Monotone Functions Research Articles

Glivenko–Cantelli theorems for integrated functionals of stochastic processes

We prove a Glivenko–Cantelli theorem for integrated functionals of latent continuous-time stochastic processes. Based on a bracketing condition via random brackets, the theorem establishes the uniform convergence of a sequence of empirical occupation measures towards the occupation measure induced by underlying processes over large classes of test functions, including indicator functions, bounded monotone functions, Lipschitz-in-parameter functions, and Hölder classes as special cases. The general Glivenko–Cantelli theorem is then applied in more concrete high-frequency statistical settings to establish uniform convergence results for general integrated functionals of the volatility of efficient price and local moments of microstructure noise.

Breakdown of superconductivity in a magnetic field with self-intersecting zero set

We prove that the lowest eigenvalue of the Laplace operator with a magnetic field having a self-intersecting zero set is a monotone function of the parameter defining the strength of the magnetic field, in a neighborhood of infinity. We apply this monotonicity result on the study of the transition from superconducting to normal states for the Ginzburg-Landau model, and prove that the transition occurs at a unique threshold value of the applied magnetic field.

МЕТОД ПОШАГОВОГО СГЛАЖИВАНИЯ ЭКСПЕРИМЕНТАЛЬНЫХ ЗАВИСИМОСТЕЙ ДЛЯ ЗАДАЧ КРАТКОСРОЧНОГО ПРОГНОЗИРОВАНИЯ

The article considers the correspondence of the step-by-step smoothing method as one of the possible algorithms for short-term forecasting of statistics of equal-current measurements of monotone functions, which represent the values of the determining parameters that evaluate the dynamics of the states of complex technical systems based on the operating time. The true value of the monitored parameter is considered unknown, and the processed measurement values are distributed normally. The measurements are processed by step-by-step smoothing. As a result of processing, a new statistic is formed, which is a forecast statistic, each value of which is a half-sum of the measurement itself and the so-called private forecast. First, the forecasts obtained in this way prove to have the same distribution law as the distribution law of a sample of equally accurate measurements. Second, the forecast trend should be the same as the measurement trend and correspond to the theoretical trend, that is, the true values of the monotone function. Third, the variance of the obtained statistics should not exceed the variance of the original sample. It is inferred that the method of step-by-step smoothing method can be proposed for short-term forecasting

Computation of Optimal Linear Strong Stability Preserving Methods Via Adaptive Spectral Transformations of Poisson–Charlier Measures

Strong stability preserving (SSP) coefficients govern the maximally allowable step-size at which positivity or contractivity preservation of integration methods for initial value problems is guaranteed. In this paper, we show that the task of computing optimal linear SSP coefficients of explicit one-step methods is, to a certain extent, equivalent to the problem of characterizing positive quadratures with integer nodes with respect to Poisson–Charlier measures. Using this equivalence, we provide sharp upper and lower bounds for the optimal linear SSP coefficients in terms of the zeros of generalized Laguerre orthogonal polynomials. This in particular provides us with a sharp upper bound for the optimal SSP coefficients of explicit Runge–Kutta methods. Also based on this equivalence, we propose a highly efficient and stable algorithm for computing these coefficients, and their associated optimal linear SSP methods, based on adaptive spectral transformations of Poisson–Charlier measures. The algorithm possesses the remarkable property that its complexity depends only on the order of the method and thus is independent of the number of stages. Our results are achieved by adapting and extending an ingenious technique by Bernstein (Acta Math 52:1–66, 1928) in his seminal work on absolutely monotonic functions. Moreover, the techniques introduced in this work can be adapted to solve the integer quadrature problem for any positive discrete multi-parametric measure supported on \({\mathbb {N}}\) under some mild conditions on the zeros of the associated orthogonal polynomials.

