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# Linear programming: formulation, simplex method, and applications
To formulate a linear programming problem, we need to define the objective function, constraints, and the feasible region. The objective function is the function we want to maximize or minimize, while the constraints are the limitations on the decision variables. The feasible region is the set of all possible solutions to the problem.
The simplex method is an algorithm for solving linear programming problems. It starts with an initial feasible solution and iteratively improves it until it reaches the optimal solution. The algorithm involves the following steps:
1. Identify the initial feasible solution.
2. Select an entering variable that can improve the objective function.
3. Select a leaving variable that is no longer needed.
4. Update the solution.
5. Repeat steps 2-4 until no further improvement is possible.
The simplex method is widely used in economics research for solving problems such as supply and demand, production and cost optimization, and portfolio management.
Consider a simple supply and demand problem with two producers and one buyer. The supply functions for the two producers are given by:
$$S_1(p_1, p_2) = 10 - p_1 - p_2$$
$$S_2(p_1, p_2) = 20 - 2p_1 - p_2$$
The demand function is given by:
$$D(p_1, p_2) = 30 - p_1 - 2p_2$$
The objective is to maximize the profit, which is the difference between the revenue and the cost.
## Exercise
1. Formulate the linear programming problem for this supply and demand scenario.
2. Solve the problem using the simplex method.
# Nonlinear programming: methods and algorithms for nonlinear optimization problems
Gradient descent is a widely used algorithm for minimizing a function iteratively. It involves computing the gradient of the function at the current point and moving in the direction of the negative gradient. The algorithm terminates when the gradient is small enough.
Newton's method is another optimization algorithm that uses the first and second derivatives of the function to find the next point. It is particularly useful when the function is well-behaved and has a unique minimum.
The trust-region method is a hybrid algorithm that combines the gradient method and Newton's method. It is useful when the function is nonconvex and has multiple local minima.
Consider the following optimization problem:
$$\min f(x) = (x - 2)^2$$
subject to the constraint:
$$x \ge 0$$
## Exercise
1. Solve the problem using the gradient method.
2. Solve the problem using Newton's method.
3. Solve the problem using the trust-region method.
# Optimal control: dynamic programming, Bellman equations, and applications in economics
Optimal control is a branch of control theory that deals with the design of control systems to minimize a cost function. It is widely used in economics research to model and solve problems with sequential decisions, such as investment, consumption, and production planning.
Dynamic programming is a mathematical method used in optimal control to break down a problem into smaller subproblems and solve them iteratively. It involves the following steps:
1. Define the state variables and control variables.
2. Define the value function, which represents the expected cost of the optimal control policy.
3. Define the Bellman equations, which relate the value function to the optimal control policy.
4. Solve the Bellman equations iteratively until the value function converges.
The Bellman equations are named after Richard Bellman, who introduced them in the 1950s. They are used to derive the optimal control policy for a given problem.
Consider a simple investment problem with two stages. At the first stage, the investor can choose between two investment options, and at the second stage, the investor can choose between two consumption options. The investor's objective is to maximize the expected utility.
## Exercise
1. Formulate the optimal control problem for this investment scenario.
2. Solve the problem using dynamic programming.
# Game theory: basic concepts, Nash equilibria, and applications in economics
Game theory is a mathematical framework used to analyze strategic interactions between rational decision-makers. It is widely used in economics research to model and analyze problems with multiple players, such as market competition, bargaining, and cooperation.
The basic concepts in game theory include players, strategies, payoffs, and Nash equilibria. A Nash equilibrium is a strategy profile in which no player has an incentive to unilaterally deviate from the equilibrium strategy.
In economics, game theory is used to analyze various market structures, such as monopoly, oligopoly, and monopolistic competition. It is also used to study problems with cooperation, such as the prisoner's dilemma and the public goods game.
Consider a simple monopoly problem with a single firm producing a homogeneous good. The firm's objective is to maximize its profit, which is the difference between the market price and the production cost.
## Exercise
1. Formulate the game theory problem for this monopoly scenario.
2. Analyze the firm's strategies and the Nash equilibria.
# Dynamic programming: single and multi-stage decision problems, value iteration, and policy iteration
Dynamic programming is a mathematical method used to solve optimization problems with sequential decisions. It involves the following steps:
1. Define the state variables and control variables.
2. Define the value function, which represents the expected cost of the optimal control policy.
3. Define the Bellman equations, which relate the value function to the optimal control policy.
4. Solve the Bellman equations iteratively until the value function converges.
Value iteration is an algorithm used to solve dynamic programming problems. It involves iteratively updating the value function until it converges to the optimal value function.
Policy iteration is another algorithm used to solve dynamic programming problems. It involves iteratively updating the control policy until it converges to the optimal control policy.
Consider a simple single-stage decision problem with two states and two actions. The transition probabilities and the reward function are given by:
$$P_{ij} = \begin{bmatrix} 0.5 & 0.5 \\ 1 & 0 \end{bmatrix}$$
$$R(s, a) = \begin{bmatrix} -1 & 0 \\ 10 & -10 \end{bmatrix}$$
## Exercise
1. Solve the problem using value iteration.
2. Solve the problem using policy iteration.
# Applications of optimization methods in economics research: supply and demand, production and cost optimization, portfolio management
In supply and demand analysis, optimization methods are used to determine the optimal production levels, prices, and resource allocation. For example, the linear programming method can be used to find the equilibrium prices and quantities that maximize the social welfare.
In production and cost optimization, optimization methods are used to determine the optimal production levels, resource allocation, and cost allocation. For example, the simplex method can be used to solve the production planning problem with multiple constraints and objectives.
In portfolio management, optimization methods are used to determine the optimal portfolio weights that maximize the expected return and minimize the risk. For example, the gradient method can be used to find the optimal portfolio weights that minimize the variance of the portfolio return.
Consider a simple portfolio management problem with two assets and a risk-averse investor. The expected returns and covariance matrix of the assets are given by:
$$\mu = \begin{bmatrix} 0.1 \\ 0.05 \end{bmatrix}$$
$$\Sigma = \begin{bmatrix} 0.01 & 0.005 \\ 0.005 & 0.0025 \end{bmatrix}$$
## Exercise
1. Solve the problem using the gradient method.
2. Determine the optimal portfolio weights.
# Introduction to the Julia programming language for optimization
Julia is a high-level, high-performance programming language for technical computing. It is particularly well-suited for optimization problems due to its ease of use, strong numerical capabilities, and extensive library support.
Consider the following Julia code that calculates the factorial of a number:
```julia
function factorial(n)
if n == 0
return 1
else
return n * factorial(n - 1)
end
end
println(factorial(5)) # Output: 120
```
## Exercise
1. Install Julia and set up the development environment.
2. Write a simple Julia program to solve a linear programming problem using the JuMP package.
# Solving linear programming problems in Julia using the JuMP package
The JuMP package is a high-level mathematical programming modeling language for Julia. It provides a simple and expressive syntax for formulating optimization problems and supports various solvers, including the open-source solver GLPK.
Consider the following Julia code that solves a simple linear programming problem using the JuMP package:
```julia
using JuMP, GLPK
# Create a JuMP model
model = Model(GLPK.Optimizer)
# Define the variables
@variable(model, x >= 0)
@variable(model, y >= 0)
# Define the objective function
@objective(model, Max, 3x + 2y)
# Define the constraints
@constraint(model, x + y <= 10)
@constraint(model, 2x + y <= 15)
# Solve the problem
optimize!(model)
# Print the results
println("Optimal value: ", objective_value(model))
println("x: ", value(x))
println("y: ", value(y))
```
## Exercise
1. Install the JuMP and GLPK packages in Julia.
2. Solve the linear programming problem using the JuMP package.
# Solving nonlinear programming problems in Julia using the NLopt package
The NLopt package is a Julia wrapper for the NLopt optimization library. It provides a simple and expressive syntax for formulating nonlinear optimization problems and supports various algorithms, including gradient-based optimization methods.
Consider the following Julia code that solves a simple nonlinear programming problem using the NLopt package:
```julia
using NLopt
# Define the objective function
function objective(x, gradient)
if length(gradient) > 0
gradient[1] = 2 * x[1]
end
return x[1]^2
end
# Define the constraints
function constraint(x, gradient)
if length(gradient) > 0
gradient[1] = 1
end
return x[1] - 2
end
# Create an NLopt optimization problem
opt = Opt(:LD_MMA, 1)
# Set the objective function and constraints
set_min_objective!(opt, objective)
set_lower_bounds!(opt, [0.0])
set_upper_bounds!(opt, [Inf])
add_inequality_constraint!(opt, constraint)
# Solve the problem
(optimum, x_opt, duals) = optimize(opt, [0.0])
# Print the results
println("Optimal value: ", optimum)
println("x: ", x_opt[1])
```
## Exercise
1. Install the NLopt package in Julia.
2. Solve the nonlinear programming problem using the NLopt package.
# Solving optimal control problems in Julia using the DifferentialEquations package
The DifferentialEquations package is a Julia package for solving differential equations, including optimal control problems. It provides a simple and expressive syntax for formulating and solving control problems and supports various algorithms, including gradient-based optimization methods.
Consider the following Julia code that solves a simple optimal control problem using the DifferentialEquations package:
```julia
using DifferentialEquations
# Define the system dynamics
function dynamics!(du, u, p, t)
du[1] = u[1] - p[1] * u[2]
du[2] = p[2] * u[1]
end
# Define the objective function
function objective(u, p, t)
return u[1]^2 + u[2]^2
end
# Define the constraints
function constraint(u, p, t)
return u[1] + u[2] - 1
end
# Create a DifferentialEquations problem
prob = ODEProblem(dynamics!, [1.0, 0.0], (0.0, 10.0), [1.0, 1.0], save_everystep=false)
# Solve the problem
sol = solve(prob, callback=DifferentialEquations.CallbackSet(DifferentialEquations.save_callback(objective, constraint)))
# Print the results
println("Optimal value: ", sol.t[end])
println("x: ", sol[1, end])
println("y: ", sol[2, end])
```
## Exercise
1. Install the DifferentialEquations package in Julia.
2. Solve the optimal control problem using the DifferentialEquations package.
# Solving game theory problems in Julia using the GameTheory.jl package
The GameTheory.jl package is a Julia package for solving game theory problems. It provides a simple and expressive syntax for formulating and solving game-theoretic problems and supports various algorithms, including the Nash equilibrium method.
Consider the following Julia code that solves a simple game theory problem using the GameTheory.jl package:
```julia
using GameTheory
# Define the game matrix
matrix = [
[0, 1],
[1, 0]
]
# Define the payoffs
payoffs = [
[1, 0],
[0, -1]
]
# Create a GameTheory problem
prob = Game(matrix, payoffs)
# Solve the problem
sol = solve(prob, :nash)
# Print the results
println("Nash equilibria: ", sol)
```
## Exercise
1. Install the GameTheory.jl package in Julia.
2. Solve the game theory problem using the GameTheory.jl package. | Textbooks |
Reconstruction and estimation in the planted partition model
Elchanan Mossel, Joe Neeman, Allan Sly
The planted partition model (also known as the stochastic blockmodel) is a classical cluster-exhibiting random graph model that has been extensively studied in statistics, physics, and computer science. In its simplest form, the planted partition model is a model for random graphs on $$n$$n nodes with two equal-sized clusters, with an between-class edge probability of $$q$$q and a within-class edge probability of $$p$$p. Although most of the literature on this model has focused on the case of increasing degrees (ie. $$pn, qn \rightarrow \infty $$pn,qn→∞ as $$n \rightarrow \infty $$n→∞), the sparse case $$p, q = O(1/n)$$p,q=O(1/n) is interesting both from a mathematical and an applied point of view. A striking conjecture of Decelle, Krzkala, Moore and Zdeborová based on deep, non-rigorous ideas from statistical physics gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if $$p = a/n$$p=a/n and $$q = b/n$$q=b/n, then Decelle et al. conjectured that it is possible to cluster in a way correlated with the true partition if $$(a - b)^2 > 2(a + b)$$(a-b)2>2(a+b), and impossible if $$(a - b)^2 < 2(a + b)$$(a-b)2<2(a+b). By comparison, the best-known rigorous result is that of Coja-Oghlan, who showed that clustering is possible if $$(a - b)^2 > C (a + b)$$(a-b)2>C(a+b) for some sufficiently large $$C$$C. We prove half of their prediction, showing that it is indeed impossible to cluster if $$(a - b)^2 < 2(a + b)$$(a-b)2<2(a+b). Furthermore we show that it is impossible even to estimate the model parameters from the graph when $$(a - b)^2 < 2(a + b)$$(a-b)2<2(a+b); on the other hand, we provide a simple and efficient algorithm for estimating $$a$$a and $$b$$b when $$(a - b)^2 > 2(a + b)$$(a-b)2>2(a+b). Following Decelle et al, our work establishes a rigorous connection between the clustering problem, spin-glass models on the Bethe lattice and the so called reconstruction problem. This connection points to fascinating applications and open problems.
Probability Theory and Related Fields
91D30
Primary 05C80
Secondary 60J85
Dive into the research topics of 'Reconstruction and estimation in the planted partition model'. Together they form a unique fingerprint.
Partition Mathematics 100%
Random Graphs Business & Economics 63%
Physics Business & Economics 50%
Clustering Mathematics 42%
Prediction Business & Economics 36%
Bethe Lattice Mathematics 34%
Graph Model Business & Economics 31%
Mossel, E., Neeman, J., & Sly, A. (2015). Reconstruction and estimation in the planted partition model. Probability Theory and Related Fields, 162(3-4), 431-461. https://doi.org/10.1007/s00440-014-0576-6
Mossel, Elchanan ; Neeman, Joe ; Sly, Allan. / Reconstruction and estimation in the planted partition model. In: Probability Theory and Related Fields. 2015 ; Vol. 162, No. 3-4. pp. 431-461.
@article{7c31dd4117d0419cb7671255f9dd58fe,
title = "Reconstruction and estimation in the planted partition model",
abstract = "The planted partition model (also known as the stochastic blockmodel) is a classical cluster-exhibiting random graph model that has been extensively studied in statistics, physics, and computer science. In its simplest form, the planted partition model is a model for random graphs on $$n$$n nodes with two equal-sized clusters, with an between-class edge probability of $$q$$q and a within-class edge probability of $$p$$p. Although most of the literature on this model has focused on the case of increasing degrees (ie. $$pn, qn \rightarrow \infty $$pn,qn→∞ as $$n \rightarrow \infty $$n→∞), the sparse case $$p, q = O(1/n)$$p,q=O(1/n) is interesting both from a mathematical and an applied point of view. A striking conjecture of Decelle, Krzkala, Moore and Zdeborov{\'a} based on deep, non-rigorous ideas from statistical physics gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if $$p = a/n$$p=a/n and $$q = b/n$$q=b/n, then Decelle et al. conjectured that it is possible to cluster in a way correlated with the true partition if $$(a - b)^2 > 2(a + b)$$(a-b)2>2(a+b), and impossible if $$(a - b)^2 < 2(a + b)$$(a-b)2<2(a+b). By comparison, the best-known rigorous result is that of Coja-Oghlan, who showed that clustering is possible if $$(a - b)^2 > C (a + b)$$(a-b)2>C(a+b) for some sufficiently large $$C$$C. We prove half of their prediction, showing that it is indeed impossible to cluster if $$(a - b)^2 < 2(a + b)$$(a-b)2<2(a+b). Furthermore we show that it is impossible even to estimate the model parameters from the graph when $$(a - b)^2 < 2(a + b)$$(a-b)2<2(a+b); on the other hand, we provide a simple and efficient algorithm for estimating $$a$$a and $$b$$b when $$(a - b)^2 > 2(a + b)$$(a-b)2>2(a+b). Following Decelle et al, our work establishes a rigorous connection between the clustering problem, spin-glass models on the Bethe lattice and the so called reconstruction problem. This connection points to fascinating applications and open problems.",
keywords = "90B15, 91D30, Primary 05C80, Secondary 60J85",
author = "Elchanan Mossel and Joe Neeman and Allan Sly",
journal = "Probability Theory and Related Fields",
Mossel, E, Neeman, J & Sly, A 2015, 'Reconstruction and estimation in the planted partition model', Probability Theory and Related Fields, vol. 162, no. 3-4, pp. 431-461. https://doi.org/10.1007/s00440-014-0576-6
Reconstruction and estimation in the planted partition model. / Mossel, Elchanan; Neeman, Joe; Sly, Allan.
In: Probability Theory and Related Fields, Vol. 162, No. 3-4, 18.08.2015, p. 431-461.
T1 - Reconstruction and estimation in the planted partition model
AU - Mossel, Elchanan
AU - Neeman, Joe
AU - Sly, Allan
N2 - The planted partition model (also known as the stochastic blockmodel) is a classical cluster-exhibiting random graph model that has been extensively studied in statistics, physics, and computer science. In its simplest form, the planted partition model is a model for random graphs on $$n$$n nodes with two equal-sized clusters, with an between-class edge probability of $$q$$q and a within-class edge probability of $$p$$p. Although most of the literature on this model has focused on the case of increasing degrees (ie. $$pn, qn \rightarrow \infty $$pn,qn→∞ as $$n \rightarrow \infty $$n→∞), the sparse case $$p, q = O(1/n)$$p,q=O(1/n) is interesting both from a mathematical and an applied point of view. A striking conjecture of Decelle, Krzkala, Moore and Zdeborová based on deep, non-rigorous ideas from statistical physics gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if $$p = a/n$$p=a/n and $$q = b/n$$q=b/n, then Decelle et al. conjectured that it is possible to cluster in a way correlated with the true partition if $$(a - b)^2 > 2(a + b)$$(a-b)2>2(a+b), and impossible if $$(a - b)^2 < 2(a + b)$$(a-b)2<2(a+b). By comparison, the best-known rigorous result is that of Coja-Oghlan, who showed that clustering is possible if $$(a - b)^2 > C (a + b)$$(a-b)2>C(a+b) for some sufficiently large $$C$$C. We prove half of their prediction, showing that it is indeed impossible to cluster if $$(a - b)^2 < 2(a + b)$$(a-b)2<2(a+b). Furthermore we show that it is impossible even to estimate the model parameters from the graph when $$(a - b)^2 < 2(a + b)$$(a-b)2<2(a+b); on the other hand, we provide a simple and efficient algorithm for estimating $$a$$a and $$b$$b when $$(a - b)^2 > 2(a + b)$$(a-b)2>2(a+b). Following Decelle et al, our work establishes a rigorous connection between the clustering problem, spin-glass models on the Bethe lattice and the so called reconstruction problem. This connection points to fascinating applications and open problems.
AB - The planted partition model (also known as the stochastic blockmodel) is a classical cluster-exhibiting random graph model that has been extensively studied in statistics, physics, and computer science. In its simplest form, the planted partition model is a model for random graphs on $$n$$n nodes with two equal-sized clusters, with an between-class edge probability of $$q$$q and a within-class edge probability of $$p$$p. Although most of the literature on this model has focused on the case of increasing degrees (ie. $$pn, qn \rightarrow \infty $$pn,qn→∞ as $$n \rightarrow \infty $$n→∞), the sparse case $$p, q = O(1/n)$$p,q=O(1/n) is interesting both from a mathematical and an applied point of view. A striking conjecture of Decelle, Krzkala, Moore and Zdeborová based on deep, non-rigorous ideas from statistical physics gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if $$p = a/n$$p=a/n and $$q = b/n$$q=b/n, then Decelle et al. conjectured that it is possible to cluster in a way correlated with the true partition if $$(a - b)^2 > 2(a + b)$$(a-b)2>2(a+b), and impossible if $$(a - b)^2 < 2(a + b)$$(a-b)2<2(a+b). By comparison, the best-known rigorous result is that of Coja-Oghlan, who showed that clustering is possible if $$(a - b)^2 > C (a + b)$$(a-b)2>C(a+b) for some sufficiently large $$C$$C. We prove half of their prediction, showing that it is indeed impossible to cluster if $$(a - b)^2 < 2(a + b)$$(a-b)2<2(a+b). Furthermore we show that it is impossible even to estimate the model parameters from the graph when $$(a - b)^2 < 2(a + b)$$(a-b)2<2(a+b); on the other hand, we provide a simple and efficient algorithm for estimating $$a$$a and $$b$$b when $$(a - b)^2 > 2(a + b)$$(a-b)2>2(a+b). Following Decelle et al, our work establishes a rigorous connection between the clustering problem, spin-glass models on the Bethe lattice and the so called reconstruction problem. This connection points to fascinating applications and open problems.
KW - 90B15
KW - 91D30
KW - Primary 05C80
KW - Secondary 60J85
JO - Probability Theory and Related Fields
JF - Probability Theory and Related Fields
Mossel E, Neeman J, Sly A. Reconstruction and estimation in the planted partition model. Probability Theory and Related Fields. 2015 Aug 18;162(3-4):431-461. https://doi.org/10.1007/s00440-014-0576-6 | CommonCrawl |
Perl Weekly Challenge 139: Long Primes
by Abigail
Write a script to generate first 5 Long Primes.
A prime number (p) is called Long Prime if (1/p) has an infinite decimal expansion repeating every (p-1) digits.
\(7\) is a long prime since \(\frac{1}{7} = 0.\overline{1428571}\). The repeating part (\(142857\)) size is \(6\) i.e. one less than the prime number \(7\).
Also \(17\) is a long prime since \(\frac{1}{17} = 0.\overline{0588235294117647}\). The repeating part (\(0588235294117647\)) size is \(16\) i.e. one less than the prime number \(17\).
Another example, \(2\) is not a long prime as \(\frac{1}{2} = 0.5\). There is no repeating part in this case.
Wikipedia call long primes Full reptend prime.
A naive method would be to take a prime number, and see whether its decimal fraction repeats. But that means, having to deal with floating point numbers, and that's hard in general.
For instance, in Perl, even a 64 bit perl, \(\frac{1}{23}\) equals 0.0434782608695652174. But that's enough to determine whether the fraction repeats with period 22 - as we only have 19 digits of precision. In many other languages, the situation is similar.
Luckily, there is an alternative way. As the Wikipedia page points out, for a full reptend prime \(p\), the quotient
\[ \frac{b^{p - 1} - 1}{p} \]
gives a cyclic number, where \(b\) is the base we are working in (so, for us \(b = 10\)).
Now, \(\frac{b^{p - 1} - 1}{p}\) becomes large quickly if \(p\) increases. For instance, if \(b = 10\) and \(p = 23\), the quotient is \(43478260869565217391\) which is larger than a 64 bit integer can hold.
We could use big integers to calculate the quotient, but that doesn't work for languages with no, or poor, support for large integers.
But there is a different way. We don't need the actual quotient. All we want to know is that the resulting number doesn't contain repeats. We can do this by performing long division and check all the intermediate results are different.
For instance, if \(b = 10\) and \(p = 7\), \(b^{p - 1} - 1\) equals \(999999\). On the left, we have the long division of those numbers. What we are interested in is the numbers left over after subtracting the appropriate multiple of \(7\), this the values below the lines (and without the dropped \(9\)). Here, they are \(9 - 7 = 2, 29 - 28 = 1, 19 - 14 = 2, 59 - 56 = 3, 39 - 35 = 4, 49 - 49 = 0\). There are no duplicates in this sequence, so the quotient doesn't repeat (\(\frac{10^6 - 1}{7} = 142857\)), and hence, \(7\) is a long prime.
But if we look at the long division of \(10^{12} - 1\) and \(13\) on the right, we see that the sequence is \(9, 8, 11, 2, 3, 0, 9, 8, 11, 2, 3, 0\). This sequence repeats, and hence the quotient repeats: \(\frac{10^{12} - 1}{13} = 076923076923\). This makes \(13\) not a long prime.
Note that in the latter case, we don't need to do the full calculation. As soon as we find an intermediate value we have seen before (here the \(9\)), we know the number is not a long prime, and there is no need to continue the calculations.
Now, we could just iterate over the primes and see if there are no repeats when doing the long division. But that would require us to generate primes. In Perl, we could use Math::Prime::Util, but not every language has such a module.
Instead, we just check every number starting from 2. Composite numbers will give a duplicate when doing the long division, so any number which passes the no-duplicates check has to be prime.
Input a number, hit Calculate, and it shows whether the number is a long prime or not.
Each solution will contain two parts: a function is_long which takes a number, and returns a true or false value depending on whether the number is a long prime or not, and a main part which counts up from 2, skipping numbers which evenly divide 10, and then calls is_long, printing the numbers which are long primes, up to the required amount.
The is_long method will perform the long division described above. Note that we do not have to calculate a number of the form \(10^{p - 1} - 1\), where \(p\) is the argument to is_long. \(10^{p - 1} - 1\) will be a string of \(p - 1\) \(9\)s, so when performing the long division, we always "drop down" a \(9\).
The sequence \(a_n\) for a given number \(p\) can be calculated as follows:
\[ a_n = \begin{cases} 0 & \text{if } n = 0 \\ (10 * a_{n - 1} + 9) \mod p & \text{if } n > 0 \\ \end{cases} \]
We need \(p - 2\) terms of this sequence. If there are no duplicates, the given number is a long prime. Note that \(\forall i: 0 \leq a_i < p\).
The is_long function is now straight forward with the formula above:
my $BASE = 10;
sub is_long ($number) {
my $rest = 0;
my %seen;
for (2 .. $number) {
return 0 if $seen {$rest = ($rest * $BASE + $BASE - 1) % $number} ++;
And the main function:
my $COUNT = 5;
my $number = 1;
while ($COUNT) {
$number ++;
next if $BASE % $number == 0;
next unless is_long $number;
say $number;
$COUNT --;
Find the full program on GitHub.
The is-long function:
(define BASE 10)
(define (is-long number)
(define seen (make-array 0 number))
(define rest 0)
(define r #t)
(do ((i 2 (1+ i)))
((> i number))
(set! rest (modulo (+ (* rest BASE) BASE -1) number))
(if (= (array-ref seen rest) 1)
(set! r #f))
(array-set! seen 1 rest))
And the main program:
(define COUNT 5)
(do ((number 2 (1+ number)))
((= COUNT 0))
(cond ((= (modulo BASE number) 0) #f)
((is-long number)
(begin (display number)(newline)
(set! COUNT (- COUNT 1))))))
Implementations in other languages are very similar to Perl solution. We also have solutions in: AWK, Bash, bc, C, Go, Java, Lua, Node.js, Pascal, Python, R, Ruby, and Tcl.
☜
Please leave any comments as a GitHub issue. | CommonCrawl |
The School Executive and Heads of Professional Services
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Professor Todd R Kaplan
Streatham Court, University of Exeter, Rennes Drive, Exeter, EX4 4PU, UK
External engagement and impact
Professor Todd Kaplan is a part-time Professor specializing in Economic Theory and Behavioral Economics. He has been a faculty member since 2000. He also has a position in Economics Department at the University of Haifa. He has received grants from the Nuffield Foundation, British Academy, iFree Foundation, ESRC, Leverhulme Foundation and the Israeli Science Foundation. He was a co-winner for the Economics Network 2009 e-learning award for developing teaching resources in a grant from HEFCE. He is an Associate Editor of the Journal of Behavioral and Experimental Economics.
Nationality: USA / Israel
BS (Caltech), MA (Minnesota), PhD (Minnesota)
Personal webpage
Behaviour, Identity and Decisions
Firms, Markets and Value
Professor Kaplan has diverse research interests that span theoretical and experimental economics. My theoretical work has mostly centered on contests, auctions and mechanism design. While contests have been used to spur innovation since the British Parliament introduced the Longitude prize in 1714, it has become in vogue to spur innovation with both non-profit initiatives such as the X-prize and company-sponsored prizes such as one by Netfix. He has published five papers on such contests and their optimal design: Journal of Industrial Economics (2002), International Journal of Industrial Organization (2003), American Economic Review (2006), RAND (2008), Economics Letters (2010), and Review of Economic Design (2015). One work on auctions (Economic Theory, 2012) solved a problem thought intractable since Vickrey claimed so (over 40 years ago) in one of his seminal Nobel-Prize winning papers. Todd continued this work by looking at multiple equilibria in first-price auctions (Economic Theory Bulletin, 2015). More recently, he applied asymmetric auctions to International Trade (Journal of International Economics, 2017). Due to his experience in auctions, Todd was invited to contribute a chapter on auctions to the Handbook of Game Theory (2014).
Todd's more recent theoretical research considers someone who needs to allocate goods among individuals who can put forth efforts, but the only value of the individuals' efforts is what they imply about the individuals' desires about the good (GEB, 2013). Related to this research, Todd published work on using voting systems to allocate goods and discovered conditions where, counter intuitively, adding a cost to voting improves welfare, Journal of Public Economics (forthcoming).
Todd also has a significant research interest in experimental and behavioural economics. Todd has a paper on Self-Serving biases in Economic Inquiry (2004), a topic for which Todd published a comment in Journal of Economic Perspectives. Todd has experimentally tested a computer recommendation system in JORS (2011). Todd's recent publication (Economic Journal, 2012) studies under what conditions cooperation will form in a particularly difficult environment. An extension to this work is forthcoming in JEBO. Todd's current research is to experimentally investigate bank runs. After an initial publication in European Economic Review (2014), Todd received an ISF grant to continue this path. This builds upon Todd's theoretical work initially done as part of his PhD dissertation and published in Economic Theory (2006).
Todd also has interdisciplinary work both in Operations Research and Meteorology that involves applying experimental economics methodology to other fields. In Meteorology, Todd worked with co-authors from the UK Met Office to examine the communication of weather risks in Met Applications (2009), Weather and Forecasting (2014), and International Journal of Disaster Risk Reduction (2018). In Operations Research, Todd tested the usefulness of decision support systems with experimental methods in Journal of the Operational Research Society (2011a, 2011b).
Earlier contributions were in the field of computational economics:
Mathematica Journal (1991) and two chapters in a best-selling book edited by Hal Varian (1993). In part of this, by desire to get a computer aid for game theory homework, Todd inadvertently discovered a new algorithm for finding Nash equilibria. Probably Todd's best-known work is the "Kaplan" strategy. This was a simple computer strategy designed to compete in a simplified stock exchange. It won a well-advertised tournament by the Santa Fe Institute.
Todd's earlier work includes looking at how a group should divide the cost from a shared production function. For instance, how do two friends divide the costs when sharing a pizza. This is not so easy since they must also decide what size of pizza to order and a poor rule may cause them to overspend or underspend. Todd answers this in International Economic Review (2000) and Journal of Mathematical Economics (1999). Todd also has written on such eclectic topics as steroid use in sports (Labour Economics, 2011) and gift giving (European Economic Review, 2009) both receiving significant attention from the press.
Professor Kaplan currently has a research grant from the iFree Foundation for studying manipulation in prediction markets. He is the supervisor of Di Mu, who is studying the communication of weather warnings.
Key publications | Publications by category | Publications by year
Kaplan TR, Ruffle B, Shtudiner Z (In Press). Cooperation through Coordination in Two Stages. Journal of Economic Behavior and Organization Full text.
Choo L, Kaplan, Zultan R (In Press). Information Aggregation in Arrow-Debreu Markets: an Experiment. Experimental Economics Full text.
Chakravarty S, Kaplan TR, Myles G (In Press). When Costly Voting is Beneficial. Journal of Public Economics Full text.
Cole M, Davies R, Kaplan TR (2017). Protection in Government Procurement Auctions. Journal of International Economics, 106, 134-142.
Protection in Government Procurement Auctions
Discrimination against foreign bidders in procurement auctions has typically been achieved by price preferences. We demonstrate that in the bidding game, each level of protection via a price preference can be achieved by an equivalent tariff. When government welfare depends only on net expenditures, this equivalence carries over to the government's decision. As such, this equivalence provides a justification that agreements to eliminate price preferences to be taken in tandem with agreements to lower tariffs; e.g. the Government Procurement Agreement (GPA) in the broader context of the WTO.
Abstract. Full text. DOI.
Chakravarty S, Fonseca MA, Kaplan TR (2014). An Experiment on the Causes of Bank Run Contagions. European Economic Review, 72, 39-51. Full text.
Kaplan TR, Zamir S (2012). Asymmetric first-price auctions with uniform distributions: Analytic solutions to the general case. Economic Theory, 50(2), 269-302.
Asymmetric first-price auctions with uniform distributions: Analytic solutions to the general case
In 1961, Vickrey posed the problem of finding an analytic solution to a first-price auction with two buyers having valuations uniformly distributed on [v{script} 2, v{script} 2] and [v{script} 2, v{script} 2]. To date, only special cases of the problem have been solved. In this paper, we solve this general problem and in addition allow for the possibility of a binding minimum bid. Several interesting examples are presented, including a class where the two bid functions are linear. © 2010 Springer-Verlag.
Publications by category
Mu D, Kaplan TR, Dankers R (2018). Decision Making with Risk-Based Weather Warnings. International Journal of Disaster Risk Reduction, 30, Part A, 59-73. Full text. DOI.
Brams SJ, Kaplan TR, Kilgour DM (2015). A Simple Bargaining Mechanism that Elicits Truthful Reservation Prices. Group Decision and Negotiation, 24(3), 401-413.
A Simple Bargaining Mechanism that Elicits Truthful Reservation Prices
© 2014, Springer Science+Business Media Dordrecht. We describe a simple 2-stage mechanism whereby for two bargainers, a Buyer and a Seller, it is a weakly dominant strategy to report their true reservation prices in the 1st stage. If the Buyer reports a higher reservation price than the Seller, then the referee announces that there is a possibility for trade, and the bargainers proceed to make offers in a 2nd stage. The average of the 2nd-stage offers becomes the settlement if they both fall into the interval between the reported reservation prices; if only one offer falls into this interval, it is the settlement, but it is implemented with probability $$\frac{1}{2}$$12; if neither offer falls into the interval, there is no settlement. Comparisons are made with other bargaining mechanisms.
Abstract. DOI.
Kaplan TR, Zamir S (2015). Advances in Auctions. , 4(1), 381-453.
Advances in Auctions
© 2015 Elsevier B.V. As a selling mechanism, auctions have acquired a central position in the free market economy all over the globe. This development has deepened, broadened, and expanded the theory of auctions in new directions. This chapter is intended as a selective update of some of the developments and applications of auction theory in the two decades since Wilson (1992) wrote the previous Handbook chapter on this topic.
Marimo P, Kaplan TR, Mylne K, Sharpe M (2015). Communication of uncertainty in temperature forecasts. Weather and Forecasting, 30(1), 5-22.
Communication of uncertainty in temperature forecasts
© 2015 American Meteorological Society. Experimental economics is used to test whether undergraduate students presented with a temperature forecast with uncertainty information in a table and bar graph format were able to use the extra information to interpret a given forecast. Participants were asked to choose the most probable temperature-based outcome between a set of "lotteries." Both formats with uncertainty information were found on average to significantly increase the probability of choosing the correct outcome. However, in some cases providing uncertainty information was damaging. Factors that influence understanding are statistically determined. Furthermore, participants who were shown the graph with uncertainty information took on average less response time compared to those who were shown a table with uncertainty information. Over time, participants improve in speed and initially improve in accuracy of choosing the correct outcome.
Kaplan TR, Wettstein D (2015). The optimal design of rewards in contests. Review of Economic Design
The optimal design of rewards in contests
Using contests to generate innovation has been and is widely used. Such contests often involve offering a prize that depends upon the accomplishment (effort). Using an all-pay auction as a model of a contest, we determine the optimal reward for inducing innovation. In a symmetric environment, we find that the reward should be set to (Formula presented.) where c is the cost of producing an innovation of level x and (Formula presented.) is the weight attached by the designer to the sum of efforts. In an asymmetric environment with two firms, we find that it is optimal to set different rewards for each firm. There are cases where this can be replicated by a single reward that depends upon accomplishments of both contestants.
Kaplan TR, Zamir S (2014). Multiple equilibria in asymmetric first-price auctions. Economic Theory Bulletin, 3(1), 65-77. DOI.
Navon D, Kaplan TR, Kasten R (2013). Egocentric framing--One way people may fail in a switch dilemma: Evidence from excessive lane switching. Acta psychologica, 144, 604-616. Full text.
Chakravarty S, Kaplan T (2013). Optimal allocation without transfer payments. Games and Economic Behavior, 77(1), 1-20.
Optimal allocation without transfer payments
Often an organization or government must allocate goods without collecting payment in
return. This may pose a difficult problem either when agents receiving those goods have
private information in regards to their values or needs. In this paper, we find an optimal
mechanism to allocate goods when the designer is benevolent. While the designer cannot
charge agents, he can receive a costly but wasteful signal from them. We find conditions
for cases in which ignoring these costly signals by giving agents equal share (or using
lotteries if the goods are indivisible) is optimal. In other cases, those that send the highest
signal should receive the goods; however, we then show that there exist cases where
more complicated mechanisms are superior. Also, we show that the optimal mechanism
is independent of the scarcity of the goods being allocated.
Abstract. Full text.
Gould ED, Kaplan TR (2013). The peer effect of Jose Canseco: a reply to J. C. Bradbury. Econ Journal Watch, 10(1), 70-86.
The peer effect of Jose Canseco: a reply to J. C. Bradbury
In this paper, we respond to J. C. Bradbury's critique of our 2011 Labour Economics paper examining the peer effect of Jose Canseco. None of Bradbury's criticisms have any merit, and many reveal a severe misunderstanding of basic econometrics. For example, Bradbury accuses us of not deleting enough years from the sample, not censoring the sample on an outcome measure, and not controlling for average performance measures for each year explicitly when we have already included dummy variables for each year. Bradbury claims that we distort our findings, but he overlooks the parts of our paper that do not fit his thesis. Bradbury reexamines the performance of Canseco's teammates empirically and argues that our results are sensitive. However, this should not be surprising because Bradbury performs a completely different and highly flawed analysis. In particular, he fails to realize that he is estimating very different parameters which are difficult, if not impossible, to interpret. His specification and estimation are based on very restrictive assumptions which are not necessary, nor are they justified or even acknowledged. After examining every one of Bradbury's attacks on our paper, we conclude that none provides a convincing reason to reject our conclusions.
Balkenborg D, Kaplan T, Miller T (2012). A simple economic teaching experiment on the hold-up problem. Journal of Economic Education, 43(4), 377-385.
A simple economic teaching experiment on the hold-up problem
The hold-up problem is central to the theory of incomplete contracts. This can occur if, after making a sunk investment in a relationship, one party can be taken advantage of by the other party, leading to inefficient underinvestment. The authors describe a simple teaching experiment that illustrates the hold-up problem, and address how to integrate it into a class. © 2012 Copyright Taylor and Francis Group, LLC.
Kaplan TR (2012). Communication of preferences in contests for contracts. Economic Theory, 51(2), 487-503.
Communication of preferences in contests for contracts
This paper models a contest where several sellers compete for a contract with a single buyer. There are several styles of possible designs with a subset of them preferred by the buyer. We examine what happens when the buyer communicates information about his preferences. If the sellers are unable to change their style, then there is no effect on the welfare of the sellers. If the sellers are able to make adjustments, extra information may either boost or damage the sellers' profits. While the chance that there will be a proposal of a style preferred by the buyer cannot decrease, the buyer's surplus may increase or decrease. © 2010 Springer-Verlag.
Kaplan TR, Ruffle BJ (2012). Which Way to Cooperate. Economic Journal, 122(563), 1042-1068.
Which Way to Cooperate
We introduce a two-player, binary-choice game in which both players have a privately known incentive to enter, yet the combined surplus is highest if only one enters. Repetition of this game admits two distinct ways to cooperate: turn taking and cutoffs, which rely on the player's private value to entry. A series of experiments highlights the role of private information in determining which mode players adopt. If an individual's entry values vary little (e.g. mundane tasks), taking turns is likely; if these potential values are diverse (e.g. difficult tasks that differentiate individuals by skill or preferences), cutoff cooperation emerges. © 2011 the Author(s). The Economic Journal © 2011 Royal Economic Society.
Balkenborg D, Ishizaka A, Kaplan T (2011). Does AHP help us make a choice? - an experimental evaluation. JORS (Journal of the Operations Research Society), 62, 1801-1812. Full text.
Balkenborg DG, Kaplan TR, Ishizaka A (2011). Does AHP help us make a choice?-An experimental evaluation. Journal of the Operational Research Society, 62(10), 1801-1812.
Ishizaka A, Balkenborg D, Kaplan T (2011). Influence of aggregation and measurement scale on ranking a compromise alternative in AHP. Journal of the Operational Research Society, 62(4), 700-710.
Influence of aggregation and measurement scale on ranking a compromise alternative in AHP
Analytic Hierarchy Process (AHP) is one of the most popular multi-attribute decision aid methods. However, within AHP, there are several competing preference measurement scales and aggregation techniques. In this paper, we compare these possibilities using a decision problem with an inherent trade-off between two criteria. A decision-maker has to choose among three alternatives: two extremes and one compromise. Six different measurement scales described previously in the literature and the new proposed logarithmic scale are considered for applying the additive and the multiplicative aggregation techniques. The results are compared with the standard consumer choice theory. We find that with the geometric and power scales a compromise is never selected when aggregation is additive and rarely when aggregation is multiplicative, while the logarithmic scale used with the multiplicative aggregation most often selects the compromise that is desirable by consumer choice theory. © 2011 Operational Research Society Ltd. All rights reserved.
Gould ED, Kaplan TR (2011). Learning unethical practices from a co-worker: the peer effect of Jose Canseco. Labour Economics, 18(3), 338-348.
Learning unethical practices from a co-worker: the peer effect of Jose Canseco
This paper examines the issue of whether workers learn productive skills from their co-workers, even if those skills are unethical. Specifically, we estimate whether Jose Canseco, a star baseball player in the late 1980's and 1990's, affected the performance of his teammates by introducing them to steroids. Using panel data, we show that a player's performance increases significantly after they played with Jose Canseco. After checking 30 comparable players from the same era, we find that no other baseball player produced a similar effect. Furthermore, the positive effect of Canseco disappears after 2003, the year that drug testing was implemented. These results suggest that workers not only learn productive skills from their co-workers, but sometimes those skills may derive from unethical practices. These findings may be relevant to many workplaces where competitive pressures create incentives to adopt unethical means to boost productivity and profits. Our analysis leads to several potential policy implications designed to reduce the spread of unethical behavior among workers. © 2010 Elsevier B.V.
Balkenborg D, Kaplan T, Miller T (2011). Teaching Bank Runs with Classroom Experiments. The Journal of Economic Education, 42(3), 224-242.
Teaching Bank Runs with Classroom Experiments
Once relegated to cinema or history lectures, bank runs have become a
modern phenomenon that captures the interest of students. In this article,
the authors explain a simple classroom experiment based on the
Diamond-Dybvig model (1983) to demonstrate how a bank run—a
seemingly irrational event—can occur rationally. They then present
possible topics for discussion including various ways to prevent bank runs
and moral hazard.
Kilgour DM, Brams SJ, Kaplan TR (2011). Three procedures for inducing honesty in bargaining. ACM International Conference Proceeding Series, 170-176.
Three procedures for inducing honesty in bargaining
A bargaining procedure, or mechanism, is a set of rules for two bargainers to follow as they make offers in order to reach a mutually satisfactory agreement on, say, a price. The efficiency of a mechanism is the expected surplus it delivers to the bargainers, relative to the surplus that a social planner would deliver, or that the bargainers themselves might achieve if they truthfully revealed their reservation prices. A theoretical limit on this efficiency is known, as is a specific procedure that achieves this maximum. But this procedure induces players to make offers that do not truly reflect their reservation prices. This paper discusses three procedures that induce honest offers, although they necessarily fail to achieve maximum efficiency. Each procedure has its own characteristics and costs, and each may have some uses in particular circumstances. © 2011 ACM.
Kaplan T (2010). Communication of Preferences in Contests for Contracts. Economic Theory
Kaplan TR, Sela A (2010). Effective contests. Economics Letters, 106(1), 38-41.
Effective contests
We find that two-stage contests could be ineffective, namely, there is a higher chance of low-ability players participating (and winning) than high-ability players. However, imposing a fee on the winner can guarantee that the contest will be effective. (C) 2009 Elsevier B.V. All rights reserved.
Balkenborg D, Kaplan T (2010). Using Economic Classroom Experiments. International Review of Economics Education, 9(2), 99-106.
Using Economic Classroom Experiments
Economic classroom experiments are an excellent way to increase student interest,
but getting started may be difficult.We attempt to aid the newcomer by
recommending which experiments to use and describing the current resources
Chakravarty S, Kaplan TR (2010). Vote or Shout. The B.E. Journal of Theoretical Economics, 10(1).
Vote or Shout
We examine an environment with n voters each with a private value over two alternatives. We compare the social surplus of two mechanisms for deciding: majority voting and shouting, that is, the voter who shouts the loudest (sends the costliest wasteful signal) chooses the outcome. We find that it is optimal to use voting in the case where n is large and value for each particular alternative of the voters is bounded. In for other cases, the superior mechanism is depends upon the order statistics of the distribution of values.
Chakravarty S, Kaplan T (2010). Vote or shout. Berkeley Press Journal of Theoretical Economics, Contributions
Roulston M, Kaplan TR (2009). A laboratory-based study of understanding of uncertainty in 5-day site-specific temperature forecasts. Meteorological Applications, 16(2), 237-244.
Kaplan TR, Ruffle BJ (2009). In Search of Welfare-Improving Gifts. European Economic Review, 53(4), 445-460. Full text.
Balkenborg D, Kaplan T, Miller T (2009). Teaching Bank Runs with Classroom Experiments.
Cohen C, Kaplan TR, Sela A (2008). Optimal rewards in contests. RAND Journal of Economics, 39(2), 434-451.
Optimal rewards in contests
We study all-pay contests with effort-dependent rewards under incomplete information. A contestant's value to winning depends not only on his type but also on the effort-dependent reward chosen by the designer. We analyze which reward is optimal for the designer when his objective is either total effort or highest effort. We find that under certain conditions the optimal reward may either be negative or even decreasing in effort; however, we find no advantage to having multiple rewards.
Kaplan TR, Wettstein D (2006). Caps on Political Lobbying: Comment. American Economic Review, 96(4), 1351-1354. Full text. DOI.
Kaplan TR (2006). Why Banks Should Keep Secrets. Economic Theory, 27(2), 341-357. Full text. DOI.
Brams SJ, Kaplan TR (2004). Dividing the indivisible - Procedures for allocating cabinet ministries to political parties in a parliamentary system. Journal of Theoretical Politics, 16(2), 143-173.
Dividing the indivisible - Procedures for allocating cabinet ministries to political parties in a parliamentary system
Political parties in Northern Ireland recently used a divisor method of apportionment to choose, in sequence, ten cabinet ministries. If the parties have complete information about each other's preferences, we show that it may not be rational for them to act sincerely by choosing their most-preferred ministry that is available. One consequence of acting sophisticatedly is that the resulting allocation may not be Pareto-optimal, making all the parties worse off. Another is non-monotonicity - choosing earlier may hurt rather than help a party. We introduce a mechanism, combining sequential choices with a structured form of trading, that results in sincere choices for two parties that avoids these problems. Although there are difficulties in extending this mechanism to more than two parties, other approaches are explored, such as permitting parties to make consecutive choices not prescribed by an apportionment method. But certain problems, such as eliminating envy, remain.
Kaplan TR, Ruffle B (2004). The Self-Serving Bias and Beliefs about Rationality. Economic Inquiry, 42(2), 237-246. DOI.
Kaplan TR, Ruffle BJ (2004). The self-serving bias and beliefs about rationality. Economic Inquiry, 42(2), 237-246.
The self-serving bias and beliefs about rationality
Most previous experiments attempting to establish the existence of the self-serving bias have confounded it with strategic behavior. We design an experiment that controls for strategic behavior (Haman effects) and isolates the bias itself. The self-serving bias that we measure concerns beliefs about the rationality of others. We find very limited support for the existence of the bias. To help understand why the bias seems to hold in some settings but not in others, we discuss a distinction between biases that are self-serving and those that are actually self-defeating.
Kaplan TR, Luski I, Wettstein D (2003). Government Policy towards Multi-National Corporations. Economics Bulletin, 6(3), 1-8. Full text.
Kaplan T, Luski I, Wettstein D (2003). Government policy towards multi-national corporations. Economics Bulletin, 6(1).
Government policy towards multi-national corporations
We analyze an environment with asymmetric information where a country tries to attract a multi-national corporation. The country can use both taxes and grants to meet its objective of maximizing net revenues. We show that when the country has private information it can often convey it via its choice of a tax-grant pair. When the tax rates are unbounded the country is able to extract the full surplus. The existence of an upper bound can in some cases reduce the payoff to a stronger country.
Kaplan TR, Luski I, Wettstein D (2003). Innovative activity and sunk cost. International Journal of Industrial Organization, 21(8), 1111-1133.
Innovative activity and sunk cost
We analyze innovative activity in a general framework with time-dependent rewards and sunk costs. When firms are identical, innovation is delayed by an increase in the number of firms or a decrease in the size of the reward. When one firm has higher profit potential, it is more likely to innovate first. Our framework generalizes an all-pay auction; however, we show that under certain conditions there is qualitatively different equilibrium behavior. (C) 2003 Elsevier B.V. All rights reserved.
Kaplan TR, Luski I, Sela A, Wettenstein D (2002). All-Pay Auctions with Variable Rewards. Journal of Industrial Economics, 50(4), 417-430. Full text. DOI.
Kaplan TR (2000). Effective price-matching: a comment. International Journal of Industrial Organization, 18(8), 1291-1294.
Effective price-matching: a comment
Corts [Economic Lett. 47 (1995) 417] showed that when allowing for price-beating policies in addition to price-matching policies, the competitive outcome prevails in lieu of monopoly pricing. I show by expanding the strategy set further to include effective price strategies, the possibility of monopoly pricing is restored. (C) 2000 Elsevier Science B.V. All rights reserved.
Kaplan TR, Wettstein D (2000). Surplus Sharing with a Two-Stage Mechanism. International Economic Review, 41(2), 339-409.
Surplus sharing with a two-stage mechanism
In this article we consider environments where agents jointly produce a private output good by contributing privately owned resources. An efficient outcome may not be realized due to strategic behavior and conflicting interests of the agents. We construct a two-stage mechanism, building on a Varian mechanism. The modified mechanism ensures an equilibrium for a large class of preferences and guarantees the feasibility of outcomes.
Kaplan TR, Wettstein D (2000). The Possibility of Mixed-Strategy Equilibria with Constant-Returns-to-Scale Technology under Bertrand Competition. Spanish Economic Review, 2(1), 65-71.
Kaplan TR, Wettstein D (1999). Cost Sharing: Efficiency and Implementation. Journal of Mathematical Economics, 32(4), 489-502.
Kaplan TR, Ruffle B (1998). Self-Serving Bias [Comment]. Journal of Economic Perspectives, 12(2), 243-244.
Kaplan TR, Dickhaut J (1992). A Program for Finding Nash Equilibria. The Mathematica Journal, 1(4), 87-93.
Kaplan TR, Dickhaut J (1993). A Program for Finding Nash Equilibria. In Varian HR (Ed) Economic and Financial Modeling with Mathematica, Springer.
Kaplan TR, Mukherji A (1993). Designing an Incentive-Compatible Contract. In Varian HR (Ed) Economic and Financial Modeling with Mathematica, Springer-Verlag.
Designing an Incentive-Compatible Contract
Fischer S, Güth W, Kaplan TR, Zultan R (2017). Auctions and Leaks: a Theoretical and Experimental Investigation Auctions and Leaks: a Theoretical and Experimental Investigation.
Auctions and Leaks: a Theoretical and Experimental Investigation Auctions and Leaks: a Theoretical and Experimental Investigation
(Revised Version of JERP 2014-027)
In first- and second-price private value auctions with sequential bidding, second movers may discover the first movers'
bid. Equilibrium behavior in the first-price auction is mostly unaffected but there are multiple equilibria in the second-
price auction. Consequently, comparative statics across price rules are equivocal. Experimentally, leaks in the first-price auction favor second movers but harm first movers and sellers, as theoretically predicted. Low to medium leak
probabilities eliminate the usual revenue dominance of first-over second-price auctions. With a high leak probability,
second-price auctions generate significantly more revenue.
Fischer S, Guth W, Kaplan TR, Zultan R (2014). AUCTIONS AND LEAKS: a THEORETICAL AND EXPERIMENTAL INVESTIGATION.
Fischer S, Güth W, Kaplan TR, Zultan R (2014). Auctions and Leaks: a Theoretical and Experimental Investigation.
Auctions and Leaks: a Theoretical and Experimental Investigation
We study first- and second-price private value auctions with sequential
bidding where second movers may discover the first movers bids. There is
a unique equilibrium in the first-price auction and multiple equilibria in the
second-price auction. Consequently, comparative statics across price rules
are equivocal. We experimentally find that in the first-price auction, leaks
benefit second movers but harm first movers and sellers. Low to medium
probabilities of leak eliminate the usual revenue dominance of first-price
over second-price auctions. With a high probability of a leak, second-price
auctions generate higher revenue.
1990: Winner of the Santa Fe Institute's Double Auction Tournament
1996: Kreitman Fellowship (Israel)
2002: Zif Fellowship (Germany)
2006: Leverhulme Fellowship (UK)
2009: Co-winner of the Economics Network e-learning award
Conferences and invited presentations
Meteorology Meets Social Science: Risk, Forecast and Decision (Conference organiser)
External positions
Associate Editor, Journal of Behavioral and Experimental Economics.
Professor Kaplan's teaching interests include microeconomics, industrial economics and game theory. He is especially interested in using classroom experiments to teach economic concepts and has recently developed an experiment to demonstrate bank runs in the classroom.
BEE3018 - Game Theory | CommonCrawl |
\begin{document}
\title{Experimental demonstration of continuous quantum error correction} \date{\today}
\author{William P. Livingston} \affiliation{Department of Physics, University of California, Berkeley, CA 94720 USA} \affiliation{Center for Quantum Coherent Science, University of California, Berkeley, California 94720, USA} \thanks{Correspondence and requests for materials should be addressed to W.P.L. (email: [email protected]).}
\author{Machiel S. Blok} \affiliation{Department of Physics, University of California, Berkeley, CA 94720 USA} \affiliation{Center for Quantum Coherent Science, University of California, Berkeley, California 94720, USA} \affiliation{Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA}
\author{Emmanuel Flurin} \affiliation{Université Paris-Saclay, CEA, CNRS, SPEC, 91191 Gif-sur-Yvette Cedex, France}
\author{Justin Dressel} \affiliation{Institute for Quantum Studies, Chapman University, Orange, CA 92866, USA} \affiliation{Schmid College of Science and Technology, Chapman University, Orange, CA 92866, USA}
\author{Andrew N. Jordan} \affiliation{Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA} \affiliation{Institute for Quantum Studies, Chapman University, Orange, CA 92866, USA}
\author{Irfan Siddiqi} \affiliation{Department of Physics, University of California, Berkeley, CA 94720 USA} \affiliation{Center for Quantum Coherent Science, University of California, Berkeley, California 94720, USA}
\begin{abstract} The storage and processing of quantum information are susceptible to external noise, resulting in computational errors that are inherently continuous\cite{Minev2019}.
A powerful method to suppress these effects is to use quantum error correction\cite{PhysRevA.52.R2493, nielsen2000quantum, steane1996multiple}.
Typically, quantum error correction is executed in discrete rounds where errors are digitized and detected by projective multi-qubit parity measurements\cite{Knill2005,Chamberland2018faulttolerant}.
These stabilizer measurements are traditionally realized with entangling gates and projective measurement on ancillary qubits to complete a round of error correction.
However, their gate structure makes them vulnerable to errors occurring at specific times in the code and errors on the ancilla qubits.
Here we use direct parity measurements to implement a continuous quantum bit-flip correction code in a resource-efficient manner, eliminating entangling gates, ancilla qubits, and their associated errors.
The continuous measurements are monitored by an FPGA controller that actively corrects errors as they are detected.
Using this method, we achieve an average bit-flip detection efficiency of up to 91\%{}.
Furthermore, we use the protocol to increase the relaxation time of the protected logical qubit by a factor of 2.7{} over the relaxation times of the bare comprising qubits.
Our results showcase resource-efficient stabilizer measurements in a multi-qubit architecture and demonstrate how continuous error correction codes can address challenges in realizing a fault-tolerant system.
\end{abstract}
\maketitle
A successful quantum error correction (QEC) code decreases logical errors by redundantly encoding information and detecting errors in a more complex physical system. Such a system includes both the qubits encoding the logical quantum information and the overhead resources to perform stabilizer measurements. In a fault-tolerant QEC code, the benefit from error correction needs to outweigh the cost of extra errors associated with this overhead. In the past decade, discrete QEC has been realized in various physical systems such as ion traps\cite{Schindler1059,Negnevitsky2018,Linkee1701074}, defects in diamonds\cite{Cramer2016}, and superconducting circuits\cite{Kelly2015,Ofek2016,Andersen2020,Bultinkeaay3050,Riste2020, Stricker2020, chen2021exponential}. The stabilizer measurements in these realizations are a dominant source of error \cite{chen2021exponential} because they are indirect and require extra resources, including ancillas and entangling gates.
We demonstrate an alternative form of QEC known as continuous QEC in which continuous stabilizer measurements eliminate the cycles of discrete error correction as well as the need for ancilla qubits and entangling gates\cite{PhysRevA.65.042301, PhysRevA.79.024305, cardona2019continuoustime}. Continuous measurements have previously been used to study the dynamics of wavefunction collapse and, with the addition of classical feedback, to stabilize qubit trajectories and correct for errors in single qubit dynamics\cite{Vijay2012, PhysRevX.3.021008, PhysRevLett.112.080501}. In systems of two or more qubits, direct measurements of parity can be used to prepare entangled states through measurement\cite{PhysRevB.67.241305,PhysRevB.73.235331, PhysRevA.78.062322, PhysRevLett.112.170501, PhysRevX.6.041052, Riste2013}. Here, we use two direct continuous parity measurements and associated filtering\cite{Mohseninia2020alwaysquantumerror} to correct bit-flip errors while maintaining logical coherence. Errors are detected on a rolling basis, with the measurement rate as the primary limitation to how quickly errors are detected.
We realize our code in a planar superconducting architecture using three transmons as the bare qubits. As depicted in Fig. \ref{fig:protocol}, we implement the $ZZ$ parity measurements using two pairs of qubits coupled to joint readout resonators\cite{PhysRevA.81.040301, Riste2013}. Each resonator is coupled to its associated qubits with the same dispersive coupling $\chi_i$ with $i$ indexing the resonator, thereby making the resonator reflection response when the associated qubit pair is in $\ket{01}$ identical to the response when the pair is in $\ket{10}$. For each resonator, we set the parity probe frequency to be at the center of this shared odd parity resonance. To approximately implement a full parity measurement, we make the line-width $\kappa_i$ (\kappaA{}, \kappaB{}) of each resonator smaller than its respective dispersive shift $\chi_i$ (2.02 MHz{}, 2.34 MHz{}). When the qubit pair is in either $\ket{00}$ or $\ket{11}$, the resonance frequency is sufficiently detuned from the odd parity probe tone to keep the cavity population low and the reflected phase responses for the two even states nearly identical. After reflecting a parity tone off a cavity, the signal is amplified by a Josephson Parametric Amplifier\cite{Castellanos-Beltran2008} in phase-sensitive mode aligned with the informational quadrature.
\begin{figure*}
\caption{ Full Parity Detection. \textbf{a}, Three qubits in two cavities, with each cavity implementing a full parity measurement. Lower right: ideal phase responses of a coherent tone reflected off each cavity for different qubit states. The parity probe tones are centered on the odd-parity resonances. The phase space (IQ) plots show the ideal steady state reflected tone for the shown qubit configuration. Dashed circles are centered on all possible steady state responses. \textbf{b}, Micrograph of the superconducting chip with three transmons and two joint readout resonators. }
\label{fig:protocol}
\end{figure*}
We implement the three qubit repetition code using two $ZZ$ parity measurements as stabilizers: $Z_0Z_1$ and $Z_1 Z_2$, with $Z_j$ being the Pauli $Z$ operator on qubit $j$. The codespace can be any of the four subspaces with definite stabilizer values, so we choose the subspace with positive parity values $(+1, +1)$ for simplicity. This choice of codespace is spanned by the logical code states $\ket{0_L} = \ket{000}$ and $\ket{1_L} = \ket{111}$. The three remaining possible stabilizer values identify error subspaces in which a qubit has a single bit-flip ($X$) error relative to the codespace. A change in parity heralds that the logical state has moved to a different subspace with a different logical state encoding.
Ideal strong measurements of both code stabilizers project the logical state into either the original codespace or one of the error spaces, effectively converting analog errors to correctable digital errors. In contrast, measurements with a finite rate of information extraction, like the homodyne detection used in this experiment, result in the qubit state undergoing stochastic evolution such that the logical subspaces are invariant attractors\cite{wiseman_milburn_2009}. The observer receives noisy voltage traces with mean values that are correlated to stabilizer eigenvalues and variances that determine the continuous measurement collapse timescales. Monitoring both parity stabilizers in this manner suppresses analog drifts away from the logical subspaces, while providing a steady stream of noisy information to help identify and correct errors that do occur.
First we experimentally investigate how to extract parity information from such noisy voltage traces. Previous work has shown that Bayesian filtering is theoretically optimal \cite{Mabuchi_2009,Mohseninia2020alwaysquantumerror}. Here, we implement a simpler technique with performance theoretically comparable to that of the Bayesian filter while using fewer resources on our FPGA controller\cite{Mohseninia2020alwaysquantumerror}. We first filter the incoming voltage signals with a 1536 ns {} exponential filter to reduce the noise inherent from measuring our system with a finite measurement rate (0.40 MHz{}) and call this signal $V_i(t)$ for resonator $i$. We normalize $V_i(t)$ such that $\braket{V_i(t)}=-1$ corresponds to the system being in an odd parity state, and $\braket{V_i(t)}=1$ corresponds the the system in an even parity state. Here we have defined expectation values as averaging over all possible noise realizations. As shown in Fig. \hyperref[fig:timing]{\ref{fig:timing}a}, we monitor the trajectories of $V_i$ for signatures of bit-flips using a thresholding scheme\cite{PhysRevA.102.022415, Mohseninia2020alwaysquantumerror,2003.11248}. Supposing we prepare an even-even parity state, a bit-flip on one of the outer qubits is detected when one of the signals goes lower than a threshold $\Theta_1=-0.50$ while the other signal stays above another threshold, $\Theta_2=0.72$. A flip of the central qubit is detected when both signal traces fall below a threshold $\Theta_3=-0.39$. These thresholds are numerically chosen based on experimental trajectories to maximize detection efficiencies of flips while minimizing dark counts and misclassification errors due to noise. When a thresholding condition is met, the controller sends out a corrective $\pi$-pulse to the qubit on which the error was detected. The controller also performs a reset operation on the voltage signals in memory to reflect the updated qubit state. As shown in Fig. \hyperref[fig:timing]{\ref{fig:timing}b}, when a deterministic flip is applied to the $\ket{000}$ state, the system is reset back to $\ket{000}$ faster with feedback than through natural $T_1$ decay.
\begin{figure*}
\caption{ Error Correction. \textbf{a}, Sample experimental voltage traces of the controller correcting induced bit flips. With no errors, both voltages remain positive. When an error occurs, one or both of the voltages flip and the cross thresholds, triggering the controller to send a corrective $\pi$ pulse to bring the system back to the codespace. \textbf{b}, Voltage responses to an induced flip on $Q_0$ with (blue) and without (red) feedback. Bold lines are averages and light lines are sample individual traces. }
\label{fig:timing}
\end{figure*}
To characterize the code, we first check the ability of the controller to correct single bit-flips. We prepare the qubits in $\ket{000}$ and apply the parity readout tones for \SFEroTime{}. After \SFEflipTime{} of readout to let the resonators reach steady state, we apply a $\pi$-pulse to one of the qubits, inducing a controlled error. We record if and when the controller detects the error and sends out a correction pulse. Errors are successfully detected on $Q_0$ with 90\%{} efficiency, $Q_1$ with 86\%{} efficiency, and $Q_2$ with 91\%{} efficiency. The primary source of inefficiency is $T_1$ decay bringing the qubits back to ground before detection can happen. On average, the controller corrects an error \SFEtCorr{} after the error occurs, with the full probability density function over time shown in Fig. \hyperref[fig:t1]{\ref{fig:t1}a}. We also characterize a dark count rate for each flip variety by measuring the rate at which the controller detects a qubit flip after preparing in the ground state (3.4{}, 1.0{}, 4.0{}) $\text{ms}^{-1}$. In comparison, the thermal excitation rates for each qubit are estimated to be (1.8{}, 1.0{}, 2.0{}) $\text{ms}^{-1}$.
\begin{figure*}
\caption{ Characterizing the time to correct an error. \textbf{a}, Histogram of time between an induced error and the correction pulse for each of the qubits, normalized such the integral of the probability density $P_{flip}(t)$ gives the detection probability. Dashed lines indicate the dark count rates for each error type. \textbf{b}, Probability of detecting certain flip sequences given a flip on $Q_0$ preceding a flip on $Q_2$. The green region is the probability of the controller correctly detecting a $Q_0$ flip and then a $Q_2$ flip. The red region is the probability of the controller detecting a $Q_1$ flip, resulting in a logical error. The dotted line indicates the dead time, when these two probabilities are equal. \textbf{c}, Population decay of the excited logical state, $\ket{101}$, of the odd-odd subspace with and without feedback. With feedback on, the lifetime of the logical basis state is longer than that of an individual bare qubit. }
\label{fig:t1}
\end{figure*}
We next investigate the dominant source of logical errors while running the code: two bit flips occurring in quick succession. When two different qubits flip close together in time relative to the inverse measurement rate, the controller may incorrectly interpret the signals as an error having occurred on the unflipped qubit. The controller then flips this remaining qubit, resulting in a logical error. For continuous error correction, this effect results in a time after an error occurs we call the dead time, when a following error cannot be reliably corrected. To characterize this behavior, we prepare the system in the ground state and apply two successive bit-flips with different times between the pulses. We then check if the controller responds with the right sequence of correction pulses. In Fig. \hyperref[fig:t1]{\ref{fig:t1}b}, we show the controller's interpretation of successive flips on $Q_0$ and $Q_2$ as a function of time between them. We mark the dead time at the point where the probability of a logical error crosses the probability of successfully correcting the state. Among the possible pairs and orderings of two qubit errors, the dead times vary from \DFEdead{}.
Although the code is designed to correct bit-flip errors, the code will also protect the logical computational basis states against qubit decay, extending the $T_1$ lifetimes of the logical system beyond that of the bare qubits. As opposed to a bit-flip, a qubit decaying loses any coherent phase of the logical state, and the system will be corrected to a mixed state with the same probability distribution in the computational basis as the initial state. For example, the state $\frac{1}{\sqrt{2}}(\ket{0_L}+\ket{1_L})$ undergoing a qubit decay and correction will be restored as the density matrix $\frac{1}{2}(\ket{0_L}\bra{0_L}+\ket{1_L}\bra{1_L})$. In the long time limit of active feedback, the system will reach a steady state described by a mixed density matrix with the majority of population (87-99.6\%{}) in the selected codespace. The $T_1$ of a codespace is defined by the exponential time constant at which population of computational basis states in the codespace approach this steady state. The different codespaces of different parities have different $T_1$ decay times, with the longest decay time of \TOneOO{} associated with the odd-odd subspace, as shown in Fig. \hyperref[fig:t1]{\ref{fig:t1}c}. The shortest lifetime, \TOneEE{}, is associated with the even-even subspace, since the higher energy level in this codespace has three bare excitations and the lower energy has no excitations. In comparison, the bare $T_1$ values of the bare qubits range from \SIrange[mode=text]{20}{24}{\micro\second}, making the logical qubit excited life 2.7{} times longer than that of a bare qubit.
Although phase errors are not protected against by this code, an ideal implementation of a bit-flip code should not increase their occurrence rate. However, with our physical realization of continuous correction, we induce extra dephasing in the logical subspace through three primary channels: continuous dephasing due to the measurement tone; dephasing when going from an odd parity subspace to an even parity subspace; and dephasing related to static $ZZ$ interactions intrinsic to the chip design.
The first source of excess dephasing is measurement-induced dephasing, where the dephasing rate is proportional to the distinguishability of different qubit eigenstates under the measurement\cite{PhysRevA.74.042318}. Distinguishability is measured as $D^{(i)}_{m,n}=\left|\alpha^{(i)}_{\ket{m}}-\alpha^{(i)}_{\ket{n}}\right|^2$ where $\ket{m}$ and $\ket{n}$ are different basis states of the two qubits coupled to resonator $i$, and $\alpha^{(i)}$ is the resonator's associated coherent state\cite{PhysRevA.74.042318}. By tuning the qubit frequencies, the dispersive shifts of the system are calibrated such that $D^{(i)}_{01,10}$ are close to zero. The parity measurement distinguishability ($D^{(i)}_{01,11} \approx D^{(i)}_{01,00}$) determines the measurement-induced dephasing rate of the code. Due to finite $\chi/\kappa$, the even subspaces are not perfectly indistinguishable, with the theoretical distinguishability ratio $D^{(i)}_{00,01}/D^{(i)}_{00,11} \approx 4(\chi_i/\kappa_i)^2$. We use this formula to calculate distinguishability ratios of 40{} and 33{} for resonator 0 and 1 respectively. We plot the measured distinguishability of various state pairs in Fig. \hyperref[fig:coherence]{\ref{fig:coherence}a}, and find agreement with these predicted values as well as low distinguishability between eigenstates of odd parity. This distinguishability could be lowered even further by increasing the ratio $\chi / \kappa$.
\begin{figure*}
\caption{
Preservation of quantum coherence. \textbf{a}, Distinguishability of various state pairs in steady state readout for each measurement tone. Pairs of states in the yellow region differ in one or both of their parities. Pairs of states in the green region share their parities. Dashed lines indicate theoretically predicted distinguishability of the even eigenstates. \textbf{b}, Relative logical coherence after preparing a logical $\ket{+X_L}$ state in each of the logical parity subspaces, applying parity measurement tones without feedback, and flipping one of the qubits. Coherences are normalized to results from the same procedure without the measurement tones applied. Error bars are statistical uncertainty from repeated runs of the measurement. Dashed lines indicate predicted relative dephasing due to an odd to even parity flip on $R_0$, $R_1$, or both. \textbf{c}, Sample coherences from preparing a logical $\ket{+X_L}$ state in the odd-odd (OO) subspace, applying an error pulse, and letting the controller correct the error. Coherences are reconstructed by time bins set by the time it takes to correct the error with error bars representing statistical uncertainty. Oscillations due to static $ZZ$ coupling are visible. }
\label{fig:coherence}
\end{figure*}
The second source of excess dephasing occurs when a pair of qubits switches from an odd parity state to an even parity state. When two qubits coupled to one of the resonators have odd parity, the resonator is resonantly driven by the measurement tone and thus reaches a steady state with a larger number of photons as compared to when the qubits have even parity. If one of these qubits undergoes a bit-flip while the system is in an odd parity state, the resonator frequency shifts and the system undergoes excess dephasing as the resonator rings down to the steady state for the even subspace. The coherence of the logical state is expected to contract by a factor of $e^{-\bar{n}}$, with $\bar{n}$ being the steady state photon number of a resonator when its qubits are in an odd parity state. From the steady state dephasing rates and the resonator parameters, we independently estimate the photon number in each resonator to be .7{} and .6{} respectively when the qubits are in the odd state. To measure this effect, we prepare a 3-qubit logical encoding of an $X$-eigenstate, $\ket{+X_{L'}}=\frac{1}{\sqrt{2}}(\ket{0_{L'}}+\ket{1_{L'}})$, where $L'$ is one of the four possible logical encodings. With the measurement tone on, but without feedback, we apply a pulse on one (or none) of the qubits, taking the state to a different (or the same) codespace, $L$. We then tomographically reconstruct the magnitude of the coherence in the new codespace, $|\rho^L_{01}|$, as shown in Fig. \hyperref[fig:coherence]{\ref{fig:coherence}b}. The coherences are normalized to the $|\rho^L_{01}|$ generated by same experiment with the measurement tones off. The system demonstrates significantly less coherence when one of the parities changes from odd to even than vice versa, with reasonable agreement to the expected dephasing based on measured photon number. Since $\bar{n}$ scales inversely with $\kappa$ for a fixed measurement rate, a larger kappa would reduce this effect.
The third source of excess dephasing is related to static $ZZ$ interactions among the qubits and the uncertainty in timing between when a bit-flip error occurs and when the correction pulse is applied. Performing a Ramsey sequence on $Q_i$ while $Q_j$ is either in the ground or excited state, we measure the coefficients of the system's intrinsic $ZZ$ Hamiltonian, $H_{ZZ} = \frac{1}{2}\sum_{i\neq j} \beta_{ij} Z_i Z_j$. Since the three qubits are in a line topology, with the joint readout resonators also acting as couplers, there is significant coupling between $Q_0$ and $Q_1$ ($\beta_{01}$ = 0.49 MHz{}) and between $Q_1$ and $Q_2$ ($\beta_{12} =$ 1.05 MHz{}) while there is almost no coupling between $Q_0$ and $Q_2$ ($\beta_{02} < $ 2 kHz{}). Due to this coupling, the definite parity subspaces have different energy splittings: In the rotating frame of the qubits, the odd-odd, odd-even, even-odd, and even-even subspaces have logical energy splittings of 0, $\beta_{12}$, $\beta_{01}$, and $\beta_{01} + \beta_{12}$ respectively. When a bit-flip occurs, the system jumps to an error space and precesses at the frequency of that error space until being corrected by the controller. Since the time from the error flip to the correction pulse is generally unknown, the state can be considered to have picked up a random unknown relative phase. The net dephasing $\zeta_{zz}$ can be calculated by averaging the potential phases over the probability distribution of time, $T$, it takes to correct an error: $e^{i\phi-\zeta_{zz}}=\braket{e^{iT\Delta\beta}}_T$ with $\Delta\beta$ being the energy difference between codespace and error space. Using the distributions in Fig. \hyperref[fig:t1]{\ref{fig:t1}a} and known $\Delta \beta$, we compute $\zeta_{zz}$ to be from $2.5$ to $5.7$ depending on the codespace and the qubit flipped. Although we don't observe this dephasing directly, we perform an experiment to capture this effect. For each of the codespaces, we prepare a $\ket{+X_L}$ state in the odd-odd codespace and induce a bit-flip error while the feedback controller is active. After \CDDtomoDelay{}, we perform tomography on all three qubits and note the time at which the correction pulse occurred. We then reconstruct the logical coherence element $\rho^L_{01}$ of the density matrix conditional on time it took the controller to apply the correction pulse. As shown in Fig. \hyperref[fig:coherence]{\ref{fig:coherence}c}, we observe oscillations with frequency corresponding to the effects of $ZZ$ coupling. This source of dephasing is not intrinsic to the protocol, and can be mitigated by reducing the $ZZ$ coupling between the qubits.
Our experiment extends the capabilities of continuous measurements, demonstrating active feedback on multiple multipartite measurement operators. We use continuous quantum error correction to detect bit flips and extend the relaxation time of a logical state. Furthermore, the protocol is implemented in a planar geometry and compatible with existing superconducting qubit architectures so can in principle be combined with other error correction methods. Future improvements could be made by reducing spurious decoherence effects through novel implementations of continuous parity measurements\cite{Royereaau1695,DiVincenzo_2013} or optimizing coupling parameters. Specifically, increasing $\chi/ \kappa$ and increasing $\kappa$ will reduce dephasing for a given measurement rate. Furthermore, lowering the static $ZZ$ coupling\cite{kandala2020demonstration} will reduce the observed dephasing from an error occurring at an indeterminate time. Additional feedback could be used to reduce the effects of measurement induced dephasing\cite{PhysRevA.85.052318}. By incorporating more qubits and continuous $XX$ measurements, this scheme could be extended to stabilize fully protected logical states\cite{PhysRevA.102.022415}.
\footnotesize \textbf{Acknowledgements} We thank A. Korotkov, J. Atalaya, R. Mohseninia, and L. Martin for discussions. We also thank J.M. Kreikebaum and T. Chistolini for technical assistance. This material is based upon work supported in part by the U.S. Army Research Laboratory and the U.S. Army Research Office under contract/grant number W911NF-17-S-0008. JD also acknowledges support from the National Science Foundation - U.S.-Israel Binational Science Foundation Grant No. 735/18.
\textbf{Author contributions} E.F., M.S.B., and W.P.L. conceived the experiment. W.P.L. and E.F. designed the chip. W.P.L fabricated the chip, constructed the experimental setup, performed measurements, and analysed data with assistance from M.S.B. J.D. and A.N.J. provided theoretical support. W.P.L. wrote the manuscript with feedback from all authors. All work was carried out under the supervision of I.S.
\textbf{Competing interests} The authors declare that they have no competing financial interests. \normalsize
\onecolumngrid \section*{Methods} \twocolumngrid \textbf{Design and fabrication} The microwave properties of the chip were simulated in Ansys high-frequency electromagnetic-field simulator (HFSS), and dispersive couplings were simulated using the energy participation method with the python package pyEPR\cite{minev2020energyparticipation}. Resonators, transmission lines, and qubit capacitors were defined by reactive ion etching of 200 nm of sputtered niobium on a silicon wafer. Al-AlOx-Al Josephson junctions were added using the bridge-free ``Manhattan style'' method\cite{959656}. The junctions were then galvanically connected to the capacitor paddles through a bandaid process\cite{doi:10.1063/1.4993577}. The middle qubit is fixed frequency, and the outer two qubits are tunable with a tuning range of 260 MHz{} and 220 MHz{}. Wire bonds join ground planes across the resonators and bus lines.
\textbf{Measurement setup} A wiring diagram of our experimental setup is show in Supplementary Information Figure 1. The Josephson Parametric Amplifiers (JPAs) are fabricated with a single step using Dolan bridge Josephson junctions. They are flux pumped at twice their resonance frequency, providing narrow-band, phase-sensitive amplification. The signals are further amplified by two cryogenic HEMT amplifiers, model LNF4\_8. In the output chain for resonator 0, we include a TWPA between the JPA and the HEMT to operate that JPA at a lower gain. Infrared filters on input lines are made with an Eccosorb dielectric. The outer qubits are flux tuned with off-chip coils. The FPGA board provides full control of the qubits and readout of the resonators. An external arbitrary waveform generator creates the cavity tones and JPA drives, as well as triggering the FGPA. The JPA modulation tone is split with one branch phase shifted before both go into an IQ mixer for single sideband modulation.
\textbf{FPGA Logic} The FPGA board we used for the feedback is an Innovative Integration X6-1000M board. We programmed a custom pulse generation core to drive qubit pulses and to demodulate and filter incoming readout signals. A control unit parses instructions loaded in an instruction register. These instructions may include 1) putting a specified number of pulse commands into a queue to await pulse timing; 2) resetting a pulse timer keeping track of time within a sequence while incrementing a trigger counter; and 3) resetting the pulse timer, the trigger counter, and the instruction pointer. When a pulse instruction enters the timing queue, it waits until a specified time and is then sent to one of three different possible locations. The first possible location is a pulse library where the instruction points to a complex pulse envelope of a given duration, which is then modulated by one of three CORDIC sine/cosine generators and sent to the correct DAC. These pulses are sent down one of three qubit control lines. The second possible location is to one of the CORDIC sine/cosine generators, where the instruction will increment the phase of the generator by a specified argument, thus implementing Z rotations in the qubit frame. The third location is a demodulation core, which, similarly to the qubit pulse block, retrieves a complex waveform from memory for a specified duration. This waveform is then multiplied against the complex incoming readout signals and low-pass filtered with a 32 ns exponential filter to generate the signal $V^{DC}_i$ for feedback as well as to readout projective measurements.
When the feedback control unit is active, it takes $V^{DC}_i$, applies a secondary 1536 ns{} ns exponential filter/accumulator to further reduce the noise, and then continuously checks these traces ($V_i$) against the threshold conditions for an error to have been detected. When an error is detected, the controller injects instructions for a corrective $\pi$-pulse into the pulse generation unit. Any voltage $V_i$ which went across a threshold is then inverted as to not trip further corrective pulses. After a delay such that the corrective pulse has taken effect on chip, the $V^{DC}_i$ is inverted before being accumulated into $V_i$. In conjunction with the previous inversion of $V_i$, this effectively resets the feedback controller while avoiding interpreting the corrective pulse as another error.
The board's I/O comprises the PCIe slot for exchanging data with the computer and the ADC/DACs on the analog front-end. The FPGA can stream from multiple sources to the computer along 4 data pipelines. The primary sources are $V^{DC}_i$ and a list of timestamped pulse commands. The timing of any corrective pulses can be obtained from this second source. Further data sources include raw ADC voltages, raw DAC voltages, and $V_i$, which are only used as diagnostics. On the analog front-end, there are two ADCs running at 1 GSa/s which take in the IF readout signals from the I and Q ports of an IQ mixer, treating the two ADC inputs as the real and imaginary parts of a complex signal. To drive the three qubit lines, there is one DAC running at 1 GSa/s and, due to board constraints, two DACs running at 500 MSa/s.
\textbf{Optimizing Filter Parameters}
To optimize threshold values, we prepare the ground state and then flip either one or none of the qubits while taking parity traces ($V^{DC}_i$). In post processing, we filter the traces with the same exponential filter as on the FPGA to recreate $V_i$, and classify the resultant traces according to whether or not they pass the different thresholds registering as a qubit flip. We thus get a confusion matrix $P_{ij}=P(i|j)$, the probability of classifying a trace as a flip on $i$ given a preparation flip $j$, where $i,j \in (\operatorname{None}, 0, 1, 2)$. The thresholds were chosen to minimize $\sum_{ij}(P_{ij}-\delta_{ij})^2$.
\renewcommand{S.\arabic{equation}}{S.\arabic{equation}} \onecolumngrid \section*{Supplementary materials} \twocolumngrid \subsection{System parameters} We summarize qubit and resonator frequencies, as well as typical qubit lifetimes in the tables below. \renewcommand{1.2}{1.2}
\begin{table}[hbt!] \centering \addtolength{\tabcolsep}{2pt}
\begin{tabular}{|l|rrr|} \hhline{----}
{} & $Q_0$ & $Q_1$ & $Q_2$ \\\hhline{----} Frequency (MHz) & 5355 & 5182 & 5392 \\ \hhline{----} Anharmonicity (MHz) & 307 & 310 & 310 \\ \hhline{----} $T_1$ (\SI{}{\micro\second}) & 22 & 23 & 23 \\ \hhline{----} $T_2^*$ (\SI{}{\micro\second}) & 18 & 26 & 20 \\ \hhline{----} $T_2^{echo}$ (\SI{}{\micro\second}) & 31 & 31 & 35 \\ \hhline{----} \end{tabular} \caption{Qubit parameters} \end{table}
\begin{table}[hbt!] \centering \addtolength{\tabcolsep}{2pt}
\begin{tabular}{|l|rr|} \hhline{---} {} & $R_0$ & $R_1$ \\ \hhline{---} Frequency (MHz) & 6314 & 6405 \\ \hhline{---} $\kappa$ (kHz) & 636 & 810 \\ \hhline{---} $\chi$ (MHz) & 2.02 & 2.34 \\ \hhline{---} Quantum Efficiency & 0.62 & 0.56 \\ \hhline{---} \end{tabular} \caption{Resonator parameters} \end{table}
\subsection{Tomographic reconstruction} We use the parity resonators to perform qubit tomography. However, due to the nature of the parity condition, not all states are distinguishable by this measurement. To perform tomography, we use single qubit pulses to map each three-qubit Pauli eigenstate to $\ket{000}$ and then measure both resonators on their respective $\ket{00}$ resonance. We then measure the probability that full qubit system is in the ground state, which corresponds to reading out both resonators as 0. We additionally include data into the tomography analysis if one of the resonators reads out 1 and the other reads out 0, since we know the final state to be in either $\ket{100}$ or ${\ket{001}}$ depending on which resonator reads 1. Using this information, we construct partial Pauli expectation values such as $\braket{X^+Y^-I}$, with $P^+,P^-$ being the plus and minus projectors for a particular Pauli $P$ such that $P=P^+-P^-$. We then apply readout correction on these probabilities to mitigate the effects of readout infidelity. From this corrected data taken over many tomographic sequences, we can reconstruct full Pauli expectation values such as $\braket{XYI}$. When reconstructing logical coherences, we only measure in the $X$ and $Y$ bases. When reconstructing populations, we only measure in the $Z$ basis.
\subsection{Ramsey heralding} Qubits 0 and 2 demonstrate a strong temporal bistability in qubit frequency, with a splitting of about 80 kHz and a typical switching time on the order of .1–10 s. When taking data to reconstruct logical coherences, we include five extra sequences in our AWG sequence table, each consisting of five repeated restless Ramsey measurements with free precession times of \SI{6}{\micro\second}. With a typical initial sequence length of 64 and a repetition rate of \SI{100}{\micro\second}, the qubit's frequency state is sampled every 7 ms, allowing us to herald data runs to only include data from runs when the qubits have a particular frequency.
\subsection{Steady state dephasing} Here we derive relative dephasing rates for two qubits in a dispersive parity measurement using a classical analysis of the resonator steady states. The measurement dephasing rate is proportional to the distinguishability of resonator responses when the coupled qubits are in different eigenstates\cite{PhysRevA.77.012112}. We set the probe frequency on resonance with the cavity when qubits are in the single-excitation subspace and assume that $\chi\gg\kappa$. We also assume the external cavity coupling is much larger than the internal cavity loss, so the cavity responds with the following scattering parameter: \begin{equation}
S(f_0:=\chi\braket{Z_0+Z_1}) = \frac{-2f_0+i\kappa}{-2f_0-i\kappa} \end{equation} Odd parity states are perfectly indistinguishable. The distinguishability between states of opposite parity is \begin{equation}
D_{01,00}=|S(0) - S(2\chi)|^2 = \left|(-1)-\frac{-4\chi+i\kappa}{-4\chi-i\kappa}\right|^2 \approx 4 \end{equation}
With $z\equiv4\chi+i\kappa$, the distinguishability between the two even parity states is: \begin{align} \begin{split}
D_{11,00} & =|S(-2\chi) - S(2\chi)|^2 = \left|\frac{4\chi+i\kappa}{4\chi-i\kappa}-\frac{-4\chi+i\kappa}{-4\chi-i\kappa}\right|^2 \\
& =\left|\frac{z}{\bar{z}}-\frac{\bar{z}}{z}\right|^2= \left| \frac{z^2-\bar{z}^2}{|z|^2} \right|^2= \left| \frac{(z+\bar{z})(z-\bar{z})}{|z|^2} \right|^2\\
& =\left(\frac{(8\chi)(2\kappa)}{16\chi^2+\kappa^2}\right)^2 \approx \left(\frac{\kappa}{\chi}\right)^2 \end{split} \end{align}
From these equations, we get the following relative dephasing ($\Gamma$) and measurement rates ($\Gamma^m$) between states of different parity and states of even parity: \begin{equation} \frac{\Gamma_{01,00}}{\Gamma_{11,00}} = \frac{\Gamma^m_{01,00}}{\Gamma^m_{11,00}} \approx \frac{4\chi^2}{\kappa^2} \end{equation}
\subsection{Dynamic Dephasing}
When the resonator is not at steady state, one can have significantly increased dephasing rates after a parity flip. Here we will consider the effect of a bit flip error taking an odd parity qubit state to an even parity state while the parity measurement is on. In this case, the measurement tone is on resonance with the cavity and the cavity field will initially be in a steady state $\alpha_0$. When the qubit parity is flipped from odd to even, the cavity evolves as two copies, one for each even parity basis state ($\alpha_{00}$ and $\alpha_{11}$). As a simplifying approximation, we assume the measurement tone is turned off at the moment the parity changes as to capture just the transient dynamics. There are two equivalent methods\cite{PhysRevA.77.012112} to calculate the net dephasing $\zeta$. The first can be obtained by integrating the rate at which information leaves the cavity, $\Gamma_{\phi}^{m} = \frac{\kappa}{2}|\alpha_{00}-\alpha_{11}|^2$. The second can be obtained by integrating the rate at which the cavity dephases the qubit, $\Gamma_{\phi}=4\chi \operatorname{Im}[\alpha_{00}\alpha_{11}^*]$, with $4\chi$ being the frequency difference between the $\ket{00}$ resonance and the $\ket{11}$ resonance. Here we use the second method to simplify the calculation. We work in the rotating frame of the odd-parity resonance and define $k\equiv \kappa/2-2i\chi$ to get two cavity equations, one associated with each basis state: \begin{align} \begin{split} \dot{\alpha}_{00} & = \left(2\chi i-\frac{\kappa}{2}\right)\alpha_{00} \\ \dot{\alpha}_{11} & = \left(-2\chi i-\frac{\kappa}{2}\right)\alpha_{11} \end{split} \end{align} \begin{align} \begin{split} \alpha_{00}(t) = \alpha_{0}e^{-kt} \\ \alpha_{11}(t) = \alpha_{0}e^{-\bar{k}t} \end{split} \end{align}
\begin{align} \begin{split} \zeta & =\int_0^{\infty} 4\chi \operatorname{Im}\left[\alpha_{00}\alpha_{11}^*\right]dt = 4\chi \operatorname{Im}\left[\int_0^{\infty}\alpha_{00}\alpha_{11}^* dt \right] \\
& =\left|\alpha_0\right|^2 4\chi \operatorname{Im}\left[\int_0^{\infty} e^{-2kt} dt \right] \\
& =\left|\alpha_0\right|^2 4\chi \operatorname{Im}\left[ \frac{1}{2k} \right] = \left|\alpha_0\right|^2 4\chi \operatorname{Im}\left[\frac{2\bar{k}}{|2k|^2}\right] \\
& = \left|\alpha_0\right|^2 \frac{16\chi^2}{\kappa^2+16\chi^2} \approx |\alpha_0|^2 \end{split} \end{align}
Therefore, the magnitude of the final coherence between $\ket{00}$ and $\ket{11}$, $|\rho_{00,11}^f |$, will be dephased from the initial coherence between $\ket{01}$ and $\ket{10}$, $|\rho_{01,10}^i|$ : \begin{equation}
\left|\rho_{00,11}^f \right|= e^{-\zeta}\left| \rho_{01,10}^i\right| = e^{-|\alpha_0|^2}\left|\rho_{01,10}^i\right| \end{equation}
\onecolumngrid \renewcommand{Extended Data Figure}{Extended Data Figure} \setcounter{figure}{0}
\begin{figure}
\caption{ Cryogenic wiring diagram. The Josephson parametric amplifiers (JPAs) operate in reflection, and additionally have off chip coils not shown. The JPAs also provide narrow-band gain, so when the readout chains are combined at room temperature, the combined noise at each cavity frequency is dominated by the noise amplified by that cavity's JPA. Each superconducting coil has its leads connected by a small piece of copper wire on the sample box, forming a low frequency $(<1 \operatorname{Hz})$ RL filter with the coil. The room temperature wiring is also shown, but with linear elements (attenuators, amplifiers, filters, isolators) removed. }
\label{fig:fridge}
\end{figure}
\begin{figure}\label{fig:SM_DFE}
\end{figure}
\begin{figure}\label{fig:SM_T1}
\end{figure}
\end{document} | arXiv |
Genomic prediction with epistasis models: on the marker-coding-dependent performance of the extended GBLUP and properties of the categorical epistasis model (CE)
Johannes W. R. Martini1,
Ning Gao1,2,
Diercles F. Cardoso1,3,
Valentin Wimmer4,
Malena Erbe1,5,
Rodolfo J. C. Cantet6 &
Henner Simianer1
BMC Bioinformatics volume 18, Article number: 3 (2017) Cite this article
Epistasis marker effect models incorporating products of marker values as predictor variables in a linear regression approach (extended GBLUP, EGBLUP) have been assessed as potentially beneficial for genomic prediction, but their performance depends on marker coding. Although this fact has been recognized in literature, the nature of the problem has not been thoroughly investigated so far.
We illustrate how the choice of marker coding implicitly specifies the model of how effects of certain allele combinations at different loci contribute to the phenotype, and investigate coding-dependent properties of EGBLUP. Moreover, we discuss an alternative categorical epistasis model (CE) eliminating undesired properties of EGBLUP and show that the CE model can improve predictive ability. Finally, we demonstrate that the coding-dependent performance of EGBLUP offers the possibility to incorporate prior experimental information into the prediction method by adapting the coding to already available phenotypic records on other traits.
Based on our results, for EGBLUP, a symmetric coding {−1,1} or {−1,0,1} should be preferred, whereas a standardization using allele frequencies should be avoided. Moreover, CE can be a valuable alternative since it does not possess the undesired theoretical properties of EGBLUP. However, which model performs best will depend on characteristics of the data and available prior information. Data from previous experiments can for instance be incorporated into the marker coding of EGBLUP.
Genomic prediction aims at forecasting qualitative or quantitative properties of individuals based on known genetic information. The genetic information can for instance be given by single-nucleotide-polymorphisms (SNPs) or other kinds of genetic data of individual animals, plant lines or humans. Applied to animals and plants, genomic prediction is of central importance for breeding within the concept of genomic selection [1, 2]. Moreover, genomic prediction can also be used in medicine or epidemiology for risk assessment or prevalence studies of (partially) genetically determined diseases (e.g. [3]). One of the standard approaches for genomic prediction of quantitative traits is based on a linear regression model in which the phenotype is described by a linear function of the genotypic markers. In more detail, the standard additive linear model is defined by the equation
$$ \mathbf{y} = \mathbf{1}\mu + \mathbf{M} \boldsymbol{\beta} + \boldsymbol{\epsilon} $$
where y is the n×1 vector of phenotypes of the n individuals, 1 the n×1 vector with each entry equal to 1, μ the fixed effect and M the n×p matrix giving the p marker values of the n individuals. Moreover, β is the p×1 vector of unknown marker effects and ε a random n×1 error vector with \(\epsilon _{i} {\overset {i.i.d.}{\sim }}\mathcal {N}(0,\sigma _{\epsilon }^{2})\). Since the number of markers p is typically much larger than the number of individuals n, the additional assumption that \(\beta _{j} \overset {i.i.d.}{\sim } \mathcal {N}(0,\sigma _{\beta }^{2})\) is usually made (and all random terms together are considered as stochastically independent). In particular, using an approach of maximizing the density of a certain distribution [4], this assumption allows us to determine the penalizing weight in a Ridge Regression approach which is known as ridge regression best linear unbiased prediction (RRBLUP) and which is fully equivalent to its relationship matrix-based counterpart genomic best linear unbiased prediction (GBLUP)1 [5, 6]. The answer to the question which type of marker coding is appropriate in M depends on the combination of the type of genotypic marker and ploidy of the organism dealt with. For instance, if haploid organisms are considered or presence/absence markers are used, a possible coding for the j-th marker value of the i-th individual M i,j is the set {0,1}. Counting the occurrence of an allele of a diploid organism, the sets {0,1,2} or {−1,0,1}, or rescaled variants can be used. If the marker effects β and the fixed effect μ are predicted/estimated as \(\boldsymbol {\hat {\beta }}\) and \(\hat {\mu }\) on the basis of a training set, the expected phenotypes of individuals from a test set, which were not used to determine \(\boldsymbol {\hat {\beta }}\) and \(\hat {\mu }\), can be predicted by using their marker information in Eq. (1) with \(\hat {\mu },\boldsymbol {\hat {\beta }}\). We will call the difference between the predicted expected phenotype and the estimated fixed effect the predicted genetic value. For the purely additive model of Eq. (1) and a diploid organism with possible genotypes aa, aA and AA for locus j, the choice of how to translate these possibilities into numbers was reported not to affect the predictive ability notably, as long as the difference between the coding of aa and aA is the same as between aA and AA and equal for all markers [5, 7–9]. However, an extension of the additive model, which we call the extended GBLUP model (EGBLUP) [10, 11]
$$ y_{i} = \mu + \sum\limits_{j=1}^{p} M_{i,j} \beta_{j} + \sum\limits_{k=1}^{p}\sum\limits_{j=k}^{p} M_{i,j}M_{i,k} h_{j,k} + \epsilon_{i}, $$
has been shown to exhibit strong coding dependent performance [12, 13]. Here, \(h_{j,k}\overset {i.i.d.}{\sim } \mathcal {N}\left (0,{\sigma ^{2}_{h}}\right)\) is the pairwise interaction effect of markers j and k and all other variables as previously defined (all terms stochastically independent). Compared to Eq. (1), this model additionally incorporates pairwise products of marker values as predictor variables and thus allows us to model interactions between markers. Moreover, the interaction of a marker with itself gives a possibility to model dominance effects (see e.g. [11, 14–16]). The epistasis model of Eq. (2) and some variations with restrictions on which markers can interact have been the main object of investigation in several publications and models incorporating epistasis have been viewed as potentially beneficial for the prediction of complex traits [10, 11, 17–19], but a marker coding dependent performance was observed [12, 13].
In this work, we investigate how the marker coding specifies the effect model for markers with two or three possible values and show how we can find the marker coding for an a priori specified model. We discuss advantages and disadvantages of different coding methods and investigate properties of alternative linear models based on categorical instead of numerical dosage variables. In particular, we show how to represent these models as genomic relationship matrices. Finally, we compare the predictive abilities of different epistasis models on simulated and publicly available data sets and demonstrate a way of using the coding-dependent performance of EGBLUP to incorporate prior information.
Data sets used for assessing predictive ability
Simulated data
A population with 10 000 bi-allelic markers spread across five chromosomes was simulated, using the QMSim software [20]. The size of the first chromosome was 140 centimorgan (cM) with 3 500 markers. Chromosomes 2 to 5 had a size of 110 cM (2 750 markers), 80 cM (2 000 markers), 50 cM (1 250 markers) and 20 cM (500 markers), receptively. In order to allow mutations and linkage disequilibrium establishment, a historical population was simulated with 5 000 individuals (2 500 males and 2 500 females) with random mating for 1 000 generations with constant population size and with a replacement rate of 0.2 for males and females. Then the population size was reduced to 1 000 individuals for 20 additional generations (generation 1 001 to 1 020). The simulated mutation rate was 2.5·10−5.
We used this simulated genotypes as basis and modeled three different types of genetic architecture (purely additive, purely dominant and purely epistatic), each with a varying number of quantitative trait loci (QTL) on top. We chose these types of genetic architecture, without additive effects in the dominance and epistasis scenarios, to make the three scenarios as different as possible. To model the phenotype, out of the 10 000 markers, 200 were drawn randomly from each of the five chromosomes to define in total 1 000 QTL for additive or dominance effects. For the purely additive scenario, the 1 000 additive effects were drawn independently from a \(\mathcal {N}(0,1)\) distribution. For the first additive trait A1, 10 out of the 1 000 QTL were drawn and the genetic values of all individuals were calculated according to the effects of these 10 loci. To define a broad sense heritability of 0.8, the genetic values were standardized to mean 0 and variance 1 and individual errors were drawn from a \(\mathcal {N}(0,0.25)\) distribution. Having added these individual errors to the genetic values, these phenotypes were again standardized to mean 0 and variance 1. For the second trait A2, additional 90 QTL were drawn from the initial 1 000 to give in total 100 QTL for this trait including the QTL of trait A1 with their corresponding effects. Analogously, for A3, all initially drawn 1 000 QTL were used. The standardization procedure was identical to the one previously described for A1. For the comparison of genomic prediction with different relationship models, these 1 000 markers were removed. The relationship matrices were based on the remaining 9 000 markers.
For the dominance scenario D1 (10 QTL), D2 (100 QTL) and D3 (1 000 QTL), we used the same QTL positions as for A1, A2, and A3, respectively, but simulated \(\mathcal {N}(0,1)\)-distributed dominance effects. The standardization procedure to a broad sense heritability of 0.8 was carried out as described before.
For the epistasis traits E1, E2 and E3, 1 000, 10 000 or 100 000 pairs of markers were drawn randomly and for each draw, one of the nine possible configurations of the pair was randomly chosen to have an \(\mathcal {N}(0,1)\)-distributed effect. For instance, having drawn the marker pair j,k, only the configuration (M i,j ,M i,k )=(0,2) was chosen to have an effect, which again was drawn randomly. This was done independently for each trait, which means trait E2 does not necessarily share causal combinations of markers with trait E1. The phenotypes were standardized as described above. Note, that the markers involved in causal combinations were not removed here, since in expectation, every marker is somehow involved in the phenotype of trait E2 and E3.
We repeated this whole procedure, including the simulation of the genotypes, 20 times and compared the different models by their average predictive ability across the 20 repetitions. The simulated data can be found in Additional file 1 of this publication.
Wheat data
The wheat data which we used to compare different methods was published by Crossa et al. [21]. The 1279 DArT markers of 599 CIMMYT inbred wheat lines indicate whether a certain allele is present (1) or not (0). The phenotypic data describes standardized records of grain yield under four environmental conditions.
Mouse data
The mouse data set we used was published and described by Solberg et al. [22] and Valdar et al. [23], and was downloaded from the corresponding website of the Wellcome Trust Centre for Human Genetics. The physical map of single nucleotide polymorphisms (SNPs) was updated to the latest version of the mouse genome (Mus musculus, assembly GRCm38.p4) with the biomaRt R package [24, 25]. Only SNPs mapped to the GRCm38.p4 were used for further analysis. For the remaining markers, the ratio of missing marker values was rather low (0.33%) and we performed a random imputation. The nucleotide coded genotypes were translated to a {0,1,2} coding, where 0 and 2 denote the two homozygous and 1 the heterozygous genotype. SNPs with minor allele frequency (MAF) smaller than 0.01 were excluded from the dataset. Imputation, recoding, and quality control of genotypes were carried out with the synbreed R package simultaneously [26]. A number of 9265 SNPs remained in the dataset for further analysis. We only used individuals with available records for all considered traits for further analysis, which reduced the number of individuals to 1 298. We focused on the provided pre-corrected residuals of 13 traits from which fixed effects of trait-specific relevant covariates such as sex, season, month, have already been subtracted. A detailed description of the traits can be found on the corresponding sites of the UCL. Moreover, the data resulting from quality control and filtering as well as the corrected phenotypes of the traits we used can be found in Additional file 1.
Genomic relationship based prediction and assessment of predictive ability
We used an approach based on relationship matrices for genomic prediction. The underlying concept of this approach is the equivalence of marker effect-based and genomic relationship-based prediction ([5, 10, 11]). Given the respective relationship matrix, the prediction is performed by Eq. (3) (for a derivation of this equation see the supporting information of [11]):
$$ \begin{aligned} {}\left(\begin{array}{c} \hat{\mathbf{g}}_{train}\\ \hat{\mathbf{g}}_{test} \end{array}\right) & =\left[\mathbf{T}_{train} - s^{-1} \left(\begin{array}{cc} \mathbf{J}_{s \times s} & 0 \\ 0& 0 \end{array}\right)\right. \\ & \quad \left. + \sigma_{\epsilon}^{2} \left(\frac{1}{\sigma^{2}_{\beta}} \mathbf{G}^{-1}\right) \right]^{-1} \!\left(\! \left(\begin{array}{c} \mathbf{y}_{train} \\ 0 \end{array}\right) \!- \left(\begin{array}{c} \mathbf{1}_{s} \bar{y}_{train}\\ 0 \end{array} \right)\! \right) \end{aligned} $$
The matrix G is the central object denoting the genomic relationship matrix of the respective model. The variables \(\hat {\mathbf {g}}_{i}\) are the predicted genetic values (expected phenotype minus the fixed effect \(\hat {\mu }\)) of the respective set (training or test set). Moreover, s is the number of genotypes in the training set, 1 s is the vector of length s with each entry equal to 1, J s×s is the analogous s×s matrix with each entry equal to 1 and \(\bar {y}_{train}\) is the empirical mean of the training set. Here, T train denotes the diagonal matrix of dimension n with 0 on the diagonal at the positions of the test set genotypes, and 1 for the training set individuals.
To assess the predictive ability of different models, we chose a test set consisting of ∼ 10% of the total number of individuals (100, 60, or 130 for the simulated, the wheat and the mouse data, respectively). We then used the remaining individuals as a training set and predicted the genetic values for all individuals using Eq. (3). The variance components \(\sigma _{\epsilon }^{2}\) and \(\sigma _{\beta }^{2}\) were estimated from the training set using version 3.1 of the R package EMMREML [27]. The relationship matrix relating the genotypes of the training set was used to estimate the variance components based on the phenotypes of the training set only. The variance components were then used with the complete relationship matrix for the prediction of the genetic values of all individuals in Eq. (3). This procedure was repeated 200 times, with independently drawn test sets. The average correlation r between observed and predicted mean phenotypes of the test set was used as a measure of predictive ability. A description of how the different effect models can be translated into relationship matrices is given in the results. For the Gaussian kernel, we used the bandwidth parameter \(b=2q_{0.5}^{-1}\), with q 0.5 the median of all squared Euclidean distances between the individuals of the respective data. For the simulated data which consisted of 20 independent data sets, we present the average predictive ability and the average standard error of the mean. For the wheat and the mouse data, we used Tukey's 'Honest Significant Difference' test to contrast the performance of the different prediction methods (TukeyHSD() and lm() of R [28]).
Incorporation of prior information by marker coding
As described above, the data we used offers records of different traits or trait ×environment combinations of the same individuals. We will illustrate that the coding-dependent performance of EGBLUP can also be used to incorporate a priori information into the model by choosing the coding for each interaction with already provided data and by using the corresponding relationship matrix for prediction under altered environmental conditions or for a correlated trait. We used for the wheat data the following procedure:
We predicted all the interactions \(\hat {h}_{k,l}\) for a given trait in a given environment, under the use of the {0,1} coding originally provided by Crossa et al. [21] (as described by Martini et al. [11]).
We changed the "orientation" of all markers at once by substituting 0 by 1, and 1 by 0 and predicted all interactions \(\tilde {h}_{k,l}\) under the use of the altered coding.
If the ratio of \( \left |\frac {\hat {h}_{k,l}}{\tilde {h}_{k,l}} \right |\) was greater than or equal to 1, we assumed that the original orientation provided by the data set describes the respective interaction better than the alternative coding.
We then calculated a relationship matrix for each interaction individually by
$$\mathbf{G}_{k,l} = \mathbf{\left(M_{\bullet, k} M_{\bullet, k}^{\prime} \right) \circ \left(M_{\bullet, l} M_{\bullet, l}^{\prime} \right)} $$
with M ∙,k denoting the n×1 vector of marker data of locus k for all individuals in the respective coding which seems to fit the interaction better according to 3) (see [11, 29]). Here, ∘ denotes the Hadamard product.
The overall relationship matrix was then defined by \(\mathbf {G}= \sum \limits _{k=1}^{p} \sum \limits _{l \geq k}^{p}\mathbf {G}_{k,l}\).
We used the data of each environment to calculate an optimally coded relationship matrix for this environment, which was used afterwards for predicting phenotypes in the other environments. The underlying heuristic of step 3) is that a small effect means that the interaction is less important in the respective coding. If the underlying effect model defined by the coding does not capture the data structure, the estimated effect should be close to zero. However, if the effect of a combination is important to describe the phenotype distribution, a larger effect should be assigned (see also Example 1, where the estimated effect is 0, if the underlying parameterization cannot describe the present effect distribution).
For the mouse data, we used the 13 considered traits to construct a relationship matrix for each of them. Each relationship matrix was afterwards used for prediction within the data of the twelve other traits. The two different codings which were compared here, were the {0,1,2} coding based on the imputed originally provided data and its inverted version with 0 and 2 permuted.
In the following, we will highlight aspects of the behavior of the additive effect model of Eq. (1) when the marker coding is altered. These properties of the additive model will afterwards be compared to those of the epistasis model of Eq. (2).
All relationship matrices will be assumed to be positive definite and thus invertible. Mathematical derivations of the illustrated properties can be found in Additional file 2.
Properties of GBLUP
We start with the effect of translations of the coding, that is the addition of a number p j to the initially chosen marker coding of marker j.
(Translation-invariance of GBLUP) Let P denote a vector whose entries give the arbitrary translations p j of the coding of the locus j. Moreover, let the ratio of \(\sigma _{\epsilon }^{2}\) and \(\sigma _{\beta }^{2}\) be known and unchanged if the marker coding is translated. Let \(\boldsymbol {\hat {\beta }}\) and \(\hat {\mu }\) denote the predicted / estimated quantities if the initial coding M is used in the Mixed Model Equation approach of Eq. (1) and let \(\boldsymbol {\tilde {\beta }}\) and \(\tilde {\mu }\) denote the corresponding quantities if the translation \(\tilde {\mathbf {M}}:=\mathbf {M}-\mathbf {1}\mathbf {P'}\) is used instead of M. Then the following statements hold:
\(\tilde {\mu }=\hat {\mu } + \mathbf {P'} \boldsymbol {\hat {\beta }}\)
\(\boldsymbol {\tilde {\beta }}=\boldsymbol {\hat {\beta }}\)
The prediction of the expected phenotype of each genotype is independent of whether M or \(\tilde {\mathbf {M}}\) is used.
The statement of Property 1 has already been discussed in literature [5, 7–9], and we will present a mathematical derivation based on the Mixed Model Equations in Additional file 2. The proof will be a blueprint for the derivation of other properties based on the Mixed Model Equations which can also be found in Additional file 2. Descriptively, we can see the presented invariance with respect to translations the following way: If we change the coding to \(\tilde {\mathbf {M}}:=\mathbf {M}-\mathbf {1P'}\), then \(\tilde {\mathbf {M}}\), \(\tilde {\mu }:=\hat {\mu } + \mathbf {P' \boldsymbol {\hat {\beta }}}\) and \(\boldsymbol {\tilde {\beta }}:=\boldsymbol {\hat {\beta }}\) will fit the phenotypes the same way as M, \(\hat {\mu }\) and \(\boldsymbol {\hat {\beta }}\) do. Thus, the prediction of the marker effects and consequently the prediction of the expected phenotypes of individuals will not be affected by the change of coding as long as the method of evaluating the "goodness of fit", that is the penalizing weight in a Ridge Regression approach remains unchanged. For this reason, it is important to note here that we made the precondition that the ratio of the variance components, which defines the penalty for effect size, will not be changed. This guarantees that the method of how to quantify the "goodness of fit" remains the same. In practice this may not exactly be the case if the vector P has non-identical entries, that is if the translation of the coding is not equal for all loci, since the variance components are usually estimated from the same data and the translation may have an effect on this estimation. However, this effect has been assessed as being negligible in practice [9]. To assess this problem from a theoretical point of view, without preconditions on the changes of \({\sigma ^{2}_{i}}\), the method for determining the variance components has to be taken into account to see whether a change in the marker coding has an influence on the ratio of the determined variance components. The next property considers the effect of rescaling the given marker coding.
Let \(\boldsymbol {\hat {\beta }}\), \(\hat {\mu }\), \(\boldsymbol {\tilde {\beta }}\) and \(\tilde {\mu }\) denote the quantities as defined in Property 1 with \(\tilde {\mathbf {M}}:=c \mathbf {M}\) for a c≠0. Moreover, let \(\sigma _{\epsilon }^{2}\) and \(\sigma _{\beta }^{2}\) for M be known and let the variance components used for the Ridge Regression approach based on \(\tilde {\mathbf {M}}\) fulfill \(\frac {\tilde {\sigma }_{\epsilon }^{2}}{\tilde {\sigma }_{\beta }^{2}}=c^{2}\frac {\sigma ^{2}_{\epsilon }}{\sigma _{\beta }^{2}}\). Then the following statements hold:
\(\tilde {\mu }=\hat {\mu } \)
\(\boldsymbol {\tilde {\beta }}=c^{-1}\boldsymbol {\hat {\beta }}\)
An important aspect of Property 2 is the precondition that the ratio of the variance components is adapted. In practice, when \(\sigma _{\beta }^{2}\) is estimated, we can assume that this circumstance will approximately be given, however, we have to highlight again that this also depends on the method of how the variance components are determined.
Epistasis models of shape of Eq. (2)
The full EGBLUP model of Eq. (2) adds interaction terms of shape h j,k M i,j M i,k to the additive model of Eq. (1). We will focus on the properties of these additional terms in the following. Evidently, the product structure of the additional covariates generates a dependence of the underlying effect model on the marker coding. In particular, the genotype coded as zero has a special role. If M i,j equals zero, the whole term h j,k M i,j M i,k will be equal to zero, independently of the values of h j,k and M i,k . Thus, the model has the implicit assumption that a certain set of combinations do not interact. The marker coding decides which interactions are different from zero a priori and which combinations are clustered. For instance, for the coding {−1,0,1} for the genotypes {a a,a A,A A} of a diploid organism, any interaction with a heterozygous locus will be zero, whereas the interactions with the homozygous locus aa will be zero if the coding {0,1,2} is used. Table 1 illustrates the differences of the two different standard codings ({−1,0,1} vs. {0,1,2}). Here we see that the marker coding {0,1,2} implies that the effect is monotonously increasing (or decreasing if h j,k is negative) with the distance from the origin, whereas the coding {−1,0,1} gives a different topology by only giving weight to the double homozygous. It is not obvious which coding is to be preferred and which reasonable assumptions on the effect of pairs can be made. In the following, we will discuss theoretical properties of the model induced by the marker coding.
Table 1 Comparison of the interaction effects which are given implicitly by the marker coding {−1,0,1} (left) and {0,1,2} (right) in the interaction terms of EGBLUP. Each entry has to be multiplied with the interaction effect h j,k
As a first important observation, we note that the codings {−1,0,1} and {0,1,2} are translations of each other. Their very different interaction effect topologies illustrate that the epistasis model is not invariant with respect to translations. This fact that translations modify the model also makes obvious that by subtracting the matrix 1 P ′ with P containing the allele frequencies of the respective marker, which is the standard normalization in the additive model [6], we will change the coding for the markers according to their frequencies and thus implicitly use different effect models for each pair of loci. We do not see a theoretical basis for this discrimination in an infinitesimal model without additional prior knowledge and therefore will consider mainly models which treat markers equally. Moreover, as gene frequencies are sometimes poorly estimated and very influential, avoiding their use seems to be appealing.
As illustrated, the epistasis model is not invariant with respect to translations, but we show now that the previously described invariance with respect to rescaling persists also for the epistatis model.
Let \(\boldsymbol {\hat {\beta }}\), \(\hat {\mu }\), \(\boldsymbol {\tilde {\beta }}\) and \(\tilde {\mu }\) denote the quantities as defined in Property 1 with \(\tilde {\mathbf {M}}:=c \mathbf {M}\) for a c≠0. Moreover, let \(\boldsymbol {\hat {h}}\) and \(\boldsymbol {\tilde {h}}\) denote the corresponding predictions for the interaction effects. Let \(\sigma _{\epsilon }^{2}\), \(\sigma _{\beta }^{2}\), \({\sigma _{h}^{2}}\) for M be known and let the variance components used for the Ridge Regression approach based on \(\tilde {\mathbf {M}}\) fulfill \(\frac {\tilde {\sigma }_{\epsilon }^{2}}{\tilde {\sigma }_{\beta }^{2}}=c^{2}\frac {\sigma ^{2}_{\epsilon }}{\sigma _{\beta }^{2}}\) and \(\frac {\tilde {\sigma }_{\epsilon }^{2}}{\tilde {\sigma }_{h}^{2}}=c^{4}\frac {\sigma ^{2}_{\epsilon }}{{\sigma _{h}^{2}}}\). Then the following statements hold:
\(\boldsymbol {\tilde {h}}=c^{-2}\boldsymbol {\hat {h}} \)
A formal derivation of this property based on the Mixed Model Equations can be found in the Additional file 2, but the statements are also plausible if we follow the descriptive argumentation for the invariance of the additive model: If \(\hat {\mu }\), \(\boldsymbol {\hat {\beta }}\) and \(\boldsymbol {\hat {h}}\) fit the phenotypic data best when marker matrix M is used, \(c^{-1}\boldsymbol {\hat {\beta }}\) and \(c^{-2}\boldsymbol {\hat {h}}\) will fit the phenotypic data the same way if M is substituted by \(\tilde {\mathbf {M}}\) in Eq. (2) (for any constant c≠0). The important precondition is that the penalizing weight, which defines which fit is "best", is adapted. A question that might come up in the context of Properties 2 and 3 is whether we could also multiply each coding for locus j with its own constant c j ≠0, similar to what we had for Property 1 and vector P. A problem that will appear here is that the variance of the marker effects will not be changed uniformly and thus, we cannot simply adapt the variance components to cancel the impact of rescaling. An individual rescaling and thus weighting of each marker [30], as well as a completely individual coding of each genotype of each locus, without the side conditions that the differences in the coding of the heterozygous and the two homozygous genotypes are identical across all loci or at least symmetric for each locus [12, 13], indeed has an impact on the predictive ability of the models, in particular also on that of GBLUP. However, the variance components \({\sigma _{i}^{2}}\) can be globally adapted to cancel the impact of a non-uniform rescaling of the marker coding, in case that some columns of M are multiplied with c and the others with −c (due to the assumption of all effects being symmetrically distributed around mean zero). An adapted sign of the effects also allows the predicted effect model to remain unchanged.
Permuting the role of the alleles at locus j . Let locus j have the possible allele configurations aa, aA and AA. The prediction performance of GBLUP is unaffected by the choice of whether the allele variant a or A is counted, since we can express a permutation of the initial coding {0,1,2} by a translation by −2 and a multiplication of the coding by −1.
Obviously, this argumentation cannot be used for the epistasis model, since we do not have the possibility to translate the marker coding. This fact raises the question under which circumstances the epistasis EGBLUP model is unaffected by a permutation of the role of the allele variants.
Let us consider locus j with alleles a and A and locus k with alleles b and B (of a diploid organism). Let us use the same coding for both loci and let the three variants of aa, aA and AA be coded by three different numbers M aa <M aA <M AA (or M aa >M aA >M AA ). The only coding for the epistasis terms, whose corresponding effect model on the tuples
$$ \left\{ (j,k) | j \in \{aa,aA,AA\}, k \in\{bb,bB,BB\} \right \} $$
is invariant with respect to a permutation of the role of allele a and A satisfies −M aa =M AA and M aA =0. Analogously, for markers with only two possible values, the coding has to satisfy −M a =M A .
Property 4 is of central theoretical importance since it implies that the only coding for {0,1} marker in EGBLUP, which is invariant with respect to a permutation of the meaning of 0 and 1 is the coding {−c,c} (c≠0). Moreover, if EGBLUP shall possess this reasonable property for markers with three possible values, we have to use the coding {−c,0,c}. We will give an example to illustrate why this property is important for determining marker effects and thus why it may also be important for the overall predictive ability of the model.
Let us consider markers with two possible variants and let us assume that for each pair of markers, the correct underlying weights of the combinations is given by a coding as {0,1}. We use a {0,1} coding, but we do not know which variants of the two loci have to be coded as 1 to capture the real effect distribution. We assume that we decide which allele is coded as zero, by drawing independently from a Bernoulli-distribution with p=0.5 for each marker. To see how good the real underlying weight distribution is captured, we measure the quadratic loss between the best possible fit and the real underlying weights. Let the coding
$$ \begin{array}{c | c | c} & a & A \\ b & 0& 0 \\ B & 0 & 1\\ \end{array} $$
be the correct underlying effect distribution, with the corresponding underlying interaction effect equal to 1 (the problem remains the same if the underlying interaction effect is multiplied with any number c≠0). With a probability of 0.25, we will code both markers j and k correctly and minimize the distance to zero by predicting \(\hat {h}_{j,k}=1\). However, with a probability of 0.75, we will make a mistake and choose an incorrect orientation, which means an incorrect underlying parametric model, such as
$$ \begin{array}{c|c|c} & a & A \\ b & 1 \cdot \; h_{j,k} & 0 \\ B & 0 & 0\\ \end{array} $$
In this situation, we can determine the optimally fitting interaction \(\hat {h}_{j,k}\), which describes the distribution of Eq. (4) best, when model Eq. (5) is used, by minimizing the quadratic Euclidean distance between both effect distributions. In more detail, using a minimal quadratic loss means we have to find an \(\hat {h}_{j,k}\) which minimizes the quadratic distance between the matrices of Eq. (4) and Eq. (5):
$$ (1h_{j,k}-0)^{2}+(0-0)^{2}+(0-0)^{2}+(0-1)^{2} $$
$$h_{j,k}^{2}+1. $$
Thus, the optimal \(\hat {h}_{j,k}\) minimizing Eq. (6) is 0 and the expected quadratic loss when the right coding with unknown orientation is used, is 0.25·0+0.75·1=0.75.
Analogously, if we use the coding {−1,1} instead of Eq. (5), we will obtain the quadratic distance
$${}3(h_{j,k}-0)^{2} + (h_{j,k}-1)^{2} \qquad \text{or} \qquad 3(h_{j,k}-0)^{2} + (h_{j,k}+1)^{2} $$
each with probability 0.5, depending on whether −1 or +1 coincides with the 1 of the real underlying effects. Consequently, the minimum quadratic distance is 0.75 with probability 1, for \(\hat {h}_{j,k}= \pm 0.25\). Thus, in this example, even though the coding {−1,1} specifies a model which is surely wrong, the average quadratic loss is equal to the situation in which we know the exact shape of the effect distribution but not its orientation. If the real underlying effect distribution deviates from the {0,1} coding of Eq. (4), the possibility to adapt the orientation might be even more important.
Example 1 illustrated that the expected quadratic loss of the estimated marker-pair weights is equal for the codings {−1,1} and {0,1} even in the case that the underlying effects are a version of the latter one but with unknown orientation. Moreover, we can observe the following: Let us assume that the real underlying interactions (j,k),(j,l) and (k,l) of the three loci j,k,l are described by certain {0,1}-codings, meaning that one certain configuration has an interaction effect but the others do not. Given the underlying effects, we can adapt the coding of j,k and l by considering the effects of the pairs (j,k),(j,l). However, then the effect distribution within the model is also determined for the pair (k,l), because the marker coding has already been fixed. This configuration does not necessarily describe the interaction of (k,l) well. This fact illustrates that due to the way of how interactions are incorporated into the model in EGBLUP, the model with an asymmetric coding lacks a full flexibility to adapt to any situation. This problem does not appear with the symmetric coding, since the model is independent of the decision which allele is coded as ±1. However, there are also good reasons for choosing other types of coding. Firstly, it is not clear whether the effect that we have illustrated on the level of marker effects and quadratic loss, also translates to the level of prediction of genetic values. In the latter approach, all effects are predicted simultaneously and thus errors of individual effects can cancel out in the sum. Secondly, from a biological point of view, the symmetric coding seems inadequate: Let us consider markers with two variants and let the two loci j and k have the possible variants a,A and b,B, respectively. The symmetric coding {−1,1} assigns the weight 1h j,k to the combinations (a,b) and (A,B), meaning that the most distant genotypes, which do not share any allele, are treated as being equal in the model. Thus, overall, it is not clear which coding will be most appropriate in general. Especially in situations in which additional information on the nature of the marker or the biology of the trait is available, this information may be used to specify the effect model. In the next paragraph, we illustrate how much freedom the marker coding gives to specify the model.
Finding the marker coding for an a priori specified model. Let us consider a model with identical marker coding M aa , M aA and M AA for each locus. Then the weights in the model are given by
$$\begin{array}{@{}rcl@{}} a_{1,1}=M_{aa}^{2} & a_{1,2}=M_{aa}M_{aA} & a_{1,3}=M_{aa}M_{AA}\\ a_{2,2}=M_{aA}^{2} & a_{2,3}=M_{aA}M_{AA} & a_{3,3}=M_{AA}^{2}. \end{array} $$
If we want to predefine the weights a r,s and calculate a corresponding coding, we see that not all choices of weights can be translated into a coding for the epistasis model of Eq. (2) since contradictions can arise. However, the following statement holds:
Let three weights a r,s of Eq. (7) which include the three variables M aa , M aA , M AA in at least one weight a r,s be given by arbitrary nonzero numbers. Then the marker codings as well as the other weights are determined up to their signs.
Categorical effect models
In the following, we discuss categorical effect models in which we do not treat the marker data as numerical dosage, but as categorical variables. The goal is to build an epistasis model without the undesired properties of EGBLUP which have been described previously. We model the effects of allele combinations as being independently drawn from a Gaussian distribution with mean zero. For instance, for an additive marker effect model, the effects of aa, aA and AA are independently originating from the same distribution. For the analogous epistasis model, the effect of each combination of the alleles of two loci is drawn independently from the same distribution. We will introduce dummy {0,1} variables to indicate which allele configuration is present and thus inflate the number of variables in our model. The important fact to notice in this context is that we can use a relationship matrix approach for genomic prediction (see "Methods") and thus do not need to handle the high number of variables. This procedure also reduces computation time compared to the effect based approach. All considered effects β j of the variables are assumed to come from the same distribution: \(\beta _{j}\overset {i.i.d.}{\sim } \mathcal {N}(0,\sigma _{\beta }^{2})\).
A categorical marker effect model (CM) The underlying concept of this model is to code the configurations a a,a A,A A of locus j as three different variables. The effect of each genotype is estimated on its own. The assumption of a constant allele substitution effect, that is that the effect of AA equals twice the effect of A, which is made in the additive numerical GBLUP model, is not made here (see Fig. 1). We translate the genotypes (a a,a A,A A) which can be found at locus j to ((0,0,1),(0,1,0),(1,0,0)). The latter triples indicate which of the three states is present. A genotype of three loci described by (2,0,1) in the numerical GBLUP coding, will here be coded by the nine-tuple (1,0,0,0,0,1,0,1,0) (a triple for each locus, describing its state). We then simply use model Eq. (1) with the new coding. Advantages of this model are that it is also invariant to an exchange of the role of a and A (as GBLUP of Eq. (1) is as well), since we will only permute the meaning of the positions in the triple but change their entries accordingly. Moreover, we can account for dominance by estimating each effect on its own. A disadvantage is the increased number of variables but this can be overcome easily by the use of relationship matrices for genomic prediction. Property 6 describes the relation between the CM model and GBLUP for markers with only two possible values:
Comparison of the parametrization of the genotypic values in GBLUP and the categorical marker effect model CM: Black dots: genotypic values of the corresponding genotype of a certain locus. GBLUP parameterizes the genotypic values by a fixed effect (red dot) and a random effect determining the slope (blue line), whereas CM parameterizes by the fixed effect (red line) and independent random effects (blue lines) for each genotype
For markers with only two possible states, let M denote the n×p marker matrix in the {−1,1} coding. The relationship matrix of GBLUP is given by (a rescaled version of) M M ′. Moreover, let C be the relationship matrix of the CM model. Then
$$ \mathbf{C}= 0.5 (\mathbf{MM'}+\mathbf{J}_{n\times n} p) $$
where p is the number of markers and J n×n the n×n matrix with each entry equal to 1.
The linear relationship of the covariance matrices demonstrated in Property 6 implies that the prediction performances of GBLUP and CM are identical for markers with only two possible values.
Let us assume that the ratio of the variance components is fixed such that Property 1 holds for the CM model. Then GBLUP and the CM model are identical for markers with only two possible values.
A categorical epistasis model (CE) Analogously to the CM model, we translate the genotype of pairs of loci, e.g. (a A,b b) into {0,1}-tuples. Here, a nine-tuple indicates which combination of alleles of two loci is present. To translate the genotype (2,0,1) of the numerical {0,1,2} coding into the CE coding, we have to translate each marker pair. Each pair is coded by a nine-tuple with only one entry equal to 1 which indicates the configuration:
$$ \left(\underbrace{\bullet}_{(2,2)},\underbrace{\bullet}_{(2,1)},\underbrace{\bullet}_{(2,0)}, \underbrace{\bullet}_{(1,2)},\underbrace{\bullet}_{(1,1)},\underbrace{\bullet}_{(1,0)}, \underbrace{\bullet}_{(0,2)},\underbrace{\bullet}_{(0,1)},\underbrace{\bullet}_{(0,0)}\right). $$
The assignment of the configuration of the respective marker pair to the position of the nine-tuple can be chosen arbitrarily but has of course to be used consistently for all individuals. Let us assume that we have three subsequent loci with genotypes (2,0,1) in the ordinary numerical coding. Then, there are three possible interactions: the first two loci have the combination (2,0) which will be coded as (0,0,1,0,0,0,0,0,0). Additionally, the second pair is (2,1) which will be coded as (0,1,0,0,0,0,0,0,0), whereas the last pair (0,1) is translated to (0,0,0,0,0,0,0,1,0). As already mentioned, an obvious disadvantage of the model is the high number of variables, but we do not have to solve the system for these variables to perform genomic prediction, since we can use equivalent genomic relationship matrices. Moreover, this model eliminates several disadvantages of EGBLUP: i) The model is invariant with respect to the decision which allele is used as reference ("orientation"), since it is based on categorical variables indicating which genotype is present, ii) the effects the model can assign to different pairs of loci are not connected between pairs by their respective codings (as described for the asymmetrically coded EGBLUP after Example 1), and iii) compared to the symmetric {−1,0,1} coding of EGBLUP, CE does not generally assign the same effects to the most different allele combinations.
Relationship matrices for the respective marker models
Let M be the marker matrix of the respective numerical coding (0,1,2 or −1,0,1). In the following, we will present the corresponding relationship matrices for each model.
GBLUP. The relationship matrix for the GBLUP model is given by M M ′ (the n×p genotype matrix multiplied with its transposed version).
Epistasis models based on Eq. (2). The relationship matrix corresponding to the interactions of Eq. (2) where j≥k is given by
$$ \mathbf{H} = 0.5 \left(\mathbf{MM' \circ MM'}\right) + 0.5 \left(\mathbf{M \circ M}\right) \left(\mathbf{M \circ M}\right)'. $$
(for a derivation of this statement see [11]). Note here again that the GBLUP model is not affected by a translation of the coding in M, but the performance of EGBLUP is affected.
The categorical marker (CM) effect model The i,l-th entry of the corresponding relationship matrix C is given by the inner product of the vectors of the genotypes of individuals i and l in the coding of the CM model. This means that we count the number of loci which have the same configuration. For markers with two possible variants and the marker data in dosage 0,1 coding, we can express the i,l-th entry of C the following way:
$$ C_{i,l} = p - \sum\limits_{j=1}^{p} \left|M_{i,j} - M_{l,j}\right| $$
Analogously, for markers with three different variants, we have to count the number of zeros in the marker vectors M i,∙−M l,∙ (For the relation of Eqs. (11) and (8), see the derivation of Eq. (8) in Additional file 2).
The categorical epistasis (CE) model The i,l-th entry of the corresponding relationship matrix C E is given by the inner product of the genotypes i, l in the coding of the categorical epistasis model. Thus, the matrix counts the number of pairs which are in identical configuration and we can express the entry C E i,l in terms of C i,l since we can calculate the number of identical pairs from the number of identical loci:
$$ {C_{E}}_{i,l}= \sum_{k=1}^{C_{i,l}} k =0.5 C_{i,l} \left(C_{i,l} + 1 \right) $$
Here, we also count the "pair" of a locus with itself by allowing k∈{1,…,C i,l }. Excluding these effects from the matrix would mean, the maximum of k equals C i,l −1. In matrix notation Eq. (12) can be written as
$$ \mathbf{C}_{E}= 0.5 \mathbf{C} \circ \mathbf{C} + 0.5 \mathbf{C} $$
Note here, that the relation between GBLUP and the epistasis terms of EGBLUP is identical to the relation of CM and CE in terms of relationship matrices: For G = M M ′ and M a matrix with entries only 0 or 1, Eq. (10) gives Eq. (13) with C=G and C E =H.
Additionally to the previously discussed EGBLUP model, a common approach to incorporate "non-linearities" is based on Reproducing Kernel Hilbert Space regression [21, 31] by modeling the covariance matrix as a function of a certain distance between the genotypes. The most prominent variant for genomic prediction is the Gaussian kernel. Here, the covariance C o v i,l of two individuals is described by
$$ {Cov}_{i,l} = \exp(-b \cdot d_{i,l}), $$
with d i,l being the squared Euclidean distance of the genotype vectors of individuals i and l, and b a bandwidth parameter that has to be chosen. This approach is independent of translations of the coding, since the Euclidean distance remains unchanged if both genotypes are translated. Moreover, this approach is also invariant with respect to a scaling factor, if the bandwidth parameter is adapted accordingly (in this context see also [ 32 ]). Thus, EGBLUP and the Gaussian kernel RKHS approach capture both "non-linearities" but they behave differently if the coding is translated.
Comparison of the performance of the models on different data sets
Results on the simulated data For 20 independently simulated populations of 1 000 individuals, we modeled three scenarios of qualitatively different genetic architecture (purely additive A, purely dominant D and purely epistatic E) with increasing number of involved QTL (see "Methods") and compared the performances of the considered models on these data. In more detail, we compared GBLUP, a model defined by the epistasis terms of EGBLUP with different codings, the categorical models and the Gaussian kernel with each other. All predictions were based on one relationship matrix only, that is in the case of EGBLUP on the interaction effects only. The use of two relationship matrices did not lead to qualitatively different results (data not shown), but can cause numerical problems for the variance component estimation if both matrices are too similar. For each of the 20 independent simulations of population and phenotypes, test sets of 100 individuals were drawn 200 times independently, and Pearson's correlation of phenotype and prediction was calculated for each test set and model. The average predictive abilities of the different models across the 20 simulations are summarized in Table 2 in terms of empirical mean of Pearson's correlation and its average standard error. Comparing GBLUP to EGBLUP with different marker codings, we see that the predictive ability of EGBLUP is very similar to that of GBLUP, if a coding which treats each marker equally is used. Only the EGBLUP version, standardized by subtracting twice the allele frequency as it is done in the commonly used standardization for GBLUP [6], shows a drastically reduced predictive ability for all scenarios (see Table 2, EGBLUP VR). Moreover, considering the categorical models, we see that CE is slightly better than CM and that both categorical models perform better than the other models in the dominance and epistasis scenarios.
Table 2 Predictive abilities of the models on the simulated data. Comparison of the predictive abilities in terms of correlations between the measured phenotypes and the predictions for the individuals of the test sets ("Pearson's correlation"; 100 test set genotypes were drawn randomly from all 1000 genotypes; 200 repeats for each simulated population; 20 independent simulations of population and phenotypes). Traits of different genetic architecture (additive A, dominant D, Epistasis E) and increasing number of QTL. Model abbreviations as introduced in the text. For EGBLUP, only the matrix based on the interactions was considered here
Results on the wheat data For EGBLUP, we used here the coding {0,1} which was originally used in the data of the publication, a translation by −1 which leads to {−1,0} representing a coding in which the meaning of 0 and 1 is permuted, and a centered version {−1,1}. Moreover, we used the standardization by allele frequencies [6] to calculate EGBLUP. Additionally, we evaluated CM, CE and reevaluated the Gaussian kernel RKHS approach, previously used by Crossa et al. [21] (we used the matrix K obtained from the supplementary of the corresponding publication). The results are summarized in Table 3. CM showed exactly identical results to those of GBLUP (which has already been stated theoretically by Property 7) and is therefore not listed separately. Considering the predictive ability of EGBLUP with different codings, a first thing to note is that the variability among the EGBLUP variants is higher than that found on the simulated data. Moreover, with the data sets of environments 1,3 and 4, EGBLUP tends to outperform GBLUP. Among them, the model with symmetric {−1,1} coding performs best and the VanRaden standardized version of EGBLUP has a significantly reduced predictive ability for the data of environments 1, 2 and 3, which is analogous to what we have already seen on the simulated data. Moreover, the predictive ability of EGBLUP with symmetric coding seems to be closest to that of the Gaussian kernel. For the data of environment 2, no big differences in the performance of the models (except for the allele frequency standardized EGBLUP) can be observed. Overall, the Gaussian kernel RKHS method performs best on this data set and the predictive ability of the CE model is on the level of the asymmetrically coded versions of EGBLUP.
Table 3 Predictive abilities of the models on the wheat data. Comparison of the predictive abilities as Pearson's correlation of the measured phenotypes and the predictions for the individuals of the test sets (60 test set genotypes, trait: grain yield)
Results on the mouse data We compared the models on 13 traits related to obesity, weight and immunology. Instead of the raw phenotypes, we used pre-corrected residuals which are publicly available (see "Methods"). Again, we compared GBLUP, EGBLUP with 0,1,2 coding as well as with inverted, symmetric and by allele frequencies standardized coding, the categorical models and the Gaussian kernel RKHS approach with each other. The results are summarized in Table 4. The general patterns observed on the previously considered data remain the same: Any EGBLUP version treating the markers equally has at least the same predictive ability as GBLUP for all traits. Among them, the symmetric coding seems to perform best. The allele frequency standardized version of EGBLUP has in three of the 13 traits a higher predictive ability than its other versions (W6W, GrowthSlope, CD8Intensity), but a smaller one in ten cases. Considering only significant differences between CM and GBLUP, CM outperforms GBLUP on the traits %CD4/CD3 and %CD8/CD3 and shows a lower predictive ability only for BMI and BodyLength. Moreover, CE outperforms CM slightly. Overall, two traits are predicted best by EGBLUP VR, three traits by CE, and five by the symmetric version of EGBLUP and the Gaussian kernel, respectively.
Table 4 Predictive abilities of the models on the mouse data. Comparison of the predictive abilities as Pearson's correlation of the measured phenotypes and the predictions for the individuals of the test set (130 test set genotypes). Here, the already for fixed effects pre-corrected residuals of the phenotypes, which are also provided by the publicly available data, were used
Incorporating prior experimental information by marker coding
The coding-dependent performance of EGBLUP also offers possibilities to incorporate additional information. He et al. [12 , 13] have already illustrated the idea of data-driven coding and we have recently shown that information on the performance of genotypes grown under different environmental conditions can be used to select variables within EGBLUP which then can be used for genome assisted prediction within another environment [11]. Here, we will demonstrate that differential coding is also appropriate to incorporate prior experimental information into EGBLUP. For this, we used the different trait (× environment) combinations and adapted the marker coding of each pair of loci to the data, following the procedure described in the "Methods" section. Important here is that we decided for each pair of markers individually, which orientation the corresponding coding of the particular pair shall have. The "orientation" of the underlying effect model is chosen for each pair. Thus, we cut the connection between the coding of different pairs. The determined relationship matrices are then used to predict within the data of other traits. The results are summarized in Tables 5 and 6 for the wheat and mouse data sets, respectively. We can see here that adapting the coding to data of previous experiments can be beneficial for the predictive ability. In the case of the wheat data set, Table 5 shows that using the data of grain yield of the genotypes grown in environments 3 and 4 to infer the marker coding for each pair of marker, improves the prediction accuracy in environment 2 to a level higher than that of all methods which do not use the data of other experiments (from 0.504±0.007 to 0.544±0.006). The situation is analogue for the predictive ability in environment 3, if the data of environment 2 is used to infer the relationship matrix. However, the gain in predictive ability resulting from this procedure is relatively small compared to the gain by means of variable selection [11]. Adapting the coding to given data also helped to increase predictive ability on the mouse data (see Tables 4 and 6). For instance, improvements from 0.285±0.006 to 0.313±0.005, from 0.536±0.004 to 0.569±0.004, and from 0.664±0.004 to 0.685±0.003 were reached for the traits BodyLength, %CD3 and %CD4/CD3, respectively.
Table 5 Predictive abilities on the wheat data when prior information is incorporated in the marker coding of EGBLUP. Predictive abilities when the coding for each interaction is determined based on records under different environmental conditions
Table 6 Predictive abilities on the mouse data when prior information is incorporated in the marker coding of EGBLUP. Predictive abilities when the coding for each interaction is determined based on the records of other traits
The effect of the choice of marker coding on EGBLUP
We recalled that GBLUP is not sensitive to certain changes of the marker coding if the variance components are adapted accordingly. Analogously, we also proved that the interaction terms of EGBLUP are invariant to factors rescaling the marker coding, but showed that a translation indeed changes the underlying marker effect model drastically. In particular, we demonstrated that the effect model of EGBLUP with the asymmetric 0,1,2 coding is affected by the decision which allele to count. Thus, an important observation concerning EGBLUP is that the only coding allowing a permutation of the roles of the alleles without changing the underlying interaction effect model for the respective marker pair is symmetric around zero. This coding solves the problem of "which allele to count", but we also argued that the symmetric coding appears to be biologically implausible since it assigns the same interaction effect to the most distant genotypes. Concerning the allele frequency adjusted version EGBLUP VR, we illustrated that the different markers are not treated equally and thus that the interaction effect models here depend on the allele frequencies of the involved alleles. On the level of predictive ability, the symmetric coding tends to outperform the asymmetric versions slightly, which can most clearly be seen from the data of environment 1 and 4 of the wheat data set (Table 3). Also with the mouse data set, the symmetric coding had a higher predictive ability than the other codings treating all loci equally for all traits, but the improvements were most often very small. Concerning the allele-frequencies standardized version EGBLUP VR, we observed a drastic reduction in the predictive ability compared to other EGBLUP versions in most of the examples. Illustratively, one reason for the comparatively poor performance can be seen in the following: the relationship matrix corresponding to the interaction effects of EGBLUP in a certain coding is basically the GBLUP relationship matrix, but with each of its entries squared (if all pairwise interactions and interactions of a marker with itself are modeled, see [10 , 11] and compare to Eq. (10)). The standardization by twice the allele frequencies (and division by a certain factor representing a variance [6]) produces a GBLUP matrix which can possess entries larger than 1 and smaller than 0. In particular, if the GBLUP matrix has negative entries, squaring them changes the order of the relationship between the individuals. For instance, if A has a relation of −0.1 with individual B and −0.3 with individual C, which means that A is more closely related to B than to C, the corresponding EGBLUP matrix states that the relation between A and C is closer than that of A and B. This argumentation is equally true for the symmetric coding, but the portion of negative entries in the corresponding additive relationship matrix was close to zero for the wheat and the mouse data set when the symmetric coding was used in our examples. Overall, in spite of a certain popularity of EGBLUP in recent literature [10 , 11 , 17] our results suggest that the use of products of marker values as predictor variables is not the best way to incorporate interactions into the GBLUP model. Moreover, contrary to the theoretical findings on the "congruency" of EGBLUP and the Gaussian kernel in a RKHS approach [10], our results show that both methods respond in a different way to a change of marker coding: a translation of the coding has an impact on the predictive ability of EGBLUP, but not on that of the Gaussian kernel. Since the Euclidean distance between two vectors will not change under a translation of both vectors, the corresponding relationship matrix remains identical. A reconsideration of the limit behavior of EGBLUP when the degree of interaction increases to n-factor interaction (and n→∞) may therefore be interesting from a theoretical point of view.
To develop an alternative to EGBLUP which does not possess the illustrated undesired theoretical properties, but which –unlike the RKHS approaches– allows to interpret the predicted quantities as "effects", we considered the categorical effect models (The effects of the categorical models can be explicitly calculated from phenotypes or genetic values under the use of the well-known Mixed Model formulas for effects with the respective design matrices). As a first step, we constructed the categorical marker effect model CM, which does not use the assumption of a constant allele substitution effect (Fig. 1) and thus gives the possibility to model (over)dominance by modeling an independent effect of each genotype at a locus. The fact that this property can also lead to an increase in predictive ability was illustrated by the simulated dominance scenario. An important result is that this categorical model can be rewritten as a relationship matrix model and thus provides an equivalent to the Ridge Regression/GBLUP duality, but based on a categorical effect model instead of a numerical dosage model. Whether this model increases predictive ability will always depend on the population structure and the influence of dominance effects on a particular trait. For instance, if a population originating from lines from different heterotic pools is considered, the prevalent heterosis effect might be a good reason to use CM instead of GBLUP, since heterosis creates a deviation from the linear dosage model. Moreover, the number of heterozygous and homozygous loci in the data set is important. If most loci are mainly present in only two of the three possible SNP genotypes, CM cannot outperform GBLUP substantially. Interestingly, comparing GBLUP and CM, CM was only significantly outperformed on the traits BMI and BodyLength. Thus, abandoning the assumption of a dosage effect of an allele, which is implemented by counting its occurrence and multiplying it with an additive effect, might not in general be a problem for prediction. Note also that there are other ways of defining marker based dominance matrices as for instance described by Su et al. [33]. Moreover, dominance can implicitly be modeled by an epistatic interaction term of a locus with itsself in Eq. (2) if j=k (see [11]).
Analogously to the relation of GBLUP and EGBLUP, we extended the categorical marker effect model CM to the categorical epistasis model CE. The disadvantage of inflating the model with a huge number of variables is solved for genomic prediction by using an equivalent relationship-matrix-based approach. Interestingly, the analogy of the relation between GBLUP and EGBLUP also translates to the level of relationship matrices, which we illustrated by the theoretical result of Eq. (13). The relationship matrix of CE has the same connection to the relationship matrix of CM as the matrix defined by the interaction terms of EGBLUP has to the genomic relationship matrix of GBLUP. Moreover, CE eliminates undesired theoretical properties of EGBLUP: the question which allele to use as reference is not raised, its structure does not lead to a dependence of the effect models of different pairs of loci, and it does not assign the same effects to the most different allele combinations as the symmetrically coded EGBLUP model does. With the wheat data which consist of markers with only two possible values and for which GBLUP coincides with CM, CE outperformed GBLUP in all environments (Table 3). Moreover, CE slightly improved the predictive ability of CM for all considered traits of the mouse data set. Overall, the CE model is a valuable alternative for modeling epistasis since it eliminates undesired properties of EGBLUP and shows convincing results in practice. However, other more realistic parametric structures of effects in between EGBLUP and CE may be of interest for future research. Important steps into this direction have already been made with the "hybrid" coding according to He et al. [12 , 13], in which the marker coding is estimated from the data under the side condition of generating a monotone effect model. Moreover, an interesting approach for future investigation may be the adaption of categorical models to other types of variables, for instance defined by haplotypes.
Incorporating prior experimental information into the coding of EGBLUP
Finally, we demonstrated that marker coding can be used to incorporate prior information. An important property of the procedure we used is that we "decoupled" the effect models for different pairs by allowing to choose the orientation of the parametric model for each pair separately (see "Methods"). In particular, this means that marker j might be coded as 0,1,2 in combination with marker k, but as −2,−1,0 in combination with marker l. The criterion to decide which coding to use, was simple here by comparing the size of the absolute interaction effect of a pair when different "orientations" were used. Note here that the improvement of prediction accuracy was smaller than by means of variable selection on the wheat data set [11]. The relatively small improvement might be a result of only giving the two possibilities of both markers being in the initial coding or both markers with inverted coding, but not choosing from all possible four orientations. We used this simplified procedure, since for other combinations of one marker with original coding and the other marker with inverted coding, the assigned effect will also depend on the orientation of other pairs and thus it is difficult to determine which orientation to choose if we will additionally change the orientation of other pairs. In this regard, the presented method can be considered as a straightforward ad hoc approach to incorporate prior knowledge into the coding, capturing some part of the covariance structure of the given data and thus improving the predicitve ability on data sets with similar covariance structure.
We illustrated that the EGBLUP model possesses several undesired properties caused by the interactions being modeled by products of marker values. We showed that the symmetrically coded EGBLUP tends to perform best, that the allele frequency standardized version tends to have the lowest predicitve ability and that the CE model can be an attractive alternative to EGBLUP. Prior information from other experiments can be incorporated into the marker coding of EGBLUP, which gives the potential to enhance predictive ability for correlated traits.
1 In literature, the expression GBLUP is used for the reformulated equivalent of Eq. (1) with genetic value g:=M β and thus \(\mathbf {g}\sim \mathcal {N}(0,\sigma ^{2}_{\beta } \mathbf {MM}')\).
CM:
Categorical marker effect model
CE:
Categorical epistais model
DArT:
Diversity Arrays Technology
EGBLUP:
Extended genomic best linear unbiased prediction
GBLUP:
Genomic best linear unbiased prediction
MAF:
Minor allele frequency
SNP:
Meuwissen T, Hayes B, Goddard M. Prediction of total genetic value using genome-wide dense marker maps. Genetics. 2001; 157(4):1819–29.
Hayes BJ, Visscher PM, Goddard ME. Increased accuracy of artificial selection by using the realized relationship matrix. Genet Res. 2009; 91(01):47–60.
Abraham G, Tye-Din JA, Bhalala OG, Kowalczyk A, Zobel J, Inouye M. Accurate and robust genomic prediction of celiac disease using statistical learning. PLoS Genet. 2014; 10(2):1004137.
Henderson CR. Best linear unbiased estimation and prediction under a selection model. Biometrics. 1975; 31(2):423–47.
Habier D, Fernando R, Dekkers J. The impact of genetic relationship information on genome-assisted breeding values. Genetics. 2007; 177(4):2389–97.
VanRaden P. Efficient methods to compute genomic predictions. J Dairy Sci. 2008; 91(11):4414–23.
Piepho HP. Ridge regression and extensions for genomewide selection in maize. Crop Sci. 2009; 49(4):1165–76.
Albrecht T, Wimmer V, Auinger HJ, Erbe M, Knaak C, Ouzunova M, Simianer H, Schön CC. Genome-based prediction of testcross values in maize. Theor Appl Genet. 2011; 123(2):339–50.
Strandén I, Christensen OF. Allele coding in genomic evaluation. Genet Sel Evol. 2011; 43(25):1–11. http://www.gsejournal.org/content/43/1/25.
Jiang Y, Reif JC. Modeling epistasis in genomic selection. Genetics. 2015; 201(2):759–68.
Martini JWR, Wimmer V, Erbe M, Simianer H. Epistasis and covariance: How gene interaction translates into genomic relationship. Theor Appl Genet. 2016; 129(5):963–76.
He D, Wang Z, Parida L. Data-driven encoding for quantitative genetic trait prediction. BMC Bioinformatics. 2015; 16(Suppl 1):10.
He D, Parida L. Does encoding matter? a novel view on the quantitative genetic trait prediction problem. BMC Bioinformatics. 2016; 17(Suppl 9):272.
Falconer DS, Mackay TF, Frankham R. Introduction to quantitative genetics.
Zeng ZB, Wang T, Zou W. Modeling quantitative trait loci and interpretation of models. Genetics. 2005; 169(3):1711–25.
Hallgrímsdóttir IB, Yuster DS. A complete classification of epistatic two-locus models. BMC Genet. 2008; 9(1):17.
Hu Z, Li Y, Song X, Han Y, Cai X, Xu S, Li W. Genomic value prediction for quantitative traits under the epistatic model. BMC Genet. 2011; 12(1):15.
Mackay TF. Epistasis and quantitative traits: using model organisms to study gene-gene interactions. Nat Rev Genet. 2014; 15(1):22–33.
Wang D, El-Basyoni IS, Baenziger PS, Crossa J, Eskridge K, Dweikat I. Prediction of genetic values of quantitative traits with epistatic effects in plant breeding populations. Heredity. 2012; 109(5):313–9.
Sargolzaei M, Schenkel FS. QMSim: a large-scale genome simulator for livestock. Bioinformatics. 2009; 25(5):680–1.
Crossa J, de Los Campos G, Pérez P, Gianola D, Burgueno J, Araus JL, Makumbi D, Singh RP, Dreisigacker S, Yan J, Arief V, Banziger M, HJ B. Prediction of genetic values of quantitative traits in plant breeding using pedigree and molecular markers. Genetics. 2010; 186(2):713–24.
Solberg LC, Valdar W, Gauguier D, Nunez G, Taylor A, Burnett S, Arboledas-Hita C, Hernandez-Pliego P, Davidson S, Burns P, et al.A protocol for high-throughput phenotyping, suitable for quantitative trait analysis in mice. Mamm Genome. 2006; 17(2):129–46.
Valdar W, Solberg LC, Gauguier D, Cookson WO, Rawlins JNP, Mott R, Flint J. Genetic and environmental effects on complex traits in mice. Genetics. 2006; 174(2):959–84.
Durinck S, Spellman PT, Birney E, Huber W. Mappingidentifiers for the integration of genomic datasets with the r/bioconductor package biomart. Nat Protoc. 2009; 4(8):1184–1191.
Durinck S, Moreau Y, Kasprzyk A, Davis S, De Moor B, Brazma A, Huber W. Biomart and bioconductor: a powerful link between biological databases and microarray data analysis. Bioinformatics. 2005; 21(16):3439–440.
Wimmer V, Albrecht T, Auinger HJ, Schoen CC. synbreed: a framework for the analysis of genomic prediction data using R. Bioinformatics. 2012; 28(15):2086–7.
Akdemir D, Godfrey OU. EMMREML: Fitting Mixed Models with Known Covariance Structures. 2015. R package version 3.1. http://CRAN.R-project.org/package=EMMREML.
R Core Team. R: A Language and Environment for Statistical Computing. Vienna: R Foundation for Statistical Computing; 2014. http://www.R-project.org/.
Ober U, Huang W, Magwire M, Schlather M, Simianer H, Mackay TF. Accounting for genetic architecture improves sequence based genomic prediction for a drosophila fitness trait. PloS ONE. 2015; 10(5):1–17: e0126880. doi:10.1371/journal.pone.0126880.
Zhang Z, Ober U, Erbe M, Zhang H, Gao N, He J, Li J, Simianer H. Improving the accuracy of whole genome prediction for complex traits using the results of genome wide association studies. PloS ONE. 2014; 9(3):93017.
Gianola D, Morota G, Crossa J. Genome-enabled prediction of complex traits with kernel methods: What have we learned? In: Proceedings of the 10th World Congress of Genetics Applied to Livestock Production. Vancouver, BC, Canada: 2014. https://asas.confex.com/asas/WCGALP14/webprogram/Paper10331.html.
Long N, Gianola D, Rosa GJ, Weigel KA. Marker-assisted prediction of non-additive genetic values. Genetica. 2011; 139(7):843–54.
Su G, Christensen OF, Ostersen T, Henryon M, Lund MS. Estimating additive and non-additive genetic variances and predicting genetic merits using genome-wide dense single nucleotide polymorphism markers. PloS ONE. 2012; 7(9):45293.
JWRM thanks Maria Emilia Barreyro for helpful discussions.
We acknowledge support by the Open Access Publication Funds of the Göttingen University. JWRM thanks KWS SAAT SE for financial support. NG thanks the China Scholarship Council (CSC) for financial support. RJCC was supported by grants FONCyT PICT 2013-1661, UBACyT 20020150100230B/ 2016 and PIP CONICET 833/2013, from Argentina.
The simulated data, the filtered and imputed genotypes of the mouse data and the corrected phenotypes can be found in Additional file 1. The raw mouse data and a detailed description of the data can be found at the corresponding UCL website (at the moment http://mtweb.cs.ucl.ac.uk/mus/www/mouse/HS/index.shtmland http://mtweb.cs.ucl.ac.uk/mus/www/GSCAN/). The wheat data is offered by the corresponding publication. See also the "Methods" section for more details.
JWRM: Wrote the manuscript, derived the theoretical proofs of the statements, proposed to consider the topic; proposed and programmed the algorithm to adapt the coding to given data; analyzed the data; NG: supported the data analysis; prepared the mouse data set; parallelized the presented algorithm to adapt the coding to given data; tested the models on different data sets and with different validation methods; DFC: supported the data analysis; reevaluated the results with different prediction pipelines; simulated the genotypes with the QMSim software. VW, ME, RJCC, HS: guided the research. All authors have read and approved the final version of the manuscript.
Department of Animal Sciences, Georg-August University, Albrecht Thaer-Weg 3, Göttingen, Germany
Johannes W. R. Martini, Ning Gao, Diercles F. Cardoso, Malena Erbe & Henner Simianer
National Engineering Research Center for Breeding Swine Industry, Guangdong Provincial Key Lab of Agro-animal Genomics and Molecular Breeding, College of Animal Science, South China Agricultural University, Guangzhou, China
Ning Gao
Departamento de Zootecnia, São Paulo State University, São Paulo, Brazil
Diercles F. Cardoso
KWS SAAT SE, Einbeck, Germany
Valentin Wimmer
Institute for Animal Breeding, Bavarian State Research Centre for Agriculture, Grub, Germany
Malena Erbe
Department of Animal Production, University of Buenos Aires, INPA-CONICET, Buenos Aires, Argentina
Rodolfo J. C. Cantet
Johannes W. R. Martini
Henner Simianer
Correspondence to Johannes W. R. Martini.
Rdata-file with two lists. The list "Mouse_Data" contains a genotype matrix of 1298 individuals and 9265 markers as well as a matrix with records of 13 traits of the individuals. The list "Simulated_Data" offers the genotypes and phenotypes of the 20 simulations. Each entry of this list is a list of two elements representing genotypes and phenotypes of the respective simulation. Genotypes are given by a matrix of 1000 individuals with 9000 markers. Phenotypes are provided as a data.frame of the 1000 individuals and the 9 different phenotypes described in the Methods section. (RDATA 64512 kb)
The file presents mathematical arguments for the statements on the properties of the models, which have been made in the main text. (PDF 149 kb)
Martini, J.W.R., Gao, N., Cardoso, D.F. et al. Genomic prediction with epistasis models: on the marker-coding-dependent performance of the extended GBLUP and properties of the categorical epistasis model (CE). BMC Bioinformatics 18, 3 (2017). https://doi.org/10.1186/s12859-016-1439-1
Epistasis model
Results and data | CommonCrawl |
Results for 'Stanislav Petrov'
Євангельські Церкви В Зовнішньополітичному Векторі Релігійної Політики Срср В 1941-1948 Рр.Stanislav Petrov - 2014 - Схід 1 (127):160-165.details
У статті проведено аналіз впливу зовнішньополітичного аспекту на процес генезису "нової релігійної політики" Й. Сталіна, її інституціоналізації та причин, що призвели до охолодження в державно-церковних відносинах наприкінці 40-х років ХХ століття. Показані роль та місце, яке займали євангельські церкви в цьому процесі.
Russian Studies in European Philosophy
Огляд історіографії політики в срср щодо євангельських церков у 1940-1960-і рр.Stanislav Petrov - 2015 - Схід 1 (133):87-96.details
Стаття є детальним історіографічним оглядом проблеми політики СРСР щодо євангельських церков у 1940-1960-і роки. У ній розглядається та аналізується науковий доробок як вітчизняної історіографії класичного радянського періоду, періоду "перебудови", так і сучасної пострадянської історіографії. Окрема увага в статті приділяється роботам західних дослідників другої половини ХХ - початку ХХІ століття, які вивчали державно-релігійні відносини в СРСР щодо євангельських церков у 1940-1960-і роки. Розглядається також конфесійна історіографія проблеми, яка з'явилась в останні роки існування СРСР та набула свого розквіту у 2000-і роки.
Field Creativity and Post-Anthropocentrism.Stanislav Roudavski - 2016 - Digital Creativity 27 (1):7-23.details
Can matter, things, nonhuman organisms, technologies, tools and machines, biota or institutions be seen as creative? How does such creativity reposition the visionary activities of humans? This article is an elaboration of such questions as well as an attempt at a partial response. It was written as an editorial for the special issue of the Digital Creativity journal that interrogates the conception of Post-Anthropocentric Creativity. However, the text below is a rather unconventional editorial. It does not attempt to provide an (...) overview of the issue's theme but, instead, samples it via a particular example. The idea of the issue was to think about post-anthropocentricism by considering (1) agents, recipients and processes of creativity alongside with its (2) purpose, value, ethics and politics. This article addresses the first subtheme by puzzling at the paradoxes of "field learning" and picks at the second by considering the texture of "automated beauty". Both of these parts use chess for an example. The narrative on chess is intermitted by a section "on creativity" that attempts to contextualize the case-based discussion in the wider context and to consider motivations and implications. (shrink)
Moral Status of Artificial Systems in Philosophy of Cognitive Science
The Consciousness Revolution: A Transatlantic Dialogue: Two Days with Stanislav Grof, Ervin Laszlo, and Peter Russell.Stanislav Grof - 1999 - Element.details
Discusses current global conditions including peace, changes in society, education, religion, spirituality, and consciousness.
Bertrand Russell in 20th Century Philosophy
$1.99 used Amazon page
Psychology of the Future: Lessons From Modern Consciousness Research.Stanislav Grof - 2000 - State University of New York Press.details
This accessible and comprehensive overview of the work of Stanislav Grof, one of the founders of transpersonal psychology, was specifically written to acquaint...
Parapsychology and Consciousness in Philosophy of Cognitive Science
A Dichotomy for Some Elementarily Generated Modal Logics.Stanislav Kikot - 2015 - Studia Logica 103 (5):1063-1093.details
In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form \. We prove that many properties of these logics, such as finite axiomatisability, elementarity, axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula, together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold.
Logics in Logic and Philosophy of Logic
Axonal Wiring in Neural Development: Target-Independent Mechanisms Help to Establish Precision and Complexity.Milan Petrovic & Dietmar Schmucker - 2015 - Bioessays 37 (9):996-1004.details
Kripke Completeness of Strictly Positive Modal Logics Over Meet-Semilattices with Operators.Stanislav Kikot, Agi Kurucz, Yoshihito Tanaka, Frank Wolter & Michael Zakharyaschev - 2019 - Journal of Symbolic Logic 84 (2):533-588.details
Our concern is the completeness problem for spi-logics, that is, sets of implications between strictly positive formulas built from propositional variables, conjunction and modal diamond operators. Originated in logic, algebra and computer science, spi-logics have two natural semantics: meet-semilattices with monotone operators providing Birkhoff-style calculi and first-order relational structures (aka Kripke frames) often used as the intended structures in applications. Here we lay foundations for a completeness theory that aims to answer the question whether the two semantics define the same (...) consequence relations for a given spi-logic. (shrink)
Integrating Children With Physical Impairments Into Sports Activities: A "Golden Sun" for All Children?Stanislav Pinter, Tjasa Filipcic, Ales Solar & Maja Smrdu - 2005 - Journal of the Philosophy of Sport 32 (2):147-154.details
Philosophy of Sport in Social and Political Philosophy
"It's Also a Kind of Adrenalin Competition" – Selected Aspects of the Sex Trade as Viewed by Clients.Stanislav Ondrášek, Zuzana Řimnáčová & Alena Kajanová - 2018 - Human Affairs 28 (1):24-33.details
The main goal of the article is to describe selected aspects of the sex trade as viewed by clients who make use of the services provided by sex workers. We use data obtained through a content analysis of selected topics discussed on an erotic forum called Nornik.net. The topics were: Can a person stop "screwing"?; what was your first contact with the sex trade and how can a person hide their visits to sex workers? In the course of the analysis, (...) we identified an additional category that featured in all the topics: "trophy collecting". The discussants perceive sex workers as a commodity to be purchased and subsequently evaluated. The discussants tended to compete among themselves to visit the most sex workers or to be first to visit the latest sex worker. The discussion forum serves as a support centre and contains hints and tips on different areas of "screwing" and related issues. The forum also displays certain features of a community. (shrink)
Foundations of ArtScience: Formulating the Problem.Francis Heylighen & Katarina Petrović - 2021 - Foundations of Science 26 (2):225-244.details
While art and science still functioned side-by-side during the Renaissance, their methods and perspectives diverged during the nineteenth century, creating a still enduring separation between the "two cultures". Recently, artists and scientists again collaborate more frequently, as promoted most radically by the ArtScience movement. This approach aims at a true synthesis between the intuitive, imaginative methods of art and the rational, rule-governed methods of science. To prepare the grounds for a theoretical synthesis, this paper surveys the fundamental commonalities and differences (...) between science and art. Science and art are united in their creative investigation, where coherence, pattern or meaning play a vital role in the development of concepts, while relying on concrete representations to experiment with the resulting insights. On the other hand, according to the standard conception, science seeks an understanding that is universal, objective and unambiguous, while art focuses on unique, subjective and open-ended experiences. Both offer prospect and coherence, mystery and complexity, albeit with science preferring the former and art, the latter. The paper concludes with some examples of artscience works that combine all these aspects. (shrink)
An Extension of Kracht's Theorem to Generalized Sahlqvist Formulas.Stanislav Kikot - 2009 - Journal of Applied Non-Classical Logics 19 (2):227-251.details
Sahlqvist formulas are a syntactically specified class of modal formulas proposed by Hendrik Sahlqvist in 1975. They are important because of their first-order definability and canonicity, and hence axiomatize complete modal logics. The first-order properties definable by Sahlqvist formulas were syntactically characterized by Marcus Kracht in 1993. The present paper extends Kracht's theorem to the class of 'generalized Sahlqvist formulas' introduced by Goranko and Vakarelov and describes an appropriate generalization of Kracht formulas.
Logic and Philosophy of Logic, Miscellaneous in Logic and Philosophy of Logic
Belnap–Dunn Modal Logics: Truth Constants Vs. Truth Values.Sergei P. Odintsov & Stanislav O. Speranski - 2020 - Review of Symbolic Logic 13 (2):416-435.details
We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding 'strong negation'. Though all four values 'truth', 'falsity', 'neither' and 'both' are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for 'neither' or/and 'both' leads to quite unexpected results. To be more (...) precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics. (shrink)
Modal and Intensional Logic in Logic and Philosophy of Logic
Nonclassical Logics in Logic and Philosophy of Logic
Modal Definability of First-Order Formulas with Free Variables and Query Answering.Stanislav Kikot & Evgeny Zolin - 2013 - Journal of Applied Logic 11 (2):190-216.details
The Lattice of Belnapian Modal Logics: Special Extensions and Counterparts.Sergei P. Odintsov & Stanislav O. Speranski - 2016 - Logic and Logical Philosophy 25 (1):3-33.details
Let K be the least normal modal logic and BK its Belnapian version, which enriches K with 'strong negation'. We carry out a systematic study of the lattice of logics containing BK based on: • introducing the classes of so-called explosive, complete and classical Belnapian modal logics; • assigning to every normal modal logic three special conservative extensions in these classes; • associating with every Belnapian modal logic its explosive, complete and classical counterparts. We investigate the relationships between special extensions (...) and counterparts, provide certain handy characterisations and suggest a useful decomposition of the lattice of logics containing BK. (shrink)
Mitochondrial/Nuclear Transfer: A Literature Review of the Ethical, Legal and Social Issues.Raphaëlle Dupras-Leduc, Stanislav Birko & Vardit Ravitsky - unknowndetails
Mitochondrial/nuclear transfer to avoid the transmission of serious mitochondrial disease raises complex and challenging ethical, legal and social issues. In February 2015, the United Kingdom became the first country in the world to legalize M/NT, making the heated debate surrounding this technology even more relevant. This critical interpretive review identified 95 relevant papers discussing the ELSI of M/NT, including original research articles, government-commissioned reports, editorials, letters to editors and research news. The review presents and synthesizes the arguments present in the (...) literature in relation to the most commonly raised themes: terminology; identity, relationships and parenthood; potential harm; reproductive autonomy; available alternatives; consent; impact on specific interest groups; resources; "slippery slope"; creation, use and destruction of human embryos; and beneficence. The review concludes by identifying those ELSI that are specific to M/NT and by calling for follow-up longitudinal clinical and psychosocial research in order to equip future ELSI debate with empirical evidence. (shrink)
Notes on the Computational Aspects of Kripke's Theory of Truth.Stanislav Speranski - 2017 - Studia Logica 105 (2):407-429.details
The paper contains a survey on the complexity of various truth hierarchies arising in Kripke's theory. I present some new arguments, and use them to obtain a number of interesting generalisations of known results. These arguments are both relatively simple, involving only the basic machinery of constructive ordinals, and very general.
Liar Paradox in Logic and Philosophy of Logic
Evidence‐Based Medicine Training and Implementation in Surgery: The Role of Surgical Cultures.Simon Kitto, Ana Petrovic, Russell L. Gruen & Julian A. Smith - 2011 - Journal of Evaluation in Clinical Practice 17 (4):819-826.details
Philosophy of Medicine in Philosophy of Science, Misc
Reasoning About Arbitrary Natural Numbers From a Carnapian Perspective.Leon Horsten & Stanislav O. Speranski - 2019 - Journal of Philosophical Logic 48 (4):685-707.details
Inspired by Kit Fine's theory of arbitrary objects, we explore some ways in which the generic structure of the natural numbers can be presented. Following a suggestion of Saul Kripke's, we discuss how basic facts and questions about this generic structure can be expressed in the framework of Carnapian quantified modal logic.
The Politics of Survival in Foundations of Education: Borderlands, Frames, and Strategies.Aaron M. Kuntz & John E. Petrovic - 2011 - Educational Studies: A Jrnl of the American Educ. Studies Assoc 47 (2):174-197.details
Reason, Liberalism, and Democratic Education: A Deweyan Approach to Teaching About Homosexuality.John E. Petrovic - 2013 - Educational Theory 63 (5):525-541.details
Teaching about homosexuality, especially in a positive light, has long been held to be a controversial issue. There is, however, a view of the capacity for reason that finds that those who deem homosexuality to be controversial will ultimately contradict themselves, becoming unreasonable. By this standard of reason, homosexuality should be treated as non controversial in schools. In this essay, John Petrovic argues that this epistemic position is problematic. Instead, he defends a Deweyan epistemology that casts reason as, in part, (...) a set of socially acquired habits of mind. People who have been socialized into heterosexist habits of mind must be exposed to counterhegemonic discourses. One such discourse can be found in the public values of liberal democracy through which the practice of reason must be pursued. Petrovic discusses the practical guidance provided by assuming a view of normative reason versus habituated reason in terms of both pedagogy and curriculum. (shrink)
A Grammar of the Ugaritic Language.Stanislav Segert, Daniel Sivan & A. F. Rainey - 1999 - Journal of the American Oriental Society 119 (1):137.details
Philosophy of Linguistics in Philosophy of Language
Career Ambition as a Way of Understanding the Relation Between Locus of Control and Self-Perceived Employability Among Psychology Students.Maja Ćurić Dražić, Ivana B. Petrović & Milica Vukelić - 2018 - Frontiers in Psychology 9.details
On Algorithmic Properties of Propositional Inconsistency-Adaptive Logics.Sergei P. Odintsov & Stanislav O. Speranski - 2012 - Logic and Logical Philosophy 21 (3):209-228.details
The present paper is devoted to computational aspects of propositional inconsistency-adaptive logics. In particular, we prove (relativized versions of) some principal results on computational complexity of derivability in such logics, namely in cases of CLuN r and CLuN m , i.e., CLuN supplied with the reliability strategy and the minimal abnormality strategy, respectively.
Paraconsistent Logic in Logic and Philosophy of Logic
K příspěvku prof. Cmoreje.Stanislav Sousedík - 2011 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 18 (2):239-244.details
Ervin Laszlo's Akashic Field and the Dilemmas of Modern Consciousness Research.Stanislav Grof - 2006 - World Futures 62 (1 & 2):86 – 102.details
Ervin Laszlo's revolutionary concept of the Akashic Field and his connectivity hypothesis offer elegant solutions for the baffling paradoxes associated with "anomalous phenomena" - otherwise unexplainable observations which many scientific disciplines encountered in the course of the 20th century. This article explores the ground-breaking contributions that Laszlo's work has made to psychology by providing a plausible conceptual framework for a large number of observations and experiences amassed by modern consciousness research, which challenge the most fundamental assumptions of the traditional scientific (...) worldview. (shrink)
Non-Finitely Axiomatisable Modal Product Logics with Infinite Canonical Axiomatisations.Christopher Hampson, Stanislav Kikot, Agi Kurucz & Sérgio Marcelino - 2020 - Annals of Pure and Applied Logic 171 (5):102786.details
Our concern is the axiomatisation problem for modal and algebraic logics that correspond to various fragments of two-variable first-order logic with counting quantifiers. In particular, we consider modal products with Diff, the propositional unimodal logic of the difference operator. We show that the two-dimensional product logic $Diff \times Diff$ is non-finitely axiomatisable, but can be axiomatised by infinitely many Sahlqvist axioms. We also show that its 'square' version (the modal counterpart of the substitution and equality free fragment of two-variable first-order (...) logic with counting to two) is non-finitely axiomatisable over $Diff \times Diff$ , but can be axiomatised by adding infinitely many Sahlqvist axioms. These are the first examples of products of finitely axiomatisable modal logics that are not finitely axiomatisable, but axiomatisable by explicit infinite sets of canonical axioms. (shrink)
The Book of Ben Sira in Modern Research: Proceedings of the First International Ben Sira Conference, 28-31 July 1996, Soesterberg, Netherlands. [REVIEW]Stanislav Segert & Pancratius C. Beentjes - 2000 - Journal of the American Oriental Society 120 (1):145.details
The Academic Spin-Offs as an Engine of Economic Transition in Eastern Europe. A Path-Dependent Approach.Ivan Tchalakov, Tihomir Mitev & Venelin Petrov - 2010 - Minerva 48 (2):189-217.details
The paper questions some of the premises in studying academic spin-offs in developed countries, claiming that when taken as characteristics of 'academic spin-offs per se,' they are of little help in understanding the phenomenon in the Eastern European countries during the transitional and post-transitional periods after 1989. It argues for the necessity of adopting a path-dependent approach, which takes into consideration the institutional and organisational specificities of local economies and research systems and their evolution, which strongly influence the patterns of (...) spin-off activity. The paper provides new findings and original arguments in support of Balazs' seminal theses (Balazs 1995, 1996) about the emergence of academic spin-offs during the early transition. It reveals key economic and policy mechanisms bearing on academic entrepreneurship in Eastern Europe, such as the tensions between economic and political nomenclatures of former Communist Parties, where the dismantling or preservation of the power of political nomenclature resulted in different patterns of development—rapid reforms in the 'first wave' of EU accession countries or the establishment of rent-seeking and assets-stripping economies in countries like Bulgaria and Romania, making the transition period especially difficult. In the latter, a specific economic environment emerged, unknown in Western Europe and in the 'champions' of transition—such as suppression of the authentic entrepreneurship in a number of economic sectors, disintegration of corporate structures, etc. Thus, the paper reveals the common ground behind the two conflicting tendencies in post-socialist academic spin-offs, partially outlined in other research (Simeonova 1995; Pavlova 2000): as an authentic form of academic entrepreneurship grasping the opportunities opened up by the economic crisis and compensating failures in science and technology policy on the one hand, and as specific rent-seeking strategy draining valuable public assets on the other (the latter, in turn, boosting the negative attitudes in local scientific communities). The paper provides new findings about the evolution of the academic spin-offs in Bulgaria along the two polar trends and their positive and negative repercussions on parent research institutions. The results were achieved in the PROKNOW Project, EC 6th Framework Program. (shrink)
Kritická poznámka k sémantickým východiskům transparentní intenzionální logiky.Stanislav Sousedík - 2010 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 17 (3):358-365.details
K ontologii intencionálních jsoucen.Stanislav Sousedík - 2011 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 18 (4):524-530.details
Platónské a aristotelské pojetí intencionálních jsoucen.Stanislav Sousedík - 2012 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 19 (1):84-95.details
Work Engagement in Serbia: Psychometric Properties of the Serbian Version of the Utrecht Work Engagement Scale.Ivana B. Petrović, Milica Vukelić & Svetlana Čizmić - 2017 - Frontiers in Psychology 8.details
Spiritual Emergency: The Understanding and Treatment of Transpersonal Crises.Christina Grof & Stanislav Grof - 2017 - International Journal of Transpersonal Studies 36 (2):30-43.details
Invasion, Alienation, and Imperialist Nostalgia: Overcoming the Necrophilous Nature of Neoliberal Schools.John E. Petrovic & Aaron M. Kuntz - 2018 - Educational Philosophy and Theory 50 (10):957-969.details
The authors present a materialist analysis of the effects of neoliberalism in education. Specifically, they contend that neoliberalism is a form of cultural invasion that begets necrophilia. Neoliberalism is necrophilous in promoting a cultural desire to fix fluid systems and processes. Such desire manufactures both individuals known and culturally felt experiences of alienation which are, it is argued, symptomatic of an imperialist nostalgia that permeates educational policy and practice. The authors point to 'unschooling in schools' as a mechanism for resisting (...) the necrophilous tendencies of contemporary formations of education. (shrink)
The Dynamics of Scaling: A Memory-Based Anchor Model of Category Rating and Absolute Identification.Alexander A. Petrov & John R. Anderson - 2005 - Psychological Review 112 (2):383-416.details
Memory and Cognitive Science in Philosophy of Mind
Stanislav Orikhovsky on Religious Tolerance.R. Mnozhynska - 2004 - Ukrainian Religious Studies 30:68-79.details
Stanislav Orikhovsky - one of the most prominent Latin-speaking Ukrainian-Polish humanists of the first half of the 16th century. For a long time he was known almost exclusively as a Polish figure. We now have every reason to include him in our culture, above all because he was a conscious Ukrainian - he invariably added the term "Ukrainian" to his last name. this is frankly stated ". In Western Europe it was called "Ukrainian Demosthenes" and "modern Cicero." His teachers (...) were famous figures of the time: the German humanist Melanchthon and the reformer Martin Luther - in the latter's house he even lived during his studies. Many prominent people of that time in Italy and Poland also had the honor of communicating with him. (shrink)
A Note on Definability in Fragments of Arithmetic with Free Unary Predicates.Stanislav O. Speranski - 2013 - Archive for Mathematical Logic 52 (5-6):507-516.details
We carry out a study of definability issues in the standard models of Presburger and Skolem arithmetics (henceforth referred to simply as Presburger and Skolem arithmetics, for short, because we only deal with these models, not the theories, thus there is no risk of confusion) supplied with free unary predicates—which are strongly related to definability in the monadic SOA (second-order arithmetic) without × or + , respectively. As a consequence, we obtain a very direct proof for ${\Pi^1_1}$ -completeness of Presburger, (...) and also Skolem, arithmetic with a free unary predicate, generalize it to all ${\Pi^1_n}$ -levels, and give an alternative description of the analytical hierarchy without × or + . Here 'direct' means that one explicitly m-reduces the truth of ${\Pi^1_1}$ -formulae in SOA to the truth in the extended structures. Notice that for the case of Presburger arithmetic, the ${\Pi^1_1}$ -completeness was already known, but the proof was indirect and exploited some special ${\Pi^1_1}$ -completeness results on so-called recurrent nondeterministic Turing machines—for these reasons, it was hardly able to shed any light on definability issues or possible generalizations. (shrink)
Areas of Mathematics in Philosophy of Mathematics
Ještě k problematice existence.Stanislav Sousedík - 2010 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 17 (1):81-84.details
The Dynamics of Perceptual Learning: An Incremental Reweighting Model.Alexander A. Petrov, Barbara Anne Dosher & Zhong-Lin Lu - 2005 - Psychological Review 112 (4):715-743.details
Aspects of Consciousness in Philosophy of Mind
Gas Fuelled Engines Like Ecological Alternative for Driving of Motor Vehicles.Stanislav Beroun, Ladislav BARTONÍČEK, Josef Laurin & Celestýn Scholz - 2005 - In Alan F. Blackwell & David MacKay (eds.), Power. Cambridge University Press. pp. 1.details
Two Remarks on Partitions of Ω with Finite Blocks.Stanislav Krajči - 1999 - Mathematical Logic Quarterly 45 (3):415-420.details
We prove that all algebras P/IR, where the IR-'s are ideals generated by partitions of W into finite and arbitrary large elements, are isomorphic and homogeneous. Moreover, we show that the smallest size of a tower of such partitions with respect to the eventually-refining preordering is equal to the smallest size of a tower on ω.
Process Philosophy in the European Cultural Tradition.Vesselin Petrov - 2017 - Annals of the University of Bucharest - Philosophy Series 66 (1).details
The subject of the present exposition is namely the discussion of the question when and how process philosophy has entered into the European cultural tradition. We could approach this question in two ways: the philosophical and the historical perspective. We shall focus of our attention on both perspectives, which concern the return of process philosophy in Europe after its original moulding as a contemporary philosophical trend by Whitehead and his immediate followers. If we trace and systematize chronologically the acts of (...) influence of Whitehead's works on eminent scientists and philosophers in Europe, we can provisionally outline three stages. The paper evaluates the manner in which process philosophy became so attractive for the European philosophical thought. At the present time we can definitely conclude that it is confirmed the existence of "a school" of representatives of process philosophy in Europe. (shrink)
Realization of the Maupertuis Principle in Morphogenesis.Ernst A. Petrov - 1998 - Acta Biotheoretica 46 (1):77-80.details
Developmental Biology in Philosophy of Biology
Philosophy of Biology, General Works in Philosophy of Biology
Ubijanje nevinih: slučaj 11. rujna 2001.Neven Petrović - 2011 - Filozofska Istrazivanja 31 (3):635-649.details
Hintikka's Independence-Friendly Logic Meets Nelson's Realizability.Sergei P. Odintsov, Stanislav O. Speranski & Igor Yu Shevchenko - 2018 - Studia Logica 106 (3):637-670.details
Inspired by Hintikka's ideas on constructivism, we are going to 'effectivize' the game-theoretic semantics for independence-friendly first-order logic, but in a somewhat different way than he did in the monograph 'The Principles of Mathematics Revisited'. First we show that Nelson's realizability interpretation—which extends the famous Kleene's realizability interpretation by adding 'strong negation'—restricted to the implication-free first-order formulas can be viewed as an effective version of GTS for FOL. Then we propose a realizability interpretation for IF-FOL, inspired by the so-called 'trump (...) semantics' which was discovered by Hodges, and show that this trump realizability interpretation can be viewed as an effective version of GTS for IF-FOL. Finally we prove that the trump realizability interpretation for IF-FOL appropriately generalises Nelson's restricted realizability interpretation for the implication-free first-order formulas. (shrink)
O čem je řeč v partikulárních větách.Stanislav Sousedík - 2012 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 19 (2):238-247.details
The Price of Query Rewriting in Ontology-Based Data Access.Georg Gottlob, Stanislav Kikot, Roman Kontchakov, Vladimir Podolskii, Thomas Schwentick & Michael Zakharyaschev - 2014 - Artificial Intelligence 213:42-59.details
Computability Issues for Adaptive Logics in Multi-Consequence Standard Format.Sergei P. Odintsov & Stanislav O. Speranski - 2013 - Studia Logica 101 (6):1237-1262.details
In a rather general setting, we prove a number of basic theorems concerning computational complexity of derivability in adaptive logics. For that setting, the so-called standard format of adaptive logics is suitably adopted, and the corresponding completeness results are established in a very uniform way.
Materialistické pojetí dejín a dialektika výrobních sil.Stanislav Adam - 1979 - Filozofia 34:70.details
1 — 50 / 335 | CommonCrawl |
Resources tagged with: Generalising
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Mathematical reasoning & proof. Indices. Generalising. Rounding. Polynomial functions and their roots. Powers & roots. Creating and manipulating expressions and formulae. Limits of Sequences. Inequalities. Calculating with fractions.
Broad Topics > Using, Applying and Reasoning about Mathematics > Generalising
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
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\begin{document}
\title[Generalized K\"ahler manifolds with split tangent bundle] {Generalized K\"ahler manifolds with split tangent bundle}
\author[V. Apostolov]{Vestislav Apostolov} \address{Vestislav Apostolov \\ D{\'e}partement de Math{\'e}matiques\\ UQAM\\ C.P. 8888 \\ Succ. Centre-ville \\ Montr{\'e}al (Qu{\'e}bec) \\ H3C 3P8 \\ Canada} \email{[email protected]} \author[M. Gualtieri]{Marco Gualtieri} \address{Marco Gualtieri\\ M.I.T. Department of Mathematics \\ 77 Massachusetts Avenue \\Cambridge MA 02139-4307\\ USA } \email{[email protected]}
\thanks{We would like to thank P. Gauduchon, G. Grantcharov and N.~J.~Hitchin for their help and stimulating discussions.} \date{May 22, 2006}
\begin{abstract} We study generalized K\"ahler manifolds for which the corresponding complex structures commute and classify completely the compact generalized K\"ahler four-manifolds for which the induced complex structures yield opposite orientations. \end{abstract}
\maketitle
\section{Introduction} The notion of a {\it generalized K\"ahler structure} was introduced and studied by the second author in \cite{gualtieri}, in the context of the theory of generalized geometric structures initiated by Hitchin in \cite{hitchin}. Recall that a generalized K\"ahler structure is a pair of commuting complex structures $(\mathcal J_1,\mathcal J_2)$ on the vector bundle $TM \oplus T^*M$ over the smooth manifold $M^{2m}$, which are: \begin{enumerate} \item[$\bullet$] integrable with respect to the (twisted) Courant bracket on $TM\oplus T^*M$, \item[$\bullet$] compatible with the natural inner-product $\langle \cdot, \cdot \rangle$ of signature $(2m,2m)$ on $TM \oplus T^*M$, \item[$\bullet$] and such that the quadratic form $\langle {\mathcal J}_1\ \cdot,\mathcal J_2\ \cdot \rangle$ is definite on $TM \oplus T^*M$. \end{enumerate}
It turns out~\cite{gualtieri} that such a structure on $TM \oplus T^*M$ is equivalent to a triple $(g,J_+,J_-)$ consisting of a Riemannian metric $g$ and two integrable almost complex structures $J_{\pm}$ compatible with $g$, satisfying the integrability relations $$d^c_+ F_+ +d^c_- F_- = 0, \ dd^c_\pm F_\pm = 0,$$ where $F_{\pm} = gJ_{\pm}$ are the fundamental 2-forms of the Hermitian structures $(g,J_{\pm})$, and $d^c_\pm$ are the $i(\overline{\partial}-\partial)$ operators associated to the complex structures $J_\pm$. The closed 3-form $H=d^c_+ F_+=-d^c_-F_-$ is called the \emph{torsion} of the generalized K\"ahler structure.
These conditions on a pair of Hermitian structures were first described by Gates, Hull and Ro\v{c}ek~\cite{physicists} as the general target space geometry for a $(2,2)$ supersymmetric sigma model.
As a trivial example we can take a K\"ahler structure $(g,J)$ on $M$ and put $J_+=J$, $J_-= \pm J$ to obtain a solution of the above equations. One can ask, more generally, the following
\noindent {\it Question 1.} {When does a compact complex manifold $(M,J)$ admit a generalized K\"ahler structure $(g,J_+,J_-)$ with $J=J_+$?}
The case of interest is when $J_+ \neq \pm J_-$, i.e.~when the generalized K\"ahler structure does not come from a genuine K\"ahler structure on $(M,J)$. In this paper, we refer to such generalized K\"ahler structures as {\it non-trivial}.
Despite a growing number of explicit constructions~\cite{AGG,gualtieri-et-al,hitchin2, kobak,Lin-Tolman}, the general existence problem for non-trivial generalized K\"ahler structures remains open. On the other hand, there are a number of known obstructions, or conditions that the existence of a generalized K\"ahler structure imposes on the underlying complex manifold, which we now describe.
Firstly, it follows from the definition that for a complex manifold $(M,J)$ to admit a compatible generalized K\"ahler structure it must also admit a Hermitian metric whose fundamental 2-form is $\partial {\bar \partial}$-closed. This condition on $(M,J)$ is familiar in Hermitian geometry. It is trivially satisfied if $(M,J)$ is of K\"ahler type (i.e.~$(M,J)$ admits a K\"ahler metric). When $M$ is compact and four-dimensional ($m=2$), a result of Gauduchon~ \cite{gauduchon1} affirms that any Hermitian conformal class contains a metric with $\partial {\bar \partial}$-closed fundamental form. Hermitian metrics with $\partial{\bar \partial}$-closed fundamental form naturally appear in the study of local index theory~\cite{bismut}, on the moduli space of stable vector bundles~\cite{lubke-teleman}, and have been much discussed in the physics literature where they are referred to as `strong K\"ahler with torsion' structures. Complex manifolds admitting such Hermitian metrics are the subject of a number of other interesting results~\cite{egidi,fino-parton-salamon,granch-granch-poon,kutche,spindel-et-al}. Examples from \cite{egidi}, together with the results of \cite{fino-granch} and \cite{fino-parton-salamon}, show that there are compact complex manifolds of any dimension $2m>4$ which do not admit any Hermitian metric with $\partial {\bar \partial}$-closed fundamental form.
Secondly, Hitchin~\cite{hitchin2} showed that if $(M,J)$ carries a generalized K\"ahler structure $(g,J_+, J_-,H)$ such that $J=J_+$ and $J_{+},J_-$ do not commute, then the commutator defines a {\it holomorphic Poisson structure} $\pi = [J_+,J_-]g^{-1}$ on $(M,J)$. In the case when $H^0(M, \wedge^2 TM) =0$, for instance, this result implies that for any compatible generalized K\"ahler structure on $(M,J)$, the complex structures $J_{+}$ and $J_-$ must commute, i.e. $J_+J _- = J_-J_+$.
Thus motivated, we study in this paper non-trivial generalized K\"ahler structures $(g,J_+, J_-)$ for which $J_+$ and $J_-$ commute. In this case $Q=J_+J_-$ is an involution of the tangent bundle $TM$, and thus gives rise to a splitting $TM = T_-M \oplus T_+M$ as a direct sum of the $(\pm 1)$-eigenspaces of $Q$. Our first result, Theorem~\ref{main}, proves an assertion first made in \cite{physicists}, which can be stated as follows: {\it the sub-bundles $T_{\pm}M$ are tangent to the leaves of two transversal holomorphic foliations ${\mathcal F}_{\pm}$ on $(M,J_+)$ and $g$ restricts to each leaf to define a K\"ahler metric}.
The fact that $T_{\pm}M$ are both {\it holomorphic} and {\it integrable} sub-bundles of $TM$ directly relates our existence problem to a conjecture by Beauville~\cite{beauville}, which states that the holomorphic tangent bundle $TM$ of a compact complex manifold $(M,J)$ of K\"ahler type splits as the direct sum of two holomorphic integrable sub-bundles if and only if $M$ is covered by the product of two complex manifolds $M_1\times M_2$ on which the fundamental group of $M$ acts {\it diagonally}. This conjecture has been confirmed in various cases~\cite{beauville,campana-peternell,druel}. Combined with Hitchin's result~\cite{hitchin2} mentioned above, we obtain a wealth of K\"ahler complex manifolds which do not admit non-trivial twisted generalized K\"ahler structures at all. As pointed out in~\cite{hitchin3}, such examples include (locally) deRham irreducible compact K\"ahler--Einstein manifolds with $c_1(M)<0$ (see Theorem~\ref{KE-bis} below).
The existence of non-trivial generalized K\"ahler structures for which $J_+$ and $J_-$ commute thus reduces to the following question:
\noindent {\it Question 2.} Let $(M,J)$ be a compact complex manifold whose holomorphic tangent bundle splits as a direct sum of two holomorphic, integrable sub-bundles $T_{\pm} M$. Define a second almost complex structure $J_-$ on $M$ to be equal to $J$ on $T_-M$ and to $-J$ on $T_+M$. Does there exist a Riemannian metric $g$ on $M$ which is compatible with $J_+:=J$ and $J_-$, and such that $(g,J_{\pm})$ is a generalized K\"ahler structure on $M$?
We note that the almost complex structure $J_-$ defined as above is automatically integrable and commutes with $J_+$.
The fact that any maximal integral submanifold of $T_{\pm}M$ must be K\"ahler with respect to a compatible generalized K\"ahler metric quickly leads to non-K\"ahler examples where the answer to Question~2 is negative (see Example~\ref{high-dim-ex}). Another obstruction comes from the fact that the fundamental 2-form of a compatible generalized K\"ahler metric must be $\partial {\bar \partial}$-closed (see Example~\ref{solvmanifold}). We are thus led to suspect that the above existence problem should be more tractable when $(M,J)$ is of K\"ahler type, and we conjecture that in this case the answer to our Question~2 is `{\it yes}'. We are able to establish this in two special cases treated by Beauville in~\cite{beauville}, namely when $(M,J)$ admits a K\"ahler--Einstein metric (Theorem~\ref{KE}), and when $(M,J)$ is four-dimensional ($m=2$).
When $M$ is four dimensional, our resuts are much sharper. In this case there are two classes of generalized K\"ahler structures, according to whether $J_{+}$ and $J_{-}$ induce the same or different orientations on $M$. In this paper we shall refer to these cases as generalized K\"ahler structures of {\it bihermitian} or {\it ambihermitian} type, respectively, though in the terminology of \cite{gualtieri} they would correspond to generalized K\"ahler structures of purely even and purely odd type, respectively. Note that generalized K\"ahler structures of {ambihermitian} type are precisely those for which $J_+$ and $J_-$ commute and $J_+ \neq \pm J_-$.
In section \ref{four}, we solve completely the existence problem of generalized K\"ahler 4-manifolds of ambihermitian type, by proving the following result.
\begin{thm}\label{main} A compact complex surface $(M,J)$ admits a generalized K\"ahler structure of ambihermitian type $(g,J_+,J_-)$ with $J_+ = J$ if and only if the holomorphic tangent bundle of $(M,J)$ splits as a direct sum of two holomorpic sub-bundles. Such a complex surface $(M,J)$ is biholomorphic to one of the following: \begin{enumerate} \item[\rm (a)] a geometrically ruled complex surface which is the projectivization of a projectively flat holomorphic vector bundle over a compact Riemann surface; \item[\rm (b)] a {bi-elliptic} complex surface, i.e.~a complex surface finitely covered by a complex torus; \item[\rm (c)] a compact complex surface of Kodaira dimension $1$ and even first Betti number, which is an elliptic fibration over a compact Riemann surface, whose only singular fibres are multiple smooth elliptic curves; \item[\rm (d)] a compact complex surface of general type, uniformized by the product of two hyperbolic planes ${\mathbb H}\times {\mathbb H}$ and with fundamental group acting diagonally on the factors. \item[\rm (e)] A Hopf surface, with universal covering space ${\mathbb C}^2 \setminus \{(0,0)\}$ and fundamental group generated by a diagonal automorphism $(z_1,z_2) \mapsto (\alpha z_1, \beta z_2)$ with
$0<|\alpha| \le |\beta|<1$, and a diagonal automorphism $(z_1,z_2) \mapsto (\lambda z_1, \mu z_2)$ with $\lambda, \mu$ primitive $\ell$-th roots of $1$. \item[\rm(f)] An Inoue surface in the family $S_{\mathcal M}$ constructed in \cite{inoue}. \end{enumerate} On any of the above complex surfaces there exists a family {\rm (}depending on one arbitrary smooth function on $M${\rm )} of generalized K\"ahler structures of ambihermitian type. \end{thm} To prove this theorem we use the fact that the commuting complex structures give rise to a splitting of the holomorphic tangent bundle of $(M,J_+)$ into two holomorphic line bundles $T_{\pm} M$. Using this splitting and the methods of \cite{gauduchon1}, we describe the set of all generalized K\"ahler structures of ambihermitian type on such a complex surface. We thus establish a one-to-one correspondence between four-manifolds admitting generalized K\"ahler structures of ambihermitian type and complex surfaces with split holomorphic tangent bundle. The latter class of complex surfaces has been studied by Beauville~\cite{beauville}. We use his classification and some results from \cite{wall} to derive Theorem~\ref{main}.
We further refine our classification by considering the \emph{untwisted} case, i.e. when $[H]=0\in H^3(M,\mathbb R)$, and the \emph{twisted} case, where $[H]$ is nonzero. We show, by using the fundamental results of Gauduchon~\cite{gauduchon1, gauduchon2}, that untwisted generalized K\"ahler structures on compact four-manifolds can only exist when the first Betti number is even; likewise in the \emph{twisted} case, any generalized K\"ahler 4-manifold must have odd first Betti number (Corollary~\ref{B1}).
\section{Hermitian geometry} In this section we present certain key properties of Hermitian manifolds which we will need in the later sections, giving special attention to the four-dimensional case. Let $M$ be an oriented $2m$-dimensional manifold. A {\it Hermitian structure} on $M$ is defined by a pair $(g,J)$ consisting of a Riemannian metric $g$ and an integrable almost complex structure $J$, which are {\it compatible} in the sense that $g(J\cdot, J\cdot) = g(\cdot, \cdot)$. The Hermitian structure $(g,J)$ is called {\it positive} if $J$ induces the given orientation on $M$ and {\it negative} otherwise.
The complex structure $J$ induces a decomposition $TM\otimes{\mathbb C}=T^{1,0}M\oplus T^{0,1}M$ of the complexified vectors into $\pm i$ eigenspaces, and hence defines the usual bi-grading of complex differential forms $$\Omega^k(M) \otimes {\mathbb C} = \bigoplus_{p+q=k} \Omega^{p,q}(M),$$ where we let $J$ act on $T^*M$ by $(J\alpha)(X)=-\alpha(JX)$, so that it commutes with the Riemannian duality between vectors and 1-forms: $(J\alpha)^{\sharp} = J \alpha^{\sharp}$.
The product structure $\wedge^2 J$ induces a splitting of the real 2-forms into $\pm 1$ eigenspaces: $$\Omega^2(M)=\Omega^{J,+}(M)\oplus\Omega^{J,-}(M),$$ whose complexification is simply $\Omega^{J,+}(M)\otimes{\mathbb C} = \Omega^{1,1}(M)$ and $\Omega^{J,-}(M)\otimes{\mathbb C} = \Omega^{2,0}(M)\oplus\Omega^{0,2}(M)$. Furthermore, the {\it fundamental 2-form} $F=gJ$, a real $(1,1)$-form of square-norm $m$, defines a $g$-orthogonal splitting $\Omega^{J,+}(M)={\mathcal C}^{\infty}(M)\cdot F \oplus \Omega^{J,+}_0(M)$. In this way we obtain the $U(m)$ irreducible decomposition of real 2-forms: $$\Omega^2(M) = {\mathcal C}^{\infty}(M)\cdot F \oplus \Omega^{J,+}_0(M) \oplus \Omega^{J,-}(M).$$
On a positive Hermitian 4-manifold, the above $U(2)$ splitting of $\Omega^2(M)$ is compatible with the $SO(4)$ decomposition $\Omega^2(M)=\Omega^+(M)\oplus\Omega^-(M)$ into self-dual and anti-self-dual forms, as follows: \begin{equation}\label{U(2)-split} \Omega^+(M)= {\mathcal C}^{\infty}(M) \cdot F \oplus \Omega^{J,-}(M); \ \ \Omega^-(M) = \Omega_0^{J,+}(M). \end{equation} For a negative Hermitian structure the r\^oles of $\Omega^+(M)$ and $\Omega^-(M)$ in the above identifications are interchanged. Thus, on an oriented Riemannian four-manifold $(M,g)$, we obtain the well-known correspondence between smooth sections in $\Omega^+(M)$ (resp. $\Omega^-(M)$) of square-norm 2 and positive (resp. negative) almost Hermitian structures $(g,J)$. Whereas the existence of such smooth sections is a purely topological problem, the existence of integrable ones depends essentially on $g$. This is measured (at least at a first approximation) by the structure of the Weyl curvature tensor $W$, cf.~\cite{AG,pontecorvo,salamon}.
The {\it Lee form} $\theta\in\Omega^1(M)$ of a Hermitian structure is defined by \begin{equation}\label{Lee0} dF\wedge F^{m-2} = \frac{1}{(m-1)}\theta \wedge F^{m-1}, \end{equation} or equivalently $\theta = J \delta^g F$ where $\delta^g$ is the co-differential with respect to the Levi--Civita connection $D^g$ of $g$. Since $J$ is integrable, $dF$ measures the deviation of $(g,J)$ from a K\"ahler structure (for which $J$ and $F$ are parallel with respect to $D^g$). We have the following expression for $D^gF$ (see e.g.~\cite[p.148]{KN}): \begin{equation}\label{Kob-Nom}\begin{split} 2g((D^g_X J)Y,Z) &= d^cF({X,Y,JZ}) + d^cF(X,JY,Z), \end{split} \end{equation} where $d^c=i(\bar\partial - \partial)$, so that $d^cF= \wedge^3 J( dF)$ is a real 3-form of type $(1,2)+(2,1)$.
In four dimensions, \eqref{Lee0} reads as \begin{equation}\label{Lee} dF = \theta \wedge F, \end{equation} and \eqref{Kob-Nom} becomes (see e.g. \cite{gauduchon1, vaisman1}) \begin{equation}\label{DF} D^g_X F = \tfrac{1}{2}( X^{\flat}\wedge J\theta + JX^{\flat}\wedge \theta), \end{equation} where $X^{\flat}=g(X)$ denotes the $g$-dual 1-form to $X$. We see from this that a Hermitian 4-manifold is K\"ahler if and only if $\theta=0$.
The existence of a K\"ahler metric on a compact complex manifold $(M^{2m},J)$ implies the Hodge decomposition of the de~Rham cohomology groups $$H^k_{dR}(M,{\mathbb C})\cong \bigoplus_{p+q=k} H^{p,q}_{\bar \partial}(M),$$ where $H^{p,q}_{\bar \partial}(M)$ denote the Dolbeault cohomology groups. This, together with the equality $H^{p,q}_{\bar \partial} (M)\cong {\overline {H^{q,p}_{\bar\partial} (M)}},$ implies that the odd Betti numbers of a complex manifold admitting a K\"ahler metric must be even. When $m=2$, it turns out that this condition is also sufficient.
\begin{thm}\label{kahlerian} \cite{buchdahl,lamari,siu,todorov} Let $M$ be a compact four-manifold endowed with an integrable almost complex structure $J$. Then there exists a compatible K\"ahler metric on $(M,J)$ if and only if $b_1(M)$ is even. \end{thm} This important result was first established by Todorov~\cite{todorov} and Siu~\cite{siu}, using the Kodaira classification of compact complex surfaces. Direct proofs were found recently by Buchdahl~\cite{buchdahl} and Lamari~\cite{lamari}.
Since we deal with complex manifolds of non-K\"ahler type (i.e.~do not admit any K\"ahler metric), we recall the definition of the $\partial {\bar \partial}$-cohomology groups: $$H^{p,q}_{\partial {\bar \partial}}(M) := \{d\textrm{-closed} \ (p,q)\textrm{-forms}\}/\partial{\bar \partial}\{(p-1,q-1)\textrm{-forms}\}.$$ Note that there is a natural map $$\iota : H^{p,q}_{\partial {\bar \partial}}(M) \to H^{p,q}_{\bar \partial}(M).$$ When $(M,J)$ is of K\"ahler type, the well-known $\partial {\bar \partial}$-lemma (see e.g.~\cite{demailly}) states that the above map is in fact an isomorphism: \begin{prop}\label{dd^c-lemma}{\rm ($\partial {\bar \partial}$-lemma)} If $(M,J)$ is a compact complex manifold admitting a K\"ahler metric, then $\iota : H^{p,q}_{\partial {\bar \partial}}(M)\to H^{p,q}_{\bar \partial}(M)$ is an isomorphism. \end{prop} The $\partial {\bar \partial}$-lemma also holds on some non-K\"ahler manifolds, for example on all non-projective Moi\u{s}ezon manifolds. In fact, the $\partial {\bar \partial}$-lemma is preserved under bimeromorphic transformations and, therefore, holds on any compact complex manifold which is bimeromorphic to a K\"ahler manifold (i.e.~is in the so-called {\it Fujiki class ${\mathcal C}$}), cf.~\cite{demailly}.
While the existence of K\"ahler metrics on a compact complex manifold $(M,J)$ is generally obstructed, a fundamental result of Gauduchon~\cite{gauduchon1} states that on any {compact} conformal Hermitian manifold $(M,c,J)$, there exists a unique (up to scale) Hermitian metric $g \in c$, such that its Lee form $\theta$ is co-closed, i.e. satisfies $\delta^g \theta =0$. Such a metric is called a {\it standard} metric of $c$. By \eqref{Lee0}, a standard metric of $(c,J)$ can be equivalently defined by the equation $$2i\partial {\bar \partial} F ^{m-1}= dd^c (F^{m-1})=0. $$
We now recall how, in four dimensions, the harmonic properties of the Lee form with respect to a standard metric are related the parity of the first Betti number (compare with Theorem~\ref{kahlerian} above). \begin{prop}\label{gauduchon}\cite{gauduchon1, gauduchon2} Let $M$ be a compact four-manifold endowed with a conformal class $c$ of Hermitian metrics, with respect to an integrable almost complex structure $J$. Let $g$ be a standard Hermitian metric in $c$. Then the following two conditions are equivalent: \begin{enumerate} \item[\rm (i)] The first Betti number $b_1(M)$ is even. \item[\rm (ii)] The Lee form $\theta$ of $g$ is co-exact. \end{enumerate} \end{prop} \begin{proof} For the sake of completeness we outline a proof of this result. Let $M$ be a compact four-manifold endowed with a standard Hermitian structure $(g,J)$, and $F$ and $\theta = J \delta^g F$ be the corresponding fundamental 2-form and Lee 1-form (with $\delta^g \theta =0$).
We first prove that if $b_1(M)$ is even, then $\theta$ is co-exact (this is \cite[Th\'eor\`eme II.1]{gauduchon1}). Applying the Hodge $*$ operator to $\theta$, this is equivalent to showing that $d^cF$ is exact. Recall that $2i \partial {\bar \partial}F= dd^c F=0$ because $g$ is standard. By Theorem~\ref{kahlerian}, there exists a K\"ahler metric on $(M,J)$ and then, by Proposition~\ref{dd^c-lemma}, $${\bar \partial} F = \partial {\bar \partial} \alpha,$$ for some $(0,1)$-form $\alpha =\xi - i J\xi$. We deduce $d^cF= dd^c\xi$, as required.
In the other direction, we have to prove that if $\theta$ is co-exact then $b_1(M)$ is even. We reproduce an argument from \cite{gauduchon2}. With respect to a standard metric $g$, the forms $\theta$ and $J\theta = - \delta^g F$ are both co-closed, and therefore the $(0,1)$-form $\theta^{ 0,1} : = \theta - i J\theta $ is ${\bar \partial}$-coclosed. In terms of Hodge decomposition, this reads as $$\theta^{0,1} = \theta^{0,1}_h + {\bar \partial} ^* \Phi,$$ where $\Phi\in\Omega^{0,2}(M)$ and $\theta^{0,1}_h$ is the $({\bar \partial} {\bar \partial}^* + {\bar \partial}^* {\bar \partial})$-harmonic part of $\theta^{0,1}$. Note that $\Phi = \alpha + i \beta$ where $\alpha, \beta \in \Omega^{J,-}(M)$ and $\alpha (\cdot, \cdot) : = - \beta (J\cdot, \cdot)$.
We first claim that if $\theta^{0,1}_h=0$, then $\phi= F + \beta$ is a harmonic self-dual 2-form. Indeed, since $J$ is integrable, it satisfies $(D^g_{JX} J)(JY) = (D^g_X J)(Y)$ (see \eqref{Kob-Nom}), and therefore $J (\delta^g \beta) = \delta^g \alpha$, i.e. $$ \theta - iJ\theta = {\bar \partial} ^* \Phi = \delta^g \Phi = \delta^g \alpha + i \delta^g \beta.$$ It follows that $J\theta = - \delta^g \beta$, and thus $\delta^g \phi = J\theta + \delta^g \beta =0.$
By a well-known result of Kodaira (see e.g.~\cite{bpv}), a compact complex surface has even $b_1(M)$ if and only if the dimension $b_+(M)$ of the space of harmonic self-dual 2-forms on $(M,g)$ is equal to $2h^{2,0}(M) + 1$, where $h^{2,0}(M) = {\rm dim}_{\mathbb C} H^{2,0}_{\bar \partial}(M)$; otherwise $b_+(M)=2h^{2,0}(M)$. It follows that $b_1(M)$ is even if and only if $b_+(M) > 2{\rm dim}_{\mathbb C} H^{2,0}_{\bar \partial}(M)$.
Therefore, it suffices to show that $\theta^{0,1}_h=0$, provided that $\theta$ is co-exact (because $\phi$ will be then a harmonic self-dual 2-form which is not a real part of a holomorphic $(2,0)$-form). To this end, we consider the natural map $\kappa : H^1_{dR}(M) \to H^{0,1}_{\bar \partial}(M) \cong H^{1}(M, {{\mathcal O}})$ from de Rham to Dolbeault cohomology given by $\xi\mapsto \xi^{0,1}$ on representatives. One easily checks that $\kappa$ is well-defined and injective. Moreover, by the Noether formula (see e.g.~\cite{bpv}), $\kappa$ is an isomorphism of (real) vector spaces if and only if $b_1(M)$ is even; otherwise, the image of $H^1_{dR}(M)$ in $H^{0,1}_{\bar \partial} (M)$ is of real codimension one.
For any element $\xi^{0,1}=\xi - iJ\xi$ in the image of $\kappa$, we calculate its $L_2$-hermitian product with $\theta^{0,1}_h$: \begin{equation*} \begin{split}
\langle \theta_h^{0,1}, \xi^{0,1} \rangle_{L_2} &= \langle \theta^{0,1}, \xi^{0,1} \rangle_{L_2} - \langle {\bar \partial}^* \Phi, \xi^{0,1} \rangle_{L_2} \\
&= \langle \theta^{0,1}, \xi^{0,1} \rangle_{L_2} - \langle \Phi, {\bar \partial} \xi^{0,1} \rangle_{L_2} \\
&= \langle \theta^{0,1}, \xi^{0,1} \rangle_{L_2} = \frac{1}{2}(\theta, \xi)_{L_2} + \frac{i}{2}(J\theta, \alpha)_{L_2} \\
&= \frac{1}{2}(\theta, \xi)_{L_2} -\frac{i}{2}(\delta^gF, \alpha)_{L_2} = \frac{1}{2}(\theta, \xi)_{L_2}.
\end{split}
\end{equation*} It follows that $\langle \theta_h^{0,1}, \xi^{0,1} \rangle_{L_2} =0$, if $\theta$ is co-exact (because $\xi$ is closed). Thus, in this case, the image of $\kappa$ is contained in the complex subspace of $H^1_{\bar \partial} (M)$ which is orthogonal to $\theta^{0,1}_h$, and therefore would have real codimension at least 2, unless $\theta_{h}^{0,1}=0$. \end{proof}
Finally, we review some natural connections which are useful in the Hermitian context. An integrable almost complex structure $J$ induces a canonical holomorphic structure on the tangent bundle $TM$, via the Cauchy--Riemann operator which acts on smooth sections $X$ and $Y$ of $TM$ by $${\bar \partial }_X Y := \tfrac{1}{2}([X,Y]+J[JX,Y])=- \tfrac{1}{2} J({\mathcal L}_{Y} J)(X).$$ Identifying $TM$ with the complex vector bundle $T^{1,0}M$, this operator may be viewed as a partial connection and has the equivalent expression \begin{equation}\label{cauchy-riemann-complex} {\bar \partial}_X Y = [X,Y]^{1,0}, \end{equation} for any complex vector fields $X$ and $Y$ of type $(0,1)$ and $(1,0)$, respectively.
In a similar way, any $J$-linear connection $\nabla$ determines a partial connection ${\bar\partial}^{\nabla}$ on $T^{1,0}$ by projection, or acting on real vector fields by \begin{equation}\label{Cauchy-Riemann} \bar \partial^{\nabla}_X Y = \tfrac{1}{2}(\nabla_X Y + J \nabla_{JX}Y). \end{equation} The operators $\bar \partial$ and $\bar \partial^{\nabla}$ have the same symbol but do not coincide in general. However, it is well-known that for any Hermitian structure $(g,J)$, there exists a unique connection $\nabla$, called the {\it Chern connection} of $(g,J)$, which preserves both $J$ and $g$, and such that $\bar \partial^{\nabla} = \bar \partial$. Note that the Chern connection $\nabla$ has torsion, unless $(g,J)$ is K\"ahler. It is related to the Levi--Civita connection $D^g$ by (see e.g. \cite{gauduchon3}): \begin{equation}\label{chern-connection-higher} g(\nabla_X Y, Z) = g(D^g_X Y, Z) + \tfrac{1}{2} d^cF({X, JY, JZ}). \end{equation}
In four dimensions, one uses \eqref{Lee} to rewrite \eqref{chern-connection-higher} in the following form (cf. \cite{gauduchon1, vaisman1}): \begin{equation}\label{chern-connection} \nabla_X - D^g_X = \tfrac{1}{2}\big( X^{\flat}\otimes \theta^{\sharp} - \theta \otimes X + J\theta(X) J\big), \end{equation} where $\theta^{\sharp}= g^{-1}(\theta)$ stands for the vector field $g$-dual to $\theta$.
\section{Generalized K\"ahler structures}\label{three}
As described in the introduction, a generalized K\"ahler structure on a manifold $M$ consists of a pair $(\mathcal J_1,\mathcal J_2)$ of commuting generalized complex structures such that $\langle {\mathcal J}_1\ \cdot,\mathcal J_2\ \cdot \rangle$ determines a definite metric on $TM\oplus T^*M$. The generalized complex structures $\mathcal J_1,\mathcal J_2$ are integrable with respect to the Courant bracket on sections of $TM\oplus T^*M$, given by \[ [X+\xi,Y+\eta]_H = [X,Y] + L_X\eta-L_Y\xi -\tfrac{1}{2}d(i_X\eta-i_Y\xi) + i_Yi_XH, \] which depends upon the choice of a closed 3-form $H$, called the \emph{torsion} or twisting. The space of 2-forms $b\in \Omega^2(M)$ acts on $TM\oplus T^*M$ by orthogonal transformations via \[ e^b(X+\xi) = X+\xi + i_Xb, \] and this action affects the Courant bracket in the following way \[ [e^b(W), e^b(Z)]_H = e^b[W,Z]_{H+db}. \] So, if $(\mathcal J_1,\mathcal J_2)$ is integrable with respect to the $H$-twisted Courant bracket, then $(e^{-b}\mathcal J_1e^{b}, e^{-b}\mathcal J_2e^{b})$ is integrable for the $(H+db)$-twisted Courant bracket.
A generalized complex structure $\mathcal J$, because it is orthogonal and squares to $-1$, lies in the orthogonal Lie algebra, and therefore may be decomposed according to the splitting \[ \mathfrak{so}(TM\oplus T^*M) = \wedge^2 TM \oplus \mathrm{End}(TM) \oplus \wedge^2 T^*M, \] or, in block matrix form, \[ \mathcal J = \left( \begin{array}{cc}
A & \pi \\
\sigma & A \\ \end{array} \right), \] where $\pi$ is a bivector field, $A$ is an endomorphism of $TM$, and $\sigma$ is a 2-form. Just as for an ordinary complex structure, the integrability of $\mathcal J$ may be expressed as the vanishing of a Nijenhuis tensor $[\mathcal J,\mathcal J]=0$ obtained by extending the Courant bracket. Restricted to $\wedge^2 TM$, this specializes to the usual Schouten bracket of bivector fields, requiring that $[\pi,\pi]=0$. This means that $\pi$ is a Poisson structure.
In \cite{gualtieri}, a complete characterization of the components of the generalized K\"ahler pair $(\mathcal J_1,\mathcal J_2)$ was given in terms of Hermitian geometry, which we now repeat here. \begin{thm}[\cite{gualtieri}, Theorem 6.37]\label{mthm} For any generalized K\"ahler structure $(\mathcal J_1,\mathcal J_2)$, there exists a unique 2-form $b$ and Riemannian metric $g$ such that \[ e^{-b}\mathcal J_{1,2}e^b=\frac{1}{2}\left( \begin{array}{cc}
J_+\pm J_- & -(F_+^{-1}\mp F_-^{-1}) \\
F_+\mp F_- & J_+\pm J_- \\ \end{array} \right), \] where $J_\pm$ are integrable $g$-compatible complex structures and $F_\pm = gJ_\pm$ satisfy \begin{equation}\label{integrabGK} d^c_+F_++d^c_-F_- = 0, \ dd^c_+ F_+ = 0. \end{equation} Conversely, any pair of $g$-compatible complex structures satisfying condition~(\ref{integrabGK}) define a generalized K\"ahler structure. Note that the pair $(\mathcal J_1,\mathcal J_2)$ is integrable with respect to the $(H-db)$-twisted Courant bracket where \[ H = d^c_+ F_+. \] \end{thm} An immediate corollary of this result and the preceding discussion is that the bivector fields \begin{equation}\label{poisstruct} \pi_1 = -F_+^{-1}+F_-^{-1},\ \ \ \ \pi_2= -F_+^{-1}-F_-^{-1} \end{equation} are both Poisson structures, a fact first derived in~\cite{LyakhovichZabzine} directly from~\eqref{integrabGK}.
We also see from the theorem that by taking a bi-Hermitian structure $(g,J_+,J_-)$ such that $J_+=\pm J_-$, one obtains $d^c_+F_+ = d^c_-F_-$ and therefore \eqref{integrabGK} reduces to $d^c_+F_+ = 0$, which is nothing but the ordinary K\"ahler condition on $(g,J_+)$.
As mentioned in the introduction, when $m>2$ the second relation in~\eqref{integrabGK} imposes a nontrivial constraint on the underlying complex manifolds $(M,J_{\pm})$: they must admit a (common) Hermitian metric $g$ for which the fundamental 2-forms are $dd^c$-closed. Furthermore, if the complex manifold $(M,J_+)$ satisfies the $\partial {\bar \partial}$-lemma (see Proposition~\ref{dd^c-lemma}), then the torsion $H=d^c_+ F_+$ of any compatible generalized K\"ahler structure must be exact. \begin{prop}\label{kahler-type} Let $(M,J)$ be a compact complex manifold such that $\iota : H^{1,2}_{\partial {\bar \partial}}(M) \to H^{1,2}_{\bar \partial}(M)$ is an isomorphism. Then any generalized K\"ahler structure on $M$ is \emph{untwisted}, i.e. $[H]=0$. \end{prop}
We now proceed with an investigation of the class of generalized K\"ahler structures $(g,J_+,J_-)$ for which the pair of complex structures \emph{commute} but are unequal, i.e. which satisfy $[J_+,J_-]=0$ and $J_+\neq \pm J_-$. In the following theorem, we show that the splitting $$TM=T_+M\oplus T_-M,$$ determined by the $\pm 1$-eigenbundles of $Q=J_+J_-$, is not only integrable, i.e. determines two transverse foliations of $M$, but is also holomorphic with respect to $J_\pm$, and that the leaves of each foliation inherit a natural K\"ahler structure.
\begin{thm}\label{main2} Let $(g,J_+,J_-)$ define a generalized K\"ahler structure with $[J_+,J_-]=0$. Then the $\pm 1$-eigenspaces of $Q=J_+J_-$ define $g$-orthogonal $J_\pm$-holomorphic foliations on whose leaves $g$ restricts to a K\"ahler metric. \end{thm} \begin{proof} Let $T_\pm M= \ker (Q\mp {\rm id})=\ker (J_+\pm J_-)$. Since $\ker (J_+\pm J_-)=\mathrm{im}(J_+\mp J_-)$, we see that $T_\pm M$ coincide with the images of the Poisson structures \[ \pi_1 = (J_+-J_-)g^{-1},\ \ \ \ \pi_2 = (J_++J_-)g^{-1} \] from~\eqref{poisstruct}. Therefore $T_\pm M$ are integrable distributions and determine transverse foliations of $M$. Since $Q$ is an orthogonal operator, we see further that the foliations defined by its $\pm 1$ eigenvalues must be orthogonal with respect to the metric $g$.
The complex structures induce decompositions $T_+M\otimes{\mathbb C} = A\oplus \overline{A}$ and $T_-M\otimes{\mathbb C} = B\oplus \overline{B}$, where \[ A = T^{1,0}_{J_+} M\cap T^{0,1}_{J_-}M,\ \ \ \ B = T^{1,0}_{J_+}M\cap T^{1,0}_{J_-}M \] are themselves integrable since they are intersections of integrable distributions. We now show that $A$ is preserved by the Cauchy-Riemann operator of $J_+$, proving that $T_+M$ is a $J_+$-holomorphic sub-bundle. Let $X$ be a $(0,1)$-vector field for $J_+$ and let $Z\in C^\infty(A)$. Then \[ \overline{\partial}_XZ = [X,Z]^{1,0}. \] Since $T^{1,0}_{J_+} M= A\oplus B$, we may project to these two components: \[ \overline{\partial}_XZ = [X,Z]_A+[X,Z]_B. \] To show that $A$ is $J_+$-holomorphic, we must show the vanishing of the second term, which upon expanding $X=X_{\overline A}+X_{\overline B}$, reads \[ [X,Z]_B=[X_{\overline A},Z]_B + [X_{\overline B},Z]_B. \] The first term vanishes since $A\oplus\overline{A}=T_+M\otimes{\mathbb C}$ is involutive, and the second term vanishes since $A\oplus\overline{B}=T^{0,1}_{J_-}M$ is involutive. Therefore $A$ is $J_+$-holomorphic. An identical argument proves that $B$ is $J_+$-holomorphic, and that both $A,B$ are $J_-$-holomorphic, as required.
To show that $g$ restricts to a K\"ahler metric on the leaves of $T_\pm M$, observe that since $J_+=J_-$ along the leaves of $T_-M$, we have upon restriction $d^c_+F_+ = d^c_-F_-$. Similarly along the leaves of $T_+M$ we have $J_+=-J_-$, so that upon restriction, $d^c_+=-d^c_-$ and $F_+=-F_-$, giving again $d^c_+F_+=d^c_-F_-$. But since the generalized K\"ahler condition forces $d^c_+F_+=-d^c_-F_-$, we conclude that both $F_\pm$ are closed upon restriction to the leaves of either foliation, therefore defining K\"ahler structures there. \end{proof}
The holomorphicity of the decomposition $TM=T_+M\oplus T_-M$ proven above together with the condition $d^c_+ F_+ + d^c_-F_-=0$ also imply that $Q$ is parallel with respect to the Chern connections $\nabla^\pm$ of $J_\pm$; in other words, for a generalized K\"ahler structure with $[J_+,J_-]=0$, the Chern connections $\nabla^\pm$ have holonomy contained in $U(m_+)\times U(m_-)$ where $\dim_{\mathbb R} T_\pm M= 2m_\pm$. We now provide an alternative proof of this fact, avoiding the use of Theorem~\ref{mthm}. \begin{prop}\label{secondprof} Let $(J_+,J_-)$ be a pair of Hermitian complex structures for the Riemannian metric $g$, such that $d^c_+F_++d^c_-F_-=0$ and $[J_+,J_-]=0$. Then $Q=J_+J_-$ is covariant constant with respect to the Chern connections $\nabla^\pm$. \end{prop} \begin{proof} Since $\nabla^+J_+=0$ by definition, it suffices to show that $\nabla^+J_-=0$. From Equation~\eqref{Kob-Nom}, we see that \[ \nabla^+ - \nabla^- = L, \] where $L\in \Omega^1(\mathrm{End}(TM))$ is given by \[ 2g(L_XY,Z)= d^c_+F_+(X,J_+Y,J_+Z) - d^c_-F_-(X,J_-Y,J_-Z). \] Consequently, $\nabla^+J_- = \nabla^-J_- + [L,J_-]$. By definition, $\nabla^-J_-=0$, and expanding the commutator we obtain \begin{equation}\label{temp2} \begin{split} 2g([L_X,J_-]Y,Z) &= d^c_+ F_+(X,J_+J_-Y,J_+Z) + d^c_- F_-({X,Y,J_-Z}) \\ & +d^c_+ F_+\big(X,J_+Y,J_+J_-Z) +d^c_- F_-\big(X,J_-Y,Z). \end{split} \end{equation} If $Y$ is taken in $T_+ M$ and $Z$ in $T_-M$, then the terms cancel since $d^c_+F_++d^c_-F_-=0$. If $Y,Z\in T_+ M$, then trivially $g((\nabla^+_X J_-) Y,Z)=g((\nabla^+_XJ_+)Y,Z)=0$ and similarly for $Y,Z\in T_-M$. Hence $\nabla^+J_-$ must vanish identically. Similarly, $\nabla^-J_+=0$, proving the result. \end{proof}
In fact, this proposition provides an alternative proof not only of the holomorphicity of $T_\pm M$ but also of their integrability, by observing that since the torsion of $\nabla^+$ vanishes upon restriction to $T_\pm M$, we have for $Y,Z\in T_+M$ or $T_-M$, \[ [Y,Z] = \nabla^+_YZ-\nabla^+_ZY, \] and since $\nabla^+Q=0$, $T_\pm M$ are involutive for the Lie bracket. Applying the same argument to $T_-M$, we obtain an alternative proof of Theorem~\ref{main2}.
\begin{rem} Along the above lines one can establish the following result: Let $J_+$ and $J_-$ be a pair of commuting almost complex structures on a $2m$-manifold $M$, such that $J_+$ is {\rm integrable}, and let $T_{\pm} M$ denote the sub-bundles of $TM$ corresponding to $(\pm 1)$-eigenspaces of $Q= J_+ J_-$. Then any two of the following three conditions imply the third. \begin{enumerate} \item[\rm (a)] $T_{\pm}M$ are integrable sub-bundles of $TM$; \item[\rm (b)] $T_{\pm}M$ are holomorphic sub-bundles of $TM$ with respect to $J_+$; \item[\rm (c)] $J_-$ is an integrable almost complex structure. \end{enumerate}
\end{rem}
Let us now return to the existence problem. According to Theorem~\ref{main2}, we must consider complex manifolds $(M,J)$ whose tangent bundle splits as a direct sum of two integrable, holomorphic sub-bundles $T_{\pm}M$; the second complex structure $J_-$ is obtained from $J_+=J$ by composing with $Q$, the product structure defining $T_\pm M$. It is then natural to ask whether there is a Riemannian metric $g$ on $M$ which is compatible with the commuting pair $(J_+,J_-)$, satisfying the generalized K\"ahler condition. (This is Question~2 of the introduction.)
Locally, the answer is always `{\it yes}'. Indeed, by using complex coordinates adapted to the transverse foliations, i.e. a neighborhood $U= V \times W \subset {\mathbb C}^{m_1}\times {\mathbb C}^{m_2}$ such that $T_{-}U = TV, \ T_+U=TW$, then for any K\"ahler metrics $g_V$ and $g_W$ on $V$ and $W$, the product metric $g_U := g_V \times g_{W}$ is K\"ahler with respect to both $J_{\pm}$, and $(g_U,J_{\pm})$ is a generalized K\"ahler structure.
We now show that if there exists one generalized K\"ahler metric $g$ on $(M,J_+,J_-)$, then there is in fact a whole family parametrized by smooth functions (This is similar to the variation of a K\"ahler metric by adding $dd^c f$). This construction is closely related to the potential theory developed in \cite{physicists, rocek-et-al}. We will use the integrable decomposition \[ TM = T_+M\oplus T_-M, \] and the associated decomposition $d = \delta_++\delta_-$ of the exterior derivative (induced by the `type' decomposition $\wedge^* T^*M = (\wedge^* T_+M^*) \otimes (\wedge^* T_-M^*)$), so that, defining $\delta^c_\pm =[J_+,\delta_\pm]$, we have \begin{equation}\label{dc} d^c_\pm = \pm\delta^c_+ + \delta^c_-. \end{equation}
\begin{prop}\label{potential} Let $(M,g,J_+,J_-)$ be a generalized K\"ahler structure. Then, for any smooth function $f\in C^\infty(M,{\mathbb R})$ and sufficiently small real parameter $t$, the 2-form \begin{equation}\label{variat} {\tilde F}_+ = F_+ + t (\delta_+\delta^c_+ f - \delta_-\delta^c_-f) \end{equation} defines a new Riemannian metric $\tilde g = -\tilde F_+ J_+$ which is compatible with both $J_{\pm}$, and such that $(\tilde g, J_{\pm})$ defines a generalized K\"ahler structure with unmodified torsion class $[H]\in H^3(M,{\mathbb R})$. \end{prop} \begin{proof} The $J_\pm$-invariant 2-form in~\eqref{variat} defines the $J_-$-fundamental form $\tilde F_- = \tilde g J_-$, or \[ {\tilde F}_- = F_- + t (-\delta_+\delta^c_+ f - \delta_-\delta^c_-f). \] We now show that $d^c_+\tilde F_+ + d^c_-\tilde F_-=0$, since \begin{align*} d^c_+(\delta_+\delta^c_+-\delta_-\delta^c_-) + d^c_-(-\delta_+\delta^c_+ - \delta_-\delta^c_-) &= -\delta^c_+\delta_-\delta^c_- + \delta_-^c\delta_+\delta^c_+\\ &\ \ \ \ +\delta_+^c\delta_-\delta^c_- - \delta^c_-\delta_+\delta^c_+\\ &= 0. \end{align*} Finally, by the identity \begin{align*} d^c_+(\delta_+\delta^c_+ - \delta_-\delta^c_-)&= \delta_-^c\delta_+\delta_+^c-\delta_+^c\delta_-\delta_-^c\\ &=(\delta_++\delta_-)\delta_+^c\delta_-^c\\ &=d \delta_+^c\delta_-^c, \end{align*} we see that $d^c_+(\tilde F_+ - F_+)$ is exact, showing that $[d^c_+F_+]=[d^c_+\tilde F_+]$, completing the proof. \end{proof}
The following example shows that the global existence question is more subtle. \begin{ex}\label{high-dim-ex} Take the product $M = M_1 \times M_2$ of two complex manifolds $(M_1,J_1)$ and $(M_2, J_2)$, where the latter admits no K\"ahler metrics at all (see Theorem~\ref{kahlerian}) and put $J_{\pm} := J_1 \pm J_2$ on $TM = TM_1 \oplus TM_2$. Then $J_+$ and $J_-$ commute and induce the obvious holomorphic splitting of $TM$, but they cannot admit a compatible generalized K\"ahler metric $g$ (see Theorem~\ref{main2}). In fact, $(M,J_+,J_-)$ cannot admit any compatible Riemannian metric $g$ with $d^c_+F_+ + d^c_-F_- =0$ (see Proposition~\ref{secondprof}). Note that while $(M,J)$ admits no K\"ahler metric, $M_1$ can be chosen so that $(M,J)$ does admit Hermitian metrics with $\partial {\bar \partial}$-closed fundamental forms. \end{ex} By contrast, if $(M,J)$ is a complex manifold of K\"ahler type, we can always find a Riemannian metric compatible with both $J_+$ and $J_-$ and such that $d^c_+F + d^c_-F_-=0$, as we now show. \begin{lemma}~\label{kahler-case}
Let $(M,J_+)$ be a complex manifold of K\"ahler type whose tangent bundle splits as a direct sum of two holomorphic, integrable sub-bundles $T_{\pm} M$, and let $J_-=-J|_{T_+M}+J|_{T_-M}$. Then $M$ admits a Riemannian metric $g$, compatible with both $J_+$ and $J_-$, satisfying $d^c_+F_+ + d^c_-F_-=0$. \end{lemma} \begin{proof} Let $g_0$ be any K\"ahler metric for $(M,J_+)$; since $J_{\pm}$ commute, the $J_-$-averaged Riemannian metric $$g(\cdot, \cdot) : = \tfrac{1}{2}(g_0(\cdot, \cdot) + g_0(J_- \cdot, J_- \cdot))$$ is compatible with both $J_{\pm}$. We claim that $g$ has the desired properties.
To see this, decompose the original K\"ahler form $F_0$ according to the splitting $\wedge^2 T^*M=\wedge^2 T_+^*M\oplus (T_+^*M\otimes T_-^*M)\oplus \wedge^2 T_-^*M$, yielding \[ F_0 = F_{++} + F_{+-}+F_{--}. \] Then the fundamental forms for $(g,J_\pm)$ are \[ F_\pm = \pm F_{++} + F_{--}, \] and using Equation~\eqref{dc} and the fact $dF_0=0$, we obtain \begin{align*} d^c_-F_- &= \delta^c_-(-F_{++}) - \delta_+^cF_{--}\\ &=-d^c_+F_+, \end{align*} as required. \end{proof}
Note that in the above Lemma, the commuting bi-Hermitian structure $(g,J_\pm)$ is not necessarily generalized K\"ahler, because although $d^c_+F_++d^c_-F_-=0$, it is not necessarily the case that $dd^c_+F_+=0$. We now provide an example where this final condition cannot be fulfilled.
\begin{ex}~\label{solvmanifold} We elaborate on an example from \cite{debartolomeis-tomassini} of a compact $6$-dimensional solvmanifold $M$ which does not admit a K\"ahler structure. $M$ is obtained as a compact quotient of a complex 3-dimensional Lie group (biholomorphic to ${\mathbb C}^3$) whose complex Lie algebra $\mathfrak{g}$ is generated by the complex $(1,0)$-forms $\sigma_1, \sigma_2, \sigma_3$, such that $$d\sigma_1=0, \ d\sigma_2 = \sigma_1 \wedge \sigma_2, \ d\sigma_3 = - \sigma_1 \wedge \sigma_3.$$ Thus, $\mathfrak{g}$ (and hence $M$) inherits a natural left-invariant complex structure $J$ with respect to which the $\sigma_i$ are holomorphic $1$-forms. Note that $(M,J)$ does not satisfy the $\partial {\bar \partial}$-lemma because $\sigma_2$ and $\sigma_3$ are holomorphic but not closed.
It is straightforward to check that there are no left-invariant Hermitian metrics $g$ on $(\mathfrak{g},J)$ such that the condition $dd^c F =0$ is satisfied. Since the volume form $v = \sigma_1 \wedge {\overline \sigma_1} \wedge \sigma_2 \wedge {\overline \sigma_2} \wedge \sigma_3 \wedge {\overline \sigma_3}$ is bi-invariant, a standard argument~\cite{belgun, fino-granch} shows that $(M,J)$ does not admit {\it any} Hermitian metrics with $dd^c$-closed fundamental form. In particular, $(M,J)$ admits no compatible generalized K\"ahler structures.
However, we can define a second left-invariant complex structure $J_-$ on $\mathfrak{g}$ (and hence also on $M$) such that $T^{1,0}_{J_-}M = {\rm span}_{{\mathbb C}} \{{\overline \sigma_1}, \sigma_2, \sigma_3\}$, so that $J_+ :=J$ and $J_-$ are both integrable, commute and define {\it holomorphic} (and therefore integrable) sub-bundles $T_{\pm}M$. Furthermore, the left-invariant metric $g_0 = \sum_{i=1}^{3} \sigma_i \otimes {\overline \sigma_i}$ on $\mathfrak{g}$ defines on $M$ a Hermitian metric which is compatible with both $J_+$ and $J_-$, and such that $d^c_+F_+ + d^c_-F_-=0.$ \end{ex}
For a compact complex manifold of K\"ahler type, $(M,J)$, Beauville conjectures~\cite{beauville} that $TM$ splits as a direct sum of two holomorphic integrable sub-bundles if and only if $M$ is covered by the product of two complex manifolds $M_+\times M_-$ on which the fundamental group of $M$ acts {\it diagonally}, i.e. $\pi_1(M)$ acts on each $M_\pm$ and its action on the product is the diagonal action. In the case when there is a K\"ahler metric on $(M,J)$ whose Levi-Civita connection preserves $T_+M$ and $T_-M$, the conjecture follows by the de~Rham decomposition theorem. It has also been confirmed in other cases~\cite{beauville,campana-peternell,druel}. We mention here the following partial result.
\begin{thm}\label{KE}\cite{beauville,kobayashi} Let $(M,J)$ be a compact complex manifold which admits a K\"ahler--Einstein metric $g$, and whose tangent bundle splits as a direct sum of two holomorphic sub-bundles $T_{\pm}M$. Then $T_{\pm}M$ are parallel with respect to the Levi-Civita connection. In particular, $g$ is K\"ahler with respect to both $J_+=J$ and
$J_-=-J|_{T_+M}+J|_{T_-M}$, and therefore $(M, J)$ admits generalized K\"ahler metrics compatible with $J_+$ and $J_-$. \end{thm} \begin{proof} This is a standard Bochner argument. Let $g$ be a K\"ahler--Einstein metric on $(M,J)$. The vector bundle $E=\mathrm{End}(TM)$ is a Hermitian holomorphic bundle with unitary connection $D$ induced by the Levi--Civita connection. The Ricci endomorphism of $E$ is defined by $$K (Q)= [R, Q],$$ where $R \in C^\infty(E)$ is the usual Ricci endomorphism of the tangent bundle and $Q\in C^\infty(E)$. Since $g$ is K\"ahler--Einstein, $K\equiv 0$.
A section $Q\in C^\infty(E)$ is holomorphic if and only if $D''Q=0$, where $D=D'+D''$ is the usual decomposition of $D$ into partial connections. The classical {\it Bochner--Kodaira} identity (see e.g.~\cite{kobayashi-0,demailly}) implies that for any {\it holomorphic} section $Q$ of $E$, \begin{equation}\label{bochner}
\int_M ||D'Q||_g^2 v_g = \int_M g(K(Q),Q) v_g= 0. \end{equation} Thus, any holomorphic section of $E$ must be parallel. Applying this to $Q=J_+J_-$, we see that $T_\pm M$ are parallel for the Levi--Civita connection. By the de Rham decomposition theorem, $(M,g,J)$ must be then a local K\"ahler product of two K\"ahler--Einstein manifolds tangent to $T_{\pm}M$, respectively. The claim follows. \end{proof}
To conclude this section, we wish to indicate that the methods of Theorem~\ref{main2} and Proposition~\ref{secondprof} can be used to prove {\it non-existence} results as follows. When $J_+$ and $J_-$ do not commute, a direct computation using \eqref{temp2} shows that the commutator $P=[J_+,J_-]$ satisfies $$ \nabla^+_{X} P + J_+ (\nabla^+_{J_+X} P) = 0,$$ provided that $d^c_+F_++d^c_-F_-=0$. It follows that for any generalized K\"ahler structure $(g,J_{\pm})$, $P$ defines a $J_\pm$-{\it holomorphic} bivector field $\pi=Pg^{-1}$. This fact was first established in \cite{AGG} for the case $m=2$, and by Hitchin~\cite{hitchin2} in general; the latter work also shows that $P$ defines a $J_\pm$-holomorphic {\it Poisson structure}, a fact which follows from the fact that $\pi_1,\pi_2$ are Poisson structures (see Equation~\eqref{poisstruct}). Therefore, if $(M,J)$ does not carry a non-trivial holomorphic Poisson structure (e.g. if $H^0(M, \wedge^2(TM)) = 0$), then for any generalized K\"ahler structure $(g,J_{\pm})$ with $J_+=J$, $J_{+}$ and $J_-$ must commute. Then, by Theorem~\ref{main2}, non-trivial generalized K\"ahler structures do not exist unless the holomorphic tangent bundle of $(M,J)$ splits. Using results of \cite{beauville,campana-peternell,druel} one finds a wealth of projective complex manifolds such that $H^0(M, \wedge^2(TM))=0$ and $TM$ does not split. This argument has been used in \cite{hitchin3} to prove that a locally de Rham irreducible K\"ahler--Einstein manifold with $c_1(M)<0$ does not admit any non-trivial generalized K\"ahler structure, thus establishing a partial converse of Theorem~\ref{KE}. \begin{thm}\label{KE-bis}\cite{hitchin3} Let $(M,J)$ be a compact complex manifold of negative first Chern class. Then it admits a non-trivial generalized K\"ahler structure $(g,J_+,J_-)$ with $J_+=J$ if and only if the holomorphic tangent bundle of $(M,J)$ splits. In this case, $J_+$ and $J_-$ commute. \end{thm} \begin{proof} By the Aubin--Yau theorem~\cite{aubin,yau}, $(M,J)$ admits a K\"ahler--Einstein metric of negative scalar curvature. A standard Bochner argument shows $H^0(M, \wedge^2(TM)) = 0$. By the preceding remarks, for any generalized K\"ahler structure $(g,J_+, J_-)$ with $J_+=J$, the complex structures must commute and the result follows from Theorem~ \ref{KE}. \end{proof}
\section{Generalized K\"ahler four-manifolds}\label{four} In dimensions divisible by four, generalized K\"ahler structures fall into two broad classes, defined by whether the complex structures $\pm J_+$ and $\pm J_-$ induce the same or different orientations on the manifold. \begin{defn} Let $M$ be a manifold of dimension $4k$. A triple $(g,J_+, J_-)$, consisting of a Riemannian metric $g$ and two $g$-compatible complex structures $J_\pm$ with $J_+\neq \pm J_-$, is called a \emph{bihermitian} structure if $J_+$ and $J_-$ induce the same orientation on $M$; otherwise, it is called \emph{ambihermitian}. Similarly, an (am)bihermitian conformal structure is a triple $(c,J_+,J_-)$, where $c=[g]$ is a conformal class of (am)bihermitian metrics. \end{defn} In this section we will concentrate on the 4-dimensional case, where we have the following characterization of the generalized K\"ahler condition in terms of the Lee forms $\theta_\pm$.
\begin{prop}\label{b1} Let $(g,J_{\pm})$ be an {\rm (}am{\rm )}bihermitian structure on a four-manifold $M$. Then the condition $d^c_+F_++d^c_-F_-=0$ is equivalent to $\theta_+ + \theta_-=0$ in the bihermitian case, and to $-\theta_+ +\theta_- =0$ in the ambihermitian case. The condition $dd^c_+F_+=0$ means that $g$ is a standard metric, i.e. $\delta^g\theta_+=0$. The twisting $[H]$ vanishes if and only if $\theta_+ = \delta^g\alpha$ for $\alpha\in\Omega^2(M)$, i.e. the Lee form is co-exact. \end{prop} \begin{proof} By \eqref{Lee}, we have $d^c_{\pm}F_{\pm} = (J_{\pm}\theta_{\pm})\wedge F_{\pm},$ so that \begin{equation}\label{temp1} d^c_+F_++d^c_-F_-=(J_+\theta_+) \wedge F_+ +(J_-\theta_-)\wedge F_-. \end{equation} Note that in the bihermitian case $F_+ \wedge F_+=F_-\wedge F_-$ is twice the volume form $v_g$, whereas in the ambihermitian case $F_+ \wedge F_+ = - F_-\wedge F_- = 2v_g$. Therefore, applying the Hodge star operator $*$ to \eqref{temp1} and using the fact that $\delta^g= -* d *$ when acting on 2-forms, we obtain the result. \end{proof}
As an immediate corollary of this result, together with Proposition~\ref{gauduchon}, we obtain\footnote{Alternatively, this result follows from the generalized Hodge decomposition for generalized K\"ahler structures proven in~\cite{gualtieri-pq}.}: \begin{cor}\label{B1} Let $M$ be a generalized K\"ahler 4-manifold. If the torsion class $[H]\in H^3(M,{\mathbb R})$ vanishes, then the first Betti number must be even (and hence $M$ is of K\"ahler type); if $[H]\neq 0$ then the first Betti number must be odd. \end{cor}
Bihermitian complex surfaces were studied in~\cite{A,AGG,dloussky,kobak,pontecorvo} and classified for even first Betti number in~\cite{AGG}, where the classification of Poisson surfaces~\cite{bartocci-macri} is used, and existence is only partially proven. In fact,~\cite{AGG} provides enough to show that in this case, any bihermitian structure is conformal to a unique generalized K\"ahler structure, up to scale. \begin{prop}\label{BH} Let $(c,J_+,J_-)$ be a bihermitian conformal structure on a compact four-manifold $M$ with $b_1(M)$ even. Then there is a unique (up to scale) metric $g\in c$ such that $(g,J_+,J_-)$ is generalized K\"ahler. \end{prop} \begin{proof} By \cite[Lemma~4]{AGG}, any standard metric $g$ of $(c,J_+)$ (which is unique up to scale~\cite{gauduchon1}) is standard for $(c,J_-)$ as well, and furthermore $\theta_+ + \theta_-=0$. By Proposition~\ref{b1}, this is equivalent to the generalized K\"ahler condition.\end{proof} Some constructions of these bihermitian structures can be found in \cite{AGG,gualtieri-et-al, hitchin2,kobak,Lin-Tolman}, and these prove existence on many (but not all) of these surfaces.
In the case where the first Betti number is odd, bihermitian structures have been studied in \cite{A, AGG, dloussky, pontecorvo}. It follows from the results there that $M$ must be a finite quotient of $(S^1\times S^3) \sharp k {\overline {{\mathbb C} P}}^2, \ k \ge 0$. It is no longer true in this case that the standard metric provides a generalized K\"ahler metric in all cases. To the best of our knowledge, the only known examples of generalized K\"ahler structures on 4-manifolds with $b_1(M)$ odd are given by standard metrics in the anti-self-dual bihermitian conformal classes described in~\cite{pontecorvo}.
We now turn to the ambihermitian case, where we establish a complete classification of generalized K\"ahler structures. We start with the following observation. \begin{lemma}\label{doubly-almost-complex} Let $M$ be a four-manifold endowed with a pair $(J_+,J_-)$ of almost complex structures inducing different orientations on $M$. Then, $M$ admits a Riemannian metric compatible with both $J_\pm$ if and only if $J_+$ and $J_-$ commute. In this case, the tangent bundle splits \begin{equation}\label{Q-split} TM = T_{+}M \oplus T_-M \end{equation} as an orthogonal direct sum of Hermitian complex line bundles defined as the $\pm 1$-eigenbundles of $Q=J_+J_-$. \end{lemma} \begin{proof} Let $g$ be a Riemannian metric on $M$, compatible with $J_+$ and $J_-$. Fix the orientation on $M$ induced by $J_+$. As discussed in \S~2, the fundamental 2-forms $F_+$ and $F_-$ are sections of $\Omega^+(M)$ and $\Omega^-(M)$, respectively. Since $\Omega^-(M)$ is in the $+1$-eigenspace of $\wedge^2 J_+$, $F_-$ is $J_+$-invariant. Hence $J_+$ and $J_-$ commute. The converse is elementary. \end{proof} The proof of the above lemma shows that the existence of commuting almost complex structures on a four-manifold is a purely topological problem (in fact, it is equivalent to the existence of a field of oriented two-planes~\cite{matsushita}). Note that a similar existence problem for pairs of {\it integrable} almost complex structures on $M$ inducing different orientations was raised in \cite{beauville0}, and has been almost completely solved in \cite{kotschick2}.
Our next step is to identify the compact complex surfaces $(M,J)$ that admit a generalized K\"ahler metric $(g,J_+,J_-)$ of ambihermitian type with $J_+ =J$. \begin{lemma}\label{split} Let $(g,J_+,J_-)$ be an ambihermitian structure on a four-manifold $M$ and let $Q=J_+J_-$ be the almost product structure it defines. Then the Lee forms satisfy $\theta_+=\theta_-$ if and only if $T_\pm M$ are holomorphic sub-bundles for $J_\pm$, i.e. $\nabla^\pm Q=0$. Then the standard metric in the conformal class defines a generalized K\"ahler metric.
As a result, any compact complex surface $(M,J)$ whose tangent bundle splits as a sum of holomorphic line bundles admits a compatible generalized K\"ahler metric. \end{lemma} \begin{proof} If $\theta_+=\theta_-$, then by Proposition~\ref{b1}, we have $d^c_+F_++d^c_-F_-=0$, and so $T_\pm M$ are holomorphic by Proposition~\ref{secondprof}.
In the other direction, we use Equation~\eqref{chern-connection} and the fact that $J_-$ is skew-symmetric to express $$\nabla^+_X J_- = D^g_X J_- - \tfrac{1}{2} ( X^{\flat} \wedge (J_-\theta_+)^{\sharp} + (J_- X)^{\flat} \wedge \theta_+^{\sharp}), $$ where $\alpha \wedge X = \alpha \otimes X - X^{\flat}\otimes \alpha^{\sharp}$ for $\alpha \in T^*M$ and $X\in TM$. Finally, by \eqref{DF}, we obtain \begin{equation}\label{calculation} \nabla^+_X J_- = \tfrac{1}{2} ( X^{\flat} \wedge J_-(\theta_- - \theta_+)^{\sharp} + J_-X^{\flat}\wedge (\theta_--\theta_+)^{\sharp}). \end{equation} It is clear from Equation~\eqref{calculation} that $\nabla^+J_-$ (and hence $\nabla^\pm Q$) vanishes if and only if $\theta_+=\theta_-$, proving the result.
To prove the final statement, we note that any holomorphic one-dimensional sub-bundle $T_\pm M \subset TM$ is automatically integrable, and therefore the almost complex structure $J_- =
-J|_{T_+M}+J|_{T_-M}$ is integrable. By definition, $J_+=J$ and $J_-$ commute, and $J_\pm$ induce different orientations. Clearly there are Riemannian metrics compatible with both $J_{\pm}$. Then we may apply the first part of the lemma. \end{proof}
Now we are ready to prove our classification results for ambihermitian generalized K\"ahler structures.
\noindent {\bf Proof of Theorem~\ref{main}.} Let $(M,g,J_+,J_-)$ be a compact generalized K\"ahler four-manifold of ambihermitian type. By Proposition~\ref{b1} and Lemma~\ref{split}, the holomorphic tangent bundle of $(M,J_+)$ must split as a direct sum of two holomorphic line bundles $(T_{\pm}M,J_+)$. Complex surfaces with split tangent bundles were studied and essentially classified by Beauville~\cite{beauville}. We use his results to retrieve the list (a)--(f).
When $b_1(M)$ is even, the cases that occur according to \cite{beauville} correspond to the surfaces listed in (a)--(d) of Theorem~\ref{main}, modulo the fact that our description of the surfaces in (a) is slightly different from the one in \cite[\S 5.5]{beauville}, and that the existence of a splitting of $TM$ on {\it any} surface in (c) is not addressed in \cite[\S 5.2]{beauville}.
To clarify these points, we notice that in the case of a ruled surface $M=P(E) \to \Sigma$, \cite[Thm.C]{beauville} implies that the universal cover is the product ${\mathbb C} P^1\times \mathbb U$, where ${\mathbb U}$ is the universal covering space of $\Sigma$, and the diagonal action of $\pi_1(M) = \pi_1(\Sigma)$ gives rise to a $PGL(2,{\mathbb C})$ representation of $\pi_1(\Sigma)$, i.e. the holomorphic bundle $E$ is projectively-flat as claimed in (a).
Note that for any an elliptic fibration $f: M \to \Sigma$ as in (c), the base curve $\Sigma$ can be given the structure of an orbifold with a $2\pi/m_i$ cone point at each point corresponding to a fibre of multiplicity $m_i$ (see, \cite[\S~5.2]{beauville} and \cite[\S~7] {wall}). Since the Kodaira dimension of $M$ is equal to $1$, the orbifold Euler characteristic of $\Sigma$ must be negative, and therefore $\Sigma$ is a good orbifold uniformized by the hyperbolic space ${\mathbb H}$. Since the first Betti number of $M$ is even, it follows from \cite[Thm.7.4]{wall} the universal covering space of $M$ is ${\mathbb C} \times {\mathbb H}$, on which the fundamental group $\pi_1(M)$ acts diagonally by isometries of the canonical product K\"ahler metric.
When $b_1(M)$ is odd, the possible cases are described in \cite[\S\S~(5.2),(5.6),(5.7),(5.8)]{beauville}. To prove that the only complex surfaces that really occur are those listed in (e) and (f) in Theorem~\ref{main} we have to exclude the possibility that $(M,J_+)$ is an elliptic fibration of Kodaira dimension $1$, odd $b_1(M)$, and with only multiple singular fibres with smooth reduction. It is shown in \cite[\S~(5.2)]{beauville} that for the holomorphic tangent bundle of such a surface to split, it must be covered by a product of simply connected Riemann surfaces on which the fundamental group acts diagonally. On the other hand, any elliptic surface $M$ with Kodaira dimension $1$ and $b_1(M)$ odd is finitely covered by an elliptic fiber bundle $M'$ over a compact Riemann surface of genus $>1$, which has trivial monodromy~(cf.~\cite[p.139]{wall}). Since $M$ (and hence $M'$) is not K\"ahler, $b_1(M')$ is odd too. Wall \cite[p.141]{wall} showed that the universal cover of such an $M'$ is ${\mathbb C} \times {\mathbb H}$ on which $\pi_1(M')$ does not act diagonally. It then follows from Beauville's result cited above that the holomorphic tangent bundle of $M'$ (and hence of $M$) does not split.
It remains to establish the existence of generalized K\"ahler metrics on the complex surfaces listed in Theorem~\ref{main}. We know by Lemma~\ref{split} that there are ambihermitian metrics $(g,J_+,J_-)$ on $M$, compatible with the holomorphic splitting of $TM$, which are parametrized by the choice of Hermitian metrics on each of the factors $T_{\pm}M$, or equivalently by two smooth functions on $M$. For any such metric $g$, we have $\theta_+^g= \theta_-^g$, where $\theta_{\pm}^g$ are the corresponding Lee forms (see Lemma~\ref{split}). Let $g_0$ be a standard metric of $([g],J_+)$, i.e.~a metric in the conformal class $[g]$ such that $\delta^{g_0} (\theta_+^{g_0})=0$. Since $\theta_+^{g_0} = \theta_-^{g_0}$, the triple $(g_0,J_+,J_-)$ defines a generalized K\"ahler structure of ambihermitian type. Finally, since the standard metric is unique up to scale in any conformal class~\cite{gauduchon1}, we eventually obtain a family of generalized K\"ahler metrics on $M$, which depend on one arbitrary smooth function, completing the proof.
\begin{rem} Some Hopf surfaces described in case (e) of Theorem~\ref{main} (e.g. those with $\alpha = \beta \in {\mathbb R}$ and $\lambda = \mu =1$) admit a Riemannian metric $g$ compatible with a pair of hyper-complex structures, ${\mathcal HC}_+$ and ${\mathcal HC}_-$, inducing different orientations on $M$, and such that for any choice $J_+ \in {\mathcal HC}_+$ and $J_- \in {\mathcal HC}_-$, $(g,J_+,J_-)$ is a twisted generalized K\"ahler structure of ambihermitian type. Such Hopf surfaces do also admit an abundance of twisted generalized K\"ahler structures of bihermitian type~\cite{AGG,pontecorvo}. \end{rem}
\end{document} | arXiv |
Golden Integral Calculus By N P Bali ((BETTER))
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What is a monomorphism?
I just started to study topology and i came across monomorphism for the first time. I have googled but couldn't find a good explanation.
Can you please explain what a monomorphism is and in which situation we use it?
$\mathcal{C}$ is a category with an initial object $0$ and $\mathcal{A}$ a category.
A monomorphism in $\mathcal{C}$ is a function $f\colon X\to Y$ such that for all $f'\colon X'\to Y$ it holds that $f'=f$ (the domain and codomain agree).
The canonical example is $X\hookrightarrow \emptyset$ and $X\to\{0\}$.
Now we can start thinking about some natural categories that contain monomorphisms:
The category $\mathcal{P}_f(\mathbb{N})$ of non-empty finite sets has all finite monomorphisms.
In $\mathcal{P}_f(\mathbb{R})$ one can take the initial object $[0,1]$ with its inclusion to $\mathbb{R}$.
In $\mathcal{P}_f(X)$ for any topological space $X$ you can take the identity map $X\to X$ or $X\to \{0\}$.
If you can describe a general purpose definition of a monomorphism you can google for it.
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Your ratios don't prove what you think they prove
Watching people discuss police bias statistics, I despair. Some claim simple calculations prove police bias, some claim the opposite. Who is right?
No one. Frankly, nobody has any clue what they are talking about. It's not that the statistics are wrong exactly. They just don't prove what they're being used to prove. In this post, I want to explain why, and give you the tools to dissect these kinds of claims.
I've made every effort to avoid politics, due to my naive dream where well-meaning people can agree on facts even if they don't agree on policy.
The obvious place to start is to look at the number of people killed by police. This is easy to find.
# in US (million) 41.3 185.5 57.1
# killed by police per year 219 440 169
# killed by police per million people 5.3 2.3 2.9
Does this prove the police are racist? Before you answer, consider a different division of the population.
# in US (million) 151.9 156.9
# killed by police per year 944 46
# killed by police per million people 6.2 0.29
And here's a third one.
<18 y/o
# in US (million) 72.9 53.6 63.2 137.3
# killed by police per year 19 283 273 263
# killed by police per million people 0.26 5.2 4.3 1.9
The first table above is often presented as an obvious "smoking gun" that proves police racism with no further discussion needed. But if that were true, then the second would be a smoking gun for police sexism and the third for police ageism. So let's keep discussing.
Of course, the second and third tables have obvious explanations: Men are different from women. The young are different from the old. Because of this, they interact with the police in different ways. Very true! But the following is also true:
average height (men) 175.5cm (5'9") 177.4cm (5'10) 169.5cm (5'7")
life expectancy 74.9 yrs 78.5 yrs 81.8 yrs
mean annual income $41.5k $65.9k $51.4k
median age 33 yrs 43 yrs 28 yrs
go to church regularly 65% 53% 45%
children in single-parent homes 65% 24% 41%
identify as LGBT 4.6% 3.6% 5.4%
live in a large urban area 82% 61% 82%
poverty 21% 8.1% 17%
men obese 41% 44% 45%
women obese 56% 39% 43%
completed high school 87% 93% 66%
completed bachelor's 22% 36% 15%
heavy drinkers 4.5% 7.1% 5.1%
Maybe it's uncomfortable, but it's a fact: In the US today, there are few traits where there aren't major statistical differences between races. (Of course this doesn't mean these differences are caused by race! This is a good example of why correlation does not imply causation.)
Suppose police were required wear augmented reality goggles. On those goggles, real-time image processing changes faces so that race is invisible. Would doing this cause police statistics to equalize with respect to race?
No. Even if race is literally invisible, young urban alcoholics will have different experiences with police than old teetotalers on farms. The fraction of these kinds of people varies between races. Thus, racial averages will still look different because of things that are associated with race but aren't race as such.
So despite the thousands of claims to the contrary, just looking at killings as a function of population size doesn't prove bias. Not does it prove a lack of bias. It really doesn't prove anything.
Why do police kill more men than women? We can't rule out police bias. But surely it's relevant that men and women behave differently? So, it might seem like we should normalize not by population size, but by behavior.
One popular suggestion is to consider the number of arrests:
# arrests for violent crimes per year (thousands) 146 230 83
# killed by police per thousand violent crime arrests 1.4 1.9 1.9
Some claim this proves the police aren't biased, or even that there is bias in favor of blacks. But that's nearly circular logic: If police are biased, that would manifest in arrests as much as killings. So what we are really calculating above is
\[\frac{\text{"Normal" killings + killings due to bias}}{\text{"Normal" arrests + arrests due to bias}}.\]
The ratio doesn't tell you much about how large the bias terms are. So, unfortunately this also doesn't prove anything.
Incidentally: There are some popular but different numbers out there for this same ratio. These have tens of thousands of re-tweets with no one questioning the math. But I've checked the source data carefully, and I'm pretty sure my numbers are right. (They reach the same basic conclusion anyway.)
The police have discretion when deciding to make an arrest. But a dead body either exists or doesn't. So why not normalize by the number of murders committed?
This turns out to be basically impossible:
Something like 40% of murders go unsolved, so the race of the murderer is unknown.
The only real source of murder statistics is the FBI. They treat hispanic/non-hispanic ethnicity as independent of race. Why not just ignore hispanics then? Well, you can't. Hispanics are still counted as white or black in an unknown way. It's impossible to compare to police shooting statistics where hispanic is an alternative race.
In around 31% of cases, the FBI has no information about race, and in 40% of cases, no information about ethnicity.
I've seen tons of articles use this version of the FBI's murder data that simply drops all the cases where data are unknown. None of these articles even acknowledge the issue of missing data or different treatment of hispanics.
Instead, let's look at murder victims. This is counterintuitive, but it's relatively rare for murders to cross racial boundaries (<20%). So this is a non-terrible proxy for the number of murders committed. Data from the CDC separates out black, white, and hispanics in a similar way as police shooting statistics.
# murder victims per year 9,908 5,747 3,186
# killed by police per murder victim 0.022 0.076 0.053
So what does this prove? Again, not much. The simple fact is that most police killings are not in the context of a murder or a murder investigation. Though there are exceptions, the precise context of police killings hasn't had enough study, and definitely not enough to get reliable statistics.
Ratios are hopeless
Really, though, it's not an issue of lacking data. Philosophically, consider the any possible ratio like
\[\frac{\text{# of people of a race killed by police}}{\text{# of times act } X \text{ committed by a member of a race}}.\]
For what act \(X\) does this really measure police bias? I think it's pretty clear that no such act exists, even if we could measure it. Races vary along too many dimensions. There are too many scenarios for police use of force. Bias interacts with the world in too many ways. You just can't learn anything meaningful with these sort of simplistic high-level statistics.
This doesn't mean we need to give up. It just means you need to get closer and try harder. In the next part of this series I'll look at some valiant attempts to do that. They will disappoint us too, but for different reasons.
Data Used:
Police shootings (average 2017-2019)
Number of people of each race / sex
Number of people by age
Data by race: Life expectancy / Income / Height / Church / Single-parent homes / Identifying LGBT / Median age / School / Drinking / Poverty / Urbanity / Obesity
Arrests for violent crime
Murder victims (p. 43)
This post is part of a series on bias in policing with several posts still to come.
Part 1: Your ratios don't prove what you think they prove (This post)
Part 2: The veil of darkness
Part 3: Policy proposals and what we don't know about them
Part 4: Why fairness is basically unobservable
A breakdown of the data on the homeless crisis across the U.S.
Statistical nihilism and culture-war island hopping
Political polarization is partly a sample bias illusion
Does the gender-equality paradox actually exist?
How the United States didn't ban the death penalty
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Potential for international spread of wild poliovirus via travelers
Annelies Wilder-Smith1,2,
Wei-Yee Leong1,
Luis Fernandez Lopez3,8,
Marcos Amaku4,
Mikkel Quam5,
Kamran Khan6,7 &
Eduardo Massad8,9
The endgame of polio eradication is hampered by the international spread of poliovirus via travelers. In response to ongoing importations of poliovirus into polio-free countries, on 5 May 2014, WHO's Director-General declared the international spread of wild poliovirus a public health emergency of international concern. Our objective was to develop a mathematical model to estimate the international spread of polio infections.
Our model took into account polio endemicity in polio-infected countries, population size, polio immunization coverage rates, infectious period, the asymptomatic-to-symptomatic ratio, and also the probability of a traveler being infectious at the time of travel. We applied our model to three scenarios: (1) number of exportations of both symptomatic and asymptomatic polio infections out of currently polio-infected countries, (2) the risk of spread of poliovirus to Saudi Arabia via Hajj pilgrims, and (3) the importation risk of poliovirus into India.
Our model estimated 665 polio exportations (>99 % of which were asymptomatic) from nine polio-infected countries in 2014, of which 78.3 % originated from Pakistan. Our model also estimated 21 importations of poliovirus into Saudi Arabia via Hajj pilgrims and 20 poliovirus infections imported to India in the same year.
The extent of importations of asymptomatic and symptomatic polio infections is substantial. For countries that are vulnerable to polio outbreaks due to poor national polio immunization coverage rates, our newly developed model may help guide policy-makers to decide whether imposing an entry requirement in terms of proof of vaccination against polio would be justified.
Polio will remain a global problem as long as there is still one case in the world. By 2012, the annual number of polio cases due to wild poliovirus (WPV) had decreased by >99 % since the polio eradication program was launched in 1988. However, three countries never interrupted WPV transmission: Afghanistan, Nigeria, and Pakistan [1]. In 2013 an upsurge of cases was observed caused by outbreaks in five previously polio-free countries, triggered by importation of WPV via travelers. Preventing the further spread of WPV into polio-free countries and the ensuing outbreaks is therefore a top priority in eradicating polio.
Given WPV's infectiousness and long period of virus excretion, epidemics in previously polio-free countries via travelers importing WPV is not surprising, and has been documented on many occasions. Reinfection of 19 polio-free African countries occurred in 2009 alone [2]. The outbreak of 199 polio cases in Somalia in 2013-2014 was due to an introduction of poliovirus of Nigerian origin [3]. A large outbreak of poliomyelitis, with 463 laboratory-confirmed cases, took place in 2010 in Tajikistan after a single importation of WPV from India in 2009, with further expansion into neighboring Kazakhstan, Russia, Turkmenistan, and Uzbekistan [4]. After being polio-free for more than 10 years, an outbreak occurred in China in 2011 in Xinjiang imported from neighboring Pakistan [5]. In more recent years, the upsurge of polio cases in Pakistan led to exportation of the virus to the Middle East with new alarming outbreaks in Syria and Iraq [6, 7]. Furthermore, a higher proportion of adults has been observed in those affected by the outbreaks as a result of importation [8]. The Hajj pilgrimage has also been considered a high risk for importation of WPV from the remaining polio-infected countries that to a large extent contain Muslim populations [9]. In summary, from 2003 to 2014, there were 191 documented new importation events into previously polio-free countries, resulting in 3,763 reported cases of paralytic polio in 43 countries and costing $1.15 billion in additional funds from international organizations and agencies alone for outbreak control [10].
In response to ongoing importations of poliovirus into polio-free countries, on 5 May 2014, the Director-General of the World Health Organization (WHO) declared the international spread of WPV a public health emergency of international concern (PHEIC). Given that such a declaration has only been made for Ebola and influenza, this declaration highlights how seriously WHO considers the threat to polio eradication due to the ongoing international spread of poliovirus. WHO announced temporary recommendations to ensure that all travelers departing from polio-infected countries receive a polio vaccination (oral or injectable polio vaccine) between 4 weeks and 12 months before international travel, and should ensure that such travelers are provided with proof of vaccination [11]. India did not even wait for this declaration. In early 2014, India was declared polio-free by WHO after a three-year interval without any polio cases [12]. This achievement came at a high price. India is now understandably concerned about reintroduction via travelers - a concern that is justified given its proximity to two countries with currently the highest number of polio cases (Pakistan and Afghanistan). To prevent reintroduction of polio, India therefore decided to tighten cross-border travel rules by introducing a new polio vaccine requirement for all travelers from polio-infected countries entering India. The Government of India's Ministry of Health and Family Welfare has announced that "Resident nationals of the currently seven polio infected countries are required to receive a dose of oral polio vaccine (OPV), regardless of age and vaccination status, at least four weeks prior to departure to India. A certificate of vaccination with OPV is required for resident nationals of these countries while applying for entry visa to India" [13].
Given WHO's recent focus on the contribution of the international spread of poliovirus via travelers, it is important and timely to assess the potential for further spread. Reliance on reported events of importation will only underestimate the true importation risk, as not every imported case will be detected and reported, especially in countries with poor surveillance systems. Furthermore, not every imported case will lead to secondary cases, and polio importations without secondary cases are likely to remain unnoticed. Both asymptomatic and symptomatic cases can contribute to transmission [14], and asymptomatic polio infections (that also remain unreported) need to be taken into account when assessing the international spread of polio. In the absence of reliable surveillance data, mathematical modeling is necessary to estimate the number of importations or exportations of polio.
Our objective was to develop a mathematical model to estimate the international spread of symptomatic and asymptomatic polio infections. We then applied our model to three scenarios: (1) number of exportations of both symptomatic and asymptomatic polio infections out of currently polio-infected countries, (2) the risk of spread of poliovirus to Saudi Arabia via Hajj pilgrims, and (3) the importation risk of poliovirus into India via travelers from polio-infected countries.
We first obtained data on the number of polio cases reported to WHO in the years 2010 to 2014 [15] including obtaining the number of polio cases classified by WHO as being the result of importation. We calculated the proportion of imported polio over all polio cases per year for the 15 years from 2010 to 2014.
We then adapted a Susceptible -> Infected -> Recovered (SIR) model and fed the following facts and parameters about polio into the model: published estimates of the ratio of inapparent to paralytic illness vary from 50:1 to 1,000:1 (usually 200:1) [14]. We assumed a ratio of 200:1. Persons infected with wild poliovirus (WPV) are most infectious from 7 to 10 days before and after the onset of symptoms, but poliovirus may be present in the stool for up to 3 to 6 weeks [14], with the mean duration of WPV type 1 excretion in fecal specimens being 24 days (median, 20 to 29 days), with a range of 1 to 114 days [16, 17]. We assumed that the duration of infectiousness for our models is 4 weeks and that the life expectancy for every country is equal to 60 years (1/μ i =60). We also assumed, for simplicity, that the probability of poliovirus infection is independent of the probability of travel and that poliovirus infections are homogeneously distributed in polio-affected countries.
In the following paragraphs we explain how the model was developed. The model variables and parameters are described in Table 1. The model equations are shown in Box 1.
Table 1 Variables and parameters of the model
The general equation describing the time variation in reported polio cases in country i , in year t, I i (t), is given by Equation (1) in Box 1.
In Equation (1), λi(t) is the force of infection (per capita number of new infections per time unit), S i (t) is the number of susceptible individuals at time t , \( \frac{1}{\mu_i} \) is the life expectancy of the population in the country i , and \( \frac{1}{\gamma_i} \) is the average duration of infection in country i . The number of susceptible individuals is given by the size of the population multiplied by one minus the proportion of polio vaccination coverage. Equation (1) describes the dynamics of infectious individuals. It expresses the time variation in the number of infective people, and it contains two terms, one income (positive) term provided by the disease incidence (with rate λi(t)Si(t)) and one outcome (negative) term provided by natural deaths (with rate μ i ), and the recovery of infected individuals (with rate γ i ).
The number of polio cases in country i at year t is given by the number of cases at the year the cohort started to be followed up I(0) times the number of individuals who survived up until year t, plus the number of new cases of infection (see Equation (2) in Box 1).
We consider the reported number of cases for the year t in country i (Equation (2) of Box 1) and assumed that this annual incidence is constant along each year, that is, there is no seasonal variation.
As we assumed a 200:1 asymptomatic-to-symptomatic ratio, we multiplied the number of reported cases at year t in country i (Λ i (t) in Equation (3) of Box 1).
Dividing Equation (3) from Box 1 by the number of susceptible individuals S i (t) at year t in country i, we obtain the prevalence p i (t) in year t in country i. Multiplying p i (t) by the number of non-immune travelers from country i , we obtain the expected number of polio cases among travelers.
We conducted a sensitivity analysis for the model, which is described in detail in Box 2. After developing the model, we applied it to three scenarios:
Exportation of polio infections (asymptomatic and symptomatic) from polio-infected countries in the years 2013-2014.
We obtained data on the countries that had polio cases reported to WHO from the website indicated in Ref. [15]. The number of travelers departing these polio-affected countries was obtained from the International Air Transport Association (IATA). We applied Equation (3) to estimate the expected number of polio exportations, taking into account the degree of polio immunization coverage rates (oral polio vaccine) for the countries, outbound travel, asymptomatic-symptomatic ratio, and population size.
International spread of poliovirus via Hajj pilgrims in the years 2013 and 2014.
As the Ministry of Health of Saudi Arabia has withdrawn its public website on the number of annual Hajj pilgrims, we estimated the number of Hajj pilgrims from earlier reports in the published literature [18]. We then applied the same equation as for scenario 1 to estimate the risk of international importation of poliovirus via Hajj pilgrims into Saudi Arabia.
Importation of poliovirus into India in the years 2010-2014:
We selected the seven countries for which India has issued the requirement for proof of vaccination against polio (Afghanistan, Ethiopia, Kenya, Nigeria, Pakistan, Somalia, and Syria) [13] and obtained the numbers of nationals from these countries traveling to India from India's tourism statistics [19]. The travelers include those arriving by air, ground, and sea. The population size per country was obtained from the World Bank. The number of wild-type polio cases and the national polio immunization coverage rates were obtained from WHO [20]. We also obtained data on travel volume and force of infection for the years 2010 to 2014, using the same assumptions for all the variables except for population size, travel volume, and force of infection, where we used the yearly available data. We also took into account that in different years additional polio-infected countries existed.
In 2013, imported polio cases accounted for the majority of all reported polio cases 256 of 416 cases (corresponding to Fig. 1) shows the proportion of imported polio cases per year from 2000-2014. Over those 15 years, the proportion of polio cases as a result of importation increased significantly, and accounted for 62 % in the year 2013. The years 2005 and 2010 were also the years that showed a very high proportion of polio as a result of importation (53 % and 83 %, respectively).
Polio cases (endemic versus imported) 2000-2014
In Table 2, we summarize the number of modeled exported cases from polio-infected countries in the years 2013 and 2014. The expected number of exported polio cases in 2013 was 462, of which half were contributed by Pakistan (n = 158) and Somalia (n = 121). Among these countries, Somalia had the highest number of polio cases in 2013 (n = 194) in that particular year and the lowest vaccination coverage rate (47 %). In the year 2014, Somalia had far fewer endemic polio cases, but polio infections tripled in Pakistan. Given the high travel volume between Pakistan and India, the estimated overall annual polio exportations increased from 2013 to 2014 to 665.
Table 2 Estimated exported polio infections from polio-infected countries for 2013 and 2014, accounting for a ratio of apparent to inapparent polio infections of 200:1
Table 3 shows the estimated number of exported cases by outbound travel of Hajj pilgrims from the countries that reported polio cases to WHO in 2013 and 2014. Our findings indicated that in 2013 a total of 20 cases were exported from these polio-infected countries. Somalia (n = 8) and Pakistan (n = 6) were the two countries with the highest exportation, contributing 70 % of the total exported cases. Although there were reported polio cases in Ethiopia, Kenya, and Cameroon, based on our model, they did not contribute to any exportation. In 2014, the number of importations via Hajj pilgrims was 21, and Pakistan contributed to most of these importations (19 out of 21).
Table 3 Estimated imported polio infections to Saudi Arabia during the Hajj pilgrimage from polio-infected countries in 2013 and 2014
Figure 2 shows the routes of potential polio importation into India in 2014. The results of the application of Equation (3) for the case of polio importation into India by travelers are shown in Tables 4 and 5. The total number of estimated polio importations from nine polio-infected countries summed up to 20 polio infections imported to India in the year 2014 (Table 4). Afghanistan and Pakistan, both countries in close geographical proximity to India, contributed all 20 of the imported polio infections (13 from Pakistan, 7 from Afghanistan). Table 5 shows that the highest importation risk (24 estimated importations) into India for the years 2010-2013 was in the year 2010 when 20 countries had at least one polio case. In 2011, there were 22 estimated importations; in 2012, 11 importations; and in 2013, 13 importations of polio infections into India.
Travelers from polio-infected countries entering India in 2014
Table 4 Estimated importations of polio infections into India in the year 2014
Table 5 Estimated importations of polio infections into India (2010-2013)
Sensitivity analysis: The sensitivity analysis in Box 2 shows that the model is about 12 times less sensitive to μ and γ than to λ. In other words, the force of infection in the epidemic source country is the main driving factor for the importation risk, combined with the travel volume. However, note that λ is covariate with μ and γ. As we do not know the exact relationship among them, it is not possible to calculate either \( \frac{\partial \lambda }{\partial \mu } \) or \( \frac{\partial \lambda }{\partial \gamma } \), and thus the sensitivity analysis incomplete.
Although the number of global polio cases has been declining over the past 15 years, as shown in Fig. 1, the proportion of polio as a result of importation is increasing, which is a major concern to polio eradication. The proportion has ranged from 0 to 62 % over the past 15 years, with the years 2005, 2010, and 2013 reporting the highest proportions. However, these numbers do not reflect importation events, but rather the overall polio cases in non-endemic polio countries as a result of importation from polio-endemic or polio-infected countries (Pakistan, Afghanistan, and Nigeria being the only endemic countries since 2010). These proportions hence cannot inform from where the importation originated and how many travelers imported polio. Here, we have developed a mathematical model to address this gap. Our model not only takes into account polio endemicity (force of infection), population size, polio vaccination coverage, and infectious period, but also the probability of a traveler being infectious at the time of travel. As both symptomatic and asymptomatic polio infections are known to transmit the virus, our model includes asymptomatic cases. As the ratio of asymptomatic to symptomatic polio is very high, most of our estimated importations of poliovirus are via asymptomatic travelers, and hence our estimated numbers of importation/exportation appear high. Our sensitivity analysis shows that the force of infection in the epidemic source country is the main driving factor for the importation risk, combined with the travel volume.
In the year 2014, out of nine polio-affected countries, we modeled that 665 polio infections (of which 662 were asymptomatic, and only 3 symptomatic) were exported globally. The majority of these poliovirus exportations originated from Pakistan (521 out of 665; 78.3 %). The number of polio exportations will vary from year to year. In 2013, for example, there were 462 estimated exportations of polio infections, and less than half can be attributed to Pakistan, with Somalia contributing almost the same amount as Pakistan, as there was a major polio outbreak in Somalia that year (Table 2). As per the sensitivity analysis, we found that for every 1 % variation in the force of infection there will be approximately 1 % variation in the time-dependent prevalence in the country.
About 600 potential poliovirus importations from nine polio-infected countries in one year into any country in the world present a sizable problem. However, our model cannot tell us how many of these importation events will lead to secondary cases or even outbreaks. It is the national coverage of polio immunization in a country that will determine its vulnerability for polio outbreaks after importation. The polio vaccination coverage rate in any given country together with the quality of its surveillance system and the speed of outbreak response will determine the magnitude of outbreaks following importation. Countries with high polio vaccination coverage rates may still see importations of polio cases (the majority of which will be asymptomatic and hence not detected), but are unlikely to see secondary cases. For example, countries such as Australia, New Zealand, the United States, and Singapore have reported importations, but given their high immunization coverage rates, no single secondary cases occurred [21, 22]. For Africa, Andre Mach et al. recently assessed the risk for polio outbreaks for 2013-2014, and the authors found 15 countries to be at high risk for WPV outbreaks, 5 at moderate-to-high risk, and 6 at low risk [23]. In 15 of the 33 African countries, less than half of the population resides in areas where surveillance performance indicators have met only minimum targets [23, 24].
Every year, the Hajj pilgrimage brings more than 2 million pilgrims from all over the world to Saudi Arabia, an event that is characterized by overcrowding [9]. Many of these pilgrims are from Nigeria, Pakistan, and Afghanistan - the three remaining polio-endemic countries - and other predominantly Moslem countries that are polio-infected. Hence, there has been long-standing concern about the potential introduction of polio into the Hajj with subsequent international spread. Our model estimated 20 importations of poliovirus into Saudi Arabia via Hajj pilgrims in the year 2013, and 21 in the year 2014. About 20 potential importations need to be taken seriously. Indeed, the Kingdom of Saudi Arabia has introduced mandatory polio vaccination at the point of entry for all pilgrims coming from polio-infected countries [9].
To prevent reintroduction of the poliovirus, in January 2014 India decided to tighten cross-border travel rules by introducing a new polio vaccine requirement for all travelers from polio-infected countries entering India [12, 25]. We previously estimated the number of travelers who would be affected by this new rule imposed by the Indian government to be approximately 233,800 travelers annually from the seven countries to India, and 346,800 Indian national residents to these seven countries, in the year 2013 [25]. Our model now adds more information: our findings estimate that 13 polio importations occurred in the year 2013, and 20 in 2014. In 2014, 100 % of these polio importations into India originated from Pakistan and Afghanistan, which is not surprising given that these two countries carry the main remaining burden of polio and have the highest travel volume to India. Although the travel volume from Pakistan to India was of the same order of magnitude as that for Afghanistan to India, the force of infection of polio was higher in Pakistan, and hence the probability of importing polio from Pakistan to India is higher compared to Afghanistan. Given the high asymptomatic-to-symptomatic ratio, most likely all 20 imported infections were asymptomatic. The total expected number of importations may seem low at first sight. However, one must consider that for the year 2014 this figure implies an incidence of 20 per 380,604 travelers (5 polio infections per 100,000 travelers), which is far from negligible. In other words, of the total number of 72,159 asymptomatic and symptomatic polio infections in the nine countries, 20 (about 0.027 %) of them were estimated to have traveled to India at the time of polio infection associated with viral shedding. The number of secondary cases will depend on the polio immunization coverage rate, which according to WHO was 70 % for India [20]. Fortunately, no single imported polio case was detected or officially reported in India in the years 2013 or 2014. We added an analysis of the preceding years (2010–2012) for comparison. The years 2010 and 2011 saw high numbers of importations mainly due to the much higher number of countries that reported at least one case of polio. As even one importation event can lead to substantial outbreaks associated with hundreds of million dollars spent for their control [3, 4], our finding of 20 importations in one year (2014) lends support to India's strategy to protect its country from reinfection via travelers. India's new vaccine policy would affect approximately 380,000 travelers, and approximately 19,000 vaccinations need to be administered per one averted poliovirus importation. Given the tragic consequences of re-infecting India with polio, the number of travelers that need to be vaccinated to avert poliovirus importation would be justified. India's emphasis should be on preventing polio importation from Pakistan and Afghanistan.
Our model had the following limitations. For lack of data, we assumed that the probability of poliovirus infection is independent of the probability of travel; and for simplicity, we assumed that poliovirus infections are homogeneously distributed in polio-affected countries. These assumptions will lead to an overestimate of our results. For the estimation of outbound travel from polio-affected countries (scenario 1, Table 2), we obtained data from IATA. These data capture an estimated 90 % of all passengers traveling on commercial airlines worldwide, but do not incorporate land travel. Hence, they tend to underestimate international travel volumes between countries that share contiguous land borders (that is, where land-based border crossings are common). The true exportation numbers may therefore be even higher than we estimated, if ground travel volume is included. Unfortunately, data on ground travel volume is difficult to obtain for many of the polio-affected countries. However, India does publish the number of travelers into its country, and these data include air and land travel. As there are a lot of border crossings via land from Pakistan to India, it was important to include such land travel data for estimating poliovirus importation into India (scenario 3, Tables 4 and 5).
As long as polio exists anywhere in the world, it can be imported anywhere. Our model can be applied to all countries and provides an additional tool to estimate the risk of importation based on the main contributing factors such as travel volume and polio endemicity. In particular, for countries that are vulnerable to polio outbreaks due to poor national polio immunization coverage rates, our model may help guide policy-makers to decide whether imposing an entry requirement in terms of proof of vaccination against polio would be justified.
Box 1. Model equations as explained in the main text
The general equation for the variation in polio cases in country i, in year t, I i (t), is given by:
$$ \frac{d{I}_i(t)}{dt}={\lambda}_i(t){S}_i(t)-\left({\mu}_i+{\gamma}_i\right){I}_i(t) $$
where λ i (t) is the force of infection (per capita number of new infections per time), S i (t) is the number of susceptible individuals at time t , \( \frac{1}{\mu_i} \) is the life expectancy of the population in country i , and \( \frac{1}{\gamma_i} \) is the average duration of infection in country i. The number of susceptible individuals is given by the size of the population multiplied by one minus the proportion of polio vaccination coverage. Equation (1) describes the dynamics of infectious individuals. It expresses the time variation in the number of infective people and it contains two terms, one income (positive) term provided by the disease incidence (with rate λ i (t)S i (t)) and one outcome (negative) term provided by natural deaths (with rate μi), and the recovery of infected individuals (with rate γ i ).
Equation (1) can be integrated by standard methods, resulting in:
$$ {I}_i(t)={I}_i(0){e}^{-\left({\mu}_i+{\gamma}_i\right)t}+{\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right){e}^{-\left({\mu}_i+{\gamma}_i\right)\left(t-{t}^{\prime}\right)}dt $$
We consider \( {\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)dt \), the reported number of cases along the year t in country i, and assume this annual incidence as constant in each year. Equation (2), therefore, simplifies to
$$ {I}_i(t)={I}_i(0){e}^{-\left({\mu}_i+{\gamma}_i\right)t}+{\varLambda}_i(t)\left\{\frac{\left[1-{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{\left({\mu}_i+{\gamma}_i\right)}\right\} $$
Where \( {\varLambda}_i(t)={\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)dt \) for each individual year t.
Dividing Equation (3) by the number of susceptible individuals S i (t) in year t in country i, we obtain the prevalence p i (t) in year t in country i:
$$ {p}_i(t)={p}_i(0){e}^{-\left({\mu}_i+{\gamma}_i\right)t}+{\varLambda}_i(t)\left\{\frac{\left[1-{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{S_i(t)\left({\mu}_i+{\gamma}_i\right)}\right\} $$
Multiplying p i (t) by the number of non-immune travelers from country i, we obtain the expected number of polio cases among travelers.
Box 2. Sensitivity of the model to the parameters
For a small variation of parameter Par i , ΔPar i , the variation in the parameter-dependent variable π, Δπ, is given by the well-known error propagation formula26:
$$ \varDelta \pi ={\displaystyle \sum_i\frac{\partial \pi }{\partial Pa{r}_i}\times \varDelta Pa{r}_i} $$
The relative variation in the risk π, Δπ/π, as a function of the relative variation in the parameters ΔPar i /Par i , is therefore:
$$ \frac{\varDelta \pi }{\pi }={\displaystyle \sum_i Pa{r}_i\frac{\partial \pi }{\partial Pa{r}_i}\times \frac{\varDelta Pa{r}_i}{Pa{r}_i}\times \frac{1}{\pi }} $$
We calculated the sensitivity of Equation (3) of Box 1 to the natural mortality rate of hosts, μ:
$$ \frac{\varDelta p(t)}{p(t)}=\frac{\partial p(t)}{\partial \mu}\times \frac{\varDelta \mu }{\mu}\times \frac{\mu }{p(t)} $$
$$ \frac{\partial p(t)}{\partial \mu }=-t{p}_i(0){e}^{-\left({\mu}_2+{\gamma}_i\right)t}-\left\{\left[{\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)d{t}^{\prime}\right]\left\{\frac{\left[1-{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{S_i(t){\left({\mu}_i+{\gamma}_i\right)}^2}\right\}+\left[{\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)d{t}^{\prime}\right]\left\{\frac{\left[{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{S_i(t)}\right\}\right\}\frac{\partial \lambda }{\partial \gamma } $$
is the partial derivative of time-dependent prevalence with respect to the natural mortality rate of hosts μ.
An identical result is obtained for \( \frac{\partial p(t)}{\partial \gamma } \).
The result of \( \frac{\partial p(t)}{\partial \gamma } \) is much simpler:
$$ \frac{\partial p(t)}{\partial \lambda }=\frac{\left[1- \exp \left(-\left(\mu +\gamma \right)t\right)\right]}{S(t)\left(\mu +\gamma \right)} $$
Therefore, the equations for the sensitivity analysis are:
$$ \begin{array}{c}\hfill \frac{\varDelta p(t)}{\mu }=-t{p}_i(0){e}^{-\left({\mu}_i+{\gamma}_i\right)t}-\left\{\left[{\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)d{t}^{\prime}\right]\left\{\frac{\left[1-{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{S_i(t){\left({\mu}_i+{\gamma}_i\right)}^2}\right\}+\left[{\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)d{t}^{\prime}\right]\left\{\frac{\left[{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{S_i(t)}\right\}\right\}\hfill \\ {}\hfill \frac{\partial \lambda }{\partial \mu}\times \left\{\frac{\varDelta \mu }{\mu}\times \frac{p(t)}{\pi}\right\}\hfill \end{array} $$
$$ \begin{array}{l}\frac{\varDelta p(t)}{\gamma }=-t{p}_i(0){e}^{-\left({\mu}_i+{\gamma}_i\right)t}-\left\{\left[{\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)d{t}^{\prime}\right]\left\{\frac{\left[1-{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{S_i(t){\left({\mu}_i+{\gamma}_i\right)}^2}\right\}+\left[{\displaystyle \underset{0}{\overset{t}{\int }}}{\lambda}_i\left({t}^{\prime}\right){S}_i\left({t}^{\prime}\right)d{t}^{\prime}\right]\left\{\frac{\left[{e}^{-\left({\mu}_i+{\gamma}_i\right)t}\right]}{S_i(t)}\right\}\right\}\\ {}\frac{\partial \lambda }{\partial \mu}\times \left\{\frac{\varDelta \mu }{\gamma}\times \frac{p(t)}{\gamma}\right\}\end{array} $$
$$ \frac{\varDelta p(t)}{\lambda }=\frac{\left[1- \exp \left(-\left(\mu +\gamma \right)t\right)\right]}{S(t)\left(\mu +\gamma \right)}\times \frac{\varDelta \mu }{\lambda}\times \frac{p(t)}{\lambda } $$
As the sensitivity of the prevalence to the parameters is an explicit function of time, as time passes (t → ∞), the values settle to:
$$ \frac{\varDelta p}{\lambda}\cong \frac{\varDelta \mu }{\lambda } $$
that is, for every 1 % variation in the force of infection there will be approximately 1 % variation in the time-dependent prevalence.
For the other two parameters μ and γ the results are:
$$ \frac{\varDelta p}{\mu}\cong \frac{1}{12}\frac{\varDelta \mu }{\mu } $$
$$ \frac{\varDelta p}{\gamma}\cong -\frac{1}{12}\frac{\varDelta \gamma }{\gamma } $$
that is, the model is about 12 times less sensitive to μ and γ than to λ.
IATA:
International Air Transport Association
OPV:
PHEIC:
Public health emergency of international concern
WPV:
Wild poliovirus
Moturi EK, Porter KA, Wassilak SG, Tangermann RH, Diop OM, Burns CC, et al. Progress toward polio eradication - worldwide, 2013-2014. MMWR Morb Mortal Wkly Rep. 2014;63:468–72.
O'Reilly KM, Chauvin C, Aylward RB, Maher C, Okiror S, Wolff C, et al. A statistical model of the international spread of wild poliovirus in Africa used to predict and prevent outbreaks. PLoS Med. 2011;8:e1001109.
Kamadjeu R, Mahamud A, Webeck J, Baranyikwa MT, Chatterjee A, Bile YN, et al. Polio outbreak investigation and response in Somalia, 2013. J Infect Dis. 2014;210:S181–6.
Yakovenko ML, Gmyl AP, Ivanova OE, Eremeeva TP, Ivanov AP, Prostova MA, et al. The 2010 outbreak of poliomyelitis in Tajikistan: epidemiology and lessons learnt. Euro Surveill. 2014;19:20706.
Wen N, Fan CX, Fu JP, Ning J, Ji YX, Luo HM, et al. Enhanced surveillance of acute flaccid paralysis following importation of wild poliovirus in Xinjiang Uygur Autonomous Region. China BMC Infect Dis. 2014;14:113.
Arie S. Polio virus spreads from Syria to Iraq. BMJ. 2014;348:g2481.
Aylward RB, Alwan A. Polio in Syria. Lancet. 2014;383:489–91.
Mach O, Tangermann R, Wassilak S, Singh S, Sutter R. Outbreaks of paralytic poliomyelitis during 1996-2012: the changing epidemiology of a disease in the final stages of eradication. J Infect Dis. 2014;210 suppl 1:275–82.
Ahmed QA, Arabi YM, Memish ZA. Health risks at the Hajj. Lancet. 2006;367:1008–15.
Cochi SL, Jafari HS, Armstrong GL, Sutter RW, Linkins RW, Pallansch MA, et al. A world without polio. J Infect Dis. 2014;210:S1–4.
Polio public health emergency: temporary recommendations to reduce international spread of poliovirus. 2014. http://www.polioeradication.org/Infectedcountries/PolioEmergency.aspx Accessed 5 May 2014.
Friedrich MJ. Polio eradicated in India. JAMA. 2014;311:666.
Requirements of polio vaccination for international travellers between India and polio infected countries. http://mohfw.nic.in/showfile.php?lid=2634 of Polio vaccination for International travellers between India & polio infected countries.pdf Accessed 15 March 2014.
Atkinson W, Wolfe S, Hamborsky J, McIntyre L, editors. Centers for Disease Control and Prevention (CDC). Epidemiology and prevention of vaccine-preventable diseases. 11th ed. Washington D.C: Public Health Foundation; 2009.
Polio Eradication: Data and Monitoring. 2014. http://www.polioeradication.org/Dataandmonitoring/Poliothisweek.aspx Accessed June 2014.
Gelfand H, Leblanc D, Fox J, Conwell D. Studies on the development of natural immunity to poliomyelitis in Louisiana. II. Description and analysis of episodes of infection observed in study group households. Am J Hygiene. 1957;65:367–85.
Plotkin S, Orenstein W, Offit P, editors. Vaccines. 5th ed. Philadelphia: Saunders/Elsevier; 2008. p. 634.
Khan K, Sears J, Hu VW, Brownstein JS, Hay S, Kossowsky D, et al. Potential for the international spread of Middle East Respiratory Syndrome in association with mass gatherings in Saudi Arabia. PLOS Currents Outbreaks. 2013. Edition 1.
India Tourism Statistics. 2012. http://tourism.gov.in/writereaddata/CMSPagePicture/file/marketresearch/publications/India%20Tourism%20Statics(2012)%20new.pdf Tourism Statics(2012) new.pdf Accessed 15 March 2014.
World Health Organization. http://apps.who.int/gho/data/view.main.80601 Accessed 2 June 2014 2014.
Wilder-Smith A, Leder K, Tambyah PA. Importation of poliomyelitis by travelers. Emerg Infect Dis. 2008;14:351–2.
Chambers ST, Dickson N. Global polio eradication: progress, but determination and vigilance still needed. N Z Med J. 2011;124:100–4.
Mach A, Wolff CG, Tangermann RH, Chenoweth P, Tallis G, Kamgang JB, et al. Assessing and mitigating the risks for polio outbreaks in polio-free countries - Africa, 2013-2014. MMWR Morb Mortal Wkly Rep. 2014;63:756–61.
Levitt A, Diop OM, Tangermann RH, et al. Surveillance systems to track progress toward global polio eradication - worldwide, 2012-2013. MMWR Morb Mortal Wkly Rep. 2014;63:356–61.
Quam M, Massad E, Wilder-Smith A. Effects of India's new polio policy on travellers. Lancet. 2014;383:1632.
The World Bank. Data (Life expectancy at birth, total (years)). http://data.worldbank.org/indicator/SP.DYN.LE00.IN.
World Health Organization. Global Health Observatory Data Repository. Polio (Pol3) data by country. http://apps.who.int/gho/data/node.main.A831?lang=en. Accessed 2 June 2014.
The World Bank. Data (Total population) http://data.worldbank.org/indicator/SP.POP.TOTL. Accessed 2 June 2014.
Ministry of Tourism. Government of India. Country wise distribution of FTAs. http://tourism.gov.in/writereaddata/CMSPagePicture/file/Primary%20Content/MR/FTAs/Country%20wise%20Distribution%20of%20FTAs%20.pdf. Accessed 2 June 2014.
World Tourism Organization (2013). Yearbook of Tourism Statistics dataset. http://resources.metapress.com/pdf-preview.axd?code=j92t36rp5xlkwk7v&size=largest. Accessed 2 June 2014.
Polio Global Eradicatio Initiative. Data and Monitoring. Wild poliovirus list 2009-2014. http://www.polioeradication.org/Portals/0/Document/Data&Monitoring/Wild_poliovirus_list_2009_2014_29Apr.pdf. Accessed 2 June 2014.
We would like to thank Wolfgang Lohr for helping us extract some of the flight data.
Lee Kong Chian School of Medicine, Nanyang Technological University, Mandalay Road 11, Singapore, 308232, Singapore
Annelies Wilder-Smith & Wei-Yee Leong
Institute of Public Health, University of Heidelberg, Heidelberg, Germany
Annelies Wilder-Smith
Florida International University, Miami, USA
Luis Fernandez Lopez
Faculty of Veterinary Medicine, University of São Paulo, São Paulo, Brazil
Marcos Amaku
Department of Public Health and Clinical Medicine, Epidemiology and Global Health, Umeå University, Umeå, Sweden
Mikkel Quam
Li Ka Shing Knowledge Institute, St. Michael's Hospital, Toronto, ON, Canada
Kamran Khan
Faculty of Medicine, Division of Infectious Diseases, University of Toronto, Toronto, ON, Canada
School of Medicine, University of São Paulo, São Paulo, Brazil
Luis Fernandez Lopez & Eduardo Massad
London School of Hygiene and Tropical Medicine, London, UK
Eduardo Massad
Wei-Yee Leong
Correspondence to Annelies Wilder-Smith.
AWS is a member of the International Health Regulations (IHR) Emergency Committee on polio. KK is the founder of BlueDot, a social benefit corporation that models the international spread of emerging and reemerging infectious diseases. The other authors have no competing interests to declare.
AWS and EM conceived the study; EM and MA developed the mathematical model, and LFL the sensitivity analysis. YWL, EM, and MA did all the final calculations. YWL obtained the majority of the data and created all the tables; MQ contributed to data collection and created the map. KK obtained and analyzed the data for scenario 1 (travel volumes and patterns out of the polio-infected countries). AWS coordinated the study and wrote the manuscript; all authors contributed to the final manuscript. All authors read and approved the final manuscript.
This article is published under license to BioMed Central Ltd. 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.
Wilder-Smith, A., Leong, WY., Lopez, L.F. et al. Potential for international spread of wild poliovirus via travelers. BMC Med 13, 133 (2015). https://doi.org/10.1186/s12916-015-0363-y
Received: 13 February 2015
DOI: https://doi.org/10.1186/s12916-015-0363-y
International spread
Secondary cases
Polio vaccination
Hajj pilgrimage
Mathematical model
Medicine for Global Health | CommonCrawl |
Organic Chemistry 8th
Reactions of Aldehydes and Ketones • More Reactions of Carboxylic Acid Derivatives
Paula Yurkanis Bruice
Give two names for each of the following:
Anupa M.
Why are numbers not used to designate the position of the functional group in propanone and butanedione?
Name the following:
Which ketone in each pair is more reactive?
a. 2 -heptanone or 4 -heptanone
b. bromomethyl phenyl ketone or chloromethyl phenyl ketone
What products are formed when the following compounds react with $\mathrm{CH}_{3} \mathrm{MgBr}$, followed by the addition of dilute acid? Disregard stereoisomers.
We saw on the previous page that 3 -methyl- 3 -hexanol can be synthesized from the reaction of 2 -pentanone with ethylmagnesium bromide. What other combinations of ketone and Grignard reagent could be used to prepare the same tertiary alcohol?
a. How many stereoisomers are obtained from the reaction of 2 -pentanone with ethylmagnesium bromide followed by the addition of dilute acid?
b. How many stereoisomers are obtained from the reaction of 2-pentanone with methylmagnesium bromide followed by the addition of dilute acid?
a. Which of the following tertiary alcohols cannot be prepared by the reaction of an ester with excess Grignard reagent?
b. For those alcohols that can be prepared by the reaction of an ester with excess Grignard reagent, what ester and what Grignard reagent should be used?
Which of the following secondary alcohols can be prepared by the reaction of methyl formate with excess Grignard reagent?
Write the mechanism for the reaction of acetyl chloride with two equivalents of ethylmagnesium bromide.
Which of the following compounds does not form an alcohol when it reacts with excess Grignard reagent?
Show how the following compounds can be synthesized from cyclohexanol.
a. Show how the following compounds can be prepared, using ethyne as one of the starting materials:
1. 1-pentyn-3-ol
2. 1 -phenyl- 2 -butyn- $1-$ ol
3. 2-methyl-3-hexyn-2-ol
b. Explain why ethyne should be alkylated before, rather than after, nucleophilic addition.
Zubair A.
What is the product of the reaction of an ester with excess acetylide ion followed by the addition of pyridinium chloride?
In the mechanism for cyanohydrin formation, why is HCN the acid that protonates the alkoxide ion instead of HCl?
Can a cyanohydrin be prepared by treating a ketone with sodium cyanide?
Explain why aldehydes and ketones react with a weak acid such as hydrogen cyanide but do not react with strong acids such as $\mathrm{HCl}$ or $\mathrm{H}_{2} \mathrm{SO}_{4}$ (other than being protonated by them).
How can the following compounds be prepared from a carbonyl compound that has one less carbon than the desired product?
a. $\mathrm{HOCH}_{2} \mathrm{CH}_{2} \mathrm{NH}_{2}$
SOLUTION TO 18 a. The starting material for the synthesis of this two-carbon compound must have one carbon. Therefore, it must be formaldehyde. Addition of hydrogen cyanide followed by addition of $\mathrm{H}_{2}$ to the triple bond of the cyanohydrin forms the target molecule.
SOLUTION TO 18 b. The starting material for the synthesis of this three-carbon compound must have two carbons. Therefore, it must be acetaldehyde. Addition of hydrogen cyanide, followed by hydrolysis of the resulting cyanohydrin, forms the target molecule.
Show two ways to convert an alkyl halide into a carboxylic acid that has one more carbon than the alkyl halide.
What alcohols are obtained from the reduction of the following compounds with sodium borohydride?
a. 2 -methylpropanal
b. cyclohexanone
c. 4-tert-butylcyclohexanone
d. acetophenone
What products are obtained from the reaction of the following compounds with LiAlH_followed by treatment with dilute acid?
a. ethyl butanoate
c. methyl benzoate
b. benzoic acid
d. pentanoic acid
What amides would you react with LiAlH $_{4}$ to form the following amines?
a. benzylmethylamine
c. diethylamine
b. ethylamine
d. triethylamine
How would you make the following compounds from $N$ -benzylbenzamide?
a. dibenzylamine
c. benzyl alcohol
What are the products of the following reactions?
What reducing agents should be used to obtain the desired target molecules?
A ketone can be prepared from the reaction of a nitrile with a Grignard reagent. Describe the intermediate formed in this reaction, and show how it can be converted to a ketone.
Why is the $\mathrm{p} K_{\mathrm{a}}$ value of protonated hydroxylamine ( 6.0 ) so much lower than the $\mathrm{p} K_{\mathrm{a}}$ value of a protonated primary amine such as protonated methylamine ( 10.7 )?
At what $\mathrm{pH}$ should imine formation be carried out if the amine's protonated form has a $\mathrm{p} K_{\mathrm{a}}$ value of $9.0 ?$
The pK $_{\mathrm{a}}$ of protonated acetone is about $-7.5,$ and the $\mathrm{p} K_{\mathrm{a}}$ of protonated hydroxylamine is $6.0 .$
a. In a reaction with hydroxylamine at pH 4.5 (Figure 16.2), what fraction of acetone is present in its acidic, protonated form? (Hint: See Section 2.10.)
b. In a reaction with hydroxylamine at $\mathrm{pH}$ 1.5, what fraction of acetone is present in its acidic, protonated form?
c. In a reaction with acetone at pH 1.5 (Figure 16.2 ), what fraction of hydroxylamine is present in its reactive basic form?
Imines can exist as stereoisomers. The isomers are named using the $E, Z$ system of nomenclature (Section 4.2 ). The lone pair has the lowest priority.
Draw the structure of each of the following compounds:
a. the ( $E$ )-hydrazone of benzaldehyde
b. the ( $Z$ )-oxime of propiophenone
a. Write the mechanism for the following reactions:
1. the acid-catalyzed hydrolysis of an imine to a carbonyl compound and a primary amine
2. the acid-catalyzed hydrolysis of an enamine to a carbonyl compound and a secondary amine
b. How do the two mechanisms differ?
What are the products of the following reactions? (A trace amount of acid is present in each case.)
a. cyclopentanone + ethylamine
c. acetophenone + hexylamine
b. cyclopentanone + diethylamine
d. acetophenone + cyclohexylamine
Excess ammonia must be used when a primary amine is synthesized by reductive amination. What product will be obtained if the reaction is carried out with excess carbonyl compound?
The compounds commonly known as "amino acids" are actually $\alpha$ -aminocarboxylic acids (Section 21.0 ). What carbonyl compounds should be used to synthesize the two amino acids shown here?
Hydration of an aldehyde is also catalyzed by hydroxide ion. Propose a mechanism for the reaction.
Which ketone forms the most hydrate in an aqueous solution?
When trichloroacetaldchyde is dissolved in water, almost all of it is converted to the hydrate. Chloral hydrate, the product of the reaction, is a sedative that can be lethal. A cocktail laced with it is known-in detective novels, at least-as a "Mickey Finn." Explain why an aqueous solution of trichloroacetaldehyde is almost all hydrate.
Which of the following are
a. hemiacetals?
b. acetals?
c. hydrates?
a. Would you expect hemiacetals to be stable in basic solutions? Explain your answer.
b. Acetal formation must be catalyzed by an acid. Explain why it cannot be catalyzed by $\mathrm{CH}_{3} \mathrm{O}^{-}$
c. Can the rate of hydrate formation be increased by hydroxide ion as well as by acid? Explain.
Explain why an acetal can be isolated but most hydrates cannot be isolated.
a. What would have been the product of the preceding reaction with $\mathrm{LiAlH}_{4}$ if the keto group had not been protected?
b. What reagent could you use to reduce only the keto group?
Explain why acetals do not react with nucleophiles.
What products would be formed from the preceding reaction if the carboxylic acid group were not protected?
a. In a six-step synthesis, what is the yield of the target molecule if each of the reactions employed gives an $80 \%$ yield?
b. What would the yield be if two more steps (each with an $80 \%$ yield) were added to the synthesis?
Show how each of the following compounds could be prepared from the given starting material. Each requires a protecting group.
What is the product of each of the following reactions?
a. What two sets of reagents (each consisting of a carbonyl compound and phosphonium ylide) can be used for the synthesis of the following alkene?
b. What alkyl halide is required to prepare each of the phosphonium ylides?
c. What is the best set of reagents to use for the synthesis?
SOLUTION TO 48 a. The atoms on either side of the double bond can come from the carbonyl compound, so two pairs of compounds could be used.
SOLUTION TO 48 b. The alkyl halide required to make the phosphonium ylide would be 1-bromobutane for the first pair of reagents or 2 -bromopropane for the second pair.
SOLUTION TO 48 c. The primary alkyl halide would be more reactive in the S $_{\mathrm{N}} 2$ reaction required to make the ylide, so the best method would be to use the first set of reagents (acetone and the ylide obtained from 1-bromobutane).
a. What two sets of reagents (each consisting of a carbonyl compound and phosphonium ylide) can be used for the synthesis of each of the following alkenes?
Using bromocyclohexane as a starting material, how could you synthesize the following compounds?
Ronald P.
What is the major product of each of the following reactions?
Draw the structure for each of the following:
a. isobutyraldehyde
b. 4-hexenal
c. disopentyl ketone
d. 3 -methylcyclohexanone
e. 2,4 -pentanedione
f. 4 -bromo- 3 -heptanone
g. $\gamma$ -bromocaproaldehyde
h. 2 -ethylcyclopentanecarbaldehyde
i. 4-methyl-5-oxohexanal
Rank the following compounds from most reactive to least reactive toward nucleophilic addition:
Draw the structure of two esters that will be reduced to propanol and butanol by LiAlH_(followed by addition of aqueous acid).
a. Show the reagents required to form the primary alcohol in each of the following reactions.
b. Which of the reactions cannot be used for the synthesis of isobutyl alcohol?
Draw the products of the following reactions. Indicate whether each reaction is an oxidation or a reduction.
Using cyclohexanone as the starting material, describe how each of the following compounds can be synthesized:
Propose a mechanism for each of the following reactions:
Show how each of the following compounds can be prepared, using the given starting material:
Fill in the boxes:
Thiols can be prepared from the reaction of thiourea with an alkyl halide, followed by hydroxide-ion-promoted hydrolysis.
a. Propose a mechanism for the reaction.
b. What thiol will be formed if the alkyl halide employed is pentyl bromide?
Identify $A$ through $O$:
The only organic compound obtained when compound $\mathbf{Z}$ undergoes the following sequence of reactions gives the $^{1} \mathrm{H}$ NMR spectrum shown. Identify compound $\mathbf{Z}$.
How many signals would the product of the following reaction show in
a. its 'H NMR spectrum?
b. its $^{13} \mathrm{C}$ NMR spectrum?
Fill in the boxes with the appropriate reagents:
How could you convert $N$ -methylbenzamide to the following compounds?
a. $N$ -methylbenzylamine
d. benzyl alcohol
What are the products of the following reactions? Show all stereoisomers that are formed.
List three different sets of reagents (each set consisting of a carbonyl compound and a Grignard reagent) that could be used to prepare each of the following tertiary alcohols:
What product is formed when 3 -methyl- 2 -cyclohexenone reacts with each of the following reagents?
a. $\mathrm{CH}_{3} \mathrm{MgBr}$ followed by $\mathrm{H}_{3} \mathrm{O}^{+}$
b. $\left(\mathrm{CH}_{3} \mathrm{CH}_{2}\right)_{2} \mathrm{CuLi}$ followed by $\mathrm{H}_{3} \mathrm{O}^{+}$
c. HBr
d. $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{SH}$
Draw structures for $\mathbf{A}-\mathbf{D}$ for each of the following:
Propose a mechanism to explain how dimethyl sulfoxide and oxalyl chloride react to form the dimethylchlorosulfonium ion used as the oxidizing agent in the Swern oxidation (see Chapter $10,$ page 475 ).
a. Propose a mechanism for the following reaction:
b. What is the product of the following reaction?
Unlike a phosphonium ylide that reacts with an aldehyde or a ketone to form an alkene, a sulfonium ylide reacts with an aldehyde or a ketone to form an epoxide. Explain why one ylide forms an alkene, whereas the other forms an epoxide.
A compound gives the following IR spectrum. Upon reaction with sodium borohydride followed by acidification, it forms the product with the 1 H NMR spectrum shown below. Identify the starting material and the product.
How can the following compounds be prepared from the given starting materials?
a. In an aqueous solution, D-glucose exists in equilibrium with two six-membered ring compounds. Draw the structures of these compounds.
b. Which of the six-membered ring compounds will be the major product?
Shown below is the $^{1} \mathrm{H}$ NMR spectrum of the alkyl bromide used to make the phosphonium ylide that reacts with a ketone in a Wittig reaction to form a compound with molecular formula $C_{11} H_{14}$. What product is obtained from the Wittig reaction?
In the presence of an acid catalyst, acetaldehyde forms a trimer known as paraldehyde. Because it induces sleep when it is administered to animals in large doses, paraldehyde is used as a sedative or hypnotic. Propose a mechanism for the formation of paraldehyde.
What carbonyl compound and what phosphonium ylide are needed to synthesize the following compounds?
Identify compounds $\mathbf{A}$ and $\mathbf{B}$ :
When a cyclic ketone reacts with diazomethane, the next larger cyclic ketone is formed. This is called a ring-expansion reaction. Draw a mechanism for the following ring-expansion reaction.
A compound reacts with methylmagnesium bromide followed by acidification to form the product with the following $^{1} \mathrm{H}$ NMR spectrum. Identify the compound.
Show how each of the following compounds can be prepared from the given starting material. In each case, you will need to use a protecting group.
Describe how 1-ethylcyclohexanol can be prepared from cyclohexane. You can use any inorganic reagents, any solvents, and any organic reagents as long as they contain no more than two carbons.
The $\mathrm{p} K_{\mathrm{a}}$ values of the carboxylic acid groups of oxaloacetic acid are 2.22 and 3.98
a. Which carboxyl group is the stronger acid?
b. The amount of hydrate present in an aqueous solution of oxaloacetic acid depends on the $\mathrm{pH}$ of the solution: $95 \%$ at $\mathrm{pH} 0,81 \%$ at $\mathrm{pH} 1.3,35 \%$ at $\mathrm{pH}$, and $6 \%$ at $\mathrm{pH}$ 12.7. Explain this pH dependence.
The Baylis-Hillman reaction is a DABCO (1,4-diazabicyclo[2.2.2]octane) catalyzed reaction of an $\alpha, \beta$ -unsaturated carbonyl compound with an aldehyde to form an allylic alcohol. Propose a mechanism for the reaction. Propose a mechanism for the reaction. (Hint: DABCO serves as both a nucleophile and as a base in the reaction.)
To solve this problem, you need to read the description of the Hammett $\sigma, \rho$ treatment given in Chapter $15,$ Problem $92 .$ When the rate constants for the hydrolysis of several morpholine enamines of para-substituted propiophenones are determined at pH 4.7, the $\rho$ value is positive; however, when the rates of hydrolysis are determined at pH $10.4,$ the $\rho$ value is negative.
a. What is the rate-determining step of the hydrolysis reaction when it is carried out in a basic solution?
b. What is the rate-determining step of the hydrolysis reaction when it is carried out in an acidic solution? | CommonCrawl |
Find $(4^4 \div 4^3) \cdot 2^8$.
Performing the arithmetic in the parentheses first, we obtain $4^4 \div 4^3 = 4$, so we have \[(4^4 \div 4^3) \cdot 2^8 = 4\cdot 2^8.\]Since $4 = 2^2$, we have \[4\cdot 2^8 = 2^2 \cdot 2^8 = 2^{10}= \boxed{1024}.\] | Math Dataset |
\begin{document}
\title{XML Navigation and Transformation by \\
Tree-Walking Automata and Transducers \\
with Visible and Invisible Pebbles}
\author{ Joost Engelfriet\thanks{Email: {\tt [email protected]}} \and Hendrik Jan Hoogeboom\thanks{Email: {\tt [email protected]}} \and \\ Bart Samwel\thanks{Email: {\tt [email protected]}} \\ \\ {\small LIACS, Leiden University, the Netherlands} }
\date{} \maketitle
\begin{abstract} The pebble tree automaton and the pebble tree transducer are enhanced by additionally allowing an unbounded number of ``invisible'' pebbles (as opposed to the usual ``visible'' ones). The resulting pebble tree automata recognize the regular tree languages (i.e., can validate all generalized DTD's) and hence can find all matches of MSO definable patterns. Moreover, when viewed as a navigational device, they lead to an XPath-like formalism that has a path expression for every MSO definable binary pattern. The resulting pebble tree transducers can apply arbitrary MSO definable tests to (the observable part of) their configurations, they (still) have a decidable typechecking problem, and they can model the recursion mechanism of XSLT. The time complexity of the typechecking problem for conjunctive queries that use MSO definable patterns can often be reduced through the use of invisible pebbles. \end{abstract}
\tableofcontents
\section{Introduction}
Pebble tree transducers, as introduced by Milo, Suciu, and Vianu \cite{MilSucVia03}, are a formal model of XML navigation and transformation for which typechecking is decidable. The pebble tree transducer is a tree-walking tree transducer with nested pebbles, i.e., it walks on the input tree, dropping and lifting a bounded number of pebbles that have nested life times, whereas it produces the output tree in a parallel top-down fashion. We enhance the power of the pebble tree transducer by allowing an unbounded number of (coloured) pebbles, still with nested life times, i.e., organized as a stack. However, apart from a bounded number, the pebbles are ``invisible'', which means that they can be observed by the transducer only when they are on top of the stack (and thus the number of observable pebbles is bounded at each moment in time). We will call \abb{v-ptt} the pebble tree transducer of \cite{MilSucVia03} (or rather, the one in \cite{EngMan03}: an obvious definitional variant), and \abb{vi-ptt} the enhanced pebble tree transducer. Moreover, \abb{i-ptt} refers to the \abb{vi-ptt} that does not use visible pebbles, which can be viewed as a generalization of the indexed tree transducer of~\cite{EngVog}. And \abb{tt} refers to the pebble tree transducer without pebbles, i.e., to the tree-walking tree transducer, cf.~\cite{Eng09} and~\cite[Section~8]{thebook}. Tree-walking transducers were introduced in~\cite{AhoUll}, where they translate trees into strings.\footnote{In~\cite[Section~8]{thebook} the \abb{tt} is called tree-walking transducer and the transducer of~\cite{AhoUll} is called tree-walking tree-to-word transducer. }
The navigational part of the \abb{v-ptt}, i.e., the behaviour of the transducer when no output is produced, is the pebble tree automaton (\abb{v-pta}), introduced in~\cite{jewels}, which is a tree-walking automaton with nested pebbles. It was shown in~\cite{jewels} that the \abb{v-pta} recognizes regular tree languages only. In \cite{expressive} the important result was proved that not all regular tree languages can be recognized by the \abb{v-pta}, and thus \cite{Don70,ThaWri} the navigational power of the \abb{v-ptt} is below Monadic Second Order (\abb{MSO}) logic, which is undesirable for a formal model of XML transformation (see, e.g., \cite{NevSch}). One of the reasons for introducing invisible pebbles is that the \abb{vi-pta}, and even the \abb{i-pta}, recognizes exactly the regular tree languages (Theorem~\ref{thm:regt}). Thus, since the regular tree grammar is a formal model of DTD (Document Type Definition) in XML, the \abb{vi-pta} can validate arbitrary generalized DTD's. We note that the \abb{i-pta} is a straightforward generalization of the two-way backtracking pushdown tree automaton of Slutzki \cite{Slu}.
Surveys on the use of tree-walking automata and transducers for XML can be found in~\cite{Nev,Sch}. For a survey on tree-walking automata see~\cite{Boj}.
It is easy to show that every regular tree language can be recognized by an \mbox{\abb{i-pta}}, just simulating a bottom-up finite-state tree automaton. The proof that all \abb{vi-pta} tree languages are regular, is based on a decomposition of the \abb{vi-ptt} into \abb{tt}'s (Theorem~\ref{thm:decomp}), similar to the one for the \abb{v-ptt} in \cite{EngMan03}. Since the inverse type inference problem is solvable for \abb{tt}'s (where a ``type'' is a regular tree language), this shows that the domain of a \abb{vi-ptt} is regular, and so even the alternating \abb{vi-pta} tree languages are regular. It also shows that the typechecking problem is decidable for \abb{vi-ptt}'s, by the same arguments as used in \cite{MilSucVia03} for \abb{v-ptt}'s. More precisely, we prove (Theorem~\ref{thm:typecheck}, based on~\cite[Theorem~3]{Eng09}) that a \abb{vi-ptt} with $k$ visible pebbles can be typechecked in \mbox{($k+3$)}-fold exponential time. For varying~$k$ the complexity is non-elementary (as in~\cite{MilSucVia03}), but it is observed in~\cite{MolSch} that ``non-elementary algorithms on tree automata have previously been seen to be feasible in practice''.
Generalizing the fact that the \abb{i-pta} can recognize the regular tree languages, we prove that the \abb{vi-pta} and the \abb{vi-ptt} can perform \abb{MSO} tests on the observable part of their configuration, i.e., they can check whether or not the observable pebbles on the input tree (i.e., the visible ones, plus the top pebble on the stack) satisfy certain \abb{MSO} requirements with respect to the current position of the reading head (Theorem~\ref{thm:mso}). If all the observable pebbles are visible this is obvious (drop an additional visible pebble, simulate an \abb{i-pta} that recognizes the regular tree language corresponding to the \abb{mso} requirements, return to the pebble and lift it), but if the top pebble is invisible (or if there is no visible pebble left) that does not work and a more complicated technique must be used. Consequently, the \abb{vi-pta} can match arbitrary \abb{MSO} definable $n$-ary patterns, using $n$~visible pebbles to find all candidate matches as in \cite[Example~3.5]{MilSucVia03}, and using invisible pebbles to perform the \abb{MSO} test; the \abb{vi-ptt} can also output the matches. In fact, instead of the $n$~visible pebbles the \abb{vi-pta} can use $n-2$ visible pebbles, one invisible pebble (on top of the stack), and the reading head (Theorem~\ref{thm:matchall}).
As the navigational part of the \abb{vi-ptt}, the \abb{vi-pta} in fact computes a binary pattern on trees, i.e., a binary relation between two nodes of a tree: the position of the reading head of the \abb{vi-ptt} before and after navigation. We prove that also as a navigational device the \abb{vi-pta} and the \abb{i-pta} have the same power as \abb{MSO} logic: they compute exactly the \abb{MSO} definable binary patterns (Theorem~\ref{thm:trips}). This improves the result in \cite{trips} (where binary patterns are called ``trips''), because the \abb{i-pta} is a more natural automaton than the one considered in \cite{trips}.
One of the research goals of Marx and ten Cate (see \cite{Mar05,GorMar05,Cat06,CatSeg} and the entertaining \cite{Mar06}) has been to combine Core XPath of \cite{GotKoc02} which models the navigational part of XPath 1.0, with regular path expressions \cite{AbiBunSuc00} (or caterpillar expressions \cite{BruWoo00}) which naturally correspond to tree-walking automata. An important feature of XPath is the ``predicate'': it allows to test the context node for the existence of at least one other node that matches a given path expression. Thus, the path expression $\alpha_1[\beta]/\alpha_2$ takes an $\alpha_1$-walk from the context node to the new context node $v$, checks whether there exists a $\beta$-walk from $v$ to some other node, and then takes an $\alpha_2$-walk from $v$ to the match node. For tree automata this corresponds to the notion of ``look-ahead'' (cf. \cite[Definition~6.5]{EngVog}). We prove (Theorem~\ref{thm:look-ahead}) that an \abb{i-pta} ${\cal A}$ can use another \abb{i-pta} ${\cal B}$ as look-ahead test, i.e., ${\cal A}$ can test whether or not ${\cal B}$ has a successful computation when started in the current configuration of ${\cal A}$ (and similarly for \abb{vi-pta} and \abb{vi-ptt}). Since XPath expressions can be nested arbitrarily, we even allow ${\cal B}$ to use yet another \abb{i-pta} as look-ahead test, etcetera (Theorem~\ref{thm:iterated}). Due to this ``iterated look-ahead'' feature, we can use Kleene's classical construction to translate the $\mbox{\abb{i-pta}}$ into an XPath-like algebraic formalism, which we call \emph{Pebble XPath}, with the same expressive power as \abb{MSO} logic for defining binary patterns (Theorem~\ref{thm:xpath}). In fact, Pebble XPath is the extension of Regular XPath \cite{Mar05,Cat06} with a stack of invisible pebbles. It is proved in \cite{CatSeg} that Regular XPath is not \abb{MSO} complete (see also~\cite{Mar06}).\footnote{To be precise, it is proved in \cite{CatSeg} that Regular XPath with ``subtree relativisation'' is not \abb{mso} complete and has the same power as first-order logic with monadic transitive closure. } Other \abb{MSO} complete extensions of Regular XPath are considered in \cite{GorMar05,Cat06}.
To explain another reason for introducing invisible pebbles we consider XQuery-like conjunctive queries of the form \[ \mbox{\tt for } x_1,\dots,x_n \mbox{ \tt where } \varphi_1\wedge\dots\wedge\varphi_m \mbox{ \tt return } r, \] where $x_1,\dots,x_n$ are variables, each $\varphi_\ell$ (with $1\leq \ell\leq m$) is an \abb{MSO} formula with two free variables $x_i$ and $x_j$, and $r$ is an output tree with variables at the leaves. As observed above, such pattern matching queries can be evaluated by a \abb{vi-ptt} with $n-2$ visible pebbles, even if the {\tt where}-clause contains an arbitrary \abb{MSO} formula. In many cases, however, a much smaller number of visible pebbles suffices (Theorem~\ref{thm:matching}). This is an enormous advantage when typechecking the query, as for the time complexity every visible pebble counts (viz. it counts as an exponential). For instance if $j=i+1$ for every $\varphi_\ell$, then \emph{no} visible pebbles are needed, i.e., the query can be evaluated by an \abb{i-ptt}: we use invisible pebbles $p_1,\dots,p_n$ on the stack (in that order), representing the variables, and move them through the input tree in document order, in a nested fashion; just before dropping pebble $p_{i+1}$, each formula $\varphi_\ell(x_i,x_{i+1})$ can be verified by an MSO test on the observable part of the configuration (which consists of the top pebble $p_i$ and the reading head position).
The pebble tree transducer transforms ranked trees. However, an XML document is not ranked; it is a forest: a sequence of unranked trees. To model XML transformation by \abb{ptt}'s, forests are encoded as binary trees in the usual way. For the input, it does not make much of a difference whether the \abb{ptt} walks on a binary tree or a forest. However, as opposed to what is suggested in \cite{MilSucVia03}, for the output it \emph{does} make a difference, as pointed out in \cite{PerSei} for macro tree transducers. For that reason we also consider pebble \emph{forest} transducers (abbreviated with \abb{pft} instead of \abb{ptt}) that walk on encoded forests, but construct forests directly, using forest concatenation as basic operation. As in \cite{PerSei}, \abb{pft} are more powerful than \abb{ptt}, but the complexity of the typechecking problem is the same, i.e., \abb{vi-pft} with $k$ visible pebbles can be typechecked in ($k+3$)-fold exponential time (Theorem~\ref{thm:typecheck-pft}). In fact, \abb{pft} have all the properties mentioned before for \abb{ptt}.
The document transformation languages \abb{DTL} and \abb{TL} were introduced in \cite{ManNev00} and \cite{ManBerPerSei}, respectively, as a formal model of the recursion mechanism in the template rules of XSLT, with \abb{MSO} logic rather than XPath to specify matching and selection. Documents are modelled as forests. The language \abb{DTL} has no variables or parameters, and its only instruction is {\tt apply-templates}. The language~\abb{TL} is the extension of \abb{DTL} with accumulating parameters, i.e., the parameters of XSLT~1.0 whose values are ``result tree fragments'' (and on which no operations are allowed). We prove that every \abb{DTL} program can be simulated, with forests encoded as binary trees, by an \abb{i-ptt} (Theorem~\ref{thm:dtl}). More importantly, we prove that \abb{TL} and \abb{i-pft} have the same expressive power (Theorem~\ref{thm:tl}). Thus, in its forest version, our new model the \abb{vi-pft} can be viewed as the natural combination of the pebble tree transducer of \cite{MilSucVia03} (\abb{v-ptt}) and the \abb{TL} program of \cite{ManBerPerSei} (\abb{i-pft}). Note that \abb{v-ptt} and \abb{TL} have incomparable expressive power. As claimed by \cite{ManBerPerSei}, \abb{TL} can ``describe many real-world XML transformations''. We show that it contains all deterministic \abb{vi-pft} transformations for which the size of the output document is linear in the size of the input document (Theorem~\ref{thm:lsi}). However, the visible pebbles seem to be a requisite for the XQuery-like queries discussed above, and we conjecture that not all such queries can be programmed in \abb{TL} (though they \emph{can}, e.g., in the case that $j=i+1$ for every~$\ell$). As shown in~\cite{bex} (for a subset of \abb{MSO}), these queries can be programmed in XSLT~1.0 using parameters that have input nodes as values; however, with such parameters even \abb{v-ptt} with \emph{non}nested pebbles can be simulated, and typechecking is no longer decidable. In XSLT~2.0 \emph{all} (computable) queries can be programmed~\cite{JanKorBus}. The main result of~\cite{ManBerPerSei} is that typechecking is decidable for \abb{tl} programs. Assuming that \abb{mso} formulas are represented by deterministic bottom-up finite-state tree automata, the above relationship between \abb{tl} and \abb{i-pft} allows us to prove that \abb{tl} programs can be typechecked in $4$-fold exponential time (Theorem~\ref{thm:tltypecheck}), which seems to be one exponential better than the algorithm in~\cite{ManBerPerSei}.
In addition to the time complexity of typechecking a \abb{vi-ptt}, also the time complexity of evaluating the queries realized by a \abb{vi-pta} or a \abb{vi-ptt} is of importance. The binary pattern (or `trip') computed by a \abb{vi-pta}, i.e., the binary relation between two nodes of the input tree, can be evaluated in polynomial time. The same is true for every (fixed) expression of Pebble XPath (see the last two paragraphs of Section~\ref{sec:xpath}). Deterministic \abb{vi-ptt}'s have exponential time data complexity, provided that the output tree can be represented by a DAG (directed acyclic graph). To be precise, for every deterministic \abb{vi-ptt} there is an exponential time algorithm that transforms any input tree of that \abb{vi-ptt} into a DAG that represents the corresponding output tree (Theorem~\ref{thm:expocom}). For the \abb{vi-ptt}'s that match \abb{mso} definable $n$-ary patterns (as discussed above) the algorithm is polynomial time (Theorem~\ref{thm:polycom}).
Apart from the above results that are motivated by XML navigation and transformation, we also prove some more theoretical results. We show that (as opposed to the \abb{v-ptt}) the \abb{i-ptt} can simulate the bottom-up tree transducer (Theorem~\ref{thm:bottomup}). We show that the composition of two deterministic \abb{tt}'s can be simulated by a deterministic \mbox{\abb{i-ptt}} (Theorem~\ref{thm:composition}). This even holds when the \abb{tt}'s are allowed to perform \abb{mso} tests on their configuration, and then also vice versa, every deterministic \abb{i-ptt} can be decomposed into two such extended \abb{tt}'s (Theorem~\ref{thm:charidptt}).
We show that every deterministic \abb{vi-ptt} can be decomposed into deterministic \abb{tt}'s (Theorem~\ref{thm:detdecomp}) and that, for the deterministic \abb{vi-ptt}, $k+1$ visible pebbles are more powerful than $k$ visible pebbles (Theorem~\ref{thm:dethier}). Pebbles have to be lifted from the position where they were dropped; however, in~\cite{fo+tc} it was convenient to consider a stronger type of pebbles that can also be retrieved from a distance. Whereas \abb{i-ptt}'s with strong invisible pebbles can recognize nonregular tree languages, we show that \abb{vi-ptt}'s with strong visible pebbles can still be decomposed into \abb{tt}'s (Theorems~\ref{thm:decompplus} and~\ref{thm:decompplusI}) and hence their typechecking is decidable (as already proved for \abb{v-ptt}'s with strong pebbles in~\cite{FulMuzFI}). Similarly, deterministic \abb{vi-ptt}'s with strong visible pebbles can be decomposed into deterministic \abb{tt}'s (Theorems~\ref{thm:detdecompplus} and~\ref{thm:detdecompplusI}).
Some of these theoretical results can be viewed as (slight) generalizations of existing results for formal models of compiler construction (in particular attribute grammars), such as attributed tree transducers~\cite{Ful}, macro tree transducers~\cite{EngVog85}, and macro attributed tree transducers~\cite{KuhVog}, see also~\cite{FulVog}. As explained in~\cite[Section~3.2]{EngMan03}, attributed tree transducers are \abb{tt}'s that satisfy an additional requirement of ``noncircularity''. Similarly, as observed in~\cite{ManBerPerSei}, macro attributed tree transducers (that generalize both attributed tree transducers and macro tree transducers) are closely related to \abb{tl} programs, and hence to \abb{i-ptt}'s by Theorem~\ref{thm:tl}. For instance, Theorem~\ref{thm:composition} slightly generalizes the fact that the composition of two attributed tree transducers can be simulated by a macro attributed tree transducer, as shown in~\cite{KuhVog}.
Most of the results of this paper were announced in the PODS'07 conference~\cite{invisible}. The remaining results are based on technical notes of the authors from the years 2004--2008. This paper has not been updated with the litterature of later years (with the exception of~\cite{thebook,Eng09,CatSeg}).
\section{Preliminaries}\label{sec:trees}
\smallpar{Sets, strings, and relations} The set of natural numbers is ${\mathbb N}=\{0,1,2,\dots\}$. For $m,n\in {\mathbb N}$, we denote the interval $\{k\in{\mathbb N}\mid m\leq k\leq n\}$ by $[m,n]$. The cardinality or size of a set $A$ is denoted by $\#(A)$, and its powerset, i.e., the set of all its subsets, by $2^A$. The set of strings over $A$ is denoted by~$A^*$. It consists of all sequences $w=a_1\cdots a_m$ with $m\in{\mathbb N}$ and $a_i\in A$ for every $i\in[1,m]$.
The length~$m$ of $w$ is denoted by $|w|$. The empty string (of length $0$) is denoted by~$\varepsilon$. The concatenation of two strings $v$ and $w$ is denoted by $v\cdot w$ or just~$vw$. Moreover, $w^0=\varepsilon$ and $w^{n+1}=w\cdot w^n$ for $n\in{\mathbb N}$. The composition of two binary relations $R\subseteq A\times B$ and $S\subseteq B\times C$ is $R\circ S = \{(a,c)\mid \exists\, b\in B: (a,b)\in R, \, (b,c)\in S\}$. The~inverse of $R$ is $R^{-1}=\{(b,a)\mid (a,b)\in R\}$, and if $A=B$ then the transitive-reflexive closure of~$R$ is $R^*=\bigcup_{n\in{\mathbb N}}R^n$ where $R^0=\{(a,a)\mid a\in A\}$ and $R^{n+1}=R\circ R^n$. The composition of two classes of binary relations ${\cal R}$ and ${\cal S}$ is ${\cal R}\circ {\cal S} = \{R\circ S\mid R\in{\cal R},\,S\in{\cal S}\}$. Moreover, ${\cal R}^1={\cal R}$ and ${\cal R}^{n+1} = {\cal R}\circ{\cal R}^n$ for $n\geq 1$.
\smallpar{Trees and forests} An alphabet is a finite set of symbols. Let $\Sigma$ be an alphabet, or an arbitrary set. Unranked trees and forests over $\Sigma$ are recursively defined to be strings over the set $\Sigma\cup\{(,)\}$ consisting of the elements of $\Sigma$, the left parenthesis, and the right parenthesis, as follows. If $\sigma\in\Sigma$ and $t_1, \dots, t_m$ are unranked trees, with $m\in{\mathbb N}$, then their concatenation $t_1 \cdots t_m$ is a forest, and $\sigma(t_1 \cdots t_m)$ is an unranked tree. For $m=0$, $t_1 \cdots t_m$ is the empty forest~$\varepsilon$. For readability we also write the tree $\sigma(t_1 \cdots t_m)$ as $\sigma(t_1, \dots, t_m)$, and even as~$\sigma$ when $m=0$. Obviously, the concatenation of two forests is again a forest. It should also be noted that every nonempty forest can be written uniquely as $\sigma(f_1)f_2$ where $\sigma$ is in $\Sigma$ and $f_1$ and $f_2$ are forests. The set of forests over~$\Sigma$ is denoted $F_\Sigma$. For an arbitrary set $A$, disjoint with $\Sigma$, we denote by $F_\Sigma(A)$ the set all forests $f$ over $\Sigma\cup A$ such that every node of $f$ that is labelled by an element of $A$, is a leaf.
As usual trees and forests are viewed as directed labelled graphs. Here we distinguish between two types of edges: ``vertical'' and ``horizontal'' ones. The root of the tree $t=\sigma( t_1, \dots, t_m )$ is labelled by $\sigma$. It has vertical edges to the roots of subtrees $t_1, \dots, t_m$, which are the children of the root of $t$ and have child number $1$ to $m$. The root of~$t$ is their parent. The roots of $t_1, \dots, t_m$ are siblings, also in the case of the forest $ t_1 \cdots t_m $. There is a horizontal edge from each sibling to the next, i.e., from the root of $t_i$ to the root of $t_{i+1}$ for every $i\in [1,m-1]$. Thus, the vertical edges represent the usual parent/child relationship, whereas the horizontal edges represent the linear order between children (and between the roots in a forest), see Fig.~\ref{fig:forest}.\footnote{In informal pictures the horizontal edges are usually omitted because they are implicit in the left-to-right orientation of the page. Similarly, the arrows of the vertical edges are omitted because of the top-down orientation of the page. }
\begin{figure}
\caption{Picture of the forest $\sigma(a,\tau(b,a),b)\,\tau(\sigma(a),b)$. Formal at the left, with dotted lines for the horizontal edges and solid lines for the vertical edges, and informal at the right.}
\label{fig:forest}
\end{figure}
For a tree $t$, its root is denoted by $\mathrm{root}_t$, which is given child number $0$ for technical convenience. Its set of nodes is denoted by $\nod t$. For a forest $f= t_1 \cdots t_m $, the set of nodes $\nod f$ is the disjoint union
of the sets $\nod{t_i}$, $i\in[1,m]$. For a node $u$ of a tree~$t$ the subtree of $t$ with root $u$ is denoted $t|_u$, and the $i$-th child of $u$ is denoted $ui$ (and similarly for a forest $f$ instead of $t$). The nodes of a tree $t$ correspond one-to-one to the positions of the elements of $\Sigma$ in the string $t$, i.e., for every $\sigma\in\Sigma$, each occurrence of $\sigma$ in $t$ corresponds to a node of $t$ with label $\sigma$. Since the positions of string $t$ are naturally ordered from left to right, this induces an order on the nodes of $t$, which is called pre-order (or document order, when viewing $t$ as an XML document). For example, the tree $\sigma(\tau(\alpha,\beta),\gamma))$ has five nodes which have the labels $\sigma$, $\tau$, $\alpha$, $\beta$, and $\gamma$ in pre-order.
A \emph{ranked} alphabet (or set) $\Sigma$ has an associated mapping $\operatorname{rank}_\Sigma : \Sigma \to {\mathbb N}$. The maximal rank of elements of $\Sigma$ is denoted ${\mathit mx}_\Sigma$. By $\Sigma^{(m)}$ we denote the elements of $\Sigma$ with rank $m$. Ranked trees over $\Sigma$ are recursively defined as above with the requirement that $m = \operatorname{rank}_\Sigma(\sigma)$. The set of ranked trees over~$\Sigma$ is denoted $T_\Sigma$. For an arbitrary set $A$, disjoint with $\Sigma$, we denote by $T_\Sigma(A)$ the set $T_{\Sigma\cup A}$ where each element of $A$ has rank~$0$. We will not consider ranked forests.
Forests over an alphabet $\Sigma$ can be encoded as binary trees, in the usual way: each node has a label in $\Sigma$, a ``vertical'' pointer to its first child, and a ``horizontal'' pointer to its next sibling; the pointer is nil if there is no such child or sibling. Such a binary tree can be modelled as a ranked tree over the ranked alphabet $\Sigma\cup\{e\}$ where every $\sigma\in\Sigma$ has rank~2 and $e$ is a symbol of rank~0 that represents the empty forest $\varepsilon$ (or nil). Formally, the encoding of the empty forest equals ${\rm enc}(\varepsilon) = e$, and recursively, the encoding ${\rm enc}(f)$ of a forest $f=\sigma(f_1)f_2$ equals $\sigma({\rm enc}(f_1),{\rm enc}(f_2))$. Obviously, ${\rm enc}$ is a bijection between forests over $\Sigma$ and ranked trees over $\Sigma\cup\{e\}$. The decoding which is its inverse will be denoted by ${\rm dec}$. For an example of ${\rm enc}(f)$ see Fig.~\ref{fig:encforest} at the left.
\begin{figure}
\caption{Encoding of the forest of Fig.~\ref{fig:forest}
by ${\rm enc}$ (at the left) and by ${\rm enc}'$ (at the right).}
\label{fig:encforest}
\end{figure}
The disadvantage of this encoding is that the tree ${\rm enc}(f)$ has more nodes than the forest $f$, viz. all nodes with label $e$. That is inconvenient when comparing the behaviour of tree-walking automata on $f$ and ${\rm enc}(f)$. Thus, we will also use an encoding that preserves the number of nodes (and thus cannot encode the empty forest). For this we use the ranked alphabet $\Sigma'$ consisting, for every $\sigma\in\Sigma$, of the symbols $\sigma^{11}$ of rank~2 (for a binary node without nil-pointers), $\sigma^{01}$ and $\sigma^{10}$ of rank~1 (for a binary node with vertical or horizontal nil-pointer, respectively), and $\sigma^{00}$ of rank~0 (for a binary node with two nil-pointers). The encoding ${\rm enc}'(f)$ of a nonempty forest $f=\sigma(f_1)f_2$ equals $\sigma^{11}({\rm enc}'(f_1),{\rm enc}'(f_2))$ or $\sigma^{01}({\rm enc}'(f_2))$ or $\sigma^{10}({\rm enc}'(f_1))$ or $\sigma^{00}$, where the first (second) superscript of~$\sigma$ equals $0$ if and only if $f_1 = e$ ($f_2 = e$). Now, ${\rm enc}'$ is a bijection between nonempty forests over $\Sigma$ and ranked trees over $\Sigma'$. The decoding which is its inverse will be denoted by ${\rm dec}'$. For an example of ${\rm enc}'(f)$ see Fig.~\ref{fig:encforest} at the right. From the point of view of graphs, we assume that ${\rm enc}'(f)$ has the same nodes as $f$, i.e., $\nod{{\rm enc}'(f)}=\nod{f}$. The label of a node $u$ of $f$ is changed from $\sigma$ to $\sigma^{ij}$ where $i=1$ if and only if $u$ has at least one child, and $j=1$ if and only if $u$ has a next sibling. If $u$ has children, then its first child in ${\rm enc}'(f)$ is its first child in $f$, and its second child in ${\rm enc}'(f)$ is its next sibling (if it has one). If $u$ has no children, then its only child in ${\rm enc}'(f)$ is its next sibling (if it has one). Although this encoding is intuitively clear, it is technically less attractive. We will use ${\rm enc}'$ for the input forest of automata and transducers, and ${\rm enc}$ for the output forest of the transducers.
We assume the reader to be familiar with the notion of a \emph{regular tree grammar}. It is a context-free grammar~$G$ of which every rule is of the form $X_0\to \sigma(X_1\cdots X_m)$ where $X_i$ is a nonterminal and $\sigma$ is a terminal symbol of rank~$m$. Thus, $G$ generates a set $L(G)$ of ranked trees, which is called a regular tree language. The class of regular tree languages will be denoted $\family{REGT}$. We define a \emph{regular forest grammar} to be a context-free grammar $G$ of which every rule is of the form $X_0\to \sigma(X_1)X_2$ or $X\to\varepsilon$, where $\sigma$ is from an unranked alphabet. It generates a set $L(G)$ of (unranked) forests, which is called a regular forest language. Obviously, $L$ is a regular forest language if and only if ${\rm enc}(L)$ is a regular tree language, and, as one can easily prove, if and only if ${\rm enc}'(L)$ is a regular tree language. The regular tree/forest grammar is a formal model of DTD (Document Type Definition) in XML.\footnote{In the litterature regular forest languages are usually defined in a different way, after which it is proved that $L$ is a regular forest language if and only if ${\rm enc}(L)$ is a regular tree language, thus showing the equivalence with our definition, see, e.g., \cite[Proposition~1]{Nev}. }
\emph{Monadic second-order logic} (abbreviated as \abb{mso} logic) is used to describe properties of forests and trees. It views each forest or tree as a logical structure that has the set of nodes as domain. As basic properties of a forest over alphabet~$\Sigma$ it uses the atomic formulas $\mathrm{lab}_\sigma(x)$, ${\rm down}(x,y)$, and $\mathrm{next}(x,y)$, meaning that node $x$ has label $\sigma\in\Sigma$, that $y$ is a child of $x$, and that $y$ is the next sibling of $x$, respectively. Thus, ${\rm down}(x,y)$ and $\mathrm{next}(x,y)$ represent the vertical and horizontal edges of the graph representation of the forest. For a ranked tree over ranked alphabet $\Sigma$ we could use the same atomic formulas, but it is customary to replace ${\rm down}(x,y)$ and $\mathrm{next}(x,y)$ by the atomic formulas ${\rm down}_i(x,y)$, for every $i\in[1,{\mathit mx}_\Sigma]$, meaning that $y$ is the $i$-th child of $x$. Additionally, \abb{mso}~logic has the atomic formulas $x=y$ and $x\in X$, where $X$ is a set of nodes. The formulas are built with the usual connectives $\neg$, $\wedge$, $\vee$, and $\to$; both node variables $x,y,\dots$ and node-set variables $X,Y,\dots$ can be quantified with~$\exists$ and~$\forall$. For a forest (or ranked tree) $f$ over $\Sigma$ and a formula $\varphi(x_1,\dots,x_n)$ with $n$ free node variables $x_1,\dots,x_n$, we write $f\models \varphi(u_1,\dots,u_n)$ to mean that $\varphi$ holds in $f$ for the nodes $u_1,\dots,u_n$ of $f$ (as values of the variables $x_1,\dots,x_n$ respectively).
We will occasionally use the following formulas: $\mathrm{root}(x)$ and $\mathrm{leaf}(x)$ test whether node $x$ is a root or a leaf, and $\mathrm{first}(x)$ and $\mathrm{last}(x)$ test whether $x$ is a first or a last sibling. Also, $\mathrm{child}_i(x)$ tests whether $x$ is an $i$-th child, ${\rm up}(x,y)$ expresses that $y$ is the parent of $x$, and ${\rm stay}(x,y)$ expresses that $y$ equals $x$. Thus, we define ${\rm stay}(x,y) \equiv x=y$ and \begin{enumerate} \item[] $\mathrm{root}(x) \equiv \neg\,\exists z({\rm down}(z,x))$,
$\mathrm{leaf}(x) \equiv \neg\,\exists z({\rm down}(x,z))$, \item[] $\mathrm{first}(x) \equiv \neg\,\exists z(\mathrm{next}(z,x))$,
$\mathrm{last}(x) \equiv \neg\,\exists z(\mathrm{next}(x,z))$, \item[] $\mathrm{child}_i(x) \equiv \exists z({\rm down}_i(z,x))$,
${\rm up}(x,y) \equiv {\rm down}(y,x)$. \end{enumerate}
\smallpar{Patterns} Let $\Sigma$ be a ranked alphabet and $n\ge 0$. An $n$-ary \emph{pattern} (or $n$-ary query) over $\Sigma$ is a set $T \subseteq \{(t,u_1,\dots,u_n)
\mid t\in T_\Sigma, \,u_1,\dots,u_n \in N(t)\}$. For $n=0$ this is a tree language, for $n=1$ it is a \emph{site} (trees with a distinguished node), for $n=2$ it is a \emph{trip} \cite{trips} (or a binary tree-node relation \cite{bloem}).
We introduce a new ranked alphabet $\Sigma \times \{0,1\}^n$, the rank of $(\sigma,\ell)$ equals that of $\sigma$ in $\Sigma$. For a tree $t$ over $\Sigma$ and $n$ nodes $u_1,\dots,u_n$ we define $\operatorname{mark}(t,u_1,\dots,u_n)$ to be the tree over $\Sigma \times \{0,1\}^n$ that is obtained by adding to the label of each node $u$ in $t$ a vector $\ell \in \{0,1\}^n$ such that the $i$-th component of $\ell$ equals $1$ if and only if $u=u_i$. The $n$-ary pattern $T$ is \emph{regular} if its marked representation is a regular tree language, i.e., $\operatorname{mark}(T)\in \family{REGT}$.
An \abb{mso} formula $\varphi(x_1,\dots,x_n)$ over $\Sigma$, with $n$ free node variables $x_1,\dots,x_n$, defines the $n$-ary pattern $T(\varphi) = \{(t,u_1,\dots,u_n)
\mid t \models \varphi(u_1,\dots,u_n)\}$. Note that $T(\varphi)$ also depends on the order $x_1,\dots,x_n$ of the free variables of $\varphi$. It easily follows from the result of Doner, Thatcher and Wright \cite{Don70,ThaWri} that a pattern is \abb{mso} definable if and only if it is regular (see~\cite[Lemma~7]{bloem}).
We will also consider patterns on forests. For an unranked alphabet $\Sigma$, a (forest) pattern over $\Sigma$ is a subset of $\{(f,u_1,\dots,u_n) \mid f\in F_\Sigma, \,u_1,\dots,u_n \in N(f)\}$. As for ranked trees, an \abb{mso} formula $\varphi(x_1,\dots,x_n)$ over $\Sigma$, defines the $n$-ary (forest) pattern $\{(f,u_1,\dots,u_n) \mid f \models \varphi(u_1,\dots,u_n)\}$.
\section{Automata and Transducers}\label{sec:autotrans}
In this section we define tree-walking automata and transducers with pebbles, and discuss some of their properties.
\smallpar{Automata} A \emph{tree-walking automaton with nested pebbles} (pebble tree automaton for short, abbreviated \abb{pta}) is a finite state device with one reading head that walks from node to node over its ranked input tree following the vertical edges in either direction. Additionally it has a supply of \emph{pebbles} that can be used to mark the nodes of the tree. The automaton may drop a pebble on the node currently visited by the reading head, but it may only lift any pebble from the current node if that pebble was the last one dropped during the computation. Thus, the life times of the pebbles on the tree are nested. Here we consider two types of pebbles. First there are a finite number of ``classical'' pebbles, which we here call \emph{visible} pebbles. Each of these has a distinct colour, and at most $k$ visible pebbles (each with a different colour) can be present on the input tree during any computation, where~$k$ is fixed. Second there are \emph{invisible} pebbles. Again, these pebbles have a finite number of colours (distinct from those of the visible pebbles), but for each colour there is an unlimited supply of pebbles that can be present on the input tree. Visible pebbles can be observed by the automaton at any moment when it visits the node where they were dropped. An invisible pebble can only be observed when it was the last pebble dropped on the tree during the computation.
The possible actions of the automaton are determined by its state, the label of the current node, the child number of the node, and the set of \emph{observable} pebbles on the current node, that is, visible pebbles and an invisible pebble when it was the last pebble dropped on the tree. Unlike the \abb{PTA} from \cite{MilSucVia03}, our automata do \emph{not} branch (i.e., are not alternating).
The \abb{pta} is specified as a tuple ${\cal A} = (\Sigma, Q, Q_0, F, C, C_\mathrm{v}, C_\mathrm{i}, R,k)$, where $\Sigma$ is a ranked alphabet of input symbols, $Q$ is a finite set of states, $Q_0 \subseteq Q$ is the set of initial states, $F \subseteq Q$ is the set of final states, $C_\mathrm{v}$ and $C_\mathrm{i}$ are the finite sets of visible and invisible colours, $C= C_\mathrm{v}\cup C_\mathrm{i}$, $C_\mathrm{v} \cap C_\mathrm{i} = \varnothing$, $R$ is a finite set of rules, and $k\in {\mathbb N}$. Each rule is of the form $\tup{q,\sigma,j,b} \to \tup{q',\alpha}$ such that $q,q'\in Q$, $\sigma \in \Sigma$, $j\in[0,{\mathit mx}_\Sigma]$, $b\subseteq C$ with $\#(b\cap C_\mathrm{v})\leq k$ and $\#(b\cap C_\mathrm{i})\leq 1$, and $\alpha$ is one of the following \emph{instructions}: \[ \begin{array}{ll} {\rm stay}, & \\ {\rm up} & \text{provided } j\neq 0, \\ {\rm down}_i & \text{with } 1 \le i \le \operatorname{rank}_\Sigma(\sigma), \\ {\rm drop}_c & \text{with } c\in C
, \text{ and} \\ {\rm lift}_c & \text{with } c\in b, \end{array} \] where the first three are \emph{move instructions} and the last two are \emph{pebble instructions}. Note that, due to the nested life times of the pebbles, at most one pebble $c$ in $b$ can actually be lifted; however, the subscript $c$ of ${\rm lift}_c$ often increases the readability of a \abb{pta}.
A \emph{situation} $\tup{u,\pi}$ of the \abb{pta} ${\cal A}$ on ranked tree $t$ over $\Sigma$ is given by the position $u$ of the head of ${\cal A}$ on $t$, and the stack $\pi$ containing the positions and colours of the pebbles on the tree in the order in which they were dropped. Formally, $u\in N(t)$ and $\pi\in (N(t)\times C)^*$. The last element of $\pi$ represents the top of the stack. The set of all situations of ${\cal A}$ on $t$ is denoted $\xp{Sit}(t)$, i.e., $\xp{Sit}(t)=N(t)\times (N(t)\times C)^*$; note that it only depends on $C$. A \emph{configuration} $\tup{q,u,\pi}$ of ${\cal A}$ on $t$ additionally contains the state $q$ of ${\cal A}$, $q\in Q$. It is \emph{final} when $q\in F$. An \emph{initial} configuration is of the form $\tup{q_0,\mathrm{root}_t,\varepsilon}$ where $q_0 \in Q_0$, $\mathrm{root}_t$ is the root of $t$, and $\varepsilon$ is the empty stack. The set of all configurations of ${\cal A}$ on~$t$ is denoted $\xp{Con}(t)$, i.e., $\xp{Con}(t)=Q\times N(t)\times (N(t)\times C)^*$.
We now define the computation steps of the \abb{pta} ${\cal A}$, which lead from one configuration to another. For a given input tree $t$ they form a binary relation on $\xp{Con}(t)$. A~rule $\tup{q,\sigma,j,b} \to \tup{q',\alpha}$ is \emph{relevant} to every configuration $\tup{q,u,\pi}$ with state $q$ and with a situation $\tup{u,\pi}$ that satisfies the tests $\sigma$, $j$, and $b$, i.e., $\sigma$ and $j$ are the label and child number of node $u$, and $b$ is the set of colours of the observable pebbles dropped on the node $u$. More precisely, $b$ consists of all $c\in C_\mathrm{v}$ such that $\pair{u,c}$ occurs in $\pi$, plus $c \in C_\mathrm{i}$ if $\pair{u,c}$ is the topmost (i.e., last) element of $\pi$. Application of the rule to such a configuration possibly leads to a new configuration $\tup{q',u',\pi'}$, in which case we write $\tup{q,u,\pi}\Rightarrow_{t,{\cal A}} \tup{q',u',\pi'}$. The new state is $q'$ and the new situation $\tup{u',\pi'}$ is obtained from the situation $\tup{u,\pi}$ by the instruction $\alpha$. For the move instructions $\alpha\!=\!{\rm stay}$, $\alpha\!=\!{\rm up}$, and $\alpha\!=\!{\rm down}_i$ the pebble stack does not change, i.e., $\pi'=\pi$, and the new node $u'$ equals $u$, is the parent of $u$, or is the $i$-th child of $u$, respectively.
For the pebble instructions the node does not change, i.e., $u'=u$. When $\alpha\!=\!{\rm drop}_c$, ${\cal A}$~drops a pebble with colour $c$ on the current node, thus the node-colour pair $\pair{u,c}$ is pushed onto the pebble stack~$\pi$, i.e., $\pi'=\pi(u,c)$, unless $c$ is a visible colour and the stack already contains a pebble of that colour or already contains $k$ visible pebbles, in which case the rule is not applicable.\footnote{To be precise, the rule is not applicable if $c\in C_\mathrm{v}$, $\pi=(u_1,c_1)\cdots(u_n,c_n)$, and there exists $i\in[1,n]$ such that $c=c_i$, or $\#(\{i\in[1,n]\mid c_i\in C_\mathrm{v}\})=k$. } When $\alpha\!=\!{\rm lift}_c$, ${\cal A}$~lifts a pebble with colour $c$ from the current node, only allowed if the topmost element of the pebble stack is the pair $\pair{u,c}$, which is subsequently popped from the stack, i.e., $\pi=\pi'(u,c)$; otherwise this rule is not applicable. We will also allow instructions like $\;{\rm lift}_c\,;{\rm up}\;$ with the obvious meaning (first lift the pebble, then move up). In this way we have defined the binary relation $\Rightarrow_{t,{\cal A}}$ on $\xp{Con}(t)$, which represents the computation steps of~${\cal M}$. We will say informally that a computation step of ${\cal M}$ \emph{halts successfully} if it leads to a final configuration.
The \emph{tree language} $L({\cal A})$ \emph{accepted by} \abb{pta} ${\cal A}$ consists of all ranked trees $t$ over~$\Sigma$ such that ${\cal A}$ has a successful computation on $t$ that starts in an initial configuration. Formally, $L({\cal A})=\{t\in T_\Sigma\mid \exists\, q_0\in Q_0, q_\infty\in F, \tup{u,\pi}\in \xp{Sit}(t): \tup{q_0,\mathrm{root}_t,\varepsilon}\Rightarrow^*_{t,{\cal A}} \tup{q_\infty,u,\pi}\}$. Note that pebbles may remain in the final configuration and that the head need not return to the root. Two \abb{PTA}'s ${\cal A}$ and ${\cal B}$ are \emph{equivalent} if $L({\cal A})=L({\cal B})$.
By \abb{v$_k$i-pta} we denote a \abb{pta} with last component $k$, i.e., that uses at most $k$ visible pebbles in its computations, and an unbounded number of invisible pebbles, and by \VIPTA{k} we denote the class of tree languages accepted by \abb{v$_k$i-pta}'s.
For $k=0$, automata that only use invisible pebbles, we also use the notation \abb{i-pta}, and for automata that only use $k$ visible pebbles we use \abb{v$_k$-pta}. Moreover, \abb{ta} is used for tree-walking automata without pebbles, i.e., \abb{v$_0$-pta}. The lower case d or \family{d} is added when we only consider \emph{deterministic} automata, which have a unique initial state, no final state in the left-hand side of a rule, and no two rules with the same left-hand side. Thus we have \abb{v$_k$i-}d\abb{pta}, \VIdPTA{k}, and variants.
\smallpar{Properties of automata} It is natural, and sometimes useful, to extend the \abb{v$_k$i-pta} with the facility to test whether its pebble stack is nonempty, and if so, to test the colour of the topmost pebble. Thus, we define a \abb{pta} \emph{with stack tests} in the same way as an ordinary \abb{pta} except that its rules are of the form $\tup{q,\sigma,j,b,\gamma} \to \tup{q',\alpha}$ with $\gamma\in C\cup\{\varepsilon\}$. Such a rule is relevant to a configuration $\tup{q,u,\pi}$ if, in addition, the pebble stack $\pi$ is empty if $\gamma=\varepsilon$, and the topmost pebble of $\pi$ has colour~$\gamma$ if $\gamma\in C$.\footnote{To be precise, for $\pi=(u_1,c_1)\cdots(u_n,c_n)$ the requirements are the following: If $\gamma=\varepsilon$ then $n=0$, i.e., $\pi=\varepsilon$. If $\gamma\in C$ then $n\geq 1$ and $c_n=\gamma$. } All other definitions are the same. Note that, obviously, we may require for the above rule that $\gamma = c$ if $\alpha={\rm lift}_c$, which ensures that relevant rules with a lift-instruction are always applicable.\footnote{Additionally, we can require the following: If $\gamma=\varepsilon$ then $b=\varnothing$. If $b\cap C_\mathrm{i}=\{c\}$ then $\gamma=c$. }
It is not difficult to see that these new tests do not extend the expressive power of the \abb{pta}. Informally we will say that the \abb{v$_k$i-pta} can \emph{perform stack tests}.
\begin{lemma}\label{lem:stacktests} Let $k\geq 0$. For every \abb{v$_k$i-pta} with stack tests ${\cal A}$ an equivalent (ordinary) \abb{v$_k$i-pta} ${\cal A}'$ can be constructed in polynomial time. The construction preserves determinism and the absence of invisible pebbles.\footnote{In other words, the statement of the lemma also holds for \abb{v$_k$i-}d\abb{pta}, \abb{v$_k$-pta} and \abb{v$_k$-}d\abb{pta}.} \end{lemma}
\begin{proof} Let ${\cal A} = (\Sigma, Q, Q_0, F, C, C_\mathrm{v}, C_\mathrm{i}, R,k)$. The new automaton ${\cal A}'$ stepwise simulates ${\cal A}$ and, additionally, stores in its finite state whether or not the pebble stack is nonempty, and if so, what is the colour in $C$ of the topmost pebble. Thus, $Q'=Q\times (C\cup\{\varepsilon\})$, $Q_0'= Q_0\times \{\varepsilon\}$, and $F'=F\times (C\cup\{\varepsilon\})$. Moreover, the colour sets of ${\cal A}'$ are $C_\mathrm{v}' = C_\mathrm{v} \times (C\cup\{\varepsilon\})$ and $C_\mathrm{i}' = C_\mathrm{i} \times (C\cup\{\varepsilon\})$. In fact, if the pebble stack of ${\cal A}$ is $\pi=(u_1,c_1)(u_2,c_2)\cdots(u_n,c_n)$, with $(u_n,c_n)$ being the topmost pebble, then the stack of ${\cal A}'$ is $\pi'=(u_1,(c_1,\varepsilon))(u_2,(c_2,c_1))\cdots(u_n,(c_n,c_{n-1}))$, where $\varepsilon$ is viewed as a bottom symbol. Thus, the new colour of a pebble contains its old colour together with the old colour of the previously dropped pebble (or $\varepsilon$ if there is none). This allows ${\cal A}'$ to update its additional finite state component when ${\cal A}$ lifts a pebble. More precisely, when ${\cal A}$ is in configuration $\tup{q,u,\pi}$, the automaton ${\cal A}'$ is in configuration $\tup{(q,\gamma),u,\pi'}$, where $\gamma=c_n$ if $n\geq 1$ and $\gamma=\varepsilon$ otherwise.
The rules of ${\cal A}'$ are defined as follows. Let $\tup{q,\sigma,j,b,\gamma} \to \tup{q',\alpha}$ be a rule of ${\cal A}$, and let $b'$ be (the graph of) a mapping from $b$ to $C\cup\{\varepsilon\}$. If $\alpha$ is a move instruction, then ${\cal A}'$ has the rule $\tup{(q,\gamma),\sigma,j,b'} \to \tup{(q',\gamma),\alpha}$. If $\alpha = {\rm drop}_c$, then ${\cal A}'$ has the rule $\tup{(q,\gamma),\sigma,j,b'} \to \tup{(q',c),{\rm drop}_{(c,\gamma)}}$. If $\alpha={\rm lift}_c$, $\gamma=c$, and $(c,\gamma')\in b'$, then ${\cal A}'$ has the rule $\tup{(q,\gamma),\sigma,j,b'} \to \tup{(q',\gamma'),{\rm lift}_{(c,\gamma')}}$.
It should be clear that the construction of ${\cal A}'$ takes polynomial time. Note that $k$ is fixed and $\#(b)\leq k+1$ in the left-hand side of the rule $\tup{q,\sigma,j,b,\gamma} \to \tup{q',\alpha}$ of ${\cal A}$. \end{proof}
\abb{PTA}'s with stack tests will only be used in Sections~\ref{sec:look-ahead} and~\ref{sec:variations}. The next two properties of \abb{PTA}'s will not be used in later sections, but are meant to clarify some of the details in the semantics of the \abb{pta}.
A rule of a \abb{v$_k$i-pta} ${\cal A}$ is \emph{progressive} if it is applicable to every reachable configuration\footnote{The configuration $\tup{q,u,\pi}$ on the tree $t$ is \emph{reachable} if $\tup{q_0,\mathrm{root}_t,\varepsilon}\Rightarrow^*_{t,{\cal A}} \tup{q,u,\pi}$ for some $q_0\in Q_0$. \label{foot:reachcon} } to which it is relevant. The \abb{v$_k$i-pta} ${\cal A}$ is progressive if all its rules are progressive. Intuitively this means that ${\cal A}$ knows that its instructions can always be executed. Clearly, according to the syntax of a \abb{pta}, every rule with a move instruction is progressive. The same is true for rules with a pebble instruction ${\rm drop}_c$ or ${\rm lift}_c$ with $c\in C_\mathrm{i}$: an invisible pebble can always be dropped and an observable invisible pebble can always be lifted. Thus, only the dropping and lifting of visible pebbles is problematic. It is easy to see that, for the \mbox{\abb{v$_k$i-pta}}~${\cal A}'$ constructed in the proof of Lemma~\ref{lem:stacktests}, every rule with a lift-instruction is progressive.
A \abb{v$_k$i-pta} ${\cal A}$ is \emph{counting} if $C_v=[1,k]$ and, in each reachable configuration, the colours of the visible pebbles on the tree are $1,\dots,\ell$ for some $\ell\in [0,k]$, in the order in which they were dropped.\footnote{To be precise, for $\pi=(u_1,c_1)\cdots(u_n,c_n)$ we require that there exists $\ell\in [0,k]$ such that $(c_{i_1},\dots,c_{i_m})=(1,\dots,\ell)$ where $\{i_1,\dots,i_m\}=\{i\in[1,n]\mid c_i\in C_\mathrm{v}\}$ and $i_1<\cdots < i_m$. } Note that in the litterature \mbox{\abb{v$_k$-pta}'s} are usually counting. We have chosen to allow arbitrarily many visible colours in a \abb{v$_k$i-pta} because we want to be able to store information in the pebbles, as in the proof of Lemma~\ref{lem:stacktests}.
It is straightforward to construct an equivalent counting \abb{v$_k$i-pta} ${\cal A}'$ for a given \abb{v$_k$i-pta} ${\cal A}$ (preserving determinism and the absence of invisible pebbles). The automaton ${\cal A}'$ stepwise simulates ${\cal A}$ and, additionally, stores in its finite state the colours of the visible pebbles that are dropped on the tree, in the order in which they were dropped. Thus, the states of ${\cal A}'$ are of the form $(q,\varphi)$ where $q$ is a state of ${\cal A}$ and $\varphi$ is a string over $C_\mathrm{v}$ without repetitions, of length at most $k$. The state $(q,\varphi)$ is final if $q$ is final. The initial states are $(q,\varepsilon)$ where $q$ is an initial state of ${\cal A}$.
The rules of ${\cal A}'$ are defined as follows. Let $\tup{q,\sigma,j,b} \to \tup{q',\alpha}$ be a rule of ${\cal A}$ and let $(q,\varphi)$ be a state of ${\cal A}'$ such that every $c\in b\cap C_\mathrm{v}$ occurs in $\varphi$. Moreover, let $b'\subseteq[1,k]\cup C_\mathrm{i}$ be obtained from $b$ by changing every $c\in C_\mathrm{v}$ into $i$, if $c$ is the $i$-th element of $\varphi$. If $\alpha$ is a move instruction, or a pebble instruction ${\rm drop}_c$ or ${\rm lift}_c$ with $c\in C_\mathrm{i}$ then ${\cal A}'$ has the rule $\tup{(q,\varphi),\sigma,j,b'} \to \tup{(q',\varphi),\alpha}$. If $\alpha = {\rm drop}_c$ with $c\in C_\mathrm{v}$,
$c$ does not occur in $\varphi$, and $|\varphi|<k$, then ${\cal A}'$ has the rule
$\tup{(q,\varphi),\sigma,j,b'} \to \tup{(q',\varphi c),{\rm drop}_{|\varphi|+1}}$. Finally, if $\alpha={\rm lift}_c$ with $c\in C_\mathrm{v}$, and $\varphi=\varphi'c$ for some $\varphi'\in C^*_\mathrm{v}$, then ${\cal A}'$ has the rule
$\tup{(q,\varphi),\sigma,j,b'} \to \tup{(q',\varphi'),{\rm lift}_{|\varphi|}}$. It should be clear that ${\cal A}'$ is counting. Note also that all rules of ${\cal A}'$ with a drop-instruction are progressive. Thus, if we first apply the construction in the proof of Lemma~\ref{lem:stacktests} and then the one above, we obtain an equivalent progressive \abb{v$_k$i-pta}. Obviously, every progressive \abb{v$_k$i-pta} can be turned into an equivalent \abb{v$_{k+1}$i-pta} by simply changing its last component $k$ into $k+1$, and hence $\VIPTA{k} \subseteq \VIPTA{k+1}$ and $\VIdPTA{k} \subseteq \VIdPTA{k+1}$.\footnote{In fact, these four classes are equal, as will be shown in Theorem~\ref{thm:regt}. }
\smallpar{Transducers} A \emph{tree-walking tree transducer with nested pebbles} (abbreviated \abb{ptt}) is a \abb{pta} without final states that additionally produces an output tree over a ranked alphabet~$\Delta$. Thus, omitting $F$, it is specified as a tuple ${\cal M} = (\Sigma, \Delta, Q, Q_0,C, C_\mathrm{v}, C_\mathrm{i}, R,k)$, where $\Sigma$, $Q$, $Q_0$, $C$, $C_\mathrm{v}$, $C_\mathrm{i}$, and~$k$ are as for the \abb{pta}. The rules of ${\cal M}$ in the finite set $R$ are of the same form as for the \abb{pta}, except that ${\cal M}$ additionally has \emph{output rules} of the form $\tup{q,\sigma,j,b} \to \delta(\,\tup{q_1,\mathrm{stay}}, \dots, \tup{q_m,\mathrm{stay}}\,)$ with $\delta\in \Delta$, and $q_1,\dots , q_m \in Q$, where~$m$ is the rank of $\delta$. Intuitively, the output tree is produced recursively. In other words, in a configuration to which the above output rule is relevant (defined as for the \abb{pta}) the \abb{ptt} ${\cal M}$ outputs $\delta$, and for each child $\tup{q_i,\mathrm{stay}}$ branches into a new process, a copy of itself started in state $q_i$ at the current node, retaining the same stack of pebbles; thus, the stack is copied $m$ times. Note that a relevant output rule is always applicable. As a shortcut we may replace the stay-instruction in any $\tup{q_i,\mathrm{stay}}$ by another move instruction or a pebble instruction, with obvious semantics.
An \emph{output form} of the \abb{ptt} ${\cal M}$ on ranked tree $t$ over $\Sigma$ is a tree in $T_\Delta(\xp{Con}(t))$, where $\xp{Con}(t)$ is defined as for the \abb{pta}. Intuitively, such an output form consists on the one hand of $\Delta$-labeled nodes that were produced by ${\cal M}$ previously in the computation, using output rules, and on the other hand of leaves that represent the independent copies of ${\cal M}$ into which the computation has branched previously, due to those output rules, where each leaf is labeled by the current configuration of that copy. Note that $\xp{Con}(t)\subseteq T_\Delta(\xp{Con}(t))$, i.e., every configuration of ${\cal M}$ is an output form.
The computation steps of the \abb{ptt} ${\cal M}$ lead from one output form to another. Let $s$ be an output form and let $v$ be a leaf of $s$ with label $\tup{q,u,\pi}\in \xp{Con}(t)$. If $\tup{q,u,\pi} \Rightarrow_{t,{\cal M}} \tup{q',u',\pi'}$, where the binary relation $\Rightarrow_{t,{\cal M}}$ on $\xp{Con}(t)$ is defined as for the \abb{pta} (disregarding the output rules of ${\cal M}$), then we write $s \Rightarrow_{t,{\cal M}} s'$ where $s'$ is obtained from $s$ by changing the label of $v$ into $\tup{q',u',\pi'}$. Moreover, for every output rule $\tup{q,\sigma,j,b} \to \delta(\,\tup{q_1,\mathrm{stay}}, \dots, \tup{q_m,\mathrm{stay}}\,)$ that is relevant to configuration $\tup{q,u,\pi}$, we write $s \Rightarrow_{t,{\cal M}} s'$ where $s'$ is obtained from $s$ by replacing the node $v$ by the subtree $\delta(\tup{q_1,u,\pi},\dots,\tup{q_m,u,\pi})$. In the particular case that $m=0$, $s'$ is obtained from $s$ by changing the label of $v$ into $\delta$. In that case we will say informally that ${\cal M}$ \emph{halts successfully}, meaning that the copy of ${\cal M}$ corresponding to the node $u$ of $s$ disappears. In this way we have extended $\Rightarrow_{t,{\cal M}}$ to a binary relation on $T_\Delta(\xp{Con}(t))$.
The \emph{transduction} $\tau_{\cal M}$ \emph{realized by} ${\cal M}$ consists of all pairs of trees $t$ over $\Sigma$ and $s$ over $\Delta$ such that ${\cal M}$ has a (successful) computation on $t$ that starts in an initial configuration and ends with $s$. Formally, we define $\tau_{\cal M} = \{(t,s)\in T_\Sigma\times T_\Delta\mid \exists\, q_0\in Q_0: \tup{q_0,\mathrm{root}_t,\varepsilon}\Rightarrow^*_{t,{\cal M}} s\}$. Two \abb{ptt}'s ${\cal M}$ and ${\cal N}$ are \emph{equivalent} if $\tau_{\cal M}=\tau_{\cal N}$.
The \emph{domain of} ${\cal M}$ is defined to be the domain of $\tau_{\cal M}$, i.e., the tree language $L({\cal M})=\{t\in T_\Sigma\mid \exists\, s\in T_\Delta: (t,s)\in\tau_{\cal M}\}$. When ${\cal M}$ is viewed as a recognizer of its domain, it is actually the same as an alternating \abb{pta}. Existential states in the alternation correspond to the nondeterminism of the \abb{ptt}, universal states correspond to the recursive way in which output trees are generated. More precisely, an output rule $\tup{q,\sigma,j,b}\to\delta(\,\tup{q_1,\mathrm{stay}}, \dots, \tup{q_m,\mathrm{stay}}\,)$ corresponds to a universal state $q$ that requires every state $q_i$ to have a successful computation (and the output symbol $\delta$ is irrelevant). An ordinary (non-alternating) \abb{pta} then corresponds to a \abb{ptt} for which every output symbol has rank 0; for $m=0$ the above output rule means that the \abb{pta} halts in a final state. We say that the \abb{ptt} ${\cal M}$ is \emph{total} if $L({\cal M})=T_\Sigma$, i.e., $\tau_{\cal M}(t)\neq\emptyset$ for every input tree $t$.
Similar to the notation \VIPTA{k} for tree languages, we use the notation \VIPTT{k} for the class of transductions defined by tree-walking tree transducers
with $k$ visible nested pebbles and an unbounded number of invisible pebbles, as well as the obvious variants \VPTT{k}, and \family{I-PTT}. Additionally \family{TT}\ denotes the class of transductions realized by tree-walking tree transducers without pebbles, i.e., \VPTT{0}. Such a transducer is specified as a tuple ${\cal M} = (\Sigma, \Delta, Q, Q_0, R)$, and the left-hand sides of its rules are written $\tup{q,\sigma,j}$, omitting $b=\varnothing$. As for \abb{pta}'s, lower case \family{d} is added for \emph{deterministic} transducers, which have a unique initial state and no two rules with the same left-hand side. Moreover, lower case \family{td} is used for \emph{total deterministic} transducers, i.e., transducers that are both total and deterministic. Note that a deterministic \abb{ptt} realizes a function, and a total deterministic \abb{ptt} a total function from $T_\Sigma$ to $T_\Delta$.
\smallpar{Properties of transducers} Stack tests are defined for the \abb{ptt} as for the \abb{pta}, and Lemma~\ref{lem:stacktests} and its proof carry over to \abb{ptt}'s. If a given \abb{ptt} ${\cal M}$ has the output rule $\tup{q,\sigma,j,b,\gamma} \to \delta(\tup{q_1,{\rm stay}},\dots,\tup{q_m,{\rm stay}})$, and $b'$ is (the graph of) a mapping from~$b$ to $C\cup\{\varepsilon\}$ as in the proof for \abb{pta}'s, then the constructed \abb{ptt}~${\cal M}'$ has the rule $\tup{(q,\gamma),\sigma,j,b'} \to \delta(\tup{(q_1,\gamma),{\rm stay}},\dots,\tup{(q_m,\gamma),{\rm stay}})$.
Progressive \abb{ptt}'s can be defined as for \abb{pta}'s, based on the notion of a reachable configuration, cf. footnote~\ref{foot:reachcon}. An output form $s$ of the \abb{ptt} ${\cal M}$ on the input tree $t$ is \emph{reachable} if $\tup{q_0,\mathrm{root}_t,\varepsilon}\Rightarrow^*_{t,{\cal M}} s$ for some $q_0\in Q_0$. A~configuration of ${\cal M}$ on $t$ is \emph{reachable} if it occurs in some reachable output form of ${\cal M}$ on $t$. Note that every \abb{i-ptt} is progressive.
Also, counting \abb{ptt}'s can be defined as for \abb{pta}'s. For every \abb{v$_k$i-ptt} ${\cal M}$ an equivalent counting \abb{v$_k$i-ptt} ${\cal M}'$ can be constructed, just as for \abb{pta}'s. If $\tup{q,\sigma,j,b,\gamma} \to \delta(\tup{q_1,{\rm stay}},\dots,\tup{q_m,{\rm stay}})$ is an output rule of ${\cal M}$, and $\varphi$ and $b'$ are as in the proof for \abb{pta}'s, then ${\cal M}'$ has the rule $\tup{(q,\varphi),\sigma,j,b'} \to \delta(\tup{(q_1,\varphi),{\rm stay}},\dots,\tup{(q_m,\varphi),{\rm stay}})$. Thus, as for \abb{pta}'s, every \abb{v$_k$i-ptt} can be turned into an equivalent progressive \abb{v$_k$i-ptt}, with determinism and the absence of invisible pebbles preserved. That implies that $\VIPTT{k} \subseteq \VIPTT{k+1}$ and $\VIdPTT{k} \subseteq \VIdPTT{k+1}$.
We end this section with an example of an \abb{i-ptt}.
\begin{example}\label{ex:siberie}\leavevmode We want to generate itineraries for a trip along the Trans-Siberian Railway, starting in Moscow and ending in Vladivostok, and optionally visiting some cities along the way. An XML document lists all the stops: \begin{small} \begin{verbatim} <stop name="Moscow" large="1" initial="1">
...
<stop name="Birobidzhan" large="0">
...
<stop name="Vladivostok" large="1" final="1" />
...
</stop>
... </stop> \end{verbatim} \end{small} The initial and final stops are marked, and for every stop the \texttt{large} attribute indicates whether or not the stop is in a large city. We want to generate a list \begin{small} \begin{verbatim} <result>it-1
<result>it-2
...
<result>it-n
<endofresults />
</result>
...
</result> </result> \end{verbatim} \end{small} where {\small\texttt{it-1,it-2,...,it-n}} are all itineraries (i.e., lists of stops) that satisfy the constraint that one does not visit a small city twice in a row. An example input XML document, with the corresponding output XML document is given in Tables~\ref{tab:input} and~\ref{tab:output} (where, e.g., {\small\verb+</stop>^3+} abbreviates {\small\verb+</stop></stop></stop>+}). A deterministic \abb{i-ptt} ${\cal M}_{\text{sib}}$ is able to perform this XML transformation by systematically enumerating all possible lists of stops, marking each stop in the list (except the initial and final stop) by a pebble. Since the pebbles are invisible, ${\cal M}_{\text{sib}}$ constructs a possible list of stops on the pebble stack \emph{in reverse}, so that the stops will appear in the output tree in the correct order.
\begin{table*}[ht] \begin{small} \begin{verbatim}
<?xml version="1.0" encoding="UTF-8"?>
<?xml-stylesheet type="text/xsl" href="transsiberie.xsl"?>
<stop name="Moscow" large="1" initial="1">
<stop name="Stop 2" large="0">
<stop name="Stop 3" large="0">
<stop name="LargeStop 4" large="1">
<stop name="Stop 5" large="0">
<stop name="Vladivostok" large="1" final="1"/>
</stop>^5 \end{verbatim} \end{small} \caption{Input}\label{tab:input} \end{table*}
\begin{table*}[p] \begin{scriptsize} \begin{verbatim} <result>
<stop name="Moscow" large="1" initial="1">
<stop name="Stop 3" large="0">
<stop name="LargeStop 4" large="1">
<stop name="Stop 5" large="0">
<stop name="Vladivostok" large="1" final="1"/>
</stop>^4
<result>
<stop name="Moscow" large="1" initial="1">
<stop name="Stop 2" large="0">
<stop name="LargeStop 4" large="1">
<stop name="Stop 5" large="0">
<stop name="Vladivostok" large="1" final="1"/>
</stop>^4
<result>
<stop name="Moscow" large="1" initial="1">
<stop name="LargeStop 4" large="1">
<stop name="Stop 5" large="0">
<stop name="Vladivostok" large="1" final="1"/>
</stop>^3
<result>
<stop name="Moscow" large="1" initial="1">
<stop name="Stop 5" large="0">
<stop name="Vladivostok" large="1" final="1"/>
</stop>^2
<result>
<stop name="Moscow" large="1" initial="1">
<stop name="Stop 3" large="0">
<stop name="LargeStop 4" large="1">
<stop name="Vladivostok" large="1" final="1"/>
</stop>^3
<result>
<stop name="Moscow" large="1" initial="1">
<stop name="Stop 2" large="0">
<stop name="LargeStop 4" large="1">
<stop name="Vladivostok" large="1" final="1"/>
</stop>^3
<result>
<stop name="Moscow" large="1" initial="1">
<stop name="LargeStop 4" large="1">
<stop name="Vladivostok" large="1" final="1"/>
</stop>^2
<result>
<stop name="Moscow" large="1" initial="1">
<stop name="Stop 3" large="0">
<stop name="Vladivostok" large="1" final="1"/>
</stop>^2
<result>
<stop name="Moscow" large="1" initial="1">
<stop name="Stop 2" large="0">
<stop name="Vladivostok" large="1" final="1"/>
</stop>^2
<result>
<stop name="Moscow" large="1" initial="1">
<stop name="Vladivostok" large="1" final="1"/>
</stop>
<endofresults/>
</result>
</result> </result>^8 \end{verbatim} \end{scriptsize} \caption{Output}\label{tab:output} \end{table*}
Since in this example the XML tags are ranked, there is no need for a binary encoding of the XML documents.
The input alphabet $\Sigma$ of ${\cal M}_{\text{sib}}$ consists of all {\small\texttt{<stop~at>}} where {\small\texttt{at}} is a possible value of the attributes. The rank of \mbox{{\small\texttt{<stop at>}}} is 0 if {\small\texttt{final="1"}} and 1 otherwise. The output alphabet $\Delta$ consists of $\Sigma$, the tag $r =\;${\small\texttt{<result>}} of rank~$2$, and the tag $e =\;${\small\texttt{<endofresults>}} of rank~0. The set of pebble colours is $C=C_{\mathrm i}=\{0, 1\}$, with $C_\mathrm{v}=\varnothing$. The transducer ${\cal M}_{\text{sib}}$ will not use the attribute {\small\texttt{initial}}, as it can recognize the root by its child number~0. Also, it will disregard the attribute {\small\texttt{large}} of the initial and the final stop, and always consider them as large cities. The set of states of ${\cal M}_{\text{sib}}$ is $Q=\{q_\mathrm{start},q_1,q_0,q_\mathrm{out},q_\mathrm{next}\}$ with $Q_0=\{q_\mathrm{start}\}$.
In the rules below the variables range over the following values: $\sigma_0 \in \Sigma^{(0)}$, $\sigma_1 \in \Sigma^{(1)}$, $j,c\in \{0,1\}$, and, for $i\in \{0,1\}$, $\lambda_i\in \{\texttt{\small{<stop at>}}\in\Sigma \mid \texttt{\small{large}="}i\texttt{"}\}$. The \abb{i-ptt} ${\cal M}_{\text{sib}}$ first walks from Moscow to Vladivostok in state $q_\mathrm{start}$:
$\tup{q_\mathrm{start},\sigma_1,j,\varnothing} \to \tup{q_\mathrm{start},\mathrm{down}_1}$
$\tup{q_\mathrm{start},\sigma_0,1,\varnothing} \to \tup{q_1,\mathrm{up}}$
\noindent State $q_c$ remembers whether the most recently marked city is small or large; when a new city is marked with a pebble, it gets the colour $c$. In states $q_0$ and $q_1$ as many cities are marked as possible (in the second rule, $c=1$ or $i=1$):
$\tup{q_0,\lambda_0,1,\varnothing} \to \tup{q_0,\mathrm{up}}$
$\tup{q_c,\lambda_i,1,\varnothing} \to \tup{q_i,\mathrm{drop}_c;\mathrm{up}}$
$\tup{q_c,\sigma_1,0,\varnothing} \to r(\tup{q_\mathrm{out},\mathrm{stay}},
\tup{q_\mathrm{next},\mathrm{down}_1})$
\noindent In state $q_\mathrm{out}$ an itinerary is generated as output, while state $q_\mathrm{next}$ continues the search for itineraries by unmarking the most recently marked city:
$\tup{q_\mathrm{out},\sigma_1,0,\varnothing} \to \sigma_1(\tup{q_\mathrm{out},\mathrm{down}_1})$
$\tup{q_\mathrm{out},\sigma_1,1,\varnothing} \to \tup{q_\mathrm{out},\mathrm{down}_1}$
$\tup{q_\mathrm{out},\sigma_1,1,\{c\}} \to \sigma_1(\tup{q_\mathrm{out},\mathrm{lift_c;down}_1})$
$\tup{q_\mathrm{out},\sigma_0,1,\varnothing} \to \sigma_0$
$\tup{q_\mathrm{next},\sigma_1,1,\varnothing} \to \tup{q_\mathrm{next},\mathrm{down}_1}$
$\tup{q_\mathrm{next},\sigma_1,1,\{c\}} \to \tup{q_c,\mathrm{lift_c;up}}$
$\tup{q_\mathrm{next},\sigma_0,1,\varnothing} \to e$
\noindent Note that this XML transformation cannot be realized by a \abb{v-ptt}, because the height of the output tree is, in general, exponential in the size of the input tree, whereas it is polynomial for \abb{v-ptt}'s (cf.~\cite[Lemma~7]{EngMan03}). \end{example}
\section{Decomposition}\label{sec:decomp}
In this section we decompose every \abb{ptt} into a sequence of \abb{tt}'s, i.e., transducers without pebbles. This is useful as it will give us information on the domain of a \abb{ptt}, see Theorem~\ref{thm:regt}, and on the complexity of typechecking the \abb{ptt}, see Theorem~\ref{thm:typecheck}.
It is possible to reduce the number of visible pebbles used, by preprocessing the input tree with a total deterministic \abb{tt}. This was shown in \cite[Lemma~9]{EngMan03} for transducers with only visible pebbles. The basic idea of that proof can be extended to include invisible pebbles.
\begin{lemma}\label{lem:decomp} Let $k\ge 1$. For every \abb{v$_k$i-ptt} ${\cal M}$ a total deterministic \abb{tt} ${\cal N}$ and a \mbox{\abb{v$_{k-1}$i-ptt}} ${\cal M}'$ can be constructed in polynomial time such that $\tau_{{\cal N}}\circ\tau_{{\cal M}'}= \tau_{{\cal M}}$. If ${\cal M}$ is deterministic, then so is ${\cal M}'$. Hence, for every $k\ge 1$, $$\VIPTT{k} \subseteq \family{tdTT} \circ \VIPTT{k-1} \text{ and } \VIdPTT{k} \subseteq \family{tdTT} \circ \VIdPTT{k-1}.$$ \end{lemma}
\begin{proof} Let ${\cal M} = (\Sigma, \Delta, Q, Q_0,C, C_\mathrm{v}, C_\mathrm{i}, R,k)$ be a \abb{ptt} with $k$ visible pebbles. The construction of the \abb{tt} ${\cal N}$ and the \abb{ptt} ${\cal M}'$ with $k-1$ visible pebbles is a straightforward extension of the one in \cite[Theorem 5]{Eng09}, which slightly differs from the one in the proof of \cite[Lemma~9]{EngMan03}, but uses the same basic idea. For completeness sake we repeat a large part of the proof of \cite[Theorem 5]{Eng09}, adapted to the current formalism. The simple idea of the proof is to preprocess the input tree $t\in T_\Sigma$ in such a way that the dropping and lifting of the first visible pebble can be simulated by walking into and out of specific areas of the preprocessed input tree ${\rm pp}(t)$. This preprocessing is independent of the given pebble tree transducer ${\cal M}$. More precisely, ${\rm pp}(t)$ is obtained from $t$ by attaching to each node $u$ of $t$, as an additional (last) subtree, a fresh copy of $t$ in which (the copy of) node $u$ is marked; let us denote this subtree by $t_u$. Thus, if $t$ has $n$ nodes, then ${\rm pp}(t)$ has $n+n^2$ nodes. The subtrees $t_u$ of ${\rm pp}(t)$ are the ``specific areas'' mentioned above. As long as there are no visible pebbles on~$t$, ${\cal M}'$ stepwise simulates ${\cal M}$ on the original nodes of $t$, which form the ``top level'' of ${\rm pp}(t)$. When ${\cal M}$ drops the first visible pebble $c$ on node $u$, ${\cal M}'$ enters $t_u$ and walks to the marked node, storing $c$ in its finite state. As long as ${\cal M}$ keeps pebble $c$ on the tree, ${\cal M}'$ stays in $t_u$, stepwise simulating ${\cal M}$ on $t_u$ rather than~$t$. Since $u$ is marked in $t_u$, ${\cal M}$'s pebble $c$ at $u$ is visible to the transducer ${\cal M}'$, not as a pebble but as a marked node. Thus, during this time, ${\cal M}'$ only uses $k-1$ visible pebbles. When ${\cal M}$ lifts pebble~$c$ from $u$ (and hence all visible pebbles are lifted), ${\cal M}'$ walks from the copy of $u$ out of $t_u$, back to the original node $u$, and continues simulating ${\cal M}$ on the top level of ${\rm pp}(t)$ until ${\cal M}$ again drops a visible pebble. There is one problem: how does ${\cal M}'$ know whether or not pebble~$c$ is on top of the stack when ${\cal M}$ tries to lift it? To solve this problem, ${\cal M}'$ uses an additional special invisible pebble $\odot$. It drops pebble~$\odot$ at the copy of $u$ and thus knows that pebble~$c$ is at the top of the stack (for ${\cal M}$) when it observes pebble~$\odot$. Thus, at any moment of time, ${\cal M}'$ has the same pebble stack as~${\cal M}$, except that $c$ is replaced by $\odot$ and, moreover, the (invisible) pebbles below $\odot$ are on the top level of ${\rm pp}(t)$, whereas $\odot$ and the pebbles above it are on $t_u$.
Unfortunately, this preprocessing cannot be realized by a \abb{tt} (though it can easily be realized by a \abb{v$_1$-ptt}). For this reason we ``fold'' $t_u$ at the node $u$, such that (the marked copy of) $u$ becomes its root; let us denote the resulting tree by $\hat{t}_u$. Roughly, $\hat{t}_u$ is obtained from $t_u$ by inverting the parent-child relationship between the ancestors of $u$ (including $u$), similarly as in the tree traversal algorithm sometimes known as ``link inversion'' \cite[p.562]{Knuth}. Appropriate information is added to the node labels of those ancestors to reflect this inversion. As these changes are local (i.e., each node keeps the same neighbours) and clearly marked in the tree, ${\cal M}'$ can easily reconstruct the unfolded $t_u$, and simulate ${\cal M}$ as before. Note also that, with this change of ${\rm pp}(t)$, dropping or lifting of the first visible pebble can be simulated by ${\cal M}'$ in one computation step, because the marked copy of $u$ is the last child of the original $u$.
Now a \abb{tt} ${\cal N}$ can compute ${\rm pp}(t)$, as follows\footnote{See also \cite[Example~3.7]{MilSucVia03} where $\hat{t}_u$ occurs as ``a complex rotation of the input tree'' $t$, albeit for leaves $u$ only.}. It copies $t$ to the output (adding primes to its labels), but when it arrives at node $u$ it additionally outputs the copy $\hat{t}_u$ of $t$ in a side branch of the computation. Copying the descendants of $u$ ``down stream'' is an easy recursive task. To invert the parent-child relationship between the nodes on the path from $u$ to $\mathrm{root}_t$, ${\cal N}$ uses a single process that walks along the nodes of that path ``up stream'' to the root, inverting the relationships in the copy. Copies of other siblings of children on the path are connected as in $t$, and their descendants are copied ``down stream''. More precisely, if in $t$ the $i$-th child $v$ of parent $w$ is on the path, then, in the output $\hat{t}_u$, $v$ has an additional (last) child that corresponds to $w$, and $w$ has the same children (with their descendants) as in $t$, except that its $i$-th child is a node that is labeled by the bottom symbol $\bot$ of rank~0. For the sake of uniformity, $\mathrm{root}_t$ is also given an additional (last) child, with label $\bot$. Note that the nodes of $t$ correspond one-to-one to the non-bottom nodes of $\hat{t}_u$; in particular, the path in $t$ from $u$ to $\mathrm{root}_t$ corresponds to the path in $\hat{t}_u$ from its root to the parent of its rightmost leaf. The bottom nodes of $\hat{t}_u$ will not be visited by ${\cal M}'$.
A picture of ${\rm pp}(t)$ is given in Fig.~\ref{fig:encpeb}, where $\hat{t}_u$ is drawn for two nodes only. Note that in this picture the root of the copy of $t$ (which is also the root of ${\rm pp}(t)$) is the top of the triangle, but the root of $\hat{t}_u$ is $u$ (and, of course, similarly for $v$).
\begin{figure}
\caption{Output tree ${\rm pp}(t)$ of the \abb{tt} ${\cal N}$ of Lemma~\ref{lem:decomp} for input tree $t$.}
\label{fig:encpeb}
\end{figure}
As a concrete example, consider $t= \sigma(\delta(a,b),c)$ where $\sigma,\delta$ have rank~2 and $a,b,c$ rank~0. We will name the nodes of $t$ by their labels. Then $${\rm pp}(t) = \sigma'(\delta'(a'(\hat{t}_a),b'(\hat{t}_b),\hat{t}_\delta),c'(\hat{t}_c),\hat{t}_\sigma)$$ where $$ \begin{array}{lll} \hat{t}_a &=& a_{0,1}(\delta_{1,1}(\bot,b,\sigma_{1,0}(\bot,c,\bot))),\\ \hat{t}_b &=& b_{0,2}(\delta_{2,1}(a,\bot,\sigma_{1,0}(\bot,c,\bot))),\\ \hat{t}_\delta &=& \delta_{0,1}(a,b,\sigma_{1,0}(\bot,c,\bot)),\\ \hat{t}_c &=& c_{0,2}(\sigma_{2,0}(\delta(a,b),\bot,\bot)), \mbox{ and}\\ \hat{t}_\sigma &=& \sigma_{0,0}(\delta(a,b),c,\bot). \end{array} $$ The subscripted node labels are on the rightmost paths of the $\hat{t}_u$'s; the subscripts contain ``reconstruction'' information, to be explained below. As another example, if $t$ is the monadic tree $a(b^m(c(d^n(e))))$ of height $m+n+3$, and $u$ is the \mbox{$c$-labelled} node, then $\hat{t}_u = c_{0,1}(s_1,s_2)$ with $s_1=d^n(e)$ and $s_2$ is the binary tree $b_{1,1}(\bot,b_{1,1}(\bot,\dots b_{1,1}(\bot,a_{1,0}(\bot,\bot))\cdots ))$ of height $m+2$. This shows more clearly that $\hat{t}_u$ is obtained by ``folding''.
We now formally define the deterministic \abb{tt} ${\cal N}$ that, for given ranked alphabet $\Sigma$, realizes the preprocessing ${\rm pp}$ (called EncPeb in \cite{EngMan03}). The definition is identical to the one in \cite[Section 6]{Eng09}. Since ${\cal N}$ has no pebbles, we abbreviate the left-hand side $\tup{q,\sigma,j,\varnothing}$ of a rule by $\langle q,\sigma,j\rangle$. To simplify the definition of~${\cal N}$ we additionally allow output rules of the form $\tup{q,\sigma,j}\to \delta(s_1,\dots,s_m)$ where $\delta$ is an output symbol of rank~$m$ and every $s_i$ is either the output symbol $\bot$ or it is of the form $\tup{q',\varphi}$ where $\varphi$ is ${\rm stay}$, ${\rm up}$, or ${\rm down}_i$ with $i\in[1,m]$. Such a rule should be replaced by the rules $\langle q,\sigma,j\rangle\to \delta(\langle p_1,{\rm stay} \rangle,\dots,\langle p_m,{\rm stay} \rangle)$ and $\langle p_j,\sigma,j\rangle\to s_j$ for all $j\in[1,m]$, where $p_1,\dots,p_m$ are new states. Obviously this replacement can be done in quadratic time.
We introduce the states and rules of ${\cal N}$ one by one; in what follows $\sigma$ ranges over $\Sigma$, with $m=\operatorname{rank}_\Sigma(\sigma)$, $j$ ranges over $[0,{\mathit mx}_\Sigma]$, and $i$ over $[1,m]$. First, ${\cal N}$~has an ``identity'' state $d$ that just recursively copies the subtree of the current node to the output, using the rules $\langle d,\sigma,j\rangle\to \sigma(\langle d,{\rm down}_1\rangle,\dots,\langle d,{\rm down}_m\rangle)$. Then, ${\cal N}$ has initial state $g$ that copies the input tree $t$ to the output (with primed labels) and at each node $u$ of $t$ ``generates'' a new copy $\hat{t}_u$ of the input tree by calling the state $f$ that computes $\hat{t}_u$ by ``folding'' $t_u$. The rules for $g$ are $$\langle g,\sigma,j\rangle\to \sigma'(\langle g,{\rm down}_1\rangle,\dots,\langle g,{\rm down}_m\rangle, \langle f,{\rm stay}\rangle).$$ Note that $\sigma'$ has rank $m+1$: the root of $\hat{t}_u$ is attached to $u$ as its last child. The rules for $f$ are $$\langle f,\sigma,j\rangle\to \sigma_{0,j}(\langle d,{\rm down}_1\rangle,\dots,\langle d,{\rm down}_m\rangle, \xi_j)$$ where $\xi_j = \langle f_j,{\rm up}\rangle$ for $j\neq 0$, and $\xi_0 = \bot$. The ``reconstruction'' subscripts of $\sigma_{0,j}$ mean the following: subscript $0$ indicates that this node is the root of some $\hat{t}_u$, and subscript $j$ is the child number of $u$ in $t$. Note that $\sigma_{0,j}$ has rank $m+1$: its last child corresponds to the parent of $u$ in $t$ (viewing $\bot$ as the ``parent'' of $\mathrm{root}_t$ in $t$). The \abb{tt} ${\cal N}$ walks up along the path from $u$ to the root of $t$ using ``folding'' states $f_i$, where the $i$ indicates that in the previous step ${\cal N}$ was at the $i$-th child of the current node. The rules for $f_i$ are $$ \begin{array}{lll} \langle f_i,\sigma,j\rangle & \to & \sigma_{i,j}( \\ && \langle d,{\rm down}_1\rangle,\dots,\langle d,{\rm down}_{i-1}\rangle, \\ && \bot, \\ && \langle d,{\rm down}_{i+1}\rangle,\dots,\langle d,{\rm down}_m\rangle, \\ && \xi_j) \end{array} $$ where $\xi_j$ is as above. If a node (in $\hat{t}_u$) with label $\sigma_{i,j}$ corresponds to the node $v$ in~$t$, then the ``reconstruction'' subscript $i$ means that its parent corresponds to the $i$-th child of $v$ in $t$ (and its own $i$-th child is $\bot$), and, as above, ``reconstruction'' subscript $j$ is the child number of $v$. Just as $\sigma_{0,j}$, also $\sigma_{i,j}$ has rank $m+1$: its last child corresponds to the parent of $v$ in $t$. Note that the copy $\hat{t}_u$ of the input tree is computed by the states $f$, $f_i$ (for every $i$) and $d$, such that $f$ copies node $u$ to the output and the other states walk from $u$ to every other node $v$ of $t$ and copy $v$ to the output. To be precise, ${\cal N}$ walks from $u$ to $v$ along the shortest (undirected) path from $u$ to $v$, from $u$ up to the least common ancestor of $u$ and $v$ (in the states $f_i$), and then down to $v$ (in the state $d$). Arriving in a node $v$ from a neighbour of $v$, the transducer ${\cal N}$ branches into a new process for every other neighbour of $v$.
This ends the description of the \abb{tt} ${\cal N}$. The output alphabet $\Gamma$ of ${\cal N}$ (which will also be the input alphabet of ${\cal M}'$) is the union of $\Sigma$, $\{\bot\}$, $\{\sigma'\mid \sigma\in\Sigma\}$, and $\{\sigma_{i,j}\mid \sigma\in\Sigma, i\in[0,\operatorname{rank}_\Sigma(\sigma)], j\in[0,{\mathit mx}_\Sigma]\}$. Thus, ${\cal N}$ has $O(n^2)$ output symbols, where $n$ is the size of $\Sigma$.\footnote{We assume here that the rank of each symbol of the ranked alphabet $\Sigma$ is specified in unary rather than decimal notation, and thus ${\mathit mx}_\Sigma\leq n$; cf. the last paragraph of \cite[Section~2]{Eng09}.} So, since ${\mathit mx}_\Gamma= {\mathit mx}_\Sigma +1$, the size of $\Gamma$ is polynomial in $n$. The set of states of ${\cal N}$ is $\{d,g,f\} \cup \{f_i\mid i\in[1,{\mathit mx}_\Sigma]\}$, with initial state $g$. Thus, it has $O(n)$ states and $O(n^3)$ rules; moreover, each of these rules is of size $O(n\log n)$. Hence, the size of ${\cal N}$ is polynomial in the size of $\Sigma$, and it can be constructed in polynomial time.
We now turn to the description of the \abb{v$_{k-1}$i-ptt} ${\cal M}'$. It has input alphabet~$\Gamma$, output alphabet $\Delta$, set of states $Q\cup (Q\times C_{\mathrm{v}})$, and the same initial states and visible colours as ${\cal M}$. Its invisible colour set is $C'_{\mathrm{i}} = C_{\mathrm{i}}\cup\{\odot\}$. It remains to discuss the set $R'$ of rules of ${\cal M}'$. Let $\langle q,\sigma,j,b\rangle\to \zeta$ be a rule of ${\cal M}$ with $\operatorname{rank}_\Sigma(\sigma)=m$. We consider four cases, depending on the variant $\sigma'$, $\sigma_{0,j}$, $\sigma_{i,j}$ with $i\neq 0$, or $\sigma$ in $\Gamma$ of the input symbol $\sigma\in\Sigma$.
In the first case, we consider the behaviour of ${\cal M}'$ in state $q$ on $\sigma'$, and we assume that $b\cap C_{\mathrm{v}}=\varnothing$. If $\zeta = \tup{q',{\rm drop}_c}$ with $c\in C_{\mathrm{v}}$, then $R'$ contains the rule $\langle q,\sigma',j,b\rangle\to \tup{(q',c),{\rm down}_{m+1};{\rm drop}_\odot}$,\footnote{To be completely formal, this rule should be replaced by the two rules $\langle q,\sigma',j,b\rangle\to \tup{p,{\rm down}_{m+1}}$ and $\langle p,\sigma_{0,j},m+1,\varnothing\rangle\to \tup{(q',c),{\rm drop}_\odot}$, where $p$ is a new state. } and otherwise $R'$ contains the rule $\langle q,\sigma',j,b\rangle\to \zeta$. Thus, ${\cal M}'$ simulates ${\cal M}$ on the original (now primed) part of the input tree $t$ in ${\rm pp}(t)$, until ${\cal M}$ drops a visible pebble~$c$ on node $u$. Then ${\cal M}'$ steps to the root of $\hat{t}_u$ where it drops the invisible pebble~$\odot$, and stores $c$ in its finite state.
Next, we let $c\in C_{\mathrm{v}}$ and we consider the behaviour of ${\cal M}'$ in state $(q,c)$ on the remaining variants of $\sigma$. Let $\zeta_c$ be the result of changing in $\zeta$ every occurrence of a state $q'$ into $(q',c)$.
In the second case we assume that $c\in b$ (corresponding to the fact that $\sigma_{0,j}$ labels the marked node of some $\hat{t}_u$). If $b=\{c\}$ and $\zeta=\tup{q',{\rm lift}_c}$, then $R'$ contains the rule $\tup{(q,c),\sigma_{0,j},m+1,\{\odot\}} \to \tup{q',{\rm lift}_\odot;{\rm up}}$.\footnote{Again, to be completely formal, this rule should be replaced by the two rules $\tup{(q,c),\sigma_{0,j},m+1,\{\odot\}} \to \tup{p,{\rm lift}_\odot}$ and $\tup{p,\sigma_{0,j},m+1,\varnothing} \to \tup{q',{\rm up}}$, where $p$ is a new state. } Thus, when ${\cal M}$ lifts visible pebble~$c$ from node $u$, ${\cal M}'$ lifts invisible pebble~$\odot$ and steps from the root of $\hat{t}_u$ back to node $u$. Otherwise, $R'$ contains the rules $$\tup{(q,c),\sigma_{0,j},m+1,b\setminus\{c\}\cup\{\odot\}} \to \zeta'_c$$ (provided $b\cap C_{\mathrm{i}}=\varnothing$) and $$\tup{(q,c),\sigma_{0,j},m+1,b\setminus\{c\}} \to \zeta'_c,$$ where $\zeta'_c$ is obtained from $\zeta_c$ by changing ${\rm up}$ into ${\rm down}_{m+1}$. These two rules correspond to whether or not the invisible pebble~$\odot$ is observable. Note that the child number in ${\rm pp}(t)$ of a node with label $\sigma_{0,j}$ is always $m+1$ (and the label of its parent is $\sigma'$).
In the remaining two cases we assume that $c\notin b$ in the above rule of ${\cal M}$. In the third case, we consider $\sigma_{i,j}$ with $i\neq 0$. Then $R'$ contains the rules $\tup{(q,c),\sigma_{i,j},j',b} \to \zeta'_c$ for every $j'\in[1,{\mathit mx}_\Gamma]$, where $\zeta'_c$ is now obtained from $\zeta_c$ by changing ${\rm up}$ into ${\rm down}_{m+1}$, and ${\rm down}_i$ into ${\rm up}$. In the fourth and final case, we consider $\sigma$ itself (in $\Gamma$). Then $R'$ contains the rule $\tup{(q,c),\sigma,j,b} \to \zeta_c$. Thus, ${\cal M}'$ stepwise simulates ${\cal M}$ on every $\hat{t}_u$.
This ends the description of the \abb{v$_{k-1}$i-ptt} ${\cal M}'$. It should now be clear that $\tau_{{\cal M}'}({\rm pp}(t)) = \tau_{{\cal M}}(t)$ for every $t\in T_\Sigma$, and hence $\tau_{{\cal N}}\circ\tau_{{\cal M}'}= \tau_{{\cal M}}$. Each rule of ${\cal M}$ is turned into at most $1+\#(C_{\mathrm{v}})\cdot(2+{\mathit mx}_\Sigma({\mathit mx}_\Sigma +1))$ rules of ${\cal M}'$, of the same size as that rule (disregarding the space taken by the occurrences of $c$ and $m+1$). Thus, ${\cal M}'$ can be computed from ${\cal M}$ in polynomial time. \end{proof}
The tree ${\rm pp}(t)$ that is used in the previous proof consists of two levels of copies of the original input tree $t$; on the first level a straightforward copy of $t$ (used until the first visible pebble is dropped) and a second level of copies $\hat{t}_u$ (used to ``store'' the first visible pebble dropped). It is tempting to add another level, meant as a way to store the next visible pebble dropped. The problem with this is that it would make the first visible pebble effectively unobservable when the next one is dropped. The idea \emph{can} be used for invisible pebbles, for arbitrarily many levels.
\begin{lemma}\label{lem:nul-decomp} For every \abb{i-ptt} ${\cal M}$ a \abb{tt} ${\cal N}$ and a \abb{tt} ${\cal M}'$ can be constructed in polynomial time such that $\tau_{{\cal N}}\circ\tau_{{\cal M}'}= \tau_{{\cal M}}$. If ${\cal M}$ is deterministic, then so is~${\cal M}'$. Hence, $\family{I-PTT} \subseteq \family{TT} \circ \family{TT}$ and $\family{I-dPTT} \subseteq \family{TT} \circ \family{dTT}$. \end{lemma}
\begin{proof} The computation of a \abb{PTT} ${\cal M}$ with invisible pebbles on tree $t$ is simulated by a \abb{TT} ${\cal M}'$ (without pebbles) on tree $t'$. The input tree $t$ is preprocessed in a nondeterministic way by a \abb{TT} ${\cal N}$ to obtain $t'$. The top level of $t'$ is a copy of $t$, as before. On the next level, since the simulating transducer ${\cal M}'$ cannot store the colours of all the pebbles in its finite state (as we did for one colour in the proof of Lemma~\ref{lem:decomp}), ${\cal N}$ does not attach one copy $\hat{t}_u$ of $t$ to each node $u$ of $t$ but $\#(C_\mathrm{i})$ such copies, one for each pebble colour. In this way, the child number in $t'$ of the root of $\hat{t}_u$ represents the pebble colour. In fact, in each node $u$ of $t$ the transducer ${\cal N}$ nondeterministically decides for each pebble colour $c$ whether or not to spawn a process that copies $t$ into $\hat{t}_u$, and this is a recursive process: in each node in each copy of $t$ it can be decided to spawn such processes that generate new copies.
In this way a ``tree of trees'' is constructed. For an ``artist impression'' of such an output tree $t'$, see Fig.~\ref{fig:artists}.
\begin{figure}
\caption{An output tree $t'$ of the \abb{tt} ${\cal N}$ of Lemma~\ref{lem:nul-decomp} for input tree $t$.}
\label{fig:artists}
\end{figure}
The child number in $t'$ of the root of each copy $\hat{t}_u$ indicates an invisible pebble of colour $c$ placed at node $u$ in the original tree $t$. In each copy only one pebble is observable, the one represented by the child number of its root, exactly as the last pebble dropped in the original computation. In the simulation, moving down or up along the tree of trees corresponds to dropping and lifting invisible pebbles.
In general there is no bound on the depth of the stack of pebbles during a computation of ${\cal M}$. The preprocessor ${\cal N}$ nondeterministically constructs $t'$. If $t'$ is not sufficiently deep, the simulating transducer ${\cal M}'$ aborts the computation. Conversely, for every computation of ${\cal M}$ a tree $t'$ of sufficient depth can be constructed nonderministically from $t$.
We now turn to the formal definitions. Let ${\cal M} = (\Sigma, \Delta, Q, Q_0,C, C_\mathrm{v}, C_\mathrm{i}, R,0)$ be an \abb{i-ptt}. Without loss of generality we assume that $C=C_\mathrm{i}$ and that $C=[1,\gamma]$ for some $\gamma\in{\mathbb N}$. This choice of $C$ simplifies the representation of colours by child numbers.
First, we define the nondeterministic \abb{tt} ${\cal N}$ that preprocesses the trees over~$\Sigma$. It is a straightforward variant of the one in the proof of Lemma~\ref{lem:decomp}. The output alphabet $\Gamma$ of ${\cal N}$ is now the union of $\{\bot\}$, $\{\sigma'\mid \sigma\in\Sigma\}$, and $\{\sigma'_{i,j}\mid \sigma\in\Sigma, i\in[0,\operatorname{rank}_\Sigma(\sigma)], j\in[0,{\mathit mx}_\Sigma]\}$ where, for every $\sigma\in\Sigma$ of rank~$m$, $\sigma'$ has rank $m+\gamma$ and $\sigma'_{i,j}$ has rank $m+\gamma+1$, because $\gamma$ processes are spawned at each node, and each of these processes generates, nondeterministically, either a copy $\hat{t}_u$ of $t$ or the bottom symbol $\bot$. The set of states of ${\cal N}$ is as before, except that the state $d$ is removed (with its rules). In the rules of ${\cal N}$ we will use $\tup{f,{\rm stay}}^\gamma$ as an abbreviation of the sequence $\tup{f,{\rm stay}},\dots,\tup{f,{\rm stay}}$ of length $\gamma$. The rules for the initial state~$g$ are \[ \begin{array}{lll} \langle g,\sigma,j\rangle & \to & \sigma'(\langle g,{\rm down}_1\rangle,\dots,\langle g,{\rm down}_m\rangle, \langle f,{\rm stay}\rangle^\gamma). \end{array} \] The rules for $f$ are \[ \begin{array}{lll} \langle f,\sigma,j\rangle & \to & \bot \\[1mm] \langle f,\sigma,j\rangle & \to & \sigma'_{0,j}(\langleg,{\rm down}_1\rangle,\dots,\langleg,{\rm down}_m\rangle,\tup{f,{\rm stay}}^\gamma,\xi_j) \end{array} \] where, as before, $\xi_j = \langle f_j,{\rm up}\rangle$ for $j\neq 0$, and $\xi_0 = \bot$. Finally, the rules for $f_i$ are \[ \begin{array}{lll} \langle f_i,\sigma,j\rangle & \to & \sigma'_{i,j}( \\ && \langle g,{\rm down}_1\rangle,\dots,\langle g,{\rm down}_{i-1}\rangle, \\ && \bot, \\ && \langle g,{\rm down}_{i+1}\rangle,\dots,\langle g,{\rm down}_m\rangle, \\ && \tup{f,{\rm stay}}^\gamma, \\ && \xi_j) \end{array} \] where $\xi_j$ is as above. This ends the definition of ${\cal N}$.
Next, we define the simulating \abb{tt} ${\cal M}'$. It has input alphabet~$\Gamma$ (the output alphabet of ${\cal N}$), output alphabet $\Delta$, and the same set of states and initial states as ${\cal M}$. The set $R'$ of rules of ${\cal M}'$ is defined as follows. Let $\langle q,\sigma,j,b\rangle\to \zeta$ be a rule of ${\cal M}$ with $\operatorname{rank}_\Sigma(\sigma)=m$. Note that $b$ is either empty or a singleton. We consider three cases, that describe the behaviour of ${\cal M}'$ on the symbols $\sigma'$, $\sigma'_{0,j}$, and $\sigma'_{i,j}$ with $i\neq 0$.
In the first case we assume that $b=\varnothing$ (and hence $\zeta$ does not contain a lift-instruction). Then $R'$ contains the rule $\langle q,\sigma',j\rangle\to \zeta'$ where $\zeta'$ is obtained from~$\zeta$ by changing ${\rm drop}_c$ into ${\rm down}_{m+c}$ for every $c\in C$.
In the second case we assume that $b=\{c\}$ for some $c\in C$. Then $R'$ contains the rule $\tup{q,\sigma'_{0,j},m+c} \to \zeta'$ where $\zeta'$ is now obtained from $\zeta$ by changing ${\rm up}$ into ${\rm down}_{m+\gamma+1}$, ${\rm lift}_c$ into ${\rm up}$, and ${\rm drop}_d$ into ${\rm down}_{m+d}$ for every $d\in C$. Note that the child number in $t'$ of a node with label $\sigma'_{0,j}$ is always $m+c$ for some $c\in C$ (and the label of its parent is $\sigma'$ or $\sigma'_{i,j}$ for some $i\in[0,m]$).
In the third case we assume (as in the first case) that $b=\varnothing$. Then $R'$ contains the rule $\langle q,\sigma'_{i,j},j'\rangle\to \zeta'$ for every $j'\in[1,{\mathit mx}_\Gamma]$, where $\zeta'$ is now obtained from $\zeta$ by changing ${\rm up}$ into ${\rm down}_{m+\gamma+1}$, ${\rm down}_i$ into ${\rm up}$, and ${\rm drop}_c$ into ${\rm down}_{m+c}$ for every $c\in C$.
This ends the definition of ${\cal M}'$. It should, again, be clear that for every $t\in T_\Sigma$ and $s\in T_\Delta$, $s\in \tau_{{\cal M}}(t)$ if and only if there exists $t'\in \tau_{{\cal N}}(t)$ such that $s\in \tau_{{\cal M}'}(t')$. Hence $\tau_{{\cal N}}\circ\tau_{{\cal M}'}= \tau_{{\cal M}}$. It is straightforward to show, as in the proof of Lemma~\ref{lem:decomp}, that ${\cal N}$ and ${\cal M}'$ can be constructed in polynomial time from~${\cal M}$. Note that ${\mathit mx}_\Gamma = {\mathit mx}_\Sigma + \#(C_{\mathrm{i}})+1$ and so the size of $\Gamma$ is polynomial in the size of ${\cal M}$. \end{proof}
Combining the previous two results we can inductively decompose tree tranducers with (visible and invisible) pebbles into tree transducers without pebbles.
\begin{theorem}\label{thm:decomp} For every $k\ge 0$, $\VIPTT{k} \subseteq \family{TT}^{k+2}$. For fixed $k$, the involved construction takes polynomial time. \end{theorem}
Observe that $\VPTT{k} \subseteq \VIPTT{k-1}$ as the topmost pebble can be replaced by an invisible one, thus $\VPTT{k} \subseteq \family{TT}^{k+1}$, which was proved in~\cite[Theorem~10]{EngMan03}, also for the deterministic case.
We do not know whether Theorem~\ref{thm:decomp} is optimal, i.e., whether or not $\VIPTT{k}$ is included in $\family{TT}^{k+1}$. The deterministic version of Theorem~\ref{thm:decomp} (for $k\neq 0$) will be proved in Section~\ref{sec:variations} (Theorem~\ref{thm:detdecomp}), and we will show that it is optimal (after Theorem~\ref{thm:dethier}).
The nondeterminism of the ``preprocessing'' transducer ${\cal N}$ in the proof of Lemma~\ref{lem:nul-decomp} is rather limited. The general form of the constructed tree is completely determined by the input tree, only the depth of the construction is nondeterministically chosen. At the same time it remains nondeterministic even when we start with a deterministic \abb{pTT} with invisible pebbles: $\family{I-dPTT} \subseteq \family{TT} \circ \family{dTT}$. However, we can obtain a deterministic transduction if the number of invisible pebbles used by the transducer is bounded (over all input trees), cf. the M.~Sc.~Thesis of the third author \cite{Sam} (where visible and invisible pebbles are called global and local pebbles, respectively).
In Section~\ref{sec:poweriptt} we will show that if we start with a deterministic tree transduction, then the inclusions of Lemma~\ref{lem:nul-decomp} also hold in the other direction (Theorem~\ref{thm:composition}). In Section~\ref{sec:variations} we will show that $\family{I-dPTT} \subseteq \family{dTT}^3$ (Corollary~\ref{cor:idptt-tt3}).
\section{Typechecking}\label{sec:typechecking}
The \emph{inverse type inference problem} is to construct, for a tree transducer ${\cal M}$ and a regular tree grammar $G_{\rm out}$, a regular tree grammar $G_{\rm in}$ such that $L(G_{\rm in}) = \tau_{\cal M}^{-1}(L(G_{\rm out}))$.
The \emph{typechecking problem} asks, for a tree transducer ${\cal M}$ and two regular tree grammars $G_{\rm in}$ and $G_{\rm out}$, whether or not $\tau_{\cal M}(L(G_{\rm in}))\subseteq L(G_{\rm out})$.
The inverse type inference problem can be used to solve the typechecking problem, because $\tau_{\cal M}(L(G_{\rm in}))\subseteq L(G_{\rm out})$ if and only if $L(G_{\rm in}) \cap \tau_{\cal M}^{-1}(L'_{\rm out})=\varnothing$, where $L'_{\rm out}$ is the complement of $L(G_{\rm out})$.
It was shown in \cite{MilSucVia03} (see also \cite[Section~7]{EngMan03}) that both problems are solvable for tree-walking tree transducers with visible pebbles, i.e., for \abb{v-ptt}'s, and hence in particular for tree-walking tree transducers without pebbles, i.e., for \abb{tt}'s.\footnote{Note however that our definition of inverse type inference differs from the one in \cite{MilSucVia03}, where it is required that $L(G_{\rm in}) = \{\;s\mid \tau_{\cal M}(s)\subseteq L(G_{\rm out})\;\}$. The reason is that our definition is more convenient when considering compositions of tree transducers. } This was extended in \cite{Eng09} to compositions of such transducers and, moreover, the time complexity of the involved algorithms was improved, using a result of~\cite{Bar} for attributed tree transducers.
We define a \emph{$k$-fold exponential} function to be a function of the form $2^{g(n)}$ where $g$ is a $(k\!-\!1)$-fold exponential function; a $0$-fold exponential function is a polynomial.
\begin{proposition}\label{prop:invtypeinf} For fixed $k\geq 0$, the inverse type inference problem is solvable \\ {\em (1)} for compositions of $k$ \abb{tt}'s in $k$-fold exponential time, and \\ {\em (2)} for \abb{v$_k$-PTT}'s in $(k\!+\!1)$-fold exponential time. \end{proposition}
\begin{proposition}\label{prop:typecheck} For fixed $k\geq 0$, the typechecking problem is solvable \\ {\em (1)} for compositions of $k$ \abb{tt}'s in $(k\!+\!1)$-fold exponential time, and \\ {\em (2)} for \abb{v$_k$-PTT}'s in $(k\!+\!2)$-fold exponential time. \end{proposition}
As also observed in \cite{Eng09}, one exponential can be taken off the results of Proposition~\ref{prop:typecheck} if we assume that $G_{\rm out}$ is a total deterministic bottom-up finite-state tree automaton, because that exponential is due to the complementation of $L(G_{\rm out})$.
It is immediate from Theorem~\ref{thm:decomp} and Propositions~\ref{prop:invtypeinf}(1) and~\ref{prop:typecheck}(1) that both problems are also solvable for tree-walking tree transducers with invisible pebbles.
\begin{theorem}\label{thm:typecheck} For fixed $k\geq 0$, the inverse type inference problem and the typechecking problem are solvable for \abb{v$_k$i-PTT}'s in $(k\!+\!2)$-fold and $(k\!+\!3)$-fold exponential time, respectively. \end{theorem}
The main conclusion from Proposition~\ref{prop:typecheck}(2) and Theorem~\ref{thm:typecheck} is that the complexity of typechecking \abb{ptt}'s basically depends on the number of visible pebbles used. Thus we can improve the complexity of the problem by changing visible pebbles into invisible ones as much as possible, see Section~\ref{sec:pattern}.
Note that the solvability of the inverse type inference problem for a tree transducer ${\cal M}$ means in particular that its domain is a regular tree language, taking $L(G_{\rm out}) = T_\Delta$ where $\Delta$ is the output alphabet of ${\cal M}$. Thus, it follows from Theorem~\ref{thm:typecheck} that the domains of \abb{PTT}'s are regular, or in other words, that every alternating \abb{pta} accepts a regular tree language.
\begin{corollary}\label{cor:domptt} For every \abb{ptt} ${\cal M}$, its domain $L({\cal M})$ is regular. \end{corollary}
\section{Trees, Tests and Trips}\label{sec:tests}
In this section we show that \abb{vi-pta}'s recognize the regular tree languages, that they compute the \abb{mso} definable binary patterns (or trips), and that they can perform \abb{mso} tests on the observable part of their configuration (which consists of the position of the head and the positions of the observable pebbles).
For ``classical'' tree-walking automata with a bounded number of visible pebbles, i.e., for \abb{v-pta}'s, it was shown in \cite[Section~5]{jewels} that these automata accept regular tree languages only. However, as proved in \cite{expressive}, they cannot accept all regular tree languages. One of the main reasons for introducing an unbounded number of invisible pebbles is that they can be used to recognize every regular tree language. Recall that $\family{REGT}$ denotes the class of regular tree languages.
\begin{lemma}\label{lem:regular} $\family{REGT} \subseteq \family{I-dPTA}$. \end{lemma}
\begin{proof} As the regular tree languages are recognized by deterministic bottom-up finite-state tree automata, it suffices to explain how the computation of such an automaton ${\cal A}$ can be simulated by a deterministic \abb{pta} ${\cal A}'$ with invisible pebbles. The computation of ${\cal A}$ on the input tree can be reconstructed by a post-order evaluation of the tree. At the current node $u$, ${\cal A}'$ uses an invisible pebble to store the states in which ${\cal A}$ arrives at the first $m$ children of $u$, for some $m$. The colour of the pebble represents the sequence of states. For each ancestor $v$ of $u$ the pebble stack contains a similar pebble for the first $i-1$ children of $v$, where $vi$ is the unique child of $v$ that is also an ancestor of $u$ (or $u$ itself). If $u$ has more than $m$ children, then ${\cal A}'$ moves to its $(m+1)$-th child and drops a pebble that represents the empty sequence of states of ${\cal A}$. Otherwise, ${\cal A}'$ computes the state assumed by ${\cal A}$ in $u$ based on the states of the children, lifts the pebble at $u$, and moves to the parent of $u$ to update its pebble with that state. The post-order evaluation ensures that pebbles are used in a nested fashion.
Formally, let ${\cal A}= (\Sigma,P,F,\delta)$ where $\Sigma$ is a ranked alphabet, $P$ is a finite set of states, $F\subseteq P$ is the set of final states, and $\delta$ is the transition function that assigns a state $\delta(\sigma,p_1,\dots,p_m)\in P$ to every $\sigma\in\Sigma$ and $p_1,\dots,p_m\in P$ with $m=\operatorname{rank}_\Sigma(\sigma)$. As pebble colours the \abb{i-pta} ${\cal A}'$ has all strings in $P^*$ of length at most ${\mathit mx}_\Sigma$. Its states and rules are introduced one by one as follows, where $\sigma$ ranges over $\Sigma$, $j$ and $m$ range over $[0,\operatorname{rank}(\sigma)]$, and $p,p_1,\dots,p_m$ range over~$P$. The initial state $q_0$ does not occur in the right-hand side of any rule. In the initial state, the automaton ${\cal A}'$ drops a pebble at the root representing the empty sequence of states of ${\cal A}$, and goes into the main state $q_\circ$. The rule is \[ \rho_1: \tup{q_0,\sigma,0,\varnothing} \to \tup{q_\circ,{\rm drop}_\varepsilon}. \] In state $q_\circ$, ${\cal A}'$ consults the pebble to see whether or not all children have been evaluated, and acts accordingly. For $m<\operatorname{rank}(\sigma)$ it has the rule \[ \rho_2: \tup{q_\circ,\sigma,j,\{p_1\cdots p_m\}} \to
\tup{q_\circ,{\rm down}_{m+1};{\rm drop}_\varepsilon}, \] which handles the case that the state of ${\cal A}$ is not yet known for all children of node $u$. For $m=\operatorname{rank}(\sigma)$ and $p=\delta(\sigma,p_1,\dots,p_m)$ it has the rules \[ \begin{array}{llll} \rho_3: \tup{q_\circ,\sigma,j,\{p_1\cdots p_m\}} & \to &
\tup{\bar{q}_p,{\rm lift}_{p_1\cdots p_m};{\rm up}}
& \text{if } j\neq 0, \\[1mm] \rho_4: \tup{q_\circ,\sigma,0,\{p_1\cdots p_m\}} & \to & \tup{q_\mathrm{yes},{\rm stay}}
& \text{if } p\in F, \\[1mm] \rho_5: \tup{q_\circ,\sigma,0,\{p_1\cdots p_m\}} & \to & \tup{q_\mathrm{no},{\rm stay}}
& \text{if } p\notin F, \end{array} \] and for $m<\operatorname{rank}(\sigma)$ it has the rule \[ \rho_6: \tup{\bar{q}_p,\sigma,j,\{p_1\cdots p_m\}} \to
\tup{q_\circ,{\rm lift}_{p_1\cdots p_m};{\rm drop}_{p_1\cdots p_mp}}. \] Thus, if the states $p_1,\dots,p_m$ of ${\cal A}$ at all the children of node $u$ are known, ${\cal A}'$~computes the state $p=\delta(\sigma,p_1,\dots,p_m)$ of ${\cal A}$ at $u$. If $u$ is not the root of the input tree, then ${\cal A}'$ stores $p$ in its own state $\bar{q}_p$, lifts the pebble $p_1\cdots p_m$, and moves up to the parent of $u$. Since the pebble at the parent is now observable, it can be updated. If $u$ is the root of the input tree, then ${\cal A}'$ knows whether or not ${\cal A}$ accepts that tree, and correspondingly goes into state $q_\mathrm{yes}$ or state $q_\mathrm{no}$, where $q_\mathrm{yes}$ is the unique final state of ${\cal A}'$. Note that there is one pebble left on the root of the tree. \end{proof}
Adding an infinite supply of invisible pebbles on the other hand does not lead out of the regular tree languages. It is possible to give a proof of this fact by reducing \abb{v$_k$i-pta}'s to the backtracking pushdown tree automata of \cite{Slu}, but here we deduce it from the results of the previous section.
\begin{theorem}\label{thm:regt} For each $k\ge 0$, $\VIPTA{k} = \VIdPTA{k} = \family{REGT}$. \end{theorem}
\begin{proof} By Lemma~\ref{lem:regular}, $\family{REGT} \subseteq \VIdPTA{k}$. Conversely, as observed before, a \abb{pta} ${\cal A}$ is easily turned into a \abb{ptt} ${\cal M}$ that outputs single node tree $\delta$ (with $\operatorname{rank}(\delta)=0$) for trees accepted by ${\cal A}$: for every final state $q$ of ${\cal A}$ add all rules $\tup{q,\sigma,j,b} \to \delta$. Then $L({\cal A})=L({\cal M})$, the domain of ${\cal M}$, which is regular by Corollary~\ref{cor:domptt}. \end{proof}
Note that an infinite supply of \emph{visible} pebbles could be used to mark $a$'s and $b$'s alternatingly and thus accept the nonregular language $\{a^nb^n\mid n\in{\mathbb N}\}$ (and similarly $\{a^nb^nc^n\mid n\in{\mathbb N}\}$). Note also that the stack of pebbles cannot be replaced by two independent stacks, one for visible and one for invisible pebbles. Then we could accept $\{a^nb^n\mid n\in{\mathbb N}\}$ with just one visible pebble: drop an invisible pebble on each $a$, and then use the visible pebble on the $b$'s to count the number of $a$'s, by lifting one invisible pebble (in fact, the unique observable one) for each $b$.
Recall from Section~\ref{sec:trees} that an $n$-ary \emph{pattern} over a ranked alphabet $\Sigma$ is a set $T \subseteq \{(t,u_1,\dots,u_n)
\mid t\in T_\Sigma, \,u_1,\dots,u_n \in N(t)\}$. Recall also that the pattern~$T$ is said to be regular if its marked representation $\operatorname{mark}(T)\subseteq T_{\Sigma\times\{0,1\}^n}$ is a regular tree language. In fact, $T$ is regular if and only if it is \abb{mso} definable, which means that there is an \abb{mso} formula $\varphi(x_1,\dots,x_n)$ over $\Sigma$ such that $T = T(\varphi)$, where $T(\varphi) = \{(t,u_1,\dots,u_n) \mid t \models \varphi(u_1,\dots,u_n)\}$. Recall finally that a unary pattern ($n=1$) is called a \emph{site}, and a binary pattern ($n=2$) is called a \emph{trip}.
With the help of an unbounded supply of invisible pebbles tree-walking automata can recognize regular tree languages, Lemma~\ref{lem:regular}.
Likewise \abb{v$_n$i-pta}'s can match arbitrary \abb{MSO} definable $n$-ary patterns $\varphi$. When $n$ visible pebbles are dropped on a sequence of $n$ nodes, the invisible pebbles can be used to evaluate the tree, and test whether it belongs to the regular tree language $\operatorname{mark}(T(\varphi))$. In Section~\ref{sec:pattern} we will consider pattern matching in detail.
Ignoring the visible pebbles, it is also possible to consider just the position of the head, and test whether the input tree together with that position belongs to a given regular ``marked'' tree language.
We say that a family ${\cal F}$ of \abb{pta}'s (or \abb{ptt}'s) can \emph{perform \abb{mso} head tests} if, for a regular site $T$ over $\Sigma$, an automaton (or transducer) in ${\cal F}$ can test whether or not $(t,h) \in T$, where $t$ is the input tree and $h$ the position of the head at the moment of the test. Admittedly, this is a very informal definition. To formalize it we have to define a \abb{pta}$^{\text{\abb{mso}}}$ (or a \abb{ptt}$^{\text{\abb{mso}}}$), i.e., a \abb{pta} (or \abb{ptt}) \emph{with \abb{mso} head tests}, that has rules of the form $\tup{q,\sigma,j,b,T} \to \zeta$ where $T$ is a regular site over $\Sigma$ (specified in some effective way). Such a rule is relevant to a configuration $\tup{q,h,\pi}$ on a tree $t$ if, in addition, $(t,h) \in T$. Since the regular tree languages are closed under complement, the complement $T^\mathrm{c}$ of~$T$ can be tested in a rule with left-hand side $\tup{q,\sigma,j,b,T^\mathrm{c}}$. Such an automaton (or transducer) is deterministic if for every two distinct rules $\tup{q,\sigma,j,b,T} \to \zeta$ and $\tup{q,\sigma,j,b,T'} \to \zeta'$, the site $T'$ is the complement of the site $T$. For a family ${\cal F}$ of \abb{pta}'s (or \abb{ptt}'s), such as the \abb{v$_k$i-pta} or \abb{v$_k$i-}{\rm d}\abb{ptt} or \abb{v$_k$-pta}, we denote by ${\cal F}^{\text{\abb{mso}}}$ the corresponding family of \abb{pta}$^{\text{\abb{mso}}}$'s (or \abb{ptt}$^{\text{\abb{mso}}}$'s). With this definition of \abb{pta}$^{\text{\abb{mso}}}$ we can formally define that a family ${\cal F}$ of \abb{pta}'s can perform \abb{mso} head tests if for every \abb{pta}$^{\text{\abb{mso}}}$ in ${\cal F}^{\text{\abb{mso}}}$ an equivalent \abb{pta} in ${\cal F}$ can be constructed, and similarly for \abb{ptt}'s.
Obviously, as \abb{v-pta}'s cannot recognize all regular tree languages, they cannot perform \abb{mso} head tests either: for any regular tree language $T$ the set $\{(t,\mathrm{root}_t) \mid t\in T \}$ is a regular site.
The next result shows that any \abb{vi-pta} that uses \abb{mso} head tests as a built-in feature (i.e., any \abb{vi-pta}$^{\text{\abb{mso}}}$) can be replaced by an equivalent \mbox{\abb{vi-pta}} without such tests. The result holds for \abb{vi-pta}'s with any fixed number of visible pebbles, either deterministic or nondeterministic, and it also holds for the corresponding \abb{vi-ptt}'s.
\begin{lemma}\label{lem:sites} For each $k\ge 0$, the \abb{v$_k$i-pta} can perform \abb{mso} head tests. The same holds for the \abb{v$_k$i-}{\em d}\abb{pta}, \abb{v$_k$i-ptt}, and \abb{v$_k$i-}{\em d}\abb{ptt}. \end{lemma}
\begin{proof} Let ${\cal A}_T$ be a deterministic bottom-up finite-state tree automaton recognizing the regular tree language $\operatorname{mark}(T)$ over $\Sigma \times \{0,1\}$, representing the site~$T$, trees with a single marked node.
We show how a deterministic \abb{i-pta} ${\cal A}'_T$ can test whether or not the input tree with current head position $h$ is accepted by~${\cal A}_T$, in a computation starting in configuration $\tup{q_0,h,\varepsilon}$ and ending in configuration $\tup{q_\mathrm{yes},h,\varepsilon}$ or $\tup{q_\mathrm{no},h,\varepsilon}$, where $q_0$ is the initial state and $\{q_\mathrm{yes},q_\mathrm{no}\}$ the set of final states of ${\cal A}'_T$. Moreover, it starts the computation by dropping a pebble on $h$, and it keeps a pebble on $h$ until the final computation step. It should be obvious that this \abb{i-pta} ${\cal A}'_T$ can be used as a subroutine by any \abb{v$_k$i-pta} or \abb{v$_k$i-ptt} ${\cal A}$, starting in configuration $\tup{(\tilde{q},q_0),h,\pi}$ and ending in configuration $\tup{(\tilde{q},q_\mathrm{yes}),h,\pi}$ or $\tup{(\tilde{q},q_\mathrm{no}),h,\pi}$, for every state $\tilde{q}$ and pebble stack $\pi$ of ${\cal A}$. Just replace each rule $\tup{q,\sigma,j,b} \to \tup{q',\alpha}$ of ${\cal A}'_T$ by all possible rules $\tup{(\tilde{q},q),\sigma,j,b\cup b'} \to \tup{(\tilde{q},q'),\alpha}$ where $b'$ is a set of visible pebble colours of ${\cal A}$ (except that in the first rule of~${\cal A}'_T$, which drops a pebble on $h$, the set $b'$ possibly contains an invisible pebble colour of~${\cal A}$).
The post-order evaluation of Lemma~\ref{lem:regular} does not work here without precautions. If we mark node $h$ with an invisible pebble the pebble becomes unobservable during the evaluation. In this way we cannot take the special ``marked'' position of $h$ into account.\footnote{Marking $h$ with a visible pebble would easily work, showing that \abb{vi-pta} can perform \abb{mso} head tests.}
Instead, we first evaluate the subtree rooted at~$h$, and subsequently the subtrees rooted at the ancestors of $h$, moving along the path from $h$ to the root of the input tree. At the start of the evaluation of a subtree, we ``paint'' its root $u$ by adding a special colour to the pebble on~$u$, and preserving that information when the pebble is updated. In this way it is always clear when the painted node is visited.
We paint node $h$ with the special additional colour $\odot$ and use the evaluation process of Lemma~\ref{lem:regular} to compute the state of ${\cal A}_T$ at $h$, viewing the label $\sigma$ of each node as $(\sigma,0)$ except for the label $\sigma$ of $h$ which is treated as $(\sigma,1)$. We paint each ancestor $u$ of $h$ with an additional colour $(j,p)$ which indicates the child number $j$ of the previous ancestor of $h$ and the state $p$ at which ${\cal A}_T$ arrives at that child of~$u$ (with $h$ as a marked node). Then we use, again, the evaluation process of Lemma~\ref{lem:regular} to compute the state of~${\cal A}_T$ at $u$ (with every $\sigma$ viewed as $(\sigma,0)$), except that the information in the pebble $(j,p)$ is used for the state~$p$ of the $j$-th child of $u$, which is the unique child that has $h$ as a descendant. Repeating this process for each ancestor, we eventually reach the root of the tree, and know the outcome of the test. Then we return to the original position~$h$ picking up the pebbles left on the path from that position to the root.
Formally, let ${\cal A}_T= (\Sigma \times \{0,1\},P,F,\delta)$. For convenience we will identify the symbols $(\sigma,0)$ and $\sigma$. The \abb{i-pta} ${\cal A}'_T$ is an extension of the \abb{i-pta} ${\cal A}'$ in the proof of Lemma~\ref{lem:regular}. It has the additional states $q_{\downarrow\mathrm{yes}}$ and $q_{\downarrow\mathrm{no}}$, and in addition to the pebble colours $p_1\cdots p_m$ of ${\cal A}'$ it has the pebble colours $(\mu,p_1\cdots p_m)$ where either $\mu=\odot$ or $\mu=(i,r)$ for some $i\in[1,{\mathit mx}_\Sigma]$ and $r\in P$. The additional pebbles are used to ``paint'' $h$ (with $\mu=\odot$) and the ancestors of $h$ (with some $\mu=(i,r)$). The automaton ${\cal A}'_T$ has all the rules of~${\cal A}'$, except that rules $\rho_4$ and~$\rho_5$ will become superfluous, and rule $\rho_1$ is replaced by the rule \[ \begin{array}{lll} \rho'_1: \tup{q_0,\sigma,j,\varnothing} & \to & \tup{q_\circ,{\rm drop}_{(\odot,\varepsilon)}}. \end{array} \] Thus, ${\cal A}'_T$ starts by evaluating the subtree rooted at $h$, with $h$ as marked node. For $m <\operatorname{rank}(\sigma)$ and every $\mu$ as above, except when $\mu=(m+1,r)$ for some $r\in P$, ${\cal A}'_T$ has the rules \[ \begin{array}{lll} \rho_2^\mu: \tup{q_\circ,\sigma,j,\{(\mu,p_1\cdots p_m)\}} & \to &
\tup{q_\circ,{\rm down}_{m+1};{\rm drop}_\varepsilon} \\[1mm] \rho_6^\mu: \tup{\bar{q}_p,\sigma,j,\{(\mu,p_1\cdots p_m)\}} & \to &
\tup{q_\circ,{\rm lift}_{(\mu,p_1\cdots p_m)};{\rm drop}_{(\mu,p_1\cdots p_mp)}} \end{array} \] which intuitively means that the pebble $(\mu,p_1\cdots p_m)$ is treated in the same way as $p_1\cdots p_m$ when not all children of the current node have been evaluated: ${\cal A}'_T$~moves to the $(m+1)$-th child and calls ${\cal A}'$, and when ${\cal A}'$ returns with the state $p$, ${\cal A}'_T$ adds $p$ to the sequence of states in the pebble. However, in the exceptional case where $m <\operatorname{rank}(\sigma)$ and $\mu=(m+1,r)$, ${\cal A}'_T$ has the rule \[ \begin{array}{lll} \rho_{2,6}^\mu: \tup{q_\circ,\sigma,j,\{(\mu,p_1\cdots p_m)\}} & \to &
\tup{q_\circ,{\rm lift}_{(\mu,p_1\cdots p_m)};{\rm drop}_{(\mu,p_1\cdots p_mr)}} \end{array} \] which means that for the $(m+1)$-th child ${\cal A}'_T$ does not call ${\cal A}'$ but uses the state~$r$ that was previously computed and stored in $\mu$.
The remaining rules of ${\cal A}'_T$ handle the situations that ${\cal A}'_T$ has just evaluated the subtrees rooted at the children of $h$ or of one of the ancestors $u$ of $h$, in state~$q_\circ$. The automaton ${\cal A}'_T$ computes the state $p$ of ${\cal A}_T$ at the marked node~$h$ or the unmarked node $u$, and drops the pebble $((j,p),\varepsilon)$ at its parent $v$, where $j$ is the child number of $h$ or $u$, thus indicating that the subtree rooted at the $j$-th child of~$v$ (with~$h$ as a marked node) evaluates to $p$. Then ${\cal A}'_T$ evaluates the subtree rooted at $v$.
For $m=\operatorname{rank}(\sigma)$ and every $\mu$ as above, ${\cal A}'_T$ has the rules \[ \begin{array}{llll} \rho_3^\mu: \tup{q_\circ,\sigma,j,\{(\mu,p_1\cdots p_m)\}} & \to &
\tup{q_\circ,{\rm up};{\rm drop}_{((j,p),\varepsilon)}}
& \text{if } j\neq 0, \\[1mm] \rho_4^\mu: \tup{q_\circ,\sigma,0,\{(\mu,p_1\cdots p_m)\}} & \to &
\tup{q_{\downarrow\mathrm{yes}},{\rm stay}}
& \text{if } p\in F, \\[1mm] \rho_5^\mu: \tup{q_\circ,\sigma,0,\{(\mu,p_1\cdots p_m)\}} & \to &
\tup{q_{\downarrow\mathrm{no}},{\rm stay}}
& \text{if } p\notin F. \end{array} \] where $p=\delta((\sigma,1),p_1,\dots,p_m)$ if $\mu=\odot$ and $p=\delta(\sigma,p_1,\dots,p_m)$ otherwise.
When ${\cal A}'_T$ arrives at the root of the input tree, it knows whether or not ${\cal A}_T$ accepts that tree (with $h$ as a marked node), and moves down to $h$. For the outcome $x\in\{\mathrm{yes},\mathrm{no}\}$ the rules are \[ \begin{array}{llll} \tup{q_{\downarrow x},\sigma,j,\{((i,r),p_1\cdots p_m)\}} & \to &
\tup{q_{\downarrow x},{\rm lift}_{((i,r),p_1\cdots p_m)};{\rm down}_i} \\[1mm] \tup{q_{\downarrow x},\sigma,j,\{(\odot,p_1\cdots p_m)\}} & \to &
\tup{q_x,{\rm lift}_{(\odot,p_1\cdots p_m)}}. \end{array} \] This ends the description of ${\cal A}'_T$. \end{proof}
This result can easily be extended, using the same proof technique: \abb{pta}'s and \abb{ptt}'s can test their \emph{visible configuration}, the position of the head together with the positions and colours of the visible pebbles.
Later we will show the more complicated result that \abb{pta}'s and \abb{ptt}'s can even test their \emph{observable configuration}, i.e., the visible configuration plus the topmost pebble (Theorem~\ref{thm:mso}).
Let $C$ be the set of colours of a \abb{pta} or \abb{ptt}. To represent the visible and observable configurations, we introduce a new ranked alphabet $\Sigma \times 2^C$, such that the rank of $(\sigma,b)$ equals that of $\sigma$ in $\Sigma$. A tree over $\Sigma \times 2^C$ is a ``coloured tree''. For each pebble stack $\pi$ on a tree $t$ over $\Sigma$ we define two coloured trees. The visible configuration tree $\xp{vis}(t,\pi)$ is obtained by adding to the label of each node~$u$ of~$t$ the set $b\subseteq C$ such that $b$ contains~$c$ if and only if $(u,c)$ occurs in $\pi$ and $c\in C_{\rm v}$. Similarly for $\xp{obs}(t,\pi)$, the observable configuration tree, $b$~contains~$c$ if and only if $(u,c)$ occurs in $\pi$ and $c$ is observable (i.e., $c\in C_{\rm v}$ or $(u,c)$ is the top element of $\pi$).
