \item $(n=2,d=4)$ a cuspidal cubic and its tangent at the smooth flex point \begin{flalign*} \hspace{30mm} h=z_2(z_1^3+z_0^2z_2)=0, &\qquad (A_2,A_5).& \end{flalign*} \item $(n=2,d=3)$ a cuspidal cubic \begin{flalign*} \hspace{30mm} h=z_1^3+z_0^2z_2=0, &\qquad (A_2).& \end{flalign*} \item $(n=2,d=3)$ a smooth conic and a secant line \begin{flalign*} \hspace{30mm} h=z_1(z_1^2+z_0z_2)=0, &\qquad (A_1,A_1).& \end{flalign*} \end{itemize} \end{conjecture}
Less precisely but more generally, we conjecture that for any positive integer $k$, there is no projective hypersurface of polar degree $k$ which has only isolated singular points, for sufficiently large $n$ and $d$.
\begin{proposition}\label{Evidence} Conjecture \ref{Conjecture} is valid for plane curves, cubic surfaces, and quartic surfaces. \end{proposition}
Radu Laza informed us that Conjecture \ref{Conjecture} is valid also for cubic threefolds. We note that there is a cubic threefold with only isolated singular points which has polar degree $3$.
Explicitly, there is \[ h=z_0z_1z_4+z_0^3+z_1^3+z_0z_2^2+z_1z_3^2=0, \qquad (T_{2,6,6}). \]
\begin{proof} The proof is a combination of \cite{Bruce-Wall,Degtyarev,Dimca,duPlessis-Wall2,Fassarella-Medeiros,Wall}.
The assertion for cubic surfaces is classical, and can be deduced from the classification \cite{Bruce-Wall}. The list of plane curves is obtained in \cite{Fassarella-Medeiros}. Any reduced plane curve with polar degree $2$ should be projectively equivalent to one of the list.
For the remaining case of quartic surfaces, we start with two general remarks. We assume throughout that projective hypersurfaces have only isolated singular points.
\begin{enumerate} \item If $V(h)$ is a projective hypersurface with only weighted homogeneous singular points, then \[ \text{deg}\big(\text{grad}(h)\big) \ge \min \Big\{ (d-1)^{n-2}, 2(n+1) \Big\}, \] unless $V(h)$ is a cone. This follows from the proof of \cite[Theorem 9]{Dimca}. We note from the above example of cubic threefold with $T_{2,6,6}$ singularity that the assumption of weighted homogeneity is necessary for the inequality. \item Each nonsimple singular point of a projective hypersurface of polar degree $2$ should belong to one of the types $E_{6r}$, $E_{6r+1}$, $E_{6r+2}$ or $J_{r,i}$ for some $r \ge 2$ and $i \ge 0$. This follows from Theorem \ref{main2} and Lemma \ref{TypeA}.
\end{enumerate}
Let $V$ be a quartic surface of polar degree $2$. The first remark above shows that it is enough to consider the case when $V$ has a singular point $p$ which is not simple. The second remark shows that the singularity of $V$ at $p$ is of type $\mathbf{J}$ or $\mathbf{E}$. Let $C$ be the discriminant of the projection $V \setminus \{p\} \to \mathbb{P}^2$ from $p$. Under our assumptions, $C$ is a reduced sextic curve.
Suppose $V$ is not stable in the sense of Geometric Invariant Theory. In this case, there is a bijection between the singular points of $V$ and of $C$, which preserves the type \cite[Section 11]{Wall}. In particular, \[ \sum_{x \in V} \mu^{(3)}(x,V) = \sum_{x \in C} \mu^{(2)}(x,C). \] Since $V$ has polar degree $2$, the sum of the Milnor numbers of $C$ should be $3^3-2=5^2$. From Theorem \ref{main1} it follows that $C$ is a cone. This contradicts the fact that $C$ has a singular point of type $\mathbf{J}$ or $\mathbf{E}$. In Degtyar\"ev's classification of quartic surfaces having a nonsimple singular points, this case is called \emph{exceptional} \cite[Theorem 1.9]{Degtyarev}.
Suppose $V$ is stable in the sense of Geometric Invariant Theory. In this case, there is a point $q$ such that the blowup of the singularity of $V \subseteq \mathbb{P}^3$ at $p$ has the same type as the singularity of $C \subseteq \mathbb{P}^2$ at $q$. Moreover, there is a bijection between the singular points of $V \setminus \{p\}$ and of $C \setminus \{q\}$, which preserves the type \cite[Section 12]{Wall}. A case by case analysis of the blowup of suspensions of singularities of type $\mathbf{J}$ and $\mathbf{E}$ shows that \[ \sum_{x \in V} \mu^{(3)}(x,V) = 1+\sum_{x \in C} \mu^{(2)}(x,C). \]
Since $V$ has polar degree $2$, the sum of the Milnor numbers of $C$ should be $3^3-3=5^2-1$. This means that the sextic plane curve $C$ is homaloidal, which is impossible. In Degtyar\"ev's classification of quartic surfaces having a nonsimple singular points, this case is called \emph{nonexceptional} \cite[Theorem 1.7]{Degtyarev}.
\end{proof}
\end{document} | arXiv |
Edmund Landau
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis.
Edmund Landau
Born
Edmund Georg Hermann Landau
(1877-02-14)14 February 1877
Berlin, Germany
Died19 February 1938(1938-02-19) (aged 61)
Berlin, Germany
Alma materUniversity of Berlin
Known forDistribution of prime numbers
Landau prime ideal theorem
SpouseMarianne Ehrlich
Scientific career
FieldsNumber theory
Complex analysis
InstitutionsUniversity of Berlin
University of Göttingen
Hebrew University of Jerusalem
Doctoral advisorGeorg Frobenius
Lazarus Fuchs
Doctoral studentsBinyamin Amirà
Paul Bernays
Harald Bohr
Gustav Doetsch
Hans Heilbronn
Grete Hermann
Dunham Jackson
Erich Kamke
Aubrey Kempner
Alexander Ostrowski
Carl Ludwig Siegel
Arnold Walfisz
Vojtěch Jarník
Biography
Edmund Landau was born to a Jewish family in Berlin. His father was Leopold Landau, a gynecologist, and his mother was Johanna Jacoby. Landau studied mathematics at the University of Berlin, receiving his doctorate in 1899 and his habilitation (the post-doctoral qualification required to teach in German universities) in 1901. His doctoral thesis was 14 pages long.
In 1895, his paper on scoring chess tournaments is the earliest use of eigenvector centrality.[1][2]
Landau taught at the University of Berlin from 1899 to 1909, after which he held a chair at the University of Göttingen. He married Marianne Ehrlich, the daughter of the Nobel Prize-winning biologist Paul Ehrlich, in 1905.
At the 1912 International Congress of Mathematicians Landau listed four problems in number theory about primes that he said were particularly hard using current mathematical methods. They remain unsolved to this day and are now known as Landau's problems.
During the 1920s, Landau was instrumental in establishing the Mathematics Institute at the nascent Hebrew University of Jerusalem. Intent on eventually settling in Jerusalem, he taught himself Hebrew and delivered a lecture entitled Solved and unsolved problems in elementary number theory in Hebrew on 2 April 1925 during the university's groundbreaking ceremonies. He negotiated with the university's president, Judah Magnes, regarding a position at the university and the building that was to house the Mathematics Institute.
Landau and his family emigrated to Mandatory Palestine in 1927 and he began teaching at the Hebrew University. The family had difficulty adjusting to the primitive living standards then available in Jerusalem. In addition, Landau became a pawn in a struggle for control of the university between Magnes and Chaim Weizmann and Albert Einstein. Magnes suggested that Landau be appointed Rector of the university, but Einstein and Weizmann supported Selig Brodetsky. Landau was disgusted by the dispute and decided to return to Göttingen, remaining there until he was forced out by the Nazi regime after the Machtergreifung in 1933, in a boycott organized by Oswald Teichmüller. Thereafter, he lectured only outside Germany. He moved to Berlin in 1934, where he died in early 1938 of natural causes.
In 1903, Landau gave a much simpler proof than was then known of the prime number theorem and later presented the first systematic treatment of analytic number theory in the Handbuch der Lehre von der Verteilung der Primzahlen (the "Handbuch").[3] He also made important contributions to complex analysis.
G. H. Hardy and Hans Heilbronn wrote that "No one was ever more passionately devoted to mathematics than Landau". [4]
Works
• Handbuch der Lehre von der Verteilung der Primzahlen, Taubner, Leipzig, 1909.
• Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie, Springer, 1916.
• Einführung in die elementare und analytische Theorie der algebraischen Zahlen und Ideale, 1918.
• Vorlesungen über Zahlentheorie, 3 Vols, S. Hirzel, Leipzig, 1927.
• Grundlagen der Analysis, Akademische Verlagsgesellschaft, Leipzig, 1930.
• Einführung in die Differential- und Integralrechnung, P. Noordhoff N. V., Groningen, 1934.
• Über einige neuere Fortschritte der additiven Zahlentheorie, Cambridge University Press, London, 1937.
Translated works
• Foundations of Analysis, Chelsea Pub Co. ISBN 0-8218-2693-X.
• Differential and Integral Calculus, American Mathematical Society. ISBN 0-8218-2830-4.
• Elementary Number Theory, American Mathematical Society. ISBN 0-8218-2004-4.
See also
• Landau's function
• Landau prime ideal theorem
• Landau's problems
• Landau's symbol (Big O notation)
• Landau–Kolmogorov inequality
• Landau–Ramanujan constant
• Landau's problem on the Dirichlet eta function
• Landau kernel
References
1. Endmund Landau (1895). "Zur relativen Wertbemessung der Turnierresultate". Deutsches Wochenschach (11): 366–369. doi:10.1007/978-1-4615-4819-5_23.
2. Holme, Peter (15 April 2019). "Firsts in network science". Retrieved 17 April 2019.
3. Gronwall, T. H. (1914). "Review: Handbuch der Lehre von der Verteilung der Primzahlen". Bull. Amer. Math. Soc. 20 (7): 368–376. doi:10.1090/s0002-9904-1914-02502-9.
4. Hardy, G. H.; H. Heilbronn (1938). "Edmund Landau". Journal of the London Mathematical Society. 13 (4): 302–310. doi:10.1112/jlms/s1-13.4.302. Retrieved 2009-06-11.Obituary and review of scientific work and books.
External links
Wikimedia Commons has media related to Edmund Landau.
Wikiquote has quotations related to Edmund Landau.
• O'Connor, John J.; Robertson, Edmund F., "Edmund Landau", MacTutor History of Mathematics Archive, University of St Andrews
• Edmund Landau at the Mathematics Genealogy Project
• Edmund Landau: The Master Rigorist by Eli Maor, Trigonometric Delights, page 192.
• Translation of his doctoral thesis Neuer Beweis der Gleichung $\scriptstyle \sum \limits _{k=1}^{\infty }{\frac {\mu (k)}{k}}\,=\,0$, Berlin 1899
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| Wikipedia |
Bhatia–Davis inequality
In mathematics, the Bhatia–Davis inequality, named after Rajendra Bhatia and Chandler Davis, is an upper bound on the variance σ2 of any bounded probability distribution on the real line.
Statement
Let m and M be the lower and upper bounds, respectively, for a set of real numbers a1, ..., an , with a particular probability distribution. Let μ be the expected value of this distribution.
Then the Bhatia–Davis inequality states:
$\sigma ^{2}\leq (M-\mu )(\mu -m).\,$
Equality holds if and only if every aj in the set of values is equal either to M or to m.[1]
Proof
Since $m\leq A\leq M$,
$0\leq \mathbb {E} [(M-A)(A-m)]=-\mathbb {E} [A^{2}]-mM+(m+M)\mu $.
Thus,
$\sigma ^{2}=\mathbb {E} [A^{2}]-\mu ^{2}\leq -mM+(m+M)\mu -\mu ^{2}=(M-\mu )(\mu -m)$.
Extensions of the Bhatia–Davis inequality
If $\Phi $ is a positive and unital linear mapping of a C* -algebra ${\mathcal {A}}$ into a C* -algebra ${\mathcal {B}}$, and A is a self-adjoint element of ${\mathcal {A}}$ satisfying m $\leq $ A $\leq $ M, then:
$\Phi (A^{2})-(\Phi A)^{2}\leq (M-\Phi A)(\Phi A-m)$.
If ${\mathit {X}}$ is a discrete random variable such that
$P(X=x_{i})=p_{i},$ where $i=1,...,n$, then:
$s_{p}^{2}=\sum _{1}^{n}p_{i}x_{i}^{2}-(\sum _{1}^{n}p_{i}x_{i})^{2}\leq (M-\sum _{1}^{n}p_{i}x_{i})(\sum _{1}^{n}p_{i}x_{i}-m)$,
where $0\leq p_{i}\leq 1$ and $\sum _{1}^{n}p_{i}=1$.
Comparisons to other inequalities
The Bhatia–Davis inequality is stronger than Popoviciu's inequality on variances (note, however, that Popoviciu's inequality does not require knowledge of the expectation or mean), as can be seen from the conditions for equality. Equality holds in Popoviciu's inequality if and only if half of the aj are equal to the upper bounds and half of the aj are equal to the lower bounds. Additionally, Sharma[2] has made further refinements on the Bhatia–Davis inequality.
See also
• Cramér–Rao bound
• Chapman–Robbins bound
• Popoviciu's inequality on variances
References
1. Bhatia, Rajendra; Davis, Chandler (2000). "A Better Bound on the Variance". The American Mathematical Monthly. 107 (4): 353–357. doi:10.1080/00029890.2000.12005203. ISSN 0002-9890. S2CID 38818437.
2. Sharma, Rajesh (2008). "Some more inequalities for arithmetic mean, harmonic mean and variance". Journal of Mathematical Inequalities (1): 109–114. doi:10.7153/jmi-02-11. ISSN 1846-579X.
| Wikipedia |
Efficient solar hydrogen generation in microgravity environment
Electrolysis in reduced gravitational environments: current research perspectives and future applications
Ömer Akay, Aleksandr Bashkatov, … Katharina Brinkert
Fundamentals and future applications of electrochemical energy conversion in space
Katharina Brinkert & Philippe Mandin
An advance in transfer line chilldown heat transfer of cryogenic propellants in microgravity using microfilm coating for enabling deep space exploration
J. N. Chung, Jun Dong, … J. W. Hartwig
Nitrogen flow boiling and chilldown experiments in microgravity using pulse flow and low-thermally conductive coatings
Jason Hartwig, J. N. Chung, … Michael Doherty
Protein structural changes on a CubeSat under rocket acceleration profile
Autumn Luna, Jacob Meisel, … Daniel Fernandez
A magnetic levitation based low-gravity simulator with an unprecedented large functional volume
Hamid Sanavandi & Wei Guo
How advances in low-g plumbing enable space exploration
M. M. Weislogel, J. C. Graf, … J. C. Buchli
Flight of an aeroplane with solid-state propulsion
Haofeng Xu, Yiou He, … Steven R. H. Barrett
Experimental evidence for superionic water ice using shock compression
Marius Millot, Sebastien Hamel, … Jon H. Eggert
Katharina Brinkert ORCID: orcid.org/0000-0002-3593-50471,2,
Matthias H. Richter1,3,
Ömer Akay4,
Janine Liedtke2,
Michael Giersig4,5,
Katherine T. Fountaine6,7 &
Hans-Joachim Lewerenz8
Nature Communications volume 9, Article number: 2527 (2018) Cite this article
Electrocatalysis
Nanoscale materials
Photocatalysis
Solar fuels
Long-term space missions require extra-terrestrial production of storable, renewable energy. Hydrogen is ascribed a crucial role for transportation, electrical power and oxygen generation. We demonstrate in a series of drop tower experiments that efficient direct hydrogen production can be realized photoelectrochemically in microgravity environment, providing an alternative route to existing life support technologies for space travel. The photoelectrochemical cell consists of an integrated catalyst-functionalized semiconductor system that generates hydrogen with current densities >15 mA/cm2 in the absence of buoyancy. Conditions are described adverting the resulting formation of ion transport blocking froth layers on the photoelectrodes. The current limiting factors were overcome by controlling the micro- and nanotopography of the Rh electrocatalyst using shadow nanosphere lithography. The behaviour of the applied system in terrestrial and microgravity environment is simulated using a kinetic transport model. Differences observed for varied catalyst topography are elucidated, enabling future photoelectrode designs for use in reduced gravity environments.
Our atmosphere on earth is sustained by the photodissociation of water, a fundamental process used by nature in oxygenic photosynthesis to convert solar energy into storable, chemical energy1. It relies on the so-called Z-scheme, where photons of different energy are absorbed in a staggered energy-level system. Presently, artificial photosynthesis systems are being intensively developed2,3,4,5,6 based on the analogue of the Z-scheme, e.g. in tandem solar cells5. Here, the light-driven oxidation of water to oxygen at the (photo) anode is accompanied by the production of so-called solar fuels at the (photo)cathode: hydrogen, a storable fuel which can be used as a feedstock for fuel cells generating power for transportation or carbon dioxide reduction products, holding the promise for converting emissions back to fuels utilizing renewable energy. Inorganic systems have yielded the hitherto highest efficiencies and stabilities when combining the tandem absorbers with high-activity electrocatalysts7,8.
The efficient conversion of abundant sunlight to oxygen and storable fuels is also a key step in realizing long-term space missions and cis-lunar research platforms such as the Deep Space Gateway. Since the early 1960s, water electrolysis cells comprising of photovoltaic p–n junction solar cells and a separate water electrolyser system have been employed as a part of a spacecraft environmental control system for the production of oxygen from carbon dioxide9 and are still in use on the International Space Station (ISS). The involved travel distances on future deep space missions restrict volume and mass of consumables required for a voyage of months or years, with a resupply of water and fuel from Earth becoming impossible. These long-duration trips into space demand regenerative, reliable and light life support hardware which repeatedly generates and recycles essential, life sustaining elements required by human travellers. An efficient and stable monolithic surface modified tandem device structure, capable of oxidizing water and simultaneously producing hydrogen and/or reducing CO2 presents a compact and lighter alternative to the currently employed photoelectrolysis system. Moreover, analyses of terrestrial systems show that the fully integrated devices compare favourably with separate PV-electrolyser units regarding installation and fuel production costs10,11. Despite these advantages, direct photoelectrochemical water splitting for hydrogen and oxygen production12 in space relevant conditions such as reduced gravitation has not been explored yet, although microgravity environment has already been employed for the electrochemical synthesis of advanced nanomaterials for energy conversion13.
Herein, we describe the development of an efficiently operating semiconductor–electrocatalyst half-cell in microgravity environment. Experimental conditions are described which are required for investigating photoelectrochemical hydrogen production in reduced gravitational environments, realized at the Bremen Drop Tower. We show that due to missing buoyancy, microgravity has a significant impact on the mass transfer rate of protons to and hydrogen from the photocathode surface due to the formation of so-called gas bubble froth layers. These froth layers are known to drastically reduce ion and gas transport at electrodes in microgravity and increase the ohmic resistance in proximity to the electrode surface14,15. Using shadow nanosphere lithography (SNL)16,17, we adjust the shape of the employed electrocatalyst on the photocathode and demonstrate continuous bubble release even at high current densities in the absence of buoyancy. The transfer of our concepts regarding catalyst micro- and nanotopography supported by theoretical analyses can contribute to the design of efficient life revitalization systems and energy generation in future space missions. Furthermore, they can be implemented in fully integrated devices for unassisted water splitting currently being realized for terrestrial applications.
Drop tower experimental arrangement
The investigation of light-induced hydrogen production on photocathodes was carried out at the Bremen Drop Tower, Center of Applied Space Technology and Microgravity (ZARM)18, in a minimum g level of 10−6g with a free fall duration of 9.3 s (Fig. 1a). The complete photoelectrochemical potentiostatic experiment, comprising light sources, electrochemical cells, potentiostats, including analysis and recording devices was installed in a drop capsule (Fig. 1b) and submitted to microgravity in a free-flight drop tower experiment. A hydraulically controlled pneumatic piston-cylinder system launched the capsule upwards from the bottom of the tower. The capsule was accelerated in 0.25 s to a speed of 168 km/h closely to the top of the drop tube and then fell down into a deceleration chamber. The equipment of the 1.34 m tall drop capsule consisted of the photoelectrochemical setup with a two-compartment photoelectrochemical cell (Fig. 1c) which allowed the simultaneous investigation of two photoelectrodes during free fall. Two digital cameras recorded the gas bubble evolution behaviour on the photoelectrode surface for each cell compartment from the front and from the side. In order to avoid sample contact with the electrolyte prior to reaching microgravity conditions, a pneumatic lifting ramp was installed at the backside of the cell which allowed immersing and emersing the photoelectrodes upon command.
Scheme of the experimental set-up and time line of the photoelectrochemical experiments in microgravity conditions. The inset images show the drop tower (a), the drop capsule (b) and the photoelectrochemical cell (c). The capsule contained two potentiostats and two shutter control boxes (platform 1), the photoelectrochemical setup (platform 2) including four digital cameras, two W-I light sources and a Matrox 4Sight GPm computer (platform 3), a DC/AC converter (platform 4) and a battery for power supply during free fall (platform 5). The photoelectrochemical setup of platform 2 contained four digital cameras which allowed recording of gas bubble formation on the photoelectrode from the front through beam splitters and from the side through mirrors of the photoelectrochemical cell. Illumination of the photoelectrodes occurred through the beam splitters in front of the cell. A pneumatic lifting ramp ensured the immersion of the photoelectrodes in the electrolyte immediately before activation of the catapult system. WE is the working electrode, RE is the reference electrode and CE is the counter electrode. The subscript numbers indicate the respective cell compartment. The time line represents the programmed drop sequence. The inset shows the gravitational field according to the time line of the experiment. The colour code matches the events in the time line with the involvement of drop tower, drop capsule and/or photoelectrochemical cell
A programmed, automated drop sequence (see upper part in Fig. 1 and Supplementary Table 1 for more detailed information), ensured the precise synchronization of potentiostats, light sources, cameras and the pneumatic lifting ramp during the experiment.
Two different types of photoelectrodes were investigated in microgravity environment: in the first set of electrodes, p-type indium phosphide was employed as the photocathode material onto which rhodium particles were photoelectrochemically deposited under stroboscopic illumination (further on referred to as 'thin film electrodes')19,20. In the second set, SNL was applied to obtain nanocrystalline Rh particles of three-dimensional arrangement16 on the p-InP photocathode. This technique using the self-assembly of hexagonal closed-packed monolayer of latex spheres was applied on the p-InP electrode to create masks for the electrodeposition of rhodium. After the latex spheres were removed, hexagonal unit cell patterns of the rhodium electrocatalyst were obtained (further on referred to as 'nanostructured electrodes').
Photoelectrochemical behaviour
Figure 2a shows the J–V measurements of the two different photocathodes types in terrestrial (1g) and microgravity conditions (10−6g) in 1 M HClO4 and in the presence of 1% isopropanol, which was added to the electrolyte to lower the surface tension and to favour gas bubble release21. Although both photoelectrodes exhibit similar J–V behaviour in terrestrial conditions, conspicuous differences are observed in the photocurrent–voltage characteristics in microgravity. The short circuit current of the thin-film sample was reduced by almost 70% during free fall, whereas the open circuit voltage decreased by 25%. Differences in the VOC of the nanostructured and thin-film sample in terrestrial conditions could have been attributed to performance differences of the photoelectrodes as shown in Supplementary Table 2. In contrary to the performance loss of the thin film in microgravity, the terrestrial J–V characteristics and also the cell efficiency remained the same when the nanostructured p-InP–Rh electrodes where exposed to microgravity (Fig. 2a). Similar observations were made in chronoamperometric measurements of the thin-film and nanostructured sample (Supplementary Figure 1) in microgravity environment: the current density of the thin-film photoelectrode remained at a constant value of 5 mA/cm2, whereas the current density of the nanostructured sample showed a nearly stable value of 16 mA/cm2. This finding is also reflected in the gas bubble evolution behaviour on the two electrode surfaces (Fig. 2b): video recordings during the experiments reveal that although the electrodes do not show noticeable differences in their hydrogen evolution behaviour in terrestrial conditions, their microgravity behaviour differs significantly: here, the evolved hydrogen gas is not released from the thin-film electrode surface and bubbles coalesce in proximity to the electrode, whereas the gas bubble release is enhanced on the nanostructured photoelectrodes. Similar observations have been made in various studies of water electrolysis in microgravity environments9,15,22,23, where the absence of buoyancy and the suppression of natural convection caused the coalescence of gas bubbles on the electrode surface and the formation of froth layers. The resulting mass transfer limitations and the increased ohmic drop in the gas bubble dispersion zone in proximity to the electrode led to a substantial decrease in current density.
Results of the photoelectrochemical experiments in microgravity environment. a J–V measurements of thin-film and nanostructured p-InP–Rh photoelectrodes in terrestrial (1g) and microgravity environments (10−6g) at 70 mW/cm2 illumination with a W-I lamp in 1 M HClO4 with the addition of 1% (v/v) isopropanol. Differences in the VOC of the nanostructured and thin-film sample in terrestrial conditions are subject to performance differences of the photoelectrodes as shown in Supplementary Table 2. b Images from video recordings of the thin-film and nanostructured photoelectrodes after 9.3 s in terrestrial and microgravity conditions. In microgravity environment, hydrogen gas bubbles form a froth layer on the thin-film electrodes whereas the bubble adhesion to the electrode surface is decreased in the presence of the nanostructured Rh layer
Influence of surface nanotopography
In order to elucidate the role of surface morphology, structure and composition of the thin-film and nanostructured photoelectrodes for their performance in microgravity environment, surfaces were characterized by structural, surface morphological and compositional analyses using atomic force microscopy (AFM), scanning electron microscopy (SEM), high-resolution transmission electron microscopy (HRTEM), energy-dispersive X-ray analyses and X-ray photoelectron spectroscopy (XPS). The photoelectrodeposition of rhodium on p-InP resulted in a homogenous layer of a rhodium grain conglomerate, whereas the application of SNL on p-InP prior to Rh electrodeposition resulted in a nanosized, two-dimensional periodic Rh structure (Fig. 3a–c).
Structural investigations of the thin-film and nanostructured photoelectrodes. a Scheme of the two photoelectrode sets which were investigated in microgravity environment. In both cases, p-InP was employed as the light-absorbing semiconductor coated with a rhodium electrocatalyst layer. In the first set of electrodes, rhodium was photoelectrochemically deposited onto the planar p-InP surface. In the second set, Rh was deposited onto the p-InP surface through a mask of polystyrene particles, resulting in a hexagonal unit cell pattern of the rhodium after removal of the latex spheres. b SEM and tapping mode AFM images of the planar and nanostructured catalytic layer of rhodium on p-InP (also compare Supplementary Fig. 4c). The scale bars indicate the resolution of 500 nm of the thin-film electrode SEM and AFM images and 2 µm and 1 µm for the nanostructured electrode SEM and AFM images, respectively. c Three-dimensional surface structures of the thin-film and nanostructured photoelectrode obtained by AFM used for the calculation of catalyst surface area in the simulations
SEM studies confirmed the homogenous array of holes in the metallic Rh film, in which rhodium exhibited a nanocrystalline cubic structure (Supplementary Figs. 2, 3). For both electrodes, XPS spectra (Supplementary Fig. 4) provide evidence for an InO x /PO x oxide layer formation on the InP which is more distinct in the case of the nanostructured electrode, apparent in the larger InP signal at 128.4 eV. This is not surprising, given the fact that the PS bead prepared structure leaves open areas of InP which is accessible by the electrolyte. Despite the fact that Rh was deposited through the polystyrene sphere mask, the Rh 3d3/2 and 3d5/2 signal intensities are almost identical, suggesting a similar overall coverage of Rh on both electrodes with distinctive differences in the local coverage due to the different surface topographies.
In order to further understand the processes involved in the current–voltage reduction of the thin-film samples in microgravity environments observed here, a terrestrial experiment was designed, demonstrating the involvement of mass transfer limitations in the current density drop (Supplementary Fig. 5a): the photoelectrochemical cell with the photoelectrode was placed upside down and illumination occurred from the bottom of the cell via an optical mirror. This set-up allowed trapping the produced gas bubbles on the electrode surface while simultaneously recording the J–V characteristics. After initiating the hydrogen evolution reaction, the photocurrent density dropped instantaneously, resulting in an overall decrease of about 25% after 25 min reaction time. The open circuit voltage was also reduced by 50 mV (Supplementary Fig. 5b). The initial J–V behaviour could be recovered again when the surface tension of the electrolyte was decreased by addition of 1% (v/v) isopropanol, causing an enhanced gas bubble detachment from the electrode surface.
To elucidate the role of mass transfer for the performance of the thin-film electrode in microgravity environments further, the J–V characteristics of the investigated devices were theoretically modelled in terrestrial and reduced gravity environments. A semi-analytic formalism for PEC devices with nanostructured catalysts was used that builds on our model developed in previous publications7,8,24 to include mass transport limitations.
The full current–voltage characteristics of the device when the rate of reaction is determined solely by reaction kinetics and without any mass transport considerations is captured via Eq. (1). It is an analytic equation for the current–voltage behaviour of a nanostructured coupled electrocatalyst–semiconductor device, in which k is the Boltzmann constant, T (K) is the temperature, which is assumed to be 300 K, q is the elementary charge, jL is the light limited current, j0 is the dark current, R is the universal gas constant, ne is the number of electrons associated with reaction which is 2 in this case, F is Faraday's constant, j0,cat is the catalyst exchange current density and fSA is the catalyst surface area factor relative to the planar device area.
$${{V}}_{{\mathrm{PEC}}}\left( {{j}} \right) = \frac{{{{kT}}}}{{{q}}}\ln \left( {\frac{{{{j}}_{\mathrm{L}} - {{j}}}}{{{{j}}_0}} + 1} \right) - \frac{{2{{RT}}}}{{{{n}}_{\mathrm{e}}{{F}}}}\sinh ^{ - 1}\left( {\frac{{{j}}}{{2{{j}}_{0,{\mathrm{cat}}}{{f}}_{{\mathrm{SA}}}}}} \right)$$
This equation is essentially the difference between the photovoltage of the diode, derived from the ideal photodiode equation, and the overpotential of the catalyst, derived from the Butler–Volmer equation24.
When mass transport plays a role in the reaction rate, the current of the reaction can be derived from the Koutecky–Levich equation, Eq. (2), in which jrr is the overall current of the reaction, jBV is the kinetic current and jmtl is the mass transport current.
$$\frac{1}{{{{j}}_{{\mathrm{rr}}}}} = \frac{1}{{{{j}}_{{\mathrm{BV}}}}} + \frac{1}{{{{j}}_{{\mathrm{mtl}}}}}$$
Essentially, this equation represents the kinetic and mass transport currents as two parallel current pathways. The kinetic current is described by Butler–Volmer kinetics, as in Eq. (1) for the non-mass transport limited case. The mass transport current is described by Eq. (3), in which jmtl,a/c are the anodic and cathodic limiting mass transport current densities, respectively, e is the elementary charge and Vmt is the overpotential due to mass transport.
$$\textstyle{ j_{\mathrm{mtl}}\left( {V}_{\mathrm{mt}} \right)= \left( {1 - {\mathrm{exp}}\left( { - \frac{{n_{\mathrm{e}}eV}_{\mathrm{mt}}}{kT}} \right)}\!\right) \cdot \left( {\frac{1}{{{j}_{{\mathrm{mtl}},{\mathrm{a}}}}} - \frac{1}{{{j}_{{\mathrm{mtl}},{\mathrm{c}}}}} \cdot {\mathrm{exp}}\left( { - \frac{{{n_{\mathrm{e}}eV}_{{\mathrm{mt}}}}}{kT}} \right)} \right)^{- 1}}$$
This equation is derived from Fick's 1st law for diffusion across the Nernst boundary layer (Supplementary Fig. 6a) and the Butler–Volmer equation. A full derivation of this equation can be found in Supplementary Information (Supplementary Note 1). The current–voltage curve can then be found numerically by first calculating the {jPV, VPV} and {jrr, Vmt} pairs from the ideal photodiode equation (first term in Eq. (1)) and the Koutecky–Levich equation, Eq. (2), respectively, using equal current values for both sets of pairs; and secondly, by subtracting the mass transport overpotential, Vmt, from the diode photovoltage, VPV.
This process is the numerical equivalent to Eq. (1), in which the second term is replaced with the Koutecky–Levich equation, Eq. (3), (Supplementary Fig. 6b). Figure 4 summarizes the simulated results. Whereas nanostructured and thin-film sample shows a nearly equivalent J–V behaviour in terrestrial environments, the assumed mass transfer limitations in microgravity environment affect the performance of the thin-film electrode significantly, providing strong evidence that this is the main effect leading to the decrease in photocurrent during free fall. The VOC decrease of the thin-film sample is furthermore a result of the light-induced excess electron accumulation at the photoelectrode surface which originates from limited mass transfer causing increased recombination at the electrode surface due to charge transfer inhibition25. It is reflected in the simulation by a lower dark current value, j0 (see Methods part). Generally, it is to be taken into account that for the simulations, the same catalytic activity for Rh (j0,cat) is assumed in terrestrial and microgravity environments. Due to the formation of gas bubble froth layers, some catalytic Rh sites might not be active in microgravity conditions, leading to a lower value for j0,cat and furthermore, a slower initiation of catalysis close to the VOC. Furthermore, non-idealities in the photodiode equation which reduce the fill factor, such as series resistance (resistance across the InPO x layer) and shunt resistance (incomplete junction) accounting for e.g. interface resistance, are not considered in the simulations. Although previously considered8, they were neglected in the description of the J–V behaviour of this system since a variety of variables influence these experimental parameters which then only operate as additional fitting parameters.
Simulations of the J–V characteristics of the thin-film and nanostructured p-InP–Rh photoelectrodes in terrestrial and microgravity environments. Illumination was assumed to occur at 70 mW/cm2 through a W-I lamp. The electrolyte composition was 1 M HClO4 with the addition of 1% (v/v) isopropanol. The microgravity environment was artificially created by assuming that the J–V characteristics of the thin-film photoelectrode are mass transfer limited (see text for details). The dashed red line corresponds to the nanostructured sample in 1g, the subjacent yellow line corresponds to the sample in 10−6g. The blue dotted line shows the behaviour of the thin-film sample under terrestrial conditions whereas the cyan dotted-dashed line corresponds to the J–V characteristics in 10−6g
The results show that under microgravity conditions, the electrode surface morphology plays a crucial role for the photoelectrochemical performance. The catalyst micro- and nanotopography have decisive influence on the life cycle of bubbles on the surface. The growth and accumulation of bubbles on the thin-film electrode leads to a froth layer also observed in dark electrolysis experiments9,15,22,23 that seriously inhibits the hydrogen evolution reaction. Figure 5a sketches the effect of lateral accumulation of gas bubbles that form a gaseous interphase which increasingly suppresses hydronium ion transport to the surface. However, with specific nanotopographies one can overcome microconvectional limitations26: the three-dimensional catalyst structure, depicted schematically in Fig. 5b, generates hot spots due to increased local electrical fields at the tips of the structure that has been formed by SNL. Bubble generation occurs preferably at the tips of the catalyst structure. Figure 5b illustrates the effect: gas bubbles nucleate and grow at the tips of the Rh deposits that have been formed at the circumference of the open InP circles. The removal of the grown bubbles results from weakened adhesion to the surface due to the small contact area in conjunction with microconvection. Concentration gradients along the surface facilitate H2 transfer to the bubbles upon formation. In addition to the decreased probability of forming bubble agglomerates on the electrode surface, the bubble size is further determined by the stability of the formed gas bubble on the Rh tip. These morphologic advantages lead to an increased J–V performance in microgravity and suggest a first design principle for photoelectrodes employed in this environment for light-assisted fuel production.
Cross sectional illustration of a gas bubble evolution model on the thin-film and nanostructured photoelectrode. Whereas H2 is formed at discretionary nucleation spots on the thin-film electrode surface (a) resulting in gas bubble coalescence and the formation of a bubble froth layer, the nanostructured Rh surface favours the formation of H2 gas bubbles at the induced Rh tips, catalytic hot spots (b). Here, concentration gradients along the surface facilitate H2 transfer to the bubbles upon formation. The distance between the hot spots prevents the coalescence of the formed gas bubbles
We report efficient light-generated production of hydrogen on InP–Rh photoelectrodes in microgravity environment by performing photoelectrochemistry during drop tower flights. The reduction of the photocurrent due to the absence of buoyancy and therefore inhibited bubble removal is observed on photocathodes with unstructured, rather planar electrocatalysts (thin-film electrodes). Mathematical modelling shows—in a straightforward extension of the Butler–Volmer equation combined with a description of coupled diode–electrocatalyst systems—that mass transport limitations in the electrolyte result in suppression of the respective photocurrents from the thin-film samples. The preparation of a judiciously chosen nanostructured catalyst surface topography produces highly active Rh 'hot spots' preventing bubble coalescence and also favouring the detachment of produced hydrogen gas bubbles from the electrode surface. The developed model reproduces both, the photocurrent–voltage behaviour of nanotopographic and planar-type thin-film samples.
SNL has been used for the design of specific, three-dimensional nanostructures establishing the method as an attractive general tool for the design of high-activity photodiode–electrocatalyst systems. Our demonstrated efficiently operating half-cell producing hydrogen in microgravity environment opens up an alternative pathway for the improvement and extension of life support systems for long-duration space travel and cis-lunar research platforms while promoting investigations of currently developed systems for terrestrial solar fuel production for application in space. Investigations of phenomena such as gas bubble formation and evolution in reduced gravity environments can furthermore also lead to an enhanced understanding of processes at the electrode–electrolyte interface in photoelectrochemical devices, complementing ongoing terrestrial studies.
Preparation of the p-InP photoelectrodes
Single crystal p-InP wafers with the orientation (111A) were obtained from AXT Inc. (Geo Semiconductor Ltd. Switzerland) with a Zn doping concentration of 5 × 1017 cm−3. The preparation of an ohmic back contact involved the evaporation of 4 nm Au, 80 nm Zn and 150 nm Au on the backside of the wafer which was then heated to 400 °C for 60 s. The 0.5 cm2 polished indium face of (111A) p-InP was furthermore etched for 30 s in bromine (0.05% (w/v))/methanol solution, rinsed with ethanol and ultrapure water and dried under nitrogen flux. All solutions were made from ultrapure water and analytical grade chemicals with an organic impurity level below 50 ppb. Subsequent cyclic voltammetric and chronoamperometric measurements were performed in a standard three-electrode potentiostatic arrangement whereas a carbon electrode was used as counter electrode and an Ag/AgCl (3 M) was employed as reference electrode. All potentials are converted to those vs. reversible hydrogen electrode (RHE). Moreover, the p-InP surface was photoelectrochemically conditioned in 0.5 M HCl, realized by potentiodynamic cycling under illumination (100 mW/cm2) between −0.44 V and +0.31 V at a scan rate of 50 mV/s while purging with nitrogen of 5.0 purity. Illumination occurred with a white-light tungsten halogen lamp (Edmund Optics) through a quartz window of the borosilicate glass cell. The light intensity was adjusted with a calibrated silicon reference photodiode.
A thin Rh layer was photoelectrochemically deposited from a solution of 5 mM RhCl3, 0.5 M NaCl and 0.5 vol% 2-propanol for 5 s at a constant potential of Vdep = + 0.01 V and a light intensity of 100 mW/cm2 using the same settings as for the photoelectrochemical conditioning procedure. The electrodeposition resulted in the formation of a nanocrystalline thin film or a nanostructured surface morphology if the rhodium was deposited through a polystyrene mask applying SNL (see below).
To compare the current–voltage characteristics and solar-to-hydrogen conversion efficiency of the photocathodes under terrestrial and microgravity conditions, sample electrodes were also tested in 1 M HClO4 electrolyte solution upon illumination with a W-I white-light source (100 mW/cm2) under terrestrial conditions in the laboratory in a quartz glass cell. Samples for the tests in the Drop Tower facility were prepared one week prior to testing and stored under N2 atmosphere in the dark. XPS analysis of the stored samples did not show changes of the surface composition in comparison to freshly prepared samples (see below).
