Monotonic Research Articles

Sufficient conditions for the majorization ordering and their applications in some quantum information comparisons

In this paper, sufficiently convenient conditions for the majorization order are given, shedding new light on the comparison of quantum systems. By using some technical lemmas, stochastic comparisons of density matrices under mixing operations and also stochastic comparisons of average magnitudes of nondecreasing energy functions are performed. Using newly defined order properties that imply the majorization property, the order property of quantum densities is determined using Kullback–Leibler relative entropy. Examples are given to illustrate the results.

Word and color impressions measured with normal and simulated deutan color stimulus sets in color vision normal and deuteranopic observers.

Impressions of nine semantic words expressing abstract meanings were measured by a color selection process using twelve hues from vivid tone in the Practical Color Coordinate System (PCCS) as well as White, Grey and Black (as a normal color stimulus set) in a paired comparison method (Experiment 1). Color impressions were rated using 35 paired words in a semantic differential (SD) method (Experiment 2). The data of ten color vision normal (CVN) and four deuteranopic observers were analyzed separately by principal component analysis (PCA). Our previous study [J. Opt. Soc. Am. A37, A181 (2020)JOAOD60740-323210.1364/JOSAA.382518] using these methods indicated that deuteranopes could understand the impression of all colors if the color names can be recognized, even if deuteranopes cannot perceive redness and greenness. In this study, we also used a simulated deutan color stimulus set in which colors were modified to simulate the color appearance in deuteranopes by using Brettel-Viénot-Mollon's model to investigate how these simulated deutan colors would be treated by the deutan observers. In the CVN and deutan observers, color distributions of the principal component (PC) loading values in Experiment 1 were close to the PCCS hue circle in the normal colors, and those of the simulated deutan colors could be fitted by ellipses, but there were wide gaps of 73.7° (CVNs) and 89.5° (deutan) where only "White" existed. The word distributions as the PC score values could also be fitted by ellipses and are moderately similar between stimulus sets, but the fitting ellipses were considerably compressed in the minor axis directions in the deutan observers, although categories of words were similar between observer groups. The word distributions in Experiment 2 were not statistically different between the observer groups and the stimulus sets. The color distribution of the PC score values were statistically different, but tendencies of the color distributions were similar between observers. The color distributions of the normal colors could be fitted by ellipses, similar to the hue circle, and those of the simulated deutan colors could be fitted by cubic function curves. These results suggest that both of the stimulus sets were perceived as one dimensional and as a monotonic series of colors by a deuteranope, but the deuteranope can recognize the difference of the stimulus sets and recall the color distributions of each set, similar to those in the CVN observers.

Maker-Breaker metric resolving games on graphs

Let [Formula: see text] denote the length of a shortest path between vertices [Formula: see text] and [Formula: see text] in a graph [Formula: see text] with vertex set [Formula: see text]. For a positive integer [Formula: see text], let [Formula: see text] and [Formula: see text]. A set [Formula: see text] is a distance-[Formula: see text] resolving set of [Formula: see text] if [Formula: see text] for distinct [Formula: see text]. In this paper, we study the maker-breaker distance-[Formula: see text] resolving game (MB[Formula: see text]RG) played on a graph [Formula: see text] by two players, Maker and Breaker, who alternately select a vertex of [Formula: see text] not yet chosen. Maker wins by selecting vertices which form a distance-[Formula: see text] resolving set of [Formula: see text], whereas Breaker wins by preventing Maker from winning. We denote by [Formula: see text] the outcome of MB[Formula: see text]RG. Let [Formula: see text], [Formula: see text] and [Formula: see text], respectively, denote the outcome for which Maker, Breaker, and the first player has a winning strategy in MB[Formula: see text]RG. Given a graph [Formula: see text], the parameter [Formula: see text] is a non-decreasing function of [Formula: see text] with codomain [Formula: see text]. We exhibit pairs [Formula: see text] and [Formula: see text] such that the ordered pair [Formula: see text] realizes each member of the set [Formula: see text]; we provide graphs [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text]. Moreover, we obtain some general results on MB[Formula: see text]RG and study the MB[Formula: see text]RG played on some graph classes.