Note that as long as a \abb{pta} does not change its pebble stack by a drop- or lift-instruction, it behaves just as a \abb{ta} on $\xp{obs}(t,\pi)$.
We say that a family ${\cal F}$ of \abb{pta}'s (or \abb{ptt}'s) can \emph{perform \abb{mso} tests on the visible configuration} if, for a regular site $T$ over $\Sigma \times 2^C$, an automaton (or transducer) in ${\cal F}$ can test whether or not $(\xp{vis}(t,\pi),h) \in T$, where $t$ is the input tree, $\pi$ the current pebble stack and $h$ the current position of the head. A similar definition can be given for \emph{\abb{mso} tests on the observable configuration}. These informal definitions could be formalized in a way explained for \abb{mso} head tests before Lemma~\ref{lem:sites}.
We now show that the \abb{vi-pta} and \abb{vi-ptt} can perform \abb{mso} tests on the visible configuration. Note that for a regular site $T$ over $\Sigma \times 2^C$, $\operatorname{mark}(T)$ is a regular tree language over $\Sigma \times 2^C \times \{0,1\}$.
\begin{lemma}\label{lem:visiblesites} For each $k\ge 0$, the \abb{v$_k$i-pta} and \abb{v$_k$i-}{\em d}\abb{pta} can perform \abb{mso} tests on the visible configuration. The same holds for the \abb{v$_k$i-ptt} and \abb{v$_k$i-}{\em d}\abb{ptt}. \end{lemma}
\begin{proof} As in the proof of Lemma~\ref{lem:sites}, let ${\cal A}_T$ be a deterministic bottom-up finite-state tree automaton recognizing the regular tree language $\operatorname{mark}(T)$ over $\Sigma \times 2^C \times \{0,1\}$, representing the site $T$, coloured trees with a single marked node. As observed in the first paragraph of that proof the \abb{i-ptt} ${\cal A}'_T$ (of that proof) can be turned into a subroutine for any \abb{v$_k$i-pta} or \abb{v$_k$i-ptt} ${\cal A}$ with visible colour set~$C_\mathrm{v}$ by replacing each rule $\tup{q,\sigma,j,b} \to \tup{q',\alpha}$ of ${\cal A}'_T$ (except $\rho'_1$) by all possible rules $\tup{(\tilde{q},q),\sigma,j,b\cup b'} \to \tup{(\tilde{q},q'),\alpha}$ with $b'\subseteq C_\mathrm{v}$. This subroutine can easily be turned into one that tests whether or not $(\xp{vis}(t,\pi),h) \in T$ as follows. For the rules corresponding in this way to $\rho_3,\rho_4,\rho_5$ (in the proof of Lemma~\ref{lem:regular}), change $p=\delta(\sigma,p_1,\dots,p_m)$ into $p=\delta((\sigma,b',0),p_1,\dots,p_m)$. Similarly, for $\rho_3^\mu,\rho_4^\mu,\rho_5^\mu$ change $p=\delta((\sigma,1),p_1,\dots,p_m)$ into $p=\delta((\sigma,b',1),p_1,\dots,p_m)$ and, again, $p=\delta(\sigma,p_1,\dots,p_m)$ into $p=\delta((\sigma,b',0),p_1,\dots,p_m)$. \end{proof}
We now turn to the \abb{pta} as a navigational device: the \emph{trip $T({\cal A})$ computed by a \abb{pta} ${\cal A}$} consists of all triples $(t,u,v)$ such that ${\cal A}$, on input tree $t$, started at node $u$ in an initial state without pebbles on the tree, walks to node $v$, and halts in a final state (possibly leaving pebbles on the tree). Formally, $T({\cal A})=\{(t,u,v)\in T_\Sigma\times \nod{t}\times \nod{t} \mid \exists\, q_0\in Q_0, q_\infty\in F, \pi\in (N(t)\times C)^*: \tup{q_0,u,\varepsilon} \Rightarrow_{\cal A}^* \tup{q_\infty,v,\pi}\}$. Two \abb{PTA}'s ${\cal A}$ and ${\cal B}$ are \emph{trip-equivalent} if $T({\cal A})=T({\cal B})$. Since clearly $L({\cal A})=\{t\in T_\Sigma\mid \exists\, u\in\nod{t}: (t,\mathrm{root}_t,u)\in T({\cal A})\}$, trip-equivalence implies (language-)equivalence. A~trip $T$ is \emph{functional} if, for every $t$, $\{ (u,v) \mid (t,u,v)\in T \}$ is a function. Note that the trip computed by a deterministic \abb{pta} is functional.
It is straightforward to check that Lemma~\ref{lem:stacktests} also holds for the \abb{pta} as navigational device, replacing equivalence by trip-equivalence. Thus, \abb{v$_k$i-pta}'s can perform stack tests also when computing a trip. Similarly, they can perform the \abb{mso} tests discussed in Lemmas~\ref{lem:sites} and~\ref{lem:visiblesites}, and to be discussed in Theorem~\ref{thm:mso}.
In \cite[Theorem~8]{bloem} it is shown that every \abb{mso} definable trip (tree-node relation) can be computed by a \abb{ta}$^{\text{\abb{mso}}}$, i.e., a tree-walking automaton with \abb{mso} head tests (and vice versa). Moreover, by (the corrected version of) \cite[Theorem~9]{bloem}, if the trip is functional, then the automaton is deterministic. We will also use the fact that, according to the proof of \cite[Theorem~8]{bloem}, the \abb{mso} definable trips can be computed in a special way.
\begin{proposition}\label{prop:trips} Every \abb{mso} definable trip can be computed by a tree-walking automaton with \abb{mso} head tests that has the following two properties:
$(1)$ it never walks along the same edge twice (in either direction), and
$(2)$ it visits each node at most twice.
\noindent If the trip is functional, then the automaton is deterministic. \end{proposition}
The first property means that, when walking from a node $u$ to a node $v$, the automaton always takes the shortest (undirected) path from $u$ to~$v$, i.e., the path that leads from $u$ up to the least ancestor of $u$ and~$v$, and then down to $v$. The second property means that the automaton does not execute two consecutive stay-instructions.
The next result provides a characterization of the \abb{mso} definable trips by pebble automata that is more elegant than the one in \cite{trips}, which uses so-called marble/pebble automata, a restricted kind of \abb{v$_1$i-pta} (marbles are invisible pebbles only dropped on the path from the root to the current position of the head; a single visible pebble may only be dropped and picked up on a tree without marbles).
\begin{theorem}\label{thm:trips} For each $k\ge 0$, the trips computed by
\abb{v$_k$i-pta}'s are exactly the \abb{mso} definable trips. Similarly for \abb{V$_k$I-}{\em d}\abb{PTA}'s and functional trips. \end{theorem}
\begin{proof} Consider a trip $T$ computed by \abb{v$_k$i-pta} ${\cal A}$. Thus, for any $(t,u,v)$ in~$T$, starting at node $u$ of input tree $t$, ${\cal A}$ walks to node $v$ and halts. Then $\operatorname{mark}(T)$ can be recognized by another \abb{v$_k$i-pta} as follows. First it searches (deterministically) for the marked starting node $u$, then it simulates ${\cal A}$, and when ${\cal A}$ halts in a final state, verifies that the marked node $v$ is reached. By Theorem~\ref{thm:regt} this tree language is regular and hence $T$ is \abb{mso} definable.
By Proposition~\ref{prop:trips} every \abb{mso} definable trip can be computed by a tree-walking automaton ${\cal B}$ with \abb{mso} head tests. Since (as observed above) Lemma~\ref{lem:sites} also holds for the \abb{pta} as a navigational device, it can therefore be computed by an \abb{I-PTA} ${\cal B}'$. Moreover, if the trip is functional, then the automata ${\cal B}$ and ${\cal B}'$ are deterministic. \end{proof}
Note that the automaton ${\cal B}'$ in the above proof always removes all its pebbles before halting. Thus, that requirement could be added to the definition of the trip computed by a \abb{v$_k$i-pta} (implying that not every \abb{v$_k$i-pta} computes a trip). This conforms to the idea that one should not leave garbage after a picknick.
Using the above result, or rather Proposition~\ref{prop:trips}, we are now able to show that the \abb{pta} and \abb{ptt} can perform \abb{MSO} tests on the observable configuration, i.e., they can evaluate \abb{mso} formulas $\varphi(x)$ on the observable configuration tree $\xp{obs}(t,\pi)$ with the variable $x$ assigned to the position of the reading head.
\begin{theorem}\label{thm:mso} For each $k \ge 0$, the \abb{v$_k$i-pta} and \abb{v$_k$i-}{\em d}\abb{pta} can perform \abb{mso} tests on the observable configuration. The same holds for the \abb{v$_k$i-ptt} and \abb{v$_k$i-}{\em d}\abb{ptt}. \end{theorem}
\begin{proof} Let $T$ be a regular site over $\Sigma\times 2^C$, and let ${\cal A}$ be a \abb{v$_k$i-pta} that uses $T$ as a test to find out whether or not $(\xp{obs}(t,\pi),h)\in T$. Our aim is to construct a trip-equivalent \abb{v$_k$i-pta} ${\cal A}'$ that does not use \abb{mso} tests on the observable configuration. The proof is exactly the same for the case where ${\cal A}$ and ${\cal A}'$ are \abb{v$_k$i-ptt} (with equivalence instead of trip-equivalence).
Essentially, ${\cal A}'$ simulates ${\cal A}$. When ${\cal A}$ uses the test $T$, there are two cases. In the first case, either the pebble stack of ${\cal A}$ is empty or the colour of the topmost pebble of ${\cal A}$ is visible. Then the observable configuration equals the visible configuration, and so ${\cal A}'$ can use the test $T$ too, by Lemma~\ref{lem:visiblesites}. The remaining, difficult case is that the colour $d$ of the topmost pebble of ${\cal A}$ is invisible. To implement the test $T$ in this case it seems that ${\cal A}'$ cannot use any additional invisible pebbles (as in the proof of Lemma~\ref{lem:visiblesites}), because they make pebble $d$ unobservable. However, this is not a problem as long as the additional pebbles carry sufficient information about the position $u$ of pebble~$d$. The solution is to view $T$ as a trip from $u$ to $h$ (the position of the head), and to keep track of an automaton ${\cal B}_d$ that computes that trip. Although ${\cal B}_d$ is nondeterministic, it is straightforward for ${\cal A}'$ to employ the usual subset construction for finite-state automata.
For every $d\in C_\mathrm{i}$, let $T_d$ be the trip over $\Sigma\times 2^C$ defined by $T_d = \{(s,u,h)\mid (s',h)\in T\}$, where $s'$ is obtained from $s$ by changing the label $(\sigma,b)$ of $u$ into $(\sigma,b\cup\{d\})$. Then $(\xp{obs}(t,\pi),h)\in T$ if and only if $(\xp{vis}(t,\pi),u,h)\in T_d$, if $(d,u)$ is the topmost element of $\pi$. It should be clear from the regularity of $T$ that $T_d$ is a regular trip. Hence, by Proposition~\ref{prop:trips}, there is a \abb{ta} with \abb{mso} head tests ${\cal B}_d$ that computes $T_d$ and that has the special properties mentioned there. Therefore (see the paragraph after Proposition~\ref{prop:trips}), to keep track of the possible computations of ${\cal B}_d$, the automaton ${\cal A}'$ uses additional invisible pebbles to cover the shortest (undirected) path from $u$ to $h$. These pebbles will be called \emph{beads} to distinguish them from ${\cal A}$'s original pebbles. Each bead carries state information on computations of ${\cal B}_d$ that start at position $u$ (in an initial state) and end at position $h$. More precisely, each bead is a triple $(S,\delta,d)$ where $S$ is a set of states of ${\cal B}_d$ and $\delta\in \{{\rm up},{\rm stay}\}\cup\{{\rm down}_i\mid i\in[1,{\mathit mx}_\Sigma]\}$. There is one such bead $(S,\delta,d)$ on every node $v$ on the path from $u$ to $h$ (including $u$ and $h$) where $S$ is the set of states $p$ of ${\cal B}_d$ such that ${\cal B}_d$ has a computation on $\xp{vis}(t,\pi)$ starting at $u$ in an initial state and ending at $v$ in state $p$. Moreover, $\delta$ indicates the node $w$ just before $v$ on the path, which is the parent or $i$-th child of $v$ if $\delta$ is ${\rm up}$ or ${\rm down}_i$, respectively, and which is nonexistent when $v=u$, if $\delta={\rm stay}$. The bead at $v$ is on top of the bead at $w$ in the pebble stack of ${\cal A}'$. Thus, the bead at $h$ is always on the top of the stack of ${\cal A}'$ and hence is always observable.
The automaton ${\cal A}'$ can still simulate ${\cal A}$ because if the bead $(S,\delta,d)$ is at head position $h$, then the invisible pebble $d$ is observable at $h$ by ${\cal A}$ if and only if $\delta={\rm stay}$. If ${\cal A}$ lifts $d$, then ${\cal A}'$ lifts both $(S,{\rm stay},d)$ and $d$. If ${\cal A}$ drops another pebble $d'$ at $h$, then so does ${\cal A}'$ (and starts a new chain of beads on top of that pebble if $d'$ is invisible). When pebble $d'$ is lifted again, the beads for pebble $d$ are still available and can be used as before.
Now, suppose that ${\cal A}$ uses the test $T$ at position $h$. If ${\cal A}'$ does not see a bead at position $h$, then it uses $T$ as a test on the visible configuration. If ${\cal A}'$ sees a bead $(S,\delta,d)$ at $h$, then ${\cal A}'$ just checks whether or not $S$ contains a final state of ${\cal B}_d$, i.e., whether or not $(\xp{vis}(t,\pi),u,h)\in T_d$.
It remains to show how ${\cal A}'$ computes the beads. The path of beads is initialized by ${\cal A}'$ when ${\cal A}$ drops invisible pebble $d$. Then ${\cal A}'$ also drops pebble $d$, computes the relevant set $S$ of states of ${\cal B}_d$, and drops bead $(S,{\rm stay},d)$. The set~$S$ contains all initial states of ${\cal B}_d$, plus all states that ${\cal B}_d$ can reach from an initial state by applying one relevant rule with a stay-instruction (cf. the second property in Proposition~\ref{prop:trips}). To find the latter states, ${\cal A}'$ just simulates all those rules. Note that the \abb{mso} head tests of ${\cal B}_d$ on $\xp{vis}(t,\pi)$ are \abb{mso} tests on the visible configuration of ${\cal A}'$. That is because during the simulation of ${\cal A}$ by ${\cal A}'$ the visible configuration $\xp{vis}(t,\pi')$ of ${\cal A}'$ equals the visible configuration $\xp{vis}(t,\pi)$ of ${\cal A}$: the pebble stack $\pi$ of ${\cal A}$ is obtained from the corresponding pebble stack~$\pi'$ of ${\cal A}'$ by removing all (invisible) beads.
The path of beads is updated as follows. If we backtrack on the path from~$u$ to $h$, i.e., the current bead is $(S,\delta,d)$ with $\delta\neq{\rm stay}$ and we move in the direction~$\delta$, we just lift the current bead before moving. If we move away from~$u$, we must compute new bead information. Suppose the current bead on $h$ is $(S,{\rm up},d)$ and we move down to the $i$-th child $hi$ of $h$. Then the bead at $hi$ is $(S',{\rm up},d)$ where $S'$ can be computed in a similar way as the set $S$ above: ${\cal A}'$~simulates all computations of ${\cal B}_d$ that start at $h$ in a state of $S$ and end at $hi$ (and note that such a computation consists of one step, possibly followed by another step with a stay-instruction). Now suppose that the current bead is $(S,{\rm down}_i,d)$, which means that $u$ is a descendant of~$h$. If we move up to the parent $v$ of $h$, then the new bead is $(S',{\rm down}_j,d)$ where $j$ is the child number of~$h$. If we move down to a child $v$ of $h$ with child number $\neq i$, then the new bead is $(S',{\rm up},d)$. In each of these cases $S'$ can be computed as before, by simulating the computations of~${\cal B}_d$ from $h$ to $v$.
In general, ${\cal A}$ can of course use several regular sites $T_1,\dots,T_n$ as tests on the observable configuration. It should be obvious how to extend the proof to handle that. The beads are then of the form $(S_1,\dots,S_n,\delta,d)$ where $S_i$ is a set of states of a \abb{ta} with \abb{mso} head tests ${\cal B}_{id}$ that computes the trip $T_{id}$. To test~$T_i$ in the presence of such a bead, ${\cal A}'$ just checks whether or not $S_i$ contains a final state of ${\cal B}_{id}$. \end{proof}
\section{The Power of the I-PTT}\label{sec:poweriptt}
In this section we discuss some applications of the fact that the \abb{i-ptt} can perform \abb{mso} head tests (Lemma~\ref{lem:sites}). We prove that it can simulate the composition of two \abb{tt}'s of which the first is deterministic (cf. Lemma~\ref{lem:nul-decomp}), and that it can simulate the bottom-up tree transducer.
\smallpar{Composition of TT's} We now prove that the inclusions of Lemma~\ref{lem:nul-decomp} also hold in the other direction, provided that we start with a deterministic \abb{tt}.
\begin{theorem}\label{thm:composition} $\family{dTT} \circ \family{dTT} \subseteq \family{I-dPTT}$ and $\family{dTT} \circ \family{TT} \subseteq \family{I-PTT}$. \end{theorem}
\begin{proof} Consider two deterministic \abb{TT}'s ${\cal M}_1$ and ${\cal M}_2$. Assume that input tree~$t$ is translated into tree $s$ by transducer ${\cal M}_1$. We will simulate the computation of~${\cal M}_2$ on $s$ directly on $t$ using a \abb{pTT} ${\cal M}$ with invisible pebbles. Any action taken by ${\cal M}_2$ on node $v$ of tree $s$ will be simulated by ${\cal M}$ on the node $u$ of~$t$ that was the position of ${\cal M}_1$ when it generated $v$. This means that if ${\cal M}_2$ moves down in the tree $s$ to one of the children of $v$, the computation of ${\cal M}_1$ is simulated until it generates that child. On the other hand, if ${\cal M}_2$ moves up in the tree $s$ to the parent of $v$, it is necessary to backtrack on the computation of ${\cal M}_1$, back to the moment that that parent was generated. In this way, tree $s$ is never fully reconstructed as a whole, but at every moment ${\cal M}$ has access to a single node of $s$. The necessary node, the current node of ${\cal M}_2$, is continuously updated by moving back and forth along the computation of ${\cal M}_1$ on $t$.
Moving forward on the computation of ${\cal M}_1$ is straightforward. To be able to retrace, ${\cal M}$ uses its pebbles to record the output-generating steps of the computation of ${\cal M}_1$ on $t$. Each output rule of ${\cal M}_1$ is represented by a pebble colour, and is put on the node $u$ of $t$ where it was applied. The pebble colour also codes the child number of the generated node $v$ in $s$. Thus the pebble stack represents a (shortest) path in $s$ from the root to $v$. For each node on that path the stack contains a pebble with the rule of ${\cal M}_1$ used to generate that node and with its child number, from bottom to top.
Note that the determinism of ${\cal M}_1$ is an essential ingredient for this construction. Simulating ${\cal M}_2$, walking along the virtual tree $s$, one has to ensure that each time a node $v$ is revisited, the same rule of ${\cal M}_1$ is applied to $u$.
The above intuitive description assumes that the input tree $t$ is in the domain $L({\cal M}_1)$ of ${\cal M}_1$. In fact, it suffices to construct an \abb{i-ptt} ${\cal M}$ such that $\tau_{\cal M}(t)=\tau_{{\cal M}_2}(\tau_{{\cal M}_1}(t))$ for every such $t$, because ${\cal M}$ can then easily be adapted to start with an \abb{mso} head test verifying that the input tree is in $L({\cal M}_1)$, which is regular by Corollary~\ref{cor:domptt}.
Let us now give the formal definitions. Let ${\cal M}_1 = (\Sigma, \Delta, P, \{p_0\}, R_1)$ be a deterministic \abb{tt} and let ${\cal M}_2 = (\Delta, \Gamma, Q, Q_0, R_2)$ be an arbitrary \abb{tt}. To define the \abb{i-ptt} ${\cal M}$ it is convenient to extend the definition of an \abb{i-ptt} with a new type of instruction: we allow the right-hand side of a rule to be of the form $\tup{q',{\rm to\text{-}top}}$, which when applied to a configuration $\tup{q,u,\pi}$ leads to the next configuration $\tup{q',v,\pi}$ where $v$ is the node in the topmost element of $\pi$. Obviously this does not extend the expressive power of the \abb{i-ptt}: it is straightforward to write a subroutine that searches for the (unique observable) pebble on the tree, by first walking to the root and then executing a depth-first search of the tree until a pebble is observed.
The \abb{i-ptt} ${\cal M}$ has input alphabet $\Sigma$ and output alphabet $\Gamma$. Its set $C_\mathrm{i}$ of pebble colours consists of all pairs $(\rho,i)$ where $\rho$ is an output rule of ${\cal M}_1$, i.e., a rule of the form $\tup{p,\sigma,j}\to \delta(\tup{p_1,{\rm stay}},\dots,\tup{p_m,{\rm stay}})$ with $p,p_1,\dots,p_m\in P$, and $i$ is a child number of $\Delta$, i.e., $i\in[0,{\mathit mx}_\Delta]$. The set of states of ${\cal M}$ is defined to be $Q\cup (P\times[0,{\mathit mx}_\Delta]\times Q)$ and the set of initial states is $\{p_0\}\times \{0\}\times Q_0$. A state $q\in Q$ is used by ${\cal M}$ when simulating a computation step of ${\cal M}_2$, and a state $(p,i,q)$ is used by ${\cal M}$ when simulating the computation of ${\cal M}_1$ that generates the $i$-th child of the current node of ${\cal M}_2$ (keeping the state $q$ of ${\cal M}_2$ in memory). Initially, ${\cal M}$ simulates ${\cal M}_1$ in order to generate the root of its output tree. The rules of ${\cal M}$ are defined as follows.
First we define the rules that simulate ${\cal M}_1$. Let $\rho: \tup{p,\sigma,j}\to\zeta$ be a rule in~$R_1$. If $\zeta=\tup{p',\alpha}$ and $\alpha$ is a move instruction, then ${\cal M}$ has the rules $\tup{(p,i,q),\sigma,j,b}\to \tup{(p',i,q),\alpha}$ for every $i\in[0,{\mathit mx}_\Delta]$, $q\in Q$, and $b\subseteq C_\mathrm{i}$ with $\#(b)\leq 1$. If $\rho$ is an output rule with $\zeta=\delta(\tup{p_1,{\rm stay}},\dots,\tup{p_m,{\rm stay}})$, then ${\cal M}$ has the rules $\tup{(p,i,q),\sigma,j,b}\to \tup{q,{\rm drop}_{(\rho,i)}}$ for every $i$, $q$, $b$ as above. Thus, ${\cal M}$ simulates ${\cal M}_1$ until ${\cal M}_1$ generates an output node, drops the corresponding pebble, and continues simulating ${\cal M}_2$.
Second we define the rules that simulate ${\cal M}_2$. Let $\tup{q,\delta,i}\to \zeta$ be a rule in~$R_2$ and let $\rho: \tup{p,\sigma,j}\to \delta(\tup{p_1,{\rm stay}},\dots,\tup{p_m,{\rm stay}})$ be an output rule in~$R_1$ (with the same $\delta$). Then ${\cal M}$ has the rule $\tup{q,\sigma,j,\{(\rho,i)\}}\to \zeta'$ where $\zeta'$ is defined as follows. If $\zeta=\tup{q',{\rm down}_\ell}$, then $\zeta'= \tup{(p_\ell,\ell,q'),{\rm stay}}$. If $\zeta=\tup{q',{\rm up}}$, then $\zeta'= \tup{q',{\rm lift}_{(\rho,i)};{\rm to\text{-}top}}$. Otherwise, $\zeta'=\zeta$. Thus, ${\cal M}$ simulates every output rule or stay rule of ${\cal M}_2$ without changing its current node and current pebble stack, because the current node of ${\cal M}_2$ stays the same. To simulate a ${\rm down}_\ell$-instruction of ${\cal M}_2$, ${\cal M}$ starts simulating ${\cal M}_1$ in state $p_\ell$ with the child number $\ell$ of the next node of~${\cal M}_2$. Finally, ${\cal M}$ simulates an ${\rm up}$-instruction of~${\cal M}_2$ by lifting its topmost pebble and walking to the new topmost pebble, where it continues the simulation of~${\cal M}_2$. \end{proof}
Taking Theorem~\ref{thm:composition} and Lemma~\ref{lem:nul-decomp} together, we obtain that $\family{dTT} \circ \family{dTT} \subseteq \family{I-dPTT} \subseteq \family{I-PTT} \subseteq \family{TT} \circ \family{TT}$. It is open whether or not the first and last inclusions are proper. A way to express $\family{I-dPTT}$ and $\family{I-PTT}$ in terms of tree-walking tree transducers (without pebbles) would be to allow those transducers to have infinite input and output trees. Let us denote by $\family{dTT}^\infty$ the class of transductions realized by deterministic \abb{tt}'s that have finite input trees but can output infinite trees. As a particular example, the \abb{tt} ${\cal N}$ in the proof of Lemma~\ref{lem:nul-decomp} can be turned into such a deterministic \abb{tt} ${\cal N}^\infty$ by removing all rules $\langle f,\sigma,j\rangle \to \bot$. This ${\cal N}^\infty$ preprocesses every input tree $t$ into a unique ``tree of trees'' $t_\infty$ consisting of top level $t$ and infinitely many levels of copies $\hat{t}_u$ of $t$. Moreover, let us denote by ${}^\infty\family{TT}$ the class of transductions realized by \abb{tt}'s that output finite trees but can walk on infinite input trees, and similarly for ${}^\infty\family{dTT}$. It should be clear that the \abb{tt} ${\cal M}'$ in the proof of Lemma~\ref{lem:nul-decomp} can also be viewed as working on input tree $t_\infty$ rather than a nondeterministically generated $t'$ (and thus never aborts its simulation of ${\cal M}$). It should also be clear that the proof of Theorem~\ref{thm:composition} still works when ${\cal M}_1$ produces an infinite output tree as input tree for ${\cal M}_2$.\footnote{To see that $L({\cal M}_1)$ is regular, construct an ordinary nondeterministic \abb{tt} ${\cal N}$ by adding to ${\cal M}_1$ all rules $\tup{q,\sigma,j}\to \bot$ such that ${\cal M}_1$ has no rule with left-hand side $\tup{q,\sigma,j}$, and all rules $\tup{q,\sigma,j}\to \top$ such that ${\cal M}_1$ has a rule with that left-hand side (where $\bot$ and $\top$ are new output symbols of rank~0). Then $L({\cal M}_1)$ is the complement of $\tau_{{\cal N}}^{-1}(R)$ where $R$ is the set of output trees of ${\cal N}$ with an occurrence of $\bot$. Now use Proposition~\ref{prop:invtypeinf}(1). } Taking these results together, we obtain that $\family{I-dPTT} = \family{dTT}^\infty \circ {}^\infty\family{dTT}$ and $\family{I-PTT} = \family{dTT}^\infty \circ {}^\infty\family{TT}$. The formal definitions are left to the reader. Other characterizations of $\family{I-dPTT}$ will be shown in Section~\ref{sec:variations} (Theorem~\ref{thm:charidptt}), where we also show that $\family{I-dPTT} \subseteq \family{dTT}^3$ (Corollary~\ref{cor:idptt-tt3}).
\smallpar{Bottom-up tree transducers} The classical top-down and bottom-up tree transducers are compared to the \abb{v-ptt} at the end of~\cite[Section~3.1]{MilSucVia03}. Obviously, \abb{tt}'s generalize top-down tree transducers. In fact, the latter correspond to \abb{tt}'s that do not use the move instructions ${\rm up}$ and ${\rm stay}$. Moreover, the classical top-down tree transducers with regular look-ahead can be simulated by \abb{tt}'s with \abb{mso} head tests, and hence by \abb{i-ptt}'s. In general, bottom-up tree transducers cannot be simulated by \abb{v-ptt}'s, because otherwise every regular tree language could be accepted by a \abb{v-pta} (see below for the details), which is false as proved in~\cite{expressive}. We will show that every bottom-up tree transducer can be simulated by an \abb{i-ptt}. This will not be used in the following sections.
A \emph{bottom-up tree transducer} is a tuple ${\cal M}=(\Sigma,\Delta,P,F,R)$ where $\Sigma$ and $\Delta$ are ranked alphabets, $P$ is a finite set of states with a subset $F$ of final states, and $R$ is a finite set of rules of the form $\sigma(p_1(x_1),\dots,p_m(x_m))\to p(\zeta)$ such that $m\in{\mathbb N}$, $\sigma\in\Sigma^{(m)}$, $p_1,\dots,p_m,p\in P$ and $\zeta\in T_\Delta(\{x_1,\dots,x_m\})$. For $p\in P$, the sets $\tau_p\subseteq T_\Sigma\times T_\Delta$ are defined inductively as follows: the pair $(\sigma(t_1,\dots,t_m),s)$ is in~$\tau_p$ if there is a rule as above and there are pairs $(t_i,s_i)\in \tau_{p_i}$ for all $i\in[1,m]$ such that $s=\zeta[s_1,\dots,s_m]$, which is the result of substituting $s_i$ for every occurrence of $x_i$ in $\zeta$. The transduction $\tau_{\cal M}$ realized by ${\cal M}$ is the union of all~$\tau_p$ with $p\in F$. The transducer ${\cal M}$ is deterministic if it does not have two rules with the same left-hand side. For more information see, e.g., \cite[Chapter~IV]{GecSte}.
For every regular tree language $L$ there is a deterministic bottom-up finite-state tree automaton ${\cal A}=(\Sigma,P,F,\delta)$ (see the proof of Lemma~\ref{lem:regular}) that recognizes $L$ and hence there is a deterministic bottom-up tree transducer ${\cal M}$ that realizes the transduction $\tau_L=\{(t,1)\mid t\in L\}\cup \{(t,0)\mid t\notin L\}$. In fact, ${\cal M}=(\Sigma,\{0,1\},P,F,R)$ where $0$ and $1$ have rank~0 and $R$ is the set of all rules $\sigma(p_1(x_1),\dots,p_m(x_m))\to p(i)$ such that $\delta(\sigma,p_1,\dots,p_m)=p$ and $i = 1$ if $p\in F$, $i=0$ otherwise. A \abb{v-ptt} that computes $\tau_L$ can be turned into a \abb{v-pta} that accepts $L$ by removing every output rule $\tup{q,\sigma,j,b}\to 0$ and changing every output rule $\tup{q,\sigma,j,b}\to 1$ into $\tup{q,\sigma,j,b}\to \tup{q_\mathrm{fin},{\rm stay}}$ where $q_\mathrm{fin}$ is the final state.
Let $\family{B}$ ($\family{dB}$) denote the class of transductions realized by (deterministic) bottom-up tree transducers.
\begin{theorem}\label{thm:bottomup} $\family{B}\subseteq \family{I-PTT}$ and $\family{dB}\subseteq \family{I-dPTT}$. \end{theorem}
\begin{proof} Let ${\cal M}=(\Sigma,\Delta,P,F,R)$ be a bottom-up tree transducer. Intuitively, for a given input tree $t$, the transducer ${\cal M}$ visits each node $u$ of $t$ exactly once. It arrives at the children of $u$ in certain states $p_1,\dots,p_m$ with certain output trees $s_1,\dots,s_m$, and applies a rule $\sigma(p_1(x_1),\dots,p_m(x_m))\to p(\zeta)$ where $\sigma$ is the label of $u$. Thus, it arrives at $u$ in state $p$ with output $\zeta[s_1,\dots,s_m]$.
We construct an \abb{i-ptt} ${\cal M}'$ with \abb{mso} head tests such that $\tau_{{\cal M}'} =\tau_{\cal M}$ (see Lemma~\ref{lem:sites}). The transducer ${\cal M}'$ uses the rules of ${\cal M}$ as pebble colours. The behaviour of ${\cal M}'$ on a given input tree $t$ is divided into two phases. In the first phase ${\cal M}'$ walks through $t$ and (nondeterministically) drops one pebble $c$ on each node $u$ of $t$, in post-order. The input symbol $\sigma$ in the left-hand side of rule~$c$ must be the label of $u$. Intuitively, $c$ is the rule $\sigma(p_1(x_1),\dots,p_m(x_m))\to p(\zeta)$ applied by ${\cal M}$ at $u$ during a possible computation. When ${\cal M}$ drops $c$ on $u$ it uses \abb{mso} head tests to check that ${\cal M}$ has a computation on $t$ that arrives at the $i$-th child $ui$ of $u$ in state $p_i$, for every $i\in[1,m]$. This can be done because the state behaviour of ${\cal M}$ on $t$ is that of a bottom-up finite-state tree automaton. Thus, the tree language $L_p = \{t\in T_\Sigma\mid \exists\, s: (t,s)\in\tau_p\}$ is regular for every $p\in P$
and hence the site $T_i=\{(t,u)\mid t|_{ui} \in L_{p_i}\}$ is also regular, as can easily be seen. Note that if ${\cal M}$ is deterministic, then this first phase of ${\cal M}'$ is deterministic too, because ${\cal M}$ arrives at each node in a unique state (during a successful computation). In the second, deterministic phase ${\cal M}'$ moves top-down through $t$, checks that the states in the guessed rules are consistent, and computes the corresponding output. First ${\cal M}'$ checks for the pebble $c=\sigma(p_1(x_1),\dots,p_m(x_m))\to p(\zeta)$ at the root $u$, that the state $p$ is in~$F$. If so, it starts a process that is the same for every node $u$ of $t$. It lifts pebble $c$ and goes into state $[c,\zeta]$, in which it will output the $\Delta$-labeled nodes of $\zeta$, without leaving $u$. In state $q=[c,\delta(\zeta_1,\dots,\zeta_n)]$, it uses the output rules $\tup{q,\sigma,j,\varnothing}\to \delta(\tup{[c,\zeta_1],{\rm stay}},\dots,\tup{[c,\zeta_n],{\rm stay}})$.
When ${\cal M}'$ is in a state $[c,x_i]$, it calls a subroutine $S_i$. Subroutine $S_i$ walks through the subtrees $t|_{um},\dots,t|_{u(i+1)}$ of $t$, depth-first right-to-left, lifts the pebbles at all the nodes of those trees in reverse post-order (which is possible because the pebbles were dropped in post-order), and returns control to ${\cal M}'$, which continues by moving in state $c$ to child~$ui$ where it observes the pebble at $ui$ (again, because of the post-order dropping). Then ${\cal M}$ checks that the state in the right-hand side of that pebble is $p_i$, and repeats the above process for node $ui$ instead of $u$. It should be clear that in this way ${\cal M}'$ simulates the computations of ${\cal M}$, and so $\tau_{{\cal M}'} =\tau_{\cal M}$. Note that the bottom-up transducer ${\cal M}$ can disregard computed output, because in a rule as above it may be that $x_i$ does not occur in $\zeta$. In such a case ${\cal M}'$ clearly does not compute that output either, in the second phase, whereas it has checked in the first phase that ${\cal M}$ indeed has a computation that arrives in state $p_i$ at the $i$-th child. Note also that if $x_i$ occurs twice in $\zeta$, then ${\cal M}'$ simulates in the second phase twice the same computation of ${\cal M}$ on the $i$-th subtree (which was guessed in the first phase). \end{proof}
\section{Look-Ahead Tests}\label{sec:look-ahead}
The results on look-ahead in this section are only needed in the next section (and in a minor way in Section~\ref{sec:pft}). They also hold for the \abb{pta} as navigational device, computing a trip.
We say that a family ${\cal F}$ of \abb{PTA}'s (or \abb{PTT}'s) can \emph{perform look-ahead tests} if an automaton (or transducer) ${\cal A}$ in ${\cal F}$ can test whether or not a \abb{PTT} ${\cal B}$ (not necessarily in ${\cal F}$) has a successful computation when started in the current situation of ${\cal A}$ (i.e., position of the head and stack of pebbles). We require that $\Sigma^{\cal A} = \Sigma^{\cal B}$, $C_\mathrm{v}^{\cal A} \subseteq C_\mathrm{v}^{\cal B}$, $C_\mathrm{i}^{\cal A} \subseteq C_\mathrm{i}^{\cal B}$, and $k^{\cal A} \leq k^{\cal B}$ (where $\Sigma^{\cal A}$ is the input alphabet of ${\cal A}$, and similarly for the other notation). Since we are only interested in the existence of a successful computation, and not in its output tree, we are actually using alternating \abb{PTA}'s as look-ahead device (cf. Section~\ref{sec:autotrans}). In particular, we also allow a \abb{PTA} to be used as look-ahead ${\cal B}$, viewing it as a \abb{PTT} as in the proof of Theorem~\ref{thm:regt}.
In the formal definition of a \abb{PTA} or \abb{PTT} \emph{with look-ahead tests} (cf. the formal definition of \abb{mso} head tests before Lemma~\ref{lem:sites}), the rules are of the form $\tup{q,\sigma,j,b,{\cal B}} \to \zeta$ or $\tup{q,\sigma,j,b,\neg\,{\cal B}} \to \zeta$ which are relevant to a given configuration $\tup{q,h,\pi}$ of ${\cal A}$ on tree $t$ if the transducer ${\cal B}$ does or does not have a successful computation on $t$ that starts in the situation $\tup{h,\pi}$, i.e., if there do or do not exist $p_0\in Q_0^{{\cal B}}$ and $s\in T_{\Delta^{{\cal B}}}$ such that $\tup{p_0,h,\pi}\Rightarrow^*_{t,{\cal B}} s$ (where $\Delta^{{\cal B}}$ is the output alphabet of ${\cal B}$), or in the case of a \abb{pta} ${\cal B}$, if there do or do not exist $p_0\in Q_0^{{\cal B}}$, $p_f\in F^{{\cal B}}$, and $\tup{u,\pi}\in \xp{Sit}^{{\cal B}}(t)$ such that $\tup{p_0,\mathrm{root}_t,\varepsilon}\Rightarrow^*_{t,{\cal B}} \tup{p_f,u,\pi}$ (where $F^{{\cal B}}$ is the set of final states of ${\cal B}$).
\begin{theorem}\label{thm:look-ahead} For each $k \ge 0$, the \abb{v$_k$i-pta} and \abb{v$_k$i-}{\em d}\abb{pta} can perform look-ahead tests. The same holds for the \abb{v$_k$i-ptt} and \abb{v$_k$i-}{\em d}\abb{ptt}. \end{theorem}
\begin{proof} Let ${\cal A}$ be a \abb{v$_k$i-pta} that performs a look-ahead test by calling some \mbox{\abb{v$_m$i-ptt}} ${\cal B}$ (with $k\leq m$). We wish to construct a trip-equivalent \abb{v$_k$i-pta} ${\cal A}'$ that does not perform such look-ahead tests. By Lemma~\ref{lem:stacktests} we may construct ${\cal A}'$ as a \abb{pta} with stack tests, i.e., a \abb{pta} that can test whether its pebble stack is empty and if so, what the colour of the topmost pebble is.
As usual, ${\cal A}'$ simulates ${\cal A}$. Suppose that ${\cal A}$ uses the look-ahead test ${\cal B}$ in situation $\tup{h,\pi}$. When no pebbles are dropped, i.e., $\pi=\varepsilon$, the test whether~${\cal B}$, started in that situation, has a successful computation, is an \abb{mso} head test. Indeed, the site $T=\{(t,h)\mid \exists\, p_0\in Q_0^{{\cal B}}, s\in T_{\Delta^{{\cal B}}}: \tup{p_0,h,\varepsilon}\Rightarrow^*_{t,{\cal B}} s\}$ is regular, as $\operatorname{mark}(T)$ is the domain of the \abb{v$_m$i-ptt} ${\cal B}'$ that starts in the root, looks for the marked node $h$, and then simulates ${\cal B}$. Domains are regular by Corollary~\ref{cor:domptt}, and ${\cal A}'$ can perform \abb{mso} head tests by Lemma~\ref{lem:sites}.
In general, one may imagine that ${\cal A}'$ implements the look-ahead test by simulating ${\cal B}$. However, when ${\cal A}'$ is ready with the simulation of ${\cal B}$, that started with the stack $\pi$ of ${\cal A}$, ${\cal A}'$ must be able to recover $\pi$ to continue the simulation of~${\cal A}$. Note that ${\cal B}$ can inspect $\pi$, thereby possibly destroying part of $\pi$ and adding something else. For this reason the computations of ${\cal B}$ starting at the position of the topmost pebble of~$\pi$ will be precomputed. With each pebble dropped by ${\cal A}$, the automaton ${\cal A}'$ stores the set $S$ of states $p$ of ${\cal B}$ for which ${\cal B}$ has a successful computation when started in state $p$ at the position $u$ of the topmost stack element (and with the current stack of ${\cal A}$). Now a successful computation of ${\cal B}$ can be safely simulated, consisting of a part where the pebbles of ${\cal B}$ are on top of the stack $\pi$ inherited from ${\cal A}$, possibly followed by a precomputed part where ${\cal B}$ inspects $\pi$, starting with a visit to~$u$. We discuss how these state sets are determined, and how they are used (by ${\cal A}'$) to perform the look-ahead test. Rather then simulating~${\cal B}$, ${\cal A}'$ will use \abb{mso} tests on the observable configuration, which is possible by Theorem~\ref{thm:mso}. The colour sets of ${\cal A}'$ are $C'_\mathrm{v} = C_\mathrm{v} \times 2^{Q^{\cal B}}$ and $C'_\mathrm{i} = C_\mathrm{i} \times 2^{Q^{\cal B}}$.
If ${\cal A}$ drops the first pebble $c$ (i.e., $\pi=(h,c)$), then ${\cal A}'$ drops the pebble $(c,S)$ where it determines for every state $p$ of ${\cal B}$ whether or not $p\in S$ using an \abb{mso} head test: construct ${\cal B}'$ as above except that it now drops $c$ at the marked node~$h$ before simulating ${\cal B}$ in state $p$. Thus, this time, the domain of ${\cal B}'$ is $\operatorname{mark}(T)$ with $T=\{(t,h)\mid \exists\, s\in T_{\Delta^{{\cal B}}}: \tup{p,h,c}\Rightarrow^*_{t,{\cal B}} s\}$.
Suppose now that ${\cal A}$ uses the look-ahead test ${\cal B}$ when it is in situation $\tup{h,\pi}$ with $\pi\neq\varepsilon$, and suppose that the topmost pebble of $\pi$ has colour $d$ and that the set of visible pebble colours that occur in $\pi$ is $C_\mathrm{v}(\pi)=\{c_1,\dots,c_\ell\}\subseteq C_\mathrm{v}$, with $\ell\in[0,k]$. Then the colour of the topmost pebble of the stack $\pi'$ of ${\cal A}'$ is $(d,S)$ for some set $S$ of states of ${\cal B}$, and the set of visible pebble colours that occur in $\pi'$ is $C_\mathrm{v}(\pi')=\{(c_1,S_1),\dots,(c_\ell,S_\ell)\}$ for some $S_1,\dots,S_\ell$. Since ${\cal A}'$ can perform stack tests, it can determine $(d,S)$. Moreover, it should be clear that ${\cal A}'$ can determine $C_\mathrm{v}(\pi')$, and hence $C_\mathrm{v}(\pi)$, by an \abb{mso} test on the visible configuration. With this topmost colour $d$, this state information $S$, and this set $C_\mathrm{v}(\pi)$ of visible pebbles, the look-ahead test can be performed by ${\cal A}'$ as an \abb{mso} test on the observable configuration, as follows. Consider the observable configuration tree $\xp{obs}(t,\pi')$ with the current node $h$ marked, see Theorem~\ref{thm:mso}. We want to show that there is a regular site $T$ over $\Sigma \times 2^{C'}$ such that $(\xp{obs}(t,\pi'),h)\in T$ if and only if there exist $p_0\in Q_0^{{\cal B}}$ and $s\in T_{\Delta^{{\cal B}}}$ such that $\tup{p_0,h,\pi}\Rightarrow^*_{t,{\cal B}} s$. Indeed, $\operatorname{mark}(T)$ is the domain of a \abb{v$_{m'}$i-ptt} ${\cal B}'$, with $m'= m-\ell$. It first searches for the position $u$ of the topmost pebble, which is the unique node of $\xp{obs}(t,\pi')$ of which the label contains the colour $(d,S)$. It drops the special invisible pebble $\odot$ on $u$, and then proceeds to the marked node $h$, starts simulating ${\cal B}$ and halts successfully when it observes pebble $\odot$ at position $u$ with ${\cal B}$ in a state of $S$, or when it never has observed $\odot$ and ${\cal B}$ halts successfully (meaning that pebbles are still on top of $\odot$ when visiting $u$). Note that ${\cal B}'$ can simulate~${\cal B}$, which walks on $t$ with pebbles rather than on $\xp{obs}(t,\pi')$, because the colours in the labels of the nodes of $\xp{obs}(t,\pi')$ contain the observable pebbles on $t$ in the stack~$\pi$. Also, ${\cal B}'$ does not apply rules of ${\cal B}$ that contain a ${\rm drop}_{c_i}$-instruction with $c_i\in C_\mathrm{v}(\pi)$. The domain $\operatorname{mark}(T)$ of ${\cal B}'$ is regular and ${\cal A}'$ can perform the \abb{mso} test $T$ on its observable configuration.
The same reasoning shows that the state set for the next pebble $c$ dropped by ${\cal A}$ can be computed by \abb{mso} tests on the observable configuration: again ${\cal B}'$ first drops the pebble $c$ on $h$ before starting the simulation of ${\cal B}$ in any state $p$.
Finally it should be clear that if ${\cal A}$ uses the look-ahead tests ${\cal B}_1,\dots,{\cal B}_n$, then state information for every ${\cal B}_i$ should be stored in the pebbles, i.e., they are of the form $(c,S_1,\dots,S_n)$ where $S_i$ is a set of states of ${\cal B}_i$. \end{proof}
A natural question is now whether Theorem~\ref{thm:look-ahead} also holds for \abb{pta}'s and \abb{ptt}'s that are allowed to perform stack tests, \abb{mso} head tests, and \abb{mso} tests on the visible and observable configuration. The answer is yes.
Let us first consider the case of stack tests. Roughly speaking, if ${\cal A}$ uses look-ahead tests ${\cal B}_1,\dots,{\cal B}_n$, then we just apply the construction of Lemma~\ref{lem:stacktests} to both ${\cal A}$ and all ${\cal B}_i$, $i\in[1,n]$, and then apply Theorem~\ref{thm:look-ahead} to the resulting equivalent (ordinary) \abb{pta} ${\cal A}'$ that calls the (ordinary) \abb{ptt}'s ${\cal B}'_1,\dots,{\cal B}'_n$. It should be noted that even if ${\cal A}$ does \emph{not} use stack tests but some ${\cal B}_i$ \emph{does}, the construction of Lemma~\ref{lem:stacktests} must be applied to ${\cal A}$ too, because the stack that ${\cal B}_i$ inherits from~${\cal A}$ must contain the necessary additional information concerning the colours of previously dropped pebbles. Vice versa, if ${\cal A}$ (or another ${\cal B}_j$) uses stack tests but ${\cal B}_i$ does not, then ${\cal B}_i$ can just ignore the additional information in the stack of ${\cal A}$, but it is also correct to apply the construction of Lemma~\ref{lem:stacktests} to~${\cal B}_i$. However, not only the additional information in the stack should be passed from ${\cal A}'$ to ${\cal B}_1',\dots,{\cal B}_n'$, but also the additional information in the finite state of~${\cal A}'$. Thus, to be more precise, if ${\cal A}$ is in state $q$ and uses the look-ahead test ${\cal B}_i$, then whenever ${\cal A}'$ is in state $(q,\gamma)$, it should use the look-ahead test ${\cal B}'_i(\gamma)$ that is obtained from ${\cal B}'_i$ by changing its set $Q_0^{{\cal B}_i} \times \{\varepsilon\}$ of initial states into $Q_0^{{\cal B}_i} \times \{\gamma\}$.
For the case of \abb{mso} head tests and \abb{mso} tests on the visible configuration the proof is easier. The constructions of Lemmas~\ref{lem:sites} and~\ref{lem:visiblesites} can be applied to ${\cal A}$ and ${\cal B}_1,\dots,{\cal B}_n$ independently, depending on whether they use such tests or not. The reason is that these tests are implemented by subroutines for which the pebble stack need not be changed.
Finally, for the case of \abb{mso} tests on the observable configuration the construction of Theorem~\ref{thm:mso} is again applied simultaneously to all of ${\cal A}$ and ${\cal B}_1,\dots,{\cal B}_n$, with beads that take care of all the regular sites $T$ that are used by both ${\cal A}$ and ${\cal B}_1,\dots,{\cal B}_n$ as tests. That ensures that the beads of ${\cal A}'$ also contain the information needed by ${\cal B}'_1,\dots,{\cal B}'_n$. Note that in this case (as opposed to the case of stack tests above) ${\cal A}'$ does not carry any additional information in its finite state and thus, whenever ${\cal A}$ uses ${\cal B}_i$ as look-ahead test, ${\cal A}'$ can use ${\cal B}'_i$ as look-ahead test.
A similar natural question is whether Theorem~\ref{thm:look-ahead} also holds for \abb{pta}'s and \abb{ptt}'s that use look-ahead, in particular whether we can allow the look-ahead transducer to use another transducer as look-ahead test. The answer is again yes, with a similar solution. In fact it can be shown that the \abb{v$_k$i-pta} (and \abb{v$_k$i-ptt}) even can perform \emph{iterated} look-ahead tests, that is, they can use look-ahead tests that use look-ahead tests that use $\dots$ look-ahead tests.
Formally, we define for $n\geq 0$ the notion of a \abb{pta} or \abb{ptt} ${\cal A}$ \emph{of (look-ahead) depth} $n$, by induction on $n$. Simultaneously we define the finite sets $\xp{test}({\cal A})$ and $\xp{test}^*({\cal A})$ of \abb{ptt}'s, where $\xp{test}({\cal A})$ contains the look-ahead tests of ${\cal A}$, and $\xp{test}^*({\cal A})$ contains its iterated look-ahead tests plus ${\cal A}$ itself. For $n=0$, a \abb{pta} or \abb{ptt} ${\cal A}$ of depth~$0$ is just a \abb{pta} or \abb{ptt} (without look-ahead tests). Moreover, $\xp{test}({\cal A})=\varnothing$ and $\xp{test}^*({\cal A})=\{{\cal A}\}$. For $n\geq 0$, a \abb{pta} or \abb{ptt} ${\cal A}$ of depth~$n+1$ uses as look-ahead tests arbitrary \abb{ptt}'s of lower depth, i.e., it has rules $\tup{q,\sigma,j,b,{\cal B}} \to \zeta$ or $\tup{q,\sigma,j,b,\neg\,{\cal B}} \to \zeta$ where ${\cal B}$ is a \abb{ptt} of depth~$m\leq n$. Furthermore, $\xp{test}({\cal A})$ is the set of all \abb{ptt}'s of depth~$m\leq n$ that ${\cal A}$ uses as look-ahead tests, and $\xp{test}^*({\cal A})=\{{\cal A}\} \cup \bigcup_{{\cal B}\in \xp{test}({\cal A})}\xp{test}^*({\cal B})$. A \abb{pta} or \abb{ptt} \emph{with iterated look-ahead tests} is one of depth $n$, for some $n\in{\mathbb N}$. Note that a \abb{pta} (or \abb{ptt}) of depth~$1$ is the same as a \abb{pta} (or \abb{ptt}) with look-ahead tests. The definition of the semantics of a \abb{pta} or \abb{ptt} with iterated look-ahead tests is by induction on the depth $n$, and is entirely analogous to the one for the case $n=1$ as given in the beginning of this section.
\begin{theorem}\label{thm:iterated} For each $k \ge 0$, the \abb{v$_k$i-pta} and \abb{v$_k$i-}{\em d}\abb{pta} can perform iterated look-ahead tests. The same holds for the \abb{v$_k$i-ptt} and \abb{v$_k$i-}{\em d}\abb{ptt}. \end{theorem}
\begin{proof} We will show that for every \abb{v$_k$i-ptt} ${\cal C}$ of depth $n\geq 1$ we can construct an equivalent \abb{v$_k$i-ptt} ${\cal C}'$ of depth $n-1$. The result then follows by induction. Since the construction generalizes the one of Theorem~\ref{thm:look-ahead} (which is the case $n=1$), we will need all \abb{ptt}'s in $\xp{test}^*({\cal C}')$ to use stack tests and \abb{mso} tests on the observable configuration. Thus, for the induction to work, we first have to prove that every \abb{v$_\ell$i-ptt} of depth $m\geq 1$ can perform such tests. For the case $m=1$ we have already argued this after Theorem~\ref{thm:look-ahead}, and the general case can be proved in a similar way. Let ${\cal D}$ be a \abb{v$_\ell$i-ptt} of depth $m$ such that all ${\cal A}\in\xp{test}^*({\cal D})$ perform stack tests. We just apply the construction of Lemma~\ref{lem:stacktests} simultaneously to every \abb{ptt} ${\cal A}\in\xp{test}^*({\cal D})$, resulting in the \abb{ptt} ${\cal A}'$. Moreover, for all ${\cal A},{\cal B}\in \xp{test}^*({\cal D})$, if ${\cal A}$ is in state $q$ and uses look-ahead test ${\cal B}$, then whenever ${\cal A}'$ is in state $(q,\gamma)$, it uses look-ahead test ${\cal B}'(\gamma)$. Obviously, every ${\cal B}'(\gamma)$ is of the same depth as ${\cal B}$, and hence the resulting \abb{v$_\ell$i-ptt} ${\cal D}'$ is of the same depth $m$ as~${\cal D}$. For the \abb{mso} tests the argument is completely analogous to the argument for $m=1$ after Theorem~\ref{thm:look-ahead}, applying the appropriate constructions simultaneously to all \abb{ptt} ${\cal A}\in\xp{test}^*({\cal D})$.
Now let ${\cal C}$ be a \abb{v$_k$i-ptt} of depth $n\geq 1$ and let us construct an equivalent \abb{v$_k$i-ptt} ${\cal C}'$ of smaller depth. The argument is similar to those above. Let $P_0$ be the set of all ${\cal B}\in \xp{test}^*({\cal C})$ of depth~0, i.e., all \abb{ptt} without look-ahead tests, and let $P_1$ contain all ${\cal A}\in \xp{test}^*({\cal C})$ of depth~$\geq 1$. We now apply the construction of Theorem~\ref{thm:look-ahead} simultaneously to every \abb{ptt} ${\cal A}\in P_1$, resulting in a \abb{ptt} ${\cal A}'$ that stores state information of every ${\cal B}\in P_0$ in the pebbles. If ${\cal A}_1\in P_1$ uses look-ahead test ${\cal A}_2\in P_1$, then ${\cal A}'_1$ uses look-ahead test ${\cal A}'_2$. Note that if ${\cal A}\in P_1$ uses look-ahead test ${\cal B}\in P_0$, then ${\cal A}'$ uses an \abb{mso} test instead. Thus, clearly, the depth of every ${\cal A}'$ is one less than the depth of ${\cal A}$, and so the depth of the resulting \abb{v$_k$i-ptt} ${\cal C}'$ is $n-1$. Finally, we remove the stack tests and \abb{mso} tests from ${\cal C}'$ and its iterated look-ahead tests as explained above for ${\cal D}$. \end{proof}
Although this result does not seem practically useful, it will become important when we propose the query language Pebble XPath in the next section, as an extension of Regular XPath. Intuitively, Pebble XPath expressions are similar to \abb{i-pta} with iterated look-ahead tests. We note that \abb{ta} with iterated look-ahead tests are used in~\cite{CatSeg} to prove that Regular XPath is not \abb{mso} complete.
\section{Document Navigation}\label{sec:xpath}
We define \emph{Pebble XPath}, an extension of Regular XPath \cite{Mar05} with pebbles. Due to its potential application to navigation in XML documents, it works on (nonempty) forests rather than trees. We prove that the trips defined by the path expressions of Pebble XPath are exactly the \abb{mso} definable trips on forests.
Pebble XPath has path expressions (denoted $\alpha,\beta$) and node expressions (denoted $\varphi,\psi$). These expressions concern forests over an (unranked) alphabet~$\Sigma$ of node labels, or tags, that can be chosen arbitrarily. Since we are mainly interested in path expressions, we view the node expressions as auxiliary. A path expression describes a walk through a given nonempty forest $f$ over $\Sigma$ during which invisible coloured pebbles can be dropped on and lifted from the nodes of~$f$, in a nested (stack-like) manner. Such a walk steps through $f$ from node to node following both the vertical and horizontal edges in either direction. The context in which a path expression is evaluated (i.e., the situation at the start of the walk) is a pair $\tup{u,\pi}$ consisting of a node $u$ of $f$ and a stack $\pi$ of pebbles that lie on the nodes of $f$. Formally, a \emph{context}, or \emph{situation}, on a forest $f$ is an element of the set $\xp{Sit}(f) = \nod{f} \times (\nod{f}\times C)^*$, where $\nod{f}$ is the set of nodes of $f$ and $C$ is the finite set of colours of the pebbles (that can be chosen arbitrarily). The walk ends in another context. Thus, the semantics of a path expression is a binary relation on $\xp{Sit}(f)$. Similarly, the semantics of a node expression is a subset of $\xp{Sit}(f)$, i.e., a test on a given context. Note that the notion of a context on a forest is entirely similar to that of a situation on a ranked tree for an \abb{i-pta} with (invisible) colour set $C$.
For the syntax of Pebble XPath, we start with the basic path expressions, with $c\in C$: $$\alpha_0 ::= {\tt child} \mid {\tt parent} \mid {\tt right} \mid {\tt left} \mid {\tt drop}_c \mid {\tt lift}_c$$ The first four expressions operate on the context node only (in the usual way, moving to a child, the parent, the next sibling, and the previous sibling, respectively), whereas the last two also operate on the pebble stack (dropping/lifting a pebble of colour $c$ on/from the context node $u$, which is modeled by pushing/popping the pair $(u,c)$ on/off the stack). The syntax of path expressions is $$\alpha ::= \alpha_0 \mid \;?\varphi \mid \alpha \cup \beta \mid \alpha/\beta \mid \alpha^*$$ where $\beta$ is an alias of $\alpha$. The three last expressions show the usual regular operations on binary relations: union, composition, and transitive-reflexive closure. The expression $?\varphi$ denotes the identity relation on the set of contexts defined by the node expression $\varphi$, i.e., it filters the current context by requiring that $\varphi$ is true.
We now turn to the node expressions and start with the basic ones, with $\sigma\in\Sigma$: $$\varphi_0 ::= {\tt haslabel}_\sigma \mid {\tt isleaf} \mid {\tt isroot} \mid {\tt isfirst} \mid {\tt islast} \mid {\tt haspebble}_c$$ The first five expressions test whether the context node has label $\sigma$, whether it is a leaf, a root, the first among its siblings, or the last among its siblings. The last expression (which is the only one that also uses the pebble stack) tests whether the topmost pebble, i.e., the most recently dropped pebble, lies on the context node and has colour $c$. The syntax of node expressions is $$\varphi ::= \varphi_0 \mid \tup{\alpha} \mid \neg\varphi \mid \varphi\wedge\psi \mid \varphi\vee\psi$$ where $\psi$ is an alias of $\varphi$. The last three expressions show the usual boolean operations. The expression $\tup{\alpha}$ is like a predicate $[\alpha]$ in XPath 1.0, which filters the context by requiring the existence of at least one successful $\alpha$-walk starting from this context. In terms of tree-walking automata it is a look-ahead test. We will also consider the language \emph{Pebble CAT}, which is obtained from Pebble XPath by dropping the filter tests $\varphi ::= \langle\alpha\rangle$. The expressions of Pebble CAT are \emph{caterpillar expressions} extended with pebbles.
The formal semantics of Pebble XPath expressions is given in Tables~\ref{tab:sempath} and~\ref{tab:semnode}. For every nonempty forest $f$ over $\Sigma$, the semantics $\semf{\alpha}\subseteq \xp{Sit}(f)\times\xp{Sit}(f)$ and $\semf{\varphi}\subseteq \xp{Sit}(f)$ of path and node expressions are defined, where $u,u'$ vary over $\nod{f}$, $\pi,\pi'$ vary over $(\nod{f}\times C)^*$, and $p$ varies over $\nod{f}\times C$. Note that $\semf{ {\tt parent} }=\semf{{\tt child}}^{-1}$, $\semf{ {\tt left} }=\semf{ {\tt right} }^{-1}$, and $\semf{ {\tt lift}_c }=\semf{ {\tt drop}_c }^{-1}$. Note also that the set $\semf{ \tup{\alpha} }$ is the domain of the binary relation $\semf{ \alpha }$.
\begin{table} \[ \begin{array}{l@{\;}cl} \semf{{\tt child}} &=& \{(\tup{u,\pi},\tup{u',\pi})\mid u' \mbox{ is a child of }u\} \\ \semf{ {\tt parent} } &=& \{(\tup{u,\pi},\tup{u',\pi})\mid u' \mbox{ is the parent of }u\} \\ \semf{ {\tt right} } &=& \{(\tup{u,\pi},\tup{u',\pi})\mid
u' \mbox{ is the next sibling of }u\} \\ \semf{ {\tt left} } &=& \{(\tup{u,\pi},\tup{u',\pi})\mid
u' \mbox{ is the previous sibling of }u\} \\ \semf{ {\tt drop}_c } &=& \{(\tup{u,\pi},\tup{u,\pi p}) \mid p=(u,c)\} \\ \semf{ {\tt lift}_c } &=& \{(\tup{u,\pi p},\tup{u,\pi}) \mid p=(u,c)\} \\[1mm] \semf{ ?\varphi } &=& \{(\tup{u,\pi},\tup{u,\pi})\mid
\tup{u,\pi}\in \semf{ \varphi }\} \\ \semf{ \alpha\cup\beta } &=& \semf{ \alpha } \cup
\semf{ \beta } \\ \semf{ \alpha/\beta } &=& \semf{ \alpha } \circ \semf{ \beta } \\ \semf{ \alpha^* } &=& \semf{ \alpha }^* \end{array} \] \caption{Semantics of Pebble XPath path expressions} \label{tab:sempath} \end{table}
\begin{table} \[ \begin{array}{l@{\;}cl} \semf{ {\tt haslabel}_\sigma } &=& \{\tup{u,\pi}\mid u \mbox{ has label }\sigma\} \\ \semf{ {\tt isleaf} } &=& \{\tup{u,\pi}\mid u \mbox{ is a leaf}\} \\ \semf{ {\tt isroot} } &=& \{\tup{u,\pi}\mid u \mbox{ is a root}\} \\ \semf{ {\tt isfirst} } &=& \{\tup{u,\pi}\mid u \mbox{ is a first sibling}\} \\ \semf{ {\tt islast} } &=& \{\tup{u,\pi}\mid u \mbox{ is a last sibling}\} \\ \semf{ {\tt haspebble}_c } &=& \{\tup{u,\pi p}
\mid p=(u,c)\} \\[1mm] \semf{ \tup{\alpha} } &=& \{\tup{u,\pi}\mid
\exists \tup{u',\pi'}\colon
(\tup{u,\pi},\tup{u',\pi'})\in \semf{ \alpha }\} \\ \semf{ \neg\varphi } &=& \xp{Sit}(f)\setminus \semf{ \varphi } \\ \semf{ \varphi\wedge\psi } &=& \semf{ \varphi } \cap \semf{ \psi } \\ \semf{ \varphi\vee\psi } &=& \semf{ \varphi } \cup \semf{ \psi } \end{array} \] \caption{Semantics of Pebble XPath node expressions} \label{tab:semnode} \end{table}
The filtering XPath expression $\alpha[\beta]$ of XPath 1.0 can here be defined as $\alpha[\beta] = \alpha/?\langle\beta\rangle$. Also, the node expression ${\tt loop}(\alpha)$ from \cite{GorMar05,Cat06} can be defined as ${\tt loop}(\alpha)= \tup{{\tt drop}_c/\alpha/{\tt lift}_c}$ where $c$ is a colour not occurring in $\alpha$. Then $\semf{{\tt loop}(\alpha)}= \{\tup{u,\pi}\mid (\tup{u,\pi},\tup{u,\pi})\in \semf{\alpha}\}= \{\tup{u,\pi}\mid (\tup{u,\varepsilon},\tup{u,\varepsilon})\in \semf{\alpha}\}$, because $\alpha$ cannot inspect the stack $\pi$ and it must return to $u$ in order to lift pebble $c$.
Two path expressions $\alpha$ and $\beta$ are \emph{equivalent}, denoted by $\alpha\equiv\beta$, if $\semf{\alpha}=\semf{\beta}$ for every nonempty forest $f$ over $\Sigma$, and similarly for node expressions. Note that $?(\varphi\wedge\psi)\equiv \;?\varphi/?\psi$ and $?(\varphi\vee\psi)\equiv \;?\varphi\;\cup \;?\psi$. Hence, using De~Morgan's laws, the syntax for node expressions can be replaced by $\varphi::= \varphi_0 \mid \neg\varphi_0 \mid \langle\alpha\rangle \mid \neg\langle\alpha\rangle$ for Pebble XPath, and $\varphi::= \varphi_0 \mid \neg\varphi_0$ for Pebble CAT. Thus, keeping only the basic node expressions, we can always assume that the syntax for path expressions is $$\alpha ::= \alpha_0 \mid \;?\varphi_0 \mid \;?\neg\varphi_0 \mid \;?\tup{\beta} \mid \;?\neg\tup{\beta} \mid \alpha \cup \beta \mid \alpha/\beta \mid \alpha^*$$ for Pebble XPath, and hence $$\alpha ::= \alpha_0 \mid \;?\varphi_0 \mid \;?\neg\varphi_0 \mid \alpha \cup \beta \mid \alpha/\beta \mid \alpha^*$$ for Pebble CAT. In that case we will say that we assume the syntax to be in \emph{normal form}.
Note also that all basic node expressions except ${\tt haslabel}_\sigma$ are redundant, because ${\tt isleaf} \equiv \neg\langle{\tt child}\rangle$ (a node is a leaf if and only if it has no children), ${\tt isroot} \equiv \neg\langle{\tt parent}\rangle$, ${\tt isfirst} \equiv \neg\langle{\tt left}\rangle$, ${\tt islast} \equiv \neg\langle{\tt right}\rangle$, and ${\tt haspebble}_c \equiv \langle{\tt lift}_c\rangle$. However, these basic node expressions were kept in the syntax, because we also wish to consider the subset Pebble CAT in which there are no filter tests $\tup{\alpha}$. Note finally that when ${\tt drop}_c$, ${\tt lift}_c$, and ${\tt haspebble}_c$ are removed from Pebble XPath, the resulting formalism is exactly Regular XPath \cite{Mar05} (and in the semantics the stack can, of course, be disregarded).
The purpose of Pebble XPath is the same as that of XPath: to define trips, i.e., binary patterns. Recall from Section~\ref{sec:trees} that a trip $T$ over an unranked alphabet $\Sigma$ is a set $T\subseteq\{(f,u,v)\mid f\in F_\Sigma, u,v\in \nod{f}\}$ where $F_\Sigma$ is the set of forests over $\Sigma$. Note that $f$ is always a nonempty forest. For a path expression~$\alpha$ (based on $\Sigma$ and some $C$) we say that $\alpha$ \emph{defines the trip} $T(\alpha) = \{(f,u,v)\mid \exists\, \pi\in(N(f)\times C)^*: (\tup{u,\varepsilon},\tup{v,\pi})\in \semf{\alpha}\}$. We now define a trip~$T$ over $\Sigma$ to be \emph{definable in Pebble XPath} if there exists a Pebble XPath path expression $\alpha$ such that $T = T(\alpha)$. And similarly for Pebble CAT. The next theorem states that Pebble XPath and Pebble CAT have the same expressive power as \abb{MSO} logic on forests.
\begin{theorem}\label{thm:xpath} A trip is definable in Pebble XPath if and only if it is definable in Pebble CAT if and only if it is \abb{MSO} definable. \end{theorem}
As such our expressions have the desirable property of being a Core (and even Regular) XPath extension that is complete for \abb{MSO} definable binary patterns. Other such extensions were considered in \cite{GorMar05} (TMNF caterpillar expressions) and \cite{Cat06} ($\mu$Regular XPath). Pebble CAT is similar to PCAT of \cite{GorMar05} which has the same expressive power as the \abb{v-pta} (and thus less than \abb{mso} by \cite{expressive}). In PCAT the nesting of pebbles is defined syntactically rather than semantically.
The proof of Theorem~\ref{thm:xpath} is given in the remainder of this section. It should be clear that Pebble CAT is closely related to the \abb{i-pta}. In fact, we will show later that their relationship can be viewed as the classical equivalence of regular expressions and finite automata. The remainder of the proof is then directly based on the fact that the \abb{i-pta} has the same expressive power as \abb{mso} logic for defining trips on trees (Theorem~\ref{thm:trips}), and on the fact that the \abb{i-pta} can perform iterated look-ahead tests (Theorem~\ref{thm:iterated}). One technical problem is that these theorems are formulated for ranked trees rather than unranked forests. Thus we start by adapting Pebble XPath to ranked trees and showing that it suffices to prove Theorem~\ref{thm:xpath} for ranked trees instead of forests.
\smallpar{Pebble XPath on ranked trees} Since ranked trees are a special case of unranked forests, we need not change Pebble XPath for its use on ranked trees. However, for its comparison to the \abb{i-pta} it is more convenient to change its basic path expressions $\alpha_0$ and basic node expressions $\varphi_0$ as follows: $$\alpha_0 ::= {\tt down}_1 \mid {\tt down}_2 \mid {\tt up} \mid {\tt drop}_c \mid {\tt lift}_c$$ $$\varphi_0 ::=
{\tt haslabel}_\sigma \mid {\tt ischild}_0 \mid {\tt ischild}_1 \mid {\tt ischild}_2 \mid {\tt haspebble}_c$$
\noindent The semantics of these basic expressions for a tree $t$ over $\Sigma$ is given in Tables~\ref{tab:sempathrank} and~\ref{tab:semnoderank}. Since we will only be interested in ranked trees that encode forests, we assume that $\Sigma$ is a ranked alphabet and that the rank of each element of $\Sigma$ is at most 2. Note that ${\tt up}$ has the same semantics as ${\tt parent}$, and that the semantics of ${\tt drop}_c$, ${\tt lift}_c$, ${\tt haslabel}_\sigma$, and ${\tt haspebble}_c$ is unchanged. The remaining expressions of Pebble XPath, and their semantics (for $t$ instead of $f$), are the same as for forests, cf. the last four lines of Tables~\ref{tab:sempath} and~\ref{tab:semnode}.
\begin{table} \[ \begin{array}{l@{\;}cl} \semt{{\tt down}_1} &=& \{(\tup{u,\pi},\tup{u',\pi})\mid u'
\mbox{ is the first child of }u\} \\ \semt{{\tt down}_2} &=& \{(\tup{u,\pi},\tup{u',\pi})\mid u'
\mbox{ is the second child of }u\} \\ \semt{ {\tt up} } &=& \{(\tup{u,\pi},\tup{u',\pi})\mid u' \mbox{ is the parent of }u\} \\ \semt{ {\tt drop}_c } &=& \{(\tup{u,\pi},\tup{u,\pi p}) \mid p=(u,c)\} \\ \semt{ {\tt lift}_c } &=& \{(\tup{u,\pi p},\tup{u,\pi}) \mid p=(u,c)\} \\[1mm] \end{array} \] \caption{Basic path expressions $\alpha_0$ for a ranked tree $t$} \label{tab:sempathrank} \end{table}
\begin{table} \[ \begin{array}{l@{\;}cl} \semt{ {\tt haslabel}_\sigma } &=& \{\tup{u,\pi}\mid u \mbox{ has label }\sigma\} \\ \semt{ {\tt ischild}_0 } &=& \{\tup{u,\pi}\mid u \mbox{ is the root}\} \\ \semt{ {\tt ischild}_1 } &=& \{\tup{u,\pi}\mid u \mbox{ is a first child}\} \\ \semt{ {\tt ischild}_2 } &=& \{\tup{u,\pi}\mid u \mbox{ is a second child}\} \\ \semt{ {\tt haspebble}_c } &=& \{\tup{u,\pi p}
\mid p=(u,c)\} \\[1mm] \end{array} \] \caption{Basic node expressions $\varphi_0$ for a ranked tree $t$} \label{tab:semnoderank} \end{table}
We first show that for every path expression $\alpha$ on forests there is a path expression $\alpha'$ that computes the same trip as $\alpha$ on the binary encoding of the forests as ranked trees. We use the encoding ${\rm enc}'$ defined in Section~\ref{sec:trees}, which encodes forests over the alphabet $\Sigma$ as ranked trees over the associated ranked alphabet $\Sigma'$. Note that for every forest~$f$, ${\rm enc}'(f)$ has the same nodes as $f$. For a trip $T$ on forests, we define the \emph{encoded trip} ${\rm enc}'(T)$ on ranked trees by ${\rm enc}'(T)=\{({\rm enc}'(f),u,v)\mid (f,u,v)\in T\}$.
\begin{lemma}\label{lem:xpathft} For every Pebble XPath path expression $\alpha$ on forests over $\Sigma$, a Pebble XPath path expression $\alpha'$ on ranked trees over $\Sigma'$ can be constructed in polynomial time such that $T(\alpha')={\rm enc}'(T(\alpha))$. If $\alpha$ is a Pebble CAT expression, then so is $\alpha'$. \end{lemma}
\begin{proof} The proof is an elementary coding exercise. Let us start with Pebble XPath. We will, in fact, define $\alpha'$ such that $\sem{\alpha'}_{{\rm enc}'(f)}=\semf{\alpha}$ for every $f\in F_\Sigma$, which implies the result. It clearly suffices to do this for basic path expressions $\alpha_0$, and similarly for basic node expressions $\varphi_0$. As observed before, all basic node expressions except ${\tt haslabel}_\sigma$ are redundant, so it suffices to define ${\tt haslabel}_\sigma' \equiv {\tt haslabel}_{\sigma^{11}}\vee{\tt haslabel}_{\sigma^{10}}\vee {\tt haslabel}_{\sigma^{01}}\vee{\tt haslabel}_{\sigma^{00}}$. We now turn to the basic path expressions. We will use the auxiliary basic path expressions ${\tt child}_1$ and ${\tt parent}_1$ with the semantics $\semf{{\tt child}_1}=\{(\tup{u,\pi},\tup{u',\pi})\mid u' \text{ is the first child of }u\}$ and $\semf{{\tt parent}_1}=\semf{{\tt child}_1}^{-1}$. Since clearly ${\tt child}\equiv {\tt child}_1/{\tt right}^*$ and ${\tt parent}\equiv {\tt left}^*/{\tt parent}_1$, it suffices to define ${\tt child}'_1$ and ${\tt parent}'_1$ instead of ${\tt child}'$ and ${\tt parent}'$, as follows: ${\tt child}'_1\equiv \text{?}\varphi_1/{\tt down}_1$ where $\varphi_1$ is the disjunction of ${\tt haslabel}_{\sigma^{11}}$ and ${\tt haslabel}_{\sigma^{10}}$ for all $\sigma\in\Sigma$, and ${\tt parent}'_1\equiv \text{?}{\tt ischild}_1/{\tt up}/\text{?}\varphi_1$. Then we define ${\tt right}'\equiv {\tt down}_2 \cup \text{?}\varphi_2/{\tt down}_1$ where $\varphi_2$ is the disjunction of all ${\tt haslabel}_{\sigma^{01}}$ for $\sigma\in\Sigma$. Since $\semf{{\tt left}}$ is the inverse of $\semf{{\tt right}}$, we define ${\tt left}'\equiv \text{?}{\tt ischild}_2/{\tt up} \cup \text{?}{\tt ischild}_1/{\tt up}/\text{?}\varphi_2$. Finally, ${\tt drop}'_c \equiv {\tt drop}_c$ and ${\tt lift}'_c \equiv {\tt lift}_c$.
To prove the result for Pebble CAT, we also have to consider the other basic node expressions $\varphi_0$. Obviously, we define ${\tt haspebble}'_c\equiv{\tt haspebble}_c$. We define ${\tt isleaf}'$ to be the disjunction of ${\tt haslabel}_{\sigma^{01}}$ and ${\tt haslabel}_{\sigma^{00}}$ for all \mbox{$\sigma\in\Sigma$}, and similarly, ${\tt islast}'$ to be the disjunction of ${\tt haslabel}_{\sigma^{10}}$ and ${\tt haslabel}_{\sigma^{00}}$ for all $\sigma\in\Sigma$. It remains to consider ${\tt isfirst}$ and ${\tt isroot}$. Since we may assume the syntax of $\alpha$ to be in normal form, it suffices to define $(?\varphi_0)'$ and $(?\neg\varphi_0)'$. We define $(?{\tt isfirst})'\equiv \text{?}{\tt ischild}_0 \cup {\tt ischild}_1/{\tt up}/{\tt child}'_1$ and $(?\neg{\tt isfirst})'\equiv {\tt up}/{\tt right}'$ where ${\tt child}'_1$ and ${\tt right}'$ are defined above. For ${\tt isroot}$, we first note that $?{\tt isroot}\equiv {\tt drop}_c/{\tt left}^*/\text{?}{\tt isroot}/\text{?}{\tt isfirst}/{\tt right}^*/{\tt lift}_c$ where $c$ is any element of $C$. Intuitively, we walk from the current node to the left until we arrive at the first root, and then walk back. Thus, since the first root of a forest $f$ is encoded as the root of ${\rm enc}'(f)$, we define $(?{\tt isroot})'\equiv {\tt drop}_c/({\tt left}')^*/\text{?}{\tt ischild}_0/({\tt right}')^*/{\tt lift}_c$. Finally, we define $(?\neg{\tt isroot})'\equiv {\tt drop}_c/{\tt parent}'/{\tt child}'/{\tt lift}_c$. \end{proof}
Next we prove the reverse direction of Lemma~\ref{lem:xpathft}, for Pebble CAT.
\begin{lemma}\label{lem:cattf} For every Pebble CAT path expression $\alpha$ on ranked trees over $\Sigma'$ there is a Pebble CAT path expression $\alpha'$ on forests over $\Sigma$ such that $${\rm enc}'(T(\alpha'))=T(\alpha).$$ \end{lemma}
\begin{proof} This is also an elementary coding exercise. We assume the syntax of $\alpha$ to be in normal form, whereas for $\alpha'$ we keep the full syntax. As in the previous lemma, we will define $\alpha'$ such that $\semf{\alpha'}=\sem{\alpha}_{{\rm enc}'(f)}$. It suffices to do this for path expressions $\alpha_0$, $?\varphi_0$, and $?\neg\varphi_0$. We start with $\alpha_0$ and we define ${\tt down}'_1 \equiv {\tt child}/\text{?}{\tt isfirst} \cup \text{?}{\tt isleaf}/{\tt right}$ and ${\tt down}'_2 \equiv \text{?}\neg{\tt isleaf}/{\tt right}$. Moreover, ${\rm up}'\equiv \text{?}{\tt isfirst}/{\tt parent} \cup {\tt left}$. Finally, ${\tt drop}'_c \equiv {\tt drop}_c$ and ${\tt lift}'_c \equiv {\tt lift}_c$. We now turn to the basic node expressions. For $\varphi_0\equiv {\tt haslabel}_{\sigma^{10}}$ we define $(?\varphi_0)'\equiv \text{?}\varphi'_0$ and $(?\neg\varphi_0)'\equiv \text{?}\neg\varphi'_0$, where $\varphi_0'\equiv {\tt haslabel}_\sigma \wedge \neg{\tt isleaf} \wedge {\tt islast}$, and similarly for ${\tt haslabel}_{\sigma^{11}}$, ${\tt haslabel}_{\sigma^{01}}$, and ${\tt haslabel}_{\sigma^{00}}$. We do this also for $\varphi_0\equiv{\tt ischild}_0$ with $\varphi'_0\equiv{\tt isroot}\wedge{\tt isfirst}$, and for $\varphi_0\equiv{\tt haspebble}_c$ with $\varphi'_0\equiv {\tt haspebble}_c$. It remains to consider ${\tt ischild}_1$ and ${\tt ischild}_2$. We define $(?{\tt ischild}_2)'\equiv {\tt left}/?\neg{\tt isleaf}/{\tt right}$ and hence $(?\neg{\tt ischild}_2)'\equiv \text{?}{\tt isfirst} \cup {\tt left}/?{\tt isleaf}/{\tt right}$. For ${\tt ischild}_1$ the definitions of $(?{\tt ischild}_1)'$ and $(?\neg{\tt ischild}_1)'$ now follow from the fact that $?{\tt ischild}_1\equiv \text{?}\neg{\tt ischild}_0/?\neg{\tt ischild}_2$ and $?\neg{\tt ischild}_1\equiv \text{?}{\tt ischild}_0 \cup \text{?}{\tt ischild}_2$. \end{proof}
Lemmas~\ref{lem:xpathft} and~\ref{lem:cattf} together show that if the first equivalence of Theorem~\ref{thm:xpath} holds for ranked trees, then it also holds for forests. To show this also for the second equivalence, we need the next elementary lemma.
\begin{lemma}\label{lem:encmso} For every trip $T$ on forests, $T$ is \abb{mso} definable if and only if ${\rm enc}'(T)$ is \abb{mso} definable. \end{lemma}
\begin{proof} (Only if) Since $f$ and ${\rm enc}'(f)$ have the same nodes, for every forest $f$ over $\Sigma$, it suffices to show that the atomic formulas $\mathrm{lab}_\sigma(x)$, ${\rm down}(x,y)$, and $\mathrm{next}(x,y)$ for forests can be expressed by an \abb{mso} formula for the ranked trees that encode the forests. Clearly, $\mathrm{lab}_\sigma(x)$ can be expressed by the disjunction of all $\mathrm{lab}_{\sigma^{k\ell}}(x)$ for $k,\ell\in\{0,1\}$, as in the proof of Lemma~\ref{lem:xpathft}. For ${\rm down}(x,y)$ we show that the trip $T=\{({\rm enc}'(f),u,v)\mid f\models {\rm down}(u,v)\}$ is \abb{mso} definable. This follows from Proposition~\ref{prop:trips} because $T=T({\cal B})$ for the \abb{ta} ${\cal B}$ that has the rules (for all $k,\ell\in\{0,1\}$, $j\in\{0,1,2\}$, and $\sigma\in\Sigma$): \[ \begin{array}{lll} \tup{p_0,\sigma^{1\ell},j} & \to & \tup{p,{\rm down}_1}, \\ \tup{p,\sigma^{11},j} & \to & \tup{p,{\rm down}_2}, \\ \tup{p,\sigma^{01},j} & \to & \tup{p,{\rm down}_1}, \\ \tup{p,\sigma^{k\ell},j} & \to & \tup{p_\infty,{\rm stay}}, \end{array} \] where $p_0$ is the initial and $p_\infty$ the final state of ${\cal B}$. Thus, there is a formula $\varphi(x,y)$ such that ${\rm enc}'(f)\models \varphi(u,v)$ if and only if $f\models {\rm down}(u,v)$, for every forest $f$, which means that $\varphi(x,y)$ expresses ${\rm down}(x,y)$ on the encoding of~$f$.\footnote{For the reader familiar with \abb{mso} logic we note that it is also easy to write down the formula $\varphi(x,y)$ using the equivalences in the proof of Lemma~\ref{lem:xpathft} and the fact that the transitive-reflexive closure of an \abb{mso} definable relation is \abb{mso} definable. } The formula $\mathrm{next}(x,y)$ can be treated in the same way, where ${\cal B}$ now has the rules $\tup{p_0,\sigma^{11},j} \to \tup{p_\infty,{\rm down}_2}$ and $\tup{p_0,\sigma^{01},j} \to \tup{p_\infty,{\rm down}_1}$, and hence $T({\cal B})=\{({\rm enc}'(f),u,v)\mid f\models \mathrm{next}(u,v)\}$.
(If) For the same reason as above, it suffices to show that the atomic formulas ${\rm down}_i(x,y)$ and $\mathrm{lab}_{\sigma^{k\ell}}(x)$ for ranked trees over $\Sigma'$ can be expressed by an \abb{mso} formula for the forests they encode. For this we consider the path expressions ${\tt down}'_i$ and ${\tt haslabel}'_{\sigma^{10}}$ in the proof of Lemma~\ref{lem:cattf}, and we define \[ \begin{array}{lll} \varphi_1(x,y) & \!\!\equiv\!\! & ({\rm down}(x,y)\wedge \mathrm{first}(y)) \vee
(\mathrm{leaf}(x) \wedge \mathrm{next}(x,y)), \\[1mm] \varphi_2(x,y) & \!\!\equiv\!\! & \neg\,\mathrm{leaf}(x) \wedge \mathrm{next}(x,y), \\[1mm] \varphi_{10}(x) & \!\!\equiv\!\! &
\mathrm{lab}_\sigma(x) \wedge \neg\,\mathrm{leaf}(x)\wedge \mathrm{last}(x), \end{array} \] and similarly for the other $\varphi_{k\ell}(x,y)$. Then ${\rm enc}'(f)\models {\rm down}_i(u,v)$ if and only if $f\models \varphi_i(u,v)$, and ${\rm enc}'(f)\models \mathrm{lab}_{\sigma^{k\ell}}(u)$ if and only if $f\models \varphi_{k\ell}(u)$. \end{proof}
From now on, when we refer to Pebble XPath or Pebble CAT we always mean their version on ranked trees.
\smallpar{Directive I-PTA's} For the purpose of the proof of Theorem~\ref{thm:xpath} on ranked trees, we formulate the \abb{i-pta} in an alternative way and, for lack of a better name, call it the \emph{directive} \abb{i-pta}. For an alphabet~$\Sigma$ (of which every element has rank at most $2$) and a finite set $C$ of colours, we define a \emph{directive} over $\Sigma$ and $C$ to be a path expression $\tau$ with the syntax $\tau ::= \alpha_0 \mid \;?\varphi_0 \mid \;?\neg\,\varphi_0$ for the same $\Sigma$ and~$C$ (where $\alpha_0$ and $\varphi_0$ are as in Tables~\ref{tab:sempathrank} and~\ref{tab:semnoderank}). The finite set of directives over $\Sigma$ and $C$ is denoted $D_{\Sigma,C}$.
A \emph{directive} \emph{\abb{i-pta}} is a tuple ${\cal A} = (\Sigma, Q, Q_0, F, C, R)$, where $\Sigma$, $Q$, $Q_0$, $F$, and~$C$ are as for an ordinary \abb{i-pta} (with $C=C_\mathrm{i}$), and $R$ is a finite set of rules of the form $\tup{q,\tau,q'}$ where $q,q'\in Q$ and $\tau\in D_{\Sigma,C}$. Thus, syntactically, ${\cal A}$ can be viewed as a finite automaton of which each state transition is labeled by a directive, i.e., either by a basic path expression of Pebble XPath, or by a basic node expression of Pebble XPath, or its negation, where the node expressions are turned into path expressions by the ?-operator. Intuitively, $?\varphi_0$ and $?\neg\,\varphi_0$ represent a basic test on the current situation, whereas $\alpha_0$ is a basic instruction to be executed on the current situation. Just as for an ordinary \abb{i-pta}, a situation on a tree $t\in T_\Sigma$ is a pair $\tup{u,\pi}\in \xp{Sit}(t)$ and a configuration is a triple $\tup{q,u,\pi}$ with $q\in Q$ and $\tup{u,\pi}\in \xp{Sit}(t)$. We write $\tup{q,u,\pi}\Rightarrow_{t,{\cal A}} \tup{q',u',\pi'}$ if there is a rule $\tup{q,\tau,q'}$ such that $(\tup{u,\pi},\tup{u',\pi'})\in \semt{\tau}$, where $\semt{\tau}$ is the semantics of path expression $\tau$ on~$t$ (cf. Tables~\ref{tab:sempathrank} and~\ref{tab:semnoderank} for $\alpha_0$ and $\varphi_0$, and Table~\ref{tab:sempath} for the \mbox{?-operator}). To indicate the directive $\tau$ that is executed by ${\cal A}$ in this computation step we also write $\tup{q,u,\pi}\Rightarrow^\tau_{t,{\cal A}} \tup{q',u',\pi'}$. Moreover, we define the semantics $\semt{{\cal A}}$ of ${\cal A}$ on tree $t$ as $\semt{{\cal A}} = \{(\tup{u,\pi},\tup{u',\pi'})\in \xp{Sit}(t)\times \xp{Sit}(t) \mid \exists\, q_0\in Q_0, \,q_\infty\in F: \tup{q_0,u,\pi}\Rightarrow^*_{t,{\cal A}} \tup{q_\infty,u',\pi'}\}$. Finally, the trip computed by~${\cal A}$ on $T_\Sigma$ is $T({\cal A})=\{(t,u,v)\mid \exists\, \pi\in(N(t)\times C)^*: (\tup{u,\varepsilon},\tup{v,\pi})\in \semt{{\cal A}}\}$.
For the sake of the proofs below we also define $\sem{{\cal A}}_t$ for an ordinary \abb{i-pta}~${\cal A}$ on a tree $t$, in entirely the same way as above for a directive \abb{i-pta}.
A directive \abb{i-pta} ${\cal A}$ \emph{with look-ahead tests} is defined similarly to the ordinary case in Section~\ref{sec:look-ahead} (restricted to automata), by additionally allowing rules of the form $\tup{q,?\tup{{\cal B}},q'}$ and $\tup{q,?\neg\,\tup{{\cal B}},q'}$ where ${\cal B}$ is another directive \abb{i-pta}. The above semantics stays the same, with (as expected) \[ \semt{?\tup{{\cal B}}} = \{(\tup{u,\pi},\tup{u,\pi})\mid
\exists \tup{u',\pi'}\colon
(\tup{u,\pi},\tup{u',\pi'})\in \semt{{\cal B}}\} \] and similarly for $\semt{?\neg\,\tup{{\cal B}}}$ (with $\neg\,\exists$). A directive \abb{i-pta} \emph{with iterated look-ahead tests} is defined as in Section~\ref{sec:look-ahead}. We will use \abb{i-pta$^\text{la}$} as an abbreviation of `\abb{i-pta} with iterated look-ahead tests'.
We now show that the directive \abb{i-pta} has the same expressive power as the \abb{i-pta} (and similarly with iterated look-ahead tests). Hence Theorems~\ref{thm:trips} and~\ref{thm:iterated} also hold for the directive \abb{i-pta}, i.e., it computes the \abb{mso} definable trips, and it can perform iterated look-ahead tests. In what follows, we only consider \abb{i-pta}'s of which every input symbol has at most rank~$2$.
\begin{lemma}\label{lem:iptaft} For every directive \abb{i-pta$^\text{la}$} ${\cal A}$ there is an \abb{i-pta$^\text{la}$} ${\cal A}'$ such that $T({\cal A}')=T({\cal A})$. \end{lemma}
\begin{proof} Let ${\cal A} = (\Sigma, Q, Q_0, F, C, R)$ be a directive \abb{i-pta}. We will, in fact, define the \abb{i-pta} ${\cal A}'$ such that $\semt{{\cal A}'}=\semt{{\cal A}}$ for every $t\in T_\Sigma$, which implies the result.
We let ${\cal A}' = (\Sigma, Q, Q_0, F, C, \varnothing, C_\mathrm{i}, R', 0)$ where $C_\mathrm{i}=C$ and $R'$ is defined as follows. If $\tup{q,\alpha_0,q'}$ is a rule of ${\cal A}$, where $\alpha_0$ is a basic path expression, then ${\cal A}'$ has all rules $\tup{q,\sigma,j,b}\to \tup{q',\alpha_0}$. We now turn to the basic node expressions. A rule $\tup{q,?{\tt haslabel}_\sigma,q'}$ is simulated by all rules $\tup{q,\sigma,j,b}\to \tup{q',{\rm stay}}$, and a rule $\tup{q,?\neg\,{\tt haslabel}_\sigma,q'}$ by all rules $\tup{q,\tau,j,b}\to \tup{q',{\rm stay}}$ with $\tau\in\Sigma\setminus\{\sigma\}$. A rule $\tup{q,?{\tt ischild}_j,q'}$ is simulated by all rules $\tup{q,\sigma,j,b}\to \tup{q',{\rm stay}}$, and a rule $\tup{q,?\neg\,{\tt ischild}_j,q'}$ by the two rules $\tup{q,\sigma,j',b}\to \tup{q',{\rm stay}}$ with $j'\in\{0,1,2\}\setminus\{j\}$. A rule $\tup{q,?{\tt haspebble}_c,q'}$ is simulated by all rules $\tup{q,\sigma,j,\{c\}}\to \tup{q',{\rm stay}}$, and a rule $\tup{q,?\neg\,{\tt haspebble}_c,q'}$ by all rules $\tup{q,\sigma,j,\varnothing}\to \tup{q',{\rm stay}}$ and all rules $\tup{q,\sigma,j,\{c'\}}\to \tup{q',{\rm stay}}$ with $c'\in C\setminus\{c\}$.
Finally we consider look-ahead. If $\tup{q,?\tup{{\cal B}},q'}$ is a rule of ${\cal A}$, and ${\cal B}'$ is an \abb{i-pta$^\text{la}$} such that $\semt{{\cal B}'}=\semt{{\cal B}}$ for every $t\in T_\Sigma$, then ${\cal A}'$ has all rules $\tup{q,\sigma,j,b,{\cal B}'}\to \tup{q',{\rm stay}}$ that use ${\cal B}'$ as a look-ahead test. Similarly, the rule $\tup{q,?\neg\,\tup{{\cal B}},q'}$ is simulated by all rules $\tup{q,\sigma,j,b,\neg\,{\cal B}'}\to \tup{q',{\rm stay}}$. \end{proof}
\begin{lemma}\label{lem:iptatf} For every \abb{i-pta} ${\cal A}$ there is a directive \abb{i-pta} ${\cal A}'$ such that $T({\cal A}')=T({\cal A})$. \end{lemma}
\begin{proof} Let ${\cal A} = (\Sigma, Q, Q_0, F, C, \varnothing, C_\mathrm{i}, R, 0)$ be an \abb{i-pta} with $C_\mathrm{i}=C$. To simplify the proof we extend the syntax of the directive \abb{i-pta} by allowing rules $\tup{q,\tau,q'}$ with $\tau ::= \alpha_0 \mid \;?\varphi_0 \mid \;?\neg\,\varphi_0 \mid \tau/\tau'$, where $\tau'$ is an alias of $\tau$. This clearly does not extend their power, because a rule $\tup{q,\tau/\tau',q'}$ can be replaced by the two rules $\tup{q,\tau,p}$ and $\tup{p,\tau',q'}$ where $p$ is a new state. We now construct ${\cal A}' = (\Sigma, Q, Q_0, F, C, R')$ where $R'$ is defined as follows. If ${\cal A}$ has a rule $\tup{q,\sigma,j,b}\to \tup{q',\alpha}$ then ${\cal A}'$ has the rule $\tup{q,\tau,q'}$ such that $\tau = \tau_\sigma/\tau_j/\tau_b/\alpha$ if $\alpha\neq{\rm stay}$, and $\tau = \tau_\sigma/\tau_j/\tau_b$ if $\alpha={\rm stay}$, where $\tau_\sigma = \text{?}{\tt haslabel}_\sigma$, $\tau_j = \text{?}{\tt ischild}_j$, $\tau_{\{c\}} = \text{?}{\tt haspebble}_c$, and $\tau_\varnothing = \text{?}\neg\,{\tt haspebble}_{c_1}/\cdots/?\neg\,{\tt haspebble}_{c_n}$, where $C=\{c_1,\dots,c_n\}$. \end{proof}
As observed before, a directive \abb{i-pta} ${\cal A}$ can be viewed as a finite automaton of which each state transition is labeled by a directive. Thus, viewing the set $D_{\Sigma,C}$ as an alphabet, ${\cal A}$ accepts a string language $L_\mathrm{str}({\cal A})\subseteq D^*_{\Sigma,C}$. We now show the (rather obvious) fact that the semantics $\semt{{\cal A}}$ of ${\cal A}$ (for every tree $t$ over $\Sigma$) depends only on the language $L_\mathrm{str}({\cal A})$, cf.~\cite[Theorem~3.11]{Eng74} and~\cite[Lemma~3]{bloem}. We do this (as in~\cite[Definition~2.7]{Eng74} and~\cite[Section~4]{bloem}) by associating a semantics $\semt{L}$ with every language $L\subseteq D^*_{\Sigma,C}$. Intuitively, a string $w=\tau_1\cdots \tau_n$ of directives can be viewed as the path expression $\tau_1/\cdots/\tau_n$ and a language $L=\{w_1,w_2,\dots\}$ of such strings can be viewed as the (possibly infinite) path expression $w_1\cup w_2\cup\cdots$. Thus, for a tree $t$ over $\Sigma$ we formally define $\semt{\varepsilon}$ to be the identity on $\xp{Sit}(t)$, $\semt{\tau_1\cdots\tau_n}=\semt{\tau_1}\circ\cdots\circ\semt{\tau_n}$, and $\semt{L}=\bigcup_{w\in L}\semt{w}$. The next lemma is a special case of~\cite[Theorem~3.11]{Eng74}. Its proof is entirely similar to the one of~\cite[Lemma~3]{bloem}.
\begin{lemma}\label{lem:langipta} $\semt{{\cal A}}=\semt{L_\mathrm{str}({\cal A})}$. \end{lemma}
\begin{proof} A string $w$ of directives induces a state transition relation $R_{\cal A}(w)\subseteq Q\times Q$ as follows. For $\tau\in D_{\Sigma,C}$, $R_{\cal A}(\tau)=\{(q,q')\mid \tup{q,\tau,q'}\in R\}$. For the empty string, $R_{\cal A}(\varepsilon)$ is the identity on $Q$, and $R_{\cal A}(\tau_1\cdots\tau_n)=R_{\cal A}(\tau_1)\circ\cdots\circ R_{\cal A}(\tau_n)$. Then $L_\mathrm{str}({\cal A})= \{w\in D^*_{\Sigma,C} \mid R_{\cal A}(w)\cap (Q_0\times F)\neq\varnothing\}$.
It is straightforward to show by induction that, for all configurations $\tup{q,u,\pi}$ and $\tup{q',u',\pi'}$ and for every $w=\tau_1\cdots\tau_n$ over $D_{\Sigma,C}$, there is a computation \[ \tup{q_1,u_1,\pi_1} \Rightarrow^{\tau_1}_{t,{\cal A}} \tup{q_2,u_2,\pi_2} \Rightarrow^{\tau_2}_{t,{\cal A}} \cdots \Rightarrow^{\tau_n}_{t,{\cal A}} \tup{q_{n+1},u_{n+1},\pi_{n+1}} \] with $\tup{q_1,u_1,\pi_1}=\tup{q,u,\pi}$ and $\tup{q_{n+1},u_{n+1},\pi_{n+1}}=\tup{q',u',\pi'}$ if and only if $(\tup{u,\pi},\tup{u',\pi'})\in \semt{w}$ and $(q,q')\in R_{\cal A}(w)$. From this equivalence it follows that $\semt{{\cal A}}$ consists of all $(\tup{u,\pi},\tup{u',\pi'})$ such that \[ \exists\, q_0\in Q_0, q_\infty\in F, w\in D^*_{\Sigma,C}: (\tup{u,\pi},\tup{u',\pi'})\in \semt{w},\;(q,q')\in R_{\cal A}(w) \] i.e., such that $\exists\, w\in L_\mathrm{str}({\cal A}): (\tup{u,\pi},\tup{u',\pi'})\in \semt{w}$, which means that it equals $\semt{L_\mathrm{str}({\cal A})}$. \end{proof}
\smallpar{Proof of Theorem~\ref{thm:xpath}} We assume the syntax for path expressions $\alpha$ of Pebble XPath and Pebble CAT to be in normal form. We also add $\alpha ::= \varnothing$ to the syntax, with $\semt{\varnothing}=\varnothing$ for every tree $t$. That is possible because, e.g., $\semt{?{\tt ischild}_0/?\neg{\tt ischild}_0}=\varnothing$.
We first show that Pebble CAT has the same power as \abb{MSO}. Let us recall that the set $D_{\Sigma,C}$ of directives $\tau$ of the directive \abb{i-pta} was defined by the syntax $\tau ::= \alpha_0 \mid \;?\varphi_0 \mid \;?\neg\varphi_0$. Thus, the path expressions of Pebble CAT are, in fact, exactly the usual regular expressions over the ``alphabet'' $D_{\Sigma,C}$. Accordingly, we define for such a path expression $\alpha$ the string language $L_\mathrm{str}(\alpha)\subseteq D^*_{\Sigma,C}$ in the obvious way, interpreting the operators $\cup$, $/$, and $^*$ as union, concatenation, and Kleene star of string languages, respectively. The next lemma is the analogue of Lemma~\ref{lem:langipta}, with a straightforward proof.
\begin{lemma}\label{lem:langcat} $\semt{\alpha}=\semt{L_\mathrm{str}(\alpha)}$. \end{lemma}
\begin{proof} It is easy to see, for string languages $L_1,L_2\subseteq D_{\Sigma,C}^*$, that $\semt{L_1\cup L_2}= \semt{L_1}\cup\semt{L_2}$, $\semt{L_1L_2}= \semt{L_1}\circ\semt{L_2}$, and $\semt{L_1^*}= \semt{L_1}^*$, cf.~\cite[Lemma~2.9]{Eng74}. Then the proof is by induction on the structure of $\alpha$. \end{proof}
By Kleene's classical theorem, a string language can be accepted by a finite automaton if and only if it can be defined by a regular expression. Thus, by Lemmas~\ref{lem:langipta} and~\ref{lem:langcat}, a trip is definable in Pebble CAT if and only if it can be computed by a directive \abb{i-pta}, and hence, by Theorem~\ref{thm:trips} (for $k=0$) and Lemmas~\ref{lem:iptaft} and~\ref{lem:iptatf}, if and only if it is \abb{mso} definable.
It remains to show that if a trip is definable in Pebble XPath, then it can be computed by a directive \abb{i-pta}. We will prove below that for every Pebble XPath path expression $\alpha$ there is a directive \abb{i-pta$^\text{la}$} ${\cal A}$, i.e., a directive \abb{i-pta} with iterated look-ahead tests, such that $\semt{{\cal A}} = \semt{\alpha}$ for every $t$. This implies that $\alpha$ and ${\cal A}$ define the same trip, and then we obtain from Theorem~\ref{thm:iterated} (and Lemmas~\ref{lem:iptaft} and~\ref{lem:iptatf}) a directive \abb{i-pta} (without look-ahead) computing that same trip.
Let $n_\alpha$ be the nesting depth of subexpressions of $\alpha$ of the form $\tup{\beta}$. The proof is by induction on $n_\alpha$, and ${\cal A}$ will be of look-ahead depth $n_\alpha$. If $n_\alpha=0$, i.e., there are no such subexpressions at all, then $\alpha$ is a Pebble CAT expression, and we are done by the first part of the proof. Now suppose that the result holds for nesting depth $n$, and let $n_\alpha=n+1$. For every subexpression $\tup{\beta}$ of~$\alpha$ that is not nested within another such subexpression, let ${\cal A}_\beta$ be a directive \abb{i-pta$^\text{la}$} of look-ahead depth $n$ (or less) such that $\semt{{\cal A}_\beta} = \semt{\beta}$ for all $t$. We now define the extended ``alphabet'' $D^n_{\Sigma,C}$ to consist of all path expressions $\tau$ with the syntax $\tau ::= \alpha_0 \mid \;?\varphi_0 \mid \;?\neg\varphi_0 \mid \;?\tup{\beta} \mid \;?\neg\tup{\beta}$ where $\tup{\beta}$ ranges over the above subexpressions of $\alpha$. Then $\alpha$ can be viewed as a regular expression over the alphabet $D^n_{\Sigma,C}$, and it should be clear that Lemma~\ref{lem:langcat} is also valid in this case. Also, using $D^n_{\Sigma,C}$ instead of $D_{\Sigma,C}$ in the rules of the directive \abb{i-pta}, and identifying each ``symbol'' $?\tup{\beta}$ with the ``symbol'' $?\tup{{\cal A}_\beta}$ (and similarly for the negated tests), we obtain a subclass of the directive \abb{i-pta$^\text{la}$} of look-ahead depth $n+1$, because the semantics of the path expression $?\tup{\beta}$ is exactly the same as the meaning of the look-ahead test $?\tup{{\cal A}_\beta}$. Again, it should be clear that Lemma~\ref{lem:langipta} is also valid for these directive \abb{i-pta}'s, which are finite automata over $D^n_{\Sigma,C}$. Hence, by the same Kleene argument as in the first part of the proof, there is a directive \abb{i-pta$^\text{la}$} ${\cal A}$ of look-ahead depth $n+1$ such that $\semt{{\cal A}} = \semt{\alpha}$ for every tree $t$.
This ends the proof of Theorem~\ref{thm:xpath}, both for ranked trees and (by Lemmas~\ref{lem:xpathft}, \ref{lem:cattf}, and~\ref{lem:encmso}) for unranked forests.
\smallpar{Two remarks} (1) Although the MSO definable trips are, of course, closed under complement and intersection, we do not know whether the XPath~2.0 operations {\tt intersect} and {\tt except} can be added to the syntax of path expressions of Pebble XPath ($\alpha ::= \alpha \cap \beta \mid \alpha \setminus \beta$). That is because it is not clear whether for every \abb{i-pta} ${\cal A}$ there is an \abb{i-pta} ${\cal B}$ such that $\semt{{\cal B}} = \xp{Sit}(t) - \semt{{\cal A}}$ for every tree $t$.
(2) The language Pebble XPath meets the requirements as listed in \cite{GorMar05}. It is \emph{simple}, defined in an algebraic language using simple operators: in particular we believe that pebbles form a user friendly concept. It is \emph{understandable}, as its expressive power can be characterized in terms of automata. It is \emph{useful} in the sense that the query evaluation problem `given path expression $\alpha$ and two nodes $u,v$ in forest $f$, is $(f,u,v)\in T(\alpha)$?' is tractable. At least, the latter property holds for Pebble CAT, as $\alpha$ can be transformed into an \abb{I-PTA} in polynomial time, and the problem `$(f,u,v)\in T(\alpha)$?' can then be translated into the emptiness problem for push-down automata. For Pebble XPath the query evaluation problem is tractable for every fixed path expression $\alpha$. This is explained in more detail in the next two paragraphs.
\smallpar{Query evaluation} For a directive \abb{i-pta} ${\cal A} = (\Sigma, Q, Q_0, C, R)$, the binary node relation $T$ computed by ${\cal A}$ on an input tree $t$ can be evaluated in polynomial time as follows. It is straightforward to construct from ${\cal A}$ and $t$ an ordinary pushdown automaton ${\cal P}$ with state set $Q \times \nod{t}$ and pushdown alphabet $\nod{t}\times C$ in such a way that ${\cal P}$ (with the empty string as input) has the same computation steps as ${\cal A}$ on $t$. Note that the configurations of ${\cal P}$ are exactly the configurations $\langle q,u,\pi\rangle$ of ${\cal A}$ on $t$. Dropping and lifting a pebble corresponds to pushing and popping a pushdown symbol. Moving around in $t$ corresponds to a change of state. To decide whether $(t,u,v)\in T$, with $u,v\in\nod{t}$, decide whether ${\cal P}$ has a computation from configuration $\langle q_0,u,\varepsilon\rangle$ (for some $q_0\in Q_0$) to some final configuration $\langle q,v,\pi\rangle$. Clearly, ${\cal P}$ can be constructed in polynomial time from ${\cal A}$ and $t$, and the existence of such a computation can be verified in polynomial time.
By Lemma~\ref{lem:xpathft}, path expressions on forests can be translated into path expressions on ranked trees in polynomial time. Since for a Pebble CAT path expression on ranked trees the corresponding directive \abb{i-pta} can be constructed in polynomial time, using Kleene's construction, Pebble CAT path expressions can be evaluated in polynomial time. This does not seem to hold for Pebble XPath, as the construction in the proof of Theorem~\ref{thm:look-ahead} (which implements a look-ahead test by calling an \abb{i-pta} ${\cal B}$) is at least 2-fold exponential (because determining the domain of the related \abb{i-pta} ${\cal B}'$ takes 2-fold exponential time by Theorem~\ref{thm:typecheck}). However, the data complexity of the problem is of course polynomial, i.e., for a fixed path expression $\alpha$ we obtain a fixed directive \abb{i-pta} ${\cal A}$ for which the binary node relation can be evaluated in polynomial time.
\section{Pattern Matching}\label{sec:pattern}
One of the basic tree transformations in the context of XML is pattern matching. The transducer must find all sequences of nodes satisfying a certain description and generate the subtrees rooted at these nodes, for each match.
More precisely, we consider queries of the form \[ \mbox{\tt for } {\cal X} \mbox{ \tt where } \varphi \mbox{ \tt return } r \] in which ${\cal X}$ is a finite set of node variables, $\varphi$ is an \abb{mso} formula with its free variables in ${\cal X}$, and $r$ is a tree of which the leaves may be labeled with the variables in ${\cal X}$. In what follows we assume that ${\cal X}$ and $r$ are fixed. Let ${\cal X}=\{x_1,\dots,x_n\}$, where $x_1,\dots,x_n$ is an arbitrary order of the elements of ${\cal X}$. The transducer must find all sequences of nodes $u_1,\dots,u_n$ of the input tree $t$ that match the pattern defined by $\varphi(x_1,\dots,x_n)$, i.e., such that $t\models \varphi(u_1,\dots,u_n)$, and for each match it must generate the output tree~$r$ in which each occurrence of the variable $x_i$ is replaced by the subtree of $t$ with root~$u_i$. Usually the variables in~${\cal X}$ are indeed specified in a specific order $\lambda=(x_1,\dots,x_n)$, and it is required that the transducer finds (and generates) the matches in the lexicographic document order induced by $\lambda$. We will, however, also consider the case where this requirement is dropped, and the most efficient order $\lambda$ can be selected.
For convenience we assume that $r$ is of the form $\mu(x_1,\dots,x_n)$ for some symbol $\mu$ of rank~$n$, and so the output for each match is
$\mu(t|_{u_1},\dots,t|_{u_n})$ where $t|_u$ is the subtree of $t$ with root $u$. For convenience we also assume that the input tree $t$ is ranked. Moreover, we assume that
the output alphabet is also ranked and contains the binary symbol $@$ that allows us to list the various output trees $\mu(t|_{u_1},\dots,t|_{u_n})$, and the nullary symbol $e$ to indicate the end of the list of output trees (similar to the binary tag \texttt{<result>} and the nullary tag \texttt{<endofresults>} of Example~\ref{ex:siberie}). In Section~\ref{sec:pft} we will consider pattern matching in forests.
We now describe a total deterministic \abb{ptt} ${\cal A}$ that executes the above query. In order to find all $n$-tuples of nodes matching the $n$-ary pattern defined by the \abb{mso} formula $\varphi(x_1,\dots,x_n)$, and generate the corresponding output, the \abb{ptt}~${\cal A}$ systematically enumerates \emph{all} $n$-tuples of nodes of the input tree $t$. To do this, ${\cal A}$~uses visible pebbles $c_1,\dots,c_n$ on the stack, representing the variables $x_1,\dots,x_n$, respectively.\footnote{It is not necessary that all pebbles are visible, as we will discuss below, but it simplifies the description of ${\cal A}$. } It drops them in this order and moves each of them through the input tree $t$ in document order (i.e., in pre-order), in a nested fashion. Inductively speaking, ${\cal A}$ moves pebble~$c_1$ in pre-order through $t$ (alternately dropping and lifting $c_1$), and for each position $u_1$ of~$c_1$ it uses pebbles $c_2,\dots,c_n$ to enumerate all possible $(n\!-\!1)$-tuples $u_2,\dots,u_n$ of nodes of $t$. For each enumerated $n$-tuple $u_1,\dots,u_n$, with pebble $c_i$ at position~$u_i$, ${\cal A}$ performs the test~$\varphi$, using an \abb{mso} test on the visible configuration (Lemma~\ref{lem:visiblesites}), and, in case of success, spawns a process that outputs the corresponding $n$-tuple of subtrees.
More precisely, if the ranked input alphabet is $\Sigma$, then $\varphi$ is an \abb{mso} formula over $\Sigma$, and
${\cal A}$ has the ranked output alphabet $\Delta=\Sigma \cup\{\mu,@,e\}$ where $\mu$ has rank~$n$, and $@$ and $e$ have rank~2 and~0 respectively. For input tree $t$, the output tree $s$ is of the form $s=@(r_1,@(r_2,\dots @(r_k,e)\cdots))$ where each $r_i$ corresponds to a match, i.e., there is a sequence of nodes $u_1,\dots,u_n$ of $t$ such that $t\models \varphi(u_1,\dots,u_n)$ and $r_i=\mu(t|_{u_1},\dots,t|_{u_n})$. Moreover, the sequence $r_1,\dots,r_k$ corresponds to the sequence of all matches, in lexicographic document order. As explained above, the visible colour set of the \abb{ptt} ${\cal A}$ is $C_\mathrm{v}=\{c_1,\dots,c_n\}$, and ${\cal A}$ generates $s$ by enumerating all sequences $u_1,\dots,u_n$ of nodes of $t$ using pebbles $c_1,\dots,c_n$. To find out whether this sequence is a match, ${\cal A}$ performs the \abb{mso} test $\psi(x)$ on the visible configuration, defined by \[ \psi(x) \equiv \forall x_1,\dots,x_n(
(\mathrm{peb}_{c_1}(x_1) \wedge \cdots\wedge \mathrm{peb}_{c_n}(x_n))\to
\varphi'(x_1,\dots,x_n)) \] where $\mathrm{peb}_c(x)$ is the disjunction of all $\mathrm{lab}_{(\sigma,b)}(x)$ such that $c\in b$, and where $\varphi'$ is obtained from $\varphi$ by changing every atomic subformula $\mathrm{lab}_\sigma(y)$ into the disjunction of all $\mathrm{lab}_{(\sigma,b)}(y)$. Note that $\psi(x)$ is an \abb{mso} formula over $\Sigma\times 2^C$, where $C$ is the colour set of ${\cal A}$. Note also that the variable $x$ (for the head position) does not, and need not, occur in $\psi(x)$. If the sequence $u_1,\dots,u_n$ is not a match, then ${\cal A}$ continues the enumeration of $n$-tuples. If the sequence \emph{is} a match, then ${\cal A}$ outputs the symbol $@$ and branches into two subprocesses (as in the 5-th rule of Example~\ref{ex:siberie}). In the second (main) branch it continues the enumeration of $n$-tuples. In the first branch it outputs the symbol $\mu$ and branches into $n$ subprocesses, where the $i$-th process
searches for visible pebble~$c_i$ and then outputs $t|_{u_i}$. Note that, in this first branch, ${\cal A}$ could easily output an arbitrary tree $r$
in which every occurrence of the variable $x_i$ is replaced by $t|_{u_i}$. This ends the description of~${\cal A}$.
As the complexity of typechecking the transducer ${\cal A}$ depends critically on the number of visible pebbles used (see Theorem~\ref{thm:typecheck}), we wish to minimize that number and use as few visible pebbles as possible for matching. It should be clear that, instead of using $n$ visible pebbles, ${\cal A}$ can also use $n\!-\!2$ visible pebbles $c_1,\dots,c_{n-2}$, one invisible pebble $c_{n-1}$ on top (which is therefore always observable), and the head instead of the last pebble $c_n$. Then ${\cal A}$ can perform the \abb{mso} test $\chi(x)$ on the observable configuration, defined by $\chi(x) \equiv $ \[ \forall x_1,\dots,x_{n-1}(
(\mathrm{peb}_{c_1}(x_1) \wedge \cdots\wedge \mathrm{peb}_{c_{n-1}}(x_{n-1}))\to
\varphi'(x_1,\dots,x_{n-1},x)) \] where $x_n$ is renamed into $x$ in $\varphi'$. Thus, from Theorem~\ref{thm:mso} we obtain the following result on the matching of arbitrary \abb{MSO} definable patterns.
\begin{theorem}\label{thm:matchall} For $n\geq 2$, every \abb{MSO} definable $n$-ary pattern can be matched by a total deterministic \abb{v$_{n-2}$i-ptt}. Moreover, and in particular, every \abb{MSO} definable unary or binary pattern can be matched by a total deterministic \abb{i-ptt}. \end{theorem}
To further reduce the number of visible pebbles, we consider the more specific case of queries of the form \[ \mbox{\tt for } {\cal X} \mbox{ \tt where } \xp{\beta}(\varphi_1, \dots, \varphi_m) \mbox{ \tt return } r \] in which $\xp{\beta}(\varphi_1, \dots, \varphi_m)$ is a boolean combination (using $\wedge,\vee,\neg$) of the \abb{mso} formulas $\varphi_1, \dots, \varphi_m$, $m\geq 2$, and each $\varphi_\ell$, $\ell\in[1,m]$, has its free variables in~${\cal X}$. We will make use of the fact that not all variables in ${\cal X}$ need actually occur in each formula~$\varphi_\ell$.
As discussed in the Introduction, the {\tt for} $\cdots$\ {\tt where} construct in XQuery often induces patterns $\varphi_1 \wedge \cdots \wedge \varphi_m$ such that each $\varphi_\ell$ contains just two free variables, cf. \cite{GotKocSch}.
Consider an arbitrary query as displayed above. Let $G_\varphi=(V_\varphi,E_\varphi)$ be the undirected graph induced by the pattern $\varphi \equiv \xp{\beta}(\varphi_1, \dots, \varphi_m)$, by which we mean that the set $V_\varphi$ of vertices of $G_\varphi$ consists of the free variables of $\varphi$, i.e., $V_\varphi={\cal X}$, and that the set $E_\varphi$ of edges of $G_\varphi$ consists of the unordered pairs $\{x,y\}$ (with $x,y\in V_\varphi$, $x\neq y$) for which there exists $\ell\in[1,m]$ such that both $x$ and $y$ occur (free) in $\varphi_\ell$. Note that $G_\varphi$ does not depend on any order of the variables in ${\cal X}$. Note also that for every finite undirected graph $G$ there exists $\varphi \equiv \varphi_1 \wedge \cdots \wedge \varphi_m$ such that $G$ is isomorphic to $G_\varphi$.
Pattern matching $\varphi$, and executing the above query, can be done by a total deterministic \abb{ptt} ${\cal A}$ as follows, similarly to the general \abb{ptt} ${\cal A}$ above (as discussed before Theorem~\ref{thm:matchall}). Again, let $\lambda=(x_1,\dots,x_n)$ be an arbitrary order of the variables in ${\cal X}$. Pebbles with distinct colours $c_1,\dots,c_{n-1}$ are used to represent $x_1, \dots, x_{n-1}$, dropping them in that order. For every $j\in[1,n]$, when pebbles $c_1,\dots,c_{j-1}$ are dropped on the tree and the head is at a candidate position $u_j$ for the variable $x_j$, all \abb{mso} tests $\varphi_\ell$ are performed of which the free variables are in $\{x_1,\dots,x_j\}$ (and that have not been tested before). Thus, when ${\cal A}$ has enumerated a sequence $u_1,\dots,u_n$, it can compute the boolean value of $\varphi(u_1,\dots,u_n)$. For each match $u_1,\dots,u_n$ the tree~$r$ is generated, such that for every occurrence of the variable $x_i$ in $r$ the subtree rooted at $u_i$ is generated, by a separate process; that is straightforward, even when $c_i$ is invisible: lift pebbles $c_{n-1},\dots,c_{i+1}$ one by one (in that order),
and then access $c_i$ and output $t|_{u_i}$. Note that, as before, the matches are generated in the lexicographic document order induced by the order $\lambda$.
It remains to determine which are the visible and invisible pebbles, keeping in mind that we wish to use as many invisible pebbles as possible for matching. To do the \abb{mso} tests at position $u_j$ all pebbles $c_i$ for which $\{x_i,x_j\}\in E_\varphi$ and $i<j$ should be observable. Hence all such pebbles under the topmost pebble $c_{j-1}$ must be visible. These are the pebbles corresponding to the set \[ \mathrm{vis}(\lambda) = \{x_i \mid
\text{there exists }\{x_i,x_j\}\in E_\varphi \text{ such that } i+1<j \}. \] Thus, for ${\cal A}$ we define $C_\mathrm{v}=\{c_i\mid x_i\in \mathrm{vis}(\lambda)\}$ and $C_\mathrm{i}= \{c_i\mid x_i\notin \mathrm{vis}(\lambda)\}$. Note that $c_{n-1}\in C_\mathrm{i}$.
In the case where the order $\lambda=(x_1,\dots,x_n)$ of the variables is irrelevant, we may want to determine an optimal order. A finite undirected graph $G=(V,E)$ will be called a \emph{union of paths} if it is acyclic and has only vertices of degree at most $2$. Intuitively this means that each connected component of $G$ is a path. Thus, clearly, there is an order $v_1,\dots,v_p$ of the vertices of $G$ such that for all $i,j\in[1,p]$ with $i< j$, if $\{v_i,v_j\}\in E$ then $i+1=j$ (repeatedly pick a vertex of degree 0 or 1, and remove it from the graph together with all its incident edges). We will call this an \emph{invisible order} of the vertices of $G$. Note that a graph is a union of paths if and only if it has an invisible order. Note also that every subgraph of $G$ is also a union of paths.
For an arbitrary finite undirected graph $G=(V,E)$, let us now say that a set $W \subseteq V$ of vertices of $G$ is a \emph{visible set} of $G$ if the subgraph of $G$ induced by $V\setminus W$, denoted by $G[V\setminus W]$, is a union of paths. By the last sentence of the previous paragraph, every superset of a visible set is also a visible set.
\begin{lemma}\label{lem:vis} A set of variables $W \subseteq V_\varphi$ is a visible set of $G_\varphi$ if and only if there is an order $\lambda$ of $V_\varphi$ such that $\mathrm{vis}(\lambda)\subseteq W$. \end{lemma}
\begin{proof} (If) It is easy to verify that every $\mathrm{vis}(\lambda)$ is a visible set of $G_\varphi$. In fact, for all $i<j$, if $x_i,x_j\notin \mathrm{vis}(\lambda)$ and $\{x_i,x_j\}\in E_\varphi$, then $i+1=j$.
(Only if) Define the order $\lambda$ on $V_\varphi$ as follows. First list the vertices of $W$ in any order. Then list the remaining vertices according to an invisible order of the vertices of $G_\varphi[V_\varphi\setminus W]$. Obviously $\mathrm{vis}(\lambda) \subseteq W$. \end{proof}
\begin{theorem}\label{thm:matching} Pattern $\varphi \equiv \xp{\beta}(\varphi_1, \dots, \varphi_m)$ can be matched by a total deterministic \abb{v$_k$i-ptt} where $k = \#(W)$ for a visible set $W$ of $G_\varphi$. In particular, if $G_\varphi$ is a union of paths, then $\varphi$ can be matched by a total deterministic \abb{i-ptt}. \end{theorem}
\begin{proof} By Lemma~\ref{lem:vis} there is an order $\lambda$ of $V_\varphi$ such that $\mathrm{vis}(\lambda)\subseteq W$. Hence at most $\#(W)$ visible pebbles suffice. If $G_\varphi$ is a union of paths, then $W=\varnothing$ is a visible set. \end{proof}
Lemma~\ref{lem:vis} shows that finding an order $\lambda$ for which $\mathrm{vis}(\lambda)$ is of minimal size, is the same as finding a visible set $W$ of minimal size. Unfortunately, this is an NP-complete problem. More precisely, the problem whether for a given graph $G=(V,E)$ and a given number $k$ there is a set of vertices $V'\subseteq V$ with \mbox{$\#(V')\geq k$} such that $G[V']$ is a union of paths, is NP-complete (see Problem~GT21 of~\cite{GJ}).
We now give some examples of visible sets of a graph $G$. It suffices to take as visible vertices those of degree $\ge 3$ in $G$ (plus one vertex in each connected component that is a cycle). But often one can choose a smaller set.
\newcommand{\ladder}{ \draw (0,2) -- (0,0) -- (4,0) -- (4,2) -- (0,2); \draw (1,2) -- (1,0); \draw (2,2) -- (2,0); \draw (3,2) -- (3,0); \draw [fill] (0,2) circle [radius=0.07]; \draw [fill] (0,1) circle [radius=0.07]; \draw [fill] (0,0) circle [radius=0.07]; \draw [fill] (1,2) circle [radius=0.07]; \draw [fill] (1,1) circle [radius=0.07]; \draw [fill] (1,0) circle [radius=0.07]; \draw [fill] (2,2) circle [radius=0.07]; \draw [fill] (2,1) circle [radius=0.07]; \draw [fill] (2,0) circle [radius=0.07]; \draw [fill] (3,2) circle [radius=0.07]; \draw [fill] (3,1) circle [radius=0.07]; \draw [fill] (3,0) circle [radius=0.07]; \draw [fill] (4,2) circle [radius=0.07]; \draw [fill] (4,1) circle [radius=0.07]; \draw [fill] (4,0) circle [radius=0.07]; }
\newcommand{\smallladder}{ \draw (0,2) -- (0,0) -- (3,0) -- (3,2) -- (0,2); \draw (1.5,2) -- (1.5,0); \draw [fill] (0,2) circle [radius=0.07]; \draw [fill] (0,1) circle [radius=0.07]; \draw [fill] (0,0) circle [radius=0.07]; \draw [fill] (1.5,2) circle [radius=0.07]; \draw [fill] (1.5,1) circle [radius=0.07]; \draw [fill] (1.5,0) circle [radius=0.07]; \draw [fill] (3,2) circle [radius=0.07]; \draw [fill] (3,1) circle [radius=0.07]; \draw [fill] (3,0) circle [radius=0.07]; }
\begin{figure}
\caption{Visible sets of different sizes.}
\label{fig:vissets}
\end{figure}
\begin{figure}
\caption{Three visible sets of minimal size.}
\label{fig:minsizevis}
\end{figure}
\begin{example} If $G$ is a cycle or a star, then it has a visible set $W$ with $\#(W)=1$ (for a cycle any singleton is a visible set, and for a star the visible set $W$ consists of the centre vertex).
In Figs.~\ref{fig:vissets} and~\ref{fig:minsizevis} we show graphs with the vertices of a visible set $W$ encircled. For the graph $G$ in Fig.~\ref{fig:vissets}, the upper left $W$ consists of all vertices of degree~3. It is not minimal, in the sense that it has a proper subset that is also a visible set, as shown at the upper right. This one \emph{is} minimal, because dropping one of the vertices from $W$ produces a vertex of degree~3 in the complement. Another minimal visible set (of the same size) is shown at the lower left: dropping the leftmost vertex of $W$ produces a cycle, and dropping one of the other vertices produces two vertices of degree~3. Finally, a visible set of size~3 is shown at the lower right. It is of minimal size, i.e., $\#(W)\geq 3$ for every visible set $W$ of $G$. In fact, removing a vertex of degree~2 from $G$ leaves a graph with two disjoint cycles that both must be broken, whereas removing a vertex of degree~3 from~$G$ either leaves a graph with two disjoint cycles or a graph with a cycle and a vertex of degree~3 of which the neighbourhood is disjoint with that cycle. Thus, any pattern $\varphi$ such that $G_\varphi$ is isomorphic to $G$ can be matched with three visible pebbles.
Visible sets of minimal size need not be unique. For the graph in Fig.~\ref{fig:minsizevis}, three different visible sets of minimal size are shown. \end{example}
If we allow matches to occur more than once in the output, then Theorem~\ref{thm:matching} is not optimal (still assuming that the order $\lambda$ is irrelevant). Using the boolean laws, the \abb{mso} formula $\varphi \equiv \xp{\beta}(\varphi_1, \dots, \varphi_m)$ can be written as a disjunction $\varphi \equiv \psi_1\vee \cdots \vee \psi_k$ where each $\psi_i$ is a conjunction of some of the formulas $\varphi_1, \dots, \varphi_m$ or their negations. Now the \abb{ptt} ${\cal A}$ can execute the queries `$\mbox{\tt for } {\cal X} \mbox{ \tt where } \psi_i \mbox{ \tt return } r$' consecutively for $i=1,\dots,k$. Obviously, $G_{\psi_i}$ is a subgraph of $G_\varphi$ for every $i\in[1,k]$. Hence every visible set of $G_\varphi$ is also a visible set of $G_{\psi_i}$, and so the minimal size of the visible sets of $G_{\psi_i}$ is at most the minimal size of the visible sets of $G_\varphi$. Thus, pattern matching formulas $\psi_1,\dots,\psi_k$ consecutively needs at most as many visible pebbles as pattern matching $\varphi$, but it may need less. As a simple example, let $\varphi \equiv \varphi_1(x,y)\wedge (\varphi_2(y,z) \vee \varphi_3(x,z))$. Then $G_\varphi$ is a triangle, which needs one visible pebble. But $\varphi \equiv \psi_1\vee \psi_2$ where $\psi_1\equiv \varphi_1(x,y)\wedge \varphi_2(y,z)$ and $\psi_2\equiv \varphi_1(x,y)\wedge \varphi_3(x,z)$. Both $G_{\psi_1}$ and $G_{\psi_2}$ are (unions of) paths, which do not need visible pebbles. Thus, $\varphi$ can be matched by an \abb{i-ptt}. However, all matches for which $\varphi_1\wedge \varphi_2\wedge \varphi_3$ holds occur twice in the output.
We finally discuss another way to reduce the number of visible pebbles. Suppose that, for some $i\in[1,m]$, the formula $\varphi_i$ has exactly two free variables $x,y\in {\cal X}$. Thus, the edge $\{x,y\}$ is in $E_\varphi$. Suppose moreover that the trip defined by $\varphi_i(x,y)$ is functional. Suppose finally that $W$ is a visible set of $G_\varphi$ with $x,y\in W$. Then all other edges of $G_\varphi$ incident with $y$ can be redirected to $x$, and $y$ can be dropped from $W$. To be precise, every formula $\varphi_j$ that contains the free variable $y$ can be changed into the formula $\forall y (\varphi_i(x,y) \to \varphi_j)$ that contains the free variable $x$ instead of $y$. The resulting query is obviously equivalent to the given one.
\section{Pebble Forest Transducers}\label{sec:pft}
The \abb{ptt} transforms ranked trees, whereas XML documents are unranked forests. However, it is not difficult to use, or slightly adapt, the \abb{ptt} for the transformation of forests. The most obvious, and well-known way to do this, is to encode the forests as binary trees. Let $\family{enc}'$ be the class of all encodings ${\rm enc}'$ (one encoding for each input alphabet $\Sigma$), and let $\family{dec}$ be the class of all decodings ${\rm dec}$ (one decoding for each output alphabet $\Delta$). Then we can view the class $\family{enc}'\circ\VIPTT{k}\circ\family{dec}$ as the class of forest transductions realized by \abb{v$_k$i-ptt}'s. For the input forest $f$ this is a natural definition, because it is quite easy to visualize a \abb{ptt} walking on ${\rm enc}'(f)$ as actually walking on $f$ itself. For the output forest $g$ this is also a natural definition, as it is, in fact, easy to transform a \abb{ptt} that outputs ${\rm enc}(g)$ into a (slightly adapted type of) \abb{ptt} that directly outputs $g$ itself: change every output rule $\tup{q,\sigma,j,b}\to \delta(\tup{q_1,{\rm stay}},\tup{q_2,{\rm stay}})$ into $\tup{q,\sigma,j,b}\to \delta(\tup{q_1,{\rm stay}})\tup{q_2,{\rm stay}}$, and every output rule $\tup{q,\sigma,j,b}\to e$ into $\tup{q,\sigma,j,b}\to \varepsilon$. The definition is also natural with respect to typechecking, because a forest language $L$ is regular if and only if the tree language ${\rm enc}(L)$ is regular, and similarly for ${\rm enc}'(L)$. Since the transformation of the involved grammars can obviously be done in polynomial time, Theorem~\ref{thm:typecheck} in Section~\ref{sec:typechecking} also holds for \abb{v$_k$i-ptt} as forest transducers.
We observe here that the class $\family{enc}'\circ\VIPTT{k}\circ\family{dec}$ does not depend on the chosen encodings and decodings, i.e., $\family{enc}'$ can be replaced by the class $\family{enc}$ of all encodings ${\rm enc}$, and $\family{dec}$ by the class $\family{dec}'$ of all decodings ${\rm dec}'$. In fact, a \abb{ptt} that walks on ${\rm enc}'(f)$ can easily be simulated by one that walks on ${\rm enc}(f)$: the original label $\sigma^{kl}$ can be determined by inspecting the children of the node with label~$\sigma$. Vice versa, a \abb{ptt} that walks on ${\rm enc}(f)$ can be simulated by one that walks on ${\rm enc}'(f)$: a node with label, e.g., $\sigma^{01}$ represents the original node and its first child with label $e$; the difference between these nodes can be stored in the finite state and in the pebble colours of the simulating \abb{ptt}. Moreover, a \abb{ptt} that outputs ${\rm enc}'(g)$ can easily be simulated by one that outputs ${\rm enc}(g)$: change, e.g., the rule $\tup{q,\sigma,j,b}\to \delta^{01}(\tup{q',{\rm stay}})$ into the two rules $\tup{q,\sigma,j,b}\to \delta(\tup{p,{\rm stay}},\tup{q',{\rm stay}})$ and $\tup{p,\sigma,j,b}\to e$ where $p$ is a new state. Vice versa, a \abb{ptt} ${\cal A}$ that outputs ${\rm enc}(g)$ can be simulated by a \abb{ptt} ${\cal A}$ that outputs ${\rm enc}'(g)$, but that requires look-ahead (Theorem~\ref{thm:look-ahead}), as follows. If ${\cal A}$ has an output rule $\tup{q,\sigma,j,b}\to \delta(\tup{q_1,{\rm stay}},\tup{q_2,{\rm stay}})$, then ${\cal A}'$ has the rule $\tup{q,\sigma,j,b,{\cal B}_{01}}\to \delta^{01}(\tup{q_2,{\rm stay}})$ where ${\cal B}_{01}$ is a look-ahead test that finds out whether ${\cal A}$ can generate $e$ when started in state $q_1$ in the current situation. To be precise, ${\cal B}_{01}$ is obtained from ${\cal A}$ by changing its set of initial states into $\{q_1\}$ and removing all output rules that do not output $e$. And of course, ${\cal A}'$ has similar rules for the other symbols~$\delta^{ij}$.
So far so good, in particular for the input forest $f$. There is, however, another natural possibility for the output forest $g$, as introduced and investigated in~\cite{PerSei} for macro tree transducers. It is quite natural to allow a \abb{ptt} that directly outputs~$g$, as discussed above, to not only have output rules with right-hand sides $\delta(\tup{q_1,{\rm stay}})\tup{q_2,{\rm stay}}$ and $\varepsilon$, but also right-hand sides $\tup{q_1,{\rm stay}}\tup{q_2,{\rm stay}}$ and $\delta(\tup{q',{\rm stay}})$ that realize the concatenation of forests and the formation of a tree out of a forest.
Accordingly we define a \emph{tree-walking forest transducer with nested pebbles} (abbreviated \abb{pft}) to be the same as a \abb{ptt} ${\cal M}$, except that its output alphabet is unranked, and its output rules are of the form $\tup{q,\sigma,b,j} \to \zeta$ with $\zeta = \delta(\tup{q',\mathrm{stay}})$ introducing a new node with label $\delta$ and generating a forest from state $q'$, or $\zeta = \tup{q_1,\mathrm{stay}}\, \tup{q_2,\mathrm{stay}}$ concatenating two forests, or $\zeta = \varepsilon$ generating the empty forest. Note that a right-hand side $\delta(\tup{q_1,{\rm stay}})\tup{q_2,{\rm stay}}$ is also allowed, as it can easily be simulated in two steps.
Formally, an output form of the \abb{pft} ${\cal M}$ on an input tree $t$ is defined to be a forest in $F_\Delta(\xp{Con}(t))$. Let $s$ be an output form and let $v$ be a leaf of $s$ with label $\tup{q,u,\pi}\in \xp{Con}(t)$. If the rule $\tup{q,\sigma,b,j} \to \zeta$ is relevant to $\tup{q,u,\pi}$ then we write $s\Rightarrow_{t,{\cal M}} s'$ where $s'$ is obtained from $s$ as follows. If the rule is not an output rule, then the label of $v$ is changed in the same way as for the \abb{pta} and \abb{ptt}. If $\;\zeta = \delta(\,\tup{q',\mathrm{stay}}\,)$ then node $v$ is replaced by the subtree $\delta(\tup{q',u,\pi})$. If $\;\zeta = \tup{q_1,\mathrm{stay}}\, \tup{q_2,\mathrm{stay}}$ then node $v$ is replaced by the two-node forest $\tup{q_1,u,\pi}\tup{q_2,u,\pi}$. And if $\zeta = \varepsilon$ then the node $v$ is removed from~$s$. The transduction realized by ${\cal M}$ consists of all $(t,s)\in T_\Sigma\times F_\Delta$ such that $\tup{q_0,\mathrm{root}_t} \Rightarrow^*_{t,{\cal M}} s$ for some $q_0\in Q_0$. Thus, we have defined the \abb{pft} as a transformer of ranked trees into unranked forests. The corresponding classes of transductions are denoted by $\VIPFT{k}$. For forest transformations one can of course consider the classes $\family{enc}'\circ \VIPFT{k}$.
\begin{lemma}\label{lem:pftvsptt} For every $k\geq 0$, \[ (1) \;\;\VIPTT{k}\circ\family{dec} \subseteq \VIPFT{k} \quad\text{and}\quad (2) \;\;\VIPFT{k}\circ\family{enc} \subseteq \VIPTT{k}\circ\family{I-dPTT} \] and similarly for the deterministic case. \end{lemma}
\begin{proof} Inclusion~(1) is obvious from the discussion above: change every rule $\tup{q,\sigma,j,b}\to \delta(\tup{q_1,{\rm stay}},\tup{q_2,{\rm stay}})$ into $\tup{q,\sigma,j,b}\to \delta(\tup{q_1,{\rm stay}})\tup{q_2,{\rm stay}}$, and every rule $\tup{q,\sigma,j,b}\to e$ into $\tup{q,\sigma,j,b}\to \varepsilon$.
The proof of inclusion~(2) is similar to the proof in~\cite{PerSei} that every macro forest transducer can be simulated by two macro tree transducers. Let ${\cal M}$ be a \abb{v$_k$i-pft} with (unranked) output alphabet $\Delta$. Let $\Delta_1$ be the ranked alphabet $\Delta\cup\{@^{(2)},e^{(0)}\}$, where every element of $\Delta$ has rank~1. We now obtain the \mbox{\abb{v$_k$i-ptt}} ${\cal M}'$ from ${\cal M}$ by changing every output rule $\tup{q,\sigma,b,j} \to \tup{q_1,{\rm stay}}\, \tup{q_2,{\rm stay}}$ into $\tup{q,\sigma,b,j} \to @(\tup{q_1,{\rm stay}},\tup{q_2,{\rm stay}})$ and every output rule $\tup{q,\sigma,b,j} \to \varepsilon$ into $\tup{q,\sigma,b,j} \to e$. Let `${\rm flat}$' be the mapping from $T_{\Delta_1}$ to $F_\Delta$ defined by ${\rm flat}(@(t_1,t_2)={\rm flat}(t_1){\rm flat}(t_2)$, ${\rm flat}(\delta(t))=\delta({\rm flat}(t))$ and ${\rm flat}(e)=\varepsilon$. Then obviously $\tau_{\cal M}=\tau_{{\cal M}'}\circ {\rm flat}$. Thus, it remains to show that the mapping ${\rm flat}\circ{\rm enc}$ is in $\family{I-dPTT}$. We will prove this after Theorem~\ref{thm:dtl}. It is, in fact, not hard to see that ${\rm flat}\circ{\rm enc}$ is even in $\family{dTT}$. \end{proof}
\smallpar{Typechecking} The inverse type inference problem and the typechecking problem are defined for \abb{pft}'s as in Section~\ref{sec:typechecking}, except that $G_\mathrm{out}$ is a regular forest grammar rather than a regular tree grammar. It follows from Lemma~\ref{lem:pftvsptt}(2), together with Lemma~\ref{lem:nul-decomp}, Theorem~\ref{thm:decomp}, and Propositions~\ref{prop:invtypeinf} and~\ref{prop:typecheck} that these problems can be solved for \abb{v$_k$i-pft}'s in $(k\!+\!4)$-fold and $(k\!+\!5)$-fold exponential time. However, it is shown in~\cite[Section~7]{Eng09} that they can be solved for \abb{v$_k$-pft}'s in the same time as for \abb{v$_k$-ptt}'s, i.e., in ($k+1$)-fold and ($k+2$)-fold exponential time, respectively. This is due to the fact (shown in~\cite[Lemma~4]{Eng09}) that inverse type inference for the mapping ${\rm flat}\circ{\rm enc}$ can be solved in polynomial time, cf. the proof of Lemma~\ref{lem:pftvsptt}. For exactly the same reason a similar result holds for \abb{v$_k$i-pft}'s. In other words, Theorem~\ref{thm:typecheck} also holds for \abb{v$_k$i-pft}'s.
\begin{theorem}\label{thm:typecheck-pft} For fixed $k\geq 0$, the inverse type inference problem and the typechecking problem are solvable for \abb{v$_k$i-pft}'s in $(k\!+\!2)$-fold and $(k\!+\!3)$-fold exponential time, respectively. \end{theorem}
\smallpar{\small{MSO} tests} It should be clear that Theorem~\ref{thm:mso} also holds for the \abb{pft}, as \abb{mso} tests only concern the input tree.
\smallpar{Pattern matching} Pattern matching for forests can be defined in exactly the same way as we did for trees in Section~\ref{sec:pattern}. Since, obviously, Lemma~\ref{lem:encmso} also holds for arbitrary $n$-ary patterns instead of trips, we may however assume that the input forest $f$ over $\Sigma$ of the query \[ \mbox{\tt for } {\cal X} \mbox{ \tt where } \varphi \mbox{ \tt return } r \] is encoded as a binary tree $t={\rm enc}'(f)$ over $\Sigma'$ for which we execute the query \[ \mbox{\tt for } {\cal X} \mbox{ \tt where } \varphi' \mbox{ \tt return } r \] where $\varphi'$ is the encoding of the formula $\varphi$ according to Lemma~\ref{lem:encmso}. Consequently, we can use a \abb{pft} to execute this query and produce for each match of $\varphi'(x_1,\dots,x_n)$ the required output $r$. We may now assume that $r$ is a forest rather than a tree, and we may for simplicity assume that $r$ is of the form $\mu(x_1\cdots x_n)$ for some output symbol $\mu$. Thus, the output for each match
$\varphi'(u_1,\dots,u_n)$ is $\mu(f|_{u_1}\cdots f|_{u_n})$, and the output forest is of the form $s= r_1r_2\cdots r_ke$ where $r_1,\dots,r_k$ are the outputs corresponding to all the matches. Note that $e$ is another output symbol, and so $\Delta=\Sigma\cup\{\mu,e\}$. It should be clear how the total deterministic \abb{ptt} ${\cal A}$ in Section~\ref{sec:pattern} can be changed into a total deterministic \abb{pft} that executes this query. The only small problem
is that ${\cal A}$ outputs the encoded subtrees $t|_{u_i}$ rather than the required
subtrees $f|_{u_i}$. However, a \abb{pft} can easily transform an encoded forest
${\rm enc}'(f|_u)$ into the forest~$f|_u$, using rules $\tup{q,\sigma^{11},j,b}\to \sigma(\tup{q,{\rm down}_1})\tup{q,{\rm down}_2}$, $\tup{q,\sigma^{01},j,b}\to \sigma\tup{q,{\rm down}_1}$, $\tup{q,\sigma^{10},j,b}\to \sigma(\tup{q,{\rm down}_1})$, and $\tup{q,\sigma^{00},j,b}\to \sigma$.
From this it should be clear that Theorems~\ref{thm:matchall} and~\ref{thm:matching} also hold for forest pattern matching and \abb{pft}.
\smallpar{Expressive power} As in~\cite{PerSei}, the \abb{pft} is more powerful than the \abb{ptt}. In particular, the \abb{i-pft} is more powerful than the \abb{i-ptt} that generates encoded forests, i.e., $\family{I-PTT}\circ \family{dec}$ is a proper subclass of $\family{I-PFT}$. In fact, it is well known (cf.~\cite[Lemma~7]{EngMan03} and~\cite[Lemma~5.40]{FulVog}), and easy to see, that the height of the output tree of a functional \abb{tt} ${\cal M}$ (which means that $\tau_{\cal M}$ is a function) is linearly bounded by the size of the input tree: otherwise ${\cal M}$ would be in a loop and would generate infinitely many output trees for that input tree. Since $\family{I-PTT} \subseteq \family{TT} \circ \family{TT}$ by Lemma~\ref{lem:nul-decomp}, this implies that for a functional \abb{i-ptt} the height of the output tree is exponentially bounded by the size of the input tree. However, the following total deterministic \abb{i-pft} ${\cal M}_\text{2exp}$ outputs, for an input tree of size $n$, a forest of length double exponential in $n$. Since the height of the encoded output forest is at least the length of that forest, this transformation cannot be realized by an \abb{i-ptt} that generates encoded forests. The transducer~${\cal M}_\text{2exp}$ is similar to the \abb{i-ptt} ${\cal M}_\text{sib}$ of Example~\ref{ex:siberie}, assuming that there are large cities only. Thus, using its pebbles, it enumerates $2^n$ itineraries (where $n$ is the number of intermediate cities). However, after marking an itinerary, it does not output the itinerary, but instead branches into two identical subprocesses that continue the enumeration. After the last itinerary, ${\cal M}_\text{2exp}$ is branched into a forest of $2^{2^n}$ copies of itself, each of which finally outputs one symbol. Imitating~${\cal M}_\text{sib}$, the \abb{i-pft} ${\cal M}_{\text{2exp}}$ first walks to the leaf:
$\tup{q_\mathrm{start},\sigma_1,j,\varnothing} \to \tup{q_\mathrm{start},{\rm down}_1}$
$\tup{q_\mathrm{start},\sigma_0,1,\varnothing} \to \tup{q_1,{\rm up}}$
\noindent Then, in state $q_1$, it marks as many cities as possible:
$\tup{q_1,\sigma_1,1,\varnothing} \to \tup{q_1,{\rm drop}_c;{\rm up}}$
$\tup{q_1,\sigma_1,0,\varnothing} \to
\tup{q_\mathrm{next},{\rm down}_1}
\tup{q_\mathrm{next},{\rm down}_1}$
\noindent In state $q_\mathrm{next}$ it continues the search for itineraries by unmarking the most recently marked city; when arriving at the leaf it outputs $e$:
$\tup{q_\mathrm{next},\sigma_1,1,\varnothing} \to \tup{q_\mathrm{next},{\rm down}_1}$
$\tup{q_\mathrm{next},\sigma_1,1,\{c\}} \to \tup{q_1,{\rm lift}_c;{\rm up}}$
$\tup{q_\mathrm{next},\sigma_0,1,\varnothing} \to e$
\noindent This ends the description of the \abb{i-pft} ${\cal M}_{\text{2exp}}$.
\section{Document Transformation}\label{sec:document}
In this section we compare the \abb{i-ptt} and \abb{i-pft} to the document transformation languages \abb{DTL} and \abb{TL}, which transform (unranked) forests. We prove that \abb{DTL} can be simulated by the \abb{i-ptt}, and that \abb{TL} has the same expressive power as the \abb{i-pft}.
The \emph{Document Transformation Language} \abb{DTL} was introduced and studied in \cite{ManNev00}. A \emph{program} in the \abb{DTL} framework is a tuple ${\cal P}=(\Sigma,\Delta,Q,Q_0,R)$ where $\Sigma$ and $\Delta$ are unranked alphabets, $Q$ is a finite set of states, $Q_0\subseteq Q$ is the set of initial states, and $R$ is a finite set of \emph{template rules} of the form $\tup{q,\varphi(x)} \to f$, where $f$ is a forest over $\Delta$, the leaves of which can additionally be labelled by a \emph{selector} of the form $\tup{q',\psi(x,y)}$; $q$ and $q'$ are states in $Q$, and $\varphi$ and $\psi$ are \abb{mso} formulas over $\Sigma$, with one and two free variables respectively. Such a rule can be applied in state $q$ at an input node $x$ that matches $\varphi$, i.e., satisfies $\varphi(x)$. Then program ${\cal P}$ outputs forest $f$, where each selector $\tup{q',\psi(x,y)}$ is recursively computed as the result of a sequence of copies of ${\cal P}$, started in state $q'$ at each of the nodes $y$ that satisfy $\psi(x,y)$, the nodes taken in pre-order (i.e., document order). Thus, ${\cal P}$ ``jumps'' from node $x$ to each node $y$, according to the trip defined by the \abb{mso} formula~$\psi$.
Formally, a configuration of ${\cal P}$ on input forest $t$ is a pair $\tup{p,u}$ where $u$ is a node of $t$ and $p$ is either a state or a selector of ${\cal P}$. An output form of ${\cal P}$ on $t$ is a forest in $F_\Delta(\xp{Con}(t))$, where $\xp{Con}(t)$ is the set of configurations of ${\cal P}$ on $t$. As usual, the computation steps of ${\cal P}$ on $t$ are formalized as a binary relation $\Rightarrow_{t,{\cal P}}$ on $F_\Delta(\xp{Con}(t))$. Let $s$ be an output form and let $v$ be a leaf of $s$ with label $\tup{q,u}\in \xp{Con}(t)$, where $q$ is a state of ${\cal P}$. Moreover, let $\tup{q,\varphi(x)} \to f$ be a template rule of ${\cal P}$ such that $t\models \varphi(u)$. Let $\theta_u(f)$ be the forest obtained from~$f$ by changing every selector $\tup{q',\psi(x,y)}$ into $\tup{\tup{q',\psi(x,y)},u}$. Then we write $s\Rightarrow_{t,{\cal P}} s'$ where $s'$ is obtained from $s$ by replacing the node $v$ by the forest $\theta_u(f)$. Now let $s$ be an output form and let $v$ be a leaf of $s$ with label $\tup{\tup{q',\psi(x,y)},u}$. Then we write $s\Rightarrow_{t,{\cal P}} s'$ where $s'$ is obtained from $s$ by replacing the node $v$ by the forest $\tup{q',u'_1}\cdots\tup{q',u'_\ell}$ where $u'_1,\dots,u'_\ell$ is the sequence of all nodes $u'$ of $t$, in document order, such that $t\models \psi(u,u')$. The transduction $\tau_{\cal P}$ realized by ${\cal P}$ is defined by $\tau_{\cal P} = \{(t,s)\in F_\Sigma\times F_\Delta \mid \exists\,q_0\in Q_0: \tup{q_0,\mathrm{root}_t} \Rightarrow^*_{t,{\cal P}} s\}$.
The \abb{dtl} program ${\cal P}$ is \emph{deterministic} if for every two rules $\tup{q,\varphi(x)} \to f$ and $\tup{q,\varphi'(x)} \to f'$ with the same state $q$, the tests $\varphi(x)$ and $\varphi'(x)$ are exclusive, meaning that the sites they define are disjoint.
We observe here that in \cite{ManNev00} the selectors have a more complicated form, which we will discuss after the next lemma.
We have defined the \abb{dtl} program such that the input $t$ is an unranked forest, and thus it can in particular be a ranked tree. It should be clear from Lemma~\ref{lem:encmso} (which also holds for sites instead of trips) that we may in fact restrict ourselves to ranked trees and assume that input forests are encoded as binary trees. Thus, \emph{from now on we assume that} in the above definition $\Sigma$ is a ranked alphabet and $t\in T_\Sigma$ is a ranked input tree. This allows us to compare \abb{dtl} programs with \abb{pft}'s.
Let $\family{DTL}$ denote the transductions realized by \abb{dtl} programs and $\family{dDTL}$ those realized by deterministic \abb{dtl} programs, from ranked trees to unranked forests. Thus, the class of forest transductions realized by \abb{dtl} programs is equal to $\family{enc}'\circ \family{DTL}$, and similarly for the deterministic case.
\begin{lemma}\label{lem:dtl} $\family{DTL} \subseteq \family{I-PFT}$ and $\family{dDTL} \subseteq \family{I-dPFT}$. \end{lemma}
\begin{proof} Let ${\cal P}=(\Sigma,\Delta,Q,Q_0,R)$ be a \abb{dtl} program. We construct an equivalent \abb{I-PFT} ${\cal M}$ with \abb{mso} tests, cf.\ Theorem~\ref{thm:mso}. It has the same alphabets $\Sigma$ and $\Delta$ as ${\cal P}$. Since ${\cal M}$ stepwise simulates ${\cal P}$, its set of states consists of the states and selectors of ${\cal P}$, plus the states that it needs to execute the subroutines discussed below. It has the same initial states $Q_0$ as ${\cal P}$. Moreover, it uses invisible pebbles of a single colour $\odot$, and never lifts its pebbles.
For an input tree $t$, the transducer ${\cal M}$ simulates a template rule $\tup{q,\varphi(x)} \to f$ in state $q$ at node $u$ of $t$ by first using an \abb{mso} head test to check whether $t\models \varphi(u)$. With a positive test result, it calls a subroutine $S$ that outputs the $\Delta$-labelled nodes of the right-hand side $f$. The subroutine $S$ is started by ${\cal M}$ in state~$[f]$. If its state is of the form $[sf']$, for a tree $s$ and a forest $f'$, it uses a rule $\tup{[sf'],\sigma,j,b} \to \tup{[s],\mathrm{stay}}\,\tup{[f'],\mathrm{stay}}$, branching the computation. If the state is of the form $[\delta(f')]$, the rule is $\tup{[\delta(f')],\sigma,j,b} \to \delta(\tup{[f'],\mathrm{stay}})$, and if it is of the form $[\varepsilon]$, the rule is $\tup{[\varepsilon],\sigma,j,b}\to \varepsilon$. If the state is of the form $[\tup{q',\psi(x,y)}]$, for a selector $\tup{q',\psi(x,y)}$, the subroutine $S$ returns control to (this copy of) ${\cal M}$ in state $\tup{q',\psi(x,y)}$. In that state, ${\cal M}$ first drops a pebble $\odot$ on the current node~$u$ and then calls a subroutine $S_{q',\psi}$ that finds all nodes $u'$ in the input tree $t$ for which $\psi(u,u')$ holds. The subroutine does this by performing a depth-first traversal of $t$, starting at the root, checking in each node $u'$ whether $t\models \psi(u,u')$ using an \abb{mso} test on the observable configuration. If true, then $S_{q',\psi}$ branches into two concatenated processes. The left branch returns control to ${\cal M}$ in state~$q'$, and the right branch continues the depth-first search. When the search ends, $S_{q',\psi}$ outputs $\varepsilon$. Thus, $S_{q',\psi}$ transforms the configuration $\tup{\tup{q',\psi(x,y)},u,\pi}$ of ${\cal M}$ into the forest of configurations $\tup{q',u'_1,\pi}\cdots\tup{q',u'_\ell,\pi}$, where $u'_1,\dots,u'_\ell$ are all such nodes $u'$, in document order. With this definition of ${\cal M}$, it should be clear that $\tau_{\cal M}=\tau_{\cal P}$. \end{proof}
The selectors in \cite{ManNev00} are more general than those defined above. They can be of the form $\tup{q'_1,\psi_1(x,y), \dots, q'_m,\psi_m(x,y)}$, such that the \abb{mso} formulas $\psi_1(x,y),\dots,\psi_m(x,y)$ are mutually exclusive, i.e., the trips they define are mutually disjoint. Let $\psi(x,y)$ be the disjunction of all $\psi_i(x,y)$, $i\in[1,m]$. The execution of the above selector at node $u$ of the input tree results in the forest $\tup{q'_{i_1},u'_1}\cdots\tup{q'_{i_\ell},u'_\ell}$ where $u'_1,\dots,u'_\ell$ is the sequence of all nodes $u'$ of $t$ in document order such that $t\models \psi(u,u')$, and for every $j\in[1,\ell]$, $i_j$ is the unique number in $[1,m]$ such that $t\models \psi_{i_j}(u,u'_j)$. It should be clear that Lemma~\ref{lem:dtl} is still valid with these more general selectors. To execute the above selector, the \abb{i-pft} ${\cal M}$ calls subroutine $S_{q'_1,\psi_1,\dots,q'_m,\psi_m}$ which in each node $u'$ tests each of the formulas $\psi_i(u,u')$; if $\psi_i(u,u')$ is true, then the subroutine branches in two, in the first branch returning control to ${\cal M}$ in state $q_i$.
To compare $\family{DTL}$ to $\family{I-PTT}$ rather than $\family{I-PFT}$ we also consider \abb{dtl} programs that transform ranked trees. A \abb{dtl} program ${\cal P}=(\Sigma,\Delta,Q,Q_0,R)$ is \emph{ranked} if $\Sigma$ and $\Delta$ are both ranked alphabets, and every rule $\tup{q,\varphi(x)}\to f$ satisfies the following two restrictions:
\begin{enumerate} \item[(R1)] $f$ is a ranked tree in $T_\Delta(S)$ where $S$ is the set of selectors, and \item[(R2)] for every selector $\tup{q',\psi(x,y)}$ that occurs in $f$, every input tree $t\in T_\Sigma$, and every node $u\in N(t)$, if $t\models \varphi(u)$ then there is a unique node $v\in N(t)$ such that $t\models \psi(u,v)$. \end{enumerate}
In other words, the trip $T(\psi(x,y))$ is functional and, for fixed input tree $t\in T_\Sigma$, it is defined for every node of $t$ that satisfies $\varphi(x)$. Thus, execution of the selector $\tup{q',\psi(x,y)}$ results in a ``jump'' from node $x$ to exactly one node $y$. This clearly implies that all reachable output forms of ${\cal P}$ are ranked trees in $T_\Delta(\xp{Con}(t))$. Thus $\tau_{\cal P}\subseteq T_\Sigma\times T_\Delta$ is a ranked tree transformation. The class of transductions realized by ranked \abb{tl} programs will be denoted by $\family{DTL$_\text{r}$}$, and by $\family{dDTL$_\text{r}$}$ in the deterministic case.
\begin{corollary}\label{cor:dtlr} $\family{DTL$_\text{r}$} \subseteq \family{I-PTT}$ and $\family{dDTL$_\text{r}$} \subseteq \family{I-dPTT}$. \end{corollary}
\begin{proof} The proof is the same as the one of Lemma~\ref{lem:dtl}, except for the subroutines~$S$ and~$S_{q',\psi}$. The states of $S$ are now of the form $[s]$ where $s$ is a subtree of a right-hand side of a rule. Instead of the rules for states $[sf']$, $[\delta(f')]$, and $[\varepsilon]$, subroutine~$S$ has rules $\tup{[\delta(s_1,\dots,s_m)],\sigma,j,b}\to \delta(\tup{[s_1],{\rm stay}},\dots,\tup{[s_m],{\rm stay}})$ for every $\delta$ of rank $m$ and all trees $s_1,\dots,s_m$ (restricted to subtrees of right-hand sides). When subroutine $S_{q',\psi}$ finds a node $u'$ such that $t\models \psi(u,u')$ (and it always finds one by restriction (R2)), it returns control to ${\cal M}$ and does not continue the depth-first search. \end{proof}
It can, in fact, be shown that when output forests are encoded as binary trees, $\family{DTL}$ is included in $\family{I-PTT}$. Thus, instead of $\family{I-PFT}$ we consider the class $\family{I-PTT} \circ \family{dec}$ (which equals the class $\family{I-PTT} \circ \family{dec}'$), cf. Section~\ref{sec:pft}. The next theorem will not be used in what follows (except in the paragraph directly after the theorem).
\begin{theorem}\label{thm:dtl} $\family{DTL} \subseteq \family{I-PTT} \circ \family{dec}$ and $\family{dDTL} \subseteq \family{I-dPTT} \circ \family{dec}$. \end{theorem}
\begin{proof} Let ${\cal P}=(\Sigma,\Delta,Q,Q_0,R)$ be a \abb{dtl} program. The main difficulty in outputting the binary encoding ${\rm enc}(f)$ of a forest $f$ as opposed to the construction in the proof of Lemma~\ref{lem:dtl} is that here the first symbol $\delta$ of $f$ has to be determined before any other output can be generated. We reconsider that construction, and here essentially make a depth-first sequential search over nodes in the computation tree (implemented using a stack of pebbled nodes) instead of the recursive approach. In that way an \abb{i-ptt} ${\cal M}$ can simulate the leftmost computations of the \abb{dtl} program ${\cal P}$.
As unranked forests with selectors can be generated by the recursive definition $f ::= \delta(f)f' \mid \tup{q,\psi} f \mid \varepsilon$, where $f'$ is an alias of $f$, \abb{DTL} rules are of the form $\tup{q,\varphi(x)} \to f$, where $f$ is $\delta(f_1)f_2$, $\tup{q,\psi} f'$, or $\varepsilon$. The set of states of the transducer ${\cal M}$ to be constructed consists of the states of ${\cal P}$ and all states $[f]$ where $f$ is a subforest of a right-hand side of a rule of ${\cal P}$, plus the states of the subroutines $S'_{q',\psi}$ and $S''_{q',\psi}$ discussed below. The state $[f]$ is used to generate the binary encoding of the subforest $f$, similarly to its use by the subroutine $S$ in the proof of Lemma~\ref{lem:dtl}. The initial states of ${\cal M}$ are those of ${\cal P}$. The pebble colours used by ${\cal M}$ are $\tup{q,\psi,f}$ where $\tup{q,\psi} f$ occurs in the right-hand side of a rule of ${\cal P}$, and the special colour $\bot$. The state and pebble stack of ${\cal M}$ store a part of the output form of ${\cal P}$ that still has to be evaluated. The output alphabet of ${\cal M}$ is $\Delta\cup\{e\}$ where each $\delta\in\Delta$ has rank~2 and $e$ has rank~$0$.
The transducer ${\cal M}$ starts by dropping $\bot$ on the root. To simulate, in state~$q$, a rule $\tup{q,\varphi(x)} \to f$ of ${\cal P}$, it uses an \abb{MSO} head test to check whether $\varphi$ holds for the current node, and goes into state $[f]$. We consider the above three cases for $[f]$.
In state $[\tup{q',\psi} f']$, pebble $\tup{q',\psi,f'}$ is dropped on the current node $u$. As in the proof of Lemma~\ref{lem:dtl}, ${\cal M}$ then calls a subroutine $S'_{q',\psi}$ which, this time, finds \\[0.6mm] the first node $u'$ (in document order) for which $\psi(u,u')$ holds, where it returns control to ${\cal M}$ in state $q'$. If $S'_{q',\psi}$ does not find such a matching node $u'$, then it moves to the topmost pebble $\tup{q',\psi,f'}$, lifts it, and returns control to ${\cal M}$ in state $[f']$.
In state $[f]=[\delta(f_1)f_2]$, the root $\delta$ of the first tree of the forest is explicitly given, and this is captured by the \abb{I-PTT} output rule $\tup{[f],\sigma,j,b} \to \delta(\, \tup{[f_1],\mathrm{drop}_\bot}, \tup{[f_2],\mathrm{stay}}\,)$. The symbol $\bot$ is pushed, and never popped afterwards, making the stack of pebbles effectively empty: the first copy of the transducer evaluates $f_1$ as left child of $\delta$. The second copy inherits the stack and evaluates $f_2$ as right child of $\delta$, together with all postponed duties as stored in the stack of pebbles. This will generate the siblings of $\delta$ in the original forest.
In state $[\varepsilon]$, the transducer ${\cal M}$ determines the colour of the topmost pebble, using an \abb{mso} test on the observable configuration. If it is $\bot$, it outputs $e$ for the empty forest. Otherwise it calls subroutine $S''_{q',\psi}$ to continue the search corresponding to the topmost pebble $\tup{q',\psi,f'}$. That subroutine finds the first node $u'$ after the current node $u$ (in document order) for which $\psi(v,u')$ holds, where $v$ is the position of the topmost pebble. Similar to $S'_{q',\psi}$, if a matching node is found it returns control to ${\cal M}$ in state $q'$, and otherwise it lifts the topmost pebble and returns control to ${\cal M}$ in state $[f']$.
This ends the description of ${\cal M}$. To understand its correctness, we show how the output forms of ${\cal M}$ represent output forms of ${\cal P}$. We disregard the output forms of ${\cal M}$ that contain states of the subroutines $S'_{q',\psi}$ and $S''_{q',\psi}$, and view the execution of such a subroutine as one big computation step of ${\cal M}$ that (deterministically) changes one configuration into another. The mapping `$\xp{rep}$' from such restricted output forms of ${\cal M}$ to output forms of ${\cal P}$ is defined as follows. The $\Delta$-labelled part of the output form of ${\cal M}$ is decoded, i.e., $\xp{rep}(e)=\varepsilon$ and $\xp{rep}(\delta(s_1,s_2))=\delta(\xp{rep}(s_1))\xp{rep}(s_2)$. It remains to define `$\xp{rep}$' for the configurations on an input tree $t$ that occur in the restricted output forms of~${\cal M}$, i.e., for every configuration $\tup{p,u,\pi}$ where $p$ is a state $q$ of ${\cal P}$ or a state $[f]$. We will write $\xp{rep}(p,u,\pi)$ instead of $\xp{rep}(\tup{p,u,\pi})$. The definition is by induction on the structure of $\pi$, of which the topmost pebble is of the form $(v,\bot)$ or $(v,\tup{q',\psi,f'})$.
For a state $[f]$, we define $\xp{rep}([f],u,\pi(v,\bot))=\theta_u(f)$ and \[ \xp{rep}([f],u,\pi(v,\tup{q',\psi,f'}))= \theta_u(f)\tup{q',u'_1}\cdots\tup{q',u'_\ell}\xp{rep}([f'],v,\pi) \] where $u'_1,\dots,u'_\ell$ are all nodes $u'$ after $u$ (in document order) such that $t\models \psi(v,u')$. Note that $\xp{rep}([f],u,\pi)=\theta_u(f)\xp{rep}([\varepsilon],u,\pi)$ because $\theta_u(\varepsilon)=\varepsilon$, and hence $\xp{rep}([f_1f_2],u,\pi)=\theta_u(f_1)\xp{rep}([f_2],u,\pi)$. For a state $q$ of ${\cal P}$ we define $\xp{rep}(q,u,\pi)=\tup{q,u}\xp{rep}([\varepsilon],u,\pi)$.
It is now straightforward to prove, for every initial state $q_0$ of ${\cal P}$, every input tree $t$, and every output form $s$ of ${\cal P}$, that $\tup{q_0,\mathrm{root}_t} \Rightarrow^*_{t,{\cal P}} s$ if and only if there exists a restricted output form $s'$ of ${\cal M}$ such that $\tup{q_0,\mathrm{root}_t,(\mathrm{root}_t,\bot)} \Rightarrow^*_{t,{\cal M}} s'$ and $\xp{rep}(s')=s$. The proof of the if-direction of this equivalence is by induction on the length of the computation, and consists of four cases, depending on the state of the configuration of ${\cal M}$ that is rewritten, as discussed above, viz., $q$, $[\tup{q',\psi} f']$, $[\delta(f_1)f_2]$, or $[\varepsilon]$. From the last two cases it follows that for every restricted output form $s'$ of ${\cal M}$ there exists a restricted output form $s''$ of ${\cal M}$ such that $s' \Rightarrow^*_{t,{\cal M}} s''$, $\xp{rep}(s'')=\xp{rep}(s')$, and the states of ${\cal M}$ that occur in $s''$ are either states $q$ of ${\cal P}$ or states of the form $[\tup{q',\psi}f']$. In the only-if-direction we only consider leftmost computations of ${\cal P}$, i.e., computations in which always the first configuration of the output form (in pre-order) is rewritten. If $\xp{rep}(s')=\xp{rep}(s'')=s$, with $s''$ as above, then the first configuration of ${\cal M}$ in $s''$ corresponds to the first configuration of ${\cal P}$ in $s$, and the proof is similar to the first two cases of the proof of the if-direction. The details are left to the reader. Since $\xp{rep}(s')={\rm dec}(s')$ for every output tree $s'$ of ${\cal M}$, the above equivalence implies that $\tau_{\cal M} \circ {\rm dec}=\tau_{\cal P}$. \end{proof}
We are now able to finish the proof of Lemma~\ref{lem:pftvsptt}(2). Consider the mapping ${\rm flat}: T_{\Delta_1}\to F_\Delta$ defined in that proof. It can be realized by the one-state deterministic \abb{dtl} program with rules $\tup{q,\mathrm{lab}_@(x)}\to \tup{q,{\rm down}_1(x,y)}\tup{q,{\rm down}_2(x,y)}$, $\tup{q,\mathrm{lab}_\delta(x)}\to \delta(\tup{q,{\rm down}_1(x,y)})$ for every $\delta\in\Delta$, and $\tup{q,\mathrm{lab}_e(x)}\to \varepsilon$. Hence, by Theorem~\ref{thm:dtl}, it is in $\family{I-dPTT} \circ \family{dec}$, which means that the mapping ${\rm flat}\circ{\rm enc}$ is in $\family{I-dPTT}$.
In \cite{ManBerPerSei} the language \abb{dtl} was extended to the \emph{Transformation Language}~\abb{tl} where the states have parameters that hold unevaluated forests, similar to macro tree transducers with outside-in parameter evaluation~\cite{EngVog85}. In a \abb{tl} program~${\cal P}=(\Sigma,\Delta,Q,Q_0,R)$, the set of states $Q$ is a ranked alphabet such that the initial states in $Q_0$ have rank~$0$. The rules of \abb{tl} program ${\cal P}$ are of the form $\tup{q,\varphi(x)}(z_1,\dots,z_n) \to f$, where $n=\operatorname{rank}_Q(q)$ and $z_1,\dots,z_n$ are the formal parameters of $q$, taken from a fixed infinite parameter set $Z=\{z_1,z_2,\dots\}$. The right-hand side $f$ of the rule is a forest of which the nodes can be labeled by a symbol from $\Delta$, by a selector $\tup{q',\psi(x,y)}$, or by a formal parameter $z_i$ with $i\in[1,n]$. A node labeled by $\tup{q',\psi(x,y)}$ must have $\operatorname{rank}(q')$ children, and a node labeled by parameter $z_i$ must be a leaf. Thus, in such a forest (called an \emph{action} in~\cite{ManBerPerSei}), selectors can be nested. We could as well allow in \abb{tl} the more general selectors discussed after Lemma~\ref{lem:dtl}, but we restrict ourselves to the usual selectors for simplicity (and because they are the selectors in~\cite{ManBerPerSei}). Determinism of program ${\cal P}$ is defined as for \abb{dtl}.
An output form of ${\cal P}$ on input forest $t$ is a forest of which the nodes can be labeled either by a symbol from $\Delta$, or by a configuration $\tup{q,u}$ or $\tup{\tup{q,\psi(x,y)},u}$ of ${\cal P}$ in which case the node must have $\operatorname{rank}(q)$ children. A node of an output form, or of a right-hand side of a rule, is said to be \emph{outermost} if all its proper ancestors are labelled by a symbol from $\Delta$. The computation steps of ${\cal P}$ are formalized as a binary relation on output forms, as follows (similar to the \abb{dtl} case). Let $s$ be an output form, and let $v$ be an outermost node of $s$ with label $\tup{q,u}$, where $q$ is a state of ${\cal P}$. Moreover, let $\tup{q,\varphi(x)}(z_1,\dots,z_n) \to f$ be a rule of ${\cal P}$ such that $t\models \varphi(u)$. Let $\theta_u(f)$ be defined as in the \abb{dtl} case. Then we write $s\Rightarrow_{t,{\cal P}} s'$
where $s'$ is obtained from $s$ by replacing the subtree $s|_v$ with root $v$ by the forest $\theta_u(f)$ in which every parameter $z_i$ is replaced by the subtree
$s|_{vi}$, for $i\in[1,\operatorname{rank}(q)]$. Intuitively, the subtree $s|_{vi}$ rooted at the $i$-th child $vi$ of $v$ is the $i$-th actual parameter of (this occurrence of) the state $q$. Now let $s$ be an output form and let $v$ be an outermost node of $s$ with label $\tup{\tup{q',\psi(x,y)},u}$ and $\operatorname{rank}(q')=m$. Then we write $s\Rightarrow_{t,{\cal P}} s'$
where $s'$ is obtained from $s$ by replacing the subtree $s|_v$ with root $v$ by the forest
$\tup{q',u'_1}(s|_{v1},\dots,s|_{vm})\cdots\tup{q',u'_\ell}(s|_{v1},\dots,s|_{vm})$ where $u'_1,\dots,u'_\ell$ is the sequence of all nodes $u'$ of $t$, in document order, such that $t\models \psi(u,u')$. Intuitively, the actual parameters of (this occurrence of) the selector $\tup{q',\psi(x,y)}$ are passed to each new occurrence of the state $q'$. As in the \abb{dtl} case, the transduction realized by ${\cal P}$ is defined by $\tau_{\cal P} = \{(t,s)\in F_\Sigma\times F_\Delta \mid \exists\,q_0\in Q_0: \tup{q_0,\mathrm{root}_t} \Rightarrow^*_{t,{\cal P}} s\}$.
In \cite{ManBerPerSei} the denotational semantics of a \abb{tl} program is given as a least fixed point. It is straightforward to show that that semantics is equivalent to the above operational semantics.\footnote{It is similar to the ``alternative'' fixed point characterization of the OI context-free tree languages mentioned after~\cite[Definition~5.19]{EngSch78}. } Also, in \cite{ManBerPerSei} the syntactic formulation of \abb{tl} is such that in the right-hand side of a rule the states can have forests as parameters rather than trees. Such a forest parameter $s_1\cdots s_m$, where each $s_i$ is a tree, can be expressed in our syntactic formulation of \abb{tl} as the tree $\tup{@_m,x=y}(s_1,\dots,s_m)$, where $@_m$ is a special state of rank $m$ that has the unique rule $\tup{@_m,x=x}(z_1,\dots,z_m)\to z_1\cdots z_m$.
\begin{example} \label{ex:tlsiberie} The transformation from Example~\ref{ex:siberie} can be computed by a deterministic \abb{TL} program ${\cal P}_\mathrm{sib}$ with the following rules, where the variables $i$, $\sigma_i$, $c$, and $\lambda_i$ range over the same values as in Example~\ref{ex:siberie}, with $c=1$ or $i=1$ in rule $\rho_4$.
$\rho_1: \tup{q_\mathrm{start},\mathrm{root}(x)} \to \tup{q_\mathrm{start},\mathrm{leaf}(y)}$
$\rho_2: \tup{q_\mathrm{start},\neg\mathrm{root}(x)\wedge\mathrm{lab}_{\sigma_0}(x)} \to \tup{q_1,{\rm up}(x,y)}(\sigma_0,e)$
$\rho_3: \tup{q_0,\neg\mathrm{root}(x)\wedge\mathrm{lab}_{\lambda_0}(x)}(z_1,z_2) \to \tup{q_0,{\rm up}(x,y)}(z_1,z_2)$
$\rho_4: \tup{q_c,\neg\mathrm{root}(x)\wedge\mathrm{lab}_{\lambda_i}(x)}(z_1,z_2)$
$\hspace{2cm}\to \tup{q_i,{\rm up}(x,y)}(\lambda_i(z_1),\tup{q_c,{\rm up}(x,y)}(z_1,z_2))$
$\rho_5: \tup{q_c,\mathrm{root}(x)\wedge\mathrm{lab}_{\sigma_1}(x)}(z_1,z_2) \to r(\sigma_1(z_1),z_2)$
\noindent Intuitively, $z_1$ represents an itinerary from some city to Vladivostok, and $z_2$ represents a list of itineraries from Moscow to Vladivostok (viz. all itineraries that do not have $z_1$ as postfix), where we only consider itineraries that do not visit a small city twice in a row.
The selectors in the right-hand sides of the rules all define functional trips, and hence select just one node. Rule $\rho_1$ jumps from the root to the leaf, and rules $\rho_2$, $\rho_3$, $\rho_4$ just move to the parent.
To show the correctness of ${\cal P}_\mathrm{sib}$, let $u$ be a node of an input tree $t$, such that $u$ is not the leaf of $t$. Moreover, let $\zeta_1$ be an output tree that is an itinerary from the child of $u$ to the leaf, of which the first stop is large ($c=1$) or small ($c=0$), and let $\zeta_2$ be an arbitrary output form. Then $\tup{q_c,u}(\zeta_1,\zeta_2)$ generates the output form $r(s_1(\zeta_1),r(s_2(\zeta_1),\dots r(s_n(\zeta_1),\zeta_2)\cdots ))$ where $s_1,\dots,s_n$ are all possible itineraries from the root to $u$ such that every $s_i(\zeta_1)$ is an itinerary from root to leaf. This can be proved by induction on the number of nodes between the root and $u$. The base of the induction is by rule $\rho_5$, which generates the root label $\sigma_1$, and the induction step is by rules $\rho_3$ and $\rho_4$. In rule $\rho_3$ a small city is skipped. In rule $\rho_4$, the outermost selector $\tup{q_i,{\rm up}(x,y)}$ generates all itineraries~$s_i$ from the root to $x$ that include $x$ (or rather, its label $\lambda_i$), whereas the innermost selector $\tup{q_c,{\rm up}(x,y)}$ generates all those that do not include $x$. Taking $c=1$, $u$ equal to the parent of the leaf, and $\sigma_0$ to the label of the leaf, shows that $\tup{q_1,u}(\sigma_0,e)$ generates all required itineraries. That implies the correctness of ${\cal P}_\mathrm{sib}$ by rule $\rho_2$.
An XSLT~1.0 program with exactly the same structure as ${\cal P}_\mathrm{sib}$ is given in Section~\ref{sec:siberie}. \end{example}
As in the case of $\family{DTL}$, we will assume that in the above definition of \abb{TL} program, the input alphabet $\Sigma$ is ranked and the input forest $t$ is a ranked tree in $T_\Sigma$. Also, \emph{ranked} \abb{tl} programs are defined as for \abb{dtl} programs. In particular, for every rule $\tup{q,\varphi(x)}(z_1,\dots,z_n)\to f$, the right-hand side $f$ is a ranked tree in $T_\Delta(S\cup Z_n)$ where $S$ is the set of selectors and $Z_n=\{z_1,\dots,z_n\}$. The program ${\cal P}_\mathrm{sib}$ of Example~\ref{ex:tlsiberie} is ranked.
Let $\family{TL}$ denote the class of transductions realized by \abb{tl} programs and $\family{dTL}$ the class of those realized by deterministic \abb{tl} programs, from ranked trees to unranked forests. Moreover, $\family{TL$_\text{r}$}$ and $\family{dTL$_\text{r}$}$ denote the classes of transductions realized by ranked programs, from ranked trees to ranked trees.
In what follows we will prove that $\family{TL} = \family{I-PFT}$, and similarly for the deterministic case and for the ranked case (Theorem~\ref{thm:tl}). Note that this also implies that \abb{TL} programs and \abb{i-pft}'s realize the same forest transductions, i.e., $\family{enc}'\circ \family{TL} = \family{enc}'\circ \family{I-PFT}$. These equalities are variants of the well-known fact that macro grammars are equivalent to indexed grammars \cite{Fis}, see also~\cite[Theorem~5.24]{EngVog}.
\begin{lemma}\label{lem:tlipft} $\family{TL} \subseteq \family{I-PFT}$ and $\family{dTL} \subseteq \family{I-dPFT}$. Moreover, $\family{TL$_\text{r}$} \subseteq \family{I-PTT}$ and $\family{dTL$_\text{r}$} \subseteq \family{I-dPTT}$. \end{lemma}
\begin{proof} The construction extends the one in the proof of Lemma~\ref{lem:dtl}. The main idea is to use pebbles to store the actual parameters. Thus, the pebble colours are of the form $([s_1],\dots,[s_m])$ where $m\geq 0$ and $s_1,\dots,s_m$ are subtrees of a right-hand side of a rule (in particular, the subtrees rooted at the children of a node that is labelled by a selector).
As in the \abb{dtl} case, for an input tree $t$, the transducer ${\cal M}$ simulates a rule $\tup{q,\varphi(x)}(z_1,\dots,z_n) \to f$ in state $q$ at node $u$ of $t$ by testing whether $t\models \varphi(u)$ and, if successful, calling subroutine $S$. In this (nested) case, $S$ outputs the outermost $\Delta$-labelled nodes of $f$, plus the outermost $\Delta$-labelled nodes of the actual parameters that are the values of the formal parameters $z_i$ that occur outermost in $f$. For the states $[sf']$, $[\delta(f')]$, and $[\varepsilon]$, the rules of $S$ are as in the proof of Lemma~\ref{lem:dtl} (and see the proof of Corollary~\ref{cor:dtlr} for the ranked case). If the state of $S$ is of the form $[\tup{q',\psi(x,y)}(s_1,\dots,s_m)]$, then it drops a pebble $([s_1],\dots,[s_m])$ on the current node $u$ to represent the parameters, and returns control to (this copy of) ${\cal M}$ in state $\tup{q',\psi(x,y)}$. In that state, ${\cal M}$ calls subroutine $S_{q',\psi}$, which works as in the \abb{dtl} case. Note that ${\cal M}$ need not drop a pebble~$\odot$, as $S_{q',\psi}$ can use the pebble $([s_1],\dots,[s_m])$ instead. Finally, if the state of $S$ is of the form $[z_i]$ for some formal parameter~$z_i$, this means that the corresponding actual parameter has to be evaluated. To do this, the subroutine~$S$ searches for the topmost pebble, which has some colour $([s_1],\dots,[s_m])$. Then $S$ lifts that pebble and changes its state to $[s_i]$, ready to evaluate $s_i$.
It is easy to show, for every $i\in{\mathbb N}$, that whenever ${\cal M}$ is in state $q$ or state $\tup{q,\psi(x,y)}$ with $i\in[1,\operatorname{rank}(q)]$, and whenever $S$ is in state $[f]$ and $z_i$ occurs in~$f$, then the top pebble with colour $([s_1],\dots,[s_m])$ satisfies $i\in[1,m]$. Hence the last sentence of the previous paragraph never fails.
To understand the correctness of ${\cal M}$, we show how the output forms of ${\cal M}$ represent output forms of ${\cal P}$, similar to the correctness proof of Theorem~\ref{thm:dtl}. We restrict ourselves to output forms in which all the states of ${\cal M}$ are states of~${\cal P}$ or selectors of ${\cal P}$ or states of the subroutine $S$, i.e., we disregard the states of the subroutines $S_{q',\psi}$ and view the execution of such a subroutine as one big step in the computation of ${\cal M}$, changing a configuration $\tup{\tup{q',\psi(x,y)},u,\pi}$ deterministically into a forest $\tup{q',u'_1,\pi}\cdots\tup{q',u'_\ell,\pi}$ (which is just a one-node tree $\tup{q',u',\pi}$ in the ranked case). Thus, we define a mapping `$\xp{rep}$' from such restricted output forms of ${\cal M}$ to the output forms of ${\cal P}$. The $\Delta$-labelled part of the output form is not changed by `$\xp{rep}$', i.e., $\xp{rep}(sf)=\xp{rep}(s)\xp{rep}(f)$, $\xp{rep}(\varepsilon)=\varepsilon$, and $\xp{rep}(\delta(f))=\delta(\xp{rep}(f))$ for $\delta\in\Delta$, where $s$ is a tree and $f$ a forest (or $\xp{rep}(\delta(s_1,\dots,s_m))= \delta(\xp{rep}(s_1),\dots,\xp{rep}(s_m))$ in the ranked case). It remains to define `$\xp{rep}$' for the configurations of ${\cal M}$ that occur in restricted output forms, i.e., for every configuration $\tup{p,u,\pi}$ where~$p$ is a state $q$ of ${\cal P}$, or a selector $\tup{q',\psi(x,y)}$ of ${\cal P}$, or a state $[f]$ of $S$ (where~$f$ is a subforest of a right-hand side of a rule of ${\cal P}$). As before, we will write $\xp{rep}(p,u,\pi)$ instead of $\xp{rep}(\tup{p,u,\pi})$. The definition is by induction on the structure of $\pi$, of which we consider the topmost pebble: let $\pi=\pi'(v,([s_1],\dots,[s_m]))$. If $p=q$ or $p=\tup{q',\psi(x,y)}$, then $\xp{rep}(p,u,\pi)=\tup{p,u}(\xp{rep}([s_1],v,\pi'),\dots,\xp{rep}([s_m],v,\pi'))$. For $p=[f]$ we define $\xp{rep}([f],u,\pi)$ to be the forest $\theta_u(f)$ in which every parameter $z_i$ is replaced by $\xp{rep}([s_i],v,\pi')$, Finally, for $\pi=\varepsilon$, we define $\xp{rep}(p,u,\varepsilon)=\tup{p,u}$ in the first case, and $\xp{rep}([f],u,\varepsilon)=\theta_u(f)$ in the second case. If we consider only reachable output forms of ${\cal M}$, then `$\xp{rep}$' is well defined (cf. the previous paragraph).
It is now straightforward to prove, for every initial state $q_0$ of ${\cal P}$, every input tree $t$, and every output form $s$ of ${\cal P}$, that $\tup{q_0,\mathrm{root}_t} \Rightarrow^*_{t,{\cal P}} s$ if and only if there exists a restricted output form $s'$ of ${\cal M}$ such that $\tup{q_0,\mathrm{root}_t,\varepsilon} \Rightarrow^*_{t,{\cal M}} s'$ and $\xp{rep}(s')=s$. In the proof one should use the rather obvious fact that for every restricted output form $s'$ of ${\cal M}$ there exists a restricted output form $s''$ of ${\cal M}$ such that $s' \Rightarrow^*_{t,{\cal M}} s''$, $\xp{rep}(s'')=\xp{rep}(s')$, and no states $[f]$ of $S$ occur in $s''$. The above equivalence implies that $\tau_{\cal M}=\tau_{\cal P}$. \end{proof}
\begin{example} \label{ex:pttsiberie} The \abb{i-ptt} ${\cal M}$ corresponding to the (ranked) \abb{tl} program ${\cal P}_\mathrm{sib}$ of Example~\ref{ex:tlsiberie}, according to the proof of Lemma~\ref{lem:tlipft}, works in essentially the same way as the \abb{i-ptt} ${\cal M}_{\text{sib}}$ of Example~\ref{ex:siberie}. Rules $\rho_1$ to $\rho_5$ are translated into rules for~${\cal M}$ that are similar to the first 5 rules of ${\cal M}_{\text{sib}}$. Rule $\rho_1$ can be translated into the first rule of ${\cal M}_{\text{sib}}$, which implements the jump to the leaf. Rule $\rho_2$ can be translated into the rule $\tup{q_\mathrm{start},\sigma_0,1,\varnothing}\to \tup{q_1,{\rm drop}_{([\sigma_0],[e])};{\rm up}}$. Thus, ${\cal M}$ drops the special pebble $([\sigma_0],[e])$ at the leaf, where ${\cal M}_{\text{sib}}$ does not drop a pebble. Rule $\rho_3$ can be translated into the rule $\tup{q_0,\lambda_0,1,\varnothing}\to \tup{q_0,{\rm drop}_{([z_1],[z_2])};{\rm up}}$. Thus, ${\cal M}$ drops the ``empty'' pebble $([z_1],[z_2])$ whenever ${\cal M}_{\text{sib}}$ does not drop a pebble. Rule $\rho_4$ can be translated into the rule $\tup{q_c,\lambda_i,1,\varnothing}\to \tup{q_i,{\rm drop}_{c(\lambda_i)};{\rm up}}$, where $c(\lambda_i)$ is the pebble $([\lambda_i(z_1)],[\tup{q_c,{\rm up}(x,y)}(z_1,z_2)])$ which is dropped by~${\cal M}$ instead of the pebble $c$. Note that the pebble colours $c(\lambda_i)$ and $([\sigma_0],[e])$ include the label ($\lambda_i$ or $\sigma_0$) of the node on which the pebble is dropped, which is of course superfluous information. Finally, rule $\rho_5$ can be translated into the rule $\tup{q_c,\sigma_1,0,\varnothing}\to r(\tup{[\sigma_1(z_1)],{\rm stay}},\tup{[z_2],{\rm stay}})$, which calls the states $[\sigma_1(z_1)]$ and $[z_2]$ of the subroutine $S$. In state $[\sigma_1(z_1)]$, $S$ outputs $\sigma_1$ and goes into state $[z_1]$. We note that at any moment of time, when ${\cal M}$ is at node $u$ of the input tree, all descendants of $u$, possibly including $u$ itself, carry a pebble. Thus, in state~$[z_i]$, $S$ moves down to the child of $u$, lifts pebble $([s_1],[s_2])$ and goes into state $[s_i]$. It is now easy to see that states $[z_1]$ and $[z_2]$ of ${\cal M}$ correspond to states $q_\mathrm{out}$ and $q_\mathrm{next}$ of ${\cal M}_{\text{sib}}$, respectively. In state $[z_1]$, $S$ moves down and outputs the labels of all nodes that are marked by some pebble $c(\lambda_i)$ or $([\sigma_0],[e])$, lifting those pebbles one by one. In state $[z_2]$, $S$ moves down to the first pebble $c(\lambda_i)$, replaces that pebble by the ``empty'' pebble $([z_1],[z_2])$, and returns control to~${\cal M}$, which then goes into state $q_c$ and moves up to the parent. When, in state $[z_2]$, $S$ reaches the leaf with pebble $([\sigma_0],[e])$, it lifts that pebble and outputs $e$. \end{example}
Lemma~\ref{lem:tlipft} and Theorem~\ref{thm:typecheck-pft} (for $k=0$) together provide an alternative proof of the main result of \cite{ManBerPerSei}: the inverse type inference problem and the typechecking problem are solvable for \abb{tl} programs. The proofs are, however, similar. In~\cite{ManBerPerSei} every \abb{tl} program is decomposed into three macro tree transducers, whereas we have decomposed every \abb{i-ptt} into two \abb{tt}'s. In general, decomposition into \abb{tt}'s leads to more efficient typechecking than decomposition into macro tree transducers, because (cf. Proposition~\ref{prop:invtypeinf}) inverse type inference of a macro tree transducer takes double exponential time, unless the number of parameters is bounded and the output type is fixed \cite{PerSei}. Let us define a \abb{tl}$^\text{\abb{db}}$ program to be a \abb{tl} program in which the \abb{mso} formulas $\varphi(x)$ and $\psi(x,y)$ in the template rules of the program are represented by deterministic bottom-up finite-state tree automata that recognize the corresponding regular sites $\operatorname{mark}(T(\varphi))$ and trips $\operatorname{mark}(T(\psi))$.
\begin{theorem}\label{thm:tltypecheck} The inverse type inference problem and the typechecking problem are solvable for \abb{tl}$^\text{\abb{db}}$ programs in $3$-fold and $4$-fold exponential time, respectively. \end{theorem}
\begin{proof} By Theorem~\ref{thm:typecheck-pft}, these problems are solvable for \abb{i-pft}'s in 2-fold and \mbox{3-fold} exponential time. Let us now assume that the regular sites and trips used in \abb{mso} tests of \abb{i-pft}'s are also represented by deterministic bottom-up finite-state tree automata. Then it is easy to see that the construction in the proof of Lemma~\ref{lem:tlipft} takes polynomial time. However, the \abb{mso} tests that are used by the resulting \abb{i-pft} have to be removed, and the construction in the proof of Theorem~\ref{thm:mso} takes exponential time, as can be checked in a straightforward way. That involves checking that the constructions in the proofs of Lemmas~\ref{lem:regular},~\ref{lem:sites}, and~\ref{lem:visiblesites} take polynomial time, and so does the construction in the proof of Proposition~\ref{prop:trips} (for the nonfunctional case), i.e., in the proof of~\cite[Theorem~8]{bloem}. The exponential in the proof of Theorem~\ref{thm:mso} is due to the use of the sets of states~$S$ of ${\cal B}_d$ in the colours of the beads. Hence, solving the above problems takes one more exponential for \abb{tl}$^\text{\abb{db}}$ programs than for \abb{i-pft}. \end{proof}
A \abb{tl} program ${\cal P}=(\Sigma,\Delta,Q,Q_0,R)$ is a \emph{macro tree transducer}, more precisely an \abb{OI} macro tree transducer (see~\cite{EngVog85}), if it is ranked, and for every rule $\tup{q,\varphi(x)}(z_1,\dots,z_n)\to f$ the following hold. First, $\varphi(x)\equiv \mathrm{lab}_\sigma(x)$ for some $\sigma\in\Sigma$. Second, for every selector $\tup{q',\psi(x,y)}$ that occurs in $f$, we have $\psi(x,y)\equiv {\rm down}_i(x,y)$ for some $i\in[1,\operatorname{rank}_\Sigma(\sigma)]$. It follows immediately from Lemma~\ref{lem:tlipft} that macro tree transducers can be simulated by \abb{i-ptt}. Let $\family{MT}_\text{\abb{OI}}$ denote the class of tree transductions realized by \abb{OI} macro tree transducers, and $\family{dMT}_\text{\abb{OI}}$ the corresponding deterministic class.
\begin{corollary}\label{cor:mtiptt} $\family{MT}_\text{\abb{OI}} \subseteq \family{I-PTT}$ and $\family{dMT}_\text{\abb{OI}} \subseteq \family{I-dPTT}$. \end{corollary}
The inclusions are proper because for every \abb{OI} macro tree transduction the height of the output tree is exponentially bounded by the height of the input tree \cite[Theorem~3.24]{EngVog85}, whereas it is not difficult to construct a deterministic \abb{i-ptt} ${\cal M}$ with input alphabet $\{\sigma,e\}$, where $\operatorname{rank}(\sigma)=2$ and $\operatorname{rank}(e)=0$, such that the height of the output tree is exponential in the \emph{size} of the input tree. The transducer~${\cal M}$ is similar to the \abb{i-ptt} ${\cal M}_\text{sib}$ of Example~\ref{ex:siberie}, viewing the nodes of the input tree as large cities that are ordered by document order; thus, the number of itineraries is indeed exponential in the size of the input tree. Note that by~\cite[Corollary~7.2]{EngMan99} and~\cite[Theorem~6.18]{EngVog85}, $\family{dMT}_\text{\abb{OI}}$ properly contains the class $\family{DMSOT}$ of deterministic \emph{\abb{mso} definable tree transductions} (see also~\cite[Section~8]{thebook}). Note also that, since $\family{dB}$ is properly contained in $\family{dMT}_\text{\abb{OI}}$ by~\cite[Corollary~6.16]{EngVog85}, the second part of Corollary~\ref{cor:mtiptt} strengthens the second part of Theorem~\ref{thm:bottomup}. It is open whether or not $\family{B}$ is contained in $\family{MT}_\text{\abb{OI}}$.
We now turn to the inclusion $\family{I-PFT}\subseteq\family{TL}$. To prove that, we need a normal form for \abb{i-pft}. We say that a rule of an \abb{i-pft} is \emph{initial} if the state in its left-hand side is an initial state. We define an \abb{i-pft} ${\cal M} = (\Sigma, \Delta, Q, Q_0, C, \varnothing, C_\mathrm{i}, R, 0)$ with $C=C_\mathrm{i}$ to be in \emph{normal form} if its rules satisfy the following five requirements:
(1) Initial states do not appear in the right-hand side of a rule.
(2) All initial rules are of the form $\tup{q_0,\sigma,0,\varnothing}\to \tup{q,{\rm drop}_c}$ for some $q_0\in Q_0$, $\sigma\in\Sigma$, $q\in Q\setminus Q_0$, and $c\in C$. Intuitively, ${\cal M}$ starts its computation by dropping a pebble on the root of the input tree.
(3) All non-initial rules have a left-hand side of the form $\tup{q,\sigma,j,\{c\}}$ with $c\in C$. Intuitively, ${\cal M}$ always observes the topmost pebble, i.e., that pebble is always at the position of the head.
(4) All non-initial non-output rules have a right-hand side $\tup{q',\alpha}$ with $q'\in Q\setminus Q_0$ and $\alpha={\rm stay}$ or $\alpha=\mu;{\rm drop}_c$ or $\alpha={\rm lift}_c;\mu$ where $c\in C$ and $\mu\in\{{\rm up},{\rm stay}\}\cup\{{\rm down}_i\mid i\in[1,{\mathit mx}_\Sigma]\}$. We will identify ${\rm stay};{\rm drop}_c$ with ${\rm drop}_c$ and ${\rm lift}_c;{\rm stay}$ with ${\rm lift}_c$. Intuitively, to force that ${\cal M}$ always observes the topmost pebble, ${\cal M}$ always drops a pebble after moving, and always moves after lifting a pebble. Note that, in a successful computation, ${\cal M}$ never lifts the pebble that it dropped with an initial rule.
(5) There is a function $\delta$ from $C$ to $\{{\rm up},{\rm stay}\}\cup\{{\rm down}_i\mid i\in[1,{\mathit mx}_\Sigma]\}$ such that (i) if a rule of ${\cal M}$ has right-hand side $\tup{q',{\rm lift}_c;\mu}$, then $\mu=\delta(c)$, and (ii)~for every rule $\tup{q,\sigma,j,\{d\}}\to \tup{q',\mu;{\rm drop}_c}$ of ${\cal M}$, if $\mu={\rm up}$ then $\delta(c)={\rm down}_j$, if $\mu={\rm stay}$ then $\delta(c)={\rm stay}$, and if $\mu={\rm down}_i$ then $\delta(c)={\rm up}$. Intuitively this means that ${\cal M}$, after lifting a pebble, always knows where to find the new topmost pebble.
This ends the definition of normal form. Obviously, it can also be defined for \abb{i-ptt}'s and for \abb{i-pta}'s. The \abb{i-pta} in normal form can be viewed as a reformulation of the two-way backtracking pushdown tree automaton of~\cite{Slu}. The \abb{i-ptt} in normal form can be viewed as a reformulation of the RT(P($S$))-transducer of~\cite{Eng86,EngVog}, where $S$ is the storage type Tree-walk of~\cite{Eng86}.\footnote{See also~\cite[Section~3.3]{EngMan03} where the \abb{tt} is related to the RT($S$)-transducer for $S =$ Tree-walk. }
\begin{lemma}\label{lem:ipftnf} For every \abb{i-pft} ${\cal M}$ an equivalent \abb{i-pft} ${\cal M}'$ in normal form can be constructed. If ${\cal M}$ is deterministic, then so is ${\cal M}'$. The same holds for \abb{i-ptt}. \end{lemma}
\begin{proof} The idea of the construction is a simplified version of the one in the proof of Theorem~\ref{thm:mso}, where ``beads'' are used to cover the shortest path between the head and the topmost pebble. Assuming that the \abb{i-pta} ${\cal A}$ in that proof starts by dropping a pebble on the root (which is never lifted), the constructed \abb{i-pta} ${\cal A}'$ satisfies the above requirements on the rules. To show the details, we will repeat that construction, in a simplified form. Here, the only information a bead has to carry is the position of the previous pebble or bead. Moreover, we do not have to drop a bead on the position of the topmost pebble.
Let ${\cal M}$ be an \abb{i-pft} with colour set $C$. We may obviously assume that~${\cal M}$ already satisfies the first two requirements above. We construct ${\cal M}'$ with the same states and initial states as ${\cal M}$, and with the colour set $C\cup B$ where $B=\{{\rm up}\}\cup\{{\rm down}_i\mid [1,{\mathit mx}_\Sigma]\}$. The function $\delta$ of requirement~(5) is defined by $\delta(d)=d$ for every $d\in B$, and $\delta(c)={\rm stay}$ for every $c\in C$. The rules of ${\cal M}'$ are obtained from those of ${\cal M}$ as follows.
The initial rules of ${\cal M}$ are also rules of~${\cal M}'$.
If $\tup{q,\sigma,j,\varnothing}\to \tup{q',{\rm up}}$ is a rule of ${\cal M}$, then ${\cal M}'$ has the rules $\tup{q,\sigma,j,\{{\rm up}\}}\to \tup{q',{\rm lift}_{\rm up};{\rm up}}$ and $\tup{q,\sigma,j,\{{\rm down}_i\}}\to \tup{q',{\rm up};{\rm drop}_{{\rm down}_j}}$ for every $i$. Also, if $\tup{q,\sigma,j,\{c\}}\to \tup{q',{\rm up}}$ is a rule of ${\cal M}$, then ${\cal M}'$ has the rule $\tup{q,\sigma,j,\{c\}}\to \tup{q',{\rm up};{\rm drop}_{{\rm down}_j}}$.
Similarly, if $\tup{q,\sigma,j,\varnothing}\to \tup{q',{\rm down}_i}$ is a rule of ${\cal M}$, then ${\cal M}'$ has the rules $\tup{q,\sigma,j,\{{\rm down}_i\}}\to \tup{q',{\rm lift}_{{\rm down}_i};{\rm down}_i}$ and $\tup{q,\sigma,j,\{\mu\}}\to \tup{q',{\rm down}_i;{\rm drop}_{\rm up}}$ for every $\mu\in\{{\rm up}\}\cup\{{\rm down}_k\mid k\neq i\}$. Also, if $\tup{q,\sigma,j,\{c\}}\to \tup{q',{\rm down}_i}$ is a rule of ${\cal M}$, then ${\cal M}'$ has the rule $\tup{q,\sigma,j,\{c\}}\to \tup{q',{\rm down}_i;{\rm drop}_{\rm up}}$.
The remaining rules of ${\cal M}$ (viz. rules with right-hand side $\tup{q',{\rm stay}}$, output rules, rules that lift, and non-initial rules that drop) are treated as follows. If $\tup{q,\sigma,j,\varnothing}\to \zeta$ is such a rule of ${\cal M}$, then ${\cal M}'$ has the rules $\tup{q,\sigma,j,\{\mu\}}\to \zeta$ for every bead $\mu\in B$. If $\tup{q,\sigma,j,\{c\}}\to \zeta$ is such a rule of ${\cal M}$, then it is also a rule of ${\cal M}'$.
It should be clear that ${\cal M}'$ is equivalent to ${\cal M}$. Whenever ${\cal M}$ observes the topmost pebble $c$, so does ${\cal M}'$. Whenever ${\cal M}$ does not observe $c$, $M'$ observes a bead that indicates the direction of the topmost pebble. Note that if ${\cal M}'$ lifts pebble $c$ of~${\cal M}$, the new topmost pebble/bead is always at the same position, because when $c$ was dropped ${\cal M}'$ was observing the topmost pebble/bead. \end{proof}
The \abb{tl} program that we will construct to simulate a given \abb{i-pft} ${\cal M}$ will use \abb{mso} formulas $\varphi(x)$ and $\psi(x,y)$ that closely resemble the tests and instructions in the left-hand and right-hand sides of the rules of ${\cal M}$, respectively. Those tests and instructions are ``local'' in the sense that they only concern the node~$x$, its parent, and its children. Thus, we say that a \abb{tl} program ${\cal P}$ is \emph{local} if in the left-hand side of a rule it only uses a formula $\varphi_{\sigma,j}(x)$ for $\sigma\in\Sigma$ and $j\in[0,{\mathit mx}_\Sigma]$, where $\varphi_{\sigma,0}(x)\equiv \mathrm{lab}_\sigma(x)\wedge \mathrm{root}(x)$ and $\varphi_{\sigma,j}(x)\equiv \mathrm{lab}_\sigma(x)\wedge \mathrm{child}_j(x)$ for $j\neq 0$, and in the right-hand side of that rule it only uses the formulas ${\rm up}(x,y)$ (provided $j\neq 0$), ${\rm stay}(x,y)$, and ${\rm down}_i(x,y)$ for $i\in[1,\operatorname{rank}_\Sigma(\sigma)]$.\footnote{Recall that $\mathrm{root}(x) \equiv \neg\,\exists z({\rm down}(z,x))$, $\mathrm{child}_i(x) \equiv \exists z({\rm down}_i(z,x))$, ${\rm up}(x,y) \equiv {\rm down}(y,x)$, and ${\rm stay}(x,y)\equiv x=y$. } Thus, ${\cal P}$~also satisfies restriction (R2) in the definition of a ranked \abb{tl} program. Note that macro tree transducers, as defined before Corollary~\ref{cor:mtiptt}, are local ranked \abb{tl} programs. The classes of transductions realized by local \abb{tl} programs will be decorated with a subscript~$\ell$.
\begin{lemma}\label{lem:ipfttl} $\family{I-PFT}\subseteq\family{TL$_\ell$}$ and $\family{I-dPFT}\subseteq\family{dTL$_\ell$}$. Moreover, $\family{I-PTT}\subseteq\family{TL$_{\ell\text{r}}$}$ and $\family{I-dPTT}\subseteq\family{dTL$_{\ell\text{r}}$}$. \end{lemma}
\begin{proof} Let ${\cal M} = (\Sigma, \Delta, Q, Q_0, C, \varnothing, C_\mathrm{i}, R, 0)$ with $C=C_\mathrm{i}$ be an \abb{i-pft} in normal form. We construct a \abb{tl} program ${\cal P}$ that is equivalent to ${\cal M}$. The set of states of ${\cal P}$ is \[ Q_0\cup ((Q\setminus Q_0)\times C) \cup \{q_\bot\}. \] Each initial state has rank~$0$, each pair $\tup{q,c}$ has rank~$\#(Q\setminus Q_0)$, and $q_\bot$ has rank~$0$. The set of initial states of ${\cal P}$ is $Q_0$. The rules of ${\cal P}$ are defined as follows, where we denote a state $\tup{q,c}$ as $q^c$. Let $Q\setminus Q_0=\{q_1,\dots,q_n\}$ where we fix the order $q_1,\dots,q_n$.
First, if $\tup{q_0,\sigma,0,\varnothing}\to \tup{q,{\rm drop}_c}$ is an initial rule of ${\cal M}$, then ${\cal P}$ has the rule $\tup{q_0,\varphi_{\sigma,0}(x)}\to \tup{q^c,{\rm stay}(x,y)}(\bot,\dots,\bot)$, where $\bot$ abbreviates $\tup{q_\bot,{\rm stay}(x,y)}$. There are no rules of ${\cal P}$ with $q_\bot$ in the left-hand side.
Second, let $\tup{q,\sigma,j,\{c\}}\to \zeta$ be a (non-initial) rule of ${\cal M}$ that does not contain a drop- or lift-instruction. Thus, $\zeta$ is of the form $\tup{p,{\rm stay}}$, $\tup{p_1,{\rm stay}}\tup{p_2,{\rm stay}}$, $\delta(\tup{p,{\rm stay}})$, or $\varepsilon$, with $p,p_1,p_2\in Q$ and $\delta\in\Delta$.\footnote{In the case where ${\cal M}$ is an \abb{i-ptt}, $\zeta$ is of the form $\tup{p,{\rm stay}}$ or $\delta(\tup{p_1,{\rm stay}},\dots,\tup{p_m,{\rm stay}})$. } Then ${\cal P}$ has the rule $\tup{q^c,\varphi_{\sigma,j}(x)}(z_1,\dots,z_n)\to\zeta'$, where $\zeta'$ is obtained from $\zeta$ by replacing every $\tup{p,{\rm stay}}$ by $\tup{p^c,{\rm stay}(x,y)}(z_1,\dots,z_n)$.
Third, let $\tup{q,\sigma,j,\{d\}}\to \tup{p,\mu;{\rm drop}_c}$ be a rule of ${\cal M}$. Note that for every $\mu\in\{{\rm up},{\rm stay}\}\cup\{{\rm down}_i\mid i\in[1,{\mathit mx}_\Sigma]\}$, there is an \abb{mso} formula $\mu(x,y)$. Then ${\cal P}$ has the rule \[ \tup{q^d,\varphi_{\sigma,j}(x)}(z_1,\dots,z_n)\to\tup{p^c,\mu(x,y)}(s_1,\dots,s_n) \] where $s_i=\tup{q_i^d,{\rm stay}(x,y)}(z_1,\dots,z_n)$ for every $i\in[1,n]$; thus, the rule is \begin{eqnarray*} \lefteqn{\tup{q^d,\varphi_{\sigma,j}(x)}(z_1,\dots,z_n)\to} \\[1mm] & & \tup{p^c,\mu(x,y)} (\tup{q_1^d,{\rm stay}(x,y)}(z_1,\dots,z_n),\dots, \tup{q_n^d,{\rm stay}(x,y)}(z_1,\dots,z_n)). \end{eqnarray*}
Fourth and final, if $\tup{q,\sigma,j,\{c\}}\to \tup{q_i,{\rm lift}_c;\mu}$ is a rule of ${\cal M}$, then ${\cal P}$ has the rule $\tup{q^c,\varphi_{\sigma,j}(x)}(z_1,\dots,z_n)\to z_i$.
Intuitively, ${\cal P}$ is in state $q^c$ when ${\cal M}$ is in state $q$ and the topmost pebble of~${\cal M}$ is $c$. The parameter $z_i$ of $q^c$ contains the continuation of ${\cal M}$'s computation just after pebble $c$ is lifted and ${\cal M}$ goes into state $q_i$. At the moment that ${\cal M}$ drops pebble~$c$, ${\cal P}$ does not know what the state $q_i$ of ${\cal M}$ will be after lifting $c$ and thus prepares the continuation for every possible state. The correct continuation is then chosen by ${\cal P}$ when it simulates ${\cal M}$'s lifting of $c$. Note that due to requirement~(5) of the normal form, when ${\cal M}$ lifts a pebble, it returns to the same node where it decided to drop the pebble (at that node, or at the parent or at one of the children of that node).
Formally, we define a mapping `$\xp{rep}$' from the output forms of ${\cal M}$ (except the initial one) to those restricted output forms of ${\cal P}$ of which the outermost nodes are labelled by a symbol from $\Delta$ or by a configuration $\tup{q,u}$ where $q$~is a state of ${\cal P}$ (thus, they are not labelled by a configuration $\tup{p,u}$ where $p$~is a selector of ${\cal P}$). As in the proof of Lemma~\ref{lem:tlipft}, the $\Delta$-labelled part of the output form is not changed. Thus, it remains to define `$\xp{rep}$' for the configurations of ${\cal M}$ that contain non-initial states, which are of the form $\tup{q,u,\pi(u,c)}$ because the topmost pebble is always at the position of the head. We define $\xp{rep}(q,u,\pi(u,c))=\tup{q^c,u} \xp{rep}'(\pi)$, where $\xp{rep}'$ maps the pebble stacks of ${\cal M}$ to sequences of output forms of ${\cal P}$, recursively as follows: $\xp{rep}'(\varepsilon)=(\bot,\dots,\bot)$ and $\xp{rep}'(\pi(u,c))=(s_1,\dots,s_n)$ where $s_i=\tup{\tup{q_i^c,{\rm stay}(x,y)},u}\xp{rep}'(\pi)$ for every $i\in[1,n]$. Note that `$\xp{rep}$' is injective.
It is now straightforward to prove, for every $q\in Q\setminus Q_0$, every $c\in C$, every input tree $t$, and every output form $s$ of ${\cal P}$ (restricted as described above), that $\tup{q^c,\mathrm{root}_t}(\bot,\dots,\bot) \Rightarrow^*_{t,{\cal P}} s$ if and only if there exists an output form $s'$ of ${\cal M}$ such that $\tup{q,\mathrm{root}_t,(\mathrm{root}_t,c)} \Rightarrow^*_{t,{\cal M}} s'$ and $\xp{rep}(s')=s$. Since `$\xp{rep}$' is injective, $s'$ is in fact unique. Note that each computation step of ${\cal M}$ is simulated by two (or three) computation steps of ${\cal P}$, where the second (and third) step executes a selector to satisfy the restriction on the output forms of ${\cal P}$. Due to its special form, the execution of such a selector $\psi(x,y)$ changes the label $\tup{\tup{q',\psi(x,y)},u}$ of a node of the output form into $\tup{q',u'}$ where $u'$ is the unique node of the input tree for which $\psi(u,u')$ holds.
Taking into account the initial rules of ${\cal M}$, it should be clear that the above equivalence proves that $\tau_{\cal P}=\tau_{\cal M}$. \end{proof}
\begin{example} We illustrate Lemma~\ref{lem:ipfttl} with the deterministic \abb{i-ptt} ${\cal M}_\mathrm{sib}$ of Example~\ref{ex:siberie}. We first construct an \abb{i-ptt} ${\cal M}'_\mathrm{sib}$ in normal form that is equivalent to ${\cal M}_\mathrm{sib}$. We also allow tuples $\tup{q',{\rm lift}_d;\mu}$ in the output rules for any colour~$d$, which can easily be handled too. The transducer ${\cal M}'_\mathrm{sib}$ has a new initial state~$q_\mathrm{in}$, in which it drops pebble $\odot$ on the root, which also serves as the pebble `${\rm up}$'. The pebble `${\rm down}_1$' is denoted by $\downarrow$. The normal form function $\delta$ is defined by $\delta(\odot)={\rm up}$, $\delta(\downarrow)={\rm down}_1$, and $\delta(c)={\rm stay}$ for $c\in\{0,1\}$. There are new states $\overline{q}_0$ and $\overline{q}_1$ in which ${\cal M}'_\mathrm{sib}$ moves up, drops pebble $\downarrow$, and goes into the corresponding unbarred state. Thus the rules for them are \\[1mm] \indent $\rho_{c,d}: \tup{\overline{q}_c,\sigma,1,\{d\}} \to \tup{q_c,{\rm up};{\rm drop}_\downarrow}$\\[1mm] \noindent with $\sigma\in\Sigma$ and $d\in\{\odot,\downarrow,0,1\}$. The other rules (with $c=1$ or $i=0$ in rule $\rho_4$ as usual) are \\[1mm] \indent $\rho_0: \tup{q_\mathrm{in},\sigma_1,0,\varnothing} \to
\tup{q_\mathrm{start},{\rm drop}_\odot}$ \\[1mm] \indent $\rho_1: \tup{q_\mathrm{start},\sigma_1,j,\{\odot\}} \to
\tup{q_\mathrm{start},\mathrm{down}_1;{\rm drop}_\odot}$ \\[1mm] \indent $\rho_2: \tup{q_\mathrm{start},\sigma_0,1,\{\odot\}} \to
\tup{\overline{q}_1,{\rm stay}}$ \\[1mm] \indent $\rho_3: \tup{q_0,\lambda_0,1,\{\downarrow\}} \to \tup{\overline{q}_0,{\rm stay}}$ \\[1mm] \indent $\rho_4: \tup{q_c,\lambda_i,1,\{\downarrow\}} \to
\tup{\overline{q}_i,\mathrm{drop}_c}$ \\[1mm] \indent $\rho_5: \tup{q_c,\sigma_1,0,\{\downarrow\}} \to r(\tup{q_\mathrm{out},\mathrm{stay}},
\tup{q_\mathrm{next},{\rm lift}_\downarrow;\mathrm{down}_1})$ \\[1mm] \indent $\rho_6: \tup{q_\mathrm{out},\sigma_1,0,\{\downarrow\}} \to \sigma_1(\tup{q_\mathrm{out},{\rm lift}_\downarrow;\mathrm{down}_1})$ \\[1mm] \indent $\rho_7: \tup{q_\mathrm{out},\sigma_1,1,\{\downarrow\}} \to \tup{q_\mathrm{out},{\rm lift}_\downarrow;\mathrm{down}_1}$ \\[1mm] \indent $\rho_8: \tup{q_\mathrm{out},\sigma_1,1,\{c\}} \to \sigma_1(\tup{q_\mathrm{out},{\rm lift}_c})$ \\[1mm] \indent $\rho_9: \tup{q_\mathrm{out},\sigma_0,1,\{\odot\}} \to \sigma_0$ \\[1mm] \indent $\rho_{10}: \tup{q_\mathrm{next},\sigma_1,1,\{\downarrow\}} \to \tup{q_\mathrm{next},{\rm lift}_\downarrow;\mathrm{down}_1}$ \\[1mm] \indent $\rho_{11}: \tup{q_\mathrm{next},\sigma_1,1,\{c\}} \to
\tup{\overline{q}_c,{\rm lift}_c}$ \\[1mm] \indent $\rho_{12}: \tup{q_\mathrm{next},\sigma_0,1,\{\odot\}} \to e$ \\[1mm] \noindent We now construct the deterministic \abb{tl} program ${\cal P}$ corresponding to ${\cal M}'_\mathrm{sib}$. The states of ${\cal M}'_\mathrm{sib}$ after lifting $\downarrow$ are $q_\mathrm{out}$ and $q_\mathrm{next}$. Thus, the states of ${\cal P}$ that are active when the topmost pebble is $\downarrow$ only need two parameters $z_1,z_2$ corresponding to $q_\mathrm{out}$ and $q_\mathrm{next}$. Similarly, the states of ${\cal P}$ that are active when the topmost pebble is $c$ only need two parameters $z_1,z_2$ corresponding to $q_\mathrm{out}$ and~$\overline{q}_c$. The states of ${\cal P}$ that are active when the topmost pebble is $\odot$ do not need parameters, because $\odot$ is never lifted. Program ${\cal P}$ has the states $q_\mathrm{in}$, $q_\mathrm{start}^\odot$, $q_c^{\downarrow}$, $\overline{q}_c^d$, $q_\mathrm{out}^d$, and $q_\mathrm{next}^d$, where $c\in\{0,1\}$ and $d\in\{\odot,\downarrow,0,1\}$. Note that the state $q_\bot$ is superfluous. The initial state $q_\mathrm{in}$ and all states with superscript $\odot$ have rank~$0$, and the other states have rank~$2$.
Program ${\cal P}$ has the following rule corresponding to rule $r_{c,d}$ of ${\cal M}'_\mathrm{sib}$, with $d\neq\odot$:
\begin{eqnarray*} \lefteqn{\rho_{c,d}: \tup{\overline{q}_c^d,\varphi_{\sigma,1}(x)}(z_1,z_2)\to} \\[1mm] & & \quad\quad\tup{q_c^{\downarrow},{\rm up}(x,y)}
(\tup{q_\mathrm{out}^d,{\rm stay}(x,y)}(z_1,z_2),
\tup{q_\mathrm{next}^d,{\rm stay}(x,y)}(z_1,z_2)) \end{eqnarray*}
and for $d=\odot$ the same rule without the parameters $(z_1,z_2)$. The other rules of ${\cal P}$ are \\[1mm] \indent $\rho_0: \tup{q_\mathrm{in},\varphi_{\sigma_1,0}(x)}\to
\tup{q_\mathrm{start}^\odot,{\rm stay}(x,y)}$ \\[1mm] \indent $\rho_1: \tup{q_\mathrm{start}^\odot,\varphi_{\sigma_1,j}(x)}\to
\tup{q_\mathrm{start}^\odot,{\rm down}_1(x,y)}$ \\[1mm] \indent $\rho_2: \tup{q_\mathrm{start}^\odot,\varphi_{\sigma_0,1}(x)}\to
\tup{\overline{q}_1^\odot,{\rm stay}(x,y)}$ \\[1mm] \indent $\rho_3: \tup{q_0^{\downarrow},\varphi_{\lambda_0,1}(x)}(z_1,z_2)\to
\tup{\overline{q}_0^\downarrow,{\rm stay}(x,y)}(z_1,z_2)$ \\[1mm] \indent $\rho_4: \tup{q_c^{\downarrow},\varphi_{\lambda_i,1}(x)}(z_1,z_2)\to$ \\[1mm] \indent $\quad\quad\quad\tup{\overline{q}_i^c,{\rm stay}(x,y)}
(\tup{q_\mathrm{out}^\downarrow,{\rm stay}(x,y)}(z_1,z_2),
\tup{\overline{q}_c^\downarrow,{\rm stay}(x,y)}(z_1,z_2))$ \\[1mm] \indent $\rho_5: \tup{q_c^{\downarrow},\varphi_{\sigma_1,0}(x)}(z_1,z_2)\to
r(\tup{q_\mathrm{out}^\downarrow,{\rm stay}(x,y)}(z_1,z_2),z_2)$ \\[1mm] \indent $\rho_6: \tup{q_\mathrm{out}^\downarrow,\varphi_{\sigma_1,0}(x)}(z_1,z_2)\to
\sigma_1(z_1)$ \\[1mm] \indent $\rho_7: \tup{q_\mathrm{out}^\downarrow,\varphi_{\sigma_1,1}(x)}(z_1,z_2)\to z_1$ \\[1mm] \indent $\rho_8: \tup{q_\mathrm{out}^c,\varphi_{\sigma_1,1}(x)}(z_1,z_2)\to \sigma_1(z_1)$ \\[1mm] \indent $\rho_9: \tup{q_\mathrm{out}^\odot,\varphi_{\sigma_0,1}(x)}\to \sigma_0$ \\[1mm] \indent $\rho_{10}: \tup{q_\mathrm{next}^\downarrow,\varphi_{\sigma_1,1}(x)}(z_1,z_2)\to z_2$ \\[1mm] \indent $\rho_{11}: \tup{q_\mathrm{next}^c,\varphi_{\sigma_1,1}(x)}(z_1,z_2)\to z_2$ \\[1mm] \indent $\rho_{12}: \tup{q_\mathrm{next}^\odot,\varphi_{\sigma_0,1}(x)}\to e$ \\[1mm] \noindent Applying rule $\rho_6$ to the right-hand side of rule $\rho_5$, we obtain the rule $\rho'_5: \tup{q_c^{\downarrow},\varphi_{\sigma_1,0}(x)}(z_1,z_2)\to
r(\sigma_1(z_1),z_2)$, which is in fact rule $\rho_5$ of program ${\cal P}_\mathrm{sib}$ of Example~\ref{ex:tlsiberie}, if we identify the states $q_c^{\downarrow}$ and $q_c$. Rules $\rho_0$ and $\rho_1$ of ${\cal P}$ correspond to rule $\rho_1$ of ${\cal P}_\mathrm{sib}$ in an obvious way (with $q_\mathrm{start}^\odot$ and $q_\mathrm{start}$ identified). Since program ${\cal P}$ is deterministic, and its states generate trees (rather than forests), we can also apply rules $\rho_7-\rho_{12}$ to the right-hand side of rule $\rho_{c,d}$, and we obtain the rules \\[1mm] \indent $\rho'_{c,\downarrow}: \tup{\overline{q}_c^\downarrow,\varphi_{\sigma_1,1}(x)}(z_1,z_2)\to
\tup{q_c^{\downarrow},{\rm up}(x,y)}(z_1,z_2)$ \\[1mm] \indent $\rho'_{i,c}: \tup{\overline{q}_i^c,\varphi_{\sigma_1,1}(x)}(z_1,z_2)\to
\tup{q_i^{\downarrow},{\rm up}(x,y)}(\sigma_1(z_1),z_2)$ \\[1mm] \indent $\rho'_{c,\odot}: \tup{\overline{q}_c^\odot,\varphi_{\sigma_0,1}(x)}\to
\tup{q_c^{\downarrow},{\rm up}(x,y)}(\sigma_0,e)$ \\[1mm] \noindent Applying $\rho'_{1,\odot}$ to the right-hand side of $\rho_2$ we obtain $\rho'_2: \tup{q_\mathrm{start}^\odot,\varphi_{\sigma_0,1}(x)}\to
\tup{q_1^{\downarrow},{\rm up}(x,y)}(\sigma_0,e)$, which is rule $\rho_2$ of ${\cal P}_\mathrm{sib}$. Applying $\rho'_{0,\downarrow}$ to the right-hand side of $\rho_3$ we obtain $\rho'_3: \tup{q_0^{\downarrow},\varphi_{\lambda_0,1}(x)}(z_1,z_2)\to
\tup{q_0^{\downarrow},{\rm up}(x,y)}(z_1,z_2)$ which is rule $\rho_3$ of ${\cal P}_\mathrm{sib}$. Finally, applying rules $\rho'_{i,c}$, $\rho_7$, and $\rho'_{c,\downarrow}$ to the selectors in the right-hand side of rule $\rho_4$, respectively, we obtain the right-hand side $\tup{q_i,{\rm up}(x,y)}(\lambda_i(z_1),\tup{q_c,{\rm up}(x,y)}(z_1,z_2))$ of rule $\rho_4$ of ${\cal P}_\mathrm{sib}$. Thus, program ${\cal P}$ is essentially the same as program ${\cal P}_\mathrm{sib}$ of Example~\ref{ex:tlsiberie}. \end{example}
Lemmas~\ref{lem:tlipft} and~\ref{lem:ipfttl} together prove that \abb{tl} programs have the same expressive power as \abb{i-pft}'s. Additionally, they prove that for every \abb{tl} program there is an equivalent local one.
\begin{theorem}\label{thm:tl} $\family{TL}=\family{TL$_\ell$}=\family{I-PFT}$ and $\family{dTL}=\family{dTL$_\ell$}=\family{I-dPFT}$. Moreover, $\family{TL$_\text{r}$}=\family{TL$_{\ell\text{r}}$}=\family{I-PTT}$ and $\family{dTL$_\text{r}$}=\family{dTL$_{\ell\text{r}}$}=\family{I-dPTT}$. \end{theorem}
Since local \abb{tl} programs satisfy restriction (R2) in the definition of a ranked \abb{tl} program, the equation $\family{TL}=\family{TL$_\ell$}$ shows that the pattern matching aspect that is involved in the execution of selectors, can be viewed as an extended feature. Moreover, even the ``jumps'' in the execution of selectors, and the arbitrary \abb{mso} head tests in the left-hand sides of rules, can be viewed as extended features of~$\family{TL$_\ell$}$.
Note that for \abb{tl}$^\text{\abb{db}}_\ell$ programs the construction in the proof of Lemma~\ref{lem:tlipft} can easily be simplified to one that takes polynomial time and that results in an \abb{i-pft} that does not use \abb{mso} tests. That implies that the inverse type inference problem for such programs is solvable in \mbox{$2$-fold} exponential time, and hence typechecking can be done in $3$-fold exponential time (cf. Theorem~\ref{thm:tltypecheck}).
The local ranked \abb{tl} program is an obvious reformulation of the ``macro tree-walking transducer'' (2-mtt) of~\cite{ManBerPerSei}. The inclusion $\family{TL$_\text{r}$}\subseteq\family{TL$_{\ell\text{r}}$}$ is a (slightly stronger) version of~\cite[Theorem~5]{ManBerPerSei}. Moreover, the local ranked \abb{tl} program is the same as the \mbox{``$0$-pebble} macro tree transducer'' of~\cite[Section~5.1]{EngMan03} and it is the CFT($S$)-transducer of \cite{EngVog} for the storage type $S=$ Tree-walk, both of which generalize the macro attributed tree transducer of~\cite{KuhVog,FulVog} which additionally satisfies a noncircularity condition. It follows from Lemma~\ref{lem:nul-decomp} and Theorem~\ref{thm:tl} that $\family{TL$_{\ell\text{r}}$}\subseteq \family{TT}^2$, which was stated as an open problem in~\cite[Section~8]{EngMan03} (where $\family{TL$_{\ell\text{r}}$}$ and $\family{TT}$ are denoted 0-PMTT and 0-PTT, respectively). In view of Lemma~\ref{lem:ipftnf}, the equality $\family{TL$_{\ell\text{r}}$}=\family{I-PTT}$ is the same as the equality CFT($S$) $=$ RT(P($S$)) of~\cite[Theorem~5.24]{EngVog} for $S= \text{Tree-walk}$, and similarly for the deterministic case.
\section{A TL Program in XSLT}\label{sec:siberie}
In Tables~\ref{tab:input} and~\ref{tab:output} we listed a possible input document and the resulting output document for the \abb{i-ptt} ${\cal M}_\mathrm{sib}$ of Example~\ref{ex:siberie}. In this section we present in Table~\ref{tab:xslt} an XSLT~1.0 program with the same structure as the \abb{TL} program ${\cal P}_\mathrm{sib}$ of Example~\ref{ex:tlsiberie}. In what follows we comment on the XSLT program and its relationship to ${\cal P}_\mathrm{sib}$, abbreviated as ${\cal P}$.
\begin{table*}[p] \begin{scriptsize} \begin{verbatim} <xsl:stylesheet xmlns:xsl="http://www.w3.org/1999/XSL/Transform" version="1.0"> <xsl:output method="xml"/>
<xsl:template match="/">
<xsl:for-each select="//stop[@final=1]">
<xsl:call-template name="start" />
</xsl:for-each>
</xsl:template>
<xsl:template name="start">
<xsl:apply-templates select="parent::stop">
<xsl:with-param name="nextstoplarge" select="@large" />
<xsl:with-param name="stoplist">
<xsl:copy>
<xsl:copy-of select="attribute::*" />
</xsl:copy>
</xsl:with-param>
<xsl:with-param name="additionalresults">
<endofresults />
</xsl:with-param>
</xsl:apply-templates>
</xsl:template>
<xsl:template match="stop">
<xsl:param name="nextstoplarge" />
<xsl:param name="stoplist" />
<xsl:param name="additionalresults" />
<xsl:if test="@initial = 1">
<result>
<xsl:copy>
<xsl:copy-of select="attribute::*" />
<xsl:copy-of select="$stoplist" />
</xsl:copy>
<xsl:copy-of select="$additionalresults" />
</result>
</xsl:if>
<xsl:if test="not(@initial = 1)">
<xsl:variable name="results">
<xsl:apply-templates select="parent::stop">
<xsl:with-param name="nextstoplarge" select="$nextstoplarge" />
<xsl:with-param name="stoplist" select="$stoplist" />
<xsl:with-param name="additionalresults" select="$additionalresults" />
</xsl:apply-templates>
</xsl:variable>
<xsl:if test="@large = 1 or $nextstoplarge = 1">
<xsl:apply-templates select="parent::stop">
<xsl:with-param name="nextstoplarge" select="@large" />
<xsl:with-param name="stoplist">
<xsl:copy>
<xsl:copy-of select="attribute::*" />
<xsl:copy-of select="$stoplist" />
</xsl:copy>
</xsl:with-param>
<xsl:with-param name="additionalresults" select="$results" />
</xsl:apply-templates>
</xsl:if>
<xsl:if test="@large = 0 and $nextstoplarge = 0">
<xsl:copy-of select="$results" />
</xsl:if>
</xsl:if>
</xsl:template>
</xsl:stylesheet> \end{verbatim} \end{scriptsize} \caption{XSLT Program}\label{tab:xslt} \end{table*}
The first rule $\rho_1$ of ${\cal P}$ corresponds to the first template of the XSLT program: this template initalizes the algorithm by matching the root of the input document, jumping to the leaf by selecting the final stop, and invoking named template \texttt{start} on it.
The second rule $\rho_2$ of ${\cal P}$ corresponds to template \texttt{start}: it moves up, using the \texttt{apply-templates} instruction which selects the parent, and thus invokes the third template on that parent, which is the only template for nonroot document elements. It invokes that template with the appropriate parameters: \texttt{nextstoplarge} is 1 because $\mathtt{large = 1}$ for the final stop, \texttt{stoplist} is a list containing only the final stop, and \texttt{additionalresults} is the single element \texttt{<endofresults />}.
The remaining rules of ${\cal P}$ correspond to the third template, which is applied to all nonfinal stops. That template takes a partial stop list \texttt{stoplist} (from the current stop to the final stop) and generates all allowed ways to complete that stop list using the stops between the current one and the initial one. Nested below the deepest element of the output, it includes the result tree fragment passed in \texttt{additionalresults}. The third template has three parameters:
\begin{enumerate} \item[] \texttt{nextstoplarge}: a boolean indicating whether or not the ``next'' stop (i.e., the stop at the front of \texttt{stoplist}) is a large stop; it corresponds to states $q_1$ and $q_0$ in ${\cal P}$, respectively, \item[] \texttt{stoplist}: a partial list of stops (taken from the current stop to the final stop) for which this template will recursively generate all (allowed) ways in which it can be completed; it corresponds to parameter $z_1$ in ${\cal P}$, \item[] \texttt{additionalresults}: results that are to be appended to the results that this template generates; it corresponds to parameter $z_2$ in ${\cal P}$, \end{enumerate}
where both \texttt{stoplist} and \texttt{additionalresults} are of type `result tree fragment'.
Corresponding to rule $\rho_5$ of ${\cal P}$, the third template, when invoked on the initial stop (for which $\mathtt{initial = 1}$), has computed a complete stop list (after adding this stop) and outputs it: it copies the initial stop and nests the remainder of the stop list (i.e., the value of its parameter \texttt{stoplist}) in it; it also includes the additional results (i.e., the value of parameter \texttt{additionalresults}).
Corresponding to rules $\rho_3$ and $\rho_4$ of ${\cal P}$, the third template, when invoked on an intermediate stop (for which $\mathtt{not(initial = 1)}$), has not yet computed a complete stop list, and now calculates all allowed ways to complete it. Intuitively, it computes two result sets: one that \emph{does not} add the current stop, and one that \emph{does}. They are combined by passing the first result set as ``additional results'' to the calculation of the second one.
Thus, the third template starts by computing the first result set, and, to abbreviate the remaining code, it assigns its value to a variable called \texttt{results}. In rules $\rho_3$ and $\rho_4$ of ${\cal P}$ this result set corresponds to the selector $\tup{q_c,{\rm up}(x,y)}(z_1,z_2)$, where $c=0$ in $\rho_3$. In the case that \texttt{large = 0} and \texttt{nextstoplarge = 0}, we are not allowed to stop here because that would create two consecutive small stops. Thus the template only outputs the results that it just stored in the variable (corresponding to rule $\rho_3$ of ${\cal P}$). In the case that \texttt{large = 1} or \texttt{nextstoplarge = 1}, the template calculates all possible ways to complete the stop list that contain this stop, and includes as additional results those that are stored in the variable (corresponding to rule $\rho_4$ of ${\cal P}$).
\section{Data Complexity}
In this section we show that the transduction of a deterministic \abb{ptt} ${\cal M}$ can be realized in (1-fold) exponential time, in the sense that there is an exponential time algorithm that, for every given input tree $t$, computes a regular tree grammar $G$ that generates the language $\{\tau_{\cal M}(t)\}$. If $t$ is in the domain of ${\cal M}$, then~$G$ can be viewed as a DAG (directed acyclic graph) that defines the output tree $\tau_{\cal M}(t)$, in the usual sense. Thus, producing the actual output tree would take 2-fold exponential time. If $t$ is not in the domain of ${\cal M}$, then $G$ generates the empty tree language (which can be decided in time linear in the size of $G$).
\begin{theorem}\label{thm:expocom} For every deterministic \abb{ptt} ${\cal M}$ there is an exponential time algorithm that, for given input tree $t$, computes a regular tree grammar $G$ such that $L(G)=\{s\mid (t,s)\in\tau_{\cal M}\}$. \end{theorem}
\begin{proof} Let ${\cal M} = (\Sigma, \Delta, Q, \{q_0\},C, C_\mathrm{v}, C_\mathrm{i}, R,k)$ be a deterministic \abb{v$_k$i-ptt}. For an input tree $t\in T_\Sigma$ in the domain of ${\cal M}$, let us consider the computation $\tup{q_0,\mathrm{root}_t,\varepsilon}\Rightarrow^*_{t,{\cal M}} s$, where $s=\tau_{\cal M}(t)$, and let $\tup{q,u,\pi}$ be a configuration of~${\cal M}$ that occurs in that computation.
We claim that the length of $\pi$ is at most $N=|Q|\cdot(|C|+1)^{k+1}\cdot n^{k+2}$, where $n$ is the size of $t$.
To prove this claim we define, as an auxiliary tool, the nondeterministic \abb{v$_k$i-pta} ${\cal A}$ that is obtained from~${\cal M}$ by changing every output rule $\tup{q,\sigma,j,b}\to \delta(\tup{q_1,{\rm stay}},\dots,\tup{q_m,{\rm stay}})$ of ${\cal M}$ into the rules $\tup{q,\sigma,j,b}\to \tup{q_i,{\rm stay}}$ for all $i\in[1,m]$. Intuitively, whenever ${\cal M}$ branches, ${\cal A}$ nondeterministically follows one of those branches. Thus, all computations of ${\cal A}$ that start with $\tup{q_0,\mathrm{root}_t,\varepsilon}$ are finite. Obviously, $\tup{q,u,\pi}$ occurs in such a computation of~${\cal A}$. Let $\pi=(v_1,c_1)\cdots (v_m,c_m)$ and suppose that $m > N$. For every $\ell\in[1,m]$ we define $\pi_\ell=(v_1,c_1)\cdots (v_\ell,c_\ell)$. Then there exist configurations $\tup{q_\ell,u_\ell,\pi_\ell}$, $\ell\in[1,m]$, such that $\tup{q_0,\mathrm{root}_t,\varepsilon} \Rightarrow^*_{t,{\cal A}} \tup{q_1,u_1,\pi_1}$ and $\tup{q_\ell,u_\ell,\pi_\ell} \Rightarrow^*_{t,{\cal A}} \tup{q_{\ell+1},u_{\ell+1},\pi_{\ell+1}}$ for every $\ell\in[1,m-1]$, and such that, moreover, every configuration occurring in the computation $\tup{q_\ell,u_\ell,\pi_\ell} \Rightarrow^*_{t,{\cal A}} \tup{q_{\ell+1},u_{\ell+1},\pi_{\ell+1}}$ has a pebble stack with prefix~$\pi_\ell$. Due to the choice of $m$, there exist $i,j\in[1,m]$ with $i<j$ such that $q_i=q_j$, $u_i=u_j$, $(v_i,c_i)=(v_j,c_j)$, and for every $v\in N(t)$ and \mbox{$c\in C_\mathrm{v}$}: $(v,c)$~occurs in $\pi_i$ if and only $(v,c)$ occurs in $\pi_j$. This implies that the computation $\tup{q_i,u_i,\pi_i} \Rightarrow^*_{t,{\cal A}} \tup{q_j,u_j,\pi_j}$ can be repeated arbitrarily many times, leading to an infinite computation of ${\cal A}$, which is a contradiction and proves the claim.
We now construct the regular tree grammar $G$. Its nonterminals are
the configurations $\tup{q,u,\pi}$ of ${\cal M}$ on $t$ such that $|\pi|\leq N$. Since $N$ is polynomial in $n$, the number of nonterminals of $G$ is exponential in $n$. The initial nonterminal of $G$ is $\tup{q_0,\mathrm{root}_t,\varepsilon}$. If $\tup{q,u,\pi} \Rightarrow^*_{t,{\cal M}} \tup{q',u',\pi'} \Rightarrow_{t,{\cal M}} \delta(\tup{q_1,u',\pi'},\dots,\tup{q_m,u',\pi'})$, then $\tup{q,u,\pi} \to \delta(\tup{q_1,u',\pi'},\dots,\tup{q_m,u',\pi'})$ is a rule of $G$. To decide whether $\tup{q',u',\pi'} \Rightarrow_{t,{\cal M}} \delta(\tup{q_1,u',\pi'},\dots,\tup{q_m,u',\pi'})$ it suffices to inspect the output rules of ${\cal M}$. To decide whether $\tup{q,u,\pi} \Rightarrow^*_{t,{\cal M}} \tup{q',u',\pi'}$ we construct from~${\cal M}$ and $t$ an ordinary pushdown automaton ${\cal P}$ that simulates the non-output behaviour of ${\cal M}$ on $t$, as in the query evaluation paragraph at the end of Section~\ref{sec:xpath}. Since, as opposed to that paragraph, ${\cal M}$ also has visible pebbles, ${\cal P}$~should keep track of those pebbles in its finite state. Let $\Gamma$ be the set of all mappings $\gamma: C_\mathrm{v}\to N(t)\cup\{\bot\}$ such that $\#(\{c\in C_\mathrm{v}\mid \gamma(c)\neq \bot\})\leq k$. During ${\cal P}$'s computation, the mapping $\gamma$ in its finite state indicates for every visible pebble whether it occurs in the current stack and, if so, on which node it is dropped. Thus, we define ${\cal P}$ to have state set $Q\times N(t)\times \Gamma$ and pushdown alphabet $N(t)\times C$. A configuration $\tup{q,u,\pi}$ of ${\cal M}$ is simulated by the configuration ${\cal P}(\tup{q,u,\pi})=\tup{p,\pi}$ of ${\cal P}$ such that $p=(q,u,\gamma)$ where, for every $c\in C_\mathrm{v}$, if $\gamma(c)\in N(t)$ then $(\gamma(c),c)$ occurs in $\pi$, and if $\gamma(c)=\bot$ then $c$ does not occur in $\pi$. The transitions of the automaton ${\cal P}$ are defined in such a way that ${\cal P}$ (with the empty string as input) has the same computation steps as ${\cal M}$ (without its output rules), i.e., such that $\tup{q,u,\pi} \Rightarrow_{t,{\cal M}} \tup{q',u',\pi'}$ if and only if ${\cal P}(\tup{q,u,\pi}) \Rightarrow_{\cal P} {\cal P}(\tup{q',u',\pi'})$, where $\Rightarrow_{\cal P}$ is the computation step relation of ${\cal P}$. For instance, let ${\cal P}$ be in state $(q,u,\gamma)$ and let the top element of its stack be $(v,c)$. Let $u$ have label $\sigma$ and child number $j$, and let $b$ consist of all $c'\in C_\mathrm{v}$ with $\gamma(c')= u$ plus $c$ if $v=u$. If $\tup{q,\sigma,j,b}\to\tup{q',{\rm drop}_d}$ is a rule of ${\cal M}$ such that $d\in C_\mathrm{v}$, $\gamma(d)=\bot$, and $\#(\{c'\in C_\mathrm{v}\mid \gamma(c')\neq \bot\}) < k$, then ${\cal P}$ pushes $(u,d)$ on its stack and goes into state $(q',u,\gamma')$ where $\gamma'(d)=u$ and $\gamma'(c')=\gamma(c')$ for all $c'\neq d$. If $\tup{q,\sigma,j,b}\to\tup{q',{\rm lift}_c}$ is a rule of ${\cal M}$ such that $c\in C_\mathrm{i}$ and $v=u$, then ${\cal P}$ pops $(v,c)$ from its stack and goes into state $(q',u,\gamma)$. The transitions of ${\cal P}$ are defined similarly for the other non-output rules of ${\cal M}$. It should be clear that ${\cal P}$ can be constructed in time polynomial in $n$. Since it can be decided in polynomial time for configurations $\tup{p,\pi}$ and $\tup{p',\pi'}$ of ${\cal P}$ whether $\tup{p,\pi} \Rightarrow^*_{\cal P} \tup{p',\pi'}$, it can be decided whether $\tup{q,u,\pi} \Rightarrow^*_{t,{\cal M}} \tup{q',u',\pi'}$ in polynomial time. Hence the total time to construct $G$ is exponential. \end{proof}
Note that the first part of the above proof also shows that for every deterministic \abb{ptt} the height of the output tree is exponential in the size of the input tree.
A natural question is whether Theorem~\ref{thm:expocom} also holds for forest transducers, i.e., for deterministic \abb{pft}'s. That is indeed the case (as the reader can easily verify), except that $G$ is not a regular forest grammar, but a forest generating context-free grammar. To be precise, $G$ is a context-free grammar of which every rule is of the form $X_0\to \delta(X_1)$ or $X_0\to X_1X_2$ or $X\to\varepsilon$ where $\delta$~is a symbol from an unranked alphabet. If $L(G)=\{f\}$, then $G$ can still be viewed as a DAG that defines the forest $f$. Thus, in this sense, by Theorem~\ref{thm:tl}, deterministic \abb{tl} programs can be executed in exponential time, in accordance with the result of~\cite{JanKorBus} that XSLT~1.0 programs can be executed in exponential time.
Another natural question is whether there exist interesting subclasses of \abb{ptt}'s that can be realized in polynomial time. Here we discuss one such subclass. We define a \abb{ptt} to be \emph{bounded} if there exists $m\in {\mathbb N}$ such that output rules can only be applied when the pebble stack contains at most $m$ pebbles. Intuitively it means that the infinitely many invisible pebbles are mainly used to check \abb{mso} properties of the observable configuration. Formally it can either be required as a dynamic property of the (successful) computations of the \abb{ptt} or be incorporated statically in the semantics of the \abb{ptt}. We now show that bounded \abb{ptt}'s can be realized in polynomial time, even in the nondeterministic case.
\begin{theorem}\label{thm:polycom} For every bounded \abb{ptt} ${\cal M}$ there is a polynomial time algorithm that, for given input tree $t$, computes a regular tree grammar~$G$ such that $L(G)=\{s\mid (t,s)\in\tau_{\cal M}\}$. \end{theorem}
\begin{proof} The construction of $G$ is exactly the same as in the proof of Theorem~\ref{thm:expocom}, except that its nonterminals are now the configurations
$\tup{q,u,\pi}$ of ${\cal M}$ on $t$ such that $|\pi|\leq m$.\footnote{Additionally, $G$ has an initial nonterminal $S$ with rules $S\to \tup{q_0,\mathrm{root}_t,\varepsilon}$ for every initial state $q_0$ of ${\cal M}$. } The number of nonterminals of $G$ is therefore polynomial in the size of $t$, and since the pushdown automaton ${\cal P}$ can also be constructed (and tested) in polynomial time, the total time to construct $G$ is polynomial. \end{proof}
Again, the same result holds for \abb{pft}'s, taking $G$ to be a forest generating context-free grammar. Note that for a nondeterministic \abb{pft} ${\cal M}$ and an input tree $t$, the set $\{s\mid (t,s)\in\tau_{\cal M}\}$ is not necessarily a regular forest language.
Also, the same result holds for bounded \abb{ptt}'s that use \abb{mso} tests on the observable configuration. That is not immediate, because the construction in the proof of Theorem~\ref{thm:mso} does not preserve boundedness, due to the use of beads. However, it is easy to adapt the construction of the pushdown automaton ${\cal P}$ in the proof of Theorem~\ref{thm:expocom} to incorporate the \abb{mso} tests of the \abb{v$_k$i-ptt} ${\cal M}$. In fact, the observable configuration tree $\xp{obs}(t,\pi)$ can be constructed from $t$, from the mapping $\gamma$ in the state of ${\cal P}$, and from the top element of its stack, and then $\xp{obs}(t,\pi)$ can be tested in linear time using a deterministic bottom-up finite-state tree automaton. An example of bounded \abb{ptt}'s (with \abb{mso} tests) are the pattern matching \abb{ptt}'s of Section~\ref{sec:pattern}. In that section, every \abb{ptt} that matches an $n$-ary pattern is bounded, with bound $n$ or even $n-1$. Hence, pattern matching \abb{ptt}'s can be evaluated in polynomial time. And the same is true for pattern matching \abb{pft}'s, see Section~\ref{sec:pft}.
\section{Variations of Decomposition}\label{sec:variations}
In this section we present a number of results the proofs of which are based on variations of the decomposition techniques used in Section~\ref{sec:decomp}. In the first part of the section we consider deterministic \abb{ptt}'s, and in the second part we consider \abb{ptt}'s with strong (visible) pebbles.
\smallpar{Deterministic PTT's} As observed at the end of Section~\ref{sec:decomp} it is open whether $\family{I-dPTT} \subseteq \family{dTT} \circ \family{dTT}$. We first show that a subclass of $\family{I-dPTT}$ is included in $\family{dTT} \circ \family{dTT}$ and then we show that $\family{I-dPTT} \subseteq \family{dTT}^3$. Hence, every deterministic \abb{ptt} can be decomposed into deterministic \abb{tt}'s.
Recall that $\family{dTT$^{\text{\abb{MSO}}}$}$ denotes the class of transductions that are realized by deterministic \abb{tt}'s with \abb{mso} head tests. By Lemma~\ref{lem:sites} it is a subclass of $\family{I-dPTT}$. We will show that such transducers can be decomposed into two deterministic \abb{tt}'s of which the first never moves up. To do this we need a lemma with an alternative proof of the inclusion $\family{dTT$^{\text{\abb{MSO}}}$} \subseteq \family{I-dPTT}$, showing that the resulting \abb{i-ptt} uses its pebbles in a restricted way. The \abb{i-ptt} that is constructed in the proof of Lemma~\ref{lem:sites} does not satisfy that restriction.
For the definition of normal form of an \abb{i-ptt} see the paragraphs before Lemma~\ref{lem:ipftnf}. We now define an \abb{i-ptt} (or \abb{i-pta}) to be \emph{root-oriented} if it satisfies requirements (1)$-$(3) of the normal form, and all non-initial non-output rules have a right-hand side of one of the following five forms: $\tup{q',{\rm down}_i;{\rm drop}_c}$, $\tup{q',{\rm lift}_c;{\rm up}}$, $\tup{q',{\rm lift}_c;{\rm drop}_d}$, or $\tup{q',{\rm stay}}$, where $q'\in Q\setminus Q_0$, $i\in{\mathbb N}$ and \mbox{$c,d\in C$}. Thus, except in an initial configuration, every pebble stack is of the form $(u_1,c_1)\cdots(u_n,c_n)$ where $u_1,\dots,u_n$ is the path from the root to the current node. The \abb{i-pta} in the proof of Lemma~\ref{lem:regular} is root-oriented.
The next lemma follows from~\cite[Theorem~8.12]{thebook}, but we provide its proof for completeness sake. Let $\family{r\,I-dPTT}$ denote the class of transductions realized by root-oriented deterministic \abb{i-ptt}'s.\footnote{In~\cite[Chapter~8]{thebook} root-oriented \abb{i-ptt}'s are called tree-walking pushdown transducers, and $\family{r\,I-dPTT}$ is denoted $\family{P-DTWT}$. They are the \abb{RT(P(TR))}-transducers of~\cite{EngVog}, also called indexed tree transducers. }
\begin{lemma}\label{lem:in-ridptt} $\family{dTT$^{\text{\abb{MSO}}}$} \subseteq \family{r\,I-dPTT}$. \end{lemma}
\begin{proof} Let ${\cal M}$ be a deterministic \abb{tt} that uses a regular site $T$ as \abb{mso} head test. For simplicity we will assume that ${\cal M}$ tests $T$ in every rule. Let ${\cal A}=(\Sigma\times\{0,1\},P,F,\delta)$ be a deterministic bottom-up finite-state tree automaton that recognizes $\operatorname{mark}(T)$. As usual we identify the symbols $(\sigma,0)$ and $\sigma$. For every tree $t\in T_\Sigma$ and every node $u\in N(t)$, we define the set $\mathrm{succ}_t(u)$ of \emph{successful states} of ${\cal A}$ at $u$ to consist of all states $p\in P$ such that ${\cal A}$ recognizes~$t$ when started at $u$ in state $p$. To be precise, $\mathrm{succ}_t(\mathrm{root}_t)=F$ and if $u$ has label $\sigma\in\Sigma^{(m)}$ and $i\in[1,m]$, then $\mathrm{succ}_t(ui)$ is the set of all states $p\in P$ such that $\delta(\sigma,p_1,\dots,p_{i-1},p,p_{i+1},\dots,p_m)\in \mathrm{succ}_t(u)$, where $p_j$ is the state in which ${\cal A}$~arrives at $uj$ for every $j\in[1,m]\setminus\{i\}$.
We construct a root-oriented deterministic \abb{i-ptt} ${\cal M}'$ that stepwise simulates~${\cal M}$ and simultaneously keeps track of $\mathrm{succ}_t(v)$ for all nodes $v$ on the path from the root to the current node $u$, by storing that information in its pebble colours. It uses the \abb{i-pta} ${\cal A}'$ of Lemma~\ref{lem:regular} (with ${\cal A}$ restricted to $\Sigma\times\{0\}$) as a subroutine to compute the states in which ${\cal A}$ arrives at the children of $u$. Using these states and $\mathrm{succ}_t(u)$, it can easily test whether $(t,u)\in T$. Morover, when moving down to a child $ui$ of $u$ it can use this information to compute $\mathrm{succ}_t(ui)$.
Formally, in addition to the pebble colours $p_1\cdots p_m$ of ${\cal A}'$, the transducer~${\cal M}'$ uses pebble colours $(S,p_1\cdots p_m)$ where $S\subseteq P$. As states it uses (apart from its initial state) the states of ${\cal M}$ and states of the form $(\tilde{q},q)$ where $\tilde{q}$ is a state of~${\cal M}$ and $q$ a state of ${\cal A}'$; in fact, $q$ is either the main state $q_\circ$ of ${\cal A}'$ or it is $\bar{q}_p$ for some $p\in P$. Initially, ${\cal M}'$ drops pebble $(F,\varepsilon)$ on the root and goes into state $(\tilde{q}_0,q_\circ)$ where $\tilde{q}_0$ is the initial state of ${\cal M}$. This incorporates rule $\rho_1$ of ${\cal A}'$. The other rules of ${\cal M}'$ that correspond to ${\cal A}'$ are as follows. First, the rule $\rho_2$ of ${\cal A}'$ together with the corresponding rule for pebble colour $(S,p_1\cdots p_m)$, both for $m<\operatorname{rank}(\sigma)$: \[ \begin{array}{lll} \tup{(\tilde{q},q_\circ),\sigma,j,\{p_1\cdots p_m\}} & \to &
\tup{(\tilde{q},q_\circ),{\rm down}_{m+1};{\rm drop}_\varepsilon} \\[1mm] \tup{(\tilde{q},q_\circ),\sigma,j,\{(S,p_1\cdots p_m)\}} & \to &
\tup{(\tilde{q},q_\circ),{\rm down}_{m+1};{\rm drop}_\varepsilon}. \end{array} \] Second, the rule $\rho_3$ of ${\cal A}'$, for $m=\operatorname{rank}(\sigma)$ and $p=\delta(\sigma,p_1,\dots,p_m)$: \[ \begin{array}{llll} \tup{(\tilde{q},q_\circ),\sigma,j,\{p_1\cdots p_m\}} & \to &
\tup{(\tilde{q},\bar{q}_p),{\rm lift}_{p_1\cdots p_m};{\rm up}}
& \text{if } j\neq 0. \end{array} \] Third, the rule $r_6$ of ${\cal A}'$ together with the corresponding rule for pebble colour $(S,p_1\cdots p_m)$, both for $m<\operatorname{rank}(\sigma)$: \[ \begin{array}{lll} \tup{(\tilde{q},\bar{q}_p),\sigma,j,\{p_1\cdots p_m\}} & \to
& \tup{(\tilde{q},q_\circ),{\rm lift}_{p_1\cdots p_m};{\rm drop}_{p_1\cdots p_mp}} \\[1mm] \tup{(\tilde{q},\bar{q}_p),\sigma,j,\{(S,p_1\cdots p_m)\}} & \to
& \tup{(\tilde{q},q_\circ),{\rm lift}_{(S,p_1\cdots p_m)};{\rm drop}_{(S,p_1\cdots p_mp)}}. \end{array} \] The subroutine ${\cal A}'$ is always called at a node $u$ where ${\cal M}'$ observes a pebble of the form $(S,\varepsilon)$, and when ${\cal A}'$ is finished ${\cal M}'$ is back at the same node $u$ and observes the pebble $(S,p_1\cdots p_m)$ where $p_1,\dots,p_m$ are the states at which ${\cal A}$~arrives at the children of $u$.
Finally we consider the simulation of a step of ${\cal M}$, which either occurs when the subroutine ${\cal A}'$ is finished (instead of its rules $\rho_4$ and $\rho_5$), or just after the simulation of another step of ${\cal M}$, in which it does not move down. Suppose that ${\cal M}$ has a rule $\tup{\tilde{q},\sigma,j,T}\to \zeta$ and that $\delta((\sigma,1),p_1,\dots,p_m)\in S$, or suppose that it has a rule $\tup{\tilde{q},\sigma,j,\neg T}\to \zeta$ and $\delta((\sigma,1),p_1,\dots,p_m)\notin S$. Then ${\cal M}'$ has the following two rules, for $m=\operatorname{rank}(\sigma)$: \[ \begin{array}{lll} \tup{(\tilde{q},q_\circ),\sigma,j,\{(S,p_1\cdots p_m)\}} & \to & \zeta' \\[1mm] \tup{\tilde{q},\sigma,j,\{(S,p_1\cdots p_m)\}} & \to & \zeta' \end{array} \] such that
\begin{enumerate} \item[(1)] if $\zeta=\tup{\tilde{q}',{\rm up}}$, then $\zeta'=\tup{\tilde{q}',{\rm lift}_{(S,p_1\cdots p_m)};{\rm up}}$, \item[(2)] if $\zeta=\tup{\tilde{q}',{\rm down}_i}$, then $\zeta'=\tup{(\tilde{q}',q_\circ),{\rm down}_i;{\rm drop}_{(S',\varepsilon)}}$ \\ where $S'=\{p\in P \mid \delta(\sigma,p_1,\dots,p_{i-1},p,p_{i+1},\dots,p_m)\in S\}$, and \item[(3)] $\zeta'=\zeta$ otherwise. \end{enumerate}
This ends the formal description of ${\cal M}'$. In general, ${\cal M}$ uses regular sites $T_1,\dots,T_n$ as \abb{mso} head tests, and correspondingly ${\cal M}'$ has pebble colours of the form $(S_1,\dots,S_n,p_1\cdots p_m)$ where $S_i$ is a set of states of an automaton ${\cal A}_i$ recognizing $\operatorname{mark}(T_i)$. \end{proof}
Let $\family{dTT}\!_\downarrow$ denote the class of transductions realized by deterministic \abb{tt}'s that do not use the ${\rm up}$-instruction. Such transducers are equivalent to classical deterministic top-down tree transducers. The next lemma is shown in~\cite[Theorem~8.15]{thebook} but we provide its proof again, to show the connection to Lemma~\ref{lem:nul-decomp}.
\begin{lemma}\label{lem:ridptt-in} $\family{r\,I-dPTT} \subseteq \family{dTT}\!_\downarrow \circ \family{dTT}$. \end{lemma}
\begin{proof} Let ${\cal M}$ be a root-oriented deterministic \abb{i-ptt}. Looking at the proof of Lemma~\ref{lem:nul-decomp}, it should be clear that, for every input tree $t$, the simulating transducer ${\cal M}'$ only visits those nodes of $t'$ that correspond to a sequence of instructions of ${\cal M}$ that starts with a drop-instruction and then consists alternatingly of a down-instruction and a drop-instruction. Consequently, the ``preprocessor'' ${\cal N}$ can be adapted so as to generate just that part of $t'$. The new ${\cal N}$ does not need the states $f_i$ any more, but just has the initial state $g$ and the state $f$. Its rules are \[ \begin{array}{lll} \tup{g,\sigma,j} & \to & \sigma'(\bot^m, \tup{f,{\rm stay}}^\gamma) \\[1mm] \langle f,\sigma,j\rangle & \to & \sigma'_{0,j}(\tup{g,{\rm down}_1},\dots,\tup{g,{\rm down}_m}, \bot^\gamma, \bot) \end{array} \] where $m$ is the rank of $\sigma$ and $\bot^n$ abbreviates the sequence $\bot,\dots,\bot$ of length~$n$. Note that the child number $j$ is irrelevant. With this new, total deterministic preprocessor ${\cal N}$ the proof of Lemma~\ref{lem:nul-decomp} is still valid. \end{proof}
The following corollary was shown in~\cite[Theorem~8.22]{thebook}, but we repeat it here for completeness sake, cf. Corollary~\ref{cor:mtiptt}.
\begin{corollary}\label{cor:ridpttmt} $\family{r\,I-dPTT} = \family{dTT}\!_\downarrow \circ \family{dTT} = \family{dMT}_\text{\abb{OI}}$. \end{corollary}
\begin{proof} The inclusion $\family{dTT}\!_\downarrow \circ \family{dTT} \subseteq \family{dMT}_\text{\abb{OI}}$ follows from the inclusions $\family{dTT}\subseteq \family{dMT}_\text{\abb{OI}}$, shown in~\cite[Theorem~35 for $n=0$]{EngMan03}, and $\family{dTT}\!_\downarrow \circ \family{dMT}_\text{\abb{OI}}\subseteq \family{dMT}_\text{\abb{OI}}$, shown in~\cite[Theorem~7.6(3)]{EngVog85}. By Lemma~\ref{lem:ridptt-in} it now suffices to show that $\family{dMT}_\text{\abb{OI}} \subseteq \family{r\,I-dPTT}$ (which strengthens the second inclusion of Corollary~\ref{cor:mtiptt}). There are two ways of proving this, which are essentially the same. First, the proof of Lemma~\ref{lem:tlipft} can be adapted in a straightforward way.\footnote{The transducer ${\cal M}$ uses an additional pebble $\odot$, which it drops initially on the root and whenever it moves down (instead of calling subroutine $S_{q',\psi}$). When necessary it replaces $\odot$ by a pebble $([s_1],\dots,[s_m])$. When subroutine $S$ is in state $[z_i]$ for some parameter $z_i$, it lifts $\odot$ and moves up where it finds a pebble $([s_1],\dots,[s_m])$. } Second, the equality $\family{r\,I-dPTT} = \family{dMT}_\text{\abb{OI}}$ is shown for total functions in~\cite[Theorem~5.16]{EngVog}. By~\cite[Theorem~6.18]{EngVog85}, every transduction $\tau\in \family{dMT}_\text{\abb{OI}}$ is of the form $\tau_1\circ\tau_2$ where $\tau_1$ is the identity on a regular tree language $R$ and $\tau_2\in\family{dMT}_\text{\abb{OI}}$ is a total function. Thus, $\tau_2$ is in $\family{r\,I-dPTT}$. This implies that $\tau_1 \circ\tau_2$ is in $\family{r\,I-dPTT}$: the \abb{i-ptt} just starts by checking that the input tree is in $R$, using the root-oriented \abb{i-ptt} ${\cal A}'$ in the proof of Lemma~\ref{lem:regular} as a subroutine. \end{proof}
We now turn to the decomposition of an arbitrary deterministic \abb{i-ptt} into deterministic \abb{tt}'s.
\begin{lemma}\label{lem:decompidptt} $\family{I-dPTT} \subseteq \family{tdTT$^{\text{\abb{MSO}}}$}\circ \family{dTT}$. \end{lemma}
\begin{proof} Let ${\cal M} = (\Sigma,\Delta,Q,\{q_0\},C,\varnothing,C_\mathrm{i},R,0)$ be a deterministic \abb{i-ptt} with $C=C_\mathrm{i}$. We may assume that there is a mapping $\chi: C\to Q$ such that $\chi(c)=q'$ for every rule $\tup{q,\sigma,j,b}\to\tup{q',{\rm drop}_c}$ of ${\cal M}$. If not, then we change $C$ into $C\times Q$ and we change every rule $\tup{q,\sigma,j,b}\to\tup{q',{\rm drop}_c}$ into $\tup{q,\sigma,j,b}\to\tup{q',{\rm drop}_{(c,q')}}$ and every rule $\tup{q,\sigma,j,\{c\}}\to\tup{q',{\rm lift}_c}$ into all the rules $\tup{q,\sigma,j,\{(c,p)\}}\to\tup{q',{\rm lift}_{(c,p)}}$. Moreover, we may assume that $C=[1,\gamma]$ for some $\gamma\in{\mathbb N}$.
As in the proof of Lemma~\ref{lem:ridptt-in} we consider the proof of Lemma~\ref{lem:nul-decomp} and adapt the preprocessor ${\cal N}$ to the needs of ${\cal M}$. Every copy of the input tree that is generated by ${\cal N}$ corresponds to a unique potential pebble stack $\pi$ of ${\cal M}$. The simulating deterministic \abb{tt} ${\cal M}'$ walks on that copy whenever ${\cal M}$ has pebble stack~$\pi$. The idea is now to construct a variation ${\cal N}'$ of ${\cal N}$ that only generates those copies of the input tree $t$ that correspond to reachable pebble stacks. A~pebble stack $\pi$ is \emph{reachable} (on $t$) if ${\cal M}$ has a reachable output form that contains a configuration $\tup{q,v,\pi}$ for some $q\in Q$ and $v\in N(t)$. For a given~$t$ in the domain of ${\cal M}$, the number of reachable stacks is finite because ${\cal M}$ is deterministic and thus has a unique computation on~$t$. Consequently ${\cal N}'$ can preprocess $t$ deterministically. Then we can define a total deterministic preprocessor ${\cal N}''$ that starts by performing an \abb{mso} head test whether or not the input tree is in the domain of ${\cal M}$ (which is regular by Corollary~\ref{cor:domptt}). If it is, then ${\cal N}''$ calls ${\cal N}'$, and if it is not, then ${\cal N}''$ outputs~$\bot$ and halts.
As an auxiliary tool, we define (as in the proof of Theorem~\ref{thm:expocom}) the nondeterministic \abb{i-pta} ${\cal A}$ that is obtained from~${\cal M}$ by changing every output rule $\tup{q,\sigma,j,b}\to \delta(\tup{q_1,{\rm stay}},\dots,\tup{q_m,{\rm stay}})$ of ${\cal M}$ into the rules $\tup{q,\sigma,j,b}\to \tup{q_i,{\rm stay}}$ for $i\in[1,m]$. Intuitively, whenever ${\cal M}$ branches, ${\cal A}$ nondeterministically follows one of those branches. Obviously a nonempty pebble stack $\pi$ with top element $(u,c)$ is reachable if and only if $\tup{\chi(c),u,\pi}$ is a reachable configuration of ${\cal A}$ (see footnote~\ref{foot:reachcon}). Note that $\tup{\chi(c),u,\pi}$ is the configuration of ${\cal M}$ just after dropping pebble~$c$ at node $u$.
For pebble colour $c$, we consider the site $T_c$ consisting of all pairs $(t,u)$ such that one-pebble stack $(u,c)$ is reachable, i.e., such that ${\cal A}$ has a computation starting in the initial configuration and ending in the configuration $\tup{\chi(c),u,(u,c)}$. It is not difficult to see that $T_c$ is a regular site. In fact, $\operatorname{mark}(T_c)$ is the domain of an \abb{i-pta} ${\cal B}$ with stack tests that simulates ${\cal A}$; whenever it arrives at the marked node $u$ in state $\chi(c)$ and it observes pebble $c$, then it may lift the pebble, check that its stack is empty, and accept. Stack tests are allowed by Lemma~\ref{lem:stacktests}, and the domain of ${\cal B}$ is regular by Corollary~\ref{cor:domptt}.
We now turn to reachable pebble stacks with more than one pebble, i.e., of the form $\pi(u,c)(v,d)$. Assuming that we already know that $\pi(u,c)$ is reachable, we can find out whether $\pi(u,c)(v,d)$ is reachable through a regular trip, as follows. For pebble colours $c$ and $d$, we consider the trip $T_{c,d}$ consisting of all triples $(t,u,v)$ such that ${\cal A}$ has a computation on $t$ starting in configuration $\tup{\chi(c),u,(u,c)}$ and ending in configuration $\tup{\chi(d),v,(u,c)(v,d)}$; moreover, in every intermediate configuration the bottom element of the pebble stack must be $(u,c)$. The trip $T_{c,d}$ is regular because $\operatorname{mark}(T)$ is the domain of an \abb{i-pta}~${\cal B}'$ with stack tests that first walks to the marked node $u$. Then ${\cal B}'$ simulates ${\cal A}$, starting in state $\chi(c)$, interpreting the mark of $u$ as pebble $c$ (which cannot be lifted). Similar to ${\cal B}$ above, whenever ${\cal B}'$ arrives at the marked node $v$ in state $\chi(d)$ and it observes pebble $d$, then it may lift the pebble, check that the stack is empty, and accept. Obviously, if $\pi(u,c)$ is reachable, then $\pi(u,c)(v,d)$ is reachable if and only if $(t,u,v)\in T_{c,d}$. Let ${\cal B}_{c,d}$ be a (nondeterministic) \abb{ta} with \abb{mso} head tests that computes $T_{c,d}$, as in Proposition~\ref{prop:trips}.
The new preprocessor ${\cal N}'$ is a deterministic \abb{tt} with \abb{mso} head tests that works in the same way as ${\cal N}$ but only creates the copies of the input tree $t$ that correspond to reachable pebble stacks. Initially it uses the test $T_c$ at node $u$ to decide whether it has to create a copy of $t$ corresponding to pebble stack $(u,c)$. If the test is positive, then, just as ${\cal N}$, it creates a copy of $t$ by walking from $u$ to every other node $v$ of $t$, copying $v$ to the output. Now recall that ${\cal N}$ walks from $u$ to $v$ along the shortest (undirected) path in $t$. Thus, by Proposition~\ref{prop:trips}, ${\cal N}'$ can simulate the behaviour of \abb{ta} ${\cal B}_{c,d}$ from $u$ to $v$, for every pebble colour $d$ (using a subset construction as in the proof of Theorem~\ref{thm:mso}). Thus, arriving at $v$ it can use the trip $T_{c,d}$ to decide whether it has to create a copy of $t$ corresponding to pebble stack $(u,c)(v,d)$. At the next level it simulates all ${\cal B}_{d,d'}$ for every $d'\in C$, etcetera.
More formally, ${\cal N}'$ has initial state $g$, and all other states are of the form $(q,c,S_1,\dots,S_\gamma)$ where $q$ is a state of ${\cal N}$, $c\in C$, and $S_d$ is a set of states of ${\cal B}_{c,d}$ for every $d\in C=[1,\gamma]$. We will call them ``extended'' states in what follows. To describe the rules of ${\cal N}'$, we first recall the rules of the transducer ${\cal N}$ from the proof of Lemma~\ref{lem:nul-decomp}. Apart from the rules $\langle f,\sigma,j\rangle \to \bot$, ${\cal N}$ has the rules \[ \begin{array}{llll} \rho_g: & \langle g,\sigma,j\rangle & \to & \sigma'(\langle g,{\rm down}_1\rangle,\dots,\langle g,{\rm down}_m\rangle,
\langle f,{\rm stay}\rangle^\gamma) \\[1mm] \rho_f: & \langle f,\sigma,j\rangle & \to & \sigma'_{0,j}(\langle g,{\rm down}_1\rangle,\dots,\langle g,{\rm down}_m\rangle,
\tup{f,{\rm stay}}^\gamma, \xi_j) \\[1mm] \rho_{f_i}: & \langle f_i,\sigma,j\rangle & \to & \sigma'_{i,j}(
\langle g,{\rm down}_1\rangle,\dots,\langle g,{\rm down}_{i-1}\rangle,
\bot, \\ &&& \quad\langle g,{\rm down}_{i+1}\rangle,\dots,\langle g,{\rm down}_m\rangle,
\tup{f,{\rm stay}}^\gamma,
\xi_j) \end{array} \] where $\xi_j = \langle f_j,{\rm up}\rangle$ for $j\neq 0$, and $\xi_0 = \bot$.
The rules of ${\cal N}'$ for state $g$ are obtained from rule $\rho_g$ by adding all possible combinations of the \abb{mso} head tests $T_c$ and their negations to the left-hand side. In the right-hand side, the sequence $\tup{f,{\rm stay}}^\gamma$ should be replaced by the sequence $\zeta_1,\dots,\zeta_\gamma$ where $\zeta_c=\tup{(f,c,I_{c,1},\dots,I_{c,\gamma}),{\rm stay}}$ if $T_c$ is true, $I_{c,d}$ being the set of initial states of ${\cal B}_{c,d}$, and $\zeta_c=\bot$ if $T_c$ is false.\footnote{More precisely, $I_{c,d}$ consists of all initial states of ${\cal B}_{c,d}$, plus all states that ${\cal B}_{c,d}$ can reach from an initial state by applying a relevant rule with a stay-instruction. } The rules of ${\cal N}'$ for an ``extended'' state $(q,c,S_1,\dots,S_\gamma)$ are obtained from rule $\rho_q$ as follows. In the left-hand side change $q$ into $(q,c,S_1,\dots,S_\gamma)$. Moreover, add all \abb{mso} head tests of ${\cal B}_{c,d}$ for every $d\in C$. In the right-hand side change every occurrence of a state $q'\neq f$ into the extended state $(q',c,S'_1,\dots,S'_\gamma)$ where the set $S'_d$ is obtained from the set $S_d$ by simulating ${\cal B}_{c,d}$ appropriately, moving down to the \mbox{$\ell$-th} child if $q'=g$ in $\tup{g,{\rm down}_\ell}$ and moving up if $q'=f_j$. Moreover, the sequence $\tup{f,{\rm stay}}^\gamma$ should be replaced by $\zeta_1,\dots,\zeta_\gamma$ where $\zeta_d=\tup{(f,d,I_{d,1},\dots,I_{d,\gamma}),{\rm stay}}$ if $S_d$ contains a final state of ${\cal B}_{c,d}$, and $\zeta_d=\bot$ otherwise (where $I_{d,d'}$ is defined similarly to $I_{c,d}$ above).
It should be clear that ${\cal N}'$ produces an output for every input tree $t$ on which ${\cal M}$ has finitely many reachable pebble stacks. Thus, ${\cal N}'$ preprocesses $t$ appropriately and the deterministic \abb{tt} ${\cal M}'$ in the proof of Lemma~\ref{lem:nul-decomp} can simulate ${\cal M}$ on $\tau_{{\cal N}'}(t)$. Hence $\tau_{{\cal M}'}(\tau_{{\cal N}'}(t))=\tau_{\cal M}(t)$ for every $t$ in the domain of ${\cal M}$. \end{proof}
It is easy to adapt the proof of Theorem~\ref{thm:composition} to the case where the first (deterministic) \abb{tt} ${\cal M}_1$ uses \abb{mso} head tests; those tests can also be executed by the constructed \abb{i-ptt} ${\cal M}$, by Lemma~\ref{lem:sites}. Moreover, that proof can also easily be adapted to the case where the second transducer ${\cal M}_2$ is a root-oriented \abb{i-ptt}. From this and Lemmas~\ref{lem:in-ridptt} and~\ref{lem:decompidptt} we obtain the following characterizations of $\family{I-dPTT}$ as a corollary. We do not know whether similar characterizations hold for $\family{I-PTT}$.
\begin{theorem}\label{thm:charidptt} $\family{I-dPTT} = \family{dTT$^{\text{\abb{MSO}}}$}\circ \family{dTT} = \family{dTT$^{\text{\abb{MSO}}}$}\circ \family{dTT$^{\text{\abb{MSO}}}$} = \family{dTT$^{\text{\abb{MSO}}}$}\circ \family{r\,I-dPTT}$. \end{theorem}
\begin{proof} Let us show for completeness sake that $\family{dTT}\circ \family{r\,I-dPTT} \subseteq \family{I-dPTT}$. The proof of Theorem~\ref{thm:composition} can easily be generalized to a root-oriented \abb{i-ptt} ${\cal M}_2$, because the path from the root of $s$ to the current node $v$ of ${\cal M}_2$ is represented by the pebble stack of the constructed transducer ${\cal M}$, and so the pebbles of ${\cal M}_2$ can also be stored in the pebble stack of ${\cal M}$. For each node on that path, the stack contains a pebble with the rule of ${\cal M}_1$ that generates that node, with its child number, and with the pebble that ${\cal M}_2$ drops on that node.
Formally, the pebble colours of ${\cal M}$ are now triples $(\rho,i,c)$ where $c$ is a pebble colour of ${\cal M}_2$, and the states of ${\cal M}$ are the states of ${\cal M}_2$ and all 4-tuples $(p,i,c,q)$ where $c$ is again a pebble colour of ${\cal M}_2$. The initial state of ${\cal M}$ is now the one of ${\cal M}_2$, and if ${\cal M}_2$ has an initial rule $\tup{q_0,\delta,0,\varnothing}\to \tup{q,{\rm drop}_c}$, then ${\cal M}$ has the rule $\tup{q_0,\delta,0,\varnothing}\to \tup{(p_0,0,c,q),{\rm stay}}$. The rules of ${\cal M}$ that simulate ${\cal M}_1$ are defined as in the proof of Theorem~\ref{thm:composition}, replacing $i$ by $i,c$ everywhere for each~$c$. The rules of ${\cal M}$ that simulate the non-initial rules of ${\cal M}_2$ are defined as follows. Let $\tup{q,\delta,i,\{c\}}\to \zeta$ be a non-initial rule of~${\cal M}_2$ and let $\rho: \tup{p,\sigma,j}\to \delta(\tup{p_1,{\rm stay}},\dots,\tup{p_m,{\rm stay}})$ be an output rule of~${\cal M}_1$. Then ${\cal M}$ has the rule $\tup{q,\sigma,j,\{(\rho,i,c)\}}\to \zeta'$ where $\zeta'$ is defined as follows. If $\zeta=\tup{q',{\rm down}_\ell;{\rm drop}_d}$, then $\zeta'= \tup{(p_\ell,\ell,d,q'),{\rm stay}}$. If $\zeta=\tup{q',{\rm lift}_c;{\rm up}}$, then $\zeta'= \tup{q',{\rm lift}_{(\rho,i,c)};{\rm to\text{-}top}}$. If $\zeta=\tup{q',{\rm lift}_c;{\rm drop}_d}$, then $\zeta'= \tup{q',{\rm lift}_{(\rho,i,c)};{\rm drop}_{(\rho,i,d)}}$. In the remaining cases, $\zeta'=\zeta$. \end{proof}
As another corollary we obtain from the three Lemmas~\ref{lem:in-ridptt}, \ref{lem:ridptt-in}, and~\ref{lem:decompidptt} that $\family{I-dPTT} \subseteq \family{dTT}^3$. Moreover, $\family{I-dPTT} \subseteq \family{dMT}_\text{\abb{OI}}^2$ by the second equality of Corollary~\ref{cor:ridpttmt}. Together with Theorem~\ref{thm:tl}, that implies that $\family{dTL$_{\ell\text{r}}$}\subseteq \family{dMT}_\text{\abb{OI}}^2$, which was stated as an open problem in~\cite[Section~8]{EngMan03} (where $\family{dTL$_{\ell\text{r}}$}$ and $\family{dMT}_\text{\abb{OI}}$ are denoted 0-DPMTT and DMTT, respectively).
\begin{corollary}\label{cor:idptt-tt3} $\family{I-dPTT} \subseteq \family{dTT}\!_\downarrow \circ \family{dTT} \circ \family{dTT} \subseteq \family{dMT}_\text{\abb{OI}}\circ\family{dMT}_\text{\abb{OI}}$. \end{corollary}
We are now able to prove the deterministic analogue of Theorem~\ref{thm:decomp} for \abb{ptt}'s with at least one visible pebble.
\begin{theorem}\label{thm:detdecomp} For every $k\ge 1$, $\VIdPTT{k} \subseteq \family{dTT}^{k+2}$. \end{theorem}
\begin{proof} Since it follows from Lemma~\ref{lem:decomp} and Corollary~\ref{cor:idptt-tt3} that $\VIdPTT{k} \subseteq \family{tdTT}^{k-1}\circ \family{tdTT}\circ \family{dTT}_\downarrow \circ \family{dTT} \circ \family{dTT}$, it suffices to show that $\family{tdTT} \circ \family{dTT}_\downarrow \subseteq \family{dTT}$. For the sake of the proof of Lemma~\ref{lem:tdttmso}, we will show more generally that for all deterministic \abb{tt}'s ${\cal M}_1$ and ${\cal M}_2$ such that ${\cal M}_2$ does not use the up-instruction, a deterministic \abb{tt} ${\cal M}$ can be constructed such that $\tau_{\cal M}(t)=\tau_{{\cal M}_2}(\tau_{{\cal M}_1}(t))$ for every input tree~$t$ in the domain of ${\cal M}_1$. This can be proved by a straightforward product construction, which is an easy adaptation of the construction in the proof of Theorem~\ref{thm:composition}. Since transducer ${\cal M}_2$ never moves up, there is no need to backtrack on the computation of ${\cal M}_1$. Therefore, the constructed transducer ${\cal M}$ only considers the topmost pebble. Since that pebble is always at the position of the head, its colour can as well be stored in the finite state of ${\cal M}$. Hence ${\cal M}$~can be turned into a \abb{tt} rather than an \abb{i-ptt}.
Formally, let ${\cal M}_1=(\Sigma,\Delta,P,\{p_0\},R_1)$ and ${\cal M}_2=(\Delta,\Gamma,Q,\{q_0\},R_2)$. The deterministic \abb{tt} ${\cal M}$ has input alphabet $\Sigma$ and output alphabet $\Gamma$. Its states are of the form $(p,i,q)$ or $(\rho,i,q)$, where $p\in P$, $i\in[0,{\mathit mx}_\Delta]$, $q\in Q$, and $\rho$ is an output rule of ${\cal M}_1$, i.e., a rule of the form $\tup{p,\sigma,j}\to \delta(\tup{p_1,{\rm stay}},\dots,\tup{p_m,{\rm stay}})$. Its initial state is $(p_0,0,q_0)$. As in the proof of Theorem~\ref{thm:composition}, state $(p,i,q)$ is used by ${\cal M}$ when simulating the computation of ${\cal M}_1$ that generates the $i$-th child of the current node of ${\cal M}_2$ (keeping the state $q$ of ${\cal M}_2$ in memory). A state $(\rho,i,q)$ is used by ${\cal M}$ when simulating a computation step of ${\cal M}_2$ on the node that ${\cal M}_1$ has generated with rule $\rho$. The rules of ${\cal M}$ are defined as follows.
First the rules that simulate ${\cal M}_1$. Let $\rho: \tup{p,\sigma,j}\to\zeta$ be a rule in~$R_1$. If $\zeta=\tup{p',\alpha}$, where $\alpha$ is a move instruction, then ${\cal M}$ has the rules $\tup{(p,i,q),\sigma,j}\to \tup{(p',i,q),\alpha}$ for every $i\in[0,{\mathit mx}_\Delta]$ and $q\in Q$. If $\rho$ is an output rule, then ${\cal M}$ has the rules $\tup{(p,i,q),\sigma,j}\to \tup{(\rho,i,q),{\rm stay}}$ for every $i$ and $q$ as above.
Second the rules that simulate ${\cal M}_2$. Let $\tup{q,\delta,i}\to \zeta$ be a rule in~$R_2$ and let $\rho: \tup{p,\sigma,j}\to \delta(\tup{p_1,{\rm stay}},\dots,\tup{p_m,{\rm stay}})$ be an output rule in~$R_1$ (with the same $\delta$). Then ${\cal M}$ has the rule $\tup{(\rho,i,q),\sigma,j}\to \zeta'$ where $\zeta'$ is obtained from~$\zeta$ by changing every $\tup{q',{\rm stay}}$ into $\tup{(\rho,i,q'),{\rm stay}}$, and every $\tup{q',{\rm down}_\ell}$ into $\tup{(p_\ell,\ell,q'),{\rm stay}}$. \end{proof}
Since the topmost pebble of a \abb{v-ptt} can be replaced by an invisible pebble, we obtain from Theorem~\ref{thm:detdecomp} that $\VdPTT{k} \subseteq \family{dTT}^{k+1}$, which was proved in~\cite[Theorem~10]{EngMan03}.
Theorem~\ref{thm:detdecomp} allows us to show that, in the deterministic case, $k+1$ visible pebbles are more powerful than $k$ visible pebbles.
\begin{theorem}\label{thm:dethier} For every $k\ge 0$, $\VIdPTT{k} \subsetneq \VIdPTT{k+1}$. \end{theorem}
\begin{proof} It follows from Theorem~\ref{thm:detdecomp} and Corollary~\ref{cor:idptt-tt3} (and the inclusion $\family{dTT}\subseteq \family{dMT}_\text{\abb{OI}}$ in Corollary~\ref{cor:ridpttmt}) that $\VIdPTT{k} \subseteq \family{dMT}_\text{\abb{OI}}^{k+2}$ for every $k\geq 0$. But it is proved in~\cite[Theorem~41]{EngMan03} that, for every $k\geq 1$, $\VdPTT{k}$ is not included in $\family{dMT}_\text{\abb{OI}}^k$. Hence, since the topmost pebble of a \abb{v-ptt} can be replaced by an invisible pebble, $\VIdPTT{k}$ is not included in $\family{dMT}_\text{\abb{OI}}^{k+1}$. \end{proof}
The above proof also shows that Theorem~\ref{thm:detdecomp} is optimal, in the sense that, for every $k\geq 1$, $\VIdPTT{k}$ is not included in $\family{dTT}^{k+1}$.
Another consequence of Theorem~\ref{thm:detdecomp} is that, by the results of~\cite{man}, all total deterministic \abb{vi-pft} transformations for which the size of the output document is linear in the size of the input document, can be programmed in $\family{TL}$. Let $\family{LSI}$ be the class of all total functions $\varphi$
for which there exists a constant $c\in{\mathbb N}$ such that $|\varphi(t)|\leq c\cdot|t|$ for every input tree $t$.
\begin{theorem}\label{thm:lsi} For every $k\geq 0$, \[ \VIdPTT{k} \cap \family{LSI} \subseteq \family{I-dPTT} = \family{dTL}_\mathrm{r} \quad\text{and}\quad \VIdPFT{k} \cap \family{LSI} \subseteq \family{I-dPFT} = \family{dTL}. \] \end{theorem}
\begin{proof} It is shown in~\cite{man} that $\family{dMT}_\text{\abb{OI}}^k \cap \family{LSI} \subseteq \family{dMT}_\text{\abb{OI}}$ for every $k\geq 1$. By Theorem~\ref{thm:detdecomp} and Corollary~\ref{cor:ridpttmt}, $\VIdPTT{k} \subseteq \family{dMT}_\text{\abb{OI}}^{k+2}$. And by Corollary~\ref{cor:mtiptt} and Theorem~\ref{thm:tl}, $\family{dMT}_\text{\abb{OI}} \subseteq \family{I-dPTT} = \family{dTL}_\mathrm{r}$. This proves the first inclusion. To prove the second inclusion, let $\varphi\in\VIdPFT{k} \cap \family{LSI}$. Obviously, $\varphi\circ{\rm enc}$ is also in $\family{LSI}$, and $\varphi\circ{\rm enc}\in \VIdPTT{k}\circ\family{I-dPTT}$ by Lemma~\ref{lem:pftvsptt}(2). Hence $\varphi\circ{\rm enc} \in \family{dMT}_\text{\abb{OI}}^{k+4}\subseteq\family{I-dPTT}$, as above. In other words, $\varphi\in \family{I-dPTT}\circ{\rm dec}$. Consequently, by Lemma~\ref{lem:pftvsptt}(1) and Theorem~\ref{thm:tl}, $\varphi\in\family{I-dPFT} = \family{dTL}$. \end{proof}
In fact, $\VIdPTT{k} \cap \family{LSI}$ is the class of total functions in the class $\family{DMSOT}$ of deterministic \abb{mso} definable tree transductions discussed after Corollary~\ref{cor:mtiptt}, and similarly, $\VIdPFT{k} \cap \family{LSI}$ is the class of total functions in the class of deterministic \abb{mso} definable tree-to-forest transductions (which equals $\family{DMSOT}\circ\family{dec}$, because both ${\rm dec}$ and ${\rm enc}$ are \abb{mso} definable).
For the reader familar with results about attribute grammars (which are a well-known compiler construction tool) and related formalisms, we now briefly discuss the relationship between those results and some of the above. As explained in detail in~\cite[Section~3.2]{EngMan03}, the total deterministic tree-walking tree transducer, i.e., the td\abb{TT}, is essentially a notational variant of the attributed tree transducer (\abb{AT}) of~\cite{Ful,FulVog}, except that the \abb{at} is in addition required to be ``noncircular'', which means that no configuration can generate an output form in which that same configuration occurs. As observed at the end of Section~\ref{sec:document}, the deterministic \abb{i-ptt} has the same expressive power as the deterministic \abb{tl} program that is local and ranked, which corresponds to the macro attributed tree transducer (\abb{MAT}) of~\cite{KuhVog,FulVog} in the same way, i.e., the \abb{MAT} is the ``noncircular'' td\abb{tl}$_{\ell {\sf r}}$ program.
Since $\family{r\,I-dPTT} = \family{dMT}_\text{\abb{OI}}$ by Corollary~\ref{cor:ridpttmt}, Lemma~\ref{lem:in-ridptt} ($\family{dTT$^{\text{\abb{MSO}}}$} \subseteq \family{r\,I-dPTT}$) is closely related to the well-known fact that \abb{at} (with look-ahead) can be simulated by deterministic macro tree transducers.
Lemma~\ref{lem:ridptt-in} ($\family{r\,I-dPTT} \subseteq \family{dTT}_\downarrow \circ \family{dTT}$) is related to the fact that every total deterministic macro tree transducer can be decomposed into a deterministic top-down tree transducer followed by a YIELD mapping, which can be realized by an \abb{at}.
Theorem~\ref{thm:charidptt} ($\family{I-dPTT} = \family{dTT$^{\text{\abb{MSO}}}$}\circ \family{dTT} = \family{dTT$^{\text{\abb{MSO}}}$}\circ \family{r\,I-dPTT}$) is closely related to the fact that every \abb{mat} can be decomposed into two \abb{at}'s, and that the composition of an \abb{at} and a total deterministic macro tree transducer can be simulated by a \abb{mat}, as shown in~\cite[Theorem~4.8]{KuhVog} and its proof (see also~\cite[Corollary~7.30]{FulVog}).
The inclusion $\family{tdTT} \circ \family{dTT}_\downarrow \subseteq \family{dTT}$ in the proof of Theorem~\ref{thm:detdecomp} is closely related to the closure of \abb{AT} under right-composition with deterministic top-down tree transducers, as shown in~\cite[Theorem~4.3]{Ful} (see also~\cite[Lemma~4.11]{KuhVog} and~\cite[Lemma~5.46]{FulVog}).
We finally mention that the class $\family{DMSOT}$ of deterministic \abb{mso} definable tree transductions is properly included in $\family{dTT$^{\text{\abb{MSO}}}$}$ (see~\cite[Theorems~8.6 and~8.7]{thebook}), as shown for attribute grammars (with look-ahead) in~\cite{EngBlo}.
\smallpar{Strong pebbles} In the litterature there are pebble automata with weak and strong pebbles. Weak pebbles (which are the pebbles considered until now) can only be lifted when the reading head is at the position where they were dropped, whereas strong pebbles can also be lifted from a distance, i.e., when the reading head is at any other position. So, strong pebbles are more like dogs that can be whistled back, or like pointers that can be erased from memory. Formally, we define a pebble colour $c$ to be \emph{strong} as follows. For a rule $\tup{q,\sigma,j,b}\to\tup{q',{\rm lift}_c}$ we do not require any more that $c\in b$. If the rule is relevant to configuration $\tup{q,u,\pi}$, then it is applicable whenever the topmost element of the pebble stack is $(v,c)$ for some node $v$ (not necessarily equal to~$u$). That top pebble is then popped from the stack, i.e., $\pi=\pi'(v,c)$ where $\pi'$ is the new stack. Strong pebbles were investigated, e.g., in~\cite{fo+tc,MusSamSeg,expressive,SamSeg,FulMuzFI}.
It turns out that strong invisible pebbles are too strong, in the sense that they allow the recognition of nonregular tree languages, cf. the paragraph after Theorem~\ref{thm:regt}. For example, the nonregular language $\{a^n\#b^n\mid n\in{\mathbb N}\}$ can be accepted by an \abb{i-pta} with strong pebbles as follows. After checking that the input string $w$ is in $a^*\#b^*$, the automaton drops a pebble on $\#$ and walks to the left, dropping a pebble on every $a$. Next it walks to the end of $w$, and then walks to the left, lifting a pebble (from a distance) for every $b$ it passes. It accepts $w$ if it arrives at $\#$ and observes a pebble on $\#$.
Thus, we will only consider the \abb{pta} and \abb{ptt} with strong \emph{visible} pebbles, abbreviated as \abb{v$^+$i-pta} and \abb{v$^+$i-ptt} (and similarly for the classes of transductions they realize). Obviously, $\VIPTT{k} \subseteq \VsIPTT{k}$ for every $k\geq 0$. We do not know whether the inclusion is proper.
Let us first show that the \abb{v$^+$i-pta} and \abb{v$^+$i-ptt} can perform stack tests.
\begin{lemma}\label{lem:stacktestsplus} Let $k\geq 0$. For every \abb{v$^+_k$i-pta} with stack tests ${\cal A}$ an equivalent (ordinary) \abb{v$^+_k$i-pta} ${\cal A}'$ can be constructed in polynomial time. The construction preserves determinism and the absence of invisible pebbles. The same holds for the corresponding \abb{ptt}'s. \end{lemma}
\begin{proof} Let ${\cal A} = (\Sigma, Q, Q_0, F, C, C_\mathrm{v}, C_\mathrm{i}, R,k)$. We construct ${\cal A}'$ in the same way as in the proof of Lemma~\ref{lem:stacktests}, except that it additionally keeps track of the visible pebbles in its \emph{own} stack, in the order in which they were dropped, cf. the construction of a counting \abb{pta} after Lemma~\ref{lem:stacktests}. Thus, its states are of the form $(q,\gamma,\varphi)$ where $q\in Q$, $\gamma\in C\cup\{\varepsilon\}$, and $\varphi\in (C'_\mathrm{v})^*=(C_\mathrm{v}\times (C\cup\{\varepsilon\}))^*$ is a string without repetitions of length $\leq k$. Its initial states are $(q,\varepsilon,\varepsilon)$ with $q\in Q_0$.
The rules of ${\cal A}'$ are defined as follows. Let $\tup{q,\sigma,j,b,\gamma} \to \tup{q',\alpha}$ be a rule of~${\cal A}$, let $\varphi$ be a string over $C'_\mathrm{v}$ as above, and let $b'$ be (the graph of) a mapping from $b$ to $C\cup\{\varepsilon\}$. If $\alpha$ is a move instruction, then ${\cal A}'$ has the rule $\tup{(q,\gamma,\varphi),\sigma,j,b'} \to \tup{(q',\gamma,\varphi),\alpha}$ (and similarly for an output rule of a \abb{ptt}). If $\alpha = {\rm drop}_c$, then ${\cal A}'$ has the rule $\tup{(q,\gamma,\varphi),\sigma,j,b'} \to \tup{(q',c,\varphi'),{\rm drop}_{(c,\gamma)}}$ where $\varphi'=\varphi$ if $c\in C_\mathrm{i}$ and $\varphi'=\varphi(c,\gamma)$
otherwise (provided $|\varphi|< k$ and $(c,\gamma)$ does not occur in $\varphi$). Now let $\alpha={\rm lift}_c$ and $\gamma=c$. If $c \in C_\mathrm{i}$ and $(c,\gamma')\in b'$, then ${\cal A}'$~has the rule $\tup{(q,\gamma,\varphi),\sigma,j,b'} \to \tup{(q',\gamma',\varphi),{\rm lift}_{(c,\gamma')}}$. If $c \in C_\mathrm{v}$, then ${\cal A}'$~has the rule $\tup{(q,\gamma,\varphi(c,\gamma')),\sigma,j,b'} \to \tup{(q',\gamma',\varphi),{\rm lift}_{(c,\gamma')}}$ for every $\gamma'\in C\cup\{\varepsilon\}$ such that $(c,\gamma')$ does not occur in $\varphi$
(with $|\varphi|< k$). \end{proof}
Using this lemma, we now show that every \abb{v$^+$-ptt} can be decomposed into \abb{tt}'s, as already shown in~\cite{FulMuzFI} in a different way.\footnote{In that paper the authors ``think that those proofs cannot be generalized for the strong pebble case because the mapping EncPeb [$\cdots$] is strongly based on weak pebble handling'', where `those proofs' mainly refers to the proof of~\cite[Lemma~9]{EngMan03} in which the preprocessor is called EncPeb, see Lemma~\ref{lem:decomp}. }
\begin{lemma}\label{lem:decompplus} For every $k\ge 1$, $\VsPTT{k} \subseteq \family{TT} \circ \VsPTT{k-1}$. For fixed $k$, the construction takes polynomial time. \end{lemma}
\begin{proof} Let ${\cal M} = (\Sigma, \Delta, Q, Q_0,C, C_\mathrm{v}, C_\mathrm{i}, R,k)$ be a \abb{v$^+_k$-ptt} with $C_\mathrm{i}=\varnothing$. The construction is similar to the one in the proof of Lemma~\ref{lem:decomp}, except that we use the nondeterministic ``multi-level'' preprocessor ${\cal N}$ of the proof of Lemma~\ref{lem:nul-decomp}, for which we assume that $C_\mathrm{v}=[1,\gamma]$.
By Lemma~\ref{lem:stacktestsplus} we may assume that the simulating transducer ${\cal M}'$ can perform stack tests. As in the proof of Lemma~\ref{lem:decomp}, ${\cal M}'$ starts by simulating ${\cal M}$ on the top level of the preprocessed version $t'$ of the input tree $t$. When ${\cal M}$ drops the first pebble $c$ on node $u$, ${\cal M}'$ enters the second level copy $\hat{t}_u$ of $t$ corresponding to $c$, but it also stores $c$ in its finite state. When ${\cal M}$ wants to lift pebble $c$ and can actually do so because the pebble stack of ${\cal M}'$ is empty, ${\cal M}'$ removes $c$ from its finite state and continues simulating ${\cal M}$ on $\hat{t}_u$. Note that since $c$ can be lifted from a distance, ${\cal M}'$ cannot return to the top level without loosing its current position. When ${\cal M}$ again drops a pebble $d$ on some second-level node that corresponds to a node $v$ of $t$, ${\cal M}'$ enters the third level copy $\hat{t}_v$ corresponding to~$d$, and stores $d$ in its finite state. Thus, whenever ${\cal M}$ drops a bottom pebble, ${\cal M}'$~moves one level down in the ``tree of trees'' $t'$.
It should be noted that we could as well have taken $\gamma=1$ for ${\cal N}$ and let ${\cal M}'$ enter the unique copy of $t$ when it drops a pebble $c$, because ${\cal M}'$ keeps~$c$ in its finite state. However, the present construction simplifies the proof of Theorem~\ref{thm:detdecompplus}.
Although the above description should be clear, let us give the formal details. As in the proof of Lemma~\ref{lem:nul-decomp}, the output alphabet $\Gamma$ of ${\cal N}$ is the union of $\{\bot\}$, $\{\sigma'\mid \sigma\in\Sigma\}$, and $\{\sigma'_{i,j}\mid \sigma\in\Sigma, i\in[0,\operatorname{rank}_\Sigma(\sigma)], j\in[0,{\mathit mx}_\Sigma]\}$ where, for every $\sigma\in\Sigma$ of rank~$m$, $\sigma'$ has rank $m+\gamma$ and $\sigma'_{i,j}$ has rank $m+\gamma+1$.
As in the proof of Lemma~\ref{lem:decomp}, the \abb{v$^+_{k-1}$-ptt} ${\cal M}'$ has input alphabet~$\Gamma$, set of states $Q\cup (Q\times C_{\mathrm{v}})$, and the same initial states and pebble colours as ${\cal M}$. The rules of~${\cal M}'$ are defined as follows. Let $\langle q,\sigma,j,b\rangle\to \zeta$ be a rule of ${\cal M}$ with $\operatorname{rank}_\Sigma(\sigma)=m$.
First, we consider the behaviour of ${\cal M}'$ in state $q$, where we assume that $b=\varnothing$. Then ${\cal M}'$ has the rules $\langle q,\sigma',j,\varnothing\rangle\to \zeta_1$, $\langle q,\sigma'_{0,j},j',\varnothing\rangle\to \zeta_2$, and $\langle q,\sigma'_{i,j},j',\varnothing\rangle\to \zeta_{3,i}$ for every $i\in[1,m]$ and $j'\in[1,{\mathit mx}_\Gamma]$, where $\zeta_1$ is obtained from $\zeta$ by changing $\tup{q',{\rm drop}_c}$ into $\tup{(q',c),{\rm down}_{m+c}}$ for every $q'\in Q$ and $c\in C_{\mathrm{v}}$, $\zeta_2$ is obtained from $\zeta_1$ by changing ${\rm up}$ into ${\rm down}_{m+\gamma+1}$, and $\zeta_{3,i}$ is obtained from $\zeta_2$ by changing ${\rm down}_i$ into ${\rm up}$. Thus, whenever the pebble stack of ${\cal M}$ is empty, ${\cal M}'$ simulates ${\cal M}$ on a copy of the input tree $t$ in $t'$, until ${\cal M}$ drops a pebble~$c\in C_{\mathrm{v}}$. Then ${\cal M}'$ steps to the next level, and stores $c$ in its finite state.
Second, we consider the behaviour of ${\cal M}'$ in state $(q,c)$, where $c\in C_{\mathrm{v}}$. Rules of ${\cal M}'$ that have $\sigma'_{0,j}$ in their left-hand side are defined under the proviso that \mbox{$c\in b$}, and the other rules under the proviso that $c\notin b$. If $\zeta=\tup{q',{\rm lift}_c}$, then ${\cal M}'$ has the rules $\langle (q,c),\sigma',j,b,\varepsilon\rangle\to \tup{q',{\rm stay}}$, $\langle (q,c),\sigma'_{0,j},m+c,b\setminus\{c\},\varepsilon\rangle\to \tup{q',{\rm stay}}$, and $\langle (q,c),\sigma'_{i,j},j',b,\varepsilon\rangle\to \tup{q',{\rm stay}}$ for every $i\in[1,m]$ and $j'\in[1,{\mathit mx}_\Gamma]$, where $\varepsilon$ is the stack test that checks emptiness of the stack of ${\cal M}'$. Thus, when ${\cal M}$ lifts pebble~$c$ (at the position of $c$ or from a distance), ${\cal M}'$~removes~$c$ from memory and knows that the pebble stack of ${\cal M}$ is empty.
Otherwise, ${\cal M}'$ has the rules $\langle (q,c),\sigma',j,b\rangle\to \zeta_{c,1}$, $\langle (q,c),\sigma'_{0,j},m+c,b\setminus\{c\}\rangle\to \zeta_{c,2}$, and $\langle (q,c),\sigma'_{i,j},j',b\rangle\to \zeta_{c,3,i}$ for every $i\in[1,m]$ and $j'\in[1,{\mathit mx}_\Gamma]$, where $\zeta_{c,1}$ is obtained from $\zeta$ by changing every occurrence of a state $q'$ into $(q',c)$, $\zeta_{c,2}$ is obtained from $\zeta_{c,1}$ by changing ${\rm up}$ into ${\rm down}_{m+\gamma+1}$, and $\zeta_{c,3,i}$ is obtained from $\zeta_{c,2}$ by changing ${\rm down}_i$ into ${\rm up}$. Thus, ${\cal M}'$ simulates ${\cal M}$ on a copy of the input tree in $t'$, assuming that $c$ is present on the node with label $\sigma'_{0,j}$ and absent on the other nodes, until $c$ is lifted by ${\cal M}$. \end{proof}
The next result is an immediate consequence of Lemma~\ref{lem:decompplus}. It was proved in~\cite[Theorem~6.5(5)]{FulMuzFI}, generalizing the same result for weak pebbles in~\cite[Theorem~10]{EngMan03} (cf. Theorem~\ref{thm:detdecomp}). It implies that Propositions~\ref{prop:invtypeinf}(2) and~\ref{prop:typecheck}(2) also hold for strong pebbles. Thus, for \abb{ptt}'s without invisible pebbles, the inverse type inference problem and the typechecking problem are solvable for strong pebbles in the same time as for weak pebbles (cf.~\cite[Theorem~6.7 and~6.9]{FulMuzFI}). Note that it also implies that the domains of \abb{v$^+$-ptt}'s are regular (cf.~Corollary~\ref{cor:domptt}), which was proved in~\cite[Corollary~6.8]{FulMuzFI} and~\cite[Theorem~4.7]{Muz}.
\begin{theorem}\label{thm:decompplus} For every $k\geq 0$, $\VsPTT{k}\subseteq \family{TT}^{k+1}$. For fixed $k$, the construction takes polynomial time. \end{theorem}
To prove a similar result for deterministic \abb{ptt}'s with strong pebbles, we need the next small lemma.
\begin{lemma}\label{lem:tdttmso} For every $k\geq 1$, $(\family{tdTT$^{\text{\abb{MSO}}}$})^k \subseteq \family{dTT}\!_\downarrow\circ \family{dTT}^k$. \end{lemma}
\begin{proof} We will show by induction on $k$ that for every $\tau\in (\family{tdTT$^{\text{\abb{MSO}}}$})^k$ there exist $\tau_0\in \family{dTT}\!_\downarrow$ and $\tau_1,\dots,\tau_k\in\family{dTT}$ such that $\tau=\tau_0\circ\tau_1\circ\cdots\circ\tau_k$. Note that since $\tau$ is a total function, every output tree of $\tau_0\circ\tau_1\circ\cdots\circ\tau_{i-1}$ is in the domain of~$\tau_i$, for every $i\in[1,k]$. For $k=1$ the statement is immediate from the inclusion $\family{dTT$^{\text{\abb{MSO}}}$}\subseteq \family{dTT}_\downarrow \circ \family{dTT}$, which follows from Lemmas~\ref{lem:in-ridptt} and~\ref{lem:ridptt-in}. Now consider $\tau\in (\family{tdTT$^{\text{\abb{MSO}}}$})^{k+1}$. By induction and the case $k=1$, $\tau=\tau_0\circ\tau_1\circ\cdots\circ\tau_k\circ \tau'_0\circ \tau'_1$ with $\tau_0,\tau'_0\in\family{dTT}\!_\downarrow$ and $\tau_1,\dots,\tau_k,\tau'_1\in\family{dTT}$. Since every output tree of $\tau_0\circ\tau_1\circ\cdots\circ\tau_{k-1}$ is in the domain of $\tau_k$, we can replace $\tau_k\circ \tau'_0$ by any transduction $\tau'$ such that $\tau'(t)=\tau'_0(\tau_k(t))$ for every $t$ in the domain of $\tau_k$. Since $\tau_k\in\family{dTT}$ and $\tau'_0\in\family{dTT}\!_\downarrow$, we can take $\tau'\in\family{dTT}$ by the proof of Theorem~\ref{thm:detdecomp}. \end{proof}
Theorem~\ref{thm:decompplus} was also shown in~\cite[Theorem~10]{EngMan03} for weak pebbles in the deterministic case. Here we need one more deterministic \abb{tt}.
\begin{theorem}\label{thm:detdecompplus} For every $k\geq 1$, $\VsdPTT{k}\subseteq \family{dTT}\!_\downarrow\circ \family{dTT}^{k+1}$. \end{theorem}
\begin{proof} By Lemma~\ref{lem:tdttmso} it suffices to show that $\VsdPTT{k} \subseteq \family{tdTT$^{\text{\abb{MSO}}}$} \,\circ \,\VsdPTT{k-1}$ for every \mbox{$k\ge 1$}. The proof of this inclusion is obtained from the proof of Lemma~\ref{lem:decompplus} by changing the preprocessor ${\cal N}$ in a similar way as in the proof of Lemma~\ref{lem:decompidptt}.
For the given deterministic \abb{v$^+_k$-ptt} ${\cal M}$ we assume that $C_\mathrm{i}=\varnothing$ and $C=C_\mathrm{v}=[1,\gamma]$. As in the proof of Lemma~\ref{lem:decompidptt}, we may assume that there is a mapping $\chi:C\to Q$ that specifies the state of ${\cal M}$ after dropping a pebble (because we may also assume that ${\cal M}$ keeps track in its finite state of the pebbles in its stack, in the order in which they were dropped, cf. the proof of Lemma~\ref{lem:stacktestsplus}).
The new preprocessor ${\cal N}'$ is constructed in the same way as in the proof of Lemma~\ref{lem:decompidptt}, with a different definition of the trips $T_{c,d}$. For $c\in C$, the site $T_c$ is defined as in that proof, i.e., it consists of all pairs $(t,u)$ such that the configuration $\tup{\chi(c),u,(u,c)}$ is reachable by the automaton ${\cal A}$ associated with~${\cal M}$, which now is a nondeterministic \abb{v$^+_k$-pta}. The automaton ${\cal B}$ recognizing $\operatorname{mark}(T_c)$ is a \abb{v$^+_k$-pta} with stack tests (see Lemma~\ref{lem:stacktestsplus}). When it arrives at the marked node $u$ in state $\chi(c)$ and observes $c$, it may check that $c$ is the top pebble, lift it, check that the stack is now empty, and accept. For $c,d\in C$, the trip $T_{c,d}$ now consists of all triples $(t,u,v)$ such that ${\cal A}$ has a computation on $t$ starting in configuration $\tup{\chi(c),u,(u,c)}$ and ending in configuration $\tup{\chi(d),v,(v,d)}$, with at least one computation step. It should be clear that there is a \abb{v$^+_k$-pta} ${\cal B}'$ with stack tests that recognizes $\operatorname{mark}(T_{c,d})$: it starts by dropping $c$ on marked node~$u$ in state $\chi(c)$, and then behaves similarly to ${\cal B}$ (with respect to $v$ and $d$).
For every input tree $t$ in the domain of ${\cal M}$, the preprocessor ${\cal N}'$ produces the appropriate output. In fact, if ${\cal N}'$ would not produce output, then there would be an infinite sequence $(u_1,c_1),(u_2,c_2),\dots$ such that $(t,u_1)\in T_{c_1}$ and $(t,u_i,u_{i+1})\in T_{c_i,c_{i+1}}$ for every $i\geq 1$. But that would imply the existence of an infinite computation of ${\cal M}$ on $t$ that starts in the initial configuration, contradicting the determinism of ${\cal M}$. \end{proof}
Next, we decompose arbitrary \abb{v$^+$i-ptt}'s. To do that we need two \abb{tt}'s at each decomposition step rather than one.
\begin{lemma}\label{lem:decompplusI} For every $k\geq 1$, $\VsIPTT{k}\subseteq \family{TT} \circ \family{TT} \circ \VsIPTT{k-1}$. For fixed $k$, the construction takes polynomial time. \end{lemma}
\begin{proof} The proof of Lemma~\ref{lem:decompplus} is also valid for \abb{v$^+$i-ptt}, provided every reachable pebble stack of the given transducer has a visible bottom pebble (for the definition of reachable pebble stack see the proof of Lemma~\ref{lem:decompidptt}). Thus, it suffices to construct for every \abb{v$^+_k$i-ptt} ${\cal M}$ a \abb{tt} ${\cal N}$ and a \abb{v$^+_k$i-ptt} ${\cal M}'$ with that property, such that $\tau_{\cal N}\circ\tau_{{\cal M}'}=\tau_{\cal M}$.
Let ${\cal M} = (\Sigma, \Delta, Q, Q_0,C, C_\mathrm{v}, C_\mathrm{i}, R,k)$. The construction is similar to the one in the proof of Lemma~\ref{lem:nul-decomp}. In particular, we assume that $C_\mathrm{i}=[1,\gamma]$ and we use the same nondeterministic ``multi-level'' preprocessor ${\cal N}$ of that proof. The simulating transducer ${\cal M}'$ works in the same way as the one in the proof of Lemma~\ref{lem:nul-decomp} as long as the pebble stack of ${\cal M}$ consists of invisible pebbles only. Thus, during that time the pebble stack of ${\cal M}'$ is empty. As soon as ${\cal M}$ drops a visible pebble $c$, ${\cal M}'$ stays in the same copy of the input tree and also drops~$c$. After that, ${\cal M}'$ just simulates ${\cal M}$ on that copy until ${\cal M}$ lifts $c$. Then ${\cal M}'$ also lifts~$c$ and returns to the first mode until ${\cal M}$ again drops a visible pebble. Note that when ${\cal M}$ drops $c$, all invisible pebbles on the input tree become unobservable until $c$ is lifted.
Formally, the set of states of ${\cal M}'$ is the union of $Q$ (used in the first mode) and $Q\times C_\mathrm{v}$ (used in the second mode). The rules for the first mode are the same as in the proof of Lemma~\ref{lem:nul-decomp}, with the empty set of pebble colours added to each left-hand side. Now let $\tup{q,\sigma,j,b}\to \zeta$ be a rule of ${\cal M}$ and $\operatorname{rank}_\Sigma(\sigma)=m$. In what follows, $i$ ranges over $[1,m]$ and $j'$ over $[1,{\mathit mx}_\Gamma]$, as usual. With the following rules ${\cal M}'$ switches from the first to the second mode, where we assume that $\zeta = \tup{q',{\rm drop}_c}$ with $c\in C_\mathrm{v}$: if $b=\{d\}$ with $d\in C_\mathrm{i}$, then it uses the rule $\tup{q,\sigma_{0,j},m+d,\varnothing}\to \tup{(q',c),{\rm drop}_c}$, and if $b=\varnothing$, then it uses the rules $\tup{q,\sigma',j,\varnothing}\to \tup{(q',c),{\rm drop}_c}$ and $\tup{q,\sigma_{i,j},j',\varnothing}\to \tup{(q',c),{\rm drop}_c}$. The rules for the second mode are as follows, for every $c\in C_\mathrm{v}$. We first assume that $\zeta$ does not contain the instruction ${\rm lift}_c$. Then ${\cal M}'$ has the rules $\tup{(q,c),\sigma',j,b}\to \zeta_1$, $\tup{(q,c),\sigma_{0,j},j',b}\to \zeta_2$, and $\tup{(q,c),\sigma_{i,j},j',b}\to \zeta_{3,i}$, where $\zeta_1$ is obtained from $\zeta$ by changing every state $q'$ into $(q',c)$, $\zeta_2$ is obtained from $\zeta_1$ by changing ${\rm up}$ into ${\rm down}_{m+\gamma+1}$, and $\zeta_{3,i}$ is obtained from $\zeta_2$ by changing ${\rm down}_i$ into ${\rm up}$. Finally, if $\zeta=\tup{q',{\rm lift}_c}$, then ${\cal M}'$ switches from the second to the first mode with the following rules: $\tup{(q,c),\sigma',j,b}\to \zeta$, $\tup{(q,c),\sigma_{0,j},j',b}\to \zeta$, and $\tup{(q,c),\sigma_{i,j},j',b}\to \zeta$. \end{proof}
The next result is immediate from Lemmas~\ref{lem:decompplusI} and~\ref{lem:nul-decomp}. It implies, by Propositions~\ref{prop:invtypeinf}(1) and~\ref{prop:typecheck}(1), that the inverse type inference problem and the typechecking problem are solvable for \abb{ptt}'s with $k$ strong visible pebbles, in $(2k+2)$-fold and $(2k+3)$-fold exponential time, respectively. It also implies that the domains of \abb{v$^+$i-ptt}'s are regular, cf. Corollary~\ref{cor:domptt}.
\begin{theorem}\label{thm:decompplusI} For every $k\geq 0$, $\VsIPTT{k}\subseteq \family{TT}^{2k+2}$. For fixed $k$, the construction takes polynomial time. \end{theorem}
Applying the techniques in the proofs of Lemma~\ref{lem:decompidptt} and Theorem~\ref{thm:detdecompplus} to the proof of Lemma~\ref{lem:decompplusI}, and using Lemmas~\ref{lem:decompidptt} and~\ref{lem:tdttmso}, we obtain that every deterministic \abb{v$^+$i-ptt} can be decomposed into deterministic \abb{tt}'s, cf. Theorem~\ref{thm:detdecomp}. The formal proof is straightforward.
\begin{theorem}\label{thm:detdecompplusI} For every $k\geq 0$, $\VsIdPTT{k}\subseteq \family{dTT}\!_\downarrow\circ \family{dTT}^{2k+2}$. \end{theorem}
We do not know whether these results are optimal, i.e., whether the exponent $2k+2$ can be lowered.
\section{Conclusion}\label{sec:conc}
We have shown in Theorem~\ref{thm:decomp} that $\VIPTT{k} \subseteq \family{TT}^{k+2}$, but we do not know whether this is optimal, i.e., whether or not $\VIPTT{k} \subseteq \family{TT}^{k+1}$. Since the results on typechecking in Section~\ref{sec:typechecking} are based on this decomposition, we also do not know whether the time bound for typechecking \abb{v$_k$i-ptt}'s, as stated in Theorem~\ref{thm:typecheck}, is optimal. Using the results of~\cite{SamSeg}, it can be shown that the time bound for inverse type inference is optimal, cf. the discussion after~\cite[Corollary~1]{Eng09}.
We have shown in Theorem~\ref{thm:matchall} that all \abb{mso} definable $n$-ary patterns can be matched by deterministic \abb{v$_{n-2}$i-ptt}'s, but we do not know whether this is optimal, i.e., whether or not it can be done with less than $n-2$ pebbles. In particular, we do not know whether or not all \abb{mso} definable ternary patterns can be matched by \abb{i-ptt}'s (or, by \abb{tl} programs), cf. Theorem~\ref{thm:lsi}. In Section~\ref{sec:pattern} we have suggested ways of reducing the number of visible pebbles in special cases. Given an \abb{mso} formula $\varphi$, can one compute the minimal number of visible pebbles that is needed to match the pattern $\varphi$?
The language \abb{tl} can be extended with visible pebbles, in an obvious way. The resulting ``pebble \abb{tl} programs'' are closely related to the \emph{pebble macro tree transducers} that were introduced in~\cite{EngMan03}. What is the relationship between the $k$-pebble macro tree transducer and the \abb{v$_k$i-ptt}? Is there an analogon of Theorem~\ref{thm:tl}? It is not clear whether the proof of Theorem~\ref{thm:tl} can be generalized to the addition of visible pebbles.
We have shown in Theorem~\ref{thm:dethier} that $\VIdPTT{k} \subsetneq \VIdPTT{k+1}$, i.e., that $k+1$ visible pebbles are more powerful than $k$, in the deterministic case. We do not know whether this holds for the nondeterministic transducers, i.e., whether or not the inclusion $\VIPTT{k} \subseteq \VIPTT{k+1}$ is proper. We also do not know whether every functional \abb{v$_k$i-ptt} can be simulated by a deterministic one, where a \abb{ptt}~${\cal M}$ is functional if $\tau_{\cal M}$ is a function. If so, then the inclusion would of course be proper.
Is it decidable for a given deterministic \abb{v$_{k+1}$i-ptt} ${\cal M}$ whether or not $\tau_{\cal M}$ is in $\VIdPTT{k}$? If so, then one could compute the minimal number of visible pebbles needed to realize the transformation $\tau_{\cal M}$ by a \abb{ptt}. Obviously, that would answer the above question for the pattern $\varphi$ in the affirmative.
It is proved in~\cite{expressive} that the \abb{v$^+_k$-pta} has the same expressive power as the \abb{v$_k$-pta}, i.e., that strong pebbles are not more powerful than weak pebbles. We do not know whether or not the \abb{v$^+_k$-ptt} is more powerful than the \abb{v$_k$-ptt}, and neither whether or not the \abb{v$^+_k$i-ptt} is more powerful than the \abb{v$_k$i-ptt}.
\end{document} | arXiv |
Consequences of ignoring clustering in linear regression
Georgia Ntani ORCID: orcid.org/0000-0001-7481-68601,2,
Hazel Inskip1,3,
Clive Osmond1 &
David Coggon1,2
BMC Medical Research Methodology volume 21, Article number: 139 (2021) Cite this article
Clustering of observations is a common phenomenon in epidemiological and clinical research. Previous studies have highlighted the importance of using multilevel analysis to account for such clustering, but in practice, methods ignoring clustering are often employed. We used simulated data to explore the circumstances in which failure to account for clustering in linear regression could lead to importantly erroneous conclusions.
We simulated data following the random-intercept model specification under different scenarios of clustering of a continuous outcome and a single continuous or binary explanatory variable. We fitted random-intercept (RI) and ordinary least squares (OLS) models and compared effect estimates with the "true" value that had been used in simulation. We also assessed the relative precision of effect estimates, and explored the extent to which coverage by 95% confidence intervals and Type I error rates were appropriate.
We found that effect estimates from both types of regression model were on average unbiased. However, deviations from the "true" value were greater when the outcome variable was more clustered. For a continuous explanatory variable, they tended also to be greater for the OLS than the RI model, and when the explanatory variable was less clustered. The precision of effect estimates from the OLS model was overestimated when the explanatory variable varied more between than within clusters, and was somewhat underestimated when the explanatory variable was less clustered. The cluster-unadjusted model gave poor coverage rates by 95% confidence intervals and high Type I error rates when the explanatory variable was continuous. With a binary explanatory variable, coverage rates by 95% confidence intervals and Type I error rates deviated from nominal values when the outcome variable was more clustered, but the direction of the deviation varied according to the overall prevalence of the explanatory variable, and the extent to which it was clustered.
In this study we identified circumstances in which application of an OLS regression model to clustered data is more likely to mislead statistical inference. The potential for error is greatest when the explanatory variable is continuous, and the outcome variable more clustered (intraclass correlation coefficient is ≥ 0.01).
Clinical and epidemiological research often uses some form of regression analysis to explore the relationship of an outcome variable to one or more explanatory variables. In many cases, the study design is such that participants can be grouped into discrete, non-overlapping subsets (clusters), such that the outcome and/or explanatory variables vary less within clusters than in the dataset as a whole. This might occur, for example, in cluster-randomised controlled trials (with the units of randomisation defining clusters), or in a multi-centre observational study (the participants from each centre constituting a cluster). The extent to which a variable is "clustered" can be quantified by the intra-class correlation coefficient (ICC), which is defined as the ratio of its variance between clusters to its total variance (both between and within clusters) [1].
Clustering has implications for statistical inference from regression analysis if the outcome variable is clustered after the effects of all measured explanatory variables are taken into account. If allowance is not made for such clustering as part of the analysis, parameter estimates and/or their precision may be biased. This possibility can be demonstrated by a hypothetical study of hearing impairment and noise exposure, in which observations are made in four different cities (clusters), as illustrated in Fig. 1. In this example, the effect of cumulative noise exposure on hearing impairment is the same within each city (i.e. the regression coefficient for hearing impairment on noise exposure is the same in each cluster), but the distribution of the exposure differs across cities (Fig. 1A). After allowance for noise exposure, hearing impairment differs by city, such that it varies more between the clusters than within them. An analysis that ignored this clustering would give a misleading estimate for the regression coefficient of hearing loss on noise exposure (Fig. 1B with cluster-unadjusted and cluster-adjusted effect estimates superimposed on Fig. 1A). Moreover, even if the distribution of noise exposures in each city were similar, so that the regression coefficient was unbiased, its precision (the inverse of its variance) would be underestimated, since variance would be inflated by failure to allow for the differences between clusters (at the intercept) (Fig. 1C).
Two hypothetical relationships of hearing impairment to cumulative noise exposure in four cities. Units for noise exposure and hearing impairment have been specified arbitrarily for ease of presentation. Data for each city are distinguished by the shading of data points. A and B Depict the same hypothetical dataset. In A, only cluster-specific regression lines are indicated, while in B summary regression lines have been added for the full dataset a) when clustering is ignored (dotted red line), and b) after adjustment for clustering (solid blue line). C Shows a second dataset in which the relationship of hearing impairment to cumulative noise exposure in each city is as in A and B, but distribution of noise exposures is the same in each city. Again, the dotted red line represents the summary regression line when clustering is ignored, and the solid blue line, that after adjustment for clustering
Where, as in the example above, the number of clusters is small relative to the total number of participants in the study sample, a categorical variable that distinguishes clusters can be treated as an additional explanatory variable in the regression model [2]. However, when the number of clusters is larger (again relative to the total number of participants), use of the cluster variable as an additional explanatory variable in the regression model can seriously reduce the precision with which effects are estimated (because more degrees of freedom are used). In such circumstances, an alternative approach is to assume that cluster effects are randomly distributed with a mean and variance that can be estimated from the data in the study sample. Random intercept models assume that the effects of explanatory variables are the same across all clusters, but that the intercepts of regression lines differ with a mean and variance which can be estimated from the study data, along with the effect estimates of primary interest. Random slope models assume that the effects of explanatory variables also differ between clusters, with a mean and variance that can be estimated.
In recognition of the potential implications of clustering for statistical inference, there has been a growth over recent years in the use of statistical techniques that allow for clustering [3,4,5]. Nevertheless, many studies still ignore clustering of observations [6,7,8,9,10]. Recent systematic reviews have reported that clustering was taken into account in only 21.5% of multicentre trials [11] and 47% of cluster randomised trials [12]. This may in part reflect computational challenges and statistical complexities [13], but, perhaps because of a lack of clarity about the effects of ignoring clustering, authors have omitted to discuss the limitations of their chosen analytical techniques.
Several studies have investigated implications of ignoring clustering in statistical inference, most being based on analysis of real data [14,15,16,17,18,19,20,21]. To date, no study has systematically investigated the extent to which bias can occur in effect estimates when clustering is ignored, the determinants of that bias, or the exact consequences for the precision of estimates according to different distributions of the explanatory variable and, in particular, the extent to which the explanatory variable varies within as compared with between clusters. Such variation can be more nuanced in observational studies (in which researchers have less control over the distribution of explanatory variables), than in clinical trials where the main explanatory variable either varies only between clusters (as in cluster randomised trials), or exhibits minimal variation between as compared with within clusters (as when individual randomisation produces balanced prevalence of the explanatory variable across clusters).
The first aim of the research described in this paper was to assess in detail the implications for effect estimates (regression coefficients), and their precision (characterised by standard errors (SEs)), when a linear regression analysis exploring the relation of a continuous outcome variable to an explanatory variable fails to account for clustering. The second aim was to describe coverage by 95% confidence intervals and rates of Type I error in the same setting. These research questions were explored through simulation studies, which were designed to cover a range of scenarios that might occur in observational research, including variable degrees of clustering in the explanatory variable.
In the simplest case, in which there is a single explanatory variable, the ordinary least squares (OLS) linear regression is specified by a model of the form:
$${y}_{i}={\beta }_{0}+{\beta }_{1}{x}_{i}+{e}_{i}$$
For a continuous outcome and a single explanatory variable, the random intercept (RI) multi-level model can be viewed as an extension of the OLS model, and is specified as:
$$\begin{array}{l}{y}_{ij}={\beta }_{0j}+{\beta }_{1}{x}_{ij}+{e}_{ij}\\ ={\beta }_{0}+{\beta }_{1}{x}_{ij}+{e}_{ij}+{u}_{j}\end{array}$$
where the index \(i\) refers to the individual and the index \(j\) to the cluster, and \({\beta }_{0j}={\beta }_{0}+{u}_{j}\), the estimate of the intercept for cluster \(j\). The term \({u}_{j}\) represents the error for cluster \(j\) around the fixed intercept value of \({\beta }_{0}\), and is assumed to be normally distributed with \({u}_{j}|{x}_{ij}\sim N\left(0,{SD}_{u}^{2}\right)\). The term \({e}_{ij}\) represents the additional error within the cluster, also referred to as the individual level error term, with \({e}_{ij}|{x}_{ij},{u}_{j}\sim N\left(0,{SD}_{e}^{2}\right)\).
As described in the introduction, ICC is a measure which characterises the extent to which the outcome variable \({y}_{ij}\) is similar within clusters, given the distribution of the explanatory variable \({x}_{ij}\) [4]. For a continuous outcome variable, and with the nomenclature used above, the ICC is defined as \(\mathrm{I}\mathrm{C}\mathrm{C}=\frac{{SD}_{u}^{2}}{{SD}_{u}^{2}+{SD}_{e}^{2}}\) [1].
To explore the study questions, simulated datasets were generated according to the assumptions of the RI model (as specified in Eq. 2). For each simulation, both the number of clusters and the number of observations per cluster were set to 100. For simplicity, the size of the effect of \({x}_{ij}\) on \({y}_{ij}\) was arbitrarily set to 1 (\({\beta }_{1}=1)\), and the average value of \({y}_{ij}\) when \({x}_{ij}\)= 0 was arbitrarily set to 0 (\({\beta }_{0}=0)\).
Separate simulations were generated for a continuous and a binary explanatory variable \({x}_{ij}\). To set values for a continuous explanatory variable, \({x}_{ij}\), in a cluster \(j\), an individual level variable generated as \({x}_{0ij}\sim N\left(0,1\right)\) was added to a cluster-specific variable generated as \({shift}_{j}\sim N\left(0, {{SD}_{shift}}^{2}\right)\), so that \({x}_{ij}={x}_{0ij}+{shift}_{j}\). A total of 1,000 values for \({SD}_{shift}\), derived as \(\sim U\left[\mathrm{0,20}\right]\), were each used to generate 100 simulated samples, giving a total of 100,000 samples.
For a binary explanatory variable \({x}_{ij}\), we set the prevalence in each of the 100 clusters within a sample to be the sum of a constant "target prevalence" (the same in all clusters) and a cluster-specific variable \({shift}_{j}\sim N\left(0, {{SD}_{shift}}^{2}\right)\). In this case, 500 values of \({SD}_{shift}\), derived as \(\sim U\left[\mathrm{0,0.05}\right]\), were each used to generate cluster prevalence rates for 100 simulated samples, giving a total of 50,000 samples for each of four values for target prevalence (0.05, 0.1, 0.2 and 0.4). Where a negative value was generated for a cluster prevalence, it was set to zero. Values for \({x}_{ij}\) within a cluster were then set to achieve the designated prevalence for that cluster (with rounding as necessary). For example, if the prevalence assigned to a cluster was 0.223, then the first 22 values for \({x}_{ij}\) in the cluster were set to 1, and the remaining 78 to 0.
For both continuous and binary \({x}_{ij}\), corresponding values for the outcome variable \({y}_{ij}\) were generated according to Eq. 2. For this purpose, the individual-level error terms were drawn from a random standard normal distribution \(\left(N\left(\mathrm{0,1}\right)\right)\), and the cluster-level error terms were drawn from a random normal distribution with mean zero and variance \({SD}_{{u}_{j}}^{2}\). We aimed to explore outcomes according to the degree of clustering within study samples, and to this end, we specified six target ranges for sample ICC (0.0005–0.00149, 0.0025–0.00349, 0.005–0.0149, 0.025–0.0349, 0.05–0.149 and 0.25–0.349). Simulated data were therefore generated for six different values for \({SD}_{{u}_{j}}\) (0.0316, 0.05485, 0.1005, 0.1759, 0.3333 and 0.6547) chosen to give expected values for the sample \(ICC\) at the mid-points of the target ranges (0.001, 0.003, 0.01, 0.03, 0.1 and 0.3 respectively). Throughout the remainder of this report, the six target ranges are labelled by these midpoint values. Where, by chance, the ICC for a sample fell outside its target range, the sample was discarded, and replaced by a new sample generated using the same value for \(s{d}_{shift}\). This process continued until the ICC fell inside the target range.
Further details of the algorithms used to generate simulated samples are presented in supplementary files (Appendices A and B for continuous and binary explanatory variable respectively).
For each simulated sample, two linear regression models were fitted; an OLS model which ignored the clustering (Eq. 1), and a RI multi-level model which allowed for clustering effects (Eq. 2). For each of the models, the regression coefficient and its standard error (SE) were estimated. To assess bias in effect estimates from the two models, we calculated differences between regression coefficients estimated by the two methods (\({\beta }_{1}^{RI}\) and \({\beta }_{1}^{OLS}\)) and the "true" value of 1 (i.e. the value used in the algorithm to generate simulated samples), as has been done previously [22]. To explore how deviations from the "true" value were affected by the clustering of the explanatory variable, they were plotted against the dispersion (expressed as standard deviation (SD)) of the cluster mean values of continuous \({x}_{ij}\) (\({\stackrel{-}{x}}_{j}\)) across the clusters of each sample, and against the dispersion (again expressed as SD) of cluster prevalence rates of binary \({x}_{ij}\) across the clusters of each sample. For both continuous and binary \({x}_{ij}\), lower dispersion indicated less variation of the explanatory variable across clusters and therefore lower clustering of the explanatory variable (since within-cluster variance of \({x}_{ij}\) remained constant). In addition, descriptive statistics were produced for the distributions of deviations from the "true" value across samples, according to ICC (ICC referring to the pattern of variation in the outcome variable after allowance for the explanatory variable), and for a binary explanatory variable, also according to the target prevalence of \({x}_{ij}\).
To compare the precision of effect estimates derived from the two models, the ratios of their SEs (\({SE}^{RI}/{SE}^{OLS}\)) were calculated. Again we explored how findings varied according to ICC, clustering of the explanatory variable, and, where the explanatory variable was binary, its target prevalence.
The coverage of the 95% confidence intervals for the regression coefficient \({\beta }_{1}\) from the two methods was assessed by calculating the percentage of the estimated confidence intervals that included the" true" value that had been used in the simulations. A method was considered to have appropriate coverage if 95% of the 95% confidence intervals included the "true" value of the effect \({\beta }_{1}\) (i.e. 1). Deviations from nominal coverage could reflect bias in estimates of effect, unsatisfactory standard errors [23], or both.
To assess impacts on Type I error, simulations were repeated using the same numbers of simulated samples (i.e. 100,000 simulations for each ICC target range for continuous \({x}_{ij}\), and 50,000 simulations for each target prevalence and ICC target range for binary \({x}_{ij}\)), this time assuming no association between \({x}_{ij}\) and \({y}_{ij}\) (i.e. we set \({\beta }_{1}=0\)).The percentage of datasets for which the null hypothesis was rejected at a 5% significance level in OLS and RI modelling were compared according to ICC.
All simulations and analyses were conducted using Stata software v12.1.
Bias in regression coefficients
Figure 2 illustrates how regression coefficients estimated from the two linear models differed from the "true" value of 1. The two subplots of the figure (A and B) correspond to the two types of explanatory variable (continuous and binary respectively), and the different shades of grey represent different ICC levels with darker shades corresponding to simulated results for higher ICCs.
Difference from "true" value of 1 of regression coefficients estimated from RI and OLS models (\({\beta }_{1}^{RI}\) and \({\beta }_{1}^{OLS}\)) plotted against dispersion (expressed as SD) of the mean value/prevalence of \({x}_{ij}\) across the clusters within each sample. Results for different levels of intraclass correlation coefficient are distinguished by shades of grey as indicated in the legend. A Continuous \({x}_{ij}\). B Binary \({x}_{ij}\)
In all cases, differences from the "true" value of 1 were on average zero, indicating that both models produced unbiased estimates of the regression coefficient. However, with a continuous explanatory variable, divergence from the "true" value tended to be greater for the OLS than for the RI model, especially for higher ICC and for lower dispersion of the mean value of \({x}_{ij}\) across the clusters within a sample (Supplementary Table 1). With a binary explanatory variable, divergence from the nominal value was again greatest for high ICCs (see also Supplementary Table 2), but there was no strong relationship to dispersion of the mean prevalence of \({x}_{ij}\) across clusters, and average divergence differed less between the two models.
Ratio of standard errors
The ratios of SEs derived from the RI and OLS models (\({SE}_{{\beta }_{1}^{RI}}/{SE}_{{\beta }_{1}^{OLS}}\)) were examined in relation to the dispersion of the mean value/prevalence of the continuous/binary explanatory variable \({x}_{ij}\) across the clusters within the sample, and are presented in Fig. 3. As in Fig. 2, levels of ICC are represented by different shades of grey, with lighter shades corresponding to lower ICCs and darker shades to higher ICCs. Subplots A and B illustrate the ratios of SEs when \({x}_{ij}\) was continuous and binary, respectively.
Ratios of standard errors estimated from RI and OLS models (\({\mathrm{S}\mathrm{E}}_{{\mathrm{\beta }}_{1}^{\mathrm{R}\mathrm{I}}}\)/\({\mathrm{S}\mathrm{E}}_{{\mathrm{\beta }}_{1}^{\mathrm{O}\mathrm{L}\mathrm{S}}}\)) plotted against dispersion (expressed as SD) of the mean value/prevalence of \({x}_{ij}\) across the clusters within each sample. Results for different levels of intraclass correlation coefficient are distinguished by shades of grey as indicated in the legend. A Continuous \({x}_{ij}\). B Binary \({x}_{ij}\)
For a continuous variable \({x}_{ij}\), the ratio took its minimum value for the smallest dispersion of cluster mean values of \({x}_{ij}\) (\({\stackrel{-}{x}}_{j}\)) and increased towards a plateau as that dispersion increased. The minimum and maximum values of the ratio of the SEs (the latter corresponding to the plateau value) were ICC-dependent, higher ICCs resulting in lower minimum and higher maximum values for the ratio. The dispersion of \({\stackrel{-}{x}}_{j}\) at which the ratio of SEs approached its plateau was also ICC-dependent, being higher for larger ICCs. For very small values of dispersion of \({\stackrel{-}{x}}_{j}\), the minimum value of the ratio of the SEs was approximately one for small levels of ICC and was less than one for higher ICCs. Particularly for small values of the dispersion of \({\stackrel{-}{x}}_{j}\) and ICC \(\cong\) 0.10 or 0.30, the ratio of SEs was < 1, meaning that SEs from RI models were smaller than from OLS models.
When \({x}_{ij}\) was binary, the ratios of the SEs were below one for most of the situations examined, indicating that the SEs of the regression coefficients estimated from the RI model were smaller than those from the OLS model in most circumstances. The ratio of the SEs achieved its minimum value for the smallest dispersion of the prevalence of \({x}_{ij}\) across the clusters within a sample, and increased progressively as that dispersion increased. For small ICCs (< 0.1), the SEs from the two models were very similar. However, as ICC increased to 0.1 or higher the ratio of the SEs decreased to values much lower than 1. For constant ICC, comparison of subplots of Fig. 3B, shows that the rate of increase of the ratio of the SEs was higher for lower target prevalence rates of the \({x}_{ij}\).
Coverage of 95% confidence intervals
Table 1 shows the extent to which 95% confidence intervals covered the "true" effect of a continuous explanatory variable on the outcome (\({\beta }_{1}\)=1), when derived from the two statistical models. Results are presented separately for different levels of ICC, and for fifths of the distribution of the dispersion of cluster means of the explanatory variable across the clusters of the sample.
Table 1 Coverage (%) of "true" effect \({\beta }_{1}\) = 1 by 95% confidence intervals derived from the RI and OLS models according to fifths of the distribution of dispersion (expressed as SD) of the cluster means of a continuous \({x}_{ij}\) within samples
Irrespective of ICC and type of explanatory variable, coverage with the RI model was approximately 95% (for continuous \({\mathrm{x}}_{\mathrm{i}\mathrm{j}}\): range across ICC levels 94.75–95.33%; for binary \({\mathrm{x}}_{\mathrm{i}\mathrm{j}}\): range across ICC levels and prevalence rates of \({\mathrm{x}}_{\mathrm{i}\mathrm{j}}\) 94.72–95.15%). For a continuous \({x}_{ij}\), coverage for the OLS model was close to 95% for very low ICC but decreased with increasing levels of ICC. For the highest ICC level examined (ICC = 0.3), OLS gave a notably poor coverage of 30%. For a given ICC, coverage of 95% confidence intervals did not vary much according to dispersion of the mean value of \({x}_{ij}\) across clusters, although it was somewhat higher in the bottom fifth as compared with the rest of the distribution of dispersions.
For a binary \({x}_{ij}\), coverage from the OLS model was close to 95% (range 94.7 to 95.2%) for ICC ≤ 0.03. However, as ICC increased, coverage from the OLS model deviated from the nominal value of 95%. As shown in Fig. 4, when ICC was 0.1 or 0.3, coverage was on average lower for lower target prevalence of \({x}_{ij}\); it fell below the nominal value of 95% for 0.05 target prevalence of \({x}_{ij}\) and it increased to values higher than 95% for 0.40 target prevalence of \({x}_{ij}\) (comparison of the four sub-plots of the figure). Also, for any given target prevalence of \({x}_{ij}\), coverage was lower for increasing dispersion of prevalence of \({x}_{ij}\) across clusters. Variation of the average coverage by target prevalence of \({x}_{ij}\) and dispersion of prevalence of \({x}_{ij}\) across clusters was higher when ICC was higher (ICC = 0.3) than when it was lower (ICC = 0.1). The smallest and the largest values of coverage were 87 and 98% and they were observed when the target prevalence of \({x}_{ij}\) was 0.05, ICC = 0.3, and in the lowest and highest thirds respectively of the distribution of dispersion of prevalence of \({x}_{ij}\) across clusters. Coverage as high as 98% was also seen in the lowest third of the distribution of dispersion of prevalence of \({x}_{ij}\) across clusters for the other prevalence rates (0.10, 0.20, and 0.40) explored when ICC was high (ICC = 0.3).
Coverage (%) by 95% confidence intervals from the OLS model for ICC = 0.1 and 0.3, according to target prevalence of \(x\) (A) 0.05, B) 0.10, C) 0.20, and D) 0.40), and thirds of the distribution of the dispersion (expressed as SD) of the prevalence of \({x}_{ij}\) across the clusters within each sample
To assess the frequency of Type I error, defined as incorrect rejection of a true null hypothesis, under the OLS and the RI multi-level models, simulations were repeated assuming no association between the explanatory variable \({x}_{ij}\) and the outcome variable \({y}_{ij}\) (\({\beta }_{1}=0\)).
Figure 5 shows the percentage of simulated samples for which the null hypothesis was rejected at a 5% significance level for varying levels of ICC, when \({x}_{ij}\) was continuous. Using the RI multi-level model, the association between \({x}_{ij}\) and \({y}_{ij}\) was statistically significant in approximately 5% of the datasets for all ICCs. However, with the OLS models, Type I error rose with ICC. For a very small ICC, Type I error was close to the nominal value of 5%, but it increased rapidly as the ICC increased, reaching \(\sim 70\%\) for ICC \(\cong\) 0.30. Type I error did not vary much by dispersion of the cluster mean values of \({x}_{ij}\) within samples, but was lowest in the lowest fifth of the distribution of dispersion (Supplementary Table 3).
Percentage (%) of simulated samples for which the null hypothesis was rejected according to level of ICC when \({x}_{ij}\) was continuous and no association was assumed between outcome and explanatory variable
When the explanatory variable \({x}_{ij}\) was binary, Type I error rates from the OLS model varied very little around the nominal level of 5% when ICC values were less than 0.1; the average value was 5% and varied from 4.8 to 5.3% for different ICC values (< 0.1), target prevalence rates of \({x}_{ij}\), and dispersion of prevalence of \({x}_{ij}\) across clusters. However, for ICC values of 0.1 and 0.3, Type I error rates diverged from 5%. This is illustrated in Fig. 6 for the four target prevalence rates of \({x}_{ij}\) (subplots A, B, C, and D of the figure), and for thirds of the distribution of dispersion of prevalence of \({x}_{ij}\) across clusters. For small dispersion of prevalence rates of \({x}_{ij}\) (bottom third of the distribution), Type I error was lower than 5%, and it increased as dispersion increased. This trend was more prominent for lower values of target prevalence of \({x}_{ij}\), and for ICC = 0.3 compared to ICC = 0.1. The smallest and the largest values of Type I error were 2 and 13% and they were observed when the prevalence of \({x}_{ij}\) was 0.05 and in the lowest and highest thirds respectively of the distribution of dispersion of prevalence of \({x}_{ij}\) across clusters.
Type I error rates (%) from the OLS model for ICC = 0.1 and 0.3, by target prevalence rates of \({x}_{ij}\) (A 0.05, B 0.10, C 0.20, and D 0.40), and thirds of the distribution of the dispersion (expressed as SD) of prevalence of \({x}_{ij}\) across the clusters within each sample
In this study we focused on the implications of ignoring clustering in statistical inference regarding the relationship between a continuous outcome and a single explanatory variable \({x}_{ij}\). For each of two types of \({x}_{ij}\) (continuous and binary), we fitted RI and OLS models and explored: the deviation of effect estimates from the "true" value that was used in generating the samples; their relative precision; the extent to which their 95% confidence intervals covered the "true" value; and the frequency of Type I error when simulations assumed that there was no association between the outcome and explanatory variable. Our interest was principally in implications for analysis of data from observational studies, and we specified the generation of simulated samples to encompass a range of scenarios that might be encountered in real observational data. In particular, we considered varying degrees of clustering not only in the outcome variable (measured by ICC), but also in the explanatory variable. The latter was quantified in terms of the SD across clusters of the mean or prevalence of the explanatory variable within each cluster (its variance within clusters being fixed).
With both continuous and binary \({x}_{ij}\), where the "true" effect was non-zero, we found that irrespective of ICC, both RI and cluster-unadjusted OLS models on average gave estimates of effect close to the "true" value (i.e. they were unbiased). However, deviations from the "true" value were greater for higher ICC. For continuous \({x}_{ij}\), they tended also to be greater for the OLS than the RI model, and when the explanatory variable was less clustered. However, with a binary explanatory variable, deviations from the "true" value showed no strong relationship to the level of clustering of the explanatory variable, and average divergence from the "true value" differed less between the two models.
SEs for effect estimates from cluster-unadjusted OLS differed from those derived from RI models, the differences being driven mainly by ICC levels and the extent to which the explanatory variable was clustered. For higher clustering of the explanatory variable, the SEs of regression coefficients from the RI model were generally larger than from the cluster-unadjusted OLS model. When \({x}_{ij}\) was continuous, the ratio of SEs (\({SE}_{{\beta }_{1}^{RI}}/{SE}_{{\beta }_{1}^{OLS}}\)) was highest (> 4) for a high ICC (0.3). However, the apparently greater precision of OLS method was not universal. For low clustering of the explanatory variable, OLS regression gave larger SEs than RI modelling, particularly for higher ICCs (> 0.03). With both continuous and binary \({x}_{ij}\), SEs from RI modelling were more than 15% lower than those from OLS regression for the highest ICC value (ICC = 0.3) and the lowest clustering of the explanatory variable.
The rates of coverage of 95% confidence intervals for estimates of effect, whether of a continuous or a binary \({x}_{ij}\), when derived from a RI model were at the nominal level of 95%, irrespective of other parameters. When \({x}_{ij}\) was binary, the cluster-unadjusted OLS model also resulted in an appropriate coverage of 95% confidence intervals provided ICC was low (\(\le 0.01\)). However, for higher values of ICC, coverage varied around the nominal value of 95% (range: 87–98%) depending on the overall prevalence and the dispersion of the cluster-specific prevalence rates of \({x}_{ij}\). In contrast, when \({x}_{ij}\) was continuous, the model that failed to account for clustering resulted in much poorer coverage rates, especially as ICC increased, and they were as low as 30% for ICC = 0.3.
Setting the effect of \({x}_{ij}\) on the outcome variable to zero allowed exploration of the frequency of Type I error. With the RI model, Type I error was close to 5% in all of the scenarios explored. When \({x}_{ij}\) was continuous, we found that failure to allow for clustering increased rates of Type I error, and that the inflation of Type I error was particularly pronounced (up to 70%) when the degree of clustering was high (ICC = 0.3). In contrast, when \({x}_{ij}\) was binary, Type I error under the OLS model was close to the expected value of 5% for low ICC (< 0.1). However, when ICC was high (0.1 or 0.3), Type I error rates varied more widely around 5%, with values as low as 2% (for low target prevalence of \({x}_{ij}\) and small dispersion of its prevalence across clusters) and as high as 13% (for low target prevalence of \({x}_{ij}\) and large dispersion of its prevalence across clusters).
The analysis for each specification of parameters (expected ICC, dispersion of mean or prevalence of \({x}_{ij}\) across clusters, and (for binary \({x}_{ij}\)) overall prevalence of \({x}_{ij}\)) was based on a large number of simulated samples (100,000 for each of six target ranges of ICC for continuous \({x}_{ij}\), and 50,000 for each of 24 combinations of target ICC and target prevalence of \({x}_{ij}\) for binary \({x}_{ij}\)), each of which comprised 10,000 observations grouped in 100 equally sized clusters. By using such a large sample size (larger than in many epidemiological investigations), we reduced random sampling variation, making it easier to characterise any systematic differences between the two methods of analysis. However, the approach may have led to underestimation of the maximum errors in effect estimates that could arise from OLS as compared with multi-level modelling. Moreover, the number of observations per cluster was the same in all simulations, making it impossible to draw conclusions about effects of ignoring clustering where cluster sizes vary (including situations where cluster size is informative [24]). Also, data were simulated following the specification of the RI regression model (as described in Eq. 2) rather than that of the random-effects model. That was done because the RI model is more frequently used, especially when there is no prior expectation of differential effects of the explanatory on the outcome variable across different clusters. Simulating data following the specification of the random effects model would have added to the complexity of the algorithm used for simulation, and to the computational time required.
Given the method by which the simulated samples were generated, it was to be expected that when multilevel RI modelling was applied, irrespective of whether the explanatory variable was continuous or binary, the rate of Type I error would be 5%, and the coverage by 95% CIs would be at the nominal level of 95%. In comparison, when cluster-unadjusted OLS models were fitted to clustered data with a continuous \({x}_{ij}\), rates of Type I error were higher, particularly when the ICC was high. For the highest level of ICC examined (0.3), Type I errors were as frequent as 70%. However, even with an ICC of only 0.01, rates of Type I error were more than 10%. Consistent with this, coverage by 95% confidence intervals was much lower than the nominal value (rates down to 30%) when ICC levels were high. In contrast to these results Huang et al. [25] have reported coverage close to 95% from the OLS model when it was applied to clustered data with a continuous explanatory variable. Differences between our findings and those of Huang et al. [25] may be explained by lack of clustering in the explanatory variable in Huang's investigation. Sensitivity analysis restricting our simulated datasets to those in which clustering of explanatory variable was minimal showed that interval coverage rates were close to 95%, independent of clustering in the outcome variable (Supplementary Table 4).
When \({x}_{ij}\) was binary and OLS regression was applied, interval coverage and rates of Type I error varied little around the nominal values of 95 and 5%, and only for ICC values higher than 0.01. Overall coverage rates were above the nominal rate for higher ICCs and decreased with greater dispersion of the prevalence of \({x}_{ij}\) across clusters, and with lower overall prevalence of the \({x}_{ij}\). A similar observation of small variation of interval coverage around 95% for higher ICC values has been reported previously [26]. Type I error when \({x}_{ij}\) was binary and its overall prevalence low, varied around 5% with values below 5% for small dispersion of prevalence of \({x}_{ij}\) across clusters, and above 5% for large dispersion. For high overall prevalence of \({x}_{ij}\) (up to 0.4), Type I error rates fell below 5%. In accordance with these findings, Galbraith et al. [27] have shown that cluster-unadjusted models resulted in relatively conservative Type I error. Also, in a context of individually randomised trials, Kahan et al. [28] have shown that Type I error increased with increasing ICC and increasing difference in the probability of assignment of patients to treatment arms.
It has been widely stated that when data are clustered, effects estimated by OLS regression are unbiased [22, 26, 29,30,31], at least where cluster size is uninformative, and there is no confounding by cluster [32]. Our results confirm that for data of the type simulated (in which the clusters were all of equal size and effect sizes did not vary by cluster), coefficients from both OLS and RI regression were on average very similar to the "true" value that had been used in generating simulated samples. Previous studies based on simulated data have shown similar results [22, 25, 26, 33]. However, for individual simulated samples, effect estimates often differed from the "true" value, with larger deviations for a continuous explanatory variable when the OLS model was fitted. For continuous \({x}_{ij}\), the potential magnitude of deviations from the "true" value depended on the extent to which the outcome variable was clustered. For an ICC of 0.3, OLS estimates of effect differed by up to 10% from the "true" value (Fig. 2A). In addition, when \({x}_{ij}\) was continuous, the error in OLS estimates of the regression coefficient was largest when the between-cluster dispersion of \({x}_{ij}\) was similar to that within-clusters (the within-cluster SD of \({x}_{ij}\) having been arbitrarily set to 1 in the simulation algorithm). When \({x}_{ij}\) was binary, the deviation of OLS estimates from the "true" value increased as the dispersion of prevalence rates across clusters increased, and when the target prevalence rate across all clusters was lower (< 10%) (Supplementary Table 2). These errors in effect estimates indicate that in an individual study, failure of regression analysis to account for clustering of observations could result in substantially higher or lower estimates of effect than those derived from multilevel analysis. This has been illustrated in numerous published papers of real data, which have shown that estimates from the two analytical methods can differ to a lesser or greater extent [10, 15, 18, 20, 34]. However, in those publications, little or no information was provided to establish how the observed error related to clustering of the explanatory variable.
The ratio of the SE of a regression coefficient estimated from the RI model to that derived when an OLS model is applied to the same sample, provides an inverse measure of their relative precision. Given that effects estimated from both RI and OLS models were unbiased, and that coverage of confidence intervals derived from RI models was consistently close to the nominal value of 95%, deviation of the measure from unity is likely to reflect bias in SEs estimated by OLS regression. It is widely stated that regression coefficients are spuriously precise when clustering is not taken into account in regression models, although authors have often failed to specify the conditions under which this applies [20, 33, 35,36,37,38]. Other authors have pointed out that when \({x}_{ij}\) is identical within each cluster (as, for example, in a cluster-randomised trial), and a cluster-unadjusted approach is followed, SEs tend to be spuriously low, and that the opposite occurs when \({x}_{ij}\) varies within clusters [25, 28, 39, 40]. Higher SEs from OLS models for effects of explanatory variables that varied within clusters have been demonstrated in analyses of both real and simulated data [34, 41]. Others, however, have reported contradictory results in which SEs of effect estimates from OLS regression where \({x}_{ij}\) varied within clusters were very similar to, or lower than, those from a multi-level model [15,16,17,18, 42]. It should be noted that the dichotomy between cluster- and individual-level variables is not clear-cut. There can be varying degrees of clustering in \({x}_{ij}\), the extremes occurring where its mean value is the same for all clusters (i.e. it is completely unclustered), and where it does not vary at all within clusters (i.e. it is a cluster-specific or cluster-constant variable). However, in observational studies, an explanatory variable can lie anywhere between these extremes. In recognition of this, an early paper focused on the level of clustering in \({x}_{ij}\) as a driver for the expected bias in the precision of effect estimates [29], rather than making a dichotomous distinction between cluster-constant and cluster-varying \({x}_{ij}\). The authors reported that as clustering in \({x}_{ij}\) decreases, the bias in SEs from a cluster-unadjusted model is expected to increase, and vice versa. Taking into consideration clustering in \({x}_{ij}\) as well as in the outcome variable, a later study using simulated data showed that for a given level of clustering in the outcome variable, increasing the clustering of the explanatory variable caused the ratio of estimated SEs (\({SE}_{\beta }^{RI}/{SE}_{\beta }^{OLS}\)) to increase from values < 1 to values \(\approx 1\) [43]. Our results for continuous explanatory variables differ slightly from this, with ratios of SEs (\({SE}_{\beta }^{RI}/{SE}_{\beta }^{OLS}\)) moving from values < 1 to values > 1, as clustering of the explanatory variable, expressed as dispersion of \({\stackrel{-}{x}}_{j}\) across clusters, increased. This accords with the finding that OLS regression will have spuriously high precision when it is used to analyse cluster-randomised trials, and spuriously low precision when it is applied to data from individually randomised trials [28, 39, 40].
Bias in the precision of effect estimates for binary \({x}_{ij}\) when clustering is ignored has received only limited attention in the published literature. Several reported studies have compared standard and multi-level models, using real data with continuous and binary \({x}_{ij}\) that varied within clusters [15, 18]. Where \({x}_{ij}\) was binary, SEs derived from the OLS model were mostly larger than those from the multi-level model. The same conclusion was drawn from a study using simulated data [26]. However, neither of the studies using real data has explored the level of bias in relation to variation in the prevalence of the binary \({x}_{ij}\), and the study of simulated data assumed constant prevalence of \({x}_{ij}\) in all clusters.
In our analyses, SEs from the multi-level model were generally lower than those from the OLS model, irrespective of the dispersion of prevalence of \({x}_{ij}\) across clusters and its overall prevalence. This contrasted with findings when \({x}_{ij}\) was continuous, where ratios of SEs (\({SE}_{\beta }^{RI}/{SE}_{\beta }^{OLS}\)) increased from < 1 to > 1, as the dispersion of \({\stackrel{-}{x}}_{j}\) across clusters increased. The explanation for the discrepancy may lie in the ranges of dispersion of \({\stackrel{-}{x}}_{j}\) across clusters that were explored for continuous as compared with binary \({x}_{ij}\). For a continuous explanatory variable \({x}_{ij}\), we allowed wide dispersion of \({\stackrel{-}{x}}_{j}\) across clusters, while for binary \({x}_{ij}\) the dispersion of \({\stackrel{-}{x}}_{j}\) across clusters was constrained to low levels in order to achieve overall prevalence rates for \({x}_{ij}\) that were close to the target values (this applied particularly where the target prevalence was 5%). The ratio of SEs (\({SE}_{\beta }^{RI}/{SE}_{\beta }^{OLS}\)) for continuous \({x}_{ij}\) only clearly exceeded one for dispersions of \({\stackrel{-}{x}}_{j}\) across clusters greater than those that were examined for binary \({x}_{ij}\).
The focus of this paper was on the association between a continuous outcome and an explanatory variable \({x}_{ij}\) that varied within clusters. We showed that when \({x}_{ij}\) was continuous, and most of its variation was within rather than between clusters, the cluster-unadjusted OLS model gave larger SEs for the regression coefficient than multi-level modelling. This accords with reports that ignoring clustering can lead to spuriously high SEs when \({x}_{ij}\) varies within clusters. The reverse occurred when most of the dispersion of \({x}_{ij}\) was between rather than within clusters, a situation approaching that of a cluster-specific explanatory variable. We additionally showed that when \({x}_{ij}\) was binary, ignoring clustering in statistical modelling in most cases resulted in higher SEs for the estimated effect than those derived from the random-intercept model, possibly reflecting low simulated dispersion of the prevalence of \({x}_{ij}\) across clusters. The SEs differed more for higher ICCs but not with the overall prevalence of \({x}_{ij}\), nor with the dispersion of its prevalence across clusters (Fig. 3B). Unlike SEs, the point estimates were unbiased for both continuous or binary \({x}_{ij}\) (Fig. 2A and B).
Thus, our results support the use of multi-level modelling to account for clustering effects in linear regression analyses of data that are hierarchically structured with characteristics similar to those that we explored (large sample size, similarly sized clusters, and no variation in the effects of explanatory variables by cluster), especially where ICCs might exceed 0.01. Failure to do so is likely to result in incorrect estimates of effect (either too high or too low) with spuriously high or low precision according to the level of clustering of explanatory variables, and thus may lead to incorrect inferences. The errors in estimates of effect of a continuous \({x}_{ij}\) will be smaller when most of its dispersion is between rather than within clusters – i.e. the variable comes closer to being cluster-specific. When \({x}_{ij}\) is binary, smaller errors in the effect estimates occur when its overall prevalence \(p\) across clusters is closer to 50%, i.e. when the variance of the binary variable is at its maximum (\(p\left(1-p\right)=25\%\)).
Additionally, we have identified circumstances in which a simpler analytical approach that does not adjust for clustering is more likely to mislead statistical inference, i.e. in which rates of Type I error and interval coverage deviate materially from the nominal values of 5 and 95% respectively. These occur when \({x}_{ij}\) is continuous, and ICC levels are greater than 0.01. It is then that Type I error rates are higher than 10% and interval coverage rates are lower than 80%. Statistical inference when a standard regression model is fitted is less likely to be problematic when \({x}_{ij}\) is binary, but again Type I error rates can sometimes be greater than 10%, and corresponding interval coverage rates lower than 90%. This occurs when ICC is high, the overall prevalence of \({x}_{ij}\) is low (approximately 5%), and the dispersion of the cluster-specific prevalence of \({x}_{ij}\) is large. In all circumstances in which the ICC is small, clustering is minimal and there is little difference between RI and OLS regression.
The simulated datasets used and analysis described in the current study are available from the corresponding author on reasonable request.
RI:
Random-intercept
OLS:
Ordinary least squares
ICC:
Intra-class correlation coefficient
Rabe-Hesketh S, Skrondal A. Multilevel and longitudinal modeling using stata. USA: Taylor & Francis; 2005.
Stimson JA. Regression in space and time: a statistical essay. Am J Pol Sci. 1985;29(4):914–47.
Bingenheimer JB, Raudenbush SW. Statistical and substantive inferences in public health: issues in the application of multilevel models. Annu Rev Public Health. 2004;25:53–77.
Goldstein H. Multilevel statistical models. United Kingdom: Wiley; 2011.
McNeish D, Kelley K. Fixed effects models versus mixed effects models for clustered data: reviewing the approaches, disentangling the differences, and making recommendations. Psychol Methods. 2019;24(1):20–35.
Bland JM. Cluster randomised trials in the medical literature: two bibliometric surveys. BMC Med Res Methodol. 2004;4(1):21.
Crits-Christoph P, Mintz J. Implications of therapist effects for the design and analysis of comparative studies of psychotherapies. J Consult Clin Psychol. 1991;59(1):20.
Lee KJ, Thompson SG. Clustering by health professional in individually randomised trials. BMJ (Clinical research ed). 2005;330(7483):142–4.
Simpson JM, Klar N, Donnor A. Accounting for cluster randomization: a review of primary prevention trials, 1990 through 1993. Am J Public Health. 1995;85(10):1378–83.
Biau DJ, Halm JA, Ahmadieh H, Capello WN, Jeekel J, Boutron I, et al. Provider and center effect in multicenter randomized controlled trials of surgical specialties: an analysis on patient-level data. Ann Surg. 2008;247(5):892–8.
Oltean H, Gagnier JJ. Use of clustering analysis in randomized controlled trials in orthopaedic surgery. BMC Med Res Methodol. 2015;15(1):1–8.
Diaz-Ordaz K, Froud R, Sheehan B, Eldridge S. A systematic review of cluster randomised trials in residential facilities for older people suggests how to improve quality. BMC Med Res Methodol. 2013;13(1):1–10.
Goldstein H. Multilevel mixed linear model analysis using iterative generalized least squares. Biometrika. 1986;73(1):43–56.
Astin AW, Denson N. Multi-campus studies of college impact: which statistical method is appropriate? Res High Educ. 2009;50(4):354–67.
Grieve R, Nixon R, Thompson SG, Normand C. Using multilevel models for assessing the variability of multinational resource use and cost data. Health Econ. 2005;14(2):185–96.
Niehaus E, Campbell C, Inkelas K. HLM behind the curtain: unveiling decisions behind the use and interpretation of HLM in higher education research. Res High Educ. 2014;55(1):101–22.
Steenbergen MR, Jones BS. Modeling multilevel data structures. Am J Pol Sci. 2002;46(1):218–37.
Wendel-Vos GCW, van Hooijdonk C, Uitenbroek D, Agyemang C, Lindeman EM, Droomers M. Environmental attributes related to walking and bicycling at the individual and contextual level. J Epidemiol Community Health. 2008;62(8):689–94.
Walters SJ. Therapist effects in randomised controlled trials: what to do about them. J Clin Nurs. 2010;19(7–8):1102–12.
Park S, Lake ET. Multilevel modeling of a clustered continuous outcome: nurses' work hours and burnout. Nurs Res. 2005;54(6):406–13.
Newman D, Newman I, Salzman J. Comparing OLS and HLM models and the questions they answer: potential concerns for type VI errors. Mult Linear Regression Viewpoints. 2010;36(1):1–8.
Clarke P. When can group level clustering be ignored? Multilevel models versus single-level models with sparse data. J Epidemiol Community Health. 2008;62(8):752–8.
Bradburn MJ, Deeks JJ, Berlin JA, Russell LA. Much ado about nothing: a comparison of the performance of meta-analytical methods with rare events. Stat Med. 2007;26(1):53–77.
Nevalainen J, Datta S, Oja H. Inference on the marginal distribution of clustered data with informative cluster size. Stat Pap. 2014;55(1):71–92.
Huang FL. Alternatives to multilevel modeling for the analysis of clustered data. J Exp Educ. 2016;84(1):175–96.
Chu R, Thabane L, Ma J, Holbrook A, Pullenayegum E, Devereaux PJ. Comparing methods to estimate treatment effects on a continuous outcome in multicentre randomized controlled trials: a simulation study. BMC Med Res Methodol. 2011;11(1):1.
Galbraith S, Daniel J, Vissel B. A study of clustered data and approaches to its analysis. J Neurosci. 2010;30(32):10601–8.
Kahan BC, Morris TP. Assessing potential sources of clustering in individually randomised trials. BMC Med Res Methodol. 2013;13(1):58.
Arceneaux K, Nickerson DW. Modeling certainty with clustered data: a comparison of methods. Polit Anal. 2009;17(2):177–90.
Scott AJ, Holt D. The effect of two-stage sampling on ordinary least squares methods. J Am Stat Assoc. 1982;77(380):848–54.
Barrios T, Diamond R, Imbens GW, Koleśar M. Clustering, spatial correlations, and randomization inference. J Am Stat Assoc. 2012;107(498):578–91.
Seaman S, Pavlou M, Copas A. Review of methods for handling confounding by cluster and informative cluster size in clustered data. Stat Med. 2014;33(30):5371–87.
Maas CJ, Hox JJ. The influence of violations of assumptions on multilevel parameter estimates and their standard errors. Comput Stat Data Anal. 2004;46(3):427–40.
Dickinson LM, Basu A. Multilevel modeling and practice-based research. Ann Fam Med. 2005;3(suppl 1):S52–60.
Austin PC, Goel V, van Walraven C. An introduction to multilevel regression models. Can J Public Health. 2001;92(2):150.
Lemeshow S, Letenneur L, Dartigues JF, Lafont S, Orgogozo JM, Commenges D. Illustration of analysis taking into account complex survey considerations: the association between wine consumption and dementia in the PAQUID study. Am J Epidemiol. 1998;148(3):298–306.
Roberts C, Roberts SA. Design and analysis of clinical trials with clustering effects due to treatment. Clin Trials. 2005;2(2):152–62.
Maas CJ, Hox JJ. Sufficient sample sizes for multilevel modeling. Methodology. 2005;1:3–86.
Chuang JH, Hripcsak G, Heitjan DF. Design and analysis of controlled trials in naturally clustered environments: implications for medical informatics. JAMIA. 2002;9(3):230–8.
Sainani K. The importance of accounting for correlated observations. PM&R. 2010;2(9):858–61.
Jones K. Do multilevel models ever give different results? 2009.
Hedeker D, McMahon SD, Jason LA, Salina D. Analysis of clustered data in community psychology: with an example from a worksite smoking cessation project. Am J Community Psychol. 1994;22(5):595–615.
Bliese PD, Hanges PJ. Being both too liberal and too conservative: the perils of treating grouped data as though they were independent. Organ Res Methods. 2004;7(4):400–17.
During completion of this work, GN was supported by the Colt Foundation (PhD scholarship) and a grant award from Versus Arthritis (formerly Arthritis Research UK) (22090). The funding bodies were not involved in the study design, data analysis, interpretation of results or in writing the manuscript.
Medical Research Council Lifecourse Epidemiology Unit, University of Southampton, Southampton, UK
Georgia Ntani, Hazel Inskip, Clive Osmond & David Coggon
Medical Research Council Versus Arthritis Centre for Musculoskeletal Health and Work, Medical Research Council Lifecourse Epidemiology Unit, University of Southampton, Southampton, UK
Georgia Ntani & David Coggon
NIHR Southampton Biomedical Research Centre, University of Southampton and University Hospital Southampton NHS Foundation Trust, Southampton, UK
Hazel Inskip
Georgia Ntani
Clive Osmond
David Coggon
GN, HI, and DC conceived the concept of this study. GN carried out the simulations, analysed the data and drafted the manuscript. CO provided expert statistical advice on aspects of results presented. DC and HI critically reviewed and made substantial contributions to the manuscript. All authors read and approved the final manuscript.
Correspondence to Georgia Ntani.
Additional file 1: Appendix A.
Data generating algorithm for clustered data with continuous outcome and explanatory variables. Appendix B. Data generating algorithm for clustered data with continuous outcome and binary explanatory variables. Supplementary Table 1. Standard deviation of derived \({\beta }_{1}^{RI}\) and \({\beta }_{1}^{OLS}\) when \({x}_{ij}\) was continuous according to fifths of the distribution of dispersion (expressed as SD) of the continuous \({\stackrel{-}{x}}_{j}\). Supplementary Table 2. Standard deviation of derived \({\beta }_{1}^{RI}\) and \({\beta }_{1}^{OLS}\) when \({x}_{ij}\) was binary according to fifths of the distribution of dispersion (expressed as SD) of the prevalence of \({x}_{j}\) across clusters, and overall target prevalence of \({x}_{ij}\). Supplementary Table 3. Percentage (%) of datasets for which the null hypothesis was rejected according to level of ICC when \({\beta }_{1}=0\) and \({x}_{ij}\) was continuous according to fifths of the distribution of dispersion (expressed as SD) of the continuous \({\stackrel{-}{x}}_{j}\). Supplementary Table 4. Interval coverage rates by ICC levels when clustering in the continuous explanatory variable was minimal (dispersion of the continuous \({\stackrel{-}{x}}_{j}\)<0.2 SDs).
Ntani, G., Inskip, H., Osmond, C. et al. Consequences of ignoring clustering in linear regression. BMC Med Res Methodol 21, 139 (2021). https://doi.org/10.1186/s12874-021-01333-7
Linear regression
Random intercept model | CommonCrawl |
Fast genomic prediction of breeding values using parallel Markov chain Monte Carlo with convergence diagnosis
Peng Guo1,2,
Bo Zhu1,
Hong Niu1,
Zezhao Wang1,
Yonghu Liang1,
Yan Chen1,
Lupei Zhang1,
Hemin Ni3,
Yong Guo3,
El Hamidi A. Hay4,
Xue Gao1,
Huijiang Gao1,
Xiaolin Wu5,6,
Lingyang Xu1 &
Junya Li1
BMC Bioinformatics volume 19, Article number: 3 (2018) Cite this article
Running multiple-chain Markov Chain Monte Carlo (MCMC) provides an efficient parallel computing method for complex Bayesian models, although the efficiency of the approach critically depends on the length of the non-parallelizable burn-in period, for which all simulated data are discarded. In practice, this burn-in period is set arbitrarily and often leads to the performance of far more iterations than required. In addition, the accuracy of genomic predictions does not improve after the MCMC reaches equilibrium.
Automatic tuning of the burn-in length for running multiple-chain MCMC was proposed in the context of genomic predictions using BayesA and BayesCπ models. The performance of parallel computing versus sequential computing and tunable burn-in MCMC versus fixed burn-in MCMC was assessed using simulation data sets as well by applying these methods to genomic predictions of a Chinese Simmental beef cattle population. The results showed that tunable burn-in parallel MCMC had greater speedups than fixed burn-in parallel MCMC, and both had greater speedups relative to sequential (single-chain) MCMC. Nevertheless, genomic estimated breeding values (GEBVs) and genomic prediction accuracies were highly comparable between the various computing approaches. When applied to the genomic predictions of four quantitative traits in a Chinese Simmental population of 1217 beef cattle genotyped by an Illumina Bovine 770 K SNP BeadChip, tunable burn-in multiple-chain BayesCπ (TBM-BayesCπ) outperformed tunable burn-in multiple-chain BayesCπ (TBM-BayesA) and Genomic Best Linear Unbiased Prediction (GBLUP) in terms of the prediction accuracy, although the differences were not necessarily caused by computational factors and could have been intrinsic to the statistical models per se.
Automatically tunable burn-in multiple-chain MCMC provides an accurate and cost-effective tool for high-performance computing of Bayesian genomic prediction models, and this algorithm is generally applicable to high-performance computing of any complex Bayesian statistical model.
Genomic predictions have been proposed as a method of providing accurate estimates of the genetic merits of breeding animals using genome-wide SNP markers [1]. This new technology does not require the actual phenotyping of breeding candidates and therefore offers great promise for traits that are difficult or expensive to measure, such as carcass traits [2]. A noted feature of genomic predictions is that selection can be performed on breeding candidates at birth or young ages, which in turn accelerates genetic improvement progress more rapidly than conventional breeding approaches in farm animals [3,4,5,6].
Many genomic prediction models have been proposed, such as the Genomic Best Linear Unbiased Prediction (GBLUP) [7] and Bayesian alphabets [1, 8]. Bayesian genomic models are widely used [8,9,10,11], although complex calculations are required for Bayesian models implemented via Markov chain Monte Carlo (MCMC) and may take hours, days or even weeks to complete. Hence, parallel computing for genomic predictions is of importance for applying genomic selection in practice [12]. However, parallelization in MCMC is difficult because the procedure is iterative in the sense that simulating the next value of the chain depends on the current value, which violates Bernstein's condition of independence for parallel computing [13]. This problem increases the difficulty of delivering parallelism for a single Markov chain. Thus, Wu et al. proposed the use of a multiple-chain MCMC method to calculate Bayesian genomic prediction models [14].
Running multiple-chain MCMC provides a naïve yet efficient form of parallel computing for Bayesian genomic prediction models, although the speedup in computing is limited by the burn-in requirement, which is included to give the Markov chain time to reach equilibrium distribution. A burn-in period corresponds to the first n samples during the MCMC, and these samples are discarded after the burn-in is initiated from a poor starting point to the period before each chain moves into a high probability state. Often, the length of the burn-in is assumed to be one-tenth or one-fifth of the entire length of the MCMC iterations or even half of the total iterations [1, 8,9,10]. This rule of thumb is often used for the sake of convenience but is not necessarily optimal for computing.
In the present paper, a multiple-chain MCMC computing strategy utilizing automatic tuning of the burn-in period was proposed and demonstrated with two Bayesian genomic prediction models (BayesA and BayesCπ). Using this strategy, a convergence diagnosis based on Gelman and Rubin [15] was conducted periodically in accordance with the multiple-chain situation. The burn-in period ended as soon as the convergence criteria were met, and posterior samples of unknown model parameters were then collected to perform statistical inferences. This strategy was assessed on a simulation data set and applied to genomic predictions of four quantitative traits in a Chinese Simmental population.
The simulated data set consisted of 1000 animals in scenario 1 and scenario 2 and 2000 animals in scenario 3, with each presenting a phenotype and genotypes on five chromosomes. The GPOPSIM software package [16] was used to generate the simulation data set, including the markers and QTLs based on a mutation-drift equilibrium model in the three scenarios. In scenario 1, the heritability of the trait was set at 0.1, and each chromosome had 4000 markers. In scenario 2, the heritability was 0.5, and each chromosome had 10,000 markers. In scenario 3, the heritability was 0.3, and each chromosome had 40,000 markers. In each of the three scenarios, 200 QTLs were simulated. The mutation rate of the markers and QTLs was set at 1.25 × 10−3 for each generation.
Real phenotype and genotype data
The experimental population consisted of 1302 Simmental cattle born between 2008 and 2013 in Ulgai, Xilingol League, Inner Mongolia, China. After weaning, all cattle were transferred to the Beijing Jinweifuren farm and raised under the same nutritional and management conditions. Each animal was evaluated regularly for growth and development traits until slaughter at between 16 and 18 months of age. At slaughter, the carcass traits and meat quality traits were assessed according to the Institutional Meat Purchase Specifications [17] for Fresh Beef Guidelines. The quantitative traits used in the present study included carcass back fat thickness (CBFT), strip loin weight (SLW), carcass weight (CW), and average daily gain (ADG). Prior to the genomic predictions, the phenotypes were adjusted for systematic environmental factors, which included the farms, seasons and years, and age at slaughter, using a linear regression model. The genetic and residual variances for each of the four traits were estimated by restricted maximum likelihood (REML) based on equivalent animal models.
Each animal was genotyped by an Illumina Bovine 770 K SNP BeadChip. SNP quality control was conducted using PLINK v1.07 software [18], which excluded SNPs under the following categories: 1) SNPs on the X and Y chromosomes, 2) SNPs with minor allele frequencies less than 0.05, 3) SNPs with > 5% missing genotypes, and 4) SNPs that violated Hardy-Weinberg equilibrium (p < 10−6). After data cleaning, 1217 Simmental cattle remained for subsequent data analyses, and each had genotypes on up to 671,220 SNPs on 29 autosomes.
Statistical model
Adjusted phenotypes were described by the following linear regression model:
$$ {y}_i=\upmu +{\sum}_{j=1}^M{X}_{ij}{\alpha}_j+{e}_i $$
where y i is an adjusted phenotype for individual i, M is the number of SNPs, μ is the overall mean of the traits, a j is the additive (association) effect of the j-th SNP, X ij is the genotype (0, 1, or 2) of the j-th SNP observed on the i-th individual, and e i is the residual term.
BayesA
The BayesA model [1] assumed a priori a normal distribution for SNP effects, with zero mean and SNP-specific variances denoted by \( {\upsigma}_j^2 \), where j = 1, 2, …, M. The variances of SNP effects were independent of one another, and each followed an identical and independently distributed (IID) scaled inverse chi-square prior distribution, \( \mathrm{p}\left({\upsigma}_j^2\right)={\upchi}^{-2}\Big({\upsigma}_j^2\mid \nu, {S}^2 \)), where ν is the degree of freedom parameter and S2 is the scale parameter, both of which are assumed to be known. Thus, the marginal prior distribution of each marker effect, \( \mathrm{p}\left({\upalpha}_j|\nu, {S}^2\right)=\int N\left({\upalpha}_j|0,{\upsigma}_j^2\right){\upchi}^{-2}\left({\upsigma}_j^2|\nu, {S}^2\right)d{\upsigma}_j^2 \), was a t-distribution [19].
BayesCπ
The BayesCπ model [8] assumed a priori that each SNP effect was null with probability π or followed a normal distribution, \( N\left(0,{\sigma}_a^2\right) \), with probability 1-π.
$$ \left.{a}_j\right|\pi, {\sigma}_a^2\sim \left\{\begin{array}{c}N\left(0,{\sigma}_a^2\right)\kern0.5em with\kern0.17em probability\left(1-\pi \right)\\ {}0\kern0.5em with\kern0.17em probability\;\pi \end{array}\right. $$
In the above, \( {\sigma}_a^2 \) is a variance common to all non-zero SNP effects, and it is assigned a scaled inverse chi-square prior distribution, \( {\chi}^{-2}\left({v}_a,{s}_a^2\right) \). The value of π in the model is unknown, and it is inferred based on the prior distribution of π, which is considered uniform between 0 and 1, or π~Uniform(0, 1).
GBLUP
GBLUP [7] can be considered a re-parameterization of the Bayesian RKHS (reproducing kernel Hilbert spaces) regression [20]. In RKHS, each SNP effect is assumed to follow a normal distribution with a zero mean and common variance; and in GBLUP, genomic estimated breeding values (GEBVs) are assumed to follow a normal distribution \( \boldsymbol{u}\sim \mathrm{N}\left(0,\mathbf{G}{\upsigma}_u^2\right) \), where G is a n × n genomic (co)variance matrix that is formulated as follows [7]:
$$ G=\frac{{\boldsymbol{XX}}^{\hbox{'}}}{2{\sum}_{i=1}^n{q}_i\left(1-{q}_i\right)}, $$
where n is the number of SNPs, q i is the frequency of an allele of SNP i, and X is a centered incidence matrix of SNP effects, which are corrected for allele frequencies. The additive genetic variances and residual variances of the four traits were estimated by REML based on an animal model equivalent to (1).
Tunable versus fixed burn-in multiple-chain MCMC
Tunable burn-in multiple-chain MCMC
In multiple-chain MCMC simulations, the following processes occur. Assume that we want to estimate some target distribution p(X) but cannot directly draw samples of X from p(X). Instead, a Markov chain X0, X1, … can be generated that converges to p(X) at equilibrium via a transition density u(Xt + 1| X t ). Now, let there be i = 1, 2, …, K parallel chains, with each initialized and burned-in independently for B i updating steps before more samples are drawn at intervals. As K → ∞ and all B i → ∞, the ensemble is ergodic (i.e., tending in the limit) to p(X) [14].
To assess the convergence of multiple parallel chains simulated for each model, both the inter-chain and within-chain variances were calculated for each selected model parameter, e.g., x. Briefly, the inter-chain variance I was calculated as follows:
$$ I=\frac{n}{m-1}{\sum}_{i=1}^m{\left({\overline{x}}_i-\overline{x}\right)}^2 $$
The within-chain variance W was determined as follows:
$$ W=\frac{1}{m}{\sum}_{i=1}^m{s}_i^2. $$
where \( {s}_i^2=\frac{1}{n-1}{\sum}_{j=1}^n{\left({x}_{ij}-{\overline{x}}_i\right)}^2 \), \( {\overline{x}}_i=\frac{1}{n}{\sum}_{j=1}^n{x}_{ij} \), and \( \overline{x}=\frac{1}{m}{\sum}_{i=1}^m{\overline{x}}_i \). Then, the marginal posterior variance of x was estimated by a weighted average of W and I as follows:
$$ \widehat{\mathit{\operatorname{var}}}(x)=\frac{n-1}{n}W+\frac{1}{n}I. $$
Under the assumption that the starting distribution of x was appropriately over-dispersed, the above quantity tended to overestimate the marginal posterior variance but was unbiased under stationarity (i.e., when the starting distribution equals the target distribution) or within the limit, n → ∞.
Following Gelman et al. [21], we assessed convergence by estimating the factor by which the scale of the current distribution for x might be reduced if the posterior simulation were continued within the limit, n → ∞. This potential scale reduction was estimated by the following shrink factor:
$$ r=\sqrt{\frac{\widehat{\mathit{\operatorname{var}}}(x)}{W}} $$
which reduced to 1 as n → ∞. A high-scale reduction indicated that proceeding with more simulations could further improve the inference about the target distribution of the model parameter. To run multiple-chain MCMC simulations, a collection of shrink factors was obtained, R = (r0, r1, …, rN − 1), where N is the length of a chain, R(j)=(r(j − 1) × p, r(j − 1) × p + 1, …, rj × p − 1) is the jth subsection of R, and p is the length of R(j). Let T be a threshold that was arbitrarily provided; the mean \( {\overline{\mathrm{r}}}^{(j)} \) and standard deviation S(j) were calculated as follows:
$$ {\overline{r}}^{(j)}=\frac{1}{p}{\sum}_{i=0}^{p-1}{r}_{\left(j-1\right)\times p+i}, $$
$$ {S}^{(j)}=\sqrt{\frac{1}{p}{\sum}_{i=0}^{p-1}{\left({r}_{\left(j-1\right)\times p+i}-{\overline{r}}^{(j)}\right)}^2}. $$
The multiple chains were considered to converge in the jth subsection when \( \left|{\overline{r}}^{(j)}-1\right|<T \) and S(j) < T.
In the simulation study, each of the parallel MCMC chains was initiated independently. Then, the convergence diagnosis during burn-in used samples from each parallel chain to determine the convergence state of these chains. The end of burn-in iterations occurred when the convergence criteria were met. Then, the simulated posterior samples were collected to calculate the posterior summary statistics of the model parameters of interest, which were subject to the thinning of the MCMC chains.
For the Simmental cattle data set, we evaluated genomic prediction accuracies (GPAs) by running up to 16 parallel chains for both TBM-BayesA and TBM-BayesCπ, and the results were compared with those obtained from GBLUP.
Speedup ratio
According to Amdahl's law [22], the speedup ratio of multiple-chain MCMC over that of single-chain MCMC was calculated as follows:
$$ \mathrm{S}\left(\mathrm{K}\right)=\frac{{\mathrm{N}}_{\mathrm{T}}}{{\mathrm{N}}_{\mathrm{burn}-\mathrm{in}}+\left({\mathrm{N}}_{\mathrm{T}}-{\mathrm{N}}_{\mathrm{burn}-\mathrm{in}}\right)/\mathrm{K}} $$
where Nburn-in is the number of burn-in iterations that cannot be parallelized, NT is the total number of MCMC iterations, and K is the number of computer cores available for running multiple-chain Markov chains in parallel. The parallel computing efficiency was assessed as follows:
$$ E=S(K)/K $$
Evidently, when E = 1, the parallel computing scales linearly with the number of cores used for computing; thus, S(K) = K. However, because of the non-parallelizable burn-ins, the parallel computing efficiency is upper bounded by N T /Nburn − in (as K → ∞).
Parameter setting
Fixed burn-in multiple-chain MCMC jobs were also run on the simulation data, with the length of burn-in set at one-tenth of the total sequential MCMC iterations in scenario 2 and at one-fifth of the total sequential MCMC iterations in scenarios 1 and 3. To assess the effect of the burn-in length, we ran 50,000 iterations for each chain of fixed burn-in multiple-chain BayesA (FBM-BayesA), which included burn-ins of 2000, 4000, 6000, 8000, and 10,000 iterations in scenario 2. Threshold T was set to 0.001 in this study.
Evaluation of genomic prediction accuracy
The GEBVs were calculated as the sum of all SNP effects of each individual (say i) as follows:
$$ {GEBV}_i={\sum}_j{X}_{ij}{g}_j $$
where X ij is a genotype (coded 0, 1, or 2) for SNP j of animal i and g j is the estimated genetic effect of the jth SNP.
The GPA relative to that of phenotypic selection was calculated as \( r/\sqrt{h^2} \), where r is Pearson's correlation between GEBVs and true breeding values in the simulation study or Pearson's correlation between GEBVs and adjusted phenotypes. This criterion of relative genomic prediction accuracy (RGPA) was used so that the GPAs were comparable regardless of their respective heritabilities [23].
A fivefold cross-validation [24] was used to evaluate the genomic predictions in the Simmental data set. Briefly, the entire data set of 1217 Simmental cattle was randomly divided into five approximately equal subsets. Then, four subsets were used to estimate the SNP effects (i.e., training), and the remaining subset was used for testing the GPA (i.e., validation). The above process was rotated five times until each subset was used for testing once and only once. For each trait, fivefold cross-validations were randomly duplicated 10 times, and the GPA for each trait was calculated as the average of GPAs across the ten replicates.
The calculations were conducted on an HP ProLiant DL585 G7 (708686-AA1) server, which was equipped with an AMD Opteron 6344 (2.6G Hz) CPU, 272 G of memory and an L2 cache size of 4 M and an L3 cache size of 16 M. The operating system was Microsoft Windows. A C program with Message Passing Interface (MPI) for parallel computing was developed to implement the aforementioned multiple-chain MCMC. MPICH2 is an open source MPI implementation and a standard for message passing in parallel computing, and it is available freely (http://www.mpich.org/downloads). The Integrated Development Environment that we used is Dev-C++ 5.1, which is available freely at the following link: http://www.bloodshed.net/index.html.
Simulation studies
Speedup ratios
Running multiple chains of genomic prediction models led to substantially reduced computing time compared with running a single chain (Additional file 1: Tables S1–S3 and Figures S9–S11). The speedups increased non-linearly with the number of parallelized chains or available computer cores (Fig. 1) because of the non-parallel burn-ins, and perfect speedups were not practically observed when calculating these Bayesian genomic prediction models. In scenario 1, for example, the speedup obtained by TBM-BayesA was 1.86 when running two parallel chains and was 13.63 when running 18 parallel chains. However, the speedup obtained by FBM-BayesA increased from 1.57 when running two chains to 3.79 when running 18 parallel chains. For the results obtained with 18 parallel chains, the speedups were approximately between 3 and 6 when running fixed burn-in MCMC and between 10 and 14 when running tunable burn-in MCMC. More precisely, TBM-BayesA had considerably greater speedups than those of FBM-BayesA, and the speedups by TBM-BayesA scaled better than those by FBM-BayesA. Similar trends were found in the comparison of computing time between TBM-BayesCπ and FBM-BayesCπ (fixed burn-in, multiple-chain BayesCπ). Thus, tunable burn-in MCMC had greater parallel computing efficiencies than fixed burn-in MCMC because the use of automatic convergence diagnosis and tuning of burn-ins effectively shortened the computing time by TBM-BayesA (or TBM-BayesCπ), resulting in increased speedups in computing time with tunable burn-ins. We also noted that the loss of parallel computing efficiency relative to an assumedly perfect speedup increased with the model dimension, which is proportional to the number of SNPs in the genomic prediction models (Fig. 1).
Speedup ratios of parallel MCMC (>1 chain) over sequential MCMC (1 chain) in simulation studies under the three scenarios: a Scenario 1 (h2 = 0.1; 4000 SNPs; 200 QTLs), b Scenario 2 (h2 = 0.1; 10,000 SNPs; 200 QTLs), and c Scenario 3 (h2 = 0.1; 40,000 SNPs; 200 QTLs). Expected speedup ratios were calculated under the assumption that MCMC chains were 100% parallelizable (i.e., without burn-in)
Theoretically, the speedups achieved by running multiple-chain MCMC are limited by the length of non-parallel parts (i.e., burn-ins). Frequently, the rule of thumb for the length of burn-in tends to result in far more burn-in iterations than are required. Thus, with automatic tuning of the convergence diagnosis on multiple-chain MCMC, the burn-in length can be drastically reduced, resulting in greater speedups in computing. With all other factors equal, the speedup obviously increased with a greater number of chains (or CPU cores) running in parallel (Fig. 1).
Estimated model (SNP) effects
Various computing forms of the same genomic prediction models essentially generated highly comparable estimated model effect results. Trace plots of the residual variance obtained by various computational approaches are shown in Fig. 2. Each chain mixed very well, and all were centered near zero.
Trace plots of the simulated residual variance using various computational forms of (a) BayesA and (b) BayesCπ. The total length of MCMC was 50,000 iterations, and the length of the fixed burn-in period was 5000 iterations
The estimated SNP effects were also highly comparable among various forms of calculating the same genomic prediction models. In parallel computing, between 2 and 18 chains (with an increment of two chains) were run for each model, and the posterior mean of a SNP effect was calculated as the average of all saved posterior samples from all the chains running for that model. In sequential computing, a SNP effect was calculated as the average of all the saved posterior samples from the single chain. The results showed that the correlations of estimated SNP effects, such as in scenario 2, were greater than 0.80 between parallel MCMC and sequential MCMC and even greater than 0.90 between tunable burn-in MCMC and fixed burn-in MCMC (Fig. 3). For example, the correlations of estimated SNP effects were from 0.901 to 0.908 between FBM-BayesA and TBM-BayesA. Similar results were obtained in scenarios 1 and 3 (data not presented). Thus, the observed differences in estimated SNP effects for the same method were attributable to Monte Carlo errors, and they were essentially made trivial as soon as the MCMC chains converged to the expected stationary distributions.
Correlations of the estimated SNP effects in simulation scenario 2. Seq-BayesA = sequential BayesA; Seq-BayesCπ = sequential BayesCπ; FBM-BayesA = fixed burn-in, multiple-chain BayesA; TBM-BayesA = tunable burn-in, multiple-chain BayesA; FBM-BayesCπ = fixed burn-in, multiple-chain BayesCπ; TBM-BayesCπ = tunable burn-in, multiple-chain BayesCπ
GEBVs and GPAs
The GEBV of each animal was calculated as the sum of all SNP effects for that animal. The results showed that the GEBVs obtained from various forms of calculating the same genomic prediction models were almost identical and presented correlations that were greater than or close to unity (> 0.99). The GEBVs obtained from different models were also highly correlated but with some noticeable differences. These differences did not result from the use of varied computational strategies but reflected the use of different statistical models (and the underlying model assumptions).
Similarly, the GPAs were also analogous between different computational forms of the same model, regardless of the number of MCMC chains and types of burn-in mechanisms, although noticeable differences were observed in the GPAs between different statistical models (Tables 1, 2 and 3). In simulation scenario 1, for example, the GPA was approximately 0.523 for the various forms of BayesA with either fixed or automatically tunable burn-in periods running between 1 and 18 chains. However, the GPA obtained by various computing forms of BayesCπ varied only slightly between 0.630 and 0.632 (Table 1). Similar trends were observed in simulation scenarios 2 and 3. As a comparison, GPAs were also obtained using GBLUP; however, these values were mostly lower than the GPAs obtained from the various computing forms of BayesA and BayesCπ, although BayesA models in scenario 1 were exceptions (Table 1). Again, the slight differences in GPA among the various computing forms of the same genomic prediction models were caused by Monte Carlo errors in the simulation of posterior samples of SNP effects, whereas the differences in GPA between the various statistical models were not necessarily computational but were attributable to intrinsic differences between the methods per se.
Table 1 Genomic predictive accuracies obtained using FBM-BayesA, TBM-BayesA, FBM-BayesCπ, TBM-BayesCπ, and GBLUP in Scenario 1
Application in Chinese Simmental beef cattle
Convergence diagnoses
Convergence diagnoses were conducted for residual variances as well as for a randomly selected number of SNP effects. Generally, the MCMC simulations of residual variances converged quickly, which primarily occurred within the first 1000 iterations, and the posterior modes were highly comparable among the various computing forms of the same genomic prediction models. Nevertheless, certain differences were observed in the posterior modes of the residual variances between different models (e.g., between TBM-BayesA and TBM-BayesCπ). These results were consistent with our observations in the simulation studies, with trivial differences in the estimated SNP effects and GEBVs (and hence GPAs) among various computing forms of the same statistical models caused by Monte Carlo errors and intrinsic differences observed between different statistical models. With the estimated residual variances of the four traits used as examples, the difference was the lowest for SLW, and the posterior mode of residual variance approached 0.16 with TBM-BayesA (Figure S1 in Additional file 1) and 0.18 with BayesCπ (Figure S2 in Additional file 1); the difference was largest for CBFT, and the posterior mode of the residual variance approached 0.19 (TBM-BayesA; Additional file 1: Figure S3) and 0.30 (TBM-BayesCπ; Additional file 1: Figure S4). Trace plots of the MCMC chains of residual variance for the remaining two traits (CW and ADG) are also provided in Additional file 1: Figures S5–S8. Trace plots of MCMC chains of selected SNP effects on SLW obtained by TMB-BayesA and TMB-BayesCπ are shown in Additional file 1: Figures S12 and S13, respectively. Evidently, these MCMC chains also all converged quickly within the first 1000 iterations.
Estimated heritabilities and GPAs
The estimated heritabilities for the four traits were within the range of previous reports [25, 26]. The differences may have been caused by differences in the genomic architectures of distinct breeds. In this Chinese Simmental beef population, CBFT had a smaller heritability compared with the other three traits. Consequently, the GPAs for CBFT were also lower (0.100 ~ 0.106) than those for the other three traits (0.202 ~ 0.271) (Table 4).
Table 4 Heritability estimates and predictive accuracies of four quantitative traits in a Chinese Simmental cattle population
Nevertheless, the RGPAs were comparable among the four traits because this criterion assessed the GPAs relative to the square root of the heritability of each trait, with the latter reflecting the selection accuracy based on phenotypes and pedigree information. The RGPAs were also roughly comparable among the three models but with slight differences: TBM-BayesA and TBM-BayesCπ had a greater RGPA for CBFT, SLW, and CW but a lower RGPA for ADG; and TBM-BayesCπ had the greatest average RGPA for the four traits calculated across the three computational-statistical models. Again, these differences might not be based on computational differences but could be intrinsic to the differences in the data and statistical models.
Parallel computing of Bayesian genomic prediction models: tunable burn-in versus fixed burn-in
Bayesian regression models are of high value for genomic prediction, although the complexity of computing of these models can be intensive [14], which is increasingly becoming the bottleneck in practical genomic selection programs. The challenges are found primarily in two aspects. First, genotype and phenotype data have been accumulating drastically in the past 10 years, and these "big data" are not managed efficiently because traditional data processing methods and tools are inadequate. Hence, high-performance computing (e.g., via parallel programming and computing strategies) is required to increase the computational efficiency and generate high computational throughputs for genomic selection. Nevertheless, the computing of Bayesian genomic prediction models is not parallelizable by the nature of the iterative algorithms, which poses the second and most likely greater challenge. Although Bayesian genomic prediction models can be calculated in parallel by running multiple MCMC chains of the same model, the speedup of computing heavily depends on the length of the burn-in period, which cannot be parallelized. Often, the length of burn-ins is set arbitrarily and thus can be too short or too long. When too short, the Markov chains are not converged, and the generated samples do not represent those drawn from the targeted posterior distributions. When too long, running a longer burn-in period after the convergence of Markov chains does not improve the accuracies of the posterior estimates of the model parameters [27, 28] but does consume more time than necessary. In the present study, we proposed a tunable, multiple-chain MCMC algorithm that is capable of automatically tuning an appropriate length of burn-ins, depending only on the actual status of MCMC convergence of the Bayesian statistical model. Our results showed that this tunable burn-in algorithm was effective and able to reduce the computing time remarkably compared with its counterpart with fixed burn-ins. In the present study, we used Gelman and Rubin's convergence diagnostic method [15] to monitor the convergence state of the multiple-chain MCMC method. The shrink factor was calculated using posterior samples of the residual variance and a selected number of SNP effects from multiple chains of each model. When the multiple chains reached convergence, the burn-in period was terminated immediately, and the posterior samples generated afterward were collected and used for statistical inferences of the model parameters of interest, and they were subject to the frequency of thinning.
In the discussion that follows, we explain numerically how tunable burn-in MCMC could achieve greater speedups than fixed burn-in MCMC. Consider again the formula of the speedup ratio as in (10). Let Nburn-in = m and NT = 5 m; in FBM-BayesA and FBM-BayesCπ, the speedup ratios are S(5) = 2.78 and S(20) = 4.16 when 5 and 20 cores are available for computing, respectively. Nevertheless, the maximal speedup ratio is \( \underset{\mathrm{K}\to \infty }{\lim}\mathrm{S}\left(\mathrm{K}\right)=5 \), regardless of how many cores are available for computing. Using our strategy, Nburn-in was adjusted to reduce unnecessary burn-in iterations based on the convergence diagnosis. If Nburn-in was shortened to m/2 by the convergence diagnosis, then the corresponding speedup ratios were doubled to S(5) = 3.84 and S(20) = 7.14. Furthermore, if Nburn-in was reduced to m/4, then the speedup ratios were quadrupled to S(5) = 4.76 and S(20) = 11.11. In the simulation studies, our results showed that TBM-BayesA (or TBM-BayesCπ) tended to a burn-in length that was half or even one-fourth as long as that of FBM-BayesA (or FBM-BayesCπ). This result demonstrated that the automatic tunable burn-in MCMC method effectively shortened the length of burn-ins compared with the fixed-burn-in MCMC; therefore, the tunable burn-in MCMC led to a greater speedup and greater parallel computing efficiency.
Our results showed that the use of tunable burn-in MCMC did not change the GPA as long as the MCMC chains converged. This conclusion is also supported by previous research [27, 28]. The GEBV is a critical concept in genomic prediction, and it was calculated as the sum of all SNP effects for each animal. Our results showed that the estimated SNP effects were highly analogous between the tunable burn-in MCMC and fixed burn-in MCMC used to calculate the same statistical model; however, the former had greater speedups than the latter. The differences were essentially caused by Monte Carlo errors. Because of such small differences in the estimated SNP effects among various forms of calculating the same genomic prediction model, the differences in the calculated GEBV for individual animals could mostly be ignored.
Genomic predictions of Chinese Simmental beef cattle: a real application
The tunable burn-in MCMC and fixed burn-in MCMC methods were applied to genomic predictions of four quantitative traits in a Chinese Simmental beef population, and the GPAs obtained were also compared with those of the GBLUP. In the Chinese Simmental data set, the RGPAs were roughly comparable among the three models GBLUP, TBM-BayesA and TBM-BayesCπ because the RGPA assesses the GPA relative to the heritability of each trait in the present study. Without adjusting for the heritability differences of these traits, our results indicated that the GPA was higher for a trait with higher heritability than for traits with lower heritability. Nevertheless, both the GPA and RGPA showed noticeable differences among the different genomic prediction models, and these differences were not necessarily based on computational factors but could be intrinsic to the varied assumptions of these models. In reality, GPAs can vary with a number of factors, such as the size of the reference populations, heritability of the trait, density of the SNP panels, level of LD, and the statistical models used for prediction [2].
In this real application, our results supported that this tunable burn-in MCMC was effective and outperformed the fixed burn-in MCMC regarding speedup and parallel computing efficiency. The GPAs were comparable among the various computing forms of BayesA and BayesCπ, and these two models had greater GPAs for three of the four traits than GBLUP.
Toward greater parallel computing efficiencies: what to consider?
Finally, several strategies deserve mention, such as adaptive MCMC algorithms [29, 30] and tempering [31, 32], which can further improve the convergence of multiple-chain MCMC. These strategies were not investigated in the present study but are worthy of investigation in future studies for further increasing parallel computing efficiency for genomic prediction. For example, Metropolis-coupled MCMC is a method that is related to simulated tempering and tempered transitions [31, 32] and simultaneously runs several different Markov chains governed by different (yet related) Markov chain transition probabilities. Occasionally, the algorithm "swaps" values between different chains, with the probability governed by the Metropolis algorithm to preserve the stationarity of the target distribution. Possibly, these swaps can speed up the convergence of the algorithm substantially. Craiu et al. proposed an ensemble of MCMC chains using the covariance of samples across all chains to adopt the proposed covariance for a set of Metropolis-Hastings chains [33]. Somewhat different from these multiple-chain methods, which use a synchronous exchange of samples to expedite convergence, Murray et al. mixed in an additional independent proposal that represents some hitherto best estimate or summary of the posterior and cooperative adapting across chains [34]. Therefore, a globally best estimate of the posterior is generated at any given step, and then this estimate is mixed as a remote component with whatever local proposal that a chain has adopted. This method does not preclude adaptive treatment or tempering of that local proposal but also permits a heterogeneous blend of remote proposals, thus allowing the ensemble of chains to mix well.
An automatically tunable burn-in MCMC method for calculating Bayesian genomic prediction models was proposed and manifested using BayesA and BayesCπ models. Our results from the simulation study showed a better speedup in computing with tunable burn-in MCMC than with fixed burn-in MCMC. However, the estimated SNPs and GPAs were highly comparable regardless of the various forms of parallel computing when the same Bayesian genomic prediction model was used. In a Chinese Simmental beef population, the average GPAs for four quantitative traits obtained by the tunable burn-in BayesA and BayesCπ models were better than those obtained by the GBLUP, and although these differences may have been caused by computational factors, they might also have been attributable to intrinsic differences in the statistical model assumptions. The proposed tunable burn-in strategy for running parallel (i.e., multiple-chain) MCMC can lead to dramatically increased computational efficiency and is applicable to the computing of all complex Bayesian models, either sequentially or in parallel.
ADG:
Average daily gain
CBFT:
Carcass back fat thickness
CW:
Carcass weight
FBM:
Fixed burn-in, multiple-chain
GBLUP:
Genomic Best Linear Unbiased Prediction
GEBVs:
Genomic estimated breeding values
Genomic prediction accuracy
MCMC:
Markov chain Monte Carlo
MPI:
Message Passing Interface
QTL:
Quantitative trait locus
REML:
Restricted maximum likelihood
RGPA:
Relative genomic prediction accuracy
SLW:
Strip loin weight
TBM:
Tunable burn-in, multiple-chain
Meuwissen TH, Hayes BJ, Goddard ME. Prediction of total genetic value using genome-wide dense marker maps. Genetics. 2001;157(4):1819–29.
Chen L, Vinsky M, Li C. Accuracy of predicting genomic breeding values for carcass merit traits in Angus and Charolais beef cattle. Anim Genet. 2015;46(1):55–9.
Moser G, Tier B, Crump RE, Khatkar MS, Raadsma HW. A comparison of five methods to predict genomic breeding values of dairy bulls from genome-wide SNP markers. Genet Sel Evol. 2009;41:56.
Neves HH, Carvalheiro R, O'Brien AM, Utsunomiya YT, do Carmo AS, Schenkel FS, Solkner J, McEwan JC, Van Tassell CP, Cole JB, et al. Accuracy of genomic predictions in Bos indicus (Nellore) cattle. Genet Sel Evol. 2014;46:17.
de Campos CF, Lopes MS, e Silva FF, Veroneze R, Knol EF, Lopes PS, Guimarães SE. Genomic selection for boar taint compounds and carcass traits in a commercial pig population. Livest Sci. 2015;174:10–7.
Duchemin SI, Colombani C, Legarra A, Baloche G, Larroque H, Astruc JM, Barillet F, Robert-Granie C, Manfredi E. Genomic selection in the French Lacaune dairy sheep breed. J Dairy Sci. 2012;95(5):2723–33.
VanRaden PM. Efficient methods to compute genomic predictions. J Dairy Sci. 2008;91(11):4414–23.
Habier D, Fernando RL, Kizilkaya K, Garrick DJ. Extension of the bayesian alphabet for genomic selection. BMC Bioinformatics. 2011;12:186.
Brondum RF, Su G, Lund MS, Bowman PJ, Goddard ME, Hayes BJ. Genome position specific priors for genomic prediction. BMC Genomics. 2012;13:543.
Yang W, Tempelman RJ. A Bayesian antedependence model for whole genome prediction. Genetics. 2012;190(4):1491–501.
Zhu B, Zhu M, Jiang J, Niu H, Wang Y, Wu Y, Xu L, Chen Y, Zhang L, Gao X, et al. The impact of variable degrees of freedom and scale parameters in Bayesian methods for genomic prediction in Chinese Simmental beef cattle. PLoS One. 2016;11(5):e0154118.
Wu XL, Beissinger TM, Bauck S, Woodward B, Rosa GJ, Weigel KA, Gatti Nde L, Gianola D. A primer on high-throughput computing for genomic selection. Front Genet. 2011;2:4.
Bernstein AJ. Analysis of programs for parallel processing. IEEE Trans Electron Comput. 1966;EC-15(5):757–63.
Wu XL, Sun C, Beissinger TM, Rosa GJ, Weigel KA, Gatti Nde L, Gianola D. Parallel Markov chain Monte Carlo - bridging the gap to high-performance Bayesian computation in animal breeding and genetics. Genet Sel Evol. 2012;44:29.
Gelman A, Rubin D. Inference from iterative simulation using multiple sequences. Stat Sci. 1992;7(4):457–511.
Zhang Z, Li X, Ding X, Li J, Zhang Q. GPOPSIM: a simulation tool for whole-genome genetic data. BMC Genet. 2015;16:10.
Service AM. Institutional meat purchase specifications. In: Fresh beef series 100; 2010. http://www.ams.usda.gov/AMSv1.0/getfile?dDocName=STELDEV3003281.
Purcell S, Neale B, Todd-Brown K, Thomas L, Ferreira MA, Bender D, Maller J, Sklar P, de Bakker PI, Daly MJ, et al. PLINK: a tool set for whole-genome association and population-based linkage analyses. Am J Hum Genet. 2007;81(3):559–75.
Gianola D. Priors in whole-genome regression: the bayesian alphabet returns. Genetics. 2013;194(3):573–96.
de Los Campos G, Gianola D, Rosa GJ. Reproducing kernel Hilbert spaces regression: a general framework for genetic evaluation. J Anim Sci. 2009;87(6):1883–7.
Gelman A, Carlin J, Stern H, Rubin D. Bayesian Data Analysis. New York: Chapman and Hall; 2004.
Amdahl GM. Validity of the single processor approach to achieving large scale computing capabilities. In: Proceeding AFIPS '67 (Spring) Proceedings of the April 18-20, 1967, spring joint computer conference; 1967. p. 483–5.
Chapter Google Scholar
Rolf MM, Garrick DJ, Fountain T, Ramey HR, Weaber RL, Decker JE, Pollak EJ, Schnabel RD, Taylor JF. Comparison of Bayesian models to estimate direct genomic values in multi-breed commercial beef cattle. Genet Sel Evol. 2015;47:23.
Saatchi M, McClure MC, McKay SD, Rolf MM, Kim J, Decker JE, Taxis TM, Chapple RH, Ramey HR, Northcutt SL, et al. Accuracies of genomic breeding values in American Angus beef cattle using K-means clustering for cross-validation. Genet Sel Evol. 2011;43:40.
Mehrban H, Lee DH, Moradi MH, IlCho C, Naserkheil M, Ibanez-Escriche N. Predictive performance of genomic selection methods for carcass traits in Hanwoo beef cattle: impacts of the genetic architecture. Genet Sel Evol. 2017;49(1):1.
Fernandes Junior GA, Rosa GJ, Valente BD, Carvalheiro R, Baldi F, Garcia DA, Gordo DG, Espigolan R, Takada L, Tonussi RL, et al. Genomic prediction of breeding values for carcass traits in Nellore cattle. Genet Sel Evol. 2016;48:7.
Rosenthal J. Parallel computing and Monte Carlo algorithms. Far East J Theor Stat. 2000;4:207–36.
Guo J, Jain R, Yang P, Fan R, Kwoh CK, Zheng J. Reliable and fast estimation of recombination rates by convergence diagnosis and parallel Markov chain Monte Carlo. IEEE/ACM Trans Comput Biol Bioinform. 2013;11:63-72.
Gilks W, Robertson G, Sahu S. Adaptive Markov chain Monte Carlo through regeneration. J Am Stat Assoc. 1998;93:1045–54.
Andrieu C, Thoms J. A tutorial on adaptive MCMC. Stat Comput. 2008;18:343–73.
Marinari E, Parisi G. Simulated tempering: a new Monte Carlo scheme. Europhys Lett. 1992;19:451–8.
Neal R. Sampling from multimodal distributions using tempered transitions. Stat Comput. 1996;6:353–66.
Craiu R, Rosenthal J, Yang C. Learn from thy neighbor: parallel-chain and regional adaptive MCMC. J Am Stat Assoc. 2009;104:1454–66.
Murray L. Distributed Markov chain Monte Carlo. In: Proceedings of neural information processing systems workshop on learning on cores, clusters and clouds: 2010. Mt. Currie South. http://lccc.eecs.berkeley.edu/Papers/dmcmc_short.pdf.
We are grateful to the editor and the two anonymous reviewers whose comments have greatly helped improve this paper.
This work was supported by the National Natural Science Foundation of China (31372294, 31201782 and 31672384), National High Technology Research and Development Program of China (863 Program 2013AA102505-4), the Agricultural Science and Technology Innovation Program (ASTIP-IAS03, ASTIP-IAS-TS-16 and ASTIP-IAS-TS-9), Cattle Breeding Innovative Research Team of Chinese Academy of Agricultural Sciences (cxgc-ias-03), Beijing Natural Science Foundation (6154032).
We confirm that all data used to generate our findings are publicly available without restriction. Data are available from the Dryad Digital Repository: http://datadryad.org/resource/doi:10.5061/dryad.4qc06.
Laboratory of Molecular Biology and Bovine Breeding, Institute of Animal Science, Chinese Academy of Agricultural Sciences, Yuanmingyuan West Road 2#, Haidian District, Beijing, 100193, China
Peng Guo, Bo Zhu, Hong Niu, Zezhao Wang, Yonghu Liang, Yan Chen, Lupei Zhang, Xue Gao, Huijiang Gao, Lingyang Xu & Junya Li
College of Computer and Information Engineering, Tianjin Agricultural University, Tianjin, China
Peng Guo
Animal Science and Technology College, Beijing University of Agriculture, Beijing, China
Hemin Ni & Yong Guo
Livestock and Range Research Laboratory, ARS, USDA, Miles City, MT, USA
El Hamidi A. Hay
Biostatistics and Bioinformatics, GeneSeek (A Neogen company), Lincoln, NE, 68504, USA
Xiaolin Wu
Department of Animal Sciences, University of Wisconsin, Madison, WI, 53706, USA
Bo Zhu
Hong Niu
Zezhao Wang
Yonghu Liang
Lupei Zhang
Hemin Ni
Yong Guo
Xue Gao
Huijiang Gao
Lingyang Xu
Junya Li
JYL, HJG, HMN and YG conceived and designed the study. PG and BZ performed statistical analyses and programming in C language. PG, LYX, XLW and EH wrote the paper. PG, ZZW and YHL participated in data analyses. HN and XG carried out quantification of phenotypic data and SNP data preprocessing. YC and LPZ participated in the design of the study and contributed to acquisition of data. All authors read, commented and approved the final manuscript.
Correspondence to Lingyang Xu or Junya Li.
Animal experiments were approved by the Science Research Department of the Institute of Animal Sciences, Chinese Academy of Agricultural Sciences (CAAS) (Beijing, China). Human participants, data or tissues were not used.
Trace plots of posterior samples of residual variance from TBM-BayesA (16 chains) for SW of the first 2000 iterations. Figure S2. Trace plots of posterior samples of residual variance from TBM-BayesCπ (16 chains) for SW of the first 2000 iterations. Figure S3. Trace plots of posterior samples of residual variance from TBM-BayesA (16 chains) for CBFT of the first 1800 iterations. Figure S4. Trace plots of posterior samples of residual variance from TBM-BayesCπ (16 chains) for CBFT of the first 2500 iterations. Figure S5. Trace plots of posterior samples of residual variance from TBM-BayesA (16 chains) for CW of the first 2000 iterations. Figure S6. Trace plots of posterior samples of residual variance from TBM-BayesCπ (16 chains) for CW of the first 2000 iterations. Figure S7. Trace plots of posterior samples of residual variance from TBM-BayesA (16 chains) for ADG of the first 1400 iterations. Figure S8. Trace plots of posterior samples of residual variance from TBM-BayesCπ (16 chains) for ADG of the first 1400 iterations. Figure S9. Plot of running time in scenario1.FBM-BayesA: Fixed burn-in multiple chains parallel BayesA, TBM-BayesA: Tunable burn-in multiple chains parallel BayesA, FBM-BayesCπ: Fixed burn-in multiple chains parallel BayesCπ, TBM-BayesCπ: Tunable burn-in multiple chains parallel BayesCπ. One chain is equivalent to sequential genomic selection. Figure S10. Plot of running time in scenario 2. Figure S11. Plot of running time in scenario 3 Figure S12. Convergence of TBM-BayesA for SW. Iteration: 50,000, initial Burn-in:10,000, threshold:0.001. Figure S13. Convergence of TBM-BayesCπ for SW. Iteration: 50,000, initial Burn-in:10,000, threshold:0.001. Table S1. Running time using five genomic prediction approaches in scenario 1. Table S2. Running time using five genomic prediction approaches in scenario 2. Table S3. Running time using five genomic prediction approaches in scenario 3. (ZIP 405 kb)
Guo, P., Zhu, B., Niu, H. et al. Fast genomic prediction of breeding values using parallel Markov chain Monte Carlo with convergence diagnosis. BMC Bioinformatics 19, 3 (2018). https://doi.org/10.1186/s12859-017-2003-3
Received: 20 June 2017
Accepted: 18 December 2017
Convergence diagnosis
Genomic prediction
Tunable burn-in
Results and data | CommonCrawl |
\begin{document}
\title{A regularized entropy-based moment method for kinetic equations \footnote{ This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (\texttt{http://energy.gov/downloads/doe-public-access-plan}). } }
\date{\today} \author{Graham W. Alldredge \thanks{Department of Mathematics and Computer Science, Freie Universit\"at Berlin, 14195 Berlin, Germany, (\texttt{[email protected]}) } \and Martin Frank \thanks{Department of Mathematics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany, (\texttt{[email protected]}).} \and Cory D. Hauck \thanks{Computational Mathematics Group, Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA, (\texttt{[email protected]}).} }
\maketitle
\section{Introduction}
Kinetic equations model systems consisting of a large number of particles that interact with each other or with a background medium. They arise in a wide variety of applications, including rarefied gas dynamics \cite{Cercignani}, neutron transport \cite{Lewis-Miller-1984}, radiative transport \cite{mihalas1999foundations}, and semiconductors \cite{markowich1990}. For charge-neutral particles, these equations evolve the \textit{kinetic density function} $f \colon [0, \infty) \times X \times V \to [0, \infty)$ according to \begin{equation}\label{eq:kinetic}
\partial_t f(t, x, v) + v \cdot \nabla_x f(t, x, v) = \cC(f(t, x, \cdot))(v). \end{equation} The function $f$ depends on time $t \in [0, \infty)$, position $x \in X \subseteq \R^d$, and a velocity variable $v \in V \subseteq \R^d$. The operator $\cC$ introduces the effects of particle collisions; at each $x$ and $t$, it is an integral operator in $v$. In order to be well-posed, \eqref{eq:kinetic} must be accompanied by appropriate initial and boundary conditions.
In this work, we present a new entropy-based moment method for the velocity discretization of \eqref{eq:kinetic}. The method relies on a regularization of the optimization problem that defines the closure in the moment equations. The key advantage of our approach is that, unlike the standard entropy-based method, the solution of the moment equations in the regularized setting is not required to take on realizable values. Roughly speaking, a vector is said to be realizable if it is the velocity moment of a scalar-valued kinetic density function that takes values in a prescribed range. Typically this range is the set of nonnegative values, but in some cases, an upper bound is also enforced. In practical applications, it is advantageous to remove the requirement of realizability because it has proven to be difficult to design numerical methods, particularly high-order ones, that can maintain it.
Before introducing the regularized method, in \secref{background} we provide the necessary background on moment methods, particularly with the entropy-based approach. In \secref{structure}, we introduce the new method and show that it retains many, though not all, of the attractive structural properties of the original approach. We then show in \secref{acc} that the new method can be used to generate accurate numerical simulations of standard entropy-based moment equations, thereby bypassing the need to design a realizable solver for them. In \secref{numerics}, we demonstrate the accuracy of such simulations using the method of manufactured solutions and a benchmark problem.
\section{Background} \label{sec:background}
In this section, we briefly review the formalism for entropy-based moment methods. The key topics are: structural properties of the kinetic equations, the general moment approach, the entropy-based closure, and the issue of realizability. Throughout the discussion and for the remainder of the paper, we rely on bracket notation for velocity integration: for any $g \in L^1(V)$, \begin{align} \Vint{g} := \int_V g(v) \intdv. \end{align}
\subsection{Structure of the kinetic equation}
The structure of the kinetic equation \eqref{eq:kinetic} plays a definitive role in the design of moment methods (and numerical methods in general). This structure is induced by properties of the collision operator $\cC$ and the advection operator $\cA = \partial_t + v \cdot \nabla_x$. We highlight the basic structural elements below, which are satisfied in many situations.
\begin{enumerate}[(i)]
\item \emph{Invariant range}: There exists a set $B \subseteq [0, \infty)$,
consistent with the physical bounds on $f$, such that
$\range(f(t,\cdot,\cdot)) \subseteq B$ whenever
$\range(f(0,\cdot,\cdot)) \subseteq B$.
In general, $f$ is expected to be nonnegative becaues it is a density; for
particles satisfying Fermi-Dirac statistics, it should also be bounded from
above.
\item \emph{Conservation}: There exist functions $\phi \colon V \to \R$,
called \textit{collision invariants}, such that
\begin{align}\label{eq:invariants}
\Vint{\phi\, \cC(g)} = 0, \qquad \text{for all } g \in \text{Dom}(\cC).
\end{align}
We denote the linear span of all collision invariants by $\bbE$.
When combined with the kinetic equation, \eqref{eq:invariants}
implies local conservation laws of the form:
\begin{align}\label{eq:conservation}
\partial_t \Vint{\phi f} + \nabla_x \cdot \Vint{v \phi f} = 0.
\end{align}
\item \emph{Hyperbolicity}: For each fixed $v$, the advection operator
is hyperbolic over $(t,x) \in [0, \infty) \times X$.
\item \emph{Entropy dissipation}:
Let $D \subseteq \R$.
There exists a twice continuously differentiable, strictly convex
function $\eta \colon D \to \R$, called the \textit{kinetic entropy density},
such that
\begin{align}\label{eq:entropy-diss}
\Vint{\eta'(g) \cC(g)} \le 0 \qquad \text{for all }
g\in \text{Dom}(\cC) \text{ such that } \text{Range}(g) \subseteq D.
\end{align}
Combined with the kinetic equation, \eqref{eq:entropy-diss}
implies the local entropy dissipation law
\begin{align}\label{eq:entropy-diss-kin-eq}
\partial_t \Vint{\eta(f)} + \nabla_x \cdot \Vint{v \eta(f)} \le 0.
\end{align}
Often $D$ is consistent with physical bounds on the range of
$f$, i.e., $B = D$.
(See Table \ref{tab:ex-ent} below.)
\item \emph{H-Theorem}: Equilibria are characterized by any of the three
equivalent statements:
\begin{align}
{\rm (a)}~\Vint{\eta'(g) \cC(g)} = 0;
\qquad
{\rm (b)}~\cC(g) = 0;
\qquad
{\rm (c)}~\eta'(g) &\in \bbE.
\end{align}
\item \emph{Galilean invariance}:
There exist Galilean transformations $\cG_{O,w}$ defined by
\begin{align}\label{eq:Galilean}
{(\cG_{O,w} g)(t, x, v) := g(t, O(x - t w), O(v - w))},
\end{align}
where $O \in \operatorname{SO}(d)$ is a $d \times d$ rotation matrix and
$w \in V$ is a translation in velocity,
that commute with the advection and collision operators. i.e.,
\begin{alignat}{3}\label{eq:invariant}
\cA(\cG_{O,w} g) &= \cG_{O,w} \cA(g) \quad && \text{for all } g \in
\operatorname{Dom}(\cA) \\
\cC(\cG_{O,w} g) &= \cG_{O,w} \cC(g) \quad && \text{for all } g \in
\operatorname{Dom}(\cC).
\end{alignat}
As a consequence, the transformed particle
density $\cG_{O,w} f$ also satisfies the kinetic equation \eqref{eq:kinetic}.
\end{enumerate}
\begin{table}
\renewcommand{2}{2}
\centering
\small
\begin{tabular}{c|c|c|c|c|c}
Entropy type & $\eta(z)$ & $\dom(\eta)$ & $\eta'(z)$ & $\eta_{\ast}(y)$ & $\eta_{\ast}'(y)$ \\ \hline
Maxwell--Boltzmann & $z \log(z) - z$ & $[0, \infty)$ & $\log(z)$ & $e^y$ & $e^y$ \\
{Bose--Einstein} & $(1 + z) \log (1 + z) - z \log(z)$ & $[0, \infty)$ & $\log \left(\dfrac{z}{1+z}\right)$ & $-\log(1-e^y)$ & $\dfrac{1}{e^y - 1}$ \\
{Fermi--Dirac} & $(1 - z) \log (1 - z) + z \log(z)$ & $[0, 1]$ & $\log \left(\dfrac{z}{1-z}\right)$ & $\log(1+e^y)$ & $\dfrac{1}{e^y + 1}$ \\
{Quadratic} & $\frac12 z^2$ & $\bbR$ & $z$ & $\frac12 y^2$ & $y$
\end{tabular}
\caption{Common entropy densities $\eta$.}
\label{tab:ex-ent} \end{table}
\subsection{Entropy-based moment methods}
Moment methods encapsulate the velocity-dependence of $f$ in a vector-valued function \begin{equation} \bu(t, x)=(u_0(t, x), u_1(t, x), \dots , u_{n - 1}(t, x))) \end{equation} that approximates the velocity averages of $f$ with respect to the vector of basis functions \begin{equation}
\bm(v) = (m_0(v), m_1(v), \dots , m_{n - 1}(v))); \end{equation} that is, $u_i(t, x) \simeq \vint{m_i f(t, x, \cdot)}$ for all $i \in \{ 0, 1, \dots n - 1 \}$.
The components of $\bm$ are typically polynomials and include the collision invariants defined in \eqref{eq:conservation}.
The entropy-based moment method is a nonlinear Galerkin discretization in the velocity variable. It has the form \begin{equation}\label{eq:proj}
\partial_t \Vint{\bm F_{\bu}}
+ \nabla_x\cdot \Vint{v \bm F_{\bu})} = \Vint{\bm
\cC(F_{\bu})}, \end{equation} where $F_{\bu} = F_{\bu(t, x)}(v)$ is an ansatz that approximates the distribution function $f$ and is consistent with the moment vector $\bu$. Unlike the trial function in a traditional (linear) Galerkin method, $F_{\bu}$ is not assumed to be a linear combination of the basis functions in $\bm$. Instead, in an entropy-based moment method, the ansatz is given by the solution of a constrained optimization problem whose objective function is defined via the kinetic entropy density $\eta$ introduced in the previous subsection. Let \begin{align}
\cH(g) := \Vint{\eta(g)}. \end{align} Then the defining optimization problem is \begin{equation}\label{eq:primal}
\minimize_{g \in \bbF(V)} \: \cH(g)
\qquad \st \: \Vint{\bm g} = \bv, \end{equation} where $\bv \in \bbR^n$ and \begin{align} \label{eq:range}
\bbF(V) = \{ g \in L^1(V) : \text{Range}(g) \subseteq D
\}. \end{align}
\begin{remark}
Throughout the paper, we reserve the symbol $\bu = \bu(t,x)$ for the solution
of a partial differential equation like \eqref{eq:proj}
(e.g., \eqref{eq:mn} and \eqref{eq:reg-mn} below).
For a generic moment vector, independent of space and time, we use $\bv$.
Thus we also use $\bv$ to label the argument of various moment-dependent
functions below.
This deviates somewhat from standard notation but makes many of the computations more precise. \end{remark}
The solution to \eqref{eq:primal}, if it exists, \footnote{In general, it may not. See \cite{Jun00,Hauck-Levermore-Tits-2008,caflisch1986equilibrium, Borwein-Lewis-1991}. } takes the form $G_{\alphahatv}$, where \begin{align}\label{eq:ansatz} G_{\bsalpha} := \eta'_*(\bsalpha \cdot \bm), \end{align} $\hat{\bsalpha} \colon \bbR^n \to \bbR^n$ maps $\bv$ to the solution of the dual problem \begin{equation}\label{eq:dual} \hat{\bsalpha}(\bv) = \argmax_{\bsalpha \in \R^n}
\left\{ \bsalpha \cdot \bv - \Vint{\eta_{\ast}(\bsalpha \cdot \bm)}\right\} \end{equation} and $\eta_{\ast}$ is the Legendre dual \footnote{ See, e.g., \cite[\S 3.3.2.]{evans2010partial} or \cite[\S 3.3]{boyd2004convex}, where what we call the Legendre dual is called the conjugate function. } of $\eta$ (see Table \ref{tab:ex-ent}). In this case, first-order necessary conditions for \eqref{eq:dual} imply that \begin{align}\label{eq:dual-grad} \Vint{\bm G_{\alphahatv}} = \bv. \end{align} Hence the function $\hat \bv \colon \bbR^n \to \bbR^n$ defined by \begin{align} \label{eq:vhat}
\vhat(\bsalpha) := \Vint{\bm G_{\bsalpha}} \end{align} is the inverse of $\hat{\bsalpha}$, and the moment equations in \eqref {eq:proj} take the form \begin{align}\label{eq:mn} \partial_t \bu + \nabla_x \cdot \bff(\bu) &= \br(\bu), \end{align} where the flux function $\bff$ and relaxation term $\br$ are given by \begin{align}\label{eq:f-and-r} \bff(\bv) := \Vint{v \bm G_{\alphahatv}} \qquand \br(\bv) := \Vint{\bm \cC(G_{\alphahatv})}. \end{align}
The appeal of the entropy-based approach to closure is that \eqref{eq:mn} inherits many of the structural properties of the kinetic equation \eqref{eq:kinetic}. We summarize these here:
\begin{enumerate}[(i)]
\item \emph{Invariant range}: The natural bounds on the kinetic
equation lead to a realizability condition on the solution $\bu$.
A vector $\bv \in \R^n$ is called
\emph{realizable (with respect to $\eta$ and $\bm$)} if there exists a
$g \in \bbF(V)$ such that $\Vint{\bm g} = \bv$.
The set of all realizable moment vectors is denoted by $\cR$.
One expects formally that the solution $\bu$ of \eqref{eq:mn} satisfies
$\bu(t, x) \in \cR$ for all $(t,x) \in [0,\infty) \times X$.
If $D = B$, then this means the solution is always consistent with the
bounds on the kinetic density function $f$.
\item \emph{Conservation}: If $m_i \in \bbE$, then
$r_i(\bv) = \vint{m_i \cC(G_{\alphahatv})} = 0$ and the $i$-th component of
\eqref{eq:mn} is
\begin{align}\label{eq:conservation-mn}
\partial_t u_i + \nabla_x \cdot \Vint{v m_i G_{\alphahatu}} = 0.
\end{align}
\item \emph{Hyperbolicity \cite{Lev96}}:
When expressed in terms of $\bsbeta(t,x):=\hat{\bsalpha}(\bu(t,x))$, \eqref{eq:mn}
takes the form of a symmetric hyperbolic balance law
\begin{align}\label{eq:mn_symhyp}
h_{\ast}''(\bsbeta) \partial_t \bsbeta
+ j_{\ast}''(\bsbeta) \cdot \nabla_x \bsbeta &= \br(\hat \bv(\bsbeta)),
\end{align}
where
\begin{equation}\label{eq:entropy-entropy-flux-potentials}
h_{\ast}(\bsalpha) = \vint{\eta_{\ast}(\bsalpha \cdot \bm)}
\quand
j_{\ast}(\bsalpha) = \vint{v \eta_{\ast}(\bsalpha \cdot \bm)}
\end{equation}
are the entropy and entropy-flux potentials, respectively.
Thus \eqref{eq:mn} is a symmetrizable hyperbolic system.
\item \emph{Entropy dissipation \cite{Lev96}}:
Assume that \eqref{eq:primal} has a solution for ever vector $\bv$ in the
image of $\bu$, and let
\begin{equation}\label{eq:entropy-entropy-flux}
h(\bv) := \Vint{\eta(G_{\alphahatv})}
\quand j(\bv) := \Vint{v \eta(G_{\alphahatv})}
\end{equation}
be the entropy and entropy flux, respectively.
Using the hyperbolic structure of the left-hand side, one can show that $h$
and $j$ are compatible with $\bff$, namely that
\begin{align}\label{eq:eeflux}
j'(\bv) = h'(\bv) \cdot \frac{\partial \bff}{\partial \bv}.
\end{align}
Furthermore, we have
$h'(\bv) \cdot \br(\bv) = \alphahat(\bv) \cdot \br(\bv) \le 0$ (where the
inequality follows immediately from \eqref{eq:entropy-diss}), and thus
the moment equations \eqref{eq:mn} inherit a semi-discrete version of the
entropy-dissipation law in \eqref{eq:entropy-diss-kin-eq}:
\begin{align}\label{eq:entropy-diss-mn}
\partial_t h(\bu) + \nabla_x \cdot j(\bu) = h'(\bv) \cdot \br(\bv) \le 0.
\end{align}
We note that the existence of the entropy and entropy flux pair satisfying
\eqref{eq:eeflux} is equivalent to symmetric hyperbolicity as in
\eqref{eq:mn_symhyp}.
The dissipation of the right hand side as stated in
\eqref{eq:entropy-diss-mn}, however, does not translate automatically.
\item \emph{H-Theorem \cite{Lev96}}: The H-Theorem for the kinetic equation
can be used to show the equivalency of the following statements for
\eqref{eq:mn}:
\begin{align}
{\rm (a)}~\alphahat(\bv) \cdot \br(\bv) = 0;
\qquad
{\rm (b)}~\br(\bv) = 0;
\qquad
{\rm (c)}~\alphahat(\bv) \cdot \bm \in \bbE.
\end{align}
\item \emph{Galilean invariance \cite{JunUnt02}}: If the kinetic equation is
invariant under a transformation $\cG_{O,w}$, defined in \eqref{eq:Galilean},
and if $\operatorname{span}\{m_0,\dots,m_{n-1}\}$ is invariant under
$\cG_{O,w}$, then system \eqref{eq:mn} is also invariant under the inherited
transformation
\begin{equation}
\label{eq:mn_Galinv}
\cT_{O,w} \bu := \vint{ \bm \cG_{O,w} F_{\bu}}.
\end{equation}
If we let $T_{O,w}$ be the $n \times n$ matrix satisfying
$\bm(O(v - w)) = T_{O,w} \bm(v)$,
\footnote{
The subscripts of $T$ are given in the reverse of the order they're
applied to be consistent with their order in matrix multiplication---i.e.,
$T_{O, w} = T_{O, 0}T_{I, w}$, where $I$ is the $d \times d$
identity matrix---so that the inverse $(T_{O, w})^{-1}$ is given by
$T_{-w, O^{-1}} = T_{-w, I}T_{0, O^{-1}}$.
}
then we can give $\cT_{O,w}$ explicitly as
\begin{align}\label{eq:T}
(\cT_{O,w} \bu)(t, x) = T^{-1}_{O,w}\bu(t, O(x - tw)).
\end{align}
Then the Galilean invariance of \eqref{eq:mn} is reflected by the identity
\begin{align}\label{eq:mult-identity-gal}
\hat{\bsalpha}(T^{-1}_{O,w} \bv) = T^T_{O,w} \alphahat(\bv)
\qquad \text{(equivalently }
T^{-1}_{O,w} \vhat(\bsalpha) = \hat \bv(T^T_{O,w} \bsalpha)
\text{),}
\end{align}
(this can be derived using the first-order necessary conditions
\eqref{eq:dual-grad})
as well as the commutability of $\cT_{O,w}$ with the operator
\begin{align}
(\partial_t + \nabla_x \cdot \bff - \br)\bu
:= \partial_t \bu + \nabla_x \cdot \bff(\bu) - \br(\bu),
\end{align}
i.e.,
\begin{align}\label{eq:mn-gal-inv}
(\partial_t + \nabla_x \cdot \bff - \br)(\cT_{O,w} \bu)
= \cT_{O,w}((\partial_t + \nabla_x \cdot \bff - \br)\bu).
\end{align}
\end{enumerate}
\subsection{Realizability and relaxation of the entropy minimization problem}
The realizability condition introduced in the previous subsection can cause serious complications for numerical methods. While it may seem advantageous (for physical reasons) to require that the solution in \eqref{eq:mn} be everywhere realizable, it can unfortunately cause the closure procedure to fail rather unforgivingly in numerical simulations. Specifically, if in the course of a simulation a numerical algorithm generates a vector $\bv \nin \cR$, then the primal problem \eqref{eq:primal} will be infeasible (i.e., the constraint set will be empty) and $\bff(\bv)$ and $\br(\bv)$ will not be well-defined. Discretization errors can easily cause the numerical solution to take on values outside of the realizable set, and in such cases, the simulation will crash.
Although several algorithms have been designed to maintain the realizability of numerical solutions, each has significant limitations. For example, two kinetic schemes have been proposed: the scheme in \cite{AllHau12} is limited to second-order, while the formally higher-order method from \cite{SchneiderAlldredge2016} relies on a limiter not rigorously shown to preserve accuracy. Both kinetic schemes have the disadvantage of requiring spatial reconstructions for every node of the quadrature in the $v$ variable, \footnote{ In practice, the velocity integrals cannot be done analytically, so a quadrature is required. } and accuracy requirements dictate that there be significantly more nodes than moment components \cite{AllHau12}. Discontinuous-Galerkin schemes have also been considered, but the scheme in \cite{Olbrant2012} is limited to first-order moment vectors and one spatial dimension, while the limiter used in \cite{AlldredgeSchneider2014} can destroy high-order accuracy and relies on an expensive approximate description of $\cR$. What's more, a deeper problem obstructs the creation of realizability-preserving methods: the concrete description of $\cR$ in general remains an open problem \cite{LasserreBook}. Finally, all second- or higher-order methods so far have been limited to explicit time integration, which cannot handle the stiffness of the equations near fluid-dynamical regimes \cite{jin1999ap,mcclarren2008semi,dimarco2013asymptotic} (although the recently developed algorithm \cite{hu2017asymptotic} may be applicable).
One way to overcome the feasibility issue in \eqref{eq:primal} is to relax the constraints. This is the approach taken in \cite{Decarreau-Hilhorst-Lemarichal-Navaza-1992}, where the authors analyzed \eqref{eq:primal} in the context of an inverse problem. Specifically, a function approximation was generated from partially observed experimental data that was given by the moment constraints. Because measurement errors may generate nonrealizeable moments, the authors relaxed the equality constraints in \eqref{eq:primal} to arrive at the unconstrained problem \begin{align}\label{eq:tik-primal} \minimize_{g \in \bbF(V)} \:\cH_\gamma(g; \bv), \end{align} with the modified objective function \begin{align} \label{eq:H-gamma} \cH_\gamma(g; \bv) := \Vint{\eta(g)} + \frac1{2\gamma}
\left\| \Vint{\bm g} - \bv \right\|^2. \end{align}
Here $\gamma \in (0, \infty)$ is a parameter and $\|\cdot\|$ is the usual Euclidean norm on $\R^n$. Unlike the original primal problem \eqref{eq:primal}, the relaxed problem \eqref{eq:tik-primal} is feasible for \emph{any} $\bv \in \R^n$ (not just $\bv \in \cR$), so we expect that it will have a solution for most, indeed perhaps all, $\bv \in \R^n$.
Whenever a solution to \eqref{eq:tik-primal} exists, it has the same form as that of the original primal problem: \begin{align} \argmin_{g \in \bbF(V)} \left\{ \cH_\gamma(g; \bv) \right\} = G_{\alphahatgv}, \end{align} where $G_{\bsalpha}$ is defined in \eqref{eq:ansatz} and $\alphahat_\gamma(\bv)$ is the solution of the new dual problem: \begin{align}\label{eq:reg-mult} \alphahatg(\bv) := \argmax_{\bsalpha \in \R^n} \left\{ \bsalpha \cdot \bv - \Vint{\eta_*(\bsalpha \cdot \bm)}
- \frac\gamma2 \|\bsalpha\|^2 \right\}. \end{align} Thus the relaxation of the constraints in the primal corresponds to a Tikhonov regularization of the dual \cite{Decarreau-Hilhorst-Lemarichal-Navaza-1992}. For this reason, we refer to $\gamma$ as the regularization parameter. Indeed, the condition number of the Hessian of the dual objective in \eqref{eq:reg-mult} is bounded from above by $1 + \gamma^{-1} c $, where $c$ is the maximum eigenvalue of the Hessian of the original dual function \eqref{eq:dual}; this bound decreases as $\gamma$ increases. The regularization provided by $\gamma$ can be helpful for vectors $\bv \in \cR$ near the boundary of $\cR$, when the original dual problem \eqref{eq:dual} can be difficult to solve \cite{AllHau12}.
The price to pay for relaxing the constraints in \eqref{eq:primal} is the mismatch between $\vint{\bm G_{\alphahatgv}}$ and $\bv$; that is, unlike \eqref{eq:dual-grad}, $\vint{\bm G_{\alphahatgv}} \ne \bv$. However, because of measurement or simulation errors, $\bv$ is not known precisely in practice anyway; nor can the dual problem \eqref{eq:dual} be solved exactly. Hence if $\gamma$ is sufficiently small, then overall accuracy can be maintained. This statement can be quantified more precisely using the following definition and theorem. \begin{defn}
\label{defn:tau-optimal}
Let $\tau>0$.
Then
\begin{align}
\bG_\gamma^\tau(\bv) := \left\{ g^* \in \bbF(V) : \cH_\gamma(g^*; \bv) \leq
\inf_{g \in \bbF(V)} \left\{ \cH_\gamma(g; \bv) \right\} + \tau \right\}
\end{align}
is the set of all \emph{$\tau$-optimal} density functions. \end{defn}
\begin{thm}[\!\!\cite{engl1989convergence,engl1993convergence}]
\label{thm:acc}
Let $\bv^\delta$ be a moment vector satisfying
$\|\bv - \bv^\delta\| \le \delta$ for some $\bv \in \cR$ and
$g \in \bG_\gamma^\tau(\bv^\delta)$.
If $\gamma \sim \delta$ {\rm (}i.e., $\gamma = \cO(\delta)$ and
$\delta = \cO(\gamma)${\rm )} and $\tau = \cO(\delta)$, then
\begin{align}\label{eq:reg-err}
\left\| \Vint{\bm g} - \bv \right\| = \cO(\delta).
\end{align} \end{thm}
\noindent Theorem \ref{thm:acc} provides a strategy for choosing $\gamma$ (and $\tau$) so that the regularized problem can be used to solve \eqref{eq:mn} without losing the order of accuracy.
\section{Regularized entropy-based closures}\label{sec:structure}
In this section, we propose a new set of closures, based on the regularization \eqref{eq:tik-primal}. We replace \eqref{eq:mn} by the system of regularized entropy-based moment equations \begin{align}\label{eq:reg-mn} \partial_t \bu + \nabla_x \cdot \bff_\gamma(\bu) &= \br_\gamma(\bu), \end{align} where (cf. \eqref{eq:f-and-r}) \begin{align}\label{eq:f-and-r-gam} \bff_\gamma(\bv) := \Vint{v \bm G_{\alphahatgv}} \qquand \br_\gamma(\bv) := \Vint{\bm \cC(G_{\alphahatgv})} \end{align} are defined even when $\bv \nin \cR$. The system \eqref{eq:reg-mn} can then used to approximate the original system \eqref{eq:mn} numerically without having to enforce realizability conditions explicitly.
In the remainder of the section, we examine the structural properties of the system of regularized moment equations \eqref{eq:reg-mn}. For most of this section (particularly in Sections \ref{sec:mult-mom} and \ref{sec:rmn-structure}) we assume the primal problem \eqref{eq:tik-primal} has a minimizer. While this assumption is necessary to rigorously justify many of the formal calculations that follow in this section, there are important cases for which it does not hold. These exceptions are subject of \secref{junk-preview} and the Appendix. Under this assumption we use Legendre duality to establish the formal relationship between a moment vector $\bv$ and its corresponding multiplier vector $\alphahatg(\bv)$. Then, as in \cite{Lev96}, this relationship allows us to investigate the structure of the regularized system \eqref{eq:reg-mn}.