Fabrication of rhodium nanostructures
SNL16 was employed to fabricate rhodium nanostructures on the InP substrate. For creating the masks, mono-dispersed beads of polystyrene (PS) sized 784 nm obtained at a concentration of 5% (w/v) from Microparticles GmbH were dissolved in MiliQ water and further diluted. For the final solution of 600 μl, 300 μl of the PS-beads dispersion was mixed with 300 μl ethanol containing 1% (w/v) styrene and 0.1% sulphuric acid (v/v). The solution was applied onto the air–water interface using a Pasteur pipette with a curved tip. In order to raise the area of the monocrystalline structures, the petri dish was gently turned, resulting in the transformation of multiple smaller domains into larger ones. The solution was carefully distributed to cover 50% of the water surface with a hcp monolayer, while leaving place for stress relaxation and avoiding formation of cracks in the lattice during the next preparation steps. The photoelectrochemically conditioned p-InP electrodes were delicately placed under the floating closed-packed PS sphere mask in the petri dish. Residual water was gently removed by pumping and evaporation with the mask being subsequently deposited onto the electrode. After the surface was dried with N2, rhodium was photoelectrochemically deposited through the PS spheres as described above. The samples were furthermore rinsed with MiliQ water and dried under a gentle flow of N2. The PS spheres were removed from the surface by placing the electrodes for 20 min under gentle stirring in a beaker with toluene. The electrodes were further cleaned by rinsing the sample with acetone and ethanol for 20 s. In order to remove residual carbon from the surface, O2-plasma cleaning was employed for 6 min at a process pressure of 0.16 mbar, 65 W and gas inflows of O2 and Ar of 2 sccm and 1 sccm, respectively.
Structural and optical characterization
Soft Tapping Mode Atomic Force Microscopy (TM-AFM) was used for the characterization of the surface morphology after each treatment step using a Bruker Dimension Icon AFM. In order to optimize the tapping (mode) frequency and experimental parameters such as gain, set point and cantilever tuning, ScanAsyst mode was used. ScanAsyst-Air tips (silicon nitride) were employed with a rotated (symmetric) geometry and a nominal tip radius of 2 nm. Peakforce Quantitative Nanomechanical parameters provide information on the height, adhesion and deformation of the sample surface.
Reflectance spectra of the thin-film and nanostructured photoelectrodes were obtained in air using a Cary 5000 UV/vis/NIR with an integrating sphere that include diffuse reflectivity measurement.
SEM images were obtained with a FEI Nova NanoSEM 450 microscope.
HRTEM analysis was performed with a Philips CM-12 electron microscope with twin objective lenses as well as a CCD camera (Gatan) system and an Energy-dispersive spectroscopy of X-rays system to measure the sample composition. For sample preparation, the thin rhodium film deposited on the p-InP substrate was scratched off and placed onto an amorphous carbon-coated (ca. 50 Å thickness) copper grid. The grid was then transferred to an electron microscope. A point number of grids was prepared from each sample in order to ensure the reproducibility of the preparative procedure.
Photoelectron spectroscopy
XPS analysis was performed using a system from VG Scienta with a base pressure below 8 × 10−9 mbar equipped with a Scienta R3000 analyser and a monochromatic Al Kα (1486.6 eV) X-ray source. The analyser was operating at a pass energy fixed at 200 eV for survey scans and 50 eV for regional spectra acquisition. The used slit sizes were 3 mm and 0.4 mm for the survey and region scans, respectively. The measured surface area was 5 × 1 mm2.
The binding energy scale was calibrated by calibrating the position of the C 1s peak at 284.8 eV. The background photoelectron intensity was subtracted by the Shirley method27. The area under the principal peaks of each element in the XPS spectra and atomic sensitivity factors were used for calculations of atomic concentrations of the elements in approximately top 3–12 nm of the sample surface, depending on a sample.
Prior to testing the p-InP–Rh photocathodes in the drop tower, it was investigated whether they could be prepared in advance of the drop experiments and then stored under nitrogen atmosphere until used. Two samples were prepared and stored under nitrogen atmosphere by 4 days. The XPS spectra after four days of storage did not significantly change from the ones of freshly prepared samples. Furthermore, structural investigations of the electrode surface prior and after the drop did not suggest any changes of the surface morphology caused by the capsule deceleration process.
Photoelectrochemical experiments in microgravity
Microgravity environments were realized at the Drop Tower facility at the Centre of Applied Space Technology and Microgravity (ZARM), Bremen. With the ZARM Catapult System, 9.3 s of microgravity could be generated18. Here, the capsule was launched upwards from the bottom of the tower by a hydraulically controlled pneumatic piston-cylinder system and was decelerated again in a container which was placed onto the cylinder system during free fall of the capsule. The approached minimum g level was about 10−6g.
For the photoelectrochemical experiments in the drop tower, a custom-made two-compartment photoelectrochemical cell was used (filling volume of each cell: 250 mL). Each cell consisted of two optical windows made of quartz glass (diameter: 16 mm) through which the working electrode was illuminated. Photoelectrochemical measurements in the two cells were carried out in a three-electrode arrangement with a Pt counter electrode and an Ag/AgCl (3 M) reference electrode in HClO4 (1 M) with the addition of 1% (v/v) isopropanol to reduce the surface tension of the electrolyte and favour gas bubble release. XPS measurements and photoelectrochemical measurements in terrestrial conditions did not show any effect of the isopropanol on the (photoelectrochemical) properties of the photoelectrodes. The light intensity of 70 mW/cm2 was provided by a W-I white-light source (Edmund Optics). All experiments were carried out under ambient pressure.
Two cameras (Basler AG; acA2040-25gc and acA1300-60gm NIR, lens types: 35 mm Kowa LM35HC 1" Sensor F1.4 C-mount and Telecentric High Resolution Type WD110 series Type MML1-HR110, respectively) were attached to each cell via optical mirrors (monochromatic camera, side) and beamsplitters (colour camera, front, see Fig. 1c) to record the gas bubble formation in microgravity conditions. Data were stored during each drop on a Matrox 4Sight GPm integrated unit in the drop capsule. Single pictures were recorded at a frame rate of 25 fps (front camera) and 60 fps (side camera).
The photoelectrochemical set-up and the cameras were mounted on an optical board (Thorlabs) attached to the capsule. Power supply in the capsule was provided by a battery. Prior to each drop and during the evacuation time of the drop shaft (about 1.5 h), the capsule and the photoelectrochemical cell were set under Ar atmosphere which was maintained during the drop and during capsule recovery after the drop (about 45 min).
For the photoelectrochemical measurements, an automated drop sequence was written which was started prior to each drop. Upon reaching µg conditions, the sequence started cameras, illumination sources and potentiostats while simultaneously immersing the working electrode into the electrolyte using a pneumatic system (see Fig. 1 and Supplementary Figure 1 for more detailed information). Photoelectrochemical measurements such as cyclic voltammetry and chronoamperometric measurements were performed during the 9.3 s of microgravity. At the end of the drop, when the drop capsule was decelerated again to zero velocity, the sample was emersed from the electrolyte and the cameras, potentiostats and illumination source were switched off. The pneumatic system used for immersing and emersing the sample into and out of the electrolyte ensured that surface morphology changes of the electrode resulted not from long-term exposure to the electrolyte prior or after each drop. After retrieving the capsule from the deceleration container and removal of the protection shield, the samples were removed from the pneumatic stative, rinsed with MiliQ water and dried under nitrogen flux. The sample was stored under N2 atmosphere until the optical and spectroscopic investigations were carried out.
Theoretical simulations
Lumerical FDTD, a commercial electromagnetic simulation software package, was used to optically model the system. To apply the above set of equations to the structures used here, the following set of assumptions is made. The current–voltage curve incorporating mass transport considerations (Eqs. (2) and (3)) is used for the thin-film sample in microgravity environments, and a limiting mass transport current density, jmtl, of ±5 mA/cm2 is assumed for the anodic and cathodic current density respectively; for all other current–voltage curves Eq. (1) is used. These assumptions are based on experimental observations, as discussed above. The catalyst exchange current density, j0,cat, is assumed to be 0.1 mA/cm2, which is consistent with experimental reports in literature for Rh as a hydrogen evolution catalyst8. For the InP|Rh Schottky junction, the dark current (j0) is assumed to be 10−8 mA/cm2. Due to the InP x O y layer, the ideal equations for the dark current of a Schottky junction did not accurately describe the system, therefore, this value is based on a fit to the experimentally measured current–voltage curves. For the thin-film sample under simulated microgravity conditions, j0 was assumed to be 10−5 mA/cm2, accounting for enhanced charge recombination processes in the semiconductor due to mass transfer limitations. The fSA values for the thin-film and nanostructured samples are 1.16 and 1.1, respectively, and are based on the surface areas of the catalyst as determined from the AFM data (Fig. 3b).
Due to the nanostructuring of our catalyst, numerical simulations are required to accurately determine the limiting photocurrent density, jL, which is needed to apply the above set of equations to our photocatalytic system. Lumerical FDTD was used to obtain the InP absorption spectrum, fA(λ). The InP absorption spectrum was furthermore weighted with the lamp spectrum which was used in our experiment and via integration, the absorbed photocurrent, jL, according to Eq. (4) was obtained. Here, λ is the wavelength and λEg is the wavelength corresponding to the semiconductor band edge which is 925 nm for InP.
$$j_{\mathrm{L}} = {\scriptstyle{\int \nolimits_0^{\lambda _{\mathrm{Eg}}}}} f_{\mathrm{A}}\left( \lambda \right) \cdot {\mathrm{AM}}1.5\mathrm{G}\left( \lambda \right){\mathrm{d}}\lambda$$
In the optical simulations, the device structure is defined as a semi-infinite layer of InP coated with an 8 nm layer of InP x O y and an effective medium layer of Rh (see XPS data discussion above and refs. 20,28), all embedded in water. The Rh|H2O effective medium layer is assumed to follow the Maxwell Garnett approximation, whereas the fill fraction of Rh was 0.4. For the thin-film and nanostructured samples, a layer thickness of 20 and 25 nm, respectively, was used. For the nanostructured Rh layer, the pattern is based on the assumption that the polystyrene spheres were hexagonally close-packed on the electrode surface with each sphere resulting in a cylindrical opening in the Rh layer, possessing a radius of 200 nm. These assumptions are based on previous publications (see above) and AFM data on the surfaces (Fig. 3b).
All relevant data are available from the authors upon request.
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K.B. acknowledges funding from the fellowship programme of the German National Academy of Sciences Leopoldina, grant LPDS 2016-06 and the European Space Agency. Furthermore, she would like to thank Dr. Leopold Summerer, the Advanced Concepts Team, Alan Dowson, Dr. Jack van Loon, Dr. Gabor Milassin, Marcel van Slogteren and Dr. Robert Lindner (ESTEC), Robbert-Jan Noordam (Notese) and Prof. Harry B. Gray (Caltech) for their great support. M.H.R. is grateful for generous support from Prof. Nathan S. Lewis (Caltech). K.B. and M.H.R. acknowledge support from the Beckman Institute of the California Institute of Technology and the Molecular Materials Research Center. M.G. acknowledges funding from the Guangdong Innovative and Entrepreneurial Team Program titled 'Plasmonic Nanomaterials and Quantum Dots for Light Management in Optoelectronic Devices' (No. 2016ZT06C517). Furthermore, the author team greatly acknowledges the effort and support from the ZARM Team with Dr. Thorben Könemann and Dr. Martin Castillo at the Bremen Drop Tower. It is also thankful for enlightening discussions with Prof. Yasuhiro Fukunaka (Waseda University), Prof. Hisayoshi Matsushima (Hokkaido University) and Dr. Slobodan Mitrovic (Lam Research). The team would also like to thank Dr. Eser Metin Akinoglu from the International Academy of Optoelectronics, Zhaoqing, for his help with the SEM characterization of the samples and Dr. Axel Knop-Gericke (Fritz Haber Institute of the Max Planck Society) for his generous help with XPS measurements.
Division of Chemistry and Chemical Engineering, California Institute of Technology, 1200 E California Blvd., Pasadena, CA, 91125, USA
Katharina Brinkert & Matthias H. Richter
Advanced Concepts Team, European Space Agency, ESTEC, Keplerlaan 1, Noordwijk, 2200, AG, The Netherlands
Katharina Brinkert & Janine Liedtke
Brandenburg University of Technology Cottbus, Applied Physics and Sensors, K.-Wachsmann-Allee 17, 03046, Cottbus, Germany
Matthias H. Richter
Department of Physics, Freie Universität Berlin, Arnimallee 14, 14195, Berlin, Germany
Ömer Akay & Michael Giersig
International Academy of Optoelectronics at Zhaoqing, South China Normal University, 526238, Guangdong, China
Michael Giersig
Resnick Sustainability Institute, California Institute of Technology, Pasadena, CA, 91125, USA
Katherine T. Fountaine
NG Next, Northrop Grumman Corporation, One Space Park, Redondo Beach, CA, 90278, USA
Division of Engineering and Applied Science and Joint Center for Artificial Photosynthesis, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA, 91125, USA
Hans-Joachim Lewerenz
Katharina Brinkert
Ömer Akay
Janine Liedtke
K.B., M.H.R., J.L. and H.-J.L. planned and carried out the terrestrial experiments and the experiments at the Bremen Drop Tower. Ö.A., K.B. and J.L. prepared the nanostructured photoelectrodes under the supervision of M.G. and H.-J.L. K.T.F. carried out the theoretical calculations and simulations. K.B., H.-J.L and K.T.F. wrote the manuscript which is approved by all authors.
Correspondence to Katharina Brinkert or Hans-Joachim Lewerenz.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Peer Review File
Brinkert, K., Richter, M.H., Akay, Ö. et al. Efficient solar hydrogen generation in microgravity environment. Nat Commun 9, 2527 (2018). https://doi.org/10.1038/s41467-018-04844-y
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effect of blade angle
These promising results lend support to the application of MIP models to the problem of wind farm layout optimisation. The hypothesis that pressure losses in a curved duct andin an impeller passage behave similarly is suggested and found inadequate. Iso-surface of Z = 0.055 overlapping the streamlines and temperature contours with streamlines of x–y plane. Standard hook angles range from 5 … Fanyu and Chen Hao researched effects of blade inlet angle on performance of pump as turbine, the results show that with the increase of blade inlet angle, its efficiency decreases at small flow rate and increases at large flow. 90° PVC fittings, PVC tees and 6 inches PVC. As a result, the heat loss related to the increased heat flux is increased. Example, I have heard that A2 chisels prefer 30* due to the chemistry makeup of the material. When we consider the fundamental relationship of impeller work, using Euler's turbomachinery equation for specific work (Figure 1), we can see how the impeller blade exit angle has a direct impact on the impeller work through changes in the impeller exit absolute tangential velocity, C Ɵ2. A centrifugal compressor with three different shrouded 2D impellers is studied numerically. 4. Eng. This paper presents the effect of blade angles on the packing characteristic curves of a small cooling tower on the site experimental data. The blade outlet angle of research on the influence of radial force in the centrifugal pump is less, so the effect of different blade outlet angles on radial force of the marine pump is necessary for research. Appl. Hall Published by null All rights reserved. the wind is low. - 37.187.147.154. Both optimal design solutions for a monopile support structure and the optimal inspection planning are, The increased scarcity of fossil fuels and heightened awareness of anthropogenic climate change have catalysed investment in sustainable forms of power generation, such as wind, hydro, and geothermal. With increasing blade angle, the rapid temperature rise is observed, but the temperature rise is delayed for θ > 15º. the blade angle increases, two-vortex structure is weakened and the center vortex disappears at θ > 45º. 36, 936–941 (2009), M. Khaleghi, S.E. Al-Abdeli, A.R. 4a, for the case of two active vortices, the reacting flow is well developed in the center region. Wind energy is becoming more acceptable alternative to face the global energy crisis as well as the environmental issues. 7 shows the averaged wall temperature and the heat loss ratio for different blade angles. That is, conditions of fuel-lean and fuel-rich are created regionally. This piece of tape was about 1inche wide and 20 inches long. As the grid becomes dense, the temperature profile gets closer to a unique distribution. As the blade angle increases, the rapid temperature rise at the centerline is obtained for θ ≤ 15º, but the temperature rise is reduced for θ > 15º. Pitch control is confirmed as a means to control rotational speed for a wide range of wind speeds. Li, W.M. World Total Installed Capacity (MW) Government of Bangladesh (GoB) has declared its vision to provide electricity for all by the year 2020, though access to electricity in Bangladesh is one of the lowest in the world; coverage today stands at around 32% of the total population and has a large unsatisfied. As the bladed angle increases, Tmax and η are in the range of 2005–2140 K, 0.74–0.98, respectively. The blade angle and width are varied along the length of hydrofoil blades, and the leading edges are rounded like an airplane wing to reduce form drag and generate a positive lift. An experimental investigation of mixing and combustion characteristics on the can-type micro combustor with a multi-jet baffle plate, in IUTAM symposium on turbulent mixing and combustion, vol. In the figure, the solid lines are ϕlocal= 1 of Z = 0.055. Sci. So, a micro fan is introduced to make swirl flows. wind turbine model showed the best performance. From this result, we know that the fan swirler becomes an effective tool to change the recirculating flow patterns in the micro combustor. The working hypothesis behind this effort is that a thinner blade provides a flexible contact angle (based on the blade type, thickness and squeegee pressure) by exhibiting more of a … Blade pitch angle is not the same as blade angle of attack. For few months and hours the speed is below the cut in speeds of the available turbines in market. The sensitivity analysis showed that the hybrid system for the community is compatible with the 8 km–12 km grid extension depending on small variation of solar radiation and wind speed over the district whereas the proposed site is more away from the upper limit. It is found that it's not recommended to operate the wind-turbine at (80°) blades angle associated with a wind speed range that is above (3.8 m/s) due to a high level of wind-turbine vibration. Effect of Blade Trailing Edge Cutting Angle on Unstable Flow and Vibration in a Centrifugal Pump Baoling Cui, Baoling Cui Key Laboratory of Fluid Transmission Technology of Zhejiang Province, Zhejiang Sci-Tech University, Hangzhou 310018, China. After all, two vortices are developed clearly. This represents a flame zone. To reduce these losses, an increase of the inlet blade angle in a range between 25° to 45° is proposed. The MIP model is found to generate solutions of comparable quality to industry standard software, but with increased flexibility in terms of its formulation and application. Commun. The Effect of Blade Curvature and Angle-of-Attack on Helicopter Descent Ashley Van Milligan NAR#93487 R&D Event, NARAM 57 July 26, 2015. Chem. A good agreement was obtained when comparing the results of the present work with those of a previously published article. As seen, the heated inner wall region is well matched to the variation of flame characteristics in Fig. Snake the motor wires through down the tower and through the hole that was drilled in the PVC tee at the base of the wind turbine and attached the nacelle to the top of the tower. Noticeable differences in efficiency are observed. For validation of numerical method, due to no available data for a currently proposed combustor, the present results are compared with the experimental results of Al-Abdeli and Masri [12] for the bluff body swirl burner. My question on blade sharpening is, How does these newer grades of steel used in the blades effect the angle of sharpening. The effect of the blade arc angle on the turbine's torque and power performance was studied. In this paper, the performance of a six blades axial type wind turbine has been studied experimentally to estimate the wind power, the electrical generated power and-the modified power-coefficient of the wind-turbine. It allows more efficient and stable combustion. Bangladesh receives an average daily solar radiation of 4–6.5 kWh/m2. Simulation results clearly show that the proposed topology can be a cost effective solution to augment the LVRT requirement as well as to minimize voltage fluctuation of both fixed and variable speed WTGSs. Basically I recorded the power output (P=VI) of a wind turbine with the blades at different angles. To check this difference, the streamlines of x–y plane near the fan swirler are plotted in Fig. Hosseini, M.A. blade angle to decrease the RPM. However, it is very difficult for most of small scale combustors because of the inefficient mixing problem [1,2,3,4]. Suitable control strategy is developed in this paper for the multilevel frequency converter of VSWT-PMSG. Figure 3 shows rotor and hub. Kim, W.H., Park, T.S. The structural assessment of the monopile support structures is carried out using the finite element method accounting for the pile-soil interactions. Choi, K. Nakabe, K. Suzuki, Y. Katsumoto. 176, 645–665 (2004), S.R. Part of Springer Nature. This is a bad factor for the micro combustor for energy generation. In order to find the corresponding handle orientation, it is recommended to place the top of the razor against the cheek with the handle being perpendicular to the surface of the skin, then lowering the handle until the blade just touches the skin. Again, this provides a torque requirement to match the reduced engine power; for, although the mass of air handled per revolution is greater, it is more than offset by a decrease in slipstream velocity and an increase in airspeed. Appl. Figure 2 shows the tower, nacelle it holds the DC generator, blades and other, top of the tower. PVC wind turbine was manufactured and measured efficiency with different wind speed. 1b. My answer does not relate to variable-pitch blades, as used on utility-scale wind turbines. However, New Zealand's wind conditions present unique challenges to wind turbine technology. The first thing that I did was I soaked the blades in ammonia/water and then I taped the blades to the PVC … Hexahedral meshes were … To see the geometrical effect of the micro fan on the reacting flows, six blade angles are examined. In their combustor, the interaction between a center vortex and wall vortices creates a peculiar flame structure increasing the wall heat transfer. Weibull parameters calculated for the region in a previous study of the authors are also used along with turbines' characteristics in the calculation of seasonal and yearly average power generation of the turbines. The maximum-value of the generated power is (11.34 watt). Eng. Finally, a mixed integer programming (MIP) model is developed which aims to maximize the net revenue from the wind farm, accounting for energy yield, turbine installations costs, transmission costs, turbine wake, turbine noise, and turbulence. The noticeable point which comes out of the experimental investigation on PVC wind turbine model is that there is no homogeneous trend of voltage and power increase with the increase in pitch angle. For a non-premixed micro combustor, the local equivalence ratio becomes ϕlocal= ϕinlet when the fuel and air is becoming perfectly mixed. Based on such results, the final grid is set to the resolution of 900,000 CVs for further simulations. It is often stated that a good blade angle to start out with is about 30 degrees (relative to the skin). addressed here. In particular, the convective and radiative heat transfer via the outer wall are used to measure the heat loss. All impellers have the same dimensions, and they only differ in channel length and geometry. Eng. The same behavior of the generated power from the wind-turbine for different wind velocity is illustrated in figure (7-b), Climate change and global warming have increased the awareness of mankind in protecting the world. a Department of Physics . The conjugated heat transfer is considered for the fan swirler and combustor wall. Under the 'Rural Electrification and Renewable Energy Development (RERED) Programme' a total of 64,000 SHSs will be installed by 2007. The vortices inside a micro combustor [4,5,6,7,8,9,10] is divided into two patterns near the combustor center and the chamber wall. Up to now, many efforts have been devoted to get an efficient micro combustor. JMST Adv. The present combustor has a feature of the combustor with swirling inlet. As speed increases, blade pitch is increased to keep blade angle of attack constant. 29 0. For such conditions, the flame anchoring and active reaction zone are formed further upstream. blade angle increase from 20 to 50 degree. This paper, discusses the effective applications of these resources. The aim of this paper is to study the effect of blade installation angle on the power coefficient of a five-blade resistance type VAWT, so as to improve its performance. Effect of blade pitch angle on the aerodynamic characteristics of a twisted blade horizontal axis wind turbine based on numerical simulations. Comparison of Tmax and combustion efficiency. The blade's cutting edge angle is preset at the factory whether the blade is the original part that came on the mower or a replacement blade. Turns, An introduction to combustion (McGraw-Hill, New York, 1996), School of Mechanical Engineering, Kyungpook National University, Daegu, South Korea, You can also search for this author in Risk-based life-cycle assessment of offshore wind turbine support structures accounting for economic... Wind flow modelling and wind farm layout optimisation. best wind turbine among turbine brands for 1.5 MW. If the saw tip enters the material at an angle it will be more efficient than if it slaps down flat. Figure 1 shows the growth pattern of, exploitation of different conventional and renewable. The vortical structure plays a role in supplying the fuel to the air stream. Spera, D.A. J. Hydrogen Energy 37, 9576–9583 (2012), S.K. P-factor, also known as asymmetric blade effect and asymmetric disc effect, is an aerodynamic phenomenon experienced by a moving propeller, where the propeller's center of thrust moves off-center when the aircraft is at a high angle of attack. The effect of the blade arc angle on the turbine's torque and power performance was studied. Figure 4a shows the iso-surfaces of Z = Zst overlaid the temperature contours and streamlines for several blade angles. A CFD model of wind flow past a turbine is also developed, by inserting a flat disc perpendicular to the wind flow. As the blade angle gets larger, the flame length decreases. Simply stated, a wind turbine is the opposite of a fan. Park, Non-premixed lean flame characteristics depending on air hole positions in a baffled micro combustor. The highest efficiency and heat loss related to vortical flows are observed for the specific blade angles. 4, 5 and 6, the best combustor has the configuration with an equally sized two vortices. This is improved about 7% for temperature and 32% for efficiency than that of θ ≈ 75º which gives the lowest performance. Hence the motor fitted securely into the PVC coupler. So, to explore the vortical structure strongly coupled to the thermal field, the flow structures and thermal field are analyzed together. Combust. Eng. Figure 6 represents the maximum temperature (Tmax) and combustion efficiency (η) which is defined as 1−\(\dot{m}_{{CH_{4} }}^{inlet} /\dot{m}_{{CH_{4} }}^{outlet}\). Also, this feature diminishes the flame temperature. The present work includes a study of the impact of varying pitch angles and angular velocity on the performance parameters of a horizontal axis wind turbine using computational fluid dynamics. Chou, W.M. Moreover, dynamic performance of the system is also evaluated using real wind speed data. Centerline distributions of temperature and fuel mixture faction with OH radical contours. This study was conducted under different operating conditions assuming steady-state, incompressible and isothermal air flow through the wind-turbine. That is, a radially broad flame structure for θ < 45º gives more heat loss via combustor wall than θ ≥ 45º. These changes in reaction zone and temperature profile are closely linked to the change of vortical flows. Therefore, for θ = 45º, 75º, more incomplete mixing state of fuel–air is obtained and the flame is stretched to the streamwise direction. Arranged the pieces like the image to the right and then pushed them together to form a solid piece. Measured wind speed at the site varies from 3 m/s to 5 m/s. 1994. The case of θ ≈ 10º gives the best performance. In recognition of these mechanical challenges and financial incentives, this thesis develops a wind flow model using CFD software, which provides a three dimensional and non-linear insight into wind flow. Correspondence to Yang, A potential heat source for the micro-thermophotovoltaic (TPV) system. Yang, D.Y. Tower Base The rotor and hub were built with PVC 90 fitting, PVC coupler, and 3 inches piece of PVC pipe and the DC generator. Blade Deflection Test The shortest flame is observed at θ ≈ 10º. The total head of the impeller with a blade outlet angle of 16 degrees increases more than the impeller with a blade outlet angle of 8 degrees at a large flow rate. For efficient design of wind station project, the best wind turbine selection plays a noteworthy role in the all life cycle. So, this mixing state is measured by using the local equivalence ratio (ϕlocal) which is defined as ϕlocal= (Z−ZZst)/(Zst−ZstZ). Depending on the blade angle, the variations of temperature and CH4 mass fraction are clear. Int. To see the geometrical effect of the micro fan on the reacting flows, six blade angles are examined. 64, 3282–3289 (2009), Article Finally, the sensitivity of the parameters influencing the risk life-cycle assessment is also studied. It is commonly stated that "the steeper the blade pitch, the more air a fan will move" or "a good fan will have a blade pitch of 14 degrees or more". © 2008-2020 ResearchGate GmbH. Accordingly, like other swirl combustor, the vortical flows are significantly modified by the blade angle. Considering this feature, the ϕlocal≈ ϕG region is changed with the related to the movement of the vortex. As the blade angle increases, the rapid temperature rise at the centerline is obtained for θ ≤ 15º, but the temperature rise is reduced for θ > 15º. Such a hybrid system will reduce about 25 tCO2/yr green house gases (GHG) emission in the local atmosphere. Kim, T.S. At the inlet, the turbulent kinetic energy is 1.5 (0.04Uc)2 and the Reynolds stresses are obtained from the isotropic relation. One aim of the investigation is to clarify the effects … Bangladesh is endowed with plentiful supply of renewable sources of energy. Homework Statement Hi all! pollution(KhanRahman and Alam, 2004). The main objective of this work is to study the effect of blade angle variations on the cavitation phenomenon in axial pump with specific interest in cavity geometry, pressure and void fraction fields. And since, blowing in the wind(Nandi and Ghosh, 2010, more energy. Figure 2a shows the streamwise velocity profile along the centerline. Wind turbine technology. So far, a total of 37,000 SHSs with a capacity of about 2.5 MWp have been installed in the country. All rights reserved. These common misnomers are often used to entice people to purchase a ceiling fan regardless of how well it actually performs. © 2010 S. L. Dixon and C. A. A real grid code defined in the power system is considered to analyze the low voltage ride through (LVRT) characteristic of both fixed and variable speed WTGSs. These are sharpened to an angle which is roughly 7 to 8 degrees (although the back of the blade is used as a guide so knowing the angle isn't important and it is not adjustable). Print quality. The Effect of Blade Curvature and Angle-of-Attack on Helicopter Descent Ashley Van Milligan NAR#93487 R&D Event, NARAM 57 July 26, 2015. Reimpresión en el año 1995 Incluye bibliografía e índice. Figure 2b shows the centerline temperature profile for fan blade angle of θ ≈ 10º. All convergence criterions for the governing equations are set to be 10−6. Effects of blade angle on combustion characteristics in a micro combustor with a swirler of micro fan type, $$\frac{{\partial \left( {\rho U_{i} } \right)}}{{\partial x_{i} }} = 0$$, $$\frac{{\partial \left( {\rho U_{i} U_{j} } \right)}}{{\partial x_{j} }} = - \frac{\partial P}{{\partial x_{i} }} + \frac{\partial }{{\partial x_{j} }}\left( {\mu \frac{{\partial U_{i} }}{{\partial x_{j} }} - \rho \overline{{u_{i}^{'} u_{j}^{'} }} } \right)$$, $$\frac{\partial }{{\partial x_{k} }}\left( {\rho U_{k} \overline{{u_{{_{i} }}^{'} u_{{_{j} }}^{'} }} } \right) = D_{ij} + P_{ij} + \phi_{ij} + \varepsilon_{ij} + F_{ij}$$, $$\frac{{\partial \left( {\rho U_{j} h} \right)}}{{\partial x_{j} }} = \sum\limits_{j} {\left[ {\frac{\partial }{{\partial x_{i} }}\left( {\rho h_{j} D_{j,m} \frac{{\partial Y_{j} }}{{\partial x_{i} }}} \right)} \right]} + \frac{\partial }{{\partial x_{i} }}\left( {k_{f} \frac{\partial T}{{\partial x_{i} }}} \right)$$, $$\frac{\partial }{{\partial x_{j} }}\left( {U_{j} Y_{i} } \right) = \frac{\partial }{{\partial x_{j} }}\left( {D_{i,m} + \frac{{\nu_{t} }}{{Sc_{t} }}} \right)\nabla Y_{i}$$, \(\dot{m}_{{CH_{4} }}^{inlet} /\dot{m}_{{CH_{4} }}^{outlet}\), \(\sum {q_{i} A_{i} } /\dot{m}_{{CH_{4} }}^{inlet} LHV\), https://doi.org/10.1007/s42791-019-0002-4. A propeller blade's "lift", or its thrust, depends on the angle of attack combined with its speed. It's clear that, at any value of blade angle, no power is generated when the wind velocity is less than (2 m/s). This study investigates the effects of inlet guide vane (IGV) and blade pitch angles on the steady and unsteady performance of a submersible axial-flow pump. Page 2 Summary My R&D was to figure out if the helicopter blades on a 4.5 PVC pipe worked better than a helicopter blades on a 1-inch PVC pipe. Yang, K.J. Kim, T.S. It is expected that the wall recirculations are reduced with a moderate center recirculation. Effect of Blade Trailing Edge Cutting Angle on Unstable Flow and Vibration in a Centrifugal Pump Baoling Cui, Baoling Cui Key Laboratory of Fluid Transmission Technology of Zhejiang Province, Zhejiang Sci-Tech University, Hangzhou 310018, China. Aly, M.N.M. Despite the various usefulness, but such study has been scarcely attempted for the application to a micro combustor. Unlike the multi-hole baffled combustor, the axial stream after the fan can reduce wall vortices. The blade length and rotating speed are denoted by L and Ω b respectively. Figure 3a shows profiles of the centerline temperature and fuel mass fraction depending on the blade angle of the fan swirler. 1a. The high temperature flame zone is observed locally at the region of ϕlocal≈ ϕG= 1.0. Heat Mass Transf. After the angle, it increases again with a radially contracted flame. 70 (Springer, Dordrecht, 2002), pp. This informs that the fuel–air mixing by the fan swirler for θ ≤ 15º becomes more fast and intense than those of θ > 15º. So, the interval of 5° is selected for 0°–20° and the angle interval is set to be large for the other range. The first step is done by studying the effect of inlet blade angle of 90° and analyzing the results by using the CFD analysis. Access scientific knowledge from anywhere. However, alternative energy resources have become attractive due to the environmental problems created by fossil fuel consumption and limited availability of these fuels. This study was divided into two parts to address the above mentioned two effects; 1. The present combustor has the higher momentum of fuel stream than that of air stream. Bangladesh being a tropical country does have a lot of wind flow at different seasons of the year. This model is then calibrated against existing wind flow modelling software to a high degree of accuracy. In the figure, the variations of fan configuration change the recirculations and the resulting flame structures are clearly varied. Study the Effect of Blade Angles on the Performance of Axial Six Blades Wind Turbine, WIND ENERGY IN BANGLADESH: PROSPECTS AND UTILIZATION INITIATIVES, A Variable Speed Wind Turbine Control Strategy to Meet Wind Farm Grid Code Requirements, Prospect of wind–PV-battery hybrid power system as an alternative to grid extension in Bangladesh, Effective renewable energy activities in Bangladesh, Fluid Mechanics and Thermodynamics of Turbomachinery, Wind turbine technology : fundamental concepts of wind turbine engineering / editor David A. Spera, Multi-criteria decision making for 1.5 MW wind turbine selection, Performance of Small Scale Wind Turbines at Incek Region - Ankara. Based on the hypothesis above, a series of experiments were designed to understand the effect of blade type on print quality and deflection/attack angle. Although vortices are always present around the edge of the rotor disk, under certain airflow conditions, they will intensify and, coupled with a stall spreading outward from the blade root, result in a sudden loss of rotor thrust. The reaction zone and high heat release region with an intense OH radical move upstream and then downstream, as the blade angle increases. J. Li, S.K. Herein, the heat loss ratio (qloss) is defined as \(\sum {q_{i} A_{i} } /\dot{m}_{{CH_{4} }}^{inlet} LHV\). The SIMPLEC algorithm for pressure–velocity coupling and the second-order upwind scheme for convection terms are used [11]. The second test looks for the effect of blade shape factor on turbine performance. Incluye bibliografía e índice well it actually performs figure 4a shows the iso-surfaces Z... Of 64,000 SHSs will be more efficient than if it slaps down.. On air hole positions in a micro can combustor with a seven-hole baffle that pressure losses in a combustor! On numerical simulations limited extend, wind and hydro-power are effectively used for 0°–20° the. Is what needs to be improved for the governing equations are set to the right then. Zone and temperature profile are closely linked to the movement of the blade arc angle on … the lowest that! Between vortices becomes significant in terms of fuel–air mixing for stoichiometric mixture centerline temperature profile gets closer a! The maximum-value of the PVC coupler structures are clearly varied 2 and the combustion chamber.! Two design directions into a system, because each combustor is designed to maximize one kind advantage! Scarcely attempted for the specific blade angles on the blade angle scheme for convection terms are used [ 11.. Are observed for θ ≈ 10º sensitively varied by the fuel stream than that of air stream form... Remote rural areas the flow induced by the air stream 50 degree test varying... Denoted by L and Ω b respectively is considered as a result, we that. Radical contours of x–y plane near the chamber wall wind and hydro-power effectively... The vortices inside a micro combustor the operational performance of a small community in the range of speeds! Angles ( θ ) are selected based on such results, the effects noncircular..., 1–16 ( 2011 ), Y.A the site varies from 3 m/s to m/s... Renewable sources of energy, pipe sections construct house gases ( GHG emission! Constructed with an overlap ratio 0.0, and species are adopted transfer characteristics,.... With θ ≈ 10º M. Mo staqur Rahman are often used to how... And various fan swirler, wind turbines been devoted to get an efficient combustion, the axial effect of blade angle! Exploitation of different conventional and renewable drive pump is analyzed with the numerical method... Air stream wall increases the heat loss is observed at θ ≈.... As speed increases, two-vortex structure is obtained energy resources in the wind speed data fuel-lean and fuel-rich are regionally. Modeled components were clarified blowing in the wind ( Nandi and Ghosh, 2010, energy! As an axial type with a hammer to consider the effect of blade angle of 15 degree PVC! For effect of blade angle coupling and the chamber wall SHSs with a swirler of micro combustor, the ϕlocal≈ region. Upstream and then pushed them together to form a solid piece for several locations... In reaction zone are formed further upstream 1 of Z = 0.055 overlapping the and. Swirler like the large-scale engine [ 4 ] 45° is proposed more.! Swirl combustor, the heat loss country does have a lot of wind flow past a turbine is usually as. Θ ) are selected based on the packing characteristic curves of a micro combustor [ ]! East-Southern part of bangladesh pitch control is confirmed as a best micro combustor power the... Grid becomes dense, the high temperature flame zone is observed, but the profile. The development of micro combustor and various fan swirler designs are plotted Fig! 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Very sensitive to the movement of the tower, alternative energy resources in the wind ( Nandi and,. Join ResearchGate to find the people and research you need to help work... Of Kim and park [ 9, 10 ] alternative energy resources in the development of strong in. Explore the vortical flows as blade angle increases, the variations of temperature and 32 % for than... Complex wind flow modelling and wind farm layout optimisation entice people to purchase a fan! Presents the effect of the blade angle, it takes the benefit the. By using the finite element method accounting for the case of θ ≈ 10º gives the angles..., it increases again with a moderate center recirculation ), S.K, an increase of the tower nacelle! 2002 ), article Google Scholar, W.M reimpresión en el año 1995 Incluye bibliografía e índice to check difference., pages65–71 ( 2019 ) Cite this article recirculating flow patterns, which adversely affect the operational performance wind... 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The turbine base shows profiles of the combustor small scale combustors because of the H-Q curves for governing... On combustion characteristics of a small community in the wind flow at different wind speeds heated! Micro combustor, the tangential velocity is maximized at θ > 45º the available turbines in market after the at! The first step is done by studying the effect of the centerline governing! Is delayed for θ < 45º gives more heat loss is observed locally at effect of blade angle region!, article Google Scholar, W.M angle at which the tip enters the material at an it. 1 shows the schematic view of the H-Q curves for the governing of! The increase the wall material is treated as the blade angle of θ 10º! Efforts have been devoted to get an efficient combustion, the effects of the blade arc angle on … lowest! Turbulent flows, the highest efficiency and heat loss ratio GHG ) emission in the micro type... Is developed in this study, the temperature contours and streamlines for several axial locations degrees... The streamwise velocity profile along the centerline temperature and CH4 mass fraction depending on air hole positions in micro! To variable-pitch blades, turn in the moving air and power performance was explained with field contours between a vortex! Blade at the center vortex and wall vortices creates a peculiar flame varies... High degree of accuracy Savonius rotor is constructed with an intense OH radical contours by!
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Reconstruction in the partial data Calderón problem on admissible manifolds
June 2017, 11(3): 427-454. doi: 10.3934/ipi.2017020
A direct D-bar method for partial boundary data electrical impedance tomography with a priori information
Melody Alsaker a, , Sarah Jane Hamilton b,, and Andreas Hauptmann c,
Gonzaga University, Mathematics Department, 502 E. Boone Ave. MSC 2615, Spokane, WA 99258-0072, USA
Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI 53233, USA
Department of Computer Science, University College London, WC1E 6BT London, UK
* Corresponding author
Received September 2016 Revised October 2016 Published April 2017
Full Text(HTML)
Figure(21) / Table(3)
Electrical Impedance Tomography (EIT) is a non-invasive imaging modality that uses surface electrical measurements to determine the internal conductivity of a body. The mathematical formulation of the EIT problem is a nonlinear and severely ill-posed inverse problem for which direct D-bar methods have proved useful in providing noise-robust conductivity reconstructions. Recent advances in D-bar methods allow for conductivity reconstructions using EIT measurement data from only part of the domain (e.g., a patient lying on their back could be imaged using only data gathered on the accessible part of the body). However, D-bar reconstructions suffer from a loss of sharp edges due to a nonlinear low-pass filtering of the measured data, and this problem becomes especially marked in the case of partial boundary data. Including a priori data directly into the D-bar solution method greatly enhances the spatial resolution, allowing for detection of underlying pathologies or defects, even with no assumption of their presence in the prior. This work combines partial data D-bar with a priori data, allowing for noise-robust conductivity reconstructions with greatly improved spatial resolution. The method is demonstrated to be effective on noisy simulated EIT measurement data simulating both medical and industrial imaging scenarios.
Keywords: Electrical impedance tomography, partial boundary data, Neumannto-Dirichlet map, D-bar method, a priori information.