On the Estimation of the Persistence Exponent for a Fractionally Integrated Brownian Motion by Numerical Simulations

For a fractionally integrated Brownian motion (FIBM) of order α∈(0,1],Xα(t), we investigate the decaying rate of P(τSα>t) as t→+∞, where τSα=inf{t>0:Xα(t)≥S} is the first-passage time (FPT) of Xα(t) through the barrier S>0. Precisely, we study the so-called persistent exponent θ=θ(α) of the FPT tail, such that P(τSα>t)=t−θ+o(1), as t→+∞, and by means of numerical simulation of long enough trajectories of the process Xα(t), we are able to estimate θ(α) and to show that it is a non-increasing function of α∈(0,1], with 1/4≤θ(α)≤1/2. In particular, we are able to validate numerically a new conjecture about the analytical expression of the function θ=θ(α), for α∈(0,1]. Such a numerical validation is carried out in two ways: in the first one, we estimate θ(α), by using the simulated FPT density, obtained for any α∈(0,1]; in the second one, we estimate the persistent exponent by directly calculating Pmax0≤s≤tXα(s)<1. Both ways confirm our conclusions within the limit of numerical approximation. Finally, we investigate the self-similarity property of Xα(t) and we find the upper bound of its covariance function.

On the monotonicity of functions constructed via z-ordinal sum construction

This is the second part of a two-part paper which discusses monotonicity of functions defined on the unit interval, constructed via (z)-ordinal. In Part I we characterized all non-decreasing functions defined on the unit interval which are constructed by a non-trivial ordinal sum of semigroups and gave necessary and sufficient conditions for a function constructed via ordinal sum to be monotone. In the present Part II we describe the structure of a monotone function defined on the unit interval which is constructed via z-ordinal sum construction with respect to a finite branching set and we give necessary and sufficient conditions for a function constructed via z-ordinal sum to be monotone in the case when the intermediate condition is fulfilled. The case when the intermediate condition is not fulfilled is discussed as well.

On the monotonicity of functions constructed via ordinal sum construction

The monotonicity of functions defined on the unit interval, constructed via (z)-ordinal sum is discussed. In Part I of this two-part paper we characterize all non-decreasing functions defined on the unit interval which are constructed by a non-trivial ordinal sum of semigroups. We also give necessary and sufficient conditions for a function constructed via ordinal sum to be monotone. In Part II we describe the structure of a monotone function defined on the unit interval which is constructed via z-ordinal sum construction with respect to a finite branching set and we give necessary and sufficient conditions for a function constructed via z-ordinal sum to be monotone in the case when intermediate condition is fulfilled. The case when intermediate condition is not fulfilled is discussed as well.

On the monotonicity property of the generalized eigenvalue for weakly-coupled cooperative elliptic systems

We consider general linear non-degenerate weak-coupled cooperative elliptic systems and study certain monotonicity properties of the generalized principal eigenvalue in Rd with respect to the potential. It is shown that monotonicity on the right is equivalent to the recurrence property of the twisted operator which is, in turn, equivalent to the minimal growth property at infinity of the principal eigenfunctions. The strict monotonicity property of the principal eigenvalue is shown to be equivalent with the exponential stability of the twisted operators. An equivalence between the monotonicity property on the right and the stochastic representation of the principal eigenfunction is also established.