\subsection{Regularized moment-multiplier relationship} \label{sec:mult-mom}
Many of the structural properties of \eqref{eq:mn} rely on duality relations, which we establish here for the regularized case. We first define the convex function ${h_\gamma : \bbR^n \to \R}$ by \begin{align}\label{eq:hg}
h_\gamma(\bv) := \inf_{g \in \bbF(V)} \left\{ \cH(g) + \frac1{2\gamma}
\| \Vint{\bm g} - \bv \|^2 \right\}. \end{align} First-order optimality conditions for the dual \eqref{eq:reg-mult} imply that \begin{align}\label{eq:reg-dual-grad} \bv = \Vint{\bm G_{\alphahatgv}} + \gamma \alphahatg(\bv). \end{align} From \eqref{eq:reg-dual-grad}, we conclude that \begin{align}\label{eq:vhatg} \vhatg(\bsalpha) := \hat \bv(\bsalpha)+ \gamma \bsalpha, \end{align} where $\hat \bv$ is defined in \eqref{eq:vhat}, is the inverse of $\alphahat_\gamma$. When $\gamma= 0$, we recover the original moment map: \begin{align}
\vhatg(\bsalpha)\big|_{\gamma = 0}
= \hat \bv(\bsalpha). \end{align} Furthermore, under the assumption that the infimum in \eqref{eq:hg} is attained, substitution of \eqref{eq:reg-dual-grad} into \eqref{eq:hg} gives \begin{equation} h_\gamma(\bv)
= \cH(G_{\alphahatgv})
+ \frac\gamma 2 \| \alphahatg(\bv) \|^2
= h(\hat \bv(\alphahatg(\bv))) + \frac\gamma 2 \| \alphahatg(\bv) \|^2 \end{equation} Thus when $\bv$ is realizable, $h_\gamma \to h$ (cf. \eqref{eq:entropy-entropy-flux}) as $\gamma \to 0$. \footnote{ This limit follows directly since (i) $h$ and $\hat \bv$ are continuous functions and (ii) $\alphahat_\gamma$ is continuous with respect to $\gamma$ for $\gamma \in [0, \infty)$ when $\bv \in \cR$. Property (ii) follows from the same continuity of $\vhat_\gamma$, the inverse of $\alphahat_\gamma$. }
Duality relations established in \cite{Borwein-Lewis-1991} imply that $h_\gamma(\bv)$ is equal the maximum of the regularized dual problem \eqref{eq:reg-mult}. Therefore $h_\gamma$ is, by definition, the Legendre dual of the convex function ${(h_\gamma)_\ast \colon \R^n \to \R}$, defined by \begin{align}\label{eq:hgd}
(h_\gamma)_\ast(\bsalpha) := \Vint{\eta_{\ast}(\bsalpha \cdot \bm)}
+ \frac\gamma2 \|\bsalpha\|^2. \end{align} Differentiating this formula gives $(h_\gamma)_\ast'(\bsalpha) = \vhatg(\bsalpha)$. According to the theory of Legendre duality $((h_\gamma)_\ast')^{-1} = h_\gamma'$, so from \eqref{eq:reg-dual-grad} we have \begin{align}
h_\gamma'(\bv) = \alphahatg(\bv). \end{align}
The Hessian matrices are now straightforwardly computed: \begin{align}
(h_\gamma)_\ast''(\bsalpha) &= \frac{\partial \vhat_\gamma}{\partial \bsalpha}
= \Vint{\bm \bm \cdot \eta_{\ast}''(\bsalpha \cdot \bm)} + \gamma I
=: H_\gamma(\bsalpha), \quad \text{and} \label{eq:hess-g} \\
h_\gamma''(\bv) &= \frac{\partial \alphahat_\gamma}{\partial \bv}
= H^{-1}_\gamma(\alphahatg(\bv)), \label{eq:hess-inv-g} \end{align} where $I$ is the $n \times n$ identity matrix.
\subsection{Structural properties of the regularized equations} \label{sec:rmn-structure}
The duality relations from the last section now allow us to check whether the regularized moment system inherits the structural properties of the underlying kinetic equation.
\begin{enumerate}[(i)]
\item \emph{Invariant range}: While the regularized equations are defined
even for nonrealizable moment vectors, the underlying ansatz $G_{\alphahatgv}$ used in
the flux and collision terms takes on the same range of values as the
original entropy ansatz.
\item \emph{Conservation}: If $m_i \in \bbE$, then
$r_{\gamma,i}(\bv) = \vint{m_i \cC(G_{\alphahatgv})} = 0$ and the $i$-th component of
\eqref{eq:mn} is
\begin{align}\label{eq:conservation-rmn}
\partial_t u_i + \nabla_x \cdot \Vint{v m_i G_{\alphahatgu}} = 0.
\end{align}
\item \emph{Hyperbolicity}:
When expressed in terms of $\bsbeta(t, x) := \alphahat_\gamma(\bu(t, x))$,
\eqref{eq:reg-mn} takes the form of a symmetric hyperbolic balance law
\begin{align}\label{eq:rmn_symhyp}
(h_\gamma)_\ast''(\bsbeta) \partial_t \bsbeta
+ j_{\ast}''(\bsbeta) \cdot \nabla_x \bsbeta &= \br_\gamma(\bu),
\end{align}
where $j_{\ast}$ is the original entropy-flux potential (see
\eqref{eq:entropy-entropy-flux-potentials}).
Thus \eqref{eq:reg-mn} is also a symmetrizable hyperbolic system.
\item \emph{Entropy dissipation}:
With the original entropy flux in mind, we define
\begin{align}
j_\gamma(\bv) := \Vint{v \eta(G_{\alphahatgv})}.
\end{align}
Then $h_\gamma$ and $j_\gamma$ are compatible with $\bff_\gamma$, i.e.,
\begin{align}
j'(\bv) = h'(\bv) \cdot \frac{\partial \bff}{\partial \bv},
\end{align}
and we also have
$h_\gamma'(\bv) \cdot \br(\bv) = \alphahatg(\bv) \cdot \br_\gamma(\bv) \le 0$ from
\eqref{eq:entropy-diss}.
Thus the regularized moment equations \eqref{eq:reg-mn} have the
entropy-dissipation law
\begin{align}\label{eq:entropy-diss-rmn}
\partial_t h_\gamma(\bu) + \nabla_x \cdot j_\gamma(\bu) = h_\gamma(\bu) \cdot \alphahatg(\bu)
\le 0.
\end{align}
\item \emph{H-Theorem}: Just as with the original equations, the
following statements are equivalent:
\begin{align}
{\rm (a)}~\alphahatg(\bv) \cdot \br_\gamma(\bv) = 0;
\qquad
{\rm (b)}~\br_\gamma(\bv) = 0;
\qquad
{\rm (c)}~\alphahatg(\bv) \cdot \bm \in \bbE.
\end{align}
However, the moment vectors $\bv$ satisfying these conditions may not be
the same as those of the original system, i.e.,
$\br_\gamma^{-1}(0) \ne \br^{-1}(0)$.
\item \emph{Galilean invariance}:
In order to take advantage of the Galilean invariance of the original
equations, we use the identity
$\bu = \hat \bv(\alphahatg(\bu)) + \gamma \alphahatg(\bu)$ and write the regularized
equations as
\begin{align}\label{eq:reg-mn-0-lhs}
0 = \gamma \partial_t \alphahatg(\bu)
+ (\partial_t + \nabla_x \cdot \bff - \br)(\hat \bv(\alphahatg(\bu))).
\end{align}
It turns out that we must consider rotations and velocity translations
separately.
Let's first consider the rotation $\cT_{O,0}$.
Note that if the matrix $T_{O,0}$ (recall \eqref{eq:T}) is orthogonal, we have
$\alphahat_\gamma(T^{-1}_{O,0}\bv) = T^{-1}_{O,0} \alphahatg(\bv)$ (from the
first-order necessary conditions \eqref{eq:reg-dual-grad}) and thus
$\partial_t \alphahat_\gamma(\cT_{O,0}\bu) = \cT_{O,0} \partial_t \alphahatg(\bu)$.
When we combine this with \eqref{eq:mult-identity-gal} and
\eqref{eq:mn-gal-inv}, we have
\begin{subequations}
\begin{align}
0 &= \gamma \partial_t \alphahat_\gamma(\cT_{O,0} \bu)
+ (\partial_t + \nabla_x \cdot \bff - \br)
(\hat \bv(\alphahat_\gamma(\cT_{O,0} \bu))) \\
&= \cT_{O,0}(\gamma \partial_t \alphahatg(\bu)
+ (\partial_t + \nabla_x \cdot \bff - \br)(\hat \bv(\alphahatg(\bu)))),
\end{align}
\end{subequations}
which shows that the regularized moment system is rotationally invariant.
One can show that the matrix $T_{O,0}$ is indeed orthogonal if there exists a
radially symmetric weight function $\omega = \omega(v)$ so that
$\vint{\bm \bm^T \omega} = I$, i.e., so that the basis functions are
orthonormal with respect to $\omega$.
\footnote{
We show this using $\vint{\bm \bm^T \omega} = I$ and computing
\begin{gather}\label{eq:T-orth-rot}
T^{-1}_{O,0} = T^{-1}_{O,0} \Vint{\bm \bm^T \omega}
= \Vint{\bm(O^{-1}v) \bm^T \omega}
= \Vint{\bm(v) (\bm(O v))^T \omega(|O v|)} \nonumber \\
= \Vint{\bm (T_{O,0}\bm)^T \omega(|v|)}
= T_{O,0}^T.
\end{gather}
(Here we use $|\cdot|$ for the Euclidean norm on $\R^d$ and reserve
$\|\cdot\|$ for the Euclidean norm for moment vectors.)
This orthonormality assumption holds, e.g., for the normalized spherical
harmonics on the unit sphere.
}
However, for a velocity translation $\cT_{I,w}$ we have
\begin{align}
\partial_t \alphahat_\gamma(\cT_{I,w}\bu)
= \frac{\partial \alphahat_\gamma}{\partial \bv}\left(T_{I,w}\left((\partial_t \bu
+ w \cdot \nabla_x \bu)\big|_{(t, x - tw)} \right) \right).
\end{align}
Even if $T_{I,w}$ is orthogonal, the additional $w \cdot \nabla_x \bu$ term
is neither part of $\cT_{I,w} \partial_t \alphahatg(\bu)$ nor is it canceled by
anything else in the right-hand side of \eqref{eq:reg-mn-0-lhs}.
Thus the regularized equations fail to be translation invariant. \end{enumerate}
\subsection{Degenerate densities} \label{sec:junk-preview}
One of the major drawbacks of entropy-based moment closures is that there exist realizable moment vectors $\bv$ for which the original primal problem \eqref{eq:primal} has no solution. For these \emph{degenerate densities}, many of the structural properties of the entropy-based formulation are lost. The geometry of these densities was investigated in detail for the Maxwell-Boltzmann entropy \cite{Junk-1998}, with $V = \R$ and ${\bm(v) = (1, v, v^2, v^3, v^4)}$; extensions to multiple dimensions and more general polynomial basis functions can be found in \cite{Jun00,schneider2004entropic,Hauck-Levermore-Tits-2008}.
Unfortunately, the regularization does not fix the problem of degeneracy. Indeed, there are also moment vectors $\bv$ for which the regularized primal problem \eqref{eq:tik-primal} does not achieve its minimum. For the original primal, the fundamental issue is that for a fixed $\bv$ the constraint set $\{g \in \bbF(V) : \vint{\bm g} = \bv \}$ is not closed when $V$ is unbounded, in particular when $V = \R^d$, because the map $g \mapsto \vint{\bm g}$ is not continuous. This issue carries over to the regularized problem, since this discontinuous map appears in the objective function $\cH_\gamma$, so that $\cH_\gamma$ is not lower-semicontinuous.
Although not exactly the same, the set of degenerate moment vectors for the regularized problem can be characterized in the same fashion as the degenerate moment vectors for the original problem. As an illustrative example, consider the Maxwell--Boltzmann entropy with
$V = \R^d$ and $m_{n - 1}(v) = |v|^N$, where $m_{n - 1}$ is the only component of $\bm$ with degree greater than or equal to $N$. (This includes the example mentioned above from \cite{Junk-1998}). Let $\cA$ be the set of multiplier vectors such that $G_{\bsalpha} \in L^1(V)$. Then the main result of \cite{Jun00} can be extended to the following:
\begin{prop}\label{prop:junk-line-g} If $\bv$ can be written as \begin{align}\label{eq:junk-form-g}
\bv = \vhat_\gamma(\overline{\bsalpha}) + \begin{pmatrix}
0 \\ \vdots \\ 0 \\ \delta
\end{pmatrix}, \end{align} for some $\overline{\bsalpha} \in \cA \cap \partial \cA$ and $\delta \in (0, \infty)$, then $\bv$ is a degenerate density for the regularized problem, i.e., $\cH_\gamma(\cdot ; \bv)$ does not achieve its minimum. \end{prop}
The results from \cite{Junk-1998,Jun00} are recovered when $\gamma = 0$. We postpone a proof and further discussion to the Appendix.
\subsection{Examples}\label{sec:ex}
Now we take a look at how the regularization affects the most well-known instances of the entropy-based moment method. The simplest case is the P$_N$ equations of radiation transport. For the case of bounded velocity domains, we also consider the M$_1$ equations. For the case of unbounded velocity domains, we study the Euler equations. For the latter two, we only consider the one-dimensional cases for simplicity. To make some computations feasible, we define a partially regularized version of \eqref{eq:primal}:
\begin{subequations}
\label{eq:p-tik-primal}
\begin{align}
\minimize_{g \in \bbF(V)} \quad & \Vint{\eta(g)}
+ \frac1{2\gamma}
\sum_{i = m + 1}^{n - 1} \left( \Vint{m_i g} - v_i \right)^2, \\
\st \quad & \Vint{m_i g}
= v_i, \quad i \in \{ 0, 1, \dots , m \},
\label{eq:u0-constraint}
\end{align}
\end{subequations}
with dual problem
\begin{align}
\maximize_{\bsalpha \in \R^n}\:
\bsalpha \cdot \bv - \Vint{\eta_*(\bsalpha \cdot \bm)}
- \frac\gamma2 \sum_{i = m + 1}^{n - 1} \alpha_i^2.
\end{align}
For the existence of a solution to the primal and dual problems, the
subvector $(v_0, v_1, \dots , v_m)$ must of course satisfy realizability
conditions.
\subsubsection{Regularized P$_N$ equations} \label{sec:reg-pn}
Consider as velocity domain the unit sphere $V = S^2$, and the spherical harmonics as basis functions. The P$_N$ equations are an entropy-based closure with the entropy density $\eta(g) = \frac12 g^2$. This function is equal to its Legendre dual, $\eta = \eta_*$.
The unregularized multipliers satisfy $\Vint{\bm\bm^T} \alphahat(\bv) =\bv$. Since the spherical harmonics are an orthonormal basis, i.e., $\Vint{ \bm \bm^T} = I$, the ansatz is $G_{\alphahat(\bv)} = \bm \cdot \bv$. The regularized multipliers satisfy \begin{align}
(\gamma I+\Vint{\bm\bm^T}) \alphahatg(\bv) =\bv, \end{align} so $G_{\alphahatg(\bv)} = \frac{1}{1+\gamma}\bm\cdot \bv$. This leads to \begin{align}
\bff_\gamma(\bv)=\frac{1}{1+\gamma} \bff(\bv). \end{align} Hence the regularization acts as a filter \cite{mcclarren2010simulating,mcclarren2010robust} that damps the flux of the original equations.
\subsubsection{Regularized M$_1$ equations}
Here the velocity domain is $V = [-1, 1]$ (i.e., the one-dimensional
slab-geometry setup) and $\bm(v) = (1, v)$. The realizable set is given by ${\cR = \{(v_0, v_1) \in \R^2 : |v_1| < v_0 \}}$. We consider the Maxwell--Boltzmann entropy. \footnote{ The M$_1$ method is also often applied to the gray equations for photon transport using the Bose--Einstein entropy. These equations have the advantage that the flux $\bff$ can be given analytically \cite{Dubroca-Feugas-1999}. Unfortunately, this property is (as far as we can tell) destroyed by the introduction of $\gamma$, so we do not discuss this particular example in further detail. }
While no analytical expression can be obtained for the multipliers, one can eliminate the zero-th order multiplier $\hat \alpha_0(v_0, v_1)$ so that the optimal first-order multiplier $\hat \alpha_1(v_0, v_1)$ satisfies the single equation \cite{Min78,BruHol01} \begin{align}\label{eq:alpha1}
\frac{v_1}{v_0} = \coth(\hat \alpha_1) - \frac1{\hat \alpha_1}, \end{align} where for clarity of exposition we suppress the dependence of the optimal multipliers on the moment components. The map $\alpha_1 \mapsto \coth(\alpha_1) - 1 / \alpha_1$ is indeed a smooth bijection between $\R$ and $(-1, 1)$, which is consistent with the existence and uniqueness of the multipliers for $(v_0, v_1) \in \cR$.
We have been unable to decouple the equations for $\hat \alpha_{\gamma, 0}$ and $\hat \alpha_{\gamma, 1}$ when regularization is applied to both moment components. However, when we only regularize the first-order moment, i.e., when we solve \begin{subequations} \begin{align}
v_0 &= \Vint{\exp(\hat \alpha_{\gamma, 0} + \hat \alpha_{\gamma, 1} \mu)}
= \frac2{\hat \alpha_{\gamma, 1}} \exp(\hat \alpha_{\gamma, 0})
\sinh(\hat \alpha_{\gamma, 1}), \\
v_1 &= \Vint{\mu \exp(\hat \alpha_{\gamma, 0} + \hat \alpha_{\gamma, 1} \mu)}
+ \gamma \hat \alpha_{\gamma, 1}1 \nonumber \\
&= \frac2{\hat \alpha_{\gamma, 1}} \exp(\hat \alpha_{\gamma, 0})
\left( \cosh(\hat \alpha_{\gamma, 1})
+ \frac{\sinh(\hat \alpha_{\gamma, 1})}{\hat \alpha_{\gamma, 1}} \right)
+ \gamma \hat \alpha_{\gamma, 1}, \end{align} \end{subequations} then we can again isolate $\hat \alpha_{\gamma, 1}$ to get \begin{align}\label{eq:alpha1gam}
\frac{v_1}{v_0} = \coth(\hat \alpha_{\gamma, 1})
- \frac1{\hat \alpha_{\gamma, 1}} + \frac\gamma{v_0} \hat \alpha_{\gamma, 1}. \end{align} The map $\alpha_1 \mapsto \coth(\alpha_1) - 1 / \alpha_1 + \gamma \alpha_1 / v_0$ is a smooth bijection from $\R$ to $\R$---under the assumption $v_0 > 0$ (which is necessary for the existence of a minimizer in the partially regularized case). Thus the partially regularized problem has a solution for $(v_0, v_1) \in \{ (v_0, v_1) : v_0 > 0 \} \supset \cR$.
\figref{m1alpha1} plots the maps \eqref{eq:alpha1} and \eqref{eq:alpha1gam}.
\begin{figure}
\caption{Comparison of regularized versus unregularized closures.}
\label{fig:comparison}
\label{fig:m1alpha1}
\label{fig:u2g}
\end{figure}
\subsubsection{Regularized Euler equations}
With $V = \R^3$ and $\bm(v) = \{1, v, |v|^2\}$, the original entropy-based moment equation gives the compressible Euler equations \cite{Lev96}. In one-dimension, i.e., $V = \R$ and ${\bm(v) = (1, v, v^2)}$. The realizable set is $\cR = \{ (v_0, v_1, v_2) \in \R^3 : v_0 v_2 > v_1^2 \}$. The moment and optimal multiplier components satisfy \begin{subequations} \begin{align}
v_0 &= \sqrt{-\frac\pi{\hat \alpha_2}} \exp \left( \hat \alpha_0
- \frac{\hat \alpha_1^2}{4 \hat \alpha_2} \right), \label{eq:a0euler} \\
v_1 &= \sqrt{-\frac\pi{\hat \alpha_2}} \exp \left( \hat \alpha_0
- \frac{\hat \alpha_1^2}{4 \hat \alpha_2} \right)
\frac{-\hat \alpha_1}{2\hat \alpha_2}, \label{eq:a1euler} \\
v_2 &= \sqrt{-\frac\pi{\hat \alpha_2}} \exp \left( \hat \alpha_0
- \frac{\hat \alpha_1^2}{4 \hat \alpha_2} \right)
\left( \frac{\hat \alpha_1^2}{4\hat \alpha_2^2}
- \frac1{2 \hat \alpha_2}\right), \label{eq:a2euler} \end{align} \end{subequations} and one can readily invert these equations.
Again, we have been unable to solve these equations analytically when all moment components are regularized. We were only able to find an analytical solution for the case when we only relax the equality constraint on $v_2$. In this case \eqref{eq:a2euler} becomes \begin{align}
v_2 = \sqrt{-\frac\pi{\hat \alpha_{\gamma, 2}}}
\exp \left( \hat \alpha_{\gamma, 0}
- \frac{\hat \alpha_{\gamma, 1}^2}{4 \hat \alpha_{\gamma, 2}} \right)
\left( \frac{\hat \alpha_{\gamma, 1}^2}{4\hat \alpha_{\gamma, 2}^2}
- \frac1{2 \hat \alpha_{\gamma, 2}} \right) + \gamma \hat \alpha_{\gamma, 2}, \end{align} and with appropriate substitutions of \eqref{eq:a0euler}--\eqref{eq:a1euler} (with the multipliers now labeled with $\gamma$), we get \begin{align}
\hat \alpha_{\gamma, 2} = \frac{v_2 v_0 - v_1^2
- \sqrt{(v_2 v_0 - v_1^2)^2 + 2\gamma v_0^3}}{2\gamma v_0}. \end{align} Then the regularized second-order moment becomes: \begin{align} \label{eq:reg-euler}
\hat v_2(\alphahatg(\bv)) = v_0 \left(
\frac{v_1^2}{v_0^2} + \frac{v_2 v_0 - v_1^2
+ \sqrt{(v_2 v_0 - v_1^2)^2 + 2\gamma v_0^3}}{2v_0^2} \right). \end{align} \figref{u2g} shows this relationship, and it behaves as expected: when $\gamma$ approaches zero, it approaches the identity map for positive $v_2$; otherwise, for nonphysical negative values of $v_2$, the map returns small positive values which get even smaller as $\gamma$ goes to zero.
\section{Accuracy of the closure} \label{sec:acc}
While the properties in Section \ref{sec:rmn-structure} provide basic structure of the regularized entropy-based moment equations \eqref{eq:reg-mn}, \thmref{acc} hints at an attractive possible application: the use the regularized system to accurately solve the \emph{original} moment system \eqref{eq:mn}. To explore this idea further, we note that $\bff_\gamma(\bv) = \bff (\hat \bv(\alphahatg(\bv)) $, where $\hat \bv(\alphahatg(\bv)) \in \cR$ is the regularization of $\bv \in \bbR^n$. (Indeed, the term $\vint{\bm g}$ in \eqref{eq:reg-err} is an approximate evaluation of $\hat \bv \circ \alphahat_\gamma$ at $\bv^\delta$.)
Thus under the assumption that the Jacobian of $\bff$ is bounded, the moment mismatch $\hat \bv(\alphahatg(\bv)) - \bv$ can be used to estimate $\bff_\gamma(\bv) - \bff(\bv)$. If the $\cO(\delta)$-accuracy in \thmref{acc} holds uniformly for all $\bv \in \cR$, then we can see $\bff_\gamma$ as way to approximately, but accurately evaluate $\bff$. The collision term $\br_\gamma$ can be considered similarly.
\subsection{Accuracy of the moment regularization map}
With the help of the relationships in \secref{mult-mom}, we can now analyze the moment regularization map $\hat \bv \circ \alphahat_\gamma$ directly.
\begin{thm}\label{thm:acc-new-no-tau} Let \begin{align}\label{eq:RM}
\bv \in \cR^M := \{ \bv : \|\alphahat(\bv)\| < M \}, \end{align} and let $\bv^\delta$ satisfy \begin{align}\label{eq:udelta}
\|\bv^\delta - \bv\| \le \delta. \end{align} Then \begin{align}\label{eq:proj-error}
\|\hat \bv(\alphahat_\gamma(\bv^\delta)) - \bv\| \le \delta + M\gamma. \end{align} \end{thm}
\begin{proof} Let $\widetilde{\bv} := \vhat_\gamma(\alphahat(\bv))$. Then $\bv = \hat \bv(\alphahat_\gamma(\widetilde{\bv}))$ so that \begin{subequations}\label{eq:udiff-int} \begin{align}
\left\|\hat \bv(\alphahat_\gamma(\bv^\delta)) - \bv \right\|
&= \left\|\hat \bv(\alphahat_\gamma(\bv^\delta)) - \hat \bv(\alphahat_\gamma(\widetilde{\bv}))
\right\| \\
&= \left\|\int_0^1 \left.
\frac{\partial (\hat \bv \circ \alphahat_\gamma)}{\partial \bv}
\right|_{\widetilde{\bv} + s (\bv^\delta - \widetilde{\bv})} (\bv^\delta - \widetilde{\bv}) \intds
\right\| \\
&= \left\|\int_0^1 \left. H(\alphahat_\gamma)(H(\alphahat_\gamma) + \gamma I)^{-1}
\right|_{\widetilde{\bv} + s (\bv^\delta - \widetilde{\bv})} (\bv^\delta - \widetilde{\bv}) \intds
\right\| \\
&\le \int_0^1 \left\| \left.
H(\alphahat_\gamma)(H(\alphahat_\gamma) + \gamma I)^{-1}
\right|_{\widetilde{\bv} + s (\bv^\delta - \widetilde{\bv})} \right\| \intds
\|\bv^\delta - \widetilde{\bv}\| \\
&= \int_0^1 \left.
\frac{\| H(\alphahat_\gamma) \|}{\| H(\alphahat_\gamma) \| + \gamma}
\right|_{\widetilde{\bv} + s (\bv^\delta - \widetilde{\bv})} \intds
\|\bv^\delta - \widetilde{\bv}\| \\
&\le \|\bv^\delta - \widetilde{\bv}\|. \end{align} \end{subequations} The inverse relationship between $\hat \bv$ and $\hat{\bsalpha}$, along with \eqref{eq:vhatg}, gives \begin{align} \label{eq:utilde-alpha} \bv^\delta - \widetilde{\bv}
= (\bv^\delta - \bv) + (\bv - \widetilde{\bv} )
&\stackrel{\hphantom{\eqref{eq:vhatg}}}{=} (\bv^\delta - \bv)
+ (\hat \bv(\alphahat(\bv)) - \vhat_\gamma(\alphahat(\bv))) \nonumber \\
&\stackrel{\eqref{eq:vhatg}}{=} (\bv^\delta - \bv) - \gamma \alphahat(\bv)). \end{align}
Altogether we have \begin{align}
\| \hat \bv(\alphahat_\gamma(\bv^\delta)) - \bv \|
\stackrel{\eqref{eq:udiff-int}}{\le} \|\bv^\delta - \widetilde{\bv}\|
\stackrel{\eqref{eq:utilde-alpha}}{\le}
\|\bv^\delta - \bv \| + \gamma \|\alphahat(\bv)\|
&\stackrel{\eqref{eq:udelta}}{\le} \delta + \gamma \|\alphahat(\bv)\| \nonumber \\
&\stackrel{\eqref{eq:RM}}{\le} \delta + M\gamma. \end{align} \ \end{proof}
As a result of Theorem \ref{thm:acc}, if $\gamma \le C \delta$ for some $C \in (0, \infty)$, then the moment regularization error
$\|\hat \bv(\alphahat_\gamma(\bv^\delta)) - \bv\|$ is $\cO(\delta)$. This result gives uniform accuracy over a large set of moment vectors, but only by bounding this set away from the boundary of the realizable set by controlling the norm of the associated multiplier vectors.
The closer the moment vectors get to the boundary of the realizable set, the larger the constant in the error bound becomes.
\subsection{Stopping criterion for the optimization} \label{sec:stopping}
As in \thmref{acc}, we would like to use an approximate solution to the optimization problem while keeping $\cO(\delta)$ accuracy. The error is quantified by the value of the primal objective function, but previous work with entropy-based moment methods has used the norm of the dual gradient for the stopping criterion (e.g., \cite{AllHau12}). The norm of the dual gradient is preferable because it is already computed by the optimizer (as part of the search-direction computation) and is easy to interpret. We will show that these two stopping criteria are closely related, but first we give our result with the gradient-based criterion.
\begin{thm} Assume $\bv \in \cR^M$; let $\bv^\delta$ satisfy
$\|\bv^\delta - \bv\| \le \delta$; and let $\bsalpha$ satisfy \begin{align}\label{eq:stopping}
\| \hat \bv(\bsalpha) - \bv^\delta + \gamma \bsalpha \| \le \tau. \end{align} Then \begin{align}\label{eq:proj-error-tau}
\|\hat \bv(\bsalpha) - \bv\| \le 2\delta + M\gamma + 2\tau. \end{align} \end{thm}
\begin{proof} We write \begin{equation} \hat \bv(\bsalpha) - \bv
= \hat \bv(\bsalpha) - \bv^\delta + \gamma \bsalpha
+ (\bv^\delta - \bv)
- \gamma \bsalpha. \end{equation} and apply the triangle inequality, using \eqref{eq:stopping}, to find \begin{equation} \label{eq:thm4-proof-triangle}
\| \hat \bv(\bsalpha) - \bv \|
\leq \tau + \delta + \gamma \| \bsalpha \|. \end{equation} To bound $\bsalpha$, let $\widetilde{\bv} := \vhat_\gamma(\bsalpha) = \hat \bv(\bsalpha) + \gamma \bsalpha$. Then \begin{align} \label{eq:vtilde_bound}
\| \widetilde{\bv} - \bv \| \le \| \widetilde{\bv} - \bv^\delta \| + \| \bv^\delta - \bv \|
\stackrel{\eqref{eq:stopping}}{\le} \tau + \delta \end{align} and, since $\bsalpha = \alphahat_\gamma(\widetilde{\bv})$, \begin{align}
\bsalpha = \alphahatg(\bv) +
\int_0^1 (H(\alphahat_\gamma(\bv + s (\widetilde{\bv} - \bv))) + \gamma I)^{-1}
(\widetilde{\bv} - \bv) \intds. \end{align}
Thus $\|\bsalpha\|$ is bounded by \begin{align}\label{eq:alphagdt-bnd-g}
\| \bsalpha \| \le
\| \alphahatg(\bv) \| + \frac1{\gamma} \| \widetilde{\bv} - \bv \|
\stackrel{\eqref{eq:vtilde_bound}}{\le} \| \alphahatg(\bv) \|
+ \frac{\tau+\delta}{\gamma}. \end{align}
The term $\| \alphahatg(\bv) \|$ can be further bounded because $\|\alphahatg(\bv)\|$ is a decreasing function of $\gamma$: \begin{subequations}\label{eq:mult-dec-with-gam} \begin{align}
\frac{\partial}{\partial \gamma}\|\alphahatg(\bv)\|^2
&= 2 \alphahatg(\bv) \cdot \frac{\partial}{\partial \gamma} \alphahatg(\bv) \\
&= 2 \alphahatg(\bv) \cdot \left( - \left.
\frac{\partial \vhat_\gamma}{\partial \bsalpha} \right|_{\alphahatg(\bv)}
\left. \frac{\partial \vhat_\gamma}{\partial \gamma} \right|_{\alphahatg(\bv)}
\right) \\
&= -2 \alphahatg(\bv) \cdot ((H_\gamma^{-1}(\alphahatg(\bv)) \alphahatg(\bv)) \\
&\le 0, \end{align} \end{subequations} The derivative of $\alphahat_\gamma$ with respect to $\gamma$ is computed by differentiating both sides of $\vhat_\gamma(\alphahatg(\bv)) = \bv$ with respect to $\gamma$, as in the implicit function theorem.
Since the continuity of $\alphahat_\gamma$ with respect to $\gamma$ at $\gamma = 0$ is a consequence of the same continuity of $\vhat_\gamma$, we can extend \eqref{eq:alphagdt-bnd-g} to \begin{align}\label{eq:alphagdt-bnd}
\| \bsalpha \| \le \| \alphahat(\bv) \| + \frac{\tau+\delta}{\gamma}
\le M + \frac{\tau+\delta}{\gamma}. \end{align} Setting the bound \eqref{eq:alphagdt-bnd} into \eqref{eq:thm4-proof-triangle} yields \eqref{eq:proj-error-tau}
\end{proof}
If $\gamma \le C\delta$ and $\tau \le C'\delta$, then the error
$\|\hat \bv(\bsalpha) - \bv\|$ of the approximate projection is $\cO(\delta)$. Thus we achieve a bound like that of \thmref{acc} but with constants independent of the specific moment vectors $\bv$ and $\bv^\delta$, as long as $\bv \in \cR^M$.
We now turn to the relationship between the stopping criterion \eqref{eq:stopping} and that of \cite{engl1989convergence,engl1993convergence}. In the latter, a distribution $g$ is called \emph{$\tau'$-optimal} if, for a given tolerance $\tau' \in (0, \infty)$, it satisfies \begin{align}\label{eq:tau-opt-tik}
\cH(g) + \frac1{2\gamma} \left\| \Vint{\bm g} - \bv \right\|^2
\le \hg(\bv) + \tau', \end{align} where $\hg(\bv)$ is the infimum of $\cH_\gamma(\cdot; \bv)$; see \eqref{eq:hg}. Because $\hg(\bv)$ is typically unknown, we cannot practically enforce \eqref{eq:tau-opt-tik} as is. However, we find a computable and stronger criterion by considering the duality gap \cite[\S 5.5.1]{boyd2004convex}. Indeed, for any $\bsalpha$ we have \eqref{eq:reg-dual-grad} \begin{equation}
\bsalpha \cdot \bv - \Vint{\eta_*(\bsalpha \cdot \bm)}
- \frac{\gamma}2 \| \bsalpha \|^2
\le \hg(\bv); \end{equation} so if the multiplier vector $\bsalpha$ further satisfies \begin{equation} \label{stopping-criteria-inequality} \cH(G_{\bsalpha})
+ \frac1{2\gamma} \| \Vint{\bm G_{\bsalpha}} - \bv \|^2
\le \bsalpha \cdot \bv - \Vint{\eta_*(\bsalpha \cdot \bm)}
- \frac\gamma2 \| \bsalpha \|^2 + \tau', \end{equation} we can conclude that it satisfies \eqref{eq:tau-opt-tik}. Since the optimal duality gap is zero, \eqref{stopping-criteria-inequality} can be achieved for any $\tau' > 0$.
Now, the form of $G_{\bsalpha}$ and the fact that $\eta$ and $\eta_{\ast}$ are Legendre duals imply that \begin{align} \label{eq:stopping-criteria-legendre}
\eta(G_{\bsalpha}) + \eta_*(\bsalpha \cdot \bm)
&= \bsalpha \cdot \bm G_{\bsalpha}. \end{align} This relation reduces \eqref{stopping-criteria-inequality} to \begin{subequations} \begin{align}
\frac1{2\gamma} \| \Vint{\bm G_{\bsalpha}} - \bv
+ \gamma\bsalpha \|^2 \le \tau'. \end{align} \end{subequations} which gives a stopping criterion equivalent to \eqref{eq:stopping}, where the tolerances are related by $\tau = \sqrt{2\gamma\tau'}$.
\subsection{Accuracy tests}\label{sec:static-tests}
To verify accuracy numerically, we consider the following curve in the realizable set: \begin{equation}\label{eq:static-moment-curve} \bu(x) := \Vint{\bm \exp(\alpha_0(x) + \alpha_1(x) \mu)}, \end{equation} where $x \in [-\pi, \pi]$ and \begin{subequations}
\begin{align}
\alpha_0(x) := - \sin(x) + c \quand
\alpha_1(x) := K + \sin(x).
\end{align} \end{subequations} The parameter $K$ is used to move the moment curve closer to the boundary of the realizable set. The constant $c$ is set to \begin{align} c &= \log\left( \cfrac{K - 1}{2\sinh(K - 1)} \right) - 1 \end{align} so that $1 = \max_x u_0(x)$.
We generate an error-contaminated moment vector $\bv^\delta$ by projecting the moment curve on the interval $[0, \dx]$ onto the space of polynomials up to degree $k - 1$. We let $\bu_{\dx}(0)$ denote the evaluation at $x = 0$ of the $(k - 1)$-th degree polynomial projection of $\bu(x)$ given in \eqref{eq:static-moment-curve} on $[0, \dx]$. We choose the edge $x = 0$ simply because it would appear in a finite-volume method.
The velocity space is $V = [-1, 1]$ (as in the numerical tests in \secref{numerics} below), and for the basis functions $\bm$ we take the Legendre polynomials up to seventh order. We compute the velocity integrals in \eqref{eq:static-moment-curve} using a forty-point Gauss--Lobatto quadrature and the spatial inner products for the orthogonal projection with a twenty-point Gauss--Lobatto quadrature.
In \tabref{static}, we plot the error \begin{align}
\| \hat \bv(\bsalpha_\gamma^\tau(\bu_{\dx}(0))) - \bu(0) \|, \end{align} where $\bsalpha_\gamma^\tau(\bu_{\dx}(0)))$ denotes the first Newton iterate satisfying the stopping criterion \eqref{eq:stopping} for the moment vector $\bu_{\dx}(0)$. The test results confirm that the appropriate choices of $\gamma$ and $\tau$ give the expected orders of convergence. We note that almost all of the moment vectors $\bu_{\dx}(0)$ generated by the polynomial projections for the table are not realizable.
\begin{table}
\small
\centering
\begin{tabular}{rrrrrrr}
& \multicolumn{2}{c}{$k = 2$}
& \multicolumn{2}{c}{$k = 3$}
& \multicolumn{2}{c}{$k = 4$} \\
\cmidrule(r){2-3} \cmidrule(r){4-5} \cmidrule(r){6-7}
$\dx$ & $E^1_h$ & $\nu$
& $E^1_h$ & $\nu$
& $E^1_h$ & $\nu$ \\ \midrule
$2^{-1}$ & 5.7e-01 & & 4.2e-01 & & 2.8e-01 & \\ $2^{-2}$ & 2.8e-01 & 0.99 & 6.9e-02 & 2.62 & 1.5e-02 & 4.23 \\ $2^{-3}$ & 6.9e-02 & 2.05 & 1.0e-02 & 2.77 & 1.3e-03 & 3.55 \\ $2^{-4}$ & 1.5e-02 & 2.19 & 1.3e-03 & 2.97 & 7.5e-05 & 4.10 \\ $2^{-5}$ & 4.5e-03 & 1.75 & 1.4e-04 & 3.18 & 4.8e-06 & 3.95 \\ $2^{-6}$ & 1.3e-03 & 1.81 & 2.0e-05 & 2.83 & 3.1e-07 & 3.97 \\ $2^{-7}$ & 3.2e-04 & 1.99 & 2.5e-06 & 3.02 & 1.9e-08 & 4.03 \\ $2^{-8}$ & 7.5e-05 & 2.11 & 3.1e-07 & 3.00 & 1.1e-09 & 4.05 \\ $2^{-9}$ & 2.0e-05 & 1.90 & 3.8e-08 & 3.03 & 7.3e-11 & 3.97
\end{tabular}
\caption{Regularization errors for $K = 200$ and $\gamma = \tau = \dx^k$.}
\label{tab:static} \end{table}
\section{Numerical results} \label{sec:numerics}
In this section we demonstrate that \eqref{eq:reg-mn} can be simulated using an off-the-shelf, high-order method for hyperbolic conservation laws. Our simulations indicate that the results of \secref{acc} can be used to guide the choice of the regularization parameter $\gamma$ and the optimization tolerance $\tau$ so that a numerical solution of \eqref{eq:reg-mn} is an accurate solution of the original entropy-based moment system \eqref{eq:mn}. We also present numerical simulations of a benchmark problem.
For numerical tests, we consider a kinetic equation that describes particles of unit speed moving through a material with slab geometry (see e.g., \cite{Lewis-Miller-1984}): \begin{align}\label{eq:slab}
\partial_t f + \mu \partial_x f + \sig{a} f = \sig{s}(\Vint{f} - f) + S. \end{align} The spatial domain is $X = (\xL, \xR)$ is one-dimensional, and the velocity variable $\mu \in [-1, 1]$ gives the cosine of the angle between the microscopic velocity and and the $x$-axis. The collision operator here is $\cC(f) := \sig{s}(\vint{f} - f)$, where $\sig{s} \ge 0$ is the \emph{scattering cross section}. This collision operator $\cC$ represents isotropic scattering, is linear, and dissipates any convex entropy $\eta$. Our equation also has a loss term $\sig{a} f$, where $\sig{a} \ge 0$ is the \emph{absorption cross section}, as well as a source $S = S(t, x, \mu)$. Equation \eqref{eq:slab} is supplemented with the initial conditions \begin{align}
f(0, x, \mu) = f_0(x, \mu) \end{align} and boundary conditions \begin{equation}
f(t, \xL, \mu >0) = f_{\rm L}(t, \mu) \qquand f(t, \xR, \mu < 0) = f_{\rm R}(t, \mu) . \end{equation}
The original entropy-based moment equations for \eqref{eq:slab} are \begin{align}\label{eq:mn-slab}
\partial_t \bu + \partial_x \bff(\bu) + \sig{a} \bu
= \sig{s}R \bu + \bs, \end{align} where $R = \diag\{0, -1, \dots , -1\}$ and $\bs := \vint{\bm S}$. The regularized entropy-based moment equations for \eqref{eq:slab} are \begin{align}\label{eq:rmn-slab}
\partial_t \bu + \partial_x \bff_\gamma(\bu) + \sig{a} \bu
= \sig{s}R \hat \bv(\alphahatg(\bu)) + \bs. \end{align}
\begin{remark} To achieve the entropy-dissipation property described in \secref{rmn-structure}, we must use the regularized moment vector $\hat \bv(\alphahatg(\bu))$ in the collision operator. This makes the collision operator nonlinear. The absorption term is not part of the collision operator and thus not part of the entropy-dissipating structure of the kinetic equation \eqref{eq:slab}, and for that reason we simply leave it as a linear decay term in the regularized moment equations. \end{remark}
The initial conditions are $\bu(0, x) = \Vint{\bm f_0(x, \cdot)}$ and and to define the boundary conditions we extend the definitions of $f_{\rm L}$ and $f_{\rm R}$ from $\mu \in [0, 1]$ and $\mu \in [-1, 0]$ respectively to all $\mu \in [-1, 1]$ to get \begin{align}
\bu(t, \xL) = \bu_{\rm L}(t) := \Vint{\bm f_{\rm L}(t, \cdot)}
\quand
\bu(t, \xR) = \bu_{\rm R}(t) := \Vint{\bm f_{\rm R}(t, \cdot)}. \end{align} While this is not technically correct (and proper treatment of boundary conditions for moment methods remains an open problem), we only consider problems where the boundary conditions have at most a negligible effect on the solution.
For our numerical tests we take the Maxwell--Boltzmann entropy because it has generic physical relevance and leads to a nonnegative entropy ansatz. We take the Legendre polynomials up to order $N$ for the basis functions in $\bm$, and so the number of moment components is $n = N + 1$.
\subsection{Numerical method} \label{sec:dg}
Two common high-order methods for hyperbolic equations are the discontinuous-Galerkin (DG) \cite{CKS2000,CockburnShuIII} and weighted-essentially-nonoscillatory (WENO) \cite{Shu1998} methods. The main consideration in selecting a method is the number of times one must compute multipliers $\alphahatg(\bu)$ via \eqref{eq:reg-mult}, since this is the most expensive part of the algorithm. For the hyperbolic component, WENO offers a more attractive choice, since it only requires multipliers at the cell edges (and cell means, if a characteristic transformation is performed) in order to compute fluxes. A DG algorithm, on the other hand, needs multipliers on a quadrature set in each cell in order to evaluate the volume term. However, for the regularized equations \eqref{eq:rmn-slab} the collision term is nonlinear, and thus both the WENO and DG methods must approximately integrate this term in space with a quadrature, and each quadrature evaluation requires knowledge of the multipliers. Thus the advantages of WENO over DG are lost. We therefore proceed with a DG implementation, a description of which (in the context of solving \eqref{eq:mn-slab}) can be found in \cite{AlldredgeSchneider2014}. The implementation here is essentially the same, except that a realizability limiter is not needed.
As in \cite{AlldredgeSchneider2014} we use the Lax-Friedrichs numerical flux. The numerical dissipation constant is set to one because the eigenvalues of the flux $\bff_\gamma$ have the bound \begin{align}
\lambda_{\max}\left( \frac{\partial \bff_\gamma}{\partial \bv} \right) \le 1, \end{align} where $\lambda_{\max}$ denotes the maximum (in absolute value) eigenvalue. This is a straightforward extension of \cite[Lemma 3.1]{AlldredgeSchneider2014}. We use SSP Runge--Kutta methods for time integration, specifically those given in \cite{ketcheson2008highly}: for the second-order results, we use the $s$-stage method with ten stages; for the third-order results, we use the $r^2$-stage method with $r = 4$; and for the fourth-order results, we use the ten-stage method. We use a regular grid with $N_x$ spatial cells with width $\dx = (\xR - \xL) / N_x$. The DG basis consists of polynomials up to degree $k - 1$ on each cell. We choose the time step as in \cite{AlldredgeSchneider2014}: \begin{align}
\dt = \frac{w_Q \dx}{1 + w_Q \dx (\sig{a} + \sig{s})}, \end{align} where $w_Q$ is the weight of the endpoints of the $Q$-point Gauss-Lobatto quadrature with $2Q - 2 \ge k$. While in \cite{AlldredgeSchneider2014} this time step was chosen in order to maintain realizability of the cell means, which is irrelevant to us, we found that trying to use smaller time steps quickly led to stability problems.
We solve dual optimization problem \eqref{eq:reg-mult} using a Levenberg-Marquardt-type algorithm and the Armijo line search.
\subsection{Convergence test using a manufactured solution}
To test how accurately the numerical solution of \eqref{eq:rmn-slab} approximates the numerical solution of the original entropy-based moment equations \eqref{eq:mn-slab}, we used the method of manufactured solutions, in particular the one proposed in \cite{AlldredgeSchneider2014}: Let \begin{align}\label{eq:man-soln-exact}
\bw(t, x) := \Vint{\bm \exp(\alpha_0(t, x) + \alpha_1(t, x) \mu)}, \end{align} where \begin{align}
\alpha_0(t, x) := - \sin(x - t) + 4 t + c
\qquand
\alpha_1(t, x) := K + \sin(x - t). \end{align} As above, the parameter $K$ is used to move the moment curve closer to the boundary of the realizable set, and the constant $c$ is set to \begin{align}
c &= \log\left( \cfrac{K - 1}{2\sinh(K - 1)} \right) - 1 - 4t_{\rm f}, \end{align} so that $1 = \max_{t, x} w_0(t, x)$. The spatial domain is $X = (-\pi, \pi)$, and we take the final time $t_{\rm f} := \pi / 5$. We use periodic boundary conditions and include neither scattering nor absorption: $\sig{a} = \sig{s} = 0$. Since the goal is to converge to the solution of the original entropy-based moment equations \eqref{eq:mn-slab}, we compute the source $s$ for the manufactured solution using $\bff$ instead of $\bff_\gamma$; that is, we set ${\bs = \partial_t \bw + \partial_x \bff(\bw)}$.
Error is measured in the $L^1$ sense: Let $\bu_{\dx}(t_{\rm f}, x)$ denote the point-wise evaluation of the DG solution at the final time; we consider the errors only in the zeroth component, which are given by \begin{align}
e_{\dx} = \int_{\xL}^{\xR} |u_{\dx,0}(t_{\rm f}, x) - w_0(t_{\rm f}, x)| \intdx. \end{align} We approximate the integral with a twenty-point Gauss--Lobatto quadrature in each spatial cell. Results are given in \tabref{manufactured}; we used a factor $10^{-1}$ in front of the $\dx^k$ for $\gamma$ and $\tau$ (unlike in \secref{static-tests}). With this factor, we see the expected orders of convergence for different values of $k$. For larger values of this factor, the observed convergence in our tests is slightly smaller than expected.
\begin{table} \footnotesize \centering \begin{tabular}{rcrcrcr}
& \multicolumn{2}{c}{$k = 2$}
& \multicolumn{2}{c}{$k = 3$}
& \multicolumn{2}{c}{$k = 4$} \\
\cmidrule(r){2-3} \cmidrule(r){4-5} \cmidrule(r){6-7}
$N_x$ & $e_{\dx}$ & $\nu$ & $e_{\dx}$ & $\nu$ & $e_{\dx}$ & $\nu$ \\ \midrule
10 & 1.7184e-01 & -- & 6.9233e-02 & -- & 9.3886e-03 & -- \\
20 & 1.1080e-01 & 0.63 & 6.6889e-03 & 3.37 & 2.9149e-04 & 5.01 \\
40 & 2.9046e-02 & 1.93 & 1.2543e-03 & 2.41 & 4.6188e-05 & 2.66 \\
80 & 7.6273e-03 & 1.93 & 1.7001e-04 & 2.88 & 5.9739e-06 & 2.95 \\
160 & 2.2065e-03 & 1.79 & 2.3744e-05 & 2.84 & 4.7288e-07 & 3.66 \\
320 & 5.5530e-04 & 1.99 & 3.0206e-06 & 2.97 & 3.0056e-08 & 3.98 \\
640 & 1.4088e-04 & 1.98 & 3.8838e-07 & 2.96 & 1.9219e-09 & 3.97 \\ 1280 & 3.6005e-05 & 1.97 & 4.8748e-08 & 2.99 & 1.1919e-10 & 4.01
\end{tabular} \caption{Errors between numerical solutions of the regularized equations to the exact solution of the original equations for the manufactured-solution test. Here, $N = 3$, $K = 5$, and we use the regularization and optimization parameters $\gamma = \tau = 10^{-1}\dx^k$.} \label{tab:manufactured} \end{table}
\subsection{Plane-source benchmark} \label{sec:plane}
The plane-source problem \cite{ganapol1977generation} tests how well a method handles strong spatial gradients and angular distributions with highly localized support. We use a slightly smoothed version of this problem in which an initial delta function in space is replaced by a narrow Gaussian. Even with this smoothing, solutions are rough and numerical convergence is slow.
The domain is $X = (-1.2, 1.2)$, and the initial conditions are given by \begin{align}
f_{t = 0}(x, \mu) = \max\left(\frac{\exp\left(-x^2 / \Sigma^2 \right)}
{\Sigma}, f_{\rm floor}\right), \end{align} where $\Sigma = 0.01$, and $f_{\rm floor} = 0.5 \times 10^{-8}$ approximates a vacuum. (The ansatz with the Maxwell--Boltzmann entropy, which has the form $\exp(\bsalpha \cdot \bm)$, cannot be exactly zero.) The boundary conditions $f_{\rm L}(t, \mu) \equiv f_{\rm R}(t, \mu) \equiv f_{\rm floor}$ are consistent with the analytical solution. We simulate the solution up to $t_{\rm f} = 1$.
For the results in this section, we first found the smallest values of $\gamma$ and $\tau$ with which we could reliably compute numerical solutions of the regularized equations without the optimizer crashing. These values were $\gamma = 10^{-6}$ and $\tau = 10^{-7}$. Then with these values, we compute a very accurate, nearly converged numerical solution using the fourth-order DG method with 4000 spatial cells. We compare this solution with a high-resolution solution of the original entropy-based moment equations, which we generate using the second-order kinetic scheme of \cite{AllHau12}. We use 13000 cells with the kinetic scheme and even for the slightly smoothed version of the problem considered here, we do have to use the technique of isotropic regularization for some moment vectors in the numerical solution (see \cite{AllHau12} for details).
\figref{plane-kin-comp} shows the results for $N = 5$. In this figure, the solutions are indistinguishable, but in Figures \ref{fig:plane-kin-comp-zoom-1} and \ref{fig:plane-kin-comp-zoom-2}, we zoom in on the solutions in two places to show that differences on the order of 0.01, or about 1\%, remain. We computed solutions for other values of $N$ and found similar results.
To get some understanding of the effect of the value of $\gamma$ on the solutions, we also present numerical solutions to the plane-source problem with two larger values of $\gamma$. (We continue to use $\tau = 10^{-7}$ and the fourth-order DG method with 4000 cells.) In \figref{plane-gam-comp} we include the results using $\gamma = 10^{-2}$. While for the larger value of $\gamma$ the first and third waves of the solution are larger in magnitude, the second wave is smaller and somewhat delayed. The front of the third wave is also slightly delayed. It seems to us that while increasing the value of $\gamma$ does not have a smoothing effect, it does seem to have a delaying effect like that predicted by the analysis of the regularized P$_N$ equations in \secref{reg-pn}. The zoomed-in plots in Figures \ref{fig:plane-gam-comp-zoom-1} and \ref{fig:plane-gam-comp-zoom-2} include a third, intermediate value of $\gamma$ which confirms this observation.
\begin{figure}
\caption{Plane-source solution using the kinetic scheme and the fourth-order
DG scheme with $\gamma = 10^{-6}$.}
\label{fig:plane-kin-comp}
\caption{Plane-source solution using two different values of $\gamma$.}
\label{fig:plane-gam-comp}
\caption{Zoom-in around $x \in [0.96, 1.01]$ of the comparison of the
solutions from the old kinetic scheme and the new regularized equations.}
\label{fig:plane-kin-comp-zoom-1}
\caption{Zoom-in around $x \in [0.96, 1.01]$ of the comparison of the
solutions for different values of $\gamma$.}
\label{fig:plane-gam-comp-zoom-1}
\caption{Zoom-in around $x \in [0.63, 0.65]$ of the comparison of the
solutions from the old kinetic scheme and the new regularized equations.}
\label{fig:plane-kin-comp-zoom-2}
\caption{Zoom-in around $x \in [0.15, 0.4]$ of the comparison of the
solutions for different values of $\gamma$.}
\label{fig:plane-gam-comp-zoom-2}
\caption{Numerical solutions of the plane-source problem.}
\label{fig:plane-all}
\end{figure}
\section{Concluding remarks}
In this work we introduce a new moment method for kinetic equations. We derive this method, dubbed the regularized entropy-based moment method, by starting with the original entropy-based moment equations and relaxing the equality constraint in the optimization defining the ansatz reconstruction for the flux and collision terms. By relaxing these constraints, we can define flux and collision terms for nonrealizable moment vectors which, while unphysical, often appear as a result of discretization error in numerical simulations. The relaxation corresponds to a Tikhonov regularization in the defining optimization problem's dual.
We showed that the regularized system keeps many of the same properties as the original system: Firstly, it dissipates entropy, albeit not the same as the original system but an approximation thereof, and is hyperbolic. When the basis functions are orthonormal, the regularized system is also rotationally symmetric. On the other hand, translational invariance is lost. The problem of degenerate densities for unbounded velocity domains also carries over to the regularized problem in the form of moment vectors for which the regularized problem has no solution.
We view these regularized equations as a tool to compute approximate solutions to the original entropy-based moment equations because the error in the regularized reconstruction can be controlled through the choice of the regularization parameter. Numerical simulations using a discontinuous-Galerkin (DG) scheme confirm this accuracy for the moment equations from a one-dimensional linear kinetic equation. We can use the DG scheme essentially off-the-shelf because relaxing the realizability requirement greatly simplifies its implementation.
For possible future work, a rigorous proof of the accuracy of the regularized equations would put the accuracy results on more solid ground. In one spatial dimension (where well-posedness theory for hyperbolic systems is available), perhaps the best route to this result is by examining the difference in the Jacobians of the fluxes $\bff_\gamma$ and $\bff$ and applying the results of \cite{bianchini2002stability}.
The scheme could be improved by using an adaptive choice of the regularization parameter. Another improvement would be the development of an asymptotic-preserving scheme to handle stiff, collision-dominated kinetic regimes. This has been long sought for entropy-based moment equations, and we believe this will be more easily attainable without the obstacle of realizability.
\appendix
\section{Degenerate densities} \label{sec:junk}
We recall that in \propref{junk-line-g} we are considering the Maxwell--Boltzmann entropy and $V = \R^d$. We let $\bm$ contain polynomials up to degree $N$, for some even $N$, such that the only component of degree greater than or equal to $N$ is
$m_{n - 1}(v) = |v|^N$. Let $\cA$ be the set of multiplier vectors such that $G_{\bsalpha} \in L^1(V)$. In particular, we know that \begin{align}
\cA \subset \{ \bsalpha \in \R^n : \alpha_{n - 1} \le 0 \}
\qquand
\cA \cap \partial \cA \subset \{ \bsalpha \in \R^n : \alpha_{n - 1} = 0 \}. \end{align}
In order to prove \propref{junk-line-g}, we need the following lemma.
\begin{lemma}\label{lem:min-vhatg} The function $\cH_\gamma(\cdot ; \bv)$ achieves a unique minimum if and only if $\bv \in \vhatg(\cA)$. The minimizer, if it exists, has the form $G_{\bsalpha}$, defined in \eqref{eq:ansatz}. \end{lemma}
\begin{proof} First assume $\bv = \vhatg(\bsalpha)$ for some $\bsalpha \in \cA$. Since $\eta$ is convex, \begin{align} \label{eq:convex-expansion-Ga}
\Vint{\eta(g)} \ge \Vint{\eta(G_{\bsalpha})}
+ \Vint{\bsalpha \cdot \bm (g - G_{\bsalpha})}, \end{align} where we use the fact that $\eta'(G_{\bsalpha}) = \bsalpha \cdot \bm$. Applying \eqref{eq:convex-expansion-Ga} to the definition of $\cH_\gamma$ in \eqref{eq:H-gamma} and using the fact that $\bv = \Vint{\bm G_{\bsalpha}} + \gamma \bsalpha$ gives \begin{subequations} \begin{align}
\cH_\gamma(g; \bv) &\ge \cH_\gamma(G_{\bsalpha}; \bv)
+ \frac1{2\gamma} \| \vint{\bm g} - \bv \|^2
- \frac1{2\gamma} \| \vint{\bm G_{\bsalpha}} - \bv \|^2 \nonumber \\
&\qquad + \Vint{\bsalpha \cdot \bm (g - G_{\bsalpha})} \\
&= \cH_\gamma(G_{\bsalpha}; \bv)
+ \frac1{2\gamma} \| \vint{\bm g} - \Vint{\bm G_{\bsalpha}}
- \gamma \bsalpha \|^2
- \frac\gamma 2 \| \bsalpha \|^2 \nonumber \\
&\qquad + \Vint{\bsalpha \cdot \bm (g - G_{\bsalpha})} \\
&= \cH_\gamma(G_{\bsalpha}; \bv)
+ \frac1{2\gamma} \| \vint{\bm g} - \Vint{\bm G_{\bsalpha}}\|^2 \\
&\ge \cH_\gamma(G_{\bsalpha}; \bv). \end{align} \end{subequations} Thus $G_{\bsalpha}$ minimizes $\cH_\gamma(\cdot ; \bv)$.
On the other hand, assume $\cH_\gamma(\cdot ; \bv)$ has a minimizer, which we denote by $g^*$. The minimizer $g^*$ also solves the problem \begin{align} \label{eq:g-star-solves}
\minimize_{g \in \bbF(V)} \: \cH_\gamma(g; \bv)
\qquad \st \: \Vint{\bm g} = \Vint{\bm g^*}. \end{align} Moreover, because the penalty term in $\cH_\gamma$ is constant on the constraint set in \eqref{eq:g-star-solves}, $g^*$ solves the original problem \eqref{eq:primal} with $\bv$ replaced by $\vint{\bm g^*}$. Thus according to \cite[Theorem 9]{Hauck-Levermore-Tits-2008}, $g^* = G_{\bsalpha}$ for some $\bsalpha \in \cA$.
Now let $\widetilde g \in L^1(V)$ be smooth and have compact support. We consider $\veps$ small enough such that $G_{\bsalpha} + \veps \widetilde g \in \bbF(V)$. Since $G_{\bsalpha}$ minimizes $\cH_\gamma(\cdot ; \bv)$, the Gateaux derivative of $\cH_\gamma$ in the direction $\widetilde g$ must be zero, i.e., \begin{subequations} \begin{align}
0 &= \lim_{\veps \to 0} \frac d {d\veps}
\cH_\gamma(G_{\bsalpha} + \veps \widetilde g; \bv) \\
&= \Vint{\eta'(G_{\bsalpha}) \widetilde g} + \frac1\gamma (\Vint{\bm G_{\bsalpha}} - \bv)
\cdot \Vint{\bm \widetilde g} \\
&= \left( \bsalpha + \frac1\gamma (\Vint{\bm G_{\bsalpha}} - \bv) \right)
\cdot \Vint{\bm \widetilde g}, \end{align} \end{subequations} from which we can conclude, using the freedom in the choice of $\widetilde g$, that $\bv = \vhatg(\bsalpha)$. \end{proof}
\begin{remark} Like the original problem, when $V$ is compact there are no degenerate densities. In this case, $\cA = \R^n$, and since the dual of the regularized problem is strongly convex, it has a maximizer for any $\bv \in \R^n$, and therefore $\vhat(\cA) = \R^n$. \end{remark}
\begin{proof}[Proof of \propref{junk-line-g}] Let \begin{align}\label{eq:dual-psi}
\psi_\gamma(\bsalpha; \bv) := \bsalpha \cdot \bv
- \Vint{\eta_{\ast}(\bsalpha \cdot \bm)}
- \frac\gamma 2 \|\bsalpha\|^2 \end{align} be the dual function for the regularized problem so that \begin{align}
\alphahatg(\bv) = \argmax_{\bsalpha \in \cA} \psi_\gamma(\bsalpha; \bv). \end{align} One can extend the arguments from \cite{Hauck-Levermore-Tits-2008} to show that $\alphahatg(\bv)$ exists for \emph{any} $\bv \in \R^n$ (although when $\alphahatg(\bv)$ is on the boundary of $\cA$, it may not satisfy the first-order necessary conditions, i.e., sometimes $\vhat_\gamma(\alphahatg(\bv)) \ne \bv$).
Suppose that \eqref{eq:tik-primal} has a minimizer $g^*$. Then by \lemref{min-vhatg}, $g^* = G_{\bsalpha^*}$ with $\bv = \vhat_\gamma(\bsalpha^*)$. Since the latter shows that $\bsalpha^*$ satisfies the first-order necessary conditions for \eqref{eq:dual-psi} and $\psi_\gamma(\cdot ; \bv)$ is strictly concave, we have $\bsalpha^* = \alphahatg(\bv)$. By rearranging terms in the first-order necessary conditions, we conclude ${\bv - \gamma \alphahatg(\bv) \in \vhat(\cA)}$.
The contrapositive is: if $\bv - \gamma \alphahatg(\bv) \nin \vhat(\cA)$, then no minimizer exists. Thus our strategy is to show that when $\bv$ has the form from \eqref{eq:junk-form-g}, we have ${\bv - \gamma \alphahatg(\bv) \nin \vhat(\cA)}$.
First, note that when $\bv$ has the form from \eqref{eq:junk-form-g}, then the $\overline{\bsalpha}$ must be $\alphahatg(\bv)$. To see this, recognize that $\overline{\bsalpha} \in \cA \cap \partial \cA$ implies that $\overline{\alpha}_{n - 1} = 0$, so that the concavity of $\psi_\gamma$ (and the requisite smoothness properties assured by \cite[Lemma 5.2]{Jun00}) gives \begin{align}
\psi_\gamma(\bsalpha; \bv) \le \psi_\gamma(\overline{\bsalpha}; \bv)
+ \psi'_\gamma(\overline{\bsalpha}; \bv) \cdot (\bsalpha - \overline{\bsalpha})
\stackrel{\eqref{eq:junk-form-g}}{=} \psi_\gamma(\overline{\bsalpha}; \bv)
+ \delta \alpha_{n - 1}
\le
\psi_\gamma(\overline{\bsalpha}; \bv). \end{align} Since the maximizer of $\psi_\gamma(\cdot ; \bv)$ is unique, $\overline{\bsalpha} = \alphahatg(\bv)$. Thus \eqref{eq:junk-form-g} can be written as \begin{align}
\bv - \gamma \alphahatg(\bv) = \hat \bv(\alphahatg(\bv))
+ \begin{pmatrix} 0 \\ \vdots \\ 0 \\ \delta \end{pmatrix}, \end{align} and by \cite{Jun00}, the right-hand side is not in $\vhat(\cA)$. \end{proof}
With a little more work, one can show that all degenerate densities ${\bv \in \R^n \setminus \vhatg(\cA)}$ have the form \eqref{eq:junk-form-g}. Furthermore, for the case of more general polynomial basis functions, one can extend the arguments of \cite{Hauck-Levermore-Tits-2008} to show that the regularized problem satisfies the analogous complimentary-slackness condition \begin{align}\label{eq:comp-slack}
\alphahatg(\bv) \cdot (\bv - \vhat_\gamma(\alphahatg(\bv))) = 0. \end{align} This complementary-slackness condition is the key to characterizing the set degenerate densities (of the original problem) in \cite{Hauck-Levermore-Tits-2008}. Indeed, \eqref{eq:comp-slack} can be used to show that the set of all degenerate densities for the regularized problem is a union of normal cones which has the same form as \cite[Eq. (180)]{Hauck-Levermore-Tits-2008} with $\hat \bv$ (in that paper's notation, $\br$) replaced by $\vhat_\gamma$.
\end{document} | arXiv |
\begin{document}
\title[Multi-scale stochastic hyperbolic-parabolic equations]{Averaging principle and normal deviations for multi-scale stochastic hyperbolic-parabolic equations}
\author{Michael R\"ockner} \address{
Fakult\"{a}t f\"{u}r Mathematik, Universit\"{a}t Bielefeld, D-33501 Bielefeld, Germany, and Academy of Mathematics and Systems Science,
Chinese Academy of Sciences (CAS), Beijing, 100190, P.R.China} \curraddr{} \email{[email protected]} \thanks{}
\author{Longjie Xie} \address{School of Mathematics and Statistics $\&$ Research Institute of Mathematical Science, Jiangsu Normal University,
Xuzhou, Jiangsu 221000, P.R.China} \curraddr{} \email{[email protected]} \thanks{This work is supported by the DFG through CRC 1283 and NSFC (No. 12071186, 11931004).}
\author{Li Yang} \address{Department of Mathematics, Shandong University,
Jinan, Shandong 250100, P.R.China} \curraddr{} \email{[email protected]} \thanks{}
\subjclass[2020]{ 60H15, 60F05, 70K70} \keywords{Stochastic hyperbolic-parabolic equations; averaging principle; strong and weak convergence; homogenization}
\date{}
\dedicatory{}
\begin{abstract} We study the asymptotic behavior of stochastic hyperbolic parabolic equations with slow and fast time scales. Both the strong and weak convergence in the averaging principe are established, which can be viewed as a functional law of large numbers. Then we study the stochastic fluctuations of the original system around its averaged equation. We show that the normalized difference converges weakly to the solution of a linear stochastic wave equation, which is a form of functional central limit theorem. We provide a unified proof for the above convergence by using the Poisson equation in Hilbert spaces. Moreover, sharp rates of convergence are obtained, which are shown not to depend on the regularity of the coefficients in the equation for the fast variable. \end{abstract}
\maketitle \section{Introduction}
Let $T>0$ and $D\subseteq{\mathbb R}^d\;(d\geqslant 1)$ be a bounded open set. Consider the following system of stochastic hyperbolic-parabolic equations: \begin{equation} \label{spde01} \left\{ \begin{aligned} &\frac{\partial^2U^{\varepsilon}_t(\xi)}{\partial t^2}=\Delta U^{\varepsilon}_t(\xi)+f(U^{\varepsilon}_t(\xi), Y^{\varepsilon}_t(\xi))+\dot W^1_t(\xi),\qquad\,\,\, (t,\xi)\in(0,T]\times D,\\ &\frac{\partial Y^{\varepsilon}_t(\xi)}{\partial t} =\frac{1}{\varepsilon}\Delta Y^{\varepsilon}_t(\xi)+\frac{1}{\varepsilon}g(U^{\varepsilon}_t(\xi), Y^{\varepsilon}_t(\xi))\!+\!\frac{1}{\sqrt{\varepsilon}} \dot W_t^2(\xi),\,\, (t,\xi)\in(0,T]\times D,\\ &U^{\varepsilon}_t(\xi)=Y^{\varepsilon}_t(\xi)=0,\quad\quad\qquad\quad\qquad\quad\qquad\quad\qquad\quad\,\, (t,\xi)\in(0,T]\times\partial D,\\
&U^{\varepsilon}_0(\xi)=u(\xi),\,\,\frac{\partial U^{\varepsilon}_t(\xi)}{\partial t}\big|_{t=0}=v(\xi),\,\,Y^{\varepsilon}_0(\xi)=y(\xi),\quad\,\,\quad\quad\quad\quad\qquad\xi\in D, \end{aligned} \right. \end{equation} where $\Delta$ is the Laplacian operator, $\partial D$ denotes the boundary of the domain $D$, $f,g :{\mathbb R}^2\to{\mathbb R}$ are measurable functions, $W^1_t$ and $W^2_t$ are two mutually independent $Q_1$- and $Q_2$-Wiener processes both defined on a complete probability space $(\Omega,\mathscr{F},\{\mathscr{F}_t\}_{t\geq0},{\mathbb P})$, and the small parameter $0<\varepsilon\ll 1$ represents the separation of time scales between the `slow' process $U_t^\varepsilon$ and the `fast' motion $Y_t^\varepsilon$ (with time order $1/\varepsilon$). Randomly perturbed hyperbolic partial differential equations are usually used to model wave propagation and mechanical vibration in a random medium, see e.g. \cite{BD,BDT,DKMNX}. If these phenomena are temperature dependent or heat generating, then the underlying hyperbolic equation will be coupled with a stochastic parabolic equation, which leads to the mathematical description of slow-fast systems through (\ref{spde01}), see e.g. \cite{CPL1,Le,RR,ZZ} and the references therein. In this respect, the question that how a thermal environment at large time scales may influence the dynamics of the whole system arises.
In the mathematical literature, powerful averaging and homogenization methods have been developed to study the asymptotic behavior of multi-scale systems as $\varepsilon\to 0$. The averaging principle can be viewed as a functional law of large numbers, which says the slow component will converge to the solution of the so-called averaged equation as $\varepsilon\to 0$. The averaged equation then captures the evolution of the original system over a long time scale, which does not depend on the fast variable any more and thus is much simpler. This theory was first studied by Bogoliubov \cite{BM} for deterministic ordinary differential equations, and extended to stochastic differential equations (SDEs for short) by Khasminskii \cite{K1}, see also \cite{BK,GR,HLi,KY,V0} and the references therein. As a rule, the averaging method requires certain smoothness on both the original and the averaged coefficients of the systems. Various assumptions have been studied in order to guarantee the above convergence. Recently, the averaging principle for two time scale stochastic partial differential equations (SPDEs for short) has attracted considerable attention. In \cite{CF}, Cerrai and Freidlin proved the averaging principle for slow-fast stochastic reaction-diffusion equations with noise only in the fast motion. Later, Cerrai \cite{Ce,C2} generalized this result to more general reaction-diffusion equations, see also \cite{BYY,CL,RXY,WR} and the references therein for further developments. We also mention that Br\'ehier \cite{Br1,Br2} studied the rate of convergence in terms of $\varepsilon\to 0$ in the averaging principle for parabolic SPDEs and obtained the $1/2$-order rate of strong convergence (in the mean-square sense) and the $1$-order rate of weak convergence (in the distribution sense), which are known to be optimal. These rates of convergence are important for the study of other limit theorems in probability theory and numerical schemes, known as the Heterogeneous Multi-scale Method for the original multi-scale system, see e.g. \cite{Br3,EL}. Concerning stochastic hyperbolic-parabolic equations, Fu ect. \cite{FWLL1} established the strong convergence in the averaging principle for system (\ref{spde01}) when $d=1$ by the classical Khasminskii time discretization method, and obtained the $1/4$-order rate of strong convergence. In \cite{FWLL}, by using asymptotic expansion arguments, the authors studied the weak order convergence for system (\ref{spde01}), but only in a not fully coupled case ($g(u,y)=g(y)$), i.e., the fast equation does not depend on the slow process.
In this paper, we shall first prove the strong and weak convergence in the averaging principle for the fully coupled system (\ref{spde01}) with singular coefficients, see {\bf Theorem \ref{main1}}. Compared with \cite{FWLL,FWLL1}, we assume that the coefficients are only $\eta$-H\"older continuous with respect to the fast variable with any $\eta>0$, and we obtain the optimal $1/2$-order rate of strong convergence as well as the $1$-order rate of weak convergence. Moreover, we find that both the strong and weak convergence rates do not depend on the regularity of the coefficients in the equation for the fast variable. This implies that the evolution of the multi-scale system (\ref{spde01}) relies mainly on the slow variable, which coincides with the intuition since in the limit equation the fast component has been totally averaged out. Furthermore, the arguments we use are different from those in \cite{Br1,Ce,C2,CF,FWLL,FWLL1}. Our method to establish the strong and weak convergence is based on the Poisson equation in Hilbert space, which is more unifying and much simpler.
The averaged equation for (\ref{spde01}) is only valid in the limit when the time scale separation between the fast and slow variables is infinitely wide. Of course, the scale separation is never infinite in reality. For small but positive $\varepsilon,$ the slow variable $U_t^\varepsilon$ will experience fluctuations around its averaged motion $\bar U_t$. These small fluctuations can be captured by studying the functional central limit theorem. Namely, we are interested in the asymptotic behavior of the normalized difference \begin{align}\label{zte} Z_{t}^{\varepsilon}:=\frac{U_t^\varepsilon-\bar U_t}{\sqrt{\varepsilon}} \end{align} as $\varepsilon$ tends to 0. Such result is known to be closely related to the homogenization behavior of singularly perturbed partial differential equations, which is of its own interest, see e.g. \cite{HP,HP2}. For the study of the functional central limit theorem for finite dimensional multi-scale systems, we refer the reader to the fundamental paper by Khasminskii \cite{K1}, see also \cite{BK,CFKM,KM,Li,P-V,P-V2,RX}. The infinite dimensional situation is more open and papers on this subject are very few. In \cite{Ce2}, Cerrai studied the normal deviations for a deterministic reaction-diffusion equation with one dimensional space variable perturbed by a fast process, and proved the weak convergence to a Gaussian process, whose covariance is explicitly described. Later, this was generalized to general stochastic reaction-diffusion equations by Wang and Roberts \cite{WR}. In both papers, the methods of proof are based on Khasminskii's time discretization argument. Recently, we \cite{RXY} studied the normal deviations for general slow-fast parabolic SPDEs by using the technique of Poisson equation.
In this paper, we further develop the argument used in \cite{RXY} to study the functional central limit theorem for the stochastic hyperbolic-parabolic system (\ref{spde01}) with H\"older continuous coefficients. More precisely, we show that the normalized difference $Z_t^\varepsilon$, defined by (\ref{zte}), converges weakly as $\varepsilon \to 0$ to the solution of a linear stochastic wave equation, see {\bf Theorem \ref{main3}}. Moreover, the optimal $1/2$-order rate of convergence is obtained. This rate also does not depend on the regularity of the coefficients in the equation for the fast variable, which again is natural since in the limit equation the fast component has been homogenized out. As far as we know, the result we obtained is completely new. The argument we use to prove the functional central limit theorem is closely and universally connected with the proof of the strong and weak convergence in the averaging principle. We note that due to the model considered in this paper, the framework we deal with is different from \cite{RXY}. Furthermore, we derive the higher order spatial-temporal convergence in the averaging principle and in the functional central limit theorem. Throughout our proof, several strong and weak fluctuation estimates will play an important role, see Lemmas \ref{strf}, \ref{wef1} and \ref{wfe2} below.
The rest of this paper is organized as follows. In Section 2, we first introduce some assumptions and state our main results. Section 3 is devoted to establish some preliminary estimates. Then we prove the strong and weak convergence results, Theorem \ref{main1}, and the normal deviation result, Theorem \ref{main3}, in Section 4 and Section 5, respectively.
{\bf Notations.} To end this section, we introduce some usual notations for convenience. Given Hilbert spaces $H_1, H_2$ and $\hat H,$ we use ${\mathscr L}(H_1,H_2)$ to denote the space of all linear and bounded operators from $H_1$ to $H_2$. If $H_1=H_2,$ we write ${\mathscr L}(H_1)={\mathscr L}(H_1,H_1)$ for simplicity. Recall that an operator $Q\in{\mathscr L}(\hat H)$ is called Hilbert-Schmidt if $$ \Vert Q\Vert_{\mathscr{L}_2(\hat H)}^2:=Tr(QQ^{*})<+\infty. $$ We shall denote the space of all Hilbert-Schmidt operators on $\hat H$ by $\mathscr{L}_2(\hat H)$. Let $L^\infty_{\ell}(H_1\times H_2,\hat H)$ denote the space of all measurable maps $\phi: H_1\times H_2\to \hat H$ with linear growth, i.e., $$
\|\phi\|_{L^\infty_{\ell}(\hat H)}:=\sup_{(x,y)\in H_1\times H_2}\frac{\|\phi(x,y)\|_{\hat H}}{1+\|x\|_{H_1}+\|y\|_{H_2}}<\infty. $$
For $k\in{\mathbb N}$, the space $C_{\ell}^{k,0}(H_1\times H_2,\hat H)$ contains all $\phi\in L^\infty_\ell(H_1\times H_2,\hat H)$ such that $\phi$ has $k$ times G\^ateaux derivatives with respect to the $x$-variable satisfying $$
\|\phi\|_{C_{\ell}^{k,0}(\hat H)}:=\sup_{(x,y)\in H_1\times H_2}\frac{\sum\limits_{\i=1}^k\|D_x^i\phi(x,y)\|_{{\mathscr L}^i(H_1,\hat H)}}{1+\|x\|_{H_1}+\|y\|_{H_2}}<\infty. $$ Similarly, the space $C^{0,k}_{\ell}(H_1\times H_2,\hat H)$ contains all $\phi\in L^\infty_\ell(H_1\times H_2,\hat H)$ such that $\phi$ has $k$ times G\^ateaux derivatives with respect to the $y$-variable satisfying \begin{align*}
\|\phi\|_{C_\ell^{0,k}(\hat H)}:=\sup_{(x,y)\in H_1\times H_2}\frac{\sum\limits_{i=1}^k\|D_y^i\phi(x,y)\|_{{\mathscr L}^i(H_2,\hat H)}}{1+\|x\|_{H_1}+\|y\|_{H_2}}<\infty. \end{align*} For $k,m\in{\mathbb N}$, let $C^{k,m}_\ell(H_1\times H_2,\hat H)$ be the space of all maps satisfying \begin{align}\label{norm}
\|\phi\|_{C_\ell^{k,m}(\hat H)}:=\|\phi\|_{L^\infty_\ell(\hat H)}+\|\phi\|_{C_\ell^{k,0}(\hat H)}+\|\phi\|_{C_\ell^{0,m}(\hat H)}<\infty, \end{align} and for $\eta\in(0,1)$, the space $C^{k,\eta}_{\ell}(H_1\times H_2,\hat H)$ consists of all $\phi\in C^{k,0}_{\ell}(H_1\times H_2,\hat H)$ satisfying $$
\|\phi(x,y_1)-\phi(x,y_2)\|_{\hat H}\leqslant C_0\|y_1-y_2\|_{H_2}^\eta\big(1+\|x\|_{H_1}+\|y_1\|_{H_2}+\|y_2\|_{H_2}\big). $$ The space $C_b^{k,\eta}(H_1\times H_2,\hat H)$ consists of all $\phi\in C_{\ell}^{k,\eta}(H_1\times H_2,\hat H)$ whose $k$ times G\^ateaux derivatives with respect to the first variable are bounded, and the space $C_B^{k,\eta}(H_1\times H_2,\hat H)$ consists of all maps in $C_b^{k,\eta}(H_1\times H_2,\hat H)$ which are bounded themselves. We also introduce the space ${\mathbb C}^{k,k}_l(H_1\times H_2,\hat H)$ consisting of all maps which have $k$ times Fr\'echet derivatives with respect to both the first variable and the second variable and satisfy (\ref{norm}).
The space ${\mathbb C}^{k,k}_b(H_1\times H_2,\hat H)$ consists of all $\phi\in{\mathbb C}^{k,k}_l(H_1\times H_2,\hat H)$ with all derivatives bounded. When $\hat H={\mathbb R}$, we will omit the letter $\hat H$ for simplicity.
\section{Assumptions and main results}
Let $H:=L^2(D)$ be the usual space of square integrable functions on a bounded open domain $D$ in ${\mathbb R}^d$ with scalar product and norm denoted by $\langle\cdot,\cdot\rangle$ and $\Vert\cdot\Vert$, respectively. Let $A$ be the realization of the Laplacian with Dirichlet boundary conditions in $H$. It is known that there exists a complete orthonormal basis $\{e_n\}_{n\in{\mathbb N}}$ of $H$ such that $$ Ae_n=-\lambda_ne_n, $$ with $0<\lambda_1\leqslant\lambda_2\leqslant\cdots\lambda_n\leqslant\cdots.$ For $\alpha\in{\mathbb R}$, let $H^\alpha:={\mathcal D}((-A)^{\frac{\alpha}{2}})$ be the Hilbert space endowed with the scalar product $$ \<x,y{\rangle}_\alpha:={\langle}(-A)^{\frac{\alpha}{2}}x,(-A)^{\frac{\alpha}{2}}y{\rangle} =\sum\limits_{n=1}^{\infty}\lambda_n^{\alpha}\<x,e_n{\rangle}\<y,e_n{\rangle},\quad\forall x,y\in H^\alpha, $$ and norm $$
\|x\|_\alpha:=\left(\sum\limits_{n=1}^{\infty}\lambda_n^{\alpha}\<x,e_n{\rangle}^2\right)^{\frac{1}{2}},\quad\forall x\in H^\alpha. $$ Then $A$ can be regarded as an operator from $H^\alpha$ to $H^{\alpha-2}$.
For the drift coefficients $f$ and $g$ given in system (\ref{spde01}), we introduce two Nemytskii operators $F, G: H\times H\to H$ by \begin{align}\label{NG} F(u,y)(\xi):=f(u(\xi),y(\xi)),\quad G(u,y)(\xi):=g(u(\xi),y(\xi)),\quad\xi\in D. \end{align} We remark that these operators are not Fr\'echet differentiable in $H$.
To give precise results, it is convenient to write system (\ref{spde01}) in the following abstract formulation in $H$: \begin{equation} \label{spde1} \left\{ \begin{aligned} &\frac{\partial^2U^{\varepsilon}_t}{\partial t^2}=A U^{\varepsilon}_t +F(U^{\varepsilon}_t, Y^{\varepsilon}_t)+\dot W^1_t,\qquad\qquad t\in(0,T],\\ &\frac{\partial Y^{\varepsilon}_t}{\partial t} =\frac{1}{\varepsilon}A Y^{\varepsilon}_t+\frac{1}{\varepsilon}G(U^{\varepsilon}_t, Y^{\varepsilon}_t)+\frac{1}{\sqrt{\varepsilon}} \dot W_t^2,\quad\,\, t\in(0,T],\\
&U^{\varepsilon}_0=u,\,\,\frac{\partial U^{\varepsilon}_t}{\partial t}\big|_{t=0}=v,\,\,Y^{\varepsilon}_0=y. \end{aligned} \right. \end{equation} For $i=1,2$, we assume that $Q_i$ are nonnegative, symmetric operators with respect to $\{e_{n}\}_{n\in{\mathbb N}}$, i.e., $$Q_ie_{n}=\beta_{i,n}e_{n},\;\;\beta_{i,n}>0,n\in{\mathbb N}.$$ In addition, we assume that \begin{align}\label{trace} Tr(Q_i)=\sum\limits_{n\in{\mathbb N}}\beta_{i,n}<+\infty,\;\;i=1,2. \end{align} Given $u\in H$, consider the following frozen equation: \begin{align}\label{froz} {\mathord{{\rm d}}} Y_t^u=AY_t^u {\mathord{{\rm d}}} t+G(u,Y_t^u){\mathord{{\rm d}}} t+{\mathord{{\rm d}}} W_t^2, \quad Y_0^u=y\in H. \end{align} Under our assumptions below, the process $Y_t^u$ admits a unique invariant measure $\mu^u({\mathord{{\rm d}}} y)$. Then, the averaged equation for system (\ref{spde1}) is \begin{equation} \label{spde22} \left\{ \begin{aligned} &\frac{\partial^2\bar U_t}{\partial t^2}=A \bar U_t+\bar F(\bar U_t)+\dot W^1_t,\;\;t\in (0,T],\\
&\bar U_0=u,\,\,\frac{\partial \bar U_t}{\partial t}|_{t=0}=v , \end{aligned} \right. \end{equation} where \begin{align}\label{df1} \bar F(u):=\int_HF(u,y)\mu^u({\mathord{{\rm d}}} y). \end{align}
Let $\dot U^{\varepsilon}_t:=\partial U^{\varepsilon}_t/\partial t$ and $\dot{\bar U}_t:=\partial\bar U_t/\partial t.$ The following is the first main result of this paper.
\begin{theorem}\label{main1}
Let $T>0$, $u\in H^1$ and $v, y\in H$. Assume that $f\in C_b^{2,\eta}({\mathbb R}\times {\mathbb R},{\mathbb R})$ and $g\in C^{2,\eta}_B({\mathbb R}\times {\mathbb R},{\mathbb R})$ with $\eta>0.$ Then we have:
\noindent (i) (strong convergence) for any $q\geqslant 1$,
\begin{align}\label{rr1}
\sup\limits_{t\in[0,T]}{\mathbb E}\left(\| U^{\varepsilon}_t-\bar U_t\|_1^2+\| \dot U^{\varepsilon}_t-\dot{\bar U}_t\|^2\right)^{q/2}\leqslant C_1\,\varepsilon^{q/2};
\end{align}
(ii) (weak convergence) for any $ \phi\in {\mathbb C}_b^3(H)$ and $\tilde\phi\in {\mathbb C}_b^3(H^{-1})$,
\begin{align}\label{rr2}
\sup\limits_{t\in[0,T]}\Big(\big|{\mathbb E}[\phi(U^{\varepsilon}_t)]-{\mathbb E}[\phi(\bar U_t)]\big|+\big|{\mathbb E}[\tilde\phi(\dot U^{\varepsilon}_t)]-{\mathbb E}[\tilde\phi(\dot{\bar U}_t)]\big|\Big)\leqslant C_2\,\varepsilon,
\end{align}
where $C_1=C(T,u,v,y)$ and $C_2=C(T,u,v,y,\phi,\tilde\phi)$ are positive constants independent of $\varepsilon$ and $\eta$.
\end{theorem}
\begin{remark} (i) The $1/2$-order rate of strong convergence in (\ref{rr1}) and the $1$-order rate of weak convergence in (\ref{rr2}) should be optimal, which coincides with the SDE case as well as the stochastic reaction-diffusion equation case. Moreover, we obtain that
both the strong and weak convergence rates do not depend on the regularity of the coefficients in the equation for the fast variable. This coincides with the intuition, since in the limit equation the fast component has been averaged out.
(ii) Note that the coefficients are assumed to be only $\eta$-H\"older continuous with respect to the fast variable, which is sufficient for us to prove the above convergence in the averaging principle. However, the pathwise uniqueness of solutions for system (\ref{spde1}) is not clear under such weak assumptions. In particular, if the system is not fully coupled in the sense that the fast motion does not depend on the slow variable (i.e., $g(u,y)=g(y)$ in (\ref{spde01})), then the well-posedness for the fast equation with only H\"older continuous coefficients has been proven in \cite[Theorem 7]{DF} by using the Zvonkin's transformation. This in turn implies the strong well-posedness of the whole system (\ref{spde01}). \end{remark}
Recall that $Z_t^\varepsilon$ is defined by (\ref{zte}). In view of (\ref{spde1}) and (\ref{spde22}), we have \begin{align*} \frac{\partial^2 Z^\varepsilon_t}{\partial t^2}&=A Z^\varepsilon_t+\frac{1}{\sqrt{\varepsilon}}\Big[F(U_t^\varepsilon,Y_t^\varepsilon)-\bar F(\bar U_t)\Big]\\ &=A Z^\varepsilon_t+\frac{1}{\sqrt{\varepsilon}}\Big[\bar F(U^\varepsilon_t)-\bar F(\bar U_t)\Big]+\frac{1}{\sqrt{\varepsilon}}\delta F(U_t^\varepsilon,Y_t^\varepsilon), \end{align*} where \begin{align*} \delta F(u,y):=F(u,y)-\bar F(u). \end{align*} To study the homogenization behavior of $Z_t^\varepsilon$, we consider the following Poisson equation: \begin{align}\label{poF} {\mathcal L}_2(u,y)\Psi(u,y)=-\delta F(u,y), \end{align} where ${\mathcal L}_2$ is the generator of the frozen equation (\ref{froz}) given by \begin{align}\label{ly2} {\mathcal L}_2(u,y)\varphi(y) &:=\<Ay+G(u,y),D_y\varphi(y){\rangle}\nonumber\\ &\,\,\,\quad+\frac{1}{2}Tr\left(D^2_{y}\varphi(y)Q_2^{\frac{1}{2}}(Q_2^{\frac{1}{2}})^*\right),\quad\forall\varphi\in C_\ell^2(H), \end{align} and $u\in H$ is regarded as a parameter. According to Theorem \ref{PP} below, there exists a unique solution $\Psi$ to equation (\ref{poF}). Then, the limit process $\bar Z_{t}$ of $Z_t^\varepsilon$ turns out to satisfy the following linear stochastic wave equation: \begin{equation} \label{spdez} \left\{ \begin{aligned} &\frac{\partial^2\bar Z_{t}}{\partial t^2}=A\bar Z_{t}+D_u\bar F(\bar U_t).\bar Z_{t}+\sigma(\bar U_t)\dot W_t,\;\;t\in (0,T],\\
&\bar Z_0=0,\,\,\frac{\partial \bar Z_t}{\partial t}|_{t=0}=0 , \end{aligned} \right. \end{equation} where $W_t$ is another cylindrical Wiener process independent of $W_t^1$, and $\sigma$ is a Hilbert-Schmidt operator satisfying \begin{align*} \frac{1}{2}\sigma(u)\sigma^*(u)=\overline{\delta F\otimes\Psi}(u):=\int_{H}\big[\delta F(u,y)\otimes\Psi(u,y)\big]\mu^u({\mathord{{\rm d}}} y). \end{align*}
Let $\dot Z^{\varepsilon}_t:=\partial Z^{\varepsilon}_t/\partial t$ and $\dot{\bar Z}_t:=\partial\bar Z_t/\partial t.$ We have the following result.
\begin{theorem}[Normal deviation]\label{main3} Let $T>0,$ $f\in C_b^{2,\eta}({\mathbb R}\times {\mathbb R},{\mathbb R})$ and $g\in C^{2,\eta}_B({\mathbb R}\times {\mathbb R},{\mathbb R})$ with $\eta>0.$ Then for any $u\in H^1$, $v, y\in H$, $\phi\in {\mathbb C}_b^3(H)$ and $\tilde\phi\in {\mathbb C}_b^3(H^{-1})$, we have \begin{align}\label{clt}
\sup_{t\in[0,T]}\Big(\big|{\mathbb E}[\phi(Z_{t}^{\varepsilon})]-{\mathbb E}[\phi(\bar Z_{t})]\big|+\big|{\mathbb E}[\tilde\phi(\dot Z_{t}^{\varepsilon})]-{\mathbb E}[\tilde\phi(\dot{\bar Z}_{t})]\big|\Big)\leqslant C_3\,\varepsilon^{\frac{1}{2}}, \end{align} where $C_3=C(T,u,\dot u,y,\phi,\tilde\phi)>0$ is a constant independent of $\varepsilon$ and $\eta.$ \end{theorem}
\begin{remark} The $1/2$-order rate of convergence in (\ref{clt}) coincides with the SDE case and should be optimal. Moreover, the convergence rate does not depend on the regularity of the coefficients in the equation for the fast variable. \end{remark}
\section{Preliminaries}
\subsection{Poisson equation} We will rewrite the system (\ref{spde1}) as an abstract evolution equation. To this end, we first introduce some notations. For $\alpha\in{\mathbb R}$, by ${\mathcal H}^\alpha:=H^\alpha\times H^{\alpha-1}$ we denote the Hilbert space endowed with the scalar product $$ \<u,v{\rangle}_{{\mathcal H}^\alpha}:=\<u_1,v_1{\rangle}_\alpha+\<u_2,v_2{\rangle}_{\alpha-1},\quad \forall u=(u_1,u_2)^T,v=(v_1,v_2)^T\in {\mathcal H}^\alpha, $$ and norm $$
\interleave u\interleave_\alpha^2:=\|u_1\|_\alpha^2+\|u_2\|_{\alpha-1}^2,\quad\forall u=(u_1,u_2)^T\in {\mathcal H}^\alpha. $$ For simplicity, we write ${\mathcal H}:=H\times H^{-1}.$ Let $\Pi_1$ be the canonical projection from ${\mathcal H}$ to $H,$ and define $$ V_t^\varepsilon:=\frac{{\mathord{{\rm d}}}}{{\mathord{{\rm d}}} t}U_t^\varepsilon\quad\text{and}\quad X_t^\varepsilon:=(U_t^\varepsilon,V_t^\varepsilon)^T. $$ Then, the system (\ref{spde1}) can be rewritten as
\begin{equation} \label{} \left\{ \begin{aligned}\label{spde11} &{\mathord{{\rm d}}} X_t^\varepsilon={\mathcal A} X^{\varepsilon}_t+{{\mathcal F}}(X^{\varepsilon}_t, Y^{\varepsilon}_t)+B{\mathord{{\rm d}}} W^1_t,\\ &{\mathord{{\rm d}}} Y^{\varepsilon}_t =\varepsilon^{-1}A Y^{\varepsilon}_t+\varepsilon^{-1}{\mathcal G}(X^{\varepsilon}_t, Y^{\varepsilon}_t)+\varepsilon^{-1/2}{\mathord{{\rm d}}} W_t^2,\\ &X^{\varepsilon}_0=x,\,Y^{\varepsilon}_0=y, \end{aligned} \right. \end{equation} where $x:=(u,v)^T$, ${\mathcal G}(x,y):=G(\Pi_1(x),y)$ and $$ {\mathcal A}:=\begin{pmatrix}0&I\\A&0\end{pmatrix},\,\, {\mathcal F}(x,y):=\begin{pmatrix}0\\F(\Pi_1(x),y)\end{pmatrix},\,\,B{\mathord{{\rm d}}} W_t^1:=\begin{pmatrix}0\\{\mathord{{\rm d}}} W_t^1\end{pmatrix}, $$ and $F, G$ are defined by (\ref{NG}). Similarly, concerning the averaged equation (\ref{spde22}), let $$ \bar V_t:=\frac{{\mathord{{\rm d}}}}{{\mathord{{\rm d}}} t}\bar U_t\quad\text{and}\quad \bar X_t:=( \bar U_t,\bar V_t)^T. $$ Then we can transfer (\ref{spde22}) into a stochastic evolution equation: \begin{align}\label{spde20} {\mathord{{\rm d}}} \bar{X}_t={\mathcal A}\bar{X}_t{\mathord{{\rm d}}} t+\bar{{\mathcal F}}(\bar{X}_t){\mathord{{\rm d}}} t+B{\mathord{{\rm d}}} W_t^1,\quad \bar X_0=x=(u,v)^T\in {\mathcal H}, \end{align} where $$ \bar {{\mathcal F}}(x):=\begin{pmatrix}0\\\bar F(\Pi_1(x))\end{pmatrix}, $$ and $\bar F$ is defined by (\ref{df1}). It is known (see e.g. \cite{BDT}) that ${\mathcal A}$ generates a strongly continuous group $\{e^{t{\mathcal A}}\}_{t\geq0}$ which is given by \begin{align}\label{group} e^{t{\mathcal A}}=\begin{pmatrix}C_t&(-A)^{-\frac{1}{2}}S_t\\-(-A)^{\frac{1}{2}}S_t&C_t\end{pmatrix}, \end{align} where $C_t:=\cos((-A)^{\frac{1}{2}})t)$ and $S_t:=\sin((-A)^{\frac{1}{2}})t).$ For any $x\in{\mathcal H}$, we have $\interleave e^{{\mathcal A} t}x\interleave_0\leqslant\interleave x\interleave_0.$ Moreover, under the assumptions on $f$ and $g$, one can check that $F\in C_b^{2,\eta}(H\times H, H)$ and $G\in C_B^{2,\eta}(H\times H, H).$ By definition, we further have ${\mathcal F}\in C_b^{2,\eta}({\mathcal H}\times H, {\mathcal H}^1)$ and ${\mathcal G}\in C_B^{2,\eta}({\mathcal H}\times H, H).$ Furthermore, according to \cite[Lemma 3.7]{RXY}, we also have that $\bar{\mathcal F}\in C_b^2({\mathcal H},{\mathcal H}^1)$.
The Poisson equation will be the crucial tool in our paper. Recall that ${\mathcal L}_2(u,y)$ is defined by (\ref{ly2}). If there is no confusion possible, we shall also write \begin{align}\label{ly22} {\mathcal L}_2\varphi(y):={\mathcal L}_2(x,y)\varphi(y):={\mathcal L}_2(\Pi_1(x),y)\varphi(y),\quad\forall\varphi\in C_{\ell}^2(H). \end{align} Consider the following Poisson equation: \begin{align}\label{pois} {\mathcal L}_2(x,y)\psi(x,y)=-\phi(x,y), \end{align} where $x\in {\mathcal H}$ is regarded as a parameter, and $\phi: {\mathcal H}\times H\rightarrow \hat H$ is measurable. To be well-defined, it is necessary to make the following ``centering" assumption on $\phi$: \begin{align}\label{cen} \int_{H}\phi(x,y)\mu^x({\mathord{{\rm d}}} y)=0,\quad\forall x\in {\mathcal H}. \end{align} The following result has been proven in \cite[Theorem 3.2]{RXY}.
\begin{theorem}\label{PP} Let $\eta>0$ and $k=0,1,2$, and assume ${\mathcal G}\in C_B^{k,\eta}({\mathcal H}\times H,H)$. Then for every $\phi(\cdot,\cdot)\in C_{\ell}^{k,\eta}({\mathcal H}\times H,\hat H)$ satisfying (\ref{cen}), there exists a unique solution $\psi(\cdot,\cdot)\in \psi\in C^{k,0}_{\ell}({\mathcal H}\times H,\hat H)\cap {\mathbb C}^{0,2}_{\ell}({\mathcal H}\times H,\hat H)$ to equation (\ref{pois}) which is given by \begin{align*} \psi(x,y)=\int_0^\infty{\mathbb E}\big[\phi(x,Y_t^x(y))\big]{\mathord{{\rm d}}} t, \end{align*} where $Y_t^x(y)=Y_t^u(y)$ satisfies the frozen equation (\ref{froz}). \end{theorem}
\subsection{Moment estimates} We prove the following estimates for the solution $X_t^\varepsilon$ and $Y_t^\varepsilon$ of system (\ref{spde11}).
\begin{lemma}\label{la41}
Let $T>0,$ $x\in {\mathcal H}^1, y\in H,$ and let $(X_t^\varepsilon,Y_t^\varepsilon)$ satisfy
\begin{equation} \label{spde112}
\left\{ \begin{aligned}
&X^{\varepsilon}_t =e^{t{\mathcal A}}x+\int_0^te^{(t-s){\mathcal A}}{\mathcal F}(X^{\varepsilon}_s, Y^{\varepsilon}_s){\mathord{{\rm d}}} s+\int_0^te^{(t-s){\mathcal A}}B{\mathord{{\rm d}}} W^1_s,\\
& Y^{\varepsilon}_t =e^{\frac{t}{\varepsilon}A}y+\varepsilon^{-1}\int_0^te^{\frac{t-s}{\varepsilon}A}{\mathcal G}(X^{\varepsilon}_s, Y^{\varepsilon}_s){\mathord{{\rm d}}} s+\varepsilon^{-1/2}\int_0^te^{\frac{t-s}{\varepsilon}A}{\mathord{{\rm d}}} W_s^2.\\
\end{aligned} \right.
\end{equation}
Then for any $q\geq1,$ we have
\begin{align*}
\sup\limits_{\varepsilon\in(0,1)}{\mathbb E}\Big(\sup\limits_{t\in[0,T]}\interleave X^{\varepsilon}_t\interleave_1^{2q}\Big)\leqslant C_{T,q}\big(1+\interleave x\interleave_1^{2q}+\| y\|^{2q}\big)
\end{align*}
and
\begin{align}\label{msy}
\sup\limits_{\varepsilon\in(0,1)}\sup\limits_{t\in[0,T]}{\mathbb E}\| Y^{\varepsilon}_t\|^{2q}+\sup\limits_{\varepsilon\in(0,1)}{\mathbb E}\left(\int_0^T\|Y_t^\varepsilon\|_1^2{\mathord{{\rm d}}} t\right)^q \leqslant C_{T,q}(1+\| y\|^{2q}),
\end{align}
where $C_{T,q}>0$ is a constant. \end{lemma} \begin{proof}
Applying It\^o's formula (see e.g. \cite[Section 4.2]{LR}) to $\|Y_t^\varepsilon\|^{2q}$ and taking expectation, we have
\begin{align*}
\frac{{\mathord{{\rm d}}}}{{\mathord{{\rm d}}} t}{\mathbb E}\|Y_t^\varepsilon\|^{2q}=\frac{2q}{\varepsilon}{\mathbb E}\left[\|Y_t^\varepsilon\|^{2q-2}\<AY_t^\varepsilon,Y_t^\varepsilon{\rangle}\right]
&+\frac{2q}{\varepsilon}{\mathbb E}\left[\|Y_t^\varepsilon\|^{2q-2}{\langle}{\mathcal G}(X_t^\varepsilon,Y_t^\varepsilon),Y_t^\varepsilon{\rangle}\right]\nonumber\\
&+\Big(\frac{q}{\varepsilon}+\frac{2q(q-1)}{\varepsilon}\Big)Tr(Q_2){\mathbb E}\|Y_t^\varepsilon\|^{2q-2}.
\end{align*}
It follows from Poincar\'e inequality, Young's inequality and (\ref{trace}) that
\begin{align*}
\frac{{\mathord{{\rm d}}}}{{\mathord{{\rm d}}} t}{\mathbb E}\|Y_t^\varepsilon\|^{2q}&\leqslant-\frac{2q\lambda_1}{\varepsilon}{\mathbb E}\|Y_t^\varepsilon\|^{2q}
+\frac{2qC_0}{\varepsilon}{\mathbb E}\|Y_t^\varepsilon\|^{2q-1}\\
&\quad+\Big(\frac{q}{\varepsilon}+\frac{2q(q-1)}{\varepsilon}\Big)Tr(Q_2){\mathbb E}\|Y_t^\varepsilon\|^{2q-2}\leqslant -\frac{qC_0}{\varepsilon}{\mathbb E}\|Y_t^\varepsilon\|^{2q}+\frac{C_0}{\varepsilon}.
\end{align*}
Using Gronwall's inequality, we obtain
\begin{align}\label{msy'}
{\mathbb E}\|Y_t^\varepsilon\|^{2q}\leqslant e^{-\frac{qC_0}{\varepsilon}t}\|y\|^{2q}+\frac{C_0}{\varepsilon}\int_0^te^{-\frac{qC_0}{\varepsilon}(t-s)} {\mathord{{\rm d}}} s\leqslant C_0(1+\|y\|^{2q}).
\end{align} Furthermore, in view of \cite[Theorem 5.3.5]{CPL}, the process $X_t^\varepsilon=(U_t^\varepsilon,V_t^\varepsilon)^T$ enjoys the following energy equality:
\begin{align*}
\interleave X_t^\varepsilon\interleave_1^2=\interleave x\interleave_1^2+2\int_0^t\<V_s^\varepsilon,F(U_s^\varepsilon,Y_s^\varepsilon){\rangle}{\mathord{{\rm d}}} s+2\int_0^t\<U_t^\varepsilon,{\mathord{{\rm d}}} W_s^1{\rangle}+\int_0^tTrQ_1{\mathord{{\rm d}}} s.
\end{align*}
Then it is easy to check that
\begin{align}\label{x2q}
\interleave X_t^\varepsilon\interleave_1^{2q}
\leqslant\! C_0\bigg(1+\interleave x\interleave_1^{2q}+\Big|\!\int_0^t\!\<V_s^\varepsilon,F(U_s^\varepsilon,Y_s^\varepsilon){\rangle}{\mathord{{\rm d}}} s\Big|^q+\Big|\int_0^t\<U_t^\varepsilon,{\mathord{{\rm d}}} W_s^1{\rangle}\Big|^q\bigg).
\end{align}
On the one hand, note that
\begin{align}\label{yf}
&{\mathbb E}\sup\limits_{0\leqslant t\leqslant T}\Big|\int_0^t\<V_s^\varepsilon,F(U_s^\varepsilon,Y_s^\varepsilon){\rangle}{\mathord{{\rm d}}} s\Big|^q\nonumber\\
&\leqslant C_1{\mathbb E}\Big(\int_0^T\|V_s^\varepsilon\|^2{\mathord{{\rm d}}} s\Big)^q+C_1{\mathbb E}\left(\int_0^T(1+\|U_s^\varepsilon\|^2+\|Y_s^\varepsilon\|^{2}){\mathord{{\rm d}}} s\right)^q\nonumber\\
&\leqslant C_1{\mathbb E}\left(\int_0^T\interleave X_s^\varepsilon\interleave_1^{2q}{\mathord{{\rm d}}} s\right)+C_1{\mathbb E}\left(\int_0^T(1+\|Y_s^\varepsilon\|^{2q}){\mathord{{\rm d}}} s\right).
\end{align}
On the other hand, in view of Burkholder-Davis-Gundy's inequality, we have
\begin{align}\label{yw}
{\mathbb E}\sup\limits_{0\leqslant t\leqslant T}\Big|\int_0^t\<U_t^\varepsilon,{\mathord{{\rm d}}} W_s^1{\rangle}\Big|^q&\leqslant C_2TrQ_1{\mathbb E}\left(\int_0^T\|U_s^\varepsilon\|^2{\mathord{{\rm d}}} s\right)^{\frac{q}{2}}\nonumber\\
&\leqslant C_2{\mathbb E}\left(\int_0^T\interleave X_s^\varepsilon\interleave_1^{2q}{\mathord{{\rm d}}} s\right).
\end{align}
Combining (\ref{yf}) and (\ref{yw}) with (\ref{x2q}), we get
\begin{align*}
{\mathbb E}\Big(\sup\limits_{0\leqslant t\leqslant T}\interleave X_t^\varepsilon\interleave_1^{2q}\Big)
&\leqslant C_3(1+\interleave x\interleave_1^{2q})+C_3{\mathbb E}\left(\int_0^T\interleave X_s^\varepsilon\interleave_1^{2q}+\|Y_s^\varepsilon\|^{2q}{\mathord{{\rm d}}} s\right).
\end{align*}
Thus, it follows from Gronwall's inequality that
\begin{align*}
{\mathbb E}(\sup\limits_{0\leqslant t\leqslant T}\interleave X_t^\varepsilon\interleave_1^{2q})
&\leqslant C_4\left(1+\interleave x\interleave_1^{2q}+\int_0^T{\mathbb E}\|Y_s^\varepsilon\|^{2q}{\mathord{{\rm d}}} s\right),
\end{align*}
which together with (\ref{msy'}) yields
\begin{align*}
{\mathbb E}\Big(\sup\limits_{0\leqslant t\leqslant T}\interleave X^{\varepsilon}_t\interleave_1^{2q}\Big)\leqslant C_5(1+\interleave x\interleave_1^{2q}+\| y\|^{2q}).
\end{align*}
In order to prove estimate (\ref{msy}), we deduce that
\begin{align*}
{\mathbb E}\left(\int_0^T\|Y_t^\varepsilon\|_1^2{\mathord{{\rm d}}} t\right)^q&\leqslant C_q\left(\int_0^T\big\|e^{\frac{t}{\varepsilon}A}y\big\|_1^2{\mathord{{\rm d}}} t\right)^q\\ &\quad+C_q\,{\mathbb E}\left(\int_0^T\Big\|\varepsilon^{-1}\int_0^te^{\frac{t-s}{\varepsilon}A}{\mathcal G}((X^{\varepsilon}_s, Y^{\varepsilon}_s){\mathord{{\rm d}}} s\Big\|_1^2{\mathord{{\rm d}}} t\right)^q\\
&\quad+C_q\,{\mathbb E}\left(\int_0^T\Big\|\varepsilon^{-1/2}\int_0^te^{\frac{t-s}{\varepsilon}A}{\mathord{{\rm d}}} W_s^2\Big\|_1^2{\mathord{{\rm d}}} t\right)^q=:\sum_{i=1}^3{\mathscr Y}_i(T,\varepsilon).
\end{align*}
For the first term, we have
\begin{align*}
{\mathscr Y}_1(T,\varepsilon)&\leqslant C_6\left(\int_0^{T/\varepsilon}\sum\limits_{k=1}^\infty\lambda_ke^{-2\lambda_kt}
\<y,e_k{\rangle}^2{\mathord{{\rm d}}} t\right)^q\\&\leqslant C_6\left(\sum\limits_{k=1}^\infty(1-e^{\frac{-2\lambda_kT}{\varepsilon}})\<y,e_k{\rangle}^2\right)^q\leqslant C_6\|y\|^{2q}.
\end{align*}
Note that
\begin{align*}
\Big\|\varepsilon^{-1}\int_0^te^{\frac{t-s}{\varepsilon}A}{\mathcal G}(X^{\varepsilon}_s, Y^{\varepsilon}_s){\mathord{{\rm d}}} s\Big\|_1&\leqslant C_7 \varepsilon^{-1}\int_0^t\Big(\frac{t-s}{\varepsilon}\Big)^{-1/2}e^{-\frac{\lambda_1(t-s)}{2\varepsilon}}\|{\mathcal G}(X^{\varepsilon}_s, Y^{\varepsilon}_s)\|{\mathord{{\rm d}}} s\\
&\leqslant C_7 \int_0^{t/\varepsilon}\frac{e^{-\frac{\lambda_1s}{2}}}{s^{1/2}} {\mathord{{\rm d}}} s\leqslant C_7,
\end{align*}
which implies that
\begin{align*}
{\mathscr Y}_2(T,\varepsilon)\leqslant C_8.
\end{align*}
For the last term, by Minkowski's inequality, Burkholder-Davis-Gundy's inequality and {(\ref{trace})}, we deduce that
\begin{align*}
{\mathscr Y}_3(T,\varepsilon)&\leqslant C_{9}\bigg\{\int_0^T\left({\mathbb E}\Big\|\varepsilon^{-1/2}\int_0^te^{\frac{t-s}{\varepsilon}A}{\mathord{{\rm d}}} W_s^2\Big\|_1^{2q}\right)^{1/q}{\mathord{{\rm d}}} t\bigg\}^q\\
&\leqslant
C_{9}\bigg\{\int_0^T\bigg({\mathbb E}\Big(\varepsilon^{-1}\int_0^t\sum\limits_{k=1}^
\infty\lambda_ke^{-2\lambda_k\frac{t-s}{\varepsilon}}\<Q_2e_k,e_k{\rangle}{\mathord{{\rm d}}} s\Big)^q\bigg)^{1/q}{\mathord{{\rm d}}} t\bigg\}^q\\
&\leqslant C_{9}\bigg\{\int_0^T\bigg({\mathbb E}\Big(\int_0^{t/\varepsilon}\sum\limits_{k=1}^\infty\lambda_k
e^{-2\lambda_ks}\<Q_2e_k,e_k{\rangle}{\mathord{{\rm d}}} s\Big)^q\bigg)^{1/q}{\mathord{{\rm d}}} t\bigg\}^q\leqslant C_{9}.
\end{align*}
Combining the above computations, we get the desired result. \end{proof}
We also need the following estimate for $ {\mathcal A} X_t^\varepsilon $.
\begin{lemma}\label{la44}
Let $T>0,$ $x=(u,v)^T\in {\mathcal H}^1$ and $y\in H$. Then for any $q\geq1$ and $t\in[0,T],$ we have
\begin{align*}
{\mathbb E}\interleave {\mathcal A} X_t^\varepsilon\interleave_0^q\leqslant C_{T,q}(1+\interleave x\interleave_1^q+\|y\|^{q}),
\end{align*}
where $C_{T,q}>0$ is a constant. \end{lemma} \begin{proof} By definition, we have
$${\mathcal A} X_t^\varepsilon=\begin{pmatrix}0&I\\A&0\end{pmatrix}
\begin{pmatrix}U_t^\varepsilon\\V_t^\varepsilon\end{pmatrix}=
\begin{pmatrix}V_t^\varepsilon\\AU_t^\varepsilon\end{pmatrix}.$$
Thus, we deduce that
\begin{align*}
\interleave {\mathcal A} X_t^\varepsilon\interleave_0^q&\leqslant C_q\left(\|V_t^\varepsilon\|^q+\|AU_t^\varepsilon\|^q_{-1}\right)\\
&=C_q\left(\|V_t^\varepsilon\|^q+\|(-A)^{\frac{1}{2}}U_t^\varepsilon\|^q\right).
\end{align*}
It then follows from (\ref{spde112}) that
\begin{align*}
{\mathbb E}\|(-A)^{\frac{1}{2}}U_t^\varepsilon\|^q\leqslant&C_q\big(\|(-A)^{\frac{1}{2}}C_tu\|+\|S_tv\|\big)^q
+C_q{\mathbb E}\Big\|\int_0^tS_{t-s}F(U_s^\varepsilon,Y_s^\varepsilon){\mathord{{\rm d}}} s\Big\|^q\\
&+C_q{\mathbb E}\Big\|\int_0^tS_{t-s}{\mathord{{\rm d}}} W_s^1\Big\|^q:=\sum_{i=1}^3{\mathscr U}_i(t,\varepsilon).
\end{align*}
For the first term, we have
$${\mathscr U}_1(t,\varepsilon)\leqslant C_1\interleave x\interleave_1^q.$$
To control the second term, by Minkowski's inequality and Lemma \ref{la41}, we get
\begin{align*}
{\mathscr U}_2(t,\varepsilon)&\leqslant C_2\Big(\int_0^t\big(1+{\mathbb E}\|U_s^\varepsilon\|^q+{\mathbb E}\|Y_s^\varepsilon\|^{q}\big)^{1/q}{\mathord{{\rm d}}} s\Big)^q\leqslant C_2(1+\interleave x\interleave_1^q+\|y\|^{q}).
\end{align*}
Finally, by Burkholder-Davis-Gundy's inequality, we obtain
$${\mathscr U}_3(t,\varepsilon)\leqslant C_3.$$
Combining the above estimates, we have
$${\mathbb E}\|(-A)^{\frac{1}{2}}U_t^\varepsilon\|^q\leqslant C_4(1+\interleave x\interleave_1^{q}+\|y\|^{q}).$$
Note that
\begin{align*}
V_t^\varepsilon=-(-A)^{\frac{1}{2}}S_tu+C_tv+\int_0^tC_{t-s}F(U_s^\varepsilon,Y_s^\varepsilon){\mathord{{\rm d}}} s+\int_0^tC_{t-s}{\mathord{{\rm d}}} W_s^1.
\end{align*}
In a similar way, we can prove that
$${\mathbb E}\|V_t^\varepsilon\|^q\leqslant C_5(1+\interleave x\interleave_1^{q}+\|y\|^{q}).$$ Combining the above, we get the desired result. \end{proof}
The following estimates for the solution of the averaged equation (\ref{spde20}) can be proved in a similar way as Lemmas \ref{la41} and \ref{la44}, hence we omit the details here.
\begin{lemma}
Let $T>0$ and $x\in {\mathcal H}^1$. The averaged equation (\ref{spde20}) admits a unique mild solution $\bar X_t$ such that for all $t\geq0,$
\begin{align}\label{msb}\bar{X}_t =e^{t{\mathcal A}}x+\int_0^te^{(t-s){\mathcal A}}\bar{{\mathcal F}}(\bar{X}_s){\mathord{{\rm d}}} s+\int_0^te^{(t-s){\mathcal A}}B{\mathord{{\rm d}}} W^1_s.\end{align} Moreover, for any $q\geq1$ we have
\begin{align*}
\sup\limits_{\varepsilon\in(0,1)}{\mathbb E}\Big(\sup\limits_{t\in[0,T]}\interleave\bar X_t\interleave_1^{2q}\Big)\leqslant C_{T,q}(1+\interleave x\interleave_1^{2q})
\end{align*} and
\begin{align*}
{\mathbb E}\interleave {\mathcal A} \bar X_t\interleave_0^q\leqslant C_{T,q}(1+\interleave x\interleave_1^q),
\end{align*}
where $C_{T,q}>0$ is a constant. \end{lemma}
\section{Strong and weak convergence in the averaging principle}
\subsection{Galerkin approximation} It\^o's formula will be used frequently below in the proof of the main result. However, due to the persence of unbounded operators in the equation, we can not apply It\^o's formula for SPDE (\ref{spde11}) directly. For this reason, we use the following Galerkin approximation scheme, which reduces the infinite dimensional setting to a finite dimensional one. For every $n\in{\mathbb N},$ let $H^n=span\{e_1,e_2,\cdots,e_n\}.$ Denote the projection of $H$ onto $H^n$ by $P_n$, and set $$ {{\mathcal F}}_n(x,y):=\begin{pmatrix}0\\P_nF(\Pi_1(x),y)\end{pmatrix},\quad {\mathcal G}_n(x,y):=P_nG(\Pi_1(x),y). $$ It is easy to check that ${\mathcal F}_n$ and ${\mathcal G}_n$ satisfy the same conditions as ${\mathcal F}$ and ${\mathcal G}$ with bounds which are uniform with respect to $n$. Consider the following finite dimensional system: \begin{equation}\label{xyzn} \left\{ \begin{aligned} &{\mathord{{\rm d}}} X^{n,\varepsilon}_t ={\mathcal A} X^{n,\varepsilon}_t{\mathord{{\rm d}}} t+{{\mathcal F}}_n(X^{n,\varepsilon}_t, Y^{n,\varepsilon}_t){\mathord{{\rm d}}} t+P_n{\mathord{{\rm d}}} W^1_t,\\ &{\mathord{{\rm d}}} Y^{n,\varepsilon}_t =\varepsilon^{-1}AY^{n,\varepsilon}_t{\mathord{{\rm d}}} t+\varepsilon^{-1}{\mathcal G}_n(X^{n,\varepsilon}_t, Y^{n,\varepsilon}_t){\mathord{{\rm d}}} t+\varepsilon^{-1/2} P_n{\mathord{{\rm d}}} W_t^2, \end{aligned} \right. \end{equation} with initial values $X_0^{n,\varepsilon}=x^n\in H^n\times H^n$ and $Y_0^{n,\varepsilon}=y^n\in H^n$. The corresponding averaged equation for system (\ref{xyz}) is given by \begin{align}\label{bxn} {\mathord{{\rm d}}} \bar{X}^n_t={\mathcal A}\bar{X}^n_t{\mathord{{\rm d}}} t+\bar{{\mathcal F}}_n(\bar{X}^n_t){\mathord{{\rm d}}} t+P_{n}{\mathord{{\rm d}}} W_t^1,\quad \bar{X}^n_0=x^{n}\in H^n\times H^n, \end{align} where \begin{align}\label{Fn} \bar{{\mathcal F}}_n(x):=\int_{H^n}{{\mathcal F}}_n(x,y)\mu^{x}_n({\mathord{{\rm d}}} y), \end{align} and $\mu^{x}_n({\mathord{{\rm d}}} y)$ is the invariant measure associated with the transition semigroup of the process $Y_t^{x,n}(y)$ which satisfies the frozen equation \begin{align*} {\mathord{{\rm d}}} Y^{x,n}_t =AY^{x,n}_t{\mathord{{\rm d}}} t+ {\mathcal G}_n(x^n, Y^{x,n}_t){\mathord{{\rm d}}} t+ P_n{\mathord{{\rm d}}} W_t^2,\;\;Y^{x,n}_0=y^{n}\in H^n. \end{align*} Recall that $Y_t^u(y)$ satisfies (\ref{froz}) and note that ${\mathcal G}(x,y)=G(u,y).$ We know that $Y_t^{x,n}(y^n)$ converges strongly to $Y_t^x(y):=Y_t^u(y)$. Let $T>0,x\in {\mathcal H}^1$ and $y\in H.$ Then as shown in the proof of \cite[Lemma 3.1]{CPL2}, for any $q\geqslant 1$ and $t\in[0,T],$ we have \begin{align*} \lim\limits_{n\to\infty}{\mathbb E}\interleave X_t^{\varepsilon}- X^{n,\varepsilon}_t\interleave_1^q=0. \end{align*} Furthermore, in view of (\ref{spde112}), (\ref{msb}) and (\ref{group}) we deduce that \begin{align*}
&{\mathbb E}\interleave\bar X^n_t-\bar X_t\interleave_1^q\leqslant {\mathbb E}\Big|\!\Big|\!\Big| \int_0^te^{(t-s){\mathcal A}}(I-P_n)B{\mathord{{\rm d}}} W_s^1\Big|\!\Big|\!\Big| _1^q\\
&+{\mathbb E}\!\left(\int_0^t\!\Big(\big\| (-A)^{-\frac{1}{2}}S_{t-s}(\bar F(\bar U_s)-\bar F_n(\bar U_s))\big\|_1\!+\!\big\| C_{t-s}(\bar F(\bar X_s)-\bar F_n(\bar U_s)\big\|\Big){\mathord{{\rm d}}} s\right)^q\\
&+{\mathbb E}\!\left(\int_0^t\!\Big(\big\| (-A)^{-\frac{1}{2}}S_{t-s}(\bar F_n(\bar U_s)-\bar F_n(\bar U_s^n))\big\|_1\!+\!\big\| C_{t-s}(\bar F_n(\bar U_s)-\bar F_n(\bar U_s^n))\big\|\Big){\mathord{{\rm d}}} s\right)^q. \end{align*}
Since $\|\bar F_n-\bar F\|\to0$ as $n\to\infty$ (see e.g. \cite[(4.4)]{Br1}), the first two terms go to 0 as $n\to\infty$ by the dominated convergence theorem. For the last term, we have \begin{align*}
&{\mathbb E}\left(\int_0^t\Big(\big\| (-A)^{-\frac{1}{2}}S_{t-s}\big(\bar F_n(\bar U_s)-\bar F_n(\bar U_s^n))\big\|_1+\big\| C_{t-s}(\bar F_n(\bar U_s)-\bar F_n(\bar U_s^n))\big\|\Big){\mathord{{\rm d}}} s\right)^q\\
&\leqslant C_1\;{\mathbb E}\left(\int_0^t\|\bar U_s-\bar U^n_s\|_1{\mathord{{\rm d}}} s\right)^q\leqslant C_1\;{\mathbb E}\left(\int_0^t\interleave\bar X_s-\bar X^n_s\interleave_1{\mathord{{\rm d}}} s\right)^q, \end{align*} which in turn yields by Gronwall's inequality that $$ \lim\limits_{n\to\infty}{\mathbb E}\interleave\bar X^n_t-\bar X_t\interleave_1^q=0. $$
Therefore, in order to prove Theorem {\ref{main1}}, we only need to show that for any $q\geq1,$ \begin{align}\label{nsx} \sup_{t\in[0,T]}{\mathbb E}\interleave X_t^{n,\varepsilon}-\bar X^n_t\interleave_1^q\leqslant C_T\,\varepsilon^{q/2}, \end{align} and for every $\varphi\in {\mathbb C}_b^3({\mathcal H})$, \begin{align}\label{nwx}
\sup_{t\in[0,T]}\big|{\mathbb E}[\varphi(X_t^{n,\varepsilon})]-{\mathbb E}[\varphi(\bar X^n_t)]\big|\leqslant C_T\,\varepsilon, \end{align} where {\bf $C_T>0$ is a constant independent of $n$}. In the rest of this section, we shall only work with the approximating system (\ref{xyzn}), and prove bounds that are uniform with respect to $n$. To simplify the notations, we omit the index $n.$ In particular, the space $H^n$ is denoted by $H$.
\subsection{Proof of Theorem \ref{main1} (strong convergence)}
For simplicity, let \begin{align}\label{L1} {\mathcal L}_1\varphi(x):={\mathcal L}_1(x,y)\varphi(x):&={\langle}{\mathcal A} x+{{\mathcal F}}(x,y),D_x\varphi(x){\rangle}_{\mathcal H}\nonumber\\&\;+\frac{1}{2}Tr\left(D^2_{x}\varphi(x)(BQ_1)^{\frac{1}{2}}((BQ_1) ^{\frac{1}{2}})^*\right),\quad\forall\varphi\in C_{\ell}^2({\mathcal H}). \end{align} As shown in Subsection 4.1, to prove the strong convergence result (\ref{rr1}), we only need to prove (\ref{nsx}). To this end, we first establish the following fluctuation estimate for an integral functional of $(X_s^\varepsilon,Y_s^\varepsilon)$ over time interval $[0,t],$ which will play an important role in proving (\ref{nsx}).
\begin{lemma}[Strong fluctuation estimate]\label{strf} Let $T,\eta>0,{x=(u,v)^T\in {\mathcal H}^1}$ and $y\in H.$ Assume that ${{\mathcal F}}\in C_b^{2,\eta}({\mathcal H}\times H,{\mathcal H}^1)$ and ${\mathcal G}\in C^{2,\eta}_B({\mathcal H}\times H,H).$ Then for any $t\in[0,T]$, $q\geq1$ and every $\tilde\phi(x,y):=\begin{pmatrix}0\\\phi(u,y))\end{pmatrix}$ satisfying (\ref{cen}) with $\phi\in C_b^{2,\eta}(H\times H,H),$ we have \begin{align*}
{\mathbb E}\Big|\!\Big|\!\Big|\int_0^t e^{(t-s){\mathcal A}}\tilde\phi(X_s^\varepsilon,Y_s^\varepsilon){\mathord{{\rm d}}} s\Big|\!\Big|\!\Big|_1^q\leqslant C_{T,q}\,\varepsilon^{q/2}, \end{align*} where $C_{T,q}>0$ is a constant independent of $\varepsilon,\eta$ and $n$. \end{lemma}
\begin{proof} Let $\psi$ solve the Poisson equation, $$ {\mathcal L}_2(u,y)\psi(u,y)=-\phi(u,y), $$ and define $$ \tilde\psi_{t}(s,x,y):= e^{(t-s){\mathcal A}}\tilde\psi(x,y):=e^{(t-s){\mathcal A}}\begin{pmatrix}0\\\psi(u,y)\end{pmatrix}. $$ Since ${\mathcal L}_2$ is an operator with respect to the $y$-variable, one can check that \begin{align}\label{ppo} {\mathcal L}_2\tilde\psi_{t}(s,x,y)=-e^{(t-s){\mathcal A}}\tilde\phi(x,y). \end{align} Applying It\^o's formula to $\tilde\psi_{t}(t, X_t^\varepsilon,Y_t^{\varepsilon})$, we get \begin{align}\label{ito1} \tilde\psi_{t}(t, X_t^\varepsilon,Y_t^{\varepsilon})&=\tilde\psi_{t}(0,x,y)+\int_0^t (\partial_s+{\mathcal L}_1)\tilde\psi_{t}(s,X_s^\varepsilon,Y_s^{\varepsilon}){\mathord{{\rm d}}} s\nonumber\\ &\quad+\frac{1}{\varepsilon}\int_0^t{\mathcal L}_2\tilde\psi_{t}(s,X_s^\varepsilon,Y_s^{\varepsilon}){\mathord{{\rm d}}} s+M_t^1+\frac{1}{\sqrt{\varepsilon}}M_t^2, \end{align} where $M_t^1$ and $M_t^2$ are defined by \begin{align*} M_t^1:=\int_0^t D_x\tilde\psi_{t}(s,X_s^\varepsilon,Y_s^{\varepsilon})B{\mathord{{\rm d}}} W_s^1\quad\text{and}\quad M_t^2:=\int_0^t D_y\tilde\psi_{t}(s,X_s^\varepsilon,Y_s^{\varepsilon}){\mathord{{\rm d}}} W_s^2. \end{align*} Multiplying both sides of (\ref{ito1}) by $\varepsilon$ and using (\ref{ppo}), we obtain \begin{align}\label{poi} &\int_0^t e^{(t-s){\mathcal A}}\tilde\phi(X_s^\varepsilon,Y_s^{\varepsilon}){\mathord{{\rm d}}} s=-\int_0^t{\mathcal L}_2\tilde\psi_{t}(s,X_s^\varepsilon,Y_s^{\varepsilon}){\mathord{{\rm d}}} s\nonumber\\ &=\varepsilon\big[\tilde\psi_{t}(0,x,y)-\tilde\psi_{t}(t,X^\varepsilon_t,Y_t^{\varepsilon})\big] +\varepsilon\int_0^t\partial_s\tilde\psi_{t}(s,X_s^\varepsilon,Y_s^{\varepsilon}){\mathord{{\rm d}}} s\nonumber\\ &\quad+\varepsilon\int_0^t{\mathcal L}_1\tilde\psi_{t}(s,X_s^\varepsilon,Y_s^{\varepsilon}){\mathord{{\rm d}}} s+\varepsilon M_t^1+\sqrt{\varepsilon}M_t^2=:\sum_{i=1}^5{\mathscr J}_i(t,\varepsilon). \end{align} According to Theorem \ref{PP} , we have that $\psi \in C_{\ell}^{2,2}(H\times H,H)$ and hence \begin{align*}
\interleave e^{(t-s){\mathcal A}}\tilde\psi(x,y)\interleave_1&=\Big|\!\Big|\!\Big| \begin{pmatrix}(-A)^{-\frac{1}{2}}S_{t-s}\psi(u
,y)\\C_{t-s}\psi(u,y)\end{pmatrix}\Big|\!\Big|\!\Big|_1
\\&=\| (-A)^{-\frac{1}{2}}S_{t-s}\psi(u,y)\|_1+\| C_{t-s}\psi(u,y)\|\\&\leq2\|\psi(u,y)\|\leqslant C_1(1+\| u\|+\|y\|). &\end{align*} As a result, by Lemma {\ref{la41}} we get
\begin{align*}
{\mathbb E}\interleave{\mathscr J}_{1}(t,\varepsilon)\interleave_1^q&\leqslant C_1\,\varepsilon^q(1+{\mathbb E}\| U_t^\varepsilon\|^q+{\mathbb E}\|Y_t^{\varepsilon}\|^{q})\leqslant C_1\,\varepsilon^q. \end{align*} Note that $$ \partial_s\tilde\psi_{t}(s,x,y)=-{\mathcal A} e^{(t-s){\mathcal A}}\tilde\psi(x,y), $$ and that \begin{align*}
\interleave {\mathcal A} e^{(t-s){\mathcal A}}\tilde\psi(x,y)\interleave_1&=\Big|\!\Big|\!\Big| \begin{pmatrix}C_{t-s}\psi(u,y)\\-(-{\mathcal A})^{\frac{1}{2}}S_{t-s}\psi(u,y)\end{pmatrix}\Big|\!\Big|\!\Big|_1
\\&=\| C_{t-s}\psi(u,y)\|_1+\| -(-A)^{\frac{1}{2}}S_{t-s}\psi(u,y)\|\\&\leq2\|\psi(u,y)\|_1\leqslant C_2(1+\| u\|_1^2+\|y\|_1^2), \end{align*} where the last inequality can be obtained as in \cite[(2.16)]{CG}. Thus, using Minkowski's inequality and Lemma \ref{la41} again, we have \begin{align*}
{\mathbb E}\interleave{\mathscr J}_{2}(t,\varepsilon)\interleave_1^q&\leqslant C_2\,\varepsilon^q\,\Big(\int_0^T\big(1+{\mathbb E}\|U_s^\varepsilon\|_1^{2q}\big)^{1/q}{\mathord{{\rm d}}} t\Big)^q\\
&\quad+C_2\,\varepsilon^q\,{\mathbb E}\Big(\int_0^T\|Y_s^{\varepsilon}\|_1^2{\mathord{{\rm d}}} s\Big)^q\leqslant C_2\,\varepsilon^q. \end{align*} For the third term, we have \begin{align*}
&|{\mathcal L}_1\tilde\psi_{t}(s,X_s^\varepsilon,Y_s^{\varepsilon})|\leqslant C_3\,\big(1+\interleave {\mathcal A} X_s^\varepsilon\interleave_0^2+\interleave X_s^\varepsilon\interleave_1^2+\|Y_s^{\varepsilon}\|^2\big), \end{align*} which together with Minkowski's inequality, Lemmas \ref{la41} and {\ref{la44}} yields that \begin{align*}
{\mathbb E}\!\interleave\!{\mathscr J}_{3}(t,\varepsilon)\interleave_1^q&\leqslant\! C_3\,\varepsilon^q\bigg(\!\int_0^t\!\Big({\mathbb E}\big(1\!+\!\interleave {\mathcal A} X_s^\varepsilon\interleave_0^2+\interleave X_s^\varepsilon\interleave_1^2+\| Y_s^{\varepsilon}\|^2\big)^q\Big)^{1/q}{\mathord{{\rm d}}} s\!\bigg)^q\!\leqslant\! C_3\,\varepsilon^q. \end{align*} Finally, by Burkholder-Davis-Gundy's inequality, Theorem {\ref{PP}}, Lemma \ref{la41} and (\ref{trace}), we have \begin{align*}
{\mathbb E}\interleave{\mathscr J}_{4}(t,\varepsilon)\interleave_1^q&\leqslant C_4\,\varepsilon^q\Big(\int_0^T{\mathbb E}\big\| e^{(t-s){\mathcal A}}D_x\tilde\psi(X_s^\varepsilon,Y_s^{\varepsilon})BQ_1^{\frac{1}{2}}\big\|^{2}_{{\mathscr L}_2({\mathcal H}^1)}{\mathord{{\rm d}}} s\Big)^{q/2}\\
&\leqslant C_4\,\varepsilon^q\left(\int_0^T(1+{\mathbb E}\interleave X_s^\varepsilon\interleave_1^{2}+{\mathbb E}\|Y_s^{\varepsilon}\|^{2}){\mathord{{\rm d}}} s\right)^{q/2}\leqslant C_4\,\varepsilon^q, \end{align*} and similarly, \begin{align*} {\mathbb E}\interleave{\mathscr J}_{5}(t,\varepsilon)\interleave_1^q \leqslant C_5\,\varepsilon^{q/2}. \end{align*} Combining the above inequalities with (\ref{poi}), we get the desired estimate. \end{proof}
We are now in the position to give:
\begin{proof}[Proof of estimate (\ref{nsx})]
Fix $T>0$ below. In view of (\ref{spde112}) and (\ref{msb}), for every $t\in[0,T]$ we have \begin{align*} X_t^\varepsilon-\bar X_t&=\int_0^t e^{(t-s){\mathcal A}}\big[\bar {\mathcal F}(X_s^\varepsilon)-\bar {\mathcal F}(\bar X_s)\big]{\mathord{{\rm d}}} s+\int_0^t e^{(t-s){\mathcal A}}\delta {\mathcal F}(X_s^\varepsilon,Y_s^{\varepsilon}){\mathord{{\rm d}}} s, \end{align*} where $\delta {\mathcal F}$ is defined by \begin{align}\label{dF} \delta {\mathcal F}(x,y):={\mathcal F}(x,y)-\bar {\mathcal F}(x)=\begin{pmatrix}0\\ \delta F(\Pi_1(x),y)\end{pmatrix}. \end{align} Thus, we have for any $q\geq1,$ \begin{align*}
{\mathbb E} \interleave X_t^\varepsilon-\bar X_t\interleave_1^q&\leqslant C_0\, {\mathbb E}\Big|\!\Big|\!\Big|\int_0^t e^{(t-s){\mathcal A}}\big[\bar {\mathcal F}(X_s^\varepsilon)-\bar {\mathcal F}(\bar X_s)\big]{\mathord{{\rm d}}} s\Big|\!\Big|\!\Big|_1^q\\
&+C_0\,{\mathbb E}\Big|\!\Big|\!\Big|\int_0^t e^{(t-s){\mathcal A}}\delta {\mathcal F}(X_s^\varepsilon,Y_s^{\varepsilon}){\mathord{{\rm d}}} s\Big|\!\Big|\!\Big|_1^q=:{\mathscr I}_1(t,\varepsilon)+{\mathscr I}_2(t,\varepsilon). \end{align*} Since $\bar{\mathcal F}\in C_b^2({\mathcal H},{\mathcal H}^1),$ by Minkowski's inequality we deduce that \begin{align*} {\mathscr I}_1(t,\varepsilon)&\leqslant C_1{\mathbb E}\Big(\int_0^t\interleave\bar {\mathcal F}(X_s^\varepsilon)-\bar {\mathcal F}(\bar X_s)\interleave_1{\mathord{{\rm d}}} s\Big)^q\leqslant C_1\int_0^t{\mathbb E}\interleave X_s^\varepsilon-\bar X_s\interleave_1^q{\mathord{{\rm d}}} s. \end{align*} For the second term, noting that $\delta {\mathcal F}(x,y)$ satisfies the centering condition (\ref{cen}), it follows by Lemma \ref{strf} directly that \begin{align*} {\mathscr I}_{2}(t,\varepsilon)\leqslant C_2 \,\varepsilon^{q/2}. \end{align*} Thus, we arrive at \begin{align*}\label{} {\mathbb E} \interleave X_t^\varepsilon-\bar X_t\interleave_1^q\leqslant C_3\,\varepsilon^{q/2}+C_3\,\int_0^t{\mathbb E}\interleave X_s^\varepsilon-\bar X_s\interleave_1^q{\mathord{{\rm d}}} s, \end{align*} which together with Gronwall's inequality yields the desired result. \end{proof}
\subsection{Proof of Theorem \ref{main1} (weak convergence)} As in the previous subsection, to prove the weak convergence result in Theorem \ref{main1} , we only need to show (\ref{nwx}). The main reason for the difference between the strong and weak convergence rates in the averaging principle can be seen through the following estimate.
\begin{lemma}[Weak fluctuation estimate]\label{wef1} Let $T,\eta>0,{x=(u,v)^T\in {\mathcal H}^1}$ and $y\in H.$ Assume that ${{\mathcal F}}\in C_b^{2,\eta}({\mathcal H}\times H,{\mathcal H}^1)$ and ${\mathcal G}\in C^{2,\eta}_B({\mathcal H}\times H,H).$ Then for any $t\in[0,T]$, $\phi\in C_{\ell}^{1,2,\eta}([0,T]\times{\mathcal H}\times H)$ satisfying (\ref{cen}) and \begin{align}\label{as01}
|\partial_t\phi(t,x,y)|\leqslant C_0(1+\interleave x\interleave_1^2+\|y\|^2), \end{align} we have \begin{align*} {\mathbb E}\left(\int_0^t\phi(s,X_s^\varepsilon,Y_s^\varepsilon){\mathord{{\rm d}}} s\right)\leqslant C_T\;\varepsilon, \end{align*} where $C_T>0$ is a constant independent of $\varepsilon,\eta$ and $n$. \end{lemma} \begin{proof}
Let $\psi$ solve the Poisson equation \begin{align}\label{pot} {\mathcal L}_2\psi(t,x,y)=-\phi(t,x,y), \end{align} where ${\mathcal L}_2$ is given by (\ref{ly22}). According to Theorem \ref{PP}, we can apply It\^o's formula to $\psi(t,X_t^\varepsilon,Y_t^{\varepsilon})$ to get that \begin{align*} {\mathbb E}[\psi(t,X_t^\varepsilon,Y_t^{\varepsilon})]&=\psi(0,x,y)+ {\mathbb E}\left(\int_0^t(\partial_s+\mathcal{L}_1)\psi(s,X_s^\varepsilon,Y_s^{\varepsilon}){\mathord{{\rm d}}} s\right)\\ &\quad+\frac{1}{\varepsilon}{\mathbb E}\left(\int_0^t\mathcal{L}_2\psi(s,X_s^\varepsilon,Y_s^{\varepsilon}){\mathord{{\rm d}}} s\right). \end{align*} Combining this with (\ref{pot}), we obtain \begin{align*} &{\mathbb E}\left(\int_0^t\phi(s,X_s^\varepsilon,Y_s^{\varepsilon}){\mathord{{\rm d}}} s\right)\\ &=\varepsilon{\mathbb E}\big[\psi(0,x,y)-\psi(t,X_t^\varepsilon,Y_t^{\varepsilon})\big] +\varepsilon{\mathbb E}\left(\int_0^t\mathcal{L}_1\psi(s,X_s^\varepsilon,Y_s^{\varepsilon}){\mathord{{\rm d}}} s\right)\\ &\quad+\varepsilon{\mathbb E}\left(\int_0^t\partial_s\psi(s,X_s^\varepsilon,Y_s^{\varepsilon}){\mathord{{\rm d}}} s\right)=:\sum_{i=1}^3{\mathscr W}_i(t,\varepsilon). \end{align*} By using exactly the same arguments as in the proof of Lemma \ref{strf}, we can get that \begin{align*} {\mathscr W}_1(t,\varepsilon)+{\mathscr W}_2(t,\varepsilon)\leqslant C_1\,\varepsilon. \end{align*} To control the third term, note that $$ {\mathcal L}_2\partial_t\psi(t,x,y)=-\partial_t\phi(t,x,y). $$ In view of condition (\ref{as01}), we have \begin{align*}
|\partial_t\psi(t,x,y)|\leqslant C_0(1+\interleave x\interleave_1^2+\|y\|^{2}), \end{align*} which together with Lemma \ref{la41} implies that \begin{align*}
{\mathscr W}_3(t,\varepsilon)&\leqslant C_2\,\varepsilon{\mathbb E}\left(\int_0^t(1+\interleave X_s^\varepsilon\interleave_1^2+\|Y_s^{\varepsilon}\|^{2}){\mathord{{\rm d}}} s\right)\leqslant C_2\,\varepsilon. \end{align*} Combining the above estimates, we get the desired result. \end{proof}
Given $T>0$, consider the following Cauchy problem on $[0,T]\times{\mathcal H}$: \begin{equation} \label{ke} \left\{ \begin{aligned} &\partial_t\bar u(t,x)=\bar {\mathcal L}_1\bar u(t,x),\quad t\in(0,T],\\ & \bar u(0,x)=\varphi(x), \end{aligned} \right. \end{equation} where $\varphi:{\mathcal H}\to{\mathbb R}$ is measurable and $\bar{\mathcal L}_1$ is formally the infinitesimal generator of the process $\bar X_t$ given by \begin{align}\label{lxb} \bar{\mathcal L}_1\varphi(x)&={\langle}{\mathcal A} x+\bar {\mathcal F}(x),D_x\varphi(x){\rangle}_{\mathcal H}\nonumber\\ &\quad+\frac{1}{2}Tr\left(D^2_{x}\varphi(x)(BQ_1)^{\frac{1}{2}}((BQ_1)^{\frac{1}{2}}) ^*\right),\quad\forall\varphi\in C_{\ell}^2({\mathcal H}). \end{align}
The following result has been proven in \cite[Lemmas A.3-A.5 and 4.3]{FWLL}. \begin{lemma}\label{th47} For every $\varphi\in {\mathbb C}_b^3({\mathcal H})$, there exists a solution $\bar u\in C_b^{1,3}([0,T]\times {\mathcal H})$ to equation (\ref{ke}) which is given by \begin{align*} \bar u(t,x)={\mathbb E}\big[\varphi(\bar X_t(x))\big]. \end{align*} Moreover, for any $t\in[0,T]$ and $x, h\in{\mathcal H}^1$, we have \begin{align*}
\vert \partial_tD_x\bar u(t,x).h|\leqslant C_T\interleave h\interleave_1(1+\interleave x\interleave_1), \end{align*} where $C_T>0$ is a constant. \end{lemma}
Now, we are in the position to give:
\begin{proof}[Proof of estimate (\ref{nwx})] Given $T>0$ and $\varphi\in {\mathbb C}_b^3({\mathcal H})$, let $\bar u$ solve the Cauchy problem (\ref{ke}). For any $t\in[0,T]$ and $x\in {\mathcal H}^1$, define $$ \tilde u(t,x):=\bar u(T-t,x). $$ Then one can check that $$ \tilde u(T,x)=\bar u(0,x)=\varphi(x)\quad\text{and}\quad\tilde u(0,x)=\bar u(T,x)={\mathbb E}[\varphi(\bar X_T(x))]. $$ Using It\^o's formula and taking expectation, we deduce that \begin{align*} {\mathbb E}[\varphi(X_T^{\varepsilon})]-{\mathbb E}[\varphi(\bar X_T)]&={\mathbb E}[\tilde u(T,X_T^{\varepsilon})-\tilde u(0,x)]\\ &={\mathbb E}\left(\int_0^T\big(\partial_t+{\mathcal L}_1\big)\tilde u(t,X_t^{\varepsilon}){\mathord{{\rm d}}} t\right)\\ &={\mathbb E}\left(\int_0^T[\mathcal{L}_1\tilde u(t,X_t^{\varepsilon})-\mathcal{\bar L}_1\tilde u(t,X_t^{\varepsilon})]{\mathord{{\rm d}}} t\right)\\ &={\mathbb E}\left(\int_0^T\langle \delta {\mathcal F}(X_t^\varepsilon,Y_t^\varepsilon), D_x\tilde u(t,X_t^{\varepsilon})\rangle_{\mathcal H} {\mathord{{\rm d}}} t\right). \end{align*} Note that the function $$ \phi(t,x,y):={\langle}\delta {\mathcal F}(x,y), D_x\tilde u(t,x){\rangle}_{\mathcal H} $$ satisfies the centering condition (\ref{cen}). Moreover, by Lemma \ref{th47} we have \begin{align*} \partial_t\phi(t,x,y)={\langle}\delta {\mathcal F}(x,y), \partial_tD_x\bar u(T-t,x){\rangle}_{\mathcal H}
\leqslant C_0(1+\interleave x\interleave_1^2+\|y\|^2). \end{align*} As a result of Lemma \ref{wef1}, we have \begin{align*} {\mathbb E}[\varphi(X_T^{\varepsilon})]-{\mathbb E}[\varphi(\bar X_T)]\leqslant C_1\,\varepsilon, \end{align*} which completes the proof. \end{proof}
\section{Normal deviations}
\subsection{Cauchy problem} Define \begin{align*} {\mathcal Z}_t^\varepsilon:=\frac{X_t^\varepsilon-\bar X_t}{\sqrt{\varepsilon}}. \end{align*}
In view of (\ref{spde11}) and (\ref{spde20}), we consider the process $(X^{\varepsilon}_t, Y^{\varepsilon}_t, \bar X_t, {\mathcal Z}_t^{\varepsilon})$ as the solution to the following system of equations: \begin{equation}\label{xyz} \left\{ \begin{aligned} &{\mathord{{\rm d}}} X^{\varepsilon}_t =\!{\mathcal A} X^{\varepsilon}_t{\mathord{{\rm d}}} t+{\mathcal F}(X^{\varepsilon}_t, Y^{\varepsilon}_t){\mathord{{\rm d}}} t+B{\mathord{{\rm d}}} W^1_t,\qquad\qquad\qquad\qquad\quad\quad\,\,\, X_0^\varepsilon=x,\\ &{\mathord{{\rm d}}} Y^{\varepsilon}_t =\!\varepsilon^{-1}AY^{\varepsilon}_t{\mathord{{\rm d}}} t+\varepsilon^{-1}{\mathcal G}(X^{\varepsilon}_t, Y^{\varepsilon}_t){\mathord{{\rm d}}} t+\varepsilon^{-1/2} {\mathord{{\rm d}}} W_t^2,\qquad\qquad\quad \quad\,\,Y^{\varepsilon}_0=y,\\ &{\mathord{{\rm d}}} \bar X_t=\!{\mathcal A}\bar X_t{\mathord{{\rm d}}} t+\bar {\mathcal F}(\bar X_t){\mathord{{\rm d}}} t+B{\mathord{{\rm d}}} W^1_t,\quad\qquad\quad\qquad\quad\qquad\quad\quad\,\,\,\,\quad\bar X_0=x,\\ &{\mathord{{\rm d}}} {\mathcal Z}_t^{\varepsilon}=\!{\mathcal A} {\mathcal Z}_t^\varepsilon {\mathord{{\rm d}}} t+\varepsilon^{-1/2}[\bar {\mathcal F}(X_t^\varepsilon)-\bar {\mathcal F}( \bar X_t)]{\mathord{{\rm d}}} t+\varepsilon^{-1/2}\delta {\mathcal F}(X_t^\varepsilon,Y_t^\varepsilon){\mathord{{\rm d}}} t, \,\, Z_0^{\varepsilon}=0, \end{aligned} \right. \end{equation} where $\delta {\mathcal F}$ is defined by (\ref{dF}). As a result of Theorem \ref{main1}, we have that for any $q\geq1,$ \begin{align}\label{zne} \sup\limits_{0\leqslant t\leqslant T}{\mathbb E}\interleave {\mathcal Z}_t^\varepsilon\interleave_1^q\leqslant C_T<\infty. \end{align} Furthermore, note that \begin{align*} {\mathcal A} {\mathcal Z}_t^\varepsilon=\begin{pmatrix}0&I\\A&0\end{pmatrix} \begin{pmatrix}\frac{U_t^\varepsilon-\bar U_t}{\sqrt{\varepsilon}}\\\frac{V_t^\varepsilon-\bar V_t}{\sqrt{\varepsilon}}\end{pmatrix}=\begin{pmatrix}\frac{V_t^\varepsilon-\bar V_t}{\sqrt{\varepsilon}}\\\frac{A(U_t^\varepsilon-\bar U_t)}{\sqrt{\varepsilon}}\end{pmatrix}, \end{align*} hence we have \begin{align}\label{azne}
{\mathbb E}\interleave {\mathcal A} {\mathcal Z}_t^\varepsilon\interleave_0^q&={\mathbb E}\left(\Big\|\frac{V_t^\varepsilon-\bar V_t}{\sqrt{\varepsilon}}\Big\|^2+\Big\|\frac{A(U_t^\varepsilon-\bar U_t)}{\sqrt{\varepsilon}}\Big\|_{-1}^2\right)^{q/2}\nonumber\\
&={\mathbb E}\left(\Big\|\frac{V_t^\varepsilon-\bar V_t}{\sqrt{\varepsilon}}\Big\|^2+\Big\|\frac{(U_t^\varepsilon-\bar U_t)}{\sqrt{\varepsilon}}\Big\|_{1}^2\right)^{q/2} \leqslant C_T<\infty. \end{align}
Similarly, we rewrite (\ref{spdez}) as \begin{align}\label{zz0} {\mathord{{\rm d}}} \bar {\mathcal Z}_t={\mathcal A}\bar {\mathcal Z}_t{\mathord{{\rm d}}} t+D_x\bar {\mathcal F}(\bar X_t).\bar {\mathcal Z}_t{\mathord{{\rm d}}} t+\Sigma(\bar X_t){\mathord{{\rm d}}} W_t, \end{align} where $\bar {\mathcal Z}_t=(\bar Z_{t},\dot{\bar Z}_{t})^T$, and $\Sigma$ is a Hilbert-Schmidt operator satisfying \begin{align}\label{sst1} \frac{1}{2}\Sigma(x)\Sigma^*(x)=\overline{\delta {\mathcal F}\otimes\tilde\Psi}(x):=\int_{H}\big[\delta {\mathcal F}(x,y)\otimes\tilde\Psi(x,y)\big]\mu^x({\mathord{{\rm d}}} y), \end{align} (see e.g. \cite[(1.6)]{Ce2} and \cite[(11)]{WR}), and $\tilde\Psi$ is the solution of the following Poisson equation: \begin{align}\label{poF1} {\mathcal L}_2(x,y)\tilde\Psi(x,y)=-\delta {\mathcal F}(x,y). \end{align}
Recall that ${\mathcal L}_2(x,y)={\mathcal L}_2(u,y)$ and $\Psi(u,y)$ solves the Poisson equation (\ref{poF}). Thus, we have $\tilde\Psi(x,y)=\Psi(\Pi_1(x),y)=\Psi(u,y).$ Combining (\ref{spde20}) and (\ref{zz0}), the process $(\bar X_t, \bar {\mathcal Z}_t)$ solves the system \begin{equation*} \left\{ \begin{aligned} &{\mathord{{\rm d}}} \bar{X}_t={\mathcal A}\bar{X}_t{\mathord{{\rm d}}} t+\bar{{\mathcal F}}(\bar{X}_t){\mathord{{\rm d}}} t+{\mathord{{\rm d}}} W_t^1,\qquad\qquad\qquad\,\,\, \bar X_0=x,\\ &{\mathord{{\rm d}}} \bar {\mathcal Z}_t={\mathcal A}\bar {\mathcal Z}_t{\mathord{{\rm d}}} t+D_x\bar {\mathcal F}(\bar X_t).\bar {\mathcal Z}_t{\mathord{{\rm d}}} t+\Sigma(\bar X_t){\mathord{{\rm d}}} W_t,\qquad\!\!\bar {\mathcal Z}_0=0. \end{aligned} \right. \end{equation*} Note that the processes $\bar X_t$ and $\bar {\mathcal Z}_{t}$ depend on the initial value $x$. Below, we shall write $\bar X_t(x)$ when we want to stress its dependence on the initial value, and use $\bar {\mathcal Z}_{t}(x,z)$ to denote the process $\bar {\mathcal Z}_{t}$ with initial point $\bar {\mathcal Z}_0=z\in {\mathcal H}$.
Given $T>0,$ consider the following Cauchy problem on $[0,T]\times{\mathcal H}\times{\mathcal H}$: \begin{equation} \label{kez} \left\{ \begin{aligned} &\partial_t\bar u(t,x,z)=\bar {\mathcal L}\bar u(t,x,z),\quad t\in(0,T],\\ & \bar u(0,x,z)=\varphi(z),\\ \end{aligned} \right. \end{equation} where $\varphi:{\mathcal H}\to{\mathbb R}$ is measurable and $\bar {\mathcal L}$ is formally the infinitesimal generator of the Markov process $(\bar X_t, \bar {\mathcal Z}_t)$, i.e., $$ \bar {\mathcal L}:=\bar {\mathcal L}_1+\bar {\mathcal L}_3,
$$
with $\bar {\mathcal L}_1$ given by (\ref{lxb}) and $\bar {\mathcal L}_3$ defined by \begin{align*} \bar {\mathcal L}_3\varphi(z):=\bar {\mathcal L}_3(x,z)\varphi(z)&:={\langle}{\mathcal A} z+D_x\bar {\mathcal F}(x).z,D_z\varphi(z){\rangle}_{{\mathcal H}}\\&\;+\frac{1}{2}\,Tr\big(D^2_{z}\varphi(z)\Sigma(x)\Sigma^*(x)\big),\quad\forall\varphi\in C_{\ell}^2({\mathcal H}). \end{align*} We have the following result.
\begin{lemma}\label{bure} For every $\varphi\in {\mathbb C}_b^3({\mathcal H})$, there exists a solution $\bar u\in C_b^{1,3,3}([0,T]\times {\mathcal H}\times {\mathcal H})$ to equation (\ref{kez}) which is given by \begin{align}\label{kol} \bar u(t,x,z)={\mathbb E}\big[\varphi(\bar {\mathcal Z}_t(x,z))\big]. \end{align} Moreover,
for any $t\in[0,T]$ and $x,z,h\in {\mathcal H}^1$, we have
\begin{align}\label{utzx}
\vert \partial_tD_z&\bar u(t,x,z).h|+\vert \partial_tD_x\bar u(t,x,z).h|\nonumber\\&\leqslant C_0\big(1+\interleave x\interleave^2_1+\interleave z\interleave_1+\interleave {\mathcal A} x\interleave_0+\interleave {\mathcal A} z\interleave_0\big)\big(\interleave h\interleave_1+\interleave {\mathcal A} h\interleave_0\big),
\end{align} where $C_0>0$ is a positive constant. \end{lemma} \begin{proof}By using the same arguments as in \cite[Section 7]{Br2}, we can prove that equation (\ref{kez}) admits a solution $\bar u\in C_b^{1,3,3}([0,T]\times {\mathcal H}\times {\mathcal H})$ which is given by (\ref{kol}), see also \cite[Section 4]{Br4}.
Moreover, for $x,z,h\in {\mathcal H}^1,$ \begin{align}\label{ptdz} \partial_tD_z\bar u(t,x,z).h=D_z\partial_t\bar u(t,x,z).h=D_z(\bar{\mathcal L}_1+\bar{\mathcal L}_3)\bar u(t,x,z).h, \end{align} On the one hand, we have \begin{align*} &D_z\bar{\mathcal L}_1\bar u(t,x,z).h\nonumber\\&=D_zD_x\bar u(t,x,z).({\mathcal A} x+\bar {\mathcal F}(x),h)+\frac{1}{2}\sum\limits_{n=1}^\infty \beta_{1,n}D_zD_x^2\bar u(t,x,z).(Be_{n},Be_{n},h), \end{align*} which together with $\bar u\in C_b^{1,3,3}([0,T]\times {\mathcal H}\times {\mathcal H})$ yields that \begin{align}\label{lxdz}
&|D_z\bar{\mathcal L}_1\bar u(t,x,z).h|\leqslant C_1(1+\interleave{\mathcal A} x\interleave_0+\interleave x\interleave_1)\interleave h\interleave_1. \end{align} On the other hand, we have \begin{align*} &D_z\bar{\mathcal L}_3\bar u(t,x,z).h\\&={\langle}{\mathcal A} h,D_z\bar u(t,x,z){\rangle}_{\mathcal H}+\<D_x\bar {\mathcal F}( x).h,D_z\bar u(t,x,z)){\rangle}_{\mathcal H}\nonumber\\&+D_z^2\bar u(t,x,z).({\mathcal A} z+D_x\bar {\mathcal F}(x).z,h)+\frac{1}{2}\sum\limits_{n=1}^{\infty} D_z^3\bar u(t,x,z).(\Sigma(x) e_{n},\Sigma(x) e_{n},h). \end{align*} Thus, \begin{align}\label{lzdz}
&|D_z\bar{\mathcal L}_3\bar u(t,x,z).h|\leqslant C_2(1+\interleave{\mathcal A} z\interleave_0+\interleave z\interleave_1+\interleave x\interleave_1^2)(\interleave h\interleave_1+\interleave {\mathcal A} h\interleave_0). \end{align} Combining (\ref{ptdz}), (\ref{lxdz}) and (\ref{lzdz}), we arrive at \begin{align*}
\vert \partial_tD_z\bar u(t,x,z).h|\leqslant C_3\big(1&+\interleave x\interleave^2_1+\interleave z\interleave_1\\
&+\interleave {\mathcal A} x\interleave_0+\interleave {\mathcal A} z\interleave_0\!\big)\!\big(\!\interleave h\interleave_1+\interleave {\mathcal A} h\interleave_0\!\big).
\end{align*} Similarly, we have \begin{align*} &\partial_tD_x\bar u(t,x,z).h=D_x^2\bar u(t,x,z).({\mathcal A} x+\bar {\mathcal F}(x),h)+{\langle}{\mathcal A} h+D_x\bar {\mathcal F}(x).h,D_x\bar u(t,x,z){\rangle}_{{\mathcal H}}\\ &\qquad+\<D^2_x\bar {\mathcal F}( x).(z,h),D_z\bar u(t,x,z)){\rangle}_{\mathcal H}+D_xD_z\bar u(t,x,z).({\mathcal A} z+D_x\bar {\mathcal F}(x).z,h)\\ &\qquad+\frac{1}{2}\sum\limits_{n=1}^\infty \beta_{1,n}D_x^3\bar u(t,x,z).(Be_{n},Be_{n},h)\\ &\qquad+\frac{1}{2}\sum\limits_{n=1}^{\infty} D_xD_z^2\bar u(t,x,z).(\Sigma(x) e_{n},\Sigma(x) e_{n},h)\\ &\qquad+\sum\limits_{n=1}^{\infty} D_z^2\bar u(t,x,z).(D_x(\Sigma(x)) e_{n},\Sigma(x) e_{n},h). \end{align*} By the same argument as above, we can obtain \begin{align*}
\vert \partial_tD_x\bar u(t,x,z).h|\leqslant C_4\big(1&+\interleave x\interleave^2_1+\interleave z\interleave_1\\
&+\interleave {\mathcal A} x\interleave_0+\interleave {\mathcal A} z\interleave_0\!\big)\!\big(\!\interleave h\interleave_1+\interleave {\mathcal A} h\interleave_0\!\big),
\end{align*} which completes the proof. \end{proof}
\subsection{Proof of Theorem \ref{main3}} As before, we reduce the infinite dimensional problem to a finite dimensional one by the Galerkin approximation. Recall that $X_t^{n,\varepsilon}$ and $\bar X_t^{n}$ are defined by (\ref{xyzn}) and (\ref{bxn}), respectively. Define $$ {\mathcal Z}_t^{n,\varepsilon}:=\frac{X_t^{n,\varepsilon}-\bar X_t^{n}}{\sqrt{\varepsilon}}. $$ Then we have \begin{align*} &{\mathord{{\rm d}}} {\mathcal Z}_t^{n,\varepsilon}={\mathcal A} {\mathcal Z}_t^{n,\varepsilon} {\mathord{{\rm d}}} t+\varepsilon^{-1/2}[\bar {\mathcal F}_n(X_t^{n,\varepsilon})-\bar {\mathcal F}_n( \bar X_t^n)]{\mathord{{\rm d}}} t+\varepsilon^{-1/2}\delta {\mathcal F}_n(X_t^{n,\varepsilon},Y_t^{n,\varepsilon}){\mathord{{\rm d}}} t, \end{align*} where $\bar {\mathcal F}_n$ is given by (\ref{Fn}), and $\delta {\mathcal F}_n(x,y):={\mathcal F}_n(x,y)-\bar {\mathcal F}_n(x)$. Let $ \bar {\mathcal Z}_t^n$ satisfy the following linear equation: \begin{align*} {\mathord{{\rm d}}} \bar {\mathcal Z}_t^n={\mathcal A}\bar {\mathcal Z}_t^n{\mathord{{\rm d}}} t+D_x\bar {\mathcal F}_n(\bar X_t^n).\bar {\mathcal Z}_t^n{\mathord{{\rm d}}} t+P_n\Sigma(\bar X_t^n){\mathord{{\rm d}}} W_t, \end{align*} where $ W_t$ is a cylindrical Wiener process in $H$, and $\Sigma(x)$ is defined by (\ref{sst1}). As in \cite[Lemma 5.4]{RXY}, one can check that \begin{align}\label{znz} \lim\limits_{n\to\infty}{\mathbb E}\Big(\interleave{\mathcal Z}^\varepsilon_t-{\mathcal Z}_t^{n,\varepsilon}\interleave_1+\interleave\bar {\mathcal Z}_t-\bar {\mathcal Z}_t^{n}\interleave_1\Big)=0. \end{align}
For any $T>0$ and $\varphi\in {\mathbb C}_b^3({\mathcal H}),$ we have for $t\in[0,T]$, \begin{align}\label{wcz}
\left|{\mathbb E}[\varphi({\mathcal Z}_t^{\varepsilon})]-{\mathbb E}[\varphi(\bar {\mathcal Z}_t)]\right|&\leqslant \left|{\mathbb E}[\varphi({\mathcal Z}_t^{\varepsilon})]-{\mathbb E}[\varphi({\mathcal Z}_t^{n,\varepsilon})]\right|\nonumber\\
&\quad+\left|{\mathbb E}[\varphi({\mathcal Z}_t^{n,\varepsilon})]-{\mathbb E}[\varphi(\bar {\mathcal Z}_t^n)]\right|+\left|{\mathbb E}[\varphi(\bar {\mathcal Z}_t^n)]-{\mathbb E}[\varphi(\bar {\mathcal Z}_t)]\right|. \end{align}
According to (\ref{znz}), the first and the last terms on the right-hand of (\ref{wcz}) converge to $0$ as $n\to\infty$ . Therefore, in order to prove Theorem {\ref{main3}}, we only need to show that \begin{align}\label{nzz}
\sup_{t\in[0,T]}\left|{\mathbb E}[\varphi({\mathcal Z}_t^{n,\varepsilon})]-{\mathbb E}[\varphi(\bar {\mathcal Z}_t^n)]\right|\leqslant C_T\,\varepsilon^{\frac{1}{2}}, \end{align} where {\bf $C_T>0$ is a constant independent of} $n.$ We shall only work with the approximating system in the following subsection, and proceed to prove bounds that are uniform with respect to $n$. To simplify the notations, we shall omit the index $n$ as before.
Define \begin{align}\label{333} &{\mathcal L}_3\varphi(z):={\mathcal L}_3(x,y,\bar x,z)\varphi(z):={\langle}{\mathcal A} z,D_z\varphi(z){\rangle}_{{\mathcal H}}\\ &\,\,+\frac{1}{\sqrt{\varepsilon}}{\langle}\bar {\mathcal F}(x)-\bar {\mathcal F}(\bar x), D_z\varphi(z){\rangle}_{{\mathcal H}}+\frac{1}{\sqrt{\varepsilon}}{\langle}\delta {\mathcal F}(x,y),D_z\varphi(z){\rangle}_{{\mathcal H}},\quad\forall\varphi\in C_{\ell}^1({\mathcal H}).\nonumber \end{align} Given a function $\phi\in C_{\ell}^{1,2,\eta,2}([0,T]\times{\mathcal H}\times H\times {\mathcal H})$ satisfying the centering condition: \begin{align}\label{cen222} \int_{H}\phi(t,x,y,z)\mu^x({\mathord{{\rm d}}} y)=0,\quad\forall t>0, x,z\in {\mathcal H}, \end{align} let $\psi(t,x,y,z)$ solve the following Poisson equation \begin{align}\label{psi1} {\mathcal L}_2(x,y)\psi(t,x,y,z)=-\phi(t,x,y,z), \end{align} where $t,x,z$ are regarded as parameters. Define \begin{align}\label{ftp} \overline{\delta {\mathcal F}\cdot\nabla_z\psi}(t,x,z):=\int_{H}\nabla_z\psi(t,x,y,z).\delta {\mathcal F}(x,y)\mu^x({\mathord{{\rm d}}} y). \end{align} We first establish the following weak fluctuation estimates for an appropriate integral functional of $(X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^{\varepsilon})$ over the time interval $[0,t]$, which will play an important role in the proof of (\ref{nzz}).
\begin{lemma}[Weak fluctuation estimates]\label{wfe2} Let $T,\eta>0$, $x\in {\mathcal H}^1$ and $y\in H$. Assume that ${{\mathcal F}}\in C_b^{2,\eta}({\mathcal H}\times H,{\mathcal H}^1)$ and ${\mathcal G}\in C^{2,\eta}_B({\mathcal H}\times H,H).$ Then for any $t\in[0,T],$ $\phi\in C_{\ell}^{1,2,\eta,2}([0,T]\times{\mathcal H}\times H\times {\mathcal H})$ satisfying (\ref{cen222}) and
\begin{align}\label{as1}
|\partial_t\phi(t,x,y,z)|&\leqslant C_0\big(1+\interleave x\interleave^2_1+\interleave z\interleave_1\nonumber\\
&\qquad\quad+\interleave {\mathcal A} x\interleave_0+\interleave {\mathcal A} z\interleave_0\!\big)\big(1+\interleave x\interleave_1+\|y\|\big), \end{align} we have \begin{align} {\mathbb E}\left(\int_0^t\phi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^{\varepsilon}){\mathord{{\rm d}}} s\right)\leqslant C_T\,\varepsilon^{\frac{1}{2}},\label{we1} \end{align} and \begin{align} {\mathbb E}\left(\frac{1}{\sqrt{\varepsilon}}\int_0^t\phi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^{\varepsilon}){\mathord{{\rm d}}} s\right)\!-\!{\mathbb E}\left(\int_0^t\overline{\delta {\mathcal F}\cdot\nabla_z\psi}(s,X_s^{\varepsilon},{\mathcal Z}_s^{\varepsilon}){\mathord{{\rm d}}} s\right)\leqslant C_T\,\varepsilon^{\frac{1}{2}},\label{we2} \end{align} where $C_T>0$ is a constant independent of $\varepsilon,\eta$ and $n$. \end{lemma} \begin{proof} The proof will be divided into two steps.
\noindent{\bf Step 1.} We first prove estimate (\ref{we1}). Applying It\^o's formula to $\psi(t,X_t^\varepsilon,Y_t^{\varepsilon},{\mathcal Z}_t^\varepsilon)$ and taking expectation, we have \begin{align*} {\mathbb E}[\psi(t,X_t^\varepsilon,Y_t^{\varepsilon},{\mathcal Z}_t^\varepsilon)]&=\psi(0,x,y,0)+ {\mathbb E}\left(\int_0^t(\partial_s+\mathcal{L}_1+{\mathcal L}_3)\psi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^\varepsilon){\mathord{{\rm d}}} s\right)\\ &\quad+\frac{1}{\varepsilon}{\mathbb E}\left(\int_0^t\mathcal{L}_2\psi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^\varepsilon){\mathord{{\rm d}}} s\right), \end{align*} where ${\mathcal L}_1$ and ${\mathcal L}_3$ are defined by (\ref{L1}) and (\ref{333}), respectively. Combining this with (\ref{psi1}), we obtain \begin{align}\label{i} &{\mathbb E}\left(\int_0^t\phi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^{\varepsilon}){\mathord{{\rm d}}} s\right) =\varepsilon{\mathbb E}\big[\psi(0,x,y,0)-\psi(t,X_t^\varepsilon,Y_t^{\varepsilon},{\mathcal Z}_t^\varepsilon)\big]\nonumber\\ &\qquad+\varepsilon{\mathbb E}\left(\int_0^t\partial_s\psi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^{\varepsilon}){\mathord{{\rm d}}} s\right)+\varepsilon{\mathbb E}\left(\int_0^t\mathcal{L}_1\psi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^{\varepsilon}){\mathord{{\rm d}}} s\right)\nonumber\\ &\qquad+\varepsilon{\mathbb E}\left(\int_0^t\mathcal{L}_3\psi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^{\varepsilon}){\mathord{{\rm d}}} s\right)=:\sum_{i=1}^4{\mathscr Q}_i(t,\varepsilon). \end{align} By Theorem \ref{PP} and Lemma {\ref{la41}}, we have \begin{align*}
{\mathscr Q}_1(t,\varepsilon)\leqslant C_1\varepsilon{\mathbb E}\big(1+\interleave X_t^\varepsilon\interleave_1+\|Y_t^{\varepsilon}\|\big)\leqslant C_1\varepsilon. \end{align*} For the second term, by using Theorem \ref{PP}, condition (\ref{as1}), Lemma \ref{la41}, (\ref{zne}) and (\ref{azne}), we get \begin{align*} {\mathscr Q}_2(t,\varepsilon)\leqslant C_2\bigg(\int_0^t{\mathbb E}\big(1&+\interleave{\mathcal A} X_s^\varepsilon\interleave_0^2+\interleave{\mathcal A} {\mathcal Z}_s^\varepsilon\interleave_0^2\\
&+\interleave X_s^\varepsilon\interleave_1^4++\|Y_s^{\varepsilon}\|^2+\interleave {\mathcal Z}_s^\varepsilon\interleave_1^2\big){\mathord{{\rm d}}} s\bigg)\leqslant C_2\,\varepsilon. \end{align*} To treat the third term, since for each $t\in [0,T]$, $\phi(t,\cdot,\cdot,\cdot)\in C_{\ell}^{2,\eta,2}({\mathcal H}\times H\times{\mathcal H})$, by Theorem \ref{PP}, we have $\psi(t,\cdot,\cdot,\cdot)\in C_{\ell}^{2,2,2}({\mathcal H}\times H\times{\mathcal H})$, hence \begin{align*}
|{\mathcal L}_1\psi(t,X_t^\varepsilon,Y_t^{\varepsilon},{\mathcal Z}_t^\varepsilon)|&\leqslant |\langle {\mathcal A} X_t^\varepsilon+{\mathcal F}(X_t^\varepsilon,Y_t^\varepsilon),D_x\psi(t,X_t^\varepsilon,Y_t^{\varepsilon},{\mathcal Z}_t^\varepsilon)\rangle_{\mathcal H}| \\&\quad+\frac{1}{2}
Tr((BQ_1^{\frac{1}{2}})(BQ_1^{\frac{1}{2}})^*)\|D^2_{x}\psi(t,X_t^\varepsilon,Y_t^{\varepsilon},{\mathcal Z}_t^\varepsilon)\|_{{\mathscr L}({\mathcal H}\times{\mathcal H})}\\&
\leqslant C_3\big(1+\interleave{\mathcal A} X_t^\varepsilon\interleave_0^2+\interleave X_t^\varepsilon\interleave_1^2+\|Y_t^{\varepsilon}\|^2\big). \end{align*} As a result of Lemmas \ref{la41} and \ref{la44}, we deduce that \begin{align*}
{\mathscr Q}_3(t,\varepsilon)&\leqslant C_3\,\varepsilon{\mathbb E}\left(\int_0^t\Big(\interleave{\mathcal A} X_s^\varepsilon\interleave_0^2+\interleave X_s^\varepsilon\interleave_1^2+\|Y_s^{\varepsilon}\|^2\Big){\mathord{{\rm d}}} s\right)\leqslant C_3\,\varepsilon. \end{align*} For the last term, we have \begin{align*} {\mathscr Q}_{4}(t,\varepsilon)&=\varepsilon{\mathbb E}\left(\int_0^t{\langle}{\mathcal A} {\mathcal Z}_s^\varepsilon,D_z\psi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^\varepsilon){\rangle}_{\mathcal H}{\mathord{{\rm d}}} s\right)\\&+\sqrt{\varepsilon}{\mathbb E}\left(\int_0^t{\langle} \bar {\mathcal F}(X_s^\varepsilon)-\bar {\mathcal F}(\bar X_s),D_z\psi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^\varepsilon){\rangle}_{\mathcal H}{\mathord{{\rm d}}} s\right)\\ &+\sqrt{\varepsilon}{\mathbb E}\left(\int_0^t{\langle}\delta {\mathcal F}(X_s^\varepsilon,Y_s^{\varepsilon}),D_z\psi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^\varepsilon){\rangle}_{\mathcal H}{\mathord{{\rm d}}} s\right) . \end{align*} It follows from (\ref{azne}) and Lemma \ref{la41} again that \begin{align*}
{\mathscr Q}_{4}(t,\varepsilon)&\leqslant C_4\sqrt{\varepsilon}{\mathbb E}\left(\int_0^t(1+\interleave {\mathcal A} {\mathcal Z}_s^\varepsilon\interleave_0^2+\interleave X_s^\varepsilon\interleave_1^2+\|Y_s^{\varepsilon}\|^2){\mathord{{\rm d}}} s\right) \leqslant C_4 \sqrt{\varepsilon}. \end{align*} Combining the above inequalities with (\ref{i}), we get the desired result.
\noindent{\bf Step 2.} We proceed to prove estimate (\ref{we2}). By following exactly the same arguments as in the proof of Step 1, we get that \begin{align*} &{\mathbb E}\left(\frac{1}{\sqrt{\varepsilon}}\int_0^t\phi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^{\varepsilon}){\mathord{{\rm d}}} s\right)\leqslant C_0\sqrt{\varepsilon}+\sqrt{\varepsilon}{\mathbb E}\left(\int_0^t\mathcal{L}_3\psi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^{\varepsilon}){\mathord{{\rm d}}} s\right). \end{align*} For the last term, by definition (\ref{333}) we have \begin{align*} &\sqrt{\varepsilon}{\mathbb E}\left(\int_0^t\mathcal{L}_3\psi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^{\varepsilon}){\mathord{{\rm d}}} s\right)\\&=\sqrt{\varepsilon}{\mathbb E}\left(\int_0^t{\langle}{\mathcal A} {\mathcal Z}_s^\varepsilon,D_z\psi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^\varepsilon){\rangle}_{\mathcal H}{\mathord{{\rm d}}} s\right)\\ &+{\mathbb E}\left(\int_0^t{\langle} \bar {\mathcal F}(X_s^\varepsilon)-\bar {\mathcal F}(\bar X_s),D_z\psi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^\varepsilon){\rangle}_{\mathcal H}{\mathord{{\rm d}}} s\right)\\ &+{\mathbb E}\left(\int_0^t{\langle}\delta {\mathcal F}(X_s^\varepsilon,Y_s^{\varepsilon}),D_z\psi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^\varepsilon){\rangle}_{\mathcal H}{\mathord{{\rm d}}} s\right) =:\sum\limits_{i=1}^3{\mathscr T}_{i}(t,\varepsilon). \end{align*} Using Lemma \ref{la41} and (\ref{azne}), we get \begin{align*}
{\mathscr T}_{1}(t,\varepsilon)\leqslant C_1\sqrt{\varepsilon}{\mathbb E}\left(\int_0^t\interleave {\mathcal A} {\mathcal Z}_s^\varepsilon\interleave_0(1+\interleave X_s^\varepsilon\interleave_1+\|Y_s^{\varepsilon}\|){\mathord{{\rm d}}} s\right) \leqslant C_1 \sqrt{\varepsilon}. \end{align*} According to H\"older's inequality, Lemma \ref{la41} and Theorem \ref{main1}, we have \begin{align*}
{\mathscr T}_{2}(t,\varepsilon)\leqslant C_2\int_0^t\big({\mathbb E}\interleave X_s^{\varepsilon}-\bar X_s\interleave_1^2\big)^{1/2}\big(1+{\mathbb E}|\| X_s^\varepsilon|\|_1^2+{\mathbb E}\| Y_s^{\varepsilon}\|^{2}\big)^{1/2} {\mathord{{\rm d}}} s\leqslant C_2\,\sqrt{\varepsilon}. \end{align*} Thus, we deduce that \begin{align*} &{\mathbb E}\left(\frac{1}{\sqrt{\varepsilon}}\int_0^t\phi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^{\varepsilon}){\mathord{{\rm d}}} s\right)-{\mathbb E}\left(\int_0^t\overline{\delta {\mathcal F}\cdot\nabla_z\psi}(s,X_s^{\varepsilon},{\mathcal Z}_s^{\varepsilon}){\mathord{{\rm d}}} s\right)\nonumber\\ &\leqslant\! C_3 \sqrt{\varepsilon}+\!{\mathbb E}\left(\int_0^t\!\!\big({\langle}\delta {\mathcal F}(X_s^\varepsilon,Y_s^{\varepsilon}),D_z\psi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^\varepsilon){\rangle}_{\mathcal H}-\overline{\delta {\mathcal F}\cdot\nabla_z\psi}(s,X_s^{\varepsilon},{\mathcal Z}_s^{\varepsilon})\big){\mathord{{\rm d}}} s\!\right), \end{align*} where $\overline{\delta {\mathcal F}\cdot\nabla_z\psi}$ is defined by (\ref{ftp}). Note that the function $$ \tilde \phi(t,x,y,z):={\langle}\delta {\mathcal F}(x,y),D_z\psi(t,x,y,z){\rangle}_1-\overline{\delta {\mathcal F}\cdot\nabla_z\psi}(t,x,z) $$ satisfies the centering condition (\ref{cen222}) and condition (\ref{as1}). Thus, using (\ref{we1}) directly, we obtain $${\mathbb E}\left(\int_0^t\big({\langle}\delta {\mathcal F}(X_s^\varepsilon,Y_s^{\varepsilon}),D_z\psi(s,X_s^\varepsilon,Y_s^{\varepsilon},{\mathcal Z}_s^\varepsilon){\rangle}_{\mathcal H}-\overline{\delta {\mathcal F}\cdot\nabla_z\psi}(s,X_s^{\varepsilon},{\mathcal Z}_s^{\varepsilon})\big){\mathord{{\rm d}}} s\right)\leqslant C_4\,\sqrt{\varepsilon},$$ which completes the proof. \end{proof}
Now, we are in the position to give: \begin{proof}[Proof of estimate (\ref{nzz})]
Fix $T>0.$ For any $t\in[0,T]$ and $x,z\in {\mathcal H}^1$, let $$ \tilde u(t,x,z)=\bar u(T-t,x,z). $$ It is easy to check that $$ \tilde u(0,x,0)=\bar u(T,x,0)={\mathbb E}[\varphi(\bar {\mathcal Z}_T)]\quad\text{and}\quad\tilde u(T,x,z)=\bar u(0,x,z)=\varphi(z). $$ Applying It\^o's formula, by (\ref{kez}) we have \begin{align*} &{\mathbb E}[\varphi({\mathcal Z}_T^{\varepsilon})]-{\mathbb E}[\varphi(\bar {\mathcal Z}_T)]={\mathbb E}[\tilde u(T,\bar X_T,{\mathcal Z}_T^{\varepsilon})-\tilde u(0,x,0)]\\& ={\mathbb E}\left(\int_0^T\big(\partial_t+{\mathcal L}_1+{\mathcal L}_3\big)\tilde u(t,X_t^{\varepsilon},{\mathcal Z}_t^{\varepsilon}){\mathord{{\rm d}}} t\right)\nonumber\\ &={\mathbb E}\left(\int_0^T(\mathcal{L}_1-\mathcal{\bar L}_1)\tilde u(t,X_t^{\varepsilon},{\mathcal Z}_t^{\varepsilon}){\mathord{{\rm d}}} t\right)+{\mathbb E}\left(\int_0^T(\mathcal{L}_3-\mathcal{\bar L}_3)\tilde u(t,X_t^{\varepsilon},{\mathcal Z}_t^{\varepsilon}){\mathord{{\rm d}}} t\right)\nonumber\\ &={\mathbb E}\left(\int_0^T\langle {\mathcal F}(X_t^\varepsilon,Y_t^\varepsilon)-\bar {\mathcal F}(X_t^\varepsilon), D_x\tilde u(t,X_t^{\varepsilon},{\mathcal Z}_t^{\varepsilon}))\rangle_{\mathcal H} {\mathord{{\rm d}}} t\right)\\ &\quad+{\mathbb E}\left(\int_0^T\Big\langle \frac{\bar {\mathcal F}(X_t^\varepsilon)-\bar {\mathcal F}(\bar X_t)}{\sqrt{\varepsilon}}-D_x\bar {\mathcal F}( X_t^\varepsilon) .{\mathcal Z}_t^\varepsilon, D_z\tilde u(t,X_t^{\varepsilon},{\mathcal Z}_t^{\varepsilon}))\Big\rangle_{\mathcal H} {\mathord{{\rm d}}} t\right)\\ &\quad+\bigg[{\mathbb E}\left(\frac{1}{\sqrt{\varepsilon}}\int_0^T\langle {\mathcal F}(X_t^\varepsilon,Y_t^\varepsilon)-\bar {\mathcal F}(X_t^\varepsilon), D_z\tilde u(t,X_t^{\varepsilon},{\mathcal Z}_t^{\varepsilon}))\rangle_{\mathcal H} {\mathord{{\rm d}}} t\right)\\ &\qquad\qquad-\frac{1}{2}{\mathbb E}\left(\int_0^T Tr(D^2_{z}\tilde u(t,X_t^{\varepsilon},{\mathcal Z}_t^{\varepsilon})\Sigma( X_t^\varepsilon)\Sigma( X_t^\varepsilon)^{*}){\mathord{{\rm d}}} t\right)\bigg]:=\sum\limits_{i=1}^3{\mathscr N}_i(T,\varepsilon). \end{align*}
For the first term, recall that $\tilde\Psi$ solves the Poisson equation (\ref{poF1}) and define $$ \psi(t,x,y,z):={\langle}\tilde\Psi(x,y),D_x\tilde u(t,x,z){\rangle}_{\mathcal H}. $$ Since ${\mathcal L}_2$ is an operator with respect to the $y$-variable, one can check that $\psi$ solves the following Poisson equation: \begin{align*} \mathcal{L}_2(x,y)\psi(t,x,y,z)=-\langle \delta {\mathcal F}(x,y),D_x\tilde u(t,x,z)\rangle_{\mathcal H}=:-\phi(t,x,y,z). \end{align*} It is obvious that $\phi$ satisfies the centering condition (\ref{cen222}). Furthermore, by (\ref{utzx}) we get \begin{align*}
|\partial_t\phi(t,x,y,z)|&=\left|{\langle}\delta {\mathcal F}(x,y), \partial_tD_{x}\bar u(T-t,x,z){\rangle}_{\mathcal H}\right|\\ &\leqslant C_1\big(1+\interleave x\interleave^2_1+\interleave z\interleave_1+\interleave {\mathcal A} x\interleave_0+\interleave {\mathcal A} z\interleave_0\big)\\ &\qquad\times\big(\interleave \delta {\mathcal F}(x,y)\interleave_1+\interleave {\mathcal A} \delta {\mathcal F}(x,y)\interleave_0\big)\\
&\leqslant C_1\big(1+\interleave x\interleave^2_1+\interleave z\interleave_1+\interleave {\mathcal A} x\interleave_0+\interleave {\mathcal A} z\interleave_0\!\big)\big(1+\interleave x\interleave_1+\|y\|\big). \end{align*} Thus, it follows from (\ref{we1}) directly that $${\mathscr N}_1(T,\varepsilon)\leqslant C_1\sqrt{\varepsilon}. $$ To control the second term, by the mean value theorem, H\"older's inequality, Lemma \ref{bure}, Theorem \ref{main1} and (\ref{zne}) we deduce that for $\vartheta\in(0,1),$ \begin{align*}
{\mathscr N}_2(T,\varepsilon)&\leqslant{\mathbb E}\bigg(\int_0^T\big|\big\langle[D_x\bar {\mathcal F}(X_t^\varepsilon+\vartheta(X_t^\varepsilon-\bar X_t))\\
&\qquad\qquad\qquad-D_x\bar {\mathcal F}( X_t^\varepsilon) ].{\mathcal Z}_t^\varepsilon, D_z\tilde u(t,X_t^{\varepsilon},{\mathcal Z}_t^{\varepsilon})\big\rangle_{\mathcal H}\big|{\mathord{{\rm d}}} t\bigg)\\ &\leqslant C_2\int_0^T\big({\mathbb E}\interleave X_t^\varepsilon-\bar X_t\interleave _1^2\big)^{1/2}\big({\mathbb E}\interleave {\mathcal Z}_t^\varepsilon\interleave_1^2\big)^{1/2}{\mathord{{\rm d}}} t\leqslant C_2\sqrt{\varepsilon}. \end{align*}
For the last term, define $$ \hat\psi(t,x,y,z):={\langle}\tilde\Psi(x,y),D_z\tilde u(t,x,z){\rangle}_{\mathcal H}. $$ Then $\hat\psi$ solves the Poisson equation \begin{align*} \mathcal{L}_2(x,y)\hat\psi(t,x,y,z)=-\langle \delta {\mathcal F}(x,y),D_z\tilde u(t,x,z)\rangle_{\mathcal H}=:-\hat\phi(t,x,y,z). \end{align*} By exactly the same arguments as above, we have that $\hat\phi$ satisfies the centering condition (\ref{cen222}) and condition (\ref{as1}). Furthermore, by the definition of $\Sigma$ in (\ref{sst1}), we have \begin{align*} &\overline{\delta {\mathcal F}\cdot\nabla_z\hat\psi}(t,x,z)=\int_{H}D_z\hat\psi(t,x,y,z).\delta {\mathcal F}(x,y)\mu^x({\mathord{{\rm d}}} y)\\&=\int_{H}D_z^2\tilde u(t,x,z).(\tilde\Psi(x,y),\delta {\mathcal F}(x,y))\mu^x({\mathord{{\rm d}}} y)=\frac{1}{2}Tr(D_z^2\tilde u(t,x,z)\Sigma(x)\Sigma^*(x)). \end{align*} Thus, it follows by (\ref{we2}) directly that $$ {\mathscr N}_3(T,\varepsilon)\leqslant C_3\,\sqrt{\varepsilon}. $$ Combining the above computations, we get the desired result. \end{proof}
\end{document} | arXiv |
As matrices
Introduction to matrices
You can view a matrix simply as a generalization of a vector, where we arrange numbers in both rows and columns. Let's keep the number of rows and columns arbitrary, letting $m$ be the number of rows and $n$ the number of columns. We refer to such a matrix as an $m \times n$ matrix and write it as \begin{align*} A= \left[ \begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1n}\\ a_{21} & a_{22} & \ldots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & \ldots & a_{mn} \end{array} \right]. \end{align*} For example, a $3 \times 2$ matrix is \begin{align*} B= \left[ \begin{array}{rr} 4 & -3\\ 7 & 9\\ -5& 0 \end{array} \right], \end{align*} and a $4 \times 7$ matrix is \begin{align*} C= \left[ \begin{array}{rrrrrrr} 9 & -9 & -8 & 1 & 4 & -3 & -3\\ 7 & -1 & 7 & -3 & -5& -2& 9\\ 11 & 1 & 8 & -5& -5 & 0 & -2\\ 9 & -2 & -8 & -1 & 3 & 10 & 0 \end{array} \right]. \end{align*}
The arrangement of a matrix in rows and columns is more than just to make it look pretty. The structure of a matrix allows us to define a fundamental operation on matrices: multiplication. This multiplication forms the basis of linear algebra. In particular, this matrix multiplication allows matrices to represent linear transformations (or linear functions) that transform vectors into other vectors. (A simple example of a linear transformation is the rotation of a vector.) Other uses of matrices involve calculating their determinant.
Vectors as matrices
The concept of matrices is so powerful, that in many cases, we make our lives simpler by viewing a vector as a special type of matrix. By comparing a vector such as $\vc{x}=(1,5,3)$ to a matrix, it initially seems that the difference between vectors and matrices is that vectors have only one row while matrices have multiple rows. However, there is one important twist (literally) that isn't apparent when writing vectors in the form $\vc{x}=(1,5,3)$ . When we view vectors as matrices, we actually view them as a rotated version of the standard form, writing an $n$-dimensional vector as a $n \times 1$ column matrix \begin{align*} \vc{x} = \left[ \begin{array}{c} x_1\\ x_2\\ x_3\\ \vdots\\ x_n \end{array} \right]. \end{align*} We often call $\vc{x}$ an $n\times 1$ column vector and use the terms "column vector" and "column matrix" synonymously. The vector $\vc{x}=(1,5,3)$ written as a $3 \times 1$ column vector would be \begin{align*} \vc{x} = \left[ \begin{array}{c} 1 \\ 5 \\ 3 \end{array} \right]. \end{align*}
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Solve for $n$: $0.03n + 0.08(20 + n) = 12.6$.
Expanding the product on the left gives $0.03n + 0.08\cdot 20 + 0.08n = 12.6$. Simplifying the left side gives $0.11n + 1.6 = 12.6$. Subtracting 1.6 from both sides gives $0.11n = 11$, and dividing by 0.11 gives $n = \boxed{100}$. | Math Dataset |
\begin{document}
\title{Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities}
\author{M. Bonforte\affil{1}{Depto. de Matem\'{a}ticas, Universidad Aut\'{o}noma de Madrid (UAM), Campus de Cantoblanco, 28049 Madrid, Spain}, J. Dolbeault\affil{2}{Ceremade (UMR CNRS nr.~7534), Universit\'e Paris-Dau\-phine, Place de Lattre de Tassigny, 75775 Paris 16, France}, G. Grillo\affil{3}{Dip. di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.}, J. L. V\'azquez\affil{1}{}\hspace{-2mm}\affil{4}{ICMAT~at~UAM} } \maketitle
\begin{article}
\begin{abstract} The goal of this note is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It relies on the identification of the optimal constants in some related Hardy-Poincar\'e inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities and rates for nonlinear diffusion equations.\end{abstract}
\keywords{Fast diffusion equation | porous media equation | Barenblatt solutions | Hardy-Poincar\'e inequalities | large time behaviour | asymptotic expansion | intermediate asymptotics | sharp rates | optimal constants}
\section{Introduction} The evolution equation \beq{Eqn} \frac{\partial u}{\partial \tau}=\nabla\cdot(u^{m-1}\,\nabla u)=\frac 1m\,\Delta u^m \end{equation} with $m\neq 1$ is a simple example of a nonlinear diffusion equation which generalizes the heat equation, and appears in a wide number of applications. Solutions differ from the linear case in many respects, notably concerning existence, regularity and large-time behaviour. We consider positive solutions $u(\tau,y)$ of this equation posed for $\tau\ge 0$ and $y\in {\mathbb R}^d$, $d\ge 1$. The parameter $m$ can be any real number. The equation makes sense even in the limit case $m=0$, where $u^m/m$ has to be replaced by $\log u$, and is formally parabolic for all $m\in{\mathbb R}$. Notice that \eqref{Eqn} is degenerate at the level $u=0$ when $m>1$ and singular when $m<1$. We consider the initial-value problem with nonnegative datum $u(\tau=0,\cdot)=u_0\in L_{{\rm loc}}^1(dx)$, where $dx$ denotes Lebesgue's measure on ${\mathbb R}^d$. Further assumptions on~$u_0$ are needed and will be specified later.
The description of the asymptotic behaviour of the solutions of \eqref{Eqn} as $\tau\to\infty$ is a classical and very active subject. If $m=1$, the convergence of solutions of the heat equation with $u_0\in L_+^1(dx)$ to the Gaussian kernel (up to a mass factor) is a cornerstone of the theory. In the case of equation \eqref{Eqn} with $m>1$, known in the literature as the \emph{porous medium equation}, the study of asymptotic behaviour goes back to \cite{MR586735}. The result extends to the exponents $m\in(m_c,1)$ with $m_c:=(d-2)/d$; see~\cite{MR1977429}. In these results the Gaussian kernel is replaced by some special self-similar solutions $U_{D, T}$ known as the \emph{Barenblatt solutions} (see~\cite{Ba52}) given by \begin{equation}\label{baren.form1}
U_{D, T}(\tau,y):=\frac 1{R(\tau)^d} \left(D+\tfrac{1-m}{2\,d\,|m-m_c|}
\big|\tfrac{y}{R(\tau)}\big|^2\right)^{-\frac{1}{1-m}}_+ \end{equation} whenever $m>m_c$ and $m\ne 1,$ with \[ R(\tau):=(T+\tau)^{\frac{1}{d\,(m-m_c)}}, \] where $T\ge 0$ and $D>0$ are free parameters. To some extent, these solutions play the role of the fundamental solution of the linear diffusion equations, since $\lim_{\tau\to 0}U_{D,0}(\tau,y)=M\,\delta$, where $\delta$ is the Dirac delta distribution, and $M$ depends on~$D$. Notice that the Barenblatt solutions converge as $m\to 1$ to the fundamental solution of the heat equation, up to the mass factor $M$. The results of \cite{MR586735,MR1977429} say that $U_{D,T}$ also describes the large time asymptotics of the solutions of equation \eqref{Eqn} as $\tau\to\infty$ provided $M=\int_{{\mathbb R}^d}u_0\,dy$ is finite, a condition that uniquely determines $D=D(M)$. Notice that in the range $m\ge m_c$, solutions of~\eqref{Eqn} with $u_0\in L_+^1(dx)$ exist globally in time and mass is conserved: $\int_{{\mathbb R}^d}u(\tau,y)\,dy=M$ for any $\tau\ge 0$.
On the other hand, when $m<m_c$, a natural extension for the Barenblatt functions can be achieved by considering the same expression~\eqref{baren.form1}, but a different form for~$R$, that is \begin{equation*} R(\tau):=(T-\tau)^{-\frac{1}{d\,(m_c-m)}}\,. \end{equation*}
The parameter $T$ now denotes the \emph{extinction time}, a new and important feature. The limit case $m=m_c$ is covered by $R(\tau)=e^\tau$, $U_{D,T}(\tau,y)=e^{-d\,\tau}\left(D+e^{-2\tau}\,|y|^2/d\right)^{-d/2}$. See \cite{BBDGV, MR2282669} for more detailed considerations.
In this note, we shall focus our attention on the case $m<1$ which has been much less studied. In this regime, \eqref{Eqn} is known as the \emph{fast diffusion} equation. We do not even need to assume $m>0$. We shall summarize and extend a series of recent results on the basin of attraction of the family of \emph{generalized Barenblatt solutions} and establish the optimal rates of convergence of the solutions of \eqref{Eqn} towards a unique attracting limit state in that family. To state such a result, it is more convenient to rescale the flow and rewrite \eqref{Eqn} in self-similar variables by introducing for $m\neq m_c$ the time-dependent change of variables \begin{equation}\label{eq:chgvariable}
t:=\tfrac{1-m}{2}\log\left(\frac{R(\tau)}{R(0)}\right)\quad\mbox{and}\quad x:=\sqrt{\tfrac{1-m}{2\,d\,|m-m_c|}}\,\frac y{R(\tau)}\,, \end{equation} with $R$ as above. If $m=m_c$, we take $t=\tau/d$ and $x=e^{-\tau}\,y/\sqrt d$. In these new variables, the generalized Barenblatt functions $U_{D, T}(\tau,y)$ are transformed into \emph{generalized Barenblatt profiles} $V_D(x)$, which are stationary: \begin{equation}\label{newBaren}
V_D(x):=\(D+|x|^2\)^\frac 1{m-1}\quad x\in {\mathbb R}^d\,. \end{equation} If $u$ is a solution to~\eqref{Eqn}, the function \begin{equation*} v(t,x):= R(\tau)^{d}\,u(\tau,y) \end{equation*} solves the equation \begin{equation}\label{FPeqn} \frac{\partial v}{\partial t}=\nabla\cdot\left[v\,\nabla\left(\frac{v^{m-1}-V_D^{m-1}}{m-1}\right)\right]\quad t> 0\,,\quad x\in {\mathbb R}^d\,, \end{equation} with initial condition $v(t=0,x)=v_0(x):=R(0)^{-d}\,u_0(y)$ where $x$ and~$y$ are related according to \eqref{eq:chgvariable} with $\tau=0$. This nonlinear Fokker-Planck equation can also be written as \[ \frac{\partial v}{\partial t}=\frac 1m\,\Delta v^m+\frac 2{1-m}\,\nabla\cdot(x\,v)\quad t> 0\,,\quad x\in {\mathbb R}^d\,. \]
\section{Main results}
Our main result is concerned with the \emph{sharp rate} at which a solution $v$ of the rescaled equation \eqref{FPeqn} converges to the \emph{generalized Barenblatt profile} $V_D$ given by formula \eqref{newBaren} in the whole range $m< 1$. Convergence is measured in terms of the relative entropy given by the formula \[ \mathcal E[v]:=\frac 1{m-1}\int_{{\mathbb R}^d}\left[\frac{v^m-V_D^m}{m}-\,V_D^{m-1}(v-V_D)\right]\,dx \] for all $m\ne 0$ (modified as mentioned for $m=0$). In order to get such convergence we need the following assumptions on the initial datum $v_0$ associated to \eqref{FPeqn}:
\noindent {\bf (H1)} \emph{$V_{D_0}\le v_0 \le V_{D_1}$ for some $D_0>D_1>0$,}
\noindent {\bf (H2)} \emph{if $d\ge 3$ and $m\le m_*$, $(v_0-V_D)$ is integrable for a suitable $D\in[D_1,D_0]$.}
\noindent The case $m=m_*:=(d-4)/(d-2)$ will be discussed later. Besides, if $m>m_*$, we define $D$ as the unique value in $[D_1,D_0]$ such that $\int_{{\mathbb R}^d}(v_0-V_D)\,dx=0$.
\par\begin{theorem}\label{Thm:Main1} Under the above assumptions, if $m<1$ and $m\ne m_*$, the entropy decays according to \begin{equation} \mathcal E[v(t,\cdot)]\le C\, e^{-2\,\Lambda\, t}\quad\forall\;t\ge 0\,. \end{equation} The sharp decay rate $\Lambda$ is equal to the best constant $\Lambda_{\alpha,d}>0$ in the Hardy--Poincar\'e inequality of Theorem~{\rm \ref{Thm:Main2}} with $\alpha:=1/(m-1)<0$. Moreover, the constant $C>0$ depends only on $m,d,D_0,D_1,D$ and $\mathcal E[v_0]$. \end{theorem}
\par\indent
The precise meaning of what \emph{sharp rate} means will be discussed at the end of this paper. As in \cite{BBDGV}, we can deduce from Theorem \ref{Thm:Main1} rates of convergence in more standard norms, namely, in $L^q(dx)$ for $q\ge \max\{1,d\,(1-m)/\,[2\,(2-m)+d\,(1-m)]\}$, or in $C^k$ by interpolation. Moreover, by undoing the time-dependent change of variables~\eqref{eq:chgvariable}, we can also deduce results on the \emph{intermediate asymptotics} for the solution of equation \eqref{Eqn}; to be precise, we can get rates of decay of $u(\tau,y)-R(\tau)^{-d}\,U_{D,T}(\tau,y)$ as $\tau\to+\infty$ if $m\in [m_c,1)$, or as $\tau\to T$ if $m\in(-\infty,m_c)$.
It is worth spending some words on the basin of attraction of the Barenblatt solutions $U_{D,T}$ given by~\eqref{baren.form1}. Such profiles have two parameters: $D$ corresponds to the \emph{mass} while $T$ has the meaning of the \emph{extinction time} of the solution for $m<m_c$ and of a time-delay parameter otherwise. Fix $T$ and $D$, and consider first the case $m_*< m<1$. The basin of attraction of $U_{D,T}$ contains all solutions corresponding to data which are trapped between two Barenblatt profiles $U_{D_0,T}(0,\cdot), U_{D_1,T}(0,\cdot)$ for the same value of $T$ and such that $\int_{{\mathbb R}^d}[u_0-U_{D,T}(0,\cdot)]\,dy=0$ for some $D\in [D_1,D_0]$. If $m<m_*$ the basin of attraction of a Barenblatt solution contains all solutions corresponding to data which, besides being trapped between $U_{D_0,T}$ and $U_{D_1,T}$, are integrable perturbations of $U_{D,T}(0,\cdot)$.
Now, let us give an idea of the proof of Theorem~\ref{Thm:Main1}. First assume that $D=1$ (this entails no loss of generality). On~${\mathbb R}^d$, we shall therefore consider the measure $d\mu_\alpha:=h_\alpha\,dx$, where the weight $h_\alpha$ is the Barenblatt profile, defined by $h_\alpha(x):= (1+|x|^2)^{\alpha}$, with $\alpha=1/(m-1)<0$, and study on the weighted space $L^2(d\mu_\alpha)$ the operator \[ \mathcal L_{\alpha,d}:=-h_{1-\alpha}\,\mathrm{div}\left[\,h_\alpha\,\nabla\cdot\,\right] \]
which is such that $\int_{{\mathbb R}^d}f\,(\mathcal L_{\alpha,d}\,f)\,d\mu_{\alpha-1}=\int_{{\mathbb R}^d}|\nabla f|^2\,d\mu_\alpha$. This operator appears in the linearization of~\eqref{FPeqn} if, at a formal level, we expand \[ v(t,x)=h_\alpha(x)\left[1+\varepsilon\,f\!\(t,x\)h_\alpha^{1-m}(x)\right] \] in terms of $\varepsilon$, small, and only keep the first order terms: \[ \frac{\partial f}{\partial t}+\mathcal L_{\alpha,d}\,f=0\,. \] \noindent The convergence result of Theorem~\ref{Thm:Main1} follows from the energy analysis of this equation based on the Hardy-Poincar\'e inequalities that are described below. Let us fix some notations. For $d\ge 3$, let us define $\alpha_*:=-(d-2)/2$ corresponding to $m=m_*$; two other exponents will appear in the analysis, namely, $m_1:=(d-1)/d$ with corresponding $\alpha_1=-d$, and $m_2:=d/(d+2)$ with corresponding $\alpha_2=-(d+2)/2$. We have $m_*<m_c<m_2<m_1<1$. Similar definitions for $d=2$ give $m_*=-\infty$ so that $\alpha_*=0$, as well as $m_c=0$, and $m_1=m_2=1/2$. For the convenience of the reader, a table summarizing the key values of the parameter $m$ and the corresponding values of $\alpha$ is given in the Appendix.
\par\begin{theorem}[Sharp Hardy-Poincar\'e inequalities]\label{Thm:Main2} Let $d\ge 3$. For any $\alpha\in(-\infty,0)\setminus\{\alpha_*\}$, there is a positive constant ${\Lambda_{\alpha,d}}$ such that \begin{equation}\label{gap}
{\Lambda_{\alpha,d}}\int_{{\mathbb R}^d}|f|^2\,d\mu_{\alpha-1}\leq \int_{{\mathbb R}^d}|\nabla f|^2\,d\mu_\alpha\quad\forall\;f\in H^1(d\mu_\alpha) \end{equation} under the additional condition $\int_{{\mathbb R}^d}f\,d\mu_{\alpha-1}=0$ if $\alpha<\alpha_*$. Moreover, the sharp constant $\Lambda_{\alpha,d}$ is given by \[ \Lambda_{\alpha,d}=\left\{\begin{array}{ll} \frac 14\,(d-2+2\,\alpha)^2&\mbox{if}\;\alpha\in\left[-\frac{d+2}{2},\alpha_*\right)\cup(\alpha_*,0)\,,
\cr -\,4\,\alpha-2\,d&\mbox{if}\;\alpha\in\left[-d,-\frac{d+2}{2}\right)\,,
\cr -\,2\,\alpha&\mbox{if}\;\alpha\in(-\infty,-d)\,.\cr \end{array}\right. \] For $d=2$, inequality \eqref{gap} holds for all $\alpha<0$, with the corresponding values of the best constant $\Lambda_{\alpha,2}=\alpha^2$ for $\alpha\in [-2,0)$ and $\Lambda_{\alpha,2}=-2\alpha$ for $\alpha\in (-\infty,-2)$. For $d=1$, \eqref{gap} holds, but the values of $\Lambda_{\alpha,1}$ are given by $\Lambda_{\alpha,1}=-2\,\alpha$ if $\alpha<-1/2$ and $\Lambda_{\alpha,1}=(\alpha-1/2)^2$ if $\alpha\in[-1/2,0)$. \end{theorem}\par\indent
The Hardy-Poincar\'e inequalities \eqref{gap} share many properties with Hardy's inequalities, because of homogeneity reasons. A simple scaling argument indeed shows that \[
{\Lambda_{\alpha,d}} \int_{{\mathbb R}^d}|f|^2\,(D+|x|^2)^{\alpha-1}\,dx\leq \int_{{\mathbb R}^d}|\nabla f|^2\,(D+|x|^2)^{\alpha}\,dx \]
holds for any $f\in H^1((D+|x|^2)^{\alpha}dx)$ and any $D\ge 0$, under the additional conditions $\int_{{\mathbb R}^d}f\,(D+|x|^2)^{\alpha-1}\,dx=0$ and $D>0$ if $\alpha<\alpha_*$. In other words, the optimal constant, $\Lambda_{\alpha,d}$, does not depend on $D>0$ and the assumption $D=1$ can be dropped without consequences. In the limit $D\to 0$, they yield weighted Hardy type inequalities, cf. \cite{CKN,hardy1934inequalities}.
Theorem \ref{Thm:Main2} has been proved in \cite{BBDGV} for $m<m_*$. The main improvement of this note compared to \cite{BBDGV-CRAS,BBDGV} is that we are able to give the value of the sharp constants also in the range $(m_*,1)$. These constants are deduced from the spectrum of the operator $\mathcal L_{\alpha,d}$, that we shall study below.
It is relatively easy to obtain the classical decay rates of the linear case in the limit $m\to 1$ by a careful rescaling such that weights become proportional to powers of the modified expression $(1+(1-m)\,|x|^2)^{-1/(1-m)}$. In the limit case, we obtain the Poincar\'e inequality for the Gaussian weight. As for the evolution equation, the time also has to be rescaled by a factor $(1-m)$. We leave the details to the reader. See \cite{BBDE} for further considerations on associated functional inequalities.
\section{A brief historical overview}
The search for \emph{sharp decay rates} in fast diffusion equations has been extremely active over the last three decades. Once plain convergence of the suitably rescaled flow towards an asymptotic profile is established (cf. \cite{MR586735,MR1977429} for $m>m_c$ and \cite{BBDGV,Daskalopoulos-Sesum2006} for $m\le m_c$), getting the rates is the next step in the asymptotic analysis. An important progress was achieved by M. Del Pino and J. Dolbeault in \cite{MR1940370} by identifying sharp rates of decay for the relative entropy, that had been introduced earlier by J. Ralston and W.I. Newman in~\cite{MR760591,MR760592}. The analysis in \cite{MR1940370} uses the optimal constants in Gagliardo-Nirenberg inequalities, and these constants are computed. J.A. Carrillo and G. Toscani in \cite{MR1777035} gave a proof of decay based on the entropy/entropy-production method of D. Bakry and M. Emery, and established an analogue of the Csisz\'ar-Kullback inequality which allows to control the convergence in $L^1(dx)$, in case $m>1$. F. Otto then made the link with gradient flows with respect to the Wasserstein distance, see \cite{MR1842429}, and D. Cordero-Erausquin, B. Nazaret and C.~Villani gave a proof of Gagliardo-Nirenberg inequalities using mass transportation techniques in \cite{MR2032031}.
The condition $m\ge(d-1)/d=:m_1$ was definitely a strong limitation to these first approaches, except maybe for the entropy/entropy-production method. Gagliardo-Nirenberg inequalities degenerate into a critical Sobolev inequality for $m=m_1$, while the displacement convexity condition requires $m\ge m_1$. It was a puzzling question to understand what was going on in the range $m_c<m<m_1$, and this has been the subject of many contributions. Since one is interested in understanding the convergence towards Barenblatt profiles, a key issue is the integrability of these profiles and their moments, in terms of $m$. To work with Wasserstein's distance, it is crucial to have second moments bounded, which amounts to request $m>d/(d+2)=:m_2$ for the Barenblatt profiles. The contribution of J. Denzler and R. McCann in \cite{MR1982656,MR2126633} enters in this context. Another, weaker, limitation appears when one only requires the integrability of the Barenblatt profiles, namely $m>m_c$. Notice that the range $[m_c,1)$ is also the range for which $L^1(dx)$ initial data give rise to solutions which preserve the mass and globally exist, see for instance~\cite{MR586735,MR2282669}.
It was therefore natural to investigate the range $m\in(m_c,1)$ with entropy estimates. This has been done first by linearizing around the Barenblatt profiles in \cite{MR1901093,MR1974458}, and then a full proof for the nonlinear flow was done by J.A. Carrillo and J.L. V\'azquez in \cite{MR1986060}. A detailed account for these contributions and their motivations can be found in the survey paper \cite{MR2065020}. Compared to classical approaches based on comparison, as in the book~\cite{MR2282669}, a major advantage of entropy techniques is that they combine very well with $L^1(dx)$ estimates if $m>m_c$, or relative mass estimates otherwise, see~\cite{BBDGV}.
The picture for $m\le m_c$ turns out to be entirely different and more complicated, and it was not considered until quite recently. First of all, many classes of solutions vanish in finite time, which is a striking property that forces us to change the concept of asymptotic behaviour from large-time behaviour to behaviour near the extinction time. On the other hand, $L^1(dx)$ solutions lose mass as time evolves. Moreover, the natural extensions of Barenblatt's profiles make sense but these profiles have two novel properties: they vanish in finite time and they do not have finite~mass.
There is a large variety of possible behaviours and many results have been achieved, such as the ones described in \cite{MR2282669} for data which decay strongly as $|x|\to\infty$. However, as long as one is interested in solutions converging towards Barenblatt profiles in self-similar variables, there were some recent results on plain convergence: a paper of P. Daskalopoulos and N. Sesum, \cite{Daskalopoulos-Sesum2006}, using comparison techniques, and in two contributions involving the authors of this note, using \emph{relative entropy methods}, see \cite{BBDGV-CRAS,BBDGV}. This last approach proceeds further into the description of the convergence by identifying a suitable weighted linearization of the relative entropy. In the appropriate space, $L^2(d\mu_{\alpha-1})$, with the notations of Theorem~\ref{Thm:Main2}, it gives rise to an exponential convergence after rescaling. This justifies the heuristic computation which relates Theorems~\ref{Thm:Main1} and \ref{Thm:Main2}, and allows to identify the sharp rates of convergence. The point of this note is to explicitly state and prove such rates in the whole range $m<1$.
\section{Relative entropy and linearization}
The strategy developed in \cite{BBDGV} is based on the extension of the \emph{relative entropy} of J. Ralston and W.I. Newmann, which can be written in terms of $w=v/V_D$ as \[ \mathcal F[w]:=\frac 1{1-m}\int_{{\mathbb R}^d}\left[w-1-\frac{1}{m}\big(w^m-1\big)\right]\,V_D^m\,dx\,. \] For simplicity, assume $m\neq 0$. Notice that ${\mathcal F}[w]={\mathcal E}[v]$. Let \[
\mathcal I[w]:=\int_{{\mathbb R}^d}\left|\frac 1{m-1}\,\nabla\left[(w^{m-1}-1)\,V_D^{m-1}\right]\,\right|^2v\,dx \] be the \emph{generalized relative Fisher information}. If $v$ is a solution of \eqref{FPeqn}, then \beq{Eqn:Entropy-EntropyProduction} \frac d{dt}\mathcal F[w(t,\cdot)]=-\,\mathcal I [w(t,\cdot)]\quad\forall\;t> 0 \ee and, as a consequence, $\lim_{t\to+\infty}\mathcal F[w(t,\cdot)]=0$ for all $m<1$. The method is based on Theorem~\ref{Thm:Main2} and uniform estimates that relate linear and nonlinear quantities. Following \cite{BBDGV,BGV} we can first estimate from below and above the entropy $\mathcal{F}$ in terms of its linearization, which appears in~ \eqref{gap}: \begin{equation}\label{Entr.lin.nonlin}
h^{m-2}\int_{{\mathbb R}^d}\kern -5pt|f|^2\,V_D^{2-m}\;dx\le 2\,\mathcal F[w]\le h^{2-m}\int_{{\mathbb R}^d}\kern -5pt|f|^2\,V_D^{2-m}\;dx \end{equation} where $f:=(w-1)\,V_D^{m-1}$, $h_1(t):=\mathrm{inf}_{{\mathbb R}^d}w(t,\cdot)$, $h_2(t):=\mathrm{sup}_{{\mathbb R}^d}w(t,\cdot)$ and $h:=\max\{h_2,1/h_1\}$. We notice that \hbox{$h(t)\to 1$} as $t\to+\infty$. Similarly, the generalized Fisher information satisfies the bounds \begin{equation}\label{Fish.lin.nonlin}
\int_{{\mathbb R}^d}|\nabla f|^2\,V_D\;dx\le [1+X(h)]\,\mathcal I[w]+Y(h)\int_{{\mathbb R}^d}|f|^2\,V_D^{2-m}\;dx \end{equation} where $h_2^{2(2-m)}/h_1\le h^{5-2m}=:1+X(h)$ and \hbox{$d\,(1-m)$} $\big[\left(h_2/h_1\right)^{2(2-m)}-1\big]\le d\,(1-m)\,\big[h^{4(2-m)}-1\big]=:Y(h)$. Notice that $X(1)=Y(1)=0$. Joining these inequalities with the Hardy-Poincar\'e inequality of Theorem~\ref{Thm:Main2} gives \begin{equation} \mathcal F[w]\le \frac{h^{2-m}\,[1+X(h)]}{2\,\big[\Lambda_{\alpha,d}-Y(h)\big]}\,\mathcal{I}[w] \end{equation} as soon as $0<h<h_*:=\min\{h>0\,:\,\Lambda_{\alpha,d}-Y(h)\ge 0\}$. On the other hand, uniform relative estimates hold, according to \cite{BGV}, formula (5.33): for some $\mathsf C=\mathsf C(d,m,D,D_0,D_1)$, \begin{equation}\label{Unif} 0\le h-1\le\mathsf C\,\mathcal{F}^\frac{1-m}{d+2-(d+1)m}\,. \end {equation} Summarizing, we end up with a system of nonlinear differential inequalities, with $h$ as above and, at least for any $t>t_*$, $t_*>0$ large enough, \beq{Ineq:Gronwall} \frac{d}{dt}\mathcal{F}[w(t,\cdot)] \le-2\,\frac{\Lambda_{\alpha,d}-Y(h)}{\big[1+X(h)\big]\,h^{2-m}}\,\mathcal{F}[w(t,\cdot)]\,. \end{equation} Gronwall type estimates then show that \[ \limsup_{t\to\infty}\,{\rm e}^{2\,\Lambda_{\alpha,d}\,t}\mathcal{F}[w(t,\cdot)]<+\infty\,. \] This completes the proof of Theorem~\ref{Thm:Main1} for $m\neq 0$. The adaptation to the logarithmic nonlinearity is left to the reader. Results in \cite{BBDGV} are improved in two ways: a time-dependent estimate of $h$ is used in place of $h(0)$, and the precise expression of the rate is established. One can actually get a slightly more precise estimate by coupling \eqref{Unif} and~\eqref{Ineq:Gronwall}.
\par\noindent\begin{corollary}\label{Cor:EntropyAndUniformEstimates}Under the assumptions of Theorem~{\rm \ref{Thm:Main1}}, if $h(0)<h_*$, then $\mathcal{F}[w(t,\cdot)]\le G\big(t,h(0),\mathcal{F}[w(0,\cdot)]\big)$ for any $t\ge 0$, where $G$ is the unique solution of the nonlinear ODE \[ \frac{dG}{dt}
=-2\frac{\Lambda_{\alpha,d}-Y(h)}{[1+X(h)]\,h^{2-m}}\,G
\quad\mbox{with}\quad h=1+\mathsf C\,G^\frac{1-m}{d+2-(d+1)m} \] and initial condition $G(0)=\mathcal F[w(0,\cdot)]$. \end{corollary}
\section{Operator equivalence. The spectrum of $\mathcal L_{\alpha,d}$}
An important point of this note is the computation of the spectrum of $\mathcal{L}_{\alpha,d}$ for any $\alpha<0$. This spectrum was only partially understood in \cite{BBDGV, BBDGV-CRAS}. In particular, the existence of a spectral gap was established for all $\alpha\ne \alpha_*= (2-d)/2$, but its value was not stated for all values of $\alpha$.
J. Denzler and R.J. McCann in \cite{MR1982656,MR2126633} formally linearized the fast diffusion flow (considered as a gradient flow of the entropy with respect to the Wasserstein distance) in the framework of mass transportation, in order to guess the asymptotic behaviour of the solutions of \eqref{Eqn}. This leads to a different functional setting, with a different linearized operator, $\mathcal H_{\alpha,d}$. They performed the detailed analysis of its spectrum for all $m\in(m_c,1)$, but the justification of the nonlinear asymptotics could not be completed due to the difficulties of the functional setting, especially in the very fast diffusion range.
Our approach is based on relative entropy estimates and the Hardy-Poincar\'e inequalities of Theorem~\ref{Thm:Main2}. The asymptotics have readily been justified in \cite{BBDGV}. The operator $\mathcal L_{\alpha,d}=-h_{1-\alpha}\,\mathrm{div}\,\left[\,h_\alpha\,\nabla\cdot\,\right]$ can be initially defined on $\mathcal D(\mathbb{R}^d)$. To construct a self-adjoint extension of such an operator, one can consider the quadratic form $f\mapsto(f,\mathcal L_{\alpha,d}\,f)$, where $(\,\cdot\,,\,\cdot\,)$ denotes the scalar product on $L^2(d\mu_{\alpha-1})$. Standard results show that such a quadratic form is closable, so that its closure defines a unique self-adjoint operator, its Friedrich's extension, still denoted by the same symbol for brevity. The operator $\mathcal H_{\alpha,d}$ is different: it is obtained by taking the operator closure of $\mathcal L_{\alpha,d}$, initially defined on $\mathcal D(\mathbb{R}^d)$, in the Hilbert space \emph{$H^{1,*}(d\mu_{\alpha}):=\big\{f\in L^2(d\mu_{\alpha-1}):$ \hbox{$\int_{{\mathbb R}^d}|\nabla f|^2\,d\mu_\alpha<\infty$} and \hbox{$\int_{{\mathbb R}^d}f\,d\mu_{\alpha-1}=0$} if $\alpha<\alpha_*\big\}$}, so that \[ \mathcal H_{\alpha,d}\,f:=h_{1-\alpha}\,\nabla\cdot\big[h_\alpha\,\nabla \big(h_{1-\alpha}\,\nabla\cdot(h_\alpha\,\nabla f) \big) \big]\,. \]
Because of the Hardy-Poin\-car\'e inequality, $\int_{{\mathbb R}^d}|\nabla f|^2\,d\mu_\alpha$ defines a~norm. Denote by $\langle\,\cdot\,,\,\cdot\,\rangle$ the corresponding scalar product and notice that $\langle\,f\,,\,g\,\rangle\!=\!(\,f\,,\,\mathcal L_{\alpha,d}\,g\,)$ for any \hbox{$f,g\in\mathcal D({\mathbb R}^d)$}.
\par\begin{proposition}\label{Prop:Equivalence} The operator $\mathcal L_{\alpha,d}$ on $L^2(d\mu_{\alpha-1})$ has the same spectrum as the operators $\mathcal H_{\alpha,d}$ on $H^{1,*}(d\mu_{\alpha})$. \end{proposition}
\par\indent
The proof is based on the construction of a suitable unitary operator $U:H^{1,*}(d\mu_{\alpha})\to L^2(d\mu_{\alpha-1})$, such that $U\,\mathcal H_{\alpha,d}\,U^{-1}=\mathcal L_{\alpha,d}$. We claim that $U=\sqrt{\mathcal L_{\alpha,d}}$ is the requested unitary operator. By definition, $\mathcal D(\mathbb{R}^d)$ is a form core of $\mathcal L_{\alpha,d}$, and as a consequence, the identity has to be established only for functions $f\in\mathcal D(\mathbb{R}^d)$. Since $\|Uf\|^2=\big(\,f\,,\,\mathcal L_{\alpha,d} f\,\big)=\int_{{\mathbb R}^d}|\nabla f|^2\,d\mu_\alpha$, we get \[ \begin{split} &\hspace*{-8pt}\big(\,U\,\mathcal H_{\alpha,d}\,U^{-1}f\,,\, g\,\big) =\big\langle\,\mathcal H_{\alpha,d}\,U^{-1}f\,,\, U^{-1}g\,\big\rangle\\ &=\big(\,\mathcal L_{\alpha,d}\,U^{-1}f\,,\, \mathcal L_{\alpha,d}\,U^{-1}g\,\big) =\big(\,U\,f\,,\,U\,g\,\big) =\big\langle\, f\,,\,g\,\big\rangle\\ \end{split} \] where we have used the properties $U^*=U^{-1}$ and $U^2=\mathcal L_{\alpha,d}$. This unitary equivalence between $\mathcal L_{\alpha,d}$ and $\mathcal H_{\alpha,d}$ implies the identity of their spectra.
We may now proceed with the presentation of the actual values of the spectrum by extending the results of \cite{MR2126633}. According to \cite{MR0282313}, the spectrum of the Laplace-Beltrami operator on $S^{d-1}$ is described by \[ -\Delta_{S^{d-1}}Y_{\ell \mu}=\ell\,(\ell+d-2)\,Y_{\ell \mu} \] with $\ell=0$, $1$, $2$, \ldots and $\mu=1$, $2$, \ldots $M_\ell:=\frac{(d+\ell-3)!\,(d+2\ell-2)}{\ell!\,(d-2)!}$ with the convention $M_0=1$, and $M_1=1$ if $d=1$. Using spherical coordinates and separation of variables, the discrete spectrum of $\mathcal L_{\alpha,d}$ is therefore made of the values of $\lambda$ for which \begin{equation}\label{EigenF} v''+\(\tfrac{d-1}r+\tfrac{2\,\alpha\,r}{1+r^2}\)\,v'+\(\tfrac\lambda{1+r^2}-\tfrac{\ell\,(\ell+d-2)}{r^2}\)\,v=0 \end{equation} has a solution on ${\mathbb R}^+\ni r$, in the domain of $\mathcal L_{\alpha,d}$. The change of variables $v(r)=r^\ell\,w(-r^2)$ allows to express $w$ in terms of the hypergeometric function $_2F_1(a,b,c;z)$ with $c=\ell+d/2$, $a+b+1=\ell+\alpha+d/2$ and $a\,b=(2\,\ell\,\alpha+\lambda)/4$, as the solution for $s=-r^2$ of \[ s\,(1-s)\,y''+[\,c-(a+b+1)\,s\,]\,y'-a\,b\,y=0\,, \] see \cite{weisstein2005hf}. Based on \cite{BBDGV-CRAS,MR2126633}, we can state the following result.
\par\begin{proposition}\label{Prop:Spectrum} The bottom of the continuous spectrum of the operator $\mathcal L_{\alpha,d}$ on $L^2(d\mu_{\alpha-1})$ is $\lambda_{\alpha,d}^{\rm cont}:=\frac 14(d+2\,\alpha-2)^2$. Moreover, $\mathcal L_{\alpha,d}$ has some discrete spectrum only for $m>m_2=d/(d+2)$. For $d\ge 2$, the discrete spectrum is made of the eigenvalues \begin{equation}\label{eigen} \lambda_{\ell k}=-2\,\alpha\,\(\ell+2\,k\)-4\,k\,\(k+\ell+\frac d2-1\) \end{equation} with $\ell$, $k=0$, $1$, \ldots provided $(\ell,k)\neq(0,0)$ and $\ell+2k-1<-(d+2\,\alpha)/2$. If $d=1$, the discrete spectrum is made of the eigenvalues $\lambda_k=k\,(1-2\,\alpha-k)$ with $k\in{\mathbb N}\cap[1,1/2-\alpha]$.\end{proposition}
\par\indent
\begin{figure}
\caption{Spectrum of $\mathcal L_{\alpha,d}$ as a function of $\alpha$, for $d=5$.}
\end{figure}
Using Persson's characterization of the continuous spectrum, see \cite{MR0133586,BBDGV-CRAS}, one can indeed prove that $\lambda_{\alpha,d}^{\rm cont}$ is the optimal constant in the following inequality: for any $f\in\mathcal D({\mathbb R}^d\setminus\{0\})$, \[
\lambda_{\alpha,d}^{\rm cont}\int_{{\mathbb R}^d}|f|^2\,|x|^{2(\alpha-1)}\;dx\le\int_{{\mathbb R}^d}|\nabla f|^2\,|x|^{2\,\alpha}dx\,. \] The condition that the solution of \eqref{EigenF} is in the domain of $\mathcal L_{\alpha,d}$ determines the eigenvalues. A more complete discussion of this topic can be found in \cite{MR2126633}, which justifies the expression of the discrete spectrum.
\noindent Since $\alpha=1/(m-1)$, we may notice that for $d\ge 2$, $\alpha=-d$ (corresponding to $-2\,\alpha=\lambda_{10}=\lambda_{01}=-4\,\alpha-2\,d$) and $\alpha=-(d+2)/2$ (corresponding to $\lambda_{01}=\lambda_0^{\rm cont}:=\frac 14\,(d+2\,\alpha-2)^2$) respectively mean $m=m_1=(d-1)/d$ and $m=m_2=d/(d+2)$.
The above spectral results hold exactly in the same form when $d=2$, see \cite{MR2126633}. Notice in particular that $\lambda^{{\rm cont}}_{\alpha,d=2}=\alpha^2$ so that there is no equivalent of $m_*$ for $d=2$. With the notations of Theorem \ref{Thm:Main2}, $\alpha_*=0$. All results of Theorem \ref{Thm:Main1} hold true under the sole assumption (H1).
In dimension $d=1$, the spectral results are different, see \cite{MR2126633}. The discrete spectrum is nonempty whenever $\alpha\le-1/2$, that is $m\ge-1$.
\section{The critical case}
Since the spectral gap of $\mathcal L_{\alpha,d}$ tends to zero as $m\to m_*$\,, the previous strategy fails when $m=m_*$ and one might expect a slower decay to equilibrium, sometimes referred as \emph{slow asymptotics.} The following result has been proved in~\cite{BGV}.
\par\begin{theorem}\label{mstar} Assume that $d\ge 3$, let $v$ be a solution of~\eqref{FPeqn} with $m= m_*$, and suppose that {\rm (H1)-(H2)} hold. If $|v_0-V_D|$ is bounded a.e. by a radial $L^1(dx)$ function, then there exists a positive constant $C^*$ such that \begin{equation} \mathcal E[v(t,\cdot)]\le C^*\,t^{-1/2}\quad\forall\;t\ge 0\,, \end{equation} where $C^*$ depends only on $m,d,D_0,D_1,D$ and $\mathcal E[v_0]$. \end{theorem}
\par\indent
Rates of convergence in $L^q(dx)$, $q\in(1,\infty]$ follow. Notice that in dimension $d=3$ and $4$, we have respectively $m_*=-1$ and $m_*=0$. In the last case, Theorem \ref{mstar} applies to the logarithmic diffusion.
The proof relies on identifying first the asymptotics of the linearized evolution. In this case, the bottom of the continuous spectrum of $\mathcal{L}_{\alpha_*,d}$ is zero. This difficulty is overcome by noticing that the operator $\mathcal{L}_{\alpha_*,d}$ on ${\mathbb R}^d$ can be identified with the Laplace-Beltrami operator for a suitable conformally flat metric on ${\mathbb R}^d$, having positive Ricci curvature. Then the on-diagonal heat kernel of the linearized generator behaves like $t^{-d/2}$ for small $t$ and like $t^{-1/2}$ for large $t$. The Hardy-Poincar\'e inequality is replaced by a weighted Nash inequality: there exists a positive continuous and monotone function $\mathcal N$ on $\mathbb R^+$ such that for any nonnegative smooth function $f$ with $M=\int_{{\mathbb R}^d}f\,d\mu_{-d/2}$ (recall that $\alpha_*-1=-d/2$), \[
\frac 1{M^2}\,\int_{{\mathbb R}^d}|f|^2\,d\mu_{-d/2}\le \mathcal N\left(\frac 1{M^2}\,\int_{{\mathbb R}^d}|\nabla f|^2\,\;d\mu_{(2-d)/2}\right). \] The function $\mathcal N$ behaves as follows: $\lim_{s\to 0^+}s^{-1/3}\,\mathcal N(s)=c_1>0$ and $\lim_{s\to\infty}s^{-d/(d+2)}\mathcal N(s)=c_2>0$. Only the first limit matters for the asymptotic behaviour. Up to technicalities, inequality \eqref{Ineq:Gronwall} is replaced by $(\mathcal{F}[w(t,\cdot)])^3\le K\,\mathcal{I}[w(t,\cdot)]$ for some $K>0$, $t\ge t_0$ large enough, which allows to complete the proof.
\section{Faster convergence}
A very natural issue is the question of improving the rates of convergence by imposing restrictions on the initial data. Results of this nature have been observed in \cite{MR1901093} in case of radially symmetric solutions, and are carefully commented in~\cite{MR2126633}. By locating the center of mass at zero, we are able to give an answer, which amounts to kill the $\lambda_{10}$ mode, whose eigenspace is generated by $x\mapsto x_i$, $i=1$, $2$\ldots $d$. This is an improvement compared to the first result in this direction, which has been obtained by R. McCann and D. Slep\v{c}ev in~\cite{MR2211152}, since we obtain an improved \emph{sharp rate} of convergence of the solution of \eqref{FPeqn}, as a consequence of the following improved Hardy-Poincar\'e inequality.
\par\begin{lemma}\label{Lem:Improved2} Let $\widetilde{\Lambda}_{\alpha,d}:=-4\,\alpha-2\,d$ if $\alpha<-d$ and $\widetilde{\Lambda}_{\alpha,d}:=\lambda_{\alpha,d}^{\rm cont}$ if $\alpha\in[-d,-d/2)$. If $d\ge 2$, for any $\alpha\in(-\infty,-d)$, we have \[
\widetilde{\Lambda}_{\alpha,d} \int_{{\mathbb R}^d}|f|^2\,d\mu_{\alpha-1}\leq\int_{{\mathbb R}^d}|\nabla f|^2\,d\mu_\alpha\quad\forall\;f\in H^1(d\mu_\alpha) \] under the conditions $\int_{{\mathbb R}^d}f\,d\mu_{\alpha-1}=0$ and $\int_{{\mathbb R}^d}x\,f\,d\mu_{\alpha-1}=0$. The constant $\widetilde{\Lambda}_{\alpha,d}$ is sharp. \end{lemma}
\par\indent
This covers the range $m\in(m_1,1)$ with \hbox{$m_1=(d-1)/d$}.
\par\begin{theorem}\label{Thm:Main3} Assume that $m\in(m_1,1)$, $d\ge 3$. Under Assumption {\rm (H1)}, if $v$ is a solution of \eqref{FPeqn} with initial datum~$v_0$ such that $\int_{{\mathbb R}^d}x\,v_0\,dx=0$ and if $D$ is chosen so that $\int_{{\mathbb R}^d} (v_0-V_D)\,dx=0$, then there exists a positive constant~$\widetilde{C}$ depending only on $m,d,D_0,D_1,D$ and $\mathcal E[v_0]$ such that the relative entropy decays like \[ \mathcal E[v(t,\cdot)]\le \widetilde{C}\, {\rm e}^{-\widetilde{\Lambda}_{\alpha,d}\,t}\quad\forall\;t\ge 0\;. \] \end{theorem}
\section{A variational approach of sharpness}
Recall that $(d-2)/d=m_c<m_1=(d-1)/d$. The entropy / entropy production inequality obtained in \cite{MR1940370} in the range $m\in[m_1,1)$ can be written as $\mathcal F\leq\frac 12\,\mathcal I$ and it is known to be sharp as a consequence of the optimality case in Gagliardo-Nirenberg inequalities. Moreover, equality is achieved if and only if $v=V_D$. The inequality has been extended in \cite{MR1986060} to the range $m\in(m_c,1)$ using the Bakry-Emery method, with the same constant $1/2$, and again equality is achieved if and only if $v=V_D$, but sharpness of $1/2$ is not as straightforward for $m\in(m_c,m_1)$ as it is for $m\in[m_1,1)$. The question of the optimality of the constant can be reformulated as a variational problem, namely to identify the value of the positive constant \[ \mathcal C=\inf\frac{\mathcal I[v]} {\mathcal E [v]} \] where the infimum is taken over the set of all functions such that $v\in\mathcal D({\mathbb R}^d)$ and $\int_{{\mathbb R}^d}v\,dx=M$. Rephrasing the sharpness results, we know that $\mathcal C=2$ if $m\in(m_1,1)$ and $\mathcal C\ge 2$ if $m\in(m_c,m_1)$. By taking $v_n=V_D\,(1+\frac 1n\,f\,V_D^{1-m})$ and letting $n\to\infty$, we get \[
\lim_{n\to\infty}\frac{\mathcal I[v_n]}{\mathcal E [v_n]}=\frac{\int_{{\mathbb R}^d}|\nabla f|^2\,V_D\,dx}{\int_{{\mathbb R}^d}|f|^2\,V_D^{2-m}\,dx}\;. \] With the optimal choice for $f$, the above limit is less or equal than $2$. Since we already know that $\mathcal C\ge 2$, this shows that $\mathcal C= 2$ for any $m>m_c$. It is quite enlightening to observe that optimality in the quotient gives rise to indetermination since both numerator and denominator are equal to zero when $v=V_D$. This also explains why it is the first order correction which determines the value of $\mathcal C$, and, as a consequence, why the optimal constant, $\mathcal C=2$, is determined by the linearized problem.
When $m\le m_c$, the variational approach is less clear since the problem has to be constrained by a uniform estimate. Proving that any minimizing sequence $(v_n)_{n\in\mathbb N}$ is such that $v_n/V_D-1$ converges, up to a rescaling factor, to a function~$f$ associated to the Hardy-Poincar\'e inequalities would be a significant step, except that one has to deal with compactness issues, test functions associated to the continuous spectrum and a uniform constraint.
\section{Sharp rates of convergence and conjectures}
In Theorem~\ref{Thm:Main1}, we have obtained that the rate $\exp(-\Lambda_{\alpha,d}\,t)$ is \emph{sharp.} The precise meaning of this claim is that \[ \Lambda_{\alpha,d}=\liminf_{h\to 0_+}\inf_{w\in\mathcal S_h}\frac{\mathcal I[w]}{\mathcal F[w]}\,, \]
where the infimum is taken on the set $\mathcal S_h$ of smooth, nonnegative bounded functions $w$ such that \hbox{$\|w-1\|_{L^\infty(dx)}\le h$} and such that $\int_{{\mathbb R}^d}(w-1)\,V_D\,dx$ is zero if $d=1,2$ and $m<1$, or if $d\ge 3$ and $m_*<m<1$, and it is finite if $d\ge 3$ and $m<m_*$. Since, for a solution $v(t,x)=w(t,x)\,V_D(x)$ of \eqref{FPeqn}, \eqref {Eqn:Entropy-EntropyProduction} holds, by \emph{sharp rate} we mean the best possible rate, which is uniform in $t\ge 0$. In other words, for any $\lambda>\Lambda_{\alpha,d}$, one can find some initial datum in $\mathcal S_h$ such that the estimate $\mathcal F[w(t,\cdot)]\le\mathcal F[w(0,\cdot)]\,\exp(-\lambda\,t)$ is wrong for some $t>0$. We did not prove that the rate $\exp(-\Lambda_{\alpha,d}\,t)$ is \emph{globally sharp} in the sense that for some special initial data, $\mathcal F[w(t,\cdot)]$ decays exactly at this rate, or that $\liminf_{t\to\infty}\exp(\Lambda_{\alpha,d}\,t)\,\mathcal F[w(t,\cdot)]>0$, which is possibly less restrictive.
However, if $m\in(m_1,1)$, $m_1=(d-1)/d$, then $\exp(-\Lambda_{\alpha,d}\,t)$ is also a \emph{globally sharp} rate, in the sense that the solution with initial datum $u_0(x)=V_D(x+x_0)$ for any $x_0\in{\mathbb R}^d\setminus\{0\}$ is such that $\mathcal F[w(t,\cdot)]$ decays exactly like $\exp(-\Lambda_{\alpha,d}\,t)$. This formally answers the dilation-persistence conjecture as formulated in \cite{MR2126633}. The question is still open when $m\le m_1$.
Another interesting issue is to understand if improved rates, that is rates of the order of $\exp(-\lambda_{\ell k}\,t)$ with $(\ell,k)\neq(0,0)$, $(0,1)$, $(1,0)$ are \emph{sharp} or \emph{globally sharp} under additional moment-like conditions on the initial data. It is also open to decide whether $\exp(-\widetilde{\Lambda}_{\alpha,d}\,t)$ is sharp or globally sharp under the extra condition $\int_{{\mathbb R}^d}x\,v_0\,dx=0$.
\section{Appendix. A table of correspondence} For the convenience of the reader, a table of definitions of the key values of $m$ when $d\ge 3$ is provided below with the correspondence for the values of $\alpha=1/(m-1)$. This note is restricted to the case $m\in(-\infty,1)$, that is $\alpha\in(-\infty,0)$. \begin{table}[ht] \begin{center}{\small
\begin{tabular}{r|cccccc}\hline $m=$ &$-\infty$&$m_*$&$m_c$&$m_2$&$m_1$&$1$\cr\hline $m=$ &$-\infty$&$\frac{d-4}{d-2}$&$\frac{d-2}d$&$\frac d{d+2}$&$\frac{d-1}d$&$1$\cr\hline $\alpha=$ &$0$&$-\frac{d-2}2$&$-\frac d2$&$-\frac{d+2}2$&$-d$&$-\infty$\cr\hline \end{tabular}} \end{center} \end{table}
\begin{acknowledgments} This work has been supported by the ANR-08-BLAN-0333-01 project CBDif-Fr and the exchange program of University Paris-Dauphine and Universidad Aut\'{o}noma de Madrid. MB and JLV partially supported by Project MTM2008-06326-C02-01 (Spain). MB, GG and JLV partially supported by HI2008-0178 (Italy-Spain).
\par
\noindent \copyright\,2009 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. \end{acknowledgments}
\def$'${$'$}
\end{article}
\end{document} | arXiv |
Jorge Luis Borges and mathematics
Jorge Luis Borges and mathematics concerns several modern mathematical concepts found in certain essays and short stories of Argentinian author Jorge Luis Borges (1899-1986), including concepts such as set theory, recursion, chaos theory, and infinite sequences,[1] although Borges' strongest links to mathematics are through Georg Cantor's theory of infinite sets, outlined in "The Doctrine of Cycles" (La doctrina de los ciclos). Some of Borges' most popular works such as "The Library of Babel" (La Biblioteca de Babel), "The Garden of Forking Paths" (El Jardín de Senderos que se Bifurcan), "The Aleph" (El Aleph), an allusion to Cantor's use of the Hebrew letter aleph ($\aleph $) to denote cardinality of transfinite sets,[2] and "The Approach to Al-Mu'tasim" (El acercamiento a Almotásim) illustrate his use of mathematics.
According to Argentinian mathematician Guillermo Martínez, Borges at least had a knowledge of mathematics at the level of first courses in algebra and analysis at a university – covering logic, paradoxes, infinity, topology and probability theory. He was also aware of the contemporary debates on the foundations of mathematics.[1]
Infinity and cardinality
His 1939 essay "Avatars of the Tortoise" (Avatares de la Tortuga) is about infinity, and he opens by describing the book he would like to write on infinity: “five or seven years of metaphysical, theological, and mathematical training would prepare me (perhaps) for properly planning that book.”[3]
In Borges' 1941 story, "The Library of Babel", the narrator declares that the collection of books of a fixed number of orthographic symbols and pages is unending.[4] However, since the permutations of twenty-five orthographic symbols is finite, the library has to be periodic and self-repeating.[2]
In his 1975 short story "The Book of Sand" (El Libro de Arena), he deals with another form of infinity; one whose elements are a dense set, that is, for any two elements, we can always find another between them. This concept was also used in the physical book the short-story came from, The Book of Sand book.[1] The narrator describes the book as having pages that are "infinitely thin", which can be interpreted either as referring to a set of measure zero, or of having infinitesimal length, in the sense of second order logic.[5]
In his 1936 essay "The Doctrine of Cycles" (La doctrina de los ciclos),[6] published in his essay anthology of the same year Historia de la eternidad, Borges speculated about a universe with infinite time and finite mass: "The number of all the atoms that compose the world is immense but finite, and as such only capable of a finite (though also immense) number of permutations. In an infinite stretch of time, the number of possible permutations must be run through, and the universe has to repeat itself. Once again you will be born from a belly, once again your skeleton will grow, once again this same page will reach your identical hands, once again you will follow the course of all the hours of your life until that of your incredible death."[7] As usual with many of Borges' ideas and constructions, this line of thought was received as metaphysical speculation, a language and philosophical game. Yet almost one century later theoretical physicists are crossing the same paths, this time as a possible consequence of string theory: "“Well, if the universe is really accelerating its expansion, then we know that it’s going to get infinitely large, and that things will happen over and over and over.” And if you have infinitely many tries at something, then every possible outcome is going to happen infinitely many times, no matter how unlikely it is.".[8]
Geometry and topology
Borges in "The Library of Babel" states that "The Library is a sphere whose exact center is any hexagon and whose circumference is unattainable". The library can then be visualized as being a 3-manifold, and if the only restriction is that of being locally euclidean, it can equally well be visualized as a topologically non-trivial manifold such as a torus or a Klein bottle.[5]
In his 1951 essay "Pascal's sphere" (La esfera de Pascal),[9] Borges writes about a "sphere with center everywhere and circumference nowhere". A realization of this concept can be given by a sequence of spheres with contained centres and increasingly large radii, which eventually encompasses the entire space. This can be compared to the special point in "The Aleph" by the process of inversion.[1]
Quantum physics
In "The Garden of Forking Paths", Borges describes a novel by the fictional Chinese scholar Ts'ui Pên, whose plot bifurcates at every point in time. The idea of the flow of time branching can be compared to the many-worlds interpretation of quantum mechanics and the notion of multiverses present in some versions of string theory.[10] Similarly, the infinitude of diverging, infinite universes in mathematical cosmology is reflected Borges' rejection of linear, absolute time.[11] Borges' writings address the nature of entity and the possibility of infinite "realities", as in his essay "New Time Refutations" (1946).[12]
Chaos theory
Bifurcation theory is a model in chaos theory of order appearing from a disordered system, and is a local theory that describes behavior of systems at local points. Borges anticipated the development of bifurcation theory in mathematics, through "The Garden of Forking Paths" in 1941. In "Garden", Borges captured the idea of a system splitting into multiple, uncorrelated states. For example, if a leaf floating in a river comes across a rock, it must flow across either side of the rock, and the two possibilities are statistically uncorrelated.[13]
References
1. Martínez, Guillermo (19 February 2003). "Borges and Mathematics". Retrieved 4 March 2012.
2. Hayles, N. Katherine (1984). The Cosmic Web: Scientific Field Models and Literary Strategies in the Twentieth Century. Ithaca: Cornell University Press. ISBN 0801492904.
3. "Los avatares de la tortuga", en Sur, nº 63, Buenos Aires, diciembre 1939, pp. 18-23. (Recogido en Discusión, Buenos Aires, Emecé, 1957) Original quote says: "Cinco, sietes años de aprendizaje metafísico, teológico, matemático, me capacitarían (tal vez), para planear decorosamente ese libro."
4. Borges, Jorge Luis (1998). Collected Fictions. Viking. ISBN 0-670-84970-7.
5. Bloch, William Goldbloom (2008). The Unimaginable Mathematics of Borges' Library of Babel. Oxford University Press. ISBN 978-0-19-533457-9.
6. La doctrina de los ciclos, en Sur, nº 20, Buenos Aires, mayo 1936, pp. 20-29. (Recogido en Historia de la eternidad, Buenos Aires, Viau y Zona, 1936. Fechado 1934)
7. The original quote at the beginning of the essay says "El número de todos los átomos que componen el mundo es, aunque desmesurado, finito, y sólo capaz como tal de un número finito (aunque desmesurado también) de permutaciones. En un tiempo infinito, el número de las permutaciones posibles debe ser alcanzado, y el universo tiene que repetirse. De nuevo nacerás de un vientre, de nuevo crecerá tu esqueleto, de nuevo arribará esta misma página a tus manos iguales, de nuevo cursaras todas las horas hasta la de tu muerte increíble."
8. Brookman, John (2014). The Universe: Leading Scientists Explore the Origin, Mysteries, and Future of the Cosmos. Harper Perennial. ISBN 978-0062296085.
9. La esfera de Pascal, en La Nación, Buenos Aires, 14 enero 1951, 2.ª sec., p. 1. (Recogido en Otras inquisiciones, Buenos Aires, Sur, 1952)
10. Merrel, Floyd (1991). Unthinking Thinking: Jorge Luis Borges, Mathematics, and the New Physics. West Lafayette: Purdue University Press. ISBN 1-55753-011-4.
11. Thiher, Allen (2005). Fiction refracts science: modernist writers from Proust to Borges. University of Missouri Press.
12. Di Marco, Oscar Antonio (2006). "Borges, the Quantum Theory and Parallel Universes" (PDF). The Journal of American Science. Retrieved 10 March 2012.
13. Hayles, N. Katherine (1991). Chaos and order: complex dynamics in literature and science. University of Chicago Press. ISBN 0226321436.
Jorge Luis Borges
Bibliography
Original
collections
A Universal History of Infamy
• "Man on Pink Corner"
• "On Exactitude in Science"
Ficciones
• "Tlön, Uqbar, Orbis Tertius"
• "The Approach to Al-Mu'tasim"
• "Pierre Menard, Author of the Quixote"
• "The Circular Ruins"
• "The Lottery in Babylon"
• "An Examination of the Work of Herbert Quain"
• "The Library of Babel"
• "The Garden of Forking Paths"
• "Funes the Memorious"
• "The Form of the Sword"
• "Theme of the Traitor and the Hero"
• "Death and the Compass"
• "The Secret Miracle"
• "Three Versions of Judas"
• "The End"
• "The Sect of the Phoenix"
• "The South"
The Aleph
• "The Immortal"
• "The Dead Man"
• "The Theologians"
• "Story of the Warrior and the Captive"
• "Emma Zunz"
• "The House of Asterion"
• "Deutsches Requiem"
• "Averroes's Search"
• "The Zahir"
• "The Writing of the God"
• "The Two Kings and the Two Labyrinths"
• "The Wait"
• "The Man on the Threshold"
• "The Aleph"
Otras Inquisiciones
(1937–1952)
• "The Analytical Language of John Wilkins"
Dreamtigers
• "Borges and I"
Dr. Brodie's Report
• "The Encounter"
• "The Gospel According to Mark"
The Book of Sand
• "The Other"
• "Ulrikke"
• "The Congress"
• "There Are More Things"
• "The Disk"
• "The Book of Sand"
Shakespeare's Memory
• "Blue Tigers"
• "Shakespeare's Memory"
Other works
• "Yo, Judío"
• Historia de la eternidad
• "A New Refutation of Time"
• Borges on Martín Fierro
• "El Golem"
• Book of Imaginary Beings
• Labyrinths
• Adrogue, con ilustraciones de Norah Borges
Related
• Leonor Acevedo Suarez (mother)
• Jorge Guillermo Borges (father)
• Norah Borges (sister)
• Celestial Emporium of Benevolent Knowledge
• H. Bustos Domecq
• Pedro Mata
• Uqbar
• Borges and mathematics
| Wikipedia |
\begin{definition}[Definition:Non-Stationary Stochastic Process]
A '''non-stationary stochastic process''' is a stochastic process which does not remain in equilibrium about a constant mean level.
That is, it is a stochastic process which is not a '''stationary stochastic process'''.
\end{definition} | ProofWiki |
Traveling wave solutions for a cancer stem cell invasion model
Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations
September 2021, 26(9): 5047-5066. doi: 10.3934/dcdsb.2020332
Stochastic modelling and analysis of harvesting model: Application to "summer fishing moratorium" by intermittent control
Xiaoling Zou 1,, and Yuting Zheng 2,
Department of Mathematics, Harbin Institute of Technology(Weihai), Weihai 264209, China
Department of Basic Course, Xingtai Polytechnic College, Xingtai 054000, China
* Corresponding author: Xiaoling Zou
Received November 2019 Revised June 2020 Published September 2021 Early access November 2020
As we all know, "summer fishing moratorium" is an internationally recognized management measure of fishery, which can protect stock of fish and promote the balance of marine ecology. In this paper, "intermittent control" is used to simulate this management strategy, which is the first attempt in theoretical analysis and the intermittence fits perfectly the moratorium. As an application, a stochastic two-prey one-predator Lotka-Volterra model with intermittent capture is considered. Modeling ideas and analytical skills in this paper can also be used to other stochastic models. In order to deal with intermittent capture in stochastic model, a new time-averaged objective function is proposed. Besides, the corresponding optimal harvesting strategies are obtained by using the equivalent method (equivalency between time-average and expectation). Theoretical results show that intermittent capture can affect the optimal harvesting effort, but it cannot change the corresponding optimal time-averaged yield, which are accord with observations. Finally, the results are illustrated by practical examples of marine fisheries and numerical simulations.
Keywords: summer fishing moratorium, intermittent control, asymptotically stable in distribution, optimal harvesting, equivalent method.
Mathematics Subject Classification: Primary: 34F05, 93E03; Secondary: 60H10.
Citation: Xiaoling Zou, Yuting Zheng. Stochastic modelling and analysis of harvesting model: Application to "summer fishing moratorium" by intermittent control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 5047-5066. doi: 10.3934/dcdsb.2020332
L. J. S. Allen and A. M. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biocsi., 163 (2000), 1-33. doi: 10.1016/S0025-5564(99)00047-4. Google Scholar
L. H. R. Alvarez, Optimal harvesting under stochastic fluctuations and critical depensation, Math. Biosci., 152 (1998), 63-85. doi: 10.1016/S0025-5564(98)10018-4. Google Scholar
L. H. R. Alvarez and L. A. Shepp, Optimal harvesting of stochastically fluctuating populations, Journal of Mathematical Biology, 37 (1998), 155-177. doi: 10.1007/s002850050124. Google Scholar
I. Barbǎlat, Système d'équations différentielles d'oscillations non linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270. Google Scholar
J. Batsleer, A. D. Rijnsdorp, K. G. Hamon, H. M. J. van Overzee and J. J. Poos, Mixed fisheries management: Is the ban on discarding likely to promote more selective and fuel efficient fishing in the dutch flatfish fishery?, Fish Res., 174 (2016), 118-128. doi: 10.1016/j.fishres.2015.09.006. Google Scholar
J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465. doi: 10.1126/science.197.4302.463. Google Scholar
A. Bottaro, Y. Yasutake, T. Nomura, M. Casadio and P. Morasso, Bounded stability of the quiet standing posture: An intermittent control model, Hum. Movement Sci., 27 (2008), 473-495. doi: 10.1016/j.humov.2007.11.005. Google Scholar
C. W. Clark, Mathematical Bioeconomics. The Optimal Management of Renewable Resources, , Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. Google Scholar
J. H. Connell, On the prevalence and relative importance of interspecific competition: Evidence from field experiments, Am. Nat., 122 (1983), 661-696. doi: 10.1086/284165. Google Scholar
[10] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, 1996. doi: 10.1017/CBO9780511662829. Google Scholar
J.-M. Ecoutin, M. Simier, J.-J. Albaret, R. Laë, J. Raffray, O. Sadio and L. T. de Morais, Ecological field experiment of short-term effects of fishing ban on fish assemblages in a tropical estuarine mpa, Ocean Coastal Manage., 100 (2014), 74-85. doi: 10.1016/j.ocecoaman.2014.08.009. Google Scholar
B. $\emptyset$ksendal, Stochastic Differential Equations, Springer-Verlag, 1985. doi: 10.1007/978-3-662-13050-6. Google Scholar
A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation sis epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902. doi: 10.1137/10081856X. Google Scholar
Y. Guo, W. Zhao and X. Ding, Input-to-state stability for stochastic multi-group models with multi-dispersal and time-varying delay, Appl. Math. Comput., 343 (2019), 114-127. doi: 10.1016/j.amc.2018.07.058. Google Scholar
S. Hong and N. Hong, H$^{\infty}$ switching synchronization for multiple time-delay chaotic systems subject to controller failure and its application to aperiodically intermittent control, Nonlinear Dyn., 92 (2018), 869-883. Google Scholar
C. Hu, J. Yu, H. Jiang and Z. Teng, Exponential stabilization and synchronization of neural networks with time-varying delays via periodically intermittent control, Nonlinearity, 23 (2010), 2369-2391. doi: 10.1088/0951-7715/23/10/002. Google Scholar
Z. Y. Huang, A comparison theorem for solutions of stochastic differential equations and its applications, Proc. Amer. Math. Soc., 91 (1984), 611-617. doi: 10.1090/S0002-9939-1984-0746100-9. Google Scholar
L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equations, 217 (2005), 26-53. doi: 10.1016/j.jde.2005.06.017. Google Scholar
B. Johnson, R. Narayanakumar, P. S. Swathilekshmi, R. Geetha and C. Ramachandran, Economic performance of motorised and non-mechanised fishing methods during and after-ban period in ramanathapuram district of tamil nadu, Indian J. Fish., 64 (2017), 160-165. doi: 10.21077/ijf.2017.64.special-issue.76248-22. Google Scholar
G. B. Kallianpur, Stochastic differential equations and diffusion processes, Technometrics, 25 (1983), 208. doi: 10.1080/00401706.1983.10487861. Google Scholar
W. Li and K. Wang, Optimal harvesting policy for stochastic logistic population model, Appl. Math. Comput., 218 (2011), 157-162. doi: 10.1016/j.amc.2011.05.079. Google Scholar
M. Liu and K. Wang, Dynamics of a two-prey one-predator system in random environments, J. Nonlinear Sci., 23 (2013), 751-775. doi: 10.1007/s00332-013-9167-4. Google Scholar
O. Ovaskainen and B. Meerson, Stochastic models of population extinction, Trends Ecol. Evol., 25 (2010), 643-652. doi: 10.1016/j.tree.2010.07.009. Google Scholar
N.-T. Shih, Y.-H. Cai and I.-H. Ni, A concept to protect fisheries recruits by seasonal closure during spawning periods for commercial fishes off taiwan and the east china sea, J. Appl. Ichthyol., 25 (2009), 676-685. doi: 10.1111/j.1439-0426.2009.01328.x. Google Scholar
L. Wang, D. Jiang and G. S. K. Wolkowicz, Global asymptotic behavior of a multi-species stochastic chemostat model with discrete delays, J. Dyn. Differ. Equ., 32 (2020), 849-872. doi: 10.1007/s10884-019-09741-6. Google Scholar
W. Xia and J. Cao, Pinning synchronization of delayed dynamical networks via periodically intermittent control, Chaos, 19 (2009), 013120, 8pp. doi: 10.1063/1.3071933. Google Scholar
B. Yang, Y. Cai, K. Wang and W. Wang, Optimal harvesting policy of logistic population model in a randomly fluctuating environment, Phys. A, 526 (2019), 120817, 17pp. doi: 10.1016/j.physa.2019.04.053. Google Scholar
Y. Ye, Assessing effects of closed seasons in tropical and subtropical penaeid shrimp fisheries using a length-based yield-per-recruit model, ICES J. Mar. Sci., 55 (1998), 1112-1124. doi: 10.1006/jmsc.1998.0415. Google Scholar
C. Zhang, W. Li and K. Wang, Graph-theoretic method on exponential synchronization of stochastic coupled networks with Markovian switching, Nonlinear Anal-Hybri., 15 (2015), 37-51. doi: 10.1016/j.nahs.2014.07.003. Google Scholar
G. Zhang and Y. Shen, Exponential synchronization of delayed memristor-based chaotic neural networks via periodically intermittent control, Neural Networks, 55 (2014), 1-10. doi: 10.1016/j.neunet.2014.03.009. Google Scholar
X. Zou and K. Wang, Optimal harvesting for a stochastic lotka-volterra predator-prey system with jumps and nonselective harvesting hypothesis, Optim. Control Appl. Methods., 37 (2016), 641-662. doi: 10.1002/oca.2185. Google Scholar
X. Zou and K. Wang, Optimal harvesting for a stochastic n-dimensional competitive lotka-volterra model with jumps, Discrete Cont. Dyn-B, 20 (2015), 683-701. doi: 10.3934/dcdsb.2015.20.683. Google Scholar
X. Zou, Y. Zheng, L. Zhang and J. Lv, Survivability and stochastic bifurcations for a stochastic Holling type II predator-prey model, Commun. Nonlinear Sci Numer. Simulat., 83 (2020), 105136, 20 pp. doi: 10.1016/j.cnsns.2019.105136. Google Scholar
Figure 1. Numerical simulations for sample paths
Figure 2. Numerical simulations for time average
Figure 3. The effects of intermittent control in one-dimensional situation
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Number of words of a given length in a regular language
Is there an algebraic characterization of the number of words of a given length in a regular language?
Wikipedia states a result somewhat imprecisely:
For any regular language $L$ there exist constants $\lambda_1,\,\ldots,\,\lambda_k$ and polynomials $p_1(x),\,\ldots,\,p_k(x)$ such that for every $n$ the number $s_L(n)$ of words of length $n$ in $L$ satisfies the equation $s_L(n)=p_1(n)\lambda_1^n+\dotsb+p_k(n)\lambda_k^n$.
It's not stated what space the $\lambda$'s live in ($\mathbb{C}$, I presume) and whether the function is required to have nonnegative integer values over all of $\mathbb{N}$. I would like a precise statement, and a sketch or reference for the proof.
Bonus question: is the converse true, i.e. given a function of this form, is there always a regular language whose number of words per length is equal to this function?
This question generalizes Number of words in the regular language $(00)^*$
formal-languages regular-languages word-combinatorics
Gilles 'SO- stop being evil'Gilles 'SO- stop being evil'
$\begingroup$ a sketch of a proof is here $\endgroup$
– Artem Kaznatcheev
$\begingroup$ @ArtemKaznatcheev Interesting, thanks. Would you consider moving your answer to this question, which it fits better? $\endgroup$
– Gilles 'SO- stop being evil'
$\begingroup$ I feel that this question is a little redundant (although more general). Generalizing my approach to the proof is a little hairy, but I will take a look after dinner. $\endgroup$
$\begingroup$ @ArtemKaznatcheev Thanks. I had trouble with the second part of your answer, extending to reducible DFAs. $\endgroup$
$\begingroup$ @vzn It is a classical fact that the generating function of the number of words in a regular language is rational, which immediately implies the OP's formula (in its correct form). The difficult part is extracting the asymptotics. For details you can check (for example) the book Analytic Combinatorics mentioned in my answer. $\endgroup$
– Yuval Filmus
Given a regular language $L$, consider some DFA accepting $L$, let $A$ be its transfer matrix ($A_{ij}$ is the number of edges leading from state $i$ to state $j$), let $x$ be the characteristic vector of the initial state, and let $y$ be the characteristic vector of the accepting states. Then $$ s_L(n) = x^T A^n y. $$
Jordan's theorem states that over the complex numbers, $A$ is similar to a matrix with blocks of one of the forms $$ \begin{pmatrix} \lambda \end{pmatrix}, \begin{pmatrix} \lambda & 1 \\ 0 & \lambda \end{pmatrix}, \begin{pmatrix} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{pmatrix}, \begin{pmatrix} \lambda & 1 & 0 & 0 \\ 0 & \lambda & 1 & 0 \\ 0 & 0 & \lambda & 1 \\ 0 & 0 & 0 & \lambda \end{pmatrix}, \ldots $$ If $\lambda \neq 0$, then the $n$th powers of these blocks are $$ \begin{pmatrix} \lambda^n \end{pmatrix}, \begin{pmatrix} \lambda^n & n\lambda^{n-1} \\ 0 & \lambda^n \end{pmatrix}, \begin{pmatrix} \lambda^n & n\lambda^{n-1} & \binom{n}{2} \lambda^{n-2} \\ 0 & \lambda^n & n\lambda^{n-1} \\ 0 & 0 & \lambda^n \end{pmatrix}, \begin{pmatrix} \lambda^n & n\lambda^{n-1} & \binom{n}{2}\lambda^{n-2} & \binom{n}{3}\lambda^{n-3} \\ 0 & \lambda^n & n\lambda^{n-1} & \binom{n}{2}\lambda^{n-2} \\ 0 & 0 & \lambda^n & n\lambda^{n-1} \\ 0 & 0 & 0 & \lambda^n \end{pmatrix}, \ldots $$ Here's how we got to these formulas: write the block as $B = \lambda + N$. Successive powers of $N$ are successive secondary diagonals of the matrix. Using the binomial theorem (using the fact that $\lambda$ commutes with $N$), $$ B^n = (\lambda + n)^N = \lambda^n + n \lambda^{n-1} N + \binom{n}{2} \lambda^{n-2} N^2 + \cdots. $$ When $\lambda = 0$, the block is nilpotent, and we get the following matrices (the notation $[n = k]$ is $1$ if $n=k$ and $0$ otherwise): $$ \begin{pmatrix} [n=0] \end{pmatrix}, \begin{pmatrix} [n=0] & [n=1] \\ 0 & [n=0] \end{pmatrix}, \begin{pmatrix} [n=0] & [n=1] & [n=2] \\ 0 & [n=0] & [n=1] \\ 0 & 0 & [n=0] \end{pmatrix}, \begin{pmatrix} [n=0] & [n=1] & [n=2] & [n=3] \\ 0 & [n=0] & [n=1] & [n=2] \\ 0 & 0 & [n=0] & [n=1] \\ 0 & 0 & 0 & [n=0] \end{pmatrix} $$
Summarizing, every entry in $A^n$ is either of the form $\binom{n}{k} \lambda^{n-k}$ or of the form $[n=k]$, and we deduce that $$ s_L(n) = \sum_i p_i(n) \lambda_i^n + \sum_j c_j [n=j], $$ for some complex $\lambda_i,c_j$ and complex polynomials $p_i$. In particular, for large enough $n$, $$ s_L(n) = \sum_i p_i(n) \lambda_i^n. $$ This is the precise statement of the result.
We can go on and obtain asymptotic information about $s_L(n)$, but this is surprisingly non-trivial. If there is a unique $\lambda_i$ of largest magnitude, say $\lambda_1$, then $$ s_L(n) = p_1(n) \lambda_1^n (1 + o(1)). $$ Things get more complicated when there are several $\lambda$s of largest magnitude. It so happens that their angle must be rational (i.e. up to magnitude, they are roots of unity). If the LCM of the denominators is $d$, then the asymptotics of $s_L$ will very according to the remainder of $n$ modulo $d$. For some of these remainders, all $\lambda$s of largest magnitude cancel, and then the asymptotics "drops", and we have to iterate this procedure. The interested reader can check the details in Flajolet and Sedgewick's Analytic Combinatorics, Theorem V.3. They prove that for some $d$, integers $p_0,\ldots,p_{d-1}$ and reals $\lambda_0,\ldots,\lambda_{d-1}$, $$ s_L(n) = n^{p_{n\pmod{d}}} \lambda_{n\pmod{d}}^n (1 + o(1)). $$
Yuval FilmusYuval Filmus
Let $L \subseteq \Sigma^*$ a regular language and
$\qquad \displaystyle L(z) = \sum\limits_{n \geq 0} |L_n|z^n$
its generating function, where $L_n = L \cap \Sigma^n$ and so $|L_n|=s_L(n)$.
It is known that $L(z)$ is rational, i.e.
$\qquad \displaystyle \frac{P(z)}{Q(z)}$
with $P,Q$ polynomials; this is easiest seen by translating a right-linear grammar for $L$ into a (linear!) equation system whose solution is $L(z)$.
The roots of $Q$ are essentially responsible for the $|L_n|$, leading to the form stated on Wikipedia. This is immediately related with the method of characteristic polynomials for solving recurrences (via the recurrence which describes $(|L_n|)_{n \in \mathbb{N}}$) .
Raphael♦Raphael
$\begingroup$ It is not clear how your answer answers the question. Also, what is $L_n$? $\endgroup$
– Dave Clarke
$\begingroup$ @Gilles Analytic Combinatorics, the books by Eilenberg, the book by Berstel, Reutenauer $\endgroup$
– uli
Apr 5 '12 at 7:22
$\begingroup$ @Gilles Automata-Theoretic Aspects of Formal Power Series. $\endgroup$
$\begingroup$ @Patrick87: 1) Right, typo; thanks! 2) For finite languages, the generating function is a polynomial (and therewith rational). As $Q(z)=1$, this approach won't work. The linked theorem starts with a linear homogeneous recurrence; I don't think those can describe sequences that are zero for all $k \geq n_0$ (and non-zero for at least one value). Not sure, though. If I am right, the statement we are talking about does indeed only hold for infinite regular languages; this would not be entirely surprising as finite languages do not have any structure. $\endgroup$
– Raphael ♦
$\begingroup$ @Raphael Yeah, my thinking was similar... that seems to be a fairly serious shortcoming in the presentation of the theorem, if it doesn't hold for finite languages, since (a) finite languages are regular, (b) the theorem implies finite languages aren't regular, and (c) determining whether a language is finite is (in general) undecidable... I mean, Myhill-Nerode and the pumping lemma don't have that problem; they work for finite languages. $\endgroup$
– Patrick87
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Generating function of non-regular language | CommonCrawl |
\begin{document}
\title{Photon echoes using atomic frequency combs in Pr:YSO --- experiment and semiclassical theory}
\author{Aditya~N.~Sharma$^1$} \author{Zachary~H.~Levine$^{2}$} \author{Martin~A.~Ritter$^1$} \author{Kumel~H.~Kagalwala$^1$} \author{Eli~J.~Weissler$^{2,3}$} \author{Elizabeth~A.~Goldschmidt$^{4,5}$} \author{Alan~L.~Migdall$^{1,2}$} \affiliation{ $^\text{1}$Joint Quantum Institute, University of Maryland, College Park, MD 20742, USA } \affiliation{ $^\text{2}$Quantum Measurement Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA }
\affiliation{ $^\text{3}$Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, CO 80309, USA } \affiliation{ $^\text{4}$Department of Physics, University of Illinois, Urbana, IL 61801, USA } \affiliation{ $^\text{5}$U. S. Army Research Laboratory, Adelphi, MD 20783, USA } \date{\today}
\begin{abstract} Photon echoes in rare-earth-doped crystals are studied to understand the challenges of making broadband quantum memories using the atomic frequency comb (AFC) protocol in systems with hyperfine structure. The hyperfine structure of Pr$^{3+}$ poses an obstacle to this goal because frequencies associated with the hyperfine transitions change the simple picture of modulation at an externally imposed frequency. The current work focuses on the intermediate case where the hyperfine spacing is comparable to the comb spacing, a challenging regime that has recently been considered. Operating in this regime may facilitate storing quantum information over a larger spectral range in such systems.
In this work, we prepare broadband AFCs using optical combs with tooth spacings ranging from 1~MHz to 16~MHz in fine steps, and measure transmission spectra and photon echoes for each. We predict the spectra and echoes theoretically using the optical combs as input to either a rate equation code or a density matrix code, which calculates the redistribution of populations. We then use the redistributed populations as input to a semiclassical theory using the frequency-dependent dielectric function. The two sets of predictions each give a good, but different account of the photon echoes.
\end{abstract}
\maketitle
\section{Introduction} \label{sec:intro}
Quantum memory is a crucially important technology. Perhaps most prominently, it will play an important role in long-range quantum networks~\cite{Acin2018,Pompili2021}, which will be required to fully realize the power of quantum computers and other quantum technologies: quantum repeaters will be necessary to overcome transmission losses in these networks~\cite{Briegel1998}, and most current approaches to quantum repeaters depend on optical quantum memory~\cite{Duan2001,Awschalom2018}, a device for storing quantum states and recovering them at a later time. A practical quantum memory would enable long-lived storage and high-fidelity, high-efficiency retrieval of broadband single photons. Rare-earth-ion-doped crystals are promising candidates for implementing quantum memory due to their long ground-state lifetimes, large inhomogeneous bandwidths, and narrow homogeneous bandwidths~\cite{simon2007quantum}. Various approaches to quantum memory have been developed for rare-earth-crystal platforms. The atomic frequency comb (AFC) protocol is one such scheme~\cite{Afzelius2009}. The central idea is to store single photons in a material with a frequency-comb absorption spectrum, with comb-tooth spacing $f_{\rm rep}$, and then to recover photon echoes after a time $T_{\rm rep}=1/f_{\rm rep}$.
Pr$^{3+}$:Y$_2$SiO$_5$ (Pr:YSO) is suitable for various optical storage techniques, and it was the first material used to demonstrate on-demand AFC quantum memory~\cite{Afzelius2010}. It has been studied extensively, including work on photon echoes~\cite{Graf1997,Graf1998}, spectral-hole burning~\cite{Nilsson2004}, optical filtering~\cite{beavan2013demonstration}, electromagnetically induced transparency~\cite{Fan2019}, and stimulated Raman adiabatic passage~\cite{Klein2007,Gao2007}. The latter technique was used for selective retrieval of pulses~\cite{Wang2008}, and further studies investigated the rephasing of these coherent populations~\cite{Mieth2012} and on-demand retrieval based on spontaneous Raman scattering~\cite{Goldschmidt2013}. Storage of optical pulses has been demonstrated with stopped light using electromagnetically induced transparency~\cite{Longdell2005,Heinze2013} and a memory has been realized at the single-photon level using stopped light in a spectral hole~\cite{Kutluer2016}. Gradient-echo quantum memory was demonstrated~\cite{Hedges2010}, and AFC quantum memory has also been demonstrated at the single-photon level~\cite{Rielander2014,Seri2019}. Recently, Pr:YSO has been used for high-rate entanglement distribution as a step towards the implementation of quantum repeaters \cite{Mannami2021} and on-demand quantum memory~\cite{Horvath2021}.
The AFC protocol has also been studied in other rare-earth-ion-doped crystals, which have similarly advantageous optical properties. Ref.~\cite{Bonarota2010} studied the effect of comb-tooth shape on echo efficiency. Storage and retrieval of single photons has been demonstrated: time-bin qubits~\cite{Usmani2010, Davidson2020} and entanglement with another photon~\cite{Saglamyurek2011} storage. Temporal multimode storage has been studied~\cite{Jobez2016}. Pulse storage and retrieval has also been demonstrated using Stark shifts~\cite{Zhong2017}, superhyperfine levels~\cite{Askarani2019} and hybridized electron-nuclear hyperfine levels~\cite{Businger2020}. Gigahertz bandwidths were demonstrated using Tm$^{3+}$~\cite{Saglamyurek2011} and Er$^{3+}$~\cite{Saglamyurek2016} ions. Er$^{3+}$ has also been studied for use in quantum repeaters~\cite{Craiciu2021}.
One attractive feature of the AFC protocol highlighted in Ref.~\cite{Afzelius2009} is that in principle it enables retrieval efficiency arbitrarily close to unity. In practice, however, experimental demonstrations have achieved limited efficiency: Refs.~\cite{Sabooni2013,Jobez2014} used cavity enhancement to reach efficiencies just over 50\%.
Many rare-earth species and isotopes, including Pr:YSO, have hyperfine structure, which presents an obstacle to preparing high-bandwidth AFCs. To circumvent this challenge, past works on AFCs in Pr:YSO have followed two approaches: (1) use AFC bandwidth smaller than the hyperfine spacing \cite{Goldschmidt2013,Rielander2014,Kutluer2016}, or (2) use AFC tooth spacing larger than the entire 37~MHz spectral width of the hyperfine levels~\cite{Nicolle2021}. (Ref.~\cite{Seri2019} used a combination of these methods.) Here, we explore the intermediate regime in which the AFC bandwidth is larger than 37~MHz and the AFC tooth spacings are comparable to the hyperfine spacing. While some past studies on AFCs worked in this regime~\cite{Askarani2019,teja2019photonic}, here we systematically investigate AFC formation and echo retrieval for a range of comb-tooth spacings. We experimentally demonstrate AFC formation and echo retrieval across a range of comb-tooth spacings.
Through this systematic study, we hope to elucidate the dynamics between AFC bandwidth and hyperfine structure, which could aid in the design of future high efficiency AFC-based quantum memories.
We also develop a semiclassical theory of photon echoes which calculates the effect of the experimental optical fields applied to the crystal. Historically, photon echo phenomena have been understood as collective emission by atoms prepared in a Dicke state~\cite{Dicke1954}, whereas here we achieve good results with a semiclassical model in which individual Pr$^{3+}$ ions, treated quantum mechanically, interact with a classical electromagnetic field. Although several authors have noted the presence of a semiclassical limit~\cite{Sangouard2007,Gorshkov2007b,Afzelius2009}, and the formalism of this limit was presented in a recent review article~\cite{Chaneliere2018}, there does not appear to be a realistic calculation of photon echoes for rare-earth-doped crystals in the literature, particularly not one which calculates the AFC based on an observed optical signal as well as the photon echoes. We calculate the atomic population shifts that occur during AFC preparation; the change to the dielectric response due to these shifts; and the echoes that arise from a pulse interacting with the highly dispersive medium. We present two sets of results, one corresponding to AFCs as predicted by the rate equations and a second set as predicted by the density matrix formalism. Given the redistribution of population which is a quantitative description of the AFC, the photon echo is described within the same formalism of highly dispersive linear optical response.
Regarding other calculations in this area, Theil et al.~\cite{thiel2014measuring} gave a theoretical account matching experimental results of pulse echo linewidths in three rare-earth crystals Tm$^{3+}$:YAG, Tm$^{3+}$:LiNbO3 and Tm3$^{3+}$:YGG. More frequently, the calculations are models~\cite{xiong2008numerical} or done to propose experiments~\cite{arslanov2017optimal,tittel2010photon} rather than to do a careful comparison.
\begin{figure}
\caption{Each Pr$^{3+}$ ion in Pr:YSO has 6 excited states and 6 ground states, all of which are two-fold degenerate in the absence of an external magnetic field. (a) Thus, each ion has 9 different transition energies $a$-$i$. (b) Conversely, any laser frequency within the 4.4~GHz inhomogeneous bandwidth addresses one of these transitions for 9 different ion classes $A$-$I$. (c) Table of transition frequencies for all ion classes addressed by a single laser frequency at zero detuning. Each row shows all 9 transitions for one ion class. Rows are grouped by common ground states. Since the laser frequency addresses transition $x$ for class $X$, the diagonal elements are zero. The bold-face diagonal-block elements indicate transitions with depopulated ground states, leading to peaks in the transmission spectrum in Fig.~\ref{fig:singleFreqResponse}, while the other elements indicate troughs. The bold-face elements depend only on the excited-state spacings, while the others also depend on the ground-state spacings.}
\label{fig:introFigure}
\end{figure}
\begin{figure}
\caption{Pr:YSO transmission spectrum after spectral-hole burning at zero detuning, as measured experimentally (black) and as predicted by density matrix calculation (red). The curves have been normalized to unit transmittance. The transition frequencies from Fig.~\ref{fig:introFigure}c are shown here.
}
\label{fig:singleFreqResponse}
\end{figure}
The YSO crystal structure has two distinct Y sites where Pr$^{3+}$ impurities can occur. Here we study Pr$^{3+}$ ions at site~1~\cite{Maksimov1969crystal,Holliday1993}, which have $^3H_4\,\rightarrow\,^1D_2~(\mathbf{g}\rightarrow\mathbf{e})$ optical transitions at 605.977~nm. The homogeneous linewidth of these optical transitions is less than 1~kHz and can be observed when the crystal is cooled to liquid helium temperatures to eliminate phonon broadening; the inhomogeneous broadening, arising from local variations in the crystal structure surrounding each Pr$^{3+}$ ion, is typically several gigahertz~\cite{Equall1995}.
In the absence of an external magnetic field, the hyperfine states are all two-fold degenerate, and the ground- and excited-state manifolds each have three distinct energy levels (Fig.~\ref{fig:introFigure}). Therefore, each ion has nine transition energies labeled $a-i$ in ascending order; conversely, for any laser frequency within the inhomogeneous bandwidth, there are nine different classes of ions $A-I$ for which one of these transitions is resonant with the laser, with the laser addressing transition $x$ for class $X$.
As a result of the hyperfine structure, spectral-hole burning at even a single frequency results in multiple features (Fig.~\ref{fig:singleFreqResponse}). The large transmittance peak at zero detuning occurs because all 9 classes become transparent at this frequency. The smaller peaks occur because some subset of the classes become transparent: for example, only 3 of the 9 classes that would normally absorb at 9.42~MHz detuning are depopulated. Due to power broadening, our measurement cannot distinguish features separated by less than 0.2~MHz, but such closely-spaced peaks may appear as broader peaks or shoulders. The transmission peaks in Fig.~\ref{fig:singleFreqResponse} correspond to population depletion at detuning frequencies shown in bold in Fig.~\ref{fig:introFigure}c. The other frequencies correspond to population enhancement, hence transmission dips in most cases.
The excited-state spacings in Pr:YSO are nearly equal, leading to a comb-like response to spectral-hole burning at a single frequency. This occurs because (1) the transparency for classes $A,D,G$ at 4.58~MHz detuning overlaps with the transparency for $B,E,H$ at 4.84~MHz, and (2) an additional transparency occurs for $A,D,G$ at 9.42~MHz, further extending the comb-like response (similar reasoning for classes $C,F,I$ and $B,E,H$ explains the peaks at $-9.42$~MHz, $-4.84$~MHz, and $-4.58$~MHz). This observation suggests that the material has a naturally periodic spectrum. Although it has been noted in previous works, for example Ref.~\cite{Askarani2019}, that this periodicity dictates which tooth spacings are compatible with AFC formation, here we carefully examine tooth spacings close to the periodicity. In fact, we find that AFC formation is optimal \textit{not} for tooth spacings exactly matching the excited-state hyperfine spacings, but rather a few hundred kilohertz detuned from those values.
\section{Experiment} \label{sec:expt} We use our experimental setup for two different measurement protocols, a comb-measurement protocol and a pulse-echo protocol, each of which proceeds in two stages. In the first stage of both protocols, we prepare an AFC. In the second stage, we either probe the transmission spectrum (the comb-measurement protocol), or observe the transmission and echoes of a pulse sent to the AFC (the pulse-echo protocol).
We use a tunable diode laser near 1212~nm, which is amplified in a fiber Raman system and then doubled, producing a beam at the $^3H_4\rightarrow{^1}\!D_2$ optical transition. To stabilize the laser frequency, we use Pound-Drever-Hall feedback~\cite{Black2001} to lock the infrared light to a reference cavity with a Zerodur (Schott Glass), spacer, held in a vacuum chamber. This beam is modulated using double-passed acousto-optic modulators (AOMs) to prepare the three different beams used in our measurements. A burn beam is produced by applying a frequency-modulated (FM) radio-frequency (RF) sine wave to the . This signal has a frequency-comb structure, centered at 100~MHz, with 60~MHz bandwidth, and a tooth spacing equal to the FM modulation frequency. After the double pass, this results in an optical frequency comb with 120~MHz bandwidth and tooth spacing equal to the FM modulation frequency (Fig.~\ref{fig:fourierComparison}). A probe beam is produced by (slowly) sweeping the AOM frequency, resulting in a frequency-swept beam covering the same bandwidth as the burn beam. Alternatively, 10~ns pulses are generated by applying RF pulses to the AOM.
\begin{figure}
\caption{Optical frequency comb generated by AOM frequency modulation and observed by heterodyne mixing with a local oscillator (black curve). Here, the modulation frequency, and thus the tooth spacing, is 4.8~MHz. Our theoretical calculations require averaging, as described in the Appendix, to obtain a strictly periodic approximation to this signal. Also shown is the Fourier transform of the signal (orange points) of Fig.~5 and the transform with a small shift required to achieve phase periodicity (green points).}
\label{fig:fourierComparison}
\end{figure}
In our setup (Fig.~\ref{fig:expLayout}), we use a Pr:YSO crystal with 0.05\% substitution of Pr for Y, housed in a cryostat maintained below 4~K. Two beams are focused to overlapping waists inside the crystal. Beam~1 is the burn beam described above: we vary the tooth spacing of this optical frequency comb in 0.1~MHz steps, from 1.0~MHz to 16~MHz. The first stage of both measurement protocols consists of 30~seconds of nearly continuous burning with Beam~1, aiming to imprint the optical frequency comb on the atomic transmission spectrum and establish an AFC. In our comb-measurement protocol, Beam~2 is the probe beam described above. During the 4~ms frequency sweep of the probe beam, we acquire the transmission spectrum of the crystal. In the pulse-echo protocol, Beam~2 is the 10~ns pulse described above. We monitor emission from the crystal to detect the transmitted pulse and its echoes. Due to the different natures of the output signals from the two measurements, detectors with different response bandwidths are required. In the second stage of each measurement protocol, the 1~s measurement cycle is synchronized to an electrical pulse indicating the cryostat pump cycle, in order to minimize effects of vibrations. We repeat the measurements for approximately 30 cycles~\cite{note:cycleNumber} and average the results. In addition to the 30~s burn in stage~1, in stage~2 of each cycle, 930~ms is spent burning the crystal to reinforce the AFC.
\begin{figure}
\caption{(a)~We prepare an AFC by burning the crystal for 30~seconds with an optical frequency comb (Beam 1). We pick off some of this beam to beat against a local oscillator for a balanced heterodyne measurement of the optical spectrum with which we drive the crystal. We use Beam~2 to make two different measurements (see text). Both measurements are repeated for approximately 30~cycles and the results are averaged. Each cycle is synchronized with the cryostat pump cycle to minimize the effect of vibrations. (b)~Timing diagram for one cycle of the AFC measurement (not to scale). During the 4~ms probe period, the laser frequency is swept across the AFC bandwidth to measure transmission. The short pulses on either side of the probe are used to demarcate the sweep duration to help automate data analysis; to minimize their effect on the measurements, they are tuned to a frequency 20~MHz outside the AFC and sweep bandwidths. (c)~Timing diagram for one cycle of the echo measurement (not to scale).
}
\label{fig:expLayout}
\end{figure}
\section{Theory} \label{sec:theory}
Most discussions of the AFC protocol rely on preparation of a Dicke state of the atomic ensemble~\cite{Sangouard2007,Gorshkov2007b,Afzelius2009,Afzelius2010,Minar2010,Jin2015,Jobez2016,Saglamyurek2016,Askarani2019,teja2019photonic,Furuya2020,Holzapfel2020,Etesse2021,Nicolle2021}; in this paper, however, we show that the AFC transmission spectra and echo pulses can be obtained using only semiclassical concepts, provided the beams are in coherent states. Not all authors invoke the Dicke state. Chaneli{\`e}re et al.~\cite{chaneliere2010efficient} also outlines a semiclassical treatment of photon echoes in AFCs, but the calculation of the susceptibility is not explicit. Burman and Le~Gouet~\cite{berman2021pulsed} have made an analysis of both uniform and random AFCs in highly dispersive media using the Maxwell-Bloch equations for a two-level system, with an emphasis on analytic modelling.
In Section~\ref{subsec:refIndex}, we describe the theory required for calculating the dielectric response function. In Section~\ref{subsec:propagation} we show that this function can be used to calculate the propagation of light pulses through the crystal. Finally, in Section~\ref{subsec:denMat} we describe our method for determining how an incident field changes the ground-state population distribution.
\subsection{Calculating the dielectric response function} \label{subsec:refIndex}
The Clausius-Mossotti relation gives the macroscopic dielectric function $\varepsilon(\omega)$ of a solid in terms of the polarizabilities $\alpha_k(\omega)$ of its constituent atoms. The relation remains valid for particles embedded in a homogeneous medium if $\varepsilon(\omega)$ is interpreted as the dielectric constant relative to the undoped crystal dielectric constant $\epsilon_0\varepsilon_{\rm bkg}$~\cite{Levy1992}: \begin{eqnarray} 3\frac{\varepsilon(\omega)-1}{\varepsilon(\omega)+2} = \frac{1}{\epsilon_0 \varepsilon_{\rm bkg}} \sum_k N_k \alpha_k(\omega) .\label{eq:Clausius} \end{eqnarray} Here, the index $k$ denotes different species of oscillators, $N_k$ is the number density of such oscillators, $\epsilon_0$ is the vacuum permittivity, and $\omega$ is the frequency~\cite{Mohr2015}. When applied to optical frequencies, Eq.~(\ref{eq:Clausius}) is often called the Lorentz-Lorenz relation~\cite{zangwill2013modern}. In our problem, values of $k$ represent the ion classes forming the inhomogeneous band and the ground-state occupancies of the ions. We do not calculate the background dielectric constant of the YSO crystal $\varepsilon_{\rm bkg}$, but instead assume the refractive index value found in the literature $n_{\rm bkg}=\sqrt{\varepsilon_{\rm bkg}}=1.8$ ~\cite{Beach1990}.
The validity of the Clausius-Mossotti equation depends on our ability to define regions within the solid where charge does not pass. If such units do not exist~\cite{Martin1974}, then intrinsically solid-state approaches such as band theory~\cite{Levine1991} or the modern theory of polarizability~\cite{Resta1994} are required to calculate the dielectric function. In the present case, the Pr$^{3+}$ impurity ions (a)~have open 4f shells which are located in the interior of the ions, (b)~are dilute, hence located far from each other on the scale of an atomic bond length, and (c)~individually have very narrow resonances, whose energies differ from ion to ion by many times the resonance width. Each of these circumstances inhibits any conventional quantum mechanical interactions such as overlap of wave functions leading to bonding-antibonding splittings or band structure. Hence, Martin's criterion~\cite{Martin1974} of a well-defined surface to separate the various oscillating charges is satisfied, building our confidence in the applicability of the Clausius-Mossotti equation. The surface in question is simply a sphere drawn around each individual Pr$^{3+}$ ion well outside of the 4f electrons.
The sum over $k$ in Eq.~(\ref{eq:Clausius}) may be written as a sum over the ground-state hyperfine levels $i$, integrated over the detuning $\Delta_0$ of their lowest-energy hyperfine transition $a$ (Fig.~\ref{fig:introFigure}). We measure $\Delta_0$ relative to the center of the inhomogeneous band $\omega_0$. We define \begin{eqnarray} B_i(\omega) = \frac{1}{\epsilon_0 \varepsilon_{\rm bkg}}\int \! \text{d}\Delta_0 \, \rho_i(\Delta_0) \, \alpha_i(\omega,\Delta_0) \label{eq:BiA} \end{eqnarray} to rewrite Eq.~(\ref{eq:Clausius}) as \begin{eqnarray} 3\frac{\varepsilon(\omega)-1}{\varepsilon(\omega)+2} = \sum_i B_i(\omega) \label{eq:sumBi} \end{eqnarray} where $\alpha_i(\omega,\Delta_0)$ is the polarizability of an ion for radiation at frequency $\omega$, assuming it is in ground-state hyperfine level $i$ and its lowest-energy transition is at frequency $\omega_0+\Delta_0$; and $\rho_i(\Delta_0)$ is the density of such ions. In our application, the frequency $\Delta_0$ runs over the inhomogeneous band, which is 4.4~GHz wide~\cite{Equall1995}, hence very narrow compared with the optical transition of 495~THz. The density $\rho_i$ includes both the assumed Gaussian background due to inhomogeneous broadening as well as any laser-induced redistribution of population, as occurs in AFC preparation. Redistribution preserves $\sum_i \rho_i(\Delta_0)$: spectral-hole burning can change the population distribution within the ground-state manifold but it cannot remove impurity ions or change their resonant frequencies. We assume there is no excited-state population. Experimentally, this is ensured by a 10~ms delay after hole burning --- about 61 times the mean time for spontaneous emission --- before the pulses generating the echoes begin.
In this work, $i$ runs over the three ground states $|\,\mathbf{g},\sfrac{1}{2}\rangle$, $|\,\mathbf{g},\sfrac{3}{2}\rangle$, and $|\,\mathbf{g},\sfrac{5}{2}\rangle$, and we consider transitions to the three excited states $|\,\mathbf{e},\sfrac{1}{2}\rangle$, $|\,\mathbf{e},\sfrac{3}{2}\rangle$, and $|\,\mathbf{e},\sfrac{5}{2}\rangle$. However, our formalism applies to arbitrary numbers of ground and excited states: this makes it readily adaptable to cases where the degenerate hyperfine levels are split due to external fields, and even to calculations for other dopant ions.
In the rotating-wave approximation (RWA), the atomic polarizability has the Drude-Lorentz form \begin{eqnarray} \alpha_i(\omega,\Delta_0) = f_0 \sum_j \frac{f_{ij}}{\omega_0+\Delta_0+\Delta^{(0)}_{ij} - \omega -\text{i}\gamma} \label{eq:alphaA} \end{eqnarray} where $j$ indexes the Pr$^{3+}$ ions' excited-state hyperfine levels, $\gamma$ is the inhomogeneous width, the $f_{ij}$ are the relative transition probabilities, and $f_0$ is a constant related to the dimensional oscillator strength. The $f_{ij}$ are constrained by conservation of probability $\sum_i f_{ij} = \sum_j f_{ij}=1$~\cite{Equall1995}. The constants $\Delta^{(0)}_{ij}$ are the transition frequencies relative to the $|\,\mathbf{g},\sfrac{1}{2}\rangle\rightarrow|\,\mathbf{e},\sfrac{1}{2}\rangle$ transition (Fig.~\ref{fig:introFigure}a). Note that the RWA leads to the near-resonance term in Eq.~(\ref{eq:alphaA}) as well as a term in the ultraviolet, which we have omitted: in our application, the neglected term is at least 5 orders of magnitude smaller than the term retained.
Ref.~\cite{Nilsson2004} gives oscillator strengths for the 9 hyperfine transitions in Pr:YSO. Within a manifold, the transition rates are proportional to these oscillator strengths. Oscillator strengths differ from squares of matrix elements by a factor proportional to the transition frequencies. This factor is $1+O(10^{-7})$ because the hyperfine splittings are less than 50~MHz and the reference transition frequency is 495~THz.
Eq.~(\ref{eq:alphaA}) allows us to write \begin{eqnarray} \alpha_i(\omega,\Delta_0) = \alpha_i(\omega-\Delta_0,0) .\label{eq:alphaB} \end{eqnarray} Using the convention $\omega=\omega_0+\Delta$, the polarizability can be written as \begin{eqnarray} \tilde\alpha_i(\Delta) = f_0 \sum_j \frac{f_{ij}}{\Delta^{(0)}_{ij} - \Delta -\text{i}\gamma} .\label{eq:alphaC} \end{eqnarray} Similarly, we define $\tilde\varepsilon(\Delta)=\varepsilon(\omega)$ and $\tilde B_i(\Delta) = B_i(\omega)$. We may rewrite Eq.~(\ref{eq:BiA}) as \begin{eqnarray} \tilde B_i(\Delta) = \frac{1}{\epsilon_0}\int \! \text{d}\Delta_0 \, \rho_i(\Delta_0) \, \tilde\alpha_i(\Delta-\Delta_0) \label{eq:BiB} \end{eqnarray} and Eq.~(\ref{eq:sumBi}) as \begin{eqnarray} 3\frac{\tilde\varepsilon(\Delta)-1}{\tilde\varepsilon(\Delta)+2} = \sum_i \tilde B_i(\Delta) .\label{eq:sumBiTilde} \end{eqnarray} Eq.~(\ref{eq:BiB}) is a convolution integral. As such, it can be evaluated rapidly by Fast Fourier transforms (FFTs), using the convolution theorem. A similar formulation was presented earlier~\cite{Sonajalg1994}.
To find the dielectric response of the unperturbed impurity band in our study, the three ground-state hyperfine levels were each assumed to have an identical Gaussian density of states with a full width at half maximum (FWHM) of 4400~MHz (standard deviation $\sigma_{\rm inh}= 1869$~MHz). When the crystal is unperturbed, all three states have equal occupancy. The frequency step is 50~kHz with $2^{20}$, i.e., over 1 million, points used in the FFT, leading to a bandwidth of 52~GHz, which is more than $28\,\sigma_{\rm inh}$. The polarization sums were formed on this grid leading to a function proportional to the polarizability, or equivalently $\varepsilon(\omega)-1$. The proportionality constant was set at the peak absorption as described in Section~\ref{subsec:propagation}.
\subsection{Pulse propagation} \label{subsec:propagation}
Taking the beam direction in the crystal to be the $+z$ direction, a scalar wave propagating in such a medium obeys \begin{eqnarray} U(z) = U_0 \text{e}^{\text{i} n k z} = U_0 \text{e}^{\text{i} n_1 k z} \text{e}^{-n_2 k z} \label{eq:scalarWave} \end{eqnarray} where $U_0$ is the incident optical amplitude, $k=2\pi n_{\rm bkg} / \lambda$ is the wavevector in an undoped YSO crystal, $n=\sqrt{\varepsilon}=n_1 + \text{i} n_2$ is the adjustment to the refractive index $n_{\rm bkg}$ due to the dopant ions, and $\lambda$ is the free-space wavelength. The transmission is given by \begin{eqnarray}
T(z) = \left| \frac{U(z)}{U_0}\right|^2 = \text{e}^{-2 n_2 k z} .\label{eq:trans} \end{eqnarray} Assuming $T=9.9$~dB, as observed in our single-frequency-burn measurements in the 10~mm long crystal, we evaluate Eq.~(\ref{eq:trans}) at the center of the inhomogeneous band, finding $n_2(\omega_0)=1.1 \times 10^{-5}$. Here $n_1(\omega_0)=1$, leading to $\varepsilon(\omega_0)-1 = 2 \, \text{i} \, n_2(\omega_0)$ where we neglect the term $[n_2(\omega_0)]^2$.
In practice, we find the sum of convolution integrals $\sum_i\tilde B_i(\Delta)$, divide by $\sum_i\tilde B_i(0)$ for the unperturbed system and multiply by $\tilde\varepsilon(0)-1$, found experimentally, to obtain $\tilde\varepsilon(\Delta)-1$. Thus, we avoid an explicit value for the density of impurity ions.
Our model of photon echoes consists of applying Eq.~(\ref{eq:scalarWave}) to each Fourier component of a pulse incident on the crystal with a waveform given by $U_0(t)$. Because the pulse has a narrow bandwidth about a central frequency, the transform variable is the detuning $\Delta$. Because the crystal has an antireflective (AR) coating, reflections are neglected. The dielectric response is given by $\tilde \varepsilon(\Delta)$. At the exit face, the frequency components are back transformed to the time domain yielding $U_1(t)$ at the exit face. Symbolically, \begin{eqnarray} U_1(t) = {\cal F}^{-1} \left\{ {\cal F} \left[ U_0(t) \right]
\exp\left[ \text{i} \tilde n(\Delta) k z \right]
\right\} \label{eq:pulseFourier} \end{eqnarray} where $\tilde n(\Delta) = [\tilde\varepsilon(\Delta)]^{1/2}$ and $\cal F$ is the Fourier transform from time domain to the domain of detuned frequency $\Delta$. A similar method was used by Teja et al.~\cite{teja2019photonic}.
\subsection{Calculations of ground-state population redistribution} \label{subsec:denMat}
We calculate the AFC spectra, i.e., the redistribution of the ground-state population using two different methods, namely by solving the rate equations and by using a density matrix formalism. Each approach has strengths and weaknesses. The density matrix formalism is more generally applicable, while the rate equations may be derived from the density matrix under certain physical conditions. When the rate equations are valid, only the populations, and not the coherences, need be calculated. In both cases, we treat each ion class independently: a more thorough analysis could consider the interplay between different classes, to include spectral diffusion effects, for example, but this is beyond the scope of our present work.
The rate equations, introduced by Einstein in 1905, have played a huge role in understanding optical systems such as lasers that depend on redistribution of population of quantum levels. The density matrix is a more sophisticated level of theory which also includes quantum coherence. While the rate equation captures the effects of absorption, stimulated emission, and spontaneous emission, it cannot capture coherent phenomena such as Rabi oscillations and coherent population redistribution mechanisms such as Stimulated Raman Adiabatic Passage (STIRAP)~\cite{Unanyan1998}. Indeed, the name ``counterintuitive pulse sequence'' associated with STIRAP is ``counterintuitive'' under the unstated assumption that the intuition was built on the rate equations.
The text of Grynberg et al.~\cite{Grynberg2010} discusses the validity conditions for the rate equations. Neither of the two conditions they discuss apply: neither do the coherences relax substantially faster than the populations are transferred, nor is the correlation time of the laser small compared to the decay time associated with the (power-broadened) absorption line. Despite a question of the region of validity, we present results of atomic frequency combs based on the rate equations. We also present a calculation of the redistribution of population using the density matrix.
We considered using the stochastic Schr{\"o}dinger equation (SSE)~\cite{Plenio1998} as well. An average of the SSE is formally equivalent to the density matrix~\cite{Johanson2012}, although the SSE has advantages for single-particle dynamics~\cite{Plenio1998}. Ref.~\cite{Johanson2012} suggests that the density matrix is favored algorithmically for small matrices, and the SSE is favored for large ones. Ref.~\cite{Johanson2012} describes a cross-over when the dimension of the Hamiltonian exceeds 200. Here, the Hamiltonian is only of dimension~6, suggesting the density matrix formalism will be more efficient. This is true, but it understates the case. Because our Hamiltonian is periodic, it is possible to obtain a solution in the logarithm of the number of periods. Attal et al.~\cite{attal2018rate} make a similar observation to accelerate a similar calculation using the rate equation. We were not able to find a comparable algorithm for the SSE, making the density matrix the better method for this application.
\subsubsection{Rate Equation Formalism}
In our rate equation calculations, we assume the entire population is initially distributed equally among the 3 ground-state levels, and describe their occupancies with a vector \begin{equation} N(t=0)= \begin{pmatrix} N_{1/2}(0) \\ N_{3/2}(0) \\ N_{5/2}(0) \end{pmatrix}= \begin{pmatrix} 1/3 \\ 1/3 \\ 1/3 \end{pmatrix}. \label{eq:rateEqStart} \end{equation}
Based on our linewidth measurements, we assign each ion class a Lorentzian absorption distribution around each of its transitions. To determine the amount of optical power addressing a given transition, we integrate the product of this distribution and the measured input optical spectrum used for hole burning. Repeating this process for all 9 transitions, we form a $3\times3$ matrix $I_{kj}$ of powers addressing the transitions from $|\,\mathbf{g},j\rangle$ to $|\,\mathbf{e},k\rangle$. We assume that the transition rates from $\mathbf{g}$ to $\mathbf{e}$ are proportional to these matrix elements, motivated by the fact that our sample has a high optical depth, so that there is a high probability of absorption regardless of the oscillator strength of a particular transition. Furthermore, we assume that the excited states decay to ground states within a time increment of our calculation, according to the oscillator strength matrix $p_{\ell k}$ describing transitions from $|\,\mathbf{e},k\rangle$ to $|\,\mathbf{g},\ell \rangle$ (the squares of the matrix elements in Table \ref{tab:amps}). The rate matrix $R_{\ell j}$ describing population redistribution from $|\,\mathbf{g},j\rangle$ to $|\,\mathbf{g},\ell\rangle$ has elements \begin{equation}
\begin{aligned}
&R_{\ell j} = \sum_{k}p_{\ell k}I_{kj}, \;\;\;\ell\neq j\\
&R_{jj} = - \! \sum_{\ell\neq j} R_{\ell j}
\end{aligned}
\label{eq:rateMatrixElements} \end{equation}
Note that the off-diagonal elements are necessarily non-negative, since ions excited from $|\,\mathbf{g},j\rangle$ can only increase the population of $|\,\mathbf{g},\ell\neq j\rangle$; since these ions do not change the total population of all the ground states, the diagonal elements $R_{jj}\leq0$. The time evolution of the populations is calculated by \begin{align}
\Delta N_\ell(t)= \sum_j R_{\ell j}N_j(t)\Delta t,\\
N(t+\Delta t)= N(t)+\Delta N(t)\nonumber.
\label{eq:timeEvolution} \end{align}
\subsubsection{Density Matrix Formulation} The Lindblad master equation for the density matrix $\rho$ is \begin{eqnarray} \frac{\text{d}\rho}{\text{d} t} = -\text{i}[H,\rho] +
\left. \frac{\text{d}\rho}{\text{d} t} \right|_{\rm relax}, \label{eq:LiouvilleRho} \end{eqnarray}
where $H$ is the Hamiltonian and $\left. \frac{\text{d}\rho}{\text{d} t} \right|_{\rm relax}$ describes the relaxation. We assume the relaxation is purely due to spontaneous emission and use the form given by Ref.~\cite{Grynberg2010}.
The further development of this equation requires an explicit form for the Hamiltonian. Unfortunately, published Hamiltonian parameters are in conflict and are internally inconsistent. Parameters for the Hamiltonian determined experimentally by Longdell et al.~\cite{Longdell2002} were later found to be in conflict with the empirical optical oscillator strengths of Nilsson et al.~\cite{Nilsson2004}. In an attempt to resolve this conflict, Lovri{\'c} et al.~\cite{Lovric2012} remeasured the Hamiltonian parameters. Unfortunately, these new parameters result in an oscillator-strength matrix with two columns swapped when compared with Nilsson et al. and even with their own work, as has been pointed out previously~\cite{Bartholomew2016}.
Since we were unable to find the Hamiltonian elsewhere in the literature, we derived a Hamiltonian that is consistent with the hyperfine level structure and oscillator strengths~\cite{Lovric2012} and that takes the conventional form $\frac{1}{2} I\cdot Q \cdot I$, where $I$ is a vector of length 3 representing the spin $\sfrac{5}{2}$ nuclear spin states and $Q$ is a rank-2 traceless symmetric tensor giving the orientation of these spins in a given manifold relative to the crystal axes. There are 10 degrees of freedom to be determined: the ground- and excited-state manifolds each have 2 energy spacings (Fig.~\ref{fig:introFigure}) and 3 Euler angles indicating their orientation with respect to the crystal axes. However, it is actually sufficient to know only the relative orientation of the manifolds, so we only need to determine 3 Euler angles. The 3$\times$3 oscillator-strength matrix given in Ref.~\cite{Equall1995} provides four constraints on these Euler angles. Using a least-squares fit we find Euler angles $\alpha_E=10.3(15)^\circ$, $\beta_E = -164.4(15)^\circ$, and $\gamma_E=-130.7(15)^\circ$ (digits in parentheses, here and throughout, are total uncertainties at one standard error, and represent the uncertainty of the least significant digits). The RMS deviation of the squares of the matrix elements in Table~\ref{tab:amps} from the oscillator strength values given in Ref.~\cite{Equall1995} is 0.013. This value may be compared with the quadrature sum of the experimental uncertainty of 0.01 in Ref.~\cite{Equall1995} and the RMS uncertainty from the above fitting procedure, leading to a combined uncertainty of 0.023.
\begin{table} \begin{center} \caption{ Transition probability amplitudes between the $^3H_4$ and $^2D_1$ ($\mathbf{g}$ and $\mathbf{e}$) manifolds (the manifolds $\mathbf{g}$ and $\mathbf{e}$ are not to be confused with the transitions $g$ and $e$). State labels are those of Fig.~1. The signs, magnitudes, and reported uncertainties are found using the Euler angle fitting procedure described in the text. }
\begin{tabular}{ cccc }
\hline
\hline
& $|\,\mathbf{e},\sfrac{1}{2}\rangle$ & $|\,\mathbf{e},\sfrac{3}{2}\rangle$ & $|\,\mathbf{e},\sfrac{5}{2}\rangle$ \\ \hline $|\,\mathbf{g},\sfrac{1}{2}\rangle$ &
$\phantom{-}0.753(23)$ & $-0.602(28)$ & $-0.265(24)$ \\ $|\,\mathbf{g},\sfrac{3}{2}\rangle$ &
$-0.634(27)$ & $-0.772(22)$ & $-0.048(08)$ \\ $|\,\mathbf{g},\sfrac{5}{2}\rangle$ &
$-0.176(17)$ & $\phantom{-}0.204(19)$ & $-0.963(07)$ \\
\hline
\hline \label{tab:amps} \end{tabular} \end{center} \end{table}
Returning to the Lindblad Master equation, we proceed to a rotating frame \begin{eqnarray} \rho_{k\gamma,\ell\delta} = \sigma_{k\gamma,\ell\delta} \text{e}^{-\text{i} [\omega_0 + (\gamma-\delta) \Delta ]t}. \label{eq:changeOfPicture} \end{eqnarray} Here, $\gamma$ and $\delta$ index the optical manifold (with 1 for $\mathbf{g}$ and 2 for $\mathbf{e}$), $k$ and $\ell$ index the hyperfine level within a given manifold, and $\Delta$ is the detuning. Whereas $\rho$ varies with an optical frequency ($\approx$ 500~THz), $\sigma$ varies with the hyperfine splitting, a radio frequency (RF) of order 10~MHz, as shown in Fig.~\ref{fig:introFigure}a. This change of variables allows the optical frequencies to be removed analytically and allows the program to solve numerically on the RF time scale. A time step of $\approx$ 1~ns is chosen, with a small adjustment often required to let the time step be commensurate with the period $T_{\rm rep}$. The density matrix in the rotating frame is given by \begin{eqnarray} \frac{\text{d} \sigma}{\text{d} t} = {\cal L} \sigma ,\label{eq:LiouvilleSigma} \end{eqnarray} defining the Lindblad matrix $\cal L$ from context. This equation is solved by discretizing time and assuming that the Lindblad matrix is piecewise-constant in time. In each time step, the solution is \begin{eqnarray} \sigma(t+\delta t) = \exp({\cal L} \delta t) \sigma(t) .\label{eq:Propagator} \end{eqnarray} In practice, the matrix exponential is found through its Taylor series in the form used in Horner's method, namely \begin{eqnarray} \exp(M) = 1 + M (1 + \frac{M}{2} ( 1 + \frac{M}{3} ( ... ))) \label{eq:MatrixExp} \end{eqnarray} for any matrix $M$. The terms are evaluated from the innermost parenthesis, starting from $\frac{M}{30}$. This method proves to be more accurate than evaluating the eigenvalues and eigenvectors with the linear algebra package LAPACK~\cite{Lapack1999} and taking the exponential, stabilizing the solutions.
To evaluate the solution for a large number of frequency-modulated periods, we make use of the identity \begin{eqnarray} \int_0^{2T_{\rm rep}} \!\!\! \text{d} t \; \exp[ {\cal L}(t) t] = \left[ \int_0^{T_{\rm rep}} \!\!\! \text{d} t \; \exp[ {\cal L}(t) t] \right]^2 , \label{eq:TwoPeriod} \end{eqnarray} where $T_{\rm rep}$ is the period of ${\cal L}(t)$. Iterating Eq.~(\ref{eq:TwoPeriod}) enables calculation for any integer multiple of the period $nT_{\rm rep}$ in a time proportional to $\log(n)$. Since the pulse duration is typically millions of times $T_{\rm rep}$, this yields an enormous savings in computation time compared to propagating in small time steps for the duration of the burn beam. The computation time to evaluate the propagator for a single period takes longer than its extension to millions of periods.
Finally, the Rabi overall coupling strength $\Omega_0$ is a fitting parameter. We compare theory to experiment for the single-frequency burn given in Fig.~\ref{fig:singleFreqResponse}. The linewidth is a function of $\Omega_0$, and the value that matches the FWHM of the central peak is shown in the plot. The five prominent transmission features are in good agreement with the experiment as well as the antiholes at positive detuning. The more complex pattern at negative detuning is in somewhat less satisfactory agreement with the experiment. For the AFCs, we present values for $\Omega_0=0.0125$. We also calculated for 8 other values between $\Omega_0=0.0032$ and $0.02$. The results are quite similar in the range of 0.005 to~0.02, although the peak coefficient of correlation with the experimental results below is maximized with the value presented.
The techniques developed in this section can be applied to periodic optical excitations of the crystal. In practice, the burn beam produced in the experiment is not exactly periodic, and the procedure for obtaining a periodic approximation to it is fairly subtle (Fig.~\ref{fig:threeSignal}). Details about this signal averaging step are in the Appendix.
In principle, spatial effects play a role in spectral-hole burning. For example, along the burn beam propagation direction, transmission peaks will become sharper as the beam is progressively filtered by the crystal. We leave these refinements to future work.
To simulate our experimental AFC preparation, we run the density matrix code at 7001 frequencies over a bandwidth of 350~MHz (in 50~kHz steps which are smaller than the narrowest features experimentally observed when applying a single-frequency burn to the crystal, as shown in Fig.~\ref{fig:singleFreqResponse}), yielding predicted ground-state population fractions for each of three hyperfine levels. Outside of the 350~MHz band, the populations are assumed to be undisturbed. We multiply all population fractions by the Gaussian density of states given in Section~\ref{subsec:refIndex} to obtain the absorbing populations throughout the inhomogeneous band. The highly dispersive dielectric function of the AFC was calculated from these populations using Eq.~(\ref{eq:sumBiTilde}) and the equations leading to it in Section~\ref{subsec:refIndex}.
\begin{figure}
\caption{Input optical signals at 4.8~MHz optical-comb tooth spacing. (a)~A portion of the output of the optical heterodyne signal showing the RF modulation of the optical input. (b)~Real part of analytic signal (Fourier transform with negative frequencies removed) obtained by a stroboscopic average of the magnitude of the analytic signal and, separately, the instantaneous frequency with a small frequency shift to ensure periodicity. (c)~Spectroscopic average of instantaneous frequency {vs.\@} time for the signal in part~(b).}
\label{fig:threeSignal}
\end{figure}
\section{Results} \label{sec:results}
\begin{figure}
\caption{
Dielectric function calculated for the AFC with $f_{\rm rep}=4.9$~MHz with the probe pulse in frequency domain superimposed.
(a) The dielectric function is given over the inhomogeneous band. (b) The dielectric function is given over the small portion of the inhomogeneous band which is modified due to the presence of the AFC. The guide lines indicate that the lower graph is a magnified version of a portion of the upper graph.
The real (solid black line) and imaginary (red dotted line) parts of the dielectric function are shown. The specific function plotted is indicated on the $y$ axis.
A 10~ns Gaussian pulse, representing the experimental pulse for photon echo generation (solid blue line) at the bottom of each panel is also shown.
}
\label{fig:dielectriFunction}
\end{figure}
\subsection{AFC transmission} \label{subsec:transmissionResults} A typical dielectric function calculated in the presence of an AFC is shown in Fig.~\ref{fig:dielectriFunction}. The AFC modifies only a small central region of the dielectric function: at large detunings, the AFC dielectric function approaches the unperturbed dielectric function. The key feature of the AFC dielectric function is its rapid oscillations with frequency.
\begin{figure}
\caption{\textit{(top 3)} Transmission spectra prepared using optical combs of varying tooth spacing $f_{\rm rep}$ from density matrix calculation, experiment, and rate-equation calculation; \textit{(center 3)} the corresponding echo pulses produced; \textit{(bottom)} the energy in the first-order echo pulse from density matrix (red), experiment (blue), and rate equation (yellow). In the first two rows, the vertical offset of each curve indicates $f_{\rm rep}$; the curves highlighted in red indicate local minima of the experimental first-order echo energy with respect to $f_{\rm rep}$. These minima are indicated by vertical dashed lines in the bottom plot.}
\label{fig:7panelPlot}
\end{figure}
The transmission spectra can be calculated from the dielectric function using Eq.~(\ref{eq:trans}). Fig.~\ref{fig:7panelPlot} shows experimental results along with the corresponding theoretical spectra for both the rate equation and density matrix. The theories capture many features of the experimental transmission spectra, such as the widths and shapes of transmission windows between comb teeth, as well as transitions to non-periodic forms. The rate equations tend to predict more regular combs than the density matrix theory. We see that combs with large fundamental Fourier components occur a few hundred kHz above and below 4.7~MHz, as opposed to at 4.7~MHz where the combs are not dominated by the fundamental~\cite{Weiner1990a,Slattery2021}. The effect of these features is discussed further in Section~\ref{subsec:echoResults}.
\begin{figure}
\caption{Echo pulses from the density matrix calculation, experiment, and rate equation calculation, for $f_{rep}$ ranging from 1~MHz to 16~MHz, in steps of 0.1~MHz. The lines are offset vertically by the repetition rate.}
\label{fig:pulse}
\end{figure}
\subsection{Photon echoes} \label{subsec:echoResults}
The input pulse was modeled as a Gaussian in the time domain with 10~ns FWHM, corresponding to 88~MHz FWHM in the frequency domain. The pulse bandwidth is compared with the AFC bandwidth in Fig.~\ref{fig:dielectriFunction}, showing that it is a good match for the oscillatory region of the AFC dielectric function. After discarding spurious points arising from the artificial periodicity of the FFT, we obtain a time span of 10~$\mu$s, several times longer than the 1~$\mu$s time period in which the pulses appear. Fig.~\ref{fig:7panelPlot} shows the experimental AFCs (top center) alongside density matrix (top left) and rate equation (top right) calculations, and the corresponding pulses (bottom). Fig.~\ref{fig:pulse} shows the echo pulses over a wider range of $f_{\rm rep}$ values. Although Fig.~\ref{fig:dielectriFunction} is given for one of the density matrix calculations at a single repetition rate, we find the dielectric function separately for each theory at each repetition rate.
The initial pulse near 0~$\mu$s is present in both experiment and theory. The echo pulses appear on hyperbolas, with the $N^{\rm th}$-order echo emission occurring at $T=N/f_{\rm rep}$. Fig.~\ref{fig:7panelPlot} highlights a number of features common to the experiment and the theory. Most notably, $f_{\rm rep}$ the largest first-order echoes do not align well with the excited-state hyperfine energy spacing values (dashed lines in panel 5): the largest echoes occur for $f_{\rm rep}$ detuned slightly above and below these energy spacings. Indeed, the echo efficiency at one of those spacings is near its lowest value. The patterns of drop-out of the first and second photon echoes is predicted well: for example, the absence of a photon echo for $f_{\rm rep}=2.5$~MHz is predicted by both theories and it is experimentally observed, despite photon echoes for both $f_{\rm rep}=2.4$~MHz and 2.6~MHz. Also common to both theories and experiment are valleys near 3.5~MHz and 4.5~MHz, although the drop-out is over-predicted by theory for the 4.5 MHz case. There are also small echoes in both experiment and density matrix theory for $f_{\rm rep}$ between 2.0 MHz and 3.5 MHz, at nearly fixed delay times between 0.20~$\mu$s and 0.25~$\mu$s. These are much less prominent in the rate equation calculation.
\section{Conclusions} \label{sec:conclusion} We present a semiclassical theory of photon echoes that gives a good account of the pulses arising in an AFC experiment in Pr:YSO with fine-grained systematic spectral coverage. The theory has two variants, one in which the AFC is predicted by the Rate Equations and a second in which the AFC is predicted by a Density Matrix. In contrast to most descriptions of AFC's in the literature, our approach does not invoke the Dicke state~\cite{Dicke1954}. The theory involves obtaining the dielectric function of the contribution of the Pr$^{3+}$ ions to the crystal from (a) a rate-equation or density-matrix description of the relevant hyperfine levels in the Pr$^{3+}$ impurity ions leading to the atomic polarizability (up to a constant) along with the input optical signals that prepared the AFC, (b) the Clausius-Mossotti relation, and (c) a fit to the maximum absorption in the case of a crystal without an AFC (thus fixing the constant). The photon echoes are modeled by letting the probe pulse pass through the AFC, here modeled as a highly dispersive medium. There does not appear to have been any similar attempt to simulate realistic atomic frequency combs based on the actual optical fields used to prepare them. Further, by generating over 100 AFCs for our systematic study, we demonstrate the predictive power of our theoretical model.
Looking ahead, having a predictive model will enable tailoring the AFC preparation protocol to optimize the efficiency of the system as a quantum memory. The fact that the predictive model is relatively simple in that it relies on long-established principles of the response of an oscillator to classical electromagnetic fields will hopefully enable applications.
\section{Acknowledgments} We are grateful to Alexey Gorshkov, Robinjeet Singh, and Eite Tiesinga for helpful discussions and to Ivan Burenkov, and Sergey Polyakov for helping to build the experimental setup. Mention of commercial products does not imply endorsement by the authors' institutions.
\section*{Appendix} \subsection*{Signal averaging} The theory requires knowledge of the periodic signal entering the crystal. Here, we describe how that periodic signal is obtained from the RF recorded after heterodyne mixing, shown in Fig.~\ref{fig:expLayout}a. Signal averaging is used to extract the periodic signal from 20~$\mu$s samples of data discretized in 0.2~ns steps. The algorithm for averaging the signals is given below.
Results for the case of a modulation frequency $f_{\rm rep} = 4.8$~MHz are shown in Fig.~\ref{fig:threeSignal}, including the data before and after the averaging procedure. The instantaneous frequency shown in Fig.~\ref{fig:threeSignal}c may be compared to the intended waveform, an ideal triangle-wave modulation: the discrepancy is due to limitations of the wave form generator.
The quality of the averaged signal may be assessed by comparing its Fourier transform to the Fourier transform of the input signal, as shown for the example of 4.8~MHz in Fig.~\ref{fig:fourierComparison}. The averaged spectrum is necessarily discrete. The experimental spectrum shows strong peaks that are broadened in analogy with x-ray-diffraction Bragg peaks broadened through thermal motions of the atoms, as described by the Debye-Waller factor. The alignment of the discrete points to the peaks of the black curve validates the averaging process. Matches of similar quality were obtained for each of the 161 modulation frequencies from 1~MHz to 16~MHz in steps of 0.1~MHz. The signal averaging algorithm has the following steps: \begin{enumerate}
\item The mean is subtracted from the sample and the sample is smoothed with a 7-point symmetric binomial filter.
\item Locations of zero crossings (i.e., sign switches in adjacent smoothed data points) are found. If there are two zero crossings in consecutive time steps, the pair is not considered in the next step.
\item The instantaneous frequency $f\approx \frac{1}{2\pi} \frac{\Delta \phi}{\Delta t}$ is calculated by assigning a phase shift of $\Delta \phi =\pi$ to adjacent zero crossings which also give the time interval $\Delta t$.
\item The instantaneous frequency is averaged stroboscopically given the known driving period.
\item The envelope is obtained by extracting the analytic signal from the smoothed data of step 2 above. The analytic signal is obtained through discrete FFT followed by zeroing out the negative frequencies, followed by the inverse FFT. The envelope is the absolute value of the analytic signal.
\item The envelope is averaged stroboscopically to create a periodic function.
\item The periodic phase is obtained by integrating the averaged instantaneous frequencies obtained in step~3 over a period.
A small linear adjustment in the phase is made to enforce strict periodicity of both amplitude and phase.
\item The signal, a complex quantity, is formed as the product of the envelope from step~6 and the phasor of the periodic phase from step~7. (The phasor of a phase $\phi$ is $\text{e}^{\text{i} \phi}$.)
\item The signal is shifted in the Fourier domain to remove the central modulating frequency. This corresponds physically to incorporating the central frequency of the AOM into the optical reference signal. The effect is to create a function that may be sampled with more widely spaced points. In practice, there is a 5:1 downsampling at this step from 0.2~ns to near 1~ns. The step sizes are not exactly equal for different repetition rates because each time step must be an integer divisor of the corresponding period. \end{enumerate}
\end{document} | arXiv |
Correlations of cap diameter (pileus width), stipe length and biological efficiency of Pleurotus ostreatus (Ex.Fr.) Kummer cultivated on gamma-irradiated and steam-sterilized composted sawdust as an index of quality for pricing
Nii Korley Kortei1,
George Tawia Odamtten2,
Mary Obodai3,
Michael Wiafe-Kwagyan2 &
Deborah Louisa Narh Mensah3
The Correction to this article has been published in Agriculture & Food Security 2018 7:36
Consumption patterns of mushrooms have increased in Ghana recently owing to its acknowledgement as a functional food. Different mushroom cultivation methods and substrate types have been linked to the quality of mushrooms produced, thereby affecting its pricing.
A comparative regression analysis was carried out to assess the correlation of stipe lengths, cap diameters and biological efficiencies of mushroom fruit bodies of Pleurotus ostreatus cultivated on steam-sterilized and gamma-irradiated sawdust after exposure to ionizing radiations of doses 0, 5, 10, 15, 20, 24 and 32 kGy from a 60CO source (SL 515, Hungary) at a dose rate of 1.7 kGy/h. Steam sterilization of composted substrates was also done at a temperature of 100–105 °C for 2 h.
Cap diameters of the mushrooms ranged 41–71.5 and 0–73 mm for gamma-irradiated samples depending on dose and steam-sterilized composted sawdust, respectively. Stipe lengths ranged between 4.4–61 and 0–58.1 for gamma-irradiated samples depending on dose and steam-treated substrates, respectively. Total yields of P. ostreatus grown on the gamma irradiation-treated composted sawdust ranged between 8.8 and 1517 g/kg, while mushrooms from steam sterilized recorded 0–1642 g/kg. Biological efficiencies of mushrooms grown on irradiated sawdust ranged 3–93.3%, while steamed sawdust ranged 0–97%. Good linear correlations were established between the cap diameter and biological efficiency (r2 = 0.70), stipe length and biological efficiency (r2 = 0.91) for mushrooms cultivated on gamma-irradiated sawdust. Similarly, good correlations were established between cap diameter and biological efficiency (r2 = 0.89) stipe length and biological efficiency (r2 = 0.95) for mushrooms cultivated on steam-sterilized sawdust.
These correlations provide the possibility to use only the cap diameter and stipe lengths to predict their biological efficiency and also use this parameter for grading and pricing of mushrooms earmarked for the consumer market.
Oyster mushrooms (Pleurotus sp.) have been widely cultivated in many different parts of the world. These mushrooms have the ability to grow at a wide range of temperatures utilizing various lignocelluloses [1] due to its powerful degrading cellulolytic and pectinolytic enzymes produced in vivo. Enzymatic efficiency of mushrooms makes their nutrition one of the most proficient biotechnological processes for lignocellulosic and organic waste recycling [2] by converting plant waste residues from agricultural activities as well as forestry debris. The technology used for its cultivation is eco-friendly since it exploits the natural ability of the fungus to degrade these complex polysaccharides to generate much simple compounds useful for human nutrition.
The current state-of-the-art research shows the usage of fungal biotechnology in fields as restoration of damaged environments (mycorestoration) via mycofiltration (i.e. use of mycelia to filter water), mycoforestry (i.e. use of mycelia to restore forests), mycoremediation (using mycelia to ameliorate heavily polluted soils), myconuclear bioremediation (the use of mycelia to sequester soils of radioactive materials), mycopesticide (use as biopesticide to control pests), and also spent composts could be used as biofertilizers to enhance the fertility of the soil [3, 4]. These methods represent fungal ability to restore the ecosystem where there are no adverse effects after fungal application.
The global economic value of fungi is now well known, the reason for the rise in consumption of mushrooms is a combination of their value as food, and their medicinal and nutraceutical properties [5] hence are gaining so much popularity in Ghana.
Several studies [2, 4, 6, 7] suggest that the methodology described for substrate preparation consists of composting agricultural residues followed by pasteurization and this could be achieved in diverse ways [8, 9]. The steam method is the most popular used for its artificial cultivation [10]. Nonetheless, the conventional method of steam sterilization of substrates has some disadvantages. It is cumbersome since factors such as precise sterilization time and temperature also depend on the residual micro- and mycoflora in a given substrate material [11]. There are also the associated discomfort and health hazards of standing by the heat for long hours not excepting the use of fuel wood for heating when natural cooking gas is not readily available. Fire wood burning depletes the forests of environmentally useful plant species.
Compost substrate sterilization by autoclaving at 121 °C or hot water dipping (pasteurization) in steel drum at 60 °C for 2.3 h has been reported by [12]. Chemical pretreatment of substrates has been reported by [13]. Oyster mushrooms have also been traditionally produced using the outdoor log technique, thereby excluding substrate sterilization [14].
In recent times, gamma irradiation has been used successfully to sterilize different substrates including mushroom compost to cultivate oyster mushrooms in Ghana [4, 15,16,17].
This study was to show correlations of growth parameters (cap diameter and stipe length) and biological efficiency in relation to the use of gamma irradiation for sterilization for substrates for mushroom cultivation.
Preparation of pure culture
Pure culture was prepared using the modified method of [18]. Malt extract agar (Oxoid, Basingstoke, Hampstead, England) was prepared according to manufacturer's instructions. The media were sterilized in an autoclave for 15 min at 121 °C with 1.5 kg/cm2 pressure. The sterilized media in the test tubes were kept in slanting/sloping positions. The stipe of the mushroom was surface-sterilized with 0.1% sodium hypochlorite. A scalpel was then dipped in alcohol and flamed until it was red hot. It was then cooled for 10 s. The stipe was cut lengthwise from the cap to the tip of the stipe downwards. Small piece of the internal tissue of the broken mushroom was cut and removed with a sterile scalpel. Using an inoculation needle, the cut tissue was then immediately inserted into test tube slant and the tissue laid on the agar surface. The mouth of the test tube was flamed before the needle was inserted. The mouth of the test tube was plugged with cotton wool and was incubated at 28 ± 2 °C for mycelia growth. After 7 days, the tissue was covered with a white mycelium that was spread on the agar surface.
Preparation of spawn
The stock culture substrate was prepared by using good quality sorghum grains and CaCO3 packed tightly in 25 × 18 cm polypropylene bag. These packets were sterilized in an autoclave for 1 h at 121 °C and 1.5 kg/cm2 atmospheric pressure and were kept 24 h for cooling before inoculation. After 7–9 days, there was a complete coverage of running mycelium. The spawn then was ready for inoculation of substrate bags.
The substrate consisted of 'wawa' (Triplochiton. scleroxylon) sawdust 80–90%, 1–2% of CaCO3 and 5–10% wheat bran. Moisture content was adjusted to 65–70% [19]. The sawdust was mixed thoroughly, heaped to a height of about 1.5 and 1.5 m base and covered with polythene and made to undergo natural fermentation for 28 days. Turning was made every 4 days to ensure homogeneity.
Composted sawdust was compressed into 0.18 m × 0.32 m heat-resistant polyethylene bags. Each bag contained approximately 1000 g (1 kg) of substrate and replicated six times.
Sterilization/pasteurization
Bagged composted sawdust substrates were sterilized with either moist heat at a temperature of 100–105 °C for 2.5 h or treated with gamma irradiation.
Gamma irradiation
Bagged composted sawdust substrates were subjected to radiation doses of 0, 5, 10, 15, 20, 24 and 32 kGy at a dose rate of 1.7 kGy per hour in air from a cobalt-60 source (SLL 515, Hungary) batch irradiator. Doses absorbed were confirmed using the conventional ethanol-chlorobenzene (ECB) dosimetry system. Radiation was carried out at the Radiation Technology Centre of the Ghana Atomic Energy Commission, Accra, Ghana.
Each treatment dose was replicated six times.
Innoculation and incubation
Each bag was closed with a plastic neck and plugged in with cotton and inoculated with 5 g sorghum spawn. The bags were then incubated at 26–28 °C and 60–65% relative humidity for 20–34 days in a well-ventilated, semi-dark mushroom growth room.
Calculations of growth and yield parameters of mushrooms
Growth and yield parameters were estimated according to methods prescribed by [20,21,22] as follows [6]:
$${\text{Stipe length}}\, = \,{\text{length of cap base to end of stalk}}$$
$${\text{Average cap diameter}} = \frac{{{\text{longest }} + {\text{ shortest cap diameters}}}}{2}$$
$${\text{Biological efficiency }}\left( {{\text{B}} . {\text{E\% }}} \right) = \;\frac{{{\text{Weight of fresh mushrooms harvested }}\left( {\text{g}} \right)}}{{{\text{Dry substrate weight }}\left( {\text{g}} \right)}} \times 100$$
$${\text{Yield }}\left( {\text{g/kg}} \right)\, = \,{\text{Weight of fresh mushrooms}}$$
Statistical and regression analysis
Regression analysis of correlation was employed using Microsoft Excel (Windows 10 version). Means of yield and growth parameters were subjected to analyses of variance (one-way ANOVA).
Differences between the means were determined using the Least Significant Difference test. All findings were considered statistically significant at P < 0.05.
The various substrate treatments resulted in different degrees of growth and yield: stipe length (mm), cap diameter (mm), yields (g/kg) and biological efficiencies (%) (Figs. 1, 2, 3, 4, 5, 6, 7, 8). Cap diameters of P. ostreatus from gamma-irradiated composted sawdust ranged between 41 and 71.5 mm, while steam sterilized ranged between 0 and 73 mm (Fig. 1). Non-sterilized substrates produced poor growth of mushrooms and were significantly (P < 0.05) low. Cap diameters of mushrooms obtained on steam sterilized were similar with those obtained on 15, 20 and 32 kGy doses. They, however, differed (P > 0.05) statistically.
Cap diameters of P. ostreatus grown on gamma-irradiated and steam-sterilized composted sawdust
Stipe lengths of P. ostreatus grown on gamma-irradiated and steam-sterilized composted sawdust
Average yields of P. ostreatus grown on gamma-irradiated and steam-sterilized composted sawdust
Biological efficiencies (%) of P. ostreatus grown on gamma-irradiated and steam-sterilized composted sawdust
Correlation of cap diameters and biological efficiency of P. ostreatus grown on gamma-irradiated composted sawdust
Correlation of stipe lengths and biological efficiency of P. ostreatus grown on gamma-irradiated composted sawdust
Correlation of cap diameters and biological efficiency of P. ostreatus grown on steam-sterilized composted sawdust
Correlation of stipe lengths and biological efficiency of P. ostreatus grown on steam-sterilized composted sawdust
Stipe lengths of P. ostreatus also ranged between 44 and 61 mm for gamma-irradiated composted sawdust, while that of steam sterilized ranged between 0 and 58.1 mm. Generally, there was no significant (P > 0.05) difference in stipe length growth for both 5 and 10 kGy likewise 20 and 24 kGy doses. Interestingly, steam sterilized and 32 kGy dose produced similar growth and were not significantly (P > 0.05) different.
Biological efficiencies of P. ostreatus from the gamma-irradiated composted sawdust ranged between 3 and 93.3%. Gamma radiation doses of 5, 10, 24 and 32 kGy produced comparable (P > 0.05) biological efficiencies. Similarly, doses of 15 and 20 kGy treatments were also comparable. Steam-sterilized mushrooms recorded a range of 0–97%.
Generally, the average yields of mushrooms recorded were of range 8.8–1517 g/kg for mushrooms depending on dose applied to the substrate, and higher doses tended to yield more mushrooms. A similar trend was observed for biological efficiency and was recorded for the yield. Steam-treated substrates produced a range of 0–1642 g/kg. Although the overall greatest average total yield of mushrooms (1642 g/kg) was obtained from the 5 kGy, the 10 kGy-treated composted sawdust produced comparable results. Yield of steam-sterilized mushrooms recorded was in a similar range as 15 and 20 kGy and showed no significant difference (P > 0.05).
The correlations between biological efficiencies (%) and stipe lengths (mm) as well as cap diameters (mm) of P. ostreatus produced on both gamma irradiation and steam sterilization substrates are presented in Figs. 5, 6, 7 and 8.
There were good correlation coefficients of determinations of r2 = 0.699 (cap diameter) and r2 = 0.914 (stipe length) and biological efficiencies for P. ostreatus from gamma-irradiated sawdust compost (Figs. 5, 6).
Furthermore, the highest scattered points were obtained from correlation of stipe length and biological efficiency (Figs. 5, 6). Very good correlation coefficients were also obtained for cap diameter (r2 = 0.886) and stipe length (r2 = 0.951) on steam-sterilized substrates with biological efficiency (Figs. 7, 8).
Generally, greater stipe lengths and cap diameters (pileus) were attended by high values of biological efficiencies (Figs. 5, 6, 7, 8). Figures 5, 6, 7 and 8 show how the regression line fits the data. The highest coefficients of determination r2 (0.95) were obtained from the regression line between biological efficiency and cap diameter (pileus width). The highest scattered points were obtained from the correlation of biological efficiency and stipe length.
Cap diameter (pileus width) and stipe length
Growth parameters of cap diameter and stipe lengths of P. ostreatus obtained on the various treatments of gamma irradiation and steam sterilization varied presumably due to the extent release of nutrients from depolymerization of substrate and subsequent mobilization by hyphae in the substrate for growth.
The performance of oyster mushroom grown on composted lignocellulosic substrates with respect to stipe length and cap diameter (pileus width) also depended on the structure, compactness and physical properties of the substrate which in turn was influenced by the type of agricultural wastes and method used in preparing the substrates. Chukwurah et al. [23] reported that substrates with higher moisture retaining capacity grow better than those with lower moisture retaining capacity and that substrates which contained mixtures of different types of agricultural wastes performed better than those with single agricultural waste [23].
Results of stipe lengths and cap diameters obtained in this experiment agree with findings of some researchers [6, 23, 24]. Conversely, Kortei and Wiafe-Kwagyan [17] reported higher values of 95 and 80 mm for cap diameter and stipe lengths, respectively, for P-31 strains cultivated on gamma-irradiated corncobs when they investigated the growth of P-31 strain on eight different gamma-irradiated substrates. Kortei [24] and Sarker et al. [25] reported a lower range of 1–5 and 1.85–6.57 cm for stipe lengths and cap diameters, respectively.
Yield and biological efficiency
According to [26], biological efficiency (B.E) is an expression of the bioconversion of dry substrate to fresh fruiting bodies and indicates the fructification ability of the fungus exploiting the substrate. Biological efficiencies (65–98%) recorded in this present study were relatively higher than literature values, and our data agree with the findings of Garo and Girma [27] who reported range of 31.98–146% from the study of responses of oyster mushrooms (P. ostreatus) as influenced by different substrates in Ethiopia. Gitte et al. [28] also reported high biological efficiency of milky mushrooms on different substrates which ranged from 51.57 to 146.3%. On the contrary, low B.E (%) values were reported by [6] ranging from 61 to 0% for P. ostreatus on different lignocellulosics. Raymond et al. [24] reported a B.E range of 8.95–62.8% for cultivated oyster mushrooms (Pleurotus HK37) on sisal wastes fractions supplemented with cow dung manure in Tanzania.
Both sterilization methods were effective in decontamination and depolymerisation of substrates as evidenced in the yield as well as biological efficiency since the two are directly linked. Yields were moderately higher for mushrooms cultivated from gamma-irradiated substrates than for those from steam treatment. This observation could be attributed to the higher degree of splitting of complex polysaccharides of the substrates into smaller utilizable units by gamma rays than what obtained for steam. Secondly, gamma rays have a high lethal effect on the genome of the micro-organisms resident in the substrate either directly or indirectly during pasteurization. It can be deduced from our results that 5 kGy produced the overall best yield. This study has confirmed that the use of different doses (5, 10, 15, 20, 24 and 32 kGy) of gamma irradiation to sterilize sawdust substrate produced similar results for that of steam treatment. Although the overall maximum yield was marginally higher for steam-treated substrates, the cost of heating (either with wood, gas or electricity) with steam will be economically more expensive and capital intensive and laborious than the use of gamma irradiation.
Correlations of cap diameter, stipe length and biological efficiency
Varied degrees of growth of P. ostreatus fruit body parts have been observed on different substrates as well as different cultivation methods by some researchers [1, 12]. In this study, there were high coefficients of correlations of stipe lengths and biological efficiency (0.91–0.95) was better as compared to that of cap diameter and biological efficiency (0.69–0.88) for both methods. Ajonina and Tatah [29] reported that the size of the stalk and pileus is positively correlated with yield and with carbohydrate and protein, respectively. They noticed variations in stalk length (2.43–3.24 cm) for P. ostreatus. Ahmed et al. [30] suggested that in the case of yield, the larger the pileus size, the higher the yield. Fruiting body weight is to a large extent influenced by the thickness and diameter of the pileus. Large-sized fruit bodies are widely perceived to be of superior quality and hence highly ranked in mushroom cultivation. In grading oyster mushroom for pricing, the use of correlation of stipe length, cap diameter (pileus width) and weight will be good criteria for grading quality [31].
Shen and Royse [32] and Mondal et al. [18] reiterated that fruit bodies were susceptible to breakage during packing and transport and so reduced their storage quality. Harvested mushrooms packaged in a rigid appropriately determined packaging material with good aeration, humidity and compactness can have extended shelf life even after grading for the market.
Our results showed that there was a positive correlation between cap diameters, stipe lengths and biological efficiencies of P. ostreatus cultivated on both steam-sterilized and gamma-irradiated composted sawdust which gives a clear indication of the possibility to predict yield or biological efficiency with growth determinants. Yields of both methods were also comparable. Gamma radiation could be used as a decontaminating agent of substrates for mushroom cultivation in countries that have access to gamma irradiation facilities to augment the dreary processes associated with the conventional steam sterilization method. Pricing can be based on a quality parameter such as yield, stipe length and pileus width.
In the original publication of this article [1], there was an error in an author name. In this correction article the correct and incorrect names are indicated.
Sanchez C. Cultivation of Pleurotus ostreatus and other edible mushrooms. Appl Microbiol Biotechnol. 2010;85:1321–37.
Mandeel QA, Al-Laith AA, Mohamed SA. Cultivation of oyster mushroom (Pleurotus sp.) on various lignocellulosic wastes. World J Microbiol Biotechnol. 2005;21:601–7.
Stamets P. Mycelium running: how mushroom can help save the world. Berkeley: Ten Speed Press; 2005. p. 574.
Kortei NK. Comparative effect of steam and gamma irradiation sterilization of sawdust compost on the yield, nutrient and shelf life of Pleurotus ostreatus (Jacq.ex. Fr) Kummer stored in two different packaging materials. Ph.D. thesis, Graduate School of Nuclear and Allied Sciences, University of Ghana, Legon. 2015.
Kortei NK, Wiafe-Kwagyan M. Comparative appraisal of the total phenolic content, flavonoids, free radical scavenging activity and nutritional qualities of Pleurotus ostreatus (EM-1) and Pleurotus eous (P-31) cultivated on rice (Oryzae sativa) straw in Ghana. J Adv Biol Biotechnol. 2015;3(4):153–64.
Obodai M, Cleland-Okine J, Vowotor KA. Comparative study on the growth and yield of Pleurotus ostreatus mushroom on different lignocellulosic by-products. J Ind Microbiol Biotechnol. 2003;30:146–9.
Raymond P, Mshandete AM, Kivaisi AM. Cultivation of oyster mushroom (Pleurotus HK-37) on sisal waste fractions supplemented with cow dung manure. J Biol Life Sci. 2013;4(1):274–86.
Stamets P. Growing gourmet and medicinal mushrooms. Berkley: Ten Speed Press; 1993. p. 552.
Balasubramanya RH, Kathe AA. An inexpensive pre-treatment of cellulosic materials for growing edible oyster mushrooms. Biol Resour Technol. 1996;57:303–5.
Apetorgbor MM, Apetorgbor AK, Nutakor E. Utilization and cultivation of edible mushrooms for rural livelihood in southern Ghana. 17th Commonwealth Forestry Conference. Sri-Lanka. 2005.
Kwon H, Sik Kim B. mushroom growers' handbook: shiitake cultivation. Mushworld: Seoul; 2004. p. 260.
Oseni TO, Dlamini SO, Earnshaw DM, Masarirambi MT. Effect of substrate pre-treatment methods on oyster mushroom (Pleurotus ostreatus) production. Int J Agric Biol. 2012;14(2):251–5.
Contreras EP, Sokolov M, Mejia G, Sanchez J. Soaking of substrate in alkaline water as a pretreatment for the cultivation of Pleurotus ostreatus. J Hortic Sci Biotechnol. 2004;79(2):234–40.
Royse DJ. Cultivation of shiitake on natural and synthetic logs. 2009. p. 2. http://www.americanmushroom.org/wp-content/uploads/2014/05/Shiitake_how_to_grow_PSU.pdf. Accessed 22 Sept 2017.
Gbedemah C, Obodai M, Sawyer LC. Preliminary investigations into the bioconversion of gamma irradiated agricultural wastes by Pleurotus spp. Radiat Phys Chem. 1998;52(6):379–82.
Kortei NK, Odamtten GT, Obodai M, Appiah V, Annan SNY, Acquah SA, Armah JO. Comparative effect of gamma irradiated and steam sterilized composted 'wawa' (Triplochiton scleroxylon) sawdust on the growth and yield of Pleurotus ostreatus (Jacq.Ex.Fr.) Kummer. Innov Rom Food Biotechnol. 2014;14:69–78.
Kortei NK, Wiafe-Kwagyan M. Evaluating the effect of gamma irradiation on eight different agro-lignocellulose waste materials for the production of oyster mushrooms (Pleurotus eous (Berk.) Sacc. Strain P-31). Croat J Food Biotechnol Nutr. 2014;9(3-4):83–90.
Mondal SR, Rehana MJ, Noman SM, Adhikary SK. Comparative study on growth and yield performance evaluation of oyster mushroom (Pleurotus florida) on different substrates. J Bangladesh Agric Univ. 2010;8:213–20.
Buswell JA. Potentials of spent mushroom substrates for bioremediation purposes. Compost. 1984;2:31–5.
Amin RSM, Alam N, Sarker NC, Hossain K, Uddin MN. Influence of different amount of rice straw per packet and rate of inocula on the growth and yield of oyster mushroom (Pleurotus ostreatus). Bangladesh J Mushroom. 2008;2:15–20.
Tisdale TE, Susan C, Miyasaka-Hemmes DE. Cultivation of the oyster mushroom (Pleurotus ostreatus) on wood substrates in Hawaii. World J Microbiol Biotechnol. 2006;22:201–6.
Morais MH, Ramos AC, Matos N, Santos-Oliveira EJ. Production of shiitake mushroom (Lentinus edodes) on ligninocellulosic residues. Food Sci Technol Int. 2000;6:123–8.
Chukwurah NF, Eze SC, Chiejina NV, Onyeonagu CC, Okezie CEA, Ugwuoke KI, Ugwu FSO, Aruah CB, Akobueze EU, Nkwonta CG. Correlation of stipe length, pileus width and stipe girth of oyster mushroom (Pleurotus ostreatus) grown in different farm substrates. J Agric Biotechnol Sustain Dev. 2013;5(3):54–60.
Kortei NK. Determination of optimal growth and yield parameters of Pleurotus ostreatus grown on composted cassava peel based formulations. M.sc.Thesis, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana. 2008.
Sarker NC, Hossain MM, Sultana N, Mian IH, Karim AJMS, Amin SMR. Performance of different substrates on the growth and yield of Pleurotus ostreatus (Jacquin ex Fr.) Kummer. Bangladesh J Mushroom. 2007;1:9–20.
Fan L, Pandey A, Mohan R, Soccol CR. Comparison of coffee industry residues for production of Pleurotus ostreatus in solid state fermentation. Acta Biotechnol. 2000;20(1):41–52.
Garo G, Girma G. Responses of oyster mushroom (Pleurotus ostreatus) as influenced by substrate difference in Gamo Gofa zone, Southern Ethiopia. IOSR J Agric Vet Sci. 2016;9(4):63–8.
Gitte VK, Priya J, Kotgire G. Selection of different substrate for the cultivation of milky mushroom (Calocybe indica P&C). Indian J Tradit Knowl. 2014;13(2):434–6.
Ajonina AS, Tatah LE. Growth performance and yield of oyster mushroom (Pleurotus ostreatus) on different substrates composition in Buea South West Cameroon. Sci J Biochem. 2012. https://doi.org/10.7237/sjbch/139.
Ahmed M, Abdullah N, Ahmed KU, Borhannuddin Bhuyan MHM. Yield and nutritional composition of oyster mushroom strains newly introduced in Bangladesh. Pesqui Agropecu Bras. 2013;48(2):197–202.
Onyango BO, Palapala VA, Arama PF, Wagai SO, Gichimu BM. Suitability of selected supplemented substrates for cultivation of Kenyan native wood ear mushrooms (Auricularia auricula). Am J Food Technol. 2011;6:395–403.
Shen Q, Royse D. Effect of nutrient supplement on biological efficiency, quality and crop cycle time on maitake (Griofola frondosa). Appl Microbiol Biotechnol. 2001;57:74–8.
NKK, GTO and MO involved in the conception of the research idea, design of the experiments and data analysis and also drafted the paper. MW-K and DLNM involved in the design of the experiments and data collection. GTO and MO provided guidance, corrections and supervision to the entire research and critically reviewed the manuscript. NKK and GTO read, reviewed and amended the manuscript. All authors read and approved the final manuscript..
We are grateful to Messers Abaabase Azinkaba, Godson Agbley and Moses Mensah of Mycology Unit, CSIR-FRI, for under taking the steam sterilization and maintenance of the farm house. Messers S.N.Y. Annan, J.N.O. Armah, S.W.N.O. Mills and S.A. Acquah of the Radiation Technology Centre, Ghana Atomic Energy Commission (G.A.E.C), Kwabenya, Accra, carried out the irradiation process.
Availability of supporting data
The data sets used and analysed during the current study are available to readers as in the manuscript.
Department of Nutrition and Dietetics, School of Allied Health Sciences, University of Health and Allied Sciences, PMB 31, Ho, Ghana
Nii Korley Kortei
Department of Plant and Environmental Biology, College of Basic and Applied Sciences, University of Ghana, P. O. Box LG 55, Legon, Ghana
George Tawia Odamtten
& Michael Wiafe-Kwagyan
Food Microbiology Division, Council for Scientific and Industrial Research - Food Research Institute, P. O. Box M20, Accra, Ghana
Mary Obodai
& Deborah Louisa Narh Mensah
Search for Nii Korley Kortei in:
Search for George Tawia Odamtten in:
Search for Mary Obodai in:
Search for Michael Wiafe-Kwagyan in:
Search for Deborah Louisa Narh Mensah in:
Correspondence to Nii Korley Kortei.
The original version of this article was revised: the name of one of the authors had been spelled incorrectly. It should be Deborah Louisa Narh Mensah, not Deborah Louisa Narh-Mensah
Kortei, N.K., Odamtten, G.T., Obodai, M. et al. Correlations of cap diameter (pileus width), stipe length and biological efficiency of Pleurotus ostreatus (Ex.Fr.) Kummer cultivated on gamma-irradiated and steam-sterilized composted sawdust as an index of quality for pricing. Agric & Food Secur 7, 35 (2018) doi:10.1186/s40066-018-0185-1
Biological efficiency
Cap diameter
Stipe length
P. ostreatus
Gamma-irradiated compost | CommonCrawl |
International Journal of System Assurance Engineering and Management
pp 1–7 | Cite as
System availability assessment using a parametric Bayesian approach: a case study of balling drums
Esi Saari
Jing Lin
Liangwei Zhang
Bin Liu
Assessment of system availability usually uses either an analytical (e.g., Markov/semi-Markov) or a simulation approach (e.g., Monte Carlo simulation-based). However, the former cannot handle complicated state changes and the latter is computationally expensive. Traditional Bayesian approaches may solve these problems; however, because of their computational difficulties, they are not widely applied. The recent proliferation of Markov Chain Monte Carlo (MCMC) approaches have led to the use of the Bayesian inference in a wide variety of fields. This study proposes a new approach to system availability assessment: a parametric Bayesian approach using MCMC, an approach that takes advantages of the analytical and simulation methods. By using this approach, mean time to failure (MTTF) and mean time to repair (MTTR) are treated as distributions instead of being "averaged", which better reflects reality and compensates for the limitations of simulation data sample size. To demonstrate the approach, the paper considers a case study of a balling drum system in a mining company. In this system, MTTF and MTTR are determined in a Bayesian Weibull model and a Bayesian lognormal model respectively. The results show that the proposed approach can integrate the analytical and simulation methods to assess system availability and could be applied to other technical problems in asset management (e.g., other industries, other systems).
Asset management System availability Reliability Maintainability Bayesian statistics Markov Chain Monte Carlo (MCMC) Mining industry
Availability represents the proportion of a system's uptime out of the total time in service and is one of the most critical aspects of performance evaluation. Availability is commonly measured as Mean Time to Failure (MTTF) and Mean Time to Repair (MTTR). However, those "mean" values are normally "averaged"; thus, some useful information (e.g., trends, system complexity) may be neglected, and some problems may even be hidden.
Assessment of system availability has been studied from the design stage to the operational stage in various system configurations (e.g., in series, parallel, k-out-of-n, stand-by, multi-state, or mixed architectures). Approaches to assessing system availability mainly use either analytic or simulation techniques.
In general, analytic techniques represent the system using direct mathematical solutions from applied probability theory to make statements on various performance measures, such as the steady-state availability or the interval availability (Dekker and Groenendijk 1995; Ocnasu 2007). Researchers tend to use Markov models to assess dynamic availability or semi-Markov models using Laplace transforms to determine average performance measures (Dekker and Groenendijk 1995; Faghih-Roohi et al. 2014). However, such approaches have been criticised as too restrictive to tackle practical problems; they assume constant failure and repair rates which is not likely to be the case in the real world (Raje et al. 2000; Marquez et al. 2005). Furthermore, the time dependent availability obtained by a Markovian assumption is actually not valid for non-Markovian processes (Raje et al. 2000).
Simulation techniques estimate availability by simulating the actual process and random behaviour of the system. The advantage is that non-Markov failures and repair processes can be modelled easily (Raje et al. 2000). Recent research is working on developing Monte Carlo techniques to model the behaviour of complex systems under realistic time-dependent operational conditions (Marquez et al. 2005; Marquez and Iung 2007; Yasseri and Bahai 2018) or to model multi-state systems with operational dependencies (Zio et al. 2007). Although simulation is more flexible, it is computationally expensive.
Traditionally, Bayesian approaches have been used to assess system availability as they can solve the problem of complicated system state changes and computationally expensive simulation data; however, their development and application were stalled by the strict assumptions on prior forms and by computational difficulties. Research is more concerned with the prior's selection or the posterior's computation than the reality (Brender 1968a, b; Kuo 1985; Sharma and Bhutani 1993; Khan and Islam 2012).
The recent proliferation of Markov Chain Monte Carlo (MCMC) simulation techniques has led to the use of the Bayesian inference in a wide variety of fields. Because of MCMC's high dimensional numerical integral calculation (Lin 2014), the selection of prior information and descriptions of reliability/maintainability can be more flexible and more realistic.
This study proposes a new approach to system availability assessment: a parametric Bayesian approach with MCMC, with a focus on the operational stage, using both analytical and simulation methods. MTTF or MTTR are treated as distributions instead of being "averaged" by point estimation, and this is closer to reality; in addition, the limitations of simulation data sample size are addressed by using MCMC techniques.
The rest of this paper is organized as follows. Section 2 describes the problem statement, the balling drum system, the data preparation, and the preliminary analysis of failure and repair data. Section 3 proposes a Bayesian Weibull model for MTTF and a Bayesian lognormal model for MTTR and explains how to use an MCMC computational scheme to obtain the parameters' posterior distributions. Section 4 presents a case study, results, and discussion. Section 5 offers conclusions and suggestions for further study.
2 Problem statement
This section presents the study problem statement, the balling drum system and its configuration, the system availability framework, and data preparation; it performs a preliminary analysis of failure and repair data based on which parametric Bayesian models are constructed subsequently.
2.1 Balling drum systems in the mining industry
Our study is motivated by a balling drum system in the mining industry. The case study mine consists of five balling drums, labelled 1–5 (see Fig. 1). All five balling drums receive their feed for production in the same manner. Each balling drum is expected to produce the same amount of pellets at its maximum. According to the working mechanism and an i.i.d test, they are regarded as independent; if one of the balling drums breaks down, it does not affect the rest of the balling drums, except that total production will be reduced. One assumption is made here that the system will fail only if all subsystems fail; therefore, it is treated as a parallel system.
Description of a balling drum and the system sketch
The availability of a single balling drum, denoted as A, can be computed by
$$A = \frac{MTTF}{MTTF + MTTR}$$
According to Fig. 1, the five balling drums are in parallel. The total system availability, \({\text{A}}_{\text{system}}\), can be calculated as
$$A_{system} = 1 - \mathop \prod \limits_{i = 1}^{5} (1 - A_{i} )$$
2.2 Data preparation and preliminary analysis
The study uses the failure and repair data of the five balling drums from January 2013 to December 2018. There are 1782 records. In the first step, the null values are removed, and the data are reduced to 1774 records.
The next step reveals there are different reasons for the TTF and TTR of individual balling drums. It is noticed that, for TTR data, if 150 shutdowns are considered normal (denoted as a threshold, see Fig. 2), then those exceeding 150 should be treated as abnormal and investigated using Root Cause Analysis (RCA).
Example of TTR data for balling drum 1
After checking the work order types of such kind of abnormal data, it is found that most of them are caused by "preventive maintenance" which may due to lack of maintenance resources. To simplify the study, we assume all maintenance resources are sufficient for "preventive maintenance"; thus, the abnormally data might be caused by shortage of spare parts or skilled personnel will not be treated specially in this paper.
To determine the baseline distribution of Time to Failure (TTF) and Time to Repair (TTR), we conduct a preliminary study of failure data and repair data using traditional analysis. In this preliminary study, several distributions are considered: exponential distribution, Weibull distribution, normal distribution, log-logistic distribution, lognormal distribution, and extreme value distribution. Table 1 lists the results.
Preliminary study of failure data and repair data
Balling drum
TTF fitness
TTR fitness
Weibull
Log-logistic
Lognormal
Based on the results, the Weibull distribution and lognormal distribution are selected for the TTF and TTR for balling drums 1–5; these are applied to the parametric Bayesian models in the next section.
3 Parametric Bayesian Models
This section proposes a Bayesian Weibull model for TTF and a Bayesian lognormal model for TTR in the proposed parametric Bayesian models and explains the procedure of MCMC computational scheme to obtain the posterior distributions.
3.1 Markov Chain Monte Carlo with Gibbs sampling
The recent proliferation of Markov Chain Monte Carlo (MCMC) approaches has led to the use of the Bayesian inference in a wide variety of fields. MCMC is essentially Monte Carlo integration using Markov chains. Monte Carlo integration draws samples from the required distribution and then forms sample averages to approximate expectations. MCMC draws out these samples by running a cleverly constructed Markov chain for a long time. There are many ways of constructing these chains. The Gibbs sampler is one of the best known MCMC sampling algorithms in the Bayesian computational literature. It adopts the thinking of "divide and conquer": i.e., when a set of parameters must be evaluated, the other parameters are assumed to be fixed and known. Let \(\uptheta_{\text{i}}\) be an i-dimensional vector of parameters, and let \({\text{f}}\left( {\uptheta_{\text{j}} } \right)\) denote the marginal distribution for the jth parameter. The basic scheme of the Gibbs sampler for sampling from \({\text{p}}\left(\uptheta \right)\) is given as follows:
Step 1. Choose an arbitrary starting point \(\theta^{\left( 0 \right)} = \left( {\theta_{1}^{\left( 0 \right)} , \ldots ,\theta_{k}^{\left( 0 \right)} } \right)\);
Step 2. Generate \(\theta_{1}^{\left( 1 \right)}\) from the conditional distribution \(f\left( {\theta_{1} |\theta_{2}^{\left( 0 \right)} , \ldots ,\theta_{k}^{\left( 0 \right)} } \right)\), and generate \(\theta_{2}^{\left( 1 \right)}\) from the conditional distribution distribution \(f\left( {\theta_{2} |\theta_{1}^{\left( 1 \right)} ,\theta_{3}^{\left( 0 \right)} , \ldots ,\theta_{k}^{\left( 0 \right)} } \right);\)
Step 3. Generate \(\theta_{j}^{\left( 1 \right)}\) from \(f\left( {\theta_{j} |\theta_{1}^{\left( 1 \right)} , \ldots ,\theta_{j - 1}^{\left( 1 \right)} ,\theta_{j + 1}^{\left( 1 \right)} \ldots ,\theta_{k}^{\left( 0 \right)} } \right)\);
Step 4. Generate \(\theta_{k}^{\left( 1 \right)}\) from \(f\left( {\theta_{k} |\theta_{1}^{\left( 1 \right)} ,\theta_{2}^{\left( 1 \right)} , \ldots ,\theta_{k - 1}^{\left( 1 \right)} } \right)\); the one-step transition from \(\theta^{\left( 0 \right)}\) to \(\theta^{\left( 1 \right)} = \left( {\theta_{1}^{\left( 1 \right)} , \ldots ,\theta_{k}^{\left( 1 \right)} } \right)\) has been completed, where \(\theta^{\left( 1 \right)}\) is a one-time accomplishment of a Markov chain.
Step 5. Go to Step2.
After \({\text{t}}\) iterations, \(\uptheta^{{\left( {\text{t}} \right)}} = \left( {\uptheta_{1}^{{\left( {\text{t}} \right)}} , \ldots ,\uptheta_{\text{k}}^{{\left( {\text{t}} \right)}} } \right)\) can be obtained. Each component of \(\uptheta\) can also be obtained. Starting from different \(\uptheta^{\left( 0 \right)}\), as \({\text{t}} \to \infty\), the marginal distribution of \(\uptheta^{{\left( {\text{t}} \right)}}\) can be viewed as a stationary distribution based on the theory of the ergodic average. Then, the chain is seen as converging, and the sampling points are seen as observations of the sample.
3.2 Bayesian Weibull model for TTF
Suppose the time to failure (TTF) data \({\text{t}} = \left( {{\text{t}}_{1} ,{\text{t}}_{2} , \ldots ,{\text{t}}_{\text{n}} } \right)^{\prime}\) for \({\text{n}}\) individuals are i.i.d, and each corresponds to a 2-parameter Weibull distribution \({\text{W}}\left( {\upalpha,\upgamma} \right)\), where \(\upalpha > 0\) and \(\upgamma > 0\). Then, the p.d.f. is \({\text{f}}\left( {{\text{t}}_{\text{i}} |\upalpha,\upgamma} \right) =\upalpha \upgamma {\text{t}}_{\text{i}}^{{{\upalpha} - 1}} { \exp }\left( { - {\upgamma \text{t}}_{\text{i}}^{{\upalpha}} } \right)\), while the c.d.f. is \({\text{F}}\left( {{\text{t}}_{\text{i}} |{\upalpha},{\upgamma}} \right) = 1 - { \exp }\left( { - {\upgamma \text{t}}_{\text{i}}^{{\upalpha}} } \right)\). The reliability function is \({\text{R}}\left( {{\text{t}}_{\text{i}} |{\upalpha},{\upgamma}} \right) = { \exp }\left( { - {\upgamma \text{t}}_{\text{i}}^{{\upalpha}} } \right)\).
Denote the observed data set as \({\text{D}}_{0} = \left( {{\text{n}},{\text{t}}} \right).\) Therefore, the likelihood function for \({\upalpha}\) and \({\upgamma}\) is
$$L\left( {\alpha ,\gamma |D_{0} } \right) = \mathop \prod \limits_{i = 1}^{n} f\left( {t_{i} |\alpha ,\gamma } \right) = \mathop \prod \limits_{i = 1}^{n} \alpha \gamma t_{i}^{\alpha - 1} exp\left( { - \gamma t_{i}^{\alpha } } \right)$$
In this study, we assume \(\upalpha\) to be a gamma distribution (Kuo 1985), denoted by \({\text{G}}\left( {{\text{a}}_{0} ,{\text{b}}_{0} } \right)\) as its prior distribution, written as \({\uppi}\left( {{\upalpha}|{\text{a}}_{0} ,{\text{b}}_{0} } \right)\); we assume \({\upgamma}\) to be a gamma distribution denoted by \({\text{G}}\left( {{\text{c}}_{0} ,{\text{d}}_{0} } \right)\) as its prior distribution, written as \({\uppi}\left( {{\upgamma}|{\text{c}}_{0} ,{\text{d}}_{0} } \right).\) This means
$$\pi \left( {\alpha |a_{0} ,b_{0} } \right) \propto \alpha^{{a_{0} - 1}} exp\left( { - b_{0} \alpha } \right)$$
$$\pi \left( {\gamma |c_{0} ,d_{0} } \right) \propto \gamma^{{c_{0} - 1}} exp\left( { - d_{0} \gamma } \right)$$
Therefore, the joint posterior distribution can be obtained according to Eqs. (3)–(5) as
$$\pi \left( {\alpha ,\gamma |D_{0} } \right) \propto L\left( {\alpha ,\gamma |D_{0} } \right) \times \pi \left( {\alpha |a_{0} ,b_{0} } \right) \times \pi \left( {\gamma |c_{0} ,d_{0} } \right),$$
and the parameters' full conditional distribution with Gibbs sampling can be written as
$$\pi \left( {\alpha_{j} |\alpha^{{\left( { - j} \right)}} ,\gamma ,D_{0} } \right) \propto L\left( {\alpha ,\gamma |D_{0} } \right) \times \alpha^{{a_{0} - 1}} exp\left( { - b_{0} \alpha } \right)$$
$$\pi \left( {\gamma_{j} |\alpha ,\gamma^{{\left( { - j} \right)}} ,D_{0} } \right) \propto L\left( {\alpha ,\gamma |D_{0} } \right) \times \gamma^{{c_{0} - 1}} exp\left( { - d_{0} \gamma } \right)$$
3.3 Bayesian Lognormal model for TTR
Suppose the time to repair (TTF) data \({\text{t}} = \left( {{\text{t}}_{1} ,{\text{t}}_{2} , \ldots ,{\text{t}}_{\text{n}} } \right)^{\prime}\) for \({\text{n}}\) individuals are i.i.d., and each \({ \ln }\left( {\text{t}} \right)\) corresponds to a normal distribution, \({\text{N}}\left( {{\upmu},{\upsigma}^{2} } \right)\). We can get \({\text{t}}_{\text{i}}\)'s lognormal distribution with parameters \({\upmu}\) and \({\upsigma}^{2}\). Then, the p.d.f. and c.d.f. are given by Eqs. (9) and (10):
$$f\left( {t_{i} |\mu ,\sigma^{2} } \right) = \frac{1}{{\sqrt {2\pi } \sigma t_{i} }}exp\left\{ { - \frac{1}{{2\sigma^{2} }}\left[ {ln\left( {t_{i} } \right) - \mu } \right]^{2} } \right\}$$
$$F\left( {t_{i} |\mu ,\sigma^{2} } \right) = {\Phi }\left[ {\frac{{ln\left( {t_{i} } \right) - \mu }}{\sigma }} \right]$$
Denote the observed data set as \({\text{D}}_{0} = \left( {{\text{n}},{\text{t}}} \right)\). Therefore, according to Eq. (9), the likelihood function for \({\upmu}\) and \({\upsigma}\) becomes
$$L\left( {\mu ,\sigma |D_{0} } \right) = \mathop \prod \limits_{i = 1}^{n} f\left( {t_{i} |\mu ,\sigma^{2} } \right)$$
In this study, we assume \({\upmu}\) to be a normal distribution denoted by \({\text{N}}\left( {{\text{e}}_{0} ,{\text{f}}_{0} } \right)\) as its prior distribution, written as \({\uppi}\left( {{\upmu}|{\text{e}}_{0} ,{\text{f}}_{0} } \right)\); we assume \({\upsigma}\) to be a gamma distribution denoted by \({\text{G}}\left( {{\text{g}}_{0} ,{\text{h}}_{0} } \right)\) as its prior distribution, written as \({\uppi}\left( {{\upsigma}|{\text{g}}_{0} ,{\text{h}}_{0} } \right).\) This means
$$\pi \left( {\mu |e_{0} ,f_{0} } \right) \propto f_{0}^{{\frac{1}{2}}} exp\left[ { - \frac{{f_{0} }}{2}\left( {\mu - e_{0} } \right)^{2} } \right]$$
$$\pi \left( {\sigma |g_{0} ,h_{0} } \right) \propto \sigma^{{g_{0} - 1}} exp\left( { - h_{0} \sigma } \right)$$
Therefore, the joint posterior distribution can be obtained according to Eqs. (11)–(13) as
$$\pi \left( {\mu ,\sigma |D_{0} } \right) \propto L\left( {\mu ,\sigma |D_{0} } \right) \times \pi \left( {\mu |e_{0} ,f_{0} } \right) \times \pi \left( {\sigma |g_{0} ,h_{0} } \right)$$
Then, the parameters' full conditional distribution with Gibbs sampling can be written as
$${\uppi}\left( {\mu_{j} |\mu^{{\left( { - j} \right)}} ,\sigma ,D_{0} } \right) \propto L\left( {\mu ,\sigma |D_{0} } \right) \times f_{0}^{{\frac{1}{2}}} exp\left[ { - \frac{{f_{0} }}{2}\left( {\mu - e_{0} } \right)^{2} } \right]$$
$${\uppi}\left( {\sigma_{j} |\mu ,\sigma^{{\left( { - j} \right)}} ,D_{0} } \right) \propto L\left( {\mu ,\sigma |D_{0} } \right) \times \sigma^{{g_{0} - 1}} exp\left( { - h_{0} \sigma } \right)$$
4 Case study
This section presents a case study; it explains the procedure, gives the results, and offers a discussion.
4.1 The procedure
The procedure applied in this case study to assess the system availability of the mine's five balling drums has a total of seven steps, as described in Table 2.
Steps in the system availability assessment
Outputs in this case
Configuration definition
System configuration and dependencies determined to calculate system availability
Five balling drum system parallel and independent (see Sect. 2.1)
Reliability and maintenance data (and information) collected
1774 records for failure and repair data of the five balling drums collected from 2013 to 2018 (see Sect. 2.2)
Data cleaned and outliers removed as needed
Null values removed and abnormal data checked (see Sect. 2.2)
Preliminary Analysis
Pre-studies for TTF and TTR data performed to decide the baseline distributions
MTTF fits a Weibull distribution; MTTR fits a lognormal distribution (see Sect. 2.2)
Parametric Bayesian model building
Prior distribution defined, and analytic models developed
Bayesian Weibull model for MTTF with gamma priors and Bayesian lognormal model with gamma and normal priors constructed (see Sect. 3)
MCMC simulation
Burn-in defined and MCMC simulation implemented; convergence diagnostics and Monte Carlo error checked to confirm the effectiveness of the results
Burn-in of 1000 samples used with an additional 10,000 Gibbs samples for each Markov chain (see Sects. 3 and 4.2)
Results and analysis
Results, calculation, and discussion
Results for parameters of interest in system availability assessment (see Sects. 4.2 and 4.3)
In this case study, the calculations are implemented with WINBUGS. A three-chain Markov chain is constructed for each MCMC simulation. A burn-in of 1000 samples is used, with an additional 10,000 Gibbs samples for each Markov chain.
Vague prior distributions are adopted as follows:
For Bayesian Weibull model using TTF data:
$$\alpha \sim G\left( {0.0001,0.0001} \right),\quad \gamma \sim G\left( {0.0001,0.0001} \right)$$
For Bayesian lognormal model using TTR data:
$$\mu \sim N\left( {0,0.0001} \right),\quad \sigma \sim G\left( {0.0001,0.0001} \right).$$
Using the convergence diagnostics [i.e. checking dynamic traces in Markov chains, determining time series and Gelman–Rubin–Brooks (GRB) statistics, and comparing MC error with standard deviation (SD)] (Lin 2014), we consider the following posterior distribution summaries for our models (see Tables 3, 4), including the parameters' posterior distribution mean, SD, Monte Carlo error (MC error), and 95% highest posterior distribution density (HPD) interval.
Posterior statistics in Bayesian Weibull model for TTF
MC error
95% HPD interval
\(\alpha\)
4.288E−4
(0.4964, 0.5867)
\(\gamma\)
Posterior statistics in Bayesian lognormal model for TTR
\(\mu\)
− 0.1842
(− 0.4015, 0.0342)
\(\sigma\)
(0.1951,0.2615)
(− 0.2845,0.2697)
(− 0.4578, − 0.2354)
Using the results from Tables 3 and 4, we calculate the availability of individual balling drums in Table 5, where MTTF = \({\text{E}}\left[ {{\text{f}}\left( {{\text{t}}_{\text{i}} |{\upalpha},{\upgamma}} \right)} \right]\), and MTTR = \({\text{E}}\left[ {{\text{f}}\left( {{\text{t}}_{\text{i}} |{\upmu},{\upsigma}^{2} } \right)} \right]\).
Statistics of individual availability
MTTR
(118.1, 178.0)
(5.284, 11.58)
(6.194, 9.622)
According to Eq. (2), the system availability of the five balling drums is
$$A_{system} = 1 - \mathop \prod \limits_{i = 1}^{5} (1 - A_{i} ) \approx 0.99.$$
Compared to the traditional method of assessing availability in Eq. (1), the proposed approach extends the method to Eq. (17), where
$$A = \frac{{E\left[ {f\left( {TTF} \right)} \right]}}{{E\left[ {f\left( {TTF} \right)} \right] + E\left[ {f\left( {TTR} \right)} \right]}} = \frac{{E\left[ {f\left( {t_{i} |\alpha ,\gamma } \right)} \right]}}{{E\left[ {f\left( {t_{i} |\alpha ,\gamma } \right)} \right] + E\left[ {f\left( {t_{i} |\mu ,\sigma^{2} } \right)} \right].}}$$
Equation (17) shows the flexibility of assessing availability according to reality. For one thing, the parametric Bayesian models using MCMC make the calculation of posteriors more feasible. More importantly, however, parametric Bayesian models can be applied to predict TTF, TTR, and system availability in the future.
In this study, since the five balling drums are relatively new, the gamma distributions and normal distributions are selected as vague priors due to lack of prior information. This could be improved with more historical data/experience.
The system configurations could be extended to other more complex architectures (series, k-out-of-n, stand-by, multi-state, or mixed) by modifying Eq. (2).
The data analysis reveals that for TTF data, the shape parameter for the Weibull distribution is less than 1. The TTFs have a decreasing trend (as in an early stage of the bathtub curve) which is not suitable for the experience of mechanical equipment. The TTF data include not only corrective maintenance but also preventive maintenance. In this case study, a high percentage of TTF work orders are for preventive maintenance. The decreasing trends also indicate that a possible way to improve TTF is to improve the preventive maintenance plan.
Among those three stages, Step 1 to Step 4 can be treated as Plan stage; Step 5 and Step 6 as Do and Check stage, while Step 7 as Action stage. The outputs from Step 7 could become input for Step 2 for the next calculation period. It means these eight steps are following the "PDCA" cycle and the results could be continuously improved.
This study proposes a parametric Bayesian approach for system availability assessment on the operational stage. MCMC is adopted to take advantages of the analytical and simulation methods.
In this approach, MTTF and MTTR are treated as distributions instead of being "averaged" by a point estimation. This better reflects the reality; in addition, the limitations of simulation data sample size are compensated for by MCMC techniques.
In the case study, TTF and TTR are determined using a Bayesian Weibull model and a Bayesian lognormal model. The results show that the proposed approach can integrate the analytical and simulation methods for system availability assessment and could be applied to other technical problems in asset management (e.g., other industries, other systems).
The motivation for the research originated from the project "Key Performance Indicators (KPI) for control and management of maintenance process through eMaintenance (In Swedish: Nyckeltal för styrning och uppföljning av underhållsverksamhet m h a eUnderhåll)", which was initiated and financed by LKAB. The authors wish to thank Ramin Karim, Peter Olofsson, Mats Renfors, Sylvia Simma, Maria Rytty, Mikael From and Johan Enbak, for their support for this research in the form of funding and work hours.
Brender DM (1968a) The Bayesian assessment of system availability: advanced applications and techniques. IEEE Trans Reliab 17(3):138–147CrossRefGoogle Scholar
Brender DM (1968b) The prediction and measurement of system availability: a Bayesian treatment. IEEE Trans Reliab 17(3):127–138CrossRefGoogle Scholar
Dekker R, Groenendijk W (1995) Availability assessment methods and their application in practice. Microelectron Reliab 35(9–10):1257–1274CrossRefGoogle Scholar
Faghih-Roohi S, Xie M, Ng KM, Yam RC (2014) Dynamic availability assessment and optimal component design of multi-state weighted k-out-of-n systems. Reliab Eng Syst Saf 123:57–62CrossRefGoogle Scholar
Khan MA, Islam H (2012) Bayesian analysis of system availability with half-normal life time. Qual Technol Quant Manag 9(2):203–209MathSciNetCrossRefGoogle Scholar
Kuo W (1985) Bayesian availability using gamma distributed priors. IIE Trans 17(2):132–140CrossRefGoogle Scholar
Lin J (2014) An integrated procedure for bayesian reliability inference using Markov Chain Monte Carlo methods. J Qual Reliab Eng 2014:1–16CrossRefGoogle Scholar
Marquez AC, Iung B (2007) A structured approach for the assessment of system availability and reliability using Monte Carlo simulatoin. J Qual Maint Eng 13(2):125–136CrossRefGoogle Scholar
Marquez AC, Heguedas AS, Iung B (2005) Monte Carlo-based assessment of system availability. A case study for cogeneration plants. Reliab Eng Syst Saf 88:273–289CrossRefGoogle Scholar
Ocnasu AB (2007) Distribution system availability assessment—Monte Carlo and antithetic variates method. In: 19th international conference on electricity distribution, ViennaGoogle Scholar
Raje D, Olaniya R, Wakhare P, Deshpande A (2000) Availability assessment of a two-unit stand-by pumping system. Reliab Eng Syst Saf 68:269–274CrossRefGoogle Scholar
Sharma K, Bhutani R (1993) Bayesian analysis of system availability. Microelectron Reliab 33(6):809–811CrossRefGoogle Scholar
Yasseri SF, Bahai H (2018) Availability assessment of subsea distribution systems at the architectural level. Ocean Eng 153:399–411CrossRefGoogle Scholar
Zio E, Marella M, Podofillini L (2007) A Monte Carlo simulation approach to the availability assessment of multi-state system with operational dependencies. Reliab Eng Syst Saf 92:871–882CrossRefGoogle Scholar
1.Division of Operation and Maintenance EngineeringLuleå University of TechnologyLuleåSweden
2.Department of Industrial EngineeringDongguan University of TechnologyDongguanChina
3.Department of Management ScienceUniversity of StrathclydeGlasgowUK
Saari, E., Lin, J., Zhang, L. et al. Int J Syst Assur Eng Manag (2019). https://doi.org/10.1007/s13198-019-00803-y
DOI https://doi.org/10.1007/s13198-019-00803-y
Publisher Name Springer India | CommonCrawl |
Physics (active)
Here we'll give a brief overview of ice physics. This won't be a substitute for reading a book or taking a class on the subject, but it should help explain some of the notation and variable names that you'll see in the demos and documentation. I'll assume you're familiar with continuum mechanics and vector calculus. In particular, we'll appeal to the idea of control volumes and conservation laws. To get all the notation out of the way, here's a table of all the symbols that we'll use:
velocity $u$
thickness $h$
surface elevation $s$
bed elevation $b$
accumulation rate $\dot a$
melt rate $\dot m$
pressure $p$
strain rate tensor $\dot\varepsilon$
deviatoric stress $\tau$
Glen flow law rate $A$
Glen flow law exponent $n$
friction coefficient $C$
outward normal vector $\nu$
These fields are all reported in units of megapascals - meters - years. This unit system is a little unusual; we borrowed it from the package Elmer/ICE. It has the advantage of making certain physical constants, like the Glen flow law rate factor, live in a fairly sensible numerical range, whereas the values are often either gigantic or tiny in MKS units.
Momentum conservation
J.D. Forbes in 1848 was the first Western scientist to correctly identify viscous deformation as the reason why glaciers flow. In his original paper on the subject, Forbes has a wonderful quote about this realization:
There is something pleasing to the imagination in the unexpected analogies presented by a torrent of fiery lava and the icy stream of a glacier.
Both lava and ice flow can be described by the exact same mathematics, and that mathematics is the Stokes equations.
The Stokes equations consist of three parts: a conservation law, a constitutive relation, and boundary conditions. First, the conservation law states that the flux of stress across the boundary of a control volume $K$ cancels out exactly the body forces $f$:
$$\int_{\partial K}(\tau - pI)\cdot\nu\, ds + \int_Kf\, dx = 0.$$
The right-hand side is zero because we're assuming that the surface and body forces are roughly in balance and thus the acceleration that a fluid parcel experiences is negligible. If we were to leave those terms in we would have the full Navier-Stokes equations. When we apply the usual continuum mechanics arguments, we get a system of PDEs
$$\nabla\cdot\tau - \nabla p + f = 0.$$
To close this set of equations, we'll also need a constitutive relation between the deviatoric stress tensor and either the velocity or its gradient. For a plain old viscous Newtonian fluid, the deviatoric stress and the strain rate tensor
$$\dot\varepsilon(u) \equiv \frac{1}{2}\left(\nabla u + \nabla u^\top\right)$$
are linearly proportional to each other. But glacier ice is not a Newtonian fluid! Glen and Nye showed through a series of laboratory experiments in the 1950s that ice is a shear-thinning material and that the strain rate is roughly a power-law function of stress:
$$\dot\varepsilon = A|\tau|^{n - 1}\tau$$
where $A$ is a temperature-dependent rate factor, $n \approx 3$ is the flow law exponent, and $|\tau|$ denotes the second invariant of a rank-2 tensor: $|\tau| \equiv \sqrt{\tau : \tau / 2}$. The constitutive relation can be inverted to give an expression for the stress tensor in terms of the strain rate tensor, which again is the symmetrized gradient of the velocity. By substituting this expression for the stress tensor into the conservation law, we arrive at a second-order quasilinear partial differential equation for the velocity field.
Finally, we have to know what conditions apply at the system boundaries. At the ice surface, there is effectively zero stress:
$$(\tau - pI)\cdot\nu|_{z = s} = -p_0\nu$$
where $p_0$ is atmospheric pressure. If we were explicitly modeling firn and snow, incorporating wind-blowing effects would be more of a problem. At the ice base things get much more interesting because there are different boundary conditions in the normal and tangential directions. In the normal direction, the ice velocity has to equal to the rate of basal melting:
$$u\cdot\nu|_{z = b} = \dot m.$$
In the tangential direction, frictional contact with the bed creates resistive stresses. The relationship between resistive stresses and the ice velocity and other fields is the content of the sliding law. One of the oldest proposed sliding laws, based on the theory of regelation, is due to work by Weertman in the 1960s. Weertman sliding is a power-law relation between stress and sliding speed:
$$(\tau - pI)\cdot\nu|_{z = b} = -C|u|^{1/m - 1}u,$$
where $m$ is the sliding exponent. In Weertman's theory, the sliding exponent is identical to the Glen flow law exponent $n$ because sliding occurs more through deformation within the ice. The Weertman sliding law makes sense for glaciers flowing over hard beds, but several discoveries in the 1980s found that Antarctic ice streams more typically flow over soft, deformable sediments, with meltwater lubricating flow. For these types of glaciers, sliding is more due to plastic failure within subglacial sediments. Plastic sliding would imply that the basal shear stress is equal to the yield stress of the sediment regardless of the sliding speed, in which case the sliding exponent $m$ is equal to $\infty$. The Schoof or regularized Coulomb friction law is a synthesis of the two types of sliding.
Rather than express the Stokes equations as one big nonlinear PDE, we assume in icepack that all of the diagnostic physics can be derived from an action principle. The action principle states that the velocity and pressure that solve the Stokes equations are really also the critical point of a certain functional, called the action. The action for the Stokes equations with the Weertman sliding law is
$$J = \int_\Omega\left(\frac{n}{n + 1}A^{-1/n}|\dot\varepsilon|^{1/n + 1} - p\nabla\cdot u - f\cdot u\right)dx + \frac{m}{m + 1}\int_{\Gamma_b}C|u|^{1/m + 1}ds.$$
We've found that expressing the diagnostic model through an action principle is advantageous because there are more and better numerical methods for solving constrained convex optimization problems than there are for general nonlinear systems of equations. On top of that, an action principle is shorter to write down.
Nearly all terrestrial glacier flows have much wider horizontal than vertical extents. By expanding the equations of motion in the aspect ratio $\delta = H/L$, it's possible to derive PDE systems that are much simpler than the Stokes equations. Eliminating terms that scale like $\delta$, the vertical component of the momentum balance becomes
$$\frac{\partial}{\partial z}(\tau_{zz} - p) - \rho g = 0.$$
By integrating this equation in the vertical direction and using the fact that $\tau_{xx} + \tau_{yy} + \tau_{zz} = 0$, we can write the pressure as a function of the surface elevation and the horizontal components of the deviatoric stress tensor. This leaves us with a 3D differential equation for the two horizontal components of the velocity. The HybridModel class in icepack describes this system, known in the literature as either the first-order equations or the Blatter-Pattyn equations. We can then depth-average them to arrive at a purely 2D system called the shallow stream equations. To see all the details, you can consult the very excellent book by Greve and Blatter. The IceStream class in icepack describes the depth-averaged system.
For completeness sake, we'll write down the action functional for the shallow stream equations. Since the vertical component of the velocity and stress tensor has been eliminated, in 2D we have a new definition of the effective strain rate:
$$|\dot\varepsilon| \equiv \sqrt{\frac{\dot\varepsilon : \dot\varepsilon + \text{tr}(\dot\varepsilon)^2}{2}}.$$
For the full 3D velocity, the trace of the strain rate tensor is zero -- this is another way of restating the divergence-free condition, which we'll discuss below. But the 2D strain rate of the depth-averaged velocity field can have non-zero divergence. The action functional then becomes
$$J = \int_\Omega\left(\frac{n}{n + 1}hA^{-1/n}|\dot\varepsilon(u)|^{1/n + 1} + \frac{m}{m + 1}C|u|^{1/m + 1} + \rho gh\nabla s\cdot u\right)dx.$$
Note how the friction terms are no longer part of a boundary integral. The optimality conditions for this functional also result in a nonlinear elliptic system of partial differential equations. But the action is purely convex instead of having a saddle point structure like the Stokes equations. Consequently, they're easier to solve numerically, as well as having fewer unknowns and being posed over a lower-dimensional domain.
Mass conservation
The other piece of the puzzle that we've left out is the mass conservation equation. Ice is roughly incompressible -- we're ignoring snow and firn here -- so this can succinctly be expressed as
$$\nabla\cdot u = 0.$$
Strictly speaking, this condition is implied by the action principle for the Stokes equations that we wrote down above. It's a constraint for which the pressure acts as a Lagrange multiplier. In principle, we could use the velocity field computed from the Stokes equations to move the upper and lower free surfaces of the ice, and this is exactly what Elmer/ICE does. For depth-averaged or simplified 3D models, however, the vertical velocity is eliminated entirely, which makes this front-tracking approach more difficult. Instead, we can integrate the divergence-free condition to arrive at the 2D equation
$$\frac{\partial}{\partial t}h + \nabla\cdot h\bar u = \dot a - \dot m$$
for the ice thickness, where $\bar u$ is the depth-averaged ice velocity. (The derivation is a little subtle when you get into it and we're leaving out some of the details here. If you want to see all of them, have a look at Greve and Blatter.) This is a linear hyperbolic equation which, by itself, should strike fear into the heart. Most applications use explicit timestepping schemes for hyperbolic problems. These methods require some care in choosing both the spatial and temporal discretization in order to guarantee stability. We've opted to instead use a more expensive implicit timestepping scheme because these are usually unconditionally stable. The additional expense of an implicit scheme is minute compared to the overall cost of solving the diagnostic equations.
The momentum and mass conservation equations are the two main components of an ice flow model, but there are several other processes at work with their own governing equations.
Temperature: the ice temperature is governed by the heat equation, and most importantly strain heating within the ice and at the bedrock interface are sources of heat. The ice temperature partly determines the rate factor $A$ in Glen's flow law and the temperature gradient partly determines how much heat can be transported through the ice and how much has to be absorbed through the latent heat of melting. The governing PDE is the heat equation and the class HeatTransport3D from icepack contains a description, together with some common simplifications for ice flow.
Damage: while the dominant mode of ice movement is viscous flow, it's also a brittle material and can form fractures. Fracture mechanics models that resolve individual cracks are impractically expensive to apply at glacier-wide scales but there are a number of phenomenological models. The class DamageTransport from icepack contains a specification of the continuum damage mechanics model of Albrecht and Levermann.
Calving: the end state of ice damage is the breaking off or calving of icebergs into the ocean. This problem is especially challenging because it means that the geometry is now dynamic. There is at present no widely-accepted physical model that predicts the rate of iceberg calving well for both Greenland- or Alaska-type events (low amplitude, high frequency) and for Antarctic-type events (high amplitude, low frequency).
Fabric: we've assumed above that the Glen flow law is purely isotropic, but sustained deformation along one axis can give ice crystal grains a preferred orientation.
Hydrology: meltwater at the ice base is ultimately transported along the hydraulic potential gradient and out the ice edge; the degree to which the subglacial hydrological system is channelized or distributed partly determines the sliding resistance.
The discussion above assumes that the inputs from the oceans and atmosphere are known and prescribed. We've adopted this limited viewpoint in order to keep the scope of the project small, but the cryosphere has non-trivial feedbacks with the atmosphere, the oceans, and the landscape. We welcome any contributions of new code to model other unresolved physical processes in glaciology or to couple glaciers to other parts of the earth system.
We've omitted a lot of details above we also haven't even attempted to describe other areas of glaciology, such as the interpretation of ice cores and climate records. The Physics of Glaciers by Cuffey and Paterson is a great reference for getting a broader picture of the field. For a focus on ice dynamics from a more mathematical perspective, Dynamics of Ice Sheets and Glaciers by Greve and Blatter is a must-read. A First Course in Continuum Mechanics by Gonzalez and Stuart is a good read if you want to brush up more on fundamental physics. | CommonCrawl |
\begin{document}
\title{Spectral symmetry in conference matrices}
\begin{abstract} A conference matrix of order $n$ is an $n\times n$ matrix $C$ with diagonal entries $0$ and off-diagonal entries $\pm 1$ satisfying $CC^\top=(n-1)I$. If $C$ is symmetric, then $C$ has a symmetric spectrum $\Sigma$ (that is, $\Sigma=-\Sigma$) and eigenvalues $\pm\sqrt{n-1}$. We show that many principal submatrices of $C$ also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel matrices with symmetric spectrum can be characterized by this construction. \\[3pt] Keywords: Conference matrix, Seidel matrix, Paley graph, signed graph, symmetric spectrum, \\ AMS subject classification: 05C50. \end{abstract}
\section{Introduction} Suppose $S$ is a symmetric matrix with zero diagonal and off-diagonal entries $0$ or $\pm 1$. Then $S$ can be interpreted as the adjacency matrix of a signed graph. Signed graphs are well studied, and a number of recent papers \cite{AMP, BCKW, GHMP, R} pay attention to signed graphs for which the adjacency matrix has symmetric spectrum, which means that the spectrum is invariant under multiplication by $-1$. If $S$ contains no $-1$, then $S$ is the adjacency matrix of an ordinary graph, which has a symmetric spectrum if and only if the graph is bipartite. For general signed graphs there exist many other examples with symmetric spectrum. Here we consider the case that no off-diagonal entries are $0$, in which case $S$ can be interpreted as the Seidel matrix of a graph ($-1$ is adjacent; $+1$ is non-adjacent). It is known that a Seidel matrix of order $n$ is nonsingular if $n \not\equiv 1~({\mbox{mod}}~4)$ (see Greaves~at~al~\cite{GKMS}). Clearly, a symmetric spectrum contains an eigenvalue $0$ if $n$ is odd, therefore there exists no Seidel matrix with symmetric spectrum if $n\equiv 3~({\mbox{mod}}~4)$. For all other orders Seidel matrices with spectral symmetry exist. Examples are often built with smaller block matrices. Here we use a different approach, and investigate Seidel matrices with spectral symmetry inside larger matrices known as conference matrices (see next section).
The spectrum of $S$ does not change if some rows and the corresponding columns are multiplied by $-1$. This operation is called switching. If ${S}'$ can be obtained from $S$ by switching and/or reordering rows and columns, $S$ and ${S}'$ are called equivalent. The corresponding graphs are called switching isomorphic, or switching equivalent.
\section{Conference matrices}\label{conf}
A conference matrix of order $n$ ($n\geq 2$) is an $n\times n$ matrix $C$ with diagonal entries $0$ and off-diagonal entries $\pm 1$ satisfying $CC^\top=(n-1)I$. If $C$ is symmetric, then the spectrum $\Sigma$ of $C$ contains the eigenvalues $\sqrt{n-1}$ and $-\sqrt{n-1}$, both with multiplicity $n/2$, and we write: \[ \Sigma=\{\pm\sqrt{n-1}^{\, n/2}\}. \] Clearly
the spectrum of $C$ is symmetric. Conference matrices are well studied (see for example Section~13 of~Seidel~\cite{S}, and Section~10.4 of~Brouwer and Haemers~\cite{BH}). The order $n$ of a conference matrix is even, and every conference matrix can be switched into a symmetric one when $n\equiv 2~({\mbox{mod}}~4)$ and into a skew-symmetric one if $n\equiv 0~({\mbox{mod}}~4)$. Here we will not consider the skew case, because every skew-symmetric matrix has a symmetric spectrum. Necessary for the existence of a symmetric conference matrix of order $n$ is that $n-1$ is the sum of two squares.
If $C$ is a symmetric conference matrix of order $n=4m+2\geq 6$, switched such that all off-diagonal entries in the first row and column are equal to $1$, then $C$ is the Seidel matrix of a graph with an isolated vertex. If we delete the isolated vertex, we have a strongly regular graph $G$ with parameters $(4m+1,2m,m-1,m)$ (this means that $G$ has order $4m+1$, is $2m$-regular, every edge is in precisely $m-1$ triangles, and any two nonadjacent vertices have precisely $m$ common neighbors). Conversely, the Seidel matrix of a strongly regular graph with these parameters extended with an isolated vertex, is a symmetric conference matrix. If $4m+1$ is a prime power $q$ (say), such a strongly regular graph can be constructed as follows. The vertices of $G$ are the elements of the finite field $\bf{F}_q$, where two vertices $x$ and $y$ $(x\neq y)$ are adjacent whenever $x-y$ is a square in $\bf{F}_q$. The construction is due to Paley, the graph $G$ is known as Paley graph, and a corresponding conference matrix $C$ is a Paley conference matrix of order $n=q+1$, which we shall abbreviate to $PC(n)$. Other constructions are known. Mathon~\cite{M} has constructed conference matrices of order $n=q p^2+1$, where $p$ and $q$ are prime powers, $q\equiv 1~({\mbox{mod}}~4)$, and $p\equiv 3~({\mbox{mod}}~4)$. For $n=6$, 10, 14 and $18$, every conference matrix of order $n$ is a $PC(n)$. There is no conference matrix of order $22$ ($21$ is not the sum of two squares), and there are exactly four non-equivalent conference matrices of order 26, one of which is a $PC(26)$. The smallest order for which existence is still undecided is 86.
\section{The tool}
\begin{thm}\label{main} Suppose \[ A= \left[ \begin{array}{cc} A_1 & M\\ M^\top & A_2 \end{array} \right] \] is a symmetric orthogonal matrix with zero diagonal. Let $n_i$ be the order, and let $\Sigma_i$ be the spectrum of $A_i$ ($i=1,2$). Assume $n_1\leq n_2$, and define $m=(n_2-n_1)/2$. Then \[ \Sigma_2 = -\Sigma_1 \cup \{\pm 1^m\}. \] \end{thm}
\noindent (Recall that $\{\pm 1^m\}$ means that $1$ and $-1$ are both repeated $m$ times.)
\begin{proof} We have \[ I = A^2= \left[ \begin{array}{cc} A_1^2+M M^\top & A_1 M+M A_2\\ M^\top A_1 + A_2 M^\top & A_2^2 + M^\top\!M \end{array} \right]. \] This implies $A_1 M = -M A_2$, $A_1^2+MM^\top\! = I$ and $A_2^2 + M^\top\! M = I$. For $i=1,2$ let $\Sigma_i'$ be the sub-multiset of $\Sigma_i$ obtained by deleting all eigenvalues equal to $\pm 1$. Suppose $\lambda\in\Sigma_1'$ is an eigenvalue of $A_1$ with multiplicity $\ell$. Define $V$ such that its columns span the eigenspace of $\lambda$. Then $A_1 V=\lambda V$, and rank$(V)=\ell$. Moreover, $\lambda M^\top\! V = M^\top\! A_1 V = -A_2 M^\top\! V$. Therefore $-\lambda$ is an eigenvalue of $A_2$ and the columns of $M^\top\! V$ are eigenvectors. Using $A_1^2+MM^\top\! = I$ and $\lambda\neq\pm 1$ we have \[ {\mbox{rank}}(V) \geq {\mbox{rank}}(M^\top\! V) \geq {\mbox{rank}}(MM^\top\! V) = {\mbox{rank}}((I-A_1^2)V) = {\mbox{rank}}((1-\lambda^2)V) = {\mbox{rank}}(V). \] Therefore ${\mbox{rank}}(M^\top V) = {\mbox{rank}}(V)$, and the multiplicity ${\ell}'$ of $-\lambda\in\Sigma_2'$ is at least $\ell$. Conversely, $\ell\geq{\ell}'$ and therefore $\ell = {\ell}'$. This implies that $\Sigma_1' = -\Sigma_2'$. Finally, trace$(A_1) = \mbox{trace}(A_2)=0$ implies that for both matrices the eigenvalues $-1$ and $1$ have the same multiplicities. \end{proof}
\begin{cor}\label{cor} Suppose \[ C= \left[ \begin{array}{cc} C_1 & N\\ N^\top & C_2 \end{array} \right] \] is a symmetric conference matrix of order $n$. \\ $(i)$ ~$C_1$ has a symmetric spectrum if and only if $C_2$ has a symmetric spectrum. \\ $(ii)$~If $C_1$ and $C_2$ have symmetric spectrum then, except for eigenvalues equal to $\pm\sqrt{n-1}$, $C_1$ and $C_2$ have the same spectrum. \end{cor}
\begin{proof} Apply Theorem~\ref{main} to $A=\frac{1}{\sqrt{n-1}}C$. \end{proof}
Theorem~\ref{main} is a special case of an old tool, which has proved to be useful in spectral graph theory. It is, in fact, a direct consequence of the inequalities of Aronszajn (see~\cite{H}, Theorem~1.3.3).
\section{Submatrices}
Clearly a Seidel matrix of order 1 or 2 has symmetric spectrum, so by Corollary~\ref{cor} we obtain Seidel matrices with spectra
$\{0,\ \pm\sqrt{n-1}^{\,(n-2)/2}\}$ and $\{\pm 1,\ \pm\sqrt{n-1}^{\,(n-4)/2}\}$ if we delete one or two rows and the corresponding columns from a symmetric conference matrix of order~$n$. In the next section we will characterise this construction.
As mentioned earlier, there is no Seidel matrix with symmetric spectrum if $n\equiv 3~({\mbox{mod}}~4)$. For $n=4$ and $5$, there is exactly one equivalence class of Seidel matrices with spectral symmetry, represented by: \[ S_4= {\small \left[ \begin{array}{rrrr} 0&1&1&1\\ 1&0&-1&1\\ 1&-1&0&-1\\ 1&1&-1&0 \end{array} \right] }, \mbox{ and } S_5= {\small \left[ \begin{array}{rrrrr} 0&1&1&1&1\\ 1&0&-1&1&1\\ 1&-1&0&-1&1\\ 1&1&-1&0&-1\\ 1&1&1&-1&0 \end{array} \right] } \] with spectra \[ \{\pm 1,\ \pm\sqrt{5}\} \mbox{ and } \{0,\ \pm\sqrt{5}^2\}. \]
\begin{prp} After suitable switching, every symmetric conference matrix of order $n\geq 6$ contains $S_4$ and $S_5$ as a principal submatrix. \end{prp}
\begin{proof} The graphs of $S_4$ and $S_5$ have an isolated vertex. If we delete the isolated vertex we obtain the paths $P_3$ and $P_4$. So, it suffices to show that a strongly regular graph $G$ with parameters $(4m+1,2m,m-1,m)$ contains $P_3$ and $P_4$ as an induced subgraph. The presence of $P_3$ in $G$ is trivial. Fix an edge $\{x,y\}$ in $G$, and let $z$ be a vertex adjacent to $y$, but not to $x$. Then there are $m$ vertices which are adjacent to $x$ and not to $y$, and at least one of them ($w$ say) is nonadjacent to $z$, since otherwise $x$ and $z$ would have $m+1$ common neighbors. Thus the set $\{w,x,y,z\}$ induces a $P_4$. \end{proof}
By Corollary~\ref{cor} and the above proposition we know that there exist Seidel matrices of order $n-4$ and $n-5$ with spectra \[\{\pm 1,\ \pm\sqrt{5},\ \pm\sqrt{n-1}^{(n-8)/2}\}, \mbox{ and } \{0,\ \pm\sqrt{5}^2,\ \pm\sqrt{n-1}^{(n-10)/2}\}, \] respectively, whenever there exists a symmetric conference matrix of order $n\geq 10$.
Next we investigate how Corollary~\ref{cor} can be applied to a $PC(n)$ for $n=10$, 14, and 18 with a submatrix of order 6, 8, or 9. Up to equivalence there exist four Seidel matrices of order 6 with symmetric spectrum (see~Van~Lint and Seidel~\cite{LS}, or Ghorbani~et~al~\cite{GHMP}). The spectra are: \[ \Sigma_1=\{\pm 1,\ \pm\sqrt{5},\ \pm 3\},~ \Sigma_2=\{\pm\sqrt{5}^3\},~ \Sigma_3=\{\pm 1,\ \pm\sqrt{7\pm 2\sqrt{5}}\},~ \Sigma_4=\{\pm 1^2,\ \pm\sqrt{13}\}. \] Only $\Sigma_1$ is the spectrum of a submatrix of a $PC(10)$, each of the spectra $\Sigma_1,~\Sigma_2$ and $\Sigma_3$ belongs to a submatrix of a $PC(14)$, and all four occur as the spectrum of a submatrix of a $PC(18)$. So by Corollary~\ref{cor} we obtain Seidel matrices of order 8 and 12 with spectra \[ \Sigma_i\cup\{\pm\sqrt{13}\} \mbox{ for } i=1,2,3 \mbox{ and } \Sigma_i\cup\{\pm\sqrt{17}^3\} \mbox{ for } i=1,\ldots,4, \] respectively. All graphs of order 8 with a symmetric Seidel spectrum are given in Figure~6 of Ghorbani~et~al~\cite{GHMP}.
Up to equivalence and taking complements there are twenty such graphs (we just found three of these). By computer we found that six of these graphs have a Seidel matrix, which is a submatrix of a $PC(18)$. So Corollary~\ref{cor} gives six possible symmetric spectra for the graphs on the remaining 10 vertices. However, it turns out that these six spectra belong to seven non-equivalent graphs, of which two have the same Seidel spectrum. These seven graphs are given in Figure~\ref{10} (the last seven graphs). The same phenomenon occurs if we delete $S_4$ from a $PC(14)$. This can be done in two non-equivalent ways, which leads to two non-equivalent graphs with spectrum $\{\pm 1,\ \pm\sqrt{5},\ \pm\sqrt{13}{\,}^3\}$ (the first two in Figure~\ref{10}). The Seidel matrices of order 9 with symmetric spectrum are also given in \cite{GHMP}. It turns out that none of these is a submatrix of a $PC(18)$. But note that we already found two non-equivalent Seidel matrices of order 9 with symmetric spectrum,
one in a $PC(10)$ and one in a $PC(14)$. Also the Seidel matrix of order 8 with spectrum $\{\pm 1,\ \pm 3^3\}$ is a submtrix of a $PC(10)$, but not of a $PC(14)$ or a $PC(18)$. Similarly, the Seidel matrix of order 8 with spectrum $\Sigma_1\cup\{\pm\sqrt{13}\}$ is a submatrix of a $PC(14)$ (as we saw above), but not of a $PC(18)$.
\begin{figure}
\caption{Graphs of order $10$ for which the Seidel matrix is a submatrix of a $PC(14)$ (first 2), or a $PC(18)$ (last 7); the numbers represent the positive part of the (symmetric) spectrum.}
\label{10}
\end{figure}
\\ It is known (see Bollobas and Thomason~\cite{BT}) that every graph of order $m$ is an induced subgraph of the Paley graph of order $q$ if $q\geq f(m)=(2^{m-2}(m-1)+1)^2$.
If the smaller graph is the Paley graph of order $m$, it follows that a $PC(m+1)$ is a principal submatrix of a $PC(q+1)$ if $q\geq f(m)$. Thus, by Corollary~\ref{cor}, we obtain Seidel matrices with symmetric spectrum containing only four distinct eigenvalues: \begin{prp}\label{P} If $q$ and $m$ are prime powers satisfying $q\equiv m\equiv 1$~{\rm (mod 4)} and $q\geq f(m)$, then there exist a Seidel matrix with spectrum $\{\pm\sqrt{q}^{\,(q-2m-1)/2},\ \pm\sqrt{m}^{\,(m+1)/2}\}$. \end{prp}
\section{Characterizations}
It is clear that a Seidel matrix with symmetric spectrum and two distinct eigenvalues is a conference matrix. The next two theorems deal with three and four distinct eigenvalues.
In a more general setting, the results of this section were already obtained by Greaves and Suda~\cite{GS}.
In the proofs below we restrict to the case which is relevant to us: spectral symmetry in a symmetric conference matrix.
\begin{thm}\label{char1} Suppose $S$ is a Seidel matrix with symmetric spectrum and three distinct eigenvalues, then $S$ can be obtained from a symmetric conference matrix by deleting one row and the corresponding column. \end{thm}
\begin{proof} Suppose $S$ has order $n-1$. It follows that $S$ has an eigenvalue $0$ of multiplicity $1$, and two eigenvalues $\pm\sqrt{n-1}$, each of multiplicity $(n-2)/2$. Define $M=(n-1)I-S^2$, then rank$(M)=1$, the diagonal entries of $M$ are equal to $1$, and $M$ is positive semi-definite. This implies that $M={\bf k}{\bf k}^\top$ for some vector ${\bf k}$ with entries $\pm 1$. It follows that ${\bf k}^\top S^2 {\bf k}={\bf k}^\top ((n-1)I-M){\bf k}=(n-1)^2-(n-1)^2=0$, hence $S{\bf k}={\bf 0}$. Define \[ C=\left[ \begin{array}{cc} 0 & {\bf k}^\top\\ {\bf k} & S \end{array} \right]. \mbox{ Then } CC^\top=C^2= \left[ \begin{array}{cc} n & {\bf k}^\top\!S\\ S{\bf k} & {\bf k}{\bf k}^\top\!+S^2 \end{array} \right] = (n-1)I, \] because $S{\bf k}={\bf 0}$ and $S^2=(n-1)I-M=(n-1)I-{\bf k}{\bf k}^\top$. \end{proof}
If $S$ is the Seidel matrix of a regular graph $G$, then $G$ is strongly regular, as we have seen in Section~\ref{conf}. Here we do not require regularity. However it follows from the above that we can always switch in $C$, such that ${\bf k}$ becomes the all-one vector, in which case the switched $G$ is regular.
\begin{thm} Suppose $S$ is a Seidel matrix with symmetric spectrum and four distinct eigenvalues, which include $1$ and $-1$ both of multiplicity $1$, then $S$ can be obtained from a symmetric conference matrix by deleting two rows and the corresponding columns. \end{thm}
\begin{proof} Suppose $S$ has order $n-2$. Clearly $n$ is even, and $S^2$ has an eigenvalue $1$ of multiplicity $2$. From trace$(S^2)=(n-2)(n-3)$ it follows that $S^2$ has one other eigenvalue equal to $n-1$ of multiplicity $n-4$. Define $M=(n-1)I-S^2$. Then rank$(M)=2$, and $M$ is positive semi-definite with an eigenvalue $n-2$ of multiplicity $2$. The diagonal entries of $M$ are equal to $2$, and the off-diagonal entries are even integers. Let $T$ be a pricipal submatrix of $M$ of order $2$, then $T$ is positive semi-definite, and therefore $T$ is one of the following: \[
\left[ \begin{array}{rr} 2 & -2\\ -2 & 2 \end{array} \right], \
\left[ \begin{array}{rr} 2 & 2\\ 2 & 2 \end{array} \right], \mbox{ or }
\left[ \begin{array}{cc} 2 & 0\\ 0 & 2 \end{array} \right]. \] Since rank$(M)=2$, $M$ has a $2\times 2$ pricipal submatrix of rank~2, so the last option $T=2I$ does occur. Consider the two rows in $M$ corresponding to $T=2I$. At each coordinate place, the two entries can only consist of one $0$ and one $\pm 2$, since all other options would create a submatrix of $M$ of rank~3. Every other row of $M$ is a linear combination of these two rows, and because $M$ is symmetric, we can conclude that the rows and columns of $S$ can be ordered such that \[ M=2\left[ \begin{array}{cc} M_1 & O\\ O & M_2 \end{array} \right], \] where $M_1$ and $M_2$ have $1$ on the diagonal, $\pm 1$ off-diagonal, and ${\mbox{rank}}(M_1)={\mbox{rank}}(M_2)=1$. This implies that there exist vectors ${\bf k}_1$ and ${\bf k}_2$ with entries $\pm 1$, such that $M_1={\bf k}_1{\bf k}_1^\top$ and $M_2={\bf k}_2{\bf k}_2^\top$. Both $M_1$ and $M_2$ have one nonzero eigenvalue $(n-2)/2$, which equals the trace, so $M_1$ and $M_2$ have the same order $(n-2)/2$. With the corresponding partition of $S$ we have \[ S=\left[ \begin{array}{cc} S_1 & R\\ R^\top & S_2 \end{array} \right], \ S^2=\left[ \begin{array}{cc} S_1^2+RR^\top & S_1 R + RS_2\\ R^\top\!S_1+ S_2 R^\top & R^\top\!R+S_2^2 \end{array} \right]=(n-1)I-2 \left[ \begin{array}{cc} M_1 & O\\ O & M_2 \end{array} \right]. \]
We conclude that $S_1 R=-RS_2$, and $S_1^2+RR^\top=(n-1)I-2M_1$. Using ${\bf k}_1^\top {\bf k}_1=(n-2)/2$, and $M_1={\bf k}_1{\bf k}_1^\top$ we obtain \[ {\bf k}_1^\top S_1^2{\bf k}_1+{\bf k}_1^\top RR^\top{\bf k}_1 = {\bf k}_1^\top(S_1^2+RR^\top){\bf k}_1 = {\bf k}_1^\top((n-1)I-2M_1){\bf k}_1=(n-2)/2. \] The entries of $S_1{\bf k}_1$ are odd integers, so ${\bf k}_1^\top S_1^2{\bf k}_1\geq(n-2)/2$. This implies ${\bf k}_1^\top RR^\top{\bf k}_1=0$ and ${\bf k}_1^\top S_1^2{\bf k}_1=(n-2)/2$, so $R^\top{\bf k}_1={\bf 0}$, and $S_1{\bf k}_1$ is a $(\pm 1)$-vector ${\bf h}_1$ (say). Next observe that
$R^\top {\bf h}_1 = R^\top S_1{\bf k}_1 = -S_2R^\top{\bf k}_1={\bf 0}$, and also $S_1{\bf h}_1 = S_1^2{\bf k}_1 = (-RR^\top + (n-1)I - 2{\bf k}_1^\top{\bf k}_1){\bf k}_1={\bf k}_1$. Similarly, ${\bf h}_2=S_2{\bf k}_2$ is a $(\pm 1)$-vector, $S_2{\bf h}_2={\bf k}_2$, and $R{\bf k}_2=R{\bf h}_2={\bf 0}$. Define \[ C = \left[ \begin{array}{crrr}
0 & 1\ & -{\bf h}_1^\top & \,\ {\bf h}_2^\top \\
1 & 0\ & -{\bf h}_1^\top & -{\bf h}_2^\top \\ {\bf k}_1 & {\bf k}_1 & S_1~ & R~~ \\ {\bf k}_2 & -{\bf k}_2 & R^\top & S_2~ \end{array} \right]. \] Then $CC^\top=(n-1)I$. Finally $C^\top C= (n-1)I$ implies that ${\bf h}_1=-{\bf k}_1$ and ${\bf h}_2={\bf k}_2$. \end{proof}
As we have seen in Proposition~\ref{P}, there exist many Seidel matrices with four distinct eigenvalues and symmetric spectrum different from the ones in the above theorem. Another example is the Seidel matrix of a complete graph of order $m$, extended with $m$ isolated vertices. (see Ghorbani~et~al~\cite{GHMP}, Theorem 2.2). It is not likely that the case of four eigenvalues can be characterised in general.
Note that the above characterizations lead to nonexistence of some Seidel spectra. For example, there exist no graphs with Seidel spectra $\{0,\ \pm\sqrt{21}{\,}^{10}\}$ and $\{\pm 1,\ \pm\sqrt{21}{\,}^9\}$, because there exist no conference matrix of order $22$.
\section{Sign-symmetry}
A graph $G$ is called sign-symmetric if $G$ is switching isomorphic to its complement. If $S$ is the Seidel matrix of a sign-symmetric graph $G$ (we also call $S$ sign-symmetric), then $S$ and $-S$ are equivalent, and therefore $S$ has symmetric spectrum.
Every $PG(n)$ is sign-symmetric, but many other conference matrices are not. Up to equivalence, there are at least two conference matrices of order 38, and at least 80 of order 46 which are not sign-symmetric (see Bussemaker, Mathon and Seidel~\cite{BMS}).
If we delete one or two rows and columns from a $PC(n)$, the obtained Seidel matrix will be sign-symmetric. But in general, when we apply Corollary~\ref{cor}, there is not much we can say about the relation between sign-symmetry of $C$, $C_1$ and $C_2$.
\end{document} | arXiv |
\begin{document}
\title{Stability of the Area Law for the Entropy of Entanglement} \author{S. Michalakis} \affiliation{California Institute of Technology, Institute for Quantum Information and Matter} \email{[email protected]} \begin{abstract} Recent results \cite{tqo_stability,short_stability,michalakis:2011} on the stability of the spectral gap under general perturbations for frustration-free Hamiltonians, have motivated the following question: Does the entanglement entropy of quantum states that are connected to states satisfying an area law along gapped Hamiltonian paths, also satisfy an area law? We answer this question in the affirmative, combining recent advances in quasi-adiabatic evolution and Lieb-Robinson bounds with ideas from the proof of the $1$D area law \cite{1d_area_law}. \end{abstract}
\maketitle \section{Introduction} Over the past decade, there has been a rapidly growing interest in the role of entanglement as a measure of complexity in simulating properties of quantum many-body systems~\cite{simulation_area_law, survey_area_law}. From a theoretical point of view, the question of how the entropy of entanglement scales with system size, has generated some spectacular results, such as Hasting's area law for one-dimensional gapped systems~\cite{1d_area_law} (see also \cite{michalakis:thesis, gs_approx} for partial generalizations), with a stronger bound for groundstates of frustration-free Hamiltonians given by Arad {\it et al. } in~\cite{new_area_law}, as well as the recent breakthrough by Brandao {\it et al. } in ~\cite{exp_decay_area_law}, where it is shown that exponential decay of correlations implies an area law for the entanglement entropy of one-dimensional states.
Here, we prove that the amount of entanglement contained in states connected along gapped, local Hamiltonian paths satisfies a certain stability property. Specifically, we show that groundstates of gapped Hamiltonians whose entanglement spectrum decays fast enough to imply an area law for the entanglement entropy, are connected via gapped, local Hamiltonian paths to states that satisfy a similar area-law bound. The techniques used to prove this result apply equally well to showing the stability of the area-law for any eigenstate of a Hamiltonian with fast enough decay in the entanglement spectrum, as long as the connection is along a gapped, quasi-local Hamiltonian path. As a consequence, groundstates of gapped, {\it frustration-free} Hamiltonians that satisfy the conditions for {\it stability} of the spectral gap under weak perturbations, found in \cite{tqo_stability,short_stability,michalakis:2011}, become central objects in the study of entanglement scaling in $2$D and in higher dimensions. This follows from the fact that the condition of {\it local indistinguishability} necessary for the stability of the spectral gap, also implies a rapid decay in the entanglement spectrum of the gapped groundstates \cite{michalakis:2011}.
The main result appears in Theorem~\ref{thm:area-law}, which makes use of Lemma~\ref{lem:ent_bound} to convert a bound on the decay rate of the entanglement spectrum into a bound for the entanglement entropy of the states we are interested in. The main technical tool appears in Lemma~\ref{unitary_decomposition}, where we show that the unitary evolution of the initial groundstate along a gapped path can be approximated, up to rapidly-decaying error, with a product of three unitaries, two of which act on complementary regions and the third acts along the boundary separating the two regions. We expect that such a decomposition will find a variety of further uses, as it provides a more rigorous view of adiabatic evolution as a quantum circuit of local unitaries with depth dictated by the accuracy of the approximation we are willing to tolerate.
\section{Quasi-adiabatic evolution for gapped Hamiltonians} We begin by introducing the central technical tool used in bounding the spread of the entanglement as the initial state is adiabatically transformed. The main idea is to simulate the true adiabatic evolution with another evolution that satisfies two important properties: \begin{enumerate} \item The simulated evolution, introduced in~\cite{hast-quasi-intro} as {\it quasi-adiabatic continuation} and further developed in~\cite{tjo,quantum_hall,hast-quasi,BMNS:2011}, is indistinguishable from the true adiabatic evolution, if we restrict our attention to the evolution of uniformly gapped eigenspaces. \item Unlike the true adiabatic evolution, the quasi-adiabatic evolution is generated by quasi-local interactions and, hence, transforms local operators into quasi-local operators. \end{enumerate} First, let us define precisely which families of Hamiltonians we are considering. \begin{defn}\label{defn:path} Let $H(s) = H_0 + \sum_{u\in \Lambda} V_u(s)$, where $\Lambda \subset \mathbb{Z}^d$ and set $b_u(r) = \{ v\in \Lambda: d(u,v) \le r\}$ to be the ball of radius $r$, centered on site $u\in \Lambda$. Then, the following assumptions hold: \begin{enumerate}[i.]
\item $H_0 = \sum_{u \in \Lambda} Q_u$, with $Q_u$ supported on $b_u(r_0)$, for $r_0 \ge 0$ and $\|Q_u\|\le J_1$, for $u \in \Lambda$.
\item $V_u(s)$ has support on $b_u(r_0)$, with $V_u(0) = 0$ and $\|V_u(s)\| \le J_1, \, \|\partial_s V_u(s)\| \le J_2$, for $s \in [0,1]$ and $u\in \Lambda$. \item $H(s)$ has spectral gap $\gamma(s) \ge \gamma > 0$, with differentiable groundstate $\ket{\psi_0(s)}$, for all $s \in [0,1]$. In particular, $H_0$ has unique groundstate $\ket{\psi_0(0)} := \ket{\psi_0}$, with spectral gap $\gamma(0) \ge \gamma$. \end{enumerate} \end{defn} For the above one-parameter family of Hamiltonians $H(s)$, there exists a family of unitaries $U_s$ \cite{hast-quasi,BMNS:2011}, satisfying: \begin{equation} \label{def:adiabatic_unitary} \ket{\psi_0(s)} = U_s \, \ket{\psi_0}, \quad \partial_{s} U_s = i \, \mathcal{D}_s\, U_s, \quad U_0=\one, \quad \forall s \in [0,1], \end{equation} where the generating dynamics, $\mathcal{D}_s$, has the following decay properties: \begin{equation}
\label{def:adiabatic_generator} \mathcal{D}_s = \sum_{u\in \Lambda} \sum_{r\ge r_0} \mathcal{D}_s(u;r),\quad \mathrm{supp}(\mathcal{D}_s(u;r)) = b_u(r), \quad \|D_s(u;r)\| \le 2\, J_2 \, f_{\gamma}(r-r_0), \end{equation} for a sub-exponentially decaying function $f_{\gamma}(r)$ (e.g. $\exp\{- c_0\, r/\ln^2r\}$, for $c_0 > 0$), with $f_{\gamma}(0)=1$ and decay rate proportional to $\gamma/v_0$, where $v_0 \sim J_1 \, r_0^{d-1}$ is the Lieb-Robinson velocity for the family of Hamiltonians $\{H(s)\}_{s\ge 0}$~\cite{hast-koma,lr3}.
\section{Decomposing the quasi-adiabatic evolution} The above decay estimates on the generator of the unitary $U_s$, allow us to decompose the quasi-adiabatic evolution into a product of three unitaries, two of which act on disjoint subsets of the lattice and a third one coupling the disjoint evolutions at the boundary of the two sets. More importantly, the error in approximating the true adiabatic evolution decays sub-exponentially in the thickness of the boundary chosen for the coupling unitary. \begin{lemma}\label{unitary_decomposition} Let $U_s$ denote the unitary corresponding to the quasi-adiabatic evolution of a one-parameter, gapped family of Hamiltonians $H(s)$, as defined above. For $A \subset \Lambda$, set: \begin{enumerate} \item $I_{A}(R) = \{x \in A: d(x,\partial A) \le R\}$, \item $E_{A}(R) = \{x \in A^c: d(x,\partial A) \le R\}$, and \item $\partial A(R) = I_A(R) \cup E_A(R)$. \end{enumerate} Then, there exist unitaries $U_s(A)$, $U_s(A^c)$ and $U_s(\partial A(2R))$ with non-trivial support on $A, A^c$ and $\partial A(2R)$, respectively, such that the following bound holds: \begin{equation}\label{bnd:decomp_error}
\|U_s(A) \otimes U_s(A^c) \, U_s(\partial A(2R)) - U_s\| \le \epsilon_s(R) := c_1 \Big(e^{c_2 (J_2/\gamma) |s|}-1\Big)\, |\partial A|\, f_{\gamma}(c_3 R), \end{equation} for dimensional constants $c_1,c_2,c_3 >0$. \begin{proof} We sketch the proof, which uses the Lieb-Robinson bounds developed in \cite{hast-quasi,BMNS:2011}. First, define the following unitary operator: \begin{equation}\label{unitary:V} V_{s}(A) = U^\dagger_s \, U_s(A) \otimes U_s(A^c), \end{equation} with $\partial_s U_s(A) = i \mathcal{D}_s(A) U_s(A), \, U_A(0) = \one$ and $\partial_s U_s(A^c) = i \mathcal{D}_s(A^c) U_s(A^c), \, U_{A^c}(0) = \one$, where we define, for $Z \subset \Lambda$: \begin{equation} \mathcal{D}_s(Z) = \sum_{u\in \Lambda,\,r\ge 0: \,b_u(r)\subset Z} \mathcal{D}_s(u;r), \end{equation} noting that $\mathrm{supp}(\mathcal{D}_s(Z)) \subset Z$ and, hence, $\mathrm{supp}(U_s(X)) \subset X$. Now, we set: \begin{equation} {\mathcal F}_s(\partial A) \equiv \mathcal{D}_s - (\mathcal{D}_s(A)+\mathcal{D}_s(A^c)) = \sum_{b_u(r) \cap \partial A \neq \emptyset} \mathcal{D}_s(u;r), \end{equation} where $X \cap \partial A \neq \emptyset \Longleftrightarrow (X \cap A \neq \emptyset) \wedge (X \cap A^c \neq \emptyset)$. The operator ${\mathcal F}_s(\partial A)$ can be well approximated by the locally supported $\mathcal{D}_s(\partial A(R))$. To see this, note that: \begin{equation} {\mathcal F}_s(\partial A) - \mathcal{D}_s(\partial A(R)) = \sum_{u\in \Lambda} \sum_{r \ge d(u)} \mathcal{D}_s(u;r), \end{equation} where $d(u) = \max\{R+1-d(u,\partial A), 1+d(u,\partial A)\}$. We may partition $\Lambda$ as follows: \begin{equation} \Lambda = \cup_{k\ge 0} B_k, \quad B_k = \{u\in \Lambda: d(u,\partial A) = k\},
\end{equation} noting, further, that $|B_k| \le \sum_{u\in \partial A} (|b_u(k)|-|b_u(k-1)|) \le c_d k^{d-1} |\partial A|$, for $k\ge 1$ and $|B_0| = |\partial A|$, where $c_d = 2d$ can be thought of as the area of the unit ball in $\mathbb{Z}^d$. This implies that: \begin{align}
&\|{\mathcal F}_s(\partial A) - \mathcal{D}_s(\partial A(R))\| \le \sum_{k\ge 0} \sum_{u \in B_k} \sum_{r \ge d(u)} \|\mathcal{D}_s(u;r)\| \\
&\le 2\,J_2 \left(\sum_{k\ge 0}^{\lfloor (R+1)/2 \rfloor} |B_k| \sum_{r \ge \lceil (R+1)/2\rceil} f_{\gamma}(r-r_0) + \sum_{k\ge 1+\lfloor (R+1)/2 \rfloor} |B_k| \sum_{r \ge 1+k} f_{\gamma}(r-r_0)\right)\\
&\le 2\,J_2 |\partial A| \Big(\left(1+(R+1)^d\right) F_{\gamma}(\lceil(R+1)/2\rceil -r_0) + c_d \sum_{k\ge 1+\lfloor (R+1)/2 \rfloor} k^{d-1} F_{\gamma}(1+k-r_0)\Big), \end{align} where $F_{\gamma}(s) := \sum_{r \ge s} f_{\gamma}(r)$.
At this point, given the decay rate of $f_{\gamma}$ \cite{hast-quasi,BMNS:2011}, it should be clear that for dimensional constants $c_0,d_0 > 0$: \begin{equation} \label{bnd:gen}
\|{\mathcal F}_s(\partial A) - \mathcal{D}_s(\partial A(R))\| \le c_0\,(J_2/\gamma) |\partial A| (1+R)^{d+d_0} f_{\gamma}(\lceil (R+1)/2 \rceil - r_0). \end{equation} By differentiating both sides of (\ref{unitary:V}), we get: $ \partial_s V_{s}(A) = -\imath \left(U^\dagger_s \,{\mathcal F}_s(\partial A)\, U_s \right) V_{s}(A),\quad V_0(A)=\one. $ We may approximate the unitary $V_s(A)$ with $W_s(A)$ generated by: $ \partial_s W_{s}(A) = -\imath\, \left(U^\dagger_s \,\mathcal{D}_s(\partial A(R))\, U_s \right) W_{s}(A), \quad W_0(A) =\one, $ such that: \begin{equation}\label{bnd:general}
\|V_s(A) - W_s(A)\| = \|W^\dagger_s(A)V_s(A) - \one\| \le \int_0^s \|\partial_t W^\dagger_t(A)V_t(A)\|\, dt \le |s| \, \sup_{t\in[0,s]} \|{\mathcal F}_t(\partial A)-\mathcal{D}_t(\partial A(R))\|. \end{equation} Combined with (\ref{bnd:gen}), the above bound implies: \begin{equation}\label{bnd:unitary}
\|V_s(A) - W_s(A)\| \le c_0\,(J_2/\gamma)\, |s|\, |\partial A| (1+R)^{d+d_0} f_{\gamma}(\lceil (R+1)/2 \rceil - r_0), \quad c_0,d_0 > 0. \end{equation} Finally, we approximate $W_s(A)$ with the unitary $W_s{(\partial A(2R))}$ given by: \begin{equation} \partial_s W_s{(\partial A(2R))} = \imath \left(U^{\dagger}_s(\partial A(2R)) \,\mathcal{D}_s(\partial A(R))\, U_s(\partial A(2R)) \right) W_s{(\partial A(2R))}, \quad W_0(\partial A(R)) = \one, \end{equation} with the unitary $U_s(\partial A(2R))$ generated by $\mathcal{D}_s(\partial A(2R))$. Following (\ref{bnd:general}), we have: \begin{equation}\label{bnd:boundary}
\|W_s(A)-W_s{(\partial A(2R))}\| \le |s| \sup_{t\in [0,s]} \| U^\dagger_t\,\mathcal{D}_t(\partial A(R))\, U_t - U^{\dagger}_t(\partial A(2R)) \,\mathcal{D}_t(\partial A(R))\, U_t(\partial A(2R))\|. \end{equation} At this point, we cannot use the Lieb-Robinson bounds developed in \cite{hast-quasi},\cite[Thm. 4.6]{BMNS:2011} directly, since we are dealing with evolutions of the form $U^\dagger_s\, O_A \, U_s$ instead of $U_s\, O_A\, U^\dagger_s$. Nevertheless, by setting: \begin{equation} F_t(O_X) = U^\dagger_t\,O_X\, U_t - U^{\dagger}_t(\partial A(2R)) \,O_X\, U_t(\partial A(2R)), \end{equation} for an operator $O_X$ with support on $X \subset \Lambda$, we get: \begin{equation} F_t(O_X) = \int_0^t \left(U^\dagger_s\, i[O_X,\mathcal{D}_s-\mathcal{D}_s(\partial A(2R))] \, U_s +F_s\left(i[O_X, \mathcal{D}_s(\partial A(2R))]\right) \right) ds, \end{equation} which implies: \begin{equation}
\|F_t(O_X)\| \le |t| \sum_{b_u(r) \subsetneq \partial A(2R)} \sup_{s\in [0,t]} \|[O_X,\mathcal{D}_s(u;r)]\| + \sum_{b_u(r) \subset \partial A(2R)} \sup_{s\in [0,t]} \|[O_X, \mathcal{D}_s(u;r)]\| \int_0^t \|F_s(O_X(u;r))\| \, ds,
\end{equation} where $O_X(u;r) = [O_X, \mathcal{D}_s(u;r)]/\|[O_X, \mathcal{D}_s(u;r)]\|$. At this point, we may use the recursive argument found in \cite{hast-koma,lr3}, setting $O_X = \mathcal{D}_t(\partial A(R))$ and recalling the sub-exponential decay of $\|\mathcal{D}_s(u;r)\|$. Setting: \begin{equation}
\epsilon_s(R) := c_1 \left(e^{c_2 (J_2/\gamma) |s|}-1\right)\, |\partial A|\, f_{\gamma}(c_3 R), \end{equation} for dimensional constants $c_1,c_2,c_3 >0$, we get from (\ref{bnd:unitary}) and (\ref{bnd:boundary}): \begin{equation}
\|U_s - U_s(A)\UEW^\dagger_s{(\partial A(2R))}\| \le \|V^\dagger_s(A)-W^\dagger_s(A) \| + \|W^\dagger_s(A) - W^\dagger_s{(\partial A(2R))} \| \le \epsilon_s(R). \end{equation} Renaming $U_s(\partial A(2R))$ as $W^\dagger_s{(\partial A(2R))}$ completes the proof. \end{proof} \end{lemma}
\section{Bounding the Entanglement Entropy:} At this point, we are ready to apply Lemma~\ref{unitary_decomposition} to study the entanglement entropy of states adiabatically connected to an initial state $\ket{\psi_0}$, whose Schmidt coefficients across a cut $A:A^c$ satisfy a certain rapid-decay condition. In particular, we assume that for a rapidly-decaying function $f_A(\cdot)$, with $f_A(0)=1$ and $A$ a convex subset of $\Lambda$: \begin{equation}\label{init_decay}
\sum_{\alpha \ge N^{R |\partial A|} + 1} \sigma_0(\alpha) \le f_A(R), \quad R \in \mathbb{N}, \end{equation} where $N$ is the maximum single-site dimension and $\ket{\psi_0} = \sum_{\alpha \ge 1} \sqrt{\sigma_0(\alpha)} \ket{\psi_{{A},0}(\alpha)}\otimes \ket{\psi_{{A^c},0}(\alpha)},$ the Schmidt decomposition of $\ket{\psi_0}$ across the cut $A:A^c$, with Schmidt coefficients in decreasing order.
The proof follows an argument similar to the one found in \cite{1d_area_law}. First, we approximate the initial state $\ket{\psi_0}$ with the family of states $\{\ket{\psi_{0,R}}\}_{R\ge 0}$, where: $$
\ket{\psi_{0,R}} = \frac{1}{\sqrt{c_R}} \sum_{\alpha = 1}^{N^{R\,|\partial A|}} \sqrt{\sigma_0(\alpha)} \ket{\psi_{{A},0}(\alpha)}\otimes \ket{\psi_{{A^c},0}(\alpha)} $$
and $c_R = \sum_{\alpha \ge 1}^{N^{R |\partial A|}} \sigma_0(\alpha) \ge 1-f_A(R)$, where we used (\ref{init_decay}).
The next step is to construct states with bounded Schmidt rank and increasing overlap to the adiabatically evolved state $\ket{\psi_0(s)} = U_s\ket{\psi_0}$. To accomplish this, we define the family of states: $$\ket{\psi_{0,R}(s)} \equiv U_s(A) \otimes U_s(A^c) \, U_s(\partial A(2R)) \ket{\psi_{0,R}}.$$
Setting the overlap $P(R) := |\braket{\psi_0(s)}{\psi_{0,R}(s)}|^2$ to be, say, at least $1/2$, fixes the minimum boundary thickness $R_0$ we consider in our approximation. Then, we use Lemma~\ref{lem:ent_bound} to show that the entanglement entropy of $\ket{\psi_0(s)}$ satisfies a non-trivial bound for all $s \in [0,1]$.
\begin{theorem}\label{thm:area-law} For the gapped family of Hamiltonians $H(s)$ of Definition~\ref{defn:path}, let $\ket{\psi_0(s)}$ denote the groundstate of $H(s)$. Then, if the groundstate $\ket{\psi_0(0)}$ satisfies the decay condition (\ref{init_decay}), the entropy of $\rho_s(A) = \mathop{\mathrm{Tr}}_{A^c} \pure{\psi_0(s)}$ is bounded as follows: \begin{equation}\label{bnd:entanglement}
S(\rho_s(A)) \le 5\, (1+c_1)\, R_0\, |\partial A| + h_1, \quad R_0 = \min\{R\in \mathbb{N}: f_A(R_0) + 2 \epsilon_s(R_0) \le 1/2\}, \end{equation} where $c_1 = \sum_{n\ge 1} n \delta(n)$ and $h_1 = - \sum_{n\ge 0} \delta(n) \, \ln \delta(n)$, with $$\delta(n) = \big(f_A(n R_0)- f_A((n+1) R_0)\big) + 2 \big(\epsilon_s(n R_0)-\epsilon_s((n+1) R_0)\big)$$ and $\epsilon_s(\cdot)$ defined in \ref{bnd:decomp_error}. \end{theorem} Note that $c_1$ is a constant, as long as $\delta(n)$ decays faster than $n^{-(2+\epsilon)}$. Moreover, $h_1 = - \sum_{n\ge 1} \delta(n) \, \ln \delta(n) \le c_1 \ln t$, for $\delta(n) \ge t^{- n},\, n \ge 1,\, t > 1$ and $h_1 \le 1/\ln t$ if $\delta(n) \le t^{-n}$ for $n\ge 1$ and $t > 1$.
\begin{proof} Fix $s \in [0,1]$ and let $\rho_s(A) = \mathop{\mathrm{Tr}}_{A^c} \pure{\psi_0(s)}$. Since we will be using the Schmidt decomposition of the state $\ket{\psi_0(s)}$, let us introduce it here: $$\ket{\psi_0(s)} = \sum_{\alpha \ge 1} \sqrt{\sigma_s(\alpha)} \ket{\psi_{{A},s}(\alpha)}\otimes \ket{\psi_{{A^c},s}(\alpha)},$$ where $\sum_{\alpha} \sigma_s(\alpha) = 1$ and $\{\ket{\psi_{A,s}(\alpha)}\}$, $\{\ket{\psi_{A^c,s}(\alpha)}\}$ form orthonormal sets supported on $A$ and $A^c$, respectively. We order the Schmidt coefficients of $\ket{\psi_0(s)}$ in decreasing order such that $\alpha<\beta \implies \sigma_s(\alpha) \geq \sigma_s(\beta)$. Moreover, we note that: \begin{equation} \rho_s(A) = \sum_{\alpha \ge 1} \sigma_s(\alpha) \pure{\psi_{A,s}(\alpha)} \qquad \mbox{ and } \qquad S(\rho_s(A)) = - \sum_{\alpha \ge 1} \sigma_s(\alpha) \ln \sigma_s(\alpha). \end{equation}
We define: \begin{equation}\label{overlap}
P(R) \equiv |\braket{\psi_0(s)}{\psi_{0,R}(s)}|^2, \quad \Delta_s(R) = U_s(A) \otimes U_s(A^c) \, U_s(\partial A(2R)) - U_s \end{equation} and use Lemma~\ref{unitary_decomposition} in order to relate $P(R)$ with the sub-exponentially decaying error $\epsilon_s(R)$: \begin{equation}\label{P:small}
P(R) = |\bra{\psi_0}\one + U^\dagger_s\Delta_s(R)\ket{\psi_{0,R}}|^2 \ge |\braket{\psi_0}{\psi_{0,R}}|^2 -2\|\Delta_s(R)\| \ge 1-(f_A(R)+2\epsilon_s(R)), \end{equation} noting that $\braket{\psi_0}{\psi_{0,R}} = \sqrt{c_R}$. Setting $R_0$ to be the smallest $R$ such that $f_A(R)+2\epsilon_s(R) \le 1/2$, the decay of $f_A(R)$ and $\epsilon_s(R)$ implies: \begin{equation}\label{P:large} R \ge R_0 \implies P(R) \ge 1/2.
\end{equation} Now, the state $\ket{\psi_{0,R}(s)}$ has Schmidt rank bounded by $k_R \cdot N^{4\,R \, |\partial A|}$, where $k_R = N^{R\, |\partial A|}$ is the Schmidt rank of $\ket{\psi_{0,R}}$. To see this, first note that the Schmidt rank of $U_s(\partial A(2R)) \ket{\psi_{{A},0}(\alpha)}\otimes \ket{\psi_{{A^c},0}(\alpha)}$ is the same as that of $U_s(A)\otimesU_s(A^c) \,U_s(\partial A(2R)) \ket{\psi_{{A},0}(\alpha)}\otimes \ket{\psi_{{A^c},0}(\alpha)}$, along the boundary of $A$. It remains to see how the action of $U_s(\partial A(2R))$ on $\ket{\psi_{{A},0}(\alpha)}\otimes \ket{\psi_{{A^c},0}(\alpha)}$ affects the Schmidt rank.
Since $U_s(\partial A(2R))$ is an operator acting non-trivially only on sites in a subset of $\partial A(2R)$, we have the following general decomposition: \begin{equation}\label{decomp} U_s(\partial A(2R)) = \sum_{\alpha,\beta=1}^{D_I} \one_{A\setminus I_{A}(2R)} \otimes E(\alpha,\beta) \otimes G(\alpha,\beta) \otimes \one_{(A^c) \setminus E_{A}(2R)}, \end{equation}
where the matrices $G(\alpha,\beta)$ act on sites in $E_{A}(2R)$ and the matrix units $E(\alpha,\beta)$, which act non-trivially on $I_{A}(2R)$, form an orthonormal basis for $D_I \times D_I$ matrices. Moreover, $D_I \le N^{|I_{A}(2R)|} \le N^{2R |\partial A|}$, for convex $A$ and $N$ the maximum dimension among the single-site state spaces (e.g. $N=2$ for a system of qubits).
To bound the Schmidt rank of $$\ket{\psi_{B,0}(\gamma)} := U_s(\partial A(2R)) \ket{\psi_{{A},0}(\gamma)}\otimes \ket{\psi_{{A^c},0}(\gamma)},$$ we trace over sites in $A$ and study the rank of the operator: \begin{equation}\label{Schmidt_bound} {\mathop{\mathrm{Tr}}}_{A} \left(\pure{\psi_{B,0}(\gamma)}\right) = \sum_{\alpha=1}^{D_I}\left[ \sum_{\beta,\beta'=1}^{D_I} c(\beta,\beta')\, \ketbra{F_{\alpha}(\beta,\gamma)}{F_{\alpha}(\beta',\gamma)}\right], \end{equation} with \begin{equation*} \ket{F_{\alpha}(\beta,\gamma)} = G(\alpha,\beta)\otimes \one_{A^c) \setminus E_{A}(2R)} \ket{\psi_{A^c,0}(\gamma)} \end{equation*} and \begin{equation*} c(\beta,\beta') = \bra{\psi_{A,0}(\gamma)} \one_{A\setminus I_{A}(2R)} \otimes E(\beta',\beta) \ket{\psi_{A,0}(\gamma)}, \end{equation*} coming from (\ref{decomp}).
Clearly, as a sum of $D_I$ matrices each with rank at most $D_I$, the above operator has rank bounded by $D_I^2$ and, hence, the Schmidt rank of $U_s(\partial A(2R))\ket{\psi_A(\gamma,\delta)}$ is bounded above by $N^{4 R \,|\partial A|}$. Now, the final step is to note that if $\ket{\psi_{0,R}}$ has Schmidt rank $k_R$, then there are exactly $k_R$ states $\ket{\psi_{B,0}(\gamma)}$, each with Schmidt rank at most $N^{4 R \,|\partial A|}$, and the bound for the Schmidt rank of the approximation $\ket{\psi_{0,R}(s)} = \frac{1}{\sqrt{c_R}}\sum_{\gamma = 1}^{k_R} \sqrt{\sigma_0(\gamma)} \, U_s(A)\otimesU_s(A^c)\ket{\psi_{B,0}(\gamma)}$ follows.
The next step is to relate the overlap $P(R)$ to the Schmidt coefficients of $\ket{\psi_0(s)}$. More specifically, we will now show that for $R \ge 0$: \begin{equation} \label{constraint1}
\sum_{\alpha \le N^{5 R |\partial A|}} \sigma_s(\alpha) \geq P(R). \end{equation} To prove this, first introduce the Schmidt decomposition of $\ket{\psi_{0,R}(s)}$: \begin{equation}
\ket{\psi_{0,R}(s)} =\sum_{\beta = 1}^r \sqrt{\tau_{\beta}} \ket{\Phi_{A}(\beta)} \otimes \ket{\Phi_{A^c}(\beta)}, \mbox{ with } \sum_{\beta=1}^r \tau_{\beta} = 1 \mbox{ and } r \le N^{5R|\partial A|}, \end{equation} as we have already demonstrated. For notational convenience, let
$M_A(\alpha,\beta) = \left|\braket{\Phi_{A}(\beta)}{\psi_{{A},s}(\alpha)}\right|$ and $M_{A^c}(\alpha,\beta) = \left|\braket{\Phi_{A^c}(\beta)}{\psi_{{A^c},s}(\alpha)}\right|.$ Note that since $\{\ket{\Phi_{A}(\beta)}\}$, $\{\ket{\Phi_{A^c}(\beta)}\}$, $\{\ket{\psi_{A,s}(\alpha)}\}$ and $\{\ket{\psi_{A^c,s}(\alpha)}\}$ are orthonormal sets, Bessel's inequality implies that $$\sum_{\beta=1}^r M_A(\alpha,\beta)^2 \le 1 \mbox{ and } \, \sum_{\alpha} M_{A^c}(\alpha,\beta)^2 \le 1,$$ as well as $$\sum_{\alpha} M_A(\alpha,\beta)^2 \le 1 \implies \sum_{\alpha, \beta} M_A(\alpha,\beta)^2 \le r.$$
Then, an application of the triangle inequality followed by Cauchy-Schwarz gives the following upper bound for $P(R) = |\braket{\psi_{0,R}(s)}{\psi_{0}(s)}|^2:$ \begin{eqnarray*}
|\braket{\psi_{0,R}(s)}{\psi_{0}(s)}|^2 &\le&\left(\sum_{\alpha,\beta} \sqrt{\sigma_s(\alpha)}\sqrt{\tau(\beta)}\, M_A(\alpha,\beta) M_{A^c}(\alpha,\beta) \right)^2 \\ &\le&\left(\sum_{\alpha,\beta}\sigma_s(\alpha) M_A(\alpha,\beta)^2 \right) \, \left(\sum_{\alpha,\beta} \tau(\beta) M_{A^c}(\alpha,\beta)^2 \right) \\ &\le&\sum_{\alpha \le r} \sigma_s(\alpha). \end{eqnarray*} The last inequality follows from Schur convexity of $f([p(\alpha)]) = \sum_{\alpha} \sigma_s(\alpha) \, p(\alpha)$ and the observation that the vector $[1,1,\ldots,1,0,\ldots,0]$, with at most $r$ ones, majorizes $\left[\sum_{\beta} M_A(1,\beta)^2,\sum_{\beta} M_A(2,\beta)^2,\ldots\right]$.
To see that $f([p(\alpha)])$ is Schur convex, note that if we set $S_p(\alpha) = \sum_{k=1}^{\alpha} p(k)$ and $\Delta(\alpha,\beta) = \sigma_s(\alpha)-\sigma_s(\beta)$ then the condition that $[p(\alpha)]$ majorizes $[q(\alpha)]$ ($p \succeq q$) becomes $p \succeq q \Leftrightarrow S_p(\alpha) \ge S_q(\alpha),\, \forall \alpha$. Moreover, $$f(\{p(\alpha)\})-f(\{q(\alpha)\}) = \sum_{\alpha} \Delta(\alpha,\alpha+1) \left(S_p(\alpha)-S_q(\alpha)\right),$$ which is non-negative since we have arranged the $\sigma_s(\alpha)$ in decreasing order so that $\Delta(\alpha,\alpha+1) \ge 0, \, \forall \alpha$.