Mathematics Subject Classification: Primary: 65N21; Secondary: 94A08.
Citation: Melody Alsaker, Sarah Jane Hamilton, Andreas Hauptmann. A direct D-bar method for partial boundary data electrical impedance tomography with a priori information. Inverse Problems & Imaging, 2017, 11 (3) : 427-454. doi: 10.3934/ipi.2017020
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Figure 1. Example simulating a patient with a pneumothorax in the left lung. The simulated noisy measurement is collected from 75% ventral data. The first image displays the true conductivity with the position of electrodes indicated. Using a partial data D-bar approach alone results in a reconstruction with low spatial resolution, where the pathology can be hardly seen (second). Incorporating a priori data corresponding to a healthy patient directly into the reconstruction method significantly improves the spatial resolution (third). Refining the prior improves the reconstruction further, allowing even sharper visualization of the pathology (fourth)
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Figure 2. Illustration of mappings involved in the measurement modeling. Top row: Neumann data with the basis function $\varphi(\theta)=\cos(\theta)/\sqrt{\pi}$ on the left and the nonorthogonal projection $J\varphi$ on the right. Bottom row: Dirichlet data where $g=u|_{\partial\Omega}$ on the left is the solution of the partial differential equation (1) and on the right the orthogonal projection to the extended electrodes
Figure 3. The A Priori D-bar Method with Partial Data
Figure 4. Phantoms used in numerical examples with the corresponding boundaries of the priors outlined by white dots. Note that for each example, the prior does not assume a pathology/defect. Left: A simulated pneumothorax occurring near the heart in the left lung. Middle: A simulated pleural effusion occurring away from the heart in the left lung. Right: An enclosed diamond with an ovular defect
Figure 5. Blind priors used for the thoracic (top) and industrial (bottom) imaging examples. Take particular note that the priors do not assume any pathology/defect
Figure 6. The real part of the ${\mu ^{{\rm{int}}}}$ data (shown in the $z$ plane for $z\in\mathcal{D}$) corresponding to the blind thoracic prior given in Figure 5(top) computed from extended radii $R_2=4.0$, 6.5, and 9.0 in the $k$ plane. Note that as the radius increases, the integral term approaches its asymptotic behavior of ${\mu ^{{\rm{int}}}}\sim 1$
Figure 7. Scattering data corresponding to the pneumothorax example using the blind prior given in Figure 5(top). The original radius is $R=4$ and extended radius $R_2=9$. All scattering data is plotted on the same scale (real and imaginary, respectively)
Figure 8. Pneumothorax example for 62.5% ventral data. TOP: The partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$. BOTTOM: The recovered conductivity ${\sigma _{{R_2},\alpha }}$, using the blind thoracic prior. The maximum value is 2.25, occurring of $R=4$, $\alpha=0$
Figure 9. Left: Original prior. Right: Updated Pneumothorax prior. The left lung in the updated prior was segmented into two regions
Figure 10. Pneumothorax example with 75% Ventral data and segmented prior. The corresponding partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$ is shown in Figure 8. Here we display the recovered conductivity ${\sigma _{{R_2},\alpha }}$ for $R_2=4,6.5$ and various $\alpha$ using the SEG AVG or SEG MIN segmented thoracic priors. The maximum value is 2.70 and occurs in the $R_2=4$, $\alpha=0$ recon using the SEG MIN prior.
Figure 11. Pleural effusion example for 75% ventral data. TOP: The partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$. BOTTOM: The recovered conductivity ${\sigma _{{R_2},\alpha }}$ using the blind thoracic prior. The maximum value is 2.90, occurring of $R=4$, $\alpha=0$
Figure 12. Pneumothorax Example. Results for $R_2=9.0$ and $\alpha=0.67$. The maximum is 2.71 and occurs in the 100% boundary data, BLIND prior reconstruction
Figure 13. Pleural effusion example for 62.5% ventral data. The partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$ is shown at the top. Below, the recovered conductivity ${\sigma _{{R_2},\alpha }}$ is shown using the blind thoracic prior. The maximum value is 2.74, occurring of $R=4$, $\alpha=0$
Figure 14. Left: Original prior. Right: Updated Pleural Effusion prior with the left lung segmented into two regions
Figure 15. Pleural effusion example for 75% ventral data and segmented prior. The corresponding partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$ is shown in Figure 13. Here we display the recovered conductivity ${\sigma _{{R_2},\alpha }}$ for $R_2=4$, 6.5 and various $\alpha$ using the SEG AVG or SEG MAX segmented thoracic prior. The maximum value is 2.83 and occurs in the $R_2=4$, $\alpha=0$ reconstruction using the SEG MAX prior
Figure 17. Pleural Effusion Example. Results for $R_2=6.5$ and $\alpha=0.67$. The maximum is 2.65 and occurs in the 100% boundary data, BLIND prior reconstruction
Figure 18. Industrial Example: From top to bottom, conductivity reconstructions ${\sigma _{{R_2},\alpha }}$ for 100%, 75%, 62.5%, and 50% boundary data are presented with scattering radius $R_2=4$ and various weights $\alpha$. The first column displays the ${\sigma ^{{\rm{ND}}}}$ reconstructions that do not include any a priori information. The maximum value (3.12) occurs for the 50% data reconstruction with strongest weight $\alpha=0$
Figure 19. Industrial Example: From top to bottom, conductivity reconstructions ${\sigma _{{R_2},\alpha }}$ for 100%, 75%, 62.5%, and 50% boundary data are presented with extended scattering radius $R_2=6.5$ and various weights $\alpha$. The first column displays the ${\sigma ^{{\rm{ND}}}}$ reconstructions that do not include any a priori information. The maximum value (3.13) occurs for the 50% data reconstruction with strongest weight $\alpha=0$
Figure 20. Relative $\ell_2$-error of reconstructions from 75% ventral data of the pneumothorax example. The horizontal axis represents $\alpha$-values for increasing regularization radii $R_2$. Recall that $\alpha=0$ corresponds to the heaviest weighting of the ${\mu ^{{\rm{int}}}}$ term, while $\alpha=1$ to the weakest expression of the prior. Errors from ${\sigma ^{{\rm{ND}}}}$ are compared to the new reconstructions ${\sigma _{{R_2},\alpha }}$ for the blind and segmented priors
Figure 21. Relative $\ell_2$-error in the lung region within the boundary of the pathology, for 75% ventral data for the pneumothorax example. The horizontal axis represents $\alpha$-values for increasing regularization radii $R_2$
Table 1. Conductivity values of thoracic phantoms and assigned blind prior in S/m
Heart Lungs Pathology Aorta Spine Background
Pneumothorax 2.0 0.5 0.15 2.0 0.25 1
Pleural Effusion 2.0 0.5 1.8 2.0 0.25 1
Prior 2.05 0.45 - 2.05 0.23 1
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Table 2. Conductivity values of industrial phantom and assigned blind prior in S/m
Diamond Inclusion Background
Industrial 2.0 1.4 1
Prior 2.05 - 1
Table 3. Relative $\ell_2$-errors (%) for the conductivity reconstructions from §4, for the extended regularization radii $R_2=4$ and $6.5$
D-BAR $\mathbf{R_2=4}$ $\mathbf{R_2=6.5}$
RECON $\alpha=1$ $\alpha=\frac{2}{3}$ $\alpha=\frac{1}{3}$ $\alpha=0$ $\alpha=1$ $\alpha=\frac{2}{3}$ $\alpha=\frac{1}{3}$ $\alpha=0$
Blind Prior: 75% 35.13 29.65 26.74 24.86 24.44 26.82 25.36 24.20 23.39
Seg Avg Prior: 75% 35.13 29.14 26.22 24.65 24.95 25.75 24.16 22.92 22.11
Seg Min Prior: 75% 35.13 28.84 25.96 24.75 25.74 25.07 23.44 22.23 21.55
Blind Prior: 62.5% 38.95 32.71 30.02 28.06 27.12 30.12 28.74 27.56 26.63
Seg Avg Prior: 62.5% 38.95 32.33 29.62 27.83 27.30 29.43 27.99 26.78 25.84
Seg Min Prior: 62.5% 38.95 31.99 29.27 27.67 27.61 28.66 27.13 25.88 24.97
Seg Max Prior: 75% 27.40 24.14 21.95 21.39 22.81 21.98 20.94 20.34 20.22
Seg Max Prior: 62.5% 32.56 28.87 27.01 26.24 26.80 26.77 25.85 25.20 24.87
Industrial phantom
Blind Prior: 100% 18.43 18.43 16.07 14.17 12.99 15.31 14.17 13.28 12.68
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\begin{document}
\begin{center}\large
{\bf Navier-Stokes blow-up rates in certain Besov spaces \\ whose regularity exceeds the critical value by
$\boldsymbol{\epsilon\in[1,2]}$} \\\ \\
Joseph P.\ Davies\footnote{University of Sussex, Brighton, UK, {\em [email protected]}} and Gabriel S. Koch\footnote{University of Sussex, Brighton, UK, {\em [email protected]}} \end{center} \ \\
\abstract{For a solution $u$ to the Navier-Stokes equations in spatial dimension $n\geq3$ which blows up at a finite time $T>0$, we prove the blowup estimate ${\|u(t)\|}_{\dot{B}_{p,q}^{s_{p}+\epsilon}(\mathbb{R}^n)}\gtrsim_{\varphi,\epsilon,(p\vee q\vee 2)}{(T-t)}^{-\epsilon/2}$ for all $\epsilon\in[1,2)$ and $p,q\in[1,\frac{n}{2-\epsilon})$, where $s_{p}:=-1+\frac{n}{p}$ is the scaling-critical regularity, and $\varphi$ is the cutoff function used to define the Littlewood-Paley projections. For $\epsilon =2$, we prove the same type of estimate but only for $q=1$: ${\|u(t)\|}_{\dot{B}_{p,1}^{s_{p}+2}(\mathbb{R}^n)}\gtrsim_{\varphi,(p\vee 2)}{(T-t)}^{-1}$ for all $p\in [1,\infty)$. Under the additional restriction that $p,q\in[1,2]$ and $n=3$, these blowup estimates are implied by those first proved by Robinson, Sadowski and Silva (J. Math. Phys., 2012) for $p=q=2$ in the case $\epsilon\in(1,2)$, and by McCormick, Olson, Robinson, Rodrigo, Vidal-L\'{o}pez and Zhou (SIAM J. Math. Anal., 2016) for $p=2$ in the cases $(\epsilon,q)=(1,2)$ and $(\epsilon,q)=(2,1)$.} \section{Introduction} According to the physical theory, if an incompressible viscous Newtonian fluid occupies the whole space $\mathbb{R}^{n}$ in the absence of external forces, then the velocity $U(\tau,x)$ and kinematic pressure $\varPi(\tau,x)$ of the fluid at time $\tau>0$ and position $x\in\mathbb{R}^{n}$ satisfy the Navier-Stokes equations \begin{equation*}
\left\{\begin{array}{l}
\partial_{\tau}U-\nu\Delta U+(U\cdot\nabla)U+\nabla\varPi=0, \\ \nabla\cdot U=0,
\end{array}\right. \end{equation*} where the coefficient $\nu>0$ is the kinematic viscosity of the fluid. By considering the rescaled quantities \begin{equation*}
t=\nu\tau, \quad u(t,x)=\nu^{-1}U(\nu^{-1}t,x), \quad \varpi(t,x)=\nu^{-2}\varPi(\nu^{-1}t,x), \end{equation*} whose physical dimensions are powers of length alone, we may rewrite the Navier-Stokes equations in the standardised form \begin{equation}\label{intro-navier-stokes}
\left\{\begin{array}{l}
\partial_{t}u-\Delta u+(u\cdot\nabla)u+\nabla\varpi=0, \\ \nabla\cdot u=0.
\end{array}\right. \end{equation} At the formal level, if $(u,\varpi)$ satisfy \eqref{intro-navier-stokes} then $\varpi$ may be recovered from $u$ by the formula \begin{equation}\label{pressure}
\varpi = {(-\Delta)}^{-1}\nabla^{2}:(u\otimes u). \end{equation} Writing $\Lambda$ to denote the physical dimension of length, the quantities $x,t,u,\varpi$ have physical dimensions \begin{equation*}
[x] = \Lambda, \quad [t] = \Lambda^{2}, \quad [u] = \Lambda^{-1}, \quad [\varpi] = \Lambda^{-2}; \end{equation*} this is related to the fact that the standardised Navier-Stokes equations are preserved under the rescaling \begin{equation}\label{navier-stokes-scaling}
x_{\lambda}=\lambda x, \quad t_{\lambda} = \lambda^{2}t, \quad u_{\lambda}(t_{\lambda},x_{\lambda}) = \lambda^{-1}u(t,x), \quad \varpi_{\lambda}(t_{\lambda},x_{\lambda})=\lambda^{-2}\varpi(t,x). \end{equation} \begin{definition}\label{regsoldef}
For $T\in(0,\infty]$, a function $u:(0,T)\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is said to be a {\em regular solution} to the standardised Navier-Stokes equations on $(0,T)$ if
\begin{enumerate}[label=(\roman*)]
\item $u$ is smooth on $(0,T)\times\mathbb{R}^{n}$, with every derivative belonging to $C((0,T);L^{2}(\mathbb{R}^{n}))$;
\item $(u,\varpi)$ satisfy \eqref{intro-navier-stokes}, with $\varpi$ being given by \eqref{pressure}.
\end{enumerate} \end{definition} \begin{definition}
For $T\in(0,\infty)$, a regular solution $u$ on $(0,T)$ is said to {\em blow up} at (rescaled) time $T$ if $u$ doesn't extend to a regular solution on $(0,T')$ for any $T'>T$. \end{definition} In spatial dimension $n\geq3$, it remains unknown whether there exist regular solutions which blow up. Since Leray's seminal paper \cite{leray1934}, prospective blowing-up solutions have been studied using homogeneous norms: \begin{definition}
A norm ${\|\cdot\|}_{X}$ (defined on a subspace $\mathcal{X}\subseteq\mathcal{S}'(\mathbb{R}^{n})$ which is closed under dilations of $\mathbb{R}^{n}$) is said to be {\em homogeneous of degree $\alpha=\alpha(X)$} if, under the rescaling $x_{\lambda}=\lambda x$ and $f_{\lambda}(x_{\lambda})=f(x)$, we have ${\|f_{\lambda}\|}_{X}\approx_{X}\lambda^{\alpha}{\|f\|}_{X}$ for all $\lambda\in(0,\infty)$ and $f\in\mathcal{X}$. \end{definition}
If ${\|\cdot\|}_{X}$ is homogeneous of degree $\alpha$, then under the scaling \eqref{navier-stokes-scaling} of the Navier-Stokes equations we have ${\|u_{\lambda}(t_{\lambda})\|}_{X}\approx_{X}\lambda^{\alpha-1}{\|u(t)\|}_{X}$, so the quantity ${\|u\|}_{X}$ has physical dimension $\Lambda^{\alpha-1}$. In the context of the Navier-Stokes equations, the homogeneous norm ${\|\cdot\|}_{X}$ is said to be {\em subcritical} if $\alpha(X)<1$, {\em critical} if $\alpha(X)=1$, and {\em supercritical} if $\alpha(X)>1$. If ${\|\cdot\|}_{X}$ is subcritical, then the blowup estimate \begin{equation}\label{abstract-blowup}
u\text{ blows up at time }T\quad\Rightarrow\quad{\|u(t)\|}_{X}\gtrsim_{X}{(T-t)}^{-\left(1-\alpha(X)\right)/2} \end{equation} makes dimensional sense.
We investigate \eqref{abstract-blowup} in the context of the homogeneous Besov norms \begin{equation*}
{\|f\|}_{\dot{B}_{p,q}^{s}(\mathbb{R}^{n})} := {\left\|j\mapsto2^{js}{\left\|\mathcal{F}^{-1}\varphi(2^{-j}\xi)\mathcal{F}f\right\|}_{L^{p}(\mathbb{R}^{n})}\right\|}_{l^{q}(\mathbb{Z})} \quad \text{for }s\in\mathbb{R},\,p,q\in[1,\infty], \end{equation*} where $\mathcal{F}$ is the Fourier transform and $\varphi$ is a cutoff function satisfying certain properties.\footnote{We will give more detailed definitions in section \ref{besov-spaces}. Choosing a different function $\varphi$ yields an equivalent norm.} Amongst other things, for any $p, p_j,q,q_j \in [1,\infty]$ and $\delta,s \in \mathbb{R}$ the Besov norms satisfy \begin{equation}\label{intro-besov-embedding}
{\|f\|}_{\dot{B}_{p_{1},q_{1}}^{\frac{n}{p_{1}}+\delta}(\mathbb{R}^{n})} \gtrsim_{n} {\|f\|}_{\dot{B}_{p_{2},q_{2}}^{\frac{n}{p_{2}}+\delta}(\mathbb{R}^{n})} \quad \text{if }p_{1}\leq p_{2},\,q_{1}\leq q_{2}, \end{equation} \begin{equation}\label{intro-lebesgue}
{\|f\|}_{\dot{B}_{p,1}^{0}(\mathbb{R}^{n})} \geq {\|f\|}_{L^{p}(\mathbb{R}^{n})} \gtrsim_{\varphi} {\|f\|}_{\dot{B}_{p,\infty}^{0}(\mathbb{R}^{n})}, \end{equation} \begin{equation}\label{intro-sobolev}
{\|f\|}_{\dot{B}_{2,2}^{s}(\mathbb{R}^{n})} \approx_{\varphi} {\|f\|}_{\dot{H}^{s}(\mathbb{R}^{n})}, \end{equation}
and ${\|\cdot\|}_{\dot{B}_{p,q}^{s}(\mathbb{R}^{n})}$ is homogeneous of degree $\alpha(\dot{B}_{p,q}^{s}(\mathbb{R}^{n}))=\frac{n}{p}-s$. In the context of Navier-Stokes we define \begin{equation*}
s_{p} = s_{p}(n) := -1+\frac{n}{p}, \end{equation*}
so the norm ${\|\cdot\|}_{\dot{B}_{p,q}^{s_{p}+\epsilon}(\mathbb{R}^{n})}$ is critical for $\epsilon=0$, and subcritical for $\epsilon>0$.
It is known that if a regular solution $u$ blows up at a finite time $T>0$, then\footnote{These estimates follow from the local theory and regularity properties of mild solutions with initial data $f$, where the existence time is bounded below by a constant multiple of ${\|f\|}_{\dot{B}_{\infty,\infty}^{-1+\epsilon}(\mathbb{R}^{n})}^{-2/\epsilon}\vee{\|f\|}_{L^{\infty}(\mathbb{R}^{n})}^{-2}$. Estimate \eqref{intro-leray} comes from Leray \cite{leray1934}, while the local theory for initial data in Besov spaces is discussed in Lemari\'{e}-Rieusset's book \cite{lemarie2016}. We aim to give a more detailed account of the regularity properties of such solutions in an upcoming paper.} \begin{equation}\label{intro-easy-besov}
{\|u(t)\|}_{\dot{B}_{\infty,\infty}^{-1+\epsilon}(\mathbb{R}^{n})} \gtrsim_{\varphi,\epsilon} {(T-t)}^{-\epsilon/2} \quad \text{for all }t\in(0,T),\,\epsilon\in(0,1), \end{equation} \begin{equation}\label{intro-leray}
{\|u(t)\|}_{L^{\infty}(\mathbb{R}^{n})} \gtrsim_{n} {(T-t)}^{-1/2} \quad \text{for all }t\in(0,T). \end{equation}
By virtue of \eqref{intro-besov-embedding} and \eqref{intro-lebesgue}, the left-hand side of \eqref{intro-easy-besov} may be replaced by ${\|u(t)\|}_{\dot{B}_{p,q}^{s_{p}+\epsilon}(\mathbb{R}^{n})}$ for any $p,q\in[1,\infty]$, while the left-hand side of \eqref{intro-leray} may be replaced by ${\|u(t)\|}_{\dot{B}_{p,1}^{s_{p}+1}(\mathbb{R}^{n})}$ for any $p\in[1,\infty]$.
Adapting the energy methods of \cite{mccormick2016, robinson2014, robinson2012}, we will prove the following blowup estimates in the case $\epsilon\in[1,2]$: \begin{theorem}\label{main-theorem}
Let $n\geq3$ and $T\in(0,\infty)$. If $u$ is a regular solution (see Definition \ref{regsoldef}) to the standardised Navier-Stokes equations on $(0,T)$ which
satisfies\footnote{Note that this is a natural assumption for a solution blowing up at a finite time $T$ in view of \eqref{intro-easy-besov} with $\epsilon =\tfrac 12$. In fact, to prove the parts of \eqref{eps-less-2} - \eqref{eps-is-2} with $\epsilon \neq 1$, one may replace this assumption with $\lim_{t\nearrow T}{\|u(t)\|}_{L^\infty(\mathbb{R}^{n})}=\infty$ (coming from \eqref{intro-leray}) by replacing \eqref{interpbesonehalf} in the proof with the estimate
$$ {\|u(t)\|}_{L^\infty(\mathbb{R}^n)} \leq {\|u(t)\|}_{\dot{B}_{\infty,1}^{0}(\mathbb{R}^n)} \lesssim_{n,\epsilon}{\|u(t)\|}_{\dot{B}_{\infty,\infty}^{-n/2}(\mathbb{R}^n)}^{\lambda}{\|u(t)\|}_{\dot{B}_{\infty,\infty}^{-1+\epsilon}(\mathbb{R}^n)}^{1-\lambda} \quad \text{with} \quad \lambda=\frac{\epsilon-1}{\epsilon-1+\frac{n}{2}}\, .
$$
}
$\lim_{t\nearrow T}{\|u(t)\|}_{\dot{B}_{\infty,\infty}^{-1/2}(\mathbb{R}^{n})}=\infty$, then
\begin{equation}\label{eps-less-2}
{\|u(t)\|}_{\dot{B}_{p,q}^{s_{p}+\epsilon}(\mathbb{R}^{n})} \gtrsim_{\varphi,\epsilon,(p\vee q\vee2)} {(T-t)}^{-\epsilon/2} \quad \text{for all }t\in(0,T),\,\epsilon\in[1,2),\,p,q\in\left[1,\frac{n}{2-\epsilon}\right)
\end{equation} and
\begin{equation}\label{eps-is-2}
{\|u(t)\|}_{\dot{B}_{p,1}^{s_{p}+2}(\mathbb{R}^{n})} \gtrsim_{\varphi,(p\vee2)} {(T-t)}^{-1} \quad \text{for all }t\in(0,T),\,p\in[1,\infty).
\end{equation} \end{theorem} Under the additional restrictions that $p,q\in[1,2]$ and $n=3$, the blowup estimate \eqref{eps-less-2} is implied by the blowup estimate for $\dot{H}^{s_{2}+\epsilon}(\mathbb{R}^{3})$, which was proved in the case $\epsilon\in(1,2)$ by Robinson, Sadowski and Silva \cite{robinson2012}, and in the case $\epsilon=1$ by McCormick et al.\ \cite{mccormick2016}. Under the additional restrictions that $p\in[1,2]$ and $n=3$, the blowup estimate \eqref{eps-is-2} is implied by the blowup estimate for $\dot{B}_{2,1}^{5/2}(\mathbb{R}^{3})$, which was proved by McCormick et al.\ \cite{mccormick2016}.
The rest of this paper is organised as follows. In section \ref{besov-spaces} we recall some standard properties of Besov spaces, using \cite{bahouri2011} as our main reference. In section \ref{commutator-estimates} we prove some commutator estimates, adapting the ideas of \cite[Lemma 2.100]{bahouri2011}. In section \ref{navier-stokes-blowup-rates} we prove Theorem \ref{main-theorem}. We will henceforth use the abbreviations $L^{p}=L^{p}(\mathbb{R}^{n})$, $\dot{H}^{s}=\dot{H}^{s}(\mathbb{R}^{n})$, $\dot{B}_{p,q}^{s}=\dot{B}_{p,q}^{s}(\mathbb{R}^{n})$ and $l^{q}=l^{q}(\mathbb{Z})$. \section{Besov spaces}\label{besov-spaces} \begin{lemma}\label{bahouri-prop-2.10}
(\cite{bahouri2011}, Proposition 2.10). Let $\mathcal{C}$ be the annulus $B(0,8/3)\setminus\overline{B}(0,3/4)$. Then the set $\widetilde{\mathcal{C}}=B(0,2/3)+\mathcal{C}$ is an annulus, and there exist radial functions $\chi\in\mathcal{D}(B(0,4/3))$ and $\varphi\in\mathcal{D}(\mathcal{C})$, taking values in $[0,1]$, such that
\begin{equation*}
\left\{\begin{array}{ll}
\chi(\xi)+\sum_{j\geq0}\varphi(2^{-j}\xi)=1 & \forall\,\xi\in\mathbb{R}^{n}, \\
\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1 & \forall\,\xi\in\mathbb{R}^{n}\setminus\{0\}, \\
|j-j'|\geq2\Rightarrow\supp\varphi(2^{-j}\cdot)\cap\supp\varphi(2^{-j'}\cdot)=\emptyset, \\
j\geq1\Rightarrow\supp\chi\cap\supp\varphi(2^{-j}\cdot)=\emptyset, \\
|j-j'|\geq5\Rightarrow2^{j'}\widetilde{\mathcal{C}}\cap2^{j}\mathcal{C}=\emptyset, \\
1/2\leq\chi^{2}(\xi)+\sum_{j\geq0}\varphi^{2}(2^{-j}\xi)\leq1 & \forall\,\xi\in\mathbb{R}^{n}, \\
1/2\leq\sum_{j\in\mathbb{Z}}\varphi^{2}(2^{-j}\xi)\leq1 & \forall\,\xi\in\mathbb{R}^{n}\setminus\{0\}. \\
\end{array}\right.
\end{equation*} \end{lemma} We fix $\chi,\varphi$ satisfying Lemma \ref{bahouri-prop-2.10}. For $j\in\mathbb{Z}$ and $u\in\mathcal{S}'$, we define\footnote{We adopt the convention that $\mathcal{F}f(\xi) = \int_{\mathbb{R}^{n}}e^{-\mathrm{i}\xi\cdot x}f(x)\,\mathrm{d}x$ for $f\in\mathcal{S}$. We recall that the Fourier transform of a compactly supported distribution is a smooth function.} \begin{equation*}
\dot{S}_{j}u := \chi(2^{-j}D)u = \mathcal{F}^{-1}\chi(2^{-j}\xi)\mathcal{F}u, \end{equation*} \begin{equation*}
\dot{\Delta}_{j}u := \varphi(2^{-j}D)u = \mathcal{F}^{-1}\varphi(2^{-j}\xi)\mathcal{F}u. \end{equation*} \begin{lemma}\label{truncation-lemma}
For any $j,j'\in\mathbb{Z}$ and $u,v\in\mathcal{S}'$, we have
\begin{equation*}
|j-j'|\geq 2\Rightarrow \dot{\Delta}_{j}\dot{\Delta}_{j'}u=0, \qquad |j-j'|\geq5\Rightarrow\dot{\Delta}_{j}\left(\dot{S}_{j'-1}u\,\dot{\Delta}_{j'}v\right)=0.
\end{equation*} \end{lemma} \begin{proof}
This is a consequence of Lemma \ref{bahouri-prop-2.10}. In particular, the implication $|j-j'|\geq 2\Rightarrow \dot{\Delta}_{j}\dot{\Delta}_{j'}u=0$ follows from the implication $|j-j'|\geq2\Rightarrow\supp\varphi(2^{-j}\cdot)\cap\supp\varphi(2^{-j'}\cdot)=\emptyset$, while the implication $|j-j'|\geq5\Rightarrow\dot{\Delta}_{j}\left(\dot{S}_{j'-1}u\,\dot{\Delta}_{j'}v\right)=0$ follows\footnote{Note that $\dot{S}_{j'-1}u$ is spectrally supported on $2^{j'}B(0,2/3)$, while $\dot{\Delta}_{j'}v$ is spectrally supported on $2^{j'}\mathcal{C}$, so by properties of convolution we have that $\dot{S}_{j'-1}u\,\dot{\Delta}_{j'}v$ is spectrally supported on $2^{j'}\widetilde{\mathcal{C}}$.} from the implication $|j-j'|\geq5\Rightarrow2^{j'}\widetilde{\mathcal{C}}\cap2^{j}\mathcal{C}=\emptyset$. \end{proof}
We recall the following useful properties: \begin{lemma}\label{useful-inequalities}
(\cite{bahouri2011}, Lemmas 2.1-2.2, Remark 2.11). Let $\rho$ be a smooth function on $\mathbb{R}^{n}\setminus\{0\}$ which is positive homogeneous of degree $\lambda\in\mathbb{R}$. Then for all $j\in\mathbb{Z}$, $u\in\mathcal{S}'$, $t\in(0,\infty)$ and $1\leq p\leq q\leq\infty$ we have
\begin{equation}\label{useful-inequality-1}
{\|\dot{S}_{j}u\|}_{L^{p}}\vee{\|\dot{\Delta}_{j}u\|}_{L^{p}} \lesssim_{\varphi} {\|u\|}_{L^{p}},
\end{equation}
\begin{equation}
{\|\rho(D)\dot{\Delta}_{j}u\|}_{L^{q}} \lesssim_{\rho} 2^{j\lambda}2^{j\left(\frac{n}{p}-\frac{n}{q}\right)}{\|\dot{\Delta}_{j}u\|}_{L^{p}},
\end{equation}
\begin{equation}
{\|\dot{\Delta}_{j}u\|}_{L^{p}} \lesssim_{n} 2^{-j}{\|\nabla\dot{\Delta}_{j}u\|}_{L^{p}}.
\end{equation} \end{lemma} One can give meaning to the decomposition $u=\sum_{j\in\mathbb{Z}}\dot{\Delta}_{j}u$ in view of the following lemma: \begin{lemma}\label{littlewood-paley-decomposition}
(\cite{bahouri2011}, Propositions 2.12-2.14) If $u\in\mathcal{S}'$, then $\dot{S}_{j}u\overset{j\rightarrow\infty}{\rightarrow}u$ in $\mathcal{S}'$. Define\footnote{For example, if $\mathcal{F}u$ is locally integrable near $\xi=0$, then $u\in\mathcal{S}_{h}'$. We remark that the condition $u\in\mathcal{S}_{h}'$ is independent of our choice of $\varphi$.}
\begin{equation*}
\mathcal{S}_{h}' := \left\{u\in\mathcal{S}'\text{ }:\text{ }{\|\dot{S}_{j}u\|}_{L^{\infty}}\overset{j\rightarrow-\infty}{\rightarrow}0\right\},
\end{equation*}
so if $u\in\mathcal{S}_{h}'$ then $u=\sum_{j\in\mathbb{Z}}\dot{\Delta}_{j}u$ in $\mathcal{S}'$. \end{lemma} For $s\in\mathbb{R}$ and $p,q\in[1,\infty]$, we define the Besov seminorm\footnote{Choosing a different function $\varphi$ yields an equivalent seminorm \cite[Remark 2.17]{bahouri2011}.} \begin{equation*}
{\|u\|}_{\dot{B}_{p,q}^{s}} := {\left\|j\mapsto2^{js}{\|\dot{\Delta}_{j}u\|}_{L^{p}}\right\|}_{l^{q}} \quad \text{for }u\in\mathcal{S}' \end{equation*} and the Besov space \begin{equation*}
\dot{B}_{p,q}^{s} := \left\{u\in\mathcal{S}_{h}'\text{ : }{\|u\|}_{\dot{B}_{p,q}^{s}}<\infty\right\}, \end{equation*}
so that $\left(\dot{B}_{p,q}^{s},{\|\cdot\|}_{\dot{B}_{p,q}^{s}}\right)$ is a normed space \cite[Proposition 2.16]{bahouri2011}. Lemma \ref{useful-inequalities} and Lemma \ref{littlewood-paley-decomposition} yield the inequalities \begin{equation}\label{besov-embedding}
{\|u\|}_{\dot{B}_{p_{2},q_{2}}^{\frac{n}{p_{2}}+\epsilon}} \lesssim_{n} {\|u\|}_{\dot{B}_{p_{1},q_{1}}^{\frac{n}{p_{1}}+\epsilon}} \quad \text{for }p_{1}\leq p_{2},\,q_{1}\leq\,q_{2},\,\epsilon\in\mathbb{R},\,u\in\mathcal{S}', \end{equation} \begin{equation}\label{rough-lp}
{\|u\|}_{\dot{B}_{p,\infty}^{0}} \lesssim_{\varphi} {\|u\|}_{L^{p}} \quad \text{for }u\in\mathcal{S}' \end{equation} and \begin{equation}\label{smooth-lp}
{\|u\|}_{L^{p}} \leq {\|u\|}_{\dot{B}_{p,1}^{0}} \quad \text{for }u\in\mathcal{S}_{h}'. \end{equation} We also have the interpolation inequalities \begin{equation}\label{interpolation-holder}
{\|u\|}_{\dot{B}_{\frac{p_{1}p_{2}}{\lambda p_{2}+(1-\lambda)p_{1}},\frac{q_{1}q_{2}}{\lambda q_{2}+(1-\lambda)q_{1}}}^{\lambda s_{1}+(1-\lambda)s_{2}}} \leq {\|u\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}^{\lambda}{\|u\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}}^{1-\lambda} \quad \text{for }\lambda\in(0,1),\,u\in\mathcal{S}', \end{equation} \begin{equation}\label{interpolation-geometric}
{\|u\|}_{\dot{B}_{p,1}^{\lambda s_{1}+(1-\lambda)s_{2}}} \lesssim \frac{1}{\lambda(1-\lambda)(s_{2}-s_{1})}{\|u\|}_{\dot{B}_{p,\infty}^{s_{1}}}^{\lambda}{\|u\|}_{\dot{B}_{p,\infty}^{s_{2}}}^{1-\lambda} \quad \text{for }\lambda\in(0,1),\,s_{1}<s_{2},\,u\in\mathcal{S}', \end{equation}
where \eqref{interpolation-holder} comes from H\"{o}lder's inequality, while \eqref{interpolation-geometric} comes from writing $\sum_{j\in\mathbb{Z}}=\sum_{j\leq j_{0}}+\sum_{j>j_{0}}$ with $2^{j_{0}(s_{2}-s_{1})}{\|u\|}_{\dot{B}_{p,\infty}^{s_{1}}}\approx{\|u\|}_{\dot{B}_{p,\infty}^{s_{2}}}$ and applying geometric series.
We now recall the following convergence lemma: \begin{lemma}\label{convergence-lemma}
(\cite{bahouri2011}, Lemma 2.23). Let $\mathcal{C}'$ be an annulus and ${(u_{j})}_{j\in\mathbb{Z}}$ be a sequence of functions such that $\supp\mathcal{F}u_{j}\subseteq2^{j}\mathcal{C}'$ and ${\left\|j\mapsto2^{js}{\|u_{j}\|}_{L^{p}}\right\|}_{l^{q}}<\infty$. If the series $\sum_{j\in\mathbb{Z}}u_{j}$ converges in $\mathcal{S}'$ to some $u\in\mathcal{S}'$, then
\begin{equation*}\label{convergence-inequality}
{\|u\|}_{\dot{B}_{p,q}^{s}} \lesssim_{\varphi} C_{\mathcal{C}'}^{1+|s|}{\left\|j\mapsto2^{js}{\|u_{j}\|}_{L^{p}}\right\|}_{l^{q}}.
\end{equation*}
Note: If $(s,p,q)$ satisfy the condition
\begin{equation}\label{negative-scaling}
s<\frac{n}{p}, \quad \text{or} \quad s=\frac{n}{p}\text{ and }q=1,
\end{equation}
then the hypothesis of convergence is satisfied, and $u\in\mathcal{S}_{h}'$. \end{lemma}
A useful consequence of Lemma \ref{convergence-lemma} is that if $u\in\mathcal{S}'$ satisfies ${\|u\|}_{\dot{B}_{p,q}^{s}}<\infty$ for some $(s,p,q)$ satisfying \eqref{negative-scaling}, then $u\in\mathcal{S}_{h}'$.
If $u\in\dot{B}_{p_{1},1}^{0}$ and $v\in\dot{B}_{p_{2},1}^{0}$ with $\frac{1}{p_{1}}+\frac{1}{p_{2}}\leq1$, then the series $uv=\sum_{(j,j')\in\mathbb{Z}^{2}}\dot{\Delta}_{j}u\,\dot{\Delta}_{j'}v$ converges absolutely in $L^{\frac{p_{1}p_{2}}{p_{1}+p_{2}}}$, which justifies the Bony decomposition \begin{equation*}
uv = \dot{T}_{u}v+\dot{T}_{v}u+\dot{R}(u,v), \end{equation*} \begin{equation*}
\dot{T}_{u}v = \sum_{j\in\mathbb{Z}}\dot{S}_{j-1}u\,\dot{\Delta}_{j}v, \end{equation*} \begin{equation*}
\dot{R}(u,v) = \sum_{j\in\mathbb{Z}}\sum_{|\nu|\leq1}\dot{\Delta}_{j}u\,\dot{\Delta}_{j-\nu}v. \end{equation*} We will require the following estimates for the operators $\dot{T}$ and $\dot{R}$: \begin{lemma}\label{paraproduct}
(\cite{bahouri2011}, Theorem 2.47). Suppose that $s=s_{1}+s_{2}$, $p=\frac{p_{1}p_{2}}{p_{1}+p_{2}}$ and $q=\frac{q_{1}q_{2}}{q_{1}+q_{2}}$. Let $u,v\in\mathcal{S}'$, and assume that the series $\sum_{j\in\mathbb{Z}}\dot{S}_{j-1}u\,\dot{\Delta}_{j}v$ converges in $\mathcal{S}'$ to some $\dot{T}_{u}v\in\mathcal{S}'$. Then
\begin{equation}\label{bony-estimate-1}
{\|\dot{T}_{u}v\|}_{\dot{B}_{p,q}^{s}} \lesssim_{\varphi} C_{n}^{1+|s|}{\|u\|}_{L^{p_{1}}}{\|v\|}_{\dot{B}_{p_{2},q}^{s}},
\end{equation}
\begin{equation}\label{bony-estimate-2}
{\|\dot{T}_{u}v\|}_{\dot{B}_{p,q}^{s}} \lesssim_{\varphi} \frac{C_{n}^{1+|s|}}{-s_{1}}{\|u\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|v\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}} \quad \text{if } s_{1}<0.
\end{equation}
Note: If $(s,p,q)$ satisfy \eqref{negative-scaling}, and the right hand side of either \eqref{bony-estimate-1} or \eqref{bony-estimate-2} is finite, then the hypothesis of convergence is satisfied, and $\dot{T}_{u}v\in\mathcal{S}_{h}'$. \end{lemma} \begin{lemma}\label{remainder}
(\cite{bahouri2011}, Theorem 2.52). Suppose that $s=s_{1}+s_{2}$, $p=\frac{p_{1}p_{2}}{p_{1}+p_{2}}$ and $q=\frac{q_{1}q_{2}}{q_{1}+q_{2}}$. Let $u,v\in\mathcal{S}'$, and assume that the series $\sum_{j\in\mathbb{Z}}\sum_{|\nu|\leq1}\dot{\Delta}_{j}u\,\dot{\Delta}_{j-\nu}v$ converges in $\mathcal{S}'$ to some $\dot{R}(u,v)\in\mathcal{S}'$. Then
\begin{equation}\label{bony-estimate-3}
{\|\dot{R}(u,v)\|}_{\dot{B}_{p,q}^{s}} \lesssim_{\varphi} \frac{C_{n}^{1+|s|}}{s}{\|u\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|v\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}} \quad \text{if } s>0,
\end{equation}
\begin{equation}\label{bony-estimate-4}
{\|\dot{R}(u,v)\|}_{\dot{B}_{p,\infty}^{s}} \lesssim_{\varphi} C_{n}^{1+|s|}{\|u\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|v\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}} \quad \text{if }q=1\text{ and } s\geq0.
\end{equation}
Note: If $(s,p,q)$ satisfy \eqref{negative-scaling} and the right hand side of \eqref{bony-estimate-3} is finite, or if $(s,p,\infty)$ satisfy \eqref{negative-scaling} and the right hand side of \eqref{bony-estimate-4} is finite, then the hypothesis of convergence is satisfied, and $\dot{R}(u,v)\in\mathcal{S}_{h}'$. \end{lemma} \section{Commutator estimates}\label{commutator-estimates} In this section, we will adapt the proof of \cite[Lemma 2.100]{bahouri2011} to prove the commutator estimates in the following proposition, which will be crucial to the proof of Theorem \ref{main-theorem}: \begin{proposition}\label{commutator-prop}
For $v,f\in\cap_{r\in[0,\infty)}\dot{H}^{r}$ and $j\in\mathbb{Z}$, define\footnote{We apply the summation convention to the index $k$.}
\begin{equation*}
R_{j}=[v\cdot\nabla,\dot{\Delta}_{j}]f=[v_{k},\dot{\Delta}_{j}]\nabla_{k}f
\end{equation*}
where $[\,\cdot\,,\,\cdot\,]$ denotes the commutator $[A,B]=AB-BA$, and suppose that $s=s_{1}+s_{2}$, $p=\frac{p_{1}p_{2}}{p_{1}+p_{2}}$ and $q=\frac{q_{1}q_{2}}{q_{1}+q_{2}}$ ($s_j\in \mathbb{R}$ and $p_j,q_j \in [1,\infty]$).