Applications of the Formula of Faà di Bruno: Combinatorial Identities and Monotonic Functions

Applications of the Formula of Faà di Bruno: Combinatorial Identities and Monotonic Functions

Indefinite matrix inequalities via matrix means

For a selfadjoint involutive matrix J we equip Cn with the indefinite inner product [x,y]=〈Jx,y〉, where 〈⋅,⋅〉 denotes the usual inner product on Cn and endow the matrix algebra Mn with a partially ordered relation ≤J. We define the notion of J-mean σf(⋅,⋅) of J-selfadjoint matrices with spectra in (0,∞) for any normal matrix monotone function f:(0,∞)→(0,∞). We establish several fundamental properties of J-means such as the power monotonicity and prove an indefinite version of the Ando–Hiai inequality.

Single-Machine Scheduling Problems with the General Sum-of-Processing-Time and Position-Dependent Effect Function

This paper considers the combination of the general sum-of-processing-time effect and position-dependent effect on a single machine. The actual processing time of a job is defined by functions of the sum of the normal processing times of the jobs processed and its position and control parameter in the sequence. We consider two monotonic effect functions: the nondecreasing function and the nonincreasing function. Our focus is the following objective functions, including the makespan, the sum of the completion time, the sum of the weighted completion time, and the maximum lateness. For the nonincreasing effect function, polynomial algorithm is presented for the makespan problem and the sum of completion time problem, respectively. The latter two objective functions can also be solved in polynomial time if the weight or due date and the normal processing time satisfy some agreeable relations. For the nondecreasing effect function, assume that the given parameter is zero. We also show that the makespan problem can remain polynomially solvable. For the sum of the total completion time problem and a 1 is the deteriorating rate of the jobs, there exists an optimal solution for a 1 ≥ M ; a V-shaped property with respect to the normal processing times is obtained for 0 < a 1 ≤ 1 . Finally, we show that the sum of the weighted completion problem and the maximum lateness problem have polynomial-time solutions for a 1 > M under some agreeable conditions, respectively.

A strongly complete axiomatization of intuitionistic temporal logic

Abstract In this paper, we consider the logic ${\textsf{ITL}}^{e}$, a variant of intuitionistic linear temporal logic that is interpreted over the class of dynamic Kripke frames. These are bi-relational structures of the form $ \langle{W, \preccurlyeq , f}\rangle $ where $\preccurlyeq $ is a partial order on $W$ and $f: W \to W$ is a $\preccurlyeq $-monotone function. Our main result answers a question recently raised by Boudou et al. (2017, A decidable intuitionistic temporal logic. In Computer Science Logic 2017, pp. 14:1–14:17. Vol. 82 of LIPIcs) about axiomatizing this logic. We provide an axiomatization of ${\textsf{ITL}}^{e}$ and prove its strong completeness with respect to the class of all dynamic Kripke frames. The proposed axiomatization is infinitary; it has two derivation rules with countably many premises and one conclusion. It should be mentioned that ${\textsf{ITL}}^{e}$ is semantically non-compact, so no finitary proof system for this logic could be strongly complete.

One Concave-Convex Inequality and Its Consequences

Our starting point is an integral inequality that involves convex, concave and monotonically increasing functions. We provide some interpretations of the inequality, in terms of both probability and terms of linear functionals, from which we further generate completely monotone functions and means. The latter application is seen from the perspective of monotonicity and convexity.

On the Communication Complexity of AND Functions

Log-rank conjecture is one of challenging problems in communication complexity. It is known to hold for some special function classes such as XOR functions f(x ⊕ y) when the outer function f is monotone, symmetric, AC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> , low F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -degree, linear threshold or read-k. However, for AND functions f(x ∧ y), Log-rank conjecture is only known to hold when the outer function is a monotone function or a function with small alternating numbers. In this paper, we show that Log-rank conjecture holds for AND functions whose outer functions are read-once F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -degree polynomials, constant-F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -degree polynomials, and symmetric functions. For read-once and symmetric AND functions, their proof are based on the elementary analysis of Möbius sparsity of the outer functions and designing efficient protocols for computing the corresponding AND functions in terms of the Möbius sparsity of their outer functions. For a constant- F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -degree AND function f(x ∧ y), we use the concept of the variable rank to design a decision tree for computing f where the tree depth is bounded above by a polynomial in the logarithm of the Möbius sparsity of f. The communication protocol can be obtained easily from the constructed decision tree. Our results deepen the understanding of the log-rank conjecture for AND functions.