Now that we have demonstrated (\ref{constraint1}), we may use it in combination with (\ref{P:large}) to show that $S(\rho_s(A))$ satisfies an entropy bound. Using (\ref{P:small}-\ref{P:large}) and (\ref{constraint1}), we have for $n \ge 1$: \begin{equation} \label{constraint2}
\sum_{\alpha\geq N^{5nR_0 \, |\partial A|}+1} \sigma_s(\alpha) \leq 1-P(nR_0) \le f_A(nR_0) + 2 \epsilon_s(nR_0). \end{equation} We now maximize the entropy $$S(\rho_s(A))=-\sum_{\alpha=1} \sigma_s(\alpha) \ln(\sigma_s(\alpha))$$
subject to the constraint (\ref{constraint2}). Following the notation of Lemma~\ref{lem:ent_bound}, set $s_n = N^{5\,nR_0|\partial A|}, \, n \ge 1$ and $f(n) = f_A(nR_0)+2\epsilon_s(nR_0),\, n \ge 1$ and $f(0) = 1$, noting that $f(1) < f(0)$, by our choice of $R_0$. Then, for $r = N^{5R_0\,|\partial A|}$ and $s_1 = N^{5\,R_0|\partial A|}$, Lemma~\ref{lem:ent_bound} implies that: \begin{equation}\label{CaseI:bound}
S(\rho_s(A)) \le 5 (1+c_1) R_0 \, |\partial A| + h_1, \end{equation} where $c_1 = \sum_{n\ge 1} n \delta(n)$ and $h_1 = - \sum_{n\ge 0} \delta(n) \, \ln \delta(n)$, with $$\delta(n) = (f_A(nR_0)-f_A((n+1)R_0)) + 2 (\epsilon_s(nR_0)-\epsilon_s((n+1)R_0)).$$ \end{proof} We turn, now, to the following Lemma, which relates the decay properties of the entanglement spectrum to a bound on the entropy of entanglement: \begin{lemma}[{\bf Entropy bound}]\label{lem:ent_bound} For $\rho$ a density matrix, let $\sum_{\alpha \ge 1} \sigma(\alpha) \pure{\psi(\alpha)}$ be its spectral decomposition, with $\sigma(\alpha)$ in decreasing order. Assume that there is an increasing sequence of integers $\{s_n\}_{n\ge 0}$ with $s_n < s_{n+1} \le r\,s_n$, for $n\ge 1$ and $s_1 > s_0 = 0$, such that the following constraint holds for a strictly-decaying function $f(n)$, with $f(0)=1$: \begin{equation} \sum_{\alpha \ge s_n + 1} \sigma(\alpha) \le f(n), \quad n \ge 0.\label{constraint} \end{equation} Then, the entropy of $\rho$, given by $S(\rho) = -\sum_{\alpha \ge 1} \sigma(\alpha) \ln \sigma(\alpha)$, satisfies the following bound: $$S(\rho) \le \ln s_1 + c_1 \ln r +h_1,$$ where $c_1 = \sum_{n\ge 1} n \delta(n)$ and $h_1 = - \sum_{n\ge 0} \delta(n) \, \ln \delta(n)$, with $\delta(n) = f(n)-f(n+1)$. \begin{proof} Since the Shannon entropy is Schur-concave, the von Neumann entropy $S(\rho)$ is bounded above by the Shannon entropy $H(\overrightarrow{\mu})$ of any probability distribution $\{\mu(\alpha)\}_{\alpha\ge 1}$ consistent with the constraint (\ref{constraint}), that is majorized by $\{\sigma(\alpha)\}_{\alpha\ge 1}$ (i.e. $\overrightarrow{\mu} \preceq \overrightarrow{\sigma}$.)
Define $\delta(n) = f(n)-f(n+1), \, n \ge 0$. For $1\le \alpha \le s_1$, the constraint $\sum_{\alpha \ge 1} \sigma(\alpha) = 1$ implies: $$\sum_{\alpha=1}^{s_1} \sigma(\alpha) \ge f(0)-f(1) \implies \sum_{\alpha=1}^{s_1} \mu(\alpha) = \delta(0) \implies \mu(\alpha) = \frac{\delta(0)}{s_1}, \quad 1\le \alpha \le s_1.$$ Similarly, for $\alpha \in [s_n+1, s_{n+1}], \, n \ge 1,$ we see that we should choose $\overrightarrow{\mu}$ to satisfy: \begin{equation*} \sum_{\alpha=s_n+1}^{s_{n+1}} \mu(\alpha)= \, f(n)-f(n+1) \implies \mu(\alpha) = \frac{\delta(n)}{s_{n+1}-s_n}. \end{equation*} Gathering terms, we decompose the Shannon entropy $H(\overrightarrow{\mu})$ as $\sum_{n = 0}^{\infty} H_n(\overrightarrow{\mu})$, where we have defined $H_n(\overrightarrow{\mu}) = - \sum_{\alpha = s_n+1}^ {s_{n+1}} \mu(\alpha) \ln \mu(\alpha),\, n\ge 0$. We can bound each $H_n(\overrightarrow{\mu})$ as follows, recalling that $s_{n+1} \le s_1 r^n$ and, hence, $s_{n+1} - s_n \le s_1 r^n$: \begin{equation} H_n(\overrightarrow{\mu}) \le - \delta(n) \, \ln \delta(n) + n \delta(n) \ln r + \delta(n) \ln s_1, \quad n\ge 0. \end{equation} Using $\sum_{n\ge 0} \delta(n) =1$, we get the bound: \begin{eqnarray*} H(\overrightarrow{\mu}) &\le& \ln s_1 + c_1 \ln r + h_1, \end{eqnarray*} where we defined $c_1 = \sum_{n\ge 1} n \, \delta(n)$ and $h_1 = - \sum_{n\ge 0} \delta(n) \, \ln \delta(n)$. This completes the proof of the Lemma. \end{proof} \end{lemma} \section{Conclusion} We have shown that states satisfying an area law are connected via gapped, local Hamiltonian paths to states that satisfy a similar bound on their entropy of entanglement. This result holds for lattice systems in any dimension. Since the above result relies only on Lieb-Robinson bounds and locality properties of the {\it quasi-adiabatic evolution}, it can be shown that even gapped paths with long-range interactions satisfy the bound~(\ref{bnd:entanglement}), as long as the decay is fast enough (i.e. power-law decay with a dimension-dependent exponent.) We hope this work will motivate interested readers to work out the details of this argument, which would have important implications for the classification of phases, given the connection of states satisfying an area-law to Matrix Product States (MPS) in $1$D and Projected Entangled Pair States (PEPS) in $2$D~\cite{class_phases_1d, schuch_class_phases}.
\noindent {\bf Acknowledgements:} The author would like to thank M. B. Hastings for helpful remarks on the use of quasi-adiabatic evolution in the context of entanglement entropy, as well as fruitful discussions with B. Nachtergaele, N. Schuch and A. Gorshkov.
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>Journal of Fluid Mechanics
>Volume 854
>The energy flux spectrum of internal waves generated...
The energy flux spectrum of internal waves generated by turbulent convection
Published online by Cambridge University Press: 10 September 2018
Louis-Alexandre Couston [Opens in a new window] ,
Daniel Lecoanet ,
Benjamin Favier [Opens in a new window] and
Michael Le Bars
Louis-Alexandre Couston*
CNRS, Aix Marseille Univ, Centrale Marseille, IRPHE, Marseille, France
Daniel Lecoanet
Princeton Center for Theoretical Science, Princeton, NJ 08544, USA
Benjamin Favier
†Email address for correspondence: [email protected]
We present three-dimensional direct numerical simulations of internal waves excited by turbulent convection in a self-consistent, Boussinesq and Cartesian model of mixed convective and stably stratified fluids. We demonstrate that in the limit of large Rayleigh number ( $Ra\in [4\times 10^{7},10^{9}]$) and large stratification (Brunt–Väisälä frequencies $f_{N}\gg f_{c}$, where $f_{c}$ is the convective frequency), simulations are in good agreement with a theory that assumes waves are generated by Reynolds stresses due to eddies in the turbulent region as described in Lecoanet & Quataert (Mon. Not. R. Astron. Soc., vol. 430 (3), 2013, pp. 2363–2376). Specifically, we demonstrate that the wave energy flux spectrum scales like $k_{\bot }^{4}\,f^{-13/2}$ for weakly damped waves (with $k_{\bot }$ and $f$ the waves' horizontal wavenumbers and frequencies, respectively), and that the total wave energy flux decays with $z$, the distance from the convective region, like $z^{-13/8}$.
JFM classification
Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Internal waves Turbulent Flows: Turbulent convection
JFM Rapids
Journal of Fluid Mechanics , Volume 854 , 10 November 2018 , R3
DOI: https://doi.org/10.1017/jfm.2018.669[Opens in a new window]
© 2018 Cambridge University Press
Alexander, M. J., Geller, M., McLandress, C., Polavarapu, S., Preusse, P., Sassi, F., Sato, K., Eckermann, S., Ern, M., Hertzog, A., Kawatani, Y., Pulido, M., Shaw, T. A., Sigmond, M., Vincent, R. & Watanabe, S. 2010 Recent developments in gravity-wave effects in climate models and the global distribution of gravity-wave momentum flux from observations and models. Q. J. R. Meteorol. Soc. 136 (650), 1103–1124.Google Scholar
Ansong, J. K. & Sutherland, B. R. 2010 Internal gravity waves generated by convective plumes. J. Fluid Mech. 648, 405.CrossRefGoogle Scholar
Bordes, G., Venaille, A., Joubaud, S., Odier, P. & Dauxois, T. 2012 Experimental observation of a strong mean flow induced by internal gravity waves. Phys. Fluids 24 (8), 086602.CrossRefGoogle Scholar
van den Bremer, T. S. & Sutherland, B. R. 2018 The wave-induced flow of internal gravity wavepackets with arbitrary aspect ratio. J. Fluid Mech. 834, 385–408.CrossRefGoogle Scholar
Burns, K. J., Vasil, G. M., Oishi, J. S., Lecoanet, D. & Brown, B. P. 2018 Dedalus: Flexible Framework for Spectrally Solving Differential Equations. Astrophysics Source Code Library.Google Scholar
Canet, L., Rossetto, V., Wschebor, N. & Balarac, G. 2017 Spatiotemporal velocity–velocity correlation function in fully developed turbulence. Phys. Rev. E 95, 023107.Google ScholarPubMed
Carruthers, D. J. & Hunt, J. C. R. 1986 Velocity fluctuations near an interface between a turbulent region and a stably stratified layer. J. Fluid Mech. 165, 475–501.CrossRefGoogle Scholar
Chen, S. & Kraichnan, R. H. 1989 Sweeping decorrelation in isotropic turbulence. Phys. Fluids A 1 (12), 2019–2024.CrossRefGoogle Scholar
Chevillard, L., Roux, S. G., Lévêque, E., Mordant, N., Pinton, J.-F. & Arnéodo, A. 2005 Intermittency of velocity time increments in turbulence. Phys. Rev. Lett. 95, 064501.CrossRefGoogle ScholarPubMed
Couston, L.-A., Lecoanet, D., Favier, B. & Le Bars, M. 2017 Dynamics of mixed convective–stably-stratified fluids. Phys. Rev. Fluids 2, 094804.CrossRefGoogle Scholar
Couston, L.-A., Lecoanet, D., Favier, B. & Le Bars, M. 2018 Order out of chaos: slowly reversing mean flows emerge from turbulently generated internal waves. Phys. Rev. Lett. 120, 244505.CrossRefGoogle ScholarPubMed
Favier, B., Godeferd, F. S. & Cambon, C. 2010 On space and time correlations of isotropic and rotating turbulence. Phys. Fluids 22 (1), 015101.CrossRefGoogle Scholar
Garaud, P. 2018 Double-diffusive convection at low Prandtl number. Annu. Rev. Fluid Mech. 50 (1), 275–298.CrossRefGoogle Scholar
Goldreich, P. & Kumar, P. 1990 Wave generation by turbulent convection. Astrophys. J. 363, 694–704.CrossRefGoogle Scholar
Grisouard, N. & Bühler, O. 2012 Forcing of oceanic mean flows by dissipating internal tides. J. Fluid Mech. 708, 250–278.CrossRefGoogle Scholar
Kunze, E. 2017 Internal-wave-driven mixing: global geography and budgets. J. Phys. Oceanogr. 47 (6), 1325–1345.CrossRefGoogle Scholar
Lecoanet, D., Le Bars, M., Burns, K. J., Vasil, G. M., Brown, B. P., Quataert, E. & Oishi, J. S. 2015 Numerical simulations of internal wave generation by convection in water. Phys. Rev. E 91 (6), 063016.CrossRefGoogle ScholarPubMed
Lecoanet, D. & Quataert, E. 2013 Internal gravity wave excitation by turbulent convection. Mon. Not. R. Astron. Soc. 430 (3), 2363–2376.CrossRefGoogle Scholar
Liot, O., Seychelles, F., Zonta, F., Chibbaro, S., Coudarchet, T., Gasteuil, Y., Pinton, J.-F., Salort, J. & Chillà, F. 2016 Simultaneous temperature and velocity Lagrangian measurements in turbulent thermal convection. J. Fluid Mech. 794, 655–675.CrossRefGoogle Scholar
Munroe, J. R. & Sutherland, B. R. 2014 Internal wave energy radiated from a turbulent mixed layer. Phys. Fluids 26 (9), 096604.CrossRefGoogle Scholar
Pinçon, C., Belkacem, K. & Goupil, M. J. 2016 Generation of internal gravity waves by penetrative convection. Astron. Astrophys. 588 (A122), 1–21.CrossRefGoogle Scholar
Rogers, T. M., Lin, D. N. C. & Lau, H. H. B. 2012 Internal gravity waves modulate the apparent misalignment of exoplanets around hot stars. Astrophys. J. Lett. 758 (1), L6.CrossRefGoogle Scholar
Rogers, T. M., Lin, D. N. C., McElwaine, J. N. & Lau, H. H. B. 2013 Internal gravity waves in massive stars: angular momentum transport. Astrophys. J. 772 (1), 21.CrossRefGoogle Scholar
Sano, M., Wu, X. Z. & Libchaber, A. 1989 Turbulence in helium-gas free convection. Phys. Rev. A 40, 6421–6430.CrossRefGoogle ScholarPubMed
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: from instabilities to turbulence. Annu. Rev. Fluid Mech. 34 (1), 559–593.CrossRefGoogle Scholar
Taylor, J. R. & Sarkar, S. 2007 Internal gravity waves generated by a turbulent bottom Ekman layer. J. Fluid Mech. 590, 331–354.CrossRefGoogle Scholar
Tennekes, H. 1975 Eulerian and Lagrangian time microscales in isotropic turbulence. J. Fluid Mech. 67 (3), 561–567.CrossRefGoogle Scholar
Thorpe, S. A. 2018 Models of energy loss from internal waves breaking in the ocean. J. Fluid Mech. 836, 72–116.CrossRefGoogle Scholar
Zhou, Y. & Rubinstein, R. 1996 Sweeping and straining effects in sound generation by high Reynolds number isotropic turbulence. Phys. Fluids 8 (3), 647–649.CrossRefGoogle Scholar
Couston et al supplementary material
Movie of (a) $w(y = 0)$, (b) $T_z-\bar{T}_z$ at $y = 0$ (overbar denotes x average), (c) $w(z = 0.7)$, (d) $w(z = 1:3)$ for simulation case $C_8^{400}$. Variables in the wave region $(z > 1)$ in (a), (b) have been multiplied by $10^4$, $10^3$, respectively.
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Burns, Keaton J. Vasil, Geoffrey M. Oishi, Jeffrey S. Lecoanet, Daniel and Brown, Benjamin P. 2020. Dedalus: A flexible framework for numerical simulations with spectral methods. Physical Review Research, Vol. 2, Issue. 2,
Augustson, K. C. Mathis, S. and Astoul, A. 2020. A Model of Rotating Convection in Stellar and Planetary Interiors. II. Gravito-inertial Wave Generation. The Astrophysical Journal, Vol. 903, Issue. 2, p. 90.
Khater, Mostafa M. A. Zheng, Qiang Qin, Haiyong Attia, Raghda A. M. and Chen, Chuanjun 2020. On Highly Dimensional Elastic and Nonelastic Interaction between Internal Waves in Straight and Varying Cross-Section Channels. Mathematical Problems in Engineering, Vol. 2020, Issue. , p. 1.
Gossan, Sarah E Fuller, Jim and Roberts, Luke F 2020. Wave heating from proto-neutron star convection and the core-collapse supernova explosion mechanism. Monthly Notices of the Royal Astronomical Society, Vol. 491, Issue. 4, p. 5376.
Léard, P. Favier, B. Le Gal, P. and Le Bars, M. 2020. Coupled convection and internal gravity waves excited in water around its density maximum at 4°C. Physical Review Fluids, Vol. 5, Issue. 2,
Bowman, D. M. Burssens, S. Simón-Díaz, S. Edelmann, P. V. F. Rogers, T. M. Horst, L. Röpke, F. K. and Aerts, C. 2020. Photometric detection of internal gravity waves in upper main-sequence stars. Astronomy & Astrophysics, Vol. 640, Issue. , p. A36.
Jermyn, Adam S Chitre, Shashikumar M Lesaffre, Pierre and Tout, Christopher A 2020. Convective differential rotation in stars and planets – II. Observational and numerical tests. Monthly Notices of the Royal Astronomical Society, Vol. 498, Issue. 3, p. 3782.
Lecoanet, Daniel 2020. Fluid Mechanics of Planets and Stars. Vol. 595, Issue. , p. 31.
Couston, Louis-Alexandre Lecoanet, Daniel Favier, Benjamin and Le Bars, Michael 2020. Shape and size of large-scale vortices: A generic fluid pattern in geophysical fluid dynamics. Physical Review Research, Vol. 2, Issue. 2,
Le Bars, Michael Couston, Louis-Alexandre Favier, Benjamin Léard, Pierre Lecoanet, Daniel and Le Gal, Patrice 2020. Fluid dynamics of a mixed convective/stably stratified system—A review of some recent works. Comptes Rendus. Physique, Vol. 21, Issue. 2, p. 151.
Kooloth, Parvathi Sondak, David and Smith, Leslie M. 2021. Coherent solutions and transition to turbulence in two-dimensional Rayleigh-Bénard convection. Physical Review Fluids, Vol. 6, Issue. 1,
Wu, Samantha and Fuller, Jim 2021. A Diversity of Wave-driven Presupernova Outbursts. The Astrophysical Journal, Vol. 906, Issue. 1, p. 3.
Couston, Louis-Alexandre and Siegert, Martin 2021. Dynamic flows create potentially habitable conditions in Antarctic subglacial lakes. Science Advances, Vol. 7, Issue. 8,
Louis-Alexandre Couston (a1), Daniel Lecoanet (a2), Benjamin Favier (a1) and Michael Le Bars (a1)
DOI: https://doi.org/10.1017/jfm.2018.669 | CommonCrawl |
# Distance metrics and their applications
One common distance metric is the Euclidean distance. It measures the straight-line distance between two points in a Euclidean space. The formula for Euclidean distance is:
$$d(x, y) = \sqrt{\sum_{i=1}^{n} (x_i - y_i)^2}$$
where $x$ and $y$ are the two points, and $n$ is the number of dimensions.
Another distance metric is the Manhattan distance, also known as the L1 distance. It measures the distance between two points in a grid-like space. The formula for Manhattan distance is:
$$d(x, y) = \sum_{i=1}^{n} |x_i - y_i|$$
where $x$ and $y$ are the two points, and $n$ is the number of dimensions.
The Minkowski distance is a generalization of both the Euclidean and Manhattan distances. It is defined as:
$$d(x, y) = \left(\sum_{i=1}^{n} |x_i - y_i|^p\right)^{1/p}$$
where $x$ and $y$ are the two points, $n$ is the number of dimensions, and $p$ is a positive integer.
When working with discrete features, such as word counts in documents, cosine similarity can be a more appropriate distance metric. Cosine similarity measures the cosine of the angle between two vectors. The formula for cosine similarity is:
$$\text{cosine similarity} = \frac{a \cdot b}{\|a\|\|b\|}$$
where $a$ and $b$ are the two vectors, and $\|a\|$ and $\|b\|$ are their magnitudes.
## Exercise
Calculate the Euclidean distance between the points (2, 3) and (5, 7).
The Euclidean distance between the points (2, 3) and (5, 7) is:
$$d((2, 3), (5, 7)) = \sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$
## Exercise
Calculate the Manhattan distance between the points (2, 3) and (5, 7).
The Manhattan distance between the points (2, 3) and (5, 7) is:
$$d((2, 3), (5, 7)) = |2 - 5| + |3 - 7| = 3 + 4 = 7$$
In conclusion, the choice of distance metric is crucial in KNN algorithms. Different metrics can have different properties and suit different types of data. It is important to understand the properties of each metric and choose the one that best fits the problem at hand.
# Classification with KNN
The KNN algorithm works by finding the K nearest neighbors of a given query point in the feature space. The query point is then assigned to the class that has the majority of its neighbors.
For example, consider a dataset with two classes of points. The KNN algorithm would find the K nearest neighbors of a query point and assign the query point to the class that has the majority of its neighbors.
The choice of K is an important parameter in the KNN algorithm. A small value of K can lead to overfitting, while a large value can lead to underfitting. The optimal value of K can be determined through cross-validation or other techniques.
In addition to the choice of K, the distance metric used in the KNN algorithm can also have a significant impact on its performance. Different metrics can be more appropriate for different types of data.
In summary, KNN is a powerful algorithm for classification tasks. The choice of K and the distance metric are important parameters that need to be carefully considered to achieve optimal performance.
# Regression with KNN
K-nearest neighbors (KNN) can also be used for regression tasks, where the goal is to predict a continuous value for a given query point.
In regression, the KNN algorithm works by finding the K nearest neighbors of a given query point and predicting the value of the query point based on the average or weighted average of its neighbors.
For example, consider a dataset with two classes of points. The KNN algorithm would find the K nearest neighbors of a query point and predict the value of the query point based on the average values of its neighbors.
The choice of K and the distance metric used in the KNN algorithm are important parameters in regression tasks as well. The optimal values can be determined through cross-validation or other techniques.
In summary, KNN can be used for regression tasks by predicting the value of a query point based on the average or weighted average of its K nearest neighbors. The choice of K and the distance metric are important parameters that need to be carefully considered to achieve optimal performance.
# Outlier detection using KNN
In outlier detection, the KNN algorithm works by finding the K nearest neighbors of a given query point and determining whether the query point is an outlier based on its distance to its neighbors.
For example, consider a dataset with two classes of points. The KNN algorithm would find the K nearest neighbors of a query point and determine whether the query point is an outlier based on its distance to its neighbors.
The choice of K and the distance metric used in the KNN algorithm are important parameters in outlier detection. The optimal values can be determined through cross-validation or other techniques.
In summary, KNN can be used for outlier detection by determining whether a query point is an outlier based on its distance to its K nearest neighbors. The choice of K and the distance metric are important parameters that need to be carefully considered to achieve optimal performance.
# Real-world examples of KNN
K-nearest neighbors (KNN) has been used successfully in various real-world applications, such as image recognition, natural language processing, and bioinformatics.
For example, KNN has been used to classify images of handwritten digits, recognize speech patterns, and classify genes based on their expression profiles.
In each of these applications, KNN has demonstrated its ability to effectively handle high-dimensional data, handle noise and outliers, and achieve competitive performance compared to other machine learning algorithms.
In conclusion, KNN has been a versatile and powerful algorithm in various real-world applications. Its ability to handle high-dimensional data, noise and outliers, and achieve competitive performance makes it a popular choice in many domains.
# Advantages and disadvantages of KNN
K-nearest neighbors (KNN) has several advantages and disadvantages.
Advantages of KNN include:
- It is a simple and intuitive algorithm that is easy to understand and implement.
- It can handle high-dimensional data and noisy data effectively.
- It does not require a training set with labels, which can be useful in unsupervised learning tasks.
- It can be used for both classification and regression tasks.
Disadvantages of KNN include:
- The choice of K and the distance metric can have a significant impact on its performance.
- KNN can be computationally expensive, especially for large datasets.
- KNN is sensitive to the presence of outliers and noise in the data.
- It can be difficult to interpret the results of the algorithm, as it does not provide a clear explanation for its predictions.
In conclusion, KNN has its strengths and weaknesses. Its simplicity and effectiveness in handling high-dimensional data and noise make it a popular choice in many applications. However, the choice of K and the distance metric, as well as its sensitivity to outliers and noise, can limit its effectiveness in certain scenarios.
# Future developments in KNN
Future developments in KNN may include:
- Developing more efficient algorithms for KNN, such as approximate nearest neighbor search and parallel KNN algorithms.
- Investigating the use of more advanced distance metrics, such as string-based distances for text data or kernel-based distances for high-dimensional data.
- Exploring the use of KNN in combination with other machine learning algorithms, such as ensemble methods or deep learning techniques, to improve its performance and robustness.
In conclusion, the future of KNN is promising, as researchers continue to explore ways to improve its performance and adapt it to new types of data and problems.
# Applications in various domains
K-nearest neighbors (KNN) has been applied to various domains, including:
- Image recognition: KNN has been used to classify images of handwritten digits, recognize objects in images, and classify genes based on their expression profiles.
- Natural language processing: KNN has been used for text classification, sentiment analysis, and machine translation.
- Bioinformatics: KNN has been used to classify genes based on their expression profiles, predict protein-protein interactions, and classify cancer subtypes based on gene expression data.
- Recommender systems: KNN has been used to recommend movies, books, or products based on user preferences and item features.
In conclusion, KNN has demonstrated its versatility and effectiveness in various domains. Its ability to handle high-dimensional data and noise, as well as its ease of implementation, make it a popular choice in many applications. | Textbooks |
interior point definition in calculus
For another example, in the real line with the usual topology every point is a limit point of $\Bbb Q$, and no point is an interior point of $\Bbb Q$. (points inside the set I mean) This article was adapted from an original article by S.M. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. See more. So shouldn't it read: We define the exterior of a set in terms of the interior of the set. A point in the interior of the domain of a function is a point of local maximum if the following holds: . $$x \in U \quad\exists \epsilon > 0 : B(x,\epsilon) \subset U?$$. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. So this is an interior point for my interval. Did something happen in 1987 that caused a lot of travel complaints? Math Open Reference. Colour rule for multiple buttons in a complex platform, What is an escrow and how does it work? If $x\in U$ is an interior point, regarding your definition, there exist $\epsilon >0$ such that $B(x,\epsilon )\subseteq U$. $r > 0$. • If A is a subset of a topological space X, then Ext ( A) ∩ Int ( A) = ϕ . However, if a set has a point inside it, surely it will always have a neighborhood (or a small ball) that will be contained in the set. Let (X, d) be a metric space with distance d: X × X → [0, ∞). Not true: consider $\Bbb R$ with Eucledian topology and a set $A = \{0\}$. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Interior angle definition, an angle formed between parallel lines by a third line that intersects them. Definition 5.1.5: Boundary, Accumulation, Interior, and Isolated Points Let S be an arbitrary set in the real line R. A point b R is called boundary point of S if every non-empty neighborhood of b intersects S and the complement of S. The set of all boundary points of S is called the boundary of S, denoted by bd (S). Maybe the clearest real-world examples are the state lines as you cross from one state to the next. yes, this is the point IMO. And for your graph, it is indeed a non-open set, since for instance $(0,1)$ is not an interior point according to your definition. So an interior point is a point that's not at the edge of my boundary. Practical example. To learn more, see our tips on writing great answers. Is interior of $A$ empty? The fmincon interior-point algorithm can accept a Hessian function as an input. Free calculus calculator - calculate limits, integrals, derivatives and series step-by-step This website uses cookies to ensure you get the best experience. A point. An exact location. It has no size, only position. When you supply a Hessian, you can obtain a faster, more accurate solution to a constrained minimization problem. Definition: The area between the rays that make up an angle, and extending away from the vertex to infinity. An interior pointis a point ~x in a set S for which there exists a ± neighborhood of ~x which only contains points which belong to S. DEFINITION: boundary point Try this Drag an orange dot. Derivatives help us! Should I tell someone that I intend to speak to their superior to resolve a conflict with them? Can light reach far away galaxies in an expanding universe? I would add "topological spaces" instead of just "spaces" to be more precise. OLS coefficients of regressions of fitted values and residuals on the original regressors. Is there a word for making a shoddy version of something just to get it working? In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. rev 2020.12.8.38145, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, An essence of $\subset$ and $\subseteq$ is the same. In fact, the set of limit points of $[0,1)$ is precisely the closed interval $[0,1]$. In general, for other spaces, it can make a difference. By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy. However, if a set has a point inside it, surely it will always have a neighborhood (or a small ball) that will be contained in the set. Why does arXiv have a multi-day lag between submission and publication? It only takes a minute to sign up. More formally, the definition of a closed interval is an interval that includes all of its limits. If I take the set $\{ (x,y)\in \mathbb{R}^n: y=1, x \in \mathbb{R}\}$, which is the constant function $y=1$ on the cartesian plane, would I say this is a We could try to find a general function that gives us the slope of the tangent line at any point. When we can say 0 and 1 in digital electronic? The derivative of a function gives the slope. The interior angles of a polygon and the method for calculating their values. We'd say it's continuous at an interior point. This is true for a subset [math]E[/math] of [math]\mathbb{R}^n[/math]. By using this website, you agree to our Cookie Policy. Points usually have a name, often a letter like "A" or "B" etc. Focus of a Parabola. What is the interior of a single point in a metric space? Then this would be the point x comma f of x. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. If x ∈ U is an interior point, regarding your definition, there exist ϵ > 0 such that B (x, ϵ) ⊆ U. No balls of positive radius around $0$ are contained in $A$. Asking for help, clarification, or responding to other answers. So let's say we have some arbitrary point. Well, if you consider all of the land in Georgia as the points belonging to the set called Georgia, then the boundary points of that set are exactly those points on the state lines, where Georgia transitions to Alabama or to South Carolina or Florida, etc. This example shows how to use derivative information to make the solution process faster and more robust. Combining 2 sections according to the reviewer's comment. There is also $\subsetneqq$. The second derivative tells us if the slope increases or decreases. In such (discrete) space every element $x\in X$ is an interior point of set $\{x\}$. such that. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. What were (some of) the names of the 24 families of Kohanim? $A^\circ$: interior of $A$. Notation Intervals are designated by writing the start point and end point as an ordered pair, within brackets. What piece is this and what is it's purpose? To answer your other question: a limit point of a set $A$ can be an interior point of $A$, but it need not be. However, if a set has a point inside it, surely it will always have a neighborhood (or a small ball) that will be contained in the set. $\frac12$ is a limit point of $[0,1)$ because every interval $\left(\frac12-\epsilon,\frac12+\epsilon\right)$ contains a point (indeed, infinitely many points) of $A$ other than $\frac12$ itself. Program to top-up phone with conditions in Python. Maybe you can say why the book's definition feels unintuitive to you. Sirota (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. then a point. $S$. So continuous at interior point, interior to my interval, means that the limit as, let's say at interior point c, so this is the point x is equal to c. Upper Limit Topology: Interior and Closure of $[0,1) \cup (2,3]$. Checking my understanding of the Interior of these intervals. When we can say 0 and 1 in digital electronic? In Brexit, what does "not compromise sovereignty" mean? With the metric $(X,d) : X = \Bbb R$ and $d(x,y) = |x| + |y|$ for $x\neq y$ and $A = \{0\}$. An inflection point (sometimes called a flex or inflection) is where a Practical example, Non-set-theoretic consequences of forcing axioms, Drawing hollow disks in 3D with an sphere in center and small spheres on the rings. Updated March 17, 2017. In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. It may be noted that an exterior point of A is an interior point of A c. Theorems. What is an escrow and how does it work? Prove that $D^\circ=A^\circ$. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. The Interior of R is the set of all interior points. An interior solution is a choice made by an agent that can be characterized as an optimum located at a tangency of two curves on a graph. Let $A=[0,1)$ with the usual topology, for instance. What were (some of) the names of the 24 families of Kohanim? The one that was introduced to us is: $$x \in U \quad \exists \epsilon > 0 : B(x,\epsilon) \subseteq U.$$, However, for the last subseteq, why isn't it just a strict subset? Basic Point-Set Topology 3 means that f(x) is not in O.On the other hand, x0 was in f −1(O) so f(x 0) is in O.Since O was assumed to be open, there is an interval (c,d) about f(x0) that is contained in O.The points f(x) that are not in O are therefore not in (c,d) so they remain at least a fixed positive distance from f(x0).To summarize: there are points Polygon Interior Angles . For your space, as was pointed out, it makes no difference. rev 2020.12.8.38145, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Also, second question: is a limit point an interior point? $r > 0$. It only takes a minute to sign up. because it is possible that $B(x,\epsilon)=U$ for some $\epsilon$. When the second derivative is negative, the function is concave downward. MathJax reference. "not-open set"? Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. A set \(S\) is open if every point in \(S\) is an interior point. A set \(S\) is closed if it contains all of its boundary points. The focus of a parabola is a fixed point on the interior of a parabola used in the formal definition of the curve.. A parabola is defined as follows: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix. Use MathJax to format equations. The definition of local extrema given above restricts the input value to an interior point of the domain. • In a topological space X, (1) Ext ( ϕ) = Int ( X) (2) Ext ( X) = Int ( ϕ). just one question, how is 1/2 a limit point of A? Interior. However, no $\epsilon>0$ can be found with $x\in B(x,\epsilon)\subsetneq\{x\}$. Definition 1.17. Did Biden underperform the polls because some voters changed their minds after being polled? points that are in R and points that are outside. What are the pros and cons of buying a kit aircraft vs. a factory-built one? Point of local maximum. $A$ be a closed set and $D = \overline{A^\circ}$. Home Contact About Subject Index. Asking for help, clarification, or responding to other answers. How Close Is Linear Programming Class to What Solvers Actually Implement for Pivot Algorithms. (1.9) Note that the interior of Ais open. What is the relation between Neighbourhood of a point,Interior point and open set? Is it illegal to market a product as if it would protect against something, while never making explicit claims? So, interior points: a set is open if all the points in the set are interior points. but in the space of integers there are many. Definition. Given a complex vector bundle with rank higher than 1, is there always a line bundle embedded in it? Thanks for contributing an answer to Mathematics Stack Exchange! if there exists an. Is it possible to lower the CPU priority for a job? Drag the points below (they are shown as dots so you can see them, but a point really has no size at all!) Reciprocally, if you have ⊂, you obviously have ⊆. The point (c, f(c)) is an inflection point of the graph of the function f at the point c. If the point (c, f(c)) is an inflection point, then c is a transition number of f. But the contrary is not true because we can have f "(c) so c is a transition point without being an inflexion point. confused on the definition on the interior point of a set in $U \subseteq \mathbb{R}^n$. When you think of the word boundary, what comes to mind? R is called Closed if all boundary points of R are in R. Christopher Croke Calculus 115 such that the ball centered at. Recall from the Interior, Boundary, and Exterior Points in Euclidean Space that if. is said to be an Interior Point of. $\mathbf {a} \in S$. $B (\mathbf {a}, r) \subseteq S$. Is there any role today that would justify building a large single dish radio telescope to replace Arecibo? In the case you mention ($\mathbb R^2$ equipped with usual topology) it can be proved that interchanging $\subseteq$ and $\subsetneq$ makes no difference. The second derivative of a function at a point , denoted , is defined as the derivative at the point of the function defined as the derivative Note that the first differentiation operation must be performed, not just at the point, but at all points near it, so that we have a function for the first derivative around the point, w… Is there a word for making a shoddy version of something just to get it working? What would be the most efficient and cost effective way to stop a star's nuclear fusion ('kill it')? For example a T_1 space for which {x} is open. And we've already seen this with the definition of the derivative. A point of local extremum refers to a point in the interior of the domain of a function that is either a point of local maximum or a point of local minimum.Both these are defined below. Adherent Point, Accumulation Point, Boundary Point, Interior Point, Interior, exterior, and boundary of deleted neighborhood. Reciprocally, if you have $\subset$, you obviously have $\subseteq$. S. if there exists a positive real number. What keeps the cookie in my coffee from moving when I rotate the cup? Calculus. Definition of Interior. Identify interior, boundary, limit and isolated points of a set. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. However the use of $B(x,\epsilon)$ indicates that you are working in metric spaces and one of them is a set $X$ equipped with metric $d$ defined by $d(x,x)=0$ and $d(x,y)=1$ if $x\neq y$ for $x,y\in X$. MathJax reference. Non-circular definition of 'interior' points and 'open' sets, Interior point $x$ characterization as sum $x+\epsilon y$, Understanding definition of Interior (in topology of $\mathbb{R}$), A Question about the Intuition Behind the Definition of an Interior Point on Baby Rudin. How much share should I get in our property. And then we could take some x plus h. a point in the interior of the domain of a function f at which f'=0 or f' does not exist is a critical point of f Definition of Concavity the graph of a differentiable function y=f(x) is a) concave up on an open interval I if y' is increasing on I b) concave down on an open interval I if y' is decreasing on I If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open ball centered at x which is completely contained in S. (This is illustrated in the introductory section to this article.) Use MathJax to format equations. How I can ensure that a link sent via email is opened only via user clicks from a mail client and not by bots? So, what keeps all the points from being interior points? (points inside the set I mean). I made mistakes during a project, which has resulted in the client denying payment to my company. This would be an end point, and this would also be an end point. Your example of the graph of the constant function $y=1$ (lets call it M) is not an open set, since $ \forall x \in M \forall\epsilon >0: B_{\epsilon}(x)\nsubseteq M$. $\overline{A}: $ closure of $A$. Let me define some arbitrary point x right over here. Why is relative interior point not equivalent to interior point under the following definition? A set \(S\) is bounded if there is an \(M>0\) such that the open disk, centered at the origin with radius \(M\), contains \(S\). is called an interior point of. So, what keeps all the points from being interior points? Example The function seen above f(x) = x 4 - … Are more than doubly diminished/augmented intervals possibly ever used? Of course it is the same to write $B(x,\epsilon)\subsetneq U$ that $B(x,\epsilon)\subseteq U$ because $\epsilon>0$ is arbitrary, that is $$B(x,\epsilon)\subseteq U\implies B(x,\epsilon/2)\subsetneq U$$, Your example, whether you mean $\{ (x,y) \in \mathbb R^2 \mid x \in \mathbb R, y = 1 \}$ or $\{ (x,y) \in \mathbb R^{n-1} \times \mathbb R \subset \mathbb R^n \mid x \in \mathbb R^{n-1}, y = 1 \},$ is. Or, drag the point K. This is essentially the same definition. $\mathbf {a} \in \mathbb {R}^n$. Thanks for contributing an answer to Mathematics Stack Exchange! Refers to an object inside a geometric figure, or the entire space inside a figure or shape. Then $\frac12$ is a limit point of $A$ that is also an interior point of $A$, and $0$ and $1$ are limit points of $A$ that are not interior points of $A$. What happens if you Shapechange whilst swallowed? But then you can consider ϵ ′ = ϵ / 2, and you have B (x, ϵ ′) ⊂ U (strict). The set Int A≡ (A¯c) (1.8) is called the interior of A. But then you can consider $\epsilon'=\epsilon /2$, and you have $B(x,\epsilon')\subset U$ (strict). , i.e., there exists an open ball centered at. How do you know how much to withold on your W-4? By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy. The point K will indicate if it is within the interior of angle ∠ ABC (shown in yellow). Making statements based on opinion; back them up with references or personal experience. $x$ is an interior point by the book's definition if and only if it's an interior point by your modified definition. Of course there is none in your space, When the second derivative is positive, the function is concave upward. A classic example of an interio solution is the tangency between a consumer's budget line (characterizing the maximum amounts of good X and good Y that the consumer can afford) and the highest possible indifference curve. $S \subseteq \mathbb {R}^n$. it does not make a difference, wether you use $ ⊆$ or ⊂ for the definition. To learn more, see our tips on writing great answers. A point x0 ∈ D ⊂ X is called an interior point in D if there is a small ball centered at x0 that lies entirely in D, x0 interior point def ⟺ ∃ε > 0; Bε(x0) ⊂ … Did something happen in 1987 that caused a lot of travel complaints? Making statements based on opinion; back them up with references or personal experience. What is the endgoal of formalising mathematics? An interior point of is one for which there exists some open set containing that is also a subset of. The definition can be extended to include endpoints of intervals. So, interior points: a set is open if all the points in the set are interior points. The Boundary of R is the set of all boundary points of R. R is called Open if all x 2R are interior points. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. It follows that x∈ Int A ⇐⇒ ∃�>0 such that U(x,�) ⊂ A. • If A is a subset of a topological space X, then (1) Ext ( A) = Int ( A c) (2) Ext ( A c) = Int ( A). @user65165: You're welcome! See the comment of Masacroso on your question. Boundary point, boundary, and exterior points in the space of integers are! An ordered pair, within brackets illegal to market a product as if is! X\ } $ I tell someone that I interior point definition in calculus to speak to their superior to resolve a conflict with?... Seen this with the usual topology, for instance a word for a... Would protect against something, while never making explicit claims lines as you cross from one state to the '! I rotate the cup drag the point K will indicate if it contains all its! 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\begin{document}
\title{Decentralized Control via Dynamic Stochastic Prices: The Independent System Operator Problem}
\author{Rahul Singh,~\IEEEmembership{Student Member,~IEEE,}
P.~R.~Kumar,~\IEEEmembership{Fellow,~IEEE,}
and~Le Xie,~\IEEEmembership{Senior Member,~IEEE} \thanks{R. Singh is at 32-D716, LIDS, MIT, Cambridge, MA 02139; P. R. Kumar and Le Xie are at Dept. of ECE, Texas A\&M Univ., 3259 TAMU, College Station, TX 77843-3259). {\tt\small [email protected], {prk,le.xie}@tamu.edu.} } \thanks{Preferred address for correspondence: P. R. Kumar, Dept. of ECE, Texas A\&M Univ., 3259 TAMU, College Station, TX 77843-3259.} \thanks{This material is based upon work partially supported by NSF under Contract Nos. ECCS-1546682, NSF Science \& Technology Center Grant CCF- 0939370, NSF ECCS-1150944 and DGE-1303378.}}
\markboth{} {Shell \MakeLowercase{\textit{et al.}}: Decentralized Control via Dynamic Stochastic Prices: The Independent System Operator Problem}
\maketitle
\begin{abstract} A smart grid connects several agents who may be electricity consumers/producers, such as wind/solar/storage farms, fossil-fuel plants, industrial/commercial loads, or load-serving aggregators, all modeled as stochastic dynamical systems. In each time period, each consumes/supplies some electrical energy. Each agent's utility is either the benefit accrued from its consumption, or the negative of its generation cost. There may also be externalities modeled as negative utilities. The sum of all these utilities, called the social welfare, is the total benefit accrued from all consumption minus the total cost of generation and externalities. The Independent System Operator is charged with maximizing the social welfare subject to total generation equalling consumption in each time period, but without the agents revealing their system states, dynamic models or utility functions. It has to announce prices after interacting with agents via bid-price interactions where agents respond with their optimal generation/consumption.
If agents observe and know the laws of uncertainties affecting other agents, then there is an iterative price-bid interaction that leads to the global maximum value of social welfare attainable if agents had pooled their information.
In the important case where agents are LQG systems, the bid-price iteration is dramatically simple and tractable, exchanging only time-vectors of future prices and consumptions/generations at each time step. Agents need not know of the existence of other agents. State-dependent bidding/pricing is not needed. If the DC Power Flow Equations are incorporated it yields the optimal stochastic dynamic locational marginal prices.
Thereby a solution is proposed for a potentially economically important decentralized stochastic control problem. The results may be of broader interest in general equilibrium theory of economics for stochastic dynamic agents. \end{abstract}
\begin{IEEEkeywords} Decentralized Stochastic Control, Social Welfare, General Equilibrium Theory, Demand Response, Renewable Energy, Power Systems, Independent System Operator, Energy Market. \end{IEEEkeywords}
\IEEEpeerreviewmaketitle
\section{Introduction} \label{sec-intro} \IEEEPARstart{I}n the electricity grid, the power generated should be equal to the power consumed at all times, neglecting line losses. Unlike other commodities, electricity cannot be stored in the grid. The task of ensuring that generation is balanced with consumption, and in the most economical way, is entrusted to the Independent System Operator (ISO) in deregulated electricity markets
\cite{wu2005power}.
In the era with fossil fuel as the dominant source of electricity it was possible to adjust generation to meet demand. In the future, as more energy from uncertain and dynamically varying renewables such as wind or solar is used, it is demand that may need to be adjusted continually to balance generation. This strategy is called ``demand response." An example of an adjustable load is an inertial thermal load such as a home with an air conditioner that can be turned off for a while, while still maintaining comfort within the stipulated band of temperatures. New business models are emerging for intermediaries such as retail power service providers, also called ``aggregators" or ``load-serving entities," to sign up a large collection of such customers and undertake their demand response opportunistically, in response to shortages or excess of renewable power that reflect themselves in higher or lower prices, respectively. Large commercial enterprises and industrial loads also will similarly adjust and optimize their energy usage and cost in-house. Therefore both demand and supply will generally be dynamic and uncertain due to external factors such as uncertain supply and ambient temperature interacting with load requirements.
The problem we address is how the ISO can perform its task in the new scenario where loads and generators are stochastic dynamic systems. The primary mechanism for coordinating all entities is by time-varying stochastic prices. However, being stochastic dynamic systems, entities will need to know the probability distribution of future prices to plan their optimal consumption/generation over time. But future prices depend on all future uncertainties affecting any of the entity. Uncertainty in wind in a certain locale may affect a wind farm, cloud cover may affect a solar farm, a broken turbine blade may affect a gas turbine, low customer traffic may affect a commercial entity, or high ambient temperature may affect a group of homes, and each of them may globally impact prices everywhere at all future times. However, an entity is generally unaware of uncertainties affecting other entities or how they will respond. So how can they optimally plan their generation/consumption in the face of dynamic uncertainty and lack of knowledge of each other?
The role played by price in coordinating agents has been explored in general equilibrium theory, initiated by Walras~\cite{walras2}. In their breakthrough work, Arrow and Debreu~\cite{arrow,debreu,Debreu1954} showed that a correct choice of prices for commodities ensures, under a quasi-concavity assumption on utility functions, that a system of individual entities, where each optimizes its own response given prices, results in a systemwide Pareto optimal solution where no entity can benefit without another losing. Subsequently, Arrow, Block and Hurwicz \cite{arrowstable} showed that the prices can be discovered by Walrasian tatonnement~\cite{walras2} under appropriate conditions such as gross substitutes. Their theory extends to allow for uncertainty by simply considering each good under a different random state of nature as a different good, as shown by Arrow~\cite{arrowsecure}. Subsequently, Radner~\cite{radneruncertain} has shown the existence of prices corresponding to an equilibrium even if different agents have different random observations.
The idea of employing prices to perform this task in the electrical power domain was introduced in seminal papers by Caramanis, Bohn and Schweppe \cite{caramanis1982optimal} and Bohn, Caramanis and Schweppe \cite{bohn1984optimal}. Hogan \cite{Hogan} further elaborated the detailed implementation of a locational marginal price-based electricity market operation.
Fundamentally based on a static dispatch with no uncertainty, today's electricity market design and corresponding price signal are simply not designed for achieving social welfare optimality for dynamic generators and loads. The current market mechanism requires participants to make decoupled bids for separate time intervals. In the day-ahead market, a generator has to bid a price-generation curve for the 8am-9am slot, another separate curve for the 9am-10am slot, and so on, for each hour of the next day. However, generators have ramping constraints, such as 50 MW/hour, which give rise to inter-temporal constraints between different time slots. These are typically handled by ad hoc out-of-market (OOM) merit order measures \cite{ERCOT-Anciliary}.The bidding procedure fundamentally does not allow a generator to bid a time function even though that is critical to its operation. Similarly, in the real-time market, the bidding process does not allow a participant to optimize with respect to stochastic process models of uncertain resources such as wind. There have been many studies on the potential problems associated with this market design, such as unnecessarily price volatility \cite{Roozbehani}, network externalities \cite{Chao}, and lack of investment signals \cite{huneault1999review}. While in conventional systems the deterministic and static approach to approximating the underlying dynamic and stochastic power system may be practically appealing without much loss of optimality, emerging resources such as demand response and intermittent renewables render such approximation invalid \cite{Xie2011windintegration}. No previous work achieves social optimality of the entire collection of all stochastic dynamic systems. Providing a theoretical foundation for achieving this fundamental goal is the target of this paper.
We address the ISO problem where each agent is an individual stochastic dynamic agent whose very nature -- its dynamic model, uncertainties affecting it, and its utility function -- are not necessarily disclosed to others. Our goal is to attain a global maximum of the social welfare.
The key issue here is not existence of a solution, since here that is simply the maximizer of social welfare, but how to arrive at it, and realize it, in a distributed way.
There are several interesting aspects to the problem faced by the ISO. We seek a global optimum of the total social welfare, not just an equilibrium. To see the difference, one can consider the work of Radner \cite{radneruncertain} that is closest to ours in its allowance of different observations for different agents. In that theory, the actions of agents are constant over their information -- if an agent does not know the states of other agents, then its action does not change unless its own observed state changes. However, a globally optimal solution will require coordination of the actions of all entities so as to be responsive to each others' states. Thus, the price stochastic process will need to provide this additional coordinating information. The issue examined in this paper is how the ISO is to determine prices and ensure such coordination.
Importantly, we will fundamentally exploit the very fact that uncertain events unfold over time in a dynamic system, to design dynamic interactive strategies for coordination. This is in contrast to Arrow's approach~\cite{arrowsecure} where the problem with uncertainty is reduced to a problem without uncertainty when the descriptions of the uncertainty states of all the agents and their utility functions are known a priori. The equilibrium prices corresponding to each uncertainty state can then be computed at the very outset itself. Such an approach therefore considers the problem in ``normal" form, where the entire dynamic system is simply formulated as a ``static" system where each agent chooses its strategy as a function of prices at states. This observation is also made by Smale~\cite{smale}.
The problem is also interesting from the viewpoint of decentralized stochastic control. Since we seek to maximize social welfare, it is a problem in team theory. However, since Witsenhausen~\cite{wsh} it is known that if agents are unaware of each other's actions but influence each other's observations, then the problem is generally intractable, even in linear quadratic Gaussian (LQG) systems. Unawareness of other's actions is the norm in any distributed stochastic system such as the ISO problem. Nevertheless, in the ISO problem, we show that a system consisting of a collection of LQG systems has an elegant and tractable solution. The tatonnement process for obtaining the global optimum is remarkably simple, even if all entities have private uncertainties. There is no need for agents to even share models of their systems, their uncertainties, the probability distributions of their uncertainties, or their utility functions. Thus, the ISO can optimally coordinate a set of distributed LQG systems very simply without knowing any of their details. This is potentially important, since LQG models are widely used in power systems, where systems are often approximable as linear systems, noises as Gaussian, and costs as quadratic in states and actions.
Importantly, the above approach extends to any number of linear constraints, besides balancing generation and consumption. An important task of the ISO problem is to ensure delivery of required power flows over a congested transmission network. A commonly used of model the transmission network, is through (somewhat misleadingly labeled) ``DC power flow" equations, where differences in bus phase angles determine the power flows along the lines~\cite{bergen}. Their popularity derives from the fact that the resulting equations are linear. Hence the LQG model extends to include the transmission network and provides a very simple solution for the ISO to obtain dynamic \emph{stochastic dynamic} locational marginal prices that attain the global maximum of the social welfare.
The paper is organized as follows. Section \ref{sec-relatedworks} surveys related work.
Section \ref{sec-dynamic} describes the broad context, Section \ref{sec-model} the system of agents,
formulates the ISO problem, and describes the fundamental challenges. We then progressively build up to more complex systems. Beginning with static deterministic systems in Section \ref{sec-static}, we show how the ISO can determine both the optimal price and optimal allocations of consumptions/generations to agents through a bid-price process that corresponds to a subgradient iteration with subsequent averaging. Then we consider deterministic dynamic systems in Section \ref{sec-deterministic} and show how prices and allocations as a function of time can be determined through the same bid-price iteration. Section \ref{sec-ibs} describes iterative bid-price schemes used subsequently in the stochastic dynamic context. Then we turn to the stochastic problem in Section \ref{sec-common} where all agents are subject to a common uncertainty and show that by viewing the system as a ``tree" the results can be extended from the deterministic case.
We next show in Section \ref{sec-private} that this approach can be extended to systems where entities have private uncertainties, by viewing the system in extended form, and iterating at each time between the ISO and the agents for price and allocation discovery. Knowledge by the agents of the probability laws of each other's uncertainties is required, though not of their dynamic models, utilities, or the semantics of the uncertainties since labels of uncertainties can be non-informatively chosen. The difficult issue is complexity, caused by the exponentially exploding joint state space of the uncertainties. However, in Section \ref{sec-lqg}, we show that the complexity disappears for distributed LQG systems, leading to a simple and implementable solution. The ISO simply discovers and announces time-varying but \emph{not} state-dependent prices for future epochs, and revises them at each time step, reminiscent of model predictive control. We show in Section \ref{sec-powerflow} that this result can be extended to include any linear constraints, e.g., the widely used DC-power flow equations. Section \ref{sec-simu} presents the results of illustrative simulations, concluding in Section \ref{sec-concluding}.
\section{Related Works} \label{sec-relatedworks} No similar results appear to be known for general decentralized stochastic control. Team problems have been extensively studied, e.g.,~\cite{radnor,sardar,van}, but those formulations do not apply here since agents need to know the system dynamics of other agents. Even when the models are known, there are still considerable difficulties in decentralized stochastic control. When agents do not share observations, severe complexity arises, even in LQG systems, as shown by Witsenhausen's counterexample of a two stage problem~\cite{wsh}. The roles of observation, signaling~\cite{sardar}, and the trade-off between communication and control are evident from Witsenhausen's counterexample~\cite{wsh}. Teneketzis~\cite{demos} considers decentralized stochastic control under the restrictive assumption that the interaction between agents is ``weak". There are some recent structural results~\cite{nayar}, and results regarding sufficient statistics~\cite{jeff} under these assumptions.
From the economics side, this work is an extension of general equilibrium theory~\cite{arrow1}. To the authors' knowledge there does not appear to be any similar result for coordinating multiple LQG systems or the efficiency of the simplified signaling. While the name may appear to be related to the issues studies here, Dynamic General Stochastic Equilibrium theory pioneered in \cite{kydland1982time} addresses issues in macroeconomics, and is not relevant for the problems of interest here.
Viewed from the power system end, there have been many efforts since the deregulation of the electricity sector on a market-based framework to clear the system. Today's locational marginal price-based nodal market design is based on seminal work in \cite{bohn1984optimal,Hogan}. This has been followed up by a large body of literature focusing on designing an efficient transmission pricing mechanism in support of an efficient market \cite{wu1996folk} \cite{kirschen1997contributions}. From the system operators' perspective, the naive belief that deregulation of electricity industry would simply work was critically re-assessed following the Enron crisis and lack of long-term investment \cite{lave2004rethinking} \cite{hogan2003transmission}. From a market participant's perspective, there has been pioneering work on game theoretic approaches to modeling the market power issues in the electricity market \cite{chuang2001game} \cite{baldick2004theory}. With increasing penetration of stochastic resources, there have been efforts at designing a market bidding mechanism that achieves the social welfare optimum. Ilic et al. \cite{Ilic2011framework} have proposed a two-layered approach that internalizes individual constraints of market participants while allowing the ISO to manage the spatial complexity. References~\cite{carpenter,carpenter1} contain some heuristic approaches. Reference~\cite{wets} applies progressive hedging to deal with uncertainties on the production side; however the solution is centralized, and does not provide any theoretical guarantees. Reference \cite{rajagopal2013risk} studies how the ISO should dispatch, i.e., purchase energy and call options in different markets, under forecast errors about future loads and renewable generation, when future decisions can mitigate current errors.
However, there has been no analytical framework that precisely leads to the social welfare optimality with dynamic, stochastic inputs from market participants. The major challenge addressed is how to elicit optimal demand response in such cases without generators/loads revealing the details of their dynamic models to the ISO.
\section{The ISO Problem of Coordinating Dynamical and Uncertain Generators, Loads and Prosumers} \label{sec-dynamic}
Generators such as wind farms, photo-voltaic farms, hydro, coal or gas turbines, need to be modeled as dynamic stochastic systems. Likewise, loads such as aggregators, commercial or industrial establishments, also need to be so modeled since their demand may be governed by a dynamic system, and random. Since environmental variables such as temperature are involved, and since human beings in the loop respond to economic incentives/prices, their response may also be uncertain. Hence loads generally will also need to modeled as stochastic dynamic systems. Storage services such as battery farms or pumped hydro can also be modeled as dynamic systems where the state is the amount of energy stored. Dynamic models can also be used to model ``prosumers" such as homes with solar panels, which can switch between consumption/generation.
The utility of a generator is the negative of its cost of generation. The utility of a load is the ``benefit" that the load accrues from the consumed power. There may also be externalities, such as pollution, that can optionally be modeled as a cost, i.e., negative utility. The total of all the utilities, called social welfare, is therefore the benefit of the power consumed minus the cost of generating it and the cost of externalities, and the goal is to operate the overall system so as to maximize it.
An agent's utility is measured not statically, but over a time interval of interest, since agents are dynamic systems. A large commercial load may accrue utility over a period of time, by maintaining the temperature within a band by switching air conditioners off and on taking time lags into account, if demand response strategies are in place. Similarly, a storage service may buy and store energy at off-peak times and sell it at peak times, again accruing value only over a time interval. Generators too may accrue utility over time by ramping up generation.
An agent's utility is affected by stochastic uncertainties of other agents, since coal shortage at a generator may affect a distant load. Each agent seeks to maximize the expected value of its own utility function, with expectation taken over all uncertainties affecting all agents.
There are important and severe constraints on the information disclosed to the ISO. It does not know the states, dynamic models, or utility functions of individual agents. Whether loads or generators, individual agents may be averse to disclosing information to others for competitive reasons or to ensure privacy. A load-serving entity may not inform the ISO of the states of its loads, e.g., the temperatures of every one of its large collection of customer's homes. A solar farm may not inform the ISO of the extent of its cloud cover. More fundamentally, the agents may not even be willing to share their individual dynamic system models with each other. The ISO may not be informed of the stochastic model of wind at a wind farm, or the detailed model of the dynamics of a coal plant. Similarly, agents may not share their individual utility functions. A load-serving entity may not be willing to disclose its contracts with its customers or its cost of operations. Many of these entities compete with each other and are sensitive to sharing their information with competitors. Recently, privacy of loads has also emerged as a major concern.\footnote{Interestingly, privacy is nowhere mentioned in the seminal paper~\cite{caramanis1982optimal} that introduced spot prices, indicative of how new issues arise.} Demand response can entail violation of privacy since it has been shown that having access to a home's real power consumption allows one to deduce the number and behavior of its occupants~\cite{molina2010private}.
Even if all agents were willing to share all their information with the ISO, it would be such an intractably large amount of information, amounting to a complete state of the world, that the ISO would not be able to handle it with acceptable complexity and delay anyway.
The ISO is nevertheless charged with maximizing the sum of the utilities all the agents, i.e., the social welfare. It plays the role of a mediator. It needs to determine how much power each agent should generate/consume in each time period over the time horizon of interest. It needs to allocate the required power generation to the generators over time in the most economical fashion. It also needs to provide the optimal amount of power over time to each load to optimize its utility. Demand and supply are intertwined, since demand is uncertain and determined by the cost of generation, while generation is also uncertain and incentivized by the price that consumers are willing to pay. All the agents are stochastic dynamic systems with their own utility functions over time. The ISO needs to do all this in the face of all the stochastic uncertainties affecting the agents over the time horizon.
The ISO would like to achieve the above through economic mechanisms, by determining prices, and by agents responding with their own selfish utility maximizations as in general equilibrium theory~\cite{arrow}. The complication is that the agents are evolving stochastically in time. Therefore the prices cannot simply be announced once and for all at the beginning of the time interval of interest. For example, a future high ambient temperature can lead to very high demand that will need high prices to incentivize extra generation or reduce deferrable demand. The prices will need to vary stochastically in response to private uncertainties affecting the agents. The price stochastic process should carry all the information that is necessary for all entities in the overall system to coordinate in a globally optimal way, since each affects others.
The fundamental question examined in this paper is whether and how this optimality can be attained given stochastic dynamical system models for the agents, and what form the mediation process or tatonnement~\cite{walras1} takes. Our contribution is to show that there are iterative interaction processes under which the ISO can indeed perform this task. We address the complexity of this task under several scenarios. The complexity is very high in the general case. However, in the case where the agents can be modeled as linear Gaussian stochastic systems and the cost functions are quadratic, we show that a much simpler scheme yields the systemwide global optimum. This scheme extends to encompass other linear constraints, e.g., those modeling the electrical network, thus providing a more comprehensive solution that takes into consideration the power flows over the network.
Beyond balancing generation and consumption, there are at least two additional problems that the ISO faces. It needs to ensure no line's capacity is exceeded in the electrical transmission network, so as to prevent overheating. This requires ensuring that in the solution of the power flow equations at the obtained generations, the current carried over every line does not exceed its capacity~\cite{bergen}.
ISO's also need to ensure reliability. If contingencies occur, such as generator tripping or line-to-ground fault, then the system's electrical state should converge to an acceptable equilibrium point \cite{gan2000stability}. ISO's verify that the solution has this property for all single event contingencies, reformulating the problem if any violations are observed. Considering multiple simultaneous contingencies is computationally demanding and not the norm \cite{wang2013risk}.
\section{System Model and ISO Problem}\label{systemmodel} \label{sec-model} We consider a smart grid consisting of $M$ agents, each of which may act as a producer, consumer or both, i.e., a prosumer, evolving over a time interval $t=0,1,\ldots,T-1$. The time horizon $T$ could be $96$, which would correspond to one day of $15$ minute slots in the real-time market. There is however considerable flexibility to model other scenarios. One can model the risk-limited dispatch of \cite{rajagopal2013risk} where purchases of forward energy are made for blocks of time, with blocks getting shorter as operations approach real time. In that case the times $t=0,1, \ldots $ can correspond to the 24 hour ahead, all the one-hour ahead, and all the 15 minute ahead, times at which decisions are made by agents. The states of the agents (below) can keep track of their past purchases, temperature forecasts, etc, so that noises can be regarded as changes to past forecasts, allowing considerable generality. \\
${\mathbf{Randomness}}$ is modeled through a probability space $\left(\Omega,\mathcal{F},\mathbb{P} \right)$. The ``state of the world" $\omega \in \Omega$ captures a multitude of random phenomena spread out temporally and spatially, for example, unpredicted weather (the wind-speed), unexpected events such as coal shortage, or a damaged wind-turbine, etc. \\ {{\bf Common and Private Uncertainties.}} The randomness $\omega$ affects an agent $i$ through the stochastic processes $N_i(\omega,t)$ and $N_c(\omega,t), 0 \leq t \leq T-1$. $N_c(t)$ is a ``common" uncertainty that affects and is known causally by all agents, e.g., the weather of a city. $N_i(t)$ is a ``private" uncertainty that is specific to agent $i$, and known causally only to agent $i$. As we will see, this decomposition of uncertainties clarifies the task of constructing interaction schemes between the agents and ISO. \\ $\mathbf{Agents}$ are modeled as stochastic dynamical systems. The state $X_i(t)$ of agent $i$ at $t$ is known to it, and evolves as $ X_i(t+1) = f_i(X_i(t),U_i(t),N_i(t),N_c(t),t) $, where $f_i$ describes the dynamics of the agent $i$. The initial condition $X_i(0)$ can be random. The common case is that $U_i(t)$ is a scalar that denotes the amount of electricity consumed (negative if supplied) from the grid by agent $i$ at time $t$, but we will allow $U_i$ to be a vector of several commodities being produced and consumed. \\ ${\mathbf{Consumption/Generation \ Constraints}}$. Let $N_i^t := (N_i(0), N_i(1), \ldots , N_i(t)$ denote the past of $N_i$, and similarly define $N_c^t$. Agent $i$'s choice has to satisfy the local capacity constraints $F_i(N_i^t,N_c^t,t)U_i(t) \leq g_i(N_i^t,N_c^t,t)) + \sum_{s=0}^{t-1} C_{i}(N_i^t,N_c^t,s,t)U_i(s)$ and $h_i(N_i^t,N_c^t,t,U_i(t)) \leq 0$ for each $t$. The affineness of the former constraints in past $U_i$'s allows for ramping, the dependence on $t$ allows for seasonality, and the dependence on $N_i,N_c$ allows random effects on capacity.
\\ ${\mathbf{Observations}}$ available to an agent $i$ until time $t$ include the realizations of its system state $X_i(s)$, common noise $N_c(s)$, and its private noise $N_i(s)$ for $0 \leq s\leq t$.\\ The {\bf{One-step Cost Function}} of an agent $i$, $1 \leq i \leq M$, denoted $c_i(x_i,u_i,t)$ (or its negative, a one-step utility function $-c_i(x_i,u_i,t)$), is a function of its state and action, in period $t$. For producers, this could be the cost of labor, coal, etc.. For consumers, this could represent the cost incurred due to the high temperature of house/business facility, or due to a delay in performing a task resulting from inadequate purchase of electricity, or the negative of some benefit of the electricity usage. \\ {\bf{Externalities}}, e.g., pollution, with one-step cost $\sum_{i=1}^M e_i(u_i,t) $, say cost of mitigation, can be considered.
By allowing $e_i(u_i,t)$ to be positive/negative ISO imposed levies, cross-subsidies can be addressed. For linear levies, $e_i u_i(t)$, ISO budget balance, $\sum_{i=1}^M e_i u_i(t) = 0$, can be enforced as shown below for energy balance. \\ {\bf{Energy Balance}} should be maintained in each period, i.e., $\sum_{i=1}^M U_i(t) = 0$ for all $t=0, 1, \ldots , T-1$. We allow {{\bf general linear vector constraints}}: $\sum_{i=1}^ME_i(t)U_i(t)=d(t)$. \\ {\bf{Total System Operating Cost}}, or its negative, the {\bf{Social Welfare}}, is the sum of the expected value of the finite horizon total of the one-step costs incurred by all the agents plus externalities, $ \mathbb{E} \sum_{t=0}^{T-1} \sum_{i=1}^{M+1}[c_i \left(X_i(t),U_i(t),t \right)+e_i \left(U_i(t),t \right)]$. It is the total electricity generation cost plus the cost of externalities minus the utility provided to the consumers. The expectation above is taken with respect to the combined uncertainty or ``noise"€ process $ N(t) := \left(N_c(t), N_1(t), N_2(t), \ldots, N_M(t)\right)$ for $t=0,1, \ldots, T-1$, consisting of all the private uncertainties and the common uncertainties, as well as the random initial conditions of all the $M$ agents. \\ The {\bf{Power Flow Equations}} are algebraic equations based on Kirchoff's laws that have to be satisfied by the electrical variables, voltage and current magnitudes and phase angles. They impose constraints on $\{U_i(t), 1 \leq i \leq M \}$.
\\ The {\bf{Independent System Operator (ISO)}}'s task is to maximize the social welfare. It solicits electricity purchase/sale bids from the agents in each time slot $t=0,1,\ldots,T-1$. Our model allows for agents and the ISO to iterate on the bids. Once the price iterations have converged, the ISO declares the market clearing prices, and the electrical energy to be consumed/generated by the agents, at the declared prices. \\ The {\bf{Bidding Schemes}} allow the ISO and agents to reach a solution for prices, generation and consumption. Depending on the assumptions made about the system model, there will be different bidding schemes.
An example is the following. Consider time $s$. The ISO announces a price sequence for current and future times, $s \leq t \leq T-1$, to all agents. Agent $i$ bids, as a function of its past information, the amount of electricity it is willing to purchase/generate at the current and future times $s \leq t \leq T-1$, at the prices indicated by the ISO. After collecting the bids, the ISO updates the price sequence. An iteration of \textit{price updates} followed by \textit{bid updates}, continues till the prices and the bids converge, and then the ISO announces the allocations of generations/consumptions to agents for the current period $s$. This entire process can be repeated in each discrete time slot $s$ in real-time.
\\
{\bf{Goal of Social Welfare Maximization}}: The goal is to maximize the negative of total system cost, i.e., social welfare. Let $\mathcal{F}_t$ be the $\sigma$-algebra generated by all the noises upto time $t$, as well as initial conditions. Now we come to the stringent goal of this paper. \emph{We would like to attain the same maximum value of the social welfare as could be attained over the class of all control laws where $U(t) := \left(U_1(t), U_2(t), \ldots , U_M(t)\right)$ is adapted to $\mathcal{F}_t$}.
This is an economically important point in that even though the agents do not all act in a centralized way and even though they do not all have access to all the observations and initial conditions of each other, we would like them to collectively attain the same optimal value of social welfare by acting in a distributed way, with each agent only using its own causal observations together with the price information provided by the ISO. In fact they do not even know each other's dynamic models or cost functions, taking this problem outside of usual stochastic control/game theory.
The resulting {\bf{ISO Problem}} is:
\begin{align} & \min \mathbb{E} \sum_{t=0}^{T-1} \sum_{i=1}^{M} [c_i \left(X_i(t),U_i(t),t\right) + e_i \left(U_i(t),t\right)] \label{p0} \\ &\mbox{such that } X_i(t+1) = f_i(X_i(t),U_i(t),N_i(t),N_c(t), t); \notag \\
& \mbox{with capacity constraints } h_i(N_i^t,N_c^t,t,U_i(t)) \leq 0, \label{constraints-at-time-t-1} \\ & F_i(N_i^t,N_c^t,t)U_i(t) \leq g_i(N_i^t,N_c^t,t) \notag \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad + \sum_{s=0}^{t-1} C_{i}(N_i^t,N_c^t,s,t)U_i(s); \label{constraints-at-time-t-2}\\ &\sum_{i=1}^ME_i(t)U_i(t)=d(t) \mbox{ for }1 \leq i \leq M, 0 \leq t \leq T-1. \label{constraint-on-balancing} \end{align}
The expectation above is taken with respect to the combined uncertainty or ``noise" process $N(t):=\left(N_c(t),N_1(t),N_2(t),\ldots,N_M(t),N_M(t)\right), t=0,1,\ldots,T-1$, as well as the random initial conditions $\left(X_1(0), X_2(0), \ldots , X_M(0)\right)$. The actions $U_i(t)$ are to be taken on the basis of the past information available to agent $i$ at time $t$, which includes the past history of its own observations of its system's state, common noise and private noise, as well as the common price information provided to all agents by the ISO. The bidding process to be studied below will determine the prices announced by the ISO to all the agents.
The central issue is the following: \emph{How should the ISO determine pricing and allocations to dynamic stochastic agents so that the overall system is as optimal as it could be through centralized control, even though agents do not know each other's dynamic models or cost functions?}
\subsection{Fundamental Challenges} \label{sec-fi} The ISO Problem
poses several challenges. It is a multi-agent problem where stochastic dynamic agents with differing objectives, ignorance of each other's systems or objectives, and separate observations, are constrained in their joint actions; yet the goal is to ensure that they function as a team and jointly maximize social welfare.
\subsubsection{Constraint on joint actions}\label{fi:id} The problem cannot be solved by considering each agent separately because energy balance (\ref{constraint-on-balancing}) constrains their joint actions. \subsubsection{Privacy constraints}\label{privacy} The agents do not disclose their system dynamics functions $f_i$ to other agents or the ISO.
In fact, the agents do not even know how many agents are present. \subsubsection{Non-classical information structure}\label{fi:dc} Even if all dynamics and probability distributions of uncertainties were known to all, the ISO Problem would still lie at the core of decentralized stochastic control with a non-classical information structure \cite{wsh,wsh2,sardar,van,demos,carpenter,nayar} since each agent has separate observations from others. Even if privacy were not an issue, sharing all observations amongst all agents requires huge communication and processing overhead, etc., and may be impossible in practice. \subsubsection{Conflicting objectives} The objectives of the agents are not all aligned and may have conflicts. \subsubsection{Signaling} In decentralized stochastic control~\cite{sandelv,nayar,van}, controllers may be able to signal some private information to other agents over a ``channel" which may even be the physical plant itself.
``Prices" can play the role of a channel, with the bidding scheme functioning as encoder-decoder. Essentially the ISO needs to construct a ``price" sufficient statistic for the problem~\cite{striebel,blackwell}.
The question we address is: \emph{Can $M$ independent systems be driven to an overall optimal operation through the ISO}? We will show that there exist ``iterative bidding schemes" (IBS) which yield the same performance as that of the optimal centralized controller.
\section{Static Deterministic Systems} \label{sec-static} We begin with the problem where all generators and consumers are static and deterministic.
The ISO has to allocate generations and consumptions so that the social welfare, the total benefit accrued from consumption minus the total cost of generation is maximized.
This can be formulated as a problem of minimizing the total cost $J(u) = \sum_{i=1}^M [c_i(u_i) + e_i(u_i)] $ of $M$ agents and the externalities, where, if agent $i$ is a generator producing $-u_i$ (negative by convention for generation), the cost of generation is $c_i(u_i)$, while if it is a consumer consuming $u_i$ the utility it obtains from consumption is $-c_i(u_i)$, and $e_i(u_i)$ is the externality associated with generation/consumption.
Each generator/consumer $i$ may also be subject to linear/nonlinear vector inequality constraints $F_i u_i \leq g_i$ and $h_i(u_i) \leq 0$. (When $u_i$ is a scalar, this will reduce to either a semi-infinite interval or interval constraint on $u_i$ under convexity of $h_i$ below).
This entails solving the following optimization problem: \begin{align} &\min_{u_1, \ldots , u_M} \sum_{i=1}^M [c_i(u_i) + e_i(u_i)], \label{simple-primal} \\ &\mbox{subject to: } F_i u_i \leq g_i, h_i( u_i) \leq 0 \quad \mbox{ for } 1 \leq i \leq M, \label{individual-constraints} \\ & \mbox{and } \sum_{i=1}^M E_i u_i = d. \label{balancing_constraint} \end{align}
Dualizing only the constraint (\ref{balancing_constraint}), and denoting $u := (u_1, u_2, \ldots, u_M)$, yields, respectively, the Lagrangian, Dual Function, and optimal reward of the Dual Problem:
\begin{align} &\mathcal{L}\left(u, \lambda \right) := \sum_{i=1}^{M} [ c_i(u_i) + + e_i(u_i) + \lambda^T E_i u_i ] -\lambda^Td, \notag \\ &D(\lambda) := \min_{ \{u: F_i u_i = g_i, h_i(u_i) \leq 0 \ \forall i \} } \mathcal{L}\left(u, \lambda \right), \notag \\ &J^\star := \max_{ \lambda } D(\lambda) = D(\lambda^\star) \label{Simple_Dual} . \end{align}
\begin{assumption}[Assumption for deterministic case]\label{finite-compact-slater} (i) $c_i( \cdot)$, $e_i( \cdot )$, $h_i( \cdot )$ are convex, $\{u_i: F_i u_i \leq g_i, h_i(u_i) \leq 0 \}$ is compact, and (\ref{simple-primal},\ref{individual-constraints},\ref{balancing_constraint}) has an optimal solution. \\ (ii) Slater's Condition: There exists a feasible $\bar{u}_i$ satisfying $h_i( \bar{u}_i) < 0$ in $\mbox{RelInt(Dom}(c_i)) \cap \mbox{RelInt(Dom}(e_i))$. \end{assumption}
From (ii), $J^\star$ is also the optimal cost of the Primal (\ref{simple-primal}).
Since $D(\lambda)$ can be decomposed agent-by-agent as \begin{align*}
D(\lambda) = \sum_{i=1}^M \min_{ \{u_i: \mbox{ \scriptsize{s.t. (\ref{individual-constraints})}} \} } [ c_i(u_i) + e_i(u_i) + \lambda^T E_i u_i ]-\lambda^Td, \end{align*}
the ISO can conceivably simply announce the ``optimal price" $\lambda^{\star}$ per unit of power as that which attains the max in (\ref{Simple_Dual}), and assess an additional levy $e_i(u_i)$ on agent $i$. (This levy could be a ``carbon tax" used to mitigate the pollution). Each agent $i$ can then respond with either its generation $-u^*_i$ or consumption $u^*_i$ that minimizes its net ``loss" $c_i(u_i) + \lambda^{\star} u_i$ over (\ref{individual-constraints}). The ISO can finally announce the generation/consumption allocations to the agents.
There are two issues that arise: \\
(i) Since agents do not disclose their cost functions, there needs to be a price discovery process, as in a Walrasian auction~\cite{walras1}. The ISO's price needs to be reduced/increased according to whether the agents' response results in
excess total generation/consumption). We consider the following
\emph{iterative price-bid process:} \begin{align*}
\lambda^{k+1} & = \lambda^k + \frac{1}{k}[\sum_{i=1}^M E_i u^k_i - d], \\ u^{k+1}_i & = \scriptsize{ \begin{matrix} \mbox{argmin} \\ { \{u_i: \mbox{ \scriptsize{s.t. (\ref{individual-constraints})}} \} }
\end{matrix} } [c_i(u_i) +e_i(u_i) + (\lambda^{k+1})^T E_i u_i )]. \notag
\end{align*}
This iteration of prices\footnote{The gain $\frac{1}{k}$ can be replaced by $\frac{\alpha}{k^\delta}$ for $\frac{1}{2} < \delta \leq 1$ with $\alpha > 0$ in Sections \ref{sec-static}--\ref{sec-private}.} and bids is a subgradient algorithm that converges to an optimal price for the Dual under Assumption \ref{finite-compact-slater} \cite{anstreicher2009two}. \\
(ii) The recovery of optimal generations/consumptions from optimal price is more problematic:
\begin{example}[Counterexample to generation/consumption recovery from optimal price] \label{counterexample} Consider one generator and one load. The generator's cost of producing $-u_1$ units of energy is $-\frac{2}{5}u_1$, with $u_1$ restricted to $[-1, 0]$, and the cost of the externality is $-\frac{1}{10}u_1$. The load's utility from consuming $u_2$ units of energy is $\log(1+u_2)$ with $u_2$ restricted to $[0,2]$, and it has no externality. Energy should be balanced. The social welfare problem is:
\begin{align*} &\mbox{Min} \quad -\frac{2}{5}u_1 -\frac{1}{10}u_1 - \log (1+u_2) \\ &\mbox{Subject to: } -1 \leq u_1 \leq 0, 0 \leq u_2 \leq 2, u_1 + u_2 =0. \end{align*} The optimal solution is $(u_1^\star,u_2^\star) = (-1, 1)$.
The Dual function of price $\lambda$ is \begin{align*} &D(\lambda) = \min_{1 \leq u_1 \leq 0}[ -\frac{1}{2}u_1 +\lambda u_1 ] + [ - \log(1+u_2) ] + \lambda u_2 ]. \end{align*} The minimizers and minimum, $(-u_1(\lambda),u_2(\lambda), D(\lambda))$, are $$ = \begin{cases} (0,\mbox{Min} \{ \frac{1}{\lambda} - 1, 2 \},1 - \lambda + \log \lambda) \quad \mbox{ if } \lambda < \frac{1}{2},\\ (\mbox{\emph{any} point in } [0,1],\frac{1}{\lambda} - 1, 1 - \lambda + \log \lambda) \quad \mbox{ if } \lambda = \frac{1}{2},\\ (1, \frac{1}{\lambda} - 1, \frac{1}{2} + \log \lambda) \quad \mbox{ if } \frac{1}{2} < \lambda \leq 1,\\ (1,0, - \lambda + \frac{1}{2}) \quad \mbox{ if } 1 < \lambda, \end{cases} $$
The optimal solution of the Dual is $\lambda^\star = \frac{1}{2}$.
However, when the price $\lambda^\star = \frac{1}{2}$ is announced by the ISO, the generator can bid $-u_1 =0$ since any point in [0,1] is optimal. The load's bid is $u_2 = 1$, and there will not be balance between generation and consumption.
$\square$ \end{example}
Therefore one cannot recover the optimal bids from the optimal prices. However, they can be recovered from the \emph{iterations of the bidding process} under Assumption \ref{finite-compact-slater} by taking weighted averages of previous bids
\cite{gustavsson2015primal}. Thus the very \emph{process} of iterative bidding is itself important.
\begin{theorem}[Determining optimal bids by generators and loads \cite{gustavsson2015primal}]\label{ergodic-theorem} Let $\theta \geq 0$. Consider the averaged bids obtained recursively as follows: \begin{align} \bar{u}^k_i = \frac{\sum_{s=1}^{k-1}s^\theta}{\sum_{s=1}^{k}s^\theta} \bar{u}^{k-1}_i + \frac{k^\theta}{\sum_{s=1}^{k}s^\theta} u^{k}_i; \quad \bar{u}^0_i = u^0_i \label{ergodic_method} \end{align} Then $\bar{u}^k_i \to u_i^\star$ which is optimal for (\ref{simple-primal}).
$\square$ \end{theorem}
A larger $\theta$ weights more recent values of the iterates for $u_i$ more heavily, while $\theta=0$ takes a plain average.
\begin{example}[Continued] \label{counterexample-continued} Choosing $\theta = 2$, one obtains: \begin{align*} &\{\lambda_k \}: 0, 0, 1,0.6667, 0.5416,
\ldots \to \frac{1}{2}, \\ &\{u^k \}: \Spvek{-1;0}, \Spvek[l]{0;2}, \Spvek[c]{-0.9412;0.1176}, \Spvek{-0.9898;0.4133} , \\ &\quad \quad \Spvek{0-0.9972;0.7263} , \Spvek[l]{-0.9990;0.9526}, \ldots \to \Spvek{-1;1}.
\quad \square \end{align*} \end{example}
\section{Dynamic deterministic systems} \label{sec-deterministic}
We consider the ISO Problem for deterministic systems: \begin{align} & \min \sum_{t=0}^{T-1} \sum_{i=1}^{M}[c_i \left(x_i(t),u_i(t),t\right) + e_i \left(u_i(t),t\right)] \label{deterministic-optimal-control} \\ &\mbox{s.t. } x_i(t+1) = f_i(x_i(t),u_i(t),t), \mbox{and (\ref{constraints-at-time-t-1},\ref{constraints-at-time-t-2},\ref{constraint-on-balancing})}. \label{systemconstraint} \end{align} Since the state variables $x_i(t)$ can be expressed in terms of the inputs $u_i:=\left(u_i(0),u_i(1),\ldots,u_i(T-1)\right)$,
the ISO problem
can be written as (\ref{simple-primal}). We assume Assumption \ref{finite-compact-slater}.
The associated Lagrangian and dual function are \begin{align} &\mathcal{L}\left(u, \lambda \right): = \sum_{i=1}^{M} \sum_{t=0}^{T-1} [c_i(x_i(t),u_i(t),t) + e_i(u_i(t),t) \notag \\ & \quad \quad \quad \quad \quad \quad \quad \quad + \lambda (t)^T E_i(t) u_i(t) ] - \sum_{t=0}^{T-1} \lambda (t)^T d(t), \notag\\ &D(\lambda) := \min_{\scriptsize{\{u: u \mbox{ satisfies } (\ref{constraints-at-time-t-1},\ref{constraints-at-time-t-2})}
(N_i,N_c \mbox{\scriptsize{ absent) }} \mbox{\scriptsize{for} } \scriptsize{0 \leq t \leq T-1, \forall i\}}} \mathcal{L}\left(u, \lambda \right), \label{dual-for-deterministic-dynamic} \end{align} where $u :=\left(u_1,\ldots,u_M \right)$, $\lambda :=\left(\lambda(0),\ldots,\lambda(T-1)\right)$, and each $x_i(t)$ is regarded as a function of $u_i$ in (\ref{dual-for-deterministic-dynamic}). The Lagrangian decomposes by agents.
Therefore, given $\lambda$,
each agent $i$ solves its own problem: \begin{align} & \mbox{Min} \sum_{t=0}^{T-1} [ c_i(x_i(t),u_i(t),t) + e_i(u_i(t),t) + \lambda(t)^T E_i(t) u_i(t) ] \label{adjoinedcost} \\ &\mbox{subject to } (\ref{systemconstraint}). \notag
\end{align}
The Bid-Price Iteration proceeds as follows. The ISO announces prices $\lambda^k = \{ \lambda^k(t): 0 \leq t \leq T-1 \}$. Each agent $i$ responds with an optimal solution $u_i^k:=\left(u_i^k(0),u_i^k(1),\ldots,u_i^k(T-1)\right)$ to (\ref{adjoinedcost}).
Since the subgradient with respect to $\lambda$ of the Dual function $D(\lambda)$ is $\left(\sum_{i=1}^M u_i(0),\sum_{i=1}^M u_i(1),\ldots,\sum_{i=1}^M u_i(T-1)\right)$, the ISO employs the price iteration over $k$, for every $t \in [0, T-1]$:
\begin{align} &\lambda^{k+1}(t) = \lambda^k(t) + \frac{1}{k} [\sum_{i=1}^M E_i(t) u^k_i(t) - d(t)]. \label{PriceIterate}
\end{align}
The agent bids are averaged by (\ref{ergodic_method}) to give $\bar{u}^k_i(t)$. The ISO announces the allocations $u_i^\star(t) := \lim_{k \to \infty} \bar{u}^k_i(t)$.
\begin{theorem} \label{theorem-for-determinstic-dynamic-case} Consider the ISO problem (\ref{deterministic-optimal-control},\ref{systemconstraint}) under Assumption \ref{finite-compact-slater}. Suppose the ISO employs the price iteration (\ref{PriceIterate}), with each agent $i$ responding with an optimal solution $u_i^k:=\left(u_i^k(0),u_i^k(1),\ldots,u_i^k(T-1)\right)$ to (\ref{adjoinedcost},\ref{systemconstraint}). Then the prices $\lambda^k$ converge to the optimal prices $\lambda^\star$ for (\ref{dual-for-deterministic-dynamic}). The ISO's final allocation of generations/consumptions, $u^\star$, exists as a limit, and is optimal for (\ref{deterministic-optimal-control},\ref{systemconstraint}). If $\lim_ku_i^k$ exists, averaging is not needed since it is equal to $u^\star$.
$\square$. \end{theorem}
In this deterministic context, the whole problem can be solved at time 0,
with actions $u^{\star} := (u^{\star}(0), u^{\star}(1), \ldots , u^{\star}(T-1))$ implemented open loop.
\section{Iterative Bidding Schemes for Stochastic Systems}\label{sec-ibs} We now turn to the stochastic case. The goal is to solve the ISO Problem~(\ref{p0}) through Iterative Bidding Schemes (IBS), as in Walrasian tatonnement~\cite{arrow}. We explain what transpires in such an IBS for the simpler common uncertainty context $N(t) \equiv N_c(t)$.
A tree visualization of the system randomness, as in Fig.~\ref{extenform}, is helpful. Suppose that $N(t)$ assumes only finitely many values. We can then construct an uncertainty tree of depth $T$, in which the root node corresponds to the initial system state, and
the sequence of transpired noises $\{ N(0), N(1), \ldots , N(s-1) \}$ corresponds to some node at the level $s$.
\begin{figure}
\caption{A tree visualization of uncertainty for a two agent system evolving over three bid times, where the uncertainty values are binary, either 0 or 1.}
\label{extenform}
\end{figure}
Since all agents know the law of $\{ N_c(t) \}$, i.e., the probability measure induced on the sample paths of the noise stochastic process $\{ N_c(t) \}$, the agents know the topology of the tree, and the transition probabilities along edges.
However, the agents do not know the system dynamics of other agents, their utility functions, or states or actions. The ISO need not know the law of $N$. We will suppose that the ISO does know the topology of the tree and the labels of the nodes.
Let $\mathcal{F}_{i,t} := \sigma(X_i(0), N_i(0), N_i(1), \ldots , N_i(t-1), N_c(0), N_c(1), \ldots , N_c(t-1))$ denote the sigma-algebra generated by agent $i$'s observations up to time $t$. (Incorporating $N_i (\cdot )$ is unnecessary since private uncertainties are absent here, but will be useful subsequently in the general case of private observations).
The IBS scheme will intertwine two processes, a \textit{Bid Update Process} and a \textit{Price Update Process}. As in Section \ref{sec-static}, information revealed during the bidding process is important to determining the final allocation. Additionally, repeating the process at each time instant is important in the stochastic dynamic case in adapting to how agents are evolving over time as uncertain events happen. \\ {\bf{Bid Update Stochastic Process $\mathbf{B_s = (U_{i,s}(s), U_{i,s}(s+1),\ldots, U_{i,s}(T-1))}$}}: The bid update stochastic process $B_s$ \emph{at the particular time} $s$ of an agent $i$ specifies how much electricity that agent intends to purchase (negative if supplying) in every time period from that time $s$ till the final time $T-1$ in response to future events. As above, for illustratory purposes, we assume that $N(\cdot)$ is observed causally by all agents. Then, this bid function of agent $i$ is a function which specifies to the ISO, at any time $s$, as a function of the past history of observed noise $N(\tau), \tau <s$, how much electricity it will purchase at each instant in the future under different future uncertainties. In Fig.~\ref{extenform}, the bid function of agent $i$ specifies, for each node in the tree, the amount of electricity that it is willing to purchase if and when the system passes through that node.
{\bf{The Price Update Stochastic Process $\mathbf{\lambda_s = (\lambda_s(s), \lambda_s(s+1), \ldots , \lambda_s(T-1))}$}} is a stochastic process announced by the ISO at time $s$. Assuming that the noise process $N( \cdot )$ is observed causally by all the agents, it specifies for each time $s \leq t \leq T-1$, as a function of the past history of observed noise $N(\tau), \tau < s$, the price $\lambda_s(t)$ at which electricity will be sold in the market at time $t$ under different future uncertainties. In the tree of Fig.~\ref{extenform}, it corresponds to a price corresponding to each node of the tree at levels $s$ through $T-1$.
{\bf{$\mathbf{k}$-th Bid Update at time $\mathbf{t}$:}} Suppose that the ISO has declared a price process $\lambda^k_s$ at time $s$, where $k$ is an index that we will use for iteration. In the \textit{Bid Update}, each agent $i$ changes its bid $B^k_s$ in response to the price function $\lambda^k_s$ by solving the following problem, dubbed Agent $i$'s Problem, \begin{align} &\min_{U_i \mbox{\scriptsize{ s.t. }} (\ref{constraints-at-time-t-1},\ref{constraints-at-time-t-2})} \mathbb{E} [ \sum_{t=s}^{T-1} [ c_i(X_i(t),U_i(t),t) + e_i(U_i(t),t) \notag \\
& \quad \quad \quad \quad\quad \quad \quad + \lambda^k_s(t)^T E_i(t) U_i(t) ] | \mathcal{F}_{i,s} ]. \label{comp-obs-prob} \end{align}
{\bf{$\mathbf{(k+1)}$-th Price Update at time $\mathbf{s}$:}} The ISO updates the price process in response to the agents' bids.
Guided by the ``excess consumption function" $\sum_{i=1}^M E_i(t) U^k_{i,s}(t) - d(t)$, it raises or lowers prices to satisfy the general linear constraint:
\begin{equation} \lambda^{k+1}_s(t) = \lambda^{k}_s(t) + \frac{1}{k} [\sum_{i=1}^M E_i(t) U^k_{i,s}(t) - d(t)], \quad s \leq t \leq T-1. \label{priceupdate}
\end{equation}
{\bf{The Final Averaged Allocations.}} At time $s$, after the prices have converged, i.e., $\lambda^{\star}_s(t) = \lim_{k \to \infty} \lambda^{k}_s(t)$ for $s \leq t \leq T-1$, the ISO announces the allocations as the limits $U^\star_{i,s}(t) := \lim_{k \to \infty}\bar{U}^k_{i,s}(t)$ for $s \leq t \leq T-1$, of the following averaged bids, \begin{align} &\bar{U}^k_{i,s}(t) = \frac{\sum_{s=1}^{k-1}s^\theta}{\sum_{s=1}^{k}s^\theta} \bar{U}^{k-1}_{i,s}(t) + \frac{k^\theta}{\sum_{s=1}^{k}s^\theta} U^{k}_{i,s}(t), \label{averaged-stock-bids} \end{align} with $\bar{U}^0_{i,s}(t)= U^0_{i,s}(t)$. If the unaveraged agent bids converge, then their limit is the same as the above. As in Section~\ref{sec-static}, under appropriate conditions the limits of the prices and the averaged bids do exist.
\section{Stochastic Systems with Common Uncertainties: State Prices and Bidding} \label{sec-common} Now we analyze how the above Iterative Bidding Scheme functions in the case of common uncertainties, i.e., $N(t) \equiv N_c(t)$, and there are no private noises $N_i$.
Denote the combined state of the system by $X(t) := (X_1(t), X_2(t), \ldots , X_M(t))$, and the combined actions by $U(t) := (U_1(t), U_2(t), \ldots , U_M(t))$.
At each time $s$, a sequence of iterative \emph{tentative} price announcements by the ISO for each node at or below the current node at level $s$, followed by \emph{tentative} bids by all agents for such nodes responding optimally to the price announcement, takes place, until they converge.
At each iteration, the ISO revises the tentative price announcement to drive the ``excess consumption" at each node towards zero, and agents respond optimally according to their own cost-to-go function. This iteration of tentative prices and tentative bids continues till the prices converge. At that point
the agents consume/generate the weighted average amount they bid for the particular node occupied at time $s$.
The system then moves forward to time $s+1$, arriving at a random node at level $s+1$ according to $N(s)$, and the entire process is repeated. This is in the same fashion as Model Predictive Control.
This process is a dynamic modification of Arrow's~\cite{arrow} approach of treating each ``good" available at a certain time and place as a separate good. Since agents do not know each other's dynamics or states or actions there is the added critical proviso of bidding for future ``time-places" by each agent, with only the current price being actually implemented a la Model Predictive Control.
\begin{assumption}[Assumption for stochastic case]\label{finite-compact-slater-stochastic} (i) There is an optimal solution of (\ref{p0}) with finite cost. \\ (ii) $ \sum_{t=0}^{T-1} c_i(X_i(t),U_i(t),t)$, $\sum_{t=0}^{T-1} e_i(U_i(t),t)$, and $h_i(N_i^t,N_c^t,t,U_i(t))$ are convex in $U_i^{T-1}$ for each noise sequence $N^{T-1}$, with $X_i(t)$ a function of $U_i^{t-1}$ and $N^t$.
\\ (iii) For each fixed noise sequence $N_c^t,N_i^t$, there exists a feasible $\bar{u}$ satisfying $h_i(,t,\bar{u}_i(t)) < 0$ in $\mbox{RelInt(Dom}(c_i)) \cap \mbox{RelInt(Dom}(e_i))$ for $\mbox{ for }1 \leq i \leq M, 0 \leq t \leq T-1$. For simplicity of exposition we suppose that the noise processes $N_c(t), N_i(t)$ assume only finitely many values, allowing them to be represented by a tree as in Fig.~\ref{extenform}. \end{assumption}
\begin{theorem} In the above common uncertainty case, the price-bid solution, with price updates~(\ref{priceupdate}), and bid updates determined as the optimal solution of (\ref{comp-obs-prob}), and allocations at each $t$ given by $\lim_{k \to \infty} \bar{U}^k$ where $\bar{U}^k$ is obtained as the averaged version of $U^k$ as in (\ref{ergodic_method}), achieves the maximum social welfare when the cost functions satisfy Assumption~\ref{finite-compact-slater-stochastic}.
\end{theorem}
\noindent \emph{Proof:}
Let us suppose that $x(0)$ is fixed, without loss of generality. Let $p_v$ denote the probability of node $v$ in the uncertainty tree. The depth of the node in the tree indicates time. Now, a Markov policy~\cite{kumar} maps the system state and time to actions, thereby specifying an action $U(v) := (U_0(v), U_1(v),\ldots, U_M(v))$ satisfying $\sum_{i=1}^M E_i(v) U_i(v) = d(v)$ for every node $v$ in the tree. This is easily seen to be true by noting that each node in the uncertainty tree also indicates the state of the system at that time. Now let us consider a more general ``\emph{tree policy}" that specifies a $U(v) := (U_0(v), U_1(v),\ldots, U_M(v))$ satisfying $\sum_{i=1}^M E_i(v) U_i(v) = d(v)$ for every node $v$ in the tree. The class of tree policies is more general than the class of Markov policies, since two nodes in the tree at the same depth may correspond to the same state $X(t)$ arrived at through differing noise realizations, but a tree policy can choose different actions for them. Since the class of Markov policies contains an optimal policy, it follows that the class of tree policies also contains one.
For every tree policy, for every node $v$, there is a unique sequence of actions $U^v := \left\{U(0),\ldots,U(t)\right\}$ taken in the preceding $t$ steps, where $t$ denotes depth of node $v$. The state $X(t)$ corresponding to $v$ is thereby determined by $(v,U^v)$. The problem~(\ref{p0}) can be written equivalently as the following optimization problem, \begin{align} & \mbox{Min} \sum_{i=1}^{M}\sum_v p_v [c_i \left(v,U^v\right)+ e_i \left(U^v\right) ] \notag \\ & F_i(v)U_i(v) \leq g_i(v) + \sum_{\{v': v' \mbox{ \scriptsize{a predecessor of} } v \}} C_{i}(v',v)U_i(v'), \label{stochastic-nodal-constraints-1} \\ &h_i(U_i(v),v) \leq 0, \label{stochastic-nodal-constraints-2} \\ & \mbox{ such that } \sum_{i=1}^M E_i(v) U_i(v) = d(v),\forall v. \notag \end{align} Under Assumption~\ref{finite-compact-slater-stochastic}, the convex programming problem has no duality gap. Let $\lambda(v)$ be the Lagrange multiplier for the constraint $\sum_{i=1}^M E_i(v) U_i(v) = d(v)$, and define the vector $\lambda : = \{\lambda(v) \}$. We obtain, \begin{align*}
\mathcal{L}\left(U, \lambda \right) := \sum_{i=1}^{M}\sum_v p_v & [ \sum_v c_i(v,u^v) + e_i(u^v) + \\ & \lambda (v)^T E_i(v) U_i(v) ] - \sum_v p_v \lambda (v)^T d(v). \end{align*} The process $\lambda(v)$ will be called the ``price process". Each agent submits a bid for each possible future realization $v$ of the noise process, while the ISO specifies a price at each $v$. The proof parallels the deterministic one.
$\square$
\section{Stochastic Systems with Private Uncertainties} \label{sec-private} Now we address the general case where agents have private uncertainties in addition to common uncertainty, i.e., $N(t) = (N_c(t), N_1(t), N_2(t), \ldots , N_M(t))$, where $N_c$ is a common uncertainty that is observed by all, but each $N_i$ is only observed by agent $i$. We will suppose that all the agents know the law of $\{ N(t) : 0 \leq t \leq T-1 \}$, but that the ISO knows only the labels of the noise. The agents do not know the dynamics or the cost/utility functions or states of the other agents.
The same Price-Bid iteration can be used, as detailed in Algorithm \ref{a2}, and the same result carries over. \begin{theorem} For the above system featuring private uncertainties as well as common uncertainties, the bidding process with ISO updating prices according to (\ref{priceupdate}), each agent $i$ updating its consumption/generation bid according to the optimal solution of (\ref{comp-obs-prob}), and allocations determined at each $t$ by the averaging as in (\ref{ergodic_method}), achieves the optimal social welfare under Assumption~\ref{finite-compact-slater-stochastic}.
\end{theorem}
\noindent \emph{Proof:} At time $0$, there are no private noises $N_i(-1)$, and so the above proof holds at time $0$. Noting this, it follows that the result also holds at each time $s \geq 1$ since the bid-price iteration is repeated at each such time, and we can simply regard $s$ as the new ``initial" time.
$\square$
\begin{algorithm} \caption{: Stochastic Dynamic Agents} \label{a2} \begin{algorithmic} \STATE \textbf{Assumption:} The law of the combined noise process $\mathcal{L}(N)$ is common knowledge of all agents and labels are known to the ISO.
\FOR{ bidding times $s=0$ to $T-1$} \STATE $k=0$ \STATE \REPEAT \STATE Each agent $i$ solves the problem \begin{align*} &\mbox{Min} \quad \mathbb{E}\sum_{t = s}^{T-1} [ c_i(X_i(t),U_i(t),t) + e_i(U_i(t),t) \\ & \quad \quad \quad \quad \quad \quad \quad \quad + \lambda^{k}(t)^T E_i(t) U_i(t) ] ,
\end{align*} with initial condition $X_i(s)$ to obtain the optimal $\{U^k_{i,s}(t),s\leq t\leq T-1\}$ subject to (\ref{stochastic-nodal-constraints-1},\ref{stochastic-nodal-constraints-2}), and submits it to ISO. \STATE The ISO declares new prices for $s \leq t \leq T-1$, i.e., \begin{align*} & \lambda^{k+1}(t) = \lambda^k(t) + \frac{1}{k} [\sum_{i=1}^M E_i(t) U^k_i(t) - d(t)]. \end{align*} \STATE $k\to k+1$ \UNTIL{$\lambda^k(t)$ converges a.s. to $\lambda^{\star}(t) \mbox{ for } s \leq t \leq T-1$.} \STATE ISO computes $\bar{U}^k_{i,s}(s)$ as in (\ref{averaged-stock-bids}), and implements $U^\star_{i,s}(s) := \lim_{k \to \infty}\bar{U}^k_{i,s}(s)$. \ENDFOR \end{algorithmic} \end{algorithm}
The assumption that the law of $\mathcal{L}(N(t))$ is common knowledge can potentially be relaxed by utilizing Stochastic Approximation~\cite{borkarbook,kushner,robbins}, so that agents can ``learn" them as the system evolves.
The major drawback of this algorithm is that it is exponentially complex in $T$ due to the number of states in the tree, even if each $N(t)$ is binary. In the next section we show that we can dramatically simplify the bidding process and solution in the LQG context.
\begin{figure}
\caption{Scheme for ISO Problem with LQG Agents.}
\label{flo}
\end{figure}
\section{The ISO Problem for LQG Agents} \label{sec-lqg} We will now show that in the LQG case one can meet both the stringent privacy and lack of knowledge constraints of other agents, and yet avoid the complexity of the solution in the general case where stochastic process bids need to be made for all future uncertainties. Only iterations between \emph{vectors of future prices} announced by the ISO, and \emph{vectors of future consumption/generation bids} by the agents are needed, similar to deterministic dynamic systems. Moreover, agents need not know the laws of the private uncertainties of other agents or anything at all about each other. In fact they do not even need to know of the existence of the others. Yet, optimal coordination can be achieved by the ISO, and that too in a tractable manner where agents bid at each time. This appears practically feasible with bid periods separated by minutes.
The only difference between the deterministic dynamic case treated in Section \ref{sec-deterministic} and the LQG case is that while in the former the bid-price iteration only needs to be carried out at time $0$, in the LQG case it needs to be carried out at each time $s$. This is similar to Model Predictive Control, where we only implement the first step of the prices and consumptions/generations at each time $s$.
The $M$ agents have linear dynamics affected by Gaussian noise and have quadratic costs. Externalities constituting a positive semidefinite quadratic plus a linear term in $U_i$ could be included, but are omitted below for simplicity. Initial conditions and noises are Gaussian: $X_i(0) \sim N(0, \Sigma_{i,0})$ and $N_i(t) \sim N(0, P_{i,t})$, and independent of all others. The cost functions of agents, are quadratic, with
$Q_i \geq 0$ and $R_i >0$. The ISO Problem is: \begin{align} & \mbox{Min} \ \mathbb{E} \sum_{t=0}^{T-1}\sum_{i=1}^{M} [ X^\intercal_i(t)Q_i X_i(t) + U^{\intercal}_i(t) R_i U_i(t) ] \label{lqgisocost} \\ &X_i(t+1) = A_iX_i(t) +B_i U_i(t) + N_i(t), \mbox{ and } (\ref{constraint-on-balancing}) \label{lqgisosystem}.
\end{align} The case of time-varying systems is entirely analogous.
Agents have no knowledge of each other's presence. Agent $i$ does not know the value of $M$, the number of agents, the matrices $\left\{ A_j,B_j,Q_j,R_j, \Sigma_{j,0}, P_{j, \cdot} ) \right\}_{j\neq i}$ of other agents, the realizations of their state processes $\{ X_j (\cdot ), j\neq i \}$ or noises $\{ N_j( \cdot ), j\neq i \}$.
We propose and prove the convergence and optimality of an Iterative Bidding Scheme which is much simpler than that of Section~\ref{sec-common} in the following critical aspect. The bid function submitted at time $s$ specifying the quantity of electricity that agent $i$ is willing to purchase at times $\{ s,s+1,\ldots,T-1 \}$ is \emph{not} a function of the outcomes of the noise sequence $\{ N(t),t>s \}$. It is simply a vector $(u_i(s),u_i(s+1),\ldots,u_i(T-1))$ comprised of $T-s+1$ entries. The same is also true for prices. The ISO just specifies a vector $(\lambda(s), \lambda(s+1), \ldots , \lambda(T-1))$ of $T-s$ entries. Both are \emph{not} specified as functions of future uncertainties.
This makes it different from Arrow~\cite{arrow}.
The complexity of specifying prices or bids for all future events is avoided. Removal of future event-based bidding and prices leads to a drastic reduction in the complexity of the iterative scheme that arises even if all uncertainties were finite valued, let alone real valued as here.
Another simplifying feature is that the ISO need not average the bids as in the (\ref{ergodic_method}). The bids of agents converge at each time instant without averaging.
Note that even though the bid function at each time $s$ is not future event-based, it is determined afresh at each time. At each time $s$, the following iteration takes place: Each agent bids a vector of future generations/consumptions in response to prices announced by the ISO for future power, and the ISO updates the prices in return, until convergence. Hence, the converged prices and consumptions/generations do depend on the system states of the $M$ agents, and are therefore stochastic.
The key to showing the existence of such a simple bidding scheme lies in utilizing the certainty equivalence property of LQG systems~\cite{kumar}. \begin{algorithm} \caption{: ISO Problem with LQG Agents} \label{a3} \begin{algorithmic}
\FOR{bidding times $s=0$ to $T-1$} \STATE $k=0$ \STATE Initialize $\{ \lambda^k_s (t) : s \leq t \leq T-1 \}$ arbitrarily.
\STATE \REPEAT
\STATE Each agent $i$ solves the problem (\ref{agentibidlqrpricecosr})
for a \emph{deterministic} system (\ref{agentibidlqrsystem})
with initial condition $x_i(s) := X_i(s)$, where $X_i(s)$ is the state of the $i$-th agent at time $s$, and submits the optimal values, denoted $u_{i,s}^k(t)$, for $s \leq t \leq T-1$ to the ISO.
\STATE ISO updates the prices according to (\ref{isolqrprice2},\ref{stepsize}). \\
Increment $k$ by $1$. \UNTIL{$u^k_{i,s}(t)$ converges to $u^{\star}_{i,s}(t)$}, \STATE Implement $( u^\star_{1,s}(s), u^\star_{2,s}(s), \ldots , u^\star_{M,s}(s) )$ \ENDFOR
\end{algorithmic}
\end{algorithm}
The iterative bidding scheme is shown in Fig.~\ref{flo} and Algorithm \ref{a3}. For simplicity, consider only balance of energy. At time $s$, in response to the $(k-1)$-th iterate announced by the ISO of the price sequence $(\lambda^k_s(s),\lambda^k_s(s+1),\ldots,\lambda^k_s(T))$, agent $i$ announces the \emph{optimal open loop sequence} $(u^k_{i,s}(s), u^k_{i,s}(s+1),\ldots,u^k_{i,s}(T-1))$ for the following \emph{deterministic} Linear Quadratic Regulator (LQR) problem: \begin{align} &\min \sum_{t=s}^{T-1} [ x_i^\intercal (t)Q_i x_i(t) + u_i(t)^\intercal R_i u_i(t) + \lambda^k_s(t)u_i(t) ] \label{agentibidlqrpricecosr} \\ &\mbox{s.t. } x_i(t+1) = A_ix_i(t) + B_iu_i(t), \quad s \leq t \leq T-1, \label{agentibidlqrsystem} \\ &\mbox{with initial condition } x_i(s) := X_i(s). \label{agentibidlqrinitial} \end{align}
The price adjustment now is just for a vector of real numbers at each time $s$: \begin{align} & \lambda^{k+1}_s(t) = \lambda^k_s(t) + \alpha^k \sum_{i=1}^M u^k_{i,s}(t), s \leq t \leq T-1. \label{isolqrprice2} \\ &\mbox{ where } \alpha^k > 0, \lim_{k} \alpha^k =0, \sum_{k=0}^{\infty} \alpha^k = + \infty. \label{stepsize} \end{align}
At time $s$, the iterations in $k$ are continued till the price iterations $(\lambda^{k}_s(s),\lambda^{k}_s(s+1),\ldots,\lambda^{k}_s(T-1))$ converge to $(\lambda^{\star}_s(s),\lambda^{\star}_s(s+1),\ldots,\lambda^{\star}_s(T-1))$. Denote the corresponding limit of the input sequence of agent $i$ by $(u^{\star}_{i,s}(s),u^{\star}_{i,s} (s+1),\ldots,u^{\star}_{i,s} (T-1))$. The price at time $s$ is then set to $\lambda^{\star}_s(s)$ and each agent $i$ applies the input $u^{\star}_{i,s}(s)$. This is repeated at time $s+1$.
\begin{theorem} The bid-price iteration scheme (\ref{agentibidlqrpricecosr},\ref{agentibidlqrsystem},\ref{agentibidlqrinitial},\ref{isolqrprice2},\ref{stepsize}) achieves the optimal social welfare for the LQG ISO Problem (\ref{lqgisocost},\ref{lqgisosystem},\ref{constraint-on-balancing}).
\noindent \emph{Proof:} Let \begin{align*} & x:=(x_1,x_2,\ldots,x_M), u:=(u_1,u_2,\ldots,u_M),\\ & A := diag(A_1, A_2,\ldots,A_M), B:=diag(B_1,B_2,\ldots,B_M),\\ & Q=diag(Q_1,Q_2,\ldots,Q_M), R=diag(R_1,R_2,\ldots,R_M), \end{align*}
and consider the following \emph{deterministic constrained} LQR problem, with no noise, and featuring energy balance: \begin{align} & \min \sum_{t=0}^{T} [x^\intercal (t)Qx(t) + u^{\intercal}(t) R u(t) ] \label{fullcost:eq} \\ &\mbox{with } x(t+1) = Ax(t) +B u(t), \mbox{ and } (\ref{constraint-on-balancing}). \label{s:eq}
\end{align}
Since the state is affine in $u$, after substituting for the states, we have a positive definite quadratic programming problem with equality constraints. The Karush-Kuhn-Tucker matrix is nonsingular (Section 10.1 of \cite{boydbook}) since $R_i > 0$, and so there are unique $u^\star,\lambda^\star$ optimal for the primal and dual, respectively The Dual function is a differentiable concave quadratic function, and the subgradient method is actually a gradient method that converges under non-summability of step-sizes, without even requiring square summability (Section 2.5 of \cite{ber87}). The bids $u_i^k$ are affine functions of the prices $\lambda^k$. Since prices satisfy balancing, so does their limit. Hence this deterministic problem can be solved by the Bid-Price iteration (\ref{adjoinedcost},\ref{PriceIterate}) between the agents and the ISO, as shown for the deterministic problem in Section \ref{sec-static}, to obtain the optimal inputs $u(t)$ for $0 \leq t \leq T-1$.
However, at the particular time $s=0$ with $x_i(0)=X_i(0)$, the Bid-Price Iteration (\ref{agentibidlqrpricecosr},\ref{agentibidlqrsystem},\ref{agentibidlqrinitial}) and (\ref{isolqrprice2}) corresponds exactly to the same Bid-Price Iteration (\ref{adjoinedcost},\ref{systemconstraint}) and (\ref{PriceIterate}) as in Section \ref{sec-static}. Hence the end result of Algorithm \ref{a3} at time $s=0$ is the optimal action for (\ref{fullcost:eq},\ref{s:eq}), \begin{equation} u(0) = (u_1(0),u_2(0),\ldots,u_M(0)). \label{initial action} \end{equation}
Now note that due to energy balance, no matter how the first $(M-1)$ agents choose their consumptions/generations, agent $M$'s choice is forced to be \begin{equation} u_M(t) = - \sum_{i=0}^{M-1}u_i(t) \mbox{ for all } t, \label{solvedconstraint} \end{equation} due to the energy balance constraint. Hence one can substitute for $u_M(t)$ and obtain an equivalent standard, i.e., \emph{unconstrained}, deterministic LQR problem featuring only $(M-1)$ inputs $u_{\scriptsize \mbox{reduced}} := (u_1, u_2, \ldots , u_{M-1})$, where there is no energy balance constraint: \begin{align} & \min \sum_{t=0}^{T} [x^\intercal (t)Qx(t) + u_{\scriptsize \mbox{reduced}}^{\intercal}(t) R_{\scriptsize \mbox{reduced}} u_{\scriptsize \mbox{reduced}}(t) ] \label{reducedcost:eq} \\ &\mbox{subject to } x(t+1) = Ax(t) +B_{\scriptsize \mbox{reduced}} u(t), \label{reduceds:eq} \end{align} the \emph{deterministic reduced unconstrained} LQR problem.
For this problem (\ref{reducedcost:eq},\ref{reduceds:eq}), which is just a standard LQR Problem, the optimal solution is given by linear feedback
$u_{\scriptsize \mbox{reduced}}(0) = \Gamma_{\scriptsize \mbox{reduced}}(0) x(0)$,
where $\Gamma_{\scriptsize \mbox{reduced}}(\cdot)$ is the optimal feedback gain.
Noting that $u_M$ is linear in $u_{\scriptsize \mbox{reduced}}$, we deduce that for the full system (\ref{fullcost:eq},\ref{s:eq}) with all $M$ agents, the optimal solution for the deterministic constrained LQR problem with the energy balance constraint, is
$u(0) = \Gamma(0) x(0)$,
where $\Gamma(\cdot)$ is the optimal feedback gain obtained from $\Gamma_{\scriptsize \mbox{reduced}}$ through (\ref{solvedconstraint}).
Now consider the corresponding \emph{reduced unconstrained stochastic} LQG problem where there is white Gaussian noise in the state equations~(\ref{s:eq}): \begin{align} & \min E \sum_{t=0}^{T} [X^\intercal (t)QX(t) + U_{\scriptsize \mbox{reduced}}^{\intercal}(t) R_{\scriptsize \mbox{reduced}} U_{\scriptsize \mbox{reduced}}(t) ] \label{stochreducedcost:eq} \\ &\mbox{with } X(t+1) = AX(t) +B_{\scriptsize \mbox{reduced}} U_{\scriptsize \mbox{reduced}}(t) + N(t).\label{stochreduceds:eq} \end{align} By Certainty Equivalence~\cite{kumar}, the same linear feedback \emph{gain} as in the deterministic reduced LQR problem is also optimal. In particular, in state $X(0)=x(0)$ at time $0$, $U(0)=\Gamma(0)x(0)$ continues to be optimal. Thus $u(0)$ given by (\ref{initial action}) is optimal for (\ref{stochreducedcost:eq},\ref{stochreduceds:eq}).
However, reduced unconstrained \emph{stochastic} LQG problem (\ref{stochreducedcost:eq},\ref{stochreduceds:eq}) is equivalent to unreduced constrained LQG problem (\ref{lqgisocost},\ref{lqgisosystem},\ref{constraint-on-balancing}), and so the same $U(0)$ is optimal.
Thus the Bid-Price iteration scheme determines the optimal actions for the agents at time $0$. Our scheme (\ref{agentibidlqrpricecosr},\ref{agentibidlqrsystem},\ref{agentibidlqrinitial}) for the LQG problem repeats such a Bid-Price scheme iteration at each time $s=0,1,\ldots,T-1$. Each $X(s)$ can be regarded as an initial state for a subsequent system re-started at time $s$, and the above argument shows that the actions $U(s)$ that it results in for the agents at all times $s$ are also optimal, completing the proof.
$\square$ \end{theorem}
This result extends to LQG systems where each agent $i$ only has noisy observations $Y_i(t) = D_i X_i(t) + V_i(t)$, where $V_i$ are independent and Gaussian.
\section{Incorporating Additional Linear Constraints: The DC Optimal Power Flow} \label{sec-powerflow} Besides energy balance, there are additional constraints of interest. An important one is ensuring that the power flows are delivered over the network.
These constraints are captured by the AC Power Flow Equations~\cite{bergen},
an
approximation of which leads to the so-called DC Power Flow equations that are linear constraints~\cite{bergen}. The bid-price iterations can be extended to encompass any such additional linear constraints. The only difference is that there are several prices, one for each constraint, that each agent needs to incorporate in choosing its actions.
\begin{theorem} Consider a system consisting of $M$ agents, where each Agent $i$'s system is a Linear Gaussian System: \begin{align*} &X_i(t+1) = A_iX_i(t) +B_i U_i(t) + N_i(t). \end{align*} Agent $i$ has a quadratic cost (negative utility): \begin{align*} & \min \mathbb{E} \left(\sum_{t=0}^{T-1} [ X^\intercal_i(t)Q_i X_i(t) + U^{\intercal}_i(t) R_i U_i(t) ] \right) . \end{align*} There are $N$ linear constraints that need to be satisfied: \begin{align*} &\sum_{i=1}^M \gamma_{i,n} U_i(t) =0 \mbox{ for } 1 \leq n \leq N, t=0,1,\ldots,T-1. \end{align*}
Neither ISO nor agents know the number $M$ or the dynamics/costs/law/states/noises of other agents.
Consider the following Bid-Multiple Price Iteration. At each time $s=0,1,\ldots,T-1$, at each iterate $k$, in response to prices $\{ \lambda^k_{n,s}(t): s \leq t \leq T-1 \}$, announced by the ISO, agent $i$ solves the deterministic LQR problem: \begin{align*} &\min \sum_{t=s}^{T-1} [ x_i^\intercal (t)Q_i x_i(t) + u_i(t)^\intercal R_i u_i(t) + \sum_{n=1}^N\lambda^k_{n,s}(t)u_i(t) ], \end{align*} with $x_i(s) = X_i(s)$, determines the optimal $\{ u^k_{s}(t): s \leq t \leq T-1 \}$, and communicates this sequence to the ISO. Upon receiving the bids at iterate $k$ from all the agents at time $s$, the ISO updates the $N$ price sequences: \begin{align*} & \lambda^{k+1}_{n,s}(t) = \lambda^k_{n,s}(t) + \alpha^k \left(\sum_{i=1}^M \gamma_{i,n} u^k_{i,s}(t)\right), \end{align*} for $1\leq n \leq N$ and $s \leq t \leq T-1$, with the step-sizes satisfying (\ref{stepsize}). The multiple iterations converge, and let $\{ \lambda^{\star}_{n,s}(t): s \leq t \leq T-1 \}$ denote the limit. Correspondingly let $\{ u^{\star}_{s}(t): s \leq t \leq T-1 \}$ denote the limits of the bids by the agents. At each time $s$, agent $i$ applies $U_i(s) = u^{\star}_{s}(s)$. Then this Bid-Multiple Price Iteration yields the maximum social welfare under the multiple constraints. \\
\noindent \emph{Proof:} The proof parallel the single constraint case.
$\square$ \end{theorem}
In the case of the DC Power Flow constraints, this yields the optimal \emph{stochastic dynamic} location marginal prices \cite{bohn1984optimal} that simultaneously take into account all the factors of location, dynamics and stochasticity.
\section{Simulation Examples} \label{sec-simu}
In the following, we use the space conditioning example from~\cite{schwepe2} for thermal inertial load agents. Let $S_1,S_2,S_3$ be sets of conditioning facilities (loads), conventional generators, and renewable suppliers, respectively, and let $i \in S_1, j \in S_2, k \in S_3$. The dynamics of the temperature $X_i(t)$ of the $i$-th facility is given by (\ref{facility}),
where $X^{O}(t)$ = the outside temperature at time $t$, $\epsilon=e^{-\tau/TC}$ = ``factor of inertia", TC = 2.5 hours = time-constant of the system, $\tau$ = time duration between control epochs, which is the same as the inter-bid duration, $\eta =2.5$ = thermal conversion efficiency, and $A=0.14 kW/\degree F$ = overall thermal conductivity. With $X_i^d(t)$ the desired facility temperature, the cost incurred is a quadratic in the temperature deviation.
For fossil-fuel generators, the unit-time conventional generation cost curves \cite{wood} for supplying energy are quadratic in generation $U_j$.
We replace hard constraints on ramp-rates $|U_j(t)-U_j(t-1)|$ by a quadratic penalty, with $C_3$ below chosen so that the hard bounds are met, with state given by (\ref{convene}).
For a renewable energy facility $k$, $B_k$ denotes its buffer capacity, $W_k(t)$ stochastic wind/solar energy, and $X_k(t)$ the renewable energy level satisfying (\ref{renlevel}).
Its operating cost is constant. The resulting ISO Problem~(\ref{p0}) is \begin{align} & \min \mathbb{E}\left\{ \sum_{i\in S_1} \sum_{t=0}^{T-1} \left(X_i(t)-X_i^d(t)\right)^2 \right.\notag \\ & \left.+\sum_{j\in S_2}\left( C_{j,1} U_j(t)+C_{j,2} U_j^2(t)+ C_{j,3} \left(U_j(t)-X_j(t)\right)^2 \right) \right\} \notag \\ &\mbox{such that } \sum_{\ell \in S_1 \cup S_2 \cup S_3}^M U_\ell(t) = 0, \mbox{ for }t=1,2,\ldots,T-1, \notag \\ & X_i(t+1) = \epsilon X_i(t) + (1-\epsilon)\left (X_i^{O}(t)+\frac{\eta}{A}U_i(t) \right ), \label{facility} \\
& X_j(t+1) = U_j(t), \label{convene} \\ & X_k(t+1) = \mbox{Min} \{X_k(t)-U_k(t)+W_k(t) ,B_k \} \label{renlevel}.
\end{align}
We will compare the performance of the proposed Stochastic Dynamic Optimal Bid-Price Iteration scheme of Sections \ref{sec-private} or \ref{sec-lqg}, called "Optimal" below, with the currently followed Static Dispatch scheme of Section~\ref{sec-static} used in dynamic situations as explained in Section~\ref{sec-intro}, under which the agents perform separate and uncoupled bid-price iterations at each time $t$ to optimize the static cost $C(X(t),U(t))$ incurred at that time $t$.
\noindent {\bf{Bidding with LQG Systems:}} A day is divided into twelve $\tau=2$ hour bid-slots, so $\epsilon= 0.4493$. There are only thermal loads,
and wind-farms which have a cost function $\frac{1}{2}X^2(t)$ and with infinite storage capacity $B$. Outside temperatures and available wind power are modeled as i.i.d. and normal. (This is only a first step towards modeling the uncertainty, and other types of distributions can potentially be similarly explored). Variance of wind energy is 1 unit for all $t$. The scenario is described in Table~\ref{table2}. At the beginning of day, the thermal loads have temperature of $70\degree F$, while wind-farms have 100 units of energy. The price vector is projected at each update onto a large compact set, and, at termination, the bid vector is projected onto the hyperplane $\sum_i U_i = 0$.
Figs.~\ref{fig11}-\ref{fig13} compare performance of the two schemes as the number of bid-price updates, the number of agents connected to the grid, and variance of wind energy process, are varied. Figs.~\ref{fig14} and \ref{fig15} show how the Optimal scheme is able to attain better social welfare for scale 2. \begin{table}[h]
\centering \includegraphics[width=0.75\linewidth]{Wind-table.pdf}
\caption{Mean outside and desired thermal load temperatures (in $\degree F$), and mean wind power for the 12 periods.} \label{table2} \end{table}
\begin{figure}
\caption{Cost, i.e., negative social welfare, vs. number of Bid-Price iterations with five thermal loads and two windfarms.}
\label{fig11}
\end{figure} \begin{figure}
\caption{Cost as number of users is scaled linearly by $i$, with ratio of thermal loads to windfarms held constant at $5/2$, and with $=15+5 i$ bid-price iterations at each time $t$.}
\label{fig12}
\end{figure} \begin{figure}
\caption{Cost vs. wind variance with five thermal loads and two windfarms, with 30 Bid-Price iterations.}
\label{fig13}
\end{figure}
\begin{figure}
\caption{Power generation: Optimal scheme ``predicts" the incumbent energy shortage in advance, thereby eliciting smoother generator response. The power production costs for the two schemes are, respectively, $4.37,28.06$ ($\times10^4$), while thermal loads disutility are $13.15,9.62$ ($\times10^4$). Adding these two costs, the net costs are $1.75,3.76$ ($\times10^5$), so that savings achieved by Optimal Scheme is $53.5\%$.}
\label{fig14}
\end{figure} \begin{figure}
\caption{Prices: Optimal scheme ``declares" the energy shortage/surplus well in advance, allowing users to react appropriately and eliciting demand response.}
\label{fig15}
\end{figure} \empty \noindent {\bf{Bidding in Tree Scenario:}} The time-horizon is $2$ and time duration between two bids is $5$ hours, roughly coinciding with morning ($7$ am)/$12$ noon, giving $\epsilon = 0.1353$.
Table~\ref{table1} lists stochasticity parameters of wind for two scenarios.
For fossil plants, $C_1=0.1,C_2=0.01,C_3=0.1$. Windfarms incur no operational cost. Bid/price vectors at $t=0$ have three entries, while they are scalar at time $t=1$. \begin{table}[h] \resizebox{8.5cm}{.65cm}{
\begin{tabular}{|l|l|l|l|l|l|l|} \hline
& $\mathbf{T^O(1),T^O(2)}$ & $\mathbf{T^d(1),T^d(2)}$& $\mathbf{W_1,W_2}$ & $\mathbf{P_1,P_2}$ & $\mathbf{S_1/S_2/S_3}$& $\mathbf{B}$ \\\hline $\mathbf{Case}$ $\mathbf{1}$ & $\mathbf{30,40}$ $\mathbf{(in}$ $\mathbf{\degree F)}$ & $\mathbf{60,80}$& $\mathbf{5,0}$ & $\mathbf{0.5,0.5}$& $\mathbf{7/1/1}$&$\mathbf{30}$ \\\hline $\mathbf{Case}$ $\mathbf{2}$ & $\mathbf{40,60}$ & $\mathbf{60,90}$& $\mathbf{10,0}$ & $\mathbf{0.95,0.05}$& $\mathbf{4/1/1}$&$\mathbf{40}$\\\hline \end{tabular} } \caption{
The only stochasticity is wind availability at time $1$, with possible realizations $W_1,W_2$ with respective probabilities $P_1,P_2$.
$|S_1|/|S_2|/|S_3|$ are the relative numbers of thermal loads, fossil plants and windmills. } \label{table1} \end{table}
Figures~\ref{fig8}-\ref{fig10} compare the costs averaged over multiple wind realizations of the two policies under various scenarios, for the two schemes. Thermal loads are allowed to become energy producers, while wind-farm operators are allowed to store energy in case there is excess energy supply in the market, showcasing how potential prosumer behavior in energy market. The particular prices and power generations for Scenario 1, are shown in Table~\ref{table3}.
\begin{figure}
\caption{Performance as a function of number of Bid-Price updates for the two scenarios in Table~\ref{table1}.}
\label{fig8}
\end{figure}
\begin{figure}
\caption{Cost as number of agents is increased linearly with scale, in the ratio $S_1/S_2/S_3$ shown in Table~\ref{table1}.}
\label{fig9}
\end{figure}
\begin{figure}
\caption{Cost as wind availability at $t=1$, and storage buffer at windfarms are increased. Buffer capacity in $i$-th simulation is $10i$, while wind energy $W(1)$ is $i$. Temperature conditions and agents are as in Table~\ref{table1}.
}
\label{fig10}
\end{figure}
\begin{table}[h]
\includegraphics[width=1\linewidth]{Market-clearing-prices-power-generation.pdf}
\caption{Prices, power generation and cost savings.} \label{table3} \end{table} \section{Concluding Remarks} \label{sec-concluding} The problem of maximizing the social welfare of a collection of dynamic stochastic agents is more complex than stochastic control since agents do not know the dynamical equations or utility functions of others. It is further complicated by its dynamic, stochastic, decentralized nature, since each agent's optimal choices depend on the probability distributions of future prices, which are affected by the unknown states and actions of all agents. Yet agents have to make decisions in real-time, as does the ISO since it needs to set prices before agents can decide.
We have exhibited iterative bidding schemes that attain the optimal performance of a centralized control policy that is aware of the dynamics, utilities, uncertainties and states of all agents, under appropriate compactness-convexity or LQG assumptions. It yields the optimal stochastic dynamic locational marginal prices.
The ISO critically exploits the sequential information obtained \emph{during} the iterative price-bid process to determine the optimal prices and generation/consumption allocations. This is the stochastic dynamic analog of bidding demand/supply curves in simple static settings, whence the ISO can simply intersect cumulative demand and production curves to determine the optimal price.
The social-welfare optimality can potentially result in significant economic benefits in energy markets. The results may be of interest to general equilibrium theory.
While the agents are all presumed to be ``price takers," the scheme can be expected to have some strategic robustness under some monotonicity assumptions. For example, in the static deterministic case, agents do not benefit from overbidding/underbidding which drives the price up/down, leading to net losses for the agent in either case. Examining this in a broader context, while shortening the bid iteration process, is an important issue.
\section*{Acknowledgment} The authors thank Pravin Varaiya for identifying a significant error in an earlier version of the paper.
\begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio,angle=270]{RahulSingh.pdf}}]{Rahul Singh} received the B.E. degree in electrical engineering from Indian Institute of Technology, Kanpur, India, in 2009, the M.Sc. degree in Electrical Engineering from University of Notre Dame, South Bend, IN, in 2011, and the Ph.D. degree in electrical and computer engineering from the Department of Electrical and Computer Engineering Texas A\&M University, College Station, TX, in 2015.
He is currently a Postdoctoral Associate at the Laboratory for Information Decision Systems (LIDS), Massachusetts Institute of Technology. His research interests include decentralized control of large-scale complex cyberphysical systems, operation of electricity markets with renewable energy, and scheduling of networks serving real time traffic.
\end{IEEEbiography}
\begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{P-R-Kumar.pdf}}]{P.~R.~Kumar} B. Tech. (IIT Madras, `73), D.Sc. (Washington University, St.~Louis, `77), was a faculty member at UMBC (1977-84) and Univ.~of Illinois, Urbana-Champaign (1985-2011). He is currently at Texas A\&M University. His current research is focused on stochastic systems, energy systems, wireless networks, security, automated transportation, and cyberphysical systems.
He is a member of the US National Academy of Engineering and The World Academy of Sciences. He was awarded a Doctor Honoris Causa by ETH, Zurich. He as received the IEEE Field Award for Control Systems, the Donald~P.~Eckman Award of the AACC, Fred~W.~Ellersick Prize of the IEEE Communications Society, the Outstanding Contribution Award of ACM SIGMOBILE, the Infocom Achievement Award, and the SIGMOBILE Test-of-Time Paper Award. He is a Fellow of IEEE and ACM Fellow. He was Leader of the Guest Chair Professor Group on Wireless Communication and Networking at Tsinghua University, is a D. J. Gandhi Distinguished Visiting Professor at IIT Bombay, and an Honorary Professor at IIT Hyderabad. He was awarded the Distinguished Alumnus Award from IIT Madras, the Alumni Achievement Award from Washington Univ., and the Daniel Drucker Eminent Faculty Award from the College of Engineering at the Univ.~of Illinois. \end{IEEEbiography}
\begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{Xie-Photo-eps-converted-to.pdf}}]{Le Xie} (S'05-M'10-SM'16) received the B.E. degree in electrical engineering from
Tsinghua University, Beijing, China, in 2004, the M.Sc. degree in engineering
sciences from Harvard University, Cambridge, MA, in 2005, and the
Ph.D. degree in electrical and computer engineering from Carnegie
Mellon University, Pittsburgh, PA, in 2009.
He is currently an Associate Professor with the Department of
Electrical and Computer Engineering, Texas A\&M University, College
Station. His
research interest includes modeling and control of large-scale
complex systems, smart grid application with renewable energy
resources, and electricity markets.
\end{IEEEbiography}
\end{document} | arXiv |
B. Parent • AE23815 Heat Transfer
2009 Heat Transfer Midterm Exam
April 23rd 2009
NO NOTES OR BOOKS; USE HEAT TRANSFER TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; TOTAL POINTS: 100 PTS.
25 pts. Fins are frequently installed on tubes by a press-fit process. Consider a circumferential aluminum fin having a thickness of 1.0 mm to be installed on a 3.0-cm-diameter aluminum tube. The fin length is 1.5 cm, and the contact conductance may be taken from the tables for a 100-$\mu$inch ground surface. The convection environment is at $30^\circ$C, and $h=140$ W/m$^2\cdot^\circ$C.
(a) Calculate the heat transfer for each fin for a tube wall temperature of $200^\circ$C.
(b) What percentage reduction in heat transfer is caused by the contact conductance?
25 pts. A plate of stainless steel (18% Chrome, 8% Nickel) has a thickness of 4.0 cm and is initially uniform in temperature at $500^\circ$C. The plate is suddenly exposed to a convection environment on both sides at $80^\circ$C with $h=250~{\rm W/m}^2 \cdot ^\circ$C.
(a) Calculate the time for the center temperature to reach $100^\circ$C.
(b) Calculate the time for the face temperatures to reach $100^\circ$C.
30 pts. A solid sphere of radius $r_0=1.0~{\rm m}$ is buried in earth. The center of the sphere is at a distance of 5.0 m below the earth's surface. The thermal conductivity of the earth is $k_{\rm earth}=1.2 ~{\rm W/m \cdot {^\circ}C}$, and its surface temperature is $25^\circ {\rm C}$. Inside the sphere, there is a volumetric rate of heat generation given by $S=S_0 [1- (r/r_0)^2 ]$ with $S_0=3750 ~{\rm W/m^3} = {\rm constant}$. The thermal conductivity of the sphere material is $k_{\rm sphere}=25 ~{\rm W/m\cdot {^\circ}C}$. At the interface between the earth and the solid sphere, there is a thermal contact conductance coefficient of $h_{\rm c}=500~{\rm W/m^2 \cdot {^\circ}C}$. Assuming that the temperature distribution inside the sphere is essentially 1-D radial, calculate the following for steady-state conditions:
(a) the temperature of the earth at the interface between the earth and the sphere;
(b) the surface temperature of the sphere;
(c) the maximum temperature inside the sphere.
20 pts. A solid cube is heated to a uniform temperature of $520^\circ$C. It is then exposed to a convective cooling environment: $h=1000~{\rm W/m^2\cdot{^\circ}C}$ and $T_\infty=20^\circ$C. The length of each side of the cube is 0.05 m. The properties of the cube material are the following: $\rho=8000~{\rm kg/m}^3$, $c=1000~{\rm J/kg \cdot {^\circ}C}$, and $k=25~{\rm W/m\cdot{^\circ}C}$.
(a) How long does it take for the centre of each face of the cube to cool down to a temperature of $68.4^\circ$C?
(b) At $t=200$ s into the cooling process, the cube is wrapped completely in excellent insulation. Calculate its equilibrium temperature.
1. 64.5 W, 26.4 %
2. 939 s, 879 s
3. 400 C, 401 C, 418.5 C
4. 200 s, 75.5 C
PDF 1✕1 2✕1 2✕2 | CommonCrawl |
Association of leukocyte DNA methylation changes with dietary folate and alcohol intake in the EPIC study
F. Perrier1,
V. Viallon1,
S. Ambatipudi2,3,
A. Ghantous2,
C. Cuenin2,
H. Hernandez-Vargas2,
V. Chajès4,
L. Baglietto5,
M. Matejcic4,6,
H. Moreno-Macias7,
T. Kühn8,
H. Boeing9,
A. Karakatsani10,11,
A. Kotanidou10,12,
A. Trichopoulou10,
S. Sieri13,
S. Panico14,
F. Fasanelli15,
M. Dolle16,
C. Onland-Moret17,
I. Sluijs17,
E. Weiderpass18,19,20,21,
J. R. Quirós22,
A. Agudo23,
J. M. Huerta24,25,
E. Ardanaz24,25,26,27,
M. Dorronsoro28,
T. Y. N. Tong29,
K. Tsilidis30,
E. Riboli30,
M. J. Gunter4,
Z. Herceg2,
P. Ferrari1 na1 &
I. Romieu4 na1
Clinical Epigenetics volume 11, Article number: 57 (2019) Cite this article
There is increasing evidence that folate, an important component of one-carbon metabolism, modulates the epigenome. Alcohol, which can disrupt folate absorption, is also known to affect the epigenome. We investigated the association of dietary folate and alcohol intake on leukocyte DNA methylation levels in the European Prospective Investigation into Cancer and Nutrition (EPIC) study. Leukocyte genome-wide DNA methylation profiles on approximately 450,000 CpG sites were acquired with Illumina HumanMethylation 450K BeadChip measured among 450 women control participants of a case-control study on breast cancer nested within the EPIC cohort. After data preprocessing using surrogate variable analysis to reduce systematic variation, associations of DNA methylation with dietary folate and alcohol intake, assessed with dietary questionnaires, were investigated using CpG site-specific linear models. Specific regions of the methylome were explored using differentially methylated region (DMR) analysis and fused lasso (FL) regressions. The DMR analysis combined results from the feature-specific analysis for a specific chromosome and using distances between features as weights whereas FL regression combined two penalties to encourage sparsity of single features and the difference between two consecutive features.
After correction for multiple testing, intake of dietary folate was not associated with methylation level at any DNA methylation site, while weak associations were observed between alcohol intake and methylation level at CpG sites cg03199996 and cg07382687, with qval = 0.029 and qval = 0.048, respectively. Interestingly, the DMR analysis revealed a total of 24 and 90 regions associated with dietary folate and alcohol, respectively. For alcohol intake, 6 of the 15 most significant DMRs were identified through FL.
Alcohol intake was associated with methylation levels at two CpG sites. Evidence from DMR and FL analyses indicated that dietary folate and alcohol intake may be associated with genomic regions with tumor suppressor activity such as the GSDMD and HOXA5 genes. These results were in line with the hypothesis that epigenetic mechanisms play a role in the association between folate and alcohol, although further studies are warranted to clarify the importance of these mechanisms in cancer.
DNA methylation is a crucial epigenetic mechanism involved in regulating important cellular processes, including gene expression, cell differentiation, genomic imprinting, and preservation of chromosome stability. DNA methylation refers to the addition of methyl groups (–CH3) to the carbon-5 position of cytosine residues in a cytosine-guanine DNA sequence (CpG) by DNA methyltransferases. DNA methylation changes can be influenced by many factors including aging [17, 19] and environmental exposure such as smoking [1, 24] or specific dietary factors [35]. Experimental evidence suggests a link between B vitamins, including folate (vitamin B9), and epigenetic modifications [3]. B vitamins, especially folate, are essential components of one-carbon metabolism (OCM), the network of interrelated biochemical reaction in which a one-carbon unit is received from methyl donor nutrients and transferred into biochemical and molecular pathways essential for DNA replication and repair. Modifications in OCM can significantly impact gene expression and thereby cellular function [53].
Absorbed folate, circulating in the bloodstream, enters the OCM cycle in the liver where it is metabolized to 5-methyltetrahydrofolate (5-methylTHF) and converted into S-adenosylmethionine (SAM) after several successive transformation steps (Fig. 1). SAM is the methyl donor for numerous methylation reactions including the methylation of DNA, RNA, and proteins. The potential role of specific dietary factors including micronutrients such as folate, alcohol, and soya intake, in modifying breast cancer risk via epigenetic mechanisms, has been proposed [54], although evidence is still scarce and inconsistent.
Diagram of the one-carbon metabolism pathway. MS methionine synthase, MTHFR methylenetetrahydrofolate reductase, THF tetrahydrofolate, SAH S-adenosylhomocysteine, SAM S-adenosylmethionine
Alcohol intake affects epigenetic profiles [32]. Ethanol metabolism generates toxins that may directly lead to OCM dysfunction by reducing folate absorption, increasing renal excretion of folate, and inhibiting methionine synthase, the key enzyme in the generation of the methyl donor in the OCM [32, 33]. This antagonistic effect of alcohol on folate could plausibly increase the need of folate intake. Inadequate folate levels may result in abnormal DNA synthesis due to a reduced availability of SAM [27] and disrupted DNA repair and may, hence, influence cancer risk, including breast cancer [4, 60].
The epidemiological evidence linking dietary folate, alcohol intake, and epigenome modifications is, however, not well documented. Therefore, we investigated the relationships between dietary folate and alcohol intake with leukocyte DNA methylation patterns in the controls from the European Prospective Investigation into Cancer and Nutrition (EPIC) study on breast cancer. We complemented standard regression analysis with techniques for the identification of relevant methylated regions.
EPIC is a multicenter study that recruited over 521,000 participants, between 1992 and 2000 in 23 regional or national centers in 10 European countries (Denmark, France, Germany, Greece, Italy, The Netherlands, Norway, Spain, Sweden, and the UK) [43]. Among the 367,903 women recruited in EPIC, and after exclusion of 19,583 participants with prevalent cancers at recruitment (except non-melanoma skin cancer), first malignant primary BC occurred for 10,713 women during follow-up between 1992 and 2010. Within a nested case-control study that included 2491 invasive BC cases [34], a subsample of 960 women who completed dietary and lifestyle questionnaires and provided blood samples at recruitment (480 cases and 480 matched controls) from Germany, Greece, Italy, The Netherlands, Spain, and the UK was selected for the DNA methylation analyses [2]. The present study included analysis of 450 controls only originally enrolled in this case-control study on breast cancer (BC) nested within the EPIC study.
Methylation acquisition
Genome-wide DNA methylation profiles in buffy coat samples were quantified using the Illumina Infinium HumanMethylation 450K (HM450K) BeadChip assay [5] in 960 biospecimens from women included in the BC nested case-control study. A total of 20 biospecimens with replicates used to compare technical inter- and intra-assay batch effects and then excluded from the main analysis together with 19 matched pairs, i.e., 38 samples, where at least one of the two samples had a low-quality bisulfite conversion efficiency (intensity signal< 4000) or did not pass all of the Illumina GenomeStudio quality control steps, which were based on built-in control probes for staining, hybridization, extension, and specificity [23]. To prevent collider bias [11], as both alcohol intake and folate intake and DNA methylation profiles are all potentially associated with causes of BC, among the 902 remaining samples from the original case-control study on BC nested within EPIC study, only cancer-free women were selected for the present study. For the 451 controls sample, probes with detection p values higher than 0.05 were assigned "missing" value. After the exclusion of 14,548 cross-reactive probes [10], 47,963 probes overlapping known SNPs with minor allele frequency (MAF) greater than 5% in the overall population (European ancestry) [10] and 1483 low-quality probes (i.e., missing in more than 5% of the samples), 421,583 probes were left for the statistical analyses [2].
For each probe, β values were calculated as the ratio of methylated intensity over the overall intensity, defined as the sum of methylated and unmethylated intensities. The following preliminary adjustment steps were applied to β values: (i) color bias normalization using smooth quantile normalization [13], (ii) quantile normalization [6], and (iii) type I and type II bias correction using the beta-mixture quantile normalization (BMIQ) [56]. Then, M values, defined as \( {M}_{\mathrm{values}}={\log}_2\left(\frac{\beta_{\mathrm{values}}}{1-{\beta}_{\mathrm{values}}}\right) \), were computed [14]. Surrogate variable analysis (SVA) [30, 31] was used to remove systematic variation due to the processing of the biospecimens during methylation acquisition such as batch, indicating groups of samples processed at the same time, and the position of the samples within the chip [40]. Then M values were standardized to have an identical variance of 1.
The percentage of white blood cell counts, i.e., T cells (CD8+T and CD4+T), natural killer (NK) cells, B cells, monocytes, and granulocytes, was quantified using Houseman's estimation method [20] and included as covariates in the analysis.
Lifestyle and dietary exposures
Data on dietary habits were collected at recruitment through validated center- or country-specific dietary questionnaires (DQ) [43]. Northern Italy (Florence, Turin, and Varese), UK, Germany, and The Netherlands used self-administered extensive quantitative food-frequency questionnaires (FFQs), whereas Southern Italy (Naples and Ragusa), Spain, and Greece's centers used interview methods. Usual consumption of alcoholic beverages (number of glasses per day or week) per type of alcoholic beverage (wine, beer, spirits, and liquors) during the 12 months before the administration of dietary questionnaires was collected at recruitment. In addition, 24-h dietary recall (R) harmonized across EPIC countries was collected from a random sample (n = 36,900) in each center to be used as reference measurements [50]. R measurements were used to improve estimation of alcohol content per specific alcoholic beverages using a country-specific estimation of average of glass volume [48]. Dietary folate intake (μg/day) was estimated using the updated EPIC Nutrient Data Base (ENDB) [49], obtained after harmonization from country-specific food composition tables [7]. No specific information on the use of folate supplements was available.
After exclusion of one outlier value of dietary folate (value larger than the third quartile plus 10 times the inter-quartile range of the distribution), a total of 450 observations from controls only were retained for statistical analyses.
The association between dietary folate, alcohol intake, and methylation levels was evaluated via (i) CpG site-specific analysis, (ii) identification of differentially methylated regions (DMRs) [41], and (iii) fused lasso (FL) regression [57].
CpG site-specific models
M values expressing methylation levels at each CpG were linearly regressed on dietary folate (log-transformed to reduce skewness) and alcohol intake. Models were adjusted for recruitment center, age at recruitment (year), menopausal status (pre- or post-menopause), and white blood cell counts (proportions of T cells, natural killer cells, B cells, and monocytes in blood). False discovery rate (FDR) was used to control statistical tests for multiple testing.
For the two CpG sites that were associated with alcohol intake, based on q values, the percentage of methylation change for 1 standard deviation (SD) increase of alcohol intake was calculated as follows:
Methylation values in site j were log-transformed and regressed on alcohol intake (Ai), for each site j, and for i = 1, … , n, as:
$$ \log \left({M}_{ij}\right)={\alpha}_{0j}+{\alpha}_{1j}{A}_i+{\gamma_j}^T{Z}_i $$
where α1j estimate the regression coefficient, Zi is a vector of confounding factors related to methylation levels through a vector of regression coefficients γj. The ratio of any two log-transformed methylation values log(Mij1) and log(Mij0) with a difference of alcohol intake of 1 SD (\( {\widehat{\sigma}}_{\mathrm{alc}} \)) was predicted as \( {\widehat{\alpha}}_{1j}{\widehat{\sigma}}_{\mathrm{alc}} \). Therefore, the average percentage of methylation change for an increase of 1-SD in alcohol intake was estimated as:
$$ \frac{M_{ij1}}{M_{ij0}}=\left(\mathit{\exp}\left({\widehat{\alpha}}_{1j}{\widehat{\sigma}}_{\mathrm{alc}}\right)-1\right)\ast 100 $$
DMR models
Differentially methylated region (DMRs) analyses were identified with the DMRcate package [41]. The rationale of this method is to use kernel smoothing to replace the t test statistics at a given CpG site by a weighted average of t test statistics across its neighboring sites on the same chromosome. More precisely, let pc express the number of sites located on a given chromosome c with c ∈ {1, … , 23} (the 23rd chromosome is chromosome X). For any site k on this chromosome, with k = 1, … , pc, the term tk2 indicates the square of the t test statistics obtained in site-specific analyses. For each site j on chromosome c, tj2 is replaced by the term \( {{\widehat{t}}_j}^2 \), defined as \( {{\widehat{t}}_j}^2=\sum \limits_{k=1}^{p_c}{K}_{jk}{t_k}^{2.} \)
where the terms Kjk express weights, with larger values for sites k closer to j. Let xk express the position of site k on the chromosome, i.e., its chromosomal coordinate in base pairs, these weights are defined using a Gaussian kernel, as
$$ {K}_{jk}=\exp \left(\frac{-{\left|{x}_j-{x}_k\right|}^2}{2{\left(\lambda /C\right)}^2}\right) $$
where parameters λ and C represent the bandwidth and the scaling factor, respectively. Here, we used λ = 1000 and C = 2, respectively, as recommended in [41].
Under the null hypothesis of no association between site j and alcohol (or folate), the distribution of \( \frac{{{\widehat{t}}_j}^2{\sum}_k^{p_c}{K}_{jk}}{\sum_k^n{K_{jk}}^2} \) can be approximated by a χ2 distribution [41] with \( {\left({\sum}_k^{p_c}{K}_{jk}\right)}^2/{\sum}_k^{p_c}{K_{jk}}^2 \) degrees of freedom [45]. Accordingly, p values were obtained for each site separately in each chromosome and q values were computed using FDR correction on all the p values to control for multiple testing. Then, DMRs were defined as regions with at least two significant sites separated by a maximal distance λ of 1000 base pairs. In line with [41], t statistics tk were obtained from regression models using an empirical Bayes method to shrink the CpG site variance [51], as implemented in the limma package [52]. For each DMR, the minimum q value, the minimum and maximum coefficients (in absolute value) of the sites included in the region were presented as qDMR, βmin, DMR, and βmax, DMR.
Fused lasso regression
Multivariate penalized regression provides an alternative to DMRs. We implemented a fused lasso (FL) regression [57], which is better suited than the standard lasso when covariates (CpGs) are naturally ordered and the objective is to identify regions on the chromosome of differentially methylated CpG sites. FL is particularly useful when the number of features (p) is way larger than the sample size (n), a situation classically known as p ≫ n.
FL is a multivariable regression method combining two penalties: (i) the lasso penalty, which introduces sparsity of the parameter vector, i.e., many elements of the estimated vector are encouraged to be set to zero, and (ii) the fused penalty, which encourages sparsity of the difference between two consecutive components in the parameter vector, thus introducing smoothness of parameter estimates in adjacent CpG sites [57].
To mimic the DMR analysis, a FL analysis was implemented where dietary folate and alcohol were, in turn, regressed on CpG methylation levels within each chromosome. The vector of methylation coefficient estimates \( \widehat{\beta} \) obtained by fused lasso regression was defined as
$$ \widehat{\beta}=\arg \min \left\{{\sum}_i{\left({y}_i-{\sum}_j{M}_{ij}{\beta}_j-{\gamma}^T{Z}_i\right)}^2+{\widehat{\lambda}}_1{\sum}_{j=1}^{p_c}{\omega}_j\left|{\beta}_j\right|+{\widehat{\lambda}}_2{\sum}_{j=2}^{p_c}{\nu}_j\left|{\beta}_j-{\beta}_{j-1}\right|\right\}, $$
where yi indicates, in turn, alcohol and dietary folate values for sample i = 1, … , n, Mij is the methylation levels at CpG site j, βj is the associated regression coefficient, Zi is a vector of confounding factors, consistently with linear regression and DMR analyses described above, γ is the corresponding non-penalized vector of coefficients, and ωj and νj are the weights associated with lasso penalty and fused penalty, respectively.
Following the rationale of the adaptive lasso [61] and the iterated lasso [8], the FL procedure was run for the first time with weights ωj and νj set to 1, which returned \( {\widehat{\beta}}_0 \), an initial estimate of \( \widehat{\beta} \). The final estimates \( \widehat{\beta} \) were obtained after running a second FL procedure with weights defined as \( {\omega}_j=\frac{1}{\left|{\widehat{\beta}}_{0,j}\right|+\varepsilon } \) and \( {\nu}_j=\frac{1}{\left|{\widehat{\beta}}_{0,j}-{\widehat{\beta}}_{0,j-1}\ \right|+\varepsilon } \), with ε = 10−4.
The FL procedure was implemented on a predefined grid of 50 × 50 = 2500 values for the pair of parameters (λ1, λ2). More precisely, the grid for λ1 consisted of 50 equally spaced values (on a log scale) between \( \frac{\lambda_{1,\max }}{1000} \) and λ1, max, where λ1, max was the lowest λ1 value for which FL returned a null \( \widehat{\beta} \) vector for λ2=0, a situation where FL reduces to a standard lasso. For each value λ1on this grid, the grid for λ2consisted of 50 equally spaced values (on a log scale) between \( \frac{\lambda_{2,\max}\left({\lambda}_1\right)}{1000} \) and λ2, max(λ1), where λ2, max(λ1) was the lowest λ2 value for which FL returned a vector \( \widehat{\beta} \) with all components equal. The optimal pair of tuning parameters (λ1, λ2) was selected as the one minimizing the prediction error estimated by 5-fold cross-validation [16], whose principle can be summarized as follows. The original sample is first partitioned into 5 equally sized subsamples. One subsample is held as the test set while the other 4 are used as a training set, on which FL estimates are computed for the 2500 values for (λ1, λ2). The prediction error is computed on the test set, and the process is repeated 5 times, and for each of the 2500 values of (λ1, λ2). The prediction error is defined as the averaged prediction error on the 5 test sets. FL analysis was implemented using the FusedLasso package.
Preprocessing steps and statistical analyses were carried out using the R software (https://www.r-project.org/) and the Bioconductor packages [21], including lumi, wateRmelon, and sva [29] for the preprocessing steps. The nominal level of statistical significance was set to 5%.
Study population characteristics
Detailed characteristics of the 450 women included in the study are shown in Table 1. The average age at blood collection was 52 years (range 26–73). Participants had an average body mass index (BMI) of 26 kg/m2 (range 16–43) and were mostly post-menopausal (59%), never-smokers (56%), and moderately physically inactive (42%). The average daily intake of dietary folate was 270 μg/day (range 91–1012), and alcohol daily intake was 8 g/day (range 0–72). Non-alcohol consumers, defined as participants consuming less than 0.1 g/day of alcohol at recruitment, represented 15% of the population. Most participants were from the Italian and the German EPIC centers (Additional file 1: Figure S1).
Table 1 Characteristics of the study population (n = 450)
After FDR correction, dietary folate intake was not significantly associated with methylation levels at any CpG sites (data not shown). Alcohol intake was inversely associated with the cg07382687 CpG site (qval = 0.048) and positively associated with the cg03199996 site (qval = 0.029) (Table 2). Both sites were located in an open sea region, i.e., a genomic region of isolated CpGs. cg07382687 was within the body region of gene CREB3L2, and cg03199996 was within the body region of gene FAM65C.
Table 2 CpG site-specific model results for the significant CpG sites for alcohol intake (adjusted for recruitment center, age at recruitment, menopausal status, and level of different lymphocyte subtypes)
DMR analysis
A total of 24 regions associated with dietary folate were identified, which included 190 CpG sites over-represented in the TSS1500 and 1st exon regions and under-represented in the body regions and regions outside any gene regions (Fig. 2a). The 15 most significant regions are described in Table 3 and the whole list provided in Additional file 2: Table S1. Among the 24 DMRs, 54% showed an inverse association with dietary folate, i.e., had a βmax, DMR < 0. The DMR most significantly associated with dietary folate (qDMR = 1.3E−13, βmax, DMR = 0.019) was DMR.F1 in chromosome 7, including 49 CpG sites, related to HOXA5 and HOXA6 genes. DMR.F5 was associated with HOXA4, another gene of the homeobox family, (qDMR = 5.8E−4, βmax, DMR = − 0.016).
Repartition of gene regions (gene region feature category describing the CpG position, from UCSC. TSS200, 200 bases upstream of the transcriptional start site (TSS); TSS1500, 1500 bases upstream of the TSS; 5′UTR, within the 5′ untranslated region, between the TSS and the ATG start site; body, between the ATG and stop codon; irrespective of the presence of introns, exons, TSS, or promoters; 3′UTR, between the stop codon and poly A signal) among DMRs compare to their repartition within the Illumina 450K (the repartition of CpG sites was done among the 421,583 sites included in this study). a DMRs significant for folate. b DMRs significant for alcohol. c Illumina 450K
Table 3 The 15 most significant DMRs associated with dietary folate out of 24 significant DMRs (adjusted for recruitment center, age at recruitment, menopausal status, and level of different lymphocyte subtypes)
Alcohol intake was associated with methylation levels in 90 DMRs, including 550 CpG sites over-represented in TSS200, 1st exon, and 5′ untranslated regions (5′UTR) and under-represented in the body regions and the regions outside any gene regions (Fig. 2b). The 15 most significant DMRs are detailed in Table 4, and the full list is described in Additional file 3: Table S2. Alcohol intake was positively associated with methylation levels in 66% of the 90 DMRs. The two sites associated with alcohol intake in the CpG site-specific analyses were not included in any DMRs. The most significant DMR associated with alcohol consumption was DMR.A1, 9 sites within the GSDMD gene, (qDMR = 4.7E−14, βmax, DMR = 0.020).
Table 4 The 15 most significant DMRs associated with alcohol out of 90 significant DMRs (adjusted for recruitment center, age at recruitment, menopausal status, and level of different lymphocyte subtypes)
Methylation levels of each CpG site located in the two most significant DMRs for folate and alcohol, i.e.DMR.F1, DMR.F2, DMR.A1 and DMR.A2, are presented in Additional file 4: Figure S2 by tertiles of dietary folate and alcohol intake, respectively. Correlation heatmaps of CpG sites in DMR.A1, DMR.A2, DMR.F1, and DMR.F2 are displayed in Additional file 5: Figure S3, showing high levels of correlation among methylation levels within the DMR.F2 of dietary folate and the DMR.A2 of alcohol. Other regions showed less correlation, including the DMR.A1 of alcohol intake.
For dietary folate, we identified 71 FL regions, 50 presenting a positive association and 21 an inverse association. Three FL regions were overlapping the 15 most significant DMRs (Table 3). Seven out of 8 sites from a FL region within the GDF7 gene were included in the DMR.F2 (βFL = − 0.0029). All sites from a FL region associated with the PRSS50 gene were part of the DMR.F4 (βFL = − 0.0069). Six out of 7 sites from the FL region within the GPR19 gene were within the DMR.F9 (βFL = 0.0076). None of the 68 other FL regions were overlapping any folate-related DMRs.
For alcohol consumption, we identified 133 FL regions, 71 regions presenting a positive association and 62 an inverse association. Twenty-one regions were included in alcohol-related DMRs. Among them, 9 were overlapping 6 of the 15 most significant DMRs (Table 4). The situation where two close FL regions were part of the same DMR was observed 3 times in the 15 most significant alcohol-related DMRs. In particular, four and three sites from two FL regions located in chromosome 22 were included in DMR.A11, associated with genes SMC1B and RIBC2. All the 9 sites from a FL region were included in DMR.A9 (βFL = − 0.474).
Graphical representations of the DMRs, the FL regions, and their overlap are illustrated for each chromosome in Additional file 6: Figure S4 for dietary folate and Additional file 7: Figure S5 for alcohol intake. For dietary folate, most of FL regions were located in chromosome 3, chromosome 22, and chromosome X. A maximum of four DMRs located in the same chromosome was observed for chromosomes 2 and 3. As for alcohol intake, DMR and FL showed overlap mostly in chromosomes 6 and 22, with, respectively, 4 and 3 DMRs overlapping FL regions.
In this study of women from a large prospective cohort, we investigated the association of dietary folate and alcohol intake with leukocyte DNA methylation via three different approaches. The site-specific analysis aimed at identifying single CpG sites independently from each other, whereas DMR and FL analyses aimed at identifying regions of CpG sites using the inter-correlation between methylation levels in close sites, thus exploiting the potential of specific regions of the epigenome to show methylation activity related to lifestyle factors.
While site-specific analysis showed a lack of association between dietary folate, alcohol intake, and individual CpG sites, DMR and FL analyses identified regions of the epigenome associated with dietary folate or alcohol intake. These two sites are located within the body region of the genes FAMB65C and CREB3L2. The FAMB65C gene, also named RIPOR3, is a non-annotated gene. The CREB3L2 gene encodes a transcriptional activator protein and plays a critical role in cartilage development by activating the transcription of SEC23A [18]. Translocation of CREB3L2 gene, located on chromosome 7, and the FUS gene (fused in sarcoma) located on the chromosome 16 has been found in some tumors, including skin cancer and soft tissue sarcoma [37, 38].
Alcohol is known to alter DNA methylation, mostly because it contributes to deregulation of folate absorption, which can lead to a dysfunction of OCM [27]. In our study, alcohol intake was associated with 90 DMRs, some of which may have a role in specific carcinogenesis processes. For example, alcohol intake was inversely associated with methylation levels in DMR.A64 related to the MLH1 gene, which is frequently mutated in hereditary nonpolyposis colon cancer (HNPCC) [39]. A positive association between alcohol intake and methylation in the DMR.A79 was related to the TSPAN32 (tetraspanin 32) gene, also known as the TSSC6 gene, which is one of the several tumor suppressor genes located at locus 11p15.5 in the imprinted gene domain of chromosome 11 [28]. This locus has been associated with adrenocortical carcinoma, lung, ovarian, and breast cancers. Methylations within DMR.A1 were positively associated with alcohol intake, and the related GSDMD gene has also been suggested to act as a tumor suppressor [44]. Alcohol intake was also positively associated with DMR.A6 related to the gene ADAM32, which encodes a protein involved in diverse biological processes, such as brain development, fertilization, tumor development, and inflammation [36].
Several genes, associated with the 24 DMRs identified in our study for dietary folate, were possibly involved in biological processes leading to carcinogenesis. For example, dietary folate was positively associated with methylation in DMR.F16 related to the RTKN (rhotekin) gene, which interacts with GTP-bound Rho proteins. Rho proteins regulate many important cellular processes, including cell growth and transformation, cytokinesis, transcription, and smooth muscle contraction. Dysregulation of the Rho signal transduction pathway has been implicated in many forms of cancer such as bladder cancer, gastric cancer, and breast cancer [9, 15]. Dietary folate was also associated with methylation levels in DMR.F1 and DMR.F5 within the HOXA4, HOXA5, and HOXA6 genes, members of the HOX family, known to be associated with cellular differentiation [46]. Perturbed HOX gene expression has been implicated in multiple cancer types [47]. In addition, HOXA5 may also regulate gene expression and morphogenesis. Methylation of this gene may result in the loss of its expression and, since the encoded protein upregulates the tumor suppressor p53, may play an important role in tumorigenesis [55].
Results from site-specific and DMR analyses were generated with different analytical strategies: methylation levels in different sites were assumed independent in the former, with linear regression models fitted separately in each CpG site, while in the latter, the physical proximity of CpGs was exploited to identify specific regions of the epigenome with similar methylation activity, under the assumption that neighboring CpG sites may share relevant epigenetic information. FL analysis revealed some overlaps with DMRs, particularly for alcohol intake, where 9 FL regions were observed within the 15 most significant DMRs. Yet, the overlap between DMR and FL analyses is relatively low and their results deserve cautious interpretations as they have differences in analytical strategies. Unlike DMRs, FL does not take into account the physical distance between consecutive sites, but rather introduce smoothness of parameters estimated in adjacent mutually adjusted CpG sites. Methylation levels within a chromosome were mutually adjusted in FL regression, while in DMR analysis t test statistics were based on independent associations of methylation levels with folate and alcohol.
The association between folate and DNA methylation has been investigated at different stages of human life, in particular during fetal development and elderly, where folate is especially needed. A meta-analysis of mother-offspring pairs estimated the association between maternal plasma folate during pregnancy and DNA methylation in cord blood [25]. After FDR correction, maternal plasma folate was positively associated with methylation level at 27 CpG sites and inversely associated with methylation level at 416 CpG sites. None of these sites was observed in any of the 24 DMRs related to dietary folate in the present study. This might be explained by the lack of power to identify specific sites due to the sample size: over 2000 samples were included in Joubert's meta-analysis against 450 in our study. Then, different methods were used to assess folate intake, i.e., plasma folate against dietary folate.
An intervention study was conducted to evaluate the effects of long-term supplementation with folic acid and vitamin B12 on white blood cell DNA methylation in elderly subjects [26]. After the intervention of 2 years, 162 sites were significantly differentially methylated compared to baseline, versus 6 sites only for the placebo group. Folate and vitamin B12 were not significantly associated with methylation level in any CpG sites. Within the same study, 173 and 425 DMRs were identified for folate and vitamin B12, respectively. The gene HOX4, which was inversely associated with dietary folate in our study in DMR.F5, was the only region overlapping with the first 10 DMRs found in the intervention study [26]. However, a higher level of folic acid was observed in the intervention study: averages blood folate of 52 and 23 nmol/L in the intervention and placebo groups, respectively, compared to an average blood folate of 15 nmol/L in our study which might partly explain the different findings.
Within a recent meta-analysis including 9643 participants of European ancestry, aged 42 to 76 years with 54% women [32], 363 CpG sites were significantly associated with alcohol consumption, with 87% of these sites showing inverse associations. In our study, site cg02711608 was part of the 363 identified sites and was also included in DMR.A25 associated with gene SLC1A5. SLC1A5 gene encodes a protein which is a sodium-dependent amino acid transporter [42]. The important difference in the number of significant sites between the meta-analysis and the present study might mostly be explained by the larger study population size and the larger levels of alcohol intake observed in the meta-analysis [32]. Indeed, in the meta-analysis, composed of 46% of men, the medians of alcohol intake ranged from 0 to 14 g/day in the 10 European cohorts, while with a median of 3.5 g/day, alcohol intake was quite low in our study, which included only women. Lastly, cohort-specific approaches were used in the meta-analysis to remove technical variability, while the SVA approach was used in our study, which was shown to produce conservative findings compared to other normalizing techniques [40].
In our study, the sample size was relatively low (n = 450), and women only were included. With a median value of 3.5 g/day, a 95th percentiles equal to 31 g/day, and a percentage of non-consumers equal to 15%, alcohol intake displayed limited variability which potentially constrained the power of the study. In addition, questionnaire measurements used to assess dietary folate and alcohol intake are prone to exposure misclassification, which likely attenuated associations between lifestyle exposures and methylation levels. These elements may alone explain the lack of significant associations in our study. Further studies including men and women, possibly with larger sample size, are needed to further investigate the relationship between dietary folate, alcohol intake, and DNA methylation.
A major strength of this study was the use of ad hoc methodology for normalization of methylation data. Technical management of samples likely introduces systematic technical variability in methylation measurements that might compromise the accuracy of the acquisition process and, if not properly taken into account, could introduce bias in the estimation of the association of interest. The population used in this study included European women from the UK, Germany, Italy, Greece, The Netherlands, and Spain, implying a diversity of diet and lifestyle habits. Three approaches were used to evaluate the relationship between dietary folate, alcohol intake, and DNA methylation. The comparison between DMR and FL analyses was particularly relevant to identify regions of the genome associated with dietary folate and alcohol intake.
Alcohol was classified as group 1 carcinogen in 2012 by the IARC Monograph [22] and was associated with cancer of the upper aero-digestive tract, female breast, liver, and colorectum. Dietary folate has been recently inversely associated with the risk of breast cancer in EPIC [12], although the evidence is not conclusive [59]. Among the DMRs identified in this study for dietary folate or alcohol intake, several regions were associated with genes potentially implicated in cancer development, such as RTKN, the HOX family of genes, and the two tumor suppressor genes GSDMD and TSPAN32. Our study provides some evidence that dietary folate and alcohol intakes may be associated with carcinogenesis through a deregulation of epigenetic mechanisms, although our findings need to be replicated in future evaluations.
In this study, site-specific analyses served as a basis to explore more complex evaluations. By addressing the high dimensionality and complexity of DNA methylation, statistical techniques used in this work may prove useful for future epigenetic studies focusing on the relationship between lifestyle exposures, DNA methylation, and the occurrence of disease outcomes. These tools presented may be adapted to suit specific features of other -omics data.
Weak associations between alcohol intake and methylation levels at two CpG sites were observed. DMR and FL analyses provided evidence that specific regions of CpG sites were associated with dietary folate and alcohol intake, assuming that neighboring features share relevant epigenetic information. Folate and alcohol are known not only to be associated with breast cancer but also to have a mutually antagonistic role in the one-carbon metabolism. In some regions identified by DMRs or FL analysis, mapped genes are known to act as tumor suppressors such as the GSDMD and HOXA5 genes. These results were in line with the hypothesis that folate- and alcohol-deregulated epigenetic mechanisms might have a role in the pathogenesis of cancer.
BMI:
DMR:
Differentially methylated region
EPIC:
European Prospective Investigation into Cancer and Nutrition
FDR:
FL:
Fused lasso
HM450K:
Illumina Infinium HumanMethylation 450K
MAF:
Minor allele frequency
NK:
OCM:
One-carbon metabolite
S-Adenosylmethionine
SVA:
Surrogate variable analysis
Ambatipudi S, Cuenin C, Hernandez-Vargas H, Ghantous A, Le Calvez-Kelm F, Kaaks R, Herceg Z. Tobacco smoking-associated genome-wide DNA methylation changes in the EPIC study. Epigenomics. 2016;8(5):599–618. https://doi.org/10.2217/epi-2016-0001.
Ambatipudi S, Horvath S, Perrier F, Cuenin C, Hernandez-Vargas H, Le Calvez-Kelm F, Herceg Z. DNA methylome analysis identifies accelerated epigenetic ageing associated with postmenopausal breast cancer susceptibility. Eur J Cancer. 2017;75:299–307. https://doi.org/10.1016/j.ejca.2017.01.014
Ba Y, Yu H, Liu F, Geng X, Zhu C, Zhu Q, Zhang Y. Relationship of folate, vitamin B12 and methylation of insulin-like growth factor-II in maternal and cord blood. Eur J Clin Nutr. 2011;65(4):480–85. https://doi.org/10.1038/ejcn.2010.294.
Baglietto L, English DR, Gertig DM, Hopper JL, Giles GG. Does dietary folate intake modify effect of alcohol consumption on breast cancer risk? Prospective cohort study. Bmj. 2005;331(7520):807. https://doi.org/10.1136/bmj.38551.446470.06.
Bibikova M, Barnes B, Tsan C, Ho V, Klotzle B, Le JM. High density DNA methylation array with single CpG site resolution. Genomics. 2011;98. https://doi.org/10.1016/j.ygeno.2011.07.007.
Bolstad B. M. Probe level quantile normalization of high density oligonucleotide array data. 2001. Retrieved from http://bmbolstad.com/stuff/qnorm.pdf
Bouckaert KP, Slimani N, Nicolas G, Vignat J, Wright AJ, Roe M, Finglas PM. Critical evaluation of folate data in European and international databases: recommendations for standardization in international nutritional studies. Mol Nutr Food Res. 2011;55(1):166–80. https://doi.org/10.1002/mnfr.201000391.
Candès EJ, Wakin MB, Boyd SP. Enhancing sparsity by reweighted ℓ 1 minimization. J Fourier Anal Appl. 2008;14(5):877–905. https://doi.org/10.1007/s00041-008-9045-x.
Chen M, Bresnick AR, O'Connor KL. Coupling S100A4 to Rhotekin alters Rho signaling output in breast cancer cells. Oncogene. 2012;32:3754. https://doi.org/10.1038/onc.2012.383 https://www.nature.com/articles/onc2012383#supplementary-information.
Chen YA, Lemire M, Choufani S, Butcher DT, Grafodatskaya D, Zanke BW, Weksberg R. Discovery of cross-reactive probes and polymorphic CpGs in the Illumina Infinium HumanMethylation450 microarray. Epigenetics. 2013;8(2):203–09. https://doi.org/10.4161/epi.23470.
Cole SR, Platt RW, Schisterman EF, Chu H, Westreich D, Richardson D, Poole C. Illustrating bias due to conditioning on a collider. Int J Epidemiol. 2010;39(2):417–20. https://doi.org/10.1093/ije/dyp334.
de Batlle J, Ferrari P, Chajes V, Park J. Y, Slimani N, McKenzie F, Romieu I. Dietary folate intake and breast cancer risk: European prospective investigation into cancer and nutrition. J Natl Cancer Inst. 2015;107(1):367. https://doi.org/10.1093/jnci/dju367.
Du P, Kibbe WA, Lin SM. Lumi: a pipeline for processing Illumina microarray. Bioinformatics. 2008;24(13):1547–8. https://doi.org/10.1093/bioinformatics/btn224.
Du P, Zhang X, Huang C-C, Jafari N, Kibbe WA, Hou L, Lin SM. Comparison of beta-value and M-value methods for quantifying methylation levels by microarray analysis. BMC Bioinformatics. 2010;11:587. https://doi.org/10.1186/1471-2105-11-587.
Fan J, Ma LJ, Xia SJ, Yu L, Fu Q, Wu CQ, Tang XD. Association between clinical characteristics and expression abundance of RTKN gene in human bladder carcinoma tissues from Chinese patients. J Cancer Res Clin Oncol. 2005;131(3):157–62. https://doi.org/10.1007/s00432-004-0638-8.
Hastie T. The elements of statistical learning: data mining, inference, and prediction; 2009.
Heyn H, Li N, Ferreira HJ, Moran S, Pisano DG, Gomez A. Distinct DNA methylomes of newborns and centenarians. Proc Natl Acad Sci U S A. 2012;109. https://doi.org/10.1073/pnas.1120658109.
Hino K, Saito A, Kido M, Kanemoto S, Asada R, Takai T, Imaizumi K. Master regulator for chondrogenesis, Sox9, regulates transcriptional activation of the endoplasmic reticulum stress transducer BBF2H7/CREB3L2 in chondrocytes. J Biol Chem. 2014;289(20):13810–820. https://doi.org/10.1074/jbc.M113.543322.
Horvath S, Raj K. DNA methylation-based biomarkers and the epigenetic clock theory of ageing. Nat Rev Genet. 2018;19(6):371–84. https://doi.org/10.1038/s41576-018-0004-3.
Houseman EA, Accomando WP, Koestler DC, Christensen BC, Marsit CJ, Nelson HH, Kelsey KT. DNA methylation arrays as surrogate measures of cell mixture distribution. BMC Bioinformatics. 2012;13(1):1–16. https://doi.org/10.1186/1471-2105-13-86.
Huber W, Carey VJ, Gentleman R, Anders S, Carlson M, Carvalho BS, Morgan M. Orchestrating high-throughput genomic analysis with bioconductor. Nat Methods. (2015);12(2):115–21. https://doi.org/10.1038/nmeth.3252.
IARC Working Group on the Evaluation of Carcinogenic Risks to Humans. (2012). Personal habits and indoor combustions. Volume 100 E. A review of human carcinogens (1017–1606). Retrieved from https://www.ncbi.nlm.nih.gov/pubmed/23193840, https://www.ncbi.nlm.nih.gov/pmc/PMC4781577/.
Illumina (Producer). (2011). GenomeStudio/BeadStudio Software Methylation Module.
Joehanes R, Just AC, Marioni RE, Pilling LC, Reynolds LM, Mandaviya PR. London SJ. Epigenetic signatures of cigarette smoking. Circ Cardiovasc Genet. 2016;9(5):436–47. https://doi.org/10.1161/circgenetics.116.001506.
Joubert BR, den Dekker HT, Felix JF, Bohlin J, Ligthart S, Beckett E, London SJ. Maternal plasma folate impacts differential DNA methylation in an epigenome-wide meta-analysis of newborns. Nat Commun. (2016);7:10577. https://doi.org/10.1038/ncomms10577.
Kok DE, Dhonukshe-Rutten RA, Lute C, Heil SG, Uitterlinden AG, van der Velde N, Steegenga WT. The effects of long-term daily folic acid and vitamin B12 supplementation on genome-wide DNA methylation in elderly subjects. Clin Epigenetics. 2015;7;121. https://doi.org/10.1186/s13148-015-0154-5.
Kruman II, Fowler AK. Impaired one carbon metabolism and DNA methylation in alcohol toxicity. J Neurochem. 2014;129(5):770–80. https://doi.org/10.1111/jnc.12677.
Lee MP, Brandenburg S, Landes GM, Adams M, Miller G, Feinberg AP. Two novel genes in the center of the 11p15 imprinted domain escape genomic imprinting. Hum Mol Genet. 1999;8(4):683–90.
Leek JT, Johnson WE, Parker HS, Jaffe AE, Storey JD. The sva package for removing batch effects and other unwanted variation in high-throughput experiments. Bioinformatics. 2012;28(6):882–3. https://doi.org/10.1093/bioinformatics/bts034.
Leek JT, Storey JD. Capturing heterogeneity in gene expression studies by surrogate variable analysis. PLoS Genet. 2007;3. https://doi.org/10.1371/journal.pgen.0030161.
Leek JT, Storey JD. A general framework for multiple testing dependence. Proc Natl Acad Sci U S A. 2008;105(48):18718–23. https://doi.org/10.1073/pnas.0808709105.
Liu C, Marioni RE, Hedman AK, Pfeiffer L, Tsai PC, Reynolds LM, Levy D. A DNA methylation biomarker of alcohol consumption. Mol Psychiatry. 2016. https://doi.org/10.1038/mp.2016.192.
Mason JB, Choi S-W. Effects of alcohol on folate metabolism: implications for carcinogenesis. Alcohol. 2005;35(3):235–41. https://doi.org/10.1016/j.alcohol.2005.03.012.
Matejcic M, de Batlle J, Ricci C, Biessy C, Perrier F, Huybrechts I, Chajes V. Biomarkers of folate and vitamin B12 and breast cancer risk: report from the EPIC cohort. Int J Cancer. 2017;140(6):1246–59. https://doi.org/10.1002/ijc.30536.
Niculescu MD, Haggarty P. Nutrition in epigenetics: Wiley; 2011.
O'Leary NA, Wright MW, Brister JR, Ciufo S, Haddad D, McVeigh R, Pruitt KD. Reference sequence (RefSeq) database at NCBI: current status, taxonomic expansion, and functional annotation. Nucleic Acids Res. 2016;44(D1):D733–D745. https://doi.org/10.1093/nar/gkv1189.
Panagopoulos I, Storlazzi CT, Fletcher CD, Fletcher JA, Nascimento A, Domanski HA, Mertens F. The chimeric FUS/CREB3l2 gene is specific for low-grade fibromyxoid sarcoma. Genes Chromosomes Cancer. 2004;40(3):218–28. https://doi.org/10.1002/gcc.20037.
Patel RM, Downs-Kelly E, Dandekar MN, Fanburg-Smith JC, Billings SD, Tubbs RR, Goldblum JR. FUS (16p11) gene rearrangement as detected by fluorescence in-situ hybridization in cutaneous low-grade fibromyxoid sarcoma: a potential diagnostic tool. Am J Dermatopathol. 2011;33(2):140–3. https://doi.org/10.1097/IAE.0b013e318176de80.
Peltomaki P, de la Chapelle A. Mutations predisposing to hereditary nonpolyposis colorectal cancer. Adv Cancer Res. 1997;71:93–119.
Perrier F, Novoloaca A, Ambatipudi S, Baglietto L, Ghantous A, Perduca V, Ferrari, P. Identifying and correcting epigenetics measurements for systematic sources of variation. Clin Epigenetics. 2018;10(1):38. https://doi.org/10.1186/s13148-018-0471-6.
Peters TJ, Buckley MJ, Statham AL, Pidsley R, Samaras K, V Lord R, Molloy PL. De novo identification of differentially methylated regions in the human genome. Epigenetics Chromatin. 2015;8(1):1–16. https://doi.org/10.1186/1756-8935-8-6.
Pochini L, Scalise M, Galluccio M, Indiveri C. Membrane transporters for the special amino acid glutamine: structure/function relationships and relevance to human health. Frontiers Chem. 2014;2(61). https://doi.org/10.3389/fchem.2014.00061.
Riboli E, Hunt KJ, Slimani N, Ferrari P, Norat T, Fahey M, Saracci R. European Prospective Investigation into Cancer and Nutrition (EPIC): study populations and data collection. Public Health Nutr. 2002;5(6B):1113–24. https://doi.org/10.1079/phn2002394.
Saeki N, Usui T, Aoyagi K, Kim DH, Sato M, Mabuchi T, Sasaki H. Distinctive expression and function of four GSDM family genes (GSDMA-D) in normal and malignant upper gastrointestinal epithelium. Genes Chromosomes Cancer. 2009;48(3):261–71. https://doi.org/10.1002/gcc.20636.
Satterthwaite FE. An approximate distribution of estimates of variance components. Biometrics. 1946;2. https://doi.org/10.2307/3002019.
Seifert A, Werheid DF, Knapp SM, Tobiasch E. Role of Hox genes in stem cell differentiation. World J Stem Cells. 2015;7(3):583–95. https://doi.org/10.4252/wjsc.v7.i3.583.
Shah N, Sukumar S. The Hox genes and their roles in oncogenesis. Nat Rev Cancer. 2010;10(5):361–71. https://doi.org/10.1038/nrc2826.
Sieri S, Agudo A, Kesse E, Klipstein-Grobusch K, San-Jose B, Welch AA, Slimani N. Patterns of alcohol consumption in 10 European countries participating in the European Prospective Investigation into Cancer and Nutrition (EPIC) project. Public Health Nutr. 2002;5(6b):1287–96. https://doi.org/10.1079/phn2002405.
Slimani N, Deharveng G, Unwin I, Southgate DA, Vignat J, Skeie G, Riboli E. The EPIC nutrient database project (ENDB): a first attempt to standardize nutrient databases across the 10 European countries participating in the EPIC study. Eur J Clin Nutr. 2007a;61(9):1037–56. https://doi.org/10.1038/sj.ejcn.1602679.
Slimani N, Kaaks R, Ferrari P, Casagrande C, Clavel-Chapelon F, Lotze G, Riboli E. European Prospective Investigation into Cancer and nutrition (EPIC) calibration study: rationale, design and population characteristics. Public Health Nutr. 2007b;5(6b):1125–45. https://doi.org/10.1079/PHN2002395.
Smyth GK. Linear models and empirical Bayes methods for assessing differential expression in microarray experiments. Stat Appl Genet and Mol Biol. 2004;3.
Smyth GK. Limma: linear models for microarray data. In: Gentleman R, Carey V, Dudoit S, Irizarry R, Huber W, editors. Bioinformatics and computational biology solutions using R. and bioconductor. New York: Springer; 2005.
Szyf M. The implications of DNA methylation for toxicology: toward toxicomethylomics, the toxicology of DNA methylation. Toxicol Sci. 2011;120(2):235–55. https://doi.org/10.1093/toxsci/kfr024.
Teegarden D, Romieu I, Lelievre SA. Redefining the impact of nutrition on breast cancer incidence: is epigenetics involved? Nutr Res Rev. 2012;25(1):68–95. https://doi.org/10.1017/s0954422411000199.
Teo WW, Merino VF, Cho S, Korangath P, Liang X, Wu Rc, Sukumar S. HOXA5 determines cell fate transition and impedes tumor initiation and progression in breast cancer through regulation of E-cadherin and CD24. Oncogene. 2016;35(42):5539–51. https://doi.org/10.1038/onc.2016.95.
Teschendorff AE, Marabita F, Lechner M, Bartlett T, Tegner J, Gomez-Cabrero D, Beck S. A beta-mixture quantile normalization method for correcting probe design bias in Illumina Infinium 450 k DNA methylation data. Bioinformatics. 2013;29(2):189–96. https://doi.org/10.1093/bioinformatics/bts680.
Tibshirani R, Saunders M, Rosset S, Zhu J, Knight K. Sparsity and smoothness via the fused lasso. J. R. Stat. Soc. Ser. B Stat Methodol. 2005;67(1):91–108. https://doi.org/10.1111/j.1467-9868.2005.00490.x.
Wareham NJ, Jakes RW, Rennie KL, Schuit J, Mitchell J, Hennings S, Day NE. Validity and repeatability of a simple index derived from the short physical activity questionnaire used in the European Prospective Investigation into Cancer and Nutrition (EPIC) study. Public Health Nutr. 2003;6(4):407–13. https://doi.org/10.1079/phn2002439.
World Cancer Research Fund International, & American Institue for Cancer Research. (2017). Continuous update project report: diet, nutrition, physical activity and breast cancer. Retrieved from https://www.wcrf.org/sites/default/files/Breast-Cancer-2017-Report.pdf
Zhang S, Hunter DJ, Hankinson SE, Giovannucci EL, Rosner BA, Colditz GA, Willett WC. A prospective study of folate intake and the risk of breast cancer. JAMA. 1999;281(17):1632–7.
Zou H. The adaptive lasso and its oracle properties. J Am Stat Assoc. 2006;101(476):1418–29. https://doi.org/10.1198/016214506000000735.
The authors would like to thank the financial support provided by La Fondation de France for a doctoral fellowship. They are also grateful for all the women who participated in the EPIC cohort and without whom this work would not have been possible.
This work was supported by a doctoral fellowship from 'Fondation de France' (grant number 2015 00060737) to FP and the grants from the Institut National du Cancer (INCa, France, 2012-070 to IR and ZH), la Ligue nationale contre le cancer (to Z. Herceg). ZH was supported by the European Commission (EC) Seventh Framework Programme (FP7) Translational Cancer Research (TRANSCAN) Framework, the Fondation Association pour la Recherche contre le Cancer (ARC, France). In addition, this study was supported by postdoctoral fellowship to SA from the International Agency for Research on Cancer, partially supported by the EC FP7 Marie Curie Actions – People – Co-funding of regional, national and international programmes (COFUND). SA's work is supported by Cancer Research UK (grant number: C18281/A19169). SA work in the Medical Research Council Integrative Epidemiology Unit at the University of Bristol which is supported by the Medical Research Council and the University of Bristol (grant number: MC_UU_00011/1, MC_UU_00011/4 and MC_UU_00011/5).
The coordination of EPIC is financially supported by the European Commission (DG-SANCO) and the International Agency for Research on Cancer. The national cohorts are supported by German Cancer Aid, German Cancer Research Center (DKFZ), Federal Ministry of Education and Research (BMBF), Deutsche Krebshilfe, Deutsches Krebsforschungszentrum and Federal Ministry of Education and Research (Germany); the Hellenic Health Foundation (Greece); Associazione Italiana per la Ricerca sul Cancro-AIRC-Italy and National Research Council (Italy); Dutch Ministry of Public Health, Welfare and Sports (VWS), Netherlands Cancer Registry (NKR), LK Research Funds, Dutch Prevention Funds, Dutch ZON (Zorg Onderzoek Nederland), World Cancer Research Fund (WCRF), Statistics Netherlands (The Netherlands); Health Research Fund (FIS), PI13/00061 to Granada, PI13/01162 to EPIC-Murcia, PI13/02633 to EPIC-Navarra), Regional Governments of Andalucía, Asturias, Basque Country, Murcia and Navarra, ISCIII RETIC (RD06/0020) (Spain); Cancer Research UK (14136 to EPIC-Norfolk; C570/A16491 and C8221/A19170 to EPIC-Oxford), Medical Research Council (1000143 to EPIC-Norfolk, MR/M012190/1 to EPIC-Oxford) (UK).
The funders of the study had no role in study design, data collection, data analysis, data interpretation or writing of the manuscript.
For information on how to submit an application for gaining access to EPIC data and/or biospecimens, please follow the instructions at http://epic.iarc.fr/access/index.php
Ferrari P and Romieu I are joint senior authors.
Nutritional Methodology and Biostatistics Group, International Agency for Research on Cancer (IARC), World Health Organization, 150, cours Albert Thomas, 69372, Lyon CEDEX 08, France
F. Perrier
, V. Viallon
& P. Ferrari
Epigenetics Group, IARC, Lyon, France
S. Ambatipudi
, A. Ghantous
, C. Cuenin
, H. Hernandez-Vargas
& Z. Herceg
MRC Integrative Epidemiology Unit, Bristol Medical School, University of Bristol, Bristol, UK
Nutritional Epidemiology Group, IARC, Lyon, France
V. Chajès
, M. Matejcic
, M. J. Gunter
& I. Romieu
Department of Clinical and Experimental Medicine, University of Pisa, Pisa, Italy
L. Baglietto
Department of Preventive Medicine, Keck School of Medicine, University of Southern California/Norris Comprehensive Cancer Center, Los Angeles, CA, USA
M. Matejcic
Universidad Autonoma Metropolitana, Mexico City, Mexico
H. Moreno-Macias
Division of Cancer Epidemiology, German Cancer Research Center (DKFZ), Heidelberg, Germany
T. Kühn
Department of Epidemiology, German Institute of Human Nutrition (DIfE), Potsdam-Rehbrücke, Germany
H. Boeing
Hellenic Health Foundation, Athens, Greece
A. Karakatsani
, A. Kotanidou
& A. Trichopoulou
2nd Pulmonary Medicine Department, School of Medicine, National and Kapodistrian University of Athens, "ATTIKON" University Hospital, Haidari, Greece
1st Department of Critical Care Medicine and Pulmonary Services, University of Athens Medical School, Evangelismos Hospital, Athens, Greece
A. Kotanidou
Epidemiology and Prevention Unit, Fondazione IRCCS Istituto Nazionale dei Tumori, Milan, Italy
S. Sieri
Dipartimento di Medicina Clinica e Chirurgia, Federico II University, Naples, Italy
Cancer Epidemiology Unit, Department of Medical Sciences, University of Turin, Via Santena 7, Turin, Italy
F. Fasanelli
National Institute of Public Health and the Environment (RIVM), Centre for Health Protection (pb12), Bilthoven, The Netherlands
M. Dolle
Department of Epidemiology, Julius Center Research Program Cardiovascular Epidemiology, Utrecht, The Netherlands
C. Onland-Moret
& I. Sluijs
Department of Research, Cancer Registry of Norway, Institute of Population-Based Cancer Research, Oslo, Norway
E. Weiderpass
Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden
Genetic Epidemiology Group, Folkhälsan Research Center and Faculty of Medicine, University of Helsinki, Helsinki, Finland
Department of Community Medicine, University of Tromsø, The Arctic University of Norway, Tromsø, Norway
Public Health Directorate, Asturias, Spain
J. R. Quirós
Unit of Nutrition and Cancer, Cancer Epidemiology Research Program, Catalan Institute of Oncology-IDIBELL, L'Hospitalet de Llobregat, Barcelona, Spain
Department of Epidemiology, Murcia Regional Health Council, IMIB-Arrixaca, Murcia, Spain
J. M. Huerta
& E. Ardanaz
CIBER Epidemiology and Public Health CIBERESP, Madrid, Spain
Navarra Public Health Institute, Pamplona, Spain
E. Ardanaz
IdiSNA, Navarra Institute for Health Research, Pamplona, Spain
Public Health Direction and Biodonostia Research Institute and CIBERESP, Basque Regional Health Department, San Sebastian, Spain
M. Dorronsoro
Cancer Epidemiology Unit, Nuffield Department of Population Health, University of Oxford, Oxford, UK
T. Y. N. Tong
Department of Epidemiology and Biostatistics, School of Public Health, Imperial College London, London, UK
K. Tsilidis
& E. Riboli
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FP performed the statistical data analysis and drafted the manuscript. IR and PF developed the concept of the study with FP and contributed to draft the manuscript. SA and CC were responsible for the technical aspects of DNA methylation acquisition. IR and ZH conceived the epigenetics study in the nested case-control study on breast cancer and critically reviewed the manuscript. SA, AG, and HHV contributed to the interpretation of the results. LB, CV, MM, and MJG were involved in the data interpretation. All authors contributed to draft the final versions of the manuscript. All authors read and approved the final manuscript.
Correspondence to P. Ferrari.
The study was approved by the Ethical Review Board of the International Agency for Research on Cancer, and by the local Ethics Committees in the participating centres. This study was also conducted in accordance with the IARC Ethic Committee (Project No 10-22).
Figure S1. Sample size by recruitment centers. (PDF 10 kb)
Table S1. DMRs associated with dietary folate (log). (DOCX 19 kb)
Table S2. DMRs associated with alcohol intake. (DOCX 34 kb)
Figure S2. Graphical representation of the most 2 significant DMR of dietary folate and alcohol intake. The x-axis represents the position (hg 19 coordinates) of the CpGs included in the plotted DMR. Each tertile of dietary folate, alcohol intake, or their interaction is represented by different colors: green for T1, blue for T2, and red for T3. For all the CpGs included in the plotted DMR, the dashed lines are their 1st and 3rd quartiles of methylation levels and the points represent their median values. (PDF 33 kb)
Figure S3. Correlation heatmap of methylation levels inside the two most significant DMR of folate and alcohol. (PDF 43 kb)
Figure S4. DMRs and FL regions of folate in each chromosome. Dark blue rectangles represent DMRs and light blue FL regions. Overlaps between the two methods are represented by red points. Positive coefficients of the two methods are represented on the top part of each graphic, and negative coefficients are on the bottom part. Positive (negative) coefficients of DMRs were set to 0.5 (− 0.5) and positive (negative) coefficients of FL regions were set to 1 (− 1) to clearly differentiate DMRs from FL regions. The x-axis represents the rank of CpG sites according to their position on the chromosome. (PDF 12 kb)
Figure S5. DMRs and FL regions of alcohol in each chromosome. Dark blue rectangles represent DMRs and light blue FL regions. Overlaps between the two methods are represented by red points. Positive coefficients of the two methods are represented on the top part of each graphic and negative coefficients are on the bottom part. Positive (negative) coefficients of DMRs were set to 0.5 (− 0.5), and positive (negative) coefficients of FL regions were set to 1 (− 1) to clearly differentiate DMRs from FL regions. The x-axis represents the rank of CpG sites according to their position on the chromosome. (PDF 58 kb)
Perrier, F., Viallon, V., Ambatipudi, S. et al. Association of leukocyte DNA methylation changes with dietary folate and alcohol intake in the EPIC study. Clin Epigenet 11, 57 (2019) doi:10.1186/s13148-019-0637-x
Dietary folate
Alcohol intake
EPIC cohort
Lifestyle epigenetics | CommonCrawl |
\begin{document}
\title{Mass and Expansion of Asymptotically Conical K\"ahler Metrics} \thispagestyle{empty} \begin{abstract}
We prove an expansion theorem on scalar-flat asymptotically conical (AC) K\"ahler metrics. The ADM mass is known to be defined in ALE Riemannian manifolds. The concept of ADM mass can be generalized to AC cases. Consider an AC K\"ahler manifold with asymptotic to a Ricci-flat K\"ahler metric cone with complex dimension $n$. Assuming the weak decay conditions required for the mass to be well-defined, then each scalar-flat AC K\"ahler metric admits an expansion that the main term is given by the standard K\"ahler metric of the metric cone and the leading error term is of $O(r^{2-2n})$ with coefficient only depending on the ADM mass and its dimension. Besides, the mass formula by Hein-LeBrun \cite{hein2016mass} also can be proved in our setting. As an interesting application, a new version of the positive mass theorem will also be discussed in the cases of the resolutions of the Ricci-flat K\"ahler cones.
\end{abstract}
\section{Introduction and Statement of Main Theorems} Schwarzschild metric is known to be a solution of Einstein's vacuum equations. In general space-time $\mathbb{R}^{n+1}$ $(n\geq 3)$, the Schwarzschild metric has the following explicit expression. \begin{align*}
g = - (1- \frac{2m}{r^{n-2}})dt^2 + (1-\frac{2m}{r^{n-2}}) dr^2 + r^2 g_{S^{n-1}}, \end{align*} where the constant $m$ is the mass of the the Schwarzschild metric. One can easily observe that, by restricting to each space slice $t=c$, the Schwarzschild metric differs from the Euclidean metric by quantity of order $O(r^{2-n})$.
Motivated by Schwarschild metrics, Arnowitt, Deser and Misner \cite{PhysRev.122.997} first generalized the definition of the mass (ADM mass) to asymptotically flat metrics. Here, we apply the definition of mass to asymptotically locally Euclidean (ALE) manifolds. Let $(X,g)$ be an complete non-compact Riemannian manifold, we say $(X,g)$ is ALE if there is a compact subset $K \subseteq X$ such that $X-K$ has finitely many components, denoted by $X_{i,\infty}$ with $X-K = \bigcup_{i} X_{i,\infty}$ and each $X_{i,\infty}$ is diffeomorphic to $ (\mathbb{R}^n-B_R)/\Gamma_i$, where $B_R$ is a closed ball of radius $R$ and $\Gamma$ is a finite subset of $O(n)$. The metric $g$ is asymptotic to the Euclidean metric with rapid decay rate. Then, the ADM mass of the end $X_{i,\infty}$ is given by \begin{align} \label{admmass1}
m_i(g)= \lim_{r \rightarrow \infty} \frac{\Gamma(\frac{n}{2})}{4(n-1)\pi^{n/2}} \int_{S_r/\Gamma_i} (g_{ij,i}-g_{ii,j})n^j d\mu \end{align} where $S_r$ is the Euclidean sphere of radius $r$, $n$ is the outward Euclidean unit normal vector and $d\mu$ is the volume form induced by standard metric on $S_r$. Bartnik \cite{bartnik1986mass} and Chru\'{s}ciel \cite{Chruciel1985BoundaryCA} give the appropriate decay condition to make the ADM mass coordinate-invariant, for Sobolev case and H\"older case respectively. Here, we list the decay condition of H\"older case for the metric $g$ on each end $X_{i,\infty}$, \begin{enumerate} \item[(i)] the scalar curvature $R$ of $g$ belongs to $L^1$. \item[(ii)] the metric $g$ is asymptotic to the Euclidean metric $\delta_{ij}$ at the end with decay rate $-\tau$ for some $\tau>(n-2)/2$, \begin{align} \label{decayale}
g_{ij}= \delta_{ij} + O(r^{-\tau}) , \qquad |\nabla (\psi^{-1})^* g) |_{g_0} = O(r^{-\tau-1}). \end{align} \end{enumerate}
In this paper, one of main results is that the scalar-flat ALE K\"ahler metric of complex dimension $n$ is of decay rate $2-2n$, which has an expansion, $g_{ij} = \delta_{ij} + O(r^{2-2n})$ and the leading error term is determined by the ADM mass of $g$ (see theorem \ref{expanthm}). Furthermore, the expansion also works in scalar-flat AC K\"ahler cases if we replace the Euclidean metric by the standard metric on K\"ahler cone.
Rather than directly introduce the expansion theorem, we first introduce an important tool in K\"ahler geometry, the ddbar lemma. The ddbar lemma is a standard result in compact K\"ahler manifolds and it can be easily proved in ALE K\"ahler manifolds if we assume a fast decay condition (see \cite[theorem 8.4.4]{joyce2000compact}). In Colon-Hein \cite[theorem 3.11]{conlon2013asymptotically}, the ddbar lemma is generalized to asymptotically conical (AC) K\"ahler manifolds with a lower decay condition $(\text{only need }-\tau <0)$, with additional assumption of non-negative Ricci curvature. In the author's previous paper \cite[Proposition 3.6]{yao2020invariant}, the ddbar lemma can be proved in negative line bundle over K\"ahler C-spaces (compact simply-connected homogeneous K\"ahler manifolds) without any decay assumption at infinity. Here, we give a new version of ddbar lemma by revising the Colon-Hein's result and dropping the non-negative Ricci curvature assumption.
The ddbar lemma on the complement of $K\subseteq X$ will also be discussed in this paper. In Goto \cite[section5]{goto2012calabi} and Conlon-Hein \cite[appendix A]{conlon2013asymptotically}, ddbar lemma is proved in the case of complex dimension $n \geq 3$ with a trivial canonical bundle. In this paper, we derive a ddbar lemma on $X\backslash K$ without making the above assumptions, but with high decay rate error terms.
Let $(L, g_L)$ be a compact Riemannian manifold, the \textit{Riemannian cone} $C_L$ associated with $L$ is defined to be $L\times \mathbb{R}_{>0}$ with Riemannian metric $g_{0} = dr^2 + r^2 g_L$. A Riemannian cone $C_L$ is said to be \textit{K\"ahler} if there exists a $g_{0}$-parallel complex structure $J_0$ such that the corresponding fundamental form $\omega_0 = g_0(J_0\cdot, \cdot)$ is closed, in particular, $\omega_0 = i\partial \overline{\partial} r^2$. If we assume that the K\"ahler cone $(C_L, g_0, J_0)$ is Ricci-flat with $\dim_\mathbb{C} C_L =n$, then, according to the standard calculation in \cite[section 1.4]{sparks2010sasaki} and \cite[section 11.1]{boyer2008sasakian}, the link $L$ is Sasaki-Einstein with $\textup{Ric} g_L = 2(n-1) g_L$.
Let $(X, J, g)$ be an AC K\"ahler manifold asymptotic to $C_L$, where $(C_L, g_0)$ is a Ricci-flat K\"ahler cone with link $L$. Throughout the paper we always assume that the manifolds only have one end, then there exists a compact subset $K\subseteq X$ and $B_R =\{x\in C_L, r(x) < R\}$, such that $\psi: X-K \rightarrow C_L-B_{R}$ is a diffeomorphism satisfying the following decay conditions, \begin{enumerate} \item[(i)] the scalar curvature $R$ of $g$ belongs to $L^1$. \item[(ii)] the complex structure $J$ on $X$ decays to $J_0$, the induced almost complex structure from $C_L$. \item[(iii)] the metric $g$ is asymptotic to the reference metric $g_L$ at the end with decay rate $-\tau $ for some $\tau>0$, for $i=0,1,\ldots,k$, \begin{align} \label{decayac}
| \nabla^i ((\psi^{-1})^* g - g_0)|_{g_0} = O(r^{-\tau-i}). \end{align} \end{enumerate}
\begin{mtheorem} (ddbar lemma) \label{ddclem} Let $X$ be AC K\"ahler manifolds asymptotic to a Ricci-flat K\"ahler cone $C_L$. Let $k$ be a large positive integer, $\alpha \in (0,1)$ and $\delta > 0$, \begin{enumerate} \item[(i)] Let $\omega$ be a $d$-exact real $(1,1)$-form on $X$ satisfying the decay condition $\omega \in \mathcal{C}^{k,\alpha}_{-\delta}$. Then, there exists a real function $\varphi \in \mathcal{C}^{k+2,\alpha}_{2-\delta}$ such that $\omega = dd^c \varphi$. \item[(ii)] Let $\omega$ be a $d$-exact real $(1,1)$-form on the end $X\backslash K$. Then there exists a real function $\varphi \in \mathcal{C}^{k+2, \alpha}_{2-\delta}(X\backslash K)$ such that, \begin{align*}
\omega = dd^c \varphi + O(r^{-2n}). \end{align*} \end{enumerate} \end{mtheorem}
We generalize the definition of mass to AC Riemannian manifolds. Let $(C_L, g_0)$ be a Riemannian metric cone of link $(L, g_L)$ and $(X,g)$, a complete Riemannian manifold asymptotic to $(C_L, g)$ at infinity. The definition in (\ref{admmass1}) requires a coordinate system at infinity, which does not exist in general AC Riemannian manifolds. Hence, we start from a coordinate-free expression of mass. In Lee's book \cite[Section 3.1.3]{lee2019geometric}, the mass is defined to be \begin{align} \label{admmass2}
\mathfrak{m}(g) = \frac{1}{2(2n-1) \operatorname{Vol}(L)}\lim_{r \rightarrow \infty} \int_{L(r)} (\overline{\nabla}^j g_{ij}- (\operatorname{tr}_{g_0}g)_i) n^i d\operatorname{Vol}_{L(r)}, \end{align} where $\overline{\nabla}$ is the Levi-Civita connection with respect to $g_0$ and $n$ is the outer normal vector field on $L(r)$. In Hein-LeBrun \cite{hein2016mass}, the ADM mass in ALE K\"ahler manifolds is a quantity determined by the total scalar curvature and its topological data. A similar mass formula also holds in AC K\"ahler manifolds asymptotic to Ricci-flat K\"ahler cones. Let $\iota: H^2_c (X) \rightarrow H^2_{dR}(X)$ be the map induced by the natural embedding of chain complex $\Omega^{\bullet}_{c} \hookrightarrow \Omega^{\bullet}_{dR}$. Generally speaking, $\iota$ is not an isomorphism, but the first Chern class of $(X,J)$ always has a pre-image under $\iota$ (see Lemma \ref{cptclasslm}). Then, we have the following mass formula.
\begin{mtheorem} Let $(X, J, g)$ be an AC K\"ahler manifolds asymptotic to a Ricci-flat K\"ahler cone $(C_L, J_0, g_0)$ satisfying the decay condition (\ref{decayac}) with $\tau = n-1+ \epsilon$. Then, we have \begin{align} \label{admmass3} \mathfrak{m}( g) = -\frac{2\pi \langle \iota^{-1} c_1, [\omega]^{n-1} \rangle}{(2n-1) (n-1)! \operatorname{Vol}(L)} + \frac{1}{2(2n-1) \operatorname{Vol}(L)} \int_{X} R_g d\operatorname{Vol}_g, \end{align} where $[\omega]$ is the K\"ahler class of $(g,J)$, $c_1$ is the first Chern class of $(X,J)$ and $R_g$ is the scalar curvature of $g$ on $X$. \end{mtheorem}
An interesting application of the mass formula (\ref{admmass3}) is the positive mass theorem. The positive mass theorem is first proved in AE manifolds of lower dimension by Schoen, Yau \cite{schoen1979complete}. Afterwards, Witten \cite{witten1981new} develop a new method to prove the positive mass theorem on spin manifolds. And in K\"ahler cases, Hein-Lebrun \cite{hein2016mass} confirms the positive mass theorem on AE K\"ahler manifolds. However, in ALE cases, Lebrun \cite{Lebrun1988CounterexamplesTT} constructs the first counter-example of positive mass theorem. Here, we discuss a new version of positive mass theorem on resolution spaces of Ricci-flat K\"ahler cones with an isolated canonical singularity at vertex.
\begin{mtheorem} (positive mass theorem)
Let $(X, J)$ be a resolution space of a Ricci-flat K\"ahler cone $C_L$ such that the only singularity $O \in C_L$ is canonical. If $(X,g)$ has scalar curvature $R \geq 0$, then the mass $\mathfrak{m}(X,g) \geq 0$, and equals zero only if $(X,J,g)$ is a crepant resolution of $C_L$ with a scalar-flat K\"ahler metric $g$. \end{mtheorem}
Now, we can introduce the following expansion theorem in AC K\"ahler case.
\begin{mtheorem} \label{expanthm} Let $(X, J)$ be a AC K\"ahler manifold asymptotic to a Ricci-flat K\"ahler cone $(C_L, J_0)$ and we assume that $k$ be a large positive integer and $\tau = n-1+\epsilon$, \begin{enumerate} \item[(i)] Let $\omega_1$, $\omega_2$ be K\"ahler forms on $X$ with the corresponding metrics satisfying decay condition (\ref{decayac}) and $R_1$, $R_2$, the scalar curvature of $\omega_1$ and $\omega_2$. If \begin{enumerate}
\item[$\bullet$] $[\omega_1] = [\omega_2]$
\item[$\bullet$] $R_1 =R_2$, \end{enumerate} then $\omega_2 = \omega_1 + dd^c \varphi$ with the potential $\varphi \in \mathcal{C}^{k+2,\alpha}_{2-2\tilde{\tau}}$, for some $\tilde{\tau} >n-1$ depending on $(n, L, \tau)$. \item[(ii)] Let $\omega$ be a scalar flat K\"ahler form on X satifying decay condition (\ref{decayac}). And the complex structure $J$ is asymptotic to $J_0$ satisfying \begin{align} \label{cxdec} J= J_0+ O(r^{2-2n-\epsilon'}), \qquad \epsilon' >0, \end{align} Then, the scalar flat K\"ahler form $\omega$ admits the following expansion at infinity. In particular, outside a compact set of $X$, for complex dimension $n\geq 3$, we have \begin{align*} \omega = \frac{1}{2} dd^c r^2 + \frac{(2n-1)\mathfrak{m}(\omega)}{2(4-2n)(n-1)} dd^c r^{4-2n} + O(r^{-2\tilde{\tau}}), \end{align*} where $m(\omega)$ is the ADM mass of $\omega$ and $\tilde{\tau} > n-1$ only depends on $(n, L, \epsilon', \tau)$. And for complex dimension $n=2$, we have \begin{align*} \omega = \frac{1}{2} dd^c r^2 +\frac{3 \mathfrak{m}(\omega)}{2} dd^c \log r + O(r^{-2\tilde{\tau}}), \end{align*} where $\tilde{\tau} = \min\{\tau, 3/2\}$. \end{enumerate} \end{mtheorem}
As special cases of AC K\"ahler metrics, all ALE K\"ahler metrics satisfy the expansion theorem. According to the statement (i) of Theorem \ref{expanthm} in ALE K\"ahler cases, we can define K\"ahler potential spaces with relatively "good" decay rate. Precisely, by fixing a K\"ahler metric $\omega$, then, we can define the following potential space of ALE K\"ahler metrics, \begin{align*}
\mathcal{H}_{-\tau} ([\omega]) = \Big\{f \in \mathcal{C}^\infty_{-2 \tilde{\tau}}: \omega + i\partial\overline{\partial} f >0, \ \tilde{\tau} = \min\{\tau, n-\frac{1}{2}\} \Big\} \end{align*} In fact, $\mathcal{H}_{-\tau} ([\omega])$ does not contain all ALE K\"ahler metrics, but it is enough for "prescribed scalar curvature" problem, as all ALE K\"ahler metrics with the same scalar curvature as $\omega$ are contained in $\mathcal{H}_{-\tau} ([\omega])$. For the statement (ii), in ALE K\"ahler case, the fall-off condition of complex structure (\ref{cxdec}) is automatically satisfied. In fact, for $n\geq 3$, in the asymptotic chart, the complex structure $J$ coincides with the standard one $J_0$, and for $n=2$, in the asymptotic chart, $J = J_0 + O(r^{-3})$. One can check \cite[Lemma 2.3, Proposition 4.5]{hein2016mass} for details.
This paper is a part of Ph.D thesis of the author. The author would like to thank Professor Hans-Joachim Hein and Professor Bianca Santoro for suggesting the problem, and for constant support, many helpful comments, as well as much enlightening conversation. This work is completed while the author is supported by scholarship from University of M\"unster, WWU.
\section{Preliminary on Analysis} \label{pres} \subsection{The Laplacian of 1-form on Riemanian cones} \label{laequ1fss}
This subsection is mainly dedicated to preparing for the proof of Theorem \ref{ddclem}. We will solve the laplacian equation of $1$-form on Riemannian cones. The fact (lemma \ref{laequ1f}) was also claimed in \cite[Lemma 3.7(ii)]{conlon2013asymptotically}. Here, we just give a detailed proof for this lemma.
Let $L$ be a closed Riemannian manifold with dimension $n-1$ ($ n\geq 4$) and $C(L) = L \times \mathbb{R}_{>0}$ with standard cone metric $g_{C}= dr^2 + r^2 g$. Throughout this paper, we will apply the weighted H\"{o}lder norms on Riemannian cones. Let $T$ be a tensor field on the Riemannian cone $(C_L, g)$ and $U$ be an open domain in $C_L$, the weighted H\"older norm with order $\rho$ is defined to be the following, \begin{align} \label{hnorm}
||T||_{C^{k,\alpha}_{\rho} (U)} = \sum_{i=0}^{k} \sup_{x \in U} \big|(r^2 +1)^{\frac{1}{2} (-\rho+i)} \nabla^i T \big|_{g} + \sup_{x, y \in U} (r^2 +1)^{\frac{1}{2}(-\rho+k+\alpha)} \frac{|\nabla^{k}T(x)- \nabla^k T(y)|_g}{|x-y|^\alpha} \end{align}
where $\nabla$ is the Levi-Civita connection of $g$ and $|T(x)-T(y)|_g$ is defined via parallel transport minimal geodesic from $x$ to $y$. A tensor field $T$ is belong to $\mathcal{C}^{k,\alpha}_\rho(U)$ if the H\"older norm $||T||_{\mathcal{C}^{k,\alpha}_\rho(U)}$ is finite. Similarly, we can define a weighted H\"older norm on each AC Riemannian manifold as in (\ref{hnorm}) by fixing a smooth radial function $\tilde{r}$, which is obtained by smoothly extending the radial function $r$ to the whole manifold.
Let $\Delta = dd^* + d^* d$ be the Hodge-Laplacian operator on $C_L$ and $\Delta_L = d_L d^*_L + d_L^* d_L$, the Hodge-Laplacian operator on $L$. Let $0= \lambda'_0 < \lambda'_1 \leq \lambda'_2 \leq \ldots $ (listed with multiplicity) be the increasing sequence of eigenvalues of $\Delta_L$ on functions and $\kappa_0$ (constant), $\kappa_1, \ldots$, the corresponding eigenfuncions satisfying $||\kappa_i||_{L^2(L)} =1$, for all $i\geq 0$. Viewing $\Delta_L$ as an operator acting on the spaces of 1-forms on $L$, we immediately get a family of eigen-1-forms, $\Delta_L d_L \kappa_i = \lambda'_i d_L \kappa_i$, $i \geq 1$. The $L^2$-normalization of $\kappa_i$ implies that $||d_L\kappa_i||_{L^2(L)}= (\lambda'_i)^{1/2} $. Besides, according to Hodge decomposition on $L$, we also have a family of coclosed eigen-1-forms, $\Delta_L \eta_j = \lambda''_j \eta_j$ $(d^*_L \eta_j =0)$, where the eigenvalues are listed as an increasing sequence, $0\leq \lambda''_1 \leq \lambda''_2 \leq \ldots$ and $||\eta_j||_{L^2(L)} =1$. In summary, we have \begin{enumerate}
\item[(i)] Exact eigenforms: $ d_L\kappa_i$ with eigenvalue $\lambda'_i$ and $||d_L\kappa_i||_{L^2(L)}= \lambda_i^{1/2} $, for $i \geq 1$.
\item[(ii)] Coclosed eigenforms: $\eta_j$ with eigenvalue $ \lambda''_j $ and $||\eta_j||_{L^2(L)}=1$. \end{enumerate} Consider an 1-form $\beta$ defined on $C(L)$, then we can write $\beta = \kappa (r, y) dr + \eta (r,y) $. By spectral decomposition of 1-form on $L$, we obtain the Fourier series of $\kappa(r,y)$ and $\eta(r,y)$, \begin{align} \label{dec1f} \begin{split} \kappa (r,y) &= \sum_{i\geq 0} f_i (r) \kappa_i (y)\\ \eta (r,y) &= \sum_{i\geq 1} g_i (r) d_L \kappa_i (y) + \sum_{j \geq 1} h_j (r) \eta_j (y). \end{split} \end{align} Based on calculation in \cite[(2.14-2.15)]{cheeger1994cone} or \cite[(3.8)]{cheeger1983spectral}, we have an explicit formula for the Laplacian of 1-form $ \beta = \kappa dr + \eta$. In particular, \begin{equation}\label{la1f} \begin{split} \Delta \beta = \ &dr \big( -\kappa'' -\frac{n-1}{r} \kappa' + \frac{n-1}{r^2} \kappa + \frac{1}{r^2} \Delta_{L} \kappa + \frac{2}{r^3} d^*_{L} \eta\big) \\
& -\eta'' -\frac{n-3}{r} \eta' -\frac{2}{r} d_L \kappa + \frac{1}{r^2} \Delta_L \eta. \end{split} \end{equation} Regarding the Laplacian equation of 1-form, we introduce exceptional sets $A,\ B,\ C$ defined in (\ref{exset1}), (\ref{exset2}), (\ref{exset3}) respectively; namely, the set of orders of homogeneous harmonic 1-forms of three different types. If we write $U(r_0) =\{x \in C_L, r(x) > r_0\}$ and $\overline{U}(r_0)$ represents its topological closure, then we have the following lemma,
\begin{lem} \label{laequ1f} Let $\theta$ be an 1-form defined on $\overline{U}(1)$ with $\theta \in \mathcal{C}^{k,\alpha}_{\rho}$. If $k \geq 2n+3$ and $\rho+2 \notin A \bigcup B \bigcup C$, then there exists a solution $\beta \in \mathcal{C}^{k+2,\alpha}_{\rho+2} (\overline{U}(1))$ satisfying the Laplacian equation, $\Delta \beta = \theta$ and $\beta$ satisfies the estimate. \begin{align*}
||\beta||_{\mathcal{C}^{k+2,\alpha}_{\rho+2} (U (2))} \leq C ||\theta||_{\mathcal{C}^{k, \alpha}_{\rho}(U (1))}. \end{align*} where $C$ only depends on $(n, L, k, \rho)$. \end{lem}
\begin{proof} One of key points to solve the Laplacian equation of 1-form is to observe that the Laplacian equation is equivalent to a system of ODEs based on (\ref{dec1f}) and (\ref{la1f}). By spectrum decomposition of the Laplacian, $\beta= \kappa dr + \eta$ can be written as in (\ref{dec1f}) and similarly, we can also represent $\theta= \theta_0 dr + \theta_1$ as, \begin{align*} \theta_0 = \sum_{i\geq 0} u_i (r) \kappa_i (y), \qquad \theta_1 = \sum_{i \geq 1} v_i (r) d _L \kappa_{i} (y) + \sum_{j\geq 1} w_j (r) \eta_j (y), \end{align*} where $v_i,\ w_j \in \mathcal{C}^{{k,\alpha}}_{\rho +1}$ and $u_i \in \mathcal{C}^{k,\alpha}_{\rho}$. According to (\ref{la1f}), by comparing the coefficients, the Laplacian equation $\Delta \beta = \theta$ is equivalent to the following system of ODEs, \begin{align} -h''_{j} - \frac{n-3}{r} h'_j + \frac{\lambda''_j}{ r^{2}} h_j = w_j, \quad (j \geq 1) \label{laequ1} \end{align} and \begin{subequations} \begin{align}
&-f''_i - \frac{n-1} {r} f'_i + \frac{ n-1 +\lambda'_i }{ r^2} f_i - \frac{2\lambda_i'}{ r^3} g_i= u_i, \label{laequ2} \\
&-g''_i - \frac{ n-3 }{ r } g'_i + \frac{ \lambda'_i }{r^2} g_i -\frac{2}{ r } f_i = v_i, \quad (i \geq 0) \label{laequ3} \end{align} \end{subequations} To solve equation (\ref{laequ1}), notice that the corresponding homogeneous equation has the solutions $r^{a_j^{\pm}}$, where the orders are given by \begin{align} \label{exset1} A= \{a_j^{\pm}-1, j\geq 1 \}, \qquad a^{\pm}_{j} = -\frac{n-4}{2} \pm \sqrt{ \Big( \frac{n-4}{2}\Big)^2 +\lambda''_j}, \end{align} where $A$ defines the first exceptional set. If we write $\displaystyle H_j= \begin{pmatrix}h_{j} \\ h'_{j} \end{pmatrix} $, then we have the following representation formula, \begin{align} \label{sol1} H_j (r) = W_{j}(r) \Big( W_j^{-1}(1) H_j(1) + \int_{1}^r W_{j}^{-1}(s) \begin{pmatrix} 0\\ w_{j} (s) \end{pmatrix} ds \Big) \end{align} where the Wronskian $W_\lambda$ is given by \begin{align*} W_{j} (r) = \begin{pmatrix} r^{a^{+}_{j}} & r^{a^-_{j}} \\ a^+_{j} r^{a^{+}_{j}-1} & a^-_{j} r^{a^{-}_{j}-1} \end{pmatrix}. \end{align*} Noting that te exceptional set can be ordered nondecreasingly as $\ldots \leq a_2^- \leq a_1^- \leq a_1^+ \leq a_2^+ \leq \ldots$. Without loss of generality, assume that $\rho > 1-n$ (to ensure $\rho +3 > a_j^-$ for $j\geq 1$). Then, we can deduce the following explicit formula of $h_{j}$, \begin{align}\label{sol2} h_j(r) = \hat{h}_j (r) + A_{+} r^{a^+_{j}}+ A_{-} r^{a^-_{j}}. \end{align} In case of $ a_j^+ > \rho+3 $, $\hat{h}_\lambda$, $A_+$ and $A_-$ are given as follows, \begin{align} \hat{h}_j &= \frac{1}{a^-_j - a^+_j} \bigg( r^{a^+_j} \int_{\infty}^r s^{1-a^+_j} w_j(s) ds - r^{a^-_j} \int_{r_0}^r s^{1-a^-_j} w_j (s) ds\bigg), \nonumber \\ A_{-} &=\frac{1}{a^{-}_{j} - a^{+}_{j} } \big( - a^+_{j} h_{j}(1) + h'_{j}(1) \big), \label{coefsol1} \\ A_{+} &= \frac{1}{a^-_{j}-a^+_{j}}\bigg( a^-_{j} h_{j}(1) - h'_{j}(1) + \int_{1}^\infty s^{1-a^+_{j}} w_j (s) ds \bigg). \nonumber \end{align} It is obvious to see that $\hat{h}_{j} (r) = O(r^{3+\rho})$. By choosing certain values $(h_j(1), h'_j (1) )$ such that the coefficient $A_{+}$ and $A_{-}$ are vanishing, then we have $h_{j}(r) = O(r^{3+\rho})$. In case that $a^{+}_{j} < 3+\rho$, $h_j(r)$ has the same expression as (\ref{sol2}) with different $\hat{h}_j $, $A_+$ as follows \begin{equation} \label{coefsol2} \begin{split} \hat{h}_j &= \frac{1}{a^-_j - a^+_j} \bigg( r^{a^+_j} \int_{1}^r s^{1-a^+_j} w_j(s) ds - r^{a^-_j} \int_{1}^r s^{1-a^-_j} w_j (s) ds\bigg),\\ A_{+} &= \frac{1}{a^-_{j}-a^+_{j}}\big( a^-_{j} h_{j}(1) - h_{j}'(1) \big) \end{split} \end{equation} and $A_-$ has the same formula as in (\ref{coefsol1}). The reason we exclude the exceptional set $A$ is that if $\rho +2 = a_j^\pm-1$, there exists some $\log$ terms appear in $\hat{h}_j$. There is only one special case remains to check; that is when $n =4$ and $\lambda''_j =0$. In this case, the solutions of the corresponding homogenous equation of (\ref{laequ1}) is generated by $1$ and $\log r$. Then, by similar computation as above, the solution $h_j$ can be written as, \begin{align*} h_j (r) = h(1) + h'(1) \log r + \log r \int_1^r s w_j (s) ds - \int_1^r \big( s \log s\big) w_j(s) ds \end{align*} The assumption $\rho+3 > 4-n =0$ ensures that $h_j (r) = O(r^{{\rho}+3})$. In conclution, if $\rho+3 \notin A$, then we have $h_j \in \mathcal{C}^{k+2,\alpha}_{\rho+3}$. It is also easy to see that $h_j = O(r^{3+\rho})$.
To solve (\ref{laequ2}) and (\ref{laequ3}), we introduce an auxiliary functions, $D_i= -g_i''-(n-1)r^{-1} g'_i + {\lambda'_i}{r^{-2}} g_i $ and $E_i = f-g'$, then the equations (\ref{laequ2}) and (\ref{laequ3}) can be rewritten as follows, \begin{subequations} \begin{align} -&E''_i - \frac{n-1}{r} E'_i +\frac{\lambda'_i+n-1}{r^2} E_i+D'_i = u_i ,\label{laequ2'} \\ &D_i - \frac{2}{r}E_i = v_i. \label{laequ3'} \end{align} \end{subequations} The system of equations can be reduces to \begin{align}\label{laequ4} -E''_i - \frac{n-3}{r} E'_i +\frac{\lambda'_i+n-3}{r^2} E_i = \vartheta_i, \end{align} where $\vartheta_i = u_i - v'_i \in C^{k-1,\alpha}_{\rho}$. The equation (\ref{laequ4}) can be solved by the same method as (\ref{laequ1}). Only to notice that the exceptional set is different from $A$, \begin{align} \label{exset2} B=\{b_i^{\pm}, i\geq 0 \}, \qquad b_i^{\pm} =-\frac{n-4}{2} \pm \sqrt{\Big(\frac{n-4}{2}\Big)^2 + \lambda'_i +n -3}. \end{align} If $\rho +2 \notin B$, by similar discussion from (\ref{sol1}) to (\ref{coefsol2}), there exists a solution $E_i \in C^{k+1,\alpha}_{\rho+2}$ satisfying (\ref{laequ4}). It suffices to solve $g$ and $f$. The equation (\ref{laequ2'}) can be rewritten as \begin{align} \label{laequ5} -g''_i- \frac{n-1}{r} g'_i + \frac{\lambda'_i}{r^2} g_i = \varpi_i, \end{align} where $\varpi_i =2r^{-1} E_i +v_i \in \mathcal{C}^{k,\alpha}_{\rho+1}$. If we introduce another exceptional set, \begin{align}\label{exset3} C=\{c_i^{\pm}-1, i\geq 1 \}, \qquad c_i^{\pm} =-\frac{n-2}{2} \pm \sqrt{\Big(\frac{n-2}{2}\Big)^2 + \lambda'_i }, \end{align} then by the same method, assuming $\rho+2 \notin B\bigcup C$, there exists a solution $g_i \in \mathcal{C}^{k+2,\alpha}_{\rho+3}$. By the definition of $E_i$, we obtain that $f_i = g'_i + E_i \in \mathcal{C}^{k+1,\alpha}_{\rho+2}$. According to previous discussion, we have found coefficients satisfying the right decay condition of Fourier series of $\beta$. It remains to show that the Fourier series converges in $\mathcal{C}^{2}_{loc}$ topology.
If $\lambda_j >0$, $j\in J$, we have the following estimte for $w_j (r)$, \begin{align} w_j(r) =\int_{L} (\theta_1(r, y), \eta_j(y))_{g_L} dy &= {(\lambda''_j)^{-\frac{k}{2}} }\int_L \big( \Delta_L^{\frac{k}{2}} \theta_1(r,y), \eta_j \big)_{g_L} dy \nonumber \\
&\leq \operatorname{Vol}(L)^{\frac{1}{2}} (\lambda''_j)^{-\frac{k}{2}} r^{\rho+1} || \theta (r,y) ||_{k,\alpha; \rho}. \label{coefest1} \end{align} By by expression of $\hat{h}_j$ in (\ref{coefsol1}), we have \begin{align} \label{coefest2}
|\hat{h}_j(r)| + r|\hat{h}_j'(r)| + r^2 |\hat{h}_j''(r)| \leq C(n, \rho) r^{\rho +3} \sup |r^{-\rho -1} w_j (r)| \end{align} Combining (\ref{coefest2}) with (\ref{coefest1}), we have the estimate \begin{align}\label{coefest3}
|\hat{h}_j(r)| + r|\hat{h}_j'(r)| + r^2 |\hat{h}_j''(r)| \leq C (n, L, \rho) (1+\lambda_j)^{-\frac{k}{2}} r^{\rho+3} ||\theta(r,y)||_{k,\alpha; \rho}. \end{align} Applying Moser iteration to $\eta_j$, we obtain that \begin{align*}
||\eta_j||_{L^\infty(L)} \leq C(n,L) (\lambda''_j)^{\frac{n-1}{2}} ||\eta_j||_{L^2(L)} = C(n, L) (\lambda'')_j^{\frac{n-1}{2}}.
\end{align*}
Then, the Schauder estimates for $\Delta_L$ implies that $||\eta_j||_{C^{2,\alpha}(L)} \leq C(n,L) (1+\lambda''_j)^{\frac{n+1}{2}}$; hence \begin{align} \label{eigen1fest1}
|\eta_j|+r |\nabla \eta_j| + r^2 | \nabla^2 \eta_j |\leq C(n ,L)(1+ \lambda''_j) ^{\frac{n+1}{2}}r^{-1}. \end{align} Recall the Weyl's law for differential forms \cite[Appendix by J. Dodziuk]{chavel1984eigenvalues}, $\lambda_j \sim j^{\frac{2}{n-1}}$. The Weyl's law together with (\ref{coefest3}) and (\ref{eigen1fest1}) can deduce that when $k \geq 2n+3 $, \begin{align*}
|\hat{h}_j(r) \eta_j(y)| + r |\nabla (\hat{h}_j(r) \eta_j (y))| + r^2| \nabla^2 (\hat{h}_j (r)\eta_j (y))| \leq C(n,L) j^{\big(-1 +\frac{2n-k}{n-1}\big)} r^{\rho+2} ||\theta (r,y)||_{k,\alpha; \rho+1}, \end{align*} which implies that the series $\sum_{j\in J}\hat{ h}_j (r) \eta_j (y)$ converges in $\mathcal{C}^2_{loc} $. By choosing certain $h_j (1)$ and $h'_j(1)$ to ensure that $A_{+}= A_{-}=0$, $\sum_{j\in J} { h}_j (r) \eta_j (y)$ converges in $\mathcal{C}^2_{loc} $. To show the convergence of $\sum_{i\in I} {f}_i (r) \kappa_i (y) dr$ and $\sum_{i\in I} {g}_i (r) d_L \kappa_i (y)$, we deduce a similar estimate as (\ref{coefest3}) for $E_i$ at first. Recall that $\vartheta_i = u_i - v'_i $, then we have \begin{align} \vartheta_i (r) &= \int_{L} \theta_0 (r,y) \kappa_i (y) dy - \lambda_i^{-1} \int_{L} (\partial_r \theta_1 (r, y), d_L \kappa_i (y) )_{g_L}dy \nonumber \\ &= \lambda_i^{-\frac{k}{2}} \int_L \big(\Delta_L^{\frac{k}{2}} \theta_0 (r, y)\big) \kappa_i (y) dy - \lambda_i^{-\frac{k+1}{2}} \int_L \big(\partial_r \big(\Delta_L^{\frac{k-1}{2}} \theta_1 (r,y) \big), d_L \kappa_i (y)\big)_{g_L} dy \nonumber \\
& \leq C(n, L) \lambda_i^{-\frac{k}{2}} r^{\rho} ||\theta ||_{k,\alpha; \rho}. \label{coefest4} \end{align} Recall the equation (\ref{laequ4}), together with (\ref{coefest4}), by the same method to derive (\ref{coefest3}), we have, \begin{align} \label{coefest5}
|E_i(r)|+r|E'_i(r)|+r^2|E''_i (r)| \leq C(n, L,\rho) (\lambda'_i)^{-\frac{k}{2}} r^{\rho+2} ||\theta||_{k,\alpha; \rho}. \end{align} Again, according to equation (\ref{laequ5}) and estimate (\ref{coefest5}), the same method shows that \begin{align*}
\varpi_i (r) = \frac{2 E_i}{r} + v_i \leq C(n, L, \rho) r^{\rho +1} (\lambda'_i)^{-\frac{k}{2}} r^{\rho+1} ||\theta||_{k,\alpha; \rho}. \end{align*} Therefore, we have \begin{align} \label{coefest6}
|g_i(r)|+r|g'_i(r)|+r^2|g''_i (r)| \leq C(n, L,\rho) (1+ \lambda'_i)^{-\frac{k}{2}} r^{\rho+3} ||\theta||_{k,\alpha; \rho}. \end{align} Combining (\ref{coefest5}), (\ref{coefest6}) and definition of $E_i$, we obtain, \begin{align} \label{coefest7}
|f_i(r)|+r|f'_i(r)|+r^2|f''_i (r)| \leq C(n, L,\rho) (1+\lambda'_i)^{-\frac{k-1}{2}} r^{\rho+2} ||\theta||_{k,\alpha; \rho}. \end{align} Also, we can derive the similar estimate for $\kappa_i$ and $d_L \kappa_i $ as in (\ref{eigen1fest1}), by Moser iteration and Schauder estimates \begin{equation} \label{eigenest2} \begin{split}
|\kappa_i|+r |\nabla \kappa_i| + r^2 | \nabla^2 \kappa_i | &\leq C(n ,L)(1+ \lambda'_i) ^{\frac{n+1}{2}}; \\
|d_L\kappa_i|+r |\nabla d_L \kappa_i| + r^2 | \nabla^2 d_L\kappa_i | &\leq C(n ,L)(1+ \lambda'_i) ^{\frac{n+3}{2}}r^{-1}. \end{split} \end{equation} Then, the estmates (\ref{coefest6}), (\ref{coefest7}) and (\ref{eigenest2}) together with Weyl's law implies that, when $k \geq 2n+3$, the series $\sum_{i\in I} {f}_i (r) \kappa_i (y) dr$ and $\sum_{i\in I} {g}_i (r) d_L \kappa_i (y)$ converge in $C^2_{loc}$ topology. And the convergence of series also implies that \begin{align*}
||\beta||_{L^\infty} \leq C (n, L,\rho, k) r^{\rho +2}||\theta||_{k ,\alpha; \rho}. \end{align*} Since the Laplacian can be viewed as an elliptic operator for vector bundle $\wedge^1 T^*X$ over $X$, according to elliptic operator theory on conical manifolds \cite[Theorem 4.12]{marshal2002deformations}, we have Schauder-type estimate for the equation, $\Delta \beta = \theta$. \begin{align*}
||\beta||_{{k+2,\alpha ;\rho+2}} \leq C (n, L,\rho, k) ( ||\theta||_{k, \alpha; \rho} + ||\beta||_{0; \rho+2} )\leq C (n, L,\rho, k) ||\theta||_{k, \alpha; \rho}. \end{align*} \end{proof}
The proof of lemma \ref{laequ1f} not only works for 1-forms, but also for functions in weighted H\"older space. Also by spectral decompsition technique, we can solve $\Delta_0 u =f$ as follows. Recall that the increasing sequence $0= \lambda'_0 <\lambda'_1 \leq \lambda'_2 \ldots$ is eigenvalues of $\delta$ acting on functions, then the exceptional set is given by, \begin{align}\label{excd} D =\{ d_i^{\pm}, \text{ for } i=0,1,\ldots \}, \qquad d^{\pm}_{i} =-\frac{n-2}{2}\pm \sqrt{\Big(\frac{n-2}{2}\Big)^2 + \lambda'_i}. \end{align}
\begin{cor} Let $f$ be a function defined on $\overline{U}(1)$ with $f \in \mathcal{C}^{k,\alpha}_{\rho-2}$. If $k \geq $ and $\rho \notin D $, then there exists a solution $u \in \mathcal{C}^{k+2,\alpha}_{\rho} (\overline{U}(1))$ satisfying the Laplacian equation, $\Delta_0 u = f$ and $u$ satisfies the estimate. \begin{align} \label{laequfestc}
||u||_{\mathcal{C}^{k+2,\alpha}_{\rho} (U (2))} \leq C ||f||_{\mathcal{C}^{k, \alpha}_{\rho-2}(U (1))}. \end{align} where $C$ only depends on $(n, L, k, \rho)$. \end{cor}
\subsection{Laplacian on asymptotically conical manifolds} \label{aclaequss}
The asymptotic behavior of Laplacian on ALE manifolds has been studied in many references. The main idea to solve the Laplacian equation on ALE manifolds is to apply the Fredholm property of the Laplacian operator. In Lockhart \cite[Corollary 6.5]{Lockhart1981FredholmPO}, Lockhart-McOwen \cite[Theorem 1.3]{ASNSP_1985_4_12_3_409_0} and Cantor \cite[Theorem 6.3]{cantor1981elliptic}, the Fredholm property is proved in Sobolev case for the elliptic operators that are asymptotic to the Euclidean Laplacian at infinity. The H\"older case has been discussed in \cite[section 4]{chaljub1979problemes} for real dimension 3. In Marshall \cite[Theorem 6.9]{marshal2002deformations}, the Fredholm property of Laplacian operator has been generalized to AC manifolds on both Sobolev and H\"older cases. In this section, we will summarize the key results of the H\"older cases on AC manifolds and prove the following proposition. Recall that $(C_L, g_0)$ is a Riemannian cone and $(X, g)$ asymptotic to $(C_L, g_0)$ at infinity with a diffeomrphsim $\psi: X_\infty = X-K \rightarrow (R_0, \infty) \times L$ and $ \displaystyle |\nabla^i((\psi^{-1})^* g - g_0)|_{g_0} = O(r^{-i-\tau})$, for integers $i = 0,1,\ldots, k $.
In $X_\infty$, let $\Delta_0$ and $\Delta$ be the Laplacian operators of metrics $g_0$ and $g$ respectively and $D$, the exceptional set given in (\ref{excd}).
\begin{pro} \label{laequac} Suppose $(X, g)$ is a complete AC manifold asymptotic to a Riemannian cone $(C_L, g)$ with $\dim_{\mathbb{R}} X = n \geq 4$ and $f \in \mathcal{C}^{k,\alpha}_{\rho-2}$. \begin{enumerate} \item[\textup(i)] Let $\rho \in (0,\infty)\backslash D$, there exists a solution $u \in \mathcal{C}^{k+2,\alpha}_{\rho}$. \item[\textup{(ii)}] Let $\rho \in (2-n,0)$, there exists a unique solution $u \in C^{k, \alpha}_{\rho+2}$ of $\Delta u = f$. \item[\textup{(iii)}] Let $\rho \in (-\infty , 2-n)\backslash D $, there exists a unique solution $u = A r^{2-n} + v$, where \begin{align} \label{constlaequ} A = \frac{1}{(n-2) \operatorname{Vol}(L)} \int_{X} f d\operatorname{Vol}_X \end{align} and $v\in \mathcal{C}^{k+2,\alpha}_{\tilde{\rho}}$ with $\tilde{\rho} = \max \{d_1^-, \rho, 2-n-\tau\}$. \end{enumerate} \end{pro}
In the following we write $\ker (\Delta, \delta+2)=\ker(\Delta: \mathcal{C}^{k+2,\alpha}_{\delta+2} \rightarrow \mathcal{C}^{k, \alpha}_{\delta})$ and $\operatorname{Im} (\Delta, \delta)=\operatorname{Im}(\Delta: \mathcal{C}^{k+2,\alpha}_{\delta+2} \rightarrow \mathcal{C}^{k, \alpha}_{\delta})$. The key observation is that the estimate (\ref{laequfestc}) can be modified to obtain a similar estimate for elliptic operators asymptotic to $\Delta_0$ on AC Riemannian manifolds; namely, the scale broken estimate referring to \cite[Theorem 6.7]{marshal2002deformations}. Precisely, fixing $X_{2R} = \{x\in X, r(x)> 2R\}$ with $R> R_0$, for all $u \in \mathcal{C}^{k+2,\alpha}_{\rho}$, we have \begin{align} \label{scalbroest}
||u||_{\mathcal{C}^{k+2,\alpha}_{\rho}} \leq C( ||\Delta u||_{\mathcal{C}^{k,\alpha}_{\rho-2}} + ||u||_{\mathcal{C}^0 (X_{2R})}). \end{align} It can be proved that the Laplacian operator $\Delta: \mathcal{C}^{k+2,\alpha}_{\rho} \rightarrow \mathcal{C}^{k,\alpha}_{\rho-2}$ on $X$ is Fredholm if $\rho \notin D$. To sketch the proof of Fredholm property, we observe that the estimate (\ref{scalbroest}) implies that the set $\operatorname{Im} (\Delta, \rho)$ is closed. To see this, let $\{f_i\}$ be a Cauchy sequence in $C^{k,\alpha}_{\rho-2}$ contained in $\operatorname{Im} (\Delta, \rho)$. Notice that there exists a closed subspace $\mathcal{B}_\rho $ such that $\mathcal{C}^{k+2,\alpha}_{\rho} = \mathcal{B}_{\rho} \oplus \ker (\Delta, \rho) $ (because $\ker (\Delta, \rho)$ is finite-dimensional), then we have a bounded sequence of preimages $\{u_i\}$ in $\mathcal{B}_{\rho}$ with $\Delta u_i =f_i$. According to Arzel\`{a}-Ascoli and (\ref{scalbroest}), by taking a subsequence of $\{u_i\}$, we obtain a Cauchy sequence in $\mathcal{C}^0(X_{2R})$. Hence, $\{u_i\}$ is also a Cauchy sequence by passing to the subsequence. Let $f$, $u$ be the convergence of $\{f_i\}$ and $\{u_i\}$ respectively, then $f =\Delta u$ is belong to $\operatorname{Im}(\Delta, \rho)$. Based on self-adjointness of $\Delta$, the Laplacian $\Delta: \mathcal{C}^{k+2,\alpha}_{\delta+2} \rightarrow \mathcal{C}^{k, \alpha}_{\delta}$ \begin{align} \label{fhdual}
\operatorname{Im} (\Delta, \delta)= \{f \in \mathcal{C}^{k,\alpha}_{\rho}, (f , h )_{L^2}=0 \text{ for all } h\in \ker ( \Delta, -n -\delta) \} \end{align} Then, the Fredholm property of $\Delta$ follows from the fact that $\ker(\Delta, \rho) $ is finite-dimensional.
\begin{proof}[Proof of Proposition \ref{laequac}] By the maximal principle of harmonic functions, we can show that $\ker(\Delta, \rho)=0$ if $\rho <0$. In the cases (i) and (ii), by Fredholm property of $\Delta$ and (\ref{fhdual}), when $\rho \in (-n,-2)$, $\operatorname{Im} (\Delta, \rho)$ is annihilated by $\ker (\Delta, -n-\rho) =0$. Hence, $\operatorname{Im} (\Delta, \rho) = \mathcal{C}^{k,\alpha}_{\rho}$ for $\rho\in (-n,-2)$. The uniqueness in the case (ii) is directly from the fact $\ker (\Delta, \rho) = 0$ for $\rho \in (-n,-2)$. .
For the case (iii), notice that $\Delta r^{2-n} = \Delta_0 r^{2-n} + (\Delta-\Delta_0) r^{2-n}= O(r^{-n-\tau})$ and \begin{align*} \int_{X} \Delta r^{2-n} d\operatorname{Vol}_X &= -\lim_{r \rightarrow \infty} \int_{L(r)} \langle\nabla r^{2-n}, n \rangle_g d\operatorname{Vol}_{L(r)}\\ & = \lim_{r \rightarrow \infty}(n-2) r^{1-n} \int_{L(r)} g(\partial r, \partial r) d\operatorname{Vol}_{L(r)} \\ & = \lim_{r \rightarrow \infty}(n-2) \operatorname{Vol}(L) +O(r^{-\tau}) = (n-2) \operatorname{Vol}(L). \end{align*} Consider the function $f- A \Delta r^{2-n}$, where the constant $A$ is chosen to be (\ref{constlaequ}). Observing that $f-A \Delta r^{2-n} = O(r^{\max\{-n-\tau, \rho-2\}})$ and $\int_{X} f-A\Delta r^{2-n}$ =0. Without loss of generality, we assume $\max\{-n-\tau, \rho-2\} \in (d_1^- -2, -n)$. Then (\ref{fhdual}) implies that the only annihilating functions are constant functions. The integration is vanishing indicates that $f-A \Delta r^{2-n}$ is in $\operatorname{Im} (\Delta, \tilde{\rho}-2)$ with $\tilde{\rho} = \max \{d^-_1, \rho, 2-n-\tau \}$; hence, we have solution $u = Ar^{2-n} +v$ with $v\in \mathcal{C}^{k+2,\alpha}_{\tilde{\rho}}$. \end{proof}
\section{Proof of $dd^c$ lemma} \label{ddclems}
\subsection{Proof of Theorem \ref{ddclem} (i)} \label{pfddbar} Recall that $(X, J, g)$ is an AC K\"ahler manifold asymptotic to a Ricci-flat K\"ahler cone $(C_L, J_0, g_0)$ at infinity with a diiffeomorphism $\psi: X_\infty = X-K \rightarrow C_L -B_{R_0}$. In the end $X_\infty $, there are two sets of differential operators. In particular, we write $d$, $\partial$, $\overline{\partial}$ and $d^*$, $\partial^*$, $\overline{\partial}^*$ as the differential opertators and the dual operators with respect to $(g,J)$, which also induces the Laplacian operators $\Delta$ in the asymptotic chart. Also let $d_0$, $\partial_0$, $\overline{\partial}_0$, $d_0^*$, $\partial_0^*$, $\overline{\partial}_0^*$ and $\Delta_0$ be the Laplacian operators with respect to $(g_0,J_0)$. Let $\omega \in \mathcal{C}^{k,\alpha}_{-\delta}$ be an exact real $(1,1)$-form on $X$. Then, by \cite[Theorem 3.11]{conlon2013asymptotically}, there exists a real 1-form $\alpha \in \mathcal{C}^{k+1,\alpha}_{1-\delta}$ such that $\omega = d\alpha$. Decompose $\alpha$ to $(1,0)$-form and $(0,1)$-form with respect to $J$, $\alpha^{1,0}$ and $\alpha^{0,1}$. Consider the following Laplacian equation for $\phi$, \begin{align} \label{ddbareq1}
\frac{1}{2}\Delta \phi = \overline{\partial}^* \alpha^{0,1}, \end{align} Based on the Proposition \ref{laequac}, there exists a solution $\phi \in \mathcal{C}^{k+2,\alpha}_{2-\delta}$ for (\ref{ddbareq1}). Define $\xi = \alpha^{0,1}-\overline{\partial} \phi $. Due to the low decay rate, it is not necessary that $\xi =0$. The idea of the proof is to reduce the decay rate of $\xi$ and to find $\hat{\phi}$ such that $\overline{\partial} \hat{\phi} = \alpha^{0,1}$. It is observed that \begin{align} \label{harcon} \overline{\partial}^* \xi = 0, \qquad \overline{\partial} \xi =0. \end{align} Hence, $\xi $ is a $J$-harmonic (0,1)-form in $\mathcal{C}^{k+1,\alpha}_{1-\delta}$. If $1-\delta$ is nonnegative, we can reduce the order of $\xi$ as follows. Assuming that $\theta = \Delta_0 \xi = (\Delta_0 - \Delta ) \xi = O(r^{-1-\delta-\tau})$, according to lemma \ref{laequ1f}, there exists a solution $\beta = O(r^{1-\delta-\tau})$ such that $\Delta_0 \beta= \theta$ in the asymptotic chart of $X$. Hence, we can write $\xi$ in the asymptotic chart of $X$ as \begin{align*} \xi = \xi_0 + \beta, \end{align*}where $\xi_0$ is a harmonic 1-form with respect to $\Delta_0$ of decay rate $1-\delta$ and $\beta$ is a 1-form of decay rate $1- \delta- \tau$. Notice that $\xi_0$ is not necessary a $(0,1)$-form with respect to $J$, but the $(1,0)$ part of $\xi_0$ has the decay rate $1-\delta-\tau$. Then, according to \cite[Lemma 2.27]{cheeger1994cone} or referring to \cite[Lemma B.1]{hein2017calabi}, the assumption that $0\leq 1-\delta <1$ implies there exists a harmonic function $\psi_0 $ with respect to $( J_0, g_0)$ such that $ d \psi_0 = \xi_0$ with $\psi_0 = \mathcal{C}^{k+2,\alpha}_{2-\delta}$ and $\partial \psi_0 =\mathcal{C}^{k+1,\alpha}_{1-\delta-\tau}$, $\overline{\partial} \psi_0 =\mathcal{C}^{k+1,\alpha}_{1-\delta}$. Extend $\psi_0$ to a smooth function $\tilde{\psi}_0$ on $X$. According to proposition \ref{laequac}, by solving equation $\Delta b = -\Delta \tilde{\psi}_0 $, there exists a harmonic function $\psi$ with respect to $(J,g)$ such that, \begin{align*}
\psi =\tilde{ \psi_0 }+ b. \end{align*} Since in asymptotic chart of $X$, $-\Delta \tilde{\psi}_0 = (\Delta_0 - \Delta) \tilde{\psi}_0 = O(r^{-\delta-\tau}) $, we have $b$ is a function of class $\mathcal{C}^{k+2,\alpha}_{2-\delta-\tau}$. Now, consider a new (0,1)-form given by \begin{align} \label{reddecay} \xi_1 = \xi - \overline{\partial} \psi, \end{align} which also satisfies conditon (\ref{harcon}) $\overline{\partial}\xi_1 = \overline{\partial}^* \xi_1=0$. And $\xi_1$ has the following decay rate, \begin{align*} \xi_1 &= \xi_0 + \beta - \overline{\partial}\psi\\ &= \beta + \partial \tilde{\psi}_0 + \overline{\partial} \tilde{\psi}_0 - \overline{\partial} \psi\\ &= \beta + \partial \tilde{\psi}_0 -\overline{\partial} b, \end{align*} where $\beta$, $\partial \psi_0 \in \mathcal{C}^{k+1,\alpha}_{1-\delta-\tau}$. Hence, $\xi_1 \in \mathcal{C}^{k+1,\alpha}_{1-\delta -\tau}$. Repeating this process until the decay rate of 1-form to be negative, precisely, we can find $\hat{\xi}= \xi- \overline{\partial}\hat{\psi}$ such that $\hat{\psi} \in \mathcal{C}^{k+2,\alpha}_{2-\delta}$ and $\hat{\xi} \in \mathcal{C}^{k+1,\alpha}_{1-\delta-n\tau}$ with $1-\delta-n\tau <0$. To simplify our notation, we replace $\xi$ with the 1-form of negative decay rate. In the following proof, we assume that $\xi\in \mathcal{C}^{k+1,\alpha}_{1-\delta}$ ($1-\delta <0$) with $\overline{\partial}^* \xi= \overline{\partial}\xi=0$. We introduce the following lemma.
\begin{lem} \label{decay1f} Let $k$ be a large positive integer and $1-\delta <0$, \begin{enumerate} \item[(i)] Let $\xi$ is a real 1-form on $X$ of class $\mathcal{C}^{k+1,\alpha}_{1-\delta}$. Assuming that $\xi$ is a harmonic 1-form on $X$, then, $\xi \in \mathcal{C}^{k+1,\alpha}_{-2n+2}$. \item[(ii)] Let $\xi$ is a harmonic (0,1)-form on $X$ of class $\mathcal{C}^{k+1,\alpha}_{1-\delta}$ with $\overline{\partial} \xi = 0$, then $\xi\in \mathcal{C}^{k+1,\alpha}_{-2n+1}$ \end{enumerate} \end{lem}
\begin{proof} Let $\xi$ be a real harmonic 1-form on $X$. According to lemma \ref{laequ1f}, $\xi$ can be represented outside a compact set of $X$ as follows, \begin{align}\label{decayred1f} \xi = \xi_0 + \beta \end{align} where $\xi_0$ is a harmonic form with respect to $\Delta_0$ of the same decay rate as $\xi$ and $\beta $ is a 1-form of decay rate at most $1-\delta-\tau$. By assumption that $X$ is asymptotic to the conical manifold $C_L$, $\xi_0$ can be expressed on basis of spectral decomposition with respect to Laplacian operator on link $L$, \begin{align*} \xi_0 &= \sum_{i \geq 0 } [ f_i (r) \kappa_i (y) dr + g_i (r) d_L \kappa_i(y) ] + \sum_{j\in J} h_j (r) \eta_j (y)\\
&= \sum_{i\geq 0} [d(g_i (r) \kappa_i (y))+ E_i (r) \kappa_i (y) dr] + \sum_{j\geq 1} h_j (r) \eta_j (y). \end{align*} where $E_i(r) = f_i(r) - g'_i(r)$. Then, the function $g_i$, $E_i$ satisfy the following equations, ($\dim_{\mathbb{R}}X =2n$) \begin{subequations} \begin{align} -&E''_i - \frac{2n-3}{r} E'_i +\frac{\lambda'_i+2n-3}{r^2} E_i= 0 ,\label{harcon1} \\ -&g_i'' - \frac{2n-1}{r} g_i' +\frac{\lambda'_i}{r^2} g_i = \frac{2}{r}E_i. \label{harcon2} \end{align} \end{subequations} By solving the equations (\ref{harcon1}) and (\ref{harcon2}), the harmonic 1-forms with respect to $\Delta_0$ of negative decay rate at infinity have the following two types, \begin{enumerate} \item[(I)] Let $\lambda_i$ be all eigenvalues of $\Delta_L$ for functions on $L$, \begin{align*} d(r^{c_i^-} \kappa_i),\quad \text{with } c^-_i = -(n-1) - \sqrt{(n-1)^2 + \lambda'_i } \end{align*} \item[(II)] Let $\lambda_i$ be all \textit{positive} eigenvalues of $\Delta_L$ for functions on $L$, \begin{align*}
B_i r^{b_i^-} \kappa_i dr + 2 d(r^{b_i^- +1} \kappa_i ), \quad \text{with } b_i^{-} =-(n-2) - \sqrt{(n-2)^2 + \lambda'_i +2n -3}, \end{align*} and $B_i$ is a constant given by $B_i = - (b_i^-)^2 - 2n b_i^- +\lambda_i +1-2n $. \end{enumerate} For the coclosed part of decomposition, consider the following equation, \begin{align} \label{harcon3} -h''_{j} - \frac{2n-3}{r} h'_j + \frac{\lambda_j}{ r^{2}} h_j =0, \end{align} By solving (\ref{harcon3}), we have the third type of harmonic 1-form, \begin{enumerate} \item[(III)] Let $\lambda_j$ be the eigenvalues of $\Delta_L$ for the co-closed 1-forms on $L$, \begin{align*} r^{a^-_j} \eta_j, \quad \text{with } a_j^- = -(n-2)- \sqrt{ (n-2)^2 +\lambda''_j} \end{align*} \end{enumerate} To estimate the eigenvalues, we notice that $L$ is a Sasakian-Einstein manifold with Ricci curvature $\textup{Ric}_{g_L} = 2(n-1) g_L $. By Lichnerowicz-Obata first eigenvalue theorem and \cite[Lemma B.2]{hein2017calabi}, let $\lambda'_1$ and $\lambda''_1$ be the first positive eigenvalue of closed and coclosed 1-form, we have the following estimate, \begin{align*} \lambda'_{1} (L) \geq 2n-1, \qquad \lambda''_{1} (L) \geq 4n-4. \end{align*} Then, we obtain the greatest possibe decay rate of harmonic 1-forms for all these three types. For type (I), the forms decays to rate at most $-2n+1$ at infinity. For type (II), the forms decays to rate at most $-2n+2$. And the forms of type (III) decays to rate at most $-2n+1$. Hence, we obtain $\xi_0 = O(r^{-2n +2})$. If $1-\delta-\tau \geq -2n+2 $, then, by expression (\ref{decayred1f}), $\xi $ is of decay rate $1-\delta-\tau$. Again, noticing that $\beta$ is the solution of $\Delta_0 \beta = (\Delta_0 -\Delta) \xi$, there exists a $d$-closed solution $\beta$ of decay rate $1-\delta-2\tau$. Repeating the process, we can finally obtain that $\xi = O(r^{2-2n})$.
For the second statement, let $\xi$ be a harmonic $(0,1)$-form on $X$ with $\overline{\partial} \xi = 0$. Notice that $\xi$ also can be written as in (\ref{decayred1f}), $\xi = \xi_0 + \beta$. According to (i), we can assume $\xi$, $\xi_0$ of rate $-d \leq 2-2n$ and $\beta$ of rate $-d -\tau$. $\xi$ can be decomposed to $(1,0)$ and $(0,1)$ form with respect to $J_0$, \begin{align*}
\xi = \xi^{1,0}_{J_0} + \xi^{0,1}_{J_0}, \quad, \xi^{1,0}_{J_0}= \frac{\xi - iJ_0 \xi}{2}, \ \xi^{0,1}_{J_0} = \frac{\xi + iJ_0 \xi}{2}. \end{align*} Since $\xi$ is a $(0,1)$-form with respect to $J$, $\xi- i J_0 \xi = i (J-J_0) \xi = O(r^{-d -\tau})$. The $(1,0)$ part of $\xi$ with respect to $J_0$ has rate $-d -\tau$, so does $\xi_0$. Then, the highest order terms of $\xi_0$ is a homogeneous harmonic $(0,1)$-form with respect to $J_0$, denoted by $\xi_0^{h}$. The fact $\overline{\partial}\xi=0 $ implies that $\overline{\partial}_0 \xi_0^h = O (r^{-d-1 -\tau})$, If we assume the rate of $\xi^h_0$ is greater than $1-2n$, based on the classification of harmonic 1-forms, $\xi^h_0$ must be of type (II). Then, $\xi_0^h$ can be written as, \begin{align*}
\xi^h_0 = C (2b^-_i + 6 - 4n) r^{b^-_i } \kappa_i \overline{\partial}_0 r + 2 r^{b^-_i +1}\overline{\partial}_0 \kappa_i, \end{align*} where $i\geq 1$ and $b_i^- \geq 1-2n$. By taking differentiation, we have \begin{align*}
\overline{\partial}_0 \xi^h_0 = C(4b_i^- +4-4n) r^{b_i^-} \overline{\partial}_0 \kappa_i \wedge \overline{\partial}_0 r. \end{align*} It is easy to check that $\overline{\partial}_0 \xi^h_0$ is a nonvanishing homogeneous form of rate $-d-1 $, which contradicts against $\overline{\partial}_0 \xi_0^h = O (r^{b^-_i-1 -\tau})$. Therefore, we have $\xi = O(r^{1-2n})$. \end{proof}
According to lemma \ref{decay1f}, we reduce the proposition \ref{ddclem} to the case of fast decay rate. Let $\hat{\omega} = 2 \operatorname{Re} \partial \xi $ be a closed real (1,1) form of decay rate at most $-2n+1$. Also notice that $\operatorname{tr}_\omega \hat{\omega}= 2 \operatorname{Re} \overline{\partial}^* \xi =0$. Recall that there exists a natural radial function defined in $X$ and let $B_{R} = \{x\in X | r(x) \leq R\}$, for some very large positive real number $R$. Then, it is easy to check that \begin{align}\label{ddcint}
2 (n-2) !\int_{B_R} |\hat{\omega}|^2 d\operatorname{Vol} =\int_{B_R} d(2\operatorname{Re} \xi \wedge \hat{\omega} \wedge \omega^{n-2}) = \int_{\partial B_R} 2 \operatorname{Re} \xi \wedge \hat{\omega} \wedge \omega^{n-2}. \end{align} The decay conditions of $\hat{\omega}$ and $\xi$ imply that the integration in (\ref{ddcint}) tends to zero as $R$ goes to infinity. Hence, we have $\hat{\omega} =0$. In conclusion, the exact real (1,1)-form $\omega$ can be expressed as \begin{align*} \omega = 2 \operatorname{Re} \partial \alpha^{0,1} = 2 \operatorname{Re} \partial ( \xi + \overline{\partial} \phi) = \hat{\omega} + 2i \partial \overline{\partial} \operatorname{Im} \phi = dd^c \operatorname{Im} \phi, \end{align*} where $\phi \in \mathcal{C}^{k+2, \alpha}_{2-\delta}$. Therefore, we complete the proof of proposition \ref{ddclem}.
\subsection{Proof of Theorem \ref{ddclem} (ii): an obstruction of ddbar lemma}\label{ddclem'ss} Let $(X,J)$ be an AC K\"ahler manifold asymptotic to a Ricci-flat K\"ahler cone $(C_L, J_0)$ and $K$ be the compact set in $X$ such that $X\backslash K \cong C_L \backslash \overline{B(R_0)}$. We apply the notation as in section \ref{pfddbar}. Let $\omega $ be the real exact (1,1)-form on $X\backslash K$ with $\omega = d\alpha$, where $\alpha$ is a real 1-form on $X\backslash K$. Then, $\alpha$ can split into the sum of a (0,1)-form and a (1,0)-form with respect to $J$, $\alpha = \alpha^{1,0} + \alpha^{0,1}$. Here $\overline{\partial}^* \alpha^{0,1}$ is a complex function defined on $X\backslash K$ of class $\mathcal{C}^{k,\alpha}_{-\delta}$. By extending $\overline{\partial}^* \alpha^{0,1}$ to a function $f \in \mathcal{C}^{k,\alpha}_{-\delta}$ on the whole manifold $X$. Then the Laplacian equation (\ref{ddbareq1}) can be rewritten as $\Delta \phi = 2 f$. According to Proposition \ref{laequac}, we have a solution $\phi \in \mathcal{C}^{k+2,\alpha}_{2-\delta}$. Hence, we can define a (0,1)-form on $X\backslash K$, $\xi= \alpha^{0,1}-\overline{\partial} \phi$. It is also easy to check $\overline{\partial} \xi = 0$, $\overline{\partial}^* \xi =0 $. Based on the proof of Theorem \ref{ddclem} (i), we can reduce the rate of $\xi$ at infinity as in (\ref{reddecay}); i.e. we can find a harmonic function $\psi \in \mathcal{C}^{k+2,\alpha}_{2-\delta}$ such that $\hat{\xi} = \xi - \overline{\partial} \psi$ is a (0,1)-form of class $\mathcal{C}^{k+1,\alpha}_{1-\delta- m \tau}$ with $1- \delta -m \tau<0$ and $\overline{\partial} \hat{\xi} =0$, $\overline{\partial}^* \hat{\xi} =0 $. Then, \begin{align} \label{ddbar'eq1}
\omega = 2 \operatorname{Re} (\partial \alpha^{0,1}) = 2 \operatorname{Re} (\partial \hat{\xi} + \partial \overline{\partial} \phi + \partial \overline{\partial} \psi) = 2 i\partial \overline{\partial} \operatorname{Im} (\phi +\psi) + 2\operatorname{Re} (\partial \hat{\xi}). \end{align} The above expression of $\omega$ (\ref{ddbar'eq1}) implies that the obstruction of ddbar lemma is the harmonic 1-form $\hat{\xi}$. Unfortunately, the obstruction term, $ \operatorname{Re} (\partial \hat{\xi})$, cannot be vanishing in general, see counter example \ref{ctexddbar'}. However, the decay rate of the term can be reduced to enough high order. According to Lemma \ref{decay1f}, the decay rate of $\hat{\xi}$ is at most $1-2n$. Hence, $\omega = i\partial \overline{\partial} f + O(r^{-2n})$ for some real function $f \in \mathcal{C}^{k+2,\alpha}_{2-\delta}$, which completes the proof of theorem \ref{ddclem} (ii).
\begin{rem} \label{remobs} Fixing $X\backslash K$ as above, we assume that $J_0 =J$ in this remark. An interesting fact we observed in the proof of theorem \ref{ddclem} (ii) is that the obstruction of ddbar lemma is described by the class $\mathcal{H}^{0,1}_{\overline{\partial}, \overline{\partial}^*}/ \overline{\partial} \mathcal{H}$, where $\overline{\partial} \mathcal{H}$ is the $\overline{\partial}$-image of complex valued harmonic functions on $X\backslash K$ asymptotic to zero at infinity and \begin{align*}
\mathcal{H}^{0,1}_{\overline{\partial},\overline{\partial}^*} = \{\xi \in \Omega^{0,1} (X \backslash K) | \ \overline{\partial} \xi = 0,\ \overline{\partial}^* \xi =0 \text{ and } \xi \text{ decays to 0 at infinty }\}.
\end{align*}
The fact directly follows from \ref{ddbar'eq1} and the classification of harmonic 1-forms on metric cones as discussed in the proof of Lemma \ref{decay1f}. \end{rem}
\begin{ex} \label{ctexddbar'}
In this example, we discuss the ddbar lemma on $\mathbb{C}^n\backslash \{0\}$. For $n\geq 3$, any exact real $(1,1)$-form $\omega$ satisfies the ddbar lemma; i.e. $\omega = dd^c f$ for a real function function $f$. In the higher dimension cases, One refers to \cite[Lemma 5.5]{goto2012calabi} or \cite[Corollary A.3]{conlon2013asymptotically} for details. However, in the case of $n=2$, the ddbar lemma doesn't hold. According to remark \ref{remobs}, the obstruction of ddbar lemma is given by $\mathcal{H}^{0,1}_{\overline{\partial}, \overline{\partial}^*}/ \overline{\partial} \mathcal{H}$. Then, we can calculate the dimension of the spaces, $\mathcal{H}^{0,1}_{\overline{\partial}, \overline{\partial}^*}$ and $\overline{\partial} \mathcal{H}$ for each growth rate. For $k\geq 0$, we have
\begin{align*}
\dim_{\mathbb{C}} (\mathcal{H}^{0,1}_{\overline{\partial},\overline{\partial}^*})_{-2-k} = k(k+1), \quad \dim_{\mathbb{C}} (\overline{\partial} \mathcal{H})_{-2-k} = k^2.
\end{align*}
Let $\omega$ be an exact real $(1,1)$-form of decay rate $-\delta$, then there exists a real function $f $ of decay rate $2-\delta$ such that,
\begin{align*}
\omega = dd^c f + \theta_{-4} + \theta_{-5} + O(r^{-6}).
\end{align*}
And we can write down $\theta_{-4}$ and $\theta_{-5}$ explicitly,
\begin{align*}
\theta_{-4} = \operatorname{Re}( C_{-3} \partial \xi_{-3} )= \operatorname{Re}\bigg\{ C_{-3} \partial \bigg( \frac{\bar{z}_2 d\bar{z}_1 - \bar{z}_1 d\bar{z}_2}{(|z_1|^2+|z_2|^2)^2} \bigg)\bigg\},
\end{align*}
where $C_{-3}$ is a complex constant and the form $\xi_{-3}$ is a representative of $\mathcal{H}^{0,1}_{\overline{\partial}, \overline{\partial}^*}/ \overline{\partial} \mathcal{H}$ of decay rate $-3$. Similarly, $\theta_{-5}$ can be written as follows,
\begin{align*}
\theta_{-5} &= \operatorname{Re} ( C_{-4,1} \xi_{-4,1} + C_{-4,2} \xi_{-4,2} ) \\
&= \operatorname{Re}\bigg\{ C_{-4,1} \partial \bigg( \frac{\bar{z}_1 \bar{z}_2 d\bar{z}_1 - \bar{z}^2_1 d\bar{z}_2}{(|z_1|^2+|z_2|^2)^3} \bigg) + C_{-4,2} \partial \bigg( \frac{ \bar{z}^2_2 d\bar{z}_1 - \bar{z}_1 \bar{z}_2 d\bar{z}_2}{(|z_1|^2+|z_2|^2)^3}\bigg)\bigg\},
\end{align*}
where $ C_{-4,1}$ and $C_{-4,2}$ are complex constant. In Joyce's book \cite[Theorem 8.9.2]{joyce2000compact}, the statement of ddbar lemma on $\mathbb{C}^2/\{0\}$, which states that $\omega= dd^c f + \theta_{-4}$, is wrong, as we can find infinite error terms. \end{ex}
\section{The Mass formula on AC K\"ahler Manifolds} In this section, we dedicate to generalize the Hein-LeBrun's mass \cite[Theorem C]{hein2016mass} formula to AC K\"ahler manifolds asymptotic to Ricci-flat K\"ahler cones. In section \ref{ACtoLCY}, we introduce some basic set-ups on the manifolds. By introducing a system of coframes at infinity, we calculate the Chern connection forms and the curvature forms of canonical line bundle. In section \ref{massacsec}, the generalized definition of mass will be given in the AC Riemannian manifolds. And we complete the proof of mass formula in AC K\"ahler manifolds asymptotic to Ricci-flat K\"ahler cones.
\subsection{AC K\"ahler manifolds Asymptotic to Locally Calabi-Yau Cones} \label{ACtoLCY}
Let $(C_L, J_0)$ be a K\"ahler cone with Ricci-flat metric $g_0$. The associated link $L$ has positive Ricci curvature $\textup{Ric}(g_L) = (2n-2) g_L$. According to Myers's Theorem, the fundamental group $\pi_1$ is finite. Hence, there exists a universal covering $\widetilde{L}$ with finite deck transformation $\Gamma$ acting on $\widetilde{L}$. Let $(C_{\widetilde{L}}, \widetilde{g}_0)$ be the metric cone of $\widetilde{L}$, then $\widetilde{C}_{L}$ is a finite covering of $C_L$ with $\widetilde{C}_{{L}}/ \Gamma \cong C_L$. By pulling back the complex structure on $C_L$, the covering space $(\widetilde{C}_{L}, \widetilde{J}_0)$ is also a K\"ahler cone with Ricci-flat metric $\widetilde{g}_0$. Since $(\widetilde{C}, \widetilde{J}_0, \widetilde{g}_0)$ is K\"ahler Ricci-flat with trivial fundamental group, then there exists a nowhere vanishing holomorphic section of canonical bundle $K_{\widetilde{C}_{L}}$, $\widetilde{\Omega}_0$ and $\tilde{\omega}_0^{n} = i^{n^2} \widetilde{\Omega}_0 \wedge \overline{\widetilde{\Omega}}_0$. $\widetilde{\Omega}_0$ induces a \textit{multi-valued} nonvanishing holomorphic section, $\Omega_0$, of $K_{C_L}$ on $C_L$ through the covering map; i.e., there exists a group representation $\kappa: \Gamma \rightarrow U(1)$ such that \begin{align} \label{mulvalue}
\gamma^*\widetilde{ \Omega}_0 = \kappa(\gamma) \widetilde{\Omega}_0, \qquad \gamma \in \Gamma. \end{align}
In other words, the multi-valued section $\Omega_0$ has $|\Gamma|$ branches and each branch differs by a constant factor $\kappa(\gamma)$, $\gamma \in \Gamma$.
Let $(C_L, J_0, g_0)$ be a Ricci-flat K\"ahler cone with the link $L$. Let $(X, J, g)$ be an AC K\"ahler manifold asymptotic to $C_L$ together with the diffeomorphism $\Psi : X_\infty= X-K \rightarrow C_L - B_R$, where $B_R = \{x \in C_L, r(x) \leq R\}$. We also assume that $g$ decays to $g_0$ in the end $X_\infty$ with rate $-\tau = 1-n-\epsilon$ and $J$ decays to $J_0$ at infinity.
In order to make analysis in the setting, we introduce a local frame system for K\"ahler cones. Recall that the link $L$ is a \textit{Sasakian} manifold with metric $g_L$. Let $\partial r$ be the normal vector field in radial direction, there exists a canonical vector field over $L$ defined by $\xi = J_0(r \partial r)$, namely, the \textit{Reeb vector field} on $L$. And the \textit{contact 1-form} $\eta$ is the dual of $\xi$ with respect to the metric $g_L$. The contact 1-form $\eta$ defines a subbundle of tangent bundle on the link $L$, namely $D = \ker \eta \rightarrow L$. Restricting $J_0$ on the subbundle $D$ defines an almost complex structure on $D$, which guarantees the existence of a local basis $(m_i, \overline{m}_i)$ for $D$, $i=1,\ldots, n-1$, where $m_i$ is an (1,0) type vector with respect to $J_0|_D$. For any point in $L$, there exists a neighborhood $U$ and local basis $(\xi; m_i, \overline{m}_i)$ for the tangent bundle $T_L (U)$. Let $m_0 = r\partial r -i \xi$, then $(m_i, \overline{m}_i)_{i=0}^{n-1}$ defines a basis on $T_{C_L}(U)$. According to the fact $r\partial r$ is a global real holomorphic vector field on $C_L$, the scaling map of $C_L$ is holomorphic. Hence, the local basis $(m_i, \overline{m}_i)_{i=0}^{n-1}$ is defined in $U \times \mathbb{R}_{>0}$ by pulling forward the scaling map. Let $\{v_i, \overline{v}_i\}_{i=0}^{n-1}$ be the normalizing basis by defining $v_i = r^{-1} m_i$ and $v_i$, $\overline{v}_i$. are (1,0), (0,1) type vectors according to $J_0$. Let 1-forms, $\mu^i$, $\overline{\mu}^i$ be the dual basis of $v_i$, $\overline{v}_i$, then $J_0$ can be written explicitly as \begin{align*} J_0 = i \mu^i \otimes v_i - i \bar{\mu}^i \otimes \bar{v}_i. \end{align*} Let $\overline{\nabla}$ be the Levi-Civita connection of $(C_L, g_0)$. One can easily check that \begin{align} \label{conncal1} \overline{\nabla}_{ \partial r}v_i =\overline{\nabla}_{\partial r} \bar{v}_i=0, \qquad \overline{\nabla}_{ \partial r}\mu^i =\overline{\nabla}_{\partial r} \bar{\mu}^i=0 \end{align} and $\overline{\nabla} J_0 =0$ also implies that $\overline{\nabla}_T v_i$ and $\overline{\nabla}_T \bar{v}_i $ are vectors of type $(1,0)$ and $(0,1)$ respectively for any tangent vector fields $T$ of $C_L$.
In the end $X_\infty$, the frames $\{\mu^i, \bar{\mu}^i\}$ and $\{v_i, \bar{v}_i\}$ are still defined by diffeomorphism $\Psi$. Precisely, there exists a finite open covering on link $L = \bigcup U_k$ such that $X_\infty \subseteq \bigcup V_k$, where $V_k = \Psi^{-1}(U_k \times \mathbb{R}_{>R})$. And on each open set $V_k$, we have local frames $\{\mu^i, \bar{\mu}^i\}$ and $\{v_i, \bar{v}_i\}$. We call the data $(V_k, \{\mu^i, \bar{\mu}^i\}, \{v_i, \bar{v}_i\})$ a system of \textit{asymptotic local frames} on $X$. A quick calculation shows that $J-J_0$ has decay rate at least $-\tau$. Precisely, if we write $\nabla$ as Levi-Civita connection of $g$, then by $J$ (resp. $J_0$) is $\nabla$-parellel (resp. $\nabla_0$-parellel), \begin{align} \label{cxconnre1}
\overline{\nabla}(J- J_0) = -(\nabla-\overline{\nabla}) J. \end{align}
If we write the difference of two Levi-Civita connection as $ A=\nabla- \overline{\nabla}$, $A$ can be viewed as a tensor on $X$. The tensor $A$ can be represented with respect to the system of asymptotic local frame $\{v_i, \bar{v}_i\}$, for instance, $(\nabla-\overline{\nabla})_{v_j}v_i = A^k_{ji} v_k + A^{\bar{k}}_{ji} \bar{v}_k $. Noticing that $|A|= O(r^{-\tau-1})$, if we also write $J$ based on a system of asymptotic local frames, then (\ref{conncal1}) implies that $|\overline{\nabla}_{\partial r} (J-J_0)|= O(r^{-\tau-1}) $. Hence, along each ray, the derivative of each coefficient of $J-J_0$ has decay rate $-\tau$. Therefore, we have $J-J_0 = O(r^{-\tau})$. Explicitly, $J$ can be written as the following tensor, \begin{align*} J= J_0 + i\mathcal{J}^i_{\bar{j}} \bar{\mu}^{j}& \otimes v_i +i\mathcal{K}^i_j \mu^j \otimes v_i\\ &-i\overline{ \mathcal{J}^i_{\bar{j}}} \mu^{j} \otimes \bar{v}_i -i\overline{ \mathcal{K}^i_j} \bar{\mu}^j \otimes \bar{v}_i, \end{align*} with $\mathcal{J}^i_{\bar{j}}$, $ \mathcal{K}^i_j = O(r^{-\tau})$. According to $J^2=-1$ and comparing the terms of decay rate $-\tau$, we have \begin{align*}
\mathcal{K}^i_j =0 \quad \mod O(r^{-2\tau}). \end{align*} Therefore, \begin{align}\label{cxfr1}
J= J_0 + i\mathcal{J}^i_{\bar{j}} \bar{\mu}^{j}& \otimes v_i - i\overline{ \mathcal{J}^i_{\bar{j}}} \mu^{j} \otimes \bar{v}_i + O(r^{-2\tau}) \end{align}
According to (\ref{cxconnre1}), we can derive the relationship between $A$ and $\overline{\nabla} \mathcal{J}$, \begin{align*}
-(\nabla- \overline{\nabla})J_0 = & -2i A^{\bar{j}}_{i\bar{k}}\mu^i \otimes \bar{\mu}^k \otimes \bar{v}_j - 2i A^{\bar{j}}_{ik} \mu^i \otimes {\mu}^k \otimes \bar{v}_j\\
& +2i A^{j}_{\bar{i}\bar{k}} \bar{\mu}^i \otimes \bar{\mu}^k \otimes {v}_j + 2i A^{j}_{\bar{i} k} \bar{\mu}^i \otimes {\mu}^k \otimes {v}_j \end{align*} and \begin{align*}
\overline{\nabla}(J-J_0) = i(\overline{\nabla}\mathcal{J})^j_{\bar{k}, i} + i (\overline{\nabla}\mathcal{J})^j_{\bar{k}, \bar{i}} - i (\overline{\nabla}{\mathcal{J}})^{\bar{j}}_{k, i}- i(\overline{\nabla}{\mathcal{J}})^{\bar{j}}_{k, \bar{i}}. \end{align*} The relation $\overline{\nabla}(J-J_0) =-(\nabla- \overline{\nabla})J_0 +O(r^{-2\tau-1})$ implies that, \begin{align}\label{cxconnre2}
(\overline{\nabla}\mathcal{J})^j_{\bar{k}, i} = 2 A^{j}_{i\bar{k}} + O(r^{-2\tau-1}), \qquad
(\overline{\nabla}\mathcal{J})^j_{\bar{k}, \bar{i}} = 2 A^{j}_{\bar{i}\bar{k}} + O(r^{-2\tau-1}) \end{align} The fact that $A$ is symmetric implies that \begin{align} \label{symoften}
(\overline{\nabla}\mathcal{J})^j_{\bar{k}, \bar{i}}- (\overline{\nabla}\mathcal{J})^j_{\bar{i}, \bar{k}} = O(r^{-2\tau-1}) \end{align} Recall that, on the cone $C_L$, there exists a nowhere vanishing multi-valued holomorphic volume form $\Omega_0$. By restricting to local charts $V_k$, $\Omega_0$ can be expressed as a single-valued form for each $\gamma \in \Gamma$ with respect to the asymptotic local frame of $V_k$, \begin{align*}
\big(\Omega_0 |_{V_k} \big)_\gamma = \kappa(\gamma) f \mu^0 \wedge \cdots \wedge \mu^{n-1}, \end{align*} where $\kappa(\gamma)$ is defined in (\ref{mulvalue}). And the condition $\overline{\nabla}_{\partial r} \Omega_0 =0 $, together with (\ref{conncal1}), implies that $f$ is bounded on $V_k$. It is possible to define a smooth $(n, 0)$ form $\Omega$ on $X_\infty$ with respect to $J$ which tends to $\Omega_0$ at infinity with decay rate $-\tau$. Let $\varpi^i = (\mu^i - iJ \mu^i)/2$ be the $(1,0)$ part of $\mu^i$ with respect to $J$, \begin{align}
\big(\Omega|_{V_k}\big)_\gamma &= \kappa(\gamma) f \varpi^0 \wedge \ldots \wedge \varpi^{n-1} \nonumber \\
&=\big(\Omega_0|_{V_k}\big)_\gamma + \frac{\kappa(\gamma)}{2}\sum_{i=0}^{n-1}(-1)^{i-1} f \mathcal{J}^i_{\bar{j}} \bar{\mu}^j \wedge \mu^0 \wedge \ldots \wedge \widehat{\mu^i} \wedge \ldots \wedge \mu^{n-1} + O(r^{-2\tau}) \label{n0expr1} \end{align}
It's easy to check that $(\Omega|_{V_k}, V_k )$ can be patched together to obtain a multi-valued form $\Omega$ on the end, as $\Omega|_{V_k}$ can be viewed as $(n,0)$ projection of $\Omega_0$ in each $(V_k, J)$. The fall-off condition of $\Omega$ to $\Omega_0$ indicates that $\Omega$ is nowhere vanishing if the radial function $r$ large enough. Without loss of generality, we assume that $\Omega$ is nowhere vanishing on $X_\infty$. We can rewrite the formula (\ref{n0expr1}) on $V_k$ as follows, \begin{align}\label{n0expr2} \Omega = \Omega_0 + i \langle J-J_0 , \Omega_0 \rangle + O(r^{-2\tau}). \end{align}
where $\langle J-J_0 , \Omega_0 \rangle$ means the contraction of tensor fields by viewing $J-J_0$ and $\Omega_0$ as tensor fields on $V_k$. Let $h$ be the standard metric on canonical bundle of $X$ defined to be $h(\mathcal{S})= |\mathcal{S}\wedge \overline{\mathcal{S}}|/ \omega^n $ for any section $\mathcal{S}$ of canonical bundle on $X$. Let $D_h$ be the Chern connection of $h$, then there exists a connection 1-form $\theta$ defined on $X_\infty$ as $D_h \mathcal{S} = \theta \otimes \mathcal{S}$. Locally, we can take $\mathcal{S} = \Omega_\gamma$, where $\Omega_\gamma$ is a single-valued local section of canonical bundle. Recall that $(0,1)$ part of the covariant derivative $D^{0,1}_h =\bar{\partial}$, then $D^{(1,0)} \Omega_\gamma =\bar{\partial} \Omega_\gamma = d\Omega_\gamma$. By writing $D^{0,1}_h \Omega_\gamma = \alpha \otimes \Omega_\gamma$, in the following lemma, we derive an explicit formula of $\alpha$, in which we can see that $\alpha$ is independent of the choice of $\gamma$ modeling the higher decay terms. \begin{lem} \label{01conn} Let $\alpha$ be a $(0,1)$ part of connection 1-form satisfying $D_h^{0,1} \Omega_\gamma = \alpha \otimes \Omega_\gamma$, then in each asymptotic local frame $\{V_k, \mu^j \}$, \begin{align*}
\alpha|_{V_k} = - \sum_{i,j} A^{i}_{i\bar{j}} \bar{\mu}^j + O(r^{-2\tau-1}) \end{align*} \end{lem}
\begin{proof}
Observe that the operator $d$ can be represented as $d= \mu^i \wedge \overline{\nabla}_{v_i} + \bar{\mu}^i \wedge \overline{\nabla}_{\overline{v}_i} $. Then, $\Omega_0$ is a holomorphic volume form on $(C_L, J_0)$ implies that \begin{align*} 0 = d \Omega_0 = \bar{\mu}^i \wedge \overline{\nabla}_{\bar{v}_i} \Omega_0. \end{align*} Hence, $\overline{\nabla}_{\bar{v}_i} \Omega_0 =0$. Together with (\ref{n0expr2}), we have \begin{align} d \Omega &=d\Omega_0 + i d \langle J- J_0, \Omega_0 \rangle + O(r^{-2\tau-1})\nonumber\\ & = \frac{i}{2} \bar{\mu}^i \wedge \langle \overline{\nabla}_{\bar{v}_i} (J-J_0), \Omega_0 \rangle \label{d10part1} \\ & \qquad + \frac{i}{2} \mu^i \wedge \big( \langle \overline{\nabla}_{v_i} (J-J_0), \Omega_0 \rangle + \langle J-J_0, \overline{\nabla}_{v_i}\Omega_0 \rangle \big) +O(r^{-2\tau-1}). \nonumber \end{align} To simplify the expression in (\ref{d10part1}), we apply (\ref{cxconnre1}) or (\ref{cxconnre2}) and (\ref{symoften}). Then, we have \begin{equation}\label{d10part2} \begin{split} i \bar{\mu}^i \wedge \langle \overline{\nabla}_{\bar{v}_i} (J-J_0), \Omega_0 \rangle &= i\bar{\mu}^i \wedge \langle (\nabla- \overline{\nabla})_{\bar{v}_i} J_0 , \Omega_0 \rangle + O(r^{-2\tau-1}) \\ &= \kappa(\gamma) \sum_{i,j,k}(-1)^i 2f A_{\bar{i} \bar{j}}^{k} \bar{\mu}^i \wedge \bar{\mu}^j \wedge \mu^0 \wedge \ldots \wedge \widehat{\mu^k} \wedge \ldots \wedge \mu^n + O(r^{-2\tau-1}) \\ &=O(r^{-2\tau-1}). \end{split} \end{equation} Let $\overline{\Gamma}$ be the Christoffel symbol of $ \overline{\nabla}$, then a quick calculation shows that \begin{align} \label{d10part3} i\mu^i \wedge \langle J-J_0, \overline{\nabla}_{v_i} \Omega_0 \rangle = \kappa(\gamma) \sum_{i,j,k} \mathcal{K}^i_{\bar{j}} (f_i - f \overline{\Gamma}^k_{ik}) \bar{\mu}^{{j}} \wedge \mu^0 \wedge \ldots \wedge \mu^{n-1}, \end{align} and \begin{align} i\mu^i \wedge \langle \overline{\nabla}_{v_i} (J-J_0), \Omega_0 \rangle &= i \mu^i \wedge \langle (\nabla - \overline{\nabla})_{v_i} J_0, \Omega_0 \rangle + O(r^{-2 \tau -1 }) \nonumber \\ &=-2 \kappa(\gamma) \sum_{i,j} f A_{i \bar{j}}^i \bar{\mu}^j \wedge \mu^0 \wedge \ldots \wedge \mu^{n-1} + O(r^{-2 \tau-1}). \label{d10part4} \end{align} Then, (\ref{d10part2})-(\ref{d10part4}) imply that $D^{0,1}_h \Omega = \alpha \otimes \Omega$ with \begin{align}\label{d10part5}
\alpha|_{V_k} = \frac{1}{2} \sum_{i,j,k} \big( \mathcal{K}^i_{\bar{j}} (\log f)_i - \mathcal{K}^i_{\bar{j}} \overline{\Gamma}^{k}_{ik}- 2 A^i_{i \bar{j}} \big)\bar{\varpi}^j + O (r^{-2\tau-1}) \end{align} The formula (\ref{d10part5}) can be simplified further by considering the holomorphic volume form condition $\nabla_{\bar{v}_i}\Omega_0=0$, \begin{align} \label{d10partcon}
\nabla_{\bar{v}_i}\Omega_0 = \kappa(\gamma) ((\log f)_{\bar{i}} - \sum_{k}\overline{\Gamma}^{k}_{\bar{i}k}) f \mu^1 \wedge \ldots \wedge \mu^n \end{align}
The base K\"ahler metric $\omega_0 = g_0 (J_0\cdot, \cdot)$ can be written as $\omega_0 = i (g_0)_{i\bar{j}} \mu^i \wedge \bar{\mu}^j$. The relation $i^{n^2}\Omega_0 \wedge \overline{\Omega}_0 =\frac{1}{n!} \omega_0^n $ implies that $|f|^2 = \det((g_0)_{i\bar{j}})$, then \begin{align*}
(\log|f|^2 )_i & = (g_0)^{j\bar{k}} (g_0)_{j\bar{k},i} \\
& = (g_0)^{j\bar{k}}\big(\overline{\Gamma}^{p}_{ij}(g_0)_{p\bar{k}}+ \overline{\Gamma}^{\bar{q}}_{i\bar{k}}(g_0)_{j\bar{q}} \big) \\
& = \sum_j \overline{\Gamma}^{j}_{ij} + \sum_{k}\overline{\Gamma}^{\bar{k}}_{i\bar{k}}. \end{align*} Therefore, combining with (\ref{d10partcon}), \begin{align*}
(\log f)_i - \sum_k \overline{\Gamma}^k_{ik} & = (\log |f|^2)_i - (\log \bar{f})_i - \sum_k \overline{\Gamma}^k_{ik} \\
& = (\log |f|^2)_i - \sum_{k} \overline{\Gamma}^{\bar{k}}_{i\bar{k}} - \sum_k \overline{\Gamma}^k_{ik} =0 \end{align*} we complete the proof of the lemma. \end{proof}
Therefore, after fixing a Ricci-flat K\"ahler cone $(C_L, J_0, g_0)$, (\ref{cxconnre2}) implies that $\alpha$ only depends on the complex structure $J$ after modeling higher decay terms. By assuming that $D^{1,0} \Omega = \beta \otimes \Omega$, we have \begin{align*} \partial h(\Omega) &= \langle D^{1,0} \Omega, \Omega\rangle + \langle \Omega, \overline{D^{0,1} \Omega} \rangle \\ &= h(\Omega) \beta +h(\Omega) \overline{\alpha}. \end{align*} Hence, we have $\beta = -\overline{\alpha} + \partial \log h(\Omega)$. Let $\theta_\omega$ be the real part of $-i\theta$, we obtain the following expression, \begin{align}\label{acconnform1}
\theta_\omega = 2 \operatorname{Im} \alpha -\frac{1}{2} d^C \log \frac{\omega^n}{|\Omega_0 \wedge \bar{\Omega}_0|} + O(r^{-2\tau-1}). \end{align} Recall that, in $X_\infty$, the Ricci form is given by $\rho = d \theta_\omega$ and the first Chern class is $c_1 = [\rho/\pi]$. By adding a copy of link $L$ to the end of $X$, we have the following long exact sequence, \begin{align} \label{lexseq1} \cdots \rightarrow H_{dR}^1(L) \rightarrow H^2_c (X) \xrightarrow{\iota} H_{dR}^2 (X) \rightarrow H_{dR}^2 (L) \rightarrow \cdots \end{align} Noting that $L$ is the link of a Ricci-flat K\"ahler cone $C_L$, It is known that $L$ is Sasakian-Einstein with $Ric_{g_L} = 2(n-1) g_L$. According to the standard Bochner technique, we have $H^1_{dR} (X)$ vanishes. \begin{lem} \label{cptclasslm} Let $ [\varrho] \in H^2_{dR} (X)$ where $\varrho$ is a closed 2-form with decay rate $-\delta$, $\delta> 2$. Then, there exists a unique preimage of $[\varrho]$ in $H^2_c(X)$, denoted by $\iota^{-1} [\varrho]$. \end{lem}
\begin{proof} The uniqueness of preimage directly follows from $H^{1}_{dR} (L) =0 $. Now, we describe the mapping $r: H^{2}_{dR} (X) \rightarrow H^2_{dR} (L)$ in (\ref{lexseq1}). Notice that, topologically, $X_\infty\cong L \times (R_0, \infty)$, then, the closed 2-form $\varrho$ can be written as $\varrho = \eta_2 + dr \wedge \eta_1$. According to the fall-off condition of $\varrho$ with decay rate $-\delta$, the norm of $\eta_1$, $\eta_2$ on $L$ satisfy \begin{align*}
|\eta_1|_{g_L} = r| \eta_1|_{g_0} = O(r^{1-\delta}), \qquad |\eta_2|_{g_L}= r^2 |\eta_2|_{g_0} = O (r^{2-\delta}) \end{align*}
By adding a copy of link $L$ at infinity, then in topological space $\overline{X}_\infty = L \times (R_0, \infty]$, $\eta_1$ and $\eta_2$ can extend to infinity by defining zero forms at infinity. Hence, we obtain an extension of $\varrho$ in the space $\overline{X}_\infty$, denoted by $\overline{\varrho}$, with $\overline{\varrho} |_{\infty} = 0$. Then, the class $[\varrho]$ under the mapping $r$ is given as follows, \begin{align*}
r([\varrho]) = \big[ \overline{ \varrho} |_{\infty} \big] =0 \end{align*} Then, the long exact sequence (\ref{lexseq1}) implies that we have a preimage of $[\varrho]$ in $H_c^2(X)$. \end{proof}
Based on the proof of lemma \ref{cptclasslm}, we can explicitly construct a closed form with compact support represents $\iota^{-1} [\varrho]$. Recall that $\varrho = \eta_2 + dr \wedge \eta_1$ in $X_\infty$. We integrate the form $dr \wedge \eta_1$ in $r$ direction; in particular $\tilde{\eta}_1 (r) = - \int_{\infty}^r dr \wedge \eta_1$. The 1-form $\tilde{\eta}_1$ is well-defined as $|\eta_1|_{g_L}$ has decay rate $1-\delta < -1$. Then $\varrho= \eta_2 - d_L \tilde{\eta}_1 + d \tilde{\eta}_1 (r)$. Let $\tilde{\eta}_2 = \eta_2-d_L \tilde{\eta}_1$, then the closedness of $\tilde{\eta}_2$ implies that \begin{align*} d \tilde{\eta}_2 = d_L \tilde{\eta}_2 + dr \wedge \partial_r \tilde{\eta}_2 =0 \quad \Rightarrow \quad d_L \tilde{\eta}_2 =0, \quad \partial_r \tilde{\eta}_2 =0. \end{align*} Observing that $ \lim_{r\rightarrow \infty}\tilde{\eta}_2 =0$, together with $\partial_r \tilde{\eta}_2 =0$, we obtain that $\tilde{\eta}_2 = 0$. Hence, in $X_\infty$, $\varrho = d \tilde{\eta}_1$. Let $\chi$ be a smooth cut-off function defined in $X_\infty$ such that \begin{align} \label{cutoff} \begin{split} f(x) =\begin{cases}
1, \quad \text{ if } |x| \geq 3 R_0;\\
0, \quad \text{ if } |x| \leq 2 R_0. \end{cases} \end{split} \end{align} Then, $\varrho-d (\chi \tilde{\eta}_1)$ represents the class $\iota^{-1} [\varrho]$. Back to the case of the first Chern class $c_1 = [\rho /2\pi]$, the expresstion (\ref{acconnform1}) shows that the Ricci form decays at infinity with order $-\tau-2 < -2$. By lemma \ref{cptclasslm}, we have $\iota^{-1} c_1 \in H^2_c (X)$ and $(\rho- d(f \theta_\omega))/2\pi$ represents this class.
\subsection{The Mass Formula} \label{massacsec}
To give a reasonable definition of mass on AC K\"ahler manifolds, we will invoke an expression of mass without applying Euclidean coordinates. As mentioned in \cite[Section 3.1.3]{lee2019geometric}, the ADM mass can be interpreted as the integral of linearization of scalar curvature at the Euclidean background metric. This idea can be generalized to the AC K\"ahler cases if we replace the background Euclidean metric with the Riemannian cone metric. If we perturb the cone metric $g_0$ and consider $g(t) = (1-t) g_0 + t g_1$ where $g_1$ is another Riemannian metric defined on $C_L$ with the corresponding scalar curvature $R(t) \in L^1 (C_L)$ and bounded near $0$, according to Appendix A, lemma \ref{linofs1} for any real local frame on $X_\infty$ \begin{align*}
\frac{d}{dt}\Big|_{t=0} R(t) = \overline{\nabla}^i \overline{\nabla}^j g_{ij} - \overline{\Delta} \operatorname{tr}_{g_0} g \end{align*} Notice that \begin{align*}
\int_{C_L} \frac{d}{dt}\Big|_{t=0} R(t) = \lim_{r \rightarrow \infty} \int_{L(r)} (\overline{\nabla}^j g_{ij}- (\operatorname{tr}_{g_0}g)_i) n^i d\operatorname{Vol}_{L(r)}, \end{align*} where $n$ is a outer normal vector field of $L(r)$ and the integrand of the right-hand side is independent of the choice of real frame. Let $(X, g)$ be an AC Riemannian manifold asymptotic to metric cone $(C_L, g_0)$, we give the abstract definition of mass on $(X,g)$ by \begin{align} \label{massdef}
\mathfrak{m}(g) = \frac{1}{2(2n-1) \operatorname{Vol}(L)}\lim_{r \rightarrow \infty} \int_{L(r)} (\overline{\nabla}^j g_{ij}- (\operatorname{tr}_{g_0}g)_i) n^i d\operatorname{Vol}_{L(r)} \end{align}
Let $(X, g, J)$ be an AC K\"ahler manifold asymptotic to a Ricci-flat K\"ahler cone $(C_L, g_0, J_0)$ of complex dimension $n$ with the metric $g$ asymptotic to $g_0$ in the end $X_\infty$, the remaining part of this section is aimed to derive a mass formula on $(X, g, J)$ which generalize the Hein-LeBrun's mass formula \cite{hein2016mass}. Observe that
\begin{align} \label{massfp1}
-\frac{1}{2} (\operatorname{tr}_{g_0} g)_i = - \frac{1}{2} \Big(\log \frac{\det g}{\det g_0} \Big)_i + O(r^{-2\tau-1})
\end{align}
and according to the calculation in Appendix (\ref{fofa})
\begin{align} \label{massfp2}
\overline{\nabla}^j g_{ij} -\frac{1}{2} (\operatorname{tr}_{g_0}g)_i = \frac{1}{2} g_0^{jk} ( \overline{\nabla}_k g_{ij} + \overline{\nabla}_j g_{ik}- \overline{\nabla}_i g_{jk}) = g_0^{jk} A_{jk, i},
\end{align} where $A_{jk,i}=g_{il} A^l_{jk}$. Also observe that the formulas (\ref{massfp1}) and (\ref{massfp2}) can relate with the terms of $\theta_\omega$ in (\ref{acconnform1}). Then, we have the following mass formula,
\begin{pro} \label{massf1} Let $(X, g, J)$ be an AC K\"ahler manifold asymptotic to a locally Calabi-Yau cone $(C_L, g_0, J_0)$ of complex dimension $n$ with $|\overline{\nabla}^i (g-g_0)|_{g_0} = O(r^{-\tau-i})$ for $i = 0,1,\ldots$, where $\tau = 1-n-\epsilon$. Let $\omega$ represent the K\"ahler form of $g$ on $X$. Let $\theta_\omega$ be the connection 1-form as in (\ref{acconnform1}). Then, the mass has the following formula, \begin{align} \label{massf2} \mathfrak{m}( g) = \frac{1}{(2n-1) (n-1)! \operatorname{Vol}(L)} \lim_{r \rightarrow \infty} \int_{L(r)} \theta_\omega \wedge \omega^{n-1}. \end{align} \end{pro}
\begin{proof} According to the observation from (\ref{massfp1}) and (\ref{massfp2}), the mass formula can be rewritten on asymptotic local coframe $\{\mu_i, \bar{\mu}_i \}$, \begin{align*}
\mathfrak{m}(g) = \frac{1}{2(2n-1) \operatorname{Vol}(L)}\lim_{r\rightarrow \infty} \bigg\{ \int_{L(r)} * 2 \operatorname{Re} \big( A_{, \bar{i}} \bar{\mu}^i\big) - \int_{L(r)} * \bigg(d \log \frac{\sqrt{det g}}{\sqrt{\det{g_0}}}\bigg) + O(r^{-2\epsilon})\bigg\} \end{align*} where $*$ is the Hodge star operator of on $(X,g)$ and $A_{,i} = 2g_0^{j\bar{k}}A_{j\bar{k},i}$. Also notice that \begin{align*}
* \bigg(d \log \frac{\sqrt{\det g}}{\sqrt{\det{g_0}}}\bigg) = -\Big(Jd \log \frac{\omega^n}{ |\Omega_0 \wedge \Omega_0|} \Big)\wedge \frac{\omega^{n-1}}{(n-1)!} = d^C \log \frac{\omega^n}{|\Omega_0 \wedge \Omega_0|} \wedge \frac{\omega^{n-1}}{(n-1)!}. \end{align*} and \begin{align*}
\operatorname{Im} \alpha \wedge \frac{\omega^{n-1}}{(n-1)!} = (J \operatorname{Re} \alpha) \wedge \frac{\omega^{n-1}}{(n-1)!} = * (\operatorname{Re} \alpha ) \end{align*} Comparing with the explicit expression of $\theta$ in (\ref{acconnform1}), then the mass formula can be rewritten as \begin{align} \label{massf2}
\mathfrak{m}(g) = \frac{1}{(2n-1) \operatorname{Vol}(L)} \lim_{r\rightarrow \infty} \bigg\{ \int_{L(r)} * \operatorname{Re}\big( A_{, \bar{i}} \mu^{\bar{i}}- 2\alpha \big) + \frac{1}{(n-1)!} \int_{L(r)} \theta_\omega \wedge \omega^{n-1} \bigg\} \end{align} It suffices to prove the difference $A_{,\bar{i}}\mu^{\bar{i}}- 2\alpha$ does not contribute to the integral. If we write the metric $g$ in terms of frames $\{\mu^i, \mu^{\bar{i}}\}$, \begin{align*}
g = g_{ij} \mu^i \otimes \mu^j + g_{i \bar{j} } \mu^i \otimes \mu^{\bar{j}} + \overline{g_{i \bar{j}}} \mu^{\bar{i}} \otimes \mu^j + \overline{g_{ij}} \mu^{\bar{i}} \otimes \mu^{\bar{j}}.
\end{align*} Noting that $\omega = g(J\cdot, \cdot)$ and inserting (\ref{cxfr1}) into the expression of $g$, we obtain that, \begin{align*} \omega = i g_{i\bar{j}} \mu^i \wedge \mu^{\bar{j}} + i( g_{i\bar{k}} \mathcal{J}^{\bar{k}}_j - g_{j \bar{k}} \mathcal{J}^{\bar{k}}_i ) \mu^i \wedge \mu^j - i ( g_{k \bar{i}} \mathcal{J}^{k}_{\bar{j}} - g_{k \bar{j}} \mathcal{J}^k_{\bar{i}}) \mu^{\bar{i}} \wedge \mu^{\bar{j}} + O(r^{-2\tau}). \end{align*} The (0,2)-part of $\omega$ with respect to $J_0$ is given by $\omega^{0,2} = -i(g_{k\bar{i}} \mathcal{J}^k_{\bar{j}}- g_{k\bar{j}}\mathcal{J}^k_{\bar{i}}) \mu^{\bar{i}}\wedge \mu^{\bar{j}} $.Now we calculate $d^* \omega^{0,2}$. Notice that $d^* = [d^C, \Lambda]$, then \begin{equation} \begin{split} \label{massfp3}
d^* \omega^{0,2} & = i \Lambda (J\circ d (g_{k\bar{i}}\mathcal{J}^k_{\bar{j}} - g_{k \bar{j}} \mathcal{J}^k_{\bar{i}})\mu^{\bar{i}}\wedge \mu^{\bar{j}} ) + O(r^{-2\tau-1})\\
& = (\overline{\nabla} \mathcal{J})^k_{\bar{j}, l}(g_0)_{k\bar{i}} (g_0)^{l \bar{i}} \mu^{\bar{j}} - (\overline{\nabla} \mathcal{J})^k_{\bar{i}, l}(g_0)_{k \bar{j}} (g_0)^{l\bar{i}} \mu^{\bar{j}} + O(r^{-2\tau-1}) \\
& = 2 A^{l}_{l \bar{j}} \mu^{\bar{j}} - 2 (g_0)^{i \bar{k}}A_{ i \bar{k}, \bar{j}} \mu^{\bar{j}} + O(r^{-2\tau-1})\\
& = 2\alpha - A_{, \bar{j}} \mu^{\bar{j}}+ O(r^{-2\tau-1}). \end{split} \end{equation} Inserting (\ref{massfp3}) to the first term of (\ref{massf2}), we have \begin{align*}
\int_{L(r)} * \operatorname{Re} \big(A_{, \bar{i}} \mu^{\bar{i}} -2 \alpha \big) & = \frac{1}{2}\int_{L(r)} * \big( d^* \omega^{0,2} + d^* \overline{\omega^{0,2}} \big) + O(r^{-2 \epsilon})\\
& =\frac{1}{2} \int_{L(r)} d * (\omega^{0,2}+ \overline{\omega^{0,2}})+ O(r^{-2 \epsilon}) = O (r^{-2\epsilon}), \end{align*} hence, the proof is completed. \end{proof}
\begin{thm}\label{massf2} Let $(X, g, J)$ satisfy the same condition as in proposition \ref{massf1}, then we have the following mass formula \begin{align} \label{massf3} \mathfrak{m}( g) = -\frac{2\pi \langle \iota^{-1} c_1, [\omega]^{n-1} \rangle}{(2n-1) (n-1)! \operatorname{Vol}(L)} + \frac{1}{2(2n-1) \operatorname{Vol}(L)} \int_{X} R_g d\operatorname{Vol}_g \end{align} where $c_1$ is the first Chern class of $(X, J)$, $[\omega]$ is the K\"ahler class of $g$, $R_g$ is the scalar curvature of $g$ and $(\cdot ,\cdot ) $ is the duality pairing between $H^{2}_c (X)$ and $H^{2n-2} (X)$. \end{thm}
\begin{proof} In proposition \ref{massf1}, we have already proved a mass formula (\ref{massf2}), where $\theta_\omega$ is a 1-form defined on $X_\infty$ with $d\theta_\omega = \rho$. Consider a smooth cut-off function $f$ as in (\ref{cutoff}) which is $\equiv 0$ on $X- X_\infty$ and $\equiv 1$ if the radius is of large enough. According to the discussion after lemma \ref{cptclasslm}, the first Chern class admits a representative with compact support $\iota^{-1} c_1 = [\rho_0 / 2\pi]$, where $\rho_0 = \rho - d(f \theta_\omega)$ is compactly supported. Let $X_{r\leq R} =\{x\in X, r(x) \leq R \}$ with $R > 3 R_0$, then $\rho_0$ has compact support on $X_{r < R}$. We have \begin{align*}
\frac{(n-1)!}{2} \int_{X_{r \leq R}} R_g d\operatorname{Vol}_g & = \int_{ X_{r \leq R}} \rho \wedge \omega^{n-1} \\
& = \int_{X_{r \leq R}} \big(\rho_0 + d(f \theta_{\omega})\big) \wedge \omega^{n-1}\\
& = {2\pi} \langle \iota^{-1} c_1, [\omega]^{n-1} \rangle + \int_{L(R)} \theta_{\omega} \wedge \omega^{n-1} \end{align*} Then, according to (\ref{massf2}), we have \begin{align*}
\mathfrak{m}(g) & = \frac{1}{(2n-1) (n-1)! \operatorname{Vol}(L)} \lim_{r\rightarrow \infty} \int_{L(r)} \theta_\omega \wedge \omega^{n-1} \\
& = -\frac{2\pi \langle \iota^{-1} c_1, [\omega]^{n-1} \rangle}{(2n-1) (n-1)! \operatorname{Vol}(L)} + \frac{1}{2(2n-1) \operatorname{Vol}(L)} \int_{X} R_g d\operatorname{Vol}_g \end{align*} which completes the proof of mass formula \end{proof} \section{Expansion of AC K\"ahler Metrics } \label{expanacsec} \subsection{Proof of Theorem \ref{expanthm} (i)} According to the conditions given in theorem \ref{expanthm} (i), $\omega_1$ and $\omega_2$ are two K\"ahler forms in the same K\"ahler class and decay to the model K\"ahler form $\omega_0$ with rate $-\tau$ ($\tau = n-1 +\epsilon$) in $X_\infty$. We also assume that their scalar curvature are identically equal. Firstly, we obtain an expression of scalar curvature $R(\omega)$ with respect to some K\"ahler metric $\omega$ satisfying the fall-off condition, based on the formula (\ref{acconnform1}). Noting that, in (\ref{acconnform1}), the leading term of $ \operatorname{Im} \alpha$ only depends on the complex structure $J$ by lemma \ref{01conn}. In particular, let $A_J = 2 \operatorname{Im} \alpha+ O(r^{-2\tau-1})$ in (\ref{acconnform1}), then \begin{align} \label{scalarcurv}
R(\omega) = \operatorname{tr}_\omega \Big(dA_J - \frac{1}{2}dd^C \log \frac{\omega^n}{|\Omega_0 \wedge \overline{\Omega}_0|} \Big) + O(r^{-2-2\tau}) \end{align} Theorem \ref{ddclem} implies that there exists a function $\varphi\in \mathcal{C}^{k+2,\alpha}_{2-2\tau}$ such that $\omega_2 = \omega_1 +dd^C \varphi$. Based on the condition $R_1 = R_2$, we have \begin{align}
0 &= \operatorname{tr}_{\omega_2} \rho (\omega_2) - \operatorname{tr}_{\omega_1} \rho(\omega_1) \nonumber\\ & = \operatorname{tr}_{\omega_1} (\rho({\omega_2 })-\rho({\omega_1})) + (\operatorname{tr}_{\omega_2}-\operatorname{tr}_{\omega_1} ) \rho({\omega_2}) \label{difsc} \end{align} By inserting (\ref{acconnform1}), we rewrite the first term of (\ref{difsc}), \begin{align*}
\operatorname{tr}_{\omega_1} (\rho({\omega_2 })-\rho({\omega_1}))&=\operatorname{tr}_{\omega_1}\bigg(dA_J-\frac{1}{2} dd^C \log \frac{(\omega_1+dd^C \varphi)^n}{|\Omega \wedge \overline{ \Omega} | }-dA_J+ \frac{1}{2} dd^C \log \frac{\omega_1^n}{|\Omega \wedge \overline{ \Omega} | }\bigg) \\ &=-\frac{1}{2} (\operatorname{tr}_{\omega_1} dd^C)^2 \varphi + O(r^{-2\tau-2})\\ & =-\frac{1}{2} \Delta_{\omega_1}^2 \varphi + O(r^{-2\tau -2}). \end{align*} By writing the second term of (\ref{difsc}) in locally asymptotic frame, it is observed that the coefficients of $\rho(\omega_2)$ belongs to $\mathcal{C}^{k-2,\alpha}_{-\tau-2}$ and $(\operatorname{tr}_{\omega_2}-\operatorname{tr}_{\omega_1}) \in \mathcal{C}^{k,\alpha}_{-\tau}$, which implies that \begin{align*} (\operatorname{tr}_{\omega_2}-\operatorname{tr}_{\omega_1})\rho(\omega_2) \in \mathcal{C}^{k-2,\alpha}_{-2\tau-2}. \end{align*} Hence, we can deduce that $\varphi$ satisfies the equation, $\Delta_{\omega_1}^2 \varphi = f $ with $f\in \mathcal{C}^{k-2,\alpha}_{-2-2\tau}$ on $X$ with $-2-2\tau < -2n$. By solving the Laplacian on asymptotic conical manifolds, proposition \ref{laequac} (iii) imples \begin{align} \label{expanpoten} \Delta_{\omega_1} \varphi = C r^{2-2n} + \phi, \qquad \phi \in \mathcal{C}^{k,\alpha}_{-2\tilde{\tau}} \text{ with } \tilde{\tau} = \min\{\tau, - d^-_1/2\}, \end{align} where $d_1^-$ is the second negative exceptional weight in $D$ defined in $\ref{excd}$ with $d_1^- < d_0^- = 2-2n$. By proposition \ref{massf1}, combining with (\ref{acconnform1}), we have \begin{align} \nonumber \mathfrak{m} (\omega_2)- \mathfrak{m} (\omega_1) = \lim_{r\rightarrow \infty} \frac{1}{(2n-1) (n-1)! \operatorname{Vol}(L)}&\bigg[ \int_{L(r)} \theta_{\omega_2} \wedge \omega_2^{n-1}-\int_{L(r)} \theta_{\omega_1} \wedge \omega_1^{n-1}\bigg] \\ \nonumber = \lim_{r\rightarrow \infty} C(n , L) &\bigg[ \int_{L(r)}\theta_{\omega_2} \wedge (\omega_2^{n-1}-\omega_1^{n-1}) \\ \nonumber & \qquad - \int_{L(r)} (\theta_{\omega_2} - \theta_{\omega_1}) \wedge \omega_1^{n-1} \bigg]. \\ \nonumber =\lim_{r\rightarrow \infty}C(n, L) & \bigg[ \int_{L(r)} \theta_{\omega_2} \wedge (\omega_2^{n-1}-\omega_1^{n-1}) \\ & \qquad - \frac{1}{2}\int_{L(r)} d^C \log \frac{(\omega_1+dd^C \varphi)^{n}}{\omega_1^n} \wedge \omega_1^{n-1} \bigg], \label{masscon} \end{align} where $C(n,L) $ is a constant given in proposition \ref{massf1}. Since $\varphi \in \mathcal{C}^{k,\alpha}_{-\tau}$ and by (\ref{acconnform1}), the coefficients of $\theta_{\omega_2}$ with respect to a local asymptotic frame are of class $\mathcal{C}^{k-1,\alpha}_{-\tau-1}$, then we have \begin{align} \label{masscon1} \int_{L(r)} \theta_{\omega_2} \wedge (\omega_2^{n-1}-\omega_1^{n-1}) = \int_{L(r)} \theta_{\omega_2}\wedge dd^C \varphi \wedge (\omega_1^{n-2} + \omega_1^{n-3} \wedge \omega_2 + \ldots + \omega^{n-2}_2 ) = O(r^{-2\epsilon}), \end{align} and \begin{align} \label{masscon2} \int_{L(r)} d^C \log \frac{(\omega_1+dd^C \varphi)^n}{\omega_1^n} \wedge \omega_1^{n-1}= \int_{L(r)} d^C (\Delta_{\omega_1} \varphi) \wedge \omega_1^{n-1} +O(r^{-2\epsilon}). \end{align} By inserting (\ref{masscon1}) and (\ref{masscon2}), the difference of mass is \begin{align*} \mathfrak{m} (\omega_2) - \mathfrak{m} (\omega_1) = -\lim_{r\rightarrow \infty} C(n,L) \int_{L(r)} d^C(\Delta_{\omega_1} \varphi) \wedge \omega_1^{n-1} \end{align*}
Notice that the standard volume form on the link $L$ of the model cone $C_L$ is given by $d \operatorname{Vol}_{g_L} = \eta \wedge \omega_0^{n-1}|_{L(1)}$, where $\eta =r^{-1} J_0 dr$ is the contact 1-form of the Sasakian manifold $L$. Then, \begin{equation} \label{masscoefac} \begin{split}
-\int_{L(r)} d^C(\Delta_{\omega_1} \varphi) \wedge \omega_1^{n-1} &= -C \int_{L(r)} J d(r^{2-2n}) \wedge \omega_1^{n-1} \\ &= C\int_{L(r)} (2n-2) r^{1-2n} (r \eta) \wedge \omega_0^{n-1} + O(r^{-\tau}) \\ &= C (2n-2){\operatorname{Vol} (L)} + O(r^{-\tau}), \end{split} \end{equation} where $C$ is the constant in (\ref{expanpoten}). According to the mass formula in theorem \ref{massf2}, the mass only depends on the K\"ahler class, the first Chern class and scalar curvature; hence, $\mathfrak{m}(\omega_1) =\mathfrak{m}(\omega_2)$. Then, we have $C=0$, which completes the proof of theorem \ref{expanthm} (i).
\subsection{ Proof of Theorem \ref{expanthm} (ii)} For the proof theorem \ref{expanthm} (ii), in the case of $\dim_{\mathbb{C}} X_{\infty} = 2$, noticing that the link of Ricci-flat K\"ahler cone is an Sasakian Einstein manifold of real dimensional 3, $L$ has constant sectional curvature. Thus, $C_L$ is biholomorphic to $\mathbb{C}^2/\Gamma$ for some finite subgroup $\Gamma \subseteq U(2)$. Hence, the discussion can be reduced to ALE cases. Since $H^2(X_\infty) = H^2(S^3/\Gamma) = 0$, $\omega-\omega_{euc}$ is exact real $(1,1)$ form on $X_\infty$ with decay rate $-\tau$. According to Theorem \ref{ddclem} (ii), there exists $\varphi \in \mathcal{C}^{k+2, \alpha}_{2-\tau}$ such that \begin{align} \label{ddcsfkform} \omega - \omega_{euc} = dd^c \varphi + O(r^{-4}). \end{align} If $\omega$ is a scalar-flat K\"ahler metric, then by formula of scalar curvature, in the end $X_\infty$, \begin{align} \label{scalflatcon} 0= \operatorname{tr}_\omega \rho &= \operatorname{tr}_\omega d A_J - \frac{1}{2} \operatorname{tr}_\omega dd^c \log \frac{\omega^n}{\omega_0^n} + O(r^{-2\tau-2}) \nonumber\\ & = \operatorname{tr}_\omega dA_J -\frac{1}{2} \Delta_{\omega}^2 \varphi + O(r^{\max\{-2\tau-2, -6\}}). \end{align} Recall that, in complex dimension $2$, the complex structure of ALE manifolds decays to the standard complex structure in Euclidean space as $J-J_0 = O(r^{-3})$ (see \cite[Proposition 4.5]{hein2016mass}). Since $A_J = 2 \operatorname{Im} \alpha$, where $\alpha$ is given in Lemma \ref{01conn}, then its differential has $dA_J = O(r^{-5})$. Then, the equation (\ref{scalflatcon}) can be rewritten as, \begin{align}
\Delta^2_\omega \varphi = f, \quad \text{ for } f \in \mathcal{C}^{k-2,\alpha}_{-2\tilde{\tau}-2}, \end{align} where $\tilde{\tau} = \min\{\tau, 3/2\}$. Also, by Proposition \ref{laequac} (iii), $\Delta_\omega \varphi = C r^{2-2n} + \phi $ with $\phi \in \mathcal{C}^{k,\alpha}_{-2\tilde{\tau}}$. According to Proposition \ref{massf1}, we have \begin{align*} \mathfrak{m} (\omega) &= \lim_{r \rightarrow \infty} \frac{1}{3\operatorname{Vol}(L)} \int_{L(r)} \theta_\omega \wedge \omega^{n-1}\\ &= \lim_{r \rightarrow \infty} \frac{1}{3\operatorname{Vol}(L)} \int_{L(r)} \Big( A_J - \frac{1}{2} d^C \log \frac{(\omega_0 + dd^C \varphi)^n}{\omega_0^n } \Big) \wedge \omega^{n-1} \\ &= -\lim_{r \rightarrow \infty} \frac{1}{3\operatorname{Vol}(L)} \int_{L(r)} d^C (\Delta_{\omega} \varphi) \wedge \omega^{n-1} \end{align*} Then, repeating the calculation in (\ref{masscoefac}), we have the constant $C$, \begin{align*}
C = 3 \mathfrak{m}(\omega). \end{align*} Notice that $\displaystyle \Delta_0 \log r = \frac{2}{r^2}$, we have the solution \begin{align*}
\varphi = \frac{3 \mathfrak{m}(\omega)}{2} \log r + O(r^{-2\tilde{\tau}+2}). \end{align*}
In the case of $\dim_{\mathbb{C}} X_{\infty} \geq 3$, the asymptotic condition of the K\"ahler form on $X_\infty$, $\omega - \omega_0 = O(r^{-\tau})$ with $\tau = n-1 +\epsilon > 2$, implies that $\omega- \omega_0$ is an exact form. One can check it by identifying $L\times [T_0, \infty)$ with $X_\infty$. Define a new parameter $t = \log r$ on cylinder $L\times [T_0, \infty) $. Consider the cylinder metric $g_{cyl} = dt^2 + g_L$, then the relation with the conical metric is $g_{cyl}= e^{-2t} g_0 $. Therefore, $|\omega - \omega_0|_{g_{cyl}} = O(e^{2-\tau})$. Now, let $\omega-\omega_0 = \eta_2 + \eta_1 \wedge dt$. If we integrate the form in $t$ direction, $\eta = -\int_\infty^t \eta_1 \wedge dt$, then \begin{align*}
d\eta &= d_L \eta - \partial_t \eta \wedge dt \\
& = - \int_\infty^t d_L \eta_1 \wedge dt + \eta_1 \wedge dt \\
& = \int^t_\infty \partial_t \eta_2 \wedge dt + \eta_1 \wedge dt = \eta_2 + \eta_1 \wedge dt, \end{align*} where the third equality is from the closedness of $\omega-\omega_0$. According to Theorem \ref{ddclem} (ii), there exists a real function $\varphi \in \mathcal{C}^{k+2,\alpha}_{2-\tau}$ such that $\omega - \omega_0 = i\partial \overline{\partial} \varphi + O(r^{-2n})$. Based on the formula (\ref{scalarcurv}) and $R(\omega)=0$, we have the equation $\Delta^2_\omega \varphi = f$, for $f \in \mathcal{C}^{k-2,\alpha}_{-2\tau'-2}$, where the decay rate of $f$ is determined by the decay rate of $dA_J$ (by assumption (\ref{cxdec})), the dimension $n$ and $\tau$; in particular, $\tau' = \min\{n, \tau, n-1+ \epsilon'\}$. Also, by proposition \ref{laequac} (ii), $\Delta_\omega \varphi = C r^{2-2n} + \phi \ (**)$ with $\phi \in \mathcal{C}^{k,\alpha}_{-2\tilde{\tau}}$, $\tilde{\tau} = \min\{\tau', -d^-_1/2\}$. Combining proposition \ref{massf1}, (\ref{acconnform1}) and (\ref{masscoefac}), we can obtain a formula for the constant $C$ in $(**)$, \begin{align*}
C = \frac{2n-1}{n-1} m(\omega). \end{align*} Notice that $\Delta_0 r^{4-2n} = 2(4-2n) r^{2-2n}.$ Therefore, we have the expansion of the solution of $(**)$, \begin{align*}
\varphi = \frac{C}{ 2(4-2n)} r^{4-2n} + O(r^{2-2\tilde{\tau}}) \end{align*} Noting the the standard K\"ahler form on $C_L$ is $\omega_0 = dd^c r^2 /2$, the K\"ahler form $\omega$ admits the following expansion, \begin{align*}
\omega = \frac{1}{2} dd^c r^2 + \frac{(2n-1) \mathfrak{m}(\omega)}{2(4-2n)(n-1)} dd^c r^{4-2n} + O(r^{-2\tilde{\tau}}), \end{align*} which completes the proof.
\section{Positive Mass Theorem on Resolution Spaces of Calabi-Yau Cones} \label{pmts}
We conclude the paper by proving the positive mass theorem based on the generalized ADM mass defined in (\ref{massdef}).
Let $(C_L, J_0, g_0)$ be a Ricci-flat K\"ahler cone with only one singularity at the vertex $O$. Recall the resolution of the Ricci-flat K\"ahler cone is a smooth complex manifold $X$ with a proper map $\pi : X \rightarrow C_L$ such that the map $\pi: X\backslash E \rightarrow C_L \backslash \{O\} $ is biholomorphic, where $E= \pi^{-1}(O)$ is the exceptional divisor. The positive mass theorem does not always hold for resolution spaces of Ricci-flat K\"ahler cone cones, for instance, the counter example constructed by LeBrun \cite{cmp/1104162166}. So it is reasonable to consider a special class of resolution spaces. In particular, we say the isolated singularity $O \in C_L$ is \textit{canonical} if the resolution spaces $(X, \pi)$ satisfies, \begin{align} \label{kx}
K_X = \pi^* K_{C_L} + \sum_{i} a_i E_i, \end{align} where $E_i$ are the irreducible hypersurfaces and $a_i \geq 0$. Then, we can prove the following positive mass theorem on canonical resolution spaces of Calabi-Yau cones.
\begin{thm} Let $(C_L, J_0, g_0)$ be a Ricci-flat K\"ahler cone such that the only singularity $O \in C_L$ is canonical. Suppose that $X$ admits an asymptotically conical K\"ahler metrics $g$ with decay rate $-\tau$, $\tau = n-1+\epsilon$ $(\epsilon >0)$. If $(X,g)$ has scalar curvature $R \geq 0$, then the mass $\mathfrak{m}(X,g) \geq 0$, and equals zero only if $(X,J,g)$ is a crepant resolution of $C_L$ with a scalar-flat K\"ahler metric $g$. \end{thm}
\begin{proof} According to the assumption that $s \geq 0$, the integral of scalar curvature in the mass formula (\ref{massf3}) is nonnegative, and equals zero only if $g$ is scalar-flat. It suffices to prove the first term of the mass formula is also nonnegative.
Firstly, recalling the map $\iota: H_c^2(X) \rightarrow H^2 (X)$ induced by inclusion of cohomology classes, we prove that $[\omega] \in H^2(X)$ has a preimage in $H_c^2(X)$. For $\dim_\mathbb{C} X =2$, the Calabi-Yau cone must be $\mathbb{C}^2/ \Gamma$ with $\Gamma$ is a finite subgroup of $U(2)$. The fact $H^{1}(S^3/ \Gamma) = H^2(S^3/ \Gamma)=0$ implies that $\iota$ is an isomorphism. For $\dim_\mathbb{C} X \geq 3$, since we have the decay condition $\omega - \frac{1}{2}dd^C r^2 = O(r^{-\tau})$ with $\tau>2$ in $X_\infty$, the lemma \ref{cptclasslm} implies that $\omega - \frac{1}{2} dd^C r^2 $ is an exact form in $X_\infty$, written by $d \theta$. Let $f$ be a smooth function identically equal to $1$ near infinity and vanishing in a compact set of $X$, for instance, the function defined in (\ref{cutoff}), then $\omega - d (f (d^C r^2/4 + \theta))$ has a compact support.
Notice that $c_1 (X) = - c_1(K_X)$, then, \begin{align*}
-\langle \iota^{-1} c_1, [\omega]^{n-1} \rangle = - \langle \iota^{-1} c_1, (\iota^{-1}[\omega])^{n-1} \rangle = \langle c_1(K_X),(\iota^{-1} [\omega])^{n-1} \rangle. \end{align*} According to (\ref{kx}) and Poincar\'{e} duality, we have, \begin{align*}
\langle c_1(K_X), (\iota^{-1}[\omega])^{n-1} \rangle = \sum_{i}a_i\int_{E_i} \omega^{n-1}\geq 0. \end{align*} The above inequality only if $a_i =0$ for all $i$, which completes the proof.
\end{proof}
\begin{ex}
Let $D$ be a compact Fano manifold with $\dim_\mathbb{C} D =n-1$ and $K_D^\times$, the space by shrinking the zero section of the canonical line line bundle of $D$. In this example, we consider the standard resolution of $(K_D^{\alpha})^\times$, where $\alpha$ is a positive fractional number. Firstly, we check that $K_D^\times$ is a Calabi-Yau cone. Let $U$ be a local chart in $D$ as well as a local trivialization of $K_D$. We assume that $U\times \mathbb{C} \subseteq K_D$ has a holomorphic coordinate system $\{ z_1, \ldots, z_{n-1}, u \}$. There exists a local holomorphic $(n,0)$ form on $U \times \mathbb{C}$, $ \Omega_U = dz_1\wedge \ldots \wedge dz_{n-1} \wedge d u$ and $\Omega_U$ can be extended naturally to be a global nonvanishing holomorphic $(n,0)$ form on $K_D$. To see this, let $V$ be another local chart of $D$ with $K_D|_V = V\times \mathbb{C}$ and $\{w_1,\ldots, w_{n-1} , v\}$, its holomorphic coordinates. Recall that the transition function of $K_D$ is given by $ v= u \cdot J(\partial z/ \partial w)$, then, \begin{align*} dw_1\wedge \ldots \wedge dw_{n-1} \wedge d v = J(\partial z/ \partial w) dw_1 \wedge \ldots \wedge dw_{n-1} \wedge du = dz_1\wedge \ldots \wedge dz_1 \wedge d u. \end{align*}
Hence, we have a global nonvanishing holomorphic $(n,0)$ form, $\Omega$ on $K_D$. By restricting $\Omega$ to the space away from the zero section of $K_D$, we show that $K_D^\times $ is a Calabi-Yau cone. Similarly, we can find a global holomorphic $(n,0)$ form, $\Omega$, on $K_D^{\alpha}$. Let $U$ be a local chart in $D$ with $(K_D)^\alpha |_{U} = U \times \mathbb{C}$ and $\{z_1, \ldots, z_{n-1}, t \}$ be a holomorphic coordinates of $\{U \times \mathbb{C}\}$. Then, locally, $\Omega$ can be written as \begin{align*}
\Omega|_{U} = \frac{1}{\alpha} t^{\frac{1-\alpha}{\alpha}} dz_1 \wedge \ldots \wedge dz_{n-1} \wedge dt. \end{align*} Also, by restricting $\Omega$ away from the zero section of $K_D^\alpha$, we obtain a nonvanishing holomorphic (possibly multi-valued) $(n,0)$ form on $(K_D^\alpha)^\times$. Note that $\Omega$ is multi-valued if the order $(1-\alpha)/\alpha$ is not an integer. Therefore, for the resolution map by shrinking zero section, $\pi: K^\alpha_D \rightarrow (K^\alpha_D)^\times$, we have the following adjunction type formula, \begin{align*}
K_{ K_D^{\alpha}} = \pi^* K_{ (K_D^\alpha)^{\times}} + \frac{1-\alpha}{\alpha} D \end{align*} Hence, $K^\alpha_D$ satisfies the positive mass theorem if and only if $\alpha \leq 1$. \end{ex}
\appendix \section{Linearization of Scalar Curvature and Mass on AE manifolds}
The goal of this appendix is to complete the detailed calculation of linearization of scalar curvature. Based on the calculation, we will see that, on AE manifolds, the ADM mass can be interpreted as the integral of linearization of scalar curvature over the sphere at infinity.
Let $X$ be a Riemannian manifold with a base metric $\overline{g}$. If we assume that $g$ is another metric over $X$, then the linearization of scalar curvature $DR(g)$ with respect to $(X, \overline{g})$ is defined to be \begin{align*}
DR(g) = \frac{d}{dt} \Big|_{t=0} R(t) \end{align*} where $R(t)$ is the scalar curvature of $g(t) = \overline{g} + t(g- \overline{g})$. Let $\{\xi_i, 1\leq i \leq n\}$ be a system of local frame on $X$. Let $\nabla$ (resp. $\overline{\nabla}$) be the Levi-Civita connection of $g$ (resp. $\overline{g}$) and $\Gamma$ (resp. $\overline{\Gamma}$) be the Christoffel symbol of $\nabla$ (resp. $\overline{\nabla}$). Also the difference of two connections is denoted by $A = \nabla - \overline{\nabla}$. Then, the Ricci curvature of $g$ and $\overline{g}$ has the following relation, \begin{lem} Let $R_{ij}$ and $\overline{R}_{ij}$ be the Ricci curvature of $g$ and $\overline{g}$ with respect to the local frame system $\{\xi_i, 1\leq i\leq n\}$ and its dual $\{\omega^i, 1\leq i \leq n\}$, then we have \begin{align*} R_{ij} = \overline{R}_{ij} + (\overline{\nabla}_k A^k_{ij} - \overline{\nabla}_i A^k_{kj}) + (A^k_{k l} A^{l}_{ij} - A^k_{jl} A^l_{ik}) + A^k_{lj} \omega^l([\xi_k, \xi_i]). \end{align*} \end{lem} \begin{proof} Notice that $R_{ij} = \omega^k (R(\xi_k, \xi_i)\xi_j) = \omega^k (\nabla_{\xi_{k}} \nabla_{\xi_i} \xi_j - \nabla_{\xi_i} \nabla_{\xi_k} \xi_j)$ and $\overline{R}_{ij}$ admits a similar formula. Then, we have \begin{align}\label{linofsp1} \begin{split}
R_{ij}- \overline{R}_{ij} = &\ \omega^k (\nabla_{\xi_{k}} \nabla_{\xi_i} \xi_j - \nabla_{\xi_i} \nabla_{\xi_k} \xi_j) - \omega^k (\overline{\nabla}_{\xi_{k}} \overline{\nabla}_{\xi_i} \xi_j - \overline{ \nabla}_{\xi_i} \overline{\nabla}_{\xi_k} \xi_j) \\
=&\ \omega^k \Big[ \overline{\nabla}_{\xi_k} (\nabla_{\xi_i} - \overline{\nabla}_{\xi_i}) \xi_j + (\nabla_{\xi_k}-\overline{\nabla}_{\xi_k}) \overline{\nabla}_{\xi_i} \xi_j + (\nabla_{\xi_k} - \overline{\nabla}_{\xi_k})(\nabla_{\xi_i} - \nabla_{\xi_i}) \xi_j \Big] \\
& -\omega^k \Big[ \overline{\nabla}_{\xi_i} (\nabla_{\xi_k} - \overline{\nabla}_{\xi_k}) \xi_j + (\nabla_{\xi_i}-\overline{\nabla}_{\xi_i}) \overline{\nabla}_{\xi_k} \xi_j + (\nabla_{\xi_i} - \overline{\nabla}_{\xi_i})(\nabla_{\xi_k} - \nabla_{\xi_k}) \xi_j \Big]\\
=& \ \big(\xi_k (A_{ij}^k) + A^l_{ij} \overline{\Gamma}^k_{lk} + A^k_{lk} \overline{\Gamma}^l_{ij} + A^l_{ij} A^k_{lk} \big) - \big( \xi_i (A_{kj}^k) + A^l_{kj} \overline{\Gamma}^k_{il} + A^k_{li} \overline{\Gamma}^l_{kj} + A^l_{kj} A^k_{li} \big) \\
=& \ \xi_k (A_{ij}^k) - \xi_i (A_{kj}^k) + A^l_{ij} \overline{\Gamma}^k_{lk} + A^k_{lk} \overline{\Gamma}^l_{ij} - A^l_{kj} \overline{\Gamma}^k_{il} - A^k_{li} \overline{\Gamma}^l_{kj} \\ &\ +A^l_{ij} A^k_{lk} - A^l_{kj} A^k_{li}.
\end{split} \end{align} Also we can compute $\overline{\nabla} A$, then \begin{align}\label{linofsp2} \begin{split}
\overline{\nabla}_k A^k_{ij} - \overline{\nabla}_i A^k_{kj} = & \ \xi_k (A_{ij}^k) +A^l_{ij} \overline{\Gamma}_{lk}^k - A^k_{lj} \overline{\Gamma}_{ki}^l - A^k_{il} \overline{\Gamma}^l_{kj} \\
& - \xi_i (A_{kj}^k) - A^l_{kj} \overline{\Gamma}^k_{il} + A^k_{lj} \overline{\Gamma}^l_{ik} + A^k_{lk} \overline{\Gamma}^l_{ij}\\
= & \ \xi_k (A_{ij}^k) - \xi_i (A_{kj}^k) + A^l_{ij} \overline{\Gamma}^k_{lk} + A^k_{lk} \overline{\Gamma}^l_{ij} - A^l_{kj} \overline{\Gamma}^k_{il} - A^k_{li}\overline{\Gamma}^l_{kj}\\
& + A^k_{lj} \overline{\Gamma}^l_{ik} - A^k_{lj} \overline{\Gamma}^l_{ki}. \end{split} \end{align} Notice that $A$ is a symmetric tensor, but $\overline{\Gamma}$ is not and \begin{align*}
\overline{\Gamma}^{l}_{ik} - \overline{\Gamma}^l_{ki} = \omega^l ([\xi_i, \xi_k]). \end{align*} By plugging (\ref{linofsp2}) into (\ref{linofsp1}), we have \begin{align*}
R_{ij}- \overline{R}_{ij} = \overline{\nabla}_k A^k_{ij} - \overline{\nabla}_i A^k_{kj} - A^k_{lj}\omega^l ([\xi_i, \xi_k]) +A^l_{ij} A^k_{lk} - A^l_{kj} A^k_{li}, \end{align*} which completes the proof. \end{proof}
\begin{lem} \label{linofs1} Let $DR(g)$ be the linearization of scalar curvature with respect to $(X, \overline{g})$. Then, on the local frame system $\{\xi_i, 1\leq i\leq n\}$, \begin{align*}
DR(g) = \langle \textup{Ric}_{\overline{g}}, g-\overline{g} \rangle_{\overline{g}} + \overline{\nabla}^j \overline{ \nabla}^i g_{ij} - \overline{\Delta} \operatorname{tr}_{\overline{g}} g. \end{align*} \end{lem} \begin{proof} Firstly, we derive a formula for $A$. Notice that \begin{align*}
\overline{\nabla}_i g_{jk} & = \xi_i (g_{jk}) - g( \overline{\nabla}_{\xi_i} \xi_j , \xi_k ) - g( \xi_j , \overline{\nabla}_{\xi_i} \xi_k ) \\
& = g\big( (\nabla_{\xi_i} - \overline{\nabla}_{\xi_i}) \xi_j , \xi_k \big) +g \big( \xi_j ,( \nabla_{\xi_i} -\overline{\nabla}_{\xi_i}) \xi_k \big) \\
& = g_{lk} A^l_{ij} + g_{jl} A^l_{ik}. \end{align*} By permutation of the indice $i,j,k$, we have \begin{align}\label{fofa}
A^k_{ij} = \frac{1}{2} g^{kl} (\overline{\nabla}_i g_{jl} + \overline{\nabla}_j g_{il} - \overline{\nabla}_l g_{ij}) \end{align} Let $A(t) = \nabla_t - \overline{\nabla}$ and $\nabla_t$ be the Levi-Civita connection of $g(t) = \overline{g} + t(g - \overline{g})$, then \begin{align*}
R(t) = g^{ij}(t) \big( \overline{R}_{ij} + \overline{\nabla}_k A^k_{ij}(t) - \overline{\nabla}_i A^k_{kj} (t) - A^k_{lj}(t) \omega^l ([\xi_i, \xi_k]) +A^l_{ij}(t) A^k_{lk}(t) - A^l_{kj}(t) A^k_{li}(t) \big) \end{align*} Notice that $A(0) =0$, we have \begin{align} \begin{split}\label{linofsp3}
\frac{d}{dt} \Big|_{t=0} R(t) = & \overline{R}_{ij} \frac{d}{dt} \Big|_{t=0} g^{ij}(t) \\ & + \overline{g}^{ij} \frac{d}{dt} \big|_{t=0} \big(\overline{\nabla}_k A^k_{ij}(t) - \overline{\nabla}_i A^k_{kj} (t)\big) - \omega^l ([\xi_i, \xi_k]) \overline{g}^{ij} \frac{d}{dt} \Big|_{t=0} A^k_{lj}(t). \end{split} \end{align} Then, we compute each term in (\ref{linofsp3}) separately. \begin{align*}
\frac{d}{dt} \Big|_{t=0} A^k_{lj}(t) & = \frac{1}{2} \overline{g}^{km} (\overline{\nabla}_l \dot{g}_{jm} + \overline{\nabla}_j \dot{g}_{ml} - \overline{\nabla}_m \dot{g}_{jl})(0)\\
& = \frac{1}{2} \overline{g}^{km} (\overline{\nabla}_l {g}_{jm} + \overline{\nabla}_j {g}_{ml} - \overline{\nabla}_m {g}_{jl}) \end{align*} Inserting into the third term of (\ref{linofsp3}), \begin{align*}
\omega^l ([\xi_i, \xi_k]) \overline{g}^{ij} \frac{d}{dt} \Big|_{t=0} A^k_{lj}(t) & = \frac{1}{2}\overline{g}^{ij} \overline{g}^{km} (\overline{\nabla}_{[i,k]} {g}_{jm} + \overline{\nabla}_j {g}_{m[i,k]} - \overline{\nabla}_m {g}_{j[i,k]}) =0 \end{align*} The above identity is vanishing because of the symmetry of indices; in particular, by permuting $(ij)$ and $(km)$, we can check that each term in above identity is vanishing. Notice that \begin{align*}
\overline{g}^{ij} \frac{d}{dt} \Big|_{t=0} \overline{\nabla}_{i} A^k_{kj}(t)
& = \frac{1}{2} \overline{g}^{ij} \overline{\nabla}_i \big( \overline{g}^{kl}(\overline{\nabla}_j g_{kl} + \overline{\nabla}_k g_{lj} - \overline{\nabla}_l g_{jk}) \big)\\
& = \frac{1}{2} \overline{g}^{ij} \overline{\nabla}_{i} \overline{\nabla}_j \big(\overline{g}^{kl} g_{kl} \big) \\
& = \frac{1}{2} \Delta \operatorname{tr}_{\overline{g}} g \end{align*} and \begin{align*}
\overline{g}^{ij} \frac{d}{dt} \Big|_{t=0} \overline{\nabla}_{k} A^k_{ij}(t)
& = \frac{1}{2}\overline{g}^{ij} \overline{\nabla}_k \big( \overline{g}^{kl}(\overline{\nabla}_j g_{il} + \overline{\nabla}_i g_{lj} - \overline{\nabla}_l g_{ji}) \big)\\
& = \overline{g}^{ij} \overline{\nabla}_k \big( \overline{g}^{kl}\overline{\nabla}_j g_{il} \big) - \frac{1}{2} \overline{g}^{ij} \overline{\nabla}_k \big( \overline{g}^{kl} \overline{\nabla}_l g_{ij} \big) \\
& = \overline{\nabla}^j \overline{ \nabla}^i g_{ij} - \frac{1}{2} \overline{\Delta} \operatorname{tr}_{\overline{g}} g, \end{align*}
combining with $ d/dt|_{t=0} g^{ij} (t) = \overline{g}^{ik}\overline{g}^{jl} (g-\overline{g})_{kl} $ and \ref{linofsp3}, then we have, \begin{align*}
\frac{d}{dt} \Big|_{t=0} R(t) = \langle \textup{Ric}_{\overline{g}}, g-\overline{g} \rangle_{\overline{g}} + \overline{\nabla}^j \overline{ \nabla}^i g_{ij} - \overline{\Delta} \operatorname{tr}_{\overline{g}} g, \end{align*} which completes the proof. \end{proof} Based on lemma \ref{linofs1}, we can quickly derive the ADM mass on AE manifolds from the integral of linearization of scalar curvature. Let $g$ be a Riemannian metric in $\mathbb{R}^n$ and $DR(g)$ be the linearization of scalar curvature with respect to the Euclidean metric on $\mathbb{R}^n$. Then, the integral of $DR(g)$ is given by \begin{align}
\int_{\mathbb{R}^n} DR(g) d\operatorname{Vol}_{euc}
& = \lim_{r \rightarrow \infty} \int_{S^{n-1}(r)} \big( \overline{\nabla}^j g_{ij} - (\operatorname{tr}_{g_0} g) \big) n^i d \operatorname{Vol}_{S^{n-1} (r)} \nonumber \\
& = \lim_{r \rightarrow \infty} \int_{S^{n-1}(r)} (g_{ij,j}- g_{jj, i}) n^i d\operatorname{Vol}_{S^{n-1} (r)}. \label{massae} \end{align} Notice that the formula (\ref{massae}) only depends on the information of metrics at infinity. Hence, up to a constant factor, (\ref{massae}) can be defined to be the mass on AE manifolds.
\end{document} | arXiv |
Intersection graph
In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an intersection graph, but some important special classes of graphs can be defined by the types of sets that are used to form an intersection representation of them.
Formal definition
Formally, an intersection graph G is an undirected graph formed from a family of sets
$S_{i},\,\,\,i=0,1,2,\dots $
by creating one vertex vi for each set Si, and connecting two vertices vi and vj by an edge whenever the corresponding two sets have a nonempty intersection, that is,
$E(G)=\{\{v_{i},v_{j}\}\mid i\neq j,S_{i}\cap S_{j}\neq \emptyset \}.$
All graphs are intersection graphs
Any undirected graph G may be represented as an intersection graph. For each vertex vi of G, form a set Si consisting of the edges incident to vi; then two such sets have a nonempty intersection if and only if the corresponding vertices share an edge. Therefore, G is the intersection graph of the sets Si.
Erdős, Goodman & Pósa (1966) provide a construction that is more efficient, in the sense that it requires a smaller total number of elements in all of the sets Si combined. For it, the total number of set elements is at most n2/4, where n is the number of vertices in the graph. They credit the observation that all graphs are intersection graphs to Szpilrajn-Marczewski (1945), but say to see also Čulík (1964). The intersection number of a graph is the minimum total number of elements in any intersection representation of the graph.
Classes of intersection graphs
Many important graph families can be described as intersection graphs of more restricted types of set families, for instance sets derived from some kind of geometric configuration:
• An interval graph is defined as the intersection graph of intervals on the real line, or of connected subgraphs of a path graph.
• An indifference graph may be defined as the intersection graph of unit intervals on the real line
• A circular arc graph is defined as the intersection graph of arcs on a circle.
• A polygon-circle graph is defined as the intersection of polygons with corners on a circle.
• One characterization of a chordal graph is as the intersection graph of connected subgraphs of a tree.
• A trapezoid graph is defined as the intersection graph of trapezoids formed from two parallel lines. They are a generalization of the notion of permutation graph, in turn they are a special case of the family of the complements of comparability graphs known as cocomparability graphs.
• A unit disk graph is defined as the intersection graph of unit disks in the plane.
• A circle graph is the intersection graph of a set of chords of a circle.
• The circle packing theorem states that planar graphs are exactly the intersection graphs of families of closed disks in the plane bounded by non-crossing circles.
• Scheinerman's conjecture (now a theorem) states that every planar graph can also be represented as an intersection graph of line segments in the plane. However, intersection graphs of line segments may be nonplanar as well, and recognizing intersection graphs of line segments is complete for the existential theory of the reals (Schaefer 2010).
• The line graph of a graph G is defined as the intersection graph of the edges of G, where we represent each edge as the set of its two endpoints.
• A string graph is the intersection graph of curves on a plane.
• A graph has boxicity k if it is the intersection graph of multidimensional boxes of dimension k, but not of any smaller dimension.
• A clique graph is the intersection graph of maximal cliques of another graph
• A block graph of clique tree is the intersection graph of biconnected components of another graph
Scheinerman (1985) characterized the intersection classes of graphs, families of finite graphs that can be described as the intersection graphs of sets drawn from a given family of sets. It is necessary and sufficient that the family have the following properties:
• Every induced subgraph of a graph in the family must also be in the family.
• Every graph formed from a graph in the family by replacing a vertex by a clique must also belong to the family.
• There exists an infinite sequence of graphs in the family, each of which is an induced subgraph of the next graph in the sequence, with the property that every graph in the family is an induced subgraph of a graph in the sequence.
If the intersection graph representations have the additional requirement that different vertices must be represented by different sets, then the clique expansion property can be omitted.
Related concepts
An order-theoretic analog to the intersection graphs are the inclusion orders. In the same way that an intersection representation of a graph labels every vertex with a set so that vertices are adjacent if and only if their sets have nonempty intersection, so an inclusion representation f of a poset labels every element with a set so that for any x and y in the poset, x ≤ y if and only if f(x) ⊆ f(y).
See also
• Contact graph
References
• Čulík, K. (1964), "Applications of graph theory to mathematical logic and linguistics", Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963), Prague: Publ. House Czechoslovak Acad. Sci., pp. 13–20, MR 0176940.
• Erdős, Paul; Goodman, A. W.; Pósa, Louis (1966), "The representation of a graph by set intersections" (PDF), Canadian Journal of Mathematics, 18 (1): 106–112, doi:10.4153/CJM-1966-014-3, MR 0186575, S2CID 646660.
• Golumbic, Martin Charles (1980), Algorithmic Graph Theory and Perfect Graphs, Academic Press, ISBN 0-12-289260-7.
• McKee, Terry A.; McMorris, F. R. (1999), Topics in Intersection Graph Theory, SIAM Monographs on Discrete Mathematics and Applications, vol. 2, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 0-89871-430-3, MR 1672910.
• Szpilrajn-Marczewski, E. (1945), "Sur deux propriétés des classes d'ensembles", Fund. Math., 33: 303–307, doi:10.4064/fm-33-1-303-307, MR 0015448.
• Schaefer, Marcus (2010), "Complexity of some geometric and topological problems" (PDF), Graph Drawing, 17th International Symposium, GS 2009, Chicago, IL, USA, September 2009, Revised Papers, Lecture Notes in Computer Science, vol. 5849, Springer-Verlag, pp. 334–344, doi:10.1007/978-3-642-11805-0_32, ISBN 978-3-642-11804-3.
• Scheinerman, Edward R. (1985), "Characterizing intersection classes of graphs", Discrete Mathematics, 55 (2): 185–193, doi:10.1016/0012-365X(85)90047-0, MR 0798535.
Further reading
• For an overview of both the theory of intersection graphs and important special classes of intersection graphs, see McKee & McMorris (1999).
External links
• Jan Kratochvíl, A video lecture on intersection graphs (June 2007)
• E. Prisner, A Journey through Intersection Graph County
| Wikipedia |
Edmond Halley
Edmond[2] (or Edmund)[3] Halley FRS (/ˈhæli/;[4][5] 8 November [O.S. 28 October] 1656[lower-alpha 1] – 25 January 1742 [O.S. 14 January 1741])[7][8] was an English astronomer, mathematician and physicist. He was the second Astronomer Royal in Britain, succeeding John Flamsteed in 1720.
Edmond Halley
FRS
Portrait of Halley (c. 1690)
Born8 November [O.S. 28 October] 1656[lower-alpha 1]
Haggerston, Middlesex, England
Died25 January 1742 [O.S. 14 January 1741] (aged 85)
Greenwich, Kent, England
Resting placeSt. Margaret's, Lee, South London
NationalityEnglish
Alma materThe Queen's College, Oxford
Spouse
Mary Tooke
(m. 1682)
ChildrenEdmond (1699—1742)
Margaret (1685—1743)
Catherine (1688—1765)[1]
Scientific career
FieldsAstronomy, mathematics, physics, cartography
InstitutionsUniversity of Oxford
Royal Observatory, Greenwich
From an observatory he constructed on Saint Helena in 1676–77, Halley catalogued the southern celestial hemisphere and recorded a transit of Mercury across the Sun. He realised that a similar transit of Venus could be used to determine the distances between Earth, Venus, and the Sun. Upon his return to England, he was made a fellow of the Royal Society, and with the help of King Charles II, was granted a master's degree from Oxford.
Halley encouraged and helped fund the publication of Isaac Newton's influential Philosophiæ Naturalis Principia Mathematica (1687). From observations Halley made in September 1682, he used Newton's law of universal gravitation to compute the periodicity of Halley's Comet in his 1705 Synopsis of the Astronomy of Comets.[lower-alpha 2] It was named after him upon its predicted return in 1758, which he did not live to see.
Beginning in 1698, Halley made sailing expeditions and made observations on the conditions of terrestrial magnetism. In 1718, he discovered the proper motion of the "fixed" stars.
Early life
Halley was born in Haggerston in Middlesex. According to Halley, his birthdate was 8 November [O.S. 28 October] 1656.[6] his father, Edmond Halley Sr., came from a Derbyshire family and was a wealthy soap-maker in London.[9] As a child, Halley was very interested in mathematics. He studied at St Paul's School,[9] where he developed his initial interest in astronomy, and was elected captain of the school in 1671.[6] On 3 November [O.S. 23 October] 1672, Halley's mother, Anne Robinson, died.[6] In July 1673,[6] he began studying at The Queen's College, Oxford.[9] Halley took a twenty-four-foot (7.3 m) long telescope with him, apparently paid for by his father.[10] While still an undergraduate, Halley published papers on the Solar System and sunspots.[11] In March 1675, he wrote to John Flamsteed, the Astronomer Royal (England's first), telling him that the leading published tables on the positions of Jupiter and Saturn were erroneous, as were some of Tycho Brahe's star positions.[12]
Career
Publications and inventions
In 1676, Flamsteed helped Halley publish his first paper, titled "A Direct and Geometrical Method of Finding the Aphelia, Eccentricities, and Proportions of the Primary Planets, Without Supposing Equality in Angular Motion", about planetary orbits, in Philosophical Transactions of the Royal Society.[12] Influenced by Flamsteed's project to compile a catalogue of stars of the northern celestial hemisphere, Halley proposed to do the same for the southern sky,[13] dropping out of school to do so. He chose the south Atlantic island of Saint Helena (west of Africa), from which he would be able to observe not only the southern stars, but also some of the northern stars with which to cross-reference them.[14] King Charles II supported his endeavour.[15] Halley sailed to the island in late 1676, then set up an observatory with a large sextant with telescopic sights.[16] Over a year, he made observations with which he would produce the first telescopic catalogue of the southern sky,[16] and observed a transit of Mercury across the Sun. Focusing on this latter observation, Halley realised that observing the solar parallax of a planet—more ideally using the transit of Venus, which would not occur within his lifetime—could be used to trigonometrically determine the distances between Earth, Venus, and the Sun.[17][lower-alpha 3]
Halley returned to England in May 1678, and used his data to produce a map of the southern stars.[19] Oxford would not allow Halley to return because he had violated his residency requirements when he left for Saint Helena. He appealed to Charles II, who signed a letter requesting that Halley be unconditionally awarded his Master of Arts degree, which the college granted on 3 December 1678.[20] Just a few days before,[21] Halley had been elected as a fellow of the Royal Society, at the age of 22.[22] In 1679, he published Catalogus Stellarum Australium ('A catalogue of the stars of the South'), which includes his map and descriptions of 341 stars.[19][23][lower-alpha 4] Robert Hooke presented the catalogue to the Royal Society.[25] In mid-1679, Halley went to Danzig (Gdańsk) on behalf of the Society to help resolve a dispute: because astronomer Johannes Hevelius' observing instruments were not equipped with telescopic sights, Flamsteed and Hooke had questioned the accuracy of his observations; Halley stayed with Hevelius and checked his observations, finding that they were quite precise.[24]
By 1681, Giovanni Domenico Cassini had told Halley of his theory that comets were objects in orbit.[26] In September 1682, Halley carried out a series of observations of what became known as Halley's Comet; his name became associated with it because of his work on its orbit and predicting its return in 1758[27] (which he did not live to see). In early 1686, Halley was elected to the Royal Society's new position of secretary, requiring him to give up his fellowship and manage correspondence and meetings, as well as edit the Philosophical Transactions.[28][lower-alpha 5] Also in 1686, Halley published the second part of the results from his Helenian expedition, being a paper and chart on trade winds and monsoons. The symbols he used to represent trailing winds still exist in most modern day weather chart representations. In this article he identified solar heating as the cause of atmospheric motions. He also established the relationship between barometric pressure and height above sea level. His charts were an important contribution to the emerging field of information visualisation.[29]
Halley spent most of his time on lunar observations, but was also interested in the problems of gravity. One problem that attracted his attention was the proof of Kepler's laws of planetary motion. In August 1684, he went to Cambridge to discuss this with Isaac Newton, much as John Flamsteed had done four years earlier, only to find that Newton had solved the problem, at the instigation of Flamsteed with regard to the orbit of comet Kirch, without publishing the solution. Halley asked to see the calculations and was told by Newton that he could not find them, but promised to redo them and send them on later, which he eventually did, in a short treatise titled On the motion of bodies in an orbit. Halley recognised the importance of the work and returned to Cambridge to arrange its publication with Newton, who instead went on to expand it into his Philosophiæ Naturalis Principia Mathematica published at Halley's expense in 1687.[30] Halley's first calculations with comets were thereby for the orbit of comet Kirch, based on Flamsteed's observations in 1680–1681.[lower-alpha 6] Although he was to accurately calculate the orbit of the comet of 1682, he was inaccurate in his calculations of the orbit of comet Kirch. They indicated a periodicity of 575 years, thus appearing in the years 531 and 1106, and presumably heralding the death of Julius Caesar in a like fashion in 45 BCE. It is now known to have an orbital period of circa 10,000 years.
In 1691, Halley built a diving bell, a device in which the atmosphere was replenished by way of weighted barrels of air sent down from the surface.[32] In a demonstration, Halley and five companions dived to 60 feet (18 m) in the River Thames, and remained there for over an hour and a half. Halley's bell was of little use for practical salvage work, as it was very heavy, but he made improvements to it over time, later extending his underwater exposure time to over 4 hours.[33] Halley suffered one of the earliest recorded cases of middle ear barotrauma.[32] That same year, at a meeting of the Royal Society, Halley introduced a rudimentary working model of a magnetic compass using a liquid-filled housing to damp the swing and wobble of the magnetised needle.[34]
In 1691, Halley sought the post of Savilian Professor of Astronomy at Oxford. While a candidate for the position, Halley faced the animosity of the Astronomer Royal, John Flamsteed, and the Anglican Church questioned his religious views,[lower-alpha 7] largely on the grounds that he had doubted the Earth's age as given in the Bible.[35][lower-alpha 8] After Flamsteed wrote to Newton to rally support against Halley, Newton wrote back in hopes of reconciliation, but was unsuccessful.[35] Halley's candidacy was opposed by both the Archbishop of Canterbury, John Tillotson, and Bishop Stillingfleet, and the post went instead to David Gregory, who had Newton's support.[38]
In 1692, Halley put forth the idea of a hollow Earth consisting of a shell about 500 miles (800 km) thick, two inner concentric shells and an innermost core.[39] He suggested that atmospheres separated these shells, and that each shell had its own magnetic poles, with each sphere rotating at a different speed. Halley proposed this scheme to explain anomalous compass readings. He envisaged each inner region as having an atmosphere and being luminous (and possibly inhabited), and speculated that escaping gas caused the aurora borealis.[40] He suggested, "Auroral rays are due to particles, which are affected by the magnetic field, the rays parallel to Earth's magnetic field."[41]
In 1693 Halley published an article on life annuities, which featured an analysis of age-at-death on the basis of the Breslau statistics Caspar Neumann had been able to provide. This article allowed the British government to sell life annuities at an appropriate price based on the age of the purchaser. Halley's work strongly influenced the development of actuarial science. The construction of the life-table for Breslau, which followed more primitive work by John Graunt, is now seen as a major event in the history of demography.
The Royal Society censured Halley for suggesting in 1694 that the story of Noah's flood might be an account of a cometary impact.[42] A similar theory was independently suggested three centuries later, but is generally rejected by geologists.[43]
In 1696, Newton was appointed as warden of the Royal Mint and nominated Halley as deputy comptroller of the Chester mint. Halley spent two years supervising coin production. While there, he caught two clerks pilfering precious metals. He and the local warden spoke out about the scheme, unaware that the local master of the mint was profiting from it.[44]
In 1698, the Czar of Russia (later known as Peter the Great) was on a visit to England, and hoped Newton would be available to entertain him. Newton sent Halley in his place. He and the Czar bonded over science and brandy. According to one disputed account, when both of them were drunk one night, Halley jovially pushed the Czar around Deptford in a wheelbarrow.[45]
Exploration years
In 1698, at the behest of King William III, Halley was given command of the Paramour, a 52 feet (16 m) pink, so that he could carry out investigations in the South Atlantic into the laws governing the variation of the compass, as well as to refine the coordinates of the English colonies in the Americas.[46] On 19 August 1698, he took command of the ship and, in November 1698, sailed on what was the first purely scientific voyage by an English naval vessel. Unfortunately problems of insubordination arose over questions of Halley's competence to command a vessel. Halley returned the ship to England to proceed against officers in July 1699. The result was a mild rebuke for his men, and dissatisfaction for Halley, who felt the court had been too lenient.[47] Halley thereafter received a temporary commission as a captain in the Royal Navy, recommissioned the Paramour on 24 August 1699 and sailed again in September 1699 to make extensive observations on the conditions of terrestrial magnetism.[9] This task he accomplished in a second Atlantic voyage which lasted until 6 September 1700, and extended from 52 degrees north to 52 degrees south.[9] The results were published in General Chart of the Variation of the Compass (1701).[9] This was the first such chart to be published and the first on which isogonic, or Halleyan, lines appeared.[48][49] The use of such lines inspired later ideas such as those of isotherms by Alexander von Humboldt in his maps.[50] In 1701, Halley made a third and final voyage on the Paramour to study the tides of the English Channel.[51] In 1702, he was dispatched by Queen Anne on diplomatic missions to other European leaders.[51]
The preface to Awnsham and John Churchill's collection of voyages and travels (1704), supposedly written by John Locke or by Halley, valourised expeditions such as these as part of a grand expansion of European knowledge of the world:
What was cosmography before these discoveries, but an imperfect fragment of a science, scarce deserving so good a name? When all the known world was only Europe, a small part of Africk, and the lesser portion of Asia; so that of this terraqueous globe not one sixth part had ever been seen or heard of. Nay so great was the ignorance of man in this particular, that learned persons made a doubt of its being round; others no less knowing imagin'd all they were not acquainted with, desart and uninhabitable. But now geography and hydrography have receiv'd some perfection by the pains of so many mariners and travelers, who to evince the rotundity of the earth and water, have sail’d and travell'd round it, as has been here made appear; to show there is no part uninhabitable, unless the frozen polar regions, have visited all other countries, tho never so remote, which they have found well peopl'd, and most of them rich and delightful…. Astronomy has receiv'd the addition of many constellations never seen before. Natural and moral history is embelish'd with the most beneficial increase of so many thousands of plants it had never before receiv'd, so many drugs and spices, such variety of beasts, birds and fishes, such rarities in minerals, mountains and waters, such unaccountable diversity of climates and men, and in them of complexions, tempers, habits, manners, politicks, and religions…. To conclude, the empire of Europe is now extended to the utmost bounds of the earth, where several of its nations have conquests and colonies. These and many more are the advantages drawn from the labours of those, who expose themselves to the dangers of the vast ocean, and of unknown nations; which those who sit still at home abundantly reap in every kind: and the relation of one traveler is an incentive to stir up another to imitate him, whilst the rest of mankind, in their accounts without stirring a foot, compass the earth and seas, visit all countries, and converse with all nations.[52]
Life as an academic
In November 1703, Halley was appointed Savilian Professor of Geometry at the University of Oxford, his theological enemies, John Tillotson and Bishop Stillingfleet having died. In 1705, applying historical astronomy methods, he published the paper Astronomiae cometicae synopsis (A Synopsis of the Astronomy of Comets); in this, he stated his belief that the comet sightings of 1456, 1531, 1607, and 1682 were of the same comet, and that it would return in 1758.[53][lower-alpha 2] Halley did not live to witness the comet's return, but when it did, the comet became generally known as Halley's Comet.
By 1706 Halley had learned Arabic and completed the translation started by Edward Bernard[55] of Books V–VII of Apollonius's Conics from copies found at Leiden and the Bodleian Library at Oxford. He also completed a new translation of the first four books from the original Greek that had been started by the late David Gregory. He published these along with his own reconstruction of Book VIII[56] in the first complete Latin edition in 1710. The same year, he received an honorary degree of doctor of laws from Oxford.[9]
In 1716, Halley suggested a high-precision measurement of the distance between the Earth and the Sun by timing the transit of Venus. In doing so, he was following the method described by James Gregory in Optica Promota (in which the design of the Gregorian telescope is also described). It is reasonable to assume Halley possessed and had read this book given that the Gregorian design was the principal telescope design used in astronomy in Halley's day.[57] It is not to Halley's credit that he failed to acknowledge Gregory's priority in this matter. In 1717–18 he discovered the proper motion of the "fixed" stars (publishing this in 1718)[58] by comparing his astrometric measurements with those given in Ptolemy's Almagest. Arcturus and Sirius were two noted to have moved significantly, the latter having progressed 30 arc minutes (about the diameter of the moon) southwards in 1800 years.[59]
In 1720, together with his friend the antiquarian William Stukeley, Halley participated in the first attempt to scientifically date Stonehenge. Assuming that the monument had been laid out using a magnetic compass, Stukeley and Halley attempted to calculate the perceived deviation introducing corrections from existing magnetic records, and suggested three dates (460 BC, AD 220 and AD 920), the earliest being the one accepted. These dates were wrong by thousands of years, but the idea that scientific methods could be used to date ancient monuments was revolutionary in its day.[60]
Halley succeeded John Flamsteed in 1720 as Astronomer Royal, a position Halley held until his death in 1742 at the age of 85.[13] He was buried in the graveyard of the old church of St Margaret's, Lee (since rebuilt), at Lee Terrace, Blackheath.[61] He was interred in the same vault as the Astronomer Royal John Pond; the unmarked grave of the Astronomer Royal Nathaniel Bliss is nearby.[62] His original tombstone was transferred by the Admiralty when the original Lee church was demolished and rebuilt – it can be seen today on the southern wall of the Camera Obscura at the Royal Observatory, Greenwich. His marked grave can be seen at St Margaret's Church, Lee Terrace.[63][64]
Despite the persistent misconception that Halley received a knighthood, it is not the case. The idea can be tracked back to American astronomical texts such as William Augustus Norton's 1839 An Elementary Treatise on Astronomy, possibly due to Halley's royal occupations and connections to Sir Isaac Newton.[65]
Personal life
Halley married Mary Tooke in 1682 and settled in Islington. The couple had three children.[11]
Named after Edmond Halley
• Halley's Comet (orbital period (approximately) 75 years)
• Halley (lunar crater)
• Halley (Martian crater)
• Halley Research Station, Antarctica
• Halley's method, for the numerical solution of equations
• Halley Street, in Blackburn, Victoria, Australia
• Edmund Halley Road, Oxford Science Park, Oxford, OX4 4DQ UK
• Edmund Halley Drive, Reston, Virginia, United States
• Edmund Halley Way, Greenwich Peninsula, London
• Halley's Mount, Saint Helena (680m high)
• Halley Drive, Hackensack, New Jersey, intersects with Comet Way on the campus of Hackensack High School, home of the Comets
• Rue Edmund Halley, Avignon, France
• The Halley Academy, a school in London, England
• Halley House School, Hackney London (2015)
• Halley Gardens, Blackheath, London.
Pronunciation and spelling
There are three pronunciations of the surname Halley. These are /ˈhæli/, /ˈheɪli/, and /ˈhɔːli/. As a personal surname, the most common pronunciation in the 21st century, both in Great Britain[4] and in the United States,[5] is /ˈhæli/ (rhymes with "valley"). This is the personal pronunciation used by most Halleys living in London today.[66] This is useful guidance but does not, of course, tell us how the name should be pronounced in the context of the astronomer or the comet. The alternative /ˈheɪli/ is much more common in the latter context than it is when used as a modern surname. Colin Ronan, one of Halley's biographers, preferred /ˈhɔːli/. Contemporary accounts spell his name Hailey, Hayley, Haley, Haly, Halley, Hawley and Hawly, and presumably pronunciations varied similarly.[67]
As for his given name, although the spelling "Edmund" is quite common, "Edmond" is what Halley himself used, according to a 1902 article,[2] though a 2007 International Comet Quarterly article disputes this, commenting that in his published works, he used "Edmund" 22 times and "Edmond" only 3 times,[68] with several other variations used as well, such as the Latinised "Edmundus". Much of the debate stems from the fact that, in Halley's own time, English spelling conventions were not yet standardised, and so he himself used multiple spellings.[3]
In popular media
• Halley is voiced by Cary Elwes in the 2014 documentary series Cosmos: A Spacetime Odyssey.
• A fictional version of Halley appears in The Magnus Archives, a horror podcast.
• Fritz Weaver portrayed a fictional version of Halley in Comet Watch, a second season episode of the anthology series Tales from the Darkside.
• Actor John Wood was cast as Edmond Halley in the TV series, Longitude in 2000.[69]
• Halley is a major figure in David Williamson's play Nearer the Gods, about Sir Isaac Newton
• The pronunciation /ˈheɪli/ was used by rock and roll singer Bill Haley, who called his backing band his "Comets" after the common pronunciation of Halley's Comet in the United States at the time.[70]
See also
• History of geomagnetism
Notes
1. This date is by Halley's own account, but is otherwise unconfirmed.[6]
2. This was perhaps the first astronomical mystery solved using Newton's laws by a scientist other than Newton.[54]
3. He wrote as late as 1716 in hopes of a future expedition to make these observations.[18]
4. This contribution caused Flamsteed to nickname Halley "the southern Tycho".[23] Tycho had catalogued the stars observed by Johannes Hevelius.[24]
5. For his payment, he was given 75 unsold copies of the Society's unsuccessful book The History of Fish, which it had depleted its funds on.[28]
6. Halley asked Newton to obtain Flamsteed's observations for him, as his own relationship with the older astronomer had deteriorated.[31]
7. "To what extent Halley's failure was due the animosity of John Flamsteed or to his stout defence [sic] of his religious belief that not every iota of scripture was necessarily divinely inspired is still a matter of some argument. All Oxford appointees had to assent to the Articles of Religion and be approved by the Church of England. Halley's religious views could not have been too outlandish because the University was happy to grant him another chair 12 years later. ... Halley held liberal religious views and was very outspoken. He believed in having a reverent but questioning attitude towards the eternal problems and had little sympathy for those who unquestioningly accepted dogma. He was certainly not an atheist." Hughes 1985, pp. 198, 201.
8. Halley had noticed that observable geological processes take much longer than implied by the Genesis flood narrative. In attempt to explain the biblical account, Halley had theorized that the gravity of a passing comet could have suddenly raised the oceans in a certain area.[36] Following his failure to obtain the professorship, he investigated ocean salinity as an indicator of the Earth's age, since salt is carried to the ocean by rivers. He estimated the Earth to be over 100 million years old.[37]
References
1. Oxford Dictionary of National Biography. Halley, Edmond (1656–1742) Alan Cook
2. The Times (London) Notes and Queries No. 254, 8 November 1902 p.36
3. Hughes, David W.; Green, Daniel W. E. (January 2007). "Halley's First Name: Edmond or Edmund" (PDF). International Comet Quarterly. Harvard University. 29: 14. Bibcode:2007ICQ....29....7H. Might we suggest... simply recogniz[ing] both forms, noting that—in the days when Halley lived—there was no rigid 'correct' spelling, and that this particular astronomer seemed to prefer the 'u' over the 'o' in his published works.
4. Jones, Daniel; Gimson, Alfred C. (1977) [1917]. Everyman's English Pronunciation Dictionary. Everyman's Reference Library (14 ed.). London: J. M. Dent & Sons. ISBN 0-460-03029-9.
5. Kenyon, John S.; Knott, Thomas A. (1953). A Pronouncing Dictionary of American English. Springfield, MA: Merriam-Webster Inc. ISBN 0-87779-047-7.
6. Sagan & Druyan 1997, p. 40.
7. The source of the dates of birth and death is a biography of Edmond Halley written shortly after his death: Biographia Britannica, vol. 4, 1757, pp. 2494–2520. On his tombstone at Lee near Greenwich his year of birth and his year of death were inscribed as follows: Natus est A.C. MDCLVI. Mortuus est A.C. MDCCXLI. Before 1752 the Julian calendar was used in England. Also, the year began on 25 March.
8. "Halley, Edmond". astro.uni-bonn.de.
9. Clerke, Agnes Mary (1911). "Halley, Edmund" . In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 12 (11th ed.). Cambridge University Press. p. 856.
10. Sagan & Druyan 1997, pp. 40–41.
11. Oxford Dictionary of National Biography (2004). "Edmond Halley". Westminster Abbey. Retrieved 3 May 2015.
12. Sagan & Druyan 1997, p. 41.
13. BBC. "Edmond Halley (1656–1742)". Retrieved 28 March 2017.
14. Sagan & Druyan 1997, p. 42.
15. Cook, Alan (2003). "Edmond Halley and Visual Representation in Natural Philosophy". In Lefèvre, Wolfgang; Renn, Jürgen; Schoepflin, Urs (eds.). The Power of Images in Early Modern Science. Basel: Birkhäuser. pp. 251–262. doi:10.1007/978-3-0348-8099-2_13. ISBN 978-3-0348-8099-2.
16. Ridpath, Ian. "Edmond Halley's southern star catalogue". Star Tales. Archived from the original on 26 October 2021. Retrieved 22 February 2022.
17. Jeremiah Horrocks, William Crabtree, and the Lancashire observations of the transit of Venus of 1639, Allan Chapman 2004 Cambridge University Press doi:10.1017/S1743921305001225
18. Sagan & Druyan 1997, p. 60.
19. Kanas, Nick (2012). Star Maps: History, Artistry, and Cartography (2nd ed.). Chichester, U.K.: Springer. p. 123. ISBN 978-1-4614-0917-5.
20. Sagan & Druyan 1997, p. 45.
21. O'Connor, J. J.; Robertson, E. F. (January 2000). "Edmond Halley - Biography". Maths History. Archived from the original on 10 August 2020. Retrieved 28 June 2021.
22. Sharp, Tim (11 December 2018). "Edmond Halley: An Extraordinary Scientist and the Second Astronomer Royal". Space.com. Archived from the original on 14 February 2014. Retrieved 28 June 2021.
23. Hughes 1985, p. 202.
24. Jones, Harold Spencer (1957). "Halley as an Astronomer". Notes and Records of the Royal Society of London. 12 (2): 175–192. doi:10.1098/rsnr.1957.0008. ISSN 0035-9149. JSTOR 530833. S2CID 202574705.
25. Sagan & Druyan 1997, p. 44.
26. Sagan & Druyan 1997, p. 48.
27. Lancaster-Brown, Peter (1985). Halley & His Comet. Blandford Press. pp. 76–78. ISBN 0-7137-1447-6.
28. Sagan & Druyan 1997, p. 56.
29. Halley E. (1686), "An Historical Account of the Trade Winds, and Monsoons, Observable in the Seas between and Near the Tropicks, with an Attempt to Assign the Phisical Cause of the Said Winds", Philosophical Transactions, 16:153–168 doi:10.1098/rstl.1686.0026
30. Peter Ackroyd. Newton. Great Britain: Chatto and Windus, 2006.
31. Sagan & Druyan 1997, p. 64.
32. Edmonds, Carl; Lowry, C; Pennefather, John. "History of diving". South Pacific Underwater Medicine Society Journal. 5 (2). Archived from the original on 14 October 2010. Retrieved 17 March 2009.{{cite journal}}: CS1 maint: unfit URL (link)
33. "History: Edmond Halley". London Diving Chamber. Retrieved 6 December 2006.
34. Gubbins, David, Encyclopedia of Geomagnetism and Paleomagnetism, Springer Press (2007), ISBN 1-4020-3992-1, ISBN 978-1-4020-3992-8, p. 67
35. Sagan & Druyan 1997, p. 62.
36. Sagan & Druyan 1997, p. 59.
37. Sagan & Druyan 1997, p. 63.
38. Derek Gjertsen, The Newton Handbook, ISBN 0-7102-0279-2, pg 250
39. Halley, E. (1692). "An account of the cause of the change of the variation of the magnetic needle; with an hypothesis of the structure of the internal parts of the earth". Philosophical Transactions of the Royal Society of London. 16 (179–191): 470–478.
40. Carroll, Robert Todd (13 February 2006). "hollow Earth". Skeptic's Dictionary. Retrieved 23 July 2006.
41. "10 Illuminating Facts about the Northern Lights". Oceanwide Expeditions. Retrieved 24 August 2018.
42. V. Clube and B. Napier, The Cosmic Serpent London: Faber and Faber, 1982.
43. Deutsch, A., C. Koeberl, J.D. Blum, B.M. French, B.P. Glass, R. Grieve, P. Horn, E.K. Jessberger, G. Kurat, W.U. Reimold, J. Smit, D. Stöffler, and S.R. Taylor, 1994, The impact-flood connection: Does it exist? Terra Nova. v. 6, pp. 644–650.
44. Sagan & Druyan 1997, p. 67.
45. Sagan & Druyan 1997, pp. 67–68.
46. Sagan & Druyan 1997, p. 68.
47. Halley, Edmond (1982). The Three Voyages of Edmond Halley in the Paramore, 1698–1701. UK: Hakluyt Society. pp. 129–131. ISBN 0-904180-02-6.
48. Cook, Alan (12 April 1997). Edmond Halley: Charting the Heavens and the Seas (1 ed.). Oxford USA: Oxford University Press. ISBN 0198500319. Retrieved 5 January 2015.
49. Cook, Alan (2001). "Edmond Halley and the Magnetic Field of the Earth". Notes and Records of the Royal Society of London. 55 (3): 473–490. doi:10.1098/rsnr.2001.0158. ISSN 0035-9149. JSTOR 531953. S2CID 122788971.
50. Robinson, A. H.; Wallis, Helen M. (1967). "Humboldt's Map of Isothermal Lines: A Milestone in Thematic Cartography". The Cartographic Journal. 4 (2): 119–123. doi:10.1179/caj.1967.4.2.119. ISSN 0008-7041.
51. Sagan & Druyan 1997, p. 70.
52. Halley or Locke,'A Collection of Voyages and Travels, some now first printed from manuscript', Preface, p.lxxiii
53. Sagan & Druyan 1997, pp. 66–67.
54. Sagan & Druyan 1997, p. 66.
55. M.B. Hall, 'Arabick Learning in the Correspondence of the Royal Society, 1660–1677', The 'Arabick' Interest of the Natural Philosophers in 17th-Century England, p.154
56. Michael N. Fried, 'Edmond Halley's Reconstruction of the Lost Book of Apollonius's Conics: Translation and Commentary', Spring 2011
57. Wakefield, Julie; Press, Joseph Henry (2005). Halley's Quest: A Selfless Genius and His Troubled Paramore. USA: National Academies Press. ISBN 0309095948. Retrieved 5 January 2015.
58. Aitken, Robert G. (October 1942). "Edmund Halley and Stellar Proper Motions". Astronomical Society of the Pacific Leaflets. SAO/NASA Astrophysics Data System (ADS). 4 (164): 108. Bibcode:1942ASPL....4..103A. Retrieved 27 June 2021 – via Harvard.edu.
59. Holberg, Jay B. (2007). Sirius: Brightest Diamond in the Night Sky. Chichester, UK: Praxis Publishing. pp. 41–42. ISBN 978-0-387-48941-4.
60. Johnson, Anthony, Solving Stonehenge, The New Key to an Ancient Enigma(Thames & Hudson 2008) ISBN 978-0-500-05155-9
61. "Location of Edmond Halley's tomb". shadyoldlady.com. The Shady Old Lady's guide to London. Retrieved 5 January 2015.
62. Halley's gravesite is in a cemetery at the junction of Lee Terrace and Brandram Road, across from the Victorian Parish Church of St Margaret. The cemetery is a 30-minute walk from the Greenwich Observatory.
63. "Photograph of Edmond Halley's Tombstone". flamsteed.org. Flamsteed Society. Retrieved 5 January 2015.
64. Redfern, Dave (Summer 2004). Doing the Halley Walk (Issue 14 ed.). London: Horizons. Retrieved 5 January 2015.
65. Rosenfeld, Randall; Edgar, James (2010). "2010JRASC.104...28R Page 28". Journal of the Royal Astronomical Society of Canada. 104 (1): 28. Bibcode:2010JRASC.104...28R. Retrieved 23 May 2022.
66. Ian Ridpath. "Saying Hallo to Halley". Retrieved 8 November 2011.
67. "Science: Q&A". The New York Times. 14 May 1985. Retrieved 8 November 2011.
68. Hughes, David W.; Green, Daniel W. E. (January 2007). "Halley's First Name: Edmond or Edmund" (PDF). International Comet Quarterly. Harvard University. 29: 7. Bibcode:2007ICQ....29....7H.
69. "Longitude © (1999)". Retrieved 22 June 2021.
70. "Guide Profile: Bill Haley". Oldies.about.com. Archived from the original on 21 January 2012. Retrieved 8 November 2011.
Sources
• Hughes, David W. (August 1985). "Edmond Halley, Scientist" (PDF). Journal of the British Astronomical Association. London, UK: British Astronomical Association. 95 (5): 193. Bibcode:1985JBAA...95..193H.
• Sagan, Carl & Druyan, Ann (1997). Comet. New York: Random House. ISBN 978-0-3078-0105-0.
Further reading
• Armitage, Angus (1966). Edmond Halley. London: Nelson.
• Coley, Noel (1986). "Halley and Post-Restoration Science". History Today. 36 (September): 10–16.
• Cook, Alan H. (1998). Edmond Halley: Charting the Heavens and the Seas. Oxford: Clarendon Press. Bibcode:1998ehch.book.....C.
• Darrigol, Olivier (2012). A History of Optics from Greek Antiquity to the Nineteenth Century. Oxford University. p. 76. ISBN 9780191627453. Halley is noted as the first to publish the algebraic version of the thin lens equation.
• Ronan, Colin A. (1969). Edmond Halley, Genius in Eclipse. Garden City, New York: Doubleday and Company.
• Seyour, Ian (1996). "Edmond Halley – explorer". History Today. 46 (June): 39–44.
• Sarah Irving (2008). "Natural science and the origins of the British empire (London,1704), 92–93". A Collection of Voyages and Travels. 3 (June): 92–93.
External links
Wikiquote has quotations related to Edmond Halley.
Wikisource has original works by or about:
Edmond Halley
Wikimedia Commons has media related to Edmond Halley.
• Edmond Halley Biography (SEDS)
• A Halley Odyssey
• The National Portrait Gallery (London) has several portraits of Halley: Search the collection Archived 19 December 2006 at the Wayback Machine
• Halley, Edmond, An Estimate of the Degrees of the Mortality of Mankind (1693)
• Halley, Edmond, Some Considerations about the Cause of the Universal Deluge (1694)
• A synopsis of the astronomy of comets By Edmund Halley, Savilian Professor of Geometry, at Oxford; And Fellow of the Royal Society. Translated from the Original, printed at Oxford. Oxford: John Senex. 1705 – via Internet Archive.
• Halley, Edmund, A Synopsis of the Astronomy of Comets (1715) annexed on pages 881 to 905 of volume 2 of The Elements of Astronomy by David Gregory
• Material on Halley's life table for Breslau on the Life & Work of Statisticians site: Halley, Edmond
• Halley, Edmund, Considerations on the Changes of the Latitudes of Some of the Principal Fixed Stars (1718) – Reprinted in R. G. Aitken, Edmund Halley and Stellar Proper Motions (1942)
• O'Connor, John J.; Robertson, Edmund F., "Edmond Halley", MacTutor History of Mathematics Archive, University of St Andrews
• Online catalogue of Halley's working papers (part of the Royal Greenwich Observatory Archives held at Cambridge University Library)
• Halley, Edmond (1724) "Some considerations about the cause of the universal deluge, laid before the Royal Society, on the 12th of December 1694" and "Some farther thoughts upon the same subject, delivered on the 19th of the same month" Philosophical Transactions, Giving Some Account of the Present Undertakings, Studies, and Labours of the Ingenious, in Many Considerable Parts of the World. Vol. 33 p. 118–125. – digital facsimile from Linda Hall Library
• Works by Edmond Halley at LibriVox (public domain audiobooks)
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We introduce the concepts of IVF irresolute mappings and IVF irresolute open mappings, and investigate characterizations for such mappings on the interval-valued fuzzy topological spaces.
Y. B. Jun. G. C. Kang and M. A. Ozturk Interval-valued fuzzy semiopen, preopen and $\alpha$-open mappings, Honam Math. J., 28 (2) (2006), 241-259.
T. K. Mondal and S. K. Samanta, Topology of interval-valued fuzzy sets, Indian J. Pure Appl. Math., 30(1) (1999), 23-38. | CommonCrawl |
The idea of a probability distribution
A random variable is a variable that is subject to variations due to random chance. One can think of a random variable as the result of a random experiment, such as rolling a die, flipping a coin, picking a number from a given interval. The idea is that, each time you perform the experiment, you obtain a sample of the random variable. Since the variable is random, you expect to get different values as you obtain multiple samples. (Some values might be more likely than others, as in an experiment of rolloing two six-sided die and recording the sum of the resulting two numbers, where obtaining a value of 7 is much more likely than obtaining value of 12.) A probability distribution is a function that describes how likely you will obtain the different possible values of the random variable.
It turns out that probability distributions have quite different forms depending on whether the random variable takes on discrete values (such as numbers from the set $\{1,2,3,4,5,6\}$) or takes on any value from a continuum (such as any real number in the interval $[0,1]$). Despite their different forms, one can do the same manipulations and calculations with either discrete or continuous random variables. The main difference is usually just whether one uses a sum or an integral.
Discrete probability distribution
A discrete random variable is a random variable that can take on any value from a discrete set of values. The set of possible values could be finite, such as in the case of rolling a six-sided die, where the values lie in the set $\{1,2,3,4,5,6\}$. However, the set of possible values could also be countably infinite, such as the set of integers $\{0, 1, -1, 2, -2, 3, -3, \ldots \}$. The requirement for a discrete random variable is that we can enumerate all the values in the set of its possible values, as we will need to sum over all these possibilities.
For a discrete random variable $X$, we form its probability distribution function by assigning a probability that $X$ is equal to each of its possible values. For example, for a six-sided die, we would assign a probability of $1/6$ to each of the six options. In the context of discrete random variables, we can refer to the probability distribution function as a probability mass function. The probability mass function $P(x)$ for a random variable $X$ is defined so that for any number $x$, the value of $P(x)$ is the probability that the random variable $X$ equals the given number $x$, i.e., \begin{align*} P(x) = \Pr(X = x). \end{align*} Often, we denote the random variable of the probability mass function with a subscript, so may write \begin{align*} P_X(x) = \Pr(X = x). \end{align*}
For a function $P(x)$ to be valid probability mass function, $P(x)$ must be non-negative for each possible value $x$. Moreover, the random variable must take on some value in the set of possible values with probability one, so we require that $P(x)$ must sum to one. In equations, the requirenments are \begin{gather*} P(x) \ge 0 \quad \text{for all $x$}\\ \sum_x P(x) = 1, \end{gather*} where the sum is implicitly over all possible values of $X$.
For the example of rolling a six-sided die, the probability mass function is \begin{gather*} P(x) = \begin{cases} \frac{1}{6} & \text{if $x \in \{1,2,3,4,5,6\}$}\\ 0 & \text{otherwise.} \end{cases} \end{gather*}
If we rolled two six-sided dice, and let $X$ be the sum, then $X$ could take on any value in the set $\{2,3,4,5,6,7,8,9,10,11,12\}$. The probability mass function for this $X$ is \begin{gather*} P(x) = \begin{cases} \frac{1}{36} & \text{if $x \in \{2,12\}$}\\ \frac{2}{36}=\frac{1}{18} & \text{if $x \in \{3,11\}$}\\ \frac{3}{36}=\frac{1}{12} & \text{if $x \in \{4,10\}$}\\ \frac{4}{36}=\frac{1}{9} & \text{if $x \in \{5,9\}$}\\ \frac{5}{36} & \text{if $x \in \{6,8\}$}\\ \frac{6}{36} =\frac{1}{6} & \text{if $x = 7$}\\ 0 & \text{otherwise.} \end{cases} \end{gather*} $P(x)$ is plotted as a bar graph in the following figure.
Continuous probability distribution
A continuous random variable is a random variable that can take on any value from a continuum, such as the set of all real numbers or an interval. We cannot form a sum over such a set of numbers. (There are too many, since such a continuum is uncountable.) Instead, we replace the sum used for discrete random variables with an integral over the set of possible values.
For a continuous random variable $X$, we cannot form its probability distribution function by assigning a probability that $X$ is exactly equal to each value. The probability distribution function we must use in the case is called a probability density function, which essentially assigns the probability that $X$ is near each value. For intuition behind why we must use such a density rather than assigning individual probabilities, see the page that describes the idea behind the probability density function.
Given the probability density function $\rho(x)$ for $X$, we determine the probability that $X$ is in any set $A$ (i.e., that $X \in A$ (confused?)) by integrating $\rho(x)$ over the set $A$, i.e., \begin{gather*} \Pr(X \in A) = \int_A \rho(x)dx. \end{gather*} Often, we denote the random variable of the probability density function with a subscript, so may write \begin{gather*} \Pr(X \in A) = \int_A \rho_X(x)dx. \end{gather*}
The definition of this probability using an integral gives one important consequence for continuous random variables. If the set $A$ contains just a single element, we can immediately see that the probability that $X$ is equal to that one value is exactly zero, as the integral over a single point is zero. For a continuous random variable $X$, the probability that $X$ is any single value is always zero.
In other respects, the probability density function of a continuous random variables behaves just like the probability mass function for a discrete random variable, where we just need to use integrals rather than sums. For a function $\rho(x)$ to be valid probability density function, $\rho(x)$ must be non-negative for each possible value $x$. Just as for discrete random variable, a continuous random variable must take on some value in the set of possible values with probability one. In this case, we require that $\rho(x)$ must integral to one. In equations, the requirenments are \begin{gather*} \rho(x) \ge 0 \quad \text{for all $x$}\\ \int \rho(x)dx = 1, \end{gather*} where the integral is implicitly over all possible values of $X$.
For examples of continuous random variables and their associated probability density functions, see the page on the idea behind the probability density function.
The idea of a probability density function
Nykamp DQ, "The idea of a probability distribution." From Math Insight. http://mathinsight.org/probability_distribution_idea
Keywords: probability
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The Legend of Abraham Wald
The story might well be true, and there is certainly, as we shall see, a solid germ of truth in it, but there is very little evidence for the best bits."...
Bill Casselman
University of British Columbia, Vancouver, Canada
Email Bill Casselman
The myth
It makes a great story.
The year is 1943. American bombers are suffering badly from German air defense. The military decides it needs some advice on how to cut losses, so they consult the wizards in the Statistical Research Group at Columbia University to see what their best options might be. One possibility is to use more armor on planes, but armor weighs a lot, and adding too much would lower performance considerably. So the Air Force brass ask the SRG, how much armor should we use for optimal results, and where should we put it?
The SRG was one of several collaborating groups of scientists formed soon after America joined the war. The story of its beginning, in the summer of 1942, is told well in W. Allen Wallis' autobiographical memoir. The SRG was staffed by a distinguished lot, including many of the most prominent statisticians of the post-war world, the economists Milton Friedman and George Stigler--who were later to receive Nobel Prizes in economics--and the mathematician Abraham Wald. Norbert Wiener was at one time a consultant to the group. Recruitment to the SRG was by an "old-boy" network (to use a phrase also applicable to that other successful war-time operation across the ocean at Bletchley Park), but it prided itself on what we would call diversity.
Wald was born in the former Austrian-Hungarian empire in 1902, in the city now called Cluj. It advertizes itself as the unoffical capital of Transylvania, which is now a part of Romania but inhabited in the past largely by Hungarians, and Hungarian was Wald's mother tongue. He started his professional life in Vienna as a pure mathematician, but became interested in the mathematics of statistics in the mid-thirties. As a Jew, he was deprived of his academic position in Austria, and like others in his situation was lucky to be able to move to the United States. At the time the SRG was founded, he was on the faculty of Columbia University, which is where the SRG was located, and he was one of its first members. By all accounts, he was impressively bright--"smartest man in the room," says one recent book (but keep in mind, most of the time there were many smart men in the room).
The problem of armoring planes is assigned to Wald. Along with the assignment, he is given a fair amount of statistical data regarding aircraft damage, for example the location of damage from hits by enemy aircraft. It happens that most of the damage is located on the fuselage and very little in the area around motors, and the military is expecting to add armor to the fuselage, where the density of hits is highest. "Not so fast," said Wald. "What you should really do is add armor around the motors! What you are forgetting is that the aircraft that are most damaged don't return. You don't see them. Hits by German shells are presumably distributed somewhat randomly. The number of damaged motors you are seeing is far less than randomness would produce, and that indicates that it is the motors that are the weak point." The advice is taken, and in fact Wald's techniques for interpreting aircraft damage statistics continue through two later conflicts.
The Internet loves this tale. Try a search for
"Abraham Wald" aircraft
to see what Mr. Google has to show you, and you will find dramatic headlines:
ABRAHAM WALD AND THE MISSING BULLET HOLES
Seeing is Disbelieving
How A Story From World War II Shapes Facebook Today
The hole story: What you don't see will kill you
The reason for this excitement is that the aircraft damage is an example of what is known as "survivorship bias." This is a technical term for what we all know well: the dead don't often get to tell their side of the story, and yet sometimes it would be better if they did. The loss is the source of all kinds of misinformation, as the Internet will tell you emphatically. Including deceptive practices in selling hot stocks, which may explain much of the buzz.
Well, it's gratifying to see a great mathematician become a legend for good reasons, rather than bad. "MATHEMATICAL GENIUS SCORES AGAINST ARMY BRASS!" reads pretty well. After all, publicity about mathematicians typically concentrates on features most of us would rather not think about. But it would be much more gratifying if there were more truth to the story, or at least more reason for believing it. Some of us prefer our history lessons to be taken from the non-fiction shelves.
The story might well be true, and there is certainly, as we shall see, a solid germ of truth in it, but there is very little evidence for the best bits. The capsule biography of Wald is accurate, and although he might not have been the smartest man in the room, he was probably nearly always the most accomplished mathematician in the room, which counted for a lot. But ... most of the rest of the story is--to use a charitable phrase--"plausible reconstruction." There is extremely little source material for what Wald had to say about aircraft damage.
The autobiographical memoir by W. Allen Wallis is the best source--practically the only source--for the operation of the SRG. It is surprisingly entertaining as well as informative, but its coverage of Wald's work at the SRG concentrates on the invention of sequential analysis, for which Wald eventually became deservedly famous. This is a technique for improving quality control in production, say of military ordnance. It was used, apparently with great success, by thousands of wartime production facilities. But it is not exactly great material for Internet headlines: "HEY! ARMY TRUCK TIRE PRODUCTION ROSE 6.37% IN AUGUST 1944!"
To be precise, regarding Wald's work on aircraft damage we have (1) two short and rather vague mentions in Wallis' memoir of work on aircraft vulnerability and (2) the collection of the actual memoranda that Wald wrote on the subject. That's it! Everything not in one of these places must be considered as fiction, not fact. Or at best, as I say, plausible reconstruction. Not to complain too much--the history of mathematics is plagued by the temptation, rarely resisted, to write as things should have been, rather than what they were. Reality is rarely as logical as one might hope. I should add, though, that it's not only mathematical reality that gets slighted in the Internet versions of this tale--you should be quite amused by the pictures that accompany the Internet headlines. Lots and lots of airplanes with bullet holes scattered all over them. One goes so far as to claim it is showing you Wald's own sketches (and we do not have any idea at all as to whether if he ever made any). Most show diagrams of aircraft that by no means match what must have been involved--my favorite is of a venerable DC3, a plane referred to by the military as the C-47. These served as cargo carriers in WWII and rarely saw real combat except by straying from route. "As long as it has motors and flies" seems to be the criterion for the art work. A few of the web sites show chilling clips of American planes being destroyed in action. These certainly show, in case you might have forgotten, what stakes were ultimately involved in the apparently abstract technology being developed in the comfort of upper Manhattan.
The vague references in Wallis' memoir are particularly interesting, since Wald is not mentioned in them. One of them (p. 323) says in entirety, "The problem of aircraft vulnerability led SRG to devise a technique for determining vulnerability from damage survived by our own planes ... " The other (p. 324) names Wallis himself as the author of a note titled Uses for Aircraft Vulnerability Figures. This, however, is one of a list of a random selection of reports from the SRG, and there might well have been other reports on the same topic. (Do these reports still exist in some deep archive?)
So the only really reliable account of Wald's work is what we find in Wald's own writings.
The true story, or at least part of it
The memoranda by Wald are severely technical. Not much drama at all. In particular, Wald says nothing about what the military should do to improve things. If I understand Wallis correctly, it was the general policy of the SRG to answer just the questions asked and never--well, hardly ever--attempt to offer advice on applications of what they discovered. Military decisions were made by the military.
The memoranda are so technical, in fact, that in the account by Jordan Ellenberg, a photograph of one page of the document is flashed at the reader with an apology for suddenly introducing a topic possibly suitable only for adults. There is, however, a very valuable guide to the memoranda by Marc Mangel and Francisco Samaniego, that appeared almost at the same time the memoranda were made available to the public by the Center for Naval Analyses.
There are eight items among the memoranda. Five of them deal with a single problem, estimating probabilities of an airplane's survival, given that it has already been hit. Its outstanding feature is that it offers a way to estimate damage on the planes that never returned. A kind of magic, indeed. One--just one--deals with the problem of vulnerability of different sections of an airplane, and this shares with the previous sections some impressive estimates. That is, as the Internet fiction suggests, both have to deal with the problem that downed planes aren't around to give evidence.
Consider the first problem. We are given data, such as the number of hits, only on returning aircraft. The question Wald asked--or perhaps the one he was asked to look at--was, "Given these data, what can we say about the probability of surviving a given number of hits?" Not a complicated question, but with a complicated answer. All we know about the planes that didn't return is ... that they didn't return. In truth, there might be a number of reasons for this, since--for example--a number of fatalities in the war were from mechanical failure. Of course Wald had to be very careful. It was in principle possible, one might suppose, that all downed airplanes ran out of gasoline. The point is that this was extremely unlikely. In other words, any answer to the question is complicated by the missing data associated to planes that were downed. Wald could only calculate his probabilities by making certain reasonable assumptions, and being very, very careful about how the assumptions played a role in results. In all his works on statistics, in fact, he was renowned for being very, very careful with assumptions.
His first simplifying assumption is that planes are downed because of enemy fire. Rather than mechanical failure, say.
What data did Wald have to work with? This seems to have varied from time to time, but at the least, in so far as this problem was concerned, he was given the number of planes sent out on missions, the number returning, and the number of hits on each plane that came back. In the example treated by Mangel and Samaniego (following Wald):
Number $\phantom{xxx}$ Ratio
Planes in the mission $400 = N$ $\phantom{xxx}$ $\phantom{s_{0} =\>}1.00$
Planes returning $380$ $\phantom{xxx}$ $\phantom{s_{0}=\>}0.95$
Number of planes downed $20$ $\phantom{xxx}$ $\phantom{s_{0}=\>}0.05$
Number returning with no hits $S_{0} = 320$ $\phantom{xxx}$ $s_{0} = 0.80$
With $1$ hit $S_{1} = \phantom{2}32$ $\phantom{xxx}$ $s_{1} = 0.080$
With $2$ hits $S_{2} = \phantom{2}20$ $\phantom{xxx}$ $s_{2} = 0.050$
With $3$ hits $S_{3} = \phantom{22}4$ $\phantom{xxx}$ $s_{3} = 0.010$
The $N$ planes on the mission divide into two major groups, the $S$ survivors and the $L$ planes that are downed. These in turn divide into groups according to how many hits they get: $N_{i}$ is the total number with exactly $i$ hits, similarly $S_{i}$ and $L_{i}$. Of course we know all the $S_{i}$, and know nothing about the $L_{i}$ except for three simple things: (1) $L = \sum L_{i} = N - S$, and (2) $L_{i} + S_{i} = N_{i}$, and (3) $L_{0} = 0$, because we have assumed that all those that are lost are lost because they have been hit. Let $N_{\ge i}$ be the sum $\sum_{j \ge i} N_{j}$, etc. Thus $$ N = N_{\lt i} + N_{\ge i} \, . $$
It seems a little crazy, but what we really want to do is figure out what all the missing numbers $L_{i}$ are, or at least estimate them in a reasonable way. It looks at first sight as though this is a task for a conjuror rather than a mathematician.
If, as Mangel and Samniego advise you to do, you think on your own about this problem, you will likely be led to come up with something rather complicated. Yet Wald's reasoning is remarkably simple. One of his best ideas is to introduce variables that we do have at least some chance of estimating, and from which all the rest can be computed. Let $p_{i}$ be the conditional probability of going down on the $i$-th hit, having survived $i-1$ hits. Thus $p_{1}$ is just the probability of going down on the first hit, and $p_{i}$ is the proportion of those who receive $\ge i$ hits who are, however, shot down by the $i$-th. In an equation: $$ p_{i} = { L_{i} \over N_{\ge i} } \, . $$
We can write this also as $$ \eqalign { L_{i} &= p_{i} \cdot \Big( \sum_{j \ge i} N_{j} \Big) \cr &= p_{i} \cdot \Big(N - \sum_{j \lt i} N_{j} \Big ) \cr &= p_{i} \cdot \Big( N - \sum_{j\lt i} S_{j} - \sum_{j\lt i} L_{j} \Big) \, . \cr } $$
Here's the basis of the magic to come: We know what the $S_{i}$ are. Therefore the last equation for $L_{i}$ is an equation that can be solved by induction for the $L_{i}$, since we know that $L_{0} = 0$, if only we know the $p_{i}$! Thus $$ \eqalign { L_{0} &= 0 \cr L_{1} &= p_{1} \cdot (N - S_{0}) \cr L_{2} &= p_{2} \cdot (N - S_{0} - S_{1} - L_{1} ) \cr & \dots \cr } $$
Of course this leads us to the question--how can we figure out what the $p_{i}$ are? The short answer is, we can't, but Wald was able to make various estimates of them, by an argument that appears to me all the more subtle the more I try to understand it.
Let $q_{i} = 1- p_{i}$, which is the conditional probability of surviving $i$ hits, given that there are at least $i$. These are a main ingredient in what I'll call Wald's basic equation: $$ \sum_{m=1}^{n} { S_{m} \over q_{1}q_{2} \ldots q_{m} } = 1 - S_{0} \, . $$
I'll try to explain in a moment where this comes from--as far as I can see it is not an obvious relation, although it is not difficult to derive, and it is even less obvious that it gets us anywhere. It is a single equation with several unknowns, so in general there will be many possible solutions. Wald's approach is to find which solutions in this large world of solutions are most likely.
But first let me give you some idea of how the equation can give us approximate values of the $p_{i}$. Following Wald and Mangel-Samaniego, let's look first at an unrealistically simple case. We expect hits to weaken a plane, or at least not improve its chances, which means that $$ q_{1} \ge q_{2} \ge \ldots \, , $$
but as a first very rough approximation we might guess that all the $q_{i}$ are equal, so that for the denominators $$ q_{1}q_{2} \ldots q_{i} = q^{i} $$
for some fixed $q$. This amounts to assuming that a hit does not weaken a plane, which does not seem to be far off the truth. With this assumption Wald's basic equation becomes $$ { s_{1} \over q } + { s_{2} \over q^{2} } + \cdots + { s_{n} \over q^{n} } = 1 - s_{0} \, $$ and in our example $$ { 0.080 \over q } + { 0.050 \over q^{2} } + { 0.010 \over q^{3} } + + { 0.005 \over q^{4} } + { 0.005 \over q^{5} } = 0.20 $$ This tells us that $q$ is the root of a relatively simple equation. I include below a graph of the function on the left, as well as the level line at $0.20$. We see that $q$ is approximately $0.85$, and a little more calculation (using Newton's method, for example) gives us the slightly more accurate $0.851$ (but the extra decimal digit is spurious, given the coarseness of the data).
From this point, Wald's memoranda go on to apply the basic equation in order to find plausible bounds on the possible values of the $q_{i}$ rather than exact guesses, by even more subtle arguments. After that he applies similar techniques to the problem of locating the most fatal hits on the planes. Abraham Wald was not a necromancer, but he was a magician. He might not able to make the dead speak, but he could pull a few rabbits out of thin air.
Mathematical magic
Rather than discuss these topics, I'll try to explain where the basic equation comes from.
Wald's own argument for his basic equation is followed by Mangel and Samaniego. It is extremely clever. Wald does something only a mathematician could love, he says in effect let's consider an imaginary scenario in which only dummy bullets are fired. I have to confess that I find the argument a bit obscure, and I think this is because its plausibility seems to depend on some probabilistic intuition I don't have. So I offer something new, if less adventurous.
I start with something that Wald mentions, and seems to think important, but doesn't use in a crucial way: The number of hits on an airplane is bounded, so that $N_{\gt n} = 0$ for some $n$. In our example, $n =5$. Now the induction formula for $p_{i}$ tells us that $$ \eqalign { p_{i} &= { L_{i} \over N_{\ge i} } \cr q_{i} &= 1 - p_{i} \cr &= { N_{\ge i} - L_{i} \over N_{\ge i} } \cr S_{i} + N_{\ge i+1} &= q_{i}N_{\ge i} \cr } $$
for all $i$. If we combine these facts, we deduce first that $$ S_{n} = q_{n} N_{\ge n} \, . $$ But we also deduce a descending inductive formula: $$ N_{\ge i} = { S_{i} \over q_{i} } + { N_{\ge i+1} \over q_{i} } $$ that leads to the sequence of formulas $$ \eqalign { N_{\ge n} &= S_{n}/q_n \cr N_{\ge n-1} &= { S_{n} \over q_{n-1}q_{n} } + { S_{n-1} \over q_{n-1} } \cr & \dots \cr N_{\ge 0} = N &= { S_{n} \over q_{1}\dots q_{n} } + \cdots + { S_{1} \over q_{1} } + S_{0}\, . \cr } $$ The last equation is Wald's basic equation! Quod erat demonstrandum!
Postcript
My indignation at how the internet dealt with Wald's work was overblown. Stephen Stigler (son of George, and a statistician at the University of Chicago) called my attention to a note by W. Allen Wallis himself in which he mentions Wald's work explicitly in connection with survivorship bias. Wallis' original article in the Journal of the American Statistical Association was followed by two very brief comments and then by a further 'rejoinder' of a bit more than one page. Towards the end of it he says, "The military was inclined to provide protection for those parts that on returning planes showed the most hits. Wald assumed, on good evidence, that hits in combat were uniformly distributed over the planes. It follows that hits on the more vulnerable parts were less likely to be found on returning planes than hits on the less vulnerable parts, since planes receiving hits on the more vulnerable parts were less likely to return to provide data. From these premises, he devised methods for estimating vulnerability of various parts."
Stephen Stigler recalled to us that both Wallis and his father George Stigler had mentioned this work of Wald in conversation several times. He called attention to Wallis' remarks in a letter published in the May 1989 issue of Nature in which he also pointed out the relevance of survivorship bias to the interpretation of the statistical record of trilobite fossils. This may have been the original seed from which the tree of subsequent comment grew.
Thanks to Marc Mangel and Phil DePoy for assistance. My thanks to Pawan Gupta for pointing out a minor error in two of the equations in the original posting of the column.
Reading further
Wald's memoranda
The originals were written by Wald around 1943, and these were later published in 1981 by the Document Center of the Center for Naval Analyses (CNA), 2000 North Beauregard St., Alexandria, Virginia 22311.
One might well wonder how the publication of Wald's memoranda, along with the near simultaneous publication of the article by Mangel and Samaniego, came about. Why the wait for nearly forty years?
Around 1980, W. Allen Wallis was in the process of leaving the University of Rochester, where he had worked for many years. In the process, he found a number of items left over from his days at the SRG, and offered them to Phil DePoy, also in Rochester at that time and working at the CNA. DePoy writes to us: "The original material was given to me by W. Allen Wallis after he moved to Washington to take a position with the State Department (that of Under Secretary of State for Economic Affairs). He came into my office one day carrying a large box which he found when he was moving out of his house in Rochester. He said that he had collected a lot of loose papers from the SRG when they closed the office down in 1946. He asked me to review everything in the box and determine if anything was worth saving. I read everything and decided that most of it was not worth preserving. The only material that I saved was a package of items by Abraham Wald. With some minimal editing, I published eight 'papers' under Wald's name in July 1980."
Recorded history hangs by thin threads. The most unfortunate fact in Wald's history is that he died in an airplane accident in the mountains of southern India in 1950, and had no chance to write his autobiography.
Abraham Wald's Work on Aircraft Survivability
A survey of Wald's work on aircraft damage by Marc Mangel and Francisco Samaniego. Originally appeared in volume 79 of the Journal of the American Statistical Association.
Mangel has told us, "In 1981 or so, in anticipation of the 40th anniversary of Operations Evaluation Group of the Center for Naval Analyses, Phil DePoy asked me to prepare a version of Wald's report that would be publishable."
Oskar Morgenstern, 'Abraham Wald, 1902-1950', Econometrica 19, 361-367, 1951.
Jacob Wolfowitz, `Abraham Wald, 1902-1950', Annals of Mathematical Statistics 23, 1-13, 1952.
Memoirs by two of Wald's closest friends and collaborators.
W. Allen Wallis, 'The Statistical Research Group, 1942-1945', Journal of the American Statistical Association 75, 320-330, 1980.
Unfortunately not publicly accessible. This is well written, with panache. Who would have thought such a dry topic could furnish so much pleasure?
Exploring the history of one's professional field is often a mark of maturity. Reminiscing about it is usually a mark of senility.
I have obtained a copy of the Final Report of SRG from the National Archives, where it and other SRG documents are kept in the Center for Polar and Scientific Archives.
The program that resulted ... eventually made a major contribution to the war effort. Its aftermath, in fact, continues to make major contributions not only to the American economy but also to the Japanese economy.
We pinched pennies ... One principal staff menber still alleges resentfully that the adminstrative assistant told him to economize by writing equations on both sides of the paper.
In 1950 I was 30 years closer to the events than I am now, and, furthermore, I had the use of a 37 year old's memory--something that now I can scarcely recall ever having had.
These are among many amusing passages. One asks immediately, is the Final Report extant? One curious fact is that the "computers" at the SRG were (in Wallis' words) "30 young women, mostly mathematics graduates of Hunter or Vassar." Again, reminiscent of Bletchley Park. The only woman on the staff seems to have been Mina Rees, then teaching at Hunter College. Like other members of the staff, she had a distinguished career ahead of her.
Jordan Ellenberg, How not to be wrong. Penguin, 2014. A non-technical account of many mathematical topics. The opening is one of the better plausible reconstructions of Wald's work.
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\begin{definition}[Definition:Category of Locally Ringed Spaces]
The '''category of locally ringed spaces''' is the category $\mathbf{LRS}$ with:
{{DefineCategory
|ob = locally ringed spaces
|mor = morphisms of locally ringed spaces
|id = identity morphisms of ringed spaces
|comp = composition of morphisms of ringed spaces
}}
\end{definition} | ProofWiki |
Mat. Zametki:
Mat. Zametki, 1998, Volume 63, Issue 5, Pages 774–784 (Mi mz1344)
A remark on sets of determining elements for reaction-diffusion systems
I. D. Chueshov
V. N. Karazin Kharkiv National University
Abstract: For a class of systems of parabolic equations, conditions represented by a finite set of linear functionals on the phase space that uniquely determine the long-time behavior of solutions are found. The cases in which it is sufficient to define these determining functionals only on a part of the components of the state vector are singled out. As examples, systems describing the Belousov–Zhabotinsky reaction and the two-dimensional Navier–Stokes equations are considered.
DOI: https://doi.org/10.4213/mzm1344
Mathematical Notes, 1998, 63:5, 679–687
UDC: 517.94
Citation: I. D. Chueshov, "A remark on sets of determining elements for reaction-diffusion systems", Mat. Zametki, 63:5 (1998), 774–784; Math. Notes, 63:5 (1998), 679–687
\Bibitem{Chu98}
\by I.~D.~Chueshov
\paper A remark on sets of determining elements for reaction-diffusion systems
\jour Mat. Zametki
\mathnet{http://mi.mathnet.ru/mz1344}
\crossref{https://doi.org/10.4213/mzm1344}
\jour Math. Notes
http://mi.mathnet.ru/eng/mz1344
https://doi.org/10.4213/mzm1344
http://mi.mathnet.ru/eng/mz/v63/i5/p774
I. D. Chueshov, "Theory of functionals that uniquely determine the asymptotic dynamics of infinite-dimensional dissipative systems", Russian Math. Surveys, 53:4 (1998), 731–776
T. Yu. Semenova, "Approximation by step functions of functions belonging to Sobolev spaces and uniqueness of solutions of differential equations of the form $u-F(x,u,u')$", Izv. Math., 71:1 (2007), 149–180
T. Yu. Semenova, "Conditions on Determining Functionals for Subsets of Sobolev Space", Math. Notes, 86:6 (2009), 831–841 | CommonCrawl |
Emergence of zero-field non-synthetic single and interchained antiferromagnetic skyrmions in thin films
Robust Formation of Ultrasmall Room-Temperature Neél Skyrmions in Amorphous Ferrimagnets from Atomistic Simulations
Chung Ting Ma, Yunkun Xie, … S. Joseph Poon
Current-driven dynamics and inhibition of the skyrmion Hall effect of ferrimagnetic skyrmions in GdFeCo films
Seonghoon Woo, Kyung Mee Song, … Joonyeon Chang
Isolated zero field sub-10 nm skyrmions in ultrathin Co films
Sebastian Meyer, Marco Perini, … Stefan Heinze
An achiral ferromagnetic/chiral antiferromagnetic bilayer system leading to controllable size and density of skyrmions
F. J. Morvan, H. B. Luo, … J. P. Liu
Antiskyrmions and their electrical footprint in crystalline mesoscale structures of Mn1.4PtSn
Moritz Winter, Francisco J. T. Goncalves, … Toni Helm
Spin photogalvanic effect in two-dimensional collinear antiferromagnets
Rui-Chun Xiao, Ding-Fu Shao, … Hua Jiang
Tuning the density of zero-field skyrmions and imaging the spin configuration in a two-dimensional Fe3GeTe2 magnet
Bei Ding, Xue Li, … Wenhong Wang
The microscopic origin of DMI in magnetic bilayers and prediction of giant DMI in new bilayers
Priyamvada Jadaun, Leonard F. Register & Sanjay K. Banerjee
Observation of Skyrmions at Room Temperature in Co2FeAl Heusler Alloy Ultrathin Film Heterostructures
Sajid Husain, Naveen Sisodia, … Sujeet Chaudhary
Amal Aldarawsheh ORCID: orcid.org/0000-0003-4163-76681,2,
Imara Lima Fernandes ORCID: orcid.org/0000-0002-5078-78041,
Sascha Brinker ORCID: orcid.org/0000-0002-7077-12441,
Moritz Sallermann1,3,4,
Muayad Abusaa5,
Stefan Blügel ORCID: orcid.org/0000-0001-9987-47331 &
Samir Lounis ORCID: orcid.org/0000-0003-2573-28411,2
Nature Communications volume 13, Article number: 7369 (2022) Cite this article
Magnetic properties and materials
Antiferromagnetic (AFM) skyrmions are envisioned as ideal localized topological magnetic bits in future information technologies. In contrast to ferromagnetic (FM) skyrmions, they are immune to the skyrmion Hall effect, might offer potential terahertz dynamics while being insensitive to external magnetic fields and dipolar interactions. Although observed in synthetic AFM structures and as complex meronic textures in intrinsic AFM bulk materials, their realization in non-synthetic AFM films, of crucial importance in racetrack concepts, has been elusive. Here, we unveil their presence in a row-wise AFM Cr film deposited on PdFe bilayer grown on fcc Ir(111) surface. Using first principles, we demonstrate the emergence of single and strikingly interpenetrating chains of AFM skyrmions, which can co-exist with the rich inhomogeneous exchange field, including that of FM skyrmions, hosted by PdFe. Besides the identification of an ideal platform of materials for intrinsic AFM skyrmions, we anticipate the uncovered knotted solitons to be promising building blocks in AFM spintronics.
Magnetic skyrmions are particle-like topologically protected twisted magnetic textures1,2,3 with exquisite and exciting properties4,5. They often result from the competition between the Heisenberg exchange and the relativistic Dzyaloshinskii-Moriya interaction (DMI)6,7, which is present in materials that lack inversion symmetry and have a finite spin orbit coupling. Since their discovery in multiple systems, ranging from bulk, thin films, surfaces to multilayers8,9,10,11,12,13,14,15,16, skyrmions are envisioned as promising candidates for bits, potentially usable in the transmission and storage of information in the next generation of spintronic devices17,18,19,20,21,22. However, requirements for future (nano-)technologies are not only limited to the generation of information bits but are also highly stringent from the point of view of simultaneous efficiency in reading, control, and power consumption21,23. Miniaturization of ferromagnetic (FM) skyrmions suffers from the presence of dipolar interactions24, while their stabilization generally requires an external magnetic field. Another drawback is the skyrmion Hall effect4, caused by the Magnus force that deflects FM skyrmions when driven with a current, which hinders the control of their motion. Additionally, FM skyrmions exhibit a rather complex dynamical behavior as function of applied currents25,26,27,28,29,30,31, under the presence of defects.
Antiferromagnetic (AFM) skyrmions are expected to resolve several of the previous issues and offer various advantages. Indeed, AFM materials being at the heart of the rapidly evolving field of AFM spintronics32,33,34 are much more ubiquitous than ferromagnets. Their compensated spin structure inherently forbids dipolar interactions, which should allow the stabilization of rather small skyrmions while enhancing their robustness against magnetic perturbations. AFM skyrmions were predicted early on using continuum models35, followed with multiple phenomenology-based studies on a plethora of properties and applications, see e.g., Refs. 36,37,38,39,40,41,42,43,44,45,46,47,48. The predicted disappearance of the Magnus force, which triggers the skyrmion Hall effect, would then enable a better control of the skyrmion's motion38,49, which has been partially illustrated experimentally in a ferrimagnet50,51.
Intrinsic AFM meronic spin-textures (complexes made of half-skyrmions) were recently detected in bulk phases52,53,54 while synthetic AFM skyrmions were found within multilayers55. However, the observation of intrinsic AFM skyrmions has so far been elusive, in particular at surfaces and interfaces, where they are highly desirable for racetrack concepts. A synthetic AFM skyrmion consists of two FM skyrmions realized in two different magnetic layers, which are antiferromagnetically coupled through a non-magnetic spacer layer. In contrast to that an intrinsic AFM skyrmion is a unique magnetic entity since it is entirely located in a single layer. Here we predict from first-principles (see Method section) intrinsic AFM skyrmions in a monolayer of Cr deposited on a surface known to host ferromagnetic skyrmions: A PdFe bilayer grown on Ir(111) fcc surface as illustrated in Fig. 1a. The AFM nature of Cr coupled antiferromagnetically to PdFe remarkably offers the right conditions for the emergence of a rich set of complex AFM textures. The ground state is collinear row-wise AFM (RW-AFM) within the Cr layer (see inset of Fig. 1c), a configuration hosted by a triangular lattice so far observed experimentally only in Mn/Re(0001)56,57. The difference to the latter, however, is that although being collinear, the Cr layer interfaces with a magnetic surface, the highly non-collinear PdFe bilayer.
Fig. 1: Interchained AFM skyrmions in CrPdFe trilayer on Ir(111).
a Schematic representation of the investigated trilayer deposited on Ir(111) following fcc stacking. b The interchaining of skyrmions is reminiscent of interpenetrating rings, which realize topologically protected phases. c The ground state being RW-AFM (see inset) can host AFM skyrmions that can be isolated or interlinked to form multimers of skyrmions, here we show examples ranging from dimers to pentamers. The AFM skyrmions can be decomposed into FM skyrmions living in sublattices illustrated in d. In case of the single AFM skyrmion, two of the sublattices, L1 and L2, are occupied by the FM skyrmions shown in e. L3 and L4 host quasi-collinear AFM spins in contrast to the FM skyrmions emerging in the case of the double AFM skyrmion presented in f. Note that the separation of sublattices L1, L2, L3, and L4 shown in e and f is only done for illustration.
A plethora of localized chiral AFM-skyrmionic spin textures (Fig. 1c) and metastable AFM domain walls (see Supplementary Fig. 1) emerge in the Cr overlayer. Besides isolated topological AFM solitons, we identify strikingly unusual interpenetrating AFM skyrmions, which are reminiscent of crossing rings (see schematic Fig. 1b), the building blocks of knot theory where topological concepts such as Brunnian links are a major concept58. The latter has far reaching consequences in various fields of research, not only in mathematics or physics but extends to chemistry and biology. For instance, the exciting and intriguing interchain process, known also as catenation, is paramount in carbon-, molecular-, protein- or DNA-based assemblies59,60,61. We discuss the mechanisms enforcing the stability of the unveiled interchained topological objects, their response to magnetic fields and the subtle dependence on the underlying magnetic textures hosted in PdFe bilayer. Our findings are of prime importance in promoting AFM localized entities as information carriers in future AFM spintronic devices.
AFM skyrmions in CrPdFe/Ir(111) surface
PdFe deposited on Ir(111) surface hosts a homo-chiral spin spiral as a ground state14,62 emerging from the interplay of the Heisenberg exchange interactions and DMI. The latter is induced by the heavy Ir substrate, which has a strong spin-orbit coupling. Upon application of a magnetic field, sub 10-nm FM skyrmions are formed14,18,62,63,64,65,66. After deposition of the Cr overlayer, the magnetic interactions characterizing Fe are strongly modified (see comparison plotted in Supplementary Fig. 2) due to changes induced in the electronic structure (Supplementary Fig. 3). The Heisenberg exchange interaction among Fe nearest neighbors (n.n.) reduces by 5.5 meV (a decrease of 33%). This enhances the non-collinear magnetic behavior of Fe, which leads to FM skyrmions even without the application of a magnetic field (see Supplementary Fig. 4). The n.n. Cr atoms couple strongly antiferromagnetically (−51.93 meV), which along with the antiferromagnetic coupling of the second n.n. (−6.69 meV) favors the Néel state. The subtle competition with the ferromagnetic exchange interactions of the third n.n. (5.32 meV) stabilizes the RW-AFM state independently from the AFM interaction with the Fe substrate (the detailed magnetic interactions are shown in Supplementary Fig. 2). As illustrated in Fig. 1c, the RW-AFM configuration is characterized by parallel magnetic moments along a close-packed atomic row, with antiparallel alignment between adjacent rows. Due to the hexagonal symmetry of the atomic lattice, the AFM rows can be rotated in three symmetrically equivalent directions. We note that the moments point out-of-plane due to the magnetic anisotropy energy (0.5 meV per magnetic atom).
The DM interactions among Cr atoms arise due to the broken inversion symmetry and is mainly induced by the underlying Pd atoms hosting a large spin-orbit coupling. The n.n. Cr DMI (1.13 meV) is of the same chiral nature and order of magnitude as that of Fe atoms (1.56 meV), which gives rise to the chiral non-collinear behavior illustrated in Fig. 1c. We note that the solitons are only observed if Cr magnetic interactions beyond the n.n. are incorporated, which signals the significance of the long-range coupling in stabilizing the observed textures. Since the Heisenberg exchange interaction among the Cr atoms is much larger than that of Fe, the AFM solitons are bigger, about a factor of three larger than the FM skyrmions found in Fe.
While the RW-AFM state is defined by two sublattices, the different AFM skyrmions, isolated or overlapped, can be decomposed into interpenetrating FM skyrmions living in four sublattices illustrated in Fig. 1d and denoted as L1, L2, L3, and L4. In the RW-AFM phase, L1 and L4 are equivalent and likewise for L2 and L3. It is evident that the moments in L1 and L4 are antiparallel to the ones in L2 and L3. Taking a closer look at the isolated AFM magnetic texture, one can dismantle it into two FM skyrmions with opposite topological charges anchored in the distinct antiparallel FM sublattices L1 and L2, while L3 and L4 carry rather collinear magnetization (Fig. 1e). In the case of the overlapped AFM skyrmions, however, no sublattice remains in the collinear state. As an example, the dimer consists of two couples of antiferromagnetically aligned skyrmions, each being embedded in one of the four sublattices (Fig. 1f).
Our study reveals that in contrast to interlinked magnetic textures, single AFM skyrmions are significantly sensitive to the magnetic environment hosted by the underlying PdFe bilayer. For brevity, we focus in the next sections on the single and two-overlapping AFM skyrmions and address the mechanisms dictating their stability.
Stabilization mechanism of the overlapping AFM skyrmions
The formation of overlapped solitons is an unusual phenomenon since FM skyrmions repel each other. It results from competing interactions among the skyrmions living in the different sublattices, which finds origin in the natural AFM coupling between the n.n. magnetic moments. Depending on the hosting sublattice (L1 to L4), the four skyrmions shown in Fig. 1f experience attraction or repulsion. The sublattices are chosen such that nearest neighbors within a sublattice are third nearest neighbors in the overall system. This choice leads to the exchange coupling preferring the parallel alignment of spins within a given sublattice. When looking at any sublattice in isolation, this effective ferromagnetic-like exchange interaction enables the existence of skyrmions in a collinear background. In the overall system, however, pairs of sublattices interact via the first and second nearest-neighbor exchange interactions, which prefers anti-parallel spin alignments. Therefore, the exchange interaction between skyrmions formed at sublattices with a parallel background, such as (L1, L4) and (L2, L3), and denoted in the following as skyrmion-skyrmion homo-interactions, are repulsive as usually experienced by FM skyrmions. In contrast, and for the same reasons, interaction between skyrmions in sublattices with oppositely oriented background spins, denoted as hetero-interactions, are attractive as it is for (L1, L2), (L2, L4), (L3, L4), and (L1, L3). Clearly, the set of possible hetero-interactions, enforced by the attractive nature induced by the DMI, outnumbers the homo ones. The interchained AFM skyrmion is simply the superposition of the sublattice skyrmions at the equilibrium distance, here 2.58 nm between the two AFM skyrmions, where both interactions (attraction and repulsion) are equal.
To substantiate the proposed mechanism, we quantify the skyrmion-skyrmion interaction. We simplify the analysis by neglecting the Cr-Fe magnetic interactions, which puts aside the impact of the rich non-collinear magnetic behavior hosted by the PdFe bilayer. In this case, single AFM skyrmions disappear and only the overlapping ones are observed. We take the skyrmion dimer illustrated in Fig. 2a and proceed to a rigid shift of the lower AFM skyrmion while pinning the upper one at the equilibrium position. We extract the skyrmion-skyrmion interaction map as a function of distance, as shown in Fig. 2b, which clearly demonstrates that as soon as the AFM skyrmions are pulled away from each other, the energy of the system increases. Note that within this procedure, the sublattice interactions (L1, L2) and (L3, L4) do not contribute to the plots since they are assigned to each of the AFM skyrmions moved apart from each other. Two minima are identified along a single direction as favored by the symmetry reduction due to the AFM arrangement of the magnetic moments in which the skyrmions are created. Indeed, one notices in Fig. 1d that due to the sublattice decomposition symmetry operations are reduced to C2, i.e., rotation by 180∘, while mirror symmetries, for example, originally present in the fcc(111) lattice are broken. Figure 2c, d depict the skyrmion-skyrmion interaction, which hosts either one or two minima, as a function of distance along two directions indicated by the dashed lines, blue and black, in Fig. 2a. The two minima found along the blue line should be degenerate and correspond to the swapping of the two AFM skyrmions. The breaking of degeneracy is an artifact of the rigid shift assumed in the simulations, which can be corrected by allowing the moments to relax (see red circle in Fig. 2c). The maximum of repulsion is realized when the two AFM skyrmions perfectly overlap (see inset).
Fig. 2: Energetics of two interchained AFM skyrmions.
a Two overlapping AFM skyrmions decoupled from the PdFe bilayer with black and blue lines representing two examples of paths along which the lower skyrmion is rigid-shifted with respect to the upper one, which is pinned. b Two-dimensional map of the total energy difference with respect to the magnetic state shown in a as a function of the distance between the skyrmion centers. c Energy profile along the blue line shown in a. A double minimum is found once the skyrmions swap their positions and become truly degenerate once the rigidity of the spin state is removed (see the red circle). d The Heisenberg exchange is the most prominent contribution to the skyrmion stabilization, as shown along the path hosting a single minimum. The total skyrmion-skyrmion repulsive homo-interaction is dominated by the attractive hetero-interaction, red curves in e and f, respectively. The DMI contribution, shown in insets, is smaller and sublattice independent. It favors the overlap of AFM skyrmions.
The interaction profile shown in Fig. 2d is decomposed into two contributions: the skyrmion-skyrmion homo- and hetero-interactions, which we plot in Fig. 2e, f, respectively. The data clearly reveals the strong repulsive nature of the homo-interaction mediated by the Heisenberg exchange, which competes with the attractive hetero-interaction driven by both the Heisenberg exchange coupling and DMI. The latter skyrmion-skyrmion interaction is strong enough to impose the unusual compromise of having strongly overlapping solitons.
Impact of magnetic field
Prior to discussing stability aspects pertaining to the single AFM skyrmion in detail, we apply a magnetic field perpendicular to the substrate and disclose pivotal ingredients for the formation of the isolated solitons. In general, the reaction of FM and AFM skyrmions to an external magnetic field is expected to be deeply different. When applied along the direction of the background magnetization, FM skyrmions reduce in size while recent predictions expect a size expansion of AFM skyrmions36,41,42, thereby enhancing their stability.
To inspect the response of AFM skyrmions to a magnetic field perpendicular to the substrate, we first remove, as done in the previous section, the Cr-Fe interaction since it gives rise to a non-homogeneous and strong effective exchange field. In this particular case, we can only explore the case of interchained AFM skyrmions. As illustrated in Fig. 3a, the size of each of the sublattice skyrmions, which together form the AFM skyrmion dimer, increases with an increasing magnetic field. The type of the hosting sublattice, with the magnetization being parallel or antiparallel to the applied field, seems important in shaping the skyrmions dimension. Strikingly, and in strong contrast to what is known for FM skyrmions, the AFM skyrmions, single and multimers, were found to be stable up to extremely large magnetic fields. Although the assumed fields are unrealistic in the lab, they can be emulated by the exchange field induced by the underlying magnetic substrate. Indeed, the magnetic interaction between Cr and its nearest neighboring Fe atoms, carrying each a spin moment of 2.51 μB, reaches −3.05 meV, which translates to an effective field of about 21 T. At this value, the average skyrmion radius is about 1.6 nm, which is 30% smaller than the one found once the Cr-Fe magnetic coupling is enabled (see Fig. 3b).
Fig. 3: Impact of magnetic field of AFM skyrmion radius.
Radius of the sublattice FM skyrmions for two-interchained AFM skyrmions a decoupled from and b coupled to the Fe magnetization. c A single case is shown for the isolated AFM skyrmion since it disappears without the inhomogeneous magnetic field emerging from the substrate. Examples of snapshots of the AFM skyrmions are illustrated as insets of the different figures. In Fe, the amount of FM skyrmions and antiskyrmions increases once applying a magnetic field, which erases the ground state spin-spiral. The coupling to the Fe magnetization affects the evolution of the AFM skyrmions as function of the magnetic field dramatically. d depicts the dependence of the AFM skyrmion under a magnetic field of 70 Tesla on the surrounding magnetic environment, by sequentially deleting one FM skyrmion or antiskyrmion in the Fe layer and relaxing the spin structure. At some point, removing any of the single FM skyrmions in the lower left of d annihilates the AFM skyrmion.
We note that since the skyrmions are not circular in shape, their radius is defined as the average distance between the skyrmion's center and the position where the spin moments lie in-plane. The significant size difference is induced by the strong inhomogenous exchange field emanating from the Fe sub-layer, which can host spirals, skyrmions and antiskyrmions.
If the Cr-Fe interaction is included, the size dependence changes completely. Instead of the rather monotonic increase with the field, the size of the skyrmion is barely affected until reaching about 50 T, which is accompanied by substantial miniaturization of the AFM skyrmions. Here, a phase transition occurs in Fe, which initially hosts spin spirals that turn into FM skyrmions (see Supplementary Fig. 5). After being squeezed down to an average radius of 1.48 nm at 140 T, the size expansion observed without the Cr-Fe interaction is recovered because the substrate magnetization is fully homogeneous and parallel to the Zeeman field. Likewise, single AFM skyrmions, found only once the coupling to the substrate is enabled, react in a similar fashion to the field as depicted in Fig. 3c. The substantial difference, however, is that fields larger than 80 T destroy the AFM skyrmions due to the annihilation of the Fe FM skyrmions. This highlights an enhanced sensitivity to the underlying magnetic environment and clearly demonstrates the robustness enabled by skyrmion interchaining.
Stabilization mechanism for single AFM skyrmions
We learned that single AFM skyrmions can be deleted after application of an external magnetic field or by switching off the exchange coupling to the magnetic substrate. Both effects find their origin in the magnetization behavior of the PdFe bilayer. To explore the underlying correlation, we consider as an example the magnetic configuration obtained with a field of 70 T and delete one after the other the skyrmions and antiskyrmions found in Fe, then check whether the AFM skyrmion in Cr survives (see example in Fig. 3d). We notice that the AFM skyrmion disappears by deleting the FM solitons located directly underneath or even a bit away. Supplementary Fig. 6 shows that when shifted across the lattice, the AFM skyrmion disappears if fixed above a magnetically collinear Fe area.
We proceed in Fig. 4 to an analysis of the Fe-Cr interaction pertaining to the lower-right snapshot presented in Fig. 3d by separating the Heisenberg exchange contribution from that of DMI and plotting the corresponding site-dependent heat maps of these two contributions for each sublattice. Here, we consider as reference energy that of the RW-AFM collinear state surrounding the non-collinear states in Fig. 3d. The building-blocks of the AFM skyrmion are shown in Fig. 4a, d, where one can recognize the underlying Fe FM skyrmions in the background. The latter are more distinguishable in the sublattices free from the AFM skyrmion as illustrated in Fig. 4g, j. The order of magnitude of the interactions clearly indicates that the DMI plays a minor role and that one can basically neglect the interactions arising in the skyrmion-free sublattices, namely L3 and L4. It is the Heisenberg exchange interaction emerging in the sublattices L1 and L2 that dictates the overall stability of the AFM skyrmion.
Fig. 4: Interaction map of the single AFM skyrmion with the magnetic substrate.
In the first row of figures, sublattice decomposition of Cr skyrmion including the underlying Fe skyrmions shown in four columns a, d, g, and j, corresponding respectively to L1, L2, L3, and L4. The AFM skyrmion is made of two FM skyrmions hosted by sublattices L1 and L2. In Fe, FM skyrmions and antiskyrmions can be found in all four lattices. The second row (b, e, h, and k) illustrates the sublattice dependent two dimensional Heisenberg exchange energy map corresponding to the areas plotted in the first row, followed by the third row (c, f, i, and l) corresponding to DMI. Note that the energy difference ΔE is defined with respect to the RW-AFM background.
In L2, the core of the magnetization of the Cr FM skyrmion points along the same direction as that of the underlying Fe atoms, which obviously is disfavored by the AFM coupling between Cr and Fe (−3.05 meV for nearest neighbors). This induces the red exchange area surrounding the core of the AFM skyrmion (black circle in Fig. 4e), which is nevertheless sputtered with blue spots induced by the magnetization of the core of the Fe FM skyrmions pointing in the direction opposite to that of the Cr moments in L2. The latter is a mechanism reducing the instability of the Cr skyrmion. Overall, the total energy cost in having the Cr skyrmion in L2 reaches +693.7 meV and is compensated by the exchange energy of −712.4 meV generated by the one living in sublattice L1. Here, the scenario is completely reversed since the core of the Cr skyrmion has its magnetization pointing in the opposite direction than that of the neighboring Fe atoms and therefore the large negative blue area with the surrounding area being sputtered by the Fe skyrmions, similar to the observation made in L2 (see Fig. 4d). Overall, the Cr AFM skyrmion arranges its building blocks such that the energy is lowered by the skyrmion anchored in sublattice L1. Here, the details of the non-collinear magnetic textures hosted by Fe play a primary role in offering the right balance to enable stabilization. This explains the sensitivity of the single AFM skyrmion to the number and location of the underlying FM Fe skyrmions. Removing non-collinearity in Fe makes both building blocks of the AFM skyrmion equivalent without any gain in energy from the Cr-Fe interaction, which facilitates the annihilation of the Cr skyrmion.
It is enlightening to explore the phase diagrams of the AFM skyrmions as function of the underlying magnetic interactions. The latter are multiplied by a factor renormalizing the initial parameters. In Fig. 5 we illustrate the impact of DMI vector's (D) magnitude, Heisenberg exchange J and anisotropy K on the formation of various phases including the one hosting double overlapped AFM skyrmions. For simplicity, we consider the case where the interaction between Cr and the underlying Fe layer is switched off. A color code is amended to follow the changes induced on the distance between the AFM skyrmions. From this study, we learn that in contrast to the DMI, which tends to increase the size of the structures, J and K tend to miniaturize the skyrmions, ultimately favoring their annihilation. The phase hosting AFM skyrmions is sandwiched between the RW-AFM state and a phase hosting stripe domains. It is convenient to analyse the unveiled overall behavior in terms of the impact of DMI. The latter protects the AFM skyrmions structure from shrinking, similarly to FM skyrmions67,68. So for small values of DMI compared to J in Fig. 5a, or compared to K in Fig. 5b, the AFM skyrmions shrink and disappear. In contrast, large values of the DMI increase the size of the skyrmions till reaching a regime where stripe domains are formed. Within the phase hosting AFM skyrmions, increasing J or K results in smaller skyrmions.
Fig. 5: Phase diagrams of the free double interchained AFM skyrmions.
a Phase diagram obtained by fixing the magnetic anisotropy energy K while changing the set of DMI and Heisenberg exchange interaction J, or b by fixing J while modifying K and DMI. The color gradient pertaining to the skyrmion phase indicates the distance between two AFM skyrmions. c Illustration of the states shown in the phase diagrams. Note that an in-plane Néel state is predicted for large DMI and small J.
Thermal stability with Geodesic nudged elastic band (GNEB) method
So far we have demonstrated that the interlinked AFM skyrmion multimers can indeed exist as local minima of the energy expression given by the Heisenberg Hamiltonian (Eq. (1)). Another important question, however, is the stability of these structures against thermal excitations. Answering this question requires knowledge about how deep or shallow these energy minima are, which can be quantified as a minimal energy barrier that the system has to overcome in order to escape a minimum, keeping in mind that the Néel temperature of the RW-AFM ground state is ≈ 310 K as obtained from our Monte Carlo simulations69,70,71. To investigate this issue, we systematically carried out a series of Geodesic nudged elastic band (GNEB) simulations71,72,73 for AFM multimers, containing initially ten interchained skyrmions, then calculating the energy barrier needed to annihilate one AFM skyrmion at a time as depicted in Fig. 6a, showing the successive magnetic states between which, the energy barrier has been calculated. Note that deleting one of the AFM skyrmions forming the dimer leads to the RW-AFM state. The energy barrier is given by the energy difference between the nth AFM skyrmions state local minimum (hosting n AFM interchained skyrmions) and the relevant saddle point located on the minimum energy path connecting the initial state with the (n−1)th AFM skyrmions state. The energy barrier increases from about 8 meV (≈90 K) for the double interchained AFM skyrmions to 13 meV (≈150 K) for three interchained ones, reaching a saturation value of ≈18.5 meV (≈214 K) for chains containing more than five AFM skyrmions, see Fig. 6b. Hence, increasing the number of interchained skyrmions enhances their stability, which is further amplified when enabling the interaction with the PdFe substrate. Instead of 8 meV pertaining to the free skyrmion dimer, the barrier reaches 45.7 meV (≈530 K) owing to the interaction with the underlying substrate while the single AFM skyrmion experiences a barrier of 10 meV (≈113 K). Thus, the exchange field emanating from the PdFe substrate promotes the use of interchained AFM skyrmions in room temperature applications. By analysing how the different interactions contribute to the barrier, we identified the DMI as a key parameter for the thermal stability of the interchained AFM skyrmions. For example, in the case of free double interchained AFM skyrmions, the Heisenberg exchange interactions contribution is −87 meV, the magnetic anisotropy contribution is −150 meV while the DMI provides a barrier of 245 meV. Interestingly and as expected, it is the magnetic exchange interaction between Cr and Fe that is mainly responsible for the thermal stability of the single AFM skyrmion.
Fig. 6: Energy barriers for chains of free interchained AFM skyrmions.
a Snapshots of the explored skyrmion chains. b The energy barrier obtained with GNEB simulations for deleting a single AFM skyrmion from the lower edge of the free (not interacting with PdFe) chains.
Following a two-pronged approach based on first-principles simulations combined with atomistic spin dynamics, we identify a thin film that can host intrinsic, i.e., non-synthetic, AFM skyrmions at zero magnetic field. A Cr monolayer deposited on a substrate known to host FM skyrmions, PdFe/Ir(111), offers the right AFM interface combination enabling the emergence of a rich set of AFM topological solitons. Owing to the AFM nature of Cr, its ground state is RW-AFM as induced by magnetic interactions beyond nearest neighbors. We strikingly discovered interchained AFM skyrmions, which can be cut and decoupled into isolated solitons via the inhomogenous exchange field emanating from PdFe bilayer. Interestingly, interchaining enhances their stability, which is largely amplified by the exchange field emanating from the substrate. The intra-overlayer skyrmion-skyrmion interaction favors the important overlap of the AFM skyrmions, and makes them robust against the rich magnetic nature of the PdFe substrate. In contrast, the single AFM skyrmion annihilates if positioned on a homogeneously magnetized substrate and it is only through the presence of various spin-textures such as spin spirals and multiples of skyrmions or antiskyrmions that one of the building-blocks of the AFM skyrmion can lower the energy enough to enable stability.
Since the experimental observation of intrinsic AFM skyrmions has so far been elusive at interfaces, our predictions open the door for their realization in well-defined materials and offer the opportunity to explore them in thin film geometries. The robustness of the interchained skyrmions qualify them as ideal particles for room temperature racetrack memory devices to be driven with currents while avoiding the skyrmion Hall effect. Preliminary work indicates that AFM skyrmions move faster than their FM partners when reacting to an applied current, with an intriguing behavior induced by the non-collinear exchange field emanating from the substrate. If the latter is collinear or non-magnetic, the AFM textures are predicted to be quasi-free from the skyrmion Hall effect. The ability to control the substrate magnetization makes the system, we studied, a rich playground to design and tune the highly sensitive single AFM skyrmion living in the overlayer. We envision patterning the ferromagnetic surface with regions hosting different magnetic textures, being trivial, such as the ferromagnetic regions, or topological, to define areas where the AFM skyrmion can be confined or driven within specific paths. We envisage a rich and complex response to in-plane currents, which could move the underlying FM solitons along specific directions subjected to the Magnus effect. The latter would affect the response of the overlaying isolated AFM skyrmions in a non-trivial fashion.
We anticipate that the proposed material and the FM-substrate on which AFM films can be deposited offer an ideal platform to explore the physics of AFM skyrmions. Noticing already that different overlapped AFM skyrmions can co-exist establishes a novel set of multi-soliton objects worth exploring in future studies. Besides their fundamental importance enhanced by the potential parallel with topological concepts known in knot theory, such unusual AFM localized entities might become exciting and useful constituents of future nanotechnology devices resting on non-collinear spin-textures and the emerging field of antiferromagnetic spintronics.
First-principles calculations
The relaxation parameters were obtained using the Quantum-Espresso computational package74. The projector augmented wave pseudo potentials from the PS Library75 and a 28 × 28 × 1 k-point grid were used for the calculations. The Cr, Pd, Fe, and Ir interface layer were fcc-stacked along the [111] direction and relaxed by 4%, 5.8%, 8.1%, and −1% with respect to the Ir ideal interlayer distance, respectively. Positive (negative) numbers refer to atomic relaxations towards (outward of) the Ir surface.
The electronic structure and magnetic properties were simulated using all electron full potential scalar relativistic Koringa -Kohn-Rostoker (KKR) Green function method76,77 in the local spin density approximation. The slab contains 30 layers (3 vacuum + 1 Cr + 1 Pd + 1 Fe + 20 Ir + 4 vacuum). The momentum expansion of the Green function was truncated at \({\ell }_{\max }=3\). The self-consistent calculations were performed with a k-mesh of 30 × 30 points and the energy contour contained 23 complex energy points in the upper complex plane with 9 Matsubara poles. The Heisenberg exchange interactions and DM vectors were extracted using the infinitesimal rotation method78,79 with a k-mesh of a 200 × 200.
Hamiltonian Model and atomistic spin dynamics
In our study, we consider a two dimensional Heisenberg model on a triangular lattice, equipped with Heisenberg exchange coupling, DMI, the magnetic anisotropy energy, and Zeeman term. All parameters were obtained from ab-initio. The energy functional reads as follows:
$$H={H}_{{{{{{{{\rm{Exc}}}}}}}}}+{H}_{{{{{{{{\rm{DMI}}}}}}}}}+{H}_{{{{{{{{\rm{Ani}}}}}}}}}+{H}_{{{{{{{{\rm{Zeem}}}}}}}}},$$
$${H}_{{{{{{{{\rm{Exc}}}}}}}}}=-\mathop{\sum}\limits_{ < i,\, j > }{J}_{ij}^{{{{{{{{\rm{Cr-Cr}}}}}}}}}{{{{{{{{\bf{S}}}}}}}}}_{i}\cdot {{{{{{{{\bf{S}}}}}}}}}_{j}-\mathop{\sum}\limits_{ < i,\, j > }{J}_{ij}^{{{{{{{{\rm{Fe-Cr}}}}}}}}}{{{{{{{{\bf{S}}}}}}}}}_{i}\cdot {{{{{{{{\bf{S}}}}}}}}}_{j}-\mathop{\sum}\limits_{ < i,\, j > }{J}_{ij}^{{{{{{{{\rm{Fe-Fe}}}}}}}}}{{{{{{{{\bf{S}}}}}}}}}_{i}\cdot {{{{{{{{\bf{S}}}}}}}}}_{j},$$
$${H}_{{{{{{{{\rm{DMI}}}}}}}}}=\mathop{\sum}\limits_{ < i,\, j > }{{{{{{{{\bf{D}}}}}}}}}_{ij}^{{{{{{{{\rm{Cr-Cr}}}}}}}}}\cdot [{{{{{{{{\bf{S}}}}}}}}}_{i}\times {{{{{{{{\bf{S}}}}}}}}}_{j}]+\mathop{\sum}\limits_{ < i,\, j > }{{{{{{{{\bf{D}}}}}}}}}_{ij}^{{{{{{{{\rm{Fe-Cr}}}}}}}}}\cdot [{{{{{{{{\bf{S}}}}}}}}}_{i}\times {{{{{{{{\bf{S}}}}}}}}}_{j}]+\mathop{\sum}\limits_{ < i,\, j > }{{{{{{{{\bf{D}}}}}}}}}_{ij}^{{{{{{{{\rm{Fe-Fe}}}}}}}}}\cdot [{{{{{{{{\bf{S}}}}}}}}}_{i}\times {{{{{{{{\bf{S}}}}}}}}}_{j}],$$
$${H}_{{{{{{{{\rm{Ani}}}}}}}}}=-{K}^{{{{{{{{\rm{Cr}}}}}}}}}\mathop{\sum}\limits_{i}{\left({S}_{i}^{z}\right)}^{2}-{K}^{{{{{{{{\rm{Fe}}}}}}}}}\mathop{\sum}\limits_{i}{\left({S}_{i}^{z}\right)}^{2},$$
$${H}_{{{{{{{{\rm{Zeem}}}}}}}}}=-\mathop{\sum}\limits_{i}{h}_{i}{S}_{i}^{z},$$
where i and j are site indices carrying each magnetic moments. S is a unit vector of the magnetic moment. \({J}_{ij}^{{{{{{{{\rm{X-Y}}}}}}}}}\) is the Heisenberg exchange coupling strength, being < 0 for AFM interaction, between an X atom on site i and a Y atom on site j. A similar notation is adopted for the DMI vector D and the magnetic anisotropy energy K (0.5 meV per magnetic atom). The latter favors the out-of-plane orientation of the magnetization, and hi = μiB describes the Zeeman coupling to the atomic spin moment μ at site i assuming an out-of-plane field.