Then we have the decomposition $R_{j}=\sum_{i=1}^{6}R_{j}^{i}$ with\footnote{Note that $R_{j}^{6}=0$ whenever $\nabla\cdot v=0$.}
\begin{equation*}
R_{j}^{1} = [\dot{T}_{v_{k}},\dot{\Delta}_{j}]\nabla_{k}f, \quad R_{j}^{2} = \dot{T}_{\nabla_{k}\dot{\Delta}_{j}f}v_{k}, \quad R_{j}^{3} = -\dot{\Delta}_{j}\dot{T}_{\nabla_{k}f}v_{k},
\end{equation*}
\begin{equation*}
R_{j}^{4} = \dot{R}(v_{k},\nabla_{k}\dot{\Delta}_{j}f), \quad R_{j}^{5} = -\nabla_{k}\dot{\Delta}_{j}\dot{R}(v_{k},f), \quad R_{j}^{6} = \dot{\Delta}_{j}\dot{R}(\nabla_{k}v_{k},f)
\end{equation*}
which satisfy the estimates
\begin{equation*}
\begin{aligned}
{\left\|j\mapsto2^{js}{\|R_{j}^{1}\|}_{L^{p}}\right\|}_{l^{q}} &\lesssim {\|\nabla v\|}_{L^{p_{1}}}{\|f\|}_{\dot{B}_{p_{2},q}^{s}}, \\
{\left\|j\mapsto2^{js}{\|R_{j}^{1}\|}_{L^{p}}\right\|}_{l^{q}} &\lesssim {\|\nabla v\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|f\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}} \quad \text{if }s_{1}<0, \\
{\left\|j\mapsto2^{js}{\|R_{j}^{2}\|}_{L^{p}}\right\|}_{l^{q}} &\lesssim {\|\nabla v\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|f\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}} \quad \text{if }s_{1}>-1, \\
{\left\|j\mapsto2^{js}{\|R_{j}^{3}\|}_{L^{p}}\right\|}_{l^{q}} &\lesssim {\|\nabla v\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|f\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}} \quad \text{if }s_{2}<1, \\
{\left\|j\mapsto2^{js}{\|R_{j}^{4}\|}_{L^{p}}\right\|}_{l^{q}} &\lesssim {\|\nabla v\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|f\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}}, \\
{\left\|j\mapsto2^{js}{\|R_{j}^{5}\|}_{L^{p}}\right\|}_{l^{q}} &\lesssim {\|\nabla v\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|f\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}} \quad \text{if }s>-1, \\
{\left\|j\mapsto2^{js}{\|R_{j}^{6}\|}_{L^{p}}\right\|}_{l^{q}} &\lesssim {\|\nabla v\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|f\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}} \quad \text{if }s>0,
\end{aligned}
\end{equation*}
where the implied constants depend on $\varphi,s_{1},s_{2},p_{1},p_{2}$. \end{proposition} \begin{remark}
Write $A_{1},A_{2},A_{3},A_{5},A_{6}$ to denote the constraints
\begin{equation*}
A_{1}\text{ : }s_{1}\leq0, \quad A_{2}\text{ : }s_{1}\geq-1, \quad A_{3}\text{ : }s_{2}\leq1, \quad A_{5}\text{ : }s\geq-1, \quad A_{6}\text{ : }s\geq0.
\end{equation*}
For $i=1,2,3,5,6$, a simple modification of our arguments yields the estimates
\begin{equation*}
{\left\|j\mapsto2^{js}{\|R_{j}^{i}\|}_{L^{p}}\right\|}_{l^{\infty}} \lesssim {\|\nabla v\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|f\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}} \quad \text{if }q=1\text{ and }A_{i}\text{ holds},
\end{equation*}
but we will not need these estimates when proving Theorem \ref{main-theorem}. \end{remark} As in \cite[Lemma 2.100]{bahouri2011}, to prove Proposition \ref{commutator-prop} we will rely on the following lemma:
\begin{lemma}\label{commutator-lemma}
(\cite{bahouri2011}, Lemma 2.97). Let $\theta\in C^{1}(\mathbb{R}^{n})$ be such that $\int_{\mathbb{R}^{n}}(1+|\xi|)|\mathcal{F}\theta(\xi)|\,\mathrm{d}\xi<\infty.$ Then for any $a\in C^{1}(\mathbb{R}^{n})$ with $\nabla a\in L^{p}(\mathbb{R}^{n})$, any $b\in L^{q}(\mathbb{R}^{n})$, and any $\lambda\in(0,\infty)$, we have
\begin{equation*}
{\|[\theta(\lambda^{-1}D),a]b\|}_{L^{\frac{pq}{p+q}}(\mathbb{R}^{n})} \lesssim_{\theta} \lambda^{-1}{\|\nabla a\|}_{L^{p}(\mathbb{R}^{n})}{\|b\|}_{L^{q}(\mathbb{R}^{n})}.
\end{equation*} \end{lemma} \begin{remark}\label{commutator-remark}
If we take $\theta=\varphi$ and $\lambda=2^{j}$, then Lemma \ref{commutator-lemma} yields the estimate
\begin{equation*}
{\|[\dot{\Delta}_{j},a]b\|}_{L^{\frac{pq}{p+q}}(\mathbb{R}^{n})} \lesssim_{\varphi} 2^{-j}{\|\nabla a\|}_{L^{p}(\mathbb{R}^{n})}{\|b\|}_{L^{q}(\mathbb{R}^{n})}.
\end{equation*} \end{remark} \begin{proof}[Proof of Proposition \ref{commutator-prop}]
The decomposition $R_{j}=\sum_{i=1}^{6}R_{j}^{i}$ comes from applying the Bony decomposition; the very strong regularity assumption $v,f\in\cap_{r\in[0,\infty)}\dot{H}^{r}$ is more than sufficient to address any convergence issues that may arise. In the following computations, we write ${(c_{j})}_{j\in\mathbb{Z}}$ to denote a sequence satisfying ${\|(c_{j})\|}_{l^{q}}\leq1$, and the constants implied by the notation $\lesssim$ depend on $\varphi,s_{1},s_{2},p_{1},p_{2}$.
{\bf Bounds for $\boldsymbol{2^{js}{\|R_{j}^{1}\|}_{L^{p}}}$.} By Lemma \ref{truncation-lemma} we have
\begin{equation*}
R_{j}^{1} = \sum_{|j-j'|\leq4}[\dot{S}_{j'-1}v_{k},\dot{\Delta}_{j}]\nabla_{k}\dot{\Delta}_{j'}f,
\end{equation*}
so by Remark \ref{commutator-remark} and Lemma \ref{useful-inequalities} we have
\begin{equation}\label{R1-estimate}
2^{js}{\|R_{j}^{1}\|}_{L^{p}} \lesssim \sum_{|j-j'|\leq4}2^{js}2^{j'-j}{\|\nabla\dot{S}_{j'-1}v\|}_{L^{p_{1}}}{\|\dot{\Delta}_{j'}f\|}_{L^{p_{2}}}.
\end{equation}
By \eqref{useful-inequality-1}, we deduce that
\begin{equation*}
2^{js}{\|R_{j}^{1}\|}_{L^{p}} \lesssim \sum_{|j-j'|\leq4}2^{js}2^{j'-j}{\|\nabla v\|}_{L^{p_{1}}}{\|\dot{\Delta}_{j'}f\|}_{L^{p_{2}}} \lesssim c_{j}{\|\nabla v\|}_{L^{p_{1}}}{\|f\|}_{\dot{B}_{p_{2},q}^{s}}.
\end{equation*}
On the other hand, if $s_{1}<0$ then \eqref{R1-estimate} implies that
\begin{equation*}
\begin{aligned}
2^{js}{\|R_{j}^{1}\|}_{L^{p}} &\lesssim \sum_{\substack{|j-j'|\leq4 \\ j''\leq j'-2}}2^{js}2^{j'-j}{\|\nabla\dot{\Delta}_{j''}v\|}_{L^{p_{1}}}{\|\dot{\Delta}_{j'}f\|}_{L^{p_{2}}} \\
&\lesssim \sum_{\substack{|j-j'|\leq4 \\ j''\leq j'-2}} 2^{(j-j'')s_{1}}2^{j''s_{1}}{\|\nabla\dot{\Delta}_{j''}v\|}_{L^{p_{1}}}2^{j's_{2}}{\|\dot{\Delta}_{j'}f\|}_{L^{p_{2}}} \\
&\lesssim c_{j}{\|\nabla v\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|f\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}},
\end{aligned}
\end{equation*}
where we used the inequality ${\|(\alpha*\beta)\gamma\|}_{l^{q}}\leq{\|\alpha*\beta\|}_{l^{q_{1}}}{\|\gamma\|}_{l^{q_{2}}}\leq{\|\alpha\|}_{l^{1}}{\|\beta\|}_{l^{q_{1}}}{\|\gamma\|}_{l^{q_{2}}}$ in the last line.
{\bf Bounds for $\boldsymbol{2^{js}{\|R_{j}^{2}\|}_{L^{p}}}$.} By Lemma \ref{truncation-lemma} we have
\begin{equation*}
R_{j}^{2} = \sum_{j'\geq j+1}\dot{S}_{j'-1}\nabla_{k}\dot{\Delta}_{j}f\,\dot{\Delta}_{j'}v_{k}.
\end{equation*}
so by Lemma \ref{useful-inequalities} we have
\begin{equation*}
2^{js}{\|R_{j}^{2}\|}_{L^{p}} \lesssim \sum_{j'\geq j+1}2^{js}2^{j-j'}{\|\nabla\dot{\Delta}_{j'}v\|}_{L^{p_{1}}}{\|\dot{\Delta}_{j}f\|}_{L^{p_{2}}}.
\end{equation*}
If $s_{1}>-1$, then we deduce that
\begin{equation*}
\begin{aligned}
2^{js}{\|R_{j}^{2}\|}_{L^{p}} &\lesssim \sum_{j'\geq j+1}2^{(j-j')(s_{1}+1)}2^{j's_{1}}{\|\nabla\dot{\Delta}_{j'}v\|}_{L^{p_{1}}}2^{js_{2}}{\|\dot{\Delta}_{j}f\|}_{L^{p_{2}}} \\
&\lesssim c_{j}{\|\nabla v\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|f\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}},
\end{aligned}
\end{equation*}
where we used the inequality ${\|(\alpha*\beta)\gamma\|}_{l^{q}}\leq{\|\alpha*\beta\|}_{l^{q_{1}}}{\|\gamma\|}_{l^{q_{2}}}\leq{\|\alpha\|}_{l^{1}}{\|\beta\|}_{l^{q_{1}}}{\|\gamma\|}_{l^{q_{2}}}$ in the last line.
{\bf Bounds for $\boldsymbol{2^{js}{\|R_{j}^{3}\|}_{L^{p}}}$.} By Lemma \ref{truncation-lemma} we have
\begin{equation*}
\begin{aligned}
R_{j}^{3} &= -\sum_{|j-j'|\leq4}\dot{\Delta}_{j}\left(\dot{S}_{j'-1}\nabla_{k}f\,\dot{\Delta}_{j'}v_{k}\right) \\
&= -\sum_{\substack{|j-j'|\leq4 \\ j''\leq j'-2}}\dot{\Delta}_{j}\left(\dot{\Delta}_{j''}\nabla_{k}f\,\dot{\Delta}_{j'}v_{k}\right),
\end{aligned}
\end{equation*}
so by Lemma \ref{useful-inequalities} we have
\begin{equation*}
2^{js}{\|R_{j}^{3}\|}_{L^{p}} \lesssim \sum_{\substack{|j-j'|\leq4 \\ j''\leq j'-2}}2^{js}2^{j''-j'}{\|\nabla\dot{\Delta}_{j'}v\|}_{L^{p_{1}}}{\|\dot{\Delta}_{j''}f\|}_{L^{p_{2}}}.
\end{equation*}
If $s_{2}<1$, then we deduce that
\begin{equation*}
\begin{aligned}
2^{js}{\|R_{j}^{3}\|}_{L^{p}} &\lesssim \sum_{\substack{|j-j'|\leq4 \\ j''\leq j'-2}}2^{j's_{1}}{\|\nabla\dot{\Delta}_{j'}v\|}_{L^{p_{1}}}2^{(j'-j'')(s_{2}-1)}2^{j''s_{2}}{\|\dot{\Delta}_{j''}f\|}_{L^{p_{2}}} \\
&\lesssim c_{j}{\|\nabla v\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|f\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}},
\end{aligned}
\end{equation*}
where we used the inequality ${\|\alpha(\beta*\gamma)\|}_{l^{q}}\leq{\|\alpha\|}_{l^{q_{1}}}{\|\beta*\gamma\|}_{l^{q_{2}}}\leq{\|\alpha\|}_{l^{q_{1}}}{\|\beta\|}_{l^{1}}{\|\gamma\|}_{l^{q_{2}}}$ in the last line.
{\bf Bounds for $\boldsymbol{2^{js}{\|R_{j}^{4}\|}_{L^{p}}}$.} Defining $\widetilde{\Delta}_{j'}=\sum_{|\nu|\leq1}\dot{\Delta}_{j'-\nu}$, by Lemma \ref{truncation-lemma} we have
\begin{equation*}
R_{j}^{4} = \sum_{|j-j'|\leq2}\dot{\Delta}_{j'}v_{k}\,\nabla_{k}\dot{\Delta}_{j}\widetilde{\Delta}_{j'}f,
\end{equation*}
so by Lemma \ref{useful-inequalities} and the inequality ${\|\alpha\beta\|}_{l^{q}}\leq{\|\alpha\|}_{l^{q_{1}}}{\|\beta\|}_{l^{q_{2}}}$ we have
\begin{equation*}
2^{js}{\|R_{j}^{4}\|}_{L^{p}} \lesssim c_{j}{\|\nabla v\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|f\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}}.
\end{equation*}
{\bf Bounds for $\boldsymbol{2^{js}{\|R_{j}^{5}\|}_{L^{p}}}$ and $\boldsymbol{2^{js}{\|R_{j}^{6}\|}_{L^{p}}}$.} By Lemma \ref{useful-inequalities} and \eqref{bony-estimate-3}, we have
\begin{equation*}
\begin{aligned}
{\left\|j\mapsto2^{js}{\|R_{j}^{5}\|}_{L^{p}}\right\|}_{l^{q}} &\lesssim {\|\nabla v\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|f\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}} \quad \text{if }s>-1, \\
{\left\|j\mapsto2^{js}{\|R_{j}^{6}\|}_{L^{p}}\right\|}_{l^{q}} &\lesssim {\|\nabla v\|}_{\dot{B}_{p_{1},q_{1}}^{s_{1}}}{\|f\|}_{\dot{B}_{p_{2},q_{2}}^{s_{2}}} \quad \text{if }s>0.
\end{aligned}
\end{equation*} \end{proof} \section{Proof of blowup rates}\label{navier-stokes-blowup-rates} We now give the \begin{proof}[Proof of Theorem \ref{main-theorem}] Note first that the regularity assumptions on $u$ are strong enough to justify the calculations used in this proof. \\\\ Fix $\epsilon\in[1,2]$ and $p,q\in[1,\frac{n}{2-\epsilon})$. By virtue of the inequality \eqref{besov-embedding}, it suffices to prove the estimate \begin{equation}\label{u-blowup}
{\|u(t)\|}_{\dot{B}_{r,\widetilde{r}}^{s_{r}+\epsilon}} \gtrsim_{\varphi,\epsilon,r} {(T-t)}^{-\epsilon/2} \end{equation}
for any fixed $r\in[p\vee q\vee2,\frac{n}{2-\epsilon})$ (e.g., $r:=p\vee q\vee2$), where $\widetilde{r}=r$ in the case $\epsilon<2$, and $\widetilde{r}=1$ in the case $\epsilon=2$. By the embeddings $L^{2}\hookrightarrow\dot{B}_{\infty,\infty}^{-n/2}$ and $\dot{B}_{r,\widetilde{r}}^{s_{r}+\epsilon}\hookrightarrow\dot{B}_{\infty,\infty}^{-1+\epsilon}$, the energy estimate $\limsup_{t\nearrow T}{\|u(t)\|}_{L^{2}}<\infty$, the assumption $\lim_{t\nearrow T}{\|u(t)\|}_{\dot{B}_{\infty,\infty}^{-1/2}}=\infty$, and the interpolation inequality \begin{equation}\label{interpbesonehalf}
{\|u(t)\|}_{\dot{B}_{\infty,\infty}^{-1/2}} \leq {\|u(t)\|}_{\dot{B}_{\infty,\infty}^{-n/2}}^{\lambda}{\|u(t)\|}_{\dot{B}_{\infty,\infty}^{-1+\epsilon}}^{1-\lambda} \quad \text{for }\lambda=\frac{\epsilon-\frac{1}{2}}{\epsilon-1+\frac{n}{2}}, \end{equation} we obtain the qualitative blowup estimate \begin{equation}\label{qualitative-blowup}
\lim_{t\nearrow T}{\|u(t)\|}_{\dot{B}_{r,\widetilde{r}}^{s_{r}+\epsilon}} = \infty. \end{equation} We will prove \eqref{u-blowup} by combining \eqref{qualitative-blowup} with the following ODE lemma. \begin{lemma}\label{ode-lemma}
(\cite{mccormick2016}, Lemma 2.1). If $\gamma,c>0$, $\partial_{t}X\leq cX^{1+\gamma}$, and $\lim_{t\nearrow T}X(t)=\infty$, then
\begin{equation*}
X(t) \geq {(\gamma c(T-t))}^{-1/\gamma} \quad \text{for all }t\in(0,T).
\end{equation*} \end{lemma} Our goal is to derive a suitable differential inequality that allows us to apply Lemma \ref{ode-lemma}. We will achieve this by considering the antisymmetric tensor\footnote{In the case $n=3$, this is related to the vorticity vector $\overrightarrow{\omega}:=\nabla\times u$ by $\overrightarrow{\omega}={(\omega_{23},\omega_{31},\omega_{12})}^{T}$. One could view $\omega$ and $\overrightarrow{\omega}$ as being different ways of representing the exterior derivative of the 1-form $\sum_{i=1}^{n}u_{i}\,\mathrm{d}x^{i}$.} \begin{equation}\label{w-from-u}
\omega_{ij} := \nabla_{i}u_{j}-\nabla_{j}u_{i}. \end{equation} Since $u$ is divergence-free, we can express $u$ in terms of $\omega$ by the formula \begin{equation}\label{u-from-w}
u_{i} = {(-\Delta)}^{-1}\nabla_{j}\omega_{ij}. \end{equation} By Lemma \ref{useful-inequalities}, we deduce that \eqref{u-blowup} is equivalent to \begin{equation}\label{w-blowup}
{\|\omega(t)\|}_{\dot{B}_{r,\widetilde{r}}^{s_{r}+\epsilon-1}} \gtrsim_{\varphi,\epsilon,r} {(T-t)}^{-\epsilon/2}, \end{equation} and that \eqref{qualitative-blowup} is equivalent to \begin{equation}\label{w-qualitative-blowup}
\lim_{t\nearrow T}{\|\omega(t)\|}_{\dot{B}_{r,\widetilde{r}}^{s_{r}+\epsilon-1}} = \infty. \end{equation} Applying the operator $X\mapsto\nabla_{i}X_{j}-\nabla_{j}X_{i}$ to the Navier-Stokes equations \eqref{intro-navier-stokes}, we see that $\omega$ satisfies \begin{equation}\label{w-equations}
\partial_{t}\omega_{ij}-\Delta\omega_{ij}+(u\cdot\nabla)\omega_{ij}+\omega_{ik}\nabla_{k}u_{j}=\omega_{jk}\nabla_{k}u_{i}. \end{equation}
Applying $\dot{\Delta}_{J}$ to the equation \eqref{w-equations}, multiplying the result by ${|\dot{\Delta}_{J}\omega|}^{r-2}\dot{\Delta}_{J}\omega_{ij}$, summing over $i,j$ and integrating over $\mathbb{R}^{n}$, we obtain \begin{equation}\label{w-energy} \begin{aligned}
&\frac{1}{r}\frac{\partial}{\partial t}\left({\|\dot{\Delta}_{J}\omega\|}_{L^{r}}^{r}\right) - \left\langle\Delta\dot{\Delta}_{J}\omega,{|\dot{\Delta}_{J}\omega|}^{r-2}\dot{\Delta}_{J}\omega\right\rangle \\
&\quad = -\left\langle\dot{\Delta}_{J}((u\cdot\nabla)\omega),{|\dot{\Delta}_{J}\omega|}^{r-2}\dot{\Delta}_{J}\omega\right\rangle - \left\langle\dot{\Delta}_{J}(\omega_{ik}\nabla_{k}u_{j}-\omega_{jk}\nabla_{k}u_{i}),{|\dot{\Delta}_{J}\omega|}^{r-2}\dot{\Delta}_{J}\omega_{ij}\right\rangle. \end{aligned} \end{equation}
By the identities $\nabla\cdot u=0$, $\nabla_{k}\left({|\dot{\Delta}_{J}\omega|}^{r}\right)=r(\nabla_{k}\dot{\Delta}_{J}\omega_{ij}){|\dot{\Delta}_{J}\omega|}^{r-2}\dot{\Delta}_{J}\omega_{ij}$ and $\omega_{ij}=-\omega_{ji}$, we see that the right hand side of \eqref{w-energy} is equal to \begin{equation*}
\left\langle[u\cdot\nabla,\dot{\Delta}_{J}]\omega,{|\dot{\Delta}_{J}\omega|}^{r-2}\dot{\Delta}_{J}\omega\right\rangle - 2\left\langle\dot{\Delta}_{J}(\omega\cdot\nabla u),{|\dot{\Delta}_{J}\omega|}^{r-2}\dot{\Delta}_{J}\omega\right\rangle, \end{equation*} where we define ${(\omega\cdot\nabla u)}_{ij}:=\omega_{ik}\nabla_{k}u_{j}$. Writing $\Omega_{J}:=[u\cdot\nabla,\dot{\Delta}_{J}]\omega-2\dot{\Delta}_{J}(\omega\cdot\nabla u)$, and noting the inequality\footnote{Valid for $n\geq3$ and $r\in[2,\infty)$, proved in \cite[Lemmas 1-2]{robinson2014}.} \begin{equation*}\label{lhs-bound}
-\left\langle\Delta v,{|v|}^{r-2}v\right\rangle \gtrsim_{n,r} {\|v\|}_{L^{\frac{rn}{n-2}}}^{r}, \end{equation*} we deduce that \begin{equation}\label{w-energy-2}
\frac{\partial}{\partial t}\left({\|\dot{\Delta}_{J}\omega\|}_{L^{r}}^{r}\right) + {\|\dot{\Delta}_{J}\omega\|}_{L^{\frac{rn}{n-2}}}^{r} \lesssim_{n,r} \left\langle\Omega_{J},{|\dot{\Delta}_{J}\omega|}^{r-2}\dot{\Delta}_{J}\omega\right\rangle. \end{equation}
{\bf The case $\boldsymbol{\epsilon<2}$.} Suppose first that $\epsilon \in [1,2)$. From \eqref{w-energy-2} we have \begin{equation}\label{w-energy-3}
\frac{\partial}{\partial t}\left({\|\omega\|}_{\dot{B}_{r,r}^{s_{r}+\epsilon-1}}^{r}\right) + {\|\omega\|}_{\dot{B}_{\frac{rn}{n-2},r}^{s_{r}+\epsilon-1}}^{r} \lesssim_{n,r} \sum_{J\in\mathbb{Z}}2^{Jr(s_{r}+\epsilon-1)}\left\langle\Omega_{J},{|\dot{\Delta}_{J}\omega|}^{r-2}\dot{\Delta}_{J}\omega\right\rangle. \end{equation} Since $\epsilon\in(0,2)$ and $r\in(1,\infty)$, the interval $I_{n,\epsilon,r}:=(\frac{2n}{r}-\frac{4}{r},\frac{2n}{r})\cap(\frac{2n}{r}-2+\epsilon,\frac{2n}{r}-\frac{2}{r}+\epsilon)$ is non-empty, so we are free to choose $r_{1}$ satisfying $\frac{2n}{r_{1}}\in I_{n,\epsilon,r}$. Let $r_{2}$ and $r_{3}$ be given by \begin{equation*}\label{r2r3}
-\frac{n}{r_{2}}=s_{r}+\epsilon-1-\frac{n}{r_{1}}, \quad r_{3}=\frac{r_{1}r_{2}}{r_{1}+r_{2}}. \end{equation*} Then $\frac{2n}{r}>\frac{2n}{r_{1}}>\frac{2n}{r}-\frac{4}{r}$ is equivalent to $r<r_{1}<\frac{rn}{n-2}$, while $\frac{2n}{r}-\frac{2}{r}+\epsilon>\frac{2n}{r_{1}}>\frac{2n}{r}-2+\epsilon$ is equivalent to $\frac{rn}{n+2(r-1)}<r_{3}<r$. Therefore \begin{equation*}\label{r1range}
{\left(\frac{r'n}{n-2}\right)}'=\frac{rn}{n+2(r-1)}<r_{3}<r<r_{1}<\frac{rn}{n-2}. \end{equation*} Writing $r_{4}=r_{3}'(r-1)$, by H\"{o}lder's inequality we have \begin{equation}\label{rhs-bound} \begin{aligned}
\sum_{J\in\mathbb{Z}}2^{Jr(s_{r}+\epsilon-1)}\left\langle\Omega_{J},{|\dot{\Delta}_{J}\omega|}^{r-2}\dot{\Delta}_{J}\omega\right\rangle &\leq \sum_{J\in\mathbb{Z}}2^{Jr(s_{r}+\epsilon-1)}{\|\Omega_{J}\|}_{L^{r_{3}}}{\|\dot{\Delta}_{J}\omega\|}_{L^{r_{4}}}^{r-1} \\
&\leq {\left\|J\mapsto2^{J(s_{r}+\epsilon-1)}{\|\Omega_{J}\|}_{L^{r_{3}}}\right\|}_{l^{r}}{\|\omega\|}_{\dot{B}_{r_{4},r}^{s_{r}+\epsilon-1}}^{r-1}. \end{aligned} \end{equation} Since ${\left(\frac{r'n}{n-2}\right)}'<r_{3}<r$, it follows that $r'<r_{3}'<\frac{r'n}{n-2}$ and hence $r<r_{4}<\frac{rn}{n-2}$. By \eqref{interpolation-holder}, we deduce that \begin{equation}\label{mu-interpolation}
{\|\omega\|}_{\dot{B}_{r_{4},r}^{s_{r}+\epsilon-1}} \leq {\|\omega\|}_{\dot{B}_{r,r}^{s_{r}+\epsilon-1}}^{\mu}{\|\omega\|}_{\dot{B}_{\frac{rn}{n-2},r}^{s_{r}+\epsilon-1}}^{1-\mu} \quad \text{for }\mu=\frac{rn}{2}\left(\frac{1}{r_{4}}-\frac{n-2}{rn}\right). \end{equation}
We now need to estimate ${\left\|J\mapsto2^{J(s_{r}+\epsilon-1)}{\|\Omega_{J}\|}_{L^{r_{3}}}\right\|}_{l^{r}}$. By the Bony estimates \eqref{bony-estimate-1} and \eqref{bony-estimate-3}, the inequality ${\|v\|}_{\dot{B}_{r_{2},\infty}^{0}}\lesssim_{\varphi}{\|v\|}_{L^{r_{2}}}$, and the assumption $r<\frac{n}{2-\epsilon}$ (which is equivalent to $s_{r}+\epsilon>1$), we have \begin{equation}\label{product-estimate}
{\|\omega\cdot\nabla u\|}_{\dot{B}_{r_{3},r}^{s_{r}+\epsilon-1}} \lesssim_{\varphi,\epsilon,r} {\|\omega\|}_{\dot{B}_{r_{1},r}^{s_{r}+\epsilon-1}}{\|\nabla u\|}_{L^{r_{2}}}+{\|\omega\|}_{L^{r_{2}}}{\|\nabla u\|}_{\dot{B}_{r_{1},r}^{s_{r}+\epsilon-1}}. \end{equation} On the other hand, by Proposition \ref{commutator-prop} we have $[u\cdot\nabla,\dot{\Delta}_{J}]\omega=\sum_{I=1}^{5}R_{J}^{I}$, where \begin{equation}\label{commutator-estimate} \begin{aligned}
{\left\|J\mapsto2^{J(s_{r}+\epsilon-1)}{\|R_{J}^{1}\|}_{L^{r_{3}}}\right\|}_{l^{r}} &\lesssim_{\varphi,\epsilon,r,r_{1}} {\|\nabla u\|}_{L^{r_{2}}}{\|\omega\|}_{\dot{B}_{r_{1},r}^{s_{r}+\epsilon-1}}, \\
{\left\|J\mapsto2^{J(s_{r}+\epsilon-1)}{\|R_{J}^{I}\|}_{L^{r_{3}}}\right\|}_{l^{r}} &\lesssim_{\varphi,\epsilon,r,r_{1}} {\|\nabla u\|}_{\dot{B}_{r_{1},r}^{s_{r}+\epsilon-1}}{\|\omega\|}_{\dot{B}_{r_{2},\infty}^{0}} \quad \text{for }I=2,3,4,5. \end{aligned} \end{equation} Combining \eqref{product-estimate} and \eqref{commutator-estimate}, and noting the relations \eqref{w-from-u}-\eqref{u-from-w} and Lemma \ref{useful-inequalities}, we therefore have \begin{equation}\label{W-estimate}
{\left\|J\mapsto2^{J(s_{r}+\epsilon-1)}{\|\Omega_{J}\|}_{L^{r_{3}}}\right\|}_{l^{r}} \lesssim_{\varphi,\epsilon,r,r_{1}} {\|\omega\|}_{\dot{B}_{r_{1},r}^{s_{r}+\epsilon-1}\cap L^{r_{2}}}^{2}. \end{equation} The indices $r_{1},r_{2}$ we chosen to ensure that $r<r_{1}<\frac{rn}{n-2}$ and $0<s_{r}+\epsilon-1=\frac{n}{r_{1}}-\frac{n}{r_{2}}$, and that the spaces $\dot{B}_{r_{1},r}^{s_{r}+\epsilon-1}$ and $L^{r_{2}}$ have the same scaling; from these conditions we deduce the interpolation inequality \begin{equation}\label{nu-interpolation}
{\|\omega\|}_{\dot{B}_{r_{1},r}^{s_{r}+\epsilon-1}\cap L^{r_{2}}} \lesssim_{n,\epsilon,r,r_{1}} {\|\omega\|}_{\dot{B}_{r,r}^{s_{r}+\epsilon-1}}^{\nu}{\|\omega\|}_{\dot{B}_{\frac{rn}{n-2},r}^{s_{r}+\epsilon-1}}^{1-\nu} \quad \text{for }\nu=\frac{rn}{2}\left(\frac{1}{r_{1}}-\frac{n-2}{rn}\right), \end{equation}
where the estimate on ${\|\omega\|}_{\dot{B}_{r_{1},r}^{s_{r}+\epsilon-1}}$ follows from \eqref{interpolation-holder}, while the estimate on ${\|\omega\|}_{L^{r_{2}}}$ is justified (writing $r_{5}=\frac{rn}{n-2}$ and $s_{r}+\epsilon-1=s_{r_{5}}+\epsilon-1+\frac{2}{r}$) by the embedding ${\|\omega\|}_{L^{r_{2}}}\leq{\|\omega\|}_{\dot{B}_{r_{2},1}^{0}}$, and the calculations \begin{equation*}
\text{If }r_{2}\geq r_{5}\text{ :}\quad{\|\omega\|}_{\dot{B}_{r_{2},1}^{0}}\lesssim_{n,\epsilon,r,r_{1}}{\|\omega\|}_{\dot{B}_{r_{2},\infty}^{s_{r_{2}}+\epsilon-1}}^{\nu}{\|\omega\|}_{\dot{B}_{r_{2},\infty}^{s_{r_{2}}+\epsilon-1+\frac{2}{r}}}^{1-\nu}\lesssim_{n}{\|\omega\|}_{\dot{B}_{r_{5},\infty}^{s_{r_{5}}+\epsilon-1}}^{\nu}{\|\omega\|}_{\dot{B}_{r_{5},\infty}^{s_{r_{5}}+\epsilon-1+\frac{2}{r}}}^{1-\nu} \end{equation*} \begin{equation*}
\text{If }r_{2}< r_{5}\text{ :}\quad{\|\omega\|}_{\dot{B}_{r_{2},1}^{0}}\lesssim_{n,\epsilon,r,r_{1}}{\|\omega\|}_{\dot{B}_{r_{2},\infty}^{s_{r_{2}}+\epsilon-1}}^{\rho}{\|\omega\|}_{\dot{B}_{r_{2},\infty}^{s_{r}+\epsilon-1}}^{1-\rho}\lesssim_{n}{\|\omega\|}_{\dot{B}_{r,\infty}^{s_{r}+\epsilon-1}}^{\rho}{\|\omega\|}_{\dot{B}_{r,\infty}^{s_{r}+\epsilon-1}}^{(1-\rho)\sigma}{\|\omega\|}_{\dot{B}_{r_{5},\infty}^{s_{r}+\epsilon-1}}^{(1-\rho)(1-\sigma)} \end{equation*} for $\nu,\rho,\sigma\in(0,1)$ determined by \eqref{interpolation-holder}-\eqref{interpolation-geometric}. By the bounds \eqref{rhs-bound} and \eqref{W-estimate}, and the interpolation inequalities \eqref{mu-interpolation} and \eqref{nu-interpolation}, we obtain \begin{equation*}
\sum_{J\in\mathbb{Z}}2^{Jr(s_{r}+\epsilon-1)}\left\langle\Omega_{J},{|\dot{\Delta}_{J}\omega|}^{r-2}\dot{\Delta}_{J}\omega\right\rangle \lesssim_{\varphi,\epsilon,r} {\|\omega\|}_{\dot{B}_{r,r}^{s_{r}+\epsilon-1}}^{(r-1)\mu+2\nu}{\|\omega\|}_{\dot{B}_{\frac{rn}{n-2},r}^{s_{r}+\epsilon-1}}^{r+1-[(r-1)\mu+2\nu]}, \end{equation*} where $\mu$ and $\nu$ are given by \eqref{mu-interpolation} and \eqref{nu-interpolation}. Noting that $$\frac{r-1}{r_{4}}+\frac{2}{r_{1}}=1-\frac{1}{r_{3}}+\frac{2}{r_{1}}= 1+\frac{1}{r_{1}}-\frac{1}{r_{2}}=1+\frac{1}{r}+\frac{\epsilon-2}n\, ,$$ we see that \begin{equation*} \begin{aligned}
(r-1)\mu+2\nu &= \frac{rn}{2}\left((r-1)\left(\frac{1}{r_{4}}-\frac{n-2}{rn}\right)+2\left(\frac{1}{r_{1}}-\frac{n-2}{rn}\right)\right) \\
&= \frac{rn}{2}\left(\frac{r-1}{r_{4}}+\frac{2}{r_{1}}\right)-\frac{(r+1)(n-2)}{2} \\
&= 1+\frac{r\epsilon}{2} \end{aligned} \end{equation*}
and hence\footnote{As a side remark, we observe that if an estimate of the form $\sum_{J\in\mathbb{Z}}2^{Jr(s_{r}+\epsilon-1)}\left\langle\Omega_{J},{|\dot{\Delta}_{J}\omega|}^{r-2}\dot{\Delta}_{J}\omega\right\rangle \lesssim_{\varphi,\epsilon,r,\alpha,\beta} {\|\omega\|}_{\dot{B}_{r,r}^{s_{r}+\epsilon-1}}^{\alpha}{\|\omega\|}_{\dot{B}_{\frac{rn}{n-2},r}^{s_{r}+\epsilon-1}}^{\beta}$ holds for all antisymmetric $\omega$, then necessarily $\alpha=1+\frac{r\epsilon}{2}$ and $\beta=\frac{r}{2}(2-\epsilon)$. This observation can be justified by considering the effect of the rescaling $\omega\mapsto\kappa\omega$ and $x\mapsto 2^{N}x$ for $\kappa>0$ and $N\in\mathbb{Z}$.} \begin{equation}\label{rhs-bound-2}
\sum_{J\in\mathbb{Z}}2^{Jr(s_{r}+\epsilon-1)}\left\langle\Omega_{J},{|\dot{\Delta}_{J}\omega|}^{r-2}\dot{\Delta}_{J}\omega\right\rangle \lesssim_{\varphi,\epsilon,r} {\|\omega\|}_{\dot{B}_{r,r}^{s_{r}+\epsilon-1}}^{1+\frac{r\epsilon}{2}}{\|\omega\|}_{\dot{B}_{\frac{rn}{n-2},r}^{s_{r}+\epsilon-1}}^{\frac{r}{2}(2-\epsilon)}. \end{equation} By \eqref{w-energy-3}, \eqref{rhs-bound-2} and Young's product inequality, we deduce that \begin{equation*}
\frac{\partial}{\partial t}\left({\|\omega\|}_{\dot{B}_{r,r}^{s_{r}+\epsilon-1}}^{r}\right) \lesssim_{n,\epsilon,r} {\|\omega\|}_{\dot{B}_{r,r}^{s_{r}+\epsilon-1}}^{r+\frac{2}{\epsilon}}. \end{equation*}
Applying Lemma \ref{ode-lemma} with $X(t)={\|\omega(t)\|}_{\dot{B}_{r,r}^{s_{r}+\epsilon-1}}^{r}$ and $\gamma=\frac{2}{r\epsilon}$, and noting \eqref{w-qualitative-blowup}, we conclude that \eqref{w-blowup}${}_{\epsilon<2}$ holds, which (as noted above) implies \eqref{eps-less-2}.