Cascade Submodular Maximization: Question Selection and Sequencing in Online Personality Quiz

Personality quiz is a powerful tool that enables costumer segmentation by actively asking them questions, and marketers are using it as an effective method of generating leads and increasing e‐commerce sales. We study the problem of how to select and sequence a group of quiz questions so as to optimize the quality of customer segmentation. We assume that the customer will sequentially scan the list of questions. After reading a question, the customer makes two, possibly correlated, random decisions: (i) she first decides whether to answer this question or not, and then (ii) decides whether to continue reading the next question or not. We further assume that the utility of questions that have been answered can be captured by a monotone and submodular function. In general, our problem falls into the category of non‐adaptive active learning‐based customer profiling. Note that under our model, the probability of a question being answered depends on the location of that question, as well as the set of other questions placed ahead of that question, this makes our problem fundamentally different from existing studies on submodular optimization. We develop a series of question selection and sequencing strategies with provable performance bound. Although we focus on the application of quiz design in this study, our results apply to a broad range of applications, including assortment optimization with position bias effect.

The Modular Inequalities for Hardy-type Operators on Monotone Functions in Orlicz Space

This paper is devoted to the study of modular inequality for general Hardy–Copson type operator restricted on the cone of monotone functions from weighted Orlicz space with general weight.

On Hardy type inequalities in grand Lebesgue spaces L p) for 0 < p ≤ 1

In this paper, we prove the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces In particular, we prove similar results for the Hardy operator in weighted Lebesgue spaces. Also, it is proved that the grand Lebesgue space is a quasi-Banach function space. Finally, we establish necessary and sufficient conditions for the boundedness of some integral operator in weighted quasi-Banach Lebesgue spaces.

New inequalities for operator monotone functions

"In this paper we prove that, if f:[0,∞)→R is operator monotone on [0,∞), then for all A, B such that 0&lt;α≤A≤β&lt;γ≤B≤δ for some positive constants α, β, γ, δ, 0≤(γ-β)((f(δ)-f(β))/(δ-β))≤f(B)-f(A)≤(δ-α)((f(γ)-f(α))/(γ-α)). In particular, we have the refinement and reverse of the celebrated Löwner-Heinz inequality 0&lt;(γ-β)((δ^{r}-β^{r})/(δ-β))≤B^{r}-A^{r}≤(δ-α)((γ^{r}-α^{r})/(γ-α)) for all r∈(0,1]."

COMBINED NEWTON’S THIRD-ORDER CONVERGENCE METHOD FOR MINIMIZE ONE VARIABLE FUNCTIONS

Contex. The article deals with the actual problem of numerical optimization of slowly computed unimodal functions of one variable. The analysis of existing methods of minimization of the first and second orders of convergence, which showed that these methods can be used to quickly solve these problems for functions, the values of which can be obtained without difficulty. For slowly computed functions, these methods give slow algorithms; therefore, the problem of developing fast methods for minimizing such functions is urgent.&#x0D; Objective. Development of a combined third-order Newtonian method of convergence to minimize predominantly slowly computed unimodal functions, as well as the development of a database, including smooth, monotonic and partially constant functions, to test the method and compare its effectiveness with other known methods.&#x0D; Method. A technique and an algorithm for solving the problem of fast minimization of a unimodal function of one variable by a combined numerical Newtonian method of the third order of convergence presented. The method is capable of recognizing strictly unimodal, monotonic and constant functions, as well as functions with partial or complete sections of a flat minimum.&#x0D; Results. The results of comparison of the proposed method with other methods, including the fast Brent method, presented. 6954 problems were solved using the combined Newtonian method, while the method turned out to be faster than other methods in 95.5% of problems, Brent’s method worked faster in only 4.5% of problems. In general, the analysis of the calculation results showed that the combined method worked 1.64 times faster than the Brent method.&#x0D; Conclusions. A combined third-order Newtonian method of convergence proposed for minimizing predominantly slowly computed unimodal functions of one variable. A database of problems developed, including smooth, monotone and partially constant functions, to test the method and compare its effectiveness with other known methods. It is shown that the proposed method, in comparison with other methods, including the fast Brent method, has a higher performance.