To explore the magnetic properties and emerging complex states we utilize the Landau–Lifshitz-equation (LLG) as implemented in the Spirit code71. We assumed periodic boundary conditions to model the extended two-dimensional system with cells containing 1002, 2002, 3002, and 4002 sites.
The data needed to evaluate the conclusions in the paper are present in the paper and the Supplementary Information.
Code availability
We used the following codes: Quantum ESPRESSO, SPIRIT can be found at https://github.com/spirit-code/spirit, and the KKR code is a rather complex ab-initio DFT-based code, which is in general impossible to use without proper training on the theory behind it and on the practical utilization of the code. We are happy to provide the latter code upon request.
Bogdanov, A. N. & Yablonskii, D. Thermodynamically stable "vortices" in magnetically ordered crystals. the mixed state of magnets. J. Exp. Theor. Phys. 95, 178 (1989).
Bogdanov, A. & Hubert, A. Thermodynamically stable magnetic vortex states in magnetic crystals. J. Magn. Magn. Mater. 138, 255–269 (1994).
Rössler, U. K., Bogdanov, A. N. & Pfleiderer, C. Spontaneous skyrmion ground states in magnetic metals. Nature 442, 797–801 (2006).
Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotechnol. 8, 899–911 (2013).
Fert, A., Cros, V. & Sampaio, J. Skyrmions on the track. Nat. Nanotechnol. 8, 152–156 (2013).
Dzyaloshinsky, I. A thermodynamic theory of "weak" ferromagnetism of antiferromagnetics. J. Phys. Chem. Solids 4, 241–255 (1958).
Moriya, T. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120, 91–98 (1960).
Mühlbauer, S. et al. Skyrmion lattice in a chiral magnet. Science 323, 915–919 (2009).
Pappas, C. et al. Chiral paramagnetic skyrmion-like phase in mnsi. Phys. Rev. Lett. 102, 197202 (2009).
Yu, X. et al. Real-space observation of a two-dimensional skyrmion crystal. Nature 465, 901–904 (2010).
Yu, X. et al. Skyrmion flow near room temperature in an ultralow current density. Nat. Commun. 3, 1–6 (2012).
Yu, X. et al. Near room-temperature formation of a skyrmion crystal in thin-films of the helimagnet fege. Nat. Mater. 10, 106–109 (2011).
Heinze, S. et al. Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions. Nat. Phys. 7, 713–718 (2011).
Romming, N. et al. Writing and deleting single magnetic skyrmions. Science 341, 636–639 (2013).
Chen, G., Mascaraque, A., N'Diaye, A. T. & Schmid, A. K. Room temperature skyrmion ground state stabilized through interlayer exchange coupling. Appl. Phys. Lett. 106, 242404 (2015).
Soumyanarayanan, A. et al. Tunable room-temperature magnetic skyrmions in ir/fe/co/pt multilayers. Nat. Mater. 16, 898–904 (2017).
Kiselev, N., Bogdanov, A., Schäfer, R. & Rößler, U. Chiral skyrmions in thin magnetic films: new objects for magnetic storage technologies? J. Phys. D-Appl. Phys. 44, 392001 (2011).
Crum, D. M. et al. Perpendicular reading of single confined magnetic skyrmions. Nat. Commun. 6, 1–8 (2015).
Wiesendanger, R. Nanoscale magnetic skyrmions in metallic films and multilayers: a new twist for spintronics. Nat. Rev. Mater. 1, 16044 (2016).
Garcia-Sanchez, F., Sampaio, J., Reyren, N., Cros, V. & Kim, J. A skyrmion-based spin-torque nano-oscillator. New J. Phys. 18, 075011 (2016).
Fert, A., Reyren, N. & Cros, V. Magnetic skyrmions: advances in physics and potential applications. Nat. Rev. Mater. 2, 17031 (2017).
Fernandes, I. L., Bouhassoune, M. & Lounis, S. Defect-implantation for the all-electrical detection of non-collinear spin-textures. Nat. Commun. 11, 1–9 (2020).
Zhang, X. et al. Skyrmion-electronics: writing, deleting, reading and processing magnetic skyrmions toward spintronic applications. J. Phys.-Condes. Matter 32, 143001 (2020).
Büttner, F., Lemesh, I. & Beach, G. S. D. Theory of isolated magnetic skyrmions: From fundamentals to room temperature applications. Sci. Rep. 8, 4464 (2018).
Lin, S.-Z., Reichhardt, C., Batista, C. D. & Saxena, A. Particle model for skyrmions in metallic chiral magnets: Dynamics, pinning, and creep. Phys. Rev. B 87, 214419 (2013).
Woo, S. et al. Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets. Nat. Mater. 15, 501–506 (2016).
Jiang, W. et al. Direct observation of the skyrmion Hall effect. Nat. Phys. 13, 162–169 (2017).
Litzius, K. et al. Skyrmion hall effect revealed by direct time-resolved x-ray microscopy. Nat. Phys. 13, 170–175 (2017).
Fernandes, I. L., Bouaziz, J., Blügel, S. & Lounis, S. Universality of defect-skyrmion interaction profiles. Nat. Commun. 9, 1–7 (2018).
Fernandes, I. L., Chico, J. & Lounis, S. Impurity-dependent gyrotropic motion, deflection and pinning of current-driven ultrasmall skyrmions in PdFe/Ir(111) surface. J. Phys.-Condes. Matter 32, 425802 (2020).
Arjana, I. G., Lima Fernandes, I., Chico, J. & Lounis, S. Sub-nanoscale atom-by-atom crafting of skyrmion-defect interaction profiles. Sci. Rep. 10, 14655 (2020).
Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J. Antiferromagnetic spintronics. Nat. Nanotechnol 11, 231–241 (2016).
Olejník, K. et al. Terahertz electrical writing speed in an antiferromagnetic memory. Sci. Adv. 4, eaar3566 (2018).
Gomonay, O., Baltz, V., Brataas, A. & Tserkovnyak, Y. Antiferromagnetic spin textures and dynamics. Nat. Phys. 14, 213–216 (2018).
Bogdanov, A., Roessler, U. K., Wolf, M. & Müller, K.-H. Magnetic structures and reorientation transitions in noncentrosymmetric uniaxial antiferromagnets. Phys. Rev. B 66, 214410 (2002).
Rosales, H. D., Cabra, D. C. & Pujol, P. Three-sublattice skyrmion crystal in the antiferromagnetic triangular lattice. Phys. Rev. B 92, 214439 (2015).
Keesman, R., Raaijmakers, M., Baerends, A., Barkema, G. & Duine, R. Skyrmions in square-lattice antiferromagnets. Phys. Rev. B 94, 054402 (2016).
Zhang, X., Zhou, Y. & Ezawa, M. Antiferromagnetic skyrmion: stability, creation and manipulation. Sci. Rep. 6, 1–8 (2016).
Zhang, X., Ezawa, M. & Zhou, Y. Thermally stable magnetic skyrmions in multilayer synthetic antiferromagnetic racetracks. Phys. Rev. B 94, 064406 (2016).
Göbel, B., Mook, A., Henk, J. & Mertig, I. Antiferromagnetic skyrmion crystals: Generation, topological hall, and topological spin hall effect. Phys. Rev. B 96, 060406 (2017).
Bessarab, P. et al. Stability and lifetime of antiferromagnetic skyrmions. Phys. Rev. B 99, 140411 (2019).
Potkina, M. N., Lobanov, I. S., Jónsson, H. & Uzdin, V. M. Skyrmions in antiferromagnets: Thermal stability and the effect of external field and impurities. J. Appl. Phys. 127, 213906 (2020).
Liu, Z., dos Santos Dias, M. & Lounis, S. Theoretical investigation of antiferromagnetic skyrmions in a triangular monolayer. J. Phys.-Condes. Matter 32, 425801 (2020).
Shen, L. et al. Dynamics of the antiferromagnetic skyrmion induced by a magnetic anisotropy gradient. Phys. Rev. B 98, 134448 (2018).
Silva, R., Silva, R., Pereira, A. & Moura-Melo, W. Antiferromagnetic skyrmions overcoming obstacles in a racetrack. J. Phys.-Condes. Matter 31, 225802 (2019).
Khoshlahni, R., Qaiumzadeh, A., Bergman, A. & Brataas, A. Ultrafast generation and dynamics of isolated skyrmions in antiferromagnetic insulators. Phys. Rev. B 99, 054423 (2019).
Díaz, S. A., Klinovaja, J. & Loss, D. Topological magnons and edge states in antiferromagnetic skyrmion crystals. Phys. Rev. Lett. 122, 187203 (2019).
Zarzuela, R., Kim, S. K. & Tserkovnyak, Y. Stabilization of the skyrmion crystal phase and transport in thin-film antiferromagnets. Phys. Rev. B 100, 100408 (2019).
Barker, J. & Tretiakov, O. A. Static and dynamical properties of antiferromagnetic skyrmions in the presence of applied current and temperature. Phys. Rev. Lett. 116, 147203 (2016).
Woo, S. et al. Current-driven dynamics and inhibition of the skyrmion hall effect of ferrimagnetic skyrmions in gdfeco films. Nat. Commun. 9, 1–8 (2018).
Hirata, Y. et al. Vanishing skyrmion hall effect at the angular momentum compensation temperature of a ferrimagnet. Nat. Nanotechnol. 14, 232–236 (2019).
Gao, S. et al. Fractional antiferromagnetic skyrmion lattice induced by anisotropic couplings. Nature 586, 37–41 (2020).
Ross, A. et al. Structural sensitivity of the spin hall magnetoresistance in antiferromagnetic thin films. Phys. Rev. B 102, 094415 (2020).
Jani, H. et al. Antiferromagnetic half-skyrmions and bimerons at room temperature. Nature 590, 74–79 (2021).
Legrand, W. et al. Room-temperature stabilization of antiferromagnetic skyrmions in synthetic antiferromagnets. Nat. Mater. 19, 34–42 (2020).
Spethmann, J. et al. Discovery of magnetic single-and triple-Q states in Mn/Re (0001). Phys. Rev. Lett. 124, 227203 (2020).
Spethmann, J., Grünebohm, M., Wiesendanger, R., von Bergmann, K. & Kubetzka, A. Discovery and characterization of a new type of domain wall in a row-wise antiferromagnet. Nat. Commun. 12, 1–8 (2021).
Wu, F. Y. Knot theory and statistical mechanics. Rev. Mod. Phys. 64, 1099–1131 (1992).
Amabilino, D. B. & Stoddart, J. F. Interlocked and intertwined structures and superstructures. Chem. Rev. 95, 2725–2828 (1995).
Dabrowski-Tumanski, P. & Sulkowska, J. I. Topological knots and links in proteins. Proc. Natl. Acad. Sci. USA. 114, 3415–3420 (2017).
Bates, A. D. et al. DNA topology (Oxford University Press, USA, 2005).
Dupé, B., Hoffmann, M., Paillard, C. & Heinze, S. Tailoring magnetic skyrmions in ultra-thin transition metal films. Nat. Commun. 5, 1–6 (2014).
Simon, E., Palotás, K., Rózsa, L., Udvardi, L. & Szunyogh, L. Formation of magnetic skyrmions with tunable properties in PdFe bilayer deposited on Ir (111). Phys. Rev. B 90, 094410 (2014).
Romming, N., Kubetzka, A., Hanneken, C., von Bergmann, K. & Wiesendanger, R. Field-dependent size and shape of single magnetic skyrmions. Phys. Rev. Lett. 114, 177203 (2015).
Bouhassoune, M. & Lounis, S. Friedel oscillations induced by magnetic skyrmions: From scattering properties to all-electrical detection. Nanomaterials 11, 194 (2021).
Fernandes, I. L., Blügel, S. & Lounis, S. Spin-orbit enabled all-electrical readout of chiral spin-textures. ArXiv:2202.11637 (2022).
Sampaio, J., Cros, V., Rohart, S., Thiaville, A. & Fert, A. Nucleation, stability and current-induced motion of isolated magnetic skyrmions in nanostructures. Nat. Nanotechnol. 8, 839–844 (2013).
Rohart, S. & Thiaville, A. Skyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii-Moriya interaction. Phys. Rev. B 88, 184422 (2013).
Hinzke, D. & Nowak, U. Monte carlo simulation of magnetization switching in a heisenberg model for small ferromagnetic particles. Comput. Phys. Commun. 121, 334–337 (1999).
Nowak, U. Thermally activated reversal in magnetic nanostructures. Annu. Rev. Comput. Phys. 9, 105–151 (2001).
Article CAS MATH Google Scholar
Müller, G. P. et al. Spirit: Multifunctional framework for atomistic spin simulations. Phys. Rev. B 99, 224414 (2019).
Bessarab, P. F., Uzdin, V. M. & Jónsson, H. Method for finding mechanism and activation energy of magnetic transitions, applied to skyrmion and antivortex annihilation. Comput. Phys. Commun. 196, 335–347 (2015).
Müller, G. P. et al. Duplication, collapse, and escape of magnetic skyrmions revealed using a systematic saddle point search method. Phys. Rev. Lett. 121, 197202 (2018).
Giannozzi, P. et al. Quantum espresso: a modular and open-source software project for quantum simulations of materials. J. Phys.-Condes. Matter 21, 395502 (2009).
Dal Corso, A. Pseudopotentials periodic table: From h to pu. Comput. Mater. Sci. 95, 337–350 (2014).
Papanikolaou, N., Zeller, R. & Dederichs, P. H. Conceptual improvements of the KKR method. J. Phys. Condens. Matter 14, 2799 (2002).
Bauer, D. S. G. Development of a relativistic full-potential first-principles multiple scattering Green function method applied to complex magnetic textures of nano structures at surfaces (Forschungszentrum Jülich Jülich, 2014).
Liechtenstein, A., Katsnelson, M., Antropov, V. & Gubanov, V. Local spin density functional approach to the theory of exchange interactions in ferromagnetic metals and alloys. JMMM 67, 65–74 (1987).
Ebert, H. & Mankovsky, S. Anisotropic exchange coupling in diluted magnetic semiconductors: Ab initio spin-density functional theory. Phys. Rev. B 79, 045209 (2009).
We thank Markus Hoffmann for fruitful discussions. This work was supported by the Federal Ministry of Education and Research of Germany in the framework of the Palestinian-German Science Bridge (BMBF grant number 01DH16027) and the Deutsche Forschungsgemeinschaft (DFG) through SPP 2137 "Skyrmionics" (Projects LO 1659/8-1, BL 444/16-2). The authors gratefully acknowledge the computing time granted through JARA on the supercomputer JURECA at Forschungszentrum Jülich.
Open Access funding enabled and organized by Projekt DEAL.
Peter Grünberg Institute and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, D-52425, Jülich, Germany
Amal Aldarawsheh, Imara Lima Fernandes, Sascha Brinker, Moritz Sallermann, Stefan Blügel & Samir Lounis
Faculty of Physics, University of Duisburg-Essen and CENIDE, 47053, Duisburg, Germany
Amal Aldarawsheh & Samir Lounis
RWTH Aachen University, 52056, Aachen, Germany
Moritz Sallermann
Science Institute and Faculty of Physical Sciences, University of Iceland, VR-III, 107, Reykjavík, Iceland
Department of Physics, Arab American University, Jenin, Palestine
Muayad Abusaa
Amal Aldarawsheh
Imara Lima Fernandes
Sascha Brinker
Stefan Blügel
Samir Lounis
S.L. initiated, designed and supervised the project. A.A. performed the simulations with support and supervision from I.L.F., S.Br. and M.S. A.A., I.L.F., S.Br., M.S., M.A., S.Bl. and S.L. discussed the results. A.A. and S.L. wrote the manuscript to which all co-authors contributed.
Correspondence to Amal Aldarawsheh or Samir Lounis.
Aldarawsheh, A., Fernandes, I.L., Brinker, S. et al. Emergence of zero-field non-synthetic single and interchained antiferromagnetic skyrmions in thin films. Nat Commun 13, 7369 (2022). https://doi.org/10.1038/s41467-022-35102-x
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Nature Communications (Nat Commun) ISSN 2041-1723 (online) | CommonCrawl |
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fMRI scan
b. tracks successive images of brain tissue to show brain function
a. tracks radioactive glucose to reveal brain activity.
c. uses magnetic fields and radio waves to show brain anatomy
Nerves from the left side of the brain are mostly linked to the ___________ side of the body, and vice versa.
In what brain region would damage be most likely to (1) disrupt your ability to skip rope?
cerebellum
In what brain region would damage be most likely to (2) disrupt your ability to hear and taste?
In what brain region would damage be most likely to (3) perhaps leave you in a coma?
reticular formation
In what brain region would damage be most likely to (4) cut off the very breath and heartbeat of life?
Electrical stimulation of a cat's amygdala provokes angry reactions. Which autonomic nervous system division is activated by such stimulation?
The sympathetic nervous system
What are the three key structures of the limbic system, and what functions do they serve?
(1) The amygdala is involved in aggression and fear responses. (2) The hypothalamus is involved in bodily maintenance, pleasurable rewards, and control of the hormonal systems. (3) The hippocampus processes conscious memory.
How do neuroscientists study the brain's connections to behavior and mind?
Clinical observations and lesioning reveal the general effects of brain damage. Electrical, chemical, or magnetic stimulation can also reveal aspects of information processing in the brain. MRI scans show anatomy. EEG, PET, and fMRI (functional MRI) recordings reveal brain function.
What structures make up the brainstem, and what are the functions of the brainstem, thalamus, reticular formation, and cerebellum?
The brainstem, the oldest part of the brain, is responsible for automatic survival functions. Its components are the medulla (which controls heartbeat and breathing), the pons (which helps coordinate movements), and the reticular formation (which affects arousal).
The thalamus, sitting above the brainstem, acts as the brain's sensory control center. The cerebellum, attached to the rear of the brainstem, coordinates muscle movement and balance and also helps process sensory information.
What are the limbic system's structures and functions?
The limbic system is linked to emotions, memory, and drives. Its neural centers include the hippocampus (which processes conscious memories); the amygdala (involved in responses of aggression and fear); and the hypothalamus (involved in various bodily maintenance functions, pleasurable rewards, and the control of the endocrine system). The hypothalamus controls the pituitary (the "master gland") by stimulating it to trigger the release of hormones.
A limbic system reward center located in front of the hypothalamus is called the____?
Nucleus accumbens
Psych Final
Principles of Psychology Module 1
Speed Test - Mix Em Up
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Suppose the Federal Reserve announced that it would pursue contractionary monetary policy to reduce inflation. For conditions, explain whether it would make the ensuing recession more or less severe. There is little confidence in the Fed's determination to reduce inflation.
No matter how hard she tries to lose weight, Anne finds it difficult to drop below 145 pounds. The weight of 145 pounds is best described as Anne's a. neophobia. b. basal metabolic rate. c. body mass index. d. PYY. e. set point.
A random sample of 20 purchases showed the following amounts (in \$):$ $$ \begin{array}{lrrr}39.05 & 2.73 & 32.92 & 47.51 \\ 37.91 & 34.35 & 64.48 & 51.96 \\ 56.95 & 81.58 & 47.80 & 11.72 \\ 21.57 & 40.83 & 38.24 & 32.98 \\ 75.16 & 74.30 & 47.54 & 65.62\end{array} $$ $The mean was$\$45.26$ and the standard deviation was $\$20.67$. For the $90 \%$ confidence interval for the mean purchases of all customers, assuming that the assumptions and conditions for the confidence interval have been met. b) How large would the sample size have to be to reduce the margin of error to $\$ 0.80$ ?
How will the economy of the future be effected by changes in productivity?
Consumer Behavior: Buying, Having, Being
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Using Fully Homomorphic Hybrid Encryption to Minimize Non-interative Zero-Knowledge Proofs
Craig Gentry, Jens Groth, Yuval Ishai, Chris Peikert, Amit Sahai, Adam Smith
A non-interactive zero-knowledge (NIZK) proof can be used to demonstrate the truth of a statement without revealing anything else. It has been shown under standard cryptographic assumptions that NIZK proofs of membership exist for all languages in NP. While there is evidence that such proofs cannot be much shorter than the corresponding membership witnesses, all known NIZK proofs for NP languages are considerably longer than the witnesses. Soon after Gentry's construction of fully homomorphic encryption, several groups independently contemplated the use of hybrid encryption to optimize the size of NIZK proofs and discussed this idea within the cryptographic community. This article formally explores this idea of using fully homomorphic hybrid encryption to optimize NIZK proofs and other related cryptographic primitives. We investigate the question of minimizing the communication overhead of NIZK proofs for NP and show that if fully homomorphic encryption exists then it is possible to get proofs that are roughly of the same size as the witnesses. Our technique consists in constructing a fully homomorphic hybrid encryption scheme with ciphertext size $$|m|+{\mathrm {poly}}(k)$$|m|+poly(k), where $$m$$m is the plaintext and $$k$$k is the security parameter. Encrypting the witness for an NP-statement allows us to evaluate the NP-relation in a communication-efficient manner. We apply this technique to both standard non-interactive zero-knowledge proofs and to universally composable non-interactive zero-knowledge proofs. The technique can also be applied outside the realm of non-interactive zero-knowledge proofs, for instance to get witness-size interactive zero-knowledge proofs in the plain model without any setup or to minimize the communication in secure computation protocols.
Zero-knowledge Proof
Homomorphic
Homomorphic Encryption
Interactive Proofs
Secure Computation
Gentry, C., Groth, J., Ishai, Y., Peikert, C., Sahai, A., & Smith, A. (2015). Using Fully Homomorphic Hybrid Encryption to Minimize Non-interative Zero-Knowledge Proofs. Journal of Cryptology, 28(4), 820-843. https://doi.org/10.1007/s00145-014-9184-y
Gentry, Craig ; Groth, Jens ; Ishai, Yuval ; Peikert, Chris ; Sahai, Amit ; Smith, Adam. / Using Fully Homomorphic Hybrid Encryption to Minimize Non-interative Zero-Knowledge Proofs. In: Journal of Cryptology. 2015 ; Vol. 28, No. 4. pp. 820-843.
@article{ad5193b66f2f4a54be64e60c73c57a38,
title = "Using Fully Homomorphic Hybrid Encryption to Minimize Non-interative Zero-Knowledge Proofs",
abstract = "A non-interactive zero-knowledge (NIZK) proof can be used to demonstrate the truth of a statement without revealing anything else. It has been shown under standard cryptographic assumptions that NIZK proofs of membership exist for all languages in NP. While there is evidence that such proofs cannot be much shorter than the corresponding membership witnesses, all known NIZK proofs for NP languages are considerably longer than the witnesses. Soon after Gentry's construction of fully homomorphic encryption, several groups independently contemplated the use of hybrid encryption to optimize the size of NIZK proofs and discussed this idea within the cryptographic community. This article formally explores this idea of using fully homomorphic hybrid encryption to optimize NIZK proofs and other related cryptographic primitives. We investigate the question of minimizing the communication overhead of NIZK proofs for NP and show that if fully homomorphic encryption exists then it is possible to get proofs that are roughly of the same size as the witnesses. Our technique consists in constructing a fully homomorphic hybrid encryption scheme with ciphertext size $$|m|+{\mathrm {poly}}(k)$$|m|+poly(k), where $$m$$m is the plaintext and $$k$$k is the security parameter. Encrypting the witness for an NP-statement allows us to evaluate the NP-relation in a communication-efficient manner. We apply this technique to both standard non-interactive zero-knowledge proofs and to universally composable non-interactive zero-knowledge proofs. The technique can also be applied outside the realm of non-interactive zero-knowledge proofs, for instance to get witness-size interactive zero-knowledge proofs in the plain model without any setup or to minimize the communication in secure computation protocols.",
author = "Craig Gentry and Jens Groth and Yuval Ishai and Chris Peikert and Amit Sahai and Adam Smith",
journal = "Journal of Cryptology",
Gentry, C, Groth, J, Ishai, Y, Peikert, C, Sahai, A & Smith, A 2015, 'Using Fully Homomorphic Hybrid Encryption to Minimize Non-interative Zero-Knowledge Proofs', Journal of Cryptology, vol. 28, no. 4, pp. 820-843. https://doi.org/10.1007/s00145-014-9184-y
Using Fully Homomorphic Hybrid Encryption to Minimize Non-interative Zero-Knowledge Proofs. / Gentry, Craig; Groth, Jens; Ishai, Yuval; Peikert, Chris; Sahai, Amit; Smith, Adam.
In: Journal of Cryptology, Vol. 28, No. 4, 30.10.2015, p. 820-843.
T1 - Using Fully Homomorphic Hybrid Encryption to Minimize Non-interative Zero-Knowledge Proofs
AU - Gentry, Craig
AU - Groth, Jens
AU - Ishai, Yuval
AU - Peikert, Chris
AU - Sahai, Amit
AU - Smith, Adam
N2 - A non-interactive zero-knowledge (NIZK) proof can be used to demonstrate the truth of a statement without revealing anything else. It has been shown under standard cryptographic assumptions that NIZK proofs of membership exist for all languages in NP. While there is evidence that such proofs cannot be much shorter than the corresponding membership witnesses, all known NIZK proofs for NP languages are considerably longer than the witnesses. Soon after Gentry's construction of fully homomorphic encryption, several groups independently contemplated the use of hybrid encryption to optimize the size of NIZK proofs and discussed this idea within the cryptographic community. This article formally explores this idea of using fully homomorphic hybrid encryption to optimize NIZK proofs and other related cryptographic primitives. We investigate the question of minimizing the communication overhead of NIZK proofs for NP and show that if fully homomorphic encryption exists then it is possible to get proofs that are roughly of the same size as the witnesses. Our technique consists in constructing a fully homomorphic hybrid encryption scheme with ciphertext size $$|m|+{\mathrm {poly}}(k)$$|m|+poly(k), where $$m$$m is the plaintext and $$k$$k is the security parameter. Encrypting the witness for an NP-statement allows us to evaluate the NP-relation in a communication-efficient manner. We apply this technique to both standard non-interactive zero-knowledge proofs and to universally composable non-interactive zero-knowledge proofs. The technique can also be applied outside the realm of non-interactive zero-knowledge proofs, for instance to get witness-size interactive zero-knowledge proofs in the plain model without any setup or to minimize the communication in secure computation protocols.
AB - A non-interactive zero-knowledge (NIZK) proof can be used to demonstrate the truth of a statement without revealing anything else. It has been shown under standard cryptographic assumptions that NIZK proofs of membership exist for all languages in NP. While there is evidence that such proofs cannot be much shorter than the corresponding membership witnesses, all known NIZK proofs for NP languages are considerably longer than the witnesses. Soon after Gentry's construction of fully homomorphic encryption, several groups independently contemplated the use of hybrid encryption to optimize the size of NIZK proofs and discussed this idea within the cryptographic community. This article formally explores this idea of using fully homomorphic hybrid encryption to optimize NIZK proofs and other related cryptographic primitives. We investigate the question of minimizing the communication overhead of NIZK proofs for NP and show that if fully homomorphic encryption exists then it is possible to get proofs that are roughly of the same size as the witnesses. Our technique consists in constructing a fully homomorphic hybrid encryption scheme with ciphertext size $$|m|+{\mathrm {poly}}(k)$$|m|+poly(k), where $$m$$m is the plaintext and $$k$$k is the security parameter. Encrypting the witness for an NP-statement allows us to evaluate the NP-relation in a communication-efficient manner. We apply this technique to both standard non-interactive zero-knowledge proofs and to universally composable non-interactive zero-knowledge proofs. The technique can also be applied outside the realm of non-interactive zero-knowledge proofs, for instance to get witness-size interactive zero-knowledge proofs in the plain model without any setup or to minimize the communication in secure computation protocols.
JO - Journal of Cryptology
JF - Journal of Cryptology
Gentry C, Groth J, Ishai Y, Peikert C, Sahai A, Smith A. Using Fully Homomorphic Hybrid Encryption to Minimize Non-interative Zero-Knowledge Proofs. Journal of Cryptology. 2015 Oct 30;28(4):820-843. https://doi.org/10.1007/s00145-014-9184-y | CommonCrawl |
\begin{document}
\title[Linking, framing and the Kontsevich integral]{Linking coefficients and the Kontsevich integral} \author[J.B. Meilhan ]{Jean-Baptiste Meilhan} \address{Univ. Grenoble Alpes, CNRS, IF, 38000 Grenoble, France} \email{[email protected]}
\keywords{Kontsevich integral, Jacobi diagrams, linking number, framing}
\begin{abstract} It is well known how the linking number and framing can be extracted from the degree $1$ part of the (framed) Kontsevich integral. This note gives a general formula expressing any product of powers of these two invariants as combination of coefficients in the Kontsevich integral. This allows in particular to express the sum of all coefficients of a given degree in terms of the linking coefficients. The proofs are purely combinatorial. \end{abstract}
\maketitle
\section{Introduction}
The \emph{Kontsevich integral} is a strong invariant of framed oriented knots and links, which dominates all finite type and quantum invariants, in the sense that any other factors through it. It takes values in a certain space of \emph{chord diagrams}, which are copies of the oriented unit circle, endowed with a number of chords, which are pairings of pairwise disjoint points on the circles; chord diagrams naturally come with a degree, which is given by the number of chords. Kontsevich defined this invariant in terms of iterated integrals, what can be seen as a far-reaching generalization of the Gauss integral for the linking number of two curves \cite{Kontsevich}. As a matter of fact, it is well-known that the linking number appears as the simplest coefficient in the Kontsevich integral. Specifically, given a framed link $L$, denoting by $C_L[D]$ the coefficient of a chord diagram $D$ in the Kontsevich integral of $L$, and denoting by $\ell_{ij}$ the linking number of the $i$th and $j$th components, we have \begin{equation}\tag{$1_1$}\label{eq:lk}
\ell_{ij}(L) = C_L[_i\dessin{0.5cm}{D12}\!\!\textrm{ }_j]. \end{equation} Denoting half the framing of the $i$th component of $L$ by $\ell_{ii}(L)$, it is also well-known that \begin{equation}\tag{$2_1$}\label{eq:fr}
\ell_{ii}(K)=\frac{1}{2} fr_i(L)= C_L[\dessin{0.5cm}{D11}\!\textrm{ }_i]. \end{equation} Hence the degree $1$ part of the Kontsevich integral of a link is fully characterized by the linking coefficients, \emph{i.e.} the coefficients of the linking matrix. The main purpose of this note is a formula generalizing these two elementary results.
Let $\mathcal{S}_m$ be the set of symmetric matrices of size $m$ with coefficients in $\mathbb{N}$. Given $S=(s_{ij})_{i,j}\in \mathcal{S}_m$, we define $\mathcal{D}_S(m)$ as the set of all possible chord diagrams on $m$ circles with exactly $s_{ij}$ chords of type $(i,j)$ for all $i,j$. Here, a \emph{type $(i,j)$ chord} is a chord whose endpoints sit on components $i$ and $j$; in particular, a type $(i,i)$ chord has both endpoints on the $i$th component.
\begin{thm}\label{thm:main} Let $L$ be an $m$-component framed oriented link in $S^3$ and let $S=(s_{ij})_{i,j}\in \mathcal{S}_m$. We have
$$ \ell_S(L):= \prod_{1\le i\le j\le m} \frac{1}{s_{ij}!} \ell_{ij}(L)^{s_{ij}} = \sum_{D \in \mathcal{D}_S(m)} C_L[D]. $$ \end{thm} This general formula has several noteworthy consequences.
On one hand, if $S$ has a single nonzero entry $n=s_{ij}$ with $i<j$, we obtain a generalization of (\ref{eq:lk}) to all powers of the linking number: \begin{equation}\tag{$1_n$}\label{eq:lkn}
\frac{1}{n!} \ell_{ij}(L)^n = \sum_{D \in \mathcal{M}^{ij}_n(m)} C_L[D], \end{equation} where $\mathcal{M}^{ij}_n(m)$ denotes the set of all possible degree $n$ chord diagrams on $m$ circles whose $n$ chords are of type $(i,j)$. \\ Similarly, if $S$ has a single nonzero entry $n=s_{ii}$ on the diagonal, we obtain that \begin{equation}\tag{$2_n$}\label{eq:frn}
\frac{1}{n!2^n} fr_i(L)^n = \sum_{D \in \mathcal{I}^i_n(m)} C_L[D], \end{equation} where $ \mathcal{I}^i_n(m)$ denotes the set of all degree $n$ chord diagrams on $m$ circles, such that all $n$ chords are on the $i$th circle.
On the other hand, the set $\mathcal{D}_{k}(m)$ of all degree $k$ chord diagrams on $m$ circles is partitioned into the sets $\mathcal{D}_S(m)$ for all matrices $S$ in $\mathcal{S}_m$ with $\vert S\vert = k$, where $\vert S\vert = \sum_{1\le i\le j\le m} s_{ij}$ is the \emph{degree} of $S$. Thus we have: \begin{equation}\tag{$3$}
\sum_{D \in \mathcal{D}_{k}(m)} C_L[D] = \sum_{S\in \mathcal{S}_m\textrm{ ; $\vert S\vert = k$}} \ell_S(L). \end{equation} \noindent This expresses the sum of \emph{all} coefficients of degree $k$ in the Kontsevich integral in terms of the linking coefficients.
More generally, Theorem \ref{thm:main} gives partial, but general (\emph{i.e.} in all degrees) identifications of the coefficients of the Kontsevich integral in terms of the coefficients of the linking matrix. A number of works investigate, in a similar way, how combinations of coefficients in the Kontsevich integral identify with classical invariants of knot theory, see for example \cite{Stanford,HM,Okamoto1997,Okamoto1998,C,CM}, although such results are often only given for low degree terms.
\begin{acknowledgments} This work is partially supported by the project AlMaRe (ANR-19-CE40-0001-01) of the ANR. The author wishes to thank Georges Abitbol and Benjamin Audoux for inspiring dicussions. \end{acknowledgments}
\section{The framed Kontsevich integral in a nutshell}\label{sec:K}
We briefly review the combinatorial definition of the framed Kontsevich integral, as given by Le and Murakami in \cite{LM}; see also \cite[\S 6]{Ohtsuki}.
A \emph{chord diagram} $D$ on the disjoint union $\dessin{0.25cm}{D10}^{\,m}$ of $m$ copies of the oriented circle, is a collection of copies of the unit interval, such that the set of all endpoints is embedded into $\dessin{0.25cm}{D10}^{\,m}$. We call \emph{chord} any of these copies of the interval. The \emph{degree} of $D$ is defined as its number of chords. \\ In figures, bold lines depict (portions of) $\dessin{0.25cm}{D10}^{\,m}$, and dashed lines are used for chords.
We denote by $\mathcal{A}(m)$ the $\mathbb{Q}$-vector space generated by all chord diagrams on $\dessin{0.25cm}{D10}^{\,m}$, modulo the \emph{4T relation}:
$$ \dessin{1cm}{4T}. $$
We now describe the source of the framed Kontsevich integral. A framed q-tangle is an oriented tangle, equipped with a framing and a parenthesization on both sets of boundary points. Any such tangle can be decomposed into copies of the q-tangles $I$, $X_{\pm}$, $C_{\pm}$ and $\Lambda_{\pm}$ shown in Figure \ref{fig:ehmanitselementarystuff}, along with those obtained by reversing the orientation on any component. \begin{figure}
\caption{The elementary q-tangles $I$, $X_{\pm}$, $C_{\pm}$ and $\Lambda_{\pm}$}
\label{fig:ehmanitselementarystuff}
\end{figure} Such a decomposition is not unique, but a complete set of relations is known, relating any two possible decompositions, see \cite[Thm.~6.5]{Ohtsuki}. The \emph{framed Kontsevich integral} $Z$ can thus be determined by specifying its values on the above q-tangles so that all relations are satisfied. This is done as follows.
We set $Z(I)$ to be the portion of diagram $\uparrow$ without chord.
For the positive and negative crossings $X_\pm$, we set
\begin{equation}\label{eq:Xk}
Z(X_\pm)= \sum_{k\ge 0} \frac{(\pm 1)^k}{2^k k!} X_k\textrm{, where }X_k = \dessin{0.7cm}{Xk}.
\end{equation}
Next, set $Z(C_\pm)=\sqrt{\nu}$, where $\nu\in \mathcal{A}(1)$ is the Kontsevich integral of the $0$-framed unknot $U_0$, which was explicitly computed in \cite{BNGT} as follows: \begin{equation}\label{eq:nu}
\nu = \chi\Big(\textrm{exp}_\sqcup\big(\sum_{n\ge 1} b_{2n} W_{2n}\big) \Big), \end{equation} where
$W_{2n}$ is a \emph{wheel}, that is a unitrivalent diagram of the form $\begin{array}{c}\includegraphics[scale=0.7]{wheel.pdf}\end{array}$
with $2n$ univalent vertices,
$b_{2n}$ is the coefficient of $x^{2n}$ in the Taylor expansion of $\frac{1}{2}\textrm{ln}\frac{\textrm{sinh(x/2)}}{x/2}$,
$\textrm{exp}_\sqcup$ stands for the exponential with respect to the disjoint union of diagrams,
and the map $\chi$ takes the sum over all possible ways of attaching the univalent vertices of a (disjoint union of) wheel(s) to $\dessin{0.25cm}{D10}$,
then applies recursively the STU relation below to produce a combination of chord diagrams:
$$ \dessin{1cm}{stu}. $$
Lastly, we set $Z(\Lambda_\pm)=\Phi^{\pm 1}$, where $\Phi$ is a \emph{Drinfeld associator}; we do not further discuss here this important ingredient of the construction, as it will play no role in our argument. The interested reader is referred to \cite[App. D]{Ohtsuki}.
\section{Linking coefficients and the Kontsevich integral}
The purpose of this section is to prove Theorem \ref{thm:main}. This will be done in Subsection \ref{sec:proof}, after setting some notation and preliminary results in the next two subsections.
\subsection{Some notation}
Let $A$ be an element of $\mathcal{A}(m)$. Let $D$ be a chord diagram on $m$ circles. We denote by $C[D](A)$ the coefficient of $D$ in $A$. In particular, we set $$C_L[D] := C[D](Z(L)),$$ for a framed oriented link $L$, that is, $C_L[D]$ denotes \lq the coefficient\rq\, of diagram $D$ in the Kontsevich integral of $L$. This quantity is of course in general not well-defined since an element of $\mathcal{A}(m)$ consists of diagrams subject to the $4T$ relation, but taking an appropriate combination of such coefficients shall yield a link invariant, see Claim
\ref{rem:inv} below.
We denote by $C[D]$ the assignment $A\mapsto C[D](A)$ and, abusing notation, we still denote by $C[D]$ the precomposition $L\mapsto C_L[D]$ with the Kontsevich integral.
Let $S=(s_{ij})_{i,j}\in \mathcal{S}_m$ be a symmetric matrix of size $m$ with entries in $\mathbb{N}$.
We set $$ \mathcal{L}_S := \sum_{D \in \mathcal{D}_S(m)} C[D], $$ where $\mathcal{D}_S(m)$ is the set of all chord diagrams on $m$ circles with exactly $s_{ij}$ chords of type $(i,j)$ for all $i\le j$, as defined in the introduction. \begin{claim}\label{rem:inv} This formula yields a well-defined map on $\mathcal{A}(m)$, and in particular defines an $m$-component link invariant. \end{claim}
\noindent This is straighforwardly checked using the following general invariance criterion: given a collection $\mathcal{D}$ of chord diagrams, the assignement $X := \sum_{D \in \mathcal{D}} C[D]$ defines a link invariant if and only if $X$ vanishes on any linear combination of chord diagrams arising from a $4T$ relation. See for example \cite{C}.
We also recall from the introduction the link invariant associated with the symmetric matrix $S$, $$ \ell_S= \prod_{1\le i\le j\le m} \frac{1}{s_{ij} !} \ell_{ij}^{s_{ij}},$$ where $\ell_{ij}$ is the linking number between components $i$ and $j$ if $i\neq j$, and $\ell_{ii}:=\frac{1}{2} fr_i$ is half the framing of the $i$th component.
\subsection{Crossing change formula for the invariant $\mathcal{L}_S$} \label{sec:var}
Let us pick some indices $a,b$ in $\{1,\cdots,m\}$ (possibly with $a=b$). Consider two $m$-component links $L_+$ and $L_-$, that are identical away from a small $3$-ball, where they look as follows: $$ L_+ = \dessin{0.7cm}{over}\quad\textrm{and}\quad L_- = \dessin{0.7cm}{under}. $$ \noindent We stress that this crossing change may involve two strands of either the same ($a=b$) or different ($a\neq b$) components.
Now let $S=(s_{ij})_{i,j}\in \mathcal{S}_m$. By the mere definition of the Kontsevich integral at a crossing, we have
$$\mathcal{L}_S(L_+) - \mathcal{L}_S(L_-) = \sum_{j\ge 0} \frac{1}{(2j+1)!2^{2j}} \mathcal{L}_S(D_{2j+1}), $$ where $D_k\in \mathcal{A}(m)$ is obtained from the Kontsevich integral of $L_\pm$ by replacing the local contribution of the crossing involved in the crossing change, as given in (\ref{eq:Xk}), by the local diagram $\dessin{0.7cm}{Xk}$ with exactly $k$ parallel chords.
Set $s:=s_{ab}=s_{ba}$, the entry of the matrix $S$ corresponding to our crossing change. In order to slightly simplify our notation, for any $p$ such that $s\ge p\ge 0$ we denote by $\mathcal{L}_{p}$ the invariant $\mathcal{L}_{S_{p}}$, where $S_{p}$ is the matrix $S$ with the coefficient $s$ replaced by $p$; in particular we have $\mathcal{L}_{s}=\mathcal{L}_{S}$.
Clearly, we have $\mathcal{L}_s(D_k)=0$ for $k>s$, hence the variation formula \begin{equation}\label{eq:F_n} \mathcal{L}_S(L_+) - \mathcal{L}_S(L_-) = \sum_{j=0}^{\lfloor\frac{s+1}{2}\rfloor} \frac{1}{(2j+1)!2^{2j}} \mathcal{L}_s(D_{2j+1}). \end{equation}
When $k\le s$, we have the following.
\begin{claim}\label{lem:key}
For all $k$ such that $s\ge k\ge 0$, we have $\mathcal{L}_s(D_k)=\mathcal{L}_{s-k}(D_{0})$. \end{claim}
\begin{proof} A degree $n$ chord diagram on $m$ circles that contributes to $\mathcal{L}_s(D_k)$ necessarily contains $k$ parallel chords of type $(a,b)$, as imposed by the definition of $D_k$. The set of all diagrams contributing to $\mathcal{L}_s(D_k)$ is thus obtained by adding $n-k$ chords, with exactly $s-k$ additional chords of type $(a,b)$, in all possible ways. But since these additional chords do not arise from the crossing change, they are attached outside a disk containing the $k$ parallel chords of type $(a,b)$. This is thus equivalent to taking the contribution of \emph{all} chord diagrams in $\mathcal{D}_{S_{s-k}}$. \end{proof}
By Claim \ref{lem:key}, computing the variation $\mathcal{L}_S(L_+) - \mathcal{L}_S(L_-)$ reduces to computing $\mathcal{L}_{k}(D_0)$ for all $k$. This is done in the next lemma. \begin{lemma}\label{lem:key2}
We have
$\mathcal{L}_k(D_0)=\sum_{p=0}^{k} \frac{(-1)^p}{p!2^p} \mathcal{L}_{k-p}(L_+)$
for all $k$; $s\ge k\ge 0$. \end{lemma} \begin{proof} The proof is by induction on $k$. The formula for $k=0$ is clear: the invariant $\mathcal{L}_{0}$ vanishes on any diagram with a chord of type $(a,b)$, so that $\mathcal{L}_{0}(D_k)=0$ for all $k\ge 1$, and by definition of the Kontsevich integral at a positive crossing (\ref{eq:Xk}), we thus have
$\mathcal{L}_{0}(L_+)=\mathcal{L}_{0}(D_0)$. For the inductive step, again by definition of the Kontsevich integral at a positive crossing, we have \begin{eqnarray*} \mathcal{L}_{k}(D_0) & = & \mathcal{L}_k(L_+)-\sum_{j=1}^{k} \frac{1}{j! 2^j} \mathcal{L}_{k}(D_j)\\
& = & \mathcal{L}_k(L_+)-\sum_{j=1}^{k} \frac{1}{j! 2^j} \mathcal{L}_{k-j}(D_0)\\
& = & \mathcal{L}_k(L_+)-\sum_{j=1}^{k} \frac{1}{j! 2^j} \sum_{i=0}^{k-j} \frac{(-1)^i}{i!2^i} \mathcal{L}_{k-j-i}(L_+), \end{eqnarray*} where the second equality follows from Claim \ref{lem:key}, while the third equality uses the induction hypothesis. For each $p$ such that $0\le p\le k$, the coefficient of $\mathcal{L}_{k-p}(L_+)$ in the above double sum is then given by \begin{eqnarray*}
-\sum_{j=1}^p \frac{1}{j! 2^j} \times \frac{(-1)^{p-j}}{(p-j)!2^{p-j}}
& = & \frac{-1}{p! 2^p} \sum_{j=1}^p (-1)^{p-j} \binom{p}{j} \\
& = & \frac{-1}{p! 2^p} \Big( \underbrace{\sum_{j=0}^p (-1)^{p-j} \binom{p}{j}}_{=0} -(-1)^p\Big)\\
& = & \frac{(-1)^p}{p! 2^p}. \end{eqnarray*} This concludes the proof. \end{proof}
\subsection{Proof of Theorem \ref{thm:main}}\label{sec:proof}
We proceed by induction on the degree $\vert S\vert=\sum_{i\le j} s_{ij}$ of $S$. The base case $n=1$ corresponds to the case where $S$ has a single nonzero entry $s_{ij}=1$ ($i\le j$), and is given by the well-known formulas (\ref{eq:lk}) and (\ref{eq:fr}) recalled in the introduction.
Now, assume that the formula holds for all matrices of $\mathcal{S}_m$ of degree $<k$. Let $S\in \mathcal{S}_m$ be a degree $k$ matrix. Choose some indices $a,b$ such that, in the matrix $S$, the entry $s=s_{ab}$ is nonzero (possibly $a=b$). Let $L_+$ and $L_-$ be two $m$-component links that differ by a crossing change between components $a$ and $b$, as in Subsection \ref{sec:var}. Combining (\ref{eq:F_n}) with Claim \ref{lem:key} and Lemma \ref{lem:key2}, we obtain $$ \mathcal{L}_S(L_+) - \mathcal{L}_S(L_-) = \sum_{j= 0}^{\lfloor\frac{s+1}{2}\rfloor} \frac{1}{(2j+1)!2^{2j}} \sum_{k=0}^{s-2j-1} \frac{(-1)^k}{k!2^k} \mathcal{L}_{s-2j-k-1}(L_+).$$ By the induction hypothesis, we have $$\mathcal{L}_{s-2j-k-1}(L_+)=\frac{1}{(s-2j-k-1)!} \ell_{ab}^{s-2j-k-1}\prod_{\{i,j\}\neq \{a,b\}} \frac{1}{s_{ij}!} \ell_{ij}^{s_{ij}}.$$ Setting $\ell_0:=\prod_{\{i,j\}\neq \{a,b\}} \frac{1}{s_{ij}!} \ell_{ij}^{s_{ij}}$, we thus have $$ \mathcal{L}_S(L_+) - \mathcal{L}_S(L_-) =
\sum_{j= 0}^{\lfloor\frac{s+1}{2}\rfloor} \frac{\ell_0}{(2j+1)!2^{2j}}
\sum_{k=0}^{s-2j-1} \frac{(-1)^k}{k!(s-2j-k-1)!2^k}\ell_{ab}(L_+)^{s-2j-k-1}. $$
The coefficient of $\ell_{ab}(L_+)^{s-i}$ in the above formula is given by \begin{eqnarray*} \sum_{j= 0}^{\lfloor\frac{i+1}{2}\rfloor} \frac{\ell_0}{(2j+1)!2^{2j}}.\frac{(-1)^{i-2j-1}}{(i-2j-1)!(s-i)!2^{i-2j-1}}
& = & \frac{(-1)^{i+1}\ell_0}{2^{i-1}(s-i)!}\sum_{j= 0}^{\lfloor\frac{i+1}{2}\rfloor} \frac{1}{i!}\binom{i}{2j+1}\\
& = & \frac{(-1)^{i+1}\ell_0}{2^{i-1}i!(s-i)!}\underbrace{\sum_{j= 0}^{\lfloor\frac{i+1}{2}\rfloor} \binom{i-1}{2j} + \binom{i-1}{2j+1}}_{=2^{i-1}}. \end{eqnarray*} This shows that $$ \mathcal{L}_S(L_+) - \mathcal{L}_S(L_-) = \ell_0\sum_{i=1}^n \frac{(-1)^{i+1}}{i!(s-i)!} \ell_{ab}^{s-i}. $$
Now, this formula coincides with the variation of the linking invariant $\ell_S$: $$ \ell_S(L_+) - \ell_S(L_-) = \ell_0\sum_{i=1}^n \frac{(-1)^{i+1}}{i!(s-i)!} \ell_{ab}^{s-i}. $$ This is easily verified using the binomial formula, noting that $\ell_{ab}(L_-)=\ell_{ab}(L_+)-1$ (in particular, if $a=b$, we indeed have $fr_a(L_-)=fr_a(L_+)-2$).
Hence we showed that the invariants $\mathcal{L}_S$ and $\ell_S$ have the same variation formula under a crossing change. By a sequence of such operations, any $m$-component link can be deformed into a split union of unknots, each with framing $0$ or $1$ depending on the parity of the framing of the component: it remains to check that both invariants take the same value on such links.
Denote by $U_0$ and $U_1$ the unknot with framing $0$ or $1$, respectively, and let $L_0$ be a split union of $m$ unknots, such that the $i$th component is a copy of $U_{\varepsilon_i}$, $\varepsilon_i\in\{0,1\}$. The framed Kontsevich integral of $L_0$ can be written in $\mathcal{A}(m)$ as a disjoint union $Z(L_0)=\sqcup_i Z(U_{\varepsilon_i})$; in particular, we may assume that it only contains type $(i,i)$ chords for $1\le i\le m$.
It follows that, if the matrix $S$ contains a nonzero coefficient away from the diagonal, then both invariants $\mathcal{L}_S$ and $\ell_S$ clearly vanish, and the proof is complete.
In the case where $S$ is a diagonal matrix, the invariant $\mathcal{L}_S$ of $L_0$ splits as $$ \mathcal{L}_S(L_0) = \prod_{i=1}^m \sum_{D \in \mathcal{S}_{s_{ii}}} C_{U_{\varepsilon_i}}[D],$$ where $\mathcal{S}_k$ denotes the set of all possible chord diagrams on $\dessin{0.25cm}{D10}$ with $k$ chords. The proof then follows readily from the following claim. \begin{claim}\label{claimfinal} For all integer $k$, we have \[\textrm{$\sum_{D \in \mathcal{S}_{k}} C_{U_{0}}[D]=0$\,\,\,\,\,\, and\,\,\,\,\,\, $\sum_{D \in \mathcal{S}_{k}} C_{U_{1}}[D]=\frac{1}{k!2^k}$.}\] \end{claim} \noindent Indeed, these are the values taken by the invariant $\dfrac{1}{k!}(\frac{1}{2}fr)^k$ on both $U_0$ and $U_1$, thus showing that the invariants $\mathcal{L}_S$ and $\ell_S$ do also coincide on $L_0$ when $S$ is diagonal. Hence it only remains to prove Claim \ref{claimfinal} to complete the proof.
\begin{proof}[Proof of Claim \ref{claimfinal}] For simplicity, set $\mathcal{F}_k(K):= \sum_{D \in \mathcal{S}_k} C_K[D]$ for a knot $K$.
Let us first compute $\mathcal{F}_k(U_0)$. We recalled in (\ref{eq:nu}) the computation of $Z(U_0)=\nu$ of \cite{BNGT}. Now, given a (disjoint union of) wheel(s) with $k$ univalent vertices attached to $\dessin{0.25cm}{D10}$ in some way, applying recursively the STU relation to get a combination of chord diagrams, as prescribed by $\chi$, produces an alternate sum with $2^k$ terms, where the coefficients add up to zero. This simple observation shows that $\mathcal{F}_k(U_0)=0$.
We now consider $\mathcal{F}_k(U_1)$. It is well-known that $\sqrt{\nu}$ commutes with any chord endpoint in a chord diagram (this is a consequence of the 4T relation). This can be used to check that $$ Z(U_1) = Z(U_0)\sharp \left(\textrm{exp}_\sharp \frac{1}{2}\dessin{0.5cm}{D11}\right) = Z(U_0)\sharp \left( \sum_{k\ge 0} \frac{1}{k!2^k} D_k \right), $$ where $D_k$ denotes the chord diagram on $\dessin{0.25cm}{D10}$ with $k$ parallel chords and where $\sharp$ is the connected sum of chord diagrams. From the above computation of $\mathcal{F}_k(U_0)$, we obtain that the only term contributing to $\mathcal{F}_k(U_1)$ is the degree $k$ diagram $D_k$ arising from $\textrm{exp}_\sharp \frac{1}{2}\dessin{0.5cm}{D11}$. Summarizing, we obtain that $\mathcal{F}_k(U_1)=\frac{1}{k!2^k}$. \end{proof}
\end{document} | arXiv |
\begin{document}
\title[Incompressible limit of MHD equations] {Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions}
\author{Song Jiang} \address{LCP, Institute of Applied Physics and Computational Mathematics, P.O.
Box 8009, Beijing 100088, P.R. China}
\email{[email protected]}
\author{Qiangchang Ju} \address{Institute of Applied Physics and Computational Mathematics, P.O.
Box 8009-28, Beijing 100088, P.R. China}
\email{qiangchang\[email protected]}
\author [Fucai Li]{Fucai Li$^*$} \thanks{$^*$Corresponding author} \address{Department of Mathematics, Nanjing University, Nanjing
210093, P.R. China}
\email{[email protected]}
\keywords{Compressible MHD equations, incompressible MHD equations, ideal incompressible MHD equations, low Mach number limit, zero viscosity limit}
\subjclass[2000]{76W05, 35B40}
\begin{abstract} This paper is concerned with the incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. It is rigorously shown that the weak solutions of the compressible magnetohydrodynamic equations converge to the strong solution of
the viscous or inviscid incompressible magnetohydrodynamic
equations as long as the latter exists both for the well-prepared initial data and general
initial data. Furthermore, the convergence rates are also obtained
in the case of the well-prepared initial data. \end{abstract}
\maketitle
\section{Introduction}
Magnetohydrodynamics (MHD) studies the dynamics of compressible quasineutrally ionized fluids under the influence of electromagnetic fields. The applications of magnetohydrodynamics cover a very wide range of physical objects, from liquid metals to cosmic plasmas. The compressible viscous MHD equations in the isentropic case take the form (see, e.g., \cite{KL,LL,PD}) \begin{align} &\partial_t \tilde{\rho} +{\rm div }(\tilde{\rho}\tilde{{\bf u}})=0, \label{a1a} \\ &\partial_t (\tilde{\rho}\tilde{{\bf u}})+{\rm div }(\tilde{\rho}\tilde{{\bf u}}\otimes\tilde{{\bf u}})+\nabla \tilde{P} =({\rm curl\, }\tilde{ {\bf H}})\times \tilde{{\bf H}}+\tilde{\mu}\Delta\tilde{{\bf u}} +(\tilde{\mu}+\tilde{\lambda})\nabla({\rm div }\tilde{{\bf u}}), \label{a1b} \\ &\partial_t \tilde{{\bf H}} -{\rm curl\, }(\tilde{{\bf u}}\times\tilde{{\bf H}})=-{\rm curl\, }(\tilde{\nu}\, {\rm curl\, }\tilde{{\bf H}}),\quad {\rm div }\tilde{{\bf H}}=0.\label{a1c} \end{align} Here $x\in \mathbb{T}^d$, a torus in $\mathbb{R}^d$, $d=2 $ or $3$, $ t>0$, the unknowns $\tilde{\rho}$ denotes the density, $\tilde{{\bf u}}=(\tilde{u}_1,\dots,\tilde{u}_d)\in
\mathbb{R}^d$ the velocity, and $\tilde{{\bf H}}=(\tilde{H}_1,\dots, \tilde{H}_d)\in \mathbb{R}^d$ the magnetic field, respectively. The constants $\tilde{\mu}$ and $\tilde{\lambda}$ are the shear and bulk viscosity coefficients of the flow, respectively, satisfying $\tilde{\mu}>0$ and
$2\tilde{\mu}+d\tilde{\lambda}>0$; the constant $\tilde{\nu}>0$ is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field. $\tilde{P}(\tilde{\rho})$ is the pressure-density function and here we consider the case \begin{equation} \tilde{P}(\tilde{\rho})=a \tilde{\rho}^\gamma, \label{aad} \end{equation} where $a>0 $ and $ \gamma >1$ are constants.
The well-posedness of the Cauchy problem and initial boundary value problems for (\ref{a1a})-(\ref{a1c}) has been investigated recently. The global existence of weak solutions to the compressible MHD equations with general initial data was obtained by Hu and Wang~\cite{HW1,HW2} (also see \cite{FY2} on ``variational solutions'').
From the physical point of view, one can formally derive the incompressible models from the compressible ones when the Mach number goes to zero and the density becomes almost constant. Based on this observation, Hu and Wang \cite{HW3} proved the convergence of the weak solutions of the compressible MHD equations \eqref{a1a}-\eqref{a1c}
to a weak solution of the viscous incompressible MHD equations. Jiang, Ju and Li \cite{JJL} obtained the convergence towards the strong solution of the ideal incompressible MHD equations in the whole space by using the dispersion property of the wave equation if both the shear viscosity and the magnetic diffusion coefficients go to zero.
In this paper, we shall extend the results on the Cauchy problem in \cite{JJL} to the periodic case. First, we consider the well-prepared initial data for which the oscillations will never appear. We will rigorously show the weak solutions of the compressible MHD equations converge to the strong solution of the ideal incompressible MHD equations in the periodic domain if both the shear viscosity and the magnetic diffusion coefficients go to zero, as well as to the strong solution of the viscous incompressible MHD equations. Furthermore, we shall also give the rates of convergence
which are not obtained in \cite{HW3, JJL}. Secondly, we consider the case of general initial data. For this case the oscillations (acoustic waves) will appear. Comparing with \cite{JJL} where the Cauchy problem was dealt with, the acoustic waves in the current situation will lose the dispersion property and will interact each other. Thus, here we have to impose more regular conditions than $L^2$ on the initial data to control the oscillating parts. In addition, we have to assume that the Sobolev norm of the oscillating parts is comparable to the magnetic diffusion coefficient in order to deal with the general initial data. We will rigorously prove the convergence of the weak solutions of the compressible MHD equations to the strong solution of the incompressible MHD equations, as well as to the strong solution of the partial viscous incompressible MHD equations.
To begin our argument, we first give some formal analysis. Formally, by utilizing the identity $$
\nabla(|\tilde{{\bf H}}|^2)=2(\tilde{{\bf H}}\cdot \nabla )\tilde{{\bf H}}+2\tilde{{\bf H}}\times {\rm curl\, } \tilde{{\bf H}}, $$ we can rewrite the momentum equation \eqref{a1b}
as \begin{equation}\label{a2a} \partial_t (\tilde{\rho}\tilde{{\bf u}})+{\rm div }(\tilde{\rho}\tilde{{\bf u}}\otimes\tilde{{\bf u}}) +\nabla \tilde{P} =(\tilde{{\bf H}}\cdot \nabla
)\tilde{{\bf H}}-\frac12\nabla(|\tilde{{\bf H}}|^2)+\tilde{\mu}\Delta\tilde{{\bf u}}
+(\tilde{\mu}+\tilde{\lambda})\nabla({\rm div }\tilde{{\bf u}}). \end{equation} By the identities $$ {\rm curl\, }{\rm curl\, } \tilde{{\bf H}}= \nabla\,{\rm div } \tilde{{\bf H}}-\Delta \tilde{{\bf H}}$$ and \begin{equation*}
{\rm curl\, }( \tilde{ {\bf u}}\times\tilde{{\bf H}}) =
\tilde{{\bf u}} ({\rm div } \tilde{{\bf H}}) - \tilde{{\bf H}} ({\rm div }\tilde{{\bf u}})
+ (\tilde{{\bf H}}\cdot \nabla)\tilde{{\bf u}} - (\tilde{{\bf u}}\cdot \nabla)\tilde{{\bf H}}, \end{equation*} together with the constraint ${\rm div }\tilde{{\bf H}}=0$, the magnetic field equation \eqref{a1c} can be expressed as \begin{equation}\label{a2b}
\partial_t\tilde{{\bf H}} +
({\rm div } \tilde{{\bf u}})\tilde{ {\bf H}}+ (\tilde{{\bf u}} \cdot \nabla)\tilde{{\bf H}}
- (\tilde{{\bf H}}\cdot \nabla)\tilde{{\bf u}}= \tilde{\nu } \Delta \tilde{{\bf H}}. \end{equation}
We introduce the scaling \begin{equation*}
\tilde{\rho} (x, t )=\rho^\epsilon (x, \epsilon t), \quad
\tilde{{\bf u}} (x, t )=\epsilon {\bf u}^\epsilon(x,\epsilon t), \quad
\tilde{{\bf H}} (x, t )=\epsilon {\bf H}^\epsilon(x,\epsilon t)
\end{equation*} and assume that the viscosity coefficients $\tilde{\mu}$, $\tilde{\xi}$, and $\tilde{\nu}$
are small constants and scaled like \begin{equation}\label{pa}
\tilde{\mu}=\epsilon \mu^\epsilon, \quad \tilde{\lambda}=\epsilon \lambda^\epsilon,
\quad \tilde{\nu}=\epsilon \nu^\epsilon, \end{equation} where $\epsilon\in (0,1)$ is a small parameter and the normalized coefficients $\mu^\epsilon$, $\lambda^\epsilon$, and $\nu^\epsilon$ satisfy $\mu^\epsilon>0$, $2\mu^\epsilon+d\lambda^\epsilon>0$, and $\nu^\epsilon>0$.
With the preceding scalings and the pressure function \eqref{aad}, the compressible MHD equations \eqref{a1a}, \eqref{a2a}, and \eqref{a2b} take the form \begin{align}
& \partial_t {\rho}^\epsilon
+\text{div}({\rho}^\epsilon{{\bf u}}^\epsilon)=0, \label{a2i}\\ & \partial_t
({\rho}^\epsilon{{\bf u}}^\epsilon)+\text{div}({\rho}^\epsilon{{\bf u}}^\epsilon\otimes
{{\bf u}}^\epsilon)+\frac{a \nabla (\rho^\epsilon)^\gamma
}{\epsilon^2}
= ({{\bf H}}^\epsilon\cdot \nabla
){{\bf H}}^\epsilon-\frac12\nabla(|{{\bf H}}^\epsilon|^2)\nonumber\\
&\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad
+{\mu}^\epsilon\Delta{{\bf u}}^\epsilon+({\mu}^\epsilon+{\lambda}^\epsilon)
\nabla({\rm div } {{\bf u}}^\epsilon), \label{a2j}\\
& \partial_t {{\bf H}}^\epsilon +
({\rm div } {{\bf u}}^\epsilon) { {\bf H}}^\epsilon+ ( {{\bf u}}^\epsilon \cdot \nabla) {{\bf H}}^\epsilon
- ( {{\bf H}}^\epsilon\cdot \nabla) {{\bf u}}^\epsilon= {\nu }^\epsilon
\Delta {{\bf H}}^\epsilon,\ \ {\rm div } {{\bf H}}^\epsilon=0.\label{a2k}
\end{align} Moreover, by replacing $\epsilon$ by $\sqrt{a\gamma}\;\epsilon$, we can always assume $a=1/\gamma$.
Now, we investigate the incompressible limit of the compressible MHD equations \eqref{a2i}-\eqref{a2k}. Formally let $\epsilon \rightarrow 0$ in the equations \eqref{a2i}-\eqref{a2k}, then we obtain from the momentum equation \eqref{a2j} that $\rho^\epsilon $ converges to some function $\bar \rho(t)\geq 0$. If we further assume that the initial datum $\rho^\epsilon_0$
is of order $1+O(\epsilon)$ (this can be guaranteed by the initial energy bound \eqref{bd} below), then we can expect that $\bar\rho=1$. Thus, the continuity equation \eqref{a2i} gives ${\rm div}\,{{\bf u}}=0$. Furthermore, using the assumption \begin{equation}\label{pam}
\mu^\epsilon\rightarrow 0, \quad \nu^\epsilon \rightarrow 0 \quad \text{as}\quad \epsilon \rightarrow 0, \end{equation}
we obtain the following ideal incompressible MHD equations (suppose that the limits ${{\bf u}}^\epsilon\rightarrow {\bf u}$ and ${{\bf H}}^\epsilon\rightarrow {\bf H}$ exist)
\begin{align} & \partial_t {\bf u}+({\bf u}\cdot \nabla){\bf u} -({{\bf H}} \cdot \nabla
){{\bf H}}+\nabla p +\frac12\nabla(|{{\bf H}} |^2) =0, \label{a2l}\\ & \partial_t {{\bf H}} + ( {{\bf u}} \cdot \nabla) {{\bf H}}
- ( {{\bf H}} \cdot \nabla) {{\bf u}} = 0, \label{a2m}\\
&{\rm div } {\bf u}=0, \quad {\rm div } {\bf H}=0.\label{a2n}
\end{align}
In Section 3 we shall rigorously prove that the weak solutions of the compressible MHD equations \eqref{a2i}-\eqref{a2k}
converge to, as $\epsilon\to 0$, the strong solution of the ideal incompressible MHD equations \eqref{a2l}-\eqref{a2n} for the well-prepared initial data in the time interval where the strong solution of \eqref{a2l}-\eqref{a2n} exists. Furthermore, the convergence rates are obtained. To show these results, since the viscosity coefficients
go to zero, we lose the spatial compactness property of the velocity and the magnetic field, and the arguments in \cite{HW3} do not work here. To overcome such difficulty, we shall carefully exploit the energy arguments.
Next, if we assume that the shear and the bulk viscosity coefficients and
the magnetic diffusivity coefficient satisfy \begin{equation}\label{pamb}
\mu^\epsilon\rightarrow \mu>0, \quad \lambda^\epsilon \rightarrow \lambda, \quad\nu^\epsilon \rightarrow \nu>0 \quad \text{as}\quad \epsilon \rightarrow 0, \end{equation}
then the compressible MHD equations \eqref{a2l}-\eqref{a2n} formally converges to the incompressible MHD equations (suppose that the limits ${{\bf u}}^\epsilon\rightarrow {\bf u}$ and ${{\bf H}}^\epsilon\rightarrow {\bf H}$ exist)
\begin{align} & \partial_t {\bf u}+({\bf u}\cdot \nabla){\bf u} -\mu\Delta {\bf u}+\nabla p-({{\bf H}} \cdot \nabla
){{\bf H}} +\frac12\nabla(|{{\bf H}} |^2) =0, \label{a2ll}\\ & \partial_t {{\bf H}} + ( {{\bf u}} \cdot \nabla) {{\bf H}}
- ( {{\bf H}} \cdot \nabla) {{\bf u}} - \nu\Delta {\bf H}= 0, \label{a2mm}\\
&{\rm div }\,{\bf u}=0, \quad {\rm div } {\bf H}=0.\label{a2nn}
\end{align} In Sections 3 and 4 we shall prove the convergence to the strong solution of the incompressible viscous MHD equations \eqref{a2ll}-\eqref{a2nn} for both the well-prepared and the general initial data. Furthermore, the convergence rates are also obtained for the well-prepared initial data. For the general initial data, we shall also show the convergence to the strong solution of the partial viscous incompressible MHD equations (that is, $\mu=0$ and $\nu>0$ in \eqref{a2ll}-\eqref{a2nn}).
There are a lot of studies on the compressible MHD equations in the literature. Besides the aforementioned results, the interested reader can see \cite{K,LY} on the global smooth solutions with small initial data and see \cite{VK,FY1} on the local strong solution with general initial data. We also mention the work \cite{ZJX} where a MHD model describing the screw pinch problem in plasma physics was discussed and the global existence of weak solutions with symmetry was obtained.
Before ending the introduction, we give the notation used throughout the current
paper. We denote the space $L^q_2(\mathbb{T}^d)$ by $$
L^q_2(\mathbb{T}^d)= \{f\in L_{loc}(\mathbb{T}^d): f 1_{\{|f|\geq 1/2\}}\in L^q, f 1_{\{|f|\leq 1/2\}}\in L^2\} . $$ We use the letters $C$ and $C_T$ to denote various positive constants independent of $\epsilon$, but $C_T$ may depend on $T$. For convenience, we denote by $H^r \equiv H^r (\mathbb{T}^d)$ ($r\in\mathbb{R}$) the standard Sobolev space. For any vector field $\mathbf{v}$, we denote by $P\mathbf{v}$ and $Q\mathbf{v}$ the divergence-free part and the gradient part of $\mathbf{v}$, respectively. Namely, $Q\mathbf{v}=\nabla \Delta^{-1}(\text{div}\mathbf{v})$ and $P\mathbf{v}=\mathbf{v}-Q\mathbf{v}$.
We state our main results in Section 2 and present the proofs for the well-prepared case
in Section 3 and the ill-prepared case in Section 4, respectively.
\section{main results}
We first recall the local existence of strong solutions to the ideal incompressible MHD equations \eqref{a2l}-\eqref{a2n} in the torus $\mathbb{T}^d$. The proof can be found in \cite{DL72,ST}.
\begin{prop}[\!\cite{DL72,ST}]\label{imhd}
Assume that the initial data $({\bf u} ,{\bf H} )|_{t=0}=({\bf u}_0,{\bf H}_0)$ satisfy ${\bf u}_0,{\bf H}_0\in {H}^s$ ($s>d/2+1$), and ${\rm div }\,{\bf u}_0=0$, ${\rm div }{\bf H}_0=0$. Then, there exist a
$T^*\in (0,\infty)$ and a unique solution
$({\bf u},{\bf H})\in L^{\infty}([0,T^*),{H}^s)$ to the ideal incompressible MHD equations \eqref{a2l}-\eqref{a2n} satisfying, for any $0<T<T^*$, ${\rm div }\,{\bf u} =0$, ${\rm div }{\bf H} =0$, and \begin{equation}\label{ba}
\sup_{0\le t\le T}\!\big\{||({\bf u},{\bf H})(t)||_{H^s}
+||(\partial_t{\bf u},\partial_t{\bf H})(t)||_{H^{s-1}}+ ||\nabla p(t)||_{H^{s-1}}\big\} \le C_T. \end{equation} \end{prop} \begin{rem} The local existence of strong solutions to the incompressible viscous MHD equations \eqref{a2ll}-\eqref{a2nn} was also established in \cite{DL72,ST}. \end{rem}
We prescribe the initial conditions to the compressible MHD equations \eqref{a2i}-\eqref{a2k} as \begin{equation}\label{bb}
\rho^\epsilon|_{t=0}=\rho^\epsilon_0(x), \quad \rho^\epsilon
{\bf u}^\epsilon|_{t=0}=\rho^\epsilon_0(x){\bf u}^\epsilon_0(x)\equiv \mathbf{m}^\epsilon_0(x),
\quad {\bf H}^\epsilon|_{t=0}={\bf H}^\epsilon_0(x),
\end{equation} and assume that \begin{equation}\label{bc}
\rho_0^\epsilon \geq 0,\, \rho^\epsilon_0\in L^\gamma,\,
\rho^\epsilon_0|{\bf u}^\epsilon_0|^2\in L^1,\,{\bf H}^\epsilon_0\in L^2,\, {\rm div }{\bf H}^\epsilon_0=0,\,
\mathbf{m}^\epsilon_0=0\;\;\text{ for a.e. }\;\; {\rho^\epsilon_0=0}. \end{equation} Moreover, we assume that the initial data also satisfy the following uniform bound
\begin{equation}\label{bd}
\int_{\mathbb{T}^d}
\Big[\frac12\rho^\epsilon_0|{\bf u}^\epsilon_0|^2+\frac12|{\bf H}^\epsilon_0|^2
+\frac{a}{\epsilon^2{(\gamma-1)}}\big((\rho^\epsilon_0)^\gamma -1-\gamma
(\rho^\epsilon_0-1)\big)
\Big] dx \leq C.
\end{equation} The initial energy inequality \eqref{bd} implies that $\rho^\epsilon_0$ is of order $1+O(\epsilon)$.
Under the above assumptions, it was proved in \cite{HW1} that the compressible MHD equations \eqref{a2i}-\eqref{a2k} with initial data \eqref{bb}-\eqref{bd} has a global weak solution. More precisely, we have
\begin{prop}[\cite{HW1}]\label{cmhd}
Let $\gamma>d/2$. Suppose that the initial data
$(\rho^\epsilon_0,{\bf u}^\epsilon_0,{\bf H}^\epsilon_0)$ satisfy the assumptions \eqref{bc} and \eqref{bd}.
Then the compressible MHD equations
\eqref{a2i}-\eqref{a2k} with the initial data \eqref{bb} enjoy
at least one global weak solution $(\rho^\epsilon, {\bf u}^\epsilon, {\bf H}^\epsilon)$
satisfying \begin{enumerate}
\item $ \rho^\epsilon\in L^\infty(0,\infty;L^\gamma)\cap
C([0,\infty),L^r)$ for all $1\leq r< \gamma$, $\rho^\epsilon
|{\bf u}^\epsilon|^2\in
L^\infty(0,\infty; L^1)$, ${\bf H}^\epsilon \in L^\infty(0,\infty; L^2)$,
and ${\bf u}^\epsilon \in L^2(0,T; H^1)$,
$\rho^\epsilon{\bf u}^\epsilon\in C([0,T], L^{\frac{2\gamma}{\gamma+1}}_{weak})$,
${\bf H}^\epsilon \in L^2(0,T; H^1)\cap C([0,T],L^{\frac{2\gamma}{\gamma+1}}_{weak})$
for all $T\in (0,\infty)$;
\item the energy inequality
\begin{equation}\label{be}
\mathcal{E}^\epsilon(t)+\int^t_0 \mathcal{D}^\epsilon(s)ds\leq
\mathcal{E}^\epsilon(0)
\end{equation}
holds with the finite total energy
\begin{equation}\label{bf}
\qquad\qquad \mathcal{E}^\epsilon(t)\equiv \int_{\mathbb{T}^d}
\Big[\frac12\rho^\epsilon|{\bf u}^\epsilon|^2+\frac12|{\bf H}^\epsilon|^2
+\frac{a}{\epsilon^2{(\gamma-1)}}\big((\rho^\epsilon)^\gamma
-1-\gamma (\rho^\epsilon-1)\big)
\Big](t)
\end{equation}
and the dissipation energy \begin{equation}\label{bff}
\mathcal{D}^\epsilon(t)\equiv \int_{\mathbb{T}^d}\big[ \mu^\epsilon|\nabla
{\bf u}^\epsilon|^2
+(\mu^\epsilon+\lambda^\epsilon) |{\rm div}{\bf u}^\epsilon|^2 + \nu^\epsilon |\nabla
{\bf H}^\epsilon|^2\big](t); \end{equation}
\item the continuity equation is satisfied in the sense of
renormalized solutions, i.e.,
\begin{equation} \label{cnsbg}
\partial_t b(\rho^\epsilon)+{ \rm div}
(b(\rho^\epsilon){\bf u}^\epsilon)+\big(b'(\rho^\epsilon)\rho^\epsilon-b(\rho^\epsilon)\big)
{\rm div}{\bf u}^\epsilon =0
\end{equation} for any $b\in C^1(\mathbb{T})$ such that $b'(z)$ is constant for $z$ large enough;
\item the equations \eqref{a2i}-\eqref{a2k} hold in
$\mathcal{D}'( \mathbb{T}^d\times(0,\infty))$. \end{enumerate} \end{prop}
The main results of this paper can be stated as follows.
\begin{thm}\label{MRa} Let $s>{d}/{2}+2$ and $\mu^\epsilon+\lambda^\epsilon>0$.
Suppose that the initial data $(\rho^\epsilon_0,{\bf u}^\epsilon_0,{\bf H}_0^\epsilon)$
satisfy the conditions presented in Proposition \ref{cmhd}.
Assume further that \begin{align}
&\int_{\mathbb{T}^d}|\rho_0^\epsilon-1|^21_{(|\rho_0^\epsilon-1|\leq
\delta)}\;dx+\int_{\mathbb{T}^d}|\rho_0^\epsilon-1|^\gamma1_{(|\rho_0^\epsilon-1|> \delta)}\;dx \leq C\epsilon^2,\label{icd1}\\ & \qquad
||\sqrt{\rho^\epsilon_0}\mathbf{u}_0^\epsilon-\mathbf{u}_0||^2_{L^2(\mathbb{T}^d)}
\leq C\epsilon, \quad ||\mathbf{H}_0^\epsilon-\mathbf{H}_0||^2_{L^2(\mathbb{T}^d)} \leq C\epsilon \label{icd2} \end{align} for any $\delta\in (0,1)$, where $\mathbf{u}_0$ and $\mathbf{H}_0$ are defined in Proposition \ref{imhd}. We assume that the shear viscosity $\mu^\epsilon$ and the magnetic diffusion coefficient $\nu^\epsilon $ satisfy \begin{align}\label{abc}
\mu^\epsilon =\epsilon^\alpha,\quad \nu^\epsilon=\epsilon^\beta \end{align} for some constants $\alpha , \beta>0$ satisfying $0<\alpha+\beta <2$.
Let $({\bf u},{\bf H})$ be the smooth solution to the ideal incompressible MHD equations \eqref{a2l}-\eqref{a2n} defined on $[0,T^*)$ with
$({\bf u} ,{\bf H} )|_{t=0}=({\bf u}_0,{\bf H}_0)$. Then, for any $0<T<T^*$, the global weak solution $(\rho^\epsilon, {\bf u}^\epsilon, {\bf H}^\epsilon)$ of the compressible MHD equations \eqref{a2i}-\eqref{a2k} established in Proposition \ref{cmhd} satisfies \begin{align}
&\int_{\mathbb{T}^d}|\rho^\epsilon-1|^21_{(|\rho^\epsilon-1|\leq
\delta)}\;dx+\int_{\mathbb{T}^d}|\rho^\epsilon-1|^\gamma1_{(|\rho^\epsilon-1|> \delta)}\;dx \leq C_T\epsilon^2, \label{icd10}\\ & \qquad
||\sqrt{\rho^\epsilon}\mathbf{u}^\epsilon-\mathbf{u}||^2_{L^2(\mathbb{T}^d)} \leq C_T\epsilon^\sigma, \quad
||\mathbf{H}^\epsilon-\mathbf{H}||^2_{L^2(\mathbb{T}^d)} \leq C_T\epsilon^\sigma \label{icd20} \end{align} for any $t\in [0,T]$, where $\sigma=\min\{\alpha,\beta, 1-(\alpha+\beta)/2 \}.$ \end{thm}
The proof of Theorem \ref{MRa} is based on the combination of the modulated energy method, motivated by Brenier \cite{B00}, the weak convergence method and the refined energy analysis. Masmoudi \cite{M01b} made use of such idea to study the incompressible, inviscid limit of the compressible Navier-Stokes equations in both the whole space and the torus. Comparing with the proof in \cite{M01b}, here we have to overcome the difficulties caused by the strong coupling of the hydrodynamic motion and the magnetic field.
Furthermore, we can use an idea similar to that described above to obtain the convergence of the compressible MHD equations \eqref{a2i}-\eqref{a2k} to the incompressible viscous MHD equations \eqref{a2ll}-\eqref{a2nn}. In fact, we have the following result.
\begin{thm}\label{MRb} Let $s>{d}/{2}+2$ and $\mu^\epsilon+\lambda^\epsilon>0$.
Suppose that the initial data $(\rho^\epsilon_0,{\bf u}^\epsilon_0,{\bf H}_0^\epsilon)$ satisfy the conditions presented in Proposition \ref{cmhd}. Assume further that \begin{align}
&\int_{\mathbb{T}^d}|\rho_0^\epsilon-1|^21_{(|\rho_0^\epsilon-1|\leq
\delta)}\;dx+\int_{\mathbb{T}^d}|\rho_0^\epsilon-1|^\gamma1_{(|\rho_0^\epsilon-1|> \delta)}\;dx \leq C\epsilon^2,\label{icd11}\\
& \qquad ||\sqrt{\rho^\epsilon_0}\mathbf{u}_0^\epsilon
-\mathbf{u}_0||_{L^2(\mathbb{T}^d)}^2 \leq C\epsilon, \quad
||\mathbf{H}_0^\epsilon-\mathbf{H}_0||_{L^2(\mathbb{T}^d)}^2 \leq C\epsilon \label{icd22} \end{align} for any $\delta\in (0,1)$ and for some $\mathbf{u}_0,\mathbf{H}_0\in H^s(\mathbb{T}^d)$, satisfying ${\rm div } {\bf u}_0=0, {\rm div } {\bf H}_0 =0.$ We also assume that the shear viscosity $\mu^\epsilon $ and the magnetic diffusion coefficient $\nu^\epsilon $ satisfy \eqref{pamb}.
Let $({\bf u},{\bf H})$ be the smooth solution to the incompressible MHD equations \eqref{a2ll}-\eqref{a2nn}
with $({\bf u} ,{\bf H} )|_{t=0}=({\bf u}_0,{\bf H}_0)$.
Then, for any $0<T<T^{**}$ ( $ T^{**}$ is the maximal time of existence for \eqref{a2ll}-\eqref{a2nn}), the global weak solution $(\rho^\epsilon, {\bf u}^\epsilon, {\bf H}^\epsilon)$ of the compressible MHD equations \eqref{a2i}-\eqref{a2k} established in Proposition \ref{cmhd} satisfies that
$\nabla {\bf u}^\epsilon$ and $\nabla {\bf H}^\epsilon$ converge strongly to $\nabla {\bf u}$
and $\nabla {\bf H}$ in $L^2 (0,T;L^2(\mathbb{T}^d))$, respectively.
Moreover, for any $t\in [0,T]$, we have \begin{align}
&\int_{\mathbb{T}^d}|\rho^\epsilon-1|^21_{(|\rho^\epsilon-1|\leq
\delta)}\;dx+\int_{\mathbb{T}^d}|\rho^\epsilon-1|^\gamma1_{(|\rho^\epsilon-1|> \delta)}\;dx \leq C_T\epsilon^2,\label{icd110}\\ &
||\sqrt{\rho^\epsilon}\mathbf{u}^\epsilon-\mathbf{u}||^2_{L^2(\mathbb{T}^d)}
+||\mathbf{H}^\epsilon-\mathbf{H}||^2_{L^2(\mathbb{T}^d)} \leq C_T\frac{\epsilon}{\sqrt{\mu\nu}}+C_T\Big(\frac{|\mu^\epsilon-\mu|}{\sqrt{\mu}}
+\frac{|\nu^\epsilon-\nu|}{\sqrt{\nu}}\Big).\label{icd220} \end{align} \end{thm}
To show Theorem \ref{MRb}, besides the techniques mentioned above, we have to employ a new technique, that is, to modulate both the total energy and the partial dissipative energy simultaneously. Moreover, the dissipative effect of the viscous terms is also carefully exploited to obtain the desired results.
\begin{rem} Comparing with Theorem \ref{MRa}, we have gotten the better convergence rates \eqref{icd220} than (\ref{icd20}) when the shear viscosity and the magnetic diffusion coefficient tend to some positive constants. \end{rem}
Some results in Theorem \ref{MRb} can be extended to the case of general initial data. More precisely, we shall obtain
the convergence of the compressible MHD equations \eqref{a2i}-\eqref{a2k}
to the incompressible MHD
equations \eqref{a2ll}-\eqref{a2nn} for the general initial data
under the conditions that the oscillating parts of the initial data have higher regularity and the Sobolev norm of the oscillation parts
is comparable to the magnetic diffusion coefficient. This implies that the influence of oscillations
on the magnetic field can be balanced by the diffusive effect of the magnetic field,
which is one of the new ingredients in our paper.
To describe the result, we write $\rho^\epsilon=1+\epsilon \varphi^\epsilon$ and denote \begin{equation*}
\Pi^\epsilon(x,t)=\frac{1}{\epsilon}\sqrt{\frac{2a}{\gamma-1}
\big((\rho^\epsilon)^\gamma-1-\gamma(\rho^\epsilon-1)\big)}. \end{equation*} We will use the above approximation $\Pi^\epsilon(x,t)$ instead of $\varphi^\epsilon$, since we can not obtain any bound for $\varphi^\epsilon$ in $L^\infty(0,T;L^2)$ directly if $\gamma <2$.
\begin{thm}\label{MRc} Let $s>2+{d}/{2}$ and $2\mu+d\lambda>0$.
Suppose that the initial data $(\rho^\epsilon_0,{\bf u}^\epsilon_0,{\bf H}_0^\epsilon)$
satisfy the conditions presented in Proposition \ref{cmhd}.
Moreover, we assume that $\sqrt{\rho^\epsilon_0}{\bf u}^\epsilon_0$ converges strongly in $L^2$ to some $\tilde{{\bf u}}_0$ satisfying $Q\tilde{{\bf u}}_0\in H^{s-1}$, ${\bf H}^\epsilon_0$ converges strongly in $L^2$ to some ${\bf H}_0$ with $\int_{\mathbb{T}^d} {\bf H}_0(x)dx=0$,
$\Pi^\epsilon|_{t=0}=\Pi^\epsilon_0$ converges strongly in $L^2$ to some $\varphi_0\in H^{s-1}$, and
\begin{align}\label{Qu}
\|\varphi_0\|_{H^2}+||Q\tilde{ {\bf u}}_0||_{H^2}\leq c_0 \nu \end{align}
for some constant $c_0>0$. Let $({\bf u},{\bf H})$ be the smooth solution to the incompressible MHD equations \eqref{a2ll}-\eqref{a2nn}
with $({\bf u} ,{\bf H} )|_{t=0}=({\bf u}_0,{\bf H}_0)\in H^s(\mathbb{T}^d)$ satisfying
${\bf u}_0=P\tilde{ {\bf u}}_0$ and ${\rm div } {\bf H}_0=0$.
Then, for any $0<T<T^{**}$ ( $ T^{**}$ is the maximal time of existence for \eqref{a2ll}-\eqref{a2nn}), the global weak solution $(\rho^\epsilon, {\bf u}^\epsilon, {\bf H}^\epsilon)$ of the compressible MHD equations \eqref{a2i}-\eqref{a2k} established in Proposition \ref{cmhd} satisfies \begin{enumerate}
\item $\rho^\epsilon$ converges strongly to $1$ in
$C([0,T],L^\gamma_2(\mathbb{T}^d))$;
\item $\nabla {\bf H}^\epsilon$ converges strongly to $\nabla {\bf H}$
in $L^2 (0,T;L^2(\mathbb{T}^d))$; \item ${\bf H}^\epsilon$ converges strongly to
${\bf H}$ in $L^\infty(0,T; L^2(\mathbb{T}^d))$;
\item $P(\sqrt{\rho^\epsilon}{\bf u}^\epsilon)$ converges strongly
to ${\bf u}$ in $L^\infty (0,T;L^2(\mathbb{T}^d))$;
\item $\sqrt{\rho^\epsilon}{\bf u}^\epsilon$ converges weakly to
${\bf u}$ in $H^{-1}(0,T; L^2(\mathbb{T}^d))$.
\end{enumerate} \end{thm}
By slightly modifying the proof of Theorem \ref{MRc}, we can obtain the convergence of compressible MHD equations to the partial viscous incompressible MHD equations when the shear viscosity goes to zero and the magnetic diffusion coefficient goes to a positive constant. The partial viscous incompressible MHD equations correspond to the case of turbulent flow with very high Reynolds number (where the viscosity of flow can be ignored, see \cite{LZ}). \begin{thm}\label{MRd} Let $s>2+{d}/{2}$.
Suppose that the conditions in Theorem \ref{MRc} hold. Moreover, we assume that \begin{align*}
\nu^\epsilon \rightarrow \nu>0, \quad 2\mu^\epsilon+\lambda^\epsilon
\rightarrow 2\theta> 0 \quad \text{as} \quad \epsilon \rightarrow 0, \end{align*} and $\mu^\epsilon =\epsilon^\alpha $ for some constant $0< \alpha<1$. Let $({\bf u},{\bf H})$ be the smooth solution to the following partially viscous incompressible MHD equations
\begin{align*} & \partial_t {\bf u}+({\bf u}\cdot \nabla){\bf u} +\nabla p-({{\bf H}} \cdot \nabla
){{\bf H}} +\frac12\nabla(|{{\bf H}} |^2) =0, \\ & \partial_t {{\bf H}} + ( {{\bf u}} \cdot \nabla) {{\bf H}}
- ( {{\bf H}} \cdot \nabla) {{\bf u}} - \nu\Delta {\bf H}= 0, \\
&{\rm div } {\bf u}=0, \quad {\rm div } {\bf H}=0,
\end{align*}
with $({\bf u} ,{\bf H} )|_{t=0}=({\bf u}_0,{\bf H}_0)\in H^s(\mathbb{T}^3)$ satisfying
${\bf u}_0=P\tilde{ {\bf u}}_0$ and ${\rm div } {\bf H}_0=0$.
Then, for any $0<T<T^{**}$ ($ T^{**}$ is the maximal time of existence for \eqref{a2ll}-\eqref{a2nn}), the global weak solution $(\rho^\epsilon, {\bf u}^\epsilon, {\bf H}^\epsilon)$ of the compressible MHD equations \eqref{a2i}-\eqref{a2k} established in Proposition \ref{cmhd} satisfies \begin{enumerate}
\item $\rho^\epsilon$ converges strongly to $1$ in
$C([0,T],L^\gamma_2(\mathbb{T}^d))$;
\item $\nabla {\bf H}^\epsilon$ converges strongly to $\nabla {\bf H}$
in $L^2 (0,T;L^2(\mathbb{T}^d))$; \item ${\bf H}^\epsilon$ converges strongly to
${\bf H}$ in $L^\infty(0,T; L^2(\mathbb{T}^d))$;
\item $P(\sqrt{\rho^\epsilon}{\bf u}^\epsilon)$ converges strongly
to ${\bf u}$ in $L^\infty (0,T;L^2(\mathbb{T}^d))$;
\item $\sqrt{\rho^\epsilon}{\bf u}^\epsilon$ converges weakly to
${\bf u}$ in $H^{-1}(0,T; L^2(\mathbb{T}^d))$.
\end{enumerate} \end{thm}
\begin{rem} The assumption that $\Pi^\epsilon_0$ converges strongly in $L^2$ to some $\varphi_0$ in fact implies that $\varphi^\epsilon_0$ converges strongly to $\varphi_0$ in $L^\gamma_2$. \end{rem}
\begin{rem} When taking ${\bf H}^\epsilon\equiv 0$ in \eqref{a1a}-\eqref{a1c}, the MHD equations reduce to the classical compressible Navier-Stokes equations. The low Mach number limit problem of the compressible Navier-Stokes equations has been investigated extensively, for instance, see \cite{G,H98,LM98,DG99,D02}. The interested reader can refer to the survey article \cite{M06} for more related results. \end{rem}
\begin{rem}
We point out that our arguments in the present paper can be applied to the case of ${\bf H}^\epsilon\equiv 0$. In this case, we obtain the convergence of the compressible Navier-Stokes equtions to the incompressible
Euler or Navier-Stokes equations with general initial data, extending thus the results
in \cite{LM98,M01b}. \end{rem}
\section{Proof of Theorems \ref{MRa} and \ref{MRb}}
In this section, we shall prove our convergence results for the case of well-prepared initial data by combining the modulated energy method, the weak convergence method, and the refined energy analysis.
\begin{proof}[Proof of Theorem \ref{MRa}]
We divide the proof into several steps.
\emph{Step 1: Basic energy estimates and compact arguments.}
By the assumptions on the initial data we obtain, from the energy inequality \eqref{be}, that the total energy $\mathcal{E}^\epsilon(t)$ has a uniform upper bound for a.e. $t\in
[0,T]$, $T>0$. This uniform bound implies that $\rho^\epsilon |{\bf u}^\epsilon|^2$ and $ \big((\rho^\epsilon)^\gamma-1-\gamma(\rho^\epsilon-1)\big)/{\epsilon^2}$ are bounded in $L^\infty(0,T;L^1)$ and $ {\bf H}^\epsilon $ is bounded in $L^\infty(0,T;L^2)$. Using the analysis in \cite{LM98}, we obtain \begin{equation}\label{L2g}
\int_{\mathbb{T}^d}\frac{1}{\epsilon^2}|\rho^\epsilon-1|^2 1_{\{|\rho^\epsilon-1|\leq \frac12\}}
+\int_{\mathbb{T}^d}\frac{1}{\epsilon^2}|\rho^\epsilon-1|^\gamma
1_{\{|\rho^\epsilon-1|\geq \frac12\}} \leq C, \end{equation} which implies \eqref{icd10}
and \begin{equation}\label{re}
\rho^\epsilon \rightarrow 1 \ \
\text{strongly in}\ \ C([0,T],L^\gamma_2(\mathbb{T}^d)). \end{equation} From the results in \cite{LM98}, we know that
$\|{\bf u}^\epsilon\|_{L^2_tL^2_x}^2\leq C+C\|\nabla{\bf u}^\epsilon\|_{L^2_tL^2_x}^2$. Furthermore, the fact that
$\rho^\epsilon |{\bf u}^\epsilon|^2$ and $ |{\bf H}^\epsilon|^2$ are bounded in $L^\infty(0,T;L^1)$ implies the following convergence (up to the extraction of a subsequence $\epsilon_n$): \begin{align*} & \sqrt{\rho^\epsilon} {\bf u}^\epsilon \ \text{converges weakly-$\ast$} \,\, \text{to some}\, \mathbf{J} \, \,\text{in}\, L^\infty(0,T;L^2(\mathbb{T}^d)),\\ & {\bf H}^\epsilon \ \text{converges weakly-$\ast$} \,\, \text{to some}\, \mathbf{K} \, \,\text{in}\, L^\infty(0,T;L^2(\mathbb{T}^d)). \end{align*}
Thus, to finish our proof, we need to
show that $\mathbf{J}={\bf u}$ and $\mathbf{K}={\bf H}$ in some sense and the inequalities \eqref{icd20} hold, where $({\bf u},{\bf H})$ is the strong solution to the ideal incompressible MHD equations \eqref{a2l}-\eqref{a2n}.
\emph{Step 2: The modulated energy functional and the uniform estimates.}
We first recall the energy inequality of the compressible MHD equations \eqref{a2i}-\eqref{a2k}, i.e., for almost all $t$, there holds
\begin{align}\label{cei}
& \frac12\int_{\mathbb{T}^d}\Big[\rho^\epsilon(t)|{\bf u}^\epsilon|^2(t)+|{\bf H}^\epsilon|^2(t)
+(\Pi^\epsilon(t))^2
\Big]
+\mu^\epsilon \int^t_0\! \int_{\mathbb{T}^d}|\nabla {\bf u}^\epsilon|^2 \nonumber\\
&\ \ +(\mu^\epsilon+\lambda^\epsilon)\int^t_0\! \int_{\mathbb{T}^d}|{\rm div}{\bf u}^\epsilon |^2
+\nu^\epsilon\int^t_0\! \int_{\mathbb{T}^d}|\nabla {\bf H}^\epsilon|^2\nonumber\\
& \ \ \ \leq \frac12\int_{\mathbb{T}^d}\Big[\rho_0^\epsilon|{\bf u}_0^\epsilon|^2
+|{\bf H}_0^\epsilon|^2+(\Pi^\epsilon_0)^2 \Big] .
\end{align} The conservation of energy for the ideal incompressible MHD equations \eqref{a2l}-\eqref{a2n} reads \begin{equation}\label{cfi}
\frac12 \int_{\mathbb{T}^d}\big[|{\bf u}|^2(t)+ |{\bf H}|^2(t)\big] =\frac12 \int_{\mathbb{T}^d}
\big[|{\bf u}_0|^2+|{\bf H}_0|^2\big]. \end{equation} Using ${\bf u}$ to test the momentum equation \eqref{a2j}, we obtain \begin{align}\label{cii} & \int_{\mathbb{T}^d}(\rho^\epsilon {\bf u}^\epsilon \cdot {\bf u})(t) +\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon \cdot \big[({\bf u}\cdot \nabla)
{\bf u}-({\bf H}\cdot \nabla){\bf H}+\nabla p+\frac12\nabla(|{\bf H}|^2)\big] \nonumber\\ & -\int^t_0\int_{\mathbb{T}^d}\big[(\rho^\epsilon {\bf u}^\epsilon \otimes {\bf u}^\epsilon)\cdot \nabla {\bf u}+ ({\bf H}^\epsilon\cdot \nabla){\bf H}^\epsilon \cdot {\bf u}
- \mu^\epsilon \nabla {\bf u}^\epsilon \cdot \nabla {\bf u}\big] = \int_{\mathbb{T}^d}
\rho^\epsilon_0 {\bf u}^\epsilon_0 \cdot {\bf u}_0. \end{align}
Similarly, using ${\bf H}$ to test the magnetic field equation \eqref{a2k}, one gets \begin{align}\label{cj1i} & \int_{\mathbb{T}^d}({\bf H}^\epsilon \cdot {\bf H})(t) +\int^t_0\int_{\mathbb{T}^d}{\bf H}^\epsilon \cdot\big[({\bf u}\cdot \nabla) {\bf H}-({\bf H}\cdot \nabla){\bf u}\big] + \nu^\epsilon \int^t_0\int_{\mathbb{T}^d} \nabla {\bf H}^\epsilon \cdot \nabla {\bf H}\nonumber\\ &\quad +\int^t_0\int_{\mathbb{T}^d}\big[({\rm div } {{\bf u}}^\epsilon) { {\bf H}}^\epsilon + ( {{\bf u}}^\epsilon \cdot \nabla) {{\bf H}}^\epsilon - ( {{\bf H}}^\epsilon\cdot \nabla)
{{\bf u}}^\epsilon\big]\cdot {\bf H} = \int_{\mathbb{T}^d} {\bf H}^\epsilon_0 \cdot {\bf H}_0. \end{align} Summing up \eqref{cei} and \eqref{cfi}, and inserting \eqref{cii} and \eqref{cj1i} into the resulting inequality,
we can deduce the following inequality by a straightforward computation \begin{align}\label{cki}
& \frac{1}{2}\int_{\mathbb{T}^d}\Big\{ |\sqrt{\rho^\epsilon} {\bf u}^\epsilon
-{\bf u}|^2(t)+ | {\bf H}^\epsilon
-{\bf H}|^2(t)
+(\Pi^\epsilon)^2(t)\Big\}\nonumber\\ &\quad
+ {\mu^\epsilon} \int^t_0\int_{\mathbb{T}^d}|\nabla {\bf u}^\epsilon|^2
+ (\mu^\epsilon+\lambda^\epsilon)\int^t_0\! \int_{\mathbb{T}^d}|{\rm div}
{\bf u}^\epsilon |^2+ {\nu^\epsilon} \int^t_0\int_{\mathbb{T}^d}|\nabla {\bf H}^\epsilon |^2\nonumber\\ & \leq {\mu^\epsilon} \int^t_0\int_{\mathbb{T}^d}\nabla {\bf u}^\epsilon\cdot\nabla {\bf u} + {\nu^\epsilon} \int^t_0\int_{\mathbb{T}^d}\nabla {\bf H}^\epsilon\cdot\nabla {\bf H}
-\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon\cdot [({\bf H}\cdot \nabla) {\bf H})] \nonumber\\ &\quad -\int^t_0\int_{\mathbb{T}^d} ({\bf H}^\epsilon\cdot \nabla) {\bf H}^\epsilon \cdot{\bf u} +\int^t_0\int_{\mathbb{T}^d}{\bf H}^\epsilon \cdot\big[({\bf u}\cdot \nabla) {\bf H}-({\bf H}\cdot \nabla){\bf u}\big] \nonumber\\ &\quad
+\int^t_0\int_{\mathbb{T}^d}\big[({\rm div } {{\bf u}}^\epsilon) { {\bf H}}^\epsilon+
( {{\bf u}}^\epsilon \cdot \nabla) {{\bf H}}^\epsilon
- ( {{\bf H}}^\epsilon\cdot \nabla) {{\bf u}}^\epsilon\big]\cdot {\bf H}
+\frac12\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon \cdot \nabla(|{\bf H}|^2)
\nonumber\\ &\quad +\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon \cdot\big[ ({\bf u}\cdot \nabla) {\bf u} +\nabla p \big]-\int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon {\bf u}^\epsilon\otimes {\bf u}^\epsilon)\cdot \nabla {\bf u}\nonumber\\ &\quad +\int_{\mathbb{T}^d} \big[(\sqrt{\rho^\epsilon}-1)\sqrt{\rho^\epsilon}{\bf u}^\epsilon\cdot {\bf u}\big](t) -\int_{\mathbb{T}^d} \big[(\sqrt{\rho^\epsilon}-1)\sqrt{\rho^\epsilon}{\bf u}^\epsilon\cdot {\bf u}\big](0)\nonumber\\
&\quad +\frac{1}{2}\int_{\mathbb{T}^d}\Big\{ |\sqrt{\rho^\epsilon} {\bf u}^\epsilon
-{\bf u}|^2(0)+|{\bf H}^\epsilon
-{\bf H} |^2(0) +(\Pi^\epsilon_0)^2\Big\}. \end{align}
We first deal with the right-hand side of the inequality \eqref{cki}. Denoting $\mathbf{w}^{\epsilon}=\sqrt{\rho^\epsilon} {\bf u}^\epsilon -{\bf u}$ and $\mathbf{Z}^{\epsilon}= {\bf H}^\epsilon -{\bf H}$, integrating by parts, and using the fact that ${\rm div}\, {\bf H}^\epsilon =0,$ $ {\rm div}\, {\bf u} =0$ and ${\rm div}\,{\bf H} =0$, we find that \begin{align}\label{ck1i} & -\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon\cdot [({\bf H}\cdot \nabla) {\bf H})]
-\int^t_0\int_{\mathbb{T}^d} ({\bf H}^\epsilon\cdot \nabla) {\bf H}^\epsilon \cdot{\bf u} \nonumber\\ & +\int^t_0\int_{\mathbb{T}^d}{\bf H}^\epsilon \cdot\big[({\bf u}\cdot \nabla) {\bf H}-({\bf H}\cdot \nabla){\bf u}\big]+\frac{1}{2}\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon
{\bf u}^\epsilon\cdot \nabla(|{\bf H}|^2)\nonumber\\ &
+\int^t_0\int_{\mathbb{T}^d}\big[({\rm div } {{\bf u}}^\epsilon) { {\bf H}}^\epsilon
+ ( {{\bf u}}^\epsilon \cdot \nabla) {{\bf H}}^\epsilon
- ( {{\bf H}}^\epsilon\cdot \nabla) {{\bf u}}^\epsilon\big]\cdot {\bf H} \nonumber\\
= & -\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon\cdot [({\bf H}\cdot \nabla) {\bf H})] +\int^t_0\int_{\mathbb{T}^d} ({\bf H}^\epsilon\cdot \nabla) {\bf u} \cdot {\bf H}^\epsilon \nonumber\\ & +\int^t_0\int_{\mathbb{T}^d}({\bf u}\cdot \nabla){\bf H}\cdot {\bf H}^\epsilon -\int^t_0\int_{\mathbb{T}^d}({\bf H}\cdot \nabla){\bf u} \cdot {\bf H}^\epsilon \nonumber\\ & - \int^t_0\int_{\mathbb{T}^d}( {{\bf u}}^\epsilon \cdot \nabla){\bf H} \cdot {{\bf H}}^\epsilon
+ \int^t_0\int_{\mathbb{T}^d}( {{\bf H}}^\epsilon\cdot \nabla) {\bf H} \cdot {{\bf u}}^\epsilon
+\frac{1}{2}\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon
{\bf u}^\epsilon\cdot \nabla(|{\bf H}|^2) \nonumber\\
= & \int^t_0\int_{\mathbb{T}^d}(1-\rho^\epsilon) {\bf u}^\epsilon\cdot [({\bf H}\cdot \nabla) {\bf H})] +\int^t_0\int_{\mathbb{T}^d}[({\bf H}^\epsilon-{\bf H})\cdot \nabla]{\bf u} \cdot ({\bf H}^\epsilon-{\bf H})\nonumber\\ &+\int^t_0\int_{\mathbb{T}^d}[({\bf H}^\epsilon-{\bf H})\cdot \nabla]{\bf H} \cdot ({\bf u}^\epsilon-{\bf u}) - \int^t_0\int_{\mathbb{T}^d}[({\bf u}^\epsilon-{\bf u})\cdot \nabla]{\bf H} \cdot ({\bf H}^\epsilon-{\bf H})\nonumber\\
& +\frac12\int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon-1){{\bf u}}^\epsilon \nabla(| {\bf H}|^2)\nonumber\\ \leq & \int^t_0\int_{\mathbb{T}^d}(1-\rho^\epsilon) {\bf u}^\epsilon\cdot [({\bf H}\cdot \nabla) {\bf H})]
+\int^t_0 ||\mathbf{Z}^\epsilon(s)||^2_{L^2}|| \nabla {\bf u}(s)||_{L^\infty} ds\nonumber\\
&+ \int^t_0 \big[||\mathbf{w}^{\epsilon}(s)||^2_{L^2}
+ ||\mathbf{Z}^\epsilon(s)||^2_{L^2}\big] || \nabla {\bf H}(s)||_{L^\infty} ds +\int^t_0\!\int_{\mathbb{T}^d}(\mathbf{Z}^\epsilon \cdot \nabla){\bf H} \cdot [(1-\sqrt{\rho^\epsilon}){\bf u}^\epsilon] \nonumber\\ &- \int^t_0\!\int_{\mathbb{T}^d}\big\{[(1-\sqrt{\rho^\epsilon}){\bf u}^\epsilon ]\cdot \nabla\big\}{\bf H} \cdot \mathbf{Z}^\epsilon
+\frac12\int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon-1){{\bf u}}^\epsilon \nabla(|{\bf H}|^2) \end{align} and \begin{align}\label{cli}
& \int^t_0\int_{\mathbb{T}^d}\big[\rho^\epsilon {\bf u}^\epsilon \cdot(({\bf u}\cdot \nabla) {\bf u}+\nabla p)\big]-\int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon {\bf u}^\epsilon\otimes {\bf u}^\epsilon)\cdot \nabla {\bf u}\nonumber\\
=& -\int^t_0\int_{\mathbb{T}^d}(\mathbf{w}^{\epsilon}\otimes
\mathbf{w}^{\epsilon})\cdot \nabla {\bf u}
+\int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon-\sqrt{\rho^\epsilon}){\bf u}^\epsilon\cdot
(({\bf u}\cdot \nabla) {\bf u})\nonumber\\
&
+\int^t_0\int_{\mathbb{T}^d}[(\sqrt{\rho^\epsilon}{\bf u}^\epsilon-{\bf u})\cdot \nabla] {\bf u}
\cdot \mathbf{w}^{\epsilon}
+ \int^t_0 \int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon \cdot \nabla p\nonumber\\
& -\int^t_0\int_{\mathbb{T}^d}(\sqrt{ \rho^\epsilon}{\bf u}^\epsilon- {\bf u}) \cdot \nabla
\big(\frac{|{\bf u}|^2}{2}\big).
\end{align}
Substituting \eqref{ck1i} and \eqref{cli} into the inequality \eqref{cki}, we conclude that \begin{align}\label{cmi}
&
||\mathbf{w}^{\epsilon}(t)||^2_{L^2}+
||\mathbf{Z}^{\epsilon}(t)||^2_{L^2}+||\Pi^\epsilon(t)||^2_{L^2}
+ 2{\mu^\epsilon} \int^t_0\int_{\mathbb{T}^d}|\nabla {\bf u}^\epsilon|^2 \nonumber\\
& + 2(\mu^\epsilon+\lambda^\epsilon)\int^t_0\! \int_{\mathbb{T}^d}|{\rm div}{\bf u}^\epsilon |^2 + 2{\nu^\epsilon} \int^t_0\int_{\mathbb{T}^d}|\nabla {\bf H}^\epsilon|^2 \nonumber\\
\leq & 2C \int^t_0 (||\mathbf{w}^{\epsilon}(\tau)||^2_{L^2}+
||\mathbf{Z}^{\epsilon}(\tau)||^2_{L^2})(||\nabla {\bf u}(\tau)||_{L^\infty}+||\nabla
{\bf H}(\tau)||_{L^\infty})d\tau\nonumber\\
& +||\mathbf{w}^{\epsilon}(0)||^2_{L^2}+||\mathbf{Z}^{\epsilon}(0)||^2_{L^2}
+||\Pi^\epsilon_0||^2_{L^2}
+2\sum^{6}_{i=1}R^{\epsilon}_i(t), \end{align} where \begin{align*} R^{\epsilon}_1(t)= & {\mu^\epsilon} \int^t_0\int_{\mathbb{T}^d}\nabla {\bf u}^\epsilon\cdot\nabla {\bf u} + {\nu^\epsilon} \int^t_0\int_{\mathbb{T}^d}\nabla {\bf H}^\epsilon\cdot\nabla {\bf H}, \\ R^{\epsilon}_2(t)=&\int_{\mathbb{T}^d} \big[(\sqrt{\rho^\epsilon}-1)\sqrt{\rho^\epsilon}{\bf u}^\epsilon\cdot
{\bf u} \big](t) -\int_{\mathbb{T}^d} \big[(\sqrt{\rho^\epsilon}-1)\sqrt{\rho^\epsilon}{\bf u}^\epsilon\cdot
{\bf u} \big](0)\nonumber\\
& +\int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon-\sqrt{\rho^\epsilon})
{\bf u}^\epsilon\cdot (({\bf u}\cdot \nabla ){\bf u}),\\ R^{\epsilon}_3(t)= & \int^t_0\!\int_{\mathbb{T}^d}(\mathbf{Z}^\epsilon \cdot \nabla) {\bf H} \cdot [(1-\sqrt{\rho^\epsilon}){\bf u}^\epsilon] - \int^t_0 \!\int_{\mathbb{T}^d}\big\{[(1-\sqrt{\rho^\epsilon}){\bf u}^\epsilon ]\cdot \nabla\big\}{\bf H} \cdot \mathbf{Z}^\epsilon, \nonumber\\ R^{\epsilon}_4(t)= & \int^t_0\int_{\mathbb{T}^d}(1-\rho^\epsilon) {\bf u}^\epsilon\cdot [({\bf H}\cdot \nabla) {\bf H})]
+\frac12 \int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon-1){\bf u}^\epsilon\cdot \nabla (| {\bf H}|^2),\\ R^{\epsilon}_5(t)=&
\int^t_0 \int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon \cdot \nabla p,\\ R^{\epsilon}_6(t)=& -\int^t_0\int_{\mathbb{T}^d}(\sqrt{ \rho^\epsilon} {\bf u}^\epsilon - {\bf u}) \cdot \nabla
\big(\frac{|{\bf u}|^2}{2}\big). \end{align*}
\emph{Step 3: Convergence of the modulated energy functional.}
To show the convergence of the modulated energy functional \eqref{cmi} and to finish our proof, we have to estimate the reminders $R^{\epsilon}_i(t)$, $i=1,\dots,6.$
First, in view of \eqref{L2g} and the following two elementary inequalities \begin{align}
&|\sqrt{x}-1|^2\leq M|x-1|^\gamma,\;\;\;|x-1|\geq\delta, \;\; \gamma\geq 1,\label{ine10}\\
&|\sqrt{x}-1|^2\leq M|x-1|^2,\;\;\;x\geq 0 \label{ine20} \end{align} for some positive constants $M$ and $0<\delta<1$, we obtain \begin{align}
\int_{\mathbb{T}^d}|\sqrt{\rho^\epsilon}-1|^2
=&\int_{|\rho^\epsilon-1|\leq
\frac12}|\sqrt{\rho^\epsilon}-1|^2+\int_{|\rho^\epsilon-1|\geq
\frac12}|\sqrt{\rho^\epsilon}-1|^2\nonumber\\
\leq&M\int_{|\rho^\epsilon-1|\leq
\frac12}|\rho^\epsilon-1|^2+M\int_{|\rho^\epsilon-1|\geq
\frac12}|\rho^\epsilon-1|^\gamma\nonumber\\ \leq &M\epsilon^2.\label{ine30} \end{align} Now, we begin to estimate the terms $R^\epsilon_i(t),i=1,\dots, 6$. For the term $R^{\epsilon}_1(t)$, by Young's inequality and the regularity of ${\bf u}$ and ${\bf H}$, we have \begin{align}\label{rr1}
|R^{\epsilon}_1(t)|\leq \frac{\mu^\epsilon}{2} \int^t_0\int_{\mathbb{T}^d}|\nabla {\bf u}^\epsilon|^2
+ \frac{\nu^\epsilon}{2}\int^t_0\int_{\mathbb{T}^d}|\nabla {\bf H}^\epsilon|^2 +C_T\mu^\epsilon+C_T\nu^\epsilon. \end{align}
For the term $R^{\epsilon}_2(t)$, by H\"older's inequality, the estimate \eqref{ine30},
the assumption on the initial data, the estimate on $\sqrt{\rho^\epsilon}{\bf u}^\epsilon$, and the regularity of ${\bf u}$, we infer that \begin{align}\label{rr2}
|R^{\epsilon}_2(t)|\leq & C\epsilon +||{\bf u}(t)||_{L^\infty}
\Big(\int_{\mathbb{T}^d} |\sqrt{\rho^\epsilon}-1|^2 \Big)^{\frac12}
\Big(\int_{\mathbb{T}^d} \rho^\epsilon |{\bf u}^\epsilon|^2\Big)^{\frac12}\nonumber\\
&+ ||[({\bf u}\cdot\nabla){\bf u}](t)||_{L^\infty}
\Big(\int^t_0\int_{\mathbb{T}^d} |\sqrt{\rho^\epsilon}-1|^2\Big)^{\frac12}
\Big(\int^t_0\int_{\mathbb{T}^d} \rho^\epsilon |{\bf u}^\epsilon|^2 \Big)^{\frac12}\nonumber\\ \leq & C_T\epsilon. \end{align}
For the term $R^{\epsilon}_3(t)$, making use of the inequality \eqref{ine30}, the basic inequality \eqref{be}, the estimates on ${\bf u}^\epsilon$ and ${\bf H}^\epsilon$, the regularity of ${\bf H}$, the assumption \eqref{abc}, and Sobolev's imbedding theorem, we get \begin{align}\label{rr3}
|R^\epsilon_3(t)|\leq &(||[({\bf H}\cdot \nabla){\bf H}](t)||_{L^\infty}
+ ||\nabla {\bf H}(t)||_{L^\infty}\cdot||{\bf H}(t)||_{L^\infty})\nonumber\\
& \times\Big(\int^t_0\int_{\mathbb{T}^d}|\sqrt{\rho^\epsilon}-1|^2\Big)^{\frac12}
\Big(\int^t_0\int_{\mathbb{T}^d}|{\bf u}^\epsilon|^2\Big)^{\frac12}\nonumber\\
& + ||\nabla {\bf H}(t)||_{L^\infty}\int^t_0\Big[\Big(\int_{\mathbb{T}^d}|\sqrt{\rho^\epsilon}-1|^2\Big)^{\frac12}
||{\bf u}^\epsilon(\tau)||_{L^6} ||{\bf H}^\epsilon(\tau)||_{L^3}\Big]d\tau\nonumber\\ \leq &
C_T\epsilon(1+(\mu^\epsilon)^{-\frac12})+||\sqrt{\rho^\epsilon}-1||_{L^\infty(0,T;L^2)}
\Big(\int^t_0||{\bf u}^\epsilon(\tau)||^2_{H^1}d\tau\Big)^{\frac12}
\Big(\int^t_0||{\bf H}^\epsilon(\tau)||^2_{H^1}d\tau\Big)^{\frac12}\nonumber\\ \leq & C_T\epsilon(1+(\mu^\epsilon)^{-\frac12}) +C_T\epsilon
(||{\bf u}^\epsilon||_{L^2(0,T;L^2)}
+||\nabla{\bf u}^\epsilon||_{L^2(0,T;L^2)})\nonumber\\
& \times (||{\bf H}^\epsilon||_{L^2(0,T;L^2)}+||\nabla{\bf H}^\epsilon||_{L^2(0,T;L^2)})\nonumber\\ \leq & C_T\epsilon(1+(\mu^\epsilon)^{-\frac12}) +C_T\epsilon\big[1+(\mu^\epsilon)^{-\frac12}\big] \cdot \big[1+(\nu^\epsilon)^{-\frac12}\big]\nonumber\\ \leq & C_T \epsilon^{1-\alpha/2} +C_T \epsilon^{\sigma}\leq C_T \epsilon^{\sigma}, \end{align} where $\sigma =1-(\alpha+\beta)/2.$
For the term $R^{\epsilon}_4(t)$, one can utilize the inequality \eqref{ine30}, the estimates on ${\bf u}^\epsilon$ and $\sqrt{\rho^\epsilon}{\bf u}^\epsilon$, the regularity of ${\bf H}$, and $\rho^\epsilon-1=\rho^\epsilon-\sqrt{\rho^\epsilon} +\sqrt{\rho^\epsilon} -1$ to deduce \begin{align}\label{rr4}
|R^\epsilon_4(t)|\leq &(||[({\bf H}\cdot \nabla){\bf H}](t)||_{L^\infty}+ ||\nabla (|{\bf H}|^2)||_{L^\infty})
\Big(\int^t_0\int_{\mathbb{T}^d}|\sqrt{\rho^\epsilon}-1|^2\Big)^{\frac12}\nonumber\\ & \times\bigg[
\Big(\int^t_0\int_{\mathbb{T}^d}|{\bf u}^\epsilon|^2\Big)^{\frac12}+
\Big(\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon|{\bf u}^\epsilon|^2\Big)^{\frac12} \bigg]\nonumber\\ \leq & C_T \epsilon(1+(\mu^\epsilon)^{-\frac12})\leq C_T \epsilon^{1-\alpha/2} . \end{align}
Using \eqref{ba}, \eqref{L2g} and \eqref{re}, the term $R^{\epsilon}_5(t)$ can be bounded as follows. \begin{align}\label{rr5}
| R^{\epsilon}_5(t)|=&
\Big|\int^t_0 \int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon \cdot \nabla p\Big|\nonumber\\
=& \Big| \int_{\mathbb{T}^d}\big\{((\rho^\epsilon-1) p)(t)-((\rho^\epsilon-1)
p)(0)\big\}- \int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon-1) \partial_t p \Big|\nonumber\\
\leq & \Big(\int_{|\rho^\epsilon-1|\leq \frac12}|\rho^\epsilon-1|^2 \Big)^{\frac12}
\Big[\Big(\int_{\mathbb{T}^3}|p(t)|^2\Big)^{\frac12}+\Big(\int_{\mathbb{T}^3}|p(0)|^2\Big)^{\frac12}\Big]\nonumber\\
& + \Big(\int_{|\rho^\epsilon-1|\geq \frac12}|\rho^\epsilon-1|^\gamma \Big)^{\frac{1}{\gamma}}
\Big[\Big(\int_{\mathbb{T}^3}|p(t)|^{\frac{\gamma}{\gamma-1}}\Big)^{\frac{\gamma-1}{\gamma}}
+\Big(\int_{\mathbb{T}^3}|p(0)|^{\frac{\gamma}{\gamma-1}}\Big)^{\frac{\gamma-1}{\gamma}}\Big]\nonumber\\
&+\int^t_0 \Big(\int_{|\rho^\epsilon-1|\leq \frac12}|\rho^\epsilon-1|^2\Big)^{\frac12}\Big(\int_{\mathbb{T}^d}|\partial_tp(t)|^2 \Big)^{\frac12} \nonumber\\
& + \int^t_0 \Big(\int_{|\rho^\epsilon-1|\geq \frac12} |\rho^\epsilon-1|^{\gamma}\Big)^{{\frac{1}{\gamma}}}
\Big(\int_{\mathbb{T}^d}| \partial_t p(t)|^{\frac{\gamma}{\gamma-1}} \Big)^{{\frac{\gamma-1}{\gamma}}}\nonumber\\ \leq & C_T(\epsilon+\epsilon^{2/\kappa})\leq C_T \epsilon, \end{align} where $\kappa=\min\{2,\gamma\}$ and we have used the conditions $s>2+d/2$ and $\gamma>1$.
Finally, to estimate the term $ R^\epsilon_6(t)$, we rewrite it as \begin{align}\label{rr6}
R^{\epsilon}_6(t)=& -\int^t_0\int_{\mathbb{T}^d}(\sqrt{ \rho^\epsilon}{\bf u}^\epsilon-
{\bf u}) \cdot \nabla \big(\frac{|{\bf u}|^2}{2}\big)\nonumber\\
=&\int^t_0\int_{\mathbb{T}^d}\sqrt{\rho^\epsilon}(\sqrt{\rho^\epsilon}-1)
{\bf u}^\epsilon \cdot \nabla \big(\frac{|{\bf u}|^2}{2}\big)
-\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon \cdot \nabla
\big(\frac{|{\bf u}|^2}{2}\big)\nonumber\\
=& R^{\epsilon}_{61}(t)+R^{\epsilon}_{62}(t), \end{align} where \begin{align*}
R^{\epsilon}_{61}(t)& = \int^t_0\int_{\mathbb{T}^d}\sqrt{\rho^\epsilon}(\sqrt{\rho^\epsilon}-1) {\bf u}^\epsilon
\cdot
\nabla \big(\frac{|{\bf u}|^2}{2}\big),\\ R^{\epsilon}_{62}(t) &=
\int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon-1)
\partial_t\big(\frac{|{\bf u}|^2}{2}\big)
- \int_{\mathbb{T}^d}\Big[\Big((\rho^\epsilon-1)
\big(\frac{|{\bf u}|^2}{2}\big)\Big)(t) -\Big((\rho^\epsilon-1) \big(\frac{|{\bf u}|^2}{2}\big)\Big)(0)\Big]. \end{align*} Applying arguments similar to those used for $R^{\epsilon}_{61}(t)$ and $R^{\epsilon}_{5}(t)$, we arrive at the following boundedness \begin{align}\label{rr66}
|R^{\epsilon}_{6}(t)|\leq |R^{\epsilon}_{61}(t)|+|R^{\epsilon}_{62}(t)|\leq C_T \epsilon. \end{align}
Inserting the estimates \eqref{rr1}-\eqref{rr66} into \eqref{cmi} and applying Gronwall's inequality, we conclude \begin{align}\label{cq}
& ||\mathbf{w}^{\epsilon}(t)||^2_{L^2}+
||\mathbf{Z}^{\epsilon}(t)||^2_{L^2}+||\Pi^\epsilon(t)||^2_{L^2}\nonumber\\ & \qquad \leq \bar C
\big[||\mathbf{w}^{\epsilon}(0)||^2_{L^2}+||\mathbf{Z}^{\epsilon}(0)||^2_{L^2}
+ ||\Pi^\epsilon_0||^2_{L^2} +C_T\epsilon^\sigma\big], \quad\mbox{for a.e.}\;\; t\in [0,T], \end{align} where
\begin{align}\label{ccc} \bar C=\exp {\Big\{C\int^T_0\big[||\nabla
{\bf u}(\tau)||_{L^\infty} +
||\nabla{\bf H}(\tau)||_{L^\infty}\big]d\tau\Big\}}<+\infty. \end{align}
Now, letting $\epsilon $ go to $0$, we obtain $\mathbf{K}={\bf H}$ in $L^\infty(0,T; L^2)$ and $\mathbf{J}={\bf u}$ in $L^\infty(0,T; L^2)$. The inequality \eqref{icd20} follows from \eqref{icd2} and \eqref{cq} directly. Thus, we complete the proof. \end{proof}
\begin{proof}[Proof of Theorem \ref{MRb}]
For simplicity we assume here that $\mu^\epsilon\equiv\mu$, $\lambda^\epsilon\equiv\lambda$, and $\nu^\epsilon\equiv\nu$ are constants, independent of $\epsilon$, satisfying $\mu>0$, $\mu+\lambda>0$, and $\nu>0$. The case \eqref{pa} can be treated similarly. The proof of Theorem \ref{MRb} is similar to that of Theorem \ref{MRa}. Since the viscosity is involved here, we have to modulate the part of dissipation energy in the energy inequality \eqref{be}. We state the main different points in the proof here.
From the basic energy inequality \eqref{be}, we obtain that, for a.e.
$t\in [0,T]$, $\rho^\epsilon |{\bf u}^\epsilon|^2$ and $\big((\rho^\epsilon)^\gamma-1-\gamma(\rho^\epsilon-1)\big)/{\epsilon^2}$ are bounded in $L^\infty(0,T;L^1)$, $ {\bf H}^\epsilon $ is bounded in $L^\infty(0,T;L^2)$, $\nabla {\bf u}^\epsilon $ is bounded in $L^2(0,T;L^2)$, and $\nabla {\bf H}^\epsilon $ is bounded in $L^2(0,T;L^2)$. Therefore, we have \begin{equation*}
\rho^\epsilon \rightarrow 1 \ \
\text{strongly in}\ \ C([0,T],L^\gamma_2(\mathbb{T}^d)), \end{equation*}
and ${\bf u}^\epsilon$ is bounded in $L^2(0,T;L^2)$. The boundedness of $\rho^\epsilon |{\bf u}^\epsilon|^2$ and $ |{\bf H}^\epsilon|^2$ in $L^\infty(0,T;L^1)$ implies the following convergence (up to the extraction of a subsequence $\epsilon_n$): \begin{align*} & \sqrt{\rho^\epsilon} {\bf u}^\epsilon \ \text{converges weakly-$\ast$} \,\, \text{to some}\, \bar{\mathbf{J}} \, \,\text{in}\, L^\infty(0,T;L^2(\mathbb{T}^d)),\\ & {\bf H}^\epsilon \ \text{converges weakly-$\ast$} \,\, \text{to some}\, \bar{\mathbf{K}} \, \,\text{in}\, L^\infty(0,T;L^2(\mathbb{T}^d)). \end{align*} Our main task in this section is to show that $ \bar{\mathbf{J}}={\bf u}$ and $ \bar{\mathbf{K}}={\bf H}$ in some sense, where $({\bf u},{\bf H})$ is the strong solution to the viscous incompressible MHD equations \eqref{a2ll}-\eqref{a2nn}.
Next, we shall also modulate the energy inequality \eqref{be}. The conservation of energy for the viscous incompressible MHD equations \eqref{a2ll}-\eqref{a2nn} reads \begin{equation}\label{cff1}
\frac12 \int_{\mathbb{T}^d}\big[|{\bf u}|^2 + |{\bf H}|^2\big](t)
+\int^t_0 \int_{\mathbb{T}^d} \big[\mu|\nabla{\bf u}|^2+ \nu|\nabla{\bf H}|^2\big]
=\frac12 \int_{\mathbb{T}^d} \big[|{\bf u}_0|^2+|{\bf H}_0|^2\big]. \end{equation} Similarly to Step 2, we use ${\bf u}$ to test the momentum equation \eqref{a2j} to deduce \begin{align}\label{civ1} & \int_{\mathbb{T}^d}(\rho^\epsilon {\bf u}^\epsilon \cdot {\bf u})(t) +\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon \cdot \big[({\bf u}\cdot \nabla)
{\bf u}-({\bf H}\cdot \nabla){\bf H}-\mu \Delta {\bf u}+\nabla p+\frac12\nabla(|{\bf H}|^2)\big] \nonumber\\ & -\int^t_0\int_{\mathbb{T}^d}\big[(\rho^\epsilon {\bf u}^\epsilon \otimes {\bf u}^\epsilon)\cdot \nabla {\bf u}+ ({\bf H}^\epsilon\cdot \nabla){\bf H}^\epsilon \cdot {\bf u}
- \mu \nabla {\bf u}^\epsilon \cdot \nabla {\bf u}\big] = \int_{\mathbb{T}^d}
\rho^\epsilon_0 {\bf u}^\epsilon_0 \cdot {\bf u}_0. \end{align} Then, we test \eqref{a2k} by ${\bf H}$ to infer that \begin{align}\label{cjj1} & \int_{\mathbb{T}^d}({\bf H}^\epsilon \cdot {\bf H})(t) +\int^t_0\int_{\mathbb{T}^d}{\bf H}^\epsilon \cdot\big[({\bf u}\cdot \nabla) {\bf H}-({\bf H}\cdot \nabla){\bf u}-\nu \Delta {\bf H}\big] + \nu \int^t_0\int_{\mathbb{T}^d} \nabla {\bf H}^\epsilon \cdot \nabla {\bf H}\nonumber\\ & \quad +\int^t_0\int_{\mathbb{T}^d}\big[({\rm div } {{\bf u}}^\epsilon) { {\bf H}}^\epsilon + ( {{\bf u}}^\epsilon \cdot \nabla) {{\bf H}}^\epsilon - ( {{\bf H}}^\epsilon\cdot \nabla)
{{\bf u}}^\epsilon\big]\cdot {\bf H} = \int_{\mathbb{T}^d}{\bf H}^\epsilon_0 \cdot {\bf H}_0. \end{align} Summing up \eqref{cei} and \eqref{cff1}, and inserting \eqref{cii}, \eqref{cj1i} with $\mu^\epsilon\equiv\mu$ and $\lambda^\epsilon\equiv\lambda$, \eqref{civ1}, and \eqref{cjj1} into the resulting inequality, we deduce the following inequality by a straightforward calculation \begin{align}\label{ckk}
& \ \ \frac{1}{2}\int_{\mathbb{T}^d}\Big\{ |\sqrt{\rho^\epsilon} {\bf u}^\epsilon
-{\bf u}|^2(t)+ | {\bf H}^\epsilon
-{\bf H}|^2(t)
+(\Pi^\epsilon)^2(t)\Big\}\nonumber\\ &\quad
+ \mu\int^t_0\int_{\mathbb{T}^d}|\nabla {\bf u}^\epsilon -\nabla {\bf u} |^2
+ \frac{\nu}{2} \int^t_0\int_{\mathbb{T}^d}|\nabla {\bf H}^\epsilon-\nabla {\bf H} |^2
+ (\mu+\lambda)\int^t_0\! \int_{\mathbb{T}^d}|{\rm div}{\bf u}^\epsilon |^2\nonumber\\ &\quad
+ \frac{\nu}{2} \int^t_0\int_{\mathbb{T}^d}|\nabla {\bf H}^\epsilon|^2
+ \frac{\nu}{2} \int^t_0\int_{\mathbb{T}^d}|\nabla {\bf H}|^2 \nonumber\\ &\leq -\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon\cdot [({\bf H}\cdot \nabla) {\bf H})] -\int^t_0\int_{\mathbb{T}^d} ({\bf H}^\epsilon\cdot \nabla) {\bf H}^\epsilon \cdot{\bf u} \nonumber\\ &\quad +\int^t_0\int_{\mathbb{T}^d} {\bf H}^\epsilon \cdot\big[({\bf u}\cdot \nabla) {\bf H}-({\bf H}\cdot \nabla){\bf H}\big] +\frac12\int^t_0\int_{\mathbb{T}^d}
\rho^\epsilon {\bf u}^\epsilon \cdot \nabla(|{\bf H}|^2)\nonumber\\ &\quad +\int^t_0\int_{\mathbb{T}^d}\big[({\rm div } {{\bf u}}^\epsilon) { {\bf H}}^\epsilon
+ ( {{\bf u}}^\epsilon \cdot \nabla) {{\bf H}}^\epsilon
- ( {{\bf H}}^\epsilon\cdot \nabla) {{\bf u}}^\epsilon\big]\cdot {\bf H} +\mu\int^t_0\int_{\mathbb{T}^d}(1-\rho^\epsilon) {\bf u}^\epsilon \Delta u
\nonumber\\ &\quad +\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon \cdot\big[ ({\bf u}\cdot \nabla) {\bf u} +\nabla p \big]-\int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon {\bf u}^\epsilon\otimes {\bf u}^\epsilon)\cdot \nabla {\bf u}\nonumber\\ &\quad +\int_{\mathbb{T}^d} \big[(\sqrt{\rho^\epsilon}-1)\sqrt{\rho^\epsilon}{\bf u}^\epsilon\cdot {\bf u}\big](t)
-\int_{\mathbb{T}^d} \big[(\sqrt{\rho^\epsilon}-1)\sqrt{\rho^\epsilon}{\bf u}^\epsilon\cdot {\bf u}\big](0)\nonumber\\
&\quad +\frac{1}{2}\int_{\mathbb{T}^d}\Big\{ |\sqrt{\rho^\epsilon} {\bf u}^\epsilon
-{\bf u}|^2(0)+|{\bf H}^\epsilon
-{\bf H} |^2(0)
+(\Pi^\epsilon_0)^2\Big\}. \end{align}
By virtue of \eqref{ck1i} and \eqref{cli}, we can rewrite the inequality \eqref{ckk} as follows \begin{align}\label{ckk1}
& \ \ \frac{1}{2}\int_{\mathbb{T}^d}\Big\{ |\sqrt{\rho^\epsilon} {\bf u}^\epsilon
-{\bf u}|^2(t)+ | {\bf H}^\epsilon
-{\bf H}|^2(t)
+(\Pi^\epsilon)^2(t)\Big\}\nonumber\\ &\quad
+ {\mu} \int^t_0\int_{\mathbb{T}^d}|\nabla {\bf u}^\epsilon -\nabla {\bf u} |^2
+ \frac{\nu}{2} \int^t_0\int_{\mathbb{T}^d}|\nabla {\bf H}^\epsilon-\nabla {\bf H} |^2\nonumber\\
&\quad + (\mu+\lambda)\int^t_0\! \int_{\mathbb{T}^d}|{\rm div}{\bf u}^\epsilon|^2
+ \frac{\nu}{2} \int^t_0\int_{\mathbb{T}^d}|\nabla {\bf H}^\epsilon|^2
+ \frac{\nu}{2} \int^t_0\int_{\mathbb{T}^d}|\nabla {\bf H}|^2 \nonumber\\ &\leq
\frac{1}{2}\int_{\mathbb{T}^d}\Big\{ |\sqrt{\rho^\epsilon} {\bf u}^\epsilon
-{\bf u}|^2(0)+|{\bf H}^\epsilon
-{\bf H} |^2(0)
+(\Pi^\epsilon_0)^2\Big\}\nonumber\\
& \quad + R^\epsilon_2(t)+ R^\epsilon_4(t)+ R^\epsilon_5(t)+ R^\epsilon_6(t)+ R^\epsilon_7(t)+ R^\epsilon_8(t), \end{align} where $R^\epsilon_2(t)$ and $R^\epsilon_i(t)$, $i=4,5,6$, are the same as before, and \begin{align*}
R^\epsilon_7(t)&=\mu\int^t_0\int_{\mathbb{T}^d}(1-\rho^\epsilon) {\bf u}^\epsilon \Delta u,\\
R^\epsilon_8(t) &=
\int^t_0\!\int_{\mathbb{T}^d}(\mathbf{Z}^\epsilon \cdot \nabla) {\bf H} \cdot [(1-\sqrt{\rho^\epsilon}){\bf u}^\epsilon] - \int^t_0\!\int_{\mathbb{T}^d}\big\{[(1-\sqrt{\rho^\epsilon}){\bf u}^\epsilon ]\cdot \nabla\big\}{\bf H} \cdot \mathbf{Z}^\epsilon. \end{align*}
Form the previous arguments on $R^\epsilon_2(t)$ and $R^\epsilon_i(t)$, $i=4,5,6$, we get \begin{align}\label{ckk2}
|R^\epsilon_2(t)|+\sum^6_{i=4}| R^\epsilon_i(t)|\leq C_T \epsilon. \end{align}
Now, we estimate $R^\epsilon_7(t)$ and $R^\epsilon_8(t)$. Using the inequality \eqref{ine30}, H\"older's inequality, the estimates on ${\bf u}^\epsilon$ and $\sqrt{\rho^\epsilon}{\bf u}^\epsilon$,
the regularity of ${\bf u}$, and $\rho^\epsilon-1=\rho^\epsilon-\sqrt{\rho^\epsilon}+\sqrt{\rho^\epsilon} -1$, we obtain \begin{align}\label{ckk3}
|R^\epsilon_7(t)|\leq & \mu ||\Delta{\bf u}(t)||_{L^\infty}
\Big(\int^t_0\int_{\mathbb{T}^d}|\sqrt{\rho^\epsilon}-1|^2\Big)^{\frac12}
\bigg[
\Big(\int^t_0\int_{\mathbb{T}^d}|{\bf u}^\epsilon|^2\Big)^{\frac12}+
\Big(\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon|{\bf u}^\epsilon|^2\Big)^{\frac12} \bigg]\nonumber\\ \leq & C_T \epsilon(1+\frac{1}{\sqrt{\mu}})\leq C_T \frac{\epsilon}{\sqrt{\mu}}. \end{align}
For the term $R^{\epsilon}_8(t)$, we can make use of \eqref{ine30}, \eqref{be}, and the estimates on ${\bf u}^\epsilon$ and ${\bf H}^\epsilon$, the regularity of ${\bf H}$, the assumption \eqref{abc}, and Sobolev's imbedding theorem to deduce \begin{align}\label{ckk4}
|R^\epsilon_8(t)|\leq &(||[({\bf H}\cdot \nabla){\bf H}](t)||_{L^\infty}
+ ||\nabla {\bf H}(t)||_{L^\infty}\cdot||{\bf H}(t)||_{L^\infty})\nonumber\\
& \times\Big(\int^t_0\int_{\mathbb{T}^d}|\sqrt{\rho^\epsilon}-1|^2\Big)^{\frac12}
\Big(\int^t_0\int_{\mathbb{T}^d}|{\bf u}^\epsilon|^2\Big)^{\frac12}\nonumber\\
& + ||\nabla {\bf H}(t)||_{L^\infty}\int^t_0\Big[\Big(\int_{\mathbb{T}^d}|
\sqrt{\rho^\epsilon}-1|^2\Big)^{\frac12}
||{\bf u}^\epsilon(\tau)||_{L^6} ||{\bf H}^\epsilon(\tau)||_{L^3}\Big]d\tau\nonumber\\ \leq &
C_T\epsilon(1+\frac{1}{\sqrt{\mu}})+||\sqrt{\rho^\epsilon-1}||_{L^\infty(0,T;L^2)}
\Big(\int^t_0||{\bf u}^\epsilon(\tau)||^2_{H^1}d\tau\Big)^{\frac12}
\Big(\int^t_0||{\bf H}^\epsilon||^2_{H^1}(\tau)d\tau\Big)^{\frac12}\nonumber\\ \leq & C_T\epsilon(1+\frac{1}{\sqrt{\mu}}) +C_T\epsilon
(||{\bf u}^\epsilon||_{L^2(0,T;L^2)}
+||\nabla{\bf u}^\epsilon||_{L^2(0,T;L^2)})\nonumber\\ & \times
(||{\bf H}^\epsilon||_{L^2(0,T;L^2)}+||\nabla{\bf H}^\epsilon||_{L^2(0,T;L^2)})\nonumber\\ \leq & C_T\epsilon(1+\frac{1}{\sqrt{\mu}}) +C_T\epsilon\big[1+\mu^{-\frac12}\big] \cdot \big[1+\nu^{-\frac12}\big]\nonumber\\ \leq & C_T \epsilon +C_T \epsilon/\sqrt{\mu\nu}\leq C_T \epsilon/\sqrt{\mu\nu}. \end{align}
Now, substituting \eqref{ckk2}-\eqref{ckk4} into \eqref{ckk1} and applying Gronwall's inequality, we conclude \begin{align}\label{ckk5}
& ||\mathbf{w}^{\epsilon}(t)||^2_{L^2}+
||\mathbf{Z}^{\epsilon}(t)||^2_{L^2}+||\Pi^\epsilon(t)||^2_{L^2}\nonumber\\ & \quad \leq \bar C
\big[||\mathbf{w}^{\epsilon}(0)||^2_{L^2}+||\mathbf{Z}^{\epsilon}(0)||^2_{L^2}
+ ||\Pi^\epsilon_0||^2_{L^2} +C_T\epsilon/\sqrt{\mu\nu}\big], \quad\mbox{for a.e.}\;\; t\in [0,T], \end{align} where $\bar C$ is defined by \eqref{ccc}. Combining \eqref{icd22} with \eqref{ckk5} we obtain \eqref {icd220}. Substituting \eqref{ckk5} into \eqref{ckk1}, we conclude that $\nabla {\bf u}^\epsilon$ converges to $\nabla {\bf u}$ strongly
in $L^2 (0,T;L^2(\mathbb{T}^d))$
and $\nabla {\bf H}^\epsilon$ to $\nabla {\bf H}$ strongly
in $L^2 (0,T; L^2(\mathbb{T}^d))$. This completes the proof of Theorem \ref{MRb}. \end{proof}
\section{Proof of Theorem \ref{MRc}}
In this section we shall study the incompressible limit of the compressible MHD equations \eqref{a2i}-\eqref{a2k} with general initial data. Compared with the case of the well-prepared initial data, the main difficulty here is to control the oscillations caused by the initial data. For simplicity, we assume here that $\mu^\epsilon\equiv\mu$, $\lambda^\epsilon\equiv\lambda$, and $\nu^\epsilon\equiv\nu$ are constants, independent of $\epsilon$, satisfying $\mu>0$, $2\mu+d\lambda>0$, and $\nu>0$.
\begin{proof}[Proof of Theorem \ref{MRc}]
As stated in the proof of Theorem \ref{MRb},
we obtain from the basic energy inequality \eqref{be} that, for a.e.
$t\in [0,T]$, $\rho^\epsilon |{\bf u}^\epsilon|^2$ and $ \big((\rho^\epsilon)^\gamma-1-\gamma(\rho^\epsilon-1)\big)/{\epsilon^2}$ are bounded in $L^\infty(0,T;L^1)$, $ {\bf H}^\epsilon $ is bounded in $L^\infty(0,T;L^2)$, $\nabla {\bf u}^\epsilon $ is bounded in $L^2(0,T;L^2)$, and $\nabla {\bf H}^\epsilon $ is bounded in $L^2(0,T;L^2)$. Therefore, we have \begin{equation}\label{rhoc}
\rho^\epsilon \rightarrow 1 \ \
\text{strongly in}\ \ C([0,T],L^\gamma_2(\mathbb{T}^d)), \end{equation}
and ${\bf u}^\epsilon$ is bounded in $L^2(0,T;L^2)$. The fact that $\rho^\epsilon |{\bf u}^\epsilon|^2$ and $ |{\bf H}^\epsilon|^2$ are bounded in $L^\infty(0,T;L^1)$ gives the following convergence (up to the extraction of a subsequence $\epsilon_n$): \begin{align*} & \sqrt{\rho^\epsilon} {\bf u}^\epsilon \ \text{converges weakly-$\ast$} \,\, \text{to some}\, { \bar{\mathbf{J}}} \, \,\text{in}\, L^\infty(0,T;L^2(\mathbb{T}^d)),\\ & {\bf H}^\epsilon \ \text{converges weakly-$\ast$} \,\, \text{to some}\, { \bar{\mathbf{K}}} \, \,\text{in}\, L^\infty(0,T;L^2(\mathbb{T}^d)). \end{align*} Our main task in this section is to show that $ \bar{\mathbf{J}} ={\bf u}$ and $ \bar{\mathbf{K}} ={\bf H}$ in some sense, where $({\bf u},{\bf H})$ is the strong solution to the incompressible viscous MHD equations \eqref{a2ll}-\eqref{a2nn}. The key point is to control the oscillations caused by the
initial data. This can be done as follows.
\emph{Step 1: Description and cancelation of the oscillations.}
In order to describe the oscillations caused by the initial data, we employ the ``filtering" method which has been used previously by several authors, see~\cite{LM98,G,M01b,D02}.
We project the momentum equation \eqref{a2j} on the ``gradient vector-fields'' to find \begin{align}\label{Pr} &\partial_t Q(\rho^\epsilon{\bf u}^\epsilon)+Q[{\rm div }(\rho^\epsilon{\bf u}^\epsilon\otimes{\bf u}^\epsilon)]
- (2\mu+\lambda)\nabla{\rm div }{\bf u}^\epsilon+\frac12\nabla(|{{\bf H}}^\epsilon|^2)\nonumber\\
&\quad -Q[({{\bf H}}^\epsilon\cdot \nabla
){{\bf H}}^\epsilon] +\frac{a}{\epsilon^2}\nabla \big((\rho^\epsilon)^\gamma
-1-\gamma (\rho^\epsilon-1)\big)+\frac{1}{\epsilon^2}\nabla (\rho^\epsilon-1)=0.
\end{align} Noticing $\rho^\epsilon=1+\epsilon\varphi^\epsilon $, we can write \eqref{a2i} and \eqref{Pr} as \begin{align} &\epsilon\partial_t \varphi^\epsilon+{\rm div } Q(\rho^\epsilon {\bf u}^\epsilon)=0,\label{ep1}\\
&\epsilon\partial_t Q(\rho^\epsilon {\bf u}^\epsilon)+ \nabla \varphi^\epsilon=\epsilon\mathbf{F}^\epsilon,\label{ep2} \end{align} where $ \mathbf{F}^\epsilon$ is given by \begin{align} \mathbf{F}^\epsilon=&-Q[{\rm div }(\rho^\epsilon{\bf u}^\epsilon\otimes{\bf u}^\epsilon)]
+(2\mu+\lambda)\nabla{\rm div }{\bf u}^\epsilon-\frac12\nabla(|{{\bf H}}^\epsilon|^2)\nonumber\\
& +Q[({{\bf H}}^\epsilon\cdot \nabla
){{\bf H}}^\epsilon] -\frac{a}{\epsilon^2}\nabla \big((\rho^\epsilon)^\gamma
-1-\gamma (\rho^\epsilon-1)\big). \label{ff} \end{align} Therefore, we introduce the following group defined by $\mathcal{L}(\tau)=e^{\tau L}$, $\tau \in \mathbb{R}$, where $L$ is the operator defined on $\mathcal{D}_0'\times (\mathcal{D}')^{d}$ with $\mathcal{D}_0'=\{\phi \in \mathcal{D}', \int_{\mathbb{T}^d}\phi(x)dx=0\}$, by $$ L\Big(\begin{array}{c}
\phi \\ \mathbf{v} \end{array} \Big) =\Big(\begin{array}{c}
-\text{div}\,\mathbf{v} \\ -\nabla \phi \end{array} \Big). $$ Then, it is easy to check that $e^{\tau L}$ is an isometry on each
$H^r\times (H^r)^d$ for all $r\in \mathbb{R}$ and for all $\tau \in
\mathbb{R}$. Denoting $$ \Big(\begin{array}{c}
\bar \phi (\tau) \\ \bar{\mathbf{v}}(\tau) \end{array} \Big) =e^{\tau L}\Big(\begin{array}{c}
\phi \\ \mathbf{v} \end{array} \Big), $$ we have $$ \frac{\partial \bar \phi}{\partial \tau}=-\text{div} \bar{\mathbf{v}}, \quad
\frac{\partial \bar{\mathbf{v}}}{\partial \tau}=-\nabla \bar \phi. $$ Thus, $\frac{\partial^2\bar \phi}{\partial \tau ^2}-\Delta \bar \phi
=0$.
In the sequel, we shall denote
$$\mathbf{U}^\epsilon =\left(\begin{array}{c}
\varphi^\epsilon \\
Q(\rho^\epsilon {\bf u}^\epsilon)
\end{array}\right),\quad \mathbf{V}^\epsilon= \mathcal{L}\Big(-\frac{t}{\epsilon}\Big) \left(\begin{array}{c}
\varphi^\epsilon\\
Q(\rho^\epsilon {\bf u}^\epsilon)
\end{array}\right)
$$ and use the following approximations
$$\bar{\mathbf{U}}^\epsilon =\left(\begin{array}{c}
\Phi^\epsilon \\
Q(\sqrt{\rho^\epsilon} {\bf u}^\epsilon)
\end{array}\right),\quad \bar{\mathbf{V}}^\epsilon= \mathcal{L}\Big(-\frac{t}{\epsilon}\Big) \left(\begin{array}{c}
\Phi^\epsilon\\
Q(\sqrt{\rho^\epsilon} {\bf u}^\epsilon)
\end{array}\right),
$$ which satisfy \begin{align}\label{UU}
||{\mathbf{U}}^\epsilon-\bar{\mathbf{U}}^\epsilon||_{L^\infty(0,T;
L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^d))}\rightarrow 0 \ \
\text{as} \ \ \epsilon \rightarrow 0. \end{align}
With this notation, we can rewrite the equations \eqref{ep1}-\eqref{ep2} as $$ \partial_t \mathbf{U}^\epsilon =\frac{1}{\epsilon} L \mathbf{U}^\epsilon +\widehat{\mathbf{F}^\epsilon}, $$ or equivalently \begin{align}\label{ep3} \partial_t \mathbf{V}^\epsilon = \mathcal{L}\Big(-\frac{t}{\epsilon}\Big) \widehat{\mathbf{F}^\epsilon}, \end{align} where (and in what follows) $\widehat{\mathbf{v}} $ denotes $(0, \mathbf{v})^\mathrm{T}$.
It is easy to check that $\mathbf{F}^\epsilon$, given by \eqref{ff}, is bounded in $L^2(0,T;H^{-s_0}(\mathbb{T}^d))$ for some $s_0$ ($ s_0\in \mathbb{R})$. Hence, $\mathbf{V}^\epsilon$ is compact in time. Moreover, by virtue of the energy inequality \eqref{be} and the boundedness of the linear projector $P$, $\mathbf{V}^\epsilon \in L^\infty(0,T; L^{\frac{2\gamma}{\gamma+1}}(\mathbb{T}^d))$ uniformly in $\epsilon$. Thus, \begin{align}\label{ep4} \mathbf{V}^\epsilon \,\, \text{converges strongly to some}\,
\, {\bar{\mathbf{V}}} \,\, \text{in}\,\, L^r(0,T;H^{-s'}(\mathbb{T}^d)) \end{align}
for all $ s'> s_0$ and $1 < r<\infty$.
Denote $\theta\equiv 2\mu+\lambda$, $\mathcal{L}_1(\tau)$ the first component of $ \mathcal{L}(\tau)$, and $\mathcal{L}_2(\tau)$ the last $d$ components of $ \mathcal{L}(\tau)$. If we had sufficient compactness in space, then we could pass the limit in \eqref{ep3} and obtain the following limit system for the oscillating parts \begin{align}\label{osci}
\partial_t \bar{\mathbf{V}} +\mathcal{Q}_1({\bf u}, \bar{\mathbf{V}})
+\mathcal{Q}_2(\bar{\mathbf{V}}, \bar{\mathbf{V}})-\frac{\theta}{2}\Delta \bar{\mathbf{V}}=0, \end{align} where ${\bf u}$ is the strong solution of the viscous incompressible MHD
equations \eqref{a2ll}-\eqref{a2nn}, $\mathcal{Q}_1$ is a linear form of ${\mathbf{V}}$ defined by \begin{align}\label{ep5} \mathcal{Q}_1({\bf v}, \mathbf{V})=\lim_{\tau \rightarrow \infty}\frac{1}{\tau} \int^\tau_0 \mathcal{L}(-s) \bigg( \begin{array}{c}
0 \\
{\rm div }({\bf v}\otimes \mathcal{L}_2(s)\mathbf{V}+\mathcal{L}_2(s)\mathbf{V}\otimes{\bf v}) \end{array} \bigg)ds, \end{align} and $\mathcal{Q}_2$ is a bilinear form of ${\mathbf{V}}$ defined by \begin{align}\label{ep6} \mathcal{Q}_2(\mathbf{V}, \mathbf{V})=\lim_{\tau \rightarrow \infty}\frac{1}{\tau} \int^\tau_0 \mathcal{L}(-s) \bigg( \begin{array}{c}
0 \\
{\rm div }( \mathcal{L}_2(s)\mathbf{V}\otimes \mathcal{L}_2(s)\mathbf{V})
+\frac{\gamma-1}{2}\nabla (\mathcal{L}_1(s)\mathbf{V})^2 \end{array} \bigg)ds \end{align} for any divergence-free vector field ${\bf v} \in L^2(\mathbb{T}^d)^d$ and any $\mathbf{V} =(\phi, \nabla q)^{\mathrm{T}} \in L^2(\mathbb{T}^{d})^{d+1}$. Actually, the convergence in \eqref{ep5} and \eqref{ep6} can be guaranteed by the following Proposition.
\begin{prop}[\cite{M01b}]\label{P2} For all ${\bf v}\in L^{r_1}(0,T;L^2)$ and $\mathbf{V}\in L^{r_2}(0,T;L^2)$, we have the following weak convergences (${r_1}$ and ${r_2}$ are such that the products are well defined) \begin{align}
& w-\lim_{\epsilon\rightarrow 0} \mathcal{L}
\Big(-\frac{t}{\epsilon}\Big)\bigg( \begin{array}{c}
0 \\
{\rm div }({\bf v}\otimes \mathcal{L}_2(\frac{t}{\epsilon})\mathbf{V}
+\mathcal{L}_2(\frac{t}{\epsilon})\mathbf{V}\otimes{\bf v}) \end{array} \bigg)= \mathcal{Q}_1({\bf v},\mathbf{V}), \label{ep13}\\
& w-\lim_{\epsilon\rightarrow 0} \mathcal{L}
\Big(-\frac{t}{\epsilon}\Big)\bigg(
\begin{array}{c}
0 \\
{\rm div }( \mathcal{L}_2(\frac{t}{\epsilon})\mathbf{V}\otimes \mathcal{L}_2(\frac{t}{\epsilon})\mathbf{V})
+\frac{\gamma-1}{2}\nabla (\mathcal{L}_1(\frac{t}{\epsilon})\mathbf{V})^2 \end{array} \bigg) = \mathcal{Q}_2(\mathbf{V},\mathbf{V}). \label{ep14} \end{align} \end{prop} The viscosity term in the oscillation equations \eqref{osci} is obtained by the following Proposition.
\begin{prop}[\cite{M01b}]\label{Pv} Suppose that the same hypothesis as in Proposition \ref{P2} on $\mathbf{V}$ holds. Then, we have \begin{align}\label{ep7}
\frac{\theta}{2}\Delta \mathbf{V}=\lim_{\tau \rightarrow \infty}\frac{1}{\tau} \int^\tau_0 \mathcal{L}(-s) \bigg( \begin{array}{c}
0 \\
\theta \Delta \mathcal{L}_2(s)\mathbf{V} \end{array} \bigg)ds. \end{align} \end{prop}
The following propositions, the proof of which can be found in \cite{M01b}, play an important role in our subsequent analysis.
\begin{prop}[\cite{M01b}]\label{P1} For all ${\bf v},\mathbf{V}, \mathbf{V}_1 $ and $ \mathbf{V}_2 $ (regular enough to define all the products), we have \begin{align}
& \int_{\mathbb{T}^d} \mathcal{Q}_1({\bf v},\mathbf{V} )\mathbf{V} =0, \ \
\int_{\mathbb{T}^d} \mathcal{Q}_2(\mathbf{V},\mathbf{V} )\mathbf{V} =0, \label{ep10}\\ & \int_{\mathbb{T}^d} [\mathcal{Q}_1({\bf v},\mathbf{V}_1)\mathbf{V}_2+
\mathcal{Q}_1({\bf v},\mathbf{V}_2)\mathbf{V}_1]=0, \label{ep11}\\
& \int_{\mathbb{T}^d} [\mathcal{Q}_2(\mathbf{V}_1,\mathbf{V}_1)\mathbf{V}_2+
2\mathcal{Q}_2(\mathbf{V}_1,\mathbf{V}_2)\mathbf{V}_1]=0. \label{ep12} \end{align} \end{prop}
\begin{prop}[\cite{M01b}]\label{P3} Using the symmetry of $\mathcal{Q}_2$, we can extend the equality \eqref{ep14} in Proposition \ref{P2} to the case: \begin{align}\label{ep77}
& w-\lim_{\epsilon\rightarrow 0} \mathcal{L}
\Big(-\frac{t}{\epsilon}\Big)\bigg\{\bigg( \begin{array}{c}
0 \\
\frac12{\rm div }\big[\mathcal{L}_2(\frac{t}{\epsilon})\mathbf{V}_1\otimes
\mathcal{L}_2(\frac{t}{\epsilon})\mathbf{V}_2 +\mathcal{L}_2(\frac{t}{\epsilon})
\mathbf{V}_2\otimes \mathcal{L}_2(\frac{t}{\epsilon})\mathbf{V}_1\big] \end{array} \bigg) \nonumber\\
&\qquad \qquad \qquad \quad \ \ + \bigg( \begin{array}{c}
0 \\
\frac{\gamma-1}{2}\nabla (\mathcal{L}_1(\frac{t}{\epsilon})\mathbf{V}_1 \otimes
\mathcal{L}_1(\frac{t}{\epsilon})\mathbf{V}_2)
\end{array}\bigg)\bigg\}
= \mathcal{Q}_2(\mathbf{V}_1,\mathbf{V}_2). \end{align} Moreover, the above identity holds for $\mathbf{V}_1\in L^q(0,T;H^r)$ and $\mathbf{V}_2\in L^p(0,T;H^{-r})$ with $r\in \mathbb{R}$ and $1/p +1/q=1$. Also, (\ref{ep77}) can be extended to the case where we replace $\mathbf{V}_2$ in the left-hand side by a sequence $\mathbf{V}^\epsilon_2$ such that $\mathbf{V}^\epsilon_2$ converges strongly to $\mathbf{V}_2$ in $ L^p(0,T;H^{-r})$. \end{prop}
\emph{Step 2: The modulated energy functional and uniform estimates.}
Let $\mathbf{V}^0$ be the solution of the following system
\begin{align}\label{ep8}
\partial_t \mathbf{V}^0 +\mathcal{Q}_1({\bf u}, \mathbf{V}^0)
+\mathcal{Q}_2(\mathbf{V}^0, \mathbf{V}^0)-\frac{\theta}{2}\Delta \mathbf{V}^0=0
\end{align}
with initial data
\begin{align}\label{ep9}
{\mathbf{V}^0}|_{t=0}=(\varphi_0,Q\tilde{{\bf u}}_0)^{\mathrm{T}},
\end{align} where ${\bf u}$ is the strong solution of the viscous incompressible MHD equations \eqref{a2ll}-\eqref{a2nn} with initial velocity ${\bf u}_0$. From \cite{M01b}, we know that the Cauchy problem \eqref{ep8}-\eqref{ep9} has a unique global strong solution.
In order to prove the convergence results in Theorem \ref{MRc}, we have to bound the term $$
\Big\|\sqrt{\rho^\epsilon} {\bf u}^\epsilon -{\bf u}-\mathcal{L}_2\Big(\frac{t}{\epsilon}\Big)\mathbf{V}^0\Big\|^2_{L^2{(\mathbb{T}^d})}
+\|{\bf H}^\epsilon-{\bf H}\|^2_{L^2{(\mathbb{T}^d})}
+\Big\|\Pi^\epsilon-\mathcal{L}_1\Big(\frac{t}{\epsilon}\Big)\mathbf{V}^0\Big\|^2_{L^2({\mathbb{T}^d})}. $$
To this end, we first recall the following energy inequality of the compressible MHD equations \eqref{a2i}-\eqref{a2k}:
\begin{align}\label{ce}
& \frac12\int_{\mathbb{T}^d}\Big[\rho^\epsilon(t)|{\bf u}^\epsilon|^2(t)+|{\bf H}^\epsilon|^2(t)
+(\Pi^\epsilon(t))^2
\Big]
+\mu \int^t_0\! \int_{\mathbb{T}^d}|\nabla {\bf u}^\epsilon|^2 \nonumber\\
&\ \ +(\mu+\lambda)\int^t_0\! \int_{\mathbb{T}^d}|{\rm div}{\bf u}^\epsilon |^2
+\nu\int^t_0\! \int_{\mathbb{T}^d}|\nabla {\bf H}^\epsilon |^2\nonumber\\ & \ \ \ \leq
\frac12\int_{\mathbb{T}^d}\Big[\rho_0^\epsilon|{\bf u}_0^\epsilon|^2
+|{\bf H}_0^\epsilon|^2+(\Pi^\epsilon_0)^2 \Big], \quad\mbox{for a.e.}\;\; t\in [0,T].
\end{align} On the other hand, the conservation of energy for the incompressible viscous MHD equations \eqref{a2ll}-\eqref{a2nn} reads \begin{equation}\label{cff}
\frac12 \int_{\mathbb{T}^d}\big[|{\bf u}|^2(t)+ |{\bf H}|^2(t)\big]
+\int^t_0 \int_{\mathbb{T}^d} \big[\mu|\nabla{\bf u} |^2+ \nu|\nabla{\bf H} |^2\big]
=\frac12 \int_{\mathbb{T}^d} \big[|{\bf u}_0|^2+|{\bf H}_0|^2\big]. \end{equation}
For the system \eqref{ep8}, Proposition \ref{P1} implies that $$ \int_{\mathbb{T}^d}\mathcal{Q}_1({\bf u}, \mathbf{V}^0) \mathbf{V}^0=0,\ \ \int_{\mathbb{T}^d}\mathcal{Q}_2(\mathbf{V}^0, \mathbf{V}^0) \mathbf{V}^0=0, $$ from which the following equality follows. \begin{align}\label{cfh}
\frac12 \int_{\mathbb{T}^d}| \mathbf{V}^0|^2+\frac{\theta}{2} \int_{\mathbb{T}^d}|\nabla \mathbf{V}^0|^2
= \frac12\int_{\mathbb{T}^d}| \mathbf{V}^0(t=0)|^2. \end{align}
Using $\mathcal{L}_1(\frac{t}{\epsilon})\mathbf{V}^0$ as a test function and noticing $\rho^\epsilon=1+\epsilon \varphi^\epsilon$, we obtain the following weak formulation of the continuity equation \eqref{a2i} \begin{align}\label{ch}
\int_{\mathbb{T}^d} \mathcal{L}_1\Big(\frac{t}{\epsilon}\Big)\mathbf{V}^0\varphi^\epsilon(t) &+\frac{1}{\epsilon}\int^t_0\int_{\mathbb{T}^d} \Big[\text{div}\Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\varphi^\epsilon+{\rm div }(\rho^\epsilon {\bf u}^\epsilon)
\mathcal{L}_1\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big]\nonumber\\
& -\int^t_0\int_{\mathbb{T}^d}
\mathcal{L}_1\Big(\frac{\tau}{\epsilon}\Big)\partial_t\mathbf{V}^0 \varphi^\epsilon
=\int_{\mathbb{T}^d} \varphi_0\varphi^\epsilon_0. \end{align} We use ${\bf u}$ and $\mathcal{L}_2 (\frac{t}{\epsilon} )\mathbf{V}^0$ to test the momentum equation \eqref{a2j} respectively, to deduce \begin{align}\label{civ} & \int_{\mathbb{T}^d}(\rho^\epsilon {\bf u}^\epsilon \cdot {\bf u})(t) +\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon \cdot \big[({\bf u}\cdot \nabla)
{\bf u}-({\bf H}\cdot \nabla){\bf H}-\mu \Delta {\bf u}+\nabla p+\frac12\nabla(|{\bf H}|^2)\big] \nonumber\\ & -\int^t_0\int_{\mathbb{T}^d}\big[(\rho^\epsilon {\bf u}^\epsilon \otimes {\bf u}^\epsilon)\cdot \nabla {\bf u}+ ({\bf H}^\epsilon\cdot \nabla){\bf H}^\epsilon \cdot {\bf u}
- \mu \nabla {\bf u}^\epsilon \cdot \nabla {\bf u}\big] = \int_{\mathbb{T}^d}
\rho^\epsilon_0 {\bf u}^\epsilon_0 \cdot {\bf u}_0 \end{align} and \begin{align}\label{cj} & \int_{\mathbb{T}^d}\Big(\rho^\epsilon {\bf u}^\epsilon \cdot \mathcal{L}_2 \Big(\frac{t}{\epsilon}\Big)\mathbf{V}^0\Big)(t) +\frac{1}{\epsilon}\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon \cdot\nabla\Big( \mathcal{L}_1\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\nonumber\\ &\quad -\int^t_0\int_{\mathbb{T}^d} \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\partial_t\mathbf{V}^0 \cdot(\rho^\epsilon {\bf u}^\epsilon) -\int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon {\bf u}^\epsilon \otimes {\bf u}^\epsilon)\cdot \nabla\Big( \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\nonumber\\ & \quad +\int^t_0\int_{\mathbb{T}^d}\Big[\mu \nabla {\bf u}^\epsilon \cdot \nabla \Big( \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) +(\mu+\lambda)\text{div}{\bf u}^\epsilon \,\text{div}
\Big( \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\Big]\nonumber\\ & \quad - \int^t_0\int_{\mathbb{T}^d}({\bf H}^\epsilon\cdot \nabla){\bf H}^\epsilon \cdot \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0
-\int^t_0\int_{\mathbb{T}^d}\frac12|{\bf H}^\epsilon|^2{\rm div }\Big(\mathcal{L}_2
\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\nonumber\\ & \quad
-\int^t_0\int_{\mathbb{T}^d} \Big(\frac{1}{\epsilon}\varphi^\epsilon+\frac{\gamma-1}{2}(\Pi^\epsilon)^2\Big)\text{div}\, \Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) = \int_{\mathbb{T}^d}\rho^\epsilon_0 {\bf u}^\epsilon_0 \cdot Q\tilde{{\bf u}}_0. \end{align} Similarly, we test \eqref{a2k} by ${\bf H}$ to get \begin{align}\label{cjj} & \int_{\mathbb{T}^d}({\bf H}^\epsilon \cdot {\bf H})(t) +\int^t_0\int_{\mathbb{T}^d}{\bf H}^\epsilon \cdot\big[({\bf u}\cdot \nabla) {\bf H}-({\bf H}\cdot \nabla){\bf u}-\nu \Delta {\bf H}\big] + \nu \int^t_0\int_{\mathbb{T}^d} \nabla {\bf H}^\epsilon \cdot \nabla {\bf H}\nonumber\\ & \quad +\int^t_0\int_{\mathbb{T}^d}\big[({\rm div } {{\bf u}}^\epsilon) { {\bf H}}^\epsilon + ( {{\bf u}}^\epsilon \cdot \nabla) {{\bf H}}^\epsilon - ( {{\bf H}}^\epsilon\cdot \nabla)
{{\bf u}}^\epsilon\big]\cdot {\bf H} = \int_{\mathbb{T}^d}{\bf H}^\epsilon_0 \cdot {\bf H}_0. \end{align} Summing up \eqref{ce}, \eqref{cff} and \eqref{cfh}, inserting \eqref{ch}-\eqref{cjj} into the resulting inequality, and using the fact \begin{align*}
\int_0^t \int_{\mathbb{T}^d} \Big(\mathcal{L}\Big(\frac{\tau}{\epsilon}
\Big)\partial_t\mathbf{V}^0\Big) \cdot \mathbf{U}^\epsilon
=\int_0^t \int_{\mathbb{T}^d} \partial_t\mathbf{V}^0 \cdot \mathbf{V}^\epsilon, \end{align*}
we deduce, after a straightforward calculation, the following inequality: \begin{align}\label{ckkl}
& \frac{1}{2}\int_{\mathbb{T}^d}\Big\{ \Big|\sqrt{\rho^\epsilon} {\bf u}^\epsilon
-{\bf u}- \mathcal{L}_2\Big(\frac{t}{\epsilon}\Big)\mathbf{V}^0\Big|^2(t)+ | {\bf H}^\epsilon
-{\bf H}|^2(t)
+\Big|\Pi^\epsilon- \mathcal{L}_1\Big(\frac{t}{\epsilon}\Big)\mathbf{V}^0\Big|^2(t)\Big\}\nonumber\\
&\quad + {\mu} \int^t_0\int_{\mathbb{T}^d}\Big|\nabla\Big( {\bf u}^\epsilon -{\bf u}-
\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\Big|^2
+ \frac{\nu}{2} \int^t_0\int_{\mathbb{T}^d}|\nabla {\bf H}^\epsilon-\nabla {\bf H} |^2 \nonumber\\
&\quad+ \frac{\mu}{2} \int^t_0\int_{\mathbb{T}^d}\big(|\nabla {\bf H}^\epsilon|^2
+ |\nabla {\bf H}|^2\big)
+ (\mu+\lambda)\int^t_0\! \int_{\mathbb{T}^d}\Big|{\rm div}\Big( {\bf u}^\epsilon
-{\bf u}- \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\Big|^2\nonumber\\ &\leq
\frac{1}{2}\int_{\mathbb{T}^d}\Big\{ \Big|\sqrt{\rho^\epsilon} {\bf u}^\epsilon -{\bf u}-
\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big|^2(0)+
| {\bf H}^\epsilon -{\bf H}|^2(0)
+\Big|\Pi^\epsilon- \mathcal{L}_1\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big|^2(0)\Big\}\nonumber\\ & \quad +\sum^8_{i=1}A^\epsilon_i(t), \end{align} where \begin{align} A^\epsilon_1(t) =&
\int_{\mathbb{T}^d} \Big[(\sqrt{\rho^\epsilon}-1)\sqrt{\rho^\epsilon}{\bf u}^\epsilon\cdot \Big({\bf u}+\mathcal{L}_2\Big(\frac{t}{\epsilon}\Big)\mathbf{V}^0 \Big)\Big](t)-\int_{\mathbb{T}^d} \Big[(\Pi^\epsilon-\varphi^\epsilon)\mathcal{L}_1\Big(\frac{t}{\epsilon}\Big)\mathbf{V}^0 \Big](t)\nonumber\\ & -\int_{\mathbb{T}^d} \Big[(\sqrt{\rho^\epsilon}-1)\sqrt{\rho^\epsilon}{\bf u}^\epsilon\cdot \Big({\bf u}+\mathcal{L}_2\Big(\frac{t}{\epsilon}\Big)\mathbf{V}^0 \Big)\Big](0) +\int_{\mathbb{T}^d} \Big[(\Pi^\epsilon-\varphi^\epsilon)\mathcal{L}_1\Big(\frac{t}{\epsilon}\Big)\mathbf{V}^0 \Big](0), \label{A1}\\ A^\epsilon_2(t) =&\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon \cdot
\nabla p-\mu\int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon-1){\bf u}^\epsilon\Delta {\bf u},\label{A2}\\ A^\epsilon_3 (t)= & -\frac{\theta}{2}\int^t_0\int_{\mathbb{T}^d}\Delta \mathbf{V}^0 \cdot \mathbf{V}^\epsilon
- \mu\int^t_0\int_{\mathbb{T}^d}\nabla {\bf u}^\epsilon\cdot\nabla
\Big( \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\nonumber\\ &- (\mu+\lambda)\int^t_0\int_{\mathbb{T}^d} {\rm div }{\bf u}^\epsilon\cdot{\rm div } \Big( \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big),\label{A3}\\ A^\epsilon_4 (t)= &-
\frac{\theta}{2}\int^t_0\int_{\mathbb{T}^d}|\nabla \mathbf{V}^0|^2
+\mu\int^t_0\int_{\mathbb{T}^d}\Big|\nabla
\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big) \mathbf{V}^0\Big|^2
+(\mu+\lambda)\int^t_0\int_{\mathbb{T}^d}\Big|{\rm div } \mathcal{L}_2
\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big|^2,\label{A4}\\ A^\epsilon_5(t)=& -\int^t_0\int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon\cdot [({\bf H}\cdot \nabla) {\bf H})] -\int^t_0\int_{\mathbb{T}^d} ({\bf H}^\epsilon\cdot \nabla) {\bf H}^\epsilon \cdot{\bf u} \nonumber\\ & +\int^t_0\int_{\mathbb{T}^d} {\bf H}^\epsilon \cdot\big[({\bf u}\cdot \nabla) {\bf H}-({\bf H}\cdot \nabla){\bf H} \big] +\frac12\int^t_0\int_{\mathbb{T}^d}
\rho^\epsilon {\bf u}^\epsilon \cdot \nabla(|{\bf H}|^2)\nonumber\\ & +\int^t_0\int_{\mathbb{T}^d}\big[({\rm div } {{\bf u}}^\epsilon) { {\bf H}}^\epsilon
+ ( {{\bf u}}^\epsilon \cdot \nabla) {{\bf H}}^\epsilon
- ( {{\bf H}}^\epsilon\cdot \nabla) {{\bf u}}^\epsilon\big]\cdot {\bf H}, \label{A5}\\ A^\epsilon_6 (t)=& \int^t_0\int_{\mathbb{T}^d}\rho^\epsilon {\bf u}^\epsilon \cdot\big[ ({\bf u}\cdot \nabla) {\bf u} \big]-\int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon {\bf u}^\epsilon\otimes {\bf u}^\epsilon)\cdot \Big(\nabla u+ \mathcal{L}_2 \Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) \nonumber\\ & - \frac{\gamma-1}{2}\int^t_0\int_{\mathbb{T}^d} (\Pi^\epsilon)^2 \text{div} \Big( \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big),\label{A6}\\ A^\epsilon_7 (t)= & \int^t_0\int_{\mathbb{T}^d}\big[\mathcal{Q}_1({\bf u}, \mathbf{V}^0)
+\mathcal{Q}_2(\mathbf{V}^0, \mathbf{V}^0)\big]\cdot\mathbf{V}^\epsilon,\label{A7}\\ A^\epsilon_8 (t)= &
-\int^t_0\int_{\mathbb{T}^d}\frac12|{\bf H}^\epsilon|^2{\rm div }\Big(\mathcal{L}_2 \Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)
- \int^t_0\int_{\mathbb{T}^d}({\bf H}^\epsilon\cdot \nabla){\bf H}^\epsilon
\cdot \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0.\label{A8} \end{align}
\emph{Step 3: Convergence of the modulated energy functional.}
To show the convergence of the modulated energy functional \eqref{ckkl},
we need to estimate the remainders $A^{\epsilon}_i(t)$, $ i=1,\dots, 8$. In the sequel, we will denote by $\omega^\epsilon(t)$ any sequence of time-dependent functions which converges to $0$ uniformly in $t$. For convenience, we also denote
$\mathbf{w}^{\epsilon}\equiv\sqrt{\rho^\epsilon} {\bf u}^\epsilon -{\bf u}-\mathcal{L}_2(\frac{t}{\epsilon})\mathbf{V}^0$, $\mathbf{Z}^{\epsilon}\equiv {\bf H}^\epsilon -{\bf H}$, and $\Psi^\epsilon \equiv \Pi^\epsilon-\mathcal{L}_1(\frac{t}{\epsilon})\mathbf{V}^0$.
For the term $A^{\epsilon}_1(t)$, we employ \eqref{be}, \eqref{ine30},
the regularity of ${\bf u}$ and \eqref{UU}, and follow a procedure similar to that for $R^\epsilon_2(t)$, to obtain \begin{align}\label{dd1}
|A^{\epsilon}_1(t)|\leq C_T \epsilon +\omega^\epsilon(t). \end{align} On the other hand, the term $A^{\epsilon}_2(t)$ has the same bound as $R^{\epsilon}_5(t)+R^{\epsilon}_7(t)$, thus \begin{align}\label{dd2}
|A^{\epsilon}_2(t)|\leq C_T \epsilon. \end{align}
To bound the term $A^{\epsilon}_3(t)$, we integrate by parts and use the fact $ \mathcal{L}_2(\frac{t}{\epsilon})\mathbf{V}^0=\nabla \tilde{q}^\epsilon$ for some function $\tilde{q}^\epsilon$ and Proposition \ref{Pv} to infer \begin{align*} - \mu\int^t_0\int_{\mathbb{T}^d}\nabla {\bf u}^\epsilon\cdot\nabla\Big( \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) &=\mu\int^t_0\int_{\mathbb{T}^d}\Delta {\bf u}^\epsilon\cdot \mathcal{L}_2 \Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\\ &=\mu\int^t_0\int_{\mathbb{T}^d}\mathcal{L}\Big(-\frac{\tau}{\epsilon}\Big) \Big(\begin{array}{c} 0\\ \Delta {\bf u}^\epsilon \end{array} \Big)\cdot \mathbf{V}^0\\ &=\frac{\mu}{2}\int^t_0\int_{\mathbb{T}^d} \Delta \bar{\mathbf{V}}\cdot \mathbf{V}^0+\omega^\epsilon(t), \\ - (\mu+\lambda)\int^t_0\int_{\mathbb{T}^d} {\rm div }{\bf u}^\epsilon\cdot{\rm div }\Big( \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) & =(\mu+\lambda)\int^t_0\int_{\mathbb{T}^d} \Delta {\bf u}^\epsilon\cdot\mathcal{L}_2 \Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\\ &=\frac{\mu+\lambda}{2}\int^t_0\int_{\mathbb{T}^d} \Delta \bar{\mathbf{V}} \cdot \mathbf{V}^0+\omega^\epsilon(t), \end{align*} and \begin{align*} -\frac{\theta}{2}\int^t_0\int_{\mathbb{T}^d}\Delta \mathbf{V}^0 \cdot \mathbf{V}^\epsilon =-\frac{\theta}{2}\int^t_0\int_{\mathbb{T}^d}\Delta \bar{\mathbf{V}}\cdot \mathbf{V}^0+\omega^\epsilon(t). \end{align*} Thus, recalling $\theta=2\mu+\lambda$, one has \begin{align}\label{dd3} A^{\epsilon}_3(t)=\omega^\epsilon(t). \end{align}
Similarly, it follows from Proposition \ref{Pv} that \begin{align}\label{dd4} A^{\epsilon}_4(t)=\omega^\epsilon(t). \end{align}
Recalling \eqref{ck1i} and using H\"older's inequality, the inequalities \eqref{be} and \eqref{ine30},
the regularity of ${\bf H}$, and Sobolev's imbedding theorem, we conclude \begin{align}\label{dd5} A^{\epsilon}_5(t) \leq & \int^t_0\int_{\mathbb{T}^d}(1-\rho^\epsilon) {\bf u}^\epsilon\cdot [({\bf H}\cdot \nabla) {\bf H})]
+\int^t_0 ||\mathbf{Z}^\epsilon(\tau)||^2_{L^2}|| \nabla {\bf u}(\tau)||_{L^\infty} d\tau\nonumber\\
&+ \int^t_0 \big[||\mathbf{w}^{\epsilon}(\tau)||^2_{L^2}
+ ||\mathbf{Z}^\epsilon(\tau)||^2_{L^2}\big] || \nabla {\bf H}(\tau)||_{L^\infty} d\tau \nonumber\\ &+\int^t_0\!\int_{\mathbb{T}^d}(\mathbf{Z}^\epsilon \cdot \nabla){\bf H} \cdot \Big[(1-\sqrt{\rho^\epsilon}){\bf u}^\epsilon+\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big] \nonumber\\ &- \int^t_0\!\int_{\mathbb{T}^d}\Big\{\Big[(1-\sqrt{\rho^\epsilon}){\bf u}^\epsilon +\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big]\cdot \nabla\Big\} {\bf H}\cdot\mathbf{Z}^\epsilon\nonumber\\
& +\frac12\int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon-1){{\bf u}}^\epsilon \nabla(|{\bf H}|^2)\nonumber\\
\leq & \int^t_0 \big[||\mathbf{w}^{\epsilon}(\tau)||^2_{L^2}
+ ||\mathbf{Z}^\epsilon(\tau)||^2_{L^2}\big] \cdot \big[|| \nabla {\bf u}(\tau)||_{L^\infty}
+|| \nabla {\bf H}(\tau)||_{L^\infty}\big] d\tau\nonumber\\ &+\int^t_0\!\int_{\mathbb{T}^d}\Big[(\mathbf{H}^\epsilon\cdot \nabla){\bf H} \cdot \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0 - \Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\cdot \nabla\Big){\bf H} \cdot\mathbf{H}^\epsilon
\Big]d\tau+C_T\epsilon. \end{align}
The term $A^\epsilon_6(t)$ can be rewritten as \begin{align}\label{dd6}
A^\epsilon_6(t) =& -\int^t_0\int_{\mathbb{T}^d}(\mathbf{w}^{\epsilon}\otimes
\mathbf{w}^{\epsilon})\cdot \nabla\Big( {\bf u} +\mathcal{L}\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) - \frac{\gamma-1}{2}\int^t_0\int_{\mathbb{T}^d}
|\Psi^\epsilon|^2 \text{div}\Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) \nonumber\\ & -\int^t_0\int_{\mathbb{T}^d}\Big\{ \sqrt{\rho^\epsilon}{\bf u}^\epsilon\otimes \Big({\bf u} +\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) +\Big({\bf u} +\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\otimes \sqrt{\rho^\epsilon}{\bf u}^\epsilon\Big\} \cdot \nabla\Big( {\bf u} +\mathcal{L}\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\nonumber\\ & +\int^t_0\int_{\mathbb{T}^d}
\Big({\bf u} +\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) \otimes \Big({\bf u} +\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\cdot \nabla\Big( {\bf u} +\mathcal{L}\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)
\nonumber\\
&+ \int^t_0\int_{\mathbb{T}^d} \frac{\gamma-1}{2}
\Big|\mathcal{L}_1\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big|^2
\text{div} \Big( \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) - {(\gamma-1)} \mathcal{L}_1\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\, \Pi^\epsilon\, \text{div} \Big( \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\nonumber\\ & +\int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon-\sqrt{\rho^\epsilon}){\bf u}^\epsilon\cdot
(({\bf u}\cdot \nabla) {\bf u}) -\int^t_0\int_{\mathbb{T}^d}(\sqrt{ \rho^\epsilon}{\bf u}^\epsilon- {\bf u}) \cdot \nabla
\big(\frac{|{\bf u}|^2}{2}\big).
\end{align} We have to bound all the terms on the right-hand side of \eqref{dd6}. Keeping in mind that ${\rm div }\,{\bf v}=0$ and applying Proposition \ref{P2}, one obtains \begin{align}\label{dd61} & \int^t_0\int_{\mathbb{T}^d}
\Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) \otimes \Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\cdot
\nabla\Big( {\bf u} +\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)
+\frac{\gamma-1}{2}\Big|\mathcal{L}_1\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big|^2 {\rm div }\Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\nonumber\\ =& -\int^t_0\int_{\mathbb{T}^d}
\Big\{{\rm div }\Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) \otimes \Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)
+\frac{\gamma-1}{2}\nabla\Big|\mathcal{L}_1\Big(\frac{\tau}{\epsilon}\Big)
\mathbf{V}^0\Big|^2\Big\} \cdot \big(\mathbf{V}^0 +\widehat{{\bf u}}\big)\nonumber\\ =& -\int^t_0\int_{\mathbb{T}^d} \mathcal{L}\Big(-\frac{\tau}{\epsilon}\Big) \bigg( \begin{array}{c} 0\\ {\rm div }\Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) \otimes \Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)
+\frac{\gamma-1}{2}\nabla\Big|\mathcal{L}_1\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big|^2 \end{array} \bigg) \cdot \big(\mathbf{V}^0+\widehat{{\bf u}}\big)\nonumber\\ =& -\int^t_0\int_{\mathbb{T}^d} \mathcal{Q}_2(\mathbf{V}^0,\mathbf{V}^0) \cdot \big(\mathbf{V}^0+\widehat{{\bf u}}\big)+\omega^\epsilon(t)= \omega^\epsilon(t). \end{align} From Proposition \ref{P3} we get \begin{align} \label{dd62} &-\int^t_0\int_{\mathbb{T}^d} \Big\{\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\otimes\sqrt{\rho^\epsilon}{\bf u}^\epsilon + \sqrt{\rho^\epsilon}{\bf u}^\epsilon\otimes\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big\} \cdot\nabla\Big({\bf v} +\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\nonumber\\ & -\int^t_0\int_{\mathbb{T}^d}{(\gamma-1)} \mathcal{L}_1\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\, \Pi^\epsilon\, \text{div} \Big( \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\nonumber\\ =&\int^t_0\int_{\mathbb{T}^d} \Big\{{\rm div }\Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\otimes\sqrt{\rho^\epsilon}{\bf u}^\epsilon +\sqrt{\rho^\epsilon}{\bf u}^\epsilon\otimes\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) +(\gamma-1) \nabla\Big( \mathcal{L}_1\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Pi^\epsilon\Big)\Big\}\nonumber\\ & \ \ \ \ \quad \cdot\Big({\bf u} +\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\nonumber\\ =&\int^t_0\int_{\mathbb{T}^d} \mathcal{L}\Big(-\frac{\tau}{\epsilon}\Big) \bigg( \begin{array}{c} 0\\ {\rm div }\Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\otimes\sqrt{\rho^\epsilon}{\bf u}^\epsilon + \sqrt{\rho^\epsilon}{\bf u}^\epsilon\otimes\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) +(\gamma-1) \nabla\Big( \mathcal{L}_1\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Pi^\epsilon\Big) \end{array} \bigg)\nonumber\\ & \ \ \ \ \quad \cdot \big(\mathbf{V}^0+\widehat{{\bf u}}\big)\nonumber\\ =&\int^t_0\int_{\mathbb{T}^d}
\mathcal{L}\Big(-\frac{\tau}{\epsilon}\Big) \bigg( \begin{array}{c} 0\\ {\rm div }\Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\otimes Q(\sqrt{\rho^\epsilon}{\bf u}^\epsilon) + Q(\sqrt{\rho^\epsilon}{\bf u}^\epsilon)\otimes\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) +(\gamma-1) \nabla\Big( \mathcal{L}_1\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Pi^\epsilon\Big) \end{array} \bigg)\nonumber\\ & \ \ \ \ \quad \cdot \big(\mathbf{V}^0+\widehat{{\bf u}}\big)\nonumber\\ &+\int^t_0\int_{\mathbb{T}^d}
\mathcal{L}\Big(-\frac{\tau}{\epsilon}\Big) \bigg( \begin{array}{c} 0\\ {\rm div }\Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\otimes P(\sqrt{\rho^\epsilon}{\bf u}^\epsilon) + P(\sqrt{\rho^\epsilon}{\bf u}^\epsilon)\otimes\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) \end{array} \bigg) \cdot \big(\mathbf{V}^0+\widehat{{\bf u}}\big)\nonumber\\ =& \int^t_0\int_{\mathbb{T}^d}\big\{ 2 \mathcal{Q}_2(\mathbf{V}^0, \bar{\mathbf{V}}) \cdot\big(\mathbf{V}^0+\widehat{{\bf u}}\big) +\mathcal{Q}_1({\bf u}, \mathbf{V}^0) \cdot\big(\mathbf{V}^0+\widehat{{\bf u}}\big)\big\}+ \omega^\epsilon(t). \end{align} Similarly, we have \begin{align}\label{dd63} &-\int^t_0\int_{\mathbb{T}^d}\big\{ \sqrt{\rho^\epsilon}{\bf u}^\epsilon\otimes {\bf u} +{\bf u}\otimes \sqrt{\rho^\epsilon}{\bf u}^\epsilon\big\} \cdot \nabla\Big( {\bf u} +\mathcal{L}\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) = \int^t_0\int_{\mathbb{T}^d}
\mathcal{Q}_1({\bf u}, \bar{\mathbf{V}}) \mathbf{V}^0 + \omega^\epsilon(t), \end{align} and \begin{align}\label{dd64} & \int^t_0\int_{\mathbb{T}^d} \Big\{{\bf u} \otimes \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0 + \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big) \otimes {\bf u} \Big\} \cdot \nabla\Big( {\bf u} +\mathcal{L}\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)
\nonumber\\ & \quad = - \int^t_0\int_{\mathbb{T}^d} \mathcal{Q}_1({\bf u}, \mathbf{V}^0) \mathbf{V}^0 + \omega^\epsilon(t). \end{align} On the other hand, using the basic energy equality \eqref{be} and the regularity of ${\bf u}$, following the same process as for $R^\epsilon_6(t)$, we obtain \begin{align}\label{dd65}
\Big| \int^t_0\int_{\mathbb{T}^d}(\rho^\epsilon-\sqrt{\rho^\epsilon}){\bf u}^\epsilon\cdot
(({\bf u}\cdot \nabla) {\bf u}) -\int^t_0\int_{\mathbb{T}^d}(\sqrt{ \rho^\epsilon}{\bf u}^\epsilon- {\bf u}) \cdot \nabla
\big(\frac{|{\bf u}|^2}{2}\big)\Big| \leq C_T \epsilon. \end{align}
The term $A^\epsilon_7(t)$ can be rewritten as \begin{align}\label{dd7}
A^\epsilon_7(t) = \int^t_0\int_{\mathbb{T}^d}\big[\mathcal{Q}_1({\bf u}, \mathbf{V}^0)
+\mathcal{Q}_2(\mathbf{V}^0, \mathbf{V}^0)\big]\cdot\bar{\mathbf{V}}+ \omega^\epsilon(t). \end{align} Substituting \eqref{dd61}-\eqref{dd7} into \eqref{dd6}, we conclude that \begin{align}\label{A6A7}
|A^\epsilon_6(t)| +|A^\epsilon_7(t)|\leq C\int^t_0(||\mathbf{w}^\epsilon||^2
+||\Psi^\epsilon||^2)\Big(\|\nabla{\bf u}\|_{L^\infty} +\Big\|
\nabla\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big\|_{L^\infty}\Big)d\tau+C_T \epsilon + \omega^\epsilon(t). \end{align}
Thus, we insert the estimates on $A^\epsilon_i(t)$ ($i=1,\cdots,7$) into \eqref{ckkl} to obtain \begin{align}\label{EE}
& \frac{1}{2}\int_{\mathbb{T}^d}\big\{ | \mathbf{w}^\epsilon|^2
+ | \mathbf{Z}^\epsilon |^2
+ |\Psi^\epsilon|^2 \big\}(t)
+ {\mu} \int^t_0\int_{\mathbb{T}^d}\Big|\nabla\Big( {\bf u}^\epsilon
-{\bf u}- \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\Big|^2\nonumber\\
&\quad+ \frac{\nu}{2} \int^t_0\int_{\mathbb{T}^d} \big(|\nabla \mathbf{Z}^\epsilon |^2
+|\nabla {\bf H}^\epsilon|^2 + |\nabla {\bf H}|^2\big)
+ (\mu+\lambda)\int^t_0\! \int_{\mathbb{T}^d}\Big|{\rm div}\Big( {\bf u}^\epsilon
-{\bf u}- \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\Big)\Big|^2\nonumber\\
\leq & \int^t_0 \big[||\mathbf{w}^{\epsilon}(\tau)||^2_{L^2}
+ ||\mathbf{Z}^\epsilon(\tau)||^2_{L^2}\big] \cdot \Big[|| \nabla
{\bf u}(\tau)||_{L^\infty}+|| \nabla {\bf H}(\tau)||_{L^\infty}
+\| \nabla\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\|_{L^\infty})\Big] d\tau \nonumber\\
& +\ \frac{1}{2}\int_{\mathbb{T}^d}\big\{ | \mathbf{w}^\epsilon|^2
+ | \mathbf{Z}^\epsilon |^2
+ |\Psi^\epsilon|^2\big\}(0)+C_T\epsilon +\omega^\epsilon(t)+
A^\epsilon_8(t)+A^\epsilon_9(t), \end{align} where \begin{align}\label{A9} A^\epsilon_9(t) =\int^t_0\!\int_{\mathbb{T}^d}\Big[(\mathbf{H}^\epsilon \cdot \nabla){\bf H} \cdot \mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0 - \Big(\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\cdot \nabla\Big){\bf H} \cdot\mathbf{H}^\epsilon
\Big]. \end{align}
Now, to deal with $A^\epsilon_8(t)$ and $A^\epsilon_9(t)$, we denote
${\bar{{\bf H}}}^\epsilon_0=\frac{1}{|\mathbb{T}^d|}\int_{\mathbb{T}^d} {\bf H}^\epsilon_0(x)dx$ to deduce from the magnetic field equation \eqref{a2k} that \begin{equation}\label{average2} \int_{\mathbb{T}^d} {\bf H}^\epsilon(x,t)dx= \int_{\mathbb{T}^d}
{\bf H}^\epsilon_0(x)dx = {\bar{{\bf H}}}^\epsilon_0 |\mathbb{T}^d|. \end{equation} The assumption that ${\bf H}^\epsilon_0$ converges strongly in $L^2$ to some ${\bf H}_0$ implies \begin{align*}
\Big|\int_{\mathbb{T}^d} {\bf H}^\epsilon_0(x)-{\bf H}_0(x)dx\Big|
& \leq \int_{\mathbb{T}^d} |{\bf H}^\epsilon_0(x)-{\bf H}_0(x)|dx \\
& \leq |\mathbb{T}^d|^{\frac12} \Big( \int_{\mathbb{T}^d}|
{\bf H}^\epsilon_0(x)-{\bf H}_0(x)|^2dx\Big)^{\frac12} \rightarrow 0\ \ \text{as} \ \ \epsilon \to 0, \end{align*} whence \begin{align}\label{average1} {\bar{{\bf H}}}^\epsilon_0 \rightarrow {\bar{{\bf H}}}_0 \ \ \text{as} \ \ \epsilon\rightarrow 0. \end{align}
Using H\"older's inequality, Sobolev's imbedding theorem, Poinc\'{a}re's inequality, the isometry property of $\mathcal{L}$, and \eqref{average2} and \eqref{average1}, we find that \begin{align}\label{A89}
|{A}^\epsilon_8 (t)|+|{A}^\epsilon_9 (t)|
\leq &\frac{\nu}{4} \int^t_0[ ||\nabla{\bf H}^\epsilon||^2+ ||\nabla{\bf H}^\epsilon||^2](\tau) d\tau+\omega(t)\nonumber\\
&+\frac{1}{\nu} (\|\varphi_0\|^2_{H^2}+||Q{\bf u}_0||^2_{H^2}) \int^t_0[
||\nabla{\bf H}^\epsilon||^2+ ||\nabla{\bf H}^\epsilon||^2](\tau)d\tau . \end{align}
Thus, substituting \eqref{A89} into \eqref{EE} and using \eqref{Qu}, we deduce by Gronwall's inequality that, for almost all $t\in [0,T]$, \begin{align}\label{cqq}
&
||\mathbf{w}^{\epsilon}(t)||^2_{L^2}+
||\mathbf{Z}^{\epsilon}(t)||^2_{L^2}+||\Psi^\epsilon(t)||^2_{L^2}\nonumber\\ & \quad \leq \tilde{C}
\Big\{||\mathbf{w}^{\epsilon}(0)||^2_{L^2}+||\mathbf{Z}^{\epsilon}(0)||^2_{L^2}
+ ||\Pi^\epsilon_0-\varphi_0||^2_{L^2} +C_T\epsilon +\sup_{0\leq s\leq t}\omega^{\epsilon}(s)\Big\}, \end{align} where
$$ \tilde{C}=\exp {\Big\{C\int^T_0 \Big[|| \nabla
{\bf u}(\tau)||_{L^\infty}+|| \nabla {\bf H}(\tau)||_{L^\infty}
+\| \nabla\mathcal{L}_2\Big(\frac{\tau}{\epsilon}\Big)\mathbf{V}^0\|_{L^\infty})\Big] d\tau\Big\}}<+\infty. $$
\emph{Step 4: End of the proof of Theorem \ref{MRc}.}
Letting $\epsilon$ go to zero in \eqref{cqq}, we see that ${\bf H}^\epsilon$ converges strongly to
${\bf H}$ in $L^\infty(0,T; L^2(\mathbb{T}^d))$. Hence, $\bar{\mathbf{K}}=\mathbf{H}$. Combining \eqref{EE} with \eqref{cqq}, we can easily prove that
$\nabla {\bf H}^\epsilon$ converges strongly to $\nabla {\bf H}$ in $L^2 (0,T;L^2(\mathbb{T}^d))$.
Next, it suffices to prove (4) and (5) in Theorem \ref{MRc}. Noting that $P(\mathcal{L}_2(\frac{t}{\epsilon})\mathbf{V}^0)=0$ and the fact that the projection operator $P$ is a bounded linear mapping form $L^2$ to $L^2$, we obtain, with the help of \eqref{cqq}, that \begin{align}\label{pes1}
\sup_{0\leq t\leq T}\|P(\sqrt{\rho^\epsilon} {\bf u}^\epsilon)
-{\bf u}\|_{L^2}
=&\sup_{0\leq t\leq T}\Big\|P\Big(\sqrt{\rho^\epsilon} {\bf u}^\epsilon-{\bf u}-\mathcal{L}_2
\Big(\frac{t}{\epsilon}\Big)\mathbf{V}^0\Big)\Big\|_{L^2}\nonumber\\
\leq &\sup_{0\leq t\leq T}\Big\|\sqrt{\rho^\epsilon} {\bf u}^\epsilon-{\bf u}-\mathcal{L}_2
\Big(\frac{t}{\epsilon}\Big)\mathbf{V}^0\Big\|_{L^2}\nonumber\\ \rightarrow&\;\; 0 \;\;\mbox{as}\;\; \epsilon \rightarrow 0. \end{align} Therefore, (4) is proved. Utilizing \eqref{rhoc}, we deduce from \eqref{a2i} that $\mbox{div}\,{\bf u}^\epsilon$ converges weakly to $0$ in $H^{-1}((0, T)\times\mathbb{T}^d)$. Thus we obtain easily that $Q{\bf u}^\epsilon$ converges weakly to $0$ in $H^{-1}(0, T; L^2(\mathbb{T}^d))$. In view of the fact that ${\bf u}^\epsilon$ is bounded in $L^2(0,T;L^2(\mathbb{T}^d))$ and $\sqrt{\rho^\epsilon}$ converges strongly to $1$ in $C([0, T], L^2(\mathbb{T}^d))$, we see that $Q(\sqrt{\rho^\epsilon}{\bf u}^\epsilon)$ converges weakly to $0$ in $H^{-1}(0, T; L^2(\mathbb{T}^d))$. Obviously, the fact $\sqrt{\rho^\epsilon}{\bf u}^\epsilon= P(\sqrt{\rho^\epsilon} {\bf u}^\epsilon)+Q(\sqrt{\rho^\epsilon}{\bf u}^\epsilon)$ implies the weak convergence of $\sqrt{\rho^\epsilon}{\bf u}^\epsilon$ to ${\bf u}$ in $H^{-1}(0, T; L^2(\mathbb{T}^d))$. The proof of Theorem \ref{MRc} is finished. \end{proof}
Theorem \ref{MRd} can be shown by slightly modifying the proof of Theorem \ref{MRc}, and therefore, we omit its proof here.
{\bf Acknowledgement:} This work was done while Fucai Li was visiting the Institute of Applied Physics and Computational Mathematics. He would like to thank the institute for hospitality. Jiang was supported by the National Basic Research Program (Grant No. 2005CB321700) and NSFC (Grant No. 40890154). Ju was supported by NSFC (Grant No. 10701011). Li was supported by NSFC (Grant No. 10501047, 10971094).
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An analysis of economic losses from cyberattacks: based on input–output model and production function
Akiyoshi Kokaji ORCID: orcid.org/0000-0002-1672-413X1 &
Atsuhiro Goto1
There has recently been a global increase in economic losses due to cyberattacks. However, research on the economic damage caused by cyberattacks has mainly focused on attacked companies, and spillover damage to other sectors has not been sufficiently investigated. This study analyzed the economic losses from cyberattacks in Japan using the production function and input–output model to improve the accuracy of damage prediction and various national measures. First, we provide an estimation method for the annual direct damage by industry using a production function. The mainstream input dataset is lost working hours owing to cyber incidents. Second, we devised a model to estimate the amount of spillover damage to the entire country using the input–output model. Third, although the cyber damage data were limited to only interview data by the JNSA and IPA, we showed the process of estimating direct and spillover damage in all sectors in Japan. As a result, we consider that our estimation method is feasible and effective at the national level. This study contributes to future research on cyber resilience by analyzing the damage caused by cyberattacks from a macroeconomic perspective using a production function and input–output model.
Several studies have examined the extent of the damage caused by natural and manufactured disasters. These studies determined the amount of damage caused in Japan and its various regions. A report by the Cabinet Office (2013) estimated direct damage from natural disasters to be between 97.6 and 169.5 trillion yen, and the full amount of damage has been estimated to be between 35.1 and 50.8 trillion yen in the Nankai Trough earthquake. Concerning the economic damage caused by cyberattacks, CSIS (2021) indicated that the damage exceeded USD 1 trillion and accounted for 1% of the global gross domestic product. Despite this huge loss, studies on the economic damage caused by cyberattacks have been restricted to micro-analyses. Few studies have performed comprehensive and quantitative damage analyses, including an analysis of spillover damage. Although the government of Japan has adopted several cyber security policies, such as supply chain security for critical infrastructures in CS (2021), it must be understood that an objective and quantitative analysis of results is the key to effective policymaking.
This study devised a method for quantitatively estimating the economic damage caused by cyberattacks in Japan and contributed to improving the accuracy of damage prediction and the formulation of various national measures. Specifically, we devised a method to analyze Japan's economic damage caused by cyberattacks using production functions and input–output analysis.
The remainder of this paper is organized as follows. Section 2 reviews research analyzing the economic damage caused by natural disasters and cyberattacks. Section 3 presents a method for estimating the direct and spillover damage. Section 4 presents our estimation results and provides a discussion of our findings. Section 5 presents the conclusions of the findings and plans for future research.
Previous research
Analysis in Japan
Most estimation studies on economic loss due to cyberattacks in Japan are restricted to microeconomics analysis and lack quantitative analysis of the economic damage from cyberattacks. Tanaka et al. reported an estimation of economic loss due to information security incidents in Japan of about 4.6 to 9.4 billion yen from 2009 to 2011, respectively, in Tanaka (2014), but their study lacks established methods of analysis. For example, no analysis of spillover damage has been conducted yet.
In the case of natural disasters, studies have shown that the amount of damage is estimated using a production function and input–output table. For example, Shimoda and Fujikawa (2012) used an input–output model to measure the damage caused by the Great East Japan Earthquake on the demand side (backward-relation effect) of production, which experienced a decline, as well as on the supply side. A supply type model is used to measure spillover damage (forward output effect) at the initial stage of the disaster, and a demand-type model is used thereafter.
In Japan, the Information Technology Promotion Agency (IPA) and the Japan Network Security Association (JNSA) have published the results of their analyses of cyberattack damage. The amount of damage was calculated based on their self-developed model. However, the amount of damage from each incident is limited to that of the victim company, and there is no model for calculating the amount for the entire country. Tanaka (2013) suggest using the Cobb–Douglas production function to estimate cyber damage in the entire country, but none showed its feasibility and effectiveness.
Analysis overseas
Japan's Ministry of Internal Affairs and Communications has analyzed several cases of cyberattacks in the MIC (2019). Emphasizing the model and data related to estimating the amount of damage, Appendix 2 summarizes the analyses.
In terms of the model for estimating the amount of damage, only two studiesFootnote 1 used the existing economic analysis model. The other models are original, unpublished models. Lloyd and the University of Cambridge's Center for Risk Studies have used the input–output model to calculate spillover damage (disruption of the power supply) from cyberattacks on a power grid on the east coast of the United States.
We found a limited number of models to calculate the amount of damage. The few existing models are unpublished, self-made models, and most damage amount calculations are based on subjective estimation, which is a general economic effect. Few quantitative estimates have been based on the objective methods used in the analysis. Appropriate analysis cannot be performed using a subjective analysis alone.
In summary, there is no established model for analyzing/estimating the damage and target data and range, among others, of the damage from cyberattacks. The most targeted damage is direct. There is only one overseas document on spillover damage. Reports on the scope of the target damage often differ depending on the literature, and the data are not unified.
Therefore, it is meaningful to estimate the direct and spillover damages using the production and input–output models, respectively. In addition, this study collected mainstream data through interviews and hearings, considering that incident data from cyberattacks are often not disclosed.
Methods for estimating direct and spillover damage
In this study, we constructed a production function and measured the decrease in production value due to a decrease in the labor force of the IT department caused by cyberattacks. In addition, we measured the negative production-inducing effect caused by this decrease in production using input–output analysis and estimated spillover damage.
The production function expresses the relationship between the production factor and the output (production value/gross value-added amount) using mathematical formulas. Capital stock and labor, considered the most universal factors of production, are usually used as explanatory variables for output. We recognized that analysis by the production function is a suitable approach for calculating the economic damage (from the viewpoint of production) in the event of a cybersecurity incident involving labor damage, as in this study.
Spillover damage (decrease in production) due to damage involving production factors leads to a further decrease in production through dependency between industries. For example, if production were stopped because of a disaster, it would also stop the production of industrial parts. The industrial suspension of industrial parts further causes the production suspension of other parts and raw materials. Input–output analysis is a powerful tool for measuring the magnitude of such spillover damage.
Direct damage estimation method using the production function
We estimated the direct damage from cyberattacks (including viral infections) to the entire country based on the system and data recovery times. Our estimation model agrees with that of Tanaka (2014).
We estimated the production function by assuming that "the net value added (Y)" can be realized by the labor force (L-Lr) after deducting the system recovery time and data recovery working times (Lr) associated with a cyberattack (Eq. (1)). Then, based on the coefficient of the production function, using the labor force (L), we were able to determine when the system recovery time and working time (Lr) could be used for the original production activity (Y+). The difference between (Y+) and (Y) was used as the direct damage amount (LS) (Eq. (2)). We assumed that K is capital stock and constant, regardless of cybersecurity incidents, systems, or data recovery. In addition, Eq. (1) is established when the relationship α + β = 1 (constant returns to scale) holds for capital allocation ratio α and labor allocation ratio β:
$$\varvec{Y}=\varvec{AK}^{\alpha } (\varvec{L}-\varvec{Lr})^{1 - \alpha}.$$
Dividing both sides of Eq. (1) by L − Lr gives Eq. (2):
$$\varvec{Y} / (\varvec{L}-{\varvec{Lr} ) = \varvec{AK}^{\alpha } (\varvec{L}}-\varvec{Lr})^{-\alpha } =\varvec{A}(\varvec{K}/(\varvec{L}-\varvec{Lr}))^{\alpha } .$$
To calculate the amount of damage directly based on Eq. (2), we estimated the production function of Eq. (1); however, A, α, and β were calculated based on the logarithmic transformation in Eq. (3):
$$\text{ln}\varvec{Y}/(\varvec{L}-\varvec{Lr})=\text{ln}\varvec{A}+\alpha \text{ln}\varvec{K}/(\varvec{L}-\varvec{Lr}) .$$
Because the relationship between Y+ and Y in Eq. (2) is Y+/Y = (L/(L − Lr))1−α, the amount of damage can be calculated directly using Eq. (4):
$$\varvec{LS}=((\varvec{L}/{\varvec{L}}-{\varvec{Lr}} )^{1 - \alpha }-\boldsymbol{1})\varvec{Y}=((\varvec{L}/{\varvec{L}}-{\varvec{Lr}} )^{\beta }-\boldsymbol{1} )Y.$$
We collected economic statistical data published by IPA, JNSA, and unpublished JNSA data. We classified the industrial sector based on 108 industries' data from the Japan Industrial Productivity (JIP) database of the Institute of Economic and Industrial Research.
We describe the following components: (1) output (Y), capital stock (K), and labor force (L); (2) number of working hours (Lr) allocated to the system or data recovery times; and (3) estimation method of A, α, and β.
1) Output (Y), capital stock (K), and labor (L)
The net value added (Y) is calculated by subtracting the intermediate input from the output using the sectoral output/intermediate input reported in the 2015 JIP data input–output table. Capital stock (K) denotes the real net capital stock of the capital sector and investment data. We used data on man-hours, given that the total number of working hours of (L) is critical to reflect the system data recovery time after cyberattacks.
2) Number of working hours (Lr) allocated for system recovery or data recovery after a cyberattack
Because there are no data on the number of working hours, we estimated (Lr) allocated for the system or data recovery by each industry using the following procedure. In addition, Lr is the time spent on system and data recovery only for the IT department:
National-level estimation of the number of working hours allocated to the system or data recovery after a cyberattack
The IPA report (2014) outlined the time the IT department took to recover the system after a cyberattack, the additional data processing time (time spent other than recovery), and the time required to resolve other incidents. For this survey, 13,000 companies with more than 21 employees were randomly selected by industry, whose number of employees was from a private company database (Teikoku Databank). The responses to this questionnaire were 1913, with a valid response rate of 14.7%.
An analysis of the valid responses shows in the case of "large companies with more than 300 employees", the IT personnel spent 18.5 h, 5.6 h, and 23.1 h (total 47.2 hours) on recovery, additional data processing, and other incidents. In the case of "companies of between 21 and 300 employees", they are 13.1 h, 3.8 h, and 23.1 h (40.0 h total), respectively.
The 2014 economic census reported that the number of large companies with 300+ employees is 15,526, and that of employees between 300 and 20 is 320,085. We estimated the lost time for IT department employees to be 728,205 h and 12,795,816 h, respectively, for a total of 13,524,021 h. For reference, this means that the average lost time of the IT department per company is 40.3 h in each cyberattack incident.
It should be noted that this time is only the time lost in the "IT department" and does not include the time lost in other departments such as sales and administration departments.
Estimation of the number of working hours by industry
The industries and number of employees are stratified sampling (proportional allocation method) based on the distribution of companies by the number of employees and by industry in the Japan Standard Industrial Classification of the 2012 Economic Census to ensure statistical validity.
Using the number of IT department working hours for the entire country estimated in (A), we estimated the number of working hours by industry based on the 2017 JNSA information security incident data (380 data).
The JNSA data were generated by collecting and analyzing the results of analyses of personal information leakage incidents reported in newspapers and the Internet in the relevant fiscal year; these data originally included cases unrelated to cyberattacks. Therefore, we sorted the contents of each incident data and identified cases of cyberattacksFootnote 2 (75 of 380 cases were cyberattacks).
Subsequently, we identified the industry (108 industries) from industry category information in the 75-incident data. Next, the amount of damage for each incident calculated independently by the JNSA was tabulated by country and industry (108 industries). Table 1 shows the amount of damage calculated by the JNSA for each incident using self-made method aggregated for each of the 108 industries.
Table 1 The amount of damage by 108 industries calculated by JNSA
Then, we calculated the ratio of the whole country and each industry regarding damage calculated by the JSNA and apportioned the direct damage amount of the whole country calculated in (A) to each industry using this ratio. Table 2 shows the results. In this study, we used 2017 JNSA data. In the JNSA data, only the 2017 data show the industry category; therefore, we directly used the industry category of the FY2017 JNSA data.
Table 2 Lr of 108 industries
3) A, α, and β (scale coefficient (A), capital share (α), and labor share (β) in the production function)
We calculated A, α, and β based on Eq. (3), which is a logarithmic transformation of Y, K, L, and Lr for the 108 sectors from 2013 to 2015. Following the method highlighted in the study by Tanaka (2014), in the JIP data, the industrial sections codes 72 (housing) and 108 (activities not elsewhere classified) for three years from 2013 to 2015, and 36 (pig iron and crude steel) for 2013 were excluded. This is because the added value, capital stock, and labor man-hours were zero, owing to a lack of data.
The results of the estimation were as follows: A = 0.23370315, α = 0.53480907, β = 0.46519093. The coefficient of determination R2 of Y/(L-Lr) on the left-hand side and K/(L-Lr) on the right-hand side of Eq. (3) was 0.5214027.
Estimating method of spillover damage by input–output model
We estimated the spillover damage by industry based on the amount of direct damage by industry, as calculated in 3–2, and using the following input–output model: specifically, the amount of damage is calculated using a competitive import model. We define input coefficient matrix A, final demand column vector F, output vector Y, n × n unit matrix I, export column vector E, and import column vector M. If \(\widehat{M}\) is a matrix with the import coefficients on the diagonal and zeros for the off-diagonal, then we can express the formula as follows:
$$\begin{aligned} &{\varvec{Y}} = \varvec{AY} + \varvec{F}+{\varvec{E}}-{\varvec{M}}={\varvec{AY}}+{\varvec{F}}+{\varvec{E}}-{\boldsymbol{\widehat{M}}}(\varvec{AY}+\varvec{F}) \\ & \quad \Leftrightarrow {\varvec{Y}}=(\varvec{I} -(\varvec{I}-{\boldsymbol{\widehat{M}}})\varvec{A})^{-1} ((\varvec{I}-{\boldsymbol{\widehat{M}}})\varvec{F}+{\varvec{E}}). \\ \end{aligned}$$
Here, F in Eq. (5) corresponds to the direct damage calculated in Eq. (4), and the direct damage in Eq. (4) is estimated based on the value-added production function. In the input–output table, estimates are made on a production value basis. Therefore, when F is inserted into Eq. (5), it is necessary to revise it to a production value basis. This revision was calculated using the ratio of the value-added amount and the production amount in the input–output table, and the amount excluding non-household consumption expenditure (accommodation, daily allowance, entertainment expenses, welfare expenses) was estimated as the value-added amount. In addition, because this research focuses on the domestic damage caused by cyberattacks in Japan, we calculated F as the product of the amount of added value by the self-sufficiency rate of each industry, where the self-sufficiency rate is obtained by subtracting the import coefficient from 1. The import coefficient is calculated by dividing the absolute value of "(less) Total imports" by "Total domestic demand" in the input–output table.
Because Y calculated using Eq. (5) includes direct damage, it is necessary to exclude direct damage from spillover damage. Therefore, spillover damage is estimated using Eq. (6):
$$\begin{aligned}& {\varvec{Y}}\left( {\text{spillover damage only}} \right) \\ &\quad= {\varvec{Y}}\left( {\text{spillover damage including direct damage}} \right) {-}{\varvec{F}}\left( {\text{direct damage}} \right). \hfill \\ \end{aligned}$$
We used the direct input damage by industry calculated in the 2015 input–output table (37 I/O sections). Based on the integrated major sections in the input–output table for Japan, we divided the industrial sections into 37 I/O sections.
In this study, we showed that it is possible to estimate not only the direct damage caused by cyberattacks, but also spillover damage at the national level using the production function and I/O model.
Estimated damage for each industrial sector
The estimation results are shown in Tables 3, 4, and 5.
Table 3 shows the direct damage, spillover damage in each of the 37 I/O sectors, and total damage (Japan, FY2015) based on the model in 3-2-1 and 3-3-1, and Table 4 shows the proportion of each industry to the total damage of the whole country.
The direct damage in Eq. (4) in [a] of Table 3 was estimated based on the value-added production function. Then, direct damage based on production value ([c] in Table 3) is calculated by using the ratio ([b] in Table 3) of the value-added amount and the production amount. Next, because this study focuses on the domestic impact of cyberattacks, F in Eq. (5) is calculated by multiplying the direct damage based on the production value ([c] in Table 3) by the self-sufficiency rate of each industry so that F in Eq. (5) is shown as [d] of Table 3. Finally, we calculate the spillover damage ([e] in Table 3) based on Eqs. (5) and (6).
Table 3 Direct, spillover, and total damages in 37 I/O industrial sectors (Japan)
Table 4 Estimated damages and proportion of damages by 37 I/O sectors
Table 5 shows the top 5 industries with the highest total losses and that the JIP industry code 59 (Information and communications) suffered damages of approximately 12,556 million yen, accounting for 40.3% of the total damage. The damages caused by industry codes 31 (business-oriented machinery), 51 (semiconductor devices and integrated circuits), 53 (finance and insurance), and 66 (business services) accounted for 5.7%, 9.3%, 12.3%, and 11.2% of the total damage, respectively. The top five industries accounted for 78.8% of the total damage.
Table 5 Summary of the top 5 I/O sectors
Discussion of estimated damage for all sectors
As shown in Table 3, we estimated damages for all sectors based on the IPA dataset in 3-2-2 A). The direct damage (based on domestic production value), spillover damage, and total amount were approximately JPY 18,785 million, JPY 12,385 million, and JPY 31,170 million, respectively.
Here, we should note that the IPA dataset only showed the lost working hours in IT departments caused by cyberattacks and does not include the lost working hours in other business sections during IT department work for IT system recovery. If the cyberattack incident survey includes lost working hours in other business sections, our model will show a larger Lr, therefore the total damage will be huge. In addition, immeasurable losses, such as the loss of business opportunities and brand damage, may occur in cyberattack victim companies.
Table 5 shows that JIP industry code 59 (Information and communications) suffered damages of approximately 12,256 million yen, accounting for 40.3. % of the total damages. While these analyses by industry are useful for cybersecurity and economic policy discussions, it is important to improve the input dataset's quantity and quality for our estimation model. Therefore, we expect to establish a framework for collecting information on cyber incidents at the national level and for data standardization in Japan, as in the case of the United States.Footnote 3
By presenting a method for analyzing the damage caused by cyberattacks from a macroeconomic perspective and using production functions and input–output tables, this study contributes to future studies on cyber resilience.
This study takes a macroeconomic viewpoint to directly estimate the economic losses from cyberattacks in Japan—the amount of direct and spillover damage. Cyberattack recovery consumed at least IT department working hours in Japan and caused damage worth approximately 31,170 million yen for the financial year 2015.
Future studies can improve the accuracy of the aforementioned estimation using data on the working hours required for recovery in each industry. As mentioned above, it is also expected to establish a framework for collecting information on cyber incidents at the national level and standardizing data in Japan.
We plan to study and analyze industrial characteristics in future research more precisely. First, we analyze a specific industry's characteristics by utilizing the information and communications input–output table published by the Ministry of Internal Affairs and Communications. Next, we analyze the forward linkage of the spillover effect in addition to the backward linkage, as in this study.
The data for output (Y), capital stock (K), and labor (L) are available from the JIP database: [hyperlink to dataset(s)/data source, e.g., "https://www.rieti.go.jp/jp/database/JIP2015/index.html"]. The data for number of working hours (Lr) allocated for system recovery or data recovery after a cyberattack are available from IPA: [hyperlink to dataset(s)/data source, e.g., "https://www.ipa.go.jp/security/fy26/reports/isec-survey/index.html"], and JSNA. Although the JNSA data are not open to the public, they were individually requested and obtained for this research and are available if requested individually. The data for the number of companies are available from the 2015 Economic Census: [hyperlink to dataset(s)/data source, e.g., "https://www.stat.go.jp/data/e-census/2016/kekka/gaiyo.html"]. The Japanese benchmark input–output table is available from the Ministry of Internal Affairs and Communication: [hyperlink to dataset(s)/data source, e.g., "https://www.e-stat.go.jp/stat-search/files?page=1&layout=datalist&toukei=00200603&tstat=000001130583&cycle=0&year=20150&month=0"].
RAND Cooperation uses the input–output table by the Organization for Economic Co-operation and Development to calculate the spillover damage of cyberattacks. The Mitsubishi Research Institute, Inc. analyzes the reputational damage of cyberattacks from the stock price effect before and after a certain event (e.g., mergers and acquisitions), by analyzing the cumulative abnormal returns. A method for analyzing changes in corporate value is used.
Incident data leakage identified cause categories; such as worm viruses, bug security holes, and unauthorized access during cyberattacks. The cause categories excluded from the target include: internal fraud, loss/misplacement, distress, and erroneous operation.
The US Department of Land Security (DHS) provides insurance companies with a data collection and analysis platform (Cyber Incident Data and Analysis Repository). This platform helps private modeling companies assess cyber risk and cyber accumulation for modeling. In one of the cases, a data standard format was integrated to the management system (CAMS: Cyber Accumulation Management System) (some public institutions (e.g., FBI) also play a role).
CSIS:
The Center for Strategic and International Studies (USA)
Information-technology Promotion Agency (Japan)
JNSA:
The Japan Network Security Association (Japan)
JIP:
The Japan Industrial Productivity
CO (Cabinet Office) (2013) Assumption of damage from the Nankai Trough earthquake (Points of the second report—Damage and economic damage of facilities, 2013.https://www.bousai.go.jp/jishin/nankai/taisaku_wg/pdf/20130318_kisha.pdf. Accessed 20 Mar 2022.
CS (Cybersecurity Strategy,The Government of Japan) (2021) Cybersecurity for All, Sept 2021. https://www.nisc.go.jp/pdf/policy/kihon-s/cs-senryaku2021-en-booklet.pdf. Accessed 18 Sept 2022.
CSIS (Center for Strategic and International Studies) (2021) The hidden costs of cybercrime, 2021. https://www.csis.org/analysis/hidden-costs-cybercrime. Accessed 20 Mar 2022.
IPA (Information Technology Promotion Agency) (2014) Investigation of damage caused by information security events, 2014 (in Japanese). https://www.ipa.go.jp/security/fy26/reports/isec-survey/index.html. Accessed 20 Mar 2022.
IPA(Information Technology Promotion Agency) (2021) Information Security 10 major Threats 2021. (in Japanese) https://www.ipa.go.jp/security/vuln/10threats2021.html. Accessed 20 Mar 2022.
JNSA (Japan Network Security Association) (2018) Survey results on information security incidents—Leakage of personal information, 2018, Available via JNSA (in Japanese) .https://www.jnsa.org/result/incident/2018.html. Accessed 20 Mar 2022.
MIC (Ministry of Internal Affairs and Communications) (2019) White paper on information and communications, section 3 Economic losses such as cyberattacks, 2019 (in Japanese). https://www.soumu.go.jp/johotsusintokei/whitepaper/ja/r01/html/nd113320.html. Accessed 20 Mar 2022
Shimoda M, Fujikawa K (2012) Industrial relation analysis model and supply constraints caused by the great East Japan Earthquake. Input–output Anal 20:133–146. https://doi.org/10.11107/papaios.20.133 (in Japanese)
Tanaka H (2013) Report of the Committee on Information Security Damage and Countermeasures—Construction of a new model of threats and damage in companies, Research Group on Information Security Damage and Countermeasures, IPA, 2013 (in Japanese). (This material is old and not currently available, but we have owned this manuscript.)
Tanaka H, Takemura T, Iitaka Y, Hanamura K, Komatsu A (2014) Research on estimation of economic loss due to information security incidents. J Econ Policy Stud 11:59–62 (in Japanese)
Institute of Information Security, 2-14-1 Tsuruya-Cho, Kanagawa-Ku, Yokohama, Kanagawa, 221-0835, Japan
Akiyoshi Kokaji & Atsuhiro Goto
Akiyoshi Kokaji
Atsuhiro Goto
AG designed the study, and AK performed the calculations. AK interpreted the results and drafted the manuscript. AK revised the manuscript and created the figures. Both authors read and approved the final manuscript.
Correspondence to Akiyoshi Kokaji.
This research does not involve human subjects, human material, or human data.
Appendix 1: Definition of term
The terms used in this study are defined as follows:
For "Cyber attack", Information-technology Promotion Agency, Japan revealed in IPA (2021) that the major organizational threats include damage from ransomware, theft of confidential information by targeted attacks, and telework. The new normal ways of working have exposed organizations to supply chain attacks, financial damage from fraudulent emails, information leakage due to internal fraud and negligence, business suspension due to IT infrastructure failure, unauthorized logins for Internet services, and increased misuse after the renewal of vulnerability countermeasure information.
Regarding the characteristics of cyberattacks, the Ministry of Defense and the Self-Defense Forces presented a list of the characteristics of cyberattacks in a report titled (2012). These characteristics include "attacker superiority", "diversity", "anonymity", "top secret (confidentiality)", and "deterrence difficulty. Therefore, it is difficult to analyze the economic damage caused by these characteristics.
This study excluded indirect damage (damage effects due to rumors on brand value). This study focuses on the following direct and spillover damages caused by cyberattacks on the economy:
Direct damage (damage to the attacked company/industry) includes a general investigation of the cause, system recovery, data corruption, leakage, damage compensation, system outage, business interruption, and opportunity loss.
Spillover damage (damage to companies/industries directly affected by direct damage) includes general damage to other companies and industries doing business with the directly affected companies and damage from the attacked companies to other companies, industries, and society. Damage to social infrastructure may affect social and economic activities and cause significant losses.
Appendix 2: Previous research: analysis of damage caused by cyberattacks
Source/year of publication/title, etc.
Target countries and regions
Target year
Overview of damage calculation
Damage calculation model
1 CSIS (Center for Strategic and International Studies, USA), McAfee (2020) The Hidden Cost of Cybercrime Worldwide 2020 $1 trillion (equivalent to 1% GDP) Unknown 1500 companies in The Overview
2 RAND Cooperation (2018) Estimating the Global Cost of Cyber Risk: Methodology and Examples 63 countries 2017 $800 billion (equivalent to 1.1% GDP) I/O model OECD data, financial data, incident data, etc.
3 Cyber Security Ventures (2020) Cybercrime To Cost The World $10.5 Trillion Annually Worldwide 2021 $6 trillion Undisclosed Undisclosed
4 Microsoft, et al. (2018) Cybersecurity Threats to Cost Organizations in Asia Pacific US$1.75 Trillion in Economic Losses Asia Pacific 2017 $1.745 trillion (equivalent to 7% GDP) Self-made model Overview, economic data
5 Accenture (2019) The Cost of Cybercrime 11 countries 2018 $13 million per company on average Self-made model 2600 people from 355 companies from interviews
6 JNSA (NPO Japan Network Security Association) (2019) Report on Information Security Incidents Japan 2018 640 million yen per company and 268.4 billion yen for Japan as a whole JO model (Self-made model) Public information
7 Trend Micro (2020) Corporate Security Trends in 2020 Japan 2018 Average 210 million yen per company Unknown 1086 companies from the interview
8 RISI (Repository of Industrial Security Incidents: operated by Security Incidents Organization, a U.S. non-profit organization) United States 28 years 6% of cases exceed $10 million Unknown Unknown
9 Ponemon Institute (2015) 2015Cost of Cyber Crime Study: United States(Cyber Crime) 8 countries 2015 $15 million per company Unknown 58 U.S. companies, 553 companies in 7 other countries
10 Ponemon Institute (2015)2015 Cost of Data Breach Study: Global Analysis(Cyber Impact) 11 countries 2015 $3.79 million per company ($154 per record) Self-made model 1500 companies in the overview
11 McAfee(2013)The Economic Impact of Cybercrime and cyber-Espionage Worldwide, USA 2013 Worldwide: $0.3 trillion to $1, U.S. $0.024-$0.12 trillion Unknown Unknown
12 AFCEA (armed forces communications and electronic association military communications and electronics association) – – (Unpublished) (Unpublished) (Unpublished)
13 Mitsubishi Research Institute, and Ministry of Economy, Trade, and Industry, (2007) Evaluation of damage to corporate value due to cybersecurity accident (MRI/ The University of Tokyo) Japan 2007 (Mitsubishi Research institute's expected amount of damage is 1.1 billion yen, etc.) CAR (Cumulative. Abnormal Return) analysis model Stock market information
Created by the authors based on published materials of the Ministry of Internal Affairs and Communications
Kokaji, A., Goto, A. An analysis of economic losses from cyberattacks: based on input–output model and production function. Economic Structures 11, 34 (2022). https://doi.org/10.1186/s40008-022-00286-4
Revised: 04 November 2022
Input–output model
Production function
Spillover damage of cyberattacks | CommonCrawl |
\begin{definition}[Definition:Choice Function/Use of Axiom of Choice]
The Axiom of Choice (abbreviated '''AoC''' or '''AC''') is the following statement:
:''All $\mathbb S$ as above have a choice function.''
It can be shown that the AoC it does not follow from the other usual axioms of set theory, and that it is relative consistent to these axioms (i.e., that AoC does not make the axiom system inconsistent, provided it was consistent without AoC).
Note that for any given set $S \in \mathbb S$, one can select an element from it (without using AoC). AoC guarantees that there is a choice function, i.e., a function that "simultaneously" picks elements of all $S \in \mathbb S$.
AoC is needed to prove statements such as "all countable unions of finite sets are countable" (for many specific such unions this
can be shown without AoC), and AoC is equivalent to many other mathematical statements such as "every vector space has a basis".
{{WIP|The above needs to be rewritten more tersely. Examples of its use should be moved either to the AoC page itself or onto the specific pages where the statements themselves are used.<br/>Note that the recent amendment to this page which added a considerable quantity of material was made by an anonymous editor and therefore we can not enter into a discussion with him/her.}}
Category:Definitions/Set Theory
\end{definition} | ProofWiki |
A multigrid based finite difference method for solving parabolic interface problem
Three types of weak pullback attractors for lattice pseudo-parabolic equations driven by locally Lipschitz noise
November 2021, 29(5): 3121-3139. doi: 10.3934/era.2021029
Global behavior of P-dimensional difference equations system
Amira Khelifa 1, and Yacine Halim 2,,
Department of Mathematics and LMAM laboratory, Mohamed Seddik Ben Yahia University, BP 98 Ouled Aissa 18000, Jijel, Algeria
Department of Mathematics and Computer Sciences, Abdelhafid Boussouf University Center, RP 26 Mila 43000, Mila, Algeria
* Corresponding author: Yacine Halim
Received January 2021 Revised March 2021 Published November 2021 Early access April 2021
The global asymptotic stability of the unique positive equilibrium point and the rate of convergence of positive solutions of the system of two recursive sequences has been studied recently. Here we generalize this study to the system of $ p $ recursive sequences $x_{n+1}^{(j)}=A+\left(x_{n-m}^{(j+1) mod (p)} \;\;/ x_{n}^{(j+1) mod (p)}\;\;\;\right) $, $ n = 0,1,\ldots, $ $ m,p\in \mathbb{N} $, where $ A\in(0,+\infty) $, $ x_{-i}^{(j)} $ are arbitrary positive numbers for $ i = 1,2,\ldots,m $ and $ j = 1,2,\ldots,p. $ We also give some numerical examples to demonstrate the effectiveness of the results obtained.
Keywords: System of difference equations, global asymptotic stability, boundedness, rate of convergence.
Mathematics Subject Classification: Primary: 39A10; Secondary: 40A05.
Citation: Amira Khelifa, Yacine Halim. Global behavior of P-dimensional difference equations system. Electronic Research Archive, 2021, 29 (5) : 3121-3139. doi: 10.3934/era.2021029
I. Dekkar, N. Touafek and Y. Yazlik, Global stability of a third-order nonlinear system of difference equations with period-two coefficients, Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A Mat., 111 (2017), 325-347. doi: 10.1007/s13398-016-0297-z. Google Scholar
R. Devault, C. Kent and W. Kosmala, On the recursive sequence $x_{n+1} = p+(x_{n-k}/x_{n})$, J. Difference Equ. Appl., 9 (2003), 721-730. Google Scholar
S. N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-9168-6. Google Scholar
E. M. Elsayed, On a system of two nonlinear difference equations of order two, Proc. Jangeon Math. Soc., 18 (2015), 353-368. Google Scholar
E. M. Elsayed, Solutions of rational difference systems of order two, Math. Comput. Modelling, 55 (2012), 378-384. doi: 10.1016/j.mcm.2011.08.012. Google Scholar
E. M. Elsayed and T. F. Ibrahim, Periodicity and solutions for some systems of nonlinear rational difference equations, Hacet. J. Math. Stat., 44 (2015), 1361-1390. Google Scholar
[7] E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman and Hall/CRC Press, Boca Raton, FL, 2005. Google Scholar
M. Gümüş, The global asymptotic stability of a system of difference equations, J. Difference Equ. Appl., 24 (2018), 976-991. doi: 10.1080/10236198.2018.1443445. Google Scholar
Y. Halim, Global character of systems of rational difference equations, Electron. J. Math. Anal. Appl., 3 (2015), 204-214. Google Scholar
Y. Halim and M. Bayram, On the solutions of a higher-order difference equation in terms of generalized Fibonacci sequences, Math. Methods. Appl. Sci., 39 (2016), 2974-2982. doi: 10.1002/mma.3745. Google Scholar
Y. Halim, N. Touafek and Y. Yazlik, Dynamic behavior of a second-order nonlinear rational difference equation, Turkish J. Math., 39 (2015), 1004-1018. doi: 10.3906/mat-1503-80. Google Scholar
M. Kara and Y. Yazlik, Solvability of a system of nonlinear difference equations of higher order, Turk. J. Math., 43 (2019), 1533-1565. doi: 10.3906/mat-1902-24. Google Scholar
V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-017-1703-8. Google Scholar
G. Papaschinopoulos and C. J. Schinas, On the behavior of the solutions of a system of two nonlinear difference equations, Commun. Appl. Nonlinear Anal., 5 (1998), 47-59. Google Scholar
G. Papaschinopoulos and C. J. Schinas, Oscillation and asymptotic stability of two systems of difference equations of rational form, J. Differ. Equations Appl., 7 (2001), 601-617. doi: 10.1080/10236190108808290. Google Scholar
M. Pituk, More on Poincaré's and Perron's theorems for difference equations, J. Difference Equ. Appl., 8 (2002), 201-216. doi: 10.1080/10236190211954. Google Scholar
D. T. Tollu, Y. Yazlik and N. Taşkara, Behavior of positive solutions of a difference equation, J. Appl. Math. Inform., 35 (2017), 217-230. doi: 10.14317/jami.2017.217. Google Scholar
D. T. Tollu, Y. Yazlik and N. Taşkara, On a solvable nonlinear difference equation of higher order, Turkish J. Math., 42 (2018), 1765-1778. doi: 10.3906/mat-1705-33. Google Scholar
Y. Yazlik, D. T. Tollu and N. Taskara, On the behaviour of solutions for some systems of difference equations, J. Comput. Anal. Appl., 18 (2015), 166-178. Google Scholar
D. Zhang, W. Ji, L. Wang and X. Li, On the symmetrical system of rational difference equation $x_{n+1}=A+y_{n-k}/y_{n}, y_{n+1}=A+x_{n-k}/x_{n}$, Appl. Math., 4 (2013), 834-837. Google Scholar
Figure 1. The plot of system (23) with $ A = 1.2>1 $
Figure 2. The plot of system (23) with $ A = 1 $
Figure 3. The plot of system (23) with $ A = 0.9<1 $
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Marcel Freitag. Global existence and boundedness in a chemorepulsion system with superlinear diffusion. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5943-5961. doi: 10.3934/dcds.2018258
Cemil Tunç. Stability, boundedness and uniform boundedness of solutions of nonlinear delay differential equations. Conference Publications, 2011, 2011 (Special) : 1395-1403. doi: 10.3934/proc.2011.2011.1395
Haibo Cui, Zhensheng Gao, Haiyan Yin, Peixing Zhang. Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4839-4870. doi: 10.3934/dcds.2016009
Yueh-Cheng Kuo, Huey-Er Lin, Shih-Feng Shieh. Asymptotic dynamics of hermitian Riccati difference equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2037-2053. doi: 10.3934/dcdsb.2020365
Andrejs Reinfelds, Klara Janglajew. Reduction principle in the theory of stability of difference equations. Conference Publications, 2007, 2007 (Special) : 864-874. doi: 10.3934/proc.2007.2007.864
Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061
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Amira Khelifa Yacine Halim | CommonCrawl |
Optimization of fermentation conditions for an Escherichia coli strain engineered using the response surface method to produce a novel therapeutic DNA vaccine for rheumatoid arthritis
Juan Long1 na1,
Xiao Zhao1 na1,
Fei Liang1,
Nan Liu1,
Yuying Sun1 &
Yongzhi Xi ORCID: orcid.org/0000-0001-9980-92451
Fermentation condition optimization and nutrients screening are of equal importance for efficient production of plasmid DNA vaccines. This directly affects the downstream purification and final quality and yield of plasmid DNA vaccines. The present study aimed to optimize the fermentation conditions for high-throughput production of therapeutic DNA vaccine pcDNA-CCOL2A1 by engineered Escherichia coli DH5α, using the response surface method (RSM).
We hypothesized that optimized fermentation conditions significantly increase the yield of pcDNA-CCOL2A1 therapeutic DNA vaccine, a novel DNA vaccine for treating rheumatoid arthritis (RA). Single-factor analysis was performed to evaluate the optimal basal culture medium from LB, 2 × YT, TB, M9 (Glycerol) and M9 (Glucose), respectively. Thereafter, the Plackett-Burman design (PBD) was used to ascertain the three most significant factors affecting the vaccine yields, followed by the paths of steepest ascent to move to the nearest region of maximum response. Initial screening through the PBD revealed that the most key factors were peptone, mannitol, and inoculum concentration. Subsequent use of RSM was further optimized for the production of therapeutic DNA vaccine pcDNA-CCOL2A1 through Box-Behnken design (BBD). The final optimized fermentation conditions were as follows: peptone, 25.86 g/L; mannitol, 8.08 g/L; inoculum concentration, OD = 0.36. Using this statistical experimental design, the yield of therapeutic DNA vaccine pcDNA-CCOL2A1 markedly increased from 223.37 mg/L to339.32 mg/L under optimal conditions, and a 51.9% increase was observed compared with the original medium.
The present results provide a basis for further production of high-quality and high-yield therapeutic DNA vaccine pcDNA-CCOL2A1 in pilot-scale and even industrial-scale.
Therapeutic DNA vaccines, especially antigen-specific tolerizing DNA vaccines, as novel therapeutic strategies for rheumatoid arthritis (RA), have displayed marked advantages compared with current therapies including disease-modifying antirheumatic drugs (DMARDs), cytotoxic agents, cytokine antagonists or monoclonal antibodies, tofacitinib, glucocorticoids, etc. [1,2,3]. These therapies generally control RA disease activity by either suppressing overall immune function or partially neutralizing individual cytokines or partially antagonizing individual cytokine receptors; however, they rarely modulate immune cell populations, except for methotrexate (MTX) [4,5,6,7,8]. Furthermore, they not only cannot cure the disease but also inadequately discontinue disease progression, especially for the invasive destruction of articular cartilage and bone. In particular, they can also increase the potential risk of severe infections and malignancy [4, 5].
Unlike current aforementioned therapies, antigen-specific tolerizing DNA vaccine pcDNA-CCOL2A1 encoding chicken type II collagen exerts its therapeutic effects through specific immune modulation, especially for inducing potent immune tolerance against RA. More precise mechanisms of action include increase in the contents of CD4+CD25+ T regulatory cells, reductions in the specific proliferative response of T lymphocytes to CII, and induction of a shift from Th1 to Th2 cells, accompanied by down-regulation of Th1-cytokine TNF-α and up-regulation of both Th2-cytokine IL10 and Th3-cytokine TGF-β. Moreover, we previously reported that DNA vaccine pcDNA-CCOL2A1 displayed efficacy comparable to those of the current "gold standard" therapy, methotrexate (MTX), in the established collagen-induced arthritis (CIA) rat model. And it is safe and well-tolerated without any abnormal clinical signs and adverse effects on normal physiological function [9,10,11,12], suggesting that this vaccine has a high drugability.
Successful establishment of a three-tier cell bank with high stability and identification of a high-yield Escherichia coli DH5α strain to produce therapeutic DNA vaccine pcDNA-CCOL2A1 would yield a sound theoretical and material basis for further pilot-scale tests and even industrial-scale production of this vaccine [13]. Furthermore, fermentation conditions including growth conditions, culture types, and culture medium composition influence the yield and productivity of plasmid DNA vaccines, by directly influencing the downstream purification, quality and yield of plasmid DNA vaccines [14,15,16]. Since many factors existed in the fermentation condition, a great number of experiments should be simultaneously conducted, and the possible interactions between these factors would be studied. So a reasonable analytical methods will lead to lower reagent consumption and considerably less laboratory work. Conventional single dimensional research gives unreliable results, inaccurate conclusion. Orthogonal testing optimum design can reduce the number of combinations, and obtain optimal factor levels. However, it cannot give a regression equation for the whole parameter tested. By contrast, the combination of Plackett-Burman design (PBD) with common optimization methodology Box-Behnken design-response surface method (BBD-RSM) can collectively eliminate these limitations and are powerful and useful in determining the key factors rapidly from a multivariable system [17]. PBD provides indication and tendency regarding the necessity of each variables in relatively few experiments, the following Box-Behnken design (BBD) provides a large amount of information and the interaction of the independent variables on the response by a small number of experiment [18,19,20,21]. The data from BBD subject to a second-order multiple regression equation showing the dependence of the response (i.e. the plasmid yield) on independent variables (i.e. the concentration of the separate components of the nutrient medium or fermentation parameters), and even give predictive results of responses and the possible levels of related independent variables. The equation of the model can clearly present the effects for binary combinations of the independent variables.
Essentially, plasmid DNA vaccine production is aimed at increasing yield and productivity and decreasing manufacturing cost. Hence, we hypothesize that optimization of fermentation conditions significantly increases the yield and productivity of therapeutic DNA vaccine pcDNA-CCOL2A1. The present study aimed to determine the effect of optimized fermentation conditionsof the engineered Escherichia coli DH5α on the yield of therapeutic DNA vaccine pcDNA-CCOL2A1. To our knowledge, this is the first study to systemically optimize fermentation conditions of the engineered Escherichia coli DH5α for producing therapeutic DNA vaccine pcDNA-CCOL2A1 through a combination of the commonly used PBD with common optimization methodology BBD-RSM.
Single-factor analysis of basal culture medium revealed optimal carbon and nitrogen sources for producing therapeutic DNA vaccine pcDNA-CCOL2A1
Several previous studies have reported the precedence of single-factor analysis before using PBD and BBD [22, 23]. Accordingly, initial screening was performed for the selection of optimum basal culture medium, wherein 2 × YT was found to be advantageous for the yield of plasmid DNA vaccine pcDNA-CCOL2A1 produced by the engineered E. coli DH5α. Further evaluation of carbon and nitrogen sources indicated that mannitol and peptone can significantly increase plasmid yield compared with the basal culture medium 2 × YT (p<0.05), as shown in Fig. 1a-b.
a, b Single-factor analysis of basal culture medium revealed optimal carbon and nitrogen sources for producing therapeutic DNA vaccine pcDNA-CCOL2A1. a Evaluation of the optimal basal culture medium among LB, 2 × YT, TB, M9 (Glycerol) and M9 (Glucose) in shaking flask culture through One-Way ANOVA. b Screen the optimal carbon and nitrogen sources in shaking flask culture using One-Way ANOVA. *p < 0.05, * * p < 0.001. Data are expressed as the mean ± standard deviation (SD) of 3 independent experiments
PBD screening elucidated the key variables affecting the yield of therapeutic DNA vaccine pcDNA-CCOL2A1
In the PBD experiment, ten variables were chosen to screen the key factors affecting the yield of plasmid DNA vaccine, as shown in Table 1. The data reported in Table 2 showed a substantial variation in plasmid yield among the 12 experimental sets, varying from 146.60 ± 15.25 mg/L to 312.86 ± 13.69 mg/L under two different levels of factors. Based on regression analysis of PBD in Table 3, the fitting model for the yield of plasmid DNA vaccine was significant (p = 0.0287). The ratio of adequate precision measures the signal-to-noise ratio, and a ratio greater than 4 is desirable. In this case, adequate precision was 84.831, confirming that the model could adequately navigate the design space. The goodness of the model was checked by the determination coefficient R2, which was 0.9999. Among these factors, peptone, yeast extract, mannitol, and inoculum concentration were the significant model terms on the response (p<0.05). The three most significant variables were peptone, mannitol, and inoculum concentration, and their contributions to the yield of plasmid DNA vaccine were 48.18%, 21.56%, and 21.55% respectively. In particular, these three variables exerted a positive effect on plasmid production. Other independent variables with p>0.05 were generally considered insignificant and would be not included in the subsequent optimizing step. Thereafter, the culture conditions were reduced to three most significant variables: peptone, mannitol, and inoculums concentration. The precise optimal values of the individual variables were still unknown but could be determined through subsequent BBD.
Table 1 Variables and their levels used in Plackett-Burman design for screening of culture conditions affecting the yield of plasmid DNA vaccine pcDNA-CCOL2A1 by the engineered E. coli DH5α
Table 2 Plackett–Burman design matrix with response value for screening of culture conditions affecting the yield of plasmid DNA vaccine pcDNA-CCOL2A1 by the engineered E. coli DH5α
Table 3 The regression analysis of variance for Plackett–Burman factorial model for the yield of plasmid DNA vaccine
The steepest ascent experiment optimized the key variables affecting the yield of therapeutic DNA vaccine pcDNA-CCOL2A1
Based on the analysis of the screening design, the path of steepest ascent was then applied to determine the most suitable direction for changing the variable ranges. As the three most significant variables exerted a positive effect on plasmid production, the direction of steepest ascent should increase their concentration to approach the optimal experimental region of maximum response. Five sets of experiments of the steepest ascent and corresponding experimental results were showed in Table 4. The yield of plasmid DNA vaccine peaked at the third step and no further improvement could be achieved in there sponse when peptone, mannitol, and inoculum concentration were selected to be 26 g/L, 8 g/L and 0.35, respectively, which suggested that it was proximal to the region of maximum response. Accordingly, these levels of the three factors in the third set were considered the center point of BBD.
Table 4 Steepest ascent experiments to move the experimental region towards the maximum yield of plasmid DNA vaccine pcDNA-CCOL2A1 by the engineered E. coli DH5α
BBD optimized the screened culture conditions for the yield of therapeutic DNA vaccine pcDNA-CCOL2A1
Preliminary trials confirmed that peptone (24–28 g/L), mannitol (7~9 g/L), and inoculum concentration (0.25~0.45) were suitable. In the present analysis, experiments were designed to obtain a second-order polynomial equation consisting of 12 trials plus 5 central points. The design matrix of the variables was showed in Table 5 along with the experimental values of response. Through multiple regression analysis of the experimental data, shown in Table 5, the following second-orderpolynomial equation was derived for the plasmid yield by only considering the significant terms:
$$ {\displaystyle \begin{array}{c}Y=338.78-3.70\times \mathrm{A}+5.06\times \mathrm{B}+3.98\times \mathrm{C}+10.52\times \mathrm{A}\times \mathrm{B}\\ {}+3.18\times \mathrm{A}\times \mathrm{C}-5.98\times \mathrm{B}\times \mathrm{C}-18.12\times {\mathrm{A}}^2-23.34\times {\mathrm{B}}^2-15.89\times {\mathrm{C}}^2\end{array}} $$
Table 5 Through BBD optimizing the screened culture conditions for the yield of plasmid DNA vaccine pcDNA-CCOL2A1 by the engineered Escherichia coli DH5α
Where Y is the predicted response of plasmid yield, A, B, and C are the coded values of peptone, mannitol, and inoculum concentration, respectively. Statistical significance of the second-order model and all the coefficient estimates were assessed using ANOVA, and the data are shown in Table 6. The quadratic regression model was highly significant, which was evident from the F-test with a very low probability value (p<0.0001). The value of adj-R2 (0.9626) suggested that the total variation of 96.26% for the yield of plasmid DNA vaccine was attributed to the independent variables. The determination coefficient (R2 = 0.9836), which is commonly used to assess the goodness of the model, exhibited an excellent correlation between the experimental and predicted response values. Alow CV (CV = 1.25%) value clearly revealed that the deviations between experimental and predicted values were low and it displayed not only a high degree of precision but also high reliability in conducted experiments. Adequate precision measures the signal-to-noise ratio, and a ratio greater than 4 is desirable. In this study,a ratio of 20.387 indicated an adequate signal. Therefore, the quadratic model was selected in this optimization study. Table 6 showed the corresponding p-value and the parameter estimate.
Table 6 Regression analysis of a full second-order polynomial model for the optimized yield of plasmid DNA vaccine by the engineered E. coli DH5α
This multiple nonlinear model resulted in three response surface graphs through canonical analysis of the response surface. Interpretation of the response surface 3D model and contour plot were the graphical representations of regression equation. They provided visual interpretations of the relationship between responses and experimental levels of each variable, and the type of interactions between two test variables. Fig. 2a was the fitted response surface 3D model and their corresponding contour plots for the yield of plasmid DNA vaccine produced by the predicted model, respectively. Fig. 1a shows that the yield of plasmid DNA vaccine significantly increased with peptone increasing from 24 to 25.86 g/L, mannitol increasing from 7 to 8.08 g/L, but decreased beyond this centerpoint, reaching a maximum yield of 339.03 mg/L. The effect of peptone and mannitolon the yield of plasmid DNA vaccine was also sensitive within the tested range, which was proved by the p-value (0.0309, 0.0078) in Table 6. Furthermore, the significant interaction of peptone and mannitol could be easily explained by its elliptical shape of the contour plot and p-value (0.001). It was also noticed in Fig. 2b-c that the response presented downward movement when the value of variables was higher than the center point, indicating the existence of the maximum predicted value of the yield of plasmid DNA vaccine. The statistical optimal values of variables were obtained when moving along the major and minor axes of the contour and the response at the center point yields the maximum plasmid production. These observations were also verified through canonical analysis of the response surface. By solving the inverse matrix from the second-order polynomial equation, the optimum values of the test variables were peptone, 25.86 g/L;mannitol, 8.08 g/L; inoculum concentration, 0.36. Under the optimal conditions, the maximum predicted the yield of plasmid DNA vaccine was 339.32 mg/L. To confirm the validity of the model for predicting the maximum yield of plasmid DNA vaccine, an additional experiment using this optimum operation conditions was performed under shake-flask culture. The average yield of plasmid DNA vaccine was 341.86 ± 10.67 mg/L (N = 3). The results were closely related to the data obtained from optimization analysis, suggesting that the RSM model was adequate for reflecting the expected optimization, and the model was satisfactory and accurate.
a, b, c Response surface 3D model (left) and contour plot (right) to assessthe effects of the three variables on the yield of plasmid DNA vaccine pcDNA-CCOL2A1 produced by the engineered E. coli DH5α. a response surface plot showing the mutual effect of peptone and mannitol on the yield of plasmid DNA vaccine pcDNA-CCOL2A1; b response surface plot showing the mutual effect of peptone and inoculum concentration on the yield of plasmid DNA vaccine pcDNA-CCOL2A1; c response surface plot showing the mutual effect of mannitol and inoculum concentration on the yield of plasmid DNA vaccine pcDNA-CCOL2A1
The final acquisition of plasmid DNA vaccines with the highest yield, purity, and quality were closely related to not only the upstream antigen-specific genes for disease targets, the most appropriate expression vectors, and the appropriate Escherichia coli strains for production, but also the optimized fermentation conditions, culture media, and scale-up as well as the downstream purification technology [24,25,26,27,28,29,30]. In the present study, we have optimized the fermentation conditions at a shake-flask level for the engineered Escherichia coli DH5α to for high yield of therapeutic DNA vaccine pcDNA-CCOL2A1 through combined PBD with BBD-RSM, by which the yield of therapeutic DNA vaccine pcDNA-CCOL2A1 was markedly increased.
In practice terms, the medium compositions such as the basal culture media, the carbon sources, the nitrogen sources, the carbon/nitrogen ratio (C/N), amino acid starvation, etc., as the essential factors for the fermentation conditionsfor the production of plasmid DNA vaccines are usually the first to be chosen and optimized in the beginning of the fermentation condition optimization for increasing plasmid DNA production in E. coli strains [14, 15]. The fermentation condition optimization, including screening of optimal medium compositions, is influenced by many factors, among which, interactions may exist. The routine single-dimensional studies changing one independent variable at a time and maintaining the others constant yields unreliable results, inaccurate conclusions, and even frequent interactions of two or more factors [17]. Thus, it is necessary to apply reasonable experimental designs and optimization methodologies in condition screening and process optimization. Because E. coli strain DH5a used in the present study was selected typically for plasmid DNA production [31], we first used single-factor analysis to evaluate several basal culture media commonly used for culturing DH5a, which include LB, 2 × YT, TB, M9 (Glycerol) and M9 (Glucose). Finally, we screened 2 × YT as the optimal basal culture medium, mannitol as the optimal carbon source, and peptone as the optimal nitrogen source. In theory, the production of plasmid DNA vaccines is also affected by varying both carbon and nitrogen concentrations [30, 32,33,34]. Thus, we applied PBD to further screen out the three most significant factors affecting the yield of therapeutic DNA vaccine pcDNA-CCOL2A1, followed by the paths of steepest ascent to move to the nearest region of maximum response. The most significant factors identified through PBD were peptone, mannitol, and inoculum concentrations. Together, our results indicate that PBD is efficient in screening medium components at the shake-flask level and has been widely used in the optimization of fermentation conditions [18, 19]. This technique cannot determine the exact quantity but can provide indication and tendency regarding the necessity of each variables in relatively few experiments.
In the present study, we used RSM to further optimize the yield of therapeutic DNA vaccine pcDNA-CCOL2A1 by BBD. RSM not only helped locate the optimum levels of the most significant factors but also proved to be useful and satisfactory in this process-optimizing practice. Through these optimization experiments, the maximum yield of plasmid DNA vaccine at 339.32 mg/L was obtained under the optimum conditions with peptone (25.86 g/L), mannitol (8.08 g/L), and inoculum concentration (OD = 0.36), which is significantly higher than those of most studies. Most current fermentation media and processes have only resulted in low yields of plasmid DNA (< 200 mg/L) [14, 35], though a few have resulted in high yields (500–1500 mg/L) [36,37,38]. Compared with the original medium, an increase of 51.9% was obtained. The predicted plasmid yield was closely related with the experimental value, which was 341.86 ± 10.67 mg/L (N = 3). Further studies are required to assess the optimization of fermentation conditions involving in several major factors such as growth conditions, culture types, culture medium compositions, etc. In the present study, we obtained a higher yield of plasmid DNA vaccine by only optimizing the two factors of the components of the nutrient medium and inoculum concentration. Hence, further optimization of fermentation conditions including growth conditions and culture types would significantly increase both the yield and productivity of therapeutic DNA vaccine pcDNA-CCOL2A1. These optimization methods for fermentation conditions are currently being investigated in our laboratory.
In summary, the fermentation medium and conditions of the engineered Escherichia coli DH5α producing a novel therapeutic DNA vaccine pcDNA-CCOL2A1 were scientifically selected and optimized by RSM. Under the optimum conditions with peptone (25.86 g/L), mannitol (8.08 g/L), and inoculum concentration (OD = 0.36), the maximum yield of plasmid DNA vaccine at 339.32 mg/L was obtained, with an increase of 51.9%. In addition to this, we conducted experiments under the optimal conditions.The experimental value was 341.86 ± 10.67 mg/L (N = 3), which was closely related with the predicted plasmid yield. The present results will provide a robust foundation for further pilot-scale tests and industrial-scale production of final high-quality and high-yield therapeutic DNA vaccine pcDNA-CCOL2A1 for RA in the near future.
Plasmid and bacterial strains
Eukaryotic expression vector for producing therapeutic DNA vaccine pcDNA-CCOL2A1 was previously constructed in our laboratory, which contains a 4837 bp cDNA sequence encoding the chicken type II procollagen gene, but lacking the N-propeptides. To obtain high levels of CCOL2A1 gene expression, both the signal sequence and the Kozak consensus sequence were inserted into pcDNA™3.1(+), a highly stable vector used for transient gene expression [9, 39]. The resulting recombinant plasmid containing an ampicillin resistance gene for selection was cloned in E. coli DH5α (CB101; Tiangen, Beijing, China).
Medium and cultivation
Media included Luria-Bertani (LB) [10 g/L tryptone, 5 g/L yeast extract, 10 g/L NaCl, pH 7.0], Tartof and Hobbs (TB)[12 g/L tryptone, 24 g/L yeast extract, 0.4% glycerol, 2.31 g/L KH2PO4, 12.54 g/L K2HPO4], M9 [0.01 g/L CaCl2, 0.24 g/L MgSO4, 12.8 g/L Na2HPO4·7H2O, 3 g/L KH2PO4, 0.5 g/L NaCl, 1 g/L NH4Cl, 20% Glucose or Glycerol], 2 × YT [16 g/L tryptone, 10 g/L yeast extract, 5 g/L NaCl, pH 7.0], and Microelement mix [0.01 g/L MnSO4·7H2O, 0.05 g/L ZnSO4·7H2O, 0.01 g/L H3BO3, 0.01 g/L CaCl2·2H2O, 0.01 g/L Na2MoO4, 0.2 g/L CoCl2·6H2O, 0.01 g/L AlK(SO4)2·12H2O, 0.001 g/L NiCl2·6H2O] [40]. Engineered Escherichia coli DH5α were cultured in 100 mL of medium in 500 mL Erlenmeyer flasks with the initial inoculum concentration of OD600 = 0.1, and incubated at 37 °C on a rotary shaker at 220 rpm. After 16 h of incubation, plasmid DNA was purified from the bacterial cell, using the WizardR Plus SV Minipreps DNA Purification System (Promega, USA). The concentrations of the plasmid pcDNA-CCOL2A1 were measured at OD260 and OD280 using Synergy™ HT Multi-Mode Microplate Reader (BioTek Instruments, Inc., Winooski, VT, USA).
Single-factor analysis
In each experiment, one factor was changed with the other factors remaining constant. The initial evaluation was performed to identify the optimal basal culture medium from LB, 2 × YT, TB, M9 (glycerol) and M9 (glucose). The effect of various carbon and nitrogen sources was also determined through single-factor analysis. Carbon sources (5 g/L glycerol, glucose, and mannitol) were evaluated, while other components were maintained constant as basal culture medium. The nitrogen sources (5 g/L peptone, NH4Cl, urea) were analyzed with other constituents as that of basal culture medium. Although this method is time-consuming, it is propitious to the selection of level in PBD, rendering the results more reasonable and credible.
Plackett–Burman design for screening
Multiple regression analysis and analysis of variance (ANOVA) were conducted for fitting the mathematical model using Design Expert software (Version 8.0.6, Stat-Ease Inc., Minneapolis, MN, USA). Ten variables (peptone, yeast extract, NaCl, Ampicillin [Amp], microelements, mannitol, rotational speed, pH, fermentation temperature, and inoculum concentration) were assessed using PBD and the model was evaluated using the F-test and goodness of fit through multiple correlations R. Each independent variable was tested at two levels, high and low, which are denoted by (+) and (−), respectively. The experimental design with the name, symbol code, and actual levels of the variables are shown in Tables 1 and 2 shows details of the design matrix. In this study, 12 experiments were conducted and the most optimal variables were selected for further evaluation. Based on regression analysis of the variables, significant levels at 95% level (p<0.05) were considered to significantly affect the yield of the plasmid vaccine.
Path of the steepest ascent experiment
After having identified the three most significant variables through the PBD, the steepest ascent experiment was performed to move the experimental region of the response in the direction of the optimum, by appropriately changing the range of the selected variables. The path initiated from the design center of the factorial design (the screening design) and receded when no further improvement in the response could be achieved. When the maximum value was gained, that point could be considered as the center point for the optimization experimental design [31]. Table 4 summarizes the experimental design, the variables, and their values.
Box-Behnken design
The RSM is a collection of statistical tools and techniques for constructing and exploring a putative functional relationship between a response variable (i.e., plasmid yield) and a set of design variables (i.e., peptone, mannitol, and inoculum concentration). It is possible to derive an expression for performance measurement on the basis of the response values obtained from experiments using a particular combination of input variables [41]. In the present study, by employing BBD and RSM, the effects of the three independent variables (peptone, 24–28 g/L; mannitol, 7–9 g/L; inoculum concentration, OD = 0.25–0.45) and three levels (high, middle, and low) on the response (plasmid yield) were investigated to determine the optimal conditions, which maximized the yield of therapeutic DNA vaccine pcDNA-CCOL2A1 from shake cultivation. Each independent variable was coded at three levels: − 1, 0, and + 1. The BBD comprised 17 experiments with five center points (to allow for estimation of pure error) and facilitated calculations of response function at intermediate levels, fitting a second-order response surface. Table 4 shows the variables and their values and the experimental design. This methodology allows for modeling of a second-order equation that describes the process. Plasmid production was analyzed through multiple regression analysis through the least squares method to fit the following equation:
$$ \mathrm{Y}={\upbeta}_0+\sum {\upbeta}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}+\sum {\upbeta}_{\mathrm{i}\mathrm{j}}{\mathrm{x}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{j}}+\sum {\upbeta}_{\mathrm{i}\mathrm{i}}{{\mathrm{x}}_{\mathrm{i}}}^2 $$
Where Y is the measured response variable; β0, βi, βij, and βii are constants and regression coefficients of the model, and xi and xj represent the independent variables in coded values. Data from the BBD for the optimization of plasmid production was subjected to second-order multiple regression analysis using the least squares method to obtain the parameter estimators of the mathematical model [18, 42]. Second-order multiple regression analysis was performed using the Design Expert software (Version 8.0.6, State-Ease Inc., Minneapolis, MN, USA) statistical package. The model was further assessed using ANOVA.
BBD-RSM:
Box-Behnken design-response surface method
CCOL2A1:
Chicken type II procollagen
CIA:
Collagen-induced arthritis
DMARDs:
Disease-modifying antirheumatic drugs
MTX:
PBD:
Plackett-Burman design
RSM:
Response surface method
Mallon S. DNA vaccines: Treatment options for autoimmune diseases. Microbiol Mol Gentics. 2008;4:99–103.
Robinson WH, Garren H, Utz PJ, Steinman L. Millennium Award. Proteomics for the development of DNA tolerizing vaccines to treat autoimmune disease. Clin Immunol. 2002;103:7–12.
Rosenthal KS, Mikecz K, Steiner HL 3rd, Glant TT, Finnegan A, Carambula RE, et al. Rheumatoid arthritis vaccine therapies: perspectives and lessons from therapeutic ligand epitope antigen presentation system vaccines for models of rheumatoid arthritis. Expert Rev Vaccine. 2015;14:891–908.
Singh JA, Saag KG, Bridges SL Jr, Akl EA, Bannuru RR, Sullivan MC, et al. 2015 American college of rheumatology guideline for the treatment of rheumatoid arthritis. Arthritis Rheumat. 2016;68:1–26.
Smolen JS, Landewé R, Bijlsma J, Burmester G, Chatzidionysiou K, Dougados M, et al. EULAR recommendations for the management of rheumatoid arthritis with synthetic and biological disease-modifying antirheumatic drugs: 2016 update. Ann Rheum Dis. 2017;76:960–77.
McInnes IB, Schett G. Mechanisms of disease: The pathogenesis of rheumatoid arthritis. NEJM. 2011;365:2205–19.
Holmdahl R, Malmström V, Burkhardt H. Autoimmune priming, tissue attack and chronic inflammation - the three stages of rheumatoid arthritis. Eur J Immunol. 2014;44:1593–9.
Song XQ, Liang F, Liu N, Luo Y, Yuan F, Xue H, et al. Therapeutic efficacy of experimental rheumatoid arthritis with low-dose methotrexate by increasing partially CD4+CD25+Treg cells and inducing Th1 toTh2 shift in both cells and cytokines. Biomed Pharmacother. 2010;64:463–71.
Song XQ, Liang F, Liu N, Luo Y, Xue H, Yuan F, et al. Construction and characterization of a novel DNA vaccine that is potent antigen-specific tolerizing therapy for experimental arthritis by increasing CD4+CD25+Treg cells and inducing Th1 to Th2 shift in both cells and cytokines. Vaccine. 2009;27:690–700.
Long J, ZhaoX YS, Zhang ZJ, Jin J, Yu K, et al. Safety and immunogenicity of a novel therapeutic DNA vaccine encoding chicken type II collagen for rheumatoid arthritis in normal rats. Hum Vaccines Immunother. 2015;11:2777–84.
Zhao X, Long J, Yun S, Zhang ZJ, Jin J, Yu K, et al. Evaluation of humoral and cellular immune responses to a DNA vaccine encoding chicken type II collagen for rheumatoid arthritis in normal rats. Hum Vaccines Immunother. 2015;11:938–45.
Zhao X, Long J, Liang F, Liu N, Sun YY, Xi YZ. Dynamic profiles, biodistribution and integration evaluation of a novel therapeutic DNA vaccine encoding CCOL2A1for rheumatoid arthritis in normal rat. Vaccine. 2018; (in press)
Long J, Zhao X, Yuan F, Liang F, Liu N, Sun YY, et al. Genetic stability of an Escherichia coli strain engineered to produce a novel therapeutic DNA vaccine encoding chicken type II collagen for rheumatoid arthritis. Process Biochem. 2017;52:86–93.
Carnes AE. Fermentation design for the manufacture of therapeutic plasmid DNA. Bio Process Intern. 2005;3:36–44.
Cai Y, Rodriguez S, Hebel H. DNA vaccine manufacture: scale and quality. Expert Rev Vaccine. 2009;8:1277–91.
Williams JA, Luke J, Langtry S, Anderson S, Hodgson CP, Carnes AE. Generic plasmid DNA production platform incorporating low metabolic burden seed-stock and fed-batch fermentation processes. Biotechnol Bioeng. 2009;103:1129–43.
Balusu R, Paduru RR, Kuravi S, Seenayya G, Reddy G. Optimization of critical medium components using response surface methodology for ethanol production from cellulosic biomass by Clostridium thermocellum SS19. Process Biochem. 2005;40:3025–30.
Cui FJ, Zhao LM. Optimization of xylanase production from Penicilliumsp.WX-Z1 by a two-step statistical strategy: Plackett-Burman and box-Behnken experimental design. Int J Mol Sci. 2012;13:10630–46.
Salihu A, Alam MZ, AbdulKarim MI, Salleh HM. Optimization of lipase production by Candida cylindracea in palm oil mill effluent based medium using statistical experimental design. J Mol Catal Enzym. 2011;69:66–73.
Cui FJ, Liu ZQ, Li Y, Ping LF, Ping LY, Zhang ZC, et al. Production of mycelial biomass and exo-polymer by Hericiumerinaceus CZ-2: Optimization of nutrients levels using response surface methodology. Biotechnol Bioproc. 2010;15:299–307.
Periyasamy AK, Muthu M, Kannan V. Optimization of nutrients for the production of RNase by Bacillus firmus VKPACU1 using response surface methodology. Biotechnol Bioprocess. 2010;14:202–6.
Kanmani P, Kumar RS, Yuvaraj N, Paari KA, Pattukumar V, Arul V. Optimization of media components for enhanced production of Streptococcus phocaePI80 and its bacteriocin using response surface methodology. Braz J Microbiol. 2011;42:716–20.
Preetha R, Jayaprakash NS, Philip R, Singh ISB. Optimization of medium for the production of a novel aquaculture probiotic, Micrococcus MCCB 104 using central composite design. Biotechnol Bioprocess Eng. 2007;12:548–55.
Goncalves GA, Prather KL, Monteiro GA, Prazeres DM. Engineering of Escherichia coli strains for plasmid biopharmaceutical production: scale-up challenges. Vaccine. 2014;32:2847–50.
Yau SY, Keshavarz-Moore E, Ward J. Host strain influences on supercoiled plasmid DNA production in Escherichia coli: implications for efficient design of large-scale processes. Biotechnol Bioeng. 2008;101:529–44.
Jaén KE, Lara AR, Ramírez OT. Effect of heating rate on pDNA production by E. coli. Biochem Eng J. 2013;79:230–8.
Lara AR, Knabben I, Regestein L, Sassi J, Caspeta L, Ramírez OT, et al. Comparison of oxygen enriched air vs. pressure cultivations to increase oxygen transfer and to scale-up plasmid DNA production fermentations. Eng Life Sci. 2011;11:382–6.
Wunderlich M, Taymaz-Nikerel H, Gosset G, Ramírez OT, Lara AR. Effect of growth rate on plasmid DNA production and metabolic performance of engineered Escherichia coli strains. J Biosci Bioeng. 2014;117:336–42.
Bohle K, Ross A. Plasmid DNA production for pharmaceutical use: role of specific growth rate and impact on process design. Biotechnol Bioeng. 2011;108:2099–106.
Zheng S, Friehs K, He N, Deng X, Li Q, He Z, et al. Optimization of medium components for plasmid production by recombinant E. coli DH5a, pUK21CMVβ1.2. Biotechnol Bioprocess Eng. 2007;12:213–21.
Gao H, Liu M, Liu JT, Dai HQ, Zhou XL. Medium optimization for the production of avermectin B1a by Streptomyces avermitilis 14−12A using response surface methodology. Bioresour Techol. 2009;100:4012–6.
Lopes MB, Martins G, Calado CRC. Kinetic modeling of plasmid bioproduction in Escherichia coli DH5a cultures over different carbon-source compositions. J Biotechnol. 2014;186:38–48.
O'Mahony K, Freitag R, Hilbrig F, Müller P, Schumacher I. Strategies for high titre plasmid DNA production in Escherichia coli DH5a. Process Biochem. 2007;42:1039–49.
Silva F, Queiroz JA, Domingues FC. Evaluating metabolic stress and plasmid stability in plasmid DNA production by Escherichia coli. Biotechnol Adv. 2012;30:691–708.
Carnes AE, Williams JA. Plasmid DNA manufacturing technology. Recent Pat Biotechnol. 2007;1:151–66.
Singer A, Eiteman MA, Altman E. DNA plasmid production in differenthost strains of Escherichia coli. J Ind Microbiol Biotechnol. 2009;36:521–30.
Listner K, Bentley L, Okonkowski J, Kistler C, Wnek R, Caparoni A, et al. Development of a highly productive and scalable plasmid DNA production platform. Biotechnol Prog. 2006;222:1335–45.
Carnes AE, Hodgson CP, Williams JA. Inducible Escherichia coli fermentation for increased plasmid DNA production. Biotechnol Appl Biochem. 2006;45:155–66.
Xi CX, Liu N, Liang F, Guo SQ, Sun YY, Yang F, et al. Molecular cloning: characterization and localization of chicken type II procollagen gene. Gene. 2006;366:67–76.
Sambrook J, Russwell DW. Molecular Cloning: A Laboratory Manual, 2001, 3rd ed. by Cold Spring Harbor Laboratory Press.
Li Q, Li YM, Han S, Liu YZ, Song DX. Optimization of fermentation conditions and properties of an exopolysaccharide from Klebsiella sp. H-207 and application in adsorption of hexavalent chromium. PLoS One. 2013;8:1–11.
Ren X, He L, Cheng J, Chang J. Optimization of the solid-state fermentation and properties of a polysaccharide from paecilomyces cicadae (Miquel) samson and its antioxidant activities in vitro. PLoS One. 2014:e87578.
The authors thank the National Scientific and Technological Commission for funding this research.
This study was supported in part by a grant from the National Major Scientific and Technological Special Project for "Significant New Drug Development" (No.2009ZX09103–624 and No.2015GKS-072/139 to YZX). The funding agency played no role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.
All data generated or analyzed during this study are included in this published article.
YZX conceived the project, obtained grant support, designed research, and analyzed and interpreted data. JL, XZ, FL, NL and YYS performed the experiments, analysis, interpretation, statistic. JL, XZ and YZX wrote the paper and YZX revised the manuscript. All authors edited and approved the manuscript.
Juan Long and Xiao Zhao contributed equally to this work.
Department of Immunology and National Center for Biomedicine Analysis, Beijing 307 Hospital, No.8, Dongda Ave, Fengtai District, Beijing, 100071, People's Republic of China
Juan Long
, Xiao Zhao
, Fei Liang
, Nan Liu
, Yuying Sun
& Yongzhi Xi
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Correspondence to Yongzhi Xi.
This is an open-access article distributed under the terms of the Creative Commons Attribution Non-Commercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0) which permits copy and redistribute the material just in non-commercial usages, provided the original work is properly cited.
The authors have no competing interest to declare in this work. The authors alone are responsible for the content and writing of the paper.
Long, J., Zhao, X., Liang, F. et al. Optimization of fermentation conditions for an Escherichia coli strain engineered using the response surface method to produce a novel therapeutic DNA vaccine for rheumatoid arthritis. J Biol Eng 12, 22 (2018) doi:10.1186/s13036-018-0110-y
Accepted: 06 August 2018
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UNSPECIFIED (1986) Microbe-mineral interactions of relevance in the hydrometallurgy of complex sulphides. In: International Conference on `Progress in Metallurgical Research: Fundamental and Applied Aspects, IIT-Kanpur, p. 105-511.
Babu, Ramesh (1986) Studies on drop formation at conical tips - a theoretical approach. In: Proceedings of International Conference on `Progress in metallurgical research: fundamental and applied aspects, IIT-Kanpur, p. 279-87, pp. 4918-4927.
Sundaramoorthy, M and Parrack, P and Sasisekharan, V (1986) Application of the precession method to fiber diffraction: structural variations in lithium B-DNA. In: Proceeding Conversation Discipline Biomolecular Stereodynamics, pp. 217-225.
Banerjee, K and Ramakrishnan, KR and Sastry, PS (1986) An SIMD Architecture for Relaxation Labelling. In: Platinum Jubilee Conference on Systems and Signal Processing, December 11-13, 1986, Bangalore.
Ganguly, P (1986) Electrical transport and magnetic properties of oxides with potassium nickel fluoride $(K_2NiF_4)$ structure. In: Adv. Solid State Chem., Proc. INSA Golden Jubilee Symp. Solid State Chem, 1985, New Delhi, pp. 135-158.
Gopalakrishnan, J (1986) Low-temperature synthesis of novel metal oxides by topochemical reactions. In: Advance Solid State Chemistry Proceedings of INSA Golden Jubilee Symposium-Solid State Chem, 1985, New Delhi, pp. 48-66.
Krishnan, V (1986) Biomimetic model reactions of cavity-bearing porphyrins. In: Proceedings of the Indian National Science Academy, Part A: Physical Sciences, 1986, pp. 909-923.
Moudgal, NR and Martin, F and Kotagi, SG and Sairam, MR and Ravindranath, N and Rao, AJ and Murthy, GS (1986) Development of oFSH as a vaccine for the male-A status report on the recent researches carried out using the bonnet monkey (M.radiata). In: Immunological Approaches to Contraception and Promotion of Fertility, 1985, Plenum, New York, pp. 103-110.
Rao, CNR (1986) Developments in solid state chemistry: a partial overview. In: INSA Golden Jubilee Symposium on Solid State Chemistry, 1985, New Delhi, pp. 1-24.
Rao, CNR (1986) Synthesis and reactions of some novel metal oxide systems. In: Indian Academy of National Sciences Part A: Physical Sciences, 1986, pp. 699-714.
Rao, KJ (1986) Chemical bond and the nature of inorganic glasses. In: INSA Golden Jubilee Symposium on Solid State Chemistry, 1985, pp. 176-191.
Rao, KJ and Rao, BG and Damodaran, RV and Selvaraj, U (1986) Novel glasses with octahedral structural groups. In: 14th International Congress on Glass, 1986, Calcutta, pp. 182-189.
Srinivasa, N and Krishnan, V and Ramakrishnan, KR and Rajgopal, K (1986) Image Reconstruction Form Truncated Projections : A Linear Predication Approach. In: 1986 IEEE International Conference on Acoustics, Speech, and Signal Processing. ICASSP '86, April, Tokyo, Vol.11, 1733-1736.
Thathachar, MAL and Sastry, PS (1986) Estimator Algorithms for Learning Automata. In: Platinum Jubilee Conference on Systems and Signal Processing, Dec. 1986, Bangalore.
Thukaram, D and Iyengar, Ramakrishna BS and Parthasarathy, K (1986) Optimum allocation of reactive power in AC/DC power systems. In: Proc. 53rd Annual Research and Development Session, 8-10 May 1986, Bhubaneswar, New Delhi, India.
Thukaram, D and Parthasarathy, K and Iyengar, Ramakrishna BS (1986) Static VAR compensators for unbalanced reactive power demands and harmonic minimization. In: Fourth National Power Systems Conference, Feb. 1986, Varanasi, India.
Gore, AP and Paranjape, Sharayu and Rajarshi, MB and Gadgil, Madhav (1986) Some Methods for Summarizing Survivorship Data in Nonstandard Situations. In: Biometrical Journal, 28 (5). 577 -586.
Adiga, BS and Shankar, P (1986) Fast Public Key Cryptosystem Based On Matrix Rings. In: Electronics Letters, 22 (22). pp. 1182-1183.
Aggarwal, Vijay and Tikekar, VG and Hsu, Lie-Fern (1986) Bottleneck assignment problems under categorization. In: Computers & Operations Research, 13 (1). pp. 11-26.
Ajitkumar, P and Cherayil, Joseph D (1986) Ammonium ions prevent methylation of uridine to ribothymidine in Azotobacter vinelandii tRNA. In: Journal of Biosciences, 10 (2). pp. 267-276.
Akila, R and Jacob, KT and Shukla, AK (1986) Concept of thermodynamic capacity. In: Bulletin of Materials Science, 8 (4). pp. 453-465.
Anand, GV and George, Mathews K (1986) Normal-mode sound propagation in an ocean with sinusoidal surface waves. In: Journal of the Acoustical Society of America, 80 (1). pp. 238-243.
Anantharamaiah, KR and Bhattacharya, D (1986) Ionized Gas towards Galactic Centre-Constraints from Low-Frequency Recombination Lines. In: Journal of Astrophysics & Astronomy, 7 (3). 141 -153.
Ananthraj, S and Varma, KBR and Rao, KJ (1986) Thermal Properties of Silver Pyrophosphate - Anomalously High Glass Heat Capacities. In: Materials Research Bulletin, 21 (11). pp. 1369-1374.
Arivoli, T and Ramkumar, K and Satyam, M (1986) Magnetoresistors based on Composites. In: Journal of Physics D Applied Physics, 19 (9). pp. 183-185.
Arjunan, P and Ramamurthy, V (1986) Selectivity in the Photochemistry of \beta-ionyl and \beta -ionylidene Derivatives in \beta-cyclodextrin: Microsolvent Effect. In: Journal of Photochemistry, 33 (1). pp. 123-134.
Arumugam, S and Khetrapal, CL (1986) Nuclear magnetic resonance spectra of oriented bicyclic systems containing heteroatom(s):the spectrum of 2-thiocoumarin. In: Canadian Journal of Chemistry / Revue canadienne de chimie, 64 (4). pp. 714-716.
Asokan, S and Gopal, ESR and Parthasarathy, G (1986) Pressure-induced polymorphous crystallization in bulk Si20Te80 glass. In: Journal of Materials Science, 21 (2). pp. 625-629.
Asokan, S and Parthasarathy, G and Gopal, ESR (1986) Crystallization Studies on bulk $Si_xTe_{100-x}$ Glasses. In: Journal of Non-Crystalline Solids, 86 (1-2). pp. 48-64.
Asokan, S and Parthasarathy, G and Gopal, ESR (1986) Crystallization studies on bulk SixTe100-x glasses. In: Journal of Non-Crystalline Solids, 86 (1-2). pp. 48-64.
Asokan, S and Parthasarathy, G and Gopal, ESR (1986) Evidence for a new metastable crystalline compound in Ge---Te system. In: Materials Research Bulletin, 21 (2). 217 -224.
Asokan, S and Parthasarathy, G and Subbanna, G N and Gopal, E S R (1986) Electrical transport and crystallization studies of glassy semiconducting Si20Te80 alloy at high pressure. In: Journal of Physics and Chemistry of Solids, 47 (4). 341 -348.
Atre, MV and Mukunda, N (1986) Classical particles with internal structure: general formalism and application to first-order internal spaces. In: Journal of Mathematical Physics, 27 (12). pp. 2908-2919.
Ayyoob, M and Hegde, MS (1986) Chlorination of silver dosed with potassium and barium in presence of oxygen: An X-ray photoelectron spectroscopy study. In: Journal of Catalysis, 97 (2). 516 -526.
Ayyoob, Mohammed and Hegde, Manjanath S (1986) Electron Spectroscopic Studies of Formic Acid Adsorption and Oxidation on Cu and Ag dosed with Barium. In: Journal of the Chemical Society, Faraday Transactions 1: Physical Chemistry in Condensed Phases, 82 . 1651 -1662.
Babau, Ramesh J and Bhatt, Vivekananda M (1986) New reagents 41. Reduction of sulphonyl chlorides and sulphoxideswith aluminum iodide. In: Tetrahedron Letters, 27 (9). pp. 1073-1074.
Babu, Ramesh S (1986) Experimental Studies on Drop Formation at the Tip of Melting Rods. In: Metallurgical and Materials Transactions B, Process Metallurgy and Materials Processing Science, 17 (3). pp. 471-477.
Babu, Ramesh S (1986) An absolute method for the determination of surface tension of liquids using pendent drop profiles. In: Bulletin of Materials Science, 90 (18). 4337 -4340.
Babu, DS and Vedavathy, TS (1986) Improving the optimum generalised stop-and-wait ARQ scheme. In: Electronics Letters, 22 (12). pp. 649-650.
Babu, Ramesh J and Bhatt, Vivekananda M (1986) New Reagents $4^1$. Reduction of Sulphonyl Chlorides and Sulphoxides with Aluminum Iodide. In: Tetrahedron Letters, 27 (9). pp. 1073-1074.
Bagchi, Biman (1986) Debye-Waller factor of the solid from the self-diffusion coefficient at the solid-liquid interface. In: Journal of Chemical Physics, 85 (8). pp. 4667-4668.
Bagchi, Biman (1986) Dynamic structure factor across the liquid-solid interface: appearance of a delta-function elastic peak. In: Chemical Physics Letters, 125 (1). 91 -96.
Bagchi, Biman (1986) Excitation wavelength and viscosity dependence of Landau-Zener electronic transitions in condensed media. In: Chemical Physics Letters, 128 (5-6). pp. 521-527.
Bagchi, Biman and Kirkpatrick, TR (1986) On the kinetics of crystal growth from a supercooled melt. In: Proceedings of the Indian Academy of Sciences - Chemical Sciences, 96 (6). 465-472.
Bai, BN Pramila and Biswas, SK (1986) Effect of Load on Dry Sliding Wear of Aluminum-Silicon Alloys. In: Tribology Transactions, 29 (1). 116 -120.
Balaji, VN and Rao, Jagannatha M and Rao, Shashidhar N and Dietrich, Stephen W and Sasisekharan, V (1986) Geometry Of Proline And Hydroxyproline .1. An Analysis Of X-Ray Crystal-Structure Data. In: Biochemical and Biophysical Research Communications, 140 (3). 895 -900.
Balakrishnan, M and Chinnaiya, GP and Nair, PG and Rao, Jagannadha A (1986) Studies on serum progesterone levels in Zebu × Holstein heifers during pre- and peripubertal periods. In: Animal Reproduction Science, 11 (1). pp. 11-15.
Balaram, Hemalatha and Sukumar, M and Balaram, P (1986) Stereochemistry of \alpha-Aminoisobutyric Acid Peptides in Solution: Conformations of Decapeptides with a Central Triplet of Contiguous L-Amino Acids. In: Biopolymers, 25 (11). pp. 2209-2223.
Balasubrahmanyam, SN (1986) Anisochrony of O-methylene protons of ethyl ester functions and structural factors in the connected chiral entities. In: Journal of Chemical Sciences, 96 (1-2). pp. 21-58.
Bardi, R and Piazzesi, AM and Toniolo, C and Raj, PA and Raghothama, S and Balaram, P (1986) Solid state and solution conformation of Boc-L-Met-Aib-L-Phe-OMe. Beta-turn conformation of a sequence related to an active chemotactic peptide analog. In: International Journal of Peptide & Protein Research, 27 (3). pp. 229-238.
Bardi, R and Piazzes, AM and Toniolo, C and Raj, Antony P and Raghothama, S and Balaram, P (1986) Conformations of the amino aterminal tetrapeptide of emerimicins and antiamoebins in solution and in the solid state. In: International Journal of Biological Macromolecules, 8 (4). pp. 201-206.
Bardi, R and Piazzesi, AM and Toniolo, C and Antony Raj, P and Raghothama, Srinivasarao and Balaram, Padmanabhan (1986) Conformations of the amino terminal tetrapeptide of emerimicins and antiamoebin in solution and in the solid state. In: International Journal of Biological Macromolecules, 8 (4). pp. 201-206.
Bardi, R and Piazzesi, AM and Toniolo, C and Balaram, Padmanabhan and Sukumar, M (1986) Stereochemistry of peptides containing 1-aminocyclopentane carboxylic acid (Acc5). Solution and solid state conformations of Boc-Acc5-Acc5-NHMe. In: Biopolymers, 25 (9). pp. 1635-1644.
Bardi, R and Piazzesi, AM and Toniolo, C and Sukumar, M and Balaram, P (1986) Stereochemistry of Peptides Containing 1-AminocyclopentanecarboxylicA cid ( Am5): Solution and Solid-state Conformations of Boc-Acc '-Acc'-NHMe. In: Biopolymers, 25 (9). pp. 1635-1644.
Bardi, R and Piazzesi, AM and Toniolo, C and Sukumar, M and Balaram, P (1986) Stereochemistry of Peptides Containing 1-Aminocyclopentanecarboxylic Acid $({Acc}^5)$ : Solution and Solid-state Conformations of $Boc-{Acc}^5-{Acc}^5-NHMe$. In: Biopolymers, 25 (9). pp. 1635-1644.
Bhanuprakash, K and Kulkarni, GV and Chandra, Asish K (1986) On calculations of intermolecular potentials. In: Journal of Computational Chemistry, 7 (6). pp. 731-738.
Bharathi, Devi B and Sarma, VVS (1986) A Fuzzy Approximation Scheme for Sequential Learning in Pattern Recognition. In: IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, 16 (5). 668 -679.
Bhashyam, AT and Deshpande, SM and Mukunda, HS and Goyal, G (1986) A Novel Operator-Splitting Technique for One-Dimensional Laminar Premixed Flames. In: Combustion Science and Technology, 46 (3-6). 223 -247.
Bhaskarwar, Ashok N and Kumar, R (1986) Oxidation of sodium sulphide in the presence of fine activated carbon particles in a foam bed contactor. In: Chemical Engineering Science, 41 (2). 399 -404.
Bhat, Vasudeva and Gopalakrishnan, J (1986) HNbWO6 and HTaWO6: Novel oxides related to ReO3 formed by ion exchange of rutile-type LiNbWO6 and LiTaWO6. In: Journal of Solid State Chemistry, 63 (2). pp. 278-283.
Bhat, Vasudeva and Gopalakrishnan, J (1986) A New Method for the Synthesis of Oxide Bronzes of Tungsten, Molybdenum and Vanadium. In: Journal of the Chemical Society, Chemical Communications . pp. 1644-1645.
Bhatia, KL and Gosain, DP and Parthasarathy, G and Gopal, ESR (1986) Bismuth-doped amorphous-germanium sulfidesemiconductors. In: Physical Review B, 34 (12). pp. 8786-8793.
Bhatia, KL and Gosain, DP and Parthasarathy, G and Gopal, ESR (1986) Morphological structure of bismuth-doped n-type amorphous germanium sulphide semiconductors. In: Journal of materials science letters, 5 (12). 1281 -1284.
Bhatia, KL and Gosain, DP and Parthasarathy, G and Gopal, ESR (1986) On the structural features of doped amorphous chalcogenide semiconductors. In: Journal of Non-Crystalline Solids, 86 (1-2). pp. 65-71.
Bhatia, KL and Gosain, DP and Parthasarathy, G and Gopal, ESR (1986) Pressure-induced first-order transition in layered crystalline semiconductor GeSe to a metallic phase. In: Physical Review B, 33 (2). pp. 1492-1494.
Bhatt, MV and BM, Hosur (1986) Electron transfer mechanism for periodic acid oxidation of aromatic substrates. In: Indian Journal of Chemistry, Section B: Organic Chemistry Including Medicinal Chemistry, 25B (10). pp. 1004-1005.
Bhatt, Vivekananda M and Shashidhar, MS (1986) Carbonyl Participation During the Hydrolysis of Aryl Benzenesulphonates. In: Tetrahedron Letters, 27 (19). pp. 2165-2166.
Bhattacharya, D and Srinivasan, G (1986) On the implication of the recently discovered 5 millisecond binary pulsar PSR 1855+09. In: Current science, 55 (7). 327 -330.
Bhattacharyya, K and Das, PK and Rao, Nageshwara B and Ramamurthy, V (1986) Laser flash photolysis study of triplets of cyclobutanethiones. In: Journal of Photochemistry, 32 (3). pp. 331-340.
Bhattacharyya, K and Ramamurthy, V and Das, PK and Sharat, S (1986) Short-lived triplets of aliphatic thioketenes. In: Journal of Photochemistry, 35 (3). pp. 299-309.
Bhattacharyya, Kankan and Das, Paritosh K and Ramamurthy, Vaidhyanathan and Rao, V Pushkara (1986) Triplet-state Photophysics and Transient Photochemistry of Cyclic Enethiones A Laser Flash Photolysis Study. In: Journal of the chemical society-faraday transactions ii, 82 . pp. 135-147.
Biswas, Margaret and Sekharudu, Y Chandra and Rao, VSR (1986) Complex carbohydrates: 1. Conformational studies on some oligosaccharides related to N-glycosyl proteins which interact with concanavalin A. In: International Journal of Biological Macromolecules, 8 (1). pp. 2-8.
Biswas, NN (1986) Foldable compatibility matrix for the folding of programmable logic arrays. In: International Journal of Electronics, 61 (1). pp. 97-103.
Biswas, SK and Mahesh, MS and lyengar, BSR (1986) Simple new PWM patterns for thyristor three-phase AC/DC convertors. In: IEE Proceedings B: Electric Power Applications, 133 (6). 354 -358.
Bokhari, SA and Balakrishnan, N (1986) A Method to Extend the Spectral Iteration Technique. In: IEEE Transations on Antennas and Propagation, AP-34 (1). pp. 51-57.
Bose, DN and Seishu, B and Parthasarathy, G and Gopal, ESR (1986) Doping Dependence of Semiconductor-Metal Transition in InP at High Pressures. In: Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences, 405 (1829). pp. 345-353.
Buttrey, DJ and Honig, JM and Rao, CNR (1986) Magnetic properties of quasi-two-dimensional La2NiO4. In: Journal of Solid State Chemistry, 64 (3). 287 -295.
Chakrabarti, A (1986) Cooling of a composite slab. In: Applied Scientific Research, 43 (3). pp. 213-225.
Chakrabarti, A (1986) Diffraction by a Dielectric Half-Plane. In: IEEE Transactions on Antennas and Propagation, 34 (6). 830 -833.
Chakrabarti, A (1986) The sputtering temperature of a cooling cylindrical rod with an insulated core. In: Applied Scientific Research, 43 (2). 107 -113.
Chakrabarti, A and Nalini, VN (1986) Hydrodynamic pressure on a dam with a periodically corrugated reservoir bed. In: Acta Mechanica, 60 (1-2). pp. 91-97.
Chakrabarti, Amaresh (1986) The sputtering temperature of a cooling cylindrical rod with an insulated core. In: Applied Scientific Research, 43 (2). 107 -113.
Chakravarthy, Purandar and Reddy, NM (1986) Theoretical study of a 16-µm CO2 downstream-mixing gasdynamic laser: A two-dimensional approach. In: Applied Physics Letters, 48 (4). 263 -265.
Chakravarty, AR (1986) Chemistry of diruthenium compounds having metal-metal multiple bonds. In: Proceedings of the Indian National Science Academy, Part A: Physical Sciences, 52 (4). pp. 749-763.
Chakravarty, Purandar and Reddy, NM and Reddy, KPJ (1986) A study of the effect of N2 reservoir temperature on a 16 μm CO2-N2 downstream-mixing gasdynamic laser. In: Optics Communications, 58 (2). pp. 130-132.
Chanda, M and Rempel, GL (1986) Cuprous oxide catalyzed air oxidation of thiosulfate and tetrathionate. In: Applied Catalysis, 23 (1). pp. 101-110.
Chandra, H Sharat (1986) Haldane's rule back in the news. In: Nature, 323 (6083). p. 20.
Chandrasekar, A and Nath, G (1986) Time dependent rotational flow of a viscous fluid over an infinite porous disk with a magnetic field. In: International Journal of Engineering Science, 24 (10). pp. 1667-1680.
Chandrasekhar, J and Rao, CNR (1986) Computer Simulation of Quenched Liquids: The Glassy State of Water. In: Chemical Physics Letters, 131 (3). pp. 267-270.
Chandrasekharaiah, HS (1986) Natural Frequencies And Transient Responses Of 3-Phase Rotating Machine Windings. In: IEEE Transactions on Energy Conversion, 1 (3). 167 -173.
Chandrasekharappa, SC and Gopalakrishnan, AS and Jacob, TM (1986) Antibodies specific to deoxythymidine 5'-phosphate. In: Indian Journal of Biochemistry & Biophysics, 23 (4). pp. 233-237.
Chandrasekharappa, SC and Jacob, TM (1986) Purification of Antibodies Specific to a Dinucleotide Using the Hapten Bound to Deae Cellulose as an Affinity Column. In: Immunological Investigations, 15 (1). pp. 1-9.
Chandrashekhara, K and Gopalakrishnan, P (1986) Analysis of an orthotropic cylindrical shell having a transversely isotropic core subjected to axisymmetric load. In: Thin-Walled Structures, 4 (3). pp. 223-237.
Changkakoti, Rupak and Pappu, Sastry V (1986) Study on the pH dependence of diffraction efficiency of phase holograms in dye sensitized dichromated gelatin. In: Applied Optics, 25 (5). pp. 798-801.
Char, Shobha and Gopinathan, Karumathil P (1986) Arginyl-tRNA Synthetase from Mycobacterium smegmatis SN2: Purification and Kinetic Mechanism. In: The Journal of Biochemistry, 100 (2). pp. 349-357.
Chary, BR and Bhat, HL and Chandrasekhar, P and Narayanan, PS (1986) Vibrational spectroscopic studies of the ferroelectric $LiRbSO_4$. In: Journal of Raman Spectroscopy, 17 (1). pp. 59-63.
Chary, BR and Bhat, HL and Chandrasekhar, P and Narayanan, PS (1986) Vibrational spectroscopic studies of the ferroelectric LiRbSO4. In: Journal of Raman Spectroscopy, 17 (1). pp. 59-63.
Chary, BR and Shashikala, MN and Bhat, HL (1986) Pressure effect on the Dielectric Properties of $LiCsSO_4$. In: Current Science, 55 (20). pp. 1021-1223.
Chattopadhyay, K and Aaronson, HI (1986) Interfacial structure and crystallographic studies of transformations in betat' and beta Cu---Zn alloys-II. martensitic formation of alpha1' plates in beta'. In: Acta Metallurgica, 34 (4). pp. 713-720.
Chattopadhyay, K and Ravi, VA and Ranganathan, S (1986) Non-equilibrium trapping in Al---In system. In: Acta Metallurgica, 34 (4). 691 -693.
Chaturvedi, VK and Kurup, CK (1986) Effect of lutein on the transport of Ca2+ across phospholipid bilayer and mitochondrial membrane. In: Biochemistry International, 12 (2). 373 -377.
Chaturvedi, VK and Kurup, Ramakrishna CK (1986) Interaction of lutein with phosphatidylcholine bilayers. In: Biochimica et Biophysica Acta, 860 (2). pp. 286-292.
Chiu, Charles B and Pasupathy, J and Wilson, Sanford J (1986) Determination of baryon magnetic moments from QCD sum rules. In: Physical Review D, 33 (7). pp. 1961-1973.
Choubey, Divaker and Gopinathan, KP (1986) Characterization of beta-lactamase from Mycobacterium smegmatis SN2. In: Current Microbiology, 13 (3). pp. 171-175.
Choubey, Divaker and Gopinathan, KP (1986) Factors affecting the synthesis and distribution of beta-lactamase in Mycobacterium smegmatis SN2 cultures. In: Current Microbiology, 13 (2). 103 -106.
Choudhuri, Arnab Rai (1986) Magnetic helicity as a constraint on coronal dissipation. In: NASA. Goddard Space Flight Center Coronal and Prominence Plasmas . pp. 451-456.
Choudhuri, Arnab Rai (1986) Magnetic energy dissipation in force-free jets. In: Astrophysical Journal, 310 (1). pp. 96-103.
Choudhuri, Arnab Rai (1986) The dynamics of magnetically trapped fluids. I - Implications for umbral dots and penumbral grains. In: Astrophysical Journal, 302 . pp. 809-825.
Conrad, Michael and Brahmachari, SK and Sasisekharan, V (1986) DNA structural variability as a factor in gene expression and evolution. In: Biosystems, 19 (2). pp. 123-126.
Das, Manoj K and Raghothama, S and Balaram, P (1986) Membrane channel forming polypeptides. Molecular conformation and mitochondrial uncoupling activity of antiamoebin,an a-aminoisobutyric acid containing peptide. In: Biochemistry, 25 (22). pp. 7110-7117.
Dasgupta, Chandan and Pandit, Rahul (1986) Kinetics of domain growth: The relevance of two-step quenches. In: Physical Review B, 33 (7). pp. 4752-4757.
Dasgupta, Dipak and Rajagopalan, Malini and Sasisekharan, V (1986) DNA-Binding Characteristics of a Synthetic Analog of Distamycin. In: Biochemical and Biophysical Research Communications, 140 (2). pp. 626-631.
Dattaguru, B and Naidu, ACB and Krishnamurthy, T and Ramamurthy, TS (1986) Development of special fastener elements. In: Computers & Structures, 24 (1). 127 -134.
Deobagkar, DN and Shankar, V and Deobagkar, DD (1986) Separation of 5-methylcytosine-rich DNA using immobilized antibody. In: Enzyme and Microbial Technology, 8 (2). pp. 97-100.
Desayi, P and Ganesan, N (1986) Fracture Behavior of Ferrocement Beams. In: Journal of Structural Engineering, 112 (7). 1509 -1525.
Devi, CD Surma and Nagaraj, M and Nath, G (1986) Axial heat conduction effects in natural convection along a vertical cylinder. In: International Journal of Heat and Mass Transfer, 29 (4). pp. 654-656.
Dhanasekaran, N and Moudgal, NR (1986) Studies on follicular atresia: role of tropic hormone and steroids in regulating cathepsin-D activity of preantral follicles of the immature rat. In: Molecular and Cellular Endocrinology, 44 (1). 77 -84.
Dinesha, KV and Iyer, Suman B and Vishveshwara, Saraswathi (1986) Energy expressions for atomic configurations in the L-S coupling scheme. In: International Journal of Quantum Chemistry, 30 (6). pp. 783-790.
DurgaKumari, B and Adiga, Radhakantha P (1986) Estrogen modulation of retinol-binding protein in immature chicks: Comparison with riboflavin carrier protein. In: Molecular and Cellular Endocrinology, 46 (2). pp. 121-130.
DurgaKumari, B and Adiga, Radhakantha P (1986) Hormonal induction of riboflavin carrier protein in the chicken oviduct and liver: a comparison of kinetics and modulation. In: Molecular and Cellular Endocrinology, 44 (3). 285-292.
Durrant, MC and Hegde, MS and Rao, CNR (1986) Electronic structures of $H_2O.BF_3$ and related n-v addition compounds: a combined EELS-UPS study in vapor phase. In: Journal of Chemical Physics, 85 (11). pp. 6356-6360.
Durrant, MC and Hegde, MS and Rao, CNR (1986) Electronic-structures of h2o.bf3 and related n-v addition-compounds - a combined eels-ups study in vapor-phase. In: Journal of Chemical Physics, 85 (11). 6356 -6360.
Elhawary, ME and Rao, RS and Christensen, GS (1986) Optimal hydrothermal load flow: Formulation and a successive approximation solution for fixed head systems. In: Optimal Control Applications and Methods, 7 (4). pp. 337-354.
Elliott, Stephen R and Rao, CNR and Thomas, John M (1986) The Chemistry of the Noncrystalline State. In: Angewandte Chemie International Edition in English, 25 (1). pp. 31-46.
Francis, Vidyasagar NK and Dwarki, Varavan IJ and Padmanaban, Govindarajan (1986) A comparative study of tbe regulation of cytochrome P-450 and glutathione transferase gene expression in rat liver. In: Nucleic Acids Research, 14 (6). 2497 -2510.
Gangadevi, T and Rao, M Subba and Kutty, TR Narayan (1986) Kinetics of thermal decomposition of barium zirconyl oxalate. In: Monatshefte für Chemie, 117 (1). pp. 21-32.
Ganguli, AK and Gopalakrishnan, J (1986) Magnetic properties of calcium iron manganese oxide $Ca_2Fe_2-\timesMn\timesO5$. In: Journal of Chemical Sciences, 97 ((5-6)). pp. 627-630.
Ganguly, P and Rao, CNR (1986) High $T_c$ superconductivity in oxides derived from lanthanum strontium copper oxide $(La_{1.8}Sr_{0.2}CuO_4)$. In: Proceedings - Indian Academy of Sciences, Chemical Sciences, 97 (5-6). pp. 631-633.
Ganguly, P and Vasanthacharya, NY (1986) Infrared and Mössbauer spectroscopic study of the metal-insulator transition in some oxides of perovskite structure. In: Journal of Solid State Chemistry, 61 (2). pp. 164-170.
Ganguly, P and Vasanthacharya, NY (1986) On the use of oxalates as starting materials for low-temperature preparation of bronzes and other reduced phases. In: Materials Research Bulletin, 21 (4). pp. 479-482.
Gehlot, Vijay and Srikantb, YN (1986) An interpreter for slips-An applicative language based on LAMBDA-Calculus. In: Computer Languages, 11 (1). pp. 1-13.
Ghosal, Dipak and Patnaik, L M (1986) Parallel polygon scan conversion algorithms: Performance evaluation on a shared bus architecture. In: Computers & Graphics, 10 (1). pp. 7-25.
Gopalakrishna, AV and Ganagi, MS (1986) Weak discontinuities in relativisitic MHD. In: Astrophysics and Space Science, 120 (1). pp. 139-149.
Gopalakrishnan, J (1986) Synthesis and structure of some interesting oxides of bismuth. In: Proceedings of the Indian Academy of Sciences - Chemical Sciences, 96 (6). pp. 449-458.
Gopalakrishnan, M and Patnaik, LM (1986) Integrating voice and data on SALAN: an experimental local area network. In: Computer Communications, 9 (4). pp. 186-194.
Gopalarao, AS and Thukaram, D and Iyengar, Ramakrishna BS (1986) Torque Angle Loop Analysis of Synchronous Machines. In: Electric Machines & Power Systems, 11 (3). pp. 215-227.
Govindarajan, S and Babu, PJ and Patil, KC (1986) Thermal analysis of metal sulfate hydrazinates and hydrazinium metal sulfates. In: Thermochimica Acta, 97 . pp. 287-293.
Govindarajan, Subbiah and Patil, Kashinath C and Manohar, Hattikudur and Werner, Per-Erik (1986) Hydrazinium as a Ligand: Structural, Thermal, Spectroscopic, and Magnetic Studies of Hydrazinium Lanthanide Di-sulphate Monohydrates; Crystal Structure of the Neodymium Compound. In: Journal of the Chemical Society, Dalton Transactions (1). pp. 119-123.
Govindarajan, Subbiah and Patil, Kashinath C and Poojary, Damodara M and Hattikudur, Manohar (1986) Synthesis, Characterization and X-ray Structure of Hexahydrazinium Diuranyl Pentaoxalate Dihydrate, ($N_2H_5)_6(UO_2)_2(C_2O_4)_5.2H_2O$. In: Inorganica Chimica Acta, 120 (1). pp. 103-107.
Govindarajan, Subbiah and Patil, Kashinath C and Poojary, MDamodara and Manohar, Hattikudur (1986) Synthesis, characterization and X-ray structure of hexahydrazinium diuranyl pentaoxalate dihydrate, (N2H5)6(UO2)2(C2O4)5·2H2O. In: Inorganica Chimica Acta, 120 (1). 103 -107.
Govindarao, Venneti MH and Chidambaram, M (1986) Hydrogenation Of alpha-Methylstyrene In A Bcsr: Effect Of Distribution Of Solids. In: Journal of Chemical Engineering of Japan, 19 (3). pp. 243-245.
Gupta, SC (1986) Axisymmetric melting of a long cylinder due to an infinite flux. In: Proceedings Mathematical Sciences, 95 (1). pp. 1-12.
Hajra, JP and Venkatraman, M and Ranganathan, S (1986) Thermodynamics and phase equilibria in the Co-Ni-Mn system. In: Transactions of the Indian Institute of Metals, 39 (3). pp. 211-218.
Hegde, MS and Ayyoob, M (1986) $o^{2-}$ and $o^{1-}$ types of oxygen species on Ni and barium-dosed Ni and Cu surfaces. In: Surface Science Letters, 173 (2-3). L635-L640.
Hegde, MS and Ayyoob, Mohammed (1986) O2- and O1- types of oxygen species on Ni and barium-dosed Ni and Cu surfaces. In: Applied Surface Science, 173 (2-3). L635-L640.
Hiriyanna, KT and Ramakrishnan, TV (1986) Deoxyribonucleic acid replication time in Mycobacterium tuberculosis H37 Rv. In: Archives of Microbiology, 144 (2). pp. 105-109.
Ilangovan, S and Vasu, KI (1986) Electro-hydrometallurgy of chalcopyrites - VII. An appraisal of ferric chloride leaching process. In: Bulletin of Electrochemistry, 2 (6). pp. 611-613.
Indusekhar, H and Kumar, V (1986) Properties of iron related quenched-in levels in p-silicon. In: Physica Status Solidi A-Applied Research, 95 (1). 269 -278.
Indusekhar, H and Kumaran, V and Sengupta, D (1986) Investigation of Deep Defects Due to -Particle Irradiation in n-Silicon. In: Physica Status Solidi A, 93 (2). pp. 645-653.
Iwase, M and Ichise, E and Jacob, KT (1986) Physical chemistry of mixed conducting zirconia ceramics. In: Sprechsaal, 119 (4). 280 -283.
Iyengar, KTS and Pandya, SK (1986) Application of the method of initial functions for the analysis of composite laminated plates. In: Archive of Applied Mechanics, 56 (6). pp. 407-416.
Iyengar, NGR and Umaretiya, JR (1986) Deflection analysis of hybrid laminated composite plates. In: Composite Structures, 5 (1). pp. 15-32.
Iyengar, NGR and Umaretiya, JR (1986) Transverse vibrations of hybrid laminated plates. In: Journal of Sound and Vibration, 104 (3). 425 -435.
Iyengar, RN (1986) A Nonlinear System Under Combined Periodic and Random Excitation. In: Journal of Statistical Physics, 44 (5-6). 907 -920.
Iyer, K Viswanathan and Patnaik, LM (1986) Performance Study of a Centralized Concurrency Control Algorithm for Distributed Database Systems using SIMULA. In: Computer Journal, 29 (2). 118 -126.
Jacob, KT (1986) Solubility and activity of oxygen in liquid nickel in equilibrium with $\alpha -Al_2o_3$Nio.(1 + x)$Al_2o_3$. In: Metallurgical and Materials Transactions B, Process Metallurgy and Materials Processing Science, 17B (4). pp. 763-70.
Jacob, KT and Hajra, JP (1986) Electromagnetic levitation study of sulfur in liquid iron, nickel, and iron-nickel alloys. In: Transactions of the Indian Institute of Metals, 39 (1). pp. 62-69.
Jacob, KT and Iyengar, GNK and Kim, WK (1986) Spinel-Corundum Phase Equilibria in the Systems Mn-Cr-Al-O and Co-Cr-Al-O at 1373 K. In: Journal of the American Ceramic Society, 69 (6). pp. 487-492.
Jacob, KT and Shukla, AK and Akila, Ramachandran and Kale, GM (1986) Gibbs' energy of formation of nickel orthosilicate $(Ni_2SiO_4)$. In: High Temperature Materials and Processes, 7 (2-3). pp. 141-148.
Jacob, KT (1986) Casting of titanium and titanium alloys. In: Defence Science Journal, 36 (2). pp. 121-141.
Jacob, KT (1986) Solubility and activity of oxygen in liquid nickel in equilibrium with -Al2O3 and NiO.(1+x)Al2O3. In: Metallurgical and Materials Transactions B, 17 (4). pp. 763-770.
Jacob, KT and Hajra, JP (1986) Oxygen content of liquid cobalt in equilibrium with CoO.(1+x)Al2O3 and -Al2O3. In: Zeitschrift für Metallkunde, 77 . pp. 673-677.
Jacob, KT and Iyengar, GNK (1986) Thermodynamic study of Fe2O3 Fe2(SO4)3 equilibrium using an oxyanionic electrolyte (Na2SO4 I). In: Metallurgical and Materials Transactions B, 17 (2). pp. 323-329.
Jacob, KT and Iyengar, GNK and Srikanth, S (1986) Phase relations and activities in the Co Ni O system at 1373 K. In: Bulletin of Materials Science, 8 (1). pp. 71-79.
Jacob, KT and Kale, GM and Iyengar, GNK (1986) Oxygen potentials, Gibbs' energies and phase relations in the Cu-Cr-o system. In: Journal of Materials Science, 21 (8). pp. 2753-2758.
Jacob, KT and Kumar, BV (1986) Thermodynamic properties of Cr Mo solid alloys. In: Zeitschrift für Metallkunde, 77 . pp. 207-212.
Jacob, KT and Shukla, AK and Ramachandran, A and Kale, GM (1986) Gibbs energy of formation of nickel orthosilicate (Ni2SiO4). In: High Temperature Materials and Processes, 7 (2-3). pp. 141-148.
Jacob, KT and Waseda, Y and Iwase, M (1986) Sensor for H2S or S2 based on alumina. In: Journal of the American Ceramic Society, 1 (3). pp. 264-270.
James, Jose and Rao, M Subba (1986) Reaction product of lime and silica from rice husk ash. In: Cement and Concrete Research, 16 (1). pp. 67-73.
James, Jose and Rao, M Subba (1986) Reactivity of rice husk ash. In: Cement and Concrete Research, 16 (3). pp. 296-302.
James, Jose and Rao, M Subba (1986) Silica from rice husk through thermal decomposition. In: Thermochimica Acta, 97 . pp. 329-336.
Jhans, H and Honig, JM and Rao, CNR (1986) Optical properties of reduced LiNbO,. In: Journal of Physics C: Solid State Physics, 19 (19). 3649 -3658.
Jiran, E and Jacob, KT (1986) Computation of thermodynamic properties of multicomponent solutions: extension of Toop model. In: Metallurgical and Materials Transactions A, 17 (6). pp. 1102-1104.
Joshi, NV (1986) Evolution of sex ratios in social hymenoptera: consequences of finite brood size. In: Journal of Genetics, 65 (1-2). 55 -64.
Juneja, JM and Abraham, KP and Iyengar, GNK (1986) Thermodynamic study of liquid magnesium-aluminium alloys by vapour pressure measurement using the boiling point method. In: Scripta Metallurgica, 20 (2). pp. 177-180.
Kaliannan, P and Vishveshwara, S and Rao, VSR (1986) Anomeric effect in carbohydrates-an ab initio study on extended model systems. In: Proceedings of Indian Academy of Sciences: Chemical Sciences, 96 (5). pp. 327-339.
Kamal, K and Durvasula, S (1986) Macromechanical behaviour of composite laminates. In: Composite Structures, 5 (4). 309 -318.
Kamal, K and Durvasula, S (1986) Some studies on free vibration of composite laminates. In: Composite Structures, 5 (3). pp. 177-202.
Kamath, P Vishnu and Hegde, MS and Rao, CNR (1986) A novel investigation of vapor-phase charge-transfer complexes of halogens with n-donors by electron energy loss spectroscopy. In: Journal of Physical Chemistry, 90 (10). pp. 1990-1992.
Karbelkar, SN (1986) On the axiomatic approach to the maximum entropy principle of inference. In: Pramana, 26 (4). 301 -310.
Karle, IL and Sukumar, M and Balaram, Padmanabhan (1986) Parallel packing of alpha-helices in crystals of the zervamicin IIA analog Boc-Trp-Ile-Ala-Aib-Ile-Val-Aib-Leu-Aib-Pro-OMe.2H2O. In: Proceedings Of The National Academy Of Sciences Of The United States Of America, 83 (24). pp. 9284-9288.
Karle, Isabella L and Sukumar, Muppalla and Balaram, Padmanabhan (1986) Parallel packing of \alpha-helices in crystals of the zervamicin IIA analog Boc-Trp-Ile-Ala-Aib-Ile-Val-Aib-Leu-Aib-Pro-OMe ${2H}_2O$. In: Proceedings of the National Academy of Sciences of the United States of America, 83 (24). pp. 9284-9288.
Kasturi, TR and Rajasekhar, B and Sivaramakrishnan, R and Reddy, Amruta P and Madhusudhan, G and Prasad, KB and Ganesha, * and Venkatesan, K and Row, Guru TN and Puranik, VG (1986) Reaction of tetrahydropyranyl ether of 1-bromomethyl-2-naphthol with tetrachlorocatechol-structures of novel products. In: Indian Journal of Chemistry, Section B: Organic Chemistry Including Medicinal Chemistry, 25B (11). pp. 1091-1092.
Katti, S and Narasimhamurthy, M and Krishna, G (1986) On the sufficient conditions for the equality of the unassignable polynomial and Davison's fixed polynomial of strongly connected systems. In: IEEE Transactions on Automatic Control, 31 (5). 443-445.
Khan, MI and Sastry, MV and Surolia, Avadhesha (1986) Thermodynamic and kinetic analysis of carbohydrate binding to the basic lectin from winged bean (Psophocarpus tetragonolobus). In: Journal of Biological Chemistry, 261 (7). pp. 3013-3019.
Khandke, Lakshmi and Gullapalli, Sharada and Patole, Milind S and Ramasarma, T (1986) Vanadate-stimulated NADH oxidation by xanthine oxidase: An intrinsic property. In: Archives of Biochemistry and Biophysics, 244 (2). pp. 742-749.
Kishore, K and Begum, A Sameena and Sankaralingam, S (1986) Changes in the calorimetric value and ignition temperature of composite solid propellants during ageing. In: Defence Science Journal, 36 (4). pp. 425-428.
Kishore, K and Dharumaraj, GV and Gayathri, V (1986) Effect of triethanolamine and benzaldehyde on the storage stability of polystyrene - ammonium perchlorate propellant. In: Defence Science Journal, 36 (4). pp. 381-387.
Kishore, K and Sankaralingam, S (1986) Effect of Pressure on Polymer Ignition. In: Journal of Fire Sciences, 4 (2). pp. 94-99.
Kishore, K and Verneke, VR Pai and Pitchaih, K and Sridhara, K (1986) Mechanism of catalytic activity of metal oxides and chromites on ammonium perchlorate pyrolysis. In: Fuel, 65 (8). pp. 1169-1171.
Kishore, Raghuvansh and Balaram, Padmanabhan (1986) Stereochemically Constrained Enkephalin Analogs Containing $\alpha$-Aminoisobutyric Acid and 1-Amino-Cyclopentane-l-Carbolylic Acid. In: NIDA Research Monograph, 69 . pp. 312-331.
Kishore, K and Joseph, Mary and Dharumaraj, Verghese and Vijayshree, MN (1986) The effect of some catalysts on the curing of oxiranes with p-phenylenediamine. In: Journal of Applied Polymer Science, 31 (8). 2829 -2837.
Kishore, K and Mukundan, T (1986) Poly(styrene peroxide): an auto-combustible polymer fuel. In: Nature, 324 . pp. 130-131.
Kishore, K and Rajalingam, P (1986) The Bonding Ability and Bonding Site of a New Ferrocene Based Silicon Compound in Composite Solid Propellants. In: Journal of Polymer Science Part C: Polymer Letters, 24 (9). pp. 471-476.
Kishore, K and Vasanthakumari, R (1986) Crystallization and Melting Behavior of Isotactic Polybutene-1 at High Pressures. In: Journal of Polymer Science: Part A: Polymer Chemistry, 24 . pp. 2011-2019.
Kishore, K and Vasanthakumari, R (1986) Crystallization behaviour of polyethylene and i-polybutene-1 blends. In: Polymer, 27 (3). 337 -343.
Kishore, Kaushal and Mallick, Ishwardas M and Annakutty, Kunnappallil S (1986) Synthesis and spectroscopic data of 3,3',5,5'-tetrabromo-4,4'-diaminodiphenyldibromomethane. In: Journal of Chemical & Engineering Data, 31 (2). pp. 262-264.
Kishore, Kaushal and Pandey, Hrishi K (1986) Indian Sapota Tree Rubber. In: Journal of Polymer Science Part C: Polymer Letters, 24 (8). pp. 393-397.
Kishore, U Sudarsan (1986) Effect of load changes on dynamic coefficient of friction of crystalline and amorphous materials. In: Journal of Materials Science Letters, 5 (2). pp. 198-200.
Krishnamurthy, SS and Cameron, TS and Vincent, BR and Kumaravel, SS (1986) Assignment of phosphorus-31 chemical shifts to isomers of 2,3-dialkoxy-$\lambda 3$-diazadiphosphetidines. Crystal and molecular structure of trans$-[PhNP(OCH_2CF_3)]2$. In: Zeitschrift fuer Naturforschung, Teil B: Anorganische Chemie, Organische Chemie, 41B (9). pp. 1067-1070.
Krishnan, Girish and Shivaprasad, AP (1986) Serial quaternary-to-analogue converters. In: International Journal of Electronics, 61 (4). 531 -538.
Krishnan, D and Patnaik, LM (1986) GEODERM: geometric shape design system using an entity-relationship model. In: Computer-Aided Design, 18 (4). pp. 207-218.
Kulkarni, GV (1986) Cluster approach to chemisorption and electrochemisorption -a critique. In: Indian Journal of Technology, 24 (8). pp. 457-464.
Kulkarni, Gopal R and Murthy, SK (1986) Induction of choline-ethanolamine kinase in chicken liver by 17-\beta -estradiol. In: Indian Journal of Biochemistry & Biophysics, 23 (5). pp. 254-257.
Kumar, MP Subodh and Srikant, YN (1986) Graphical simulation of Petri Nets. In: Computers & Graphics, 10 (3). 225 -228.
Kumar, N (1986) Quantum-Ohmic Resistance Fluctuation In Disordered Conductors - An Invariant Imbedding Approach. In: Pramana, 27 (1-2). pp. 33-42.
Kumar, Sampath TS and Hegde, MS (1986) Electron Spectroscopic Study of Surface Segregation and Oxidation of Cu-In and Cu-Au Alloys. In: Applied Surface Science, 26 (2). pp. 219-229.
Kumar, Sampath TS and Rao, Kameswara L and Hegde, MS (1986) Stabilization Of Geo Phase On n-Ge And Laser Irradiated Ge Films. In: Applied Surface Science, 27 (3). pp. 255-261.
Kumari, Durga B and Adiga, PR (1986) Correlation between riboflavin carrier protein induction and its $mRNA$ activity in estrogen stimulated chicken liver and oviduct. In: Journal of Biosciences, 10 (2). pp. 193-202.
Kumari, M (1986) Unsteady Incompressible Two-Dimensional and Axisymmetric Turbulent Boundary Layer Flows. In: Acta Mechanica, 59 (3-4). 251 -268.
Kumari, M and Nath, G (1986) Unsteady Self-similar Stagnation Point Boundary Layers for Micropolar Fluids. In: Indian Journal of Pure and Applied Mathematics, 17 (2). pp. 231-244.
Kumari, M and Nath, G (1986) Unsteady free Convection MHD Boundary Layer Flow Near a Three-Dimensional Stagnation Point. In: Indian Journal of Pure and Applied Mathematics, 17 (7). pp. 957-968.
Kurny, ASW and Rao, Mohan M and Mallya, RM (1986) Design and construction of a laboratory model ion-nitriding unit. In: Indian Journal of Technology, 24 (10). pp. 671-675.
Kurny, ASW and Mallya, RM and Rao, M Mohan (1986) A study on the nature of the compound layer formed during the ion nitriding of En40B steel. In: Materials Science and Engineering, 78 (1). pp. 95-100.
Kutty, TRN and Devi, L'Gomathi (1986) Photoelectrochemical behavior of titanate perovskite solid solutions. In: Indian Journal of Technology, 24 (7). pp. 391-398.
Kutty, TRN and Murthy, SRN and Anantha, GV (1986) Ree Geochemistry and Petrogenesis of Ultramafic Rocks of Chalk-Hills, Salem. In: Journal of The Geological Society of India, 28 (6). pp. 449-466.
Kutty, TRN (1986) Behaviour of acceptor states in semiconducting BaTiO3 and SrTiO3. In: Proceedings of the Indian Academy of Sciences - Chemical Sciences, 96 (6). 581 -597.
Kutty, TRN and Devi, Gomathi L and Murugaraj, P (1986) The Change in Oxidation State of Mn Ions in Semiconducting $BaTiO_3$ and $SrTiO_3$ Around the Phase Transition Temperatures. In: Materials Research Bulletin, 21 (9). pp. 1093-1102.
Lagisetty, JS and Das, PK and Kumar, R and Gandhi, KS (1986) Breakage of viscous and non-Newtonian drops in stirred dispersions. In: Chemical Engineering Science, 41 (1). pp. 65-72.
Lakshmanan, VS and Madhavan, Veni CE (1986) Binary decompositions and acyclic schemes. In: Lecture Notes in Computer Science, 241 . pp. 214-238.
Lalitha, R and Kalpana, GV and Ramasarma, T (1986) Inhibition of mevalonate kinase by disulfide compounds. In: Indian Journal of Biochemistry & Biophysics, 23 (4). pp. 204-207.
Lalitha, R and Ramasarma, T (1986) Mevalonate phosphorylation in lemon grass leaves. In: Indian Journal of Biochemistry & Biophysics, 23 (5). pp. 249-253.
Latha, PK and Brahmachari, Samir K (1986) B to Z transitions in DNA and their biological implications. In: Journal of Scientific & Industrial Research, 45 (12). pp. 521-533.
Lord, Eric A (1986) Gauge theory of a group of diffeomorphisms. II. The conformal and de Sitter groups. In: Journal of Mathematical Physics, 27 (12). pp. 3051-3054.
Lord, Eric A and Goswami, P (1986) Gauge theory of a group of diffeomorphisms. I. General principles. In: Journal of Mathematical Physics, 27 (9). 2415 -2422.
Madyastha, K Madhava and Krishnamachary, N (1986) Purification and partial characterization of microsomal cytochrome b555 from the higher plant Catharanthus-Roseus. In: Biochemical and Biophysical Research Communications, 136 (2). 570 -576.
Madyastha, KMadhava and Chadha, Anju (1986) Metabolism of 1,8-Cineole in Rat: Its Effects on Liver and Lung Microsomal Cytochrome P-450 Systems. In: Bulletin of Environmental Contamination and Toxicology, 37 (5). 759 -766.
Madyastha, Madhava K and Chadha, Anju (1986) Metabolism of 1,8-Cineole in Rat: Its Effects on Liver and Lung Microsomal Cytochrome P-450 Systems. In: Bulletin of Environmental Contamination and Toxicology, 37 (1). pp. 759-766.
Mahesh, GV and Ravindranathan, P and Patil, KC (1986) Preparation, characterization and thermal analysis of rare earth and uranyl hydrazinecarboxylate derivatives. In: Journal of Chemical Sciences, 97 (2). pp. 117-123.
Mahesh, GV and Patil, KC (1986) Thermal reactivity of metal acetate hydrazinates. In: Thermochimica Acta, 99 . pp. 153-158.
Majumder, Kumud and Brahmachari, Samir K and Sasisekharan, V (1986) Sequence dependence and role of 5'-phosphate in the B to Z transition. In: FEBS Letters, 198 (2). 240 -244.
Majumder, Kumud and Latha, PK and Brahmachari, Samir K (1986) Use of a volatile buffer at ambient temperature *1: Versatile approach to the purification of self-complementary synthetic deoxyoligonucleotides by reversed-phase high-performance liquid chromatography. In: Journal of Chromatography A, 355 (1). pp. 328-334.
Mangalgiri, PD and Dattaguru, B (1986) A large orthotropic plate with misfit pin under arbitrarily oriented biaxial loading. In: Composite Structures, 6 (4). 271 -281.
Manne, Veeraswamy and Kutty, Krishnan R and Pillarisetti, Subba Rao V (1986) Purification and properties of synephrinase from Arthrobacter synephrinum. In: Archives of Biochemistry and Biophysics, 248 (1). 324 -334.
Marmo, G and Mukunda, N (1986) Symmetries and constants of the motion in the Lagrangian formalism on $TQ$: beyond point transformations. In: Il Nuovo Cimento - B, 92 (1). pp. 1-12.
Mathialagan, N and Rao, Jagannadha A (1986) Gonadotropin-releasing hormone (GnRH) stimulates both secretion and synthesis of human chorionic gonadotropin (hCG) by first trimester human placental minces in vitro. In: Biochemistry International, 13 (5). pp. 757-765.
Mathialagan, N and Rao, Jagannadha A (1986) Gonadotropin-releasing hormone in first trimester human placenta: isolation, partial characterization and in vitro biosynthesis. In: Journal of Biosciences, 10 (4). pp. 429-441.
Mathialagan, N and Rao, Jagannadha A (1986) Plasma levels of gonadotropin releasing hormone during menstrual cycle of Macaca radiata. In: Journal of Biosciences, 10 (4). pp. 423-428.
Mishra, AK and Rangarajan, SK (1986) Theory of electron transfer processes - an overview of concepts (I). In: Indian Journal of Technology, 24 (11). pp. 727-736.
Mohanakrishnan, P and Easwaran, KRK (1986) Theoretical investigations of the two-bond proton-carbon-13 coupling constants. Angular variations of the couplings involving carboxyl carbon. In: Chemical Physics, 104 (3). 409 -414.
Mohanty, Bani P and Subramanian, S and Hajra, JP (1986) Electro slag refining of commercial aluminum. In: Transactions of the Indian Institute of Metals, 39 (6). pp. 646-647.
Mudakavi, JR and Ramaswamy, YS (1986) Extraction-spectrophotometric determination of traces of mercury(II) with bromide and Rhodamine 6G. In: Journal of the Indian Institute of Science, 66 (3). pp. 155-162.
Mukherjee, A and Venkatesha, YV (1986) Digital color reproduction on color television monitors. In: Computer Vision, Graphics, and Image Processing, 36 (1). 114 -132.
Mukhopadhyay, NK and Subbanna, GN and Ranganathan, S and Chattopadhyay, K (1986) An electron microscopic study of quasicrystals in a quaternary alloy : Mg32(Al, Zn, Cu)49. In: Scripta Metallurgica, 20 (4). pp. 525-528.
Mukhopadhyay, PK and Raychaudhuri, AK (1986) Easy to build four-terminal AC bridge. In: Journal of Physics E - Scientific Instruments, 19 (10). 792 -793.
Mukunda, N and Sudarshan, ECG (1986) The three faces of Maxwell�s equations. In: Pramana, 27 (1-2). pp. 1-18.
Muniyappa, K and Radding, CM (1986) The homologous recombination system of phage lambda. Pairing activities of beta protein. In: The American Society for Biochemistry and Molecular Biology, 261 (16). pp. 7472-7478.
Munjal, ML and Prasad, MG (1986) On plane-wave propagation in a uniform pipe in the presence of a mean flow and a temperature gradient. In: Journal of the Acoustical Society of America, 80 (5). 1501 -1506.
Munshi, SK and Murthy, MRN (1986) Strategies for collecting screen-less oscillation data. In: Journal of Applied Crystallography, 19 . pp. 61-62.
Murali, N and Chandrasekhar, K and Kumar, Anil (1986) Use of $45^o$ Pulse Pair as a Filter for Pure-Phase Two-Dimensional NMR Spectroscopy. In: Journal of Magnetic Resonance, 70 (1). pp. 153-156.
Murali, N and Chandrasekhar, K and Kumar, Anil (1986) Use of 45° pulse pair as a filter for pure-phase two-dimensional NMR spectroscopy. In: Journal of Magnetic Resonance, 70 (1). 153 -156.
Murali, N and Kumar, Ami (1986) Multiple-quantum artifacts in single-quantum two-dimensional correlated NMR spectra of strongly coupled spins. In: Chemical Physics Letters, 128 (1). pp. 58-61.
Muralidharan, K and Shaila, MS and Gadagkar, Raghavendra (1986) Evidence for Multiple mating in the primitively eusocial wasp Ropalidia Marginata (Lep.) (Hymenoptera : Vespidae). In: Journal of Genetics, 65 (3). pp. 153-158.
Murthy, MS and Raghavendrachar, P and Sriram, SV (1986) Thermal decomposition of doped calcium hydroxide for chemical energy storage. In: Solar Energy, 36 (1). pp. 53-62.
Murthy, VSR and Kishore, * and Seshan, S (1986) Morphology Of Flake, Ductile And Compacted Graphite. In: Journal of Metals, 38 (12). pp. 24-28.
Murthy, GS and Moudgal, NR (1986) Use of epoxysepharose for protein immobilisation. In: Journal of Biosciences, 10 (3). pp. 351-358.
Murthy, Kumari and Ramesh, Usha and Bhat, SV (1986) EPR Investigations of Phase Transitions in Lithium Potassium Sulfate: $LiKSO_4$. In: Journal of Physics and Chemistry of Solids, 47 (9). pp. 927-931.
Murugaraj, P and Kutty, TRN and Rao, Subba M (1986) Diffuse phase transformations in neodymium-doped $BaTiO_3$ ceramics. In: Journal of Materials Science, 21 (10). pp. 3521-3527.
Mytri, VD and Shivaprasad, AP (1986) Constant factor incremental delta modulator. In: International Journal of Electronics, 61 (1). pp. 129-135.
Mytri, VD and Shivaprasad, AP (1986) Improving the dynamic range of a CVSD coder. In: Electronics Letters, 22 (8). 429 -430.
Mytri, VD and Shivaprasad, AP (1986) Hybrid constant factor incremental delta modulators. In: IEE Proceedings F: Radar & Signal Processing, 133 (6). 522 -525.
Nagaraj, TS and Murthy, BR Srinivasa (1986) Prediction of Compressibility of Overconsolidated Uncemented Soils. In: Journal of Geotechnical and Geoenvironmental Engineering, 112 (4). pp. 484-488.
Naik, Hemamalini and Subramanyam, SV (1986) Non-ohmic conduction and electrical switching under pressure of the charge transfer complexo-tolidine-iodine. In: Pramana, 26 (1). pp. 61-66.
Nair, Nandini and Ramakrishna, CK (1986) A comparative study of the effects of administration of diethylhexyl phthalate on hepatic mitochondria of the rat and the mouse. In: Indian Journal of Biochemistry & Biophysics, 23 (5). pp. 270-273.
Nair, Nandini and Kurup, Ramakrishna CK (1986) Investigations on the Mechanism of the Hypocholesterolemic Action of Diethylhexyl Phthalate in Rats. In: Biochemical Pharmacology, 35 (20). pp. 3441-3447.
Nandy, SK and Patnaik, LM (1986) Linear time geometrical design rule checker based on quadtree representation of VLSI mask layouts. In: Computer-Aided Design, 18 (7). 380 -388.
Nanjundaswamy, KS and Murthy, MN Sankarshana (1986) Low-temperature stabilization of pure Ni(IT)O. In: Materials Chemistry and Physics, 15 (1). pp. 37-44.
Narahari, Y and Viswanadham, N (1986) On the invariants of coloured Petri Nets. In: Lecture Notes in Computer Science, 222 . 330 -345.
Narasimhamurthy, N and Samuelson, AG (1986) Synthesis of Aryl Orthocarbonates. In: Tetrahedron Letters, 27 (8). pp. 991-992.
Narasimhamurthy, N and Samuelson, AG (1986) Thiocarbonyl to carbonyl group transformation using CuCl and NaOH. In: Tetrahedron Letters, 27 (33). pp. 3911-3912.
Narasimhan, S and Vithayathil, PJ (1986) A new reaction of o-benzoquinone with N-acetyl-DL-tryptophan (a 3-substituted indole) and characterization of the product. In: Indian Journal of Biochemistry & Biophysics, 23 (4). pp. 215-219.
Narasimhan, Lakshmi V and Ramachandra, JK and Anvekar, Dinesh K (1986) Design and evaluation of a dual-microcomputer shared memory system with a shared I/O bus. In: Microprocessors and Microsystems, 10 (1). pp. 3-10.
Narasu, Lakshmi M and Gopinathan, KP (1986) Purification of Larvicidal Protein from Bacillus Sphaericus 1593. In: Biochemical and Biophysical Research Communications, 141 (2). pp. 756-761.
Narayana Rao, K and Munjal, ML (1986) Noise reduction with perforated three-duct muffler components. In: Sadhana : Academy Proceedings in Engineering Sciences, 9 (4). pp. 255-269.
Narayanan, AS and Devanathan, R (1986) Note on the Similarity Solutions of Unsteady Jets in Rotating Fluids. In: Acta Mechanica, 60 (3-4). pp. 241-250.
Natarajan, KA and Upadhyaya, Ramesh (1986) Flocculation studies on iron ore fines using synthetic polymeric flocculants. In: Transactions of the Indian Institute of Metals, 39 (6). pp. 627-636.
Nayyar, AH and Scadron, MD and Sinha, KP (1986) Universal jellium BCS picture for Peierls and spin-Peierls phase transitions. In: Physics Letters A, 113 (8). 442 -444.
Padma, Doddaballapur K (1986) A gravimetric procedure for the determination of wet precipitated sulphur, dissolved sulphur, soluble sulphides and hydrogen sulphide. In: Talanta, 33 (6). pp. 550-552.
Padmanabhan, Kaillathe and Dopp, Dietrich and Venkatesan, Kailasam and Ramamurthy, Vaidyanathan (1986) Solid-state Photochemistry of Nitro Compounds: Structure-Reactivity Correlations. In: Journal of the Chemical Society, Perkin Transactions 2 (24). 897 -906.
Pandit, Shashidhara S and Jacob, KT (1986) Vanadium-oxygen equilibrium in liquid cobalt at 1873 K. In: Transactions of the Indian Institute of Metals, 39 (6). pp. 556-561.
Pandita, TK (1986) Evaluation of Thimet 10-G for mutagenicity by 4 different genetic systems. In: Mutation Research, 171 (2-3). pp. 131-138.
Papavinasam, E and Natarajan, S and Shivaprakash, NC (1986) Reinvestigation of the crystal-structure of beta-alanine. In: International Journal of Peptide & Protein Research, 28 (5). pp. 525-528.
Parthasarathy, G and Gopal, ESR and Krishnamurthy, HR and Pandit, R and Sekhar, JA (1986) Quasi Crystalline A1-Mn Alloys: Pressure Induced Crystallization and Structural Studies. In: Current Science, 55 (11). pp. 517-520.
Parthasarathy, G and Asokan, S and Gopal, ESR (1986) Pressure induced polymorphous crystallization in bulk Ge20Te80 glass. In: Physica A: Statistical Mechanics and its Applications, 139 (1-3). pp. 266-268.
Parthasarathy, G and Ramakrishna, R and Asokan, S and Gopal, ESR (1986) Effect of pressure on the electrical resistivity of bulk amorphous Al23Te77 alloy under various stages of crystallization. In: Journal of Materials Science Letters, 5 (8). pp. 809-811.
Patel, Mukul N and Gopinathan, Karumathil P (1986) Lysozyme-Sensitive Bioemulsifier for Immiscible Organophosphorus Pesticides. In: Applied and Environmental Microbiology, 52 (5). pp. 1224-1226.
Patil, KC (1986) Metal-hydrazine complexes as precursors to oxide materials. In: Chemical Sciences, 96 (6). pp. 459-464.
Patnaik, LM and Basu, JK (1986) Two Tools for Interprocess Communication in Distributed Data-Flow Systems. In: Computer Journal, 29 (6). pp. 506-521.
Patnaik, LM and Govindarajan, R and Ramadoss, NS (1986) Design and Performance Evaluation of EXMAN: An EXtended MANchester Data Flow Computer. In: IEEE Transactions on Computers, 35 (3). pp. 229-244.
Patnaik, LM and Shenoy, RS and Krishnan, D (1986) Set theoretic operations on polygons using the scan-grid approach. In: Computer-Aided Design, 18 (5). pp. 275-279.
Patnaik, LM and Sundararaman, K (1986) Performance evaluation of a distributed concurrency control algorithm. In: Computers & Electrical Engineering, 12 (1-2). pp. 73-88.
Patole, Milind S and Kurup, Ramakrishna CK and Ramasarma, T (1986) Reduction of vanadate by a microsomal redox system. In: Biochemical and Biophysical Research Communications, 141 (1). pp. 171-175.
Paul, PKC and Sukumar, M and Bardi, R and Piazzesi, AM and Valle, G and Toniolo, C and BaIaram, P (1986) Stereochemically Constrained Peptides. Theoretical and Experimental Studies on the Conformations of Peptides Containing 1-Aminocyclohexanecarboxylic Acid. In: Journal of the American Chemical Society, 108 (20). pp. 6363-6370.
Paul, PKC and Sukumar, M and Bardi, R and Piazzesi, AM and Valle, G and Toniolo, C and Balaram, Padmanabhan (1986) Stereochemically constrained peptides. Theoretical and experimental studies on the conformations of peptides containing 1-aminocyclohexane carboxylic acid. In: journal of the American chemical Society, 108 (20). pp. 6363-6370.
Pillai, PRSaseendran and Muralidhara, MK and Naidu, PS (1986) Wideband acoustic absorption characteristics of rubberized coir for underwater applications. In: Ultrasonics, 24 (6). 363 -367.
Poojary, M Oamodara and Manohar, Hattikudur (1986) Interaction of Metal Ions with 2'-Deoxyribonucleotides. Crystal and Molecular Structure of a Cobalt(l1) Complex with 2'-Deoxyinosine 5'-Monophosphate. In: Dalton Transactions (2). 309-312.
Prabhakaran, K and Sen, P and Rao, CNR (1986) Hydroxylation of oxygen-covered Cu(110) and Zn(0001) surfaces by interaction with CH3OH, (CH3)2NH, H2S, HCl and other proton donor molecules. In: Surface Science, 169 (2-3). L301-L306.
Prabhakaran, K and Sen, P and Rao, CNR (1986) Studies of Molecular Oxygen Adsorbed on Cu Surfaces. In: Surface Science, 177 (2). L971-L977.
Prabhu, R and Rao, GR and Jamaluddin, M and Ramakrishnan, T (1986) N-[2-naphthyl]-glycine hydrazide, a potent inhibitor of dna-dependent rna-polymerase of mycobacterium-tuberculosis h37rv. In: Journal of Biosciences, 10 (1). 163 -166.
Prasad, Durga M and Sathyanarayana, S (1986) Electrode Kinetics of a Nickel-Cadmium cell and Failure-mode prediction- Constant voltage Charging. In: Indian Journal of Chemical Technology, 24 (7). pp. 361-371.
Prasad, S Narendra and Hegde, Malati (1986) Phenology and seasonality in the tropical deciduous forest of Bandipur, South India. In: Proceedings of the Indian Academy of Sciences - Plant Sciences, 96 (2). 121-133.
Prasad, GL and Adiga, PR (1986) Decarboxylation of arginine and ornithine by arginine decarboxylase purified from cucumber (Cucumis sativus) seedlings. In: Journal of Biosciences, 10 (2). pp. 203-213.
Prasad, GL and Adiga, PR (1986) Purification and characterization of putrescine synthase from cucumber seedlings. A multifunctional enzyme involved in putrescine biosynthesis. In: Journal of Biosciences, 10 (3). pp. 373-391.
Prasad, Madhu and Rao, Radhika Rani and Chaudhuri, Ray AK (1986) A versatile AC mutual inductance bridge. In: Journal of Physics E - Scientific Instruments, 19 (12). pp. 1013-1016.
Prasad, Ramesh GK and Chattopadhyay, K and Rao, Mohan M (1986) Structure-property correlation in dual-phase copper-tin alloys. In: Journal of Materials Science Letters, 5 (10). pp. 991-994.
R, Ramesham and S, Sathyanarayana (1986) Kinetics of corrosion of passive metals. Part III: applicability of the new technique of transient Tafel polarization and its decay. In: Indian Journal of Technology, 24 (8). pp. 536-544.
RS, Subrahmanya (1986) Kinetic currents in polarography. Part II. Spherical diffusion to the DME for pseudo-first order catalyzed processes and modification of Ilkovic equation. In: Journal of the Electrochemical Society of India, 35 (3). pp. 157-162.
Radha, TS and Ramprasad, BS (1986) Speckle based fiber optic sensor for the measurement of the change in refractive index in liquid mixtures. In: Journal of the Electrochemical Society of India, 35 (2). pp. 135-136.
Rague Schleyer, Paul von and Kaufmann, Elmar and Kos, Alexander J and Mayr, Herbert and Chandrasekhard, Jayaraman (1986) Stabilization of the Alleged 'Bishomoaromatic' Bicyclo[3.2.l]octa-2,6-dienyl Anion by Counterion Interactions and by Hyperconjugation. In: Journal of the Chemical Society - Series Chemical Communications (21). 1583 -1585.
Rajasekar, N (1986) Synthesis and Spectroscopic Investigations of Complexes of Lanthanide Nitrates with Isoquinoline-2-oxide. In: Synthesis and Reactivity in Inorganic, Metal-Organic, and Nano-Metal Chemistry, 16 (8). pp. 1109-1119.
Rajashekara, KS and Joseph, Vithyathil and Rajagopalan, V (1986) Protection and Switching-Aid Networks for Transistor Bridge Inverters. In: IEEE Transactions on Industrial Electronics, 33 (2). pp. 185-192.
Raju, Ramaswamy and Jacob, Mathai T (1986) The Rna Binding Subset of Adenosine Antibodies. In: Immunological Investigations, 15 (5). 405 -417.
Rajumon, MK and Hegde, MS and Rao, CNR (1986) Electronic structure and oxidation of aluminium-modified Ni and Cu surfaces. In: Solid State Communications, 60 (3). pp. 267-270.
Ram, Mohan RA and Ganapathi, L and Ganguly, P and Rao, CNR (1986) Evolution of three-dimensional character across the Lan+1NinO3n+1 homologous series with increase in n1. In: Journal of Solid State Chemistry, 63 (2). pp. 139-147.
Ram, RA Mohan and Gopalakrishnan, Jagannatha (1986) Mixed valency in the high-temperature phases of transition metal molybdates,AMoO4 (A=Fe, Co, Ni). In: Proceedings of the Indian Academy of Sciences - Chemical Sciences, 96 (5). 291 -296.
Ramadurai, S (1986) Supernova induced star formation. In: Bulletin of the Astronomical Society of India, 14 (4). pp. 207-210.
Ramadurai, S and Thejappa, G (1986) Preferential acceleration of $^3He$ by lower hybrid waves. In: Advances in Space Research, 6 (6). pp. 281-284.
Ramakrishnan, V and Dhas, Arul G and Narayanan, PS (1986) Raman spectra of $NaLa{(MoO_4)}_2$ single crystal. In: Journal of Raman Spectroscopy, 17 (3). pp. 273-275.
Ramakrishnan, V and Dhas, Arul G and Narayanan, PS (1986) Raman Spectra of NaLa(MoO,), Single Crystal. In: Journal of Raman Spectroscopy, 17 (3). pp. 273-275.
Ramamurthy, TS and Krishnamurthy, T and Narayana, K Badari and Vijayakuma, K and Dattaguru, B (1986) Modified crack closure integral method with quarter point elements. In: Mechanics Research Communications, 13 (4). 179 -186.
Ramamurthy, V (1986) Organic Photochemistry in Organized Media. In: Tetrahedron, 42 (21). pp. 5753-5839.
Ramaraj, N and Rajaram, R and Parthasarathy, K (1986) A new analytical approach to optimize a generation schedule. In: Electric Power Systems Research, 11 (2). pp. 147-152.
Ramasesha, S (1986) A Diagrammatic Valence Bond Method for Configuration Interaction Calculations in Atoms and Molecules. In: Chemical Physics Letters, 130 (6). pp. 522-525.
Ramasesha, S (1986) Electron-electron interactions in polyacetylene. In: Proceedings of the Indian Academy of Sciences - Chemical Sciences, 96 (6). pp. 509-521.
Ramesh, R and Ravikumar, K and Kishore, * (1986) Wear of En 31 steel. In: Transactions of the Indian Institute of Metals, 39 (4). pp. 329-334.
Ramesh, N and Shouche, Yogesh S and Brahmachari, Samir K (1986) Recognition of B and Z forms of DNA by Escherichia coli DNA polymerase I. In: Journal of Molecular Biology, 190 (4). pp. 635-638.
Ramesham, R and Sathyanarayana, S (1986) Kinetics of corrosion of passive metals. Part II: Failure of linear polarization technique. In: Indian Journal of Technology, 24 (8). pp. 529-535.
Rangarajan, SK (1986) High Amplitude periodic signal theory: part II-nonlinear analysis and phenomenological Decoupling. In: Indian Journal of Chemical Technology, 24 (7). pp. 352-356.
Ranjith, K and Narasimhan, R (1986) Asymptotic and finite element analyses of mode III dynamic crack growth at a ductile-brittle interface. In: International Journal of Fracture, 76 (1). pp. 61-77.
Rao, CNR and Roberts, MW and Weightman, P (1986) Studies of Solids and Surfaces by Auger Electron Spectroscopy. In: Philosophical Transactions of the Royal Society of London - Series A: Mathematical and Physical Sciences, 318 (1541). 37 -50.
Rao, Jagannadha A and Moudgal, NR and Hao, Choh Li (1986) \beta -Endorphin: intranasal administration increases the serum prolactin level in monkey. In: International Journal of Peptide & Protein Research, 28 (5). pp. 546-548.
Rao, Krishna GS and Sarma, Prasad MS (1986) Studies in terpenoids. Part LXV. Synthesis of 4-(2-methoxy-5-methylphenyl)-6-methylheptan-2-one, a secosesquiterpene structural analog of sesquichamaenol and himasecolone. In: . Indian Journal of Chemistry, Section B: Organic Chemistry Including Medicinal Chemistry, 25B (7). pp. 752-753.
Rao, M and Yelloji, K and Natarajan, KA (1986) Electrochemical behavior of sulfide minerals under open-circuit conditions. In: Transactions of the Indian Institute of Metals, 39 (6). pp. 582-591.
Rao, Subba GSR and Vijaybhaskar, K and Srikrishna, A (1986) Regioselective synthesis of 1-methylbicyclo[2.2.2]octene derivatives. In: Indian Journal of Chemistry, Section B: Organic Chemistry Including Medicinal Chemistry, 25B (8). pp. 785-786.
Rao, BG and Rao, KJ (1986) Electron spin resonance studies of d5 ions (Fe3+ and Mn2+) in lead oxide-lead halide glasses. In: Chemical Physics, 102 (1-2). pp. 121-132.
Rao, BG and Rao, KJ (1986) The study of oxidation state of manganese in lead oxyhalide glasses by optical spectroscopy. In: Journal of Materials Science Letters, 5 (2). pp. 141-143.
Rao, BG and Vasanthacharya, NY and Rao, KJ (1986) Magnetic susceptibility studies of lead oxyhalide glasses containing transition metal oxides. In: Proceedings of the Indian Academy of Sciences - Chemical Sciences, 96 (5). pp. 383-388.
Rao, CNR and Ganguly, P (1986) A New Criterion for the Metallicity of Elements. In: Solid State Communications, 57 (1). pp. 5-6.
Rao, CNR and Gopalakrishnan, J and Vidyasagar, K and Ganguli, AK and Ramanan, A and Ganapathi, L (1986) Novel metal oxides prepared by ingenious synthetic routes. In: Journal of Materials Research, 1 (2). pp. 280-294.
Rao, CNR and Rajumon, MK and Prabhakaran, K and Hegde, MS and Kamath, PV (1986) Precursor Species of Carbon Monoxide Before its Dissociation on Aluminium-Promoted Ni and Cu Surfaces. In: Chemical Physics Letters, 129 (2). pp. 130-134.
Rao, Kameswara L and Harshavardhan, Solomon K and Selvarajan, A and Hegde, MS (1986) Novel laser induced image storage by chemical modification of surfaces in in situ textured amorphous Ge films. In: Applied Physics Letters, 49 (13). pp. 826-828.
Rao, Kusuma G (1986) Sensible heat fluxes during the active and break phases of the southwest monsoon over the Indian region. In: Boundary-Layer Meteorology, 36 (3). 283 -294.
Rao, R Vittal and Sukavanam, N (1986) Kac-Akhiezer formula for normal integral operators. In: Journal of Mathematical Analysis and Applications, 114 (2). pp. 458-467.
Rao, R Vittal and Sukavanam, N (1986) Spectral analysis of finite section normal integral operators. In: Journal of Mathematical Analysis and Applications, 115 (1). pp. 23-45.
Rao, Ramachandra A and Deshikachar, KS (1986) MHD Oscillatory Flow of Blood Through Channels of Variable Cross Section. In: International Journal of Engineering Science, 24 (10). pp. 1615-1628.
Rao, Ranga G and Prabhakaran, K and Rao, CNR (1986) Nitrogen Adsorbed on Clean and Promoted Ni Surfaces. In: Surface Science, 176 (1-2). L835-L840.
Rao, S P Sudhakara and Varughese, KI and Manohar, H (1986) Ternary metal complexes of anionic and neutral pyridoxine (vitamin B6) with 2,2'-bipyridine. Syntheses and x-ray structures of (pyridoxinato)bis(2,2'-bipyridyl)cobalt(III) perchlorate and chloro(2,2'-bipyridyl)(pyridoxine)copper(II) perchlorate hydrate. In: Inorganic Chemistry, 25 (6). pp. 734-740.
Rao, SP Sudhakara and Manohar, Hattikudur and Aoki, Katsuyuki and Yamazaki, Hiroshi and Bau, Robert (1986) Novel Oxidation of Pyridoxal in Ternary Metal Complexes. An X-Ray Study of the Products. In: Journal of the Chemical Society - Series Chemical Communications (1). pp. 4-6.
Rao, Sankar M (1986) An overview of climate models. In: Earth and Planetary Sciences, 95 (3). pp. 447-484.
Rao, Sankara K (1986) Plantlets from somatic callus tissue of the East Indian Rosewood (Dalbergia latif olia Roxb.). In: Plant Cell Reports, 5 (3). pp. 199-201.
Rao, Shashidhar N and Sasisekharan, V (1986) Conformations of Dinucleoside Monophosphates in Relation to Duplex DNA Structures. In: Biopolymers, 25 (1). pp. 17-30.
Ravikumar, C and Balakrishnan, N (1986) A low cost microprocessor based multiple pressure measuring system. In: Journal of Microcomputer Applications, 9 . pp. 319-326.
Ravindram, M and Kalvinskas, John J (1986) Coal desulfurization in a fluidized bed reactor. In: Environmental Progress, 5 (4). pp. 264-272.
Ravindranath, N and Moudgal, NR (1986) Antifertility effect of tamoxifen as tested in the female bonnet monkey (Macaca radiata). In: Journal of Biosciences, 10 (1). 167 -170.
Ravindranath, NH and Chanakya, HN (1986) Biomass based energy system for a South Indian village. In: Biomass, 9 (3). 215 -233.
Ravindranathan, P and Patil, KC (1986) A one-step process for the preparation ofγ-Fe2O3. In: Journal of Materials Science Letters, 5 (2). pp. 221-222.
Raviprasad, K and Tenwick, M and Davies, HA and Chattopadhyay, K (1986) The Nature of Ordered Structures in Melt Spun Iron-Silicon Alloys. In: Scripta Metallurgica, 20 (9). pp. 1265-1270.
Ravishankar, Malavalli K and Pappu, Sastry V (1986) Fiber-optic sensor-based refractometer-cum-liquid level indicator. In: Applied Optics, 25 (4). pp. 480-482.
Ray, AK and Dwarakadasa, ES and Raman, KS (1986) Effect of porosity and slag inclusions on the fatigue fracture behaviour of 15CDV6 butt welds. In: Journal of Materials Science Letters, 5 (8). pp. 765-768.
Ray, Arabinda and Mohammad, SK Noor and Kulkarni, GV (1986) Reinvestigation of the antifungal activities of some aromatic N-oxides. In: Proceedings of the Indian Academy of Sciences - Chemical Sciences, 96 (1-2). pp. 67-71.
Raychaudhuri, AK (1986) Low temperature Properties of glasses-unsolved problems. In: Proceedings of the Indian Academy of Sciences - Chemical Sciences, 96 (6). pp. 559-564.
Reddy, Bramham A and Gopinathan, KP (1986) Existence of single-strand interruptions in the genomic DNA of mycobacteriophage I3. In: FEMS Microbiology Letters, 37 (2). pp. 163-167.
Reddy, NM and Reddy, KPJ and Prasad, Krishna MR (1986) Theoretical Gain Optimization in $CO_2-N_2-H_2$ Gasdynamic Lasers with Two-Dimensional Wedge Nozzles. In: AIAA Journal, 24 (12). pp. 2045-2046.
Reddy, NM and Reddy, KPJ and Prasad, Krishna MR (1986) Theoretical gain optimization in CO2-N2-H2 gasdynamic lasers with two-dimensional wedge nozzles. In: AIAA Journal, 24 (12). pp. 2045-2046.
Reddy, Bramham A and Gopinathan, Karumathil P (1986) Presence of random single-strand gaps in mycobacteriophage 13 DNA. In: Gene, 44 (2-3). pp. 227-234.
Reddy, KPJ (1986) Stability criteria for single pulse solutions of cw passive mode-locked lasers. In: Optics Communications, 56 (6). 433 -434.
Rukmani, K and Ramakrishna, J (1986) Chlorine-35 Nuclear Quadrupole Resonance Studies in 2,3- and 3,5-Dichloroanisoles. In: Journal of the chemical society-faraday transactions II, 82 (3). pp. 291-298.
Sachdev, PL and Philip, V (1986) Invariance group properties and exact solutions of equations describing time-dependent free surface flows under gravity. In: Quarterly of Applied Mathematics, 43 (4). 465 -482.
Sachdev, PL and Tikekar, VG and Nair, KRC (1986) Evolution and decay of spherical and cylindrical N waves. In: Journal of Fluid Mechanics, 172 . pp. 347-371.
Sachdev, PL and Nair, KRC and Tikekar, VG (1986) Generalized Burgers equations and Euler-Painlev transcendents. I. In: Journal of Mathematical Physics, 27 (6). pp. 1506-1522.
Sadhu, C and Dutta, S and Gopinathan, KP (1986) Presence of mycobacteriophage 13-1ike DNA sequences in the genome of its host Mycobacterium smegmatis. In: Archives of Microbiology, 146 (2). pp. 166-169.
Sahal, Dinkar and Balaram, P (1986) Peptide Models of Electrostatic Interactions in Proteins: $NMR$ Studies on Two \beta - Turn Tetrapeptides Containing $Asp-His$ and $Asp-Lys$ Salt Bridges. In: Biochemistry, 25 (20). pp. 6004-6013.
Sampath, DS and Balaram, P (1986) Rapid procedure for the resolution of racemic gossypol. In: Journal of the Chemical Society, Chemical Communications (9). pp. 649-650.
Sampath, DS and Balaram, Padmanabhan (1986) Resolution of racemic gossypol and interaction of individual enantiomers with serum albumins and model peptides. In: Biochimica et Biophysica Acta, 882 (2). pp. 183-186.
Samuel, Manoharan T and Madhava, Madyastha K (1986) A novel conversion of narcotine into a macrolide. In: Indian Journal of Chemistry, Section B: Organic Chemistry Including Medicinal Chemistry, 25 (3). p. 227.
Sangunni, KS and Ravi, R and Bhat, HL and Narayanan, PS (1986) Switching Process in Ferroelectric Triglycine Selenate. In: Japanese Journal of Applied Physics, 25 (3). 380 -382.
Sankar, G and Vasudevan, S and Rao, CNR (1986) Analysis of EXAFS data of multiphasic heterogeneous catalysts. In: Chemical Physics Letters, 127 (6). 620 -626.
Sankar, G and Vasudevan, S and Rao, CNR (1986) An EXAFS investigation of $Cu-ZnO$ methanol synthesis catalysts. In: Journal of Chemical Physics, 85 (4). pp. 2291-2299.
Sankar, G and Vasudevan, S and Rao, CNR (1986) Extended X-ray Absorption Fine Structure Studies of Bimetallic ${Cu-Ni/ \gamma -$Al_2O_3$}$ Catalysts. In: Journal of Physical Chemistry, 90 (21). pp. 5325-5328.
Sankar, Gopinathan and Rao, Ramachandra CN (1986) Nature of Ni and Cu Species in Reduced Bimetallic $Ni-Cu/Al_20_3$ Catalysts. In: Angewandte Chemie International Edition in English, 25 (8). pp. 753-754.
Sankar, Gopinathan and Rao, Ramachandra CN (1986) Nature of Ni and Cu Species in Reduced Bimetallic Ni-Cu/Alz03 Catalysts. In: Angewandte Chemie (English Edition), 25 (8). 753 -754.
Sarma, Prasad MS and Rao, Krishna GS (1986) Studies in terpenoids. Part LXVIII. Synthesis of sesquiterpenic dimethylisobutyl- and di- and trimethylisopropylindans. In: Indian Journal of Chemistry, Section B: Organic Chemistry Including Medicinal Chemistry, 25B (9). pp. 951-952.
Sarma, Prasad MS and Rao, Krishna GS (1986) Studies in terpenoids. Part LXVII. Sesquiterpenic indans of bicyclonidorellane skeleton: synthesis of 1,6-diisopropylindan and ethyl 2-(6-isopropyl-1-indanyl)propionate. In: Indian Journal of ChemistrySection B: Organic Chemistry Including Medicinal Chemistry, 25B (9). pp. 953-954.
Sarma, BS and Ramakrishna, J (1986) Proton magnetic relaxation in (TMA)2HgBr4 and (TMA)2HgI4. In: Pramana, 26 (3). pp. 263-268.
Sarma, Kandula VN and Sridharan, K and Rao, A Achutha and Sarma, CSS (1986) Computer model for vedavati ground water basin. Part 1. Well field model. In: Sadhana : Academy Proceedings in Engineering Sciences, 9 . pp. 31-42.
Sarode, PR (1986) EXAFS in Niobium Dichalcogenides Intercalated With First-Row Transition Metals. In: Physica Status Solidi A: Applied Research, 98 (2). pp. 391-397.
Sasisekharan, V (1986) A new method for generation of quasi-periodic structures withn fold axes: Application to five and seven folds. In: Pramana, 26 (3). L283-L293.
Sastri, P and Lahiri, AK (1986) "Central Atoms" models for ternary silicate and alumino-silicate melts. In: Metallurgical and Materials Transactions B, Process Metallurgy and Materials Processing Science, 17 (1). pp. 105-110.
Sastry, DH and Murthy, Gunturi S (1986) Impression creep behavior of metals at high temperatures. In: Transactions of the Indian Institute of Metals, 39 (4). pp. 369-379.
Sastry, Krishna MV and Surolia, A (1986) Intrinsic Fluorescence Studies on Saccharide Binding to Artocarpus Integrifolia Lectin. In: Bioscience Reports, 6 (10). pp. 853-860.
Sastry, MV and Banarjee, Probal and Patanjali, Sankhavaram R and Swamy, Joginadha M and Swarnalatha, GV and Surolia, Avadhesha (1986) Analysis of Saccharide Binding to Artocarpus integrifolia Lectin Reveals Specific Recognition of T-antigen (\beta D-Gal(1\rightarrow 3)GalNAc. In: Journal of Biological Chemistry, 261 (25). pp. 11726-11733.
Sathiakumar, S and Biswas, SK and Vithayathil, Joseph (1986) Microprocessor-Based Field-Oriented Control of A CSI-Fed Induction Motor Drive. In: IEEE Transactions on Industrial Electronics, 33 (1). 39 -43.
Sathiakumar, S and Vithayathil, Joseph and Biswas, SK (1986) Microprocessor-Based Field-Oriented Control of A CSI-Fed Induction Motor Drive. In: IEEE Transactions on Industrial Electronics, IE-33 (1). pp. 39-43.
Sathish, S and Chaterjee, S and Awasthi, ON and Gopal, ESR (1986) Electron-electron scattering and ultrasonic attenuation in potassium. In: Journal of Low Temperature Physics, 63 (5-6). pp. 423-429.
Sathyanarayana, S and Ramesham, R (1986) Kinetics of Corrosion of passive metals - part I: Concepts and Theory. In: Indian Journal of Technology, 24 (7). pp. 447-455.
Sathyaprakash, BS and Goswami, P and Sinha, KP (1986) Singularity-free cosmology: A simple model. In: Physical Review D – Particles and Fields, 33 (8). pp. 2196-2200.
Satyabhama, S and Seelan, R Sathiagnana and Padmanaban, G (1986) Expression of Cytochrome P-450 and Albumin Genes in Rat Liver: Effect of Xenobioticst. In: Biochemistry, 25 (16). pp. 4508-4512.
Schleye, Paul von Rague and Spitznagel, Gunther H and Chandrasekhar, Jayaraman (1986) The ethyl, 1- and 2-propyl, and other simple alky carbanions do not exist. In: Tetrahedron Letters, 27 (37). 4411 -4414.
Sekhar, JA and Rajasekharan, T and Rao, Rama P and Parthasarathy, G and Ramkumar, S and Gopal, ESR and Lakshmi, CS and Mallya, RM (1986) Electron and x-ray diffraction studies on Al86Fe14, Al82Fe18 and Al75Fe25 quasicrystals. In: Pramana, 27 (1-2). pp. 267-273.
Sekharudu, Y Chandra and Biswas, Margaret and Rao, VSR (1986) Complex carbohydrates: 2. The modes of binding of complex carbohydrates to concanavalin A - a computer modelling approach. In: International Journal of Biological Macromolecules, 8 (1). pp. 9-19.
Sen, P and Rao, CNR (1986) An eels study of water, methanol, formaldehyde and formic acid adsorbed on clean and oxygen-covered zinc(0001) surfaces. In: Surface Science, 172 (2). pp. 269-280.
Sen, P and Rao, CNR and Thomas, JM (1986) Structure of Alkali Metal Ionic Clusters. In: Journal of Molecular Structure, 146 . pp. 171-174.
Seralathan, M and Rangarajan, SK (1986) Fluctuation phenomena in electrochemistry part I. The formalism. In: Journal of Electroanalytical Chemistry, 208 (1). pp. 13-28.
Seralathan, M and Rangarajan, SK (1986) Fluctuation phenomena in electrochemistry part II. Modelling the noise sources. In: Journal of Electroanalytical Chemistry, 208 (1). pp. 29-56.
Seshan, S and Vijayalakshmi, D (1986) Heat pipes-concepts, materials and applications. In: Energy Conversion and Management, 26 (1). 1 -9.
Shukla, AK and Ramesh, KV and Kannan, AM (1986) Fuel cells: Problems and prospects. In: Journal of Chemical Sciences, 97 (3-4). pp. 513-527.
Simon, R and Sudarshan, ECG and Mukunda, N (1986) Gaussian-Maxwell beams. In: Journal of the Optical Society of America A: Optics and Image Science Vision, 3 (4). pp. 536-540.
Singh, Sharat and Usha, Govindarajan and Tung, Chen Ho and Turro, Nicholas J and Ramamurthy, Vaidhyanathan (1986) Modification of chemical reactivity by cyclodextrins. Observation of moderate effects on Norrish type I and type II photobehavior. In: Journal of Organic Chemistry, 51 (6). pp. 941-944.
Sinha, KP and Sudarshan, ECG and Vigier, JP (1986) Superfluid vacuum carrying real Einstein-de Broglie waves. In: Physics Letters A, 114 (6). 298 -300.
Sita, Lakshmi G and Chattopadhyay, S and Tejavathi, DH (1986) Plant regeneration from shoot callus of rosewood (Dalbergia latifolia Roxb). In: Plant Cell Reports, 5 (4). pp. 266-268.
Soman, KV and Ramakrishnan, C (1986) Identification and analysis of extended strands and beta-sheets in globular proteins. In: International Journal of Biological Macromolecules, 8 (2). pp. 89-96.
Somasekharan, KN and Kalpagam, V (1986) Use of a compensation parameter in the Thermal Decomposition of Copolymers. In: Thermochimica Acta, 107 . pp. 379-382.
Somasundaram, T and Ganguly, P and Rao, CNR (1986) Photoacoustic investigation of phase transitions in solids. In: Journal of Physics C: Solid State Physics, 19 (13). 2137 -2151.
Somasundaram, T and Rao, Sanjay SR and Maheshwari, R (1986) Pigments in Thermophilic fungi. In: Current Science, 55 (19). pp. 957-960.
Sridharan, A and Rao, Sudhakar M and Gajarajan, VS (1986) Influence of adsorbed sulfate on the engineering properties of kaolinite and bentonite. In: India Clay Research, 5 (2). pp. 74-81.
Srinivasan, J and Basu, Biswajit (1986) A numerical study of thermocapillary flow in a rectangular cavity during laser melting. In: International Journal of Heat and Mass Transfer, 29 (4). pp. 563-572.
Srinivasan, R and Usha, S (1986) Auxiliary coils for generating magnetic field gradients for a Faraday magnetometer. In: Journal of physics e-scientific instruments, 19 (11). 930 -932.
Subbanna, GN and Kutty, TRN and Iyer, Anantha GV (1986) Structural intergrowth of brucite in anthophyllite. In: American Mineralogist, 71 . pp. 1198-1200.
Subbanna, GN and Rao, CNR (1986) Metal-Ceramic Composites: A Study Of Small Metal Particles (Divided Metals). In: Materials Research Bulletin, 21 (12). pp. 1465-1471.
Subrahmanya, RS (1986) Kinetic currents in polarography. Part I. Linear diffusion to the DME for pseudo first order catalyzed processes. In: Journal of the Electrochemical Society of India, 35 (3). pp. 151-155.
Subrahmanyam, HN and Subramanyam, SV (1986) Accurate measurement of thermal expansion of solids between 77 K and 350 K by 3-terminal capacitance method. In: Pramana, 27 (5). pp. 647-660.
Subramanyam, SV and Naik, Hemamalini (1986) Nonlinear conduction and electrical switching in one-dimensional conductors. In: Proceedings of the Indian Academy of Sciences - Chemical Sciences, 96 (6). 499 -508.
Sudha, LV and Sathyanarayana, DN and Manogaran, S (1986) $^{13}C$ NMR spectra of 1,3-dipyridyl- and pyridylphenylthioureas. Chemical shift assignments and conformational implications. In: Spectrochimica Acta, Part A: Molecular and Biomolecular Spectroscopy, 42 (12). pp. 1373-1378.
Sudha, Lalgudi V and Sathyanarayana, Dixit N (1986) Proton and Carbon-13 Nuclear Magnetic Resonance Studies of Conformations of 1,3-Dipyridylthioureas. In: Journal of the Chemical Society, Perkin Transactions 2: Physical Organic Chemistry (1972-1999), 11 . pp. 1647-1650.
Sundararaman, Narayanaswamy and Ravindram, M and Bhatt, MV (1986) Kinetic Modeling of Vapor-Phase Oxidation of 1-Phenylethanol over Thorium Molybdate. In: Industrial & Engineering Chemistry Product Research and Development, 25 (4). pp. 512-517.
Suresh, BS and Padma, DK (1986) Reactivities of Tetrahalosilanes and Silane with Sulphur Trioxide. In: Polyhedron, 5 (10). pp. 1579-1580.
Suresh, BS and Padma, DK (1986) Reactivities of tetrahalosilanes and silane with sulphur trioxide. In: Polyhedron, 5 (10). 1579 -1580.
Suresh, CG and Ramaswamy, Jayanthi and Vijayan, M (1986) X-ray Studies on Crystalline Complexes Involving Amino Acids and Peptides. XIII. Effect of Chirality on Molecular Aggregation: The Crystal Structures of $L$-Arginine $D$-Aspartate and $L$-Arginine $D$-Glutamate Trihydrate. In: Acta Crystallographica, B42 (5). pp. 473-478.
Suresh, CG and Ramaswamy, Jayanthi and Vijayan, M (1986) X-ray Studies on Crystalline Complexes Involving Amino Acids and Peptides. XIII. Effect of Chirality on Molecular Aggregation: The Crystal Structures of L-Arginine D-Aspartate and L-Arginine D-Glutamate Trihydrate. In: Acta Crystallographica Section B Structural Science, 42 (5). pp. 473-478.
Surma Devi, CD and Takhar, HS and Nath, G (1986) Unsteady, three-dimensional, boundary-layer flow due to a stretching surface. In: International Journal of Heat and Mass Transfer, 29 (12). pp. 1996-1999.
Suryanarayana, VVS and Rao, BU and Padayatty, JD (1986) Expression in E. coli of the cloned cDNA for the major antigen of foot and mouth disease virus Asia 1 63/72. In: Journal of Genetics, 65 (1-2). pp. 19-30.
Suryaprakash, N and Khetrapal, CL (1986) The use of $^{13}C$ satellites in the proton NMR spectra of Oriented systems for the Determination of Molecular Structure. In: Magnetic Resonance in Chemistry, 24 (3). pp. 247-250.
Swamy, KC Kumara and Krishnamurthy, SS (1986) Studies of phosphazenes. 28. Reactions of pentachloro- and pentafluoro(triphenylphosphazenyl)cyclotriphosphazenes with sodium methoxide. Mechanistic aspects and their implications for nucleophilic displacement at a tetrahedral phosphorus(V) center. In: Inorganic Chemistry, 25 (7). 920 -928.
Swamy, MJ and Sastry, M V Krishna and Khan, MI and Surolia, A (1986) Thermodynamic and kinetic studies on saccharide binding to soya-bean agglutinin. In: Biochemical Journal, 234 (3). 515 -522.
Syamala, MS and Devanathan, S and Ramamurthy, V (1986) Modification of the Photochemical Behaviour of Organic Molecules by Cyclodextrin: Geometric Isomerization of Stilbenes and Alkyl Cinnamates. In: Journal of Photochemistry, 34 (2). pp. 219-229.
Syamala, MS and Ramamurthy, V (1986) Consequences of Hydrophobic Association in Photoreactions: Photodimerization of Stilbenes in Water. In: Journal of Organic Chemistry, 51 (19). pp. 3712-3715.
Syamala, MS and Reddy, Dasaratha G and Rao, Nageswara B and Ramamurthy, V (1986) Chemistry in Cavities. In: Current Science, 55 (18). pp. 875-886.
Takhar, HS and Devi, CDSurma and Nath, G (1986) MHD Flow with Heat and Mass Transfer Due to a Point Sink. In: Indian Journal of Pure and Applied Mathematics, 17 (10). 1242 -1247.
Thathachar, Mandayam AL and Sastry, PS (1986) Relaxation Labeling with Learning Automata. In: IEEE Transactions on Pattern Analysis and Machine Intelligence, 8 (2). pp. 256-268.
Thirumaleshwar, M and Subramanyam, SV (1986) Exergy analysis of a Gifford-McMahon cycle cryorefrigerator. In: Cryogenics, 26 (4). 248 -251.
Thirumaleshwar, M and Subramanyam, SV (1986) Gifford-McMahon cycle- a theoretical analysis. In: Cryogenics, 26 (3). pp. 177-188.
Thirumaleshwar, M and Subramanyam, SV (1986) Heat balance analysis of single stage Gifford-McMahon cycle cryorefrigerator. In: Cryogenics, 26 (3). pp. 189-195.
Thirumaleshwar, M and Subramanyama, SV (1986) Two stage Gifford-McMahon cycle cryorefrigerator operating at 20 K. In: Cryogenics, 26 (10). 547 -555.
Thomas, Thresia and Nandi, US and Poddar, SK (1986) Thermodynamic characterization of oligo dG. poly rC hybrid helixes. In: Indian Journal of Biochemistry & Biophysics, 23 (4). pp. 192-196.
Thukaram, D and Iyengar, Ramakrishna BS and Parthasarathy, K (1986) An algorithm for optimum control of static VAR compensators to meet phase-wise unbalanced reactive power demands. In: Electric Power Systems Research, 11 (2). pp. 129-137.
Uberoi, C (1986) On the Kelvin-Helmholtz instability of structured plasma layers in the magnetosphere. In: Planetary and Space Science, 34 (12). pp. 1223-1227.
Uberoi, C and Narayanan, Satya A (1986) Effect of Variation of Magnetic Field Direction on Hydromagnetic Surface Waves. In: Plasma Physics and Controlled Fusion, 28 (11). pp. 1635-1643.
Ubgade, R and Sarode, PR (1986) Study of strontium compounds and minerals by X-ray absorption spectroscopy. I. K-edge shifts. In: Physica Status Solidi A, 99 (1). pp. 295-301.
Unni, Emmanual and Rao, MRS (1986) Androgen Binding Protein Levels And Fsh Binding To Testicular Membranes In Vitamin A Deficient Rats And During Subsequent Replenishment With Vitamin A. In: Journal of Steroid Biochemistry, 25 (4). pp. 579-583.
Usha, MG and Rao, Subba M and Kutty, Narayanan TR (1986) Preparation and thermal stability of ammonium alkaline earth trioxalatocobaltate(III) hydrates:$NH_4M^{2+}[Co(C_2O_4)_3].xH_2O$. In: Journal of Thermal Analysis and Calorimetry, 31 (1). pp. 7-14.
Usha, R and Murthy, MRN (1986) Protein structural homology: A metric approach. In: International Journal of Peptide and Protein Research, 28 (4). pp. 364-369.
Usha, G and Rao, BNageswer and Chandrasekhar, Jayaraman and Ramamurthy, V (1986) The Origin of Regioselectivity in a-Cleavage Reactions of Cyclopropenethiones: Potential Role of Pseudo-Jahn-Teller Effect in Substituted Cyclopropenyl Systems. In: Journal of Organic Chemistry, 51 (19). pp. 3630-3635.
Vaidehi, N and Akila, R and Shukla, AK and Jacob, KT (1986) Enhanced Ionic Conduction in Dispersed Solid Electrolyte Systems $CaF_2-Al_2O_3$ and $CaF_2-CeO_2$. In: Materials Research Bulletin, 21 (8). pp. 909-916.
Vaidehi, N and Akila, R and Shukla, AK and Jacob, KT (1986) Enhanced ionic conduction in dispersed solid electrolyte systems CaF2---Al2O3 and CaF2---CeO2. In: Materials Research Bulletin, 21 (8). pp. 909-916.
Vani, VC and Guha, S and Gopal, ESR (1986) Coexistence curve of acetonitrile and cyclohexane liquid system. In: Journal of Chemical Physics, 84 (7). 3999 -4007.
Varalakshmi, K and Savithri, HS and Rao, NA (1986) Identification of amino acid residues essential for enzyme activity of sheep liver 5,10-methylenetetrahydrofolate reductase. In: Biochemical Journal, 236 (1). 295 -298.
Vasantha, R and Nath, G (1986) Second-order boundary layers for steady, incompressible, three-dimensional stagnation point flows. In: International Journal of Heat and Mass Transfer, 29 (12). pp. 1993-1996.
Vasantha, R and Nath, G (1986) Unsteady compressible second-order boundary layers at the stagnation point of two-dimensional and axisymmetric bodies. In: Heat and Mass Transfer, 20 (4). pp. 273-281.
Vedula, S and Ramasesha, CS and Rao, A Achuta and Prasad, B Shyam (1986) Computer-model for vedavati groundwater basin .3. Irrigation potential. In: Sadhana : Academy Proceedings in Engineering Sciences, 9 . pp. 57-68.
Vellareddy, Anantharam and Patanjali, Sankhavaram R and Swam, Joginadha M and Sanadiq, Ashok R and Goldstein, Irwin J and Surolia, Avadhesha (1986) Isolation, Macromolecular Properties, and Combining Site of a Chitooligosaccharide-specific Lectin from the Exudate of Ridge Gourd(Luffaa cutangzda). In: Journal of Biological Chemistry, 261 (31). pp. 14621-14627.
Venkatesh, A and Murthy, Ramana PV and Rao, KP (1986) Finite element analysis of bimodulus composite stiffened thin shells of revolution. In: Computers & Structures, 22 (1). pp. 13-24.
Venkateswara, R and Rao, Sankara S and Vaidyanathan, CS (1986) Phytochemical constituents of cultured cells of Eucalyptus tereticornis SM. In: Plant Cell Reports, 5 (3). pp. 231-233.
Venkateswaran, S and Mallya, RM and Seshadri, MR (1986) Effect of trace elements on the fluidity of eutectic aluminum-silicon alloy using the vacuum suction technique. In: Transactions of the American Foundrymen's Society, 94 . pp. 701-708.
Venu, K and Sastry, VSS and Ramakrishna, J (1986) Proton magnetic resonance study of molecular dynamics in (NH4)2CdI4. In: Chemical Physics, 107 (1). 123 -127.
Verneker, Pai VR and Shaha, B (1986) Dual Role of Metallic Lithium in the Initiation of Vinyl Polymerization. In: Journal of Polymer Science Part C: Polymer Letters, 24 (1). pp. 1-5.
Verneker, VR Pai and Shahat, B (1986) On Coloration of Polyacrylonitrile: A Nuclear Magnetic Resonance Study. In: Macromolecules, 19 (7). 1851 -1856.
Vidyasagar, M and Levy, BC and Viswanadham, N (1986) A Note on the Genericity of Simultaneous Stabilizability and Pole Assignability. In: Circuits, Systems, and Signal Processing, 5 (3). 371 -387.
Vidyasagar, M and Viswanadham, N (1986) Construction of inverses with prescribed zero minors and applications to decentralized stabilization. In: Linear Algebra and its Applications, 83 . pp. 103-115.
Vijayamohanan, K and Shukla, AK and Sathyanarayana, S (1986) Statistical Optimization of Iron Electrodes for Alkaline Storage Batteries. In: Indian Journal of Technology, 24 (7). 430 -434.
Vijayaraju, K and Dwarakadasa, ES (1986) Computer-aided composition-treatment-structure-property correlation studies in steels. In: Bulletin of Materials Science, 8 (2). pp. 193-198.
Vijayaraju, K and Dwarakadasa, ES and Panchapagesan, TS (1986) Role of vacancies in the ductile fracture of commercially pure aluminum. In: Journal of Materials Science Letters, 5 (10). pp. 1000-1002.
Vyas, K and Manohar, H (1986) Solid state acyl migration in salicylamides. An x-ray study of the O- and N-propionyl derivatives. In: Molecular Crystals and Liquid Crystals, 137 (1). pp. 37-43.
Vyas, K and Manohar, H (1986) Solid-State Acyl Migration In Salicylamides - An X-Ray Study Of The Ortho-Propionyl And Normal-Propionyl Derivatives. In: Molecular Crystals and Liquid Crystals, 137 (1-4). pp. 37-43.
Yaparpalvi, R and Das, PK and Mukherjee, AK and Kumar, R (1986) Drop Formation under Pulsed Conditions. In: Chemical Engineering Science, 41 (10). pp. 2547-2553.
Yashonath, S and Rao, CNR (1986) Structural Changes Accompanying the Formation of Isopentane Glass. In: Journal of Physical Chemistry, 90 (12). 2581 -2584.
Yashonath, S and Rao, CNR (1986) An investigation of solid adamantane by a modified isothermal-isobaric ensemble Monte Carlo simulation. In: Journal of Physical Chemistry, 90 (12). 2552 -2554.
Editorials/Short Communications
Jagadeeswaran, P and Cherayil, Joseph D (1986) A general model for the conformational switch in 5S RNA during protein synthesis. In: Journal of Theoretical Biology, 83 (2). pp. 369-375.
Kumar, N and Jayannavar, AM (1986) Resistance fluctuation at the mobility edge. In: Journal of Physics C: Solid State Physics, 19 (4). L85-L89.
Mugeraya, Sridhar and Prabhakar, BR (1986) Measurement of resistivity and dielectric constant of beach-sand minerals. In: Journal of Electrostatics, 18 (1). 109 -112.
Rao, Narasimha K and Vijayalakshmi, D and Baskaran, N (1986) Mechanism of interaction of reversible and irreversible inhibitors with human-liver serine hydroxymethyltransferase. In: Journal of Protein Chemistry, 5 (4). 291 -292.
Ravichandrana, KS and Dwarakadasaa, ES (1986) Some Considerations on the Occurrence of Intergranular Fracture during Fatigue Crack growth in Steels. In: Materials Science and Engineering A, 83 (1). L11 -L16.
Surolia, A and Ramprasad, MP (1986) Immunotoxins to combat AIDS. In: Nature, 322 (6075). pp. 119-120.
Vasantha, R and Pop, I and Nath, G (1986) Non-darcy natural convection over a slender vertical frustum of a cone in a saturated porous medium. In: International Journal of Heat and Mass Transfer, 29 (1). pp. 153-156.
Goodenough, John Bannister and Shukla, Ashok Kumar and Silvapaliteiro, Carlos Antonio da and Jamieson, Keith Roderick and Hamnett, Andrew and Manoharan, Ramasamy (1986) Electrode for reducing oxygen. Patent Number(s) WO 8601642 A1. Patent Assignee(s) National Research Development Corporation . | CommonCrawl |
Fractability for partially ordered groups and Boolean algebras
Ján Jakubík, Judita Lihová
Abstract. In the present paper there are introduced and investigated the notions of fractal and semifractal lattice ordered groups or Riesz groups, respectively. The definitions are related to those which have been applied in the lattice theory. Besides, there is shown the existence of a proper class of Boolean algebras which are semifractal lattices but fail to be fractal lattices.
AMS Subject Classification (1991): 06F15, 06D35
Keyword(s): lattice ordered group ($\ell $-group, for short), fractal, semifractal, homogeneous Boolean algebra, Carathéodory functions
Received April 19, 2010, and in revised form November 23, 2010. (Registered under 30/2010.)
Minimal clones with many majority operations
Mike Behrisch, Tamás Waldhauser
Abstract. We present two minimal clones containing 26 and 78 majority operations respectively, more than any other previously known example.
Keyword(s): clone, minimal clone, majority operation
Received March 29, 2010, and in revised form August 17, 2010. (Registered under 21/2010.)
An extended lemma of Dobrowolski and Smyth's congruence
Artūras Dubickas
Abstract. We give a short proof of a recent result of Hare, McKinnon and Sinclair on divisibility of the resultant of two polynomials whose roots are prime powers of a given monic polynomial. Our proof is based on Smyth's congruence involving powers of conjugate algebraic integers. An application to the Mahler measure of a polynomial is also given.
AMS Subject Classification (1991): 11C08, 11R04, 11R09
Keyword(s): polynomial, resultant, Mahler's measure
Indefinite extreme points of the unit ball in a polynomial space
Lozko Milev, Nikola Naidenov
Abstract. The present paper continues work started by G. A. Muñoz-Fernández, Sz. Gy. Révész and J. B. Seoane-Sepúlveda [10] (degree 2 homogeneous polynomials, description of all extreme points) and L. Milev, N. Naidenov [8] (degree 2 algebraic polynomials, definite extreme points) by describing the indefinite extreme points of the unit ball of the space of degree 2 bivariate algebraic polynomials equipped with the maximum norm on the standard triangle of the plane. The main motivation for taking up this work is the hope that via the Krein--Milman theorem, this description will be useful in deriving the exact constants in certain inequalities, including the multivariate Bernstein inequality over general, non-symmetric convex bodies.
AMS Subject Classification (1991): 52A21, 26C05, 26B25
Keyword(s): convexity, extreme points, polynomials
Received January 8, 2010, and in final form May 18, 2010. (Registered under 3/2010.)
Convergence of generalized Nevanlinna functions
Heinz Langer, Annemarie Luger, Vladimir Matsaev
Abstract. Let $\kappa $ be a positive integer. A sequence $(f_n)$ of generalized Nevanlinna functions of the class ${\bf N}_\kappa $, which converges locally uniformly on some nonempty open subset of the complex plane to a function $f$, need not converge on any larger set, and $f$ can belong to any class $\bf N_{\kappa '}$ with $0\le\kappa '\le\kappa $. In this note we show that if it is a priori known that $f$ belongs to the same class ${\bf N}_\kappa $ then the sequence $(f_n)$ converges locally uniformly on the set $({\msbm C}\setminus{\msbm R})\cap{\rm hol}f$, and the sets of poles or generalized poles of nonpositive type of $f_n$ converge to the set of poles or generalized poles of nonpositive type of $f$. Moreover, a compactness result for generalized Nevanlinna functions is proved.
AMS Subject Classification (1991): 30E20, 30C15, 46C20, 46G99
Keyword(s): generalized Nevanlinna functions, rational functions, locally uniform convergence
Received February 25, 2011, and in revised form July 8, 2011. (Registered under 12/2011.)
Meromorphic solutions of second order Briot--Bouquet differential equations which are obtained not through the first order case
Yukitaka Abe, Atsuko Kogie
Abstract. We give a new proof of the fact that any meromorphic solution of a second order Briot--Bouquet differential equation in the whole plane is a degenerate or non-degenerate elliptic function. Our argument does not depend on the first order case.
AMS Subject Classification (1991): 34A20, 30D05
Keyword(s): Briot--Bouquet differential equations
Received April 28, 2009, and in revised form August 17, 2010. (Registered under 59/2009.)
Slight extensions of four celebrated Tandori theorems
László Leindler
Abstract. In this note we return to some problems of general orthogonal series examined most intensively at the fifties. We show that the monotonicity conditions in four fundamental theorems of K. Tandori can be replaced by weaker assumptions.
AMS Subject Classification (1991): 40A30, 40G05
Keyword(s): orthogonal series, convergence, summability
Received March 2, 2010. (Registered under 16/2010.)
On the convergence of Cesàro means of negative order of double trigonometric Fourier series of functions of bounded partial generalized variation
Ushangi Goginava, Artur Sahakian
Abstract. The convergence of Cesàro means of negative order of double trigonometric Fourier series of functions of bounded partial $\Lambda $-variation is investigated. The sufficient and neccessary conditions on the sequence $\Lambda =\{\lambda_{n}\} $ are found for the convergence of Cesàro means of Fourier series of functions of bounded partial $\Lambda $-variation.
Keyword(s): Fourier series, $\Lambda $-variation, generalized variation, Cesàro means
Received March 12, 2010, and in revised form May 17, 2010. (Registered under 19/2010.)
On the strong logarithmic summability of the double trigonometric Fourier series
Rostom Getsadze
Abstract. We prove the following theorem: {\it Suppose that $E\subset[0,2\pi )^2$ is any Lebesgue measurable set, $\mu_{2}E >0,$ and $\phi(u)$ is a nonnegative, continuous and nondecreasing function on $[0,\infty )$ such that $u\phi(u)$ is a convex function on $[0,\infty )$ and $ \phi(u) = o(\ln u), u \to\infty. $ Then there exists a function $g \in L_1([0,2\pi )^2)$ such that $ \int_{[0,2\pi )^2} | g(x,y) |\phi(| g(x,y) |)dx dy < \infty_{\strut }^{\strut } $ and the sequence of the strong logarithmic means by squares of the double trigonometric Fourier series of $g$, that is, the sequence $ \left\{{1\over\ln N}\sum_{k=1}^N {| S_{k,k}(g;x,y) - g (x,y)| \over k}, N=2,3,\ldots\right \} _{\strut }^{\strut } $ is not bounded in measure on $E$.}
AMS Subject Classification (1991): 42C15, 42C10
Keyword(s): double Fourier series, strong logarithmic means, bounded in measure
Received April 7, 2010. (Registered under 23/2010.)
Commutators of pseudo-differential operators on local Hardy spaces
Yasuo Komori-Furuya
Abstract. We consider the commutator operator $[B, \sigma(x,D)]$ of the multiplication operator by a function $B$ and a pseudo-differential operator $\sigma(x,D)$, and prove that $[B, \sigma(x,D)]$ is bounded on the local Hardy spaces $h^p({\msbm R}^n)$. We also show that our result is optimal.
Keyword(s): pseudo-differential operator, commutator, Hardy space, local Hardy space
Received January 13, 2009, and in revised form April 11, 2011. (Registered under 7/2009.)
Double Hilbert transform in ${\msbm R}^2$
Abstract. Let $P(s,t)$ denote a real-valued polynomial of real variables $s$ and $t$. For $f \in{\cal S}$ (i.e., a Schwartz class function), define the operator $T$ by (1) $ Tf(x,y) = \lim_{\epsilon,\eta\to 0}\int_{\epsilon\le |s| \le1} \int_{\eta\le |t|\le1 }f (x-s, y-P(s,t))_{\strut }^{\strut } {ds dt\over st}. $ We determine a necessary and sufficient condition on $P(s,t)$ so that the operator $T$ is bounded on $L^p({\msbm R}^2)$ for $1 < p < \infty $.
Keyword(s): Newton diagram, multiplier, van der Corput's lemma
Received January 20, 2010, and in revised form September 9, 2010. (Registered under 4/2010.)
On the duality property for semi-compact operators on Banach lattices
Belmesnaoui Aqzzouz, Aziz Elbour
Abstract. We investigate some properties of the class of semi-compact operators on Banach lattices and we give some interesting consequences. In particular we are interested in Banach lattices $E$ and $F$ such that (a) every order bounded operator $T\colon E\rightarrow F$ possessing a semi-compact adjoint is semi-compact as well, (b) every order bounded semi-compact operator $T\colon E\rightarrow F$ has a semi-compact adjoint.
AMS Subject Classification (1991): 46A40, 46B40, 46B42
Keyword(s): semi-compact operator, order continuous norm, discrete Banach lattice
Received March 5, 2010, and in revised form May 10, 2010. (Registered under 18/2010.)
Composition operators on Besov and Dirichlet type spaces of the ball
Matthew A. Pons
Abstract. Various operator theoretic properties of composition operators with linear fractional symbol acting on the Dirichlet space of the unit ball are discussed. Furthermore, we use Calderón's complex interpolation to investigate the spectrum of composition operators with automorphic symbol acting on the analytic Besov spaces of the ball and on the weighted Dirichlet spaces of the ball, which include the Dirichlet, Arveson, Hardy and Bergman spaces.
Keyword(s): composition operator, Dirichlet space, Besov space, spectrum, complex interpolation
Received December 1, 2009. (Registered under 6408/2009.)
A representation of unitary operators in ${\Pi}_{\kappa}$ in terms of isometric operators and orthogonal projections
Muhamed Borogovac
Abstract. Let $\Pi_{\kappa }$ be a Pontryagin space with decomposition $\Pi_{\kappa }=\Pi_{+}(+)\Pi_{-}$, $\kappa =\dim\Pi _{+}< \infty $. Let $U$ be a unitary operator in $\Pi_{\kappa }$ and let $U= \left[{A B\atop C D}\right ]$ be its matrix representation that corresponds to the given decomposition of $\Pi_{\kappa }$. In this note operators $A$, $B$, $C$, and $D$ are given in terms of isometric operators and orthogonal projections in a way that those expressions are necessary and sufficient conditions for the operator $U$ to be unitary. The results are more specific and intuitive than the results from the last chapter of [2]. The obtained representation of $U$ is applied to study operator $T$ that has $\kappa $-dimensional positive invariant subspace $J_{+}$ and allows a J-polar decomposition. The radius of the spectrum $\sigma(T\mid J_{+}) $ is estimated.
AMS Subject Classification (1991): 47B50, 46C20
Keyword(s): Pontryagin space, unitary operator in $\Pi_{\kappa }$, polar decomposition
Received April 16, 2010, and in revised form May 6, 2011. (Registered under 29/2010.)
Compact weighted composition operators on Orlicz--Lorentz spaces
S. C. Arora, Gopal Datt
Abstract. The paper characterizes the continuity and compactness of the weighted composition operators $W_{(u,T)}$ on Orlicz--Lorentz spaces $L_{\varphi,w}$.
AMS Subject Classification (1991): 47B38, 46E30
Keyword(s): distribution function, Lorentz space, Orlicz space, Orlicz--Lorentz space, compact operator, weighted composition operator
Received December 11, 2009, and in revised form March 25, 2011. (Registered under 6415/2009.)
On operators with closed ranges
Zsigmond Tarcsay
Abstract. We characterize those subpositive operators for which their Krein--von Neumann extension has closed range, moreover we construct their Moore--Penrose inverse. Our treatment follows as a tool the factorization approach to the extension theory of positive operators. As addition we give a short proof of Dixmier's theorem that a bounded positive operator $A$ and its square root $A^{1/2}$ have the same range if and only if $A$ has closed range and of Banach's closed range theorem for Hilbert space operators.
AMS Subject Classification (1991): 47A20, 47B65, 47A05
Keyword(s): characterization, positive operator, closed range, Krein--von Neumann extension, Moore--Penrose inverse
Received December 23, 2009, and in revised form February 1, 2011. (Registered under 6468/2009.)
On power bounded operators that are quasiaffine transforms of singular unitaries
Maria F. Gamal'
Abstract. In [9] a question is raised: if a power bounded operator is quasisimilar to a singular unitary operator, is it similar to this unitary operator? For polynomially bounded operators, a positive answer to this question is known [1], [13]. In this paper a positive answer is given in some particular cases, but in general an answer rests unknown.
Keyword(s): power bounded operator, singular unitary operator, similarity, quasisimilarity, quasiaffine transform
Received April 8, 2010, and in revised form October 12, 2010. (Registered under 24/2010.)
On unitary dilations of two-parameter semigroups of contractions and continuous commutant lifting
Ramón Bruzual, Marisela Domínguez, Mayra Montilla
Abstract. It is proved that every semigroup of contractions with parameter on ${\msbm Q}_{+} \times{\msbm Q}_{+}$ or ${\msbm Q}_{+} \times{\msbm N}$ has a unitary dilation. The dilation result about ${\msbm Q}_{+} \times{\msbm Q}_{+}$ is used to obtain a new proof of the Slociński dilation theorem, which says that every strongly continuous semigroup of contractions, with parameter on ${\msbm R}_{+} \times{\msbm R}_{+}$, has a strongly continuous unitary dilation. The result about ${\msbm Q}_{+} \times{\msbm N}$ is used to obtain a new proof of the continuous version of the commutant lifting theorem.
Keyword(s): unitary dilation, semigroup of contractions, commutant lifting
Received November 29, 2009, and in revised form April 13, 2010. (Registered under 6233/2009.)
Composition operators on spaces of Lipschitz functions
Fernanda Botelho, James Jamison
Abstract. We provide a characterization of compact weighted composition operators on spaces of vector-valued Lipschitz functions. We also give estimates of the essential norm of composition operators on these spaces.
Keyword(s): weighted composition operators, essential norm
Absorptive continuous ${\msbm R}$-group actions on locally compact spaces
Gabriel Nguetseng
Abstract. We introduce the notion of an ${\msbm R}$-group of which the classical groups ${\msbm R}$, ${\msbm Z}$ and ${\msbm R}_{+}^{\ast }$ are typical examples, and we study flows $( X,{\cal H}) $, where $X$ is a locally compact space and ${\cal H}$ is a continuous ${\msbm R}$-group action on $X$ with the further property that any compact set is \hbox{\it absorbed }(in the ordinary meaning in use in the theory of topological vector spaces) by any neighbourhood of some characteristic point in $X$ called the center of ${\cal H}$. The case where $X$ is a locally compact abelian group is also considered. We are particularly interested in discussing the asymptotic properties of ${\cal H}$, which is made possible by proving a deep theorem about the existence of nontrivial ${\cal H}$-homogeneous positive measures on $X$. Also, a close connection with homogenization theory is pointed out. It appears that the present paper lays the foundation of the mathematical framework that is needed to undertake a systematic study of homogenization problems on manifolds, Lie groups included.
AMS Subject Classification (1991): 37B05, 43A07, 46J10, 28A25, 28A50, 26E60, 54D45
Keyword(s): locally compact space, group actions
Received May 1, 2010, and in revised form February 8, 2011. (Registered under 36/2010.)
Equality in László Fejes Tóth's triangle bound for hyperbolic surfaces
Christophe Bavard, Károly J. Böröczky, Borbála Farkas, István Prok, Lluis Vena
Abstract. For $k\geq7$, we determine the minimal area of a compact hyperbolic surface, and an oriented compact hyperbolic surface that can be tiled by embedded regular triangles of angle $2\pi /k$. Based on this, all the cases of equality in László Fejes Tóth's triangle bound for hyperbolic surfaces are described.
AMS Subject Classification (1991): 51M10, 52C22
Keyword(s): tiling of hyperbolic surfaces
Received April 14, 2010, and in revised form January 11, 2011. (Registered under 28/2010.)
Asymptotics of nearly critical Galton--Watson processes with immigration
Péter Kevei
Abstract. We investigate the inhomogeneous Galton--Watson processes with immigration, where $\rho_n$, the offspring means in the $n^{\rm th}$ generation, tends to $1$. We show that if the second derivatives of the offspring generating functions go to $0$ rapidly enough, then the asymptotics are the same as in the INAR(1) case, treated in [4]. We also determine the limit if this assumption does not hold showing the optimality of the conditions.
AMS Subject Classification (1991): 60J80
Keyword(s): nearly critical Galton--Watson process, immigration, compound Poisson distribution, negative binomial distribution
Received October 20, 2010, and in revised form July 10, 2011. (Registered under 72/2010.)
Non-parametric bootstrap tests for parametric distribution families
Gábor Szűcs
Abstract. Durbin's estimated empirical process is a widely used tool to testing goodness of fit for parametric distribution families. In general, statistical methods based on the process are not distribution free and the critical values can not always be calculated in a theoretical way. One can avoid these difficulties by applying the parametric or the non-parametric bootstrap procedure. Although the parametric bootstrapped estimated empirical process is well investigated, only a few papers dealt with the non-parametric version. Recently, Babu and Rao pointed out that in the latter case a bias correction is needed, and they proved the weak convergence of the bootstrapped process in continuous distribution families. Our paper presents a weak approximation theorem for the non-parametric bootstrapped estimated empirical process using similar conditions under which Durbin's non-bootstrapped process converges. The result covers the most important continuous and discrete distribution families. Simulation studies in the Poisson and the normal distribution are also reported.
AMS Subject Classification (1991): 62E20, 62F40, 62G30
Keyword(s): bootstrap, parametric estimation, empirical process, approximation, convergence in distribution
Received December 30, 2010, and in revised form April 7, 2011. (Registered under 90/2010.) | CommonCrawl |
Ruziewicz problem
In mathematics, the Ruziewicz problem (sometimes Banach–Ruziewicz problem) in measure theory asks whether the usual Lebesgue measure on the n-sphere is characterised, up to proportionality, by its properties of being finitely additive, invariant under rotations, and defined on all Lebesgue measurable sets.
This was answered affirmatively and independently for n ≥ 4 by Grigory Margulis and Dennis Sullivan around 1980, and for n = 2 and 3 by Vladimir Drinfeld (published 1984). It fails for the circle.
The problem is named after Stanisław Ruziewicz.
References
• Lubotzky, Alexander (1994), Discrete groups, expanding graphs and invariant measures, Progress in Mathematics, vol. 125, Basel: Birkhäuser Verlag, ISBN 0-8176-5075-X.
• Drinfeld, Vladimir (1984), "Finitely-additive measures on S2 and S3, invariant with respect to rotations", Funktsional. Anal. i Prilozhen., 18 (3): 77, MR 0757256.
• Margulis, Grigory (1980), "Some remarks on invariant means", Monatshefte für Mathematik, 90 (3): 233–235, doi:10.1007/BF01295368, MR 0596890.
• Sullivan, Dennis (1981), "For n > 3 there is only one finitely additive rotationally invariant measure on the n-sphere on all Lebesgue measurable sets", Bulletin of the American Mathematical Society, 4 (1): 121–123, doi:10.1090/S0273-0979-1981-14880-1, MR 0590825.
• Survey of the area by Hee Oh
| Wikipedia |
Chronology of computation of π
The table below is a brief chronology of computed numerical values of, or bounds on, the mathematical constant pi (π). For more detailed explanations for some of these calculations, see Approximations of π.
Part of a series of articles on the
mathematical constant π
3.1415926535897932384626433...
Uses
• Area of a circle
• Circumference
• Use in other formulae
Properties
• Irrationality
• Transcendence
Value
• Less than 22/7
• Approximations
• Madhava's correction term
• Memorization
People
• Archimedes
• Liu Hui
• Zu Chongzhi
• Aryabhata
• Madhava
• Jamshīd al-Kāshī
• Ludolph van Ceulen
• François Viète
• Seki Takakazu
• Takebe Kenko
• William Jones
• John Machin
• William Shanks
• Srinivasa Ramanujan
• John Wrench
• Chudnovsky brothers
• Yasumasa Kanada
History
• Chronology
• A History of Pi
In culture
• Indiana Pi Bill
• Pi Day
Related topics
• Squaring the circle
• Basel problem
• Six nines in π
• Other topics related to π
The last 100 decimal digits[1] of the latest 2022 world record computation are:[2]
4658718895 1242883556 4671544483 9873493812 1206904813 2656719174 5255431487 2142102057 7077336434 3095295560
Before 1400
Date Who Description/Computation method used Value Decimal places
(world records
in bold)
2000? BCЕAncient Egyptians[3] 4 × (8⁄9)2 3.1605...1
2000? BCЕAncient Babylonians[3] 3 + 1⁄8 3.1251
2000? BCЕ Ancient Sumerians[4] 3 + 23/216 3.1065 1
1200? BCЕAncient Chinese[3] 3 30
800–600 BCEShatapatha Brahmana – 7.1.1.18 [5] Instructions on how to construct a circular altar from oblong bricks:
"He puts on (the circular site) four (bricks) running eastwards 1; two behind running crosswise (from south to north), and two (such) in front. Now the four which he puts on running eastwards are the body; and as to there being four of these, it is because this body (of ours) consists, of four parts 2. The two at the back then are the thighs; and the two in front the arms; and where the body is that (includes) the head."[6]
25⁄8 = 3.1251
800? BCЕShulba Sutras[7]
[8][9]
(6⁄(2 + √2))2 3.088311 ...0
550? BCЕBible (1 Kings 7:23)[3] "...a molten sea, ten cubits from the one brim to the other: it was round all about,... a line of thirty cubits did compass it round about"30
434 BCEAnaxagoras attempted to square the circle[10] compass and straightedgeAnaxagoras didn't offer any solution0
400 BCE to 400 CEVyasa[11]
verses: 6.12.40-45 of the Bhishma Parva of the Mahabharata offer:
"...
The Moon is handed down by memory to be eleven thousand yojanas in diameter. Its peripheral circle happens to be thirty three thousand yojanas when calculated.
...
The Sun is eight thousand yojanas and another two thousand yojanas in diameter. From that its peripheral circle comes to be equal to thirty thousand yojanas.
..."
30
c. 250 BCEArchimedes[3] 223⁄71 < π < 22⁄7 3.140845... < π < 3.142857...2
15 BCEVitruvius[8] 25⁄8 3.1251
between 1 and 5Liu Xin[8][12][13] Unknown method giving a figure for a jialiang which implies a value for π π ≈ 162⁄(√50+0.095)2. 3.1547...1
130Zhang Heng (Book of the Later Han)[3] √10 = 3.162277...
736⁄232
3.1622...1
150Ptolemy[3] 377⁄120 3.141666...3
250Wang Fan[3] 142⁄45 3.155555...1
263Liu Hui[3] 3.141024 < π < 3.142074
3927⁄1250
3.14163
400He Chengtian[8] 111035⁄35329 3.142885...2
480Zu Chongzhi[3] 3.1415926 < π < 3.1415927
355⁄113
3.14159267
499Aryabhata[3] 62832⁄20000 3.14163
640Brahmagupta[3] √10 3.162277...1
800Al Khwarizmi[3] 3.14163
1150Bhāskara II[8] 3927⁄1250 and 754⁄240 3.14163
1220Fibonacci[3] 3.1418183
1320Zhao Youqin[8] 3.1415926
1400–1949
Date Who Note Decimal places
(world records in bold)
All records from 1400 onwards are given as the number of correct decimal places.
1400Madhava of Sangamagrama Discovered the infinite power series expansion of π,
now known as the Leibniz formula for pi[14]
10
1424Jamshīd al-Kāshī[15] 16
1573Valentinus Otho 355⁄113 6
1579François Viète[16] 9
1593Adriaan van Roomen[17] 15
1596Ludolph van Ceulen 20
1615 32
1621Willebrord Snell (Snellius) Pupil of Van Ceulen 35
1630Christoph Grienberger[18][19] 38
1654 Christiaan Huygens Used a geometrical method equivalent to Richardson extrapolation 10
1665Isaac Newton[3] 16
1681Takakazu Seki[20] 11
16
1699Abraham Sharp[3] Calculated pi to 72 digits, but not all were correct 71
1706John Machin[3] 100
1706William Jones Introduced the Greek letter 'π'
1719Thomas Fantet de Lagny[3] Calculated 127 decimal places, but not all were correct 112
1722Toshikiyo Kamata 24
1722Katahiro Takebe 41
1739Yoshisuke Matsunaga 51
1748Leonhard Euler Used the Greek letter 'π' in his book Introductio in Analysin Infinitorum and assured its popularity.
1761Johann Heinrich Lambert Proved that π is irrational
1775Euler Pointed out the possibility that π might be transcendental
1789Jurij Vega[21] Calculated 140 decimal places, but not all were correct 126
1794Adrien-Marie Legendre Showed that π2 (and hence π) is irrational, and mentioned the possibility that π might be transcendental.
Late 18th centuryAnonymous manuscript Turns up at Radcliffe Library, in Oxford, England, discovered by F. X. von Zach, giving the value of pi to 154 digits, 152 of which were correct[22] 152
1824William Rutherford[3] Calculated 208 decimal places, but not all were correct 152
1844Zacharias Dase and Strassnitzky[3] Calculated 205 decimal places, but not all were correct 200
1847Thomas Clausen[3] Calculated 250 decimal places, but not all were correct 248
1853Lehmann[3] 261
1853Rutherford[3] 440
1853William Shanks[23] Expanded his calculation to 707 decimal places in 1873, but an error introduced at the beginning of his new calculation rendered all of the subsequent digits incorrect (the error was found by D. F. Ferguson in 1946). 527
1882Ferdinand von Lindemann Proved that π is transcendental (the Lindemann–Weierstrass theorem)
1897The U.S. state of Indiana Came close to legislating the value 3.2 (among others) for π. House Bill No. 246 passed unanimously. The bill stalled in the state Senate due to a suggestion of possible commercial motives involving publication of a textbook.[24] 0
1910Srinivasa Ramanujan Found several rapidly converging infinite series of π, which can compute 8 decimal places of π with each term in the series. Since the 1980s, his series have become the basis for the fastest algorithms currently used by Yasumasa Kanada and the Chudnovsky brothers to compute π.
1946 D. F. Ferguson Most digits ever calculated by hand. 620
1947 Ivan Niven Gave a very elementary proof that π is irrational
January 1947 D. F. Ferguson Made use of a desk calculator 710
September 1947 D. F. Ferguson Desk calculator 808
1949 Levi B. Smith and John Wrench Made use of a desk calculator 1,120
1949–2009
Date Who Implementation Time Decimal places
(world records in bold)
All records from 1949 onwards were calculated with electronic computers.
1949 G. W. Reitwiesner et al. The first to use an electronic computer (the ENIAC) to calculate π [25] 70 hours 2,037
1953Kurt Mahler Showed that π is not a Liouville number
1954 S. C. Nicholson & J. Jeenel Using the NORC[26] 13 minutes 3,093
1957 George E. Felton Ferranti Pegasus computer (London), calculated 10,021 digits, but not all were correct[27][28] 33 hours 7,480
January 1958 Francois Genuys IBM 704[29] 1.7 hours 10,000
May 1958 George E. Felton Pegasus computer (London) 33 hours 10,021
1959 Francois Genuys IBM 704 (Paris)[30] 4.3 hours 16,167
1961 Daniel Shanks and John Wrench IBM 7090 (New York)[31] 8.7 hours 100,265
1961 J.M. Gerard IBM 7090 (London) 39 minutes 20,000
February 1966 Jean Guilloud and J. Filliatre IBM 7030 (Paris)[28] 41.92 hours 250,000
1967 Jean Guilloud and M. Dichampt CDC 6600 (Paris) 28 hours 500,000
1973 Jean Guilloud and Martine Bouyer CDC 7600 23.3 hours 1,001,250
1981 Kazunori Miyoshi and Yasumasa Kanada FACOM M-200[28] 137.3 hours 2,000,036
1981 Jean Guilloud Not known 2,000,050
1982 Yoshiaki Tamura MELCOM 900II[28] 7.23 hours 2,097,144
1982 Yoshiaki Tamura and Yasumasa Kanada HITAC M-280H[28] 2.9 hours 4,194,288
1982 Yoshiaki Tamura and Yasumasa Kanada HITAC M-280H[28] 6.86 hours 8,388,576
1983 Yasumasa Kanada, Sayaka Yoshino and Yoshiaki Tamura HITAC M-280H[28] <30 hours 16,777,206
October 1983 Yasunori Ushiro and Yasumasa Kanada HITAC S-810/20 10,013,395
October 1985 Bill Gosper Symbolics 3670 17,526,200
January 1986 David H. Bailey CRAY-2[28] 28 hours 29,360,111
September 1986 Yasumasa Kanada, Yoshiaki Tamura HITAC S-810/20[28] 6.6 hours 33,554,414
October 1986 Yasumasa Kanada, Yoshiaki Tamura HITAC S-810/20[28] 23 hours 67,108,839
January 1987 Yasumasa Kanada, Yoshiaki Tamura, Yoshinobu Kubo and others NEC SX-2[28] 35.25 hours 134,214,700
January 1988 Yasumasa Kanada and Yoshiaki Tamura HITAC S-820/80[32] 5.95 hours 201,326,551
May 1989 Gregory V. Chudnovsky & David V. Chudnovsky CRAY-2 & IBM 3090/VF 480,000,000
June 1989 Gregory V. Chudnovsky & David V. Chudnovsky IBM 3090 535,339,270
July 1989 Yasumasa Kanada and Yoshiaki Tamura HITAC S-820/80 536,870,898
August 1989 Gregory V. Chudnovsky & David V. Chudnovsky IBM 3090 1,011,196,691
19 November 1989 Yasumasa Kanada and Yoshiaki Tamura HITAC S-820/80[33] 1,073,740,799
August 1991 Gregory V. Chudnovsky & David V. Chudnovsky Homemade parallel computer (details unknown, not verified) [34][33] 2,260,000,000
18 May 1994 Gregory V. Chudnovsky & David V. Chudnovsky New homemade parallel computer (details unknown, not verified) 4,044,000,000
26 June 1995 Yasumasa Kanada and Daisuke Takahashi HITAC S-3800/480 (dual CPU) [35] 3,221,220,000
1995 Simon Plouffe Finds a formula that allows the nth hexadecimal digit of pi to be calculated without calculating the preceding digits.
28 August 1995 Yasumasa Kanada and Daisuke Takahashi HITAC S-3800/480 (dual CPU) [36][37] 56.74 hours? 4,294,960,000
11 October 1995 Yasumasa Kanada and Daisuke Takahashi HITAC S-3800/480 (dual CPU) [38][37] 116.63 hours 6,442,450,000
6 July 1997 Yasumasa Kanada and Daisuke Takahashi HITACHI SR2201 (1024 CPU) [39][40] 29.05 hours 51,539,600,000
5 April 1999 Yasumasa Kanada and Daisuke Takahashi HITACHI SR8000 (64 of 128 nodes) [41][42] 32.9 hours 68,719,470,000
20 September 1999 Yasumasa Kanada and Daisuke Takahashi HITACHI SR8000/MPP (128 nodes) [43][44] 37.35 hours 206,158,430,000
24 November 2002 Yasumasa Kanada & 9 man team HITACHI SR8000/MPP (64 nodes), Department of Information Science at the University of Tokyo in Tokyo, Japan[45] 600 hours 1,241,100,000,000
29 April 2009 Daisuke Takahashi et al. T2K Open Supercomputer (640 nodes), single node speed is 147.2 gigaflops, computer memory is 13.5 terabytes, Gauss–Legendre algorithm, Center for Computational Sciences at the University of Tsukuba in Tsukuba, Japan[46] 29.09 hours 2,576,980,377,524
2009–present
Date Who Implementation Time Decimal places
(world records in bold)
All records from Dec 2009 onwards are calculated and verified on servers and/or home computers with commercially available parts.
31 December 2009 Fabrice Bellard
• Core i7 CPU at 2.93 GHz
• 6 GiB (1) of RAM
• 7.5 TB of disk storage using five 1.5 TB hard disks (Seagate Barracuda 7200.11 model)
• 64 bit Red Hat Fedora 10 distribution
• Computation of the binary digits: 103 days
• Verification of the binary digits: 13 days
• Conversion to base 10: 12 days
• Verification of the conversion: 3 days
• Verification of the binary digits used a network of 9 Desktop PCs during 34 hours, Chudnovsky algorithm, see [47] for Bellard's homepage.[48]
131 days 2,699,999,990,000
2 August 2010 Shigeru Kondo[49]
• using y-cruncher[50] by Alexander Yee
• the Chudnovsky algorithm was used for main computation
• verification used the Bellard & BBP (Plouffe) formulas on different computers; both computed 32 hexadecimal digits ending with the 4,152,410,118,610th.
• with 2× Intel Xeon X5680 @ 3.33 GHz – (12 physical cores, 24 hyperthreaded)
• 96 GiB DDR3 @ 1066 MHz – (12× 8 GiB – 6 channels) – Samsung (M393B1K70BH1)
• 1 TB SATA II (Boot drive) – Hitachi (HDS721010CLA332), 3× 2 TB SATA II (Store Pi Output) – Seagate (ST32000542AS) 16× 2 TB SATA II (Computation) – Seagate (ST32000641AS)
• Windows Server 2008 R2 Enterprise x64
• Computation of binary digits: 80 days
• Conversion to base 10: 8.2 days
• Verification of the conversion: 45.6 hours
• Verification of the binary digits: 64 hours (primary), 66 hours (secondary)
• Verification of the binary digits were done simultaneously on two separate computers during the main computation.[51]
90 days 5,000,000,000,000
17 October 2011 Shigeru Kondo[52]
• using y-cruncher by Alexander Yee
• the Chudnovsky algorithm was used for main computation
• Verification using the Bellard & BBP (Plouffe) formulas: 1.86 days and 4.94 days
371 days 10,000,000,000,050
28 December 2013 Shigeru Kondo[53]
• using y-cruncher by Alexander Yee
• with 2× Intel Xeon E5-2690 @ 2.9 GHz – (16 physical cores, 32 hyperthreaded)
• 128 GiB DDR3 @ 1600 MHz – 8× 16 GiB – 8 channels
• Windows Server 2012 x64
• the Chudnovsky algorithm was used for main computation
• Verification using Bellard's variant of the BBP formula: 46 hours
94 days 12,100,000,000,050
8 October 2014 Sandon Nash Van Ness "houkouonchi"[54]
• using y-cruncher by Alexander Yee
• with 2× Xeon E5-4650L @ 2.6 GHz
• 192 GiB DDR3 @ 1333 MHz
• 24× 4 TB + 30× 3 TB
• the Chudnovsky algorithm was used for main computation
• Verification using the BBP formula: 182 hours
208 days 13,300,000,000,000
11 November 2016 Peter Trueb[55][56]
• using y-cruncher by Alexander Yee
• with 4× Xeon E7-8890 v3 @ 2.50 GHz (72 cores, 144 threads)
• 1.25 TiB DDR4
• 20× 6 TB
• the Chudnovsky algorithm was used for main computation
• Verification using Bellard's variant of the BBP formula: 28 hours[57]
105 days 22,459,157,718,361
= ⌊πe × 1012⌋
14 March 2019 Emma Haruka Iwao[58]
• using y-cruncher v0.7.6
• Computation: 1× n1-megamem-96 (96 vCPU, 1.4TB) with 30TB of SSD
• Storage: 24× n1-standard-16 (16 vCPU, 60GB) with 10TB of SSD
• the Chudnovsky algorithm was used for main computation
• Verification: 20 hours using Bellard's 7-term BBP formula, and 28 hours using Plouffe's 4-term BBP formula
121 days 31,415,926,535,897
= ⌊π × 1013⌋
29 January 2020 Timothy Mullican[59][60]
• using y-cruncher v0.7.7
• Computation: 4x Intel Xeon CPU E7-4880 v2 @ 2.50 GHz
• 320GB DDR3 PC3-8500R ECC RAM
• 48× 6TB HDDs (Computation) + 47× LTO Ultrium 5 1.5TB Tapes (Checkpoint Backups) + 12× 4TB HDDs (Digit Storage)
• the Chudnovsky algorithm was used for main computation
• Verification: 17 hours using Bellard's 7-term BBP formula, 24 hours using Plouffe's 4-term BBP formula
303 days 50,000,000,000,000
14 August 2021 Team DAViS of the University of Applied Sciences of the Grisons[61][62]
• using y-cruncher v0.7.8
• Computation: AMD Epyc 7542 @ 2.9 GHz
• 1 TiB of memory
• 38x 16 TB HDDs (Of those, 34 are used for swapping and 4 used for storage)
• the Chudnovsky algorithm was used for main computation
• Verification: 34 hours using Bellard's 4-term BBP formula
108 days 62,831,853,071,796
= ⌊2π × 1013⌋
21 March 2022 Emma Haruka Iwao[63][64]
• using y-cruncher v0.7.8
• Computation: n2-highmem-128 (128 vCPU and 864 GB RAM)
• Storage: 663 TB
• the Chudnovsky algorithm was used for main computation
• Verification: 12.6 hours using BBP formula
158 days 100,000,000,000,000
See also
• History of pi
• Approximations of π
References
1. The last digit shown here is the 100,000,000,000,000th digit of π.
2. "Validation File". Retrieved 2022-06-09.{{cite web}}: CS1 maint: url-status (link)
3. David H. Bailey; Jonathan M. Borwein; Peter B. Borwein; Simon Plouffe (1997). "The quest for pi" (PDF). Mathematical Intelligencer. 19 (1): 50–57. doi:10.1007/BF03024340. S2CID 14318695.
4. "Origins: 3.14159265…". Biblical Archaeology Society. 2022-03-14. Retrieved 2022-06-08.
5. Eggeling, Julius (1882–1900). The Satapatha-brahmana, according to the text of the Madhyandina school. Princeton Theological Seminary Library. Oxford, The Clarendon Press. pp. 302–303.{{cite book}}: CS1 maint: date and year (link)
6. The Sacred Books of the East: The Satapatha-Brahmana, pt. 3. Clarendon Press. 1894. p. 303. This article incorporates text from this source, which is in the public domain.
7. "4 II. Sulba Sutras". www-history.mcs.st-and.ac.uk.
8. Ravi P. Agarwal; Hans Agarwal; Syamal K. Sen (2013). "Birth, growth and computation of pi to ten trillion digits". Advances in Difference Equations. 2013: 100. doi:10.1186/1687-1847-2013-100.
9. Plofker, Kim (2009). Mathematics in India. Princeton University Press. p. 18. ISBN 978-0691120676.
10. https://www.math.rutgers.edu/~cherlin/History/Papers2000/wilson.html
11. Jadhav, Dipak (2018-01-01). "On The Value Implied In The Data Referred To In The Mahābhārata for π". Vidyottama Sanatana: International Journal of Hindu Science and Religious Studies. 2 (1): 18. doi:10.25078/ijhsrs.v2i1.511. ISSN 2550-0651. S2CID 146074061.
12. 趙良五 (1991). 中西數學史的比較. 臺灣商務印書館. ISBN 978-9570502688 – via Google Books.
13. Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd. Volume 3, 100.
14. Bag, A. K. (1980). "Indian Literature on Mathematics During 1400–1800 A.D." (PDF). Indian Journal of History of Science. 15 (1): 86. π ≈ 2,827,433,388,233/9×10−11 = 3.14159 26535 92222..., good to 10 decimal places.
15. approximated 2π to 9 sexagesimal digits. Al-Kashi, author: Adolf P. Youschkevitch, chief editor: Boris A. Rosenfeld, p. 256 O'Connor, John J.; Robertson, Edmund F., "Ghiyath al-Din Jamshid Mas'ud al-Kashi", MacTutor History of Mathematics Archive, University of St Andrews Azarian, Mohammad K. (2010). "Al-Risāla Al-Muhītīyya: A Summary". Missouri Journal of Mathematical Sciences. 22 (2): 64–85. doi:10.35834/mjms/1312233136.
16. Viète, François (1579). Canon mathematicus seu ad triangula : cum adpendicibus (in Latin).
17. Romanus, Adrianus (1593). Ideae mathematicae pars prima, sive methodus polygonorum (in Latin). apud Ioannem Keerbergium. hdl:2027/ucm.5320258006.
18. Grienbergerus, Christophorus (1630). Elementa Trigonometrica (PDF) (in Latin). Archived from the original (PDF) on 2014-02-01.
19. Hobson, Ernest William (1913). 'Squaring the Circle': a History of the Problem (PDF). Cambridge University Press. p. 27.
20. Yoshio, Mikami; Eugene Smith, David (2004) [1914]. A History of Japanese Mathematics (paperback ed.). Dover Publications. ISBN 0-486-43482-6.
21. Vega, Géorge (1795) [1789]. "Detérmination de la demi-circonférence d'un cercle dont le diameter est = 1, exprimée en 140 figures decimals". Supplement. Nova Acta Academiae Scientiarum Petropolitanae. 11: 41–44.
Sandifer, Ed (2006). "Why 140 Digits of Pi Matter" (PDF). Southern Connecticut State University. Archived from the original (PDF) on 2012-02-04.
22. Benjamin Wardhaugh, "Filling a Gap in the History of π: An Exciting Discovery", Mathematical Intelligencer 38(1) (2016), 6-7
23. Hayes, Brian (September 2014). "Pencil, Paper, and Pi". American Scientist. Vol. 102, no. 5. p. 342. doi:10.1511/2014.110.342. Retrieved 13 February 2022.
24. Lopez-Ortiz, Alex (February 20, 1998). "Indiana Bill sets value of Pi to 3". the news.answers WWW archive. Department of Information and Computing Sciences, Utrecht University. Archived from the original on 2005-01-09. Retrieved 2009-02-01.
25. Reitwiesner, G. (1950). "An ENIAC determination of π and e to more than 2000 decimal places". MTAC. 4: 11–15. doi:10.1090/S0025-5718-1950-0037597-6.
26. Nicholson, S. C.; Jeenel, J. (1955). "Some comments on a NORC computation of π". MTAC. 9: 162–164. doi:10.1090/S0025-5718-1955-0075672-5.
27. G. E. Felton, "Electronic computers and mathematicians," Abbreviated Proceedings of the Oxford Mathematical Conference for Schoolteachers and Industrialists at Trinity College, Oxford, April 8–18, 1957, pp. 12–17, footnote pp. 12–53. This published result is correct to only 7480D, as was established by Felton in a second calculation, using formula (5), completed in 1958 but apparently unpublished. For a detailed account of calculations of π see Wrench, J. W. Jr. (1960). "The evolution of extended decimal approximations to π". The Mathematics Teacher. 53 (8): 644–650. doi:10.5951/MT.53.8.0644. JSTOR 27956272.
28. Arndt, Jörg; Haenel, Christoph (2001). Pi - Unleashed. Springer. ISBN 978-3-642-56735-3.
29. Genuys, F. (1958). "Dix milles decimales de π". Chiffres. 1: 17–22.
30. This unpublished value of x to 16167D was computed on an IBM 704 system at the French Alternative Energies and Atomic Energy Commission in Paris, by means of the program of Genuys
31. Shanks, Daniel; Wrench, John W. J.r (1962). "Calculation of π to 100,000 decimals". Mathematics of Computation. 16 (77): 76–99. doi:10.1090/S0025-5718-1962-0136051-9.
32. Kanada, Y. (November 1988). "Vectorization of multiple-precision arithmetic program and 201,326,000 decimal digits of pi calculation". Proceedings Supercomputing Vol.II: Science and Applications. pp. 117–128 vol.2. doi:10.1109/SUPERC.1988.74139. ISBN 0-8186-8923-4. S2CID 122820709.
33. "Computers". Science News. 24 August 1991. Retrieved 2022-08-04.
34. Bigger slices of Pi (determination of the numerical value of pi reaches 2.16 billion decimal digits) Science News 24 August 1991 http://www.encyclopedia.com/doc/1G1-11235156.html
35. ftp://pi.super-computing.org/README.our_last_record_3b%5B%5D
36. ftp://pi.super-computing.org/README.our_last_record_4b%5B%5D
37. "GENERAL COMPUTATIONAL UPDATE". www.cecm.sfu.ca. Retrieved 2022-08-04.
38. ftp://pi.super-computing.org/README.our_last_record_6b
39. ftp://pi.super-computing.org/README.our_last_record_51b%5B%5D
40. "Record for pi : 51.5 billion decimal digits". 2005-12-24. Archived from the original on 2005-12-24. Retrieved 2022-08-04.
41. ftp://pi.super-computing.org/README.our_last_record_68b%5B%5D
42. https://www.plouffe.fr/simon/constants/Pi68billion.txt
43. ftp://pi.super-computing.org/README.our_latest_record_206b%5B%5D
44. "Record for pi : 206 billion decimal digits". www.cecm.sfu.ca. Retrieved 2022-08-04.
45. "Archived copy". Archived from the original on 2011-03-12. Retrieved 2010-07-08.{{cite web}}: CS1 maint: archived copy as title (link)
46. "Archived copy". Archived from the original on 2009-08-23. Retrieved 2009-08-18.{{cite web}}: CS1 maint: archived copy as title (link)
47. "Fabrice Bellard's Home Page". bellard.org. Retrieved 28 August 2015.
48. http://bellard.org/pi/pi2700e9/pipcrecord.pdf
49. "PI-world". calico.jp. Archived from the original on 31 August 2015. Retrieved 28 August 2015.
50. "y-cruncher – A Multi-Threaded Pi Program". numberworld.org. Retrieved 28 August 2015.
51. "Pi – 5 Trillion Digits". numberworld.org. Retrieved 28 August 2015.
52. "Pi – 10 Trillion Digits". numberworld.org. Retrieved 28 August 2015.
53. "Pi – 12.1 Trillion Digits". numberworld.org. Retrieved 28 August 2015.
54. "y-cruncher – A Multi-Threaded Pi Program". numberworld.org. Retrieved 14 March 2018.
55. "pi2e". pi2e.ch. Retrieved 15 November 2016.
56. Alexander J. Yee. "y-cruncher – A Multi-Threaded Pi Program". numberworld.org. Retrieved 15 November 2016.
57. "Hexadecimal Digits are Correct! – pi2e trillion digits of pi". pi2e.ch. 31 October 2016. Retrieved 15 November 2016.
58. "Google Cloud Topples the Pi Record". Retrieved 14 March 2019.
59. "The Pi Record Returns to the Personal Computer". Retrieved 30 January 2020.
60. "Calculating Pi: My attempt at breaking the Pi World Record". 26 June 2019. Retrieved 30 January 2020.
61. "Pi-Challenge - world record attempt by UAS Grisons - University of Applied Sciences of the Grisons". www.fhgr.ch. 2021-08-14. Archived from the original on 2021-08-17. Retrieved 2021-08-17.
62. "Die FH Graubünden kennt Pi am genauesten – Weltrekord! - News - FH Graubünden". www.fhgr.ch (in German). 2021-08-16. Archived from the original on 2021-08-17. Retrieved 2021-08-17.
63. "Calculating 100 trillion digits of pi on Google Cloud". Google Cloud Blog. Retrieved 2022-06-10.
64. "News (2019)". numberworld.org. Retrieved 2022-06-10.
External links
• Borwein, Jonathan, "The Life of Pi"
• Kanada Laboratory home page
• Stu's Pi page
• Takahashi's page
| Wikipedia |
Artificial Intelligence Stack Exchange is a question and answer site for people interested in conceptual questions about life and challenges in a world where "cognitive" functions can be mimicked in purely digital environment. It only takes a minute to sign up.
What is a "surrogate model"?
In the following paragraph from the book Automated Machine Learning: Methods, Systems, Challenges (by Frank Hutter et al.)
In this section we first give a brief introduction to Bayesian optimization, present alternative surrogate models used in it, describe extensions to conditional and constrained configuration spaces, and then discuss several important applications to hyperparameter optimization.
What is an "alternative surrogate model"? What exactly does "alternative" mean?
terminology definitions hyperparameter-optimization bayesian-optimization surrogate-model
yousef yeganeyousef yegane
What is Bayesian optimization?
Bayesian optimization (BO) is an optimization technique used to model an unknown (usually continuous) function $f: \mathbb{R}^d \rightarrow Y$, where typically $d \leq 20$, so it can be used to solve regression and classification problems, where you want to find an approximation of $f$. In this sense, BO is similar to the usual approach of training a neural network with gradient descent combined with the back-propagation algorithm, so that to optimize an objective function. However, BO is particularly suited for regression or classification problems where the unknown function $f$ is expensive to evaluate (that is, given the input $\mathbf{x} \in \mathbb{R}^d$, the computation of $f(x) \in Y$ takes a lot of time or, in general, resources). For example, when doing hyper-parameter tuning, we usually need to first train the model with the new hyper-parameters before evaluating the specific configuration of hyper-parameters, but this usually takes a lot of time (hours, days or even months), especially when you are training deep neural networks with big datasets. Moreover, BO does not involve the computation of gradients and it usually assumes that $f$ lacks properties such as concavity or linearity.
How does Bayesian optimization work?
There are three main concepts in BO
the surrogate model, which models an unknown function,
a method for statistical inference, which is used to update the surrogate model, and
the acquisition function, which is used to guide the statistical inference and thus it is used to update the surrogate model
The surrogate model is usually a Gaussian process, which is just a fancy name to denote a collection of random variables such that the joint distribution of those random variables is a multivariate Gaussian probability distribution (hence the name Gaussian process). Therefore, in BO, we often use a Gaussian probability distribution (the surrogate model) to model the possible functions that are consistent with the data. In other words, given that we do not know $f$, rather than finding the usual point estimate (or maximum likelihood estimate), like in the usual case of supervised learning mentioned above, we maintain a Gaussian probability distribution that describes our uncertainty about the unknown $f$.
The method of statistical inference is often just an iterative application of the Bayes rule (hence the name Bayesian optimization), where you want to find the posterior, given a prior, a likelihood and the evidence. In BO, you usually place a prior on $f$, which is a multivariate Gaussian distribution, then you use the Bayes rule to find the posterior distribution of $f$ given the data.
What is the data in this case? In BO, the data are the outputs of $f$ evaluated at certain points of the domain of $f$. The acquisition function is used to choose these points of the domain of $f$, based on the computed posterior distribution. In other words, based on the current uncertainty about $f$ (the posterior), the acquisition function attempts to cleverly choose points of the domain of $f$, $\mathbf{x} \in \mathbb{R}^d$, which will be used to find an updated posterior. Why do we need the acquisition function? Why can't we simply evaluate $f$ at random domain points? Given that $f$ is expensive to evaluate, we need a clever way to choose the points where we want to evaluate $f$. More specifically, we want to evaluate $f$ where we are more uncertain about it.
There are several acquisition functions, such as expected improvement, knowledge-gradient, entropy search, and predictive entropy search, so there are different ways of choosing the points of the domain of $f$ where we want to evaluate it to update the posterior, each of which deals with the exploration-exploitation dilemma differently.
What can Bayesian optimization be used for?
BO can be used for tuning hyper-parameters (also called hyper-parameter optimisation) of machine learning models, such as neural networks, but it has also been used to solve other problems.
What is an alternative surrogate model?
In the book Automated Machine Learning: Methods, Systems, Challenges (by Frank Hutter et al.) that you are quoting, the authors say that the commonly used surrogate model Gaussian process scales cubically in the number of data points, so sparse Gaussian processes are often used. Moreover, Gaussian processes also scale badly with the number of dimensions. In section 1.3.2.2., the authors describe some alternative surrogate models to the Gaussian processes, for example, alternatives that use neural networks or random forests.
nbro♦nbro
A surrogate model is a simplified model. It is a mapping $y_S=f_S(x)$ that approximates the original model $y=f(x)$, in a given domain, reasonably well. Source: Engineering Design via Surrogate Modelling: A Practical Guide
In the context of Bayesian optimization, one wants to optimize a function $y=f(x)$ which is expensive (very time consuming) to evaluate, therefore one optimizes the surrogate model $y_S=f_S(x)$ which is cheaper (faster) to evaluate.
Javier-AcunaJavier-Acuna
$\begingroup$ Forgive my ignorance, but why exactly is yS=fS(x) faster to evaluate? $\endgroup$
– Goose
$\begingroup$ Imagine that the original model is computed from Finite Element simulations (x would be some geometric parameter or material constant for instance and f(x) some quantity of interest) and f_S is a polynomial approximation like a0 + a1x + a2x^2. f(x) can take some hours to evaluate whereas f_S(x) can be calculated pretty fast $\endgroup$
– Javier-Acuna
Recently, I've been thinking this question as well. After reading several papers, finally came up with some thoughts about the surrogate model. In FEM(finite element method), we try to find a weak form to approximate the strong form so that we can solve the weak form analytically. (weak form: approximation equation; strong form: PDE in real world) In my opinion, the surrogate model can be regarded as 'weak form'. There are many methods can form a surrogate model. And if we use a NN model as the surrogate model, the training process is equivalent to 'solving analytically'.
T.C. LiuT.C. Liu
Not the answer you're looking for? Browse other questions tagged terminology definitions hyperparameter-optimization bayesian-optimization surrogate-model or ask your own question.
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What exactly is a Parzen? | CommonCrawl |
Bill Foster (politician)
Assumed office
Judy Biggert (Redistricting)
March 8, 2008 – January 3, 2011
Dennis Hastert
George William Foster
(1955-10-07) October 7, 1955 (age 65)
Madison, Wisconsin, U.S.
Ann (Divorced 1996)
Aesook Byon
(m. 2008)
University of Wisconsin–Madison (BS)
Harvard University (MS, PhD)
House website
George William Foster (born October 7, 1955) is an American businessman, physicist, and U.S. Representative for Illinois's 11th congressional district, winning the seat in 2012.[1] He was previously the U.S. Representative for Illinois's 14th congressional district from 2008 to 2011. He is a member of the Democratic Party.
1 Early life, education, and business career
2 Physics career
3 U.S. House of Representatives
3.1 Elections
3.2 Tenure
3.3 Committee assignments
3.4 Caucus memberships
4 Electoral history
Early life, education, and business career
Foster was born in 1955 in Madison, Wisconsin. As a teenager, he attended James Madison Memorial High School He received his bachelor's degree in physics from the University of Wisconsin–Madison in 1976 and his Ph.D. in physics from Harvard University in 1983.[2] The title of his doctoral dissertation is "An experimental limit on proton decay: p → p o s i t r o n + π 0 {\displaystyle p\rightarrow \mathrm {positron} +\pi ^{0}} ."[3]
At age 19, Foster and his younger brother Fred started a business in their basement with $500 from their parents. The company, Electronic Theatre Controls (ETC), now has over 650 employees worldwide and manufactures over half of the theater lighting equipment in the United States. Installations include Broadway shows, Rolling Stones tours, opera houses, Super Bowl halftime shows, and at schools, churches, and community centers around the world.[4]
Physics career
After completing his Ph.D., Foster moved to the Fox Valley with his family to pursue a career in high-energy (particle) physics at Fermilab, a Department of Energy National Laboratory. During Foster's 22 years at Fermilab he participated in several projects, including the design of equipment and data analysis software for the CDF Detector, which were used in the discovery of the top quark, and the management of the design and construction of a 3 km Anti-Proton Recycler Ring for the Main Injector.[5][6]
He has been elected a fellow of the American Physical Society, was on the team receiving the 1989 Bruno Rossi Prize for cosmic ray physics for the discovery of the neutrino burst from the supernova SN 1987A, received the Particle Accelerator Technology Prize from the Institute of Electrical and Electronics Engineers, and was awarded an Energy Conservation award from the United States Department of Energy for his application of permanent magnets for Fermilab's accelerators.[7]
2008 special
Main article: 2008 Illinois's 14th congressional district special election
On November 26, 2007, former House Republican Speaker J. Dennis Hastert resigned as the Representative from Illinois' 14th congressional district. Foster announced his candidacy to fill the vacancy on May 30, 2007.[8] In the March special election, Foster defeated Republican nominee and Hastert-endorsed candidate Jim Oberweis 53%–47%.[9][10]
Main article: 2008 United States House of Representatives elections in Illinois § District 14
In November, Oberweis ran against Foster again in a rematch. Foster won re-election to a full term 58%–42%.[11]
Foster was challenged by Republican nominee State Senator Randy Hultgren and Green Party nominee Daniel Kairis. Despite winning the endorsements from the Chicago Tribune,[12] the Chicago Sun-Times[13] and The Daily Herald,[14] Foster lost to Hultgren 51%–45%.[15][16]
In May 2011, Foster sold his home in Geneva, moved to Naperville and announced plans to run for Congress in the 11th district, which encompasses Aurora, Joliet, Lisle in addition to Naperville. It also includes roughly a quarter of his old district.[17][18] The district had previously been the 13th, represented by seven-term Republican Judy Biggert. Although Biggert's home in Hinsdale had been shifted to the Chicago-based 5th district, Biggert opted to seek election in the 11th, which contained half of her old territory.[19]
On November 6, 2012, Foster won the election for the 11th district with 58% of the vote; Biggert conceded the race at 9:45pm.[20]
Foster ran again and was unopposed in the Democratic primary in March 2014.[21] For the general election, he faced Republican nominee, State Representative Darlene Senger, and defeated her with 53.5% of the vote to her 46.5% of the vote.[22]
Although it was initially thought that Foster would not be sworn in until April due to the need to count absentee ballots before the first election would be certified, he took the oath of office on March 11.[23]
Foster joined Vern Ehlers (R-MI) and Rush Holt Jr. (D-NJ) as the only research physicists ever to be elected to Congress.[24] On his first day in office, he cast the deciding vote to keep from tabling an ethics bill that would create an independent outside panel to investigate ethics complaints against House members.[25][26]
According to the Center for Responsive Politics, Bill Foster received $637,050 from labor related political action committees during his runs for Congress. $180,000 of this money came from PACs linked to public sector unions. $110,000 of these donations came from PACs linked to industrial labor unions.
According to the Federal Election Commission, Nancy Pelosi gave $4,000 to Bill Foster's 2012 campaign committee. PACs under the control of Pelosi have donated $10,000 to his 2012 campaign.
Foster supported allowing the Bush tax cuts to expire. During a debate with his opponent in the 2012 election Foster said, "The tax cuts were promised to generate job growth, but did not. If you follow the money, when you give a dollar to a very wealthy person, they won't typically put it back into the local economy." He said the tax benefits ended up in overseas accounts and spent on luxury purchases.[27]
Bill Foster has opposed efforts to repeal the estate tax. On 31 August 2005, U.S. Newswire reported that Foster said, "The proponents of estate tax repeal are fond of calling it the 'death tax'. It's not a death tax, it's a Rich Kids' tax." In 2009, just before the estate tax was scheduled for a one-year repeal, Foster voted to permanently extend the then current estate tax rate of 45%.
Card check
According to the official Thomas website, Bill Foster co-sponsored the Employee Free Choice Act of 2009, which would enable unionization of small businesses of less that 50 employees. On 25 February 2012, the Daily Herald reported, "Foster pointed to his support for the Employee Free Choice Act while serving at the congressman in the 14th District as proof of his union support."
Stimulus spending
Foster voted for the American Recovery and Reinvestment Act of 2009[28]
Foster voted for the Patient Protection and Affordable Care Act, commonly referred to as Obamacare.[29] On June 29, 2012, the Chicago Tribune reported that Foster said the following about his vote for Obamacare, "I'm proud of my vote, and I would be proud to do it again."
He also voted for the Dodd-Frank Wall Street Reform and Consumer Protection Act, with all ten of the amendments he proposed being added to the final bill.[30]
He voted against the American Clean Energy and Security Act, which would create a Cap and trade system.[31]
Asked if the Second Amendment should be up for reinterpretation, Foster said "It always has been up for reinterpretation. The technology changes, and the weapons thought to be too dangerous to be in private hands change. A civil war cannon is frankly much less dangerous than weapons we are allowed to carry on the streets in many of the states and cities in our country today. This is something where technology changes and public attitude changes and both are important in each of the generations."[32]
Committee on Financial Services (2008–2011; 2013–present)
Subcommittee on Capital Markets and Government-Sponsored Enterprises
Subcommittee on Monetary Policy and Trade
Committee on Science, Space, & Technology
Subcommittee on Environment
Select Subcommittee on the Coronavirus Crisis[33]
Committee on Oversight and Government Reform (2008–2011)
Caucus memberships
New Democrat Coalition[34]
Congressional Arts Caucus[35]
U.S.-Japan Caucus[36]
Illinois 14th Congressional District Special Democratic Primary, 2008[37]
Democratic Bill Foster 32,982 49.60
Democratic John Laesch 28,433 42.76
Democratic Jotham Stein 5,082 7.64
66,497 100.0
Illinois 14th Congressional District Democratic Primary, 2008[38]
Democratic Joe Serra 6,033 7.90
Illinois 14th Congressional District Special Election, 2008[39]
Republican Jim Oberweis 47,180 47.47
Illinois 14th Congressional District General Election, 2008[40]
Democratic Bill Foster (incumbent) 185,404 57.75
Republican Jim Oberweis 135,653 42.25
321,057 100.0
Democratic Bill Foster (incumbent) 25,446 100.0
Democratic Bobby G. Rose 1 0.00
Republican Randall M. "Randy" Hultgren 112,369 51.31
Democratic Bill Foster (incumbent) 98,645 45.04
Green Daniel J Kairis 7,949 3.63
Write-in votes Doug Marks 50 0.02
Democratic Juan Thomas 5,212 25.13
Democratic Jim Hickey 3,399 16.39
Democratic Bill Foster 148,928 58.57
Republican Judy Biggert (incumbent) 105,348 41.43
Write-in votes Chris Michel 19 0.01
Republican Darlene Senger 81,335 46.54
Write-in votes Constant "Connor" Vlakancic 1 0.00
Republican Tonia Khouri 108,995 39.55
Republican Nick Stella 82,358 36.16
Foster and his wife, Aesook Byon, live in Naperville, Illinois.[48][49] He has two adult children from his first marriage.[18] Foster has lived and worked in northern Illinois (Naperville, Geneva, Batavia, and St. Charles) since 1984.
^ "Judy Biggert Concedes Race To Bill Foster". CBS Chicago. Nov 6, 2012.
^ "Bill Foster - Who Runs Government". The Washington Post. Retrieved August 10, 2018.
^ Foster, George William (1983). A Experimental Limit on Proton Decay: Proton ---> Positron + Neutral Pion. Harvard University. Bibcode:1983PhDT........48F.