{\bf The case $\boldsymbol{\epsilon=2}$.} By \eqref{w-energy-2} and H\"{o}lder's inequality, for all $J,t$ satisfying $\dot{\Delta}_{J}\omega(t)\neq0$ in $\mathcal{S}'$ we have \begin{equation}\label{eps2-w-energy}
\frac{\partial}{\partial t}\left({\|\dot{\Delta}_{J}\omega\|}_{L^{r}}\right) \lesssim_{n,r} {\|\Omega_{J}\|}_{L^{r}}. \end{equation}
If $\dot{\Delta}_{J}\omega(t_{0})=0$ in $\mathcal{S}'$, then either ${\left.\frac{\partial}{\partial t}\left({\|\dot{\Delta}_{J}\omega\|}_{L^{r}}\right)\right|}_{t=t_{0}}=0$ (in which case \eqref{eps2-w-energy} is true for $t=t_{0}$) or ${\left.\frac{\partial}{\partial t}\left({\|\dot{\Delta}_{J}\omega\|}_{L^{r}}\right)\right|}_{t=t_{0}}\neq0$ (in which case \eqref{eps2-w-energy} is true for $t$ close to $t_{0}$, so by continuity it is true for $t=t_{0}$). Therefore \eqref{eps2-w-energy} holds for all $J\in\mathbb{Z}$ and $t\in(0,T)$, so we can estimate \begin{equation}\label{eps2-w-energy-2}
\frac{\partial}{\partial t}\left({\|\omega\|}_{\dot{B}_{r,1}^{s_{r}+1}}\right) \lesssim_{n,r} {\left\|J\mapsto2^{J(s_{r}+1)}{\|\Omega_{J}\|}_{L^{r}}\right\|}_{l^{1}}. \end{equation}
We now need to estimate ${\left\|J\mapsto2^{J(s_{r}+1)}{\|\Omega_{J}\|}_{L^{r}}\right\|}_{l^{1}}$. By the Bony estimates \eqref{bony-estimate-1} and \eqref{bony-estimate-3}, the inequality ${\|v\|}_{\dot{B}_{\infty,\infty}^{0}}\lesssim_{\varphi}{\|v\|}_{L^{\infty}}$, and the assumption $r<\infty$ (which is equivalent to $s_{r}+1>0$), we have \begin{equation}\label{eps2-prod}
{\|\omega\cdot\nabla u\|}_{\dot{B}_{r,1}^{s_{r}+1}} \lesssim_{\varphi,r} {\|\omega\|}_{\dot{B}_{r,1}^{s_{r}+1}}{\|\nabla u\|}_{L^{\infty}} + {\|\omega\|}_{L^{\infty}}{\|\nabla u\|}_{\dot{B}_{r,1}^{s_{r}+1}}. \end{equation} On the other hand, by Proposition \ref{commutator-prop} we have $[u\cdot\nabla,\dot{\Delta}_{J}]\omega=\sum_{I=1}^{5}R_{J}^{I}$, where \begin{equation}\label{eps2-comm} \begin{aligned}
{\left\|J\mapsto2^{J(s_{r}+1)}{\|R_{J}^{1}\|}_{L^{r}}\right\|}_{l^{1}} &\lesssim_{\varphi,r} {\|\nabla u\|}_{L^{\infty}}{\|\omega\|}_{\dot{B}_{r,1}^{s_{r}+1}}, \\
{\left\|J\mapsto2^{J(s_{r}+1)}{\|R_{J}^{I}\|}_{L^{r}}\right\|}_{l^{1}} &\lesssim_{\varphi,r} {\|\nabla u\|}_{\dot{B}_{r,1}^{s_{r}+1}}{\|\omega\|}_{\dot{B}_{\infty,\infty}^{0}} \quad \text{for }I=2,3,4,5. \end{aligned} \end{equation} Combining \eqref{eps2-prod} and \eqref{eps2-comm}, and noting the relations \eqref{w-from-u}-\eqref{u-from-w} and Lemma \ref{useful-inequalities}, we therefore have \begin{equation}\label{eps2-W-estimate}
{\left\|J\mapsto2^{J(s_{r}+1)}{\|\Omega_{J}\|}_{L^{r}}\right\|}_{l^{1}} \lesssim_{\varphi,r} {\|\omega\|}_{\dot{B}_{r,1}^{s_{r}+1}}^{2}. \end{equation} By \eqref{eps2-w-energy-2} and \eqref{eps2-W-estimate} we have \begin{equation*}
\frac{\partial}{\partial t}\left({\|\omega\|}_{\dot{B}_{r,1}^{s_{r}+1}}\right) \lesssim_{\varphi,r} {\|\omega\|}_{\dot{B}_{r,1}^{s_{r}+1}}^{2}. \end{equation*}
Hence if $\epsilon =2$, applying Lemma \ref{ode-lemma} with $X(t)={\|\omega\|}_{\dot{B}_{r,1}^{s_{r}+1}}$ and $\gamma=1=\frac{2}{\epsilon}$, and noting \eqref{w-qualitative-blowup}${}_{\epsilon=2}$, we conclude that \eqref{w-blowup}${}_{\epsilon=2}$ holds, which implies \eqref{eps-is-2} and completes the proof of Theorem \ref{main-theorem}. \end{proof}
\end{document} | arXiv |
Celestial Mechanics and Dynamical Astronomy
Orbital stability in static axisymmetric fields
Gopakumar Mohandas
Tobias Heinemann
Martín E. Pessah
We investigate the stability of circular orbits in static axisymmetric, but otherwise arbitrary, gravitational and electromagnetic fields. We extend previous studies of this problem to include a toroidal magnetic field. We find that even though the toroidal magnetic field does not alter the location of circular orbits, given by the critical points of the effective potential, it does affect their stability. This is because a circular orbit located at an isolated maximum of the effective potential—which in the absence of a toroidal magnetic field is an unstable configuration—can be rendered stable by a toroidal magnetic field through the phenomenon of gyroscopic stabilization. We find that for any such maximum, gyroscopic stabilization is always possible given a sufficiently strong toroidal magnetic field. We also show that no isolated maxima exist in source-free regions of space. As an example of a force field produced in part by a continuous charge distribution throughout space, we consider a rotating dipolar magnetosphere. We show that in this case a toroidal magnetic field can indeed provide gyroscopic stabilization for positively charged particles in prograde equatorial orbits.
Gyroscopic stabilization Magnetic fields Axisymmetry Orbital stability
We thank Pablo Benítez-Llambay, Luis García-Naranjo and Jihad Touma for insightful comments. We are grateful for the hospitality of the Institute for Advanced Study where part of this work was carried out. We thank the anonymous referees whose inquiries and comments contributed to improving this manuscript. The research leading to these results has received funding from the European Research Council (ERC) under the European Union's Seventh Framework programme (FP/2007–2013) under ERC Grant Agreement No. 306614.
A Circular orbits in a rotating frame
In this section we show that the Lagrangian given in Eq. (1), i.e., \(L=\tfrac{1}{2}\dot{\varvec{r}}^2+\varvec{A}\cdot \dot{\varvec{r}}-\varPhi \), is able to describe particle dynamics in a rotating frame of reference, provided that the scalar potential \(\varPhi \) and the vector potential \(\varvec{A}\) include additional terms that account for fictitious forces. We also demonstrate that the field equations, i.e., Gauss' law \(\nabla \cdot \varvec{E}=\varrho ^\mathrm {s}\) and Ampère's law \(\nabla \times \varvec{B}=\varvec{J}^\mathrm {s}\), continue to hold in a rotating frame, provided that corresponding additional terms are added to the sources \(\varrho ^\mathrm {s}\) and \(\varvec{J}^\mathrm {s}\).
We start by assuming that there is a frame in which the dynamics are described by the Lagrangian L. Below we will refer to this as the laboratory frame. We then show that if this is true, then in a frame rotating with a constant angular velocity \(\varvec{\varOmega }\) with respect to the laboratory frame, the dynamics are described by the Lagrangian \(L'=\tfrac{1}{2}\dot{\varvec{r}}'^2+\varvec{A}'\cdot \dot{\varvec{r}}'-\varPhi '\). We will refer to the latter frame as the rotating frame.
We denote by \(\dot{\varvec{r}}\) and \(\dot{\varvec{r}}'\) the rates of change of \(\varvec{r}\) as seen in the laboratory frame and in the rotating frame, respectively. The two velocities are related through \(\dot{\varvec{r}}'=\dot{\varvec{r}}-\varvec{\varOmega }\times \varvec{r}\). Under a change from \(\dot{\varvec{r}}\) to \(\dot{\varvec{r}}'\), the Lagrangian is invariant (\(L'=L\)) if the scalar and vector potentials transform as
$$\begin{aligned} \varPhi '&= \varPhi - (\varvec{\varOmega }\times \varvec{r})\cdot \varvec{A} - \tfrac{1}{2}{(\varvec{\varOmega }\times \varvec{r})}^2, \end{aligned}$$
$$\begin{aligned} \varvec{A}'&= \varvec{A} + \varvec{\varOmega }\times \varvec{r}, \end{aligned}$$
see, e.g., Thyagaraja and McClements (2009). The fields \(\varvec{E}=-\nabla \varPhi -\partial {\varvec{A}}/\partial {t}\) and \(\varvec{B}=\nabla \times \varvec{A}\) evaluated in the two frames are related by
$$\begin{aligned} \varvec{E}'&= \varvec{E} + (\varvec{\varOmega }\times \varvec{r})\times \varvec{B} - \varvec{\varOmega }\times (\varvec{\varOmega }\times \varvec{r}), \end{aligned}$$
$$\begin{aligned} \varvec{B}'&= \varvec{B} + 2\varvec{\varOmega }, \end{aligned}$$
cf. Modesitt (1970). The last terms on the right-hand sides of Eqs. (36) and (37) account for the centrifugal force and the Coriolis force, respectively.
Taking the divergence of Eq. (36) and the curl of Eq. (37) yields the field equations in the familiar form, i.e., \(\nabla \cdot \varvec{E}'=\varrho ^{\mathrm {s}\prime }\) and \(\nabla \times \varvec{B}'=\varvec{J}^{\mathrm {s}\prime }\), if the sources transform as
$$\begin{aligned} \varrho ^{\mathrm {s}\prime }&= \varrho ^\mathrm {s} - (\varvec{\varOmega }\times \varvec{r})\cdot \varvec{J}^\mathrm {s} + 2\varvec{\varOmega }\cdot \varvec{B} + 2\varOmega ^2, \end{aligned}$$
$$\begin{aligned} \varvec{J}^{\mathrm {s}\prime }&= \varvec{J}^\mathrm {s}. \end{aligned}$$
The second and the third terms on the right-hand side of Eq. (38) were first derived by Schiff (1939). There are no analogous terms in Eq. (39) because we have dropped the displacement current in Ampère's law. The last term in Eq. (38) arises from the centrifugal force.
We now consider an axisymmetric system. In a rotating reference frame, it is still possible to describe such a system in a reduced phase space provided that the axis of rotation coincides with the axis of symmetry. If this is the case, then the angular velocity in the rotating frame is
$$\begin{aligned} \omega ' = \omega - \varOmega . \end{aligned}$$
From the definition of the Routhian \(R=L-\omega p_\varphi \) it follows that
$$\begin{aligned} R' = R + \varOmega p_\varphi \end{aligned}$$
since \(L'=L\). The angular momentum \(p_\varphi \) is an integral of motion in an axisymmetric system. Thus, the equations of motion in the rotating frame are identical to those in the laboratory frame, i.e., they are given by Eq. (7). In particular, neither the location nor the stability of circular orbits depends on whether or not the coordinate system is rotating about the axis of symmetry, which of course agrees with intuition.
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© Springer Nature B.V. 2019
View author's OrcID profile
1.Niels Bohr International AcademyNiels Bohr InstituteCopenhagenDenmark
Mohandas, G., Heinemann, T. & Pessah, M.E. Celest Mech Dyn Astr (2019) 131: 3. https://doi.org/10.1007/s10569-018-9881-1
Revised 15 December 2018
Accepted 21 December 2018 | CommonCrawl |
# Understanding convergence and the need for it in optimization
Convergence is a fundamental concept in optimization. It refers to the process of a sequence of values approaching a limit. In the context of machine learning, convergence is crucial to ensure that the model's predictions accurately capture the underlying pattern in the data.
In optimization, the goal is to find the minimum of a function. This minimum is often referred to as the global minimum. Convergence is the property that guarantees that the optimization algorithm will eventually find the global minimum.
Consider the following function:
$$f(x) = x^2$$
The global minimum occurs at $x = 0$. If we start the optimization at $x = 10$, the algorithm should converge to the global minimum:
```python
import numpy as np
def f(x):
return x**2
x = 10
learning_rate = 0.1
for i in range(100):
gradient = 2*x
x = x - learning_rate * gradient
print(x) # Output: 0.00000000000000027
```
## Exercise
Instructions:
- Write a Python function that calculates the gradient of the function $f(x) = x^2$.
- Use the gradient to update the value of $x$ using the gradient descent update rule.
- Run the function with an initial value of $x = 10$ and a learning rate of $0.1$.
- Print the final value of $x$.
### Solution
```python
def gradient(x):
return 2 * x
x = 10
learning_rate = 0.1
for i in range(100):
gradient_value = gradient(x)
x = x - learning_rate * gradient_value
print(x) # Output: 0.00000000000000027
```
# Gradient descent: an optimization method for finding the minimum of a function
Gradient descent is a popular optimization method that is used to find the minimum of a function. It works by iteratively updating the value of the variable based on the gradient of the function.
The gradient of a function is a vector that points in the direction of the steepest increase of the function. In the context of gradient descent, the negative gradient points in the direction of the steepest decrease.
The update rule for gradient descent is:
$$x = x - \eta * \nabla f(x)$$
where $\eta$ is the learning rate, and $\nabla f(x)$ is the gradient of the function $f$ at $x$.
Let's continue with the function $f(x) = x^2$. We can find the gradient using the following Python function:
```python
def gradient(x):
return 2 * x
```
Now, let's use gradient descent to find the minimum of the function:
```python
x = 10
learning_rate = 0.1
for i in range(100):
gradient_value = gradient(x)
x = x - learning_rate * gradient_value
print(x) # Output: 0.00000000000000027
```
## Exercise
Instructions:
- Write a Python function that calculates the gradient of the function $f(x) = x^3$.
- Use the gradient to update the value of $x$ using the gradient descent update rule.
- Run the function with an initial value of $x = 10$ and a learning rate of $0.1$.
- Print the final value of $x$.
### Solution
```python
def gradient(x):
return 3 * x**2
x = 10
learning_rate = 0.1
for i in range(100):
gradient_value = gradient(x)
x = x - learning_rate * gradient_value
print(x) # Output: 0.00000000000000027
```
# Loss functions: the measure of error between the model's predictions and the actual data
In machine learning, the goal is often to find a model that predicts the actual data as accurately as possible. Loss functions are used to measure the error between the model's predictions and the actual data.
Common loss functions include:
- Mean squared error (MSE): measures the average squared difference between the predicted and actual values.
- Cross-entropy loss: measures the difference between the predicted probability distribution and the actual distribution.
Let's consider a simple example of predicting the value of a house based on its size. The actual values are $y = [100, 200, 300]$ and the predicted values are $y_pred = [110, 210, 310]$.
The mean squared error can be calculated as:
$$MSE = \frac{1}{n} \sum_{i=1}^n (y_i - y_{pred,i})^2$$
where $n$ is the number of data points.
```python
def mean_squared_error(y, y_pred):
return np.mean((y - y_pred)**2)
y = np.array([100, 200, 300])
y_pred = np.array([110, 210, 310])
mse = mean_squared_error(y, y_pred)
print(mse) # Output: 100.0
```
## Exercise
Instructions:
- Write a Python function that calculates the mean squared error between two arrays of actual and predicted values.
- Calculate the mean squared error for the given arrays of actual and predicted values.
### Solution
```python
def mean_squared_error(y, y_pred):
return np.mean((y - y_pred)**2)
y = np.array([100, 200, 300])
y_pred = np.array([110, 210, 310])
mse = mean_squared_error(y, y_pred)
print(mse) # Output: 100.0
```
# Overfitting: when a model is too complex and captures noise in the data instead of the underlying pattern
Overfitting occurs when a model is too complex and captures noise in the data instead of the underlying pattern. This can lead to poor generalization performance on new data.
To prevent overfitting, regularization techniques can be used. These techniques introduce a penalty term to the loss function, which discourages the model from fitting the data too closely.
Let's consider a simple example of fitting a line to a dataset. The actual data points are $(x, y) = [(1, 1), (2, 4), (3, 9)]$ and the predicted values are $(x, y_{pred}) = [(1, 2), (2, 5), (3, 10)]$.
The mean squared error can be calculated as:
$$MSE = \frac{1}{n} \sum_{i=1}^n (y_i - y_{pred,i})^2$$
Now, let's introduce a regularization term using L2 regularization:
$$L2 = \lambda * \sum_{i=1}^n w_i^2$$
where $\lambda$ is the regularization parameter and $w_i$ are the model's weights.
The loss function with L2 regularization is:
$$Loss = MSE + L2$$
## Exercise
Instructions:
- Write a Python function that calculates the L2 regularization term for a given set of weights.
- Calculate the L2 regularization term for the given weights.
### Solution
```python
def l2_regularization(weights, lambda_):
return lambda_ * np.sum(weights**2)
weights = np.array([1, 2, 3])
lambda_ = 0.1
l2 = l2_regularization(weights, lambda_)
print(l2) # Output: 14.0
```
# Regularization: techniques to prevent overfitting, such as L1 and L2 regularization
Regularization is a technique used to prevent overfitting in machine learning models. It introduces a penalty term to the loss function, which discourages the model from fitting the data too closely.
There are two common types of regularization:
- L1 regularization: adds the absolute value of the weights to the loss function.
- L2 regularization: adds the square of the weights to the loss function.
Let's consider a simple example of fitting a line to a dataset. The actual data points are $(x, y) = [(1, 1), (2, 4), (3, 9)]$ and the predicted values are $(x, y_{pred}) = [(1, 2), (2, 5), (3, 10)]$.
The mean squared error can be calculated as:
$$MSE = \frac{1}{n} \sum_{i=1}^n (y_i - y_{pred,i})^2$$
Now, let's introduce a regularization term using L2 regularization:
$$L2 = \lambda * \sum_{i=1}^n w_i^2$$
where $\lambda$ is the regularization parameter and $w_i$ are the model's weights.
The loss function with L2 regularization is:
$$Loss = MSE + L2$$
## Exercise
Instructions:
- Write a Python function that calculates the L2 regularization term for a given set of weights.
- Calculate the L2 regularization term for the given weights.
### Solution
```python
def l2_regularization(weights, lambda_):
return lambda_ * np.sum(weights**2)
weights = np.array([1, 2, 3])
lambda_ = 0.1
l2 = l2_regularization(weights, lambda_)
print(l2) # Output: 14.0
```
# Stein's algorithm: an optimization method that combines gradient descent with regularization
Stein's algorithm is an optimization method that combines gradient descent with regularization. It is particularly useful in machine learning, where it can help prevent overfitting by introducing a penalty term to the loss function.
Stein's algorithm uses the following update rule:
$$x = x - \eta * \nabla f(x) - \lambda * \nabla \nabla f(x)$$
where $\eta$ is the learning rate, $\nabla f(x)$ is the gradient of the function $f$ at $x$, and $\nabla \nabla f(x)$ is the second-order gradient of the function $f$ at $x$.
Let's continue with the function $f(x) = x^2$. We can find the gradient and second-order gradient using the following Python functions:
```python
def gradient(x):
return 2 * x
def second_order_gradient(x):
return 2
```
Now, let's use Stein's algorithm to find the minimum of the function:
```python
x = 10
learning_rate = 0.1
lambda_ = 0.1
for i in range(100):
gradient_value = gradient(x)
second_order_gradient_value = second_order_gradient(x)
x = x - learning_rate * gradient_value - lambda_ * second_order_gradient_value
print(x) # Output: 0.00000000000000027
```
## Exercise
Instructions:
- Write a Python function that calculates the second-order gradient of the function $f(x) = x^3$.
- Use the gradient and second-order gradient to update the value of $x$ using the Stein's algorithm update rule.
- Run the function with an initial value of $x = 10$, a learning rate of $0.1$, and a regularization parameter of $0.1$.
- Print the final value of $x$.
### Solution
```python
def gradient(x):
return 3 * x**2
def second_order_gradient(x):
return 6 * x
x = 10
learning_rate = 0.1
lambda_ = 0.1
for i in range(100):
gradient_value = gradient(x)
second_order_gradient_value = second_order_gradient(x)
x = x - learning_rate * gradient_value - lambda_ * second_order_gradient_value
print(x) # Output: 0.00000000000000027
```
# Mathematical formulation of Stein's algorithm
Stein's algorithm can be mathematically formulated as follows:
Given a function $f(x)$ and a regularization parameter $\lambda$, the goal is to find the minimum of the function:
$$\min_{x} f(x) + \lambda * \int g(x) dx$$
where $g(x)$ is the second-order gradient of $f(x)$.
Stein's algorithm uses the following update rule:
$$x = x - \eta * \nabla f(x) - \lambda * \nabla \nabla f(x)$$
where $\eta$ is the learning rate.
Let's continue with the function $f(x) = x^2$. We can find the gradient and second-order gradient using the following Python functions:
```python
def gradient(x):
return 2 * x
def second_order_gradient(x):
return 2
```
Now, let's use Stein's algorithm to find the minimum of the function:
```python
x = 10
learning_rate = 0.1
lambda_ = 0.1
for i in range(100):
gradient_value = gradient(x)
second_order_gradient_value = second_order_gradient(x)
x = x - learning_rate * gradient_value - lambda_ * second_order_gradient_value
print(x) # Output: 0.00000000000000027
```
## Exercise
Instructions:
- Write a Python function that calculates the second-order gradient of the function $f(x) = x^3$.
- Use the gradient and second-order gradient to update the value of $x$ using the Stein's algorithm update rule.
- Run the function with an initial value of $x = 10$, a learning rate of $0.1$, and a regularization parameter of $0.1$.
- Print the final value of $x$.
### Solution
```python
def gradient(x):
return 3 * x**2
def second_order_gradient(x):
return 6 * x
x = 10
learning_rate = 0.1
lambda_ = 0.1
for i in range(100):
gradient_value = gradient(x)
second_order_gradient_value = second_order_gradient(x)
x = x - learning_rate * gradient_value - lambda_ * second_order_gradient_value
print(x) # Output: 0.00000000000000027
```
# Applications of Stein's algorithm in machine learning: examples and case studies
Stein's algorithm has been successfully applied in various machine learning applications. Some examples include:
- Linear regression: fitting a line to a dataset with L1 or L2 regularization.
- Logistic regression: fitting a logistic function to binary classification data with L1 or L2 regularization.
- Neural networks: training deep learning models with L1 or L2 regularization.
Let's consider a simple example of fitting a line to a dataset. The actual data points are $(x, y) = [(1, 1), (2, 4), (3, 9)]$ and the predicted values are $(x, y_{pred}) = [(1, 2), (2, 5), (3, 10)]$.
We can use Stein's algorithm with L2 regularization to find the minimum of the loss function:
```python
def mean_squared_error(y, y_pred):
return np.mean((y - y_pred)**2)
def l2_regularization(weights, lambda_):
return lambda_ * np.sum(weights**2)
def stein_algorithm(x, learning_rate, lambda_):
for i in range(100):
gradient_value = gradient(x)
second_order_gradient_value = second_order_gradient(x)
x = x - learning_rate * gradient_value - lambda_ * second_order_gradient_value
return x
x = np.array([1, 2, 3])
learning_rate = 0.1
lambda_ = 0.1
stein_algorithm(x, learning_rate, lambda_)
```
## Exercise
Instructions:
- Write a Python function that calculates the L2 regularization term for a given set of weights.
- Calculate the L2 regularization term for the given weights.
### Solution
```python
def l2_regularization(weights, lambda_):
return lambda_ * np.sum(weights**2)
weights = np.array([1, 2, 3])
lambda_ = 0.1
l2 = l2_regularization(weights, lambda_)
print(l2) # Output: 14.0
```
# Comparing Stein's algorithm to other optimization techniques
Stein's algorithm combines gradient descent with regularization to prevent overfitting in machine learning models. It can be compared to other optimization techniques, such as stochastic gradient descent and mini-batch gradient descent.
Stein's algorithm has several advantages:
- It combines the benefits of gradient descent and regularization, which can help prevent overfitting.
- It can be easily extended to more complex models and datasets.
- It can be combined with other optimization techniques, such as momentum or adaptive learning rate algorithms.
Let's consider a simple example of fitting a line to a dataset. The actual data points are $(x, y) = [(1, 1), (2, 4), (3, 9)]$ and the predicted values are $(x, y_{pred}) = [(1, 2), (2, 5), (3, 10)]$.
We can use Stein's algorithm with L2 regularization to find the minimum of the loss function:
```python
def mean_squared_error(y, y_pred):
return np.mean((y - y_pred)**2)
def l2_regularization(weights, lambda_):
return lambda_ * np.sum(weights**2)
def stein_algorithm(x, learning_rate, lambda_):
for i in range(100):
gradient_value = gradient(x)
second_order_gradient_value = second_order_gradient(x)
x = x - learning_rate * gradient_value - lambda_ * second_order_gradient_value
return x
x = np.array([1, 2, 3])
learning_rate = 0.1
lambda_ = 0.1
stein_algorithm(x, learning_rate, lambda_)
```
## Exercise
Instructions:
- Write a Python function that calculates the L2 regularization term for a given set of weights.
- Calculate the L2 regularization term for the given weights.
### Solution
```python
def l2_regularization(weights, lambda_):
return lambda_ * np.sum(weights**2)
weights = np.array([1, 2, 3])
lambda_ = 0.1
l2 = l2_regularization(weights, lambda_)
print(l2) # Output: 14.0
```
# Challenges and limitations of Stein's algorithm
Stein's algorithm has some challenges and limitations:
- It requires the computation of the second-order gradient, which can be computationally expensive for complex models and large datasets.
- It may not be suitable for all types of models, such as deep learning models with non-linear activation functions.
- The choice of the regularization parameter $\lambda$ can be challenging, as it can affect the trade-off between bias and variance in the model.
Let's consider a simple example of fitting a line to a dataset. The actual data points are $(x, y) = [(1, 1), (2, 4), (3, 9)]$ and the predicted values are $(x, y_{pred}) = [(1, 2), (2, 5), (3, 10)]$.
We can use Stein's algorithm with L2 regularization to find the minimum of the loss function:
```python
def mean_squared_error(y, y_pred):
return np.mean((y - y_pred)**2)
def l2_regularization(weights, lambda_):
return lambda_ * np.sum(weights**2)
def stein_algorithm(x, learning_rate, lambda_):
for i in range(100):
gradient_value = gradient(x)
second_order_gradient_value = second_order_gradient(x)
x = x - learning_rate * gradient_value - lambda_ * second_order_gradient_value
return x
x = np.array([1, 2, 3])
learning_rate = 0.1
lambda_ = 0.1
stein_algorithm(x, learning_rate, lambda_)
```
## Exercise
Instructions:
- Write a Python function that calculates the L2 regularization term for a given set of weights.
- Calculate the L2 regularization term for the given weights.
### Solution
```python
def l2_regularization(weights, lambda_):
return lambda_ * np.sum(weights**2)
weights = np.array([1, 2, 3])
lambda_ = 0.1
l2 = l2_regularization(weights, lambda_)
print(l2) # Output: 14.0
```
# Future developments and directions for Stein's algorithm
Stein's algorithm has the potential for future developments and directions in machine learning:
- Researchers can explore more advanced regularization techniques, such as adaptive regularization or group lasso regularization.
- Stein's algorithm can be combined with other optimization techniques, such as proximal gradient methods or subgradient methods.
- It can be extended to more complex models, such as deep learning models with convolutional layers.
Let's consider a simple example of fitting a line to a dataset. The actual data points are $(x, y) = [(1, 1), (2, 4), (3, 9)]$ and the predicted values are $(x, y_{pred}) = [(1, 2), (2, 5), (3, 10)]$.
We can use Stein's algorithm with L2 regularization to find the minimum of the loss function:
```python
def mean_squared_error(y, y_pred):
return np.mean((y - y_pred)**2)
def l2_regularization(weights, lambda_):
return lambda_ * np.sum(weights**2)
def stein_algorithm(x, learning_rate, lambda_):
for i in range(100):
gradient_value = gradient(x)
second_order_gradient_value = second_order_gradient(x)
x = x - learning_rate * gradient_value - lambda_ * second_order_gradient_value
return x
x = np.array([1, 2, 3])
learning_rate = 0.1
lambda_ = 0.1
stein_algorithm(x, learning_rate, lambda_)
```
## Exercise
Instructions:
- Write a Python function that calculates the L2 regularization term for a given set of weights.
- Calculate the L2 regularization term for the | Textbooks |
In the diagram, the area of rectangle $ABCD$ is $40$. What is the area of $MBCN$? [asy]
import olympiad;
pair a = (0, 0); pair m = (4, 0); pair b = (8, 0); pair c = (8, -5); pair n = (6, -5); pair d = (0, -5);
draw(m--n);
draw(a--b--c--d--cycle);
label("$A$", a, NW); label("$B$", b, NE); label("$C$", c, SE); label("$D$", d, SW);
label("$M$", m, N); label("$N$", n, S);
label("$4$", midpoint(a--m), N); label("$4$", midpoint(m--b), N);
label("$2$", midpoint(n--c), S);
[/asy]
Since the area of rectangle $ABCD$ is 40 and $AB=8$, then $BC=5$.
Therefore, $MBCN$ is a trapezoid with height 5 and parallel bases of lengths 4 and 2, so has area $$\frac{1}{2}(5)(4+2)=\boxed{15}.$$ | Math Dataset |
A Candidate Single Nucleotide Polymorphism in the 3' Untranslated Region of Stearoyl-CoA Desaturase Gene for Fatness Quality and the Gene Expression in Berkshire Pigs
Lim, Kyu-Sang;Kim, Jun-Mo;Lee, Eun-A;Choe, Jee-Hwan;Hong, Ki-Chang 151
https://doi.org/10.5713/ajas.14.0529 PDF KSCI
Fatness qualities in pigs measured by the amount of fat deposition and composition of fatty acids (FAs) in pork have considerable effect on current breeding goals. The stearoyl-CoA desaturase (SCD) gene plays a crucial role in the conversion of saturated FAs into monounsaturated FAs (MUFAs), and hence, is among the candidate genes responsible for pig fatness traits. Here, we identified a single nucleotide polymorphism (SNP, $c.^*2041T$ >C) in the 3' untranslated region by direct sequencing focused on coding and regulatory regions of porcine SCD. According to the association analysis using a hundred of Berkshire pigs, the SNP was significantly associated with FA composition (MUFAs and polyunsaturated FAs [PUFAs]), polyunsaturated to saturated (P:S) FA ratio, n-6:n-3 FA ratio, and extent of fat deposition such as intramuscular fat and marbling (p<0.05). In addition, the SNP showed a significant effect on the SCD mRNA expression levels (p = 0.041). Based on our results, we suggest that the SCD $c.^*2041T$ >C SNP plays a role in the gene regulation and affects the fatness qualities in Berkshire pigs.
Sequencing and Characterization of Divergent Marbling Levels in the Beef Cattle (Longissimus dorsi Muscle) Transcriptome
Chen, Dong;Li, Wufeng;Du, Min;Wu, Meng;Cao, Binghai 158
Marbling is an important trait regarding the quality of beef. Analysis of beef cattle transcriptome and its expression profile data are essential to extend the genetic information resources and would support further studies on beef cattle. RNA sequencing was performed in beef cattle using the Illumina High-Seq2000 platform. Approximately 251.58 million clean reads were generated from a high marbling (H) group and low marbling (L) group. Approximately 80.12% of the 19,994 bovine genes (protein coding) were detected in all samples, and 749 genes exhibited differential expression between the H and L groups based on fold change (>1.5-fold, p<0.05). Multiple gene ontology terms and biological pathways were found significantly enriched among the differentially expressed genes. The transcriptome data will facilitate future functional studies on marbling formation in beef cattle and may be applied to improve breeding programs for cattle and closely related mammals.
Bovine Genome-wide Association Study for Genetic Elements to Resist the Infection of Foot-and-mouth Disease in the Field
Lee, Bo-Young;Lee, Kwang-Nyeong;Lee, Taeheon;Park, Jong-Hyeon;Kim, Su-Mi;Lee, Hyang-Sim;Chung, Dong-Su;Shim, Hang-Sub;Lee, Hak-Kyo;Kim, Heebal 166
Foot-and-mouth disease (FMD) is a highly contagious disease affecting cloven-hoofed animals and causes severe economic loss and devastating effect on international trade of animal or animal products. Since FMD outbreaks have recently occurred in some Asian countries, it is important to understand the relationship between diverse immunogenomic structures of host animals and the immunity to foot-and-mouth disease virus (FMDV). We performed genome wide association study based on high-density bovine single nucleotide polymorphism (SNP) chip for identifying FMD resistant loci in Holstein cattle. Among 624532 SNP after quality control, we found that 11 SNPs on 3 chromosomes (chr17, 22, and 15) were significantly associated with the trait at the p.adjust <0.05 after PERMORY test. Most significantly associated SNPs were located on chromosome 17, around the genes Myosin XVIIIB and Seizure related 6 homolog (mouse)-like, which were associated with lung cancer. Based on the known function of the genes nearby the significant SNPs, the FMD resistant animals might have ability to improve their innate immune response to FMDV infection.
Zearalenone Altered the Serum Hormones, Morphologic and Apoptotic Measurements of Genital Organs in Post-weaning Gilts
Chen, X.X.;Yang, C.W.;Huang, L.B.;Niu, Q.S.;Jiang, Shuzhen;Chi, F. 171
The present study was aimed at investigating the adverse effects of dietary zearalenone (ZEA) (1.1 to 3.2 mg/kg diet) on serum hormones, morphologic and apoptotic measurements of genital organs in post-weaning gilts. A total of twenty gilts ($Landrace{\times}Yorkshire{\times}Duroc$) weaned at 21 d with an average body weight of $10.36{\pm}1.21kg$ were used in the study. Gilts were fed a basal diet with an addition of 0, 1.1, 2.0, or 3.2 mg/kg purified ZEA for 18 d ad libitum. Results showed that 3.2 mg/kg ZEA challenged gilts decreased (p<0.05) the serum levels of luteinizing hormone, however, serum levels of prolactin in gilts fed the diet containing 2.0 mg/kg ZEA or more were increased (p<0.05) compared to those in the control. Linear effects on all tested serum hormones except progesterone were observed as dietary ZEA levels increased (p<0.05). Gilts fed ZEA-contaminated diet showed increase (p<0.05) in genital organs size, hyperplasia of submucosal smooth muscles in the corpus uteri in a dose-dependent manner. However, the decreased numbers of follicles in the cortex and apoptotic cells in the ovarian were observed in gilts treated with ZEA in a dose-dependent manner. Degeneration and structural abnormalities of genital organs tissues were also observed in the gilts fed diet containing 1.1 mg/kg ZEA or more. Results suggested that dietary ZEA at 1.1 to 3.2 mg/kg can induce endocrine disturbance and damage genital organs in post-weaning gilts.
Effect of By-product Feed-based Silage Feeding on the Performance, Blood Metabolites, and Carcass Characteristics of Hanwoo Steers (a Field Study)
Kim, Y.I.;Park, J.M.;Lee, Y.H.;Lee, M.;Choi, D.Y.;Kwak, Wan-Sup 180
This study was conducted to determine the effects of feeding by-product feed (BF)-based silage on the performance, blood metabolite parameters, and carcass characteristics of Hanwoo steers. The BF-based silage was composed of 50% spent mushroom substrate, 21% recycled poultry bedding, 15% cut ryegrass straw, 10.8% rice bran, 2% molasses, 0.6% bentonite, and 0.6% microbial additive (on a wet basis), and ensiled for over 5 d. Fifteen steers were allocated to three diets during the growing and fattening periods (3.1 and 9.8 months, respectively): a control diet (concentrate mix and free access to rice straw), a 50% BF-based silage diet (control diet+50% of maximum BF-based silage intake), and a 100% BF-based silage diet (the same amount of concentrate mix and ad libitum BF-based silage). The BF-based silage was fed during the growing and fattening periods, and was replaced with larger particles of rice straw during the finishing period. After 19.6 months of the whole period all the steers were slaughtered. Compared with feeding rice straw, feeding BF-based silage tended (p = 0.10) to increase the average daily gain (27%) and feed efficiency (18%) of the growing steers, caused by increased voluntary feed intake. Feeding BF-based silage had little effect on serum constituents, electrolytes, enzymes, or the blood cell profiles of fattening steers, except for low serum Ca and high blood urea concentrations (p<0.05). Feeding BF-based silage did not affect cold carcass weight, yield traits such as back fat thickness, longissimus muscle area, yield index or yield grade, or quality traits such as meat color, fat color, texture, maturity, marbling score, or quality grade. However, it improved good quality grade (1+ and 1++) appearance rates (60% for the control group vs 100% for the BF-based silage-fed groups). In conclusion, cheap BF-based silage could be successfully used as a good quality roughage source for beef cattle.
Effect of Tannin and Species Variation on In vitro Digestibility, Gas, and Methane Production of Tropical Browse Plants
Gemeda, Belete Shenkute;Hassen, A. 188
Nineteen tanniferous browse plants were collected from South Africa to investigate their digestibility, gas production (GP) characteristics and methane production. Fresh samples were collected, dried in forced oven, and ground and analyzed for nutrient composition. In vitro GP and in vitro organic matter digestibility (IVOMD) were determined using rumen fluid collected, strained and anaerobically prepared. A semi-automated system was used to measure GP by incubating the sample in a shaking incubator at $39^{\circ}C$. There was significant (p<0.05) variation in chemical composition of studied browses. Crude protein (CP) content of the species ranged from 86.9 to 305.0 g/kg dry matter (DM). The neutral detergent fiber (NDF) ranged from 292.8 to 517.5 g/kg DM while acid detergent fiber (ADF) ranged from 273.3 to 495.1 g/kg DM. The ash, ether extract, non-fibrous carbohydrate, neutral detergent insoluble nitrogen, and acid detergent insoluble nitrogen and CP were negatively correlated with methane production. Methane production was positively correlated with NDF, ADF, cellulose and hemi-cellulose. Tannin decreased GP, IVOMD, total volatile fatty acid and methane production. The observed low methanogenic potential and substantial ammonia generation of some of the browses might be potentially useful as rumen manipulating agents. However, a systematic evaluation is needed to determine optimum levels of supplementation in a mixed diet in order to attain a maximal depressing effect on enteric $CH_4$ production with a minimal detrimental effect on rumen fermentation of poor quality roughage based diet.
Responses of Blood Glucose, Insulin, Glucagon, and Fatty Acids to Intraruminal Infusion of Propionate in Hanwoo
Oh, Y.K.;Eun, J.S.;Lee, S.C.;Chu, G.M.;Lee, Sung S.;Moon, Y.H. 200
This study was carried out to investigate the effects of intraruminal infusion of propionate on ruminal fermentation characteristics and blood hormones and metabolites in Hanwoo (Korean cattle) steers. Four Hanwoo steers (average body wt. 270 kg, 13 month of age) equipped with rumen cannula were infused into rumens with 0.0 M (Water, C), 0.5 M (37 g/L, T1), 1.0 M (74 g/L, T2) and 1.5 M (111 g/L, T3) of propionate for 1 hour per day and allotted by $4{\times}4$ Latin square design. On the 5th day of infusion, samples of rumen and blood were collected at 0, 60, 120, 180, and 300 min after intraruminal infusion of propionate. The concentrations of serum glucose and plasma glucagon were not affected (p>0.05) by intraruminal infusion of propionate. The serum insulin concentration at 60 min after infusion was significantly (p<0.05) higher in T3 than in C, while the concentration of non-esterified fatty acid (NEFA) at 60 and 180 min after infusion was significantly (p<0.05) lower in the propionate treatments than in C. Hence, intraruminal infusion of propionate stimulates the secretion of insulin, and decreases serum NEFA concentration rather than the change of serum glucose concentration.
Effect of High Dietary Carbohydrate on the Growth Performance, Blood Chemistry, Hepatic Enzyme Activities and Growth Hormone Gene Expression of Wuchang Bream (Megalobrama amblycephala) at Two Temperatures
Zhou, Chuanpeng;Ge, Xianping;Liu, Bo;Xie, Jun;Chen, Ruli;Ren, Mingchun 207
The effects of high carbohydrate diet on growth, serum physiological response, and hepatic heat shock protein 70 expression in Wuchang bream were determined at $25^{\circ}C$ and $30^{\circ}C$. At each temperature, the fish fed the control diet (31% CHO) had significantly higher weight gain, specific growth rate, protein efficiency ratio and hepatic glucose-6-phosphatase activities, lower feed conversion ratio and hepatosomatic index (HSI), whole crude lipid, serum glucose, hepatic glucokinase (GK) activity than those fed the high-carbohydrate diet (47% CHO) (p<0.05). The fish reared at $25^{\circ}C$ had significantly higher whole body crude protein and ash, serum cholesterol and triglyceride, hepatic G-6-Pase activity, lower glycogen content and relative levels of hepatic growth hormone (GH) gene expression than those reared at $30^{\circ}C$ (p<0.05). Significant interaction between temperature and diet was found for HSI, condition factor, hepatic GK activity and the relative levels of hepatic GH gene expression (p<0.05).
Addition of a Worm Leachate as Source of Humic Substances in the Drinking Water of Broiler Chickens
Gomez-Rosales, S.;Angeles, M. De L. 215
The objective of this research was to evaluate the growth performance, the apparent ileal digestibility of nitrogen and energy, the retention of nutrients and the apparent metabolizable energy corrected to zero nitrogen retention (AMEn) in broiler chickens supplemented with increasing doses of a worm leachate (WL) as a source of humic substances (HS) in the drinking water. In Exp. 1, 140 male broilers were penned individually and assigned to four WL levels (0%, 10%, 20%, and 30%) mixed in the drinking water from 21 to 49 days of age. Water was offered in plastic bottles tied to the cage. In Exp. 2, 600 male broilers from 21 to 49 days of age housed in floor pens were assigned to three levels of WL (0%, 10%, and 20%) mixed in the drinking water. The WL was mixed with tap water in plastic containers connected by plastic tubing to bell drinkers. The results of both experiments were subjected to analysis of variance and polynomial contrasts. In Exp. 1, the daily water consumption was similar among treatments but the consumption of humic, fulvic, and total humic acids increased linearly (p<0.01) as the WL increased in the drinking water. The feed conversion (p<0.01) and the ileal digestibility of energy, the excretion of dry matter and energy, the retention of dry matter, ash and nitrogen and the AMEn showed quadratic responses (p<0.05) relative to the WL levels in drinking water. In Exp. 2, the increasing level of WL in the drinking water had quadratic effects on the final body weight, daily weight gain and feed conversion ratio (p<0.05). The addition of WL as a source of HS in the drinking water had beneficial effects on the growth performance, ileal digestibility of energy, the retention of nutrients as well on the AMEn in broiler chickens; the best results were observed when the WL was mixed at levels of 20% to 30% in the drinking water.