ANALYSIS OF THE PROPERTIES OF A NONLINEAR SHEAR FLOW MODEL OF THIXOTROPIC MEDIA TAKING INTO ACCOUNT THE MUTUAL INFLUENCE OF STRUCTURAL EVOLUTION AND DEFORMATION PROCESS

A systematic analytical study was conducted on the mathematical properties of the previously proposed prototype of a nonlinear Maxwell-type constitutive equation for describing the shear flow of thixotropic media (viscous liquid polymers, viscoelastic melts, concentrated solutions, pastes, emulsions), which takes into account the mutual influence of the deformation process and structural evolution (the kinetics of formation and breaking of intermolecular bonds) on viscosity and shear modulus and the effect of the deformation process on this kinetics. In the uniaxial case, the constitutive equation is governed by a nondecreasing material function and six positive parameters. The equation is reduced to a system of two nonlinear autonomous differential equations for the stress and the structuredness parameter. It is proved that the equilibrium position of this system is unique. The dependences of the position coordinates on all material parameters and on the shear rate for an arbitrary nondecreasing material function are investigated in general form, and all the dependences are proved to be monotonic. Equations for the flow and viscosity curves are derived and investigated. It is proved that the model leads to an increasing dependence of the equilibrium stress on the shear rate and to a decreasing apparent viscosity curve, which reflect the typical properties of the experimental flow curves of pseudoplastic media. Using six arbitrary material parameters and a material function that control the model, we analytically study the phase portrait of the nonlinear system of two differential equations, to which the model is reduced, for dimensionless stress and the degree of structuredness in the vicinity of its only equilibrium position. It is proved that the equilibrium position is always stable and can be of three kinds only: a stable node, a degenerate node, or a stable focus. Criteria for each kind are found in the form of explicit constraints on the material function, model parameters, and shear rate. The existence of a stable focus indicates the nonmonotonicity of the system solutions and the existence of deformation modes with (damped) fluctuations of stress and structuredness when stationary values are reached. The influence of the material parameters and material function on the type of equilibrium point and on the behavior of the model integral curves is analyzed.

Existence of entropy solutions for some quasilinear anisotropic elliptic unilateral problems with variable exponents

In this paper, we shall be concerned with the study of the following quasilinear anisotropic elliptic Dirichlet problems of the type \begin{document}$\begin{equation} -\mbox{div }\;a(x,u,\nabla u) = f -\mbox{div } F \qquad \mbox{in}\quad \Omega,\quad\quad\quad\quad\quad(1) \end{equation}$ \end{document} where $ f\in L^{1}(\Omega) $ and $ F \in \prod_{i = 1}^{N} L^{p'_{i}(\cdot)}(\Omega), $ and $ a_{i}(x,u,\xi) $ are Carathéodory functions from $ \Omega \times I\!\!R \times I\!\!R^{N} $ into $ I\!\!R $, which satisfy assumptions of growth, coercivity and strict monotonicity. We prove the existence of entropy solutions for the quasilinear elliptic equation associated to the unilateral problem by relying on the penalization method, in the anisotropic variable exponent Sobolev spaces. Our approach is also based on the techniques of monotone operators in Banach spaces, the existence of weak solutions, and some approximations methods. The problems of the type $ (1) $ are very interesting from the purely mathematical point of view. On the other hand, such equations $ (1) $ appear in different contexts, in particular, the mathematical description of motions of the non-newtonien fluids; we quote for instance the electro-rheological fluids; the deformation of membrane constrained by an obstacle, the image processing and other various physical applications.

Strict monotonicity, continuity, and bounds on the Kertész line for the random-cluster model on Zd

Ising and Potts models can be studied using the Fortuin–Kasteleyn representation through the Edwards–Sokal coupling. This adapts to the setting where the models are exposed to an external field of strength h > 0. In this representation, which is also known as the random-cluster model, the Kertész line is the curve that separates two regions of the parameter space defined according to the existence of an infinite cluster in Zd. This signifies a geometric phase transition between the ordered and disordered phases even in cases where a thermodynamic phase transition does not occur. In this article, we prove strict monotonicity and continuity of the Kertész line. Furthermore, we give new rigorous bounds that are asymptotically correct in the limit h → 0 complementing the bounds from the work of Ruiz and Wouts [J. Math. Phys. 49, 053303 (2008)], which were asymptotically correct for h → ∞. Finally, using a cluster expansion, we investigate the continuity of the Kertész line phase transition.