^ Electronic Theatre Controls (2008). "Lighting Solutions from ETC". Archived from the original on 25 February 2008. Retrieved 2008-03-11.
^ Foster, G. William (May 12–16, 1997). "[4C.01] The Fermilab Permanent Magnet Antiproton Recycler Ring". The 1997 Particle Accelerator Conference Meeting Program Vancouver BC, Canada. Fermilab. Archived from the original on July 18, 2003. Retrieved February 24, 2008.
^ Spotts, Peter N. (2004-05-01). "Physicists hope to win support for new subatomic smasher". The Christian Science Monitor. Retrieved 2008-03-11.
^ American Astronomical Society – High Energy Astrophysics Division (1989). "HEAD AAS Rossi Prize Winners". Archived from the original on 6 April 2008. Retrieved 2008-03-11.
^ "Geneva man seeks position in Congress". Courier News (Elgin, IL). 2007-05-31. Retrieved 2008-03-11.
^ "General election results". Chicago Tribune. 2008-03-08.
^ "IL – District 14 – Special Election". Our Campaigns.
^ "IL – District 14". Our Campaigns.
^ "For the US House". Chicago Tribune. 2010-10-07. Archived from the original on 2010-10-09. Retrieved 2010-10-09.
^ "Foster for 14th District". Chicago Sun-Times. 2010-10-06. Archived from the original on October 12, 2010.
^ "Congress, 14th District: Foster". The Daily Herald. 2010-10-16.
^ "Our Campaigns – IL – District 14 Race – Nov 02, 2010". ourcampaigns.com. Retrieved 7 September 2015.
^ "Clout St. Democrat Foster concedes defeat in 14th District". Chicago Tribune. 2010-11-02.
^ Lynn Sweet (31 May 2011). "Illinois Congress 2012: Bill Foster running in new 11th district". Chicago Sun Times. Archived from the original on 2 June 2011. Retrieved 1 June 2011.
^ a b Katherine Skiba (31 May 2011). "In wake of remap plan, ex-lawmaker to run again". Chicago Tribune. Archived from the original on 26 October 2012. Retrieved 1 June 2011.
^ Mike Flannery, Dane Placko (Aug 9, 2012). "FOX Chicago Sunday: Biggert, Foster debate to represent 11th Congressional District". Fox Chicago. Archived from the original on 2012-11-26.
^ Matt Hanley, Jenette Sturges (November 6, 2012). "Foster returns to Congress with win over Biggert". The Herald-News. Archived from the original on February 3, 2013.
^ "Official Illinois State Board of Elections Results – March 18, 2014 Primary Election (P. 31)" (PDF). Archived from the original (PDF) on October 21, 2014. Retrieved December 28, 2014.
^ "Illinois General Election 2014". Illinois State Board of Elections. 2014-11-04. Archived from the original on 2014-12-15. Retrieved 2014-12-28.
^ Hague, Leslie (2008-03-11). "Foster sworn into Congress". Daily Herald. Retrieved 2008-03-12.
^ Cornelia Dean (2008-07-10). "Physicists in Congress Calculate Their Influence". The New York Times. Retrieved 2010-02-11.
^ "Final Vote Results for Roll Call 121". 2008-03-11.
^ Jim Tankersley. "First day, swing vote for new Rep. Bill Foster". The Baltimore Sun. Archived from the original on 2008-03-17.
^ Dauskurdas, Sherri (September 2, 2012). "Biggert, Foster sit down for first debate of new 11th district". The Bugle. Archived from the original on 2012-09-10.
^ http://clerk.house.gov/evs/2009/roll046.xml
^ "Final Vote Results for Roll Call 165". HR 3590 Recorded Vote : Bill Title: Patient Protection and Affordable Care Act. U.S. House of Representatives. 21 Mar 2010.
^ "Bill's Congressional Career". Billfoster.com. Bill Foster for Congress. Archived from the original on 2012-03-25. Retrieved 2012-04-13.
^ "Final Vote Results for Roll Call 477: HR 2454". Recorded Vote; Question: On Passage; Bill Title: American Clean Energy and Security Act. U.S. House of Representatives. 26 Jun 2009.
^ Hegarty, Erin. "Rep. Bill Foster: Second Amendment meant to be reinterpreted by each generation". chicagotribune.com. Retrieved 2020-04-23.
^ "Pelosi Names Select Members to Bipartisan House Select Committee on the Coronavirus Crisis". Speaker Nancy Pelosi. 2020-04-29. Retrieved 2020-05-11.
^ "Members". New Democrat Coalition. Archived from the original on 8 February 2018. Retrieved 6 February 2018.
^ "Membership". Congressional Arts Caucus. Archived from the original on 12 June 2018. Retrieved 13 March 2018.
^ "Members". U.S. - Japan Caucus. Retrieved 11 December 2018.
^ "Election Results 2008 SPECIAL PRIMARY". Illinois State Board of Elections. Retrieved October 25, 2019.
^ "Election Results 2008 GENERAL ELECTION". Illinois State Board of Elections. Retrieved October 25, 2019. [permanent dead link]
^ "Election Results 2008 SPECIAL GENERAL ELECTION". Illinois State Board of Elections. Retrieved October 25, 2019.
^ "Election Results 2010 GENERAL PRIMARY". Illinois State Board of Elections. Retrieved October 25, 2019. [permanent dead link]
^ "Three House Members Wearing New Rings in the 111th". The Washington Post.
^ "Foster, Bill – Statement of Candidacy". Federal Elections Commission. 2011-09-29. Archived from the original on 2012-07-17. Retrieved 2011-10-04.
Biggert, Foster square off in 11th Dist. debate, ABC 7 Chicago, October 13, 2012, complete video
2012 candidate questionnaire at the Daily Herald
2012 candidate questionnaire at the Northwest Herald
2012 candidate questionnaire at the Chicago Sun-Times
2012 candidate questionnaire at WTTW Chicago Tonight
2012 candidate questionnaire and video at ABC 7 Chicago
Wikimedia Commons has media related to Bill Foster.
Congressman Bill Foster official U.S. House website
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Bill Foster at Curlie
Biography at the Biographical Directory of the United States Congress
Profile at Vote Smart
Financial information (federal office) at the Federal Election Commission
Legislation sponsored at the Library of Congress
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Dennis Hastert Member of the U.S. House of Representatives
from Illinois's 14th congressional district
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How To Find Factors Of A Large Number In C
Factor equation calculator, best algebra 2 book, How to convert from fraction notations to decimal notations by first find percent notation, HOW TO DETERMINE THE EQUATION OF A QUADRATIC FUNCTION WHEN GIVE THE VERTEX AND INTERCEPT, 3rd degree factor solver, Algabra1, Worksheets on rotation. Factors are the numbers that are multiplied together to get another number. The reality of choosing a new place to live encompasses an incredibly large series of factors, all competing for your attention. If you double the standard deviation (s), the sample size goes up by a factor of four. Consequently people looked in the names of the emperors Nero and Diokletian for 666 and they found it, because they persecuted the Christians. SOLUTION 4 : (Algebraically simplify the fractions in the numerator using a common denominator. Recall that when we factor a number, we are looking for prime factors that multiply together to give the number; for example. SUBJECT: Enforcement Guidance on the Consideration of Arrest and Conviction Records in Employment Decisions Under Title VII of the Civil Rights Act of 1964, as amended, 42 U. Factorize the numbers and identify all common factors. Tap into Daltile's extensive knowledge of tile through tools and resources designed to provide you with information throughout your project and beyond. This will save lots of time calculating the factors. C program to find HCF and LCM: The code below find the highest common factor and the least common multiple of two integers. Find the least common multiple of two whole numbers less than or equal to 12. Calculating factors of very large integers can take a long time. We now wish to look at the special case of multiplying two binomials and develop a pattern for this type of multiplication. In other words, it is a process of searching for and obtaining applicants for jobs so that the right people in right number can be selected. A number expressed as the product of a number between 1 and 10 (including 1) and a power of 10 is said to be in scientific form or scientific notation. 5x to get the. Prime Number Program In C - Any whole number which is greater than 1 and has only two factors that is 1 and the number itself, is called a prime number. Thus there are 9 positive integers that are factors to 36. Write a C program to input a number from user and find Prime factors of the given number. Any number of the form p a q b r c will have (a + 1) (b + 1) (c + 1) factors, where p, q, r are prime. This is a module and command-line utility for factoring integers. Usually, the factor with more digits is written first. The number and types of organisms that an ecosystem can support depends on the resources available (food sources) and on environmental factors, such as the amount of available sunlight, water, and the temperature. Factors I Really Like joyThese game of poker is perhaps one of many classiest and then long-term culture for taking 100 % free time. Popular and trusted online dictionary with over 1 million words. The numbers must be separated by commas, spaces or tabs or may be entered on separate lines. 24 = 2 3 × 3 60 = 2 2 × 3 × 5. scales can be compared between different maps. Prime factors of a number are those factors which are prime in nature and by which the number itself is completely divisible (1 will not be taken as prime number). The highest common factor (HCF) of two or more numbers is the largest number that is a factor of all of the given numbers. Find a partial fractions decomposition for. Process of finding all factors of x in efficient way;. In this case, the plant is the subgroup. Learn: How to find factorial of large numbers in C++ using array, this program will explain finding the factorial of large number. FREE with a 30 day free trial. How Do I Find My Medicare Number Online A couple of companies (though not the majors) implement "captured agents". Algorithms with numbers One of the main themes of this chapter is the dramatic contrast between two ancient problems that at rst seem very similar: Factoring: Given a number N, express it as a product of its prime factors. This means that 4 is the largest whole number that is a factor of both 8 and 12. Combinations are used in a large number of game type problems. There are some exceptions to this. * * It is easier to read 1kB than 1024 bytes and 1MB than 1048576 bytes. To understand this example, you should have the knowledge of following C programming topics:. In other words, it is a process of searching for and obtaining applicants for jobs so that the right people in right number can be selected. While hundreds of residents were last night waiting to find out the extent of the blaze that damaged at least one home and destroyed a number of other buildings, authorities deemed it safe enough. $\endgroup$ - David K Feb 21 '17 at 9:28. It provides a factorization technique that helps with factoring trinomials with large numbers. the number of ways in which x heads can occur in n flips, divided by the number of different possible results of the series of flips, measured by number of heads. Given an integer N, there is a simple way to find the total number of its factors. Strahan Queen's University Despite the widespread use of exploratory factor analysis in psychological research,. A number like 36, on the other hand, has _lots_ of factors: 36 = 1 * 36 = 2 * 18 = 3 * 12 = 4 * 9 = 6 * 6 Now,. 8 inches per hour) yields an adjusted square footage of 7,480 square feet. primefac version 1. Prime Factors of a Number calculator finds all prime factors of a positive integer. Some of the risk factors for congenital CP are: Low birthweight―Children who weigh less than 5 1/2 pounds (2,500 grams) at birth, and especially those who weigh less than 3 pounds, 5 ounces (1,500 grams) have a greater chance of having CP. Download the set (5 Worksheets). If the number of factors is c, a composite number, then it could be said that there is more than an infinite number of integers with that many factors because the infinite list of integers will include {2ᶜ⁻¹, 3ᶜ⁻¹, 5ᶜ⁻¹,. The freely accessible information on the Web, in consort with the escalating cost of library materials, jeopardizes the traditional mission of libraries to create and sustain large self-sufficient collections for their users. Notice that you raise the number 2. How to find the GCF of 3+ Numbers — FAST … no prime factorizing March 26, 2015 Suppose you need to find the GCF of three or more numbers, and you'd really prefer to avoid prime factorizing. Finding Modulus of a Very Large Number with a Normal Number I recently encountered this problem in a C++ program running on a 32-bit UNIX platform. The CORRECT answer: must be only PRIME numbers must multiply together to give the specified quantity. It is a specification language and not a development process so there are no details given on the features and how and when you link them together during systems development. This factor, which captures most of the variance in those three variables, could then be used in other analyses. We can test for divisibility by 3 (see table above) to quickly find a factor of 621 other than 1 and itself. Easy to use online maths calculators and solvers for various topics. In this article, we will show you, How to write a C Program to find Factors of a Number using FOR LOOP, WHILE LOOP, Pointers and FUNCTIONS C Program to. Scale factor - MCQ. The numbers must be separated by commas, spaces or tabs or may be entered on separate lines. The zeroth element of the sequence is 0. Pioneers in Language Reference for 200 years. - calculate sqrt(n) in a variable befor. However when confronted with 600851475143, it returns 4370432, definitely not prime. Therefore, 3 is a factor of any of those nine digit numbers and none of them are prime. I'm aware that I can find any number of articles on the Internet that explain how the RSA algorithm works to encrypt and decrypt messages, but I can't seem to find any article that explains the algorithm used to generate the p and q large and distinct prime numbers that are used in that algorithm. Find definitions, meanings, synonyms, pronunciations, translations, origin and examples. ) (Factor the denominator. In modern browsers, this calculator does most factorizations within a second. It also covers Prime Number Definition, prime factorization, calculating prime factors, finding all prime numbers between 1 and 100 in a simple way, list of all prime numbers upto 1000. Again, we can loop till 4 and find all the factors of 16. k, kilo, M, mega, and the rest, from yotta to yocto. , {small, medium, large} and {pen, brush, pencil}. There are several ways to calculate the smallest or largest number in a range. For example, say we want to find the number of factors of 12. Repetitive Strain Injury: a Computer User's Guide, by Pascarelli and Quilter (John Wiley and Sons, 1994), is a comprehensive source of information on how to prevent, live with, and recover from RSI. The structural model for two-way ANOVA with interaction is that each combi-. Part of the power of ANOVA is the ability to estimate and test interaction effects. Well, we can certainly find huge values for R that are difficult to factor. Factorials of N>20 can't be stored even in a 64-bit long long variable. It also covers Prime Number Definition, prime factorization, calculating prime factors, finding all prime numbers between 1 and 100 in a simple way, list of all prime numbers upto 1000. FindLaw's Learn About the Law section is the perfect starting point. Divide by 2. The concentrating effects that occur with blood and tissue products have inadvertently disseminated infections unrecognized at the time, such as HIV and hepatitis B and C. Repeat this until one of the numbers reaches 1 (in which case the gcd is 1, and the numbers are relatively prime) or until one of the numbers reaches 0 (in which case the gcd is the remaining value. Streiner and Norman observe that plausibly small adjustments in the initial estimate can have big effects on the calculation! Which is why statisticians are so successful at making the calculated sample size exactly equal the number of available patients. This is a whole number, so 1 is a factor at the low end. If it is possible, continue dividing this quotient successively by the same prime number. Beyond Meat initiated at sell as large competitors make gains in plant-based food rguably the number one reason for investors not to buy Beyond Meat shares at this time is the fact that Beyond. Reshape the area rectangle to see different factorizations of the number. And finally, your progam is not right. 3 and assesses the ability to divide whole numbers less than 100 when solving word problems in situations involving equal groups. } as well as many other integers. The common factors or of 12 and 18 are 1, 2, 3 and 6. Just too many factors - reject. For temperatures of 300 or 400 K, the compression factor is close to 1 over quite a large pressure range. So the largest factor on our factors list is going to be 120. asked by Luna on September 11, 2017; math. This method seems like it would be taxing if the number was large. Let's find the LCM of 30 and 45. Find real-time PEWCX - PNC Multi-Factor Large Cap Growth Fund Class C stock quotes, company profile, news and forecasts from CNN Business. These are the "tools" of a chemist, and to use these tools effectively, we must organize them in a sensible manner and look for patterns of reactivity that permit us make plausible predictions. As an exercise, write a function that returns the nth Fibonacci number using recursion. 50 are medium, and. b Write down 432 as a product of its prime factors, in index form. Find all divisors of a natural number | Set 2 This article is contributed by Ashutosh Kumar. A bit of groups will simply take really healthy persons and will not take you are not a current condition. 4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. There are several ways to calculate the smallest or largest number in a range. Factorials of N>20 can't be stored even in a 64-bit long long variable. Minitab's make patterned data capability can be helpful when entering numeric factor levels. This was coined by C. Now, the best way to find the prime factorization will be to store the sieve of prime factors initially. Downloading Large Adobe PDF Files. ), or by using methods in the System. This multiplication and simplification explains why, to factor a quadratic, we'll need to start by finding the two numbers (being the p and the q above) that add up to equal b, where those numbers also multiply to equal c. To find the factors of a number, begin with 1 and the number itself, then divide the number by 2, 3, 4, etc. A number of possible adverse health effects of very large doses of vitamin C have been identified, mainly based on in vitro experiments or isolated case reports, and include genetic mutations, birth defects, cancer, atherosclerosis, kidney stones, "rebound scurvy," increased oxidative stress, excess iron absorption, vitamin B 12 deficiency, and. The factors listed below will shift the supply curve either out or in. Find the prime factorization using a factor tree. Athenaeus gives us a detailed description of a very large warship, built by Ptolemy Philopator (c. But lucky for you, you have a number with lots of easy small prime factors. Popular and trusted online dictionary with over 1 million words. Much of the information from this webpage comes from this book, and the book contains far more detail than I was able to include here. (if some factor is greater than sqrt(n), then there has to be a factor which is smaller than that. After we compiled our list of startup failure post-mortems, one of the most frequent requests we got was to use these posts to figure out. Powers a 0 and a-n are defined as follows: a 0 = a (a ≠ 0) and. Large groups function differently in a number of important respects to smaller groups. Factors are important when working with fractions, as well as when trying to find patterns within numbers. This protein is transported out of cells into the extracellular matrix, which is an intricate lattice of proteins and other molecules that forms in the spaces between cells. Both the American and British naming conventions. C program to reverse a number: This program reverses a number entered by a user and then print it on the screen. The numbers that are completely divisible by the given number (it means the remainder should be 0) are called as factors of a given number in C. Nitrogen approximates to ideal behaviour at ordinary pressures. Generally speaking, a business's grade is a function of the level of. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. To find the prime factorization of a number, you need to break that number down to its prime factors. This is the smallest number divisible by 1, 2, 3, , 20 Note that number 9,699,690 only has 1 factor of 2, while numbers 4, 12, and 20 have 2 factors of 2, number 8 has 3 factors of 2, and number 16 has 4 factors of 2. The GCD of two integers X and Y is the largest number that divides both of X and Y (without leaving a remainder). Find factors of a number using an area model. The tag is an six character hashcode tag that is displayed for very large numbers. In order to be effective, a critical success factor must: Be vital to the organization's success. To find other factors, start dividing the number starting from two and working your way up until you reach that number divided by 2. They are all described in this chapter. Write down all factors of $$ \red c $$ which multiply to $$\red { \fbox {4}} $$ (Note: since $$\red 4 $$ is positive we only need to think about pairs that are either both positive or both negative. When an editor needs to decide whether to run with a particular story, s/he will ask how well the story meets each of these criteria. In this lesson, we will discuss a very interesting Mathematical shortcut: How to check whether a number is a perfect square or not. Pioneers in Language Reference for 200 years. 2) A number that divides a given set of numbers is called a common factor of the numbers. Working with large numbers in C/C++ is always a problem. For most industrial companies, 1. There are some exceptions to this. The prime factorization does not include 1, but does include every copy of every prime factor. SUBJECT: Enforcement Guidance on the Consideration of Arrest and Conviction Records in Employment Decisions Under Title VII of the Civil Rights Act of 1964, as amended, 42 U. You can start playing for free! Greatest Common Factor - Sample Math Practice Problems The math problems below can be generated by MathScore. primefac version 1. Active 3 years, When analyzing arithmetic with large numbers, we usually. Complexity of finding factors of a number. Repeat this until one of the numbers reaches 1 (in which case the gcd is 1, and the numbers are relatively prime) or until one of the numbers reaches 0 (in which case the gcd is the remaining value. txt, with size as. People with factor V Leiden thrombophilia have a higher than average risk of developing a type of blood clot called a deep venous thrombosis (DVT). Any quotient that does not have a remainder means that both the divisor and the quotient are factors of that number. Any number can be written as a product of prime numbers in a unique way (except for the order). 2 = the index_num. What other factors can influence effect size? Although effect size is a simple and readily interpreted measure of effectiveness, it can also be sensitive to a number of spurious influences, so some care needs to be taken in its use. Few such numbers are: Prime Factors of 24 are 2, 2, 2, 3 Prime Factors of 6 are 2, 3. C program for swapping of two numbers 14. C ifelse Statement. Divisibility Test for 3: if the sum of the digits of a number is divisible by 3, then the number is divisible by 3. Large animals, such as ourselves, follow this strategy. How to: Find the HCF or LCM of numbers using prime factors Key ideas: Any integer greater than 1 can be written as a product of primes. If you have a large data file (even 1,000 cases is large for clustering) or a mixture of continuous and categorical variables, you should use the SPSS two-step procedure. Re: is there a formula to calculate the number of factors? by jp. Access tax forms, including Form Schedule C, Form 941, publications, eLearning resources, and more for small businesses with assets under $10 million. A number like 36, on the other hand, has _lots_ of factors: 36 = 1 * 36 = 2 * 18 = 3 * 12 = 4 * 9 = 6 * 6 Now,. Paul Starick, They run large enterprises – or not so large, in. Check the prime factors page to learn how to find prime factors of an integer. Scale is another relative term meaning "size" in relationship to some system of measurement. But we can find factorial for large numbers using simple multiplication method. Factoring Polynomials. From that we would like to obtain a curved or scaled grade which is again a score between 0 and 100 (or occasionally a number over 100). IF the value of A>that of B THEN. Learn How To Find a Prime Numbers with simple maths aptitude tricks. 8 inches per hour) yields an adjusted square footage of 7,480 square feet. Galpin in relation to delineating rural communities in terms of the trade and service areas surrounding a central village (Harper and Dunham 1959: 19). The world's largest digital library. Or another way of thinking about it-- a number is a factor of 154 if 154 is a multiple of that number. This video. This is a whole number, so 1 is a factor at the low end. is a perfect cube. Converse is not necessarily true). If you type an integer into the box, this page will show you the factors of your number. Cancel Anytime. Learn how to do just about everything at eHow. Is there a formula to calculate the sum of all proper divisors of a number? Is there a genral formula to find if a number X has it's factor such that they add. All factors of big numbers in C/C++. How to find whether a given number is prime or not? What is a prime number A number is greater than 1 is called a prime number, if it has only two factors, namely 1 and the number itself. Note that 4 has some other factors besides 1 and 4: 4 = 1 * 4 4 = 2 * 2 But these are the only factors of 4. The highest common factor is the product of the common prime factors. Numbers with four factors have two types of rectangular array: 8 has 1 x 8 and 2 x 4. This value is handy when exchanging large values or rechecking calculations. Factors in R come in two varieties: ordered and unordered, e. Drug In Outpatient Or Inpatient With guests insurance in USA, they will receive significant protection when ever moving to the and find the benefit in packages that offer essential coverage cheaply. The usual assumptions of Normality, equal variance, and independent errors apply. In some cases it may be helpful to use the identity. Step 1 : Split the given number as prime factors using prime factorization method or tree method. Usually that integer will be large (441 is the object of one question that comes to mind), since the question wouldn't be very challenging otherwise. a) only one prime factor: you can as well take 2, b) two prime factors: 2 and 3, c) three prime factors: 2, 3 and 5, d) four prime factors: 2, 3, 5 and 7; this still works because 2*3*5*7 = 210, e) five prime factors: 2, 3, 5, 7 and 11, because 2*3*5*7*11 = 2310 which is less than 5000. Your account. The problem with using approximations and rounding with numbers on a calculator isn't limited to working with two numbers of very different size. Factors that Shift the Supply Curve. C Program to Display Factors of a Number Example to find all factors of an integer (entered by the user) using for loop and if statement. GOTO step 4. Developing the Concept. Write an algorithm and draw the flowchart to find whether a given number is even or odd? Write an algorithm and draw the flowchart to Swap two integers? Write an algorithm an draw the flowchart to compute the average of the three numbers? Write an algorithm and draw the flowchart to find the largest number amoug two numbers? write a c program. In order to check whether a number is a perfect cube or not, we find its prime factors and group together triplets of the prime factors. Similarly, 5 is a factor 20 and 20=5x4 so we need only test the smaller number 4 as a factor. You may enter between two and ten non-zero integers between -2147483648 and 2147483647. In general, square numbers seem to have an odd number of factors, while numbers with four, six, and eight factors are rectangular. If you scan any organic textbook you will encounter what appears to be a very large, often intimidating, number of reactions. Use, for example, the input 10403131221399539746 and it takes quite a while. Recall that when we factor a number, we are looking for prime factors that multiply together to give the number; for example. bi_factor - prime-factor large numbers The program will compute the prime factors of each number you give it as an argument. RSA's main security foundation relies upon the fact that given two large prime numbers, a composite number (in this case \(n\)) can very easily be deduced by multiplying the two primes together. To understand this example, you should have the knowledge of following C programming topics:. Occasionally a GMAT question will ask you for the number of factors of a certain integer. Tax Withholding Estimator. We are an industrial property rental leasing agent specializing in commercial office spaces and corporate lease negotiation. The Fullness Factor is a calculated rather than measured value, and has a few distinct advantages over the Glycemic Index: Fullness Factors are instantly determinable for ALL foods Knowledge of the nutrient information contained on a standard nutrition facts label is all that is required to determine the Fullness Factor. In order for the number to be a perfect cube a, b. That's its actual smallest factor, and its largest factor is 120. Department of Health & Human Services. Plants are also subject to the same sorts of forces as animals. A ratio is an expression that tells us the quotient of two numbers. A large number of bottles in this category have relatively little value, but there are several types that bring good money. correct number of factors. Others live in a quite stable environment, such as a climax forest. FACTORING LARGE SEMI-PRIMES It is well known that it is difficult to factor a large semi-prime number N into its two prime components. 3 times 7 is 21. Factors are whole numbers that are multiplied together to produce another number. The standard online $0 commission does not apply to large block transactions requiring special handling, restricted stock transactions, trades placed directly on a foreign exchange, transaction-fee mutual funds, futures, or fixed income investments. Machine cycles and resources that could be used for computation are instead used to package and transmit data. 8 million injuries or illnesses in 2018. Alice is frightened of big numbers and hence is asking you for help. We will refer to the number of observations in each group as n and the total number of observations as N. To find the GCF, first write the factors of each number. The rate of nonfatal occupational injuries and illnesses among private industry employees was unchanged for the first time since 2012 at 2. That services features health related concerns like eye caution, oral alignment, urgent treatment, comes to visit to the doctor, costs from prescribed prescription drugs and so forth. 84 ÷ 2 = 42 84 is even. Then we will move on to find the GCD of a list of numbers and code it in both C++ as well as Python. It would take a long time to write out all the factors and multiples of. Multiplier: The multiplier is the number (factor) that you are multiplying by. For example, June 30, 2018, to December 31, 2018, is 184 days which is half a year. C++ program to find factorial of a number. C program for swapping of two numbers 14. Repetitive Strain Injury: a Computer User's Guide, by Pascarelli and Quilter (John Wiley and Sons, 1994), is a comprehensive source of information on how to prevent, live with, and recover from RSI. Look at the sign of c and your lists from Steps 1 and 2 to see if you want a sum or difference. Find the greatest common factor of two whole numbers less than or equal to 100. You could also use factoring algorithms (like pollard's rho, or others ). To find your conversion factor, simply divide the desired number of servings by the original number of servings. Any system where human lives are at stake must place extreme emphasis on reliability and integrity. Find your yodel. How Do I Find My Medicare Number Online A couple of companies (though not the majors) implement "captured agents". Select a cell below or to the right of the numbers for which you want to find the smallest number. Example to find all factors of an integer (entered by the user) using for loop and if statement. For example, an increase in the width of a heat sink by a factor of two would increase the heat dissipation capability by a factor of two,whereas and increase the heat dissipation capability by a factor of 1. Once you have established the relationships between factors on your diagram, you can look to see if you can put numbers to the relationships. For example, since 3 x 4 = 12, we say that the pair of numbers 3 and 4 are factors of 12. Factorials of N>20 can't be stored even in a 64-bit long long variable. Find the greatest common factor of two whole numbers less than or equal to 100. The BHF's vision is a world free from the fear of heart and circulatory diseases. Divide by 2. Some areas have a high population density while others have a low population density. Factors that Shift the Supply Curve. Sorry we couldn't find a match for that, please try again It's time to slash council numbers and CEO pay packets. When a user types a query, Google tries to find the most relevant answer from its index based on many factors. Almost every competitive examination has 2-3 medium to difficult level questions based on factors of a number. 8 a Write down 240 as a product of its prime factors, in index form. The large number of similar terms with different definitions has caused much confusion with both coaches and athletes. 24 = 2 3 × 3 60 = 2 2 × 3 × 5. Find the Prime Factors of a Number: A prime number is any number with no divisors other than itself and 1, such as 2 and 11. This tells the formula the number of columns away from the left most column to return in case of match. In many cases, we can easily determine the minimum sample size needed to estimate a process parameter, such as the population mean. C program for swapping of two numbers 14. In this article we will show you, How to write a Java Program to find Factors of a Number using For Loop, While Loop, Do While Loop and Functions. Big integers must be used for such calculations. As the number approaches or falls below 1 (which means the company has a negative working capital), you will need to take a close look at the business and make sure there are no liquidity issues. The following guidelines are very important in writing a successful swot analysis. However if one of the prime factors is a single factor or a double factor then the number is not a perfect cube. We have only these three cases since there can be at max two prime factors of Y. The Fullness Factor is a calculated rather than measured value, and has a few distinct advantages over the Glycemic Index: Fullness Factors are instantly determinable for ALL foods Knowledge of the nutrient information contained on a standard nutrition facts label is all that is required to determine the Fullness Factor. One way to find the least common multiple of two numbers is to first list the prime factors of each number. A shift in the supply curve (for example from A to C) is caused by a factor other than the price of the good and results in a different quantity supplied at each price. The tag is an six character hashcode tag that is displayed for very large numbers. 7 ÷ 7 = 1 7 is prime. This gcd of two numbers in C program allows the user to enter two positive integer values and we are going to calculate the Highest Common Factor of those two values. With a little amount of work you find that $2,453 = 11 \times 223$. The disease likely develops from multiple factors, such as genetics, lifestyle and environment. The numbers 0 and 1 are neither prime nor composite. Factors can divide evenly into a number with no remainder or decimal. Cancel Anytime. Factors in R are stored as a vector of integer values with a corresponding set of character values to use when the factor is displayed. Next, determine whether those 2 numbers can be factored again. Since factors typically have quite a small number of levels, for large vectors x it is helpful to supply nmax as an upper bound on the number of unique values. We are unable to ask the form for the current scale factor in effect, so we need to calculate it ourselves. In some cases it may be helpful to use the identity. This definition has two main features: Firstly, the fact that globalisation is not an end result, but is a continuing process that keeps. 21 ÷ 3 = 7 3 is a prime factor of 21, divide by 3. Using the number 3784 as an example, start by dividing it by the smallest prime factor (bigger than 1) that goes into it evenly with no remainder. We can create a file of required size using this tool. You could also use factoring algorithms (like pollard's rho, or others ). Developing the Concept. The non-ideal behaviour gets worse at lower temperatures. Form W-9 Request for Taxpayer Identification Number. Put on your thinking caps to find the answer that best fits the problem in these MCQ worksheets. Sorry we couldn't find a match for that, please try again It's time to slash council numbers and CEO pay packets. GOTO step 4. is square of a prime number: F(Y) = 3. Multiply your common factors together and you end up with the greatest common factor for both numbers!. The problem is to figure out what the current scale factor in use on the form is. Factor equation calculator, best algebra 2 book, How to convert from fraction notations to decimal notations by first find percent notation, HOW TO DETERMINE THE EQUATION OF A QUADRATIC FUNCTION WHEN GIVE THE VERTEX AND INTERCEPT, 3rd degree factor solver, Algabra1, Worksheets on rotation. ), it means that the R. Ex: 6 has three factors 2, 3, 1. Grade A will break down the steps for you, show you simple examples with visual illstrations, and also give you some clever tips and tricks. 121 will not go into 120. Find all divisors of a natural number | Set 2 This article is contributed by Ashutosh Kumar. Sign in to review and manage your activity, including things you've searched for, websites you've visited, and videos you've watched. The disease likely develops from multiple factors, such as genetics, lifestyle and environment. In this section, we will consider in detail two classical algorithms for sorting and searching—binary search and mergesort—along with several applications where their efficiency plays a critical role. Fortunatly, there is a method that every scale operation must go through. Repetitive Strain Injury: a Computer User's Guide, by Pascarelli and Quilter (John Wiley and Sons, 1994), is a comprehensive source of information on how to prevent, live with, and recover from RSI. For example, if you want to factor 12, you could use 4 and 3 since they multiply to make 12. Learn how to find the greatest common factor using factoring, prime factorization and the Euclidean Algorithm. Trying to find a black lesbian or bi woman in. In general, square numbers seem to have an odd number of factors, while numbers with four, six, and eight factors are rectangular. 54 cm or 1 gallon = 231 cubic inches or 1 foot = 12 inches or 1 meter = 100 centimeters. The freely accessible information on the Web, in consort with the escalating cost of library materials, jeopardizes the traditional mission of libraries to create and sustain large self-sufficient collections for their users. That services features health related concerns like eye caution, oral alignment, urgent treatment, comes to visit to the doctor, costs from prescribed prescription drugs and so forth. To find the prime factorization of a number, you need to break that number down to its prime factors. Program in c to print 1 to 100 without using loop 13. The only factors for 21 are 1, 21, 3, and 7. This video. a) only one prime factor: you can as well take 2, b) two prime factors: 2 and 3, c) three prime factors: 2, 3 and 5, d) four prime factors: 2, 3, 5 and 7; this still works because 2*3*5*7 = 210, e) five prime factors: 2, 3, 5, 7 and 11, because 2*3*5*7*11 = 2310 which is less than 5000. To model division with fractions, we more or less reverse the process used for multiplication. It is a quick way to find factors of large numbers. The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two (or more) numbers, is the largest number which is a factor of each. Split number into digits in c programming 16. | CommonCrawl |
Project End: May 2015
Presented at ICRA Humanoid Application Challenge, May 2015 in Seattle, USA
Won first place
Highlights of our research into small-scale humanoid robotic skiing.
This video was used during our presentation at the 2015 ICRA conference.
After the 2014 ICRA Humanoid Application Challenge was cancelled, the competition returned for the 2015 conference to be held in Seattle.
Ever since our success with the hockey project in 2012 the idea of trying other winter sports lingered in the back of my mind. I had already tried snowshoeing, but I felt that this was too similar to standard walking to be sufficiently novel for the competition.
However, alpine skiing was one of the winter sports I wanted to try that was novel (there have been some attempts at alpine skiing robots, but all are large-scale and use complex sensors), had complex challenges (balancing on uneven terrain, control and release of stored energy, low-contrast environment for vision), and was visually-appealing.
Making the Skis
Jennifer posing with her skis
With the project settled on, my father and I once again descended into his workshop to build robot-sized skis and poles.
The skis were made by steam-bending lengths of wood to form the curved tips, with final shaping done by hand on a bandsaw. We drilled and counter-sunk holes in the bottom of the skis so that we could screw them directly to the robot's feet; the skis do not have a free-heel binding like those used for cross-country or telemark skiing. Finally we treated the skis with wax, just as one would a real wooden ski.
The poles were a similarly simple construction; a length of dowel was sharpened to a point using a pencil sharpener, with a large-diameter disc affixed just above the point to form the snow basket.
Given its superficial similarity to walking and skating, we chose to tackle cross-country skiing first. We used the same basic walking gait that we use for HuroCup and RoboCup (and indeed for our skating gait as well), only with the stride height lowered and the stride length and period dramatically increased. The result was a long, fluid motion that mimicked traditional cross-country skiing fairly well.
A comparison of walking, skating, and skiing gaits
The biggest challenge with the cross-country skiing was controlling the robot's balance; because snow compacts unevenly underfoot the robot was constantly losing its balance. The small size of the robot meant that small changes in terrain elevation (just a few cm) were enough to topple the robot over. However, by adapting the PID controllers developed for the bongo board to work in three dimensions we were able to improve the robot's balance. (More on this below.)
However, on hard, icy snow, the robot could cross-country ski very well without the need for any additional active balancing, as this early video shows:
Our early attempts at cross-country skiing on hard snow
Alpine skiing (aka downhill skiing) was the primary focus of this research. While cross-country served as a small introduction, the majority of our efforts were on allowing the robot to
ski down a hill without falling over,
demonstrate simple steering, and
autonomously navigate a simple obstacle course using vision
As with the hockey project, we started with small, static tests; equip the robot with skis, design a fixed pose to test gliding/steering/braking, go to a nearby hill, push the robot down said hill, and see what happens. Conveniently there was a decorative embankment with a steep incline just outside E2-EITC, the building on campus in which our lab was located. Packing up the robot, some spare batteries, and my laptop, we ventured out into the snow and tested out whether the robot's skis could glide downhill, and whether or not they could provide enough surface area along the edges to allow the robot to carve left/right and brake.
Our keyframe tests proved successful. With time running out to get our entry submitted for the competition, we braved the cold once more to film our progress so far. Our official submission video, showing cross-country skiing on different kinds of snow and alpine skiing using static keyframes is below.
Our qualification video for the 2015 ICRA Humanoid Application Challenge
The apline skiing portions of this video used static keyframes
With our competition entry locked in, work continued on improving the robot's alpine skiing algorithms. We quicky identified balance as the key concern; if the robot cannot stay upright on its skis it cannot hope to steer, let alone navigate around obstacles.
Based on my thesis research on active balancing on dynamic terrain -- which by this point was essentially complete and I was into the final revisions of my thesis -- I developed a three-dimensional version of the PID controller we used at the 2013 ICRA challenge.
Frontal views of the robot showing compensation for lateral inclination
Side views of the robot showing compensation for different hill inclinations
The new system used two independent PID systems; one to control the robot's pitch, and a second to control the robot's roll. By using lower P and D gains (and an I gain of zero) we were able to develop a system that would allow the robot to smoothly adjust its pose based on the inclination of the hill (i.e. leaning back more on steeper portions, standing straighter on flatter portions), as well as react to sudden disturbances.
Demonstration of the active balancing techniques
Vision-Based Steering
With the balancing problem largely tackled, our last obstacle -- pun marginally intended -- was to integrate some kind of reactive, vision-based steering to let the robot navigate around mock slalom gates.
Because of the small size of our hills -- Manitoba is notoriously flat, being on the prairies -- we opted to create a very simple obstacle course. A designated skiing area would be marked with pink markers on one side and blue markers on the other. The robot must stay between those markers, similar to a giant slalom.
Based on our obstacle course software for HuroCup, I implemented another PID system to control the robot's steering left and right. For every pink marker in sight the robot would turn to the left an amount proportional to the distance from the right edge of the frame (i.e. an object on the right edge would result in minimal turning, but an object on the left edge would result in maximal turning). Similarly, for every blue marker the robot would turn right an amount proportionl to the object's distance from the left edge:
(Mathjax requires javascript. Without it, instead of pretty equations all you'll see is the raw LaTeX source code)
\[ \begin{array}{l} B \mbox{: the set of blue markers currently visible} \\ P \mbox{: the set of pink markers currently visible} \\ l(m) \mbox{: the distance from the left frame edge to the marker $m$} \\ r(m) \mbox{: the distance from the right frame edge to the marker $m$} \\ h(m) \mbox{: the distance from the bottom of the frame to the marker $m$} \\ k \mbox{: the turning constant} \\ turnR_{i} = -kl(b_i)\frac{1}{h(b_i)} \forall b_i \in B \\ turnL_{i} = kr(p_i)\frac{1}{h(p_i)} \forall p_i \in P \\ T = \sum{turnR} + \sum{turnL} \\ \epsilon = turn - t \\ turn' = k_p\epsilon + k_i\int{\epsilon}d\epsilon + k_d\frac{d\epsilon}{dt} \end{array} \]
By using a minimal D-gain and a modest P-gain (our I-gain was set to zero) we were able to develop a system that could smoothly adjust the robot's turn as markers moved in and out of its field of view.
The actual turn was performed by using the robot's inverse-kinematic (IK) modules for the arms and legs to enter a "steering stance." When turning the skis are angled such that the edges along the inside of the turn are lower, digging into the snow. The robot shifts its entire CoM over to the inside of the turn, extending the outward arm for balance. Finally, the outward arm is rotated such that the pole does not drag in the snow.
From left: left turn stance, straight stance, right turn stance
This use of simple PID controllers for controlling the robot's pitch, roll, and steering is -- to our knowledge -- a novel development in the world on autonomous skiing humanoids. Despite the simplicity of the algorithms involved, the results proved to be a robust system for navigating simple obstacle courses when alpine skiing.
ICRA Presentation
With the snow long-gone (well, not so long; we had a late flurry in April, but it melted quickly) we packed up our robot and equipment and flew to Seattle for ICRA.
Our presentation did well; we did a combination live demo and video, since it was impossible to show actual alpine skiing indoors, in May, in Seattle for all the reasons you expect.
The judging this year was slightly different that the 2012 and 2013 competitions. There was no panel of judges in Seattle; the entries were scored entirely based on peer review from the other competitors.
As with previous years, there was a good variety of entries, including (among others) our control-theory focused skiing, to the use of a robot as an autonomous actor (complete with a live skit of "Indiana Darwin" -- I have video of that, I just need to upload it), and a cloud-based robotic testing suite where users can upload their code to a remote robot and watch it perform in real-time.
In the end however, there can be only one winner, and for the second time in four years UofM finished on top. From a personal perspective, this was my last robotics competition before graduating, so I was very happy to finish on a high note.
Related and Future Work
As with our hockey project, work never technically ended on the skiing. Eventually we'd like to adapt our small-scale algorithms for use on a larger robot. Such work has not been done yet, but as new students come to the lab one of them may take up the challenge
This work is very closely tied to my own thesis research on active balancing, and the afore-mentioned hockey project.
Jennifer on the slopes
(Image from Winnipeg Free Press)
Active Balancing and Turning for Alpine Skiing Robots. Chris Iverach-Brereton, Brittany Postnikoff, Jacky Baltes, Amirhossein Hosseinmemar. Knowledge Engineering Review (KER) Special Issue. Accepted, publication pending
Radio7 Deutchland
Winnipeg Free Press
Gadgetify
Motherboard (Vice)
Tomorrow Daily (CNET)
Gizmodo (This may well be my favourite article about this project) | CommonCrawl |
Greenberg's conjectures
Greenberg's conjecture is either of two conjectures in algebraic number theory proposed by Ralph Greenberg. Both are still unsolved as of 2021.
Invariants conjecture
The first conjecture was proposed in 1976 and concerns Iwasawa invariants. This conjecture is related to Vandiver's conjecture, Leopoldt's conjecture, Birch–Tate conjecture, all of which are also unsolved.
The conjecture, also referred to as Greenberg's invariants conjecture, firstly appeared in Greenberg's Princeton University thesis of 1971 and originally stated that, assuming that $F$ is a totally real number field and that $F_{\infty }/F$ is the cyclotomic $\mathbb {Z} _{p}$-extension, $\lambda (F_{\infty }/F)=\mu (F_{\infty }/F)=0$, i.e. the power of $p$ dividing the class number of $F_{n}$ is bounded as $n\rightarrow \infty $. Note that if Leopoldt's conjecture holds for $F$ and $p$, the only $\mathbb {Z} _{p}$-extension of $F$ is the cyclotomic one (since it is totally real).
In 1976, Greenberg expanded the conjecture by providing more examples for it and slightly reformulated it as follows: given that $k$ is a finite extension of $\mathbf {Q} $ and that $\ell $ is a fixed prime, with consideration of subfields of cyclotomic extensions of $k$, one can define a tower of number fields $k=k_{0}\subset k_{1}\subset k_{2}\subset \cdots \subset k_{n}\subset \cdots $ such that $k_{n}$ is a cyclic extension of $k$ of degree $\ell ^{n}$. If $k$ is totally real, is the power of $l$ dividing the class number of $k_{n}$ bounded as $n\rightarrow \infty $? Now, if $k$ is an arbitrary number field, then there exist integers $\lambda $, $\mu $ and $\nu $ such that the power of $\ell $ dividing the class number of $k_{n}$ is $\ell ^{e_{n}}$, where $e_{n}={\lambda }n+\mu ^{\ell _{n}}+\nu $ for all sufficiently large $n$. The integers $\lambda $, $\mu $, $\nu $ depend only on $k$ and $\ell $. Then, we ask: is $\lambda =\mu =0$ for $k$ totally real?
Simply speaking, the conjecture asks whether we have $\mu _{\ell }(k)=\lambda _{\ell }(k)=0$ for any totally real number field $k$ and any prime number $\ell $, or the conjecture can also be reformulated as asking whether both invariants λ and µ associated to the cyclotomic $Z_{p}$-extension of a totally real number field vanish.
In 2001, Greenberg generalized the conjecture (thus making it known as Greenberg's pseudo-null conjecture or, sometimes, as Greenberg's generalized conjecture):
Supposing that $F$ is a totally real number field and that $p$ is a prime, let ${\tilde {F}}$ denote the compositum of all $\mathbb {Z} _{p}$-extensions of $F$. (Recall that if Leopoldt's conjecture holds for $F$ and $p$, then ${\tilde {F}}=F$.) Let ${\tilde {L}}$ denote the pro-$p$ Hilbert class field of ${\tilde {F}}$ and let ${\tilde {X}}=\operatorname {Gal} ({\tilde {L}}/{\tilde {F}})$, regarded as a module over the ring ${\tilde {\Lambda }}={\mathbb {Z} _{p}}[[\operatorname {Gal} ({\tilde {F}}/F)]]$. Then ${\tilde {X}}$ is a pseudo-null ${\tilde {\Lambda }}$-module.
A possible reformulation: Let ${\tilde {k}}$ be the compositum of all the $\mathbb {Z} _{p}$-extensions of $k$ and let $\operatorname {Gal} ({\tilde {k}}/k)\simeq \mathbb {Z} _{p}^{n}$, then $Y_{\tilde {k}}$ is a pseudo-null $\Lambda _{n}$-module.
Another related conjecture (also unsolved as of yet) exists:
We have $\mu _{\ell }(k)=0$ for any number field $k$ and any prime number $\ell $.
This related conjecture was justified by Bruce Ferrero and Larry Washington, both of whom proved (see: Ferrero–Washington theorem) that $\mu _{\ell }(k)=0$ for any abelian extension $k$ of the rational number field $\mathbb {Q} $ and any prime number $\ell $.
p-rationality conjecture
Another conjecture, which can be referred to as Greenberg's conjecture, was proposed by Greenberg in 2016, and is known as Greenberg's $p$-rationality conjecture. It states that for any odd prime $p$ and for any $t$, there exists a $p$-rational field $K$ such that $\operatorname {Gal} (K/\mathbb {Q} )\cong (\mathbb {Z} /\mathbb {2Z} )^{t}$. This conjecture is related to the Inverse Galois problem.
Further reading
• R. Greenberg, On some questions concerning the lwasawa invariants, Princeton University thesis (1971)
• R. Greenberg, "On the lwasawa invariants of totally real number fields", American Journal of Mathematics, issue 98 (1976), pp. 263–284
• R. Greenberg, "Iwasawa Theory — Past and Present", Advanced Studies in Pure Mathematics, issue 30 (2001), pp. 335–385
• R. Greenberg, "Galois representations with open image", Annales mathématiques du Québec, volume 40, number 1 (2016), pp. 83–119
• B. Ferrero and L. C. Washington, "The Iwasawa Invariant $\mu _{p}$ Vanishes for Abelian Number Fields", Annals of Mathematics (Second Series), volume 109, number 2 (May, 1979), pp. 377–395
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CBSE Notes for Class 6 Maths Chapter 7: Fractions
Fraction Definition
Types of Fractions
Proper Fraction
Mixed Fraction
Improper Fraction
Mixed Fractions to Improper Fractions
Improper Fractions to Mixed Fraction
How to Represent Fraction on Number Line?
Simplest form of Fraction
Like and Unlike Fractions
How to add Fractions
How to Subtract Fractions
Comparison, Addition and Subtraction of Mixed Fractions
A fraction is a number representing a part of a whole. The whole may be a single object or a group of objects.
Suppose Ramesh has a chocolate and we want to equally share with his Friend Amit. He will divide the chocolate into two pieces and keep one piece with him and give another piece to Amit. So basically they each have got 1 part out of 2 parts i.e 1/2 of the chocolate. Similarly if they have another suresh also, then they will divide chocolate in three equal parts,then each of them will have 1 out of 3 parts i.e 1/3 of the chocolate
$\frac {3}{11}$
3 out of 11 parts
3 -> Called Numerator
11 -> Called Denominator
When expressing a situation of counting parts to write a fraction, it must be ensured that all parts are equal
Identify the Numerator and Denominator from below fraction
a. $\frac {2}{15}$
b. $\frac {11}{13}$
c. $\frac {3}{2}$
d. $\frac {1}{8}$
Types of Fraction
Fractions are of three types.
a. Proper Fraction
b. Improper Fraction
c. Mixed Fraction.
Let dive into each of them in detail in below
What is proper Fraction
Proper Fraction is the fraction which is less than 1 or where Numerator is less than Denominator
Here the Denominator shows the part whole has been divided and numerator shows the part which has been considered.
This is the same fraction which we discussed with fraction definition
$\frac {1}{3}$
What is Improper Fraction
Improper Fraction is the fraction which is greater than 1 or where Numerator is greater than Denominator
Lets understand this with example . Suresh has 5 chocolates and he has to divide those chocolate among four friends.We can divide each chocolate into fours parts and each one can have one-quarter part of the each chocolate. So Each friend will be having 5 parts of the one-quarter part.Now 4 parts make one whole. So basically each one of them is getting 1 whole and 1 part. So this can be written as 5/4
Here numerator is more than denominator
$\frac {11}{5}$
What is Mixed Fraction
It is combination of whole number and proper fraction
$8 \frac {4}{9}$
Lets the example of improper fraction only. The division can made in another way. We give one chocolate to each of them and divide the fifth chocolate into four pieces. So each of them got 1 full chocolate and one-quarter part of last chocolate. So this can written as
$1 + \frac {1}{4} =1 \frac {1}{4}$
This is mixed fraction.
Also $1 + \frac {1}{4} =1 \frac {1}{4} = \frac {5}{4}$
How to convert Mixed Fractions to Improper Fractions
Obtain the mixed fraction. Let the mixed fraction be 52/6
Identify the whole number and the numerator (top) and denominator (bottom) of the proper fraction.
Whole Number=5
Numerator=2
Denominator=6
Apply the formula
$\frac {(Whole \times Denominator) + Numerator}{Denominator}$
$ 5\frac {2}{6}$
=$\frac {32}{6}$
So $1\frac {1}{3}=\frac {4}{3}$
$ 1 \frac {1}{4}$
5$\frac {2}{3}$
3$\frac {5}{11}$
2$\frac {17}{44}$
How to convert Improper Fractions to Mixed Fraction
Obtain the improper fraction.
Divide the numerator by the denominator and obtain the quotient and remainder.
Write the mixed fraction as
$Quotient \frac {Reminder}{denominator}$
Here Numerator is greater than denominator, So Improper fraction
Now dividing 11 by 3, we get reminder as 2
So $\frac {11}{3}=3 \frac {2}{3}$
$\frac {37}{18}$
$\frac {103}{9}$
We can show fractions on a number line. In order to represent 1/2 on the number line, draw the number line and look for the portion between 0 and 1
Now, divide the gap between 0 and 1 into two equal parts. The point of division represents 1/2.
To represent 1/4 on a number line, we divide the gap between 0 and 1 into 4 equal parts
First point will represent ½
Second point will represent 2/4 =1/2
Third point will represent 3/4
Simplest Form of a Fraction
If the numerator and the denominator of a fraction have no common factor except, then it is said to be in its simplest form or lowest form.
Checkout Simplifying Fraction calculator
Equivalent fractions are fractions that have the same value in its simplest form.
$\frac {2}{6} \;,\; \frac {1}{3} \;,\; \frac {6}{18}$ are equivalent fractions as they have same value
The equivalent fraction of a given fraction is obtained by multiplying both the numerator and the denominator of the given fraction by the same number.
Checkout Equivalent Fraction calculator
Equivalent Fraction can be obtained by multiplying both the numerator and the denominator of the given fraction by the same number.
$\frac {2}{3} = \frac {2 \times 3}{3 \times 3}= \frac {6}{9}$
$\frac {2}{3} = \frac {2 \times 5}{3 \times 5}= \frac {10}{15}$
Like Fractions and Unlike Fractions
Like Fractions
Fractions with the same denominators are called like fractions.
$\frac {1}{3}$, $\frac {2}{3}$ are Like Fractions
$\frac {1}{4}$ ,$\frac {3}{4}$ are like Fractions
Unlike Fraction
Fractions with different denominators are called unlike fractions.
$\frac {1}{3}$ , $\frac {2}{5}$ are Unlike Fractions
Identify which ones are like fractions and unlike fractions
a. $\frac {2}{3}$ and $\frac {4}{3}$
b. $\frac {2}{5}$ and $\frac {1}{3}$
c. $\frac {11}{5}$ and $\frac {17}{4}$
d. $\frac {2}{5}$ and $\frac {9}{5}$
a. These are like fractions
b. These are unlike fractions
c. These are unlike fractions
d. These are like fractions
We often come across a situation where we need to compare fractions.There is systematic procedure available for Comparing Fractions. It is divided into two Comparing Like Fraction which is easy and other is Comparing unlike Fraction. lets take a deep dive into it
Comparing Like Fraction
The numerator value decides the larger value.
$ \frac {5}{6} > \frac {2}{6}$
$ \frac {3}{6} > 0 $
$ \frac {1}{6} < \frac {6}{6} $
So, $ \frac {1}{10} < \frac {2}{10} < \frac {3}{10} < \frac {4}{10} $
Comparing Unlike Fraction
We can further divide into two parts.
a. Comparing fractions with same numerator
For fractions having same numerator, the fraction with the lowest denominator is the greater number
2/5 and 2/7
Here 5 < 7
So 2/5 > 2/7
b. Comparing fractions with different denominator and numerator
Here we would be using the technique of equivalent fractions. We would convert each of the fraction into equivalent fraction such that they become like fractions.Then comparison is simple. So here are the steps
1. Find the LCM of the denominators
2. Convert each fraction into equivalent fraction such that denominator is the LCM.
3. Now both the fraction are converted into Like fraction,so we can do the comparison easily
Let us check few example to make it clear
1)Compare $\frac {2}{3}$ and $\frac {2}{7}$
LCM of denominator is 3 and 7 is 21
So converting them equivalent Like fractions
$\frac {2}{3} = \frac {2 \times 7}{3 \times 7} = \frac {14}{21}$
Now $\frac {15}{21} > \frac {14}{21}$
So $\frac {5}{7} > \frac {2}{3}$
2)Compare $\frac {5}{6}$ and $\frac {13}{15}$
LCM of denominator is 6 and 15 is 30
$\frac {13}{15} = \frac {13 \times 2}{15 \times 2} = \frac {26}{30}$
So $\frac {13}{15} > \frac {5}{6}$
We often come across a situation where we need to add fractions.There is systematic procedure available for adding Fractions. It is divided into two parts Adding Like Fraction which is easy and other is Adding unlike Fraction. lets take a deep dive into it
Adding Like Fraction
Addition: The numerator adds to provide the final fraction value.
$\frac {1}{5} + \frac {1}{5} = \frac {2}{5}$
$\frac {9}{11} + \frac {1}{11} = \frac {10}{11}$
Adding UnLike Fraction
First we need to convert the unlike fraction to like fraction using the LCM of the denominators and convert each fraction into like fraction using the LCM
And then it works like "like" Fraction
Perform the below Addition
$\frac {1}{2} + \frac {1}{3}$
LCM of 2 and 3 is 6
So converting them into equivalent Like Fractions
$\frac {1}{2} =\frac {1 \times 3}{2 \times 3} =\frac {3}{6}$
$\frac {1}{2} + \frac {1}{3} = \frac {3}{6} + \frac {2}{6} =\frac {5}{6}$
$\frac {1}{5} - \frac {1}{6}$
$\frac {1}{2} + \frac {1}{3} + \frac {1}{4}$
Checkout Adding Fraction calculator
We often come across a situation where we need to subtract fractions.There is systematic procedure available for subtract Fractions. It is divided into two parts Subtracting Like Fraction which is easy and other is Subtracting unlike Fraction. lets take a deep dive into it
Subtracting Like Fraction
Subtraction: The Numerator subtract to provide the final fraction value.
$\frac {4}{5} - \frac {1}{5} = \frac {3}{5}$
$\frac {9}{11} - \frac {1}{11} = \frac {8}{11}$
Subtracting UnLike Fraction
Perform the below Subtraction
$\frac {1}{2} - \frac {1}{3}$ Solution
$\frac {1}{2} - \frac {1}{3} = \frac {3}{6} - \frac {2}{6} =\frac {1}{6}$
$\frac {2}{11} - \frac {1}{12}$
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Two mixed fractions can be added or subtracted by adding or subtracting the whole number of the two fractions and then adding or subtracting the fractional parts together.
Two mixed fractions can also be converted into improper fractions and then added or subtracted.
1)Perform the below Addition
$1\frac {1}{2} + 2\frac {1}{3}$
First way
We add the whole number and add the fractional part
$1\frac {1}{2} + 2\frac {1}{3}$ =$1 + 2+ \frac {1}{2} +\frac {1}{3}$ =$3 +\frac {1}{2} +\frac {1}{3}$ Now LCM of 2 and 3 is 6
=$3+ \frac {1}{2} + \frac {1}{3} = 3 + \frac {3}{6} + \frac {2}{6} =3 +\frac {5}{6} =3\frac {5}{6}$
Second way
We convert them into improper fraction
=$\frac {3}{2} + \frac {7}{3}$
Now LCM of 2 and 3 is 6
$\frac {7}{3} =\frac {7 \times 2}{3 \times 2} =\frac {14}{6}$
=$\frac {9}{6} + \frac {14}{6}$
=$\frac {23}{6} =3\frac {5}{6}$
1$\frac {1}{5} + 1\frac {1}{6}$
3$\frac {1}{5} - 2\frac {1}{6}$
1$\frac {1}{2} + 2\frac {1}{3} + 3\frac {1}{4}$
Question 1 Which of these fraction is greatest?
A) $\frac {11}{39}$
B)$\frac {10}{39}$
C)$\frac {1}{3}$
D)$\frac {9}{39}$
Question 2 Which of these fraction is lowest?
A)$\frac {13}{24}$
B)$\frac {1}{2}$
C)$\frac {16}{24}$
D)$\frac {17}{39}$
Question 3 which is of these is proper fraction
A)$\frac {10}{3}$
B)$\frac {17}{3}$
D)$1\frac {1}{3}$
Question 4 Which of the following is in the lowest form
A) 2/10
B) 11/121
C) 4/76
D) 11/13
Question 5 The sum $\frac {1}{11} + \frac {9}{11}$?
C)$\frac {8}{11}$
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Local stability implies global stability for the planar Ricker competition model
Nonlocal convection-diffusion volume-constrained problems and jump processes
March 2014, 19(2): 353-372. doi: 10.3934/dcdsb.2014.19.353
Isotropic realizability of electric fields around critical points
Marc Briane 1,
Institut de Recherche Mathématique de Rennes & INSA de Rennes, 20 avenue des Buttes de Cöesmes, CS 70839, 35708 Rennes Cedex 7, France
Received June 2013 Revised September 2013 Published February 2014
In this paper we study the isotropic realizability of a given regular gradient field $\nabla u$ as an electric field, namely when $u$ is solution to the equation div$\left(\sigma\nabla u\right)=0$ for some isotropic conductivity $\sigma>0$. The case of a smooth function $u$ without critical point was investigated in [7] thanks to a dynamical system approach which yields a global isotropic realizability result in $\mathbb{R}^d$. The presence of a critical point $x^*$ needs a specific treatment according to the behavior of the gradient flow in the neighborhood of $x^*$. The case where the hessian matrix $\nabla^2 u(x^*)$ is invertible with both positive and negative eigenvalues is the most favorable: the anisotropic realizability is a consequence of Morse's lemma, while the Hadamard-Perron theorem leads us to a characterization of the isotropic realizability around $x^*$ through some boundedness condition involving the laplacian of $u$ along the gradient flow. When the matrix $\nabla^2 u(x^*)$ has $d$ positive eigenvalues or $d$ negative eigenvalues, we get a strong maximum principle under the same boundedness condition. However, when the matrix $\nabla^2 u(x^*)$ is not invertible, the derivation of the isotropic realizability is much more intricate: the Hartman-Wintner theorem gives necessary conditions for the isotropic realizability in dimension two, while the dynamical system approach provides a criterion of non realizability in any dimension. The two methods are illustrated by a two-dimensional and a three-dimensional example.
Keywords: critical point., gradient system, isotropic conductivity, Electric field.
Mathematics Subject Classification: 35B27, 78A30, 37C1.
Citation: Marc Briane. Isotropic realizability of electric fields around critical points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 353-372. doi: 10.3934/dcdsb.2014.19.353
G. Alessandrini, Critical points of solutions of elliptic equations in two variables,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14 (1987), 229. Google Scholar
G. Alessandrini & R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions,, SIAM J. Math. Anal., 25 (1994), 1259. doi: 10.1137/S0036141093249080. Google Scholar
G. Alessandrini & V. Nesi, Univalent $\sigma$-harmonic mappings,, Arch. Rational Mech. Anal., 158 (2001), 155. doi: 10.1007/PL00004242. Google Scholar
D. V. Anosov, S. K. Aranson, V. I. Arnold, I. U. Bronshtejn and V. Z. Grines, Dynamical Systems. I,, translated from the Russian, 1 (1988). doi: 10.1007/978-3-642-61551-1. Google Scholar
V. I. Arnold, Ordinary Differential Equations,, translated from the third Russian edition by R. Cooke, (1992). Google Scholar
P. Bauman, A. Marini and V. Nesi, Univalent solutions of an elliptic system of partial differential equations arising in homogenization,, Indiana Univ. Math. J., 50 (2001), 747. doi: 10.1512/iumj.2001.50.1832. Google Scholar
M. Briane, G. W. Milton & A. Treibergs, Which electric fields are realizable in conducting materials?, to appear in Math. Mod. Num. Anal., (1301). Google Scholar
D. Gilbarg & N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, reprint of the 1998 edition, (1998). Google Scholar
P. Hartman & A. Wintner, On the local behavior of solutions of non-parabolic partial differential equations (I),, Amer. J. Math., 75 (1953), 449. doi: 10.2307/2372496. Google Scholar
M. Hirsch, S. Smale & R. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos,, second edition, 60 (2004). Google Scholar
J. Milnor, Morse Theory,, based on lecture notes by M. Spivak and R. Wells, 51 (1963). Google Scholar
F. Schulz, Regularity Theory for Quasilinear Elliptic Systems and Monge-Ampère Equations in Two Dimensions,, Lecture Notes in Mathematics 1445, 1445 (1990). Google Scholar
Mohameden Ahmedou, Mohamed Ben Ayed, Marcello Lucia. On a resonant mean field type equation: A "critical point at Infinity" approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1789-1818. doi: 10.3934/dcds.2017075
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Antonio Ambrosetti, Massimiliano Berti. Applications of critical point theory to homoclinics and complex dynamics. Conference Publications, 1998, 1998 (Special) : 72-78. doi: 10.3934/proc.1998.1998.72
Anna Maria Candela, Giuliana Palmieri. Some abstract critical point theorems and applications. Conference Publications, 2009, 2009 (Special) : 133-142. doi: 10.3934/proc.2009.2009.133
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Urszula Foryś, Yuri Kheifetz, Yuri Kogan. Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models. Mathematical Biosciences & Engineering, 2005, 2 (3) : 511-525. doi: 10.3934/mbe.2005.2.511
Salvatore A. Marano, Sunra Mosconi. Non-smooth critical point theory on closed convex sets. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1187-1202. doi: 10.3934/cpaa.2014.13.1187
J. K. Krottje. On the dynamics of a mixed parabolic-gradient system. Communications on Pure & Applied Analysis, 2003, 2 (4) : 521-537. doi: 10.3934/cpaa.2003.2.521
Frank Jochmann. Decay of the polarization field in a Maxwell Bloch system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 663-676. doi: 10.3934/dcds.2003.9.663
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Xavier Perrot, Xavier Carton. Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 971-995. doi: 10.3934/dcdsb.2009.11.971
Wenjing Chen, Louis Dupaigne, Marius Ghergu. A new critical curve for the Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2469-2479. doi: 10.3934/dcds.2014.34.2469
Marc Briane | CommonCrawl |
\begin{document}
\title{Existence of subcritical percolation phases for generalised weight-dependent random connection models} \begin{spacing}{0.9} \begin{abstract} We derive a sufficient condition for the existence of a subcritical percolation phase for a wide range of continuum percolation models where each vertex is embedded into Euclidean space and carries an independent weight. In contrast to many established models, the presence of an edge is not only allowed to depend on the distance and weights of its end vertices but can also depend on the surrounding vertex set. Our result can be applied in particular to models combining heavy-tailed degree distributions and long-range effects, which are typically well connected. Moreover, we establish bounds on the tail-distribution of the number of points and the diameter of the subcritical component of a typical point. The proofs rest on a multi-scale argument.
\noindent\footnotesize{{\textbf{AMS-MSC 2020}: 60K35}
\noindent\textbf{Key Words}: Phase transition, component size, geometric random graph, random connection model, boolean model, scale-free percolation, long-range percolation, interference graphs} \end{abstract} \end{spacing}
\section{Introduction} The standard objects studied in continuum percolation theory are random graphs \(\mathscr{G}^\lambda\) on the points of a homogeneous Poisson point process on \(\mathbb{R}^d\) of intensity \(\lambda>0\). The spatial embedding of the vertices enters the connection probability in a way that vertices at a short distance are likelier connected by an edge than far apart vertices. Many well established models belong to that framework, i.e., Gilbert's disc model~\cite{Gilbert61}, the random connection model~\cite{MeesterPenroseSarkar1997,Penrose2016}, the Poisson--Boolean model~\cite{Hall85,Gouere08} and its soft version~\cite{GGM2022}, continuum scale-free percolation~\cite{DeijfenHofstadHooghiemstra2013,DeprezWuthrich2019} or the age-dependent random connection model~\cite{GraLuMo2022}. The standard question in percolation theory then is whether there exists a critical Poisson intensity \(\lambda_c\in(0,\infty)\) such that the connected component of the origin (added to the graph if necessary) is infinite with a positive probability for all \(\lambda>\lambda_c\) but is finite almost surely for \(\lambda<\lambda_c\). We call the regime \((0,\lambda_c)\) the \emph{subcritical percolation phase} and \((\lambda_c,\infty)\) the \emph{supercritical percolation phase}. Often, by ergodicity of the underlying Poisson point process and the way edges are drawn, if \(\lambda>\lambda_c\) then \(\mathscr{G}^\lambda\) contains an infinite connected component almost surely and if \(\lambda<\lambda_c\) there cannot be an infinite connected component somewhere in the graph. Moreover, under very mild assumptions on the distribution of \(\mathscr{G}^\lambda\), an existing infinite component is almost surely unique~\cite{BurtonKeane89}.
In dimension \(d\geq 2\) percolation models typically contain a supercritical phase \cite{MeesterRoy1996} which is essentially a consequence of the existence of a supercritical percolation phase in nearest-neighbour Bernoulli percolation on \(\mathbb{Z}^2\)~\cite{Grimmett1999}. Therefore, for \(d\geq 2\), the proof of existence of a non-trivial phase-transition \(\lambda_c\in(0,\infty)\) reduces to prove the existence of a subcritical phase. In this article we present sufficient conditions for the existence of a subcritical percolation phase in a quite general setting. Under these conditions we are able to exhibit estimates for the tail behavior of the distribution of the Euclidean diameter and the number of points of the component of a typical point in the subcritical regime.
In a recent paper, Gracar et al.\ introduce a new coefficient \(\delta_{\rm eff}\) which the authors use to identify whether one-dimensional percolation models contain a supercritical phase~\cite{GraLuMo2022}. We show how this coefficient can be used to derive the existence of a subcritical phase in all dimensions by generalising arguments of Gou\'{e}r\'{e} for the Poisson--Boolean model~\cite{Gouere08}. In~\cite{GraLuMo2022}, the coefficient~\(\delta_{\rm eff}\) is derived for the \emph{weight-dependent random connection model}~\cite{GHMM2022}. In this class of models, containing all the aforementioned models, each vertex carries an independent mark. The connection mechanism is such that edges are drawn independently given the vertex locations and their marks. Additionally connections to spatially close vertices or vertices with small marks are preferred, where the first preference leads to clustering and the latter can be used to get heavy-tailed degree distributions. This is done in a way that clustering and degree-distribution are modelled independently, i.e., the degree distribution depends only on the way the vertex marks enter the connection probability whereas the strength of clustering is determined by the geometric restrictions alone. We build on their work but extend the setting to models where both effects are allowed to depend on each other. Moreover, we additionally allow the edges to depend on local neighbourhoods of their end vertices.
\subsection{Framework} We consider graphs where the vertex set is given by a standard Poisson point process on \(\mathbb{R}^d\) of intensity \(\lambda>0\). Each vertex carries an independent mark distributed uniformly on \((0,1)\). We denote a vertex by \(\mathbf{x}=(x,u_x)\) and refer to \(x\in\mathbb{R}^d\) as the vertex's \emph{location} and to \(u_x\in(0,1)\) as the vertex's \emph{mark}. We denote the set of all marked vertices by \(\mathcal{X}=\mathcal{X}^\lambda\) and remark that \(\mathcal{X}\) is a standard Poisson point process on \(\mathbb{R}^d\times(0,1)\) of intensity \(\lambda>0\) \cite{LastPenrose2017}. We denote by \(\mathbf{N}=\mathbf{N}(\mathbb{R}^d\times(0,1))\) the set of all at most countably subsets of \(\mathbb{R}^d\times(0,1)\) so that \(\mathcal{X}\) is a random element of \(\mathbf{N}\). We denote its law and expectation by \(\mathbb{P}^\lambda\) and \(\mathbb{E}^\lambda\). Given \(\mathcal{X}\), a pair of vertices \(\mathbf{x}=(x,u_x), \mathbf{y}=(y,u_y)\in\mathcal{X}\) is connected by an edge with probability \(\mathbf{p}(\mathbf{x},\mathbf{y},\mathcal{X}\setminus\{\mathbf{x},\mathbf{y}\})=\mathbf{p}(\mathbf{y},\mathbf{x},\mathcal{X}\setminus\{\mathbf{x},\mathbf{y}\})\). That is, whether an edge is drawn depends not only on the potential end vertices of the edge but may also depend on all other vertices. We assume that \(\mathbf{p}\) fulfills the following homogeneity condition when integrating with respect to the underlying Poisson process: For two deterministically given vertices \(\mathbf{x}\) and \(\mathbf{y}\), we have \begin{equation}
\mathbb{E}^\lambda[\mathbf{p}(\mathbf{x},\mathbf{y},\mathcal{X})]= \mathbb{E}^\lambda[\varphi(u_x, u_y, |x-y|^d,\mathcal{X})] \label{eq:varphi}, \end{equation} where \(\varphi:(0,1)\times(0,1)\times(0,\infty)\times\mathbf{N}\to[0,1]\) is measurable. Note that the deterministically given vertices are no elements of the Poisson point process with probability one. We make the following assumptions on the function \(\varphi\): \begin{enumerate}[(i)]
\item \(\varphi\) is symmetric in the first two arguments and non-increasing in the first three arguments. As a result, connections to spatially close vertices or vertices with a small mark are preferred. We also assume that \(\varphi\) is translation invariant and isotropic jointly in the third and fourth argument.
\item The integral
\[\int_0^1\int_0^1\int_0^\infty\mathbb{E}^\lambda[\varphi (s,t,r,\mathcal{X})] \mathrm{d} r\, \mathrm{d} s \, \mathrm{d} t\]
is finite. This then ensures that expected degrees are finite. \end{enumerate}
Condition~\eqref{eq:varphi} essentially says that the annealed probability of two given vertices being connected only depends on the given vertices' distance and marks. Indeed, since \(\mathcal{X}\) is a homogeneous process and \(\varphi\) is translation invariant and isotropic jointly in the third and fourth argument, the Poisson point process and its influence on the connection probability looks in expectation everywhere the same.
We denote the resulting undirected graph by \(\mathscr{G}=\mathscr{G}^\lambda\) and also write now $\mathbb{P}^\lambda$ and $\mathbb{E}^\lambda$ for the underlying probability measure and its expectation. We denote the event that \(\mathbf{x}\) and \(\mathbf{y}\) are connected by an edge by \(\mathbf{x}\sim\mathbf{y}\) and that they belong to the same connected component by \(\mathbf{x}\leftrightarrow\mathbf{y}\). For a given vertex we denote by \(\mathscr{C}(\mathbf{x})\) the connected component of \(\mathbf{x}\).
\subsection{Main result} To formulate our main result, we work on the Palm version \cite{LastPenrose2017} of the model. That is, a distinguishable typical vertex \(\mathbf{o}=(o,U_o)\) is placed at the origin, marked with an independent uniform random variable \(U_0\) and then added to the vertex set. The graph \(\mathscr{G}_o=\mathscr{G}_o^\lambda\) is then constructed as above but now with the additional vertex \(\mathbf{o}\). We denote its law by \(\mathbb{P}_o=\mathbb{P}_o^\lambda\). We define the random variables \begin{equation*}
\begin{aligned}
& \mathscr{C}=\mathscr{C}(\mathbf{o}) := \{\mathbf{x}\in\mathcal{X}: \mathbf{o}\leftrightarrow\mathbf{x}\}, \\
& \mathscr{M}= \mathscr{M}(\mathbf{o}):= \sup\{|x|^d: \mathbf{x}\in\mathscr{C}(\mathbf{o})\} \ \text{ and } \\
& \mathscr{N}= \mathscr{N}(\mathbf{o}):= \sharp\mathscr{C}(\mathbf{o}),
\end{aligned} \end{equation*} where \(\sharp A\) denotes the number of elements in a countable set \(A\). To ensure the existence of a subcritical percolation phase, we rely on two features our graph has to provide: First, the number of 'long edges` should be sufficiently small which then yields that percolation must happen locally in some sense. Second, the influence of the whole vertex set on the connection mechanism should be driven by spatially close vertices only to ensure that local percolation in two balls at a large distance can be seen as independent. To measure the intensity of 'long edges` in \(\mathscr{G}\), the key quantity is given by the following limit
\begin{equation*}
\begin{aligned}
-\lim_{\mu\downarrow 0}\liminf_{n\to\infty} \frac{\log \int\limits_{n^{-1-\mu}}^1 \int\limits_{n^{-1-\mu}}^1 \mathbb{E}^\lambda[\varphi(s,t,n,\mathcal{X})]\mathrm{d} s \mathrm{d} t}{\log n}.
\end{aligned}
\end{equation*} Choosing subsequences if needed, it is no loss of generality to assume in the following that this limit exists. We then write \begin{equation} \label{eq:deltaEff} \begin{aligned} \psi(\mu)&:=\psi(\mu,\varphi)=\lim_{n\to\infty} \frac{\log \int\limits_{n^{-1-\mu}}^1 \int\limits_{n^{-1-\mu}}^1 \mathbb{E}^\lambda[\varphi(s,t,n, \mathcal{X})]\mathrm{d} s \mathrm{d} t}{\log n}\quad\text{ and }\\
\delta_{\rm eff} &:= \delta_{\rm eff}(\varphi) =-\lim_{\mu\downarrow 0}\psi(\mu,\varphi)=-\inf_{\mu>0}\psi(\mu,\varphi) \end{aligned} \end{equation}
and call $\delta_{\rm eff}$ the \emph{effective decay exponent} (associated with \(\varphi\)). The effective decay exponent \(\delta_{\text{eff}}\) quantifies the occurrence of 'long edges` in a way comparable to classical long-range percolation where each pair of vertices \(x\) and \(y\) is connected by an edge independently with probability proportional to \(|x-y|^{-d\delta}\) for some \(\delta>1\). Indeed, in that scenario we have \(\delta_{\text{eff}}=\delta\). For more background on \(\delta_\text{eff}\), we refer to~\cite{GraLuMo2022}.
Let us write \(\mathcal{B}_r(x)\) for the open ball of radius \(r\) centered in \(x\) and \(\mathcal{B}_r:=\mathcal{B}_r(o)\). To specify local percolation and quantify the influence of far apart vertices on the connection mechanism, we need to introduce some notation. For measureable domains \(D\subset\mathbb{R}^d\) and \(I\subset(0,1)\) we write \[
\mathcal{X}(D\times I)=\{\mathbf{x}=(x,u_x)\in\mathcal{X}\colon x\in D, u_x\in I\}. \] If \(I=(0,1)\), we simply write \(\mathcal{X}(D)=\mathcal{X}(D\times(0,1))\). Further, we denote by \(\mathscr{C}_D(\mathbf{x})\) the connected component of \(\mathbf{x}\) restricted to the vertices located in \(D\). For a given location \(x\in\mathbb{R}^d\) and \(\alpha>1\), we define the event \begin{equation} \label{eq:GEvent}
G_\alpha(x)= \big\{\exists \mathbf{y}\in\mathcal{X}(\mathcal{B}_{\alpha^{1/d}}(x)):\mathscr{C}_{\mathcal{B}_{10\alpha^{1/d}}(x)}(\mathbf{y})\not\subset\mathcal{X}(\mathcal{B}_{8\alpha^{1/d}}(x))\big\}. \end{equation} That is, the vertex \(\mathbf{y}\) located close to \(x\) reaches with a path a vertex located in the annulus \(\mathcal{B}_{10\alpha^{1/d}}(x)\setminus\mathcal{B}_{8\alpha^{1/d}}\) without using vertices located outside \(\mathcal{B}_{10\alpha^{1/d}}(x)\). We abbreviate \(G_\alpha = G_\alpha(o)\). \begin{comment} To specify local percolation and quantify the influence of far apart vertices on the connection mechanism, we define for a given vertex \(\mathbf{x}=(x,u_x)\) and \(\alpha>0\) the event \(G_\alpha(\mathbf{x})\) that \(\mathbf{x}\) is connected by a path to a vertices \(\mathbf{y}\) located in the annulus \(\mathcal{B}_{(10\alpha)^{1/d}}(x)\setminus\mathcal{B}_{(8\alpha)^{1/d}}\) using only vertices located in \(\mathcal{B}_{(10\alpha)^{1/d}}\). We abbreviate \(G_\alpha = G_\alpha(\mathbf{o})\). \end{comment}
We say that \(\mathscr{G}^\lambda\) is \emph{mixing} if there exist \(\zeta>0\) and \(C_\text{mix}>0\) such that for all \(\lambda>0\) and all \(|x|>30\alpha^{1/d}\), we have \begin{equation}\label{eq:mixing}
\big|\operatorname{Cov}\big(\mathbbm{1}_{G_{\alpha}},\mathbbm{1}_{G_\alpha(x)}\big)\big| \leq C_\text{mix} \, \lambda \alpha^{-\zeta}. \end{equation} Note that, in the examples mentioned in the introduction, \(\mathbf{p}(\mathbf{y},\mathbf{x},\mathcal{X}\setminus\{\mathbf{x},\mathbf{y}\})=\mathbf{p}(\mathbf{y},\mathbf{x})\) and hence all of them are mixing in our sense.
\begin{theorem}[Existence of a subcritical phase] \label{thm:Subcritical}
If \(\delta_{\rm eff}>2\) and $\mathscr{G}$ is mixing,
then there exists a critical intensity \(\lambda_c>0\) such that for all \(\lambda<\lambda_c\)
\[
\mathbb{P}_o^\lambda(\mathscr{N}<\infty)=1\qquad\text{ and }\qquad \mathbb{P}_o^\lambda(\mathscr{M}<\infty)=1.
\] \end{theorem}
Let us remark that for the proof of Theorem~\ref{thm:Subcritical} it suffices if the right-hand side in~\eqref{eq:mixing} is replaced by \(\lambda g(\alpha)\) where \(g(\alpha)\) tends to zero at an arbitrary speed. However, to derive bounds on the decay of $\mathbb{P}_o^\lambda(\mathscr{N}\ge y)$ and $\mathbb{P}_o^\lambda(\mathscr{M}\ge y)$ we need that the graph mixes fast enough.
\begin{theorem}[Decay properties in the subcritical phase] \label{thm:SubcriticalDecay} Let $s>0$ and assume that $\psi(s+1)< -(s+3)$ as well as $\zeta> s+1$, then there exists \(\lambda'_c>0\) such that for all $t<s$ and all \(\lambda<\lambda'_c\), \begin{equation*}
\int_1^\infty y^t\mathbb P_o(\mathscr{M}\ge y)\mathrm{d} y<\infty\qquad\text{ and }\qquad \int_1^\infty y^t\mathbb P_o(\mathscr{N}\ge y)\mathrm{d} y<\infty. \end{equation*} \end{theorem}
We present the proofs of both theorems in Section~\ref{sec:Proofs}.
\subsection{Examples} \paragraph{The weight-dependent random connection model} This model was introduced in~\cite{GHMM2022} and further studied in~\cite{GLM2021, GraLuMo2022,GGM2022}. Here, edges are drawn conditionally independent given \(\mathcal{X}\) and we have \begin{align}\label{ex_1}
\mathbf{p}(\mathbf{x},\mathbf{y},\mathcal{X}\setminus\{\mathbf{x},\mathbf{y}\})=\rho\big(\beta^{-1} g(u_x,u_y)|x-y|^d\big), \end{align}
where \(\rho:(0,\infty)\to[0,1]\) is an integrable and non increasing \emph{profile function} and \(g:(0,1)^2\to(0,\infty)\) is a non increasing \emph{kernel function} which is symmetric in both arguments. Here, \(\beta>0\) controls the edge intensity of the graph by scaling the vertices' distance in the connection probability. By the Poisson point process mapping theorem~\cite{LastPenrose2017} it is no loss of generality to fix \(\beta=1\) and only vary the Poisson intensity \(\lambda\) or doing the opposite and fixing \(\lambda=1\) whilst varying \(\beta\). Two types of profile functions have been established in the literature. The \emph{long-range} profile function \(\rho(x):=p(1\wedge |x|^{-\delta})\) for \(\delta> 1\) or the \emph{short-range} profile function \(\rho(x):=p\mathbbm{1}\{0\le x\le 1\}\) for some \(p\in(0,1]\). These profile functions together with the \emph{interpolation-kernel}
\[
g(s,t):= (s\wedge t)^{\gamma}(s\vee t)^{\gamma'}, \text{ for } \gamma\in[0,1), \gamma'\in[0,2-\gamma),
\] introduced in~\cite{GraLuMo2022}, represent many of the literature's model such as the Poisson--Boolean model and its soft version, scale-free percolation and the age-dependent random connection model, cf.~Table~\ref{tab:interPol}.
\begin{table} \begin{center} \caption{Various choices for \(\gamma\), \(\gamma'\) and \(\delta\) for the weight-dependent random connection model and the models they represent in the literature. Here, to shorten notation, \(\delta=\infty\) represents models constructed with \(\rho\) being the indicator function}
\begin{tabular}{l|l}
\textbf{Parameters} & \textbf{Names and references} \\ \hline
\(\gamma = 0, \gamma' = 0, \delta = \infty\) & random geometric graph, Gilbert's disc model \cite{Gilbert61} \\
\(\gamma = 0, \gamma' =0, \delta<\infty\) & random connection model \cite{MeesterPenroseSarkar1997,Penrose2016}, \\ & long-range percolation \cite{Schulman1983}\\
\(\gamma>0, \gamma'=0,\delta=\infty\) & Boolean model \cite{Hall85,Gouere08}, scale-free Gilbert graph \cite{Hirsch2017} \\
\(\gamma>0, \gamma' = 0, \delta<\infty\) & soft Boolean model \cite{GGM2022} \\
\(\gamma = 0, \gamma'>1, \delta = \infty\) & ultra-small scale-free geometric network \cite{Yukich2006} \\
\(\gamma>0, \gamma'=\gamma, \delta\leq \infty\) & scale-free percolation \cite{DeijfenHofstadHooghiemstra2013,DeprezWuthrich2019}, \\ & geometric inhomogeneous random graphs \cite{BringmannKeuschLengler2019} \\
\(\gamma>0, \gamma'=1-\gamma, \delta\leq\infty\) & age-dependent random connection model \cite{GGLM2019} \end{tabular} \label{tab:interPol} \end{center} \end{table}
The model~\eqref{ex_1} has also been studied under the name \emph{geometric inhomogeneous random graphs} in a similar yet slightly different parametrisation in~\cite{KomjathyLapinskasLengler2021,BringmannKeuschLengler2019,HofstadHoornMaitra2023}. Since \(\delta_\text{eff}>2\) in case that \(\delta>2\), \(\gamma<1-\nicefrac{1}{\delta}\) and \(\gamma'<1-\gamma\), there always exists a subcritical percolation phase in these cases, cf.\ Figure~\ref{fig:Percolation}. The model~\eqref{ex_1} also shows that the question of non existence of a subcritical phase cannot be answered with \(\delta_\text{eff}\) alone. Indeed, in~\cite{GLM2021} it is shown that for \(\gamma<\nicefrac{\delta}{(\delta+1)}\) and \(\gamma'\leq 1-\gamma\) there always exists a subcritical phase for all \(\delta>1\) which also includes parameter regimes with \(\delta_{\rm eff}\leq 2\).
Let us mention that, in order to get bounds for the decay of \(\mathscr{M}\) and \(\mathscr{N}\), we identify for \(\delta>2\) and \(\gamma'>\nicefrac{1}{\delta}\) that \begin{equation*}
\begin{aligned}
\delta_{\rm eff} > s+3\quad \Leftrightarrow\quad \gamma < \frac{\delta(1-\gamma')-s-1}{\delta}.
\end{aligned} \end{equation*} Since we also know from above that \(\gamma<1-\nicefrac{1}{\delta}\) this inequality has a valid solution whenever \(\gamma'>\nicefrac{s}{\delta}\). This is particularly satisfied for all \(s\leq 1\). If however \(\gamma'<\nicefrac{1}{\delta}\), we infer for \(\gamma>\nicefrac{1}{\delta}\) that \begin{equation*}
\begin{aligned}
\delta_{\rm eff} > s+3\quad \Leftrightarrow\quad \gamma <\frac{\delta-s-2}{\delta},
\end{aligned} \end{equation*} which has a valid solution whenever \(s\leq 1\). This case includes in particular the soft Boolean model (\(\gamma'=0\)). Finally, if also \(\gamma<\nicefrac{1}{\delta}\), then simply \(\delta_{\rm eff}=\delta\).
\begin{figure}
\caption{Phase diagram in \(\gamma\) and \(\gamma'\) for the weight-dependent random connection model constructed with the interpolation kernel and a profile function of polynomial decay at rate \(\delta>2\). The solid lines marks the phase transition \(\delta_{\rm eff}=2\). Dashed lines represent no change of behaviour.}
\label{fig:Percolation}
\end{figure}
\paragraph{Soft Boolean model with local interference} We also present an example of a mixing graph where the edge probabilities indeed dependent on the surrounding point cloud. The idea is to combine the soft Boolean model~\cite{GGM2022} with local inference and noise in the spirit of SINR percolation~\cite{DousseEtAl2006, Tobias2020}. To formulate the model let us denote, for a given vertex \(\mathbf{y}=(y,s)\) and \(\xi\geq 0\), the random variable \[
N^\xi(\mathbf{y},\mathcal{X}):= \sharp \big\{\mathbf{z}\in\mathcal{X} : |z-y|^d\leq s^{-\xi}\big\}. \] The graph \(\mathscr{G}=\mathscr{G}^\lambda\) is then generated by connecting \(\mathbf{x}=(x,t)\) and \(\mathbf{y}=(y,s)\) with probability \begin{align}\label{ex_2}
\mathbf{p}(\mathbf{x},\mathbf{y},\mathcal{X}\setminus\{\mathbf{x},\mathbf{y}\}) = \mathbbm{1}_{\{s<t\}}\frac{1\wedge s^{-\gamma\delta}|x-y|^{-d\delta}}{1+N^{\xi}(\mathbf{y},\mathcal{X}\setminus \{\mathbf{x},\mathbf{y}\})}+\mathbbm{1}_{\{s\geq t\}}\frac{1\wedge t^{-\gamma\delta}|x-y|^{-d\delta}}{1+N^{\xi}(\mathbf{x},\mathcal{X}\setminus \{\mathbf{x},\mathbf{y}\})}, \end{align} where again \(\gamma\in(0,1)\) and \(\delta>2\). Since \(N_\lambda^\xi((y,s),\mathcal{X})\overset{d}{\sim}N_\lambda^\xi((o,s),\mathcal{X})\), Condition~\eqref{eq:varphi} is satisfied with \[
\varphi(s,t,r,\mathcal{X})= \mathbbm{1}_{\{s<t\}}\frac{1\wedge s^{-\gamma\delta}r^{-\delta}}{1+N^{\xi}((o,s),\mathcal{X})}+\mathbbm{1}_{\{s\geq t\}}\frac{1\wedge t^{-\gamma\delta}r^{-\delta}}{1+N^{\xi}((o,t),\mathcal{X})}. \] Let us note that the model~\eqref{ex_2} is a combination of the soft Boolean model, a special instance of the weight-dependent random connection model above, with random interference coming from the vertices surrounding the vertices that are to be connected. More precisely, each vertex \(\mathbf{y}=(y,s)\) has a sphere of influence of radius \(s^{-\gamma/d}\) and a sphere of interference of radius \(s^{-\xi/d}\). The mark \(s\) can be understood as an inverse attraction parameter and the smaller \(s\) the more attractive the vertex is. Note that either both spheres are big (for small \(s\)) or small (for large \(s\)). Now, the vertex \(\mathbf{y}\) likes to connect to each vertex located within its sphere of influence, enlarged by an independent Pareto random variable for each candidate to include long-range effects (cf.\ the description of the soft Boolean model in~\cite{GGM2022}). However, \(\mathbf{y}\) gets distracted by all vertices contained in its sphere of interference which makes it more difficult to form edges. For \(\xi=0\), each sphere of interference is of radius one and the model reduces to a version of the soft Boolean model with some additional yet on large scales insignificant fluctuations.
We start by showing that \(\mathscr{G}\) is mixing for \(\xi<1\). To this end, observe that the left-hand side in~\eqref{eq:mixing} is bounded by some constant times the probability of the complement of the largest event on which the covariance is zero. Further observe that \(G_\alpha\) and \(G_\alpha(x)\) are independent whenever there is no pair of vertices in the involved balls such that their spheres of interference intersect. In other words, the covariance in \eqref{eq:mixing} differs from zero when there are vertices \(\mathbf{y}\in\mathcal{X}(\mathcal{B}_{10\alpha^{1/d}})\) and \(\mathbf{z}\in\mathcal{X}(\mathcal{B}_{10\alpha^{1/d}}(x))\) such that \[
u_y^{-\xi}+u_z^{-\xi}\geq |z-y|^d. \]
We recall that \(|x|^d\geq 30^d\alpha\) and hence \(|z-y|^d\geq c\alpha\) for some constant \(c\). Moreover, \(u_y^{-\xi}+u_z^{-\xi}<2(u_y^{-\xi}\vee u_z^{-\xi})\). Hence, the question reduces to the question whether there exists a vertex in one of the two balls with a mark smaller than \(c\alpha^{-\nicefrac{1}{\xi}}\). But, the expected number of such vertices and hence also the probability of existence of at least one such vertex is bounded by \[
C \lambda \alpha^{1-\nicefrac{1}{\xi}} \] for some constant $C$. Now, since \(\xi<1\), the graph is mixing with exponent \(\zeta=\nicefrac{1}{\xi}-1\). Let us further calculate some values of \(\delta_{\rm eff}\), see Figure~\ref{fig:DistractBool}. We write \(f\asymp g\) for positive functions if \(f/g\) is uniformly bounded from zero and infinity. Since \(N^\xi((o,s),\mathcal{X})\) is Poisson distributed with parameter of order \(s^{-\xi}\), we infer \begin{equation*}
\begin{aligned}
n^{-\delta}\int_{1/n}^1 s^{-\gamma\delta} \mathbb{E}^\lambda\Big[\frac{1}{1+ N^\xi((o,s),\mathcal{X})}\Big] \mathrm{d} s &\asymp n^{-\delta}\int_{1/n}^1 s^{-\gamma\delta+\xi} \mathrm{d} s \asymp n^{-\delta}\vee n^{-\delta(1-\gamma)-1-\xi}.
\end{aligned} \end{equation*} Hence, if \(\gamma<\nicefrac{(1+\xi)}{\delta}\), then \(\delta_{\rm eff}=\delta\). This in particular always true if \(\xi>\delta-2\). In the case \(\gamma>\nicefrac{(1+\xi)}{\delta}\), we have \begin{equation*}
\begin{aligned}
\delta_{\rm eff}>2 & \quad \Leftrightarrow\quad \delta(1-\gamma)+\xi>1\quad \Leftrightarrow \quad\gamma<\frac{\delta+\xi-1}{\delta}.
\end{aligned} \end{equation*} For \(\xi=0\) we recover the bound for the soft Boolean model found in the previous paragraph. For \(\xi>0\) we observe that the local distraction indeed makes it harder to percolate. To get bounds on the tail distribution of \(\mathscr{M}\) and \(\mathscr{N}\) when \(\delta_{\rm eff}\neq\delta\), we observe for the mixing parameter \[
\frac{1}{\xi}-1>s+1 \quad\Leftrightarrow \quad\xi>\frac{1}{\delta+2} \] and for \(\gamma>\nicefrac{(1+\xi)}{\delta}\), we infer \[
\delta_{\rm eff}>s+3\quad\Leftrightarrow \quad\gamma<\frac{\delta+\xi-s-2}{\delta}, \] which has a valid solution if the right-hand side is positive, which is equivalent to \( s<\delta+\xi-2\). \begin{figure}
\caption{Phase diagram for \(\gamma\) and \(\delta\) for the soft Boolean model with local interference. The dashed, dashed-dotted, dotted and dashed-double dotted lines represent the phase transition for \(\delta_{\rm eff}>2\) for \(\xi=0,0.3,0.6\) and \(0.9\).}
\label{fig:DistractBool}
\end{figure}
\section{Proofs}\label{sec:Proofs} We employ a multiscale argument similar to the one used for the Poisson--Boolean model in~\cite{Gouere08}. Recall the notation of \[
\mathcal{X}(D\times I)=\{\mathbf{x}=(x,u_x)\in\mathcal{X}\colon x\in D, u_x\in I\} \] and \(\mathcal{X}(D)=\mathcal{X}(D\times(0,1))\). Further recall the event \(G_\alpha(x)\) and \(G_\alpha\) introduced in~\eqref{eq:GEvent}. \begin{comment} Further, we write \(\mathbf{x}\xleftrightarrow[D\times I]{}\mathbf{y}\) (resp.\ \(\mathbf{x}\xleftrightarrow[D]{}\mathbf{y}\)) for the event that \(\mathbf{x}\) and \(\mathbf{y}\) are connected by a path using only vertices from \(\mathcal{X}(D\times I)\) (resp.\ \(\mathcal{X}(D)\)). For a given vertex \(\mathbf{x}=(x,u_x)\) and \(\alpha>0\), recall the event \begin{equation*}
G_\alpha(\mathbf{x})=\Big\{\exists \mathbf{y}\in\mathcal{X}\big(\mathcal{B}_{(10\alpha)^{1/d}}(x)\setminus\mathcal{B}_{(8\alpha)^{1/d}}(x)\big)\colon \mathbf{x}\xleftrightarrow[\mathcal{B}_{(10\alpha)^{1/d}}(x)]{}\mathbf{y}\Big\} \end{equation*} and the abbreviation \(G_\alpha = G_\alpha(\mathbf{o})\). \end{comment} Let us write $A^c=\mathbb{R}^d\setminus A$, for the complement of any $A\subset \mathbb{R}^d$, and define two further events \begin{equation*}
\begin{aligned}
H_\alpha&=\{\exists \mathbf{x}\in \mathcal{X}(\mathcal{B}^c_{10\alpha^{1/d}}) \text{ and } \mathbf{y}\in \mathcal{X}(\mathcal{B}_{9\alpha^{1/d}})\colon \mathbf{x}\sim \mathbf{y}\} \quad \text{ and}\\
F_\alpha&=\{\exists\mathbf{y}\in \mathcal{X}(\mathcal{B}_{100\alpha^{1/d}}) \text{ and } \mathbf{x}\in \mathcal{X}\colon \mathbf{x}\sim \mathbf{y}\text{ and }|x-y|^d\geq \alpha\}.
\end{aligned} \end{equation*}
We start by proving that both \(\mathbb{P}(H_\alpha)\) and \(\mathbb{P}(F_\alpha)\) tend to zero as \(\alpha\to\infty\) whenever \(\delta_\text{eff}>2\).
\begin{lemma}\label{lem:Errors} ~\ \begin{enumerate}[(i)]
\item If \(\delta_{\rm eff}>2\), then for all \(\varepsilon>0\) such that \(\delta_{\rm eff}- \varepsilon>2\) there exists \(\mu>0\) and a constant \(C\) only depending on the dimension \(d\) and the choice of \(\varepsilon\) and \(\mu\) such that
\begin{equation*}
\mathbb{P}(H_\alpha) \leq C (\lambda \vee \lambda^2) \alpha^{-\eta}
\end{equation*}
with \(\eta = (\delta_{\rm eff}-2-\varepsilon)\wedge \mu\).
\item If \(\delta_{\rm eff}>2\), then for all \(\varepsilon>0\) such that \(\delta_{\rm eff}- \varepsilon>2\) there exists \(\mu>0\) and a constant \(C'\) only depending on the dimension \(d\) and the choice of \(\varepsilon\) and \(\mu\) such that
\begin{equation*}
\mathbb{P}(F_\alpha) \leq C' (\lambda\vee \lambda^2) \alpha^{-\eta}
\end{equation*}
with \(\eta = (\delta_{\rm eff}-2-\varepsilon)\wedge \mu\). \end{enumerate} \end{lemma} \begin{proof}
Define the event
\[
\widetilde{H}_\alpha := \{\exists \mathbf{x}\in \mathcal{X}(\mathcal{B}^c_{10\alpha^{1/d}}) \text{ and } \mathbf{y}\in \mathcal{X}(\mathcal{B}_{9\alpha^{1/d}})\colon u_x\geq |x|^{-d(1+\mu)}, u_y\geq |x|^{-d(1+\mu)} \text{ and }\mathbf{x}\sim \mathbf{y}\}
\]
and note that
\[H_\alpha\subset\widetilde{H}_\alpha\cup\{\exists\mathbf{x}\in\mathcal{X}(\mathcal{B}^c_{10\alpha^{1/d}})\colon |u_x|<|x|^{-d(1+\mu)}\}\cup\{\exists\mathbf{y}\in\mathcal{X}(\mathcal{B}_{9\alpha^{1/d}})\colon |u_y|<10^d\alpha^{-(1+\mu)}\}.\]
By standard Poisson process properties, the last two events have probabilities of order
\[
\lambda \int\limits_{|x|^d>10^d\alpha} \mathrm{d} x \ |x|^{-d(1+\mu)} = \tfrac{\pi(d)}{d\mu 10^{\mu}} \lambda \alpha^{-\mu}\qquad\text{ and }\qquad \lambda\int\limits_{|y|^d<9^d\alpha} \mathrm{d} y \ (10^d\alpha)^{-(1+\mu)} = \tfrac{9\pi(d)}{10^{d(1+\mu)}} \lambda\alpha^{-\mu},
\]
where \(\pi(d)\) denotes the volume of the \(d\)-dimensional unit ball. For the event \(\widetilde{H}_\alpha\), for any $\varepsilon>0$ and all sufficiently large $\alpha$ and sufficiently small $\mu$, we calculate using the Mecke-equation~\cite{LastPenrose2017}, \eqref{eq:varphi} and the definition of \(\delta_{\rm eff}\),
\begin{equation*}
\begin{aligned}
\mathbb{P}(\widetilde{H}_\alpha) & \leq \mathbb{E}^\lambda\Big[\sum_{\mathbf{y}\in\mathcal{X}(\mathcal{B}_{9\alpha^{1/d}})}\sum_{\mathbf{x}\in\mathcal{X}(\mathcal{B}_{10\alpha^{1/d}}^c)}\mathbbm{1}\{u_x,u_y>|x|^{-d(1+\mu)}\}\mathbf{p}(\mathbf{x},\mathbf{y},\mathcal{X}\setminus\{\mathbf{x},\mathbf{y}\})\Big] \\
& = \lambda^2\int\limits_{|y|^d<9^d\alpha}\mathrm{d} y \int\limits_{|x|^d>10^d\alpha}\mathrm{d} x\int\limits_{|x|^{-d(1+\mu)}}\mathrm{d} u_y \int\limits_{|x|^{-d(1+\mu)}}\mathrm{d} u_x \; \mathbb{E}^\lambda[\mathbf{p}(\mathbf{x},\mathbf{y},\mathcal{X})] \\
& = \lambda^2\int\limits_{|y|^d<9^d\alpha}\mathrm{d} y \int\limits_{|x|^d>10^d\alpha} \mathrm{d} x \int\limits_{|x|^{-d(1+\mu)}}^{1}\mathrm{d} u_y \int\limits_{|x|^{-d(1+\mu)}}^1\mathrm{d} u_x \mathbb{E}^\lambda\big[\varphi(u_x,u_y,|x-y|^d,\mathcal{X})\big] \\
& \leq C \lambda^2 \pi(d) \alpha \int\limits_{|x|^d> \alpha} \mathrm{d} x \int\limits_{|x|^{-d(1+\mu)}}^{1}\mathrm{d} u_y \int\limits_{|x|^{-d(1+\mu)}}^1\mathrm{d} u_x \ \mathbb{E}^\lambda[\varphi(u_x,u_y,|x|^d,\mathcal{X})] \\
& \leq C\lambda^2 \pi(d) \alpha\int\limits_{|x|^d> \alpha} |x|^{-d(\delta_{\rm eff}-\varepsilon)} \mathrm{d} x \\
&\leq \tfrac{C\pi(d)^2}{\delta_\text{eff}-\varepsilon-1} \lambda^2 \alpha^{2-\delta_{\rm eff}+\varepsilon}.
\end{aligned}
\end{equation*}
The proof of (ii) works analogous to the one of (i). We again use the same cut-off of the vertex marks and consider the coinciding event \(\widetilde{F}_\alpha\) and have
\begin{equation*}
\begin{aligned}
\mathbb{P}(\widetilde{F}_\alpha) & \leq \lambda^2\int\limits_{|y|^d<100^d\alpha}\mathrm{d} y \int\limits_{|x-y|^d>\alpha} \mathrm{d} x \int\limits_{|x|^{-d(1+\mu)}}^{1}\mathrm{d} u_y \int\limits_{|x|^{-d(1+\mu)}}^1\mathrm{d} u_x \ \mathbf{E}^\lambda[\varphi(s,t,|x-y|^d,\mathcal{X})] \\
& \leq C\lambda^2 \pi(d) \alpha \int\limits_{|x|^d>\alpha} \mathrm{d} x \int\limits_{|x|^{-d(1+\mu)}}^{1}\mathrm{d} u_y \int\limits_{|x|^{-d(1+\mu)}}^1\mathrm{d} u_x \ \mathbf{E}^\lambda[\varphi(s,t,|x|^d,\mathcal{X})] \\
&\leq C' \lambda^2 \alpha^{2-\delta_{\rm eff}+\varepsilon},
\end{aligned}
\end{equation*}
as desired. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:Subcritical}] Note that, as in~\cite[Proposition 3.1]{Gouere08}, for the diameter we have that \begin{equation}
\mathbb{P}_o(\mathscr{M}\ge 9^d\alpha)\le \mathbb{P}(G_\alpha)+\mathbb{P}(F_\alpha). \label{eq:MBound} \end{equation} On the other hand, for the number of points, observe that \[
G_\alpha^c\cap H_\alpha^c\subset \{\mathscr{C}\subset \mathcal X(\mathcal B_{10\alpha^{1/d}})\}\subset \{\mathscr N\le \# \mathcal X(\mathcal B_{10\alpha^{1/d}})\}. \] Moreover, by a standard Chernoff bound for the Poisson point process, there exists constants \(c, c'>0\) such that \[
\mathbb{P}^\lambda (\sharp\mathcal{X}(B_{10\alpha^{1/d}})>c\lambda \alpha) \leq e^{-\nicefrac{c'\alpha}{\lambda}} \] and hence \begin{equation}\label{eq:NBound}
\begin{aligned}
\mathbb{P}_o(\mathscr{N} >c\lambda \alpha)&\leq \mathbb{P}_o(\mathscr{N}> \sharp \mathcal X(\mathcal{B}_{10\alpha^{1/d}})) + \mathbb{P}^\lambda(\sharp\mathcal{X}(B_{10\alpha^{1/d}})>c\lambda \alpha) \\ & \leq \mathbb{P}(G_\alpha)+\mathbb{P}(H_\alpha)+e^{-\nicefrac{c'\alpha}{\lambda}}.
\end{aligned} \end{equation}
Hence, it suffices to prove that $\mathbb{P}(G_\alpha)$ tends to zero as $\alpha$ tends to infinity for all sufficiently small $\lambda$. From now on, we always assume that \(\lambda<1\) and therefore in particular \(\lambda^2<\lambda\). We write \(S_r\) for the sphere of radius \(r\) and define two finite sets \(\mathcal{K},\mathcal{L}\subset\mathbb{R}^d\) satisfying \(\mathcal{K}\subset S_{10}\) and \(\mathcal{L}\subset S_{80}\) as well as \[
S_{10}\subset \mathcal{K}+\mathcal{B}_1 \text{ and } S_{80}\subset \mathcal{L}+\mathcal{B}_1. \] The key observation for the remaining proof is that \[
G_{10^d\alpha}\setminus F_\alpha\subset \Big(\bigcup_{k\in \mathcal{K}}G_\alpha(\alpha k)\Big)\cap\Big(\bigcup_{{l}\in \mathcal{L}}G_\alpha(\alpha{l})\Big). \] This is because on \(G_{10^d\alpha}\setminus F_\alpha\) there exists a path from a vertex located in \(\mathcal{B}_{10\alpha^{1/d}}\) to some vertex located in the annulus \(\mathcal{B}_{100\alpha^{1/d}}\setminus\mathcal{B}_{80\alpha^{1/d}}\) using only vertices located in \(\mathcal{B}_{100\alpha^{1/d}}\) and edges no longer than \(\alpha^{1/d}\). Obviously, the sphere \(S_{10\alpha^{1/d}}\) as well as the sphere \(S_{80\alpha^{1/d}}\) are crossed by an edge. By the covering property of \(\mathcal{K}\) and the fact that all edges are shorter than \(\alpha^{1/d}\), one of the end vertices of the edge crossing \(S_{10\alpha^{1/d}}\) is located in \(\mathcal{B}_{\alpha^{1/d}}(\alpha k)\) for some \(k\in\mathcal{K}\). Let's denote this vertex by \(\mathbf{x}_k\). As the path needs to reach \(\mathbf{x}_k\) and all edges are shorter than \(\alpha^{1/d}\), the path also has to pass a vertex \(\mathbf{x}_{\text{annul}}\) located in \(\mathcal{B}_{10\alpha^{1/d}}(\alpha k)\setminus\mathcal{B}_{8\alpha^{1/d}}(\alpha k)\) that is connected by a path to \(\mathbf{x}_k\) using only vertices located in \(\mathcal{B}_{10\alpha^{1/d}}\). Put differently, \(G_\alpha(\alpha k)\) occurs, see Figure~\ref{fig:my_label}. Using the same arguments, \(G_{\alpha}(\alpha l)\) occurs for some \(l\in\mathcal{L}\) and as a result, \[
\mathbb{P}(G_{10\alpha}\setminus F_\alpha) \leq \sum_{k\in \mathcal{K},\, l\in \mathcal{L}} \mathbb{P}(G_\alpha (\alpha k)\cap G_\alpha(\alpha l)). \] Therefore, there exists a constant \(C_1\) depending on the choice of \(\mathcal{K}\) and \(\mathcal{L}\) such that \[
\mathbb{P}(G_{10\alpha})\le C_1 \ \mathbb{P}(G_\alpha)^2 + \mathbb{P}(F_\alpha) + C_1\max_{k\in\mathcal{K},\, l\in\mathcal{L}}\Big(\mathbb{P}\big(G_\alpha(\alpha k)\cap G_{\alpha}(\alpha l)\big)-\mathbb{P}(G_\alpha)^2\Big). \] Now, by the mixing assumption~\eqref{eq:mixing} and translation invariance, we have \[
\max_{k\in \mathcal{K},\, l \in\mathcal{L}}\Big(\mathbb{P}\big(G_\alpha(\alpha k)\cap G_{\alpha}(\alpha l)\big)-\mathbb{P}(G_\alpha)^2\Big) \leq C_{\text{mix}}\lambda\alpha^{-\zeta}. \] Combining this with Lemma~\ref{lem:Errors}, we find a constant \(C_2\) such that \begin{equation}\label{eq:P(G)bound}
\mathbb{P}(G_{10\alpha})\leq C_2 \mathbb{P}(G_\alpha)^2 + C_2\lambda \alpha^{-(\eta\wedge \zeta)}. \end{equation} Let us define the functions \(f(\alpha)=C_2\mathbb{P}(G_\alpha)\) and \(g(\alpha)=C_2^2 \lambda \alpha^{-(\eta\wedge\zeta)}\) and set \[
\lambda_0=\frac{1}{2\cdot 10^d C_2\pi(d)} \wedge \frac{1}{4 C_2^2}. \] Then, we have for all \(\lambda<\lambda_0\) that \[
C_2\mathbb{P}(G_\alpha) \leq C_2 \mathbb{E}^\lambda[\sharp\mathcal{X}(\mathcal{B}_{10\alpha^{1/d}})] = 10^d C_2 \pi(d) \lambda\alpha \leq 1/2, \] for all \(\alpha\in[1,10^d]\) as well as \(g(\alpha)\leq \nicefrac{1}{4}\) for all \(\alpha\geq 1\). Moreover, by~\eqref{eq:P(G)bound} \[
f(\alpha)\leq f(\nicefrac{\alpha}{10^d})^2 + g(\alpha) \] and therefore \(\mathbb{P}(G_\alpha)\to 0\) as \(\alpha\to\infty\) by~\cite[Lemma 3.7]{Gouere08} as desired. \end{proof}
\begin{figure}\label{fig:my_label}
\end{figure}
\begin{proof}[Proof of Theorem~\ref{thm:SubcriticalDecay}] First note that, since $\mu\mapsto\psi(\mu)$ is increasing, we have $$ \delta_{\rm eff}\ge -\psi(s+1)> s+3 $$ and hence, for $\varepsilon=\delta_{\rm eff} -3-s>0$, it holds that $\delta_{\rm eff}-2-\varepsilon= s+1$. Now, define $$ \mu(\varepsilon):=\sup\left\{\mu>0\colon \psi(\mu)+\delta_{\rm eff}\le \varepsilon \right\} $$ and note that $\psi(s+1)+\delta_{\rm eff}=\psi(s+1)+s+3+\varepsilon<\varepsilon$ and hence, by monotonicity, also $\mu(\epsilon)\ge s+1$. But then, $\eta=(\delta_{\rm eff}-2-\varepsilon)\wedge \mu(\varepsilon)\wedge \zeta\ge s+1$. Now, recall the functions \(f(\alpha)=C_2\mathbb{P}(G_\alpha)\) and \(g(\alpha)=C_2^2\lambda\alpha^{-\eta}\) from the previous proof and fix \(\lambda<\lambda_0\). By the choice of \(\eta\) we have for all \(t<s\) that \[
\int_1^\infty \alpha^t g(\alpha)\mathrm{d} \alpha \leq C \int_1^\infty \alpha^{t-(s-1)}\mathrm{d} \alpha <\infty \] and also \[
\int_1^\infty\alpha^t\big(\mathbb P(H_\alpha)+\mathbb P(F_\alpha))\mathrm{d} \alpha<\infty. \] From the integrability of \(g\) we derive by~\cite[Lemma 3.7]{Gouere08} \[
\int_1^\infty \alpha^t f(\alpha)\mathrm{d} \alpha <\infty \] and therefore \(\int_1^\infty \alpha^t\mathbb{P}(G_\alpha)<\infty\). From~\eqref{eq:MBound} we conclude \[
\int_1^\infty \alpha^t\mathbb{P}_o(\mathscr{M}\geq \alpha) \mathrm{d}\alpha\leq C\int_1^\infty \alpha^t\big(\mathbb{P}(G_\alpha)+\mathbb{P}(F_\alpha)\big)\mathrm{d}\alpha <\infty. \] For \(\mathscr{N}\), we derive with~\eqref{eq:NBound} \begin{equation*}
\begin{aligned}
\int_1^\infty \alpha^t\mathbb{P}_o(\mathscr{N}\geq \alpha)\mathrm{d}\alpha & \leq C\lambda \int_1^\infty \alpha^t\big(\mathbb{P}(G_\alpha)+\mathbb{P}(H_\alpha)+e^{-\nicefrac{c'\alpha}{\lambda}}\big)\mathrm{d}\alpha <\infty.
\end{aligned} \end{equation*} This finishes the proof. \end{proof}
\paragraph{Acknowledgement} We gratefully received support by the Leibniz Association within the Leibniz Junior Research Group on \textit{Probabilistic Methods for Dynamic Communication Networks} as part of the Leibniz Competition.
\footnotesize{\printbibliography}
\end{document} | arXiv |
Tutorial #10: SAT Solvers II: Algorithms
Authors: S. Prince, C. Srinivasa
In part I of this tutorial, we introduced the SAT problem and discussed its applications. The SAT problem operates on Boolean logical formulae and we discussed how to convert these to conjunctive normal form. Here, a set of clauses are logical $\text{AND}$ed together. Each clause logically $\text{OR}$s a set of literals (variables and their complements). For example:
\begin{equation}\label{eq:example_cnf_conditioning}
\phi:= (x_{1} \lor x_{2} \lor x_{3}) \land (\overline{x}_{1} \lor x_{2} \lor x_{3}) \land (x_{1} \lor \overline{x}_{2} \lor x_{3}) \land (x_{1} \lor x_{2} \lor \overline{x}_{3}). \tag{1}
\end{equation}
where the notation $\lor$ represents a $\text{OR}$ operation and $\land$ represents a $\text{AND}$ operation. The satisfiability problem establishes whether there is any way to set the variables $x_{1},x_{2},x_{3}\in$$\{$$\text{true}$,$\text{false}$$\}$ so that the formula $\phi$ evaluates to $\text{true}$.
In this tutorial we focus exclusively on the SAT solver algorithms that are applied to this problem. We'll start by introducing two ways to manipulate Boolean logic formulae. We'll then exploit these manipulations to develop algorithms of increasing complexity. We'll conclude with an introduction to conflict-driven clause learning which underpins most modern SAT solvers.
Operations on Boolean formulae
SAT solvers rely on repeated algebraic manipulation of the formula that we wish to test for satisfiability. Two such manipulations are conditioning and resolution. In this section we will discuss each in turn.
In conditioning, we set a variable $x_{i}$ to a concrete value (i.e., $\text{true}$ or $\text{false}$). When we set $x_{i}$ to $\text{true}$, we can simplify the formula using two rules:
All clauses containing $x_{i}$ can be removed from the formula. These clauses are now satisfied.
Any terms $\overline{x}_{i}$ in the remaining clauses can be removed. These must now evaluate to $\text{false}$ and hence cannot be used to satisfy their clauses.
For example, consider the formula:
\begin{equation}\label{eq:example_cnf_conditioning_s}
When we set $x_{1}=$ $\text{true}$, this becomes
\begin{equation}\label{eq:example_cnf_conditioning2}
\phi \land x_{1} := (x_{2} \lor x_{3}). \tag{3}
where the first, third and fourth clause have been removed as they are now satisfied (by rule 1) and the term $\overline{x}_{1}$ has been removed from the second clause as this term is now $\text{false}$ (by rule 2).
Similarly, when we condition by setting a variable to $\text{false}$ all clauses containing $\overline{x}_{i}$ disappear, as do any terms $x_{i}$ in the remaining clauses. Setting $x_{1}$ to $\text{false}$ in equation 2 gives:
\begin{equation}
\phi \land \overline{x}_{1} := (x_{2} \lor x_{3}) \land (\overline{x}_{2} \lor x_{3}) \land (x_{2} \lor \overline{x}_{3}). \tag{4}
Note that variable $x_{i}$ must be either $\text{true}$ or $\text{false}$ and so:
\begin{eqnarray}
\phi &=& (\phi \land x_{i}) \lor (\phi \land \overline{x}_{i})\\
&=& (x_{2} \lor x_{3}) \lor ((x_{2} \lor x_{3}) \land (\overline{x}_{2} \lor x_{3}) \land (x_{2} \lor \overline{x}_{3})).\nonumber \tag{5}
\end{eqnarray}
Here we apply the conditioning operation twice and the result is to remove the variable $x_{i}$ from the formula $\phi$ to yield two simpler formulae, $(\phi \land x_{i})$ and $(\phi \land \overline{x}_{i})$ which are logically $\text{OR}$ed together. Note though, that the result is not in conjunctive normal form.
The second common operation applied to Boolean formulae is resolution. Consider two clauses $c_{1}$ and $c_{2}$ where $x_{i}\in c_{1}$ and $\overline{x}_{i}\in c_{2}$. When we resolve by $x_{i}$, we replace these two clauses with a single clause $(c_{1}\setminus x_{i})\lor (c_{2}\setminus\overline{x}_{i})$. This clause is known as the resolvent and contains the remaining terms in $c_{1}$ and $c_{2}$ after $x_{i}$ and $\overline{x}_{i}$ are removed.
This is best illustrated with an example. Consider the formula:
\begin{equation}\label{eq:example_cnf_resolution}
\phi:= (x_{1}\lor x_{2} \lor \overline{x}_{3}) \land (\overline{x}_{2} \lor x_{4}) \land (x_{2} \lor x_{4}\lor x_{5}). \tag{6}
We note that $x_{2}$ is in the first clause and $\overline{x}_{2}$ is in the second clause and so we can resolve with respect to $x_{2}$ by combining the remaining terms from the first and second clause:
\phi:= (x_{1}\lor \overline{x}_{3} \lor x_{4}) \land (x_{2} \lor x_{4}\lor x_{5}). \tag{7}
Note that the third clause is unaffected by this operation.
The underlying logic is as follows. If $x_{2}$ is $\text{false}$, then for the first clause to be satisfied we must have $x_{1}\lor \overline{x}_{3}$. However, if $x_{2}$ is $\text{true}$, then for the second clause to be satisfied, we must have $x_{4}$. Since either $x_{2}$ or $\overline{x}_{2}$ must be the case, it follows that we must have $x_{1}\lor \overline{x}_{3} \lor x_{4}$.
Unit resolution
An important special case is unit resolution. Here, at least one of the clauses that we are resolving with respect to is a unit clause (i.e., only contains a single literal). For example,
\phi:= (x_{1}\lor \overline{x}_{3} \lor \overline{x}_{4}) \land x_{4} \tag{8}
Resolution between these two clauses works as normal. However, we can go further. Since we know that $x_{4}$ must be $\text{true}$ from the second clause, the effect of resolution here is the same as conditioning. We can remove all clauses containing $x_{4}$ and remove all terms $\overline{x}_{4}$ from the remaining clauses. So unit resolution can be seen as either a special case of resolution or as a conditioning operation depending how you look at it.
Unit propagation
A unit resolution operation may create more unit clauses. In this case, we can repeatedly apply unit resolution to the expression and at each stage we eliminate one of the variables from consideration. This procedure is known as unit propagation.
SAT solving algorithms based on resolution
We now present a series of learning algorithms that use conditioning and resolution to solve the satisfiability problem. In this section, we will use resolution to solve the 2-SAT problem and show why this can be solved in polynomial time. Then we'll introduce the directional resolution algorithm which uses resolution to solve 3-SAT problems and above, but we'll see that this becomes more computationally complex. In the next section, we'll move to algorithms that primarily exploit the conditioning algorithm to solve SAT problems.
Solving 2-SAT by unit propagation
To solve a 2-SAT problem we first condition on an arbitrarily chosen variable. This sets off a unit propagation process (a chain of unit resolutions) in which variables are removed one-by-one. This continues until either the formula is satisfied or we are left with a contradiction $x_{i}\land \overline{x}_{i}$.
Worked example: This process is easiest to understand using a concrete example. Consider the following 2-SAT problem in four variables:
\phi:= (x_{1}\lor \overline{x}_{2}) \land (\overline{x}_{1}\lor \overline{x}_{3}) \land (x_{2} \lor x_{3}) \land (\overline{x}_{2} \lor x_{4}) \land (x_{3}\lor \overline{x}_{4}). \tag{9}
We start with a single step of conditioning on an arbitrarily chosen variable. Here we'll choose $x_{1}$ and apply the formula $\phi = (\phi \land x_{1}) \lor (\phi \land \overline{x}_{1})$. We could work directly with this cumbersome expression, but in practice we set $x_{1}$ to $\text{true}$ and test for satisfiability. If this is not satisfiable, then we set $x_{1}$ to $\text{false}$ and try again and if neither are satisfiable, then the expression is not satisfiable as a whole.
Let's work through this process explicitly. Setting $x_{1}$ to $\text{true}$ gives:
\phi \land x_{1} =(x_{1}\lor \overline{x}_{2}) \land (\overline{x}_{1}\lor \overline{x}_{3}) \land (x_{2} \lor x_{3}) \land (\overline{x}_{2} \lor x_{4}) \land (x_{3}\lor \overline{x}_{4}) \land x_{1}. \tag{10}
We now perform unit resolution with respect to $x_{1}$ which means removing any clauses that contain $x_{1}$ and removing $\overline{x}_{1}$ from the rest of the formula to get:
\phi \land x_{1} = \overline{x}_{3} \land (x_{2} \lor x_{3}) \land (\overline{x}_{2} \lor x_{4}) \land (x_{3} \lor \overline{x}_{4}). \tag{11}
Notice that we are left with another unit clause $\overline{x}_{3}$ so we know $x_{3}$ must be $\text{false}$ and we can perform unit resolution again to yield:
\phi \land x_{1}\land \overline{x}_{3} = x_{2} \land (\overline{x}_{2} \lor x_{4})\land \overline{x}_{4}. \tag{12}
This time, we have two unit clauses. We can perform unit resolution with respect to either. We'll choose $x_{2}$ so we now now that $x_{2}$ is $\text{true}$ and we get:
\phi \land x_{1}\land\overline{x}_{3}\land x_{2} = x_{4} \land \overline{x}_{4} =\text{false}. \tag{13}
Clearly this is a contradiction, and so we conclude that the formula is not satisfiable if we set $x_{1}$ to $\text{true}$.
We now repeat this process with $x_{1}$ = $\text{false}$, which gives
\phi \land \overline{x}_{1} &=&(x_{1}\lor \overline{x}_{2}) \land (\overline{x}_{1}\lor \overline{x}_{3}) \land (x_{2} \lor x_{3}) \land (\overline{x}_{2} \lor x_{4})\land (\overline{x}_{4} \lor x_{3}) \land \overline{x}_{1} \nonumber \\
&=& \overline{x}_{2} \land (x_{2} \lor x_{3}) \land (\overline{x}_{2} \lor x_{4}) \land (\overline{x}_{4} \lor x_{3}). \tag{14}
Once more, this leaves a unit clause $\overline{x}_{2}$, so we set $x_{2}$ to $\text{false}$ and perform unit resolution again to get
\phi \land \overline{x}_{1}\land \overline{x}_{2} = x_{3} \land (\overline{x}_{4} \lor x_{3}) \tag{15}
which gives the unit clause $x_{3}$ and so we set $x_{3}$ to $\text{true}$. Now something different happens. The entire of the right hand side disappears. Since there are no clauses left to be satisfied, the formula is satisfiable:
\phi \land \overline{x}_{1}\land \overline{x}_{2} \land x_{3}= \text{true} \tag{16}
Note that the formula is satisfiable regardless of the value of $x_{4}$ (it is on neither side of the equation) so we have found two satisfiable solutions $\{\overline{x}_{1},\overline{x}_{2},x_{3},x_{4}\}$ and $\{\overline{x}_{1},\overline{x}_{2},x_{3},\overline{x}_{4}\}$.
Complexity: If there are $V$ variables, there are at most $V$ rounds of unit resolution for each of the two values of the initial conditioned variable. Each unit resolution procedure is linear in the number of clauses $C$ so the algorithm has total complexity $\mathcal{O}[CV]$.
It's possible to reach a case where the chain of unit propagation stops and we have to condition on one of the remaining variables to start it again. However, this only occurs when subsets of the variables have no interaction with one another and so it does not add to the complexity.
Directional resolution
Now consider what happens if we apply the unit resolution approach above to a 3-SAT problem. When we condition on the first variable $x_{i}$, we remove clauses that contain $x_{i}$ and remove $\overline{x}_{i}$ from the rest of the clauses. Unfortunately, this doesn't create another unit clause (at best it just changes a subset of the 3-clauses to 2-clauses), and so it's not clear how to proceed.
Directional resolution is a method that uses resolution to tackle $3$-SAT and above. The idea is to choose an ordering of the variables and then perform all possible resolution operations with each variable in turn before moving on. We continue until we find a contradiction or reach the end. In the latter case, we work back in the reverse order to find the values that satisfy the expression.
Worked example: Again, this is best understood via a worked example. Consider the formula:
&\phi:= &(x_{1}\lor \overline{x}_{2} \lor \overline{x}_{4}) \land (\overline{x}_{1}\lor x_{3}\lor \overline{x}_{5}) \land \\
&&\hspace{1cm}(x_{2} \lor x_{3}\lor \overline{x}_{4}) \land (\overline{x}_{2} \lor \overline{x}_{4}\lor x_{5}) \land (\overline{x}_{3} \lor x_{4}\lor x_{5}).\nonumber \tag{17}
We sort the clauses into bins. Those containing $x_{1}$ or $\overline{x}_{1}$ are put in the bin 1 and any remaining clauses containing $x_{2}$ or $\overline{x}_{2}$ are put in bin 2 and so on:
&& x_{1}: (x_{1}\lor \overline{x}_{2} \lor \overline{x}_{4}), (\overline{x}_{1}\lor x_{3}\lor \overline{x}_{5}) \nonumber \\
&& x_{2}: (x_{2} \lor x_{3}\lor \overline{x}_{4}), (\overline{x}_{2} \lor \overline{x}_{4}\lor x_{5}) \nonumber \\
&& x_{3}: (\overline{x}_{3} \lor x_{4}\lor x_{5}) \nonumber \\
&& x_{4}: \nonumber \\
&& x_{5}: \tag{18}
We work through these bins in turn. For each bin we perform all possible resolutions and move the resulting generated clauses into subsequent bins. So for bin 1 we resolve the clauses $(x_{1}\lor \overline{x}_{2} \lor \overline{x}_{4})$ and $(\overline{x}_{1}\lor x_{3}\lor \overline{x}_{5})$ with respect to $x_{1}$ to get the new clause $\color{BurntOrange} (\overline{x}_{2}\lor \overline{x}_{4}\lor x_{3}\lor \overline{x}_{5})$. We add this to bin 2 as it contains a $x_{2}$ term:
&& x_{2}: (x_{2} \lor x_{3}\lor \overline{x}_{4}), (\overline{x}_{2} \lor \overline{x}_{4}\lor x_{5}), \color{BurntOrange}(\overline{x}_{2}\lor \overline{x}_{4}\lor x_{3}\lor \overline{x}_{5}) \nonumber \\
We then consider bin 2 and resolve the clauses with respect to $x_{2}$ in all possible ways. In bin 2 there is one clause containing $x_{2}$ and we can resolve it against the two clauses containing $\overline{x}_{2}$. This creates two new clauses that we simplify and add to bin 3 since they contain terms in $x_{3}$:
&& x_{1}: (x_{1}\lor \overline{x}_{2} \lor \overline{x}_{4}), (\overline{x}_{1}\lor \overline{x}_{3}\lor \overline{x}_{5}) \nonumber \\
&& x_{2}: (x_{2} \lor x_{3}\lor \overline{x}_{4}), (\overline{x}_{2} \lor \overline{x}_{4}\lor x_{5}), (\overline{x}_{2}\lor \overline{x}_{4}\lor x_{3}\lor \overline{x}_{5}) \nonumber \\
&& x_{3}: (\overline{x}_{3} \lor x_{4}\lor x_{5}), \color{BurntOrange} (x_{3}\lor \overline{x}_{4} \lor x_{5})\color{Black}, \color{BurntOrange}(x_{3} \lor \overline{x}_{4}\lor \overline{x}_{5}) \nonumber \\
Now we consider bin 3. Again, there are three clauses here and combining them with resolution creates two new clauses. Resolving the first and second clause with respect to $x_{3}$ creates $(x_{4}\lor \overline{x}_{4} \lor x_{5})$ which evaluates to $\text{true}$ since either $x_{4}$or $\overline{x}_{4}$ must always be $\text{true}$. Similarly, combining the first and third clause creates the clause $x_{4}\lor x_{5}\lor \overline{x}_{4}\lor \overline{x}_{5}$ which evaluates to $\text{true}$ and so we are done. At this point, we can say that the formula is $\text{SAT}$ as we have not created any contradictions of the form $x_{i}\land\overline{x}_{i}$ during this resolution process
Finding the certificate: To find an example that satisfies the expression, we work backwards, setting the bin value to $\text{true}$ or $\text{false}$ in such a way that it satisfies the clause. There are no clauses in bin 5 and so we are free to choose either value. We'll set $x_{5}$ to be $\text{true}$. Similarly, there are no clauses in bin 4 and so we will arbitrarily set $x_{4}$ to $\text{true}$ as well. After these changes we have:
&& x_{2}: (x_{2} \lor x_{3}\lor \overline{x}_{4}), (\overline{x}_{2} \lor \overline{x}_{4}\lor x_{5}), (\overline{x}_{2}\lor \overline{x}_{4}\lor \overline{x}_{3}\lor \overline{x}_{5}) \nonumber \\
&& x_{3}: (\overline{x}_{3} \lor x_{4}\lor x_{5}), (x_{3}\lor \overline{x}_{4} \lor x_{5}) \nonumber \\
&& x_{4}: \text{true} \nonumber \\
&& x_{5}: \text{true} \tag{21}
Now we consider the third bin. We substitute in the values for $x_{4}$ and $x_{5}$ and see that both clauses evaluate to $\text{true}$, regardless of the value of $x_{3}$, so again, we can choose any value that we want. We'll set $x_{3}$ to $\text{false}$ to give:
&& x_{3}: \text{false}\nonumber \\
Progressing to the second bin, we observe that the second and third clause are already satisfied by the previous assignments, but the first clause is not since $x_{3}$ is $\text{false}$ and $x_{4}$ is $\text{true}$. Consequently, we must satisfy this clause by setting $x_{2}$ to $\text{true}$:
&& x_{3}: \text{false} \nonumber \\
Finally, we consider the first bin. We note that the second clause is satisfied because $x_{3}$ is $\text{false}$ but the first clause is not and so to satisfy it, we must set $x_{1}$ to $\text{true}$ and now we have a satisfying example.
Complexity: The directional resolution procedure works, but is not especially efficient. For large problems, the number of clauses can expand very quickly: if there were $C$ clauses and half contain $x_{1}$ and the other half $\overline{x}_{1}$ then we could create $C^{2}/4$ new clauses in the first step. For a $K$-SAT problem, each of these clauses are larger than the original ones with size $2(K-1)$.
It is possible to improve the efficiency. Any time we generate a unit clause, we can perform unit propagation which may eliminate many variables. Also in our example we organized the bins by the variable index, but this was an arbitrary choice. This order can have a big effect on the total computational cost and so careful selection can improve efficiency. However, even with these improvements, this approach is not considered viable for large problems.
SAT solving algorithms based on conditioning
In this section, we will develop algorithms that are fundamentally centered around the conditioning operation (although they also have unit resolution embedded). We'll describe both the DPLL algorithm and clause learning algorithms which underpin most modern SAT solvers. To understand these methods, we first need to examine the connection between conditioning and tree search.
SAT as binary search
We'll use the running example of the following Boolean formula with $C=7$ clauses and $V=4$ variables:
\begin{eqnarray}\label{eq:SAT_working_example}
\phi&:=&(x_{1} \lor x_{2}) \land (x_{1} \lor \overline{x}_{2} \lor \overline{x}_{3} \lor x_{4}) \land (x_{1} \lor \overline{x}_{3} \lor \overline{x}_{4}) \land \\
&&\hspace{0.5cm} (\overline{x}_{1} \lor x_{2} \lor \overline{x}_{3}) \land (\overline{x}_{1} \lor x_{2} \lor \overline{x}_{4}) \land (\overline{x}_{1} \lor x_{3} \lor x_{4}) \land (\overline{x}_{2} \lor x_{3}) \nonumber, \tag{24}
Consider conditioning on variable $x_{1}$ so that we have:
\phi = (\phi \land \overline{x}_{1}) \lor (\phi \land x_{1}). \tag{25}
This equation makes the obvious statement that in any satisfying solution $x_{1}$ is either $\text{true}$ or $\text{false}$. We could first investigate the case where $x_{1}$ is $\text{false}$. If we establish this is $\text{SAT}$ then we are done, and if not we consider the case where $x_{1}$ is $\text{true}$. Taking this one step further, we could condition each of these two cases on $x_{2}$ to get:
\phi = ((\phi \land \overline{x}_{1}) \land \overline{x}_{2}) \lor ((\phi \land \overline{x}_{1}) \land \overline{x}_{2}) \lor ((\phi \land x_{1})\land \overline{x}_{2} ) \lor ((\phi \land x_{1})\land x_{2} ). \tag{26}
and now we could consider each of the four combinations $\{\overline{x}_{1}\overline{x}_{2}\},\{\overline{x}_{1}x_{2}\},\{x_{1}\overline{x}_{2}\}$ and $\{x_{1}x_{2}\}$ in turn, terminating when we find a solution that is $\text{SAT}$.
One way to visualise this process is as searching through a binary tree (figure 1). At each node of the tree we branch on one of the variables. When we reach a leaf, we have known values for each variable and we can just check if the solution is $\text{SAT}$.
Figure 1. SAT as binary search. At each node of the search tree we condition on a variable, splitting into a left sub-tree in which this variable is set to $\text{false}$ and a right sub-tree in which it is set to $\text{true}$. There is one level in the tree for each variable so that at each leaf all the variables are set and we can test if the formula is satisfied. For the example in equation 24, the formula evaluates to $\text{false}$ for the first 14 leaves and the individual clauses that are violated are indicated in grey at each leaf. The last two leaves are both satisfying solutions. In practice, we would stop searching when we found the first satisfying solution and so we would only need to test 15 leaves in this example.
This example was deliberately constructed to be pathological in that the first 14 combinations (or equivalently leaves of the tree) all make the formula evaluate to $\text{false}$. These are signified in the plot by red crosses. We number the clauses:
&& 1:(x_{1} \lor x_{2}) \nonumber\\
&& 2:(x_{1} \lor \overline{x}_{2} \lor \overline{x}_{3} \lor x_{4}) \nonumber\\
&& 3:(x_{1} \lor \overline{x}_{3} \lor \overline{x}_{4}) \nonumber\\
&& 4:(\overline{x}_{1} \lor x_{2} \lor \overline{x}_{3}) \nonumber\\
&& 6:(\overline{x}_{1} \lor x_{3} \lor x_{4}) \nonumber\\
&& 7:(\overline{x}_{2} \lor x_{3}) \tag{27}
and for each leaf of the tree in figure 1, the clauses that were contradicted are indicated in grey. In this case, both of the last two combinations (leaves) satisfy the formula, and once we find the first one ($x_{1}, x_{2}, x_{3},\overline{x}_{4})$ we can return SAT.
Note, we have not yet obviously made the algorithm more efficient. We might still have to search all $2^{V}$ combinations of variables to establish satisfiability or lack thereof. However, viewing SAT solving as tree search is the foundation that supports more efficient algorithms.
Efficient binary search
We can immediately improve the efficiency of the binary search method by some simple bookkeeping. As we pass through the tree we keep track of which clauses are satisfied and which are not. As soon as we find one that is not satisfied, we do not need to explore further and we can backtrack. Similarly, if we find a situation where all of the clauses are already satisfied before we reach a leaf then we can return $\text{SAT}$ without exploring further. This means that the variables below this point can take any value.
In our worked example, when we pass down the first branch and set $x_{1}$ to $\text{false}$ and $x_{2}$ to $\text{false}$ we have already contradicted clause 1 which was $(x_{1} \lor x_{2})$, and so there is no reason to proceed further. Continuing in this way we only need to search a subset of the full tree (figure 2). We find the first satisfying solution when $x_{1},x_{2},x_{3}=$$\text{true}$,$\text{true}$,$\text{true}$ and need not continue to the leaf. As we saw from the full tree in figure 1, the setting of $x_{4}$ is immaterial.
Figure 2. Efficient binary search. With some simple bookkeeping, tree search can be made much more efficient. If we track the status of each clause then we can backtrack as soon as one of the clauses is violated. Again, the index of the violated clause is shown in grey. Similarly, when we have satisfied all of the clauses we can return $\text{SAT}$ even though we have not yet reached a leaf. The variables below this can be set to any value.
DPLL
We can also consider the tree search from an algebraic point of view. Each time we make a decision at a node in the tree, we are conditioning on a given variable. So when we set $x_{1}$ to $\text{false}$, the resulting formula is
\begin{eqnarray}\label{eq:sat_tree_cond}
\phi\land \overline{x}_{1} := x_{2} \land (\overline{x}_{2} \lor \overline{x}_{3} \lor x_{4}) \land (\overline{x}_{3} \lor \overline{x}_{4}) \land (\overline{x}_{2} \lor x_{3}), \tag{28}
where we have used the usual recipe of removing all clauses containing $\overline{x}_{1}$ and removing the term $x_{1}$ from the remaining clauses.
The Davis–Putnam–Logemann-Loveland (DPLL) algorithm takes tree search one step further, by embedding unit propagation into the search algorithm (figure 3). For example, when we condition on $\overline{x}_{1}$ and yield the new expression in equation 28, we generate the unit clause $x_{2}$. We can perform unit resolution using $x_{2}$ to get:
\phi\land \overline{x}_{1}\land x_{2} := (\overline{x}_{3} \lor x_{4}) \land (\overline{x}_{3} \lor \overline{x}_{4}) \land x_{3}, \tag{29}
which creates another unit clause $x_{3}$. Applying unit resolution again we yield the contradiction $x_{4}\land \overline{x}_{4}$ and need proceed no further.
Figure 3. DPLL algorithm. By performing unit propagation where possible, we can eliminate many variables very efficiently. In this case, once we condition on $\overline{x}_{1}$, this creates a unit clause in $x_{2}$, which starts a chain of unit resolution operations that establishes a contradiction. Similar effects happen at other points in the tree.
To summarize, the DPLL algorithm consists of tree search, where we perform unit propagation whenever unit clauses are produced. Since unit resolution can be done in linear time, this is much more efficient than the tree search that it replaces.
Note that in our worked example, the unit propagation process always generated a contradiction or a $\text{SAT}$ solution. However, this is not necessarily the case in a larger problem. After unit resolution there will usually be non-unit clauses left containing the remaining variables have neither been conditioned on, nor eliminated using unit resolution. At this point, we condition on the next available variable and continue down the tree, performing unit resolution when we can (figure 4).
Figure 4. DPLL algorithm in practice. In a real problem the DPLL algorithm will alternate between conditioning on variables and performing unit resolution. The effect of this is that we condition on different variables in different paths of the tree.
Conflict Driven Clause learning
The DPLL algorithm makes SAT solving by tree search much more efficient, but there can still be considerable wasted computation. Consider the case where we have set $x_{1}$ to $\text{false}$ and then set $x_{2}$ to $\text{false}$ (figure 5). However, imagine that there are clauses that mean that when $x_{2}$ is $\text{false}$, there is no way to set the variables $x_{3}$ and $x_{4}$ in a valid way. For example, the following combination of clauses will achieve this:
(x_{2} \lor x_{3}\lor x_{4}) \land (x_{2} \lor x_{3}\lor \overline{x}_{4}) \land (x_{2} \lor \overline{x}_{3}\lor x_{4}) \land (x_{2} \lor \overline{x}_{3}\lor \overline{x}_{4}). \tag{30}
As we work through the sub-tree in the blue region in figure 5, we duly establish that there is no possible solution.
As we search through the tree, we will eventually come to another place where we set $x_{2}$ to $\text{false}$ and now we must work through exactly the same calculations again to establish that there is no valid solution (yellow region in figure 5). In a large problem this may happen many times.
Figure 5. Motivation for conflict-driven clause learning. Consider the case where setting $x_{2}$ to $\text{false}$ inevitably results in a conflict where there is no way to set $x_{3}$ and $x_{4}$. Without taking action, we will have to repeat the computations to find this conflict in every sub-tree where $x_{2}$ is $\text{false}$ (blue and yellow rectangles). When conflict-driven clause learning algorithms find a conflict in a sub-tree, they add a new clause to the original expression that prevents redundant exploration of sub-trees.
Conflict-driven clause learning aims to reduce this redundancy. When a conflict occurs, the cause is found and we add a new clause to the original statement that prevents exploration of redundant sub-trees. For example, in this simple case, we could add the clause $(x_{2})$ which would prevent exploration of trees where $x_{2}$ is $\text{false}$.
Unfortunately, the causes of a conflict are usually more complex than a single variable. To find the combinations of variables that are ultimately responsible for the conflict, we build a structure called an implication graph as we search through the tree.
Figure 6a provides a concrete example of a SAT problem where there are 11 clauses and 10 variables. Figure 6b illustrates the situation where we are mid-way through the DPLL search in which we have interleaved processes of conditioning (blue shaded areas) and unit resolution (yellow shaded areas). We have just established a conflict at clause 11 (at the blue arrow) which cannot be satisfied when we set $x_{5}$ to $\text{true}$.
Figure 6. Implication graphs. a) A SAT problem with 11 clauses in 10 variables. b) We are mid-way though the DPLL process (at the blue arrow) and have just found a conflict at clause 11. c) The implication graph representing the current state. Each vertex corresponds to a variable (blue if conditioned, yellow if inferred by unit resolution). The incoming edges to yellow vertices correspond to the variables that caused the vertex variable to be inferred and are labelled with the relevant clause. So, for example, $x_{6}$ is set to $\text{false}$ by clause $c_{2}$ because $x_{1}$ is $\text{false}$ and $x_{2}$ is $\text{true}$. The conflict results when setting variable $x_{5}$ implies both $x_{7}$ and it's complement $\overline{x}_{7}$.
Figure 6c is the implication graph associated with this point in the search, which contains all of the variables that we have established so far. The literals $\overline{x}_{1}, x_{2},x_{3}, x_{5}$ that we conditioned on are depicted with blue vertices and the literals $x_{4}, \overline{x}_{6},x_{10},x_{9}, x_{7}$ and $\overline{x}_{7}$ that resulted from unit propagation are shown as yellow vertices. Each edge depicts a contribution to the unit resolution process. For example, the edge between $\overline{x}_{1}$ and $x_{4}$ represents the fact that when $x_{1}$ is set to $\text{false}$, we must set $x_{4}$ to $\text{true}$. This is due to clause 1 and the edge is accordingly labelled with $c_{1}$. Similarly, we can see that $x_{10}$ has become $\text{true}$ by clause $c_{3}$ because previously in the search process $x_{1}$ and $x_{6}$ were both set to $\text{false}$.
You can see from this implication graph exactly how the conflict happened. When we condition on $x_{5}$, clause $c_{6}$ implied that $x_{7}$ must be $\text{true}$ given that $x_{2}$ and $x_{5}$ were both $\text{true}$, but clause $c_{11}$ implied that $x_{7}$ must be $\text{false}$ given that $x_{5}$ and $x_{10}$ were both $\text{true}$. So, one interpretation is that the conflict is inevitable given states $x_{2},x_{5}$ and $x_{10}$ that were inputs to these contradictory clauses.
However, this is not the only interpretation. For example, if $x_{10}$ is one of the proximal causes, then this was only set to $\text{true}$ because we previously set $x_{1}$ and $x_{6}$ to $\text{false}$. So maybe we should attribute the contradiction not to variables $x_{2},x_{5},x_{10}$ but to variables $x_{2},x_{5},\overline{x}_{6},\overline{x}_{1}$. We can use the implication graph to find alternative explanations. Any cut of the graph that separates the conditioning (blue) variables from the conflict defines an explanation (figure 7). The explanatory variables are the source vertices of the edges that were cut.
Figure 7. Cutting the implication graph to find the causes of a conflict. Any cut that separates the conditioned nodes (in blue) from the conflict can be interpreted as a possible cause. a) In this case, the cause of the conflict is attributed to variables $x_{2}, x_{5}, x_{10}$, since the edges from these variables are cut. b) In this second case, the cause is attributed to variables $\overline{x}_{1}, x_{2}, x_{5}$.
Having established a cause, we must now derive a new clause that prevents the SAT solver from exploring similar dead-ends in the future. If the case was attributed to $\overline{x}_{1}, x_{2}, x_{5}$, then we would add the clause $(x_{1}\lor \overline{x}_{2} \lor \overline{x}_{5})$ to prevent this combination happening. We continue exploring the tree, by jumping back up the tree structure to a sensible point and resuming with this new constraint.
Machine leaning powered CDCL
The previous discussion outlined the main ideas of conflict-driven clause learning algorithms, but there are many additional choices to be made in a modern system. For example, we must decide the order of variables to condition on. In our examples, we have done this in numerical order, but this choice was arbitrary and there is no particular reason to evaluate them in the same order as we go down different branches in the tree. Much work is devoted to developing heuristics to making this choice. For example, we might prioritize variables that are in short clauses, with the goal of triggering unit propagation earlier. Alternatively, we might prioritize variables that are in lots of clauses as this will simplify the expression a great deal.
There are also many other decisions to make. In CDCL, we must choose which of many potential explanations for a conflict is superior and decide exactly where we should jump back to in the tree. Some solvers periodically restart the solution process to avoid wasting the available computation time fruitlessly searching a single branch, and we must decide when exactly to perform these restarts. One approach to making these decisions is to use machine learning to guide the choices.
For example, Liang et al., (2016) developed uses a reward function to choose the order in which variables in a CDCL solver are considered. A reward function $r[i]$ is defined for each variable $x_{i}$:
r[i] \propto \frac{1}{numConflicts-lastConflict[i])} \tag{31}
Here $numConflicts$ keeps track of the total number of conflicts the solver has encountered so far whereas $lastConflict[i]$ keeps track of the last time variable $x_{i}$ was involved in a conflict. From this we can see that a variable which was recently involved in a conflict would get a high reward. This reward is then incorporated into a score function:
Q[i] \longleftarrow (1-\alpha)Q[i] + \alpha r[i] \tag{32}
At any iteration where a new variable must be selected for conditioning, the variable with the highest score is picked, provided that it is not currently already conditioned on. This is known as the conflict history branching heuristic.
In the above formulation, the terms appearing in the reward will always be known and can hence be computed exactly. However, Liang et al., (2016) also dealt with a case where the reward function is defined such that the terms appearing in it have associated uncertainty. This new reward definition improves the branching heuristic and the authors show how tools from a Multi-Armed Bandit framework in RL can be used to estimate the uncertainty and further improve performance.
In further work Liang et al., 2017 the same authors show how gradient based methods can be used to optimize another branching heuristic, one based on how many learnt clauses can be obtained from each decision. Other examples of machine learning in the SAT community include Nejati et al., (2017) where reinforcement learning is used to decide when to restart the solver.
In this blog, we introduced resolution and conditioning operations. We then developed a series of algorithms based on these operations. We started with using resolution (via unit propagation) to efficiently solve 2-SAT and then investigated the directional resolution for 3-SAT and above. We re-framed the SAT solving problem as tree search where conditioning is used at each branch in the tree. This led to the DPLL and CDCL algorithms.
For further information about SAT solving including the DPLL and CDCL algorithms, consult the Handbook of Satisfiability. Most chapters are available on the internet if you search for their titles individually. A second useful resource is Donald Knuth's Facsicle 6.
In part III of this article, we'll investigate a completely different approach to SAT solving that relies on belief propagation in factor graphs. Finally, we'll show how the machinery of SAT solving can be extended to continuous variables by introducing satisfiability module theory (SMT) solvers.
S. Prince
C. Srinivasa
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Metallurgical and Materials Transactions A
January 2015 , Volume 46, Issue 1, pp 377–395 | Cite as
Multigrain and Multiphase Mathematical Model for Equiaxed Solidification
Marcelo Aquino Martorano
Davi Teves Aguiar
Juan Marcelo Rojas Arango
A deterministic multigrain and multiphase model of equiaxed solidification of binary alloys is proposed, implemented, and analyzed. An important feature of the present model is the creation of classes of dendritic and globulitic grains according to their instantaneous sizes during solidification. Globulitic and dendritic grain growth, coarsening of secondary dendrite arms, distribution of nucleation undercoolings, and equiaxed eutectic growth are consistently included in the model equations. Important model assumptions are uniform temperature, negligible liquid convection, and negligible grain movement. Calculated cooling curves, solid fraction evolution, average grain sizes, and eutectic fractions agree well with predictions of previous models for dendritic and globulitic equiaxed grains. Predicted grain sizes decrease with an increase in cooling rate for an Al-2.12 pct Cu alloy and with an increase in Si concentration up to 3 pct for Al-Si alloys, agreeing quantitatively with experimental results. Simulations for an Al-7 pct Si alloy predict that an increase in grain size correlates with an increase in the magnitude of the recalescence observed in cooling curves. These calculations agree well with experimental results when the transition from a globulitic to a dendritic morphology occurs in the model before the minimum temperature of recalescence is reached.
Representative Elementary Volume Growth Velocity Interdendritic Liquid Eutectic Cell Eutectic Solidification
Area of interface (m2) or constant of coarsening law (m s−a )
Area of REV boundary (m2)
Constant of eutectic growth law (m s−1 K−2)
Constant of coarsening law
Boundary of REV
Mass fraction of solute (–)
\( \bar{C} \)
Surface average of solute mass fraction (–)
Initial solute mass fraction (–)
\( \left\langle {C_{di} } \right\rangle^{di} \)
Average of solute mass fraction in interdendritic liquid of grains in class i (–)
Mass fraction of solute in the external liquid (–)
Cl∞
Mass fraction of solute in bulk liquid (–)
\( \left\langle {C_{l} } \right\rangle^{l} \)
Average of solute mass fraction in external liquid (–)
Solute mass fraction of the liquidus line (–)
Cliq,gi
Solute mass fraction of the liquidus line for globulitic grains of class i (–)
Volumetric specific heat (J m−3 K−1)
\( \left\langle {C_{si} } \right\rangle^{si} \)
Average of solute mass fraction in the solid of grains in class i (–)
Interface between eutectic and interdendritic liquid in grains of class i
Coefficient of solute diffusion (m2 s−1)
erfc
Complementary error function
IV−1
Inverse of the Ivantsov function
\( \vec{j}_{k} \)
Diffusive flux of solute (kg m−2 s−1)
k, j
Constituents of the multiphase model
Solute partition coefficient (–)
\( \bar{l} \)
Final average grain size (m)
Volumetric latent heat (J m−3)
Slope of the liquidus line (K pct mass−1)
Total number density of grains of primary solid (m−3)
\( \vec{n}_{k} \)
Normal unit vector at the interfaces of k and pointing out of V k (–)
Number density of eutectic cells (m−3)
neext
Extended number density of eutectic cells (m−3)
Number density of grains in class i (m−3)
niext
Extended number density of grain nuclei in class i (m−3)
Total number density of substrate particles for heterogeneous nucleation (m−3)
Number of existing grain classes
Maximum number of possible grain classes
Peclet number for grains in class i (–)
Average heat flux out of REV (J m−2 s−1)
Radial coordinate (m)
\( \dot{R} \)
Cooling rate of the liquid before solidification (K s−1)
Representative elementary volume
Reext
Extended radius of eutectic cells (m)
Radius of spherical unit cell for grains in class i (m)
Radius of grain envelopes in class i (m)
Riext
Extended radius of grains in class i (m)
Concentration of interfacial area (m−1)
Time (s)
tne
Time of eutectic nucleation (s)
Time of nucleation of grains in class i (s)
Temperature (K)
Initial temperature (K)
Eutectic temperature (K)
Melting point of the pure metal (K)
Volume of the REV (m3)
Volume of constituent k (m3)
\( \bar{w} \)
Average normal interface velocity (m s−1)
\( \vec{w}_{k} \)
Velocity vector of an interface of constituent k (m s−1)
Greek symbols
Thickness of effective diffusion layer (m)
Gibbs–Thomson coefficient (m K)
Δt
Time step of the numerical method (s)
Undercooling of the external liquid (K)
ΔTne
Undercooling for nucleation of eutectic cells (K)
ΔTni
Undercooling for nucleation of the grains in class i (K)
ΔTnucl
Undercooling range for nucleation of the primary phase (K)
\( \overline{{\Delta T_{N} }} \)
Average nucleation undercooling (K)
ΔTσ
Standard deviation of the nucleation undercooling distribution (K)
Volume fraction (–)
λi
Average spacing between secondary dendrite arms in grains of class i (m)
λi0
Initial spacing between secondary dendrite arms in grains of class i (m)
Mass density (kg m−3)
σϕ
Standard deviation of particle size distribution (–)
ϕ0
Geometrical mean diameter of particle size distribution (μm)
ψk
General field variable defined in constituent k
Dimensionless undercooling (–)
Subscripts
Interdendritic liquid of grains in class i
Eutectic
External eutectic
Grains of class i
Index of a grain class
External liquid
Interface between external liquid and interdendritic liquid of grains in class i
Interface between external liquid and eutectic
Interface between external liquid and grain envelopes of class i
Interface between external liquid and the solid of grains in class i
Interface between interdendritic liquid and solid of grains in class i
Primary solid
Primary solid of grains in class i
Manuscript submitted February 6, 2014.
The authors thank the financial support from FAPESP (03/08576-7) and CNPq (475451/04-0).
Macroscopic Conservation Equations
The general transport theorem[19] applied to the volume V k of constituent k within a REV of constant volume V 0 can be written as
$$ \frac{\text{d}}{{{\text{d}}t}}\left( {\varepsilon_{k} \frac{1}{{V_{k} }}\int\limits_{{V_{k} }} {\psi_{k} {\text{d}}V} } \right) = \frac{1}{{V_{0} }}\int\limits_{{V_{k} }} {\frac{{\partial \psi_{k} }}{\partial t}} {\text{d}}V + \frac{1}{{V_{0} }}\oint\limits_{{A_{kj} + A_{kb} }} {\psi_{k} \vec{w}_{k} \cdot \vec{n}_{k} {\text{d}}A}, $$
where ψ k is a field variable defined in constituent k; A kj is the total interfacial area between constituent k and all adjacent constituents within V 0; A kb is the boundary area formed by the REV boundary cutting through constituent k; \( \vec{w}_{k} \) is the velocity of all interfaces and boundaries of k, being zero for its boundaries coinciding with the REV boundary (A kb ), which is motionless; \( \vec{n}_{k} \) is a normal unit vector at the interfaces of k and pointing out of V k .
For ψ k = ρ, which is the mass density and is assumed constant and equal for all constituents, Eq. [A1] gives
$$ \frac{{{\text{d}}\varepsilon_{k} }}{{{\text{d}}t}} = \frac{1}{{V_{0} }}\int\limits_{{A_{kj} }} {\vec{w}_{k} \cdot \vec{n}_{k} {\text{d}}A} $$
For \( \psi_{k} = \rho {\kern 1pt} C_{k}, \)where C k is the mass fraction of solute in constituent k, the species conservation equation in differential form, namely \( \partial \left( {\rho C_{k} } \right)/\partial t = - \vec{\nabla } \cdot \vec{j}_{k} \) (where \( \vec{j}_{k} \) is the diffusive flux of solute), can be substituted on the right-hand side of Eq. [A1]. After application of Gauss theorem and consideration that no solute flows through the REV boundary, the equation becomes
$$ \rho \frac{{{\text{d}}\left( {\varepsilon_{k} \left\langle {C_{k} } \right\rangle^{k} } \right)}}{{{\text{d}}t}} = \frac{1}{{V_{0} }}\int\limits_{{A_{kj} }} {\left( {\rho C_{k} \vec{w}_{k} - \vec{j}_{k} } \right) \cdot \vec{n}_{k} } {\text{d}}A $$
Some of the surface integrals in these equations can be simplified by the following two approximations[43, 44, 45]
$$ \frac{1}{{V_{0} }}\int\limits_{{A_{kj} }} {\vec{w}_{k} \cdot \vec{n}_{k} {\text{d}}A} = S_{kj} \bar{w}_{kj} $$
$$ - \frac{1}{{V_{0} }}\int\limits_{{A_{kj} }} {\vec{j}_{k} \cdot \vec{n}_{k} {\text{d}}A} = \rho D_{k} \frac{{S_{kj} }}{{\delta_{k} }}\left( {\bar{C}_{k} - \left\langle {C_{K} } \right\rangle^{k} } \right), $$
where \( \bar{w}_{kj} \) is the average normal interface velocity; S kj = A kj /V 0 is an interfacial area concentration; \( \bar{C}_{k} \) and δ k are, respectively, the surface average solute concentration and the effective diffusion length in constituent k at the interface between constituents k and j; and D k is the diffusion coefficient of solute in k. In the absence of convection, the following two integrated jump condition can be used[19]
$$ \int\limits_{{A_{kj} }} {\vec{w}_{k} \cdot \vec{n}_{k} {\text{d}}A} = - \int\limits_{{A_{kj} }} {\vec{w}_{j} \cdot \vec{n}_{j} {\text{d}}A} $$
$$ \int\limits_{{A_{kj} }} {\left( {\rho C_{k} \vec{w}_{k} - \vec{j}_{k} } \right) \cdot \vec{n}_{k} } {\text{d}}A = - \int\limits_{{A_{kj} }} {\left( {\rho C_{j} \vec{w}_{j} - \vec{j}_{j} } \right) \cdot \vec{n}_{j} } {\text{d}}A $$
Development of the Main Equations
The general equations developed in the previous section were applied to the primary solid (si) and interdendritic liquid (di) of all grains belonging to an arbitrary class i and also to the eutectic (e) and external liquid (l), shown in Figure 1. Since the constituents si and di are defined for each of the N classes, there can be at most 2N + 2 different constituents in the model. As shown in Figure 1, the possible interfacial areas between each pair of constituents are A le , A sdi , A ldi , A lsi , A die , and A sie , where subscripts indicate the interface between constituents.
Mass Conservation
Equation [A2] could be applied to the eutectic (k = e), although there are dividing surfaces of discontinuities (interfaces between the solid phases) inside it, because ρ is constant. When \( A_{kj} = A_{le} + \sum\nolimits_{i = 1}^{N} {\left( {A_{sie} + A_{die} } \right)} \), \( \vec{w}_{e} = 0 \) on A sie , and Eq. [A4] is substituted, Eq. [A2] becomes
$$ \frac{{{\text{d}}\varepsilon_{e} }}{{{\text{d}}t}} = \frac{1}{{V_{0} }}\int\limits_{{A_{le} + \sum\nolimits_{i = 1}^{N} {A_{die} } }} {\vec{w}_{{e{\kern 1pt} }} \cdot \vec{n}_{e} {\text{d}}A} = \left( {S_{le} + \sum\limits_{i = 1}^{N} {S_{die} } } \right)\bar{w}_{e} $$
where S die = A die /V 0 and S le = A le /V 0. The volume fraction of the external eutectic (ɛ ee ), i.e., the eutectic formed in the external liquid is calculated from
$$ \frac{{{\text{d}}\varepsilon_{ee} }}{{{\text{d}}t}} = \frac{1}{{V_{0} }}\int\limits_{{A_{le} }} {\vec{w}_{e} \cdot \vec{n}_{e} {\text{d}}A} = S_{le} \bar{w}_{e}. $$
Application of Eq. [A2] to the primary solid (k = si) implies \( A_{kj} = A_{sdi} + A_{lsi} \) and gives the following important equation
$$ \frac{{{\text{d}}\varepsilon_{si} }}{{{\text{d}}t}} = \frac{1}{{V_{0} }}\int\limits_{{A_{lsi} }} {\vec{w}_{si} \cdot \vec{n}_{si} {\text{d}}A} + \frac{1}{{V_{0} }}\int\limits_{{A_{sdi} }} {\vec{w}_{si} \cdot \vec{n}_{si} {\text{d}}A} $$
(A10)
Equation [A2] was applied to the interdendritic liquid (\( k = d{\kern 1pt} i \)) using \( A_{kj} = A_{sdi} + A_{ldi} \) and added to Eq. [A10], canceling the integral over A sdi owing to Eq. [A6]. The simplification given by Eq. [A4] is also used giving
$$ \frac{{{\text{d}}\varepsilon_{gi} }}{{{\text{d}}t}} = \frac{1}{{V_{0} }}\int\limits_{{A_{ldi} }} {\vec{w}_{di} \cdot \vec{n}_{di} {\text{d}}A} + \frac{1}{{V_{0} }}\int\limits_{{A_{lsi} }} {\vec{w}_{si} \cdot \vec{n}_{si} {\text{d}}A} = S_{ldi} \bar{w}_{ldi} + S_{lsi} \bar{w}_{lsi}, $$
where ɛ gi = ɛ di + ɛ si , which is the volume fraction of all grains in class i, before the eutectic reaction occurs; S ldi = A ldi /V 0; and S lsi = A lsi /V 0.
For the external liquid (l), the following constitutive relation is valid
$$ \varepsilon_{l} = 1 - \varepsilon_{ee} - \sum\limits_{i = 1}^{N} {\varepsilon_{gi} } $$
Local thermodynamic equilibrium is assumed at the solid–liquid interface, implying that C si = KC di at A sdi (dendritic grains) and that C si = KC l at A lsi (globulitic grains), where K is the solute partition coefficient. Also C di or C l adjacent to solid–liquid interfaces can be related to T using the liquidus line, indicated for dendritic grains as C di = C liq(T) and for globulitic grains as C l = C liq,gi (T, R i ext ), where R i ext is the radius of curvature of the globulitic grains in class i (Section II–C–2). Equation [A3] was applied to the primary solid (k = si) adopting \( A_{kj} = A_{sdi} + A_{lsi} \), giving
$$ \rho \frac{{{\text{d}}\left( {\varepsilon_{si} \left\langle {C_{si} } \right\rangle^{si} } \right)}}{{{\text{d}}t}} = \frac{1}{{V_{0} }}\int\limits_{{A_{sdi} }} {\left( {\rho KC_{\text{liq}} \vec{w}_{si} - \vec{j}_{si} } \right) \cdot \vec{n}_{si} } {\text{d}}A + \frac{1}{{V_{0} }}\int\limits_{{A_{lsi} }} {\left( {\rho KC_{{{\text{liq}},gi}} \vec{w}_{si} - \vec{j}_{si} } \right) \cdot \vec{n}_{si} } {\text{d}}A $$
Substituting Eq. [A5], considering the local thermodynamic equilibrium and noting that C liq and C liq,gi are uniform within the REV give
$$ \begin{aligned} \rho \frac{{{\text{d}}\left( {\varepsilon_{si} \left\langle {C_{si} } \right\rangle^{si} } \right)}}{{{\text{d}}t}} & = \rho KC_{liq} \left( {\frac{1}{{V_{0} }}\int\limits_{{A_{sdi} }} {\vec{w}_{si} \cdot \vec{n}_{si} } {\text{d}}A} \right) + \rho KC_{liq,gi} \left( {\frac{1}{{V_{0} }}\int\limits_{{A_{lsi} }} {\vec{w}_{si} \cdot \vec{n}_{si} } {\text{d}}A} \right) + \\ & \quad + \rho D_{s} \frac{{S_{sdi} }}{{\delta_{sdi} }}\left( {KC_{\text{liq}} - \left\langle {C_{si} } \right\rangle^{si} } \right) + \rho D_{s} \frac{{S_{lsi} }}{{\delta_{lsi} }}\left( {KC_{{{\text{liq}},gi}} - \left\langle {C_{si} } \right\rangle^{si} } \right) \\ \end{aligned}. $$
The integral over A sdi is replaced with Eq. [A10] and all integrals over A lsi are simplified using Eq. [A4] to give
$$ \begin{aligned} \frac{{{\text{d}}\left( {\varepsilon_{si} \left\langle {C_{si} } \right\rangle^{si} } \right)}}{{{\text{d}}t}} & = KC_{\text{liq}} \left( {\frac{{d\varepsilon_{si} }}{dt} - S_{lsi} \bar{w}_{lsi} } \right) + KC_{{{\text{liq}},gi}} S_{lsi} \bar{w}_{lsi} + \\ & \quad + D_{s} \frac{{S_{sdi} }}{{\delta_{sdi} }}\left( {KC_{\text{liq}} - \left\langle {C_{si} } \right\rangle^{si} } \right) + D_{s} \frac{{S_{lsi} }}{{\delta_{lsi} }}\left( {KC_{{{\text{liq}},gi}} - \left\langle {C_{si} } \right\rangle^{si} } \right) \\ \end{aligned} $$
Equation [A3] is applied to the interdendritic liquid (\( k = d{\kern 1pt} i \)), considering \( A_{kj} = A_{sdi} + A_{ldi} \) and substituting Eq. [A7]. Owing to local equilibrium and uniformity of the interdendritic liquid composition, \( C_{l} = C_{\text{liq}} \) at A ldi , the equation becomes
$$ \rho \frac{\text{d}}{{{\text{d}}t}}\left( {\varepsilon_{{d{\kern 1pt} i}} \left\langle {C_{di} } \right\rangle^{{d{\kern 1pt} i}} } \right) = - \frac{1}{{V_{0} }}\int\limits_{{A_{sdi} }} {\left( {\rho KC_{\text{liq}} \vec{w}_{si} - \vec{j}_{si} } \right) \cdot \vec{n}_{si} } {\text{d}}A - \frac{1}{{V_{0} }}\int\limits_{{A_{ldi} }} {\left( {\rho C_{\text{liq}} \vec{w}_{l} - \vec{j}_{l} } \right) \cdot \vec{n}_{l} } {\text{d}}A. $$
Substituting again Eq. [A5] and noting that C liq is uniform in A sdi and A ldi gives
$$ \begin{aligned} \frac{\text{d}}{{{\text{d}}t}}\left( {\varepsilon_{{d{\kern 1pt} i}} \left\langle {C_{{d{\kern 1pt} i}} } \right\rangle^{{d{\kern 1pt} i}} } \right) & = - KC_{\text{liq}} \left( {\frac{1}{{V_{0} }}\int\limits_{{A_{sdi} }} {\vec{w}_{si} \cdot \vec{n}_{si} } {\text{d}}A} \right) - C_{\text{liq}} \left( {\frac{1}{{V_{0} }}\int\limits_{{A_{ldi} }} {\vec{w}_{l} \cdot \vec{n}_{l} } {\text{d}}A} \right) \\ & \quad - D_{s} \frac{{S_{sdi} }}{{\delta_{sdi} }}\left( {KC_{\text{liq}} - \left\langle {C_{si} } \right\rangle^{si} } \right) - D_{l} \frac{{S_{ldi} }}{{\delta_{li} }}\left( {C_{\text{liq}} - \left\langle {C_{l} } \right\rangle^{l} } \right) \\ \end{aligned} $$
In the integral over A ldi , \( \vec{n}_{l} = - \vec{n}_{di} \), \( \vec{w}_{l} = \vec{w}_{di} \), and Eq. [A11] is substituted. The integral over A sdi is again replaced with Eq. [A10], forcing the integral on A lsi to appear, which is finally approximated by Eq. [A4], resulting in
$$ \begin{aligned} \left\langle {C_{{d{\kern 1pt} i}} } \right\rangle^{di} \frac{{{\text{d}}\varepsilon_{di} }}{{{\text{d}}t}} & = - \varepsilon_{{d{\kern 1pt} i}} \frac{{{\text{d}}\left\langle {C_{{d{\kern 1pt} i}} } \right\rangle^{{d{\kern 1pt} i}} }}{{{\text{d}}t}} - K{\kern 1pt} C_{\text{liq}} \left( {\frac{{{\text{d}}\varepsilon_{si} }}{{{\text{d}}t}} - S_{lsi} \bar{w}_{lsi} } \right) + C_{\text{liq}} \left( {\frac{{{\text{d}}\varepsilon_{gi} }}{{{\text{d}}t}} - S_{lsi} \bar{w}_{lsi} } \right) \\ & \quad - D_{s} \frac{{S_{sdi} }}{{\delta_{sdi} }}\left( {KC_{\text{liq}} - \left\langle {C_{si} } \right\rangle^{si} } \right) - D_{l} \frac{{S_{ldi} }}{{\delta_{li} }}\left( {C_{\text{liq}} - \left\langle {C_{l} } \right\rangle^{l} } \right) \\ \end{aligned}. $$
Considering \( \varepsilon_{{d{\kern 1pt} i}} = \varepsilon_{gi} - \varepsilon_{si} \) and 〈C di 〉 di = C liq, after some arrangement the final equation is
$$ \begin{aligned} \left( {1 - K} \right)C_{\text{liq}} \frac{{{\text{d}}\varepsilon_{si} }}{{{\text{d}}t}} & = \left( {\varepsilon_{{g{\kern 1pt} i}} - \varepsilon_{{s{\kern 1pt} i}} } \right)\frac{{{\text{d}}C_{\text{liq}} }}{{{\text{d}}t}} + \left( {1 - K} \right)C_{\text{liq}} S_{lsi} \bar{w}_{lsi} \\ & \quad + D_{s} \frac{{S_{sdi} }}{{\delta_{sdi} }}\left( {KC_{\text{liq}} - \left\langle {C_{si} } \right\rangle^{si} } \right) + D_{l} \frac{{S_{ldi} }}{{\delta_{li} }}\left( {C_{\text{liq}} - \left\langle {C_{l} } \right\rangle^{l} } \right) \\ \end{aligned}. $$
Equation [A3] is applied to the external liquid (l), implying that the surface integrals are done over A ldi and A lsi of all N grain classes. Also C l = C liq at A ldi , and C l = C liq,gi at A lsi , giving
$$ \rho \frac{{{\text{d}}\left( {\varepsilon_{l} \left\langle {C_{l} } \right\rangle^{l} } \right)}}{{{\text{d}}t}} = \sum\limits_{i = 1}^{N} {\left[ {\frac{1}{{V_{0} }}\int\limits_{{A_{ldi} }} {\left( {\rho C_{\text{liq}} \vec{w}_{l} - \vec{j}_{l} } \right) \cdot \vec{n}_{l} } {\text{d}}A + \frac{1}{{V_{0} }}\int\limits_{{A_{lsi} }} {\left( {\rho C_{{{\text{liq}},gi}} \vec{w}_{l} - \vec{j}_{l} } \right) \cdot \vec{n}_{l} } {\text{d}}A} \right]}. $$
As before, C liq and C liq,gi are taken out of the integral sign, \( \vec{w}_{l} = \vec{w}_{di} \) and \( \vec{n}_{l} = - \vec{n}_{di} \) are substituted in the integral over A ldi , while \( \vec{w}_{l} = \vec{w}_{si} \) and \( \vec{n}_{l} = - \vec{n}_{si} \) are substituted in the integral over A lsi . Also, substituting Eq. [A5] gives
$$ \begin{aligned} \frac{{{\text{d}}\left( {\varepsilon_{l} \left\langle {C_{l} } \right\rangle^{l} } \right)}}{{{\text{d}}t}} & = - \sum\limits_{i = 1}^{N} {\left[ {C_{\text{liq}} \left( {\frac{1}{{V_{0} }}\int\limits_{{A_{ldi} }} {\vec{w}_{di} \cdot \vec{n}_{di} } {\text{d}}A} \right) + C_{{{\text{liq}},gi}} \left( {\frac{1}{{V_{0} }}\int\limits_{{A_{lsi} }} {\vec{w}_{si} \cdot \vec{n}_{si} } {\text{d}}A} \right) } \right.} \\ & \quad \left. { - D_{l} \frac{{S_{ldi} }}{{\delta_{li} }}\left( {C_{\text{liq}} - \left\langle {C_{l} } \right\rangle^{l} } \right) - D_{l} \frac{{S_{lsi} }}{{\delta_{li} }}\left( {C_{{{\text{liq}},gi}} - \left\langle {C_{l} } \right\rangle^{l} } \right)} \right] \\ \end{aligned}. $$
The integrals over A ldi and A lsi are approximated by Eqs. [A11] and [A4], and \( \sum\nolimits_{i = 1}^{N} {{\text{d}}\varepsilon_{gi} /{\text{d}}t = - {\text{d}}\varepsilon_{l} /{\text{d}}t} \) (valid before the eutectic) is used, giving the final equation
$$ \begin{aligned} \frac{{{\text{d}}\left( {\varepsilon_{l} \left\langle {C_{l} } \right\rangle^{l} } \right)}}{{{\text{d}}t}} & = C_{\text{liq}} \frac{{{\text{d}}\varepsilon_{l} }}{{{\text{d}}t}} + \sum\limits_{i = 1}^{N} {\left[ {\left( {C_{\text{liq}} - C_{{{\text{liq}},gi}} } \right)S_{lsi} \bar{w}_{lsi} } \right]} + D_{l} \left( {\sum\limits_{i = 1}^{N} {\frac{{S_{ldi} }}{{\delta_{li} }}} } \right)\left( {C_{\text{liq}} - \left\langle {C_{l} } \right\rangle^{l} } \right) \\ & \quad + D_{l} \sum\limits_{i = 1}^{N} {\left[ {\frac{{S_{lsi} }}{{\delta_{li} }}\left( {C_{{{\text{liq}},gi}} - \left\langle {C_{l} } \right\rangle^{l} } \right)} \right]} \\ \end{aligned} $$
W. Oldfield: ASM Trans. Q., 1966, vol. 59, pp. 945–61.Google Scholar
M. Rappaz, P. Thévoz, Z. Jie, J.P. Gabathuler, and H. Lindscheid: in State of the Art of Computer Simulation of Casting and Solidification Processes, H. Fredriksson ed., Les Éditions de Physique, 1986, pp. 277–84.Google Scholar
I. Maxwell and A. Hellawell: Acta Metall., 1975, vol. 23, pp. 229–37.CrossRefGoogle Scholar
A.L. Greer, A.M. Bunn, A. Tronche, P.V. Evans, and D.J. Bristow: Acta Mater., 2000, vol. 48, pp. 2823–35.CrossRefGoogle Scholar
M. Rappaz and P. Thevoz: Acta Metall., 1987, vol. 35, pp. 1487–97.CrossRefGoogle Scholar
P. Thevoz, J.L. Desbiolles, and M. Rappaz: Metall. Trans. A, 1989, vol. 20, pp. 311–22.CrossRefGoogle Scholar
Z. Kolenda, J. Donizak, and J.C.E. Bocardo: Metall. Mater. Trans. B, 1999, vol. 30B, pp. 505–13.CrossRefGoogle Scholar
C.Y. Wang and C. Beckermann: Metall. Trans. A, 1993, vol. 24A, pp. 2787–2802.CrossRefGoogle Scholar
C.Y. Wang and C. Beckermann: Mater. Sci. Eng. A, 1993, vol. 171, pp. 199–211.CrossRefGoogle Scholar
C. Gandin, S. Mosbah, T. Volkmann, and D. Herlach: Acta Mater., 2008, vol. 56, pp. 3023–35.CrossRefGoogle Scholar
M. Wu and A. Ludwig: Acta Mater., 2009, vol. 57, pp. 5621–31.CrossRefGoogle Scholar
A.K. Dahle, K. Nogita, J.W. Zindel, S.D. McDonald, and L.M. Hogan: Metall. Mater. Trans. A, 2001, vol. 32A, pp. 949–60.CrossRefGoogle Scholar
S. Nafisi, R. Ghomashchi, and H. Vali: Mater. Charact., 2008, vol. 59, pp. 1466–73.CrossRefGoogle Scholar
S.D. McDonald, A.K. Dahle, J.A. Taylor, and D.H. StJohn: Metall. Mater. Trans. A, 2004, vol. 35A, pp. 1829–37.CrossRefGoogle Scholar
S.D. McDonald, K. Nogita, and A.K. Dahle: Acta Mater., 2004, vol. 52, pp. 4273–80.CrossRefGoogle Scholar
T.W. Clyne: Mater. Sci. Eng., 1984, vol. 65, pp. 111–24.CrossRefGoogle Scholar
D.D. Goettsch and J.A. Dantzig: Metall. Mater. Trans. A, 1994, vol. 25A, pp. 1063–79.CrossRefGoogle Scholar
J.C. Slattery: Advanced Transport Phenomena, Cambridge University Press, Cambridge, UK, 1999, pp. 585–86.CrossRefGoogle Scholar
W.W. Mullins and R.F. Sekerka: J. Appl. Phys., 1963, vol. 34, pp. 323–29.CrossRefGoogle Scholar
M. Qian: Acta Mater., 2006, vol. 54, pp. 2241–52.CrossRefGoogle Scholar
M.A. Martorano, C. Beckermann, and C.A. Gandin: Metall. Mater. Trans. A, 2003, vol. 34A, pp. 1657–74.CrossRefGoogle Scholar
J. Lipton, M.E. Glicksman, and W. Kurz: Mater. Sci. Eng., 1984, vol. 65, pp. 57–63.CrossRefGoogle Scholar
T.Z. Kattamis, J.C. Coughlin, and M.C. Flemings: Trans. AIME, 1967, vol. 239, pp. 1504–11.Google Scholar
H.B. Aaron, D. Fainstei, and G.R. Kotler: J. Appl. Phys., 1970, vol. 41, pp. 4404–10.CrossRefGoogle Scholar
J.W. Christian: The Theory of Transformation in Metals and Alloys, Pergamon, New York, NY, 2002, pp. 16–22.Google Scholar
T.E. Quested and A.L. Greer: Acta Mater., 2004, vol. 52, pp. 3859–68.CrossRefGoogle Scholar
M.C. Flemings: Mater. Trans., 2005, vol. 46, pp. 895–900.CrossRefGoogle Scholar
D. Eskin, Q. Du, D. Ruvalcaba, and L. Katgerman: Mater. Sci. Eng. A, 2005, vol. 405, pp. 1–10.CrossRefGoogle Scholar
P. Thévoz, J.L. Desbiolles, and M. Rappaz: Metall. Trans. A, 1989, vol. 20A, pp. 311–22.CrossRefGoogle Scholar
C.A. Gandin and M. Rappaz: Acta Metall. Mater., 1994, vol. 42, pp. 2233–46.CrossRefGoogle Scholar
Y. Du, Y.A. Chang, B.Y. Huang, W.P. Gong, Z.P. Jin, H.H. Xu, Z.H. Yuan, Y. Liu, Y.H. He, and F.Y. Xie: Mater. Sci. Eng. A, 2003, vol. 363, pp. 140–51.CrossRefGoogle Scholar
S.I. Fujikawa, K.I. Hirano, and Y. Fukushima: Metall. Trans. A, 1978, vol. 9A, pp. 1811–15.CrossRefGoogle Scholar
C.A. Gandin: Acta Mater., 2000, vol. 48, pp. 2483–2501.CrossRefGoogle Scholar
P. Magnin, J.T. Mason, and R. Trivedi: Acta Metall. Mater., 1991, vol. 39, pp. 469–80.CrossRefGoogle Scholar
M. Bamberger, B.Z. Weiss, and M.M. Stupel: Mater. Sci. Technol., 1987, vol. 3, pp. 49–56.CrossRefGoogle Scholar
J.A. Sarreal and G.J. Abbaschian: Metall. Trans. A, 1986, vol. 17A, pp. 2063–73.CrossRefGoogle Scholar
M. Johnsson: Thermochim. Acta, 1995, vol. 256, pp. 107–21.CrossRefGoogle Scholar
L.A. Tarshis, J.L. Walker, and J.W. Rutter: Metall. Trans., 1971, vol. 2, pp. 2589–97.CrossRefGoogle Scholar
M. Easton and D. StJohn: Metall. Mater. Trans. A, 1999, vol. 30A, pp. 1613–23.CrossRefGoogle Scholar
J.A. Spittle: Int. J. Cast. Met. Res., 2006, vol. 19, pp. 210–22.CrossRefGoogle Scholar
D. Apelian, G.K. Sigworth, and K.R. Whaler: Trans. AFS, 1984, vol. 92, pp. 297–307.Google Scholar
J. Ni and C. Beckermann: Metall. Trans. B, 1991, vol. 22B, pp. 349–61.CrossRefGoogle Scholar
S. Whitaker: Chem. Eng. Sci., 1973, vol. 28, pp. 139–47.CrossRefGoogle Scholar
W.G. Gray: Chem. Eng. Sci., 1975, vol. 30, pp. 229–33.CrossRefGoogle Scholar
© The Minerals, Metals & Materials Society and ASM International 2014
1.Department of Metallurgical and Materials EngineeringUniversity of São PauloSão PauloBrazil
2.Materials DepartmentPetróleo Brasileiro S.A.Rio De JaneiroBrazil
3.University of AntioquiaMedellínColombia
Martorano, M.A., Aguiar, D.T. & Arango, J.M.R. Metall and Mat Trans A (2015) 46: 377. https://doi.org/10.1007/s11661-014-2620-7 | CommonCrawl |
\begin{document}
\title{New Results on Vector and Homing Vector Automata\footnote{Many of these results were obtained during Yakary\i lmaz's visit to Bo\u{g}azi\c{c}i University in July-August 2017.}}
\author{\"{O}zlem Salehi}
\address{ Bo\u{g}azi\c{c}i University, Department of Computer Engineering \\ Bebek 34342, \.{I}stanbul, Turkey\\
\email{[email protected]}}
\author{Abuzer Yakary{\i}lmaz}
\address{University of Latvia, Center for Quantum Computer Science\\ R\={\i}ga, Latvia \\
\email{[email protected]} }
\author{A. C. Cem Say}
\address{ Bo\u{g}azi\c{c}i University, Department of Computer Engineering \\ Bebek 34342, \.{I}stanbul, Turkey\\
\email{[email protected]}}
\markboth{\"{O}. Salehi, A. Yakary{\i}lmaz, A. C. C. Say} {New Results on Vector and Homing Vector Automata}
\maketitle
\begin{abstract} We present several new results and connections between various extensions of finite automata through the study of vector automata and homing vector automata. We show that homing vector automata outperform extended finite automata when both are defined over $ 2 \times 2 $ integer matrices. We study the string separation problem for vector automata and demonstrate that generalized finite automata with rational entries can separate any pair of strings using only two states. Investigating stateless homing vector automata, we prove that a language is recognized by stateless blind deterministic real-time version of finite automata with multiplication iff it is commutative and its Parikh image is the set of nonnegative integer solutions to a system of linear homogeneous Diophantine equations. \end{abstract}
\keywords{Homing vector automata; group automata; finite automata.} \section{Introduction}
Extending finite automata with an external storage is a well studied topic of automata theory. In this manner, various extensions of finite automata such as quantum and probabilistic models can be viewed as finite automata augmented with a vector as memory and the computation process can be seen as a series of vector and matrix multiplications.
With the motivation of the matrix multiplication view of programming, we have previously introduced vector automata (VA) \cite{SYS13}, which are finite automata equipped with a vector that is multiplied with an appropriate matrix at each step. Only one of the entries of the vector can be checked for equivalence to a rational number and the computation is successful if it ends in an accept state with a successful equivalence test. We have shown that this model is closely related to various extensions of finite automata including counter automata \cite{FMR67}, finite automata with multiplication (FAM) \cite{ISK76}, and generalized finite automata \cite{Tu68}.
In many of the models which extend finite automata, a computation is successful if the register is equal to its initial value at the end of the computation. Applying this idea to vector automata, we have introduced the homing vector automaton (HVA) \cite{SS15}, which can check its vector for equivalence to its initial value and accepts a string if the computation ends with the initial vector in an accept state. We have shown that there exist close connections between homing vector automata \cite{SSD16} and extended finite automata \cite{DM00}. In fact, homing vector automata can be seen as a special case of rational monoid automata \cite{RK10}.
The close connections among the models exhibited so far motivates us to further investigate vector automata and homing vector automata. In this respect, the aim of this paper is to present some new results and establish further relationships among the various extensions of finite automata, through the study of these two models. We also provide some additional results about vector automata and homing vector automata, which may further advance the research in the topic.
We first focus on extended finite automata and show that allowing postprocessing does not change the computational power of the model. It is an open question whether the class of languages recognized by extended finite automata over $ 3 \times 3 $ integer matrices includes that of one-way nondeterministic blind three counter automata. We prove that one-way three-dimensional nondeterministic blind homing vector automata (NBHVA) with integer entries can simulate any nondeterministic blind multicounter machine. We answer an open question from \cite{SSD16}, by proving that one-way NBHVAs are more powerful than extended finite automata \cite{DM00} when both are defined on the set of $ 2 \times 2$ integer matrices.
Given two strings, a finite automaton (DFA) is said to separate them if it accepts one and rejects the other. The string separation problem, which asks for the minimum number of states needed for accomplishing this task, was introduced by Goral{\v{c}}{\'\i}k and Koubek \cite{GK86}, and the best known upper bound was given in \cite{Ro89}. The results from \cite{GK86} provide a logarithmic lower bound in the length of the strings. The problem of finding a generic tight bound is still open \cite{DESW11}.
We study the problem of string separation, and show that deterministic blind vector automata with vectors of size two can distinguish any string from any other string in blind mode (without the need to check the entries of the vector during the computation) even when they are restricted to be stateless (i.e. with only one state). This result implies that generalized finite automata with rational entries can distinguish any pair of strings by using only two states. We also present some results on finite language recognition.
Stateless machines \cite{YDI08,IKO10,KMO09} have been investigated by many researchers, motivated by their relation to membrane computing and P systems \cite{Pau00}, which are stateless models inspired from biology. While vector automata can simulate their classical states in their vectors by using longer vectors, this is not the case for homing vector automata. This leads us to investigate stateless homing vector automata in more detail.
Our study on stateless homing vector automata yields a characterization for the class of languages recognized by stateless real-time deterministic FAMs without equality (0-DFAMW) \cite{ISK76}. It turns out that a language is recognized by a 0-DFAMW iff it is commutative and its Parikh image is the set of nonnegative solutions to a system of linear homogeneous Diophantine equations. When the computation is nondeterministic, then any language recognized by a stateless real-time nondeterministic FAM without equality is commutative. We conclude by providing some further examples and observations about language recognition power of stateless homing vector automata.
\section{Background} \label{sec: back}
\subsection{Preliminaries}
The following notation will be used throughout the paper. The set of states is $Q = \{ q_1,\ldots,q_n \}$ for some $ n \geq 1 $, where $ q_1 $ is the initial state, unless otherwise specified. $Q_a \subseteq Q$ is the set of accept state(s). The input alphabet is denoted by $\Sigma$ and we assume that it never contains $ \dollar $ (the right end-marker). For a given string $ w \in \Sigma^* $, $w^r$ denotes its reverse, $ |w| $ denotes its length, $ w[i] $ denotes its $i$'th symbol, $ |w|_{\sigma} $ denotes the number of occurrences of symbol $ \sigma $ in $ w $ and $ \mathtt{L}_w $ denotes the singleton language containing only $ w $. For a given language $L$, its complement is denoted by $\overline{L}$. The power set of a set $S$ is denoted $ \mathcal{P}(S) $.
For a string $ w \in \Sigma^* $ where $ \Sigma=\{\sigma_1,\sigma_2,\dots,\sigma_k\} $ is an ordered alphabet, the Parikh image of $ w $ is defined as $ \phi(w)=(|w|_{\sigma_1}, |w|_{\sigma_2},\dots, |w|_{\sigma_k}) $. For a language $ L $, $ \phi(L) = \{\phi(w)| w \in L\} $.
A subset $ S \subseteq \mathbb{N}^n $ is a \textit{linear} set if $ S=\{v_0 + \Sigma_{i=1}^k c_iv_i | c_1,\dots,c_k \in \mathbb{N}\} $ for some $ v_0,\dots,v_k \in\mathbb{N}^n $. A \textit{semilinear} set is a finite union of linear sets. A language is called \textit{semilinear} if $ \phi(L) $ is semilinear.
For a machine model $A$, $\mathfrak{L}(A)$ denotes the class of all languages recognized by machines of type $A$. We denote the class of context-free languages by $ \mathsf{CF} $ and the set of recursively enumerable languages by $ \mathsf{RE} $.
For a given row vector $ v $, $ v[i] $ denotes its $i$'th entry. Let $ A_{k \times l} $ be a $ k \times l $ dimensional matrix. $ A[i,j] $ denotes the entry in the $ i $'th row and $ j $'th column of $ A $. Given matrices $ A_{k\times l} $ and $ B_{m \times n} $, their tensor product is defined as
\[ A \otimes B _{km \times ln} =\mymatrix{ccc}{ A[1,1]B & \cdots & A[1,l]B \\ \vdots & & \vdots \\ A[k,1]B &\dots &A[k,l]B}. \]
\subsection{Machine definitions}
In this section, we give definitions of the various models which will appear throughout the paper.
An input string $ w $ is placed on a tape in the form $ w\dollar$. Most of the time, we focus on real-time computation, where the tape head moves right at each step. A machine is \textit{deterministic} if the next move of the machine is uniquely determined, and \textit{nondeterministic} if there may exist more than one possible move at each step.
When we want to specify the number of states of a machine, we add an $ n $- (or $(n)$- to avoid any confusion) to the front of the model name where $ n $ is the number of states. We will examine stateless models, i.e. one-state automata where the single state is accepting, in more detail. It is clear that if the single state is non-accepting, then the automaton can recognize only the empty language. When a machine is stateless, we denote this by adding 0- to the front of the abbreviation for the model name.
A \textit{(real-time) deterministic $k$-counter automaton} (D$ k $CA) \cite{FMR67} is a deterministic finite automaton equipped with $ k $ counters. The counters are initially set to 0 and updated at each step by $c \in \{-1,0,1\}^k$, based on the current state, scanned symbol and the current status of the counters. The status of the counters is given by $\theta \in \{=,\neq\}^k $, where $ = $ and $ \neq $ denote whether the corresponding counter values equal zero or not, respectively. An input string is accepted if the machine is in an accept state at the end of the computation. A \textit{(real-time) deterministic blind $k$-counter automaton} \cite{Gr78} D$ k $BCA is a D$ k $CA which can check the value of its counters only at the end of the computation. An input string is accepted by a blind counter automaton if the machine is in an accept state and all counter values are equal to 0 at the end of the computation.
A \textit{generalized finite automaton} (GFA) \cite{Tu69} is a 5-tuple $ G=(Q,\Sigma,\{A_{\sigma \in \Sigma}\}, v_0, f),$
where the $A_{\sigma \in \Sigma}$'s are $|Q|\times |Q|$ real valued transition matrices, and $v_0$ and $f$ are the real valued initial row vector and final column vector, respectively. The acceptance value for an input string $w
\in \Sigma^*$ is defined as $f_{G}(w)=v_oA_{w[1]}\cdots A_{w[|w|]}f$.
There are various ways to define the language recognized by a generalized finite automaton. In this paper, we are interested in the class $\textup{S}^=$, which contains languages of the form $L=(G,=\lambda)=\{w\in \Sigma^* \mid f_{\mathcal{G }}(w)=\lambda\}$, where $ \lambda \in \mathbb{R}$ is called the cutpoint \cite{Tu69}.
A GFA whose components are restricted to be rational numbers is called a \textit{Turakainen finite automaton} (TuFA) in \cite{Yak12} and the class of languages recognized by a TuFA in the same language recognition mode with a rational cutpoint is denoted by $\textup{S}^=_{\mathbb{Q}}$.
A \textit{(real-time) deterministic finite automaton with multiplication} (DFAM) \cite{ISK76} is a deterministic finite automaton equipped with a register holding a positive rational number. The register is initialized to 1 at the beginning of the computation and multiplied with a positive rational number at each step, based on the current state, scanned symbol and whether the register is equal to 1 or not. An input string is accepted if the computation ends in an accept state with the register value being equal to 1, after processing the right end-marker. A DFAM \textit{without equality} (DFAMW) is a DFAM which cannot check whether or not the register has value 1 during computation. Nondeterministic versions NFAM and NFAMW are defined analogously.\footnote{The original definition of FAMs is given for \textit{one-way} machines (1DFAM, 1NFAM, 1DFAMW, 1NFAMW) where the tape head is allowed to stay on the same input symbol for more than one step. The computation halts and the string is accepted when the machine enters an accept state with the tape head on the end-marker $\dollar$.}
Let $ M $ be a monoid. An \textit{extended finite automaton over $ M $} (\textit{$ M $-automaton}, \textit{monoid automaton}) \cite{DM00, Co05,Ka09} is a nondeterministic finite automaton equipped with a register which holds an element of the monoid $ M $. The register is initialized with the identity element of $ M $ and multiplied (operation is applied) by an element of the monoid based on the current state and the scanned symbol (or the empty string). An input string is accepted if the value of the register is equal to the identity element of the monoid and the computation ends in an accept state. When $ M $ is a group, then the model is called \textit{group automaton}.
Note that extended finite automata are nondeterministic by definition and they are blind in the sense that the register cannot be checked until the end of the computation. Computation of an extended finite automaton is not real-time since the machine is allowed to make $ \varepsilon $ transitions. The class of languages recognized by $ M $-automata is denoted by $ \mathfrak{L}(M) $.
\subsection{Vector automata and homing vector automata}
A \textit{ (real-time) $k$-dimensional deterministic vector automaton} (DVA($k$)) \cite{SYS13} is a 7-tuple
\[ V = (Q,\Sigma,M,\delta,q_1,Q_a,v_0), \] where $ v_0 $ is the initial ($k$-dimensional, rational-valued) row vector, $ M $ is a finite set of $k \times k$-dimensional rational-valued matrices, and $ \delta $ is the transition function (described below) defined as \[ \delta:Q \times \Sigma \cup \{\dollar\} \times \{=,\neq\} \rightarrow Q \times M. \] Let $ w\in \Sigma^* $ be a given input. The automaton $ V $ reads the sequence $ w\dollar $ from left to right symbol by symbol. It uses its states and its vector to store and process the information. In each step, it can check whether the first entry of the vector is equal ($=$) to 1 or not ($ \neq $). We call this feature the ``status" of the first entry.
The details of the transition function are as follows. When $ V $ is in state $ q \in Q $, reads symbol $ \sigma \in \Sigma \cup \{\dollar\} $, and the first entry status is $ \tau \in \{ =,\neq \} $, the transition \[ \delta(q,\sigma,\tau) = (q',A) \] results in $V$ entering state $ q' \in Q $, and its vector being multiplied by $ A \in M $ from the right.
At the beginning of the computation, $ V $ is in state $ q_1 $ and the vector is $ v_0 $. Then, after reading each symbol, the state and vector are updated according to the transition function as described above. The input $ w $ is accepted if the final state is an accept state and the first entry of the final vector is 1 after processing the right end-marker $\dollar$. Otherwise, the input is rejected. The set of all accepted strings is said to be the language recognized by $ V $.
A \textit{ (real-time) $k$-dimensional deterministic homing vector automaton} (DHVA($k$)) is defined in \cite{SS15} as being different from vector automata in two ways: (1) Homing vector automata do not read the right end-marker after reading the input, so there is no chance of postprocessing and, (2) instead of checking the status of the first entry, a homing vector automaton checks whether the complete current vector is identical to the initial vector or not. Formally, the transition function $\delta$ is defined as $$\delta: Q \times \Sigma \times \{=,\neq\} \rightarrow Q\times M,$$ where $ = $ indicates equality to the initial vector $ {v_0} $, and $ \neq $ indicates inequality. An input string is accepted if the computation ends in an accept state and the vector is equal to its initial value.
The blind versions of these models, \textit{(real-time) $k$-dimensional deterministic blind vector automaton} (DBVA($ k $)) and \textit{(real-time) $k$-dimensional deterministic blind homing vector automaton} (DBHVA($ k $)) cannot check the status of the vector until the end of the computation. Therefore, the domain of the transition function changes to $ Q \times \Sigma \cup \{\dollar\} $ and $ Q \times \Sigma$ for vector automata and homing vector automata respectively. The acceptance condition is the same as in the non-blind case.
The definitions of \textit{(real-time) $k$-dimensional nondeterministic vector automaton}, abbreviated NVA($k$), and \textit{(real-time) $k$-dimensional nondeterministic homing vector automaton}, abbreviated NHVA($k$), are almost the same as that of the deterministic versions, except that the range of the transition function $ \delta $ is now defined as $ \mathcal{P}(Q \times M)$, which allows the machine to follow more than one computational path. An input string is accepted if and only if there is a path ending with the acceptance condition.
Abbreviations used for some model variants discussed so far are given in Table \ref{tab: abb}.
\begin{table} \caption{Abbreviations for some model names.}\label{tab: abb}{\footnotesize \begin{tabular}{|p{2.1cm}|p{1.7cm}|p{1.7cm}|p{2.2cm}|p{2.2cm}|}
\hline & Deterministic & Deterministic & Nondeterministic & Nondeterministic \\
& blind & & blind & \\
\hline Vector \hspace{0.8cm} automaton & \hspace{1.7cm} DBVA($k$) & \hspace{1.7cm} DVA($ $k$ $) & \hspace{1.7cm} NBVA($k$) & \hspace{1.7cm}NVA($k$) \\
\hline Homing vector automaton &\hspace{0.7cm} DBHVA($k$) & \hspace{1.7cm} DHVA($k$) & \hspace{1.7cm}NBHVA($k$) & \hspace{1.7cm}NHVA($k$) \\
\hline Counter automaton &\hspace{0.7cm} D$k$BCA & \hspace{1.7cm} D$k$CA & \hspace{1.7cm}N$k$BCA & \hspace{1.7cm}N$k$CA \\
\hline Finite automata \hspace{0.9cm} with\hspace{1cm} multiplication & \hspace{6cm}DFAMW & \hspace{1.7cm} DFAM & \hspace{1.7cm} NFAMW & \hspace{1.7cm} NFAM \\
\hline
\end{tabular}}
\end{table}
When we do not want to explicitly state the dimension of the vector or the number of the counters, then we may omit $ k $ in the abbreviation when we talk about vector automata, homing vector automata and counter automata. Similarly, we may omit $ D $ and $ N $ in the abbreviation when we talk about a statement that is true for both cases.
\section{New results on homing vector automata and extended finite automata} \label{sec: relation } In this section, we will focus on the relationship between extended finite automata and homing vector automata.
\subsection{Some lemmas on homing vector automata} \label{sec: endmarker}
We will start by proving a lemma that will be used in the rest of the section.
Homing vector automata are not allowed to perform postprocessing by definition, since they do not read the right end-marker. In this section, we show that allowing postprocessing does not bring any additional power to nondeterministic blind homing vector automata, as the postprocessing step can be handled by using some extra states. We prove the result for the more general case of 1NBHVAs, NBHVAs that are capable of making $ \varepsilon $ transitions. The computation of a 1NBHVA using end-marker ends once the machine processes the end-marker $ \dollar $.
HVAs using end-marker will be denoted by the abbreviation $\textup{HVA}_\$ $.
\begin{lemma}
\label{thm: NBHVA-endmarker}
Let $ L $ be a language recognized by an $(n)$-$\textup{1NBHVA}_\$(k)$ $ V $. Then, $ L $ is also recognized by an $(n+2)$-$\textup{1NBHVA}(k)$ $ V' $. \end{lemma} \begin{proof}
We construct $V'$ such that $V'$ mimics the transitions of $V$ on every possible symbol $\sigma \in \Sigma \cup \{\varepsilon\}$. In addition, we create new transitions to handle the postprocessing, which emulate $ V $'s last action before reading the end-marker (which would end up in an accept state) and the end-marker ($ \sigma \dollar $) at once: At any point during the scanning, if reading $ \sigma $ would cause $V$ to switch to a state from which the end-marker would lead to an accept state, a new nondeterministic transition takes $ V' $ to the additional state, which is an accept state. During this transition, the vector is multiplied by a matrix of the form $ A_\sigma A_\dollar $, where $ A_\sigma $ and $ A_\dollar $ are the corresponding matrices defined in the transition function of $ V $. All other states of $V'$, which are inherited from $V$, are designated to be non-accept states.
Thus, $ V' $ simulates the computation of $ V $ on any non-empty string, and accepts the input in some computational path if and only if $ V $ accepts it.
If $ L $ contains the empty string, we add one more state that has the following properties: (i) it becomes the new initial state of the resulting machine, (ii) it is an accept state, (iii) it causes the machine to behave (i.e. transition) like the original initial state of $ V $ upon reading the first symbol, and (iv) there is no transition coming in to this state. \end{proof}
The idea given in the proof of Lemma \ref{thm: NBHVA-endmarker} does not apply for non-blind models since the status of the vector may be changed after reading the last symbol of the input (just before reading the right end-marker). In fact, one can show that DHVAs using end-marker are more powerful than ordinary DHVAs in terms of language recognition by the witness language $\mathtt{NEQ}=\{ a^ib^j : i \neq j\}$ . \\ \begin{comment} \begin{theorem} \label{thm: DHVAendmarker}
$ \bigcup_k\mathfrak{L}(\textup{DHVA($ k $)}) \subsetneq \bigcup_k \mathfrak{L}(\textup{DHVA($ k $)}_\$)$. \end{theorem} \begin{proof}
The subset inclusion is immediate, since the postprocessing may very well involve multiplication with the identity matrix. For the inequality, consider the language $\mathtt{NEQ}=\{ a^ib^j : i \neq j\}$. Suppose for a contradiction that there exists a DHVA($ k $) $H$ recognizing $\mathtt{NEQ}$ for some $k$. Let $v_0$ be the initial vector of $H$. There exist sufficiently long strings $w_1=a^mb^n$ and $w_2=a^mb^o$, $m \neq n$, $m \neq o$, $n < o$ such that $H$ is in the same accept state
after reading $w_1$ and $w_2$ and the vector is equal to $v_0$, since the strings belong to $\mathtt{NEQ}$. When both strings are extended with $b^{m-n}$, $a^mb^nb^{m-n} \notin \mathtt{NEQ}$ whereas $a^mb^o b^{m-n} \in \mathtt{NEQ}$. Since the same vector is being multiplied with the same matrices associated with the same states during the processing of the string $b^{m-n}$, it is not possible for $H$ to give different responses.
Now let us prove that the language $\mathtt{NEQ}$ can be recognized by a $\textup{DHVA}_\$(2)$ $H'$. The initial vector of $H'$ is $v_0'=\mypar{1~~1} $, and the matrices to be used are as follows.
\[A=\mymatrix{ cc }{
1 & 0 \\
1 & 1
}~~
B=\mymatrix{ rc }{
1 & 0 \\
-1 & 1
}~~
C_= =\mymatrix{ cc }{
1 & 0 \\
0 & 0
}~~
C_{\neq}=\mymatrix{ cc }{
0 & 0 \\
1 & 1
}
\]
For each $a$, the first entry is increased by 1 and for each $b$ the first entry is decreased by 1 using the second entry of the vector, which is equal to 1 throughout the computation. The increment and decrement operations are performed by the matrices $A$ and $B$.
When reading the end-marker, if the value of the vector is equal to its initial value, meaning that the number of $a$'s and $b$'s were equal to each other, the vector is multiplied with $C_=$, which sets the second entry to 0, so that the input string is not accepted. Otherwise, if the vector is not equal to its initial value, meaning that the number of $a$'s and $b$'s were not equal, the vector is multiplied with $C_{\neq}$, which sets the first entry to 1. This returns the vector to its initial value, and the input
string is accepted. \end{proof}
\end{comment} In the following lemma, we show that any 1NBHVA with end-marker whose matrices are rational valued can be simulated by integer valued matrices in the cost of increasing the size of the vector by 2. The lemma is also valid for the deterministic and real-time models.
\begin{lemma}
\label{thm: ratint} For any given rational-valued $(n)$-$ \textup{1NBHVA}_\$(k)$ $ V $, there exists an integer-valued $(n)$-$ \textup{1NBHVA}_\$(k+2)$ $ V' $ that recognizes the same language. \end{lemma} \begin{proof}
It is easy to see that, for any HVA, the initial vector $ v_0 $ can be replaced with the vector $ t v_0 $ for any rational $ t \neq 0 $ without changing the recognized language. Therefore, any HVA can be assumed to be defined with an integer-valued initial vector without loss of generality.
Let the initial vector of $ V $ be
$
v_0 = \mypar{ i_1 ~~ i_2 ~~ \cdots ~~ i_k } \in \mathbb{Z}^k$. Furthermore, let $ S_{i_a}, S_{i_b},S_{i_\varepsilon}$ and $S_{i_\dollar} $
be the sets of $ k \times k $ rational-valued matrices of $ V $ to be applied when $ V $ is in state $q_i$ and reads the symbols $ a $, $ b $, $ \varepsilon $ and $\dollar$, respectively.
The automaton $ V' $ is obtained from $V$ by modifying the initial vector and matrices. We pick an integer $ c \in \mathbb{Z}$ such that when multiplied with $ c $, the matrices in $S_{i_a},S_{i_b},S_{i_\varepsilon}$ and $ S_{i_\dollar} $ contain only integer values, where $ 1 \leq i \leq n $. Then, we define the set $ S'_{i_\sigma} $ by letting
\[
v_0' = \mypar{ v_0 ~~ 1 ~~ 1 } \mbox{ ~and~ }
A'_{i_{\sigma_j}} = \mymatrix{c|cc}{ c A_{i_{\sigma_j}} & 0 & 0 \\ \hline 0 & c & 0 \\ 0 & 0 & 1 },
\]
where $ A_{i_{\sigma_j}} \in S_{i_{\sigma}} $, $ \sigma \in \{a,b,\varepsilon,\dollar\} $ and $ j $ enumerates the number of possible transitions for $ \sigma $ in state $ q_i $. $ v'_0 $ is the initial vector of $ V' $, and when in state $ q_i $, $V'$ multiplies the vector with an integer-valued matrix from the set $ S_{i_{a}}'$, $ S_{i_{b}}' $ or $ S_{i_{\varepsilon}}' $ upon reading the inputs $a$, $b$, and $ \varepsilon $ respectively. When in state $q_i$ and reading symbol $\dollar$, the vector is multiplied with a matrix from the set $ S''_{i_\dollar} $, which is obtained by multiplying the matrices in $ S_{i_\dollar}' $ with
\[
\mymatrix{ c|cc }{
& 0 & 0 \\
~~~ - I ~~~~~ & \vdots & \vdots
\\
& 0 & 0
\\ \hline
i_1 ~~ i_2 ~~ \cdots ~~ i_k & 0 & 0 \\
i_1 ~~ i_2 ~~ \cdots ~~ i_k & 1 & 1
},
\]
where $ I $ denotes the identity matrix.
Let $ w \in \{a,b\}^* $ be a string of length $ l $, and let's consider a single computation path of length $ p+1 \geq l+1 $ for $ w $. The final vector is calculated as
\[
v_f = v_0 M_1 M_2 \cdots M_{p} M_{p+1}
=
\mypar{ j_1 ~~ j_2 ~~ \cdots ~~ j_k } \in \mathbb{Q}^k ,
\]
where $ M_i $ is the matrix used at the $ i $'th transition step.
It is easy to see that
\[
v_f' = v_0' M_1' \cdots M_p' M_{p+1}'
=
\mypar{ c^{p+1} v_f ~~~ c^{p+1} ~~~ 1 }.
\]
(Note that the equation above contains $ M_{p+1}' $, but not $ M_{p+1}'' $.) By setting $ c'= c^{p+1} $, we can rewrite $ v_f' $ as
\[
\mypar{ c'j_1 ~~~ c'j_2~~~ \cdots ~~~ c'j_k~~~ c'~~~ 1 }
\in \mathbb{Z}^{k+2}.
\]
For accepted input strings, $ v_f' $ holds $ c'$ times the initial vector. The postprocessing step, which is accomplished by a matrix from the set $ S''_{i_\dollar} $, helps setting the vector back to its initial value to satisfy the acceptance criteria.
Hence, $ v_f'' = v_0' M_1' \cdots M_p' M_{p+1}'' $ is
\[
v''_f
=
\mypar{ c'i_1- c'j_1+ i_1 ~~~~
c'i_2 - c'j_2 + i_2 ~~~~
\cdots
~~~~ c'i_k- c'j_k + i_k
~~~~ 1 ~~~~ 1}.
\]
It is clear that $ v_0 = v_f $ if and only if $ v'_0 = v''_f $. Thus, rational-valued $ V $ and integer-valued $ V' $ recognize the same language. \end{proof}
\begin{corollary}
Rational-valued \textup{1NBHVA}s and integer-valued \textup{1NBHVA}s recognize the same class of languages. \end{corollary} \begin{proof}
By using Lemma \ref{thm: ratint}, we can conclude that any rational-valued 1NBHVA can be simulated by an integer-valued 1NBHVA using the end-marker. Then, by using Lemma \ref{thm: NBHVA-endmarker}, we can remove the end-marker. \end{proof}
\subsection{Extended finite automata and homing vector automata defined over integer matrices}
Before presenting the main results of the section, we observe that postprocessing does not bring additional power to extended finite automata.
\begin{lemma} Let $ L $ be a language recognized by an extended finite automaton $ V $ using end-marker. Then $ L $ is also recognized by an extended finite automaton $ V' $ that does not use the end-marker. \end{lemma} \begin{proof}
Extended finite automata are one-way and nondeterministic by definition. Hence, the proof idea of Lemma \ref{thm: NBHVA-endmarker} applies here as well. With the help of the additional transitions and extra states, the postprocessing step can be handled by an ordinary extended finite automaton without end-marker. \end{proof}
Let $ M $ be a multiplicative monoid of $ k \times k $ integer matrices. We denote by $\textup{HVA}(k)_{M} $ a $ k $-dimensional homing vector automaton whose matrices belong to $ M $. For the rest of the section, we are going to investigate $ M $-automata and $\textup{HVA}(k)_{M} $. $ \mathbb{Z}^{k \times k} $ denotes the multiplicative monoid of $ k \times k$ matrices with integer entries.
In \cite{SDS16}, it is shown that $\mathsf{RE} \subseteq \mathfrak{L}\textup{(1NBHVA(4)}_{\mathbb{Z}^{4 \times 4}})$. Therefore, Lemma \ref{thm: ratint} says something new only about the class of languages recognized by 3-dimensional 1NBHVAs with integer entries. Note that any blind $ k $-counter automaton can be simulated by a one-dimensional homing vector automaton whose register is multiplied by $ k $ distinct primes and their multiplicative inverses. Using the fact that any 1N$ k $BCA can be simulated by a 1NBHVA(1), we show that the same result can be achieved by a 1NBHVA(3) whose matrices are restricted to have integer entries. Note that the result is also true for real-time NBHVAs.
\begin{theorem}\label{cor: z3}
$ \bigcup_k\mathfrak{L}(\textup{1N$ k $BCA}) \subsetneq \mathfrak{L}\textup{(1NBHVA(3)}_{\mathbb{Z}^{3 \times 3}}) $. \end{theorem} \begin{proof}
Any 1N$ k $BCA can be simulated by a 1NBHVA(1)$ _\mathbb{Q^+} $ and any 1NBHVA(1)$ _\mathbb{Q^+} $ can be simulated by a 1NBHVA$_ \dollar $(3)$ _{\mathbb{Z}^{3 \times 3}} $ by Lemma $ \ref{thm: ratint} $. By using additional states, we can obtain an equivalent $\textup{1NBHVA(3)}_{\mathbb{Z}^{3 \times 3}} $ without end-marker by Lemma \ref{thm: NBHVA-endmarker}. The inclusion is proper since the unary nonregular language $ \mathtt{UPOW'=\{a^{n+2^n}\} | n\geq 0} $ can be recognized by a $\textup{NBHVA(3)}_{\mathbb{Z}^{3 \times 3}} $ \cite{SDS16}, which is not possible for 1N$ k $BCAs \cite{Gr78,Ib78}. \end{proof}
It is an open question whether $ \mathfrak{L}(\mathbb{Z}^{3 \times 3}) $ includes the class of languages recognized by 1N3BCAs. We cannot adapt Theorem \ref{cor: z3} to $ \mathbb{Z}^{3 \times 3} $-automata since the product of the matrices multiplied by the register in Lemma \ref{thm: NBHVA-endmarker} is not equal to the identity matrix and the acceptance condition for extended finite automata is not satisfied.
In \cite{SSD16} it is shown that the non-context-free language $ \mathtt{POW_r}=\{a^{2^n}b^n | n \geq 0\} $ can be recognized by a $ \textup{DBHVA(2)}_{\mathbb{Z}^{2 \times 2}} $. It is left open whether there exists a 1NBHVA with integer entries recognizing $ \mathtt{POW_r} $. Theorem \ref{thm: z22} answers two open questions from \cite{SSD16}, by demonstrating a 1NBHVA with integer entries recognizing $ \mathtt{POW_r} $ and revealing that 1NBHVAs are more powerful than the corresponding extended finite automata, when the matrices are restricted to the set $ \mathbb{Z}^{2 \times 2} $. By $ \mathbf{F}_2 $ we denote the free group of rank 2.
\begin{theorem}\label{thm: z22}
$ \mathfrak{L}(\mathbb{Z}^{2 \times 2}) \subsetneq \mathfrak{L}(\textup{1NBHVA(2)} _{\mathbb{Z}^{2 \times 2}}) $. \end{theorem} \begin{proof} In \cite{SSa18}, it is proven that any $ \mathbb{Z}^{2 \times 2} $-automaton can be converted to a $ \mathbf{F}_2 $-automaton. By setting the initial vector to be $ (1~~0) $, any $ \mathbf{F}_2 $-automaton can be simulated by a 1NBHVA(2) whose matrices have determinant 1 with integer entries \cite{SSD16} and the inclusion follows.
Now we are going to prove that the inclusion is proper. Since $ \mathbf{F}_2 $-automata recognize exactly the class of context-free languages \cite{DM00,Co05}, $ \mathfrak{L}(\mathbb{Z}^{2 \times 2})= \mathsf{CF}$. Let us construct a DBHVA$ _\dollar $(2)$ _{\mathbb{Z}^{2 \times 2}} $ $ V $ recognizing the non-context-free language $ \mathtt{POW_r}=\{a^{2^n}b^n | n \geq 0\} $. The state diagram of $ V $ is given in Figure \ref{fig: machine}.
\begin{figure}\label{fig: machine}
\end{figure}
\[
A_a=\mymatrix{ cc }{
1 & 0 \\
1 & 1
} ~~
A_b=\mymatrix{ cc }{
1 & 0 \\
0 & 2
}~~
A_{\dollar}=\mymatrix{ rr }{
1 & 1 \\
-1 & -1
}
\]
The initial vector of $ V $ is $ v_0=(1~~1) $. While reading $ a $ in $ q_1 $, $ V $ multiplies its vector with the matrix $ A_a $. It moves to $ q_2 $ when it scans the first $ b $ and multiplies its vector with the matrix $ A_b $ as it reads each $ b $. When $ V $ reads the end-marker, it multiplies its vector with the matrix $ A_{\dollar} $ and moves to $ q_3 $.
When the vector is multiplied by $ A_a $, the first entry of the vector is increased by 1 and when the vector is multiplied by $ A_b $, the second entry of the vector is multiplied by 2. Hence, after reading an input string of the form $ a^ib^j $, the vector is equal to $ (i+1~~2^j) $, as a result of the multiplication by the matrix product $ {A_a}^i{A_b}^j $. After multiplication by $ A_{\dollar} $, the second entry of the vector is subtracted from the first entry and this value is stored in both entries of the vector. The value of the final vector is equal to $ (1~~1) $ iff $ i+1-2^j=1 $. Hence, the accepted strings are those which are of the form $ a^{2^j}b^{j} $. As it is proven in Lemma \ref{thm: NBHVA-endmarker} that a NBHVA without end-marker recognizing the same language as a given NBHVA with end-marker can be constructed just by increasing the number of states but not the vector size, we can conclude that $ \mathtt{POW_r} $ can be recognized by a $ \textup{NBHVA(2)} _{\mathbb{Z}^{2 \times 2}} $.
\end{proof}
\section{The string separation problem} \label{sec:distinguish}
In this section we investigate the string separation problem for vector and homing vector automata. Recently, the same question has been investigated in \cite{BMY16,BMY17} for different models such as probabilistic, quantum, and affine finite automata (respectively, PFA, QFA, AfA). (We refer the reader to \cite{SayY14,DCY16A} for details of these models.) As these models are capable of storing information in their probabilities or amplitudes, they can be much more state-efficient than DFAs. For example, nondeterministic QFAs and zero-error AfAs can distinguish any pair by using only two states.
The storage mechanism of vector automata and homing vector automata enables these two models to separate strings by encoding the string in their vector entries. When measuring efficiency of these two models, it is fairer to measure succinctness in terms of both the number of states and the vector size.\footnote{In \cite{SYS13}, the size of an $ n $-DBVA($ k $) is defined as $ nk $, and a hierarchy theorem is given based on the size of DBVAs.}
We first show that VAs can simulate their classical states by using longer vectors and conclude that their stateless counterparts have the same computational power.
\begin{lemma}
\label{lem:1-state-VA}
Any given $n$$\textup{-DVA($k$)}$ $ V $ can be simulated by a $\textup{0-DVA($nk$+1)}$, which recognizes the same language. \end{lemma} \begin{proof}
Note that the following modification to any DVA does not change the recognized language: If the last transition yields a non-accept state, then the vector is multiplied by the zero matrix, i.e. we guarantee that the first entry of the vector is set to 0 if the final state is non-accepting. Assume that $V$ is an $ n $-state $k$-dimensional DVA with this property.
We describe a 0-DVA$(nk+1)$ $ V' $. We will call the vector of $V'$ `the new vector', and index its last $ nk $ entries by pairs as
\[
\underbrace{(q_1,1), (q_1,2), \ldots, (q_1,k)}_{1st~block}, \underbrace{(q_2,1), (q_2,2), \ldots, (q_2,k)}_{2nd~block}, \ldots,\ldots, \underbrace{(q_n,1), (q_n,2), \ldots, (q_n,k)}_{n\mbox{-}th~block},
\]
where we have $ n $ blocks, and the $i$'th block is associated with $q_i \in Q$ ($1 \leq i \leq n$). Based on this representation, $V'$ will keep track of the state and the vector of $ V $ at each step. The remaining first entry of the new vector holds the sum of all of its entries at indices of the form $ (q,1) $, for any $q$.
During the simulation, all but one of the blocks will be forced to contain zeros in all of its entries. The only exception is the block corresponding to the current state of $ V $, which will contain the values in $V$'s vector.
Thus, the first entry of the new vector keeps the value of the first entry of $ V $'s vector for each step, and so, $V'$ can easily implement the vector test done by $ V $, i.e. testing whether the first entry of $V$'s vector is 1 or not.
The new vector is initially set to
\[
( v_0[1], v_0[1], v_0[2],\ldots,v_0[n],0,\ldots,0 ),
\]
where the first entry is set to $ v_0[1] $ and then the first block is set to $ v_0 $ since $ V $ starts its computation in $ q_1 $, and, all other blocks are set to zeros.
Since $ V' $ is stateless, it has a single $ (1+nk) \times (1+nk)$ matrix for every ($ \sigma $, $\tau$) pair.
This big matrix is designed so that
each transition of the form $ \delta(q_i,\sigma,\tau) = (q_j,A) $ of $ V $
is implemented by multiplying the $ i $'th block by $ A $ and transferring the result to the $ j $'th block. Thus, both the simulated state and the vector of $ V $ are updated accordingly. Moreover, the first entry of the new vector is set to the summation mentioned above. In this way, the computation of $ V $ is simulated by $ V' $ and both of them make the same decision on each input, i.e. the first entry of the new vector is set to 1 if and only if the final state of $ V $ is accepting and the first entry of $ V $'s vector is 1. \end{proof}
Stateless models are advantageous, as they allow leaving behind the details of the state transitions and focusing on the vector matrix multiplications. In the following series of results, we are going to show that stateless DBVAs can separate any pair of strings.
In \cite{SYS13}, it is proven that $ \bigcup_k\mathfrak{L}$(DBVA($ k $)) = $S^=_{\mathbb{Q}}$. The proof involves construction of a 0-DBVA($ k $) to simulate a TuFA with $ k $ states. Therefore, any result about 0-DBVA($ k $)s is also true for TuFAs with $ k $ states and more generally for generalized finite automata with $ k $ states.
We start by distinguishing unary strings.
\begin{theorem}
For any $ i \geq 0 $, there exists a \textup{0-DBVA(1)} that distinguishes the string $ a^i $ from any other string in $\{a^*\}$. \end{theorem} \begin{proof}
The initial vector is $ v_0=(2^i) $. For each symbol $a$, the vector is multiplied with $\bigl ( \frac{1}{2} \bigr) $. For any other input symbol, the vector is multiplied by (0). The end-marker is associated with multiplication by (1). Therefore, the final vector is $ (1) $ if and only if the automaton reads $ a^i $. \end{proof}
\begin{corollary}
The language $ \mathtt{L}_\varepsilon $ is recognized by a \textup{0-DBHVA(1)}
. \end{corollary} \begin{proof}
If one removes the transition associated with the end-marker from the machine described in the proof above for $i=0$, one obtains a \textup{0-DBHVA(1)} recognizing $ \mathtt L_\varepsilon $. \end{proof}
For any $ x \in \{1,2\}^+ $, let $ e(x) $ be the number encoded by $ x $ in base 3: \[
e(x) = 3^{|x|-1} x[1] + 3^{|x|-2} x[2] + \cdots + 3^1 x[|x|-1] + 3^0 x[|x|] . \]
The encoding $e(x)$ can be easily obtained by using vector-matrix multiplications. Starting with the initial vector $v_0= (1~~~0 )$, and multiplying the vector with \[ A_1 = \mymatrix{rr}{ 1 & ~ 1\\ 0 & ~ 3 } \mbox{ and } A_2 = \mymatrix{rr}{ 1 & ~ 2 \\ 0 & ~ 3}, \] respectively for each scanned $1$ and $2$ from left to right, one obtains $(1~~~e(x) )$ at the end. We can easily extend this encoding for any generic alphabet. For any $ x \in \{1,2,\ldots,m-1\}^+ $ for some $m>3$, $e_m(x)$ is the number encoded by $ x $ in base $m$: \[
e_m(x) = m^{|x|-1} x[1] + m^{|x|-2} x[2] + \cdots + m^1 x[|x|-1] + m^0 x[|x|] . \] To encode $e_m(x)$, $v_0$ is multiplied with \[ A_i = \mymatrix{rr}{ 1 & ~ i \\ 0 & ~ m }, \] for each symbol $i$. Note that the results proven in this section for binary alphabets can be easily extended to any alphabet using the encoding $ e_m(\cdot ) $.
Now we show that 2-dimensional vector automata can distinguish strings without using any states.
\begin{theorem}
\label{thm: VA-dist-x}
The string $ x \in \{1,2\}^+ $ is distinguished from any other string in $\{1,2\}^*$ by a \textup{0-DBVA(2)}. \end{theorem} \begin{proof}
We build the machine in question. The initial vector is
\[
v_0 = (1 ~~~ e(x^r)).
\]
The matrices applied to the vectors for symbols $ 1 $ and $ 2 $ are
\[
A_1^{-1} = \mymatrix{rr}{ 1 & ~ -\frac{1}{3} \\ 0 & ~ \frac{1}{3} }
\mbox{ and }
A_2^{-1} = \mymatrix{rr}{ 1 & ~ -\frac{2}{3} \\ 0 & ~ \frac{1}{3} },
\]
respectively. When applied on the vector, the matrix $ A_1^{-1} $ (resp., $ A_2^{-1} $) leaves the first entry unchanged at value 1, and subtracts 1 (resp., 2) from the second entry and then divides the result of this subtraction by 3.
Let $ y \in \{1,2\}^* $ be the given input. Suppose that $y \neq \varepsilon$. After reading $ y $, the vector changes as
\[
( 1 ~~~ e(x^r))
\rightarrow
\mypar{1 ~~~ \frac{e(x^r) - y[1]}{3}}
\rightarrow
\mypar{ 1 ~~~ \frac{e(x^r) - y[1]-3y[2]}{3^2} }
\]
\[
\rightarrow
\mypar{ 1 ~~~ \frac{e(x^r) - 3^0y[1]-3^1y[2]-3^2y[3]}{3^3} }
\]
for the first three steps, and, at the $ i $'th step, we have
\[
\mypar{ 1 ~~~ \frac{e(x^r) - \sum_{j=1}^i 3^{j-1}y[j]}{3^i} }
\rightarrow
\mypar{ 1 ~~~ \frac{ \frac{e(x^r) - \sum_{j=1}^i 3^{j-1}y[j]} {3^i} - y[i+1]}{3} },
\]
which is equal to
\[
\mypar{ 1 ~~~ \frac{e(x^r) - \sum_{j=1}^{i+1} 3^{j-1}y[j]}{3^{i+1}} }.
\]
Thus, after reading $ y $, the vector is equal to
\[
\mypar{1 ~~~ e(x^r) - e(y^r)}.
\]
After reading the end-marker, the final vector is set as
\[
v_f = \mypar{1 + e(x^r) - e(y^r) ~~~ e(x^r)}
\]
by using the matrix
\[
A_{\$}=\mymatrix{cc}{ 1 & ~e(x^r) \\ 1 & 0 }.
\]
Thus, if $ x=y $, then $ v_f = v_0 $ and the first entry is 1. Otherwise, the first entry of $ v_f $ is not equal to 1.
If $y=\varepsilon$, then the final vector is obtained by multiplying the initial vector with $A_{\$}$ upon reading the end-marker: $$v_f=\mypar{1 + e(x^r) ~~~ e(x^r)}.$$
Since the first entry of the vector is never equal to 1, $\varepsilon$ is never accepted.
\end{proof}
We now focus on recognizing finite languages containing multiple strings. Stateless VAs can accomplish this task while no stateless HVA can recognize any finite language as we show in Section \ref{sec: stateless}.
\begin{theorem}\label{thm: VA-fin}
Any finite language $ X =\{ x_1,\ldots,x_k \} \subseteq \{1,2\}^+ $
can be recognized by a \textup{0-DBVA($2^{k}+1$)}. \end{theorem} \begin{proof}
We know from the proof of Theorem \ref{thm: VA-dist-x}
that for any $ x_i \in X $, there is a 0-DBVA(2) $ V_{x_i} $ that starts in $ (1~~~e(x_i^r)) $ and ends in $ ( 1+e(x_i^r)-e(y^r) ~~~ e(x_i) ) $ for the input string $ y $. Let $ V'_{x_i} $ be a 0-DBVA(2) obtained by modifying $ V_{x_i} $ such that the vector equals $ (e(x_i^r)-e(y^r) ~~~ 0) $ at the end.
We build a \textup{0-DBVA($2^{k}+1$)} $ V_X $ that
executes each $ V'_{x_i} $ in parallel (by employing a tensor product of all those machines) in the last $ 2^k $ entries of its vector, and then performs an additional postprocessing task. The initial vector of $ V_X $ is
\[
v_0=(1~~~\otimes_{i=1}^k (1~~~e(x^r_i)) ),
\]
that is, each $ V'_{x_i} $ is initialized with its initial vector. In order to execute each $ V'_{x_i} $, the matrices
\[
\mymatrix{c|c}{1 & 0 \cdots 0 \\ \hline 0 & \\ \vdots & \otimes_{i=1}^k A_1^{-1} \\ 0 & }
\mbox{ and }
\mymatrix{c|c}{1 & 0 \cdots 0 \\ \hline 0 & \\ \vdots & \otimes_{i=1}^k A_2^{-1} \\ 0 & }
\]
are respectively used for symbols $ 1 $ and $ 2 $. Before reading $ \dollar $, the vector is equal to
\[
(1~~~\otimes_{i=1}^k (1~~~e(x^r_i)- e(y^r) ).
\] The transition matrix used upon scanning the end-marker symbol $\dollar$ can be described as the product of two matrices $ A_{\dollar_1} $ and $ A_{\dollar_2} $:
\[ A_{\dollar_1} =
\mymatrix{c|c}{1 & 0 \cdots 0 \\ \hline 0 & \\ \vdots & \otimes_{i=1}^k A_0 \\ 0 }
\mbox{ and }
A_{\dollar_2} =
\mymatrix{c|c}{1 & 1~~~\otimes_{i=1}^k (1~~~e(x^r_i)) \\ \hline 1 & \\0 & \\ \vdots & 0\\ 0 & }
\]
where
\[ A_{0} =
\mymatrix{cc}{0 & 0 \\ 1 & 0 }.
\]
After multiplication with $ A_{\dollar_1} $, the vector becomes
\[
(1 ~~~\otimes_{i=1}^k(e(x_i^r)-e(y^r)~~~0 )),
\]
which is identical to
\[
\mypar{1 ~~~ \Pi_{i=1}^k(e(x_i^r)-e(y^r)) ~~~ 0 ~~~ \cdots ~~~ 0 }.
\]
If $ y \in X $, then the value of the second entry in this vector is also equal to 0 since one of the factors is zero in the multiplication. Otherwise, the value of the second entry is non-zero. $ A_{\dollar_2} $ in the product for $\dollar$ sets the final vector to
\[
v_f = \mypar{1+\Pi_{i=1}^k(e(x_i^r)-e(y^r))~~~\otimes_{i=1}^k (1~~~e(x^r_i))}.
\]
Here, the last $ 2^k-1 $ entries can be set to those values easily, since $ v_0 $ is a fixed a vector. Thus, if $ y \in X $, then $ v_f = v_0 $ with $ v_f[1] = 1 $, and so the input is accepted. Otherwise, $ v_f \neq v_0 $ with $ v_f[1] \neq 1 $, and so the input is rejected.
\end{proof}
\begin{comment} \begin{theorem} \label{thm:HVA-dist-x} The string $ x \in \{1,2\}^+ $ with length $l$ is distinguished from any other string by a \textup{2-DBHVA(3)} $ H_x $. \end{theorem} \begin{proof} The initial vector of $H_x$ is $ v_0=\mypar{1 ~~ 0 ~~ 0}$. The initial state is $ q_1 $ and the only accept state is $ q_2 $. We define two matrices for symbols $ 1 $ and $ 2 $ as: \[ A_1 = \dfrac{1}{5} \mymatrix{rrr}{4 & -3 & ~0 \\ 3 & 4 & 0 \\ 0 & 0 & 5} \mbox{ and } A_2 = \dfrac{1}{5} \mymatrix{rrr}{4 & ~0 & -3 \\ 0 & 5 & 0 \\ 3 & 0 & 4}. \]
Let $ A^{-1}_{x^r} = A^{-1}_{x[l]} A^{-1}_{x[l-1]} \cdots A^{-1}_{x[2]} A^{-1}_{x[1]} $. Let $ y \in \{1,2\}^* $ be the given input. If $ y = \varepsilon $, then the state is never changed and so the input is never accepted. Therefore, we assume $ y \in \{1,2\}^+ $ in the remaining part.
When in $ q_1 $, the vector is updated as \[ v_1 = v_0 A^{-1 }_{x^r} A_{y[1]} \] upon reading the first symbol of $y$ and the state is set to $ q_2 $. From now on, the state is never changed but the vector is updated by multiplying with $ A_\sigma $ for any scanned symbol $ \sigma \in \{1,2\} $: \[
v_f = v_0 A^{-1}_{x^r} A_{y[1]} \cdots A_{y[|y|-1]} A_{y[|y|]}. \] Thus, as shown in \cite{AW02}, if $ y = x $, then $ v_f = v_0 $, and if $ y \neq x $, $ v_f \neq v_0 $.
\end{proof}
\abu{Teorem 6yi istedigin gibi degistirebilirsin.}
\abu{Bu arada asagidaki aciklama ya da Teorem 6, Teorem 5i gereksiz hale getirmiyor mu? Toerem 5, 3 boyutlu vektor kullaniyor ve Teorem 6, $k$ boyutun yeterli oldugunu soyluyor. Teorem 6'da $k$ icin minimum bir deger belirtmemisiz. Eger okumayi zorlastirmayacaksa, belki Teorem 5i ve 6yi birlikte dusunup, tek bir sonuc yazilabilir. BURAYI SANA BIRAKTIM.
\ozlem{Teorem 5 alfabe boyutunın 2'den büyük olduğu duruma genellenmiyor yani değil mi? O zaman Teorem 6 daha genel bir şey söylüyor, 5i silip 6yı bırakabiliriz.} \abu{5SUBAT: Bu kismi sana birakiyorum Ozlem. Istegin sekilde kisaltabilirsin.}
Any string belonging to alphabet $\Sigma$ where $|\Sigma|=k$ can be distinguished from any other string by a 2-DBHVA($k$), by using generalized Stern-Brocot encoding defined in \cite{SSD16}. Choose $v_0=\mypar{1 ~~ 1 ~~ \cdots ~~ 1}$ and let $A_{a_i}$ be the $k$ dimensional identity matrix whose $i$'th column is replaced with a column of 1's for symbol $a_i$ in the above proof. }
\end{comment}
Now we move on to string separation by homing vector automata. We show that any nonempty string can be distinguished from any other string by a 2-DBHVA($2$).
\begin{theorem}
\label{thm:HVA-dist-sb}
The string $ x \in \{1,2\}^+ $ is distinguished from any other string in $\{1,2\}^*$ by a \textup{2-DBHVA($2$)}. \end{theorem} \begin{proof}
Let us construct a 2-DBHVA(2) distinguishing $x$. The initial state is named $ q_1 $, and the only accept state is named $ q_2 $. We are going to use the encoding described above and the associated matrices. The proof is similar to the proof of Theorem \ref{thm: VA-dist-x}. The initial vector is $$ v_0=\mypar{1 ~~ 0}.$$
Let $ y \in \{1,2\}^* $ be the given input. If $ y = \varepsilon $, then the state is never changed, and so the input is never accepted. Therefore, we assume $ y \in \{1,2\}^+ $ in the remaining part.
To encode $x^r$, we will use the matrix $$ A_{x^r} = A_{x[|x|]}A_{x[|x|-1]} \cdots A_{x[2]} A_{x[1]}. $$
When in $ q_1 $, the vector is updated as $$ v_1=v_0 A_{x^r} A_{y[1]}^{-1} $$ upon reading the first symbol of $y$, and the state is set to $ q_2 $. The value of the vector is equal to $$\mypar{1~~~~ \frac{e(x^r)-y[1]}{3}}.$$ From now on, the state is never changed, but the vector is updated by multiplication with $ A_1^{-1} $ and $A_2^{-1}$ for each scanned symbol 1 and 2 respectively. Thus, as in Theorem \ref{thm: VA-dist-x}, after reading $y$ the vector is equal to
$$ v_f=(1 ~~e(x^r)-e(y^r)).$$
Hence, we conclude that $v_f=v_0$ iff $x=y$. \end{proof}
It turns out that using two states in the above proof is essential, and we cannot obtain a similar result by using just one state, regardless of the dimension of the vector. More precisely, if $ x $ is accepted by a 0-NHVA, then $ xx $ is also accepted by the same automaton, since any further repetition of the same input naturally carries the vector through a trajectory that ends up in its initial value. We are therefore motivated to examine the limitations imposed by statelessness on homing vector automata in more detail in the next section.
The stateless vector automata we propose in Theorem \ref{thm: VA-fin} for recognizing a finite language require a vector of exponential size in the number of strings. Note that the machine $ V_X $ built in the proof of the theorem is deterministic and homing with end-marker. If we allow nondeterminism, then Lemma \ref{thm: NBHVA-endmarker} helps us obtain the same result with just two dimensions and using three states for ordinary NBHVAs.
\begin{corollary}
Any finite language $ X=\{x_1,\dots, x_k\} \subseteq \{1,2\}^+ $ can be recognized by a \textup{3-NBHVA(2)} $H_X$. \end{corollary} \begin{proof}
We first construct a 0-$\textup{NBHVA}_\dollar(2)$ $N_X$. $N_X$ nondeterministically picks an $ x_i $ and then executes the deterministic automaton given in the proof of Theorem \ref{thm: VA-dist-x}. Since $N_X$ is homing using end-marker, we conclude that there exists a
3-NBHVA(2) $H_X$ recognizing $X$ by Lemma \ref{thm: NBHVA-endmarker}. (A 2-NBHVA(2) is in fact sufficient if $X$ does not contain the empty string, as can be seen in the proof of Lemma \ref{thm: NBHVA-endmarker}.) \end{proof}
\section{Stateless homing vector automata} \label{sec: stateless}
In this section, we investigate the computation power of stateless HVAs. In these machines, only the vector is updated after reading an input symbol.
\subsection{Remarks} The limitation of having a single state for homing vector automata leads to the acceptance of the string $ xx $, whenever the string $ x $ is accepted. This is true since further repetition of the same input naturally carries the vector through a trajectory that ends up in its initial value. Based on this observation, we can list the following consequences:
\begin{itemize}
\item If string $ x $ is accepted by a \textup{0-NHVA} $H$, then any member of $ \{x\}^* $ is also accepted by $H$.
\item If all members of a language $L $ are accepted by a \textup{0-NHVA} $H$, then any string in $ L^* $ is also accepted by $H$.
\item If language $L $ is recognized by a \textup{0-NHVA}, then $ L = L^* $.
\item \textup{0-NHVA}s cannot recognize any finite language except $ \mathtt{L}_\varepsilon $. \end{itemize}
All of the results given above are also valid for deterministic or blind models. We can further make the following observation.
\begin{lemma}
\label{lemma: diff}
If the strings $ w_1 $ and $ w_1w_2 $ are accepted by a \textup{0-DHVA} $ H$, then the string $ w_2$ is also accepted by $ H $. \end{lemma} \begin{proof}
After reading $ w_1 $, the value of the vector is equal to its initial value. Since $ w_1w_2 $ is also accepted by $ H $, reading $ w_2 $ results in acceptance when started with the initial vector.
\end{proof}
For the unary case we have the following.
\begin{theorem}\label{thm: gcd}
If the strings $ a^i $ and $ a^j $ ($ 1<i<j $) are accepted by a \textup{0-DHVA} $ H$, then the string $ a^{\gcd(i,j)} $ is also accepted by $ H $. \end{theorem} \begin{proof}
It is well known that for any positive integers $ i,j $, there are two integers $ l_i $ and $ l_j $ such that $ i l_i + j l_j = \gcd(i,j) $. Assume that $ l_i $ is positive and $ l_j $ is non-positive. (The other case is symmetric.) Note that $ il_i \geq j(-l_j) $. The strings $ a^{j(-l_j)} $ and $ a^{i l_i} $ are accepted by $ H $. By Lemma \ref{lemma: diff}, the string $ a^{i l_i-j(-l_j)} $, which is $ a^{\gcd(i,j)} $, is also accepted by $ H $.
\end{proof}
\begin{corollary}
\label{cor: gcd}
If the strings $ a^i $ and $ a^j $ ($ 1<i<j $) are accepted by a \textup{0-DHVA} $ H$ and $ \gcd(i,j)=1 $, then $ H $ recognizes $ a^* $. \end{corollary}
Let us now investigate the case where the set of matrices is commutative.
Let $ L \in \Sigma^* $ be a language. The \textit{commutative closure} of $ L $ is defined as $ com(L)=\{x \in \Sigma^* | \phi(x) \in \phi(L) \} $. A language is \textit{commutative} if $ com(L)=L $.
\begin{theorem}\label{thm: comm} If a language $L $ is recognized by a \textup{0-NBHVA} $ H $ with a commutative set of matrices, then $L$ is commutative. \end{theorem} \begin{proof}
Let $w \in L $ and suppose that the string $w=w_{[1]}w_{[2]}\cdots w_{[n]}$ is accepted by $ H $. Let $A_1 A_2\cdots A_n$ be the product of the matrices labeling the computation such that $$v_0 A_1 A_2 \cdots A_n=v_0$$ where $v_0$ is the initial vector of $ H $. Since the matrices are commutative, then for any permutation $\tau$, $$A_1A_2 \cdots A_n=A_{\tau(1)} A_{\tau(2)} \cdots A_{\tau(n)}.$$
This leads to the acceptance of the string $w'=w_{[\tau(1)]} w_{[\tau(2)]} \cdots w_{[\tau(n)]} $ since $$v_0A_{\tau(1)} A_{\tau(2)} \cdots A_{\tau(n)}=v_0.$$ Hence, if $w$ is accepted by $ H $, then any string obtained from $ w $ by permuting its letters is also accepted by $ H $. Any string $ x $ with $ \phi(x)=\phi(w) $ is in $ L $ and we conclude that $ L $ is commutative.
\end{proof}
When the computation is not blind, then the class of languages recognized is no longer commutative. The language of balanced strings of brackets $ \mathtt{DYCK} $ can be recognized by 0-DHVA(1) as follows. Starting with the initial vector $ (1) $, for each left bracket the vector is multiplied by $ (2) $. As long as the vector is not equal to (1), for each right bracket, the vector is multiplied by ($\frac{1}{2}$). If the vector is equal to $ (1) $ and the next symbol is a right bracket, then the vector is set to (0).
\begin{corollary}\label{cor: commreverse} If a language $L $ is recognized by a \textup{0-NBHVA} $ H $ with a commutative set of matrices, then $L=L^r$. \end{corollary} \begin{proof}
Suppose that $w \in L$. Then it is clear that $w^r$ will be also accepted by $ H $ by Theorem \ref{thm: comm} and $w^r \in L$. Since for every string $ w $ it is true that $w \in L $ if and only if $ w^r \in L$, we conclude that $L=L^r$.
\end{proof}
\begin{comment} In a homing vector automaton, a string may be accepted if the product of the matrices multiplied with the register is the identity matrix. When that is the case, the following theorem is true.
\begin{theorem}\label{thm:identity}
Let $c(w)$ be the product of invertible matrices labeling a successful computation of a \textup{0-NBHVA} $ H $ upon reading the string $w=w_{[1]}w_{[2]}\cdots w_{[n]}$. If $c(w)$ is equal to the identity matrix, the string $w_{[k+1]} w_{{[k+2]}} \cdots w_{[n]}w_{[1]} w_{[2]} \cdots w_{[k]} $ is also accepted by $ H $ for any $k$ such that $1 \leq k \leq n-1 $. \end{theorem} \begin{proof}
Suppose the string $w=w_{[1]}w_{[2]}\cdots w_{[n]}$ is accepted by $ H $ and let $c(w)=A_1 A_2\cdots A_n$ be the product of the matrices labeling the computation such that $v_0A_1 A_2 \cdots A_n=v_0$ where $ v_0 $ is the initial vector of $ H $. If $A_1A_2 \dots A_n=I$, then for any $ k \in \{1,\ldots,k-1\} $ we clearly have
\begin{align*}
A_1 A_2 \cdots A_k &= {A_n}^{-1}{A_{n-1}}^{-1}\cdots {A_{k+1}}^{-1}\\
A_{k+1}A_{k+2}\dots A_{n}A_1 A_2 \dots A_k &= A_{k+1}A_{k+2}\dots A_{n} {A_n}^{-1}{A_{n-1}}^{-1}\cdots {A_{k+1}}^{-1} = I.
\end{align*}
This leads to the acceptance of the string $w_{[k+1]} w_{{[k+2]}} \cdots w_{[n]}w_{[1]} w_{[2]} \cdots w_{[k]} $, since $$v_0A_{k+1}A_{k+2}\cdots A_{n}A_1 A_2 \cdots A_k=v_0.$$ \end{proof}
A language $ L \in \Sigma^* $ is said to be \textit{cyclic} if $ \forall u,v \in \Sigma^*$, $ uv \in L \iff vu \in L $. We can state the following corollary about the languages recognized by stateless group automata.
\begin{corollary}
If $ L \in \Sigma^*$ is recognized by a stateless group automaton, then $ L $ is cyclic. \end{corollary} \begin{proof}
In a group automaton, a successful computation is the one in which the register is multiplied by the identity matrix. For all $ u,v \in \Sigma^* $, $ uv\in L \iff vu \in L $ by Theorem \ref{thm:identity}, and it follows that $ L $ is cyclic. \end{proof} \end{comment}
We now focus on stateless HVAs whose vectors have dimension 1 and demonstrate some results on stateless FAMWs. Note that stateless FAMs do not process the end-marker $\dollar$ by definition, since their single state is also an accept state and the computation ends once $\dollar$ is scanned in an accept state.
We start by comparing the class of languages recognized by stateless versions of deterministic and blind finite automata with multiplication, and 1-dimensional homing vector automata. The capability of multiplication with negative rational numbers brings additional power to the stateless DBHVA(1)s.
\begin{theorem}
$ \mathfrak{L} \textup{(0-DFAMW)} \subsetneq \mathfrak{L}\textup{(0-DBHVA(1))}.$ \end{theorem}
\begin{proof}Let $ \mathtt{EVENAB}=\{a^nb^n |~n = 2k \mbox{ for some }k\geq 0 \}$. The following 0-DBHVA(1) recognizes $ \mathtt{EVENAB} $: The register is multiplied by $ (-2) $ and $ (\frac{1}{2}) $ when the machine reads an $ a $ and $ b $ respectively.
Suppose that there exists some 0-1DFAMW recognizing $ \mathtt{EVENAB} $. Let $ m_a $ and $ m_b $ be the positive rational numbers that are multiplied by the register upon reading $ a $ and $ b $. Since $ aabb\in \mathtt{EVENAB} $, it is true that $ m_a^2m_b^2=1 $. Since both $ m_a $ and $ m _b$ are positive, it is not possible that $ m_am_b=-1 $. It follows that $ m_am_b=1 $, in which case the string $ ab $ is accepted and we get a contradiction. Hence we conclude that $ \mathtt{EVENAB} $ cannot be recognized by any 0-DFAMW. \end{proof}
When the register is multiplied with only positive rational numbers, then we have $ \mathfrak{L} $(0-DFAMW)=$\mathfrak{L}$(0-DBHVA(1))$ _\mathbb{Q^+} $. The equivalences also hold for the nondeterministic models.
For real-time, nondeterministic and blind machines, we can state the following theorem.
\begin{theorem}\label{thm: commu}
If $ L \in \mathfrak{L}(\textup{0-NFAMW})$, then $ L $ is commutative. \end{theorem} \begin{proof}
A 0-NFAMW is a 0-NBHVA(1) whose register is multiplied with only positive rational numbers. Therefore, $ L $ is also accepted by a 0-NBHVA(1). Since multiplication in dimension 1 is commutative, the result follows by Theorem \ref{thm: comm}. \end{proof}
It is known that a bounded language is accepted by a 1NFAMW iff it is semilinear \cite{ISK76}. In the next theorem, we prove a similar result and characterize the class of languages recognized by 0-DFAMWs. We show that any language recognized by a 0-DFAMW is commutative and semilinear. Furthermore, any commutative language whose Parikh image is the set of nonnegative solutions to a system of linear homogeneous Diophantine equations can be recognized by a 0-DFAMW.
\begin{theorem}
$ L \in \mathfrak{L}(\textup{0-DFAMW}) $ iff $ L $ is commutative and $\phi(L) $ is the set of nonnegative integer solutions to a system of linear homogeneous Diophantine equations. \end{theorem} \begin{proof} Let $ L $ be a language over the alphabet $ \Sigma=\{\sigma_1,\dots,\sigma_n\} $ recognized by a 0-DFAMW $ V $. Let $M=\{m_1,m_2,\dots,m_n\}$ be the set of rational numbers such that the register is multiplied with $m_i$ upon reading $\sigma_i$. Let $P=\{p_1,p_2,\dots ,p_k\}$ be the set of prime factors of the denominators and the numerators of the rational numbers in $M$. Then each $ m_i $ can be expressed as
$$
m_i=\frac{p_1^{x_{1_i}}p_2^{x_{2_i}}\cdots p_k^{x_{k_i}}}{p_1^{y_{1_i}}p_2^{y_{2_i}}\cdots p_k^{y_{k_i}}} .
$$
If a string $w$ is accepted by $V$, then the value of the register should be equal to 1 after reading $ w $, which is possible only if
$$
m_1^{w_{|\sigma_1|}}m_2^{w_{|\sigma_2|}}\cdots m_n^{w_{|\sigma_n|}}=1.
$$
This leads to the following system of linear Diophantine equations in $ n$ variables.
\begin{align*}
(x_{1_1}-y_{1_1})w_{|\sigma_1|}+(x_{1_2}-y_{1_2})w_{|\sigma_2|}+\dots + (x_{1_n}-y_{1_n})w_{|\sigma_n|}&=0\\
(x_{2_1}-y_{2_1})w_{|\sigma_1|}+(x_{2_2}-y_{2_2})w_{|\sigma_2|}+\dots + (x_{2_n}-y_{2_n})w_{|\sigma_n|}&=0\\
&\vdots\\
(x_{k_1}-y_{k_1})w_{|\sigma_1|}+(x_{k_2}-y_{k_2})w_{|\sigma_2|}+\dots + (x_{k_n}-y_{k_n})w_{|\sigma_n|}&=0\\
\end{align*}
For $ j=1,\dots ,k $, the $ j $'th equation is stating that the exponent of $ p_j $ is equal to 0 after reading $ w$.
\begin{comment}
In matrix form, we can state it as
$$
\mymatrix{llll}{
x_{1_1}-y_{1_1}&x_{1_2}-y_{1_2}&\dots & x_{1_n}-y_{1_n}\\
x_{2_1}-y_{2_1}&x_{2_2}-y_{2_2}&\dots & x_{2_n}-y_{2_n} \\
\vdots&&& \\
x_{k_1}-y_{k_1}&x_{k_2}-y_{k_2}&\dots & x_{k_n}-y_{k_n} \\
}\mymatrix{c}{
w_{|\sigma_1|}\\
w_{|\sigma_2|}\\
\vdots \\
w_{|\sigma_n|}\\
}=\mymatrix{c}{
0\\
0\\
\vdots \\
0\\
},
$$
\noindent or $ As=0 $.
\end{comment}
Hence, we can conclude that the Parikh images of the accepted strings are the nonnegative solutions to a system of linear homogeneous Diophantine equations. $ L $ is commutative by Theorem \ref{thm: commu}. (One can further conclude that $ L $ is semilinear.)
For the converse, suppose that we are given a commutative language $ L $ over the alphabet $ \Sigma = \{\sigma_1,\dots,\sigma_n\} $. Let $ S $ be the set of Parikh images of the strings in $ L $. $ S $ is the set of nonnegative solutions to a system of, say, $ k $ linear homogeneous Diophantine equations in $ n $ unknowns,
\begin{align*} a_{11}s_1 + a_{12}s_2+\dots +a_{1n}s_n&=0\\ a_{21}s_2 + a_{22}s_2+\dots +a_{2n} s_n&=0\\ &\vdots\\ a_{k1}s_1+a_{k2}s_2+\dots + a_{kn}s_n&=0\\ \end{align*} where $ \mypar{s_1 ~~~s_2 ~~~\dots~~~ s_n} \in S$.
We construct a 0-DFAMW $ V $ recognizing $ L $ as follows. We choose a set of $ k $ distinct prime numbers, $ P=\{p_1,p_2,\dots,p_k\} $. When $ V$ reads $ \sigma_i $, the register is multiplied by $$ m_i= p_1^{a_{1i}}p_2^{a_{2i}}\cdots p_k^{a_{ki}}.$$ Suppose that a string $ w $ is accepted by $ V $. Then
$$
m_1^{w_{|\sigma_1|}}m_2^{w_{|\sigma_2|}}\cdots m_n^{w_{|\sigma_n|}}=1.
$$
The product is equal to 1 if all of the exponents of the prime factors are equal to 0, that is when $ \mypar{w_{|\sigma_1|} ~~~w_{|\sigma_2|}~~~ \dots ~~~ w_{|\sigma_n|} }\in S$. Hence we see that the set of accepted strings are those with Parikh image in $ S $. Since $ L $ is commutative, any string $ w $ with $ \phi(w) \in S $ belongs to $ L $ and we conclude that $ V $ recognizes $ L $.
\end{proof}
Note that $ \mathfrak{L} $(0-DFAMW)=$\mathfrak{L}$(0-1DFAMW), since a 0-1DFAMW that has an instruction to stay on some input symbol cannot read the rest of the string.
\subsection{Regular languages}
Let $ D $ be an $ n $-state deterministic finite automaton. Without loss of generality, we can enumerate its states as $ q_1,\ldots,q_n $ where $q_1$ is the initial state. Moreover we can denote $ q_i $ by $ e_i $, which is the basis vector in dimension $ n $ having 1 in its $ i $'th entry, and 0 otherwise. Besides, for each symbol $\sigma $, we can design a zero-one matrix, say $ A_\sigma $, that represents the transitions between the states, i.e. $ A_\sigma(i,j) = 1 $ if and only if $ D $ switches from $ q_i $ to $ q_j $ when reading symbol $\sigma $. Thus, the computation of $ D $ on an input, say $ w $, can be traced by matrix-vector multiplication: \[
e_j = e_1 A_{w[1]} A_{w[2]} \cdots A_{w[|w|]} \] if and only if $ D $ ends its computation in $ q_j $ after reading $ w $.
Based on this representation, we can easily observe that if a language $ L $ is recognized by an $ n $-DFA whose initial state is the single accept state, then $ L $ is recognized by a 0-DBHVA($n$).
Let us give some examples of stateless HVAs recognizing regular languages. \\
\noindent \textbf{Example 1:} \label{ex: 2} Let $\mathtt{AB_k}^*=\{a^kb^k\}^*$ for a given $k>1$, and let us construct a 0-DBHVA(2$ k $) $ H_k $ recognizing $\mathtt{AB_k}^*$. The initial vector of $ H_k $ is $ v_0=(1 ~~ 0 ~~ \cdots ~~ 0) $. For each $ a $, the value in the $ i $'th entry of the vector is transferred to the $ (i+1) $'st entry when $ 1 \leq i \leq k $, and, the rest of the entries are set to 0. For each $ b $, the value in the $ (i+k) $'th entry of the vector is transferred to the $ (i+k+1 \mod 2k) $'th entry, and the rest of the entries are set to 0, ($ 1 \leq i \leq k $). Thus, we return to the initial vector if and only if after some number of $ (a^kb^k) $ blocks have been read. \\
\noindent \textbf{Example 2:} \label{ex: 1}The language $ \mathtt{MOD_m} $ is defined as $ \mathtt{MOD_m} = \{ a^i \mid i \mod m \equiv 0 \}. $ It is easy to observe that any unary $n$-state DFA whose initial state is the single accept state can recognize either $ L_\varepsilon $ or $ \mathtt{MOD_m} $ for some $m\leq n$. Hence, for any $ m>0 $, the language $ \mathtt{MOD_m} $ is recognized by a \textup{0-DBHVA($m$)}. \\
Note that if it is allowed to use arbitrary algebraic numbers in the transition matrices, then for every $ m>0 $, the language $ \mathtt{MOD_m} $ is recognized by a \textup{0-DBHVA(2)} with initial vector $ v_0 = \mypar{1 ~~ 0} $ that multiplies its vector with the matrix $$ A_m = \mymatrix{cc}{ \cos \frac{2 \pi}{m} & -\sin \frac{2 \pi}{m} \\ \sin \frac{2 \pi}{m} & \cos \frac{2 \pi}{m} } $$ for each scanned symbol. (The entries of $A_m$ are algebraic for any $m$ \cite{Le93}.) \\
One may ask whether all regular languages $ L $ satisfying $ L= L^* $ can be recognized by stateless HVAs. We provide a negative answer for the deterministic model. The language $ \mathtt{MOD23} $ is defined as $ \{ a^2,a^3 \}^* = a^* \setminus \{a\} $, satisfying that $ \mathtt{MOD23} = \mathtt{MOD23}^* $. $ \mathtt{MOD23} $ cannot be recognized by any \textup{0-DHVA} since a 0-DHVA which accepts $ a^2 $ and $ a^3 $, also accepts $ a $ by Lemma \ref{lemma: diff}.
\subsection{Additional results on stateless homing vector automata}
One sees that the nonregular language $\mathtt{AB} = \{ a^nb^n | n \geq 0 \} $ cannot be recognized by any 0-NHVA($ k $) for any $ k $, since $ \mathtt{AB} \neq \mathtt{AB}^* $. On the other hand, $ \mathtt{EQ} = \{ x \in \{a,b\}^* \mid |x|_a = |x|_b \} $ can be recognized by a 0-DBHVA(1) with initial vector $ (1) $, and transition matrices $ A_a = (2) $ and $ A_b = \myvector{\frac{1}{2}} $. In the next theorem, we show that it is possible to recognize the star of $\mathtt{AB} $, even in the deterministic and blind computation mode with a stateless homing vector automaton. Note that $ \mathtt{AB}^* $ cannot be recognized by any 1NFAMW \cite{ISK76}.
\begin{theorem}
\label{thm:0HVA-upal}
The language $\mathtt{AB}^*= \{ \{a^nb^n\}^* \mid n \geq 0 \}$ is recognized by a \textup{0-DBHVA(10)}. \end{theorem} \begin{proof} We are going to construct a \textup{0-DBHVA(10)} recognizing $\mathtt{AB}^*$. We start with a 0-DBHVA(3) named $H $, whose initial vector is $ v_0 = (1 ~~ 0 ~~ 0) $. When reading a symbol $a$, $ H $ applies the matrix
$$ A_a = \mymatrix{ccc}{1 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0},
$$
which maps $ ( 1 ~~ t_2 ~~ t_3 ) $ to $ ( 1 ~~ 2t_2 + 1 ~~ 0 ) $. This operator encodes the count of $ a$'s in the value of the second entry. Started with the initial vector, if $ H $ reads $ i $ $a$'s, the vector becomes $ ( 1 ~~ 2^i -1 ~~ 0 ) $.
When reading a symbol $ b $, $ H $ applies the matrix
$$ A_b = \mymatrix{rrr}{1 & 0 & -\frac{1}{2} \\ 0 & 0 & \frac{1}{2} \\ 0 & 0 & \frac{1}{2}},
$$
that sets the value of the second entry to 0 and decrements the counter mentioned above, which it maintains in
the third entry. A vector starting as $ ( 1 ~~ 2^i -1 ~~ 0 ) $
would get transformed to $ ( 1 ~~ 0 ~~ 2^{i-j}-1 ) $
after reading $ j $ $ b $'s.
Note that the third entry is different than 0 when the number of $ a $'s and $ b $'s do not match in a block. If symbol $ a $ is read after a block of $ b $'s, the counting restarts from 0, and the value of the third entry is again set to 0.
By tensoring $ H $ with itself, we obtain a 0-DBHVA(9). The initial vector of this new machine is $ v_0 \otimes v_0 $, and its operators are $ A_a \otimes A_a $ and $ A_b \otimes A_b $.
Based on this tensor machine, we define our final 0-DBHVA(10), say, $ H' $. The initial vector is $v_0'= (v_0 \otimes v_0 ~~ 0) = (1 ~~ 0 ~~ \cdots ~~ 0).$
$ H' $ applies the matrices $A_a'$ and $ A_b' $ for each scanned $ a $ and $ b $, respectively:
\[
A_a' = \mymatrix{c|c}{ A_a \otimes A_a & \begin{array}{c}0 \\ \vdots \\ 0 \\ 1 \end{array} \\ \hline 0 ~~ \cdots ~~ 0 & 1 }~~~A_b' = \mymatrix{c|c}{ A_b \otimes A_b & \begin{array}{c}0 \\ \vdots \\ 0\\0 \end{array} \\ \hline 0 ~~ \cdots ~~ 0 & 1 }.
\]
The matrix $ A_a' $ implements $ A_a \otimes A_a $ in its first nine entries. Additionally, it adds the value of the ninth entry of the vector to the tenth one. The operator $ A_b' $ implements $ A_b \otimes A_b $ in its first nine entries. Additionally, it preserves the value of the tenth entry. Suppose that the vector of $ H $ is equal to $ (1~~ a_2 ~~ a_3) $ at some point of the computation. Due to the property of the tensor product, the vector entries of $ H' $ hold the values $ (1 ~~a_2~~ a_3~~ a_2~~ a_2a_2~~ a_2a_3~~ a_3~~ a_3a_2~~ a_3a_3~~ 1) $.
The value of the third entry is important as it becomes nonzero when the number of $ a $'s and $ b $'s are different in a block. Hence, the ninth entry holds some positive value when this is the case. When $ H' $ starts reading a new block of $ a $'s, it adds the ninth entry to the tenth one with the help of the matrix $ A_a' $. Once this entry is set to a positive value, it never becomes 0 again, since its value can only be changed by the addition of the ninth entry, which is always a square of a rational number. Therefore, such an input is never accepted.
If this is not the case and the number of $ a $'s and $ b $'s are equal to each other in every block, then the third and therefore the ninth entry are always equal to 0 before $ H' $ starts reading a new block. We can conclude that the value of the tenth entry will be equal to 0 at the end of the computation. As the input string ends with a $ b $, the second entry is also set to 0 upon multiplication with $ A_b' $. Hence, all the entries except the first one are equal to 0 and the vector is equal to its initial value which leads to the acceptance of the input string.
Thus, the only accepted inputs are the members of $ \mathtt{AB}^* $. \end{proof}
Nondeterministic HVAs are more powerful than their deterministic variants in terms of language recognition in general. In the next theorem, we show that this is also true for the stateless models.
\begin{theorem}
\begin{enumerate}[i.]
\item $\mathfrak{L}
\textup{(0-DBHVA)} \subsetneq \mathfrak{L} \textup{(0-NBHVA)}. $
\item $ \mathfrak{L}
\textup{(0-DHVA)} \subsetneq \mathfrak{L} \textup{(0-NHVA)}. $
\end{enumerate} \end{theorem} \begin{proof}
Let us construct a 0-NBHVA(1) $ H $ recognizing $\mathtt{LEQ}=\{x \in \{a,b\}^* \mid |x|_a \leq |x|_b\}$. Starting with the initial vector $(1) $, $ H $ multiplies its vector with $ A=(2) $ for each $a$ and with $B_1=(\frac{1}{2})$ or $B_2=(1) $ for each $b$ nondeterministically.
Suppose that $\mathtt{LEQ}$ can be recognized by a 0-DHVA($k$) $ H' $. The strings $ w_1=b $ and $ w_2=ba $ are accepted by $ H' $. By Lemma \ref{lemma: diff}, the string $w_3=a$ is also accepted by $ H' $. We obtain a contradiction, and conclude that $\mathtt{LEQ}$ cannot be recognized by any 0-DHVA($k$). \end{proof}
\begin{comment} Now let us compare the langauge recognition power of 1 and 2 dimensional stateless HVAs.
\begin{theorem}\label{thm:12}
\begin{enumerate}[i.]
\item $ \mathfrak{L}
\textup{(0-DBHVA(1))} \subsetneq \mathfrak{L} \textup{(0-DBHVA(2))}. $
\item $ \mathfrak{L}
\textup{(0-NBHVA(1))} \subsetneq \mathfrak{L} \textup{(0-NBHVA(2))}. $
\end{enumerate} \end{theorem} \begin{proof}
Note that the language $ \mathtt{AB_1}^*=(ab)^* $ can be recognized by a 0-DBHVA(2) by Example \ref{ex: 2}. Assume that $ \mathtt{AB_1}^* $ is recognized by a 0-NBHVA($1$). Since $\mathtt{AB_1}^*$ is not equal to its reverse, by Corollary \ref{cor: commreverse} we get a contradiction. We conclude that $\mathtt{AB_1}^*$ cannot be recognized by any 0-NBHVA(1). \end{proof}
Now, we compare the blind and non-blind versions of one dimensional HVAs.
\begin{theorem}
\begin{enumerate}[i.]
\item $ \mathfrak{L}
\textup{(0-DBHVA(1))} \subsetneq \mathfrak{L} \textup{(0-DHVA(1))}. $
\item $ \mathfrak{L}
\textup{(0-NBHVA(1))} \subsetneq \mathfrak{L} \textup{(0-NHVA(1))}. $
\end{enumerate} \end{theorem} \begin{proof}
Let us construct a 0-DHVA(1) $ H $ recognizing $ \mathtt{AB_1}^*=(ab)^* $. The initial vector is equal to (1). For each scanned $ a $, $ H $ multiplies its vector with $ A_==(2) $ if it is equal to its initial value, and with $ A_{\neq}=(0) $ otherwise. For each scanned $ b $, $ H $ multiplies its vector with $ B_==(0) $ if it is equal to its initial value, and with $ B_{\neq}=(\frac{1}{2}) $ otherwise. $\mathtt{AB_1}^*$ cannot be recognized by any 0-NBHVA(1) as we saw in the proof of Theorem \ref{thm:12}. \end{proof} \end{comment}
Let us look at some closure properties for the stateless models. All of the stateless models are closed under the star operation since for any language recognized by a stateless homing vector automaton, it is true that $ L=L^* $.
\begin{theorem}
i. $ \mathfrak{L}\textup{(0-DBHVA)} $ is closed under the following operation:\\
\indent a) intersection\\
ii. $ \mathfrak{L}\textup{(0-DBHVA)} $ and $ \mathfrak{L}\textup{(0-DHVA)} $ are not closed under the following operations:\\
\indent a) union\\
\indent b) complement\\
\indent c) concatenation \end{theorem} \begin{proof}
\textit{i. a)} The proof in \cite{SDS16} that DBHVAs are closed under intersection is also valid for stateless models.\\
\textit{ii. a)} The languages $ \mathtt{MOD_2}$ and $ \mathtt{MOD_3}$ are recognized by 0-DBHVAs by Example \ref{ex: 1}. Their union cannot be recognized by any 0-DHVA, since a 0-DHVA accepting the strings $a^2$ and $a^3$ should also accept the non-member string $a$ by Lemma \ref{lemma: diff}.\\
\textit{b)} Complement of the language $ \mathtt{MOD_m} $ ($m>1$)is not recognized by any 0-NHVA, since $ \overline{\mathtt{MOD_m}} $ contains $ a $ and any 0-NHVA accepting $ a $ accepts any member of $a^*$.\\
c) The languages $ \mathtt{MOD_2}$ and $ \mathtt{MOD_3}$ are recognized by 0-DBHVAs by Example \ref{ex: 1}. Their concatenation $\mathtt{MOD23}$ cannot be recognized by any 0-DHVA.~~~ \end{proof}
$ \mathfrak{L}\textup{(0-NBHVA)} $ is closed under intersection. $ \mathfrak{L}\textup{(0-NBHVA)} $ and $ \mathfrak{L}\textup{(0-NHVA)} $ are not closed under complement. The proofs are identical.
\section{Open questions and future work}
We proved that 1NBHVAs are more powerful than extended finite automata when both are defined over $ 2 \times 2 $ integer matrices. Is this result still true when both models are defined over $ 3 \times 3 $ integer matrices?
Do 0-NHVAs recognize every regular language $L$ satisfying $ L=L^*$? Is there any nonregular language $L$ satisfying $L=L^*$ that cannot be recognized by any stateless HVA?
We proved that any language recognized by a 0-NFAMW is commutative. What can we say about the non-blind case?
We gave a characterization for the class of languages recognized by 0-DFAMW. Can we give a similar characterization for the non-blind and nondeterministic models?
\section*{Acknowledgments} This research was supported by Bo\u{g}azi\c{c}i University Research Fund (BAP) under grant number 11760. Salehi is partially supported by T\"{U}B\.{I}TAK (Scientific and Technological Research Council of Turkey). Yakary{\i}lmaz is partially supported by ERC Advanced Grant MQC. We thank Flavio D'Alessandro for his helpful answers to our questions, and the anonymous reviewers for their constructive comments.
\end{document} | arXiv |
\begin{document}
\begin{abstract} We prove the existence and give estimates of the fundamental solution (the heat kernel) for the equation $\partial_t ={\mathcal L}^{\kappa}$ for non-symmetric non-local operators $$
{\mathcal L}^{\kappa}f(x):= \int_{\mathbb{R}^d}( f(x+z)-f(x)- {\bf 1}_{|z|<1} \left<z,\nabla f(x)\right>)\kappa(x,z)J(z)\, dz\,, $$ under broad assumptions on $\kappa$ and $J$. Of special interest is the case when the order of the operator ${\mathcal L}^{\kappa}$ is smaller than or equal to 1. Our approach rests on imposing suitable cancellation conditions on the internal drift coefficient $$
\int_{r\leqslant |z|<1} z \kappa(x,z)J(z)dz\,,\qquad 0<r\leqslant 1\,, $$ which allows us to handle the non-symmetry of
$z\mapsto \kappa(x,z)J(z)$. The results are new even for the $1$-stable L{\'e}vy measure $J(z)=|z|^{-d-1}$. \end{abstract}
\title{Fundamental solution for super-critical non-symmetric L{\'e}
\noindent {\bf AMS 2010 Mathematics Subject Classification}: Primary 60J35, 47G20; Secondary 60J75, 47D03.
\noindent {\bf Keywords and phrases:} heat kernel estimates,
L\'evy-type operator, non-symmetric operator, non-local operator, non-symmetric Markov process, Feller semigroup, Levi's parametrix method.
\section{Introduction}\label{sec:intr}
In recent years, there has been a lot of interest in constructing semigroups for L{\'e}vy-type operators \cite{MR3353627}, \cite{MR3652202}, \cite{MR3500272}, \cite{MR3817130}, \cite{GS-2018}, \cite{MR2163294}, \cite{MR2456894}, \cite{FK-2017}, \cite{BKS-2017}, \cite{KR-2017}, \cite{MR3294616}, \cite{MR3544166}, \cite{PJ}, \cite{CZ-new}, \cite{CZ-survey}, \cite{MR3965398}. Such operators arise naturally due to the Courr{\`e}ge-Waldenfels theorem \cite[Theorem~4.5.21]{MR1873235}, \cite[Theorem~2.21]{MR3156646}. In general, they are not symmetric, so the $L^2$-theory or Dirichlet forms and the corresponding $L^2$-semigroups of operators does not apply in this context. We shall discuss operators of the form \begin{align}
{\mathcal L}^{\kappa}f(x)&:= \int_{{\R^{d}}}( f(x+z)-f(x)- {\bf 1}_{|z|<1} \left<z,\nabla f(x)\right>)\kappa(x,z)J(z)\, dz \,, \label{e:intro-operator-a1-crit1} \end{align} and allow for non-symmetric measures $\kappa(x,z)J(z) dz$. The paper is a continuation of the research conducted in \cite{GS-2018}, where operators of the form \eqref{e:intro-operator-a1-crit1} were considered under stronger conditions. We also improve and extend the results of \cite{PJ}, and part of those in~\cite{CZ-new} and \cite{MR3652202}.
We now introduce our setting, notation and motivations. Let $d\in{\mathbb N}$ and $\nu:[0,\infty)\to[0,\infty]$ be a non-increasing function ($\nu \not\equiv 0$) with $$
\int_{{\R^{d}}} (1\land |x|^2) \nu(|x|)dx<\infty\,. $$ We consider $J: {\R^{d}} \to [0, \infty]$
such that for some $c_{\!\scriptscriptstyle J} \in [1,\infty)$ and
all $x\in {\R^{d}}$, \begin{equation}\label{e:psi1}
c_{\!\scriptscriptstyle J}^{-1} \nu(|x|)\leqslant J(x) \leqslant c_{\!\scriptscriptstyle J} \,\nu(|x|)\,. \end{equation} Furthermore, suppose that $\kappa(x,z)$ is a Borel function on $\mathbb{R}^d\times {\R^{d}}$ such that \begin{equation}\label{e:intro-kappa} 0<\kappa_0\leqslant \kappa(x,z)\leqslant \kappa_1<\infty\, , \end{equation} for some numbers $\kappa_0$, $\kappa_1$, and there is $\beta\in (0,1)$ and a number $\kappa_2\geqslant 0$ with \begin{equation}\label{e:intro-kappa-holder}
|\kappa(x,z)-\kappa(y,z)|\leqslant \kappa_2|x-y|^{\beta}\, . \end{equation} The following concentration functions play a prominent role in the paper, $$
h(r):= \int_{{\R^{d}}} \left(1\land \frac{|x|^2}{r^2}\right) \nu(|x|)dx\,,\qquad \quad K(r):=r^{-2} \int_{|x|<r}|x|^2 \nu(|x|)dx\,,\qquad r>0\,. $$ We say that \emph{the weak scaling condition} at the origin holds if there are $\alpha_h\in (0,2]$ and $C_h \in [1,\infty)$ such that \begin{equation}\label{eq:intro:wlsc}
h(r)\leqslant C_h\,\lambda^{\alpha_h}\,h(\lambda r)\, ,\qquad r, \lambda \in (0,1]\,. \end{equation} In a similar fashion, we consider the existence of $\beta_h\in (0,2]$ and $c_h\in (0,1]$ such that \begin{equation}\label{eq:intro:wusc}
h(r)\geqslant c_h\,\lambda^{\beta_h}\,h(\lambda r)\, ,\qquad r, \lambda \in (0,1]\,.\\ \end{equation} We propose two more conditions on $\kappa(x,z)$ and $J(z)$. Suppose there are numbers $\kappa_3, \kappa_4\geqslant 0$ such that \begin{align}\label{e:intro-kappa-crit}
\sup_{x\in{\R^{d}}}\left| \int_{r\leqslant |z|<1} z\, \kappa(x,z) J(z)dz \right| &\leqslant \kappa_3 rh(r)\,,
\qquad r\in (0,1],\\
\left| \int_{r\leqslant |z|<1} z\, \big[ \kappa(x,z)- \kappa(y,z)\big] J(z)dz \right| &\leqslant \kappa_4 |x-y|^{\beta} rh(r)\,, \qquad r\in (0,1]. \label{e:intro-kappa-crit-H} \end{align}
\noindent We are ready to specify our framework. The {\it dimension} $d$ and {\it the profile function $\nu$} are fixed. We will use alternatively
two sets of assumptions. \begin{enumerate} \item[] \begin{enumerate} \item[$\Qa$:] \quad \eqref{e:psi1}--\eqref{eq:intro:wlsc} hold, $\alpha_h=1$; \eqref{e:intro-kappa-crit} and \eqref{e:intro-kappa-crit-H} hold; \item[$\Qb$:] \quad \eqref{e:psi1}--\eqref{eq:intro:wusc} hold, $0<\alpha_h \leqslant \beta_h <1$ and $1-\alpha_h<\beta \land \alpha_h$; \eqref{e:intro-kappa-crit} and \eqref{e:intro-kappa-crit-H} hold. \end{enumerate} \end{enumerate}
For the sake of the discussion, we assume that \eqref{e:psi1}--\eqref{eq:intro:wlsc} hold. We notify that the internal non-symmetry of the operator \eqref{e:intro-operator-a1-crit1} may result in a non-zero {\it internal drift} coefficient \begin{align*}
\int_{{\R^{d}}} z \left( {\bf 1}_{|z|<r}-{\bf 1}_{|z|<1} \right) \kappa(x,z)J(z) dz\,. \end{align*} Accordingly, \begin{align}\label{eq:L_split} {\mathcal L}^{\kappa}f(x)=
\int_{{\R^{d}}}( f(x+z)-f(x)- {\bf 1}_{|z|<r} \left<z,\nabla f(x)\right>)\,\kappa(x,z)J(z) dz \nonumber \\
+\left(\int_{{\R^{d}}} z \left( {\bf 1}_{|z|<r}-{\bf 1}_{|z|<1} \right) \kappa(x,z)J(z) dz\right) \cdot \nabla f(x)\,. \end{align} The influence of the {\it internal drift} term \eqref{eq:L_split} may be different depending on the order of the operator ${\mathcal L}^{\kappa}$ measured by $\alpha_h$ and $\beta_h$ in \eqref{eq:intro:wlsc} and~\eqref{eq:intro:wusc}. Below we will usually let $r=h^{-1}(1/t)$. \begin{fact}\label{fact:1} If \eqref{e:psi1}--\eqref{eq:intro:wlsc}
hold and $\alpha_h>1$, then \eqref{e:intro-kappa-crit} and \eqref{e:intro-kappa-crit-H} hold. \end{fact} \noindent The fact follows from Lemma~\ref{lem:int_J}. Thus, we have that if $\alpha_h>1$ (the sub-critical case), then the inequalities \eqref{e:intro-kappa-crit} and \eqref{e:intro-kappa-crit-H} are automatically satisfied. It explains a posteriori the success of the analysis of the sub-critical non-symmetric case in \cite{GS-2018}, see also \cite{MR3652202}, \cite{PJ}, \cite{CZ-new}, \cite{MR1744782}. On the other hand, if $\alpha_h=1$ (the critical case) or $\alpha_h< 1$ (the super-critical case), the study of the operator \eqref{e:intro-operator-a1-crit1} is harder and we need the conditions \eqref{e:intro-kappa-crit} and \eqref{e:intro-kappa-crit-H}. This resolves a question about natural conditions for critical and sub-critical non-symmetric operators that lead to strong results, similar to those in \cite{GS-2018}, and which provide a thorough analysis of the operator, the fundamental solution for the corresponding parabolic equation,
and the associated semigroup. We obtain such results if either of the set of assumptions $\Qa$ or $\Qb$ is satisfied, see Section~\ref{sec:main_res}. Note that under the symmetry condition, i.e., when $J(z)=J(-z)$ and $\kappa(x,z)=\kappa(x,-z)$, $x,z\in{\R^{d}}$, which is usually assumed in the literature, the problematic terms involving $\nabla f$ disappear after rewriting the operator ${\mathcal L}^{\kappa}$ as \begin{align*} \frac12 \int_{{\R^{d}}}( f(x+z)+f(x-z)-2f(x))\,\kappa(x,z)J(z)\, dz \,. \end{align*} In fact, under the symmetry, we have \begin{align}\label{eq:0}
\sup_{r\in (0,1]} \sup_{x\in{\R^{d}}}\left| \int_{r\leqslant |z|<1} z \kappa(x,z)J(z)dz \right| =0\,. \end{align} The condition \eqref{eq:0} was used in a non-symmetric case in \cite{PJ} and \cite{CZ-new} for $\nu(r)=r^{-d-1}$. The conditions \eqref{e:intro-kappa-crit} and \eqref{e:intro-kappa-crit-H} are much less restrictive.
\begin{example}\label{ex:1} Let $\nu(r)=r^{-d-1}$. Then \eqref{eq:intro:wlsc} and \eqref{eq:intro:wusc} hold with $\alpha_h=\beta_h=1$. The inequalities \eqref{e:intro-kappa-crit} and \eqref{e:intro-kappa-crit-H} are read as \begin{align*}
\sup_{x\in{\R^{d}}}\left| \int_{r\leqslant |z|<1} z\, \kappa(x,z) J(z)dz \right| &\leqslant c \,,
\qquad r\in (0,1],\\
\left| \int_{r\leqslant |z|<1} z\, \big[ \kappa(x,z)- \kappa(y,z)\big] J(z)dz \right| &\leqslant c |x-y|^{\beta}\,, \qquad r\in (0,1]. \end{align*} Hence, if $J(z)$ and $\kappa(x,z)$ are such that \eqref{e:psi1}, \eqref{e:intro-kappa}, \eqref{e:intro-kappa-holder} and the above inequalities hold, then the assumptions $\Qa$ are satisfied. We also note that
$\int_{r \leqslant|z|<1}|z| \nu(|z|)dz= c \log(1/r)$. \end{example}
Due to our general setting, we can deal with other interesting operators.
\begin{example} Let $\nu(r)=r^{-d-1}\log(2+1/r)$. Then \eqref{eq:intro:wlsc} holds with $\alpha_h=1$, but not with any $\alpha_h>1$. Furthermore, \eqref{eq:intro:wusc} holds for every $\beta_h>1$, but not with $\beta_h=1$. We also have that $\nu(r)$ is comparable to $r^{-d}h(r)$, see \cite[Lemma~5.3 and~5.4]{GS-2018}. Thus \eqref{e:intro-kappa-crit} and \eqref{e:intro-kappa-crit-H} allow, respectively, logarithmic growth as $r\to 0$ as follows \begin{align*}
\sup_{x\in{\R^{d}}}\left| \int_{r\leqslant |z|<1} z\, \kappa(x,z) J(z)dz \right| &\leqslant c \log(2+1/r) \,,
\qquad r\in (0,1],\\
\left| \int_{r\leqslant |z|<1} z\, \big[ \kappa(x,z)- \kappa(y,z)\big] J(z)dz \right| &\leqslant c |x-y|^{\beta} \log(2+1/r)\,, \qquad r\in (0,1]. \end{align*} Thus, if $J(z)$ and $\kappa(x,z)$ are such that \eqref{e:psi1}, \eqref{e:intro-kappa}, \eqref{e:intro-kappa-holder}
and the above inequalities hold, then the assumptions $\Qa$ are satisfied. Noteworthy, here $\int_{r \leqslant|z|<1}|z|\nu(|z|)dz$ is comparable to \mbox{$[\log(2+1/r)]^2$} for small $r$. \end{example}
\begin{example} Let $\nu(r)=r^{-d-\alpha}$ and $\alpha \in (1/2,1)$. Then \eqref{eq:intro:wlsc} and \eqref{eq:intro:wusc} hold with $\alpha_h=\beta_h=\alpha$. Note that for $J(z)$ and $\kappa(x,z)$ to satisfy $\Qb$, we need {\it the balance condition} $\alpha+\beta>1$ to hold. Furthermore, we have $r h(r)=r^{1-\alpha}h(1)$, while
$\int_{r \leqslant|z|<1}|z|\nu(|z|)dz$ is comparable to a positive constant for small $r$. \end{example}
\noindent Given a profile function $\nu$, it is not hard to find $J(z)$ and $\kappa(x,z)$ such that \eqref{e:psi1}--\eqref{e:intro-kappa-holder} hold, and \begin{align*}
\sup_{x\in{\R^{d}}}\left| \int_{r\leqslant |z|<1} z \kappa(x,z)J(z)dz \right| \geqslant
c \int_{r\leqslant |z|<1} |z|\nu(|z|)\,dz\,,\qquad r\in (0,1], \end{align*} for some $c>0$. Therefore, in each of the above examples, such choice of $J(z)$ and $\kappa(x,z)$ is not admissible, because the condition \eqref{e:intro-kappa-crit} fails. In fact, they can be chosen so that \eqref{e:intro-kappa-crit-H} fails as well. Put differently, similarly to \eqref{eq:0}, the conditions \eqref{e:intro-kappa-crit} and \eqref{e:intro-kappa-crit-H} require certain cancellations to take place.
The success of our approach based on the usage of \eqref{e:intro-kappa-crit} and \eqref{e:intro-kappa-crit-H} suggests that also in other studies and contexts where a counterpart of \eqref{eq:0} plays a role (see \cite{CZ-Dini}) a relaxation of assumptions to proper counterparts of \eqref{e:intro-kappa-crit} and \eqref{e:intro-kappa-crit-H} should be possible.
As further applications, our results allow solving uniquely {\it the martingale problem} for the operator $({\mathcal L}^{\kappa}, C_c^{\infty}({\R^{d}}))$. They also have applications to {\it the Kato class} of the semigroup $(P^{\kappa}_t)_{t\geqslant 0}$ corresponding to ${\mathcal L}^{\kappa}$, given in \eqref{e:intro-semigroup}. For details see \cite[Remarks~1.5 and~1.6]{GS-2018}.
Throughout the paper we make an effort to control how constants depend on the initial parameters of our model. The reason for that is twofold. First of all, it is necessary in the preliminaries, like Sections~\ref{sec:analysis_LL} and~\ref{sec:analysis_LL_2}, to be able to execute the construction and to find the key properties of a candidate for the solution in Section~\ref{sec:constr}. The second reason is more application-oriented: uniform results for families of operators or processes are desired in such areas as mixing property, multiscale models, homogenization, stationary distribution, see \cite{MR2779833}, \cite{MR1988467}, \cite{MR4238225}. The operators we consider resemble those investigated in the study of mean field games \cite{MR4309434}.
The main tool used in this paper is the parametrix method, proposed by E. Levi \cite{zbMATH02644101} to solve elliptic Cauchy problems. It was successfully applied in the theory of partial differential equations \cite{zbMATH02629782}, \cite{MR1545225}, \cite{MR0003340}, \cite{zbMATH03022319}, with an overview in the monograph \cite{MR0181836}, as well as in the theory of pseudo-differential operators \cite{MR2093219}, \cite{MR3817130}, \cite{MR3652202}, \cite{FK-2017}, \cite{MR3294616}. In particular, operators comparable in a sense to the fractional Laplacian were intensively studied by this method \cite{MR0492880}, \cite{MR616459}, \cite{MR972089}, \cite{MR1744782}, \cite{MR2093219}, also very recently \cite{MR3500272}, \cite{PJ}, \cite{CZ-new}, \cite{KR-2017}. More detailed historical comments on the development of the method can be found in \cite[Bibliographical Remarks]{MR0181836} and in the introductions of \cite{MR3652202} and \cite{BKS-2017}.
Basically we follow the scheme of \cite{GS-2018}, which in turn extends and strengthens
\cite{MR3817130} and \cite{MR3500272}. The results in the present paper are of the same type as in \cite{GS-2018} with the main progress being the recognition and usage of the conditions \eqref{e:intro-kappa-crit} and \eqref{e:intro-kappa-crit-H}.
Other related papers treat, for instance, (symmetric) singular L{\'e}vy measures \cite{BKS-2017}, \cite{KR-2017} or
(symmetric) exponential L{\'e}vy measures \cite{KL-2018}. We also list some papers that use different techniques to associate a semigroup with an operator by symbolic calculus \cite{MR0367492}, \cite{MR0499861}, \cite{MR666870}, \cite{MR1659620}, \cite{MR1254818}, \cite{MR1917230}, \cite{MR2163294}, \cite{MR2456894}, Dirichlet forms \cite{MR2778606}, \cite{MR898496}, \cite{MR2492992}, \cite{MR2443765}, \cite{MR2806700} or perturbation series \cite{MR1310558}, \cite{MR2283957}, \cite{MR2643799}, \cite{MR2876511}, \cite{MR3550165}, \cite{MR3295773}. For probabilistic methods and applications, we refer the reader to \cite{MR3022725}, \cite{MR3544166}, \cite{MR1341116},
\cite{MR3765882}, \cite{K-2015}, \cite{KR-2017}.
As stated in the abstract, when the present paper was first made public on \href{https://arxiv.org/abs/1807.04257v1}{arXiv:1807.04257v1}, the results were new even for the operators discussed in Example~\ref{ex:1}. Those operators are now included in the recent paper \cite{KKS-2020}.
\section{Main results}\label{sec:main_res}
We start by giving an exact meaning to \eqref{e:intro-operator-a1-crit1}. We apply the operator \eqref{e:intro-operator-a1-crit1}, in a strong or weak sense, only when it is well defined according to the following definitions. Let $f\colon {\R^{d}}\to \mathbb{R}$ be a Borel measurable function.
\begin{defn}[\textbf{Strong operator}] We say that the operator ${\mathcal L}^{\kappa}f$ is well defined if the gradient $\nabla f(x)$ exists and the corresponding integral in \eqref{e:intro-operator-a1-crit1} converges absolutely for every $x\in{\R^{d}}$. \end{defn}
We denote by ${\mathcal L}^{\kappa,\varepsilon}f$ the expression \eqref{e:intro-operator-a1-crit1} with $J(z)$ replaced by $
J(z){\bf 1}_{|z|>\varepsilon}$, $\varepsilon \in [0,1]$.
\begin{defn}[\textbf{Weak operator}] We let \begin{equation*} {\mathcal L}^{\kappa,0^+}f(x):=\lim_{\varepsilon \to 0^+}{\mathcal L}^{\kappa,\varepsilon}f(x)\,, \end{equation*} if the (strong) operators ${\mathcal L}^{\kappa,\varepsilon}f$ are well defined for $\varepsilon \in (0,1]$, and the limit exists for every~$x\in{\R^{d}}$. \end{defn}
It is clear that the operator ${\mathcal L}^{\kappa,0^+}$ is an extension of ${\mathcal L}^{\kappa,0}= {\mathcal L}^{\kappa}$, meaning that if ${\mathcal L}^{\kappa}f$ is well defined, then so is ${\mathcal L}^{\kappa,0^+}f$ and ${\mathcal L}^{\kappa,0^+}f={\mathcal L}^{\kappa}f$. Therefore, it is desired to prove the existence of of solutions to the equation $\partial_t={\mathcal L}^{\kappa}$ and the uniqueness of a solution to $\partial_t={\mathcal L}^{\kappa,0^+}$.
Here are our main results. \begin{theorem}\label{t:intro-main} Assume $\Qa$ or $\Qb$. Let $T>0$. There is a unique function $p^{\kappa}(t,x,y)$ on $(0,T]\times {\R^{d}}\times {\R^{d}}$ such~that \begin{itemize} \item[(i)] For all $t\in(0,T]$, $x,y\in {\R^{d}}$, $x\neq y$, \begin{equation}\label{e:intro-main-1} \partial_t p^{\kappa}(t,x,y)={\mathcal L}_x^{\kappa,0^+}p^{\kappa}(t,x, y)\,. \end{equation} \item[(ii)] The function $p^{\kappa}(t,x,y)$ is jointly continuous on $(0,T]\times {\R^{d}}\times {\R^{d}}$ and for any $f\in C_c^{\infty}({\R^{d}})$, \begin{equation}\label{e:intro-main-5}
\lim_{t\to 0^+}\sup_{x\in {\R^{d}}}\left| \int_{{\R^{d}}}p^{\kappa}(t,x,y)f(y)\, dy-f(x)\right|=0\, . \end{equation} \noindent \item[(iii)]
For every $t_0\in (0,T)$ there are $c>0$ and $f_0\in L^{1}({\R^{d}})$ such that for all $t\in (t_0,T]$, $x,y\in{\R^{d}}$, \begin{equation}\label{e:intro-main-2}
|p^{\kappa}(t,x,y)|\le c f_0(x-y)\,, \end{equation} and \begin{equation}\label{e:intro-main-4}
|{\mathcal L}_x^{\kappa, \varepsilon}p^{\kappa}(t,x,y)|\leqslant c \,,\qquad \varepsilon \in (0,1]\,. \end{equation} \item[(iv)] For every $t\in (0,T]$ there is $c>0$ such that for all $x,y\in{\R^{d}}$, \begin{equation}\label{e:intro-main-a1}
|\nabla_x p^{\kappa}(t,x,y)|\leqslant c\,. \end{equation} \end{itemize} \end{theorem}
In the next theorem, we collect more qualitative properties of $p^{\kappa}(t,x,y)$. To this end, for $t>0$ and $x\in \mathbb{R}^d$ we define {\it the bound function}, \begin{equation}\label{e:intro-rho-def}
\Upsilon_t(x):=\left( [h^{-1}(1/t)]^{-d}\land \frac{tK(|x|)}{|x|^{d}} \right) . \end{equation} It is an integrable function, which may provide sharp estimates for the heat kernel, extending the well known two-sided bounds
$t^{-d/\alpha}\land t/|x|^{d+\alpha}$ of the fundamental solution to $\partial_t = \Delta^{\alpha/2}$, where $\Delta^{\alpha/2}:= -(-\Delta^{\alpha/2})$ is the fractional Laplacian, see \cite[Thorem~1.1]{GS-2017} as well as more detailed discussion provided in \cite[Section~5]{GS-2017}. Properties of the bound function can be found also in \cite[Section~5]{GS-2018}.
\begin{theorem}\label{t:intro-further-properties} Assume $\Qa$ or $\Qb$. The following hold true. \begin{enumerate} \item[\rm (1)] (Non-negativity) The function $p^{\kappa}(t,x,y)$ is non-negative on $(0,\infty)\times{\R^{d}}\times{\R^{d}}$. \item[\rm (2)] (Conservativeness) For all $t>0$, $x\in{\R^{d}}$, \begin{equation*} \int_{{\R^{d}}}p^{\kappa}(t,x,y) dy =1\, . \end{equation*} \item[\rm (3)] (Chapman-Kolmogorov equation) For all $s,t > 0$, $x,y\in \mathbb{R}^d$, \begin{equation*} \int_{\mathbb{R}^d}p^{\kappa}(t,x,z)p^{\kappa}(s,z,y)\, dz =p^{\kappa}(t+s,x,y)\, . \end{equation*} \item[\rm (4)] (Upper estimate) For every $T>0$ there is $c>0$ such that for all $t\in (0,T]$, $x,y\in {\R^{d}}$, \begin{equation*} p^{\kappa}(t,x,y) \leqslant c \Upsilon_t(y-x)\, . \end{equation*} \item[\rm (5)] (Fractional derivative) For every $T>0$ there is $c>0$ such that for all $t\in (0,T]$, $x,y\in{\R^{d}}$, \begin{align*}
|{\mathcal L}_x^{\kappa } p^{\kappa}(t, x, y)|\leqslant c t^{-1}\Upsilon_t(y-x)\,. \end{align*} \item[\rm (6)] (Gradient) For every $T>0$ there is $c>0$ such that for all $t\in (0,T]$, $x,y\in{\R^{d}}$, \begin{equation*}
\left|\nabla_x p^{\kappa}(t,x,y)\right|\leqslant c\! \left[h^{-1}(1/t)\right]^{-1} \Upsilon_t(y-x)\,. \end{equation*} \item[\rm (7)] (Continuity) The function ${\mathcal L}_x^{\kappa} p^{\kappa}(t,x,y)$ is jointly continuous on $(0,\infty)\times {\R^{d}}\times{\R^{d}}$. \item[\rm (8)] (Strong operator) For all $t>0$, $x,y\in{\R^{d}}$, \begin{equation*} \partial_t p^{\kappa}(t,x,y)= {\mathcal L}_x^{\kappa}\, p^{\kappa}(t,x,y)\,. \end{equation*} \item[\rm (9)] (H\"older continuity) For all $T>0$, $\gamma \in [0,1] \cap[0,\alpha_h)$, there is $c>0$ such that for all $t\in (0,T]$ and $x,x',y\in {\R^{d}}$, \begin{equation*}
\left|p^{\kappa}(t,x,y)-p^{\kappa}(t,x',y)\right| \leqslant c
(|x-x'|^{\gamma}\land 1) \left[h^{-1}(1/t)\right]^{-\gamma} \big( \Upsilon_t(y-x)+ \Upsilon_t(y-x') \big). \end{equation*} \item[\rm (10)] (H\"older continuity) For all $T>0$, $\gamma \in [0,\beta)\cap [0,\alpha_h)$, there is $c>0$ such that for all $t\in (0,T]$ and $x,y,y'\in {\R^{d}}$, \begin{equation*}
\left|p^{\kappa}(t,x,y)-p^{\kappa}(t,x,y')\right| \leqslant c
(|y-y'|^{\gamma}\land 1) \left[h^{-1}(1/t)\right]^{-\gamma} \big( \Upsilon_t(y-x)+ \Upsilon_t(y'-x) \big). \end{equation*} \end{enumerate} The constants in {\rm (4) -- (6)} may be chosen to depend only on $d, c_{\!\scriptscriptstyle J}, \kappa_0, \kappa_1, \kappa_2, \kappa_3, \kappa_4, \beta, \alpha_h, C_h, h, T$. The same holds for {\rm (9)} and {\rm (10)} but with additional dependence on $\gamma$. \end{theorem}
For $t>0$, we define \begin{equation}\label{e:intro-semigroup} P_t^{\kappa}f(x)=\int_{{\R^{d}}} p^{\kappa}(t,x,y)f(y)\, dy\, ,\quad x\in {\R^{d}}\, , \end{equation} whenever the integral exists in the Lebesgue sense. We also put $P_0^{\kappa}=\mathrm{Id}$, the identity operator.
\begin{theorem}\label{thm:onC0Lp} Assume $\Qa$ or $\Qb$. The following hold true. \begin{enumerate} \item[\rm (1)] $(P^{\kappa}_t)_{t\geqslant 0}$ is an analytic strongly continuous positive contraction semigroup
on \mbox{$(C_0({\R^{d}}),\|\cdot\|_{\infty})$.}
\item[\rm (2)] $(P^{\kappa}_t)_{t\geqslant 0}$ is an analytic strongly continuous semigroup on every $(L^p({\R^{d}}),\|\cdot\|_p)$, \mbox{$p\in [1,\infty)$.} \item[\rm (3)] Let $(\mathcal{A}^{\kappa},D(\mathcal{A}^{\kappa}))$ be the
generator of $(P_t^{\kappa})_{t\geqslant 0}$ on $(C_0({\R^{d}}),\|\cdot\|_{\infty})$.\\
Then \begin{enumerate} \item[\rm (a)] $C_0^2({\R^{d}}) \subseteq D(\mathcal{A}^{\kappa})$ and $\mathcal{A}^{\kappa}={\mathcal L}^{\kappa}$ on $C_0^2({\R^{d}})$, \item[\rm (b)] $(\mathcal{A}^{\kappa},D(\mathcal{A}^{\kappa}))$ is the closure of $({\mathcal L}^{\kappa}, C_c^{\infty}({\R^{d}}))$, \item[\rm (c)] the function $x\mapsto p^{\kappa}(t,x,y)$ belongs to $D(\mathcal{A}^{\kappa})$ for all $t>0$, $y\in{\R^{d}}$, and $$ \mathcal{A}^{\kappa}_x\, p^{\kappa}(t,x,y)= {\mathcal L}_x^{\kappa}\, p^{\kappa}(t,x,y)=\partial_t p^{\kappa}(t,x,y)\,,\qquad x\in{\R^{d}}\,. $$ \end{enumerate} \item[\rm{(4)}] Let $(\mathcal{A}^{\kappa},D(\mathcal{A}^{\kappa}))$ be the
generator of $(P_t^{\kappa})_{t\geqslant 0}$ on $(L^p({\R^{d}}),\|\cdot\|_p)$, $p\in [1,\infty)$.\\
Then \begin{enumerate} \item[\rm (a)] $C_c^2({\R^{d}}) \subseteq D(\mathcal{A}^{\kappa})$ and $\mathcal{A}^{\kappa}={\mathcal L}^{\kappa}$ on $C_c^2({\R^{d}})$, \item[\rm (b)] $(\mathcal{A}^{\kappa},D(\mathcal{A}^{\kappa}))$ is the closure of $({\mathcal L}^{\kappa}, C_c^{\infty}({\R^{d}}))$, \item[\rm (c)] the function $x\mapsto p^{\kappa}(t,x,y)$ belongs to $D(\mathcal{A}^{\kappa})$ for all $t>0$, $y\in{\R^{d}}$, and in $L^p({\R^{d}})$, $$ \mathcal{A}^{\kappa} \, p^{\kappa}(t,\cdot,y)= {\mathcal L}^{\kappa}\, p^{\kappa}(t,\cdot,y)=\partial_t p^{\kappa}(t,\cdot,y)\,. $$ \end{enumerate} \end{enumerate} \end{theorem}
Finally, we provide a lower bound for the heat kernel $p^{\kappa}(t,x,y)$. It is quite typical that one first proves a lower bound in terms of $h^{-1}$ and $\nu$, as we do in the first two statements of Theorem~\ref{thm:lower-bound}, cf. \cite[Remark~5.7 and Section~4]{GS-2017}, \cite[Section~5]{MR4140542}. In our setting, we have that $\nu(r) \leqslant c\, r^{-d}K(r)$, see \cite[Lemma~7.1]{GS-2017}, which indicates a possible difference between those lower bounds and the upper bound by $\Upsilon_t(y-x)$. The converse inequality $\nu(r) \geqslant c r^{-d}K(r)$ is well understood, see
\cite[Lemma~7.3]{GS-2017}, and leads to sharp two-sided bounds in the third statement of the theorem. For abbreviation, we write $\varpi$ to denote the collection of $d, c_{\!\scriptscriptstyle J},\kappa_0,\kappa_1,\kappa_2,\kappa_3,\kappa_4,\beta,\alpha_h, C_h, h$.
\begin{theorem} \label{thm:lower-bound} Assume $\Qa$ or $\Qb$. The following hold true. \begin{itemize} \item[(i)] There are $T_0=T_0(\nu,\varpi)>0$ and $c=c(\nu,\varpi)>0$ such that for all $t\in (0,T_0]$, $x,y\in{\R^{d}}$, \begin{equation}\label{e:intro-main-11} p^{\kappa}(t,x,y)\geqslant c\left(
[h^{-1}(1/t)]^{-d}\wedge t \nu \left( |x-y|\right)\right). \end{equation} \item[(ii)] If additionally $\nu$ is positive, then for every $T>0$ there is $c=c(T,\nu,\varpi)>0$ such that \eqref{e:intro-main-11} holds for $t\in(0,T]$ and $x,y\in{\R^{d}}$.\\ \item[(iii)] If additionally there are $\bar{\beta}\in [0,2)$ and $\bar{c}>0$ such that $\bar{c} \lambda^{d+\bar{\beta}} \nu (\lambda r) \leqslant \nu(r)$, $\lambda \leqslant 1$, $r>0$, then for every $T >0$ there is $c=c(T,\nu,\bar{c},\bar{\beta},\varpi)>0$ such that for all $t\in(0,T]$ and $x,y\in{\R^{d}}$,
\begin{equation}\label{e:intro-main-111} p^{\kappa}(t,x,y) \geqslant c \Upsilon_t(y-x)\,. \end{equation} \end{itemize} \end{theorem}
\begin{remark}\label{rem:smaller_beta} If
\eqref{e:intro-kappa}, \eqref{e:intro-kappa-holder} hold, then $|\kappa(x,z)-\kappa(y,z)|\leqslant (2\kappa_1 \vee \kappa_2)|x-y|^{\beta_1}$ for every $\beta_1 \in [0,\beta]$. \end{remark}
\noindent According to the parametrix method, the fundamental solution $p^{\kappa}$ is expected to be given by \begin{align*} p^{\kappa}(t,x,y)= p^{\mathfrak{K}_y}(t,x,y)+\int_0^t \int_{{\R^{d}}}p^{\mathfrak{K}_z}(t-s,x,z)q(s,z,y)\, dzds\,, \end{align*} where $q(t,x,y)$ solves the equation \begin{align*} q(t,x,y)=q_0(t,x,y)+\int_0^t \int_{{\R^{d}}}q_0(t-s,x,z)q(s,z,y)\, dzds\,, \end{align*} and $$q_0(t,x,y)=\big({\mathcal L}_x^{{\mathfrak K}_x}-{\mathcal L}_x^{{\mathfrak K}_y}\big) p^{\mathfrak{K}_y}(t,x,y)\,.$$ Here $p^{\mathfrak{K}_w}$ is the heat kernel corresponding to the L{\'e}vy operator ${\mathcal L}^{\mathfrak{K}_w}$ obtained from the operator ${\mathcal L}^{\kappa}$ by freezing its coefficients: $\mathfrak{K}_w(z)=\kappa(w,z)$. In our setting, we draw the initial knowledge on $p^{{\mathfrak K}_w}$ from \cite{GS-2017}, which we then exploit in Sections~\ref{sec:analysis_LL} and~\ref{sec:analysis_LL_2} to establish further properties. Already in this preliminary part we essentially incorporate \eqref{e:intro-kappa-crit} and \eqref{e:intro-kappa-crit-H}, which differs from \cite{GS-2018}. We also see the effect of the internal drift and the fact that the order of the operator does not have to be strictly larger than one, e.g., Proposition~\ref{thm:delta_crit}, \eqref{e:delta-difference-abs-crit1}, Lemma~\ref{l:some-estimates-3b-crit1}. In Section~\ref{sec:q} we carry out the construction of $q$ and so also $p^{\kappa}$. In view of future developments, the following remark is notable. \begin{remark} We emphasize that the construction of $p^{\kappa}$ is possible, and many preliminary facts hold true under a weaker assumption \begin{enumerate} \item[] \begin{enumerate} \item[$\Qzero$:] \quad \eqref{e:psi1}--\eqref{eq:intro:wlsc} hold, $\alpha_h\in (0,1]$; \eqref{e:intro-kappa-crit} and \eqref{e:intro-kappa-crit-H} hold. \end{enumerate} \end{enumerate} In particular, see Lemma~\ref{l:estimates-q0-crit1}, Theorem~\ref{t:definition-of-q-crit1}, Lemma~\ref{lem:phi_cont_xy-crit1}, Proposition~\ref{l:phi-y-abs-cont-crit1}
and \eqref{e:p-kappa}, \eqref{e:def-phi-y-2}. \end{remark}
\noindent The subsequent non-trivial step is to verify that $p^{\kappa}$ is the actual solution. To this end, in Section~\ref{sec:phi} we need extra constraints which eventually result in $\Qa$ and $\Qb$, see for instance Lemma~\ref{e:L-on-phi-y-crit1} and the comments preceding Lemma~\ref{lem:phi_pomoc-crit1} and Lemma~\ref{lem:some-est_gen_phi_xy-crit1}. In Section~\ref{sec:p_kappa} we collect the initial properties of $p^{\kappa}$. In Section~\ref{sec:Main} we establish a nonlocal maximum principle, analyze the semigroup $(P_t^{\kappa})_{t\geqslant 0}$, complement the fundamental properties of $p^{\kappa}$, and prove Theorems~\ref{t:intro-main}--\ref{thm:lower-bound}. Section~\ref{sec:appA} contains auxiliary results. We give a final comment on the connection with \cite{GS-2018}. \begin{remark} The structure of the present paper is similar to that of \cite{GS-2018} to keep the same train of thought, but also to facilitate the transition between the papers while comparing and identifying the corresponding results. The reason for doing the latter is that the proofs that are the same as in \cite{GS-2018} are reduced to a minimum, we only list which facts are needed, occasionally give general ideas, and refer the reader to \cite{GS-2018} for details. We deliberately focus on and explain those aspects that are different from \cite{GS-2018}. We believe that such a presentation makes the content more comprehensible. In Lemma~\ref{lem:p-kappa-final-prop-crit1} we also give a correction of a part of the proof of \cite[Lemma~4.10]{GS-2018}. \end{remark}
In what follows, the function $\nu$ and the constants $d$, $c_{\!\scriptscriptstyle J}$, $\kappa_0$, $\kappa_1$, $\kappa_2$, $\beta$, $\kappa_3$, $\kappa_4$, $\alpha_h$, $C_h$, $\beta_h$, $c_h$ can be regarded as fixed. Apart from Sections~\ref{sec:Main} and~\ref{sec:appA}, we explicitly formulate all assumptions in lemmas, corollaries, propositions, and theorems. On the other hand, {\bf in Section~\ref{sec:Main} we assume that either $\Qa$ or $\Qb$ holds}.
\section{Notation}\label{sec:notation}
For the reader's convenience, we collect inhere the notation repeatedly used in the paper. By $c(d,\ldots)$ we denote a positive number that depends only on the listed parameters $d,\ldots$. By $\sigma$ we represent the collection of $c_{\!\scriptscriptstyle J},\kappa_0,\kappa_1,\kappa_3,\alpha_h, C_h, h$. Throughout the article, $\omega_d=2\pi^{d/2}/\Gamma(d/2)$ is the surface measure of the unit sphere in ${\R^{d}}$. We use ``$:=$" to denote the definition. As usual, $a\land b:=\min\{a,b\}$ and $a\vee b := \max\{a,b\}$.
The operator ${\mathcal L}^{\kappa}$ is given in \eqref{e:intro-operator-a1-crit1}. The functions $h(r)$, $K(r)$, and $\Upsilon_t(x)$ were introduced in Section~\ref{sec:intr} and Section~\ref{sec:main_res}. Here is a glossary of symbols to be used (and explained) below. For $$\mathfrak{K}\colon {\R^{d}} \to [0,\infty)\,,$$ we introduce the operator \begin{align}\label{op:aux} {\mathcal L}^{\mathfrak{K}}f(x):=
\int_{{\R^{d}}}( f(x+z)-f(x)- {\bf 1}_{|z|<1} \left<z,\nabla f(x)\right>)\,\mathfrak{K}(z)J(z)\, dz \,. \end{align} The corresponding heat kernel is denoted by \begin{align}\label{heat_kernel:aux} p^{\mathfrak{K}}(t,x,y)=p^{\mathfrak{K}}(t,y-x)\,. \end{align} We let \begin{align}\label{e:delta-f-def}
\delta_{1.r}^{\mathfrak{K}} (t,x,y;z)&:=p^{\mathfrak{K}}(t,x+z,y)-p^{\mathfrak{K}}(t,x,y)-{\bf 1}_{|z|<r}\left< z,\nabla_x p^{\mathfrak{K}}(t,x,y)\right>, \end{align} and \begin{align*} \delta^{\mathfrak{K}}(t,x,y;z)&:=\delta_{1.1}^{\mathfrak{K}}(t,x,y;z)\,. \end{align*} Thus for $\mathfrak{K}_1$ and $\mathfrak{K}_2$ we have \begin{align}\label{eq:L_delta} {\mathcal L}_x^{\mathfrak{K}_1} \,p^{\mathfrak{K}_2}(t,x,y) &=\int_{{\R^{d}}}\delta^{\mathfrak{K}_2} (t,x,y;z)\, \mathfrak{K}_1(z)J(z)dz\,, \end{align} and \begin{equation}\label{eq:L_delta_r} \begin{aligned} {\mathcal L}_x^{\mathfrak{K}_1} \,p^{\mathfrak{K}_2}(t,x,y) &=\int_{{\R^{d}}}\delta_{1.r}^{\mathfrak{K}_2} (t,x,y;z)\, \mathfrak{K}_1(z)J(z)dz \\
&\quad +\left(\int_{{\R^{d}}} z \left( {\bf 1}_{|z|<r}-{\bf 1}_{|z|<1} \right) \mathfrak{K}_1(z) J(z)\, dz\right) \cdot \nabla_x p^{\mathfrak{K}_2}(t,x,y) \,. \end{aligned} \end{equation} Starting from Section~\ref{sec:analysis_LL_2}, we shall use \begin{align}\label{def:k-frozen} \mathfrak{K}_w(z):=\kappa(w,z)\,, \end{align} which defines ${\mathcal L}^{\mathfrak{K}_w}f(x)$, $p^{\mathfrak{K}_w}(t,x,y)$ and $\delta_{1.r}^{\mathfrak{K}_w}(t,x,y;z)$. The main objects in the paper are \begin{align}\label{e:q0-definition} q_0(t,x,y):= \big({\mathcal L}_x^{{\mathfrak K}_x}-{\mathcal L}_x^{{\mathfrak K}_y}\big) p^{\mathfrak{K}_y}(t,x,y) =
\int_{{\R^{d}}}\delta^{\mathfrak{K}_y}(t,x,y;z)\left(\kappa(x,z)-\kappa(y,z)\right)J(z)dz \,, \end{align} \begin{align}\label{e:qn-definition} q_n(t,x,y):=\int_0^t \int_{{\R^{d}}}q_0(t-s,x,z)q_{n-1}(s,z,y)\, dzds\,, \end{align} \begin{align}\label{def:q} q(t,x,y):=\sum_{n=0}^{\infty}q_n(t,x,y)\,, \end{align} and \begin{align}\label{e:p-kappa} p^{\kappa}(t,x,y):=p^{\mathfrak{K}_y}(t,x,y)+\int_0^t \int_{{\R^{d}}}p^{\mathfrak{K}_z}(t-s,x,z)q(s,z,y)\, dzds\,. \end{align} The integral part of \eqref{e:p-kappa} is of special interest and to investigate its properties we introduce \begin{align}\label{e:phi-y-def} \phi_y(t,x,s):=\int_{{\R^{d}}} p^{\mathfrak{K}_z}(t-s,x,z)q(s,z,y)\, dz\,, \end{align} and \begin{align}\label{e:def-phi-y-2} \phi_y(t,x):=\int_0^t \phi_y(t,x,s)\, ds =\int_0^t \int_{{\R^{d}}}p^{\mathfrak{K}_z}(t-s,x,z)q(s,z,y)\, dzds\, . \end{align} Our estimates shall be frequently presented by means of \begin{align}\label{def:err}
\err{\gamma}{\beta}(t,x):= \left[h^{-1}(1/t)\right]^{\gamma} \left(|x|^{\beta}\land 1\right) t^{-1} \Upsilon_t(x)\,. \end{align} To shorten the notation in Section~\ref{sec:analysis_LL} we shall use the expressions \begin{align*}
\mathcal{F}_{1}&:=\Upsilon_t(y-x-z){\bf 1}_{|z|\geqslant h^{-1}(1/t)}+ \left[ \left(\frac{|z|}{h^{-1}(1/t)} \right)^2 \land \left(\frac{|z|}{h^{-1}(1/t)} \right) \right] \Upsilon_t(y-x),\\
\mathcal{F}_{2}&:=\Upsilon_t(y-x-z){\bf 1}_{|z|\geqslant h^{-1}(1/t)}+ \left[ \left(\frac{|z|}{h^{-1}(1/t)}\right)\wedge 1\right] \Upsilon_t(y-x). \end{align*} Thus, $\mathcal{F}_1=\mathcal{F}_1(t,x,y;z)$ and $\mathcal{F}_2=\mathcal{F}_2(t,x,y;z)$. We shall also need the non-increasing function $$ \Theta(t):= 1+\ln\left(1 \vee \left[h^{-1}(1/t)\right]^{-1}\right),\qquad t>0\,. $$
We use the following function spaces: $L^p({\R^{d}})$ denotes the Lebesgue space with $p\in [1,\infty)$, $C(D)$ are continuous functions on $D\subseteq {\mathbb R}^n$, $n\in \mathbb{N}$. Furthermore, $C_b({\R^{d}})$, $C_0({\R^{d}})$, $C_c({\R^{d}})$ are subsets of $C({\R^{d}})$ of functions that are bounded,
vanish at infinity, and have compact support, respectively. We write $f\in C^k({\R^{d}})$ if the function and all its derivatives up to (including if finite) order $k\in \mathbb{N}\cup \{\infty\}$ are elements of $C({\R^{d}})$; we similarly understand $C_b^k({\R^{d}})$, $C_0^k({\R^{d}})$, $C_c^k({\R^{d}})$. In particular, $C_c^{\infty}({\R^{d}})$ are smooth functions with compact support. The set $C^{k,\eta}({\R^{d}})$ consists of functions in $C^{k}({\R^{d}})$ such that all the derivatives of order $k$ are (uniformly) H{\"o}lder continuous with exponent $0<\eta<1$; we similarly define $C_b^{k,\eta}({\R^{d}})$, $C_0^{k,\eta}({\R^{d}})$, $C_c^{k,\eta}({\R^{d}})$.
\section{Analysis of the heat kernel of ${\mathcal L}^{\mathfrak{K}}$} \label{sec:analysis_LL}
In this section, we consider ${\mathcal L}^{\mathfrak{K}}$ given by \eqref{op:aux} with $J(z)$ satisfying \eqref{e:psi1} and \eqref{eq:intro:wlsc}, and a function~$\mathfrak{K}(z)$ such that \begin{align}\label{ineq:k-bounded} 0<\kappa_0 \leqslant \mathfrak{K}(z) \leqslant \kappa_1\,, \end{align} and \begin{align}\label{ineq:k-int_control}
\left| \int_{r\leqslant |z|<1} z\, \mathfrak{K}(z) J(z)dz \right|\leqslant \kappa_3 rh(r)\,, \qquad r\in (0,1]. \end{align} The operator ${\mathcal L}^{\mathfrak{K}}f$ is well defined for functions $f\in C_c^{\infty}({\R^{d}})$ and uniquely determines a L{\'e}vy process and its transition density $p^{\mathfrak{K}}(t,x,y)$ as represented in \eqref{heat_kernel:aux}. Then for all $t>0$, $x,y\in{\R^{d}}$, \begin{equation}\label{eq:p_gen_klas} \partial_t p^{\mathfrak{K}}(t,x,y)= {\mathcal L}_x^{\mathfrak{K}}\, p^{\mathfrak{K}}(t,x,y)\,. \end{equation} For more information we refer the reader to \cite[Section~6]{GS-2018}; in particular, the condition \cite[(96)]{GS-2018} is satisfied, see \eqref{e:psi1}, \cite[(86)]{GS-2018}
and \eqref{eq:intro:wlsc}.
Clearly, \eqref{ineq:k-bounded}
corresponds to \eqref{e:intro-kappa}, while \eqref{ineq:k-int_control} corresponds to \eqref{e:intro-kappa-crit}. We want to emphasize the role of \eqref{e:intro-kappa-crit} and \eqref{ineq:k-int_control}. In particular, \eqref{ineq:k-int_control} yields the following fundamental upper bound for $p^{\mathfrak{K}}$ and its derivatives.
\begin{proposition}\label{prop:gen_est_crit} Assume \eqref{e:psi1}, \eqref{eq:intro:wlsc}, \eqref{ineq:k-bounded}, \eqref{ineq:k-int_control}. For every $T>0$ and $\bbbeta\in \mathbb{N}_0^d$ there exists a constant $c=c(d,T,\bbbeta,\sigma)$ such that for all $t\in (0,T]$, $x,y\in{\R^{d}}$, \begin{align*}
|\partial_x^{\bbbeta} p^{\mathfrak{K}}\left(t,x,y\right)|\leqslant
c \left[h^{-1}(1/t) \right]^{-|\bbbeta|} \Upsilon_t(y-x)\,. \end{align*} \end{proposition} \noindent{\bf Proof.} The result follows from \cite[Proposition~5.4 ii)]{GS-2017} with $r_*=1$. {
$\Box$
}
Here is a lower bound for $p^{\mathfrak{K}}$.
\begin{lemma}\label{prop:gen_est_low-crit1} Assume \eqref{e:psi1}, \eqref{eq:intro:wlsc}, \eqref{ineq:k-bounded}, \eqref{ineq:k-int_control}.
For every $T,\theta>0$ there exists a constant $\tilde{c}=\tilde{c}(d,T,\theta,\nu,\sigma)$
such that for all $t\in (0,T]$ and $|x-y|\leqslant \theta h^{-1}(1/t)$, \begin{align*} p^{\mathfrak{K}}\left(t,x,y\right)\geqslant \tilde{c} \left[ h^{-1}(1/t)\right]^{-d}\,. \end{align*} \end{lemma} \noindent{\bf Proof.} We use \cite[Corollary~5.5]{GS-2017} with $x-y- tb_{[h_0^{-1}(1/t)]}$ in place of $x$, which is allowed since by \eqref{ineq:k-int_control} we have
$|tb_{[h_0^{-1}(1/t)]}|\leqslant a h_0^{-1}(1/t)$ for $a=a(d,T,\sigma)$. {
$\Box$
}
Proposition~\ref{prop:gen_est_crit} enables analysis of the increments of the heat kernel.
\begin{lemma}\label{lem:pk-collected} Assume \eqref{e:psi1}, \eqref{eq:intro:wlsc}, \eqref{ineq:k-bounded}, \eqref{ineq:k-int_control}.
For every $T>0$ there exists a constant $c=c(d,T,\sigma)$ such that for all $r>0$, $t\in (0,T]$, $x,x',y,z\in{\R^{d}}$ we have \begin{align}
\left|p^{\mathfrak{K}}(t,x+z,y)-p^{\mathfrak{K}}(t,x,y)\right|&\leqslant c\, \mathcal{F}_2(t,x,y;z)\,,\label{ineq:est_diff_1} \\
\left|\nabla_x p^{\mathfrak{K}}(t,x+z,y)-\nabla_x p^{\mathfrak{K}}(t,x,y)\right|&\leqslant c \left[h^{-1}(1/t)\right]^{-1} \mathcal{F}_2(t,x,y;z)\,,\label{ineq:est_grad_1}\\
|\delta_{1.r}^{\mathfrak{K}}(t,x,y;z)| &\leqslant c \big(
\mathcal{F}_{1}(t,x,y;z){\bf 1}_{|z|<r}+\mathcal{F}_{2}(t,x,y;z){\bf 1}_{|z|\geqslant r}\big)\,, \label{ineq:est_delta_1_crit} \end{align}
and whenever $|x'-x|<h^{-1}(1/t)$, then \begin{align}\label{ineq:diff_delta_1_crit}
|\delta_{1.r}^{\mathfrak{K}}(t,x',y;z)-\delta_{1.r}^{\mathfrak{K}}(t,x,y;z)|
\leqslant c\left(\frac{|x'-x|}{h^{-1}(1/t)}\right) \big(
\mathcal{F}_{1}(t,x,y;z){\bf 1}_{|z|<r}+\mathcal{F}_{2}(t,x,y;z){\bf 1}_{|z|\geqslant r}\big)\,. \end{align} \end{lemma} \noindent{\bf Proof.} The inequalities follow from Proposition~\ref{prop:gen_est_crit} and \cite[Corollary~5.10]{GS-2018}, cf. \cite[Lemma 2.3 -- 2.8]{GS-2018}. The idea of the proof is to represent the differences as integrals of derivatives in all cases when the absolute value of the argument increment is smaller than $h^{-1}(1/t)$. {
$\Box$
}
Due to \cite[Corollary~5.10]{GS-2018}, the inequality \eqref{ineq:est_diff_1} can be written equivalently as \begin{align}\label{ineq:est_diff_1*}
\left|p^{\mathfrak{K}}(t,x',y)-p^{\mathfrak{K}}(t,x,y)\right|&\leqslant c \left(\frac{|x'-x|}{h^{-1}(1/t)} \land 1\right) \big( \Upsilon_t(y-x') + \Upsilon_t(y-x)\big)\,. \end{align} The form \eqref{ineq:est_diff_1} is useful for estimating integrals, whereas \eqref{ineq:est_diff_1*} easily yields what follows.
\begin{lemma}\label{lem:pkw_holder} Assume \eqref{e:psi1}, \eqref{eq:intro:wlsc}, \eqref{ineq:k-bounded}, \eqref{ineq:k-int_control}. For every $T>0$ there exists a constant $c=c(d,T,\sigma)$ such that for all $t\in(0,T]$, $x,x',y,w \in {\R^{d}}$ and $\gamma\in [0,1]$, \begin{align*}
|p^{\mathfrak{K}}(t,x',y)-p^{\mathfrak{K}}(t,x,y) |
\leqslant c (|x-x'|^{\gamma}\land 1) \left[h^{-1}(1/t)\right]^{-\gamma}
\big( \Upsilon_t(y-x') + \Upsilon_t(y-x)\big). \end{align*} \end{lemma} In the next result we estimate ${\mathcal L}_x^{\mathfrak{K}_1} p^{\mathfrak{K}_2}(t,x,y)$, crucially using \eqref{ineq:k-int_control}.
\begin{lemma}\label{lem:Lkp_abs} Assume \eqref{e:psi1}, \eqref{eq:intro:wlsc} and let $\mathfrak{K}_1$, $\mathfrak{K}_2$ satisfy \eqref{ineq:k-bounded}, \eqref{ineq:k-int_control}. For every $T>0$ there exists a constant $c=c(d,T,\sigma)$ such that for all $t\in (0,T]$, $x,y\in{\R^{d}}$ we have \begin{align}\label{ineq:Lkp_abs}
\left| \int_{{\R^{d}}} \delta^{\mathfrak{K}_2} (t,x,y;z) \, \mathfrak{K}_1(z) J(z)dz
\right| \leqslant c t^{-1} \Upsilon_t(y-x)\,. \end{align} \end{lemma} \noindent{\bf Proof.} Let ${\rm I}$ be the left hand side of \eqref{ineq:Lkp_abs}. We note that the integral defining ${\rm I}$ converges absolutely. Using \eqref{eq:L_delta_r} with $r=h^{-1}(1/t)$, and \eqref{ineq:est_delta_1_crit}, \begin{align*}
{\rm I} &\leqslant c\int_{|z|\geqslant h^{-1}(1/t)} \mathcal{F}_{2} (t,x,y;z) \, \mathfrak{K}_1(z) J(z)dz
+c \int_{|z|< h^{-1}(1/t)} \mathcal{F}_{1} (t,x,y;z) \, \mathfrak{K}_1(z) J(z)dz\\
&\quad + \left| \int_{{\R^{d}}} z \left({\bf 1}_{|z|<h^{-1}(1/t)} - {\bf 1}_{|z|<1}\right) \mathfrak{K}_1(z) J(z)dz\right| |\nabla_x p^{\mathfrak{K}_2}(t,x,y)|\,. \end{align*} By \eqref{ineq:k-int_control} and Proposition~\ref{prop:gen_est_crit},
the last term is bounded by $c\, t^{-1}\Upsilon_t(y-x)$. The same holds for the first two terms, because of \eqref{e:psi1}, \eqref{ineq:k-bounded}, \cite[Lemma~5.1 (8) and~5.9]{GS-2018}. {
$\Box$
}
In what follows, we shall see a difference in the estimates compared to \cite{GS-2018}. The forthcoming result is an analogue of \cite[Theorem~2.9]{GS-2018} suitable for the present development.
\begin{proposition}\label{thm:delta_crit} Assume \eqref{e:psi1}, \eqref{eq:intro:wlsc}, \eqref{ineq:k-bounded}, \eqref{ineq:k-int_control}. For every $T>0$, the inequalities \begin{align}
&\int_{{\R^{d}}} |\delta^{\mathfrak{K}} (t,x,y;z)|\, J(z)dz \leqslant c\, \vartheta(t)\, t^{-1} \Upsilon_t(y-x)\,, \label{e:fract-der-est1-crit}\\
\int_{{\R^{d}}} |\delta^{\mathfrak{K}} (t,x',y;z)-&\delta^{\mathfrak{K}} (t,x,y;z)|\, J(z)dz \leqslant c \left(\frac{|x'-x|}{h^{-1}(1/t)} \land 1\right) \vartheta(t)\, t^{-1} \big( \Upsilon_t(y-x') + \Upsilon_t(y-x)\big),\nonumber \end{align} hold for all $t\in(0,T]$, $x,x',y\in{\R^{d}}$ with \begin{enumerate} \item[(a)] $\vartheta(t)=\Theta(t)$ and $c=c(d,T,\sigma)$ if $\alpha_h=1$, \item[(b)] $\vartheta(t)=t \,[h^{-1}(1/t)]^{-1}$ and $c=c(d,T,\sigma,\beta_h,c_h)$ if \eqref{eq:intro:wusc} holds for $0<\alpha_h \leqslant \beta_h<1$. \end{enumerate} \end{proposition} \noindent{\bf Proof.} By \eqref{ineq:est_delta_1_crit} with $r=h^{-1}(1/t)$ we get \begin{align*}
\int_{{\R^{d}}} &|\delta^{\mathfrak{K}} (t,x,y;z)|\, J(z)dz \\
&\leqslant \int_{{\R^{d}}} |\delta_{1.r}^{\mathfrak{K}} (t,x,y;z)|\, J(z)dz + \int_{{\R^{d}}} |z| \left| {\bf 1}_{|z|<r}-{\bf 1}_{|z|<1} \right| J(z)\, dz\, |\nabla_x p^{\mathfrak{K}}(t,x,y)| \\
&\leqslant c \int_{|z|\geqslant h^{-1}(1/t)} \mathcal{F}_{2} (t,x,y;z) \, J(z)dz
+c \int_{|z|< h^{-1}(1/t)} \mathcal{F}_{1} (t,x,y;z) \, J(z)dz\\
&+\int_{{\R^{d}}} |z| \left| {\bf 1}_{|z|<h^{-1}(1/t)}-{\bf 1}_{|z|<1} \right| J(z) dz\, \left[h^{-1}(1/t)\right]^{-1} \Upsilon_t(y-x)\,. \end{align*} The first inequality in the statement follows from \eqref{e:psi1},
\cite[Lemma~5.1 (8) and~5.9]{GS-2018} and Lemma~\ref{lem:int_J}. Now we prove the second inequality. If $|x'-x|\geqslant h^{-1}(1/t)$, then \begin{align*}
\int_{{\R^{d}}} \left(|\delta^{\mathfrak{K}} (t,x',y;z)|+|\delta^{\mathfrak{K}} (t,x,y;z)|\right)J(z)dz \leqslant c \, \vartheta(t) t^{-1} \left( \Upsilon_t(y-x')+\Upsilon_t(y-x) \right)\,. \end{align*}
If $|x'-x|< h^{-1}(1/t)$, we use \eqref{ineq:diff_delta_1_crit}, \eqref{ineq:est_grad_1} and, again, \cite[Lemma~5.1 and~5.9]{GS-2018} and Lemma~\ref{lem:int_J}.
{
$\Box$
}
The next result is a tool to relate the heat kernels corresponding to two coefficients $\mathfrak{K}_1$, $\mathfrak{K}_2$. \begin{lemma}\label{lem:rozne_1} Assume \eqref{e:psi1}, \eqref{eq:intro:wlsc} and let \eqref{ineq:k-bounded}, \eqref{ineq:k-int_control} hold for $\mathfrak{K}_1$, $\mathfrak{K}_2$, $\mathfrak{K}_3$. For all $t>0$, $x,y\in{\R^{d}}$ and $s\in (0,t)$, \begin{align*} \frac{d}{d s} \int_{{\R^{d}}} &p^{\mathfrak{K}_1}(s,x,z) p^{\mathfrak{K}_2}(t-s,z,y)\,dz\\ &= \int_{{\R^{d}}} {\mathcal L}_x^{\mathfrak{K}_1}p^{\mathfrak{K}_1}(s,x,z) \, p^{\mathfrak{K}_2}(t-s,z,y)\,dz - \int_{{\R^{d}}} p^{\mathfrak{K}_1}(s,x,z)\, {\mathcal L}_z^{\mathfrak{K}_2} p^{\mathfrak{K}_2}(t-s,z,y) \,dz\,, \end{align*} and \begin{align*} \int_{{\R^{d}}} {\mathcal L}^{\mathfrak{K}_3}_x p^{\mathfrak{K}_1}(s,x,z)\, p^{\mathfrak{K}_2}(t-s,z,y)\,dz = &\int_{{\R^{d}}} p^{\mathfrak{K}_1}(s,x,z) \, {\mathcal L}_z^{\mathfrak{K}_3} p^{\mathfrak{K}_2}(t-s,z,y)\,dz\,. \end{align*} \end{lemma} \noindent{\bf Proof.} The proof is the same as in \cite[Lemma~2.10]{GS-2018}. The first part follows by the dominated convergence theorem, which justifies the differentiation under the integral sign, and then by applying \eqref{eq:p_gen_klas}. The second identity is obtained after changing the order of integration and integrating by parts, see \eqref{e:delta-f-def} and \eqref{eq:L_delta}. In both cases we use the fact that for all $0<t_0<T<\infty$ there exists a constant $c=c(d,T,t_0,\sigma)$ such that for all $t\in[t_0,T]$, $x,y\in{\R^{d}}$, \begin{align}\label{ineq:aux_Q0}
\int_{{\R^{d}}} |\delta^{\mathfrak{K}_1} (t,x,y;z)|\, J(z)dz \leqslant c\, t^{-1}\Upsilon_t(y-x)\leqslant c t_0^{-1}\Upsilon_{t_0}(y-x)\,, \end{align} which is valid under the assumptions of the lemma, see \eqref{e:fract-der-est1-crit}. {
$\Box$
}
\section{Analysis of the heat kernel of ${\mathcal L}^{\mathfrak{K}_w}$} \label{sec:analysis_LL_2}
In this section we work under $\Qzero$. In particular, we consider $J(z)$ satisfying \eqref{e:psi1} and \eqref{eq:intro:wlsc}, and $\kappa(x,z)$ such that \eqref{e:intro-kappa} and \eqref{e:intro-kappa-crit} hold. For a fixed $w\in {\R^{d}}$ we let $\mathfrak{K}_w(z)=\kappa(w,z)$, as in \eqref{def:k-frozen}. Since \eqref{ineq:k-bounded} and \eqref{ineq:k-int_control} hold for $\mathfrak{K}_w$, the results of Section~\ref{sec:analysis_LL} remain in force. Like in Section~\ref{sec:analysis_LL} we let $p^{\mathfrak{K}_w}(t,x,y)$ be the heat kernel of ${\mathcal L}^{{\mathfrak K}_w}$. This procedure is known as freezing coefficients of the operator ${\mathcal L}^{\kappa}$ given in~\eqref{e:intro-operator-a1-crit1}.
First, we estimate $ \big( {\mathcal L}_x^{\mathfrak{K}_{w'}}-{\mathcal L}_x^{\mathfrak{K}_w} \big)p^{\mathfrak{K}}(t,x,y) $. \begin{lemma}\label{lem:Lkp_abs-H} Assume $\Qzero$ and let \eqref{ineq:k-bounded}, \eqref{ineq:k-int_control} hold for $\mathfrak{K}$. For every $T>0$ there exists a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4)$ such that for all $t\in (0,T]$, $x,y, w, w'\in{\R^{d}}$ we have \begin{align}\label{ineq:Lkp_abs-H}
\left| \int_{{\R^{d}}} \delta^{\mathfrak{K}} (t,x,y;z) \, \left( \kappa(w',z)- \kappa(w,z)\right) J(z)dz
\right| \leqslant c \left( |w'-w|^{\beta}\land 1 \right) t^{-1} \Upsilon_t(y-x)\,. \end{align} \end{lemma}
\noindent{\bf Proof.} If $|w'-w|\geqslant 1$ we apply \eqref{ineq:Lkp_abs}. Let ${\rm I}$ be the left hand side of \eqref{ineq:Lkp_abs-H}
and $|w'-w|< 1$. We also note that the integral defining ${\rm I}$ converges absolutely. Using \eqref{eq:L_delta_r} with $r=h^{-1}(1/t)$, and~\eqref{ineq:est_delta_1_crit}, \begin{align*}
{\rm I} &\leqslant c\int_{|z|\geqslant h^{-1}(1/t)} \mathcal{F}_{2} (t,x,y;z) \, |\kappa(w',z)- \kappa(w,z)| J(z)dz\\
&\quad+c \int_{|z|< h^{-1}(1/t)} \mathcal{F}_{1} (t,x,y;z) \, |\kappa(w',z)- \kappa(w,z)| J(z)dz\\
&\quad + \left| \int_{{\R^{d}}} z \left({\bf 1}_{|z|<h^{-1}(1/t)} - {\bf 1}_{|z|<1}\right) \big( \kappa(w',z)- \kappa(w,z)\big) J(z)dz\right| |\nabla_x p^{\mathfrak{K}}(t,x,y)|\,. \end{align*} By \eqref{e:intro-kappa-crit-H} and
Proposition~\ref{prop:gen_est_crit}
the last term is bounded by $|w'-w|^{\beta}t^{-1}\Upsilon_t(y-x)$. The same is true for the first two terms by \eqref{e:psi1}, \eqref{e:intro-kappa-holder}, \cite[Lemma~5.1 (8) and~5.9]{GS-2018}. {
$\Box$
}
Now we estimate $\big({\mathcal L}_{x'}^{\mathfrak{K}_{w'}}-{\mathcal L}_{x'}^{\mathfrak{K}_w} \big) p^{\mathfrak{K}}(t,x',y)-\big({\mathcal L}_{x}^{\mathfrak{K}_{w'}}-{\mathcal L}_{x}^{\mathfrak{K}_w} \big) p^{\mathfrak{K}}(t,x,y)$. \begin{lemma}\label{lem:Lkp_abs-H-H} Assume $\Qzero$ and let \eqref{ineq:k-bounded}, \eqref{ineq:k-int_control} hold for $\mathfrak{K}$. For every $T>0$ there exists a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4)$ such that for all $t\in(0,T]$, $x,x',y, w, w'\in{\R^{d}}$, \begin{align}\label{ineq:Lkp_abs-H-H}
&\left| \int_{{\R^{d}}} \left( \delta^{\mathfrak{K}} (t,x',y;z)-\delta^{\mathfrak{K}} (t,x,y;z) \right) \left( \kappa(w',z)- \kappa(w,z)\right) J(z)dz \right| \nonumber \\
&\qquad \leqslant c \left(\frac{|x'-x|}{h^{-1}(1/t)} \land 1\right) \left( |w'-w|^{\beta}\land 1 \right) t^{-1} \big( \Upsilon_t(y-x') + \Upsilon_t(y-x)\big)\,. \end{align} \end{lemma}
\noindent{\bf Proof.} If $|x'-x|\geqslant h^{-1}(1/t)$ we apply \eqref{ineq:Lkp_abs-H}.
Let ${\rm I}$ be the left hand side of \eqref{ineq:Lkp_abs-H-H} and $|x'-x|< h^{-1}(1/t)$. By \eqref{eq:L_delta_r} with $r=h^{-1}(1/t)$, and \eqref{ineq:diff_delta_1_crit}, \begin{align*}
{\rm I} &\leqslant c \left(\frac{|x'-x|}{h^{-1}(1/t)}\right) \int_{|z|\geqslant h^{-1}(1/t)} \mathcal{F}_{2} (t,x,y;z) \, |\kappa(w',z)- \kappa(w,z)| J(z)dz\\
&\quad+c \left(\frac{|x'-x|}{h^{-1}(1/t)}\right) \int_{|z|< h^{-1}(1/t)} \mathcal{F}_{1} (t,x,y;z) \, |\kappa(w',z)- \kappa(w,z)| J(z)dz\\
&\quad + \left| \int_{{\R^{d}}} z \left({\bf 1}_{|z|<h^{-1}(1/t)} - {\bf 1}_{|z|<1}\right) \big( \kappa(w',z)- \kappa(w,z)\big) J(z)dz\right| |\nabla_{x'} p^{\mathfrak{K}}(t,x',y)-\nabla_{x}p^{\mathfrak{K}}(t,x,y)|\,. \end{align*} By \eqref{e:intro-kappa-crit-H} and \eqref{ineq:est_grad_1},
we bound the last expression by
$(|w'-w|^{\beta}\land 1 ) (|x'-x|/h^{-1}(1/t)) t^{-1}\Upsilon_t(y-x)$. For the first two terms we rely on \eqref{e:psi1}, \eqref{e:intro-kappa-holder}, \cite[Lemma~5.1 (8) and~5.9]{GS-2018}. {
$\Box$
}
In Lemma~\ref{lem:Lkp_abs-H} and~\ref{lem:Lkp_abs-H-H} our assumptions \eqref{ineq:k-bounded} and \eqref{ineq:k-int_control} play an important role. They also influence Proposition~\ref{prop:Hcont_kappa_crit1} and other results. We note a difference in the estimates \eqref{e:delta-difference-abs-crit1} in comparison to the corresponding bound in \cite[Theorem~2.11]{GS-2018}.
In what follows we provide several results on the regularity of the heat kernel $p^{\mathfrak{K}_{w}}(t,x,y)$ in~$w\in{\R^{d}}$.
\begin{proposition}\label{prop:Hcont_kappa_crit1} Assume $\Qzero$. For every $T>0$ there exists a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4)$ such that for all $t\in (0,T]$, $x,y,w,w'\in{\R^{d}}$, \begin{align*}
|p^{\mathfrak{K}_{w'}}(t,x,y)-p^{\mathfrak{K}_w}(t,x,y)| &\leqslant c\, (|w'-w|^{\beta}\land 1)\,\Upsilon_t(y-x)\,,\\
|\nabla_x p^{\mathfrak{K}_{w'}}(t,x,y)-\nabla_x p^{\mathfrak{K}_w}(t,x,y)| &\leqslant c (|w'-w|^{\beta}\land 1) \left[h^{-1}(1/t)\right]^{-1} \Upsilon_t(y-x) \,,\\
\left| {\mathcal L}_x^{\mathfrak{K}_x} p^{\mathfrak{K}_{w'}}(t,x,y)- {\mathcal L}_x^{\mathfrak{K}_x} p^{\mathfrak{K}_{w}}(t,x,y)\right|
&\leqslant c (|w'-w|^{\beta}\land 1)\,
t^{-1}\Upsilon_t(y-x) \,. \end{align*} Moreover, for every $T>0$, the inequality \begin{align}
\int_{{\R^{d}}} |\delta^{\mathfrak{K}_{w'}} (t,x,y;z)-\delta^{\mathfrak{K}_w} (t,x,y;z)| \,J(z)dz &\leqslant c (|w'-w|^{\beta}\land 1) \,\vartheta(t)\,
t^{-1}\Upsilon_t(y-x) \,, \label{e:delta-difference-abs-crit1} \end{align} holds for all $t\in (0,T]$, $x,y,w,w'\in{\R^{d}}$ with \begin{enumerate} \item[(a)] $\vartheta(t)=\Theta(t)$ and $c=c(d,T,\sigma,\kappa_2,\kappa_4)$ if $\alpha_h=1$, \item[(b)] $\vartheta(t)=t \,[h^{-1}(1/t)]^{-1}$ and $c=c(d,T,\sigma,\kappa_2,\kappa_4,\beta_h,c_h)$ if \eqref{eq:intro:wusc} holds for $0<\alpha_h \leqslant \beta_h<1$. \end{enumerate} \end{proposition} \noindent{\bf Proof.} In what follows, we use \cite[Corollary~5.14, Lemma~5.6]{GS-2018} and the monotonicity of $h^{-1}$ without further comment. The proof resembles that of \cite[Theorem~2.11]{GS-2018}, but in parts (ii), (iii) and (iv) different adjustments are needed to use our assumptions. \\ (i) Using Lemma~\ref{lem:rozne_1}, we get \begin{align*} p^{\mathfrak{K}_{w'}}(t,x,y)-p^{\mathfrak{K}_{w}}(t,x,y) &= \lim_{\varepsilon_1 \to 0^+} \int_{\varepsilon_1}^{t/2} \int_{{\R^{d}}} p^{\mathfrak{K}_{w'}}(s,x,z) \left( {\mathcal L}_z^{\mathfrak{K}_{w'}} - {\mathcal L}_z^{\mathfrak{K}_w}\right) p^{\mathfrak{K}_w}(t-s,z,y)\,dzds\\ &+ \lim_{\varepsilon_2\to 0^+ } \int_{t/2}^{t-\varepsilon_2} \int_{{\R^{d}}} \left( {\mathcal L}_x^{\mathfrak{K}_{w'}} - {\mathcal L}_x^{\mathfrak{K}_{w}}\right) p^{\mathfrak{K}_{w'}}(s,x,z) p^{\mathfrak{K}_w}(t-s,z,y)\,dzds \,. \end{align*} By Proposition~\ref{prop:gen_est_crit} and \eqref{ineq:Lkp_abs-H}, \begin{align*} &\int_{\varepsilon}^{t/2}
\int_{{\R^{d}}} p^{\mathfrak{K}_{w'}}(s,x,z)\, | \!\left( {\mathcal L}_z^{\mathfrak{K}_{w'}}
- {\mathcal L}_z^{\mathfrak{K}_w}\right) p^{\mathfrak{K}_w}(t-s,z,y)|\,dzds\\
&\leqslant c\, (|w'-w|^{\beta}\land 1) \int_{\varepsilon}^{t/2} \int_{{\R^{d}}} \Upsilon_s (z-x)\, (t-s)^{-1}\Upsilon_{t-s}(y-z) \,dzds\\
&\leqslant c\, (|w'-w|^{\beta}\land 1)\, \Upsilon_t(y-x) \int_{\varepsilon}^{t/2} t^{-1}ds\,. \end{align*} Similarly, \begin{align*} &\int_{t/2}^{t-\varepsilon}
\int_{{\R^{d}}} |\!\left( {\mathcal L}_x^{\mathfrak{K}_{w'}}
- {\mathcal L}_x^{\mathfrak{K}_w}\right) p^{\mathfrak{K}_{w'}}(s,x,z) |\, p^{\mathfrak{K}_w}(t-s,z,y)\,dzds
\leqslant c \, (|w'-w|^{\beta}\land 1)\, \Upsilon_t(y-x)\,. \end{align*}
\noindent (ii) Let $w_0\in{\R^{d}}$ be fixed. Define $\mathfrak{K}(z)=(\kappa_0/(2\kappa_1)) \kappa(w_0,z)$ and $\widehat{\mathfrak{K}}_w (z)=\mathfrak{K}_w(z)- \mathfrak{K}(z)$. By the construction of the L{\'e}vy process, we have \begin{align}\label{eq:przez_k_0-impr} p^{\mathfrak{K}_w}(t,x,y)=\int_{{\R^{d}}} p^{\mathfrak{K}}(t,x,\xi) p^{\widehat{\mathfrak{K}}_w}(t,\xi,y)\,d\xi\,. \end{align} Then by \eqref{ineq:est_diff_1} we can differentiate under the integral in \eqref{eq:przez_k_0-impr}. By Proposition~\ref{prop:gen_est_crit} we get \begin{align*}
|\nabla_x p^{\mathfrak{K}_{w'}}(t,x,y)-\nabla_x p^{\mathfrak{K}_w}(t,x,y)|
& \leqslant \int_{{\R^{d}}} \left| \nabla_x p^{\mathfrak{K}}(t, x,\xi) \right| \left| p^{\widehat{\mathfrak{K}}_{w'}}(t,\xi,y)-p^{\widehat{{\mathfrak{K}}}_w}(t,\xi,y)\right| d\xi\\
&\leqslant c (|w'-w|\land 1) \left[h^{-1}(1/t)\right]^{-1} \Upsilon_t(y-x)\,. \end{align*}
\noindent (iii) By \eqref{eq:przez_k_0-impr} we have \begin{align*} \delta^{\mathfrak{K}_{w'}} (t,x,y;z)-\delta^{\mathfrak{K}_w} (t,x,y;z) = \int_{{\R^{d}}} \delta^{\mathfrak{K}}(t,x,\xi;z) \left(p^{\widehat{\mathfrak{K}}_{w'}}(t,\xi,y)-p^{\widehat{{\mathfrak{K}}}_w}(t,\xi,y)\right)
d\xi. \end{align*} Then by \eqref{ineq:Lkp_abs}, \begin{align*}
\left| {\mathcal L}_x^{\mathfrak{K}_x} p^{\mathfrak{K}_{w'}}(t,x,y)- {\mathcal L}_x^{\mathfrak{K}_x} p^{\mathfrak{K}_{w}}(t,x,y)\right| &\leqslant
\int_{{\R^{d}}} \left| {\mathcal L}_x^{\mathfrak{K}_x}p^{\mathfrak{K}}(t,x,\xi ) \right|
\left| p^{\widehat{\mathfrak{K}}_{w'}}(t,\xi,y)-p^{\widehat{{\mathfrak{K}}}_w}(t,\xi,y)\right|
d\xi\\
&\leqslant c (|w'-w|\land 1)\, t^{-1} \Upsilon_{t}(y-x)\,. \end{align*}
\noindent (iv) By Proposition~\ref{thm:delta_crit}, \begin{align*}
\int_{{\R^{d}}} &|\delta^{\mathfrak{K}_{w'}} (t,x,y;z)-\delta^{\mathfrak{K}_w} (t,x,y;z)| \,J(z)dz \\ &\leqslant
\int_{{\R^{d}}} \left( \int_{{\R^{d}}} |\delta^{\mathfrak{K}}(t,x,\xi;z)| \,J(z)dz\right)
\left| p^{\widehat{\mathfrak{K}}_{w'}}(t,\xi,y)-p^{\widehat{{\mathfrak{K}}}_w}(t,\xi,y)\right|
d\xi\\ &\leqslant
c (|w'-w|\land 1)\int_{{\R^{d}}} \vartheta(t) t^{-1}\Upsilon_t(\xi-x) \Upsilon_t(y-\xi)\, d\xi \\
&\leqslant c(|w'-w|\land 1) \vartheta(t) t^{-1} \Upsilon_t(y-x)\,. \end{align*} {
$\Box$
}
Our results mostly have the same form as those in \cite{GS-2018}, and similarly as in \cite{GS-2018} we are able to deduce the joint continuity, the concentration of mass, and cancellations.
\begin{lemma}\label{lem:cont_frcoef} Assume $\Qzero$. The functions $p^{\mathfrak{K}_w}(t,x,y)$ and $\nabla_x p^{\mathfrak{K}_w}(t,x,y)$ are jointly continuous in $(t, x, y,w) \in (0,\infty)\times ({\R^{d}})^3$. The function ${\mathcal L}_x^{\mathfrak{K}_{v}} p^{\mathfrak{K}_{w}}(t,x,y)$ is jointly continuous in $(t,x,y,w,v)\in (0,\infty)\times ({\R^{d}})^4$. Furthermore, \begin{align}\label{e:some-estimates-2c-crit1}
\lim_{t \to 0^+ } \sup_{x\in{\R^{d}}} \left| \int_{{\R^{d}}} p^{\mathfrak{K}_y}(t,x,y)\, dy -1\right|=0 \end{align} \end{lemma} \noindent{\bf Proof.} The result follows from Proposition~\ref{prop:Hcont_kappa_crit1}, \cite[Lemma~6.1]{GS-2018}, \eqref{ineq:est_delta_1_crit} and Lemma~\ref{l:convolution}, cf. \cite[Lemma~3.1, 3.2 and~3.4]{GS-2018}. {
$\Box$
}
\begin{lemma}\label{e:some-estimates-2bb-crit1} Assume $\Qzero$. Let $\beta_1\in [0,\beta]\cap [0,\alpha_h)$. For every $T>0$ there exists a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4,\beta_1)$ such that for all $t\in (0,T]$, $x\in{\R^{d}}$, \begin{equation*}
\left|\int_{{\R^{d}}} \nabla_x p^{\mathfrak{K}_y} (t,x,y)\,dy \right| \leqslant c\! \left[h^{-1}(1/t)\right]^{-1+\beta_1}\,. \end{equation*} \end{lemma} \noindent{\bf Proof.} The inequality stems from \eqref{ineq:est_diff_1}, Proposition~\ref{prop:Hcont_kappa_crit1} and Lemma~\ref{l:convolution}, cf. \cite[Lemma~3.4]{GS-2018}.
{
$\Box$
}
Compared to \cite{GS-2018}, the two expressions in Lemma~\ref{l:some-estimates-3b-crit1} and Lemma~\ref{l:some-estimates-3b-crit1-impr} need to be estimated separately.
\begin{lemma}\label{l:some-estimates-3b-crit1} Assume $\Qzero$. Let $\beta_1\in [0,\beta]\cap [0,\alpha_h)$. For every $T>0$, the inequality \begin{align*}
\int_{{\R^{d}}} \left|\int_{{\R^{d}}} \delta^{\mathfrak{K}_y} (t,x,y;z) \,dy \right| J(z)dz &\leqslant c\, \vartheta(t)\, t^{-1}\left[h^{-1}(1/t)\right]^{\beta_1}, \end{align*} holds for all $t\in (0,T]$, $x\in{\R^{d}}$ with \begin{enumerate} \item[(a)] $\vartheta(t)=\Theta(t)$ and $c=c(d,T,\sigma,\kappa_2,\kappa_4,\beta_1)$ if $\alpha_h=1$, \item[(b)] $\vartheta(t)=t \,[h^{-1}(1/t)]^{-1}$ and $c=c(d,T,\sigma,\kappa_2,\kappa_4,\beta_1,\beta_h,c_h)$ if \eqref{eq:intro:wusc} holds for $0<\alpha_h \leqslant \beta_h<1$. \end{enumerate} \end{lemma} \noindent{\bf Proof.} We subtract zero and use \eqref{e:delta-difference-abs-crit1}, to get \begin{align*}
\int_{{\R^{d}}} \left|\int_{{\R^{d}}} \delta^{\mathfrak{K}_y} (t,x,y;z)
-\delta^{\mathfrak{K}_x} (t,x,y;z) \,dy \right| J(z)dz \leqslant c \int_{{\R^{d}}} \vartheta(t) \err{0}{\beta_1}(t,x-y) \,dy\,. \end{align*} The result follows from Lemma~\ref{l:convolution}(a). {
$\Box$
}
\begin{lemma}\label{l:some-estimates-3b-crit1-impr} Assume $\Qzero$. Let $\beta_1\in [0,\beta]\cap [0,\alpha_h)$. For every $T>0$ there exists a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4,\beta_1)$ such that for all $t\in (0,T]$, $x\in{\R^{d}}$, \begin{align*}
\left| \int_{{\R^{d}}} {\mathcal L}_x^{\mathfrak{K}_x} p^{\mathfrak{K}_y} (t,x,y) \,dy \right| &\leqslant c t^{-1}\left[h^{-1}(1/t)\right]^{\beta_1}\,. \end{align*} \end{lemma} \noindent{\bf Proof.} By Proposition~\ref{prop:Hcont_kappa_crit1}, we have \begin{align*}
\left| \int_{{\R^{d}}} {\mathcal L}_x^{\mathfrak{K}_x} p^{\mathfrak{K}_y} (t,x,y) \,dy \right|=
\left| \int_{{\R^{d}}}\left( {\mathcal L}_x^{\mathfrak{K}_x} p^{\mathfrak{K}_y} (t,x,y)- {\mathcal L}_x^{\mathfrak{K}_x} p^{\mathfrak{K}_x} (t,x,y) \right)dy \right| \leqslant c \int_{{\R^{d}}} \err{0}{\beta_1}(t,x-y)\, dy\,. \end{align*} The result follows from Lemma~\ref{l:convolution}(a). {
$\Box$
}
\section{Levi's construction of the heat kernel}\label{sec:constr}
In this section, we focus on the objects introduced in Section~\ref{sec:notation} by the formulae \eqref{e:q0-definition}--\eqref{e:def-phi-y-2}. The estimates are stated using a short notation proposed in \eqref{def:err}.
\subsection{Construction of $q(t,x,y)$}\label{sec:q}
\begin{lemma}\label{l:estimates-q0-crit1} Assume $\Qzero$. For every $T>0$ there exists a constant $c=c(d,T,\sigma,\kappa_2, \kappa_4)\geqslant 1$ such that for all $\beta_1\in[0,\beta]$, $t\in (0,T]$ and $x,x',y,y'\in{\R^{d}}$ \begin{align}\label{e:q0-estimate-crit1}
|q_0(t,x,y)|\leqslant c \err{0}{\beta_1}(t,y-x)\,, \end{align} and for every $\gamma\in [0,\beta_1]$, \begin{align}
&|q_0(t,x,y)-q_0(t,x',y)|\nonumber\\
&\leqslant c \left(|x-x'|^{\beta_1-\gamma}\land 1\right)\left\{\left(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\right)(t,x-y) +\left(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\right)(t,x'-y)\right\},\label{e:estimate-step3-crit1} \end{align} and \begin{align}
&|q_0(t,x,y)-q_0(t,x,y')|\nonumber \\
&\leqslant c \left(|y-y'|^{\beta_1-\gamma}\land 1\right)\left\{\left(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\right)(t,x-y) +\left(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\right)(t,x-y')\right\}. \label{e:estimate-q0-2-crit1} \end{align} \end{lemma} \noindent{\bf Proof.} (i) \eqref{e:q0-estimate-crit1} follows from \eqref{ineq:Lkp_abs-H}.\\
(ii) For $|x-x'|\geqslant 1$ the inequality holds by \eqref{e:q0-estimate-crit1} and \cite[(92)]{GS-2018}: \begin{align*}
|q_0(t,x,y)| \leqslant c \err{0}{\beta_1}(t,y-x) \leqslant c\left[ h^{-1}(1/T)\vee 1\right]^{\beta_1-\gamma} \err{\gamma-\beta_1}{\beta_1}(t,y-x)\,. \end{align*}
For $1\geqslant |x-x'|\geqslant h^{-1}(1/t)$ the result follows from \eqref{e:q0-estimate-crit1} and \begin{align*}
|q_0(t,x,y)| \leqslant c \err{0}{\beta_1}(t,y-x) = c \left[ h^{-1}(1/t)\right]^{\beta_1-\gamma} \err{\gamma-\beta_1}{\beta_1}(t,y-x)
\leqslant c |x-x'|^{\beta_1-\gamma} \err{\gamma-\beta_1}{\beta_1}(t,y-x)\,. \end{align*} Now, \eqref{ineq:Lkp_abs-H} and \eqref{ineq:Lkp_abs-H-H} yield \begin{align*}
& |q_0(t,x,y)-q_0(t,x',y)|=\left|\int_{{\R^{d}}} \delta^{\mathfrak{K}_y} (t,x,y;z)(\kappa(x,z)-\kappa(y,z))\,J(z)dz\right.\\
& \hspace{0.1\linewidth} -\left. \int_{{\R^{d}}}\delta^{\mathfrak{K}_y}(t,x',y;z)(\kappa(x',z)-\kappa(y,z))\,J(z)dz\right|\\
& \hspace{0.05\linewidth} \leqslant \left| \int_{{\R^{d}}}\left( \delta^{\mathfrak{K}_y}(t,x,y;z)-\delta^{\mathfrak{K}_y}(t,x',y;z)\right) \left(\kappa(x,z)-\kappa(y,z)\right) J(z)dz\right| \\
& \hspace{0.1\linewidth} + \left| \int_{{\R^{d}}}\left( \delta^{\mathfrak{K}_y} (t,x',y;z)\right) \left(\kappa(x,z)-\kappa(x',z)\right) J(z)dz\right|\\
& \hspace{0.1\linewidth} + c \left(|x-x'|^{\beta_1}\land 1\right)\int_{{\R^{d}}}|\delta^{\mathfrak{K}_y}(t,x',y;z)|\,J(z)dz\\
\leqslant c &\left(|x-y|^{\beta_1}\land 1\right)
\left(\frac{|x-x'|}{h^{-1}(1/t)} \land 1\right) \big(\err{0}{0} (t,x-y)+\err{0}{0}(t,x'-y)\big)
+ c \left(|x-x'|^{\beta_1}\land 1\right) \err{0}{0}(t,x'-y). \end{align*} Applying
$(|x-y|^{\beta_1}\land 1)\leqslant (|x-x'|^{\beta_1}\land 1) + (|x'-y|^{\beta_1}\land 1)$ we obtain \begin{align*}
|q_0(t,x,y)-q_0(t,x',y)|\leqslant \ &c \left(\frac{|x-x'|}{h^{-1}(1/t)} \land 1\right) \big(\err{0}{\beta_1} (t,x-y)+\err{0}{\beta_1}(t,x'-y)\big)\\
&+c \left(|x-x'|^{\beta_1}\land 1\right)\err{0}{0}(t,x'-y). \end{align*}
Thus, in the last case $|x-x'|\leqslant h^{-1}(1/t)\land 1$ we have
$|x-x'|/ h^{-1}(1/t)\leqslant |x-x'|^{\beta_1-\gamma} \left[h^{-1}(1/t)\right]^{\gamma-\beta_1}$
and $|x-x'|^{\beta_1}\leqslant |x-x'|^{\beta_1 -\gamma} \left[h^{-1}(1/t)\right]^{\gamma}$.\\
(iii) We treat the cases $|y-y'|\geqslant 1$ and $1\geqslant |y-y'|\geqslant h^{-1}(1/t)$ like in part (ii). Note that by $\delta^{\mathfrak{K}}(t,x,y;z)=\delta^{\mathfrak{K}}(t,-y,-x;z)$, \eqref{ineq:Lkp_abs-H}, \eqref{ineq:Lkp_abs-H-H} and Proposition~\ref{prop:Hcont_kappa_crit1}, \begin{align*}
&|q_0(t,x,y)-q_0(t,x,y')|\\
&\leqslant \left| \int_{{\R^{d}}} \delta^{\mathfrak{K}_y}(t,x,y;z)\left(\kappa(y',z)-\kappa(y,z)\right)J(z)dz\right| \\
& \ \ \ +\left| \int_{{\R^{d}}}\left(\delta^{\mathfrak{K}_y}(t,x,y;z)-\delta^{\mathfrak{K}_y}(t,x,y';z)\right)\left(\kappa(x,z)-\kappa(y',z)\right)J(z)dz \right|\\
&\ \ \ +\left|\int_{{\R^{d}}}\left(\delta^{\mathfrak{K}_y}(t,x,y';z)-\delta^{\mathfrak{K}_{y'}}(t,x,y';z)\right)\kappa(x,z) J(z)dz\right|\\
&\ \ \ +\left|-\int_{{\R^{d}}}\left(\delta^{\mathfrak{K}_y}(t,x,y';z)-\delta^{\mathfrak{K}_{y'}}(t,x,y';z)\right)\kappa(y',z) J(z)dz \right|\\
&\leqslant c \left( |y-y'|^{\beta_1}\land 1\right) \err{0}{0}(t,x-y)\\
&\quad +c \left( |x-y'|^{\beta_1}\land 1\right) \left(\frac{|y-y'|}{h^{-1}(1/t)} \land 1\right) \left(\err{0}{0}(t,x-y)+\err{0}{0}(t,x-y')\right)\\
&\quad + c \left( |y-y'|^{\beta_1}\land 1\right) \err{0}{0}(t,x-y') \,. \end{align*} Applying
$(|x-y'|^{\beta_1}\land 1)\leqslant (|x-y|^{\beta_1}\land 1) + (|y-y'|^{\beta_1}\land 1)$ we obtain \begin{align*}
|q_0(t,x,y)-q_0(t,x,y')|\leqslant \ & c
\left(\frac{|y-y'|}{h^{-1}(1/t)} \land 1\right) \big(\err{0}{\beta_1} (t,x-y)+\err{0}{\beta_1}(t,x-y')\big)\\
&+c \left(|y-y'|^{\beta_1}\land 1\right) \big( \err{0}{0}(t,x-y)+\err{0}{0}(t,x-y')\big). \end{align*}
This proves \eqref{e:estimate-q0-2-crit1} in the case $|y-y'|\leqslant h^{-1}(1/t)\land 1$. {
$\Box$
}
We thus estimated $q_0$. The estimates are of the same form as in \cite[Lemma~3.6]{GS-2018}. Using Lemma~\ref{l:convolution} they propagate to functions $q_n$ and $q$ defined in \eqref{e:qn-definition} and \eqref{def:q}, respectively, see the proof of Theorem~\ref{t:definition-of-q-crit1}. We also note that $\beta_1$ is used in Lemma~\ref{l:estimates-q0-crit1} merely for technical convenience, but becomes relevant when estimating $q_n$ and $q$.
We stress that among others,
the inequality \eqref{e:difference-q-estimate-crit1} plays a special role and is often used to improve the integrability or bounds of singular functions, sometimes
along with cancellations like those proved in Lemma~\ref{e:some-estimates-2bb-crit1} or Lemma~\ref{l:some-estimates-3b-crit1-impr}, see comments ahead of Lemma~\ref{lem:int_grad_phi} and Lemma~\ref{lem:some-est_gen_phi_xy-crit1}.
\begin{theorem}\label{t:definition-of-q-crit1} Assume $\Qzero$. The series in \eqref{def:q} is locally uniformly absolutely convergent on $(0, \infty)\times \mathbb{R}^d \times \mathbb{R}^d$ and solves the integral equation \begin{align}\label{e:integral-equation-crit1} q(t,x,y)=q_0(t,x,y)+\int_0^t \int_{{\R^{d}}}q_0(t-s,x,z)q(s,z,y)\, dzds\, . \end{align} Moreover, for every $T> 0$ and $\beta_1\in (0,\beta]\cap (0,\alpha_h)$ there is a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4, \beta_1)$
such that on $(0,T]\times{\R^{d}}\times{\R^{d}}$, \begin{align}\label{e:q-estimate-crit1}
|q(t,x,y)|\leqslant c \big(\err{0}{\beta_1}+\err{\beta_1}{0}\big)(t,x-y)\,, \end{align} and for any $\gamma\in (0,\beta_1]$ there is $c=c(d,T,\sigma,\kappa_2,\kappa_4, \beta_1,\gamma)$ such that on $(0, T]\times {\R^{d}} \times {\R^{d}}$, \begin{align}
&|q(t,x,y)-q(t,x',y)|\nonumber\\
&\leqslant c \left(|x-x'|^{\beta_1-\gamma}\land 1\right) \left\{\big(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(t,x-y)+\big(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(t,x'-y)\right\}\,, \label{e:difference-q-estimate-crit1} \end{align} and \begin{align}
&|q(t,x,y)-q(t,x,y')|\nonumber\\
&\leqslant c \left(|y-y'|^{\beta_1-\gamma}\land 1\right) \left\{\big(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(t,x-y)+\big(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(t,x-y')\right\}\,. \label{e:difference-q-estimate_1-crit1} \end{align} \end{theorem} \noindent{\bf Proof.} The proof follows from Lemmas~\ref{l:estimates-q0-crit1} and~\ref{l:convolution} -- it is the same as for \cite[Theorem~3.7]{GS-2018}.
{
$\Box$
}
\subsection{Properties of $\phi_y(t,x,s)$ and $\phi_y(t,x)$}\label{sec:phi}
We shall prove estimates for the integral part of~ \eqref{e:p-kappa}. We use the notation introduced in \eqref{e:phi-y-def} and \eqref{e:def-phi-y-2}.
\begin{lemma}\label{lem:phi_cont_xy-crit1} Assume $\Qzero$. Let $\beta_1\in (0,\beta]\cap (0,\alpha_h)$. For every $T>0$ there exists a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4, \beta_1)$ such that for all $t\in (0,T]$, $x,y\in{\R^{d}}$, \begin{align*}
|\phi_y(t,x)|\leqslant c t \big(\err{0}{\beta_1}+\err{\beta_1}{0}\big)(t,x-y)\,. \end{align*} For any $T>0$ and $\gamma \in [0,1]\cap [0,\alpha_h)$ there exists a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4, \beta_1,\gamma)$
such that for all $t\in (0,T]$, $x,x',y\in {\R^{d}}$, \begin{align*}
|\phi_{y}(t,x)-\phi_{y}(t,x')|&\leqslant c (|x-x'|^{\gamma}\land 1) \, t \left\{ \big( \err{\beta_1-\gamma}{0}+\err{-\gamma}{\beta_1}\big)(t,x-y)+ \big( \err{\beta_1-\gamma}{0}+\err{-\gamma}{\beta_1}\big)(t,x'-y) \right\}. \end{align*} For any $T>0$ and $\gamma \in (0,\beta)$ there exists a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4, \beta_1,\gamma)$
such that for all $t\in (0,T]$, $x,y,y'\in {\R^{d}}$, \begin{align*}
|\phi_{y}(t,x)-\phi_{y'}(t,x)|&\leqslant c (|y-y'|^{\beta_1-\gamma}\land 1)\, t \left\{ \big( \err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(t,x-y)+ \big( \err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(t,x-y') \right\}. \end{align*} \end{lemma} \noindent{\bf Proof.} The proof follows from Lemma~\ref{lem:pkw_holder}, Proposition~\ref{prop:gen_est_crit}, Theorem~\ref{t:definition-of-q-crit1} and Lemma~\ref{l:convolution}, and is the same as in \cite[Lemma~3.8]{GS-2018}. {
$\Box$
}
\begin{lemma}\label{lem:phi_cont_joint-crit1} Assume $\Qzero$. The function $\phi_y(t,x)$ is jointly continuous in $(t,x,y)\in (0,\infty)\times {\R^{d}} \times {\R^{d}}$. \end{lemma} \noindent{\bf Proof.} The idea of the proof is the same as that of \cite[Lemma~3.9]{GS-2018} and relies on Proposition~\ref{prop:gen_est_crit}, \eqref{e:q-estimate-crit1}, \cite[(94)]{GS-2018}, Lemma~\ref{l:convolution} and~ \ref{lem:cont_frcoef}, \cite[Lemma~5.6 and~5.15]{GS-2018}. {
$\Box$
}
From this moment on, the major effort is to obtain sufficient regularity of the integral part of \eqref{e:p-kappa}. Recall that we need $\beta_1<\alpha_h$ in order to apply Lemma~\ref{l:convolution}. The additional condition $1<\beta_1+\alpha_h$, known in certain contexts as the balance condition, which shall appear below in our assumptions, makes it possible to differentiate \eqref{e:def-phi-y-2}, that is, to calculate and estimate its gradient, see Lemma~\ref{l:gradient-phi-y-crit1}. We need such a result if we want to apply either the strong or the weak operator \eqref{e:intro-operator-a1-crit1} to the candidate of a solution defined by \eqref{e:p-kappa}.
\begin{lemma}\label{lem:phi_pomoc-crit1} Assume $\Qzero$. For all $0<s<t$, $x,y\in{\R^{d}}$, \begin{align}\label{eq:grad_phi_pomoc-crit1} \nabla_x \phi_y(t,x,s)=\int_{{\R^{d}}} \nabla_x p^{\mathfrak{K}_z}(t-s,x,z)q(s,z,y)\, dz\,, \end{align} \begin{align}\label{e:L-on-phi-y2-crit1} {\mathcal L}_x^{\mathfrak{K}_x}\phi_y(t,x,s) =\int_{{\R^{d}}} {\mathcal L}_x^{\mathfrak{K}_x}p^{\mathfrak{K}_z}(t-s,x,z)q(s,z,y)\, dz\,. \end{align} \end{lemma} \noindent{\bf Proof.} We get \eqref{eq:grad_phi_pomoc-crit1} by \eqref{ineq:est_diff_1}, \eqref{e:q-estimate-crit1}, Lemma~\ref{l:convolution}, and the dominated convergence theorem. Now, by \eqref{e:phi-y-def} and \eqref{eq:grad_phi_pomoc-crit1}, \begin{align} {\mathcal L}_x^{\mathfrak{K}_x}\phi_y(t,x,s) =\int_{{\R^{d}}} \left(\int_{{\R^{d}}} \delta^{\mathfrak{K}_z} (t-s, x,z;w) q(s,z, y) \,dz\right) \kappa(x,w)J(w) dw\,. \label{e:L-on-phi-y2-first-crit1} \end{align} Finally, we use Fubini's theorem justified by \eqref{ineq:aux_Q0}, \eqref{e:q-estimate-crit1} and Lemma~\ref{l:convolution}(b). {
$\Box$
}
In the proof of the next result we recognize a typical {\it modus operandi} when dealing with integrals of functions that at first glance
seem to be too singular: we add and subtract $q(s,x,y)$, use its regularity
\eqref{e:difference-q-estimate-crit1} (which reduces part of the singularity) and profit from cancellations, this time from Lemma~\ref{e:some-estimates-2bb-crit1}.
\begin{lemma}\label{lem:int_grad_phi} Assume $\Qzero$ and $1-\alpha_h <\beta\land \alpha_h$. Let $\beta_1\in (0,\beta]\cap (0,\alpha_h)$. For every $T>0$
there exists a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4, \beta_1)$ such that for all $t\in (0,T]$, $x\in{\R^{d}}$, \begin{align*}
\int_{{\R^{d}}} \int_0^t \left| \nabla_x \phi_y(t,x,s) \right| ds \, dy\leqslant c \left[h^{-1}(1/t)\right]^{-1+\beta_1}\,. \end{align*} \end{lemma} \noindent{\bf Proof.} By the monotonicity of $h^{-1}$, if we prove the statement for some value of $\beta_1$, then it also holds for smaller values. We assume that $1-\alpha_h<\beta_1$ and we let
$\gamma \in (0,\beta_1)$ satisfying $1-\alpha_h<\beta_1-\gamma$. By \eqref{eq:grad_phi_pomoc-crit1}, Proposition~\ref{prop:gen_est_crit}, \eqref{e:difference-q-estimate-crit1}, Lemma~\ref{e:some-estimates-2bb-crit1} and \eqref{e:q-estimate-crit1}, \begin{align*}
\left| \nabla_x \phi_y(t,x,s) \right|
&\leqslant \int_{{\R^{d}}} \left| \nabla_x p^{\mathfrak{K}_z}(t-s,x,z) \right| \left| q(s,z,y) -q(s,x,y)\right| dz\\
&\quad + \left| \int_{{\R^{d}}} \nabla_x p^{\mathfrak{K}_z}(t-s,x,z)\, dz \right| \left| q(s,x,y)\right|\\ &\leqslant \int_{{\R^{d}}} (t-s)\err{-1}{\beta_1-\gamma}(t-s, x-z) \big(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(s,z-y)\,dz \\ &\quad +\int_{{\R^{d}}} (t-s)\err{-1}{\beta_1-\gamma}(t-s, x-z) \,dz\, \big(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(s,x-y) \\ &\quad+ \left[ h^{-1}(1/(t-s))\right]^{-1+\beta_1} \big(\err{0}{\beta_1}+\err{\beta_1}{0}\big)(s,x-y)\,. \end{align*} Finally, we integrate in $y$ over ${\R^{d}}$ using Lemma~\ref{l:convolution}(a) and then in $s$ over $(0,t)$ using \cite[Lemma~5.15]{GS-2018}. Note that in the last step we integrate $[h^{-1}(1/(t-s))]^{-1+\beta_1-\gamma}$, which requires a condition $(-1+\beta_1-\gamma)/\alpha_h +1 >0$, equivalently $\alpha_h+\beta_1>1+\gamma$, and is fulfilled thanks to our assumptions. {
$\Box$
}
\begin{lemma}\label{l:gradient-phi-y-crit1} Assume $\Qzero$ and $1-\alpha_h <\beta\land \alpha_h$. For every $T>0$ there exists a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4,\beta)$
such that for all $t \in(0,T]$, $x,y\in{\R^{d}}$, \begin{align}\label{e:gradient-phi-y-crit1} &\nabla_x\phi_y(t,x)=\int_0^t \int_{{\R^{d}}} \nabla_x p^{\mathfrak{K}_z}(t-s,x,z) q(s,z,y)\, dzds\,,\\ \nonumber\\ \label{e:gradient-phi-y-estimate-crit1}
&\left|\nabla_x\phi_y(t,x) \right|\leqslant c \!\left[ h^{-1}(1/t)\right]^{-1} t \,\err{0}{0}(t,x-y)\,. \end{align} \end{lemma} \noindent{\bf Proof.} The proof is like in \cite[Lemma~3.10]{GS-2018} and rests on \eqref{eq:grad_phi_pomoc-crit1}, Proposition~\ref{prop:gen_est_crit}, \eqref{e:q-estimate-crit1}, Lemma~\ref{l:convolution}, \cite[(93), (94), Lemma~5.3 and~5.15, Proposition~5.8]{GS-2018}, \eqref{e:difference-q-estimate-crit1}, Lemma~\ref{e:some-estimates-2bb-crit1}, and the fact that $\alpha_h>1/2$. {
$\Box$
}
So far, in Lemma~\ref{l:gradient-phi-y-crit1} we managed to calculate and estimate the gradient of $\phi_y(t,x)$, which is the integral part of \eqref{e:p-kappa}. Now we shall treat in a similar fashion the operator \eqref{e:intro-operator-a1-crit1} acting on $\phi_y(t,x)$. The first step is to show that the operator can actually be applied (that the respective integrals converge) and to find a formula -- Lemma~\ref{e:L-on-phi-y-crit1}. The second step is to prove the estimates in Lemma~\ref{lem:I_0_oszagorne-crit1-impr}. To achieve that, for the first step, in Lemma~\ref{lem:some-est_gen_phi_xy-crit1} and~\ref{ineq:I_0_oszagorne-crit1}, we prove auxiliary bounds justifying the use of Fubini's theorem, however those technical results do not provide the desired estimates for the second step. The reason is that when using \eqref{e:difference-q-estimate-crit1}, due to the position of the absolute value, we cannot make use of additional cancellations and we merely rely on \eqref{e:fract-der-est1-crit} and Lemma~\ref{l:some-estimates-3b-crit1}, which causes extra growth. Therefore, contrary to \cite{GS-2018}, we are forced to distinguish between the two steps. An improvement of the estimates, taking cancellations into account, is given in Lemma~\ref{lem:some-est_gen_phi_xy-crit1-impr}, Corollary~\ref{cor:int_Lphi} and Lemma~\ref{lem:I_0_oszagorne-crit1-impr}.
In (a) below we address the critical case. In (b) we deal with the super-critical case.
\begin{lemma}\label{lem:some-est_gen_phi_xy-crit1} Assume $\Qzero$. Let $\beta_1\in (0,\beta]\cap (0,\alpha_h)$. For all $T>0$, $\gamma \in(0,\beta_1]$ the inequalities \begin{align}
\int_{{\R^{d}}}\left(\int_{{\R^{d}}} |\delta^{\mathfrak{K}_z} (t-s, x,z;w)||q(s,z, y)| \,dz\right) \kappa(x,w)J(w) dw\nonumber \hspace{0.15\linewidth}\\
\leqslant c_1\int_{{\R^{d}}} \vartheta(t-s)\err{0}{0}(t-s, x-z) \big(\err{0}{\beta_1}+\err{\beta_1}{0}\big)(s,z-y)\,dz\,, \label{e:Fubini1-crit1} \\ \nonumber \\
\int_{{\R^{d}}} \left| \int_{{\R^{d}}} \delta^{\mathfrak{K}_z}(t-s,x,z;w)q(s,z,y)\,dz \right| \kappa(x,w)J(w)dw \leqslant c_2 \big( {\rm I}_1+{\rm I}_2+{\rm I}_3 \big), \label{ineq:some-est_gen_phi_xy-crit1} \end{align} where \begin{align*} {\rm I}_1+{\rm I}_2+{\rm I}_3:= & \int_{{\R^{d}}} \vartheta(t-s) \err{0}{\beta_1-\gamma}(t-s,x-z) \big(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(s,z-y) \,dz \\ & + \vartheta(t-s) (t-s)^{-1} \left[h^{-1}(1/(t-s))\right]^{\beta_1-\gamma} \big(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(s,x-y) \\ & + \,\vartheta(t-s) (t-s)^{-1}\left[h^{-1}(1/(t-s))\right]^{\beta_1} \big(\err{0}{\beta_1}+\err{\beta_1}{0}\big)(s,x-y)\,, \end{align*} hold for all $0<s<t\leqslant T$, $x,y\in{\R^{d}}$ with \begin{enumerate} \item[(a)] $\vartheta(t)=\Theta(t)$ and $c_1=c_1(d,T,\sigma,\kappa_2,\kappa_4,\beta_1)$, $c_2=c_2(d,T,\sigma,\kappa_2,\kappa_4,\beta_1,\gamma)$ if $\alpha_h=1$, \item[(b)] $\vartheta(t)=t \,[h^{-1}(1/t)]^{-1}$ and $c_1=c_1(d,T,\sigma,\kappa_2,\kappa_4,\beta_1,\beta_h,c_h)$, $c_2=c_2(d,T,\sigma,\kappa_2,\kappa_4,\beta_1,\gamma,\beta_h,c_h)$ if \eqref{eq:intro:wusc} holds for $0<\alpha_h \leqslant \beta_h<1$. \end{enumerate} \end{lemma} \noindent{\bf Proof.} The inequality \eqref{e:Fubini1-crit1} follows from \eqref{e:intro-kappa}, \eqref{e:fract-der-est1-crit} and \eqref{e:q-estimate-crit1}. Next, let ${\rm I}_0$ be the left hand side of \eqref{ineq:some-est_gen_phi_xy-crit1}. By \eqref{e:difference-q-estimate-crit1}, \eqref{e:q-estimate-crit1}, \eqref{e:intro-kappa}, \eqref{e:fract-der-est1-crit}, Lemma~\ref{l:some-estimates-3b-crit1} and~\ref{l:convolution}(a), \begin{align*}
{\rm I}_0&\leqslant \int_{{\R^{d}}} \int_{{\R^{d}}} |\delta^{\mathfrak{K}_z}(t-s,x,z;w)| |q(s,z,y)-q(s,x,y)|\,dz\, \kappa(x,w)J(w)dw\\
&\quad + \int_{{\R^{d}}} \left| \int_{{\R^{d}}} \delta^{\mathfrak{K}_z}(t-s,x,z;w) \,dz\right| \kappa(x,w)J(w)dw \, |q(s,x,y)|\\
&\leqslant c \int_{{\R^{d}}} \left( \int_{{\R^{d}}} |\delta^{\mathfrak{K}_z}(t-s,x,z;w)|\,J(w)dw \right) \left(|x-z|^{\beta_1-\gamma}\land 1\right) \big(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(s,z-y) \,dz \\
&\quad + c \int_{{\R^{d}}} \left( \int_{{\R^{d}}} |\delta^{\mathfrak{K}_z}(t-s,x,z;w)|\,J(w)dw \right) \left(|x-z|^{\beta_1-\gamma}\land 1\right) dz\, \big(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(s,x-y) \\ &\quad + c \, \vartheta(t-s)\, (t-s)^{-1}\left[h^{-1}(1/(t-s))\right]^{\beta_1} \big(\err{0}{\beta_1}+\err{\beta_1}{0}\big)(s,x-y) \leqslant c ({\rm I}_1+{\rm I}_2+{\rm I}_3)\,. \end{align*} {
$\Box$
}
The inequality \eqref{ineq:some-est_gen_phi_xy-crit1} looks a bit rough, but it is left in such form on purpose: it is used not only to prove Lemma~\ref{ineq:I_0_oszagorne-crit1} and Lemma~\ref{lem:some-est_p_kappa-crit1}, but also to
shorten the argument in the proof of Lemma~\ref{lem:I_0_oszagorne-crit1-impr}.
\begin{lemma} \label{ineq:I_0_oszagorne-crit1} Assume $\Qzero$ and $1-\alpha_h <\beta\land \alpha_h$. For any $\beta_1\in (0,\beta]$ such that $1-\alpha_h<\beta_1<\alpha_h$ and $0<\gamma_1\leqslant \gamma_2\leqslant \beta_1$ satisfying $$ 1-\alpha_h<\beta_1-\gamma_1\,,\qquad\qquad 2\beta_1-\gamma_2<\alpha_h\,, $$ the inequality \begin{align*}
\int_{{\R^{d}}} \int_0^t &\left| \int_{{\R^{d}}} \delta^{\mathfrak{K}_z}(t-s,x,z;w)q(s,z,y)\,dz \right| ds\,\kappa(x,w)J(w)dw\nonumber \\ &\hspace{0.3\linewidth}\leqslant c \,\vartheta(t) \big(\err{0}{\beta_1}+\err{\gamma_1}{\beta_1-\gamma_1}+\err{\beta_1+\gamma_1-\gamma_2}{0}\big)(t,x-y)\,, \end{align*} holds for all $t\in(0,T]$, $x,y\in{\R^{d}}$ with \begin{enumerate} \item[(a)] $\vartheta(t)=\Theta(t)$ and $c=c(d,T,\sigma,\kappa_2,\kappa_4,\beta_1,\gamma_1,\gamma_2)$ if $\alpha_h=1$, \item[(b)] $\vartheta(t)=t \,[h^{-1}(1/t)]^{-1}$ and $c=c(d,T,\sigma,\kappa_2,\kappa_4,\beta_1,\gamma_1,\gamma_2,\beta_h,c_h)$ if \eqref{eq:intro:wusc} holds for $0<\alpha_h \leqslant \beta_h<1$. \end{enumerate} \end{lemma} \noindent{\bf Proof.} Let ${\rm I}_0$ be the left hand side of \eqref{ineq:some-est_gen_phi_xy-crit1}. In the two cases discussed below, we apply Lemma~\ref{l:convolution}(b), the monotonicity of $h^{-1}$ and $\Theta$ (see also Lemma~\ref{lem:cal_TCh}), and $\Ab$ of \cite[Lemma~5.3]{GS-2018}. For $s\in (0,t/2]$ we use \eqref{e:Fubini1-crit1} to get \begin{align*} {\rm I}_0& \leqslant c\, \vartheta(t-s)\bigg\{\left( (t-s)^{-1}\left[h^{-1}(1/(t-s))\right]^{\beta_1}+(t-s)^{-1}\left[h^{-1}(1/s)\right]^{\beta_1} +s^{-1}\left[h^{-1}(1/s)\right]^{\beta_1} \right) \\ &\hspace{0.52\linewidth} \times \,\err{0}{0}(t,x-y) + (t-s)^{-1}\err{0}{\beta_1}(t,x-y) \bigg\}\\ &\leqslant c\,\vartheta(t) \bigg\{\left( t^{-1}\left[h^{-1}(1/t)\right]^{\beta_1} +s^{-1} \left[h^{-1}(1/s)\right]^{\beta_1} \right) \err{0}{0}(t,x-y) + t^{-1}\err{0}{\beta_1}(t,x-y) \bigg\}. \end{align*} For $s\in (t/2,t)$ we use \eqref{ineq:some-est_gen_phi_xy-crit1} with $\gamma=\gamma_1$. While estimating the expression \begin{align*} \vartheta(t-s) \int_{{\R^{d}}} \err{0}{\beta_1-\gamma}(t-s,x-z) \err{\gamma-\beta_1}{\beta_1}(s,z-y) \,dz\,, \end{align*} we use Lemma~\ref{l:convolution}(b) with $n_1=n_2=2\beta_1-\gamma_2$, $m_1=\beta_1-\gamma_1$, $m_2=\beta_1$ and later Lemma~\ref{lem:cal_TCh}, so our assumptions concerning the choice of $\gamma_1$, $\gamma_2$ are used. More precisely, we have \begin{align*} {\rm I}_1 & \leqslant c \vartheta(t-s)\bigg\{ (t-s)^{-1} \left[h^{-1}(1/(t-s))\right]^{\beta_1-\gamma_1} \left[h^{-1}(1/s)\right]^{\gamma_1} + s^{-1} \left[h^{-1}(1/s)\right]^{\beta_1+\gamma_1-\gamma_2} \\ &\hspace{0.32\linewidth}+ (t-s)^{-1} \left[h^{-1}(1/(t-s))\right]^{2\beta_1-\gamma_2} \left[h^{-1}(1/s)\right]^{\gamma_1-\beta_1} \bigg\} \err{0}{0}(t,x-y)\\ &\quad + c \vartheta(t-s) (t-s)^{-1} \left[h^{-1}(1/(t-s))\right]^{\beta_1-\gamma_1} \left[h^{-1}(1/s)\right]^{\gamma_1-\beta_1} \err{0}{\beta_1}(t,x-y)\\ &\quad +c \vartheta(t-s) s^{-1} \left[h^{-1}(1/s)\right]^{\gamma_1} \err{0}{\beta_1-\gamma_1}(t,x-y) \\ & \leqslant c \vartheta(t-s) \bigg\{ (t-s)^{-1} \left[h^{-1}(1/(t-s))\right]^{\beta_1-\gamma_1} \left[h^{-1}(1/t)\right]^{\gamma_1} + t^{-1} \left[h^{-1}(1/t)\right]^{\beta_1+\gamma_1-\gamma_2}\\ &\hspace{0.32\linewidth}+(t-s)^{-1} \left[h^{-1}(1/(t-s))\right]^{2\beta_1-\gamma_2} \left[h^{-1}(1/t)\right]^{\gamma_1-\beta_1} \bigg\} \err{0}{0}(t,x-y)\\ &\quad + c \vartheta(t-s) (t-s)^{-1} \left[h^{-1}(1/(t-s))\right]^{\beta_1-\gamma_1} \left[h^{-1}(1/t)\right]^{\gamma_1-\beta_1} \err{0}{\beta_1}(t,x-y)\\ &\quad +c \vartheta(t-s) t^{-1} \left[h^{-1}(1/t)\right]^{\gamma_1} \err{0}{\beta_1-\gamma_1}(t,x-y)\,. \end{align*} Next, like above with \cite[(94)]{GS-2018}, \begin{align*} {\rm I}_2 & \leqslant c \, \vartheta(t-s) (t-s)^{-1} \left[h^{-1}(1/(t-s))\right]^{\beta_1-\gamma_1} \big(\err{\gamma_1}{0}+\err{\gamma_1-\beta_1}{\beta_1}\big)(t,x-y)\,. \end{align*} Similarly, ${\rm I}_3\leqslant c \,\vartheta(t-s) (t-s)^{-1} \left[h^{-1}(1/(t-s))\right]^{\beta_1} \big(\err{0}{\beta_1}+\err{\beta_1}{0}\big)(t,x-y)$. Finally, by \cite[Lemma~5.15]{GS-2018} and Lemma~\ref{lem:cal_TCh}, and a fact that $\alpha_h>1/2$, \begin{align*} \int_0^t {\rm I}_0\,ds \leqslant c \,\vartheta(t) \big(\err{0}{\beta_1}+\err{\gamma_1}{\beta_1-\gamma_1}+\err{\beta_1+\gamma_1-\gamma_2}{0}\big)(t,x-y)\,. \end{align*} {
$\Box$
}
We can now successfully apply \eqref{e:intro-operator-a1-crit1} to \eqref{e:def-phi-y-2}.
\begin{lemma}\label{e:L-on-phi-y-crit1} Assume $\Qa$ or $\Qb$. We have for all $t >0$, $x,y\in{\R^{d}}$, \begin{equation*} {\mathcal L}_x^{\mathfrak{K}_x} \phi_y(t,x)= \int_0^t \int_{{\R^{d}}} {\mathcal L}_x^{\mathfrak{K}_x} p^{\mathfrak{K}_z}(t-s,x,z) q(s,z,y)\, dzds\,. \end{equation*} \end{lemma} \noindent{\bf Proof.} By \eqref{e:def-phi-y-2} and \eqref{e:gradient-phi-y-crit1} in the first equality, and Lemma~\ref{ineq:I_0_oszagorne-crit1} and \eqref{e:Fubini1-crit1} in the second (allowing us to change the order of integration twice) the proof is as follows \begin{align*} {\mathcal L}_x^{\mathfrak{K}_x} \phi_y(t,x) &=\int_{{\R^{d}}} \left( \int_0^t \int_{{\R^{d}}} \delta^{\mathfrak{K}_z}(t-s,x,z;w)q(s,z,y)\,dzds\right) \kappa(x,w)J(w)dw\\ &= \int_0^t \int_{{\R^{d}}}\left( \int_{{\R^{d}}} \delta^{\mathfrak{K}_z}(t-s,x,z;w)\, \kappa(x,w)J(w)dw\right) q(s,z,y)\,dzds\,. \end{align*} {
$\Box$
}
We improve the estimates.
\begin{lemma}\label{lem:some-est_gen_phi_xy-crit1-impr} Assume $\Qzero$. Let $\beta_1\in (0,\beta]\cap (0,\alpha_h)$. For all $T>0$, $\gamma \in(0,\beta_1]$ there exist constants $c_1=c_1(d,T,\sigma,\kappa_2,\kappa_4,\beta_1)$ and $c_2=c_2(d,T,\sigma,\kappa_2,\kappa_4,\beta_1,\gamma)$ such that for all $0<s<t\leqslant T$, $x,y\in{\R^{d}}$, \begin{align}
\left| {\mathcal L}_x^{\mathfrak{K}_x}\phi_y(t,x,s)\right| & \leqslant c_1\int_{{\R^{d}}} \err{0}{0}(t-s, x-z) \big(\err{0}{\beta_1}+\err{\beta_1}{0}\big)(s,z-y)\,dz\,, \label{e:Fubini1-crit1-impr} \\ \nonumber \\
\left| {\mathcal L}_x^{\mathfrak{K}_x}\phi_y(t,x,s)\right| &\leqslant c_2 \big( {\rm I}_1+{\rm I}_2+{\rm I}_3 \big), \label{ineq:some-est_gen_phi_xy-crit1-impr} \end{align} where \begin{align*} {\rm I}_1+{\rm I}_2+{\rm I}_3:= & \int_{{\R^{d}}} \err{0}{\beta_1-\gamma}(t-s,x-z) \big(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(s,z-y) \,dz \\ & + (t-s)^{-1} \left[h^{-1}(1/(t-s))\right]^{\beta_1-\gamma} \big(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(s,x-y) \\ & + \, (t-s)^{-1}\left[h^{-1}(1/(t-s))\right]^{\beta_1} \big(\err{0}{\beta_1}+\err{\beta_1}{0}\big)(s,x-y)\,. \end{align*} \end{lemma} \noindent{\bf Proof.} The first inequality follows from \eqref{ineq:Lkp_abs} and \eqref{e:q-estimate-crit1}. By \eqref{e:L-on-phi-y2-crit1}, \eqref{e:difference-q-estimate-crit1}, \eqref{e:q-estimate-crit1}, \eqref{ineq:Lkp_abs}, Lemma~\ref{l:some-estimates-3b-crit1-impr} and~\ref{l:convolution}(a), \begin{align*}
\left| {\mathcal L}_x^{\mathfrak{K}_x}\phi_y(t,x,s)\right|&\leqslant \int_{{\R^{d}}}
\left| {\mathcal L}_x^{\mathfrak{K}_x}p^{\mathfrak{K}_z}(t-s,x,z)\right|
|q(s,z,y)-q(s,x,y)|\,dz \\
&\quad + \left| \int_{{\R^{d}}}{\mathcal L}_x^{\mathfrak{K}_x}p^{\mathfrak{K}_z}(t-s,x,z) \,dz\right| \, |q(s,x,y)|\\
&\leqslant c \int_{{\R^{d}}} \err{0}{0}(t-s,x-z) \left(|x-z|^{\beta_1-\gamma}\land 1\right) \big(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(s,z-y) \,dz \\
&\quad + c \int_{{\R^{d}}} \err{0}{0}(t-s,x-z) \left(|x-z|^{\beta_1-\gamma}\land 1\right) dz\, \big(\err{\gamma}{0}+\err{\gamma-\beta_1}{\beta_1}\big)(s,x-y) \\ &\quad + c \, (t-s)^{-1}\left[h^{-1}(1/(t-s))\right]^{\beta_1} \big(\err{0}{\beta_1}+\err{\beta_1}{0}\big)(s,x-y) \leqslant c ({\rm I}_1+{\rm I}_2+{\rm I}_3)\,. \end{align*} {
$\Box$
}
Here is a consequence of \eqref{e:Fubini1-crit1-impr}, \eqref{ineq:some-est_gen_phi_xy-crit1-impr}, Lemma~\ref{l:convolution}(a) and \cite[Lemma~5.15]{GS-2018}. \begin{corollary}\label{cor:int_Lphi} Assume $\Qzero$ and $1-\alpha_h <\beta\land \alpha_h$. Let $\beta_1 \in (0,\beta]\cap (0,\alpha_h)$.
For every $T>0$ there exists a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4, \beta_1)$
such that for all $t\in(0,T]$, $x\in{\R^{d}}$, \begin{align*}
\int_{{\R^{d}}} \int_0^t \left| {\mathcal L}_x^{\mathfrak{K}_x}\phi_y(t,x,s) \right| ds \,dy \leqslant c t^{-1} \left[h^{-1}(1/t)\right]^{\beta_1} \,. \end{align*} \end{corollary}
\begin{lemma}\label{lem:I_0_oszagorne-crit1-impr} Assume $\Qzero$. Let $\beta_1\in (0,\beta]\cap (0,\alpha_h)$. For all $T>0$, $0<\gamma_1\leqslant \gamma_2\leqslant \beta_1$ satisfying $$ 0<\beta_1-\gamma_1\,,\quad \qquad 2\beta_1-\gamma_2<\alpha_h\,, $$ there exists a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4,\beta_1,\gamma_1,\gamma_2)$ such that for all $t\in(0,T]$, $x,y\in{\R^{d}}$, \begin{align}
\int_0^t \left| {\mathcal L}_x^{\mathfrak{K}_x}\phi_y(t,x,s)\right|ds \leqslant c \big(\err{0}{\beta_1}+\err{\gamma_1}{\beta_1-\gamma_1}+\err{\beta_1+\gamma_1-\gamma_2}{0}\big)(t,x-y)\,. \label{ineq:I_0_oszagorne-crit1-impr} \end{align} \end{lemma} \noindent{\bf Proof.} The proof goes by the same lines as the proof of Lemma~\ref{ineq:I_0_oszagorne-crit1} but with $\vartheta$ replaced by $1$, and Lemma~\ref{lem:some-est_gen_phi_xy-crit1-impr} in place of Lemma~\ref{lem:some-est_gen_phi_xy-crit1}. {
$\Box$
} \begin{lemma}\label{lem:Lphi_cont-crit} Assume $\Qa$ or $\Qb$. The function ${\mathcal L}_x^{\mathfrak{K}_x} \phi_y(t,x)$ is jointly continuous in $(t,x,y)\in (0,\infty)\times {\R^{d}}\times {\R^{d}}$. \end{lemma} \noindent{\bf Proof.} The proof is the same as in \cite[Lemma~3.13]{GS-2018} and requires Lemma~\ref{e:L-on-phi-y-crit1}, \eqref{e:L-on-phi-y2-crit1}, \eqref{ineq:some-est_gen_phi_xy-crit1-impr}, \eqref{ineq:Lkp_abs}, \eqref{e:difference-q-estimate_1-crit1}, \eqref{e:q-estimate-crit1}, \cite[(94), Lemma~5.15]{GS-2018}, Lemmas~\ref{l:convolution} and~\ref{lem:cont_frcoef}. {
$\Box$
}
In the final results of that section, we prepare to calculate the time derivative of~\eqref{e:p-kappa}.
\begin{proposition}\label{l:phi-y-abs-cont-crit1} Assume $\Qzero$. For all $t>0$, $x,y\in {\R^{d}}$, $x\neq y$, we have \begin{align*} \phi_y(t,x) =\int_0^t \left(q(r,x,y)+ \int_0^r \int_{{\R^{d}}} {\mathcal L}_x^{\mathfrak{K}_z} p^{\mathfrak{K}_z}(r-s,x,z) q(s,z,y)\, dzds\right) dr\,. \end{align*} \end{proposition} \noindent{\bf Proof.} The idea of the proof is that
differentiating \eqref{e:def-phi-y-2} in $t>0$, we expect to get \begin{align*} \partial_t \phi_y(t,x) = q(t,x,y)+
\int_0^t \int_{{\R^{d}}}\partial_t \, p^{\mathfrak{K}_z}(t-s,x,z) q(s,z,y)\, dzds\,. \end{align*} Actually, we intend to prove the following integral counterpart, \begin{align*} \phi_y(t,x) =\int_0^t \left(q(r,x,y)+ \int_0^r \int_{{\R^{d}}} \partial_r\, p^{\mathfrak{K}_z}(r-s,x,z) q(s,z,y)\, dzds\right) dr\,, \end{align*} see \eqref{eq:p_gen_klas}. Therefore, the aim is to justify \begin{align*} \int_0^t & \int_0^r \int_{{\R^{d}}} \partial_r\, p^{\mathfrak{K}_z}(r-s,x,z) q(s,z,y)\, dzds\,dr= \int_0^t \int_0^r \partial_r \phi_y(r ,x,s)\, dsdr\\ &= \int_0^t \int_s^t \partial_r \phi_y(r ,x,s)\, drds = \int_0^t \left( \phi_y(t ,x,s)- \lim_{\varepsilon \to 0^+} \phi_y(s+\varepsilon ,x,s) \right)ds\\ &= \phi_y(t,x) - \int_0^t \lim_{\varepsilon \to 0^+} \phi_y(s+\varepsilon ,x,s)ds\,, \end{align*} and prove that $\lim_{\varepsilon \to 0^+} \phi_y(s+\varepsilon ,x,s) = q(s,x,y)$. Details are like in the proof of \cite[Lemma~3.14]{GS-2018}: we use \eqref{e:phi-y-def}, \eqref{eq:p_gen_klas}, \eqref{ineq:Lkp_abs}, \eqref{e:q-estimate-crit1}, \eqref{e:L-on-phi-y2-crit1}, \eqref{ineq:I_0_oszagorne-crit1-impr}, \eqref{e:q0-estimate-crit1}, Lemma~\ref{l:convolution}, \cite[(92), (93), (94)]{GS-2018}, \eqref{e:some-estimates-2c-crit1}, \eqref{e:estimate-step3-crit1}, Proposition~\ref{prop:gen_est_crit}, \cite[Lemma~5.6]{GS-2018}, \cite[Theorem~7.21]{MR924157}. {
$\Box$
}
If we combine \eqref{e:integral-equation-crit1} and Lemma~\ref{e:L-on-phi-y-crit1}, then we can represent the integrand in the statement of Proposition~\ref{l:phi-y-abs-cont-crit1} as $q_0(s,x,y)+ {\mathcal L}_x^{\mathfrak{K}_x} \phi_y (s,x)$. Hence we conclude what follows.
\begin{corollary}\label{e:phi-y-partial_1-crit} Assume $\Qa$ or $\Qb$. For all $x,y\in{\R^{d}}$, $x\neq y$, the function $\phi_y(t,x)$ is differentiable in $t>0$ and \begin{align*}
\partial_t \phi_y(t,x) = q_0(t,x,y)+ {\mathcal L}_x^{\mathfrak{K}_x} \phi_y (t,x)\,. \end{align*} \end{corollary}
\subsection{Properties of $p^\kappa(t, x, y)$}\label{sec:p_kappa}
We collect what can already be said about $p^\kappa(t, x, y)$.
\begin{lemma}\label{lem:some-est_p_kappa-crit1} Assume $\Qzero$ and $1-\alpha_h <\beta\land \alpha_h$. Let $\beta_1\in (0,\beta]\cap (0,\alpha_h)$. For every $T>0$ the inequalities \begin{align}
\int_{{\R^{d}}} | \delta^{\kappa}(t,x,y;z) | \,\kappa(x,z)J(z)dz &\leqslant c_1\, \vartheta(t) \err{0}{0}(t,x-y)\,, \label{ineq:some-est_p_kappa-crit1} \\
\int_{{\R^{d}}} \left|\int_{{\R^{d}}} \delta^{\kappa}(t,x,y;z)\, dy \right|\kappa(x,z)&J(z)dz \leqslant c_2\, \vartheta(t) t^{-1} \left[h^{-1}(1/t)\right]^{\beta_1}\,, \label{ineq:some-est_p_kappa_1-crit1} \end{align} hold for all $t\in(0,T]$, $x,y\in{\R^{d}}$ with \begin{enumerate} \item[(a)] $\vartheta(t)=\Theta(t)$ and $c_1=c_1(d,T,\sigma,\kappa_2,\kappa_4,\beta)$, $c_2=c_2(d,T,\sigma,\kappa_2,\kappa_4,\beta_1)$ if $\alpha_h=1$, \item[(b)] $\vartheta(t)=t \,[h^{-1}(1/t)]^{-1}$ and $c_1=c_1(d,T,\sigma,\kappa_2,\kappa_4,\beta,\beta_h,c_h)$, $c_2=c_2(d,T,\sigma,\kappa_2,\kappa_4,\beta_1,\beta_h,c_h)$ if \eqref{eq:intro:wusc} holds for $0<\alpha_h \leqslant \beta_h<1$. \end{enumerate} \end{lemma} \noindent{\bf Proof.} By \eqref{e:p-kappa} and \eqref{e:gradient-phi-y-crit1}, \begin{align*} \delta^{\kappa}(t,x,y;w)=\delta^{\mathfrak{K}_y}(t,x,y;w)+\int_0^t\int_{{\R^{d}}} \delta^{\mathfrak{K}_z}(t-s,x,z;w)q(s,z,y)\,dzds\,. \end{align*} We deduce \eqref{ineq:some-est_p_kappa-crit1} from \eqref{e:fract-der-est1-crit}, Lemma~\ref{ineq:I_0_oszagorne-crit1}, \cite[(92), (93)]{GS-2018}. The inequality \eqref{ineq:some-est_p_kappa_1-crit1} results from \eqref{ineq:some-est_gen_phi_xy-crit1}, \cite[Lemma~5.15]{GS-2018}, Lemma~\ref{l:some-estimates-3b-crit1}, \ref{l:convolution}(a) and~\ref{lem:cal_TCh}. {
$\Box$
}
\begin{lemma}\label{e:fract-der-p-kappa-2b-crit1} Assume $\Qzero$ and $1-\alpha_h <\beta\land \alpha_h$. Let $\beta_1\in (0,\beta]\cap (0,\alpha_h)$.
For every $T>0$ there exists a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4,\beta_1)$ such that for all $t\in (0,T]$, $x\in{\R^{d}}$, \begin{equation*}
\left| \int_{{\R^{d}}}\nabla_x p^{\kappa}(t,x,y)\,dy\right|\leqslant c \left[h^{-1}(1/t)\right]^{-1+\beta_1} \,, \end{equation*} \end{lemma} \noindent{\bf Proof.} We get the inequality from Lemma~\ref{e:some-estimates-2bb-crit1}, \eqref{e:gradient-phi-y-crit1}, \eqref{eq:grad_phi_pomoc-crit1} and
Lemma~\ref{lem:int_grad_phi}. {
$\Box$
}
\begin{lemma}\label{l:p-kappa-difference-crit-1} Assume $\Qa$ or $\Qb$.\\ \noindent (a) The function $p^{\kappa}(t,x,y)$ is jointly continuous on $(0, \infty)\times {\R^{d}} \times {\R^{d}}$.
\noindent (b) For every $T> 0$ there is a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4, \beta)$
such that for all $t\in (0,T]$ and $x,y\in {\R^{d}}$, $$
|p^{\kappa}(t,x,y)|\leqslant c t \err{0}{0}(t,x-y). $$
\noindent (c) For all $t>0$, $x,y\in{\R^{d}}$, $x\neq y$, $$ \partial_t p^{\kappa}(t,x,y)= {\mathcal L}_x^{\kappa}\, p^{\kappa}(t,x,y)\,. $$
\noindent (d) For every $T>0$ there is a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4,\beta)$
such that for all $t\in (0,T]$, $x,y\in{\R^{d}}$, \begin{align}\label{e:fract-der-p-kappa-1b-crit1}
|{\mathcal L}_x^{\kappa} p^{\kappa}(t, x, y)|\leqslant c \err{0}{0}(t,x-y)\,, \end{align} and
\begin{align}\label{e:fract-der-p-kappa-2-crit1}
\left|\nabla_x p^{\kappa}(t,x,y)\right|\leqslant c\! \left[h^{-1}(1/t)\right]^{-1} t \err{0}{0}(t,x-y)\,. \end{align}
\noindent (e) For all $T>0$, $\gamma \in [0,1]\cap [0,\alpha_h)$, there is a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4, \beta,\gamma)$
such that for all $t\in (0,T]$ and $x,x',y\in {\R^{d}}$, \begin{align*}
\left|p^{\kappa}(t,x,y)-p^{\kappa}(t,x',y)\right| \leqslant c
(|x-x'|^{\gamma}\land 1) \,t \left( \err{-\gamma}{0} (t,x-y)+ \err{-\gamma}{0}(t,x'-y) \right). \end{align*}
\noindent For all $T>0$, $\gamma \in [0,\beta)\cap [0,\alpha_h)$, there is a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4, \beta,\gamma)$ such that for all $t\in (0,T]$ and $x,y,y'\in {\R^{d}}$, \begin{align*}
\left|p^{\kappa}(t,x,y)-p^{\kappa}(t,x,y')\right| \leqslant c
(|y-y'|^{\gamma}\land 1)\, t \left( \err{-\gamma}{0}(t,x-y)+ \err{-\gamma}{0}(t,x-y') \right). \end{align*}
\noindent (f) The function ${\mathcal L}_x^{\kappa}p^{\kappa}(t,x,y)$ is jointly continuous on $(0,\infty)\times {\R^{d}}\times {\R^{d}}$. \end{lemma} \noindent{\bf Proof.} The statement of (a) follows from Lemmas~\ref{lem:cont_frcoef} and~\ref{lem:phi_cont_joint-crit1}. Part (b) is a result of Proposition~\ref{prop:gen_est_crit} and Lemma~\ref{lem:phi_cont_xy-crit1}.
The equation in (c) is a consequence of \eqref{e:p-kappa}, \eqref{eq:p_gen_klas} and Corollary~\ref{e:phi-y-partial_1-crit}: $\partial_t p^{\kappa}(t,x,y)={\mathcal L}_x^{\mathfrak{K}_x} p^{\mathfrak{K}_y}(t,x,y)+ {\mathcal L}_x^{\mathfrak{K}_x} \phi_y(t,x)={\mathcal L}_x^{\mathfrak{K}_x} p^{\kappa}(t,x,y)$. We get \eqref{e:fract-der-p-kappa-1b-crit1} by \eqref{e:p-kappa}, \eqref{ineq:Lkp_abs}, \eqref{ineq:I_0_oszagorne-crit1-impr}, \cite[(92), (93)]{GS-2018} (see also Lemma~\ref{e:L-on-phi-y-crit1} and \eqref{e:L-on-phi-y2-crit1}). For the proof of \eqref{e:fract-der-p-kappa-2-crit1} we use Proposition~\ref{prop:gen_est_crit} and \eqref{e:gradient-phi-y-estimate-crit1}. The first inequality of part (e) follows from
Lemmas~\ref{lem:pkw_holder} and~\ref{lem:phi_cont_xy-crit1}, and \cite[(92), (93)]{GS-2018}. The same argument suffices for the second inequality of part (e) when supported by $$
|p^{\mathfrak{K}_y}(t,x,y)-p^{\mathfrak{K}_{y'}}(t,x,y')|
\leqslant |p^{\mathfrak{K}_y}(t,-y,-x)-p^{\mathfrak{K}_{y}}(t,-y',-x)|
+ |p^{\mathfrak{K}_y}(t,x,y')-p^{\mathfrak{K}_{y'}}(t,x,y')| $$ and Proposition~\ref{prop:Hcont_kappa_crit1}. Part (f) follows from Lemmas~\ref{lem:cont_frcoef} and~\ref{lem:Lphi_cont-crit}. {
$\Box$
}
\section{Main Results and Proofs}\label{sec:Main} In the whole section, we assume that either $\Qa$ or $\Qb$ holds. \subsection{A nonlocal maximum principle} Recall that ${\mathcal L}^{\kappa,0^+}f:=\lim_{\varepsilon \to 0^+}{\mathcal L}^{\kappa,\varepsilon}f$ is an extension of ${\mathcal L}^{\kappa}f:={\mathcal L}^{\kappa,0}f$. Moreover, the well-posedness of those operators requires the existence of the gradient $\nabla f$. The uniqueness of solutions to \eqref{e:nonlocal-max-principle-4-crit} stated in Corollary~\ref{cor:jedn_max-crit} will be used, for instance, in the next subsection. For the proofs of the following, see \cite[Theorem~4.1]{GS-2018}. \begin{theorem} \label{t:nonlocal-max-principle-crit} Let $T>0$ and $u\in C([0,T]\times {\R^{d}})$ be such that \begin{align}\label{e:nonlocal-max-principle-1-crit}
\| u(t,\cdot)-u(0,\cdot) \|_{\infty} \xrightarrow {t\to 0^+} 0\,, \qquad \qquad \sup_{t\in [0,T]} \| u(t,\cdot){\bf 1}_{|\cdot|\geqslant r} \|_{\infty} \xrightarrow {r\to \infty}0\,. \end{align} Assume that $u(t,x)$ satisfies the following equation: for all $(t,x)\in (0,T]\times {\R^{d}}$, \begin{align}\label{e:nonlocal-max-principle-4-crit} \partial_t u(t,x)={\mathcal L}_x^{\kappa,0^+}u(t,x)\, . \end{align} If $\sup_{x\in{\R^{d}}} u(0,x)\geqslant 0$, then for every $t\in (0,T]$, \begin{align}\label{e:nonlocal-max-principle-5-crit} \sup_{x\in \mathbb{R}^d}u(t,x)\leqslant \sup_{x\in {\R^{d}}}u(0,x)\, . \end{align} \end{theorem} \begin{corollary}\label{cor:jedn_max-crit} If $u_1, u_2 \in C([0,T]\times {\R^{d}})$ satisfy \eqref{e:nonlocal-max-principle-1-crit}, \eqref{e:nonlocal-max-principle-4-crit}
and $u_1(0,x)=u_2(0,x)$, then $u_1\equiv u_2$ on $[0,T]\times {\R^{d}}$. \end{corollary}
\subsection{Properties of the semigroup $(P^{\kappa}_t)_{t\ge 0}$} Define $$ P_t^{\kappa}f(x)=\int_{\mathbb{R}^d}p^\kappa(t,x, y)f(y)dy. $$ We first collect some properties of $\Upsilon_t*f$. \begin{remark}\label{rem:conv_Lp-crit} We have $\Upsilon_t*f \in C_b({\R^{d}})$ for any $f\in L^p({\R^{d}})$, $p\in [1,\infty]$. Moreover, $\Upsilon_t*f\in C_0({\R^{d}})$ for any $f\in L^p({\R^{d}})\cup C_0({\R^{d}})$, $p\in [1,\infty)$. Furthermore, there is $c=c(d)$ such that
$\|\Upsilon_t*f \|_p\leqslant c \|f\|_p$ for all $t>0$, $p\in [1,\infty]$. The above follows from $\Upsilon_t\in L^1({\R^{d}})\cap L^{\infty}({\R^{d}})\subseteq L^q({\R^{d}})$ for every $q\in [1,\infty]$ (see \cite[Lemma~5.6]{GS-2018}), and from properties of the convolution. \end{remark}
\begin{lemma}\label{lem:bdd_cont-crit1} (a) We have $P_t^{\kappa} f \in C_b({\R^{d}})$ for any $f\in L^p({\R^{d}})$, $p\in [1,\infty]$. Moreover, $P_t^{\kappa} f \in C_0({\R^{d}})$ for any $f\in L^p({\R^{d}})\cup C_0({\R^{d}})$, $p\in [1,\infty)$. For every $T>0$ there exists a constant $c=c(d,T,\sigma,\kappa_2,\kappa_4, \beta)$ such that for all $t\in(0,T]$ we get $$
\|P^{\kappa}_t f\|_p\leqslant c \|f\|_p\,. $$ (b) $P^{\kappa}_t\colon C_0({\R^{d}})\to C_0({\R^{d}})$, $t>0$, and for any bounded uniformly continuous function $f$, $$
\lim_{t\to 0^+} \|P^{\kappa}_t f -f \|_{\infty}=0\,. $$ (c) $P^{\kappa}_t\colon L^p({\R^{d}})\to L^p({\R^{d}})$, $t>0$, $p\in [1,\infty)$, and for any $f\in L^p({\R^{d}})$, $$
\lim_{t\to 0^+} \|P_t^{\kappa}f -f \|_p=0\,. $$ \end{lemma} \noindent{\bf Proof.} The proof is like that for \cite[Lemma~4.4]{GS-2018} and uses Remark~\ref{rem:conv_Lp-crit}, Lemma~\ref{l:p-kappa-difference-crit-1}, \ref{lem:phi_cont_xy-crit1}, \ref{l:convolution}(a), \eqref{e:some-estimates-2c-crit1}, \cite[Lemma~5.6]{GS-2018} and Proposition~\ref{prop:gen_est_crit}. {
$\Box$
}
Our aim now is to prove Proposition~\ref{lem:gen_sem_step1-crit1} and Lemma~\ref{lem:p-kappa-final-prop-crit1}. The first one provides an important link between $P_t^{\kappa}$ and the operator ${\mathcal L}^{\kappa}$. The other complements the fundamental properties of $p^\kappa(t,x, y)$. They are both obtained by virtue of Corollary~\ref{cor:jedn_max-crit}, but beforehand we have to make sure that certain functions satisfy \eqref{e:nonlocal-max-principle-4-crit}. The necessary results are prepared in Lemma~\ref{lem:grad_Pt-crit1} -- \ref{l:L-int-commute-crit1}. In particular, we show that we can apply the operator ${\mathcal L}^{\kappa}$ to $ \int_0^t P^{\kappa}_s f(x)\,ds$.
The formula \eqref{eq:grad_Pt_1-crit1} below is the same as \cite[(69)]{GS-2018}, however the proof given there fully relies on the condition $\alpha_h>1$, while we assume that $\alpha_h\leqslant 1$. Here we obtain \eqref{eq:grad_Pt_1-crit1} under an additional restriction on $f$, which suffices for our purposes.
\begin{lemma}\label{lem:grad_Pt-crit1} For any $f\in L^p({\R^{d}})$, $p\in [1,\infty]$, we have for all $t>0$, $x\in{\R^{d}}$, \begin{align}\label{eq:grad_Pt-crit1} \nabla_x \,P_t^{\kappa} f(x)= \int_{{\R^{d}}} \nabla_x\, p^{\kappa}(t,x,y) f(y)dy\,. \end{align} For any bounded (uniformly) H\"older continuous function $f \in C^\eta_b({\R^{d}})$, $1-\alpha_h<\eta$, and all $t>0$, $x\in{\R^{d}}$, \begin{align}\label{eq:grad_Pt_1-crit1} \nabla_x \left( \int_0^t P^{\kappa}_s f(x)\,ds \right)= \int_0^t \nabla_x P^{\kappa}_s f(x)\,ds\,. \end{align} \end{lemma} \noindent{\bf Proof.}
By \eqref{e:fract-der-p-kappa-2-crit1} and \cite[Corollary~5.10]{GS-2018}
for $|\varepsilon|<h^{-1}(1/t)$, \begin{align*}
\left| \frac1{\varepsilon}( p^{\kappa}(t,x+\varepsilon e_i,y)-p^{\kappa}(t,x,y)) \right| |f(y)| \leqslant c \left[h^{-1}(1/t)\right]^{-1} \Upsilon_t (x-y) |f(y)|\,. \end{align*} The right hand side is integrable by Remark~\ref{rem:conv_Lp-crit}. We can use the dominated convergence theorem, which gives \eqref{eq:grad_Pt-crit1}.
For $f \in C^\eta_b({\R^{d}})$ (we can assume that $\eta<\alpha_h$)
we let $\widetilde{x}=x+\varepsilon\theta e_i$ and by \eqref{e:fract-der-p-kappa-2-crit1}, Lemma~\ref{e:fract-der-p-kappa-2b-crit1} and~\ref{l:convolution}(a) we have \begin{align*}
&\left| \int_{{\R^{d}}}
\frac1{\varepsilon}( p^{\kappa}(s,x+\varepsilon e_i,y)-p^{\kappa}(s,x,y)) f(y)\,dy \right|
\leqslant\left| \int_{{\R^{d}}} \int_0^1 \partial_{x_i} p^{\kappa}(s,\widetilde{x},y) \, d\theta\, f(y)\,dy\right|\\
& \leqslant \left| \int_{{\R^{d}}} \int_0^1 \partial_{x_i} p^{\kappa}(s,\widetilde{x},y) \big[ f(y)-f(\widetilde{x})\big] \, d\theta \,dy\right| + \left| \int_{{\R^{d}}} \int_0^1 \partial_{x_i} p^{\kappa}(s,\widetilde{x},y)f(\widetilde{x})\, d\theta \,dy\right|\\ &\leqslant c \left[h^{-1}(1/s)\right]^{-1} \int_0^1 \int_{{\R^{d}}} s\err{0}{\eta}(s, \widetilde{x}-y) \,dy\, d\theta + c\left[h^{-1}(1/s)\right]^{-1+\beta_1}\\ &\leqslant c \left[h^{-1}(1/s)\right]^{-1+\eta}+ c \left[h^{-1}(1/s)\right]^{-1+\beta_1}\,. \end{align*} The right hand side is integrable over $(0,t)$ by \cite[Lemma~5.15]{GS-2018}.
Finally, \eqref{eq:grad_Pt_1-crit1} follows by the dominated convergence theorem. {
$\Box$
}
\begin{lemma}\label{l:L-int-commute0-crit1} For any function $f\in L^p({\R^{d}})$, $p\in [1,\infty]$, and all $t>0$, $x\in{\R^{d}}$, \begin{align}\label{e:L-int-commute-2-crit1} {\mathcal L}_x^{\kappa}P_t^{\kappa} f(x)=\int_{{\R^{d}}}{\mathcal L}_x^{\kappa} \,p^{\kappa}(t,x, y)f(y)dy\, . \end{align}
Furthermore, for every $T>0$ there exists a constant $c>0$ such that for all $f\in L^p({\R^{d}})$, $t\in (0,T]$, \begin{align}\label{e:LP-p-estimate-crit1}
\| {\mathcal L}^{\kappa}P_t^{\kappa} f\|_p\leqslant c t^{-1} \|f\|_p\,. \end{align} \end{lemma} \noindent{\bf Proof.} By the definition and \eqref{eq:grad_Pt-crit1}, \begin{align}\label{eq:LPf-crit1} {\mathcal L}_x^{\kappa} P_t^{\kappa} f(x) =\int_{{\R^{d}}} \left( \int_{{\R^{d}}} \delta^{\kappa}(t,x,y;z) f(y)dy \right) \kappa(x,z)J(z)dz\,. \end{align} The equality follows from an application of Fubini's theorem, justified by \eqref{ineq:some-est_p_kappa-crit1} and Remark~\ref{rem:conv_Lp-crit}. The inequality follows then from \eqref{e:L-int-commute-2-crit1}, \eqref{e:fract-der-p-kappa-1b-crit1}, Remark~\ref{rem:conv_Lp-crit}. {
$\Box$
}
\begin{lemma}\label{lem:for_max-crit1} Let $f\in C_0({\R^{d}})$. For $t>0$, $x\in{\R^{d}}$ we define $u(t,x)=P^{\kappa}_t f(x)$ and $u(0,x)=f(x)$. Then $u\in C([0,T]\times {\R^{d}})$, \eqref{e:nonlocal-max-principle-1-crit} holds and $\partial_t u(t,x)={\mathcal L}_x^{\kappa}u(t,x)$ for all $t,T>0$, $x\in{\R^{d}}$. \end{lemma} \noindent{\bf Proof.} The proof is exactly like that of \cite[Lemma~4.7]{GS-2018}. {
$\Box$
}
\begin{lemma}\label{l:L-int-commute-crit1} For any bounded (uniformly) H\"older continuous function
$f \in C^\eta_b({\R^{d}})$, $1-\alpha_h<\eta$, and all $t>0$, $x\in{\R^{d}}$, we have $\int_0^t | {\mathcal L}_x^{\kappa} P_s^{\kappa}f(x)|ds <\infty$ and \begin{align}\label{e:L-int-commute-crit1} {\mathcal L}_x^{\kappa}\left( \int_0^t P_s^{\kappa}f(x)\,ds\right) =\int_0^t {\mathcal L}_x^{\kappa} P_s^{\kappa}f(x)\,ds\,. \end{align} \end{lemma} \noindent{\bf Proof.} By definition and Lemma~\ref{lem:grad_Pt-crit1}, \begin{align*} {\mathcal L}_x^{\kappa} \int_0^t P_s^{\kappa}f(x)\,ds &=\int_{{\R^{d}}} \left( \int_0^t \int_{{\R^{d}}} \delta^{\kappa} (s,x,y;z) f(y)dy ds \right) \kappa(x,z)J(z)dz\,. \end{align*}
Note that the proof will be finished by \eqref{eq:LPf-crit1} if we can change the order of integration from $dsdz$ to $dzds$. To this end we use Fubini's theorem, which is justified by the following. We have $|f(y)-f(x)|\leqslant c (|y-x|^{\eta} \land 1)$ and we can assume that $\eta<\alpha_h$. Then \begin{align*}
\int_{{\R^{d}}} \int_0^t &\left| \int_{{\R^{d}}} \delta^{\kappa} (s,x,y;z) f(y)dy \right| ds \, \kappa(x,z)J(z)dz\\
&\leqslant \int_{{\R^{d}}} \int_0^t \left| \int_{{\R^{d}}} \delta^{\kappa} (s,x,y;z) \big[f(y)-f(x)\big] dy\right| ds \,\kappa(x,z)J(z)dz\\
&\quad+\int_{{\R^{d}}} \int_0^t \left| \int_{{\R^{d}}} \delta^{\kappa} (s,x,y;z) f(x) dy\right| ds\, \kappa(x,z)J(z)dz=: {\rm I}_1+{\rm I}_2\,. \end{align*} By \eqref{ineq:some-est_p_kappa-crit1}, we get ${\rm I}_1\leqslant c \int_0^t \int_{{\R^{d}}} \vartheta(s) \err{0}{\eta}(s,y-x) dyds$, while by \eqref{ineq:some-est_p_kappa_1-crit1} ${\rm I}_2\leqslant c \int_0^t \vartheta(s) s^{-1} \left[h^{-1}(1/s)\right]^{\beta_1}ds$. The integrals are finite by \cite[Lemma~5.15]{GS-2018}, Lemma~\ref{l:convolution}(a) and~\ref{lem:cal_TCh}. {
$\Box$
}
\begin{proposition}\label{lem:gen_sem_step1-crit1} For any $f\in C_b^{2}({\R^{d}})$ and all $t>0$, $x\in{\R^{d}}$, \begin{align}\label{eq:gen_sem_step1-crit1} P_t^{\kappa}f(x)-f(x)=\int_0^t P_s^{\kappa}{\mathcal L}^{\kappa} f(x)\,ds\,. \end{align} \end{proposition} \noindent{\bf Proof.} We outline the main steps, for details, see the proof of \cite[Proposition~4.9]{GS-2018}.
\noindent (i) Note that ${\mathcal L}^{\kappa}f \in C_0({\R^{d}})$ for any $f\in C_0^2({\R^{d}})$.
\noindent (ii) Show that ${\mathcal L}^{\kappa}f \in C_b^{\eta}({\R^{d}})$ for any $f\in C_0^{2,\eta}({\R^{d}})$.
\noindent (iii) Show that \eqref{eq:gen_sem_step1-crit1} holds for any $f\in C_0^{2,\eta}({\R^{d}})$ if $1-\alpha_h<\eta\leqslant \beta$. It is achieved by using Corollary~\ref{cor:jedn_max-crit} to prove that the following functions are equal \begin{align*} u_1(t,x)= \begin{cases} P_t^{\kappa} f(x), \quad &t>0\\ f(x), & t=0 \end{cases} \,, \qquad u_2(t,x)= \begin{cases} f(x)+\int_0^t P_s^{\kappa}{\mathcal L}^{\kappa} f(x)\,ds, \quad &t>0 \\ f(x), &t=0 \end{cases} \,. \end{align*} Here we use Lemmas~\ref{lem:for_max-crit1}, \ref{l:L-int-commute-crit1} and \cite[Theorem~7.21]{MR924157}.
\noindent
(iv) We extend \eqref{eq:gen_sem_step1-crit1} to $f\in C_b^2({\R^{d}})$ by approximating it with $f_n=(f*\phi_n)\cdot \varphi_n \in C_c^{\infty}({\R^{d}})$, where $\phi_n$ is a standard mollifier while $\varphi_n(x)=\varphi(x/n)$ for $\varphi\in C_c^{\infty}({\R^{d}})$ satisfying $\varphi(x)=1$ if $|x|\leqslant 1$, and $\varphi(x)=0$ if $|x|\geqslant 2$. {
$\Box$
}
\begin{lemma}\label{lem:p-kappa-final-prop-crit1} The function $p^{\kappa}(t,x,y)$ is non-negative, $\int_{{\R^{d}}} p^{\kappa}(t,x,y)dy= 1$ and $p^{\kappa}(t+s,x,y)=\int_{{\R^{d}}}p^{\kappa}(t,x,z)p^{\kappa}(s,z,y)dz$ for all $s,t>0$, $x,y\in{\R^{d}}$. \end{lemma} \noindent{\bf Proof.} The proof is like that of \cite[Lemma~4.10]{GS-2018} and we use Lemma~\ref{lem:for_max-crit1}, Theorem~\ref{t:nonlocal-max-principle-crit}, Lemma~\ref{l:p-kappa-difference-crit-1}, Corollary~\ref{cor:jedn_max-crit}, Proposition~\ref{lem:gen_sem_step1-crit1}. However, the proof of the convolution property in \cite{GS-2018} contains a gap: at that stage it is not clear why the function $p^{\kappa}(t+s,x,y)$ should satisfy the equation \eqref{e:nonlocal-max-principle-4-crit} for all $x\in{\R^{d}}$. Here we present a correction that is valid for both papers.
Let $T,s>0$ and $\varphi\in C_c^{\infty}({\R^{d}})$. For $t>0$, $x\in{\R^{d}}$ define $$ u_1(t,x)=P_t^{\kappa}f(x)\,,\qquad u_1(0,x)=f(x)=P_s^{\kappa}\varphi(x)\,, $$ and $$ u_2(t,x)=P_{t+s}^{\kappa}\varphi(x)\,, \qquad u_2(0,x)=P_s^{\kappa}\varphi(x)\,. $$ By Lemma~\ref{lem:bdd_cont-crit1}(b) $f\in C_0({\R^{d}})$ and thus by Lemma~\ref{lem:for_max-crit1} $u_1$ satisfies the assumptions of Corollary~\ref{cor:jedn_max-crit}. Now, since $\varphi$ has compact support by Lemma~\ref{l:p-kappa-difference-crit-1}(a) we get $u_2\in C([0,T]\times {\R^{d}})$. We will use \cite[(94)]{GS-2018} several times in what follows. By Lemma~\ref{l:p-kappa-difference-crit-1} (c) and (d), \begin{align*}
\|u_2(t,\cdot)-u_2(0,\cdot)\|_{\infty}
&\leqslant \sup_{x\in{\R^{d}}}\int_{{\R^{d}}}\int_0^t |{\mathcal L}_x^{\kappa}\,p^{\kappa}(u+s,x,y)|du \,|\varphi(y)|dy\\
&\leqslant c t \err{0}{0}(s,0)\int_{{\R^{d}}}|\varphi(y)|dy \to 0\,, \quad \mbox{as } t\to 0^+\,. \end{align*} Furthermore, by Lemma~\ref{l:p-kappa-difference-crit-1}(b) \begin{align*}
\sup_{t\in[0,T]}\|u_2(t\cdot){\bf 1}_{|\cdot|\geqslant r}\|_{\infty}
&\leqslant c (T+s) \sup_{x\in{\R^{d}}} {\bf 1}_{|x|\geqslant r}\int_{{\R^{d}}}\err{0}{0}(s,x-y)|\varphi(y)|dy\\
&=c (T/s+1)\sup_{x\in{\R^{d}}} {\bf 1}_{|x|\geqslant r} \left(\Upsilon_t*|\varphi|\right)(x)\to 0\,, \quad \mbox{as } r\to \infty\,, \end{align*} because
$\Upsilon_t*|\varphi|\in C_0({\R^{d}})$ (see Remark~\ref{rem:conv_Lp-crit}). Finally, by the mean value theorem, Lemma~\ref{l:p-kappa-difference-crit-1}(c), \eqref{e:fract-der-p-kappa-1b-crit1} and the dominated convergence theorem $\partial_t u_2(t,x)=\int_{{\R^{d}}}\partial_t p^{\kappa}(t+s,x,y)\varphi(y) dy$. Then we apply Lemma~\ref{l:p-kappa-difference-crit-1}(c) and Lemma~\ref{l:L-int-commute0-crit1} to obtain $\partial_t u_2(t,x)={\mathcal L}_x^{\kappa}u(t,x)$. Therefore, by Corollary~\ref{cor:jedn_max-crit} $u_1=P_t^{\kappa}P_s^{\kappa}\varphi = P_{t+s}^{\kappa}\varphi = u_2$. The convolution property now follows by Fubini theorem, the arbitrariness of $\varphi$ and by Lemma~\ref{l:p-kappa-difference-crit-1}(a). {
$\Box$
}
\subsection{Proofs of Theorems~\ref{t:intro-main}--\ref{thm:onC0Lp}}
At this point we have all necessary tools to proceed exactly like in \cite{GS-2018}. A slight difference in the proof of Theorem~\ref{thm:lower-bound} is explained below.\\
\noindent {\bf Proof of Theorem~\ref{thm:onC0Lp}}. It is the same as in \cite{GS-2018} and relies on Lemma~\ref{lem:bdd_cont-crit1}, \ref{lem:p-kappa-final-prop-crit1}, Proposition~\ref{lem:gen_sem_step1-crit1}, \eqref{eq:grad_Pt-crit1}, \eqref{e:fract-der-p-kappa-2-crit1}, Remark~\ref{rem:conv_Lp-crit}, \eqref{ineq:some-est_p_kappa-crit1}, \eqref{eq:LPf-crit1}, \eqref{e:intro-kappa}, \eqref{e:intro-kappa-holder}, \eqref{e:L-int-commute-2-crit1}, Lemma~\ref{l:p-kappa-difference-crit-1}(f) and~(d), \cite[Corollary~5.10]{GS-2018},
\eqref{e:LP-p-estimate-crit1}, \cite[Chapter~1, Theorem~2.4(c) and~2.2]{MR710486}, \cite[Chapter~2, Theorem~5.2(d)]{MR710486}. {
$\Box$
}
\noindent {\bf Proof of Theorem~\ref{t:intro-further-properties}}. All the results are collected in Lemma~\ref{l:p-kappa-difference-crit-1} and~\ref{lem:p-kappa-final-prop-crit1}, except for part (8), which is given in Theorem~\ref{thm:onC0Lp} part 3(c). {
$\Box$
}
\noindent {\bf Proof of Theorem~\ref{t:intro-main}}. The same as in \cite{GS-2018}. {
$\Box$
}
\noindent {\bf Proof of Theorem~\ref{thm:lower-bound}}. The argument is the same as in \cite{GS-2018} except for the following modification of the prove of the estimate $$
\sup_{|\xi|\leqslant 1/r} |q(z,\xi)| \leqslant c_2 h(r). $$ Since for
$\varphi \in {\mathbb R}$ we have $\left|e^{i\varphi}-1-i\varphi \right|\leqslant |\varphi|^2$, by \eqref{e:intro-kappa-crit} we get \begin{align*}
|q(z,\xi)|&\leqslant
\int_{{\R^{d}}} \left| e^{i\left<\xi, w\right>}-1 - i\left<\xi,w\right> {\bf 1}_{|w|<1\land 1/|\xi|} \right|\kappa(x,w)J(w)dw\\
&\quad +\left| i \left<\xi,\int_{{\R^{d}}} w \left({\bf 1}_{|w|<1\land 1/|\xi|}
- {\bf 1}_{|w|<1}\right)
\kappa(x,w)J(w)dw \right>\right|\\ &\leqslant
|\xi|^2 \int_{|w|<1\land 1/|\xi|} |w|^2 \kappa(x,w)J(w)dw
+ \int_{|w|\geqslant 1\land 1/|\xi|} 2\, \kappa(x,w)J(w)dw\\
&\quad + |\xi| \left| \int_{{\R^{d}}} w \left({\bf 1}_{|w|<1\land 1/|\xi|}
- {\bf 1}_{|w|<1}\right)
\kappa(x,w)J(w)dw\right|
\leqslant c h(1\land 1/|\xi|)\,. \end{align*} {
$\Box$
}
\section{Appendix - Unimodal L{\'e}vy processes}\label{sec:appA}
Let $d\in{\mathbb N}$ and $\nu:[0,\infty)\to[0,\infty]$ be a non-increasing function satisfying
$$\int_{{\R^{d}}} (1\land |x|^2) \nu(|x|)dx<\infty\,.$$ For any such $\nu$, there exists a unique
pure-jump isotropic unimodal L{\'e}vy process $X$ (see \cite{MR3165234}, \cite{MR705619}). We define $h(r)$, $K(r)$ and $\Upsilon_t(x)$ as in the introduction. At this point we refer the reader to \cite[Section~5]{GS-2018} for various important properties of those functions. Following \cite[Section~5]{GS-2018} in the whole section {\bf we assume that} $h(0^+)=\infty$. We consider the scaling conditions:
\noindent there are $\alpha_h \in(0,2]$, $C_h\in[1,\infty)$ and $\theta_h\in(0,\infty]$ such that \begin{equation}\label{eq:wlsc:h}
h(r)\leqslant C_h\lambda^{\alpha_h }h(\lambda r),\qquad \lambda\leqslant 1,\, r< \theta_h;
\end{equation} there are $\beta_h \in (0,2]$, $c_h\in (0,1]$ and $\theta_h \in (0,\infty]$ such that \begin{equation}\label{eq:wusc:h}
c_h\,\lambda^{\beta_h}\,h(\lambda r)\leqslant h(r)\, ,\quad \lambda\leqslant 1, \,r< \theta_h. \end{equation}
The first and the latter inequality in the next lemma are taken from \cite[Section~5]{GS-2018}. We keep them here for easier reference.
\begin{lemma}\label{lem:int_J} Let $h$ satisfy \eqref{eq:wlsc:h} with $\alpha_h>1$, then \begin{align*}
\int_{r \leqslant |z|< \theta_h } |z|\nu(|z|)dz \leqslant \frac{(d+2) C_h}{\alpha_h-1} \, r h(r)\,, \qquad r>0\,. \end{align*} Let $h$ satisfy \eqref{eq:wlsc:h} with $\alpha_h=1$, then \begin{align*}
\int_{r \leqslant |z|< \theta_h } |z|\nu(|z|)dz \leqslant [(d+2)C_h] \, \ln(\theta_h/r)\, r h(r)\,, \qquad r>0\,. \end{align*} Let $h$ satisfy \eqref{eq:wusc:h} with $\beta_h<1$, then \begin{align*}
\int_{|z|< r} |z| \nu(|z|)dz\leqslant \frac{d+2}{c_h(1-\beta_h)}\, r h(r)\,,\qquad r< \theta_h \,. \end{align*} \end{lemma} \noindent{\bf Proof.} Under \eqref{eq:wlsc:h} with $\alpha_h=1$ we have \begin{align*}
\int_{r \leqslant |z|< \theta_h } |z|\nu(|z|)dz \leqslant (d+2) \int_r^{\theta_h} h(s)ds \leqslant (d+2) C_h\int_r^{\theta_h} (r/s)h(r)ds\,, \end{align*} which ends the proof. {
$\Box$
}
\begin{lemma}\label{lem:cal_TCh} Assume \eqref{eq:intro:wlsc}. Let $k,l \geqslant 0$ and $\theta, \eta, \beta,\gamma\in\mathbb{R}$ satisfy $(\beta/2)\land (\beta/\alpha_h)+1-\theta>0$, $(\gamma/2)\land(\gamma/\alpha_h)+1-\eta>0$. For every $T>0$ there exists a constant $c=c(\alpha_h,C_h,h^{-1}(1/T)\vee 1, \theta, \eta,\beta,\gamma,k,l)$ such that for all $t\in (0,T]$, \begin{align*} \int_0^t [\Theta(u)]^{l}\, u^{-\eta}\left[ h^{-1}(1/u)\right]^{\gamma}[\Theta(t-u)]^{k}\, (t-u)^{-\theta}\left[ h^{-1}(1/(t-u))\right]^{\beta}du \\ \leqslant c\, [\Theta(t)]^{l+k}\, t^{1-\eta-\theta}\left[h^{-1}(1/t)\right]^{\gamma+\beta}\,. \end{align*} Furthermore, $\Theta(t/2)\leqslant c\, \Theta(t)$, $t\in (0,T]$. \end{lemma} \noindent{\bf Proof.} The last part of the statement follows from \cite[Lemma~5.3 and Remark~5.2]{GS-2018}. Note that it suffices to consider the integral over $(0,t/2)$. Again by \cite[Lemma~5.3 and Remark~5.2]{GS-2018} we have for $c_0=C_h [h^{-1}(1/T)\vee 1]^2$ and $s\in (0,1)$, $$ \left[ h^{-1}(t^{-1}s^{-1})\right]^{-1} \leqslant c_0\, s^{-1/\alpha_h}\left[h^{-1}(1/t)\right]^{-1}, \qquad h^{-1}(t^{-1}s^{-1})\leqslant s^{1/2}h^{-1}(t^{-1})\,. $$ Thus for $u\in (0,t/2)$ we get $$ [\Theta(t-u)]^{k}(t-u)^{-\theta}[h^{-1}(1/(t-u))]^{\beta}\leqslant c \,[\Theta(t)]^{k}\, t^{-\theta}\,[h^{-1}(1/t)]^{\beta}\,, $$ and we concentrate on $$\int_0^{t/2} [\Theta(u)]^{l} u^{-\eta} \left[h^{-1}(1/u)\right]^{\gamma}du \leqslant c\,t^{1-\eta} \left[h^{-1}(1/t)\right]^{\gamma} \int_0^{1/2} [\Theta(ts)]^{l} s^{(\gamma/2)\land (\gamma/\alpha_h)-\eta} ds \,.$$ Furthermore, we have \begin{align*} &\int_0^{1/2} \left[ \ln\left(1\vee\left[ h^{-1}(t^{-1}s^{-1})\right]^{-1}\right)\right]^{l} s^{(\gamma/2)\land (\gamma/\alpha_h)-\eta}ds\\ &\leqslant \int_0^{1/2} 2^{l} \left\{ [\ln (c_0 s^{-1})]^{l} +\left[ \ln \left(1\vee\left[ h^{-1}(t^{-1}\right]^{-1}\right)\right]^{l} \right\} s^{(\gamma/2)\land (\gamma/\alpha_h)-\eta}ds \leqslant c \,[\Theta(t)]^{l}\,. \end{align*} Finally, \begin{align*}
\int_0^{1/2} [\Theta(ts)]^{l} s^{(\gamma/2)\land (\gamma/\alpha_h)-\eta} ds \leqslant c\, [\Theta(t)]^{l}. \end{align*} {
$\Box$
}
The next lemma is taken from \cite[Section~5]{GS-2018} and complemented with part (d). It is one of the most often used technical result in the paper. Let $B(a,b)$ be the beta function, i.e., $B(a,b)=\int_0^1 s^{a-1} (1-s)^{b-1}ds$, $a,b>0$.
\begin{lemma}\label{l:convolution} Assume \eqref{eq:intro:wlsc} and let $\beta_0\in(0,\alpha_h\land 1)$. \begin{itemize} \item[(a)] For every $T>0$ there exists a constant $c_1=c_1(d,\beta_0,\alpha_h,C_h,h^{-1}(1/T)\vee 1)$ such that for all $t\in(0,T]$ and $\beta\in [0,\beta_0]$, $$ \int_{{\R^{d}}} \err{0}{\beta}(t,x)\,dx \leqslant c_1 t^{-1} \left[h^{-1}(1/t)\right]^{\beta}\,. $$ \item[(b)] For every $T>0$ there exists a constant $c_2=c_2(d,\beta_0,\alpha_h,C_h,h^{-1}(1/T)\vee 1) \ge 1$ such that for all $\beta_1,\beta_2,n_1,n_2,m_1,m_2 \in[0,\beta_0]$ with $n_1, n_2 \leqslant \beta_1+\beta_2$, $m_1\leqslant \beta_1$, $m_2\leqslant \beta_2$ and all $0<s<t\leqslant T$, $x\in{\R^{d}}$, \begin{align*} \int_{{\R^{d}}} \err{0}{\beta_1}(t-s&,x-z)\err{0}{\beta_2}(s,z) \,dz\\ \leqslant c_2 &\Big[ \left( (t-s)^{-1} \left[h^{-1}(1/(t-s))\right]^{n_1} + s^{-1}\left[h^{-1}(1/s)\right]^{n_2}\right) \err{0}{0}(t,x)\\ &+(t-s)^{-1}\left[ h^{-1}(1/(t-s))\right]^{m_1} \err{0}{\beta_2}(t,x) + s^{-1}\left[ h^{-1}(1/s)\right]^{m_2} \err{0}{\beta_1}(t,x)\Big]. \end{align*} \item[(c)] Let $T>0$. For all $\gamma_1, \gamma_2\in{\mathbb R}$, $\beta_1,\beta_2,n_1,n_2,m_1,m_2 \in[0,\beta_0]$ with $n_1, n_2 \leqslant \beta_1+\beta_2$, $m_1\leqslant \beta_1$, $m_2\leqslant \beta_2$ and $\theta,\eta \in [0,1]$, satisfying \begin{align*} (\gamma_1+n_1\land m_1)/2 \land (\gamma_1+n_1\land m_1)/\alpha_h +1-\theta>0\,,\\ (\gamma_2+n_2\land m_2)/2\land (\gamma_2+n_2\land m_2)/\alpha_h+1-\eta>0\,, \end{align*} and all $0<s<t\leqslant T$, $x\in{\R^{d}}$, we have \begin{align} \int_0^t\int_{{\R^{d}}}& (t-s)^{1-\theta}\, \err{\gamma_1}{\beta_1}(t-s,x-z) \,s^{1-\eta}\,\err{\gamma_2}{\beta_2}(s,z) \,dzds \nonumber\\ & \leqslant c_3 \, t^{2-\eta-\theta}\Big( \err{\gamma_1+\gamma_2+n_1}{0} +\err{\gamma_1+\gamma_2+n_2}{0} +\err{\gamma_1+\gamma_2+m_1}{\beta_2} +\err{\gamma_1+\gamma_2+m_2}{\beta_1} \Big)(t,x)\,,\label{e:convolution-3} \end{align} where $ c_3= c_2 \, (C_h[h^{-1}(1/T)\vee 1]^2)^{-(\gamma_1\land 0+\gamma_2 \land 0)/\alpha_h}
B\left( k+1-\theta, \,l+1-\eta\right) $ and \begin{align*} k=\left(\frac{\gamma_1+n_1\land m_1}{2}\right)\land \left(\frac{\gamma_1+n_1\land m_1}{\alpha_h}\right), \quad l=\left(\frac{\gamma_2+n_2\land m_2}{2}\right)\land \left(\frac{\gamma_2+n_2\land m_2}{\alpha_h}\right). \end{align*} \item[(d)] Let $T>0$. For all $k,l\geqslant 0$, $\gamma_1, \gamma_2\in{\mathbb R}$, $\beta_1,\beta_2,n_1,n_2,m_1,m_2 \in[0,\beta_0]$ with $n_1, n_2 \leqslant \beta_1+\beta_2$, $m_1\leqslant \beta_1$, $m_2\leqslant \beta_2$ and $\theta,\eta \in [0,1]$, satisfying \begin{align*} (\gamma_1+n_1\land m_1)/2 \land (\gamma_1+n_1\land m_1)/\alpha_h +1-\theta>0\,,\\ (\gamma_2+n_2\land m_2)/2\land (\gamma_2+n_2\land m_2)/\alpha_h+1-\eta>0\,, \end{align*} and for all $0<s<t\leqslant T$, $x\in{\R^{d}}$, we have \begin{align} \int_0^t\int_{{\R^{d}}}& [\Theta(t-s)]^k (t-s)^{1-\theta}\, \err{\gamma_1}{\beta_1}(t-s,x-z) [\Theta(s)]^{l}\,s^{1-\eta}\,\err{\gamma_2}{\beta_2}(s,z) \,dzds \nonumber\\ & \leqslant c_4 \, [\Theta(t)]^{k+l}\, t^{2-\eta-\theta}\Big( \err{\gamma_1+\gamma_2+n_1}{0} +\err{\gamma_1+\gamma_2+n_2}{0} +\err{\gamma_1+\gamma_2+m_1}{\beta_2} +\err{\gamma_1+\gamma_2+m_2}{\beta_1} \Big)(t,x)\,,\label{e:convolution-3-crit} \end{align} where $c_4=c_4(d,\beta_0,\alpha_h,C_h,h^{-1}(1/T)\vee 1,k,l,\gamma_1,\gamma_2,n_1,n_2,m_1,m_2,\theta,\eta)$. \end{itemize} \end{lemma} \noindent{\bf Proof.} For the proof of part (d) we multiply the result of part (b) by $$ [\Theta(t-s)]^{k} (t-s)^{1-\theta} \left[h^{-1}(1/(t-s))\right]^{\gamma_1} [\Theta(s)]^{\gamma_1} s^{1-\eta} \left[h^{-1}(1/s)\right]^{\gamma_2}\,, $$ and apply Lemma~\ref{lem:cal_TCh}. {
$\Box$
}
\begin{remark} When using Lemma~\ref{l:convolution} without specifying the parameters, we apply the usual case, i.e., $n_1=n_2=\beta_1+\beta_2$ ($\leqslant \beta_0$), $m_1=\beta_1$, $m_2=\beta_2$. Similarly, if only $n_1$, $n_2$ are specified, then $m_1=\beta_1$, $m_2=\beta_2$. \end{remark}
\end{document} | arXiv |
\begin{document}
\title{A Classically Impossible Task Done by Using Quantum Resources}
\author{Won-Young Hwang }
\affiliation{Department of Physics Education, Chonnam National University, Gwangju 61186, Republic of Korea }
\begin{abstract} We propose a task that cannot be done by using any classical mechanical means but can be done with quantum resources. The task is closely related to the violation of Bell's inequality. \pacs{03.65.Ud, 03.67.-a} \end{abstract}
\maketitle
\section{Introduction}
Quantum resources can perform some tasks that cannot be done by classical ones, e.g., quantum nonlocality \cite{Bel87}, quantum key distribution \cite{Scar09}, and quantum computing \cite{Nie00}. In particular, quantum entangled states show marvelous correlations between two remotely separated observers, violating Bell's inequality \cite{Bel87}. In this paper, we propose another classically impossible task that can be done by using quantum mechanics. This is motivated by a recent quantum key distribution protocol with post-selection \cite{Li14} and is closely related to the violation of Bell's inequality. In the task, two remotely separated persons, Alice and Bob, prepare some (either classical or quantum) physical entities under certain conditions. The physical entities are sent to another person, Charlie, who selects some subsets of the physical entities. Correlations between Alice's and Bob's data corresponding to the selected ones, if the physical entities are classical, can be shown to obey a constraint that is identical to Bell's inequality \cite{Bel87}. However, one can show that the constraint can be violated by using quantum entities.