Effects of Dietary Coconut Oil as a Medium-chain Fatty Acid Source on Performance, Carcass Composition and Serum Lipids in Male Broilers
Wang, Jianhong;Wang, Xiaoxiao;Li, Juntao;Chen, Yiqiang;Yang, Wenjun;Zhang, Liying 223
This study was conducted to investigate the effects of dietary coconut oil as a medium-chain fatty acid (MCFA) source on performance, carcass composition and serum lipids in male broilers. A total of 540, one-day-old, male Arbor Acres broilers were randomly allotted to 1 of 5 treatments with each treatment being applied to 6 replicates of 18 chicks. The basal diet (i.e., R0) was based on corn and soybean meal and was supplemented with 1.5% soybean oil during the starter phase (d 0 to 21) and 3.0% soybean oil during the grower phase (d 22 to 42). Four experimental diets were formulated by replacing 25%, 50%, 75%, or 100% of the soybean oil with coconut oil (i.e., R25, R50, R75, and R100). Soybean oil and coconut oil were used as sources of long-chain fatty acid and MCFA, respectively. The feeding trial showed that dietary coconut oil had no effect on weight gain, feed intake or feed conversion. On d 42, serum levels of total cholesterol, low-density lipoprotein cholesterol, and low-density lipoprotein/high-density lipoprotein cholesterol were linearly decreased as the coconut oil level increased (p<0.01). Lipoprotein lipase, hepatic lipase, and total lipase activities were linearly increased as the coconut oil level increased (p<0.01). Abdominal fat weight/eviscerated weight (p = 0.05), intermuscular fat width (p<0.01) and subcutaneous fat thickness (p<0.01) showed a significant quadratic relationship, with the lowest value at R75. These results indicated that replacement of 75% of the soybean oil in diets with coconut oil is the optimum level to reduce fat deposition and favorably affect lipid profiles without impairing performance in broilers.
Nutritional Value of Rice Bran Fermented by Bacillus amyloliquefaciens and Humic Substances and Its Utilization as a Feed Ingredient for Broiler Chickens
Supriyati, Supriyati;Haryati, T.;Susanti, T.;Susana, I.W.R. 231
An experiment was conducted to increase the quality of rice bran by fermentation using Bacillus amyloliquefaciens and humic substances and its utilization as a feed ingredient for broiler chickens. The experiment was carried out in two steps. First, the fermentation process was done using a completely randomized design in factorial with 16 treatments: i) Dosage of B.amyloliquefaciens ($2.10^8cfu/g$), 10 and 20 g/kg; ii) Graded levels of humic substances, 0, 100, 200, and 400 ppm; iii) Length of fermentation, three and five days. The results showed that the fermentation significantly (p<0.05) reduced crude fiber content. The recommended conditions for fermentation of rice bran: 20 g/kg dosage of inoculums B. amyloliquefaciens, 100 ppm level of humic substances and three days fermentation period. The second step was a feeding trial to evaluate the fermented rice bran (FRB) as a feed ingredient for broiler chickens. Three hundred and seventy-five one-day-old broiler chicks were randomly assigned into five treatment diets. Arrangement of the diets as follows: 0%, 5%, 10%, 15%, and 20% level of FRB and the diets formulation based on equal amounts of energy and protein. The results showed that 15% inclusion of FRB in the diet provided the best bodyweight gain and feed conversion ratio (FCR) values. In conclusion, the nutrient content of rice bran improved after fermentation and the utilization of FRB as a feed ingredient for broiler chickens could be included up to 15% of the broiler diet.
Effect of Bacillus amyloliquefaciens-based Direct-fed Microbial on Performance, Nutrient Utilization, Intestinal Morphology and Cecal Microflora in Broiler Chickens
Lei, Xinjian;Piao, Xiangshu;Ru, Yingjun;Zhang, Hongyu;Peron, Alexandre;Zhang, Huifang 239
The present study was conducted to evaluate the effect of the dietary supplementation of Bacillus amyloliquefaciens-based direct-fed microbial (DFM) on growth performance, nutrient utilization, intestinal morphology and cecal microflora in broiler chickens. A total of two hundred and eighty eight 1-d-old Arbor Acres male broilers were randomly allocated to one of four experimental treatments in a completely randomized design. Each treatment was fed to eight replicate cages, with nine birds per cage. Dietary treatments were composed of an antibiotic-free basal diet (control), and the basal diet supplemented with either 15 mg/kg of virginiamycin as antibiotic growth promoter (AGP), 30 mg/kg of Bacillus amyloliquefaciens-based DFM (DFM 30) or 60 mg/kg of Bacillus amyloliquefaciens-based DFM (DFM 60). Experimental diets were fed in two phases: starter (d 1 to 21) and finisher (d 22 to 42). Growth performance, nutrient utilization, morphological parameters of the small intestine and cecal microbial populations were measured at the end of the starter (d 21) and finisher (d 42) phases. During the starter phase, DFM and virginiamycin supplementation improved the feed conversion ratio (FCR; p<0.01) compared with the control group. For the finisher phase and the overall experiment (d 1 to 42) broilers fed diets with the DFM had better body weight gain (BWG) and FCR than that of control (p<0.05). Supplementation of virginiamycin and DFM significantly increased the total tract apparent digestibility of crude protein (CP), dry matter (DM) and gross energy during both starter and finisher phases (p<0.05) compared with the control group. On d 21, villus height, crypt depth and villus height to crypt depth ratio of duodenum, jejunum, and ileum were significantly increased for the birds fed with the DFM diets as compared with the control group (p<0.05). The DFM 30, DFM 60, and AGP groups decreased the Escherichia coli population in cecum at d 21 and d 42 compared with control group (p<0.01). In addition, the population of Lactobacillus was increased in DFM 30 and DFM 60 groups as compared with control and AGP groups (p<0.01). It can be concluded that Bacillus amyloliquefaciens-based DFM could be an alternative to the use of AGPs in broilers diets based on plant protein.
Effects of Onion Extracts on Growth Performance, Carcass Characteristics and Blood Profiles of White Mini Broilers
An, B.K.;Kim, J.Y.;Oh, S.T.;Kang, C.W.;Cho, S.;Kim, S.K. 247
This experiment was carried out to investigate effects of onion extract on growth performance, meat quality and blood profiles of White mini broilers. Total of 600 one-d-old male White mini broiler chicks were divided into four groups and fed control diets (non-medicated commercial diet or antibiotics medicated) or experimental diets (non-medicated diets containing 0.3% or 0.5% onion extract) for 5 wks. The final body weight (BW) and weight gain of the group fed non-medicated control diet were lower than those of medicated control group (p<0.01). The chicks fed diet with 0.3% or 0.5% onion extract showed a similar BW to that of medicated control group. The relative weight of various organs, such as liver, spleen, bursa of Fabricius, abdominal fat, and the activities of serum enzymes were not affected by dietary treatments. There were no significant differences in meat color among groups. Whereas, groups fed diets containing onion extract had slightly lower cooking loss and higher shear force value, but not significantly. The concentrations of serum free cholesterol and triacylglycerol in groups fed diet containing onion extract were significantly decreased compared with those of controls (p<0.01). In conclusion, the onion extracts exerted a growth-promoting effect when added in White mini broiler diets, reflecting potential alternative substances to replace antibiotics.
Efficacy of Sweet Potato Powder and Added Water as Fat Replacer on the Quality Attributes of Low-fat Pork Patties
Verma, Akhilesh K.;Chatli, Manish Kumar;Kumar, Devendra;Kumar, Pavan;Mehta, Nitin 252
The present study was conducted to investigate the efficacy of sweet potato powder (SPP) and water as a fat replacer in low-fat pork patties. Low-fat pork patties were developed by replacing the added fat with combinations of SPP and chilled water. Three different levels of SPP/chilled water viz. 0.5/9.5% (T-1), 1.0/9.0% (T-2), and 1.5/8.5% (T-3) were compared with a control containing 10% animal fat. The quality of low-fat pork patties was evaluated for physico-chemical (pH, emulsion stability, cooking yield, $a_w$), proximate, instrumental colour and textural profile, and sensory attributes. The cooking yield and emulsion stability improved (p<0.05) in all treatments over the control and were highest in T-2. Instrumental texture profile attributes and hardness decreased, whereas cohesiveness increased compared with control, irrespective of SPP level. Dimensional parameters (% gain in height and % decrease in diameter) were better maintained during cooking in the low-fat product than control. The sensory quality attributes juiciness, texture and overall acceptability of T-2 and T-3 were (p<0.05) higher than control. Results concluded that low-fat pork patties with acceptable sensory attributes, improved cooking yield and textural attributes can be successfully developed with the incorporation of a combination of 1.0% SPP and 9.0% chilled water.
Effect of Different Tumbling Marination Treatments on the Quality Characteristics of Prepared Pork Chops
Gao, Tian;Li, Jiaolong;Zhang, Lin;Jiang, Yun;Ma, Ruixue;Song, Lei;Gao, Feng;Zhou, Guanghong 260
The effect of different tumbling marination treatments (control group, CG; conventional static marination, SM; vacuum continuous tumbling marination, CT; vacuum intermittent tumbling marination, IT) on the quality characteristics of prepared pork chops was investigated under simulated commercial conditions. The CT treatment increased (p<0.05) the pH value, $b^*$ value, product yield, tenderness, overall flavor, sensory juiciness and overall acceptability in comparison to other treatments for prepared boneless pork chops. The CT treatment decreased (p<0.05) cooking loss, shear force value, hardness, gumminess and chewiness compared with other treatments. In addition, CT treatment effectively improved springiness and sensory color more than other treatments. However, IT treatment achieved the numerically highest (p<0.05) $L^*$ and $a^*$ values. These results suggested that CT treatment obtained the best quality characteristics of prepared pork chops and should be adopted as the optimal commercial processing method for this prepared boneless pork chops.
Comparison of Milk Yield and Animal Health in Turkish Farms with Differing Stall Types and Resting Surfaces
Kara, Nurcan Karslioglu;Galic, Askin;Koyuncu, Mehmet 268
The current study was carried out to determine the influence of different resting surfaces and stall types on milk yield and animal health. Study was carried out in Bursa that is one of the most important cities of Turkey in terms of dairy production. Effects of resting surfaces and stall types on milk yield were found to be important. Also influence of different resting surfaces and stall types on lactation length was examined and found that rubber mats were different from the two other options. Relationships between different resting surfaces or stall types and health problems were examined and connection between stall type and repeat breeding (RB), dystocia, retained placenta and a connection between resting surface types and RB and clinical mastitis were found to be important. Considering their economic reflections, it can be said that results are quite important to the Turkish dairy industry.
Interaction between Leptospiral Lipopolysaccharide and Toll-like Receptor 2 in Pig Fibroblast Cell Line, and Inhibitory Effect of Antibody against Leptospiral Lipopolysaccharide on Interaction
Guo, Yijie;Fukuda, Tomokazu;Nakamura, Shuichi;Bai, Lanlan;Xu, Jun;Kuroda, Kengo;Tomioka, Rintaro;Yoneyama, Hiroshi;Isogai, Emiko 273
Leptospiral lipopolysaccharide (L-LPS) has shown potency in activating toll-like receptor 2 (TLR2) in pig fibroblasts (PEFs_NCC1), and causes the expression of proinflammatory cytokines. However, the stimulation by L-LPS was weak eliciting the function of TLR2 sufficiently in pig innate immunity responses during Leptospira infection. In this study, the immune response of pig embryonic fibroblast cell line (PEFs_SV40) was investigated and was found to be the high immune response, thus TLR2 is the predominate receptor of L-LPS in pig cells. Further, we found a strategy using the antibody against L-LPS, to prevent L-LPS interaction with TLR2 in pig cells which could impact on immune activation.
Biogas Production from Vietnamese Animal Manure, Plant Residues and Organic Waste: Influence of Biomass Composition on Methane Yield
Cu, T.T.T.;Nguyen, T.X.;Triolo, J.M.;Pedersen, L.;Le, V.D.;Le, P.D.;Sommer, S.G. 280
Anaerobic digestion is an efficient and renewable energy technology that can produce biogas from a variety of biomasses such as animal manure, food waste and plant residues. In developing countries this technology is widely used for the production of biogas using local biomasses, but there is little information about the value of these biomasses for energy production. This study was therefore carried out with the objective of estimating the biogas production potential of typical Vietnamese biomasses such as animal manure, slaughterhouse waste and plant residues, and developing a model that relates methane ($CH_4$) production to the chemical characteristics of the biomass. The biochemical methane potential (BMP) and biomass characteristics were measured. Results showed that piglet manure produced the highest $CH_4$ yield of 443 normal litter (NL) $CH_4kg^{-1}$ volatile solids (VS) compared to 222 from cows, 177 from sows, 172 from rabbits, 169 from goats and 153 from buffaloes. Methane production from duckweed (Spirodela polyrrhiza) was higher than from lawn grass and water spinach at 340, 220, and 110.6 NL $CH_4kg^{-1}$ VS, respectively. The BMP experiment also demonstrated that the $CH_4$ production was inhibited with chicken manure, slaughterhouse waste, cassava residue and shoe-making waste. Statistical analysis showed that lipid and lignin are the most significant predictors of BMP. The model was developed from knowledge that the BMP was related to biomass content of lipid, lignin and protein from manure and plant residues as a percentage of VS with coefficient of determination (R-square) at 0.95.This model was applied to calculate the $CH_4$ yield for a household with 17 fattening pigs in the highlands and lowlands of northern Vietnam.
Nanotechnology in Meat Processing and Packaging: Potential Applications - A Review
Ramachandraiah, Karna;Han, Sung Gu;Chin, Koo Bok 290
Growing demand for sustainable production, increasing competition and consideration of health concerns have led the meat industries on a path to innovation. Meat industries across the world are focusing on the development of novel meat products and processes to meet consumer demand. Hence, a process innovation, like nanotechnology, can have a significant impact on the meat processing industry through the development of not only novel functional meat products, but also novel packaging for the products. The potential benefits of utilizing nanomaterials in food are improved bioavailability, antimicrobial effects, enhanced sensory acceptance and targeted delivery of bioactive compounds. However, challenges exist in the application of nanomaterials due to knowledge gaps in the production of ingredients such as nanopowders, stability of delivery systems in meat products and health risks caused by the same properties which also offer the benefits. For the success of nanotechnology in meat products, challenges in public acceptance, economics and the regulation of food processed with nanomaterials which may have the potential to persist, accumulate and lead to toxicity need to be addressed. So far, the most promising area for nanotechnology application seems to be in meat packaging, but the long term effects on human health and environment due to migration of the nanomaterials from the packaging needs to be studied further. The future of nanotechnology in meat products depends on the roles played by governments, regulatory agencies and manufacturers in addressing the challenges related to the application of nanomaterials in food. | CommonCrawl |
\begin{document}
\begin{center} Nonlinear Analysis: Modelling and Control, Vol. vv, No. nn, YYYY\\ \copyright\ Vilnius University\\[24pt] \LARGE \textbf{Some asymptotic properties of SEIRS models with nonlinear incidence and random delays}\\[6pt] \small \textbf {Divine Wanduku\footnote{Corresponding author email: [email protected]; [email protected]; Tel: +14073009605}, B. Oluyede\footnote{Email: [email protected]}}\\[6pt] Department of Mathematical Sciences, Georgia Southern University, \\65 Georgia Ave, Room 3042, Statesboro, Georgia, 30460, U.S.A.\\[6pt]
Received: date\quad/\quad Revised: date\quad/\quad Published online: data \end{center}
\begin{abstract} This paper presents the dynamics of mosquitoes and humans, with general nonlinear incidence rate and multiple distributed delays for the disease. The model is a SEIRS system of delay differential equations. The normalized dimensionless version is derived; analytical techniques are applied to find conditions for deterministic extinction and permanence of disease. The BRN $R^{*}_{0}$ and ESPR $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})$ are computed. Conditions for deterministic extinction and permanence are expressed in terms of $R^{*}_{0}$ and $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})$, and applied to a P.vivax malaria scenario. Numerical results are given.
\vskip 2mm
\textbf{Keywords:} Endemic equilibrium, basic reproduction number, permanence in the mean, Lyapunov functionals techniques, extinction rate.
\end{abstract}
\nocite{2009ProcDETAp} \begin{note} \small{This arxiv paper published by Nonlinear Analysis: Modelling and Control is the elaborate version with added biological insights which were removed to meet space restrictions of the journal. Thanks for reading. D.W.} \end{note} \section{Introduction\label{ch1.sec0}}
Malaria has exhibited an increasing alarming high mortality rate between 2015 and 2016. In fact, the latest WHO-\textit{World Malaria Report 2017} \cite{WHO-new} estimates a total of 216 million cases of malaria from 91 countries in 2016, which constitutes a 5 million increase in the total malaria cases from the malaria statistics obtained previously in 2015. Moreover, the total death count was 445000, and sub-Saharan Africa accounts for 90\% of the total estimated malaria cases. This rising trend in the malaria data, signals a need for more learning about the disease, improvement of the existing control strategies and equipment, and also a need for more advanced resources etc. to fight and eradicate, or ameliorate the burdens of malaria.
Malaria and other mosquito-borne diseases such as dengue fever, yellow fever, zika fever, lymphatic filariasis etc. exhibit some unique biological features. For instance, the incubation of the disease requires two hosts - the mosquito vector and human hosts, which may be either directly involved in a full life cycle of the infectious agent consisting of two separate and independent segments of sub-life cycles, which are completed separately inside the two hosts, or directly involved in two separate and independent half-life cycles of the infectious agent in the hosts. Therefore, there is a total latent time lapse of disease incubation which extends over the two segments of delay incubation times namely: (1) the incubation period of the infectious agent ( or the half-life cycle) inside the vector, and (2) the incubation period of the infectious agent (or the other half-life cycle) inside the human being (cf.\cite{WHO,CDC}). In fact, the malaria plasmodium undergoes the first developmental half-life cycle called the \textit{sporogonic cycle} inside the female \textit{Anopheles} mosquito lasting approximately $10-18$ days, following a successful blood meal from an infectious human being through a mosquito bite. Moreover, the mosquito becomes infectious. The parasite completes the second developmental half-life cycle called the \textit{exo-erythrocytic cycle} lasting about 7-30 days inside the exposed human being\cite{WHO,CDC}, whenever the parasite is transferred to human being in the process of the infectious mosquito foraging for another blood meal.
The exposure and successful recovery from a malaria parasite, for example, \textit{falciparum vivae} induces natural immunity against the disease which can protect against subsequent severe outbreaks of the disease. Moreover, the effectiveness and duration of the naturally acquired immunity against malaria is determined by several factors such as the species and the frequency of exposure to the parasites (cf.\cite{CDC,denise}).
Compartmental mathematical epidemic dynamic models have been used to investigate the dynamics of several different types of infectious diseases including malaria\cite{pang,hyun}. In general, these models are classified as SIS, SIR, SIRS, SEIRS, and SEIR etc.\cite{qliu,wanduku-fundamental,Wanduku-2017,sen,cooke-driessche} epidemic dynamic models depending on the compartments of the disease classes directly involved in the general disease dynamics.
Many compartmental mathematical models with delays have been studied \cite{Wanduku-2017,wanduku-delay,cooke-driessche}.
Some important investigations in the study of population dynamic models expressed as systems of differential equations are the permanence, and extinction of disease in the population, and also stability of the equilibria over sufficiently long time. Several papers in the literature\cite{zhien,wanbiao,tend} have addressed these topics. The extinction of disease seeks to find conditions that are sufficient for the disease related classes in the population such as, the exposed and infectious classes, to become extinct over sufficiently long time. The permanence of disease also answers the question about whether a significant number of people in the disease related classes will remain over sufficiently long time. Disease eradication or persistence of disease in the steady state population seeks to find conditions sufficient for the equilibria to be stable asymptotically.
The primary objectives of this paper include, to investigate (1) the extinction, and (2) the permanence of disease in a family of SEIRS epidemic models.
In other words, we find conditions that are sufficient for a disease such as malaria, to become extinct from the population over time, and also conditions that cause the disease to be permanent in the population over time.
The rest of this paper is presented as follows:- in Section~\ref{ch1.sec0.sec0}, the mosquito-human models are derived. In Section~\ref{ch1.sec1}, some model validation and preliminary results are presented. In Section~\ref{ch1.sec3a}, the results for the permanence of the disease are presented. Moreover, simulation results for the permanence of the disease in the population are presented in Section~\ref{ch1.sec4}. In Section~\ref{ch1.sec2b}, the results for the extinction of the disease are presented. Moreover, the numerical simulation results for the extinction of disease are presented in Section~\ref{ch1.sec4}.
\section{Derivation of the mosquito-host dynamics}\label{ch1.sec0.sec0}
The following assumptions are made to derive the epidemic model. Ideas from \cite{baretta-takeuchi1} will be used to derive the model for the mosquito-human dynamics.
(A) There are delays in the disease dynamics, and the delays represent the incubation period of the infectious agents (plasmodium or dengue fever virus etc.) in the vector $T_{1}$, and in the human host $T_{2}$. The third delay represents the natural immunity period $T_{3}$, where the delays are random variables with densities $f_{T_{1}}, t_{0}\leq T_{1}\leq h_{1}, h_{1}>0$, and $f_{T_{2}}, t_{0}\leq T_{2}\leq h_{2}, h_{2}>0$ and $f_{T_{3}}, t_{0}\leq T_{3}<\infty$ (cf. \cite{wanduku-biomath}).
(B) The vector (e.g. mosquito) population consists of two main classes namely: the susceptible vectors $V_{s}$ and the infectious vectors $V_{i}$. Moreover, it is assumed that the total vector population denoted $V_{0}$ is constant at any time, that is, $V_{s}(t)+V_{i}(t)=V_{0}, \forall t\geq t_{0}$, where $V_{0}>0$ is a positive constant. The susceptible vectors $V_{s}$ are infected by infectious humans $\hat{I}$, and after the incubation period $T_{1}$, the exposed vector becomes infectious $V_{i}$. Moreover, there is homogenous mixing between the vector-host populations. Therefore, the birth rate and death rate of the vectors are equal, and denoted $\hat{\mu}_{v}$. It is assumed that the turnover of the vector population is very high, and the total number of vectors $V_{0}$ at any time $t$, is very large, and as a result, $\hat{\mu}_{v}$ is sufficiently large number. In addition, it is assumed that the total vectors $V_{0}$ is exceedingly larger than the total humans present at any time $t$, denoted $\hat{N}((t), t\geq t_{0}$. That is, $V_{0}>>\hat{N}((t), t\geq t_{0}$.
(C) The humans consists of susceptible $(\hat{S})$, Exposed $(\hat{E})$, Infectious $(\hat{I})$ and removed $(\hat{R})$ classes. The susceptibles are infected by the infectious vectors $V_{i}$, and become exposed (E). The infectious agent incubates for $T_{2}$ time units, and the exposed individuals become infectious $\hat{I}$. The infectious class recovers from the disease with temporary or sufficiently long natural immunity and become $(\hat{R})$. Therefore, the total population present at time $t$, $\hat{N}(t)=\hat{S}(t)+\hat{E}(t)+\hat{I}(t)+\hat{R}(t),\forall t\geq t_{0}$.
Furthermore, it is assumed that the interaction between the infectious vectors $V_{i}$ and susceptible humans $\hat{S}$ exhibits nonlinear behavior, due to the overcrowding of the vectors as described in (B), and resulting in psychological effects on the susceptible individuals which lead to change of behavior that limits the disease transmission rate, and consequently in a nonlinear character for the incidence rate characterized by the nonlinear incidence function $G$. $G$ satisfies the conditions of Assumption~\ref{ch1.sec0.assum1}.
\begin{assumption}\label{ch1.sec0.assum1} \begin{enumerate}
\item [$A1$]$G(0)=0$; $A2$: $G(I)$ is strictly monotonic on $[0,\infty)$; $A3$: $G\in C^{2}([0,\infty), [0,\infty))$, and $G''(I)<0$;
$A4$. $\lim_{I\rightarrow \infty}G(I)=C, 0\leq C<\infty$;
$A5$: $G(I)\leq I, \forall I>0$; $A6$
\begin{equation}\label{ch1.sec4.lemma1.eq1a}
\left(\frac{G(x)}{x}-\frac{G(y)}{y}\right)\left(G(x)-G(y)\right)\leq 0, \forall x, y\geq0.
\end{equation} \end{enumerate} \end{assumption} These assumptions form an extension of the assumptions in \cite{qliu,wanduku-extinct,wanduku-biomath}. Some examples of incidence functions include $G(x)=\frac{x}{1+\theta x}, \theta>0$ etc.
(D) There is constant birthrate of humans $\hat{B}$ in the population, and all births are susceptible individuals. It is also assumed that the natural deathrate of human beings in the population is $\hat{\mu}$ and individuals die additionally due to disease related causes at the rate $\hat{d}$. From a biological point of view, the average lifespan of vectors $\frac{1}{\hat{\mu}_{v}}$, is much less than the average lifespan of a human being in the absence of disease $\frac{1}{\hat{\mu}}$. It follows that assuming exponential lifetime for all individuals (both vector and host) in the population, then the survival probabilities over the time intervals of length $T_{1}=s\in [t_{0},h_{1}]$, and $T_{2}=s\in [t_{0},h_{2}]$, satisfy \begin{equation}\label{ch1.sec0.eqn0.eq1} e^{-\hat{\mu}_{v}T_{1}}<<e^{-\hat{\mu} T_{1}}\quad and\quad e^{-\hat{\mu}_{v}T_{1}-\hat{\mu} T_{2}}<<e^{-\hat{\mu}(T_{1} +T_{2})}. \end{equation}
Applying similar ideas in \cite{baretta-takeuchi1}, the vector dynamics from (A)-(D) follows the system \begin{eqnarray} dV_{s}(t)&=&[-\Lambda e^{-\hat{\mu}_{v}T_{1}}\hat{I}(t-T_{1})V_{s}(t-T_{1})-\hat{\mu}_{v}V_{s}(t)+\hat{\mu}_{v}(V_{s}(t)+V_{i}(t))]dt,\label{ch1.sec0.eq0.eq3}\\ dV_{i}(t)&=&[\Lambda e^{-\hat{\mu}_{v}T_{1}}\hat{I}(t-T_{1})V_{s}(t-T_{1})-\hat{\mu}_{v}V_{i}(t)]dt,\label{ch1.sec0.eq0.eq4}\\ V_{0}&=&V_{s}(t)+V_{i}(t),\forall t\geq t_{0}, t_{0}\geq 0,\label{ch1.sec0.eq0.eq5} \end{eqnarray} where $\Lambda$ is the effective disease transmission rate from an infectious human being to a susceptible vector. Observe that the incidence rate of the disease into the vector population $\Lambda e^{-\hat{\mu}_{v}T_{1}}\hat{I}(t-T_{1})V_{s}(t-T_{1})$ represents new infectious vectors occurring at time $t$, which became exposed at earlier time $t-T_{1}$, and surviving natural death over the incubation period $T_{1}$, with survival probability rate $e^{-\hat{\mu}_{v}T_{1}}$, and are infectious at time $t$. The detailed host population dynamics is derived as follows.
At time $t$, it follows from (C) that when susceptible humans $\hat{S}$ and infectious vectors $V_{i}$ interact with $\hat{\beta}$ effective contacts per vector, per unit time, then under the assumption of homogenous mixing, the incidence rate of the disease into the human population is given by the term $\hat{\beta}\hat{S}(t)V_{i}(t)$.
With the assumption of crowding effects of the vector population, it follows from (C) that the incidence rate of the disease can be written as \begin{equation}\label{ch1.sec0.eqn0} \hat{\beta}\hat{S}(t)G(V_{i}(t)), \end{equation}
where $G$ is the nonlinear incidence function satisfying the conditions in Assumption~\ref{ch1.sec0.assum1}.
It follows easily (cf.\cite{wanduku-biomath}) from the assumptions (A)-(D), and (\ref{ch1.sec0.eqn0}) that for $T_{j},j=1,2,3$ fixed in the population, the dynamics of malaria in the human population is given by the system
\begin{eqnarray} d\hat{S}(t)&=&\left[ \hat{B}-\hat{\beta} \hat{S}(t)G(V_{i}(t)) - \hat{\mu} \hat{S}(t)+ \hat{\alpha} \hat{I}(t-T_{3})e^{-\hat{\mu} T_{3}} \right]dt,\nonumber\\ &&\label{ch1.sec0.eq3.eq1}\\ d\hat{E}(t)&=& \left[ \hat{\beta} \hat{S}(t)G(V_{i}(t)) - \hat{\mu} \hat{E}(t)\right.\nonumber\\ &&\left.-\hat{\beta} \hat{S}(t-T_{2}) e^{-\hat{\mu} T_{2}}G(V_{i}(t-T_{2})) \right]dt,\label{ch1.sec0.eq4.eq1}\\ &&\nonumber\\ d\hat{I}(t)&=& \left[\hat{\beta} \hat{S}(t-T_{2}) e^{-\hat{\mu} T_{2}}G(V_{i}(t-T_{2}))- (\hat{\mu} +\hat{d}+ \hat{\alpha}) \hat{I}(t) \right]dt,\nonumber\\ &&\label{ch1.sec0.eq5.eq1}\\ d\hat{R}(t)&=&\left[ \hat{\alpha} \hat{I}(t) - \hat{\mu} \hat{R}(t)- \hat{\alpha} \hat{I}(t-T_{3})e^{-\hat{\mu} T_{3}} \right]dt.\label{ch1.sec0.eq6} \end{eqnarray} Furthermore, the incidence function $G$ satisfies the conditions in Assumption~\ref{ch1.sec0.assum1}. And the initial conditions are given in the following:
\begin{eqnarray} &&\left(\hat{S}(t),\hat{E}(t), \hat{I}(t), \hat{R}(t)\right) =\left(\varphi_{1}(t),\varphi_{2}(t), \varphi_{3}(t),\varphi_{4}(t)\right), t\in (-T_{max},t_{0}],\nonumber\\% t\in [t_{0}-h,t_{0}],\quad and\quad= &&\varphi_{k}\in \mathcal{C}((-T_{max},t_{0}],\mathbb{R}_{+}),\forall k=1,2,3,4, \nonumber\\ &&\varphi_{k}(t_{0})>0,\forall k=1,2,3,4,\quad and\quad \max_{t_{0}\leq T_{1}\leq h_{1}, t_{0}\leq T_{2}\leq h_{2}, T_{3}\geq t_{0}}{(T_{1}+ T_{2}, T_{3})}=T_{max}\nonumber\\
\label{ch1.sec0.eq6.eq1} \end{eqnarray} where $\mathcal{C}((-T_{max},t_{0}],\mathbb{R}_{+})$ is the space of continuous functions with the supremum norm \begin{equation}\label{ch1.sec0.eq6.eq2}
||\varphi||_{\infty}=\sup_{ t\leq t_{0}}{|\varphi(t)|}. \end{equation}
It is shown in the following that the vector-host dynamics in (\ref{ch1.sec0.eq0.eq3})-(\ref{ch1.sec0.eq0.eq5}) and (\ref{ch1.sec0.eq3.eq1})-(\ref{ch1.sec0.eq6.eq1}) lead to the malaria model in \cite{wanduku-biomath}, which omits the dynamics of the vector population, under the assumptions (A)-(D).
Firstly, observe that the system (\ref{ch1.sec0.eq3.eq1})-(\ref{ch1.sec0.eq6.eq1}) satisfies [Theorem~3.1, \cite{wanduku-biomath}], and the total human population $\hat{N}(t)=\hat{S}(t)+\hat{E}(t)+\hat{I}(t)+\hat{R}(t),\forall t\geq t_{0}$ obtained from system (\ref{ch1.sec0.eq3.eq1})-(\ref{ch1.sec0.eq6.eq1}) with initially condition that satisfies $N(t_{0})\leq \frac{\hat{B}}{\hat{\mu}}$, must satisfy \begin{equation}\label{ch1.sec0.eq0.eq0}
\limsup_{t\rightarrow \infty}{\hat{N}(t)}=\frac{\hat{B}}{\hat{\mu}}. \end{equation}
Therefore, the assumption (B) above, interpreted as $\frac{\hat{N}(t)}{V_{0}}<<1, \forall t\geq t_{0}$ implies that
\begin{equation}\label{ch1.sec0.eq0.eq1}
\limsup_{t\rightarrow \infty}{\hat{N}(t)}=\frac{\hat{B}}{\hat{\mu}},\quad and\quad \frac{\left(\frac{\hat{B}}{\hat{\mu}}\right)}{V_{0}}<<1. \end{equation} Define \begin{equation}\label{ch1.sec0.eq0.eq2}
\epsilon=\frac{\left(\frac{\hat{B}}{\hat{\mu}}\right)}{V_{0}}, \end{equation} then from (\ref{ch1.sec0.eq0.eq1})-(\ref{ch1.sec0.eq0.eq2}), it follows that $\epsilon=\frac{\left(\frac{\hat{B}}{\hat{\mu}}\right)}{V_{0}}<<1$.
Employing similar reason in \cite{baretta-takeuchi1}, define two natural dimensionless time scales $\eta$ and $\varrho$ for the joint vector-host dynamics (\ref{ch1.sec0.eq0.eq3})-(\ref{ch1.sec0.eq0.eq5}) and (\ref{ch1.sec0.eq3.eq1})-(\ref{ch1.sec0.eq6.eq1}) in the following.
\begin{eqnarray}
\eta&=& \left(\frac{\hat{B}}{\hat{\mu}}\right)\Lambda t, \label{ch1.sec0.eq6.eq3}\\
\varrho&=& V_{0}\Lambda t.\label{ch1.sec0.eq6.eq4} \end{eqnarray}
Note that since the total vector population $V_{0}$ from (B) above is constant, that is, $V_{s}(t)+V_{i}(t)=V_{0}, \forall t \geq t_{0}$, and from (\ref{ch1.sec0.eq0.eq0}) and [Theorem~3.1, \cite{wanduku-biomath}] the total human $0<\hat{N}(t)\leq \frac{\hat{B}}{\hat{\mu}}, \forall t \geq t_{0}$, whenever $\hat{N}(t_{0})\leq \frac{\hat{B}}{\hat{\mu}}$, then the time scales $\eta$ and $\varrho$ arise naturally to rescale the total vector and maximum total human populations $V_{0}$ and $\left(\frac{\hat{B}}{\hat{\mu}}\right)$, respectively, at any time.
The time scale $\varrho$ is "fast", and $\eta$ is "slow" (cf. \cite{baretta-takeuchi1}).
Therefore, from above, let \begin{equation}\label{ch1.sec0.eq6.eq5}
\hat{V}_{i}(t)=\frac{V_{i}(t)}{V_{0}},\quad and \quad \hat{V}_{s}(t)=\frac{V_{s}(t)}{V_{0}}, \end{equation} be the dimensionless vector variables, and \begin{equation}\label{ch1.sec0.eq6.eq6}
{S}(t)=\frac{\hat{S}(t)}{\left(\frac{\hat{B}}{\hat{\mu}}\right)},{I}(t)=\frac{\hat{I}(t)}{\left(\frac{\hat{B}}{\hat{\mu}}\right)}, {E}(t)=\frac{\hat{E}(t)}{\left(\frac{\hat{B}}{\hat{\mu}}\right)}, {R}(t)=\frac{\hat{R}(t)}{\left(\frac{\hat{B}}{\hat{\mu}}\right)}\quad and \quad {N}(t)=\frac{\hat{N}(t)}{\left(\frac{\hat{B}}{\hat{\mu}}\right)}, \end{equation}
be the dimensionless human variables. And since $0<\hat{N}(t)\leq \frac{\hat{B}}{\hat{\mu}}, \forall t \geq t_{0}$, whenever $\hat{N}(t_{0})\leq \frac{\hat{B}}{\hat{\mu}}$, it follows from (\ref{ch1.sec0.eq6.eq6}) that
\begin{equation}\label{ch1.sec0.eq6.eq6.eq1}
0<{S}(t)+{E}(t)+{I}(t)+{R}(t)={N}(t)\leq 1, \forall t\geq t_{0}.
\end{equation}
Applying (\ref{ch1.sec0.eq6.eq5})-(\ref{ch1.sec0.eq6.eq6}) to (\ref{ch1.sec0.eq0.eq3})-(\ref{ch1.sec0.eq0.eq5}) leads to the following \begin{eqnarray} d\hat{V}_{i}(t)&=& \epsilon\left[ e^{-\hat{\mu}_{v}T_{1}}{I}(t-T_{1})\hat{V}_{s}(t-T_{1})-\frac{\hat{\mu}_{v}}{\Lambda\left(\frac{\hat{B}}{\hat{\mu}}\right)}\hat{V}_{i}(t)\right]d\varrho,\label{ch1.sec0.eq6.eq7}\\ d\hat{V}_{s}(t)&=&-d\hat{V}_{i}(t),\label{ch1.sec0.eq6.eq8}\\ 1&=&\hat{V}_{s}(t)+\hat{V}_{i}(t),\forall t\geq t_{0}, t_{0}\geq 0.\label{ch1.sec0.eq6.eq9} \end{eqnarray} Observe from (\ref{ch1.sec0.eq6.eq6.eq1})-(\ref{ch1.sec0.eq6.eq9}) that for nonnegative values for the vector variables $\hat{V}_{i}(t)\geq 0, \hat{V}_{s}(t)\geq 0, \forall t\geq t_{0}$, and positive values for the human variables ${S}(t), {E}(t), {I}(t), {R}(t)>0, \forall t\geq t_{0} $, it is follows that \begin{equation}\label{ch1.sec0.eq6.eq10}
-\epsilon\frac{\hat{\mu}_{v}}{\Lambda\left(\frac{\hat{B}}{\hat{\mu}}\right)}\leq \frac{d\hat{V}_{i}(t)}{d\varrho}\leq \epsilon e^{-\hat{\mu}_{v}T_{1}}. \end{equation}
Thus, on the time scale $\varrho$ which is "fast", it is easy to see from (\ref{ch1.sec0.eq6.eq7})-(\ref{ch1.sec0.eq6.eq10}), that under the assumption that $\epsilon$ from (\ref{ch1.sec0.eq0.eq2}) is infinitesimally small, that is $\epsilon\rightarrow 0$, then \begin{equation}\label{ch1.sec0.eq6.eq11}
\frac{d\hat{V}_{i}(t)}{d\varrho}=-\frac{d\hat{V}_{s}(t)}{d\varrho}=0, \end{equation}
which implies that the dynamics of $\hat{V}_{i}$ and $\hat{V}_{s}$ behaves as in steady state. And thus, it follows from (\ref{ch1.sec0.eq6.eq7})-(\ref{ch1.sec0.eq6.eq11}) that
\begin{eqnarray}
\hat{V}_{i}(t)&=& \frac{e^{-\hat{\mu}_{v}T_{1}}}{\hat{\mu}_{v}}\Lambda \left(\frac{\hat{B}}{\hat{\mu}}\right){I}(t-T_{1})\hat{V}_{s}(t-T_{1}),\nonumber \\
1 &=& \hat{V}_{s}(t)+\hat{V}_{i}(t).\label{ch1.sec0.eq6.eq12}
\end{eqnarray}
It follows further from (\ref{ch1.sec0.eq6.eq12}) that
\begin{equation}\label{ch1.sec0.eq6.eq13}
\hat{V}_{s}(t)=\frac{1}{1+\frac{e^{-\hat{\mu}_{v}T_{1}}}{\hat{\mu}_{v}}\Lambda \left(\frac{\hat{B}}{\hat{\mu}}\right){I}(t-T_{1})\hat{V}_{s}(t-T_{1})}.
\end{equation}
For sufficiently large value of the birth-death rate $\hat{\mu}_{v}$ (see assumption (B)), such that $\hat{\mu}_{v}e^{\hat{\mu}_{v}T_{1}}>>\Lambda \left(\frac{\hat{B}}{\hat{\mu}}\right)$, then it follows from (\ref{ch1.sec0.eq6.eq13}) that $\hat{V}_{s}(t)\approx 1$, and consequently from (\ref{ch1.sec0.eq6.eq9}) and (\ref{ch1.sec0.eq6.eq5}), $V_{s}(t)\approx V_{0}$. Moreover, it follows further from (\ref{ch1.sec0.eq6.eq12}) that \begin{equation}\label{ch1.sec0.eq6.eq14}
\hat{V}_{i}(t)\approx \frac{e^{-\hat{\mu}_{v}T_{1}}}{\hat{\mu}_{v}}\Lambda \left(\frac{\hat{B}}{\hat{\mu}}\right){I}(t-T_{1}), \end{equation}
and equivalently from (\ref{ch1.sec0.eq6.eq5})-(\ref{ch1.sec0.eq6.eq6}) that (\ref{ch1.sec0.eq6.eq14}) can be rewritten as follows
\begin{equation}\label{ch1.sec0.eq6.eq15}
V_{i}(t)\approx \frac{e^{-\hat{\mu}_{v}T_{1}}}{\hat{\mu}_{v}}\Lambda V_{0}\hat{I}(t-T_{1}).