Existence of minimizers for non-quasiconvex functionals by strict monotonicity

Existence of minimizers for non-quasiconvex functionals by strict monotonicity

Kritičke napomene o članku "Postojanje rešenja diferencijalne jednačine drugog reda pomoću rezultata o nepokretnoj tački F-kontrakcija korištenjem ω0-distancu"

Introduction/purpose: In this paper, several critical remarks are presented concerning the paper of Iqbal & Rizwan: Existence of the solution to second order differential equation through fixed point results for nonlinear F-contractions involving w0-distance from 2020. Methods: Conventional theoretical methods of functional analysis. Results: It is shown that their use of the non-decreasing "control" function F instead of a strictly increasing one in Wardowski-type results usually produces contradictions. Conclusion: It is shown that such results can be obtained in a more general class of metric-like spaces, where strict monotonicity is the only assumption that has to be imposed on the function F. An example is presented showing that the obtained results are stronger than the classic ones.

Global bifurcation of positive solutions for a class of superlinear first-order differential systems

We are concerned with the first-order differential system of the form $$\left\{ \begin{array}{ll} u'(t)+a(t)u(t)=\lambda b(t) f(v(t-\tau(t))), &t\in\mathbb{R},\\ v'(t)+a(t)v(t)=\lambda b(t)g(u(t-\tau(t))), &t\in\mathbb{R},\\ \end{array} \right. $$ where $\lambda\in\mathbb{R}$~is a parameter. $a,b\in C(\mathbb{R},[0,+\infty))$ are two $\omega$-periodic functions such that $\int_0^\omega a(t)\text{d}t>0$,~$\int_0^\omega b(t)\text{d}t>0$,~$\tau\in C(\mathbb{R},\mathbb{R})$ is an $\omega$-periodic function. The nonlinearities~$f,g\in C(\mathbb{R},(0,+\infty))$~are two nondecreasing continuous functions and satisfy superlinear conditions at infinity.~By using the bifurcation theory,~we will show the existence of an unbounded component of positive solutions, which is bounded in positive $\lambda$-direction.

Approximation by pseudo-linear discrete operators

<p style='text-indent:20px;'>In this note, we construct a pseudo-linear kind discrete operator based on the continuous and nondecreasing generator function. Then, we obtain an approximation to uniformly continuous functions through this new operator. Furthermore, we calculate the error estimation of this approach with a modulus of continuity based on a generator function. The obtained results are supported by visualizing with an explicit example. Finally, we investigate the relation between discrete operators and generalized sampling series.</p>

Nonstationary homoclinic solutions for infinite-dimensional fractional reaction-diffusion system with two types of superlinear nonlinearity

This paper is dedicated to studying nonstationary homoclinic solutions with the least energy for a class of fractional reaction-diffusion system \begin{document}$ \begin{eqnarray*} \label{1.1} \left\{\begin{array}{lll} \partial_t u+ (-\Delta)^s u+V(x)u+W(x)v = H_v(t, x, u, v), \\ - \partial_t v + (-\Delta)^s v+V(x)v+W(x)u = H_u(t, x, u, v), \\ |u(t, x)|+|v(t, x)|\rightarrow 0, \ \ \text{as}\ \ |t|+|x|\rightarrow \infty, \end{array} \right. \end{eqnarray*} $\end{document} where $ 0<s<1, \ z = (u, v): \mathbb{R}\times \mathbb{R}^N\rightarrow \mathbb{R}^{2} $, which originate from a wide variety of fields such as theoretical physics, optimal control, chemistry and biology. We obtain ground state solutions of Nehari-Pankov type under mild conditions on the nonlinearity by further developing non-Nehari method with two types of superlinear nonlinearity. If in addition the corresponding functional is even, we also obtain infinitely many geometrically distinct solutions by using some arguments about deformation type and Krasnoselskii genus. Nevertheless, we need to overcome some difficulties: one is that the associated functional is strongly indefinite, the second is due to the absence of strict monotonicity condition, a key ingredient of seeking the ground state solution on suitable manifold, we need some new methods and techniques. The third lies that some delicate analysis are needed for the dropping of classical super-quadratic assumption on the nonlinearity and in verifying the link geometry and showing the boundedness of Cerami sequences.