In Section II, we describe the task in detail. In Section III, we explain why the task cannot be performed by using classical entities while it can be done by using quantum resources. In Section IV, we discuss related issues and present conclusions.
\section{The task}
(i) Alice generates a random number $a$, which we call the basis. Here, $a= 0,1$, and the probability that $a=0$ and $a=1$ are $1/2$ and $1/2$, respectively. She also generates a random number $x$, which we call the state. Here, $x= 0,1$, and the probability that $x=0$ and $x=1$ are $p_{a0}$ and $p_{a1}= 1-p_{a0}$, respectively, for basis $a$. To each combination of the basis $a$ and the state $x$, a physical entity is assigned. Let $(a,x)$ denote the physical entity corresponding to $a$ and $x$. Here, the physical entities may be any physical one, either classical or quantum, and the physical entities may be statistical mixtures of mutually different ones. Bob does the same things. He generates a random number $b$, which we call the basis. Here $b= 0,1$, and the probability that $b=0$ and $b=1$ are $1/2$ and $1/2$, respectively. Bob also generates a random number $y$, which we call the state. Here, $y= 0,1$, and the probability that $y=0$ and $y=1$ are $p^{\prime}_{b0}$ and $p^{\prime}_{b1}= 1- p^{\prime}_{b0}$, respectively, for basis $b$. To each combination of the basis $b$ and the state $y$, a physical entity is assigned. Let $(b,y)^{\prime}$ denote the physical entity corresponding to $b$ and $y$.
However, the physical entities must satisfy a condition of basis independence. Suppose that Alice and Bob repeated step (i) many times, generating an ensemble of the physical entities. Let us hypothetically separate the ensemble of the physical entities according to the basis. Denote the ensemble of Alice's (Bob's) physical entities with basis $a$ ($b$) by $\rho^A_a$ ($\rho^B_b$). For example, $\rho^A_0$ is a statistical mixture of the physical entities $(0,0)$ and $(0,1)$ with corresponding probabilities $p_{00}$ and $p_{01}=1-p_{00}$. {\it The condition is that the statistical mixtures of physical entities corresponding to different bases cannot be discriminated by any physical means; that is, $\rho^A_0$ and $\rho^B_0$ cannot be discriminated from $\rho^A_1$ and $\rho^B_1$, respectively.} Let us take an example of the physical entities satisfying the condition. Suppose that Alice prepares physical entities $(0,0)= |0\rangle$, $(0,1) = |1\rangle$, $(1,0)= |+\rangle$ and $(1,1)= |-\rangle$ with the probabilities $p_{00}= p_{01}= p_{10}= p_{11}= 1/2$. Here, $|0\rangle$ and $|1\rangle$ are two mutually orthogonal states of quantum bits and $|\pm\rangle= (1/\sqrt{2})(|0\rangle \pm |1\rangle)$. Then, the mixtures $\rho^A_0$ and $\rho^A_1$ are given by quantum mixed states described by density operators $(1/2) (|0\rangle \langle 0|+ |1\rangle \langle 1|)$ and $(1/2) (|+\rangle \langle +|+ |-\rangle \langle -|)$, respectively. Because the two density operators are identical, clearly the two mixtures $\rho^A_0$ and $\rho^A_1$ cannot be discriminated by using any physical means; thus, the condition is obeyed.
(ii) Alice and Bob send their respective physical entities $(a,x)$ and $(b,y)$ to Charlie. Then, Charlie may do any physical measurement, including doing nothing, on the physical entities. Based on the measurement's outcomes, Charlie chooses a bit $c$ between $0$ and $1$ and announces it to Alice and Bob.
(iii) Steps (i) and (ii) are repeated many times. Alice and Bob select only the data for the cases when Charlie announced $c=1$, discarding the others. Let us denote the total number of incidents for $x$ and $y$ with bases $a$ and $b$ by $n(x,y;a,b)$. For example, the total number of incidents when $x=0$ and $y=1$ with bases $a=1$ and $b=0$ is $n(01;10)$. The probabilities for $x$ and $y$ conditioned with bases $a$ and $b$ is given by
\begin{equation}
p(x,y|a,b)= \frac{n(x,y;a,b)}{\sum_{x,y}n(x,y;a,b)}. \label{1} \end{equation}
The correlation between the bases $a$ and $b$ is given by $E(a,b)= p(0,0|a,b)+ p(1,1|a,b)- p(0,1|a,b)- p(1,0|a,b)$. Now, the Bell function $S= E(0,0)+ E(0,1)+ E(1,0)- E(1,1)$ is calculated. If $|S|>2$, then the task is completed.
\section{Tasks that classical entities cannot perform but quantum ones can}
Now, let us see how the task cannot be performed with classical entities. Consider the condition of basis independence that the statistical mixtures of physical entities corresponding to different bases cannot be discriminated by using any physical means; that is, $\rho^A_0$ ($\rho^B_0$) cannot be discriminated from $\rho^A_1$ ($\rho^B_1$). For classical entities, this means that the mixtures $\rho^A_0$ and $\rho^A_1$ are actually identical. Otherwise, the two mixtures can be discriminated because, in principle, classical entities can be directly measured. We can describe classical entities by using a variable $\lambda$, where $0 \leq \lambda \leq 1$, without loss of generality. Mixtures can be characterized by thier probability distributions $P(\lambda)$ with $\int P(\lambda) d\lambda =1 $. Now, the two mixtures $\rho^A_0$ and $\rho^A_1$ must have exactly the same probability distribution $P(\lambda)$.
Let us consider Alice's physical entities $(a,0)$ and $(a,1)$. As said above, they can be mixtures. Let us denote the probability distributions corresponding to $(a,0)$ and $(a,1)$ by $P_{a0}(\lambda)$ and $P_{a1}(\lambda)$, respectively. (Now we have $P(\lambda)= p_{a0} P_{a0}(\lambda)+ p_{a1} P_{a1}(\lambda)$.)
First, we consider the case when $P_{a0}(\lambda)$ and $P_{a1}(\lambda)$ do not overlap. The other case will be dealt with later. Let us consider a $\lambda$ for which $P(\lambda)>0$. Then, we have either $P_{a0}(\lambda)>0$ or $P_{a1}(\lambda)>0$. Clearly, we can see that in the former (latter) case the physical entity $\lambda$ is from the mixture $(a,0)$ ($(a,1)$). Now, we can classify the set of all $\lambda$'s into four sets: $\Lambda_{ij}= \{\lambda| P_{0i}(\lambda)>0 \hspace{2mm} \mbox{and} \hspace{2mm} P_{1j}(\lambda)>0 \}$, where $i,j=0,1$. Namely, $\Lambda_{ij}$ is the set of $\lambda$'s that must have come from mixture $(0,i)$ ($(1,j)$) if the basis is $0$ ($1$). For Bob's mixtures $(b,0)^{\prime}$ and $(b,1)^{\prime}$, the same thing can be said. Let us describe Bob's physical entities by using a variable $\lambda^{\prime}$ with $0 \leq \lambda^{\prime} \leq 1$ and characterize the mixtures by using the probability distribution $P^{\prime}(\lambda^{\prime})$ with $\int P^{\prime}(\lambda^{\prime}) d\lambda^{\prime}= 1$. Let us denote the probability distributions corresponding to $(b,0)^{\prime}$ and $(b,1)^{\prime}$ by $P^{\prime}_{b0}(\lambda^{\prime})$ and $P^{\prime}_{b1}(\lambda^{\prime})$, respectively. Here, we also assume that $P^{\prime}_{b0}(\lambda^{\prime})$ and $P^{\prime}_{b1}(\lambda^{\prime})$ do not overlap. Now, we also can classify the set of all $\lambda^{\prime}$'s into four sets: $\Lambda^{\prime}_{kl}= \{\lambda^{\prime}| P^{\prime}_{0k}(\lambda^{\prime})>0 \hspace{2mm} \mbox{and} \hspace{2mm} P^{\prime}_{1l}(\lambda^{\prime})>0 \}$ where $k,l=0,1$.
What happens during the task is that, regardless of basis, Alice and Bob send Charlie certain physical entities $\lambda$ and $\lambda^{\prime}$ chosen according to their probability distributions $P(\lambda)$ and $P^{\prime}(\lambda^{\prime})$, respectively. (See Fig.1)
\begin{figure}
\caption{Alice and Bob send Charlie certain physical entities $\lambda$ and $\lambda^{\prime}$ chosen according to their probability distributions $P(\lambda)$ and $P^{\prime}(\lambda^{\prime})$, respectively, regardless of the basis. What is considered in this figure is a case when $\lambda \otimes \lambda^{\prime} \in \Lambda_{01}^{00}$, which means that if $\lambda$ came from basis $0$ ($1$), then $\lambda$ is from the $0$ ($1$) state, and that if $\lambda^{\prime}$ came from basis $0$ ($1$), then $\lambda^{\prime}$ is from the $0$ ($0$) state. For a certain $\lambda \otimes \lambda^{\prime}$, Charlie has no information about the basis, so the chosen incidents are statistically evenly distributed over four possible combinations of the basis.}
\label{Fig-1}
\end{figure}
Then, Charlie selects some subset of the product $\lambda \otimes \lambda^{\prime}$ among those he received. Here, we can divide the set of all $\lambda \otimes \lambda^{\prime}$'s into 16 subsets, ${\bf \Lambda}_{ij}^{kl}= \{\lambda \otimes \lambda^{\prime}| \lambda \in \Lambda_{ij} \hspace{2mm} \mbox{and} \hspace{2mm} \lambda^{\prime} \in \Lambda^{\prime}_{kl} \}$. Suppose that steps (i) and (ii) are repeated $N$ times in total, and from among them, $M$ incidents are selected by Charlie in step (iii). Let us denote the number of incidents when $\lambda \in \Lambda_{ij}$ and $\lambda^{\prime} \in \Lambda^{\prime}_{kl}$ among the selected ones by $m(ij;kl)$. The relative frequency is given by
\begin{equation}
\tilde{m}(ij;kl)= \frac{m(ij;kl)}{\sum_{i,j,k,l}m(ij;kl)}=\frac{m(ij;kl)}{M}. \label{2} \end{equation}
Now, let us consider a $\lambda \otimes \lambda^{\prime}$, which is an element of ${\bf \Lambda}_{ij}^{kl}$ with certain $i,j,k,l$. Note that the $\lambda$ and the $\lambda^{\prime}$ have come from either one of the two bases. $\lambda \otimes \lambda^{\prime} \in {\bf \Lambda}_{ij}^{kl}$ means that if the $\lambda$ is from basis $0$ ($1$), then the $\lambda$ must be from the $i$ ($j$) state, and if the $\lambda^{\prime}$ is from basis $0$ ($1$), then the $\lambda^{\prime}$ must be from the $k$ ($l$) state. However, because of the condition of basis independence, at the step (iii), Charlie has no information about the bases from which the $\lambda$ and the $\lambda^{\prime}$ came. Thus, the probability that a certain $\lambda \otimes \lambda^{\prime}$ is chosen by Charlie is independent of the bases. Combined with the fact that $a,b$ are random, this implies that the $m(ij;kl)$ incidents are statistically evenly distributed over the four combinations of bases; that is, statistically $m(ij;kl)/4$ incidents are the case when the $\lambda \otimes \lambda^{\prime} \in {\bf \Lambda}_{ij}^{kl}$ came from bases $a$ and $b$. Therefore, the relative frequency $\tilde{m}(ij;kl|ab)$ within those incidents when $\lambda \otimes \lambda^{\prime}$ came from the bases $a$ and $b$, is independent of the bases and is equal to the relative frequency $\tilde{m}(ij;kl)$. Now, let us derive the probabilities for $x$ and $y$ conditioned with bases $a$ and $b$, $p(x,y|a,b)$; We can observe that $p(i,k|0,0)= \sum_{j,l} \tilde{m}(ij;kl)$,
$p(i,l|0,1)= \sum_{j,k} \tilde{m}(ij;kl)$, $p(j,k|1,0)= \sum_{i,l} \tilde{m}(ij;kl)$, and $p(j,l|1,1)= \sum_{i,k} \tilde{m}(ij;kl)$. We calculate the correlation $E(0,0)=p(0,0|0,0)+ p(1,1|0,0)- p(0,1|0,0)- p(1,0|0,0)= \sum_{i,j,k,l}\hspace{1mm} (1-2i)(1-2k) \hspace{1mm}\tilde{m}(ij;kl)$, and similarly $E(0,1)= \sum_{i,j,k,l}\hspace{1mm} (1-2i)(1-2l) \hspace{1mm}\tilde{m}(ij;kl)$, $E(1,0)= \sum_{i,j,k,l}\hspace{1mm} (1-2j)(1-2k) \hspace{1mm}\tilde{m}(ij;kl)$, and
$E(1,1)= \sum_{i,j,k,l}\hspace{1mm} (1-2j)(1-2l) \hspace{1mm}\tilde{m}(ij;kl)$. Now, we have the Bell function $S= \sum_{i,j,k,l}\hspace{1mm} \{(1-2i)[(1-2k)+(1-2l)]+(1-2j)[(1-2k)-(1-2l)]\} \hspace{1mm}\tilde{m}(ij;kl)$. Because the absolute value of $\{(1-2i)[(1-2k)+(1-2l)]+(1-2j)[(1-2k)-(1-2l)]\}$ is equal to or less than $2$ in any case and $ \sum_{i,j,k,l} \tilde{m}(ij;kl)=1$, we obtain $|S| \leq 2$. Note that, with respect to the calculations, the derivation here is the same as the one for Bell's inequality.
Now, let us discuss the cases when $P_{a0}(\lambda)$ and $P^{\prime}_{b0}(\lambda^{\prime})$ overlap $P_{a1}(\lambda)$ and $P^{\prime}_{b1}(\lambda^{\prime})$, respectively. Here, a given physical entity $\lambda \otimes \lambda^{\prime}$ cannot determine which state the physical entities belong to if $\lambda \otimes \lambda^{\prime}$ is in overlapping region. Thus, this case corresponds to the indeterministic local realistic model \cite{Fin82, Sta80}, which still satisfies the Bell inequality. Similarly to the indeterministic case, we can see that the Bell inequality is fulfilled even by the overlapping case, $S= E(0,0)+ E(0,1)+ E(1,0)- E(1,1)= \int \{\bar{f}(0,\lambda \otimes \lambda^{\prime})[ \bar{g}(0,\lambda \otimes \lambda^{\prime})+ \bar{g}(1,\lambda \otimes \lambda^{\prime})] + \bar{f}(1,\lambda \otimes \lambda^{\prime})[ \bar{g}(0,\lambda \otimes \lambda^{\prime})- \bar{g}(1,\lambda \otimes \lambda^{\prime})]\} P(\lambda) P^{\prime}(\lambda^{\prime}) d (\lambda \otimes \lambda^{\prime}) \leq 2.$
Here, $\bar{f}(a,\lambda \otimes \lambda^{\prime})$ ($ \bar{g}(b,\lambda \otimes \lambda^{\prime})$) is the average of $1-2x$ ($1-2y$) conditioned for basis $a$ ($b$) and $\lambda \otimes \lambda^{\prime}$, respectively. That is, $\bar{f}(a,\lambda \otimes \lambda^{\prime})= \sum_x (1-2x) p(x|a,\lambda \otimes \lambda^{\prime})$ and $\bar{g}(b,\lambda \otimes \lambda^{\prime})= \sum_y (1-2y) p(y|b,\lambda \otimes \lambda^{\prime})$, where, for example, $p(x|a,\lambda \otimes \lambda^{\prime})$ is the probability that the state is $x$ conditioned with that the basis is $a$ and physical entity is $\lambda \otimes \lambda^{\prime}$. At the second equality of this derivation, the condition of basis independence is used.
That quantum resources can perform the task is easy to see. Suppose that Alice prepares the physical entities $(0,0)= |0\rangle$, $(0,1) = |1\rangle$, $(1,0)= |+\rangle$ and $(1,1)= |-\rangle$ with the probabilities $p_{00}= p_{01}= p_{10}= p_{11}= 1/2$, respectively, and that Bob prepares $(0,0)^{\prime}= |\theta= \pi/4 \rangle$, $(0,1)^{\prime} = |\theta= 5\pi/4\rangle$, $(1,0)^{\prime}= |\theta= 3\pi/4\rangle$ and $(1,1)^{\prime}= |\theta= 7\pi/4\rangle$ with the probabilities $p^{\prime}_{00}= p^{\prime}_{01}= p^{\prime}_{10}= p^{\prime}_{11}= 1/2$, respectively, where $|\theta= \theta \rangle= \cos(\theta/2)|0\rangle+ \sin(\theta/2)|1\rangle$. This satisfies the condition of basis independence as we have seen in Section II. Charlie performs a measurement composed of $|\varphi^+ \rangle \langle \varphi^+ |$ and $1- |\varphi^+ \rangle \langle \varphi^+|$, where the Bell state is $|\varphi^+ \rangle = (1/\sqrt{2})(|00\rangle + |11\rangle)$, and announces $1$ and $0$ when the outcome is $|\varphi^+ \rangle \langle \varphi^+ |$ and $1- |\varphi^+ \rangle \langle \varphi^+|$, respectively. With the fact that the probability for Charlie to get outcome $|\varphi^+ \rangle \langle \varphi^+ |$ for $|\theta=A\rangle |\theta=A+ \Delta \rangle$ is $(1/2) \cos^2 (\Delta/2)$, one can easily see the $S= 2\sqrt{2} >2$.
\section{Discussion and conclusion.}
If Alice and Bob can perfectly prepare the quantum states as prescribed, then the condition of basis independence is fulfilled because density operators corresponding to different bases are identical. In practice, however, state preparation cannot be perfect although the prepared states may be close to the prescribed ones. Thus, the condition cannot be satisfied perfectly. However, even in practice, a way exists to prepare states satisfying the condition of basis-independence perfectly; Alice prepares an imperfect state that is close to the Bell state $|\varphi^+ \rangle = (1/\sqrt{2})(|0\rangle_{\alpha} |0 \rangle_{\beta} + |1\rangle_{\alpha} |1\rangle_{\beta})$. Then, she either performs an imperfect measurement that is close the $Z$ measurement composed of $|0\rangle \langle0|$ and $|1\rangle \langle1|$ or performs an imperfect measurement that is close the $X$ measurement composed of $|+\rangle \langle+|$ and $|-\rangle \langle-|$ on the quantum state at the $\alpha$ site. Then, the quantum state prepared at the $\beta$ site, although it is not in the prescribed state perfectly, satisfies the condition of basis independence perfectly in order to avoid faster-than-light communication, as is well known \cite{Nie00}. Now, similarly Bob prepares his (imperfect) states obeying the condition of basis independence, and Charlie does his (imperfect) measurement that is close to the prescribed one. Then, the $S$ value can still be close to $2\sqrt{2}$ completing the task because the operations are close to the prescribed ones although not perfect. Here, we can see that an experiment on entanglement swapping \cite{Zuk93,Jen02} can be transformed to that on the task proposed here if Alice and Bob perform their measurements earlier than Charlie. (The other case when Charlie performs the measurement earlier corresponds to an experiment that violates Bell's inequality. The two cases are equivalent to each other, as is well known.) Thus, practical implementation of entanglement swapping can be immediately used for the task.
The detection loophole \cite{Pea70} also applies to the task; Let us change the rule of the game by newly introducing a case when the values of the $x$ and $y$ are two. That is, Alice (Bob) chooses one of $0$,$1$, and $2$ for $x$ ($y$) in step (i). Following the same reasonings as before, the set of physical entities $\lambda \otimes \lambda^{\prime}$ are divided into $3^4=81$ subsets ${\bf \Lambda}_{ij}^{kl}$'s. The rule is also changed such that incidents with value $2$ are discarded by Alice and Bob; for example, if $\lambda \otimes \lambda^{\prime} \in {\bf \Lambda}_{02}^{01}$ and Alice's basis $a=1$, then the incidents, even if selected by Charlie, are discarded by Alice and Bob regardless of Bob's basis $b$. Let us take a illustrating example that even achieves $S=4$; the statistical measures of the sets ${\bf \Lambda}_{02}^{02}$, ${\bf \Lambda}_{02}^{20}$, ${\bf \Lambda}_{20}^{02}$, and ${\bf \Lambda}_{20}^{21}$ are all $1/4$ while the statistical measures of other sets are zero.
Let us discuss the potential advantage of our proposal over the Bell test. Because only the incidents selected by Charlie are considered, the task can be performed regardless of Charlie's detector efficiency. Thus, if we can prepare perfect states as prescribed, then we can perform a classically impossible task regardless of the detector's efficiency. If this is the case, then it is an advantage because in the case of a loophole-free violation of Bell inequality, high efficiency detectors are necessary \cite{Ebe93,Hwa96}. Because of the imperfection of the states, however, the condition of basis independence is also imperfectly obeyed. Thus, saying that the task has been performed is difficult. In order to satisfy the condition of basis independence perfectly, we need to adapt the method above, which uses highly entangled states. In this case, however, the detection loophole still exists. Thus, high-efficiency detectors are needed to perform the task.
To summarize, we proposed a task that could not be done by using any classical mechanical means but could be done with quantum resources. Under the condition of basis independence, Alice and Bob prepare some physical entities. They send the entities, while keeping data about the entities, to Charlie, who selects some of the entities. If only classical resources are allowed, the correlation between Alice's and Bob's data corresponding to the selected ones can be shown to satisfy an inequality, that is the same as Bell's inequality. It was shown that quantum resources can violate the inequality. Then, we discussed the issues of imperfect states and the potential advantage of the task.
\begin{references} \bibitem{Bel87} J. S. Bell, {\it Speakable and Unspeakable in Quantum Mechanics} (Cambridge University Press, Cambridge, UK, 1987). \bibitem{Scar09}
V. Scarani, B.-P. Helle, N. J. Cerf, M. Dusek, N. L\"utkenhaus, and M. Peev, Rev. Mod. Phys. {\bf 81}, 1301 (2009).
\bibitem{Nie00} M. A. Nielsen and I. L. Chuang, {\it Quantum Computation and Quantum Information} (Cambridge
University Press, Cambridge, UK, 2000). \bibitem{Li14} H.-W. Li, Z.-Q. Yin, W. Chen, S. Wang, G.-C. Guo, and Z.-F. Han, Phys. Rev. A {\bf 89}, 032302 (2014). \bibitem{Fin82} A. Fine, Phys. Rev. Lett. {\bf 48}, 291 (1982). \bibitem{Sta80} H. P. Stapp, Found. Phys. {\bf 10}, 767 (1980). \bibitem{Pea70} P. M. Pearle, Phys. Rev. D {\bf 2}, 1418 (1970). \bibitem{Ebe93} P. H. Eberhard, Phys. Rev. A {\bf 47}, 747 (1993). \bibitem{Hwa96} W.-Y. Hwang, I. G. Koh, and Y. D. Han, Phys. Lett. A {\bf 212}, 309 (1996). \bibitem{Zuk93} M. Zukonwski, A. Zeilinger, M. A. Horne, and A. K. Ekert, Phys. Rev. Lett. {\bf 93}, 4287 (1993). \bibitem{Jen02} T. Jennewein, G. Weihs, J.-W. Pan, and A. Zeilinger, Phys. Rev. Lett. {\bf 88}, 017903 (2002).
\end{references}
\end{document} | arXiv |
\begin{document}
\title{Generalizing Downsampling from Regular Data to Graphs
\thanks{\iftoggle{arxiv}
\begin{abstract}
Downsampling produces coarsened, multi-resolution representations of data and it is used, for example, to produce lossy compression and visualization of large images, reduce computational costs, and boost deep neural representation learning.
Unfortunately, due to their lack of a regular structure, there is still no consensus on how downsampling should apply to graphs and linked data.
Indeed reductions in graph data are still needed for the goals described above, but reduction mechanisms do not have the same focus on preserving topological structures and properties, while allowing for resolution-tuning, as is the case in regular data downsampling.
In this paper, we take a step in this direction, introducing a unifying interpretation of downsampling in regular and graph data. In particular, we define a graph coarsening mechanism which is a graph-structured counterpart of controllable equispaced coarsening mechanisms in regular data. We prove theoretical guarantees for distortion bounds on path lengths, as well as the ability to preserve key topological properties in the coarsened graphs. We leverage these concepts to define a graph pooling mechanism that we empirically assess in graph classification tasks, providing a greedy algorithm that allows efficient parallel implementation on GPUs, and showing that it compares favorably against pooling methods in literature.
\end{abstract}
\section{Introduction}
The concept of information coarsening is fundamental in the adaptive processing of data, as it provides a simple, yet effective, means to obtain multi-resolution representations of information at different levels of abstraction. In large scale problems coarsening also serves to provide computational speed-ups by solving tasks on the reduced representation, ideally with a contained loss in precision with respect to solving the original problem.
Coarsening is key in Convolutional Neural Networks~\cite[CNNs,][]{fukushima_neocognitron_1980,lecun_backpropagation_1989}, where pooling is often used to repeatedly subsample an image to extract visual feature detectors at increasing levels of abstraction (e.g., blobs, edges, parts, objects, etc). Downsampling is also popular in the adaptive processing of timeseries where, for instance, it is used in clockwork-type Recurrent Neural Networks~\cite{koutnik_clockwork_2014,carta_incremental_2021} to store information extracted at different frequencies and timescales. More recently, the Graph Convolutional Networks~\cite[GCNs,][]{micheli_neural_2009,gori_new_2005,bacciu_gentle_2020} community popularized graph reduction mechanisms as a structured counterpart of the image pooling mechanism in \emph{classical} CNNs.
The definition of a reduction mechanism that downsamples information at regular intervals between data points (e.g., a sample, a pixel, a timestamped observation, etc) is straightforward when working with images and time series. It can be achieved simply by picking up a data point every $k$ ones, where $k$ is a given reduction factor defining the distance between the sampled points in the original data, possibly aggregating the properties of non-selected point with appropriate functions. The same approach cannot be straightforwardly applied to graphs, which lack regularity and a consistent ordering among their constituent data points, i.e., the nodes. Therefore, defining a well-formed notion of downsampling for graphs becomes non-trivial.
The research community has been tackling this issue by a number of approaches, including differentiable clustering of node embeddings~\cite{ying_hierarchical_2018,bianchi_spectral_2020},
graph reductions~\cite{shuman_multiscale_2016,loukas_graph_2019}, and node ranking~\cite{cangea_towards_2018,gao_graph_2019}.
Notably, approaches like the latter select important nodes in a graph and simply discard the rest without protecting the linked structure of the network, while reduction methods typically focus on preserving structure without accounting for the role or relevance of nodes involved.
What is yet an open problem is how to define a controllable graph coarsening method, which reduces the size while preserving the overall structure by sampling \textit{representative} yet \emph{evenly} spaced elements, similarly to the approaches discussed above for image and time series reduction.
This paper provides a first approach introducing such a topology-preserving graph coarsening and its use in graph pooling. We provide mechanisms which are the graph equivalent of pooling and striding operators on regular data, accompanying our intuition with formal proofs (in the Supplementary Material) of the equivalence of such operators on graphs which model regular data.
Central to our contribution is the definition of a mechanism to find a set of nodes that are approximately equally spaced (at distance no less than $k$) in the original graph. We build on the graph-theoretic concept of Maximal $k$-Independent Sets (\kmis), that also comes with the ability to pin-point important nodes in each area of the graph. The selected nodes are then used as vertices of the reduced graph whose topology is defined in such a way that key structural properties of the original graph are well preserved. To this end, we provide theoretical guarantees regarding distance distortions between a graph and its reduction.
Additionally, we prove the reduced graph has the same number of connected components as the original.
The latter point is particularly relevant for a graph pooling mechanism as it guarantees that the structure is not broken in disconnected fragments, which can hinder the performance of neural message passing in the GCN layers.
Such properties are fundamental to ensure that the original graph is downsampled evenly throughout its structure, preserving distances and sparseness of the key focal points in the graph. By this means, the reduced graph can be used as an accurate fast estimator of the distances between nodes in the original graph, where the amount of compression can be easily regulated through the choice of the $k$ reduction factor.
Concurrently, we borrow from node-ranking methods~\cite{gao_graph_2019} to produce {\kmis}s that maximize the total weights associated to the selected nodes,
in order to preserve relevant nodes without compromising structure.
In summary, our contributions
are the following: \begin{itemize}
\item We introduce a graph coarsening method leveraging \kmis that is the graph-structured counterpart of equispaced sampling in flat data.
We provide a greedy parallel algorithm to efficiently compute the \kmis reduction, which is well suited to use in GPU accelerators
(\cref{sect:kmis}).
\item We give formal proof of equivalence of our approach to regular downsampling in convolutional neural networks, when applied to diagonal grid graphs (\cref{sect:theo} and Supplementary Material).
\item We prove theoretical guarantees on the distance distortions between a graph and its reduction. We provide also a formal complexity analysis of the introduced algorithms, proving, both theoretically and experimentally, their scalability on large real-world graphs (\cref{sect:theo} and Supplementary Material).
\item We integrate \kmis reduction both as a pooling layer and as a downsampling operator for GCNs, providing an empirical confirmation of its advantages over literature approaches on graph classification benchmarks (\cref{sect:exp}).
\end{itemize}
\section{Notation and Definitions}\label{sect:back}
We represent a graph $\gr{G}$ as a pair of disjoint sets $(V, E)$, where $V = \{1, \dots, n\}$
is its node set and $E \subset V\times V$ its edge set, with $|E| = m$.
A graph can also be represented as a symmetric matrix $\ensuremath{\mat{A}} \in \ensuremath\mathds{R}_+^{n\times n}$, such that $\ensuremath{\mat{A}}_{uv} = \ensuremath{\mat{A}}_{vu}$ is equal to a weight associated to the edge $uv \in E$ or zero if $uv \not\in E$.
The neighborhood $\ensuremath N(v)$ of $v$ is the set of nodes adjacent to it (denoted $\ensuremath N[v]$ if includes $v$ itself), and the degree $\ensuremath \operatorname{deg}(v)$ of $v$ is defined as the number of its neighbors, i.e., $\ensuremath \operatorname{deg}(v) = |\ensuremath N(v)|$.
The \emph{unweighted} distance between two nodes $u, v \in V$, denoted as $\ensuremath\operatorname{d}(u, v)$, is defined as the length of the shortest path between the two nodes. If there is no path between the two nodes, then $\ensuremath\operatorname{d}(u,v) =
\infty$.
The $k$-hop neighborhood $\ensuremath N_k(v)$ of $v$ ($\ensuremath N_k[v]$ if inclusive) is the set of nodes that can be reached by a path in $\gr{G}$ of length at most $k$. The \emph{$k$-th power} of a graph $\gr{G}^k$ is the graph where each node of $\gr{G}$ is connected to its $k$-hop neighbors.
To avoid confusion, any function may be denoted with a subscript to specify the graph on which is defined (e.g.,
$\ensuremath\operatorname{d}_\gr{G}$).
An \emph{independent set}, is a set of nodes $S \subseteq V$ such that no two of which are adjacent in $\gr{G}$. An independent set is \emph{maximal} if
is not a subset of another one in $\gr{G}$.
A (maximal) $k$-\emph{independent set} is a (maximal) independent set of $\gr{G}^k$.
\section{Graph Coarsening with {$\bm{k}$}-MWIS}\label{sect:kmis}
When dealing with signals, images, or other kinds of Euclidean data, \emph{downsampling} often amounts to \emph{keeping every $k$-th} data point, where $k$ is a given reduction factor. This means, for a generic discrete $n$-dimensional Euclidean datum, keeping a subset of its points such that every two of them are \emph{exactly} $k$ points far from each other on every of its dimensions. On graph-structured data, we lose this regularity along with the concept of dimensionality, and hence defining a new notion of downsampling that applies to graph becomes non-trivial.
Here we define a graph coarsening method that, similarly to {\it classical} downsampling, reduces the size of a graph $\gr{G}$ by a given ``factor'', by finding a set of \emph{almost} evenly spaced nodes within $\gr{G}$. These nodes will form the node set of the reduced graph, while its topology will be constructed starting from $\gr{G}$ in a way in which some of its key properties will be preserved, such as connectivity, or approximated, such as pairwise node distances.
\paragraph{Coarsening algorithm.}
Given a graph $\gr{G} = (V, E)$ and a distance $k$, we want to obtain a coarsen representation of $\gr{G}$ by first selecting a set of nodes $S \subseteq V$, that we refer to as \emph{centroids}, such that every two centroids are more than $k$ hops distant from each other, and such that no area of the graph remains unsampled; in other words, a \emph{maximal $k$-independent sets} (\kmis) of $\gr{G}$: this way, each centroid will be more than $k$ hops from every other, while the \emph{maximality} ensures every node of $G$ is within $k$ hops from a centroid.
Any MIS of a graph $\gr{G}^k$ is a \kmis of $\gr{G}$~\cite{agnarsson_powers_2003}, thus a \kmis could be na\"ively computed by known MIS algorithms, such as \citet{luby_simple_1985} or \citet{blelloch_greedy_2012}, on the $k$-th power of the adjacency matrix of $\gr{G}$. Using this approach will require $\bigO{n^2}$ space since the density of $\gr{G}^k$ increases rapidly with $k$, becoming rapidly impractical for real world graphs with millions or billions of nodes.
To overcome this problem, we introduce \cref{alg:k-mis} that efficiently computes a \kmis of $\gr{G}$ without explicitly computing its $k$-th power.
\begin{table}[tb] \setlength{\intextsep}{0pt}
\begin{algorithm}[H]
\begin{algorithmic}[1]
\Function{\kmis}{$\gr{G}$, $U$, $\ensuremath{\pi}$}
\State\algorithmicif\ $\lvert U \rvert = 0$ \algorithmicthen\ \Return $\emptyset$\label{line:k-mis-check}
\State $\ensuremath{\pi}_0 \gets \ensuremath{\pi}$\label{line:k-hop-1}
\For{$i = 1, \dots, k$}\label{line:k-hop-2}
\ParFor{$v \in U$}\label{line:k-hop-2-1}
\State $\ensuremath{\pi}_i(v) \gets \min_{u \in \ensuremath N[v]\cap U}\ \ensuremath{\pi}_{i-1}(u)$\label{line:k-hop-2-2}
\EndParFor
\EndFor
\State $S_0 \gets \{v \in U \mid \ensuremath{\pi}(v) = \ensuremath{\pi}_k(v)\}$\label{line:k-hop-3}
\For{$i = 1, \dots, k$}\label{line:k-restrict-2}
\State $S_i \gets \bigcup_{v \in S_{i-1}} \ensuremath N[v]$\label{line:k-restrict-2-1}
\EndFor
\State $R \gets U \setminus S_k$\label{line:k-restrict-3}
\State \Return $S_0 \cup {}$\Call{\kmis}{$G$, $R$, $\ensuremath{\pi}$}\label{line:k-mis-rec}
\EndFunction
\end{algorithmic}
\caption{ Parallel Greedy $k$-\textsc{MIS} algorithm, adapted from~\citet{blelloch_greedy_2012}. Given a graph $\gr{G}$, a subset of its nodes $U \subseteq V$, and a node ranking $\ensuremath{\pi}$, returns a maximal $k$-independent set in $\gr{G}$, with $k \in \ensuremath\mathds{N}$.}
\label{alg:k-mis}
\end{algorithm}
\begin{algorithm}[H]
\begin{algorithmic}[1]
\Function{Cluster}{$\gr{G} = (V, E)$, $k$, $\ensuremath{\pi}$}
\State $S \gets$ \Call{\kmis}{$\gr{G}$, $V$, $\ensuremath{\pi}$}
\State $\ensuremath{\pi}_0 \gets \ensuremath{\pi}$
\ParFor{$v \in V \setminus S$}
\State $\ensuremath{\pi}_0(v) \gets +\infty$
\EndParFor
\For{$i = 1, \dots, k$}
\ParFor{$v \in V$}
\State $\ensuremath{\pi}_i(v) \gets \min_{u \in \ensuremath N[v]}\ \ensuremath{\pi}_{i-1}(u)$
\EndParFor
\EndFor
\State \Return $\{\{u \in V \mid \ensuremath{\pi}_k(u) = \ensuremath{\pi}(v)\}\}_{v \in S}$
\EndFunction
\end{algorithmic}
\caption{Parallel \kmis partitioning algorithm. Given a graph $\gr{G}$, $k\in \ensuremath\mathds{N}$, and a node ranking $\ensuremath{\pi}$, returns a partition of $\gr{G}$.
}
\label{alg:cluster}
\end{algorithm} \end{table}
Once the \kmis $S \subseteq V$ is computed with \cref{alg:k-mis}, we construct the coarsened graph $\gr{H} = (S, E')$ as follows:
\begin{enumerate}
\item using \cref{alg:cluster}, we compute a partition $\mathcal{P}$ of $V$ of size $\lvert S \rvert$, such that
\begin{enumerate}
\item every $P\in\mathcal{P}$ contains exactly one centroid and (a subset of) its $k$-hop neighbors, and
\item for every node in $P$ there is a centroid in $P$ at distance at most $k$-hops;
\end{enumerate}
\item\label{item:edges} for every edge in $E$ we add an edge in $E'$ joining the two nearest centroids in the partitions containing the source and destination nodes. If this generates multiple edges, we coalesce them into a single one, and we aggregate their weights according to a predefined aggregation function (e.g., sum);
\item \emph{(pooling, optional)} in case of weights/labels associated to the nodes, these can also be aggregated according to the partitioning $\mathcal{P}$.
\end{enumerate}
A detailed discussion of \cref{alg:k-mis,alg:cluster} will be provided later in \cref{sect:theo}.
\begin{figure}
\caption{\emph{(first column)} Average pooling and \emph{(second to fourth columns)} our method using different ranking and aggregation functions, for varying values of $k$.}
\label{fig:mnist}
\end{figure}
\paragraph{Node ordering.}
A key property of our \kmis algorithm (similarly to the one of~\citet{blelloch_greedy_2012}) is that it is \emph{deterministic}: given a graph $\gr{G}$ and a \emph{ranking} of its nodes $\ensuremath{\pi}: V \to \{1,\dots,n\}$, that defines the position of the nodes in a given ordering, \cref{alg:k-mis} will always produce the same \kmis, for any $k \ge 0$. This property has some interesting consequences:
\begin{itemize}
\item The ranking $\ensuremath{\pi}$ can be used to lead \cref{alg:k-mis} to greedily include nodes having a higher rank under a given order of importance, such as a centrality measure, a task-dependent relevance, or a (possibly learned) \emph{scoring value}. (Note that the computation of the ranking can impact the complexity of the algorithm.)
\item If the ranking can be \emph{uniquely} determined by the nodes themselves (e.g., in function
of their attributes or
their neighbors), \cref{alg:k-mis,alg:cluster} become injective and hence, permutation invariant.\footnote{Notice that, in our setting, if $\ensuremath{\pi}: V \to \{1, \dots, n\}$ is injective, then it is also bijective and, hence, a permutation.} This can be obtained
by ranking the nodes with respect to a score computed by means of a (sufficiently expressive) GCN, as learning injective functions over the nodes in a graph is a problem strictly related to the one of graph isomorphism, a topic that is gaining a lot of traction in the graph learning community~\cite{morris_weisfeiler_2019,xu_how_2019,maron_provably_2019,loukas_how_2020,geerts_expressiveness_2021,papp_theoretical_2022}.
\item A properly chosen ranking can produce a marginally greater total score of the selected nodes with respect to the one that we would get by greedily selecting the top scoring ones. This aspect
will be discussed more in detail in \cref{sect:theo}.
\end{itemize}
We now provide two examples on how we can change the ranking of the nodes to prioritize salient aspects according to a specific preference. Examples are conducted on the graph defined by the first sample of the MNIST dataset~\cite{lecun_mnist_2010}, a $28\times 28$ monochromatic image (first row of \cref{fig:mnist}) where every pixel is connected to the ones in the same pixel row, column or diagonal.
First, we simulate the typical downsampling on images (also known as \emph{average pooling}~\cite{fukushima_neocognitron_1980}), where squared partitions of $p\times p$ pixels are averaged together (first column of \cref{fig:mnist}). To do this, we set the ranking $\ensuremath{\pi}$ of \cref{alg:cluster} as the \emph{lexicographic} ordering: given $(i, j)$ the coordinate of a pixel, we rank the nodes in decreasing order of $28i + j$. The resulting reduction is in the second column of \cref{fig:mnist}: averaging intensities of pixels in the same partition produces a coarsened graph which is identical to classical downsampling. Note that this result is partly due to the fact that \cref{alg:cluster} also makes use of $\ensuremath{\pi}$ to define the clustering, such that the nodes in a partition have always a lower rank with respect to the centroid in the same partition.
Secondly, we rank nodes in decreasing order of \emph{intensity}, thus prioritizing the pixels (i.e., the nodes) belonging to the drawn digit. Here we show two different results: the first, where we average the lightness and coordinates of the nodes in the same clusters (third column of \cref{fig:mnist}), and a second one, where we just keep the ones belonging to the nodes in the \kmis (fourth column). We see that the reduced graphs indeed prioritized the digit against other pixels, producing a coarsened representation where the digit is also remarkably recognizable.
\section{Theoretical Analysis and Results}\label{sect:theo}
\paragraph{Regular downsampling.}
Downsampling plays a key role in Convolutional Neural Networks~\cite[CNNs,][]{goodfellow_deep_2016}, where it is adopted, for instance, in \emph{strided} convolutions and \emph{pooling} layers. In strided convolutions, an input tensor (e.g., a time series, an image, or a voxel grid) is reduced by applying the convolved filter every $s$-th of its entries, along every dimension, while skipping the others. In pooling layers, instead, tensors are reduced by summarizing every $p$-sided sub-tensors, taken at regular intervals. (More specific reductions are also possible, where distinct intervals are used for every dimension.)
We can show that, on $n$-dimensional diagonal grid graphs (i.e., grids where nodes are also diagonally adjacent), \cref{alg:k-mis,alg:cluster} behave \emph{exactly} as the aforementioned downsampling strategies, if we rank their nodes by their position in lexicographic order. This is of particular interest as the adjacencies in these graphs can represent the receptive fields of a single convolutional layer when applied to a some regular data of the same shape, like
images (2-dimensional) or voxel grids (3-dimensional). Specifically, if $\gr{G} = (V, E)$ is a diagonal grid constructed using the entries of a given tensor as nodes, and $\ensuremath{\pi}$ is the ranking of these entries in lexicographic order of their position indices, we have that
\begin{enumerate}
\item $k\text{-MIS}(\gr{G}, V, \ensuremath{\pi})$ selects the same entries of a strided convolution with $s=k+1$,
\item $\textsc{Cluster}(\gr{G}, k, \ensuremath{\pi})$ partitions the tensor as a pooling layer with $p = k+1$, and
\item the reduced graph obtained by contracting the resulting partition is again a diagonal grid of the same dimensions of their output tensor. \end{enumerate}
A formal restatement and proof of these properties are provided in the Supplementary Material, while in \cref{fig:mnist} we show an example of the equivalence between pooling (first column) and our reduction method (second column).
\paragraph{Connectivity of the reduced graph.}
For the sake of conciseness, hereafter we denote with $(\gr{H}, \rho) = \ensuremath\mathcal{R}(\gr{G}, k)$ the function reducing a graph $\gr{G}$ by contracting the clusters obtained with \cref{alg:cluster}, as described in \cref{sect:kmis}. The term $\gr{H} = (S, E')$ denotes the reduced graph, where $S$ is the \kmis of $\gr{G}$, while $\rho: V \to S$ is the function mapping every node to the (exactly one) centroid in its cluster. The following results are invariant with respect to the ranking parameter and the aggregation function used to reduce the edges or the nodes.
We follow a simple observation: for every edge in $uv \in E'$ with $u \neq v$, the nodes $u$ and $v$ are within $2k + 1$ hops in $\gr{G}$, since two nodes in $S$ are connected in $\gr{H}$ only if an edge in $\gr{G}$ crosses their two clusters. This property, combined with the lower bound implicitly defined by the \kmis, yields the following bounds.
\begin{remark}\label{rmk:centroid_hops} For any $uv \in E(\gr{H})$ such that $u \neq v$, we have that $ k + 1 \le \ensuremath\operatorname{d}_\gr{G}(u, v) \le 2k + 1$. \end{remark}
An example of this property is shown in \cref{fig:minnesota}, where bounds in \cref{rmk:centroid_hops} apply for the Minnesota road network~\cite{davis_university_2011} reduced with different values of $k$.
\begin{figure*}
\caption{Minnesota road network~\cite{davis_university_2011} reduced with different values of $k$. For $k = 0$, the two bounds coincide, as the graph is not reduced at all. For $k = 1$, the real distance covered by an edge is polarized (is either 2 or 3). For greater values of $k$, the edges' real distance span over all the range $[k +1, 2k + 1] \cap \ensuremath\mathds{N}$.}
\label{fig:minnesota}
\end{figure*}
From the above observation, we can obtain the two following properties.
\begin{proposition}\label{thm:length}
Let $\gr{G}$ be a connected graph and $(\gr{H}, \ensuremath\rho) = \ensuremath\mathcal{R}\big(\gr{G}, k\big)$, with $k \ge 0$. Then, $\forall u, v \in V(\gr{G})$,
\[
\ensuremath\operatorname{d}_{\gr{H}}(\ensuremath\rho(u), \ensuremath\rho(v)) \le \ensuremath\operatorname{d}_\gr{G}(u, v) \le (2k + 1)\ensuremath\operatorname{d}_{\gr{H}}(\ensuremath\rho(u), \ensuremath\rho(v)) + 2k.
\] \end{proposition}
\begin{corollary}\label{th:cc}
For any $k\ge 0$, $\gr{G}$ and $\gr{H} = \mathcal{R}\big(\gr{G}, k\big)$ have the same number of connected components. \end{corollary}
The full proofs are provided in the Supplementary Material.
Both \cref{thm:length,th:cc} are fundamental in our proposal of using \kmis reduction as a pooling method in Graph Neural Networks. In particular:
\begin{enumerate*}[label=\emph{(\roman*)}]
\item differently from several other pooling techniques~\cite{cangea_towards_2018,gao_graph_2019,knyazev_understanding_2019,lee_self-attention_2019,zhang_structure-feature_2020,ranjan_asap_2020,ma_path_2020}, we can guarantee that the input graph is not divided in multiple components, and that, if applied repeatedly, our method will eventually produce a single representation node for the whole graph;
\item when training with batches of graphs at a time, our method guarantees also that different graphs are not joined together. \end{enumerate*}
\paragraph{Algorithm discussion and complexity.}
In order to avoid computing the $k$-th graph power of a possibly large-scale graph, \cref{alg:k-mis} modifies the one by \citet[][
Algorithm~2, also restated in the Supplementary Material]{blelloch_greedy_2012} to compute the \kmis without explicitly generating every $k$-hop neighborhood. Given a graph $\gr{G} = (V, E)$, a subset of its nodes $U \subseteq V$, and a (injective) node mapping $\ensuremath{\pi}: V \to \{1, \dots, n\}$ (that we can consider as a {ranking} of the nodes under a given permutation), \Cref{alg:k-mis} works as follows:
\begin{enumerate}
\item\label{item:step-1} if $U\subseteq V$ is not empty, in
\cref{line:k-hop-1,line:k-hop-2,line:k-hop-2-1,line:k-hop-2-2,line:k-hop-3}
we find the set of nodes $S_0$ with minimum rank among their $k$-hop neighbors (i.e., their neighbors in $\gr{G}^k$). This is done with $k$ steps of label propagation such that, at each step, every node takes the minimum label found within their
($1$-hop) neighbors. We only propagate labels belonging to nodes still in $U$;
\item\label{item:step-2} in \cref{line:k-restrict-2,line:k-restrict-2-1,line:k-restrict-3} we remove from $U$ all the nodes that are at most $k$-hops from a node in $S_0$ (i.e., all their neighbors in $\gr{G}^k$). This is also done with $k$ steps of label propagation starting from the nodes in $S_0$, where this time the propagated label is a flag signaling that the node shall be removed;
\item\label{item:step-3} finally, the algorithm makes a recursive call in \cref{line:k-mis-rec} using only the remaining nodes. The resulting set is merged with $S_0$ and returned. \end{enumerate}
It is easy to see that, if $k=1$, steps \labelcref{item:step-1,item:step-2,item:step-3} become exactly Blelloch's algorithm,
whereas by taking a general $k$ every step is extended to consider $k$-hop neighbors of $\gr{G}$, thus efficiently emulating Blelloch's algorithm on $\gr{G}^k$.
As for complexity, \citet{blelloch_greedy_2012} propose several trade-offs between \emph{work} and \emph{depth} on a \emph{concurrent-read/concurrent-write} PRAM model (CRCW, with minimum priority concurrent write). Here, we consider one version (Algorithm~2 from \citet{blelloch_greedy_2012})
which allows an efficient parallel implementation with $\bigO{m}$ work and $\bigO{\log^3 n}$ depth with high probability~\cite[see][Lemma~4.2]{blelloch_greedy_2012}, and most closely resembles the structure of \cref{alg:k-mis}.
Our algorithm introduces a factor $k$ (compared to the one of \citet{blelloch_greedy_2012})
on the operations performed on lines~\cref{line:k-hop-1,line:k-hop-2,line:k-hop-2-1,line:k-hop-2-2,line:k-hop-3} and~\cref{line:k-restrict-2,line:k-restrict-2-1,line:k-restrict-3} to compute the $k$-hop neighborhood. It follows that the work and depth of \cref{alg:k-mis} are bounded by $k$ times that of Blelloch's algorithm, i.e., $\bigO{k(n+m)}$ work and $\bigO{k \log^3 n}$ depth w.h.p., where an extra $\bigO{n}$ work is needed to generate the additional vector of labels, which is modified every $k$ iterations.
Regarding \cref{alg:cluster}, after computing the \kmis, the algorithm performs $k$ steps of label propagation, which add $\bigO{k(n+m)}$ work and $\bigO{k\log n}$ depth to the total computation. Total space consumption is $\bigO{n+m}$, comprising input and $\bigO{1}$ label vectors of size $\bigO{n}$.
\begin{proposition} Given a graph $\gr{G}$, an integer $k \in \ensuremath\mathds{N}$, and a random ranking of the nodes $\ensuremath{\pi}$, both \cref{alg:k-mis,alg:cluster} can be implemented to run on a CRCW PRAM using $\bigO{k(n + m)}$ work, $\bigO{k\log^3 n}$ depth, and $\bigO{n+m}$ space. The depth bound holds w.h.p. \end{proposition}
\paragraph{Bounds on the total weight.}
In any greedy MIS algorithm, whenever we add a node to the independent set we have to remove all of its neighbors from the graph. Having observed this, a typical heuristic to compute larger-weight independent sets is to select nodes with high weight and low degree~\cite{caro_new_1979,wei_lower_1981}. Following this intuition, \citet{sakai_note_2003} proposed the following rules: given $\vec{x} \in \ensuremath\mathds{R}_+^n$ a vector of positive weights associated to each node, add to the independent set the node $v$ maximizing either
\begin{enumerate*}[label=\emph{(\roman*)}]
\item\label{item:sakai-1} $\vec{x}_v/(\ensuremath \operatorname{deg}(v) + 1)$, or
\item\label{item:sakai-2} $\vec{x}_v/(\sum_{u \in \ensuremath N[v]} \vec{x}_u)$. \end{enumerate*}
Both rules can be trivially extended to $k$-hop neighborhoods by computing $\gr{G}^k$, which would however require $\bigO{n^2}$ space, unless done sequentially. Parallel computation of the \emph{neighborhood function} $\ensuremath \operatorname{deg}_k(v) = \lvert\ensuremath N_k(v)\rvert$ in limited space can be achieved only by resorting to approximations, e.g. using Monte Carlo methods~\cite{cohen_size-estimation_1997} or approximate sets representations~\cite{palmer_anf_2002,boldi_hyperanf_2011}, and still this would not extend to approximate rule~\ref{item:sakai-2}.
To overcome these limitations, we overestimate the sum of the weights in the $k$-hop neighborhood of each node, by computing instead $\vec{c}_k = (\ensuremath{\mat{A}} + \ensuremath\mat{I})^k\vec{x} \in \ensuremath\mathds{R}_+^n$, where $\ensuremath{\mat{A}}, \ensuremath\mat{I} \in \{0, 1\}^{n\times n}$ are, respectively, the adjacency and the identity matrices. The matrix $(\ensuremath{\mat{A}} + \ensuremath\mat{I})^k \in \ensuremath\mathds{N}_0^{n\times n}$ represents the number of $k$-walks (i.e., sequences of adjacent nodes of length $k$) from every pair of nodes in the graph. Clearly, $[(\ensuremath{\mat{A}} + \ensuremath\mat{I})^k]_{uv} \ge 1$ if $v \in \ensuremath N_k[u]$, while the equality holds for every pair of nodes for $k=1$. When $\vec{x} = \vec{1}$, $\vec{c}_k$ is equal to the $k$-path centrality~\cite{sade_sociometrics_1989,borgatti_graph-theoretic_2006}. Notice that we do not need to compute $(\ensuremath{\mat{A}} + \ensuremath\mat{I})^k$ explicitly, as $\vec{c}_k$ can be obtained with a sequence of $k$ matrix-vector products, that can be computed in $\bigO{n+m}$ space, $\bigO{k(n+m)}$ work and $\bigO{k\log n}$ depth.
In the following, we provide a generalization of the bounds of \citet{sakai_note_2003} when a \kmis is computed by \cref{alg:k-mis} with the ranking defined by rules \ref{item:sakai-1}-\ref{item:sakai-2} approximated by the $k$-walk matrix $\vec{c}_k$. We remark that, for $k=1$, the following theorems are providing the same bounds as the one given by \citet{sakai_note_2003}. The full proofs can be found in the Supplementary Material.
\begin{theorem}\label{th:bound-k-degree} Let $\gr{G} = (V, E)$ be a graph, with (unweighted) adjacency matrix $\ensuremath{\mat{A}} \in \{0, 1 \}^{n\times n}$ and with $\vec{x} \in \ensuremath\mathds{R}_+^n$ representing a vector of positive node weights. Let $k \in \ensuremath\mathds{N}$ be an integer, then define $w: V \to \ensuremath\mathds{R}_+$ as
\begin{align}\label{eq:rank-k-degree}
w(v) = \frac{\vec{x}_v}{[(\ensuremath{\mat{A}} + \ensuremath\mat{I})^k\vec{1}]_v}, \end{align}
and $\ensuremath{\pi}_{w}$ as the ranking of the nodes in decreasing order of $w$. Then, $k\textnormal{-MIS}(\gr{G}, V, \ensuremath{\pi}_{w})$ outputs a maximal $k$-independent set $S$ such that $\sum_{u \in S} \vec{x}_u \ge \sum_{v \in V} w(v)$. \end{theorem}
\begin{theorem}\label{th:bound-k-weights} Let $\gr{G} = (V, E)$ be a graph, with (unweighted) adjacency matrix $\ensuremath{\mat{A}} \in \{0, 1 \}^{n\times n}$ and with $\vec{x} \in \ensuremath\mathds{R}_+^n$ representing a vector of positive node weights. Let $k \in \ensuremath\mathds{N}$ be an integer, then define $w: V \to \ensuremath\mathds{R}_+$ as
\begin{align}\label{eq:rank-k-weights}
w(v) = \displaystyle\frac{\vec{x}_v}{[(\ensuremath{\mat{A}} + \ensuremath\mat{I})^k\vec{x}]_v}, \end{align}
and $\ensuremath{\pi}_{w}$ as the ranking of the nodes in decreasing order of $w$. Then, $k\textnormal{-MIS}(\gr{G}, V, \ensuremath{\pi}_{w})$ outputs a maximal $k$-independent set $S$ such that $\sum_{u \in S} \vec{x}_u \ge \sum_{v \in V} w(v)\cdot\vec{x}_v$. \end{theorem}
\begin{theorem}\label{th:ratio} Let $\gr{G} = (V, E)$ be a non-empty graph with positive node weights $\vec{x} \in \ensuremath\mathds{R}_+^n$, and let $\ensuremath{\pi}_w$ be a ranking defined as in \cref{th:bound-k-degree} or \ref{th:bound-k-weights} for any given $k\in \ensuremath\mathds{N}$. Then
$ \sum_{v\in S} \vec{x}_v \ge \alpha(\gr{G}^k)/\Delta_k, $
where $S = k\textnormal{-MIS}(\gr{G}, V, \ensuremath{\pi}_w)$ and $\Delta_k = \max_{v \in V}\ [(\ensuremath{\mat{A}} + \ensuremath\mat{I})^k\vec{1}]_v$. \end{theorem}
Recalling that $\alpha(\gr{G}^k)$ is the optimal solution, \cref{th:ratio} shows that our heuristics guarantee a $\Delta_k^{-1}$ approximation. This bound degrades very quickly as the value of $k$ increases, since the number of $k$-walks may exceed the total number of nodes in the graph. In the Supplementary Material we show that, in practice, the total weight produced by \cref{alg:k-mis} is on par with respect to the one obtained using the exact neighborhood function for low values of $k$. This aspect is of practical value as in general the $k$ values used for graph pooling are on the low-end.
\section{Related Works} \label{sect:relate}
\paragraph{Maximal $\bm{k}$-Independent Sets.}
Computing
a \kmis can be trivially done in (superlinear) polynomial time and space using matrix powers~\cite{agnarsson_powers_2003} and any greedy MIS algorithm~\cite[e.g.,][]{luby_simple_1985,blelloch_greedy_2012}.
\citet{koerts_k-independent_2021} proposed a formulation of the problem both as an integer linear program and as a semi-definite program, but still relying on the $k$-th power of the input graph.
Several papers propose efficient algorithms to solve the {maximum} (weighted or unweighted) $k$-IS problem on specific classes of graphs~\cite{agnarsson_powers_2000,agnarsson_powers_2003,eto_distance-d_2014,bhattacharya_generalized_1999,duckworth_large_2003,pal_sequential_1996,hsiao_efficient_1992,hota_efficient_2001,saha_maximum_2003}, which fall beyond the scope of this article.
To the best of our knowledge, the only other parallel algorithm for computing a maximal $k$-independent set was proposed by \citet{bell_exposing_2012} as a generalization of the one of \citet{luby_simple_1985} for $k \ge 1$. This algorithm is essentially the same as \cref{alg:k-mis,alg:cluster}, but without the ranking argument, making the algorithm non-deterministic, as the nodes are always extracted in a random order.
\paragraph{Graph Coarsening and Reduction.}
MISs (i.e., with $k=1$) were adopted as a first sampling step in \citet{barnard_fast_1994}, although their final reduction step may not preserve the connectivity of the graph.
Using MIS was also suggested by \citet{shuman_multiscale_2016} as an alternative sampling step for their graph reduction method. The spectral reduction proposed by \citet[neighborhood variant]{loukas_graph_2019} does not use sampling as a first reduction step, but sequentially contracts node neighborhoods until a halting condition is reached, performing similar steps to the classical greedy algorithm for maximum-weight independent sets.
\paragraph{Graph Pooling.} In a contemporary and independent work, \citet{stanovic_maximal_2022} introduced a pooling mechanism based on maximal independent (vertex) sets, named MIVSPool. Their method is analogous to ours, but restricted to the case of \kmis[1], that they compute using the parallel algorithm of \citet{meer_stochastic_1989}. Another related model is \textsc{EdgePool}~\cite{diehl_towards_2019}, which computes instead a maximal matching, i.e., a maximal independent set of edges, selecting the edges depending on a learned scoring function. \citet{nouranizadeh_maximum_2021} also proposed a pooling method that constructs an independent set maximizing the mutual information between the original and the reduced graph. To do so, the authors leverage on a sequential algorithm with cubic time complexity and also no guarantees that the resulting set is maximal. Apart from a few other cases~\cite{dhillon_weighted_2007,luzhnica_clique_2019,ma_graph_2019,wang_haar_2020,bacciu_non-negative_2019,bacciu_k-plex_2021,bianchi_hierarchical_2020}, pooling in Graph Neural Networks (GNNs) usually entails an adaptive approach, typically realized by means of another neural network. These pooling methods can be divided in two types: \emph{dense} and \emph{sparse}. Dense methods, such as \textsc{DiffPool}~\cite{ying_hierarchical_2018}, \textsc{MinCutPool}~\cite{bianchi_spectral_2020,bianchi_hierarchical_2020}, \textsc{MemPool}~\cite{khasahmadi_memory-based_2019}, \textsc{StructPool}~\cite{yuan_structpool_2019}, and \textsc{DMoN}~\cite{tsitsulin_graph_2022}, compute for each node a soft-assignment to a fixed number of clusters defined by a reduction factor $r \in (0, 1)$, thus generating a matrix requiring $\bigO{rn^2}$ space. Sparse methods, such as \textsc{gPool/TopKPool}~\cite{gao_graph_2019,cangea_towards_2018}, \textsc{SAGPool}~\cite{lee_self-attention_2019,knyazev_understanding_2019}, \textsc{GSAPool}~\cite{zhang_structure-feature_2020}, \textsc{ASAPool}~\cite{ranjan_asap_2020}, \textsc{PANPool}~\cite{ma_path_2020}, \textsc{iPool}~\cite{gao_ipoolinformation-based_2021}, and \textsc{TAGPool}~\cite{gao_topology-aware_2021}, instead, compute a score for each node (requiring $\bigO{n}$ space), and reduce the graph by keeping only the top $\lceil rn \rceil$ scoring ones and dropping the rest. Although scalable, these methods provide no theoretical guarantees regarding the preservation of connectivity of the reduced graph, as the $n - \lceil rn \rceil$ dropped nodes may disconnect the graph.
\section{Experimental Analysis} \label{sect:exp}
\begin{table*}[tb]
\centering
\small \begin{tabular}{lr@{$\,\pm\,$}lr@{$\,\pm\,$}lr@{$\,\pm\,$}lr@{$\,\pm\,$}lr@{$\,\pm\,$}l} \toprule Model & \multicolumn{2}{c}{DD} & \multicolumn{2}{c}{REDDIT-B} & \multicolumn{2}{c}{REDDIT-5K} & \multicolumn{2}{c}{REDDIT-12K} & \multicolumn{2}{c}{GITHUB} \\ \midrule \model{Baseline} & 75.51 & 1.07 & 78.40 & 8.68 & \ \, 48.32 & 2.38 & \ 45.04 & 6.63 & {\bfseries 69.89} & {\bfseries 0.28} \\
\model{BDO} & \textit{\bfseries 76.69} & \textit{\bfseries 1.79} & 85.63 & 1.43 & 45.95 & 5.49 & 41.89 & 7.14 & 65.64 & 0.90 \\ \model{Graclus} & 75.17 & 2.11 & 84.05 & 5.81 & 43.22 & 12.24 & 43.08 & 9.32 & 67.64 & 0.57 \\ \model{EdgePool} & 74.70 & 1.57 & 85.98 & 1.57 & 52.44 & 1.11 & {\bfseries 47.58} & {\bfseries 0.78} & \textit{\bfseries 68.72} & \textit{\bfseries 0.52} \\ \model{TopKPool} & 74.92 & 2.03 & 81.10 & 3.82 & 45.28 & 3.88 & 38.55 & 2.35 & 65.93 & 0.45\\ \model{SAGPool} & 73.26 & 2.26 & 84.90 & 3.94 & 46.29 & 5.61 & 42.30 & 3.70 & 64.29 & 5.70\\ \model{ASAPool} & 73.73 & 2.18 & 78.37 & 5.22 & 39.53 & 7.76 & 39.14 & 3.58 & 66.98 & 0.96\\ \model{PANPool} & 73.26 & 1.94 & 77.44 & 4.95 & 46.04 & 3.78 & 40.97 & 3.02 & 62.48 & 2.84\\\midrule \kmis \emph{(strided)} & 76.44 & 1.50 & 86.32 & 1.90 & {\bfseries 54.30} & {\bfseries 0.53} & 46.06 & 0.58 & 67.87 & 0.48 \\ \kmis \emph{(max-pool)} & {\bfseries 76.91} & {\bfseries 1.06} & {\bfseries 87.57} & {\bfseries 1.96} & 53.44 & 1.52 & \textit{\bfseries 47.51} & \textit{\bfseries 0.99} & 68.24 & 0.94 \\ \kmis \emph{(mean-pool)} & 73.56 & 1.19 & \textit{\bfseries 86.98} & \textit{\bfseries 1.13} & \textit{\bfseries 54.00} & \textit{\bfseries 0.94} & 46.73 & 0.85 & 68.60 & 0.67 \\ \bottomrule \end{tabular}
\caption{Classification accuracy on selected benchmark datasets \emph{(mean{\footnotesize $\bm{\,\pm\,}$}std)}} \label{tab:accuracy} \end{table*}
\cref{tab:accuracy} summarizes the average classification accuracy obtained on selected classification benchmarks using the same underlying Graph Neural Networks (GNNs) and different kinds of pooling mechanisms. For classification tasks, we chose those datasets having the highest number of nodes from in the \emph{TUDataset}~\cite{morris_tudataset_2020} collection (i.e., DD~\cite{dobson_distinguishing_2003}, GITHUB-STARGAZERS~\cite{rozemberczki_karate_2020}, REDDIT-BINARY and -MULTI-5K/12K~\cite{yanardag_deep_2015}),
where pooling layers may prove more useful. All datasets were divided in \emph{training} (70\%), \emph{validation} (10\%), and \emph{test} (20\%) sets using a randomized stratified split with fixed seed. All models have the same general architecture: 3 GNNs (optionally) interleaved by 2 layers of pooling, a global pooling method (\emph{sum} and \emph{max}), and a final MLP with dropout~\cite{srivastava_dropout_2014} as classifier. All models were trained using Adam optimizer~\cite{kingma_adam_2017}. We performed a model selection using the training and validation split, and then we computed the average test set classification accuracy obtained by the best configuration, on 10 inference runs using different seed values. The hyper-parameter concerning the reduction factor ($k$ in our case, or $r$ for other methods) has been chosen among the other parameters during the model selection phase. All models have been implemented and trained using PyTorch~\cite{paszke_pytorch_2019} and PyTorch Geometric~\cite{fey_fast_2019}. A detailed description of the datasets, models, and experimental settings are provided in the Supplementary Material, together with additional experiments regarding the controlled scaling and efficiency of our method, showing that it can reduce graphs with over 100 million edges in less than 10 seconds.
We compared our reduction method against different kinds of pooling layers readily available on the PyG library (we avoided \textsc{DiffPool}-like dense methods as they do not scale well on the selected datasets), and also against our own method using random rankings of the nodes, as done in the aggregation scheme of \citet[BDO,][]{bell_exposing_2012}. This method, along with the baseline (with no pooling) and \textsc{Graclus}, are the only compared architectures that require no additional parameters.
For our method, the node scoring function is computed by means of a sigmoid-activated linear layer having as input the features of the nodes.
As in the other parametric methods, the feature vectors of the nodes are multiplied by the final score beforehand,
to make the scoring function end-to-end trainable. The computed scores constitute the node weights and, as described in \cref{sect:theo}, the resulting ranking is obtained according to \cref{eq:rank-k-weights}. The reduction is performed in two settings: \emph{strided}, in which we perform no aggregation,
and \emph{pool}, in which we aggregate the feature vectors in each partition using, respectively, \emph{max} and \emph{mean} aggregations.
Looking at \cref{tab:accuracy} (where the top two accuracy scores for each dataset are in boldface) it is immediately evident how the proposed \kmis-based approaches obtain consistently high accuracy, suggesting that the
\emph{evenly-spaced} centroid selection is indeed able to capture essential properties of each graph.
On the other hand, the random permutation variant of \citet{bell_exposing_2012},
seem to overall perform worse than the other \kmis-based strategies, while still obtaining a considerable result on DD and REDDIT-B. This suggests that exploiting the ranking function of \cref{alg:k-mis} to select relevant nodes is indeed able to improve the representativeness of the downsampled graph.
It is also particularly noteworthy how one of the best performing model, \textsc{EdgePool}, is also the only other parametric pooling method to preserve the connectivity of the graph, as its reduction step consists of contracting a maximal matching. This highlights the importance of preserving the connectivity of the network when performing pooling in GNNs, while also suggesting that evenly-spaced reductions can benefit graph representation learning tasks.
Finally, we observe a remarkable performance of the baseline algorithm (no pooling) on the GITHUB
dataset: we may speculate that the graphs are simple enough to not require pooling, yet at the same time \kmis approaches obtains competitive accuracy, suggesting it is a reliable and versatile choice.
\section{Conclusions}
We introduced a new general graph coarsening approach that
aims to preserve fundamental topological properties of the original graph, acting like a structured counterpart of downsampling methods for regular data.
The coarsening reduction can be regulated by the parameter $k$, going from the original graph, when $k=0$, to up to a single node as $k$ approaches the graph's diameter, shrinking graphs uniformly as pairwise distances maintain a stretch controlled by $k$. Furthermore, we showed how this parameter generalizes to the pooling and stride intervals when applied to diagonal grid graphs.
The algorithm is designed to provide such guarantees while at the same time allowing a scalable parallel implementation, which processes graphs with up to 100 million edges in just a few seconds on a single GPU.
The empirical analysis provided evidence of effectiveness of our \kmis pooling in several graph classification benchmarks, showing superior performance with respect to related parametric and non-parametric methods from the literature.
This approach fills a methodological gap between reduction techniques for structured data and their rigorous counterparts on regular data. Given its generality and scalability, it has potential of positively impacting a plethora of com\-pu\-ta\-tion\-al\-ly-intense applications for large scale networks, such as graph visualization, 3D mesh simplification, and classification.
\begin{small}
\end{small}
\iftoggle{arxiv}{
\break \appendix
\begin{center}
\LARGE\bf Supplementary Material \end{center}
\section{Blelloch's Algorithm}\label{sec:mis-alg}
In \cref{alg:seq-mis,alg:mis} we restate, respectively, the sequential and its equivalent parallel greedy MIS algorithm, as proposed by \citet[Algorithm~1 and 2]{blelloch_greedy_2012}.
\begin{table}[h] \setlength{\intextsep}{0pt} \begin{algorithm}[H] \begin{algorithmic}[1]
\Function{MIS}{$\gr{G} = (V, E)$, $\ensuremath{\pi}$}
\State\algorithmicif\ $\lvert V \rvert = 0$ \algorithmicthen\ \Return $\emptyset$
\State $v \gets \operatorname{argmin}_{u \in V} \ensuremath{\pi}(u)$
\State $R \gets V \setminus \ensuremath N[v]$
\State\Return $\{v\} \cup {}$\Call{MIS}{$\gr{G}[R]$, $\ensuremath{\pi}$}
\EndFunction \end{algorithmic} \caption{Sequential Greedy \textsc{MIS} algorithm, from~\citet{blelloch_greedy_2012}. Given a graph $\gr{G}$ and a node ranking $\ensuremath{\pi}$, returns a maximal independent set in $\gr{G}$.} \label{alg:seq-mis} \end{algorithm}
\begin{algorithm}[H] \begin{algorithmic}[1]
\Function{MIS}{$\gr{G} = (V, E)$, $\ensuremath{\pi}$}
\State\algorithmicif\ $\lvert V \rvert = 0$ \algorithmicthen\ \Return $\emptyset$\label{line:mis-check}
\State $S \gets \{v \in V \mid\forall u \in \ensuremath N(v).\ \ensuremath{\pi}(v) < \ensuremath{\pi}(u) \}$\label{line:mis-hop}
\State $R \gets V \setminus \bigcup_{v \in S} \ensuremath N[v]$\label{line:mis-restrict}
\State\Return $S \cup {}$\Call{MIS}{$\gr{G}[R]$, $\ensuremath{\pi}$}\label{line:mis-rec}
\EndFunction \end{algorithmic} \caption{Parallel Greedy \textsc{MIS} algorithm, from~\citet{blelloch_greedy_2012}. Given a graph $\gr{G}$ and a node ranking $\ensuremath{\pi}$, returns a maximal independent set in $\gr{G}$.} \label{alg:mis} \end{algorithm} \end{table}
\section{Deferred Proofs}
\subsection{Equivalence with downsampling on regular data}
Given the tensors $\tens{V} \in \ensuremath\mathds{R}^{d_1 \times \dots \times d_n \times f}$, $\tens{K} \in \ensuremath\mathds{R}^{c_1 \times \dots \times c_n \times f \times g}$, and $\tens{Z}\in \ensuremath\mathds{R}^{(d_1 - c_1 + 1) \times \dots \times (d_n - c_n + 1) \times g}$, a $n$-dimensional multi-channel convolution~\cite{goodfellow_deep_2016}, can be defined as
\begin{align*}
\tens{Z}_{i_1, \dots, i_n, k} &= \big[\!\operatorname{conv}(\tens{V}, \tens{K})\big]_{i_1, \dots, i_n, k} \\
&=\sum_{j_1, \dots, j_n, h} \tens{V}_{i_1 + j_1, \dots, i_n + j_n, h} \cdot \tens{K}_{j_1, \dots, j_n, h, k}, \end{align*}
where $\tens{V}$, $\tens{Z}$, and $\tens{K}$, represent, respectively, the \emph{input}, the \emph{output}, and the \emph{kernel} tensors. (All tensor indices start from~0.) Typically, a zero-padding is also applied to the input tensor, in a way to produce an output tensor with the same shape of the input one~\cite[also known as ``same'' convolution,][]{goodfellow_deep_2016}.
A \emph{strided} convolution, with stride $s \ge 1$, applies the kernel every $s$-th entry of the input tensor, along every (non-channel) dimension, skipping the other entries, i.e.,
\begin{align*}
\tens{Z}_{i_1, \dots, i_n, k} &= \big[\!\operatorname{conv}(\tens{V}, \tens{K}, s)\big]_{i_1, \dots, i_n, k} \\
&= \sum_{j_1, \dots, j_n, h} \tens{V}_{s\cdot i_1 + j_1, \dots, s\cdot i_n + j_n, h} \cdot \tens{K}_{j_1, \dots, j_n, h, k}. \end{align*}
Notice that the strided convolution can be obtained also by \emph{downsampling} the standard one, as
\begin{align*} \big[\!\operatorname{conv}(\tens{V}, \tens{K}, s)\big]_{i_1, \dots, i_n, k} = \big[\!\operatorname{conv}(\tens{V}, \tens{K})\big]_{s\cdot i_1, \dots, s\cdot i_n, k}. \end{align*}
On the other hand, a \emph{pooling} layer, with pooling size $p \ge 1$, reduces the input tensor by aggregating every $p$-sided sub-tensor in $\tens{V}$ taken at regular intervals of length $p$, that is,
\begin{align*}
\tens{Z}_{i_1, \dots, i_n, k} &= \big[\!\operatorname{pool}(\tens{V}, p)\big]_{i_1, \dots, i_n, k} \\
&= \bigoplus_{j_1, \dots, j_n \in \{0, \dots, p-1\}} \tens{V}_{p\cdot i_1 + j_1, \dots, p\cdot i_n + j_n, k}, \end{align*}
where $\oplus$ is a permutation invariant aggregation function (e.g., \emph{max} or \emph{mean}).
We can model the adjacencies of a tensor's entries by means of a \emph{diagonal grid}, that we define as follows.
\begin{definition}[Diagonal grid]\label{def:diagonal-grid} Let $d_1, \dots, d_n \in \ensuremath\mathds{N}$. A \emph{diagonal grid} $\ensuremath\operatorname{G}_{\boxtimes}(d_1, \dots, d_n) = (V, E)$ is the graph with vertices \[
V = \{0,\dots, d_1 -1\} \times \dots \times \{0,\dots, d_n - 1\}\ (\subset \ensuremath\mathds{N}_0^n) \] that has an edge joining every two vectors at unit Chebyshev distance, i.e.,
$
E = \{(\vec{i},\, \vec{j}) \in V^2 \mid \lVert \vec{i} - \vec{j}\rVert_\infty = 1\}. $
\end{definition} \begin{remark}\label{rmk:grid-dist} In a diagonal grid $\ensuremath\operatorname{G}_{\boxtimes}(d_1, \dots, d_n)$, the distance between two elements $\vec{i}, \vec{j}$ is given by their Chebyshev distance, i.e., $\ensuremath\operatorname{d}(\vec{i}, \vec{j}) = \lVert \vec{i} - \vec{j}\rVert_\infty = \max_h \lvert i_h - j_h\rvert.$ \end{remark}
Notice that the nodes of a diagonal grid $\gr{G} = (V, E) = \ensuremath\operatorname{G}_{\boxtimes}(d_1, \dots, d_n)$ can be used to index the first $n$ dimensions of a tensor $\tens{V} \in \ensuremath\mathds{R}^{d_1 \times \dots \times d_n \times f}$. As a consequence, this tensor can be adopted as a \emph{feature tensor} of the graph $\gr{G}$, and, for any node $\vec{i}\in V$, we can retrieve its feature vector $\vec{x} \in \ensuremath\mathds{R}^f$ with
$ \vec{x}_k = \tens{V}_{i_1,\dots,i_n,k}. $
In \cref{thm:stride} we will prove that, for any $k\ge0$, if $\ensuremath{\pi}$ is the ranking of the nodes in $V$ in \emph{lexicographic order} (see \cref{def:lex} below), we have that
\begin{enumerate}
\item $k\text{-MIS}(\gr{G}, V, \ensuremath{\pi})$ selects the same entries of a strided convolution with $s=k+1$, that is, it will select all the entries of $\tens{V}$ having indices that are multiple of $k+1$ in all the first $n$ dimensions (\cref{thm:stride}.\ref{thm:stride-1}).
\item $\mathcal{P} = \textsc{Cluster}(\gr{G}, k, \ensuremath{\pi})$ partitions the tensor as a pooling layer with $p = k+1$, that is, the entries indexed by the vectors in every partition $P\in \mathcal{P}$ will form a $(k+1)$-sided sub-tensor of $\tens{V}$ (\cref{thm:stride}.\ref{thm:stride-2a}).
\item Finally, the reduced graph obtained by contracting $\mathcal{P}$ (that is, the \emph{quotient graph} $\gr{G}/\mathcal{P}$) is isomorphic to the diagonal grid having the same shape (in the first $n$-dimensions) of the output tensor $\tens{Z} = \operatorname{pool}(\tens{V}, k+1)$. Moreover, every partition in $\mathcal{P}$ can be mapped to the aggregated entry in $\tens{Z}$ by a properly defined isomorphism (\cref{thm:stride}.\ref{thm:stride-2b}). \end{enumerate}
\begin{definition}[Lexicographic order]\label{def:lex} Given a finite set of vectors $X \subset \ensuremath\mathds{R}^n$, the \emph{lexicographic order} on $X \times X$ is defined as
\[
\vec{x} \prec_{\text{lex}} \vec{y} \iff \bigvee_{i} \Big( (x_i < y_i) \wedge \bigwedge_{j < i} (x_j = y_j) \Big). \] \end{definition}
\begin{proposition}\label{thm:stride} Let $\gr{G} = (V, E) = \ensuremath\operatorname{G}_{\boxtimes}(d_1, \dots, d_n)$ be a diagonal grid, and $\ensuremath{\pi}$ the ranking of its vertices in lexicographic order.
Then, the following propositions hold for any $k \ge 0\!:$
\begin{enumerate}
\item\label{thm:stride-1} $S = k\textnormal{-MIS}(\gr{G}, V, \ensuremath{\pi}) = \{ \vec{i} \in V \!\mid \forall h.\, i_h \bmod k + 1 = 0\};$
\item\label{thm:stride-2} Let $\mathcal{P} = \textsc{Cluster}(\gr{G}, k, \ensuremath{\pi})$. Then:
\begin{enumerate}
\item\label{thm:stride-2a} $\mathcal{P} = \{\{\vec{i} + \vec{h} \in V \mid \vec{h}\in \{0,\dots,k\}^n \}\}_{\vec{i} \in S};$
\item\label{thm:stride-2b} $\gr{G}/\mathcal{P} \cong \ensuremath\operatorname{G}_{\boxtimes}\!\big(\big\lfloor\frac{d_1}{k+1}\big\rfloor, \dots, \big\lfloor\frac{d_n}{k+1}\big\rfloor\big)$ with isomorphism
\[
\phi(P) = \tfrac{1}{k+1}\cdot \vec{i} \quad\text{such that}\quad \{\vec{i}\} = P \cap S.
\]
\end{enumerate} \end{enumerate} \end{proposition} \begin{proof} \begin{enumerate}
\item We can prove the first point by well-founded induction on the sequence of nodes ordered by their ranking $\ensuremath{\pi}$. Specifically, we prove the following proposition for every $u \in V$:
\begin{align*}
P(u) &\equiv S_u = \{ \vec{i} \in V_u \mid \forall h.\ {i}_h \bmod k + 1 = 0\},
\end{align*}
where $V_u = \{v \in V \mid \ensuremath{\pi}(v) \le \ensuremath{\pi}(u)\}$ and $S_u = k\textnormal{-MIS}(\gr{G}, V_u, \ensuremath{\pi})$. Notice that $V_u$ represents also the set of the first $\lvert V_u\rvert$ vertices processed by \cref{alg:k-mis}, in its equivalent sequential implementation, and hence $S_u \subseteq S$ for every $u\in V$.
\begin{description}
\item[Base case.] The minimal node with respect to the lexical ordering is $\vec{0} \in V$. $P(u)$ is satisfied, since
$u =\vec{0}$ is the first and only element in $V_u$, which is then selected and returned as singleton $k$-independent set by \cref{alg:k-mis}.
\item[Induction.] For any $u \in V,$ we assume by induction that $P(w)$ holds for any other $w \in V_u \setminus \{u\}$. Let $v$ be the maximal element in this set, i.e.,
\[
v = \operatorname{argmax}_{w \in V_u\setminus \{u\}} \ensuremath{\pi}(w).
\]
Since \cref{alg:k-mis} selects the vertices in lexicographic order, $u$ will be the last one extracted from $V_u$. Hence, the returned $k$-independent set will be either
\begin{align*}
k\textnormal{-MIS}(\gr{G}, V_u, \ensuremath{\pi}) = \begin{cases}
S_v & \text{if } \ensuremath\operatorname{d}(u, S_v) \le k,\\
S_v \cup \{u\} & \text{otherwise.}
\end{cases}
\end{align*}
In the first case, $P(u)$ is satisfied, since any node in $S$
is at least $k+1$ hops from any other node in $S_v\, (\subseteq S)$.
In the second case, assume by contraposition that $u =\vec{j}\not\in S$. Then, there exists at least an index $h$ such that $j_h \bmod k+1 \neq 0$. Let $\vec{j}' \in S$ be defined as
\begin{align}\label{eq:nearest-centroid}
\forall i.\ j'_i = {j}_i - (j_i \bmod k + 1) \le j_k.
\end{align}
This index precedes lexicographically $\vec{j}$ and must belong to a node in $S_v$ by inductive assumption. This is impossible since $\ensuremath\operatorname{d}(\vec{j}, \vec{j}') = \lVert \vec{j} - \vec{j}'\rVert_\infty\le k$ violates the $k$-independence condition, hence $\vec{j}$ must be also in $S$, and $P(u)$ is satisfied.
\end{description}
\item[\ref{thm:stride-2a}.] \cref{alg:cluster} associates every node in $V$ to the $k$-hop neighbor in $S = k\textnormal{-MIS}(\gr{G}, V, \ensuremath{\pi})$ with lowest rank. We know by {\cref{thm:stride}.\ref{thm:stride-1}} that $S = \{ \vec{i} \in V \mid \forall h.\ i_h \bmod k + 1 = 0\}$ and, by \cref{rmk:grid-dist}, that the (inclusive) $k$-hop neighborhood of a node $\vec{i}$ is given by
\begin{align*}
\ensuremath N_{\gr{G}}[\vec{i}] &= \{\vec{j} \mid \lVert \vec{i} - \vec{j}\rVert_\infty \le k \} \\
&= \{\vec{i} + \vec{h} \mid \vec{h} \in \mathbb{Z}^n\ \wedge\ \lVert\vec{h}\rVert_\infty \le k \}.
\end{align*}
For every node in $V$, the $k$-hop neighbor in $S$ with lowest rank will always be the one having offset $\vec{h}$ with all non-positive entries (there is always one, and can be retrieved as in \cref{eq:nearest-centroid}). Dually, every $\vec{i} \in S$ will form a cluster with all of its $k$-hop neighbors having offset $\vec{h} \in \{0,\dots,k\}^n.$
\item[\ref{thm:stride-2b}.] Let $\gr{G} / \mathcal{P} = (\mathcal{P}, \mathcal{E})$ and $\ensuremath\operatorname{G}_{\boxtimes}\!\big(\big\lfloor\frac{d_1}{k+1}\big\rfloor, \dots, \big\lfloor\frac{d_n}{k+1}\big\rfloor\big) = (V', E')$.
By \cref{thm:stride}.\ref{thm:stride-1} and \ref{thm:stride}.\ref{thm:stride-2a}, the cluster of any node $\vec{i} \in S$ will be connected to the ones of any other node $\vec{j}\in S$ such that $\lVert \vec{i} - \vec{j} \rVert_\infty = k+1$. These are also the only edges in the quotient graph since, by maximality of the $k$-independent set, there is an edge between two clusters $P, Q \in \mathcal{P}$ if and only if the two vertices $\{\vec{i},\vec{j}\} = S \cap (P \cup Q)$ are at most $2k+1 < 2(k+1)$ apart from each other (otherwise a node between $\vec{i}$ and $\vec{j}$ would belong to a different cluster). The function $\phi: \mathcal{P} \to V'$ is a bijection since
\begin{enumerate}
\item any cluster in $\mathcal{P}$ has one and only one node in $S$ ($\mathcal{P} \leftrightarrow S$), and
\item by scaling the vectors in $S$ by $k+1$ we obtain exactly the node set $V'$ ($S \leftrightarrow V'$).
\end{enumerate}
Hence, we can reach the conclusion as following,
\begin{align*}
(P,\, Q) \in \mathcal{E} &\Leftrightarrow \lVert \vec{i} - \vec{j} \rVert_\infty = k + 1\tag{$\vec{i}, \vec{j}\in S\cap(P\cup Q)$}\\
&\Leftrightarrow \lVert (k+1) \cdot (\phi(P) - \phi(Q))\rVert_\infty = k + 1 \\
&\Leftrightarrow \lVert \phi(P) - \phi(Q)\rVert_\infty = 1 \\
&\Leftrightarrow (\phi(P),\, \phi(Q)) \in E'.
\end{align*}
\end{enumerate} \end{proof}
\subsection{Connectivity}\label{sec:connectivity-proofs}
\begin{lemma}\label{lemma:length}
Let $\gr{G}$ be a connected graph and $\gr{H} = \ensuremath\mathcal{R}\big(\gr{G}; k\big)$, with $k \in \ensuremath\mathds{N}$. Then, $\forall u, v \in V(\gr{H})$,
\[
\ensuremath\operatorname{d}_{\gr{H}}(u, v) \le \ensuremath\operatorname{d}_\gr{G}(u, v) \le (2k + 1)\ensuremath\operatorname{d}_{\gr{H}}(u, v).
\]
\end{lemma}
\begin{proof} The first inequality is proven by construction of $H$: for each node $w$ in the shortest path $\ensuremath\gr{P} = u{\sim}v \subseteq \gr{G}$ (including $u$ and $v$) there exist a node $w' \in V(\gr{H})$ such that $\ensuremath\operatorname{d}_{\gr{G}}(w, w') \le k$;
let $R$ the set of these nodes. Each pair of adjacent nodes in $\ensuremath\gr{P}$ either belong in the same $k$-hop neighborhood for some node $w' \in R$, or belong in two distinct ones $w',w''$ which are made adjacent by the coarsening process. Hence the path induced by $R$ is connected and of size at most $e(\ensuremath\gr{P}) = \ensuremath\operatorname{d}_\gr{G}(u, v)$, so $\ensuremath\operatorname{d}_\gr{H}(u, v) \le |R| - 1 \le \ensuremath\operatorname{d}_{\gr{G}}(u, v)$.
For the second inequality, let $\ensuremath\gr{P}' = u{\sim}v$ a shortest path in $\gr{H}$, of length $\ensuremath\operatorname{d}_\gr{H}(u, v)$; recalling Remark~\ref{rmk:centroid_hops}, any two consecutive nodes $u, w$ on $\ensuremath\gr{P}'$ are at (hop) distance at most $2k + 1$ in $\gr{G}$, so there is a $u,v$-path in $\gr{G}$ of length at most $(2k + 1)\ensuremath\operatorname{d}_\gr{H}(u, v)$.
\end{proof}
\begin{proof}[Proof of \cref{thm:length}] The first inequality holds by the same arguments used for the fist inequality of \cref{lemma:length}. The second inequality follows from \cref{lemma:length} and triangle inequality. Namely,
\begin{align*}
\ensuremath\operatorname{d}_\gr{G}(u, v) \le \ensuremath\operatorname{d}_\gr{G}(u, \ensuremath\rho(u)) + \underbrace{\ensuremath\operatorname{d}_{\gr{G}}(\ensuremath\rho(u), \ensuremath\rho(v))}_{\hspace{-1em}\le (2k + 1)\ensuremath\operatorname{d}_{\gr{H}}(\ensuremath\rho(u), \ensuremath\rho(v))\hspace{-1em}} + \ensuremath\operatorname{d}_\gr{G}(\ensuremath\rho(v), v). \end{align*}
Finally, $\forall x\in V,\ \ensuremath\operatorname{d}_\gr{G}(\ensuremath\rho(x), x)\le k$ by construction of $H$. \end{proof}
\begin{proof}[Proof of \cref{th:cc}] If $\gr{G}$ is connected, then also $\gr{H}$ is connected, by \cref{lemma:length}. Otherwise, let $C \subseteq V(\gr{G})$ the nodes of a connected component of $\gr{G}$. Since $V(\gr{H})$ is a \kmis of $\gr{G}$, $C \cap V(H)$ is a \kmis of $\gr{G}[C]$. Applying \cref{alg:cluster} to $\gr{G}[C]$ (with the same ordering used on $\gr{G}$) will produce the reduced graph $\gr{H}' = \gr{H}[C \cap V(\gr{H})]$, which by \cref{lemma:length} is connected. Finally, \cref{alg:cluster} joins two nodes in $V(\gr{H})$ only if there exists and edge in $\gr{G}$ intersecting their $k$-hop neighborhood, hence \cref{alg:cluster} does not connect different components of $\gr{G}$. \end{proof}
\subsection{Lower bounds}\label{sec:bound-proofs}
\begin{definition}[Restricted Neighborhood]\label{def:rneigh} Let $\gr{G} = (V, E)$ be a graph, $\ensuremath{\pi}$ be a ranking of its nodes, and $k \in \ensuremath\mathds{N}$ an integer. Compute $S = k\textnormal{-MIS}(\gr{G}, V, \ensuremath{\pi})$ as in \cref{alg:k-mis}, and let $V_0 \supseteq \dots \supseteq V_r$ and $S_0 \supseteq \dots \supseteq S_r$ the filtration of nodes produced respectively by the inputs and the outputs of its $r$ recursive calls, with $V_0 = V$, $S_0 = S$, and $V_r = S_r = \emptyset$. We define the \emph{restricted $k$-hop neighborhood} of a node $v \in S$, denoted $\ensuremath\widehat{\neigh}_k(v)$ (resp.\ $\widehat{\ensuremath N}_k[v]$ if inclusive), as
\begin{align*}
\ensuremath\widehat{\neigh}_k(v) = \ensuremath N_k(v) \cap V_{i^\ast} \quad\text{ with }\quad i^\ast = \max \{ i \mid v \in S_i\}. \end{align*} \end{definition}
\begin{remark}\label{th:rneigh-partition} Let $S = k\textnormal{-MIS}(\gr{G}, V, \ensuremath{\pi})$. Then, $\mathcal{P} = \{\ensuremath\widehat{\neigh}_k[v]\}_{v\in S}$ forms a partition of $V$. \end{remark}
\begin{remark}\label{th:deg-plus-one} Let $\gr{G} = (V, E)$ a graph, $\ensuremath{\mat{A}} \in \{0, 1 \}^{n\times n}$ its (unweighted) adjacency matrix, and $\vec{x} \in \ensuremath\mathds{R}_+^n$ a vector of non-negative values. Then, for any $v \in V$ and $k \in \ensuremath\mathds{N}$, \[ \sum_{u \in \ensuremath N_{k}[v]} \vec{x}_u \le [(\ensuremath{\mat{A}} + \ensuremath\mat{I})^k\vec{x}]_v. \] \end{remark}
\begin{proof}[Proof of \cref{th:bound-k-degree}] \begin{align*}
\textstyle\sum_{v \in S} \vec{x}_v &= \textstyle\sum_{v \in S} w(v)\cdot [(\ensuremath{\mat{A}} + \ensuremath\mat{I})^k\vec{1}]_v \\
&\ge \textstyle\sum_{v \in S} w(v) \cdot\big\lvert \ensuremath N_k[v]\big\rvert\tag{\cref{th:deg-plus-one}} \\
&\ge \textstyle\sum_{v \in S} w(v) \cdot\big\lvert \ensuremath\widehat{\neigh}_k[v]\big\rvert\tag{$\ensuremath\widehat{\neigh}_k[v] \subseteq \ensuremath N_k[v]$} \\
&\ge \textstyle\sum_{v \in S} \textstyle\sum_{u \in \ensuremath\widehat{\neigh}_k[v]} w(u)\tag{$\forall u\in\ensuremath\widehat{\neigh}_k[v].\ w(v) \ge w(u)$} \\
&= \textstyle\sum_{v\in V} w(v).\tag{\cref{th:rneigh-partition}} \end{align*}
\end{proof}
\begin{proof}[Proof of \cref{th:bound-k-weights}] \begin{align*}
\textstyle\sum_{v \in S} \vec{x}_v &= \textstyle\sum_{v \in S} w(v)\cdot [(\ensuremath{\mat{A}} + \ensuremath\mat{I})^k\vec{x}]_v \\
&\ge \textstyle\sum_{v \in S} w(v) \cdot \sum_{u \in \ensuremath N_k[v]}\vec{x}_u\tag{\cref{th:deg-plus-one}} \\
&\ge \textstyle\sum_{v \in S} w(v) \cdot \textstyle\sum_{u \in \ensuremath\widehat{\neigh}_k[v]}\vec{x}_u\tag{$\ensuremath\widehat{\neigh}_k[v] \subseteq \ensuremath N_k[v]$} \\
&\ge \textstyle\sum_{v \in S} \textstyle\sum_{u \in \ensuremath\widehat{\neigh}_k[v]} w(u) \cdot \vec{x}_u\tag{$\forall u\in\ensuremath\widehat{\neigh}_k[v].\ w(v) \ge w(u)$} \\
&= \textstyle\sum_{v\in V} w(v) \cdot \vec{x}_v.\tag{\cref{th:rneigh-partition}} \end{align*} \end{proof}
\begin{remark}\label{th:associativity} Let $\mat{A} \in \ensuremath\mathds{R}^{n\times n}$ be a symmetric matrix and $\vec{x} \in \ensuremath\mathds{R}^n$ be a vector. Then
\[ \sum_{i = 1}^n \sum_{j = 1}^n \mat{A}_{ij}\cdot\vec{x}_j = \sum_{i = 1}^n \vec{x}_i \cdot \sum_{j = 1}^n \mat{A}_{ij}. \] \end{remark}
\begin{proposition}[\citet{kako_approximation_2009}]\label{th:kako-prop} Assume that $a_i > 0$ and $b_i > 0$ for $1 \le i \le n$. Then
\[ \sum_{i = 1}^n\frac{{b_i}^2}{a_i} \ge \frac{\big(\sum_{i = 1}^n b_i\big)^2}{\sum_{i = 1}^n a_i}. \] \end{proposition}
\begin{proof}[Proof of \cref{th:ratio}] Let $\Delta_k = \max_{v \in V}\ [(\ensuremath{\mat{A}} + \ensuremath\mat{I})^k\vec{1}]_v$. When using a ranking induced by \cref{eq:rank-k-degree} we have that \begin{align*}
\textstyle\sum_{v\in S}\vec{x}_v &\ge \textstyle\sum_{v \in V} w(v)\tag{\cref{th:bound-k-degree}}\\
&\ge \frac{\sum_{v \in V} \vec{x}_v}{\Delta_k}\tag{$\forall v. \in V.\ \Delta_k \ge [(\ensuremath{\mat{A}} + \ensuremath\mat{I})^k\vec{1}]_v$}\\
&\ge \frac{\alpha(G^k)}{\Delta_k}.\tag{$\sum_{v \in V} \vec{x}_v \ge \alpha(\gr{G}^k)$} \end{align*}
When using the rank induced by \cref{eq:rank-k-weights}, instead, we have \begin{align*}
\sum_{v\in S}\vec{x}_v &\ge\sum_{v \in V} \frac{{\vec{x}_v}^2}{[(\ensuremath{\mat{A}} + \ensuremath\mat{I})^k\vec{x}]_v}\tag{\cref{th:bound-k-degree}}\\
&\ge \frac{\big(\sum_{v \in V} \vec{x}_v\big)^2}{\sum_{v \in V} [(\ensuremath{\mat{A}} + \ensuremath\mat{I})^k\vec{x}]_v}\tag{\cref{th:kako-prop}}\\
&= \frac{\big(\sum_{v \in V} \vec{x}_v\big)^2}{\sum_{v \in V} \vec{x}_v \cdot [(\ensuremath{\mat{A}} + \ensuremath\mat{I})^k\vec{1}]_v}\tag{\cref{th:associativity}}\\
&\ge \frac{\big(\sum_{v \in V} \vec{x}_v\big)^2}{\Delta_k\sum_{v \in V} \vec{x}_v }\tag{$\forall v. \in V.\ \Delta_k \ge [(\ensuremath{\mat{A}} + \ensuremath\mat{I})^k\vec{1}]_v$}\\
&\ge \frac{\alpha(G^k)}{\Delta_k}.\tag{$\sum_{v \in V} \vec{x}_v \ge \alpha(\gr{G}^k)$} \end{align*}
\end{proof}
\section{Experimental Setting}
\subsection{Description of the datasets}\label{subsec:graph-descr}
We run benchmark experiments on real-world undirected graphs from different domains, namely
\begin{itemize}
\item \emph{Orkut}, \emph{LiveJournal}, \emph{Youtube}, and \emph{Brightkite}, are social networks from the SNAP dataset~\cite{leskovec_snap_2014}, where every node represents a {user} and every edge a {friendship} relation. Like most social networks, those networks have an small effective diameter, that in this case amounts at most to $6.5$.
\item \emph{Skitter}, which is an Internet topology graph built from traceroutes, where every autonomous system (AS) is represented as a node which is connected by an edge to other ASs if there there has been reported an exchange of information between two of them. This graph was also retrieved from the SNAP dataset~\cite{leskovec_snap_2014}. Its diameter is $6$.
\item \emph{Enron} is an email communication network from SNAP~\cite{leskovec_snap_2014}, where two nodes are email addresses and there is an edge between two of them if they exchanged at least an email (sent or received). It has an effective diameter of $4.8$.
\item \emph{DBLP} and \emph{AstroPh} are two co-authorship networks, respectively from the 10th DIMACS Challenge~\cite{bader_10th_2011} and from the SNAP~\cite{leskovec_snap_2014} dataset. In \emph{DBLP}, every node in the networks represents a paper and there is an edge if two papers share at least an author, while in \emph{AstroPh} every node represents an author and there is an edge between two of them if they co-authored a paper. Their effective diameters are $6.8$, and $4.8$, respectively.
\item \emph{Europe} and \emph{Luxembourg} are two road networks from the 10th DIMACS Challenge dataset~\cite{bader_10th_2011}. In these networks every edge represents a road (of some kind) and a node a crossing. These are the only weighted graphs, where every weight represents the Euclidean distance between the coordinates of the two endpoints. Differently from the previous networks, street maps are (almost) planar graphs and, as such, can be expected to have a high diameter. \end{itemize}
All the graphs were retrieved from the University of Florida Sparse Matrix Collection~\cite{davis_university_2011}.
For the classification tasks, we used the following benchmark datasets:
\begin{itemize}
\item \emph{DD}~\cite{dobson_distinguishing_2003}, a dataset of graphs representing protein structures, in which the nodes are (labeled) amino acids and two nodes are connected by an edge if they are less than 6 Angstroms apart. The task consists in classifying enzymes and non-enzymes.
\item \emph{REDDIT}~\cite[\emph{-BINARY}, \emph{-MULTI-5K}, and \emph{-MULTI-12K},][]{yanardag_deep_2015}, are social networks where there is an edge between two users if there was reported an interaction between them (in the form of comments in a discussion thread). The task consists in classifying different kinds of communities.
\item \emph{GITHUB-STARGAZERS}~\cite{rozemberczki_karate_2020}, a social network of developers, where every edge is a ``following'' relation. The task is to decide if a community belongs to web or machine-learning developers. \end{itemize}
Since REDDIT and GITHUB datasets have no node labels, we set them to a constant value (fixed to 1) for every node. All the benchmark datasets were retrieved from the \emph{TUDataset} collection~\cite{morris_tudataset_2020}. Further information regarding the graphs and datasets are summarized in \cref{tab:graph-info}.
\begin{table}[htb]
\centering
\resizebox{\columnwidth}{!}{
\footnotesize \begin{tabular}{ll@{\ \ }rrrr} \toprule Graph & Type & $n$ & $m$ & $\ensuremath \operatorname{deg}$ & $\ensuremath\operatorname{d}_{90}$\\ \midrule Orkut & Social & 3072441 & 117185083 & $76.3$ & $4.8$ \\ {LiveJournal} & Social & 3997962 & 69362378 & $34.7$ & $6.5$ \\ {Youtube} & Social & 1134890 & 5975248 & $10.5$ & $6.5$ \\ {Brightkite} & Social & 58228 & 214078 & $7.4$ & $6.0$ \\ {Skitter} & Web & 1696415 & 22190596 & $26.2$ & $6.0$ \\ {Enron} & Email & 36692 & 367662 & $20.0$ & $4.8$ \\ {AstroPh} & Auth. & 18772 & 396160 & $42.2$ & $4.8$ \\ {DBLP} & Auth. & 540486 & 30491458 & $112.8$ & $6.8$ \\
{Europe} & Road & 50912018 & 108109320 & $4.2$ & $>$$10^3$ \\ {Luxembourg} & Road & 114599 & 239332 & $4.2$ & $\approx$$10^3$ \\
\midrule\midrule
Dataset & Type & $n$ (avg.) & $m$ (avg.) & Size & Class \\\midrule
DD & Protein & 284.32 & 715.66 & 1178 & 2 \\
REDDIT-B & Social & 429.63 & 497.75 & 2000 & 2 \\
REDDIT-5K & Social & 508.52 & 594.87 & 4999 & 5 \\
REDDIT-12K & Social & 391.41 & 456.89 & 11929 & 11 \\
GITHUB & Social & 113.79 & 234.64 & 12725 & 2 \\ \bottomrule
\end{tabular}}
\caption{Benchmark graphs and dataset information}
\label{tab:graph-info} \end{table}
\subsection{Ablation studies}
\begin{table*}[h]
\centering
\small \begin{tabular}{llr@{$\,\pm\,$}lr@{$\,\pm\,$}lr@{$\,\pm\,$}lr@{$\,\pm\,$}lr@{$\,\pm\,$}l}
\toprule Model & Sampling & \multicolumn{2}{c}{DD} & \multicolumn{2}{c}{REDDIT-B} & \multicolumn{2}{c}{REDDIT-5K} & \multicolumn{2}{c}{REDDIT-12K} & \multicolumn{2}{c}{GITHUB} \\ \midrule TopKPool & {Top-$k$} & {\bfseries 74.92} & {\bfseries 2.03} & 81.10 & 3.82 & 45.28 & 3.88 & 38.55 & 2.35 & 65.93 & 0.45 \\
& {$k$-MIS} & 74.75 & 1.64 & {\bfseries 86.85} & {\bfseries 0.41} & {\bfseries 52.80} & {\bfseries 0.62} & {\bfseries 46.42} & {\bfseries 0.43} & {\bfseries 67.32} & {\bfseries 0.81} \\\midrule SAGPool & {Top-$k$} & 73.26 & 2.26 & 84.90 & 3.94 & 46.29 & 5.61 & 42.30 & 3.70 & 64.29 & 5.70 \\
& {$k$-MIS} & {\bfseries 75.97} & {\bfseries 1.59} & {\bfseries 88.87} & {\bfseries 2.18} & {\bfseries 52.77} & {\bfseries 1.82} & {\bfseries 47.69} & {\bfseries 0.68} & {\bfseries 68.75} & {\bfseries 0.39} \\\midrule ASAPool & {Top-$k$} & 73.73 & 2.18 & {\bfseries 78.37} & {\bfseries 5.22} & 39.53 & 7.76 & {\bfseries 39.14} & {\bfseries 3.58} & {\bfseries 66.98} & {\bfseries 0.96} \\
& {$k$-MIS} & {\bfseries 74.75} & {\bfseries 1.20} & 76.36 & 6.25 & {\bfseries 48.20} & {\bfseries 1.57} & 34.23 & 6.29 & 66.30 & 0.33 \\\midrule PANPool & {Top-$k$} & 73.26 & 1.94 & 77.44 & 4.95 & 46.04 & 3.78 & {40.97} & {3.02} & 62.48 & 2.84 \\
& {$k$-MIS} & {\bfseries 76.23} & {\bfseries 1.97} & {\bfseries 85.85} & {\bfseries 1.06} & {\bfseries 49.63} & {\bfseries 1.47} & {\bfseries 45.10} & {\bfseries 0.84}
& {\bfseries 64.82} & {\bfseries 1.41} \\ \bottomrule \end{tabular} \caption{Classification accuracy on selected benchmark datasets \emph{(mean{\footnotesize $\bm{\,\pm\,}$}std)}.} \label{tab:ablation} \end{table*}
The \kmis selection (\cref{alg:k-mis}) and reduction (\cref{alg:cluster}) can also be used as a \emph{plug-in replacement} of the more trivial \topk selection, where, in the first case, $k$ regulates the reduction of the graph as discussed in \cref{sect:kmis}, while in the latter it specifies the number of nodes to be selected from the graphs (usually defined in terms of a fraction of the size of the graphs). In \cref{tab:ablation} we show how changing the sampling mechanism from \topk to \kmis in pooling methods from the literature affects their performance on the selected benchmark datasets. Apart from \model{TopKPool}, where by sampling the \topk nodes it achieves better results only on the DD dataset (the smallest one), we can see instead that substituting the \topk with \kmis selections boost the performance of the consdidered pooling methods. The only methods in which the different sampling seem to provide little to no benefit is \model{ASAPool}. We argue that this is caused by the more complex (and more expensive) reductions adopted by this method: while \topk methods, once selceted the $S \subseteq V$ nodes from the input graph (with $\lvert S \rvert = k = \lceil rn\rceil$ for some ratio $r \in(0, 1)$), they reduce the adjacency matrix of the graph $\ensuremath{\mat{A}}\in \ensuremath\mathds{R}^{n\times n}$ by selecting its rows and column $\ensuremath{\mat{A}}_\text{out} = \ensuremath{\mat{A}}_{S, S} \in \ensuremath\mathds{R}^{k\times k}$, \model{ASAPool} reduces the input matrix $\ensuremath{\mat{A}}$ by selecting \emph{also the neighborhoods} of the $S$ selected nodes, defining $\ensuremath\mat{S} = \ensuremath{\mat{A}}_S$, which itself could require $O(rn^2)$ space, and then reducing the adjacency matrix by following \citet{ying_hierarchical_2018}, thus computing $\ensuremath{\mat{A}}_{\text{out}} = \ensuremath\mat{S}\ensuremath{\mat{A}}\ensuremath\mat{S}^{\ensuremath{\!\mat{\top}\!}}$. When $\ensuremath{\mat{A}}$ is symmetric, this is equivalent as computing $\ensuremath{\mat{A}}_{\text{out}}=\ensuremath{\mat{A}}_{S,S}^3 \in \ensuremath\mathds{R}^{k\times k}$, that is, selecting the $S$ nodes on the \emph{third power} of the input graph, which can become dense at will (or even full). As an example, if \model{ASAPool} selects $k = \lceil rn\rceil$ peripheral nodes of a star graph of $n + 1$ nodes and $n$ edges, the resulting reduced graph becomes complete (i.e., a graph with $k^2 \ll n$ edges). This property may facilitate the exchange of information between nodes in the reduced graph, but will hinder the scalability of the method and, hence, \model{ASAPool} should not be even be considered as a ``sparse'' pooling method, differently from the other \topk ones or \kmis.
\subsection{Other benchmarks}\label{subsec:exp-time}
\begin{figure*}
\caption{Reduction ratio (\emph{top}) and running time (\emph{bottom}, in log scale) of our approach for varying $k$ (also in log scale).}
\label{fig:time}
\end{figure*}
\begin{table}[th]
\centering
\resizebox{\linewidth}{!}{
\small \begin{tabular}{llrrrr} \toprule
& & \multicolumn{2}{c}{Ranking with \cref{eq:rank-k-degree}} & \multicolumn{2}{c}{ Ranking with \cref{eq:rank-k-weights}} \\\cmidrule(lr){3-4}\cmidrule(lr){5-6} Graph & $k$ & Greedy & Ours & Greedy & Ours \\ \midrule AstroPh & 1 & 392347.4 & 392355.1 & 392533.8 & 392528.4 \\
& 2 & 149701.3 & 148257.2 & 149625.2 & 147953.5 \\
& 3 & 72923.3 & 71929.1 & 72857.1 & 71847.4 \\ \midrule Enron & 1 & 1220157.4 & 1220194.0 & 1219789.6 & 1219789.3 \\
& 2 & 217197.2 & 215428.5 & 217131.7 & 215016.6 \\
& 3 & 148230.3 & 147460.3 & 148175.6 & 147392.6 \\ \midrule Brightkite & 1 & 1931765.8 & 1931884.3 & 1932375.7 & 1932378.2 \\
& 2 & 824662.6 & 821815.2 & 824654.3 & 820564.1 \\
& 3 & 456965.4 & 457761.3 & 456838.0 & 457368.3 \\ \midrule Luxembourg & 1 & 3286607.7 & 3286587.3 & 3309810.9 & 3309836.2 \\
& 2 & 2267010.7 & 2265353.6 & 2290198.3 & 2284708.0 \\
& 3 & 1721185.9 & 1717789.7 & 1740735.1 & 1725635.3 \\
& 4 & 1379199.0 & 1373352.9 & 1393949.6 & 1371500.4 \\
& 5 & 1147810.9 & 1141633.0 & 1159119.1 & 1133255.1 \\
& 6 & 978606.1 & 970727.6 & 988133.8 & 959172.7 \\
& 7 & 851061.6 & 840502.3 & 858769.6 & 830110.7 \\
& 8 & 751680.4 & 739632.1 & 757523.8 & 728799.9 \\ \bottomrule \end{tabular}} \caption{Weight comparison of ${k}$-MISs obtained with our relaxation and the classical greedy algorithm.}
\label{tab:mwis-results} \end{table}
\Cref{fig:time} reports the average reduction ratio (\emph{top row}) and average running time (\emph{bottom row}, in \emph{user} seconds) of ten runs of our method on selected benchmark graphs, using different values of $k$ and a ranking induced by \cref{eq:rank-k-weights} with constant weights. Times refer to the ones needed respectively to compute the ranking (as described in \cref{sect:theo}), the \kmis (\cref{alg:k-mis}), and then reducing the graph (\cref{alg:cluster} and edge reduction, as described in \cref{sect:kmis}). We can clearly observe the linearity of the time complexity of our algorithm, since the execution time increases linearly with $k$. We can also see that, for the graphs with small diameter (see \cref{tab:graph-info}), the running time decreases once $k$ approaches their effective diameter, since most of the nodes will be assigned to the same centroid during the first recursive call of \cref{alg:k-mis}, thus decreasing the expected depth of the algorithm and also its execution time.
\cref{tab:mwis-results} reports the average total weight obtained by computing the greedy sequential MIS algorithm on $\gr{G}^k$ \emph{(Greedy)}, compared to \cref{alg:k-mis} on $\gr{G}$ \emph{(Ours)}. For both algorithms we used \cref{eq:rank-k-degree,eq:rank-k-weights}, fixing $k=1$ for the greedy one (thus applying the original rules of \citet{sakai_note_2003} on $\gr{G}^k$). Results are averaged on ten runs with node weights extracted uniformly at random in the interval $[1, 100]$. The rationale behind these results is two-fold. First, differently from \cref{alg:k-mis}, in sequential greedy algorithms \cite[see, e.g.,][]{sakai_note_2003,kako_approximation_2009} those nodes maximizing \cref{eq:rank-k-degree} or (\ref{eq:rank-k-weights}) are iteratively added to the independent set and their neighbors are removed from the graph. \cref{eq:rank-k-degree,eq:rank-k-weights} are computed on the remaining subgraph, possibly producing a different value with respect to the one computed in the previous step. Nonetheless, in \cref{tab:mwis-results} we can see that, for $k=1$, the difference in the results between computing \cref{eq:rank-k-degree,eq:rank-k-weights} at every step \emph{(Greedy)} or just once \emph{(Ours)} is minimal (we actually obtain a better approximation on certain configurations). Secondly, even if \cref{eq:rank-k-degree,eq:rank-k-weights} for $k > 1$ are a loose overestimation of the same formula computed on $\gr{G}^k$, we show that the performance deteriorates very slowly with the increase of $k$, obtaining a total weight similar to the one computed by the greedy algorithm on the (denser) graph $\gr{G}^k$ for small values of $k$.
\subsection{Model selection}\label{sec:model-selection}
The model architecture, which is shared among all the trained models, can be summarized as follows:
\begin{small} \begin{align*} \text{GNN}_{h} \to \text{Pool} &\to \text{GNN}_{2h} \to \text{Pool} \to \text{GNN}_{4h} \to \text{Glob} \to \text{MLP}, \end{align*} \end{small}
where $h$ is number of output features, $c$ is the number of classes in the dataset, Glob is the global \emph{sum} and \emph{max} aggregation of the remaining node features concatenated, while MLP is defined as
\begin{small} \[ \text{Dropout}_{p=0.3} \to \text{NN}_{2h} \to \text{Dropout}_{p=0.3} \to \text{NN}_{c}. \] \end{small}
Every GNN/NN is followed by a \emph{ReLU} activation function. The pooling method (Pool) is changed for every method (and removed for the baseline), while the GNN layer is chosen among a set of layers during the model selection phase. The possible GNN layers are \textsc{GCN}~\cite{kipf_semi-supervised_2017}, \textsc{GATv2}~\cite{brody_how_2021}, and \textsc{GIN}~\cite{xu_how_2019}, unless the model is \textsc{PANPool}, which requires a specific GNN layer \cite[\textsc{PANConv},][]{ma_path_2020}. The best model for each poling layer was chosen using a grid search, with hyper-parameter space defined as in \cref{tab:hyper-params}, where $\eta$ refers to the learning rate, $b$ to the batch size, $r$ to the reduction ratio, $L$ to the maximal path length of \textsc{PANConv}, and $\sigma$ to the scoring activation function of \textsc{EdgePool}, as proposed by its authors~\cite{diehl_towards_2019}. To increase the parallelization of the grid search, every configuration run had a cap of 8GB of GPU memory to compute the training. Runs that could not fit into this limit were discarded from the model selection process.
\begin{table}[htb]
\centering
\resizebox{\linewidth}{!}{
\small
\begin{tabular}{lll}
\toprule
Models & Param. & Values \\\midrule
All & $\eta$ & 0.001, 0.0001 \\
& $h$ & 32, 64, 128 \\
& $b$ & 32, 64, 128 \\\midrule
All except \textsc{PANPool} & GNN & \textsc{GCN}, \textsc{GATv2}, \textsc{GIN} \\\midrule
\textsc{PANPool} & GNN & \textsc{PANConv}\\
& $L$ & 1, 2, 3 \\\midrule
\textsc{TopK}-, SAG-, ASAP-, \textsc{PANPool} & $r$ & 0.5, 0.2, 0.1 \\\midrule
\textsc{EdgePool} & $\sigma$ & \emph{tanh}, \emph{softmax} \\\midrule
\kmis, BDO & $k$ & 1, 2, 3 \\\bottomrule
\end{tabular}}
\caption{Hyper-parameter space}
\label{tab:hyper-params} \end{table}
\subsection{Hardware and software}
Our method has been implemented using PyTorch~\cite{paszke_pytorch_2019} and PyTorch Geometric~\cite{fey_fast_2019}, to allow high-level scripting of CUDA code. Every label propagation required by the algorithms has been implemented in form of message-passing (i.e., gather-scatter) to run exclusively on GPU. All the experiments has been executed on a machine running Ubuntu Linux with an AMD EPYC 7742 64-Core processor with 1TB of RAM, and a NVIDIA A100 with 40GB of on-board memory.
}{}
\end{document} | arXiv |
Direct numerical simulations of two-phase flow in an inclined pipe
Fangfang Xie, Xiaoning Zheng, Michael S. Triantafyllou, Yiannis Constantinides, Yao Zheng, George Em Karniadakis
Journal: Journal of Fluid Mechanics / Volume 825 / 25 August 2017
Print publication: 25 August 2017
We study the instability mechanisms leading to slug flow formation in an inclined pipe subject to gravity forces. We use a phase-field approach, where the Cahn–Hillard model is used to model the interface. At the inlet, a stratified flow is imposed with a specified velocity profile. We validate our numerical results by comparing against previous theoretical models and by predicting the various flow regimes for horizontal and inclined pipes, including stratified flow, slug flow, dispersed bubble flow and annular flow. Subsequently, we focus on slug formation in an inclined pipe and connect its appearance with specific vortical dynamics. A two-dimensional channel geometry is first considered. When the heavy fluid is injected as the top layer, inverted vortex shedding emerges, which periodically impacts on the bottom wall, as it develops further downstream. The accumulation of heavy fluid in the bottom wall causes a back flow that induces rolling waves and interacts with the upstream jet. When the heavy fluid is placed as the bottom layer, the heavy fluid accumulates and initially forms a massive slug at the bottom region, close to the inlet. Subsequently, the heavy fluid slug starts to break into smaller pieces, some of which translate along the pipe. During the accumulation phase, a back flow forms also generating rolling waves. Occasionally, a rolling wave can reach the top of the pipe and form a new slug. To describe the generation of vorticity from the two-phase interface and pipe walls in the slug formation, we study the circulation dynamics and connect it with the resulting two-phase flow patterns. Finally, we conduct three-dimensional (3-D) simulations in a circular pipe and compare the differences between the 3-D flow patterns and its circulation dynamics against the 2-D simulation results.
The flow dynamics of the garden-hose instability
Fangfang Xie, Xiaoning Zheng, Michael S. Triantafyllou, Yiannis Constantinides, George Em Karniadakis
We present fully resolved simulations of the flow–structure interaction in a flexible pipe conveying incompressible fluid. It is shown that the Reynolds number plays a significant role in the onset of flutter for a fluid-conveying pipe modelled through the classic garden-hose problem. We investigate the complex interaction between structural and internal flow dynamics and obtain a phase diagram of the transition between states as function of three non-dimensional quantities: the fluid-tension parameter, the dimensionless fluid velocity and the Reynolds number. We find that the flow patterns inside the pipe strongly affect the type of induced motion. For unsteady flow, if there is symmetry along a direction, this leads to in-plane motion whereas breaking of the flow symmetry results in both in-plane and out-of-plane motions. Hence, above a critical Reynolds number, complex flow patterns result for the vibrating pipe as there is continuous generation of new vorticity due to the pipe wall acceleration, which is subsequently shed in the confined space of the interior of the pipe.
U-shaped fairings suppress vortex-induced vibrations for cylinders in cross-flow
Fangfang Xie, Yue Yu, Yiannis Constantinides, Michael S. Triantafyllou, George Em Karniadakis
We employ three-dimensional direct and large-eddy numerical simulations of the vibrations and flow past cylinders fitted with free-to-rotate U-shaped fairings placed in a cross-flow at Reynolds number $100\leqslant \mathit{Re}\leqslant 10\,000$ . Such fairings are nearly neutrally buoyant devices fitted along the axis of long circular risers to suppress vortex-induced vibrations (VIVs). We consider three different geometric configurations: a homogeneous fairing, and two configurations (denoted A and AB) involving a gap between adjacent segments. For the latter two cases, we investigate the effect of the gap on the hydrodynamic force coefficients and the translational and rotational motions of the system. For all configurations, as the Reynolds number increases beyond 500, both the lift and drag coefficients decrease. Compared to a plain cylinder, a homogeneous fairing system (no gaps) can help reduce the drag force coefficient by 15 % for reduced velocity $U^{\ast }=4.65$ , while a type A gap system can reduce the drag force coefficient by almost 50 % for reduced velocity $U^{\ast }=3.5,4.65,6$ , and, correspondingly, the vibration response of the combined system, as well as the fairing rotation amplitude, are substantially reduced. For a homogeneous fairing, the cross-flow amplitude is reduced by about 80 %, whereas for fairings with a gap longer than half a cylinder diameter, VIVs are completely eliminated, resulting in additional reduction in the drag coefficient. We have related such VIV suppression or elimination to the features of the wake flow structure. We find that a gap causes the generation of strong streamwise vorticity in the gap region that interferes destructively with the vorticity generated by the fairings, hence disorganizing the formation of coherent spanwise cortical patterns. We provide visualization of the incoherent wake flow that leads to total elimination of the vibration and rotation of the fairing–cylinder system. Finally, we investigate the effect of the friction coefficient between cylinder and fairing. The effect overall is small, even when the friction coefficients of adjacent segments are different. In some cases the equilibrium positions of the fairings are rotated by a small angle on either side of the centreline, in a symmetry-breaking bifurcation, which depends strongly on Reynolds number. | CommonCrawl |
8.3: Kinetic Energy
[ "article:topic", "authorname:openstax", "Kinetic energy", "license:ccby", "showtoc:no", "transcluded:yes", "source-phys-4007" ]
Physics 201 - Fall 2019v2
Book: Custom Physics textbook for JJC
Calculate the kinetic energy of a particle given its mass and its velocity or momentum
Evaluate the kinetic energy of a body, relative to different frames of reference
It's plausible to suppose that the greater the velocity of a body, the greater effect it could have on other bodies. This does not depend on the direction of the velocity, only its magnitude. At the end of the seventeenth century, a quantity was introduced into mechanics to explain collisions between two perfectly elastic bodies, in which one body makes a head-on collision with an identical body at rest. The first body stops, and the second body moves off with the initial velocity of the first body. (If you have ever played billiards or croquet, or seen a model of Newton's Cradle, you have observed this type of collision.) The idea behind this quantity was related to the forces acting on a body and was referred to as "the energy of motion." Later on, during the eighteenth century, the name kinetic energy was given to energy of motion.
Newton's cradle in motion. One ball is set in motion and soon collides with the rest, conveying the energy through the rest of the balls and eventually to the last ball, which in turn is set in motion. Image used with permission (CC SA-BY 3.0; Dominique Toussaint).
With this history in mind, we can now state the classical definition of kinetic energy. Note that when we say "classical," we mean non-relativistic, that is, at speeds much less that the speed of light. At speeds comparable to the speed of light, the special theory of relativity requires a different expression for the kinetic energy of a particle, as discussed in Relativity. Since objects (or systems) of interest vary in complexity, we first define the kinetic energy of a particle with mass m.
The kinetic energy of a particle is one-half the product of the particle's mass m and the square of its speed \(v\):
\[K = \frac{1}{2} mv^{2} \ldotp \label{7.6}\]
We then extend this definition to any system of particles by adding up the kinetic energies of all the constituent particles:
$$K =\sum \frac{1}{2} mv^{2} \ldotp \label{7.7}$$
Note that just as we can express Newton's second law in terms of either the rate of change of momentum or mass times the rate of change of velocity, so the kinetic energy of a particle can be expressed in terms of its mass and momentum (\(\vec{p}\) = m \(\vec{v}\)), instead of its mass and velocity. Since v = \(\frac{p}{m}\), we see that
$$K = \frac{1}{2} m \left(\dfrac{p}{m}\right)^{2} = \frac{p^{2}}{2m}$$
also expresses the kinetic energy of a single particle. Sometimes, this expression is more convenient to use than Equation \(\ref{7.6}\). The units of kinetic energy are mass times the square of speed, or kg • m2/s2. But the units of force are mass times acceleration, kg • m/s2, so the units of kinetic energy are also the units of force times distance, which are the units of work, or joules. You will see in the next section that work and kinetic energy have the same units, because they are different forms of the same, more general, physical property.
Example \(\PageIndex{1}\): Kinetic Energy of an Object
What is the kinetic energy of an 80-kg athlete, running at 10 m/s?
The Chicxulub crater in Yucatan, one of the largest existing impact craters on Earth, is thought to have been created by an asteroid, traveling at 22 km/s and releasing 4.2 x 1023 J of kinetic energy upon impact. What was its mass?
In nuclear reactors, thermal neutrons, traveling at about 2.2 km/s, play an important role. What is the kinetic energy of such a particle?
To answer these questions, you can use the definition of kinetic energy in Equation \(\ref{7.6}\). You also have to look up the mass of a neutron.
Do not forget to convert km into m to do these calculations, although, to save space, we omitted showing these conversions.
$$K = \frac{1}{2} (80\; kg)(10\; m/s)^{2} = 4.0\; kJ \ldotp \nonumber $$
$$m = \frac{2K}{v^{2}} = \frac{2(4.2 \times 10^{23}\; J)}{22\; km/s)^{2}} = 1.7 \times 10^{15}\; kg \ldotp \nonumber$$
$$K = \frac{1}{2} (1.68 \times 110^{-27}\; kg) (2.2\; km/s)^{2} = 4.1 \times 10^{-21}\; J \ldotp \nonumber$$
In this example, we used the way mass and speed are related to kinetic energy, and we encountered a very wide range of values for the kinetic energies. Different units are commonly used for such very large and very small values. The energy of the impactor in part (b) can be compared to the explosive yield of TNT and nuclear explosions, 1 megaton = 4.18 x 1015 J. The Chicxulub asteroid's kinetic energy was about a hundred million megatons. At the other extreme, the energy of subatomic particle is expressed in electron-volts, 1 eV = 1.6 x 10−19 J. The thermal neutron in part (c) has a kinetic energy of about one fortieth of an electronvolt.
Exercise \(\PageIndex{1}\)
A car and a truck are each moving with the same kinetic energy. Assume that the truck has more mass than the car. Which has the greater speed?
A car and a truck are each moving with the same speed. Which has the greater kinetic energy?
Because velocity is a relative quantity, you can see that the value of kinetic energy must depend on your frame of reference. You can generally choose a frame of reference that is suited to the purpose of your analysis and that simplifies your calculations. One such frame of reference is the one in which the observations of the system are made (likely an external frame). Another choice is a frame that is attached to, or moves with, the system (likely an internal frame). The equations for relative motion, discussed in Motion in Two and Three Dimensions, provide a link to calculating the kinetic energy of an object with respect to different frames of reference.
Example \(\PageIndex{2}\): Kinetic Energy Relative to Different Frames
A 75.0-kg person walks down the central aisle of a subway car at a speed of 1.50 m/s relative to the car, whereas the train is moving at 15.0 m/s relative to the tracks.
What is the person's kinetic energy relative to the car?
What is the person's kinetic energy relative to the tracks?
What is the person's kinetic energy relative to a frame moving with the person?
Since speeds are given, we can use \(\frac{1}{2}\)mv2 to calculate the person's kinetic energy. However, in part (a), the person's speed is relative to the subway car (as given); in part (b), it is relative to the tracks; and in part (c), it is zero. If we denote the car frame by C, the track frame by T, and the person by P, the relative velocities in part (b) are related by \(\vec{v}_{PT}\) = \(\vec{v}_{PC}\) + \(\vec{v}_{CT}\). We can assume that the central aisle and the tracks lie along the same line, but the direction the person is walking relative to the car isn't specified, so we will give an answer for each possibility, vPT = vCT ± vPC, as shown in Figure \(\PageIndex{1}\).
Figure \(\PageIndex{1}\): The possible motions of a person walking in a train are (a) toward the front of the car and (b) toward the back of the car.
$$K = \dfrac{1}{2} (75.0\; kg)(11.50\; m/s)^{2} = 84.4\; J \ldotp \nonumber$$
$$v_{PT} = (15.0 \pm 1.50)7; m/s \ldotp \nonumber$$ Therefore, the two possible values for kinetic energy relative to the car are $$K = \dfrac{1}{2} (75.0\; kg)(13.5\; m/s)^{2} = 6.83\; kJ \nonumber $$ and $$K = \frac{1}{2} (75.0\; kg)(16.5\; m/s)^{2} = 10.2\; kJ \ldotp \nonumber$$
In a frame where vP = 0, K = 0 as well.
You can see that the kinetic energy of an object can have very different values, depending on the frame of reference. However, the kinetic energy of an object can never be negative, since it is the product of the mass and the square of the speed, both of which are always positive or zero.
You are rowing a boat parallel to the banks of a river. Your kinetic energy relative to the banks is less than your kinetic energy relative to the water. Are you rowing with or against the current?
The kinetic energy of a particle is a single quantity, but the kinetic energy of a system of particles can sometimes be divided into various types, depending on the system and its motion. For example:
If all the particles in a system have the same velocity, the system is undergoing translational motion and has translational kinetic energy.
If an object is rotating, it could have rotational kinetic energy.
If it is vibrating, it could have vibrational kinetic energy.
The kinetic energy of a system, relative to an internal frame of reference, may be called internal kinetic energy. The kinetic energy associated with random molecular motion may be called thermal energy. These names will be used in later chapters of the book, when appropriate. Regardless of the name, every kind of kinetic energy is the same physical quantity, representing energy associated with motion.
Example \(\PageIndex{3}\): Special Names for Kinetic Energy
A player lobs a mid-court pass with a 624-g basketball, which covers 15 m in 2 s. What is the basketball's horizontal translational kinetic energy while in flight?
An average molecule of air, in the basketball in part (a), has a mass of 29 u, and an average speed of 500 m/s, relative to the basketball. There are about 3 x 1023 molecules inside it, moving in random directions, when the ball is properly inflated. What is the average translational kinetic energy of the random motion of all the molecules inside, relative to the basketball?
How fast would the basketball have to travel relative to the court, as in part (a), so as to have a kinetic energy equal to the amount in part (b)?
In part (a), first find the horizontal speed of the basketball and then use the definition of kinetic energy in terms of mass and speed, K = \(\frac{1}{2} mv^{2}\). Then in part (b), convert unified units to kilograms and then use K = \(\frac{1}{2} mv^{2}\) to get the average translational kinetic energy of one molecule, relative to the basketball. Then multiply by the number of molecules to get the total result. Finally, in part (c), we can substitute the amount of kinetic energy in part (b), and the mass of the basketball in part (a), into the definition K = \(\frac{1}{2} mv^{2}\), and solve for v.
The horizontal speed is \(\frac{(15\; m)}{(2\; s)}\), so the horizontal kinetic energy of the basketball is $$\frac{1}{2} (0.624\; kg)(7.5\; m/s)^{2} = 17.6\; J \ldotp \nonumber$$
The average translational kinetic energy of a molecule is $$\frac{1}{2} (29\; u) (1.66 \times 10^{-27}\; kg/u) (500\; m/s)^{2} = 6.02 \times 10^{-21}\; J, \nonumber $$ and the total kinetic energy of all the molecules is $$(3 \times 10^{23})(6.02 \times 10^{-21}\; J) = 1.80\; kJ \ldotp \nonumber$$
$$v = \sqrt{\frac{2(1.8\; kJ)}{(0.624\; kg)}} = 76.0\; m/s \ldotp \nonumber$$
In part (a), this kind of kinetic energy can be called the horizontal kinetic energy of an object (the basketball), relative to its surroundings (the court). If the basketball were spinning, all parts of it would have not just the average speed, but it would also have rotational kinetic energy. Part (b) reminds us that this kind of kinetic energy can be called internal or thermal kinetic energy. Notice that this energy is about a hundred times the energy in part (a). How to make use of thermal energy will be the subject of the chapters on thermodynamics. In part (c), since the energy in part (b) is about 100 times that in part (a), the speed should be about 10 times as big, which it is (76 compared to 7.5 m/s).
8.2: Work
8.4: Work-Energy Theorem | CommonCrawl |
\begin{document}
\nolabels \begin{center}{\bfseries On the Baker's map and the Simplicity \\ of the Higher Dimensional Thompson Groups \(nV\) }\end{center}
\begin{center}{MATTHEW G. BRIN}\end{center}
\noindent {\bfseries Abstract}. {\itshape We show that the baker's map is a product of transpositions (particularly pleasant involutions), and conclude from this that an existing very short proof of the simplicity of Thompson's group \(V\) applies with equal brevity to the higher dimensional Thompson groups \(nV\).}
\ifthenelse{\equal{paper}{paper}}{ \relax }{ \tableofcontents
}
\makelabels \ifthenelse{\equal{paper}{draft}}{ \showlabels }{\relax}
Of the original groups \(F\subseteq T\subseteq V\) of Richard Thompson (see \cite{CFP}), all are infinite and finitely presented, and the last two are simple. An infinite family of groups \(nV\), \(n\in \{1, 2, 3, \ldots, \omega\}\), of which \(1V=V\), is introduced in \cite{brin:hd3} where it is shown that \(2V\) is infinite and simple and not isomorphic to \(V\). A finite presentation for \(2V\) is given in \cite{brin:hd4}, it is shown that \(nV\) and \(mV\) are isomorphic only when \(m=n\) in \cite{bleak+lanoue}, and metric properties of \(2V\) are studied in \cite{burillo:hdmetric}.
A very short argument that \(V=1V\) is generated by transpositions is given in Section 12 of \cite{brin:hd3}, followed by an equally short argument based on this fact (due to Rubin) that \(V\) is simple. It is also shown in that section that the baker's map (an element of \(2V\)) prevents the first argument from showing that \(2V\) is generated by transpositions. As a result, the proof in \cite{brin:hd3} of the simplicity of \(2V\) is rather involved and is based on calculations which show that the abelianization of \(2V\) is trivial.
Here we give a short proof that the baker's maps in \(2V\) are products of transpositions in \(2V\). From there it is an easy exercise to combine this with the material in Section 12 of \cite{brin:hd3} to give a short proof of the simplicity of \(2V\) and also to extend all the results to all of the \(nV\), \(n\le \omega\).
Longer arguments for simplicity exist. Presentations (finite when \(n<\omega\)) for the \(nV\), \(n\le\omega\), are given in \cite{matucci+hennig}, and one can calculate from these presentations that each of the \(nV\), \(n \le \omega\), has trivial abelianization. From the arguments in Section 3 of \cite{brin:hd3} (which are about \(2V\) but, as noted in 4.1 of \cite{brin:hd3}, they apply as well to the \(nV\), \(n \le \omega\)) it then follows that each of the \(nV\), \(n \le \omega\), is simple.
To keep this paper brief, we use notation, terminology and graphics from \cite{brin:hd3}, and from this point we assume that the reader is familiar with their meanings.
\begin{lemma} Any baker's map in \(2V\) is a product of finitely many proper transpositions from \(2V\). \end{lemma}
The (primary) baker's map is given by the following.
\[ \xy (-9,-9); (-9,9)**@{-}; (9,9)**@{-}; (9,-9)**@{-}; (-9,-9)**@{-}; (0,-9); (0,9)**@{-}; (-4.5,0)*{\scriptstyle0}; (4.5,0)*{\scriptstyle1}; \endxy \quad \longrightarrow \quad \xy (-9,-9); (-9,9)**@{-}; (9,9)**@{-}; (9,-9)**@{-}; (-9,-9)**@{-}; (-9,0); (9,0)**@{-}; (0,-4.5)*{\scriptstyle0}; (0,4.5)*{\scriptstyle1}; \endxy \]
A (secondary) baker's map is given by a pair of patterns that are identical and identically numbered with one exception: for one singly divided rectangle in the domain and for the corresponding singly divided rectangle in the range, the division is vertical in the domain and horizontal in the range. An example is below.
\[ \xy (0,0); (0,24)**@{-}; (24,24)**@{-}; (24,0)**@{-}; (0,0)**@{-}; (0,12); (24,12)**@{-}; (12,12); (12,24)**@{-}; (12,18); (24,18)**@{-}; (18,12); (18,18)**@{-}; (15,15)*{\scriptstyle2}; (21,15)*{\scriptstyle3}; (18,21)*{\scriptstyle4}; (6,18)*{\scriptstyle1}; (12,6)*{\scriptstyle0}; \endxy \quad \xy (0,0)*{\ }; (0,12)*{\longrightarrow}; \endxy \quad \xy (0,0); (0,24)**@{-}; (24,24)**@{-}; (24,0)**@{-}; (0,0)**@{-}; (0,12); (24,12)**@{-}; (12,12); (12,24)**@{-}; (12,18); (24,18)**@{-}; (12,15); (24,15)**@{-}; (18,13.5)*{\scriptstyle2}; (18,16.5)*{\scriptstyle3}; (18,21)*{\scriptstyle4}; (6,18)*{\scriptstyle1}; (12,6)*{\scriptstyle0}; \endxy \]
We refer to the rectangle containing the non-identity part of the baker's map as the {\itshape support} of the baker's map.
A transposition is given by a pair of identical patterns that are numbered identically except for a switch of two of the numbers. The transposition is {\itshape proper} if there are more than two rectangles in each pattern.
In the following, we say that two elements are identical modulo transpositions if each is a product of elements and the two products are identical if the transpositions are removed.
1. {\itshape Any baker's map \(\beta\) is, modulo transpositions, a product of a baker's map whose support is the left half of the support of \(\beta\), with a baker's map whose support is the right half of the support of \(\beta\).} In the pictures below, we show only the support of \(\beta\). The first arrow is the baker's map \(\beta\) in a reducible form. The second is a proper transposition. The composition is the promised product of two baker's maps.
\[ \xy (-10,-10); (-10,10)**@{-}; (10,10)**@{-}; (10,-10)**@{-}; (-10,-10)**@{-}; (0,-10); (0,10)**@{-}; (-8,-8)*{0}; (2,-8)*{2}; (-5,-10); (-5,10)**@{-}; (-3,-8)*{1}; (7,-8)*{3}; (5,-10); (5,10)**@{-}; \endxy \rightarrow \xy (-10,-10); (-10,10)**@{-}; (10,10)**@{-}; (10,-10)**@{-}; (-10,-10)**@{-}; (-10,0); (10,0)**@{-}; (-8,-8)*{0}; (-8,2)*{2}; (0,-10); (0,10)**@{-}; (2,-8)*{1}; (2,2)*{3}; \endxy \rightarrow \xy (-10,-10); (-10,10)**@{-}; (10,10)**@{-}; (10,-10)**@{-}; (-10,-10)**@{-}; (-10,0); (10,0)**@{-}; (-8,-8)*{0}; (-8,2)*{1}; (0,-10); (0,10)**@{-}; (2,-8)*{2}; (2,2)*{3}; \endxy \]
2. {\itshape Any baker's map \(\beta\) is, modulo transpositions, a product of a baker's map whose support is the upper half of the support of \(\beta\), with a baker's map whose support is the lower half of the support of \(\beta\).} The relevant pictures follow and the comments are as in 1.
\[ \xy (-10,-10); (-10,10)**@{-}; (10,10)**@{-}; (10,-10)**@{-}; (-10,-10)**@{-}; (-10,0); (10,0)**@{-}; (-8,-8)*{0}; (-8,2)*{1}; (0,-10); (0,10)**@{-}; (2,-8)*{2}; (2,2)*{3}; \endxy \rightarrow \xy (-10,-10); (-10,10)**@{-}; (10,10)**@{-}; (10,-10)**@{-}; (-10,-10)**@{-}; (-10,0); (10,0)**@{-}; (-8,-8)*{0}; (-8,-3)*{1}; (-10,-5); (10,-5)**@{-}; (-8,2)*{2}; (-8,7)*{3}; (-10,5); (10,5)**@{-}; \endxy \rightarrow \xy (-10,-10); (-10,10)**@{-}; (10,10)**@{-}; (10,-10)**@{-}; (-10,-10)**@{-}; (-10,0); (10,0)**@{-}; (-8,-8)*{0}; (-8,-3)*{2}; (-10,-5); (10,-5)**@{-}; (-8,2)*{1}; (-8,7)*{3}; (-10,5); (10,5)**@{-}; \endxy \]
In the following, ``arbitrarily small'' means having support with diameter smaller than an arbitrarily chosen positive real.
3. {\itshape Any baker's map is, modulo transpositions, a product of arbitrarily small baker's maps.} This follows from 1 and 2.
4. {\itshape A product of a baker's map and an inverse of a baker's map with disjoint supports is a product of transpositions.} Let \(A\) and \(B\) be the disjoint supports. We refer to the rectangles in figures below to describe a sequence of transpositions. (a) Switch \(A_0\) with \(B_0\). (b) Switch \(A_1\) with \(B_1\). (c) Switch \(A\) with \(B\). The composition of (a) with (b) with (c) in that order is the desired result.
\[ A=\xy (-9,-9); (-9,9)**@{-}; (9,9)**@{-}; (9,-9)**@{-}; (-9,-9)**@{-}; (0,-9); (0,9)**@{-}; (-4.5,0)*{A_0}; (4.5,0)*{A_1}; \endxy \quad , \quad B=\xy (-9,-9); (-9,9)**@{-}; (9,9)**@{-}; (9,-9)**@{-}; (-9,-9)**@{-}; (-9,0); (9,0)**@{-}; (0,-4.5)*{B_0}; (0,4.5)*{B_1}; \endxy \]
5. {\itshape If \(R\) is a rectangle in a pattern so that neither
side of \(R\) has length more than \(\frac12\), then the baker's
map with support \(R\) is a product of transpositions.} The assumptions make \(R\) one half of a rectangle \(A\) that is not all of the unit square. Thus there is a rectangle \(B\) that is disjoint from \(A\). Let \(S\) be the rectangle that is the ``other half'' of \(A\). Let \(\alpha\) be a product of a baker's map on \(A\) with an inverse of a baker's map on \(B\). By 4, this is a product of transpositions. By 1 or 2 we can modify \(\alpha\) so that it is still a product of transpositions, and is a baker's map on each of \(R\) and \(S\) and an inverse of a baker's map on \(B\). Let \(\beta\) be a product of a baker's map on \(B\) and an inverse of a baker's map on \(S\). By 4 this is a product of transpositions. Now the composition of \(\alpha\) with \(\beta\) gives the desired result.
The lemma follows from 3 and 5. As discussed, this implies the following.
\begin{thm} The \(nV\), \(n\le \omega\), are generated by transpositions and are simple. \end{thm}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2}
\noindent Binghamton University, Binghamton, NY 13902-6000, USA
\end{document} | arXiv |
# Audio processing fundamentals
Consider the following example:
An audio signal is represented by the function:
$$f(t) = A \sin(2\pi f_0 t + \phi)$$
Where $A$ is the amplitude, $f_0$ is the frequency, and $\phi$ is the phase.
The Fourier transform of this signal is given by:
$$F(\omega) = A \frac{1}{2}\left(\delta(\omega - f_0) + \delta(\omega + f_0)\right)$$
This shows that the frequency content of the signal is concentrated at $f_0$ and $-f_0$.
## Exercise
1. Consider an audio signal with a frequency of 440 Hz. Calculate the period of the signal in seconds.
### Solution
The period of a signal with a frequency of 440 Hz is given by $T = \frac{1}{f} = \frac{1}{440} = 0.00227$ seconds.
# Fourier transform and its application in audio processing
## Exercise
1. Calculate the inverse Fourier transform of the following function:
$$F(\omega) = A \frac{1}{2}\left(\delta(\omega - f_0) + \delta(\omega + f_0)\right)$$
### Solution
The inverse Fourier transform of the given function is:
$$f(t) = A \sin(2\pi f_0 t + \phi)$$
Where $A$ is the amplitude, $f_0$ is the frequency, and $\phi$ is the phase.
# Introduction to Java programming
## Exercise
1. Write a Java program that calculates the sum of the first 100 integers using a loop.
```java
public class Sum {
public static void main(String[] args) {
int sum = 0;
for (int i = 1; i <= 100; i++) {
sum += i;
}
System.out.println("The sum of the first 100 integers is: " + sum);
}
}
```
# Data structures and algorithms in Java
## Exercise
1. Write a Java program that sorts an array of integers in ascending order using the bubble sort algorithm.
```java
public class BubbleSort {
public static void main(String[] args) {
int[] array = {5, 3, 8, 1, 6, 9, 2, 7, 4};
bubbleSort(array);
for (int i : array) {
System.out.print(i + " ");
}
}
public static void bubbleSort(int[] array) {
for (int i = 0; i < array.length - 1; i++) {
for (int j = 0; j < array.length - i - 1; j++) {
if (array[j] > array[j + 1]) {
int temp = array[j];
array[j] = array[j + 1];
array[j + 1] = temp;
}
}
}
}
}
```
# Integrating Java with audio processing libraries
## Exercise
1. Write a Java program that uses the TarsosDSP library to read an audio file and calculate its Fourier transform.
```java
import be.tarsos.dsp.AudioDispatcher;
import be.tarsos.dsp.AudioEvent;
import be.tarsos.dsp.AudioProcessor;
import be.tarsos.dsp.io.AudioInputStream;
import be.tarsos.dsp.io.AudioReader;
import be.tarsos.dsp.util.fft.FFT;
public class FourierTransform {
public static void main(String[] args) {
String audioFilePath = "path/to/audio/file.wav";
AudioReader reader = new AudioReader();
AudioInputStream audioInputStream = reader.getStream(audioFilePath);
AudioDispatcher dispatcher = new AudioDispatcher(audioInputStream, 2048, 0);
dispatcher.addAudioProcessor(new AudioProcessor() {
@Override
public void process(AudioEvent audioEvent) {
float[] audioData = audioEvent.getFloatBuffer();
FFT fft = new FFT(audioData.length);
fft.forward(audioData);
// Process the Fourier transform
}
});
dispatcher.run();
}
}
```
# Designing and implementing a Fourier transform algorithm in Java
## Exercise
1. Write a Java program that uses the TarsosDSP library to calculate the Fourier transform of an audio signal.
```java
import be.tarsos.dsp.AudioDispatcher;
import be.tarsos.dsp.AudioEvent;
import be.tarsos.dsp.AudioProcessor;
import be.tarsos.dsp.io.AudioInputStream;
import be.tarsos.dsp.io.AudioReader;
import be.tarsos.dsp.util.fft.FFT;
public class FourierTransform {
public static void main(String[] args) {
String audioFilePath = "path/to/audio/file.wav";
AudioReader reader = new AudioReader();
AudioInputStream audioInputStream = reader.getStream(audioFilePath);
AudioDispatcher dispatcher = new AudioDispatcher(audioInputStream, 2048, 0);
dispatcher.addAudioProcessor(new AudioProcessor() {
@Override
public void process(AudioEvent audioEvent) {
float[] audioData = audioEvent.getFloatBuffer();
FFT fft = new FFT(audioData.length);
fft.forward(audioData);
// Process the Fourier transform
}
});
dispatcher.run();
}
}
```
# Signal processing techniques for audio data
## Exercise
1. Write a Java program that uses the TarsosDSP library to calculate the pitch of an audio signal.
```java
import be.tarsos.dsp.AudioDispatcher;
import be.tarsos.dsp.AudioEvent;
import be.tarsos.dsp.AudioProcessor;
import be.tarsos.dsp.io.AudioInputStream;
import be.tarsos.dsp.io.AudioReader;
import be.tarsos.dsp.pitch.PitchDetectionHandler;
import be.tarsos.dsp.pitch.PitchDetectionResult;
import be.tarsos.dsp.pitch.PitchProcessor;
public class PitchDetection {
public static void main(String[] args) {
String audioFilePath = "path/to/audio/file.wav";
AudioReader reader = new AudioReader();
AudioInputStream audioInputStream = reader.getStream(audioFilePath);
AudioDispatcher dispatcher = new AudioDispatcher(audioInputStream, 2048, 0);
PitchDetectionHandler pitchDetectionHandler = new PitchDetectionHandler() {
@Override
public void handlePitch(PitchDetectionResult pitchDetectionResult, float[] audioData) {
// Process the pitch detection result
}
};
PitchProcessor pitchProcessor = new PitchProcessor(PitchProcessor.PitchEstimationAlgorithm.YIN, 2048, 44100, pitchDetectionHandler);
dispatcher.addAudioProcessor(pitchProcessor);
dispatcher.run();
}
}
```
# Visualization of frequency domain analysis results
## Exercise
1. Write a Java program that uses the TarsosDSP library to create a spectrogram of an audio signal.
```java
import be.tarsos.dsp.AudioDispatcher;
import be.tarsos.dsp.AudioEvent;
import be.tarsos.dsp.AudioProcessor;
import be.tarsos.dsp.io.AudioInputStream;
import be.tarsos.dsp.io.AudioReader;
import be.tarsos.dsp.util.fft.FFT;
import be.tarsos.dsp.util.fft.Spectrogram;
public class Spectrogram {
public static void main(String[] args) {
String audioFilePath = "path/to/audio/file.wav";
AudioReader reader = new AudioReader();
AudioInputStream audioInputStream = reader.getStream(audioFilePath);
AudioDispatcher dispatcher = new AudioDispatcher(audioInputStream, 2048, 0);
dispatcher.addAudioProcessor(new AudioProcessor() {
@Override
public void process(AudioEvent audioEvent) {
float[] audioData = audioEvent.getFloatBuffer();
FFT fft = new FFT(audioData.length);
fft.forward(audioData);
Spectrogram spectrogram = new Spectrogram(audioData, fft);
// Process the spectrogram
}
});
dispatcher.run();
}
}
```
# Real-world applications of frequency domain analysis in Java
## Exercise
1. Write a Java program that uses the TarsosDSP library to transcribe music from an audio signal.
```java
import be.tarsos.dsp.AudioDispatcher;
import be.tarsos.dsp.AudioEvent;
import be.tarsos.dsp.AudioProcessor;
import be.tarsos.dsp.io.AudioInputStream;
import be.tarsos.dsp.io.AudioReader;
import be.tarsos.dsp.pitch.PitchDetectionHandler;
import be.tarsos.dsp.pitch.PitchDetectionResult;
import be.tarsos.dsp.pitch.PitchProcessor;
public class MusicTranscription {
public static void main(String[] args) {
String audioFilePath = "path/to/audio/file.wav";
AudioReader reader = new AudioReader();
AudioInputStream audioInputStream = reader.getStream(audioFilePath);
AudioDispatcher dispatcher = new AudioDispatcher(audioInputStream, 2048, 0);
PitchDetectionHandler pitchDetectionHandler = new PitchDetectionHandler() {
@Override
public void handlePitch(PitchDetectionResult pitchDetectionResult, float[] audioData) {
// Process the pitch detection result to transcribe the music
}
};
PitchProcessor pitchProcessor = new PitchProcessor(PitchProcessor.PitchEstimationAlgorithm.YIN, 2048, 44100, pitchDetectionHandler);
dispatcher.addAudioProcessor(pitchProcessor);
dispatcher.run();
}
}
```
# Advanced topics in Java-based frequency domain analysis
In conclusion, this textbook has provided a comprehensive introduction to frequency domain analysis with Java. It has covered the fundamentals of audio processing, the Fourier transform and its application in audio processing, the Java programming language and its features, data structures and algorithms in Java, integration with audio processing libraries, designing and implementing Fourier transform algorithms, signal processing techniques for audio data, visualization of frequency domain analysis results, real-world applications of frequency domain analysis in Java, and advanced topics in Java-based frequency domain analysis. | Textbooks |
You are given $$$n$$$ rectangles on a plane with coordinates of their bottom left and upper right points. Some $$$(n-1)$$$ of the given $$$n$$$ rectangles have some common point. A point belongs to a rectangle if this point is strictly inside the rectangle or belongs to its boundary.
Find any point with integer coordinates that belongs to at least $$$(n-1)$$$ given rectangles.
The first line contains a single integer $$$n$$$ ($$$2 \le n \le 132\,674$$$) — the number of given rectangles.
Each the next $$$n$$$ lines contains four integers $$$x_1$$$, $$$y_1$$$, $$$x_2$$$ and $$$y_2$$$ ($$$-10^9 \le x_1 < x_2 \le 10^9$$$, $$$-10^9 \le y_1 < y_2 \le 10^9$$$) — the coordinates of the bottom left and upper right corners of a rectangle.
Print two integers $$$x$$$ and $$$y$$$ — the coordinates of any point that belongs to at least $$$(n-1)$$$ given rectangles.
The picture below shows the rectangles in the first and second samples. The possible answers are highlighted.
The picture below shows the rectangles in the third and fourth samples.
Server time: Apr/22/2019 04:08:36 (g1). | CommonCrawl |
Recycling and storage of nitrogen from breakdown of macromolecules?
If nitrogen is so important (it's a constituent of both proteins and nucleic acids), why are animals still throwing them out, and needing to eat it again? Is it true that in almost a billion years of evolution, life still haven't found a way to recycle nitrogen inside the cells or inside the body of multi-cellular organisms? Why does this seem so hard to achieve? Or are there animals that don't need to excrete nitrogen compounds (like ammonia, urea and uric acid), and instead reuse them?
evolution biochemistry excreta nitrogen-cycle
RodrigoRodrigo
$\begingroup$ Ammonia, urea, and uric acid are toxic to the body; why would we want to keep them around? Asking why didn't "x" evolve to happen when what actually has happened is working fairly to extremely well is fairly pointless. $\endgroup$
– anongoodnurse
$\begingroup$ Also the necessary nitrogen-containing compounds seem to be in more than adequate supply in the diet, so that eating enough calories, proteins, &c provides more than enough nitrogen. What to do with the excess, except excrete it? $\endgroup$
– jamesqf
$\begingroup$ @anongoodnurse yes they are toxic, but they are created as a by-product. Why not recycle them back into protein and nucleic acids? Then there would be no toxic residuals asking to be excreted. $\endgroup$
$\begingroup$ @jamesqf we store fat and glycogen for energy, but we must eat nitrogen to replace what was excreted. I guess we could pass more time without eating if we were able to recycle nitrogen. This could be an advantage in lots of animal groups, but as far as I know, all animal groups have nitrogen excretion mechanisms. This makes me think nitrogen recycling is somehow very hard for animals to do. The question is: why? $\endgroup$
$\begingroup$ @tomd - Yes, and in nature (before all this population) we used to return that fixed nitrogen to plants which could use them (whereas we can't). Benefits all around. I have two dogs who have not yet learned to use the toilet. You should see my back yard... $\endgroup$
The premise of the question is incorrect. Mammalian organisms do recycle nitrogen. They only excrete excess nitrogen.
Ammonia from deamination of amino acids can be incorporated into glutamate and glutamine:
Transamination can then transfer the amino group from glutamate, for example, to other ketoacids for the synthesis of other amino acids.
The capacity for recycling nitrogen is presumably determined by requirement. If there is excess to requirement, then there is no evolutionary advantage in retaining the excess and instead a mechanism has evolved to neutralize it and then excrete it.
Addendum: Recycling, yes, but why not also storage?
Although nitrogen from the degradation of amino acids is recycled, the excess is excreted, rather than stored in some form. Why has no storage form (analogous to glycogen or triglyceride) evolved? I will consider this from two inter-related viewpoints.
Yes, nitrogen is important for the proteins required for growth and maintenance of organisms, but the effects of depravation are only seen in the long-term. Carbon is of more immediate importance, i.e. the supply of carbohydrates to provide energy. Indeed, in extreme starvation of mammals metabolism treats muscle protein as expendable, breaking it down for the carbon skeleton of the amino acids. Thus, the selective advantage the evolution of a nitrogen storage system might give to an organism is not so clear.
If there were a nitrogen storage system one needs to consider what form it might take. Obviously not as ammonia, because of its toxicity, but it could be condensed into a neutral form. However a small metabolite, such as glutamate (above), would disrupt the metabolic balance, and a dedicated small molecule such as urea would be excluded because of the effect on osmosis and general water balance. So we are thinking of a macromolecule — a conventional protein like ovalbumin (which may, in fact, be a storage protein), or some special 'new' branched protein. The trouble with this method of storing nitrogen is that involves tying up large amounts of carbon — a price, as I argued in 1, that the organism cannot afford to pay.
DavidDavid
$\begingroup$ I wonder who downvoted that and why? $\endgroup$
$\begingroup$ @Rodrigo — Don't worry. I don't any more. I actually plan to add some ideas on why the excess nitrogen is excreted rather than stored, which I hope will improve the answer further. $\endgroup$
$\begingroup$ Well, it will be great if you do that, because it's really interesting. $\endgroup$
$\begingroup$ You might want to add that this process does not offer a way of long-term storage of nitrogen and is only for immediate usage of released ammonia due to side effects of its products and requirement of substrates in other metabolic activities. See my answer. $\endgroup$
– another 'Homo sapien'
$\begingroup$ @another'Homosapien' — Not that I can think of at the moment. Busy week ahead and am away at the weekend, so low profile until next week. $\endgroup$
Short answer: This is because the most common form of nitrogen found in multicellular organisms (not incorporated into any other compound) i.e. ammonia is too toxic to be stored or recycled.
Background: Ammonia, the product of deamination of amino acids, is basic in nature ($NH_3~+~H_2O \rightarrow NH_4OH \rightleftharpoons NH_4^+ + OH^-$) and thus disturbs intracellular pH (Ohmori et al, 1986). Though the mechanisms through which ammonia exerts its toxic effects are not known, hyperammonemia is known to alter several amino acid pathways and neurotransmitter systems, cerebral energy metabolism, nitric oxide synthesis, oxidative stress and signal transduction pathways, leading to irreversible damage to the developing central nervous system: cortical atrophy, ventricular enlargement and demyelination; causing cognitive impairment, seizures and cerebral palsy (Braissant et al, 2013).
Talking about how it is produced (just for information, skip this part if you are aware of urea cycle): Amino acids are deaminated for storage since the carbon skeleton of these amino acids can be easily converted to glucose or fatty acids for storage. This reaction yields ammonia which, in liver, is converted to urea (via urea cycle) which is not only less toxic, but also requires lesser water for excretion. See this page for more information and the diagram below.
Talking about why it is not recycled, there has always been a constant supply of amino acids to organisms (through nitrogen fixation or food). Indeed, amino acids are deaminated for storage i.e. we already have more than enough of it. Also, storing urea is also not easy since it is toxic too (it is just less toxic than ammonia). And since these products are so much harmful, it seems difficult for an organism to evolve itself so that it can store toxins rather than just throwing it away so that it can later be ingested in a useful form (amino acids from plants or herbivores). Finally, evolution works as decent with modification, meaning if something is a better alternative of current situation, it is not necessary that organisms will evolve towards that (unless there is strong selection pressure for it). In short, evolving to recycle nitrogen is (most likely) just not worth it.
P.S. you might be interested in this question to know more about evolution. Another answer here talks about storing excess nitrogen by adding ammonia into amino acids, yielding glutamate and glutamine. However, this process can be used only to a limited extent due to limited amount of substrate ($\alpha$-keto glutaric acid and glutamate) in the body. Also, excess glutamine in the body can have many major and minor side effects like skin rash, vomiting, etc. (see here for full list). Also, excess glutamate has much more pronounced effects. Since glutamate is a neurotransmitter, its excess causes cellular damage. This is what makes it an excitotoxin (see this for more details). Apart from this, the substrate ($\alpha$-keto glutaric acid) is an intermediate of Krebs cycle and is important in many other cellular activities (see here for details). So, its deficiency (caused due to conversion into glutamate) can easily affect many crucial metabolic activities. Thus, though this process seems an effective way of recycling nitrogen, it has many side-effects and is not much reliable i.e. can only be seen as an immediate solution. This is why when glutamate and glutamine build up in body, they are converted back to $\alpha$-keto glutaric acid and glutamate and the ammonia released via urea cycle.
another 'Homo sapien'another 'Homo sapien'
$\begingroup$ Thank you! I know how evolution works, just thought that, in the huge variability of animal kingdom, there might be some animal able to recycle nitrogen. I disagree that it seems difficult for an organism to evolve itself so that it can store toxins. What about all the venom stored in snakes, wasps, scorpions, spiders, rays...? $\endgroup$
$\begingroup$ @rodrigo venoms are specific to some special proteins (see this), and their effect can thus be prevented by storing them in structures lacking those proteins. Since ammonia disrupts as basic cellular properties as pH, it doesn't seem easy to store it inside a cell/tissue/organ. Yet, there might be organisms who can do so, about whom we don't know yet... $\endgroup$
$\begingroup$ @Rodrigo: But for animals, the problem is not storing nitrogen, but getting rid of the excess. It's rather like asking why we haven't evolved to store carbon dioxide :-) $\endgroup$
$\begingroup$ @another'Homosapien' Right. But we can convert ammonia into urea and store urea (or uric acid, for the animals who use that). Maybe we could convert urea (or uric acid) back into a form of nitrogen that we could use to build more proteins, instead of throwing it out. But it seems there's really an excess of nitrogen in animal diets. $\endgroup$
$\begingroup$ @Rodrigo, if you want to go in that direction, you might ask why we don't just store it in the form of protein (the same way we store fat). I guess you could argue that muscles are in fact big stores of protein :) $\endgroup$
– Victor Chubukov
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Why is ammonia converted to urea rather than secreted in the urine? | CommonCrawl |
\begin{document}
\title{On the absence of percolation in a line-segment based lilypond model} \def\mathbb{A}{\mathbb{A}}
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\author{Christian Hirsch} \thanks{Institute of Stochastics, Ulm University, 89069 Ulm, Germany; E-mail: {\tt [email protected]}.}
\begin{abstract} We prove the absence of percolation in a directed Poisson-based random geometric graph with out-degree $1$. This graph is an anisotropic variant of a line-segment based lilypond model obtained from an asymmetric growth protocol, which has been proposed by Daley and Last. In order to exclude backward percolation, one may proceed as in the lilypond model of growing disks and apply the mass-transport principle. Concerning the proof of the absence of forward percolation, we present a novel argument that is based on the method of sprinkling. \end{abstract}
\keywords{{lilypond model}, {mass-transport principle}, {percolation}, {random geometric graph}, sprinkling} \subjclass[2010]{Primary 60K35; Secondary 82B43}
\maketitle
\section{Introduction} \label{intSec} The classical lilypond model describes a hard-sphere system defined by the following growth-stopping protocol. Start with a planar homogeneous Poisson point process $X$ whose atoms serve as germs of a growth process. At time $0$ and with the same speed at each element $x\in X$ a spherical grain begins to grow. As soon as one such grain touches another both cease to grow. This model and various generalizations have been intensively studied for almost $20$ years, so that today an entire family of results concerning existence, uniqueness, stabilization and absence of percolation is known. We refer the reader to the original articles \cite{lilypond,lilypond2,lilypond5,nnhs,lilypond3,lilypond4} for details. The purpose of the present paper is to further advance the completion of this picture by adding a result on the absence of percolation in a lilypond model based on an asymmetric growth-stopping protocol. To be more precise, we consider a model where from each atom of a planar Poisson point process a line segment starts to grow in one of the directions $\pm e_1=(\pm1,0)$, $\pm e_2=(0,\pm1)$ and as soon as a line segment touches an already existing one, the former ceases to grow. This model is an anisotropic variant of one of the two line-segment based lilypond models introduced in~\cite{lilypond6}. A realization of the anisotropic lilypond model is shown in Figure~\ref{sfbmFig}.
The lilypond model gives rise to a directed graph on $X$, where an edge is drawn from $x$ to $y$ if the growth of the line segment at $x$ is stopped by the line segment at $y$. We prove that with probability $1$, this graph exhibits neither forward nor backward percolation, thus verifying~\cite[Conjecture 7.1]{lilypond6} in an anisotropic, one-sided setting. The investigation of the absence of percolation in lilypond-type models has been initiated in~\cite{nnhs} and we briefly review the main idea to establish the absence of percolation in models using spherical grains. After that, we explain which parts of the proof have to be modified in the line-segment setting.
In lilypond models based on spherical grains, the notion of doublets plays a crucial role, where a \emph{doublet} consists of a pair of disk-shaped grains $B_1,B_2\subset\mathbb{R}^2$ such that $B_1$ stops the growth of $B_2$ and $B_2$ stops the growth of $B_1$. This notion allows to subdivide the proof for the absence of percolation provided in~\cite{lilypond} into two steps. In the first step it is shown that a.s. each connected component of the lilypond model contains at most one doublet. In the second step, the a.s. absence of descending chains for homogeneous Poisson point processes is used to show that every connected component also contains at least one doublet. In particular, by mapping each connected component to the midpoint of the two doublet centers we are able construct a locally finite set from the family of connected components in a translation-covariant way. Therefore, an application of the mass-transport principle implies the absence of infinite connected components.
\begin{figure}
\caption{Realization of the lilypond line-segment model (cutout)}
\label{sfbmFig}
\end{figure}
In our setting, the notion of doublets is replaced by cycles, where a \emph{cycle} consists of a sequence of line segments $L_1,L_2,\ldots,L_k\subset\mathbb{R}^2$ such that each $L_{i+1}$ ($i=1,\ldots,k-1$) stops the growth of $L_i$ and $L_1$ stops the growth of $L_k$. As in the setting of spherical grains, it is clear that any connected component contains at most one cycle. Furthermore, another application of the mass-transport principle proves the absence of infinite connected components containing a cycle. On the other hand, to show that every connected component contains at least one cycle, we use the \emph{sprinkling} technique developed in~\cite{sprinkling}. In other words, we first express the planar homogeneous Poisson point process $X$ as superposition of two independent homogeneous Poisson point processes $X=X^{(1)}\cup X^{(2)}$, where the intensity of $X^{(1)}$ is only slightly smaller than the intensity of $X$. When considering the lilypond model on $X^{(1)}$, this graph could contain connected components without a cycle, a priori. The idea of the proof is to show that sprinkling the remaining centers $X^{(2)}$ has the effect of stopping every semi-infinite directed path in the lilypond model based on the point process $X^{(1)}$ and that if the sprinkling intensity is chosen sufficiently small, then no additional semi-infinite paths appear. One key step in the formalization of this idea is to combine a stabilization result for the lilypond model at hand with a standard result on dependent percolation~\cite{domProd} to ensure that, except for small exceptional islands in the plane, one has good control on the effects of the sprinkling.
Our method uses only rather general properties of the line-segment based lilypond model and it might be useful to prove the absence of percolation in further directed Poisson-based random geometric graphs with out-degree $1$. Indeed, the sprinkling technique applies if the underlying graph satisfies a suitable shielding condition and there is a positive probability of modifying the graph locally inside large square so that any path entering the square is stopped.
The paper is organized as follows. In Section~\ref{defSec}, we provide a precise description of the lilypond model under consideration and state the main result of this paper, Theorem~\ref{mainProp}, which deals with the absence of percolation. In Section~\ref{bPercSec}, we explain how the mass-transport principle can be used to deduce the absence of backward percolation from the absence of forward percolation, and we also state several auxiliary results, which are important in the proof of the absence of forward percolation. Assuming these auxiliary results, in Section~\ref{ideaSec}, we prove the absence of forward percolation using the sprinkling technique. Section~\ref{lilySec} is devoted to the proof of the auxiliary results. Finally, in Section~\ref{extSec}, we discuss possible extensions of the sprinkling technique to other directed Poisson-based random geometric graphs of outdegree at most $1$.
\section{Model definition and statement of main result} \label{defSec} The purpose of this section is two-fold. First, we provide a formal definition of the line-segment based asymmetric lilypond model which shall be the topic of our considerations. Second, we state the main result of the present paper, Theorem~\ref{mainProp}.
In the lilypond model under consideration, at time $0$ from every point of an independently marked homogeneous planar Poisson point process $X$ with intensity $1$, a line segment starts to grow in one of the four directions $\mathbb{M}=\lcu\pm e_1,\pm e_2\rcu$ (which is chosen uniformly at random). It stops growing as soon as it hits another line segment. The growth-stopping protocol is asymmetric in the sense that in contrast to the hitting line segment, the segment being hit does \emph{not} stop growing (provided of course that its growth had not already stopped before the collision). Although the growth dynamics of this lilypond model admits a very intuitive description, providing a rigorous mathematical definition is not entirely trivial. Nevertheless, by now this problem has been investigated for many variants of the classical lilypond model from~\cite{nnhs} and existence as well as uniqueness are guaranteed if the underlying point process does not admit a suitable form of descending chains. These chains are usually easy to exclude for independently marked homogeneous Poisson point processes, see e.g.~\cite{lilypond6,lilypond,lilypond5}. For our purposes the correct variant is the following. \begin{definition} \label{aniDescChainDef} Let $b>0$ and $\varphi\subset\mathbb{R}^{2}$ be locally finite. A (finite or infinite) set $\{\xi_i\}_{i\ge1}\subset\varphi$ is said to form a \emph{$b$-bounded anisotropic descending chain} if
$\left|\xi_1-\xi_2\right|_\infty\le b$ and
$\left|\xi_i-\xi_{i+1}\right|_\infty<\left|\xi_{i-1}-\xi_i\right|_\infty$ for all $i\ge2$. A set $\{\xi_i\}_{i\ge1}\subset\varphi$ is said to define an \emph{anisotropic descending chain} if it forms a \emph{$b$-bounded anisotropic descending chain} for some $b>0$. \end{definition} In particular, one can derive the following result whose proof is obtained by a straightforward adaptation of the arguments in~\cite{lilypond6}, where we write $\mathbb{N}_\mathbb{M}$ for the family of all locally finite subsets of $\mathbb{R}^{2,\mathbb{M}}=\mathbb{R}^2\times\mathbb{M}$. \begin{proposition} \label{lilyPreDef} Let $\varphi\in\mathbb{N}_\mathbb{M}$ be an $\mathbb{M}$-marked locally finite set that does not contain anisotropic descending chains and such that $(\xi-\eta)/\left\lvert \xi-\eta \right\rvert\not\in\lcu\pm e_1,\pm e_2,(\pm e_1\pm e_2)/\sqrt{2}\rcu $ for all $x=(\xi,v),y=(\eta,w)\in\varphi$ with $x\ne y$. Then, there exists a unique function $f:\varphi\to [0,\infty]$ with the following properties. \begin{enumerate} \item $[\xi,\xi+f(x)v)\cap [\eta,\eta+f(y)w)=\emptyset$ for all $x=(\xi,v),y=(\eta,w)\in\varphi$ with $x\ne y$ (hard-core property), and \item for every $x\in\varphi$ with $f(x)<\infty$ there exists a unique $y=(\eta,w)\in\varphi$ such that $\xi+f(x)v\in [\eta,\eta+f(y)w)$ and $\left\lvert \xi+f(x)v-\eta\right\rvert<f(x)$ (existence of stopping neighbors). \end{enumerate} \end{proposition}
In the following, we denote by $\mathbb{N}^\prime$ the family of all non-empty $\varphi\in\mathbb{N}_\mathbb{M}$ such that $f(x)<\infty$ for all $x\in\varphi$, such that $(\xi-\eta)/\left\lvert \xi-\eta \right\rvert\not\in\lcu\pm e_1,\pm e_2,(\pm e_1\pm e_2)/\sqrt{2}\rcu $ for all $x=(\xi,v),y=(\eta,w)\in\varphi$ with $x\ne y$, and such that $\varphi$ does not contain anisotropic descending chains. Furthermore, it will be convenient to introduce functions $h_{\mathsf{c}}:\mathbb{N}^\prime\times \mathbb{R}^{2,\mathbb{M}}\to \mathbb{R}^{2,\mathbb{M}}$ and $h_{\mathsf{g}}:\mathbb{N}^\prime\times\mathbb{R}^{2,\mathbb{M}}\to\mathbb{R}^{2}$, where $h_{\mathsf{c}}(\varphi,x)$ denotes the uniquely determined stopping neighbor of $x$ (in the sense of point $2.$ in Proposition~\ref{lilyPreDef}), and where $h_{\mathsf{g}}(\varphi,(\xi,v))=\xi+f(x)v$. In other words, $h_{\mathsf{c}}(\varphi,x)$ denotes the element of $\varphi$ stopping the growth of $x$, whereas $h_{\mathsf{g}}(\varphi,x)$ denotes the actual endpoint of the segment emanating from $x$. Therefore, we call $h_{\mathsf{c}}(\varphi,x)$ the \emph{combinatorial descendant} and $h_{\mathsf{g}}(\varphi,x)$ the \emph{geometric descendant} of $x$. If $x=(\xi,v)\not\in\varphi$, we put $h_{\mathsf{c}}(\varphi,x)=x$ and $h_{\mathsf{g}} (\varphi,x)=\xi$.
Our results on the absence of percolation can be stated using only the notion of combinatorial descendants. Still, tracing the path described by following iteratively the geometric descendants is in a sense much closer to the geometry of the underlying lilypond model than tracing the path of iterated combinatorial descendants. Thus, it is not surprising that geometric descendants play a crucial role in the analysis of percolation properties. This justifies the introduction of separate notation despite the fact that $h_{\mathsf{g}}(\varphi,x)$ could be easily recovered from $x$, $\varphi$ and $h_{\mathsf{c}}(\varphi,x)$.
In order to state our main result, it is convenient to introduce for any $x\in X$ the set $h_{\mathsf{c}}^{(\infty)}(X,x)=\big\{ h_{\mathsf{c}}^{(n)}(X,x):n\ge0\big\}$, where we recursively define $h_{\mathsf{c}}^{(0)}(X,x)=x$ and $h_{\mathsf{c}}^{(n)}(X,x)=h_{\mathsf{c}}\big(X,h_{\mathsf{c}}^{(n-1)}(X,x)\big)$, $n\ge1$. \begin{theorem} \label{mainProp} With probability $1$, the lilypond line-segment model does not percolate, i.e., \begin{enumerate} \item for every $x\in X$ the set $h_{\mathsf{c}}^{(\infty)}(X,x)$ is finite, and\label{part1} \item for every $x\in X$ there exist only finitely many $y\in X$ with $x\in h_{\mathsf{c}}^{(\infty)}(X,y)$.\label{part2} \end{enumerate} \end{theorem}
\section{Absence of backward percolation and statement auxiliary results} \label{bPercSec} The goal of the present section is two-fold. First, we show how the absence of backward percolation (part 2. of Theorem~\ref{mainProp}) can be derived from the absence of forward percolation (part 1. of Theorem~\ref{mainProp}) using the mass-transport principle. Second, we highlight three important properties of the lilypond model, which will be verified in Section~\ref{lilySec}. The benefit of introducing these properties in the present section is that these are the main properties of the lilypond model that will be used in the proof for the absence of forward percolation in Section~\ref{ideaSec}. In this way, we separate the presentation of the sprinkling method from the rather technical verification of the three properties.
In the following, for $r>0$ and $\xi\in\mathbb{R}^2$, we denote by $Q_r(\xi)=[-r/2,r/2]^2+\xi$ the square of side length $r$ centered at $\xi$. We also put $Q_r^\mathbb{M}(\xi)=Q_r(\xi)\times\mathbb{M}$. To begin with, we deduce the absence of backward percolation from the absence of forward percolation. \begin{proof}[Proof of Theorem~{\hyperref[part2]{\ref*{mainProp}.\ref*{part2}}} assuming Theorem~{\hyperref[part2]{\ref*{mainProp}.\ref*{part1}}}] Similar to the arguments used in~\cite{lilypond,lilypond5}, we use the mass-transport principle. Loosely speaking, to define a translation-covariant mass transport, we first note that from the absence of forward percolation, we deduce that starting from any point of the Poisson point process and taking iterated combinatorial descendants we arrive at a cycle. Transporting one unit of mass from that point to the center of gravity of the cycle, we see that choosing a discretization of the Euclidean space into squares, the expected total outgoing mass from any square is finite, whereas the occurrence of backward percolation would result in some square receiving an infinite amount of mass.
To be more precise, for every $x\in X$ we denote by $V(x)$ the set of all $y\in X$ such that $h_{\mathsf{c}}^{(n)}(\varphi,x)=y$ for infinitely many $n\ge1$ and by $C(x)$ the center of gravity of the spatial coordinates in $V(x)$. Since Theorem~{\hyperref[part2]{\ref*{mainProp}.\ref*{part1}}} implies that $V(x)$ is finite, this point is well-defined. Next, we introduce a function $\psi:\mathbb{Z}^2\times\mathbb{Z}^2\to [0,\infty)$ by putting $$\psi(z_1,z_2)=\#\{x\in X\cap Q^\mathbb{M}_1(z_1):C(x)\in Q_1(z_2)\},$$ so that $\psi(z_1,z_2)$ denotes the number of elements of $x\in X\cap Q^\mathbb{M}_1(z_1)$ such that $C(x)$ is contained in $Q_1(z_2)$. Note that if $x\in X$ is such that $C(x)\in Q_1(o)$ and there exist infinitely $y\in X$ with $x\inh_{\mathsf{c}}^{(\infty)}(X,y)$, then $\sum_{z\in\mathbb{Z}^2}\psi(z,o)=\infty$. Furthermore, for any $z\in\mathbb{Z}^d$ the random variables $\psi(z,o)$ and $\psi(o,-z)$ have the same distribution, so that the assumption from Section~\ref{defSec} that $X$ is a homogeneous Poisson point process with intensity $1$ yields \begin{align*} \mathbb{E}\sum_{z\in\mathbb{Z}^2}\psi(z,o)=\sum_{z\in\mathbb{Z}^2}\mathbb{E}\psi(z,o)=\sum_{z\in\mathbb{Z}^2}\mathbb{E}\psi(o,-z)=\mathbb{E}\sum_{z\in\mathbb{Z}^2}\psi(o,-z)=\mathbb{E}\#(X\cap Q^\mathbb{M}_1(o))=1. \end{align*} In particular, $\sum_{z\in\mathbb{Z}^2}\psi(z,o)$ is a.s. finite, so that with probability $1$ there does not exist $x\in X$ such that $C(x)\in Q_1(o)$ and such that $x\inh_{\mathsf{c}}^{(\infty)}(X,y)$ for infinitely many $y\in X$. Using stationarity once more completes the proof of Theorem~{\hyperref[part2]{\ref*{mainProp}.\ref*{part2}}}. \end{proof}
In the proof of the absence of forward percolation the sprinkling method~\cite{sprinkling} is used. In the first step, we form the graph based on all but a tiny fraction of $X$, while in the second step the remaining points of $X$ are added independently in order to stop any of the possibly existing infinite paths. In order to turn this rough description into a rigorous proof, we make use of three important properties of the lilypond model. In the present section, we state these properties and provide explanations and illustrations in order to make the reader familiar with them. Next, in Section~\ref{ideaSec}, we provide a proof for the absence of forward percolation based on these properties. Finally, in Section~\ref{lilySec}, we verify these properties for the specific lilypond model under consideration. We present the three properties in order of increasing complexity.
First, we note that the combinatorial descendant function $h_{\mathsf{c}}$ satisfies a continuity property in the sense that if $\varphi\in\mathbb{N}^\prime$ and $\varphi_1\subset\varphi_2\subset\cdots$ is an increasing family of elements of $\mathbb{N}^\prime$ with $\bigcup_{n\ge1}\varphi_n=\varphi$, then for every $x\in\varphi$ the combinatorial descendant of $x$ in $\varphi$ agrees with combinatorial descendant of $x$ in $\varphi_i$, for all sufficiently large $i\ge1$. \begin{proposition} \label{contProp} The considered lilypond line-segment model satisfies the continuity property. \end{proposition} Section~\ref{contSec} is devoted to the proof of this proposition. Next, when considering the process of passing iteratively to combinatorial descendants, we need some control of distances between the corresponding geometric descendants. To be more precise, we consider a discretization of the Euclidean plane into large squares and call some of these squares good. Loosely speaking, if we start from any finite family of squares with the property that all adjacent squares are good, then these good squares should act as a shield: if starting from some point whose geometric descendant lies in the initial finite family of cubes, then the following geometric descendant is located either also in a square of that family or in an adjacent one. To be more precise, we say that the lilypond model satisfies the shielding condition (SH) if there exists a family of events $(A_s)_{s\ge1}$ on $\mathbb{N}_\mathbb{M}$ with $\lim_{s\to\infty}\mathbb{P}(X^{(1)}\cap Q_{3s}^\mathbb{M}(o) \in A_s)=1$ and such that the following condition is satisfied, where we write $B_1\oplus B_2=\{b_1+b_2:b_1\in B_1,\,b_2\in B_2\}$ for the Minkowski sum of $B_1,B_2\subset\mathbb{R}^2$. \begin{enumerate} \item[(SH)]
Consider the lattice $\mathbb{Z}^2$ with edges given by $\lcu \{ z_1,z_2\}:\left\lvert z_1-z_2\right\rvert_\infty\le1\rcu$, let $B\subset \mathbb{Z}^2$ and denote by $B^\prime=\{z\in\mathbb{Z}^2\setminus B:|z-z^\prime|_\infty= 1 \text{ for some }z^\prime\in B\}$ the outer boundary of $B$. If $\varphi\in \mathbb{N}^\prime$ is such that $(\varphi-sz^\prime)\cap Q_{3s}^\mathbb{M}(o)\in A_s$ for all $z^\prime\in B^\prime$, then $$h_{\mathsf{g}}(\varphi,h_{\mathsf{c}}(\varphi,x))\in sB\oplus Q_{3s}(o)$$
for all $x=(\xi,m)\in\varphi$ with $h_{\mathsf{g}}(\varphi,x)\in sB\oplus Q_s(o)$. \end{enumerate} A site $z\in\mathbb{Z}^2$ with $(X^{(1)}-sz)\cap Q^\mathbb{M}_{3s}(o)\in A_s$, is called \emph{$s$-good}. See Figure~\ref{shFig} for an illustration of the shielding condition (SH). In Section~\ref{lilySec}, we verify that this condition is satisfied in the present setting. \begin{proposition} \label{shieldCond} The considered lilypond line-segment model satisfies condition \emph{(SH)}. \end{proposition}
\begin{figure}
\caption{Possible configuration as in condition (SH); set $sB\oplus Q_s(o)$ in gray}
\label{shFig}
\end{figure}
Finally, we need to know that in the lilypond model under consideration, sprinkled germs can be used to stop already existing segments from growing. To explain this property in greater detail, we first introduce the precise form of sprinkling that will be use in the following. For every $s>1$ the Poisson point process $X$ can be represented as $X=X^{(1)}(s)\cup X^{(2)}(s)$, where $X^{(1)}(s)$ is independent of $X^{(2)}(s)$ and both point processes are independently $\mathbb{M}$-marked homogeneous Poisson point processes with intensities $1-s^{-3}$ and $s^{-3}$, respectively. In particular, $\lim_{s\to\infty}\mathbb{P}(X^{(2)}(s)\cap Q^\mathbb{M}_s(o)=\emptyset)=1$. Usually, the value of $s$ is clear from the context and then we write $X^{(i)}$ instead of $X^{(i)}(s)$.
Having introduced the sprinkling, we now discuss a third important property of the lilypond model, which will be called \emph{uniform stopping property}. We assume that there exists a family of positive real numbers $(p_s)_{s\ge1}$ (possibly tending to $0$ as $s\to\infty$) with the following property. Ideally, we would like to achieve that, conditioned on $X^{(1)}\cap Q^\mathbb{M}_{3s}(o)$, with a probability at least $p_s$ adding the sprinkling $X^{(2)}\cap Q^\mathbb{M}_s(o)$ will cause all segments entering $Q_s(o)$ to become stuck in a cycle in $Q_s(o)$, whereas the structure of the lilypond model outside $Q_s(o)$ is left largely unchanged. However, this goal is too ambitious. Indeed, in some pathological cases, we can encounter realizations of $X^{(1)}\cap Q^\mathbb{M}_{3s}(o)$ for which the probability of observing a suitable sprinkling is much lower than $p_s$. Still, to prove the absence of forward percolation, it suffices to impose that the probability of such pathological configurations tends to $0$ as $s\to\infty$. In order to state this property precisely, for $s>0$, $\varphi\in\mathbb{N}^\prime$ and $z\in\mathbb{Z}^2$ it is convenient to denote by $$\partial^{\mathsf{in}}_{z,s}\(\varphi\)=\big\{ x\in\varphi\setminus Q^\mathbb{M}_s(o):h_{\mathsf{g}}(\varphi,x)\in Q_s(sz)\big\}$$ the subset of all points $x\in\varphi\setminus Q^\mathbb{M}_s(o)$ whose geometric descendant is contained in $Q_s(sz)$.
Now, we say that the lilypond satisfies the uniform stopping condition (condition (US)) if there exist a family of positive real numbers $(p_s)_{s\ge1}$, $p_s\in(0,1]$ and a family of events $(A^\prime_s)_{s\ge1}$ on $\mathbb{N}^\prime\times \mathbb{N}^\prime$ such that $A^\prime_s\subset A_s\times\mathbb{N}^\prime$, \begin{align} \label{posChanceLem} \mathbb{P}\(\big(X^{(1)}\cap Q^\mathbb{M}_{3s}(o),X^{(2)}\cap Q^\mathbb{M}_s(o)\big)\in A_s^\prime\mid X^{(1)}\cap Q^\mathbb{M}_{3s}(o)\)\ge p_s 1_{X^{(1)}\cap Q^\mathbb{M}_{3s}(o)\in A_s}\text{ a.s.}, \end{align}
and such that the following condition is satisfied. \begin{enumerate} \item[(US)] Let $\varphi_1,\varphi_2\in \mathbb{N}^\prime$ be such that $\varphi_2\subset Q^\mathbb{M}_s(o)$ and $(\varphi_1\cap Q_{3s}^\mathbb{M}(o),\varphi_2)\in A^\prime_s$. Moreover, let $\psi\subset\mathbb{R}^{2,\mathbb{M}}\setminus Q^{\mathbb{M}}_{s}(o)$ be a finite set such that for every $z\in\mathbb{Z}^2$ either $\psi\cap Q^\mathbb{M}_{s}(sz)=\emptyset$ or $\((\varphi_1-sz)\cap Q^\mathbb{M}_{3s}(o),(\psi-sz)\cap Q^\mathbb{M}_{s}(o)\)\in A^\prime_s$. If $\varphi_1\cup\psi^\prime\in\mathbb{N}^\prime$ for all $\psi^\prime\subset\varphi_2\cup\psi$, then the following stabilization properties are true. \begin{enumerate} \item If $x\in \varphi_2$, then $h_{\mathsf{c}}(\varphi_{1}\cup\varphi_2\cup\psi,x)=h_{\mathsf{c}}(\varphi_2,x).$ \item If $x\in \varphi_1$, then either $h_{\mathsf{c}}(\varphi_1\cup\varphi_2\cup\psi,x)\in\varphi_2$ or \par\hspace*{-\leftmargin}\parbox{\textwidth}{$$h_{\mathsf{c}}\(\varphi_1\cup\varphi_2\cup\psi,x\)=h_{\mathsf{c}}\(\varphi_1\cup\psi,x\)\text{ and }x\not\in \partial^{\mathsf{in}}_o\(\varphi_1\cup\psi\).$$} \end{enumerate} \end{enumerate} If a site $z\in\mathbb{Z}^2$ is such that $\((X^{(1)}-sz)\cap Q^\mathbb{M}_{3s}(o),(X^{(2)}-sz)\cap Q^\mathbb{M}_{s}(o)\)\in A^\prime_s$, then we also say that the site $z$ (or the sprinkling at this site) is \emph{$s$-perfect}. See Figure~\ref{uStopFig} for an illustration of the uniform stopping condition. Again, the verification of condition (US) is postponed to Section~\ref{lilySec}. \begin{proposition} \label{stopCond} The considered lilypond line-segment model satisfies condition \emph{(US)}. \end{proposition}
\begin{figure}
\caption{Configuration before addition of $\varphi_2$}
\label{uStopFig1}
\caption{Configuration after addition of $\varphi_2$ (red)}
\label{uStopFig2}
\caption{Possible configurations as in condition (US); $\psi=\emptyset$}
\label{uStopFig}
\end{figure}
\begin{remark} Properties (a) and (b) seem complicated at first sight, but allow for a simple heuristic description.
Property (a) can be rephrased as stating that the lilypond model on $\varphi_2$ is autonomous in the sense that changes outside of $Q^\mathbb{M}_s(o)$ cannot alter this sub-configuration. To be more precise, as $h_{\mathsf{c}}(\varphi_{1}\cup\varphi_2\cup\psi,x)=h_{\mathsf{c}}(\varphi_2,x)$ and as $\varphi_2$ is assumed to be contained in $Q^\mathbb{M}_s(o)$, we conclude that $h_{\mathsf{c}}(\varphi_2,x)\in Q^\mathbb{M}_s(o)$, so that the iterates of $x\in\varphi_2$ stay in $Q^\mathbb{M}_s(o)$. It is also useful to note that from $x\in\varphi_2$ and $h_{\mathsf{c}}(\varphi_2,x)\in Q^\mathbb{M}_s(o)$ we can deduce that $h_{\mathsf{g}}(\varphi_2,x)\in Q_s(o)$.
Property (b) yields the existence of suitable configurations such that any line segment that enters $Q_s(o)$ has a descendant in the sprinkled set (and therefore stops inside this square), whereas for any other line segment the addition of the sprinkled germs does not change the descendant. Also note that in the first case of part (b) knowing that $h_{\mathsf{g}}(\varphi_{1}\cup\varphi_2\cup\psi,x)\in Q_s(o)$ for all $x\in\varphi_2$ allows us to deduce from $h_{\mathsf{c}}\(\varphi_1\cup\varphi_2\cup\psi,x\)\in \varphi_2$ that $h_{\mathsf{g}}\(\varphi_1\cup\varphi_2\cup\psi,x\)\in Q_s(o)$. \end{remark} \begin{remark} Of course, the strong degree of internal stability that is required in properties (a) and (b) occurs rather rarely, but condition (US) only requires that for most configurations induced by $X^{(1)}$ it occurs with a positive probability that is bounded away from $0$. \end{remark}
A rough sketch of the proof of Theorem~{\hyperref[part2]{\ref*{mainProp}.\ref*{part1}}} goes as follows. We start by considering the lilypond model on $X^{(1)}$. For large $s$, all but a sub-critical set of sites are $s$-good and therefore the configuration in the corresponding squares will only be influenced by sprinkling close to these squares. First, we add those sprinkled points whose effects we cannot control well in the sense that $A^\prime_s$ is not satisfied. This will increase slightly the sub-critical clusters formed by those squares for which we only have little information about the behavior of the lilypond model. However, since the sprinkling is of very low intensity, these enlarged clusters are still sub-critical. So far, we have held back the sprinkling inside the squares satisfying $A^\prime_s$ and due to their special nature we can precisely control the effects of adding them to the system. In particular, any purported infinite path must also be infinite before adding the final sprinkling. However, a path in the lattice that is killed with probability bounded away from $0$ each time it hits a site in the super-critical cluster, will be killed eventually. The preceding argument is made rigorous in Section~\ref{ideaSec}. Moreover, it can also be used to see that the number iterations until a cycle is reached exhibits at least exponentially decreasing tail probabilities.
\section{Absence of forward percolation} \label{ideaSec} In this section, we provide the details for the proof of Theorem~{\hyperref[part2]{\ref*{mainProp}.\ref*{part1}}}, which is based on a sprinkling argument. The main difficulty arises from the observation that the sprinkling has to be analyzed in a rather delicate way because two essential properties must be satisfied. On the one hand, we have to guarantee that if $x\in X^{(1)}$ is an element with $\# h_{\mathsf{c}}^{(\infty)}({X^{(1)}},x)=\infty$, then after the sprinkling we must have $\# h_{\mathsf{c}}^{(\infty)}({X},x)<\infty$. On the other hand, the sprinkling should not influence the lilypond model too strongly, since for all $x\in X^{(1)}$ with $\# h_{\mathsf{c}}^{(\infty)}({X^{(1)}},x)<\infty$ it has to be ensured that, after adding the sprinkling, the set of descendants $h_{\mathsf{c}}^{(\infty)}({X},x)$ is still finite. As indicated above, we solve this problem by adding the sprinkling in two steps. First, we construct a point process $X^{(3)}\subset\mathbb{R}^{2,\mathbb{M}}$ with $X^{(1)}\subset X^{(3)}\subset X$ by adding only those germs of $X^{(2)}$ for which we have little knowledge as of how their addition would affect the existing directed graph. In a second step we add the remaining germs of $X^{(2)}$ for which we have precise information about their impact on the already existing model.
To construct $X^{(3)}$, we introduce a discrete site process that allows us to determine whether there is either \begin{enumerate} \item no sprinkling inside the corresponding square, or \item an $s$-perfect sprinkling, or \item some other sprinkling. \end{enumerate} To be more precise, we define a $\{0,1,2\}$-valued site process $\lcu Y_z\rcu_{z\in\mathbb{Z}^2}$ as follows. If $X^{(2)}\cap Q^\mathbb{M}_s(sz)=\emptyset$ and $z$ is $s$-good, then $Y_z=0$. Next, $Y_z=1$ if $z$ is $s$-perfect, i.e., if $$\big((X^{(1)}-sz)\cap Q_{3s}^\mathbb{M}(o),(X^{(2)}-sz)\cap Q^\mathbb{M}_s(o)\big)\in A^{\prime}_s.$$ If neither $Y_z=0$ nor $Y_z=1$, then $Y_z=2$. Since we assumed that $\mathbb{N}^\prime$ does not contain the empty configuration, $z$ being $s$-perfect implies that $X^{(2)}\cap Q^\mathbb{M}_s(sz)\ne\emptyset$, so that there is no ambiguity in the definition of $Y$. Also note that conditioned on $X^{(1)}$ the site process $\{ Y_z\}_{z\in\mathbb{Z}^2}$ is an inhomogeneous independent site process.
In the next step we identify a large set of sites for which we have good control over the effect of the sprinkling. We recursively construct sets of revealed sites $(R^{(i)})_{i\ge0}$, $R^{(i)}\subset{\mathbb{Z}^2}$ and bad sites $(B^{(i)})_{i\ge0}$, $B^{(i)}\subset{\mathbb{Z}^2}$ as follows. Initially, put $R^{(0)}=B^{(0)}=\{ z\in\mathbb{Z}^2:Y_z=2\}$. Now suppose $i\ge0$ and that $R^{(i)}\subset\mathbb{Z}^2$ as well as $B^{(i)}\subset\mathbb{Z}^2$ have already been constructed. For $z\in\mathbb{Z}^2$ write $S(z)=\{z^\prime\in\mathbb{Z}^2: \left|z-z^\prime\right|_\infty\le1\}$. Choose the closest bad site $z\in\mathbb{Z}^2$ to the origin with the property that its neighborhood $S(z)$ is not already completely revealed, i.e., $z\in B^{(i)}$ but $S(z)\not\subset R^{(i)}$. If several sites have this property, we choose the lexicographically smallest one. Put $R^{(i+1)}=R^{(i)}\cup S(z)$ and $B^{(i+1)}=B^{(i)}\cup\{z^\prime\in S(z): X^{(2)}\cap Q^\mathbb{M}_s(sz^\prime)\ne\emptyset\}$. Finally, put $R=\bigcup_{i\ge0}R^{(i)}$ and see Figure~\ref{rConstr} for an illustration of the construction of the set $R$. \begin{figure}
\caption{Initial set of bad sites}
\label{rFig1}
\caption{New bad site revealed}
\label{abcabc}
\caption{No new bad sites revealed}
\label{abcabd}
\caption{Construction of $R$}
\label{rConstr}
\end{figure}
We first note that for sufficiently large $s\ge1$ only very few sites are revealed at all.
\begin{lemma} \label{revLem} There exists $s\ge1$ such that with probability $1$, the revealed sites $R\subset\mathbb{Z}^2$ are dominated from above by a sub-critical Bernoulli site-percolation process. \end{lemma} \begin{proof} First, $R\subset Y^{(a)}\cup Y^{(b)}$, where $Y^{(a)}\subset\mathbb{Z}^2$ denotes the set of sites $z\in\mathbb{Z}^2$ whose neighborhood $S(z)$ contains a site which is not $s$-good and where $Y^{(b)}\subset\mathbb{Z}^2$ consists of those $z\in\mathbb{Z}^2$ with $X^{(2)}\cap Q^\mathbb{M}_{3s}(sz)\ne\emptyset$. We note that $Y^{(a)}$ and $Y^{(b)}$ are $5$-dependent site processes that are independent of each other. Moreover, the probability that a given site is contained in $Y^{(a)}\cup Y^{(b)}$ tends to $0$ by the definition of $X^{(2)}$ and the assumption $\lim_{s\to\infty}\mathbb{P}\(X^{(1)}\cap Q_{3s}^\mathbb{M}(o) \in A_s\)=1$. Hence, the claim follows from~\cite[Theorem 0.0]{domProd}. \end{proof} In the remaining part of this section, we fix $s\ge1$ such that $R$ is dominated by a sub-critical Bernoulli site-percolation process. Then, we define $X^{(3)}=X^{(1)}\cup\(X^{(2)}\cap (sR\oplus Q_s(o))\)$. In other words, to create the original point process $X$ from $X^{(3)}$ we only have to add the sprinkling $X^{(2)}$ in the unrevealed region $sR^c\oplus Q_s(o)$, where $R^c= \mathbb{Z}^2\setminus R$ denotes the complement of $R$ in $\mathbb{Z}^2$.
In order to compare $h_{\mathsf{c}}^{(\infty)}\({X^{(3)}},x\)$ and $h_{\mathsf{c}}^{(\infty)}(X,x)$, it is important to understand the effect of adding the sprinkled nodes $X^{(2)}\cap (sR^c\oplus Q_s(o))$ in a step-by-step manner. \begin{lemma} \label{itLem} Let $R^c=\{z_1,z_2,\ldots\}$ be an arbitrary enumeration of $ R^c$ and put $X^{(2,i)}=\bigcup_{j=1}^i (X^{(2)}\cap Q^\mathbb{M}_s(sz_j))$ as well as $X^{(3,i)}=X^{(3)}\cup X^{(2,i)}$. Then, for every $i\ge0$ the following properties are satisfied. \begin{enumerate} \item Let $x\in X^{(3)}$. Then, either $h_{\mathsf{c}}(X^{(3,i)},x)\in X^{(2,i)}\cap Q^\mathbb{M}_s(sz)$ for some $z\in R^c$ or $h_{\mathsf{c}}(X^{(3,j)},x)=h_{\mathsf{c}}(X^{(3)},x)$ for all $j\in\{0,\ldots,i\}$. \item Let $z\in R^c$, $x\in \partial^{\mathsf{in}}_z(X^{(3)})$ and $h_{\mathsf{c}}(X^{(3,j)},x)=h_{\mathsf{c}}(X^{(3)},x)$ for all $j\in\{0,\ldots,i\}$. Then $X^{(2,i)}\cap Q^\mathbb{M}_s(sz)=\emptyset$. \end{enumerate} \end{lemma} \begin{proof} We prove the desired properties by induction on $i$, the case $i=0$ being clear. If $X^{(2)}\cap Q^\mathbb{M}_s\(s{z_{i+1}}\)=\emptyset$, then we deduce immediately that $X^{(3,i+1)}=X^{(3,i)}$ and can apply the induction hypothesis to conclude the proof. Therefore, we may assume $X^{(2)}\cap Q^\mathbb{M}_s(s{z_{i+1}})\ne\emptyset$.
After these preliminary observations, we begin with the proof of the first statement. Applying the definition of $s$-perfectness for $z=z_{i+1}$, $\varphi_1=X^{(3)}$ and $\psi =X^{(2,i)}$, we note that if $h_{\mathsf{c}}(X^{(3,i+1)},x)=h_{\mathsf{c}}(X^{(3,i)},x)$, then this statement also follows from the induction hypothesis. Observing that property 1. is also true in the remaining case, where $h_{\mathsf{c}}(X^{(3,i+1)},x)\in X^{(2)}\cap Q^\mathbb{M}_s\(s{z_{i+1}}\)$
completes the proof of the first statement.
Next, we verify the second statement. For $z\ne z_{i+1}$ the claim follows directly from the induction hypothesis, so that we can concentrate on the case $z=z_{i+1}$ and $X^{(2)}\cap Q^\mathbb{M}_s\(s{z_{i+1}}\)\ne\emptyset$. From $h_{\mathsf{c}}(X^{(3,i)},x)=h_{\mathsf{c}}(X^{(3)},x)$ we conclude that $x\in \partial^{\mathsf{in}}_z(X^{(3,i)})$, so that $s$-perfectness of $z_{i+1}$ implies $h_{\mathsf{c}}(X^{(3,i+1)},x)\in X^{(2)}\cap Q^\mathbb{M}_{s}(s{z_{i+1}})$, contradicting the assumption $h_{\mathsf{c}}(X^{(3,i+1)},x)=h_{\mathsf{c}}(X^{(3)},x)$. \end{proof}
The following result allows us to pass to the limit $i\to\infty$.
\begin{lemma} \label{contLem}
Let $x\in X^{(3)}$. Then, either $h_{\mathsf{c}}(X,x)=h_{\mathsf{c}}(X^{(3,i)},x)$ for all $i\ge0$ or $h_{\mathsf{c}}(X,x)\in X^{(2)}\cap Q^\mathbb{M}_s(sz)$ for some $z\in R^c$. Moreover, if $z\in R^c$, $x\in \partial^{\mathsf{in}}_z(X^{(3)})$ and $h_{\mathsf{c}}(X,x)=h_{\mathsf{c}}(X^{(3)},x)$, then $X^{(2)}\cap Q^\mathbb{M}_s(sz)=\emptyset$. \end{lemma}
\begin{proof}
By continuity, $h_{\mathsf{c}}(X,x)=h_{\mathsf{c}}(X^{(3,i)},x)$ for all sufficiently large $i\ge1$. In particular, part $(i)$ of Lemma~\ref{itLem} implies that either $h_{\mathsf{c}}(X,x)=h_{\mathsf{c}}(X^{(3,i)},x)=h_{\mathsf{c}}(X^{(3,j)},x)$ for all $j\in\{0,\ldots,i\}$ or $h_{\mathsf{c}}(X,x)=h_{\mathsf{c}}(X^{(3,i)},x)\in X^{(2)}\cap Q^\mathbb{M}_s(sz)$ for some $z\in R^c$. Combining this result with part $(ii)$ of Lemma~\ref{itLem} yields the second part of the assertion. \end{proof}
After this preliminary work, it is straightforward to establish the following comparison between the sets $h_{\mathsf{c}}^{(\infty)}({X^{(3)}},x)$ and $h_{\mathsf{c}}^{(\infty)}(X,x)$. \begin{lemma} \label{auxLem2} If $x\in X$ is such that $\# h_{\mathsf{c}}^{(\infty)}(X,x)=\infty$, then $h_{\mathsf{c}}^{(\infty)}(X,x)\subset X^{(3)}$ and $h_{\mathsf{c}}^{(n)}(X,x)=h_{\mathsf{c}}^{(n)}(X^{(3)},x)$ for all $n\ge0$. \end{lemma} \begin{proof} Let $x\in X$ be such that $\#h_{\mathsf{c}}^{(\infty)}(X,x)=\infty$.
If there exists $n\ge0$ with $h_{\mathsf{c}}^{(n)}(X,x)\in X^{(2)}\cap Q^\mathbb{M}_s(sz)$ for some $z\in R^c$, then $\#h_{\mathsf{c}}^{(\infty)}(X,x)<\infty$. Indeed, choosing $i\ge0$ so that $h_{\mathsf{c}}(X^{(3,i)},h_{\mathsf{c}}^{(n)}(X,x))=h_{\mathsf{c}}^{(n+1)}(X,x)$, we can apply part (a) of property (US) to $\varphi^{(1)}=X^{(3)}$, $\psi=X^{(2,i)}\setminus Q^\mathbb{M}_s(sz)$ and $\varphi_2=X^{(2)}\cap Q^\mathbb{M}_s(sz)$ to deduce that $h_{\mathsf{c}}^{(n+1)}(X,x)\in X^{(2)}\cap Q^\mathbb{M}_s(sz)$. Hence, using induction, we conclude that $h_{\mathsf{c}}^{(m)}(X,x)\in X^{(2)}\cap Q^\mathbb{M}_s(sz)$ for all $m\ge n$, which implies that $\#h_{\mathsf{c}}^{(\infty)}(X,x)<\infty$. This observation yields $h_{\mathsf{c}}^{(\infty)}(X,x)\subset X^{(3)}$ and the first statement in Lemma~\ref{contLem} allows us to conclude that $h_{\mathsf{c}}^{(n)}(X,x)=h_{\mathsf{c}}^{(n)}(X^{(3)},x)$ for all $n\ge0$, as desired. \end{proof} \begin{proof}[Proof of Theorem~{\hyperref[part2]{\ref*{mainProp}.\ref*{part1}}}] Assume the contrary. Then, by Lemma~\ref{auxLem2}, there exists $x\in X^{(3)}$ with
$\# h_{\mathsf{c}}^{(\infty)}(X^{(3)},x)=\infty$ and $h_{\mathsf{c}}^{(n)}(X,x)=h_{\mathsf{c}}^{(n)}({X^{(3)}},x)$ for all $n\ge0$. Denote this event by $A^{*}_x$. It suffices to show that $\mathbb{P}\(A^{*}_x\mid X^{(1)},X^{(3)}, R\)=0$ for all $x\in X^{(3)}$.
Putting $h_{\mathsf{g}}^{(n)}(X^{(3)},x)=h_{\mathsf{g}}\big(X^{(3)},h_{\mathsf{c}}^{(n-1)}(X^{(3)},x)\big)$, we consider the sequence of geometric descendants $\big(h_{\mathsf{g}}^{(n)}(X^{(3)},x)\big)_{n\ge1}$. First, we assert that $\big(h_{\mathsf{g}}^{(n)}(X^{(3)},x)\big)_{n\ge1}$ hits infinitely many squares of the form $Q_s(sz)$ with $z\in R^c$. If $z,z^\prime\in\mathbb{Z}^2$ are such that $h_{\mathsf{g}}^{(n)}(X^{(3)},x)\in Q_s(sz)$ and $h_{\mathsf{g}}^{(n+1)}(X^{(3)},x)\in Q_s(sz^\prime)$, then applying condition (SH) with the set $B$ chosen as the connected component of $\{z\}\cup R$ containing $z$ shows that
$z^\prime\in B\oplus Q_3(o)$ (noting that Lemma~\ref{revLem} implies the finiteness of $B$). In particular, if $h_{\mathsf{g}}^{(n+1)}(X^{(3)},x)$ does not lie in $sB\oplus Q_s(o)$, then $z^\prime$ is contained in the outer boundary of $B$, which forms a subset of $R^c$. Hence, if $\#h_{\mathsf{c}}^{(\infty)}\({X^{(3)}},x\)=\infty$, then after performing finitely many steps we obtain a geometric descendant contained in $sR^c\oplus Q_s(o)$. Since $\big(h_{\mathsf{g}}^{(n)}(X^{(3)},x)\big)_{n\ge1}$ hits each bounded Borel set only a finite number of times, this proves the assertion. Hence, there exist infinitely many $z_{1},z_{2},\ldots\in R^c$ and $i_1,i_2,\ldots\ge1$ such that $h_{\mathsf{c}}^{(i_j)}(X^{(3)},x)\in\partial^{\mathsf{in}}_{z_j}(X^{(3)})$ for all $j\ge1$. Moreover, we note that Lemma~\ref{contLem} implies $X^{(2)}\cap Q^\mathbb{M}_s(s{z_{j}})=\emptyset$ for all $j\ge1$.
However, we also observe that conditioned on $X^{(1)}$, $X^{(3)}$ and $R$ the restriction of the site process $\{ Y_z\}_{z\in\mathbb{Z}^2}$ to $ R^c$ defines a $\{0,1\}$-valued Bernoulli site process such that for $z\in R^c$ the (conditional) probability of the event $\{ Y_z=1\}$ is given by $\mathbb{P}\(Y_z=1\mid Y_z\in\{0,1\},X^{(1)} \)$. In particular, the events $\{ Y_{z_j}=1\}$, $j\ge1$ occur independently given $X^{(1)}$, $X^{(3)}$ and $R$ and by~\eqref{posChanceLem} we have $\mathbb{P}\big(Y_z=1\mid Y_z\in\{0,1\},X^{(1)} \big)\ge p_s$ a.s. Therefore, with probability $1$, there exists $j_0\ge1$ with $Y_{z_{j_0}}=1$ contradicting the previously derived $X^{(2)}\cap Q^\mathbb{M}_s(sz_{j_0})=\emptyset$. \end{proof}
\section{Proofs of auxiliary results} \label{lilySec} In the present section we provide the proof of the three auxiliary results that were used in the proof of the absence of forward percolation, Propositions~\ref{contProp},~\ref{shieldCond} and~\ref{stopCond}.
\subsection{Proof of Proposition~\ref{shieldCond}} To begin with, we show that with high probability the length $\nu_1(Q_s(o)\cap I(X,x))$ of the intersection of a given square $Q_s(o)$ with any segment of the form $I(X,x)=[\xi,h_{\mathsf{g}}(X,x)]$ is not too large. \begin{lemma} \label{extInfLem} Let $\alpha>0$. Then, there exists a family of events $\big(A^{(1,\alpha)}_s\big)_{s\ge1}$ with $$\lim_{s\to\infty}\mathbb{P}\big(X^{(1)}\cap Q^\mathbb{M}_{s}(o)\in A^{(1,\alpha)}_s\big)=1$$ and such that the following property is satisfied. If $\varphi\subset Q^\mathbb{M}_s(o)$ is such that $\varphi\in A^{(1,\alpha)}_s$, then for every locally finite $\psi\subset \mathbb{R}^{2,\mathbb{M}}\setminus Q^\mathbb{M}_s(o)$ with $\varphi\cup\psi\in \mathbb{N}^\prime$ and every $x\in \varphi\cup\psi$, $$\nu_1\(I\(\varphi\cup \psi,x\)\cap Q_s(o)\)\le s^\alpha.$$ \end{lemma}
\begin{proof} Without loss of generality, we may assume $\alpha<1$. We consider the cases $x\in \psi$ and $x\in \varphi$ separately and start with the case $x\in\psi$. By rotational and reflection symmetry, it suffices to prove that with high probability for every locally finite $\psi\subset\mathbb{R}^{2,\mathbb{M}}\setminus Q^\mathbb{M}_{s}(o)$ and $x=(\xi,v)\in\psi$ with $v=e_1$, $\pi_1(\xi)<0$ and $\pi_2\(\xi\)\in [0,s/2]$ we have $$\nu_1\big(I(X^{(1)}\cap Q^\mathbb{M}_{s}(o)\cup\psi,x)\cap Q_s(o)\big)\le s^\alpha.$$ Here $\pi_i:\mathbb{R}^{2}\to \mathbb{R}$ denotes the projection to the $i$th coordinate. Put $\delta=s/\lfloor s^{1-\alpha/2}\rfloor$, $R_1=[-2.5\delta,2.5\delta]\times[-\delta,\delta]$ and $R_2=[-\delta/2,\delta/2]\times[-\delta,0]$. For $\xi\in\mathbb{R}^2$ we denote by $E^{}_{\xi}$ the intersection of the events $\varphi\cap ((\xi+R_2)\times \{e_2\})\ne\emptyset$ and $\varphi\cap ((\xi+R_1)\times \mathbb{M})\subset(\xi+R_2)\times \{e_2\}$. See Figure~\ref{extInfFig} for an illustration.
\begin{figure}
\caption{Occurrence of $E_{o}$}
\label{extInfFig}
\end{figure} Then, $$\mathbb{P}(X^{(1)}\cap Q^\mathbb{M}_{s}(o)\in E_\xi)\ge \text{exp}\(-10(1-s^{-3})\delta^2\)\(1-\text{exp}\(-(1-s^{-3})\delta^2/8\)\)\ge {\delta^2}/16$$ for all $\xi\in\mathbb{R}^2$ and all $s>0$ sufficiently large. For $\sigma\in\mathbb{R}$ denote by $M^{}_{\sigma}\subset\mathbb{R}^2$ the set $\{(-s/2+5i\delta) e_1+\sigma e_2: 0\le i\le \lfloor s^\alpha/\(5\delta\)\rfloor\}$, so that $$ 1-\mathbb{P}\big(X^{(1)}\cap Q^\mathbb{M}_{s}(o)\in \cup_{\xi\in M_\sigma^{}} E_{\xi}\big)\le \(1-\delta^2/16\)^{\lfloor s^\alpha/\(5\delta\)\rfloor+1}. $$ Since $s^\alpha\delta^{}\ge s^{\alpha/2}$, we see that $\mathbb{P}\(X^{(1)}\cap Q^\mathbb{M}_{s}(o)\in \bigcap_{\xi\in M_{\sigma}^{}} E_{\xi}^c\)$ decays sub-exponentially fast as $s\to\infty$. Therefore also $\mathbb{P}\(X^{(1)}\cap Q^\mathbb{M}_{s}(o)\in \bigcup_{j=1}^{s/\delta-1}\bigcap_{\xi\in M_{-s/2+j\delta}^{}} E_{\xi}^c\)$ decays sub-exponentially fast in $s$. Note that $\varphi\in \bigcup_{\xi\in M_{-s/2+j\delta}} E_{\xi}$ implies that for every locally finite $\psi\subset\mathbb{R}^{2,\mathbb{M}}\setminus Q^\mathbb{M}_s(o)$ with $\varphi\cup\psi\in \mathbb{N}^\prime$ and every $(\xi,v)\in\psi$ with $v=e_1$, $\pi_1(\xi)<0$ and $\pi_2(\xi)\in[-s/2+j\delta,-s/2+(j+1)\delta]$ we have $\nu_1\(I\(\varphi\cup\psi,x\)\cap Q_s(o)\)\le s^\alpha$. This proves the first case of the claim.
Next, consider the case $x\in \varphi$. Using the Slivnyak-Mecke formula this part can be proven similarly as the case $x\in\psi$, but we include some details for the convenience of the reader. Again, by symmetry it suffices to prove that with high probability for every locally finite $\psi\subset\mathbb{R}^{2,\mathbb{M}}\setminus Q^\mathbb{M}_{s}(o)$ with $X^{(1)}\cap Q^\mathbb{M}_{s}(o)\cup\psi\in\mathbb{N}^\prime$ and $x=(\xi,v)\in X^{(1)}\cap Q^\mathbb{M}_{s}(o)$ with $v=e_1$, $\pi_1(\xi)<s/2-s^\alpha$ and $\pi_2(\xi)\in (0,s/2)$ we have $$\nu_1\big(I(X^{(1)}\cap Q^\mathbb{M}_{s}(o)\cup\psi,x)\cap Q_s(o)\big)\le s^\alpha.$$
For $\xi\in\mathbb{R}^2$ denote by $M^{\prime}_{\xi}\subset\mathbb{R}^2$ the set $\{\xi+5i\delta e_1: 0\le i\le \lfloor s^\alpha/\(5\delta\)\rfloor \}$. Note that $\varphi\in \bigcup_{\eta\in M_{\xi}^{\prime}} E_{\eta}$ implies that for every locally finite $\psi\subset\mathbb{R}^{2,\mathbb{M}}\setminus Q^\mathbb{M}_s(o)$ such that $\varphi\cup\psi\in \mathbb{N}^\prime$ we have $\nu_1\(I\(\varphi\cup\psi,x\)\cap Q_s(o)\)\le s^\alpha$.
Moreover, by the Slivnyak-Mecke formula the expectation of the number $N$ of points $x=(\xi,e_1)\in X^{(1)}\cap Q^\mathbb{M}_{s}(o)$ for which the event $X^{(1)}\cap Q^\mathbb{M}_{s}(o)\in\bigcup_{\eta\in M_{\xi}^{}} E_{\eta}$ occurs is given by $$\mathbb{E} N=\lambda \int_{Q^\mathbb{M}_{s}(o)}\mathbb{P}\Big(\big(X^{(1)}\cap Q^\mathbb{M}_{s}(o)\cup \{ (\xi,v)\}\big)\in\cup_{\eta\in M_{\xi}^{}} E_{\eta} \Big) {\rm d} (\xi,v).$$ By a similar argument as in the case $x\in \psi$, we see that the probability inside the integrand decays sub-exponentially fast in $s$ (uniformly over all $(\xi,v)\in Q^\mathbb{M}_{s}(o)$), which completes the proof Lemma~\ref{extInfLem}. \end{proof} \begin{remark} A suitable analog of Lemma~\ref{extInfLem} can also be shown for isotropic line-segment models, but the proof becomes more involved. Indeed, a similar construction can be used, but now instead of four directions, one considers the shielding property seen from a set of directions with size growing polynomially in $s$. \end{remark} This auxiliary result immediately verifies condition (SH), when using the family of events $\big(A^{(1,\alpha)}_s\big)_{s\ge1}$ for some arbitrary $\alpha\in(0,1/2)$. \begin{proof}[Proof of Proposition~\ref{shieldCond}] Let $B\subset \mathbb{Z}^2$ be a finite set of sites and denote by $B^\prime$ the outer boundary of $B$. Moreover, let $\varphi\in \mathbb{N}^\prime$ be such that $(\varphi-sz^\prime)\cap Q_{s}^\mathbb{M}(o)\in A^{(1,\alpha)}_s$ for all $z^\prime\in B^\prime$ and $x=(\xi,v)\in\varphi$ be such that $h_{\mathsf{g}}(\varphi, x)\in sB\oplus Q_s(o)$. Put $(\eta,w)=h_{\mathsf{c}}(\varphi,x)$ and $D=\mathbb{R}^2\setminus (sB\oplus Q_{3s}(o))$. Using Lemma~\ref{extInfLem} twice implies that
$\mathsf{dist}(\eta,D)\ge s/2$ and $\mathsf{dist}(h_{\mathsf{g}}\(\varphi,(\eta,w)\),D)\ge s/4$, as desired. \end{proof}
\subsection{Proof of Proposition~\ref{contProp}} \label{contSec} Next, we consider the property of continuity. This has already been investigated for a large class of lilypond models and is typically based on suitable descending chains arguments, see e.g.~\cite{lilypond6,lilypond5}. This approach also yields the desired result for the present anisotropic model, but for the convenience of the reader, we provide a detailed proof.
In the following, for $\varphi\in\mathbb{N}^\prime$ and $x=(\xi,v)\in\varphi$, it is convenient to write $f_\varphi(x)$ instead of $|\xi-h_{\mathsf{g}}(\varphi,x)|$. First, we investigate how the behavior of $f_\varphi$ is related to the existence of long descending chains. \begin{lemma} \label{descChainDiffLem2} Let $\varphi,\varphi^{\prime}\in \mathbb{N}^\prime$ and suppose that $x_1\in\varphi^{}\cap\varphi^{\prime}$ is such that $f_{\varphi^{}}(x_1)< f_{\varphi^{\prime}}(x_1)$. Furthermore, define recursively $x_{i+1}\in\varphi^{}\cup\varphi^{\prime}$ by $$x_{i+1}=\begin{cases}h_{\mathsf{c}}\({\varphi^{}},x_i\)&\text{ if }x_i\in\varphi^{}\\ x_i&\text{else}\end{cases}$$ if $i$ is odd and by $$x_{i+1}=\begin{cases}h_{\mathsf{c}}\({\varphi^{\prime}},x_i\)&\text{ if }x_i\in\varphi^{\prime}\\ x_i&\text{else}\end{cases}$$ if $i$ is even. Finally, put $i_0=\min\{i\ge1: x_i\not\in \varphi^{}\cap\varphi^{\prime}\}$. Then, $(x_i)_{1\le i\le i_0}$ constitutes an $f_{\varphi^{}}(x_1)$-bounded anisotropic descending chain of pairwise distinct elements. Additionally, $f_{\varphi^{}}(x_i)<f_{\varphi^{\prime}}(x_i)$ if $i\in\{1,\ldots i_0-1\}$ is odd and $f_{\varphi^{\prime}}(x_i)<f_{\varphi^{}}(x_i)$ if $ i\in\{1,\ldots,i_0-1\}$ is even. In particular, the absence of infinite anisotropic descending chains in $\varphi^{}$ and $\varphi^{\prime}$ implies $i_0<\infty$. \end{lemma} \begin{proof}
At first glance, it might not be obvious how it is possible to have $f_{\varphi}(x_1)< f_{\varphi^\prime}(x_1)$ and $x_2\in\varphi\cap\varphi^\prime$. In other words, how can it be that the segment at $x_2$ stops the growth of the segment at $x_1$ in the lilypond model built from $\varphi$, but not in the one built from $\varphi^\prime$. A more thorough thought reveals that this effect occurs if in the configuration $\varphi^\prime$ the segment at $x_3$ stops the growth of the segment at $x_2$ before the latter can stop the growth of the segment at $x_1$. Next, we extend this observation into a rigorous proof of the lemma and write $x_i=(\xi_i,v_i)$, $i\ge1$. The relation $\left|\xi_1-\xi_2\right|_\infty=f_{\varphi^{}}(x_1)$ follows immediately from the definition of stopping neighbors. Now, assume that $i\in\{2,\ldots,i_0-1\}$ is odd. From $x_i=h_{\mathsf{c}}({\varphi^{\prime}},x_{i-1})$ and $x_{i+1}=h_{\mathsf{c}}(\varphi^{},x_{i})$ we conclude $|\xi_{i}-\xi_{i-1}|_\infty=f_{\varphi^{\prime}}(x_{i-1})$ and $\left|\xi_{i+1}-\xi_{i}\right|_\infty=f_{\varphi^{}}(x_{i})$. Furthermore, by induction $f_{\varphi^{\prime}}(x_{i-1})<f_{\varphi^{}}(x_{i-1})$, so that \begin{align} \label{descEq}
f_{\varphi^{}}(x_i)<\left|\xi_i-h_{\mathsf{g}}(\varphi^{\prime},x_{i-1})\right|<\min\(f_{\varphi^{\prime}}(x_{i-1}), f_{\varphi^{\prime}}(x_i)\). \end{align} The case of even $i\in\{2,\ldots,i_0-1\}$ is analogous.
It remains to show that the $\{x_i\}_{1\le i\le i_0}$ are pairwise disjoint and by the definition of $i_0$, it suffices to prove this claim for $\{x_i\}_{1\le i<i_0}$. Furthermore, as $f_{\varphi^{}}(x_i)<f_{\varphi^{\prime}}(x_i)$ if $1\le i<i_0$ is odd and $f_{\varphi^{\prime}}(x_i)<f_{\varphi^{}}(x_i)$ if $1\le i<i_0$ is even, it suffices to consider the pairwise disjointness of the points in $\{x_i\}_{\substack{1\le i<i_0\\i\text{ even}}}$ and the points in $\{x_i\}_{\substack{1\le i<i_0\\i\text{ odd}}}$ separately. We consider for instance the case of even $i\in\{1,\ldots,i_0-1\}$. Then, we note that an application of~\eqref{descEq} and its analog for even parity yields \begin{align*} f_{\varphi^{\prime}}(x_i)>f_{\varphi^{}}(x_{i+1})>f_{\varphi^{\prime}}(x_{i+2}), \end{align*} so that by induction $f_{\varphi^{\prime}}(x_i)>f_{\varphi^{\prime}}(x_j)$ for all even $i,j\in\{1,\ldots,i_0-1\}$ with $i<j$. \end{proof} As corollary, we verify the continuity property of the lilypond model. \begin{proof}[Proof of Proposition~\ref{contProp}] Let $x\in\varphi$ be arbitrary and choose $n_0^\prime\ge1$ such that $x\in\varphi_{n_0^\prime}$. Lemma~\ref{descChainDiffLem2} implies that if $n\ge1$ and $s>0$ are such that $\varphi\cap Q^\mathbb{M}_s(o)=\varphi_n\cap Q^\mathbb{M}_s(o)$, but $h_{\mathsf{c}}(\varphi,x)\neh_{\mathsf{c}}(\varphi_n,x)$, then there exists a $f_{\varphi}(x)$-bounded anisotropic descending chain starting in $x$ and leaving $Q_s(o)$. In particular, if $n_1<n_2<\cdots$ is an increasing sequence with $h_{\mathsf{c}}(\varphi,x)\neh_{\mathsf{c}}(\varphi_{n_i},x)$ for all $i\ge1$, then there exist arbitrarily long $f_{\varphi}(x)$-bounded anisotropic chains starting at $x$. Since $\varphi$ is locally finite, this would produce an infinite $f_{\varphi}(x)$-bounded anisotropic chain starting at $x$, thereby contradicting the definition of $\mathbb{N}^\prime$. \end{proof} We conclude the investigation of the continuity property by showing that if $X\subset\mathbb{R}^{2,\mathbb{M}}$ is an independently and uniformly marked homogeneous Poisson point process, then $\mathbb{P}(X\in\mathbb{N}^\prime)=1$. To obtain bounds for the probability of observing long anisotropic descending chains, the following auxiliary computation is useful. \begin{lemma} \label{intCompLem} Let $b\ge0$, $k\ge0$ and $\xi_0\in\mathbb{R}^2$. Then,
$\int_{\mathbb{R}^2} 1_{|\xi-\xi_0|_\infty\le b}|\xi-\xi_0|_\infty^{2k} {\rm d} x={4}b^{2k+2}/({k+1}).$ \end{lemma} \begin{proof} We may assume $\xi_0=o$ and put $\xi=(\xi^{(1)},\xi^{(2)})$. Then, by symmetry, \begin{align*}
\int 1_{|\xi|_\infty\le b}|\xi|_\infty^{2k} {\rm d} \xi&=4\int_0^b\int_0^b\max\{\xi^{(1)},\xi^{(2)}\}^{2k}{\rm d}\xi^{(2)} {\rm d}\xi^{(1)}\\ &=8\int_0^b(\xi^{(1)})^{2k}\int_0^{\xi^{(1)}}{\rm d}\xi^{(2)}{\rm d}\xi^{(1)}\\ &=8\int_0^b(\xi^{(1)})^{2k+1}{\rm d}\xi^{(1)}\\ &={8}b^{2k+2}/({2k+2}).&\hspace{6.4cm}\qedhere \end{align*} \end{proof}
Using Lemma~\ref{intCompLem}, we can bound the probability of seeing long descending chains.
\begin{lemma} \label{descChainCompLem} Let $n\ge1$, $b,s>0$ and let $X$ be a homogeneous Poisson point process in $\mathbb{R}^2$ with intensity $\lambda>0$. Denote by $A^{(2)}_{s,b,n}$ the event that there exist pairwise distinct $X_0,\ldots,X_n\in X$, with $X_0\in Q_s(o)$ and such that $\{X_i\}_{0\le i\le n}$ forms a $b$-bounded anisotropic descending chain. Then, $\mathbb{P}\big(A^{(2)}_{s,b,n}\big)\le{s^2(4b^2)^n\lambda^{n+1}}/n!$. \end{lemma}
\begin{proof} Denote by $N$ the number of $(n+1)$-tuples of pairwise distinct elements $X_0,\ldots,X_n\in X$ such that $X_0\in Q_s(o)$ and $\{X_i\}_{0\le i\le n}$ forms a $b$-bounded anisotropic descending chain. Then, using Lemma~\ref{intCompLem} and the Campbell formula, \begin{align*}
\mathbb{E} N&\le\lambda^{n+1} \int \cdots\int 1_{\xi_0\in Q_s(o)}1_{b\ge \left|\xi_0-\xi_1\right|_\infty\ge\cdots\ge\left|\xi_{n-1}-\xi_n\right|_\infty}{\rm d} \xi_n\cdots {\rm d} \xi_0\\
&=4\lambda^{n+1}\int \cdots\int 1_{\xi_0\in Q_s(o)}1_{b\ge \left|\xi_0-\xi_1\right|_\infty\ge\cdots\ge\left|\xi_{n-2}-\xi_{n-1}\right|_\infty}\left|\xi_{n-2}-\xi_{n-1}\right|_\infty^{2}{\rm d} \xi_{n-1}\cdots {\rm d} \xi_0\\ &=\cdots\\ &=\frac{\(4b^2\)^n\lambda^{n+1}}{n!}\int 1_{\xi_0\in Q_s(o)}{\rm d} \xi_0=\frac{s^2\(4b^2\)^n\lambda^{n+1}}{n!}.&\hspace{1.6cm}\qedhere \end{align*} \end{proof} Lemma~\ref{descChainCompLem} implies the absence of infinite anisotropic descending chains under Poisson assumptions. \begin{corollary} Let $X$ be an independently and uniformly $\mathbb{M}$-marked homogeneous Poisson point process in $\mathbb{R}^2$. Then, $\mathbb{P}(X\in\mathbb{N}^\prime)=1$. \end{corollary}
\subsection{Proof of Proposition~\ref{stopCond}} Finally, we verify the uniform stopping condition (US). In order to achieve this goal, it is first of all crucial to note that the configuration of the lilypond model in a given square is determined by the line segments entering this square. To make this more precise, it is convenient to introduce a variant of $\partial^{\mathsf{in}}_{z,s} \varphi$ that also takes into account line segments intersecting the square $Q_{s}(sz)$. Hence, for $s>0$, $z\in\mathbb{Z}^2$ and $\varphi\in\mathbb{N}^\prime$ we put $$\partial^{\mathsf{in},*}_{z,s}\(\varphi\) =\{x\in \varphi\setminus Q_{s}^\mathbb{M}(sz): I(\varphi,x)\cap Q_{s}(sz)\ne\emptyset\}.$$ Furthermore, also line segments leaving a square will play an important role, so that for $s>0$, $z\in\mathbb{Z}^2$ and $\varphi\in\mathbb{N}^\prime$ we put $$\partial^{\mathsf{out}}_{z,s}\(\varphi\) =\{x\in \varphi\cap Q_{s}^\mathbb{M}(sz): h_{\mathsf{g}}(\varphi,x)\not\in Q_s(sz)\}.$$ Since the value of $s$ is usually clear from the context, we often write $\partial^{\mathsf{in},*}_z\(\varphi\)$ and $\partial^{\mathsf{out}}_z\(\varphi\)$ instead of $\partial^{\mathsf{in},*}_{z,s}\(\varphi\)$ and $\partial^{\mathsf{out}}_{z,s}\(\varphi\)$. Using these definitions, we now obtain the following auxiliary result. \begin{lemma} \label{inAuxLem} Let $s>0$ and $\varphi^{},\varphi^{\prime}\in\mathbb{N}^\prime$ be such that $\varphi^{}\cap Q^\mathbb{M}_s(o)\cup \partial^{\mathsf{in},*}_{o}\({\varphi^{}} \)=\varphi^{\prime}\cap Q^\mathbb{M}_s(o)\cup \partial^{\mathsf{in},*}_{o}\({\varphi^{\prime}}\).$ Then, \begin{enumerate} \item $h_{\mathsf{c}}\({\varphi^{}},x\)=h_{\mathsf{c}}\({\varphi^{\prime}},x\)$ for all $x\in\varphi^{}\cap Q^\mathbb{M}_s(o)\cup\partial^{\mathsf{in},*}_{z}\({\varphi^{}}\)$ with $\{h_{\mathsf{g}}({\varphi^{}},x),h_{\mathsf{g}}({\varphi^{\prime}},x)\}\cap Q_s(o)\ne\emptyset$, \item $\partial^{\mathsf{in},*}_{o}\(\varphi^{}\)=\partial^{\mathsf{in}}_o\(\varphi^{}\)$ if and only if $\partial^{\mathsf{in},*}_{o}\(\varphi^{\prime}\)=\partial^{\mathsf{in}}_o\(\varphi^{\prime}\)$, \item $\partial^{\mathsf{out}}_{o}\(\varphi^{\prime}\)=\partial^{\mathsf{out}}_o\(\varphi^{}\)$. \end{enumerate} \end{lemma} \begin{proof} For readability, we write $f$, $f^\prime$ instead of $f_{\varphi^{}}$, $f_{\varphi^{\prime}}$. Put $$\varphi^{\prime\prime}=\lcu x\in \varphi^{}\cap Q^\mathbb{M}_s(o)\cup\partial^{\mathsf{in},*}\varphi: \lcu h_{\mathsf{g}}(\varphi^{},x), h_{\mathsf{g}}(\varphi^{\prime},x)\rcu \cap Q_s(o)\ne\emptyset \rcu.$$ Our first goal is to show $f(x)=f^\prime(x)$ for all $x\in\varphi^{\prime\prime}$. Suppose, for the sake of deriving a contradiction, that there exists $x_1\in\varphi^{\prime\prime}$ with $f(x_1)\ne f^\prime(x_1)$, e.g. $f(x_1)<f^\prime(x_1)$. By Lemma~\ref{descChainDiffLem2} it suffices to show that for any such $x_1$ we have $x_2\in\varphi^{\prime\prime}$, where $x_2=h_{\mathsf{c}}(\varphi^{},x_1)$.
First, we assert that $h_{\mathsf{g}}(\varphi^{},x_1)\in Q_s(o)$. Assuming the contrary, we could conclude from $x_1\in\varphi^{\prime\prime}$ that $h_{\mathsf{g}}(\varphi^{\prime},x_1)\in Q_s(o)$. Since $f(x_1)<f^\prime(x_1)$, we deduce that $h_{\mathsf{g}}(\varphi,x_1)$ is contained on the line segment connecting $x_1$ and $h_{\mathsf{g}}(\varphi^\prime,x_1)$, which is only possible if $x_1\not\in Q^\mathbb{M}_s(o)$. In particular, $x_1\in\partial^{\mathsf{in},*}_{o}(\varphi^{\prime})$. However, as $h_{\mathsf{g}}(\varphi^{},x_1)$ is not contained in $Q_s(o)$, we obtain that $x_1\not\in\partial^{\mathsf{in},*}_{o}(\varphi^{})$ contradicting our assumption $\partial^{\mathsf{in},*}_{o}(\varphi^{})=\partial^{\mathsf{in},*}_{o}(\varphi^{\prime})$. This proves the assertion, which implies that $x_2\in \varphi^{}\cap Q^\mathbb{M}_s(o)\cup \partial^{\mathsf{in},*}_{o}(\varphi^{})$. From the assumption $f(x_1)<f^\prime(x_1)$, we then conclude $f^\prime(x_2)<f(x_2)$. We claim that $h_{\mathsf{g}}(\varphi^{\prime},x_2)\in Q_s(o)$ and assume the contrary for the sake of deriving a contradiction. Then, we conclude from $h_{\mathsf{g}}(\varphi^\prime,x_2)\in [x_2,h_{\mathsf{g}}(\varphi^{},x_1)]$ and $h_{\mathsf{g}}(\varphi,x_1)\in [x_2,h_{\mathsf{g}}(\varphi,x_2)]$ that $x_2\not\in Q^\mathbb{M}_s(o)$, $x_2\not\in\partial^{\mathsf{in},*}_o(\varphi^\prime)$ and $x_2\in\partial^{\mathsf{in},*}_o(\varphi)$, contradicting our assumption. This completes the proof of $f(x)=f^\prime(x)$ for all $x\in\varphi^{\prime\prime}$. Property 2. is an immediate consequence of property 1. To prove the third claim, let $x\in \varphi^{}\cap Q^\mathbb{M}_s(o)$. If $x\not\in\partial^{\mathsf{out}}_o(\varphi)$, then $h_{\mathsf{g}}(\varphi,x)\in Q_s(o)$ and therefore also $h_{\mathsf{g}}(x,\varphi^\prime)=h_{\mathsf{g}}(x,\varphi)\in Q_s(o)$. In other words, $\varphi\cap Q^\mathbb{M}_s(o)\setminus \partial^{\mathsf{out}}_o(\varphi)\subset\varphi^{\prime}\cap Q^\mathbb{M}_s(o)\setminus \partial^{\mathsf{out}}_o\(\varphi^\prime\)$ and the other inclusion follows by symmetry. \end{proof}
Using Lemmas~\ref{extInfLem} and~\ref{descChainDiffLem2}, we show that the set $\partial^{\mathsf{in},*}_{o,s} (X)$ stabilizes with high probability. \begin{lemma} \label{extStab} There exists a family of events $\big(A_s^{(3)}\big)_{s\ge1}$ such that $$\lim_{s\to\infty}\mathbb{P}\big(X^{(1)}\cap Q^\mathbb{M}_{3s}(o)\in A_s^{(3)}\big)=1$$
and such that if $\varphi\in\mathbb{N}^\prime$ is such that $\varphi\cap Q^\mathbb{M}_{3s}(o)\in A_s^{(3)}$, then $\partial^{\mathsf{in},*}_o \(\varphi\) =\partial^{\mathsf{in},*}_o\(\varphi\cup\psi\)$ for all locally finite $\psi\subset\mathbb{R}^{2,\mathbb{M}}\setminus Q^\mathbb{M}_{3s}(o)$ with $\varphi\cup \psi \in \mathbb{N}^\prime$. \end{lemma}
\begin{proof}In the proof, we make use of the events $A^{(1,1/8)}_s$ and $A^{(2)}_{s,b,n}$ introduced in Lemmas~\ref{extInfLem} and~\ref{descChainCompLem}, respectively. Furthermore, put $S=\{z\in\mathbb{Z}^2:|z|_\infty=2\}$ and denote by $$A^{(3^\prime)}_s=A^{(1,1/8)}_s\cap \bigcap_{z\in S} \big\{\varphi\in \mathbb{N}_\mathbb{M}: \(\varphi-sz/3\)\cap Q^\mathbb{M}_{s/3}(o) \in A^{(1,1/8)}_{s/3}\big\} $$ the event that $A^{(1,1/8)}_s$ occurs in $Q_s(o)$ intersected with the event that $A^{(1,1/8)}_{s/3}$ occurs in each of the $(s/3)$-squares surrounding $Q_{s}(o)$. Now assume that $\varphi\in\mathbb{N}^\prime$ is such that $\varphi\cap Q^\mathbb{M}_{3s}(o)\in A^{(3^\prime)}_s$ and that there exists $\psi\subset\mathbb{R}^{2,\mathbb{M}}\setminus Q^\mathbb{M}_{3s}(o)$ with $\varphi\cup \psi\in \mathbb{N}^\prime$ and $\partial^{\mathsf{in},*}_o \(\varphi\) \ne\partial^{\mathsf{in},*}_o\(\varphi\cup\psi\).$ Then, there exists $x\in Q^\mathbb{M}_{2s}(o)$ with $h_{\mathsf{c}}\(\varphi,x\)\ne h_{\mathsf{c}}\(\varphi\cup\psi,x\)$. By Lemma~\ref{descChainDiffLem2}, there exists an $s^{1/4}$-bounded anisotropic descending chain of pairwise distinct elements of $\varphi$ starting in $x$ and ending in $\mathbb{R}^2\setminus Q_{3s}(o)$. Furthermore, by our assumption this chain consists of at least $n_s=\lfloor s/(2s^{1/4})\rfloor= \lfloor s^{3/4}/2\rfloor $ hops. Hence, $\varphi\cap Q^\mathbb{M}_{3s}(o) \in A^{(2)}_{2s, s^{1/4},n_s}$ and therefore, we put $A^{(3)}_s=A^{(3^\prime)}_s\setminus A^{(2)}_{2s, s^{1/4},n_s}$. By Lemma~\ref{descChainCompLem}, the probability for the occurrence of $X^{(1)}\cap Q^{\mathbb{M}}_{3s}(o)\in A^{(2)}_{2s,s^{1/4},n_s}$ is bounded from above by $4s^2(4s^{1/2})^{n_s}/ {n_s}!$. By Stirling's formula, the latter expression tends to $0$ as $s\to\infty$. \end{proof}
First, we provide a definition of $A_s$ such that if $\varphi\subset Q^\mathbb{M}_{3s}(o)$ is such that $\varphi\in A_s$, then \begin{enumerate} \item $\varphi$ satisfies the shielding property, i.e., $\varphi\cap Q^\mathbb{M}_{s}(o)\in A^{(1,1/2)}_s,$ \item $\varphi$ satisfies the external stabilization property, i.e., $\varphi\in A^{(3)}_s,$ and \item the points of $\varphi$ do not come too close to each other and also not too close to the boundary of $Q_s(o)$. \end{enumerate} To be more precise, for $s\ge1$ we put $$A_s=A^{(1,1/2)}_s\cap A^{(3)}_s\cap A^{(4)}_s,$$
where $A^{(4)}_s$ denotes the family of all $\varphi\in \mathbb{N}_\mathbb{M}$ such that $\varphi\subset Q^\mathbb{M}_{3s}(o)$ and $\varphi$ is \emph{$s^{-4}$-separated}, i.e., $|\pi_k(\xi)-\pi_k(\eta)|\ge s^{-4}$ for all $k\in\{1,2\}$ and all $$x=(\xi,v),y=(\eta,w)\in (\varphi\cap Q^\mathbb{M}_{3s}(o))\cup\big(\{\pm(s/2,s/2)\}\times \mathbb{M}\big)\text{ with }x\ne y,$$ where we recall that $\pi_k(\xi)$, $k\in\{1,2\}$ denotes the $k$th coordinate of $\xi$. Note that the set $\{\pm(s/2,s/2)\}\times \mathbb{M}$ is added, since it is important to have some room at the boundary of $Q_s(o)$, where we can add sprinklling used to stop incoming segments. Taking into account Lemmas~\ref{extInfLem} and~\ref{extStab}, in order to show that $\lim_{s\to\infty}\mathbb{P}(X^{(1)}\cap Q^\mathbb{M}_{3s}(o)\in A_s)=1$ it suffices to prove $\lim_{s\to\infty}\mathbb{P}\big(X^{(1)}\cap Q^\mathbb{M}_{3s}(o)\in A^{(4)}_s\big)=1$, which is achieved in the following result. \begin{lemma} As $s\to\infty$ the probability $\mathbb{P}\big(X^{(1)}\cap Q^\mathbb{M}_{3s}(o)\in A^{(4)}_s\big)$ tends to $1$. \end{lemma} \begin{proof}
The expected number of distinct points $x=(\xi,v)$, $y=(\eta,w)\in X^{(1)}\cap Q^\mathbb{M}_{3s}(o)$ satisfying $|\pi_k(\xi)-\pi_k(\eta)|\le s^{-4}$ for some $k\in\{1,2\}$ is of order at most $s^3\cdot s^{-4}$ and therefore tends to $0$ as $s\to\infty$. Similarly, the expected number of $x=(\xi,v)\in X^{(1)}\cap Q^\mathbb{M}_{3s}(o)$ with $|\pi_k(\xi)-\zeta|\le s^{-4}$ for some $k\in\{1,2\}$ and $\zeta=\pm s/2$ is of order $s\cdot s^{-4}$, so that it also tends to $0$ as $s\to\infty$. \end{proof}
The next step is to introduce the family of events $\(A^\prime_s\)_{s\ge1}$, i.e., to define suitable sprinkling configurations. Here, small four-cycles play an important role. \begin{definition} \label{4cycle} Let $\delta>0$ and $\xi\in\mathbb{R}^2$. We say that $D=\{ x_{1},\ldots,x_{4}\}=\{(\xi_{1},v_1),\ldots,(\xi_4,v_{4})\}\subset Q^{\mathbb{M}}_\delta(\xi)$ forms a \emph{$(\xi,\delta)$-cycle} if the following conditions are satisfied, where we put formally $x_5=x_1$ and $v_5=v_1$. \begin{enumerate} \item $v_j=\rho_{\pi/2}\(v_{j+1}\)$ for all $j\in\{1,\ldots,4\}$,
where $\rho_{\pi/2}:\mathbb{R}^2\to\mathbb{R}^2$ denotes rotation by $\pi/2$, \item $h_{\mathsf{g}}\(D,x_j\)\in Q_{\delta}(\xi)$ for all $j\in\{1,\ldots,4\}$, \item $h_{\mathsf{g}}\(D,x_j\)\in I\(D,x_{j+1}\)$ for all $j\in\{1,\ldots,4\}$, \item $\min\(\langle \xi-\xi_{j}, v_j\rangle,\langle \xi-\xi_{j}, v_{j+1}\rangle\)>0$ for all $j\in\{1,\ldots,4\}$. \end{enumerate} The fourth condition ensures that $\xi$ belongs to the inner part delimited by the $(\xi,\delta)$-cycle. In particular, if we consider a line segment starting from $\mathbb{R}^2\setminus Q_\delta(\xi)$ and whose corresponding ray contains $\xi$, then this ray must hit the cycle. See Figure~\ref{4cycleFig} for an illustration of a $(\xi,\delta)$-cycle.
\end{definition} \begin{figure}
\caption{Example of a $(\xi,\delta)$-cycle}
\label{4cycleFig}
\end{figure} An important feature of $(\xi,\delta)$-cycles is the following strong external stabilization property. \begin{lemma} \label{stabCycle} Let $\delta>0$, $\xi\in\mathbb{R}^2$ and $\varphi\in \mathbb{N}^\prime$ be such that $\varphi\cap Q^\mathbb{M}_{3\delta}(\xi) =\emptyset$. Furthermore, let $D=\{x_1,\ldots,x_4\}=\{(\xi_1,v_1),\ldots,(\xi_4,v_4)\}\subset\mathbb{R}^{2,\mathbb{M}}$ be a $(\xi,\delta)$-cycle. Then, $h_{\mathsf{c}}\({D\cup\varphi},x_i\)=h_{\mathsf{c}}\(D,x_i\)$ for all $i\in\{1,\ldots,4\}$. \end{lemma} \begin{proof} Suppose there exists $i\in\{1,\ldots,4\}$ with $h_{\mathsf{c}}\(D\cup\varphi,x_i\)\ne h_{\mathsf{c}}\(D,x_i\)$. By the hard-core property we see that we cannot have $f_{D\cup\varphi}(x_j)\ge f_D(x_j)$ for all $j\in\{1,\ldots,4\}$ and we choose $j_1\in\{1,\ldots,4\}$ with $f_{D\cup\varphi}(x_{j_1})<f_{D}(x_{j_1})$. Since all segments grow at the same speed (which is equal to 1), we deduce that \begin{align*}
|\eta-\xi|_\infty&\le |\eta-\xi_{j_1}|_\infty+|\xi_{j_1}-\xi|_\infty\le f_{D\cup\varphi}(x_{j_1}) +\delta/2\le 2\delta, \end{align*} where $(\eta,w)=h_{\mathsf{c}}\({D\cup\varphi},x_{j_1}\)$. Thus, $h_{\mathsf{c}}\({D\cup\varphi},x_{j_1}\)\in Q^\mathbb{M}_{3\delta}(\xi)$, violating the assumption $\varphi\cap Q^\mathbb{M}_{3\delta}(\xi)=\emptyset$. \end{proof} The notion of $(\xi,\delta)$-cycles can be used to define configurations $X^{(2)}\cap Q^\mathbb{M}_s(o)$ satisfying the relation $\# h_{\mathsf{c}}^{(\infty)}\({X^{(1)}\cup X^{(2)}\cap Q^\mathbb{M}_s(o)},x\)<\infty$ for all $x\in\partial^{\mathsf{in},*}_o\({X^{(1)}}\)$. These cycles are used to stop all segments intersecting $Q_s(o)$ except for those leaving the square. More precisely, we make the following definition, where for $\psi\in\mathbb{N}^\prime$ we write $\psi\in A^{*}_{\xi,\delta}$ if the configuration of $\psi\cap Q_\delta(\xi)$ consists precisely of one $(\xi,\delta)$-cycle. \begin{definition} \label{deviceDef} Let $s>0$, $\varphi\in\mathbb{N}_\mathbb{M}$ and $\psi\in\mathbb{N}^\prime$ be such that $\varphi\subset Q^\mathbb{M}_{3s}(o)$, $\varphi\in A^{(3)}_s$ and $\psi\subset Q^\mathbb{M}_s(o)$. Let $\varphi^\prime\in\mathbb{N}^\prime$ be any locally finite set with $\varphi^\prime\cap Q^\mathbb{M}_{3s}(o)=\varphi$. Then, we put $\delta=s^{-4}$, \begin{align*} M_1&=\{\xi+{3\delta}v/{8}:(\xi,v)\in \varphi\cap Q^\mathbb{M}_s(o)\setminus\partial^{\mathsf{out}}_o(\varphi^\prime)\},\\ M_2&=\{P_{(\xi,v)}+{3\delta}v/{8}:(\xi,v)\in \partial^{\mathsf{in}}_o(\varphi^\prime)\},\text{ and }\\ M_3&=\{(-s/2+3\delta/8)(e_1+e_2)\}. \end{align*} Here $P_{(\xi,v)}$ denotes the first intersection point of $I(\varphi,(\xi,v))$ and $\partial Q_s(o)$. Note that since $\varphi\in A^{(3)}_s$, the definition of $M_1$, $M_2$ and $M_3$ does not depend on the choice of $\varphi^\prime$. Then, we define $(\varphi,\psi)\in A^\prime_s$ to be the intersection of the events $\lcu \psi\subset\((M_1\cup M_2\cup M_3)\oplus Q_{\delta/16}(o)\)\times\mathbb{M}\rcu$, $\varphi\in A_s$ and $\psi\in \bigcap_{(\xi,v)\in M_1\cup M_2\cup M_3} A^{*}_{\xi,\delta/16}$. See Figure~\ref{sprinkleFig} for an illustration of the effect on the lilypond model when adding a set of germs $\psi$ satisfying $(\varphi,\psi)\in A^{\prime}_{s}$. \end{definition} \begin{figure}
\caption{Configuration before addition of $\psi$}
\label{sprinkleFig1}
\caption{Configuration after addition of $\psi$}
\label{sprinkleFig2}
\caption{Impact of the addition of $\psi$ with $(\varphi,\psi)\in A^\prime_s$}
\label{sprinkleFig}
\end{figure}
\begin{lemma} \label{posChanceLemLil} The events $\(A_s^\prime\)_{s\ge1}$ introduced in Definition~\emph{\ref{deviceDef}} satisfy condition~\eqref{posChanceLem}. \end{lemma} \begin{proof} Assume that $X^{(1)}\cap Q^\mathbb{M}_{3s}(o)\in A_s$ and let $M_1,M_2,M_3\subset Q_s(o)$ be as in the definition of the event $A^\prime_s$. Furthermore, put $\delta=s^{-4}$ and $M_{1,2,3}=M_1\cup M_2\cup M_3$. We conclude from $\delta$-separatedness that for all $x_1,x_2\in M_{1,2,3}$ with $x_1\ne x_2$ we have $Q_{\delta/16}(x_1)\cap Q_{\delta/16}(x_2)=\emptyset$. In particular, for every $(\xi,v)\in M_{1,2,3}$ the event $X^{(2)}\in A^{*}_{\xi,\delta/16}$ is independent of the family of events $X^{(2)}\in A^{*}_{\eta,\delta/16}$ for $(\eta,w)\in M_{1,2,3}$ with $\eta\ne \xi$. Furthermore, from $\delta$-separatedness we also conclude $\#M_1+\#M_2 \le 3\lceil s\delta^{-1}\rceil$. Finally, note that $\mathbb{P}\(X^{(2)}\cap Q^\mathbb{M}_s(o)\subset (M_{1,2,3}\oplus Q_{\delta/16}(o))\times\mathbb{M}\)\ge \mathbb{P}\(X^{(2)}\cap Q^\mathbb{M}_s(o)=\emptyset\)$. Hence, we may choose $p_s=\mathbb{P}\(X^{(2)}\cap Q^\mathbb{M}_s(o)=\emptyset\)\mathbb{P}\big(X^{(2)}\in A^{*}_{o,\delta/16}\big(X^{(2)}_o\big)\big)^{3\lceil s\delta^{-1}\rceil+1}.$ \end{proof}
Finally, we verify condition (US). Note that if $(\varphi_1,\varphi_2)\in A^\prime_s$, then $\varphi_2\ne\emptyset$ is an immediate consequence of the definition of $A^\prime_s$. Moreover, part (a) of the condition follows from Lemma~\ref{stabCycle}. Hence, it remains to verify part (b). This will be achieved in the following two results. As a first step, we provide a precise description of the combinatorial descendant function $h_{\mathsf{c}}(\varphi_1\cup\varphi_2\cup\psi,\cdot)$ under the additional assumption that $\partial^{\mathsf{in}}_z(\varphi_{1}\cup\psi)=\partial^{\mathsf{in}}_z (\varphi_{1})$ and $\partial^{\mathsf{out}}_z(\varphi_{1}\cup\psi)=\partial^{\mathsf{out}}_z(\varphi_{1})$ for all $z\in\mathbb{Z}^d$. \begin{lemma} \label{usLem1} Let $z_0\in\mathbb{Z}^2$, $\varphi_1,\varphi_2\in \mathbb{N}^\prime$ be such that $\varphi_2\subset Q^\mathbb{M}_s(sz_0)$ and $((\varphi_1-sz_0)\cap Q_{3s}^\mathbb{M}(o),\varphi_2-sz_0)\in A^\prime_s$. Moreover, let $\psi\subset\mathbb{R}^{2,\mathbb{M}}\setminus Q^{\mathbb{M}}_{s}(sz_0)$ be a finite set such that for every $z\in\mathbb{Z}^2$ either $((\varphi_1-sz)\cap Q^\mathbb{M}_{3s}(o),(\psi-sz)\cap Q^\mathbb{M}_{s}(o))\in A^\prime_s$ or $\psi\cap Q^\mathbb{M}_{s}(sz)=\emptyset$. Furthermore, assume that $\varphi_1\cup\psi^\prime\in\mathbb{N}^\prime$ for all $\psi^\prime\subset\varphi_2\cup\psi$ and also that $\partial^{\mathsf{in}}_z (\varphi_{1}\cup \psi)=\partial^{\mathsf{in}}_z (\varphi_{1})$ and $\partial^{\mathsf{out}}_z( \varphi_{1}\cup\psi)=\partial^{\mathsf{out}}_z (\varphi_{1})$ for all $z\in\mathbb{Z}^2$. Then, for every $x\in \varphi_1\cup\varphi_2\cup\psi$, \begin{align*} h_{\mathsf{c}}(\varphi_1\cup\varphi_2\cup\psi,x)= \begin{cases} h_{\mathsf{c}}(\varphi_2,x) &\text{if }x\in\varphi_2,\\ h_{\mathsf{c}}(\varphi_1\cup\varphi_2,x)&\text{if }h_{\mathsf{g}}(\varphi_{1}\cup\psi,x)\in Q_s(sz_0),\\ h_{\mathsf{c}}(\varphi_{1}\cup\psi,x) &\text{otherwise.} \end{cases} \end{align*} \end{lemma} \begin{proof} Without loss of generality, we may assume that $z_0=o$. Define a function $h_{\mathsf{c}}^\prime: \mathbb{R}^{2,\mathbb{M}}\to \mathbb{R}^{2,\mathbb{M}}$ by the right hand side of the asserted identity in the statement of the lemma. We show that $h_{\mathsf{c}}^\prime$ satisfies the characteristic properties of the lilypond model on $\varphi_1\cup\varphi_2\cup\psi$ and therefore coincides with $h_{\mathsf{c}}(\varphi_{1}\cup\varphi_2\cup\psi,\cdot)$. The geometric descendant function corresponding to $h_{\mathsf{c}}^\prime$ is denoted by $h_{\mathsf{g}}^\prime$. First, we note that by definition of $A^\prime_s$, the hard-core property and the existence of stopping neighbors is clearly satisfied for every $x\in \varphi_2$. Next, we claim that for every $x\in\varphi_{1}\cup\psi$, \begin{align} \label{shorterHC}
|\xi - h_{\mathsf{g}}^\prime(x)|\le |\xi -h_{\mathsf{g}}(\varphi_{1}\cup\psi,x)|. \end{align}
This will imply the hard-core property. To prove~\eqref{shorterHC} it suffices to consider the case where $h_{\mathsf{g}}(\varphi_{1}\cup\psi,x)\in Q_s(o)$. Assume the contrary, i.e., that $|\xi - h_{\mathsf{g}}^\prime(x)|>|\xi -h_{\mathsf{g}}\(\varphi_{1}\cup\psi,x\) |$. Then, by properties $\partial^{\mathsf{in}}_o(\varphi_{1}\cup\psi)=\partial^{\mathsf{in}}_o(\varphi_{1})$, $\partial^{\mathsf{out}}_o(\varphi_{1}\cup\psi)=\partial^{\mathsf{out}}_o(\varphi_{1})$ and the definition of $A_s^\prime$, this would imply that $(\eta,w)=h_{\mathsf{c}}(\varphi_1\cup\psi,x)\in \mathbb{R}^{2,\mathbb{M}}\setminus Q^\mathbb{M}_{3s}(o)$, contradicting
$|\eta-h_{\mathsf{g}}(\varphi_1\cup\psi,x)|<|\xi-h_{\mathsf{g}}(\varphi_1\cup\psi,x)|.$
Next, we consider the issue of existence of stopping neighbors and put $y=h_{\mathsf{c}}\(\varphi_{1}\cup\psi,x\)$. If $h_{\mathsf{g}}\(\varphi_{1}\cup\psi,x\)\in Q_s(o)$, then this follows again from the properties $\partial^{\mathsf{in}}_o \(\varphi_{1}\cup\psi\)=\partial^{\mathsf{in}}_o \(\varphi_{1}\)$ and $\partial^{\mathsf{out}}_o \(\varphi_{1}\cup\psi\)=\partial^{\mathsf{out}}_o\( \varphi_{1}\)$. If $x\in \partial^{\mathsf{out}}_o\(\varphi_{1}\cup\psi\)$, then an elementary geometric argument shows that $h_{\mathsf{g}}(\varphi_{1}\cup\psi,y)\not\in Q_s(o)$, so that $h_{\mathsf{g}}(\varphi_{1}\cup\psi,x)\in I(\varphi_{1}\cup\psi,y)$. It remains to consider the case, where $x\in \mathbb{R}^{2,\mathbb{M}}\setminus Q^\mathbb{M}_s(o)$, but $x\not\in \partial^{\mathsf{in}}_o\(\varphi_{1}\cup\psi\)$. Then, $y$ is clearly a stopping neighbor of $x$ with respect to $h_{\mathsf{c}}^\prime$ if $h_{\mathsf{g}}(\varphi_{1}\cup\psi,y)\not\in Q_s(o)$. Furthermore, the case $y\in\(\varphi_{1}\cup\psi\)\cap Q^\mathbb{M}_s(o)\setminus \partial^{\mathsf{out}}_o \(\varphi_{1}\cup\psi\)$ is not possible, as it would imply $h_{\mathsf{g}}(\varphi_1\cup\xi,x)\in Q_s(o)$. Finally, consider the case $y\in\partial^{\mathsf{in}}_o \(\varphi_{1}\cup\psi\)$ and denote by $P_y$ the first intersection point of $I(\varphi_1\cup\psi,y)$ and $Q_s(o)$. Then, the claim follows from the observation $h_{\mathsf{g}}(\varphi_{1}\cup\xi,x)\in [\eta,P_y)$. \end{proof}
Using Lemma~\ref{usLem1}, we can now complete the verification of condition (US).
\begin{lemma} \label{usLem2} Let $z_0\in\mathbb{Z}^2$, $\varphi_1,\varphi_2\in \mathbb{N}^\prime$ be such that $\varphi_2\subset Q^\mathbb{M}_s(sz_0)$ and $((\varphi_1-sz_0)\cap Q_{3s}^\mathbb{M}(o),\varphi_2-sz_0)\in A^\prime_s$. Moreover, let $\psi\subset\mathbb{R}^{2,\mathbb{M}}\setminus Q^{\mathbb{M}}_{s}(sz_0)$ be a finite set such that for every $z\in\mathbb{Z}^2$ either $((\varphi_1-sz)\cap Q^\mathbb{M}_{3s}(o),(\psi-sz)\cap Q^\mathbb{M}_{s}(o))\in A^\prime_s$ or $\psi\cap Q^\mathbb{M}_{s}(sz)=\emptyset$. Furthermore, assume that $\varphi_1\cup\psi^\prime\in\mathbb{N}^\prime$ for all $\psi^\prime\subset\varphi_2\cup\psi$. Then, for every $z\in\mathbb{Z}^2$, \begin{enumerate} \item $\partial^{\mathsf{in}}_z(\varphi_1\cup\varphi_2\cup\psi)=\partial^{\mathsf{in}}_z(\varphi_{1})$. \item $\partial^{\mathsf{out}}_z(\varphi_1\cup\varphi_2\cup\psi)=\partial^{\mathsf{out}}_z(\varphi_{1})$. \item Let $x\in \varphi_1$. Then, either $h_{\mathsf{c}}(\varphi_{1}\cup\varphi_2\cup\psi,x)\in\varphi_2$ or \par\hspace*{-\leftmargin}\parbox{\textwidth}{$$h_{\mathsf{c}}(\varphi_{1}\cup\varphi_2\cup\psi,x)=h_{\mathsf{c}}(\varphi_1\cup \psi,x)\text{ and }x\not\in \partial^{\mathsf{in}}_{z_0}(\varphi_1\cup\psi).$$} \end{enumerate} \end{lemma} \begin{proof} The proof is obtained by using induction on the number of squares of the form $Q_s(sz)$ that admit a non-empty intersection with $\psi$. If $\psi=\emptyset$, then the conditions of Lemma~\ref{usLem1} are satisfied and we can use the description of $h_{\mathsf{c}}(\varphi_1\cup\varphi_2,\cdot)$ given there. In order to verify items $1$ and $2$ suppose that $x\in\varphi_1\cup\varphi_2$ is contained in the symmetric difference of $\partial^{\mathsf{in}}_z(\varphi_1\cup\varphi_2)$ and $\partial^{\mathsf{in}}_z(\varphi_{1})$ or in the symmetric difference of $\partial^{\mathsf{out}}_z(\varphi_1\cup\varphi_2)$ and $\partial^{\mathsf{out}}_z(\varphi_{1})$. By the representation of $h_{\mathsf{c}}$ in Lemma~\ref{usLem1}, this can only happen if $h_{\mathsf{g}}(\varphi_{1},x)\in Q_s(sz_0)$. But in this case, the definition of the event $A_s^\prime$ guarantees that also $h_{\mathsf{g}}(\varphi_{1}\cup\varphi_2,x)\in Q_s(sz_0)$, so that $x$ cannot lie in either of the symmetric differences. Concerning item $3$ if $h_{\mathsf{c}}(\varphi_{1}\cup\varphi_2,x)\not\in\varphi_2$, then we are in the third case of the representation in Lemma~\ref{usLem1}, and the assertion follows.
Next, we proceed with the induction step and decompose $\psi$ as $\psi=\psi^{(1)}\cup\psi^{(2)}$, where $\emptyset\ne\psi^{(1)}\subset Q^\mathbb{M}_s(sz^\prime)$ and $\psi^{(2)}\subset\mathbb{R}^{2,\mathbb{M}}\setminus Q^\mathbb{M}_s(sz^\prime)$ for some $z^\prime\in \mathbb{Z}^2$. Applying the induction hypothesis with $\psi^{(2)}$ instead of $\psi$ and $\psi^{(1)}$ instead of $\varphi_2$, we see that that $\partial^{\mathsf{in}}_z(\varphi_{1}\cup \psi)=\partial^{\mathsf{in}}_z(\varphi_{1})$ and $\partial^{\mathsf{out}}_z( \varphi_{1}\cup\psi)=\partial^{\mathsf{out}}_z (\varphi_{1})$ for all $z\in\mathbb{Z}^2$. Hence, we may again use the description of $h_{\mathsf{c}}(\varphi_1\cup\varphi_2,\cdot)$ provided in Lemma~\ref{usLem1}. Items $1$-$3$ can now be checked using similar arguments as in the case $\psi=\emptyset$, but for the convenience of the reader, we give some details. Concerning items $1$ and $2$ suppose that $x\in\varphi_1\cup\varphi_2\cup\psi$ is contained in the symmetric difference of $\partial^{\mathsf{in}}_z(\varphi_1\cup\varphi_2\cup\psi)$ and $\partial^{\mathsf{in}}_z(\varphi_{1})$ or in the symmetric difference of $\partial^{\mathsf{out}}_z(\varphi_1\cup\varphi_2\cup\psi)$ and $\partial^{\mathsf{out}}_z(\varphi_{1})$. This is only possible if $x\in\varphi_1\cup\psi$. Like in the case $\psi=\emptyset$, we can exclude the option $h_{\mathsf{g}}(\varphi_{1}\cup\psi,x)\in Q_s(sz_0)$. Finally, in the remaining case, we have $h_{\mathsf{c}}(\varphi_1\cup\varphi_2\cup\psi,x)=h_{\mathsf{c}}(\varphi_1\cup\psi,x)$, and we may use the induction hypothesis to conclude that $x$ cannot be contained in either of the symmetric differences. The third item can now be verified using precisely the same argumentation as in the case $\psi=\emptyset$. \end{proof} \begin{proof}[Proof of Proposition~\ref{stopCond}] It just remains to observe that part (a) of condition (US) follows from Lemma~\ref{stabCycle}, whereas part (b) follows from Lemma~\ref{usLem2}. \end{proof}
\section{Possible extensions} \label{extSec} The present section concludes the paper by discussing possible extensions of the sprinkling approach to other Poisson-based directed random geometric graphs of out-degree at most $1$. The aim of the organization of the proof for the absence of forward percolation (Theorem~{\hyperref[part2]{\ref*{mainProp}.\ref*{part1}}}) was to highlight that the arguments depend on the specific model only via three crucial properties: continuity, the shielding condition (SH) and the uniform stopping condition (US). Additionally, to simplify the presentation, we used sometimes that the geometric descendant of $x\in X$ lies on the line segment connecting $h_{\mathsf{c}}(X,x)$ and $h_{\mathsf{g}}(X,h_{\mathsf{c}}(X,x))$, but removing this condition for a specific example should only be a minor issue.
Apart from the anisotropic lilypond model that we have discussed in detail, another example to which the sprinkling technique applies is given by the directed random geometric graph on a homogeneous Poisson point process, where for some fixed $k\ge1$ for each point a descendant is chosen among the $k$ nearest neighbors according to some distribution. In fact, the verification of the crucial conditions for this example is far less involved than in the lilypond setting.
Moreover, it would be interesting to extend the sprinkling technique to further models of lilypond type. The common characteristic of the lilypond model considered in this paper and the lilypond model introduced in~\cite{lilypond6} is an asymmetry in the growth-stopping protocol. When two line segments hit only one of the two ceases its growth. In both models this asymmetry prevents one from using the classical argumentation for proving absence of percolation, which is based on the absence of suitable descending chains. Although the sprinkling approach seems to be sufficiently strong to deal also with the example considered in~\cite{lilypond6}, there are two important differences that make the verification of conditions (SH) and (US) considerably more involved. First, the latter model is isotropic, so that the shielding property now has to prevent trespassings in all directions simultaneously. Second, it features two-sided growth so that the sprinkling has to stop line segments entering a square at roughly the same point in time as before in order to ensure that the configuration of the lilypond model outside the square remains largely unchanged. D.~J.~Daley also proposed to investigate a higher-dimensional analog in $\mathbb{R}^d$, where the two-sided line segments are replaced by the intersection of balls with isotropic codimension $1$ hyperplanes. Since the sprinkling approach is a priori not restricted to the planar setting, it would be very interesting to investigate whether it is also applicable for proving the absence of percolation in these higher-dimensional lilypond models.
\subsection*{Acknowledgments} The author is grateful for the detailed reports by the anonymous referees that helped to substantially improve the quality of earlier versions of the manuscript and, in particular, for correcting an error in condition (SH). The author thanks D.~J.~Daley for proposing the percolation problem concerning the line-segment model and the generalization to percolation in directed graphs with out-degree $1$. The author also thanks S.~Ziesche and G.~Bonnet for interesting discussions and useful remarks on earlier versions of the manuscript. This work has been supported by a research grant from DFG Research Training Group 1100 at Ulm University.
\begin{comment} \section{Examples} \label{exSec} In the present section, we verify conditions (SH) and (US) for two specific examples. The first example concerns walks on $K$-nearest neighbor graphs. The absence of percolation is a new result, but we primarily included this example as a warming-up exercise, in order to develop a good intuition on the crucial condition (US). Our second example, a line-segment based lilypond model constitutes the original motivation behind the present paper and makes use of the assumptions in their entire generality (in the sense that for instance $h_{\mathsf{c}}$ and $h_{\mathsf{g}}$ do not coincide).
\subsection{Walks on $K$-nearest neighbor graphs}
In the following, we fix an integer $K\ge1$ and a probability measure $\pi$ on $\lcu 1,\ldots, K\rcu$. Moreover, we denote by $\mathbb{N}^\prime\subset \mathbb{N}_\mathbb{M}$ the family of all $\varphi\in \mathbb{N}_\mathbb{M}$ such that $\#\varphi\ge K+1$ and $\left\lvert\xi_1-\xi_2\right\rvert\ne\left\lvert \eta_1-\eta_2\right\rvert$ for all $(\xi_1,k_1),(\xi_2,k_2),(\eta_1,\ell_1),(\eta_2,\ell_2)\in \varphi$ with $\xi_1\ne \xi_2$ and $\{\xi_1,\xi_2\}\ne \{\eta_1,\eta_2\}$. Then we consider DWREs obtained as subgraphs of the $K$-nearest neighbor graph, where the descendant of a given point is chosen at random from the $K$ nearest neighbors according to the distribution $\pi$. To be more precise, for $\varphi\in\mathbb{N}^\prime$ and $x=(\xi,k)\in\varphi$ we put $h_{\mathsf{c}}(\varphi,x)=y$ and $h_{\mathsf{g}}(\varphi,x)=\eta$ where $y=(\eta,\ell)\in \varphi$ is determined by $\#\(\varphi\cap B^\mathbb{M}_{\left\lvert \xi-\eta\right\rvert}\(\xi\)\)=k+1$ and where $$B^\mathbb{M}_{\left\lvert \xi-\eta\right\rvert}(\xi)=\lcu \xi^\prime\in\mathbb{R}^d: \left\lvert \xi-\xi^\prime \right\rvert\le \left\lvert \xi-\eta \right\rvert \rcu \times \mathbb{M}$$ denotes the product of the ball of radius $\left\lvert \xi-\eta\right\rvert$ centered at $\xi$ with the mark space $\mathbb{M}$.
In the following, we assume that $X\subset\mathbb{R}^{d,\mathbb{M}}$ denotes an independently marked homogeneous Poisson point process, where the distribution of the marks is given by $\pi$. First note that the stabilization property is clearly satisfied, so that it remains to verify conditions (SH) and (US).
Let $A^{(1)}_s$ be the following event. First subdivide the cube $Q^\mathbb{M}_{3s}(o)$ into $k=(3(4d+1))^d$ congruent subcubes $Q_{s,1},\ldots, Q_{s,k}$ of side length $s/(4d+1)$ and then denote by $A^{(1)}_s$ the family of all $\varphi \in \mathbb{N}^\prime$ such that each subcube $Q_{s,i}$ contains at least $K+1$ points of $\varphi$, i.e., \begin{align} \label{kNAPrelDef} A^{(1)}_s=\bigcap_{i=1}^K\lcu\varphi\in\mathbb{N}^\prime: \#\(\varphi\cap Q_{s,i}\)\ge K+1\rcu. \end{align} We claim that using the family of events $\(A^{(1)}_s\)_{s\ge1}$ one can verify condition (SH). In order to achieve this goal, for $\varphi\in\mathbb{N}^\prime$ and $x=(\xi,k)\in\varphi$ it is convenient to introduce the \emph{environment of stabilization} $\mathsf{env}_{\mathsf{stab}}(\varphi,x)$ as the smallest ball centered at $\xi$ containing precisely $K+1$ elements, i.e., $$\mathsf{env}_{\mathsf{stab}}(\varphi,x)=B^\mathbb{M}_{R_{\mathsf{stab}}(\varphi,x)}(\xi),$$ where $$R_{\mathsf{stab}}(\varphi,x)=\inf\lcu r\ge0: \#\(\varphi\cap B^\mathbb{M}_r(\xi)\)=K+1\rcu,$$ denotes the \emph{radius of stabilization}. \begin{proposition} \label{shProp} Condition \emph{(SH)} is satisfied when using the family of events $(A^{(1)}_s)_{s\ge1}$ defined in~\eqref{kNAPrelDef}. \end{proposition} \begin{proof} Let $B,B^\prime\subset \mathbb{Z}^d$ be finite subsets such that no vertex in $B$ is contained in the infinite connected component of $\mathbb{Z}^d\setminus B^\prime$. Moreover, let $\varphi\in \mathbb{N}^\prime$ be such that $(\varphi-sz^\prime)\cap Q_{3s}^\mathbb{M}(o)\in A^{(1)}_s$ for all $z^\prime\in B^\prime$ and let $x\in\varphi$ be such that $y=h_{\mathsf{c}}\(\varphi,x\)\in Q^\mathbb{M}_s(sz)$ for some $z\in B$. We claim that $\lcu x,h_{\mathsf{c}}(\varphi,y)\rcu\subset s\(B\cup B^\prime\)\oplus Q^\mathbb{M}_s(o)$. Indeed, if $h_{\mathsf{c}}(\varphi,y)\not\in s\(B\cup B^\prime\)\oplus Q^\mathbb{M}_s(o)$, then $\mathsf{env}_{\mathsf{stab}}(\varphi,y)$ would contain strictly more than $K+1$ from $\varphi$, as it contains $h_{\mathsf{c}}(\varphi,y)$ and also encompasses one of the subcubes described by the event $A^{(1)}_s$. This yields a contradiction to the definition of $R_{\mathsf{stab}}\(\varphi,x\)$ and the assertion $x\in s\(B\cup B^\prime\)\oplus Q^\mathbb{M}_s(o)$ can be shown similarly. \end{proof}
For the uniform stopping condition we refine $A^{(1)}_s$ by imposing suitable additional conditions. To be more precise, the event $A_s$ is represented as the intersection of four events \begin{align} \label{kNADef} A_s=A^{(1)}_s\cap A^{(2)}_s\cap A^{(3)}_s\cap A^{(4)}_s, \end{align} which are defined as follows. For $A^{(2)}_s$ we require that close to the boundary of $Q_s(o)$ there should be no points of the point process, i.e., $$A^{(2)}_s=\lcu\varphi\in\mathbb{N}_\mathbb{M}: \varphi\cap \(Q^\mathbb{M}_{s}(o)\setminus Q^\mathbb{M}_{s-s^{-d}}(o)\)=\emptyset\rcu.$$ Next, for $A^{(3)}_s$ we require that for any point $(\xi,k)$ in $\varphi \cap Q^\mathbb{M}_{3s}(o)$ there are no neighbors whose distance to $\xi$ is in $(\rho-s^{-2d},\rho+s^{-2d})$, where $\rho=\mathsf{dist}(\xi,\partial Q_s(o))$, i.e., $$A^{(3)}_s=\lcu\varphi\in\mathbb{N}_\mathbb{M}: \varphi\cap \(B^\mathbb{M}_{\rho+s^{-2d}}(\xi)\setminus B^\mathbb{M}_{\rho-s^{-2d}}(\xi)\)=\emptyset, \text{ for all }x=(\xi,k)\in Q^\mathbb{M}_{3s}(o)\rcu.$$ Finally, $A^{(4)}_s$ encodes the event that no point of $Q^\mathbb{M}_s(o)$ should be simultaneously close to two neighboring cubes, i.e., $$A^{(4)}_s=\bigcap_{\substack{z,z_1,z_2\in\mathbb{Z}^d\cap Q_3(o)\\ \left\lvert z-z_1\right\rvert_1=\left\lvert z-z_2\right\rvert_1=1}}\lcu\varphi\in\mathbb{N}_\mathbb{M}: \min_{(\xi,k)\in \varphi\cap Q^\mathbb{M}_s(sz)}\left\lvert \mathsf{dist}(\xi,Q_s(sz_1))- \mathsf{dist}(\xi,Q_s(sz_2))\right\rvert \ge s^{-d}\rcu.$$
First, we observe that for $s>1$ the events $A_s$ occur with high probability. \begin{proposition} Let $(A_s)_{s\ge1}$ be the family of events defined in~\eqref{kNADef}. Then $$\lim_{s\to\infty}\mathbb{P}(X^{(1)}\cap Q^\mathbb{M}_{3s}(o)\in A_s)=1.$$ \end{proposition} \begin{proof} The Poisson concentration inequality implies that $\lim_{s\to\infty}\mathbb{P}\(X^{(1)}\cap Q^\mathbb{M}_{3s}(o)\in A^{(1)}_s\)=1$. For $A^{(2)}_s$, we note that the number of points in $X^{(1)}\cap \(Q^\mathbb{M}_{s}(o)\setminus Q^\mathbb{M}_{s-s^{-d}}(o)\)$ is Poisson-distributed with parameter of order $s^d-(s-s^{-d})^d$ and the latter expression tends to $0$ as $s\to\infty$. Concerning $A^{(3)}_s$, the expected number of elements $(\xi,k)\in X^{(1)}\cap Q^\mathbb{M}_{3s}(o)\setminus Q^\mathbb{M}_s(o)$ such that $X^{(1)}\cap \(B^\mathbb{M}_{\rho+s^{-2d}}(\xi)\setminus B^\mathbb{M}_{\rho}(\xi)\)\ne\emptyset$ is of order at most $s^{d}s^{d-1}s^{-2d}$, and again this expression tends to $0$ as $s\to\infty$. Finally, for $z_1,z_2\in\mathbb{Z}^d$ with $\left\lvert z_1\right\rvert_1=\left\lvert z_2\right\rvert_1=1$ we note that the expected number of $x=(\xi,k)\in X^{(1)}\cap Q^\mathbb{M}_s(sz)$ with $\left\lvert \mathsf{dist}(\xi,Q_s(sz_1))- \mathsf{dist}(\xi,Q_s(sz_2))\right\rvert \le s^{-d}$ is of order at most $s^{d-1}s^{-d}$, yielding the desired result for the events $A^{(4)}_s$. \end{proof} In order to verify the final condition (US), we first need to define a suitable family of events $\(A_s^\prime\)_{s\ge1}$. Broadly speaking, we say that $(\varphi_1,\varphi_2)\in A_s^\prime$ if $\varphi_1\in A_s$ and $\varphi_2$ consists of a densely filled shell contained between $Q_s(o)$ and $Q_{s-s^{-d}}(o)$. To be more precise, for $s>0$ put $a_s=((s-s^{-3d})s^{4d})/2$ and define the discrete shell $S_s=\lcu z\in \mathbb{Z}^d: \left\lvert z\right\rvert_\infty\in [a_s,a_s+1)\rcu.$
Then we put $A_s^\prime=A_s\times A_s^{\prime\prime}$, where \begin{align} \label{kNApDef} A_s^{\prime\prime}=\lcu \varphi_2\in\mathbb{N}_\mathbb{M}: \varphi_2\subset s^{-4d}\(sS_s\oplus Q^\mathbb{M}_{1}(o)\)\text{ and }\min_{z\in S_s}\#\(\varphi_2\cap Q^\mathbb{M}_{s^{-4d}}\(s^{-4d}z\)\)\ge K+1\rcu. \end{align} First, we note that clearly~\eqref{posChanceLem} is satisfied. Hence, it remains to verify condition (US). \begin{proposition} Condition \emph{(US)} is satisfied when using the family of events $(A^\prime_s)_{s\ge1}=(A_s\times A^{\prime\prime}_s)_{s\ge1}$, where $A^{\prime\prime}_s$ is defined in~\eqref{kNApDef}. \end{proposition} \begin{proof} Let $\varphi_1,\varphi_2\in \mathbb{N}^\prime$ be such that $\varphi_2\subset Q^\mathbb{M}_s(o)$ and $\(\varphi_1\cap Q_{3s}^\mathbb{M}(o),\varphi_2\)\in A^\prime_s$. Moreover, let $\psi\subset\mathbb{R}^{d,\mathbb{M}}\setminus Q^{\mathbb{M}}_{s}(o)$ be a finite set such that for every $z\in\mathbb{Z}^d$ either $\(\(\varphi_1-sz\)\cap Q^\mathbb{M}_{3s}(o),\(\psi-sz\)\cap Q^\mathbb{M}_{s}(o)\)\in A^\prime_s$ or $\psi\cap Q^\mathbb{M}_{s}(sz)=\emptyset$. Finally, assume that $\varphi_1\cup\psi,\varphi_1\cup\varphi_2\cup\psi\in\mathbb{N}^\prime$.
First note that by $A^{(2)}_s$ for every $x=(\xi,k)\in \varphi_2$, $$\mathsf{env}_{\mathsf{stab}}(\varphi_{1}\cup\varphi_2\cup\psi,x)\subset Q^\mathbb{M}_{s}(o)\setminus Q^\mathbb{M}_{s-s^{-d}}(o),$$ which yields part (a) of condition (US).
To show condition (b), let $x\in \varphi_1$ be arbitrary and choose $z\in\mathbb{Z}^d$ such that $x\in Q^\mathbb{M}_{s}(sz)$. If $\mathsf{env}_{\mathsf{stab}}(\varphi_{1}\cup\psi,x)\cap \varphi_2=\emptyset$, then clearly $h_{\mathsf{c}}(\varphi_{1}\cup\varphi_2\cup\psi,x)=h_{\mathsf{c}}(\varphi_{1}\cup\psi,x)$ and $x\not\in \partial^{\mathsf{in}}_o\(\varphi_{1}\cup\psi\).$ Next, assume that $\mathsf{env}_{\mathsf{stab}}(\varphi_{1}\cup\psi,x)\cap \varphi_2\ne\emptyset$. Then we conclude from the event $A^{(4)}_s$, that $x$ is such that $\mathsf{env}_{\mathsf{stab}}(\varphi_{1},x)\cap \psi=\emptyset$, so that $h_{\mathsf{c}}(\varphi_1,x)=h_{\mathsf{c}}(\varphi_{1}\cup\psi,x)$. Write $x=(\xi,k)$ and define $k_0\in\lcu 1,\ldots, K\rcu$ by the property that the $k_0$th neighbor of $x$ in $\varphi_{1}\cup\varphi_2\cup\xi$ is contained in $\varphi_2$, and for each $i\in\{1,\ldots,k_0-1\}$ the $i$th neighbor of $x$ in $\varphi_{1}\cup\varphi_2\cup\xi$ is contained in $\varphi_1$. Then we conclude from the event $A^{(3)}_s$ that for each $i\in\{k_0,\ldots,K\}$ the $i$th neighbor of $x$ is contained in $\varphi_2$. Hence, the proof is complete if $k\ge k_0$ and it remains to consider the case $k<k_0$. Then $$h_{\mathsf{c}}(\varphi_{1}\cup\varphi_2\cup\psi,x)=h_{\mathsf{c}}(\varphi_{1},x)=h_{\mathsf{c}}(\varphi_{1}\cup\psi,x),$$ and, we conclude from $A^{(2)}_s$ that if $x\not\in Q^\mathbb{M}_s(o)$, then also $h_{\mathsf{c}}\(\varphi_1,x\)\not\in Q^\mathbb{M}_s(o)$, as required. \end{proof} \end{comment}
\end{document}
\end{document} | arXiv |
Taylor Series beyond 2-3 terms?
im looking to understand the tangent taylor series, but im struggling to understand how to use long division to divide one series (sine) into the other (cosine). I also can't find examples of the Tangent series much beyond X^5 (wikipedia and youtube videos both stop at the second or third term), which is not enough for me to see any pattern. (x^3/3 + 2x^5/15 tells me nothing).
Wiki says Bernouli Numbers which i plan on studying next, but seriously, i could really use an example of tangent series out to 5-6 just to get a ballpark of what's going on before i start plug and pray. If someone can explain why the long division of the series spits out x^3/3 instead of x^3/3x^2, that would help too,
because I took x^3/6 divided by x^2/2 and got 2x^3/6x^2, following the logic that 4/2 divided by 3/5 = 2/0.6 or 20/6. So I multiplied my top and bottom terms for the numerator, and my two middle terms for the denominator (4x5)/(2x3) = correct.
But when i do that with terms in the taylor series I'm doing something wrong. does that first x from sine divided by that first 1 from cosine have anything to do with it?
Completely lost.
sequences-and-series taylor-expansion
TristianTristian
$\begingroup$ The power series coefficients for the tangent (expanded at the origin) are somewhat mysterious, and yes, the Bernoulli numbers are worth studying if you really want to know what they are and how to compute them. The process of dividing two power series can also produce a continued fraction, with a much more easily recognized pattern. $\endgroup$ – hardmath Jul 5 '18 at 21:03
$\begingroup$ An odd question: When you say 'understand', what particularly are you aiming to understand? If your interest is in calculation then there are much better ways of calculating tangent than its Taylor series; if your interest is in understanding the coefficients themselves, then I don't know that computing them past the first few terms will help particularly much with that understanding (but studying Bernoulli numbers will). What's your ultimate goal? $\endgroup$ – Steven Stadnicki Jul 5 '18 at 22:54
$\begingroup$ not sure this will even be seen, but note on the series long division below: in 5th row of the long division, when i derive (x^3/3 * x^6/720 = x^9/2160)'' i get (9 * 8 * x^7/2160) = x^7/30, NOT x^7/72. How did they get 72? I divided 2160 by 72 and got 30 $\endgroup$ – Tristian Jul 7 '18 at 4:31
$$\tan(x) = x+{\frac{1}{3}}{x}^{3}+{\frac{2}{15}}{x}^{5}+{\frac{17}{315}}{x}^{7}+ {\frac{62}{2835}}{x}^{9}+{\frac{1382}{155925}}{x}^{11}+{\frac{21844}{ 6081075}}{x}^{13}+\ldots$$
EDIT: Long division:
$$ \matrix{& & x &+ \frac{x^3}{3} &+ \frac{2 x^5}{15} &+ \frac{17 x^7}{315}&+ \ldots\cr& &---&---&---&---&--- \cr 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \ldots & | & x &- \frac{x^3}{6} &+ \frac{x^5}{120} &- \frac{x^7}{5040} &+ \ldots\cr & & x &- \frac{x^3}{2} &+ \frac{x^5}{24} &- \frac{x^7}{720} &+ \ldots\cr & & ---&---&---&---&---\cr & & &\frac{x^3}{3} &- \frac{x^5}{30} &+ \frac{x^7}{840} &+ \ldots\cr & & & \frac{x^3}{3} & - \frac{x^5}{6} & + \frac{x^7}{72} &+\ldots\cr & & & --- & --- & --- & ---\cr & & & & \frac{2 x^5}{15} & - \frac{4 x^7}{315} & +\ldots\cr & & & & \frac{2 x^5}{15} & - \frac{2 x^7}{30} & +\ldots\cr & & & & --- & --- & ---\cr & & & & & \frac{17 x^7}{315} & + \ldots }$$
Robert IsraelRobert Israel
$\begingroup$ wow!! how did you get those? and how do I get those? $\endgroup$ – Tristian Jul 5 '18 at 20:03
$\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review $\endgroup$ – max_zorn Jul 5 '18 at 20:37
$\begingroup$ @max_zorn: With respect, it actually does answer the Question. OP asks about the Taylor series for "the tangent" and says after looking at Wikipedia's mention of Bernoulli numbers that (s)he "plan(s) on studying next, but seriously, [I] could really use an example of tangent series out to 5-6 [terms]". $\endgroup$ – hardmath Jul 5 '18 at 20:43
$\begingroup$ @hardmath: I interpret this as if the OP wants to see details on the division of the two power series. How does dumping a solution show Tristian anything? How were the coefficients obtained? By Wolfram Alpha or some other tool or by some paper-and-pencil work? $\endgroup$ – max_zorn Jul 5 '18 at 20:50
$\begingroup$ @max_zorn: I like your idea of showing the OP how to form the ratio of two power series, because this matches what they seem to have tried. But the OP seems overwhelmed by the details of what such a computation entails and specifically asks for "a solution" (I assume so that they can know what the answer should look like). Certainly there is an opportunity to write up something further on this Question, if you are so inclined. $\endgroup$ – hardmath Jul 5 '18 at 20:57
You might find it conceptually easier to set up the identity of power series and compare the first few coefficients, and solve. This is algebraically equivalent to long division, though the order of some of the arithmetic operations is somewhat rearranged.
Write the desired Taylor series at $x = 0$ as $$\tan x \sim a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots .$$ Since $\tan$ is odd, all of the coefficients of the even terms vanish, i.e., $0 = a_0 = a_2 = a_4 = \cdots$. (This insight isn't necessary---we'd recover this fact soon anyway---but it does make the next computation easier.)
Replacing the functions in $$\cos x \tan x = \sin x$$ with their Taylor series gives $$\left(1 - \frac{1}{2!} x^2 + \frac{1}{4!} x^4 - \cdots\right)(a_1 x + a_3 x^3 + a_5 x^5 \cdots) = x - \frac{1}{3!} x^3 + \frac{1}{5!} x^5 - \cdots .$$
Now, comparing the coefficients of the terms $x, x^3, x^5, \ldots$, on both sides respectively gives $$\begin{align*} a_1 &= 1 \\ a_3 - \tfrac{1}{2} a_1 &= -\tfrac{1}{6} \\ a_5 - \tfrac{1}{2} a_3 + \tfrac{1}{24} a_1 &= \tfrac{1}{120} \\ & \,\,\vdots \end{align*}$$ and successively solving and substituting gives $$\tan x \sim x + \tfrac{1}{3} x^3 + \tfrac{2}{15} x^5 + \cdots .$$ Of course, it's straightforward (if eventually tedious) to compute as many terms as you want this way.
An efficient proof of the formula you mentioned involving the Bernoulli numbers for the general coefficient is given in this answer.
TravisTravis
My impression is that it's kind of backwards, in a numerical sense, to think about the coefficients of the $\tan$ series in terms of the Bernoulli numbers because it's simple and numerically stable to calculate the $\tan$ coefficients directly and in fact provides a reasonable method for computing the Bernoulli numbers given the formula in @RobJohn's post. Since $y(x)=\tan x$ is an odd function of $x$ analytic at $x=0$, $$y=\sum_{n=0}^{\infty}a_nx^{2n+1}$$ Then $y^{\prime}=\sec^2x=\tan^2x+1=y^2+1$ so $$\sum_{n=0}^{\infty}(2n+1)a_nx^{2n}=1+\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}a_ia_jx^{2i+2j+2}=1+\sum_{n=1}^{\infty}\left(\sum_{i=0}^{n-1}a_ia_{n-i-1}\right)x^{2n}$$ The constant term reads $$a_0=1$$ The terms in $x^{4n}$ are $$a_{2n}=\frac1{4n+1}\sum_{i=0}^{2n-1}a_ia_{2n-i-1}=\frac2{4n+1}\sum_{i=0}^{n-1}a_ia_{2n-i-1}$$ While the terms in $x^{4n+2}$ are $$a_{2n+1}=\frac1{4n+3}\sum_{i=0}^{2n}a_ia_{2n-i}=\frac1{4n+3}\left(a_n^2+2\sum_{i=0}^{n-1}a_ia_{2n-i}\right)$$ The numerical stability arises because all terms in the formulas for $a_n$ have the same sign.
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An alternative and straightforward method is: $$\begin{align}y&=\tan x \ (=0)\\ y'&=\frac{1}{\cos^2 x}=1+\tan^2x=1+y^2 \ (=1)\\ y''&=2yy'=2y(1+y^2)=2y+2y^3 \ (=0) \\ y'''&=2y'+6y^2y'=2+8y^2+6y^4 \ (=2)\\ y^{(4)}&=16yy'+24y^3y'=16y+40y^3+24y^5 \ (=0)\\ y^{(5)}&=16+120y^2y'+120y^4y'=16+136y^2+240y^4+120y^6 \ (=16)\\ y^{(6)}&=272yy'+960y^3y'+720y^5y'=272y+1232y^3+1680y^5+720y^7 \ (=0)\\ y^{(7)}&=272y'+3696y^2y'+8400y^4y'+5040y^6y'=272+3968y^2+\cdots \ (=272)\end{align}$$ Hence: $$\begin{align}\tan x&=0+\frac{1}{1!}x+\frac{0}{2!}x^2+\frac{2}{3!}x^3+\frac{0}{4!}x^4+\frac{16}{5!}x^5+\frac{0}{6!}x^6+\frac{272}{7!}x^7+\cdots\\ &=x+\frac{1}{3}x^3+\frac{2}{15}x^5+\frac{17}{315}x^7+\cdots\end{align}$$ Note: You can continue as far as you want, though the computation gets tedious. WA shows the expansion to many more terms (press on "More terms" button).
farruhotafarruhota
Write $\frac{\sin x}{x}=\frac{\tan x}{x}\cos x$ as a power series in $x^2$, with $\frac{\tan x}{x}=t_0+t_1 x^2+t_2 x^4+\cdots$. Equating coefficients of powers of $x^2$ one by one gives $1=t_0,\,-\frac{1}{6}=-\frac{t_0}{2}+t_1,\,\frac{1}{120}=\frac{t_0}{24}-\frac{t_1}{2}+t_2$ etc. Write down as many of those as you like. Thus $t_0=1,\,t_1=\frac{1}{3},\,t_2=\frac{2}{15}$ etc.
J.G.J.G.
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Chapter 15: Raising Capital
Landrie_Rich
Whether a firm obtains capital by debt or equity financing depends on _____.
a. the firm's growth prospects
b. the firm's life-cycle stage
c. the name of the firm
d. the size of the firm
a, b, d
Since most banks will not loan to startup companies with no assets, most startup ventures need _____.
a. OPM
b. EVA
c. POM
Which of the following are important considerations when choosing between venture capitalists?
a. SEC affiliation
b. Financial strength
c. Exit strategy
d. Tombstones
e. Style
b, c, e
Access to venture capital is very limited and it is estimated that only ___________ company is funded for every 100 proposals received.
Crowdfunding typically uses which of the following to raise small amounts of capital from a large number of people?
a. Investment banks
b. Commercial banks
c. Internet
How a firm raises capital depends on the size of the firm, its growth prospects, and its _____.
a. inflation rate
b. employee skill level
c. life-cycle stage
Many startup companies are now choosing to raise funds through a(n) _____ rather than the traditional venture capital methods.
a. ICO
b. ECO
c. PCO
d. CCO
The large payoff for a venture capital firm typically comes when the company is either sold to another company or goes __________.
Private equity firms provide financing for firms that otherwise would have difficulty raising capital such as _____ firms.
a. Fortune 500
b. closely held private
c. startup
d. distressed
b, c, d
Which of the following are true about the venture capital (VC) market?
a. Personal contacts are important in gaining access to the VC market.
b. Access to venture capital is very limited.
c. It is relatively inexpensive to access VC.
d. Any good idea will easily attract VC.
In order to issue a security to the public, management's first step is to ___.
a. file a registration statement
b. prepare the tombstone advertisement
c. obtain board approval
d. distribute copies of the prospectus
The first public equity issue made by a firm is called a(n) ___.
a. rights offering
b. initial public offering
c. red herring
d. initial dividend offering
The initial sale of a token on a digital currency platform is called _____.
a. a coin-commitment offering
b. an interest-only offering
c. a penny launch
d. an initial coin offering
If a cash offer is a public offer, a(n) ________ is usually involved.
a. underwriter
b. mortgage broker
c. venture capitalist
d. commercial bank
A venture capitalist will most likely experience a big payoff with a successful startup company when the start-up _____.
a. files for bankruptcy protection
b. goes public
c. pays back the venture capitalist in full
d. sells its first product or service
Which is true regarding the difference between competitive and negotiated underwriting?
a. Competitive underwriting is typically cheaper than negotiated underwriting.
b. Both types of underwriting typically come at the similar costs.
c. Negotiated underwriting is typically cheaper than competitive underwriting.
The market for venture capital refers to the _____.
a. financial market reserved only for new firms
b. market for initial public offerings
c. private financial marketplace for new or distressed firms
d. bond market for new firms
A company must file a registration statement with the SEC unless the _____.
a. issue is less than $5 million
b. issue is greater than $50 million
c. loan matures in 1 year
d. loan's maturity exceeds nine months
A firm commitment offer is one in which the underwriter ____________ the entire offer.
An initial public offering (IPO) is also referred to as a(n) ___.
a. unseasoned new issue
b. seasoned new issue
c. leveraged buyout
d. rights offering
T/F: During the aftermarket period, is it typical for members of the underwriting syndicate to sell securities for less than the offering price.
Investment firms that act as intermediaries between the company selling securities and the public are called ______________.
A Green Shoe provision is used to ___.
a. provide an incentive to banks to underwrite a risky issue
b. give an incentive to investors to purchase a risky issue
c. cover excess demand and oversubscriptions
d. dispose of excess shares of stock
With the ______ method of selecting a syndicate, the issuing firm offers its securities to the highest bidding underwriter.
a. competitive offer
b. rights offering
c. negotiated offer
d. Dutch auction underwriting
An agreement in an underwriting contract that prohibits insider shares from being sold immediately following an IPO is called a _______ period.
a. peaceful
b. respite
c. quiet
d. lockup
Financing from wealthy individuals or private investment groups is referred to as ______ capital.
a. venture
b. hedge
c. stealth
d. riskless
An investment bank that underwrites a security issue by buying the securities for less than the offering price and accepting the risk that the securities won't sell is using the ______ method.
a. best efforts
b. firm commitment
c. general cash offering
d. competitive offer
The period of time before and after an IPO when communication with the public is limited is known as the ______ period.
a. quiet
b. red herring
c. lockup
d. no call
The period after a new issue is initially sold to the public is called the ___________.
In a(n) _______________ listing, a firm arranges for its stock to be listed on an exchange without marketing and other help from an underwriter.
T/F: The most difficult part of the underwriting process for an initial public offering is determining the correct offer price.
The lockup period in an underwriting contract _____.
a. prohibits communication with the public
b. expires after one business day
c. prohibits insider shares from being sold immediately following an IPO
d. is the time of first issuance of the red herring
In the 1999-2000 time period, companies missed out on $_____ because of underpricing.
a. 47 billion
b. 67 million
c. 30 billion
d. 67 billion
The quiet period ends ________ calendar days after an IPO.
The available evidence indicates that there are pronounced cycles in the degree of IPO underpricing and the _____.
a. length of the quiet period
b. number of IPOs
c. length of the lockup period
In a direct listing, a firm arranges for its stock to be listed on an exchange _____.
a. through a crowdfunding company
b. through a Green Shoe company
c. with daily assistance from an underwriter
d. without marketing and other help from an underwriter
T/F: The partial adjustment phenomenon refers to the fact that firms only raise their IPO offer prices partially.
______ helps new shareholders earn a higher return on the shares they buy.
a. The best efforts method
b. Underpricing
c. The Green Shoe provision
d. The Dutch auction method
The ____________ curse describes how average investors in an IPO receive their full allocation of new shares because those in the know avoided the issue.
winner's
Potential reasons for stock price declines after the announcement of new equity issues include debt usage, issue costs, and _____.
a. managerial information
b. investor attraction
c. media attention
d. insurance requirements
In the 1999-2000 time period, companies missed out on $67 billion because of ___.
a. corruption
b. taxes
c. underpricing
d. overpricing
The costs associated with new issues are known as ___.
a. unbearable costs
b. flotation costs
c. sunk costs
d. opportunity costs
The available evidence indicates that there are pronounced cycles in which of the following?
(Select 2)
a. The number of IPOs
b. The degree of IPO underpricing
c. The length of the quiet period
d. The length of the lockup period
Which of the following are costs of issuing new securities (select all that apply)?
a. The gross spread
b. The Green Shoe option
c. Economies of scale
d. Underpricing
The _____ phenomenon refers to the fact that most firms may raise their IPO offer prices, but they typically do not move the price high enough.
a. cheap selling
c. tombstone
d. partial adjustment
A rights offering grants _____.
a. existing shareholders the right to buy new shares
b. existing shareholders the right to receive a fixed dividend
c. new shareholders the right to buy new shares at old prices
d. existing shareholders the right to convert their shares into debt
When average investors in an IPO receive their full allocation of new shares because the smart money avoided the issue, they fall victim to ____.
a. the Green Shoe effect
b. the efficient market theory
c. the winner's curse
d. the law of diminishing returns
To take advantage of a rights offering, a shareholder may order some or all of the rights to be sold, exercise the right, or _____.
a. keep the rights indefinitely
b. exchange the right at a local bank for cash
c. let the right expire
Possible explanations of the drop in a stock's price after an announcement of a new equity issue are that the announcement is an indication that ___.
a. the firm has too much equity
b. the firm has too much debt
c. management believes the firm is overvalued
d. there is too much information available
b, c
The number of rights needed to buy one share of stock is found by dividing the _________ shares by the ___________ shares.
old; new
The flotation costs are the costs associated with __________ issues.
A right is basically a ___.
a. call option
b. put option
Which new issue cost results from a stock initially being sold for less than its true value?
a. Hubris pricing
c. Inherent discounting
A stock typically goes ex rights __________ trading day(s) before the holder-of-record date.
Another name for a rights offering is a(n) ______________ subscription.
In a rights offering, when an existing stockholder is notified that they have been given one right for each share of stock owned, they can do which of the following?
a. Order all the rights to be sold
b. Keep the rights indefinitely
c. Do nothing and let the rights expire
d. Subscribe to the full number of entitled shares
a, c, d
The funds to be raised divided by the subscription price is the equation for _____.
a. dividend income valuation
b. prospectus determination
c. the number of new shares
A standby underwriting arrangement in conjunction with a rights offering gives the ___.
a. firm an alternative avenue of sale to ensure the success of the rights offering
b. company a way to cancel the offering
c. stockholders the right to buy unsold shares
d. firm an alternate investment banker if there is a conflict between the issuer and the agent
The main difference between an ordinary call option and a right is that _____.
a. calls closely resemble warrants
b. rights are issued by the firm
c. calls are issued by the firm
It is impossible to underprice a(n) ______.
a. initial public offering
b. seasoned equity issue
c. rights offering
Dilution refers to a loss in _________ shareholders' value.
a. new
b. existing
c. any
d. potential
Dilution of the ownership of existing shareholders can be ______ with a rights offering.
a. accomplished
b. defended
c. avoided
d. maintained
The type of underwriting that requires the underwriter to purchase unsubscribed shares is known as ____________ underwriting.
T/F: Any decrease in market value when new shares are issued is attributable to the company using the proceeds to invest in negative NPV projects.
Most debt is ___.
a. privately issued
b. issued through the SEC
c. issued with stock shares attached
d. publicly issued
The subscription price must be ____________ (below/above) the market price of the stock in a rights offer.
A shelf registration allows firms to issue new equity securities using the ______ method.
a. free throw
b. three-point
c. dribble
d. traveling
A rights offering provides the main benefit of avoiding ___________, or loss in value, of ownership for existing shareholders.
_______ value dilution is more important than ______ value dilution.
a. Book; market
b. Market; book
c. Historic; accounting
d. Accounting; market
Debt that is issued privately accounts for _____ of all debt.
a. over half
b. 25 percent
c. less than half
d. nearly 100 percent
A firm can use a shelf registration if ___.
a. it is rated investment grade
b. it has never violated the 1934 Securities Act
c. its aggregate market value is more than $150 million
d. it has not defaulted on debt in the past 3 years
e. it has an aggregate equity market value of $100 million or more
FIN 357 Chapter 15
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Recall that the mentioned figure gives data concerning **(1)** the dependent variable Coupon, which equals $1$ if a Williams Apparel customer redeems a coupon and $0$ otherwise; **(2)** the independent variable Purchases, which is last year's purchases (in hundreds of dollars) by the customer; and **(3)** the independent variable Card, which equals $1$ if the customer has a Williams Apparel credit card and $0$ otherwise. Identify and interpret the odds ratio estimates for *Purchases* and *Card*.
A pharmaceutical firm's annual report states that company sales of over-the-counter medicines increased by $48.1 \%$. b. To a competitor, what type of information would this represent?
**Use the table feature of a graphing calculator to predict each limit. Check your work by using either a graphical or an algebraic approach.** $\lim _{x \rightarrow 4} f(x)$, where $$ f(x)=\left\{\begin{array}{cc}12-\frac{3}{4} x & \text { if } x \leq 4 \\ x^2-7 & \text { if } x>4\end{array}\right. $$
Determine whether $A=B, A \subseteq B$, $B \subseteq A, A \subset B, B \subset A$, or if none of these applies. (There may be more than one answer.) \ $A=\{$ penny, nickel, dime, quarter $\}$ $B=\{$ penny, quarter $\}$
Century 21 Accounting: General Journal
11th Edition•ISBN: 9781337623124Claudia Bienias Gilbertson, Debra Gentene, Mark W Lehman
Fundamentals of Financial Management, Concise Edition
10th Edition•ISBN: 9781337902571 (1 more)Eugene F. Brigham, Joel Houston
Essentials of Investments
9th Edition•ISBN: 9780078034695 (3 more)Alan J. Marcus, Alex Kane, Zvi Bodie
4th Edition•ISBN: 9781259730948 (2 more)Don Herrmann, J. David Spiceland, Wayne Thomas | CommonCrawl |
\begin{document}
\title{Lie contact structures and chains\hbox to5pt{\hss$\cdot$\hss}} \author{Vojt\hbox to5pt{\hss$\cdot$\hss} ech \hbox to5pt{\hss$\cdot$\hss} Z\'adn\'ik} \begin{abstract}
Lie contact structures generalize the classical Lie sphere geometry of
oriented hyperspheres in the standard sphere.
They can be equivalently described as parabolic geometries corresponding
to the contact grading of orthogonal real Lie algebra.
It follows the underlying geometric structure can be interpreted in
several equivalent ways.
In particular, we show this is given by a split-quaternionic structure on
the contact distribution, which is compatible with the Levi bracket.
In this vein, we study the geometry of chains, a distinguished family of
curves appearing in any parabolic contact geometry.
Also to the system of chains there is associated a canonical parabolic
geometry of specific type.
Up to some exceptions in low dimensions,
it turns out this can be obtained by an extension of the parabolic geometry
associated to the Lie contact structure if and only if the latter is locally
flat.
In that case we can show that chains are never geodesics of an affine
connection, hence, in particular, the path geometry of chains is always
non-trivial.
Using appropriately this fact, we conclude that the path geometry of chains
allows to recover the Lie contact structure, hence, in particular,
transformations preserving chains must preserve the Lie contact structure. \end{abstract} \address{Masaryk University, Brno, Czech Republic} \email{[email protected]}
\subjclass[2000]{53C15, 53C05, 53D10} \keywords{Lie contact structures, parabolic geometries, chains}
\maketitle
\section{Introduction} \label{1} The Lie sphere geometry is the geometry of oriented hyperspheres in the standard sphere established by S.~Lie. A generalization of the corresponding geometric structure to general smooth manifold is provided by \cite{SY} and further studied by other authors. Here we briefly present the basic ideas and outline the purposes of this paper.
\subsection{Classics} Let us consider the vector space $\Bbb R^{n+4}$ with an inner product of signature $(n+2,2)$. The projectivization of the cone of non-zero null-vectors in $\Bbb R^{n+4}$ is a hyperquadric in the projective space $\Bbb R\mathbb P^{n+3}$, which is called the \textit{Lie quadric} and denoted by $Q^{n+2}$. The standard sphere $S^{n+1}$ is then realized as the intersection of $Q^{n+2}$ with a hyperplane in $\Bbb R\mathbb P^{n+3}$. There is a bijective correspondence between the Lie quadric $Q^{n+2}$ and the set of the so called kugels of $S^{n+1}$. The kugel of $S^{n+1}$ is an oriented hypersphere or a point (called a point sphere) in $S^{n+1}$. The point is that kugels corresponding to the same projective line are in oriented contact, with the point sphere as the common contact point. Hence each projective line in $Q^{n+2}$ is uniquely represented by the common contact point and the common unit normal of the family of kugels in contact. This establishes a bijective correspondence between the set of projective lines in $Q^{n+2}$ (i.e. isotropic planes in $\Bbb R^{n+4}$) and the unit tangent sphere bundle of $S^{n+1}$, denoted by $T_1(S^{n+1})$. Either of the two is then understood as the model Lie contact structure in dimension $2n+1$.
By definition, the Lie transformation group $G$ is the group of projective transformations of $\Bbb R\mathbb P^{n+3}$ preserving the Lie quadric. Hence $G$ is isomorphic to $PO(n+2,2)$, the quotient of $O(n+2,2)$ by its center which is $\hbox to5pt{\hss$\cdot$\hss}\pm\operatorname{id}\hbox to5pt{\hss$\cdot$\hss}$. The group $G$ acts transitively (and effectively) on the set of projective lines in $Q^{n+2}$, and hence on $T_1(S^{n+1})$, and preserves the canonical contact structure. As a homogeneous space, $T_1(S^{n+1})\cong G/P$ where $P\subset G$ is the stabilizer of some element. It turns out $P$ is a parabolic subgroup of $G$.
The generalization of the concepts above to general contact manifold yields the notion of the Lie contact structure, which is defined in section 3 in \cite{SY} as a reduction of the adapted frame bundle to an appropriate subgroup of the structure group. Applying the Tanaka's theory, the equivalence problem for Lie contact manifolds is solved in principle by Theorem 4.3 in \cite{SY} and rather explicitly in \cite{Miy}. Anyway, there is established an equivalence between Lie contact structures and normal Cartan geometries of type $(G,P)$. Important examples of Lie contact structures are observed on the unit tangent sphere bundles of Riemannian manifolds.
\subsection{General signature} The ideas above allow a natural generalization considering the inner product on $\Bbb R^{n+4}$ to have an arbitrary signature $(p+2,q+2)$, where $p+q=n$. Following the previous approach, we still consider the space of projective lines in the Lie quadric $Q^{n+2}$, i.e.\ the space of isotropic planes in $\Bbb R^{n+4}$, as the model. As a homogeneous space, this is isomorphic to $G/P$, where $G=PO(p+2,q+2)$ and $P\subset G$ is the stabilizer of an isotropic plane in $\Bbb R^{n+4}$. It is then natural to define the \textit{Lie contact structure of signature $(p,q)$} as the underlying structure of a parabolic geometry of type $(G,P)$ which is the meaning of the definition in \ref{2.2}, which we adopt from section 4.2.5 in \cite{CS}. Note that in low dimensions the Lie contact structure may have a specific flavour, which is briefly discussed in remark \ref{rem1}.
First of all, the Lie contact structure on $M$ involves a contact structure $H\subset TM$ so that $H\cong L^*\otimes R$, where $L$ and $R$ are auxiliary vector bundles over $M$ of rank 2 and $n$. Note that the auxiliary bundles has almost no intrinsic geometrical meaning, however, for $H\cong L^*\otimes R$, there is a distinguished subset in each $H_x$ consisting of all the elements of rank one. This is the \textit{Segre cone} which plays a role in the sequel. The maximal linear subspaces contained in the cone have dimension $n$ and it turns out they are isotropic with respect to the Levi bracket. Characterization of the Lie contact structure in these terms is provided by Proposition \ref{prop1}.
On the other hand, the tensor product structure of $H_x$ can be naturally rephrased as a \textit{split-quaternionic structure}. The compatibility with the Levi bracket is easy to express and it turns out these data also characterize the Lie contact structure entirely, Proposition \ref{prop2}. This is the convenient interpretation of the Lie contact structure we further use below. (Note that in both cases there is minor additional input involved, namely a fixed trivialization of a line bundle over $M$.)
\subsection{Chains} As for any parabolic contact structure, there is a general concept of \textit{chains} which form a distinguished family of curves generalizing the Chern--Moser chains on CR manifolds of hypersurface type. As unparametrized curves (paths), chains are uniquely determined by a tangent direction, transverse to the contact distribution, in one point. A path geometry of chains can be equivalently described as a regular normal parabolic geometry of a specific type over the open subset of $\operatorname{\mathcal P} TM$ consisting of all non-contact directions in $TM$. It turns out it is possible to relate the parabolic contact geometry and the path geometry of chains on the level of Cartan geometries directly, i.e.\ without the prolongation. This was done for Lagrangean contact structures and CR structures of hypersurface type in \cite{CZ}.
In the rest of present paper we follow the mentioned construction and the consequences for Lie contact structures. First, dealing with the homogeneous model, one explicitly describes the data allowing the direct relation between the Lie contact geometry and the path geometry of chains, section \ref{3.4}. Second, the construction in general is compatible with the normality condition if and only if the parabolic contact structure is torsion free. For Lie contact structures in dimension grater than or equal to 7, the normality and torsion freeness imply the structure is locally flat, see Theorem \ref{th2}.
Comparing to earlier studies in \cite{CZ,CZ2}, this brings a rather strong restriction. Under this assumption the tools we use may seem a bit non-proportional, however, to our knowledge there is no elementary argument covering the results below, not even in the homogeneous model. Analyzing the curvature of the induced Cartan geometry, we conclude with some applications for locally flat Lie contact structures: Chains are never geodesics of an affine connection, hence, in particular, the path geometry of chains is never flat, Theorem \ref{th3}. Therefore the harmonic curvature of the path geometry of chains (and an efficient interpretation of the structure) allows one to recover the Lie contact structure. Hence, contact diffeomorphisms preserving chains must preserve the Lie contact structure, Theorem \ref{th4}.
\section{Lie contact structures} \label{2} In this section we bring the general definition of Lie contact structures with the alternative interpretations of the underlying geometric structure. We start with a necessary background.
\subsection{Parabolic contact structures} \label{2.0} For a semisimple Lie group $G$ and a parabolic subgroup $P\subset G$, \textit{parabolic geometry} of type $(G,P)$ on a smooth manifold $M$ consists of a principal $P$-bundle ${\Cal G}\to M$ and a Cartan connection $\omega\in\Omega^1({\Cal G},\frak g)$, where $\frak g$ is the Lie algebra of $G$. The Lie algebra $\frak g$ is always equipped with the grading of the form $\frak g=\frak g_{-k}\oplus \dots \oplus \frak g_{0} \oplus \dots \oplus\frak g_{k}$ such that the Lie algebra $\frak p$ of $P$ is $\frak p = \frak g_0 \oplus \dots \oplus \frak g_k$. By $G_0$ we denote the subgroup in $P$, with the Lie algebra $\frak g_0$, consisting of all elements in $P$ whose adjoint action preserves the grading of $\frak g$. The grading of $\frak g$ induces a $P$-invariant filtration of $\frak g$, which gives rise to a filtration of the tangent bundle $TM$, due to the usual identification $TM\cong{\Cal G}\hbox to5pt{\hss$\cdot$\hss}_P\frak g/\frak p$ via the Cartan connection $\omega$. On the graded vector bundle corresponding to this filtration, there is an algebraic bracket induced by the Lie bracket of vector fields, which is called the \textit{Levi bracket}. The parabolic geometry is \textit{regular} if the Levi bracket corresponds to the bracket in $\frak g$ under the identification above.
\textit{Parabolic contact geometry} is a parabolic geometry whose underlying geometric structure consists of a contact distribution $H\subset TM$ and some additional structure on $H$. These correspond to contact gradings of simple Lie algebras as follows. The \textit{contact grading} of a simple Lie algebra is a grading $\frak g=\frak g_{-2}\oplus\frak g_{-1}\oplus\frak g_0\oplus\frak g_1\oplus\frak g_2$ such that $\frak g_{-2}$ is one dimensional and the Lie bracket $[\hbox to5pt{\hss$\cdot$\hss},\hbox to5pt{\hss$\cdot$\hss}]:\frak g_{-1}\hbox to5pt{\hss$\cdot$\hss}\frak g_{-1} \to\frak g_{-2}$ is non-degenerate. Let us consider regular parabolic geometry of type $(G,P)$ such that the Lie algebra of $G$ admits a contact grading. Then the corresponding filtration of the tangent bundle of $M$ is just a distribution $H\subset TM$, which turns out to be contact, the Levi bracket ${\Cal L}:H\hbox to5pt{\hss$\cdot$\hss} H\to TM/H$ is non-degenerate, and the reduction of $\operatorname{gr}(TM):=(TM/H)\oplus H$ to the structure group $G_0$ gives rise to the additional structure on $H$.
For general parabolic geometry $({\Cal G}\to M,\omega)$, the curvature is often described by the so called \textit{curvature function} $\kappa:{\Cal G}\to\Lambda^2(\frak g/\frak p)^*\otimes\frak g$, which is given by \begin{equation*}
\kappa(u)(X+\frak p,Y+\frak p)=d\omega(\omega^{-1}(X)(u),\omega^{-1}(Y)(u))+[X,Y]. \end{equation*} The Killing form on $\frak g$ provides an identification $(\frak g/\frak p)^*$ with $\frak p_+$, hence the curvature function is viewed as having values in $\Lambda^2\frak p_+\otimes\frak g$. The grading of $\frak g$ induces grading also to this space, which brings the notion of homogeneity. In particular, parabolic geometry is regular if and only if the curvature function has values in the part of positive homogeneity. Parabolic geometry is called \textit{torsion free} if $\kappa$ has values in $\Lambda^2\frak p_+\otimes \frak p$; note that torsion free parabolic geometry is automatically regular. Next, parabolic geometry is called \textit{normal} if $\partial^* \circ \kappa =0$, where $\partial^*:\Lambda^2\frak p_+\otimes\frak g\to\frak p_+\otimes\frak g$ is the differential in the standard complex computing the homology $H_*(\frak p_+,\frak g)$ of $\frak p_+$ with coefficients in $\frak g$. For a regular normal parabolic geometry, the \textit{harmonic curvature} $\kappa_H$ is the composition of $\kappa$ with the natural projection $\ker(\partial^*)\to H_2(\frak p_+,\frak g)$. By definition, $\kappa_H$ is a section of ${\Cal G}\hbox to5pt{\hss$\cdot$\hss}_P H_2(\frak p_+,\frak g)$ and the point is it can be interpreted in terms of the underlying structure. For all details on parabolic geometries we primarily refer to \cite{CS}.
\subsection{Contact grading of $\mathfrak{so}(p+2,q+2)$} \label{2.1} Consider the inner product on $\Bbb R^{n+4}$ given by the matrix $$ \pmat{0&0&-\Bbb I_2\hbox to5pt{\hss$\cdot$\hss} 0&\Bbb I_{p,q}&0\hbox to5pt{\hss$\cdot$\hss} -\Bbb I_2&0&0}, $$ where $\Bbb I_{p,q}=\pmat{\Bbb I_p&0\\hbox to5pt{\hss$\cdot$\hss}&-\Bbb I_q}$ and $\Bbb I_r$ is the unit matrix of rank $r$. According to this choice, the Lie algebra $\frak g=\mathfrak{so}(p+2,q+2)$ has got the following form with blocks of sizes 2, $n$, and 2: \begin{equation*}
\pmat{A&U&w\mathbb J\hbox to5pt{\hss$\cdot$\hss} X&D&\Bbb I_{p,q}U^t\hbox to5pt{\hss$\cdot$\hss} z\mathbb J&X^t\Bbb I_{p,q}&-A^t}, \end{equation*} with $\mathbb J:=\pmat{0&1\hbox to5pt{\hss$\cdot$\hss}-1&0}$, where $z,w\in\Bbb R$, $D\in\mathfrak{so}(p,q)$, and $X,A, U$ are real matrices of size $n\hbox to5pt{\hss$\cdot$\hss} 2, 2\hbox to5pt{\hss$\cdot$\hss} 2, 2\hbox to5pt{\hss$\cdot$\hss} n$, respectively. The contact grading of $\frak g$ is read along the diagonals so that $z$ parametrizes $\frak g_{-2}$, $X$ corresponds to $\frak g_{-1}$, the pair $(A,D)$ to $\frak g_0$, $U$ to $\frak g_1$, and $w$ to $\frak g_2$. In particular, $X\in\frak g_{-1}$ is understood as an element of $\Bbb R^{2*}\otimes\Bbb R^n$, the space of linear maps from $\Bbb R^2$ to $\Bbb R^n$.
If we write $X\in\frak g_{-1}$ as the matrix $(X_1,X_2)$ with columns $X_1,X_2\in\Bbb R^n$, and similarly for $Y$, then the Lie bracket $[\hbox to5pt{\hss$\cdot$\hss},\hbox to5pt{\hss$\cdot$\hss}]:\frak g_{-1}\hbox to5pt{\hss$\cdot$\hss}\frak g_{-1}\to\frak g_{-2}$ is explicitly given by \begin{equation} \label{eq1}
[X,Y]=(\span{X_1,Y_2}-\span{X_2,Y_1})\.e, \end{equation} where $\span{\hbox to5pt{\hss$\cdot$\hss},\hbox to5pt{\hss$\cdot$\hss}}$ denotes the inner product on $\Bbb R^n$ corresponding to $\Bbb I_{p,q}$ and $e$ is the generator of $\frak g_{-2}$ corresponding to $z=1$ in the description above. Indeed, this bracket is non-degenerate. Since $\frak g_{-1}$ is identified with the space of linear maps from $\Bbb R^2$ to $\Bbb R^n$, an endomorphism of $\Bbb R^m$ acts also on $\frak g_{-1}$ by the composition. In matrices, this is given by the left multiplication $X\mapsto CX$. From \eqref{eq1} it is obvious the bracket $\frak g_{-1}\hbox to5pt{\hss$\cdot$\hss}\frak g_{-1}\to\frak g_{-2}$ is invariant under the action of endomorphisms on $\Bbb R^n$ preserving the inner product, i.e.\hbox to5pt{\hss$\cdot$\hss} \begin{equation} \label{eq2}
[CX,CY]=[X,Y] \end{equation} for any $C\in O(p,q)$ and any $X,Y\in\frak g_{-1}$. Similarly, an endomorphism $A$ of $\Bbb R^2$ acts on $\frak g_{-1}$ from the right by $X\mapsto XA$. An easy direct calculation shows that this is compatible with the bracket so that \begin{equation} \label{eq3}
[XA,YA]=\det A\hbox to5pt{\hss$\cdot$\hss}[X,Y] \end{equation} for any $A\in\mathfrak{gl}(2,\Bbb R)$ and any $X,Y\in\frak g_{-1}$.
Following the introductory section, let the group to the Lie algebra $\frak g=\mathfrak{so}(p+2,q+2)$ be $G:=PO(p+2,q+2)$. The parabolic subgroup $P\subset G$ with the Lie algebra $\frak p=\frak g_0\oplus\frak g_1\oplus\frak g_2$ is represented by block upper triangular matrices in $G=PO(p+2,q+2)$ with blocks of sizes 2, $n$, and 2. Obviously, $P$ stabilizes the plane spanned by the first two vectors of the standard basis in $\Bbb R^{n+4}$ (which is indeed isotropic). The subgroup $G_0\subset P$ is represented by block diagonal matrices in $P$ with blocks of the same size. Explicitly, $G_0$ is formed by the classes of matrices of the form \begin{equation*}
\pmat{B&0&0\hbox to5pt{\hss$\cdot$\hss} 0&C&0\hbox to5pt{\hss$\cdot$\hss} 0&0&(B^{-1})^t}, \end{equation*} where $B\in GL(2,\Bbb R)$ and $C\in O(p,q)$. Each class has just two elements which differ by the sign, hence $G_0\cong (GL(2,\Bbb R)\hbox to5pt{\hss$\cdot$\hss} O(p,q))/\hbox to5pt{\hss$\cdot$\hss}\pm\operatorname{id}\hbox to5pt{\hss$\cdot$\hss}$. Let us represent an element of $G_0$ by the pair $(B,C)$ and an element of $\frak g_-$ by $(z,X)$ as above. Then a direct computation shows the adjoint action of $G_0$ on $\frak g_-$ is explicitly given by $$
\operatorname{Ad}(B,C)(z,X)=(\det B^{-1}\.z,CXB^{-1}), $$ which indeed does not depend on the representative matrix. This shows the restriction to $\frak g_{-1}\cong\Bbb R^{2*}\otimes\Bbb R^n$ comes from the product of the standard representation of $GL(2,\Bbb R)$ on $R^2$ and of $O(p,q)$ on $\Bbb R^n$. Combining with \eqref{eq2} and \eqref{eq3}, the action of $G_0$ on $\frak g_-$ is compatible with the Lie bracket $\frak g_{-1}\hbox to5pt{\hss$\cdot$\hss}\frak g_{-1}\to\frak g_{-2}$, i.e.\ the bracket is indeed $G_0$-equivariant.
\subsection{Lie contact structures} \label{2.2}
\textit{Lie contact structure of signature $(p,q)$} on a smooth manifold $M$
of dimension $2n+1$, $n=p+q$, consists of the following data:
\begin{itemize}
\item a contact distribution $H\subset TM$,
\item two auxiliary vector bundles $L\to M$ and $R\to M$ of rank 2 and $n$,
\item a bundle metric of signature $(p,q)$ on $R$, where $p+q=n$,
\item an isomorphism $H\cong L^*\otimes R$,
\end{itemize}
such that the Levi bracket is invariant under the action of $O(R_x)\cong
O(p,q)$ on $H_x$, for each $x\in M$, i.e.\
${\Cal L}(\xi,\eta)={\Cal L}(\gamma\circ\xi,\gamma\circ\eta)$ for any $\gamma\in O(R_x)$ and any
$\xi,\eta\in L_x^*\otimes R_x$.
As we announced above, the Lie contact structure should coincide with the underlying structure corresponding to the parabolic geometry of type $(G,P)$, i.e. with the parabolic contact structure corresponding to the contact grading of $\frak g=\mathfrak{so}(p+2,q+2)$ . This is really the case and the equivalence is formulated as Proposition 4.2.5 in \cite{CS}: \begin{thm*} \label{th0}
Let $G=PO(p+2,q+2)$ and $P\subset G$ be the stabilizer of an isotropic plane.
Then the category of regular normal parabolic geometries of type $(G,P)$
is equivalent to the category of Lie contact structures of signature $(p,q)$. \end{thm*}
\subsection*{Remarks} \label{rem1} (1) Note that the section 4.2.5 in \cite{CS} develops according to the choice $G=O(p+2,q+2)$. However, different choices of the Lie group to the given Lie algebra coincide generally up to a (usually finite) cover. In particular, the freedom in the choice does not affect the local description of the underlying structure.
(2) Although we define the Lie contact structure for general $n=p+q$, there are some specific features in low dimensions. For instance, the Lie contact structure of signature $(2,0)$ (on 5-dimensional manifold) is basically equivalent to a non-degenerate CR structure with indefinite Levi form, which is discussed e.g.\ in section 5 in \cite{SY} in some detail. In our terms, the equivalence is provided by the Lie algebra isomorphism $\mathfrak{so}(4,2)\cong\mathfrak{su}(2,2)$ and the uniqueness of the contact grading. Similarly, for $\mathfrak{so}(3,3)\cong\mathfrak{sl}(4,\Bbb R)$, the Lie contact structure of signature $(1,1)$ is equivalent to the Lagrangean contact structure (on a 5-manifold). The interpretation of $\mathfrak{so}(3,2)\cong\mathfrak{sp}(4)$ is a bit different, since the Lie contact structure of signature $(1,0)$ is rather trivial and usually excluded from the considerations. However, the structure corresponding to the contact grading of $\mathfrak{sp}(4)$ is the contact projective structure (on a 3-manifold). If dimension of the base manifold is grater then or equal to 7, the Lie contact structure starts to work with no specific issue, up to the only exception in dimension 9, which is due to $\mathfrak{so}(6,2)\cong\mathfrak{so}^*(8)$. An interested reader may investigate the equivalent structure behind that isomorphism.
\subsection{Segre cone} \label{2.3} For $H_x\cong L^*_x\otimes R_x$, there is a distinguished subset $\mathcal C_x\subset H_x$, the so called \textit{Segre cone}, consisting of all the linear maps $L_x\to R_x$ of rank one. On the level of the Lie algebra $\frak g=\mathfrak{so}(p+2,q+2)$, the cone $\mathcal C_x\subset H_x$ corresponds to the $G_0$-invariant subset of $\frak g_{-1}\cong\Bbb R^{2*}\otimes\Bbb R^n$ consisting of the elements of the from $f\otimes u$ for some $f\in R^{2*}$ and $u\in\Bbb R^n$. It follows that $f\otimes\Bbb R^n$ is a maximal linear subspace in the cone (consisting of the linear maps $\Bbb R^2\to\Bbb R^n$ with the common kernel $\ker f$). For arbitrary elements $X=f\otimes u_1$ and $Y=f\otimes u_2$ from $f\otimes\Bbb R^n$, the substitution into \eqref{eq1} yields that $[X,Y]=0$, i.e.\ the subspace $f\otimes\Bbb R^n\subset\frak g_{-1}$ is isotropic with respect to the Lie bracket $\frak g_{-1}\hbox to5pt{\hss$\cdot$\hss}\frak g_{-1}\to\frak g_{-2}$. Hence for general rank-one elements $f_1\otimes u_1,f_2\otimes u_2$ from $\Bbb R^{2*}\otimes\Bbb R^n=\frak g_{-1}$, it follows that $$
[f_1\otimes u_1,f_2\otimes u_2]=|f_1,f_2|\span{u_1,u_2}\.e, $$
where $|\hbox to5pt{\hss$\cdot$\hss},\hbox to5pt{\hss$\cdot$\hss}|$ denotes the standard exterior product on $\Bbb R^{2*}$, $\span{\hbox to5pt{\hss$\cdot$\hss},\hbox to5pt{\hss$\cdot$\hss}}$ is the standard inner product on $\Bbb R^n$ of signature $(p,q)$, and $e$ is the generator of $\frak g_{-2}$ as in \ref{2.1}. This shows the Lie bracket is actually of the form
$\Lambda^2\Bbb R^2\otimes S^2\Bbb R^{n*}\otimes\frak g_{-2}$ and it determines the inner product on $\Bbb R^n$ provided we fix an identification $\Lambda^2\Bbb R^2\otimes\frak g_{-2}\cong\Bbb R$. This is provided by the $G_0$-invariant mapping $f_1\wedge f_2\otimes ze\mapsto z|f_1,f_2|$, which also yields a trivialization of the line bundle $\Lambda^2L\otimes TM/H$ over any Lie contact manifold $M$. \begin{prop*} \label{prop1}
A Lie contact structure on a smooth manifold $M$ of dimension $2n+1$
is equivalent to the following data:
\begin{itemize}
\item a contact distribution $H\subset TM$,
\item two auxiliary vector bundles $L\to M$ and $R\to M$ of rank 2 and $n$,
\item an isomorphism $H\cong L^*\otimes R$,
\item an isomorphism $\Lambda^2L^*\cong TM/H$,
\end{itemize}
such that for $\varphi\in L_x^*$ the subspace $\varphi\otimes R_x\subset H_x$ is
isotropic with respect to the Levi bracket, i.e.\
${\Cal L}(\varphi\otimes\upsilon_1,\varphi\otimes\upsilon_2)=0$ for any
$\upsilon_1,\upsilon_2\in R_x$. \end{prop*} \begin{proof}
According to the discussion above, we only need to construct a Lie contact
structure on $M$ from the later data.
In general, the Levi bracket is a section of the bundle $\Lambda^2 H^*\otimes TM/H$,
which decomposes according to the isomorphism $H\cong L^*\otimes R$ as
$(\Lambda^2L\otimes S^2R^*\otimes TM/H)\oplus(S^2L\otimes\Lambda^2R^*\otimes TM/H)$.
Since we assume $\varphi\otimes R$ is isotropic with respect to ${\Cal L}$, for any $\varphi$,
the Levi bracket ${\Cal L}$ factorizes through the first summand only.
Since we assume a fixed identification of line bundles $\Lambda^2L^*$ and
$TM/H$, i.e. a trivialization of $\Lambda^2L\otimes TM/H$, the Levi
bracket determines a (non-degenerate) bundle metric on $R$, which accomplishes
the Lie contact structure on $M$.
In particular, if $O(R_x)$ is the group of orthogonal transformations of
$R_x$ according to this inner product, then the Levi bracket is by construction
invariant under the action of $O(R_x)$, over each $x\in M$. \end{proof}
\subsection{Split quaternions} \label{2.4} The algebra of \textit{split quaternions} $\Bbb H_s$ is the four-dimensional real algebra admitting a basis $(1,i,j,k)$ such that $i^2=j^2=1$ and $k=ij=-ji$ (consequently $k^2=-1$, $kj=-jk=i$, etc.). As an associative algebra, $\Bbb H_s$ is isomorphic to the space of $2\x2$ real matrices so that the norm on $\Bbb H_s$ corresponds to the determinant. This correspondence is explicitly given by \begin{equation} \label{eq5}
1\mapsto\pmat{1&0\\hbox to5pt{\hss$\cdot$\hss}&1},\ i\mapsto\pmat{1&0\\hbox to5pt{\hss$\cdot$\hss}&-1},\hbox to5pt{\hss$\cdot$\hss}
j\mapsto\pmat{0&1\\hbox to5pt{\hss$\cdot$\hss}&0},\ k\mapsto\pmat{0&1\hbox to5pt{\hss$\cdot$\hss}-1&0}. \end{equation} In particular, the norm squared of $i,j$, and $k$, equals $-1,-1$, and 1, respectively. For general imaginary quaternion $q=ai+bj+ck$, the norm squared is
$|q|^2=-a^2-b^2+c^2$, which can also be given by $q^2=-|q|^2$.
The \textit{split-quaternionic structure} on a vector bundle $H$ over $M$ is defined as a 3-dimensional subbundle $\mathcal Q\subset\operatorname{End} H$ admiting a basis $(I,J,K)$ such that $I^2=J^2=-\operatorname{id}$ and $K=I\circ J=-J\circ I$. Note that, in contrast to the true quaternionic structures, split-quaternionic structure may exist on vector budles of any even rank.
\begin{lem*} \label{lem0}
Let $H\to M$ be a vector bundle of rank $2n$.
(1) A split-quaternionic structure $\mathcal Q$ on $H$ is equivalent to an isomorphism
$H\cong L^*\otimes R$, where $L\to M$ and $R\to M$ are vector bundles of
rank 2 and $n$, respectively.
(2) In particular, a non-zero element $\xi$ in $H_x\cong L^*_x\otimes R_x$ has rank one
if and only if there is $A\in\mathcal Q_x$ such that $A(\xi)=\xi$ (consequently,
$A$ is a product structure).
(3) Moreover, the maximal linear subspaces contained in the cone of
rank-one elements in $H_x$ are the $+1$-eigenspaces of the product
structures in $\mathcal Q_x$. \end{lem*}
\begin{proof}
(1) Let $H\cong L^*\otimes R$.
Since $L_x\cong\Bbb R^2$ and $R_x\cong\Bbb R^n$, for each $x\in M$, the natural
action of $\Bbb H_s$ on $\Bbb R^2$ extends, by the composition, to the action on
$\Bbb R^{2*}\otimes\Bbb R^n$.
Denoting by $I,J$, and $K$ the endomorphisms of $H_x\cong\Bbb R^{2*}\otimes\Bbb R^n$
corresponding to the elements $i,j$, and $k$ from \eqref{eq5}, we obtain a
split-quaternionic structure $\mathcal Q_x:=\span{I,J,K}$.
Conversely, let a split-quaternionic structure $\mathcal Q=\span{I,J,K}$ be given.
For each $x\in M$, the subspace $H_x\subset T_xM$ decomposes to the eigenspaces
$H_x^+\oplus H_x^-$ with respect to the product structure $I$.
Let us define the auxiliary vector bundles over $M$ to be $R:=H^+\subset H$ and
$L:=\span{I,J}^*$, the dual of $\span{I,J}\subset\mathcal Q$.
(Obviously, $R$ and $L$ have got rank $n$ and 2, respectively.)
We claim that $L^*\otimes R\cong H$ under the mapping $\varphi\otimes X\mapsto
\varphi(X)$:
By definition, $\varphi\in L^*$ is a linear combination of the endomorphisms $I,J$
and the restriction of $I$ to $R=H^+$ is the identity.
Since $IJ=-JI$, it follows that $J$ swaps the subspaces $H^+$ and $H^-$.
Hence any $\xi\in H_x$ can be uniquely written as $\xi=\upsilon_1+J\upsilon_2$, for
$\upsilon_1,\upsilon_2\in R_x$, which is the image of
$I\otimes\upsilon_1+J\otimes\upsilon_2$ under the mapping above.
Because it is a linear map between the spaces of the same dimension, the
claim follows.
(2)
As above, let us interpret $\xi\in H_x$ as a
linear map $\Bbb R^2\to\Bbb R^n$ and $A\in\mathcal Q_x$ as an endomorphisms
$\Bbb R^2\to\Bbb R^2$.
Hence $A(\xi)$ is interpreted as the composition $\xi\circ A$.
Easily, $\xi\circ A=\xi$ for some $A$ if and only if $A$ is the
identity or $\ker\xi$ is non-trivial and invariant under $A$ and there
further is a non-zero vector fixed by $A$.
In other words, the later condition means that $A$ has two real eigenvalues
1 and $\lambda$ so that the eigenspace corresponding to $\lambda$ coincides with
$\ker\xi$.
Considering an endomorphisms $A=aI+bJ+cK$, the eigenvalues of $A$ turn
out to be the solutions of $\lambda^2+\det A=0$.
Hence 1 is an eigenvalue of $A$ if and only if $\det A=-1$.
Consequently, the second eigenvalue is $-1$ and $A$ is a skew reflection.
In terms of split quaternions, $|A|^2=-1$ and $A$ corresponds to a product
structure on $H_x$.
Note that for any one-dimensional subspace $\ell$ in $\Bbb R^2$ there is a
skew reflection $A=aI+bJ+cK$ whose eigenspace to $-1$ is $\ell$.
Altogether, since $A$ is never the identity,
the equivalence follows.
(3)
For the last statement, note that the maximal linear subspaces in the Segre
cone $\mathcal C_x$ have dimension $n$ and all of them are parametrized by
one-dimensional subspaces in $\Bbb R^2$ as follows.
For any line $\ell\subset\Bbb R^2$, let us consider the set $W_\ell$ of all linear maps
$\xi:\Bbb R^2\to \Bbb R^n$ so that $\ker\xi=\ell$.
Indeed, $W_\ell\subset H_x$ is a linear subspace of dimension $n$ whose
each element has got rank one.
As before, there is a skew reflection $A$ in $\Bbb R^2$ so that $A|_\ell=-\operatorname{id}$.
Hence $\xi\circ A=\xi$ for any $\xi\in W_\ell$, i.e. $W_\ell$ is the eigenspace
to 1 of the corresponding product structure on $H_x$.
The converse is evident as well. \end{proof}
For any $A\in\mathcal Q_x$, $A\circ A$ is a multiple of the identity, hence there is a natural inner product in each fibre $\mathcal Q_x$; the corresponding norm is
defined by $A\circ A=-|A|^2\operatorname{id}$, the signature is $(1,2)$. Let us denote by $\mathcal S$ the line bundle over $M$, whose fiber consists of
all real multiples of this inner product on $\mathcal Q_x$. For a Lie contact manifold $M$, the corresponding split-quaternionic structure $\mathcal Q$ on $H\subset TM$ has to be compatible with the Levi bracket, which is expressed by \eqref{eq3} on the level of Lie algebra. Since $\det A$ corresponds to $|A|^2$ under the above identifications, we conclude with the following description of the Lie contact structure. \begin{prop*} \label{prop2}
A Lie contact structure on a smooth manifold $M$ of dimension $2n+1$
is equivalent to the following data:
\begin{itemize}
\item a contact distribution $H\subset TM$,
\item a split-quaternionic structure $\mathcal Q=\span{I,J,K}\subset\operatorname{End} H$,
\item an identification $\mathcal S\cong TM/H$,
\end{itemize}
such that the Levi bracket is compatible as
${\Cal L}(A(\xi),A(\eta))=|A|^2\hbox to5pt{\hss$\cdot$\hss}{\Cal L}(\xi,\eta)$, for any $\xi,\eta\in H_x$ and
$A\in\mathcal Q_x$ in each $x\in M$. \end{prop*} \begin{proof}
According to the previous discussion, we just construct a Lie contact
structure from the later data.
By the lemma above, the split-quaternionic structure $\mathcal
Q$ provides an identification $H\cong L^*\otimes R$ where $L$ and $R$ are
appropriate vector bundles.
(Namely, $R_x\subset H_x$ is defined as the $+1$-eigenspace of the product
structure $I$ and $L_x$ as the dual of $\span{I,J}$.)
Now, for $\varphi\in L^*_x$, let us consider the subset
$W:=\varphi\otimes R_x\subset H_x$.
This is a linear $n$-dimensional subspace contained in the Segre cone of
$H_x$ and there is a product structure
$A\in\mathcal Q_x$ such that $A|_{W}=\operatorname{id}$.
Hence for any $\xi,\eta\in W$, the compatibility
${\Cal L}(A(\xi),A(\eta))=|A|^2\hbox to5pt{\hss$\cdot$\hss}{\Cal L}(\xi,\eta)$ yields ${\Cal L}(\xi,\eta)=-{\Cal L}(\xi,\eta)$
and so ${\Cal L}(\xi,\eta)=0$.
In other words, $\varphi\otimes R_x$ is an isotropic subspace in $H_x$ for any $\varphi$.
According to Proposition \ref{prop1}, in order to determine a Lie contact
structure on $M$ we further need to identify the line bundles
$\Lambda^2L^*$ and $TM/H$ over $M$.
However, this is provided by the identification $\mathcal S\cong TM/H$ assumed above
and the identification $\mathcal S\cong\Lambda^2L^*$ which follows:
By definition, $L^*_x$ is identified with $\span{I,J}\subset\mathcal Q_x$.
Hence the correspondence between the volume forms on $\span{I,J}$ and the
multiples of the natural inner product on $\mathcal Q_x$ is obvious. \end{proof}
\section{Chains} \label{3}
\subsection{Path geometry of chains} \label{3.1} For general parabolic contact geometry $({\Cal G}\to M,\omega)$ of type $(G,P)$, the \textit{chains} are defined as projections of flow lines of constant vector fields on ${\Cal G}$ corresponding to non-zero elements of $\frak g_{-2}$, where $\frak g_{-2}$ is the 1-dimensional subspace from the contact grading of $\frak g$ as above. In the homogeneous model $G/P$, the chains are the curves of type $t\mapsto g\operatorname{exp}(tX)P$, for $g\in G$ and $X\in\frak g_{-2}$. As unparametrized curves, chains are uniquely determined by a tangent direction in a point, \cite[section 4]{CSZ}. By definition, chains are defined only for directions which are transverse to the contact distribution $H\subset TM$. In classical terms, the family of chains defines a path geometry on $M$ restricted, however, only to the directions transverse to $H$.
A \emph{path geometry} on $M$ is a smooth family of unparametrized curves (paths) on $M$ with the property that for any point $x\in M$ and any direction $\ell\subset T_xM$ there is unique path $C$ from the family such that $T_xC=\ell$. It turns out that path geometry on $M$ can be equivalently described as a parabolic geometry over $\operatorname{\mathcal P} TM$ of type $(\tilde G,\tilde P)$, where $\tilde G=PGL(m+1,\Bbb R)$, $m=\dim M$, and $\tilde P$ is the parabolic subgroup as follows. Let us consider the grading of $\tilde\hbox to5pt{\hss$\cdot$\hss}=\mathfrak{sl}(m+1,\Bbb R)$ which is schematically described by the block decomposition with blocks of sizes 1, 1, and $m-1$: $$
\pmat{\tilde\hbox to5pt{\hss$\cdot$\hss}_0&\tilde\hbox to5pt{\hss$\cdot$\hss}^E_1&\tilde\hbox to5pt{\hss$\cdot$\hss}_2\hbox to5pt{\hss$\cdot$\hss}\tilde\hbox to5pt{\hss$\cdot$\hss}^E_{-1}&\tilde\hbox to5pt{\hss$\cdot$\hss}_0&\tilde\hbox to5pt{\hss$\cdot$\hss}^V_1\hbox to5pt{\hss$\cdot$\hss}\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}&\tilde\hbox to5pt{\hss$\cdot$\hss}^V_{-1}&\tilde\hbox to5pt{\hss$\cdot$\hss}_0}. $$ Then $\tilde\hbox to5pt{\hss$\cdot$\hss}=\tilde\hbox to5pt{\hss$\cdot$\hss}_0\oplus\tilde\hbox to5pt{\hss$\cdot$\hss}_1\oplus\tilde\hbox to5pt{\hss$\cdot$\hss}_2$ is a parabolic subalgebra of $\tilde\hbox to5pt{\hss$\cdot$\hss}$ and $\tilde P\subset\tilde G$ is the subgroup represented by block upper triangular matrices so that its Lie algebra is $\tilde\hbox to5pt{\hss$\cdot$\hss}$. As usual, the parabolic geometry associated to the path geometry on $M$ is uniquely determined (up to isomorphism) provided we consider it is regular and normal.
Denote by $\tilde M=\operatorname{\mathcal P}_0TM$ the open subset of $\operatorname{\mathcal P} TM$ consisting of all lines in $TM$ which are transverse to the contact distribution $H\subset TM$. Now the family of chains on $M$ gives rise to a parabolic geometry $(\tilde{\Cal G}\to\tilde M,\tilde\om)$ of type $(\tilde G,\tilde P)$ which we call the \textit{path geometry of chains} and which we are going to describe in a direct way extending somehow the Cartan geometry $({\Cal G}\to M,\omega)$. For this reason it is crucial to observe that $\tilde M$ is naturally isomorphic to the quotient bundle ${\Cal G}/Q$, where $Q\subset P$ is the stabilizer of the 1-dimensional subspace $\frak g_{-2}\subset\frak g_-$ under the action induced from the adjoint representation. Evidently, the Lie algebra of $Q$ is $\frak q=\frak g_0\oplus\frak g_2$. Hence the couple $({\Cal G}\to\tilde M,\omega)$ forms a Cartan (but not parabolic) geometry of type $(G,Q)$.
\subsection{Induced Cartan connection} \label{3.2} Starting with the homogeneous model $G/P$ of the parabolic contact geometry, let $(\tilde{\Cal G}\to\widetilde{G/P},\tilde\om)$ be the regular normal parabolic geometry of type $(\tilde G,\tilde P)$ corresponding to the path geometry of chains. Since $\widetilde{G/P}=\operatorname{\mathcal P}_0T(G/P)$ is isomorphic to the homogeneous space $G/Q$, the Cartan geometry $(\tilde{\Cal G}\to G/Q,\tilde\om)$ is homogeneous under the action of the group $G$. Hence it is known there is a pair $(i,\alpha)$ of maps, consisting of a Lie group homomorphism $i:Q\to\tilde P$ and a linear map $\alpha:\frak g\to\tilde\hbox to5pt{\hss$\cdot$\hss}$, so that $\tilde{\Cal G}\cong G\hbox to5pt{\hss$\cdot$\hss}_Q\tilde P$ and $j^*\tilde\om=\alpha\circ\mu$, where $j$ is the canonical inclusion ${\Cal G}\hookrightarrow{\Cal G}\hbox to5pt{\hss$\cdot$\hss}_Q\tilde P$ and $\mu$ is the Maurer--Cartan form on $G$. In particular, the pair $(i,\alpha)$ has to satisfy the following conditions: \begin{enumerate} \item $\alpha\circ\operatorname{Ad}(h)=\operatorname{Ad}(i(h))\circ\alpha$ for all $h\in Q$,
\item the restriction $\alpha|_\frak q$ coincides with $i':\frak q\to\tilde\hbox to5pt{\hss$\cdot$\hss}$, the derivative of $i$, \item the map $\underline{\alpha}:\frak g/\frak q\to\tilde\hbox to5pt{\hss$\cdot$\hss}/\tilde\hbox to5pt{\hss$\cdot$\hss}$ induced by
$\alpha$ is a linear isomorphism. \end{enumerate}
On the other hand, any pair of maps $(i,\alpha)$ which are compatible in the above sense gives rise to a functor from Cartan geometries of type $(G,Q)$ to Cartan geometries of type $(\tilde G,\tilde P)$. There is an easy control over the natural equivalence of functors associated to different pairs, see section 3 in \cite{CZ}.
If $\kappa$ is the curvature function of the Cartan geometry $({\Cal G}\to M,\omega)$, then the curvature function $\tilde\kappa$ of the Cartan geometry induced by $(i,\alpha)$ is completely determined by $\kappa$ and the contribution of $\alpha$. More specifically, this is given by \begin{equation} \label{eq6}
\tilde\kappa\circ j=\alpha\circ\kappa\circ\underline{\alpha}^{-1} +\Psi_\alpha, \end{equation} where $\Psi_\alpha:\Lambda^2(\tilde\hbox to5pt{\hss$\cdot$\hss}/\tilde\hbox to5pt{\hss$\cdot$\hss})\to\tilde\hbox to5pt{\hss$\cdot$\hss}$ is defined as follows. Consider a bilinear map $\frak g\hbox to5pt{\hss$\cdot$\hss}\frak g\to\tilde\hbox to5pt{\hss$\cdot$\hss}$ defined by \begin{equation*}
(X,Y)\to [\alpha(X),\alpha(Y)]-\alpha([X,Y]). \end{equation*} The map is obviously skew symmetric and, due to the compatibility conditions on $(i,\alpha)$, it factorizes to a well-defined $Q$-equivalent map $\frak g/\frak q\hbox to5pt{\hss$\cdot$\hss}\frak g/\frak q\to\tilde\hbox to5pt{\hss$\cdot$\hss}$. The map $\Psi_\alpha$ is then obtained according to the isomorphism $\underline{\alpha}^{-1}:\tilde\hbox to5pt{\hss$\cdot$\hss}/\tilde\hbox to5pt{\hss$\cdot$\hss}\to\frak g/\frak q$. Note that $\Psi_\alpha$ vanishes if and only if $\alpha$ is a homomorphism of Lie algebras.
\subsection{The pair} \label{3.4} Here we describe the pair $(i,\alpha)$ for Lie contact structures explicitly. Considering the dimension of the base manifold $M$ is $m=2n+1$, we have to consider $G=PO(p+2,q+2)$, $p+q=n$, and $\tilde G=PGL(2n+2,\Bbb R)$. According to the definition in \ref{3.1}, the subgroup $Q\subset P$ is represented by the block matrices of the form \begin{equation*}
\pmat{B&0&wC\mathbb J\hbox to5pt{\hss$\cdot$\hss} 0&C&0\hbox to5pt{\hss$\cdot$\hss} 0&0&(B^{-1})^t}, \end{equation*} where $\mathbb J=\pmat{0&1\hbox to5pt{\hss$\cdot$\hss}-1&0}$, $w\in\Bbb R$, $B\in GL(2,\Bbb R)$, and $C\in O(p,q)$. If we denote $\beta:=\det B$ and substitute $B=\pmat{p&r\\hbox to5pt{\hss$\cdot$\hss}&q}$, then the two maps $i:Q\to\tilde P$ and $\alpha:\frak g\to\tilde\hbox to5pt{\hss$\cdot$\hss}$ are defined explicitly by \begin{eqnarray*} i\pmat{ p&r&0&-rw&pw\hbox to5pt{\hss$\cdot$\hss} s&q&0&-qw&sw\hbox to5pt{\hss$\cdot$\hss} 0&0&C&0&0\hbox to5pt{\hss$\cdot$\hss} 0&0&0&\frac q\beta&-\frac s\beta\hbox to5pt{\hss$\cdot$\hss} 0&0&0&-\frac r\beta&\frac p\beta} &:=& \pmat{
\sqrt{|\beta|}&-w\sqrt{|\beta|}&0&0\hbox to5pt{\hss$\cdot$\hss}
0&\frac1{\sqrt{|\beta|}}&0&0\hbox to5pt{\hss$\cdot$\hss}
0&0&\frac q{\sqrt{|\beta|}} C&-\frac s{\sqrt{|\beta|}} C\hbox to5pt{\hss$\cdot$\hss}
0&0&-\frac r{\sqrt{|\beta|}} C&\frac p{\sqrt{|\beta|}} C}, \end{eqnarray*} \begin{eqnarray*} \alpha\pmat{ a&b&U_1&0&w\hbox to5pt{\hss$\cdot$\hss} c&d&U_2&-w&0\hbox to5pt{\hss$\cdot$\hss} X_1&X_2&D&\Bbb I_{p,q} U_1^t&\Bbb I_{p,q} U_2^t\hbox to5pt{\hss$\cdot$\hss} 0&z&X_1^t\Bbb I_{p,q}&-a&-c\hbox to5pt{\hss$\cdot$\hss} -z&0&X_2^t\Bbb I_{p,q}&-b&-d} &:=& \pmat{ \frac{a+d}2&-w&\frac12U&\frac12V\hbox to5pt{\hss$\cdot$\hss} z&-\frac{a+d}2&-\frac12Y^t\Bbb I_{p,q}&\frac12X^t\Bbb I_{p,q}\hbox to5pt{\hss$\cdot$\hss} X_1&-\Bbb I_{p,q} U_2^t&D+\frac{d-a}2\Bbb I_n&-c\Bbb I_n\hbox to5pt{\hss$\cdot$\hss} X_2&\Bbb I_{p,q} U_1^t&-b\Bbb I_n&D+\frac{a-d}2\Bbb I_n}. \end{eqnarray*}
\begin{prop*} \label{lem1}
The map $i:Q\to\tilde P$ is an injective Lie group homomorphism,
$\alpha:\frak g\to\tilde\hbox to5pt{\hss$\cdot$\hss}$ is linear, and the pair $(i,\alpha)$ satisfies the conditions
(1)--(3) from \ref{3.2}.
\end{prop*}
Hence the pair $(i,\alpha)$ gives rise to an extension functor from Cartan
geometries of type $(G,Q)$ to Cartan geometries of type $(\tilde G,\tilde P)$. \begin{proof}
The map $i$
is obviously well defined, i.e.\ the image of an element of $Q\subset PO(p+2,q+2)$
does not depend on the representative matrix.
Further, $i$ is smooth and injective and a direct computation shows this is a
homomorphism of Lie groups.
The map $\alpha$ is linear and the compatibility of the pair
$(i,\alpha)$ follows as follows:
(1) involves a little tedious but very straightforward checking,
(2) is an easy exercise,
(3) follows from $\frak g/\frak q\cong\frak g_-\oplus\frak g_1$ and $\tilde\hbox to5pt{\hss$\cdot$\hss}_-\cong\tilde\hbox to5pt{\hss$\cdot$\hss}/\tilde\hbox to5pt{\hss$\cdot$\hss}$,
restricting $\alpha$ to $\frak g_-\oplus\frak g_1$. \end{proof}
\subsection{The properties} \label{3.5} Here we analyze the map $\Psi_\alpha:\Lambda^2(\tilde\hbox to5pt{\hss$\cdot$\hss}/\tilde\hbox to5pt{\hss$\cdot$\hss})\to\tilde\hbox to5pt{\hss$\cdot$\hss}$ from \ref{3.2}. For this reason we need a bit of notation:
As a linear space, $\tilde\hbox to5pt{\hss$\cdot$\hss}/\tilde\hbox to5pt{\hss$\cdot$\hss}$ can be identified with $\tilde\hbox to5pt{\hss$\cdot$\hss}_-=\tilde\hbox to5pt{\hss$\cdot$\hss}_{-1}^E\oplus\tilde\hbox to5pt{\hss$\cdot$\hss}_{-1}^V\oplus\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}$. Using brackets in $\tilde\hbox to5pt{\hss$\cdot$\hss}$, we may identify $\tilde\hbox to5pt{\hss$\cdot$\hss}_{-1}^V$ with $\tilde\hbox to5pt{\hss$\cdot$\hss}_1^E\otimes\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}$ if necessary. We will further view $\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}$ as $\Bbb R^{2n}=\Bbb R^n\hbox to5pt{\hss$\cdot$\hss}\Bbb R^n$ and correspondingly write $X\in\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}$ as $\pmat{X_1\\frak X_2}$ for $X_1,X_2\in\Bbb R^n$. Next, the semi-simple part $\tilde\hbox to5pt{\hss$\cdot$\hss}_0^{ss}$ of $\tilde\hbox to5pt{\hss$\cdot$\hss}_0$ is isomorphic to $\mathfrak{sl}(2n,\Bbb R)$ and the restriction of the adjoint representation of $\tilde\hbox to5pt{\hss$\cdot$\hss}_0^{ss}$ on $\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}$ coincides with the standard representation of $\mathfrak{sl}(2n,\Bbb R)$ on $\Bbb R^{2n}$. Finally, by $\span{\hbox to5pt{\hss$\cdot$\hss},\hbox to5pt{\hss$\cdot$\hss}}$ we denote the standard inner product of signature $(p,q)$ on $\Bbb R^n$ as above, i.e.\hbox to5pt{\hss$\cdot$\hss}$\span{X_1,X_2}=X_1^t\Bbb I_{p,q}X_2$ for any $X_1,X_2\in\Bbb R^n$.
\begin{lem*} \label{lem2}
Viewing the map $\Psi_\alpha$ from \ref{3.2} as an element of
$(\tilde\hbox to5pt{\hss$\cdot$\hss}_-)^*\wedge(\tilde\hbox to5pt{\hss$\cdot$\hss}_-)^*\otimes\tilde\hbox to5pt{\hss$\cdot$\hss}$, then
it lies in the subspace $(\tilde\hbox to5pt{\hss$\cdot$\hss}_{-1}^V)^*\wedge(\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2})^*\otimes\tilde\hbox to5pt{\hss$\cdot$\hss}_0^{ss}$.
Denoting by $W_0$ a non-zero element of $\tilde\hbox to5pt{\hss$\cdot$\hss}_1^E$, then the trilinear map
$\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}\hbox to5pt{\hss$\cdot$\hss} \tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}\hbox to5pt{\hss$\cdot$\hss} \tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}\to\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}$
defined by $(X,Y,Z)\mapsto [\Psi_{\alpha}(X,[Y,W_0]),Z]$ is (up to a non-zero
multiple) the complete symmetrization of the map
$$
(X,Y,Z)\mapsto \pmat{\span{X_1,Y_2}Z_1-\span{X_1,Y_1}Z_2\hbox to5pt{\hss$\cdot$\hss}
\span{X_2,Y_2}Z_1-\span{X_1,Y_2}Z_2}.
$$ \end{lem*} \begin{proof}
For $X,Y\in\tilde\hbox to5pt{\hss$\cdot$\hss}_-$, let $\hat X,\hat Y$ be the unique
elements in $\frak g_-\oplus\frak g_1$ so that $\alpha(\hat X)$ and $\alpha(\hat Y)$ is congruent to
$X$ and $Y$ modulo $\tilde\hbox to5pt{\hss$\cdot$\hss}$, respectively.
By the definition, one computes directly
$\Psi_\alpha(X,Y)=[\alpha(\hat X),\alpha(\hat Y)]-\alpha([\hat X,\hat Y])$ which has always
values in the lower right $2n\x2n$ block of $\tilde\hbox to5pt{\hss$\cdot$\hss}$ with vanishing trace, i.e.\
in the semi-simple part of $\tilde\hbox to5pt{\hss$\cdot$\hss}_0$.
Further, $\Psi_\alpha$ vanishes whenever both the entries $X,Y$ lie
in $\tilde\hbox to5pt{\hss$\cdot$\hss}^V_{-1}$ or in $\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}$ or any of them lies in $\tilde\hbox to5pt{\hss$\cdot$\hss}^E_{-1}$.
Hence the map $\Psi_\alpha$ is indeed of the form $\tilde\hbox to5pt{\hss$\cdot$\hss}^V_{-1}\wedge\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}\to\frak g_0^{ss}$.
Considering $X,Y\in\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}$, the element $[Y,W_0]$ lies in $\tilde\hbox to5pt{\hss$\cdot$\hss}^V_{-1}$ and the
non-zero $2n\hbox to5pt{\hss$\cdot$\hss} 2n$ block of $\Psi_\alpha(X,[Y,W_0])$ looks explicitly like
$$
\pmat{\frac12(R_{12}+\operatorname{tr} R_{12}\hbox to5pt{\hss$\cdot$\hss}\Bbb I_n)-R_{21} &\frac12(R_{11}-\operatorname{tr} R_{11}\hbox to5pt{\hss$\cdot$\hss}\Bbb I_n) \hbox to5pt{\hss$\cdot$\hss}
\frac12(R_{22}-\operatorname{tr} R_{22}\hbox to5pt{\hss$\cdot$\hss}\Bbb I_n) & R_{12}-\frac12(R_{21}+\operatorname{tr} R_{21}\hbox to5pt{\hss$\cdot$\hss}\Bbb I_n) },
$$
where $R_{11}=(X_1Y_1^t+Y_1X_1^t)\Bbb I_{p,q}$,
$R_{22}=(X_2Y_2^t+Y_2X_2^t)\Bbb I_{p,q}$,
$R_{12}=(X_1Y_2^t+Y_1X_2^t)\Bbb I_{p,q}$, and $R_{21}=R_{12}^t$.
The value of $[\Psi_{\alpha}(X,[Y,W_0]),Z]$ is then obtained by applying the above
matrix to the vector $Z=\pmat{Z_1\\hbox to5pt{\hss$\cdot$\hss}_2}$ from $\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}=\Bbb R^n\hbox to5pt{\hss$\cdot$\hss}\Bbb R^n$.
The result turns out to be the cyclic sum of
$$
\frac12(\span{Y_1,X_2}+\span{Y_2,X_1})\pmat{Z_1\hbox to5pt{\hss$\cdot$\hss}-Z_2}
+\pmat{-\span{X_1,Y_1}Z_2\hbox to5pt{\hss$\cdot$\hss}\span{X_2,Y_2}Z_1},
$$
which is the complete symmetrization of
$$
\span{X_1,Y_2}\pmat{Z_1\hbox to5pt{\hss$\cdot$\hss}-Z_2}+\pmat{-\span{X_1,Y_1}Z_2\hbox to5pt{\hss$\cdot$\hss}\span{X_2,Y_2}Z_1}
$$
up to a non-zero multiple. \end{proof}
In \ref{3.2} we motivated the definition of the pair $(i,\alpha)$. In order to justify the choice we made in \ref{3.4}, we have to check that starting with the homogeneous model, the associated Cartan geometry determined by $(i,\alpha)$ is regular and normal, i.e.\ it is the canonical Cartan geometry describing the path geometry of chains on $G/P$. This is provided by the following Theorem. \begin{thm} \label{th2}
Let $\dim M\ge 7$ and $(\mathcal G\to M,\omega)$ be a regular normal parabolic geometry of
type $(G,P)$ and let $(\tilde{\Cal G}:=\mathcal G\hbox to5pt{\hss$\cdot$\hss}_Q\tilde P\to\operatorname{\mathcal P}_0TM,\tilde\omega_\alpha)$
be the parabolic geometry obtained using the
extension functor associated to the pair $(i,\alpha)$ defined in \ref{3.4}.
Then this geometry is regular and normal if and only if $(\mathcal G\to M,\omega)$ is locally flat.
In that case the induced Cartan geometry is non-flat and torsion free. \end{thm} \begin{proof}
If $(\mathcal G\to M,\omega)$ is locally flat parabolic geometry of type $(G,P)$ then
by \eqref{eq6} the curvature of the induced Cartan geometry is determined only by
$\Psi_\alpha$.
Since $\Psi_\alpha$ in non-trivial, the induced Cartan geometry in non-flat and
since the values of $\Psi_\alpha$ are in $\tilde\hbox to5pt{\hss$\cdot$\hss}_0\subset\tilde\hbox to5pt{\hss$\cdot$\hss}$, it is torsion free
and hence regular.
In order to prove the normality, one has to show that $\del^*\Psi_\alpha=0$.
This can be proved either by a direct computation or by a more conceptual
argument as in the proof of Theorem 3.6 in \cite{CZ}.
Conversely, let us suppose the induced Cartan geometry
$(\tilde{\Cal G}\to\operatorname{\mathcal P}_0TM,\tilde\om_\alpha)$ is regular and normal.
Then, analyzing the harmonic curvature of $\tilde\om_\alpha$ as in the first part of the
proof of Theorem 3.8 in \cite{CZ}, it easily follows the Cartan geometry $(\mathcal G\to
M,\omega)$ is necessarily torsion free.
If $\dim M\ge 7$, different from 9, there are two harmonic curvature
components of $\omega$; if $\dim M=9$, one of the two components above is
further split into two parts.
In any case, all the components are of homogeneity one,
i.e.\ all are torsions.
Hence for $\dim M\ge 7$, torsion free and normal parabolic geometry of
type $(G,P)$ is flat, i.e.\ locally isomorphic to the homogeneous model. \end{proof}
\section{Applications} \label{4}
The applications in this section are based on the normality of the induced Cartan geometry. Hence according to Theorem \ref{th2} we consider the Lie contact structure is locally flat.
\begin{thm} \label{th3}
Let $M$ be a contact manifold of dimension $\ge 7$ with locally flat
Lie contact structure.
Then there is no linear connection on $TM$ which has the chains among
its geodesics. \end{thm} \begin{proof}
Since we deal with locally flat Lie contact structures, the induced path
geometry of chains is regular and normal by Theorem \ref{th2} and its
curvature is completely determined by the map $\Psi_\alpha$.
From lemma \ref{lem2} we know that $\Psi_\alpha$ is of homogeneity three, hence
it must coincide with the unique harmonic curvature component
which is there in this homogeneity for generalized path geometries,
see e.g.\ the summary in section 3.7 in \cite{CZ}.
However by the second part of Theorem 4.7 in \cite{C1}, the vanishing of
this component is equivalent to the fact the path geometry comes from a
projective structure on $M$.
Since $\Psi_\alpha\ne 0$, the claim follows. \end{proof}
In other words, even working with the locally flat Lie contact structures, the family of chains forms a rather complicated system of curves. In terms of second order ODE's, the chain equation is never locally equivalent either to the trivial equation or to the geodesic equation. In particular, considering the homogeneous model, the chain equation provides an example of non-trivial torsion free second order ODE with a reasonably large automorphism group. More specifically, the automorphism group obviously contains $G=PO(p+2,q+2)$, i.e.\ it has dimension at least $\frac{n^2+5n+6}2$, provided $n=p+q$ as before. We will see in the next section the dimension actually equals to the dimension of $G$.
\subsection{The reconstruction} As we know from \ref{3.5}, the harmonic curvature $\tilde\kappa_H$ of the induced Cartan geometry is a section of the bundle associated to $(\tilde\hbox to5pt{\hss$\cdot$\hss}_{-1}^V)^*\otimes(\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2})^*\otimes\tilde\hbox to5pt{\hss$\cdot$\hss}_0^{ss}$. We are going to interpret this quantity geometrically, which will allow us to reconstruct the Lie contact structure from the path geometry of chains.
Let $({\Cal G}\to M,\omega)$ be the parabolic geometry corresponding to a locally flat Lie contact structure on $M$. Let $\tilde M=\operatorname{\mathcal P}_0TM$ and let $(\tilde{\Cal G}\to\tilde M,\tilde\om)$ be the parabolic geometry induced by the path geometry of chains. Let us denote by $\tilde E$ and $\tilde V$ the subbundles in $T\tilde M$ corresponding to the subspaces $\tilde\hbox to5pt{\hss$\cdot$\hss}_{-1}^E$ and $\tilde\hbox to5pt{\hss$\cdot$\hss}_{-1}^V$ in $\tilde\hbox to5pt{\hss$\cdot$\hss}_{-1}$, respectively. Let us further denote $\tilde F:=T\tilde M/(\tilde E\oplus\tilde V)$; as an associated bundle over $\tilde M$ this corresponds to $\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}\cong\tilde\hbox to5pt{\hss$\cdot$\hss}_-/\tilde\hbox to5pt{\hss$\cdot$\hss}_{-1}$. As before, we can replace the space $(\tilde\hbox to5pt{\hss$\cdot$\hss}_{-1}^V)^*\otimes(\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2})^*\otimes\tilde\hbox to5pt{\hss$\cdot$\hss}_0^{ss}$ by $\tilde\hbox to5pt{\hss$\cdot$\hss}_{-1}^E\otimes(\otimes^3(\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2})^*)\otimes\tilde\hbox to5pt{\hss$\cdot$\hss}_{-2}$; the corresponding associated bundle is $\tilde E\otimes(\otimes^3\tilde F^*)\otimes\tilde F$. Altogether, since $\tilde E\subset T\tilde M$ is a line bundle, we can view the harmonic curvature $\tilde\kappa_H$ as a section of the bundle $\otimes^3\tilde F^*\otimes\tilde F\to\tilde M$ determined up to a non-zero multiple. (More specifically, lemma \ref{lem2} shows it is actually completely symmetric and trace free.)
In order to express $\tilde\kappa_H$ in terms of the underlying Lie contact structure on $M$, let us fix $x\in M$ and $\ell\in\pi^{-1}(x)$, where $\pi:\tilde M\to M$ is the natural projection. (By definition, $\ell$ is a line in $T_xM$ which is transverse to the contact distribution $H_x\subset T_xM$.) For each $\xi\in T_xM$, let $\tilde\xi\in T_\ell\tilde M$ be any lift and consider its class in $\tilde F_\ell=T_\ell\tilde M/(\tilde E_\ell\oplus\tilde V_\ell)$. Since $\tilde V\subset T\tilde M$ is the vertical subbundle of the projection $\pi$, this class is independent of the choice of the lift. From the explicit description of the tangent map of the projection $\pi$ it easily follows that its appropriate restriction yields a linear isomorphism $H_x\cong\tilde F_\ell$. Altogether, for a fixed $x$ and $\ell$, the $\tilde\kappa_H$ gives rise to an element of $\otimes^3H_x^*\otimes H_x$, denoted by $S_x$, which is determined up to a non-zero multiple. Its explicit description and the geometrical meaning are as follows. \begin{lem*} \label{lem3}
Let $(M,H=L^*\otimes R)$ be a locally flat Lie contact manifold and let
$\mathcal Q=\span{I,J,K}$ be the corresponding split-quaternionic structure on
$H$ as in \ref{2.4}.
Let $S_x$ be the element of $\otimes^3H_x^*\otimes H_x$ constructed from
the harmonic curvature of the associated path geometry of chains as above.
(1) Then, up to a non-zero multiple, $S$ is the cyclic sum of the mapping
$$
(\xi,\eta,\zeta)\mapsto{\Cal L}(\xi,I\eta)I\zeta+{\Cal L}(\xi,J\eta)J\zeta-{\Cal L}(\xi,K\eta)K\zeta,
$$
which is independent of the choice of basis of $\mathcal Q$.
(2) A non-zero element $\xi$ in $H_x=L_x^*\otimes R_x$ is of rank one if and only if
$S(\xi,\xi,\xi)=0$ or there is $\eta\in H_x$ such that $S(\xi,\xi,\eta)$ is
a non-zero multiple $\xi$. \end{lem*} Although the Levi bracket ${\Cal L}$ has values in $TM/H$, we view it here as a real valued bilinear form, because of the non-zero multiple freedom. \begin{proof}
(1) Since $\tilde\kappa_H$ is given by $\Psi_\alpha$, we just need to compare the
map above with the one described in lemma \ref{lem2}.
On the level of Lie algebra $\frak g$, the Levi bracket ${\Cal L}$ corresponds to $[\
,\hbox to5pt{\hss$\cdot$\hss}]:\frak g_{-1}\hbox to5pt{\hss$\cdot$\hss}\frak g_{-1}\to\frak g_{-2}$, which is explicitly described by
\eqref{eq1} in \ref{2.1}.
The operators $I,J$, and $K$ correspond to the multiplication by
matrices \eqref{eq5} on $\frak g_{-1}=\Bbb R^{2*}\otimes\Bbb R^n$ from the right.
Representing $\xi\in H_x$ by a matrix $(X_1,X_2)$ with columns
$X_1,X_2\in\Bbb R^n$, the images $I\xi$, $J\xi$ and $K\xi$ corresponds to
$(X_1,-X_2)$, $(X_2,X_1)$, and $(-X_2,X_1)$, respectively.
Representing also $\eta$ and $\zeta$ by $(Y_1,Y_2)$ and $(Z_1,Z_2)$,
the expressions ${\Cal L}(\xi,I\eta)$, ${\Cal L}(\xi,J\eta)$, and ${\Cal L}(\xi,K\eta)$,
correspond then to
$-\span{X_1,Y_2}-\span{X_2,Y_1}$, $\span{X_1,Y_1}-\span{X_2,Y_2}$, and
$\span{X_1,Y_1}+\span{X_2,Y_2}$, respectively.
Hence the direct substitution yields that the two mappings correspond each
other up to a non-zero multiple.
The independence of the choice of basis of $\mathcal Q$ follows by a
straightforward checking.
(2) In the above terms, $\xi$ is of rank one if and only if the
corresponding vectors $X_1$ and $X_2$ are linearly dependent.
Using this it is then easy to check that $S(\xi,\xi,\xi)=0$ and, moreover,
that $S(\xi,\xi,\eta)$ equals to a multiple of $\xi$, for any $\eta$.
For the converse statement, let us distinguish the two cases:
Firstly, suppose ${\Cal L}(\xi,I\xi)$, ${\Cal L}(\xi,J\xi)$, and ${\Cal L}(\xi,K\xi)$ does
not vanish simultaneously.
Since the assumption $S(\xi,\xi,\xi)=0$ is equivalent to
${\Cal L}(\xi,I\xi)I\xi+{\Cal L}(\xi,J\xi)J\xi-{\Cal L}(\xi,K\xi)K\xi=0$, it follows that
if two of the summands vanish then $\xi$ will be 0.
Hence if $\xi\ne 0$ and $S(\xi,\xi,\xi)=0$ then at most one of the three
summands vanishes and this yields that $\xi=A\xi$ for a specific element
$A\in\span{I,J,K}$.
Hence $\xi$ has got rank one by lemma \ref{lem0}.
Secondly, suppose that ${\Cal L}(\xi,I\xi)={\Cal L}(\xi,J\xi)={\Cal L}(\xi,K\xi)=0$.
Then $S(\xi,\xi,\xi)$ vanishes trivially, however, it also turns out that
$S(\xi,\xi,\eta)$ is a non-zero multiple of
${\Cal L}(\xi,I\eta)I\xi+{\Cal L}(\xi,J\eta)J\xi-{\Cal L}(\xi,K\eta)K\xi$, for any $\eta$.
Hence the assumption there is $\eta$ so that $S(\xi,\xi,\eta)=\xi$ yields that
there is $A\in\span{I,J,K}$ so that $A\xi=\xi$.
The rest follows again by lemma \ref{lem0}. \end{proof}
According to the development in \ref{2.3}, to recover the Lie contact structure on $M$ it is enough to determine the rank-one elements in $H_x$. The above lemma provides this in terms of $S_x$, hence we conclude by the following interesting result: \begin{thm*} \label{th4}
Let $M$ be a locally flat Lie contact manifold.
Then the Lie contact structure can be reconstructed from the harmonic curvature
of the regular normal
parabolic geometry associated to the path geometry of chains.
Consequently, a contact diffeomorphism on $M$ which maps chains to chains
is an automorphism of the Lie contact structure. \end{thm*} \begin{proof}
A contact diffeomorphism $f$ on $M$ lifts to a diffeomorphism $\tilde f$ on
$\tilde M=\mathcal P_0TM$.
The assumption that $f$ preserves chains yields that $\tilde f$ is an
automorphism of the associated path geometry of chains.
In particular, $\tilde f$ is compatible with the harmonic curvature
$\tilde\kappa_H$, which by assumption corresponds to the mapping $S$ above.
Hence $f$ is compatible with $S$ and the rest follows. \end{proof}
\subsection{Final remarks} As we noted in remark \ref{rem1}, the Lie contact structures in dimension 5 and 3 are basically equivalent to other parabolic contact structures. The procedure of previous sections is in effect independent of the dimension, however, the difference in these cases is that there are harmonic curvature components with higher homogeneity. Hence the torsion freeness condition does not imply the local flatness as in \ref{th2} and the results are more general. Consult the corresponding sections in \cite{CZ} and \cite{CZ2} for details.
\end{document} | arXiv |
pdgLive Home > Supersymmetric Particle Searches > ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$, ${{\widetilde{\mathit \chi}}_{{2}}^{\pm}}$ (Charginos) mass limits
${{\widetilde{\boldsymbol \chi}}_{{1}}^{\pm}}$, ${{\widetilde{\boldsymbol \chi}}_{{2}}^{\pm}}$ (Charginos) mass limits INSPIRE search
Charginos are unknown mixtures of w-inos and charged higgsinos (the supersymmetric partners of ${{\mathit W}}$ and Higgs bosons). A lower mass limit for the lightest chargino (${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$) of approximately 45 GeV, independent of the field composition and of the decay mode, has been obtained by the LEP experiments from the analysis of the ${{\mathit Z}}$ width and decays. These results, as well as other now superseded limits from ${{\mathit e}^{+}}{{\mathit e}^{-}}$ collisions at energies below 136$~$GeV, and from hadronic collisions, can be found in the 1998 Edition (The European Physical Journal C3 1 (1998)) of this Review.
Unless otherwise stated, results in this section assume spectra, production rates, decay modes and branching ratios as evaluated in the MSSM, with gaugino and sfermion mass unification at the GUT scale. These papers generally study production of ${{\widetilde{\mathit \chi}}_{{1}}^{0}}{{\widetilde{\mathit \chi}}_{{2}}^{0}}$ , ${{\widetilde{\mathit \chi}}_{{1}}^{+}}{{\widetilde{\mathit \chi}}_{{1}}^{-}}$ and (in the case of hadronic collisions) ${{\widetilde{\mathit \chi}}_{{1}}^{+}}{{\widetilde{\mathit \chi}}_{{2}}^{0}}$ pairs, including the effects of cascade decays. The mass limits on ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ are either direct, or follow indirectly from the constraints set by the non-observation of ${{\widetilde{\mathit \chi}}_{{2}}^{0}}$ states on the gaugino and higgsino MSSM parameters $\mathit M_{2}$ and $\mu $. For generic values of the MSSM parameters, limits from high-energy ${{\mathit e}^{+}}{{\mathit e}^{-}}$ collisions coincide with the highest value of the mass allowed by phase-space, namely ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}{ {}\lesssim{} }\sqrt {s }$/2. The still unpublished combination of the results of the four LEP collaborations from the 2000 run of LEP2 at $\sqrt {\mathit s }$ up to $\simeq{}209~$GeV yields a lower mass limit of 103.5$~$GeV valid for general MSSM models. The limits become however weaker in certain regions of the MSSM parameter space where the detection efficiencies or production cross sections are suppressed. For example, this may happen when: (i)$~$the mass differences $\Delta \mathit m_{+}$= ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}–{\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ or $\Delta {\mathit m}_{{{\mathit \nu}}}$= ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}–{\mathit m}_{{{\widetilde{\mathit \nu}}}}$ are very small, and the detection efficiency is reduced; (ii)$~$the electron sneutrino mass is small, and the ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ production rate is suppressed due to a destructive interference between ${{\mathit s}}$ and ${{\mathit t}}$ channel exchange diagrams. The regions of MSSM parameter space where the following limits are valid are indicated in the comment lines or in the footnotes.
VALUE (GeV)
$> 1050$ 95 1
SIRUNYAN
CMS ${}\geq{}1{{\mathit \gamma}}$ + $\not E_T$, Tchi1chi1F, ${{\widetilde{\mathit \chi}}_{{1}}^{0}}$ $\rightarrow$ ${{\mathit \gamma}}{{\widetilde{\mathit G}}}$
$> 825$ 95 1
CMS ${}\geq{}1{{\mathit \gamma}}$ + $\not E_T$, Tchi1chi1G, ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ $\rightarrow$ ${{\widetilde{\mathit \chi}}_{{1}}^{0}}{+}$ soft
CMS ${}\geq{}1{{\mathit \gamma}}$ + $\not E_T$, Tchi1n12-GGM, 120 GeV $<$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ $<$ 720 GeV
AABOUD
2019 AU
ATL 0, 1, 2 or more ${{\mathit \ell}}$, ${{\mathit H}}$ ( $\rightarrow$ ${{\mathit \gamma}}{{\mathit \gamma}}$ , ${{\mathit b}}{{\mathit b}}$ , ${{\mathit W}}{{\mathit W}^{*}}$ , ${{\mathit Z}}{{\mathit Z}^{*}}$ , ${{\mathit \tau}}{{\mathit \tau}}$ ) (various searches), Tchi1n2E, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$=0 GeV
$>112$ 95 3
2019 BU
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\widetilde{\mathit \chi}}_{{1}}^{+}}{{\widetilde{\mathit \chi}}_{{2}}^{0}}$ +2 jets, ${{\widetilde{\mathit \chi}}_{{1}}^{+}}$ $\rightarrow$ ${{\mathit \ell}^{+}}{{\mathit \nu}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , heavy sleptons, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{+}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 1 GeV, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{+}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{2}}^{0}}}$
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\widetilde{\mathit \chi}}_{{1}}^{+}}{{\widetilde{\mathit \chi}}_{{2}}^{0}}$ + 2 jets, ${{\widetilde{\mathit \chi}}_{{1}}^{+}}$ $\rightarrow$ ${{\mathit \ell}^{+}}{{\mathit \nu}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , heavy sleptons, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{+}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 30 GeV, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{+}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{2}}^{0}}}$
2019 CI
CMS ${}\geq{}$1 ${{\mathit H}}$ ( $\rightarrow$ ${{\mathit \gamma}}{{\mathit \gamma}}$ ) + jets + $\not E_T$, Tchi1n2E, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 1 GeV
CMS ${{\mathit \gamma}}$ + lepton + $\not E_T$, Tchi1n1A
2018 AY
ATLS 2${{\mathit \tau}}+\not E_T$, Tchi1chi1D and ${{\widetilde{\mathit \tau}}_{{L}}}$-only, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
ATLS 2${{\mathit \tau}}+\not E_T$, Tchi1n2D and ${{\widetilde{\mathit \tau}}_{{L}}}$-only, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
ATLS 2${{\mathit \ell}}+\not E_T$, Tchi1chi1C, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$=0 GeV
ATLS 2,3${{\mathit \ell}}+\not E_T$, Tchi1n2C, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$=0 GeV
$> 580$ 95 10
ATLS 2,3${{\mathit \ell}}+\not E_T$, Tchi1n2F, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$=0 GeV
$\text{none 130 - 230, 290 - 880}$ 95 11
2018 CK
ATLS 2${{\mathit H}}$ ( $\rightarrow$ ${{\mathit b}}{{\mathit b}}$ )+$\not E_T$,Tn1n1A, GMSB
$\text{none 220 - 600}$ 95 12
2018 CO
ATLS 2,3${{\mathit \ell}}$ + $\not E_T$, recursive jigsaw, Tchi1n2F, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
ATLS 2${{\mathit \ell}}$ (soft) + $\not E_T$, Tchi1n2F, wino, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 10 GeV
ATLS 2${{\mathit \ell}}$ (soft) + $\not E_T$, Tchi1n2G, higgsino, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 5 GeV
$> 1060$ 95 15
ATLS 2${{\mathit \gamma}}$ + $\not E_T$, GGM, Tchi1chi1A, any NLSP mass
ATLS ${}\geq{}4{{\mathit \ell}}$, RPV, ${{\mathit \lambda}_{{12k}}}{}\not=$0, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ $>$ 500 GeV
ATLS ${}\geq{}4{{\mathit \ell}}$, RPV, ${{\mathit \lambda}_{{12k}}}{}\not=$0, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ $>$ 50 GeV
ATLS ${}\geq{}4{{\mathit \ell}}$, RPV, ${{\mathit \lambda}_{{i33}}}{}\not=$0, 400 GeV $<$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ $<$ 700 GeV
CMS ${}\geq{}1{{\mathit \gamma}}$ + $\not E_T$, GGM, wino-like ${{\widetilde{\mathit \chi}}_{{2}}^{0}}{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ pair production, nearly degenerate wino and bino masses
CMS ${}\geq{}1{{\mathit \gamma}}$ + $\not E_T$, Tchi1n1A
CMS ${}\geq{}1{{\mathit \gamma}}$ + $\not E_T$, Tchi1chi1A
2018 AJ
CMS 2${{\mathit \ell}}$ (soft) + $\not E_T$, Tchi1n2F, wino, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{2}}^{0}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 20 GeV
2018 AO
CMS ${{\mathit \ell}^{\pm}}{{\mathit \ell}^{\pm}}$ or ${}\geq{}3{{\mathit \ell}}$ , Tchi1n2A, ${\mathit m}_{{{\widetilde{\mathit \ell}}}}$ = ${\mathit m}_{{{\widetilde{\mathit \nu}}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$+ 0.5 (${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$), ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
CMS ${{\mathit \ell}^{\pm}}{{\mathit \ell}^{\pm}}$ or ${}\geq{}3{{\mathit \ell}}$ , Tchi1n2A, ${\mathit m}_{{{\widetilde{\mathit \ell}}}}$ = ${\mathit m}_{{{\widetilde{\mathit \nu}}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$+ 0.05 (${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$), ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
CMS ${{\mathit \ell}^{\pm}}{{\mathit \ell}^{\pm}}$ or ${}\geq{}3{{\mathit \ell}}$ , Tchi1n2H, ${\mathit m}_{{{\widetilde{\mathit \ell}}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$+ 0.5 (${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$), ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
CMS ${{\mathit \ell}^{\pm}}{{\mathit \ell}^{\pm}}$ or ${}\geq{}3{{\mathit \ell}}$ , Tchi1n2H, ${\mathit m}_{{{\widetilde{\mathit \ell}}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$+ 0.05 (${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$), ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
CMS ${{\mathit \ell}^{\pm}}{{\mathit \ell}^{\pm}}$ or ${}\geq{}3{{\mathit \ell}}$ , Tchi1n2D, ${\mathit m}_{{{\widetilde{\mathit \tau}}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$+ 0.5 (${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$), ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
CMS ${{\mathit \ell}^{\pm}}{{\mathit \ell}^{\pm}}$ or ${}\geq{}3{{\mathit \ell}}$ , Tchi1n2E, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
CMS ${{\mathit \ell}^{\pm}}{{\mathit \ell}^{\pm}}$ or ${}\geq{}3{{\mathit \ell}}$ , Tchi1n2F, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
2018 AP
CMS Combination of searches, Tchi1n2E, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
CMS Combination of searches, Tchi1n2F, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
CMS Combination of searches, Tchi1n2I, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
2018 AR
CMS ${{\mathit \ell}^{\pm}}{{\mathit \ell}^{\mp}}$ + jets + $\not E_T$, Tchi1n2F, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
2018 DN
CMS ${{\mathit \ell}^{\pm}}{{\mathit \ell}^{\mp}}$ , Tchi1chi1E, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 1 GeV
$\bf{> 810}$ 95 22
CMS ${{\mathit \ell}^{\pm}}{{\mathit \ell}^{\mp}}$ , Tchi1chi1C, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
2018 DP
CMS 2${{\mathit \tau}}+\not E_T$, Tchi1chi1D,${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$=0 GeV
CMS 2${{\mathit \tau}}+\not E_T$, Tchi1n2D, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
CMS ${}\geq{}$1 ${{\mathit H}}$ ( $\rightarrow$ ${{\mathit \gamma}}{{\mathit \gamma}}$ ) + jets + $\not E_T$, Tchi1n2E, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ $<$ 25 GeV
KHACHATRYAN
CMS 2${{\mathit \tau}}+\not E_T$, Tchi1chi1C and ${{\widetilde{\mathit \tau}}}$-only, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$=0 GeV
CMS 1${{\mathit \ell}}$ + 2${{\mathit b}}$-jets + $\not E_T$, Tchi1n2E, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
$>500$ 95 27
ATLS 2${{\mathit \ell}^{\pm}}+\not E_T$,Tchi1chi1B,${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$=0 GeV
ATLS 2${{\mathit \ell}^{\pm}}+\not E_T$, Tchi1chi1C, low ${{\mathit \Delta}}$m for ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$, ${{\widetilde{\mathit \chi}}_{{1}}^{0}}$
ATLS 3,4${{\mathit \ell}}+\not E_T$,Tchi1n2B, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$=0 GeV
ATLS 3,4${{\mathit \ell}}+\not E_T$, Tchi1n2C, ${\mathit m}_{{{\widetilde{\mathit \ell}}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$+ 0.5 (or 0.95) (${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$)
ATLS 2 hadronic ${{\mathit \tau}}+\not E_T$ $\&$ 3${{\mathit \ell}}+\not E_T$ combination,Tchi1n2D,${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$=0 GeV
CMS ${}\geq{}1{{\mathit \gamma}}$ + 1 ${{\mathit e}}$ or ${{\mathit \mu}}$ + $\not E_T$, Tchi1n1A
ATLS ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{2}}^{0}}}$, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
ATLS ${}\geq{}$2 ${{\mathit \gamma}}$ + $\not E_T$, GGM, bino-like NLSP, any NLSP mass
ATLS ${}\geq{}$1 ${{\mathit \gamma}}$ + ${{\mathit e}},{{\mathit \mu}}$ + $\not E_T$, GGM, wino-like NLSP
ATLS ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ ${{\widetilde{\mathit \chi}}_{{2}}^{0}}$ $\rightarrow$ ${{\mathit \ell}^{\pm}}{{\mathit \nu}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}{{\mathit \ell}^{\pm}}{{\mathit \ell}^{\mp}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , simplified model, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{2}}^{0}}}$, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
ATLS ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ ${{\widetilde{\mathit \chi}}_{{2}}^{0}}$ $\rightarrow$ ${{\mathit W}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}{{\mathit Z}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , simplified model, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{2}}^{0}}}$, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
ATLS ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ ${{\widetilde{\mathit \chi}}_{{2}}^{0}}$ $\rightarrow$ ${{\mathit W}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}{{\mathit H}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , simplified model, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{2}}^{0}}}$, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
ATLS ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ ${{\widetilde{\mathit \chi}}_{{2}}^{0}}$ $\rightarrow$ ${{\mathit \tau}^{\pm}}{{\mathit \nu}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}{{\mathit \tau}^{\pm}}{{\mathit \tau}^{\mp}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , simplified model, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{2}}^{0}}}$, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
ATLS RPV, ${}\geq{}4{{\mathit \ell}^{\pm}}$, ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ $\rightarrow$ ${{\mathit W}^{(*)\pm}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , ${{\widetilde{\mathit \chi}}_{{1}}^{0}}$ $\rightarrow$ ${{\mathit \ell}^{\pm}}{{\mathit \ell}^{\mp}}{{\mathit \nu}}$
CMS ${{\widetilde{\mathit \chi}}_{{2}}^{0}}$ $\rightarrow$ ${{\mathit H}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ and ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ $\rightarrow$ ${{\mathit W}^{\pm}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ simplified models, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{2}}^{0}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
ATLS 3${{\mathit \ell}^{\pm}}$ + $\not E_T$, pMSSM, SMS
2012 CT
ATLS ${}\geq{}4{{\mathit \ell}^{\pm}}$, RPV, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ $>$ 300 GeV
CHATRCHYAN
2012 BJ
CMS ${}\geq{}$2 ${{\mathit \ell}}$, jets + $\not E_T$, ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}{{\widetilde{\mathit \chi}}_{{2}}^{0}}$
$\bf{>94}$ 95 39
DLPH ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$, tan $\beta {}\leq{}$40, $\Delta {\mathit m}_{{+} }>$3~GeV,all
CMS ${}\geq{}1{{\mathit \gamma}}$ + jets + $\not E_T$, Tchi1chi1A
CMS ${}\geq{}1{{\mathit \gamma}}$ + jets + $\not E_T$, Tchi1n1A
CMS ${}\geq{}1{{\mathit \gamma}}$ + jets + $\not E_T$, GGM, ${{\widetilde{\mathit \chi}}_{{2}}^{0}}$ ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ pair production, wino-like NLSP
CMS ${}\geq{}1{{\mathit \gamma}}$ + 1 ${{\mathit e}}$ or ${{\mathit \mu}}$ + $\not E_T$, Tglu1F, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{2}}^{0}}}$ $>$ 200 GeV
CMS 1,2 soft ${{\mathit \ell}^{\pm}}$+jets+$\not E_T$, Tchi1n2A, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}−{\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$=20 GeV
ATLS ${}\geq{}$2 ${{\mathit \tau}}$ + $\not E_T$, direct ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}{{\widetilde{\mathit \chi}}_{{2}}^{0}}$ , ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}{{\widetilde{\mathit \chi}}_{{1}}^{\mp}}$ production, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{2}}^{0}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
ATLS ${}\geq{}$2 ${{\mathit \tau}}$ + $\not E_T$, direct ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}{{\widetilde{\mathit \chi}}_{{1}}^{\mp}}$ production, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
$\text{none 100 - 105, 120 - 135, 145 - 160}$ 95 45
ATLS ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ ${{\widetilde{\mathit \chi}}_{{1}}^{\mp}}$ $\rightarrow$ ${{\mathit W}^{+}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}{{\mathit W}^{-}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , simplified model, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
ATLS ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ ${{\widetilde{\mathit \chi}}_{{1}}^{\mp}}$ $\rightarrow$ ${{\mathit \ell}^{+}}{{\mathit \nu}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}{{\mathit \ell}^{-}}{{\overline{\mathit \nu}}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , simplified model, ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV
CDF 3${{\mathit \ell}^{\pm}}$+ $\not E_T$, ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ $\rightarrow$ ${{\mathit \ell}}{{\mathit \nu}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , mSUGRA with ${\mathit m}_{\mathrm {0}}$=60 GeV
CMS ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ $\rightarrow$ ${{\mathit W}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , ${{\mathit \ell}}{{\widetilde{\mathit \nu}}}$ , ${{\widetilde{\mathit \ell}}}{{\mathit \nu}}$ , simplified model
2013 Q
CDF ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ $\rightarrow$ ${{\mathit \tau}}{{\mathit X}}$ , simplified gravity- and gauge-mediated models
ATLS 3${{\mathit \ell}^{\pm}}$ + $\not E_T$, pMSSM
ATLS ${{\mathit \ell}^{\pm}}{{\mathit \ell}^{\mp}}$ + $\not E_T$, ${{\mathit \ell}^{\pm}}{{\mathit \ell}^{\pm}}$ + $\not E_T$, ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}{{\widetilde{\mathit \chi}}_{{2}}^{0}}$
CMS ${{\widetilde{\mathit W}}^{0}}$ $\rightarrow$ ${{\mathit \gamma}}{{\widetilde{\mathit G}}}$ , ${{\widetilde{\mathit W}}^{\pm}}$ $\rightarrow$ ${{\mathit \ell}^{\pm}}{{\widetilde{\mathit G}}}$ ,GMSB
CMS tan ${{\mathit \beta}}$=3, ${{\mathit m}_{{0}}}$=60 GeV, ${{\mathit A}_{{0}}}$=0, ${{\mathit \mu}}>$0
1 SIRUNYAN 2020B searched in 35.9 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for events with at least one photon and large $\not E_T$. No significant excess above the Standard Model expectations is observed. Limits are set on chargino masses in a general gauge-mediated SUSY breaking (GGM) scenario Tchi1n12-GGM, see Figure 4. Limits are also set on the NLSP mass in the Tchi1chi1F and Tchi1chi1G simplified models, see their Figure 5. Finally, limits are set on the gluino mass in the Tglu4A simplified model, see Figure 6.
2 AABOUD 2019AU searched in 36.1 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for direct electroweak production of charginos and next-to-lightest neutralinos decaying into lightest neutralinos and a ${{\mathit W}}$, and a Higgs boson, respectively. Fully hadronic, semileptonic, diphoton, and multilepton (electrons, muons) final states with missing transverse momentum are considered in this search. Observations are consistent with the Standard Model expectations, and 95$\%$ confidence-level limits of up to 680 GeV on the chargino/next-to-lightest neutralino masses are set (Tchi1n2E model). See their Figure 14 for an overlay of exclusion contours from all searches.
3 SIRUNYAN 2019BU searched for pair production of gauginos via vector boson fusion assuming the gaugino spectrum is compressed, in 35.9 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV. The final states explored included zero leptons plus two jets, one lepton plus two jets, and one hadronic tau plus two jets. A similar bound is obtained in the light slepton limit.
4 SIRUNYAN 2019CI searched in 77.5 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for events with one or more high-momentum Higgs bosons, decaying to pairs of photons, jets and $\not E_T$. No significant excess above the Standard Model expectations is observed. Limits are set on the sbottom mass in the Tsbot4 simplified model, see Figure 3, and on the wino mass in the Tchi1n2E simplified model, see their Figure 4. Limits are also set on the higgsino mass in the Tn1n1A and Tn1n1B simplified models, see their Figure 5.
5 SIRUNYAN 2019K searched in 35.9 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for events with a photon, an electron or muon, and large $\not E_T$. No significant excess above the Standard Model expectations is observed. In the framework of GMSB, limits are set on the chargino and neutralino mass in the Tchi1n1A simplified model, see their Figure 6. Limits are also set on the gluino mass in the Tglu4A simplified model, and on the squark mass in the Tsqk4A simplified model, see their Figure 7.
6 AABOUD 2018AY searched in 36.1 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for direct electroweak production of charginos as in Tchi1chi1D models in events characterised by the presence of at least two hadronically decaying tau leptons and large missing transverse energy. No significant deviation from the expected SM background is observed. In the Tchi1chi1D model, assuming decays via intermediate ${{\widetilde{\mathit \tau}}_{{L}}}$, the observed limits rule out ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ masses up to 630 GeV for a massless ${{\widetilde{\mathit \chi}}_{{1}}^{0}}$. See their Fig.7 (left). Interpretations are also provided in Fig 8 (top) for different assumptions on the ratio between ${\mathit m}_{{{\widetilde{\mathit \tau}}}}$ and ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ + ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$.
7 AABOUD 2018AY searched in 36.1 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for direct electroweak production of charginos and neutralinos as in Tchi1n2D models, in events characterised by the presence of at least two hadronically decaying tau leptons and large missing transverse energy. No significant deviation from the expected SM background is observed. Assuming decays via intermediate ${{\widetilde{\mathit \tau}}_{{L}}}$ and ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{2}}^{0}}}$, the observed limits rule out ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ masses up to 760 GeV for a massless ${{\widetilde{\mathit \chi}}_{{1}}^{0}}$. See their Fig.7 (right). Interpretations are also provided in Fig 8 (bottom) for different assumptions on the ratio between ${\mathit m}_{{{\widetilde{\mathit \tau}}}}$ and ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ + ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$.
8 AABOUD 2018BT searched in 36.1 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for direct electroweak production of charginos, chargino and next-to-lightest neutralinos and sleptons in events with two or three leptons (electrons or muons), with or without jets and large missing transverse energy. No significant excess above the Standard Model expectations is observed. Limits are set on the chargino mass up to 750 GeV for massless neutralinos in the Tchi1chi1C simplified model exploiting 2${{\mathit \ell}}$ + 0 jets signatures, see their Figure 8(a).
9 AABOUD 2018BT searched in 36.1 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for direct electroweak production of charginos, chargino and next-to-lightest neutralinos and sleptons in events with two or three leptons (electrons or muons), with or without jets, and large missing transverse energy. No significant excess above the Standard Model expectations is observed. Limits are set on the chargino mass up to 1100 GeV for massless neutralinos in the Tchi1n2C simplified model exploiting 3${{\mathit \ell}}$ signature, see their Figure 8(c).
10 AABOUD 2018BT searched in 36.1 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for direct electroweak production of charginos, chargino and next-to-lightest neutralinos and sleptons in events with two or three leptons (electrons or muons), with or without jets, and large missing transverse energy. No significant excess above the Standard Model expectations is observed. Limits are set on the chargino mass up to 580 GeV for massless neutralinos in the Tchi1n2F simplified model exploiting 2${{\mathit \ell}}$+2 jets and 3${{\mathit \ell}}$ signatures, see their Figure 8(d).
11 AABOUD 2018CK searched for events with at least 3 ${{\mathit b}}$-jets and large missing transverse energy in two datasets of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV of 36.1 ${\mathrm {fb}}{}^{-1}$ and 24.3 ${\mathrm {fb}}{}^{-1}$ depending on the trigger requirements. The analyses aimed to reconstruct two Higgs bosons decaying to pairs of ${{\mathit b}}$-quarks. No significant excess above the Standard Model expectations is observed. Limits are set on the Higgsino mass in the T1n1n1A simplified model, see their Figure 15(a). Constraints are also presented as a function of the BR of Higgsino decaying into an higgs boson and a gravitino, see their Figure 15(b).
12 AABOUD 2018CO searched in 36.1 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for direct electroweak production of mass-degenerate charginos and next-to-lightest neutralinos in events with two or three leptons (electrons or muons), with or without jets, and large missing transverse energy. The search channels are based on recursive jigsaw reconstruction. Limits are set on the chargino mass up to 600 GeV for massless neutralinos in the Tchi1n2F simplified model exploiting the statistical combination of 2${{\mathit \ell}}$+2 jets and 3${{\mathit \ell}}$ channels. Chargino masses below 220 GeV are not excluded due to an excess of events above the SM prediction in the dedicated regions. See their Figure 13(d).
13 AABOUD 2018R searched in 36.1 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for electroweak production in scenarios with compressed mass spectra in final states with two low-momentum leptons and missing transverse momentum. The data are found to be consistent with the SM prediction. Results are interpreted in Tchi1n2G wino models and ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ masses are excluded up to 175 GeV for ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ $−$ ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 10 GeV. The exclusion limits extend down to mass splittings of 2 GeV, see their Fig. 10 (bottom).
14 AABOUD 2018R searched in 36.1 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for electroweak production in scenarios with compressed mass spectra in final states with two low-momentum leptons and missing transverse momentum. The data are found to be consistent with the SM prediction. Results are interpreted in Tchi1n2G higgsino models and ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ masses are excluded up to 145 GeV for ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}} - {\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 5 GeV. The exclusion limits extend down to mass splittings of 2.5 GeV, see their Fig. 10 (top).
15 AABOUD 2018U searched in 36.1 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV in events with at least one isolated photon, possibly jets and significant transverse momentum targeting generalised models of gauge-mediated SUSY breaking. No significant excess of events is observed above the SM prediction. Results of the diphoton channel are interpreted in terms of lower limits on the masses of gauginos Tchi1chi1A models, which reach as high as 1.3 TeV. Gaugino masses below 1060 GeV are excluded for any NLSP mass, see their Fig. 10.
16 AABOUD 2018Z searched in 36.1 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for events containing four or more charged leptons (electrons, muons and up to two hadronically decaying taus). No significant deviation from the expected SM background is observed. Limits are set on the Higgsino mass in simplified models of general gauge mediated supersymmetry Tn1n1A/Tn1n1B/Tn1n1C, see their Figure 9. Limits are also set on the wino, slepton, sneutrino and gluino mass in a simplified model of NLSP pair production with R-parity violating decays of the LSP via ${{\mathit \lambda}_{{12k}}}$ or ${{\mathit \lambda}_{{i33}}}$ to charged leptons, see their Figures 7, 8.
17 SIRUNYAN 2018AA searched in 35.9 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for events with at least one photon and large $\not E_T$. No significant excess above the Standard Model expectations is observed. Limits are set on wino masses in a general gauge-mediated SUSY breaking (GGM) scenario with bino-like ${{\widetilde{\mathit \chi}}_{{1}}^{0}}$ and wino-like ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ and ${{\widetilde{\mathit \chi}}_{{2}}^{0}}$ , see Figure 7. Limits are also set on the NLSP mass in the Tchi1n1A and Tchi1chi1A simplified models, see their Figure 8. Finally, limits are set on the gluino mass in the Tglu4A and Tglu4B simplified models, see their Figure 9, and on the squark mass in the Tskq4A and Tsqk4B simplified models, see their Figure 10.
18 SIRUNYAN 2018AJ searched in 35.9 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for events containing two low-momentum, oppositely charged leptons (electrons or muons) and $\not E_T$. No excess over the expected background is observed. Limits are derived on the wino mass in the Tchi1n2F simplified model, see their Figure 5. Limits are also set on the stop mass in the Tstop10 simplified model, see their Figure 6. Finally, limits are set on the Higgsino mass in the Tchi1n2G simplified model, see Figure 8 and in the pMSSM, see Figure 7.
19 SIRUNYAN 2018AO searched in 35.9 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for direct electroweak production of charginos and neutralinos in events with either two or more leptons (electrons or muons) of the same electric charge, or with three or more leptons, which can include up to two hadronically decaying tau leptons. No significant excess above the Standard Model expectations is observed. Limits are set on the chargino/neutralino mass in the Tchi1n2A, Tchi1n2H, Tchi1n2D, Tchi1n2E and Tchi1n2F simplified models, see their Figures 14, 15, 16, 17 and 18. Limits are also set on the higgsino mass in the Tn1n1A, Tn1n1B and Tn1n1C simplified models, see their Figure 19.
20 SIRUNYAN 2018AP searched in 35.9 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for direct electroweak production of charginos and neutralinos by combining a number of previous and new searches. No significant excess above the Standard Model expectations is observed. Limits are set on the chargino/neutralino mass in the Tchi1n2E, Tchi1n2F and Tchi1n2I simplified models, see their Figures 7, 8, 9 an 10. Limits are also set on the higgsino mass in the Tn1n1A, Tn1n1B and Tn1n1C simplified models, see their Figure 11, 12, 13 and 14.
21 SIRUNYAN 2018AR searched in 35.9 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for events containing two opposite-charge, same-flavour leptons (electrons or muons), jets and $\not E_T$. No significant excess above the Standard Model expectations is observed. Limits are set on the gluino mass in the Tglu4C simplified model, see their Figure 7. Limits are also set on the chargino/neutralino mass in the Tchi1n2F simplified models, see their Figure 8, and on the higgsino mass in the Tn1n1B and Tn1n1C simplified models, see their Figure 9. Finally, limits are set on the sbottom mass in the Tsbot3 simplified model, see their Figure 10.
22 SIRUNYAN 2018DN searched in 35.9 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for direct electroweak production of charginos and for pair production of top squarks in events with two leptons (electrons or muons) of the opposite electric charge. No significant excess above the Standard Model expectations is observed. Limits are set on the chargino mass in the Tchi1chi1C and Tchi1chi1E simplified models, see their Figure 8. Limits are also set on the stop mass in the Tstop1 and Tstop2 simplified models, see their Figure 9.
23 SIRUNYAN 2018DP searched in 35.9 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for direct electroweak production of charginos and neutralinos or of chargino pairs in events with a tau lepton pair and significant missing transverse momentum. Both hadronic and leptonic decay modes are considered for the tau lepton. No significant excess above the Standard Model expectations is observed. Limits are set on the chargino mass in the Tchi1chi1D and Tchi1n2 simplified models, see their Figures 14 and 15. Also, excluded stau pair production cross sections are shown in Figures 11, 12, and 13.
24 SIRUNYAN 2018X searched in 35.9 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for events with one or more high-momentum Higgs bosons, decaying to pairs of photons, jets and $\not E_T$. The razor variables (${{\mathit M}_{{R}}}$ and ${{\mathit R}^{2}}$) are used to categorise the events. No significant excess above the Standard Model expectations is observed. Limits are set on the sbottom mass in the Tsbot4 simplified model and on the wino mass in the Tchi1n2E simplified model, see their Figure 5. Limits are also set on the higgsino mass in the Tn1n1A and Tn1n1B simplified models, see their Figure 6.
25 KHACHATRYAN 2017L searched in about 19 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for events with two ${{\mathit \tau}}$ (at least one decaying hadronically) and $\not E_T$. In the Tchi1chi1C model, assuming decays via intermediate ${{\widetilde{\mathit \tau}}}$ or ${{\widetilde{\mathit \nu}}_{{\tau}}}$ with equivalent mass, the observed limits rule out ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ masses up to 420 GeV for a massless ${{\widetilde{\mathit \chi}}_{{1}}^{0}}$. See their Fig.5.
26 SIRUNYAN 2017AW searched in 35.9 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 13 TeV for events with a charged lepton (electron or muon), two jets identified as originating from a ${{\mathit b}}$-quark, and large $\not E_T$. No significant excess above the Standard Model expectations is observed. Limits are set on the mass of the chargino and the next-to-lightest neutralino in the Tchi1n2E simplified model, see their Figure 6.
27 AAD 2016AA summarized and extended ATLAS searches for electroweak supersymmetry in final states containing several charged leptons, $\not E_T$, with or without hadronic jets, in 20 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV. The paper reports the results of new interpretations and statistical combinations of previously published analyses, as well as new analyses. Exclusion limits at 95$\%$ C.L. are set on the ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ mass in the Tchi1chi1B and Tchi1chi1C simplified models. See their Fig. 13.
28 AAD 2016AA summarized and extended ATLAS searches for electroweak supersymmetry in final states containing several charged leptons, $\not E_T$, with or without hadronic jets, in 20 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV. The paper reports the results of new interpretations and statistical combinations of previously published analyses, as well as new analyses. Exclusion limits at 95$\%$ C.L. are set on mass-degenerate ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ and ${{\widetilde{\mathit \chi}}_{{2}}^{0}}$ masses in the Tchi1n2B, Tchi1n2C, and Tchi1n2D simplified models. See their Figs. 16, 17, and 18. Interpretations in phenomenological-MSSM, two-parameter Non Universal Higgs Masses (NUHM2), and gauge-mediated symmetry breaking (GMSB) models are also given in their Figs. 20, 21 and 22.
29 KHACHATRYAN 2016R searched in 19.7 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for events with one or more photons, one electron or muon, and $\not E_T$. No significant excess above the Standard Model expectations is observed. Limits are set on wino masses in a general gauge-mediated SUSY breaking model (GGM), for a wino-like neutralino NLSP scenario, see Fig. 5. Limits are also set in the Tglu1D and Tchi1n1A simplified models, see Fig. 6. The Tchi1n1A limit is reduced to 340 GeV for a branching ratio reduced by the weak mixing angle.
30 AAD 2015BA searched in 20.3 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for electroweak production of charginos and neutralinos decaying to a final state containing a ${{\mathit W}}$ boson and a 125 GeV Higgs boson, plus missing transverse momentum. No excess beyond the Standard Model expectation is observed. Exclusion limits are derived in simplified models of direct chargino and next-to-lightest neutralino production, with the decays ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ $\rightarrow$ ${{\mathit W}^{\pm}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ and ${{\widetilde{\mathit \chi}}_{{2}}^{0}}$ $\rightarrow$ ${{\mathit H}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ having 100$\%$ branching fraction, see Fig. 8. A combination of the multiple final states for the Higgs decay yields the best limits (Fig. 8d).
31 AAD 2015CA searched in 20.3 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for events with one or more photons and $\not E_T$, with or without leptons (${{\mathit e}}$, ${{\mathit \mu}}$). No significant excess above the Standard Model expectations is observed. Limits are set on wino masses in the general gauge-mediated SUSY breaking model (GGM), for wino-like NLSP, see Fig. 9, 12
32 AAD 2014H searched in 20.3 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for electroweak production of charginos and neutralinos decaying to a final sate with three leptons and missing transverse momentum. No excess beyond the Standard Model expectation is observed. Exclusion limits are derived in simplified models of direct chargino and next-to-lightest neutralino production, with decays to the lightest neutralino via either all three generations of leptons, staus only, gauge bosons, or Higgs bosons, see Fig. 7. An interpretation in the pMSSM is also given, see Fig. 8.
33 AAD 2014X searched in 20.3 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for events with at least four leptons (electrons, muons, taus) in the final state. No significant excess above the Standard Model expectations is observed. Limits are set on the wino-like chargino mass in an R-parity violating simplified model where the decay ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ $\rightarrow$ ${{\mathit W}^{(*)\pm}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , with ${{\widetilde{\mathit \chi}}_{{1}}^{0}}$ $\rightarrow$ ${{\mathit \ell}^{\pm}}{{\mathit \ell}^{\mp}}{{\mathit \nu}}$ , takes place with a branching ratio of 100$\%$, see Fig. 8.
34 KHACHATRYAN 2014L searched in 19.5 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for evidence of chargino-neutralino ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}{{\widetilde{\mathit \chi}}_{{2}}^{0}}$ pair production with Higgs or ${{\mathit W}}$-bosons in the decay chain, leading to ${{\mathit H}}{{\mathit W}}$ final states with missing transverse energy. The decays of a Higgs boson to a photon pair are considered in conjunction with hadronic and leptonic decay modes of the ${{\mathit W}}$ bosons. No significant excesses over the expected SM backgrounds are observed. The results are interpreted in the context of simplified models where the decays ${{\widetilde{\mathit \chi}}_{{2}}^{0}}$ $\rightarrow$ ${{\mathit H}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ and ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ $\rightarrow$ ${{\mathit W}^{\pm}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ take place 100$\%$ of the time, see Figs. $22 - 23$.
35 AAD 2013 searched in 4.7 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV for charginos and neutralinos decaying to a final state with three leptons (${{\mathit e}}$ and ${{\mathit \mu}}$) and missing transverse energy. No excess beyond the Standard Model expectation is observed. Exclusion limits are derived in the phenomenological MSSM, see Fig. 2 and 3, and in simplified models, see Fig. 4. For the simplified models with intermediate slepton decays, degenerate ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ and ${{\widetilde{\mathit \chi}}_{{2}}^{0}}$ masses up to 500 GeV are excluded at 95$\%$ C.L. for very large mass differences with the ${{\widetilde{\mathit \chi}}_{{1}}^{0}}$. Supersedes AAD 2012AS.
36 AAD 2013B searched in 4.7 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV for gauginos decaying to a final state with two leptons (${{\mathit e}}$ and ${{\mathit \mu}}$) and missing transverse energy. No excess beyond the Standard Model expectation is observed. Limits are derived in a simplified model of wino-like chargino pair production, where the chargino always decays to the lightest neutralino via an intermediate on-shell charged slepton, see Fig. 2(b). Chargino masses between 110 and 340 GeV are excluded at 95$\%$ C.L. for ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 10 GeV. Exclusion limits are also derived in the phenomenological MSSM, see Fig. 3.
37 AAD 2012CT searched in 4.7 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV for events containing four or more leptons (electrons or muons) and either moderate values of missing transverse momentum or large effective mass. No significant excess is found in the data. Limits are presented in a simplified model of R-parity violating supersymmetry in which charginos are pair-produced and then decay into a ${{\mathit W}}$-boson and a ${{\widetilde{\mathit \chi}}_{{1}}^{0}}$, which in turn decays through an RPV coupling into two charged leptons ( ${{\mathit e}^{\pm}}{{\mathit e}^{\mp}}$ or ${{\mathit e}^{\pm}}{{\mathit \mu}^{\mp}}$ ) and a neutrino. In this model, chargino masses up to 540 GeV are excluded at 95$\%$ C.L. for ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ above 300 GeV, see Fig. 3a. The limit deteriorates for lighter ${{\widetilde{\mathit \chi}}_{{1}}^{0}}$. Limits are also set in an R-parity violating mSUGRA model, see Fig. 3b.
38 CHATRCHYAN 2012BJ searched in 4.98 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV for direct electroweak production of charginos and neutralinos in events with at least two leptons, jets and missing transverse momentum. No significant excesses over the expected SM backgrounds are observed and 95$\%$ C.L. limits on the production cross section of ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}{{\widetilde{\mathit \chi}}_{{2}}^{0}}$ pair production were set in a number of simplified models, see Figs. 7 to 12.
39 ABDALLAH 2003M uses data from $\sqrt {s }$ = $192 - 208$ GeV to obtain limits in the framework of the MSSM with gaugino and sfermion mass universality at the GUT scale. An indirect limit on the mass of charginos is derived by constraining the MSSM parameter space by the results from direct searches for neutralinos (including cascade decays), for charginos and for sleptons. These limits are valid for values of $\mathit M_{2}<$ 1 TeV, $\vert {{\mathit \mu}}\vert {}\leq{}$2 TeV with the ${{\widetilde{\mathit \chi}}_{{1}}^{0}}$ as LSP. Constraints from the Higgs search in the $\mathit m{}^{{\mathrm {max}}}_{h}$ scenario assuming ${\mathit m}_{{{\mathit t}}}$= 174.3$~$GeV are included. The quoted limit applies if there is no mixing in the third family or when ${\mathit m}_{{{\widetilde{\mathit \tau}}_{{1}}}}\text{-}{\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}>$ 6 GeV. If mixing is included the limit degrades to 90 GeV. See Fig.~43 for the mass limits as a function of tan $\beta $. These limits update the results of ABREU 2000W.
40 KHACHATRYAN 2016AA searched in 7.4 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for events with one or more photons, hadronic jets and $\not E_T$. No significant excess above the Standard Model expectations is observed. Limits are set on wino masses in the general gauge-mediated SUSY breaking model (GGM), for a wino-like neutralino NLSP scenario and with the wino mass fixed at 10 GeV above the bino mass, see Fig. 4. Limits are also set in the Tchi1chi1A and Tchi1n1A simplified models, see Fig. 3.
41 KHACHATRYAN 2016R searched in 19.7 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for events with one or more photons, one electron or muon, and $\not E_T$. No significant excess above the Standard Model expectations is observed. Limits are also set in the Tglu1F simplified model, see Fig. 6.
42 KHACHATRYAN 2016Y searched in 19.7 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for events with one or two soft isolated leptons, hadronic jets, and $\not E_T$. No significant excess above the Standard Model expectations is observed. Limits are set on the ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ mass (which is degenerate with the ${{\widetilde{\mathit \chi}}_{{2}}^{0}}$) in the Tchi1n2A simplified model, see Fig. 4.
43 AAD 2014AV searched in 20.3 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for the direct production of charginos, neutralinos and staus in events containing at last two hadronically decaying ${{\mathit \tau}}$-leptons, large missing transverse momentum and low jet activity. The quoted limit was derived for direct ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}{{\widetilde{\mathit \chi}}_{{2}}^{0}}$ and ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}{{\widetilde{\mathit \chi}}_{{1}}^{\mp}}$ production with ${{\widetilde{\mathit \chi}}_{{2}}^{0}}$ $\rightarrow$ ${{\widetilde{\mathit \tau}}}{{\mathit \tau}}$ $\rightarrow$ ${{\mathit \tau}}{{\mathit \tau}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ and ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ $\rightarrow$ ${{\widetilde{\mathit \tau}}}{{\mathit \nu}}$( ${{\widetilde{\mathit \nu}}_{{\tau}}}{{\mathit \tau}}$) $\rightarrow$ ${{\mathit \tau}}{{\mathit \nu}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{2}}^{0}}}$ = ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$, ${\mathit m}_{{{\widetilde{\mathit \tau}}}}$ = 0.5 (${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ + ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$), ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV. No excess over the expected SM background is observed. Exclusion limits are set in simplified models of ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}{{\widetilde{\mathit \chi}}_{{1}}^{\mp}}$ and ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}{{\widetilde{\mathit \chi}}_{{2}}^{0}}$ pair production, see their Figure 7. Upper limits on the cross section and signal strength for direct di-stau production are derived, see Figures 8 and 9. Also, limits are derived in a pMSSM model where the only light slepton is the ${{\widetilde{\mathit \tau}}_{{R}}}$, see Figure 10.
44 AAD 2014AV searched in 20.3 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for the direct production of charginos, neutralinos and staus in events containing at last two hadronically decaying ${{\mathit \tau}}$-leptons, large missing transverse momentum and low jet activity. The quoted limit was derived for direct ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}{{\widetilde{\mathit \chi}}_{{1}}^{\mp}}$ production with ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}$ $\rightarrow$ ${{\widetilde{\mathit \tau}}}{{\mathit \nu}}$( ${{\widetilde{\mathit \nu}}_{{\tau}}}{{\mathit \tau}}$) $\rightarrow$ ${{\mathit \tau}}{{\mathit \nu}}{{\widetilde{\mathit \chi}}_{{1}}^{0}}$ , ${\mathit m}_{{{\widetilde{\mathit \tau}}}}$ = 0.5 (${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{\pm}}}$ + ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$), ${\mathit m}_{{{\widetilde{\mathit \chi}}_{{1}}^{0}}}$ = 0 GeV. No excess over the expected SM background is observed. Exclusion limits are set in simplified models of ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}{{\widetilde{\mathit \chi}}_{{1}}^{\mp}}$ and ${{\widetilde{\mathit \chi}}_{{1}}^{\pm}}{{\widetilde{\mathit \chi}}_{{2}}^{0}}$ pair production, see their Figure 7. Upper limits on the cross section and signal strength for direct di-stau production are derived, see Figures 8 and 9. Also, limits are derived in a pMSSM model where the only light slepton is the ${{\widetilde{\mathit \tau}}_{{R}}}$, see Figure 10.
45 AAD 2014G searched in 20.3 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for electroweak production of chargino pairs, or chargino-neutralino pairs, decaying to a final sate with two leptons (${{\mathit e}}$ and ${{\mathit \mu}}$) and missing transverse momentum. No excess beyond the Standard Model expectation is observed. Exclusion limits are derived in simplified models of chargino pair production, with chargino decays to the lightest neutralino via either sleptons or gauge bosons, see Fig 5.; or in simplified models of chargino and next-to-lightest neutralino production, with decays to the lightest neutralino via gauge bosons, see Fig. 7. An interpretation in the pMSSM is also given, see Fig. 10.
46 AALTONEN 2014 searched in 5.8 fb${}^{-1}$ of ${{\mathit p}}{{\overline{\mathit p}}}$ collisions at $\sqrt {s }$ = 1.96 TeV for evidence of chargino and next-to-lightest neutralino associated production in final states consisting of three leptons (electrons, muons or taus) and large missing transverse momentum. The results are consistent with the Standard Model predictions within 1.85 $\sigma $. Limits on the chargino mass are derived in an mSUGRA model with ${\mathit m}_{\mathrm {0}}$ = 60 GeV, tan ${{\mathit \beta}}$ = 3, ${{\mathit A}_{{0}}}$ = 0 and ${{\mathit \mu}}$ $>$0, see their Fig. 2.
47 KHACHATRYAN 2014I searched in 19.5 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV for electroweak production of chargino pairs decaying to a final state with opposite-sign lepton pairs (${{\mathit e}}$ or ${{\mathit \mu}}$) and missing transverse momentum. No excess beyond the Standard Model expectation is observed. Exclusion limits are derived in simplified models, see Fig. 18.
48 AALTONEN 2013Q searched in 6.0 fb${}^{-1}$ of ${{\mathit p}}{{\overline{\mathit p}}}$ collisions at $\sqrt {s }$ = 1.96 TeV for evidence of chargino-neutralino associated production in like-sign dilepton final states. One lepton is identified as the hadronic decay of a tau lepton, while the other is an electron or muon. Good agreement with the Standard Model predictions is observed and limits are set on the chargino-neutralino cross section for simplified gravity- and gauge-mediated models, see their Figs. 2 and 3.
49 AAD 2012AS searched in 2.06 fb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV for charginos and neutralinos decaying to a final state with three leptons (${{\mathit e}}$ and ${{\mathit \mu}}$) and missing transverse energy. No excess beyond the Standard Model expectation is observed. Exclusion limits are derived in the phenomenological MSSM, see Fig. 2 (top), and in simplified models, see Fig. 2 (bottom).
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52 CHATRCHYAN 2011V looked in 35 pb${}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 7 TeV for events with ${}\geq{}$3 isolated leptons (${{\mathit e}}$, ${{\mathit \mu}}$ or ${{\mathit \tau}}$), with or without jets and $\not E_T$. No evidence for an excess over the expected background is observed. Limits are derived in the CMSSM (${{\mathit m}_{{0}}}$, ${{\mathit m}_{{1/2}}}$) plane for tan ${{\mathit \beta}}$ = 3 (see Fig. 5).
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\begin{document}
\centerline{\bf Structure of root graded Lie algebras}
\centerline{Malihe Yousofzadeh\footnote{This research was in part supported by a grant from IPM (No. 89170030).}}
\parbox{5in}{Abstract. We give a complete description of Lie algebras graded by an infinite irreducible locally finite root system.}
\section{introduction} In 1992, S. Berman and R. Moody \cite{BM} introduced the notion of a Lie algebra graded by an irreducible reduced finite root system. Their definition was motivated by a construction appearing in the classification of finite dimensional simple Lie algebras containing nonzero toral subalgebras \cite{Se}. The classification of root graded Lie algebras in the sense of S. Berman and R. Moody was given, in part, by S. Bermen and R. Moody themselves and was completed by G. Benkart and E. Zelmanov \cite{BZ} in 1996. This classification has been based on a type-by-type approach; for each type $X,$ the authors give a recognition theorem for centerless Lie algebras graded by a root system of type $X.$ In 1996, E. Neher \cite{N} generalized the notion of root graded Lie algebras by switching from fields of characteristic zero to rings containing $1/6$ and working with locally finite root systems instead of finite root systems. Roughly speaking, according to him, a Lie algebra $\mathcal{L}$ over a ring containing $1/6$ is graded by a reduced locally finite root system $R$ if $\mathcal{L}$ is a $Q(R)-$graded Lie algebra generated by homogenous submodules of nonzero degrees and that for any nonzero root $\alpha\in R,$ there are homogenous elements $e$ and $f$ of degrees $\alpha$ and $-\alpha$ respectively such that $[e,f]$ acts diagonally on $\mathcal{L}.$ He realized root graded Lie algebras for reduced types other than $F_4,$ $G_2$ and $E_8$ as central extensions of Tits-Kantor-Koecher algebras of certain Jordan pairs. Finally in 2002, B. Allison, G. Benkart and Y. Gao \cite{ABG2} defined a Lie algebra graded by an irreducible finite root system of type $BC$ and studied root graded Lie algebras of type $BC_n$ for $n\geq2.$ In 2003, G. Benkart and O. Smirnov \cite{BS} studied Lie algebras graded by a finite root system of type $BC_1$ and finalized the classification of Lie algebras graded by an irreducible finite root system.
A Lie algebra $\mathcal{L}$ graded by an irreducible finite root system $R$ has a weight space decomposition with respect to a splitting Cartan subalgebra of a finite dimensional split simple Lie subalgebra $\mathfrak{g}$ of $\mathcal{L},$ whose set of weights is contained in $R.$ This feature allows us to decompose $\mathcal{L}$ as $\mathcal{L}=\mathcal{M}_1\oplus\mathcal{M}_2$ in which $\mathcal{M}_1$ is a direct sum of finite dimensional irreducible nontrivial $\mathfrak{g}-$submodules and $\mathcal{M}_2$ is a trivial $\mathfrak{g}-$submodule of $\mathcal{L}.$ One can derive a specific vector space $\frak{b}$ from the $\mathfrak{g}-$module structure of $\mathcal{M}_1.$ This vector space is equipped with an algebraic structure which is induced by the Lie algebraic structure of $\mathcal{L}.$ Moreover the Lie algebra $\mathcal{L}$ can be reconstructed from the algebra $\frak{b}$ in a prescribed way, see \cite{ABG1} and \cite{ABG2}. This construction led to finding a finite presentation for the universal central extension of a Lie torus of a finite type other than $A$ and $C,$ see \cite{You}, \cite{AYY}. This motivates us to generalize this construction for Lie algebras graded by infinite root systems.
We give a complete description of the structure of root graded Lie algebras. We fix an infinite irreducible locally finite root system $R$ and show that a Lie algebra $\mathcal{L}$ graded by $R$ can be described in terms of a locally finite split simple Lie subalgebra ${\mathcal G},$ some natural representations of ${\mathcal G}$ and a certain algebra called the {\it coordinate algebra}. We also give the Lie bracket on $\mathcal{L}$ in terms of the Lie bracket on ${\mathcal G},$ the action of the representations and the product on $\frak{b}.$ More precisely, depending on type of $R,$ we consider a quadruple $\mathfrak{c}$ so called {\it coordinate quadruple}. We next correspond to $\mathfrak{c},$ a specific algebra $\frak{b}_{\mathfrak{c}}$ and a specific Lie algebra $\{\frak{b}_{\mathfrak{c}},\frak{b}_{\mathfrak{c}}\}.$ Then for each subspace $\mathcal K$ of the center of $\{\frak{b}_{\mathfrak{c}},\frak{b}_{\mathfrak{c}}\}$ satisfying a certain property called {\it the uniform property}, we define a Lie algebra $\mathcal{L}(\frak{b}_{\mathfrak{c}},\mathcal K)$ and show that it is a Lie algebra graded by $R.$ Conversely, given a Lie algebra $\mathcal{L}$ graded by $R,$ we prove that $\mathcal{L}$ can be decomposed as $\mathcal{M}_1\oplus\mathcal{M}_2$ where $\mathcal{M}_1$ is a direct sum of certain irreducible nontrivial $\mathfrak{g}-$submodules for a locally finite spilt simple Lie subalgebra $\mathfrak{g}$ of $\mathcal{L}$ and $\mathcal{M}_2$ is a specific subalgebra of $\mathcal{L}.$ We derive a quadruple $\mathfrak{c}$ from the $\mathfrak{g}-$module structure of $\mathcal{M}_1$ and show that it is a coordinate quadruple. We also prove that there is a subspace $\mathcal K$ of $\{\frak{b}_{\mathfrak{c}},\frak{b}_{\mathfrak{c}}\}$ satisfying the uniform property such that $\mathcal{M}_2$ is isomorphic to the quotient algebra $\{\frak{b}_{\mathfrak{c}},\frak{b}_{\mathfrak{c}}\}/\mathcal K$ and moreover $\mathcal{L}$ is isomorphic to $\mathcal{L}(\frak{b}_{\mathfrak{c}},\mathcal K).$ If the root system $R$ is reduced, our method suggests another approach to characterize Lie algebras graded by $R$ compared with what is offered by E. Neher \cite{N}.
The author wishes to thank the hospitality of Mathematics and Statistics Department, University of Ottawa, where some parts of this work were carried out. The author also would like to express her sincere gratitude to Professor Saeid Azam and Professor Erhard Neher for some fruitful discussions.
\section{Preliminaries} Throughout this work, $\mathbb{N}$ denotes the set of nonnegative integers and $\mathbb{F}$ is a field of characteristic zero. Unless otherwise mentioned, all vector spaces are considered over
$\mathbb{F}.$ We denote the dual space of a vector space $V$ by $V^*.$ For a linear transformation $T$ on a vector space $V,$ if the trace of $T$ is defined, we denote it by $tr(T).$ Also for a nonempty set $S,$ by $id_{_S}$ (or $id$ if there is no confusion), we mean the identity map on $S$ and by $|S|,$ we mean the cardinality of $S.$ Finally for an index set $I,$ by a conventional notation, we take $\bar I:=\{\bar i\mid i\in I\}$ to be a disjoint copy of $I$ and for each subset $J$ of $I,$ by $\bar J,$ we mean the subset of $\bar I$ corresponding to $J.$
\subsection{Locally Finite Split Simple Lie Algebras} In this subsection, we recall the structure of infinite dimensional locally finite split simple Lie algebras from \cite{NS} and state some facts which play key roles in this work. Let us start with the following definition. \begin{Definition} {\em Let ${\mathcal H}$ be a Lie algebra. We say an ${\mathcal H}-$module $\mathcal{M}$ has a {\it weight space decomposition with respect to ${\mathcal H},$} if \begin{equation*} \mathcal{M}=\oplus_{\alpha\in {\mathcal H}^*}\mathcal{M}_\alpha \;\hbox{where}\; \mathcal{M}_\alpha:=\{x\in\mathcal{M}\mid h\cdot x=\alpha(h)x;\;\;\forall\; h\in {\mathcal H}\} \end{equation*} for all $\alpha\in{\mathcal H}^*.$ The set $R:=\{\alpha\in{\mathcal H}^*\mid \mathcal{M}_\alpha\neq \{0\}\}$ is called the {\it set of weights} of $\mathcal{M}$ (with respect to ${\mathcal H}$). For $\alpha\in R,$ $\mathcal{M}_\alpha$ is called a {\it weight space}, and any element of $\mathcal{M}_\alpha$ is called a {\it weight vector} of {\it weight} $\alpha.$ If a Lie algebra $\mathcal{L}$ has a weight space decomposition with respect to a nontrivial subalgebra $H$ of $\mathcal{L}$ via the adjoint representation, $H$ is called a {\it split toral subalgebra}. The set of weights of $\mathcal{L}$ is called the {\it root system} of $\mathcal{L}$ with respect to $H,$ and the corresponding weight spaces are called {\it root spaces of $\mathcal{L}.$} A Lie algebra $\mathcal{L}$ is called {\it split} if it contains a {\it splitting Cartan subalgebra}, that is a split toral subalgebra $H$ of $\mathcal{L}$ with $\mathcal{L}_0=H.$ } \end{Definition}
The root system of a locally finite split simple Lie algebra with respect to a splitting Cartan subalgebra is a reduced irreducible locally finite root system in the following sense (see \cite{Bo} and \cite{NS}): \begin{Definition}\label{def-root}\cite{LN} {\rm Let ${\mathcal U}$ be a nontrivial vector space and $R$ be a subset of ${\mathcal U},$ $R$ is said to be a {\it locally finite root system in ${\mathcal U}$} of {\it rank} $dim({\mathcal U})$ if the following are satisfied:
(i) $R$ is locally finite, contains zero and spans ${\mathcal U}.$
(ii) For every $\alpha\in R^\times:=R\setminus\{0\},$ there exists $\check\alpha\in{\mathcal U}^*$ such that $\check\alpha(\alpha)=2$ and $s_\alpha(\beta)\in R$ for $\alpha,\beta\in R$ where $s_\alpha:{\mathcal U}\longrightarrow{\mathcal U}$ maps $u\in{\mathcal U}$ to $u-\check\alpha(u)\alpha.$ We set by convention $\check0$ to be zero.
(iii) $\check\alpha(\beta)\in{\mathbb Z},$ for $\alpha,\beta\in R.$
Set $R_{sdiv}:=(R\setminus\{\alpha\in R\mid 2\alpha\in R\})\cup \{0\}$ and call it the {\it semi-divisible subsystem} of $R.$ The root system $R$ is called {\it reduced} if $R=R_{sdiv}.$ } \end{Definition}
Suppose that $R$ is a locally finite root system. A nonempty subset $S$ of $R$ is said to be a {\it subsystem} of $R$ if $S$ contains zero and $s_\alpha(\beta)\in S$ for $\alpha,\beta\in S\setminus\{0\}.$ A subsystem $S$ of $R$ is called {\it full} if $\hbox{span}_\mathbb{F} S\cap R=S.$ Following \cite[\S 2.6]{LN}, we say two nonzero roots $\alpha,\beta$ are {\it connected} if there exist finitely many roots $\alpha_1=\alpha,\alpha_2,\ldots,\alpha_n=\beta$ such that $\check\alpha_{i+1}(\alpha_{i})\neq 0,$ $1\leq i\leq n-1.$ Connectedness defines an equivalence relation on $R^\times$ and so $R^\times$ is the disjoint union of its equivalence classes called {\it connected components} of $R.$ A nonempty subset $X$ of $R$ is called {\it irreducible,} if each two nonzero elements $x,y\in X$ are connected and it is called {\it closed} if $(X+X)\cap R\subseteq X.$
It is easy to see that if $X$ is a connected component of a locally finite root system $R,$ then $X\cup\{0\}$ is a closed subsystem of $R.$ For the locally finite root system $R,$ take $\{R_\lambda\mid \lambda\in\Gamma\}$ to be the class of all finite subsystems of $R,$ and say $\lambda\preccurlyeq\mu$ $(\lambda,\mu\in\Gamma)$ if $R_\lambda$ is a subsystem of $R_\mu,$ then $(\Gamma,\preccurlyeq)$ is a directed set and $R$ is the direct union of $\{R_\lambda\mid \lambda\in\Gamma\}.$ Furthermore, if $R$ is irreducible, it is the direct union of its irreducible finite subsystems.
Two locally finite root systems $( R,{\mathcal U})$
and $(S,{\mathcal V})$ are said to be isomorphic if there is a linear transformation $f:{\mathcal U}\longrightarrow {\mathcal V}$ such that $f(R)= S.$
Suppose that $I$ is a nonempty index set and ${\mathcal U}:=\oplus_{i\in I}\mathbb{F}\epsilon_i$ is the free $\mathbb{F}-$module over the set $I.$ Define the form $$\begin{array}{c}(\cdot,\cdot):{\mathcal U}\times{\mathcal U}\longrightarrow\mathbb{F}\\ (\epsilon_i,\epsilon_j)=\delta_{i,j}, \hbox{ for } i,j\in I \end{array}$$ and set \begin{equation}\label{locally-finite} \begin{array}{l} \dot A_I:=\{\epsilon_i-\epsilon_j\mid i,j\in I\},\\ D_I:=\dot A_I\cup\{\pm(\epsilon_i+\epsilon_j)\mid i,j\in I,\;i\neq j\},\\ B_I:=D_I\cup\{\pm\epsilon_i\mid i\in I\},\\ C_I:=D_I\cup\{\pm2\epsilon_i\mid i\in I\},\\ BC_I:=B_I\cup C_I. \end{array} \end{equation} One can see that these are irreducible locally finite root systems in their $\mathbb{F}-$span's which we refer to as {\it type} $A,B,C,D$ and $BC$ respectively. Moreover every irreducible locally finite root system of infinite rank is isomorphic to one of these root systems (see \cite[\S4.14 \S8]{LN}). Now we suppose $R$ is an irreducible locally finite root system as above and note that $(\alpha,\alpha)\in\mathbb{N}$ for all $\alpha\in R.$ This allows us to define $$\begin{array}{l} R_{sh}:=\{\alpha\in R^\times\mid (\alpha,\alpha)\leq(\beta,\beta);\;\;\hbox{for all $\beta\in R$} \},\\ R_{ex}:=R\cap2 R_{sh},\\ R_{lg}:= R^\times\setminus( R_{sh}\cup R_{ex}). \end{array}$$ The elements of $R_{sh}$ (resp. $R_{lg},R_{ex}$) are called {\it short roots} (resp. {\it long roots, extra-long roots}) of $R$.