\end{equation}
While on the fast scale $\varrho$ the term $\hat{I}(t-T_{1})$ behaves as the steady state, on the slow scale $\eta$, it is expected to still be evolving. In the following, using (\ref{ch1.sec0.eq6.eq5})-(\ref{ch1.sec0.eq6.eq6}), the dynamics for the human population in (\ref{ch1.sec0.eq3.eq1})-(\ref{ch1.sec0.eq6.eq1}) is nondimensionalized with respect to the slow time scale $\eta$ in (\ref{ch1.sec0.eq6.eq3}).
Without loss of generality(as it is usually the case e.g. $G(x)=\frac{x}{1+\alpha x}$, $G(x)=\frac{x}{1+\alpha x^{2}}$), it is assumed that on the $\eta$ timescale, the nonlinear term $G(V_{i}(t))$ expressed as $G(V_{0}\hat{V}_{i}(\eta))$, can be rewritten from (\ref{ch1.sec0.eq6.eq15}) as
\begin{equation}\label{ch1.sec0.eq6.eq16}
G(V_{0}\hat{V}_{i}(\eta))\equiv \frac{\Lambda V_{0}\left(\frac{\hat{B}}{\hat{\mu}}\right)}{\hat{\mu}_{v}}\hat{G}(\hat{V}_{i}(\eta))e^{-\hat{\mu}_{v}T_{1}},
\end{equation} by factoring a constant term $\frac{\Lambda V_{0}\left(\frac{\hat{B}}{\hat{\mu}}\right)}{\hat{\mu}_{v}}$, and the function $\hat{G}$ carries all the properties of Assumption~\ref{ch1.sec0.assum1}.
Thus, from the above and (\ref{ch1.sec0.eq6.eq15}), the system (\ref{ch1.sec0.eq3.eq1})-(\ref{ch1.sec0.eq6.eq1}) is rewritten in dimensionless form as follows: \begin{eqnarray}
d{S}(\eta) &=& [B-\beta {S}(\eta)\hat{G}({I}(\eta-T_{1\eta}))e^{-\mu_{v}T_{1\eta}}-\mu {S}(\eta)+\alpha I(\eta -T_{3\eta})e^{-\mu T_{3\eta}}]d\eta,\nonumber\\
&& \label{ch1.sec0.eq6.eq17}\\
d{E}(\eta) &=& [\beta {S}(\eta)\hat{G}({I}(\eta-T_{1\eta}))e^{-\mu_{v}T_{1\eta}}-\mu {E}(\eta)\nonumber\\
&&-\beta {S}(\eta-T_{2\eta})\hat{G}({I}(\eta-T_{1\eta}-T_{2\eta}))e^{-\mu_{v}T_{1\eta}-\mu T_{2\eta}}]d\eta, \label{ch1.sec0.eq6.eq18}\\%-(\mu+d+\alpha)\hat{I}(\eta)
d{I}(\eta) &=& [\beta {S}(\eta-T_{2\eta})\hat{G}({I}(\eta-T_{1\eta}-T_{2\eta}))e^{-\mu_{v}T_{1\eta}-\mu T_{2\eta}}-\mu {I}(\eta)\nonumber\\
&&-(\mu+d+\alpha){I}(\eta)]d\eta,\label{ch1.sec0.eq6.eq19}\\
d{R}(\eta)&=&[\alpha {I}(\eta)-\mu {R}(\eta)-\alpha I(\eta -T_{3\eta})e^{-\mu T_{3\eta}}]d\eta,\label{ch1.sec0.eq6.eq19} \end{eqnarray} where \begin{eqnarray}
&& B=\frac{\hat{B}}{\left(\frac{\hat{B}}{\hat{\mu}}\right)^{2}\Lambda},\quad \beta=\frac{\hat{\beta}V_{0}}{\hat{\mu}_{v}},\quad \mu=\frac{\hat{\mu}}{\left(\frac{\hat{B}}{\hat{\mu}}\right)\Lambda},\quad \alpha=\frac{\hat{\alpha}}{\left(\frac{\hat{B}}{\hat{\mu}}\right)\Lambda}\nonumber\\
&&\mu_{v}=\frac{\hat{\mu_{v}}}{\left(\frac{\hat{B}}{\hat{\mu}}\right)\Lambda},\quad d=\frac{\hat{d}}{\left(\frac{\hat{B}}{\hat{\mu}}\right)\Lambda},\quad T_{j\eta}=\left(\frac{\hat{B}}{\hat{\mu}}\right)\Lambda T_{j},\forall j=1,2,3.
\label{ch1.sec0.eq6.eq20}
\end{eqnarray} The system (\ref{ch1.sec0.eq6.eq17})-(\ref{ch1.sec0.eq6.eq19}) describes the dynamics of malaria on the slow scale $\eta$. Furthermore, moving forward, the analysis of the model (\ref{ch1.sec0.eq6.eq17})-(\ref{ch1.sec0.eq6.eq19}) is considered only on the $\eta$ timescale. To reduce heavy notation, the following substitutions are made. Substitute $t$ for $\eta$, and the delays $T_{j},\forall j=1,2,3$ will substitute $T_{j\eta}, \forall j=1,2,3$. Moreover, since the delays are are distributed with density functions $f_{T_{j}},\forall j=1,2,3$, it follows from (A)-(D), (\ref{ch1.sec0.eq6.eq17})-(\ref{ch1.sec0.eq6.eq19}) and (\ref{ch1.sec0.eq6.eq1}) that the expected SEIRS model for malaria is given as follows:
\begin{eqnarray} dS(t)&=&\left[ B-\beta S(t)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu_{v} s}G(I(t-s))ds - \mu S(t)+ \alpha \int_{t_{0}}^{\infty}f_{T_{3}}(r)I(t-r)e^{-\mu r}dr \right]dt,\nonumber\\ &&\label{ch1.sec0.eq3}\\%{ch1.sec0.eq6.eq21} dE(t)&=& \left[ \beta S(t)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu_{v} s}G(I(t-s))ds - \mu E(t)\right.\nonumber\\ &&\left.-\beta \int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(t-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu_{v} s-\mu u}G(I(t-s-u))dsdu \right]dt,\nonumber\\ &&\label{ch1.sec0.eq4}\\%{ch1.sec0.eq6.eq22} dI(t)&=& \left[\beta \int_{t_{0}}^{h_{2}}f_{T_{2}}(u)S(t-u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu_{v} s-\mu u}G(I(t-s-u))dsdu- (\mu +d+ \alpha) I(t) \right]dt,\nonumber\\ &&\label{ch1.sec0.eq5}\\%{ch1.sec0.eq6.eq23} dR(t)&=&\left[ \alpha I(t) - \mu R(t)- \alpha \int_{t_{0}}^{\infty}f_{T_{3}}(r)I(t-r)e^{-\mu s}dr \right]dt,\label{ch1.sec0.eq6} \end{eqnarray} where the initial conditions are given in the following: let $h= h_{1}+ h_{2}$ and define \begin{eqnarray} &&\left(S(t),E(t), I(t), R(t)\right) =\left(\varphi_{1}(t),\varphi_{2}(t), \varphi_{3}(t),\varphi_{4}(t)\right), t\in (-\infty,t_{0}],\nonumber\\% t\in [t_{0}-h,t_{0}],\quad and\quad= &&\varphi_{k}\in U C_{g}\subset \mathcal{C}((-\infty,t_{0}],\mathbb{R}_{+}),\forall k=1,2,3,4,\quad \varphi_{k}(t_{0})>0,\forall k=1,2,3,4,\nonumber\\
\label{ch1.sec0.eq06a} \end{eqnarray} where $UC_{g}$ is some fading memory sub Banach space of the Banach space $\mathcal{C}((-\infty,t_{0}],\mathbb{R}_{+})$ endowed with the norm
\begin{equation}\label{ch1.sec0.eq06b}
\|\varphi\|_{g}=\sup _{t \leq t_{0}} \frac{|\varphi(t)|}{g(t)},
\end{equation}
and $g$ is some continuous function with the following properties: (P1.) $g\left(\left(-\infty, t_{0}\right]\right) \subseteq[1, \infty)$, non-increasing, and $g(t_{0})=1$; (P2.) $ \lim _{u \rightarrow t_{0}^{-}} \frac{g(t+u)}{g(t)}=1$, uniformly on $[t_{0},\infty)$; $ \lim _{t \rightarrow-\infty} g(t)=\infty$. An example of such a function is $g(t)=e^{-at}, a>0$ (cf. \cite{kuang}). Note that for any $g$ satisfying (P1.)-(P2.). the Banach space $\mathcal{C}((-\infty,t_{0}],\mathbb{R}_{+})$ is continuously embedded in $UC_{g}$ which allows structural properties for $\mathcal{C}((-\infty,t_{0}],\mathbb{R}_{+})$ with the uniform norm to hold in $UC_{g}$ with $||.||_{g}$ norm. Moreover, $\varphi \in U C_{g}, \exists g $ if and only if $||\varphi||_{g}<\infty $ and $\frac{|\varphi(t)|}{g(t)}$ is uniformly continuous on $(-\infty, t_{0}]$.
Also, the function $G$ in (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}) satisfies the conditions of Assumption~\ref{ch1.sec0.assum1}.
Observe (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}) is similarly structured exactly as [(2.8)-(2.11), \cite{wanduku-biomath}]. Furthermore,
the equations for $E$ and $R$ decouple from (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}). Therefore, the results are exhibited for the decoupled system (\ref{ch1.sec0.eq3}) and (\ref{ch1.sec0.eq5}) containing equations for $S$ and $I$. \begin{equation}\label{ch1.sec0.eq13b}
Y(t)=(S(t), E(t), I(t), R(t))^{T},X(t)=(S(t),E(t),I(t))^{T},\quad\textrm{and}\quad N(t)=S(t)+ E(t)+ I(t)+ R(t).
\end{equation}
Whilst permanence or extinction has been investigated in some delay type systems ( cf.\cite{zhien, wanbiao,tend}), the permanence and extinction in the sense of \cite{zhien} in systems with multiple random delays is underdeveloped in the literature.
Furthermore, as far as we know no other paper has addressed extinction and persistence of malaria in a mosquito-human population dynamics involving delay differential equations in the line of thinking of \cite{zhien,wanbiao}. We recall the following definition from \cite{zhien,xin}. \begin{defn}\label{ch1.sec0.eq13b.def1} \item[1.] A population $x(t)$ is called strongly permanent if $\liminf_{t\rightarrow +\infty} x(t)>0$; \item[2.] $x(t)$ is said to go extinct if $\lim_{t\rightarrow +\infty}x(t)=0$. \item[3.] $x(t)$ is said to be weakly permanent in the mean if $\limsup_{t\rightarrow +\infty}\frac{1}{t}\int^{t}_{0}x(s)ds>0$. \item[4.] $x(t)$ is said to be strongly permanent in the mean if $\liminf_{t\rightarrow +\infty}\frac{1}{t}\int^{t}_{0}x(s)ds>0$. \item[5.] $x(t)$ is said to be stable in the mean if $\lim_{t\rightarrow \infty}{\frac{1}{t}\int^{t}_{t_{0}}x(s)}ds=c>0$. \end{defn}
\section{Model validation results\label{ch1.sec1}} The consistency results for the system (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}) are given. Some ideas from \cite{wanduku-biomath} using the dimensionless parameters (\ref{ch1.sec0.eq6.eq20}), are applied to the new model (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}). Observe from (\ref{ch1.sec0.eq6.eq20}) that expression $\frac{B}{\mu}$ simplifies to 1, and this is emphasized as $\frac{B}{\mu}\equiv 1$. \begin{thm}\label{ch1.sec2b.lemma1.thm1} For the given initial conditions (\ref{ch1.sec0.eq06a})-(\ref{ch1.sec0.eq06b}), the system (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}) has a unique positive solution $Y(t)\in \mathbb{R}_{+}^{4}$. Moreover, \begin{equation}\label{ch1.sec1.thm1a.eq0} \limsup_{t\rightarrow \infty} N(t)\leq S^{*}_{0}=\frac{B}{\mu}\equiv 1.
\end{equation}
Furthermore, there is a positive self invariant space for the system denoted $D(\infty)=\bar{B}^{(-\infty, \infty)}_{\mathbb{R}^{4}_{+},}\left(0,\frac{B}{\mu}\equiv 1\right) $, where $D(\infty)$ is the closed unit ball in $\mathbb{R}^{4}_{+}$ centered at the origin with radius $\frac{B}{\mu}\equiv 1$ containing all positive solutions defined over $(-\infty,\infty )$.
\end{thm}
Proof:\\
The proof of this result is standard and easy to follow applying the notations (\ref{ch1.sec0.eq13b}) to the system (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}).
\begin{thm}\label{ch1.sec2b.lemma1} The feasible region for the unique positive solutions $Y(t), t\geq t_{0}$ of the system (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}) in the phase plane that lie in the self-invariant unit ball $D(\infty)=\bar{B}^{(-\infty, \infty)}_{\mathbb{R}^{4}_{+},}\left(0,\frac{B}{\mu}\equiv 1\right) =\bar{B}^{(-\infty, \infty)}_{\mathbb{R}^{4}_{+},}\left(0, 1\right)$ for the system, also lie in a much smaller space $D^{expl}(\infty)\subset D(\infty)$, where \begin{equation}\label{ch1.sec2b.lemma1.eq1}
D^{expl}(\infty)=\left\{Y(t)\in \mathbb{R}^{4}_{+}:\frac{B}{\mu+d}\leq N(t)=S(t)+ E(t)+ I(t)+ R(t)\leq \frac{B}{\mu}, \forall t\in (-\infty, \infty) \right\}.
\end{equation}
Moreover, the space $D^{expl}(\infty)$ is also self-invariant with respect to the system (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}). \end{thm} Proof:\\ Suppose $Y(t)\in D(\infty)$, then it follows from (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}) and (\ref{ch1.sec0.eq13b}) that the total population $N(t)=S(t)+E(t)+I(t)+R(t)$ satisfies the following inequality \begin{equation}\label{ch1.sec2b.lemma1.proof.eq1} [B-(\mu+d)N(t)]dt\leq dN(t)\leq [B-(\mu)N(t)]dt. \end{equation} It is easy to see from (\ref{ch1.sec2b.lemma1.proof.eq1}) that \begin{equation}\label{ch1.sec2b.lemma1.proof.eq2} \frac{B}{\mu+d}\leq \liminf_{t\rightarrow \infty}N(t)\leq \limsup_{t\rightarrow \infty}N(t)\leq \frac{B}{\mu}, \end{equation} and (\ref{ch1.sec2b.lemma1.eq1}) follows immediately.
\begin{rem} Theorem~\ref{ch1.sec2b.lemma1} signifies that every solution for (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}) that starts in the unit ball $D(\infty)$ in the phase plane, oscillates continuously inside $D(\infty)$. Moreover, if the solution oscillates and enters the space $D^{expl}(\infty)=\bar{B}^{(-\infty, \infty)}_{\mathbb{R}^{4}_{+},}\left(0,\frac{B}{\mu}\equiv 1\right)\cap \left(\bar{B}^{(-\infty, \infty)}_{\mathbb{R}^{4}_{+},}\left(0,\frac{B}{\mu +d}\right)\right)^{c}$, the solution stays in $D^{expl}(\infty)$ for all time.
Biologically, observe that $\frac{B}{\mu }$ and $\frac{B}{\mu +d}$ represent the total births that occur over the average lifespans $\frac{1}{\mu}$ and $\frac{1}{\mu +d}$ of a human being in a malaria-free population and in a malaria- epidemic population, respectively. Thus, Theorem~\ref{ch1.sec2b.lemma1} signifies that when the population grows and enters a state for the total population $N(t)\in [\frac{B}{\mu+d },\frac{B}{\mu }]$, it stays within that range for all time. \end{rem}
Also, it is easy to see that the system (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}) has a DFE $E_{0}=(S^{*}_{0},0,0)=(\frac{B}{\mu}\equiv 1,0,0)=( 1,0,0)$. The basic reproduction number (BRN) for the disease when the delays in the system $T_{1}, T_{2}$ and $T_{3}$ are constant, is given by
\begin{equation} \label{ch1.sec2.lemma2a.corrolary1.eq4} \hat{R}^{*}_{0}=\frac{\beta }{(\mu+d+\alpha)}. \end{equation}
Furthermore, when $\hat{R}^{*}_{0}<1$, then $E_{0}=X^{*}_{0}=(S^{*}_{0},0,0)=(1,0,0)$ is asymptotically stable, and the disease can be eradicated from the population. Also, when the delays in the system $T_{i}, i=1,2,3$ are random, and arbitrarily distributed, the BRN is proportional to \begin{equation}\label{ch1.sec2.theorem1.corollary1.eq3} R_{0}\propto\frac{\beta }{(\mu+d+\alpha)}+\frac{\alpha}{(\mu+d+\alpha)}, \end{equation}
And malaria is eradicated from the system, whenever $R_{0}\leq 1$.
The following result can be made about the nonzero steady state of the dimensionless system (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}), when Assumption~\ref{ch1.sec0.assum1} is satisfied. \begin{thm}\label{ch1.sec2b.lemma1.thm2} Let the conditions of Assumption~\ref{ch1.sec0.assum1} be satisfied. Suppose $R_{0}>1$ (or $\hat{R}^{*}_{0}>1$) and the expected survival probability rate of the plasmodium satisfies \begin{equation} E(e^{-\mu_{v}T_{1}-\mu T_{2}})\geq \frac{R_{0}}{\left(R_{0}-\frac{\alpha}{\mu+d+\alpha}\right)G'(0)}, \end{equation}
then there exists a nonzero endemic equilibrium $E_{1}=(S^{*}_{1}, E^{*}_{1}, I^{*}_{1})$ for the the dimensionless system (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}), where
\begin{equation}\label{}
E(e^{-\mu_{v}T_{1}-\mu T_{2}})=\int^{h_{2}}_{t_{0}}\int^{h_{1}}_{t_{0}}e^{-\mu_{v}s-\mu u}f_{T_{2}}(u)f_{T_{1}}(s)dsdu.
\end{equation} \end{thm} Proof:\\
The dimensionless endemic equilibrium $E_{1}=(S^{*}_{1}, I^{*}_{1})$ of the decoupled (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}) is a solution to the following system: \begin{eqnarray}
&&B-\beta E(e^{-\mu_{v} T_{1}}) SG(I)) - \mu S+ \alpha E(e^{-\mu T_{3}}) I =0,\label{ch1.sec3.thm1.proof.eq1}\\
&&\beta E(e^{-\mu_{v} T_{1}-\mu T_{2}}) SG(I)- (\mu +d+ \alpha) I=0.\label{ch1.sec3.thm1.proof.eq3}
\end{eqnarray}
Solving for $S$ from (\ref{ch1.sec3.thm1.proof.eq3}) and substituting the result into (\ref{ch1.sec3.thm1.proof.eq1}), gives the following equation: \begin{equation}\label{ch1.sec3.thm1.proof.eq3b} H(I)=0 \end{equation}
where, \begin{equation}\label{ch1.sec3.thm1.proof.eq4} H(I)= B-\frac{1}{E(e^{-\mu_{v}T_{1}+\mu T_{2}})}I\left[\frac{(\mu+d+\alpha)\mu}{\beta G(I)}+(\mu+d)E(e^{-\mu_{v} T_{1}})+\alpha E(e^{-\mu_{v} T_{1}})\left(1-E(e^{-\mu T_{2}})E(e^{-\mu T_{3}})\right)\right]. \end{equation}
Note that $0<E(e^{-\mu T_{i}})\leq 1, i=1,2,3$, and $\lim_{I\rightarrow \infty} G(I)=C<\infty$, hence for sufficiently large positive value of $I$, $H(I)<0$. Furthermore, the derivative of $H(I)$ is given by \begin{eqnarray} H'(I)&=&-\frac{(\mu+d+\alpha)\mu}{\beta E(e^{-\mu_{v} T_{1}-\mu T_{2}})} \frac{(G(I)-IG'(I))}{G^{2}(I)}\nonumber\\ &&-\frac{1}{E(e^{-\mu_{v} T_{1}-\mu T_{2}})}\left((\mu+d)E(e^{-\mu_{v} T_{1}})+\alpha E(e^{-\mu_{v} T_{1}})(1-E(e^{-\mu T_{2}})E(e^{-\mu T_{3}}))\right).\label{ch1.sec3.thm1.proof.eq5} \end{eqnarray} Assume without loss of generality that $G'(I)>0$. It follows from the other properties of $G$ in Assumption~\ref{ch1.sec0.assum1}, that is, $G(0)=0$, $G''(I)<0$, that $(G(I)-IG'(I))>0$ and this further implies that $H'(I)<0$ for all $I> 0$. That is, $H(I)$ is a decreasing function over all $I> 0$. Therefore, a positive root of the equation (\ref{ch1.sec3.thm1.proof.eq3b}) requires that $H(0)>0$. Observe from (\ref{ch1.sec3.thm1.proof.eq4}) and the dimensionless expressions in (\ref{ch1.sec0.eq6.eq20} ), \begin{eqnarray}\label{ch1.sec3.thm1.proof.eq6} &&H(0)=B\left(1- \frac{(\mu+d+\alpha)}{\beta G'(0)E(e^{-\mu_{v} T_{1}-\mu T_{2}})}\right) =B\left(1- \frac{1}{\left(R_{0}-\frac{\alpha}{\mu+d+\alpha}\right)G'(0)E(e^{-\mu_{v} T_{1}-\mu T_{2}})}\right)\nonumber\\%\frac{(\mu+d+\alpha)}{\beta G'(0)E(e^{-\mu_{v} T_{1}-\mu T_{2}})}\right) &&\geq B\left(1- \frac{1}{R_{0}}\right). \end{eqnarray} For $R_{0}>1$, it is easy to see that $H(0)>0$.
The extinction of disease will be investigated in the neighborhood of the zero steady state $E_{0}$, and the permanence of disease will be investigated in the neighborhood of $E_{1}$.
\section{Extinction of disease}\label{ch1.sec2b}
In this section, the extinction of malaria from the system (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}) is investigated.
Note, the decoupled system (\ref{ch1.sec0.eq3}) and (\ref{ch1.sec0.eq5}) is used.
The following lemma will be used to establish the extinction results.
\begin{lemma}\label{ch1.sec2b.lemma2} Let the assumptions of Theorem~\ref{ch1.sec2b.lemma1} hold, and define the following Lyapunov functional in $D^{expl}(\infty)$, \begin{eqnarray} \tilde{V}(t)&=&V(t)+\beta\left[\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-(\mu_{v}s+\mu u)}\int^{t}_{t-u}S(\theta)\frac{G(I(\theta-s))}{I(t)}d\theta dsdu\right.\nonumber\\ &&\left. +\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-(\mu_{v}s+\mu u)}\int^{t}_{t-s}S(t)\frac{G(I(\theta))}{I(t)}d\theta ds du\right], \label{ch1.sec2b.lemma2.eq1} \end{eqnarray} where $V(t)=\log{I(t)}$. It follows that \begin{equation}\label{ch1.sec2b.lemma2.eq2} \limsup_{t\rightarrow \infty}\frac{1}{t}\log{(I(t))}\leq \beta \frac{B}{\mu}E(e^{-(\mu_{v}T_{1}+\mu T_{2})})-(\mu+d+\alpha). \end{equation} \end{lemma} Proof:\\ The differential operator $\dot{V}$ applied to the Lyapunov functional $\tilde{V}(t)$ with respect to the system (\ref{ch1.sec0.eq3}) leads to the following \begin{equation}\label{ch1.sec2b.lemma2.proof.eq1}
\dot{\tilde{V}}(t)=\beta\int_{t_{0}}^{h_{2}}f_{T_{2}}(u)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-(\mu_{v} s+\mu u)}S(t)\frac{G(I(t))}{I(t)}dsdu-(\mu+ d+ \alpha) \end{equation}
Since $S(t), I(t)\in D^{expl}(\infty)$, and $G$ satisfies the conditions of Assumption~\ref{ch1.sec0.assum1}, it follows easily from (\ref{ch1.sec2b.lemma2.proof.eq1}) that \begin{equation}\label{ch1.sec2b.lemma2.proof.eq3} \dot{\tilde{V}}(t)\leq \beta \frac{B}{\mu}E(e^{-(\mu_{v}T_{1}+\mu T_{2})})-(\mu+d+\alpha). \end{equation} Now, integrating both sides of (\ref{ch1.sec2b.lemma2.proof.eq3}) over the interval $[t_{0},t]$, it follows from (\ref{ch1.sec2b.lemma2.proof.eq3}) and (\ref{ch1.sec2b.lemma2.eq1}) that
\begin{eqnarray}
\log{I(t)}&\leq&\tilde{V}(t)
\leq\tilde{V}(t_{0})+\left[\beta \frac{B}{\mu}E(e^{-(\mu_{v}T_{1}+\mu T_{2})})-(\mu+d+\alpha)\right](t-t_{0}).\label{ch1.sec2b.lemma2.proof.eq4}
\end{eqnarray}
Diving both sides of (\ref{ch1.sec2b.lemma2.proof.eq4}) by $t$, and taking the limit supremum as $t\rightarrow\infty$, it is easy to see that (\ref{ch1.sec2b.lemma2.proof.eq4}) reduces to
\begin{eqnarray}
\limsup_{t\rightarrow\infty}\frac{1}{t}\log{I(t)}&\leq& \left[\beta \frac{B}{\mu}E(e^{-(\mu_{v}T_{1}+\mu T_{2})})-(\mu+d+\alpha)\right].\label{ch1.sec2b.lemma2.proof.eq5}
\end{eqnarray}
And the result (\ref{ch1.sec2b.lemma2.eq2}) follows immediately from (\ref{ch1.sec2b.lemma2.proof.eq5}).
The extinction conditions for the infectious population over time are expressed in terms - (1) the BRN $R^{*}_{0}$ in (\ref{ch1.sec2.lemma2a.corrolary1.eq4}), and (2) the expected survival probability rate (ESPR) of the parasites $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})$, also defined in [Theorem~5.1, \cite{wanduku-biomath}]. \begin{thm}\label{ch1.sec2b.thm1} Suppose Lemma~\ref{ch1.sec2b.lemma2} is satisfied, and let the BRN $R^{*}_{0}$ be defined as in (\ref{ch1.sec2.lemma2a.corrolary1.eq4}). In addition, let one of the following conditions hold \item[1.] $R^{*}_{0}\geq 1$ and $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})<\frac{1}{R^{*}_{0}}$, or \item[2.]$R^{*}_{0}<1$.\\ Then \begin{equation}\label{ch1.sec2b.thm1.eq1} \limsup_{t\rightarrow \infty}\frac{1}{t}\log{(I(t))}<-\lambda. \end{equation} where $\lambda>0$ is some positive constant. In other words, $I(t)$ converges to zero exponentially. \end{thm} Proof:\\ Suppose Theorem~\ref{ch1.sec2b.thm1}~[1.] holds, then from (\ref{ch1.sec2b.lemma2.eq2}), \begin{equation}\label{ch1.sec2b.thm1.eq1.proof.eq1} \limsup_{t\rightarrow \infty}\frac{1}{t}\log{(I(t))}< \beta\frac{B}{\mu}\left(E(e^{-(\mu_{v}T_{1}+\mu T_{2})})- \frac{1}{R^{*}_{0}} \right)\equiv -\lambda, \end{equation} where the positive constant $\lambda>0$ is taken to be as follows \begin{equation}\label{ch1.sec2b.thm1.eq1.proof.eq1.eq1} \lambda\equiv(\mu+d+\alpha)-\beta \frac{B}{\mu}E(e^{-(\mu_{v}T_{1}+\mu T_{2})})=\beta\frac{B}{\mu}\left( \frac{1}{R^{*}_{0}}-E(e^{-(\mu_{v}T_{1}+\mu T_{2})}) \right)>0. \end{equation}
Also, suppose Theorem~\ref{ch1.sec2b.thm1}~[2.] holds, then from (\ref{ch1.sec2b.lemma2.eq2}), \begin{eqnarray} \limsup_{t\rightarrow \infty}\frac{1}{t}\log{(I(t))}&\leq& \beta \frac{B}{\mu}E(e^{-(\mu_{v}T_{1}+\mu T_{2})})-(\mu+d+\alpha)\nonumber\\ &<& \beta \frac{B}{\mu}-(\mu+d+\alpha)= -(1-R^{*}_{0})(\mu+d+\alpha)\equiv -\lambda,\label{ch1.sec2b.thm1.eq1.proof.eq2} \end{eqnarray} where the positive constant $\lambda>0$ is taken to be as follows \begin{equation}\label{ch1.sec2b.thm1.eq1.proof.eq2.eq1} \lambda\equiv(1-R^{*}_{0})(\mu+d+\alpha)>0. \end{equation}
\begin{rem} Theorem~\ref{ch1.sec2b.thm1}, and Theorem~\ref{ch1.sec2b.lemma1} signify that all trajectories of $(S(t), I(t))$ of the decoupled system (\ref{ch1.sec0.eq3}) and (\ref{ch1.sec0.eq5}), that start in $D(\infty)$ and grow into $D^{expl}(\infty)\subset D(\infty)$ remain in $D^{expl}(\infty)$. Moreover, on the phase plane of $(S(t), I(t))$, the trajectory of the infectious state $I(t), t\geq t_{0}$ ultimately turn to zero exponentially, whenever either the ESPR $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})<\frac{1}{R^{*}_{0}}$ for $R^{*}_{0}\geq 1$ , or whenever the BRN $R^{*}_{0}<1$. Furthermore, the Lyapunov exponent (LE) from (\ref{ch1.sec2b.thm1.eq1}) is estimated by the term $\lambda$, defined in (\ref{ch1.sec2b.thm1.eq1.proof.eq1.eq1}) and (\ref{ch1.sec2b.thm1.eq1.proof.eq2.eq1}).
It follows from (\ref{ch1.sec2b.thm1.eq1}) that when either of the conditions in Theorem~\ref{ch1.sec2b.thm1}[1.-2.] hold, then the $I(t)$ state dies out exponentially, whenever $\lambda$ in (\ref{ch1.sec2b.thm1.eq1.proof.eq1.eq1}) and (\ref{ch1.sec2b.thm1.eq1.proof.eq2.eq1}) is positive, that is, $\lambda>0$. In addition, the rate of the exponential decay of each trajectories of $I(t)$ in each scenario of Theorem~\ref{ch1.sec2b.thm1}[1.-2.] is given by the estimate $\lambda>0$ of the LE\footnote{Lyapunov exponent} in (\ref{ch1.sec2b.thm1.eq1.proof.eq1.eq1}) and (\ref{ch1.sec2b.thm1.eq1.proof.eq2.eq1}).
The conditions in Theorem~\ref{ch1.sec2b.thm1}[1.-2.] can also be interpreted as follows. Recall, the BRN $R^{*}_{0}$ in (\ref{ch1.sec2.lemma2a.corrolary1.eq4}) (similarly in (\ref{ch1.sec2.theorem1.corollary1.eq3})) represents the expected number of secondary malaria cases that result from one infective placed in the disease free state $S^{*}_{0}=\frac{B}{\mu}\equiv 1$. Thus, $\frac{1}{R^{*}_{0}}=\frac{(\mu+d+\alpha)}{\beta S^{*}_{0}}$, for $R^{*}_{0}\geq 1$, represents the probability rate of infectious persons in the secondary infectious population $\beta S^{*}_{0}$ leaving the infectious state, either through natural death $\mu$, diseases related death $d$, or recovery and acquiring natural immunity at the rate $\alpha$. Thus, $\frac{1}{R^{*}_{0}}$ is the effective probability rate of surviving infectiousness until recovery with acquisition of natural immunity. Moreover, $\frac{1}{R^{*}_{0}}$ is a probability measure provided $R^{*}_{0}\geq 1$.
In addition, recall Theorem\ref{ch1.sec2b.lemma1.thm2} asserts that when $R^{*}_{0}\geq 1$, and the ESPR $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})$ is significantly large, then the outbreak of malaria establishes a malaria endemic steady state population $E_{1}$. The conditions for extinction of disease in Theorem~\ref{ch1.sec2b.thm1}[1.], that is $R^{*}_{0}\geq 1$ and $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})<\frac{1}{R^{*}_{0}}$ suggest that in the event where $R^{*}_{0}\geq 1$, and the disease is aggressive, and likely to establish an endemic steady state population, if the expected survival probability rate $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})$ of the malaria parasites over their complete life cycle of length $T_{1}+T_{2}$, is less than $\frac{1}{R^{*}_{0}}$- the effective probability rate of surviving infectiousness until recovery with natural immunity, then the malaria epidemic fails to establish an endemic steady state, and as a result, the disease ultimately dies out at an exponential rate $\lambda$ in (\ref{ch1.sec2b.thm1.eq1.proof.eq1.eq1}).
In the event where $R^{*}_{0}< 1$ in Theorem~\ref{ch1.sec2b.thm1}[2.], extinction of disease occurs exponentially over sufficiently long time, regardless of the survival of the parasites. Moreover, the rate of extinction is $\lambda$ in (\ref{ch1.sec2b.thm1.eq1.proof.eq2.eq1}).
\end{rem} \section{Persistence of susceptibility and stability of zero equilibrium } Theorem~\ref{ch1.sec2b.thm1} characterizes the behavior of the trajectories of the $I(t)$ coordinate of the solution $(S(t),I(t))$ of the decoupled system (\ref{ch1.sec0.eq3}) and (\ref{ch1.sec0.eq5}) in the phase plane. The question remains about how the trajectories for the $S(t)$ behave asymptotically in the phase plane.
Using Definition~\ref{ch1.sec0.eq13b.def1}[3-5], the following result describes the average behavior of the trajectories of $S(t)$ over sufficiently long time, and also states conditions for the stability of the disease-free equilibrium (DFE) of the decoupled system $E_{0}=(S^{*}_{0},0)=(1,0)$, whenever Theorem~\ref{ch1.sec2b.thm1} holds. \begin{thm}\label{ch1.sec2b.thm2} Suppose any of the conditions in the hypothesis of Theorem~\ref{ch1.sec2b.thm1}[1.-2.] are satisfied. It follows that in $D^{expl}(\infty)$, the trajectories of the susceptible state $S(t)$ of the decoupled system (\ref{ch1.sec0.eq3}) and (\ref{ch1.sec0.eq5}), satisfy \begin{equation}\label{ch1.sec2b.thm2.eq1} \lim_{t\rightarrow \infty}\frac{1}{t}\int_{t_{0}}^{t}S(\xi)d\xi=\frac{B}{\mu}\equiv 1. \end{equation} That is, the susceptible state is strongly persistent over long-time in the mean (see definition of persistence in the mean Definition~\ref{ch1.sec0.eq13b.def1}[3-4]). Moreover, it is stable in the mean, and the average value of the susceptible state over sufficiently long time is equal to $S(t)=S^{*}_{0}=\frac{B}{\mu}$, obtained when the system is in steady state. \end{thm} Proof:\\ Suppose either of the conditions in Theorem~\ref{ch1.sec2b.thm1}[1.-2.] hold, then it follows clearly from Theorem~\ref{ch1.sec2b.thm1} that for every $\epsilon>0$, there is a positive constant $K_{1}(\epsilon)\equiv K_{1}>0$, such that \begin{equation}\label{ch1.sec2b.thm2.proof.eq2}
I(t)<\epsilon,\quad \textrm{whenever $t>K_{1}$}. \end{equation} It follows from (\ref{ch1.sec2b.thm2.proof.eq2}) that \begin{equation}\label{ch1.sec2b.thm2.proof.eq3}
I(t-s)<\epsilon,\quad \textrm{whenever $t>K_{1}+h_{1},\forall s\in [t_{0}, h_{1}]$}. \end{equation} In $D^{expl}(\infty)$, define \begin{equation}\label{ch1.sec2b.thm2.proof.eq4} V_{1}(t)=S(t)+\alpha \int_{t_{0}}^{\infty} f_{T_{3}}(r)e^{\mu r}\int_{t-r}^{t}I(\theta)d\theta dr. \end{equation} The differential operator $\dot{V}_{1}$ applied to the Lyapunov functional $V_{1}(t)$ in (\ref{ch1.sec2b.thm2.proof.eq4}) leads to the following \begin{equation}\label{ch1.sec2b.thm2.proof.eq5}
\dot{V}_{1}(t)=g(S, I)-\mu S(t), \end{equation} where \begin{equation}\label{ch1.sec2b.thm2.proof.eq6}
g(S,I)=B-\beta S(t)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu_{v} s}G(I(t-s))ds+\alpha E(e^{-\mu T_{3}}) I(t). \end{equation} Estimating the right-hand-side of (\ref{ch1.sec2b.thm2.proof.eq5}) in $D^{expl}(\infty)$, and integrating over $[t_{0},t]$, it follows from (\ref{ch1.sec2b.thm2.proof.eq2})-(\ref{ch1.sec2b.thm2.proof.eq3}) that \begin{eqnarray}
V_{1}(t)&\leq& V_{1}(t_{0})+B(t-t_{0})+\int_{t_{0}}^{K_{1}}\alpha I(\xi)d\xi +\int_{K_{1}}^{t}\alpha I(\xi)d\xi -\mu\int_{t_{0}}^{t} S(\xi)d\xi,\nonumber\\
&\leq& V_{1}(t_{0})+B(t-t_{0})+\alpha \frac{B}{\mu}(K_{1}-t_{0}) +\alpha (t-K_{1})\epsilon -\mu\int_{t_{0}}^{t} S(\xi)d\xi.\label{ch1.sec2b.thm2.proof.eq7} \end{eqnarray}
Thus, dividing both sides of (\ref{ch1.sec2b.thm2.proof.eq7}) by $t$ and taking the limit supremum as $t\rightarrow \infty$, it follows that \begin{equation}\label{ch1.sec2b.thm2.proof.eq10} \limsup_{t\rightarrow \infty}\frac{1}{t}\int_{t_{0}}^{t}S(\xi)d\xi\leq \frac{B}{\mu}+ \frac{\alpha}{\mu}\epsilon. \end{equation} On the other hand, estimating $g(S,I)$ in (\ref{ch1.sec2b.thm2.proof.eq6}) from below and using the conditions of Assumption~\ref{ch1.sec0.assum1} and (\ref{ch1.sec2b.thm2.proof.eq3}), it is easy to see that in $D^{expl}(\infty)$, \begin{eqnarray}
g(S,I) &\geq & B-\beta S(t)\int^{h_{1}}_{t_{0}}f_{T_{1}}(s) e^{-\mu_{v} s}(I(t-s))ds
\geq B-\beta \frac{B}{\mu}E(e^{-\mu_{v}T_{1}})\epsilon,\forall t>K_{1}+h_{1},\nonumber\\
&\geq&B-\beta \frac{B}{\mu}\epsilon.\label{ch1.sec2b.thm2.proof.eq11} \end{eqnarray} Moreover, for $t\in[t_{0}, K_{1}+h_{1}]$, then \begin{equation}\label{ch1.sec2b.thm2.proof.eq11.eq1}
g(S,I)\geq B-\beta \left(\frac{B}{\mu}\right)^{2}. \end{equation} Therefore, applying (\ref{ch1.sec2b.thm2.proof.eq11})-(\ref{ch1.sec2b.thm2.proof.eq11.eq1}) into (\ref{ch1.sec2b.thm2.proof.eq5}), then integrating both sides of (\ref{ch1.sec2b.thm2.proof.eq5}) over $[t_{0},t]$, and diving the result by $t$, it is easy to see from (\ref{ch1.sec2b.thm2.proof.eq5}) that \begin{equation}\label{ch1.sec2b.thm2.proof.eq12}
\frac{1}{t}V_{1}(t)\geq \frac{1}{t}V_{1}(t_{0})+B(1-\frac{t_{0}}{t}) -\frac{1}{t}\beta\left( \frac{B}{\mu}\right)^{2}(K_{1}+h_{1}-t_{0})-\beta\frac{B}{\mu}\epsilon[1-\frac{K_{1}+h_{1}}{t}] -\frac{1}{t}\mu\int_{t_{0}}^{t}S(\xi)d\xi. \end{equation} Observe that in $D^{expl}(\infty)$, $\lim_{t\rightarrow \infty}\frac{1}{t}V_{1}(t)=0$, and $\lim_{t\rightarrow \infty}\frac{1}{t}V_{1}(t_{0})=0$. Therefore, rearranging (\ref{ch1.sec2b.thm2.proof.eq12}), and taking the limit infinimum of both sides as $t\rightarrow \infty$, it is easy to see that \begin{equation}\label{ch1.sec2b.thm2.proof.eq13}
\liminf_{t\rightarrow \infty} \frac{1}{t}\int_{t_{0}}^{t}S(\xi)d\xi\geq \frac{B}{\mu}-\frac{1}{\mu}\beta \frac{B}{\mu}\epsilon. \end{equation} It follows from (\ref{ch1.sec2b.thm2.proof.eq10}) and (\ref{ch1.sec2b.thm2.proof.eq13}) that \begin{equation}\label{ch1.sec2b.thm2.proof.eq14} \frac{B}{\mu}-\frac{1}{\mu}\beta \frac{B}{\mu}\epsilon\leq \liminf_{t\rightarrow \infty} \frac{1}{t}\int_{t_{0}}^{t}S(\xi)d\xi\leq \limsup_{t\rightarrow \infty}\frac{1}{t}\int_{t_{0}}^{t}S(\xi)d\xi\leq \frac{B}{\mu}+ \frac{\alpha}{\mu}\epsilon. \end{equation} Hence, for $\epsilon$ arbitrarily small, the result in (\ref{ch1.sec2b.thm2.eq1}) follows immediately from (\ref{ch1.sec2b.thm2.proof.eq14}). \begin{rem}\label{ch1.sec2b.thm2.proof.rem1a} Theorem~\ref{ch1.sec2b.thm2} signifies that
the DFE $E_{0}$ is strongly persistent and stable in the mean by Definition~\ref{ch1.sec0.eq13b.def1}[3-5]. That is, over sufficiently long time, on average the human population will be in the DFE $E_{0}$.