Learning of monotone functions with single error correction

Abstract Learning of monotone functions is a well-known problem. Results obtained by V. K. Korobkov and G. Hansel imply that the complexity φM (n) of learning of monotone Boolean functions equals C n ⌊ n / 2 ⌋ $\begin{array}{} \displaystyle C_n^{\lfloor n/2\rfloor} \end{array}$ + C n ⌊ n / 2 ⌋ + 1 $\begin{array}{} \displaystyle C_n^{\lfloor n/2\rfloor+1} \end{array}$ (φM (n) denotes the least number of queries on the value of an unknown monotone function on a given input sufficient to identify an arbitrary n-ary monotone function). In our paper we consider learning of monotone functions in the case when the teacher is allowed to return an incorrect response to at most one query on the value of an unknown function so that it is still possible to correctly identify the function. We show that learning complexity in case of the possibility of a single error is equal to the complexity in the situation when all responses are correct.

Learning of monotone functions with single error correction

Abstract Learning of monotone functions is a well-known problem. Results obtained by V. K. Korobkov and G. Hansel imply that the complexity φM (n) of learning of monotone Boolean functions equals C n ⌊ n / 2 ⌋ $\begin{array}{} \displaystyle C_n^{\lfloor n/2\rfloor} \end{array}$ + C n ⌊ n / 2 ⌋ + 1 $\begin{array}{} \displaystyle C_n^{\lfloor n/2\rfloor+1} \end{array}$ (φM (n) denotes the least number of queries on the value of an unknown monotone function on a given input sufficient to identify an arbitrary n-ary monotone function). In our paper we consider learning of monotone functions in the case when the teacher is allowed to return an incorrect response to at most one query on the value of an unknown function so that it is still possible to correctly identify the function. We show that learning complexity in case of the possibility of a single error is equal to the complexity in the situation when all responses are correct.

Hyperspaces through regular and meager subcontinua

Given a metric continuum X, let C(X) and F1(X) be the hyperspaces of subcontinua and the one-point sets of X, respectively. Let D(X) be the collection of all regular subsets in X belonging to C(X) and let M(X) be all the subcontinua of X with empty interior. In the first part of this paper we are interested in the problem to classify continua that satisfies one of the following equalities: F1(X)=M(X), D(X)=D(X)−F1(X), D(X)=C(X)−M(X) and C(X)=D(X)∪M(X). We show that for hereditarily locally connected continua these equalities and the property of not contain a continuum called dendrite D1 are equivalent each other. In the second part, we consider a continuum X for which there exists a continuous surjective and monotone function f:X→[0,1]. We show that the conditions that X is a λ-type continuum and every t in [0,1] is a cohesion point are equivalent to the equalities D(X)={A∈C(X):f(A)∈D([0,1])} and M(X)={A∈C(X):f(A)∈M([0,1])}. Throughout this paper we pose open problems.

Upper and Lower Bounds for Noncommutative Perspectives of Operator Monotone Functions: the Case of Second Variable

Upper and Lower Bounds for Noncommutative Perspectives of Operator Monotone Functions: the Case of Second Variable

Discounted optimal stopping problems in continuous hidden Markov models

We study a two-dimensional discounted optimal stopping problem related to the pricing of perpetual commodity equities in a model of financial markets in which the behaviour of the underlying asset price follows a generalized geometric Brownian motion and the dynamics of the convenience yield are described by an unobservable continuous-time Markov chain with two states. It is shown that the optimal time of exercise is the first time at which the commodity spot price paid in return to the fixed coupon rate hits a lower stochastic boundary being a monotone function of the running value of the filtering estimate of the state of the chain. We rigorously prove that the optimal stopping boundary is regular for the stopping region relative to the resulting two-dimensional diffusion process and the value function is continuously differentiable with respect to the both variables. It is verified by means of a change-of-variable formula with local time on surfaces that the value function and the boundary are determined as a unique solution of the associated parabolic-type free-boundary problem. We also give a closed-form solution to the optimal stopping problem for the case of an observable Markov chain.