Memory-efficient model-based deep learning with convergence and robustness guarantees.

Computational imaging has been revolutionized by compressed sensing algorithms, which offer guaranteed uniqueness, convergence, and stability properties. Model-based deep learning methods that combine imaging physics with learned regularization priors have emerged as more powerful alternatives for image recovery. The main focus of this paper is to introduce a memory efficient model-based algorithm with similar theoretical guarantees as CS methods. The proposed iterative algorithm alternates between a gradient descent involving the score function and a conjugate gradient algorithm to encourage data consistency. The score function is modeled as a monotone convolutional neural network. Our analysis shows that the monotone constraint is necessary and sufficient to enforce the uniqueness of the fixed point in arbitrary inverse problems. In addition, it also guarantees the convergence to a fixed point, which is robust to input perturbations. We introduce two implementations of the proposed MOL framework, which differ in the way the monotone property is imposed. The first approach enforces a strict monotone constraint, while the second one relies on an approximation. The guarantees are not valid for the second approach in the strict sense. However, our empirical studies show that the convergence and robustness of both approaches are comparable, while the less constrained approximate implementation offers better performance. The proposed deep equilibrium formulation is significantly more memory efficient than unrolled methods, which allows us to apply it to 3D or 2D+time problems that current unrolled algorithms cannot handle.

Groundstates and infinitely many solutions for the Schrödinger-Poisson equation with magnetic field

<p style='text-indent:20px;'>In this paper, we investigate the nonlinear Schrödinger-Poisson equation with magnetic field. By combining non-Nehari manifold method and some new energy estimate inequalities, we obtain the existence of a ground state solution, where the strict monotonicity condition and the Ambrosetti-Rabinowitz growth condition are not needed. Moreover, when both the potential and the nonlinearity are sign-changing, by applying the Fountain Theorem and some analytical techniques, we prove the existence of infinitely many solutions. Our results extend and improve the present ones in the literature.</p>

Expected utility for probabilistic prospects and the common ratio property

We prove the existence of an expected utility function for preferences over probabilistic prospects satisfying Strict Monotonicity, Indifference, the Common Ratio Property, Substitution and Reducibility of Extreme Prospects. The example in [1] that is inconsistent with the existence of a von Neumann-Morgenstern for preferences over probabilistic prospects, violates the Common Ratio Property. Subsequently, we prove the existence of expected utility functions with piecewise linear Bernoulli utility functions for preferences that are piece-wise linear. For this case a weaker version of the Indifference Assumption that is used in the earlier existence theorems is sufficient. We also state analogous results for probabilistic lotteries. We do not require any compound prospects or mixture spaces to prove any of our results. In the third last section of this paper, we ?argue? that the observations related to Allais paradox, do not constitute a violation of expected utility maximization by individuals, but is a likely manifestation of individuals assigning (experiment or menu-dependent?) subjective probabilities to events which disagree with their objective probabilities.

Subexponentialiy of densities of infinitely divisible distributions

We show the equivalence of three properties for an infinitely divisible distribution: the subexponentiality of the density, the subexponentiality of the density of its Lévy measure and the tail equivalence between the density and its Lévy measure density, under monotonic-type assumptions on the Lévy measure density. The key assumption is that tail of the Lévy measure density is asymptotic to a non-increasing function or is almost decreasing. Our conditions are natural and cover a rather wide class of infinitely divisible distributions. Several significant properties for analyzing the subexponentiality of densities have been derived such as closure properties of [ convolution, convolution roots and asymptotic equivalence ] and the factorization property. Moreover, we illustrate that the results are applicable for developing the statistical inference of subexponential infinitely divisible distributions which are absolutely continuous.

Hardy Inequality in Variable Grand Lebesgue Spaces for Nonincreasing Functions

Исследуется ограниченность оператора усреднения Харди между взвешенными гранд-пространствами Лебега с переменным показателем для невозрастающих функций. Библиография: 26 названий.