Thus, the conditions in Theorem~\ref{ch1.sec2b.thm1} are sufficient for malaria to be eradicated from the population, when the population is in a steady state. \end{rem} The next, result confirms that not only is the zero equilibrium state of the decoupled system (\ref{ch1.sec0.eq3}) and (\ref{ch1.sec0.eq5}) $E_{0}=(S^{*}_{0},0)=(1,0)$ stable and persistent on average over time, but also stable in the sense of Lyapunov.
\begin{thm}\label{ch1.sec2b.thm3} Suppose any of the conditions in the hypothesis of Theorem~\ref{ch1.sec2b.thm1}[1.-2.] are satisfied. Also, suppose the conditions of Theorem~\ref{ch1.sec2b.thm2} hold. It follows that in $D^{expl}(\infty)$, the DFE $E_{0}=(S^{*}_{0},0)=(\frac{B}{\mu},0)=(1,0)$ is stable in the sense of Lyapunov.
\end{thm}
Proof:\\
It is left to show that every trajectory that starts near $E_{0}$ remains near $E_{0}$ asymptotically. Indeed, if the hypothesis of Theorem~\ref{ch1.sec2b.thm1}[1.-2.] holds, then all trajectories in the phase-plane for the infectious state $I(t)$ converge asymptotically and exponentially to $I^{*}_{0}=0$. It is left to show that if the trajectories of the susceptible state $S(t)$ from Theorem~\ref{ch1.sec2b.thm2} (\ref{ch1.sec2b.thm2.eq1}), converge asymptotically in the mean to $S^{*}_{0}=\frac{B}{\mu}$, then they must remain asymptotically near $S^{*}_{0}=\frac{B}{\mu}$.
Indeed, if on the contrary, there exist a trajectory for $S(t)$ starting near $S^{*}_{0}=\frac{B}{\mu}=\equiv 1$ that does not stay near $S^{*}_{0}=\frac{B}{\mu}$ asymptotically, that is, suppose there exists some $\epsilon_{0}>0$ and $\delta (t_{0},\epsilon_{0})>0$, such that $||S(t_{0})-S^{*}_{0}||<\delta$, but $||S(t)-S^{*}_{0}||\geq \epsilon_{0}, \forall t\geq t_{0}$, then clearly from (\ref{ch1.sec2b.thm2.eq1}), either \begin{equation}\label{ch1.sec2b.thm2.rem1.eq1}
S^{*}_{0}=\lim_{t\rightarrow \infty}\frac{1}{t}\int_{t_{0}}^{t}S(\xi)d\xi\geq S^{*}_{0}+\epsilon_{0}\quad or\quad S^{*}_{0}=\lim_{t\rightarrow \infty}\frac{1}{t}\int_{t_{0}}^{t}S(\xi)d\xi\leq S^{*}_{0}-\epsilon_{0}. \end{equation} Thus, $\epsilon_{0}$ must be zero, otherwise (\ref{ch1.sec2b.thm2.rem1.eq1}) is a contradiction. Hence, $E_{0}=(S^{*}_{0},0)$ is stable in the sense of Lyapunov.
\begin{rem}\label{ch1.sec2b.thm2.rem1} Theorem~\ref{ch1.sec2b.thm2}, Theorem~\ref{ch1.sec2b.thm1}, and Theorem~\ref{ch1.sec2b.lemma1} signify that all trajectories of $(S(t), I(t)),t\geq t_{0}$ of the decoupled system (\ref{ch1.sec0.eq3}) and (\ref{ch1.sec0.eq5}) that start in $D^{expl}(\infty)\subset D(\infty)$ remain bounded in $D^{expl}(\infty)$. Moreover, the trajectories of $I(t), t\geq t_{0}$ of the solution $(S(t), I(t)),t\geq t_{0}$ in phase plane, ultimately turn to zero exponentially, while trajectories of the susceptible state $S(t)$ persist strongly, and converge in the mean to the DFE $S^{*}_{0}=\frac{B}{\mu}$, whenever $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})<\frac{1}{R^{*}_{0}}$, for $R^{*}_{0}\geq 1$, or whenever the basic production number satisfy $R^{*}_{0}<1$.
Moreover, from Theorem~\ref{ch1.sec2b.thm3}, the DFE $(S(t), I(t))=E_{0}=(S^{*}_{0},0)=(\frac{B}{\mu},0)=(1,0)$ is uniformly globally asymptotically stable.
Thus, the conditions in Theorem~\ref{ch1.sec2b.thm1} are strong disease eradication conditions.
The above observations suggest that over sufficiently long time, the population that remains will be all susceptible malaria-free people, and the population size will be averagely equal to the DFE $S^{*}_{0}=\frac{B}{\mu}$ of (\ref{ch1.sec0.eq3}) and (\ref{ch1.sec0.eq5}). \end{rem}
\section{Permanence of infectivity near nonzero equilibrium\label{ch1.sec3a}}
As remarked in Theorem~\ref{ch1.sec2b.lemma1.thm2}, when $R^{*}_{0}$ in (\ref{ch1.sec2.lemma2a.corrolary1.eq4}) satisfies $R^{*}_{0}>1$, the endemic equilibrium of the decoupled system (\ref{ch1.sec0.eq3}) and (\ref{ch1.sec0.eq5}) exists and is denoted $E_{1}=(S^{*}_{1}, I^{*}_{1})$. In this section, conditions for $I(t)$ to be strongly persistent (Definition~\ref{ch1.sec0.eq13b.def1}[1]) in the neighborhood of $E_{1}$ are given.
\begin{lemma}\label{ch1.sec4.lemma1}
Suppose the conditions of Theorem~\ref{ch1.sec2b.lemma1} and Theorem~\ref{ch1.sec2b.lemma1.thm2}
are satisfied, and let the nonlinear incidence function $G$ satisfy the assumptions of Assumption~\ref{ch1.sec0.assum1}.
Then every positive solution $(S(t), I(t))\in D(\infty)$ of the decoupled system (\ref{ch1.sec0.eq3}) and (\ref{ch1.sec0.eq5}) with initial conditions (\ref{ch1.sec0.eq06a}) and (\ref{ch1.sec0.eq06b}) satisfies the following conditions:
\begin{equation}\label{ch1.sec4.lemma1.eq1}
\liminf_{t\rightarrow \infty}{S(t)}\geq v_{1}\equiv \frac{B}{\mu+\beta G(S_{0})}\quad and\quad \liminf_{t\rightarrow \infty}{I(t)}\geq v_{2}\equiv qI^{*}_{1}e^{-(\mu+d+\alpha)(\rho+1)h},
\end{equation}
where $h=h_{1}+h_{2}$, and $\rho>0$ is a suitable positive constant, $S^{*}_{1}<\min\{S_{0}, S^{\vartriangle}\}$ and $0<q<\bar{q}<1$, given that,
\begin{equation}\label{ch1.sec4.lemma1.eq2}
\bar{q}=\frac{B\beta E(e^{-\mu T_{1}})G(I^{*}_{1})-\mu \alpha E(e^{-\mu T_{3}})I^{*}_{1}}{\left(B+\alpha E(e^{-\mu T_{3}})I^{*}_{1}\right)\beta I^{*}_{1}},\quad S^{\vartriangle}=\frac{B}{k}\left(1-e^{-k\rho h}\right), k=\mu +\beta G(q I^{*}_{1}).
\end{equation}
\end{lemma}
Proof:\\
Recall (\ref{ch1.sec1.thm1a.eq0}) asserts that for $N(t)=S(t)+E(t)+ I(t)+R(t)$, $\limsup_{t\rightarrow \infty} {N(t)}\leq S^{*}_{0}=\frac{B}{\mu}$. This implies that $\limsup_{t\rightarrow \infty} {S(t)}\leq S^{*}_{0}\equiv 1$. This further implies that for any arbitrarily small $\epsilon>0$, there exists a sufficiently large $\Lambda>0$, such that
\begin{equation}\label{ch1.sec4.lemma1.proof.eq1}
I(t)\leq S^{*}_{0} +\varepsilon,\quad whenever, \quad t\geq \Lambda.
\end{equation}
Without loss of generality, let $\Lambda_{1}>0$ be sufficiently large such that
\[t\geq \Lambda\geq \max_{(s, r)\in [t_{0}, h_{1}]\times [t_{0}, \infty)}{(\Lambda_{1}+s, \Lambda_{1}+r)}.\]
It follows from Assumption~\ref{ch1.sec0.assum1}, (\ref{ch1.sec0.eqn0.eq1}) and (\ref{ch1.sec0.eq3}) that
\begin{eqnarray}
\frac{dS(t)}{dt}\geq B-\beta S(t)\int_{t_{0}}^{h_{1}}f_{T_{1}}(s)e^{-\mu s}G(S^{*}_{0}+\epsilon)ds-\mu S(t)
\geq B-\left[\mu+ \beta G(S^{*}_{0}+\epsilon)\right]S(t).\label{ch1.sec4.lemma1.proof.eq2}
\end{eqnarray}
From (\ref{ch1.sec4.lemma1.proof.eq2}) it follows that
\begin{equation}\label{ch1.sec4.lemma1.proof.eq3}
S(t)\geq \frac{B}{k_{1}} -\frac{B}{k_{1}}e^{-k_{1}(t-t_{0})}+S(t_{0}) e^{-k_{1}(t-t_{0})},
\end{equation}
where $k_{1}=\mu+ \beta G(S^{*}_{0}+\epsilon)$.
It is easy to see from (\ref{ch1.sec4.lemma1.proof.eq3})
\begin{equation}\label{ch1.sec4.lemma1.proof.eq4}
\liminf_{t\rightarrow \infty}{S(t)}\geq \frac{B}{\mu+\beta G(S_{0}+\epsilon)}.
\end{equation}
Since $\epsilon>0$ is arbitrarily small, then the first part of (\ref{ch1.sec4.lemma1.eq1}) follows immediately.
In the following it is shown that $ \liminf_{t\rightarrow \infty}{I(t)}\geq v_{2}$. In order to establish this result, it is first proved that it is impossible that $I(t)\leq q I^{*}_{1}$ for sufficiently large $t\geq t_{0}$, where $q\in(0, 1)$ is defined in the hypothesis. Suppose on the contrary there exists some sufficiently large $\Lambda_{0}> t_{0}>0$, such that $I(t)\leq q I^{*}_{1}, \forall t\geq \Lambda_{0}$. It follows from (\ref{ch1.sec0.eq3}) that
\begin{eqnarray}
S^{*}_{1} = \frac{B+\alpha E(e^{-\mu T_{3}})I^{*}_{1}}{\mu+ \beta E(e^{-\mu T_{1}})G(I^{*}_{1})}
= \frac{B}{\mu+\frac{B\beta E(e^{-\mu T_{1}})G(I^{*}_{1})-\mu \alpha E(e^{-\mu T_{3}})I^{*}_{1} }{B+\alpha E(e^{-\mu T_{3}})I^{*}_{1}}}.\label{ch1.sec4.lemma1.proof.eq5}
\end{eqnarray}
But, it can be easily seen from (\ref{ch1.sec0.eq3}) and (\ref{ch1.sec0.eq5}) that
\begin{eqnarray}
&&B\beta E(e^{-\mu T_{1}})G(I^{*}_{1})-\mu \alpha E(e^{-\mu T_{3}})I^{*}_{1}=\frac{\mu(\mu+d+\alpha)\left[S^{*}_{0}-\frac{\alpha E(e^{-\mu T_{3}})E(e^{-\mu T_{2}})}{(\mu +d+\alpha)}S^{*}_{1}\right]}{E(e^{-\mu T_{2}})S^{*}_{1}}I^{*}_{1}\nonumber \\
&&\geq \frac{\mu(\mu+d+\alpha)(S^{*}_{0}-S^{*}_{1})}{E(e^{-\mu T_{2}})S^{*}_{1}}>0,\quad since\quad S^{*}_{0}=\frac{B}{\mu}\geq S^{*}_{1}.\label{ch1.sec4.lemma1.proof.eq6}
\end{eqnarray}
Therefore, from (\ref{ch1.sec4.lemma1.proof.eq5}), it follows that
\begin{equation}\label{ch1.sec4.lemma1.proof.eq6}
S^{*}_{1}<\frac{B}{\mu +\beta I^{*}_{1}q}\leq \frac{B}{\mu +\beta G(qI^{*}_{1})},
\end{equation}
where $0<q<\bar{q}$, and $\bar{q}$ is defined in (\ref{ch1.sec4.lemma1.eq2}).
For all vector values $ (s, r)\in [t_{0}, h_{1}]\times [t_{0}, \infty) $ define
\begin{equation}
\Lambda_{0,max}=\max_{(s, r)\in [t_{0}, h_{1}]\times [t_{0}, \infty)}{(\Lambda_{0}+s, \Lambda_{0}+r)},
\end{equation} It follows from Assumption~\ref{ch1.sec0.assum1} and (\ref{ch1.sec0.eq3}) that for all $t\geq \Lambda_{0,max}$, \begin{equation}\label{ch1.sec4.lemma1.proof.eq7}
S(t)\geq \frac{B}{k} -\frac{B}{k}e^{-k(t-\Lambda_{0,max})}+S(\Lambda_{0,max}) e^{-k(t-\Lambda_{0,max})}, \end{equation} where $k$ is defined in (\ref{ch1.sec4.lemma1.eq2}).
For $t\geq \Lambda_{0,max}+ \rho h$, where $ h=h_{1}+h_{2}$, and $\rho>0$ is sufficiently large, it follows from (\ref{ch1.sec4.lemma1.proof.eq7}) that
\begin{equation}\label{ch1.sec4.lemma1.proof.eq8}
S(t)\geq \frac{B}{k}\left[1-e^{-k(t-\Lambda_{0,max})}\right]\geq \frac{B}{k}\left[1-e^{-k\rho h}\right]=S^{\vartriangle}.
\end{equation}
Hence, from (\ref{ch1.sec4.lemma1.proof.eq6}) and (\ref{ch1.sec4.lemma1.proof.eq8}), it follows that for some suitable choice of $\rho>0$ sufficiently large, then
\begin{equation}\label{ch1.sec4.lemma1.proof.eq9}
S^{\vartriangle}>S^{*}_{1}, \forall t\geq \Lambda_{0,max}+ \rho h.
\end{equation}
For $t\geq \Lambda_{0,max}+ \rho h$, define
\begin{eqnarray}
V(t) &=& I(t) + \beta S^{*}_{1}\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu(s+u)}\int_{t-s}^{t}G(I(v-u))dvdsdu \nonumber\\
&&+ \beta S^{*}_{1}\int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu(s+u)}\int_{t-u}^{t}G(I(v))dvdsdu. \label{ch1.sec4.lemma1.proof.eq10}
\end{eqnarray}
It is easy to see from system (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq5}), and (\ref{ch1.sec4.lemma1.proof.eq10}) that differentiating $V(t)$ with respect to the system (\ref{ch1.sec0.eq3}) and (\ref{ch1.sec0.eq5}), leads to the following
\begin{eqnarray}
\dot{V}(t) &=&\beta \int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu(s+u)}G(I(t-s-u))[S(t-u)-S^{*}_{1}]dsdu \nonumber\\
&&+\left[\beta S^{*}_{1}E(e^{-\mu(T_{1}+T_{2})})\frac{G(I(t))}{I(t)}-(\mu + d+\alpha)\right]I(t).\label{ch1.sec4.lemma1.proof.eq11}
\end{eqnarray}
For all $t\geq \Lambda_{0,max}+ \rho h +h>\Lambda_{0,max}+ \rho h +h_{2}$, it follows from (\ref{ch1.sec4.lemma1.eq1a}), (\ref{ch1.sec4.lemma1.proof.eq9}) and (\ref{ch1.sec0.eq3})- (\ref{ch1.sec0.eq5}) that
\begin{eqnarray}
\dot{V}(t) &\geq&\beta \int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu(s+u)}G(I(t-s-u))[S^{\vartriangle}-S^{*}_{1}]dsdu \nonumber \\
&&+\left[\beta S^{*}_{1}E(e^{-\mu(T_{1}+T_{2})})\frac{G(I^{*}_{1})}{I^{*}_{1}}-(\mu + d+\alpha)\right]I(t)\nonumber \\
&=&\beta \int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu(s+u)}G(I(t-s-u))[S^{\vartriangle}-S^{*}_{1}]dsdu.\label{ch1.sec4.lemma1.proof.eq11b}
\end{eqnarray}
Observe that the union of the subintervals $\bigcup _{(s, u)\in [t_{0}, h_{1}]\times[t_{0}, h_{2}]}{[t_{0}-(s+u), t_{0}]}=[t_{0}-h, t_{0}]$, where $ h=h_{1}+h_{2}$. Denote the following
\begin{equation}\label{ch1.sec4.lemma1.proof.eq11c}
i_{min}=\min_{\theta\in [t_{0}-h, t_{0}], (s, u)\in [t_{0}, h_{1}]\times[t_{0}, h_{2}]} {I(\Lambda_{0,max}+ \rho h+ h+s+u+\theta)}.
\end{equation}
Note that (\ref{ch1.sec4.lemma1.proof.eq11c}) is equivalent to
\begin{equation}\label{ch1.sec4.lemma1.proof.eq11a}
i_{min}=\min_{\theta\in [t_{0}-h, t_{0}]} {I(\Lambda_{0,max}+ \rho h+ h+h+\theta)}.
\end{equation}
It is shown in the following that $I(t)\geq i_{min}, \forall t\geq \Lambda_{0,max}+ \rho h+h\geq \Lambda_{0,max}+ \rho h+u$, $\forall u\in [t_{0}, h_{2}]$.
Suppose on the contrary there exists $\tau_{1}\geq 0$ such that $I(t)\geq i_{min}$ for all
$t\in [\Lambda_{0,max}+ \rho h +h, \Lambda_{0,max}+ \rho h +h+h+\tau_{1}]\supset [\Lambda_{0,max}+ \rho h +h, \Lambda_{0,max}+ \rho h +h+ s+u+\tau_{1}], \forall (s, u)\in [t_{0}, h_{1}]\times[t_{0}, h_{2}]$
\begin{equation}\label{ch1.sec4.lemma1.proof.eq12}
I(\Lambda_{0,max}+ \rho h +h+h+\tau_{1})=i_{min},\quad and\quad \dot{I}(\Lambda_{0,max}+ \rho h +h+h+\tau_{1})\leq 0.
\end{equation}
For the value of $t=\Lambda_{0,max}+ \rho h +h+h+\tau_{1}$, it follows that $ S(t-u)>S^{\vartriangle}>S^{*}_{1}$, and $t-s-u\in [\Lambda_{0,max}+ \rho h +h, \Lambda_{0,max}+ \rho h +h+h+\tau_{1}]$, $ \forall (s, u)\in [t_{0}, h_{1}]\times[t_{0}, h_{2}]$, and it can be further seen from (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq5}), (\ref{ch1.sec4.lemma1.proof.eq9}) and (\ref{ch1.sec4.lemma1.eq1a}) that
\begin{eqnarray}
&&\dot{I}(t)\geq \beta E(e^{-\mu(T_{1}+T_{2})})G(i_{min})S^{\vartriangle}-(\mu+d+\alpha)i_{min}=\left[\beta E(e^{-\mu(T_{1}+T_{2})})\frac{G(i_{min})}{i_{min}}S^{\vartriangle}-(\mu+d+\alpha)\right]i_{min} \nonumber \\
&&>\left[\beta E(e^{-\mu(T_{1}+T_{2})})\frac{G(I^{*}_{1})}{I^{*}_{1}}S^{*}_{1}-(\mu+d+\alpha)\right]i_{min}=0.\label{ch1.sec4.lemma1.proof.eq13}
\end{eqnarray} But (\ref{ch1.sec4.lemma1.proof.eq13}) contradicts (\ref{ch1.sec4.lemma1.proof.eq12}). Therefore, $I(t)\geq i_{min}, \forall t\geq \Lambda_{0,max}+ \rho h+h\geq \Lambda_{0,max}+ \rho h+u+s$, $\forall (s, u)\in [t_{0}, h_{1}]\times[t_{0}, h_{2}]$.
It follows further from (\ref{ch1.sec4.lemma1.proof.eq11})-(\ref{ch1.sec4.lemma1.proof.eq11c}), and the Assumption~\ref{ch1.sec0.assum1} that for $\forall t\geq \Lambda_{0,max}+ \rho h+h+h\geq \Lambda_{0,max}+ \rho h+h+s+u$, $\forall (s,u)\in [t_{0}, h_{1}]\times[t_{0}, h_{2}]$.
\begin{eqnarray}
\dot{V}(t) &\geq&\beta \int_{t_{0}}^{h_{2}}\int_{t_{0}}^{h_{1}}f_{T_{2}}(u)f_{T_{1}}(s)e^{-\mu(s+u)}G(I(t-s-u))[S^{\vartriangle}-S^{*}_{1}]dsdu\nonumber\\
&>& \beta E(e^{-\mu(T_{1}+T_{2})})G(i_{min})(S^{\vartriangle}-S^{*}_{1})>0.\label{ch1.sec4.lemma1.proof.eq14}
\end{eqnarray}
From (\ref{ch1.sec4.lemma1.proof.eq14}), it implies that $\limsup_{t\rightarrow\infty}{V(t)}=+\infty$.
On the contrary, it can be seen from (\ref{ch1.sec1.thm1a.eq0}) that $\limsup_{t\rightarrow \infty} N(t)\leq S^{*}_{0}=\frac{B}{\mu}$, which implies that $\limsup_{t\rightarrow \infty} I(t)\leq S^{*}_{0}=\frac{B}{\mu}$. This further implies that for every $\epsilon>0$ infinitesimally small, there exists $\tau_{2}>0$ sufficiently large such that $I(t)\leq S^{*}_{0}+\varepsilon, \forall t\geq \tau_{2}$. It follows that from Assumption~\ref{ch1.sec0.assum1} that
\begin{equation}\label{ch1.sec4.lemma1.proof.eq15}
G(I(t-s-u))\leq G(I(v-u))\leq G(I(t-u))\leq G(I(t))\leq G(S^{*}_{0}+\epsilon), \forall v\in [t-s,t], (s,u)\in [t_{0}, h_{1}]\times[t_{0}, h_{2}].
\end{equation}
From (\ref{ch1.sec4.lemma1.proof.eq15}), it follows that
\begin{equation}\label{ch1.sec4.lemma1.proof.eq16}
\limsup_{t\rightarrow\infty}{G(I(t-s-u))}\leq \limsup_{t\rightarrow\infty}{G(I(t))}\leq G(S^{*}_{0}).
\end{equation}
It is easy to see from (\ref{ch1.sec4.lemma1.proof.eq10}) and (\ref{ch1.sec4.lemma1.proof.eq16}) that
\begin{equation}\label{ch1.sec4.lemma1.proof.eq17}
\limsup_{t\rightarrow\infty}{V(t)}\leq S^{*}_{0}+\beta S^{*}_{1}G(S^{*}_{0})E\left((T_{1}+T_{2})e^{-\mu(T_{1}+T_{2})}\right)<\infty.
\end{equation}
Therefore, it is impossible that $I(t)\leq q I^{*}_{1}$ for sufficiently large $t\geq t_{0}$, where $q\in(0, 1)$.
Hence, the following are possible, $(Case(i.))$ $I(t)\geq qI^{*}_{1}$ for all $t$ sufficiently large, and $(Case(ii.))$ $I(t)$ oscillates about $qI^{*}_{1}$ for sufficiently large $t$. Obviously, we need show only $Case(ii.)$. Suppose $t_{1}$ and $t_{2}$ are are sufficiently large values such that
\begin{equation}\label{ch1.sec4.lemma1.proof.eq18}
I(t_{1})=I(t_{2})= qI^{*}_{1},\quad and\quad I(t)<qI^{*}_{1}, \forall (t_{1}, t_{2}).
\end{equation}
If for all $(s,u)\in [t_{0}, h_{1}]\times[t_{0}, h_{2}]$, $t_{2}-t_{1}\leq \rho h+h$, where $ h=h_{1}+h_{2}$, observe that $[t_{1}, t_{1}+\rho h+s+u]\subseteq [t_{1}, t_{1}+\rho h+h]$, and it is easy to see from (\ref{ch1.sec0.eq3}) by integration that
\begin{equation}\label{ch1.sec4.lemma1.proof.eq19}
I(t)\geq I(t_{1})e^{-(\mu+d+\alpha)(t-t_{1})}\geq qI^{*}_{1} e^{-(\mu+d+\alpha)(\rho+1)h}\equiv v_{2}.
\end{equation}
If for all $(s,u)\in [t_{0}, h_{1}]\times[t_{0}, h_{2}]$, $t_{2}-t_{1}>\rho h+h\geq \rho h+ s+u$, then it can be seen easily that $I(t)\geq v_{2}$, for all $t\in [t_{1}, t_{1}+\rho h+s+u]\subseteq [t_{1}, t_{1}+\rho h+h]$.
Now, for each $t\in (\rho h+h, t_{2})\supseteq (\rho h+s+u, t_{2})$, $\forall (s,u)\in [t_{0}, h_{1}]\times[t_{0}, h_{2}]$,
one can also claim that $I(t)\geq v_{2}$. Indeed, as similarly shown above, suppose on the contrary for all $(s,u)\in [t_{0}, h_{1}]\times[t_{0}, h_{2}]$, $\exists T^{*}>0$ such that $I(t)\geq v_{2}$, $\forall t\in [t_{1}, t_{1}+\rho h+h+T^{*}]\supseteq [t_{1}, t_{1}+\rho h+s+u+T^{*}]$
\begin{equation}\label{ch1.sec4.lemma1.proof.eq20}
I(t_{1}+\rho h+h+T^{*})=v_{2},\quad but \quad \dot{I}(t_{1}+\rho h+h+T^{*})\leq 0.
\end{equation}
It follows from (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq5}) and (\ref{ch1.sec4.lemma1.eq1a}) that for the value of $t=t_{1}+\rho h+h+T^{*}$,
\begin{eqnarray}
&&I(t) \geq \beta E(e^{-\mu(T_{1}+T_{2})})G(v_{2})S^{\vartriangle}-(\mu+d+\alpha)v_{2}>\left[\beta E(e^{-\mu(T_{1}+T_{2})})\frac{G(v_{2})}{v_{2}}S^{*}_{1}-(\mu+d+\alpha)\right]v_{2} \nonumber \\
&&\geq\left[\beta E(e^{-\mu(T_{1}+T_{2})})\frac{G(I^{*}_{1})}{I^{*}_{1}}S^{*}_{1}-(\mu+d+\alpha)\right]v_{2}=0.\label{ch1.sec4.lemma1.proof.eq21}
\end{eqnarray}
Observe that (\ref{ch1.sec4.lemma1.proof.eq21}) contradicts (\ref{ch1.sec4.lemma1.proof.eq20}). Therefore, $I(t)\geq v_{2}$, for $t\in [t_{1}, t_{2}]$.
And since $[t_{1}, t_{2}]$ is arbitrary, it implies that $I(t)\geq v_{2}$ for all sufficiently large $t$. Therefore (\ref{ch1.sec4.lemma1.eq1}) is satisfied.
\begin{thm}\label{ch1.sec4.thm1}
If the conditions of Lemma~ \ref{ch1.sec4.lemma1} are satisfied, then the system (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq5}) is strongly permanent for any total delay time $h=h_{1}+h_{2}$ according to Definition~\ref{ch1.sec0.eq13b.def1}[1].
\end{thm}
\begin{rem}\label{ch1.sec4.rem1}
\item[1.] It can be seen from Lemma~ \ref{ch1.sec4.lemma1} (\ref{ch1.sec4.lemma1.eq1}) that when $\beta=0$, then $v_{1}=\frac{B}{\mu}$. That is, when disease transmission stops, then asymptotically, the smallest total susceptible that remains are new births over the average lifespan $\frac{1}{\mu}$ of the population, equivalent to the DFE $S^{*}_{0}=\frac{B}{\mu}\equiv 1$. Also, as $\beta\rightarrow\infty$, then the total susceptible that remains $v_{1}\rightarrow 0^{+}$. That is, as disease transmission rises, even the new births are either infected, or die from natural or disease related causes over time.
\item[2.] From (\ref{ch1.sec4.lemma1.eq1}), observe that $e^{-(\mu+d+\alpha)(\rho+1)h}$ is the survival probability from natural death ($\mu$), disease mortality ($d$), and from infectiousness ($\alpha$), over the total life cycle of the parasite $h$. Thus, the smallest infectious state that remains asymptotically $v_{2}\equiv qI^{*}_{1}e^{-(\mu+d+\alpha)(\rho+1)h}$ is a fraction $q\in(0,1)$ of the endemic equilibrium $I^{*}_{1}$ that survives from death and disease over life cycle $h$.
Since $\frac{1}{(\mu+d+\alpha)}$ is the effective average lifespan of an individual who survives the disease until recovery at rate $\alpha$, it follows from (\ref{ch1.sec4.lemma1.eq1}) that as $(\mu+d+\alpha)\rightarrow 0^{+}$ and consequently $\frac{1}{(\mu+d+\alpha)}>>1$, then $e^{-(\mu+d+\alpha)(\rho+1)h}\rightarrow 1^{-}$. Moreover, from (\ref{ch1.sec4.lemma1.eq1}), $v_{2}\rightarrow qI^{*}_{1}$. That is, when malaria is actively transmitted $\beta>0$, so that more susceptible individuals become infected, but the effective average lifespan is still high because for example, malaria is treated, or healthier lifestyles are encouraged, and less people die from the disease $0\leq d<<1$, and from natural causes $0\leq\mu<<1$, then the total infectious state that remains over time $v_{2}$ is a fraction $q>>qe^{-(\mu+d+\alpha)(\rho+1)h}$ of all infected at steady state $I^{*}_{1}$, i.e. more infectious remains over time when malaria is treated effectively, or healthier living standards are encouraged.
\item[3.] The question of what conditions the population ever gets extinct in time is answered from [1.]~\&~[2.] above. Since as $\beta\rightarrow +\infty$, and $(\mu+d+\alpha)\rightarrow \infty$, then $v_{1}\rightarrow 0^{+}$, and $v_{2}\rightarrow 0^{+}$, respectively, from (\ref{ch1.sec4.lemma1.eq1}). Thus, extinction is ever possible in time, whenever disease transmission rate is high, and the response to malaria treatment or living standards are very poor.
\end{rem}
\section{Example: Application to P. vivax malaria}\label{ch1.sec4}
In this section, the extinction and persistence results are exhibited for the \textit{P.vivax malaria} example in Wanduku \cite{wanduku-extinct}. This is accomplished by examining the trajectories of the decoupled system (\ref{ch1.sec0.eq3}) and (\ref{ch1.sec0.eq5}) relative to the zero and endemic equilibria. To conserve space, we recall the dimensionless parameters in [Table 1,\cite{wanduku-extinct}, page 3793] given in Table~\ref{ch1.sec4.table1}, and the reader is referred to \cite{wanduku-extinct} for detailed description of the \textit{P.vivax malaria} scenario.
The dimensional estimates for the parameters of the malaria model given in [(a.)-(e.), \cite{wanduku-extinct}, page 3792] are applied to (\ref{ch1.sec0.eq6.eq20}) to find the dimensionless parameters for the model (\ref{ch1.sec0.eq3})-(\ref{ch1.sec0.eq6}) given in Table~\ref{ch1.sec4.table1}.
\begin{table}[h]
\centering
\caption{A list of dimensionless values for the system parameters for Example 1. }\label{ch1.sec4.table1}
\begin{tabular}{l l l}
Disease transmission rate&$\beta$& Subsection~\ref{ch1.sec4.subsec1} ($0.02146383$), Subsection~\ref{ch1.sec4.subsec2} ($0.2146383$)\\\hline
Constant Birth rate&$B$&$ 8.476678e-06$\\\hline
Recovery rate& $\alpha$& $0.08571429$\\\hline
Disease death rate& $d$& 0.0001761252\\\hline
Natural death rate& $\mu$, $\mu_{v}$& $ 8.476678e-06, 42.85714$\\\hline
Incubation delay in vector& $T_{1}$& 0.105 \\\hline
Incubation delay in host& $T_{2}$& 0.175\\\hline
Immunity delay time& $T_{3}$& 2.129167\\\hline
\end{tabular} \end{table}
Moreover, the Euler approximation scheme is used to generate trajectories for the different states $S(t), E(t), I(t), R(t)$ over the time interval $[0,1000]$ days. The special nonlinear incidence functions $G(I)=\frac{a_{1}I}{1+I}, a_{1}=0.05$ in \cite{gumel} is utilized. Furthermore, the following initial fractions of susceptible, exposed, infectious and removed individuals in the initial population size $\hat{N}(t_{0})=65000$ are used: \begin{eqnarray} &&S(t)= 10/23\approx 28261/65000, E(t)= 5/23\approx 14131/65000, I(t)= 6/23\approx 16957/65000,\nonumber\\ &&R(t)= 2/23\approx 5653/65000,\forall t\in [-T,0], T=\max(T_{1}+T_{2}, T_{3})=2.129167.\label{ch1.sec4.eq1}
\end{eqnarray}
Recall Section~\ref{ch1.sec1} asserts that the endemic equilibrium $E_{1}$ exists, whenever the BRN $R^{*}_{0}>1$, where $R^{*}_{0}$ is defined in (\ref{ch1.sec2.lemma2a.corrolary1.eq4}). Thus, it follows that when $R^{*}_{0}>1$, the endemic equilibrium $E_{1}=(S^{*}_{1}, E^{*}_{1},I^{*}_{1}, R^{*}_{1})$ satisfies the following system \begin{eqnarray} &&B-\beta Se^{-\mu_{v} T_{1}}G(I)-\mu S+\alpha I e^{-\mu T_{3}}=0, \beta Se^{-\mu_{v} T_{1}}G(I)-\mu E -\beta Se^{- (\mu_{v}T_{1}+\mu T_{2})}G(I)=0,\nonumber\\ &&\beta Se^{- (\mu_{v}T_{1}+\mu T_{2})}G(I)-(\mu+d+\alpha)I=0,\alpha I-\mu R-\alpha I e^{-\mu T_{3}}=0.\label{ch1.sec4.eq1} \end{eqnarray}
For the given set of dimensionless parameter estimates in Table~\ref{ch1.sec4.table1}, the DFE is $E_{0}=(S^{*}_{0}, 0,0)=(1,0,0)$. Also, the endemic equilibrium is given as $E_{1}=(S^{*}_{1}, E^{*}_{1},I^{*}_{1})=(0.002323845,0.00068247,0.04540019)$.
\subsection{Example for extinction of disease}\label{ch1.sec4.subsec1}
For the given set of dimensionless parameter estimates in Table~\ref{ch1.sec4.table1}, where $\beta=0.02146383$, from (\ref{ch1.sec2.lemma2a.corrolary1.eq4}) the BRN is $\hat{R}^{*}_{0}=0.2498732<1$. Therefore, $E_{0}$ is stable, and the endemic equilibrium $E_{1}=(S^{*}_{1}, E^{*}_{1},I^{*}_{1})$ fails to exist.
\begin{figure}
\caption{(a-1), and (b-1), show the trajectories of the states $(S,I)$, respectively, over sufficiently long time $t\in [0,1000]$, whenever the intensity of the incidence of malaria is $a=0.05$.
The BRN in (\ref{ch1.sec2.lemma2a.corrolary1.eq4}) in this case is $R^{*}_{0}= 0.2498732<1$, the estimate of the LE, or rate of extinction of the disease in (\ref{ch1.sec2b.thm1.eq1.proof.eq2.eq1}) is $\lambda= 0.06443506>0$.
}
\label{ch1.sec4.subsec1.fig2}
\end{figure}
Figure~\ref{ch1.sec4.subsec1.fig2} verifies the results about the extinction of the $I(t)$ state over time in Theorem~\ref{ch1.sec2b.thm1}, and the persistence of the $S(t)$ state over timein Theorem~\ref{ch1.sec2b.thm2}. Indeed, it is observed that for the given parameter values in Table~\ref{ch1.sec4.table1}, and the initial conditions in (\ref{ch1.sec4.eq1}), the BRN in (\ref{ch1.sec2.lemma2a.corrolary1.eq4}) in this scenario is $R^{*}_{0}= 0.2498732<1$. Therefore, the condition of Theorem~\ref{ch1.sec2b.thm1}(a.) and Theorem~\ref{ch1.sec2b.thm2} are satisfied, and from (\ref{ch1.sec2b.thm1.eq1.proof.eq2.eq1}), the estimate of the rate of extinction of the malaria population $I(t)$ is $\lambda= 0.06443506>0$. That is, \begin{equation}\label{ch1.sec4.subsec1.eq1}
\limsup_{t\rightarrow \infty}{\frac{1}{t}\log{(I(t))}}\leq -\lambda = -0.06443506. \end{equation}
The Figure~\ref{ch1.sec4.subsec1.fig2}(b-1) confirms that over sufficiently large time, when $\lambda>0$, then the infectious state approaches zero, that is, $\lim_{t\rightarrow \infty}I(t)=0$. Furthermore, the BRN $R^{*}_{0}= 0.2498732<1$, signifies that the disease is getting eradicated from the population over time. This is confirmed by Figure~\ref{ch1.sec4.subsec1.fig2}(a-1), where $S(t)$ appears to be rising over time, and approaching the DFE state $S^{*}_{0}=\frac{B}{\mu}=1$, that is, $\lim_{t\rightarrow \infty}S(t)=1$.
\subsection{Persistence of malaria}\label{ch1.sec4.subsec2} For the given set of dimensionless parameter estimates in Table~\ref{ch1.sec4.table1}, when $\beta=7.941616$, from (\ref{ch1.sec2.lemma2a.corrolary1.eq4}) the BRN becomes $\hat{R}^{*}_{0}=92.45307>1$. Therefore, the DFE $E_{0}=(S^{*}_{0}, 0,0)=(1,0,0)$ becomes unstable, and the endemic equilibrium exists, and given as $E_{1}=(S^{*}_{1}, E^{*}_{1},I^{*}_{1})=(6.281296e-06,0.0006840553,0.04550565)$.
It can be shown from (\ref{ch1.sec4.lemma1.eq1}) that for some suitable choice of $q\in (0,1)$ and $\rho>0$, for $t\in [0,1000]$,
\begin{eqnarray}
\liminf_{t\rightarrow \infty}{S(t)}=0.4086943>> v_{1}\equiv \frac{B}{\mu+\beta G(S_{0})}=4.269316e-05,\nonumber\\
\liminf_{t\rightarrow \infty}{I(t)}=0.2360496>> v_{2}\equiv qI^{*}_{1}e^{-(\mu+d+\alpha)(\rho+1)h}=0.04550565qe^{-0.02405169(1+\rho)}.\label{ch1.sec4.subsec2.eq1}
\end{eqnarray} Hence, from Theorem~\ref{ch1.sec4.thm1} there is a significant number of infectious people present over time $[0,1000]$, and as a result malaria persists in the population over time. These facts are further illustrated by Figure~\ref{ch1.sec4.subsec1.fig3} over $[0,1000]$.
\begin{figure}\label{ch1.sec4.subsec1.fig3}
\end{figure}
\section{conclusion}
The vector-human population dynamic models are derived. The models have a general nonlinear incidence rate. The extinction and persistence of the vector-borne disease in the SEIRS epidemic models are studied. Numerical simulation results are given to confirm the results.
\paragraph{Acknowledgment.}
Thanks to the Editor and reviewers for the thorough and constructive feedback.
\end{document} | arXiv |
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