A locally finite split simple Lie algebra is said to be of type $A,B,C$ or $D$ if its corresponding root system with respect to a splitting Cartan subalgebra is of type $A,B,C$ or $D$ respectively. In what follows, we recall from \cite{NS} the classification of infinite dimensional locally finite split simple Lie algebras.
Suppose that $J$ is an index set and ${\mathcal V}={\mathcal V}_J$ is a vector space with a fixed basis $\{v_j\mid j\in J\}.$ One knows that $\mathfrak{gl}({\mathcal V}):=End({\mathcal V})$ together with $$[\cdot,\cdot]:\mathfrak{gl}({\mathcal V})\times\mathfrak{gl}({\mathcal V})\longrightarrow \mathfrak{gl}({\mathcal V});\; (X,Y)\mapsto XY-YX;\;\; X,Y\in\mathfrak{gl}({\mathcal V})$$is a Lie algebra. Now for $j,k\in J,$ define \begin{equation}\label{elementary2}e_{j,k}:{\mathcal V}\longrightarrow{\mathcal V};\;\; v_i\mapsto \delta_{k,i}v_j,\;\;\; (i\in J),\end{equation} then $\mathfrak{gl}(J):=\hbox{span}_\mathbb{F}\{e_{j,k}\mid j,k\in J\}$ is a Lie subalgebra of $\mathfrak{gl}({\mathcal V})$.
\begin{Lemma}[{Classical Lie algebras of type $A$}]\label{type-a-alg} Suppose that $I$ is a non-empty index set of cardinality greater than $1$, $I_0$ is a fixed subset of
$I$ with $|I_0|>1$ and ${\mathcal V}$ is a vector space with a basis $\{v_i\mid i\in I\}.$ Take $\Lambda$ to be an index set containing $0$ and $\{I_\lambda\mid \lambda\in \Lambda\}$ to be the class of all finite subsets of $I$ containing $I_0.$ Set $${\mathcal G}:=\mathfrak{sl}(I):=\{\phi\in\mathfrak{gl}(I)\mid tr(\phi)=0\},$$ and for $\lambda\in\Lambda,$ take \begin{equation*}\label{simple-a-alg} \begin{array}{l}{\mathcal G}_{_{I_\lambda}}:={\mathcal G}^\lambda:={\mathcal G}\cap\hbox{span}\{e_{r,s}\mid r,s\in I_\lambda\}. \end{array}\end{equation*}
Then $\mathfrak{sl}(I)$ is a locally finite split simple Lie subalgebra of $\mathfrak{gl}(I)$ with splitting Cartan subalgebra ${\mathcal H}:=\hbox{span}\{e_{i,i}-e_{j,j}\mid i,j\in I\}$ and corresponding root system isomorphic to $\dot A_I.$ Moreover for $i,j\in I$ with $i\neq j,$ we have $${\mathcal G}_{\epsilon_i-\epsilon_j}=\mathbb{F} e_{i,j}.$$ Also for each $\lambda\in \Lambda,$ ${\mathcal G}^\lambda$ is a finite dimensional split simple Lie subalgebra of ${\mathcal G}$ with splitting Cartan subalgebra ${\mathcal H}^\lambda:={\mathcal H}\cap{\mathcal G}^\lambda$ and ${\mathcal G}$ is the direct union of $\{{\mathcal G}^\lambda\mid \lambda \in \Lambda\}.$
\end{Lemma}
In the following lemma, we see that locally finite split simple Lie algebras of type $B$ can be described in terms of derivations of {\it Clifford Jordan algebras} which are defined as following: \begin{Definition}[\cite{Yos}]\label{yoshii}
{\rm Suppose that $A$ is a unital commutative associative algebra over $\mathbb{F}$ and ${\mathcal W}$ is a unitary $A$-module. Suppose that $g:{\mathcal W}\times {\mathcal W}\longrightarrow A$ is a symmetric $A$-bilinear form and set
$\mathcal{J}=\mathcal{J}(g,{\mathcal W}):=A\oplus{\mathcal W}.$ The vector space $\mathcal{J}$ together with the following multiplication $$(a_1+w_1)(a_2+w_2)=a_1a_2+g(w_1,w_2)+a_1w_2+a_2w_1$$ for $a_1,a_2\in A$ and $w_1,w_2\in{\mathcal W}$ is a Jordan algebra called a {\it Clifford Jordan algebra.} For $a,b\in\mathcal{J},$ define $D_{a,b}:=-[{\bf L}_a,{\bf L}_b]:={\bf L}_b{\bf L}_a-{\bf L}_a{\bf L}_b$ where ${\bf L}_a,{\bf L}_b$ are left multiplications by $a$ and $b$ respectively. For a subspace $V$ of $\mathcal{J},$ set $D_{V,V}$ to be the subspace of endomorphisms of $\mathcal{J}$ spanned by $D_{a,b}$ for $a,b\in V.$ One can see that for $w_1,w_2\in{\mathcal W},$ $D_{w_1,w_2}$ can be identified with
$D_{w_1,w_2}|_{_{\mathcal W}}.$ This allows us to consider $D_{{\mathcal W},{\mathcal W}}$ as a subalgebra of $\mathfrak{gl}({\mathcal W}).$} \end{Definition}
\begin{Lemma}[{Classical Lie algebras of type $B$}]
\begin{comment}
Now suppose that $0$ is a symbol and $I$ is an index set. Suppose $\bar I$ is a copy of $I$ and denote its elements with $\bar i,$ $i\in I.$ Let ${\mathcal V}$ be a vector space with a basis $\{v_j\mid j\in J:=\{0\}\cup I\cup\bar{I}\}.$ Consider the bilinear form $(\cdot,\cdot)$ on ${\mathcal V}$ defined by $$\begin{array}{c}(v_j,v_{\bar k})=(v_{\bar k},v_j)=2\delta_{j,k},\; (v_0,v_0)=2,\\ (v_j,v_k)=(v_j,v_0)=(v_0,v_j)=(v_0,v_{\bar j})=(v_{\bar j},v_0)=(v_{\bar j},v_{\bar k})=0;\;j,k\in I.\end{array}$$ Set $$\mathfrak{o}_B({\mathcal V}):=\{\phi\in\mathfrak{gl}({\mathcal V})\mid (\phi(v),w)=-(v,\phi(w)),\; \hbox{for all $v,w\in{\mathcal V}$}\}.$$
Also if ${\mathcal V}$ is a vector space with a basis $\{v_j\mid j\in J:= I\cup\bar{I}\}.$ Consider the bilinear form $(\cdot,\cdot)$ on ${\mathcal V}$ defined by $$(v_j,v_{\bar k})=(v_{\bar k},v_j)=2\delta_{j,k},\; (v_j,v_k)=0,\; (v_{\bar j},v_{\bar k})=0$$ and set $$\mathfrak{o}_D({\mathcal V}):=\{\phi\in\mathfrak{gl}({\mathcal V})\mid (\phi(v),w)=-(v,\phi(w)),\; \hbox{for all $v,w\in{\mathcal V}$}\}.$$
Finally suppose that ${\mathcal V}$ is a vector space with a basis $\{v_j\mid j\in J:= I\cup\bar{I}\}$ and consider the bilinear form $(\cdot,\cdot)$ on ${\mathcal V}$ defined by $$(v_j,v_{\bar k})=(v_{\bar k},v_j)=-2\delta_{j,k},\; (v_j,v_k)=0,\; (v_{\bar j},v_{\bar k})=0$$ and set $$\mathfrak{sp}({\mathcal V}):=\{\phi\in\mathfrak{gl}({\mathcal V})\mid (\phi(v),w)=-(v,\phi(w)),\; \hbox{for all $v,w\in{\mathcal V}$}\}.$$ \end{comment}
\label{type-b-alg} Suppose that $I$ is a non-empty index set. Take $J:=\{0\}\uplus I\uplus\bar{I}$ and consider the vector space ${\mathcal V}:={\mathcal V}_J$ as before. Define the bilinear form $(\cdot,\cdot)$ on ${\mathcal V}$ by \begin{equation}\label{form-b-alg}\begin{array}{c}(v_j,v_{\bar k})=(v_{\bar k},v_j)=2\delta_{j,k},\; (v_0,v_0)=2,\\ (v_j,v_k)=(v_j,v_0)=(v_0,v_j)=(v_0,v_{\bar j})=(v_{\bar j},v_0)=(v_{\bar j},v_{\bar k})=0;\;j,k\in I,\end{array}\end{equation} and set $${\mathcal G}:=\mathfrak{o}_B(I):=\{\phi\in\mathfrak{gl}(J)\mid (\phi(v),w)=-(v,\phi(w)),\; \hbox{for all $v,w\in{\mathcal V}$}\}.$$ Then we have the following:
(i) ${\mathcal G}$ is a locally finite split simple Lie subalgebra of $\mathfrak{gl}(J)$ with splitting Cartan subalgebra ${\mathcal H}:=\hbox{span}_\mathbb{F}\{h_i:=e_{i,i}-e_{\bar i,\bar i}\mid i\in I\}$ and corresponding root system isomorphic to $B_I.$ Moreover for $i,j\in J$ with $i\neq j,$ we have $$\begin{array}{c}{\mathcal G}_{\epsilon_i-\epsilon_j}=\mathbb{F} (e_{i,j}-e_{\bar j,\bar i}),\;{\mathcal G}_{\epsilon_i+\epsilon_j}=\mathbb{F} (e_{i,\bar j}-e_{ j,\bar i}),\;{\mathcal G}_{-\epsilon_i-\epsilon_j}=\mathbb{F} (e_{\bar i,j}-e_{\bar j,i})\\ {\mathcal G}_{\epsilon_i}=\mathbb{F} (e_{i,0}-e_{0,\bar i}),\;{\mathcal G}_{-\epsilon_i}=\mathbb{F} (e_{\bar i,0}-e_{0,i}).\end{array}$$
(ii) For the Clifford Jordan algebra $\mathcal{J}((\cdot,\cdot),{\mathcal V}),$ we have ${\mathcal G}=D_{{\mathcal V},{\mathcal V}}.$ \begin{comment} $D_{v_i,v_j}$ corresponds to $e_{i,\bar j}-e_{j,\bar i}$ $D_{v_i,v_{\bar j}}$ corresponds to -$e_{i, j}+e_{\bar j,\bari}$ $D_{v_{\bar i},v_{\bar j}}$ corresponds to $e_{\bar i, j}-e_{\bar j, i}$ $D_{v_{\bar i},v_{0}}$ corresponds to $e_{\bar i, 0}-e_{0 ,i}$ $D_{v_{i},v_{0}}$ corresponds to $e_{ i, 0}-e_{0 ,\bar i}$ $D_{v_{i},v_{\bar i}}$ corresponds to $e_{\bar i, i}-e_{i, i}$ \end{comment}
(iii) For a fixed subset $I_0$ of $I,$ take $\Lambda$ to be an index set containing $0$ such that $\{I_\lambda\mid \lambda\in\Lambda\}$ is the class of all finite subsets of $I$ containing $I_0.$ For each $\lambda\in \Lambda,$ set \begin{equation}\label{simple-b-alg}\begin{array}{l}{\mathcal G}_{_{I_\lambda}}:={\mathcal G}^\lambda:={\mathcal G}\cap\hbox{span}\{e_{r,s}\mid r,s\in \{0\}\cup I_\lambda\cup\bar I_\lambda\}. \end{array}\end{equation} Then ${\mathcal G}^\lambda$ ($\lambda\in\Lambda$) is a finite dimensional split simple Lie subalgebra of ${\mathcal G}$ of type $B,$ with splitting Cartan subalgebra ${\mathcal H}^\lambda:={\mathcal H}\cap{\mathcal G}^\lambda$ and ${\mathcal G}$ is the direct union of $\{{\mathcal G}^\lambda\mid \lambda \in \Lambda\}.$
\end{Lemma} \begin{Lemma} [Classical Lie algebras of type $D$] \label{type-d-alg} Suppose that $I$ is a non-empty index set and $I_0$ is a fixed subset of $I.$ Set $J:=I\uplus\bar{I}$ and take $\{I_\lambda\mid \lambda\in\Lambda\},$ where $\Lambda$ is an index set containing $0,$ to be the class of all finite subsets of $I$ containing $I_0.$ Define the bilinear form $(\cdot,\cdot)$ on ${\mathcal V}={\mathcal V}_J$ by \begin{equation}\label{form-d-alg}(v_j,v_{\bar k})=(v_{\bar k},v_j)=2\delta_{j,k},\; (v_j,v_k)=(v_{\bar j},v_{\bar k})=0;\;(j,k\in I),\end{equation} and set $$\begin{array}{l}{\mathcal G}:=\mathfrak{o}_D(I):=\{\phi\in\mathfrak{gl}(J)\mid (\phi(v),w)=-(v,\phi(w)),\; \hbox{for all $v,w\in{\mathcal V}$}\},\\ {\mathcal H}:=\hbox{span}_\mathbb{F}\{h_i:=e_{i,i}-e_{\bar i,\bar i}\mid i\in I\}.\end{array} $$Also for $\lambda\in\Lambda,$ take $${\mathcal G}_{_{I_\lambda}}:={\mathcal G}^\lambda:={\mathcal G}\cap\hbox{span}\{e_{r,s}\mid r,s\in I_\lambda\cup\bar I_\lambda\}.$$ Then ${\mathcal G}$ is a locally finite split simple Lie subalgebra of $\mathfrak{gl}(J)$ with splitting Cartan subalgebra ${\mathcal H}$ and corresponding root system isomorphic to $D_I.$ Moreover for $i,j\in J$ with $i\neq j,$ we have $${\mathcal G}_{\epsilon_i-\epsilon_j}=\mathbb{F} (e_{i,j}-e_{\bar j,\bar i}),\;{\mathcal G}_{\epsilon_i+\epsilon_j}=\mathbb{F} (e_{i,\bar j}-e_{ j,\bar i}),\;{\mathcal G}_{-\epsilon_i-\epsilon_j}=\mathbb{F} (e_{\bar i,j}-e_{\bar j,i}).$$ Also for each $\lambda\in \Lambda,$ ${\mathcal G}^\lambda$ is a finite dimensional split simple Lie subalgebra of ${\mathcal G},$ of type $D,$ with splitting Cartan subalgebra ${\mathcal H}^\lambda:={\mathcal H}\cap{\mathcal G}^\lambda,$ and ${\mathcal G}$ is the direct union of $\{{\mathcal G}^\lambda\mid \lambda \in \Lambda\}.$
\end{Lemma}
\begin{Lemma} [Classical Lie algebras of type $C$]\label{type-c-alg} Suppose that $I$ is a non-empty index set and $J:= I\uplus\bar{I}.$ Consider the bilinear form $(\cdot,\cdot)$ on ${\mathcal V}={\mathcal V}_J$ defined by\begin{equation}\label{form-c}(v_j,v_{\bar k})=-(v_{\bar k},v_j)=2\delta_{j,k},\; (v_j,v_k)=0,\; (v_{\bar j},v_{\bar k})=0,\;\;\; (j,k\in I),\end{equation}and set $${\mathcal G}:=\mathfrak{sp}(I):=\{\phi\in\mathfrak{gl}(J)\mid (\phi(v),w)=-(v,\phi(w)),\; \hbox{for all $v,w\in{\mathcal V}$}\}.$$ Also for a fixed subset $I_0$ of $I,$ take $\{I_\lambda\mid \lambda\in\Lambda\}$ to be the class of all finite subsets of $I$ containing $I_0,$ in which $\Lambda$ is an index set containing $0,$ and for each $\lambda\in \Lambda,$ set \begin{equation}\label{simple-c-alg}{\mathcal G}_{_{I_\lambda}}:={\mathcal G}^\lambda:={\mathcal G}\cap\hbox{span}\{e_{r,s}\mid r,s\in I_\lambda\cup\bar I_\lambda\}. \end{equation}
Then ${\mathcal G}$
is a locally finite split simple Lie subalgebra of $\mathfrak{gl}(J)$ with splitting Cartan subalgebra ${\mathcal H}:=\hbox{span}_\mathbb{F}\{h_i:=e_{i,i}-e_{\bar i,\bar i}\mid i\in I\}.$ Moreover for $i,j\in I$ with $i\neq j,$ we have $$\begin{array}{c}{\mathcal G}_{\epsilon_i-\epsilon_j}=\mathbb{F} (e_{i,j}-e_{\bar j,\bar i}),\;{\mathcal G}_{\epsilon_i+\epsilon_j}=\mathbb{F} (e_{i,\bar j}+e_{ j,\bar i}),\;{\mathcal G}_{-\epsilon_i-\epsilon_j}=\mathbb{F} (e_{\bar i,j}+e_{\bar j,i})\\ {\mathcal G}_{2\epsilon_i}=\mathbb{F} e_{i,\bar i},\;{\mathcal G}_{-2\epsilon_i}=\mathbb{F} e_{\bar i,i}.\end{array}$$ Also for $\lambda\in\Lambda,$ ${\mathcal G}^\lambda$ is a finite dimensional split simple Lie subalgebra of type $C,$ with splitting Cartan subalgebra ${\mathcal H}^\lambda:={\mathcal H}\cap{\mathcal G}^\lambda,$ and
${\mathcal G}$ is the direct union of $\{{\mathcal G}^\lambda\mid \lambda\in\Lambda\}.$ \end{Lemma} \begin{Proposition}\label{locally-alg}\cite[Theorem VI.7]{NS} Suppose that $I$ is an infinite index set, then $\mathfrak{o}_B(I)$ is isomorphic to $\mathfrak{o}_D(I).$ Moreover if ${\mathcal G}$ is an infinite dimensional locally finite split simple Lie algebra, then ${\mathcal G}$ is isomorphic to exactly one of the Lie algebras $\mathfrak{sl}(I),$ $\mathfrak{o}_B(I)$ or $\mathfrak{sp}(I).$ \end{Proposition}
\begin{Lemma}\label{final3} Suppose that $R$ is an irreducible locally finite root system and $S$ is an irreducible closed subsystem of $R.$ Suppose ${\mathcal G}$ is a locally finite split simple Lie algebra with a splitting Cartan subalgebra ${\mathcal H}$ and the root system $R_{sdiv}.$ Set $\mathfrak{g}:=\sum_{\alpha\in S_{sdiv}^\times}{\mathcal G}_\alpha\oplus\sum_{\alpha\in S_{sdiv}^\times}[{\mathcal G}_\alpha,{\mathcal G}_{-\alpha}]$ and $\mathfrak{h}:={\mathcal H}\cap\mathfrak{g},$ then the restriction of
$$\pi:{\mathcal H}^*\longrightarrow \mathfrak{h}^*;\;\;f\mapsto f|_{\mathfrak{h}},\;\; f\in {\mathcal H}^*$$ to $S$ is injective. Identify $\alpha\in S$ with $\pi(\alpha)$ via $\pi,$ then $\mathfrak{g}$ is a locally finite split simple Lie subalgebra of ${\mathcal G}$ with splitting Cartan subalgebra $\mathfrak{h}$ and corresponding root system $S_{sdiv}.$ \end{Lemma}
\noindent{\bf Proof. } We first claim that \begin{equation}\label{crutial}\parbox{3in}{\it\begin{center} if $\alpha,\beta\in S$ and $\alpha-\beta\not \in R,$ then there is $h\in \mathfrak{h}$ such that $\alpha(h)>0$ and $\beta(h)\leq0.$\end{center}}\end{equation} To prove this, we note that since $\alpha-\beta\not\in R,$ we have $\alpha\neq 0$ and $\beta\neq0 .$ Moreover, it follows from the theory of locally finite root systems that $\beta-2\alpha\not\in R$ and $\alpha-2\beta\not\in R,$ also if $2\alpha\in R$ or $2\beta\in R,$ then $2\alpha-2\beta\not \in R.$ Therefore setting $$\alpha':=\left\{\begin{array}{ll}\alpha&\hbox{ if $2\alpha\not\in R$} \\ 2\alpha&\hbox{ if $2\alpha\in R,$} \end{array}\right.\quad\hbox{and}\quad \beta':=\left\{\begin{array}{ll}\beta&\hbox{ if $2\beta\not\in R$} \\ 2\beta&\hbox{ if $2\beta\in R,$}\end{array}\right. $$ we have $\alpha',\beta'\in S_{sdiv}^\times$ and $\alpha'-\beta'\not\in R.$ Next we fix $e\in{\mathcal G}_{\alpha'}$ and $f\in{\mathcal G}_{-\alpha'}$ such that $(e,h:=[e,f],f)$ is an $\mathfrak{sl}_2$-triple. Since $\alpha'-\beta'\not\in R_{sdiv},$ one knows from $\mathfrak{sl}_2-$module theory that $\beta'(h)\leq 0$ while $\alpha'(h)=2>0.$ Therefore $h\in[{\mathcal G}_{\alpha'},{\mathcal G}_{-\alpha'}]\subseteq\mathfrak{h},$ $\alpha(h)>0$ and $\beta(h)\leq 0.$ \begin{comment} $\sum_{k=-\infty}^\infty{\mathcal G}_{\beta+k\alpha}$ is a ${\mathcal G}_\alpha+[{\mathcal G}_\alpha,{\mathcal G}_{-\alpha}]+{\mathcal G}_{-\alpha}-$module. Now use $\mathfrak{sl}_2-$module theory. \end{comment} This completes the proof of the claim. Now suppose $\alpha,\beta\in S$ with $\pi(\alpha)=\pi(\beta).$ We must show $\alpha=\beta.$ We prove this through the following three cases:
\underline{\bf Case 1. $\alpha,\beta\in S_{sdiv}:$} If $\gamma:=\alpha-\beta\in R^\times,$ then since $S$ is a closed subsystem of $R$ and $S_{sdiv}$ is a closed subsystem of $S,$ we get $\gamma\in S_{sdiv}^\times.$ Thus there is $t\in[{\mathcal G}_\gamma,{\mathcal G}_{-\gamma}]\subseteq\mathfrak{h}$ with $\gamma(t)=2,$ so $(\alpha-\beta)(t)=2$ which contradicts the fact that $\alpha\mid_{\mathfrak{h}}=\beta\mid_{\mathfrak{h}}.$ Therefore $\alpha-\beta\not\in R^\times. $ Now if $\alpha-\beta\neq0,$ then $\alpha-\beta\not\in R$ and so using (\ref{crutial}), one finds $h\in\mathfrak{h}$ with $\alpha(h)>0$ and $\beta(h)\leq0.$ This is again a contradiction. Therefore $\alpha=\beta.$
\underline{\bf Case 2. $\alpha,\beta\not\in S_{sdiv}:$} In this case $2\alpha,2\beta\in S_{sdiv}$ and so by Case 1, $2\alpha=2\beta$ which in turn implies that $\alpha=\beta.$
\underline{\bf Case 3. $\alpha\in S_{sdiv}, \beta\not\in S_{sdiv}:$} If $\alpha-\beta\not\in R,$ then by (\ref{crutial}), there is $h\in \mathfrak{h}$ such that $\alpha(h)>0$ and $\beta(h)\leq 0$ which contradicts the fact that $\pi(\alpha)=\pi(\beta).$ Also if $\alpha=2\beta,$ then since $\alpha\in S_{sdiv}^\times,$ there is $h\in[{\mathcal G}_\alpha,{\mathcal G}_{-\alpha}]\subseteq\mathfrak{h}$ with $\alpha(h)=2.$ Thus $\alpha(h)\neq \beta(h)$ which is again a contradiction. Therefore $\alpha-\beta\in R$ and $\alpha\neq 2\beta.$ Now if $\alpha-\beta\neq0,$ we get that $\alpha-\beta\in R_{sh},$ $\alpha\in R_{lg},$ $\beta\in R_{sh},$ $\gamma:=\alpha-2\beta\in R_{lg},$ $\alpha+\gamma,\alpha-\gamma\in R_{ex}$ and $\alpha+2\gamma,\alpha-2\gamma\not\in R.$ Now since $\gamma\in S_{sdiv}^\times,$ there is $h\in[{\mathcal G}_\gamma,{\mathcal G}_{-\gamma}]\subseteq\mathfrak{h}$ with $\gamma(h)=2.$ Also since $\alpha+2\gamma,\alpha-2\gamma\not\in R,$ one concludes form $\mathfrak{sl}_2-$module theory that $\alpha(h)=0.$ \begin{comment} $\sum_{k=-\infty}^\infty{\mathcal G}_{k\gamma+\alpha}$ is a ${\mathcal G}_\gamma+[{\mathcal G}_\gamma,{\mathcal G}_{-\gamma}]+{\mathcal G}_{-\gamma}-$module. Now $\mathfrak{sl}_2-$module theory implies that $\alpha(h)=0.$ \end{comment} So we have $\beta(h)=\alpha(h)=0.$ But this gives that $2=\gamma(h)=(\alpha-2\beta)(h)=0,$ a contradiction. Thus we have $\alpha=\beta.$ This completes the proof of the first assertion.
For the last assertion, we note that $S$ is a closed subsystem of $R$ and $S_{sdiv}$ is a closed subsystem of $S,$ so it is easily seen that $\mathfrak{g}$ is a subalgebra of ${\mathcal G}.$ Now this together with the fact that
$\pi|_S$ is injective completes the proof.\qed
\begin{Definition}\label{direct-limit} {\rm Take ${\mathcal G},$ $\Lambda,$ ${\mathcal G}^\lambda,$ and ${\mathcal H}^\lambda$ $(\lambda\in\Lambda)$ to be as in one of Lemmas \ref{type-a-alg}, \ref{type-b-alg}, \ref{type-d-alg} and \ref{type-c-alg}. For $\lambda,\mu\in\Lambda,$ we say $\lambda\preccurlyeq\mu$ if ${\mathcal G}^\lambda$ is a subalgebra of ${\mathcal G}^\mu.$ Let $\chi$ be a representation of ${\mathcal G}$ in a vector space $\mathcal{M}.$ We say $\mathcal{M}$ is a {\it direct limit ${\mathcal G}$-module with directed system} $\{\mathcal{M}^\lambda\mid \lambda\in\Lambda\}$ if \begin{itemize} \item for $\lambda,\mu\in\Lambda$ with $\lambda\preccurlyeq\mu,$ $\mathcal{M}^\lambda\subseteq\mathcal{M}^\mu\subseteq\mathcal{M}$ and as a vector space, $\mathcal{M}$ is the direct union of $\{\mathcal{M}^\lambda\mid\lambda\in\Lambda\},$ \item for $\lambda\in\Lambda,$ $\mathcal{M}^\lambda$ is finite dimensional and for all $x\in{\mathcal G}^\lambda,$ $\mathcal{M}^\lambda$ is invariant under $\chi(x),$ \item for $\lambda\in\Lambda,$ $\chi\mid_{{\mathcal G}^\lambda}$ defines a nontrivial finite dimensional irreducible ${\mathcal G}^\lambda-$module in $\mathcal{M}^\lambda$ having a weight space decomposition with respect to ${\mathcal H}^\lambda$ whose weight spaces corresponding to nonzero weights are one dimensional, \item for $\lambda,\mu\in\Lambda$ with $\lambda\preccurlyeq\mu,$ the set of weights of ${\mathcal G}^\lambda-$module $\mathcal{M}^\lambda$ is contained in the set of weights of ${\mathcal G}^\mu-$module $\mathcal{M}^\mu$ restricted to ${\mathcal H}^\lambda$ and
$(\mathcal{M}^\lambda)_{p|_{_{{\mathcal H}^\lambda}}}=(\mathcal{M}^\mu)_p$ for each weight $p$ of $\mathcal{M}^\mu$ for which $p|_{_{{\mathcal H}^\lambda}}$ is a nonzero weight
of $\mathcal{M}^\lambda.$ \end{itemize}} \end{Definition} Using standard techniques, one can verify the following propositions: \begin{Proposition}\label{dir-lim-mod} Consider ${\mathcal G},$ $\Lambda,$ ${\mathcal G}^\lambda$ and ${\mathcal H}^\lambda$ $(\lambda\in\Lambda)$ as in Definition \ref{direct-limit}. Suppose that $\mathcal{M}$ is a direct limit ${\mathcal G}-$module with directed system $\{\mathcal{M}^\lambda\mid\lambda\in\Lambda\}.$ Then we have the followings:
(i) $\mathcal{M}$ is an irreducible ${\mathcal G}-$module.
(ii) If ${\mathcal W}$ is another direct limit ${\mathcal G}-$module with directed system $\{{\mathcal W}^\lambda\mid \lambda\in\Lambda\},$ and for each $\lambda\in\Lambda,$ ${\mathcal G}^\lambda-$modules $\mathcal{M}^\lambda$ and ${\mathcal W}^\lambda$ are isomorphic, then as two ${\mathcal G}-$modules, $\mathcal{M}$ and ${\mathcal W}$ are isomorphic. \end{Proposition}
\begin{Proposition} \label{rep-local} Suppose that $I$ is a nonempty index set.
(a) Take ${\mathcal G}:=\mathfrak{o}_B(I)$ and use the same notations as in Lemma \ref{type-b-alg}. Define $$\pi:{\mathcal G}\longrightarrow End({\mathcal V});\;\pi(\phi)(v)=\phi(v);\;\;\phi\in{\mathcal G},\; v\in {\mathcal V},$$ then
(i) $\pi$ is an irreducible representation of ${\mathcal G}$ in ${\mathcal V}$ equipped with a weight space decomposition with respect to ${\mathcal H}$ whose set of weights is $\{0,\pm\epsilon_i\mid i\in I\}$ with ${\mathcal V}_0=\mathbb{F} v_0,$ ${\mathcal V}_{\epsilon_i}=\mathbb{F} v_i$ and ${\mathcal V}_{-\epsilon_i}=\mathbb{F} v_{\bar i}$ for $i\in I,$
(ii) for each $\lambda\in \Lambda,$ set \begin{equation}\label{simple-b-mod}{\mathcal V}_{_{I_\lambda}}:={\mathcal V}^\lambda:=\hbox{span}_\mathbb{F}\{v_r\mid r\in \{0\}\cup I_\lambda\cup\bar I_\lambda\},\end{equation} then ${\mathcal V}$ is a direct limit ${\mathcal G}-$module with directed system $\{{\mathcal V}^\lambda\mid \lambda \in \Lambda\}.$
(b) Use the same notations as in Lemma \ref{type-c-alg} and take ${\mathcal G}:=\mathfrak{sp}(I).$
(i) Define $$\pi_1:{\mathcal G}\longrightarrow End({\mathcal V});\;\pi(\phi)(v):=\phi(v);\;\;\phi\in{\mathcal G},\; v\in {\mathcal V}.$$ Then $\pi_1$ is an irreducible representation of ${\mathcal G}$ in ${\mathcal V}$ equipped with a weight space decomposition with respect to ${\mathcal H}$ whose set of weights is $\{\pm\epsilon_i\mid i\in I\}$ with ${\mathcal V}_{\epsilon_i}=\mathbb{F} v_i$ and ${\mathcal V}_{-\epsilon_i}=\mathbb{F} v_{\bar i}$ for $i\in I.$ Also for $J:=I\cup\bar{I}$ and \begin{equation}\label{module-s-c}\mathcal{S}:=\{\phi\in \mathfrak{gl}(J)\mid tr(\phi)=0,(\phi(v),w)=(v,\phi(w)),\; \hbox{for all $v,w\in{\mathcal V}$}\},\end{equation} $$\pi_2:{\mathcal G}\longrightarrow End(\mathcal{S});\; \pi_2(X)(Y):=[X,Y];\;\; X\in {\mathcal G},\; Y\in \mathcal{S}$$ is an irreducible representation of ${\mathcal G}$ in $\mathcal{S}$ equipped with a weight space decomposition with respect to ${\mathcal H}$ whose set of weights is $\{0,\pm(\epsilon_i\pm\epsilon_j)\mid i,j\in I,\; i\neq j\}$ with
$\mathcal{S}_0=\hbox{span}_\mathbb{F}\{e_{r,r}+e_{\bar r,\bar r}-\frac{1}{|I_\lambda|}\sum_{i\in I_\lambda} (e_{i,i}+e_{\bar i,\bar i})\mid \lambda\in\Lambda,r\in I_\lambda\},$ $\mathcal{S}_{\epsilon_i+\epsilon_j}=\mathbb{F}(e_{i,\bar j}-e_{j,\bar i}),$ $\mathcal{S}_{-\epsilon_i-\epsilon_j}=\mathbb{F}(e_{\bar i,j}-e_{\bar j, i})$ and $\mathcal{S}_{\epsilon_i-\epsilon_j}=\mathbb{F}(e_{i,j}+e_{\bar j,\bar i})$ ($i,j\in I, i\neq j$).
(ii) For $\lambda\in \Lambda,$ set \begin{equation}\label{simple-c}\begin{array}{l} {\mathcal V}_{_{I_\lambda}}:={\mathcal V}^\lambda:=\hbox{span}_\mathbb{F}\{v_r\mid r\in I_\lambda\cup\bar I_\lambda\},\\\mathcal{S}_{_{I_\lambda}}:=\mathcal{S}^\lambda:=\mathcal{S}\cap\hbox{span}_\mathbb{F}\{e_{r,s}\mid r,s\in I_\lambda\cup\bar I_\lambda\}.\end{array}\end{equation} Then ${\mathcal V}$ is a direct limit ${\mathcal G}-$module with directed system $\{{\mathcal V}^\lambda\mid \lambda \in \Lambda\}$ and $\mathcal{S}$ is a direct limit ${\mathcal G}-$module with directed system $\{\mathcal{S}^\lambda\mid \lambda\in \Lambda\}.$ \end{Proposition}
\subsection{Finite Dimensional Case} In this subsection, we state a proposition on representation theory of finite dimensional split simple Lie algebras. This proposition is an essential tool for the proof of our results in the next section. We start with an elementary but important fact about finite dimensional representations of a finite dimensional split semisimple Lie algebra. \begin{Lemma}\label{elementary} Suppose that ${\mathcal G}$ is a finite dimensional split semisimple Lie algebra with a splitting Cartan subalgebra ${\mathcal H}$ and the root system $R.$ Let ${\mathcal V}$ be a finite dimensional ${\mathcal G}$-module equipped with a weight space decomposition with respect to ${\mathcal H}.$ Take $\Pi$ to be the set of weights of ${\mathcal V}.$ If $\alpha\in R^\times$ and $\lambda\in \Pi$ are such that $\alpha+\lambda\in\Pi,$ then ${\mathcal G}_\alpha\cdot{\mathcal V}_\lambda\neq\{0\}.$ In particular if ${\mathcal V}_{\lambda+\alpha}$ is one dimensional, then ${\mathcal G}_\alpha\cdot{\mathcal V}_\lambda={\mathcal V}_{\lambda+\alpha}.$ \end{Lemma} \noindent{\bf Proof. } Take $e\in{\mathcal G}_\alpha$ and $f\in{\mathcal G}_{-\alpha}$ to be such that $(e,h:=[e,f],f)$ is an $\mathfrak{sl}_2$-triple and define $\mathfrak{s}:=\hbox{span}_\mathbb{F}\{e,h,f\}.$ Set ${\mathcal W}:=\sum_{k=-\infty}^{\infty}{\mathcal V}_{\lambda+k\alpha},$ then ${\mathcal W}$ is a finite dimensional $\mathfrak{s}$-module and so by Weyl theorem, it is decomposed into finite dimensional irreducible $\mathfrak{s}$-modules, say ${\mathcal W}=\oplus_{i=1}^n{\mathcal W}_i$ where $n$ is a positive integer and ${\mathcal W}_i,$ $1\leq i\leq n,$ is a finite dimensional irreducible $\mathfrak{s}$-module. We note that the set of weights of ${\mathcal W}$ with respect to $\mathbb{F} h$ is $\Pi'=\{\lambda(h)+2k\mid k\in{\mathbb Z}\quad\hbox{and}\quad \lambda+k\alpha\in\Pi\}$ and that for $k\in{\mathbb Z}$ with $ \lambda+k\alpha\in\Pi,$ ${\mathcal W}_{\lambda(h)+2k}={\mathcal V}_{\lambda+k\alpha}.$ Now as $\lambda,\lambda+\alpha\in\Pi,$ we have $\lambda(h),\lambda(h)+2\in\Pi'$ and so by $\mathfrak{sl}_2$-module theory, there is $1\leq i\leq n$ such that $\lambda(h),\lambda(h)+2$ are weights for ${\mathcal W}_i.$ Now again using $\mathfrak{sl}_2$-module theory, we get that $$0\neq e\cdot ({\mathcal W}_i)_{\lambda(h)}\subseteq e\cdot {\mathcal W}_{\lambda(h)}= e\cdot {\mathcal V}_\lambda\subseteq{\mathcal G}_\alpha\cdot{\mathcal V}_\lambda$$ showing that ${\mathcal G}_\alpha\cdot{\mathcal V}_\lambda\neq\{0\}.$ The last statement is derived simply from the first assertion.\qed
\begin{Lemma}\label{gen-fact}Suppose that $\{e_i,f_i,h_i\mid 1\leq i\leq n\}$ is a set of Chevalley generators for a finite dimensional split simple Lie algebra ${\mathcal G}$ and ${\mathcal V}$ is a ${\mathcal G}-$module equipped with a weight space decomposition with respect to the Cartan subalgebra ${\mathcal H}:=\hbox{span}\{h_i\mid 1\leq i\leq n\}.$
Let $v$ be a weight vector, $m$ be a positive integer and $1\leq i, j_1,\ldots,j_m\leq n.$ Let the set $\{k\in\{1,\ldots,m\}\mid j_k=i\}$ be a nonempty set and $k_1<\cdots<k_p$ be such that $\{k\in\{1,\ldots,m\}\mid j_k=i\}=\{k_1,\ldots,k_p\}.$ Then if $f_i\cdot v=0,$ we have $$f_i\cdot e_{j_m}\cdot\cdots\cdot e_{j_1}\cdot v\in\sum_{t=1}^p\mathbb{F} e_{j_m}\cdot\cdots e_{j_{k_p}}\cdot\cdots \cdot \widehat{e_{j_{k_t}}}\cdot\cdots\cdot e_{j_{k_1}}\cdot\cdots \cdot e_{j_1}\cdot v,$$ in which $"\widehat{\;\;}\;"$ means omission. \end{Lemma}
\noindent{\bf Proof. } Using Induction on $p,$ we are done.
\qed
\begin{Proposition} \label{gen1} Suppose $R_1$ is an irreducible finite root system and $R_2$ is an irreducible full subsystem of $R_1$ of rank greater that 1. Let ${\mathcal G}_1$ be a finite dimensional split simple Lie algebra with a splitting Cartan subalgebra ${\mathcal H}_1$ and corresponding root system $(R_1)_{sdiv}.$ Set ${\mathcal G}_2:=\sum_{\alpha\in (R_2)_{sdiv}^\times}({\mathcal G}_1)_\alpha\oplus\sum_{\alpha\in (R_2)_{sdiv}^\times}[({\mathcal G}_1)_\alpha,({\mathcal G}_1)_{-\alpha}]$ and ${\mathcal H}_2:={\mathcal H}_1\cap{\mathcal G}_2.$ For $i=1,2,$ assume ${\mathcal V}_i$ is a ${\mathcal G}_i$-module equipped with a weight space decomposition with respect to ${\mathcal H}_i$ and take $\Lambda_i$ ($i=1,2$) to be the set of wights of ${\mathcal V}_i$ with respect to ${\mathcal H}_i.$ Suppose that
(i) $R_1$ and $R_2$ are of the same type $X\neq G_2,F_4,E_{6,7,8}$,
(ii) $\Lambda_1\subseteq R_1$ and $\Lambda_2\subseteq\{\alpha|_{_{{\mathcal H}_2}}\mid \alpha\in R_2\},$
(iii) ${\mathcal V}_2\subseteq{\mathcal V}_1$ with $({\mathcal V}_2)_{\alpha|_{_{{\mathcal H}_2}}}\subseteq({\mathcal V}_1)_\alpha,$
for $\alpha\in \{\beta\in R_2\mid\beta|_{{\mathcal H}_2}\in\Lambda_2\}\setminus\{0\}.$
Let ${\mathcal W}$ be a nontrivial finite dimensional irreducible ${\mathcal G}_2$-submodule of ${\mathcal V}_2$ and take ${\mathcal U}$ to be the ${\mathcal G}_1$-submodule of ${\mathcal V}_1$ generated by ${\mathcal W},$ then ${\mathcal U}$ is a finite dimensional irreducible ${\mathcal G}_1-$module equipped with a weight space decomposition with respect to ${\mathcal H}_1$ whose set of nonzero weights is $(R_1)_{sh},$ (resp. $(R_1)^\times_{sdiv},$ or $((R_1)_{sdiv})_{sh}$) if the set of nonzero weights of ${\mathcal W}$ is the set of elements of $(R_2)_{sh}$ (resp. $(R_2)^\times_{sdiv},$ or $((R_2)_{sdiv})_{sh}$) restricted to ${\mathcal H}_2.$ \end{Proposition} \begin{comment} \begin{rem} {\rm Use the same notations as in Proposition \ref{gen1} and suppose that $\pi:{\mathcal G}_1\longrightarrow End (W)$ is a finite dimensional representation of ${\mathcal G}_1$ in $W,$ then using a standard technic, one can see that if $\lambda$ is a weight for $W,$ $\alpha\in (R_1)_{sdiv}^\times,$ and $e_\alpha\in{\mathcal G}_\alpha,$ $f_\alpha\in{\mathcal G}_{-\alpha}$ are such that $(e_\alpha,[e_\alpha,f_\alpha],f_\alpha)$ is an $\mathfrak{sl}_2-$triple, then $\pi(e_\alpha)$ and $\pi(f_\alpha)$ are nilpotent endomorphisms and \begin{equation}\label{rev3}exp(\pi(e_\alpha))exp(-\pi(f_\alpha))exp(\pi(e_\alpha)) W_\lambda\subseteq W_{\lambda-\lambda(h)\alpha}.\end{equation} Now as $\Lambda_1\subseteq R_1,$ ${\mathcal V}_1$ is decomposed into finite dimensional irreducible ${\mathcal G}_1-$submodule and so if $\lambda$ is a weight for ${\mathcal V}_1,$ (\ref{rev3}) implies that
$$exp(\pi(e_\alpha))exp(-\pi(f_\alpha))exp(\pi(e_\alpha)) ({\mathcal V}_1)_\lambda\subseteq ({\mathcal V}_1)_{\lambda-\lambda(h)\alpha}.$$ Now it follows that if $\gamma\in R_2$ is such that $\gamma|_{{\mathcal H}_2}$ is a weight for ${\mathcal W},$ then for each $\eta\in R_2 $ with the same length as $\gamma,$ $\eta|_{{\mathcal H}_2}$ is a weight for ${\mathcal W}.$}
Suppose that $W$ is a finite dimensional irreducible ${\mathcal G}_1-$module. For each $1\leq i\leq n,$ $ad(e_i)$ is locally nilpotent as the set of weights of $W$ is finite and as $W$ is finite dimensional $ad(e_i)$ is nilpotent on $W.$ Similarly, $ad(f_i)$ is nilpotent on $W.$ Therefore if $\phi$ is the corresponding representation of the ${\mathcal G}_1-$module $W,$ we have for $h\in{\mathcal H}_1$ $$exp(\phi(e_i))exp(\phi(-f_i))exp(\phi(e_i))\phi(h)exp(\phi(-e_i))exp(\phi(f_i))exp(\phi(-e_i))=\phi(s{\alpha_i}(h))$$ [Hum]. This in particular implies that $$exp(\phi(e_i))exp(\phi(-f_i))exp(\phi(e_i)) W_\lambda\subseteq W_{s_{\alpha_i}(\lambda)}.$$ Now it leads to $exp(\phi(e_i))exp(\phi(-f_i))exp(\phi(e_i)) (V_1)_\lambda\subseteq (V_1)_{s_{\alpha_i}(\lambda)}.$
Suppose that we use the notations of this proposition. If $\gamma\in R_2$ and $\gamma|_{{\mathcal H}_2}$ is the highest weight for ${\mathcal W}$ and $\gamma$ is a short root, then ${\mathcal W}_{\gamma|_{{\mathcal H}_2}}\subseteq (V_2)_{\gamma|_{{\mathcal H}_2}}\subseteq (V_1)_{\gamma}.$ So for each $1\leq i\leq \ell,$ $$\{0\}\neq exp(\phi(e_i))exp(\phi(-f_i))exp(\phi(e_i))({\mathcal W}_{\gamma|_{{\mathcal H}_2}})\in {\mathcal W}\cap(V_1)_{s_{\alpha_i}(\gamma)}.$$ Now as $exp(\phi(e_i))exp(\phi(-f_i))exp(\phi(e_i))({\mathcal W}_{\gamma|_{{\mathcal H}_2}})$ is a weight space, $$exp(\phi(e_i))exp(\phi(-f_i))exp(\phi(e_i))({\mathcal W}_{\gamma|_{{\mathcal H}_2}})\subseteq {\mathcal W}_{s_{\alpha_i}(\gamma)|_{{\mathcal H}_2}}.$$ This means that the restriction of any short root to ${\mathcal H}_2$ is a weight. Now as $\gamma|_{{\mathcal H}_2}$ is identified with $\gamma$ for each $\gamma\in R_2,$ the highest weight $W$ is the restriction of the highest short root to ${\mathcal H}_2.$
\label{rem2} \end{rem} \end{comment} \noindent{\bf Proof. } Take $n:=n_1$ and $\ell:=n_2$ to be the rank of $R_1$ and $R_2$
respectively. Using Lemma \ref{final3}, we identify $\beta\in R_2$ with $\beta|_{{\mathcal H}_2}.$ Also without loss of generality, we assume $R_1,R_2$ and bases $\Delta _1,\Delta_2$ for $(R_1)_{sdiv}$ and $(R_2)_{sdiv}$ respectively are as in the following tables:$${\small
\begin{tabular}{|c|c|} \hline \hbox{ Type }& \hbox{ $R_k(k=1,2)$ }\\ \hline $ A$& $\{\pm(\varepsilon_i-\varepsilon_j)\mid 1\leq i<j\leq n_k+1\}\cup\{0\},$ $n_k\geq2$\\ \hline $B$&$\{\pm\varepsilon_i,\pm(\varepsilon_i\pm \varepsilon_j)\mid 1\leq i<j\leq n_k\}\cup\{0\},$ $n_k\geq2$\\ \hline $C$&$\{\pm2\varepsilon_i,\pm(\varepsilon_i\pm \varepsilon_j)\mid 1\leq i<j\leq n_k\}\cup\{0\},$ $n_k\geq3$\\ \hline $D$ &$\{\pm(\varepsilon_i\pm \varepsilon_j)\mid 1\leq i<j\leq n_k\}\cup\{0\},$ $n_k\geq4$\\ \hline $BC$&$\{\pm\varepsilon_i,\pm(\varepsilon_i\pm \varepsilon_j)\mid 1\leq i,j\leq n_k\},$ $n_k\geq2$\\ \hline \end{tabular} }$$
$$ {\small
\begin{tabular}{|c|c|} \hline \hbox{ Type }& \hbox{ $\Delta_k(k=1,2)$ }\\ \hline $ A$& $\{\alpha_i:=\epsilon_{i+1}-\epsilon_{i}\mid 1\leq i\leq n_k\}$\\ \hline $B$&$\{\alpha_1:=\epsilon_1,\alpha_i:=\epsilon_i-\epsilon_{i-1}\mid 2\leq i\leq n_k\}$\\ \hline $C$&$\{\alpha_1:=2\epsilon_1,\alpha_i:=\epsilon_i-\epsilon_{i-1}\mid 2\leq i\leq n_k\}$\\ \hline $D$ &$\{\alpha_1:=\epsilon_1+\epsilon_2,\alpha_i:=\epsilon_i-\epsilon_{i-1}\mid 2\leq i\leq n_k\}$\\ \hline $BC$&$\{\alpha_1:=2\epsilon_1,\alpha_i:=\epsilon_i-\epsilon_{i-1}\mid 2\leq i\leq n_k\}$\\ \hline \end{tabular} } $$
\begin{comment} Since the set of weights of ${\mathcal V}_1$ is a subset of $ R_1.$ If $\{e_i,h_i,f_i\mid 1\leq i\leq n\}$ is a set of Chevalley generators and $\lambda$ is a weight and $v\in({\mathcal V}_1)_\lambda,$ the the $\mathfrak{g}_1-$module generated by $v$ is \begin{equation}\sum_{t,s\in\mathbb{N}}\mathbb{F} (f_{i_t}\cdot\cdots\cdot f_{i_1}\cdot e_{j_s}\cdot \cdots\cdot e_{j_1}\cdot v)\end{equation}
Now if $\beta\in R_1,$ $\mu$ is a weight and $w\in ({\mathcal V}_1)_\mu$ and $ e_{j_s}\cdot \cdots\cdot e_{j_1}\cdot w\in ({\mathcal V}_1)_{\beta},$ then $\beta=\mu+\sum_{k=1}^s\alpha_{j_k}.$ Now since $\{\alpha_i\mid 1\leq i\leq n\}$ is a basis for $\hbox{span}_\mathbb{F} R_1,$ there are a finitely many $\{\alpha_{j_1},\cdots,\alpha_{j_s}\}$ such that $\beta=\lambda+\sum_{k=1}^s\alpha_{j_k}.$ Similarly $ f_{i_t}\cdot \cdots\cdot f_{i_1}\cdot w)\in ({\mathcal V}_1)_{\beta},$ then $\beta=\mu-\sum_{k=1}^t\alpha_{i_k}.$ Now since $\{\alpha_i\mid 1\leq i\leq n\}$ is a basis for $\hbox{span}_\mathbb{F} R_1,$ there are a finitely many $\{\alpha_{i_1},\cdots,\alpha_{i_t}\}$ such that $\beta=\lambda-\sum_{k=1}^t\alpha_{i_k}.$ Therefore for any $\gamma\in R_1,$ there are finitely many $\{\alpha_{j_1},\cdots,\alpha_{j_s}\}$ such that $e_{j_s}\cdot \cdots\cdot e_{j_1}\cdot v\in {\mathcal V}_{\gamma}$ and for each $\eta\in R_1$ and each $\{\alpha_{j_1},\cdots,\alpha_{j_s}\},$ there are finitely many $\{\alpha_{i_1},\cdots,\alpha_{i_t}\}$ such that $f_{i_t}\cdot\cdots\cdot f_{i_1}\cdot e_{j_s}\cdot \cdots\cdot e_{j_1}\cdot v\in {\mathcal V}_\eta.$ Therefore \begin{equation}\sum_{t,s\in\mathbb{N}}\mathbb{F} (f_{i_t}\cdot\cdots\cdot f_{i_1}\cdot e_{j_s}\cdot \cdots\cdot e_{j_1}\cdot v)\end{equation} is finite dimensional. So ${\mathcal V}$ can be written as a direct sum of finite dimensional irreducible ${\mathcal G}_1-$module. \end{comment}
Suppose that ${\mathcal W}$ is of highest pair $(v,\alpha)$ with respect to $\Delta_2.$ Since the set of weights of ${\mathcal W}$ is permuted by the Weyl group of $R_2,$ one gets that $\alpha=\alpha_{*}^2$ for $*\in\{sh,lg,ex\}$ where for $i=1,2,$ $\alpha^i_{sh}$ (resp. $\alpha^i_{lg},$ or $\alpha^i_{ex}$) denotes the highest short (resp. long, or extra long) root of $R_i$ with respect to $\Delta_i.$ Next suppose that $\{e_i,f_i,h_i\mid 1\leq i\leq n\}$ is a set of Chevalley generators for ${\mathcal G}_1$ with respect to $\Delta_1,$ then $\{e_i,f_i,h_i\mid 1\leq i\leq \ell\}$ is a set of Chevalley generators for ${\mathcal G}_2$ with respect to $\Delta_2.$ Now as ${\mathcal U}$ is a ${\mathcal G}_1-$submodule of ${\mathcal V}_1$ generated by $v,$ we have \begin{equation}\label{tavakol1}{\mathcal U}=\sum_{t,s\in\mathbb{N}}\mathbb{F} (f_{i_t}\cdot\cdots\cdot f_{i_1}\cdot e_{j_s}\cdot \cdots\cdot e_{j_1}\cdot v)\end{equation} where $i_1,\ldots,i_t,j_1,\ldots,j_s\in\{1,\ldots,n\}.$ This implies that ${\mathcal U}$ is finite dimensional as $\Lambda_1$ is a subset of the finite root system $ R_1.$ So there are a positive integer $p$ and irreducible finite dimensional ${\mathcal G}_1-$submodules ${\mathcal U}_j$ ($1\leq j\leq p$) of ${\mathcal U}$ such that ${\mathcal U}=\oplus_{j=1}^p{\mathcal U}_j.$ But we know that $v$ generates the ${\mathcal G}_1-$submodule $ {\mathcal U},$ and that $v\in {\mathcal U}\cap({\mathcal V}_2)_\alpha\subseteq{\mathcal U}\cap({\mathcal V}_1)_\alpha={\mathcal U}_\alpha= \oplus_{j=1}^p({\mathcal U}_j)_\alpha,$ so
\begin{equation}\label{gen3} \parbox{2.6in}{for any $1\leq j\leq p,$ there is a nonzero element $u_j\in({\mathcal U}_j)_\alpha$ such that $v=\sum_{j=1}^pu_j$.} \end{equation} This in particular implies that each ${\mathcal U}_j$ $(1\leq j\leq p)$ is a nontrivial irreducible ${\mathcal G}_1-$module. But we know that for $1\leq j\leq p,$ the set of weights of ${\mathcal U}_j$ is a subset of $R_1,$ and that it is permuted by the Weyl group of $R_1,$ so the highest weight of ${\mathcal U}_j$ is $\alpha_{*}^1$ for $*=sh,lg,ex.$ Therefore using the finite dimensional theory, one knows that \begin{equation}\label{one-dim}\parbox{4.5in}{ \begin{center}{\it the weight spaces of ${\mathcal U}_j$ ($1\leq j\leq p$ ) with respect to ${\mathcal H}_1$ corresponding to nonzero weights are one dimensional.}\end{center}} \end{equation}
Now we are ready to proceed with the proof in the following three steps: \begin{itemize} \item{Step 1.} $ {\mathcal U}_t$ ($1\leq t\leq p$) is a finite dimensional irreducible ${\mathcal G}_1-$module of highest weight $\alpha_*^1$ if $\alpha=\alpha_*^2$ for $*=sh,lg,ex.$ \item{Step 2.} $\dim( {\mathcal U}_\alpha)=1,$ \item{Step 3.} $p=1.$ \end{itemize}
\noindent\textbf{Step 1:} We use a case-by-case argument to prove the desired point. We note that there is nothing to show if $R_1$ is of type $A$ or $D,$ and continue as following:
\textbf{\emph{\underline{Type $B:$}}} One can see that in this case $\alpha_{sh}^1=\epsilon_n,$ $\alpha_{sh}^2=\epsilon_\ell,$ $\alpha_{lg}^1=\epsilon_{n}+\epsilon_{n-1}$ and $\alpha_{lg}^2=\epsilon_\ell+\epsilon_{\ell-1}.$ We first assume $\alpha=\alpha^2_{sh}$ and show that the highest weight of ${\mathcal U}_t$ is the highest short root of $R_1$. For this, it is enough to show that no long root is a weight for ${\mathcal U}_t.$ Suppose to the contrary that the set of weights of ${\mathcal U}_t$ contains a long root or equivalently contains all long roots. Setting $\beta:=\epsilon_{\ell-1},$ we get $\alpha+\beta$ is a long root of $R_1$ and so $\alpha+\beta$ is a weight for ${\mathcal U}_t.$ Now fix $x\in({\mathcal G}_2)_{\beta}=({\mathcal G}_1)_\beta$ and note that $\alpha+\beta$ is a weight for $ {\mathcal U}_t;$ applying Lemma \ref{elementary} together with (\ref{gen3}) and (\ref{one-dim}), we have $x\cdot u_t\neq 0.$ This gives that $0\neq \sum_{j=1}^px\cdot u_j=x\cdot v\in ({\mathcal G}_2)_\beta\cdot{\mathcal W}_\alpha\subseteq {\mathcal W}_{\alpha+\beta}$ which is a contradiction as $\alpha+\beta$ is a long root and cannot be a weight for ${\mathcal W}.$ Therefore ${\mathcal U}_t$ has no long root as a weight and so we are done in the case $\alpha=\alpha_{sh}^2$. Next suppose $\alpha=\alpha^2_{lg}$ and note that by (\ref{gen3}), $u_t$ is a weight vector of ${\mathcal U}_t$ of weight $\alpha.$ Since $\alpha$ is a long root, the set of weights of ${\mathcal U}_t$ contains all long roots and so the highest long root is the highest weight of ${\mathcal U}_t.$
\textbf{\emph{\underline{Type $C:$}}} In this case, we have $\alpha_{sh}^1=\epsilon_{n-1}+\epsilon_n,$ $\alpha_{sh}^2=\epsilon_{\ell-1}+\epsilon_\ell,$ $\alpha_{lg}^1=2\epsilon_{n}$ and $\alpha_{lg}^2=2\epsilon_\ell.$ Setting $\beta:=\epsilon_{\ell-1}-\epsilon_\ell,$ we get $\alpha_{sh}^2+\beta\in(R_1)_{lg}.$ Now using the same argument as in Type $B,$ we are done.
\textbf{\emph{\underline{Type $BC:$}}} In this case, we have $\alpha_{sh}^1=\epsilon_n,$ $\alpha_{sh}^2=\epsilon_\ell,$ $\alpha_{lg}^1=\epsilon_{n-1}+\epsilon_n,$ $\alpha_{lg}^2=\epsilon_{\ell-1}+\epsilon_\ell,$ $\alpha_{ex}^1=2\epsilon_{n}$ and $\alpha_{ex}^2=2\epsilon_\ell.$ One can also easily see that $$\{\gamma+\beta\mid \gamma\in(R_1)_{sh},\beta\in(R_1)_{lg}\cup (R_{1})_{ex}\}\cap R_1\subseteq(R_1)_{sh}.$$ This together with (\ref{tavakol1}), (\ref{gen3}) and the fact that $\Delta_1\subseteq (R_1)_{ex}\cup(R_1)_{lg}$ proves the claim stated in Step 1 in the case that $\alpha=\alpha_{sh}^2.$ Now suppose that $\alpha=\alpha_{lg}^2.$ Setting $\beta:=\epsilon_{\ell-1}-\epsilon_\ell,$ we get that $\alpha+\beta$ is an extra long root. Now we are done using the same argument as in Type $B.$ Next suppose $\alpha=\alpha_{ex}^2,$ then by (\ref{gen3}), $u_t$ is a weight vector of weight $\alpha$ which is an extra long root. Therefore any extra long root is a weight for ${\mathcal U}_t$ and so the highest weight of ${\mathcal U}_t$ is $\alpha_{ex}^1.$ This completes the proof of Step 1.
\noindent{\bf Step 2:} We first note that depending on the type of $R_2,$ $\alpha$ is one of $\epsilon_\ell,2\epsilon_\ell,\epsilon_\ell+\epsilon_{\ell-1}$ or $\epsilon_{\ell+1}-\epsilon_{1}.$ If $\alpha=\epsilon_{\ell}+\epsilon_{\ell-1},$ then either $R_2$ is of type $B$ or $D$ and by Step 1, the set of nonzero weights of ${\mathcal U}$ coincides with $R^\times,$ or $R$ is of type $C$ or $BC$ and the set of nonzero weights of ${\mathcal U}$ coincides with $(R_{sdiv})_{sh}.$ In both cases, using induction on $r\in\mathbb{N}\setminus\{0\},$ one can see that \begin{equation}\label{rev1} \parbox{4in}{
if $1\leq m_1,\ldots,m_r\leq n$ and for each $1\leq p\leq r,$ $\alpha_{m_p}+\cdots+\alpha_{m_1}+\alpha$ is a weight for ${\mathcal U},$ then $\{m_1,\ldots,m_r\}\subseteq\{\ell,\ldots,n\}$ and $\alpha_{m_r}+\cdots+\alpha_{m_1}+\alpha=\epsilon_{q}+\epsilon_{q'}$ for some $\ell-1\leq q\neq q'\leq n.$ }\end{equation} Also if $\alpha=\epsilon_{\ell+1}-\epsilon_1,\epsilon_\ell,2\epsilon_{\ell},$ one can see that \begin{equation}\label{rev2}\parbox{3.5in}{ if $r$ is a positive integer and $1\leq m_1,\ldots,m_r\leq n$ are such that for each $1\leq p\leq r,$ $\alpha_{m_p}+\cdots+\alpha_{m_1}+\alpha$ is a weight for ${\mathcal U},$ then $\{m_1,\ldots,m_r\}\subseteq\{\ell+1,\ldots,n\}.$ }\end{equation}
Now suppose that $0\neq u\in {\mathcal U}_{\alpha}, $ we shall show that $u$ is a scalar multiple of $v.$ Since $u\in {\mathcal U},$ by (\ref{tavakol1}), $u$ is written as a linear combination of weight vectors of the form $f_{i_t}\cdot \cdots\cdot f_{i_1},e_{j_s}\cdot \cdots\cdot e_{j_1}\cdot v,$ $t,s\in\mathbb{N},\;1\leq i_1,\ldots,i_t,j_1,\ldots,j_s\leq n.$ So without loss of generality, we suppose $$u=f_{i_t}\cdot\cdots\cdot f_{i_1}\cdot e_{j_s}\cdot \cdots\cdot e_{j_1}\cdot v$$ where $t,s\in\mathbb{N},$ and $1\leq i_1,\ldots, i_t,j_1,\ldots,j_s\leq n.$ Since $u$ is of weight $\alpha,$ we get that $\alpha+\alpha_{j_1}+\cdots+\alpha_{j_s}-\alpha_{i_1}-\cdots-\alpha_{i_t}=\alpha.$ This implies that \begin{equation} \label{coin} s=t\quad\hbox{and}\quad (j_1,\ldots,j_s)=(\sigma(i_1),\ldots,\sigma(i_t)) \end{equation} for a permutation $\sigma$ of $\{i_1,\ldots,i_t\}.$ We note that $\alpha$ is an element of $R_{2}^+$ and so it is written as a linear combination of $\{\alpha_i\mid 1\leq i\leq \ell\}$ with nonnegative rational coefficients not all equal to zero. Now since $\{\alpha_i\mid 1\leq i\leq n\}$ is a base of $(R_{1})_{sdiv},$ $\alpha-\alpha_{j}$ ($\ell+1\leq j\leq n$) is not a root of $R_1$ and so it is not an element of $\Lambda_1.$ Therefore \begin{equation}\label{gen5}f_{j}\cdot v=0,\;\;\ell+1\leq j\leq n. \end{equation} Now this
implies that \begin{equation} \label{eq1}\begin{array}{c} f_{j}\cdot e_{j}\cdot v=e_j\cdot f_j\cdot v-h_{j}\cdot v=0-h_{j}\cdot v\in\mathbb{F} v,\\ (\ell+1 \leq j\leq n).\end{array} \end{equation} We also note that as $v$ is a highest vector of ${\mathcal G}_2$-module ${\mathcal W},$ $e_j\cdot v=0$ for $ 1\leq j\leq \ell.$ Therefore one gets that \begin{equation}\label{js} \hbox{$j_1\in\{\ell+1,\ldots, n\}$ provided that $s\neq0.$ } \end{equation}
Now we are ready to prove that $u$ is a scalar multiple of $v.$ If $s=0,$ there is nothing to prove. So we suppose $s\geq1,$ and use induction on $s$ to prove. If $s=1,$ we get the result appealing (\ref{coin}), (\ref{js}) and (\ref{eq1}). Now suppose $s>1.$ If $\alpha=\epsilon_{\ell}+\epsilon_{\ell-1},$ then using (\ref{rev1}) together with (\ref{coin}), we get that $i_1\in\{\ell,\ldots,n\}.$ This together with (\ref{gen5}) and the fact that $2\epsilon_{\ell-1}$ is not a weight for ${\mathcal U}$ implies that $f_{i_1}\cdot v=0.$ Next take $1\leq k_1<\ldots< k_r\leq s$ to be the only indices with $j_{k_1}=\cdots=j_{k_r}=i_1$ and use Lemma \ref{gen-fact} to get $$f_{i_t}\cdot\cdots\cdot f_{i_1}\cdot e_{j_s}\cdot\cdots\cdot e_{j_1}\cdot v\in\sum_{q=1}^r\mathbb{F} f_{i_t}\cdot\cdots\cdot f_{i_2}\cdot e_{j_s}\cdot\cdots e_{j_{k_r}}\cdot\cdots \cdot \widehat{e_{j_{k_q}}}\cdot\cdots\cdot e_{j_{k_1}}\cdot\cdots \cdot e_{j_1}\cdot v.$$ Now induction hypothesis completes the proof of this step in the case that $\alpha=\epsilon_{\ell-1}+\epsilon_\ell.$ Next suppose $\alpha\in\{\epsilon_\ell,2\epsilon_{\ell},\epsilon_{\ell+1}-\epsilon_{1}\},$ then using (\ref{rev2}) together with (\ref{coin}), (\ref{gen5}), Lemma \ref{gen-fact} and the same argument as before, we get the result.
{\bf Step 3:} It is immediate using Step 2 together with (\ref{gen3}).\qed
\section{Root graded Lie algebras} The structure of Lie algebras graded by an irreducible finite root system have been studied in \cite{BM}, \cite{BZ}, \cite{ABG1}, \cite{N}, \cite{ABG2} and \cite{BS}. A Lie algebra $\mathcal{L}$ graded by an irreducible finite root system $R$ contains a finite dimensional split simple Lie algebra ${\mathcal G}$ and with respect to a splitting Cartan subalgebra, it is equipped with a weight space decomposition whose set of weights is contained in $R.$ This feature allows us to decompose $\mathcal{L}$ into finite dimensional irreducible ${\mathcal G}-$submodules whose set of nonzero weights is $R_{sh},$ $R_{sdiv}^\times$ or $(R_{sdiv})_{sh}.$ Collecting the components of the same highest weight results in the decomposition \begin{equation}\label{*}\mathcal{L}=({\mathcal G}\otimes \mathcal A)\oplus(\mathcal{S}\otimes\mathcal{B})\oplus({\mathcal V}\otimes {\mathcal C})\oplus \mathcal D\end{equation} in which $\mathcal D$ is a trivial submodule of $\mathcal{L}$ and $\mathcal{S}$ (resp. ${\mathcal V}$) is the finite dimensional irreducible ${\mathcal G}-$module whose set of nonzero weights is $(R_{sdiv})_{sh}$ (resp. $R_{sh}$). The Lie algebraic structure on $\mathcal{L}$ induces an algebraic structure on $\frak{b}:=\mathcal A\oplus\mathcal{B}\oplus{\mathcal C}$ which we refer to as the {\it coordinate algebra} of $\mathcal{L}.$ The Lie bracket on $\mathcal{L}$ can be rewritten using the ingredients involved in describing the product defined on the algebra $\frak{b}.$ In this section, we have a compromising view on the coordinate algebras of root graded subalgebras of $\mathcal{L}.$ We devote this section to two subsections. In the first subsection, we illustrate the structure of a specific Lie algebra which we shall frequently use in the sequel of the paper. In the second subsection, we consider a Lie algebra $\mathcal{L}$ graded by an irreducible finite root system $R$ and for a full irreducible subsystem $S$ of $R$ which is of the same type as $R,$ we take $\mathcal{L}^{^S}$ to be the Lie subalgebra of $\mathcal{L}$ generated by homogeneous spaces in correspondence to $S^\times.$ We show that the coordinate algebra of the Lie subalgebra $\mathcal{L}^{^S},$ which is an algebra graded by $S,$ does not depend on $S.$ In fact, we prove that the coordinate algebra of $\mathcal{L}^{^S}$ coincides with the coordinate algebra of $\mathcal{L}.$ Moreover, we describe the Lie bracket on $\mathcal{L}$ in terms of the ingredients involved in describing the Lie bracket on $\mathcal{L}^{^S}$ with respect to its coordinate algebra. Our method is based on a type-by-type approach. Since the proofs for different types are quite similar, we go through the proofs in details if $R$ is of type $BC$ and for other types, we just report the results and leave the proofs to the readers.
\subsection{A specific Lie algebra}\label{subsect2-1} By a {\it star algebra} $(\mathfrak{a},*),$ we mean an algebra $\mathfrak{a}$ together with a self-inverting antiautomorphism $*$ which is referred to as an {\it involution}. We call a quadruple $(\mathfrak{a},*,{\mathcal C},f)$ a {\it coordinate quadruple} if one of the following holds: \begin{itemize} \item (Type $A$) $\mathfrak{a}$ is a unital associative algebra, $*=id_\mathfrak{a},$ ${\mathcal C}=\{0\}$ and $f:{\mathcal C}\times{\mathcal C}\longrightarrow \mathfrak{a}$ is the zero map.
\item (Type $B$) $\mathfrak{a}=\mathcal A\oplus\mathcal{B}$ where $\mathcal A$ is a unital commutative associative algebra and $\mathcal{B}$ is a unital associative $\mathcal A-$module equipped with a symmetric bilinear form and $\mathfrak{a}$ is the corresponding Clifford Jordan algebra, $*$ is a linear transformation fixing the elements of $\mathcal A$ and skew fixing the elements of $\mathcal{B},$ ${\mathcal C}=\{0\}$ and $f:{\mathcal C}\times{\mathcal C}\longrightarrow \mathfrak{a}$ is the zero map.
\item (Type $C$) $\mathfrak{a}$ is a unital associative algebra, $*$ is an involution, ${\mathcal C}=\{0\}$ and $f:{\mathcal C}\times{\mathcal C}\longrightarrow \mathfrak{a}$ is the zero map.
\item (Type $D$) $\mathfrak{a}$ is a unital commutative associative algebra $*=id_\mathfrak{a},$ ${\mathcal C}=\{0\}$ and $f:{\mathcal C}\times{\mathcal C}\longrightarrow \mathfrak{a}$ is the zero map.
\item (Type $BC$) $\mathfrak{a}$ is a unital associative algebra, $*$ is an involution, ${\mathcal C}$ is a unital associative $\mathfrak{a}-$module and $f:{\mathcal C}\times{\mathcal C}\longrightarrow \mathfrak{a}$ is a skew-hermitian form.
\end{itemize}
Suppose that $(\mathfrak{a},*,{\mathcal C},f)$ is a coordinate quadruple. Denote by $\mathcal A$ and $\mathcal{B},$ the fixed and the skew fixed points of $\mathfrak{a}$ under $*,$ respectively.
Set $\frak{b}:=\frak{b}(\mathfrak{a},*,{\mathcal C},f):=\mathfrak{a}\oplus{\mathcal C}$ and define \begin{equation}\label{probinbc-n} \begin{array}{c}\cdot:\frak{b}\times\frak{b}\longrightarrow \frak{b}\\ (\alpha_1+c_1,\alpha_2+c_2)\mapsto(\alpha_1\cdot \alpha_2)+f(c_1,c_2)+\alpha_1\cdot c_2+\alpha_2^*\cdot c_1, \end{array} \end{equation} for $\alpha_1,\alpha_2\in\mathfrak{a}$ and $c_1,c_2\in{\mathcal C}.$ Also for $\beta,\beta'\in\frak{b},$ set \begin{equation}\label{end9-n} \beta\circ\beta':=\beta\cdot\beta'+\beta'\cdot\alpha\quad\hbox{and}\quad [\beta,\beta']:=\beta\cdot \beta'-\beta'\cdot\beta \end{equation} and for $c,c'\in{\mathcal C},$ define \begin{equation}\label{diamond-heart-n} \begin{array}{c} \begin{array}{ll} \diamond:{\mathcal C}\times{\mathcal C}\longrightarrow\mathcal A,& (c,c')\mapsto\frac{f(c,c')-f(c',c)}{2};\;c,c'\in{\mathcal C}, \end{array}
\\ \begin{array}{ll} \hbox{\tiny$\heartsuit$}:{\mathcal C}\times{\mathcal C}\longrightarrow\mathcal{B},& (c,c')\mapsto\frac{f(c,c')+f(c',c)}{2};\;c,c'\in{\mathcal C}. \end{array} \end{array} \end{equation}
Now suppose that $\ell$ is a positive integer and for $\alpha,\alpha'\in\mathfrak{a}$ and $c,c'\in{\mathcal C},$ consider the following endomorphisms \begin{equation}\label{derivbc} \begin{array}{l} d_{\alpha,\alpha'}^{\ell,\frak{b}}:\frak{b}\longrightarrow\frak{b},\\ \beta\mapsto\left\{\begin{array}{ll} \frac{1}{\ell+1}[[\alpha,\alpha'],\beta]& X=A_\ell,\; \beta\in \frak{b},
\\ \alpha'(\alpha\beta)-\alpha(\alpha'\beta)&X=B_\ell,\;\beta\in\frak{b},
\\
\frac{1}{4\ell}[[\alpha,\alpha']+[\alpha^{*},\alpha'^{*}],\beta]&X=C_\ell,BC_\ell,\;\;\beta\in\mathfrak{a},
\\ \frac{1}{4\ell}([\alpha,\alpha']+[\alpha^{*},\alpha'^{*}])\cdot \beta&X=C_\ell,BC_\ell,\;\;\beta\in{\mathcal C},
\\ 0& X=D_\ell,\; \beta\in\frak{b},\end{array}\right.\\delta^{\ell,\frak{b}}_{c,c'}:\frak{b}\longrightarrow\frak{b},\\ \beta\mapsto\left\{\begin{array}{ll}\frac{-1}{2\ell}[c\hbox{\tiny$\heartsuit$} c',\beta]&\; X=BC_\ell,\;\beta\in\mathfrak{a},
\\ \frac{-1}{2\ell}(c\hbox{\tiny$\heartsuit$} c')\cdot\beta-\frac{1}{2}(f(\beta,c')\cdot c+f(\beta,c)\cdot c')&X=BC_\ell,\;\beta\in{\mathcal C},
\\ 0&\hbox{otherwise},
\end{array}\right.\\ d^{\ell,\frak{b}}_{\alpha,c}:=d^{\ell,\frak{b}}_{c,\alpha}:=0,
\\ d^{\ell,\frak{b}}_{\alpha+c,\alpha'+c'}:=d^{\ell,\frak{b}}_{\alpha,\alpha'}+d^{\ell,\frak{b}}_{c,c'}. \end{array} \end{equation} One can see that for $\beta,\beta'\in\frak{b},$ $d^{\ell,\frak{b}}_{\beta,\beta'}\in Der(\frak{b}).$ Next take $K$ to be a subspace of $\frak{b}\otimes \frak{b}$ spanned by $$\begin{array}{l} \alpha\otimes c,\;\;c\otimes\alpha,\;\;a\otimes b,\\ \alpha\otimes\alpha'+\alpha'\otimes\alpha,\;\;c\otimes c'-c'\otimes c,\\ (\alpha\cdot \alpha')\otimes\alpha''+(\alpha''\cdot\alpha)\otimes\alpha'+(\alpha'\cdot\alpha'')\otimes\alpha,\\ f(c,c')\otimes\alpha+( \alpha^*\cdot c')\otimes c-(\alpha\cdot c)\otimes c' \end{array}$$ for $\alpha,\alpha',\alpha''\in\mathfrak{a},$ $a\in\mathcal A,$ $b\in\mathcal{B},$ and $c,c'\in{\mathcal C}.$ Then $(\frak{b}\otimes\frak{b})/K$ is a Lie algebra under the Lie bracket \begin{equation}\label{last4} [(\beta_1\otimes \beta_2)+K,(\beta'_1\otimes \beta'_2)+K]_\ell=((d^{\ell,\frak{b}}_{\beta_1,\beta_2}(\beta'_1)\otimes \beta'_2)+K)+(\beta'_1\otimes d^{\ell,\frak{b}}_{\beta_1,\beta_2} (\beta'_2))+K)\end{equation} for $\beta_1,\beta_2,\beta'_1,\beta'_2\in\frak{b}$ (see \cite[Proposition 5.23]{ABG2} and \cite{ABG1}). We denote this Lie algebra by $\{\frak{b},\frak{b}\}_\ell$ (or $\{\frak{b},\frak{b}\}$ if there is no confusion) and for $\beta_1,\beta_2\in\frak{b},$ we denote $(\beta_1\otimes\beta_2)+K$ by $\{\beta_1,\beta_2\}_\ell$ (or $\{\beta_1,\beta_2\}$ if there is no confusion). We recall the {\it full skew-dihedral homology group} $$FH(\frak{b}):=\{\sum_{i=1}^n\{\beta_i,\beta'_i\}_\ell\in\{\frak{b},\frak{b}\}_\ell\mid \sum_{i=1}^nd^{\ell,\frak{b}}_{\beta_i,\beta'_i}=0\}$$ of $\frak{b}$ (with respect to $\ell$) from \cite{ABG2} and \cite{ABG1} and note that it is a subset of the center of $\{\frak{b},\frak{b}\}_\ell. $ For $\beta_1=a_1+b_1+c_1\in\frak{b}$ and $\beta_2=a_2+b_2+c_2\in\frak{b}$ with $a_1,a_2\in\mathcal A,$ $b_1,b_2\in\mathcal{B}$ and $c_1,c_2\in{\mathcal C},$ set \begin{equation}\label{beta*}\beta_{_{\beta_1,\beta_2}}^*:=[a_1,a_2]+[b_1,b_2]-c_1\hbox{\tiny$\heartsuit$} c_2.\end{equation} We say a subset $\mathcal K$ of the full skew-dihedral homology group of $\frak{b}$ satisfies the ``{\it uniform property on $\frak{b}$}" if for $\beta_1,\beta'_1,\ldots,\beta_n,\beta'_n\in\frak{b},$ $\sum_{i=1}^n\{\beta_i,\beta'_i\}_\ell\in\mathcal K$ implies that $\sum_{i=1}^n\beta^*_{\beta_i,\beta'_i}=0.$ \begin{rem}\label{rem3}{\rm Suppose that $\ell,\ell'$ are two positive integers. If $\mathcal K$ is a subset of the full skew-dihedral homology group of $\frak{b}(\mathfrak{a},*,{\mathcal C},f)$ with respect to $\ell$ satisfying the uniform property on $\frak{b}(\mathfrak{a},*,{\mathcal C},f)$, it is a subset of the full skew-dihedral homology group of $\frak{b}(\mathfrak{a},*,{\mathcal C},f)$ with respect to $\ell'$ satisfying the uniform property on $\frak{b}(\mathfrak{a},*,{\mathcal C},f).$ In other words, the uniform property on $\frak{b}(\mathfrak{a},*,{\mathcal C},f)$ dose not depend on $\ell.$} \end{rem}
\subsection{Lie algebras graded by a finite root system}\label{subsect2-2} In this work, we study root graded Lie algebras in the following sense:
\begin{Definition}\label{root-graded}{\rm Suppose that $R$ is an irreducible locally finite root system. We say a Lie algebra $\mathcal{L}$ is an {\it $R-$graded Lie algebra with graded pair $({\mathcal G},{\mathcal H})$} if the followings are satisfied:
i) ${\mathcal G}$ is a locally finite split simple Lie subalgebra of $\mathcal{L}$ with splitting Cartan subalgebra ${\mathcal H}$ and corresponding root system $R_{sdiv}.$
ii) $\mathcal{L}$ has a weight space decomposition $\mathcal{L}=\oplus_{\alpha\in R}\mathcal{L}_\alpha$ with respect to ${\mathcal H}$ via the adjoint representation.
iii) $\mathcal{L}_0=\sum_{\alpha\in R^{\times}}[\mathcal{L}_\alpha,\mathcal{L}_{-\alpha}].$} \end{Definition}
The following lemma easily follows from Lemma \ref{final3}. \begin{Lemma}\label{final1} Suppose that $R$ is an irreducible locally finite root system and $\mathcal{L}$ is a Lie algebra graded by $R$ with grading pair $({\mathcal G},{\mathcal H}).$ Let $S$ be an irreducible full subsystem of $R$ and set \begin{equation}\label{final2} \begin{array}{l} \displaystyle{\mathcal{L}^{^S}:=\sum_{\alpha\in S^\times}\mathcal{L}_\alpha\oplus\sum_{\alpha\in S^\times}[\mathcal{L}_\alpha,\mathcal{L}_{-\alpha}],}\\ \displaystyle{{\mathcal G}^{^S}:=\sum_{\alpha\in S_{sdiv}^\times}{\mathcal G}_\alpha\oplus\sum_{\alpha\in S_{sdiv}^\times}[{\mathcal G}_\alpha,{\mathcal G}_{-\alpha}].} \end{array} \end{equation} Then $\mathcal{L}^{^S}$ is an $S-$graded Lie subalgebra of $\mathcal{L}$ with grading pair $({\mathcal G}^{^S},{\mathcal H}^{^S}:={\mathcal H}\cap{\mathcal G}^{^S}).$ \end{Lemma}
Before going through the main body of this subsection, we want to fix a notation. If $A$ is a subspace of a vector space $V_1$ and $B$ is a subspace of a vector space $V_2,$ by a conventional notation, we take $A\dot\otimes B$ to be the vector subspace of $V_1\otimes V_2$ spanned by $a\otimes b$ for $a\in A$ and $b\in B.$
\subsubsection{\textbf{Type $BC$}}\label{subsub1} Suppose that $I$ is a nonempty index set of cardinality $m_n:=n>3$ and $I_0$ is a nonempty subset of $I$ of cardinality $m_\ell:=\ell>3.$ Take ${\mathcal V}:={\mathcal V}^n$ to be a vector space with a basis $\{v_i\mid i\in I\cup\bar I\}$ equipped with a nondegenerate symmetric bilinear form $(\cdot,\cdot)$ as in (\ref{form-c}). Set ${\mathcal G}^n:=\mathfrak{sp}(I)$ and take $\mathcal{S}:=\mathcal{S}^n$ to be as in (\ref{module-s-c}). Consider (\ref{simple-c}), (\ref{simple-c-alg}) and set $${\mathcal V}^\ell:={\mathcal V}_{_{I_0}},\;\;{\mathcal G}^\ell:={\mathcal G}_{_{I_0}},\;\;\mathcal{S}^\ell:=\mathcal{S}_{_{I_0}}.$$
We take $Id_{_{\mathcal V}}$ to be the identity map on ${\mathcal V}$ and define the linear endomorphism $Id_{_{{\mathcal V}^\ell}}$ on ${\mathcal V}$ by $$\begin{array}{c}Id_{_{{\mathcal V}^{\ell}}}:{\mathcal V}\longrightarrow {\mathcal V}\\ v_i\mapsto v_i,\;v_{\bar i}\mapsto v_{\bar i},\;v_{j}\mapsto 0,\;v_{\bar j}\mapsto 0;\;\; (i\in I_0,\; j\in I\setminus I_0). \end{array}$$ Also for $\lambda=\ell,n$ and $x,y\in{\mathcal G}^\lambda\oplus\mathcal{S}^\lambda,$ set \begin{equation} \label{end7} x\circ_\lambda y:=xy+yx-(1/m_\lambda)tr(xy)Id_{_{{\mathcal V}^\lambda}}. \end{equation}
Next for $u,v\in{\mathcal V},$ define \begin{equation}\label{u,v}\begin{array}{l} \;[u, v] :{\mathcal V}\longrightarrow{\mathcal V};\;w\mapsto\frac{1}{2}((v,w)u+(w,u)v)+\frac{1}{2\ell }(u,v)Id_{_{{\mathcal V}^\ell}}(w);\;\;w\in{\mathcal V},
\\ \;u\circ v:{\mathcal V}\longrightarrow{\mathcal V};\;w\mapsto\frac{1}{2}((v,w)u+(u,w)v);\;\;w\in{\mathcal V},
\\ \;[u, v]_n :{\mathcal V}\longrightarrow{\mathcal V};\;w\mapsto\frac{1}{2}((v,w)u+(w,u)v)+\frac{1}{2n }(u,v)Id_{_{\mathcal V}}(w);\;\;w\in{\mathcal V}. \end{array} \end{equation}
One can easily see that up to isomorphism $$\begin{array}{ll}{\mathcal G}^\ell=\hbox{span}\{u\circ v\mid u,v\in{\mathcal V}^\ell\},& \mathcal{S}^\ell=\hbox{span}\{[u,v]\mid u,v\in{\mathcal V}^\ell\},\\{\mathcal G}^n=\hbox{span}\{u\circ v\mid u,v\in{\mathcal V}^n\},& \mathcal{S}^n=\hbox{span}\{[u,v]_n\mid u,v\in{\mathcal V}^n\}.\end{array}$$
Suppose that $R$ is an irreducible finite root system of type $BC_I$ and $S$ is the irreducible full subsystem of $R$ of type $BC_{I_0}.$ Suppose that $\mathcal{L}$ is an $R-$graded Lie algebra with grading pair $(\mathfrak{g},\mathfrak{h})$ and take $\mathcal{L}^{^S},$ $\mathfrak{g}^{^S}$ and $\mathfrak{h}^{^S}$ to be as in Lemma \ref{final1}. In order to simplify the using of the notations, we set \begin{equation}\label{general} \mathcal{L}^n:=\mathcal{L},\;\; \mathcal{L}^{\ell}:=\mathcal{L}^{^S},\;\;[u, v]_\ell:=[u,v];\;\;(u,v\in{\mathcal V}). \end{equation}
One knows that as a $\mathfrak{g}^{^S}-$module, $\mathcal{L}^\ell$ can be decomposed into finite dimensional irreducible $\mathfrak{g}^{^S}-$submodules, each of which is a finite dimensional irreducible $\mathfrak{g}^{^S}-$module with highest weight contained in $S.$ Take \begin{equation}\label{last3-fin}\mathcal{L}^\ell=\mathcal{L}^{^S}=\bigoplus_{i\in {\mathcal I}_0}\mathfrak{g}_i\oplus\bigoplus_{j\in\mathcal{J}_0}\mathfrak{s}_j\oplus\bigoplus_{t\in {\mathcal T}_0}V_t\oplus E\end{equation} to be the decomposition of $\mathcal{L}^\ell$ into finite dimensional irreducible $\mathfrak{g}^{^S}-$modules in which ${\mathcal I}_0,\mathcal{J}_0,{\mathcal T}_0$ are (possibly empty) index sets and for $i\in{\mathcal I}_0,$ $j\in \mathcal{J}_0,$ and $t\in{\mathcal T}_0,$ $\mathfrak{g}_i$ is isomorphic to $\mathfrak{g}^{^S}(\simeq{\mathcal G}^\ell),$ $\mathfrak{s}_j$ is isomorphic to $\mathcal{S}^\ell,$ $V_t$ is isomorphic to ${\mathcal V}^\ell$ and $E$ is a trivial $\mathfrak{g}^{^S}-$submodule. \begin{Lemma}\label{divide1} Use the notation as in the text and consider $\mathcal{L}=\mathcal{L}^n$ as a $\mathfrak{g}-$module. Then there exist index sets ${\mathcal I},$ $\mathcal{J},$ ${\mathcal T}$ with ${\mathcal I}_0\subseteq{\mathcal I},\mathcal{J}_0\subseteq\mathcal{J},{\mathcal T}_0\subseteq{\mathcal T},$ and a class $\{\mathcal D_n,{\mathcal G}_i,\mathcal{S}_j,{\mathcal V}_t\mid i\in {\mathcal I},j\in \mathcal{J},t\in {\mathcal T}\}$ of finite dimensional $\mathfrak{g}-$submodules of $\mathcal{L}$ such that
\begin{itemize} \item $\mathcal D_n$ is a trivial $\mathfrak{g}-$module, ${\mathcal G}_i$ is isomorphic to $\mathfrak{g},$ $\mathcal{S}_j$
is isomorphic to $\mathcal{S},$ and ${\mathcal V}_t$ is isomorphic to ${\mathcal V},$ for $i\in{\mathcal I},j\in \mathcal{J},$ $t\in {\mathcal T},$ \\ \item $\mathfrak{g}_i\subseteq{\mathcal G}_i,$ $\mathfrak{s}_j\subseteq\mathcal{S}_j,$ $V_t\subseteq{\mathcal V}_t$ ($i\in{\mathcal I}_0,$ $j\in\mathcal{J}_0,$ $t\in{\mathcal T}_0$),\\ \item $\mathcal{L}^n=\mathcal{L}=\bigoplus_{i\in{\mathcal I}}{\mathcal G}_i\oplus\bigoplus_{j\in\mathcal{J}}\mathcal{S}_j\oplus\bigoplus_{t\in{\mathcal T}}{\mathcal V}_t\oplus\mathcal D_n.$ \end{itemize}We make a convention that we refer to \begin{equation}\label{last6}(\mathcal{I},\mathcal{J},{\mathcal T},\{\mathfrak{g}_i\},\{{\mathcal G}_i\},\{\mathfrak{s}_j\},\{\mathcal{S}_j\},\{V_t\},\{{\mathcal V}_t\},E,\mathcal D_n)\end{equation} as an $(R,S)-$datum for the pair $(\mathcal{L}^n,\mathcal{L}^\ell).$
\end{Lemma}
\noindent{\bf Proof. } For $i\in \mathcal{I}_0,$ by Proposition \ref{gen1}, the ${\mathcal G}-$submodule ${\mathcal G}_i$ of $\mathcal{L} $ generated by $\mathfrak{g}_i$ is a finite dimensional ${\mathcal G}-$module isomorphic to ${\mathcal G}.$ For $\alpha\in S_{sdiv}^\times$ and $0\neq x\in(\mathfrak{g}_i)_\alpha\subseteq\mathcal{L}_\alpha,$ we have $x\in\mathcal{L}_\alpha\cap\mathfrak{g}_i\subseteq\mathcal{L}_\alpha\cap{\mathcal G}_i=({\mathcal G}_i)_\alpha.$ Now as $\dim({\mathcal G}_i)_\alpha=\dim(\mathfrak{g}_i)_\alpha=1,$ we get \begin{equation}\label{final4}(\mathfrak{g}_i)_\alpha=({\mathcal G}_i)_\alpha;\;\;\;\alpha\in S^\times_{sdiv}.\end{equation}
\underline{\noindent\textbf{Claim 1.}} The sum $\sum_{i\in \mathcal{I}_0 }{{\mathcal G}} _i$ is a direct sum: Suppose that $i_0,i_1,\ldots,i_n$ are distinct elements of $\mathcal{I}_0$ and $0\neq x \in{\mathcal G}_{i_0} \cap\sum_{t=1}^n{\mathcal G}_{i_t}.$ Then as ${\mathcal G}_{i_0} $ is an irreducible ${\mathcal G} -$module, we get that $${\mathcal G}_{i_0} \subseteq\sum_{t=1}^n{\mathcal G}_{i_t}.$$ This together with (\ref{final4}) implies that for $\alpha\in S_{sdiv}^\times\subseteq R_{sdiv}^\times,$ $$(\mathfrak{g}_{i_0})_\alpha=({\mathcal G}_{i_0})_\alpha\subseteq\sum_{t=1}^n({\mathcal G}_{i_t})_\alpha=\sum_{t=1}^n(\mathfrak{g}_{i_t})_\alpha$$ which contradicts the fact that $\sum_{i\in \mathcal{I}_0}{\mathfrak{g}}_i$ is direct. This completes the proof of Claim 1.
Now for $j\in\mathcal{J}_0$ and $t\in\mathcal{T}_0,$ take $\mathcal{S} _j$ and ${\mathcal V} _t$ to be the finite dimensional irreducible ${\mathcal G}-$submodules of $\mathcal{L} $ generated by $\mathfrak{s}_j$ and $V_t$ respectively. Using the same argument as above, one can see that the summations $\sum_{j\in\mathcal{J}_0}\mathcal{S} _j$ and $\sum_{t\in\mathcal{T}_0}{\mathcal V} _t$ are direct. Set $${\mathcal G}(n):=\oplus_{i\in \mathcal{I}_0 }{{\mathcal G}} _i,\;\;\mathcal{S}(n):=\oplus_{j\in\mathcal{J}_0}\mathcal{S} _j,\;\;{\mathcal V}(n):=\oplus_{t\in\mathcal{T}_0}{\mathcal V} _t.$$ We note that ${\mathcal G}(n)$ (resp. $\mathcal{S}(n)$ and ${\mathcal V}(n)$) is a ${\mathcal G}-$submodule of $\mathcal{L}$ whose set of weights is $R_{sdiv}$ (resp. $R_{lg}\cup\{0\}$ and $R_{sh}$).
\underline{\noindent\textbf{Claim 2.}} For $\alpha\in R_{lg},$ $x\in {\mathcal G}(n)_\alpha,$ and $y\in \mathcal{S}(n)_\alpha,$ we have $x+y=0$ if and only if $x=y=0:$ Suppose that $x+y=0.$ Since $x\in{\mathcal G}(n)_\alpha=\sum_{i\in\mathcal{I}_0}({\mathcal G}_i )_\alpha,$ we get $x=\sum_{i\in\mathcal{I}_0}x_i$ with finitely many nonzero terms $x_i\in({\mathcal G}_i)_\alpha,$ for $i\in\mathcal{I}_0.$ Similarly $y=\sum_{j\in\mathcal{J}_0}y_j$ with finitely many nonzero terms $y_j\in(\mathcal{S}_j)_\alpha,$ for $j\in\mathcal{J}_0.$ Now we recall that $\ell,n>3$ and $R,S$ are root systems of type $BC_n$ and $BC_\ell$ respectively. This allows us to pick $\beta_1,\beta_2\in R_{lg}$ such that $\beta:=\alpha+\beta_1+\beta_2\in S_{lg}$ and $\alpha+\beta_1\in R_{lg}.$ Fix $a_1\in{\mathcal G}_{\beta_1}$ and $a_2\in{\mathcal G}_{\beta_2}.$ Using Lemma \ref{elementary}, we get that \begin{equation}\label{eq2} \parbox{3.6in}{ if $x_i$ $(i\in\mathcal{I}_0)$ is nonzero, then $ a_2\cdot a_1\cdot x_i$ is a nonzero element of $({\mathcal G}_i )_\beta=(\mathfrak{g}_i)_\beta$ and similarly if $j\in\mathcal{J}_0$ and $y_j\neq0,$ $ a_2\cdot a_1\cdot y_j$ is a nonzero element of $(\mathcal{S}_j )_\beta=(\mathfrak{s}_j)_\beta.$ }\end{equation}Now since $x+y=0,$ we get that $\sum_{i\in\mathcal{I}_0}x_i=-\sum_{j\in\mathcal{J}_0}y_j$ which in turn implies that $\sum_{i\in\mathcal{I}_0}a_2\cdot a_1\cdot x_i=-\sum_{j\in\mathcal{J}_0} a_2\cdot a_1\cdot y_j.$ But the right hand side is an element of $\oplus_{j\in\mathcal{J}_0}(\mathfrak{s}_j)_\beta$ and the left hand side is an element of $\oplus_{i\in\mathcal{I}_0}(\mathfrak{g}_i)_\beta.$
Therefore $a_2\cdot a_1\cdot x_i=0$ and $a_2\cdot a_1\cdot y_j=0$ for $i\in\mathcal{I}_0$ and $j\in\mathcal{J}_0.$ This together with (\ref{eq2}) implies that for $i\in \mathcal{I}_0$ and $j\in\mathcal{J}_0,$ $x_i=0$ and $y_j=0.$ This completes the proof of Claim 2.
\underline{\noindent\textbf{Claim 3.}} For $x\in{\mathcal G}(n)_0$ and $y\in\mathcal{S}(n)_0,$ $x+y=0$ if and only if $x=y=0:$ Suppose that $x+y=0$ and $x\neq0.$ Since ${\mathcal G}(n)_0= \sum_{i\in \mathcal{I}_0}({\mathcal G}_i)_0,$ we have $x=\sum_{i\in\mathcal{I}_0}x_i$ with finitely many nonzero terms $x_i\in ({\mathcal G}_i)_0,$ $i\in\mathcal{I}_0.$ Fix $t\in\mathcal{I}_0$ such that $x_t\neq 0.$ Since $x_t$ is a nonzero element of the irreducible nontrivial $\mathfrak{g} -$module ${\mathcal G}_t ,$ there is
$\alpha\in R^\times$ and $0\neq a\in\mathfrak{g}_\alpha$ such that $a\cdot x_t\neq
0.$ We note that $x\in{\mathcal G}(n)_0$ and $y\in\mathcal{S}(n)_0,$ therefore we have $a\cdot x\in{\mathcal G}(n)_\alpha$ and $a\cdot y\in\mathcal{S}(n)_\alpha.$ Now as $0=a\cdot x +a\cdot y,$ Claim 2 together with the fact that the set of weights of $\mathcal{S}(n)$ is $R_{lg}\cup\{0\}$ implies that $a\cdot x=0$ and $a\cdot y=0.$ So $\sum_{i\in\mathcal{I}_0}a\cdot x_i=0.$ But by Claim 1, $\sum_{i\in\mathcal{I}_0}{\mathcal G}_i$ is a direct sum, so $a\cdot x_t=0$ which is a contradiction. Therefore $x=0$ and so $y=0$ as well. This completes the proof of Claim 3.
\underline{\textbf{Claim 4.}} The sum ${\mathcal G}(n)+\mathcal{S}(n)+{\mathcal V}(n)$ is a direct sum: Suppose that $x\in{\mathcal G}(n),$ $y\in\mathcal{S}(n)$ and $z\in{\mathcal V}(n)$ are such that $x+y+z=0.$ We have $x=\displaystyle{\sum_{\alpha\in R_{sdiv}}x_\alpha}$ with $x_\alpha\in{\mathcal G}(n)_\alpha\subseteq\mathcal{L}_\alpha$ for $\alpha\in R_{sdiv},$ $y=\displaystyle{\sum_{\alpha\in R_{lg}\cup\{0\}}y_\alpha}$ with $y_\alpha\in\mathcal{S}(n)_\alpha\subseteq\mathcal{L}_\alpha$ for $\alpha\in R_{lg}\cup\{0\},$ and $z=\displaystyle{\sum_{\alpha\in R_{sh}}z_\alpha}$ with $z_\alpha\in{\mathcal V}(n)_\alpha\subseteq\mathcal{L}_\alpha$ for $\alpha\in R_{sh}.$ Therefore one gets that $$ \begin{array}{c} x_0+y_0=0,\; z_\alpha=0,\;x_\beta+y_\beta=0, \;x_\gamma=0;\\( \alpha\in R_{sh},\;\beta\in R_{lg},\;\gamma\in R_{ex}). \end{array}$$ Now using Claims 2,3, we are done
To complete the proof, we note that as a $\mathfrak{g}-$module, $\mathcal{L} $ can be decomposed into finite dimensional irreducible $\mathfrak{g}-$submodules with the set of weights contained in $R .$ Now as $\bigoplus_{i\in\mathcal{I}_0}{\mathcal G} _i\oplus\bigoplus_{j\in\mathcal{J}_0}\mathcal{S} _j\oplus\bigoplus_{t\in\mathcal{T}_0}{\mathcal V} _t$ is a submodule of $\mathcal{L} ,$ one can find index sets $\mathcal{I} ,\mathcal{J} ,\mathcal{T}$ with $$\mathcal{I}_0\subseteq\mathcal{I} ,\;\mathcal{J}_0\subseteq\mathcal{J} ,\;\mathcal{T}_0\subseteq\mathcal{T} $$ and a class $\{\mathcal D_n,{\mathcal G}_i,\mathcal{S}_j,{\mathcal V}_t\mid i\in\mathcal{I}\setminus\mathcal{I}_0,j\in\mathcal{J}\setminus\mathcal{J}_0,t\in{\mathcal T}\setminus{\mathcal T}_0\}$ of finite dimensional $\mathfrak{g}-$submodules such that $\mathcal D_n$ is a trivial $\mathfrak{g}-$module, ${\mathcal G}_i $ is isomorphic to ${\mathcal G}$ $(i\in\mathcal{I}\setminus\mathcal{I}_0),$ $\mathcal{S}_j$ is isomorphic to $\mathcal{S}$ ($j\in\mathcal{J}\setminus \mathcal{J}_0$), ${\mathcal V} _t$ is isomorphic to ${\mathcal V} $ $(t\in\mathcal{T} \setminus\mathcal{T}_0)$ and \begin{eqnarray*} \mathcal{L} &=&(\bigoplus_{i\in\mathcal{I}_0}{\mathcal G} _i\oplus\bigoplus_{j\in\mathcal{J}_0}\mathcal{S} _j\oplus\bigoplus_{t\in\mathcal{T}_0}{\mathcal V} _t) \oplus(\bigoplus_{i\in\mathcal{I}\setminus\mathcal{I}_0}{\mathcal G} _i\oplus\bigoplus_{j\in\mathcal{J}\setminus\mathcal{J}_0}\mathcal{S} _j\oplus\bigoplus_{t\in\mathcal{T}\setminus{\mathcal T}_0}{\mathcal V} _t \oplus\mathcal D_n)\\&=&\bigoplus_{i\in\mathcal{I}}{\mathcal G} _i\oplus\bigoplus_{j\in\mathcal{J}}\mathcal{S} _j\oplus\bigoplus_{t\in\mathcal{T}}{\mathcal V} _t \oplus\mathcal D_n. \end{eqnarray*} This completes the proof.\qed
From now on, we use the data appeared in the $(R,S)-$datum (\ref{last6}). We take $\mathcal A_n $ to be a vector space with a basis $\{a_i\mid i\in \mathcal{I} \},$ $\mathcal{B}_n $ to be a vector space with a basis $\{b_j\mid t\in\mathcal{J} \}$ and ${\mathcal C}_n $ to be a vector space with a basis $\{c_t\mid t\in \mathcal{T}\}.$ Then as a ${\mathcal G}^n-$module, $\mathcal{L} $ can be identified with \begin{equation}\label{last1}({\mathcal G}^n \otimes \mathcal A_n )\oplus(\mathcal{S}^n \otimes \mathcal{B}_n )\oplus({\mathcal V}^n \otimes{\mathcal C}_n )\oplus\mathcal D_n .\end{equation} Take \begin{equation}\label{identification}\varphi:\mathcal{L}\longrightarrow ({\mathcal G}^n \otimes \mathcal A_n )\oplus(\mathcal{S}^n \otimes \mathcal{B}_n )\oplus({\mathcal V}^n \otimes{\mathcal C}_n )\oplus\mathcal D_n \end{equation} to be the canonical identification. Next define $\mathcal A_\ell$ to be the vector subspace of $\mathcal A_n $ spanned by $\{a_i\mid i\in\mathcal{I}_0\},$ $\mathcal{B}_\ell$ to be the vector subspace of $\mathcal{B}_n $ spanned by $\{b_j\mid j\in\mathcal{J}_0\}$ and ${\mathcal C}_\ell$ to be the vector subspace of ${\mathcal C}_n $ spanned by $\{c_t\mid t\in \mathcal{T}_0\}.$ Then it follows from (\ref{last3-fin}) that as a ${\mathcal G}^\ell-$module, $\mathcal{L}^\ell=\mathcal{L}^{^S}$ can be identified with \begin{equation}\label{last2}({\mathcal G}^\ell\dot\otimes \mathcal A_\ell)\oplus(\mathcal{S}^\ell\dot\otimes \mathcal{B}_\ell)\oplus({\mathcal V}^\ell\dot\otimes{\mathcal C}_\ell)\oplus \mathcal D_\ell\end{equation} where $\mathcal D_\ell:=\varphi(E).$ \begin{comment} $\varphi(\mathfrak{g}_i)={\mathcal G}^\ell\otimes a_i.$ Indeed $\varphi((\mathfrak{g}_i)_\alpha)={\mathcal G}^n_\alpha\otimes a_i={\mathcal G}^\ell_\alpha\otimes a_i$ for all nonzero root $\alpha,$ in particular if $\alpha=\theta,$ the highest weight of ${\mathcal G}^S.$ Now as $\varphi$ is a $\mathfrak{g}-$module isomorphism and $\mathfrak{g}_i$ as a $\mathfrak{g}^S-$module is generated by $(\mathfrak{g}_i)_\theta,$ we are done. \end{comment}
In what follows using \cite[Thm. 2.48]{ABG2}, for $\mu=\ell,n,$ we give the algebraical structure of $\mathcal{L}^{^\mu}$ in terms of the ingredients involved in the decomposition of $\mathcal{L}^\mu$ into finite dimensional irreducible ${\mathcal G}^\mu-$modules. Set $\mathfrak{a}_\mu:=\mathcal A_\mu\oplus\mathcal{B}_\mu.$ Then there are a bilinear map $\cdot_\mu:\mathfrak{a}_\mu\times\mathfrak{a}_\mu\longrightarrow\mathfrak{a}_\mu$
and a linear map $*_\mu:\mathfrak{a}_\mu\longrightarrow\mathfrak{a}_\mu$ such that $(\mathfrak{a}_\mu,\cdot_\mu)$ is a unital associative algebra and $*_\mu$ is an involution on $\mathfrak{a}_\mu$ with $*_\mu$-fixed points $\mathcal A_\mu$ and $*_\mu$-skew fixed points $\mathcal{B}_\mu.$ Also there is a bilinear map $\cdot_\mu:\mathfrak{a}_\mu\times {\mathcal C}_\mu\longrightarrow{\mathcal C}_\mu$ such that $({\mathcal C}_\mu,\cdot_\mu)$ is a left unital associative $\mathfrak{a}_\mu$-module equipped with a skew-hermitian form $f_\mu:{\mathcal C}_\mu\times{\mathcal C}_\mu\longrightarrow \mathfrak{a}_\mu.$ Take $\frak{b}_\mu:=\frak{b}(\mathfrak{a}_\mu,*_\mu,{\mathcal C}_\mu,f_\mu)$ to be defined as in Subsection \ref{subsect2-1} and set $\cdot_\mu,$ $\circ_\mu,$ $[\cdot,\cdot]_\mu,$ $\diamond_\mu$ and $\hbox{\tiny$\heartsuit$}_\mu$ to be the corresponding features as $\cdot,$ $\circ,$ $[\cdot,\cdot],$ $\diamond$ and $\hbox{\tiny$\heartsuit$}$ defined in Subsection \ref{subsect2-1}. Also for $\beta,\beta'\in\frak{b}_\mu,$ set $d_{\beta,\beta'}^\mu:=d_{\beta,\beta'}^{\mu,\frak{b}_\mu}.$
By \cite[Theorems 2.48, 5.34]{ABG2}, $\mathcal D_\mu$ is a subalgebra of $\mathcal{L}^\mu$ and there is a subspace $\mathcal K_\mu$ of the full skew-dihedral homology group $$FH(\frak{b}_\mu)=\{\sum_{i}\{\beta_i,\beta'_i\}_\mu\mid \sum_{i}d^\mu_{\beta_i,\beta'_i}=0\}$$ of $\frak{b}_\mu$ such that $\mathcal D_\mu$ is isomorphic to the quotient algebra $\{\frak{b}_\mu,\frak{b}_\mu\}_\mu/\mathcal K_\mu.$ \begin{comment}note that $\mathcal D_\mu$ is nothing but the centeralizer of ${\mathcal G}^\mu$ in $\mathcal{L}_\mu.$\end{comment} For $\beta_1,\beta_2,$ take $\langle\beta_1,\beta_2\rangle_\mu$ to be the element of $\mathcal D_\mu$ corresponding to $\{\beta_1,\beta_2\}_\mu+\mathcal K_\mu$, then one has $\langle\mathcal A_\mu,\mathcal{B}_\mu\rangle_\mu=\langle\mathcal A_\mu,{\mathcal C}_\mu\rangle_\mu=\langle\mathcal{B}_\mu,{\mathcal C}_\mu\rangle_\mu=\{0\}$ and $\mathcal D_\mu=\langle\mathcal A_\mu,\mathcal A_\mu\rangle_\mu+\langle\mathcal{B}_\mu,\mathcal{B}_\mu\rangle_\mu+\langle{\mathcal C}_\mu,{\mathcal C}_\mu\rangle_\mu.$ Moreover the Lie bracket on $\mathcal{L}^\mu$ which is an extension of the Lie bracket on $\mathcal D_\mu$ is given by \begin{equation}\label{probc-gen-mu} \begin{array}{l} \;[x\otimes a,y\otimes a']=[x,y]\otimes\frac{1}{2}(a\circ_\mu a')+ (x\circ_\mu y)\otimes\frac{1}{2}[a,a']_\mu+tr(xy)\langle a,a'\rangle_\mu,
\\ \;[x\otimes a,s\otimes b]= (x\circ_\mu s)\otimes\frac{1}{2}[a,b]_\mu+[x,s]\otimes\frac{1}{2}(a\circ_\mu b)=-[s\otimes b,x\otimes a],
\\ \;[s\otimes b,t\otimes b']=[s,t]\otimes\frac{1}{2}(b\circ_\mu b')+ (s\circ_\mu t)\otimes\frac{1}{2}[b,b']_\mu+tr(st)\langle b,b'\rangle_\mu,
\\ \;[x\otimes a,u\otimes c]=xu\otimes a\cdot_\mu c=-[u\otimes c,x\otimes a],
\\ \;[s\otimes b,u\otimes c]=su\otimes b\cdot_\mu c=-[u\otimes c,s\otimes b],
\\ \;[u\otimes c,v\otimes c']=(u\circ v)\otimes (c\diamond_\mu c')+ [u, v]_\mu\otimes (c\hbox{\tiny$\heartsuit$}_\mu c')+(u,v)\langle c,c'\rangle_\mu,
\\ \;[\langle\beta,\beta'\rangle_\mu,x\otimes a]=x\otimes d^\mu_{\beta,\beta'}(a)=-[x\otimes a,\langle\beta,\beta'\rangle_\mu],
\\ \;[\langle\beta,\beta'\rangle_\mu,s\otimes b]=s\otimes d^\mu_{\beta,\beta'}(b)=-[s\otimes b,\langle\beta,\beta'\rangle_\mu],
\\ \;[\langle\beta,\beta'\rangle_\mu,u\otimes c]=u\otimes d^\mu_{\beta,\beta'}(c)=-[u\otimes c,\langle\beta,\beta'\rangle_\mu],
\\ \;[\langle\beta_1,\beta_2\rangle_\mu,\langle\beta'_1,\beta'_2\rangle_\mu]=\langle d^\mu_{\beta_1,\beta_2}(\beta'_1),\beta'_2\rangle_\mu+\langle\beta'_1,d^\mu_{\beta_1,\beta_2}(\beta'_2)\rangle_\mu, \end{array}
\end{equation} for $x,y\in{\mathcal G}^\mu,$ $s,t\in\mathcal{S}^\mu,$ $u,v\in{\mathcal V}^\mu,$ $a,a'\in\mathcal A_\mu,$ $b,b'\in\mathcal{B}_\mu,$ $c,c'\in{\mathcal C}_\mu$ and $\beta,\beta'\in\frak{b}_\mu.$
\begin{Lemma}\label{divide2} We have ${\mathcal I}={\mathcal I}_0,$ $\mathcal{J}=\mathcal{J}_0$ and ${\mathcal T}={\mathcal T}_0.$\end{Lemma} \noindent{\bf Proof. } It follows from (\ref{probc-gen-mu}), (\ref{last1}) and (\ref{last2}) that $$ \mathcal{L}_\alpha=\left\{ \begin{array}{ll} {\mathcal V}^n_\alpha\dot\otimes {\mathcal C}_n &\hbox{if $\alpha\in R_{sh}$} \\ ({\mathcal G}^n_\alpha\dot\otimes\mathcal A_n )\oplus(\mathcal{S}^n_\alpha\dot\otimes\mathcal{B}_n )& \hbox{if $\alpha\in R_{lg}$}\\ {\mathcal G}^n_{\alpha}\dot\otimes\mathcal A_n &\hbox{if $\alpha\in R_{ex}$} \end{array} \right.$$ and$$ (\mathcal{L}^{^S})_\alpha=\left\{ \begin{array}{ll} ({\mathcal V}^\ell)_\alpha\dot\otimes{\mathcal C}_\ell&\hbox{if $\alpha\in S_{sh}$} \\ ({\mathcal G}^\ell_\alpha\dot\otimes\mathcal A_\ell)\oplus(\mathcal{S}^\ell_\alpha\dot\otimes\mathcal{B}_\ell)& \hbox{if $\alpha\in S_{lg}$}\\ {\mathcal G}^\ell_{\alpha}\dot\otimes\mathcal A_\ell&\hbox{if $\alpha\in S_{ex}.$} \end{array} \right. $$
Now fix $\alpha\in S_{ex},$ then $$({\mathcal G}^\ell)_{\alpha}\dot\otimes\mathcal A_\ell=(\mathcal{L}^{^S})_\alpha=\mathcal{L}_\alpha={\mathcal G}^n_{\alpha}\dot\otimes\mathcal A_n .$$ This together with the fact that ${\mathcal G}^\ell_{\alpha}={\mathcal G}^n_{\alpha}$ is a one dimensional vector space, implies that the vector space $\mathcal A_\ell$ equals the vector space $\mathcal A_n. $ In particular we get ${\mathcal I}={\mathcal I}_0.$ Next fix $\alpha\in S_{sh},$ then we have $${\mathcal V}^\ell_{\alpha}\dot\otimes{\mathcal C}_\ell=\mathcal{L}^{^S}_\alpha=\mathcal{L}_\alpha={\mathcal V}^n_{\alpha}\dot\otimes{\mathcal C}_n. $$ This as above, implies that
${\mathcal T}={\mathcal T}_0.$ Finally fix $\alpha\in S_{lg},$ then $$({\mathcal G}^\ell_{\alpha}\dot\otimes\mathcal A_\ell)\oplus(\mathcal{S}^\ell_\alpha\dot\otimes\mathcal{B}_\ell)=\mathcal{L}^{^S}_\alpha=\mathcal{L}_\alpha= ({\mathcal G}^n_{\alpha}\dot\otimes\mathcal A_n )\oplus(\mathcal{S}^n_\alpha\dot\otimes\mathcal{B}_n ).$$ Now as $\mathcal{S}^\ell_\alpha=\mathcal{S}^n_\alpha$ is a one dimensional vector space, ${\mathcal G}^\ell_{\alpha}={\mathcal G}^n_{\alpha},$ $\mathcal{B}_\ell\subseteq\mathcal{B}_n$ and $\mathcal A_\ell=\mathcal A_n ,$ we get that the two vector spaces $\mathcal{B}_\ell$ and $\mathcal{B}_n $ are equal and so $\mathcal{J}=\mathcal{J}_0.$\qed
As we have already seen, on the vector space level, we have $$\mathcal A_\ell=\mathcal A_n, \mathcal{B}_\ell=\mathcal{B}_n, {\mathcal C}_\ell={\mathcal C}_n$$ which in turn implies that the vector space $\frak{b}_\ell$ equals the vector space $\frak{b}_n.$ In the following lemma, we show, in addition, that the algebras $\frak{b}_\ell$ and $\frak{b}_n$ have the same algebraic structure.
\begin{Lemma}\label{divide3} The algebraic structure of $\frak{b}_n$ coincides with the algebraic structure of $\frak{b}_\ell.$ \end{Lemma} \noindent{\bf Proof. } Using Lemma \ref{divide2}, we set \begin{equation} \label{end8} \mathcal A:=\mathcal A_\ell=\mathcal A_n,\;\;\mathcal{B}:=\mathcal{B}_\ell=\mathcal{B}_n,\;\;{\mathcal C}:={\mathcal C}_\ell={\mathcal C}_n . \end{equation}
Suppose that $i,j,k$ are distinct elements of $I_0.$ Take $x:=e_{i,j}-e_{\bar j,\bar i}\in{\mathcal G}^\ell$ and $y:=e_{ j,k}-e_{\bar k,\bar j}\in{\mathcal G}^\ell,$ then $$tr(xy)=0,\;[x,y]=e_{i,k}-e_{\bar k,\bar i},\;x\circ_n y=x\circ_\ell y=e_{i,k}+e_{\bar k,\bar i}.$$ Now for $a,a'\in\mathcal A ,$ by (\ref{probc-gen-mu}), we have {\small\begin{eqnarray*} [x,y]\otimes\frac{1}{2}(a\circ_n a')+ (x\circ_n y)\otimes\frac{1}{2}[a,a']_n \hspace{-.2cm}&=&\hspace{-.2cm}[x\otimes a,y\otimes a']\\ \hspace{-.2cm}&=&\hspace{-.2cm}[x,y]\otimes\frac{1}{2}(a\circ_\ell a')+ (x\circ_\ell y)\otimes\frac{1}{2}[a,a']_\ell. \end{eqnarray*}} This in turn implies that $$[x,y]\otimes(\frac{1}{2}(a\circ_n a')-\frac{1}{2}(a\circ_\ell a'))= (x\circ_n y)\otimes(\frac{1}{2}[a,a']_\ell-\frac{1}{2}[a,a']_n),$$ but the left hand side is an element of ${\mathcal G}\otimes \mathcal A$ and the right hand side is an element of $\mathcal{S}\otimes \mathcal{B}.$ Therefore as $[x,y]\neq 0$ and $x\circ_n y\neq 0,$ we get that $$\frac{1}{2}[a,a']_\ell-\frac{1}{2}[a,a']_n =0\quad\hbox{and}\quad \frac{1}{2}(a\circ_n a')-\frac{1}{2}(a\circ_\ell a')=0.$$ This now implies that \begin{equation}\label{gen-bc1} a\cdot_\ell a'=a\cdot_n a';\;\; a,a'\in \mathcal A. \end{equation}
Next take $i$ and $ j$ to be two distinct elements of $I_0.$ Set $s:=e_{i,j}+e_{\bar j,\bar i}\in\mathcal{S}^\ell$ and $x:=e_{j,\bar j}\in{\mathcal G}^\ell,$ then we have $$tr(xs)=0,\;[x,s]=e_{j,\bar i}-e_{i,\bar j},\;x\circ_\ell s=x\circ_n s=e_{j,\bar i}+e_{i,\bar j}.$$
Now for $a\in\mathcal A$ and $b\in\mathcal{B},$ by (\ref{probc-gen-mu}), we have {\small\begin{eqnarray*} (x\circ_\ell s)\otimes \frac{1}{2}[a,b]_\ell+[x,s]\otimes \frac{1}{2} (a\circ_\ell b)&=&[x\otimes a,s\otimes b]\\ &=&(x\circ_n s)\otimes \frac{1}{2}[a,b]_n +[x,s]\otimes \frac{1}{2} (a\circ_n b). \end{eqnarray*}} This implies that $$(x\circ_\ell s)\otimes( \frac{1}{2}[a,b]_\ell-\frac{1}{2}[a,b]_n )=[x,s]\otimes (\frac{1}{2} (a\circ_n b)-\frac{1}{2} (a\circ_\ell b)).$$ Now as before,
one gets that $$\frac{1}{2}[a,b]_\ell-\frac{1}{2}[a,b]_n =0\quad\hbox{and}\quad \frac{1}{2} (a\circ_n b)-\frac{1}{2} (a\circ_\ell b)=0.$$ This in particular implies that \begin{equation}\label{coin-bc2} a\cdot_\ell b=a\cdot_n b\quad\hbox{and}\quad b\cdot_\ell a=b\cdot_n a;\;\;\;\;\;\;\;\;(a\in\mathcal A,\; b\in\mathcal{B}). \end{equation}
Finally, for distinct fixed elements $i,j,k$ of $I_0,$ set $s:=e_{i,\bar j}-e_{j,\bar i},t=e_{\bar i,k}-e_{\bar k,i}\in\mathcal{S}^\ell.$ Then $$tr(st)=0,\;[s,t]=-e_{j,k}+e_{\bar k,\bar j},\;s\circ_\ell t=s\circ_n t=-e_{j,k}-e_{\bar k,\bar j}.$$ Therefore for $b,b'\in\mathcal{B} ,$ by (\ref{probc-gen-mu}), we have {\small\begin{eqnarray*} ([s,t]\otimes\frac{1}{2}(b\circ_\ell b'))+ ((s\circ_\ell t)\otimes \frac{1}{2}[b,b']_\ell)\hspace{-2mm}&=&\hspace{-2mm}[s\otimes b,t\otimes b']\\\hspace{-2mm}&=&\hspace{-2mm}([s,t]\otimes\frac{1}{2}(b\circ_n b'))+ ((s\circ_n t)\otimes \frac{1}{2}[b,b']_n ). \end{eqnarray*}} This implies that $$[s,t]\otimes(\frac{1}{2}(b\circ_\ell b')-\frac{1}{2}(b\circ_n b'))=(s\circ_n t)\otimes (\frac{1}{2}[b,b']_n -\frac{1}{2}[b,b']_\ell).$$ Therefore we get $$\frac{1}{2}(b\circ_\ell b')-\frac{1}{2}(b\circ_n b')=0\quad\hbox{and}\quad \frac{1}{2}[b,b']_n -\frac{1}{2}[b,b']_\ell=0$$ and so we have $b\cdot_\ell b'=b\cdot_n b'$ which together with (\ref{gen-bc1}) and (\ref{coin-bc2}) implies that \begin{equation}\label{gen-bc3} \parbox{4in}{\begin{center}$\mathfrak{a}_\ell=\mathfrak{a}_n $ (as two algebras).\end{center}} \end{equation}
Now take $x\in{\mathcal G}^\ell,$ $s\in\mathcal{S}^\ell$ and $u,v\in{\mathcal V}^\ell$ to be such that $xu\neq0$ and $sv\neq0.$ Then for $a\in\mathcal A,$ $b\in\mathcal{B}$ and $c\in{\mathcal C},$ we get using (\ref{probc-gen-mu}) that \begin{eqnarray*} xu\otimes a\cdot_\ell c&=&[x\otimes a, u\otimes c]=xu\otimes a\cdot_n c,\\ sv\otimes b\cdot_\ell c&=&[s\otimes b,v\otimes c]=sv\otimes b\cdot_n c. \end{eqnarray*} This implies that $$a\cdot_\ell c=a\cdot_n c\quad\hbox{and}\quad b\cdot_\ell c=b\cdot_n c$$ for $a\in\mathcal A,$ $b\in\mathcal{B}$ and $c\in{\mathcal C}.$ Therefore we have \begin{equation}\label{gen-coin7} {\mathcal C}_\ell={\mathcal C}_n \hbox{ (as two $\mathfrak{a}_n-$modules). } \end{equation} Now we are done thanks to (\ref{gen-bc3}) and (\ref{gen-coin7}). \qed
Now using Lemma \ref{divide3}, we set $$\mathfrak{a}:=\mathfrak{a}_n=\mathfrak{a}_\ell\quad\hbox{and}\quad \frak{b}:=\frak{b}_\ell=\frak{b}_n.$$ Also for $\beta,\beta'\in\frak{b},$ we take \begin{equation}\label{setare}\begin{array}{l}\;\beta\cdot\beta':=\beta\cdot_n\beta'=\beta\cdot_\ell\beta',\\ \;[\beta,\beta']:=\beta\cdot\beta'-\beta'\cdot\beta,\\ \;\beta\circ\beta':=\beta\cdot\beta'+\beta'\cdot\beta.\end{array}\end{equation}
\begin{Lemma}\label{divide4} For $a,a'\in\mathcal A,$ and $b,b'\in\mathcal{B},$ we have $$\begin{array}{l}\langle a,a'\rangle_n =(\frac{-1}{\ell}Id_{_{{\mathcal V}^{^\ell}}}+\frac{1}{n }Id_{_{\mathcal V}} )\otimes\frac{1}{2}[a,a'])+\langle a,a'\rangle_\ell,\\ \langle b,b'\rangle_n =(\frac{-1}{\ell}Id_{_{{\mathcal V}^{^\ell}}}+\frac{1}{n }Id_{_{\mathcal V}} )\otimes\frac{1}{2}[b,b'])+\langle b,b'\rangle_\ell. \end{array}$$
Also for $c,c'\in {\mathcal C},$ $f_\ell(c,c')=f_n(c,c'),$ $c\diamond_\ell c'=c\diamond_n c'$ and $c\hbox{\tiny$\heartsuit$}_\ell c'=c\hbox{\tiny$\heartsuit$}_n c'.$ Moreover we have \begin{eqnarray*}\langle c,c'\rangle_n &=&((\frac{1}{\ell}Id_{_{{\mathcal V}^{^\ell}}}-\frac{1}{n }Id_{_{\mathcal V}} )\otimes\frac{1}{2} c\hbox{\tiny$\heartsuit$}_n c')+\langle c,c'\rangle_\ell\\&=&((\frac{1}{\ell}Id_{_{{\mathcal V}^{^\ell}}}-\frac{1}{n }Id_{_{\mathcal V}} )\otimes\frac{1}{2} c\hbox{\tiny$\heartsuit$}_\ell c')+\langle c,c'\rangle_\ell.\end{eqnarray*}\end{Lemma} \noindent{\bf Proof. } Fix $x,y\in{\mathcal G}^{\ell}$ such that $tr(xy)\neq 0.$ For $a,a'\in\mathcal A,$ consider (\ref{setare}) and use (\ref{probc-gen-mu}) to get \begin{eqnarray*} &&([x,y]\otimes\frac{1}{2}(a\circ a'))+((x\circ_n y)\otimes\frac{1}{2}[a,a'])+tr(xy)\langle a,a'\rangle_n =[x\otimes a,y\otimes a']=\\ &&([x,y]\otimes\frac{1}{2}(a\circ a'))+((x\circ_\ell y)\otimes\frac{1}{2}[a,a'])+tr(xy)\langle a,a'\rangle_\ell. \end{eqnarray*} This implies that $$(\frac{-tr(xy)}{n }Id_{_{\mathcal V}} \otimes\frac{1}{2}[a,a'])+tr(xy)\langle a,a'\rangle_n =(\frac{-tr(xy)}{\ell}Id_{_{{\mathcal V}^{^\ell}}}\otimes\frac{1}{2}[a,a'])+tr(xy)\langle a,a'\rangle_\ell.$$ Therefore we have \begin{equation}\label{coincide1}\langle a,a'\rangle_n =(\frac{-1}{\ell}Id_{_{{\mathcal V}^{^\ell}}}+\frac{1}{n }Id_{_{\mathcal V}} )\otimes\frac{1}{2}[a,a'])+\langle a,a'\rangle_\ell;\;\;\;\;\;\; (a,a'\in\mathcal A).\end{equation}
Next fix $s,t\in\mathcal{S}^\ell$ such that $tr(st)\neq 0,$ then for $b,b'\in\mathcal{B},$ by (\ref{probc-gen-mu}), we have \begin{eqnarray*} &&([s,t]\otimes \frac{1}{2} b\circ b')+((s\circ_n t)\otimes \frac{1}{2}[b,b'])+tr(st)\langle b,b'\rangle_n =[s\otimes b,t\otimes b']=\\ &&([s,t]\otimes \frac{1}{2} b\circ b')+((s\circ_\ell t)\otimes \frac{1}{2}[b,b'])+tr(st)\langle b,b'\rangle_\ell. \end{eqnarray*} This implies that $$(\frac{-tr(st)}{n } Id_{_{{\mathcal V}}} \otimes \frac{1}{2}[b,b'])+tr(st)\langle b,b'\rangle_n =(\frac{-tr(st)}{\ell} Id_{_{{\mathcal V}^\ell}}\otimes \frac{1}{2}[b,b'])+tr(st)\langle b,b'\rangle_\ell$$ which in turn implies that \begin{equation}\label{coin-gen1} \langle b,b'\rangle_n =((\frac{-1}{\ell} Id_{_{{\mathcal V}^\ell}}+\frac{1}{n }Id_{_{\mathcal V}} )\otimes \frac{1}{2}[b,b'])+\langle b,b'\rangle_\ell;\;\;\;\;\;(b,b'\in\mathcal{B}) .\end{equation}
Now suppose that $i$ and $j$ are two distinct elements of $I_0.$ Take $u:=v_i$ and $v:=v_{\bar j},$ then $(u,v)=0$ and so $[u,v]_n=[u,v] .$ Therefore for all $c,c'\in{\mathcal C},$ by (\ref{probc-gen-mu}), we have \begin{eqnarray*} (u\circ v )\otimes (c\diamond_n c')+[u,v]\otimes (c\hbox{\tiny$\heartsuit$}_n c')&=&[u\otimes c,v\otimes c']\nonumber\\ &=&(u\circ v)\otimes (c\diamond_\ell c')+[u,v] \otimes c\hbox{\tiny$\heartsuit$}_\ell c'.\nonumber \end{eqnarray*} But $u\circ v\in{\mathcal G},$ $[u,v] $ is a nonzero element of $\mathcal{S},$ $c\diamond_\ell c',c\diamond_n c'\in\mathcal A$ and $c\hbox{\tiny$\heartsuit$}_\ell c',c\hbox{\tiny$\heartsuit$}_n c'\in \mathcal{B},$ so we get that \begin{equation}\label{last5} c\diamond c':=c\diamond_\ell c'=c\diamond_n c' \quad\hbox{and}\quad c\hbox{\tiny$\heartsuit$} c':=c\hbox{\tiny$\heartsuit$}_\ell c'=c\hbox{\tiny$\heartsuit$}_n c';\;\;\;\;\;\; (c,c'\in{\mathcal C}). \end{equation} This in turn implies that \begin{equation} \label{gen-coin5} f(c,c'):= f_\ell(c,c')=f_n(c,c');\;\;\;\;\; (c,c'\in{\mathcal C}) . \end{equation} Next for an element $i$ of $I,$ take $u:=v_i$ and $v=v_{\bar i}.$ Then for $c,c'\in{\mathcal C},$ by (\ref{probc-gen-mu}), we have \begin{eqnarray*} &&(u\circ v\otimes c\diamond c')+([u,v]_n\otimes c\hbox{\tiny$\heartsuit$} c')+\langle c,c'\rangle_n=[u\otimes c,v\otimes c']=\nonumber\\ &&(u\circ v\otimes c\diamond c')+([u,v]_\ell \otimes c\hbox{\tiny$\heartsuit$} c')+\langle c,c'\rangle_\ell ,\nonumber \end{eqnarray*} using which, one concludes that \begin{equation}\label{gen-coin6} \langle c,c'\rangle_n =\langle c,c'\rangle_\ell+(\frac{1}{2\ell}Id_{_{\mathcal V}}-\frac{1}{2n }Id_{_{{\mathcal V}^\ell}} )\otimes c\hbox{\tiny$\heartsuit$} c';\;\; \;\;\;\;(c,c'\in{\mathcal C}). \end{equation} This completes the proof.\qed
\begin{cor}\label{cor1} Let $\ell<n$ and suppose that $t\in\mathbb{N},$ $a_i,a'_i\in\mathcal A,$ $b_i,b'_i\in\mathcal{B}$ and $c_i,c'_i\in{\mathcal C}$ for $1\leq i\leq t.$ Then $\sum_{i=1}^t(\langle a_i,a'_i\rangle_\ell+\langle b_i,b'_i\rangle_\ell+\langle c_i,c'_i\rangle_\ell)=0$ if and only if $\sum_{i=1}^t([a_i,a'_i]+[b_i,b_i']-c_i\hbox{\tiny$\heartsuit$} c_i')=0$ and $\sum_{i=1}^t(\langle a_i,a'_i\rangle_n+\langle b_i,b'_i\rangle_n+\langle c_i,c'_i\rangle_n)=0.$ \end{cor} \proof By Lemma \ref{divide4}, we have \begin{eqnarray*} &&\sum_{i=1}^t(\langle a_i,a'_i\rangle_\ell+\langle b_i,b'_i\rangle_\ell+\langle c_i,c'_i\rangle_\ell)=\\ &&\sum_{i=1}^t(\langle a_i,a'_i\rangle_n+\langle b_i,b'_i\rangle_n+\langle c_i,c'_i\rangle_n)-\\ &&(\frac{-1}{\ell}Id_{_{{\mathcal V}^{^\ell}}}+\frac{1}{n }Id_{_{\mathcal V}} )\otimes\frac{1}{2}\sum_{i=1}^t([ a_i,a'_i]+[b_i,b'_i]-c_i\hbox{\tiny$\heartsuit$} c'_i). \end{eqnarray*} Now as $\sum_{i=1}^t(\langle a_i,a'_i\rangle_n+\langle b_i,b'_i\rangle_n+\langle c_i,c'_i\rangle_n)\in\mathcal D_n$ and $(\frac{-1}{\ell}Id_{_{{\mathcal V}^{^\ell}}}+\frac{1}{n }Id_{_{\mathcal V}} )\otimes\frac{1}{2}\sum_{i=1}^t([ a_i,a'_i]+[b_i,b'_i]-c_i\hbox{\tiny$\heartsuit$} c'_i)\in\mathcal{S}\otimes\mathcal{B},$ we are done.\qed
\begin{rem}\label{rem1} {\rm Consider the decomposition (\ref{last3-fin}) for $\mathcal{L}^{^S}=\mathcal{L}^\ell$ into finite dimensional irreducible $\mathfrak{g}^{^S}-$submodules and the decomposition of $\mathcal{L}=\mathcal{L}^n$ into finite dimensional irreducible $\mathfrak{g}-$submodules as in Lemma \ref{divide1}, then contemplating the identification (\ref{identification}), we have using Lemma \ref{divide4} that $$\mathcal{L}=(\bigoplus_{i\in{\mathcal I}}{\mathcal G}_i\oplus\bigoplus_{j\in\mathcal{J}}\mathcal{S}_j\oplus\bigoplus_{t\in{\mathcal T}}{\mathcal V}_t)+ E.$$ Moreover setting $\langle\beta,\beta'\rangle^n:=\varphi^{-1}(\langle\beta,\beta'\rangle_n)$ and $\langle\beta,\beta'\rangle^\ell:=\varphi^{-1}(\langle\beta,\beta'\rangle_\ell)$ for $\beta,\beta'\in \frak{b},$ we get that $\{\langle\beta,\beta'\rangle^n\mid \beta,\beta'\in\frak{b}\}$ spans $\mathcal D_n$ and that $\{\langle\beta,\beta'\rangle^\ell\mid \beta,\beta'\in\frak{b}\}$ spans $E.$ Furthermore, thanks to Corollary \ref{cor1}, for $t\in\mathbb{N},$ $a_i,a'_i\in\mathcal A,$ $b_i,b'_i\in\mathcal{B}$ and $c_i,c'_i\in{\mathcal C}$ ($1\leq i\leq t$), $\sum_{i=1}^t(\langle a_i,a'_i\rangle^\ell+\langle b_i,b'_i\rangle^\ell+\langle c_i,c'_i\rangle^\ell)=0$ if and only if $\sum_{i=1}^t([a_i,a'_i]+[b_i,b_i']-c_i\hbox{\tiny$\heartsuit$} c_i')=0$ and $\sum_{i=1}^t(\langle a_i,a'_i\rangle^n+\langle b_i,b'_i\rangle^n+\langle c_i,c'_i\rangle^n)=0.$ } \end{rem}
\begin{Proposition} \label{divide5} For $e,f\in{\mathcal G}\cup\mathcal{S},$ set $$e\circ f:=ef+fe-\frac{tr(ef)}{\ell}Id_{_{{\mathcal V}^\ell}}.$$ Also for $\beta_1=a_1+b_1+c_1\in\frak{b}$ and $\beta_2=a_2+b_2+c_2\in\frak{b}$ with $a_1,a_2\in\mathcal A,$ $b_1,b_2\in\mathcal{B}$ and $c_1,c_2\in{\mathcal C},$ we recall from (\ref{beta*}) that $\beta_{_{\beta_1,\beta_2}}^*:=[a_1,a_2]+[b_1,b_2]-c_1\hbox{\tiny$\heartsuit$} c_2$ and set $$\langle\beta_1,\beta_2\rangle:=\langle\beta_1,\beta_2\rangle_\ell,\;\;\beta_1^*=c_1,\;\;\beta_2^*=c_2$$ and take $$\mathcal D:=\hbox{span}\{\langle a,a'\rangle,\langle b,b'\rangle,\langle c,c'\rangle\mid a,a'\in\mathcal A,\;b,b'\in\mathcal{B},c,c'\in{\mathcal C}\},$$ then contemplating (\ref{last5}), we have \begin{eqnarray*}\mathcal{L}&=& ({\mathcal G}\otimes \mathcal A)\oplus(\mathcal{S}\otimes \mathcal{B})\oplus({\mathcal V}\otimes{\mathcal C})\oplus\langle\frak{b},\frak{b}\rangle_n\nonumber\\&=& (({\mathcal G}\otimes \mathcal A)\oplus(\mathcal{S}\otimes \mathcal{B})\oplus({\mathcal V}\otimes{\mathcal C}))+\mathcal D\end{eqnarray*} with the Lie bracket given by{\small \begin{equation}\label{probc-fin} \begin{array}{l} \;[x\otimes a,y\otimes a']=[x,y]\otimes\frac{1}{2}(a\circ a')+ (x\circ y)\otimes\frac{1}{2}[a,a']+tr(xy)\langle a,a'\rangle,
\\ \;[x\otimes a,s\otimes b]= (x\circ s)\otimes\frac{1}{2}[a,b]+[x,s]\otimes\frac{1}{2}(a\circ b)=-[s\otimes b,x\otimes a],
\\ \;[s\otimes b,t\otimes b']=[s,t]\otimes\frac{1}{2}(b\circ b')+ (s\circ t)\otimes\frac{1}{2}[b,b']+tr(st)\langle b,b'\rangle,
\\ \;[x\otimes a,u\otimes c]=xu\otimes a\cdot c=-[u\otimes c,x\otimes a],
\\ \;[s\otimes b,u\otimes c]=su\otimes b\cdot c=-[u\otimes c,s\otimes b],
\\ \;[u\otimes c,v\otimes c']=(u\circ v)\otimes (c\diamond c')+ [u, v]\otimes (c\hbox{\tiny$\heartsuit$} c')+(u,v)\langle c,c'\rangle,
\\ \;[\langle \beta_1,\beta_2\rangle,x\otimes a]= \frac{-1}{4\ell}(x\circ Id_{_{{\mathcal V}^\ell}}\otimes[a,\beta_{_{\beta_1,\beta_2}}^*]+[x,Id_{_{{\mathcal V}^\ell}}]\otimes a\circ \beta_{_{\beta_1,\beta_2}}^*),
\\ \;[\langle \beta_1,\beta_2\rangle,\hspace{-1mm}s\otimes b]\hspace{-1mm}=\hspace{-1mm}\frac{-1}{4\ell}([s,Id_{_{{\mathcal V}^\ell}}\hspace{-.5mm}]\hspace{-1mm}\otimes\hspace{-1mm} (b\circ \beta_{_{\beta_1,\beta_2}}^*\hspace{-1mm})\hspace{-1mm}+\hspace{-1mm}(s\circ Id_{_{{\mathcal V}^\ell}})\hspace{-1mm}\otimes \hspace{-1mm}[b, \beta_{_{\beta_1,\beta_2}}^*\hspace{-1mm}]\hspace{-1mm}+\hspace{-1mm}2tr(sId_{{\mathcal V}^\ell})\langle b,\beta_{_{\beta_1,\beta_2}}^*\hspace{-.5mm}\rangle),
\\ \;[\langle \beta_1,\beta_2\rangle,v\otimes c]=\frac{1}{2\ell}Id_{_{{\mathcal V}^\ell}}v\otimes (\beta_{_{\beta_1,\beta_2}}^*\cdot c)-\frac{1}{2}v\otimes (f(c,\beta^*_2)\cdot \beta^*_1+f(c,\beta^*_1)\cdot \beta^*_2)\\ \;[\langle\beta_1,\beta_2\rangle,\langle\beta'_1,\beta'_2\rangle]=\langle d^\ell_{\beta_1,\beta_2}(\beta'_1),\beta'_2\rangle+\langle\beta'_1,d^\ell_{\beta_1,\beta_2}(\beta'_2)\rangle \end{array}
\end{equation}} for $x,y\in{\mathcal G},$ $s,t\in\mathcal{S},$ $u,v\in{\mathcal V},$ $a,a'\in\mathcal A,$ $b,b'\in\mathcal{B},$ $c,c'\in{\mathcal C},$ $\beta_1,\beta_2,\beta_1',\beta'_2\in\frak{b}.$ \end{Proposition}
\noindent{\bf Proof. } Suppose that $x,y\in{\mathcal G},$ $s,t\in\mathcal{S}, $ $u,v\in{\mathcal V}, $ $a,a'\in\mathcal A,$ $b,b'\in\mathcal{B} $ and $c,c'\in{\mathcal C},$ then (\ref{probc-gen-mu}) (for $\mu=n$) together with Lemma \ref{divide4} implies that {\small\begin{eqnarray} [x\otimes a,y\otimes a']&=&([x,y]\otimes \frac{1}{2}(a\circ a'))+((x\circ_n y)\otimes \frac{1}{2}[a,a'])+tr(xy)\langle a,a'\rangle_{n}\nonumber\\ &=&([x,y]\otimes \frac{1}{2}(a\circ a'))+((x\circ_n y)\otimes \frac{1}{2}[a,a'])\nonumber\\&+&tr(xy)(\frac{-1}{\ell} Id_{_{{\mathcal V}^{^\ell}}}+\frac{1}{n }Id_{_{\mathcal V}} )\otimes\frac{1}{2}[a,a'])+tr(xy)\langle a,a'\rangle_\ell\label{pro-bc-gen-a}\\ &=&([x,y]\otimes \frac{1}{2}(a\circ a'))\nonumber\\&+&(((x\circ_n y)+tr(xy)(\frac{-1}{\ell}Id_{_{{\mathcal V}^{^\ell}}}+\frac{1}{n } Id_{_{\mathcal V}} ))\otimes \frac{1}{2}[a,a'])+tr(xy)\langle a,a'\rangle_\ell\nonumber\\ &=&([x,y]\otimes \frac{1}{2}(a\circ a'))+((x\circ y)\otimes \frac{1}{2}[a,a'])+tr(xy)\langle a,a'\rangle.\nonumber \end{eqnarray}} Similarly we have $$[s\otimes b,t\otimes b']=([s,t]\otimes \frac{1}{2}(b\circ b'))+((s\circ t)\otimes \frac{1}{2}[b,b'])+tr(st)\langle b,b'\rangle$$ and $$[u\otimes c,v\otimes c']=((u\circ v)\otimes c\diamond c')+([u,v]\otimes (c\hbox{\tiny$\heartsuit$} c'))+(u,v)\langle c,c'\rangle.$$
Now for $a_1,a_2\in\mathcal A,$ $b_1,b_2\in\mathcal{B}$ and $c_1,c_2\in{\mathcal C},$ set $b^*:=[a_1,a_2]+[b_1,b_2]-c_1\hbox{\tiny$\heartsuit$} c_2]$ and take $s_n:=(1/\ell)Id_{_{{\mathcal V}^\ell}}-(1/n)Id_{_{{\mathcal V}}}.$ Then for $x\in{\mathcal G},$ $s\in\mathcal{S},$ $v\in{\mathcal V},$
$a\in\mathcal A,$ $b\in \mathcal{B},$ and $c\in{\mathcal C},$ one can see that {\small $$[x\otimes a,s_n\otimes b^*]=\frac{1}{2\ell}(x\circ Id_{_{{\mathcal V}^\ell}}\otimes[a,b^*]+[x,Id_{_{{\mathcal V}^\ell}}]\otimes a\circ b^*)-\frac{1}{n}x\otimes[a,b^*],$$}
{\small $$[s\otimes b,s_n\otimes b^*]=\frac{1}{2\ell}([s,Id_{_{{\mathcal V}^\ell}}]\otimes b\circ b^*+(s\circ Id_{_{{\mathcal V}^\ell}})\otimes [b, b^*])-\frac{1}{n}s \otimes [b, b^*]\\ +tr(ss_n)\langle b,b^*\rangle$$ and $$[s_n\otimes b^*,v\otimes c]= \frac{1}{\ell}Id_{_{{\mathcal V}^\ell}}v\otimes b^*\cdot c-\frac{1}{n}v\otimes b^*\cdot c.$$}
We next note that {\small \begin{eqnarray*}[\langle a_1,a_2\rangle_n+\langle b_1,b_2\rangle_n+\langle c_1,c_2\rangle_n,x\otimes a]&=&x\otimes (d^n_{a_1,a_2}+d^n_{b_1,b_2}+d^n_{c_1,c_2})(a)\\ &=&\frac{1}{2n}x\otimes[b^*,a]\\ \;[\langle a_1,a_2\rangle_n+\langle b_1,b_2\rangle_n+\langle c_1,c_2\rangle_n,s\otimes b]&=&s\otimes (d^n_{a_1,a_2}+d^n_{b_1,b_2}+d^n_{c_1,c_2})(b)\\ &=&\frac{1}{2n}s\otimes[b^*,b]\\ \;[\langle a_1,a_2\rangle_n+\langle b_1,b_2\rangle_n+\langle c_1,c_2\rangle_n,v\otimes c]&=&v\otimes (d^n_{a_1,a_2}+d^n_{b_1,b_2}+d^n_{c_1,c_2})(c)\\ &=&\frac{1}{2n}v\otimes b^*\cdot c\\&-&v\otimes \frac{1}{2}(f(c,c_2)\cdot c_1+f(c,c_1)\cdot c_2). \end{eqnarray*}} Therefore using Lemma \ref{divide4}, an easy verification gives that {\small $$[\langle a_1,a_2\rangle+\langle b_1,b_2\rangle+\langle c_1,c_2\rangle,x\otimes a]=-\frac{1}{4\ell}(x\circ Id_{_{{\mathcal V}^\ell}}\otimes[a,b^*]+[x,Id_{_{{\mathcal V}^\ell}}]\otimes a\circ b^*),$$
\begin{eqnarray*}[\langle a_1,a_2\rangle+\langle b_1,b_2\rangle+\langle c_1,c_2\rangle,s\otimes b]&=&-\frac{1}{4\ell}([s,Id_{_{{\mathcal V}^\ell}}]\otimes b\circ b^*+(s\circ Id_{_{{\mathcal V}^\ell}})\otimes [b, b^*])\\&-&\frac{1}{2\ell}tr(sId_{{\mathcal V}^\ell})\langle b,b^*\rangle,\end{eqnarray*} and $$[\langle a_1,a_2\rangle+\langle b_1,b_2\rangle+\langle c_1,c_2\rangle,v\otimes c]=\frac{1}{2\ell}Id_{_{{\mathcal V}^\ell}}v\otimes b^*\cdot c-\frac{1}{2}v\otimes (f(c,c_2)\cdot c_1+f(c,c_1)\cdot c_2).$$}
These together with (\ref{probc-gen-mu}) complete the proof.\qed
\subsubsection{\textbf{Types $A$ and $D$}} Suppose that $I$ is an index set of cardinality $n+1>5$ and $I_0$ is a subset of $I$ of cardinality $\ell+1>5.$ Suppose that $R$ is an irreducible finite root system of type $X=\dot A_I$ or $D_I.$ Suppose that ${\mathcal V}$ is a vector space with a basis $\{v_i\mid i\in I\}$ and take ${\mathcal G}$ to be the finite dimensional split simple Lie algebra of type $X$ as in Lemma \ref{type-a-alg} or Lemma \ref{type-d-alg} respectively and set ${\mathcal G}^\ell:={\mathcal G}_{_{I_0}}.$ Suppose that ${\mathcal V}^\ell$ is the subspace of ${\mathcal V}$ spanned by $\{v_i\mid i\in I_0\}.$ We take $Id_{_{\mathcal V}}$ to be the identity map on ${\mathcal V}$ and define $Id_{_{{\mathcal V}^\ell}}$ as follows: $$\begin{array}{c}Id_{_{{\mathcal V}^{\ell}}}:{\mathcal V}\longrightarrow {\mathcal V}\\ v_i\mapsto v_i,\;v_{j}\mapsto 0;\;\; (i\in I_0,\; j\in I\setminus I_0). \end{array}$$
\begin{Theorem}\label{type-a} Suppose that $\mathcal{L}$ is a Lie algebra graded by the irreducible finite root system $R$ of type $X=\dot A_I$ or $D_I$ with grading pair $(\mathfrak{g},\mathfrak{h})$ and let $S$ be the irreducible full subsystem of $R$ of type $\dot A_{I_0}$ or $D_{I_0}$ respectively.
(i) Consider $\mathcal{L}^{^S}$ as a $\mathfrak{g}^{^S}-$module and take \begin{equation}\label{last3-a}\mathcal{L}^{^S}=\bigoplus_{i\in {\mathcal I}}\mathfrak{g}_i\oplus E\end{equation} to be the decomposition of $\mathcal{L}^{^S}$ into finite dimensional irreducible $\mathfrak{g}^{^S}-$ submodules in which ${\mathcal I}$ is an index set and for $i\in{\mathcal I},$ $\mathfrak{g}_i$ is isomorphic to $\mathfrak{g}^{^S}(\simeq{\mathcal G}^\ell),$
and $E$ is a trivial $\mathfrak{g}^{^S}-$submodule. Then there exists a class $\{\mathcal D_n,{\mathcal G}_i\mid i\in {\mathcal I}\}$ of finite dimensional $\mathfrak{g}-$submodules of $\mathcal{L}$ such that \begin{itemize} \item $\mathcal D_n$ is a trivial $\mathfrak{g}-$module and ${\mathcal G}_i$ is isomorphic to $\mathfrak{g}\simeq{\mathcal G}$ for $i\in{\mathcal I},$ \\ \item $\mathfrak{g}_i\subseteq{\mathcal G}_i,$ ($i\in{\mathcal I}$),\\ \item $\mathcal{L}=\bigoplus_{i\in{\mathcal I}}{\mathcal G}_i\oplus\mathcal D_n.$ \end{itemize}
(ii) Take $\mathcal A$ to be a vector space with basis $\{a_i\mid i\in{\mathcal I}\}$and identify $\mathcal{L}$ with $({\mathcal G}\otimes \mathcal A)\oplus\mathcal D_n,$ say via the natural identification $$\varphi:\mathcal{L}\longrightarrow ({\mathcal G}\otimes\mathcal A)\oplus\mathcal D_n .$$ Transfer the Lie algebraic structure of $\mathcal{L}$ to $({\mathcal G}\otimes\mathcal A)\oplus\mathcal D_n.$ Then $\mathcal D_\ell:=\varphi(E)$ is a subalgebra of $\varphi(\mathcal{L}^{^S})=({\mathcal G}^{^\ell}\dot\otimes \mathcal A)\oplus\mathcal D_\ell$ and $\mathcal D_n$ is a subalgebras of $({\mathcal G}\otimes\mathcal A)\oplus\mathcal D_n.$ Moreover the vector space $\mathcal A$ is equipped with an associative algebraic structure if $X=\dot A_I$ and with a commutative associative algebraic structure if $X=D_I.$
(iii) There is a subspace $\mathcal K_1$ of the full skew-dihedral homology group of $\mathcal A$ with respect to $n$ and a subspace $\mathcal K_2$ of the full skew-dihedral homology group of $\mathcal A$ with respect to $\ell$ such that $\mathcal D_n$ and $\mathcal D_\ell$ are isomorphic to the quotient algebras $\{\mathcal A,\mathcal A\}_n/\mathcal K_1$ and $\{\mathcal A,\mathcal A\}_\ell/\mathcal K_2$ respectively, say via $$\psi_1:\{\mathcal A,\mathcal A\}_n/\mathcal K_1\longrightarrow \mathcal D_n\quad\hbox{and}\quad\psi_2:\{\mathcal A,\mathcal A\}_\ell/\mathcal K_2\longrightarrow \mathcal D_\ell.$$
(iv) For $a,a'\in\mathcal A,$ take $$\langle a,a'\rangle_n:=\psi_1(\{a,a'\}_n+\mathcal K_1)\quad\hbox{and}\quad\langle a,a'\rangle_\ell:=\psi_2(\{a,a'\}_\ell+\mathcal K_2). $$ Then for $a,a'\in \mathcal A,$ we have $$\langle a,a'\rangle_n=\langle a,a'\rangle_\ell+((\frac{1}{n+1} Id_{_{{\mathcal V}}}-\frac{1}{\ell+1} Id_{_{{\mathcal V}^\ell}})\otimes (aa'-a'a).$$
(v) For $a,a'\in\mathcal A,$ set $$\langle a,a'\rangle^n:=\varphi^{-1}(\langle a,a'\rangle_n)\quad\hbox{and}\quad\langle a,a'\rangle^\ell:=\varphi^{-1}(\langle a,a'\rangle_\ell).$$ If $\ell < n,$ then for $a_1,a'_1,\ldots,a_t,a'_t\in\mathcal A,$ we have $\sum_{i=1}^t\langle a_i,a'_i\rangle^\ell=0$ if and only if $\sum_{i=1}^t\langle a_i,a'_i\rangle^n=0$ and $\sum_{i=1}^t[a_i,a'_i]=0.$
(vi) For $x,y\in {\mathcal G},$ set $x\circ y:= xy +yx - \frac{2tr(xy)}{\ell+1}Id_{{\mathcal V}^\ell}$ and for $a,a'\in\mathcal A,$ set $\langle a,a'\rangle:=\langle a,a'\rangle_\ell.$ The Lie bracket on $({\mathcal G}\otimes\mathcal A)\oplus \mathcal D_n=({\mathcal G}\otimes\mathcal A)+ \mathcal D_\ell$ is given by \begin{equation}\label{probc-fin-a} \begin{array}{l} \hbox{\small$[x\otimes a,y\otimes a']$}=\left\{\begin{array}{ll}\hbox{\small$\;[x,y]\otimes\frac{1}{2}(a\circ a')+ (x\circ y)\otimes\frac{1}{2}[a,a']+tr(xy)\langle a,a'\rangle$}& \hbox{\small$X=\dot A_I$},
\\\hbox{\small$[x,y]\otimes aa'+tr(xy)\langle a,a'\rangle$}&\hbox{\small$X=D_I,$}\end{array}\right. \\ \hbox{\small$[\langle a_1,a_2\rangle,x\otimes a]=$}\left\{\begin{array}{ll} \hbox{\small$\frac{-1}{2(\ell+1)}(x\circ Id_{_{{\mathcal V}^\ell}}\otimes[a,[a_1,a_2]]$}&\\\hbox{\small$+[x,Id_{_{{\mathcal V}^\ell}}]\otimes a\circ [a_1,a_2]+2tr(Id_{_{{\mathcal V}^\ell}}x)\langle a,[a_1,a_2]\rangle)$},&\hbox{\small$X=\dot A_I,$}\\ 0&\hbox{\small$X=D_I,$}\end{array}\right.
\\ \hbox{\small$[\langle a_1,a_2\rangle,\langle a'_1,a'_2\rangle]$}=\left\{\begin{array}{ll}\hbox{\small$\langle d^{\ell,\mathcal A}_{a_1,a_2}(a'_1),a'_2\rangle+\langle a'_1,d^{\ell,\mathcal A}_{a_1,a_2}(a'_2)\rangle,$}& \hbox{\small$X=\dot A_I,$}\\ 0&\hbox{\small$X=D_I,$}\end{array}\right. \end{array}
\end{equation} for $x,y\in{\mathcal G},$ $a,a',a_1,a_2,a'_1,a'_2\in\mathcal A.$ \end{Theorem}
\subsubsection{\textbf{Types $B$ and $C$}}\label{subsub b-c} Suppose that $I$ is a nonempty index set of cardinality $n$ greater than 4 and $I_0$ is a subset of $I$ of cardinality $\ell>4.$ Take ${\mathcal G}$ to be either $\mathfrak{o}_B(I)$ or $\mathfrak{sp}(I).$ Suppose that ${\mathcal V}$ is a vector space with a basis $\{v_0,v_i,v_{\bar i}\mid i\in I\}$ equipped with a nondegenerate symmetric bilinear form $(\cdot,\cdot)$ as in (\ref{form-b-alg}) if ${\mathcal G}=\mathfrak{o}_B(I)$ and it is a vector space with a basis $\{v_i,v_{\bar i}\mid i\in I\}$ equipped with a nondegenerate skew-symmetric bilinear form $(\cdot,\cdot)$ as in (\ref{form-c}) if ${\mathcal G}:=\mathfrak{sp}(I).$ Consider (\ref{simple-b-mod}) and (\ref{simple-c}) and set ${\mathcal V}^\ell:={\mathcal V}_{_{I_0}}.$ Set $$J:=\left\{\begin{array}{ll}I_0\cup\bar{I_0}\cup\{0\}& \hbox{if ${\mathcal G}=\mathfrak{o}_B(I)$}\\ I_0\cup\bar{I_0}&\hbox{if ${\mathcal G}=\mathfrak{sp}(I)$}\end{array}\right.$$ and define $Id_{_{{\mathcal V}^\ell}}:{\mathcal V}\longrightarrow {\mathcal V}$ to be the linear transformation defined by $$v_i\mapsto \left\{\begin{array}{ll}v_i& \hbox{if $i\in J$}\\ 0& \hbox{if $i\in I\cup\bar{I} \setminus J.$}\end{array}\right. $$ Finally set $\mathcal{S}:={\mathcal V}$ and $\mathcal{S}^\ell:={\mathcal V}^\ell$ if ${\mathcal G}:=\mathfrak{o}_B(I)$ and take $\mathcal{S}$ and $\mathcal{S}^\ell:=\mathcal{S}_{_{I_0}}$ to be as in (\ref{module-s-c}) and (\ref{simple-c}) respectively if ${\mathcal G}=\mathfrak{sp}(I).$
\begin{Theorem} \label{type-fini-b} Suppose that $\mathcal{L}$ is a Lie algebra graded by a root system $R$ of type $X=B_I $ or $C_I$ with grading pair $(\mathfrak{g},\mathfrak{h})$ and let $S$ be the irreducible full subsystem of $R$ of type $B_{I_0}$ or $C_{I_0}$ respectively.
(i) Consider $\mathcal{L}^{^S}$ as a $\mathfrak{g}^{^S}-$module and take \begin{equation}\label{last3-b}\mathcal{L}^{^S}=\bigoplus_{i\in {\mathcal I}}\mathfrak{g}_i\oplus\bigoplus_{j\in\mathcal{J}} \mathfrak{s}_j\oplus E\end{equation} to be the decomposition of $\mathcal{L}^{^S}$ into finite dimensional irreducible $\mathfrak{g}^{^S}-$ submodules in which ${\mathcal I},\mathcal{J}$ are index sets and for $i\in{\mathcal I}$ and $j\in \mathcal{J},$ $\mathfrak{g}_i$ is isomorphic to $\mathfrak{g}^{^S}(\simeq{\mathcal G}^\ell),$ $\mathfrak{s}_j$ is isomorphic to $\mathcal{S}^\ell,$
and $E$ is a trivial $\mathfrak{g}^{^S}-$submodule. Then there exists a class $\{\mathcal D_n,{\mathcal G}_i,\mathcal{S}_j\mid i\in {\mathcal I},j\in \mathcal{J}\}$ of finite dimensional $\mathfrak{g}-$submodules of $\mathcal{L}$ such that \begin{itemize} \item $\mathcal D_n$ is a trivial $\mathfrak{g}-$module, ${\mathcal G}_i$ is isomorphic to $\mathfrak{g}(\simeq{\mathcal G})$ and $\mathcal{S}_j$
is isomorphic to $\mathcal{S},$ for $i\in{\mathcal I},j\in \mathcal{J},$ \\ \item $\mathfrak{g}_i\subseteq{\mathcal G}_i,$ $\mathfrak{s}_j\subseteq\mathcal{S}_j,$
($i\in{\mathcal I},$ $j\in\mathcal{J}$),\\ \item $\mathcal{L}=\bigoplus_{i\in{\mathcal I}}{\mathcal G}_i\oplus\bigoplus_{j\in\mathcal{J}}\mathcal{S}_j\oplus\mathcal D_n.$ \end{itemize}
(ii) Take $\mathcal A$ and $\mathcal{B}$ to be vector spaces with bases $\{a_i\mid i\in{\mathcal I}\}$ and $\{b_j\mid j\in\mathcal{J}\}$ respectively and identify $\mathcal{L}$ with $({\mathcal G}\otimes \mathcal A)\oplus({\mathcal V}\otimes \mathcal{B})\oplus\mathcal D_n,$ say via the natural identification $$\varphi:\mathcal{L}\longrightarrow \bigoplus_{i\in{\mathcal I}}{\mathcal G}_i\oplus\bigoplus_{j\in\mathcal{J}}\mathcal{S}_j\oplus\mathcal D_n .$$ Transfer the Lie algebraic structure of $\mathcal{L}$ to $\bigoplus_{i\in{\mathcal I}}{\mathcal G}_i\oplus\bigoplus_{j\in\mathcal{J}}\mathcal{S}_j\oplus\mathcal D_n.$ Then $\mathcal D_\ell:=\varphi(E)$ and $\mathcal D_n$ are subalgebras of $\mathcal{L}.$
(iii) Set $\mathfrak{a}:=\mathcal A\oplus\mathcal{B}.$ If ${\mathcal G}=\mathfrak{o}_B(I),$ $\mathcal A$ is equipped with a unital commutative associative algebraic structure and the vector space $\mathcal{B}$ is equipped with a unital $\mathcal A-$module structure. Also there is a symmetric $\mathcal A-$bilinear form $f:\mathcal{B}\times \mathcal{B}\longrightarrow \mathcal A$ and $\mathfrak{a}=\mathcal{J}(f,\mathcal{B}).$ Also if ${\mathcal G}=\mathfrak{sp}(I),$ $\mathfrak{a}$ is equipped with a star algebraic structure with an involution $*$ such that $\mathcal A$ (resp. $\mathcal{B}$) is the set of $*-$fixed (resp. $*$-skew fixed) points of $\mathfrak{a}.$
(iv) There is a subspace $\mathcal K_1$ of the full skew-dihedral homology group of $\mathfrak{a}$ with respect to $n$ and a subspace $\mathcal K_2$ of the full skew-dihedral homology group of $\mathfrak{a}$ with respect to $\ell$ such that $\mathcal D_n$ and $\mathcal D_\ell$ are isomorphic to the quotient algebras $\{\mathfrak{a},\mathfrak{a}\}_n/\mathcal K_1$ and $\{\mathfrak{a},\mathfrak{a}\}_\ell/\mathcal K_2$ respectively, say via $\psi_1:\{\mathfrak{a},\mathfrak{a}\}_n/\mathcal K_1\longrightarrow \mathcal D_n$ and $\psi_2:\{\mathfrak{a},\mathfrak{a}\}_\ell/\mathcal K_2\longrightarrow \mathcal D_\ell.$
(v) For $\alpha,\alpha'\in\mathfrak{a},$ take $$\langle \alpha,\alpha'\rangle_n:=\{\alpha,\alpha'\}_n+\mathcal K_1\quad\hbox{and}\quad\langle \alpha,\alpha'\rangle_\ell:=\{\alpha,\alpha'\}_\ell+\mathcal K_2. $$ Then if $\alpha,\alpha'\in \mathcal A$ or $\alpha,\alpha'\in\mathcal{B},$ we have $$\langle \alpha,\alpha'\rangle_n=\langle \alpha,\alpha'\rangle_\ell+((\frac{1}{n} Id_{_{\mathcal V}}-\frac{1}{\ell} Id_{_{{\mathcal V}^\ell}})\otimes (1/2)(\alpha\alpha'-\alpha'\alpha)).$$
(vi) For $\alpha,\alpha'\in\mathfrak{a},$ set $$\langle \alpha,\alpha'\rangle^n:=\varphi^{-1}(\langle \alpha,\alpha'\rangle_n)\quad\hbox{and}\quad\langle \alpha,\alpha'\rangle^\ell:=\varphi^{-1}(\langle \alpha,\alpha'\rangle_\ell).$$ If $\ell < n,$ then for $a_1,a'_1,\ldots,a_t,a'_t\in\mathcal A$ and $b_1,b'_1,\ldots,b_t,b'_t\in\mathcal{B},$ we have $\sum_{i=1}^t\langle a_i,a'_i\rangle^\ell+\sum_{i=1}^t\langle b_i,b'_i\rangle^\ell=0$ if and only if $$\sum_{i=1}^t\langle a_i,a'_i\rangle^n+\sum_{i=1}^t\langle b_i,b'_i\rangle^n=0 \quad\hbox{and}\quad \sum_{i=1}^t[a_i,a'_i]+\sum_{i=1}^t[b_i,b'_i]=0.$$
(vii) For $e,f\in{\mathcal G}\cup\mathcal{S},$ set $$e\circ f:=ef+fe-\frac{tr(ef)}{\ell}Id_{_{{\mathcal V}^\ell}},$$ and for $\alpha,\alpha'\in\mathfrak{a},$ set $$\langle\alpha,\alpha'\rangle:=\langle\alpha,\alpha'\rangle_\ell,$$ the Lie bracket on $({\mathcal G}\otimes\mathcal A)\oplus (\mathcal{S}\otimes \mathcal{B}) \oplus \mathcal D_n=(({\mathcal G}\otimes\mathcal A)\oplus (\mathcal{S}\otimes \mathcal{B}))+ \mathcal D_\ell$ is given by \begin{equation}\label{probc-fin-b} \begin{array}{l} \;[x\otimes a,y\otimes a']=[x,y]\otimes a a'+ tr(xy)\langle a,a'\rangle,
\\ \;[x\otimes a,s\otimes b]=xs\otimes ab,
\\ \;[s\otimes b,t\otimes b']=D_{s,t}\otimes f(b,b')+(s,t)\langle b,b'\rangle,
\\ \;[\langle\alpha,\alpha'\rangle,x\otimes a]=x\otimes d^{\ell,\mathfrak{a}}_{\alpha,\alpha'}(a),
\\ \;[\langle\alpha,\alpha'\rangle,s\otimes b]=s\otimes d^{\ell,\mathfrak{a}}_{\alpha,\alpha'}(b),
\\ \;[\langle \alpha_1,\alpha_2\rangle,\langle \alpha'_1,\alpha'_2\rangle]=\langle d^{\ell,\mathfrak{a}}_{\alpha_1,\alpha_2}(\alpha'_1),\alpha'_2\rangle+\langle\alpha'_1,d^{\ell,\mathfrak{a}}_{\alpha_1,\alpha_2}(\alpha'_2)\rangle. \end{array}
\end{equation} (see Definition \ref{yoshii}) for $x,y\in{\mathcal G},$ $s,t\in\mathcal{S},$ $a,a'\in\mathcal A,$ $b,b'\in\mathcal{B},$ and $\alpha,\alpha',\alpha_1,\alpha_2,$ $\alpha'_1,\alpha'_2\in\mathfrak{a}$ if ${\mathcal G}=\mathfrak{o}_B(I)$ and it is given by \begin{equation}\label{probc-fin-c} \hbox{\small$\begin{array}{l} \;[x\otimes a,y\otimes a']=[x,y]\otimes\frac{1}{2}(a\circ a')+ (x\circ y)\otimes\frac{1}{2}[a,a']+tr(xy)\langle a,a'\rangle,
\\ \;[x\otimes a,s\otimes b]=(x\circ s)\otimes\frac{1}{2}[a,b]+[x,s]\otimes \frac{1}{2} (a\circ b),
\\ \;[s\otimes b,t\otimes b']=[s,t]\otimes\frac{1}{2}(b\circ b') + (s\circ t)\otimes \frac{1}{2}[b,b']+tr(st)\langle b,b'\rangle,
\\ \;[\langle \alpha,\alpha'\rangle,x\otimes a]= \frac{-1}{4\ell}((x\circ Id_{_{{\mathcal V}^\ell}})\otimes[a,\beta_{\alpha,\alpha'}^*]+[x,Id_{_{{\mathcal V}^\ell}}]\otimes (a\circ \beta_{\alpha,\alpha'}^*)),
\\ \;[\langle \alpha,\alpha'\rangle,s\otimes b]\hspace{-1mm}=\hspace{-1mm}\frac{-1}{4\ell}([s,Id_{_{{\mathcal V}^\ell}}]\hspace{-1mm}\otimes\hspace{-1mm} (b\circ \beta_{\alpha,\alpha'}^*)\hspace{-1mm}+\hspace{-1mm}(s\circ Id_{_{{\mathcal V}^\ell}})\hspace{-1mm}\otimes\hspace{-1mm} [b, \beta_{\alpha,\alpha'}^*]+2tr(sId_{_{{\mathcal V}^\ell}})\langle b,\beta^*_{\alpha,\alpha'}\rangle),
\\ \;[\langle \alpha_1,\alpha_2\rangle,\langle \alpha'_1,\alpha'_2\rangle]=\langle d^{\ell,\mathfrak{a}}_{\alpha_1,\alpha_2}(\alpha'_1),\alpha'_2\rangle+\langle\alpha'_1,d^{\ell,\mathfrak{a}}_{\alpha_1,\alpha_2}(\alpha'_2)\rangle. \end{array}$}
\end{equation} (see (\ref{beta*})) for $x,y\in{\mathcal G},$ $s,t\in\mathcal{S},$ $a,a'\in\mathcal A,$ $b,b'\in\mathcal{B},$ $\alpha,\alpha',\alpha_1,\alpha_2,\alpha'_1,\alpha'_2\in\mathfrak{a}$ if ${\mathcal G}=\mathfrak{sp}(I).$
\end{Theorem}
\section{Root graded Lie algebras - general case} In this section, we discuss certain recognition theorems to characterize Lie algebras graded by an infinite irreducible locally finite root system. The main target of the present section is to generalize the decomposition (\ref{*}) for Lie algebras graded by infinite root systems. For a Lie algebra $\mathcal{L}$ graded by an infinite locally finite root system with grading pair $(\mathfrak{g},\mathfrak{h})$, we first decompose $\mathcal{L}$ as a direct sum of a certain subalgebra of $\mathcal{L}$ and a certain locally finite completely reducible $\mathfrak{g}-$submodule. This in particular results in a generalized decomposition for $\mathcal{L}$ as in (\ref{*}). We next reconstruct the structure of $\mathcal{L}$ in terms of the ingredients involved in this decomposition. Moreover we prove that any Lie algebra graded by an irreducible locally finite root system arises in this way. As in the previous section, we concentrate our attention on type $BC$ and for other types, we just report the results. \subsection{Recognition theorem for type $BC$} Suppose that $I$ is an infinite index set and $\ell $ is a positive integer greater than 3. We assume $R$ is an irreducible locally finite root system of type $BC_I$ and take ${\mathcal G},$ $\mathcal{S}$ and ${\mathcal V}$ to be as in Lemmas \ref{type-c-alg} and \ref{rep-local}. We show that an $R-$graded Lie algebra $\mathcal{L}$ can be decomposed into \begin{equation}\label{**}({\mathcal G}\otimes \mathcal A)\oplus(\mathcal{S}\otimes\mathcal{B})\oplus({\mathcal V}\otimes {\mathcal C})\oplus\mathcal D\end{equation} in which $\mathcal A,\mathcal{B}$ and ${\mathcal C}$ are vector spaces and $\mathcal D$ is a subalgebra of $\mathcal{L}.$ We equip $\frak{b}:=\mathcal A\oplus\mathcal{B}\oplus{\mathcal C}$ with a unital associative star algebraic structure and show that $\mathcal D$ can be expressed as a quotient of the algebra $\{\frak{b},\frak{b}\}_\ell$ by a subspace of the full skew-dihedral homology group of $\frak{b}$ with respect to $\ell$ satisfying the uniform property on $\frak{b}.$ Conversely, for vector spaces $A,B,C$ and $D$ with specific natures, we form the decomposition (\ref{**}), equip it with a Lie bracket and show that it is an $R-$graded Lie algebra.
\begin{Theorem}\label{typebc} Suppose that $I$ is an infinite index set and $\ell$ is an integer greater than 3. Assume $R$ is an irreducible locally finite root system of type $BC_I$ and ${\mathcal V}$ is a vector space with a basis $\{v_i\mid i\in I\cup\bar I\}.$ Suppose that $(\cdot,\cdot)$ is a bilinear form as in (\ref{form-c}), set ${\mathcal G}:=\mathfrak{sp}(I)$ and consider $\mathcal{S}$ as in (\ref{module-s-c}). Fix a subset $I_0$ of $I$ of cardinality $\ell$ and take $R_0$ to be the full irreducible subsystem of $R$ of type $BC_{I_0}.$ Suppose that $\{R_\lambda\mid\lambda\in\Lambda\}$ is the class of all finite irreducible full subsystems of $R$ containing $R_0,$ where $\Lambda$ is an index set containing zero. For $\lambda\in\Lambda,$ take ${\mathcal G}^\lambda$ as in Lemma \ref{simple-c-alg} and ${\mathcal V}^\lambda,\mathcal{S}^\lambda$ as in (\ref{simple-c}), also define $$\begin{array}{c}\mathfrak{I}_\lambda:{\mathcal V}\longrightarrow{\mathcal V}\\ v_i\mapsto\left\{\begin{array}{ll}v_i& i\in I_\lambda\cup \bar I_{\lambda}\\ 0&\hbox{otherwise.}\end{array}\right.\end{array}$$ For $e,f\in{\mathcal G}\cup\mathcal{S},$ define $$e\circ f:=ef+fe-\frac{tr(ef)}{l}\mathfrak{I}_0.$$
(i) Suppose that $(\mathfrak{a},*,{\mathcal C},f)$ is a coordinate quadruple of type $BC$ and $\mathcal A,$ $\mathcal{B}$ are $*$-fixed and $*$-skew fixed points of $\mathfrak{a}$ respectively. Set $\frak{b}:=\frak{b}(\mathfrak{a},*,{\mathcal C},f)$ and take $[\cdot,\cdot], \circ,\hbox{\tiny$\heartsuit$},\diamond$ to be as in Subsection \ref{subsect2-1}. For $\beta_1,\beta_2\in\frak{b},$ consider $d_{\beta_1,\beta_2}^{\ell,\frak{b}}$ as in (\ref{derivbc}) and take $\beta^*_{\beta_1,\beta_2},\beta_1^*$ and $\beta_2^*$ as in Proposition \ref{divide5}. For a subset $\mathcal K$ of $FH(\frak{b})$ satisfying the uniform property on $\frak{b},$ set $$\mathcal{L}(\frak{b},\mathcal K):=({\mathcal G}\otimes\mathcal A)\oplus(\mathcal{S}\otimes \mathcal{B})\oplus({\mathcal V}\otimes{\mathcal C})\oplus(\{\frak{b},\frak{b}\}_\ell/\mathcal K).$$ Then setting $\langle \beta,\beta'\rangle:=\{\beta,\beta'\}+\mathcal K,$ $\beta,\beta'\in \frak{b},$ $\mathcal{L}(\frak{b},\mathcal K)$ together with {\small\begin{equation}\label{probc-gen} \begin{array}{l} \;[x\otimes a,y\otimes a']=[x,y]\otimes\frac{1}{2}(a\circ a')+ (x\circ y)\otimes\frac{1}{2}[a,a']+tr(xy)\langle a,a'\rangle,
\\ \;[x\otimes a,s\otimes b]= (x\circ s)\otimes\frac{1}{2}[a,b]+[x,s]\otimes\frac{1}{2}(a\circ b)=-[s\otimes b,x\otimes a],
\\ \;[s\otimes b,t\otimes b']=[s,t]\otimes\frac{1}{2}(b\circ b')+ (s\circ t)\otimes\frac{1}{2}[b,b']+tr(st)\langle b,b'\rangle,
\\ \;[x\otimes a,u\otimes c]=xu\otimes a\cdot c=-[u\otimes c,x\otimes a],
\\ \;[s\otimes b,u\otimes c]=su\otimes b\cdot c=-[u\otimes c,s\otimes b],
\\ \;[u\otimes c,v\otimes c']=(u\circ v)\otimes (c\diamond c')+ [u, v]\otimes (c\hbox{\tiny$\heartsuit$} c')+(u,v)\langle c,c'\rangle,
\\ \;[\langle \beta_1,\beta_2\rangle,x\otimes a]= \frac{-1}{4\ell}((x\circ \mathfrak{I}_0)\otimes[a,\beta_{_{\beta_1,\beta_2}}^*]+[x,\mathfrak{I}_0]\otimes (a\circ \beta_{_{\beta_1,\beta_2}}^*)),
\\ \;[\langle \beta_1,\beta_2\rangle,s\otimes b]\hspace{-1mm}=\hspace{-1mm}\frac{-1}{4\ell}([s,\mathfrak{I}_0\hspace{-.5mm}]\hspace{-1mm}\otimes\hspace{-.5mm}( b\circ \beta_{_{\beta_1,\beta_2}}^*)\hspace{-1mm}+\hspace{-1mm}(s\circ \mathfrak{I}_0)\hspace{-1mm}\otimes \hspace{-.5mm}[b, \beta_{_{\beta_1,\beta_2}}^*\hspace{-1mm}]\hspace{-1mm}+\hspace{-1mm}2tr(s\mathfrak{I}_0)\langle b,\beta_{_{\beta_1,\beta_2}}^*\hspace{-.5mm}\rangle),
\\ \;[\langle \beta_1,\beta_2\rangle,v\otimes c]=\frac{1}{2\ell}\mathfrak{I}_0v\otimes \beta_{_{\beta_1,\beta_2}}^*\cdot c-\frac{1}{2}v\otimes (f(c,\beta^*_2)\cdot \beta^*_1+f(c,\beta^*_1)\cdot \beta^*_2)\\ \;[\langle\beta_1,\beta_2\rangle,\langle\beta'_1,\beta'_2\rangle]=\langle d^\ell_{\beta_1,\beta_2}(\beta'_1),\beta'_2\rangle+\langle\beta'_1,d^\ell_{\beta_1,\beta_2}(\beta'_2)\rangle \end{array}
\end{equation}} for $x,y\in{\mathcal G},$ $s,t\in\mathcal{S},$ $u,v\in{\mathcal V},$ $a,a'\in\mathcal A,$ $b,b'\in\mathcal{B},$ $c,c'\in{\mathcal C},$ $\beta_1,\beta_2,\beta_1',\beta'_2\in\frak{b},$ is an $R-$graded Lie algebra with grading pair $({\mathcal G},{\mathcal H})$ where ${\mathcal H}$ is the splitting Cartan subalgebra of ${\mathcal G}$ defined in Lemma \ref{type-c-alg}.
$(ii)$ If $\mathcal{L}$ is an $R-$graded Lie algebra with grading pair $(\mathfrak{g},\mathfrak{h}),$ then there is a coordinate quadruple $(\mathfrak{a},*,{\mathcal C},f)$ of type $BC$ and a subspace $\mathcal K$ of $\frak{b}:=\frak{b}(\mathfrak{a},*,{\mathcal C},f)$ satisfying the uniform property on $\frak{b}$ such that $\mathcal{L}$ is isomorphic to $\mathcal{L}(\frak{b},\mathcal K).$
\end{Theorem}
\noindent{\bf Proof. } $(i)$ We prove that $\mathcal{L}(\frak{b},\mathcal K)$ together with
(\ref{probc-gen}) is a Lie algebra. For $\lambda\in \Lambda$ set $n_\lambda:=|I_\lambda|$ and $\mathcal{L}^\lambda:=({\mathcal G}^\lambda\dot\otimes\mathcal A)\oplus(\mathcal{S}^\lambda\dot\otimes \mathcal{B})\oplus({\mathcal V}^\lambda\dot\otimes{\mathcal C})\oplus\langle\frak{b},\frak{b}\rangle.$ Also for $a,a'\in\mathcal A,b,b'\in\mathcal{B},$ and $c,c'\in{\mathcal C},$ set $$\begin{array}{l}\langle a,a'\rangle_\lambda: =(((\frac{-1}{\ell}\mathfrak{I}_0+\frac{1}{n_\lambda}\mathfrak{I}_\lambda)\otimes\frac{1}{2}[a,a'])+\langle a,a'\rangle,\\ \langle b,b'\rangle_\lambda :=((\frac{-1}{\ell}\mathfrak{I}_0+\frac{1}{n_\lambda}\mathfrak{I}_\lambda)\otimes\frac{1}{2}[b,b'])+\langle b,b'\rangle,\\ \langle c,c'\rangle_\lambda :=((\frac{1}{\ell}\mathfrak{I}_0-\frac{1}{n_\lambda}\mathfrak{I}_\lambda)\otimes\frac{1}{2} c\hbox{\tiny$\heartsuit$} c')+\langle c,c'\rangle,\\ \langle a,b\rangle_\lambda=\langle b,c\rangle_\lambda=\langle a,c\rangle_\lambda:=0.\end{array}$$ Take $\langle\frak{b},\frak{b}\rangle_\lambda:=\hbox{span}\{\langle a,a'\rangle_\lambda,\langle b,b'\rangle_\lambda,\langle c,c'\rangle_\lambda\mid a,a'\in\mathcal A,b,b'\in\mathcal{B},c,c'\in{\mathcal C}\}$ and note that as $\mathcal K$ satisfies the uniform property on $\frak{b},$ we have $$\mathcal{L}^\lambda:=({\mathcal G}^\lambda\dot\otimes\mathcal A)\oplus(\mathcal{S}^\lambda\dot\otimes \mathcal{B})\oplus({\mathcal V}^\lambda\dot\otimes{\mathcal C})\oplus\langle\frak{b},\frak{b}\rangle_\lambda.$$ For $\beta=a+b+c,\beta'=a'+b'+c'\in\frak{b},$ set $\langle \beta,\beta'\rangle_\lambda:=\langle a,a'\rangle_\lambda+\langle b,b'\rangle_\lambda+\langle c,c'\rangle_\lambda.$
Now consider the linear transformation $\psi:\frak{b}\otimes\frak{b}\longrightarrow\langle\frak{b},\frak{b}\rangle_\lambda $ mapping $\beta\otimes\beta'$ to $\langle\beta,\beta'\rangle_\lambda.$ It is not difficult to see that for the subspace $K$ of $\frak{b}\otimes\frak{b}$ defined in Subsection \ref{subsect2-1}, $\psi(K)=\{0\}.$ So $\psi$ induces a linear transformation $\dot\psi:\{\frak{b},\frak{b}\}_{n_\lambda}\longrightarrow\langle\frak{b},\frak{b}\rangle_\lambda $ mapping $\{\beta,\beta'\}_{n_\lambda}$ to $\langle\beta,\beta'\rangle_\lambda.$ Take $\mathcal K_\lambda$ to be the kernel of $\dot\psi.$ If $t\in\mathbb{N},$ $a_i,a'_i\in\mathcal A, $ $b_i,b'_i\in\mathcal{B}$ and $c_i,c'_i\in{\mathcal C}$ ($1\leq i\leq t$) are such that $\sum_{i=1}^t(\{a_i,a'_i\}_{n_\lambda}+\{b_i,b'_i\}_{n_\lambda}+\{c_i,c'_i\}_{n_\lambda})\in\mathcal K_{n_\lambda},$ then $\sum_{i=1}^t(\langle a_i,a'_i\rangle_\lambda+\langle b_i,b'_i\rangle_\lambda+\langle c_i,c'_i\rangle_\lambda)=0.$ This implies that {\small \begin{equation}\label{the-final}((\frac{-1}{\ell}\mathfrak{I}_0+\frac{1}{n_\lambda}\mathfrak{I}_\lambda)\otimes\frac{1}{2}\sum_{i=1}^t([ a_i,a'_i]+[b_i,b'_i]-(c_i\hbox{\tiny$\heartsuit$} c'_i))+\sum_{i=1}^t(\langle a_i,a'_i\rangle+\langle b_i,b'_i\rangle+\langle c_i,c'_i\rangle)=0.\end{equation}} This in turn implies that $\sum_{i=1}^t(\langle a_i,a'_i\rangle+\langle b_i,b'_i\rangle+\langle c_i,c'_i\rangle)=0.$
Therefore we get that $\mathcal K_\lambda$ is a subset of the full skew-dihedral homology group of $\frak{b}$ with respect to $\ell.$ But if $\lambda\neq 0,$ (\ref{the-final}) implies that $\sum_{i=1}^t([ a_i,a'_i]+[b_i,b'_i]-(c_i\hbox{\tiny$\heartsuit$} c'_i))=0.$ Now one gets using this together with the fact that $\mathcal K_\lambda$ is a subset of the full skew-dihedral homology group of $\frak{b}$ with respect to $\ell,$ that $\mathcal K_\lambda$ is a subset of the full skew-dihedral homology group of $\frak{b}$ with respect to $n_\lambda.$ Now it follows from \cite[Chapter $V$]{ABG2} that $\mathcal{L}^\lambda$ together with the product introduced in (\ref{probc-gen}) restricted to $\mathcal{L}^\lambda\times\mathcal{L}^\lambda$ defines a Lie algebra. Therefore $\mathcal{L}$ together with $[\cdot,\cdot]$ is a Lie algebra as $\mathcal{L}=\cup_{\lambda\in\Lambda}\mathcal{L}^\lambda.$ Now one can easily see that $\mathcal{L}$ has a weight space decomposition $\mathcal{L}=\oplus_{\alpha\in R}\mathcal{L}_\alpha$ with respect to ${\mathcal H}$ in which $$\mathcal{L}_\alpha=\left\{\begin{array}{ll}{\mathcal V}_\alpha\otimes {\mathcal C}& \hbox{if } \alpha\in R_{sh}\\ ({\mathcal G}_\alpha\otimes\mathcal A)\oplus(\mathcal{S}_\alpha\otimes \mathcal{B})& \hbox{if } \alpha\in R_{lg}\\ {\mathcal G}_\alpha\otimes\mathcal A&\hbox{if } \alpha\in R_{ex}\\ ({\mathcal G}_0\otimes \mathcal A)\oplus(\mathcal{S}_0\otimes \mathcal{B})\oplus\langle\frak{b},\frak{b}\rangle& \hbox{if } \alpha=0\end{array}\right.$$ and that $\mathcal{L}$ is an $R-$graded Lie algebra with grading pair $({\mathcal G},{\mathcal H}).$
$(ii)$ For $\lambda\in\Lambda,$ set $$\begin{array}{l}\mathcal{L}^\lambda:=\sum_{\alpha\in R_\lambda^\times}\mathcal{L}_\alpha\oplus\sum_{\alpha\in R_\lambda^\times}[\mathcal{L}_\alpha,\mathcal{L}_{-\alpha}],\\\\ \mathfrak{g}^{\lambda}:=\sum_{\alpha\in (R_\lambda)_{sdiv}^\times}\mathfrak{g}_\alpha\oplus\sum_{\alpha\in (R_\lambda)_{sdiv}^\times}[\mathfrak{g}_\alpha,\mathfrak{g}_{-\alpha}] \end{array}$$ and note that $\mathfrak{g}^\lambda$ is isomorphic to ${\mathcal G}^\lambda.$ We know by Lemma \ref{final1} that $\mathcal{L}^\lambda$ is an $R_\lambda-$graded Lie algebra with grading pair $(\mathfrak{g}^\lambda,\mathfrak{h}^\lambda:=\mathfrak{g}^\lambda\cap\mathfrak{h}).$
Consider $\mathcal{L}^0$ as a $\mathfrak{g}^0-$module and suppose that $\{{\mathcal G}_i^0,\mathcal{S}^0_j,{\mathcal V}^0_t,\mathcal D_0\mid i\in \mathcal{I},j\in\mathcal{J},t\in{\mathcal T}\}$ is a class of finite dimensional $\mathfrak{g}^0-$submodules of $\mathcal{L}^0$ such that \begin{itemize} \item $\mathcal{L}^0=\sum_{i\in \mathcal{I}}{\mathcal G}_i^0\oplus\sum_{j\in\mathcal{J}}\mathcal{S}^0_j\oplus\sum_{i\in {\mathcal T}}{\mathcal V}^0_t\oplus\mathcal D_0,$ \item $\mathcal D_0$ is a trivial $\mathfrak{g}^0-$submodule of $\mathcal{L}^0,$ \item for $i\in \mathcal{I},j\in \mathcal{J}$ and $t\in {\mathcal T},$ ${\mathcal G}_i^0$ is isomorphic to ${\mathcal G}^0,$ $\mathcal{S}^0_j$ is isomorphic to $\mathcal{S}^0,$ and ${\mathcal V}^0_t$ is isomorphic to ${\mathcal V}^0.$ \end{itemize}
Now for $\lambda\in\Lambda,$ consider $\mathcal{L}^\lambda$ as a $\mathfrak{g}^\lambda-$module via the adjoint representation. Using Lemmas \ref{divide1} and \ref{divide2}, one finds finite dimensional irreducible $\mathfrak{g}^\lambda-$submodules ${\mathcal G}_i^\lambda,$ $\mathcal{S}^\lambda_j,$ ${\mathcal V}^\lambda_t$ ($i\in\mathcal{I},j\in\mathcal{J},t\in {\mathcal T}$) of $\mathcal{L}^\lambda$ and a trivial $\mathfrak{g}^\lambda-$submodule $\mathcal D_\lambda$ such that
$$(\mathcal{I},\mathcal{J},{\mathcal T},\{{\mathcal G}^0_i\},\{{\mathcal G}^\lambda_i\},\{\mathcal{S}^0_j\},\{\mathcal{S}^\lambda_j\},\{{\mathcal V}^0_t\},\{{\mathcal V}^\lambda_t\}, \mathcal D_0,\mathcal D_\lambda)$$ is an $(R^\lambda,R^0)-$datum for the pair $(\mathcal{L}^\lambda,\mathcal{L}^0)$ (see (\ref{last6})).
we know from Subsection \ref{subsect2-2} that there is a coordinate quadruple $(\mathfrak{a},*,{\mathcal C},f)$ of type $BC$ and a subspace $\mathcal K_\lambda$ of the full skew-dihedral homology group of $\frak{b}:=\frak{b}(\mathfrak{a},*,{\mathcal C},f)$ with respect to $n_\lambda=|I_\lambda|$ such that $\mathcal D_\lambda$ is a subalgebra of $\mathcal{L}^\lambda$ isomorphic to the quotient algebra $\{\frak{b},\frak{b}\}_{n_\lambda}/\mathcal K_\lambda,$ say via $\phi_\lambda:\{\frak{b},\frak{b}\}_{n_\lambda}/\mathcal K_\lambda\longrightarrow\mathcal D_\lambda.$ Now for $\beta,\beta'\in\frak{b},$ set \begin{equation}\label{2setare}\langle\beta,\beta'\rangle^\lambda:=\phi_\lambda(\{\beta,\beta'\}_\lambda+\mathcal K_\lambda).\end{equation} Take $\mathcal A$ and $\mathcal{B}$ to be the $*-$fixed and $*-$skew fixed points of $\mathfrak{a}$ respectively and note that $$\mathcal D_\lambda=\hbox{span}\{\langle a,a'\rangle^\lambda,\langle b,b'\rangle^\lambda,\langle c,c'\rangle^\lambda\mid a,a'\in\mathcal A,b,b'\in\mathcal{B},c,c'\in{\mathcal C}\}.$$ We now proceed with the proof in the following steps:
\noindent\underline{\textbf{Step 1:}} For $i\in \mathcal{I},$ $j\in \mathcal{J},$ $t\in {\mathcal T}$ and $\lambda,\mu\in\Lambda$ with $\lambda\preccurlyeq\mu,$ ${\mathcal G}_i^\mu$ is the $\mathfrak{g}^\mu-$submodule of $\mathcal{L}^\mu$ generated by ${\mathcal G}_i^\lambda,$ $\mathcal{S}_j^\mu$ is the $\mathfrak{g}^\mu-$submodule of $\mathcal{L}^\mu$ generated by $\mathcal{S}_j^\lambda$ and ${\mathcal V}^\mu_t$ is the $\mathfrak{g}^\mu-$submodule of $\mathcal{L}^\mu$ generated by ${\mathcal V}_t^\lambda.$ In other words, $$(\mathcal{I},\mathcal{J},{\mathcal T},\{{\mathcal G}^\lambda_i\},\{{\mathcal G}^\mu_i\},\{\mathcal{S}^\lambda_j\},\{\mathcal{S}^\mu_j\},\{{\mathcal V}^\lambda_t\},\{{\mathcal V}^\mu_t\}, \mathcal D_\lambda,\mathcal D_\mu)$$ is an $(R^\mu,R^\lambda)-$datum for the pair $(\mathcal{L}^\mu,\mathcal{L}^\lambda):$ It is immediate using the facts that $\mathfrak{g}^\lambda$ is a subalgebra of $\mathfrak{g}^\mu.$
\noindent\underline{\textbf{Step 2:}} For $\lambda\in\Lambda,$ $\mathcal{L}^\lambda=\sum_{i\in \mathcal{I}}{\mathcal G}_i^\lambda\oplus\sum_{j\in\mathcal{J}}\mathcal{S}^\lambda_j\oplus\sum_{i\in {\mathcal T}}{\mathcal V}^\lambda_t\oplus\mathcal D_0:$ By Step 1 and Remark \ref{rem1}, $\mathcal{L}^\lambda=(\sum_{i\in \mathcal{I}}{\mathcal G}_i^\lambda\oplus\sum_{j\in\mathcal{J}}\mathcal{S}^\lambda_j\oplus\sum_{i\in {\mathcal T}}{\mathcal V}^\lambda_t)+\mathcal D_0.$ Suppose $d\in\mathcal D_0,$ $x\in\sum_{i\in \mathcal{I}}{\mathcal G}_i^\lambda\oplus\sum_{j\in\mathcal{J}}\mathcal{S}^\lambda_j\oplus\sum_{i\in {\mathcal T}}{\mathcal V}^\lambda_t$ and $x+d=0.$ Since $d\in \mathcal D_0,$ there are $t\in\mathbb{N},$ $a_i,a'_i\in\mathcal A,$ $b_i,b'_i\in\mathcal{B}$ and $c_i,c'_i\in{\mathcal C}$ $(1\leq i\leq t)$ such that $d=\sum_{i=1}^t\langle a_i,a'_i\rangle^0+\langle b_i,b'_i\rangle^0+\langle c_i,c'_i\rangle^0.$ It follows from Step 1 and Lemma \ref{divide4} that there is $y\in\sum_{i\in \mathcal{I}}{\mathcal G}_i^\lambda\oplus\sum_{j\in\mathcal{J}}\mathcal{S}^\lambda_j\oplus\sum_{i\in {\mathcal T}}{\mathcal V}^\lambda_t$ such that $d=y+\sum_{i=1}^t\langle a_i,a'_i\rangle^\lambda+\langle b_i,b'_i\rangle^\lambda+\langle c_i,c'_i\rangle^\lambda.$ Now as $0=x+d=x+y+\sum_{i=1}^t\langle a_i,a'_i\rangle^\lambda+\langle b_i,b'_i\rangle^\lambda+\langle c_i,c'_i\rangle^\lambda,$ we get that $x+y=0$ and $\sum_{i=1}^t(\langle a_i,a'_i\rangle^\lambda+\langle b_i,b'_i\rangle^\lambda+\langle c_i,c'_i\rangle^\lambda)=0.$ Take $\mu\in \Lambda $ to be such that $\lambda\preccurlyeq\mu,$ then using Step 1, one gets that the pairs $(\mathcal{L}^\lambda,\mathcal{L}^\mu)$ and $(\mathcal{L}^0,\mathcal{L}^\lambda)$ play the same role as the pair $(\mathcal{L}^\ell,\mathcal{L}^n)$ in Subsection \ref{subsect2-2}. Using Remark \ref{rem1} for the pair $(\mathcal{L}^\lambda,\mathcal{L}^\mu),$ one gets that $\sum_{i=1}^t(\langle a_i,a'_i\rangle^\mu+\langle b_i,b'_i\rangle^\mu+\langle c_i,c'_i\rangle^\mu)=0$ and $\sum_{i=1}^t([a_i,a'_i]+[b_i,b'_i]-c_i\hbox{\tiny$\heartsuit$} c'_i)=0.$ Next using Remark \ref{rem1} for the pair $(\mathcal{L}^0,\mathcal{L}^\mu),$ we get that $d=\sum_{i=1}^t(\langle a_i,a'_i\rangle^0+\langle b_i,b'_i\rangle^0+\langle c_i,c'_i\rangle^0)=0.$
\noindent\underline{\textbf{Step 3:}} $\mathcal K_0$ satisfies the uniform property on $\frak{b}:$ Suppose that $$\sum_{i=1}^t(\{a_i,a'_i\}_\ell+\{b_i,b'_i\}_\ell+\{c_i,c'_i\}_\ell)\in\mathcal K_0,$$ for $a_1,a'_1,\ldots,a_t,a'_t\in\mathcal A,$ $b_1,b'_1,\ldots,b_t,b'_t\in\mathcal{B},$ and $c_1,c'_1,\ldots,c_n,c'_n\in{\mathcal C},$ so $\sum_{i=1}^n(\langle a_i,a'_i\rangle^0+\langle b_i,b'_i\rangle^0+\langle c_i,c'_i\rangle^0)=0.$ Now take $\lambda\in\Lambda\setminus\{0\},$ then by Step 1, $(\mathcal{L}^0,\mathcal{L}^\lambda)$ plays the same role as the pair $(\mathcal{L}^\ell,\mathcal{L}^n)$ in Subsection \ref{subsect2-2} and so an argument analogous to the proof of Step 2 shows that $\sum_{i=1}^t([a_i,a'_i]+[b_i,b'_i]-c_i\hbox{\tiny$\heartsuit$} c'_i)=0.$ This completes the proof.
\noindent\underline{\textbf{Step 4:}} {\small$$\displaystyle{ \bigcup_{\lambda\in\Lambda}\sum_{i\in \mathcal{I}}{\mathcal G}_i^\lambda=\sum_{i\in\mathcal{I}}\bigcup_{\lambda\in\Lambda}{\mathcal G}_i^\lambda}, \displaystyle{\bigcup_{\lambda\in\Lambda}\sum_{j\in\mathcal{J}}\mathcal{S}^\lambda_j=\sum_{j\in\mathcal{J}}\bigcup_{\lambda\in\Lambda}\mathcal{S}^\lambda_j,}\; \displaystyle{\bigcup_{\lambda\in\Lambda}\sum_{t\in {\mathcal T}}{\mathcal V}^\lambda_t} \displaystyle{=\sum_{t\in {\mathcal T}}\bigcup_{\lambda\in\Lambda}{\mathcal V}^\lambda_t}:$$}
We just prove the first equality; the other ones may be proved using an analogous argument. Suppose that $\displaystyle{x\in \sum_{i\in\mathcal{I}}\bigcup_{\lambda\in\Lambda}{\mathcal G}_i^\lambda,}$ then there are $i_1,\ldots,i_n\in I,$ $\mu_1,\ldots,\mu_n\in \Lambda$ and
$x_1\in{\mathcal G}_{i_1}^{\mu_1},\ldots,x_n\in{\mathcal G}_{i_n}^{\mu_n}$ with $x=x_1+\cdots+x_n.$ Take $\mu\in\Lambda$ to be such that $\mu_t\preccurlyeq\mu$ for all $1\leq t\leq n.$ Then by Step 1, for $1\leq t\leq n,$ ${\mathcal G}^{\mu_t}_{i_t}\subseteq{\mathcal G}_{i_t}^\mu.$ Therefore $x=x_1+\cdots+x_n\in\sum_{i\in I}{\mathcal G}^\mu_i\subseteq\bigcup_{\lambda\in\Lambda}\sum_{i\in I}{\mathcal G}^\lambda_i.$ This completes the proof.
\noindent\underline{\textbf{Step 5:}} We have $$\bigcup_{\lambda\in\Lambda}\mathcal{L}^\lambda=\bigcup_{\lambda\in\Lambda}(\sum_{i\in \mathcal{I}}{\mathcal G}_i^\lambda)+\bigcup_{\lambda\in\Lambda}(\sum_{j\in\mathcal{J}}\mathcal{S}^\lambda_j)+\bigcup_{\lambda\in\Lambda}\sum_{t\in {\mathcal T}}({\mathcal V}^\lambda_t)+\mathcal D_0:$$
Using Step 2, one gets that the left hand side of the above equality is a subset of the right hand side. So we need to show the other side inequality. Suppose $a$ is an element of the right hand side, then there are $\mu_1,\mu_2,\mu_3,$ $x\in\sum_{i\in \mathcal{I}}{\mathcal G}_i^{\mu_1}\subseteq\mathcal{L}^{\mu_1},y\in\sum_{j\in\mathcal{J}}\mathcal{S}^{\mu_2}_j\subseteq\mathcal{L}^{\mu_2},z\in\sum_{t\in {\mathcal T}}{\mathcal V}^{\mu_3}_t\subseteq\mathcal{L}^{\mu_3}$ and $d\in\mathcal D_{0}\subseteq \mathcal{L}^{0}$ such that $a=x+y+z+d.$ Fix an upper bound $\mu\in\Lambda$ for $\{0,\mu_1,\mu_2,\mu_3\},$ then
we have $a=x+y+z+d\in\mathcal{L}^\mu\subseteq\cup_{\lambda\in\Lambda}\mathcal{L}^\lambda.$
\noindent\underline{\textbf{Step 6:}} $(\sum_{i\in \mathcal{I}}\bigcup_{\lambda\in\Lambda}{\mathcal G}_i^\lambda)+(\sum_{j\in\mathcal{J}}\bigcup_{\lambda\in\Lambda}\mathcal{S}^\lambda_j)+(\sum_{t\in {\mathcal T}}\bigcup_{\lambda\in\Lambda}{\mathcal V}^\lambda_t) +\mathcal D_0$ is a direct sum: Suppose that $\lambda_1,\ldots,\lambda_n,\mu_1,\ldots,\mu_m,\eta_1,\ldots,\eta_p\in\Lambda$ and $i_1,\ldots,i_n\in \mathcal{I},$ $j_1,\ldots,j_m\in\mathcal{J}$ and $t_1,\ldots,t_p\in{\mathcal T}$ are distinct. Let $x_{i_1}\in{\mathcal G}^{\lambda_1}_{i_1},\ldots,x_{i_n}\in{\mathcal G}^{\lambda_n}_{i_n},$ $y_{j_1}\in\mathcal{S}_{j_1}^{\mu_1},\ldots,y_{j_m}\in\mathcal{S}_{j_m}^{\mu_m},$ $z_{t_1}\in{\mathcal V}_{t_1}^{\eta_1},\ldots,z_{t_p}\in{\mathcal V}_{t_p}^{\eta_p}$ and $d\in\mathcal D_0$ be such that $$x_{i_1}+\cdots+x_{i_n}+y_{j_1}+\cdots+y_{j_m}+z_{t_1}+\cdots+z_{t_p}+ d=0.$$ Now take $\lambda\in \Lambda$ to be an upper bound for $\{\lambda_1,\ldots,\lambda_n,\mu_1,\ldots,\mu_m,\eta_1,\ldots,\eta_p\}$ with respect to the partial ordering on $\Lambda,$ then we get using Steps 1,2 that $x_{i_1}+\cdots+x_{i_n}+y_{j_1}+\cdots+y_{j_m}+z_{t_1}+\cdots+z_{t_p}+d$ is a summation in $(\oplus_{i\in \mathcal{I}}{\mathcal G}^\lambda_i)\bigoplus(\oplus_{j\in \mathcal{J}}\mathcal{S}^\lambda_j)\bigoplus(\oplus_{t\in {\mathcal T}}{\mathcal V}^\lambda_t)\oplus\mathcal D_0.$ Therefore we have $$x_{i_1}=0,\ldots,x_{i_n}=0,y_{j_1}=0,\ldots,y_{j_m}=0,z_{t_1}=0,\ldots,z_{t_p}=0,d=0.$$This completes the proof of this step.
\noindent\underline{\textbf{Step 7:}} The assertion stated in $(ii)$ is true: Take $\mathcal A$ to be a vector space with a basis $\{a_i\mid i\in\mathcal{I}\},$ $\mathcal{B}$ to be a vector space with a basis $\{b_j\mid j\in\mathcal{J}\},$ and ${\mathcal C}$ to be a vector space with a basis $\{c_t\mid t\in{\mathcal T}\}.$ Using Steps 1-2,4-6, we get that \begin{eqnarray*}\mathcal{L}=\bigcup_{\lambda\in\Lambda}\mathcal{L}_\lambda&=&\bigcup_{\lambda\in\Lambda}(\sum_{i\in \mathcal{I}}{\mathcal G}_i^\lambda)+\bigcup_{\lambda\in\Lambda}(\sum_{j\in\mathcal{J}}\mathcal{S}^\lambda_j)+\bigcup_{\lambda\in\Lambda}(\sum_{t\in {\mathcal T}}{\mathcal V}^\lambda_t)+\mathcal D_0\\ &=&(\bigoplus_{i\in \mathcal{I}}\bigcup_{\lambda\in\Lambda}{\mathcal G}_i^\lambda)\oplus(\bigoplus_{j\in\mathcal{J}}\bigcup_{\lambda\in\Lambda}\mathcal{S}^\lambda_j)\oplus(\bigoplus_{t\in {\mathcal T}}\bigcup_{\lambda\in\Lambda}{\mathcal V}^\lambda_t)\oplus\mathcal D_0. \end{eqnarray*}
Now consider $\mathcal{L}$ as a $\mathfrak{g}-$module via the adjoint representation and for $i\in \mathcal{I},$ $j\in\mathcal{J}$ and $t\in{\mathcal T},$ set $${\mathcal G}^{(i)}:=\bigcup_{\lambda\in\Lambda}{\mathcal G}_i^\lambda,\; \mathcal{S}^{(j)}:=\bigcup_{\lambda\in\Lambda}\mathcal{S}^\lambda_j,\; {\mathcal V}^{(t)}:=\bigcup_{\lambda\in\Lambda}{\mathcal V}^\lambda_t,$$then by Propositions \ref{dir-lim-mod} and \ref{rep-local}, ${\mathcal G}^{(i)}$ is a $\mathfrak{g}-$submodule of $\mathcal{L}$ isomorphic to $\mathfrak{g}\simeq{\mathcal G},$ $\mathcal{S}^{(j)}$ is a $\mathfrak{g}-$submodule isomorphic to $\mathcal{S}$ and ${\mathcal V}^{(t)}$ is a $\mathfrak{g}-$submodule isomorphic to ${\mathcal V}.$ Therefore as a vector space, we can identify $\mathcal{L}$ with $$({\mathcal G}\otimes\mathcal A)\oplus(\mathcal{S}\otimes\mathcal{B})\oplus({\mathcal V}\otimes{\mathcal C})\oplus\mathcal D_0.$$
such that for each $\lambda\in\Lambda,$ $\mathcal{L}^\lambda$ is identified with $$({\mathcal G}^\lambda\dot\otimes\mathcal A)\oplus(\mathcal{S}^\lambda\dot\otimes\mathcal{B})\oplus({\mathcal V}^\lambda\dot\otimes{\mathcal C})\oplus\mathcal D_0.$$ Now for $\lambda\in\Lambda,$ $(\mathcal{L}^\lambda,\mathcal{L}^0)$ plays the same role as $(\mathcal{L}^n,\mathcal{L}^\ell)$ in \S \ref{subsub1} and so we are done using Step 3 together with Proposition \ref{divide5}.\qed \begin{Theorem}\label{type-a-d} Suppose that $I$ is an infinite index set and $I_0$ is a subset of $I$ of cardinality $\ell>5.$ Let $R$ be an irreducible locally finite root system of type $X=D_I$ or $X=\dot{A}_I.$ Suppose that ${\mathcal V}$ is a vector space with a basis $\{v_i\mid i\in I\}$ and take ${\mathcal G}$ to be the finite dimensional split simple Lie algebra of type $X$ as in Lemmas \ref{type-a-alg} or \ref{type-d-alg} respectively. Also define $$\mathfrak{I}_0:{\mathcal V}\longrightarrow{\mathcal V}\;\;\;\; v_i\mapsto\left\{\begin{array}{ll}v_i& i\in I_0\\ 0&\hbox{otherwise.}\end{array}\right.$$ For $x,y\in{\mathcal G},$ define $$x\circ y:=xy+yx-\frac{2tr(xy)}{l+1}\mathfrak{I}_0.$$
Suppose that $(\mathcal A,id_{\mathcal A},\{0\},{\bf 0})$ is a coordinate quadrable of type $X$ and $\mathcal K$ is a subset of the full skew-dihedral homology group of $\mathcal A$ satisfying the uniform property on $\mathcal A.$ Set $$\mathcal{L}(\mathcal A,\mathcal K):=({\mathcal G}\otimes\mathcal A)\oplus\langle\mathcal A,\mathcal A\rangle,$$ in which $\langle\mathcal A,\mathcal A\rangle$ is the quotient space $\{\mathcal A,\mathcal A\}_\ell/\mathcal K$ (see Subsection \ref{subsect2-1}) and for $a,a'\in\mathcal A,$ take $\langle a,a'\rangle:=\{a,a'\}_\ell+\mathcal K,$ then $\mathcal{L}(\mathcal A,\mathcal K)$ together with \begin{equation}\label{proa-d-gen} \begin{array}{l} \hbox{\small$[x\otimes a,y\otimes a']=$}\left\{\begin{array}{ll}\hbox{\small$\;[x,y]\otimes\frac{1}{2}(a\circ a')+ (x\circ y)\otimes\frac{1}{2}[a,a']+tr(xy)\langle a,a'\rangle$}& \hbox{\small $X=\dot A_I,$}
\\\hbox{\small$\;[x,y]\otimes aa'+tr(xy)\langle a,a'\rangle$}&\hbox{\small$X=D_I,$}\end{array}\right. \\ \hbox{\small$[\langle a_1,a_2\rangle,x\otimes a]=$}\left\{\begin{array}{ll} \hbox{\small$-\frac{1}{2(\ell+1)}((x\circ Id_{_{{\mathcal V}^\ell}})\otimes[a,[a_1,a_2]]$}\\ \hbox{\small $ +[x,Id_{_{{\mathcal V}^\ell}}]\otimes (a\circ [a_1,a_2])+2tr(Id_{_{{\mathcal V}^\ell}}x)\langle a,[a_1,a_2]\rangle)$},&\hbox{\small $X=\dot A_I,$}\\ 0&\hbox{\small $X=D_I,$}\end{array}\right.
\\ \hbox{\small $[\langle a_1,a_2\rangle,\langle a'_1,a'_2\rangle]=$}\left\{\begin{array}{ll}\hbox{\small $\langle d^{\ell,\mathcal A}_{a_1,a_2}(a'_1),a'_2\rangle+\langle a'_1,d^{\ell,\mathcal A}_{a_1,a_2}(a'_2)\rangle,$}& \hbox{\small $X=\dot A_I,$}\\ 0&\hbox{\small $X=D_I,$}\end{array}\right. \end{array}
\end{equation} for $x,y\in{\mathcal G},$ $a,a',a_1,a_2,a'_1,a'_2\in\mathcal A,$ is a Lie algebra graded by $R.$ Moreover any $R-$graded Lie algebra gives rise in this manner. \end{Theorem} \begin{Theorem}\label{type b-c} Suppose that $I$ is an infinite index set and $I_0$ is a subset of $I$ of cardinality $\ell>4.$ Take ${\mathcal G}$ to be either $\mathfrak{o}_B(I)$ or $\mathfrak{sp}(I).$ Suppose that ${\mathcal V}$ is a vector space with a basis $\{v_0,v_i,v_{\bar i}\mid i\in I\}$ equipped with a nondegenerate symmetric bilinear form $(\cdot,\cdot)$ as in (\ref{form-b-alg}) if ${\mathcal G}=\mathfrak{o}_B(I)$ and it is a vector space with a basis $\{v_i,v_{\bar i}\mid i\in I\}$ equipped with a nondegenerate skew-symmetric bilinear form $(\cdot,\cdot)$ as in (\ref{form-c}) if ${\mathcal G}:=\mathfrak{sp}(I).$ Set $$J:=\left\{\begin{array}{ll}I_0\cup\bar{I_0}\cup\{0\}& \hbox{if ${\mathcal G}=\mathfrak{o}_B(I)$}\\ I_0\cup\bar{I_0}&\hbox{if ${\mathcal G}=\mathfrak{sp}(I)$}\end{array}\right.$$ and define $\mathfrak{I}_0:{\mathcal V}\longrightarrow {\mathcal V}$ to be the linear transformation defined by $$v_i\mapsto \left\{\begin{array}{ll}v_i& \hbox{if $i\in J$}\\ 0& \hbox{if $i\in I\cup\bar{I} \setminus J.$}\end{array}\right. $$ Next set $\mathcal{S}:={\mathcal V}$ if ${\mathcal G}:=\mathfrak{o}_B(I)$ and take $\mathcal{S}$ to be as in (\ref{module-s-c}) if ${\mathcal G}=\mathfrak{sp}(I).$ For $e,f\in{\mathcal G}\cup\mathcal{S},$ set $$e\circ f:=ef+fe-\frac{tr(ef)}{\ell}\mathfrak{I}_0.$$ Suppose that $R$ is an irreducible locally finite root system of type $X=B_I$ or $X=C_I$ and $(\mathfrak{a},*,{\mathcal C},f)$ is a coordinate quadrable of type $X.$ Take $\mathcal A$ and $\mathcal{B}$ to be the set of $*-$fixed and $*-$skew fixed points of $\mathfrak{a}$ respectively. For a subset $\mathcal K$ of the full skew-dihedral homology group of $\mathfrak{a}$ satisfying the uniform property on $\mathfrak{a},$ set $$\mathcal{L}(\mathfrak{a},\mathcal K):=({\mathcal G}\otimes\mathcal A)\oplus(\mathcal{S}\otimes \mathcal{B})\oplus\langle\mathfrak{a},\mathfrak{a}\rangle,$$ in which $\langle\mathfrak{a},\mathfrak{a}\rangle$ is the quotient space $\{\mathfrak{a},\mathfrak{a}\}_\ell/\mathcal K$ (see Subsection \ref{subsect2-1}), and for $\alpha,\alpha'\in\mathfrak{a},$ take $\langle \alpha,\alpha'\rangle:=\{\alpha,\alpha'\}+\mathcal K.$ Then $\mathcal{L}(\mathfrak{a},\mathcal K)$ together with \begin{equation}\label{prod-gen} \begin{array}{l} \;[x\otimes a,y\otimes a']=[x,y]\otimes aa'+tr(xy)\langle a,a'\rangle,
\\ \;[x\otimes a,s\otimes b]=xs\otimes ab,
\\ \;[s\otimes b,t\otimes b']=D_{s,t}\otimes f(b,b')+(s,t)\langle b,b'\rangle\\ \;[\langle \alpha_1,\alpha_2\rangle,x\otimes a]= x\otimes d^{\ell,\mathfrak{a}}_{\alpha_1,\alpha_2}(a),
\\ \;[\langle \alpha_1,\alpha_2\rangle,s\otimes b]= s\otimes d^{\ell,\mathfrak{a}}_{\alpha_1,\alpha_2}(b),
\\ \;[\langle \alpha_1,\alpha_2\rangle,\langle \alpha'_1,\alpha'_2\rangle]=\langle d^{\ell,\mathfrak{a}}_{\alpha_1,\alpha_2}(\alpha'_1),\alpha'_2\rangle+\langle \alpha_1,d^{\ell,\mathfrak{a}}_{\alpha'_1,\alpha'_2}(\alpha_2)\rangle. \end{array}
\end{equation} (see Definition \ref{yoshii}) for $x,y\in{\mathcal G},$ $s,t\in\mathcal{S},$ $a,a'\in\mathcal A,$ $b,b'\in\mathcal{B},$ $\alpha,\alpha',\alpha_1,\alpha_2,\alpha'_1,\alpha'_2\in\mathfrak{a},$ if ${\mathcal G}=\mathfrak{o}_B(I)$ and
\begin{equation}\label{prob-c-gen} \begin{array}{l} \hbox{\small$[x\otimes a,y\otimes a']=[x,y]\otimes\frac{1}{2}(a\circ a')+ (x\circ y)\otimes\frac{1}{2}[a,a']+tr(xy)\langle a,a'\rangle,$}
\\ \hbox{\small$[x\otimes a,s\otimes b]=(x\circ s)\otimes\frac{1}{2}[a,b]+[x,s]\otimes \frac{1}{2} (a\circ b),$}
\\ \hbox{\small$[s\otimes b,t\otimes b']=[s,t]\otimes\frac{1}{2}b\circ b' + (s\circ t)\otimes \frac{1}{2}[b,b']+tr(st)\langle b,b'\rangle,$}
\\ \hbox{\small$[\langle \alpha,\alpha'\rangle,x\otimes a]= \frac{-1}{4\ell}((x\circ {\mathfrak{I}_0})\otimes[a,\beta_{\alpha,\alpha'}^*]+[x,{\mathfrak{I}_0}]\otimes (a\circ \beta_{\alpha,\alpha'}^*)),$}
\\ \hbox{\small$[\langle \alpha,\alpha'\rangle,s\otimes b]=\frac{-1}{4\ell}([s,{\mathfrak{I}_0}]\hspace{-1mm}\otimes\hspace{-1mm} (b\circ \beta_{\alpha_1,\alpha_2}^*)\hspace{-1mm}+\hspace{-1mm}(s\circ {\mathfrak{I}_0})\hspace{-1mm}\otimes\hspace{-1mm} [b, \beta_{\alpha_1,\alpha_2}^*]\hspace{-1mm}+\hspace{-1mm}2tr(s{\mathfrak{I}_0})\langle b,\beta^*_{\alpha,\alpha'}\rangle),$}
\\ \hbox{\small$[\langle \alpha_1,\alpha_2\rangle,\langle \alpha'_1,\alpha'_2\rangle]=\langle d^{\ell,\mathfrak{a}}_{\alpha_1,\alpha_2}(\alpha'_1),\alpha'_2\rangle+\langle\alpha'_1,d^{\ell,\mathfrak{a}}_{\alpha_1,\alpha_2}(\alpha'_2)\rangle.$} \end{array}
\end{equation} (see (\ref{beta*})) for $x,y\in{\mathcal G},$ $s,t\in\mathcal{S},$ $a,a'\in\mathcal A,$ $b,b'\in\mathcal{B},$ $\alpha,\alpha',\alpha_1,\alpha_2,\alpha'_1,\alpha'_2\in\mathfrak{a},$ if ${\mathcal G}=\mathfrak{sp}(I)$ is a Lie algebra graded by $R.$ Moreover any $R-$graded Lie algebra gives rise in this manner. \end{Theorem}
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran.\\ \\
Department of Mathematics, University of Isfahan, Isfahan, Iran, P.O.Box 81745-163.\\ [email protected]
\end{document} | arXiv |
For $n \ge 0$, let $F_n$ denote the $n$th Fibonacci number (that is, $F_0 = 0, F_1 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for all $n \ge 2$). What is the greatest possible value of the greatest common divisor of two consecutive Fibonacci numbers?
We want to find the maximum possible value of $\text{gcd}\,(F_{n}, F_{n-1})$. Since $F_{n} = F_{n-1} + F_{n-2},$ by the Euclidean algorithm, this is equivalent to finding \begin{align*}
\text{gcd}\,(F_{n-1} + F_{n-2}, F_{n-1}) &= \text{gcd}\,(F_{n-1} + F_{n-2} - F_{n-1}, F_{n-1}) \\
&= \text{gcd}\,(F_{n-1}, F_{n-2}).
\end{align*}It follows that \begin{align*}
\text{gcd}\,(F_n, F_{n-1}) &= \text{gcd}\,(F_{n-1}, F_{n-2})\\
&= \cdots = \text{gcd}\,(F_2, F_1)\\
&= \text{gcd}\,(1,1)\\
&= \boxed{1}.
\end{align*} | Math Dataset |
Find $\Delta S_{\text { rxn }}^{\circ}$ for the r…
Find $\Delta S_{\text { rxn }}^{\circ}$ for the reaction of nitrogen monoxide with hydrogen to form ammonia and water vapor. Is the sign of $\Delta S_{\mathrm{rxn}}^{\circ}$ as expected?
Find $\Delta S_{\mathrm{rxn}}^{\delta}$ for the combustion of ammonia to nitrogen dioxide and water vapor. Is the sign of $\Delta S_{\mathrm{rxn}}^{\circ}$ as expected?
(a) Find $\Delta S_{\text { rxn }}^{\circ}$ for the formation of $\mathrm{Cu}_{2} \mathrm{O}(s)$ from its elements.
(b) Calculate $\Delta S_{\text { univ }}$ , and state whether the reaction is spontaneous at 298 $\mathrm{K} .$
(a) Find $\Delta S_{\mathrm{rxn}}^{\circ}$ for the formation of HI(g) from its elements.
(b) Calculate $\Delta S_{\text { univ }},$ and state whether the reaction is spontaneous
at 298 $\mathrm{K}$ .
(a) Find $\Delta S_{\mathrm{rxn}}^{\circ}$ for the formation of $\mathrm{CH}_{3} \mathrm{OH}(l)$ from its elements.
(b) Calculate $\Delta S_{\text { univ }},$ and state whether the reaction is spontaneous at 298 $\mathrm{K}$ .
Lucas P.
Find $\Delta S_{\text { rxn }}^{\circ}$ for the combustion of methane to carbon dioxide and liquid water. Is the sign of $\Delta S_{\text { rxn }}^{\circ}$ as expected?
Given the values of $\Delta H_{\mathrm{rxn}}, \Delta S_{\mathrm{rxn}}$ and $T,$ determine $\Delta S_{\text { univ }}$ and predict whether or not each reaction is spontaneous. (Assume that all reactants and products are in their standard states.)
\begin{equation}\begin{array}{l}{\text { a. } \Delta H_{\mathrm{rxn}}^{\circ}=-95 \mathrm{kJ} ; \Delta S_{\mathrm{rxn}}^{\circ}=-157 \mathrm{J} / \mathrm{K} ; T=298 \mathrm{K}} \\ {\text { b. } \Delta H_{\mathrm{rxn}}^{\circ}=-95 \mathrm{kJ} ; \Delta S_{\mathrm{rxn}}^{\circ}=-157 \mathrm{J} / \mathrm{K} ; T=855 \mathrm{K}} \\ {\text { c. } \Delta H_{\mathrm{rxn}}^{\circ}=+95 \mathrm{kJ} ; \Delta S_{\mathrm{rxn}}^{\circ}=-157 \mathrm{J} / \mathrm{K} ; T=298 \mathrm{K}} \\ {\text { d. } \Delta H_{\mathrm{rxn}}^{\circ}=-95 \mathrm{kJ} ; \Delta S_{\mathrm{rxn}}^{\circ}=+157 \mathrm{J} / \mathrm{K} ; T=398 \mathrm{K}}\end{array}\end{equation}
In photosynthesis, plants form glucose $\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)$ and oxygen from carbon dioxide and water. Write a balanced equation for photosynthesis and calculate $\Delta H_{\mathrm{rxn}}^{\circ}, \Delta S_{\mathrm{rxn}}^{\circ},$ and $\Delta G_{\mathrm{ren}}^{\circ}$ at $25^{\circ} \mathrm{C}$ . Is photosynthesis spontaneous? | CommonCrawl |
\begin{document}
\title{Adversarial Intrinsic Motivation for Reinforcement Learning}
\begin{abstract} Learning with an objective to minimize the mismatch with a reference distribution has been shown to be useful for generative modeling and imitation learning. In this paper, we investigate whether one such objective, the Wasserstein-1 distance between a policy's state visitation distribution and a target distribution, can be utilized effectively for reinforcement learning (RL) tasks. Specifically, this paper focuses on goal-conditioned reinforcement learning where the idealized (unachievable) target distribution has full measure at the goal. This paper introduces a quasimetric specific to Markov Decision Processes (MDPs) and uses this quasimetric to estimate the above Wasserstein-1 distance. It further shows that the policy that minimizes this Wasserstein-1 distance is the policy that reaches the goal in as few steps as possible. Our approach, termed Adversarial Intrinsic Motivation (\textsc{aim}), estimates this Wasserstein-1 distance through its dual objective and uses it to compute a supplemental reward function. Our experiments show that this reward function changes smoothly with respect to transitions in the MDP and directs the agent's exploration to find the goal efficiently. Additionally, we combine \textsc{aim}\ with Hindsight Experience Replay (\textsc{her}) and show that the resulting algorithm accelerates learning significantly on several simulated robotics tasks when compared to other rewards that encourage exploration or accelerate learning. \end{abstract}
\section{Introduction}
Reinforcement Learning (RL) \citep{sutton2018reinforcement} deals with the problem of learning a policy to accomplish a given task in an optimal manner. This task is typically communicated to the agent by means of a reward function. If the reward function is sparse \citep{Arumugam2021AnIP} (e.g., most transitions yield a reward of $0$), much random exploration might be needed before the agent experiences any signal relevant to learning \citep{bellemare2016unifying,arjona2019rudder}.
Some of the different ways to speed up reinforcement learning by modifying or augmenting the reward function are shaped rewards \citep{ng1999policy}, redistributed rewards \citep{arjona2019rudder}, intrinsic motivations \citep{baldassarre_intrinsic_2013,singh_intrinsically_2010, sorg2010reward, sorg2010internal, niekum2010evolved, oudeyer2009intrinsic}, and learned rewards \citep{zheng_learning_2018, niekum2010evolved}. Unfortunately, the optimal policy under such modified rewards might sometimes be different than the optimal policy under the task reward \cite{ng1999policy,clark_faulty_2016}. The problem of learning a reward signal that speeds up learning by communicating \emph{what to do} but does not interfere by specifying \emph{how to do it} is thus a useful and complex one \citep{zheng_what_2020}.
This work considers whether a task-dependent reward function learned based on the distribution mismatch between the agent's state visitation distribution and a target task (expressed as a distribution) can guide the agent towards accomplishing this task. Adversarial methods to minimize distribution mismatch have been used with great success in generative modeling \citep{goodfellow_generative_2014} and imitation learning \citep{ho2016generative, fu_learning_2018,xiao_wasserstein_2019, torabi_generative_2019, ghasemipour2020divergence}. In both these scenarios, the task is generally to minimize the mismatch with a target distribution induced by the data or expert demonstrations. Instead, we consider the task of goal-conditioned RL, where the ideal target distribution assigns full measure to a goal state. While the agent can never match this idealized target distribution perfectly unless starting at the goal, intuitively, minimizing the mismatch with this distribution should lead to trajectories that maximize the proportion of the time spent at the goal, thereby prioritizing transitions essential to doing so.
The theory of optimal transport \citep{villani2008optimal} gives us a way to measure the distance between two distributions (called the Wasserstein distance) even if they have disjoint support. Previous work \citep{arjovsky_wasserstein_2017, gulrajani_improved_2017} has shown how a neural network approximating a potential function may be used to estimate the Wasserstein-1 distance using its dual formulation, but assumes that the metric space this distance is calculated on is Euclidean. A Euclidean metric might not be the appropriate metric to use in more general RL tasks however, such as navigating in a maze or environments where the state features change sharply with transitions in the environment.
This paper introduces a quasimetric tailored to Markov Decision Processes (MDPs), the time-step metric, to measure the Wasserstein distance between the agent's state visitation distribution and the idealized target distribution. While this time-step metric could be an informative reward on its own, estimating it is a problem as hard as policy evaluation \citep{guillot2020stochastic}. Instead, we show that the dual objective, which maximizes difference in potentials while utilizing the structure of this quasimetric for the necessary regularization, can be optimized through sampled transitions.
We use this dual objective to estimate the Wasserstein-1 distance and propose a reward function based on this estimated distance. An agent that maximizes returns under this reward minimizes this Wasserstein-1 distance. The competing objectives of maximizing the difference in potentials for estimating the Wasserstein distance and minimizing it through reinforcement learning on the subsequent reward function leads to our algorithm, Adversarial Intrinsic Motivation (\textsc{aim}).
Our analysis shows that if the above Wasserstein-1 distance is computed using the time-step metric, then minimizing it leads to a policy that reaches the goal in the minimum expected number of steps. It also shows that if the environment dynamics are deterministic, then this policy is the optimal policy.
In practice, minimizing the Wasserstein distance works well even when the environment dynamics are stochastic. Our experiments show that \textsc{aim}\ learns a reward function that changes smoothly with transitions in the environment. We further conduct experiments on the family of goal-conditioned reinforcement learning problems \cite{andrychowicz_hindsight_2018, Schaul2015UniversalVF} and show that \textsc{aim}\ when used along with hindsight experience replay (\textsc{her}) greatly accelerates learning of an effective goal-conditioned policy compared to learning with \textsc{her} and the sparse task reward. Further, our experiments show that this acceleration is similar to the acceleration observed by using the actual distance to the goal as a dense reward.
\section{Related Work}
We highlight the related work based on the various aspects of learning that this work touches, namely intrinsic motivation, goal-conditioned reinforcement learning, and adversarial imitation learning.
\subsection{Intrinsic Motivation}
Intrinsic motivations \citep{baldassarre_intrinsic_2013, oudeyer2009intrinsic, oudeyer2008can} are rewards presented by an agent to itself in addition to the external task-specific reward. Researchers have pointed out that such intrinsic motivations are a characteristic of naturally intelligent and curious agents \citep{gottlieb2013information, baldassarre2011intrinsic, baldassarre2014intrinsic}. Intrinsic motivation has been proposed as a way to encourage RL agents to learn skills \citep{barto2004intrinsically, barto2005intrinsic, singh2005intrinsically, santucci2013best} that might be useful across a variety of tasks, or as a way to encourage exploration \citep{bellemare2016unifying, csimcsek2006intrinsic, baranes2009r, forestier2017intrinsically}. The optimal reward framework \citep{singh_intrinsically_2010, sorg2010internal} and shaped rewards \citep{ng1999policy} (if generated by the agent itself) also consider intrinsic motivation as a way to assist an RL agent in learning the optimal policy for a given task. Such an intrinsically motivated reward signal has previously been learned through various methods such as evolutionary techniques \citep{niekum2010evolved, schembri2007evolving}, meta-gradient approaches \citep{sorg2010reward, zheng_learning_2018,zheng_what_2020}, and others. The Wasserstein distance has been used to present a valid reward for imitation learning \citep{xiao_wasserstein_2019, dadashi_primal_2020} as well as program synthesis \citep{ganin2018synthesizing}.
\subsection{Goal-Conditioned Reinforcement Learning} \label{sec:rel_goal}
Goal-conditioned reinforcement learning \citep{kaelbling1993learning} can be considered a form of multi-task reinforcement learning \citep{caruana1997multitask} where the agent is given the goal state it needs to reach at the beginning of every episode, and the reward function is sparse with a non-zero reward only on reaching the goal state. \textsc{uvfa} \citep{Schaul2015UniversalVF}, \textsc{her} \citep{andrychowicz_hindsight_2018}, and others \citep{zhang_automatic_2020,Ding2019GoalconditionedIL} consider this problem of reaching certain states in the environment. Relevant to our work, \citet{Venkattaramanujam2019SelfsupervisedLO} learns a distance between states using a random walk that is then used to shape rewards and speed up learning, but requires goals to be visited before the distance estimate is useful. DisCo RL \citep{Nasiriany:EECS-2020-151} extends the idea of goal-conditioned RL to distribution-conditioned RL.
Contemporaneously, \citet{ eysenbach2021replacing,eysenbach2021clearning} has proposed a method which considers goals and examples of success and tries to predict and maximize the likelihood of seeing those examples under the current policy and trajectory. For successful training, this approach needs the agent to actually experience the goals or successes. Their solution minimizes the Hellinger distance to the goal, a form of $f$-divergence. \textsc{aim}\ instead uses the Wasserstein distance which is theoretically more informative when considering distributions that are disjoint, and does not require the assumption that the agent has already reached the goal through random exploration. Our experiments in fact verify the hypothesis that \textsc{aim}\ induces a form of directed exploration in order to reach the goal.
\subsection{Adversarial Imitation Learning and Minimizing Distribution Mismatch} \label{sec:AIL}
Adversarial imitation learning \citep{ho2016generative,fu_learning_2018,xiao_wasserstein_2019,torabi_generative_2019,ghasemipour2020divergence} has been shown to be an effective method to learn agent policies that minimize distribution mismatch between an agent's state-action visitation distribution and the state-action visitation distribution induced by an expert's trajectories. In most cases this distribution that the expert induces is achievable by the agent and hence these techniques aim to match the expert distribution exactly. In the context of goal-conditioned reinforcement learning, GoalGAIL \citep{Ding2019GoalconditionedIL} uses adversarial imitation learning with a few expert demonstrations to accelerate the learning of a goal-conditioned policy. In this work, we focus on unrealizable target distributions that cannot be completely matched by the agent, and indeed, are not induced by any trajectory distribution.
\textsc{fairl} \citep{ghasemipour2020divergence} is an adversarial imitation learning technique which minimizes the Forward KL divergence and has been shown experimentally to cover some hand-specified state distributions, given a smoothness regularization as used by WGAN \citep{gulrajani_improved_2017}. $f$-\textsc{irl} \citep{ni2020fIRL} learns a reward function where the optimal policy matches the expert distribution under the more general family of $f$-divergences. Further, techniques beyond imitation learning \citep{lee2019efficient, hazan2019provably} have looked at matching a uniform distribution over states to guarantee efficient exploration.
\section{Background}
In this section we first set up the goal-conditioned reinforcement learning problem, and then give a brief overview of optimal transport.
\subsection{Goal-Conditioned Reinforcement Learning} \label{sec:goal}
Consider a goal-conditioned MDP as the tuple $\langle \sset, \aset, \gset, P, \rho_0, \sigma, \gamma \rangle$ with discrete state space $\sset$, discrete action space $\aset$, a subset of states which is the goal set $\gset \subseteq \sset$, and transition dynamics $P: \sset \times \aset \times \gset \longmapsto \Delta(\sset)$ ($\Delta(\cdot)$ is a distribution over a set) which might vary based on the goal (see below). $\rho_0: \Delta(\sset)$ is the starting state distribution, and $\sigma: \Delta(\gset)$ is the distribution a goal is drawn from. $\gamma \in [0, 1)$ is the discount factor. We use discrete states and actions for ease of exposition, but our idea extends to continuous states and actions, as seen in the experiments.
At the beginning of an episode, the starting state is drawn from $\rho_0$ and the goal for that episode is drawn from $\sigma$. The reward function $r: \sset \times \aset \times \sset \times \gset \longmapsto \mathbb{R}$ is deterministic, and $r(s_t, a_t, s_{t+1}|s_g) \defd \mathbb{I}[s_{t+1}=s_g]$. That is, there is a positive reward when an agent reaches the goal ($s_{t+1} = s_g$), and $0$ everywhere else. Since the goal is given to the agent at the beginning of the episode, in goal-conditioned RL the agent knows what this task reward function is (unlike the more general RL problem). The transition dynamics are goal-conditioned as well, with an automatic transition to an absorbing state $\Bar{s}$ on reaching the goal $s_g$ and then staying in that state with no rewards thereafter ($P(\Bar{s}|s_g, a, s_g) = 1 \;\forall\; a \in \aset$ and $P(\Bar{s}|\Bar{s}, a, s_g) = 1 \;\forall\; a \in \aset$). In short, the episode terminates on reaching the goal state.
The agent takes actions in this environment based on a policy $\pi \in \Pi: \sset \times \gset \longmapsto \Delta(\aset)$. The return $H_g$ for an episode with goal $s_g$ is the discounted cumulative reward over that episode $H_g = \sum_{t=0}^{\infty} \gamma^t r(s_t, a_t, s_{t+1} | s_g)$, where $s_0 \sim \rho_0$, $a_t \sim \pi(\cdot|s_t, s_g)$, and $s_{t+1} \sim P(\cdot|s_t, a_t, s_g)$. The agent aims to find the policy $\pi^* = \argmax_{\pi \in \Pi} \mathbb{E}_{g \in \mathcal{G}} \mathbb{E}_{s_0 \sim \rho_0} \mathbb{E}_{\pi}[H_g]$ that maximizes the expected returns in this goal-conditioned MDP. For a policy $\pi$, the agent's goal-conditioned state distribution $\rho_{\pi}(s|s_g) = \mathbb{E}_{s_0 \sim \rho_0}[(1 - \gamma)\sum_{t=0}^{\infty} \gamma^t P(s_t=s|\pi, s_g)]$. Overloading the terminology a bit, we also define the goal-conditioned target distribution $\rho_g(s| s_g) = \delta(s_g)$, a Dirac measure at the goal state $s_g$.
While learning using traditional RL paradigms is possible in goal-conditioned RL, there has also been previous work (Section \ref{sec:rel_goal}) on leveraging the structure of the problem across goals. Hindsight Experience Replay (\textsc{her}) \citep{andrychowicz_hindsight_2018} attempts to speed up learning in this sparse reward setting by taking episodes of agent interactions, where they might not have reached the goal specified for that episode, and relabeling the transitions with the goals that \emph{were} achieved during the episode. Off-policy learning algorithms are then used to learn from this relabeled experience.
\subsection{Optimal Transport and Wasserstein-1 Distance} \label{sec:opt}
The theory of optimal transport \citep{villani2008optimal,bousquet_optimal_2017} considers the question of how much work must be done to transport one distribution to another optimally, where this notion of work is defined by the use of a ground metric $d$. More concretely, consider a metric space ($\mathcal{M}$, $d$) where $\mathcal{M}$ is a set and $d$ is a metric on $\mathcal{M}$ (Definitions in Appendix \ref{sec:metrics}). For two distributions $\mu$ and $\nu$ with finite moments on the set $\mathcal{M}$, the Wasserstein-$p$ distance is denoted by: \begin{align} \label{eqn:w-p}
W_p(\mu, \nu) \defd \inf_{\zeta \in Z(\mu, \nu)} \mathbb{E}_{(X,Y)\sim \zeta}[d(X, Y)^p]^{1/p} \end{align}
where $Z$ is the space of all possible couplings, i.e.\ joint distributions $\zeta \in \Delta(\mathcal{M}\times\mathcal{M})$ whose marginals are $\mu$ and $\nu$ respectively. Finding this optimal coupling tells us what is the least amount of work, as measured by $d$, that needs to be done to convert $\mu$ to $\nu$. This Wasserstein-$p$ distance can then be used as a cost function (negative reward) by an RL agent to match a given target distribution \citep{xiao_wasserstein_2019,dadashi_primal_2020,ganin2018synthesizing}.
\begin{wrapfigure}{l}{0.32\textwidth}
\begin{center}
\includegraphics[width=0.3\textwidth]{Figures/example.png}
\end{center}
\caption{Grid world example} \label{fig:example}
\end{wrapfigure}
Finding the ideal coupling above is generally considered intractable. However, if what we need is an accurate estimate of the Wasserstein distance and not the optimal transport plan we can turn our attention to the dual form of the Wasserstein-1 distance. The Kantorovich-Rubinstein duality \citep{villani2008optimal, peyre_computational_2020} for the Wasserstein-1 distance (which we refer to simply as the Wasserstein distance hereafter) on a ground metric $d$ gives us: \begin{align} \label{eqn:KRdual}
W_1(\mu, \nu) = \sup_{\text{Lip}(f) \leq 1} \mathbb{E}_{y \sim \nu}\left[f(y)\right] - \mathbb{E}_{x \sim \mu} \left[f(x)\right] \end{align}
where the supremum is over all $1$-Lipschitz functions $f: \mathcal{M} \longmapsto \mathbb{R}$ in the metric space. Importantly, \citet{jevtic2018combinatorial} has recently shown that this dual formulation extends to quasimetric spaces as well. More details, such as the definition of the Lipschitz constant of a function and special cases used in WGAN \citep{arjovsky_wasserstein_2017, gulrajani_improved_2017, ghasemipour2020divergence} are elaborated in Appendix \ref{app:opt}. One last note of importance is that the Lipschitz constant of the potential function $f$ is computed based on the ground metric $d$.
\section{Time-Step Metric} \label{sec:cmetric}
The choice of the ground metric $d$ is important when computing the Wasserstein distance between two distributions. That is, if we want the Wasserstein distance to give an estimate of the work needed to transport the agent's state visitation distribution to the goal state, the ground metric should incorporate a notion of this work.
Consider the grid world shown in \autoref{fig:example}, where a wall (bold line) marks an impassable barrier in part of the state space. If the states are specified by their Cartesian coordinates on the grid, the Manhattan distance between the states specified by the blue and red circles is not representative of the optimal cost to go from one to the other. This mismatch would lead to an underestimation of the work involved if the two distributions compared were concentrated at those two circles. Similarly, there will be errors in estimating the Wasserstein distance if the grid world is toroidal (where an agent is transported to the opposite side of the grid if it walks off one side) or if the transitions are asymmetric (windy grid world \citep{sutton2018reinforcement}).
To estimate the work needed to transport measure in an MDP when executing a policy $\pi$, we consider a \emph{quasimetric} -- a metric that does not need to be symmetric -- dependent on the number of transitions experienced before reaching the goal when executing that policy.
\begin{definition} \label{def:tmetric} The \textbf{time-step metric} $d^\pi_T$ in an MDP with state space $\sset$, action space $\aset$, transition function $P$, and agent policy $\pi$ is a quasimetric where the distance from state $s \in \sset$ to state $s_g \in \sset$ is based on the expected number of transitions under policy $\pi$. \begin{align*}
d^\pi_T(s, s_g) \defd \mathbb{E}\; \left[T(s_g |\pi, s)\right] \end{align*}
where $T(s_g| \pi, s)$ is the random variable for the first time-step that state $s_g$ is encountered by the agent after starting in state $s$ and following policy $\pi$. \end{definition}
This quasimetric has the property that the per step cost is uniformly $1$ for all transitions except ones from the goal to the absorbing state (and the absorbing state to itself), which are $0$. Thus, it can be written recursively as: \begin{align} \label{eqn:rec_time_d}
d^\pi_T(s, s_g) = \begin{cases} 0 & \text{if } s = s_g \\
1 + \mathop{\mathbb{E}}_{a \sim \pi(\cdot|s, s_g)}\mathop{\mathbb{E}}_{s' \sim P(\cdot|s, a, s_g)}\left[ d^\pi_T(s', s_g) \right]& \text{otherwise} \end{cases} \end{align}
Recall that in order to estimate the Wasserstein distance using the dual (\autoref{eqn:KRdual}) in a metric space where the ground metric $d$ is this time-step metric, the potential function $f: \sset \longmapsto \mathbb{R}$ needs to be $1$-Lipschitz with respect to $d^\pi_T$. In \autoref{app:quasi_lip} we prove that $L$-Lipschitz continuity can be ensured by enforcing that the difference in values of $f$ on expected transitions from every state are bounded by $L$, implying \begin{align} \label{eqn:lip_mdp}
\text{Lip}(f) \leq \sup_{s \in \sset}\left\{\mathop{\mathbb{E}}_{a \sim \pi(\cdot|s, s_g)} \mathop{\mathbb{E}}_{s' \sim P(\cdot|s, a, s_g)}\left[\left\lvert f(s) - f(s')\right\rvert\right] \right\} . \end{align} Note that finding a proper way to enforce the Lipschitz constraint in adversarial methods remains an open problem \citep{liu2020lipschitz}. However, for the time-step metric considered here, \eqref{eqn:lip_mdp} is one elegant way of doing so. By ensuring that the Kantorovich potentials do not drift too far from each other on expected transitions under agent policy $\pi$ in the MDP, the conditions necessary for the potential function to estimate the Wasserstein distance can be maintained \citep{villani2008optimal,arjovsky_wasserstein_2017}. Finally, the minimum distance $d^\blacklozenge_T$ from state $s$ to a given goal state $s_g$ (corresponding to policy $\pi^{\blacklozenge}$) is defined by the Bellman optimality condition (\autoref{eqn:opt_dt} in \autoref{app:proofs}).
Consider how the time-step distance to the goal and the value function for goal-conditioned RL relate to each other. When the reward is $0$ everywhere except for transitions to the goal state, the value becomes $V^\pi(s|s_g) = \mathbb{E} \left[ \gamma^{T(s_g|\pi, s)} \right]$.
$d^\pi_T(s_0, s_g)$ and $V(s_0|s_g)$ are related as follows.
\begin{restatable}{proposition}{lowerbound} \label{prop:lower_b}
A lower bound on the value of any state under a policy $\pi$ can be expressed in terms of the time-step distance from that state to the goal: $V(s_0|s_g) \geq \gamma^{d^\pi_T(s_0, s_g)}$. \end{restatable}
The proofs for all theoretical results are in \autoref{app:proofs}. The Jensen gap $\Delta^\pi_{\text{Jensen}}(s):=V^\pi(s|s_g) -\gamma^{d^\pi_T(s, s_g)} $ describes the sharpness of the lower bound in the proposition above and it is zero if and only if $\mathrm{Var}(T(s_g|\pi, s))=0$ \citep{liao2018sharpening}. From this line of reasoning, we deduce the following proposition: \begin{restatable}{proposition}{policyCorrespondence} \label{prop:corr} If the transition dynamics are deterministic, the policy that maximizes expected return is the policy that minimizes the time-step metric ($\pi^* = \pi^\blacklozenge$). \end{restatable}
\section{Wasserstein-1 Distance for Goal-Conditioned Reinforcement Learning} \label{sec:wass_theory}
In this section we consider the problem of goal-conditioned reinforcement learning. In Section \ref{sec:wass_opt} we analyze the Wasserstein distance computed under the time-step metric $d^\pi_T$. Section \ref{sec:alg} proposes an algorithm, Adversarial Intrinsic Motivation (\textsc{aim}), to learn the potential function for the Kantorovich-Rubinstein duality used to estimate the Wasserstein distance, and giving an intrinsic reward function used to update the agent policy in tandem.
\subsection{Wasserstein-1 Distance under the Time-Step Metric} \label{sec:wass_opt}
From Sections \ref{sec:opt} and \ref{sec:cmetric} the Wasserstein distance under the time-step metric $d^\pi_T$ of an agent policy $\pi$ with visitation measure $\rho_\pi$ to a particular goal $s_g$ and its distribution $\rho_g$ can be expressed as: \begin{align} \label{eqn:dt_W1}
W_1^\pi(\rho_\pi, \rho_g) = \textstyle{\sum_{s \in \sset}} \rho_\pi(s|s_g) d^\pi_T(s, s_g) \end{align} where $W_1^\pi$ refers to the Wasserstein distance with the ground metric $d^{\pi}_T$.
The following proposition shows that the Wasserstein distance decreases as $d^{\pi}_T(s, s_g)$ decreases, while also revealing a surprising connection with the Jensen gap.
\begin{restatable}{proposition}{analytical} \label{prop:analytical} For a given policy $\pi$, the Wasserstein distance of the state visitation measure of that policy from the goal state distribution $\rho_g$ under the ground metric $d^{\pi}_T$ can be written as \begin{align}
W_{1}^{\pi}(\rho_\pi, \rho_g) = \mathop{\mathbb{E}}_{s_0 \sim \rho_0} \left[ h(d^{\pi}_T(s_0, s_g)) + \frac{\gamma}{1 - \gamma}(\Delta^\pi_{\text{Jensen}}(s_0) - 1)\right] \end{align} where $h$ is an increasing function of $d_T^\pi$. \textbf{}\end{restatable}
The first component in the above analytical expression shows that the Wasserstein distance depends on the expected number of steps, decreasing if the expected distance decreases. The second component shows the risk-averse nature of the Wasserstein distance. Concretely, the bounds for the Jensen inequality given by \citet{liao2018sharpening} imply that there are non-negative constants $C_1=C_1(d_T^\pi, \gamma)$ and $C_2=C_2(d_T^\pi, \gamma)$ depending only on the expected distance and discount factor such that $$
C_1\mathrm{Var}(T(s_g|\pi, s)) \leq \Delta^\pi_{\text{Jensen}}(s) \leq C_2 \mathrm{Var}(T(s_g|\pi, s)). $$ From the above, we can deduce that a policy with lower variance will have lower Wasserstein distance when compared to a policy with same expected distance from the start but higher variance. The relation between the optimal policy in goal-conditioned RL and the Wasserstein distance can be made concrete if we consider deterministic dynamics. \begin{restatable}{theorem}{aimsearch} \label{thm:aim_search} If the transition dynamics are deterministic, the policy that minimizes the Wasserstein distance over the time-step metrics in a goal-conditioned MDP (see \eqref{eqn:dt_W1}) is the optimal policy. \end{restatable}
\subsection{Adversarial Intrinsic Motivation to minimize Wasserstein-1 Distance} \label{sec:alg}
The above section makes it clear that minimizing the Wasserstein distance to the goal will lead to a policy that reaches the goal in as few steps as possible in expectation. If the dynamics of the MDP are deterministic, this policy will also be optimal. Note that the dual form (\autoref{eqn:KRdual}) can be used to estimate the distance, \emph{even if the ground metric $d^\pi_T$ is not known}. The smoothness requirement on the potential function $f$ can be ensured with the constraint in \autoref{eqn:lip_mdp} on all states and subsequent transitions expected under the agent policy.
Now consider the full problem. The reinforcement learning algorithm aims to learn a goal-conditioned policy with parameters $\theta \in \Theta$ whose state visitation distribution $\rho_\theta$ minimizes the Wasserstein distance to a goal-conditioned target distribution $\rho_g$ for a given goal $s_g \sim \sigma$. \textsc{aim}\ leverages the presence of the set of goals that the agent should be capable of reaching with a goal-conditioned potential function $f_{\phi}: \sset \times \gset \longmapsto \mathbb{R}$ with parameters $\phi \in \Phi$. These objectives of the potential function and the agent can be expressed together using the following adversarial objective: \begin{align} \label{eqn:pol_search}
\min_{\theta \in \Theta} \max_{\phi \in \Phi} \mathop{\mathbb{E}}_{s_g \sim \sigma}\left[f_{\phi}(s_g,s_g) - \mathop{\mathbb{E}}_{s \sim \rho_{\theta}} [f_{\phi}(s, s_g)]\right] \end{align} where the potential function $f_\phi$ is $1$-Lipschitz over the state space. Combining the objectives in Equations \ref{eqn:pol_search} and \ref{eqn:lip_mdp}, the loss for the potential function $f_\phi$ then becomes: \begin{align}
L_f \defd &\mathop{\mathbb{E}}_{s_g \sim \sigma} \left[ - f_{\phi}(s_g, s_g) + \mathop{\mathbb{E}}_{s \sim \rho_{\theta}} [f_\phi(s, s_g) ]\right] + \nonumber \\
\lambda & \mathop{\mathbb{E}}_{(s, a, s', s_g) \sim \mathcal{D}} \left[(\max(|f_\phi(s, s_g) - f_\phi(s', s_g)| - 1, 0))^2 \right] \label{eqn:f_loss} \end{align}
Where the distribution $\mathcal{D}$ should ideally contain all states in $\sset$, expected goals in $\gset$, and the transitions according to the agent policy $\pi_\theta$ and transition function $P$. Such a distribution is difficult to obtain directly. \textsc{aim}\ approximates it with a small replay buffer of transitions from recent episodes experienced by the agent, and relabels these episodes with achieved goals (similar to \textsc{her} \citep{andrychowicz_hindsight_2018}). Such an approximation does not respect the discounted measure of states later on in an episode, but is consistent with how other approaches in deep reinforcement learning tend to approximate the state visitation distribution, especially for policy gradient approaches \citep{nota2020PolicyGradient}. While it does not include all states and all goals, we see empirically that the above approximation works well.
Now we turn to the reward function that should be presented to the agent such that maximizing the return will minimize the Wasserstein distance. The Wasserstein discriminator is a potential function \citep{ng1999policy} (its value depends on the state). It can thus be used to create a shaped reward $\hat{r}(s, a, s', s_g) = r(s, a, s'|s_g) + \gamma f_\phi(s', s_g) - f_\phi(s, s_g)$ without risk of changing the optimal policy. Alternatively, we can explicitly minimize samples of the Wasserstein distance: $\hat{r}(s, a, s', s_g) = f_\phi(s', s_g) - f_\phi(s_g, s_g)$. Finally, instead of the second term $f_\phi(s_g, s_g)$, we can just use a constant bias term. In practice, all these choices work well, and the experiments use the latter (with $b = \max_{s \in \sset} f_\phi(s,s_g)$) to reduce variance in $\hat{r}$. \begin{align} \label{eqn:reward}
\hat{r}(s, a, s', s_g) = f_\phi(s', s_g) - b \end{align}
The basic procedure to learn and use adversarial intrinsic motivation (\textsc{aim}) is laid out in Algorithm \ref{alg}, and also includes how to use this algorithm in conjunction with \textsc{her}. If not using \textsc{her}, Line \ref{alg:her} where hindsight goals are added to the replay buffer can be skipped.
\section{Experiments} \label{sec:exp}
Our experiments evaluate the extent to which the reward learned through \textsc{aim} is useful as a proxy for the environment reward signal, or in tandem with the environment reward signal. In particular, we ask the following questions: \begin{itemize}[noitemsep,topsep=0pt,leftmargin=5.5mm]
\item Does \textsc{aim}\ speed up learning of a policy to get to a single goal compared to learning with a sparse reward?
\item Does the learned reward function qualitatively guide the agent to the goal?
\item Does \textsc{aim}\ work well with stochastic transition dynamics or sharp changes in the state features?
\item Does \textsc{aim}\ generalize to a large set of goals and continuous state and action spaces? \end{itemize} Our experiments suggest that the answer to all 4 questions is ``yes'', with the first three questions tested in the grid world presented in Figure \ref{fig:example} where the goal is within a room, and the agent has to go around the room from its start state to reach the goal. Goal-conditioned tasks in the established Fetch robot domain show that \textsc{aim} also accelerates learning across multiple goals in continuous state and action spaces.
This section compares an agent learning with a reward learned through \textsc{aim}\ with other intrinsic motivation signals that induce general exploration or shaped rewards that try to guide the agent to the goal. The experiments show that \textsc{aim}\ guides the agent's exploration more efficiently and effectively than a general exploration bonus, and adapts to the dynamics of the environment better than other techniques we compare to. As an overview, the baselines we compare to are: \begin{itemize}[noitemsep,topsep=0pt,leftmargin=5.5mm]
\item \textbf{RND}: with random network distillation (\textsc{rnd}) \citep{burda2018rnd} used to provide a general exploration bonus.
\item \textbf{MC}: with the distance between states learned through regression of Monte Carlo rollouts of the agent policy, similar to \citet{hartikainen2019ddl}.
\item \textbf{SMiRL}: \textsc{sm}i\textsc{rl} \citep{berseth2019smirl} is used to provide a bonus intrinsic motivation reward that minimizes the overall surprise in an episode.
\item \textbf{DiscoRL} The DiscoRL \citep{Nasiriany:EECS-2020-151} approach presents a reward to maximize the likelihood of a target distribution (normal distribution at the goal). In practice this approach is equivalent to a negative L2 distance to the goal, which we compare to in the grid world domain.
\item \textbf{GAIL}: additional \textsc{gail} \citep{ho2016generative} rewards using trajectories relabeled with achieved goals considered as having come from the expert in hindsight. This baseline is compared to in the Fetch robot domain, since that is the domain where we utilize hindsight relabeling. \end{itemize}
\paragraph{Grid World} In this task, the goal is inside a room and the agent's starting position is such that it needs to navigate around the room to find the doorway and be able to reach the goal. The agent can move in the $4$ cardinal directions unless blocked by a wall or the edge of the grid. The agent policy is learned using soft Q-learning \citep{haarnoja2017reinforcement} with no hindsight goals used for this experiment.
The agent's state visitation distribution after just $100$ Q-function updates when using \textsc{aim}-learned rewards is shown in Figure \ref{fig:grid_policy} and the learned rewards for each state are plotted in Figure \ref{fig:reward}. The state visitation when learning with the true task reward shows that the agent is unable to learn a policy to the goal (Figure \ref{fig:task_policy}). These figures show that \textsc{aim}\ enables the agent to reach the goal and learn the required policy quickly, while learning with the sparse task reward fails to do so.
\begin{figure}
\caption{Agent state visitation learning with \textsc{aim} rewards (100 iterations)}
\label{fig:grid_policy}
\caption{Learned Reward (100 training iterations)}
\label{fig:reward}
\caption{Agent state visitation learning with task reward (500 iterations)}
\label{fig:task_policy}
\caption{Grid world experiments. Agent's undiscounted state visitation (\ref{fig:grid_policy}, \ref{fig:task_policy}): Blue circle indicates agent's start state. Red circle is the goal. Blue bubbles indicate relative time agent's policy causes it to spend in respective states. Learned reward function (\ref{fig:reward}): \textsc{aim}\ reward at each state of the grid world. Bold black (or white) lines indicate walls the agent cannot transition through.}
\label{fig:gridworld}
\end{figure}
In Appendix \ref{app:grid} we also compare to the baselines described above and show that \textsc{aim}\ learns a reward that is more efficient at directing the agent's exploration and more flexible to variations of the environment dynamics, such as stochastic dynamics or transitions that cause a sharp change in the state features. None of the baselines compared to were able to direct the agent to the goal in this grid world even after given up to $5\times$ more interactions with the environment to train. \textsc{aim}'s use of the time-step metric also enabled it to adapt to variations of the environment dynamics better than the gradient penalty based regularization used in Wasserstein GANs \citep{gulrajani_improved_2017} and adversarial imitation learning \citep{ghasemipour2020divergence} approaches.
\paragraph{Fetch Robot} The generalization capability of \textsc{aim}\ across multiple goals in goal-conditioned RL tasks with continuous states and actions is tested in the MuJoCo simulator \citep{todorov2012mujoco}, on the Fetch robot tasks from OpenAI gym \citep{brockman2016openai} which have been used to evaluate learning of goal-conditioned policies previously \citep{andrychowicz_hindsight_2018, zhang_automatic_2020}. Descriptions of these tasks and their goal space is in Appendix \ref{app:fetch}. We soften the Dirac target distribution for continuous states to instead be a Gaussian with variance of $0.01$ of the range of each feature.
The goals in this setting are not the full state, but rather the dimensions of factored states relevant to the given goal. The task wrapper additionally returns the features of the agent's state in this reduced goal space, and so \textsc{aim}\ can use it to learn our reward function, rather than the full state space. It is unclear how this smaller goal space might affect \textsc{aim}. While the smaller goal space might make learning easier for potential function $f_\phi$, the partially observable nature of the goals might lead to a less informative reward.
We combine \textsc{aim}\ with \textsc{her} (refer \autoref{sec:goal}) and refer to it as [\textsc{aim}\ + \textsc{her}]. We compare this agent to the baselines we referred to above, as well as the sparse environment reward (\textsc{r} + \textsc{her}) and the dense reward derived from the negative Euclidean ($L2$) distance to the goal ($-L2$ + \textsc{her}). The $L2$ distance is proportional to the number of steps it should take the agent to reach the goal in this environment, and so the reward based on it can act as an oracle reward that we can use to test how efficiently \textsc{aim}\ can learn a reward function that helps the agent learn its policy. We used the \textsc{her} implementation using Twin Delayed DDPG (\textsc{td3}) \citep{fujimoto2018TD3} as the underlying RL algorithm from the stable baselines repository \citep{stable-baselines}. We did an extensive sweep of the hyperparameters for the baseline \textsc{her} + \textsc{r} (laid out in Appendix \ref{app:fetch}), with a coarser search on relevant hyperparameters for \textsc{aim}.
\begin{figure}
\caption{Reach}
\label{fig:reach}
\caption{Pick and Place}
\label{fig:pick}
\caption{Push}
\label{fig:push}
\caption{Slide}
\label{fig:slide}
\caption{Evaluating \textsc{aim} with \textsc{her} on some goal-conditioned RL tasks in the Fetch domain. \textsc{aim}\ learns the reward function in tandem with the policy updates. The ``$-L2$'' reward is the true negative distance to goal, acting as a oracle reward in this domain. The other baselines are detailed above.}
\label{fig:fetch}
\end{figure}
Figure \ref{fig:fetch} shows that using the \textsc{aim}-learned reward speeds up learning in three of the four Fetch domains, even without the environment reward. This improvement is very close to what we would see if we used the dense reward (based on the actual distance function in this domain). An additional comparison with an agent learning with both the \textsc{aim}-learned reward as well as the task reward (\textsc{aim}\ + \textsc{r} + \textsc{her}) can be seen in \autoref{fig:app_fetch} in the Appendix, showing that using both signals accelerates learning even more. These results also highlight that \textsc{aim}\ continues to work in continuous state and action spaces, even though our analysis focuses on discrete states and actions. Results are averaged across the 6 different seeds, with shaded regions showing standard error across runs. Statistical analysis using a mixed effects ANOVA and a Tukey test at a significance level of $95\%$ (more detail in \autoref{app:fetch_test}) show that in three of the four environments \textsc{aim}\ and \textsc{aim} + \textsc{r} have similar odds of reaching the goal as the dense shaped reward, and in all four environments \textsc{aim}\ and \textsc{aim} + \textsc{r} have higher odds of reaching the goal compared to the sparse reward.
The other baselines compare well to \textsc{aim}\ in the Fetch Reach domain (Figure \ref{fig:reach}), but do not do as well on the other problems. In fact, none of the other baselines outperform the vanilla baseline [\textsc{r} + \textsc{her}] in all the domains. The \textsc{rnd} rewards help the agent to start learning faster in the Push domain (Figure \ref{fig:push}), but lead to worse performance in Pick and Place (Figure \ref{fig:pick}). On the other hand, learning the distance function through \textsc{mc} regression helps in the Pick and Place domain, but slows down learning when dealing with Push. Most notably, both these approaches cause learning to fail in the Slide domain (Figure \ref{fig:slide}), where the credit assignment problem is especially difficult. \textsc{gail} works as well as \textsc{aim}\ and the vanilla baseline in Slide, but underperforms in the other domains. We hypothesize that the additional rewards in these baselines conflict with the task reward. Additionally, none of the three new baselines work well if we do not provide the task reward in addition to the specific bonus for that algorithm.
We did not find any configuration in the Fetch Reach domain where [\textsc{sm}i\textsc{rl} + \textsc{r} + \textsc{her}] was able to accomplish the task in the given training budget. Since SMiRL did not work on the grid world or Fetch Reach, we did not try it out on any of the other domains.
\textsc{fairl} \citep{ghasemipour2020divergence} (which has been shown to learn policies that cover hand-specified state distributions) was also applied on these $4$ domains but it failed to learn at all. Interestingly, scaling the reward such that it is always negative led to similar performance to (but not better than) \textsc{aim}. We hypothesize that \textsc{fairl}, as defined and presented, fails in these domains because the environments are episodic, and the episode ends earlier if the goal is reached. Since the \textsc{fairl} reward is positive closer to the target distribution, the agent can get close to the target, but refrain from reaching it (and ending the episode) to collect additional positive reward.
The domain where \textsc{aim}\ does not seem to have a large advantage (Slide) is one where the agent strikes an object initially and that object has to come to rest near the goal. In fact, \textsc{aim}-learned rewards, the vanilla environment reward R, and the oracle $-L2$ rewards all lead to similar learning behavior, indicating that this particular task does not benefit much from shaped rewards. The reason for this invariance might be that credit assignment has to propagate back to the single point when the agent strikes the object regardless of how dense the subsequent reward is.
\section{Discussion and Future Work} \label{sec:disc}
Approaches for estimating the Wasserstein distance to a target distribution by considering the dual of the Kantorovich relaxation have been previously proposed \citep{arjovsky_wasserstein_2017,gulrajani_improved_2017,xiao_wasserstein_2019}, but assume that the ground metric is the $L2$ distance. We improve upon them by choosing a metric space more suited to the MDP and notions of optimality in the MDP. This choice allows us to leverage the structure introduced by the dynamics of the MDP to regularize the Kantorovich potential using a novel objective.
Previous work \cite{Bellemare2017CramerD} has pointed out that the gradients from sample estimates of the Wasserstein distance might be biased. This issue is mitigated in our implementation through multiple updates of the discriminator, which they found to be empirically useful in reducing the bias. Additionally, recent work has pointed out that the discriminator in WGAN might be bad at estimating the Wasserstein distance \citep{Stanczuk2021WassersteinGW}. While our experiments indicate that the potential function in \textsc{aim}\ is learned appropriately, future work could look more deeply to verify possible inefficiencies in this estimation.
The process of learning the Wasserstein distance through samples of the environment while simultaneously estimating the cost of the full path is reminiscent of the $A^*$ algorithm \citep{hart1968formal}, where the optimistic heuristic encourages the agent to explore in a directed manner, and adjusts its estimates based on these explorations.
The discriminator objective (Equation \ref{eqn:f_loss}) also bears some resemblance to a linear program formulation of the RL problem \citep{puterman1990markov}. The difference is that this formulation minimizes the value function on states visited by the agent, while \textsc{aim}\ additionally maximizes the potential at the goal state. This crucial difference has two main consequences. First, the potential function during learning is not equivalent to the value of the agent’s policy (verified by using this potential as a critic). Second, increasing the potential of the goal state in \textsc{aim}\ directs the agent exploration in a particular direction (namely, the direction of sharpest increase in potential).
In the goal-conditioned RL setting, \textsc{aim}\ seems to be an effective intrinsic reward that balances exploration and exploitation for the task at hand. The next step is to consider whether the Wasserstein distance can be estimated similarly for more general tasks, and whether minimizing this distance in those tasks leads to the optimal policy. A different potential avenue for future work is the problem of more general exploration \citep{hazan2019provably,lee2019efficient} by specifying a uniform distribution as the target, or using this directed exploration as an intermediate step for efficient exploration \citep{Jinnai2020Exploration}.
Finally, reward design is an important aspect of practical reinforcement learning. Not only do properly shaped reward speed up learning \citep{ng1999policy}, but reward design can also subtly influence the kinds of behaviors deemed acceptable for the RL agent \citep{knox2021reward} and could be a potential safety issue keeping reinforcement learning from being deployed on real world problems. Learning-based approaches that can assist in specifying reward functions safely given alternative approaches for communicating the task could be of value in such a process of reward design, and an avenue for future research. \section*{Acknowledgements and Funding Information}
We thank Caroline Wang, Garrett Warnell, and Elad Liebman for discussion and feedback on this work. We also thank the reviewers for their thoughtful comments and suggestions that have helped to improve this paper.
This work has taken place in part in the Learning Agents Research Group (LARG) at the Artificial Intelligence Laboratory, and in part in the Personal Autonomous Robotics Lab (PeARL) at The University of Texas at Austin. LARG research is supported in part by the National Science Foundation (CPS-1739964, IIS-1724157, FAIN-2019844), the Office of Naval Research (N00014-18-2243), Army Research Office (W911NF-19-2-0333), DARPA, Lockheed Martin, General Motors, Bosch, and Good Systems, a research grand challenge at the University of Texas at Austin. PeARL research is supported in part by the NSF (IIS-1724157, IIS-1638107, IIS-1749204, IIS-1925082), ONR (N00014-18-2243), AFOSR (FA9550-20-1-0077), and ARO (78372-CS). This research was also sponsored by the Army Research Office under Cooperative Agreement Number W911NF-19-2-0333. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. Peter Stone serves as the Executive Director of Sony AI America and receives financial compensation for this work. The terms of this arrangement have been reviewed and approved by the University of Texas at Austin in accordance with its policy on objectivity in research.
\appendix
\section{Metrics and Quasimetrics} \label{sec:metrics}
A metric space ($\mathcal{M}, d$) is composed of a set $\mathcal{M}$ and a metric $d: \mathcal{M} \times \mathcal{M} \longmapsto \mathbb{R}^+ \cup \{\infty\}$ that compares two points in that set. Here $\mathbb{R}^+$ is the set of non-negative real numbers.
\begin{definition} A metric $d: \mathcal{M} \times \mathcal{M} \longmapsto \mathbb{R}^+ \cup \{\infty\}$ compares two points in set $\mathcal{M}$ and satisfies the following axioms $\forall m_1, m_2, m_3 \in \mathcal{M}$: \begin{itemize}
\item $d(m_1, m_2) = 0 \iff m_1 = m_2$ (identity of indiscernibles)
\item $d(m_1, m_2) = d(m_2, m_1)$ (symmetry)
\item $d(m_1, m_2) \leq d(m_1, m_3) + d(m_3, m_2)$ (triangle inequality) \end{itemize} \end{definition}
A variation on metrics that is important to this paper is \emph{quasimetrics}.
\begin{definition} A quasimetric \citep{Smyth1987QuasiUR} is a function that satisfies all the properties of a metric, with the exception of symmetry $d(m_1, m_2) \neq d(m_2, m_1)$. \end{definition}
As an example, consider an MDP where the actions and transition dynamics allow an agent to navigate from any state to any other state. Let $T(s_2| \pi, s_1)$ be the random variable for the first time-step that state $s_2$ is encountered by the agent after starting in state $s_1$ and following policy $\pi$. The time-step metric $d^\pi_T$ for this MDP can then be defined as
\begin{align*}
d^\pi_T(s_1, s_2) \defd \mathbb{E}\; \left[T(s_2 |\pi, s_1)\right] \end{align*}
$d^\pi_T$ is a quasimetric, since the action space and transition function need not be symmetric, meaning the expected minimum time needed to go from $s_1$ to $s_2$ need not be the same as the expected minimum time needed to from $s_2$ to $s_1$. The diameter of an MDP \citep{jaksch_near-optimal_2010,kearns2002near} is generally calculated by taking the maximum time-step distance between over all pairs of states in the MDP either under a random policy or a policy that travels from any state to any other state in as few steps as possible.
\section{Optimal Transport and Wasserstein-1 Distance} \label{app:opt}
The theory of optimal transport \citep{villani2008optimal,bousquet_optimal_2017} considers the question of how much work must be done to transport one distribution to another optimally. More concretely, suppose we have a metric space ($\mathcal{M}$, $d$) where $\mathcal{M}$ is a set and $d$ is a metric on $\mathcal{M}$. See the definitions of metrics and quasimetrics in Appendix \ref{sec:metrics}. For two distributions $\mu$ and $\nu$ with finite moments on the set $\mathcal{M}$, the Wasserstein-$p$ distance is denoted by:
\begin{align} \label{appeqn:w-p}
W_p(\mu, \nu) \defd \inf_{\zeta \in Z(\mu, \nu)} \mathbb{E}_{(X,Y)\sim \zeta} \left[d(X, Y)^p\right]^{1/p} \end{align}
where $Z$ is the space of all possible couplings between $\mu$ and $\nu$. Put another way, $Z$ is the space of all possible distributions $\zeta \in \Delta(\mathcal{M}\times\mathcal{M})$ whose marginals are $\mu$ and $\nu$ respectively. Finding this optimal coupling tells us what is the least amount of work, as measured by $d$, that needs to be done to convert $\mu$ to $\nu$. This Wasserstein-$p$ distance can then be used as a cost function (negative reward) by an RL agent to match a given target distribution \citep{xiao_wasserstein_2019,dadashi_primal_2020}.
Finding the ideal coupling (meaning finding the optimal transport plan from one distribution to the other) which gives us an accurate distance is generally considered intractable. However, if what we need is an accurate estimate of the Wasserstein distance and not the optimal transport plan (as is the case when we mean to use this distance as part of our intrinsic reward) we can turn our attention to the dual form of this distance. The Kantorovich-Rubinstein duality \citep{villani2008optimal} for the Wasserstein-1 distance on a ground metric $d$ is of particular interest and gives us the following equality:
\begin{align} \label{appeqn:KRdual}
W_1(\mu, \nu) = \sup_{\text{Lip}(f) \leq 1} \mathbb{E}_{y \sim \nu}\left[f(y)\right] - \mathbb{E}_{x \sim \mu} \left[f(x)\right] \end{align}
where the supremum is over all $1$-Lipschitz functions $f: \mathcal{M} \longmapsto \mathbb{R}$ in the metric space, and the Lipschitz constant of a function $f$ is defined as: \begin{align} \label{eqn:lip}
\text{Lip}(f) &\defd \sup \left\{ \frac{|f(y) - f(x)|}{d(x, y)} \forall (x, y) \in \mathcal{M}^2, x \neq y \right\} \end{align}
That is, the Lipschitz condition of this function $f$ (called the Kantorovich potential function) is measured according to the metric $d$. Recently, \citet{jevtic2018combinatorial} has shown that this dual formulation where the constraint on the potential function is a smoothness constraint extends to quasimetric spaces as well. If defined over a quasimetric space, the Wasserstein distance also has properties of a quasimetric (specifically, the distances are not necessarily symmetric).
If the given metric space is a Euclidean space ($d(x, y) = \|y - x\|_2$), the Lipschitz bound in Equation \ref{eqn:KRdual} can be computed locally as a uniform bound on the gradient of $f$.
\begin{align} \label{eqn:KRdual_l2}
W_1(\mu, \nu) = \sup_{\|\nabla f\| \leq 1} \mathbb{E}_{y \sim \nu}\left[f(y)\right] - \mathbb{E}_{x \sim \mu} \left[f(x)\right] \end{align}
meaning that $f$ is the solution to an optimization objective with the restriction that $\|\nabla f(x)\| \leq 1$ for all $x \in \mathcal{M}$. This strong bound on the dual in Euclidean space is the one that has been used most in recent implementations of the Wasserstein generative adversarial network \citep{arjovsky_wasserstein_2017,gulrajani_improved_2017} to regularize the learning of the discriminator function. Such regularization has been found to be effective for stability in other adversarial learning approaches such as adversarial imitation learning \citep{ghasemipour2020divergence}.
Practically, the Kantorovich potential function $f$ can be approximated using samples from the two distributions $\mu$ and $\nu$, regularization of the potential function to ensure smoothness, and an expressive function approximator such as a neural network. A more in depth treatment of the Kantorovich relaxation and the Kantorovich-Rubinstein duality, as well as their application in metric and Euclidean spaces using the Wasserstein-1 distance we lay out above, is provided by \citet{peyre_computational_2020}.
Now consider the problem of goal-conditioned reinforcement learning. Here the target distribution $\nu$ is the goal-conditioned target distribution $\rho_g$ which is a Dirac at the given goal state. Similarly, the distribution to be transported $\mu$ is the agent's goal-conditioned state distribution $\rho_\pi$.
The Wasserstein-1 distance of an agent executing policy $\pi$ to the goal $s_g$ can be expressed in a fairly straightforward manner as: \begin{align} \label{eqn:goal_W1}
W_1(\rho_\pi, \rho_g) = \sum_{s \in \sset} \rho_\pi(s | s_g) d(s, s_g) \end{align}
The above is a simplification of Equation \ref{eqn:w-p}, where $p=1$ and the joint distribution is easy to specify since the target distribution $\rho_g$ is a Dirac distribution.
\section{Lipschitz constant of Potential function} \label{app:quasi_lip}
For a given goal $s_g$ and all states $s_0 \in \sset$, recall that function $f$ is $L$-Lipschitz if it follows the Lipschitz condition as follows.
\begin{align}
\lvert f(s_g) - f(s_0)\rvert \leq L d^\pi_T(s_0, s_g)\; \forall s_0 \in \sset \end{align}
\begin{restatable}{proposition}{lipschitz_cond} \label{prop:lipschitz_cond} If transitions from the agent policy $\pi$ are guaranteed to arrive at the goal in finite time and $f$ is $L$-bounded in expected transitions, i.e., \begin{align*} \sup_{s\in S} \mathop{\mathbb{E}}_{s'\sim \pi, P} \left[\lvert f(s') - f(s)\rvert \right] \leq L, \end{align*} then $f$ is $L$-Lipschitz. \end{restatable} \begin{proof}
Since $f(s_g) - f(s_0)$ is a scalar quantity, we may write $f(s_g) - f(s_0) = \mathbb{E}_{\pi, P}[f(s_g) - f(s_0)]$. Using this fact and that $P(T(s_0) < \infty)=1$ where $T(s_0)=T^\pi(s_g | \pi, s_0)$ for notation simplicity, the LHS of the expression above becomes a telescopic sum \begin{align*} \lvert f(s_g) - f(s_0)\rvert &= \mathop{\mathbb{E}}_{\pi, P} \left[f(s_g) - f(s_0)\right] \\
&= \mathop{\mathbb{E}}_{\pi, P} \left[\left\lvert \sum_{t=0}^{T(s_0) - 1} (f(s_{t+1}) - f(s_t))\right\rvert \right].
\\
&\leq \mathop{\mathbb{E}}_{\pi, P} \left[ \sum_{t=0}^{T(s_0) - 1} \lvert f(s_{t+1}) - f(s_t)\rvert \right]. \\ \end{align*} Now let us assume that for all transitions $(s, a, s')$, $\mathbb{E}[\lvert f(s') - f(s)\rvert] \leq L$. Then \begin{align*}
\mathop{\mathbb{E}}_{\pi, P} \left[ \sum_{t=0}^{T(s_0) - 1} \vert f(s_{t+1}) - f(s_t)\rvert\right]&= \mathop{\mathbb{E}}_{T(s_0)}\left[\mathop{\mathbb{E}}_{\pi, P} \left[ \sum_{t=0}^{T(s_0) - 1} \lvert f(s_{t+1}) - f(s_t)\rvert \Big\vert T(s_0) \right] \right] \\
&\leq \mathop{E}_{T(s_0)} \left[ \sum_{t=0}^{T(s_0) - 1} L \right] \\
&= L\mathop{\mathbb{E}}_{T(s_0)}\left[ T(s_0) \right] \\
&= L d_T^\pi(s_0, s_g), \end{align*}
showing that $|f(s_g) - f(s_0)| \leq L d_T^\pi(s_0, s_g)$ as desired. \end{proof}
\section{Proofs of Claims} \label{app:proofs}
The Bellman optimality condition gives us the following optimal distance to goal: \begin{align} \label{eqn:opt_dt}
d^{\blacklozenge}_T(s, s_g) = \begin{cases} 0 & \text{if } s = s_g \\
1 + \min_{a \in \aset} \sum_{s' \in \sset} P(s'|s, a, s_g) d^{\blacklozenge}_T(s', s_g) & \text{otherwise}\end{cases} \end{align}
\lowerbound* \begin{proof} \begin{align*}
V^\pi(s|s_g) &= \mathbb{E} \left[ \gamma^{T(s_g|\pi, s)} \right] \geq \gamma^{d^\pi_T(s, s_g)} \quad \forall \ s \in \sset \end{align*} where the inequality follows as a consequence of Jensen's inequality and the convex nature of the value function. \end{proof}
\policyCorrespondence*
\begin{proof}
Consider the value of a state $s$ given goal $s_g$. If the transitions are deterministic and the agent policy $\pi$ is deterministic (as is the case for the optimal policy), then the time to reach the goal satisfies $\mathrm{Var}(T(s_g|\pi, s))=0$, implying that $\Delta_{\text{Jensen}}$ vanishes and therefore \begin{align*}
V^\pi(s|s_g) &= \gamma^{d^{\pi}_T(s, s_g)}. \end{align*} Since $\gamma \in [0, 1)$, $V^\pi$ is monotonically decreasing with $d_T^\pi$ \begin{align*}
\argmax_{\pi}V^\pi(s|s_g) = \argmin_\pi d_T^\pi(s, s_g) \; \forall \ s \in \sset \end{align*}
That is, in the deterministic transition dynamics scenario, $\pi^* = \pi^\blacklozenge$. \end{proof}
\analytical* \begin{proof}
The first step of the proof is to obtain an analytical expression for the the expected distance to the goal after $t$ steps as a function of the expected distance at $t=0$. To reduce the notation burden, denote $T(s_0)=T(s_g|\pi, s_0)$ and let $s_t(s_0)$ be the state after $t$ steps conditional on some starting state $s_0$ where actions are taken according to $\pi$. We have excluded $s_g$ and $\pi$ from the notation since they are fixed for the purpose of this proposition. Using the law of total expectation we have that for every initial $s_0$ \begin{align*} \mathbb{E}_{s_t}[d(s_t(s_0), s_g)] &= \mathbb{E}_{T(s_0)}[\mathbb{E}_{s_t}[d(s_t(s_0), s_g) \mid T(s_0)]] = \mathbb{E}_{T(s_0)}[\max(T(s_0) - t, 0)], \end{align*} Now, by expanding the definition of $\rho_\pi(s\mid s_g)$ in \eqref{eqn:dt_W1}, exchanging the order of summation, and using the previous equation we may write \begin{equation*} \begin{aligned} W_1^\pi(\rho_\pi, \rho_g) &= \sum_{s \in \sset} \sum_{t=0}^\infty (1 - \gamma) \gamma^t \mathbb{E}_{s_0}[P(s_t=s \mid \pi, s_g)] d^\pi_T(s, s_g)\\ &=\mathbb{E}_{s_0}\left[ (1 - \gamma) \sum_{t=0}^\infty \gamma^t \mathbb{E}_{s_t}[d(s_t(s_0), s_g) \mid s_0]\right] \\ &= \mathbb{E}_{s_0}\left[\mathbb{E}_{T(s_0)}\left[(1 - \gamma) \sum_{t=0}^\infty \gamma^t \max(T(s_0) - t, 0) \Big\vert s_0\right]\right] \end{aligned} \end{equation*}
Standard but tedious algebraic manipulations given in Lemma \ref{prop:analytical:support1} in the Appendix show that \begin{align*} \sum_{t=0}^\infty (1 - \gamma) \gamma^t \max(T(s_0) - t, 0) = T(s_0) - \frac{\gamma}{1-\gamma}(1 - \gamma^{T(s_0)}). \end{align*} Combining the two identities above we arrive at \begin{equation}\label{eqn:analytical} \begin{aligned} W_1^\pi(\rho_\pi, \rho_g) &= \mathbb{E}_{s_0}\left[\mathbb{E}_{T(s_0)}\left[T(s_0) - \frac{\gamma}{1 - \gamma}(1 - \gamma^{T(s_0)}) \Big\vert s_0\right]\right] \\&= \mathbb{E}_{s_0}\left[d(s_0, s_g) - \frac{\gamma}{1 - \gamma}(1 - \mathbb{E}[\gamma^{T(s_0)} \mid s_0])\right] \\
& =\mathbb{E}_{s_0}\left[d(s_0, s_g) + \frac{\gamma}{1-\gamma} \gamma^{d(s_0,s_g)} - \frac{\gamma}{1 - \gamma}(1 - \mathbb{E}[\gamma^{T(s_0)}\mid s_0] + \gamma^{d(s_0,s_g)} ) \right]\\
& =\mathbb{E}_{s_0}\left[d(s_0, s_g) + \frac{\gamma}{1-\gamma} \gamma^{d(s_0,s_g)} + \frac{\gamma}{1 - \gamma}(\Delta^\pi_{\text{Jensen}}(s_0) - 1)\right]. \end{aligned} \end{equation}
To finalize the proof, we only need to show that the function $h(\mu) = \mu + \frac{\gamma}{1 - \gamma}\gamma^\mu$ is monotonically increasing for every $\gamma\in [0, 1)$. This is a standard calculus exercise that we show in Lemma \ref{prop:analytical:support2} in Appendix \ref{appendix:aux-prop3}. \end{proof}
\aimsearch*
\begin{proof} Proposition \ref{prop:corr} shows that the Jensen gap vanishes for the optimal policy of an MDP with deterministic transitions and that it minimizes the expected distance from start for all initial states. Proposition \ref{prop:analytical}, on the other hand, implies that when the Jensen gap vanishes, the Wasserstein distance is monotonically increasing in the expected distance from the start. Together, the two propositions show that $\pi^*$ minimizes the Wasserstein distance. \end{proof}
\begin{algorithm2e}[t] \caption{\textsc{aim} + \textsc{her}} \label{alg} \SetAlgoLined \KwIn{Agent policy $\pi_{\theta}$, discriminator $f_{\phi}$, environment $env$, \\ number of Epochs $N$, number of time-steps per epoch $K$, \\ policy update period $k$, discriminator update period $m$, episode length $T$, \\ replay buffer (for HER), smaller replay buffer (for discriminator)} Initialize discriminator parameters $\phi$\; Initialize policy parameters $\theta$\; \For{$n = 0, 1, \ldots , N - 1$}{ $t=0$\; goal\_reached = True\; \While{$t < K$}{ \If{goal\_reached or episode\_over}{ Sample goal $s_g \sim \sigma(\mathcal{G})$\; Sample start state $s \sim \rho_0(\sset)$\; goal\_reached = False\; episode\_over = False\; $t_{start} = K$\; }
Sample action $a \sim \pi_\theta(\cdot|s, s_g)$\; $s' = env.step(a)$\; \If{$s' = s_g$}{
goal\_reached = True\; } \tcp{end episode if goal not reached in $T$ steps} \If{$t - t_{start} = T$}{
episode\_over = True\; }
Add $(s, a, s', s_g, goal\_reached)$ to replay buffer and smaller replay buffer\; \If{goal\_reached or episode\_over}{ \label{alg:her}
Add hindsight goals to both buffers\; } \tcp{Update policy parameters $\theta$ every $k$ steps} \If{$t \% k = 0$}{ Sample tuples $(s, a, s', s_g, goal\_reached)$ from replay buffer\; Get intrinsic reward (Equation \ref{eqn:reward})\; Update policy parameters $\theta$ using any off-policy learning algorithm\; } \tcp{Update discriminator parameters $\phi$ every $m$ steps} \If{$t \% m = 0$}{ Sample tuples $(s, a, s', s_g, goal\_reached)$ from smaller replay buffer\; Update discriminator parameters $\phi$ using Equation \ref{eqn:f_loss}\; } $t = t + 1$\; } Evaluate agent policy\; } \end{algorithm2e}
\section{Auxiliary results for Proposition \ref{prop:analytical}}\label{appendix:aux-prop3}
\begin{lemma}\label{prop:analytical:support1} Let $T$ be a positive integer. Then \begin{align*} \sum_{t=0}^\infty (1 - \gamma) \gamma^t \max(T - t, 0) = T - \frac{\gamma}{1-\gamma}(1 - \gamma^{T}). \end{align*} \end{lemma} \begin{proof} Direct computation gives \begin{align*}
(1 - \gamma) \sum_{t=0}^\infty\gamma^t \max(T - t, 0) & = (1 - \gamma) \sum_{t=0}^{T -1} \gamma^t (T - t) \\ & = (1 - \gamma)T \sum_{t=0}^{T -1} \gamma^t - (1 - \gamma)\sum_{t=0}^{T -1} t \gamma^t \end{align*} We will now simplify the two terms of the last expression. For the first one, have \begin{align*} (1 - \gamma)T \sum_{t=0}^{T -1} \gamma^t = (1 - \gamma)T\frac{1 - \gamma^T}{1 - \gamma} = T - T\gamma^T. \end{align*} For the second one, the computations are a bit more involved \begin{align*} (1 - \gamma)\sum_{t=0}^{T -1} t \gamma^t &= (1 - \gamma)\gamma \sum_{t=1}^{T -1} t \gamma^{t - 1} \\ & =(1 - \gamma) \sum_{t=1}^{T -1} \gamma \frac{d}{d\gamma}\gamma^{t} \\ & = \gamma(1 - \gamma) \frac{d}{d\gamma}\sum_{t=0}^{T -1} \gamma^{t}\\ & = \gamma(1 - \gamma) \frac{d}{d\gamma} \frac{1 - \gamma^T}{1 - \gamma} \\ & = \frac{\gamma}{(1 - \gamma)}\left(- T \gamma^{T - 1}(1 - \gamma) + (1 - \gamma^T)\right) = - T \gamma^T + \frac{\gamma}{(1 - \gamma)}(1 - \gamma^T). \end{align*} When combining the two simplified expressions the terms with $T \gamma^T$ will cancel out, yielding the desired expression. \end{proof}
\begin{lemma} \label{prop:analytical:support2} The function $h_\gamma(\mu) = \mu + \frac{\gamma}{1 - \gamma}\gamma^\mu$ is monotonically increasing for every $\gamma\in [0, 1)$. \end{lemma} \begin{proof} We must show that $\frac{d}{d\mu}h_\gamma(\mu) > 0$ for every $\gamma\in[0,1)$ and every $\mu > 0$. Computing the derivative directly we obtain \begin{align*} \frac{d}{d\mu}h_\gamma(\mu) = 1 + \frac{\log(\gamma)\gamma^{\mu + 1}}{1 - \gamma}. \end{align*} Thus, it will suffice to show that the second term above is greater than -1. For this purpose, first note that $\log(\gamma)\gamma^{\mu + 1} > \log(\gamma)$ since $\gamma < 1$. Now, we use the fact that $\log(\gamma) < 1 - \gamma$ for $\gamma < 1$. This can be verified noting that $1 - \gamma$ is the tangent line to the concave curve $\log(\gamma)$ and the curves meet at $\gamma=1$. And therefore $\log(\gamma) / (1 - \gamma) > -1$. Putting these observation together, \begin{align*} \frac{d}{d\mu}h_\gamma(\mu) = 1 + \frac{\log(\gamma)\gamma^{\mu + 1}}{1 - \gamma} > 1 + \frac{\log(\gamma)}{1 - \gamma} > 1 - 1 = 0, \end{align*} concluding the proof. \end{proof}
\section{Grid World Experiments} \label{app:grid}
\begin{figure}
\caption{Grid world with wind affecting transitions in last 6 columns}
\label{fig:windy_grid}
\caption{Learned Reward (50 training iterations)}
\label{fig:windy_reward}
\caption{Agent state distribution learning with AIM reward (50 training iterations)}
\label{fig:windy_policy}
\caption{Windy grid world (Figure \ref{fig:windy_grid}) experiments. The columns with arrows at the top and bottom have stochastic and asymmetric transitions induced by wind blowing from the top.
Learned reward function (Figure \ref{fig:windy_reward}). Reward at each state of the grid world after training for 50 iterations with \textsc{aim}. Hollow red circle indicates the goal state. White lines indicate the walls the agent cannot transition through.
The agent's state visitation (Figure \ref{fig:windy_policy}): The hollow blue circle indicates agent's start state. The hollow red circle is the goal. Blue bubbles indicate relative time the agent's policy causes it to spend in respective states. Black lines indicate walls.}
\label{fig:windy_gridworld}
\end{figure}
\paragraph{Basic experiment} The environment is a $10 \times 10$ grid with $4$ discrete actions that take the agent in the $4$ cardinal directions unless blocked by a wall or the edge of the grid. The agent policy is learned using soft Q-learning \citep{haarnoja2017reinforcement}, with an entropy coefficient of $0.1$ and a discount factor of $\gamma=0.99$. We do not use hindsight goals for this experiment, and use a single buffer with size $5000$ for both the policy as well as the discriminator training. The results are discussed in the main text. The compute used to conduct these experiments was a personal laptop with an Intel i7 Processor and 16 GB of RAM.
\paragraph{Additional experiments} We conducted variations form the basic experiment in the grid world to show that \textsc{aim}\ and its novel regularization can learn a reward function which guides the agent to the goal even in the presence of stochastic transitions as well as transitions where the state features vary wildly from one step to the next.
First, we evaluate \textsc{aim}'s ability to learn in the presence of stochastic and asymmetric transitions in a windy version (Figure \ref{fig:windy_grid}) of the above grid world. Transitions in the last six columns of the grid are affected by a wind blowing from the top. Actions that try to move upwards only succeed $60\%$ of the time, and actions attempting to move sideways cause a transition diagonally downwards $40\%$ of the time. Movements downwards are unaffected. The rest of the experiment is carried out in the same way as above, but with $128$ hidden units in the hidden layer of the agent's Q function approximator (the reward function architecture is unchanged from the previous experiment). In Figure \ref{fig:windy_gridworld} we see that \textsc{aim}\; learns a reward function that is still useful and interpretable, and leads to a policy that can confidently reach the goal, regardless of these stochastic and asymmetric transitions. Notice the effect of the stochastic transitions in the increased visitation in the sub-optimal states in the bottom two rows of column number $4$.
The next experiment tests what happens when the transition function causes the agent to jump between states where the state features vary sharply. As an example consider a toroidal grid world, where if an agent steps off one side of the grid it is transported to the other side. The distance function here should be smooth across such transitions, but might be hampered by the sharp change in input features. In Figure \ref{fig:toroid} we see show the policy and reward for a $10\times10$ toroidal grid world with start state at $(2,2)$ and goal at $(7, 7)$. Transitions are deterministic but wrap around the edges of the grid as described above: a \textbf{down} action in row $0$ will transport the agent to the same column but row $9$. The start and the goal state are set up so that there are multiple optimal paths to the goal. The entropy maximizing soft Q-learning algorithm should take these paths with almost equal probability. From Figure \ref{fig:toroid} it is evident that \textsc{aim}\; learns a reward function that is smooth across the actual transitions in the environment and allows the agent to learn a Q-function that places near equal mass on multiple trajectories.
\begin{figure}
\caption{Reward function with \textsc{aim}}
\label{fig:r_toroid}
\caption{Policy distribution under \textsc{aim}}
\label{fig:p_toroid}
\caption{Reward function estimated with WGAN loss}
\label{fig:r_gnorm}
\caption{The reward function (Figure \ref{fig:r_toroid}) learned with \textsc{aim}\; and subsequent policy distribution (Figure \ref{fig:p_toroid}) in a toroidal grid world, where an agent can transition from one edge of the grid across to the other. The hollow blue circle denotes the start state and the hollow red circle is the goal state. The reward function respects the sharp transitions from one end of the grid to the other. Conversely, if the reward function is learned using the WGAN objective \citep{gulrajani_improved_2017} (Figure \ref{fig:r_gnorm}), it does not respect the environment dynamics.}
\label{fig:toroid}
\end{figure}
Finally, we compare learning with \textsc{aim}\ to the baselines mentioned in Section \ref{sec:exp}. RND, SMiRL, and MC were implemented and debugged on the grid world domain with a goal that is easier to reach before being used on the Fetch robot tasks. Hyper-parameters for the algorithms in both domains were determined through sweeps. In the Fetch domains, the hyperparameters for all three new baselines were decided on through sweeps on the FetchReach task, similar to how they were evaluated for \textsc{aim}\ and the other baselines.
Figure \ref{fig:add_grid} shows the results of executing these additional baselines on the grid world domain we use to motivate \textsc{aim}. All the plots are taken after the techniques have had the same number of training iterations. However none of the baselines reach the goal even after providing additional time. We show the negative L2 distance to goal as a reward in the grid world domain to highlight that the DiscoRL \citep{Nasiriany:EECS-2020-151} objective should not be considered equivalent to an oracle of the distance to goal. Note that RND (Figure \ref{fig:grid_rnd}) explores most of the larger room early on, and then converges to the state distribution seen in the figure when it does not encounter the task reward. The SMiRL reward encourages the agent to minimize surprise, and the policy trained with this reward keeps the agent in the bottom left near its start state (Figure \ref{fig:grid_smirl}).
\begin{figure}
\caption{State visitation with $- L2$ reward (DiscoRL)}
\label{fig:grid_disco}
\caption{State visitation with MC distance estimation}
\label{fig:grid_ddl}
\caption{State visitation with added RND reward}
\label{fig:grid_rnd}
\caption{State visitation with SMiRL rewards}
\label{fig:grid_smirl}
\caption{DDL reward function}
\label{fig:grid_rew_ddl}
\caption{RND reward function}
\label{fig:grid_rew_rnd}
\caption{The state of the state visitation and reward functions for the new baselines. For camparison, Figure \ref{fig:grid_policy} shows the state visitation of policy trained using \textsc{aim}. All algorithms are compared after 100 training iterations.}
\label{fig:add_grid}
\end{figure}
\begin{figure}
\caption{Reach}
\label{fig:app_reach}
\caption{Pick and Place}
\label{fig:app_pick}
\caption{Push}
\label{fig:app_push}
\caption{Slide}
\label{fig:app_slide}
\caption{Comparing [\textsc{aim}\ + \textsc{her}] with an additional baseline which also uses the external task reward [\textsc{aim}\ + R + \textsc{her}]. The additional grounding provided by the external task reward allows the agent's learning to accelerate even further.}
\label{fig:app_fetch}
\end{figure}
\section{Statistical Analysis of the Results on Fetch Robot Tasks} \label{app:fetch_test}
To compare the performance of each method with statistical rigor, we used a repeated measures ANOVA design for binary observation where an observation is successful if an agent reaches the goal within an episode. We then conducted a Tukey test to compare the effects of each method, i.e., the estimated odds of reaching the goal given the algorithm. The goal of the statistical analysis presented here is twofold \begin{enumerate}[itemsep=0pt]
\item Separate the uncertainty on the performance of each method from the variation due to random seeds.
\item Adjust the probability of making a false discovery due to multiple comparisons. This extra step is necessary to avoid detecting a large fraction of falsely ``significant" differences since typical tests are designed to control the error rate of only one experiment. \end{enumerate}
The data for statistical analysis comes from $N_{\text{episodes}}=100$ evaluation episodes per each one of $N_{\text{seeds}}=6$ seeds. For all environments but FetchReach, these data is collected after 1 million environment interactions; and for FetchReach it is taken after 2000 interactions.
The repeated measures ANOVA design is formulated as a mixed effects generalized linear model and fitted separately for each one of the four environments \begin{align*} y_{ijk} &\overset{\text{\tiny iid}}{\sim} \mathrm{Bernoulli}(p_{ij}), \quad &k\in\{1,\hdots, N_{\text{episodes}}\} \\ \text{logit}(p_{ij}) &= r_{\text{seed}_i} + \beta_{\text{algorithm}_j}\quad & i\in\{1,\hdots,N_{\text{seeds}}\}, j\in\{1,\hdots,N_{\text{algorithms}}\} \\ r_{\text{seed}_i} &\overset{\text{\tiny iid}}{\sim} \sim \mathrm{Normal}(0, \sigma^2) \quad & \end{align*} The variation due to the seed effects is measured by $\sigma^2$, whereas the uncertainty about the odds of reaching the goal using each algorithm is measured by the standard errors of the coefficients $\beta_{\text{algorithm}_j}$. The Tukey test evaluates all null hypotheses $H_0\colon \beta_{\text{algorithm}_j} = \beta_{\text{algorithm}_{j'}}$ for all combinations of $j,j'$. To adjust for multiple comparisons each Tukey tests uses the Holm method. Since we are also doing a Tukey test for each environment, we further apply a Bonferroni adjustment with a factor of four. These types of adjustments are fairly common for dealing with multiple comparison in the literature of experimental design; the interested reader may consult \citep{montgomery2017design}.
The results, shown in Table \ref{tab:anova}, signal strong statistical evidence of the improvements from using the \textsc{aim}\ learned rewards. In three of the four environments \textsc{aim}\ and \textsc{aim} + \textsc{r} have similar odds of reaching the goal as the dense shaped reward ($H_0$ is not rejected,) and in all four environments \textsc{aim}\ and \textsc{aim} + \textsc{r} have statistically significant higher odds of reaching the goal than the sparse reward ($H_0$ is rejected and $\beta$ is higher.)
\begin{table}[thb]
\centering
\footnotesize
\begin{tabular}{c|r|r|r|r}
Contrast & Slide & Push & PickAndPlace & Reach\\ \midrule
$\beta_{\text{\textsc{aim}+\textsc{r}}} - \beta_{\text{\textsc{her}+dense}}$
& 0.34 (0.14) & -1.74 (0.77) & -0.10 (0.45) & *-3.43 (0.34)
\\
$\beta_{\text{\textsc{aim}}} - \beta_{\text{\textsc{her}+dense}}$
& 0.21 (0.14) & -2.19 (0.75) & *-1.50 (0.37) & *-5.01 (0.35)
\\ \midrule
$\beta_{\text{\textsc{aim}}+\textsc{r}} - \beta_{\text{\textsc{her}+sparse}}$
& *0.69 (0.13) & *5.32 (0.35) & *4.71 (0.33) & *4.75 (0.25)
\\
$\beta_{\text{\textsc{aim}}} - \beta_{\text{\textsc{her}+sparse}}$
& *0.57 (0.13) & *4.86 (0.30) & *3.31 (0.19) & *3.17 (0.24)
\end{tabular}
\caption{Results of the Tukey test on the evaluation of Fetch tasks. The table entries are log odds ratios with standard deviations shown in parentheses. Positive values mean that \textsc{aim}\ or \textsc{aim}+\textsc{r} perform better than the method with negative sign in the contrast and viceversa. Asterisks mark statistical significance at 95\%. If there is no asterisk, then $H_0$ is not rejected in which case the differences could be due to random chance.}
\label{tab:anova}
\end{table}
\section{Details of Experiments on Fetch Robot} \label{app:fetch}
The Fetch robot domain in OpenAI gym has four tasks available for testing. They are named Reach, Push, Slide, and Pick And Place. The Reach task is the simplest, with the goal being the 3-d coordinates where the end effector of the robot arm must be moved to. The Push task requires pushing an object from its current position on the table to the given target position somewhere else on the table. Slide is similar to Push, except the coefficient of friction on the table is reduced (causing pushed objects to slide) and the potential targets are over a larger area, meaning that the robot needs to learn to hit objects towards the goal with the right amount of force. Finally, Pick And Place is the task where the robot actuates it's gripper, picks up an object from its current position on the table and moves it through space to a given target position that could be at some height above the table. The goal space for the final three tasks are the required position of the object, and the goal the current state represents is the current position of that object.
Next, we note the hyperparameters used for various baselines as well as our implementation. The names of the hyperparameters are as specified in the stable baselines repository and used in the RL Zoo \citep{rl-zoo} codebase which we use for running experiments. Both the stable baselines repository and RL Zoo are available under the MIT license. These experiments were run on a compute cluster with each experiment assigned an Nvidia Titan V GPU, a single CPU and 12 GB of RAM. Each run of the TD3 baseline \textsc{her} + \textsc{r} or \textsc{her} + dense required $18$ hours to execute, and each run which included \textsc{aim}\ required $24$ hours to complete execution.
TD3 \citep{fujimoto2018TD3}, like its predecessor DDPG \citep{lillicrap2015continuous}, suffers from the policy saturating to extremes of its parameterization. \citet{hausknecht2016deep} have suggested various techniques to mitigate such saturation. We use a quadratic penalization for actions that exceed $80\%$ of the extreme value at either end, which is sufficient to not hurt learning and prevent saturation. Assuming the policy network predicts values between $-1$ and $1$ (as is the case when using the tanh activation function), the regularization loss is: \begin{align*}
L_a = \frac{1}{N} \sum_{i=1}^{N} \left[max(|\pi_\theta(s_i)| - 0.8, 0)\right] ^ 2 \end{align*} where $N$ is the mini-batch size and $s_i$ is the state for the $i^{\text{th}}$ transition in the batch.
The other modification made to the stable baselines code is to use the Huber loss instead of the squared loss for Q-learning.
For evaluation, in the Reach domain the agent policy is evaluated for 100 episodes every 2000 steps. For the other three domains, the experiment is run for 1 million timesteps, and evaluated at every 20,000 steps for 100 episodes.
\subsection{TD3 and HER (R + HER)}
\begin{tabular}{||c | c||}
\hline
Hyperparameter & Value \\ [0.5ex]
\hline\hline
n_sampled_goal & 4 \\
\hline
goal_selection_strategy & future \\
\hline
buffer_size & $10^6$ \\
\hline
batch_size & 256 \\
\hline
$\gamma$ (discount factor) & $0.95$ \\
\hline
random_exploration & 0.3 \\
\hline
target_policy_noise & 0.2 \\
\hline
learning_rate & $1^{-3}$ \\
\hline
noise_type & normal \\
\hline
noise_std & $0.2$ \\
\hline
MLP size of agent policy and Q function & $[256, 256, 256]$ \\
\hline
learning_starts & 1000 \\
\hline
train_freq & 10 \\
\hline
gradient_steps & 10 \\
\hline
$\tau$ (target policy update rate) & 0.05 \\ [1ex]
\hline \end{tabular}
\subsection{Dense reward TD3 and HER (dense + HER)}
\begin{tabular}{||c | c||}
\hline
Hyperparameter & Value \\ [0.5ex]
\hline\hline
n_sampled_goal & 4 \\
\hline
goal_selection_strategy & future \\
\hline
buffer_size & $10^6$ \\
\hline
batch_size & 256 \\
\hline
$\gamma$ (discount factor) & $0.95$ \\
\hline
random_exploration & 0.3 \\
\hline
target_policy_noise & 0.2 \\
\hline
learning_rate & $1^{-3}$ \\
\hline
noise_type & normal \\
\hline
noise_std & $0.2$ \\
\hline
MLP size of agent policy and Q function & $[256, 256, 256]$ \\
\hline
learning_starts & 1000 \\
\hline
train_freq & 100 \\
\hline
gradient_steps & 200 \\
\hline
policy_delay & 5 \\
\hline
$\tau$ (target policy update rate) & 0.05 \\ [1ex]
\hline \end{tabular}
\subsection{TD3 and HER with AIM (AIM + HER) and (AIM + R + HER)}
\begin{tabular}{||c | c||}
\hline
Hyperparameter & Value \\ [0.5ex]
\hline\hline
n_sampled_goal & 4 \\
\hline
goal_selection_strategy & future \\
\hline
buffer_size & $10^6$ \\
\hline
batch_size & 256 \\
\hline
$\gamma$ (discount factor) & $0.9$ \\
\hline
random_exploration & 0.3 \\
\hline
target_policy_noise & 0.2 \\
\hline
learning_rate & $1^{-3}$ \\
\hline
noise_type & normal \\
\hline
noise_std & $0.2$ \\
\hline
MLP size of agent policy and Q function & $[256, 256, 256]$ \\
\hline
learning_starts & 1000 \\
\hline
train_freq & 100 \\
\hline
gradient_steps & 200 \\
\hline
disc_train_freq & 100 \\
\hline
disc_steps & 20 \\
\hline
$\tau$ (target policy update rate) & 0.1 \\ [1ex]
\hline \end{tabular}
\end{document} | arXiv |
Trying to understand this Quicksort Correctness proof
This proof is a proof by induction, and goes as follows:
P(n) is the assertion that "Quicksort correctly sorts every input array of length n."
Base case: every input array of length 1 is already sorted (P(1) holds)
Inductive step: fix n => 2. Fix some input array of length n.
Need to show: if P(k) holds for all k < n, then P(n) holds as well
He then draws an array A partitioned around some pivot p. So he draws p, and calls the part of the array that is < p as the 1st part, and the part that is > p is the second part. The length of part 1 = k1, and the length of part 2 is k2. By the correctness proof of the Partition subroutine (proved earlier), the pivot p winds up in the correct position.
By inductive hypothesis: 1st, 2nd parts get sorted correctly by recursive calls. (Using P(K1),P(k2))
So: after recursive calls, entire array is correctly sorted.
My confusion: I have a lot of problem seeing exactly how this proves the correctness of it. So we assume that P(k) does indeed hold for all natural numbers k < n.
Most of the induction proofs I had seen so far go something like: Prove base case, and show that P(n) => P(n+1). They usually also involved some sort of algebraic manipulation. This proof seems very different, and I don't understand how to apply the concept of Induction to it. I can somewhat reason that the correctness of the Partition subroutine is the key. So is the reasoning for its correctness as follows: We know that each recursive call, it will partition the array around a pivot. This pivot will then be in its rightful position. Then each subarray will be further partitioned around a pivot, and that pivot will then be in its rightful position. This goes on and on until you get an subarray of length 1, which is trivially sorted.
But then we're not assuming that P(k) holds for all k < n....we are actually SHOWING it does (since the Partition subroutine will always place one element in its rightful position.) Are we not assuming that P(k) holds for all k
correctness-proof induction quicksort
FrostyStraw
FrostyStrawFrostyStraw
$\begingroup$ What is "QUE"? Did you mean "QED"? (the Latin Quod Erat Demonstrandum which does not contain any word starting for U) $\endgroup$
– Bakuriu
$\begingroup$ I did indeed mean QED. I guess my confusion led to me writing "WHAT?" in spanish $\endgroup$
– FrostyStraw
We are indeed assuming $P(k)$ holds for all $k < n$. This is a generalization of the "From $P(n-1)$, we prove $P(n)$" style of proof you're familiar with.
The proof you describe is known as the principle of strong mathematical induction and has the form
Suppose that $P(n)$ is a predicate defined on $n\in \{1, 2, \dotsc\}$. If we can show that
$P(1)$ is true, and
$(\forall k < n \;[P(k)])\Longrightarrow P(n)$
Then $P(n)$ is true for all integers $n\ge 1$.
In the proof to which you refer, that's exactly what's going on. To use quicksort to sort an array of size $n$, we partition it into three pieces: the first $k$ subarray, the pivot (which will be in its correct place), and the remaining subarray of size $n-k-1$. By the way partition works, every element in the first subarray will be less than or equal to the pivot and every element in the other subarray will be greater than or equal to the pivot, so when we recursively sort the first and last subarrays, we will wind up having sorted the entire array.
We show this is correct by strong induction: since the first subarray has $k<n$ elements, we can assume by induction that it will be correctly sorted. Since the second subarray has $n-k-1<n$ elements, we can assume that it will be correctly sorted. Thus, putting all the pieces together, we will wind up having sorted the array.
Rick DeckerRick Decker
$\begingroup$ The cool part about the principle of strong induction is that the base case $P(1)$ is not necessary! If we take $n=1$ in the induction step, then the antecedent $\forall k<1,P(k)$ is vacuous, so we have $P(1)$ unconditionally. $\endgroup$
– Mario Carneiro
$\begingroup$ Okay so...to be clear...We ASSUME P(k) is true for all k < n. And the way we SHOW that P(k) ==> P(n) (for all k < n) is through the combination of knowing that the pivot will for sure be in its correct position, and through the assumption (the inductive hypothesis) that the left and right subarrays are also sorted. Combine that with the base case (in which k = 1 < n), and the proof is complete? $\endgroup$
$\begingroup$ well I guess it wouldn't be enough to know that the pivot is in its correct position, but also that the right subarray is all greater than the pivot and the left one is all less than $\endgroup$
$\begingroup$ @FrostyStraw It's Chinese whisperes. $\endgroup$
– Raphael ♦
$\begingroup$ @FrostyStraw Hehe, I meant the proof strategy. :) $\endgroup$
This proof uses the principle of complete induction:
Suppose that:
Base case: $P(1)$
Step: For every $n > 1$, if $P(1),\ldots,P(n-1)$ hold (induction hypothesis) then $P(n)$ also holds.
Then $P(n)$ holds for all $n \geq 1$.
You can prove this principle using the usual induction principle by considering the property $$ Q(m) \Leftrightarrow P(1) \text{ and } P(2) \text{ and } \cdots \text{ and } P(m) $$ I leave you the details.
Now, let's use complete induction to prove that the following version of Quicksort sorts its input correctly:
Quicksort(A, n)
if n = 1 then:
let X[1...x] consist of all elements of A[2],...,A[n] which are at most A[1]
let Y[1...y] consist of all elements of A[2],...,A[n] which are larger than A[1]
call Quicksort(X, x)
call Quicksort(Y, y)
set A to the concatenation of X, A[1], Y
Here A[1],...,A[n] is the input array, and n is its length. The statement that we want to prove is as follows:
Let $A$ be an array of length $n \geq 1$. Denote the contents of $A$ after calling Quicksort by $B$. Then:
Quicksort terminates on $A$.
There is a permutation $\pi_1,\ldots,\pi_n$ of $1,\ldots,n$ such that $B[i] = A[\pi_i]$.
$B[1] \leq B[2] \leq \cdots \leq B[n]$.
I will only prove the third property, leaving the rest to you. We thus let $P(n)$ be the following statement:
If $A$ is an array of length $n \geq 1$, and $B$ is its contents after running Quicksort(A, n), then $B[1] \leq B[2] \leq \cdots \leq B[n]$.
The proof is by complete induction on $n$. If $n = 1$ then there is nothing to prove so suppose that $n > 1$. Let $X,x,Y,y$ be as in the procedure Quicksort. Since $x,y < n$, the induction hypothesis shows that $$ X[1] \leq X[2] \leq \cdots \leq X[x] \\ Y[1] \leq Y[2] \leq \cdots \leq Y[y] $$ Moreover, from the way we formed arrays $X$ and $Y$, it follows that $X[x] \leq A[1] < Y[1]$. Thus $$ X[1] \leq \cdots \leq X[x] \leq A[1] < Y[1] \leq \cdots \leq Y[y]. $$ It follows immediately that $B[1] \leq \cdots \leq B[n]$. Thus $P(n)$ holds.
The missing part of the argument is transitivity of '<' - i.e the property that if a < b and b < c, then a < c. The proof that the final array is sorted goes something like this:
Let A[i], A[j] be elements of the array post-sort, where i < j. Then A[i] < A[j] follows from one of the following placement cases (and there are no other cases):
(a) i and j are in the first partition - A[i] < A[j] follows by recursion/induction.
(b) i and j are in the second partition - A[i] < A[j] follows by recursion/induction.
(c) i is in the first partition and j is the index of the pivot - A[i] < A[j] follows by proof of partition procedure.
(c) i is the index of the pivot and j is in the second partition - A[i] < A[j] follows by proof of partition procedure.
(e) i is in first partition and j is in second partition - then by partition procedure, A[i] < pivot, and pivot < A[j]. So by transitivity, A[i] < A[j].
PMarPMar
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Proof of QuickSort algorithm correctness
How does Hoare's quicksort work, even if the final position of the pivot after partition() is not what its position is in the sorted array?
Array contains elements that differ by K correctness proof
Understanding how quicksort operates
Worst Case Scenario for Quicksort algorithm with pivot element n/2
prove correctness of in-order tree traversal subroutine
Understanding the upper bound proof for quick sort | CommonCrawl |
When is a statement provable?
We all know that Gödel showed that there, in a formal system, are true statements that are non-provable (undecidable). In ZFC, there's Kaplansky's Conjecture, the Whitehead problem, etc.
We can also agree that we're sure to find more non-provable statements in ZFC. What I'm curious about is the following:
What is the defining characteristic of a non-provable statement in ZFC? Are they all "strong" in some sense? Is it a necessary condition that they are strong, then? What future theorems might turn out to be non-provable? Is it, through the characteristics of the theorems known to be non-provable possible to make a fairly accurate guess if a theorem will turn out to be non-provable in ZFC?
mathematical-philosophy lo.logic proof-theory
DedalusDedalus
Not all statements that are not provable in ZFC are "strong", if by strong you mean that ZFC + the statement in question is stronger than ZFC in the sense that it implies the consistency of ZFC.
The typical example is the Continuum Hypothesis (CH). ZFC + CH is consistent iff ZFC is, and in this sense, CH is not strong since ZFC + CH itself does not imply the consistency of ZFC.
However, if one wants to prove that the existence of a measurable cardinal is not provable in ZFC, the argument is that ZFC + there is a measurable cardinal implies the consistency of ZFC and hence, by the second incompleteness thm, the existence of a measurable cardinal cannot be proved in ZFC.
Note that the situation in set theory is different from number theory. In order to show that something is not provable in Peano Arithmetic (something which is consistent with PA), one usually uses the Incompleteness Theorems, i.e., one shows that the statement implies the consistency of PA. So in some sense, number theoretic statements known to be independent over PA are strong (over PA), statements known to be independent over ZFC are not necessarily strong over ZFC.
Stefan GeschkeStefan Geschke
$\begingroup$ What about (in PA) the existence of an infinite element c, such that c>0, c>1, c>2, etc>? Does this prove Con(PA)? $\endgroup$ – Ross Millikan Dec 30 '10 at 22:10
$\begingroup$ No. Since Con(PA) is not provable from PA (assuming PA is consistent), we have an $M \vDash PA + \lnot CON(PA)$. The extension of $Th(M)$ (theory of $M$) by the set of axioms {$c > n| n \in \mathbb{N}$} is finitely satisfiable so it's satisfiable by the compactness theorem. Thus we have a model of $Th(M) + $ {$c > n| n \in \mathbb{N}$}, which will be a model of $PA + \lnot CON(PA) + ${$c > n| n \in \mathbb{N}$}. Therefore, by soundness, $PA + $ {$c > n| n \in \mathbb{N}$} cannot prove $CON(PA)$. In fact, any model $M$ of $\lnot CON(PA)$ already has such a $c$. $\endgroup$ – Jason Dec 31 '10 at 0:28
$\begingroup$ @Jason: I thought so, and that this was an example of something independent under PA that didn't prove Con(PA). Thanks. $\endgroup$ – Ross Millikan Dec 31 '10 at 3:48
$\begingroup$ Sentences $\phi$ that are independent of an axiom system $A$, such that neither $\phi$ nor $\neg \phi$ increases the consistency strength of $A$, are called "Orey sentences", and they can be constructed for arithmetical theories like PA as well as set theory. So number theory and set theory seem comparable in this regard. $\endgroup$ – Shivaram Lingamneni Oct 14 '14 at 4:33
$\begingroup$ @Ross Millikan: The existence of an infinite element cannot be expressed by a single sentence! However, if you introduce a new constant symbol, then there is a consistent extension of PA that says that there is an infinite element. So this is not quite the same as an independent sentence. $\endgroup$ – Stefan Geschke Oct 14 '14 at 16:47
Often, you can classify characteristics of a collection of independence results through the use of forcing axioms. For example, if you assume Martin's axiom MA$(\aleph_1)$, which states that for every $\aleph_1$ many dense subsets of a partial order having the countable chain condition, there is a filter meeting all of them, then you can derive a number of statements that are not true in the constructible universe. For example, MA$(\aleph_1)$ implies $2^{\aleph_0} = 2^{\aleph_1}$, the nonexistence of Suslin lines, and the existence of a Whitehead group that isn't free.
It is a current focus of research in set theory to study more powerful forcing axioms that rely on the relative consistency of certain large cardinals with ZFC. For example, the proper forcing axiom, which strengthens MA$(\aleph_1)$ by only requiring the partial order to be proper, has its relative consistency following from the existence of a supercompact cardinal while implying the relative consistency of a Woodin cardinal. These types of forcing axioms give us a flavor of a number of statements that can be true. Consequently, if a statement implied by such an axiom contradicts what is true in some canonical inner model, then we can believe in the statement's independence with ZFC provided we believe in the relative consistency of a large cardinal that implied the possibility of the forcing axiom.
JasonJason
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Theories of arithmetic from recursively inseparable sets | CommonCrawl |
R-S Integrability of Cts. Functions with Integrators of Bounded Variation
Riemann-Stieltjes Integrability of Continuous Functions with Integrators of Bounded Variation
Recall from the Riemann-Stieltjes Integrals with Integrators of Bounded Variation page that if $f$ is a bounded function defined on $[a, b]$, $\alpha$ is a function of bounded variation on $[a, b]$ and $V(x) = V_{\alpha}(a, x)$ is the total variation function on $\alpha$ then if $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ then $f$ is Riemann-Stieltjes integrable with respect to $V$ and $V - f$.
We will now look at a whole class of functions that are Riemann-Stieltjes integrable - namely continuous integrands accompanied by integrators of bounded variation.
Theorem 1: Let $f$ be a continuous function on $[a, b]$ and let $\alpha$ be a function of bounded variation on $[a, b]$. Then $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$.
Proof: Since $f$ is continuous on the closed bounded interval $[a, b]$ we have that $f$ is also bounded on $[a, b]$ by the Boundedness Theorem.
Since $\alpha$ is of bounded variation on $[a, b]$ we have that $\alpha = V - (V - \alpha)$ where $V$ and $V - \alpha$ are increasing functions with respect to $V$ be the total variation function of $\alpha$. If we show that the theorem holds for increasing functions, then we have shown that it holds for $V$ and $V - \alpha$ and hence holds for any function of bounded variation.
Assume that $\alpha$ is an increasing function. If $\alpha(a) = \alpha(b)$ then $\Delta \alpha_k = 0$ and the theorem is trivially true since $\alpha$ will be a constant integrator and we've already dealt with this case on the Riemann-Stieltjes Integrals with Constant Integrators page.
Instead assume that $\alpha(a) < \alpha(b)$. Then $\alpha(b) - \alpha(a) > 0$. We will show that Riemann's condition holds under these hypotheses.
Since $f$ is continuous on $[a, b]$ we have that $f$ is also uniformly continuous on $[a, b]$, i.e., for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $\mid x - y \mid < \delta$ then $\mid f(x) - f(y) \mid < \epsilon$.
So for $\epsilon_1 = \frac{\epsilon}{\alpha(b) - \alpha(a)} > 0$ there exists a $\delta_1 > 0$ such that if $\mid x - y \mid < \delta_1$ then:
\begin{align} \quad \mid f(x) - f(y) \mid < \epsilon_1 = \frac{\epsilon}{\alpha(b) - \alpha(a)} \quad \end{align}
Now let $P \in \mathscr{P}[a, b]$ be any partition and consider the difference of the upper and lower Riemann-Stieltjes sums:
\begin{align} \quad U(P, f, \alpha) - L(P, f, \alpha) = \sum_{k=1}^{n} [M_k(f) - m_k(f)] \Delta \alpha_k \end{align}
Since $f$ is continuous on $[a, b]$ it is also continuous on each subinterval $[x_{k-1}, x_k]$ of $[a, b]$. Furthermore, each of these subintervals $[x_{k-1}, x_k]$ are bounded and so there exists $t_k, t_k' \in [x_{k-1}, x_k]$ such that:
\begin{align} \quad M_k = \sup \{ f(x) : x \in [x_{k-1}, x_k] \} = f(t_k) \quad \mathrm{and} \quad m_k(f) = \inf \{ f(x) : x \in [x_{k-1}, x_k] \} = f(t_k') \end{align}
Hence the difference of the upper and lower Riemann-Stieltjes sums can be written as:
\begin{align} \quad U(P, f, \alpha) - L(P, f, \alpha) = \sum_{k=1}^{n} [f(t_k) - f(t_k')] \Delta \alpha_k \end{align}
Now make $P$ finer and finer such that $\| P \| < \delta$. Then the length of each subinterval will be less than $\delta$, i.e., $\mid x_k - x_{k-1} \mid < \delta$, and since $t_k, t_k' \in [x_{k-1}, x_k]$ we see that $\mid t_k - t_k' \mid < \delta$. Hence:
\begin{align} \quad \mid f(t_k) - f(t_k') \mid < \epsilon_1 = \frac{\epsilon}{\alpha(b) - \alpha(a)} \quad (*) \end{align}
So choose $P_{\epsilon}$ such that $\| P_{\epsilon} \| < \delta$. Then for all $P$ finer than $P_{\epsilon}$ we have that $(*)$ holds and so:
\begin{align} \quad U(P, f, \alpha) - L(P, f, \alpha) &= \sum_{k=1}^{n} [M_k(f) - m_k(f)] \Delta \alpha_k \\ \quad &= \sum_{k=1}^{n} [f(t_k) - f(t_k')] \Delta \alpha_k \\ \quad &< \sum_{k=1}^{n} \epsilon_1 \Delta \alpha_k \\ \quad &< \sum_{k=1}^{n} \frac{\epsilon}{\alpha(b) - \alpha(a)} \Delta \alpha_k \\ \quad &< \frac{\epsilon}{\alpha(b) - \alpha(a)} \sum_{k=1}^{n} \Delta \alpha_k \\ \quad &< \frac{\epsilon}{\alpha(b) - \alpha(a)} [\alpha(b) - \alpha(a)] \\ \quad &< \epsilon \end{align}
So for every $\epsilon > 0$ there exists a partition $P_{\epsilon} \in \mathscr{P}[a, b]$ such that if $P$ is finer than $P_{\epsilon}$ then $U(P, f, \alpha) - L(P, f, \alpha) < \epsilon$, so Riemann's condition is satisfied and $f$ is Riemann-Stieltjes integrable with respect to $\alpha$. $\blacksquare$ | CommonCrawl |
Arithmetic Fuchsian group
Arithmetic Fuchsian groups are a special class of Fuchsian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. The prototypical example of an arithmetic Fuchsian group is the modular group $\mathrm {PSL} _{2}(\mathbb {Z} )$. They, and the hyperbolic surface associated to their action on the hyperbolic plane often exhibit particularly regular behaviour among Fuchsian groups and hyperbolic surfaces.
Definition and examples
Quaternion algebras
Main article: Quaternion algebra
A quaternion algebra over a field $F$ is a four-dimensional central simple $F$-algebra. A quaternion algebra has a basis $1,i,j,ij$ where $i^{2},j^{2}\in F^{\times }$ and $ij=-ji$.
A quaternion algebra is said to be split over $F$ if it is isomorphic as an $F$-algebra to the algebra of matrices $M_{2}(F)$.
If $\sigma $ is an embedding of $F$ into a field $E$ we shall denote by $A\otimes _{\sigma }E$ the algebra obtained by extending scalars from $F$ to $E$ where we view $F$ as a subfield of $E$ via $\sigma $.
Arithmetic Fuchsian groups
A subgroup of $\mathrm {PSL} _{2}(\mathbb {R} )$ is said to be derived from a quaternion algebra if it can be obtained through the following construction. Let $F$ be a totally real number field and $A$ a quaternion algebra over $F$ satisfying the following conditions. First there is a unique embedding $\sigma :F\hookrightarrow \mathbb {R} $ such that $A\otimes _{\sigma }\mathbb {R} $ is split over $\mathbb {R} $ ; we denote by $\phi :A\otimes _{\sigma }\mathbb {R} \to M_{2}(\mathbb {R} )$ an isomorphism of $\mathbb {R} $-algebras. We also ask that for all other embeddings $\tau $ the algebra $A\otimes _{\tau }\mathbb {R} $ is not split (this is equivalent to its being isomorphic to the Hamilton quaternions). Next we need an order ${\mathcal {O}}$ in $A$. Let ${\mathcal {O}}^{1}$ be the group of elements in ${\mathcal {O}}$ of reduced norm 1 and let $\Gamma $ be its image in $M_{2}(\mathbb {R} )$ via $\phi $. Then the image of $\Gamma $ is a subgroup of $\mathrm {SL} _{2}(\mathbb {R} )$ (since the reduced norm of a matrix algebra is just the determinant) and we can consider the Fuchsian group which is its image in $\mathrm {PSL} _{2}(\mathbb {R} )$.
The main fact about these groups is that they are discrete subgroups and they have finite covolume for the Haar measure on $\mathrm {PSL} _{2}(\mathbb {R} ).$ Moreover, the construction above yields a cocompact subgroup if and only if the algebra $A$ is not split over $F$. The discreteness is a rather immediate consequence of the fact that $A$ is only split at one real embedding. The finiteness of covolume is harder to prove.[1]
An arithmetic Fuchsian group is any subgroup of $\mathrm {PSL} _{2}(\mathbb {R} )$ which is commensurable to a group derived from a quaternion algebra. It follows immediately from this definition that arithmetic Fuchsian groups are discrete and of finite covolume (this means that they are lattices in $\mathrm {PSL} _{2}(\mathbb {R} )$).
Examples
The simplest example of an arithmetic Fuchsian group is the modular $\mathrm {PSL} _{2}(\mathbb {Z} ),$ which is obtained by the construction above with $A=M_{2}(\mathbb {Q} )$ and ${\mathcal {O}}=M_{2}(\mathbb {Z} ).$ By taking Eichler orders in $A$ we obtain subgroups $\Gamma _{0}(N)$ for $N\geqslant 2$ of finite index in $\mathrm {PSL} _{2}(\mathbb {Z} )$ which can be explicitly written as follows:
$\Gamma _{0}(N)=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \mathrm {PSL} _{2}(\mathbb {Z} ):c=0{\pmod {N}}\right\}.$
Of course the arithmeticity of such subgroups follows from the fact that they are finite-index in the arithmetic group $\mathrm {PSL} _{2}(\mathbb {Z} )$ ; they belong to a more general class of finite-index subgroups, congruence subgroups.
Any order in a quaternion algebra over $\mathbb {Q} $ which is not split over $\mathbb {Q} $ but splits over $\mathbb {R} $ yields a cocompact arithmetic Fuchsian group. There is a plentiful supply of such algebras.[2]
More generally, all orders in quaternion algebras (satisfying the above conditions) which are not $M_{2}(\mathbb {Q} )$ yield cocompact subgroups. A further example of particular interest is obtained by taking $A$ to be the Hurwitz quaternions.
Maximal subgroups
A natural question is to identify those among arithmetic Fuchsian groups which are not strictly contained in a larger discrete subgroup. These are called maximal Kleinian groups and it is possible to give a complete classification in a given arithmetic commensurability class. Note that a theorem of Margulis implies that a lattice in $\mathrm {PSL} _{2}(\mathbb {C} )$ is arithmetic if and only if it is commensurable to infinitely many maximal Kleinian groups.
Congruence subgroups
A principal congruence subgroup of $\Gamma =\mathrm {SL} _{2}(\mathbb {Z} )$ is a subgroup of the form :
$\Gamma (N)=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \mathrm {PSL} _{2}(\mathbb {Z} ):a,d=1{\pmod {N}},\,b,c=0{\pmod {N}}\right\}$
for some $N\geqslant 1.$ These are finite-index normal subgroups and the quotient $\Gamma /\Gamma (N)$ is isomorphic to the finite group $\mathrm {SL} _{2}(\mathbb {Z} /N\mathbb {Z} ).$ A congruence subgroup of $\Gamma $ is by definition a subgroup which contains a principal congruence subgroup (these are the groups which are defined by taking the matrices in $\Gamma $ which satisfy certain congruences modulo an integer, hence the name).
Notably, not all finite-index subgroups of $\mathrm {SL} _{2}(\mathbb {Z} )$ are congruence subgroups. A nice way to see this is to observe that $\mathrm {SL} _{2}(\mathbb {Z} )$ has subgroups which surject onto the alternating group $A_{n}$ for arbitrary $n,$ and since for large $n$ the group $A_{n}$ is not a subgroup of $\mathrm {SL} _{2}(\mathbb {Z} /N\mathbb {Z} )$ for any $N$ these subgroups cannot be congruence subgroups. In fact one can also see that there are many more non-congruence than congruence subgroups in $\mathrm {SL} _{2}(\mathbb {Z} )$.[3]
The notion of a congruence subgroup generalizes to cocompact arithmetic Fuchsian groups and the results above also hold in this general setting.
Construction via quadratic forms
There is an isomorphism between $\mathrm {PSL} _{2}(\mathbb {R} )$ and the connected component of the orthogonal group $\mathrm {SO} (2,1)$ given by the action of the former by conjugation on the space of matrices of trace zero, on which the determinant induces the structure of a real quadratic space of signature (2,1). Arithmetic Fuchsian groups can be constructed directly in the latter group by taking the integral points in the orthogonal group associated to quadratic forms defined over number fields (and satisfying certain conditions).
In this correspondence the modular group is associated up to commensurability to the group $\mathrm {SO} (2,1)(\mathbb {Z} ).$[4]
Arithmetic Kleinian groups
Main article: Arithmetic hyperbolic 3-manifold
The construction above can be adapted to obtain subgroups in $\mathrm {PSL} _{2}(\mathbb {C} )$: instead of asking for $F$ to be totally real and $A$ to be split at exactly one real embedding one asks for $F$ to have exactly one complex embedding up to complex conjugacy, at which $A$ is automatically split, and that $A$ is not split at any embedding $F\hookrightarrow \mathbb {R} $. The subgroups of $\mathrm {PSL} _{2}(\mathbb {C} )$ commensurable to those obtained by this construction are called arithmetic Kleinian groups. As in the Fuchsian case arithmetic Kleinian groups are discrete subgroups of finite covolume.
Trace fields of arithmetic Fuchsian groups
The invariant trace field of a Fuchsian group (or, through the monodromy image of the fundamental group, of a hyperbolic surface) is the field generated by the traces of the squares of its elements. In the case of an arithmetic surface whose fundamental group is commensurable with a Fuchsian group derived from a quaternion algebra over a number field $F$ the invariant trace field equals $F$.
One can in fact characterise arithmetic manifolds through the traces of the elements of their fundamental group, a result known as Takeuchi's criterion.[5] A Fuchsian group is an arithmetic group if and only if the following three conditions are realised:
• Its invariant trace field $F$ is a totally real number field;
• The traces of its elements are algebraic integers;
• There is an embedding $\sigma :F\to \mathbb {R} $ such that for any $\gamma $ in the group, $t=\mathrm {Trace} (\gamma ^{2})$ and for any other embedding $\sigma \neq \sigma ':F\to \mathbb {R} $ we have $|\sigma '(t)|\leqslant 2$.
Geometry of arithmetic hyperbolic surfaces
The Lie group $\mathrm {PSL} _{2}(\mathbb {R} )$ is the group of positive isometries of the hyperbolic plane $\mathbb {H} ^{2}$. Thus, if $\Gamma $ is a discrete subgroup of $\mathrm {PSL} _{2}(\mathbb {R} )$ then $\Gamma $ acts properly discontinuously on $\mathbb {H} ^{2}$. If moreover $\Gamma $ is torsion-free then the action is free and the quotient space $\Gamma \setminus \mathbb {H} ^{2}$ is a surface (a 2-manifold) with a hyperbolic metric (a Riemannian metric of constant sectional curvature −1). If $\Gamma $ is an arithmetic Fuchsian group such a surface $S$ is called an arithmetic hyperbolic surface (not to be confused with the arithmetic surfaces from arithmetic geometry; however when the context is clear the "hyperbolic" specifier may be omitted). Since arithmetic Fuchsian groups are of finite covolume, arithmetic hyperbolic surfaces always have finite Riemannian volume (i.e. the integral over $S$ of the volume form is finite).
Volume formula and finiteness
It is possible to give a formula for the volume of distinguished arithmetic surfaces from the arithmetic data with which it was constructed. Let ${\mathcal {O}}$ be a maximal order in the quaternion algebra $A$ of discriminant $D_{A}$ over the field $F$, let $r=[F:\mathbb {Q} ]$ be its degree, $D_{F}$ its discriminant and $\zeta _{F}$ its Dedekind zeta function. Let $\Gamma _{\mathcal {O}}$ be the arithmetic group obtained from ${\mathcal {O}}$ by the procedure above and $S$ the orbifold $\Gamma _{\mathcal {O}}\setminus \mathbb {H} ^{2}$. Its volume is computed by the formula[6]
$\operatorname {vol} (S)={\frac {2|D_{F}|^{\frac {3}{2}}\cdot \zeta _{F}(2)}{(2\pi )^{2r-2}}}\cdot \prod _{{\mathfrak {p}}\mid D_{A}}(N({\mathfrak {p}})-1);$
the product is taken over prime ideals of $O_{F}$ dividing $(D_{A})$ and we recall the $N(\cdot )$ is the norm function on ideals, i.e. $N({\mathfrak {p}})$ is the cardinality of the finite ring $O_{F}/{\mathfrak {p}}$). The reader can check that if ${\mathcal {O}}=M_{2}(\mathbb {Z} )$ the output of this formula recovers the well-known result that the hyperbolic volume of the modular surface equals $\pi /3$.
Coupled with the description of maximal subgroups and finiteness results for number fields this formula allows to prove the following statement:
Given any $V>0$ there are only finitely many arithmetic surfaces whose volume is less than $V$.
Note that in dimensions four and more Wang's finiteness theorem (a consequence of local rigidity) asserts that this statement remains true by replacing "arithmetic" by "finite volume". An asymptotic equivalent for the number if arithmetic manifolds of a certain volume was given by Belolipetsky—Gelander—Lubotzky—Mozes.[7]
Minimal volume
Main article: Hurwitz surface
The hyperbolic orbifold of minimal volume can be obtained as the surface associated to a particular order, the Hurwitz quaternion order, and it is compact of volume $\pi /21$.
Closed geodesics and injectivity radii
A closed geodesic on a Riemannian manifold is a closed curve that is also geodesic. One can give an effective description of the set of such curves in an arithmetic surface or three—manifold: they correspond to certain units in certain quadratic extensions of the base field (the description is lengthy and shall not be given in full here). For example, the length of primitive closed geodesics in the modular surface corresponds to the absolute value of units of norm one in real quadratic fields. This description was used by Sarnak to establish a conjecture of Gauss on the mean order of class groups of real quadratic fields.[8]
Arithmetic surfaces can be used[9] to construct families of surfaces of genus $g$ for any $g$ which satisfy the (optimal, up to a constant) systolic inequality
$\operatorname {sys} (S)\geqslant {\frac {4}{3}}\log g.$
Spectra of arithmetic hyperbolic surfaces
Laplace eigenvalues and eigenfunctions
Main article: Spectral geometry
If $S$ is an hyperbolic surface then there is a distinguished operator $\Delta $ on smooth functions on $S$. In the case where $S$ is compact it extends to an unbounded, essentially self-adjoint operator on the Hilbert space $L^{2}(S)$ of square-integrable functions on $S$. The spectral theorem in Riemannian geometry states that there exists an orthonormal basis $\phi _{0},\phi _{1},\ldots ,\phi _{n},\ldots $ of eigenfunctions for $\Delta $. The associated eigenvalues $\lambda _{0}=0<\lambda _{1}\leqslant \lambda _{2}\leqslant \cdots $ are unbounded and their asymptotic behaviour is ruled by Weyl's law.
In the case where $S=\Gamma \setminus \mathbb {H} ^{2}$ is arithmetic these eigenfunctions are a special type of automorphic forms for $\Gamma $ called Maass forms. The eigenvalues of $\Delta $ are of interest for number theorists, as well as the distribution and nodal sets of the $\phi _{n}$.
The case where $S$ is of finte volume is more complicated but a similar theory can be established via the notion of cusp form.
Selberg conjecture
Main article: Selberg's 1/4 conjecture
The spectral gap of the surface $S$ is by definition the gap between the smallest eigenvalue $\lambda _{0}=0$ and the second smallest eigenvalue $\lambda _{1}>0$; thus its value equals $\lambda _{1}$ and we shall denote it by $\lambda _{1}(S).$ In general it can be made arbitrarily small (ref Randol) (however it has a positive lower bound for a surface with fixed volume). The Selberg conjecture is the following statement providing a conjectural uniform lower bound in the arithmetic case:
If $\Gamma \subset \mathrm {PSL} _{2}(\mathbb {R} )$ is lattice which is derived from a quaternion algebra and $\Gamma '$ is a torsion-free congruence subgroup of $\Gamma ,$ then for $S=\Gamma '\setminus \mathbb {H} ^{2}$ we have $\lambda _{1}(S)\geqslant {\tfrac {1}{4}}.$
Note that the statement is only valid for a subclass of arithmetic surfaces and can be seen to be false for general subgroups of finite index in lattices derived from quaternion algebras. Selberg's original statement[10] was made only for congruence covers of the modular surface and it has been verified for some small groups.[11] Selberg himself has proven the lower bound $\lambda _{1}\geqslant {\tfrac {1}{16}},$ a result known as "Selberg's 1/16 theorem". The best known result in full generality is due to Luo—Rudnick—Sarnak.[12]
The uniformity of the spectral gap has implications for the construction of expander graphs as Schreier graphs of $\mathrm {SL} _{2}(\mathbb {Z} ).$[13]
Relations with geometry
Main article: Selberg's trace formula
Main article: Cheeger constant
Selberg's trace formula shows that for an hyperbolic surface of finite volume it is equivalent to know the length spectrum (the collection of lengths of all closed geodesics on $S$, with multiplicities) and the spectrum of $\Delta $. However the precise relation is not explicit.
Another relation between spectrum and geometry is given by Cheeger's inequality, which in the case of a surface $S$ states roughly that a positive lower bound on the spectral gap of $S$ translates into a positive lower bound for the total length of a collection of smooth closed curves separating $S$ into two connected components.
Quantum ergodicity
Main article: Quantum ergodicity
The quantum ergodicity theorem of Shnirelman, Colin de Verdière and Zelditch states that on average, eigenfunctions equidistribute on $S$. The unique quantum ergodicity conjecture of Rudnick and Sarnak asks whether the stronger statement that individual eigenfunctions equidistribure is true. Formally, the statement is as follows.
Let $S$ be an arithmetic surface and $\phi _{j}$ be a sequence of functions on $S$ such that
$\Delta \phi _{j}=\lambda _{j}\phi _{j},\qquad \int _{S}\phi _{j}(x)^{2}\,dx=1.$
Then for any smooth, compactly supported function $\psi $ on $S$ we have
$\lim _{j\to \infty }\left(\int _{S}\psi (x)\phi _{j}(x)^{2}\,dx\right)=\int _{S}\psi (x)\,dx.$
This conjecture has been proven by E. Lindenstrauss[14] in the case where $S$ is compact and the $\phi _{j}$ are additionally eigenfunctions for the Hecke operators on $S$. In the case of congruence covers of the modular some additional difficulties occur, which were dealt with by K. Soundararajan.[15]
Isospectral surfaces
Main article: Isospectral
The fact that for arithmetic surfaces the arithmetic data determines the spectrum of the Laplace operator $\Delta $ was pointed out by M. F. Vignéras[16] and used by her to construct examples of isospectral compact hyperbolic surfaces. The precise statement is as follows:
If $A$ is a quaternion algebra, ${\mathcal {O}}_{1},{\mathcal {O}}_{2}$ are maximal orders in $A$ and the associated Fuchsian groups $\Gamma _{1},\Gamma _{2}$ are torsion-free then the hyperbolic surfaces $S_{i}=\Gamma _{i}\setminus \mathbb {H} ^{2}$ have the same Laplace spectrum.
Vignéras then constructed explicit instances for $A,{\mathcal {O}}_{1},{\mathcal {O}}_{2}$ satisfying the conditions above and such that in addition ${\mathcal {O}}_{2}$ is not conjugated by an element of $A$ to ${\mathcal {O}}_{1}$. The resulting isospectral hyperbolic surfaces are then not isometric.
Notes
1. Katok 1992.
2. Katok 1992, section 5.6.
3. Lubotzky, Alexander; Segal, Dan (2003). "Chapter 7". Subgroup growth. Birkhäuser.
4. Calegari, Danny (May 17, 2014). "A tale of two arithmetic lattices". Retrieved 20 June 2016.
5. Katok 1992, Chapter 5.
6. Borel, Armand (1981). "Commensurability classes and volumes of hyperbolic 3-manifolds". Ann. Scuola Norm. Sup.Pisa Cl. Sci. 8: 1–33.
7. Belolipetsky, Misha; Gelander, Tsachik; Lubotzky, Alexander; Shalev, Aner (2010). "Counting arithmetic lattices and surfaces". Ann. of Math. 172 (3): 2197–2221. arXiv:0811.2482. doi:10.4007/annals.2010.172.2197.
8. Sarnak, Peter (1982). "Class numbers of indefinite binary quadratic forms". J. Number Theory. 15 (2): 229–247. doi:10.1016/0022-314x(82)90028-2.
9. Katz, M.; Schaps, M.; Vishne, U. (2007). "Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups". J. Differential Geom. 76 (3): 399–422. arXiv:math.DG/0505007. doi:10.4310/jdg/1180135693.
10. Selberg, Atle (1965), "On the estimation of Fourier coefficients of modular forms", in Whiteman, Albert Leon (ed.), Theory of Numbers, Proceedings of Symposia in Pure Mathematics, vol. VIII, Providence, R.I.: American Mathematical Society, pp. 1–15, ISBN 978-0-8218-1408-6, MR 0182610
11. Roelcke, W. "Über die Wellengleichung bei Grenzkreisgruppen erster Art". S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1953/1955 (in German): 159–267.
12. Kim, H. H. (2003). With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. "Functoriality for the exterior square of $GL_{4}$ and the symmetric fourth of $GL_{2}$". J. Amer. Math. Soc. 16: 139–183. doi:10.1090/S0894-0347-02-00410-1.
13. Lubotzky, Alexander (1994). Discrete groups, expanding graphs and invariant measures. Birkhäuser.
14. Lindenstrauss, Elon (2006). "Invariant measures and arithmetic quantum unique ergodicity". Ann. of Math. 163: 165–219. doi:10.4007/annals.2006.163.165.
15. Soundararajan, Kannan (2010). "Quantum unique ergodicity for $\mathrm {SL} _{2}(\mathbb {Z} )\setminus {\mathcal {H}}$" (PDF). Ann. of Math. 172: 1529–1538. doi:10.4007/annals.2010.172.1529. JSTOR 29764647. MR 2680500.
16. Vignéras, Marie-France (1980). "Variétés riemanniennes isospectrales et non isométriques". Ann. of Math. (in French). 112 (1): 21–32. doi:10.2307/1971319. JSTOR 1971319.
References
• Katok, Svetlana (1992). Fuchsian groups. Univ. of Chicago press.
| Wikipedia |
\begin{document}
\preprint{PHYSICAL REVIEW A {\bf 83}, 032119 (2011); 85, 049904(E) (2012)} \author{A. A. Semenov} \email[E-mail address: ]{[email protected]}
\affiliation{Institut f\"ur Physik, Universit\"{a}t Rostock, Universit\"{a}tsplatz 3, D-18051 Rostock, Germany} \affiliation{Institute of Physics, National Academy of Sciences of Ukraine, Prospect Nauky 46, UA-03028 Kiev, Ukraine} \affiliation{Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine,\\ Vul. Metrologichna 14-b, UA-03680 Kiev, Ukraine} \author{W. Vogel} \affiliation{Institut f\"ur Physik, Universit\"{a}t Rostock, Universit\"{a}tsplatz 3, D-18051 Rostock, Germany}
\title{Fake violations of the quantum Bell-parameter bound}
\begin{abstract} Shortcomings of experimental techniques are usually assumed to diminish nonclassical properties of quantum systems. Here it is demonstrated that this standard assumption is not true in general. It is theoretically shown that the inability to resolve different photon numbers in photodetection may pseudo-increase a measured Bell parameter. Under proper conditions one even pseudo-violates the quantum Cirel'son bound of the Bell parameter, the corresponding density operator fails to be positive semi-definite. This paradox can be resolved by appropriate squash models. \end{abstract}
\pacs{03.65.Ud, 42.50.Xa, 42.65.Lm}
\maketitle
\section{Introduction} \label{Introduction}
Quantum mechanics is not a local realistic theory; this means that values of observables may not be predefined. This conclusion from the famous work by Einstein, Podolsky, and Rosen~\cite{EPR} still attracts a great deal of attention from physicists. Bell \cite{Bell} has proposed a formal framework for this discussion. He has formulated inequalities that are valid for the local realism, but they are violated by quantum physics. According to this, the Bell parameter is less than the value of $2$ for any local realistic theory. In the quantum world this parameter may exceed this threshold up to the level of $2\sqrt{2}$, known as the Cirel'son bound~\cite{Tsirelson}. The increasing interest in this subject has also stimulated a variety of experiments, see e.g.~\cite{Aspect}.
An intriguing question is whether the Bell parameter may exceed the Cirel'son quantum bound. Popescu and Rohrlich have considered the consequences of two axioms: nonlocality and relativistic causality~\cite{Popesku}. They have shown that as a consequence of the axioms, quantum mechanics appears as a particular representative of a more general theory. The latter includes the possibility of violating the Cirel'son quantum bound, with the maximum value of the Bell parameter being $4$. Another situation has been considered by Cabello~\cite{Cabello}. He has demonstrated that two qubits of a three-qubit system may also violate the Cirel'son quantum bound up to the level of $4$ based on the standard quantum theory.
The shortcomings of experimental techniques, including losses, noise, etc., play two different roles in Bell-type experiments. First, they lead to a decrease of the Bell parameter; even for the Bell states the violation is no longer maximal (see, e.g., \cite{Fedrizzi}). Second, small values of the detection efficiency are the subject of a loophole for local realism in Bell inequalities~\cite{Pearle}. For a discussion of other loopholes we refer the reader to~\cite{Kwiat}. The mentioned disadvantages also result in problems with the implementation of quantum-key distribution (QKD) protocols (cf., e.g., \cite{QuantumInf}).
A special example of such shortcomings is the impossibility to resolve between different numbers of photons. For on-off detectors it is only possible to distinguish between the presence and absence of detected photons. In addition, some standard experiments restrict the interpretation to low-dimensional Hilbert spaces, while the real electromagnetic field is characterized by an infinite-dimensional Hilbert space. In this case, the measurement procedure and further postprocessing in fact maps or squashes the higher-dimensional Hilbert space onto a low-dimensional one. If such a procedure is consistent, an appropriate squash model exists~\cite{squashing}.
The aim of this paper is to show that the straightforward measurement procedure in Bell-type experiments may result in a fake enhancement of nonclassical properties. Moreover, we predict that even the limits of quantum physics may be pseudoviolated and that the Bell parameter exceeds the quantum Cirel'son bound. In such cases, the reconstructed density operator fails to be positive semidefinite. Of course, such fake effects do not mean that we predict violations of quantum physics. Nevertheless, the correct postprocessing, which is consistent with a properly defined squash model, resolves this problem and yields acceptable results. A clear understanding of this problem is of importance for the security of quantum communication.
Our paper is organized as follows. In Sec.~\ref{BellTypeExperiment} we consider the fake violation of the Cirel'son bound of the Bell parameter by using on-off detectors. A resolution of this paradoxial result by a proper squash model is given in Sec.~\ref{DoubleClickEvents}. An alternative way of demonstrating the fake violation of quantum physics under study by the reconstruction of ill-defined two-qubit density operators is considered in Sec.~\ref{TwoQubitDensityOperator}. In Sec.~\ref{SummaryAndConclusions} we give a summary and some conclusions.
\section{Bell-type experiment} \label{BellTypeExperiment}
Let us start with a standard experimental setup briefly sketched in Fig.~\ref{Fig1}. The source of entangled photons irradiates into four modes: the horizontal and vertical polarized modes at site $A$ and similar modes at site $B$. Typically each polarization analyzer consists of a half-wave plate, which can change the polarization direction to the angles $\theta_\mathrm{A}$ and $\theta_\mathrm{B}$ at $A$ and $B$ sites, respectively, a polarization beam splitter, and two detectors for the reflected and transmitted modes.
\begin{figure}
\caption{ A typical experimental setup for checking the violation of Bell inequalities. The source $S$ produces entangled photon pairs. The polarization analyzers $A$ and $B$ consist of half-wave plates ($HWP$) polarizing beam splitters ($PBS$), and two pairs of detectors: $D_\mathrm{T_A}$ ($D_\mathrm{T_B}$) for the transmitted signal and $D_\mathrm{R_A}$ ($D_\mathrm{R_B}$) for the reflected signal at the $A$ site ($B$ site).}
\label{Fig1}
\end{figure}
Let $P_\mathrm{i_\mathrm{A}, i_\mathrm{B}}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)$ be the probability to get a click in one of the detectors $i_\mathrm{A}=\{\mathrm{T_A},\mathrm{R_A}\}$ at site $A$ and in one of the detectors $i_\mathrm{B}=\{\mathrm{T_B},\mathrm{R_B}\}$ at site $B$ for the angles of the polarization analyzers $\theta_\mathrm{A}$ and $\theta_\mathrm{B}$, respectively. The correlation coefficient is given by \begin{equation} E\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right) = \frac{P_\mathrm{same}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)-P_\mathrm{different}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)}{P_\mathrm{same}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)+P_\mathrm{different}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)},\label{correlation} \end{equation} where \begin{equation} P_\mathrm{same}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)=P_\mathrm{\mathrm{T_A}, \mathrm{T_B}}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)+P_\mathrm{\mathrm{R_A}, \mathrm{R_B}}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right), \end{equation} is the probability to get clicks on both detectors in the transmission channels or both detectors in the reflection channels, and \begin{equation} P_\mathrm{different}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)=P_\mathrm{\mathrm{T_A}, \mathrm{R_B}}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)+P_\mathrm{\mathrm{R_A}, \mathrm{T_B}}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right) \label{pdiff} \end{equation} is the probability to get clicks on the detectors in the transmission channel at one site and the reflection channel at another site. The Clauser-Horne-Shimony-Holt (CHSH) Bell-type inequality \cite{CHSH} states that the parameter \setlength\arraycolsep{0pt} \begin{eqnarray}
\mathcal{B}&=&\left|E\left(\theta_\mathrm{A}^{(1)}, \theta_\mathrm{B}^{(1)}\right)-E\left(\theta_\mathrm{A}^{(1)},
\theta_\mathrm{B}^{(2)}\right)\right|\label{BellParameter}
\\&+&\left|E\left(\theta_\mathrm{A}^{(2)}, \theta_\mathrm{B}^{(2)}\right)+E\left(\theta_\mathrm{A}^{(2)},
\theta_\mathrm{B}^{(1)}\right)\right|\nonumber, \end{eqnarray} also referred to as the Bell parameter, cannot exceed the value of $2$ for local-realistic theories. In quantum theory it is bounded by the Cirel'son bound of $2\sqrt{2}$. As shown below, this bound may be violated in the presence of experimental imperfections.
According to the photodetection theory \cite{PhotoDetection}, the probability $P_\mathrm{i_\mathrm{A}, i_\mathrm{B}}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)$ for on-off detectors is given by \begin{equation} P_\mathrm{i_\mathrm{A}, i_\mathrm{B}}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)=\sum\limits_{n,m=1}^{+\infty}\mbox{\rm Tr}\left( \hat{\Pi}_{i_\mathrm{A}}^{(n)}\hat{\Pi}_{i_\mathrm{B}}^{(m)} \hat{\Pi}_{j_\mathrm{A}}^{(0)}\hat{\Pi}_{j_\mathrm{B}}^{(0)}\hat\varrho\right), \label{Probability} \end{equation} $i_\mathrm{A}\neq j_\mathrm{A}$, $i_\mathrm{B}\neq j_\mathrm{B}$, where $\hat\varrho$ is the density operator of the light at the input ports of the polarization analyzers and \begin{equation} \hat{\Pi}_{i_\mathrm{A(B)}}^{(n)}=:\frac{\left(\eta\, \hat{n}_{i_\mathrm{A(B)}}\,+N_\mathrm{nc}\right)^n}{n!}\, \exp\left(-\eta\,\hat{n}_{i_\mathrm{A(B)}}-N_\mathrm{nc}\right):\label{POVM} \end{equation} is the positive operator-valued measure in the presence of non-unit efficiency $\eta$ and the mean number of noise counts $N_\mathrm{nc}$ (originated from the internal dark counts and the background radiation), $::$ means normal ordering; see \cite{Semenov}. For simplicity we assume the detection efficiencies and noise-count rates to be equal for all detectors. The photon-number operator in the channel $i_\mathrm{A(B)}$, $\hat{n}_{i_\mathrm{A(B)}}=\hat{a}^\dag_{i_\mathrm{A(B)}}\hat{a}_{ i_\mathrm{A(B)}}$ can be expressed by the horizontal and vertical modes, $\hat{a}_{\scriptscriptstyle \mathrm{H_{A(B)}}}$ and $\hat{a}_{\scriptscriptstyle \mathrm{V_{A(B)}}}$, respectively, using the input-output relations for the polarization analyzers: \begin{eqnarray} \hat{a}_{\scriptscriptstyle T_\mathrm{A(B)}}=\hat{a}_{\scriptscriptstyle \mathrm{H_{A(B)}}}\cos\theta_\mathrm{A(B)}+ \hat{a}_{\scriptscriptstyle \mathrm{V_{A(B)}}}\sin\theta_\mathrm{A(B)}\label{IOR1},\\ \hat{a}_{\scriptscriptstyle R_\mathrm{A(B)}}=-\hat{a}_{\scriptscriptstyle \mathrm{H_{A(B)}}}\sin\theta_\mathrm{A(B)}+ \hat{a}_{\scriptscriptstyle \mathrm{V_{A(B)}}}\cos\theta_\mathrm{A(B)}\label{IOR2}. \end{eqnarray}
In the case when the source $S$ generates the perfect Bell state, the Bell parameter $\mathcal{B}$, calculated by using Eqs.~(\ref{correlation})-(\ref{IOR2}), cannot exceed the Cirel'son bound of $2\sqrt{2}$. However, realistic sources produce, as a rule, more complicated states. Consider a parametric down-conversion (PDC) process of entangled-photon generation~\cite{Kwiat, Ma, Kok, Popescu}. Such a source, for example, has been recently used for transferring entanglement over a long-distance free-space channel~\cite{Fedrizzi}. The state
$\hat\varrho=\left|\Psi\right\rangle\left\langle\Psi\right|$ , emitted by the PDC source, is of the form~\cite{Ma, Kok} \begin{equation}
\left|\Psi\right\rangle=(\cosh\chi)^{-2}\sum\limits_{n=0}^{+\infty}
\sqrt{n+1}\tanh^n\chi\left|\Phi_n\right\rangle,\label{PDC1} \end{equation} where $\chi$ is the squeezing parameter and \begin{eqnarray}
&&\left|\Phi_n\right\rangle=\label{PDC2}\\ &&\frac{1}{\sqrt{n+1}}\sum\limits_{m=0}^{n}\left(-1\right)^m
\left|n-m\right\rangle_\mathrm{H_A}\left|m\right\rangle_\mathrm{V_A}
\left|m\right\rangle_\mathrm{H_B}\left|n-m\right\rangle_\mathrm{V_B}\nonumber. \end{eqnarray} For small $\chi$, one often restricts the consideration to $\mbox{n=1}$, representing a Bell state. Note that it was already mentioned in~\cite{Kwiat}, that in order to close loopholes for local realism one should discriminate between photon numbers while using such sources.
An analysis of higher-term contributions in~\cite{Kok, Popescu} consists of the restriction to a few terms in the series~(\ref{PDC1}). However, since this state is Gaussian, we may obtain the exact analytical expression for Eq.~(\ref{Probability}) which enables us to provide a strict analysis of the problem, as done in Ref.~\cite{Semenov2}. For simplicity, we suppose that the losses for all four modes are equal. It is possible to show that in this case one can include them into the detection losses and describe all the losses by the efficiency $\eta$ in Eq.~(\ref{POVM}).
After straightforward algebra (cf.~Ref.~\cite{Semenov2}) by using the state~(\ref{PDC1}) in Eq.~(\ref{Probability}), the probability $P_\mathrm{i_\mathrm{A}, i_\mathrm{B}}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)$ for the state~(\ref{PDC1}) can be written as \begin{eqnarray} P_\mathrm{i_\mathrm{A}, i_\mathrm{B}}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)&=&\left(1-\tanh^2\chi\right)^4e^{-4N_\mathrm{nc}} \label{ProbabilitySpecial}\\ &\times&\left[\frac{e^{2N_\mathrm{nc}}}{C_\mathrm{0}+2C_\mathrm{1}+C_\mathrm{ i_\mathrm { A },i_\mathrm{B}} }- \frac{2e^{N_\mathrm{nc}}}{C_\mathrm{0}+C_\mathrm{1}} +\frac{1}{C_\mathrm{0}}\right].\nonumber \end{eqnarray} Here \begin{equation} C_\mathrm{0}=\left\{\eta^2\tanh^2\chi- \left[1+\left(\eta-1\right)\tanh^2\chi\right]^2\right\}^2,\label{C0} \end{equation} \begin{eqnarray} C_\mathrm{1}&=&\eta\left(1-\eta\right)\left(1-\tanh^2\chi\right)\tanh^2\chi\label{C1} \\ &\times&\left\{\eta^2\tanh^2\chi- \left[1+\left(\eta-1\right)\tanh^2\chi\right]^2\right\},\nonumber \end{eqnarray} \begin{eqnarray} C_\mathrm{T_A,T_B}=C_\mathrm{R_A,R_B}&=&\eta^2\tanh^2\chi\left(1-\tanh^2\chi\right) ^2\label{Csame}\\ &\times&\left[\left(1-\eta\right)^2 \tanh^2\chi-\sin^2 \left(\theta_\mathrm{A}-\theta_\mathrm{B}\right)\right],\nonumber \end{eqnarray} \begin{eqnarray} C_\mathrm{T_A,R_B}=C_\mathrm{R_A,T_B}&=&\eta^2\tanh^2\chi\left(1-\tanh^2\chi\right) ^2\label{Cdifferent}\\ &\times&\left[\left(1-\eta\right)^2 \tanh^2\chi-\cos^2 \left(\theta_\mathrm{A}-\theta_\mathrm{B}\right)\right].\nonumber \end{eqnarray} Now we may insert Eq.~(\ref{ProbabilitySpecial}) in Eq.~(\ref{correlation}) and apply the result in Eq.~(\ref{BellParameter}) for the analysis of the Bell parameter.
In Fig.~\ref{Fig2} we show with solid lines the dependence of the maximal value of the Bell parameter $\mathcal{B}$ on the parameter $\tanh\chi$ for different values of the efficiency $\eta$ and for $N_\mathrm{nc}=10^{-6}$. For small losses this parameter exceeds the Cirel'son bound of $2\sqrt{2}$, representing a fake violation of quantum physics. This originates from a straightforward but inconsistent postprocessing of the data~\cite{Lo}.
\begin{figure}
\caption{ (Color online) Maximal value of the Bell parameter $\mathcal{B}$, obtained for on-off (solid lines) and photon-number-resolving (dashed lines) detectors, vs the parameter $\tanh\chi$ for the mean value of noise counts $N_\mathrm{nc}=10^{-6}$ and the efficiencies (a) $\eta=0.9$, (b) $\eta=0.6$, and (c) $\eta=0.4$. The minimum at $\tanh\chi=0$ is caused by noise counts.}
\label{Fig2}
\end{figure}
The effect of fake violations of the quantum Bell-parameter bound does not occur when one uses photon-number-resolving detectors, for which Eq.~(\ref{Probability}) for the probability $P_\mathrm{i_\mathrm{A}, i_\mathrm{B}}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)$ should be rewritten as \begin{equation} P_\mathrm{i_\mathrm{A}, i_\mathrm{B}}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)=\mbox{\rm Tr}\left( \hat{\Pi}_{i_\mathrm{A}}^{(1)}\hat{\Pi}_{i_\mathrm{B}}^{(1)} \hat{\Pi}_{j_\mathrm{A}}^{(0)}\hat{\Pi}_{j_\mathrm{B}}^{(0)}\hat\varrho\right). \label{Probability1} \end{equation} In this case, Eq.~(\ref{ProbabilitySpecial}) is replaced by \begin{eqnarray} P_\mathrm{i_\mathrm{A}, i_\mathrm{B}}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)&=& \left(1-\tanh^2\chi\right)^4 e^{-4 N_\mathrm{nc}} \label{ProbabilitySpecialPNR}\\ &\times&\left[2\frac{C_1^2} {C_\mathrm{0}^3} -\frac{C_\mathrm{i_\mathrm{A},i_\mathrm{B}}}{C_\mathrm{0} ^2}- 2N_\mathrm{nc}\frac{C_1}{C_\mathrm{0}^2} +N_\mathrm{nc}^2\frac{1}{C_\mathrm{0}} \right].\nonumber \end{eqnarray} In addition, results obtained with photon-number-resolving detectors demonstrate, as a rule, smaller values of the Bell parameter compared with on-off detectors; see Fig.~\ref{Fig2} and Ref.~\cite{Semenov2} for details of calculations.
It is worth noting that photocounting with the resolution of photon numbers is approximately possible by splitting an initial light beam into a large number of low-intensity beams~\cite{ArrayDetectors}. Detection of photons in each beam by an array of photodiodes allows one to get more insight into the number statistics of the detected photons. Similarly, one can use time multiplexing in a fiber loop with one photodiode~\cite{LoopDetectors}. However, such measurement techniques only partly allow one to infer the photon number statistics. The question remains whether or not such detection schemes are suited to completely eliminate the fake effects under study. This would require a more detailed analysis, which is beyond the scope of the present paper.
\section{Double-click events} \label{DoubleClickEvents}
In the following we will show that one can overcome the violation of the Cirel'son bound, even when using on-off detectors. A consistent result can be obtained by the application of a proper squash model~\cite{squashing}. The point is that the straightforward scheme considered above discards so-called double-click events when both detectors at the receiver station $A(B)$ click~\cite{Lo}.
Let us assign to the double-click events random bits with a probability $1/2$. This changes the situation completely. In this case, Eq.~(\ref{Probability}) for the probability $P_\mathrm{i_\mathrm{A}, i_\mathrm{B}}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)$ is replaced by \begin{eqnarray} P_\mathrm{i_\mathrm{A}, i_\mathrm{B}}\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)=\sum\limits_{n,m=1}^{+\infty}\mbox{\rm Tr}\left( \hat{\Pi}_{i_\mathrm{A}}^{(n)}\hat{\Pi}_{i_\mathrm{B}}^{(m)} \hat{\Pi}_{j_\mathrm{A}}^{(0)}\hat{\Pi}_{j_\mathrm{B}}^{(0)} \hat\varrho\right)\nonumber\\ +\frac{1}{2}\sum\limits_{n,m,k=1}^{+\infty}\mbox{\rm Tr}\left( \hat{\Pi}_{i_\mathrm{A}}^{(n)}\hat{\Pi}_{i_\mathrm{B}}^{(m)} \hat{\Pi}_{j_\mathrm{A}}^{(k)}\hat{\Pi}_{j_\mathrm{B}}^{(0)} \hat\varrho\right)\nonumber\\ +\frac{1}{2}\sum\limits_{n,m,k=1}^{+\infty}\mbox{\rm Tr}\left( \hat{\Pi}_{i_\mathrm{A}}^{(n)}\hat{\Pi}_{i_\mathrm{B}}^{(m)} \hat{\Pi}_{j_\mathrm{A}}^{(0)}\hat{\Pi}_{j_\mathrm{B}}^{(k)} \hat\varrho\right)\nonumber\\ +\frac{1}{4}\sum\limits_{n,m,k,l=1}^{+\infty}\mbox{\rm Tr}\left( \hat{\Pi}_{i_\mathrm{A}}^{(n)}\hat{\Pi}_{i_\mathrm{B}}^{(m)} \hat{\Pi}_{j_\mathrm{A}}^{(k)}\hat{\Pi}_{j_\mathrm{B}}^{(l)}\hat\varrho\right). \label{ProbabilitySquash} \end{eqnarray}
For the sake of simplicity, in the following, we only consider the most critical case of lossless detectors, $\eta=1$, with no noise counts, $N_\mathrm{nc}=0$. Under these conditions the violation of the quantum Cirel'son bound, as considered in the previous section, attains its maximum. Substituting state~(\ref{PDC1}) into Eqs.~(\ref{correlation}) and (\ref{ProbabilitySquash}), one gets \begin{equation} E\left(\theta_\mathrm{A}, \theta_\mathrm{B}\right)=-\frac{\cos\left[ 2\left(\theta_\mathrm{A}- \theta_\mathrm{B}\right)\right]}{D},\label{correlation_squash} \end{equation} where \begin{eqnarray} D&=&1-\frac{1}{2}\tanh^2 \chi\sin^{2}\left[2\left(\theta_\mathrm{A}- \theta_\mathrm{B}\right)\right]\label{D_on-off}\\ &+&\frac{9+3\left(1-\tanh^2\chi\right)}{2\tanh^2\chi\left(1-\tanh^2\chi\right)^2} \nonumber\\ &\times&\left\{1-\tanh^2\chi+\frac{1}{4}\tanh^4\chi\sin^{2}\left[2\left(\theta_\mathrm { A}- \theta_\mathrm{B}\right)\right]\right\}\nonumber\\ &+&\frac{\big[1-2\left(1-\tanh^2\chi\right)^2\big]\big(2-\tanh^2\chi\big)} {\tanh^2\chi\left(1-\tanh^2\chi\right)^2}.\nonumber \end{eqnarray} By applying the squash model in this way, the maximum value of the Bell parameter no longer exceeds the Cirel'son bound of $2\sqrt{2}$, see.~Fig.~\ref{Fig3}. This result demonstrates the importance of consistent squash models. In particular, it is necessary to include the double-click events into the data postprocessing.
\begin{figure}
\caption{ (Color online) Maximal value of the Bell parameter $\mathcal{B}$, obtained for lossless on-off detectors without noise counts.}
\label{Fig3}
\end{figure}
\section{Two-qubit density operator} \label{TwoQubitDensityOperator}
In order to demonstrate the inconsistency of the straightforward postprocessing when double-click events are ignored, we provide the following argument. Any density operator $\hat\rho$ of a two-qubit system, with zero mean values of all spin projections, can be expanded into a nonorthogonal and linearly independent basis as \begin{equation} \hat\rho=\frac{\hat{I}\otimes\hat{I}}{4}+\frac{1}{4}\sum\limits_{i,j=1}^{3} E\left(\theta^{(i)}_\mathrm{A},\varphi^{(i)}_{\mathrm{A}}; \theta^{(j)}_\mathrm{B},\varphi^{(j)}_{\mathrm{B}}\right) \hspace{0.3em} \hat{\Xi}^{(i)}_\mathrm{A}\!\otimes\hat{\Xi}^{(j)}_\mathrm{B},\label{2QB} \end{equation} where $\hat{I}$ is the $2\times 2$ identity matrix. $E$ denotes the correlation coefficients for spin projections with the Euler angles $\theta^{(i)}_\mathrm{\scriptscriptstyle A(B)},\varphi^{(i)}_\mathrm{\scriptscriptstyle A(B)}$ $(i=1,2,3)$. The angles $\varphi^{(i)}_\mathrm{\scriptscriptstyle A(B)}$ have been introduced to obtain a linear-independent basis of the matrix \begin{equation} \hat{\Xi}^{(i)}_\mathrm{\scriptscriptstyle A(B)}=\sum\limits_{k=1}^{3}g^{(k,i)}\left(\begin{array}{cc} \cos 2\theta^{(k)}_\mathrm{\scriptscriptstyle A(B)} & e^{-i\varphi_\mathrm{\scriptscriptstyle A(B)}^{(k)}} \sin 2\theta^{(k)}_\mathrm{\scriptscriptstyle A(B)}\\ e^{i\varphi_\mathrm{\scriptscriptstyle A(B)}^{(k)}} \sin 2\theta^{(k)}_\mathrm{\scriptscriptstyle A(B)}& -\cos 2\theta^{(k)}_\mathrm{\scriptscriptstyle A(B)} \end{array} \right), \end{equation} where $g^{(k,i)}$ is the metric tensor, which is inverse to \begin{eqnarray} \left(g^{-1}\right)_{(k,i)}&&=\cos 2\theta^{(i)}_\mathrm{\scriptscriptstyle A(B)}\cos 2\theta^{(k)}_\mathrm{\scriptscriptstyle A(B)}\\&&+ \sin 2\theta^{(i)}_\mathrm{\scriptscriptstyle A(B)}\sin 2\theta^{(k)}_\mathrm{\scriptscriptstyle A(B)}\cos\left(\varphi_\mathrm{\scriptscriptstyle A(B)}^{(i)}-\varphi_\mathrm{\scriptscriptstyle A(B)}^{(k)}\right).\nonumber \end{eqnarray}
The above-discussed mapping onto a two-qubit state implies the identification of the coefficients $E\left(\theta^{(i)}_\mathrm{A},\varphi^{(i)}_{\mathrm{A}}; \theta^{(j)}_\mathrm{B},\varphi^{(j)}_{\mathrm{B}}\right)$ in Eq.~(\ref{2QB}) with relation~(\ref{correlation}), as is usually done in experiments. To include the dependencies on the angles $\varphi^{(i)}_\mathrm{\scriptscriptstyle A(B)}$ in the experimental setup (see~Fig.~\ref{Fig1}), one should complete it with phase shifters for the horizontal polarized modes. In this case one must replace $\hat{a}_{\scriptscriptstyle \mathrm{H_{A(B)}}}$ with $\hat{a}_{\scriptscriptstyle \mathrm{H_{A(B)}}}\exp\left(i \varphi_{\scriptscriptstyle \mathrm{{A(B)}}}^{(i)}\right)$ in Eqs.~(\ref{IOR1}) and (\ref{IOR2}). In the case of using on-off detectors without considering double-click events, Eq.~(\ref{2QB}) no longer represents a correctly defined density operator for a two-qubit system. First, it is not uniquely defined since it depends on the chosen angles $\theta^{(i)}_\mathrm{\scriptscriptstyle A(B)}$ and $\varphi_\mathrm{\scriptscriptstyle A(B)}^{(i)}$, which represent different choices of the basis. Second and most importantly, for some values of the angles, operator~(\ref{2QB}) appears not to be positive semidefinite, which contradicts the fundamental properties of a quantum state.
In Fig.~\ref{Fig4} we show the minimum eigenvalue of the two-qubit density operator $\hat \rho$ inferred with on-off detectors without considering double-click events. It is clearly seen that for different choices of the angles the minimum eigenvalue of $\hat \rho$ can be either non-negative or it may become negative. In one of the cases, the mapping onto a two-qubit system clearly fails: the associated effective state does not show the properties of a correctly defined quantum state. As a consequence, the Bell parameter~(\ref{BellParameter}) may even exceed the quantum Cirel'son bound of $2\sqrt{2}$. Equation~(\ref{2QB}) can be used for reconstructing a two-qubit density operator by using the measured correlation coefficients. If the result of this reconstruction is not positive semidefinite, it fails to describe a quantum state. This is an alternative possibility to demonstrate the fake violation of quantum physics.
\begin{figure}
\caption{ (Color online) The minimum eigenvalue of the density operator $\hat{\rho}$ [see~Eq.~(\ref{2QB})] vs the parameter $\tanh \chi$ for the mean value of noise counts $N_\mathrm{nc}=10^{-6}$, the efficiency $\eta=0.6$, and the following sets of angles: (a) $\theta^{(1)}_\mathrm{\scriptscriptstyle A}=\theta^{(1)}_\mathrm{\scriptscriptstyle B}=\pi/4$, $\varphi_\mathrm{\scriptscriptstyle A}^{(1)}=\varphi_\mathrm{\scriptscriptstyle B}^{(1)}=0$, $\theta^{(2)}_\mathrm{\scriptscriptstyle A}=\theta^{(2)}_\mathrm{\scriptscriptstyle B}=\pi/4$, $\varphi_\mathrm{\scriptscriptstyle A}^{(2)}=\varphi_\mathrm{\scriptscriptstyle B}^{(2)}=\pi/2$, $\theta^{(3)}_\mathrm{\scriptscriptstyle A}=\theta^{(3)}_\mathrm{\scriptscriptstyle B}=0$, $\varphi_\mathrm{\scriptscriptstyle A}^{(3)}=\varphi_\mathrm{\scriptscriptstyle B}^{(3)}=0$; (b) $\theta^{(1)}_\mathrm{\scriptscriptstyle A}=\pi/8$, $\varphi_\mathrm{\scriptscriptstyle A}^{(1)}=0$, $\theta^{(2)}_\mathrm{\scriptscriptstyle A}=9\pi/4$, $\varphi_\mathrm{\scriptscriptstyle A}^{(2)}=\pi/2$, $\theta^{(3)}_\mathrm{\scriptscriptstyle A}=\pi$, $\varphi_\mathrm{\scriptscriptstyle A}^{(3)}=0$, $\theta^{(1)}_\mathrm{\scriptscriptstyle B}=3\pi/15$, $\varphi_\mathrm{\scriptscriptstyle B}^{(1)}=0$, $\theta^{(2)}_\mathrm{\scriptscriptstyle B}=-\pi/24$, $\varphi_\mathrm{\scriptscriptstyle B}^{(2)}=\pi/2$, $\theta^{(3)}_\mathrm{\scriptscriptstyle B}=\pi$, $\varphi_\mathrm{\scriptscriptstyle B}^{(3)}=0$.}
\label{Fig4}
\end{figure}
\section{Summary and Conclusions} \label{SummaryAndConclusions}
Our analysis of biphoton Bell-type experiments has demonstrated that the impossibility of discriminating between different photon numbers is a kind of imperfection leading to surprising results. First, it leads to a pseudoincrease of the measured Bell parameter. Second, we also predict that for high detection efficiency the measured Bell parameter can even show a pseudo-violation of the fundamental limit of quantum physics, the Cirel'son bound.
The reason for such fake violations is shown to consist in mappings of a continuous-variable quantum state onto an ill-defined two-qubit state that fails to be positive semidefinite. Such fake effects disappear when one includes in the consideration a consistent postprocessing of the measured data by a proper squash model. The knowledge of such effects plays an important role in the analysis of some quantum-key-distribution protocols.
\acknowledgements
The authors gratefully acknowledge support by the Deutsche Forschungsgemeinschaft through SFB 652. We also strongly appreciate useful discussions with Hoi-Kwong Lo.
\end{document} | arXiv |
\begin{definition}[Definition:Cumulative Distribution Function]
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.
The '''cumulative distribution function''' (or '''c.d.f.''') of $X$ is denoted $F_X$, and defined as:
:$\forall x \in \R: \map {F_X} x := \map \Pr {X \le x}$
\end{definition} | ProofWiki |
\begin{definition}[Definition:Cartesian Product/Coordinate]
Let $\ds \prod_{i \mathop \in I} S_i$ be a cartesian product.
Let $j \in I$, and let $s = \sequence {s_i}_{i \mathop \in I} \in \ds \prod_{i \mathop \in I} S_i$.
Then $s_j$ is called the '''$j$th coordinate of $s$'''.
If the indexing set $I$ consists of ordinary numbers $1, 2, \ldots, n$, one speaks about, for example, the '''first''', '''second''', or '''$n$th coordinate'''.
For an element $\tuple {s, t} \in S \times T$ of a binary cartesian product, $s$ is the '''first coordinate''', and $t$ is the '''second coordinate'''.
\end{definition} | ProofWiki |
What is significant about the half-sum of positive roots?
I apologize for the somewhat vague question: there may be multiple answers but I think this is phrased in such a way that precise answers are possible.
Let $\mathfrak{g}$ be a semisimple Lie algebra (say over $\mathbb{C}$) and $\mathfrak{h} \subset \mathfrak{g}$ a Cartan subalgebra. All the references I have seen which study the representation theory of $\mathfrak{g}$ in detail make use of the half-sum of positive roots, which is an element of $\mathfrak{h}^\ast$: e.g. Gaitsgory's notes on the category O introduce the "dotted action" of the Weyl group on $\mathfrak{h}^\ast$, the definition of which involves this half-sum.
Is there a good general explanation of why this element of $\mathfrak{h}^\ast$ is important? The alternative, I suppose, is that it is simply convenient in various situations, but this is rather unsatisfying.
rt.representation-theory lie-algebras
Will Jagy
Justin CampbellJustin Campbell
$\begingroup$ I don't why the math won't display properly: the code looks fine to me. Maybe someone who knows more about LaTeX and/or MathOverflow can fix it. $\endgroup$ – Justin Campbell Nov 5 '11 at 20:56
$\begingroup$ By itself, the asterisk is a control character, to display one use \ast in Latex. Wow, little earthquake just now. 14:52 PST. $\endgroup$ – Will Jagy Nov 5 '11 at 21:53
$\begingroup$ earthquake.usgs.gov/earthquakes/recenteqscanv/FaultMaps/… $\endgroup$ – Will Jagy Nov 5 '11 at 21:56
$\begingroup$ This question and the wonderful answers are a really great example of why MO is good. $\endgroup$ – Mariano Suárez-Álvarez Nov 9 '11 at 2:32
$\begingroup$ I hope this very good question is not closed yet. I have heard that $\rho$ of $G$ can be related to the curvature of the homogenous space $G/H$, where $H$ is closed subgroup of $G$. I am interested in particular when $G$ is noncompact semisimple Lie group and $H$ is its maximal compact subgroup. Can anybody elaborate on this.. please? $\endgroup$ – spr Jun 30 '12 at 5:15
I don't think there is a one-line answer to this question, since it depends a lot on the direction from which you approach semi-simple Lie theory. For one thing, it's probably best at first to emphasize just integral weights, among which the dominant ones parametrize irreducible finite dimensional representations. Here the weight usually denoted $\rho$ plays a ubiquitous role in the classical Weyl theory, but that too can be developed in a number of different ways. (There was some early experimentation with the notation; the alternative symbol $\delta$ also had widespread use before the Bourbaki preference for $\rho$ started to take over in 1968.)
While it's important in proofs of the Weyl character formula to view $\rho$ as the half-sum of positive roots (given a fixed positive or simple system), it's also essential to identify it with the sum of fundamental dominant weights for many purposes. In this guise it's the smallest regular dominant weight, fixed by no element of the Weyl group except the identity. When passing from integral weights to line bundles on an associated flag variety $G/B$ (with $B$ a Borel subgroup associated to positive roots relative to a fixed maximal torus which it contains), the weight $\rho$ has the distinction of defining an ample line bundle. This property is crucial in geometric approaches to Weyl's formula, as well as in spin-offs in prime characteristic due to Andersen and others.
Ultimately the importance of the weight $\rho$ is probably appreciated best in the setting of representation theory, where the finite dimensional theory is enriched by treatment of highest weight modules in more generality and the shift by $\rho$ is again ubiquitous. By the way, the convenient "dot" notation $w \cdot \lambda := w(\lambda +\rho) - \rho$ is apparently due to Robert Moody. In the earlier literature the more awkward full notation appears, or else is replaced in the Paris notation by a hidden $\rho$-shift.
None of what I've said is a complete answer to the question asked, but in any case it's more than a matter of "convenience" to emphasize $\rho$.
edited Nov 6 '11 at 0:58
Joseph O'Rourke
Jim HumphreysJim Humphreys
$\begingroup$ Note that for Kac-Moody groups, where there are infinitely many positive roots, one still defines $\rho$ as the sum of the fundamental weights. $\endgroup$ – Allen Knutson Nov 8 '11 at 1:21
$\begingroup$ @Allen: Yes, there's a lot more to be said in that direction. The two ways of looking at $\rho$ come up classically in the Weyl denominator formula, which Kac creatively generalized to certain representations of symnmetrizable Kac-Moody algebras (though in that setting $\rho$ is not quite uniquely determined). $\endgroup$ – Jim Humphreys Nov 8 '11 at 13:46
From the point of view of geometry, the crucial fact about $\rho$ is that the corresponding line bundle on the flag manifold is (upt to a sign) a (the) square-root of the canonical bundle (top exterior power of $T^*_B G/B \simeq b_- $ is the sum of the negative roots). This is of course equivalent to Alain Valette's description in terms of the modular character of the Borel. In other words its sections in the real world are half-densities (things for which we can define the $L^2$ inner product).
It is a universal fact about passage from the classical world to the quantum world (in particular the geometric construction of representations) involves a shift by the square root of the canonical bundle. There are many ways to explain or motivate this. For example if we seek unitary representations we need to be able to define an $L^2$ inner product, which means considering not sections of the bundle we might have expected but sections times half-densities (again this is Alain's answer restated). From the point of view of rings of differential operators, the adjoint of a differential operator acting on functions (or on sections of a bundle $L$) is invariantly not another diffop (on $L$) but a differential operator acting on volume forms (or on sections of $L$ tensor the canonical bundle) --- so the self dual twist of differential operators is by half-forms, ie $\rho$-shifted. (Put another way, Serre duality is a reflection centered at half-forms!)
My favorite explanation is in Beilinson-Bernstein's Proof of Jantzen Conjectures and doesn't involve self-adjointness or unitarity: it's a consistency condition for deformation quantization of symbols (functions on the cotangent bundle): if you want this deformation quantization to be correctly normalized to order two (this is not the right question to go into that) you find you need to look at differential operators twisted by half-forms, not functions. On the flag variety this means a $\rho$-shift, and from the D-module POV on representation theory this is one fundamental place where that shift is forced on you, independent of thinking of inner products. This is in particular one way to see why it comes up in the Weyl character formula, through the geometric proof via Atiyah-Bott or via the BGG resolution, both of which involve the geometry of the flag variety.
David Ben-ZviDavid Ben-Zvi
This is actually a fairly deep question. Your suspicion that there may be multiple answers is correct, but there might be some surprising connections between seemingly unrelated answers. Let me give one possible thread of explanation. The underlying principle is that the appearance of $\rho$ and the "dot" action $w\cdot\lambda=w(\lambda+\rho)-\rho$ in representation theory is closely related to the geometry of the flag variety.
One of the first places one meets $\rho$ (and the dot action) is in the Weyl character formula. A theorem of Kostant shows that the formula can be written as the ratio of two Lie algebra cohomology Euler characteristics. From this perspective, the appearance $w \cdot \lambda$ and $w\cdot0$ in the WCF is ultimately explained by the fact that these are the weights appearing in the weight space decomposition of the relevant Lie algebra cohomology modules, namely $H^*(\mathfrak n, V^\lambda)$ and $H^\ast(\mathfrak n, V^0)$, where $\mathfrak n = \bigoplus_{\alpha>0} \mathfrak g_\alpha$ and $V^\mu$ denotes the irrep of highest weight $\mu$.
We can rephrase this in geometric terms by invoking the "geometric analogue" of Kostant's theorem, i.e. the Borel–Weil–Bott theorem. Kostant's description of the Lie algebra cohomology of $\mathfrak n = \mathfrak g /\mathfrak b^-$ with coefficients in an irrep translates into a representation-theoretic description of the sheaf cohomology of certain line bundles $L_\lambda$ (constructed using integral weights $\lambda$) over the flag variety $G/B^-$ of $\mathfrak g$. Consequently, the dot action shows up in this description, and this time it's accompanied by a shift in degree. This in turn can be explained by Serre duality; the key fact is that canonical bundle of $G/B^-$ turns out to be $L_{-2\rho}$.
So, in some sense, the appearance of $\rho$ and the dot action in the WCF can be thought of as a manifestation of Serre duality.
[N.B. This is a condensed version of my lengthy original answer. The old version can be found in the edit history.]
122 silver badges33 bronze badges
FaisalFaisal
While I appreciate Dave Ben-Zvi's half-densities answer, I'm going to put forth the contrary opinion that it's largely a bookkeeping artifact.
The most familiar place that $\rho$ shows up is in the WCF of the irrep $V$ with highest weight $\lambda$, $$ Tr(t|_{V}) = \frac{\sum_w t^{w(\lambda+\rho)-\rho}}{\prod_{\Delta_+} (1-t^{-\beta})}. $$
This version of WCF is good for suggesting the existence of the BGG resolution, or for taking the Fourier transform of and obtaining the Kostant multiplicity formula. But otherwise, I claim that that it's the worst way to write it down, and suggest instead $$ Tr(t|_V) = \sum_w w \cdot \frac{t^{\lambda}}{\prod_{\Delta_+} (1-t^{-\beta})}. $$ Hurray, it's manifestly $W$-invariant, and no $\rho$ in sight! This is the natural version that one obtains by applying the Atiyah-Bott-Riemann-Roch-Lefschetz Woods Hole localization formula to the flag manifold $G/B$, as A&B mention in their paper.
You really notice it if you try to write down a WCF for nonregular weights, which corresponds to applying the localization formula to a partial flag manifold $G/P$. Then you can no longer flip weights to put everything over the same denominator, so the first version is badly broken. The second, $W$-invariant, version works just fine in this case (the denominator is a product over only part of $\Delta_+$).
EDIT: I suppose it's too strong to say it's badly broken. It's just that it's not in lowest terms.
Allen KnutsonAllen Knutson
$\begingroup$ A general remark: You get that formula if you apply A–B to the $\overline\partial$ operator acting on $\Omega^{(0,q)}(G/B,L_\lambda)$. In another paper Bott mentions that you can make the other formula for the Weyl denom (the one with $\rho$ in it!) show up if you approach things differently: $G/B$ is spin, so we can work with its (elliptic) Dirac operator $S^+ \to S^-$. Incidentally, the existence of a spin structure on $G/B$ is related to $\rho$: the canonical bundle of $G/B$ is $L_{-2\rho}$ so a holomorphic square root is given by $L_{-\rho}$, and this determines the spin structure. $\endgroup$ – Faisal Nov 8 '11 at 4:11
$\begingroup$ This is the remark also made by Bott in (1988, eqs (17, 28, 30)), right? $\endgroup$ – Francois Ziegler May 14 '19 at 21:57
Let $G=KAN$ be the Iwasawa decomposition of a semi-simple Lie group $G$. Then the modular function of the Borel subgroup (= minimal parabolic) $B=MAN$ is $\Delta_B(man)=e^{2\rho(\log a)}$ where $\rho$ is half the sum of the positive roots of the root system $\Delta(\mathfrak{g},\mathfrak{a})$.
This is relevant for the definition of the principal series of representations of $G$, say in the compact picture: for $\nu\in i\mathfrak{a}^*$, the Hilbert space of $Ind_B^G(1\otimes e^\nu\otimes 1)$ is $L^2(K/M)$, with action given by $(\pi_\nu(g)f)(k)=e^{-(\nu+\rho)H(g^{-1}k)}f(\kappa(g^{-1}k))$, where $g=\kappa(g)e^{H(g)}n$ in the Iwasawa decomposition. So $e^{-\rho}$ appears as the square root of the Radon-Nikodym cocycle, needed to make the representation unitary, since the measure on $G/B=K/M$ is not $G$-invariant.
For more on this, see sections 5.6 and 7.1 in A.W. Knapp, Representation theory of semisimple groups (an overview based on examples), Princeton MAth. Series 36, 1986.
Alain ValetteAlain Valette
Well, no one's explicitly talked about the relevance of spin structures to this story yet, so here's a sketch of the story as I understand it. For references see, for example, the nLab. I'll be blithely ignoring the difference between algebraic and topological K-theory, and $B$ denotes a positive Borel.
Roughly speaking, the Borel-Weil-Bott story is about constructing representations of $G$ by inducing them from $1$-dimensional representations of $B$. A geometric avatar of a finite-dimensional representation of $B$ is a $G$-equivariant vector bundle on $G/B$; in fact, this is an equivalence of categories, and so there is a natural isomorphism
$$K_G(G/B) \cong K_B(\text{pt}) \cong R(B)$$
from the equivariant K-theory $K_G(G/B)$ to the Grothendieck group of finite-dimensional representations of $B$. Similarly, there is a natural isomorphism
$$\text{Pic}_G(G/B) \ni L_{\lambda} \leftrightarrow \lambda \in \Lambda$$
from the group of $G$-equivariant line bundles on $G/B$ to the group of $1$-dimensional representations of $B$, which in turn can be identified with the weight lattice $\Lambda$.
In this language, induction can be interpreted as an attempt to construct a pushforward
$$K_G(G/B) \to K_G(\text{pt}) \cong R(G)$$
in equivariant K-theory from $G/B$ to a point; these kinds of pushforwards are one language for talking about geometric quantization. In algebraic geometry, the pushforward in K-theory to a point is given by taking sheaf cohomology, and Borel-Weil-Bott tells us exactly what happens when we push forward equivariant line bundles this way.
We might ask how we can take pushforwards in K-theory purely topologically, though. Recall, for example, that we can take the pushforward in cohomology of a map between compact oriented manifolds using Poincare duality. The analogous statement for real resp. complex K-theory is that we can take the pushforward of a map between compact oriented manifolds equipped with spin resp. complex spin structure. In both cases, the pushforward to a point is given by taking the index of a suitable Dirac operator. I believe all of this continues to be true equivariantly.
The happy fact about the algebro-geometric setting is that almost complex structures canonically induce complex spin structures; the corresponding Dirac operator is built from the Dolbeault operator, and with suitable hypotheses pushforward to a point is given by taking sheaf cohomology computed as Dolbeault cohomology.
What is the relevance of the Weyl vector to this story? Any complex spin structure has associated to it a canonical complex line bundle $\omega$. If we start with an almost complex structure, then $\omega$ is the canonical bundle. Then a choice of spin structure compatible with a complex spin structure is equivalent to a choice of square root $\sqrt{\omega}$. In our case, under the identification of $G$-equivariant vector bundles on $G/B$ with representations of $B$, we have
$$T(G/B) \mapsto \mathfrak{g}/\mathfrak{b}$$
where the latter has the adjoint action of $B$. This breaks up into a direct sum of $1$-dimensional weight spaces, one for each negative root, and hence under the identification of $G$-equivariant line bundles on $G/B$ with the weight lattice $\Lambda$, we have
$$\omega \mapsto 2 \rho.$$
Since $\Lambda$ is torsion-free, the isomorphism class of the square root $\sqrt{\omega}$ is unique, and as an element of $\Lambda$ it is precisely $\rho$.
So $\rho$ is special in this story because it represents the unique square root of the canonical bundle. From this perspective, the dot action
$$w \cdot \lambda = w (\rho + \lambda) - \rho$$
naturally arises as follows. Complex spin structures are canonically a torsor over complex line bundles; if $L$ is a complex line bundle, the action on complex spin structures modifies the canonical bundle by
$$\omega \mapsto \omega \otimes L^{\otimes 2}.$$
I again believe this continues to be true equivariantly, and so $G$-equivariant complex spin structures on $G/B$ are canonically a torsor over the weight lattice $\Lambda$. So we can noncanonically identify the two via
$$\lambda \mapsto \omega \otimes L_{\lambda}^{\otimes 2} \mapsto 2 \rho + 2 \lambda$$
where $\omega \otimes L_{\lambda}^{\otimes 2}$ denotes a complex spin structure, not just the corresponding canonical line bundle (and in particular it can be different from the complex spin structure given by $\omega$ even if $L_{\lambda}^{\otimes 2}$ is trivial, although that doesn't happen here). If $\lambda \in \Lambda$ is a weight, the pushforward of $L_{\lambda}$ with respect to $\omega$ can be identified with the pushforward of the trivial line bundle with respect to $\omega \otimes L_{\lambda}^{\otimes 2}$.
Now pick a maximal compact $K$ and identify $G/B$ with $K/T$, where $T = K \cap B$ is a maximal torus in $K$. Then the Weyl group $W = N_K(T)/T$ naturally acts on $K/T$; in fact it is precisely the $K$-equivariant automorphism group of $K/T$. This induces an action on $K$-equivariant line bundles, identified with characters of $T$, identified with the weight lattice $\Lambda$, which is the usual non-dot action.
The Weyl group also acts on $K$-equivariant complex spin structures on $K/T$, and this action is compatible with the torsor structure above, as well as with the map sending a complex spin structure to its canonical bundle. It is not compatible with the noncanonical identification between $K$-equivariant complex spin structures and $\Lambda$ above: instead, writing $2 \rho + 2 \lambda$ for the complex spin structure corresponding to $\omega \otimes L_{\lambda}^{\otimes 2}$, the Weyl group action is
$$w(2 \rho + 2 \lambda) = 2 \rho + 2 \left( w(\rho + \lambda) - \rho \right)$$
and using the noncanonical identification again we precisely recover the dot action! So, to summarize:
Geometrically, while the usual action of $W$ on $\Lambda$ is the natural action of $W$ on $K$-equivariant complex line bundles on $K/T$, the dot action is the natural action on $K$-equivariant complex spin structures on $K/T$; the Weyl vector $\rho$ appears when relating these because it corresponds to a distinguished $K$-equivariant complex spin structure associated to a choice of positive roots. It is the pushforward in K-theory with respect to such spin structures which lets us construct representations of $K$ from representations of $T$.
Qiaochu YuanQiaochu Yuan
I don't quite have in mind any construction that fully singles out this element, which is called the Weyl element $\rho$, as the most important one. To an extent you don't even expect that in the non-simply-laced case, because a dual elements, half of the sum of the coroots, is sometimes comparably important. However, I know that the Weyl element has long been important for $q$-analogue" reasons, which in more modern work has become more and more important because it points to quantum groups and eventually even categorification.
Consider the elementary fact that there are $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ subsets of size $k$ of the set $\{1,\ldots,n\}$. This counting fact is a special case of the Weyl dimension formula for the dimension of an irreducible representation of a complex simple Lie algebra. In this model case the representation is $\mathfrak{sl}(n,\mathbb{C})$ acting on $\Lambda^k(\mathbb{C}^n)$. You could look at the same counting problem again with multiplicative weights for the elements of $\{1,\ldots,n\}$. If you make the weight of $j$ be some variables $x_j$, the total weight is an irreducible polynomial --- equivalent to the full character of $\Lambda^k(\mathbb{C}^n)$. Magically, if you let $x_j = q^j$ for a single variable $q$, you get (up to a power of $q$) the Gaussian binomial coefficient $\binom{n}{k}_q$. This is a special case of the Weyl $q$-dimension formula which gives the character of the Weyl element. That is, the dimension of an irreducible representation $V_\lambda$ of $\mathfrak{g}$ is given by a tidy ratio, and the $q$-dimension still is. The full character is not as simple, and therefore neither are most 1-variable specializations.
Bourbaki, and maybe Weyl himself, used the $q$-dimension to prove the dimension formula by plugging in $q=1$. (It can happen in combinatorics that a $q$-analogue is easier than the original question.) In modern representation theory the $q$-dimension is even more important, because it's also (after centering the powers of $q$) the quantum dimension of the same representation $V_\lambda$ (or we can say, the same-named representation) of the quantum group $U_q(\mathfrak{g})$. The Weyl element also arises in many other ways in the representation theory of the quantum group. Actually, that's an understatement: This $q$ is the variable of the Jones polynomial and its generalizations. All of this $q$-structure remains important in the even newer categorifications of quantum groups.
(A caveat: Because half-integer powers of $q$ commonly arise, there is a substitution of $q^2$ for $q$ in passing from $q$-analogues to quantum groups. I never liked this mismatch of conventions, in fact as a student I didn't even realize/believe it, but it is an established standard.)
There is another formula for the Weyl element: It's the sum of the fundamental weights. It's already interesting that these two formulas agree.
Greg KuperbergGreg Kuperberg
I fear this question is getting a little crowded, but I do have my own hobby-horse to ride, so why hold back:
For a holomorphic symplectic variety with nice enough behavior, quantizations of said variety are in canonical bijection with power series in $H^2(X)$ (this follows from work of Bezrukavnikov and Kaledin). Furthermore, for $X=T^*G/B$, there is a canonical isomorphism of $H^2(T^*G/B)$ with $\mathfrak{h}^*$, the dual abstract Cartan of $\mathfrak{g}$.
So, picking any deformation quantization gives a power series in $H^2(T^*G/B)$, and there is one that we know and love the best: differential operators (of course, David Ben-Zvi was arguing above that maybe you shouldn't love this one best, but set that aside for a moment). What power series does this correspond to?
Of course, it's $\rho$ (this is essentially because the differential operators twisted in half-forms really are the most canonical thing, and thus correspond to 0). So, if you believe that differential operators in functions are particularly important as compared to other TDO's, you think $\rho$ is important.
Ben Webster♦Ben Webster
$\begingroup$ If I'm not mistaken this correspondence with power series is, up to order two, precisely what Beilinson-Bernstein discuss: i.e. they say we should normalize a quantization by requiring it gives the zero power series to order two, resulting in half-forms.. They do this via an elementary observation: a deformation quantization to order two, when symmetrized, still gives a commutative associative algebra (this fails to higher order), so one gets a canonical 1-jet of a path into Poisson structures. $\endgroup$ – David Ben-Zvi Nov 8 '11 at 5:10
$\begingroup$ Right. I don't claim that this is really that different from your answer. It just emphasizes in a slightly different way just how canonical $\rho$ really is. $\endgroup$ – Ben Webster♦ Nov 10 '11 at 18:54
One answer that hasn't appeared here yet is that $\rho$ is the highest weight of a spinor representation. Instead of the Dolbeault operator on the flag manifold, one can work with the Dirac operator. This leads to the subject of "Dirac Cohomology", see the book "Dirac Operators in Representation Theory". Here, instead of working with the Universal Enveloping Algebra and modules for it, one works with the tensor product of this with a Clifford algebra. Irreducible modules then acquire a spinor representation factor. This explains nicely the shift by $\rho$.
Hopefully this spring I'll finish this, which explains all this in more detail:
http://www.math.columbia.edu/~woit/brstdirac.pdf
Peter WoitPeter Woit
Parts of what I will explain here have been said in other answers, but not quite in this way.
One thing about the weight $\rho$ is that the places it tends to show up, it is really the weight $-\rho$ that is the important one. Note that the "dot" action is just a shift to make $-\rho$ the new "$0$".
So why would we like to have $-\rho$ be our new "$0$"? Because it turns out to be the weight that behaves like a $0$ should when we consider cohomology.
This becomes a bit clearer when we go to the closely related setting of a semisimple simply connected algebraic group $G$ with Borel subgroup $B$. Here we consider the functor $\operatorname{Ind}_B^G$ inducing from $B$-modules to $G$-modules (for full details of definitions and proofs, one can consult for example Jantzen's book).
To make things simpler, for a weight $\lambda$ we denote also by $\lambda$ the $1$-dimensional $B$-module where the maximal torus in $B$ acts as $\lambda$, and $B$ acts via the projection to this torus.
For $i\geq 0$ write $H^i(\lambda) = R^i\operatorname{Ind}_B^G(\lambda)$ for the $i$'th right derived functor of induction applied to the $B$-module $\lambda$. Note that this is the $i$'th cohomology of the flag variety $G/B$ with coefficients in the line bundle defined by $\lambda$.
Now the reason for wanting $-\rho$ to be our "$0$" starts to be clear, as $H^i(-\rho) = 0$ for all $i\geq 0$. But there are other weights satisfying this, so what makes $-\rho$ extra special?
What makes $-\rho$ so special is that the vanishing of the above cohomology groups in some sense happens "as early as possible". More precisely, if $P = P(\alpha)$ is a parabolic subgroup corresponding to a simple root $\alpha$, then already $R^i\operatorname{Ind}_B^P(-\rho) = 0$, and we can factor $\operatorname{Ind}_B^G$ as $\operatorname{Ind}_P^G\operatorname{Ind}_B^P$.
Moreover, if $\lambda\neq -\rho$ then there is some simple root $\alpha$ such that either $\operatorname{Ind}_B^{P(\alpha)}(\lambda)\neq 0$ or $R^1\operatorname{Ind}_B^{P(\alpha)}(\lambda)\neq 0$.
Tobias KildetoftTobias Kildetoft
If you have any free abelian group with an integral bilinear form embedded in the Lorentz space $\mathbb{R}^{n,1}$, you may consider the group of automorphisms generated by roots, i.e., reflections in vectors. The reflection hyperplanes will split the corresponding hyperbolic $n$-space into fundamental domains, and if you fix a chamber, you can choose simple roots corresponding to its walls. This setting includes all finite, affine, and hyperbolic Weyl groups. If there is a vector $\rho$ in the span of the roots satisfying $\Vert r - \rho \Vert = \Vert \rho \Vert$ for all simple roots $r$, then it is called a Weyl vector. This always exists in the finite and affine cases, and the other answers on this page give some description of the relevant geometry.
The existence of a Weyl vector gives a hyperbolic reflection group some arithmetic significance, and non-existence is generic - Lorentzian lattices of rank greater than 26 can't have Weyl vectors. From Barnard's thesis (and earlier work of Gristenko and Nikulin in small rank cases), if one has a lattice generated by roots with a Weyl vector, one may attach a vector-valued modular form, whose coefficients describe the root multiplicities of a Borcherds-Kac-Moody Lie algebra whose real simple roots are precisely those of the reflection group. The Lie algebra in turn has a Weyl denominator product that is a cusp expansion of an automorphic form on $O(n+1,2)$.
In the most extreme case, one may start with the 26-dimensional even unimodular Lorentzian lattice $I\! I_{25,1}$, and choose a chamber for its reflection group. The Dynkin diagram is naturally an affine space on the Leech lattice (by a theorem of Conway), and there is a norm zero Weyl vector $\rho$. There is an action of Leech, identified with the lattice quotient $\rho^\perp/\mathbb{Z}\rho$, on the fundamental domain by parabolic translation. Because there is a Weyl vector, one has a modular form whose coefficients control root multiplicities of a Lie algebra. In this case, we have the weight -12 form $1/\Delta$, and the Lie algebra is the fake monster Lie algebra, which apparently describes bosonic strings propagating in a 26-torus. The roots of norm $2n$ have multiplicity $p_{24}(1-n)$, i.e., the number of partitions in 24 colors.
In a different direction, there is a generalization of the Weyl character formula that holds for any Borcherds-Kac-Moody Lie algebra (not just hyperbolic), and $\rho$ appears here as any vector in the root space that satisfies the relation $\Vert r - \rho \Vert = \Vert \rho \Vert$ (equivalently, $r-2\rho \perp r$) for all simple roots $r$. In the BGG interpretation (worked out in detail in Jurisich's thesis), we find that $H_k(\mathfrak{n}_+, \mathbb{C})$ is spanned by the elements of $\bigwedge^k \mathfrak{n}_+$ whose weight $r$ satisfies $\Vert r - \rho \Vert = \Vert \rho \Vert$. In other words, when we throw away finiteness (and hence well-behaved flag varieties), $\rho$ still plays a role of selecting the part of the exterior power that contributes to the homology.
S. Carnahan♦S. Carnahan
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\begin{document}
\title{Realizability and Internal Model Control on Networks}
\author{Anders Rantzer
\thanks{The author is affiliated with Automatic Control LTH, Lund
University, Box 118, SE-221 00 Lund, Sweden.}} \maketitle \begin{abstract} It is proved that network realizability of controllers can be enforced without conservatism using convex constraints on the closed loop transfer function. Once a network realizable closed loop transfer matrix has been found, a corresponding controller can be implemented using a network structured version of Internal Model Control.
\end{abstract}
\section{Introduction} The importance of closed loop convexity in the theory for control system design has long been recognized \cite{Boyd/B91}. A large number of specifications, both in time and frequency domain, can be stated as convex constraints on the closed loop system. The convexity opens up for efficient synthesis algorithms, as well as for computation of rigorous bounds on achievable performance. However, there are also many important specifications that cannot be expressed in a closed loop convex manner. A notable example is controller complexity, measured by the number of states needed in the realization.
During the past decade, growing attention has been paid to large scale networks and distributed control. In this context, it is common to consider controller transfer matrices with a pre-specified sparsity pattern. Such sparsity constraints are generally not closed loop convex, but a number of important closed loop convex structural constraints have been derived and summarized under the framework known as quadratic invariance \cite{Rotkowitz06}. However, with exception for positive systems \cite{rantzer2018tutorial}, solutions based on sparsity restricted transfer matrices have a tendency to become computationally expensive and poorly scalable. It was therefore an important discovery in the theory for large-scale control when \cite{wang2016system} recently proved that optimization with finite impulse response constraints can be used for scalable synthesis of distributed controllers.
The objective of this short note is to isolate an idea used in ``system level synthesis'' \cite{wang2016system,anderson2017structured}, to show that network realizable controllers in the sense of \cite{vamsi2016optimal} can be synthesized using convex optimization. Moreover, we will demonstrate that the classical idea of internal model control \cite{garcia1982internal} is useful to convert the optimization outcome into a network compatible controller realization.
\section{Notation} A \emph{transfer matrix} denotes a matrix of rational functions that can be written on the form $\mathbf{G}(z)=C(zI-A)^{-1}B+D$, where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{p\times n}$, $C\in\mathbb{R}^{n\times m}$ and $D\in\mathbb{R}^{p\times m}$. It is said to be \emph{strictly proper} if $D=0$.
\section{Network Realizability}
Following \cite{vamsi2016optimal}, we make the following definition. \begin{dfn}
Given a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ with $N$ nodes, a transfer matrix $\mathbf{G}$ is said to be \emph{network realizable on} $\mathcal{G}$ if it has a stabilizable and detectable realization
\begin{align}
\left[\begin{array}{c|c}
A&B\\\hline C&D
\end{array}\right]
&=\left[\begin{array}{ccc|ccc}
A_{11}&\ldots&A_{1N}&B_{1}&&0\\
\vdots&&\vdots&&\ddots\\
A_{N1}&\ldots&A_{NN}&0&&B_{N}\\\hline
C_{11}&\ldots&C_{1n}&D_{1}&&0\\
\vdots&&\vdots&&\ddots\\
C_{N1}&\ldots&C_{NN}&0&&D_{N}
\end{array}\right]
\label{eqn:ABCD}
\end{align}
where $A_{ij}=0$ and $C_{ij}=0$ for $(i,j)\not\in\mathcal{E}$. Such a realization is said to be \emph{compatible with} $\mathcal{G}$. We need $A_{ij}\in\mathbb{R}^{n_i\times n_j}$, $B_i\in\mathbb{R}^{n_i\times m_i}$, $C_{ij}\in\mathbb{R}^{m_i\times n_j}$ and $D_i\in\mathbb{R}^{p_i\times m_i}$, where $n_i$, $p_i$ and $m_i$ are the number of states, outputs and inputs in node $i$ respectively. \end{dfn}
Given a transfer matrix and a graph, no simple test for network realizability is known. However, given a realization, it is of course straightforward to verify the conditions of Definition~1.
\begin{thm} Let the transfer matrices $\mathbf{G}_1$ and $\mathbf{G}_2$ be network realizable on $\mathcal{G}$. Then the following statements hold: \begin{description}
\item[($i$) ] $\mathbf{G}_1+\mathbf{G}_2$ is network realizable on $\mathcal{G}$.
\item[($ii$) ] If $\mathbf{G}_1$ and $\mathbf{G}_2$ are stable, then $\mathbf{G}_1\mathbf{G}_2$ is network realizable on $\mathcal{G}$.
\item[($iii$) ] If $\mathbf{G}_1(\infty)$ is invertible, then $\mathbf{G}_1^{-1}$ is network realizable on $\mathcal{G}$.
\end{description} \label{thm:add} \end{thm}
\begin{rmk}
Consider the graph with $\mathcal{V}=\{1,2,3,4\}$ and $\mathcal{E}=\{(1,1),(2,2),(3,3),(4,4),(1,3),(1,4),(2,3),(2,4)\}$. Notice that
both the two transfer matrices
\begin{align*}
\mathbf{G}_1(z)&=
\begin{bmatrix}
0&0&0&0\\
0&0&0&0\\
\frac{1}{z-2}&1&0&0\\
\frac{1}{z-2}&1&0&0
\end{bmatrix}&
\mathbf{G}_2(z)&=
\begin{bmatrix}
1&0&0&0\\
0&\frac{1}{z-2}&0&0\\
0&0&0&0\\
0&0&0&0
\end{bmatrix}
\end{align*}
are network realizable on $\mathcal{G}$, but, as was pointed out in \cite{3994145}, this is not the case with their product
\begin{align*}
\mathbf{G}_1(z)\mathbf{G}_2(z)
=\begin{bmatrix}
0&0&0&0\\
0&0&0&0\\
\frac{1}{z-2}&\frac{1}{z-2}&0&0\\
\frac{1}{z-2}&\frac{1}{z-2}&0&0
\end{bmatrix}.
\end{align*}
This shows that the stability assumption is essential for statement ($ii$) in Theorem~\ref{thm:add}. \end{rmk}
\begin{pf*}{Proof of Theorem~\ref{thm:add}}
Let $\mathbf{G}_i(z)=C^i(zI-A^i)^{-1}B^i+D^i$ where \begin{align*}
\left[\begin{array}{c|c}
A^i&B^i\\\hline C^i&D^i
\end{array}\right]
&=\left[\begin{array}{ccc|ccc}
A^i_{11}&\ldots&A^i_{1N}&B^i_{1}&&0\\
\vdots&&\vdots&&\ddots\\
A^i_{N1}&\ldots&A^i_{NN}&0&&B^i_{N}\\\hline
C^i_{11}&\ldots&C^i_{1n}&D^i_{1}&&0\\
\vdots&&\vdots&&\ddots\\
C^i_{N1}&\ldots&C^i_{NN}&0&&D^i_{N}
\end{array}\right], \end{align*} Then $\mathbf{G}_3(z)=\mathbf{G}_1(z)+\mathbf{G}_2(z)$ provided that \begin{align*}
A^3_{kl}&=\begin{bmatrix}A^1_{kl}&0\\0&A^2_{kl}\end{bmatrix}&
B^3_{kl}&=\begin{bmatrix}B^1_{k}\\B^2_{k}\end{bmatrix}\\
C^3_{kl}&=\begin{bmatrix}C^1_{kl}&C^2_{kl}\end{bmatrix} \end{align*} for all $k$ and $l$. The sparsity conditions, as well as stabilizability and detectability, follow trivially and ($i$) holds.
Similarly, $\mathbf{G}_3(z)=\mathbf{G}_2(z)\mathbf{G}_1(z)$ provided that \begin{align}
\left[\begin{array}{c|c}
A^3_{kl}&B^3_{k}\\\hline
C^3_{kl}&D^3_{k}
\end{array}\right]
&=\left[\begin{array}{cc|c}
A^1_{kl}&0&B^1_{k}\\
B^2_{k}C^1_{kl}&A^2_{kl}&B^2_{k}D^1_{k}\\\hline
D^2_{k}C^1_{kl}&C^2_{kl}&D^2_{k}D^1_{k}
\end{array}\right], \label{eqn:product} \end{align} so $\mathbf{G}_3$ satisfies the sparsity conditions. Both factors are assumed to be stable, so stabilizability and detectability hold trivially. Hence ($ii$) follows.
If $\mathbf{G}_1(z)=C(zI-A)^{-1}B+D$ and $D$ is invertible, then $\mathbf{G}_1^{-1}$ has the realization \begin{align*}
\left[\begin{array}{c|c}
A-BD^{-1}C&BD^{-1}\\\hline -D^{-1}C&D^{-1}
\end{array}\right]. \end{align*} where the needed sparsity structure, as well as stabilizability and detectability, follow from network realizability of $\mathbf{G}_1$. This proves ($iii$) .
\end{pf*}
\section{Network Realizable Controllers}
The following theorem shows that in a number of cases, network realizability conditions on the controller can be mapped into similar conditions on closed loop transfer functions. From Theorem~1, we know that such constraints are convex, so they can be conveniently included in synthesis procedures based on convex optimization.
\begin{thm}
Consider $\mathbf{P}$ and $\mathbf{C}$ such that $\mathbf{P}$ is strictly proper and define the closed loop matrix
\begin{align*}
\mathbf{H}=\begin{bmatrix}
I&-\mathbf{P}\\
\mathbf{C}&I
\end{bmatrix}^{-1}=\begin{bmatrix}
(I+\mathbf{P}\mathbf{C})^{-1}&
\mathbf{P}(I+\mathbf{C}\mathbf{P})^{-1}\\
-\mathbf{C}(I+\mathbf{P}\mathbf{C})^{-1}&
(I+\mathbf{C}\mathbf{P})^{-1}
\end{bmatrix}.
\end{align*}
Then the following two statements are equivalent:
\begin{description}
\item[($i$) ]Both $\mathbf{P}$ and $\mathbf{C}$ are network realizable on $\mathcal{G}$.
\item[($ii$) ]$\mathbf{H}$ is network realizable on $\mathcal{G}$.
\end{description}
The following two statements are also equivalent:
\begin{description}
\item[($iii$) ]Both $\mathbf{P}\mathbf{C}$ and $\mathbf{C}$ are network realizable on $\mathcal{G}$.
\item[($iv$) ]Both $(I+\mathbf{P}\mathbf{C})^{-1}$ and $\mathbf{C}(I+\mathbf{P}\mathbf{C})^{-1}$ are network realizable on $\mathcal{G}$.
\end{description}
Suppose in addition that $\mathbf{P}$ is stable and network realizable on $\mathcal{G}$, while $\mathbf{H}$ is stable. Then the following are equivalent:
\begin{description}
\item[($v$) ]$\mathbf{C}$ is network realizable on $\mathcal{G}$.
\item[($vi$) ]$\mathbf{C}(I+\mathbf{P}\mathbf{C})^{-1}$ is network realizable on $\mathcal{G}$.
\end{description} \label{thm:iff} \end{thm}
\begin{rmk}
The presentation in \cite{wang2016system} is focusing on finite impulse response representations of the closed loop maps. However, nothing excludes the use of other denominators when finite-dimensional parametrizations of closed loop dynamics are needed for computations. In fact, there is a rich literature on heuristics for selection of closed loop poles. \end{rmk}
\begin{rmk}
The statement and proof of Theorem~\ref{thm:iff} is completely independent of how the set of stabilizing controllers is parametrized. The Youla-Kucera parametrization is the most well known option, but the parametrization suggested in \cite{wang2016system} appears to give simpler formulas for unstable plants. \end{rmk}
\begin{rmk}
Stability of the closed loop transfer matrix $\mathbf{H}$ means that all poles should be strictly inside the left half plane. In most applications the poles can actually restricted to a smaller subset $\Omega$ of the complex plane. Such a stronger assumption can be used to also get a stronger conclusion, namely that the controller has a network compatible realization with no uncontrollable or unobservable modes corresponding to poles outside $\Omega$. \end{rmk}
\begin{pf*}{Proof of Theorem~\ref{thm:iff}.} The strict properness of $\mathbf{P}$ implies that $\mathbf{H}(0)$ is invertible, so the equivalence between ($i$) and ($ii$) follows immediately from Theorem~\ref{thm:add}.
The equivalence between ($iii$) and ($iv$) follows from statement ($iii$) in Theorem~\ref{thm:add} and the identity \begin{align*}
\begin{bmatrix}
I+\mathbf{P}\mathbf{C}&0\\
\mathbf{C}&I
\end{bmatrix}^{-1}=\begin{bmatrix}
(I+\mathbf{P}\mathbf{C})^{-1}&0\\
-\mathbf{C}(I+\mathbf{P}\mathbf{C})^{-1}&I
\end{bmatrix}, \end{align*} since the strict properness of $\mathbf{P}$ gives both matrices an invertible direct term.
Assume that ($v$) holds. Then ($iii$) follows and therefore also ($iv$) \!\!. This proves ($vi$) \!\!.
Conversely, suppose that ($vi$) holds. Then $\mathbf{C}(I+\mathbf{P}\mathbf{C})^{-1}$ and $\mathbf{P}$ are both stable and network realizable on $\mathcal{G}$, so the same holds for their product $\mathbf{P}\mathbf{C}(I+\mathbf{P}\mathbf{C})^{-1}$. The identity $(I+\mathbf{P}\mathbf{C})^{-1}=I-\mathbf{P}\mathbf{C}(I+\mathbf{P}\mathbf{C})^{-1}$ gives stability and network realizable on $\mathcal{G}$ for $(I+\mathbf{P}\mathbf{C})^{-1}$. Hence ($iv$) holds and the equivalence with ($iii$) proves that $\mathbf{C}$ is network realizable on $\mathcal{G}$, so the proof is complete. \end{pf*}
A common situation in applications is that a network realizable $\mathbf{Q}=\mathbf{C}(I+\mathbf{P}\mathbf{C})^{-1}$ has been designed and a corresponding controller needed. The equivalence between ($v$) and ($vi$) proves existence, but the proof of Theorem~\ref{thm:iff} is not convenient for construction of a corresponding controller. Instead, as will be seen in the next section, the classical Internal Model Control \cite{garcia1982internal} approach is useful for this task.
\section{Internal Model Control on Networks} Given $\mathbf{P}$ and $\mathbf{Q}$, consider a map from process output $y$ and reference value $r$ to control input $u$, defined by the equation \begin{align}
u=\mathbf{Q}[r+\mathbf{P}u-y]. \label{eqn:IMC} \end{align} See Figure~\ref{fig:IMC}. \begin{figure}
\caption{If $\mathbf{P}$ and $\mathbf{Q}$ are networks realizable, then a network realization of the controller can be obtained using Internal Model Control.}
\label{fig:IMC}
\end{figure} Here $\mathbf{P}$ is the ``internal model'' that is used by the controller to predict the measured process output. The difference $\mathbf{P}u-y$ denotes a comparison between the predicted output $\mathbf{P}u$ and the measurement $y$. In the ideal case that the difference is zero, the control law reduces to $u=\mathbf{Q}r$, so $\mathbf{Q}$ defines the desired map from reference to input and $\mathbf{P}\mathbf{Q}$ is the resulting map from reference $r$ to output $y$. The transfer matrix from $r-y$ to $u$, given by (\ref{eqn:IMC}), is $\mathbf{C}=\mathbf{Q}(I-\mathbf{P}\mathbf{Q})^{-1}$.
Our interest in Internal Model Control stems from the fact it generates network realizable controllers in a very natural manner. Suppose that \begin{align*}
\mathbf{P}(z)&=C(zI-A)^{-1}B\\
\mathbf{Q}(z)&=G(zI-E)^{-1}F+H \end{align*} Then the controller $\mathbf{C}=\mathbf{Q}(I-\mathbf{P}\mathbf{Q})^{-1}$ has the realization \begin{align*}
\begin{bmatrix}\hat{x}^+\\\xi^+\end{bmatrix}
&=\begin{bmatrix}A+BHC&BG\\FC&E\end{bmatrix}
\begin{bmatrix}\hat{x}\\\xi\end{bmatrix}
+\begin{bmatrix}BH\\F\end{bmatrix}(r-y)\\
u&=\begin{bmatrix}HC&G\end{bmatrix}+H(r-y). \end{align*} It is easy to see that if $B$, $F$ and $H$ are diagonal, while $A$, $C$, $E$ and $G$ have a sparsity structure compatible with the graph $\mathcal{G}$, this realization is compatible with $\mathcal{G}$ after proper ordering of the states and block partitioning of the matrices. More specifically, if node $i$ of the graph is hosting the process state $x_i$, it should also host the controller state $(\hat{x}_i,\xi_i)$.
\begin{ex} Consider a simple model for control of water levels in dams along a river: \begin{align*} \begin{cases}
x_1(t+1)=0.9x_1(t)-u_1(t)\\
x_2(t+1)=0.1x_1(t)+0.8x_2(t)+u_1(t)-u_2(t)\\
x_3(t+1)=0.2x_2(t)+0.7x_3(t)+u_2(t)-u_3(t)
\end{cases}
\end{align*} Each state represents the water level in a dam and the control variables are used to control the release of water from one dam to the next. In this case, the transfer function from $(u_1,u_2,u_3)$ to $(x_1,x_2,x_3)$ is $\mathbf{P}(z)=(zI-A)^{-1}B$ where \begin{align*}
A&=\begin{bmatrix}
0.9&0&0\\
0.1&0.8&0\\
0&0.2&0.7
\end{bmatrix}&
B&=\begin{bmatrix}
-1&0&0\\
1&-1&0\\
0&1&-1
\end{bmatrix} \end{align*} A graph $\mathcal{G}$, corresponding to downwards flow of information, is defined by the node set $\mathcal{V}=\{1,2,3\}$ and the link set $\mathcal{E}=\{(1,1),(2,1),(2,2),(3,2),(3,3)\}$. The realization above does not have diagonal $B$-matrix, but the transfer function from $u$ to $x$ is still network realizable on $\mathcal{G}$, as shown by the (non-minimal) realization \begin{align*}
\bar{A}&={\small\left[\begin{array}{cc|cc|c}
0.9&0&0&0&0\\
0&0.8&0&0&0\\\hline
0.1&0&0.8&0&0\\
0&0.2&0&0.7&0\\\hline
0&0&0.2&0&0.7
\end{array}\right]}&
\bar{B}&={\small\left[\begin{array}{r|r|r}
-1&0&0\\
1&0&0\\\hline
0&-1&0\\
0&1&0\\\hline
0&0&-1
\end{array}\right]}\\
\bar{C}&={\small\left[\begin{array}{cc|cc|c}
1&0&0&0&0\\\hline
0&1&1&0&0\\\hline
0&0&0&1&1
\end{array}\right]}&
\bar{D}&=0. \end{align*} Theorem~\ref{thm:iff} tells us that in order to find river dam controllers that only exchange information along the graph $\mathcal{G}$, it sufficient to consider $\mathbf{Q}=(I+\mathbf{C}\mathbf{P})^{-1}\mathbf{C}$ that are network realizable on $\mathcal{G}$. In particular, let $\mathbf{Q}$ have the state realization \begin{align*}
\left[\begin{array}{c|c}
E&F\\\hline G&H
\end{array}\right]
&=\left[\begin{array}{ccc|ccc}
E_{11}&0&0&F_{1}&0&0\\
E_{21}&E_{22}&0&0&F_2&0\\
0&E_{32}&E_{33}&0&0&F_3\\\hline
G_{1}&0&0&0&0&0\\
0&G_{2}&0&0&0&0\\
0&0&G_{3}&0&0&0
\end{array}\right] \end{align*} Then $\mathbf{C}=(I-\mathbf{Q}\mathbf{P})^{-1}\mathbf{Q}$ mapping $(e_1,e_2,e_3)$ to $(u_1,u_2,u_3)$ can be implemented as the Internal Model Controller \begin{align*}{\footnotesize
\!\!\!\begin{bmatrix}
\hat{x}_1^+\\\xi_1^+\\\hat{x}_2^+\\\xi_2^+\\\hat{x}_3^+\\\xi_3^+
\end{bmatrix}}
&=\left[{\footnotesize\begin{array}{cc|cc|cc}
0.9&-G_{1}&0 & 0& 0& 0\\
F_1 &E_{11}&0 & 0& 0& 0\\\hline
0.1&G_{1}&0.8&-G_2& 0& 0\\
0 &E_{21}&F_2 &E_{22}& 0& 0\\\hline
0 &0 &0.2& G_2&0.7&-G_3\\
0 &0 &0 &E_{32}&F_3 &E_{33}
\end{array}}\right] {\footnotesize\begin{bmatrix} \hat{x}_1\\\xi_1\\\hat{x}_2\\\xi_2\\\hat{x}_3\\\xi_3 \end{bmatrix} -\begin{bmatrix} 0\\e_1\\0\\e_2\\0\\e_3 \end{bmatrix}} \end{align*} This realization has a structure compatible with the graph $\mathcal{G}$ (in spite of the fact that it is based on the original state matrices $A,B$ rather than $\bar{A},\bar{B}$). \end{ex}
\end{document} | arXiv |
APS Home
Meeting Announcement
Using the Scheduler
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2008 APS March Meeting
Monday–Friday, March 10–14, 2008; New Orleans, Louisiana
Session X40: Self Assembled Protein Cages
Sponsoring Units: DBP
Chair: William Klug, University of California, Los Angeles
Room: Morial Convention Center 232
X40.00001: A minimal model for protein coat dynamics in intracellular vesicular transport
Ranjan Mukhopadhyay, Hui Wang, Greg Huber
Within eukaryotic cells, proteins are transported by vesicles formed from coated regions of membranes. The assembly of coat proteins deforms the membrane patch and drives vesicle formation. Once the vesicle has pinched off, the protein coat rapidly disassembles. Motivated by recent experimental results, we propose a minimal model for the dynamics of coat assembly and disassembly and study the spatio-temporal behavior of the system. We will show that for a range of parameters, our model can robustly generate a steady state distribution of protein clusters with characteristic sizes and will obtain the scaling behavior of average cluster size with the parameters of the model. We will also discuss the coupling of coat dynamics to sorting of cargo proteins. [Preview Abstract]
X40.00002: The study of viral assembly with fluorescence fluctuation spectroscopy
Joachim Mueller, Bin Wu, Yan Chen
Enveloped viruses contain an encapsulating membrane that the virus acquires from the host cell during the budding process. The presence of the enveloping lipid membrane complicates the physical characterization of the proteins assembled within the virus considerably. Here we present a method based on fluorescence fluctuations that quantifies the copy number of proteins within an enveloped viral particles. We choose the viral protein Gag of the human immunodeficiency virus (HIV) type 1 as a model system, because Gag expressed in cells is sufficient to produce viral-like particles (VLPs) of the same size as authentic virions. VLPs harvested from cells that express fluorescently labeled Gag were investigated by two-photon fluorescence fluctuation spectroscopy. The autocorrelation functions of the fluctuations revealed a hydrodynamic size of the fluorescent VLPs consistent with previous results based on electron microscopy. Further analysis of the fluctuations revealed a copy number of Gag per virion that is inconsistent with the prevailing model of HIV assembly. We will discuss the implications of our experimental results for the assembly process of VLPs. [Preview Abstract]
X40.00003: Spherical Proteins and Viral Capsids Studied by Theory of Elasticity
Zheng Yang, Ivet Bahar, Michael Widom
Coarse-grained elastic network models have been successful in elucidating the fluctuation dynamics of proteins around their native conformations. It is well established that the low-frequency collective motions derived by simplified normal mode analysis depend on the overall 3-dimensional shape of the biomolecule. Given that the large scale collective motions are usually involved in biological function, our objective in this work is to gain more insights into large scale collective motions of spherical proteins and virus capsids by considering a continuous model with perfect spherical symmetry. To this end, we compare the global dynamics of proteins and the analytical solutions from an elastic wave equation with spherical boundary conditions. In addition, an icosahedral discrete model is generated and analyzed for validating our continuous model. Applications to lumazine synthase, satellite tobacco mosaic virus and other viruses shows that the spherical elastic model can efficiently provide insights on collective motions that are otherwise obtained by detailed elastic network models. [Preview Abstract]
X40.00004: Low frequency mechanical modes of viruses with atomic detail
Eric Dykeman, Otto Sankey
The low frequency mechanical modes of viruses can provide important insights into the large global motions that a virus may exhibit. Recently it has been proposed that these large global motions may be excited using impulsive stimulated Raman scattering producing permanent damage to the virus. In order to understand the coupling of external probes to the capsid, vibrational modes with atomic detail are essential. The standard approach to find the atomic modes of a molecule with $N$ atoms requires the formation and diagonlization of a $3N\times 3N$ matrix. As viruses have $10^5$ or more atoms, the standard approach is difficult. Using ideas from electronic structure theory, we have developed a method to construct the mechanical modes of large molecules such as viruses with atomic detail. Application to viruses such as the cowpea chlorotic mottle virus, satellite tobacco necrosis virus, and M13 bacteriophage show a fairly complicated picture of the mechanical modes. [Preview Abstract]
X40.00005: Diversity of in-vivo assembled HIV-1 capsids
Se Il Lee, Toan Nguyen
Understanding the capsid assembly process of Human Immunodeficiency Virus (HIV), the causative agent of Acute Immuno Deficiency Syndrom (AIDS), is very important because of recent intense interest in capsid-oriented viral therapy. The unique conical shapes of mature HIV-1 capsid have drawn significant interests in the biological community and started to attract attention from the physics community. Previous studies showed that in a free assembly process, the HIV-1 conical shape is not thermodynamically stable. However, if the volume of the capsid is constrained during assembly and the capsid protein shell has high spontaneous curvature, the conical shape is stable. In this work, we focus on in-vivo HIV-1 capsid assembly. For this case, the viral envelope membrane present during assembly imposes constraint on the length of the capsid. We use an elastic continuum shell theory to approximate the energies of various HIV-1 capsid shapes (spherical, cylindrical and conical). We show that for certain range of viral membrane diameter, the conical and cylindrical shapes are both thermodynamically stable. This result is supported by experimental observation that in-vivo assembled HIV-1 capsids are very heterogeneous in shapes and sizes. Numerical calculation is also performed to improve theoretical approximation. [Preview Abstract]
X40.00006: An elastic model of partial budding of retroviruses
Rui Zhang, Toan Nguyen
Retroviruses are characterized by their unique infection strategy of reverse transcription, in which the genetic information flows from RNA back to DNA. The most well known representative is the human immunodeficiency virus (HIV). Unlike budding of traditional enveloped viruses, retrovirus budding happens together with the formation of spherical virus capsids at the cell membrane. Led by this unique budding mechanism, we proposed an elastic model of retrovirus budding in this work. We found that if the lipid molecules of the membrane are supplied fast enough from the cell interior, the budding always proceeds to completion. In the opposite limit, there is an optimal size of partially budded virions. The zenith angle of these partially spherical capsids, $\alpha$, is given by $\alpha\simeq(\tau^2/\kappa\sigma)^{1/4}$, where $\kappa$ is the bending modulus of the membrane, $\sigma$ is the surface tension of the membrane, and $\tau$ characterizes the strength of capsid protein interaction. If $\tau$ is large enough such that $\alpha\sim\pi$, the budding is complete. Our model explained many features of retrovirus partial budding observed in experiments. [Preview Abstract]
X40.00007: Calibrating elastic parameters from molecular dynamics simulations of capsid proteins
Stephen Hicks, Christopher Henley
Virus capsids are modeled with elastic network models in which a handful of parameters determine transitions in assembly [1] and morphology [2]. We introduce an approach to compute these parameters from the microscopic structure of the proteins involved. We consider each protein as one or a few rigid bodies with very general interactions, which we parameterize by fitting the simulated equilibrium fluctuations (relative translations and rotations) of a pair of proteins (or fragments) to a 6-dimensional Gaussian. We can then compose these generalized springs into the global capsid structure to determine the continuum elastic parameters. We demonstrate our approach on HIV capsid protein and compare our results with the observed lattice structure (from cryo-EM [3] and AFM indentation studies).\\{} [1] R. Zandi et al, PNAS 101 (2004) 15556.\\{} [2] J. Lidmar, L. Mirny, and D. R. Nelson, PRE 68 (2003) 051910.\\{} [3] B. K. Ganser-Pornillos et al, Cell 131 (2007) 70. [Preview Abstract]
X40.00008: Coarse-grained mechanics of viral shells
William S. Klug, Melissa M. Gibbons
We present an approach for creating three-dimensional finite element models of viral capsids from atomic-level structural data (X-ray or cryo-EM). The models capture heterogeneous geometric features and are used in conjunction with three-dimensional nonlinear continuum elasticity to simulate nanoindentation experiments as performed using atomic force microscopy. The method is extremely flexible; able to capture varying levels of detail in the three-dimensional structure. Nanoindentation simulations are presented for several viruses: Hepatitis B, CCMV, HK97, and $\phi$29. In addition to purely continuum elastic models a multiscale technique is developed that combines finite-element kinematics with MD energetics such that large-scale deformations are facilitated by a reduction in degrees of freedom. Simulations of these capsid deformation experiments provide a testing ground for the techniques, as well as insight into the strength-determining mechanisms of capsid deformation. These methods can be extended as a framework for modeling other proteins and macromolecular structures in cell biology. [Preview Abstract]
X40.00009: ABSTRACT HAS BEEN MOVED TO SESSION C1
[Preview Abstract]
X40.00010: Biochemistry in the Nanopores
Samir M. Iqbal, Bala Murali Venkatesan, Demir Akin, Rashid Bashir
Solid-state technology is fast advancing novel nano-structures for biomolecular detection. The solid-state nanopores have emerged as potential replacement of the Sanger's method for DNA sequencing. While the passage of the DNA molecule through the nanopore has been reported extensively, little has been done to identify the individual base pairs or sequences within the molecule. Learning from the mechanics of ion-channels on the cell surface, we functionalized the solid-state nanopores to recognize and selectively regulate the flow of molecules though the pore. The probe DNA was immobilized by chemical adsorption, and target DNA was passed under electrophoretic bias. The single base mismatch selectivity was achieved by using a hairpin loop in the probe. We could thus identify between the perfect complementary and mismatched target molecules. We will expand on the theoretical framework that governs the interactions of the probe and target molecules, as observed from the pulse behavior. [Preview Abstract]
X40.00011: Poisson pulsed control of particle escape
Marie McCrary, Lora Billings, Ira Schwartz, Mark Dykman
We consider the problem of escape in a double well potential. With a weak background Gaussian noise, the escape rate is well known and follows an exponential scaling with the noise intensity $D$. Here, we consider adding a small Poisson noise to the Gaussian noise. We compute the change in escape time as we add Poisson distributed pulses of a given duration and amplitude. The escape rate acquires an extra factor which is determined by the characteristic functional of the Poisson noise calculated for a function, which is determined by the system dynamics and is inversely proportional to $D$. As a result, for small $D$ even weak Poisson pulses can lead to a significant change of the escape rate. The Poisson noise induced factor depends sensitively on the interrelation between the noise correlation time and the relaxation time of the system. We compare analytical results with extensive numerical simulations. The numerical computation of escape rates for multiple interacting particles in a well will also be shown. [Preview Abstract]
X40.00012: Quorum sensing and biofilm formation investigated using laser-trapped bacterial arrays
Vernita Gordon, John Butler, Ivan Smalyukh, Matthew Parsek, Gerard Wong
Studies of individual, free-swimming (planktonic) bacteria have yielded much information about their genetic and phenotypic characteristics and about ``quorum sensing,'' the autoinducing process by which bacteria detect high concentrations of other bacteria. However, in most environments the majority of bacteria are not in the planktonic form but are rather in biofilms, which are highly-structured, dynamic communities of multiple bacteria that adhere to a surface and to each other using an extracellular polysaccharide matrix. Bacteria in biofilms are phenotypically very different from their genetically-identical planktonic counterparts.~ Among other characteristics, they are much more antibiotic-resistant and virulent.~ Such biofilms form persistent infections on medical implants and in the lungs of cystic fibrosis patients, where Pseudomonas aeruginosa biofilms are the leading cause of lung damage and, ultimately, death.~ To understand the importance of different extracellular materials, motility mechanisms, and quorum sensing for biofilm formation and stability, we use single-gene knockout mutants and an infrared laser trap to create a bacterial aggregate that serves as a model biofilm and allows us to measure the importance of these factors as a function of trapping time, surface, and nutritional environment. [Preview Abstract]
X40.00013: Self-Polarization of Cells in Elastic Gels
Assaf Zemel, Samuel Safran
The shape of a cell as well as the rigidity and geometry of its surroundings play an important role in vital cellular processes. The contractile activity of cells provides a generic means by which cells may sense and respond to mechanical features. The matrix stresses, that depend on the elasticity and geometry of cells, feedback on the cells and influence their activity. This suggests a mechanical mechanism by which cells control their shape and forces. We present a quantitative, mechanical model that predicts that cells in an elastic medium can self-polarize to form well ordered stress fibers. We focus on both single cells in a gel, as well as on an ensemble of cells that is confined to some region within the gel. While the \textit{magnitude} of the cellular forces is found to increase monotonically with the matrix rigidity the \textit{anisotropy} of the forces, and thus the ability of the cells to polarize, is predicted to depend non-monotonically on the medium's rigidity. We discuss these results with experimental findings and with the observation of an optimal medium elasticity for cell function and differentiation. [Preview Abstract]
X40.00014: Active suspensions in shear flow
A. Ahmadi, M.C. Marchetti, T.B. Liverpool
We report on the structure and rheology of an active suspension of cytoskeletal filaments and motor proteins in shear flow. Hydrodynamics equations for an active suspension were derived earlier by us [arXiv:q-bio.CB/0703029v1] by coarse-graining the Smoluchowski equation for a model of filaments and motors. The model incorporates the coupling of orientational order to flow and accounts for the exchange of momentum between filaments and solvent. In the present study we investigate the role of active crosslinkers on the formation and stability of ordered states (polar and nematic) under external shear flow. We also study the effect of motor activity on the rheological behavior of the ordered states away from boundaries. This work may also be relevant for the understanding of the flow-driven reorientation of endothelial cells under the shear stress imposed by blood flow. [Preview Abstract]
X40.00015: Selective advantage for sexual replication with random haploid fusion
Emmanuel Tannenbaum
This talk develops a simplified set of models describing asexual and sexual replication in unicellular diploid organisms. The models assume organisms whose genomes consist of two chromosomes, where each chromosome is assumed to be functional if and only if it is equal to some master sequence. The fitness of an organism is determined by the number of functional chromosomes in its genome. For a population replicating asexually, a cell replicates both of its chromosomes, and then divides and splits its genetic material evenly between the two cells. For a population replicating sexually, a given cell first divides into two haploids, which enter a haploid pool. Within the haploid pool, haploids fuse into diploids, which then divide via the normal mitotic process. When the cost for sex is small, as measured by the ratio of the characteristic haploid fusion time to the characteristic growth time, we find that sexual replication with random haploid fusion leads to a greater mean fitness for the population than a purely asexual strategy. The results of this talk are consistent with previous studies suggesting that sex is favored at intermediate mutation rates, for slowly replicating organisms, and at high population densities. [Preview Abstract] | CommonCrawl |
Exhaustion, method of
A method of proof used by mathematicians of antiquity in order to determine areas and volumes. The name "method of exhaustion" was introduced in the 17th century.
The typical scheme of proof by the method of exhaustion can, in modern terms, be explained as follows. In order to determine a quantity $ A $ one constructs a certain sequence of quantities $ C _ {1} , C _ {2} \dots $ such that
$$ \tag{1 } C _ {n} < A ; $$
one assumes that a $ B $ is known such that
$$ \tag{2 } C _ {n} < B , $$
and that for any integer $ K $ and all sufficiently large $ n $ the inequalities
$$ \tag{3 } K ( A - C _ {n} ) < D ,\ \ K ( B - C _ {n} ) < D $$
are fulfilled, with $ D $ a constant. From the modern point of view, to transfer (3) to
$$ \tag{4 } A = B $$
one only has to notice that (1)–(3) imply
$$ \lim\limits _ {n \rightarrow \infty } ( A - C _ {n} ) = 0 ,\ \ \lim\limits _ {n \rightarrow \infty } ( B - C _ {n} ) = 0 , $$
$$ A = \lim\limits _ {n \rightarrow \infty } C _ {n} = B . $$
The mathematicians of antiquity, not having developed the theory of limits (cf. Limit), used a reductio ad absurdum argument here: they proved that neither of the inequalities $ A < B $, $ A > B $ is possible. To disprove the first one, they established by the Archimedean axiom that for $ R = B - A $ there exists a $ K $ such that $ K R > D $, and (1) then led to
$$ K ( B - C _ {n} ) > \ K ( B - A ) > D , $$
which contradicts the second inequality in (3). The other assertion is disproved in a similar way. Hence (4) remains.
The introduction of the method of exhaustion and of the axiom that lies at its foundation is ascribed to Eudoxus of Cnidus. The method was extensively used by Eudoxus, while Archimedes used it with extraordinary skill and variety. E.g., in order to determine the area $ A $ of a segment of a parabola, Archimedes constructs the areas $ C _ {1} , C _ {2} \dots $ of segments that are stepwise "exhausting" the area $ A $.
$$ C _ {2} = C _ {1} + \frac{1}{4} C _ {1} , $$
$$ {\dots \dots \dots \dots } $$
$$ C _ {n} = C _ {1} + \frac{1}{4} C _ {1} + \dots + \frac{1}{4 ^ {n-} 1 } C _ {1} . $$
Instead of the limit transition
$$ A = \lim\limits _ {n \rightarrow \infty } \ C _ {n} = \ \left ( 1 + \frac{1}{4} + \frac{1}{16} + \dots \right ) C _ {1} = \frac{4}{3} C _ {1} , $$
Archimedes proves geometrically that for any $ n $,
$$ A - C _ {n} < \ \frac{1}{4 ^ {n-} 1 } C _ {1} . $$
Introducing the area
$$ B = \frac{4}{3} C _ {1} , $$
he obtains
$$ B - C _ {n} = \ \frac{1}{3 \cdot 4 ^ {n-} 1 } C _ {1} , $$
and, following the reasoning explained above, finishes his proof with
$$ A = B = \ \frac{4}{3} C _ {1} . $$
[a1] C.B. Boyer, "A history of mathematics" , Wiley (1968) pp. 100; 142–146
Exhaustion, method of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exhaustion,_method_of&oldid=46872
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